11:34:44 code $ python vs.py -compare Tie-breaker values (smaller is better): A:0 B:1 C:2 D:3 E:4 Number of candidates = 5 Number of ballots per election trial = 100 ballot_distribution: ('hypersphere', 3) ballot min/max lengths: None Allow profiles with Condorcet winners: True Trial 0: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (17) C B E D A (9) A D E B C (9) C B A E D (8) A C B D E (8) B C E D A (7) D A E B C (6) D E B C A (4) D E A B C (4) C B E A D (4) A D C B E (4) E D B C A (2) E D A B C (2) E B C D A (2) C B A D E (2) A D C E B (2) A C B E D (2) E A D B C (1) D A C E B (1) C B D E A (1) C A B E D (1) C A B D E (1) B E C D A (1) B C E A D (1) A E D B C (1) Total count = 100 A B C D E A 0 18 14 22 24 B -18 0 -20 -6 -2 C -14 20 0 -6 2 D -22 6 6 0 18 E -24 2 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 14 22 24 B -18 0 -20 -6 -2 C -14 20 0 -6 2 D -22 6 6 0 18 E -24 2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 C=26 D=15 B=9 E=7 so E is eliminated. Round 2 votes counts: A=44 C=26 D=19 B=11 so B is eliminated. Round 3 votes counts: A=44 C=37 D=19 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:239 D:204 C:201 E:179 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 14 22 24 B -18 0 -20 -6 -2 C -14 20 0 -6 2 D -22 6 6 0 18 E -24 2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 14 22 24 B -18 0 -20 -6 -2 C -14 20 0 -6 2 D -22 6 6 0 18 E -24 2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 14 22 24 B -18 0 -20 -6 -2 C -14 20 0 -6 2 D -22 6 6 0 18 E -24 2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) E B D C A (7) E D C B A (5) E D B C A (5) D C E A B (4) A B E D C (4) A B C E D (4) D E C A B (3) D C A E B (3) D A C E B (3) C D E B A (3) B E C D A (3) B A C E D (3) A C B D E (3) A B E C D (3) A B C D E (3) E D A B C (2) E B C D A (2) D C E B A (2) C B E D A (2) B E C A D (2) B E A D C (2) B E A C D (2) B C E D A (2) B C E A D (2) A D E B C (2) A D C E B (2) E D B A C (1) E C B D A (1) E B D A C (1) E A B D C (1) C D E A B (1) C D A E B (1) C B A D E (1) C A B D E (1) B C A E D (1) B A E C D (1) A E B D C (1) A D C B E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -18 -18 -22 B 12 0 10 6 -16 C 18 -10 0 -16 -14 D 18 -6 16 0 -14 E 22 16 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -18 -18 -22 B 12 0 10 6 -16 C 18 -10 0 -16 -14 D 18 -6 16 0 -14 E 22 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 D=23 B=18 C=9 so C is eliminated. Round 2 votes counts: D=28 A=26 E=25 B=21 so B is eliminated. Round 3 votes counts: E=40 A=32 D=28 so D is eliminated. Round 4 votes counts: E=61 A=39 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 D:207 B:206 C:189 A:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -18 -18 -22 B 12 0 10 6 -16 C 18 -10 0 -16 -14 D 18 -6 16 0 -14 E 22 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -18 -22 B 12 0 10 6 -16 C 18 -10 0 -16 -14 D 18 -6 16 0 -14 E 22 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -18 -22 B 12 0 10 6 -16 C 18 -10 0 -16 -14 D 18 -6 16 0 -14 E 22 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (15) D C A B E (9) E B C D A (8) A D C B E (8) D A C B E (6) A D E B C (5) E A B C D (4) D C B A E (4) A D E C B (4) E B A C D (3) A E D B C (3) A E B D C (3) E C B D A (2) D A C E B (2) C E B D A (2) B E C D A (2) A D B C E (2) A B C D E (2) E D A C B (1) E C D B A (1) E A D C B (1) E A B D C (1) D C E B A (1) D C B E A (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C A D (1) B C E D A (1) B C A D E (1) A D C E B (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -2 14 6 B -10 0 8 -2 -14 C 2 -8 0 -8 -14 D -14 2 8 0 -2 E -6 14 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305835 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 10 -2 14 6 B -10 0 8 -2 -14 C 2 -8 0 -8 -14 D -14 2 8 0 -2 E -6 14 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=31 D=23 C=5 B=5 so C is eliminated. Round 2 votes counts: E=38 A=31 D=24 B=7 so B is eliminated. Round 3 votes counts: E=43 A=32 D=25 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:212 D:197 B:191 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 14 6 B -10 0 8 -2 -14 C 2 -8 0 -8 -14 D -14 2 8 0 -2 E -6 14 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 14 6 B -10 0 8 -2 -14 C 2 -8 0 -8 -14 D -14 2 8 0 -2 E -6 14 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 14 6 B -10 0 8 -2 -14 C 2 -8 0 -8 -14 D -14 2 8 0 -2 E -6 14 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) C E B A D (6) A B E C D (6) C D E B A (5) D E B A C (4) C E D B A (4) B E A D C (4) B E A C D (4) A B E D C (4) E B C D A (3) E B C A D (3) C D A B E (3) A D C B E (3) A D B E C (3) E C B A D (2) E B D A C (2) D E B C A (2) D C E B A (2) D C A B E (2) C E B D A (2) C B E A D (2) C B A E D (2) C A B E D (2) A C D B E (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D C A (1) E B A D C (1) E B A C D (1) D E C B A (1) D E A B C (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B E C (1) B E C A D (1) A D B C E (1) A C B E D (1) A C B D E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -10 2 -10 B 10 0 -6 0 -12 C 10 6 0 2 0 D -2 0 -2 0 -10 E 10 12 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.742503 D: 0.000000 E: 0.257497 Sum of squares = 0.617615575355 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.742503 D: 0.742503 E: 1.000000 A B C D E A 0 -10 -10 2 -10 B 10 0 -6 0 -12 C 10 6 0 2 0 D -2 0 -2 0 -10 E 10 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 A=23 E=16 B=9 so B is eliminated. Round 2 votes counts: D=26 C=26 E=25 A=23 so A is eliminated. Round 3 votes counts: E=35 D=34 C=31 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:209 B:196 D:193 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 2 -10 B 10 0 -6 0 -12 C 10 6 0 2 0 D -2 0 -2 0 -10 E 10 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 2 -10 B 10 0 -6 0 -12 C 10 6 0 2 0 D -2 0 -2 0 -10 E 10 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 2 -10 B 10 0 -6 0 -12 C 10 6 0 2 0 D -2 0 -2 0 -10 E 10 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C A D B E (9) E B D A C (8) A C B E D (8) D E B C A (6) C D E B A (6) E B D C A (5) A B E D C (5) C E B D A (4) B E D A C (4) B E A D C (4) C E B A D (3) C D E A B (3) E D B C A (2) E B A D C (2) E B A C D (2) D B A E C (2) C A E B D (2) C A D E B (2) A D B C E (2) E D C B A (1) D E B A C (1) D C E B A (1) D A B E C (1) C E D B A (1) C E A B D (1) C D A B E (1) B D E A C (1) A C D B E (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 6 -12 -28 B 24 0 12 4 0 C -6 -12 0 -10 -8 D 12 -4 10 0 -6 E 28 0 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.265400 C: 0.000000 D: 0.000000 E: 0.734600 Sum of squares = 0.610074737681 Cumulative probabilities = A: 0.000000 B: 0.265400 C: 0.265400 D: 0.265400 E: 1.000000 A B C D E A 0 -24 6 -12 -28 B 24 0 12 4 0 C -6 -12 0 -10 -8 D 12 -4 10 0 -6 E 28 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=20 D=20 A=19 B=9 so B is eliminated. Round 2 votes counts: C=32 E=28 D=21 A=19 so A is eliminated. Round 3 votes counts: C=42 E=34 D=24 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:220 D:206 C:182 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 6 -12 -28 B 24 0 12 4 0 C -6 -12 0 -10 -8 D 12 -4 10 0 -6 E 28 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 6 -12 -28 B 24 0 12 4 0 C -6 -12 0 -10 -8 D 12 -4 10 0 -6 E 28 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 6 -12 -28 B 24 0 12 4 0 C -6 -12 0 -10 -8 D 12 -4 10 0 -6 E 28 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) B D C A E (7) D B A C E (6) B C D E A (6) D B C A E (5) C E B A D (5) A E C D B (5) E A C B D (4) D A E B C (4) C B E A D (4) A D E C B (4) D A E C B (3) B D C E A (3) B C E D A (3) E C A B D (2) E A C D B (2) D A C E B (2) D A B C E (2) C E A B D (2) C B D E A (2) C A E D B (2) B D A E C (2) A E D C B (2) A E D B C (2) E C A D B (1) E B A C D (1) E A B C D (1) D C B A E (1) D C A E B (1) D A B E C (1) C E A D B (1) C B E D A (1) B D E C A (1) B D E A C (1) B D A C E (1) B C E A D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 4 -18 18 B 14 0 10 -6 8 C -4 -10 0 -12 12 D 18 6 12 0 20 E -18 -8 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -18 18 B 14 0 10 -6 8 C -4 -10 0 -12 12 D 18 6 12 0 20 E -18 -8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=25 C=17 A=15 E=11 so E is eliminated. Round 2 votes counts: D=32 B=26 A=22 C=20 so C is eliminated. Round 3 votes counts: B=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:213 A:195 C:193 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 4 -18 18 B 14 0 10 -6 8 C -4 -10 0 -12 12 D 18 6 12 0 20 E -18 -8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -18 18 B 14 0 10 -6 8 C -4 -10 0 -12 12 D 18 6 12 0 20 E -18 -8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -18 18 B 14 0 10 -6 8 C -4 -10 0 -12 12 D 18 6 12 0 20 E -18 -8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (5) B E A C D (5) B C E D A (5) B C D E A (5) B C D A E (5) A D E C B (5) E D A C B (4) E B A C D (4) D C A E B (4) C D E A B (4) C D B A E (4) A D C E B (4) E A B D C (3) C D B E A (3) C D A B E (3) A E D B C (3) E D C A B (2) D C E A B (2) C D E B A (2) C D A E B (2) B E C D A (2) B A D C E (2) A E D C B (2) A E B D C (2) A D B C E (2) A B E D C (2) E C D A B (1) E B C A D (1) E A D C B (1) D A E C B (1) C E D B A (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A D C (1) B A E D C (1) B A E C D (1) B A C E D (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 2 -14 4 B -8 0 -4 -12 -8 C -2 4 0 4 14 D 14 12 -4 0 14 E -4 8 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.000000 Sum of squares = 0.540000000137 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -14 4 B -8 0 -4 -12 -8 C -2 4 0 4 14 D 14 12 -4 0 14 E -4 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.000000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=22 C=21 E=16 D=12 so D is eliminated. Round 2 votes counts: B=29 A=28 C=27 E=16 so E is eliminated. Round 3 votes counts: A=36 B=34 C=30 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:218 C:210 A:200 E:188 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 2 -14 4 B -8 0 -4 -12 -8 C -2 4 0 4 14 D 14 12 -4 0 14 E -4 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.000000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -14 4 B -8 0 -4 -12 -8 C -2 4 0 4 14 D 14 12 -4 0 14 E -4 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.000000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -14 4 B -8 0 -4 -12 -8 C -2 4 0 4 14 D 14 12 -4 0 14 E -4 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.000000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E A D (7) A E D B C (7) A E B C D (7) D C A E B (5) D A E C B (5) C D B E A (5) B C E A D (5) E A B C D (4) C B D E A (4) B E A C D (4) D B E A C (3) C E A B D (3) A E B D C (3) D C B E A (2) D C B A E (2) C E B A D (2) C D E A B (2) C D A E B (2) C A E D B (2) B C D E A (2) A E C B D (2) E B A C D (1) D B A E C (1) D A C E B (1) C E A D B (1) C B E D A (1) B E D A C (1) B E C A D (1) B D E A C (1) B A E D C (1) A E D C B (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 14 8 6 2 B -14 0 4 0 -18 C -8 -4 0 10 -8 D -6 0 -10 0 -8 E -2 18 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 6 2 B -14 0 4 0 -18 C -8 -4 0 10 -8 D -6 0 -10 0 -8 E -2 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=29 C=29 A=22 B=15 E=5 so E is eliminated. Round 2 votes counts: D=29 C=29 A=26 B=16 so B is eliminated. Round 3 votes counts: C=37 A=32 D=31 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:215 C:195 D:188 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 6 2 B -14 0 4 0 -18 C -8 -4 0 10 -8 D -6 0 -10 0 -8 E -2 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 6 2 B -14 0 4 0 -18 C -8 -4 0 10 -8 D -6 0 -10 0 -8 E -2 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 6 2 B -14 0 4 0 -18 C -8 -4 0 10 -8 D -6 0 -10 0 -8 E -2 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) B D C A E (6) C E B D A (5) C B E D A (5) B C D E A (5) A D E B C (5) E C A D B (4) B A D E C (4) E A D C B (3) E A C D B (3) D A E C B (3) C E D B A (3) C E D A B (3) C E B A D (3) C D E A B (3) B D A C E (3) B C E D A (3) A E D C B (3) A D B E C (3) E C B A D (2) E C A B D (2) D B C A E (2) D B A C E (2) B C D A E (2) A E D B C (2) E D A C B (1) E B A C D (1) E A C B D (1) E A B C D (1) D E A C B (1) D C B A E (1) D A C E B (1) D A C B E (1) D A B E C (1) C D E B A (1) C D B A E (1) C B D E A (1) B E A C D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 -4 -20 -4 B 2 0 -4 -8 -6 C 4 4 0 0 14 D 20 8 0 0 6 E 4 6 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.621293 D: 0.378707 E: 0.000000 Sum of squares = 0.529424200702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.621293 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -20 -4 B 2 0 -4 -8 -6 C 4 4 0 0 14 D 20 8 0 0 6 E 4 6 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=24 E=18 D=18 A=15 so A is eliminated. Round 2 votes counts: D=27 C=25 E=24 B=24 so E is eliminated. Round 3 votes counts: C=37 D=36 B=27 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:211 E:195 B:192 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -20 -4 B 2 0 -4 -8 -6 C 4 4 0 0 14 D 20 8 0 0 6 E 4 6 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -20 -4 B 2 0 -4 -8 -6 C 4 4 0 0 14 D 20 8 0 0 6 E 4 6 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -20 -4 B 2 0 -4 -8 -6 C 4 4 0 0 14 D 20 8 0 0 6 E 4 6 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) C B A E D (7) A C B D E (7) E D B C A (6) E D B A C (5) C A B D E (5) B E D C A (5) A B C D E (5) D E B A C (4) D E A C B (4) D E A B C (4) B C A D E (4) E D C B A (3) E D C A B (3) C A B E D (3) B A C D E (3) A C D B E (3) E D A C B (2) B C E A D (2) E C D B A (1) E B D C A (1) D A E C B (1) D A E B C (1) C E B D A (1) C B E A D (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D E B (1) B E C D A (1) B D A E C (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -14 14 14 B 12 0 4 16 16 C 14 -4 0 16 14 D -14 -16 -16 0 -6 E -14 -16 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999396 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 14 14 B 12 0 4 16 16 C 14 -4 0 16 14 D -14 -16 -16 0 -6 E -14 -16 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=21 C=21 A=18 D=14 so D is eliminated. Round 2 votes counts: E=33 B=26 C=21 A=20 so A is eliminated. Round 3 votes counts: E=36 C=33 B=31 so B is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:220 A:201 E:181 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 14 14 B 12 0 4 16 16 C 14 -4 0 16 14 D -14 -16 -16 0 -6 E -14 -16 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 14 14 B 12 0 4 16 16 C 14 -4 0 16 14 D -14 -16 -16 0 -6 E -14 -16 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 14 14 B 12 0 4 16 16 C 14 -4 0 16 14 D -14 -16 -16 0 -6 E -14 -16 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 10: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) D A C E B (6) C E B D A (6) B A D C E (6) A D B C E (5) E C B A D (4) E B C A D (4) E A D C B (4) C E D A B (4) B D A C E (4) B A D E C (4) E C B D A (3) E C A D B (3) C E D B A (3) B D C A E (3) B C D A E (3) A D E C B (3) A B D E C (3) E C A B D (2) E A B D C (2) D A B C E (2) A E D C B (2) A D B E C (2) D C A E B (1) D C A B E (1) D A E C B (1) D A C B E (1) C D E B A (1) C D E A B (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A D C (1) B E A C D (1) B C E D A (1) B A E D C (1) B A C D E (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 2 2 4 B -2 0 -8 6 -14 C -2 8 0 -8 4 D -2 -6 8 0 4 E -4 14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 2 4 B -2 0 -8 6 -14 C -2 8 0 -8 4 D -2 -6 8 0 4 E -4 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=26 C=17 A=17 D=12 so D is eliminated. Round 2 votes counts: E=28 A=27 B=26 C=19 so C is eliminated. Round 3 votes counts: E=43 A=29 B=28 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:205 D:202 C:201 E:201 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 2 4 B -2 0 -8 6 -14 C -2 8 0 -8 4 D -2 -6 8 0 4 E -4 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 4 B -2 0 -8 6 -14 C -2 8 0 -8 4 D -2 -6 8 0 4 E -4 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 4 B -2 0 -8 6 -14 C -2 8 0 -8 4 D -2 -6 8 0 4 E -4 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 11: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A B C E (8) E C B A D (6) C B E A D (6) E C B D A (5) E C D B A (4) D E A B C (4) D C A B E (4) A D B C E (4) C E B D A (3) C E B A D (3) C B A E D (3) B C E A D (3) A B E D C (3) E D C B A (2) E D A B C (2) E B C D A (2) D E C A B (2) D A E B C (2) D A C E B (2) D A B E C (2) C B A D E (2) B C A E D (2) B A C E D (2) A D C B E (2) E D B A C (1) E D A C B (1) E A D B C (1) D E A C B (1) D C E A B (1) D A C B E (1) C D A B E (1) C B E D A (1) C A B D E (1) B E C A D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -24 2 -18 B 12 0 -4 10 -4 C 24 4 0 14 2 D -2 -10 -14 0 -22 E 18 4 -2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -24 2 -18 B 12 0 -4 10 -4 C 24 4 0 14 2 D -2 -10 -14 0 -22 E 18 4 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=27 C=20 A=11 B=8 so B is eliminated. Round 2 votes counts: E=35 D=27 C=25 A=13 so A is eliminated. Round 3 votes counts: E=38 D=33 C=29 so C is eliminated. Round 4 votes counts: E=61 D=39 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:222 E:221 B:207 D:176 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -24 2 -18 B 12 0 -4 10 -4 C 24 4 0 14 2 D -2 -10 -14 0 -22 E 18 4 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -24 2 -18 B 12 0 -4 10 -4 C 24 4 0 14 2 D -2 -10 -14 0 -22 E 18 4 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -24 2 -18 B 12 0 -4 10 -4 C 24 4 0 14 2 D -2 -10 -14 0 -22 E 18 4 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 12: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) E C A B D (6) C A E B D (6) D B E C A (5) D C A B E (4) D B E A C (4) D B A E C (4) C A D E B (4) B D E A C (4) A C E B D (4) E B A C D (3) D B A C E (3) D A B C E (3) C A E D B (3) B E D A C (3) B E A D C (3) E C B D A (2) E B C A D (2) D C E B A (2) C E D A B (2) A E C B D (2) A E B C D (2) A B D E C (2) E C D B A (1) E C B A D (1) E B D C A (1) E B C D A (1) D E C B A (1) D C B A E (1) D C A E B (1) D B C A E (1) D A C B E (1) C E D B A (1) C E A D B (1) C D E B A (1) B E D C A (1) B E A C D (1) B D A E C (1) A D B C E (1) A C E D B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -12 4 -8 B -6 0 -8 8 -12 C 12 8 0 8 0 D -4 -8 -8 0 -10 E 8 12 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.303269 D: 0.000000 E: 0.696731 Sum of squares = 0.577406256039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.303269 D: 0.303269 E: 1.000000 A B C D E A 0 6 -12 4 -8 B -6 0 -8 8 -12 C 12 8 0 8 0 D -4 -8 -8 0 -10 E 8 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=26 E=17 A=14 B=13 so B is eliminated. Round 2 votes counts: D=35 C=26 E=25 A=14 so A is eliminated. Round 3 votes counts: D=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:214 A:195 B:191 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 4 -8 B -6 0 -8 8 -12 C 12 8 0 8 0 D -4 -8 -8 0 -10 E 8 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 4 -8 B -6 0 -8 8 -12 C 12 8 0 8 0 D -4 -8 -8 0 -10 E 8 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 4 -8 B -6 0 -8 8 -12 C 12 8 0 8 0 D -4 -8 -8 0 -10 E 8 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 13: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) E D A B C (5) C D B A E (5) E A D B C (4) A C D E B (4) E D A C B (3) D E A C B (3) D C E B A (3) D A E C B (3) C D B E A (3) C A D E B (3) B E A C D (3) B C D E A (3) E B A D C (2) D E C B A (2) D E B C A (2) D C E A B (2) D A C E B (2) C D A B E (2) C B D A E (2) C B A D E (2) B E D C A (2) B D C E A (2) B C A E D (2) B C A D E (2) A E D C B (2) A B C E D (2) E D B C A (1) E D B A C (1) E B D C A (1) E B D A C (1) E A B D C (1) D E C A B (1) D C B E A (1) C D A E B (1) C B D E A (1) C A B D E (1) B E D A C (1) B E C D A (1) B E A D C (1) B A C E D (1) A E D B C (1) A E C D B (1) A E C B D (1) A D C E B (1) A C D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 -8 0 B -10 0 -2 -14 -20 C -10 2 0 -14 -6 D 8 14 14 0 6 E 0 20 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -8 0 B -10 0 -2 -14 -20 C -10 2 0 -14 -6 D 8 14 14 0 6 E 0 20 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=20 E=19 D=19 B=18 so B is eliminated. Round 2 votes counts: E=27 C=27 A=25 D=21 so D is eliminated. Round 3 votes counts: E=35 C=35 A=30 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:210 A:206 C:186 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -8 0 B -10 0 -2 -14 -20 C -10 2 0 -14 -6 D 8 14 14 0 6 E 0 20 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -8 0 B -10 0 -2 -14 -20 C -10 2 0 -14 -6 D 8 14 14 0 6 E 0 20 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -8 0 B -10 0 -2 -14 -20 C -10 2 0 -14 -6 D 8 14 14 0 6 E 0 20 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 14: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) D B C E A (5) C A E D B (5) B D E A C (5) A E C B D (5) E B D A C (4) B D C A E (4) E A C D B (3) E A B D C (3) C E D A B (3) C E A D B (3) A E B D C (3) A C E D B (3) A B D E C (3) E D B C A (2) E D B A C (2) E C D B A (2) E C A D B (2) D C B E A (2) C E D B A (2) C A D E B (2) B E D A C (2) B D E C A (2) B D C E A (2) A E C D B (2) A C E B D (2) A C B E D (2) A C B D E (2) E D C B A (1) E A D C B (1) E A D B C (1) E A C B D (1) D E B C A (1) D C E B A (1) C D E B A (1) C D B E A (1) C D B A E (1) C A D B E (1) B D A E C (1) B A D E C (1) B A D C E (1) B A C D E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 0 -16 B 0 0 4 -6 -10 C -2 -4 0 -8 -8 D 0 6 8 0 -12 E 16 10 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 0 -16 B 0 0 4 -6 -10 C -2 -4 0 -8 -8 D 0 6 8 0 -12 E 16 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=22 C=19 B=19 D=15 so D is eliminated. Round 2 votes counts: B=30 A=25 E=23 C=22 so C is eliminated. Round 3 votes counts: B=34 E=33 A=33 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:223 D:201 B:194 A:193 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 0 -16 B 0 0 4 -6 -10 C -2 -4 0 -8 -8 D 0 6 8 0 -12 E 16 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 0 -16 B 0 0 4 -6 -10 C -2 -4 0 -8 -8 D 0 6 8 0 -12 E 16 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 0 -16 B 0 0 4 -6 -10 C -2 -4 0 -8 -8 D 0 6 8 0 -12 E 16 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 15: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (14) E B D A C (9) C A D B E (5) B E D C A (5) C D E B A (4) C A D E B (4) A C E D B (4) A C E B D (4) A C B E D (4) D E B C A (3) C D B E A (3) B E D A C (3) B E A D C (3) B D C E A (3) A C D E B (3) D B E C A (2) C D E A B (2) C D B A E (2) A C B D E (2) A B E D C (2) E D B C A (1) E D A C B (1) E D A B C (1) E B A D C (1) E A D C B (1) E A D B C (1) E A B D C (1) D E C B A (1) D C E B A (1) D B C E A (1) C D A E B (1) C D A B E (1) C B D E A (1) C B D A E (1) C A B D E (1) A E D C B (1) A E B D C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -12 -20 -24 B 16 0 4 12 10 C 12 -4 0 -12 -4 D 20 -12 12 0 12 E 24 -10 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 -20 -24 B 16 0 4 12 10 C 12 -4 0 -12 -4 D 20 -12 12 0 12 E 24 -10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 A=23 E=16 D=8 so D is eliminated. Round 2 votes counts: B=31 C=26 A=23 E=20 so E is eliminated. Round 3 votes counts: B=45 A=28 C=27 so C is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:216 E:203 C:196 A:164 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 -20 -24 B 16 0 4 12 10 C 12 -4 0 -12 -4 D 20 -12 12 0 12 E 24 -10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -20 -24 B 16 0 4 12 10 C 12 -4 0 -12 -4 D 20 -12 12 0 12 E 24 -10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -20 -24 B 16 0 4 12 10 C 12 -4 0 -12 -4 D 20 -12 12 0 12 E 24 -10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 16: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) C E A D B (11) B D A E C (10) D B C A E (7) D B A C E (7) A E C B D (6) E A C B D (5) D B E C A (3) C E A B D (3) A E B C D (3) A B D E C (3) E C D A B (2) E C A B D (2) D C B E A (2) C E D A B (2) B D A C E (2) B A E D C (2) A C E B D (2) A B E D C (2) E C B D A (1) E C B A D (1) E C A D B (1) E A B C D (1) D E B C A (1) D C B A E (1) C D E B A (1) C D B E A (1) C A E D B (1) C A E B D (1) C A D B E (1) B E A D C (1) B A D E C (1) Total count = 100 A B C D E A 0 -8 -10 -6 -2 B 8 0 12 -8 12 C 10 -12 0 -10 10 D 6 8 10 0 6 E 2 -12 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 -2 B 8 0 12 -8 12 C 10 -12 0 -10 10 D 6 8 10 0 6 E 2 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=21 B=16 A=16 E=13 so E is eliminated. Round 2 votes counts: D=34 C=28 A=22 B=16 so B is eliminated. Round 3 votes counts: D=46 C=28 A=26 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:212 C:199 A:187 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 -2 B 8 0 12 -8 12 C 10 -12 0 -10 10 D 6 8 10 0 6 E 2 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 -2 B 8 0 12 -8 12 C 10 -12 0 -10 10 D 6 8 10 0 6 E 2 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 -2 B 8 0 12 -8 12 C 10 -12 0 -10 10 D 6 8 10 0 6 E 2 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 17: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) E B C D A (6) D B A C E (6) B D E A C (6) A D B E C (5) A D B C E (5) C E B A D (4) C E A B D (4) C A E D B (4) B D E C A (4) E C B A D (3) D B A E C (3) A E C B D (3) E C B D A (2) E B D C A (2) D A B C E (2) C E B D A (2) C A E B D (2) C A D E B (2) B E D C A (2) B E D A C (2) A C E D B (2) A C D E B (2) E C A B D (1) E B C A D (1) E B A D C (1) E A C B D (1) E A B C D (1) D B E C A (1) D B C A E (1) D A B E C (1) C E D B A (1) C E D A B (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A B E (1) C A D B E (1) B E A D C (1) B D C E A (1) B A D E C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 2 6 2 B 6 0 8 4 6 C -2 -8 0 -6 2 D -6 -4 6 0 6 E -2 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 6 2 B 6 0 8 4 6 C -2 -8 0 -6 2 D -6 -4 6 0 6 E -2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 E=18 B=17 D=14 so D is eliminated. Round 2 votes counts: A=29 B=28 C=25 E=18 so E is eliminated. Round 3 votes counts: B=38 C=31 A=31 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:202 D:201 C:193 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 6 2 B 6 0 8 4 6 C -2 -8 0 -6 2 D -6 -4 6 0 6 E -2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 6 2 B 6 0 8 4 6 C -2 -8 0 -6 2 D -6 -4 6 0 6 E -2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 6 2 B 6 0 8 4 6 C -2 -8 0 -6 2 D -6 -4 6 0 6 E -2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 18: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) C E D A B (6) C E A B D (6) E C A D B (5) E C D A B (4) D B E A C (4) A E C B D (4) E D C B A (3) E D A B C (3) D E B C A (3) C E A D B (3) B D C A E (3) B D A E C (3) B D A C E (3) A C B E D (3) A B D E C (3) E D C A B (2) D B C E A (2) C D E B A (2) C A E B D (2) B C A D E (2) A B E D C (2) A B C E D (2) A B C D E (2) E C D B A (1) E A D C B (1) E A D B C (1) E A C D B (1) D E C B A (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E C A (1) D B A E C (1) C D B E A (1) C B E A D (1) C B D E A (1) C A B E D (1) B A D E C (1) B A D C E (1) A E D B C (1) A E B D C (1) A E B C D (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -18 -8 -24 B -12 0 -16 -12 -22 C 18 16 0 12 4 D 8 12 -12 0 -24 E 24 22 -4 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -18 -8 -24 B -12 0 -16 -12 -22 C 18 16 0 12 4 D 8 12 -12 0 -24 E 24 22 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=21 A=21 D=15 B=13 so B is eliminated. Round 2 votes counts: C=32 D=24 A=23 E=21 so E is eliminated. Round 3 votes counts: C=42 D=32 A=26 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:233 C:225 D:192 A:181 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -18 -8 -24 B -12 0 -16 -12 -22 C 18 16 0 12 4 D 8 12 -12 0 -24 E 24 22 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -18 -8 -24 B -12 0 -16 -12 -22 C 18 16 0 12 4 D 8 12 -12 0 -24 E 24 22 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -18 -8 -24 B -12 0 -16 -12 -22 C 18 16 0 12 4 D 8 12 -12 0 -24 E 24 22 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 19: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B E D A C (8) A C D E B (8) E B D A C (5) E A C D B (5) D C A B E (5) C A D E B (5) B D E A C (4) B D C A E (4) E C A B D (3) E A C B D (3) D B C A E (3) C E A B D (3) C A E B D (3) C A D B E (3) B E D C A (3) A C E D B (3) E C B A D (2) E B A D C (2) E B A C D (2) D C B A E (2) D B A C E (2) D A C B E (2) C D A B E (2) E D B A C (1) E A D C B (1) E A B D C (1) E A B C D (1) D B E A C (1) D B A E C (1) D A E C B (1) B D E C A (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 18 6 10 8 B -18 0 -22 -10 -18 C -6 22 0 4 8 D -10 10 -4 0 -10 E -8 18 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 10 8 B -18 0 -22 -10 -18 C -6 22 0 4 8 D -10 10 -4 0 -10 E -8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=24 B=20 D=17 A=13 so A is eliminated. Round 2 votes counts: C=35 E=27 B=20 D=18 so D is eliminated. Round 3 votes counts: C=45 E=28 B=27 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:221 C:214 E:206 D:193 B:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 10 8 B -18 0 -22 -10 -18 C -6 22 0 4 8 D -10 10 -4 0 -10 E -8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 10 8 B -18 0 -22 -10 -18 C -6 22 0 4 8 D -10 10 -4 0 -10 E -8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 10 8 B -18 0 -22 -10 -18 C -6 22 0 4 8 D -10 10 -4 0 -10 E -8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 20: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) D A B E C (6) B E C D A (6) B D E A C (6) C E B A D (5) C A E D B (5) B E D A C (5) B D A E C (5) A D C B E (5) A C D E B (5) A D C E B (4) A D B C E (4) E C B D A (3) E C B A D (3) E B D C A (3) C E A B D (3) C A D B E (3) D B E A C (2) D B A E C (2) C E A D B (2) C A E B D (2) C A B E D (2) E D C A B (1) E C D B A (1) D A E C B (1) D A E B C (1) D A B C E (1) C B E A D (1) C B A E D (1) C B A D E (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -4 0 B 8 0 4 14 4 C 2 -4 0 6 -8 D 4 -14 -6 0 -4 E 0 -4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -4 0 B 8 0 4 14 4 C 2 -4 0 6 -8 D 4 -14 -6 0 -4 E 0 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 E=20 A=19 D=13 so D is eliminated. Round 2 votes counts: A=28 B=27 C=25 E=20 so E is eliminated. Round 3 votes counts: B=39 C=33 A=28 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:204 C:198 A:193 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 0 B 8 0 4 14 4 C 2 -4 0 6 -8 D 4 -14 -6 0 -4 E 0 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 0 B 8 0 4 14 4 C 2 -4 0 6 -8 D 4 -14 -6 0 -4 E 0 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 0 B 8 0 4 14 4 C 2 -4 0 6 -8 D 4 -14 -6 0 -4 E 0 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 21: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) C A E B D (7) B A C E D (6) D A B C E (5) E C A D B (4) D B E A C (4) C E A B D (4) B D A E C (4) E D C A B (3) E C D B A (3) D E C B A (3) D E B C A (3) D B A E C (3) B A D C E (3) A B C E D (3) E D C B A (2) E C B D A (2) E C A B D (2) D E C A B (2) D E B A C (2) D A E C B (2) C E A D B (2) C A B E D (2) B C A E D (2) B A C D E (2) A B D C E (2) E B D C A (1) D E A B C (1) D C E A B (1) D B A C E (1) D A B E C (1) C E D A B (1) C B A E D (1) C A E D B (1) B E C A D (1) B D E A C (1) B D A C E (1) B A D E C (1) A D C B E (1) A D B C E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -8 -6 2 B -8 0 -4 -8 -8 C 8 4 0 4 -4 D 6 8 -4 0 -8 E -2 8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428571 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 A B C D E A 0 8 -8 -6 2 B -8 0 -4 -8 -8 C 8 4 0 4 -4 D 6 8 -4 0 -8 E -2 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428572 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 B=21 C=18 A=9 so A is eliminated. Round 2 votes counts: D=30 B=26 E=24 C=20 so C is eliminated. Round 3 votes counts: E=39 D=31 B=30 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:206 D:201 A:198 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -8 -6 2 B -8 0 -4 -8 -8 C 8 4 0 4 -4 D 6 8 -4 0 -8 E -2 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428572 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -6 2 B -8 0 -4 -8 -8 C 8 4 0 4 -4 D 6 8 -4 0 -8 E -2 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428572 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -6 2 B -8 0 -4 -8 -8 C 8 4 0 4 -4 D 6 8 -4 0 -8 E -2 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428572 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 22: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E A D B C (6) B C E A D (6) B C A E D (6) D A E C B (5) D A C E B (5) B E A C D (5) A D E B C (5) D E A C B (3) C D E B A (3) C B E D A (3) B C E D A (3) B A E C D (3) E B A D C (2) E B A C D (2) E A B D C (2) D C A E B (2) C E D B A (2) C D E A B (2) C D B A E (2) C D A E B (2) C D A B E (2) C B A D E (2) B E C A D (2) A E D B C (2) A B E D C (2) E D C A B (1) E C D B A (1) D E C A B (1) D C E A B (1) C B D E A (1) B E A D C (1) B C A D E (1) B A E D C (1) B A C D E (1) A D C B E (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -2 6 6 B 10 0 6 4 6 C 2 -6 0 14 8 D -6 -4 -14 0 0 E -6 -6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 6 6 B 10 0 6 4 6 C 2 -6 0 14 8 D -6 -4 -14 0 0 E -6 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=27 D=17 E=14 A=13 so A is eliminated. Round 2 votes counts: B=32 C=27 D=25 E=16 so E is eliminated. Round 3 votes counts: B=38 D=34 C=28 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:209 A:200 E:190 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 6 6 B 10 0 6 4 6 C 2 -6 0 14 8 D -6 -4 -14 0 0 E -6 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 6 6 B 10 0 6 4 6 C 2 -6 0 14 8 D -6 -4 -14 0 0 E -6 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 6 6 B 10 0 6 4 6 C 2 -6 0 14 8 D -6 -4 -14 0 0 E -6 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 23: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) B C D A E (6) B C A D E (6) E A D C B (5) D E C B A (5) D B C A E (5) D A E B C (4) A E D B C (4) A E C B D (4) A D B E C (4) E C A B D (3) C B D E A (3) C B D A E (3) A E B C D (3) E D A C B (2) E C D B A (2) D E B C A (2) D C B E A (2) D B A C E (2) D A B C E (2) C E B A D (2) C B E D A (2) C B E A D (2) C B A E D (2) B D C A E (2) A D E B C (2) A D B C E (2) A B D C E (2) E D C B A (1) E D C A B (1) E D A B C (1) E C B A D (1) E A C D B (1) D B C E A (1) D A B E C (1) C E B D A (1) A C E B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 4 14 B 0 0 2 2 -2 C 4 -2 0 -2 -6 D -4 -2 2 0 8 E -14 2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.276705 B: 0.723295 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.599721335561 Cumulative probabilities = A: 0.276705 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 4 14 B 0 0 2 2 -2 C 4 -2 0 -2 -6 D -4 -2 2 0 8 E -14 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555594775 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=23 C=15 B=14 so B is eliminated. Round 2 votes counts: C=27 D=26 A=24 E=23 so E is eliminated. Round 3 votes counts: A=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:207 D:202 B:201 C:197 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -4 4 14 B 0 0 2 2 -2 C 4 -2 0 -2 -6 D -4 -2 2 0 8 E -14 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555594775 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 14 B 0 0 2 2 -2 C 4 -2 0 -2 -6 D -4 -2 2 0 8 E -14 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555594775 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 14 B 0 0 2 2 -2 C 4 -2 0 -2 -6 D -4 -2 2 0 8 E -14 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555594775 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 24: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) A B E C D (8) D C E B A (7) B A C E D (6) D C B E A (4) A B D E C (4) E C D A B (3) E C B A D (3) E C A B D (3) D E C A B (3) C E B D A (3) B A E C D (3) A D B E C (3) A B E D C (3) D A B C E (2) C E D B A (2) C E D A B (2) C D E B A (2) B E A C D (2) B C E A D (2) B C A E D (2) B A D C E (2) A E B C D (2) E D C A B (1) E C B D A (1) E A C D B (1) E A C B D (1) E A B C D (1) D B C A E (1) D B A C E (1) D A E C B (1) D A E B C (1) C B E D A (1) B E C A D (1) B D C A E (1) B C D E A (1) B C A D E (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 8 -8 6 -8 B -8 0 0 4 -4 C 8 0 0 4 0 D -6 -4 -4 0 -10 E 8 4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.465789 D: 0.000000 E: 0.534211 Sum of squares = 0.50234077347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.465789 D: 0.465789 E: 1.000000 A B C D E A 0 8 -8 6 -8 B -8 0 0 4 -4 C 8 0 0 4 0 D -6 -4 -4 0 -10 E 8 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=25 B=21 E=14 C=10 so C is eliminated. Round 2 votes counts: D=32 A=25 B=22 E=21 so E is eliminated. Round 3 votes counts: D=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:211 C:206 A:199 B:196 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 6 -8 B -8 0 0 4 -4 C 8 0 0 4 0 D -6 -4 -4 0 -10 E 8 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 6 -8 B -8 0 0 4 -4 C 8 0 0 4 0 D -6 -4 -4 0 -10 E 8 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 6 -8 B -8 0 0 4 -4 C 8 0 0 4 0 D -6 -4 -4 0 -10 E 8 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 25: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) C D B E A (8) E B A C D (6) D C A B E (6) C D A B E (6) A D C E B (5) B E D C A (4) E B C A D (3) E B A D C (3) E A B D C (3) D C B A E (3) C D A E B (3) B E C D A (3) B E A C D (3) E A B C D (2) D B C A E (2) C D B A E (2) C B D E A (2) B E C A D (2) B E A D C (2) B D C E A (2) B C D E A (2) A E C D B (2) E C B A D (1) E C A B D (1) E A C B D (1) D A C E B (1) D A C B E (1) C E D A B (1) C D E A B (1) B D E C A (1) B D A E C (1) B C E D A (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E C B (1) A D B E C (1) A C D E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -24 -16 -4 B 2 0 -10 -8 8 C 24 10 0 2 10 D 16 8 -2 0 16 E 4 -8 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -24 -16 -4 B 2 0 -10 -8 8 C 24 10 0 2 10 D 16 8 -2 0 16 E 4 -8 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 D=21 B=21 E=20 A=15 so A is eliminated. Round 2 votes counts: D=28 E=25 C=24 B=23 so B is eliminated. Round 3 votes counts: E=40 D=33 C=27 so C is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:223 D:219 B:196 E:185 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -24 -16 -4 B 2 0 -10 -8 8 C 24 10 0 2 10 D 16 8 -2 0 16 E 4 -8 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -24 -16 -4 B 2 0 -10 -8 8 C 24 10 0 2 10 D 16 8 -2 0 16 E 4 -8 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -24 -16 -4 B 2 0 -10 -8 8 C 24 10 0 2 10 D 16 8 -2 0 16 E 4 -8 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 26: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) A D C B E (6) A C B D E (6) E B D C A (5) C B D A E (5) A D E C B (5) A C D B E (5) C D B A E (4) B C E D A (4) A E B C D (4) E A D B C (3) E A B C D (3) A E D B C (3) A B C E D (3) E D C B A (2) E B C D A (2) E B A C D (2) D C B E A (2) D C A B E (2) D A E C B (2) B C E A D (2) B C A E D (2) B C A D E (2) E D A B C (1) E B C A D (1) E A B D C (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B A E (1) D B C E A (1) D A C E B (1) C D B E A (1) C B D E A (1) C B A D E (1) B D C E A (1) B C D E A (1) B A C E D (1) A E D C B (1) A D C E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 8 14 B 2 0 2 -2 6 C 0 -2 0 4 10 D -8 2 -4 0 2 E -14 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 8 14 B 2 0 2 -2 6 C 0 -2 0 4 10 D -8 2 -4 0 2 E -14 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998419 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=27 B=13 D=12 C=12 so D is eliminated. Round 2 votes counts: A=39 E=29 C=18 B=14 so B is eliminated. Round 3 votes counts: A=40 C=31 E=29 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:206 B:204 D:196 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 8 14 B 2 0 2 -2 6 C 0 -2 0 4 10 D -8 2 -4 0 2 E -14 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998419 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 8 14 B 2 0 2 -2 6 C 0 -2 0 4 10 D -8 2 -4 0 2 E -14 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998419 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 8 14 B 2 0 2 -2 6 C 0 -2 0 4 10 D -8 2 -4 0 2 E -14 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998419 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 27: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) B E D A C (9) E B D A C (6) C A D E B (6) C A B E D (5) C B E A D (4) C A D B E (4) C A B D E (4) B E D C A (4) A C D E B (4) A C D B E (4) D E B A C (3) A D C E B (3) E D A B C (2) E B D C A (2) E B C D A (2) D E A B C (2) D A E C B (2) C B A E D (2) B E C D A (2) B C E A D (2) B C A E D (2) B A D E C (2) A D C B E (2) E D A C B (1) E C D A B (1) D B E A C (1) D A E B C (1) C E B A D (1) C B A D E (1) B D E A C (1) B D A E C (1) B C E D A (1) B A C D E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 14 -2 -8 B 14 0 10 6 8 C -14 -10 0 -6 -4 D 2 -6 6 0 -12 E 8 -8 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 14 -2 -8 B 14 0 10 6 8 C -14 -10 0 -6 -4 D 2 -6 6 0 -12 E 8 -8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=25 E=24 A=15 D=9 so D is eliminated. Round 2 votes counts: E=29 C=27 B=26 A=18 so A is eliminated. Round 3 votes counts: C=41 E=32 B=27 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:208 A:195 D:195 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 14 -2 -8 B 14 0 10 6 8 C -14 -10 0 -6 -4 D 2 -6 6 0 -12 E 8 -8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 14 -2 -8 B 14 0 10 6 8 C -14 -10 0 -6 -4 D 2 -6 6 0 -12 E 8 -8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 14 -2 -8 B 14 0 10 6 8 C -14 -10 0 -6 -4 D 2 -6 6 0 -12 E 8 -8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 28: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) D E C A B (6) E D B A C (5) A C B D E (5) D E B A C (4) B A C E D (4) A B C D E (4) E B D A C (3) D C E A B (3) D A C B E (3) C E A B D (3) C D E A B (3) B E A D C (3) B A E C D (3) B A C D E (3) E D C B A (2) E C D B A (2) E C D A B (2) D E C B A (2) D C A B E (2) C D A B E (2) C A D E B (2) C A B E D (2) C A B D E (2) B E C A D (2) B E A C D (2) B D A E C (2) A B D C E (2) E D C A B (1) E D B C A (1) E B D C A (1) E B C A D (1) E B A C D (1) D E A C B (1) D E A B C (1) D A C E B (1) C E A D B (1) C B A E D (1) B E D A C (1) B C A E D (1) B A D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 4 -2 B -10 0 -8 -4 8 C 0 8 0 10 8 D -4 4 -10 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.425216 B: 0.000000 C: 0.574784 D: 0.000000 E: 0.000000 Sum of squares = 0.511185292954 Cumulative probabilities = A: 0.425216 B: 0.425216 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 4 -2 B -10 0 -8 -4 8 C 0 8 0 10 8 D -4 4 -10 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=22 E=19 A=12 so A is eliminated. Round 2 votes counts: C=29 B=29 D=23 E=19 so E is eliminated. Round 3 votes counts: B=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:206 D:202 B:193 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 0 4 -2 B -10 0 -8 -4 8 C 0 8 0 10 8 D -4 4 -10 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 4 -2 B -10 0 -8 -4 8 C 0 8 0 10 8 D -4 4 -10 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 4 -2 B -10 0 -8 -4 8 C 0 8 0 10 8 D -4 4 -10 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 29: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (12) A B E C D (10) A C E B D (9) B E A C D (8) D C E B A (6) D C A E B (5) C A D E B (4) A E B C D (4) A C D E B (4) C D A E B (3) B D E A C (3) B A E C D (3) D B E A C (2) D B C E A (2) B E D C A (2) B E D A C (2) A D B C E (2) A C E D B (2) A C D B E (2) E B D C A (1) E B A C D (1) E A B C D (1) D E C B A (1) D C E A B (1) D C B E A (1) D C A B E (1) C E A B D (1) C D E A B (1) C A E D B (1) C A E B D (1) B E C A D (1) B E A D C (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 6 12 14 6 B -6 0 12 -2 8 C -12 -12 0 12 -6 D -14 2 -12 0 4 E -6 -8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 14 6 B -6 0 12 -2 8 C -12 -12 0 12 -6 D -14 2 -12 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=31 B=21 C=11 E=3 so E is eliminated. Round 2 votes counts: A=35 D=31 B=23 C=11 so C is eliminated. Round 3 votes counts: A=42 D=35 B=23 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:206 E:194 C:191 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 14 6 B -6 0 12 -2 8 C -12 -12 0 12 -6 D -14 2 -12 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 14 6 B -6 0 12 -2 8 C -12 -12 0 12 -6 D -14 2 -12 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 14 6 B -6 0 12 -2 8 C -12 -12 0 12 -6 D -14 2 -12 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 30: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) B C D A E (7) A D E B C (7) E A D C B (6) C B D A E (6) B D C A E (6) E A D B C (5) D B A E C (5) C B D E A (5) C E B A D (4) E C A D B (3) E C A B D (3) D B A C E (3) C E A B D (3) C B E D A (3) D A B E C (2) C A E B D (2) C A B D E (2) B D A C E (2) A E D C B (2) A E D B C (2) E D A B C (1) D E B A C (1) D E A B C (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) D A E B C (1) C E B D A (1) C B A D E (1) B D C E A (1) B A D C E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 0 2 4 B 0 0 -2 -4 -2 C 0 2 0 0 2 D -2 4 0 0 10 E -4 2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.465108 B: 0.000000 C: 0.534892 D: 0.000000 E: 0.000000 Sum of squares = 0.502434936486 Cumulative probabilities = A: 0.465108 B: 0.465108 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 2 4 B 0 0 -2 -4 -2 C 0 2 0 0 2 D -2 4 0 0 10 E -4 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 D=17 B=17 A=13 so A is eliminated. Round 2 votes counts: E=30 C=29 D=24 B=17 so B is eliminated. Round 3 votes counts: C=36 D=34 E=30 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:206 A:203 C:202 B:196 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 2 4 B 0 0 -2 -4 -2 C 0 2 0 0 2 D -2 4 0 0 10 E -4 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 4 B 0 0 -2 -4 -2 C 0 2 0 0 2 D -2 4 0 0 10 E -4 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 4 B 0 0 -2 -4 -2 C 0 2 0 0 2 D -2 4 0 0 10 E -4 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 31: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) D E A C B (7) B C A E D (7) A B C D E (7) C E B D A (5) B A D E C (5) B A C D E (5) E D C B A (4) B C A D E (4) E D B A C (3) D E C A B (3) D A E C B (3) C A B D E (3) A D B E C (3) E C D B A (2) D E A B C (2) C D E A B (2) B E C A D (2) B C E D A (2) A D E B C (2) A C B D E (2) E D A B C (1) E B D A C (1) E B C D A (1) D E B A C (1) D A E B C (1) C E D B A (1) C B E A D (1) C B A E D (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A D C (1) B E A C D (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 0 -2 B -2 0 6 4 0 C 2 -6 0 -2 -10 D 0 -4 2 0 12 E 2 0 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.608696 B: 0.108696 C: 0.086957 D: 0.173913 E: 0.021739 Sum of squares = 0.420604914765 Cumulative probabilities = A: 0.608696 B: 0.717391 C: 0.804348 D: 0.978261 E: 1.000000 A B C D E A 0 2 -2 0 -2 B -2 0 6 4 0 C 2 -6 0 -2 -10 D 0 -4 2 0 12 E 2 0 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.608696 B: 0.108696 C: 0.086957 D: 0.173913 E: 0.021739 Sum of squares = 0.420604914933 Cumulative probabilities = A: 0.608696 B: 0.717391 C: 0.804348 D: 0.978261 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=22 D=17 A=17 C=16 so C is eliminated. Round 2 votes counts: B=31 E=28 A=22 D=19 so D is eliminated. Round 3 votes counts: E=43 B=31 A=26 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:205 B:204 E:200 A:199 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 0 -2 B -2 0 6 4 0 C 2 -6 0 -2 -10 D 0 -4 2 0 12 E 2 0 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.608696 B: 0.108696 C: 0.086957 D: 0.173913 E: 0.021739 Sum of squares = 0.420604914933 Cumulative probabilities = A: 0.608696 B: 0.717391 C: 0.804348 D: 0.978261 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -2 B -2 0 6 4 0 C 2 -6 0 -2 -10 D 0 -4 2 0 12 E 2 0 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.608696 B: 0.108696 C: 0.086957 D: 0.173913 E: 0.021739 Sum of squares = 0.420604914933 Cumulative probabilities = A: 0.608696 B: 0.717391 C: 0.804348 D: 0.978261 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -2 B -2 0 6 4 0 C 2 -6 0 -2 -10 D 0 -4 2 0 12 E 2 0 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.608696 B: 0.108696 C: 0.086957 D: 0.173913 E: 0.021739 Sum of squares = 0.420604914933 Cumulative probabilities = A: 0.608696 B: 0.717391 C: 0.804348 D: 0.978261 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 32: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (15) C B D A E (11) E A D C B (8) C B D E A (6) A E B D C (5) B C D A E (4) A E D B C (4) E D A C B (3) E A C D B (3) C B E A D (3) A D E B C (3) E A B D C (2) E A B C D (2) D C B A E (2) D B C A E (2) C E D A B (2) C E B D A (2) C D B E A (2) C D B A E (2) B A D E C (2) A B D E C (2) E A C B D (1) D E A B C (1) D C E A B (1) D B A C E (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C B E D A (1) C B A E D (1) C B A D E (1) B C A D E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 16 14 14 -8 B -16 0 -2 -6 -10 C -14 2 0 -12 -10 D -14 6 12 0 -10 E 8 10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 14 14 -8 B -16 0 -2 -6 -10 C -14 2 0 -12 -10 D -14 6 12 0 -10 E 8 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=33 A=16 D=10 B=7 so B is eliminated. Round 2 votes counts: C=38 E=34 A=18 D=10 so D is eliminated. Round 3 votes counts: C=43 E=35 A=22 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:218 D:197 B:183 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 14 14 -8 B -16 0 -2 -6 -10 C -14 2 0 -12 -10 D -14 6 12 0 -10 E 8 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 14 -8 B -16 0 -2 -6 -10 C -14 2 0 -12 -10 D -14 6 12 0 -10 E 8 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 14 -8 B -16 0 -2 -6 -10 C -14 2 0 -12 -10 D -14 6 12 0 -10 E 8 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 33: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) E B A C D (8) B A E D C (8) B E A C D (6) A B E D C (6) E B C A D (4) E A B C D (4) A D C E B (4) D C B A E (3) D A B C E (3) C D E A B (3) B A D E C (3) A E B C D (3) E C A D B (2) E A C D B (2) C E D B A (2) C D E B A (2) C D A E B (2) B E A D C (2) A E C D B (2) A E B D C (2) E C D B A (1) E C D A B (1) E B C D A (1) E A C B D (1) D B C A E (1) D A C B E (1) C E D A B (1) C B E D A (1) B D C E A (1) B D A C E (1) B C D E A (1) B A D C E (1) A D E C B (1) A D E B C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 22 24 14 B -8 0 14 8 6 C -22 -14 0 -6 -18 D -24 -8 6 0 -14 E -14 -6 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 22 24 14 B -8 0 14 8 6 C -22 -14 0 -6 -18 D -24 -8 6 0 -14 E -14 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=23 D=21 A=21 C=11 so C is eliminated. Round 2 votes counts: D=28 E=27 B=24 A=21 so A is eliminated. Round 3 votes counts: D=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:234 B:210 E:206 D:180 C:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 22 24 14 B -8 0 14 8 6 C -22 -14 0 -6 -18 D -24 -8 6 0 -14 E -14 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 22 24 14 B -8 0 14 8 6 C -22 -14 0 -6 -18 D -24 -8 6 0 -14 E -14 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 22 24 14 B -8 0 14 8 6 C -22 -14 0 -6 -18 D -24 -8 6 0 -14 E -14 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 34: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) B D E C A (7) B D C A E (7) E B C A D (6) E A C D B (6) D B A C E (6) E C A B D (5) B E D C A (5) B D A C E (5) A C E D B (5) E C A D B (4) D A C B E (4) B D E A C (4) A D C B E (4) E B D C A (3) A C D B E (3) E C B A D (2) E B C D A (2) B D C E A (2) E B D A C (1) E B A D C (1) E A C B D (1) D A B C E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 8 2 -2 B 6 0 4 6 0 C -8 -4 0 -4 2 D -2 -6 4 0 12 E 2 0 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.780877 C: 0.000000 D: 0.000000 E: 0.219123 Sum of squares = 0.657783444062 Cumulative probabilities = A: 0.000000 B: 0.780877 C: 0.780877 D: 0.780877 E: 1.000000 A B C D E A 0 -6 8 2 -2 B 6 0 4 6 0 C -8 -4 0 -4 2 D -2 -6 4 0 12 E 2 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555566732 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=31 A=24 D=11 C=2 so C is eliminated. Round 2 votes counts: B=32 E=31 A=26 D=11 so D is eliminated. Round 3 votes counts: B=38 E=31 A=31 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:204 A:201 E:194 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 2 -2 B 6 0 4 6 0 C -8 -4 0 -4 2 D -2 -6 4 0 12 E 2 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555566732 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 2 -2 B 6 0 4 6 0 C -8 -4 0 -4 2 D -2 -6 4 0 12 E 2 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555566732 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 2 -2 B 6 0 4 6 0 C -8 -4 0 -4 2 D -2 -6 4 0 12 E 2 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555566732 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 35: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) A E B C D (8) E D A C B (5) D E C A B (5) D C E A B (5) B E A D C (4) A E D C B (4) A E C D B (4) E A B D C (3) D C B E A (3) D B C E A (3) C D B A E (3) C D A E B (3) B A C E D (3) E D C A B (2) E D B A C (2) E B A D C (2) D C E B A (2) C B D A E (2) C B A D E (2) B E D A C (2) B D C E A (2) B C D A E (2) B C A D E (2) A C D E B (2) A B C E D (2) E B D C A (1) E A D C B (1) E A D B C (1) D E C B A (1) D E B C A (1) C D B E A (1) C A D B E (1) B E D C A (1) B D E C A (1) B C D E A (1) A E B D C (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 12 4 2 B -2 0 0 0 -10 C -12 0 0 -4 -14 D -4 0 4 0 -14 E -2 10 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 4 2 B -2 0 0 0 -10 C -12 0 0 -4 -14 D -4 0 4 0 -14 E -2 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=25 D=20 E=17 C=12 so C is eliminated. Round 2 votes counts: B=30 D=27 A=26 E=17 so E is eliminated. Round 3 votes counts: D=36 B=33 A=31 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:218 A:210 B:194 D:193 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 4 2 B -2 0 0 0 -10 C -12 0 0 -4 -14 D -4 0 4 0 -14 E -2 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 4 2 B -2 0 0 0 -10 C -12 0 0 -4 -14 D -4 0 4 0 -14 E -2 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 4 2 B -2 0 0 0 -10 C -12 0 0 -4 -14 D -4 0 4 0 -14 E -2 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 36: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (17) C D A B E (11) D A C B E (10) E B A D C (6) E D A B C (5) E A B D C (5) C D A E B (5) E A D B C (4) E B A C D (3) C B E D A (3) C B D A E (3) D A E C B (2) D A C E B (2) C E B D A (2) B C E A D (2) B A D C E (2) A D B E C (2) A D B C E (2) E D A C B (1) E C B D A (1) E B C D A (1) C E D A B (1) C D B A E (1) C B A D E (1) C A B D E (1) B E A D C (1) B E A C D (1) B C A E D (1) B C A D E (1) A D E B C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -2 4 -6 B -8 0 10 4 -12 C 2 -10 0 10 -2 D -4 -4 -10 0 -8 E 6 12 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -2 4 -6 B -8 0 10 4 -12 C 2 -10 0 10 -2 D -4 -4 -10 0 -8 E 6 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 C=28 D=14 B=8 A=7 so A is eliminated. Round 2 votes counts: E=43 C=28 D=20 B=9 so B is eliminated. Round 3 votes counts: E=45 C=32 D=23 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:202 C:200 B:197 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -2 4 -6 B -8 0 10 4 -12 C 2 -10 0 10 -2 D -4 -4 -10 0 -8 E 6 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 4 -6 B -8 0 10 4 -12 C 2 -10 0 10 -2 D -4 -4 -10 0 -8 E 6 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 4 -6 B -8 0 10 4 -12 C 2 -10 0 10 -2 D -4 -4 -10 0 -8 E 6 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 37: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) D A C B E (9) E C B A D (6) B C E A D (5) D A B C E (4) E D A C B (3) E C A B D (3) E A C D B (3) D E B C A (3) D A C E B (3) B E C A D (3) B C D A E (3) A E C D B (3) A C B E D (3) E D B C A (2) E B C D A (2) D E A C B (2) D E A B C (2) D B E C A (2) D B C A E (2) D B A C E (2) D A E B C (2) B D C E A (2) B D C A E (2) B C A E D (2) A D C E B (2) A D C B E (2) E B D C A (1) E A C B D (1) D E B A C (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) C A B E D (1) B E D C A (1) B E C D A (1) B C D E A (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 8 -18 6 B -12 0 -14 -14 -2 C -8 14 0 -12 0 D 18 14 12 0 10 E -6 2 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -18 6 B -12 0 -14 -14 -2 C -8 14 0 -12 0 D 18 14 12 0 10 E -6 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=21 B=20 A=13 C=5 so C is eliminated. Round 2 votes counts: D=41 E=23 B=22 A=14 so A is eliminated. Round 3 votes counts: D=46 E=27 B=27 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:204 C:197 E:193 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -18 6 B -12 0 -14 -14 -2 C -8 14 0 -12 0 D 18 14 12 0 10 E -6 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -18 6 B -12 0 -14 -14 -2 C -8 14 0 -12 0 D 18 14 12 0 10 E -6 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -18 6 B -12 0 -14 -14 -2 C -8 14 0 -12 0 D 18 14 12 0 10 E -6 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 38: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) D B C A E (7) C A D E B (7) A C D E B (7) D C A B E (6) B D E A C (6) A C E D B (5) E B A D C (4) E A C B D (4) C A D B E (4) B D E C A (4) E B D A C (3) E B A C D (3) E A B C D (3) D B A C E (3) C D A B E (3) B E D A C (3) A E C B D (3) E C B A D (2) E C A B D (2) E B C D A (1) E B C A D (1) D C B A E (1) D B C E A (1) D A C B E (1) C D B A E (1) B E D C A (1) B E A D C (1) A E D B C (1) A E C D B (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 16 2 18 22 B -16 0 -16 -18 -12 C -2 16 0 12 14 D -18 18 -12 0 8 E -22 12 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 18 22 B -16 0 -16 -18 -12 C -2 16 0 12 14 D -18 18 -12 0 8 E -22 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=23 C=23 A=20 D=19 B=15 so B is eliminated. Round 2 votes counts: D=29 E=28 C=23 A=20 so A is eliminated. Round 3 votes counts: C=36 E=33 D=31 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:229 C:220 D:198 E:184 B:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 18 22 B -16 0 -16 -18 -12 C -2 16 0 12 14 D -18 18 -12 0 8 E -22 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 18 22 B -16 0 -16 -18 -12 C -2 16 0 12 14 D -18 18 -12 0 8 E -22 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 18 22 B -16 0 -16 -18 -12 C -2 16 0 12 14 D -18 18 -12 0 8 E -22 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 39: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) A D B E C (9) E B D C A (8) C A D B E (6) A D B C E (6) C E B D A (5) A D C B E (5) A C D B E (4) E B C D A (3) C E A B D (3) E B D A C (2) E B A D C (2) D A B E C (2) D A B C E (2) C B E D A (2) C A E D B (2) B E D A C (2) A E C B D (2) A E B D C (2) A D E B C (2) A C E D B (2) E C B A D (1) E C A B D (1) E A B D C (1) E A B C D (1) D B A C E (1) D A C B E (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D E A (1) B E D C A (1) B D E A C (1) A E D B C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 12 0 8 2 B -12 0 -8 -2 -8 C 0 8 0 4 -4 D -8 2 -4 0 -10 E -2 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.812341 B: 0.000000 C: 0.187659 D: 0.000000 E: 0.000000 Sum of squares = 0.695114039285 Cumulative probabilities = A: 0.812341 B: 0.812341 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 8 2 B -12 0 -8 -2 -8 C 0 8 0 4 -4 D -8 2 -4 0 -10 E -2 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555814265 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=29 C=26 D=6 B=4 so B is eliminated. Round 2 votes counts: A=35 E=32 C=26 D=7 so D is eliminated. Round 3 votes counts: A=41 E=33 C=26 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:210 C:204 D:190 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 8 2 B -12 0 -8 -2 -8 C 0 8 0 4 -4 D -8 2 -4 0 -10 E -2 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555814265 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 8 2 B -12 0 -8 -2 -8 C 0 8 0 4 -4 D -8 2 -4 0 -10 E -2 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555814265 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 8 2 B -12 0 -8 -2 -8 C 0 8 0 4 -4 D -8 2 -4 0 -10 E -2 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555814265 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 40: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) A C D E B (7) E B A C D (6) C D A E B (6) B E D C A (6) B E A D C (6) E B C A D (4) D C B A E (4) D C A B E (4) D A B C E (4) B E C D A (4) A E C B D (4) A C E D B (4) D B C E A (3) C D E A B (3) C A E D B (3) B D E C A (3) B D E A C (3) E B C D A (2) C E A B D (2) C D E B A (2) A B E D C (2) E C B A D (1) E B A D C (1) E A B C D (1) D C B E A (1) D B E A C (1) D A C B E (1) C E D B A (1) C E B D A (1) B E D A C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 0 0 0 -4 B 0 0 10 8 -12 C 0 -10 0 16 -6 D 0 -8 -16 0 -12 E 4 12 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 0 -4 B 0 0 10 8 -12 C 0 -10 0 16 -6 D 0 -8 -16 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=23 D=18 C=18 E=15 so E is eliminated. Round 2 votes counts: B=36 A=27 C=19 D=18 so D is eliminated. Round 3 votes counts: B=40 A=32 C=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 B:203 C:200 A:198 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 0 -4 B 0 0 10 8 -12 C 0 -10 0 16 -6 D 0 -8 -16 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 -4 B 0 0 10 8 -12 C 0 -10 0 16 -6 D 0 -8 -16 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 -4 B 0 0 10 8 -12 C 0 -10 0 16 -6 D 0 -8 -16 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 41: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) E B C A D (8) D A C E B (6) D A C B E (5) A D B E C (5) D E C B A (4) D C E B A (4) D C A E B (4) C D E B A (4) A C D B E (4) A C B E D (4) E B D C A (3) B C E A D (3) B A E C D (3) A D C B E (3) D E B A C (2) D C E A B (2) D A B E C (2) C E B D A (2) C D A E B (2) C A B E D (2) A D B C E (2) A B C E D (2) E D B C A (1) E C B A D (1) E B C D A (1) D E A B C (1) C E B A D (1) C D E A B (1) C B E A D (1) B E C A D (1) B E A C D (1) B A E D C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -12 -10 -4 B 4 0 0 -26 -16 C 12 0 0 -16 4 D 10 26 16 0 26 E 4 16 -4 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -10 -4 B 4 0 0 -26 -16 C 12 0 0 -16 4 D 10 26 16 0 26 E 4 16 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=23 E=14 C=13 B=9 so B is eliminated. Round 2 votes counts: D=41 A=27 E=16 C=16 so E is eliminated. Round 3 votes counts: D=45 A=28 C=27 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:239 C:200 E:195 A:185 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -12 -10 -4 B 4 0 0 -26 -16 C 12 0 0 -16 4 D 10 26 16 0 26 E 4 16 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -10 -4 B 4 0 0 -26 -16 C 12 0 0 -16 4 D 10 26 16 0 26 E 4 16 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -10 -4 B 4 0 0 -26 -16 C 12 0 0 -16 4 D 10 26 16 0 26 E 4 16 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 42: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) E C B D A (5) D C A E B (5) C E D A B (5) B E A D C (5) A D B C E (5) B E D A C (4) B E A C D (4) B A D E C (4) E B C D A (3) D A C B E (3) C D A E B (3) C A D E B (3) B E D C A (3) B A E D C (3) A D C B E (3) A C D E B (3) A C B E D (3) A B D C E (3) E D C B A (2) E B D C A (2) C E A D B (2) C D E A B (2) B A E C D (2) A B C E D (2) A B C D E (2) E C B A D (1) D C E B A (1) D A B C E (1) C E B A D (1) C E A B D (1) C A E D B (1) C A E B D (1) B E C D A (1) B E C A D (1) B D E A C (1) A D C E B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 2 6 0 B -2 0 -8 6 2 C -2 8 0 8 6 D -6 -6 -8 0 -18 E 0 -2 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.845375 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.154625 Sum of squares = 0.738567694459 Cumulative probabilities = A: 0.845375 B: 0.845375 C: 0.845375 D: 0.845375 E: 1.000000 A B C D E A 0 2 2 6 0 B -2 0 -8 6 2 C -2 8 0 8 6 D -6 -6 -8 0 -18 E 0 -2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000004218 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=24 E=19 C=19 D=10 so D is eliminated. Round 2 votes counts: B=28 A=28 C=25 E=19 so E is eliminated. Round 3 votes counts: C=39 B=33 A=28 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:205 E:205 B:199 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 6 0 B -2 0 -8 6 2 C -2 8 0 8 6 D -6 -6 -8 0 -18 E 0 -2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000004218 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 6 0 B -2 0 -8 6 2 C -2 8 0 8 6 D -6 -6 -8 0 -18 E 0 -2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000004218 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 6 0 B -2 0 -8 6 2 C -2 8 0 8 6 D -6 -6 -8 0 -18 E 0 -2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000004218 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 43: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E D A B C (7) E A D B C (6) D E A C B (5) C B A D E (5) B C A E D (5) B A E C D (5) B A C E D (5) E B A D C (4) E A B D C (4) D C E B A (4) D C A E B (4) C D B A E (4) E D B A C (3) D E C A B (3) D E B C A (3) D E A B C (3) C D A B E (3) C D B E A (2) C D A E B (2) C B A E D (2) A E B C D (2) A B E C D (2) E D B C A (1) E A B C D (1) D E C B A (1) B E A C D (1) B C E A D (1) B C D E A (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 0 -10 -18 B -6 0 10 -22 -28 C 0 -10 0 -16 -4 D 10 22 16 0 0 E 18 28 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.757599 E: 0.242401 Sum of squares = 0.632713999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.757599 E: 1.000000 A B C D E A 0 6 0 -10 -18 B -6 0 10 -22 -28 C 0 -10 0 -16 -4 D 10 22 16 0 0 E 18 28 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=26 C=18 B=18 A=6 so A is eliminated. Round 2 votes counts: D=32 E=29 B=20 C=19 so C is eliminated. Round 3 votes counts: D=44 E=29 B=27 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:225 D:224 A:189 C:185 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -10 -18 B -6 0 10 -22 -28 C 0 -10 0 -16 -4 D 10 22 16 0 0 E 18 28 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -10 -18 B -6 0 10 -22 -28 C 0 -10 0 -16 -4 D 10 22 16 0 0 E 18 28 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -10 -18 B -6 0 10 -22 -28 C 0 -10 0 -16 -4 D 10 22 16 0 0 E 18 28 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 44: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B D E C A (8) B D E A C (7) A D E B C (7) C B E D A (6) C E A D B (5) A C E D B (5) E D B C A (4) B C E D A (4) A D B E C (4) A C D E B (4) C E D A B (3) C E B D A (3) A B D E C (3) D B E A C (2) C E D B A (2) B D A E C (2) B A D E C (2) A D E C B (2) A C B D E (2) E D B A C (1) E B D C A (1) E A D B C (1) D E B A C (1) D E A B C (1) C E B A D (1) C B D E A (1) C B A D E (1) C A E B D (1) C A B E D (1) C A B D E (1) B C D E A (1) B A D C E (1) A E C D B (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 6 -4 B -4 0 4 -4 -2 C 2 -4 0 4 6 D -6 4 -4 0 6 E 4 2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999915 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 6 -4 B -4 0 4 -4 -2 C 2 -4 0 4 6 D -6 4 -4 0 6 E 4 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999953 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=31 B=25 E=7 D=4 so D is eliminated. Round 2 votes counts: C=33 A=31 B=27 E=9 so E is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:204 A:202 D:200 B:197 E:197 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 6 -4 B -4 0 4 -4 -2 C 2 -4 0 4 6 D -6 4 -4 0 6 E 4 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999953 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 -4 B -4 0 4 -4 -2 C 2 -4 0 4 6 D -6 4 -4 0 6 E 4 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999953 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 -4 B -4 0 4 -4 -2 C 2 -4 0 4 6 D -6 4 -4 0 6 E 4 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999953 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 45: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) A E D C B (9) A C B D E (7) E D B C A (6) B C D E A (6) E D B A C (5) D E B C A (4) B D C E A (4) E D C B A (3) E D A B C (3) C B D A E (3) C A B D E (3) B D E C A (3) A C E D B (3) A C D E B (3) E A D B C (2) D E C B A (2) D B E C A (2) C D E A B (2) C B D E A (2) C B A D E (2) A E B D C (2) D E C A B (1) D C E B A (1) C D E B A (1) C D B E A (1) B E D A C (1) B E A D C (1) B A E D C (1) B A E C D (1) B A C D E (1) A E C D B (1) A C B E D (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 8 0 0 B 0 0 10 -18 -16 C -8 -10 0 -20 -16 D 0 18 20 0 -2 E 0 16 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.480233 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.519767 Sum of squares = 0.500781460176 Cumulative probabilities = A: 0.480233 B: 0.480233 C: 0.480233 D: 0.480233 E: 1.000000 A B C D E A 0 0 8 0 0 B 0 0 10 -18 -16 C -8 -10 0 -20 -16 D 0 18 20 0 -2 E 0 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=19 B=18 C=14 D=10 so D is eliminated. Round 2 votes counts: A=39 E=26 B=20 C=15 so C is eliminated. Round 3 votes counts: A=42 E=30 B=28 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:218 E:217 A:204 B:188 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 0 0 B 0 0 10 -18 -16 C -8 -10 0 -20 -16 D 0 18 20 0 -2 E 0 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 0 0 B 0 0 10 -18 -16 C -8 -10 0 -20 -16 D 0 18 20 0 -2 E 0 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 0 0 B 0 0 10 -18 -16 C -8 -10 0 -20 -16 D 0 18 20 0 -2 E 0 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 46: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) A C B D E (7) A E B D C (6) A C B E D (5) D E C B A (4) B D E C A (4) B C A D E (4) E A D C B (3) D E B C A (3) A E C D B (3) A B E D C (3) E D C A B (2) E D A C B (2) E B D A C (2) D B E C A (2) C D E B A (2) C D B A E (2) C A D B E (2) C A B D E (2) B E D A C (2) B D C E A (2) B C D E A (2) B A E D C (2) A E D B C (2) A C E D B (2) A B C D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D A B (1) E A D B C (1) D E C A B (1) D C B E A (1) C D B E A (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D C A (1) B E A D C (1) B A C E D (1) B A C D E (1) A E D C B (1) A E C B D (1) A E B C D (1) A C E B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 16 18 10 6 B -16 0 4 6 0 C -18 -4 0 -10 -16 D -10 -6 10 0 -8 E -6 0 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 10 6 B -16 0 4 6 0 C -18 -4 0 -10 -16 D -10 -6 10 0 -8 E -6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=21 B=20 C=12 D=11 so D is eliminated. Round 2 votes counts: A=36 E=29 B=22 C=13 so C is eliminated. Round 3 votes counts: A=41 E=31 B=28 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:209 B:197 D:193 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 10 6 B -16 0 4 6 0 C -18 -4 0 -10 -16 D -10 -6 10 0 -8 E -6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 10 6 B -16 0 4 6 0 C -18 -4 0 -10 -16 D -10 -6 10 0 -8 E -6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 10 6 B -16 0 4 6 0 C -18 -4 0 -10 -16 D -10 -6 10 0 -8 E -6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 47: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) A B C D E (7) B A E D C (5) A B E C D (5) E C D B A (4) D E B C A (4) C E D A B (4) E D B C A (3) E D B A C (3) D C B A E (3) C E D B A (3) C D E A B (3) C D A B E (3) B E A D C (3) A C B D E (3) A B D E C (3) A B D C E (3) E B D A C (2) E B A D C (2) E A B D C (2) D B E A C (2) C D E B A (2) C A E D B (2) C A E B D (2) C A D B E (2) B E D A C (2) B A D E C (2) A C B E D (2) A B C E D (2) D E C B A (1) D E B A C (1) D C E B A (1) D B E C A (1) D B C E A (1) C A D E B (1) C A B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 0 -4 -6 B 8 0 8 -6 2 C 0 -8 0 -8 -10 D 4 6 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.459999999988 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 A B C D E A 0 -8 0 -4 -6 B 8 0 8 -6 2 C 0 -8 0 -8 -10 D 4 6 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.459999999962 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 C=23 D=14 B=12 so B is eliminated. Round 2 votes counts: A=33 E=30 C=23 D=14 so D is eliminated. Round 3 votes counts: E=39 A=33 C=28 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 B:206 D:203 A:191 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -4 -6 B 8 0 8 -6 2 C 0 -8 0 -8 -10 D 4 6 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.459999999962 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -4 -6 B 8 0 8 -6 2 C 0 -8 0 -8 -10 D 4 6 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.459999999962 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -4 -6 B 8 0 8 -6 2 C 0 -8 0 -8 -10 D 4 6 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.459999999962 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 48: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (23) E D B C A (20) E D B A C (12) D E B C A (6) C A B D E (6) A C B E D (6) B C A D E (4) E D A B C (3) D B E C A (3) C B A D E (3) A E C D B (2) A C E B D (2) E A D C B (1) E A B D C (1) D C B E A (1) B E D A C (1) B A C D E (1) A E C B D (1) A E B D C (1) A C E D B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 14 8 4 B 2 0 6 0 -2 C -14 -6 0 2 -2 D -8 0 -2 0 -2 E -4 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999989 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 14 8 4 B 2 0 6 0 -2 C -14 -6 0 2 -2 D -8 0 -2 0 -2 E -4 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999938 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=37 D=10 C=9 B=6 so B is eliminated. Round 2 votes counts: A=39 E=38 C=13 D=10 so D is eliminated. Round 3 votes counts: E=47 A=39 C=14 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:203 E:201 D:194 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 14 8 4 B 2 0 6 0 -2 C -14 -6 0 2 -2 D -8 0 -2 0 -2 E -4 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999938 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 8 4 B 2 0 6 0 -2 C -14 -6 0 2 -2 D -8 0 -2 0 -2 E -4 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999938 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 8 4 B 2 0 6 0 -2 C -14 -6 0 2 -2 D -8 0 -2 0 -2 E -4 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999938 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 49: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) E C B D A (6) B E D A C (5) E D B C A (4) A B D E C (4) D E C B A (3) D C E A B (3) C A B E D (3) B E D C A (3) B D A E C (3) B A E D C (3) B A C E D (3) A C D E B (3) A C D B E (3) E D C B A (2) E B C D A (2) D E C A B (2) D E B C A (2) D B E A C (2) C E D B A (2) C E A B D (2) C D E A B (2) B A E C D (2) B A D E C (2) A D B E C (2) A D B C E (2) A C B D E (2) A B D C E (2) A B C D E (2) E B D C A (1) E B D A C (1) D E B A C (1) D E A C B (1) D A C E B (1) D A B E C (1) C E D A B (1) C E B D A (1) C D A E B (1) C A E D B (1) C A E B D (1) B E C A D (1) B D E A C (1) B C E A D (1) A D C E B (1) A D C B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 0 -14 -10 B 14 0 2 6 0 C 0 -2 0 -6 -20 D 14 -6 6 0 -6 E 10 0 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.575048 C: 0.000000 D: 0.000000 E: 0.424952 Sum of squares = 0.511264327676 Cumulative probabilities = A: 0.000000 B: 0.575048 C: 0.575048 D: 0.575048 E: 1.000000 A B C D E A 0 -14 0 -14 -10 B 14 0 2 6 0 C 0 -2 0 -6 -20 D 14 -6 6 0 -6 E 10 0 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 E=22 D=16 C=14 so C is eliminated. Round 2 votes counts: A=29 E=28 B=24 D=19 so D is eliminated. Round 3 votes counts: E=42 A=32 B=26 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 B:211 D:204 C:186 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -14 -10 B 14 0 2 6 0 C 0 -2 0 -6 -20 D 14 -6 6 0 -6 E 10 0 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -14 -10 B 14 0 2 6 0 C 0 -2 0 -6 -20 D 14 -6 6 0 -6 E 10 0 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -14 -10 B 14 0 2 6 0 C 0 -2 0 -6 -20 D 14 -6 6 0 -6 E 10 0 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 50: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) D E C B A (9) B C D A E (6) D C B E A (5) B C A E D (4) A E C B D (4) A B D C E (4) E D C A B (3) D A B E C (3) C E B D A (3) B D C A E (3) B C A D E (3) A B E C D (3) E C A D B (2) E A C B D (2) D E C A B (2) D E A C B (2) D E A B C (2) D B C A E (2) D B A C E (2) C B D E A (2) B A C E D (2) E D A C B (1) E C D B A (1) E C D A B (1) E C A B D (1) D C E B A (1) D A E B C (1) C E D B A (1) C D E B A (1) C B E D A (1) C B E A D (1) C B A E D (1) B C D E A (1) B A C D E (1) A E D C B (1) A E D B C (1) A D B E C (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 -6 16 B 0 0 8 14 22 C 8 -8 0 12 20 D 6 -14 -12 0 6 E -16 -22 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.270572 B: 0.729428 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.605274587792 Cumulative probabilities = A: 0.270572 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -6 16 B 0 0 8 14 22 C 8 -8 0 12 20 D 6 -14 -12 0 6 E -16 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499501 B: 0.500499 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000497052 Cumulative probabilities = A: 0.499501 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=29 B=20 E=11 C=10 so C is eliminated. Round 2 votes counts: D=30 A=30 B=25 E=15 so E is eliminated. Round 3 votes counts: D=37 A=35 B=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:222 C:216 A:201 D:193 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -6 16 B 0 0 8 14 22 C 8 -8 0 12 20 D 6 -14 -12 0 6 E -16 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499501 B: 0.500499 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000497052 Cumulative probabilities = A: 0.499501 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -6 16 B 0 0 8 14 22 C 8 -8 0 12 20 D 6 -14 -12 0 6 E -16 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499501 B: 0.500499 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000497052 Cumulative probabilities = A: 0.499501 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -6 16 B 0 0 8 14 22 C 8 -8 0 12 20 D 6 -14 -12 0 6 E -16 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499501 B: 0.500499 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000497052 Cumulative probabilities = A: 0.499501 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 51: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (6) C A D E B (5) B D E C A (5) A B D C E (5) E B D C A (4) E B A C D (4) E A C B D (4) D C B A E (4) D C A B E (4) B E A D C (4) A E C B D (4) C D E A B (3) C D A B E (3) B E D C A (3) B D A C E (3) A C E D B (3) E A B C D (2) D C B E A (2) C D A E B (2) B D C A E (2) A E C D B (2) A C D E B (2) A C D B E (2) E D B C A (1) E C A D B (1) E C A B D (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A D C (1) D B E C A (1) D B C E A (1) C E D A B (1) C D E B A (1) C D B E A (1) B D C E A (1) B D A E C (1) B A D E C (1) A E B C D (1) A D C B E (1) A C E B D (1) A B E D C (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 -2 12 B -2 0 6 8 8 C 8 -6 0 -6 8 D 2 -8 6 0 16 E -12 -8 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999984 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -2 12 B -2 0 6 8 8 C 8 -6 0 -6 8 D 2 -8 6 0 16 E -12 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000317 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 B=20 D=18 C=16 so C is eliminated. Round 2 votes counts: A=30 D=28 E=22 B=20 so B is eliminated. Round 3 votes counts: D=40 A=31 E=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:208 A:202 C:202 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -8 -2 12 B -2 0 6 8 8 C 8 -6 0 -6 8 D 2 -8 6 0 16 E -12 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000317 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 12 B -2 0 6 8 8 C 8 -6 0 -6 8 D 2 -8 6 0 16 E -12 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000317 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 12 B -2 0 6 8 8 C 8 -6 0 -6 8 D 2 -8 6 0 16 E -12 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000317 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 52: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) C E A D B (9) D B C E A (8) C E D A B (8) A E C B D (7) A C E D B (6) D C E A B (5) B A E C D (5) D B A C E (4) B A D E C (4) A B E C D (4) B D E C A (3) E C B A D (2) E C A B D (2) D E C B A (2) D C E B A (2) C D E A B (2) B E C D A (2) B E A C D (2) E C D B A (1) E C A D B (1) E A C B D (1) D C B E A (1) D B E C A (1) D A C E B (1) D A B C E (1) C E D B A (1) B A D C E (1) A E B C D (1) A C D E B (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -4 C 0 4 0 12 2 D 4 8 -12 0 -6 E 6 4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.194143 B: 0.000000 C: 0.805857 D: 0.000000 E: 0.000000 Sum of squares = 0.687097017971 Cumulative probabilities = A: 0.194143 B: 0.194143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -4 C 0 4 0 12 2 D 4 8 -12 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000097335 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 A=21 C=20 E=7 so E is eliminated. Round 2 votes counts: B=27 C=26 D=25 A=22 so A is eliminated. Round 3 votes counts: C=41 B=34 D=25 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 E:207 D:197 A:196 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -4 C 0 4 0 12 2 D 4 8 -12 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000097335 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -4 C 0 4 0 12 2 D 4 8 -12 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000097335 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -4 C 0 4 0 12 2 D 4 8 -12 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000097335 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 53: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C B A D E (9) B D E C A (9) B C D E A (8) B C D A E (6) E D B A C (5) E A D C B (5) C A B D E (4) A E C D B (4) A C E D B (4) E A D B C (3) B E D C A (3) A E D C B (3) A C D E B (3) E B D A C (2) D E B A C (2) D B E C A (2) C A B E D (2) B C A E D (2) A D E C B (2) A D C E B (2) E D A C B (1) D E A B C (1) C B A E D (1) C A D B E (1) B E D A C (1) B E C D A (1) B C E D A (1) B C A D E (1) Total count = 100 A B C D E A 0 -6 0 -8 -12 B 6 0 18 0 2 C 0 -18 0 -6 -12 D 8 0 6 0 0 E 12 -2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.435959 C: 0.000000 D: 0.564041 E: 0.000000 Sum of squares = 0.508202414071 Cumulative probabilities = A: 0.000000 B: 0.435959 C: 0.435959 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -8 -12 B 6 0 18 0 2 C 0 -18 0 -6 -12 D 8 0 6 0 0 E 12 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999879 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=28 A=18 C=17 D=5 so D is eliminated. Round 2 votes counts: B=34 E=31 A=18 C=17 so C is eliminated. Round 3 votes counts: B=44 E=31 A=25 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:211 D:207 A:187 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -8 -12 B 6 0 18 0 2 C 0 -18 0 -6 -12 D 8 0 6 0 0 E 12 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999879 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -8 -12 B 6 0 18 0 2 C 0 -18 0 -6 -12 D 8 0 6 0 0 E 12 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999879 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -8 -12 B 6 0 18 0 2 C 0 -18 0 -6 -12 D 8 0 6 0 0 E 12 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999879 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 54: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (12) D E C A B (6) A B C E D (6) D E A C B (5) B A C E D (5) D A E B C (4) B C A E D (4) A D B E C (4) A B D C E (4) E C D B A (3) E C D A B (3) D B E C A (3) C E B D A (3) C E B A D (3) B A D C E (3) E D C A B (2) E A D C B (2) D E B C A (2) D A E C B (2) C E A B D (2) C B E D A (2) B C E A D (2) B C D E A (2) B A C D E (2) A E C D B (2) E C A D B (1) D B A E C (1) C E D B A (1) C B E A D (1) B D C A E (1) B D A C E (1) B C D A E (1) B C A D E (1) A E C B D (1) A E B D C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -10 -8 -10 B 6 0 -6 -6 -12 C 10 6 0 -6 -8 D 8 6 6 0 8 E 10 12 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -8 -10 B 6 0 -6 -6 -12 C 10 6 0 -6 -8 D 8 6 6 0 8 E 10 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=22 A=20 C=12 E=11 so E is eliminated. Round 2 votes counts: D=37 B=22 A=22 C=19 so C is eliminated. Round 3 votes counts: D=44 B=31 A=25 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:211 C:201 B:191 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 -10 B 6 0 -6 -6 -12 C 10 6 0 -6 -8 D 8 6 6 0 8 E 10 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 -10 B 6 0 -6 -6 -12 C 10 6 0 -6 -8 D 8 6 6 0 8 E 10 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 -10 B 6 0 -6 -6 -12 C 10 6 0 -6 -8 D 8 6 6 0 8 E 10 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 55: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) D B A C E (7) D A B C E (7) E C B A D (6) E A B C D (5) C E B D A (5) A B E D C (4) E C D B A (3) E C A B D (3) E A D B C (3) E A B D C (3) D B C A E (3) D A B E C (3) C E D B A (3) C B D A E (3) A E D B C (3) A E B D C (3) E C D A B (2) E A C B D (2) D C E A B (2) C E B A D (2) C D E B A (2) C B E D A (2) B C D A E (2) B A D C E (2) A B D E C (2) E C B D A (1) D E A C B (1) D C B A E (1) D C A B E (1) C D B E A (1) B D C A E (1) B C A D E (1) B A D E C (1) A D E B C (1) Total count = 100 A B C D E A 0 8 12 0 8 B -8 0 20 -4 0 C -12 -20 0 -14 -10 D 0 4 14 0 0 E -8 0 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.556456 B: 0.000000 C: 0.000000 D: 0.443544 E: 0.000000 Sum of squares = 0.506374594658 Cumulative probabilities = A: 0.556456 B: 0.556456 C: 0.556456 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 0 8 B -8 0 20 -4 0 C -12 -20 0 -14 -10 D 0 4 14 0 0 E -8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=25 A=22 C=18 B=7 so B is eliminated. Round 2 votes counts: E=28 D=26 A=25 C=21 so C is eliminated. Round 3 votes counts: E=40 D=34 A=26 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:209 B:204 E:201 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 0 8 B -8 0 20 -4 0 C -12 -20 0 -14 -10 D 0 4 14 0 0 E -8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 0 8 B -8 0 20 -4 0 C -12 -20 0 -14 -10 D 0 4 14 0 0 E -8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 0 8 B -8 0 20 -4 0 C -12 -20 0 -14 -10 D 0 4 14 0 0 E -8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 56: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) C A B D E (6) B C A D E (6) E D C A B (5) A C B E D (5) E C A D B (4) E B D A C (4) B D C A E (4) A C E B D (4) E A C B D (3) E A B C D (3) D B C E A (3) C A E B D (3) B D E A C (3) A E B C D (3) E A C D B (2) D E C B A (2) D E B A C (2) D C B A E (2) D B C A E (2) C D A B E (2) C A B E D (2) B A E C D (2) B A C D E (2) E D B A C (1) E C D A B (1) E B A D C (1) E B A C D (1) E A B D C (1) D E B C A (1) D C E A B (1) D C B E A (1) D C A B E (1) D B E A C (1) C E D A B (1) C D A E B (1) C A D B E (1) B D A E C (1) B A C E D (1) A E C B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -14 6 2 B -4 0 2 16 10 C 14 -2 0 12 0 D -6 -16 -12 0 2 E -2 -10 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999992 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 6 2 B -4 0 2 16 10 C 14 -2 0 12 0 D -6 -16 -12 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999993 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 B=19 C=16 A=15 so A is eliminated. Round 2 votes counts: E=30 C=26 D=24 B=20 so B is eliminated. Round 3 votes counts: C=36 E=32 D=32 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:212 C:212 A:199 E:193 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 6 2 B -4 0 2 16 10 C 14 -2 0 12 0 D -6 -16 -12 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999993 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 6 2 B -4 0 2 16 10 C 14 -2 0 12 0 D -6 -16 -12 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999993 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 6 2 B -4 0 2 16 10 C 14 -2 0 12 0 D -6 -16 -12 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999993 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 57: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (18) C B D A E (13) C B D E A (12) E A D C B (7) A E D B C (6) D A B E C (5) A D E B C (5) E C A B D (4) E A C D B (4) B C D A E (4) D B A C E (3) C E B A D (3) C B E D A (3) D B A E C (2) C E A B D (2) B D C A E (2) A E C D B (2) E C B A D (1) E A B D C (1) D B C A E (1) C E B D A (1) B D A C E (1) Total count = 100 A B C D E A 0 8 8 6 -12 B -8 0 -4 -6 -8 C -8 4 0 -2 -10 D -6 6 2 0 -4 E 12 8 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 6 -12 B -8 0 -4 -6 -8 C -8 4 0 -2 -10 D -6 6 2 0 -4 E 12 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=34 A=13 D=11 B=7 so B is eliminated. Round 2 votes counts: C=38 E=35 D=14 A=13 so A is eliminated. Round 3 votes counts: E=43 C=38 D=19 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:205 D:199 C:192 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 6 -12 B -8 0 -4 -6 -8 C -8 4 0 -2 -10 D -6 6 2 0 -4 E 12 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 -12 B -8 0 -4 -6 -8 C -8 4 0 -2 -10 D -6 6 2 0 -4 E 12 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 -12 B -8 0 -4 -6 -8 C -8 4 0 -2 -10 D -6 6 2 0 -4 E 12 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 58: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) E C D B A (7) A C B D E (7) E D B C A (5) D B A E C (5) A B D C E (5) D B E A C (4) C E D A B (4) C A E B D (4) E D C B A (3) E D B A C (3) E C A B D (3) B E D A C (3) B D A E C (3) A B E D C (3) A B D E C (3) E B D A C (2) D A B C E (2) C E A D B (2) C A D E B (2) B A D E C (2) A D B C E (2) A C E B D (2) E C D A B (1) E C B D A (1) E B D C A (1) E B A D C (1) E B A C D (1) E A C B D (1) D E B C A (1) D C E B A (1) D B A C E (1) C D A B E (1) C A D B E (1) C A B E D (1) C A B D E (1) A E C B D (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 6 4 -4 B -12 0 -4 8 -8 C -6 4 0 -2 -8 D -4 -8 2 0 -14 E 4 8 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 6 4 -4 B -12 0 -4 8 -8 C -6 4 0 -2 -8 D -4 -8 2 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=25 C=24 D=14 B=8 so B is eliminated. Round 2 votes counts: E=32 A=27 C=24 D=17 so D is eliminated. Round 3 votes counts: A=38 E=37 C=25 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:209 C:194 B:192 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 6 4 -4 B -12 0 -4 8 -8 C -6 4 0 -2 -8 D -4 -8 2 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 4 -4 B -12 0 -4 8 -8 C -6 4 0 -2 -8 D -4 -8 2 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 4 -4 B -12 0 -4 8 -8 C -6 4 0 -2 -8 D -4 -8 2 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 59: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (7) E A D C B (6) D A E C B (6) A D E B C (6) C B E D A (5) B C E A D (5) B C D A E (5) D E A C B (4) A E D B C (4) A E B C D (4) C D B E A (3) B C E D A (3) B C A E D (3) B C A D E (3) B A C E D (3) A E D C B (3) A D E C B (3) E D A C B (2) D C E B A (2) A E B D C (2) A D B E C (2) A B D C E (2) E D C A B (1) E C B D A (1) E B A C D (1) D E C B A (1) D C B E A (1) D C B A E (1) D A E B C (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E B A (1) C B D A E (1) B E A C D (1) B C D E A (1) B A C D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 10 2 8 B 0 0 0 2 2 C -10 0 0 0 0 D -2 -2 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300836 B: 0.699164 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.579332830067 Cumulative probabilities = A: 0.300836 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 2 8 B 0 0 0 2 2 C -10 0 0 0 0 D -2 -2 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 D=18 C=18 E=11 so E is eliminated. Round 2 votes counts: A=34 B=26 D=21 C=19 so C is eliminated. Round 3 votes counts: B=41 A=34 D=25 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:202 D:201 C:195 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 2 8 B 0 0 0 2 2 C -10 0 0 0 0 D -2 -2 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 2 8 B 0 0 0 2 2 C -10 0 0 0 0 D -2 -2 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 2 8 B 0 0 0 2 2 C -10 0 0 0 0 D -2 -2 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 60: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (5) C A B E D (5) C A B D E (5) B D A E C (5) B A D C E (5) A C B D E (5) E D B C A (4) D B E A C (4) D B A E C (4) C A E B D (4) E D B A C (3) E B D A C (3) D E B A C (3) A B D C E (3) A B C D E (3) E C A D B (2) E C A B D (2) E B D C A (2) D C A E B (2) D B A C E (2) C E A D B (2) C E A B D (2) C A D B E (2) B D A C E (2) B A C D E (2) E C D B A (1) E C D A B (1) E C B A D (1) E B A C D (1) D E C A B (1) D E B C A (1) D C E A B (1) D A C B E (1) D A B C E (1) C D A E B (1) C B A E D (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E A C (1) B A E C D (1) B A D E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 8 4 16 B 8 0 8 16 14 C -8 -8 0 -10 4 D -4 -16 10 0 10 E -16 -14 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 4 16 B 8 0 8 16 14 C -8 -8 0 -10 4 D -4 -16 10 0 10 E -16 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 D=20 B=19 A=13 so A is eliminated. Round 2 votes counts: C=29 B=26 E=25 D=20 so D is eliminated. Round 3 votes counts: B=37 C=33 E=30 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:210 D:200 C:189 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 4 16 B 8 0 8 16 14 C -8 -8 0 -10 4 D -4 -16 10 0 10 E -16 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 4 16 B 8 0 8 16 14 C -8 -8 0 -10 4 D -4 -16 10 0 10 E -16 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 4 16 B 8 0 8 16 14 C -8 -8 0 -10 4 D -4 -16 10 0 10 E -16 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 61: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (7) B A D E C (7) C A D B E (6) C A B D E (6) B C A E D (6) E D B A C (5) C B E A D (5) E B D A C (4) B E D A C (4) A D B C E (4) E B C D A (3) B A C D E (3) E D B C A (2) E D A B C (2) E C D A B (2) D E C A B (2) D E A C B (2) D E A B C (2) C D A E B (2) C B A D E (2) B C E A D (2) B C A D E (2) A C B D E (2) E D C A B (1) E D A C B (1) E C B D A (1) D C A E B (1) D A C E B (1) D A B E C (1) C E B D A (1) C B A E D (1) C A B E D (1) B E A D C (1) B E A C D (1) B A E D C (1) A D C B E (1) A D B E C (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 12 4 B 2 0 8 12 22 C 6 -8 0 10 12 D -12 -12 -10 0 -2 E -4 -22 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 12 4 B 2 0 8 12 22 C 6 -8 0 10 12 D -12 -12 -10 0 -2 E -4 -22 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 E=21 A=12 D=9 so D is eliminated. Round 2 votes counts: C=32 E=27 B=27 A=14 so A is eliminated. Round 3 votes counts: C=37 B=36 E=27 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:210 A:204 D:182 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 12 4 B 2 0 8 12 22 C 6 -8 0 10 12 D -12 -12 -10 0 -2 E -4 -22 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 12 4 B 2 0 8 12 22 C 6 -8 0 10 12 D -12 -12 -10 0 -2 E -4 -22 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 12 4 B 2 0 8 12 22 C 6 -8 0 10 12 D -12 -12 -10 0 -2 E -4 -22 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 62: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (14) A B C E D (10) D E B C A (7) C A B E D (6) D A C B E (5) D E C B A (4) D E B A C (3) D C E A B (3) C E A B D (3) B E A C D (3) A B E C D (3) D C A E B (2) C A E B D (2) C A D E B (2) B C A E D (2) B A E D C (2) B A E C D (2) A C D B E (2) A C B D E (2) E D B A C (1) E C B A D (1) E B D C A (1) E B C A D (1) D E A B C (1) D C E B A (1) D B E A C (1) D B A E C (1) D A C E B (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E B A (1) C D E A B (1) C D A E B (1) C A E D B (1) B E C A D (1) B E A D C (1) B D A E C (1) B A D E C (1) B A C E D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 24 16 26 28 B -24 0 -8 18 22 C -16 8 0 22 28 D -26 -18 -22 0 -14 E -28 -22 -28 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 16 26 28 B -24 0 -8 18 22 C -16 8 0 22 28 D -26 -18 -22 0 -14 E -28 -22 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=30 C=19 B=14 E=4 so E is eliminated. Round 2 votes counts: A=33 D=31 C=20 B=16 so B is eliminated. Round 3 votes counts: A=43 D=33 C=24 so C is eliminated. Round 4 votes counts: A=63 D=37 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:247 C:221 B:204 E:168 D:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 16 26 28 B -24 0 -8 18 22 C -16 8 0 22 28 D -26 -18 -22 0 -14 E -28 -22 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 16 26 28 B -24 0 -8 18 22 C -16 8 0 22 28 D -26 -18 -22 0 -14 E -28 -22 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 16 26 28 B -24 0 -8 18 22 C -16 8 0 22 28 D -26 -18 -22 0 -14 E -28 -22 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 63: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (17) C B E A D (9) C A D E B (9) E B D A C (8) C A D B E (6) D A E B C (5) B C E D A (5) A D C E B (5) E B C D A (4) B E C D A (4) E D B A C (3) C B A D E (3) E D A C B (2) E D A B C (2) C B A E D (2) B E D C A (2) A D E C B (2) D E B A C (1) D B A E C (1) D A E C B (1) D A B E C (1) C E B D A (1) C E B A D (1) C A E D B (1) B D A E C (1) B C A D E (1) A D E B C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -26 4 -16 -18 B 26 0 12 16 6 C -4 -12 0 -6 -10 D 16 -16 6 0 -22 E 18 -6 10 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 4 -16 -18 B 26 0 12 16 6 C -4 -12 0 -6 -10 D 16 -16 6 0 -22 E 18 -6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=30 E=19 A=10 D=9 so D is eliminated. Round 2 votes counts: C=32 B=31 E=20 A=17 so A is eliminated. Round 3 votes counts: C=39 B=32 E=29 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:230 E:222 D:192 C:184 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 4 -16 -18 B 26 0 12 16 6 C -4 -12 0 -6 -10 D 16 -16 6 0 -22 E 18 -6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 4 -16 -18 B 26 0 12 16 6 C -4 -12 0 -6 -10 D 16 -16 6 0 -22 E 18 -6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 4 -16 -18 B 26 0 12 16 6 C -4 -12 0 -6 -10 D 16 -16 6 0 -22 E 18 -6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 64: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (9) D B E A C (9) C A E B D (7) C A B D E (6) E B D A C (5) C A D E B (5) D E B A C (4) D B C E A (4) C A D B E (4) C A B E D (3) B E D A C (3) B D E A C (3) E D B A C (2) E B A D C (2) E A B D C (2) D C B E A (2) C B D A E (2) C A E D B (2) A E C B D (2) A C E D B (2) E A C D B (1) E A C B D (1) E A B C D (1) D E B C A (1) C D A E B (1) C D A B E (1) B E A D C (1) B D C E A (1) B D C A E (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 -2 2 4 -2 B 2 0 0 6 2 C -2 0 0 2 6 D -4 -6 -2 0 6 E 2 -2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.619542 C: 0.380458 D: 0.000000 E: 0.000000 Sum of squares = 0.528580450345 Cumulative probabilities = A: 0.000000 B: 0.619542 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 4 -2 B 2 0 0 6 2 C -2 0 0 2 6 D -4 -6 -2 0 6 E 2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500181 C: 0.499819 D: 0.000000 E: 0.000000 Sum of squares = 0.500000065664 Cumulative probabilities = A: 0.000000 B: 0.500181 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=29 A=15 E=14 B=11 so B is eliminated. Round 2 votes counts: D=34 C=31 E=18 A=17 so A is eliminated. Round 3 votes counts: C=45 D=34 E=21 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:205 C:203 A:201 D:197 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 4 -2 B 2 0 0 6 2 C -2 0 0 2 6 D -4 -6 -2 0 6 E 2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500181 C: 0.499819 D: 0.000000 E: 0.000000 Sum of squares = 0.500000065664 Cumulative probabilities = A: 0.000000 B: 0.500181 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -2 B 2 0 0 6 2 C -2 0 0 2 6 D -4 -6 -2 0 6 E 2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500181 C: 0.499819 D: 0.000000 E: 0.000000 Sum of squares = 0.500000065664 Cumulative probabilities = A: 0.000000 B: 0.500181 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -2 B 2 0 0 6 2 C -2 0 0 2 6 D -4 -6 -2 0 6 E 2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500181 C: 0.499819 D: 0.000000 E: 0.000000 Sum of squares = 0.500000065664 Cumulative probabilities = A: 0.000000 B: 0.500181 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 65: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) D C E B A (7) C D B E A (7) C B D A E (7) C D B A E (6) B C D A E (5) E A D B C (4) E A B D C (4) D E C B A (4) B A C D E (4) A E B C D (4) E D C A B (3) C D E A B (3) B C A D E (3) A E B D C (3) A B E D C (3) E D A C B (2) A C E B D (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B A C (1) E A D C B (1) D C E A B (1) D C B E A (1) C D E B A (1) C D A B E (1) C B A D E (1) C A D B E (1) B E A D C (1) B D C A E (1) B A E D C (1) B A C E D (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -6 -2 18 B 4 0 -4 8 14 C 6 4 0 24 18 D 2 -8 -24 0 14 E -18 -14 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -2 18 B 4 0 -4 8 14 C 6 4 0 24 18 D 2 -8 -24 0 14 E -18 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=27 E=16 B=16 D=13 so D is eliminated. Round 2 votes counts: C=36 A=28 E=20 B=16 so B is eliminated. Round 3 votes counts: C=45 A=34 E=21 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:211 A:203 D:192 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 18 B 4 0 -4 8 14 C 6 4 0 24 18 D 2 -8 -24 0 14 E -18 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 18 B 4 0 -4 8 14 C 6 4 0 24 18 D 2 -8 -24 0 14 E -18 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 18 B 4 0 -4 8 14 C 6 4 0 24 18 D 2 -8 -24 0 14 E -18 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 66: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (12) A D B E C (8) C E D A B (7) D A E B C (6) C B E A D (6) E C D A B (5) B E A D C (5) E B C D A (4) D A E C B (4) C D A E B (4) B A D E C (4) A D B C E (4) E C B D A (3) B A D C E (3) E D A B C (2) D A C E B (2) C D E A B (2) C B A D E (2) C A D B E (2) B C E A D (2) E D C A B (1) E D B A C (1) E D A C B (1) E C D B A (1) E B D A C (1) D E A C B (1) C B A E D (1) B E C D A (1) B E C A D (1) B C A E D (1) B C A D E (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -12 -16 -12 B -2 0 -10 -4 -14 C 12 10 0 10 2 D 16 4 -10 0 -10 E 12 14 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -16 -12 B -2 0 -10 -4 -14 C 12 10 0 10 2 D 16 4 -10 0 -10 E 12 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=19 B=18 A=14 D=13 so D is eliminated. Round 2 votes counts: C=36 A=26 E=20 B=18 so B is eliminated. Round 3 votes counts: C=40 A=33 E=27 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:217 D:200 B:185 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -16 -12 B -2 0 -10 -4 -14 C 12 10 0 10 2 D 16 4 -10 0 -10 E 12 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -16 -12 B -2 0 -10 -4 -14 C 12 10 0 10 2 D 16 4 -10 0 -10 E 12 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -16 -12 B -2 0 -10 -4 -14 C 12 10 0 10 2 D 16 4 -10 0 -10 E 12 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 67: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) A E C B D (13) C B D E A (6) D B E C A (5) A D C B E (5) D C B E A (4) C B E D A (4) A E D B C (3) A E B C D (3) E D B C A (2) E D A B C (2) D E B C A (2) D E B A C (2) D C A B E (2) C D B E A (2) C B A E D (2) C B A D E (2) B C E D A (2) B C D E A (2) A C D B E (2) A C B E D (2) E C A B D (1) E B C D A (1) E B C A D (1) E A D B C (1) E A C B D (1) E A B D C (1) E A B C D (1) D E A B C (1) D A E B C (1) D A B E C (1) C D B A E (1) C B E A D (1) C A B D E (1) B D E C A (1) A E D C B (1) A E C D B (1) A E B D C (1) A D E B C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -10 -8 -12 B 6 0 -6 -6 16 C 10 6 0 0 6 D 8 6 0 0 10 E 12 -16 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399406 D: 0.600594 E: 0.000000 Sum of squares = 0.52023819063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399406 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -8 -12 B 6 0 -6 -6 16 C 10 6 0 0 6 D 8 6 0 0 10 E 12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=31 C=19 E=11 B=5 so B is eliminated. Round 2 votes counts: A=34 D=32 C=23 E=11 so E is eliminated. Round 3 votes counts: A=38 D=36 C=26 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:211 B:205 E:190 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 -12 B 6 0 -6 -6 16 C 10 6 0 0 6 D 8 6 0 0 10 E 12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 -12 B 6 0 -6 -6 16 C 10 6 0 0 6 D 8 6 0 0 10 E 12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 -12 B 6 0 -6 -6 16 C 10 6 0 0 6 D 8 6 0 0 10 E 12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 68: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) E C D B A (7) D C B E A (6) E A B C D (5) B E A D C (5) A B D C E (5) D C B A E (4) C E D A B (4) B A E D C (4) A B E D C (4) E C D A B (3) C D E B A (3) A B E C D (3) E C A D B (2) E A C D B (2) D C A B E (2) D B C E A (2) C D E A B (2) C D A B E (2) B D C E A (2) B D A C E (2) E D B C A (1) E C A B D (1) E B D C A (1) E B A D C (1) E B A C D (1) E A C B D (1) E A B D C (1) D C E B A (1) D B C A E (1) D B A C E (1) C D B A E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D C A E (1) B D A E C (1) A E C D B (1) A E B D C (1) A E B C D (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 4 4 -4 B 12 0 10 4 20 C -4 -10 0 -16 4 D -4 -4 16 0 0 E 4 -20 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 4 -4 B 12 0 10 4 20 C -4 -10 0 -16 4 D -4 -4 16 0 0 E 4 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=26 B=26 D=17 A=17 C=14 so C is eliminated. Round 2 votes counts: E=30 B=26 D=25 A=19 so A is eliminated. Round 3 votes counts: B=39 E=33 D=28 so D is eliminated. Round 4 votes counts: B=60 E=40 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:204 A:196 E:190 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 4 -4 B 12 0 10 4 20 C -4 -10 0 -16 4 D -4 -4 16 0 0 E 4 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 4 -4 B 12 0 10 4 20 C -4 -10 0 -16 4 D -4 -4 16 0 0 E 4 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 4 -4 B 12 0 10 4 20 C -4 -10 0 -16 4 D -4 -4 16 0 0 E 4 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 69: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C D A E B (8) A C E D B (8) D C B A E (6) D C A E B (5) D B C E A (5) B E D A C (5) A C D E B (5) D C B E A (4) C D A B E (4) A E C B D (4) E B A C D (3) E A B C D (3) D C A B E (3) C A D E B (3) B E D C A (3) B D E C A (3) B D C E A (3) A E B C D (3) E B A D C (2) B C D E A (2) E A B D C (1) D A C E B (1) C D B A E (1) C A D B E (1) A E D C B (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -2 -6 10 B -4 0 -12 -12 2 C 2 12 0 -6 20 D 6 12 6 0 8 E -10 -2 -20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -6 10 B -4 0 -12 -12 2 C 2 12 0 -6 20 D 6 12 6 0 8 E -10 -2 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=24 A=23 C=17 E=9 so E is eliminated. Round 2 votes counts: B=32 A=27 D=24 C=17 so C is eliminated. Round 3 votes counts: D=37 B=32 A=31 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:214 A:203 B:187 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -6 10 B -4 0 -12 -12 2 C 2 12 0 -6 20 D 6 12 6 0 8 E -10 -2 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -6 10 B -4 0 -12 -12 2 C 2 12 0 -6 20 D 6 12 6 0 8 E -10 -2 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -6 10 B -4 0 -12 -12 2 C 2 12 0 -6 20 D 6 12 6 0 8 E -10 -2 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 70: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) B D E C A (10) A C E D B (9) A E C D B (6) B E D C A (5) C D A E B (4) B D C A E (4) E C D A B (3) E B D C A (3) E B A C D (3) E A C B D (3) D C A B E (3) B E D A C (3) A E C B D (3) E A C D B (2) D C E A B (2) B E A D C (2) B E A C D (2) B D C E A (2) B A C D E (2) A B C D E (2) E D C A B (1) E D B C A (1) E A B C D (1) D C E B A (1) D C B E A (1) D C A E B (1) D B C E A (1) C A D E B (1) B D E A C (1) B D A E C (1) B D A C E (1) B A E C D (1) A E B C D (1) A C E B D (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 14 4 6 B -12 0 -6 4 -14 C -14 6 0 14 -4 D -4 -4 -14 0 -2 E -6 14 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 4 6 B -12 0 -6 4 -14 C -14 6 0 14 -4 D -4 -4 -14 0 -2 E -6 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=34 E=17 D=9 C=5 so C is eliminated. Round 2 votes counts: A=36 B=34 E=17 D=13 so D is eliminated. Round 3 votes counts: A=44 B=36 E=20 so E is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:207 C:201 D:188 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 4 6 B -12 0 -6 4 -14 C -14 6 0 14 -4 D -4 -4 -14 0 -2 E -6 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 4 6 B -12 0 -6 4 -14 C -14 6 0 14 -4 D -4 -4 -14 0 -2 E -6 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 4 6 B -12 0 -6 4 -14 C -14 6 0 14 -4 D -4 -4 -14 0 -2 E -6 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 71: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) B C D A E (8) A E D C B (6) A E C D B (5) E D A C B (4) C D B E A (4) C B D E A (4) B D C E A (4) A E D B C (4) A B C E D (4) E D C B A (3) E D C A B (3) E A D B C (3) B A C D E (3) A E B C D (3) D E C B A (2) D B C E A (2) C E A D B (2) C B D A E (2) B C A D E (2) A E C B D (2) A E B D C (2) A C E B D (2) A B C D E (2) E D B C A (1) E D A B C (1) E A C D B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) C E D B A (1) C E D A B (1) C A E B D (1) B D E A C (1) B D A E C (1) B D A C E (1) B C D E A (1) B A D C E (1) Total count = 100 A B C D E A 0 10 10 4 -2 B -10 0 -8 -12 -16 C -10 8 0 -4 -6 D -4 12 4 0 -16 E 2 16 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 10 4 -2 B -10 0 -8 -12 -16 C -10 8 0 -4 -6 D -4 12 4 0 -16 E 2 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 B=22 C=15 D=8 so D is eliminated. Round 2 votes counts: A=30 E=28 B=25 C=17 so C is eliminated. Round 3 votes counts: B=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:211 D:198 C:194 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 10 4 -2 B -10 0 -8 -12 -16 C -10 8 0 -4 -6 D -4 12 4 0 -16 E 2 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 4 -2 B -10 0 -8 -12 -16 C -10 8 0 -4 -6 D -4 12 4 0 -16 E 2 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 4 -2 B -10 0 -8 -12 -16 C -10 8 0 -4 -6 D -4 12 4 0 -16 E 2 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 72: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) B C E A D (7) D A E B C (6) C D B A E (5) C B D E A (5) A E D B C (5) B D E A C (4) B D C E A (4) B C D E A (4) E A B D C (3) D C B A E (3) D A B E C (3) C E A B D (3) A E D C B (3) A E C D B (3) A D E C B (3) E A C D B (2) E A C B D (2) E A B C D (2) D C A E B (2) D B C A E (2) D B A E C (2) D B A C E (2) D A E C B (2) C D A E B (2) C B E D A (2) C A D E B (2) B E A C D (2) E C B A D (1) D A C E B (1) C E A D B (1) C B D A E (1) C A E D B (1) B D C A E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -8 -2 0 B 6 0 -6 -2 10 C 8 6 0 8 14 D 2 2 -8 0 10 E 0 -10 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -2 0 B 6 0 -6 -2 10 C 8 6 0 8 14 D 2 2 -8 0 10 E 0 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=23 B=22 A=15 E=10 so E is eliminated. Round 2 votes counts: C=31 A=24 D=23 B=22 so B is eliminated. Round 3 votes counts: C=42 D=32 A=26 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:204 D:203 A:192 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -2 0 B 6 0 -6 -2 10 C 8 6 0 8 14 D 2 2 -8 0 10 E 0 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -2 0 B 6 0 -6 -2 10 C 8 6 0 8 14 D 2 2 -8 0 10 E 0 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -2 0 B 6 0 -6 -2 10 C 8 6 0 8 14 D 2 2 -8 0 10 E 0 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 73: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) E D B C A (9) D E C B A (9) C A D B E (9) B E A D C (7) C D A E B (6) C A B D E (6) E D C B A (4) D C E A B (4) B E A C D (4) A B C E D (4) C D A B E (3) B A C E D (3) A C B E D (3) E D B A C (2) E B D A C (2) E B A D C (2) D C A E B (2) C D E A B (2) A B C D E (2) E C D B A (1) E B C D A (1) E B A C D (1) D E C A B (1) D E B C A (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -16 6 0 B 12 0 -2 -6 6 C 16 2 0 12 -8 D -6 6 -12 0 -6 E 0 -6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -12 -16 6 0 B 12 0 -2 -6 6 C 16 2 0 12 -8 D -6 6 -12 0 -6 E 0 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999559 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 E=22 D=17 A=11 so A is eliminated. Round 2 votes counts: B=31 C=30 E=22 D=17 so D is eliminated. Round 3 votes counts: C=36 E=33 B=31 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:211 B:205 E:204 D:191 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -16 6 0 B 12 0 -2 -6 6 C 16 2 0 12 -8 D -6 6 -12 0 -6 E 0 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999559 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 6 0 B 12 0 -2 -6 6 C 16 2 0 12 -8 D -6 6 -12 0 -6 E 0 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999559 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 6 0 B 12 0 -2 -6 6 C 16 2 0 12 -8 D -6 6 -12 0 -6 E 0 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999559 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 74: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E C A D B (7) D A C E B (6) E B C A D (5) E A C D B (5) B D A C E (5) B E D C A (4) B E A D C (4) B D C A E (4) A D C E B (4) E C A B D (3) D C A B E (3) C D A E B (3) B E D A C (3) A C D E B (3) E A C B D (2) D C A E B (2) C E A D B (2) B E C D A (2) B E C A D (2) B E A C D (2) B D A E C (2) B C D E A (2) A E C D B (2) E C B A D (1) E B C D A (1) E B A C D (1) E A B D C (1) E A B C D (1) D A B C E (1) C D E A B (1) C A E D B (1) C A D E B (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 16 6 -4 -4 B -16 0 -12 -2 -6 C -6 12 0 -2 0 D 4 2 2 0 0 E 4 6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.664335 E: 0.335665 Sum of squares = 0.554011863618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.664335 E: 1.000000 A B C D E A 0 16 6 -4 -4 B -16 0 -12 -2 -6 C -6 12 0 -2 0 D 4 2 2 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=27 D=21 A=10 C=8 so C is eliminated. Round 2 votes counts: B=34 E=29 D=25 A=12 so A is eliminated. Round 3 votes counts: B=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:207 E:205 D:204 C:202 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 -4 -4 B -16 0 -12 -2 -6 C -6 12 0 -2 0 D 4 2 2 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -4 -4 B -16 0 -12 -2 -6 C -6 12 0 -2 0 D 4 2 2 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -4 -4 B -16 0 -12 -2 -6 C -6 12 0 -2 0 D 4 2 2 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 75: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A C D E B (8) E B D C A (7) B E D C A (6) D E C A B (5) E D C B A (4) E B D A C (4) D C A E B (4) B A E C D (4) A C D B E (4) E D B A C (3) C A D E B (3) B E D A C (3) A C B D E (3) A B E C D (3) E D C A B (2) C D A E B (2) B E A C D (2) B A C E D (2) B A C D E (2) A C E D B (2) E D A C B (1) E B A D C (1) E A D C B (1) E A B D C (1) E A B C D (1) D E B C A (1) D C E A B (1) D C B A E (1) D A C E B (1) C D B A E (1) C A D B E (1) B E A D C (1) B D C E A (1) B D C A E (1) B C D A E (1) B C A D E (1) A E B C D (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -14 -6 B 8 0 10 -8 -24 C 0 -10 0 -16 -20 D 14 8 16 0 -16 E 6 24 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 -14 -6 B 8 0 10 -8 -24 C 0 -10 0 -16 -20 D 14 8 16 0 -16 E 6 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=24 A=23 D=13 C=7 so C is eliminated. Round 2 votes counts: E=33 A=27 B=24 D=16 so D is eliminated. Round 3 votes counts: E=40 A=34 B=26 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 D:211 B:193 A:186 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -14 -6 B 8 0 10 -8 -24 C 0 -10 0 -16 -20 D 14 8 16 0 -16 E 6 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -14 -6 B 8 0 10 -8 -24 C 0 -10 0 -16 -20 D 14 8 16 0 -16 E 6 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -14 -6 B 8 0 10 -8 -24 C 0 -10 0 -16 -20 D 14 8 16 0 -16 E 6 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 76: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (14) C A D E B (12) B E D A C (10) D A C E B (7) C A D B E (7) B E C D A (7) C B E A D (6) E B D A C (5) E D B A C (4) A D C E B (4) D A E C B (3) B E C A D (3) A C D E B (3) C D A B E (2) C A B D E (2) B E D C A (2) A D E B C (2) E D A B C (1) E B A C D (1) D E B A C (1) D E A B C (1) C B E D A (1) C B D A E (1) B C E A D (1) Total count = 100 A B C D E A 0 16 12 -18 14 B -16 0 4 -22 -16 C -12 -4 0 -8 -8 D 18 22 8 0 18 E -14 16 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 -18 14 B -16 0 4 -22 -16 C -12 -4 0 -8 -8 D 18 22 8 0 18 E -14 16 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=26 B=23 E=11 A=9 so A is eliminated. Round 2 votes counts: C=34 D=32 B=23 E=11 so E is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:233 A:212 E:196 C:184 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 12 -18 14 B -16 0 4 -22 -16 C -12 -4 0 -8 -8 D 18 22 8 0 18 E -14 16 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 -18 14 B -16 0 4 -22 -16 C -12 -4 0 -8 -8 D 18 22 8 0 18 E -14 16 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 -18 14 B -16 0 4 -22 -16 C -12 -4 0 -8 -8 D 18 22 8 0 18 E -14 16 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 77: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) C E D B A (9) E D B A C (8) A C B D E (8) D B E A C (6) A B D C E (6) C E D A B (5) C A B D E (5) B D A E C (5) B A D E C (5) C E A D B (4) C A E B D (4) E D C B A (3) E D B C A (3) D E B A C (3) A B C D E (3) E C D B A (2) D B A E C (2) C A B E D (2) B D E A C (2) C E B D A (1) C E A B D (1) C B E D A (1) A D B E C (1) Total count = 100 A B C D E A 0 0 20 0 4 B 0 0 10 8 14 C -20 -10 0 -10 -2 D 0 -8 10 0 14 E -4 -14 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.348011 B: 0.651989 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.546201513054 Cumulative probabilities = A: 0.348011 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 20 0 4 B 0 0 10 8 14 C -20 -10 0 -10 -2 D 0 -8 10 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=29 E=16 B=12 D=11 so D is eliminated. Round 2 votes counts: C=32 A=29 B=20 E=19 so E is eliminated. Round 3 votes counts: C=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:212 D:208 E:185 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 20 0 4 B 0 0 10 8 14 C -20 -10 0 -10 -2 D 0 -8 10 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 20 0 4 B 0 0 10 8 14 C -20 -10 0 -10 -2 D 0 -8 10 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 20 0 4 B 0 0 10 8 14 C -20 -10 0 -10 -2 D 0 -8 10 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 78: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) B A E C D (8) D C E A B (7) D E C A B (6) C A D E B (5) E B D A C (4) D E C B A (4) B E D C A (4) B E A D C (4) A B C E D (4) E D B C A (3) A C D B E (3) E D A C B (2) E A D C B (2) D C E B A (2) C D B E A (2) C A D B E (2) B A E D C (2) B A C E D (2) B A C D E (2) A B E C D (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A B C (1) E A B D C (1) D E B C A (1) D E A C B (1) C D A B E (1) C B D A E (1) B E D A C (1) B E A C D (1) B D E C A (1) B C D A E (1) B C A D E (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -8 -12 0 B -8 0 -8 -16 -10 C 8 8 0 0 -2 D 12 16 0 0 12 E 0 10 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.440247 D: 0.559753 E: 0.000000 Sum of squares = 0.507140751311 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.440247 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -12 0 B -8 0 -8 -16 -10 C 8 8 0 0 -2 D 12 16 0 0 12 E 0 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=22 D=21 E=16 A=14 so A is eliminated. Round 2 votes counts: B=35 C=27 D=22 E=16 so E is eliminated. Round 3 votes counts: B=40 D=33 C=27 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:207 E:200 A:194 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -12 0 B -8 0 -8 -16 -10 C 8 8 0 0 -2 D 12 16 0 0 12 E 0 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -12 0 B -8 0 -8 -16 -10 C 8 8 0 0 -2 D 12 16 0 0 12 E 0 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -12 0 B -8 0 -8 -16 -10 C 8 8 0 0 -2 D 12 16 0 0 12 E 0 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 79: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) B C E D A (7) B D C A E (6) A D E C B (6) A D C B E (6) E A C D B (5) D A B C E (5) A D C E B (5) E C B D A (4) E C B A D (4) C E B D A (4) B C D E A (4) E C A D B (3) A D E B C (3) E C A B D (2) E A D B C (2) E A C B D (2) D B A C E (2) D A C B E (2) C E D A B (2) C B E D A (2) A D B E C (2) E A D C B (1) E A B C D (1) D C B A E (1) C E D B A (1) C D B A E (1) C D A E B (1) C A D E B (1) B E D C A (1) B E C D A (1) B D E A C (1) B D C E A (1) B D A E C (1) B C D A E (1) A D B C E (1) Total count = 100 A B C D E A 0 16 6 4 4 B -16 0 -22 -16 -10 C -6 22 0 -8 6 D -4 16 8 0 0 E -4 10 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 4 4 B -16 0 -22 -16 -10 C -6 22 0 -8 6 D -4 16 8 0 0 E -4 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998332 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 B=23 C=12 D=10 so D is eliminated. Round 2 votes counts: A=38 B=25 E=24 C=13 so C is eliminated. Round 3 votes counts: A=40 E=31 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:210 C:207 E:200 B:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 4 4 B -16 0 -22 -16 -10 C -6 22 0 -8 6 D -4 16 8 0 0 E -4 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998332 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 4 4 B -16 0 -22 -16 -10 C -6 22 0 -8 6 D -4 16 8 0 0 E -4 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998332 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 4 4 B -16 0 -22 -16 -10 C -6 22 0 -8 6 D -4 16 8 0 0 E -4 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998332 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 80: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (15) D E A C B (9) E D B C A (8) E D A C B (8) E B C A D (8) B C A D E (8) D A C E B (6) B E C A D (6) A C B D E (6) A D C B E (5) A C D B E (5) E B D C A (3) D A E C B (3) E D A B C (2) C B A D E (2) E D C A B (1) E B C D A (1) D E C B A (1) C A B D E (1) B C E A D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -10 16 4 B 6 0 4 4 0 C 10 -4 0 8 0 D -16 -4 -8 0 -8 E -4 0 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.722131 C: 0.000000 D: 0.000000 E: 0.277869 Sum of squares = 0.59868447992 Cumulative probabilities = A: 0.000000 B: 0.722131 C: 0.722131 D: 0.722131 E: 1.000000 A B C D E A 0 -6 -10 16 4 B 6 0 4 4 0 C 10 -4 0 8 0 D -16 -4 -8 0 -8 E -4 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500004 C: 0.000000 D: 0.000000 E: 0.499996 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.500004 C: 0.500004 D: 0.500004 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=30 D=19 A=17 C=3 so C is eliminated. Round 2 votes counts: B=32 E=31 D=19 A=18 so A is eliminated. Round 3 votes counts: B=40 E=31 D=29 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:207 A:202 E:202 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 16 4 B 6 0 4 4 0 C 10 -4 0 8 0 D -16 -4 -8 0 -8 E -4 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500004 C: 0.000000 D: 0.000000 E: 0.499996 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.500004 C: 0.500004 D: 0.500004 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 16 4 B 6 0 4 4 0 C 10 -4 0 8 0 D -16 -4 -8 0 -8 E -4 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500004 C: 0.000000 D: 0.000000 E: 0.499996 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.500004 C: 0.500004 D: 0.500004 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 16 4 B 6 0 4 4 0 C 10 -4 0 8 0 D -16 -4 -8 0 -8 E -4 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500004 C: 0.000000 D: 0.000000 E: 0.499996 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.500004 C: 0.500004 D: 0.500004 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 81: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) C E A B D (6) C D E B A (6) E B A D C (5) D B A C E (5) C E D A B (5) C D E A B (5) D C B A E (4) C E D B A (4) B A E D C (4) A B E D C (4) A B D E C (4) E B D A C (3) D C B E A (3) D B A E C (3) C D A B E (3) E C D B A (2) E C B D A (2) E C B A D (2) C D A E B (2) B E A D C (2) A E B C D (2) A B E C D (2) E D C B A (1) E C A B D (1) E B C A D (1) E A B C D (1) D E C B A (1) D E B A C (1) D C E B A (1) D B E A C (1) D A B C E (1) C A D E B (1) C A B E D (1) B A D E C (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -2 -6 -20 B 18 0 0 0 -18 C 2 0 0 8 0 D 6 0 -8 0 -10 E 20 18 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.553665 D: 0.000000 E: 0.446335 Sum of squares = 0.505759913502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.553665 D: 0.553665 E: 1.000000 A B C D E A 0 -18 -2 -6 -20 B 18 0 0 0 -18 C 2 0 0 8 0 D 6 0 -8 0 -10 E 20 18 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=25 D=20 A=15 B=7 so B is eliminated. Round 2 votes counts: C=33 E=27 D=20 A=20 so D is eliminated. Round 3 votes counts: C=41 E=30 A=29 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:224 C:205 B:200 D:194 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -2 -6 -20 B 18 0 0 0 -18 C 2 0 0 8 0 D 6 0 -8 0 -10 E 20 18 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 -6 -20 B 18 0 0 0 -18 C 2 0 0 8 0 D 6 0 -8 0 -10 E 20 18 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 -6 -20 B 18 0 0 0 -18 C 2 0 0 8 0 D 6 0 -8 0 -10 E 20 18 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 82: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) E C B D A (7) E C B A D (7) E A C B D (7) A D E B C (7) A D B C E (7) E A D B C (6) E A D C B (5) D B C A E (4) D A B C E (4) C B D E A (4) C B D A E (4) A E D B C (4) A E D C B (3) E C A B D (2) D B A C E (2) C E B D A (2) B C D E A (2) B C D A E (2) E B C D A (1) D A E B C (1) D A B E C (1) C B A D E (1) B E D C A (1) A D E C B (1) A D C B E (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 2 2 6 -12 B -2 0 -14 6 -6 C -2 14 0 4 -8 D -6 -6 -4 0 -14 E 12 6 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 6 -12 B -2 0 -14 6 -6 C -2 14 0 4 -8 D -6 -6 -4 0 -14 E 12 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=25 C=23 D=12 B=5 so B is eliminated. Round 2 votes counts: E=36 C=27 A=25 D=12 so D is eliminated. Round 3 votes counts: E=36 A=33 C=31 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:204 A:199 B:192 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 6 -12 B -2 0 -14 6 -6 C -2 14 0 4 -8 D -6 -6 -4 0 -14 E 12 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 6 -12 B -2 0 -14 6 -6 C -2 14 0 4 -8 D -6 -6 -4 0 -14 E 12 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 6 -12 B -2 0 -14 6 -6 C -2 14 0 4 -8 D -6 -6 -4 0 -14 E 12 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 83: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) B C E D A (12) A D E B C (8) C B E D A (7) D E A C B (6) B C E A D (6) B C A E D (4) E D C A B (3) D A E C B (3) C E B D A (3) B C D E A (3) A B D E C (3) E D A C B (2) E C D A B (2) D E C A B (2) D A E B C (2) B A C E D (2) B A C D E (2) A D B E C (2) A B D C E (2) E C D B A (1) D E C B A (1) D E B A C (1) D B E C A (1) B D C A E (1) B D A C E (1) B C D A E (1) B C A D E (1) B A D C E (1) A E D C B (1) A E B D C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 -4 0 B -4 0 10 4 0 C -4 -10 0 -8 -4 D 4 -4 8 0 8 E 0 0 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -4 0 B -4 0 10 4 0 C -4 -10 0 -8 -4 D 4 -4 8 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=32 D=16 C=10 E=8 so E is eliminated. Round 2 votes counts: B=34 A=32 D=21 C=13 so C is eliminated. Round 3 votes counts: B=44 A=32 D=24 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:208 B:205 A:202 E:198 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 -4 0 B -4 0 10 4 0 C -4 -10 0 -8 -4 D 4 -4 8 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -4 0 B -4 0 10 4 0 C -4 -10 0 -8 -4 D 4 -4 8 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -4 0 B -4 0 10 4 0 C -4 -10 0 -8 -4 D 4 -4 8 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 84: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) A B E D C (6) E A B C D (5) D C A E B (5) A B D E C (5) A B D C E (5) D C A B E (4) C D E B A (4) B A C D E (4) E B A C D (3) D C E A B (3) D C B A E (3) A E B D C (3) E D A C B (2) E C D B A (2) E C D A B (2) E B C A D (2) E A D B C (2) E A B D C (2) D C B E A (2) C E D B A (2) C D B E A (2) C D B A E (2) B C D A E (2) B A D C E (2) A E B C D (2) A D E B C (2) A B E C D (2) E C A B D (1) E A D C B (1) E A C D B (1) D E C A B (1) D C E B A (1) D A C E B (1) C B E D A (1) C B D E A (1) B E C A D (1) B D C A E (1) B C A D E (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 12 14 18 18 B -12 0 18 14 6 C -14 -18 0 -4 -6 D -18 -14 4 0 4 E -18 -6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 18 18 B -12 0 18 14 6 C -14 -18 0 -4 -6 D -18 -14 4 0 4 E -18 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 D=20 B=19 C=12 so C is eliminated. Round 2 votes counts: D=28 A=26 E=25 B=21 so B is eliminated. Round 3 votes counts: A=41 D=32 E=27 so E is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:231 B:213 E:189 D:188 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 18 18 B -12 0 18 14 6 C -14 -18 0 -4 -6 D -18 -14 4 0 4 E -18 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 18 18 B -12 0 18 14 6 C -14 -18 0 -4 -6 D -18 -14 4 0 4 E -18 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 18 18 B -12 0 18 14 6 C -14 -18 0 -4 -6 D -18 -14 4 0 4 E -18 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 85: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) A E C D B (8) A C D B E (8) E B D C A (6) E B D A C (4) E A C B D (3) E A B D C (3) D A C B E (3) C D A B E (3) C B D A E (3) C A B D E (3) B E D C A (3) B C D A E (3) A E D C B (3) A C B D E (3) E D B A C (2) E D A B C (2) D C B A E (2) D B E C A (2) D B C E A (2) D B C A E (2) C D B A E (2) C B A E D (2) B D E C A (2) B D C E A (2) B D C A E (2) B C D E A (2) A E C B D (2) E D B C A (1) E B C D A (1) E B C A D (1) E B A D C (1) E B A C D (1) E A D C B (1) D E A B C (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 12 0 4 B -8 0 2 -4 -2 C -12 -2 0 -6 -10 D 0 4 6 0 -6 E -4 2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.771535 B: 0.000000 C: 0.000000 D: 0.228465 E: 0.000000 Sum of squares = 0.647462404008 Cumulative probabilities = A: 0.771535 B: 0.771535 C: 0.771535 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 0 4 B -8 0 2 -4 -2 C -12 -2 0 -6 -10 D 0 4 6 0 -6 E -4 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028997 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=27 B=14 C=13 D=12 so D is eliminated. Round 2 votes counts: E=35 A=30 B=20 C=15 so C is eliminated. Round 3 votes counts: A=36 E=35 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:207 D:202 B:194 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 0 4 B -8 0 2 -4 -2 C -12 -2 0 -6 -10 D 0 4 6 0 -6 E -4 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028997 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 0 4 B -8 0 2 -4 -2 C -12 -2 0 -6 -10 D 0 4 6 0 -6 E -4 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028997 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 0 4 B -8 0 2 -4 -2 C -12 -2 0 -6 -10 D 0 4 6 0 -6 E -4 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028997 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 86: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) A B D C E (9) A E D B C (8) E C A B D (5) D B C E A (5) C E B D A (5) A E C B D (5) A D B E C (5) C E D B A (4) E A C D B (3) D C B E A (3) D B A C E (3) A C B E D (3) A B D E C (3) A B C D E (3) E D C B A (2) E C D A B (2) E A D C B (2) D E A B C (2) C D B E A (2) A D B C E (2) E D A B C (1) E A D B C (1) E A C B D (1) D B A E C (1) D A E B C (1) D A B C E (1) C B E D A (1) C B D E A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 16 12 2 -4 B -16 0 -2 -18 -8 C -12 2 0 -4 -6 D -2 18 4 0 -10 E 4 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000490 D: 0.000000 E: 0.999510 Sum of squares = 0.999020654709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000490 D: 0.000490 E: 1.000000 A B C D E A 0 16 12 2 -4 B -16 0 -2 -18 -8 C -12 2 0 -4 -6 D -2 18 4 0 -10 E 4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=28 D=16 C=13 B=4 so B is eliminated. Round 2 votes counts: A=39 E=28 D=19 C=14 so C is eliminated. Round 3 votes counts: A=39 E=38 D=23 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:213 D:205 C:190 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 12 2 -4 B -16 0 -2 -18 -8 C -12 2 0 -4 -6 D -2 18 4 0 -10 E 4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 2 -4 B -16 0 -2 -18 -8 C -12 2 0 -4 -6 D -2 18 4 0 -10 E 4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 2 -4 B -16 0 -2 -18 -8 C -12 2 0 -4 -6 D -2 18 4 0 -10 E 4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 87: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (5) E B D A C (5) D E C A B (5) B C D A E (4) B A C E D (4) A E D C B (4) E D B C A (3) E D B A C (3) E A D C B (3) E A D B C (3) E A B D C (3) C D B A E (3) C B D A E (3) B E A D C (3) A C D E B (3) A B E C D (3) D E C B A (2) D E B C A (2) D C E A B (2) D C A E B (2) D B C E A (2) C D A E B (2) C A D B E (2) C A B D E (2) B E A C D (2) B D C E A (2) B C D E A (2) A E C D B (2) A E C B D (2) A C B E D (2) E D C B A (1) E B D C A (1) E B A D C (1) D E A C B (1) D C E B A (1) D B E C A (1) C B A D E (1) B E D C A (1) B C A D E (1) B A E C D (1) B A C D E (1) A E B C D (1) A C E D B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 10 -6 -8 B 0 0 0 -6 -16 C -10 0 0 -12 -16 D 6 6 12 0 -10 E 8 16 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 10 -6 -8 B 0 0 0 -6 -16 C -10 0 0 -12 -16 D 6 6 12 0 -10 E 8 16 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=21 A=20 D=18 C=13 so C is eliminated. Round 2 votes counts: E=28 B=25 A=24 D=23 so D is eliminated. Round 3 votes counts: E=41 B=31 A=28 so A is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:207 A:198 B:189 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 -6 -8 B 0 0 0 -6 -16 C -10 0 0 -12 -16 D 6 6 12 0 -10 E 8 16 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -6 -8 B 0 0 0 -6 -16 C -10 0 0 -12 -16 D 6 6 12 0 -10 E 8 16 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -6 -8 B 0 0 0 -6 -16 C -10 0 0 -12 -16 D 6 6 12 0 -10 E 8 16 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 88: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (15) C B E D A (9) B C D A E (9) B C E A D (7) B A D C E (7) A D E B C (7) E C D A B (5) A D B E C (5) C E B D A (4) A E D B C (4) E A D B C (3) E C A D B (2) D E A C B (2) C B D A E (2) E D A C B (1) E C A B D (1) E B C A D (1) E A C B D (1) D C E A B (1) D A E C B (1) D A E B C (1) D A B E C (1) C E D B A (1) C B E A D (1) C B D E A (1) B D A C E (1) B C D E A (1) B C A D E (1) B A D E C (1) B A C D E (1) A E B D C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 8 20 -12 B -6 0 6 0 -2 C -8 -6 0 -6 -6 D -20 0 6 0 -12 E 12 2 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 20 -12 B -6 0 6 0 -2 C -8 -6 0 -6 -6 D -20 0 6 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=28 A=19 C=18 D=6 so D is eliminated. Round 2 votes counts: E=31 B=28 A=22 C=19 so C is eliminated. Round 3 votes counts: B=41 E=37 A=22 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:211 B:199 C:187 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 20 -12 B -6 0 6 0 -2 C -8 -6 0 -6 -6 D -20 0 6 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 20 -12 B -6 0 6 0 -2 C -8 -6 0 -6 -6 D -20 0 6 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 20 -12 B -6 0 6 0 -2 C -8 -6 0 -6 -6 D -20 0 6 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 89: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (18) A E B C D (12) E A B C D (6) E A B D C (5) E B A D C (4) D C B A E (4) B E D C A (4) E A D B C (3) D B C E A (3) C D B A E (3) C D A B E (3) B E C A D (3) B C D E A (3) D B E C A (2) C D B E A (2) C A D B E (2) A E B D C (2) A C D E B (2) A C B E D (2) E D A B C (1) E B D A C (1) D C A E B (1) D C A B E (1) D A C E B (1) C D A E B (1) C B D E A (1) B E D A C (1) B E C D A (1) B E A C D (1) B D C E A (1) B C E A D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -8 -4 -22 B 6 0 8 0 12 C 8 -8 0 -6 2 D 4 0 6 0 -4 E 22 -12 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.692357 C: 0.000000 D: 0.307643 E: 0.000000 Sum of squares = 0.574002367003 Cumulative probabilities = A: 0.000000 B: 0.692357 C: 0.692357 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -4 -22 B 6 0 8 0 12 C 8 -8 0 -6 2 D 4 0 6 0 -4 E 22 -12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 E=20 B=15 C=12 so C is eliminated. Round 2 votes counts: D=39 A=25 E=20 B=16 so B is eliminated. Round 3 votes counts: D=44 E=31 A=25 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:206 D:203 C:198 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -4 -22 B 6 0 8 0 12 C 8 -8 0 -6 2 D 4 0 6 0 -4 E 22 -12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -4 -22 B 6 0 8 0 12 C 8 -8 0 -6 2 D 4 0 6 0 -4 E 22 -12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -4 -22 B 6 0 8 0 12 C 8 -8 0 -6 2 D 4 0 6 0 -4 E 22 -12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 90: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) D C E A B (7) B E A D C (7) E B D C A (6) C D A E B (6) A C B E D (5) A C B D E (5) E D C B A (4) C D E A B (4) B A D E C (4) D C E B A (3) D A C B E (3) B E D A C (3) A D C B E (3) A C D B E (3) A B D E C (3) A B C D E (3) D E C B A (2) D E B C A (2) B E A C D (2) A B D C E (2) E D B C A (1) E C B D A (1) E B C A D (1) D B E A C (1) C E D B A (1) C E A B D (1) C D E B A (1) C A D E B (1) C A D B E (1) C A B E D (1) B A E C D (1) A D B C E (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 14 10 6 B -4 0 -4 12 18 C -14 4 0 -22 4 D -10 -12 22 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 10 6 B -4 0 -4 12 18 C -14 4 0 -22 4 D -10 -12 22 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 D=18 C=16 E=13 so E is eliminated. Round 2 votes counts: B=32 A=28 D=23 C=17 so C is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:211 D:205 C:186 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 6 B -4 0 -4 12 18 C -14 4 0 -22 4 D -10 -12 22 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 6 B -4 0 -4 12 18 C -14 4 0 -22 4 D -10 -12 22 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 6 B -4 0 -4 12 18 C -14 4 0 -22 4 D -10 -12 22 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 91: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) E C A B D (10) D B A C E (8) E A D B C (5) B D A C E (5) E C A D B (4) E A C D B (4) D B A E C (4) C E B A D (4) C E A B D (4) A B D C E (4) E C D B A (3) C E B D A (3) B D C A E (3) A D B E C (3) E D C B A (2) E D B A C (1) E D A C B (1) E C D A B (1) E C B D A (1) E C B A D (1) E A C B D (1) D E B A C (1) D B E A C (1) D B C A E (1) D A B C E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B E D (1) C A B D E (1) B C D A E (1) A E D C B (1) A E C B D (1) A D E B C (1) A D B C E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -10 0 4 B 8 0 -20 14 0 C 10 20 0 14 8 D 0 -14 -14 0 0 E -4 0 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 0 4 B 8 0 -20 14 0 C 10 20 0 14 8 D 0 -14 -14 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=28 D=16 A=13 B=9 so B is eliminated. Round 2 votes counts: E=34 C=29 D=24 A=13 so A is eliminated. Round 3 votes counts: E=36 D=33 C=31 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:226 B:201 E:194 A:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 0 4 B 8 0 -20 14 0 C 10 20 0 14 8 D 0 -14 -14 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 0 4 B 8 0 -20 14 0 C 10 20 0 14 8 D 0 -14 -14 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 0 4 B 8 0 -20 14 0 C 10 20 0 14 8 D 0 -14 -14 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 92: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (16) E D A B C (7) E B C D A (7) D E A B C (7) C B A E D (6) D A E C B (5) C A B D E (5) A D C B E (5) E B D C A (4) C B E A D (4) B E C D A (4) A C D B E (4) D A E B C (3) C B A D E (3) C A D B E (3) A D C E B (3) E D B A C (2) E D A C B (1) E B D A C (1) D E A C B (1) C E B D A (1) B E D C A (1) B C E D A (1) B C A E D (1) B C A D E (1) A D E C B (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 10 -14 B 4 0 14 12 12 C 14 -14 0 14 12 D -10 -12 -14 0 -12 E 14 -12 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 10 -14 B 4 0 14 12 12 C 14 -14 0 14 12 D -10 -12 -14 0 -12 E 14 -12 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=22 C=22 D=16 A=16 so D is eliminated. Round 2 votes counts: E=30 B=24 A=24 C=22 so C is eliminated. Round 3 votes counts: B=37 A=32 E=31 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:213 E:201 A:189 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -14 10 -14 B 4 0 14 12 12 C 14 -14 0 14 12 D -10 -12 -14 0 -12 E 14 -12 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 10 -14 B 4 0 14 12 12 C 14 -14 0 14 12 D -10 -12 -14 0 -12 E 14 -12 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 10 -14 B 4 0 14 12 12 C 14 -14 0 14 12 D -10 -12 -14 0 -12 E 14 -12 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 93: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) D C A E B (6) E A B D C (5) D A E C B (5) C B D A E (5) C B A D E (5) C D A E B (4) B C E D A (4) E D A B C (3) E A D B C (3) D E A B C (3) C D B A E (3) C B A E D (3) C A D E B (3) B C E A D (3) B C D E A (3) A E D B C (3) A E B C D (3) E D B A C (2) E B D A C (2) D A C E B (2) C B D E A (2) B E A D C (2) A E D C B (2) E B A D C (1) D E B A C (1) D B E A C (1) D B C E A (1) D A E B C (1) C A E D B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B C A E D (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 14 6 -2 20 B -14 0 -6 -10 -22 C -6 6 0 -6 2 D 2 10 6 0 10 E -20 22 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -2 20 B -14 0 -6 -10 -22 C -6 6 0 -6 2 D 2 10 6 0 10 E -20 22 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=20 A=20 B=17 E=16 so E is eliminated. Round 2 votes counts: A=28 C=27 D=25 B=20 so B is eliminated. Round 3 votes counts: C=39 A=32 D=29 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:214 C:198 E:195 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -2 20 B -14 0 -6 -10 -22 C -6 6 0 -6 2 D 2 10 6 0 10 E -20 22 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -2 20 B -14 0 -6 -10 -22 C -6 6 0 -6 2 D 2 10 6 0 10 E -20 22 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -2 20 B -14 0 -6 -10 -22 C -6 6 0 -6 2 D 2 10 6 0 10 E -20 22 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 94: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) B C E D A (7) A D E C B (7) B C D E A (6) A E D C B (6) D A C E B (5) B E C A D (5) B E A C D (5) E C A D B (4) C E B D A (4) E C A B D (3) D C A E B (3) D A B C E (3) C E D A B (3) C D E A B (3) B D C A E (3) B A D E C (3) A B D E C (3) B A D C E (2) A D E B C (2) A D B E C (2) E C D A B (1) E C B A D (1) E A C D B (1) E A C B D (1) E A B C D (1) D C B E A (1) D B A C E (1) C B E D A (1) B D C E A (1) B A E D C (1) B A E C D (1) A E B D C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 8 2 4 B -2 0 12 16 6 C -8 -12 0 -4 0 D -2 -16 4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 2 4 B -2 0 12 16 6 C -8 -12 0 -4 0 D -2 -16 4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 A=23 D=13 E=12 C=11 so C is eliminated. Round 2 votes counts: B=42 A=23 E=19 D=16 so D is eliminated. Round 3 votes counts: B=44 A=34 E=22 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:216 A:208 D:195 E:193 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 2 4 B -2 0 12 16 6 C -8 -12 0 -4 0 D -2 -16 4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 2 4 B -2 0 12 16 6 C -8 -12 0 -4 0 D -2 -16 4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 2 4 B -2 0 12 16 6 C -8 -12 0 -4 0 D -2 -16 4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 95: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) D E B A C (6) B C A E D (6) E A D B C (5) D E A C B (5) D C B E A (5) E D A B C (4) D E B C A (4) C B D A E (4) C A B E D (4) C D B E A (3) C B A E D (3) C B A D E (3) A E D C B (3) D E C A B (2) D C E A B (2) D B E C A (2) C D B A E (2) B C D A E (2) B C A D E (2) A E D B C (2) A E C B D (2) A E B C D (2) A C E B D (2) A B E C D (2) E D A C B (1) E A D C B (1) E A B D C (1) D C E B A (1) D B C E A (1) C D A E B (1) C D A B E (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C E A (1) B A E D C (1) B A C E D (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 4 8 B -2 0 -8 -4 8 C 0 8 0 0 8 D -4 4 0 0 -4 E -8 -8 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.774292 B: 0.000000 C: 0.225708 D: 0.000000 E: 0.000000 Sum of squares = 0.650471930068 Cumulative probabilities = A: 0.774292 B: 0.774292 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 4 8 B -2 0 -8 -4 8 C 0 8 0 0 8 D -4 4 0 0 -4 E -8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=23 C=22 B=15 E=12 so E is eliminated. Round 2 votes counts: D=33 A=30 C=22 B=15 so B is eliminated. Round 3 votes counts: D=35 A=33 C=32 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:208 A:207 D:198 B:197 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 4 8 B -2 0 -8 -4 8 C 0 8 0 0 8 D -4 4 0 0 -4 E -8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 4 8 B -2 0 -8 -4 8 C 0 8 0 0 8 D -4 4 0 0 -4 E -8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 4 8 B -2 0 -8 -4 8 C 0 8 0 0 8 D -4 4 0 0 -4 E -8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 96: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) D E C A B (10) C D E B A (9) E D A C B (5) C E D A B (5) C D E A B (5) B A D E C (5) B A C D E (5) E D C A B (4) C E A D B (3) C D B E A (3) B A C E D (3) A E C B D (3) E A D B C (2) D C E B A (2) C B A E D (2) C A E B D (2) C A B E D (2) B A E D C (2) A E B D C (2) A B E C D (2) E D A B C (1) E C D A B (1) E A D C B (1) D E B A C (1) D E A C B (1) D E A B C (1) D C E A B (1) D B A E C (1) C B A D E (1) B D C A E (1) A E D B C (1) A B C E D (1) Total count = 100 A B C D E A 0 30 -2 -2 -10 B -30 0 -20 -14 -20 C 2 20 0 -6 -10 D 2 14 6 0 -8 E 10 20 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 30 -2 -2 -10 B -30 0 -20 -14 -20 C 2 20 0 -6 -10 D 2 14 6 0 -8 E 10 20 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=21 D=17 B=16 E=14 so E is eliminated. Round 2 votes counts: C=33 D=27 A=24 B=16 so B is eliminated. Round 3 votes counts: A=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:224 A:208 D:207 C:203 B:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 30 -2 -2 -10 B -30 0 -20 -14 -20 C 2 20 0 -6 -10 D 2 14 6 0 -8 E 10 20 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 -2 -2 -10 B -30 0 -20 -14 -20 C 2 20 0 -6 -10 D 2 14 6 0 -8 E 10 20 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 -2 -2 -10 B -30 0 -20 -14 -20 C 2 20 0 -6 -10 D 2 14 6 0 -8 E 10 20 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 97: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (12) B E C D A (9) A D C E B (9) D A E B C (5) D A C E B (5) C B E A D (5) E B C A D (4) C A E B D (4) B C E D A (4) E B D A C (3) E B A C D (3) C A D B E (3) B E D C A (3) A C D E B (3) E A B D C (2) D B E A C (2) D B C A E (2) D B A E C (2) D A B E C (2) C E B A D (2) C A E D B (2) A E C B D (2) A D E B C (2) E B C D A (1) E A D B C (1) D E A B C (1) D A B C E (1) C B E D A (1) C B D E A (1) C B D A E (1) C A D E B (1) B E C A D (1) A E C D B (1) Total count = 100 A B C D E A 0 12 12 -10 14 B -12 0 -4 -8 -2 C -12 4 0 -4 12 D 10 8 4 0 4 E -14 2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 -10 14 B -12 0 -4 -8 -2 C -12 4 0 -4 12 D 10 8 4 0 4 E -14 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=20 B=17 A=17 E=14 so E is eliminated. Round 2 votes counts: D=32 B=28 C=20 A=20 so C is eliminated. Round 3 votes counts: B=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:213 C:200 B:187 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 12 -10 14 B -12 0 -4 -8 -2 C -12 4 0 -4 12 D 10 8 4 0 4 E -14 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 -10 14 B -12 0 -4 -8 -2 C -12 4 0 -4 12 D 10 8 4 0 4 E -14 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 -10 14 B -12 0 -4 -8 -2 C -12 4 0 -4 12 D 10 8 4 0 4 E -14 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 98: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (6) B C D A E (6) B C A D E (6) E D A B C (5) A E D C B (5) E D A C B (4) D E A C B (4) C B A D E (4) B D C E A (4) B C A E D (4) A E C D B (4) D E B A C (3) D B C E A (3) B E D A C (3) B A C E D (3) A C E D B (3) A C B E D (3) E D B A C (2) E A D B C (2) D E B C A (2) D C E A B (2) C A B D E (2) B D E C A (2) B A E C D (2) E A D C B (1) D E C A B (1) D E A B C (1) D C E B A (1) D C B E A (1) C D B E A (1) C D B A E (1) C D A E B (1) C B D A E (1) C B A E D (1) C A D B E (1) C A B E D (1) B E A D C (1) A E B C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -2 -8 2 B 14 0 14 4 14 C 2 -14 0 6 12 D 8 -4 -6 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -8 2 B 14 0 14 4 14 C 2 -14 0 6 12 D 8 -4 -6 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=18 A=18 E=14 C=13 so C is eliminated. Round 2 votes counts: B=43 A=22 D=21 E=14 so E is eliminated. Round 3 votes counts: B=43 D=32 A=25 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:203 D:202 A:189 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -8 2 B 14 0 14 4 14 C 2 -14 0 6 12 D 8 -4 -6 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -8 2 B 14 0 14 4 14 C 2 -14 0 6 12 D 8 -4 -6 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -8 2 B 14 0 14 4 14 C 2 -14 0 6 12 D 8 -4 -6 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 99: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) E C A D B (5) E A D C B (5) B D A E C (5) B C D A E (5) B D C A E (4) B C E D A (4) A D E C B (4) E C B A D (3) D C A B E (3) D A C E B (3) D A B E C (3) C D A B E (3) B E C D A (3) B C D E A (3) E B A D C (2) E A D B C (2) D B A C E (2) D A B C E (2) C E A D B (2) C B D E A (2) C A D E B (2) B E D A C (2) B D E A C (2) A E D C B (2) A D C E B (2) E C A B D (1) E B A C D (1) E A C D B (1) D C B A E (1) D B A E C (1) D A E B C (1) D A C B E (1) C E A B D (1) C D B A E (1) C B E D A (1) C B E A D (1) C A E D B (1) B E C A D (1) B D C E A (1) B C E A D (1) A E D B C (1) A E C D B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 -6 0 B 4 0 -6 -2 4 C 10 6 0 0 6 D 6 2 0 0 6 E 0 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.431678 D: 0.568322 E: 0.000000 Sum of squares = 0.50933566505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.431678 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 0 B 4 0 -6 -2 4 C 10 6 0 0 6 D 6 2 0 0 6 E 0 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=20 C=20 D=17 A=12 so A is eliminated. Round 2 votes counts: B=31 E=24 D=24 C=21 so C is eliminated. Round 3 votes counts: B=35 E=34 D=31 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 D:207 B:200 E:192 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 0 B 4 0 -6 -2 4 C 10 6 0 0 6 D 6 2 0 0 6 E 0 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 0 B 4 0 -6 -2 4 C 10 6 0 0 6 D 6 2 0 0 6 E 0 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 0 B 4 0 -6 -2 4 C 10 6 0 0 6 D 6 2 0 0 6 E 0 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 100: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (13) B A E D C (9) B A C E D (7) D E C A B (5) D E A C B (5) D E A B C (5) A E D B C (5) E D A C B (4) C E D A B (4) B C A E D (4) B C A D E (4) B A D E C (4) A E D C B (4) C D E B A (3) C B A E D (3) C B A D E (3) B C D E A (3) E D A B C (2) A C E D B (2) A B E D C (2) E A D B C (1) D E C B A (1) D C E A B (1) C B D E A (1) C A B E D (1) B A E C D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 8 6 4 B -14 0 -2 -10 -10 C -8 2 0 4 2 D -6 10 -4 0 -4 E -4 10 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 6 4 B -14 0 -2 -10 -10 C -8 2 0 4 2 D -6 10 -4 0 -4 E -4 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=28 D=17 A=16 E=7 so E is eliminated. Round 2 votes counts: B=32 C=28 D=23 A=17 so A is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:216 E:204 C:200 D:198 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 6 4 B -14 0 -2 -10 -10 C -8 2 0 4 2 D -6 10 -4 0 -4 E -4 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 6 4 B -14 0 -2 -10 -10 C -8 2 0 4 2 D -6 10 -4 0 -4 E -4 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 6 4 B -14 0 -2 -10 -10 C -8 2 0 4 2 D -6 10 -4 0 -4 E -4 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 101: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) C A E B D (9) A C B E D (9) C D E B A (6) A B E C D (5) E B D A C (4) D E B C A (4) D B E A C (4) C A D E B (4) E B D C A (3) C E B D A (3) C D A E B (3) B E D A C (3) A B D E C (3) E B A C D (2) D E C B A (2) D C E B A (2) D A B E C (2) C E D B A (2) C D E A B (2) B D E A C (2) A C E B D (2) A C D B E (2) E B C D A (1) E B A D C (1) E A B C D (1) D E B A C (1) D C A E B (1) D B E C A (1) D B A E C (1) C E B A D (1) C A E D B (1) C A D B E (1) B A D E C (1) Total count = 100 A B C D E A 0 12 8 6 10 B -12 0 0 22 -10 C -8 0 0 8 -4 D -6 -22 -8 0 -16 E -10 10 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 6 10 B -12 0 0 22 -10 C -8 0 0 8 -4 D -6 -22 -8 0 -16 E -10 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=32 A=32 D=18 E=12 B=6 so B is eliminated. Round 2 votes counts: A=33 C=32 D=20 E=15 so E is eliminated. Round 3 votes counts: A=37 C=33 D=30 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:210 B:200 C:198 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 6 10 B -12 0 0 22 -10 C -8 0 0 8 -4 D -6 -22 -8 0 -16 E -10 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 6 10 B -12 0 0 22 -10 C -8 0 0 8 -4 D -6 -22 -8 0 -16 E -10 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 6 10 B -12 0 0 22 -10 C -8 0 0 8 -4 D -6 -22 -8 0 -16 E -10 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 102: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (14) C B A D E (7) B C E D A (7) E D A B C (6) B C A D E (5) C A B D E (4) B E D A C (4) A C D E B (4) E D B A C (3) E D A C B (3) E A D B C (3) C B D A E (3) A D C E B (3) E D C A B (2) D A E C B (2) C B E D A (2) B E D C A (2) B E C D A (2) B E A D C (2) B C E A D (2) A E D B C (2) E D B C A (1) E B D C A (1) E B D A C (1) E A D C B (1) E A B D C (1) D E C A B (1) D E A C B (1) C D E A B (1) C D B E A (1) C D A E B (1) C D A B E (1) C A D E B (1) C A D B E (1) B E A C D (1) B C A E D (1) B A E D C (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 8 10 4 B -8 0 -8 -6 -4 C -8 8 0 -10 -10 D -10 6 10 0 2 E -4 4 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 10 4 B -8 0 -8 -6 -4 C -8 8 0 -10 -10 D -10 6 10 0 2 E -4 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=25 E=22 C=22 D=4 so D is eliminated. Round 2 votes counts: B=27 A=27 E=24 C=22 so C is eliminated. Round 3 votes counts: B=40 A=35 E=25 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:204 E:204 C:190 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 10 4 B -8 0 -8 -6 -4 C -8 8 0 -10 -10 D -10 6 10 0 2 E -4 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 10 4 B -8 0 -8 -6 -4 C -8 8 0 -10 -10 D -10 6 10 0 2 E -4 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 10 4 B -8 0 -8 -6 -4 C -8 8 0 -10 -10 D -10 6 10 0 2 E -4 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 103: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) E D C B A (8) A D E C B (7) A D C B E (7) D E A C B (5) B E C A D (5) D A E C B (4) C B D A E (3) B E C D A (3) B C E D A (3) B C E A D (3) B A C D E (3) A D B E C (3) E D B C A (2) E D A B C (2) D C A E B (2) D A C B E (2) B C A E D (2) A E D B C (2) A D E B C (2) A D B C E (2) A B C D E (2) E D C A B (1) E D A C B (1) E C D B A (1) E B D C A (1) E B A D C (1) E B A C D (1) E A D B C (1) D E C A B (1) D A C E B (1) C E D B A (1) C E B D A (1) C B E D A (1) C A B D E (1) B A E C D (1) A D C E B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 0 -8 -6 B 2 0 2 -12 -14 C 0 -2 0 -14 -28 D 8 12 14 0 -6 E 6 14 28 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 -8 -6 B 2 0 2 -12 -14 C 0 -2 0 -14 -28 D 8 12 14 0 -6 E 6 14 28 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=28 B=20 D=15 C=7 so C is eliminated. Round 2 votes counts: E=32 A=29 B=24 D=15 so D is eliminated. Round 3 votes counts: E=38 A=38 B=24 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:214 A:192 B:189 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 -8 -6 B 2 0 2 -12 -14 C 0 -2 0 -14 -28 D 8 12 14 0 -6 E 6 14 28 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -8 -6 B 2 0 2 -12 -14 C 0 -2 0 -14 -28 D 8 12 14 0 -6 E 6 14 28 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -8 -6 B 2 0 2 -12 -14 C 0 -2 0 -14 -28 D 8 12 14 0 -6 E 6 14 28 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 104: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) A D B C E (10) E C B D A (7) E C B A D (7) B E C D A (7) D A B C E (6) B E C A D (5) A D C E B (5) D C E A B (4) C E D B A (4) B A E C D (3) B A D E C (3) A C E D B (3) A B D E C (3) D B A E C (2) D A B E C (2) C E D A B (2) C E B A D (2) C E A B D (2) B D A E C (2) E C D B A (1) E B C A D (1) D C A E B (1) C E A D B (1) C A E D B (1) B D E C A (1) A D B E C (1) A C E B D (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 8 2 12 B -10 0 -4 -8 -6 C -8 4 0 -2 10 D -2 8 2 0 4 E -12 6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 2 12 B -10 0 -4 -8 -6 C -8 4 0 -2 10 D -2 8 2 0 4 E -12 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 B=21 E=16 C=12 so C is eliminated. Round 2 votes counts: E=27 A=27 D=25 B=21 so B is eliminated. Round 3 votes counts: E=39 A=33 D=28 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:206 C:202 E:190 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 2 12 B -10 0 -4 -8 -6 C -8 4 0 -2 10 D -2 8 2 0 4 E -12 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 2 12 B -10 0 -4 -8 -6 C -8 4 0 -2 10 D -2 8 2 0 4 E -12 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 2 12 B -10 0 -4 -8 -6 C -8 4 0 -2 10 D -2 8 2 0 4 E -12 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 105: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) A C D E B (8) C D A E B (7) B E D C A (7) B D C E A (7) E B A C D (6) E A C D B (6) D C A B E (5) E A C B D (4) E A B C D (3) D C B A E (3) A C E D B (3) E C A D B (2) D B A C E (2) D A C B E (2) C D E A B (2) C A D E B (2) B E A C D (2) B D E C A (2) B D A C E (2) A E C D B (2) E C B D A (1) E C A B D (1) E B C A D (1) D C B E A (1) D A C E B (1) D A B C E (1) C E D B A (1) B E C D A (1) B E A D C (1) B D A E C (1) B A E D C (1) A E B C D (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -2 -8 4 B -6 0 -4 0 -2 C 2 4 0 6 14 D 8 0 -6 0 12 E -4 2 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -8 4 B -6 0 -4 0 -2 C 2 4 0 6 14 D 8 0 -6 0 12 E -4 2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=24 A=17 D=15 C=12 so C is eliminated. Round 2 votes counts: B=32 E=25 D=24 A=19 so A is eliminated. Round 3 votes counts: D=36 B=33 E=31 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:207 A:200 B:194 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -8 4 B -6 0 -4 0 -2 C 2 4 0 6 14 D 8 0 -6 0 12 E -4 2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -8 4 B -6 0 -4 0 -2 C 2 4 0 6 14 D 8 0 -6 0 12 E -4 2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -8 4 B -6 0 -4 0 -2 C 2 4 0 6 14 D 8 0 -6 0 12 E -4 2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 106: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) A C E B D (8) E C D B A (7) C A E B D (7) C A E D B (6) B D E A C (5) A C D B E (5) B D A E C (4) A C B E D (4) E B D C A (3) A C B D E (3) E D C B A (2) E D B C A (2) E C B D A (2) D E B C A (2) D C E A B (2) D B E A C (2) D B A C E (2) C E A D B (2) C E A B D (2) C D A E B (2) C A D E B (2) B D E C A (2) A B D C E (2) E C B A D (1) E C A B D (1) E B A C D (1) D E C B A (1) D C E B A (1) D B C A E (1) C E D A B (1) B E D A C (1) B E A D C (1) B A E D C (1) B A D C E (1) A D B C E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -14 0 2 B 0 0 -18 2 -10 C 14 18 0 12 6 D 0 -2 -12 0 -6 E -2 10 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 0 2 B 0 0 -18 2 -10 C 14 18 0 12 6 D 0 -2 -12 0 -6 E -2 10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=22 E=19 D=19 B=15 so B is eliminated. Round 2 votes counts: D=30 A=27 C=22 E=21 so E is eliminated. Round 3 votes counts: D=38 C=33 A=29 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:204 A:194 D:190 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 0 2 B 0 0 -18 2 -10 C 14 18 0 12 6 D 0 -2 -12 0 -6 E -2 10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 0 2 B 0 0 -18 2 -10 C 14 18 0 12 6 D 0 -2 -12 0 -6 E -2 10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 0 2 B 0 0 -18 2 -10 C 14 18 0 12 6 D 0 -2 -12 0 -6 E -2 10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 107: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (11) E D C A B (10) A C D E B (8) B C A D E (7) E B D C A (6) B E D C A (6) B E C D A (6) A D C E B (6) D E A C B (4) E D A C B (3) C D E A B (3) A C B D E (3) D C E A B (2) D A C E B (2) C A D E B (2) B E A D C (2) B C E D A (2) B C E A D (2) B C A E D (2) B A C E D (2) E D B A C (1) E C D B A (1) E C D A B (1) D E C A B (1) C E D A B (1) C B A D E (1) B E C A D (1) A D E C B (1) A D C B E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 2 -4 B 0 0 -2 4 -4 C 8 2 0 10 14 D -2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 2 -4 B 0 0 -2 4 -4 C 8 2 0 10 14 D -2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=22 A=21 D=9 C=7 so C is eliminated. Round 2 votes counts: B=42 E=23 A=23 D=12 so D is eliminated. Round 3 votes counts: B=42 E=33 A=25 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:217 B:199 D:196 A:195 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 2 -4 B 0 0 -2 4 -4 C 8 2 0 10 14 D -2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 -4 B 0 0 -2 4 -4 C 8 2 0 10 14 D -2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 -4 B 0 0 -2 4 -4 C 8 2 0 10 14 D -2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 108: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) E D A C B (5) E A B D C (5) B C A D E (5) A E D C B (5) E B A D C (4) E A D C B (4) B D C E A (4) A E C B D (4) E A D B C (3) D C B E A (3) C D B A E (3) C D A B E (3) C B A D E (3) B E D C A (3) B C D E A (3) B A C E D (3) D C A E B (2) C D B E A (2) C D A E B (2) C A B D E (2) B C A E D (2) B A E C D (2) A E C D B (2) A E B C D (2) A C D E B (2) A B E C D (2) E D B C A (1) E B D A C (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E B A (1) D B E C A (1) C B D A E (1) C A D E B (1) B E C D A (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -4 8 12 B 2 0 0 12 2 C 4 0 0 10 4 D -8 -12 -10 0 -4 E -12 -2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.324536 C: 0.675464 D: 0.000000 E: 0.000000 Sum of squares = 0.561575565559 Cumulative probabilities = A: 0.000000 B: 0.324536 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 8 12 B 2 0 0 12 2 C 4 0 0 10 4 D -8 -12 -10 0 -4 E -12 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=23 A=20 C=17 D=10 so D is eliminated. Round 2 votes counts: B=31 E=26 C=23 A=20 so A is eliminated. Round 3 votes counts: E=39 B=33 C=28 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:208 A:207 E:193 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 8 12 B 2 0 0 12 2 C 4 0 0 10 4 D -8 -12 -10 0 -4 E -12 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 12 B 2 0 0 12 2 C 4 0 0 10 4 D -8 -12 -10 0 -4 E -12 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 12 B 2 0 0 12 2 C 4 0 0 10 4 D -8 -12 -10 0 -4 E -12 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 109: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) B C E A D (5) D A E C B (4) C B D E A (4) B C D E A (4) A D E B C (4) A B D E C (4) E C A B D (3) D E C A B (3) C B E D A (3) B A E C D (3) B A C E D (3) A E B C D (3) A D E C B (3) A D B E C (3) A B E D C (3) E C B A D (2) D C A E B (2) D B C A E (2) D A E B C (2) D A B C E (2) C E B D A (2) C D B E A (2) C B E A D (2) B C D A E (2) B C A E D (2) B A D C E (2) E D C A B (1) E C A D B (1) E B C A D (1) E A C B D (1) E A B C D (1) D E A C B (1) D C E A B (1) D C B E A (1) D C A B E (1) D B A C E (1) C E D B A (1) B E C A D (1) B E A C D (1) B D C A E (1) B D A C E (1) B A C D E (1) A E D B C (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 6 4 18 B -6 0 24 14 24 C -6 -24 0 -2 -10 D -4 -14 2 0 16 E -18 -24 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 4 18 B -6 0 24 14 24 C -6 -24 0 -2 -10 D -4 -14 2 0 16 E -18 -24 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 A=23 C=14 E=10 so E is eliminated. Round 2 votes counts: D=28 B=27 A=25 C=20 so C is eliminated. Round 3 votes counts: B=40 D=31 A=29 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:217 D:200 C:179 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 4 18 B -6 0 24 14 24 C -6 -24 0 -2 -10 D -4 -14 2 0 16 E -18 -24 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 18 B -6 0 24 14 24 C -6 -24 0 -2 -10 D -4 -14 2 0 16 E -18 -24 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 18 B -6 0 24 14 24 C -6 -24 0 -2 -10 D -4 -14 2 0 16 E -18 -24 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 110: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (5) B A E D C (5) A B E C D (5) A B D C E (5) E C D B A (4) D B A C E (4) C D E A B (4) C D A E B (4) D C E B A (3) D C B A E (3) C E D B A (3) B E A D C (3) A E B C D (3) A B E D C (3) E C A B D (2) E B C D A (2) E B A D C (2) E B A C D (2) D B E C A (2) D B C A E (2) D A B C E (2) C E D A B (2) C E A D B (2) C E A B D (2) C D A B E (2) B D A E C (2) B A D E C (2) A B C D E (2) E C B D A (1) E C A D B (1) E B D C A (1) E B D A C (1) E A C B D (1) E A B C D (1) D E B C A (1) D C B E A (1) D C A B E (1) D A C B E (1) B E D A C (1) B A D C E (1) A E C B D (1) A C E B D (1) A C D E B (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 4 -4 6 B 2 0 8 2 0 C -4 -8 0 6 6 D 4 -2 -6 0 0 E -6 0 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.833542 C: 0.000000 D: 0.000000 E: 0.166458 Sum of squares = 0.722500139379 Cumulative probabilities = A: 0.000000 B: 0.833542 C: 0.833542 D: 0.833542 E: 1.000000 A B C D E A 0 -2 4 -4 6 B 2 0 8 2 0 C -4 -8 0 6 6 D 4 -2 -6 0 0 E -6 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000592 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 D=20 E=18 B=14 so B is eliminated. Round 2 votes counts: A=32 C=24 E=22 D=22 so E is eliminated. Round 3 votes counts: A=41 C=34 D=25 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:206 A:202 C:200 D:198 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 -4 6 B 2 0 8 2 0 C -4 -8 0 6 6 D 4 -2 -6 0 0 E -6 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000592 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -4 6 B 2 0 8 2 0 C -4 -8 0 6 6 D 4 -2 -6 0 0 E -6 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000592 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -4 6 B 2 0 8 2 0 C -4 -8 0 6 6 D 4 -2 -6 0 0 E -6 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000592 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 111: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) A E B C D (10) C D B A E (8) E A B C D (5) C D A B E (5) E D B A C (4) E B A D C (4) E A C B D (4) D C B A E (4) E B D A C (3) E D B C A (2) D E B C A (2) C A D B E (2) C A B D E (2) B E A D C (2) B D E A C (2) B D C A E (2) B D A C E (2) B A E D C (2) A E C B D (2) A E B D C (2) E D A B C (1) E C A D B (1) E A D B C (1) D C E B A (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C E A (1) C D B E A (1) C D A E B (1) C A E B D (1) C A D E B (1) C A B E D (1) B D A E C (1) B A C D E (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 26 14 0 B -10 0 28 24 -18 C -26 -28 0 -4 -28 D -14 -24 4 0 -22 E 0 18 28 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.295684 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.704316 Sum of squares = 0.583489909447 Cumulative probabilities = A: 0.295684 B: 0.295684 C: 0.295684 D: 0.295684 E: 1.000000 A B C D E A 0 10 26 14 0 B -10 0 28 24 -18 C -26 -28 0 -4 -28 D -14 -24 4 0 -22 E 0 18 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=22 A=17 B=12 D=11 so D is eliminated. Round 2 votes counts: E=40 C=28 A=17 B=15 so B is eliminated. Round 3 votes counts: E=46 C=31 A=23 so A is eliminated. Round 4 votes counts: E=64 C=36 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:234 A:225 B:212 D:172 C:157 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 26 14 0 B -10 0 28 24 -18 C -26 -28 0 -4 -28 D -14 -24 4 0 -22 E 0 18 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 26 14 0 B -10 0 28 24 -18 C -26 -28 0 -4 -28 D -14 -24 4 0 -22 E 0 18 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 26 14 0 B -10 0 28 24 -18 C -26 -28 0 -4 -28 D -14 -24 4 0 -22 E 0 18 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 112: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (12) B E C D A (9) A D C B E (8) E B C D A (6) D C A B E (6) B D C A E (6) B C D E A (6) A E D C B (6) E B A C D (4) E A D C B (4) B C D A E (4) D C B A E (3) D A C B E (3) C D B A E (3) C D A B E (3) A C D E B (3) E A C D B (2) E A B D C (2) E A B C D (2) E B C A D (1) D B A C E (1) C D B E A (1) C B D E A (1) B E D C A (1) B C E D A (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 6 -2 -8 20 B -6 0 -12 -14 12 C 2 12 0 -8 22 D 8 14 8 0 22 E -20 -12 -22 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -8 20 B -6 0 -12 -14 12 C 2 12 0 -8 22 D 8 14 8 0 22 E -20 -12 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=27 E=21 D=13 C=8 so C is eliminated. Round 2 votes counts: A=31 B=28 E=21 D=20 so D is eliminated. Round 3 votes counts: A=43 B=36 E=21 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:226 C:214 A:208 B:190 E:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 -8 20 B -6 0 -12 -14 12 C 2 12 0 -8 22 D 8 14 8 0 22 E -20 -12 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -8 20 B -6 0 -12 -14 12 C 2 12 0 -8 22 D 8 14 8 0 22 E -20 -12 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -8 20 B -6 0 -12 -14 12 C 2 12 0 -8 22 D 8 14 8 0 22 E -20 -12 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 113: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) C E D A B (6) E C B A D (5) B E C D A (5) B E A D C (5) E C A D B (4) E B A C D (4) E A D B C (4) D A B C E (4) C D A B E (4) B D A C E (4) C E A D B (3) B C E D A (3) B A D E C (3) E C B D A (2) E C A B D (2) E A D C B (2) E A B D C (2) D A C B E (2) C E B D A (2) B D A E C (2) A D E C B (2) A D E B C (2) A D C E B (2) A B D E C (2) E B C D A (1) E B A D C (1) E A C D B (1) D C B A E (1) D C A B E (1) C E D B A (1) C D E A B (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D A E (1) C A D E B (1) B D E A C (1) B D C A E (1) B C D E A (1) A E D C B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -8 8 -26 B 2 0 2 8 -14 C 8 -2 0 14 -14 D -8 -8 -14 0 -22 E 26 14 14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -8 8 -26 B 2 0 2 8 -14 C 8 -2 0 14 -14 D -8 -8 -14 0 -22 E 26 14 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=25 C=22 A=11 D=8 so D is eliminated. Round 2 votes counts: E=34 B=25 C=24 A=17 so A is eliminated. Round 3 votes counts: E=39 B=31 C=30 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:238 C:203 B:199 A:186 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 8 -26 B 2 0 2 8 -14 C 8 -2 0 14 -14 D -8 -8 -14 0 -22 E 26 14 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 8 -26 B 2 0 2 8 -14 C 8 -2 0 14 -14 D -8 -8 -14 0 -22 E 26 14 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 8 -26 B 2 0 2 8 -14 C 8 -2 0 14 -14 D -8 -8 -14 0 -22 E 26 14 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 114: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) D B C A E (7) B D C A E (7) B C D A E (7) E A D C B (6) C B A E D (6) C B A D E (5) E D B A C (4) E D A C B (4) A C E B D (4) E A C B D (3) D E B A C (3) B C E D A (3) D E A B C (2) D B A E C (2) C B E A D (2) C A E B D (2) C A B E D (2) B C E A D (2) B C A D E (2) A E D C B (2) A E C D B (2) A C E D B (2) E C A B D (1) E B D C A (1) E B C A D (1) E A C D B (1) D B A C E (1) C E B A D (1) C A B D E (1) B E D C A (1) B E C D A (1) B D C E A (1) A D E C B (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -30 -22 -4 10 B 30 0 6 22 18 C 22 -6 0 14 30 D 4 -22 -14 0 -2 E -10 -18 -30 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -22 -4 10 B 30 0 6 22 18 C 22 -6 0 14 30 D 4 -22 -14 0 -2 E -10 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=21 C=19 D=15 A=13 so A is eliminated. Round 2 votes counts: B=32 C=26 E=25 D=17 so D is eliminated. Round 3 votes counts: B=42 E=31 C=27 so C is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:238 C:230 D:183 A:177 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -22 -4 10 B 30 0 6 22 18 C 22 -6 0 14 30 D 4 -22 -14 0 -2 E -10 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -22 -4 10 B 30 0 6 22 18 C 22 -6 0 14 30 D 4 -22 -14 0 -2 E -10 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -22 -4 10 B 30 0 6 22 18 C 22 -6 0 14 30 D 4 -22 -14 0 -2 E -10 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 115: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (14) A D B E C (14) D A B E C (8) C E B A D (7) D B A E C (5) A D B C E (5) E C B D A (4) B D E A C (4) B D A E C (4) E B C D A (3) C E A B D (3) B E D C A (3) A C D E B (3) C D A B E (2) C B E D A (2) C A D E B (2) B E D A C (2) B D E C A (2) A D C E B (2) A D C B E (2) A C D B E (2) E C B A D (1) D C A B E (1) D A C B E (1) C E A D B (1) C D B E A (1) B C E D A (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 6 -14 4 B 6 0 2 0 18 C -6 -2 0 -6 0 D 14 0 6 0 16 E -4 -18 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.378785 C: 0.000000 D: 0.621215 E: 0.000000 Sum of squares = 0.529385947053 Cumulative probabilities = A: 0.000000 B: 0.378785 C: 0.378785 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -14 4 B 6 0 2 0 18 C -6 -2 0 -6 0 D 14 0 6 0 16 E -4 -18 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=29 B=16 D=15 E=8 so E is eliminated. Round 2 votes counts: C=37 A=29 B=19 D=15 so D is eliminated. Round 3 votes counts: C=38 A=38 B=24 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:218 B:213 A:195 C:193 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -14 4 B 6 0 2 0 18 C -6 -2 0 -6 0 D 14 0 6 0 16 E -4 -18 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -14 4 B 6 0 2 0 18 C -6 -2 0 -6 0 D 14 0 6 0 16 E -4 -18 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -14 4 B 6 0 2 0 18 C -6 -2 0 -6 0 D 14 0 6 0 16 E -4 -18 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 116: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) D A B E C (8) C E B A D (8) A D E C B (8) B D A E C (6) B C D E A (6) A E D C B (6) C E A D B (5) C B E A D (5) B D C E A (4) B C E D A (4) E C A D B (3) D B A E C (3) D A E C B (3) B D C A E (3) B C E A D (3) D B A C E (2) D A E B C (2) C D E B A (2) B D A C E (2) E C A B D (1) E A C D B (1) D C E A B (1) D C A E B (1) C E D A B (1) C B E D A (1) B E A C D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -14 2 -10 B 0 0 -10 6 -2 C 14 10 0 0 14 D -2 -6 0 0 4 E 10 2 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.569928 D: 0.430072 E: 0.000000 Sum of squares = 0.509779970123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.569928 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 2 -10 B 0 0 -10 6 -2 C 14 10 0 0 14 D -2 -6 0 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=29 D=20 A=15 E=5 so E is eliminated. Round 2 votes counts: C=35 B=29 D=20 A=16 so A is eliminated. Round 3 votes counts: C=36 D=35 B=29 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 D:198 B:197 E:197 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 2 -10 B 0 0 -10 6 -2 C 14 10 0 0 14 D -2 -6 0 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 2 -10 B 0 0 -10 6 -2 C 14 10 0 0 14 D -2 -6 0 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 2 -10 B 0 0 -10 6 -2 C 14 10 0 0 14 D -2 -6 0 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 117: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (13) B C E D A (11) A D E C B (10) B C D E A (6) B A C D E (4) A B E C D (4) E A D C B (3) D E C A B (3) D C E B A (3) C B E D A (3) B A C E D (3) A E D B C (3) A E B C D (3) E D A C B (2) E C D B A (2) D E C B A (2) D A E C B (2) C E D B A (2) B C E A D (2) A D E B C (2) A B E D C (2) E C D A B (1) E A C D B (1) D C B E A (1) D B C E A (1) D B A C E (1) C E B D A (1) B D C A E (1) B D A C E (1) B C D A E (1) B C A E D (1) B C A D E (1) A E B D C (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 16 12 12 B -6 0 2 -8 -8 C -16 -2 0 -6 -10 D -12 8 6 0 -16 E -12 8 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 12 12 B -6 0 2 -8 -8 C -16 -2 0 -6 -10 D -12 8 6 0 -16 E -12 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 B=31 D=13 E=9 C=6 so C is eliminated. Round 2 votes counts: A=41 B=34 D=13 E=12 so E is eliminated. Round 3 votes counts: A=45 B=35 D=20 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:211 D:193 B:190 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 12 12 B -6 0 2 -8 -8 C -16 -2 0 -6 -10 D -12 8 6 0 -16 E -12 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 12 12 B -6 0 2 -8 -8 C -16 -2 0 -6 -10 D -12 8 6 0 -16 E -12 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 12 12 B -6 0 2 -8 -8 C -16 -2 0 -6 -10 D -12 8 6 0 -16 E -12 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 118: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) A D B E C (7) D A B C E (6) B D A C E (6) E C A D B (5) C E B D A (5) B D C A E (5) A E D C B (5) B D C E A (4) A D E B C (4) E C B A D (3) E A C D B (3) D B A C E (3) A D E C B (3) E C D A B (2) D B C A E (2) C E D B A (2) C D E B A (2) B C E A D (2) B C D E A (2) A E D B C (2) E C A B D (1) E B C A D (1) E A D C B (1) E A C B D (1) D C E B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D A E C B (1) D A C B E (1) C E B A D (1) C B E D A (1) C B D E A (1) B C D A E (1) B A D E C (1) A E C B D (1) A D B C E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -4 -12 4 B 6 0 14 -8 10 C 4 -14 0 -14 14 D 12 8 14 0 10 E -4 -10 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -12 4 B 6 0 14 -8 10 C 4 -14 0 -14 14 D 12 8 14 0 10 E -4 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=25 E=17 D=17 C=12 so C is eliminated. Round 2 votes counts: B=31 E=25 A=25 D=19 so D is eliminated. Round 3 votes counts: B=38 A=33 E=29 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:211 C:195 A:191 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -12 4 B 6 0 14 -8 10 C 4 -14 0 -14 14 D 12 8 14 0 10 E -4 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -12 4 B 6 0 14 -8 10 C 4 -14 0 -14 14 D 12 8 14 0 10 E -4 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -12 4 B 6 0 14 -8 10 C 4 -14 0 -14 14 D 12 8 14 0 10 E -4 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 119: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) C E D A B (5) C E B D A (5) E C B D A (4) B E D A C (4) B E C D A (4) A D B C E (4) E C D B A (3) E B D C A (3) C B E A D (3) B E D C A (3) B C E A D (3) B A E D C (3) B A C E D (3) A D C B E (3) A C D E B (3) A B D E C (3) A B D C E (3) D A C E B (2) C E B A D (2) C E A D B (2) B E C A D (2) B D E A C (2) B A D E C (2) A D C E B (2) A C D B E (2) E D C B A (1) D E B A C (1) D E A B C (1) D C A E B (1) D B E A C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E D B A (1) C E A B D (1) C D E A B (1) C A E B D (1) C A D E B (1) B E A C D (1) B C A E D (1) A D B E C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -8 -4 -20 B 18 0 10 24 2 C 8 -10 0 14 2 D 4 -24 -14 0 -26 E 20 -2 -2 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 -4 -20 B 18 0 10 24 2 C 8 -10 0 14 2 D 4 -24 -14 0 -26 E 20 -2 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=23 C=22 E=18 D=9 so D is eliminated. Round 2 votes counts: B=29 A=28 C=23 E=20 so E is eliminated. Round 3 votes counts: B=40 C=31 A=29 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:221 C:207 A:175 D:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -8 -4 -20 B 18 0 10 24 2 C 8 -10 0 14 2 D 4 -24 -14 0 -26 E 20 -2 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -4 -20 B 18 0 10 24 2 C 8 -10 0 14 2 D 4 -24 -14 0 -26 E 20 -2 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -4 -20 B 18 0 10 24 2 C 8 -10 0 14 2 D 4 -24 -14 0 -26 E 20 -2 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 120: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) E C B D A (8) E B C D A (5) D A E B C (5) C A B E D (5) C E D A B (4) B C A E D (4) A C B D E (4) E D C B A (3) E C D B A (3) D B A E C (3) D A E C B (3) C E B A D (3) A D C B E (3) A B D E C (3) E D B C A (2) D E C A B (2) C E B D A (2) B A E C D (2) B A C E D (2) A D B E C (2) A C D B E (2) A B D C E (2) E B D C A (1) E B D A C (1) D E C B A (1) D E B A C (1) D E A B C (1) D B E A C (1) D A B C E (1) C E D B A (1) C E A B D (1) C D E A B (1) C B E A D (1) C A E B D (1) C A B D E (1) B E D A C (1) B E C A D (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 -20 12 B -8 0 0 0 2 C 0 0 0 6 -18 D 20 0 -6 0 -4 E -12 -2 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.024000 B: 0.576000 C: 0.016000 D: 0.288000 E: 0.096000 Sum of squares = 0.424767999838 Cumulative probabilities = A: 0.024000 B: 0.600000 C: 0.616000 D: 0.904000 E: 1.000000 A B C D E A 0 8 0 -20 12 B -8 0 0 0 2 C 0 0 0 6 -18 D 20 0 -6 0 -4 E -12 -2 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.024000 B: 0.576000 C: 0.016000 D: 0.288000 E: 0.096000 Sum of squares = 0.424767999742 Cumulative probabilities = A: 0.024000 B: 0.600000 C: 0.616000 D: 0.904000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 C=20 A=19 B=10 so B is eliminated. Round 2 votes counts: D=28 E=25 C=24 A=23 so A is eliminated. Round 3 votes counts: D=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:205 E:204 A:200 B:197 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -20 12 B -8 0 0 0 2 C 0 0 0 6 -18 D 20 0 -6 0 -4 E -12 -2 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.024000 B: 0.576000 C: 0.016000 D: 0.288000 E: 0.096000 Sum of squares = 0.424767999742 Cumulative probabilities = A: 0.024000 B: 0.600000 C: 0.616000 D: 0.904000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -20 12 B -8 0 0 0 2 C 0 0 0 6 -18 D 20 0 -6 0 -4 E -12 -2 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.024000 B: 0.576000 C: 0.016000 D: 0.288000 E: 0.096000 Sum of squares = 0.424767999742 Cumulative probabilities = A: 0.024000 B: 0.600000 C: 0.616000 D: 0.904000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -20 12 B -8 0 0 0 2 C 0 0 0 6 -18 D 20 0 -6 0 -4 E -12 -2 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.024000 B: 0.576000 C: 0.016000 D: 0.288000 E: 0.096000 Sum of squares = 0.424767999742 Cumulative probabilities = A: 0.024000 B: 0.600000 C: 0.616000 D: 0.904000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 121: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (16) D C A B E (13) B E D C A (10) A C D E B (10) D C A E B (8) D C B E A (4) A C E B D (4) E B A D C (3) C D A E B (3) B E D A C (3) B E A C D (3) E B D A C (2) D E B C A (2) D C E B A (2) D B E C A (2) B E A D C (2) A E B C D (2) A C D B E (2) A B E C D (2) E A B C D (1) D B C E A (1) C D A B E (1) C A D E B (1) C A D B E (1) B D E C A (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 2 -4 -4 B 2 0 0 -2 -10 C -2 0 0 -8 2 D 4 2 8 0 4 E 4 10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 -4 B 2 0 0 -2 -10 C -2 0 0 -8 2 D 4 2 8 0 4 E 4 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=22 A=21 B=19 C=6 so C is eliminated. Round 2 votes counts: D=36 A=23 E=22 B=19 so B is eliminated. Round 3 votes counts: E=40 D=37 A=23 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:204 A:196 C:196 B:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -4 -4 B 2 0 0 -2 -10 C -2 0 0 -8 2 D 4 2 8 0 4 E 4 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 -4 B 2 0 0 -2 -10 C -2 0 0 -8 2 D 4 2 8 0 4 E 4 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 -4 B 2 0 0 -2 -10 C -2 0 0 -8 2 D 4 2 8 0 4 E 4 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 122: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) D C E A B (7) B A E D C (7) C D E B A (6) D C A B E (5) B A E C D (5) A E B D C (5) C D E A B (4) C D B E A (4) A B E D C (4) E C D A B (3) E A B C D (3) C B D E A (3) B E A C D (3) B D A C E (3) D C B E A (2) C E D A B (2) B E C A D (2) B A D E C (2) A B D E C (2) E C B A D (1) E C A D B (1) E C A B D (1) E B A C D (1) E A D C B (1) E A C B D (1) D E C A B (1) D C A E B (1) D B C A E (1) C E D B A (1) C E B A D (1) C D B A E (1) C B E D A (1) B C D E A (1) B A C D E (1) A E D B C (1) A E B C D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -16 -10 0 B 10 0 -12 -2 16 C 16 12 0 -4 6 D 10 2 4 0 8 E 0 -16 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -10 0 B 10 0 -12 -2 16 C 16 12 0 -4 6 D 10 2 4 0 8 E 0 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=24 C=23 A=15 E=12 so E is eliminated. Round 2 votes counts: C=29 D=26 B=25 A=20 so A is eliminated. Round 3 votes counts: B=41 C=30 D=29 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:212 B:206 E:185 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -16 -10 0 B 10 0 -12 -2 16 C 16 12 0 -4 6 D 10 2 4 0 8 E 0 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -10 0 B 10 0 -12 -2 16 C 16 12 0 -4 6 D 10 2 4 0 8 E 0 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -10 0 B 10 0 -12 -2 16 C 16 12 0 -4 6 D 10 2 4 0 8 E 0 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 123: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (19) E D C A B (10) D E C A B (10) B A D C E (5) D B A E C (4) D B A C E (4) B A C D E (4) E C D A B (3) D A C B E (3) C A E B D (3) E D B A C (2) E C A B D (2) E B C A D (2) D E B A C (2) D C A B E (2) C D A E B (2) C A E D B (2) C A B E D (2) E D B C A (1) E C B A D (1) E B D C A (1) D E C B A (1) D E B C A (1) D E A B C (1) D B E A C (1) C E D A B (1) C B A E D (1) C A B D E (1) B E A C D (1) B D A C E (1) B A E C D (1) B A D E C (1) A D C B E (1) A C D B E (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 8 0 20 B 6 0 6 -4 10 C -8 -6 0 -2 10 D 0 4 2 0 -8 E -20 -10 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363740 B: 0.000000 C: 0.000000 D: 0.636260 E: 0.000000 Sum of squares = 0.53713345079 Cumulative probabilities = A: 0.363740 B: 0.363740 C: 0.363740 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 0 20 B 6 0 6 -4 10 C -8 -6 0 -2 10 D 0 4 2 0 -8 E -20 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.000000 D: 0.600001 E: 0.000000 Sum of squares = 0.520000250798 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 0.399999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=29 E=22 C=12 A=5 so A is eliminated. Round 2 votes counts: B=34 D=30 E=22 C=14 so C is eliminated. Round 3 votes counts: B=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:211 B:209 D:199 C:197 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 0 20 B 6 0 6 -4 10 C -8 -6 0 -2 10 D 0 4 2 0 -8 E -20 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.000000 D: 0.600001 E: 0.000000 Sum of squares = 0.520000250798 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 0.399999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 0 20 B 6 0 6 -4 10 C -8 -6 0 -2 10 D 0 4 2 0 -8 E -20 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.000000 D: 0.600001 E: 0.000000 Sum of squares = 0.520000250798 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 0.399999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 0 20 B 6 0 6 -4 10 C -8 -6 0 -2 10 D 0 4 2 0 -8 E -20 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.000000 D: 0.600001 E: 0.000000 Sum of squares = 0.520000250798 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 0.399999 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 124: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (7) D C B A E (6) D C A E B (5) E A C D B (4) E A B C D (4) C B D E A (4) B D C E A (4) B C E D A (4) B C D E A (4) A D E C B (4) D A E C B (3) C D B A E (3) B E A D C (3) B E A C D (3) A E D C B (3) A E C D B (3) E C A B D (2) D C A B E (2) D B C A E (2) D B A C E (2) D A E B C (2) C D B E A (2) C D A E B (2) B C E A D (2) A E D B C (2) A E B D C (2) E B C A D (1) E B A C D (1) E A C B D (1) D A C E B (1) D A C B E (1) C E D A B (1) C E B A D (1) C D E B A (1) C D A B E (1) C B E A D (1) C A D E B (1) B E C D A (1) B D E C A (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -16 -4 -4 B 6 0 -8 -8 6 C 16 8 0 12 4 D 4 8 -12 0 6 E 4 -6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -4 -4 B 6 0 -8 -8 6 C 16 8 0 12 4 D 4 8 -12 0 6 E 4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=24 C=17 A=17 E=13 so E is eliminated. Round 2 votes counts: B=31 A=26 D=24 C=19 so C is eliminated. Round 3 votes counts: B=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:220 D:203 B:198 E:194 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -4 -4 B 6 0 -8 -8 6 C 16 8 0 12 4 D 4 8 -12 0 6 E 4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -4 -4 B 6 0 -8 -8 6 C 16 8 0 12 4 D 4 8 -12 0 6 E 4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -4 -4 B 6 0 -8 -8 6 C 16 8 0 12 4 D 4 8 -12 0 6 E 4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 125: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) E C A D B (5) C E D A B (5) B D A C E (5) B A D E C (5) E A D C B (4) D A B E C (4) C E B D A (4) C E B A D (4) A D B E C (4) E A C D B (3) C E A D B (3) C D B A E (3) C B E D A (3) C B D E A (3) C B D A E (3) B D A E C (3) B C D E A (3) A D E C B (3) E A D B C (2) E A C B D (2) D A E C B (2) D A B C E (2) B C D A E (2) A E D B C (2) E C B A D (1) E C A B D (1) E A B D C (1) E A B C D (1) D C A B E (1) D B A E C (1) C D E A B (1) C D A B E (1) B E C A D (1) B C E D A (1) B C E A D (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 14 8 6 2 B -14 0 -6 -10 -6 C -8 6 0 2 -10 D -6 10 -2 0 10 E -2 6 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 6 2 B -14 0 -6 -10 -6 C -8 6 0 2 -10 D -6 10 -2 0 10 E -2 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=21 E=20 A=19 D=10 so D is eliminated. Round 2 votes counts: C=31 A=27 B=22 E=20 so E is eliminated. Round 3 votes counts: A=40 C=38 B=22 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:206 E:202 C:195 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 6 2 B -14 0 -6 -10 -6 C -8 6 0 2 -10 D -6 10 -2 0 10 E -2 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 6 2 B -14 0 -6 -10 -6 C -8 6 0 2 -10 D -6 10 -2 0 10 E -2 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 6 2 B -14 0 -6 -10 -6 C -8 6 0 2 -10 D -6 10 -2 0 10 E -2 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 126: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) D E B C A (7) D A B E C (7) D C E B A (5) D B E C A (5) D B A E C (5) D C A E B (4) B E A C D (4) A B E D C (4) C E D B A (3) C E B A D (3) B E D C A (3) B E A D C (3) A D C B E (3) A D B E C (3) A C E B D (3) A B D E C (3) E B C A D (2) D E C B A (2) D B E A C (2) C E B D A (2) C E A B D (2) C D E A B (2) C D A E B (2) A C D B E (2) E C B D A (1) E B D C A (1) E B C D A (1) C D E B A (1) C A E B D (1) B E D A C (1) B A E D C (1) B A E C D (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 6 -8 0 B 8 0 28 -6 16 C -6 -28 0 -20 -32 D 8 6 20 0 8 E 0 -16 32 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 -8 0 B 8 0 28 -6 16 C -6 -28 0 -20 -32 D 8 6 20 0 8 E 0 -16 32 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=28 C=16 B=14 E=5 so E is eliminated. Round 2 votes counts: D=37 A=28 B=18 C=17 so C is eliminated. Round 3 votes counts: D=45 A=31 B=24 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:223 D:221 E:204 A:195 C:157 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 6 -8 0 B 8 0 28 -6 16 C -6 -28 0 -20 -32 D 8 6 20 0 8 E 0 -16 32 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -8 0 B 8 0 28 -6 16 C -6 -28 0 -20 -32 D 8 6 20 0 8 E 0 -16 32 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -8 0 B 8 0 28 -6 16 C -6 -28 0 -20 -32 D 8 6 20 0 8 E 0 -16 32 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 127: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) E B A C D (6) D C A B E (6) D C B E A (5) D C B A E (5) C D A E B (5) A E C B D (5) A E B C D (5) D C A E B (4) C D E A B (4) B E A C D (4) B D E C A (4) E A C B D (3) C E D A B (3) C A D E B (3) B D A E C (3) A C E D B (3) D C E A B (2) D B C A E (2) B E D C A (2) B E D A C (2) B E A D C (2) E C B D A (1) E A C D B (1) E A B C D (1) D C E B A (1) D B E C A (1) B E C A D (1) B D E A C (1) B D C E A (1) B D A C E (1) B A E D C (1) B A D E C (1) A D C B E (1) A C D E B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -14 -20 -4 B 2 0 -6 -8 4 C 14 6 0 -6 10 D 20 8 6 0 18 E 4 -4 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 -20 -4 B 2 0 -6 -8 4 C 14 6 0 -6 10 D 20 8 6 0 18 E 4 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=23 A=17 C=15 E=12 so E is eliminated. Round 2 votes counts: D=33 B=29 A=22 C=16 so C is eliminated. Round 3 votes counts: D=45 B=30 A=25 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:212 B:196 E:186 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -14 -20 -4 B 2 0 -6 -8 4 C 14 6 0 -6 10 D 20 8 6 0 18 E 4 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -20 -4 B 2 0 -6 -8 4 C 14 6 0 -6 10 D 20 8 6 0 18 E 4 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -20 -4 B 2 0 -6 -8 4 C 14 6 0 -6 10 D 20 8 6 0 18 E 4 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 128: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B E D A C (9) C A D E B (6) B E C A D (6) B A C D E (5) A D C E B (5) E B C D A (4) D A C E B (4) B A D C E (4) E B D C A (3) B E D C A (3) B E C D A (3) B D A E C (3) A D C B E (3) E D B A C (2) C E A D B (2) C D E A B (2) B E A D C (2) B A D E C (2) A D B C E (2) E D C B A (1) E D C A B (1) E C B D A (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) D B A C E (1) D A B C E (1) C E B A D (1) C B E A D (1) C B A E D (1) C A E D B (1) C A D B E (1) B E A C D (1) B C A E D (1) B C A D E (1) B A E D C (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -2 -2 -8 B 12 0 10 6 6 C 2 -10 0 -4 -6 D 2 -6 4 0 -6 E 8 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -2 -8 B 12 0 10 6 6 C 2 -10 0 -4 -6 D 2 -6 4 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=21 C=15 A=13 D=10 so D is eliminated. Round 2 votes counts: B=42 E=23 A=18 C=17 so C is eliminated. Round 3 votes counts: B=44 E=29 A=27 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:207 D:197 C:191 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 -2 -8 B 12 0 10 6 6 C 2 -10 0 -4 -6 D 2 -6 4 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -2 -8 B 12 0 10 6 6 C 2 -10 0 -4 -6 D 2 -6 4 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -2 -8 B 12 0 10 6 6 C 2 -10 0 -4 -6 D 2 -6 4 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 129: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) A B D C E (7) E C D B A (6) D E C A B (6) D E A B C (6) D A B C E (6) C E D B A (6) C E B A D (6) B A C D E (6) A B D E C (5) E D C A B (4) C B A E D (4) C B A D E (4) B A D C E (4) E B A C D (3) D A B E C (3) B A E C D (3) E C D A B (2) A D B C E (2) E D A B C (1) E C B D A (1) E A B D C (1) D C E A B (1) D A E B C (1) C D B A E (1) B A E D C (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 12 -2 B 8 0 2 10 -4 C -2 -2 0 2 -4 D -12 -10 -2 0 4 E 2 4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.407407407277 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 A B C D E A 0 -8 2 12 -2 B 8 0 2 10 -4 C -2 -2 0 2 -4 D -12 -10 -2 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.40740740731 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 C=21 B=15 A=15 so B is eliminated. Round 2 votes counts: A=30 E=26 D=23 C=21 so C is eliminated. Round 3 votes counts: E=38 A=38 D=24 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:208 E:203 A:202 C:197 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 12 -2 B 8 0 2 10 -4 C -2 -2 0 2 -4 D -12 -10 -2 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.40740740731 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 12 -2 B 8 0 2 10 -4 C -2 -2 0 2 -4 D -12 -10 -2 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.40740740731 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 12 -2 B 8 0 2 10 -4 C -2 -2 0 2 -4 D -12 -10 -2 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.40740740731 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 130: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) D E B A C (7) D E A C B (7) A C D B E (7) A C B D E (7) B E D C A (5) E D B C A (4) C A D E B (4) C A B E D (4) B E D A C (4) B E C A D (4) A C D E B (4) E B D C A (3) D E C A B (3) B D E A C (3) D E B C A (2) D A C E B (2) C A E D B (2) C A E B D (2) A B C E D (2) E D C B A (1) E C A B D (1) E B C A D (1) D C E A B (1) D C A E B (1) D B E A C (1) D B A C E (1) D A E C B (1) C A B D E (1) B E A C D (1) B D A E C (1) B C E A D (1) B C A E D (1) A D C B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 18 6 2 B -4 0 0 2 8 C -18 0 0 4 2 D -6 -2 -4 0 10 E -2 -8 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 6 2 B -4 0 0 2 8 C -18 0 0 4 2 D -6 -2 -4 0 10 E -2 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=26 A=23 C=13 E=10 so E is eliminated. Round 2 votes counts: B=32 D=31 A=23 C=14 so C is eliminated. Round 3 votes counts: A=37 B=32 D=31 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:203 D:199 C:194 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 18 6 2 B -4 0 0 2 8 C -18 0 0 4 2 D -6 -2 -4 0 10 E -2 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 6 2 B -4 0 0 2 8 C -18 0 0 4 2 D -6 -2 -4 0 10 E -2 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 6 2 B -4 0 0 2 8 C -18 0 0 4 2 D -6 -2 -4 0 10 E -2 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 131: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) E C D A B (9) C D B A E (7) C D E A B (6) E B A D C (5) B D C A E (5) A B D C E (5) E A B D C (4) B A E D C (4) B A D E C (4) E C A D B (3) E B A C D (3) D C A B E (3) E C B D A (2) E A C D B (2) D A C B E (2) C E D B A (2) C D E B A (2) C B D A E (2) E C D B A (1) E A D C B (1) E A D B C (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C A E (1) D B A C E (1) C E D A B (1) C D A E B (1) C D A B E (1) B E C D A (1) B E A D C (1) B C D E A (1) B C D A E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 0 0 6 B 14 0 6 8 10 C 0 -6 0 -6 12 D 0 -8 6 0 16 E -6 -10 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 0 6 B 14 0 6 8 10 C 0 -6 0 -6 12 D 0 -8 6 0 16 E -6 -10 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=30 C=22 D=8 A=7 so A is eliminated. Round 2 votes counts: B=36 E=33 C=22 D=9 so D is eliminated. Round 3 votes counts: B=39 E=33 C=28 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:207 C:200 A:196 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 0 6 B 14 0 6 8 10 C 0 -6 0 -6 12 D 0 -8 6 0 16 E -6 -10 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 0 6 B 14 0 6 8 10 C 0 -6 0 -6 12 D 0 -8 6 0 16 E -6 -10 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 0 6 B 14 0 6 8 10 C 0 -6 0 -6 12 D 0 -8 6 0 16 E -6 -10 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 132: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) C B E A D (9) D A B C E (8) E C B A D (6) D A E B C (6) D E A B C (5) D A B E C (4) C E B A D (4) B A C D E (4) A B C D E (4) E D A B C (3) E C D B A (3) E A D B C (3) D E C B A (3) D C B A E (3) A B D C E (3) E D C B A (2) E A B C D (2) C B A D E (2) B A C E D (2) E D C A B (1) D E C A B (1) D E A C B (1) D C E A B (1) D A C B E (1) C E B D A (1) B C E A D (1) B C D A E (1) B C A E D (1) A D B C E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -2 12 8 B 8 0 0 8 16 C 2 0 0 6 18 D -12 -8 -6 0 -4 E -8 -16 -18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.865908 C: 0.134092 D: 0.000000 E: 0.000000 Sum of squares = 0.767777115828 Cumulative probabilities = A: 0.000000 B: 0.865908 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 12 8 B 8 0 0 8 16 C 2 0 0 6 18 D -12 -8 -6 0 -4 E -8 -16 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=28 E=20 A=10 B=9 so B is eliminated. Round 2 votes counts: D=33 C=31 E=20 A=16 so A is eliminated. Round 3 votes counts: C=42 D=37 E=21 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:216 C:213 A:205 D:185 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 12 8 B 8 0 0 8 16 C 2 0 0 6 18 D -12 -8 -6 0 -4 E -8 -16 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 12 8 B 8 0 0 8 16 C 2 0 0 6 18 D -12 -8 -6 0 -4 E -8 -16 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 12 8 B 8 0 0 8 16 C 2 0 0 6 18 D -12 -8 -6 0 -4 E -8 -16 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 133: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (6) B E D A C (6) A E C D B (6) B C A E D (5) E A D B C (4) D C A E B (4) B C D A E (4) A E C B D (4) A C E D B (4) E A D C B (3) E A B D C (3) D B E C A (3) C D A B E (3) C A E B D (3) B D C E A (3) A E D C B (3) E D A B C (2) D E B A C (2) D C B E A (2) C D A E B (2) C A E D B (2) B E A D C (2) B C D E A (2) E B D A C (1) E A C D B (1) E A C B D (1) E A B C D (1) D E A C B (1) D E A B C (1) D C B A E (1) D B E A C (1) D B C E A (1) D B C A E (1) D A E C B (1) D A C E B (1) C B D A E (1) C B A D E (1) C A D E B (1) C A B E D (1) B E A C D (1) B C E D A (1) B A C E D (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 6 0 16 B -10 0 -10 -14 -6 C -6 10 0 8 6 D 0 14 -8 0 -14 E -16 6 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.745922 B: 0.000000 C: 0.000000 D: 0.254078 E: 0.000000 Sum of squares = 0.620955389756 Cumulative probabilities = A: 0.745922 B: 0.745922 C: 0.745922 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 0 16 B -10 0 -10 -14 -6 C -6 10 0 8 6 D 0 14 -8 0 -14 E -16 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204108088 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=20 A=20 D=19 E=16 so E is eliminated. Round 2 votes counts: A=33 B=26 D=21 C=20 so C is eliminated. Round 3 votes counts: A=40 D=32 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:209 E:199 D:196 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 0 16 B -10 0 -10 -14 -6 C -6 10 0 8 6 D 0 14 -8 0 -14 E -16 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204108088 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 0 16 B -10 0 -10 -14 -6 C -6 10 0 8 6 D 0 14 -8 0 -14 E -16 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204108088 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 0 16 B -10 0 -10 -14 -6 C -6 10 0 8 6 D 0 14 -8 0 -14 E -16 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204108088 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 134: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (9) A B D E C (8) A D B C E (7) E C D B A (6) C E D B A (6) A D B E C (6) C E B D A (5) E C B D A (4) D A C E B (4) D A B E C (4) D A B C E (4) D C E B A (3) C D E B A (3) B E C A D (3) E C B A D (2) C E B A D (2) B C E A D (2) B A C E D (2) E B C A D (1) D E C B A (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C B E (1) B E A C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 -8 2 B 0 0 2 -16 0 C -2 -2 0 -4 6 D 8 16 4 0 12 E -2 0 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -8 2 B 0 0 2 -16 0 C -2 -2 0 -4 6 D 8 16 4 0 12 E -2 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=24 B=17 C=16 E=13 so E is eliminated. Round 2 votes counts: D=30 C=28 A=24 B=18 so B is eliminated. Round 3 votes counts: A=36 C=34 D=30 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:220 C:199 A:198 B:193 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -8 2 B 0 0 2 -16 0 C -2 -2 0 -4 6 D 8 16 4 0 12 E -2 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -8 2 B 0 0 2 -16 0 C -2 -2 0 -4 6 D 8 16 4 0 12 E -2 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -8 2 B 0 0 2 -16 0 C -2 -2 0 -4 6 D 8 16 4 0 12 E -2 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 135: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) E B D C A (5) D E A B C (5) C D B E A (5) A E D B C (5) E B A C D (4) C B D E A (4) A C B E D (4) D E B C A (3) D E B A C (3) D C E B A (3) C D B A E (3) B C E D A (3) B C E A D (3) A C D B E (3) E D B A C (2) E A B D C (2) D E A C B (2) D C A E B (2) C D A B E (2) B E C A D (2) A E B C D (2) A D E C B (2) A D E B C (2) E B D A C (1) E B C D A (1) E A D B C (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E A D (1) C B D A E (1) C B A E D (1) C B A D E (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B E A C D (1) B C D E A (1) B C A E D (1) B A E C D (1) A E B D C (1) A D C E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 6 -4 -20 B 16 0 18 4 -12 C -6 -18 0 -2 -10 D 4 -4 2 0 -2 E 20 12 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 6 -4 -20 B 16 0 18 4 -12 C -6 -18 0 -2 -10 D 4 -4 2 0 -2 E 20 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=22 D=22 A=22 C=20 B=14 so B is eliminated. Round 2 votes counts: C=28 E=27 A=23 D=22 so D is eliminated. Round 3 votes counts: E=40 C=34 A=26 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:213 D:200 A:183 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 6 -4 -20 B 16 0 18 4 -12 C -6 -18 0 -2 -10 D 4 -4 2 0 -2 E 20 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -4 -20 B 16 0 18 4 -12 C -6 -18 0 -2 -10 D 4 -4 2 0 -2 E 20 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -4 -20 B 16 0 18 4 -12 C -6 -18 0 -2 -10 D 4 -4 2 0 -2 E 20 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 136: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) A C B D E (9) D E B C A (7) E D A B C (6) E A C B D (5) D B C E A (5) E A D C B (4) C B A D E (4) E D B C A (3) E A D B C (3) C A B D E (3) A E C B D (3) E D B A C (2) D E B A C (2) D E A C B (2) D B E C A (2) D B C A E (2) C B A E D (2) B C E A D (2) B C D A E (2) B C A D E (2) A E C D B (2) A D C B E (2) A C E B D (2) E D A C B (1) E B C D A (1) E B A C D (1) E A B C D (1) D E A B C (1) D C B A E (1) D C A B E (1) D A C B E (1) D A B E C (1) B E D C A (1) B D E C A (1) B C D E A (1) B C A E D (1) A E D C B (1) Total count = 100 A B C D E A 0 16 18 14 -2 B -16 0 -6 2 6 C -18 6 0 2 0 D -14 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.513888888871 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 16 18 14 -2 B -16 0 -6 2 6 C -18 6 0 2 0 D -14 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.513888888741 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 D=25 B=10 C=9 so C is eliminated. Round 2 votes counts: A=32 E=27 D=25 B=16 so B is eliminated. Round 3 votes counts: A=41 E=30 D=29 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:223 E:199 C:195 B:193 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 14 -2 B -16 0 -6 2 6 C -18 6 0 2 0 D -14 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.513888888741 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 14 -2 B -16 0 -6 2 6 C -18 6 0 2 0 D -14 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.513888888741 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 14 -2 B -16 0 -6 2 6 C -18 6 0 2 0 D -14 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.513888888741 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 137: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (15) B E A D C (8) E B C D A (7) C D A B E (7) D A C B E (5) B C D A E (5) B A D C E (5) E C D A B (4) E D C A B (3) E B C A D (3) C B D A E (3) B E C D A (3) A D C B E (3) E D A C B (2) E C D B A (2) E C B D A (2) E A D C B (2) E A D B C (2) D A C E B (2) B C E D A (2) B C A D E (2) B A C D E (2) A B D C E (2) E B A C D (1) C E B D A (1) C D B A E (1) B E C A D (1) B C D E A (1) B A E D C (1) B A D E C (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -32 6 0 -18 B 32 0 22 30 4 C -6 -22 0 -6 -16 D 0 -30 6 0 -20 E 18 -4 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 6 0 -18 B 32 0 22 30 4 C -6 -22 0 -6 -16 D 0 -30 6 0 -20 E 18 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959143 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 B=31 C=12 D=7 A=7 so D is eliminated. Round 2 votes counts: E=43 B=31 A=14 C=12 so C is eliminated. Round 3 votes counts: E=44 B=35 A=21 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:244 E:225 A:178 D:178 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -32 6 0 -18 B 32 0 22 30 4 C -6 -22 0 -6 -16 D 0 -30 6 0 -20 E 18 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959143 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 6 0 -18 B 32 0 22 30 4 C -6 -22 0 -6 -16 D 0 -30 6 0 -20 E 18 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959143 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 6 0 -18 B 32 0 22 30 4 C -6 -22 0 -6 -16 D 0 -30 6 0 -20 E 18 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959143 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 138: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) E A C B D (5) B D A C E (5) E C A D B (4) E B D C A (4) E B A C D (4) B D E C A (4) B A D C E (4) A C E D B (4) E B D A C (3) E A C D B (3) D B C A E (3) C D E A B (3) C D A B E (3) B E A D C (3) A B E D C (3) E C D B A (2) E A B C D (2) D C B A E (2) D C A B E (2) D A C B E (2) C E D A B (2) C D B A E (2) C D A E B (2) B E D C A (2) A E C B D (2) A D C B E (2) A C D B E (2) E C D A B (1) E C A B D (1) E B C D A (1) E B C A D (1) D E B C A (1) D C B E A (1) D B C E A (1) C E A D B (1) C D E B A (1) C A D B E (1) B D A E C (1) B A E D C (1) B A D E C (1) Total count = 100 A B C D E A 0 -10 10 4 -16 B 10 0 4 10 -10 C -10 -4 0 -6 -14 D -4 -10 6 0 -14 E 16 10 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 10 4 -16 B 10 0 4 10 -10 C -10 -4 0 -6 -14 D -4 -10 6 0 -14 E 16 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=21 C=15 A=13 D=12 so D is eliminated. Round 2 votes counts: E=40 B=25 C=20 A=15 so A is eliminated. Round 3 votes counts: E=42 C=30 B=28 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:207 A:194 D:189 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 10 4 -16 B 10 0 4 10 -10 C -10 -4 0 -6 -14 D -4 -10 6 0 -14 E 16 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 4 -16 B 10 0 4 10 -10 C -10 -4 0 -6 -14 D -4 -10 6 0 -14 E 16 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 4 -16 B 10 0 4 10 -10 C -10 -4 0 -6 -14 D -4 -10 6 0 -14 E 16 10 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 139: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) D A E C B (6) D E B A C (5) C B A D E (5) D E A B C (4) C B A E D (4) C A B E D (4) B E D C A (4) A C E D B (4) A C D E B (4) E D A B C (3) C B E A D (3) B E C D A (3) B D E C A (3) B D C A E (3) A D C E B (3) E D B A C (2) D A E B C (2) C E A B D (2) C A E B D (2) C A D B E (2) C A B D E (2) B E C A D (2) B C E A D (2) B C D E A (2) B C D A E (2) A C D B E (2) E B D C A (1) E B C A D (1) E A D C B (1) E A C B D (1) D C B A E (1) D B E A C (1) D A C E B (1) D A C B E (1) C E B A D (1) C B D A E (1) C A D E B (1) B D C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 -18 -6 2 B 8 0 -4 12 10 C 18 4 0 14 20 D 6 -12 -14 0 4 E -2 -10 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 -6 2 B 8 0 -4 12 10 C 18 4 0 14 20 D 6 -12 -14 0 4 E -2 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=27 D=21 A=14 E=9 so E is eliminated. Round 2 votes counts: B=31 C=27 D=26 A=16 so A is eliminated. Round 3 votes counts: C=38 D=31 B=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:213 D:192 A:185 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -18 -6 2 B 8 0 -4 12 10 C 18 4 0 14 20 D 6 -12 -14 0 4 E -2 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -6 2 B 8 0 -4 12 10 C 18 4 0 14 20 D 6 -12 -14 0 4 E -2 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -6 2 B 8 0 -4 12 10 C 18 4 0 14 20 D 6 -12 -14 0 4 E -2 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 140: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) A D B E C (9) C E B D A (8) E A D B C (6) C E A B D (6) B D A E C (6) E C B A D (5) D A B E C (4) D B A C E (3) D A B C E (3) C E A D B (3) C B E D A (3) A D E B C (3) E C A B D (2) E B A D C (2) D B A E C (2) C E D B A (2) C D A B E (2) C B D A E (2) B D C A E (2) A D C E B (2) E A D C B (1) E A C D B (1) E A B D C (1) D B C A E (1) C D B E A (1) C B D E A (1) B E D C A (1) B D C E A (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D B C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -6 8 -12 B 8 0 0 10 -10 C 6 0 0 -2 8 D -8 -10 2 0 -8 E 12 10 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.346744 C: 0.653256 D: 0.000000 E: 0.000000 Sum of squares = 0.546974852101 Cumulative probabilities = A: 0.000000 B: 0.346744 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 8 -12 B 8 0 0 10 -10 C 6 0 0 -2 8 D -8 -10 2 0 -8 E 12 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444443 C: 0.555557 D: 0.000000 E: 0.000000 Sum of squares = 0.506173268842 Cumulative probabilities = A: 0.000000 B: 0.444443 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=18 A=17 B=14 D=13 so D is eliminated. Round 2 votes counts: C=38 A=24 B=20 E=18 so E is eliminated. Round 3 votes counts: C=45 A=33 B=22 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:211 C:206 B:204 A:191 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 8 -12 B 8 0 0 10 -10 C 6 0 0 -2 8 D -8 -10 2 0 -8 E 12 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444443 C: 0.555557 D: 0.000000 E: 0.000000 Sum of squares = 0.506173268842 Cumulative probabilities = A: 0.000000 B: 0.444443 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 8 -12 B 8 0 0 10 -10 C 6 0 0 -2 8 D -8 -10 2 0 -8 E 12 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444443 C: 0.555557 D: 0.000000 E: 0.000000 Sum of squares = 0.506173268842 Cumulative probabilities = A: 0.000000 B: 0.444443 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 8 -12 B 8 0 0 10 -10 C 6 0 0 -2 8 D -8 -10 2 0 -8 E 12 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444443 C: 0.555557 D: 0.000000 E: 0.000000 Sum of squares = 0.506173268842 Cumulative probabilities = A: 0.000000 B: 0.444443 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 141: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) A D C B E (6) A D B E C (5) E B C D A (4) D E C B A (4) D A C E B (4) C B E A D (4) B E C D A (4) B C E A D (4) B A E D C (4) E D C B A (3) D A E C B (3) D A E B C (3) C E D B A (3) C D E A B (3) B E C A D (3) A C D E B (3) A B C E D (3) E D B C A (2) E C D B A (2) E B D C A (2) E B D A C (2) D C E A B (2) A B D E C (2) E C B D A (1) D E B C A (1) D E A C B (1) D C E B A (1) C D A E B (1) C B A E D (1) C A B E D (1) B E D A C (1) B E A D C (1) B A E C D (1) A D C E B (1) A D B C E (1) A C D B E (1) A C B E D (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 -8 -10 B 10 0 -6 0 -6 C 6 6 0 -2 -2 D 8 0 2 0 -12 E 10 6 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 -8 -10 B 10 0 -6 0 -6 C 6 6 0 -2 -2 D 8 0 2 0 -12 E 10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=20 D=19 B=18 E=16 so E is eliminated. Round 2 votes counts: A=27 B=26 D=24 C=23 so C is eliminated. Round 3 votes counts: B=39 D=33 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:215 C:204 B:199 D:199 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 -8 -10 B 10 0 -6 0 -6 C 6 6 0 -2 -2 D 8 0 2 0 -12 E 10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -8 -10 B 10 0 -6 0 -6 C 6 6 0 -2 -2 D 8 0 2 0 -12 E 10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -8 -10 B 10 0 -6 0 -6 C 6 6 0 -2 -2 D 8 0 2 0 -12 E 10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 142: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (15) A B E C D (8) D C E A B (6) B A E C D (5) B A C D E (5) E D C A B (4) E C D B A (4) D B C A E (4) C E D B A (4) C D E B A (4) A E D C B (4) E A D C B (3) D E C A B (3) D C B E A (3) A D E C B (3) A B E D C (3) B E C A D (2) B A C E D (2) A E B C D (2) A B D E C (2) A B D C E (2) E C D A B (1) E C A B D (1) E A C D B (1) E A B C D (1) D C B A E (1) C D B E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B A D C E (1) A E B D C (1) A D C B E (1) Total count = 100 A B C D E A 0 -8 -12 -4 -10 B 8 0 -18 -24 -14 C 12 18 0 -14 4 D 4 24 14 0 4 E 10 14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -4 -10 B 8 0 -18 -24 -14 C 12 18 0 -14 4 D 4 24 14 0 4 E 10 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=26 B=18 E=15 C=9 so C is eliminated. Round 2 votes counts: D=37 A=26 E=19 B=18 so B is eliminated. Round 3 votes counts: A=39 D=38 E=23 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:210 E:208 A:183 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 -4 -10 B 8 0 -18 -24 -14 C 12 18 0 -14 4 D 4 24 14 0 4 E 10 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -4 -10 B 8 0 -18 -24 -14 C 12 18 0 -14 4 D 4 24 14 0 4 E 10 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -4 -10 B 8 0 -18 -24 -14 C 12 18 0 -14 4 D 4 24 14 0 4 E 10 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 143: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (7) A D E C B (7) A C B D E (7) D E A B C (6) C B A E D (6) E D B C A (5) E B C D A (5) C B A D E (5) B C E D A (4) E D A C B (3) D A B C E (3) B C A D E (3) A C D B E (3) E D A B C (2) E C B A D (2) E A C D B (2) D A B E C (2) C A B E D (2) C A B D E (2) B C D E A (2) A E D C B (2) A D C B E (2) E D B A C (1) E C B D A (1) E C A D B (1) E B D C A (1) D E B A C (1) D E A C B (1) D B E C A (1) D B C A E (1) D A E B C (1) B E C D A (1) B D C E A (1) B D C A E (1) B C E A D (1) A D B C E (1) A C E B D (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -4 14 4 B -2 0 -10 8 14 C 4 10 0 14 10 D -14 -8 -14 0 6 E -4 -14 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 14 4 B -2 0 -10 8 14 C 4 10 0 14 10 D -14 -8 -14 0 6 E -4 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 C=22 D=16 B=13 so B is eliminated. Round 2 votes counts: C=32 A=26 E=24 D=18 so D is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:208 B:205 D:185 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 14 4 B -2 0 -10 8 14 C 4 10 0 14 10 D -14 -8 -14 0 6 E -4 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 14 4 B -2 0 -10 8 14 C 4 10 0 14 10 D -14 -8 -14 0 6 E -4 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 14 4 B -2 0 -10 8 14 C 4 10 0 14 10 D -14 -8 -14 0 6 E -4 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 144: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (7) E B A C D (6) D C A B E (6) B D E C A (6) A C E D B (6) E B C A D (5) E A B C D (5) D C A E B (4) D A C B E (4) D C B A E (3) C E A B D (3) C D A E B (3) C A D E B (3) B E C D A (3) B E C A D (3) A E C B D (3) A E B C D (3) E C B A D (2) D C B E A (2) D B C E A (2) D B A E C (2) C D B E A (2) B E D C A (2) B E D A C (2) B E A D C (2) E B C D A (1) D B A C E (1) D A B E C (1) C E B D A (1) C A E D B (1) B E A C D (1) B D A E C (1) B C D E A (1) B A E D C (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -6 6 2 B -2 0 -4 2 -10 C 6 4 0 6 0 D -6 -2 -6 0 0 E -2 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.662346 D: 0.000000 E: 0.337654 Sum of squares = 0.552712282394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.662346 D: 0.662346 E: 1.000000 A B C D E A 0 2 -6 6 2 B -2 0 -4 2 -10 C 6 4 0 6 0 D -6 -2 -6 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=22 A=21 E=19 C=13 so C is eliminated. Round 2 votes counts: D=30 A=25 E=23 B=22 so B is eliminated. Round 3 votes counts: D=38 E=36 A=26 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:208 E:204 A:202 B:193 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 6 2 B -2 0 -4 2 -10 C 6 4 0 6 0 D -6 -2 -6 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 6 2 B -2 0 -4 2 -10 C 6 4 0 6 0 D -6 -2 -6 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 6 2 B -2 0 -4 2 -10 C 6 4 0 6 0 D -6 -2 -6 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 145: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) C E A B D (7) B D E C A (6) E C A D B (4) E B C D A (4) B D E A C (4) B C E D A (4) D E A B C (3) D A E B C (3) C B A E D (3) C A E D B (3) B E C D A (3) B D C A E (3) B C A D E (3) E D A C B (2) E C B D A (2) D B E A C (2) D A E C B (2) C E B A D (2) C B E A D (2) C A E B D (2) B E D C A (2) B D C E A (2) A C D B E (2) A C B D E (2) E D C B A (1) E C A B D (1) E B D C A (1) E A C D B (1) D E B C A (1) D E B A C (1) D B A E C (1) D A B E C (1) C E A D B (1) C A B E D (1) C A B D E (1) B D A C E (1) B C D A E (1) B A D C E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -20 0 -12 B 0 0 -6 18 -8 C 20 6 0 20 4 D 0 -18 -20 0 -14 E 12 8 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 0 -12 B 0 0 -6 18 -8 C 20 6 0 20 4 D 0 -18 -20 0 -14 E 12 8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=22 A=18 E=16 D=14 so D is eliminated. Round 2 votes counts: B=33 A=24 C=22 E=21 so E is eliminated. Round 3 votes counts: B=40 C=30 A=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:225 E:215 B:202 A:184 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -20 0 -12 B 0 0 -6 18 -8 C 20 6 0 20 4 D 0 -18 -20 0 -14 E 12 8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 0 -12 B 0 0 -6 18 -8 C 20 6 0 20 4 D 0 -18 -20 0 -14 E 12 8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 0 -12 B 0 0 -6 18 -8 C 20 6 0 20 4 D 0 -18 -20 0 -14 E 12 8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 146: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (15) C B A E D (13) E D B C A (9) E D C B A (8) A C B D E (7) C B E D A (6) D E C B A (5) D E A B C (5) A D E B C (4) D E A C B (3) B C E A D (3) A C D B E (3) A B C E D (3) A D C B E (2) E B C D A (1) D E C A B (1) D A E C B (1) C E B D A (1) C D E B A (1) C A B D E (1) B E C D A (1) B C A E D (1) A D B E C (1) A D B C E (1) A C B E D (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 16 12 B -2 0 -6 12 22 C 2 6 0 14 18 D -16 -12 -14 0 4 E -12 -22 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 16 12 B -2 0 -6 12 22 C 2 6 0 14 18 D -16 -12 -14 0 4 E -12 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=22 E=18 D=15 B=5 so B is eliminated. Round 2 votes counts: A=40 C=26 E=19 D=15 so D is eliminated. Round 3 votes counts: A=41 E=33 C=26 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:220 A:214 B:213 D:181 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 16 12 B -2 0 -6 12 22 C 2 6 0 14 18 D -16 -12 -14 0 4 E -12 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 16 12 B -2 0 -6 12 22 C 2 6 0 14 18 D -16 -12 -14 0 4 E -12 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 16 12 B -2 0 -6 12 22 C 2 6 0 14 18 D -16 -12 -14 0 4 E -12 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 147: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) C A D B E (7) B C E A D (7) C B A E D (5) A D E C B (5) D A E C B (4) D A C E B (4) B E C A D (4) A C D E B (4) E B D C A (3) E B D A C (3) E A B D C (3) D E A B C (3) E D B A C (2) E B C A D (2) E B A C D (2) D C B E A (2) D B E C A (2) D A E B C (2) D A C B E (2) C A B E D (2) C A B D E (2) B E D C A (2) B E C D A (2) A E C B D (2) A D C E B (2) A C E B D (2) A C D B E (2) E D A B C (1) E B A D C (1) D E B A C (1) D C B A E (1) D C A B E (1) D B C E A (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -4 28 0 B -2 0 -16 2 2 C 4 16 0 8 8 D -28 -2 -8 0 -8 E 0 -2 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 28 0 B -2 0 -16 2 2 C 4 16 0 8 8 D -28 -2 -8 0 -8 E 0 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=23 A=20 E=17 B=15 so B is eliminated. Round 2 votes counts: C=32 E=25 D=23 A=20 so A is eliminated. Round 3 votes counts: C=41 D=31 E=28 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:213 E:199 B:193 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 28 0 B -2 0 -16 2 2 C 4 16 0 8 8 D -28 -2 -8 0 -8 E 0 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 28 0 B -2 0 -16 2 2 C 4 16 0 8 8 D -28 -2 -8 0 -8 E 0 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 28 0 B -2 0 -16 2 2 C 4 16 0 8 8 D -28 -2 -8 0 -8 E 0 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 148: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (12) A E B D C (10) E B A C D (9) E B C D A (7) C D B E A (7) B E A D C (7) C D E B A (5) C D A E B (5) C D A B E (5) A D C E B (5) A D C B E (5) B E C D A (4) B E A C D (4) A B E D C (4) D A C B E (3) E C B D A (1) D A C E B (1) C E B D A (1) C B E D A (1) B E C A D (1) B A E D C (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 4 2 -4 6 B -4 0 -2 2 8 C -2 2 0 2 0 D 4 -2 -2 0 -2 E -6 -8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999984 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -4 6 B -4 0 -2 2 8 C -2 2 0 2 0 D 4 -2 -2 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000102 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 E=17 B=17 D=16 so D is eliminated. Round 2 votes counts: C=36 A=30 E=17 B=17 so E is eliminated. Round 3 votes counts: C=37 B=33 A=30 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:204 B:202 C:201 D:199 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 2 -4 6 B -4 0 -2 2 8 C -2 2 0 2 0 D 4 -2 -2 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000102 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -4 6 B -4 0 -2 2 8 C -2 2 0 2 0 D 4 -2 -2 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000102 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -4 6 B -4 0 -2 2 8 C -2 2 0 2 0 D 4 -2 -2 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000102 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 149: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (15) C D B A E (14) E A B C D (11) E D C A B (5) D E C B A (5) A B E C D (5) D C B A E (4) B A E C D (4) C D A B E (3) B A E D C (3) A E B D C (3) E B A D C (2) E A D B C (2) C D E A B (2) B E A D C (2) B A C D E (2) A B C E D (2) E C D A B (1) E C A D B (1) E A C D B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B C A E (1) C D E B A (1) C D B E A (1) C B A D E (1) B C D A E (1) B A D C E (1) B A C E D (1) A E B C D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 14 18 -4 B -8 0 16 12 -4 C -14 -16 0 6 -26 D -18 -12 -6 0 -22 E 4 4 26 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 14 18 -4 B -8 0 16 12 -4 C -14 -16 0 6 -26 D -18 -12 -6 0 -22 E 4 4 26 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=22 B=14 D=13 A=13 so D is eliminated. Round 2 votes counts: E=44 C=28 B=15 A=13 so A is eliminated. Round 3 votes counts: E=48 C=29 B=23 so B is eliminated. Round 4 votes counts: E=63 C=37 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:218 B:208 C:175 D:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 14 18 -4 B -8 0 16 12 -4 C -14 -16 0 6 -26 D -18 -12 -6 0 -22 E 4 4 26 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 18 -4 B -8 0 16 12 -4 C -14 -16 0 6 -26 D -18 -12 -6 0 -22 E 4 4 26 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 18 -4 B -8 0 16 12 -4 C -14 -16 0 6 -26 D -18 -12 -6 0 -22 E 4 4 26 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 150: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (11) B A D E C (10) C E D A B (7) E C D A B (6) C E D B A (6) C B E D A (5) E D A C B (4) E C A D B (4) B A E D C (4) D E A C B (3) C B D E A (3) B C A D E (3) A D B E C (3) E A D C B (2) D A E C B (2) C E B D A (2) C D E A B (2) C D A E B (2) B C E A D (2) E B C A D (1) E A D B C (1) E A C D B (1) E A C B D (1) D C A E B (1) D A C B E (1) D A B C E (1) C B D A E (1) B D C A E (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) B A C D E (1) A E D B C (1) A E B D C (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 2 -2 -4 B 8 0 -8 2 2 C -2 8 0 0 6 D 2 -2 0 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.347477 D: 0.652523 E: 0.000000 Sum of squares = 0.546526744323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.347477 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -2 -4 B 8 0 -8 2 2 C -2 8 0 0 6 D 2 -2 0 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499857 D: 0.500143 E: 0.000000 Sum of squares = 0.500000040624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=28 E=20 D=8 A=8 so D is eliminated. Round 2 votes counts: B=36 C=29 E=23 A=12 so A is eliminated. Round 3 votes counts: B=41 C=30 E=29 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:202 D:201 E:197 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 2 -2 -4 B 8 0 -8 2 2 C -2 8 0 0 6 D 2 -2 0 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499857 D: 0.500143 E: 0.000000 Sum of squares = 0.500000040624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499857 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 -4 B 8 0 -8 2 2 C -2 8 0 0 6 D 2 -2 0 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499857 D: 0.500143 E: 0.000000 Sum of squares = 0.500000040624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 -4 B 8 0 -8 2 2 C -2 8 0 0 6 D 2 -2 0 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499857 D: 0.500143 E: 0.000000 Sum of squares = 0.500000040624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499857 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 151: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) D C A B E (8) E B A C D (6) E D B A C (5) E A C B D (5) D B C A E (5) E B A D C (4) D B E C A (4) C A D E B (4) C A D B E (4) C A B D E (4) A C E B D (4) D C B A E (3) B E A C D (3) B D E C A (3) E D A C B (2) E A C D B (2) E A B C D (2) D E C A B (2) D E B C A (2) D C A E B (2) C B A D E (2) B E D A C (2) A E C B D (2) A C E D B (2) D E C B A (1) D B C E A (1) C D B A E (1) C D A B E (1) B D C E A (1) B D C A E (1) B C A D E (1) B A E C D (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 0 -4 -6 B 10 0 0 2 -6 C 0 0 0 -8 -8 D 4 -2 8 0 0 E 6 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.346298 E: 0.653702 Sum of squares = 0.547248536528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.346298 E: 1.000000 A B C D E A 0 -10 0 -4 -6 B 10 0 0 2 -6 C 0 0 0 -8 -8 D 4 -2 8 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=28 C=16 B=13 A=9 so A is eliminated. Round 2 votes counts: E=36 D=28 C=23 B=13 so B is eliminated. Round 3 votes counts: E=42 D=33 C=25 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:210 D:205 B:203 C:192 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 0 -4 -6 B 10 0 0 2 -6 C 0 0 0 -8 -8 D 4 -2 8 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -4 -6 B 10 0 0 2 -6 C 0 0 0 -8 -8 D 4 -2 8 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -4 -6 B 10 0 0 2 -6 C 0 0 0 -8 -8 D 4 -2 8 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 152: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C E D B A (6) A B D C E (5) D B E A C (4) C D E A B (4) C A E B D (4) B A E D C (4) A B C D E (4) E D B C A (3) D E C B A (3) D C E B A (3) D B A E C (3) C E D A B (3) C E A D B (3) C E A B D (3) A B E C D (3) A B D E C (3) E C B A D (2) E B D A C (2) E B C A D (2) D C A B E (2) D A B C E (2) C D A E B (2) A B C E D (2) E C A B D (1) E B D C A (1) E B C D A (1) D E B C A (1) D C E A B (1) D B E C A (1) D B A C E (1) D A C B E (1) C D A B E (1) C A E D B (1) C A D B E (1) C A B E D (1) C A B D E (1) B E A D C (1) B D E A C (1) B A E C D (1) B A D E C (1) A E B C D (1) A D B E C (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -16 -6 -6 B -4 0 -2 -10 -8 C 16 2 0 10 4 D 6 10 -10 0 -6 E 6 8 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -6 -6 B -4 0 -2 -10 -8 C 16 2 0 10 4 D 6 10 -10 0 -6 E 6 8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 A=21 E=19 B=8 so B is eliminated. Round 2 votes counts: C=30 A=27 D=23 E=20 so E is eliminated. Round 3 votes counts: C=43 D=29 A=28 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:208 D:200 A:188 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 -6 -6 B -4 0 -2 -10 -8 C 16 2 0 10 4 D 6 10 -10 0 -6 E 6 8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -6 -6 B -4 0 -2 -10 -8 C 16 2 0 10 4 D 6 10 -10 0 -6 E 6 8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -6 -6 B -4 0 -2 -10 -8 C 16 2 0 10 4 D 6 10 -10 0 -6 E 6 8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 153: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (13) E D B C A (11) C E A D B (11) E C D B A (9) C A E B D (5) A B D C E (5) E D C B A (4) C A E D B (4) B D E A C (4) E B D C A (3) D E B A C (3) D B E A C (3) D B A E C (3) B D A E C (3) E C D A B (2) C E D A B (2) C A B E D (2) C A B D E (2) B E D A C (2) A C D B E (2) E C B D A (1) E C B A D (1) E B D A C (1) C E A B D (1) C A D E B (1) C A D B E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 -20 -2 -16 B -4 0 -22 -14 -18 C 20 22 0 14 0 D 2 14 -14 0 -18 E 16 18 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.418498 D: 0.000000 E: 0.581502 Sum of squares = 0.513285025514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.418498 D: 0.418498 E: 1.000000 A B C D E A 0 4 -20 -2 -16 B -4 0 -22 -14 -18 C 20 22 0 14 0 D 2 14 -14 0 -18 E 16 18 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=29 A=21 D=9 B=9 so D is eliminated. Round 2 votes counts: E=35 C=29 A=21 B=15 so B is eliminated. Round 3 votes counts: E=44 C=29 A=27 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:226 D:192 A:183 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -20 -2 -16 B -4 0 -22 -14 -18 C 20 22 0 14 0 D 2 14 -14 0 -18 E 16 18 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -2 -16 B -4 0 -22 -14 -18 C 20 22 0 14 0 D 2 14 -14 0 -18 E 16 18 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -2 -16 B -4 0 -22 -14 -18 C 20 22 0 14 0 D 2 14 -14 0 -18 E 16 18 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 154: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) E B C D A (9) E B C A D (6) D A B C E (6) B C E A D (6) B C A D E (5) A D C E B (5) A C B D E (5) E C B A D (4) E D A B C (3) D E A B C (3) D A E C B (3) D A E B C (3) B C D A E (3) A C D B E (3) E D B C A (2) E D A C B (2) C B A E D (2) C B A D E (2) B E C D A (2) B C E D A (2) A D E C B (2) E B D C A (1) E A D C B (1) D B A E C (1) D A C B E (1) C A E B D (1) C A B D E (1) B E C A D (1) B D C A E (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 4 2 14 14 B -4 0 10 4 8 C -2 -10 0 8 14 D -14 -4 -8 0 16 E -14 -8 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999314 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 14 14 B -4 0 10 4 8 C -2 -10 0 8 14 D -14 -4 -8 0 16 E -14 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996371 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 B=21 D=17 C=6 so C is eliminated. Round 2 votes counts: A=30 E=28 B=25 D=17 so D is eliminated. Round 3 votes counts: A=43 E=31 B=26 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:209 C:205 D:195 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 14 14 B -4 0 10 4 8 C -2 -10 0 8 14 D -14 -4 -8 0 16 E -14 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996371 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 14 14 B -4 0 10 4 8 C -2 -10 0 8 14 D -14 -4 -8 0 16 E -14 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996371 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 14 14 B -4 0 10 4 8 C -2 -10 0 8 14 D -14 -4 -8 0 16 E -14 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996371 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 155: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) C B E D A (9) E B D C A (7) D A C E B (7) A D C E B (7) A D E B C (6) A C D B E (6) C A D B E (5) E B D A C (4) E B A D C (4) D A E C B (3) C D A B E (3) B E C A D (3) E D B A C (2) D C E B A (2) D C A E B (2) D A E B C (2) C D E B A (2) C D B E A (2) C A B E D (2) B E A C D (2) E B C D A (1) D E C B A (1) D E A B C (1) C D B A E (1) C B E A D (1) C B A E D (1) B E A D C (1) B C E A D (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -8 -18 -6 B 10 0 -8 -4 -2 C 8 8 0 0 4 D 18 4 0 0 0 E 6 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.653399 D: 0.346601 E: 0.000000 Sum of squares = 0.547062197391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.653399 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -18 -6 B 10 0 -8 -4 -2 C 8 8 0 0 4 D 18 4 0 0 0 E 6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=20 E=18 D=18 B=18 so E is eliminated. Round 2 votes counts: B=34 C=26 D=20 A=20 so D is eliminated. Round 3 votes counts: B=36 A=33 C=31 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 C:210 E:202 B:198 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -18 -6 B 10 0 -8 -4 -2 C 8 8 0 0 4 D 18 4 0 0 0 E 6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -18 -6 B 10 0 -8 -4 -2 C 8 8 0 0 4 D 18 4 0 0 0 E 6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -18 -6 B 10 0 -8 -4 -2 C 8 8 0 0 4 D 18 4 0 0 0 E 6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 156: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) D C E B A (6) E D C B A (4) D E C A B (4) D E A C B (4) C D B E A (4) E D A B C (3) E C B A D (3) E B A C D (3) D C B A E (3) D A E B C (3) D A C B E (3) D A B C E (3) C E D B A (3) B C A E D (3) B A C E D (3) A B C E D (3) E C B D A (2) E A D B C (2) D E A B C (2) D C A B E (2) C D B A E (2) C B E A D (2) A E B C D (2) A D E B C (2) A D B E C (2) A B E D C (2) A B C D E (2) E D C A B (1) E C D B A (1) E B C A D (1) E A B C D (1) D E C B A (1) D C E A B (1) D C A E B (1) C E B A D (1) C D E B A (1) C B D E A (1) C B A E D (1) B A E C D (1) B A C D E (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -10 -10 -2 B 8 0 -16 -18 -4 C 10 16 0 -4 10 D 10 18 4 0 16 E 2 4 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -10 -2 B 8 0 -16 -18 -4 C 10 16 0 -4 10 D 10 18 4 0 16 E 2 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=22 E=21 A=16 B=8 so B is eliminated. Round 2 votes counts: D=33 C=25 E=21 A=21 so E is eliminated. Round 3 votes counts: D=41 C=32 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:216 E:190 A:185 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -10 -2 B 8 0 -16 -18 -4 C 10 16 0 -4 10 D 10 18 4 0 16 E 2 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -10 -2 B 8 0 -16 -18 -4 C 10 16 0 -4 10 D 10 18 4 0 16 E 2 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -10 -2 B 8 0 -16 -18 -4 C 10 16 0 -4 10 D 10 18 4 0 16 E 2 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 157: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) D C E B A (7) D C B A E (7) E C D A B (6) B A E D C (6) D C B E A (5) D B A C E (5) E C A D B (4) E C A B D (4) E A C B D (4) C D E A B (4) E C D B A (3) E A B C D (3) C E D B A (3) C E D A B (3) B A D E C (3) A B E C D (3) E B A D C (2) D E C B A (2) A E B C D (2) A B E D C (2) A B C D E (2) E D C B A (1) E B A C D (1) D B C A E (1) C E A B D (1) C D E B A (1) C D A B E (1) B E A D C (1) B D E A C (1) B D A E C (1) B D A C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -6 -4 -12 B 20 0 -12 -6 -2 C 6 12 0 -8 2 D 4 6 8 0 0 E 12 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.444932 E: 0.555068 Sum of squares = 0.506065029978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.444932 E: 1.000000 A B C D E A 0 -20 -6 -4 -12 B 20 0 -12 -6 -2 C 6 12 0 -8 2 D 4 6 8 0 0 E 12 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 B=22 C=13 A=10 so A is eliminated. Round 2 votes counts: E=30 B=30 D=27 C=13 so C is eliminated. Round 3 votes counts: E=37 D=33 B=30 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:209 C:206 E:206 B:200 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -6 -4 -12 B 20 0 -12 -6 -2 C 6 12 0 -8 2 D 4 6 8 0 0 E 12 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -4 -12 B 20 0 -12 -6 -2 C 6 12 0 -8 2 D 4 6 8 0 0 E 12 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -4 -12 B 20 0 -12 -6 -2 C 6 12 0 -8 2 D 4 6 8 0 0 E 12 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 158: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) B D C E A (8) C D B A E (7) C A D E B (6) B E A D C (6) A E D C B (6) D C B E A (5) E A D C B (4) E A D B C (4) B C D E A (4) A C E D B (4) D C A E B (3) D B C E A (3) C D A E B (3) A E C D B (3) D E B A C (2) D C B A E (2) C B D A E (2) B E D A C (2) B E A C D (2) E D B A C (1) E B A D C (1) D E A C B (1) D C E A B (1) D B E C A (1) C D B E A (1) B D E C A (1) B D E A C (1) B C E D A (1) B C E A D (1) B C D A E (1) B C A D E (1) B A E C D (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -2 0 -18 B 8 0 0 -16 0 C 2 0 0 -22 8 D 0 16 22 0 8 E 18 0 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.182708 B: 0.000000 C: 0.000000 D: 0.817292 E: 0.000000 Sum of squares = 0.701348677879 Cumulative probabilities = A: 0.182708 B: 0.182708 C: 0.182708 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 0 -18 B 8 0 0 -16 0 C 2 0 0 -22 8 D 0 16 22 0 8 E 18 0 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.000000 D: 0.692308 E: 0.000000 Sum of squares = 0.573964531307 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.307692 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=19 C=19 D=18 A=15 so A is eliminated. Round 2 votes counts: E=29 B=29 C=24 D=18 so D is eliminated. Round 3 votes counts: C=35 B=33 E=32 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:223 E:201 B:196 C:194 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 0 -18 B 8 0 0 -16 0 C 2 0 0 -22 8 D 0 16 22 0 8 E 18 0 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.000000 D: 0.692308 E: 0.000000 Sum of squares = 0.573964531307 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.307692 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 -18 B 8 0 0 -16 0 C 2 0 0 -22 8 D 0 16 22 0 8 E 18 0 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.000000 D: 0.692308 E: 0.000000 Sum of squares = 0.573964531307 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.307692 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 -18 B 8 0 0 -16 0 C 2 0 0 -22 8 D 0 16 22 0 8 E 18 0 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.000000 D: 0.692308 E: 0.000000 Sum of squares = 0.573964531307 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.307692 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 159: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) C D E B A (8) A B E D C (8) E B C D A (5) D C A E B (5) B E C D A (5) A B D C E (5) E B C A D (4) E B A C D (4) D C E A B (4) C D E A B (4) B A E D C (4) A E B D C (3) A D C E B (3) E C B D A (2) D C A B E (2) D A C E B (2) D A C B E (2) B E C A D (2) A E D C B (2) A D C B E (2) A D B E C (2) A B D E C (2) C E B D A (1) C D B E A (1) C B D E A (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 6 8 -10 B 6 0 22 24 6 C -6 -22 0 4 -10 D -8 -24 -4 0 -6 E 10 -6 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 8 -10 B 6 0 22 24 6 C -6 -22 0 4 -10 D -8 -24 -4 0 -6 E 10 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997733 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 E=15 D=15 C=15 so E is eliminated. Round 2 votes counts: B=40 A=28 C=17 D=15 so D is eliminated. Round 3 votes counts: B=40 A=32 C=28 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 E:210 A:199 C:183 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 8 -10 B 6 0 22 24 6 C -6 -22 0 4 -10 D -8 -24 -4 0 -6 E 10 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997733 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 8 -10 B 6 0 22 24 6 C -6 -22 0 4 -10 D -8 -24 -4 0 -6 E 10 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997733 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 8 -10 B 6 0 22 24 6 C -6 -22 0 4 -10 D -8 -24 -4 0 -6 E 10 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997733 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 160: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) C B A E D (5) C A B E D (5) B D A E C (5) B C D E A (5) A E C D B (5) B C A E D (4) B C A D E (4) A D E B C (4) D E A C B (3) D E A B C (3) D A E C B (3) C E D B A (3) C B E A D (3) A C E B D (3) E D A C B (2) E A D C B (2) E A C D B (2) D E B C A (2) D B A E C (2) C B E D A (2) C A E B D (2) B D C E A (2) A E D C B (2) E C D A B (1) E C A D B (1) D E B A C (1) D B E C A (1) D B E A C (1) C E B D A (1) C E B A D (1) C E A D B (1) C D B E A (1) C A E D B (1) B D C A E (1) B D A C E (1) B C E A D (1) B A C E D (1) B A C D E (1) A E C B D (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 6 4 22 B -4 0 -2 2 -4 C -6 2 0 12 -2 D -4 -2 -12 0 -4 E -22 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 22 B -4 0 -2 2 -4 C -6 2 0 12 -2 D -4 -2 -12 0 -4 E -22 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=25 B=25 D=24 A=18 E=8 so E is eliminated. Round 2 votes counts: C=27 D=26 B=25 A=22 so A is eliminated. Round 3 votes counts: C=39 D=34 B=27 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:218 C:203 B:196 E:194 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 22 B -4 0 -2 2 -4 C -6 2 0 12 -2 D -4 -2 -12 0 -4 E -22 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 22 B -4 0 -2 2 -4 C -6 2 0 12 -2 D -4 -2 -12 0 -4 E -22 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 22 B -4 0 -2 2 -4 C -6 2 0 12 -2 D -4 -2 -12 0 -4 E -22 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 161: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) D B E C A (8) E C B D A (7) D B A C E (7) A D B C E (7) C E A B D (6) A C E B D (6) E D B C A (3) E C B A D (3) D B A E C (3) C A B E D (3) A C D E B (3) E B D C A (2) E A C D B (2) D B E A C (2) D B C E A (2) D A B E C (2) D A B C E (2) C B A E D (2) A E C D B (2) A C E D B (2) A C B D E (2) E D B A C (1) E C D B A (1) D E B C A (1) D A E B C (1) C A B D E (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A C E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 8 -10 8 -6 B -8 0 -10 -2 -6 C 10 10 0 8 -4 D -8 2 -8 0 -6 E 6 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -10 8 -6 B -8 0 -10 -2 -6 C 10 10 0 8 -4 D -8 2 -8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=28 A=24 C=12 B=4 so B is eliminated. Round 2 votes counts: E=32 D=32 A=24 C=12 so C is eliminated. Round 3 votes counts: E=38 D=32 A=30 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:211 A:200 D:190 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -10 8 -6 B -8 0 -10 -2 -6 C 10 10 0 8 -4 D -8 2 -8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 8 -6 B -8 0 -10 -2 -6 C 10 10 0 8 -4 D -8 2 -8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 8 -6 B -8 0 -10 -2 -6 C 10 10 0 8 -4 D -8 2 -8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 162: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) D E C B A (7) C D E B A (5) C B E D A (5) A C D B E (5) E D C B A (4) A D E B C (4) A B E D C (4) E D B A C (3) A B E C D (3) E D B C A (2) E B D A C (2) D E C A B (2) D E A C B (2) D E A B C (2) D C E B A (2) C D A E B (2) C B E A D (2) C B A E D (2) C A B D E (2) B A C E D (2) A D B E C (2) A C B D E (2) A B D E C (2) A B C D E (2) E D A B C (1) E B D C A (1) E B C D A (1) D E B C A (1) D E B A C (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D B A (1) C E B D A (1) C D E A B (1) C D B E A (1) C A B E D (1) B E D A C (1) B E C D A (1) B E A C D (1) B C E D A (1) B C E A D (1) B C A E D (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 6 -6 -4 B -2 0 2 -2 2 C -6 -2 0 4 0 D 6 2 -4 0 -2 E 4 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.052632 B: 0.368421 C: 0.210526 D: 0.263158 E: 0.105263 Sum of squares = 0.263157894741 Cumulative probabilities = A: 0.052632 B: 0.421053 C: 0.631579 D: 0.894737 E: 1.000000 A B C D E A 0 2 6 -6 -4 B -2 0 2 -2 2 C -6 -2 0 4 0 D 6 2 -4 0 -2 E 4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.368421 C: 0.210526 D: 0.263158 E: 0.105263 Sum of squares = 0.263157894741 Cumulative probabilities = A: 0.052632 B: 0.421053 C: 0.631579 D: 0.894737 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=23 D=20 E=14 B=8 so B is eliminated. Round 2 votes counts: A=37 C=26 D=20 E=17 so E is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:202 D:201 B:200 A:199 C:198 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 6 -6 -4 B -2 0 2 -2 2 C -6 -2 0 4 0 D 6 2 -4 0 -2 E 4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.368421 C: 0.210526 D: 0.263158 E: 0.105263 Sum of squares = 0.263157894741 Cumulative probabilities = A: 0.052632 B: 0.421053 C: 0.631579 D: 0.894737 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -6 -4 B -2 0 2 -2 2 C -6 -2 0 4 0 D 6 2 -4 0 -2 E 4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.368421 C: 0.210526 D: 0.263158 E: 0.105263 Sum of squares = 0.263157894741 Cumulative probabilities = A: 0.052632 B: 0.421053 C: 0.631579 D: 0.894737 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -6 -4 B -2 0 2 -2 2 C -6 -2 0 4 0 D 6 2 -4 0 -2 E 4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.368421 C: 0.210526 D: 0.263158 E: 0.105263 Sum of squares = 0.263157894741 Cumulative probabilities = A: 0.052632 B: 0.421053 C: 0.631579 D: 0.894737 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 163: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D C E A (8) A E C B D (8) D B A E C (7) A E C D B (7) C E A D B (6) B C D E A (6) D B C E A (5) B C A E D (5) C E A B D (4) B D C A E (4) B D A E C (4) A E D C B (3) D E A C B (2) D C E A B (2) D A E B C (2) C B E A D (2) B C E A D (2) B A E C D (2) E D A C B (1) E A D C B (1) D B E A C (1) D B C A E (1) C D B E A (1) C A E B D (1) B D A C E (1) B C D A E (1) B A E D C (1) B A D E C (1) A E D B C (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 4 8 0 B 4 0 6 2 4 C -4 -6 0 10 -2 D -8 -2 -10 0 -8 E 0 -4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 8 0 B 4 0 6 2 4 C -4 -6 0 10 -2 D -8 -2 -10 0 -8 E 0 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=20 A=20 C=14 E=11 so E is eliminated. Round 2 votes counts: B=35 A=30 D=21 C=14 so C is eliminated. Round 3 votes counts: A=41 B=37 D=22 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:204 E:203 C:199 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 8 0 B 4 0 6 2 4 C -4 -6 0 10 -2 D -8 -2 -10 0 -8 E 0 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 8 0 B 4 0 6 2 4 C -4 -6 0 10 -2 D -8 -2 -10 0 -8 E 0 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 8 0 B 4 0 6 2 4 C -4 -6 0 10 -2 D -8 -2 -10 0 -8 E 0 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 164: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) B E A C D (10) D C A E B (7) E B C A D (6) B E A D C (6) E B C D A (5) C D A E B (5) B E D A C (5) B A D E C (5) A D C B E (5) E C B A D (4) C A D E B (4) A C D E B (4) A B D C E (4) E C B D A (3) D C A B E (3) D B A C E (2) C D E A B (2) B E D C A (2) B E C D A (2) B D A E C (2) E D C B A (1) E B A C D (1) C E D A B (1) C E A D B (1) Total count = 100 A B C D E A 0 -8 8 0 2 B 8 0 0 10 12 C -8 0 0 -4 -4 D 0 -10 4 0 6 E -2 -12 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.655256 C: 0.344744 D: 0.000000 E: 0.000000 Sum of squares = 0.548209078456 Cumulative probabilities = A: 0.000000 B: 0.655256 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 0 2 B 8 0 0 10 12 C -8 0 0 -4 -4 D 0 -10 4 0 6 E -2 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500229 C: 0.499771 D: 0.000000 E: 0.000000 Sum of squares = 0.500000105204 Cumulative probabilities = A: 0.000000 B: 0.500229 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=22 E=20 C=13 A=13 so C is eliminated. Round 2 votes counts: B=32 D=29 E=22 A=17 so A is eliminated. Round 3 votes counts: D=42 B=36 E=22 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:201 D:200 C:192 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 0 2 B 8 0 0 10 12 C -8 0 0 -4 -4 D 0 -10 4 0 6 E -2 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500229 C: 0.499771 D: 0.000000 E: 0.000000 Sum of squares = 0.500000105204 Cumulative probabilities = A: 0.000000 B: 0.500229 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 0 2 B 8 0 0 10 12 C -8 0 0 -4 -4 D 0 -10 4 0 6 E -2 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500229 C: 0.499771 D: 0.000000 E: 0.000000 Sum of squares = 0.500000105204 Cumulative probabilities = A: 0.000000 B: 0.500229 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 0 2 B 8 0 0 10 12 C -8 0 0 -4 -4 D 0 -10 4 0 6 E -2 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500229 C: 0.499771 D: 0.000000 E: 0.000000 Sum of squares = 0.500000105204 Cumulative probabilities = A: 0.000000 B: 0.500229 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 165: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) C E B A D (6) E C A B D (5) D C B A E (5) A D E B C (5) E A C B D (4) E A B C D (4) C E A B D (4) C B E D A (4) B A E D C (4) A E B D C (4) D B C A E (3) C D E A B (3) C D B E A (3) B C E A D (3) E B A C D (2) D C A E B (2) C E D A B (2) C D E B A (2) B C D E A (2) A E B C D (2) A D B E C (2) A B D E C (2) E C B A D (1) D C E A B (1) D C B E A (1) D A E B C (1) D A C B E (1) C E D B A (1) C E A D B (1) B E C A D (1) B E A C D (1) B D C A E (1) B D A E C (1) B A E C D (1) A E D C B (1) A B E D C (1) Total count = 100 A B C D E A 0 16 -2 8 -2 B -16 0 6 6 -2 C 2 -6 0 4 -10 D -8 -6 -4 0 -4 E 2 2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -2 8 -2 B -16 0 6 6 -2 C 2 -6 0 4 -10 D -8 -6 -4 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=26 A=17 E=16 B=14 so B is eliminated. Round 2 votes counts: C=31 D=29 A=22 E=18 so E is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:210 E:209 B:197 C:195 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -2 8 -2 B -16 0 6 6 -2 C 2 -6 0 4 -10 D -8 -6 -4 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 8 -2 B -16 0 6 6 -2 C 2 -6 0 4 -10 D -8 -6 -4 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 8 -2 B -16 0 6 6 -2 C 2 -6 0 4 -10 D -8 -6 -4 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 166: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) E B A C D (7) A B D E C (7) B E A C D (6) B A E D C (6) D C B A E (5) E B C A D (4) D C A E B (4) C D E B A (4) C D E A B (4) E C A B D (3) D C E A B (3) D A B C E (3) A E B C D (3) A D B C E (3) E C D B A (2) E C D A B (2) E C B D A (2) E C B A D (2) D C B E A (2) A B E C D (2) A B D C E (2) E A C D B (1) E A B C D (1) D C A B E (1) D B C A E (1) D A C B E (1) C E D B A (1) C E D A B (1) C E B D A (1) B E C A D (1) B D C A E (1) B A E C D (1) B A D E C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 6 12 24 6 B -6 0 20 20 8 C -12 -20 0 -4 -24 D -24 -20 4 0 -12 E -6 -8 24 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 24 6 B -6 0 20 20 8 C -12 -20 0 -4 -24 D -24 -20 4 0 -12 E -6 -8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 D=20 B=16 C=11 so C is eliminated. Round 2 votes counts: A=29 D=28 E=27 B=16 so B is eliminated. Round 3 votes counts: A=37 E=34 D=29 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:221 E:211 D:174 C:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 24 6 B -6 0 20 20 8 C -12 -20 0 -4 -24 D -24 -20 4 0 -12 E -6 -8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 24 6 B -6 0 20 20 8 C -12 -20 0 -4 -24 D -24 -20 4 0 -12 E -6 -8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 24 6 B -6 0 20 20 8 C -12 -20 0 -4 -24 D -24 -20 4 0 -12 E -6 -8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 167: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (7) B D C A E (7) E B A C D (6) E A B C D (6) E A C D B (5) D C A B E (5) B E A C D (5) E C A D B (4) C D B E A (4) B C D E A (4) A E D C B (4) A E C D B (4) D C B A E (3) D B C A E (3) C E D B A (3) C D E A B (3) C D A E B (3) B E C A D (3) C E D A B (2) C D B A E (2) B E D A C (2) B E C D A (2) B A E D C (2) E A C B D (1) C A D E B (1) B E D C A (1) B D E C A (1) B D C E A (1) B D A C E (1) B A D E C (1) A E B D C (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -4 6 -20 B 16 0 8 2 8 C 4 -8 0 18 -12 D -6 -2 -18 0 -16 E 20 -8 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 6 -20 B 16 0 8 2 8 C 4 -8 0 18 -12 D -6 -2 -18 0 -16 E 20 -8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=22 C=18 A=12 D=11 so D is eliminated. Round 2 votes counts: B=40 C=26 E=22 A=12 so A is eliminated. Round 3 votes counts: B=40 E=32 C=28 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:220 B:217 C:201 A:183 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 6 -20 B 16 0 8 2 8 C 4 -8 0 18 -12 D -6 -2 -18 0 -16 E 20 -8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 6 -20 B 16 0 8 2 8 C 4 -8 0 18 -12 D -6 -2 -18 0 -16 E 20 -8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 6 -20 B 16 0 8 2 8 C 4 -8 0 18 -12 D -6 -2 -18 0 -16 E 20 -8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 168: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) E D C B A (8) A B C E D (8) C B A E D (5) B C A D E (5) E D A B C (4) C B D E A (4) C B D A E (4) A B D C E (4) E C D B A (3) D E C B A (3) D E A B C (3) D C B E A (3) C D B E A (3) A B E C D (3) E A D B C (2) D C E B A (2) D C B A E (2) C E D B A (2) C D E B A (2) C B A D E (2) B D C A E (2) B A C D E (2) A E B D C (2) E D A C B (1) E C D A B (1) E C A B D (1) E A C B D (1) E A B D C (1) E A B C D (1) D E B C A (1) D E B A C (1) C E B A D (1) B C D A E (1) B A D C E (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 -10 0 4 B 14 0 2 16 20 C 10 -2 0 18 26 D 0 -16 -18 0 10 E -4 -20 -26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999285 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 0 4 B 14 0 2 16 20 C 10 -2 0 18 26 D 0 -16 -18 0 10 E -4 -20 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 C=23 D=15 B=11 so B is eliminated. Round 2 votes counts: A=31 C=29 E=23 D=17 so D is eliminated. Round 3 votes counts: C=38 E=31 A=31 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:226 C:226 A:190 D:188 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -10 0 4 B 14 0 2 16 20 C 10 -2 0 18 26 D 0 -16 -18 0 10 E -4 -20 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 0 4 B 14 0 2 16 20 C 10 -2 0 18 26 D 0 -16 -18 0 10 E -4 -20 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 0 4 B 14 0 2 16 20 C 10 -2 0 18 26 D 0 -16 -18 0 10 E -4 -20 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 169: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) A D E B C (8) E B D A C (7) E B D C A (6) C B E D A (6) A C D B E (6) C A D B E (5) B E C D A (5) A E B D C (5) D E B A C (4) D B E C A (4) E B A D C (3) A C E B D (3) C D A B E (2) C B E A D (2) C B D E A (2) C A B E D (2) B E D C A (2) A E D B C (2) A E B C D (2) A C D E B (2) E B C D A (1) E B A C D (1) E A B D C (1) D E B C A (1) D C A B E (1) D A C E B (1) C D B A E (1) C A B D E (1) B E C A D (1) B C E D A (1) A E C D B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -4 -6 -12 B 12 0 8 2 2 C 4 -8 0 6 -8 D 6 -2 -6 0 -2 E 12 -2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -6 -12 B 12 0 8 2 2 C 4 -8 0 6 -8 D 6 -2 -6 0 -2 E 12 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=30 E=19 D=11 B=9 so B is eliminated. Round 2 votes counts: C=31 A=31 E=27 D=11 so D is eliminated. Round 3 votes counts: E=36 C=32 A=32 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:212 E:210 D:198 C:197 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 -12 B 12 0 8 2 2 C 4 -8 0 6 -8 D 6 -2 -6 0 -2 E 12 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 -12 B 12 0 8 2 2 C 4 -8 0 6 -8 D 6 -2 -6 0 -2 E 12 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 -12 B 12 0 8 2 2 C 4 -8 0 6 -8 D 6 -2 -6 0 -2 E 12 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 170: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (18) B A E C D (18) E C D A B (5) D E C B A (4) B D A C E (4) E C A D B (3) E A C B D (3) D C B E A (3) B A D C E (3) E C A B D (2) E B C A D (2) D B C E A (2) D B A C E (2) D A C E B (2) C E D A B (2) B E C D A (2) B A E D C (2) A E C B D (2) A C E D B (2) A B E C D (2) E D C B A (1) E C D B A (1) E C B A D (1) E B A C D (1) D C E B A (1) D A B C E (1) C E A D B (1) C A E D B (1) B E C A D (1) B D E C A (1) B D E A C (1) B D C A E (1) B D A E C (1) B A D E C (1) B A C E D (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -4 -4 -10 B 8 0 -6 0 -4 C 4 6 0 2 -8 D 4 0 -2 0 -8 E 10 4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -4 -4 -10 B 8 0 -6 0 -4 C 4 6 0 2 -8 D 4 0 -2 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=33 E=19 A=8 C=4 so C is eliminated. Round 2 votes counts: B=36 D=33 E=22 A=9 so A is eliminated. Round 3 votes counts: B=39 D=34 E=27 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:215 C:202 B:199 D:197 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -4 -10 B 8 0 -6 0 -4 C 4 6 0 2 -8 D 4 0 -2 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -4 -10 B 8 0 -6 0 -4 C 4 6 0 2 -8 D 4 0 -2 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -4 -10 B 8 0 -6 0 -4 C 4 6 0 2 -8 D 4 0 -2 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 171: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (23) E C B D A (20) C B E D A (5) E D B C A (3) E C A B D (3) D A B C E (3) A E D C B (3) A D E B C (3) A D B E C (3) E C B A D (2) E A D B C (2) E A C B D (2) D B C E A (2) C E B D A (2) B C D E A (2) B C A D E (2) A E D B C (2) A C B D E (2) E D C B A (1) E D A C B (1) E C D B A (1) E A D C B (1) E A C D B (1) D B A C E (1) D A B E C (1) C B E A D (1) C B A D E (1) C A E B D (1) B D C E A (1) B D C A E (1) B C D A E (1) A D C B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 10 0 B -8 0 4 -4 4 C -2 -4 0 -6 2 D -10 4 6 0 -2 E 0 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.632621 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.367379 Sum of squares = 0.535176415548 Cumulative probabilities = A: 0.632621 B: 0.632621 C: 0.632621 D: 0.632621 E: 1.000000 A B C D E A 0 8 2 10 0 B -8 0 4 -4 4 C -2 -4 0 -6 2 D -10 4 6 0 -2 E 0 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500217 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499783 Sum of squares = 0.500000094568 Cumulative probabilities = A: 0.500217 B: 0.500217 C: 0.500217 D: 0.500217 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=37 C=10 D=7 B=7 so D is eliminated. Round 2 votes counts: A=43 E=37 C=10 B=10 so C is eliminated. Round 3 votes counts: A=44 E=39 B=17 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:199 B:198 E:198 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 10 0 B -8 0 4 -4 4 C -2 -4 0 -6 2 D -10 4 6 0 -2 E 0 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500217 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499783 Sum of squares = 0.500000094568 Cumulative probabilities = A: 0.500217 B: 0.500217 C: 0.500217 D: 0.500217 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 10 0 B -8 0 4 -4 4 C -2 -4 0 -6 2 D -10 4 6 0 -2 E 0 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500217 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499783 Sum of squares = 0.500000094568 Cumulative probabilities = A: 0.500217 B: 0.500217 C: 0.500217 D: 0.500217 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 10 0 B -8 0 4 -4 4 C -2 -4 0 -6 2 D -10 4 6 0 -2 E 0 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500217 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499783 Sum of squares = 0.500000094568 Cumulative probabilities = A: 0.500217 B: 0.500217 C: 0.500217 D: 0.500217 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 172: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) A D B E C (8) D A B E C (7) E C B D A (6) E C A D B (6) B D A C E (6) A D E C B (6) D A E C B (5) B A D C E (5) E C D A B (4) C B E A D (4) B C E D A (4) C E B D A (3) B C E A D (3) A D B C E (3) E A D C B (2) D A E B C (2) B C A D E (2) A D C E B (2) E D C A B (1) E C D B A (1) E A C D B (1) D E A C B (1) D A B C E (1) B D C A E (1) B C D A E (1) B A C D E (1) A D E B C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 6 14 6 B -4 0 -8 -4 -4 C -6 8 0 -6 -2 D -14 4 6 0 8 E -6 4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 14 6 B -4 0 -8 -4 -4 C -6 8 0 -6 -2 D -14 4 6 0 8 E -6 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 A=22 E=21 C=18 D=16 so D is eliminated. Round 2 votes counts: A=37 B=23 E=22 C=18 so C is eliminated. Round 3 votes counts: A=37 E=36 B=27 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:202 C:197 E:196 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 14 6 B -4 0 -8 -4 -4 C -6 8 0 -6 -2 D -14 4 6 0 8 E -6 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 14 6 B -4 0 -8 -4 -4 C -6 8 0 -6 -2 D -14 4 6 0 8 E -6 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 14 6 B -4 0 -8 -4 -4 C -6 8 0 -6 -2 D -14 4 6 0 8 E -6 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 173: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) B A D E C (11) D E C A B (6) B D E A C (6) D E A C B (4) D E A B C (4) C B A E D (4) A B C E D (4) E D C A B (3) D E B C A (3) B D A E C (3) B C A E D (3) B C A D E (3) B A C D E (3) A E D B C (3) C E D A B (2) C D E B A (2) C B E D A (2) B D E C A (2) A C E B D (2) A B E C D (2) E D A C B (1) D E C B A (1) D E B A C (1) D C E A B (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B E A (1) C B D E A (1) C A E D B (1) C A E B D (1) C A B E D (1) A C E D B (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 20 10 14 B 18 0 24 26 22 C -20 -24 0 -2 -4 D -10 -26 2 0 8 E -14 -22 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 20 10 14 B 18 0 24 26 22 C -20 -24 0 -2 -4 D -10 -26 2 0 8 E -14 -22 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 D=20 C=18 A=15 E=4 so E is eliminated. Round 2 votes counts: B=43 D=24 C=18 A=15 so A is eliminated. Round 3 votes counts: B=51 D=27 C=22 so C is eliminated. Round 4 votes counts: B=63 D=37 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:245 A:213 D:187 E:180 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 20 10 14 B 18 0 24 26 22 C -20 -24 0 -2 -4 D -10 -26 2 0 8 E -14 -22 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 20 10 14 B 18 0 24 26 22 C -20 -24 0 -2 -4 D -10 -26 2 0 8 E -14 -22 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 20 10 14 B 18 0 24 26 22 C -20 -24 0 -2 -4 D -10 -26 2 0 8 E -14 -22 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 174: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (21) B E C A D (13) D A C E B (11) C A D B E (10) D A C B E (7) C B A D E (7) E B D A C (5) E B C D A (4) B C A D E (4) D A E C B (3) E D A B C (2) B C A E D (2) A C D B E (2) E D B A C (1) E D A C B (1) E B A D C (1) E B A C D (1) D E A C B (1) C A B D E (1) B C E A D (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 -22 -28 28 -2 B 22 0 12 22 -2 C 28 -12 0 34 -6 D -28 -22 -34 0 -4 E 2 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -22 -28 28 -2 B 22 0 12 22 -2 C 28 -12 0 34 -6 D -28 -22 -34 0 -4 E 2 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=22 B=21 C=18 A=3 so A is eliminated. Round 2 votes counts: E=36 D=23 B=21 C=20 so C is eliminated. Round 3 votes counts: E=36 D=35 B=29 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:227 C:222 E:207 A:188 D:156 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -22 -28 28 -2 B 22 0 12 22 -2 C 28 -12 0 34 -6 D -28 -22 -34 0 -4 E 2 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -28 28 -2 B 22 0 12 22 -2 C 28 -12 0 34 -6 D -28 -22 -34 0 -4 E 2 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -28 28 -2 B 22 0 12 22 -2 C 28 -12 0 34 -6 D -28 -22 -34 0 -4 E 2 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 175: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (7) D A C B E (6) B A E D C (6) B A D E C (6) A B D E C (6) E C D B A (5) D C A E B (5) D A C E B (4) C E D B A (4) C E D A B (4) B E C A D (4) B E A C D (4) A D B E C (4) A B D C E (4) E C B D A (3) E B C D A (2) E B A D C (2) D C E A B (2) C E B D A (2) B E A D C (2) A D E B C (2) A D B C E (2) E D C A B (1) E B C A D (1) E B A C D (1) D E A C B (1) D C A B E (1) C D B A E (1) C D A E B (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C E D (1) B A C D E (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 10 2 8 B -4 0 2 -4 4 C -10 -2 0 -14 -2 D -2 4 14 0 12 E -8 -4 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 2 8 B -4 0 2 -4 4 C -10 -2 0 -14 -2 D -2 4 14 0 12 E -8 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=20 D=19 C=19 E=15 so E is eliminated. Round 2 votes counts: B=33 C=27 D=20 A=20 so D is eliminated. Round 3 votes counts: C=36 B=33 A=31 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 A:212 B:199 E:189 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 2 8 B -4 0 2 -4 4 C -10 -2 0 -14 -2 D -2 4 14 0 12 E -8 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 2 8 B -4 0 2 -4 4 C -10 -2 0 -14 -2 D -2 4 14 0 12 E -8 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 2 8 B -4 0 2 -4 4 C -10 -2 0 -14 -2 D -2 4 14 0 12 E -8 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 176: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) E B D A C (7) A D C B E (7) E B D C A (6) A C E D B (5) E C A B D (4) B D E C A (4) D B A C E (3) D A B E C (3) C E B A D (3) A C D B E (3) E D B A C (2) E C B A D (2) E B C D A (2) E B C A D (2) E A C D B (2) D A C B E (2) D A B C E (2) C E A B D (2) C B A D E (2) C A E B D (2) B D E A C (2) B C D E A (2) B C D A E (2) A C D E B (2) E A D C B (1) E A D B C (1) E A C B D (1) E A B D C (1) D C B A E (1) D B E A C (1) D B C A E (1) C B D A E (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A C E (1) A E D C B (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -2 8 4 B -4 0 -8 4 2 C 2 8 0 0 10 D -8 -4 0 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.917936 D: 0.082064 E: 0.000000 Sum of squares = 0.849340941662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.917936 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 8 4 B -4 0 -8 4 2 C 2 8 0 0 10 D -8 -4 0 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=21 A=20 B=15 D=13 so D is eliminated. Round 2 votes counts: E=31 A=27 C=22 B=20 so B is eliminated. Round 3 votes counts: E=40 A=31 C=29 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:210 A:207 B:197 D:196 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 8 4 B -4 0 -8 4 2 C 2 8 0 0 10 D -8 -4 0 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 8 4 B -4 0 -8 4 2 C 2 8 0 0 10 D -8 -4 0 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 8 4 B -4 0 -8 4 2 C 2 8 0 0 10 D -8 -4 0 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 177: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) B E C D A (11) A D C E B (8) B C E A D (7) E B D A C (5) D E A B C (5) D A E C B (5) C A B D E (5) B E D C A (5) A D E C B (5) E D B A C (4) C B A E D (4) E D A B C (3) C B A D E (3) B C E D A (3) A C D E B (3) D A E B C (2) C A B E D (2) B E C A D (2) E A D B C (1) D A C E B (1) C D A B E (1) B D E C A (1) B C D E A (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -14 4 4 B -6 0 2 0 16 C 14 -2 0 10 2 D -4 0 -10 0 6 E -4 -16 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305796 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 4 4 B -6 0 2 0 16 C 14 -2 0 10 2 D -4 0 -10 0 6 E -4 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305794 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=26 A=17 E=13 D=13 so E is eliminated. Round 2 votes counts: B=36 C=26 D=20 A=18 so A is eliminated. Round 3 votes counts: B=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:212 B:206 A:200 D:196 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 4 4 B -6 0 2 0 16 C 14 -2 0 10 2 D -4 0 -10 0 6 E -4 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305794 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 4 4 B -6 0 2 0 16 C 14 -2 0 10 2 D -4 0 -10 0 6 E -4 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305794 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 4 4 B -6 0 2 0 16 C 14 -2 0 10 2 D -4 0 -10 0 6 E -4 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305794 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 178: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) E B D A C (10) D B E A C (8) E B D C A (7) C A E D B (7) D B A E C (6) C E A B D (6) C A E B D (5) B D E A C (5) A C D B E (5) A D C B E (3) E C B D A (2) E C B A D (2) E C A B D (2) D B A C E (2) C E A D B (2) A D B C E (2) A C B E D (2) A B D C E (2) E C D B A (1) E B C D A (1) E B A C D (1) D E B C A (1) D C B E A (1) D B E C A (1) D A C B E (1) C A D E B (1) B E D A C (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 6 -2 B 0 0 -2 -4 4 C 0 2 0 -4 0 D -6 4 4 0 0 E 2 -4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.285714 D: 0.071429 E: 0.214286 Sum of squares = 0.234693877551 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.714286 D: 0.785714 E: 1.000000 A B C D E A 0 0 0 6 -2 B 0 0 -2 -4 4 C 0 2 0 -4 0 D -6 4 4 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.285714 D: 0.071429 E: 0.214286 Sum of squares = 0.234693877551 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.714286 D: 0.785714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=26 D=20 A=15 B=7 so B is eliminated. Round 2 votes counts: C=32 E=27 D=25 A=16 so A is eliminated. Round 3 votes counts: C=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:202 D:201 B:199 C:199 E:199 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 6 -2 B 0 0 -2 -4 4 C 0 2 0 -4 0 D -6 4 4 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.285714 D: 0.071429 E: 0.214286 Sum of squares = 0.234693877551 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.714286 D: 0.785714 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 6 -2 B 0 0 -2 -4 4 C 0 2 0 -4 0 D -6 4 4 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.285714 D: 0.071429 E: 0.214286 Sum of squares = 0.234693877551 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.714286 D: 0.785714 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 6 -2 B 0 0 -2 -4 4 C 0 2 0 -4 0 D -6 4 4 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.285714 D: 0.071429 E: 0.214286 Sum of squares = 0.234693877551 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.714286 D: 0.785714 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 179: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) B E D C A (6) D B E A C (5) C A E B D (5) B C E A D (5) C B A D E (4) C A E D B (4) B D E C A (4) B C A E D (4) E A C D B (3) D E B A C (3) B D E A C (3) B D C A E (3) A E C D B (3) A D C E B (3) E C A B D (2) E A D C B (2) D E A B C (2) D B A C E (2) D A E C B (2) D A C E B (2) D A C B E (2) C B A E D (2) C A B E D (2) B E C A D (2) A D C B E (2) A C E D B (2) A C D E B (2) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) D E A C B (1) D B A E C (1) D A B E C (1) D A B C E (1) C E A B D (1) C A D B E (1) C A B D E (1) B E C D A (1) B C A D E (1) Total count = 100 A B C D E A 0 2 2 2 0 B -2 0 -4 -4 6 C -2 4 0 -10 -2 D -2 4 10 0 -8 E 0 -6 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.826242 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.173758 Sum of squares = 0.712867321723 Cumulative probabilities = A: 0.826242 B: 0.826242 C: 0.826242 D: 0.826242 E: 1.000000 A B C D E A 0 2 2 2 0 B -2 0 -4 -4 6 C -2 4 0 -10 -2 D -2 4 10 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000051615 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=22 C=20 E=17 A=12 so A is eliminated. Round 2 votes counts: B=29 D=27 C=24 E=20 so E is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:203 D:202 E:202 B:198 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 2 0 B -2 0 -4 -4 6 C -2 4 0 -10 -2 D -2 4 10 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000051615 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 0 B -2 0 -4 -4 6 C -2 4 0 -10 -2 D -2 4 10 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000051615 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 0 B -2 0 -4 -4 6 C -2 4 0 -10 -2 D -2 4 10 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000051615 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 180: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) E B A D C (6) C D B A E (5) B E D A C (5) C D A B E (4) C A D E B (4) B D C E A (4) B C D E A (4) A D E C B (4) E A B D C (3) E A B C D (3) D B C A E (3) D A C E B (3) D A C B E (3) A E D C B (3) A D E B C (3) A D C E B (3) E B A C D (2) E A C B D (2) D C B A E (2) C E A B D (2) C B E D A (2) C B E A D (2) C A E B D (2) B E D C A (2) B E C A D (2) B C E D A (2) A E D B C (2) A C D E B (2) E C A B D (1) E B C A D (1) D A B C E (1) C E B A D (1) C B D A E (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -6 0 -6 B 6 0 4 12 2 C 6 -4 0 6 2 D 0 -12 -6 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 0 -6 B 6 0 4 12 2 C 6 -4 0 6 2 D 0 -12 -6 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 A=19 E=18 D=12 so D is eliminated. Round 2 votes counts: B=31 A=26 C=25 E=18 so E is eliminated. Round 3 votes counts: B=40 A=34 C=26 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:205 E:205 A:191 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 0 -6 B 6 0 4 12 2 C 6 -4 0 6 2 D 0 -12 -6 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 0 -6 B 6 0 4 12 2 C 6 -4 0 6 2 D 0 -12 -6 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 0 -6 B 6 0 4 12 2 C 6 -4 0 6 2 D 0 -12 -6 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 181: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) B A D E C (7) E C D B A (6) E B A C D (6) C D E A B (6) E C A B D (5) D C E B A (5) A B D C E (5) D B A C E (4) B A E D C (4) C E D A B (3) C E A B D (3) C A E B D (3) A B E C D (3) E D C B A (2) D C B A E (2) D C A B E (2) C E A D B (2) C D A B E (2) E D B C A (1) E C D A B (1) E C B D A (1) E B C D A (1) E B A D C (1) E A C B D (1) D E B C A (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B C E (1) C D A E B (1) C A E D B (1) C A D B E (1) C A B E D (1) B D E A C (1) A E B C D (1) A C D B E (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -20 10 -12 B 8 0 -16 6 -16 C 20 16 0 16 -6 D -10 -6 -16 0 -10 E 12 16 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -20 10 -12 B 8 0 -16 6 -16 C 20 16 0 16 -6 D -10 -6 -16 0 -10 E 12 16 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=23 D=20 A=13 B=12 so B is eliminated. Round 2 votes counts: E=32 A=24 C=23 D=21 so D is eliminated. Round 3 votes counts: E=36 C=34 A=30 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:223 E:222 B:191 A:185 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -20 10 -12 B 8 0 -16 6 -16 C 20 16 0 16 -6 D -10 -6 -16 0 -10 E 12 16 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 10 -12 B 8 0 -16 6 -16 C 20 16 0 16 -6 D -10 -6 -16 0 -10 E 12 16 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 10 -12 B 8 0 -16 6 -16 C 20 16 0 16 -6 D -10 -6 -16 0 -10 E 12 16 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 182: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E B A D C (6) D C E B A (5) E C B D A (4) D C B E A (4) C E A B D (4) C A D B E (4) A C D B E (4) E C D B A (3) C D E B A (3) C D E A B (3) B A E D C (3) E D B C A (2) E A B C D (2) D C A B E (2) D A B C E (2) C E D B A (2) C D A E B (2) C A D E B (2) B E A D C (2) A C E B D (2) A C B D E (2) A B E D C (2) A B D E C (2) A B D C E (2) E D C B A (1) E C B A D (1) E B D C A (1) E B D A C (1) E B C A D (1) E B A C D (1) D E C B A (1) D E B C A (1) D B E C A (1) D B E A C (1) D B A E C (1) D B A C E (1) D A C B E (1) C E A D B (1) C D B A E (1) C A E D B (1) C A E B D (1) B E D A C (1) B D E A C (1) B D A E C (1) B A D E C (1) A E B C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -16 -6 -6 B 2 0 -22 -12 -4 C 16 22 0 8 16 D 6 12 -8 0 12 E 6 4 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -6 -6 B 2 0 -22 -12 -4 C 16 22 0 8 16 D 6 12 -8 0 12 E 6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 D=20 A=17 B=9 so B is eliminated. Round 2 votes counts: C=31 E=26 D=22 A=21 so A is eliminated. Round 3 votes counts: C=41 E=32 D=27 so D is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:231 D:211 E:191 A:185 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 -6 -6 B 2 0 -22 -12 -4 C 16 22 0 8 16 D 6 12 -8 0 12 E 6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -6 -6 B 2 0 -22 -12 -4 C 16 22 0 8 16 D 6 12 -8 0 12 E 6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -6 -6 B 2 0 -22 -12 -4 C 16 22 0 8 16 D 6 12 -8 0 12 E 6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 183: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) B A D C E (9) A B D E C (7) E C A B D (6) C E D B A (5) D C E B A (4) E C D A B (3) D E C A B (3) D A E C B (3) C E B D A (3) C D E B A (3) A E C B D (3) A E B C D (3) A B E C D (3) E D C A B (2) D E A C B (2) D C E A B (2) D C B E A (2) D B C A E (2) D A B C E (2) B D C A E (2) B C D E A (2) B A E C D (2) B A C E D (2) A B E D C (2) E D A C B (1) E C B A D (1) E C A D B (1) E A C B D (1) D B A C E (1) D A E B C (1) C E D A B (1) C D B E A (1) C B E A D (1) B D A C E (1) B C E D A (1) B C E A D (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 12 8 -12 8 B -12 0 4 4 2 C -8 -4 0 -12 -8 D 12 -4 12 0 14 E -8 -2 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -12 8 B -12 0 4 4 2 C -8 -4 0 -12 -8 D 12 -4 12 0 14 E -8 -2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=21 A=19 E=15 C=14 so C is eliminated. Round 2 votes counts: D=35 E=24 B=22 A=19 so A is eliminated. Round 3 votes counts: D=35 B=34 E=31 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:217 A:208 B:199 E:192 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -12 8 B -12 0 4 4 2 C -8 -4 0 -12 -8 D 12 -4 12 0 14 E -8 -2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -12 8 B -12 0 4 4 2 C -8 -4 0 -12 -8 D 12 -4 12 0 14 E -8 -2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -12 8 B -12 0 4 4 2 C -8 -4 0 -12 -8 D 12 -4 12 0 14 E -8 -2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 184: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (18) D C E B A (15) C E D B A (7) E C B A D (6) C D E B A (5) B A E C D (5) D C E A B (4) D A C B E (4) B E A C D (3) A D B E C (3) A D B C E (3) A B E D C (3) E B C A D (2) D C A E B (2) D C A B E (2) B E C D A (2) A B D E C (2) E C B D A (1) E C A B D (1) E B A C D (1) E A C B D (1) D C B A E (1) D B C E A (1) D A C E B (1) D A B C E (1) C E D A B (1) C E B D A (1) C B D E A (1) A E B C D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 -4 2 -4 B 2 0 -10 -4 -2 C 4 10 0 12 0 D -2 4 -12 0 -6 E 4 2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.259734 D: 0.000000 E: 0.740266 Sum of squares = 0.615455403223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.259734 D: 0.259734 E: 1.000000 A B C D E A 0 -2 -4 2 -4 B 2 0 -10 -4 -2 C 4 10 0 12 0 D -2 4 -12 0 -6 E 4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=31 C=15 E=12 B=10 so B is eliminated. Round 2 votes counts: A=37 D=31 E=17 C=15 so C is eliminated. Round 3 votes counts: D=37 A=37 E=26 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:213 E:206 A:196 B:193 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 2 -4 B 2 0 -10 -4 -2 C 4 10 0 12 0 D -2 4 -12 0 -6 E 4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 2 -4 B 2 0 -10 -4 -2 C 4 10 0 12 0 D -2 4 -12 0 -6 E 4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 2 -4 B 2 0 -10 -4 -2 C 4 10 0 12 0 D -2 4 -12 0 -6 E 4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 185: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) E B A D C (6) D B E C A (6) C A B D E (6) A E B C D (6) E A B D C (5) D E B C A (5) C D A B E (4) C A D B E (4) E A D B C (3) D C E B A (3) C D A E B (3) B E D A C (3) A E C B D (3) E D A C B (2) E D A B C (2) E A D C B (2) D C E A B (2) C D B A E (2) C A E D B (2) B A E C D (2) A E C D B (2) A C E B D (2) E D B A C (1) E B D A C (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C E A (1) C D E A B (1) C B A D E (1) C A B E D (1) B E A D C (1) B D E A C (1) B D C E A (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -8 -2 -18 B -6 0 -6 -18 -8 C 8 6 0 -16 -8 D 2 18 16 0 6 E 18 8 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -2 -18 B -6 0 -6 -18 -8 C 8 6 0 -16 -8 D 2 18 16 0 6 E 18 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=24 E=23 A=15 B=8 so B is eliminated. Round 2 votes counts: D=32 E=27 C=24 A=17 so A is eliminated. Round 3 votes counts: E=41 D=32 C=27 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:214 C:195 A:189 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -2 -18 B -6 0 -6 -18 -8 C 8 6 0 -16 -8 D 2 18 16 0 6 E 18 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -2 -18 B -6 0 -6 -18 -8 C 8 6 0 -16 -8 D 2 18 16 0 6 E 18 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -2 -18 B -6 0 -6 -18 -8 C 8 6 0 -16 -8 D 2 18 16 0 6 E 18 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 186: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) D C B E A (9) A B E C D (8) E A B C D (6) D C E B A (6) E C D B A (4) A B E D C (4) A B D C E (4) E C D A B (3) D C E A B (3) C D E B A (3) C D B E A (3) E C B D A (2) E B C A D (2) E A C B D (2) D C B A E (2) B E C D A (2) B D C A E (2) B D A C E (2) B A E C D (2) A E D C B (2) E D C A B (1) E C B A D (1) E B C D A (1) E B A C D (1) D B C A E (1) D A E C B (1) D A C B E (1) C E D B A (1) C B E D A (1) B E C A D (1) B E A C D (1) B D C E A (1) B A D C E (1) B A C D E (1) A E C D B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 2 2 -8 B 0 0 8 16 -4 C -2 -8 0 16 -16 D -2 -16 -16 0 -16 E 8 4 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 2 -8 B 0 0 8 16 -4 C -2 -8 0 16 -16 D -2 -16 -16 0 -16 E 8 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=23 D=23 B=13 C=8 so C is eliminated. Round 2 votes counts: A=33 D=29 E=24 B=14 so B is eliminated. Round 3 votes counts: A=37 D=34 E=29 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:222 B:210 A:198 C:195 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 2 -8 B 0 0 8 16 -4 C -2 -8 0 16 -16 D -2 -16 -16 0 -16 E 8 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 -8 B 0 0 8 16 -4 C -2 -8 0 16 -16 D -2 -16 -16 0 -16 E 8 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 -8 B 0 0 8 16 -4 C -2 -8 0 16 -16 D -2 -16 -16 0 -16 E 8 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 187: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) B D A E C (7) A E C B D (7) E A C B D (6) C E A D B (6) C E A B D (6) C D B E A (5) A B D E C (5) E A B D C (4) D C B A E (4) D B C E A (4) C D B A E (4) C A E D B (4) E C A B D (3) D B C A E (3) A E C D B (3) E A B C D (2) D B A E C (2) D B A C E (2) A E D B C (2) E B C D A (1) E B A D C (1) C E D B A (1) C E B D A (1) C D E A B (1) C A D B E (1) B E C D A (1) B D E A C (1) B A E D C (1) B A D E C (1) A E B C D (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 22 10 26 14 B -22 0 -6 12 -16 C -10 6 0 8 -14 D -26 -12 -8 0 -18 E -14 16 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 10 26 14 B -22 0 -6 12 -16 C -10 6 0 8 -14 D -26 -12 -8 0 -18 E -14 16 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=28 E=17 D=15 B=11 so B is eliminated. Round 2 votes counts: A=30 C=29 D=23 E=18 so E is eliminated. Round 3 votes counts: A=43 C=34 D=23 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:236 E:217 C:195 B:184 D:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 10 26 14 B -22 0 -6 12 -16 C -10 6 0 8 -14 D -26 -12 -8 0 -18 E -14 16 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 10 26 14 B -22 0 -6 12 -16 C -10 6 0 8 -14 D -26 -12 -8 0 -18 E -14 16 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 10 26 14 B -22 0 -6 12 -16 C -10 6 0 8 -14 D -26 -12 -8 0 -18 E -14 16 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 188: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (12) E B D C A (8) E B C D A (6) D A C E B (6) E B D A C (5) D A C B E (5) C E B A D (5) C A D B E (5) D C A E B (4) C A E D B (4) C A D E B (4) B E C A D (4) D A B E C (3) B E D A C (3) A C D B E (3) D E B A C (2) C E A D B (2) C D A E B (2) C A E B D (2) C A B E D (2) B E A D C (2) B E A C D (2) E C B A D (1) D E C A B (1) D B A E C (1) D A E B C (1) C E A B D (1) C A B D E (1) B A E C D (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -28 6 -8 B 4 0 2 12 -32 C 28 -2 0 16 -4 D -6 -12 -16 0 -22 E 8 32 4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -28 6 -8 B 4 0 2 12 -32 C 28 -2 0 16 -4 D -6 -12 -16 0 -22 E 8 32 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=28 D=23 B=12 A=5 so A is eliminated. Round 2 votes counts: E=32 C=31 D=24 B=13 so B is eliminated. Round 3 votes counts: E=44 C=32 D=24 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:233 C:219 B:193 A:183 D:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -28 6 -8 B 4 0 2 12 -32 C 28 -2 0 16 -4 D -6 -12 -16 0 -22 E 8 32 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -28 6 -8 B 4 0 2 12 -32 C 28 -2 0 16 -4 D -6 -12 -16 0 -22 E 8 32 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -28 6 -8 B 4 0 2 12 -32 C 28 -2 0 16 -4 D -6 -12 -16 0 -22 E 8 32 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 189: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) D C A E B (6) D B E C A (6) D B C A E (6) B E A C D (6) B D E C A (6) B D E A C (6) C A E D B (5) B D A C E (5) E B A C D (4) E A B C D (4) D B C E A (4) A C B E D (4) A C E D B (3) E D B C A (2) E C A D B (2) E A C D B (2) D E C A B (2) D C B A E (2) B E D A C (2) B D C A E (2) E D A C B (1) E D A B C (1) E B D A C (1) D C A B E (1) C E D A B (1) C E A D B (1) C D A E B (1) C D A B E (1) C A D E B (1) B E A D C (1) B A E C D (1) B A D E C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -12 -26 -18 B 10 0 18 -10 8 C 12 -18 0 -24 0 D 26 10 24 0 14 E 18 -8 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -26 -18 B 10 0 18 -10 8 C 12 -18 0 -24 0 D 26 10 24 0 14 E 18 -8 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=30 E=17 C=10 A=9 so A is eliminated. Round 2 votes counts: D=34 B=30 E=19 C=17 so C is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:237 B:213 E:198 C:185 A:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -12 -26 -18 B 10 0 18 -10 8 C 12 -18 0 -24 0 D 26 10 24 0 14 E 18 -8 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -26 -18 B 10 0 18 -10 8 C 12 -18 0 -24 0 D 26 10 24 0 14 E 18 -8 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -26 -18 B 10 0 18 -10 8 C 12 -18 0 -24 0 D 26 10 24 0 14 E 18 -8 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 190: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) C E A D B (7) A D B C E (7) E C B D A (6) E B C D A (6) E B D C A (5) D B A C E (5) D A B C E (4) C E D A B (4) B E D C A (4) B D A E C (4) B A D E C (4) C A D E B (3) A C D E B (3) D B E C A (2) C A E D B (2) B D E C A (2) B D E A C (2) A C E B D (2) A C D B E (2) A B D C E (2) E D C B A (1) E C D B A (1) E B C A D (1) D E B C A (1) D C E B A (1) D C E A B (1) D C A E B (1) D C A B E (1) D B C E A (1) D B A E C (1) C D E B A (1) C D E A B (1) B E D A C (1) B E A D C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -4 -12 2 B 0 0 8 -18 4 C 4 -8 0 -20 14 D 12 18 20 0 16 E -2 -4 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -12 2 B 0 0 8 -18 4 C 4 -8 0 -20 14 D 12 18 20 0 16 E -2 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=20 D=18 C=18 B=18 so D is eliminated. Round 2 votes counts: A=30 B=27 C=22 E=21 so E is eliminated. Round 3 votes counts: B=40 C=30 A=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:233 B:197 C:195 A:193 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -12 2 B 0 0 8 -18 4 C 4 -8 0 -20 14 D 12 18 20 0 16 E -2 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -12 2 B 0 0 8 -18 4 C 4 -8 0 -20 14 D 12 18 20 0 16 E -2 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -12 2 B 0 0 8 -18 4 C 4 -8 0 -20 14 D 12 18 20 0 16 E -2 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 191: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (13) A D E C B (9) C B D E A (8) A E D B C (7) A B E D C (6) A B C D E (6) C D E A B (3) B C D E A (3) A C D E B (3) E D C B A (2) E D A C B (2) E D A B C (2) D E C B A (2) C D A E B (2) C B A D E (2) C A D B E (2) B C A D E (2) B A C E D (2) A D E B C (2) A C D B E (2) A C B D E (2) E D B C A (1) E B D C A (1) E B D A C (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E A B (1) C D E B A (1) C A D E B (1) C A B D E (1) B E D C A (1) B E C D A (1) B E A D C (1) B C A E D (1) B A E D C (1) B A C D E (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 2 10 8 B -12 0 8 8 12 C -2 -8 0 14 16 D -10 -8 -14 0 10 E -8 -12 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 10 8 B -12 0 8 8 12 C -2 -8 0 14 16 D -10 -8 -14 0 10 E -8 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=26 C=20 E=11 D=4 so D is eliminated. Round 2 votes counts: A=39 B=26 C=21 E=14 so E is eliminated. Round 3 votes counts: A=45 B=29 C=26 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:210 B:208 D:189 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 10 8 B -12 0 8 8 12 C -2 -8 0 14 16 D -10 -8 -14 0 10 E -8 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 10 8 B -12 0 8 8 12 C -2 -8 0 14 16 D -10 -8 -14 0 10 E -8 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 10 8 B -12 0 8 8 12 C -2 -8 0 14 16 D -10 -8 -14 0 10 E -8 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 192: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (15) E C D A B (9) D B A E C (9) C E A B D (8) B A D E C (8) D E C B A (7) E C D B A (5) B A D C E (4) B A C E D (4) E D C B A (3) D C E A B (3) C E D A B (3) C E A D B (3) C A E B D (3) B A E C D (3) B A C D E (3) D E C A B (2) A B C D E (2) D E B C A (1) D C A E B (1) D A B C E (1) B D A E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 10 12 B -4 0 6 4 4 C -4 -6 0 16 4 D -10 -4 -16 0 -12 E -12 -4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 10 12 B -4 0 6 4 4 C -4 -6 0 16 4 D -10 -4 -16 0 -12 E -12 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=23 A=19 E=17 C=17 so E is eliminated. Round 2 votes counts: C=31 D=27 B=23 A=19 so A is eliminated. Round 3 votes counts: B=41 C=31 D=28 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:205 C:205 E:196 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 10 12 B -4 0 6 4 4 C -4 -6 0 16 4 D -10 -4 -16 0 -12 E -12 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 10 12 B -4 0 6 4 4 C -4 -6 0 16 4 D -10 -4 -16 0 -12 E -12 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 10 12 B -4 0 6 4 4 C -4 -6 0 16 4 D -10 -4 -16 0 -12 E -12 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 193: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (23) C E D A B (19) A B D E C (7) C B E D A (4) B C A D E (4) E D A B C (3) D E A B C (3) C E D B A (3) C A E D B (3) A C B D E (3) E D C B A (2) E D C A B (2) E D B A C (2) E D A C B (2) C B A E D (2) C B A D E (2) B A C D E (2) A D E B C (2) E D B C A (1) E C D A B (1) D E B A C (1) C B E A D (1) C A B E D (1) C A B D E (1) B E C D A (1) B D E A C (1) B D A E C (1) B C A E D (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 8 8 B 2 0 6 10 10 C -4 -6 0 -2 -6 D -8 -10 2 0 4 E -8 -10 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 8 8 B 2 0 6 10 10 C -4 -6 0 -2 -6 D -8 -10 2 0 4 E -8 -10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995359 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=33 A=14 E=13 D=4 so D is eliminated. Round 2 votes counts: C=36 B=33 E=17 A=14 so A is eliminated. Round 3 votes counts: B=41 C=39 E=20 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:209 D:194 E:192 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 8 8 B 2 0 6 10 10 C -4 -6 0 -2 -6 D -8 -10 2 0 4 E -8 -10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995359 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 8 8 B 2 0 6 10 10 C -4 -6 0 -2 -6 D -8 -10 2 0 4 E -8 -10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995359 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 8 8 B 2 0 6 10 10 C -4 -6 0 -2 -6 D -8 -10 2 0 4 E -8 -10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995359 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 194: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (5) B D C E A (5) B D A E C (5) D E C A B (4) C E D A B (4) B C D E A (4) B A D E C (4) A E C D B (4) E D A C B (3) D E A C B (3) D C B E A (3) D B A E C (3) C E A D B (3) A E B C D (3) A B E D C (3) A B D E C (3) C E B D A (2) C D E B A (2) C B E D A (2) C B D E A (2) C A E B D (2) B D C A E (2) B C D A E (2) B C A E D (2) A E D C B (2) A D E B C (2) E D C A B (1) E C A D B (1) E A D C B (1) E A C D B (1) D E C B A (1) D B E C A (1) D B E A C (1) D B C E A (1) D A E B C (1) D A B E C (1) C D B E A (1) C B E A D (1) C A E D B (1) C A B E D (1) B A E C D (1) B A D C E (1) B A C E D (1) A E C B D (1) A E B D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 -18 -4 B 4 0 -2 0 4 C 6 2 0 -14 -4 D 18 0 14 0 14 E 4 -4 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.613931 C: 0.000000 D: 0.386069 E: 0.000000 Sum of squares = 0.525960474829 Cumulative probabilities = A: 0.000000 B: 0.613931 C: 0.613931 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -18 -4 B 4 0 -2 0 4 C 6 2 0 -14 -4 D 18 0 14 0 14 E 4 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=24 C=21 A=21 E=7 so E is eliminated. Round 2 votes counts: D=28 B=27 A=23 C=22 so C is eliminated. Round 3 votes counts: D=35 B=34 A=31 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:223 B:203 C:195 E:195 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -18 -4 B 4 0 -2 0 4 C 6 2 0 -14 -4 D 18 0 14 0 14 E 4 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -18 -4 B 4 0 -2 0 4 C 6 2 0 -14 -4 D 18 0 14 0 14 E 4 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -18 -4 B 4 0 -2 0 4 C 6 2 0 -14 -4 D 18 0 14 0 14 E 4 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 195: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (10) E D B A C (9) A C B D E (8) E D B C A (7) B D C A E (7) D B E C A (6) C B D A E (5) D B C A E (4) E D A B C (3) E A D C B (3) E A C B D (3) A D B C E (3) E A D B C (2) E A C D B (2) D B E A C (2) D B C E A (2) C A E B D (2) A E D B C (2) A C E B D (2) A C D B E (2) E C A B D (1) E B D C A (1) D E A B C (1) D B A C E (1) C E B A D (1) C B D E A (1) C B A D E (1) B D E C A (1) B D C E A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C B D (1) A D B E C (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -4 -6 6 B 2 0 12 -4 16 C 4 -12 0 -16 8 D 6 4 16 0 18 E -6 -16 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -6 6 B 2 0 12 -4 16 C 4 -12 0 -16 8 D 6 4 16 0 18 E -6 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=22 C=20 D=16 B=11 so B is eliminated. Round 2 votes counts: E=31 D=25 C=22 A=22 so C is eliminated. Round 3 votes counts: A=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:213 A:197 C:192 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 6 B 2 0 12 -4 16 C 4 -12 0 -16 8 D 6 4 16 0 18 E -6 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 6 B 2 0 12 -4 16 C 4 -12 0 -16 8 D 6 4 16 0 18 E -6 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 6 B 2 0 12 -4 16 C 4 -12 0 -16 8 D 6 4 16 0 18 E -6 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 196: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) E D B C A (8) B A C E D (6) C A D B E (5) C A B D E (5) A C D B E (5) A B C E D (5) E B D C A (4) D E C A B (4) B C A E D (4) A D C E B (4) E D B A C (3) E B D A C (3) D C A E B (3) B E A C D (3) B A E C D (3) E D A C B (2) D E C B A (2) D E A C B (2) D A E C B (2) D A C E B (2) A C B E D (2) E D A B C (1) E A B D C (1) D E B C A (1) D C E B A (1) C D A B E (1) C A D E B (1) B E D C A (1) B E C A D (1) B C D E A (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 18 16 18 24 B -18 0 -8 4 10 C -16 8 0 12 16 D -18 -4 -12 0 4 E -24 -10 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 18 24 B -18 0 -8 4 10 C -16 8 0 12 16 D -18 -4 -12 0 4 E -24 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=22 B=19 D=17 C=12 so C is eliminated. Round 2 votes counts: A=41 E=22 B=19 D=18 so D is eliminated. Round 3 votes counts: A=49 E=32 B=19 so B is eliminated. Round 4 votes counts: A=62 E=38 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:238 C:210 B:194 D:185 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 18 24 B -18 0 -8 4 10 C -16 8 0 12 16 D -18 -4 -12 0 4 E -24 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 18 24 B -18 0 -8 4 10 C -16 8 0 12 16 D -18 -4 -12 0 4 E -24 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 18 24 B -18 0 -8 4 10 C -16 8 0 12 16 D -18 -4 -12 0 4 E -24 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 197: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) E B A C D (11) A C B E D (9) E B D A C (6) B E A C D (6) D E B C A (5) D C E B A (5) D C A E B (5) D C A B E (5) C A D B E (5) A C D B E (4) C D A B E (3) B A E C D (3) A B E C D (3) E D B C A (2) D E B A C (2) C A B E D (2) A C B D E (2) E D B A C (1) E B D C A (1) E B C A D (1) D E C B A (1) D C E A B (1) C E B A D (1) C A B D E (1) B E A D C (1) B A E D C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 24 26 -10 B 16 0 12 20 -6 C -24 -12 0 4 -12 D -26 -20 -4 0 -20 E 10 6 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 24 26 -10 B 16 0 12 20 -6 C -24 -12 0 4 -12 D -26 -20 -4 0 -20 E 10 6 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=24 A=20 C=12 B=11 so B is eliminated. Round 2 votes counts: E=40 D=24 A=24 C=12 so C is eliminated. Round 3 votes counts: E=41 A=32 D=27 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:221 A:212 C:178 D:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 24 26 -10 B 16 0 12 20 -6 C -24 -12 0 4 -12 D -26 -20 -4 0 -20 E 10 6 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 24 26 -10 B 16 0 12 20 -6 C -24 -12 0 4 -12 D -26 -20 -4 0 -20 E 10 6 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 24 26 -10 B 16 0 12 20 -6 C -24 -12 0 4 -12 D -26 -20 -4 0 -20 E 10 6 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 198: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) E D C A B (5) C A E B D (5) E D C B A (4) D E A C B (4) D B E A C (4) B A C E D (4) A B C D E (4) E C B A D (3) E B C D A (3) D E C A B (3) D A C E B (3) D A C B E (3) D A B C E (3) C A B E D (3) B E D C A (3) B C A E D (3) B A C D E (3) E D B C A (2) E C D B A (2) E B D C A (2) D E B C A (2) D A B E C (2) C E A B D (2) B A D C E (2) D E A B C (1) D A E B C (1) C E B A D (1) C E A D B (1) C D E A B (1) C B A E D (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E A D (1) B A E C D (1) A D C B E (1) A C D E B (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 0 -16 -8 B 4 0 6 -2 -8 C 0 -6 0 -8 -6 D 16 2 8 0 -4 E 8 8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 -16 -8 B 4 0 6 -2 -8 C 0 -6 0 -8 -6 D 16 2 8 0 -4 E 8 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=22 E=21 C=16 A=9 so A is eliminated. Round 2 votes counts: D=33 B=27 E=21 C=19 so C is eliminated. Round 3 votes counts: D=36 B=33 E=31 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:211 B:200 C:190 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -16 -8 B 4 0 6 -2 -8 C 0 -6 0 -8 -6 D 16 2 8 0 -4 E 8 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -16 -8 B 4 0 6 -2 -8 C 0 -6 0 -8 -6 D 16 2 8 0 -4 E 8 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -16 -8 B 4 0 6 -2 -8 C 0 -6 0 -8 -6 D 16 2 8 0 -4 E 8 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 199: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) C A D E B (7) B E A C D (7) D C A E B (6) C A E D B (6) D C A B E (5) A C E B D (5) E B A C D (4) D B C A E (4) A C B E D (4) E A C D B (3) E A C B D (3) D E B C A (3) D B E C A (3) D B C E A (3) B D E C A (3) A C E D B (3) D E C A B (2) D C E A B (2) C A D B E (2) B E A D C (2) B D C A E (2) A E C B D (2) E B A D C (1) E A B C D (1) D C B E A (1) C E A D B (1) C D A B E (1) B D E A C (1) B A E C D (1) B A C E D (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -2 10 2 B -10 0 -6 -4 0 C 2 6 0 6 8 D -10 4 -6 0 -10 E -2 0 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 10 2 B -10 0 -6 -4 0 C 2 6 0 6 8 D -10 4 -6 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 C=17 A=16 E=12 so E is eliminated. Round 2 votes counts: B=31 D=29 A=23 C=17 so C is eliminated. Round 3 votes counts: A=39 B=31 D=30 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:210 E:200 B:190 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 10 2 B -10 0 -6 -4 0 C 2 6 0 6 8 D -10 4 -6 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 10 2 B -10 0 -6 -4 0 C 2 6 0 6 8 D -10 4 -6 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 10 2 B -10 0 -6 -4 0 C 2 6 0 6 8 D -10 4 -6 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 200: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (14) E C B D A (9) D A B E C (5) E C B A D (4) B D C E A (4) E C A B D (3) C E B A D (3) C E A B D (3) B C E D A (3) A D E C B (3) A D B E C (3) E C D B A (2) E C A D B (2) E A C D B (2) D E B C A (2) D B E C A (2) D B A C E (2) D A E B C (2) C B E A D (2) C B A E D (2) B D A C E (2) B C D A E (2) A E D C B (2) A D E B C (2) A C E B D (2) E D C B A (1) E D C A B (1) E D A C B (1) E C D A B (1) E A D C B (1) E A C B D (1) D E B A C (1) D B E A C (1) D B C E A (1) D B A E C (1) C E B D A (1) C B E D A (1) C A E B D (1) B D C A E (1) A E C D B (1) A D C E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -2 8 -4 B -6 0 -2 -8 -4 C 2 2 0 -6 -6 D -8 8 6 0 0 E 4 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.147518 E: 0.852482 Sum of squares = 0.748486889327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.147518 E: 1.000000 A B C D E A 0 6 -2 8 -4 B -6 0 -2 -8 -4 C 2 2 0 -6 -6 D -8 8 6 0 0 E 4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555561531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 D=17 C=13 B=12 so B is eliminated. Round 2 votes counts: A=30 E=28 D=24 C=18 so C is eliminated. Round 3 votes counts: E=41 A=33 D=26 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:204 D:203 C:196 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 8 -4 B -6 0 -2 -8 -4 C 2 2 0 -6 -6 D -8 8 6 0 0 E 4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555561531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 8 -4 B -6 0 -2 -8 -4 C 2 2 0 -6 -6 D -8 8 6 0 0 E 4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555561531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 8 -4 B -6 0 -2 -8 -4 C 2 2 0 -6 -6 D -8 8 6 0 0 E 4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555561531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 201: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) E B A C D (7) A B E D C (7) C D E B A (6) B A E D C (6) E B C A D (4) D C B A E (4) C E B D A (4) C D E A B (4) A E B C D (4) E C B A D (3) D C A B E (3) D B A C E (3) E A B C D (2) D C E B A (2) D C E A B (2) D C B E A (2) D A B C E (2) B E C D A (2) B E A C D (2) B A E C D (2) B A D E C (2) E C A B D (1) E A C B D (1) D A C B E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D B E A (1) C D A E B (1) B E D C A (1) B E D A C (1) B E C A D (1) B D A E C (1) A E C D B (1) A E B D C (1) A D C E B (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -6 -2 2 B 12 0 2 10 -12 C 6 -2 0 0 -4 D 2 -10 0 0 -8 E -2 12 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888881 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -12 -6 -2 2 B 12 0 2 10 -12 C 6 -2 0 0 -4 D 2 -10 0 0 -8 E -2 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=19 E=18 B=18 A=17 so A is eliminated. Round 2 votes counts: D=30 B=27 E=24 C=19 so C is eliminated. Round 3 votes counts: D=42 E=31 B=27 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:206 C:200 D:192 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -6 -2 2 B 12 0 2 10 -12 C 6 -2 0 0 -4 D 2 -10 0 0 -8 E -2 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -2 2 B 12 0 2 10 -12 C 6 -2 0 0 -4 D 2 -10 0 0 -8 E -2 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -2 2 B 12 0 2 10 -12 C 6 -2 0 0 -4 D 2 -10 0 0 -8 E -2 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 202: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) D B E A C (5) C A E B D (5) C A D E B (5) A E B C D (5) D C A B E (4) D B E C A (4) D B C E A (4) E B A C D (3) D A C B E (3) D A B C E (3) C E B A D (3) C D E B A (3) C A E D B (3) B E D C A (3) A C E D B (3) A C E B D (3) A B E C D (3) D A B E C (2) C E D B A (2) C D A E B (2) B E D A C (2) B E A D C (2) B D E A C (2) A E C B D (2) A D C B E (2) A D B E C (2) A C D E B (2) A B D E C (2) E C B D A (1) E C B A D (1) D B A E C (1) C E B D A (1) C E A D B (1) C E A B D (1) B E A C D (1) B D E C A (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 10 0 2 8 B -10 0 -8 -16 6 C 0 8 0 0 14 D -2 16 0 0 8 E -8 -6 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.635208 B: 0.000000 C: 0.364792 D: 0.000000 E: 0.000000 Sum of squares = 0.536562416018 Cumulative probabilities = A: 0.635208 B: 0.635208 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 2 8 B -10 0 -8 -16 6 C 0 8 0 0 14 D -2 16 0 0 8 E -8 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999774 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=26 A=26 B=11 E=5 so E is eliminated. Round 2 votes counts: D=32 C=28 A=26 B=14 so B is eliminated. Round 3 votes counts: D=40 A=32 C=28 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:211 A:210 B:186 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 2 8 B -10 0 -8 -16 6 C 0 8 0 0 14 D -2 16 0 0 8 E -8 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999774 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 2 8 B -10 0 -8 -16 6 C 0 8 0 0 14 D -2 16 0 0 8 E -8 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999774 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 2 8 B -10 0 -8 -16 6 C 0 8 0 0 14 D -2 16 0 0 8 E -8 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999774 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 203: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) D E A C B (8) C B E D A (6) B C E A D (5) A E D B C (5) A B C D E (5) E B C A D (4) D A E C B (4) E A B C D (3) D A C B E (3) C B A D E (3) B C A E D (3) A B E C D (3) E D C B A (2) E D A C B (2) E D A B C (2) E B A C D (2) E A B D C (2) D E C A B (2) D C B A E (2) D A E B C (2) C E B D A (2) C D B A E (2) B C A D E (2) A E B C D (2) A D B E C (2) A D B C E (2) A B D C E (2) E D B C A (1) E B C D A (1) D E A B C (1) D C A B E (1) C D E B A (1) C D B E A (1) B A C E D (1) A E B D C (1) A D C B E (1) Total count = 100 A B C D E A 0 24 24 14 10 B -24 0 20 -6 -12 C -24 -20 0 -8 -16 D -14 6 8 0 6 E -10 12 16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 24 14 10 B -24 0 20 -6 -12 C -24 -20 0 -8 -16 D -14 6 8 0 6 E -10 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 E=19 C=15 B=11 so B is eliminated. Round 2 votes counts: A=33 C=25 D=23 E=19 so E is eliminated. Round 3 votes counts: A=40 D=30 C=30 so D is eliminated. Round 4 votes counts: A=62 C=38 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:236 E:206 D:203 B:189 C:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 24 14 10 B -24 0 20 -6 -12 C -24 -20 0 -8 -16 D -14 6 8 0 6 E -10 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 24 14 10 B -24 0 20 -6 -12 C -24 -20 0 -8 -16 D -14 6 8 0 6 E -10 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 24 14 10 B -24 0 20 -6 -12 C -24 -20 0 -8 -16 D -14 6 8 0 6 E -10 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 204: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (15) D C E A B (8) C D E B A (7) C E D B A (6) B A D C E (5) B A C E D (5) D C E B A (4) C D B E A (4) B A E C D (4) B A D E C (4) E D C A B (3) E C D A B (3) A B E C D (3) E A D C B (2) E A D B C (2) E A C D B (2) C D E A B (2) C B D A E (2) B C A E D (2) B A E D C (2) B A C D E (2) E A B D C (1) E A B C D (1) D E C A B (1) D E A C B (1) D B C A E (1) C E D A B (1) B D A C E (1) B C E A D (1) B C D A E (1) B C A D E (1) A E B D C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 6 10 2 B 4 0 8 6 8 C -6 -8 0 -4 6 D -10 -6 4 0 -10 E -2 -8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 10 2 B 4 0 8 6 8 C -6 -8 0 -4 6 D -10 -6 4 0 -10 E -2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=22 A=21 D=15 E=14 so E is eliminated. Round 2 votes counts: A=29 B=28 C=25 D=18 so D is eliminated. Round 3 votes counts: C=41 A=30 B=29 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:207 E:197 C:194 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 10 2 B 4 0 8 6 8 C -6 -8 0 -4 6 D -10 -6 4 0 -10 E -2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 10 2 B 4 0 8 6 8 C -6 -8 0 -4 6 D -10 -6 4 0 -10 E -2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 10 2 B 4 0 8 6 8 C -6 -8 0 -4 6 D -10 -6 4 0 -10 E -2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 205: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) A D C E B (8) D B E C A (6) B E C D A (5) D E B A C (4) D B E A C (4) D A E C B (4) C E A B D (4) B E D C A (4) A D C B E (4) A C E B D (4) D A B E C (3) B D E C A (3) B C E D A (3) A C E D B (3) A C B D E (3) E C B A D (2) E B D C A (2) D E B C A (2) D E A B C (2) D B A C E (2) D A B C E (2) C B E A D (2) C A B E D (2) A C D E B (2) A C D B E (2) E D B C A (1) E C A B D (1) E B C D A (1) D B C A E (1) D B A E C (1) D A C B E (1) B E C A D (1) B D C E A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 8 16 -20 2 B -8 0 14 -22 2 C -16 -14 0 -28 -10 D 20 22 28 0 28 E -2 -2 10 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 -20 2 B -8 0 14 -22 2 C -16 -14 0 -28 -10 D 20 22 28 0 28 E -2 -2 10 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=27 B=18 C=8 E=7 so E is eliminated. Round 2 votes counts: D=41 A=27 B=21 C=11 so C is eliminated. Round 3 votes counts: D=41 A=34 B=25 so B is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:249 A:203 B:193 E:189 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 16 -20 2 B -8 0 14 -22 2 C -16 -14 0 -28 -10 D 20 22 28 0 28 E -2 -2 10 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -20 2 B -8 0 14 -22 2 C -16 -14 0 -28 -10 D 20 22 28 0 28 E -2 -2 10 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -20 2 B -8 0 14 -22 2 C -16 -14 0 -28 -10 D 20 22 28 0 28 E -2 -2 10 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 206: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (17) E C B A D (14) D B A E C (5) B E C D A (5) B D E C A (5) A C E D B (5) A C E B D (5) D B E C A (4) E C B D A (3) E C A B D (3) D B A C E (3) D A C E B (3) B E C A D (3) A D B C E (3) D A E B C (2) B D A C E (2) A D C B E (2) E C A D B (1) E B C D A (1) D E C A B (1) D B E A C (1) D A C B E (1) D A B E C (1) C E A B D (1) C A E D B (1) B D A E C (1) B C E A D (1) B C A E D (1) B A D C E (1) A D C E B (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 12 -10 14 B 0 0 16 -4 16 C -12 -16 0 -8 0 D 10 4 8 0 10 E -14 -16 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 -10 14 B 0 0 16 -4 16 C -12 -16 0 -8 0 D 10 4 8 0 10 E -14 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=22 B=19 A=19 C=2 so C is eliminated. Round 2 votes counts: D=38 E=23 A=20 B=19 so B is eliminated. Round 3 votes counts: D=46 E=32 A=22 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:214 A:208 C:182 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 12 -10 14 B 0 0 16 -4 16 C -12 -16 0 -8 0 D 10 4 8 0 10 E -14 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -10 14 B 0 0 16 -4 16 C -12 -16 0 -8 0 D 10 4 8 0 10 E -14 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -10 14 B 0 0 16 -4 16 C -12 -16 0 -8 0 D 10 4 8 0 10 E -14 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 207: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) C D B E A (7) E C B A D (5) D A E C B (4) B A E C D (4) A B E D C (4) E C A D B (3) E A B C D (3) D A B C E (3) C E D A B (3) C D E B A (3) B C E A D (3) A E B D C (3) E C A B D (2) E A D C B (2) E A C D B (2) E A C B D (2) E A B D C (2) D C E A B (2) D C A B E (2) D B A C E (2) D A C E B (2) D A B E C (2) C D E A B (2) C D B A E (2) B C D A E (2) B C A D E (2) B A E D C (2) B A D C E (2) A E D B C (2) E B C A D (1) E B A C D (1) D E C A B (1) D E A C B (1) D B C A E (1) D A C B E (1) C E D B A (1) C E B D A (1) C B E D A (1) C B E A D (1) C B D A E (1) B E A C D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -6 -2 0 B 0 0 -16 -12 4 C 6 16 0 6 2 D 2 12 -6 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -2 0 B 0 0 -16 -12 4 C 6 16 0 6 2 D 2 12 -6 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 C=22 B=16 A=11 so A is eliminated. Round 2 votes counts: D=29 E=28 C=22 B=21 so B is eliminated. Round 3 votes counts: E=39 D=32 C=29 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:205 A:196 E:196 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 -2 0 B 0 0 -16 -12 4 C 6 16 0 6 2 D 2 12 -6 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -2 0 B 0 0 -16 -12 4 C 6 16 0 6 2 D 2 12 -6 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -2 0 B 0 0 -16 -12 4 C 6 16 0 6 2 D 2 12 -6 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 208: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) C A D E B (7) C D E A B (6) E D C A B (5) B E C D A (5) B E A D C (5) A D C E B (5) A C D E B (5) C D A E B (4) B E D C A (4) B C E D A (4) B A E D C (4) A B D E C (4) B E D A C (3) A E D B C (3) E D A C B (2) C E D B A (2) C D E B A (2) A E D C B (2) A B C D E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C D A B (1) E B D C A (1) E B D A C (1) E A D B C (1) D C E A B (1) C B E D A (1) C B A D E (1) B E C A D (1) B C D E A (1) B A E C D (1) B A C E D (1) B A C D E (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 18 2 8 2 B -18 0 -8 -18 -20 C -2 8 0 -8 -14 D -8 18 8 0 -2 E -2 20 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 8 2 B -18 0 -8 -18 -20 C -2 8 0 -8 -14 D -8 18 8 0 -2 E -2 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=30 C=23 E=14 D=1 so D is eliminated. Round 2 votes counts: A=32 B=30 C=24 E=14 so E is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 A:215 D:208 C:192 B:168 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 8 2 B -18 0 -8 -18 -20 C -2 8 0 -8 -14 D -8 18 8 0 -2 E -2 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 8 2 B -18 0 -8 -18 -20 C -2 8 0 -8 -14 D -8 18 8 0 -2 E -2 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 8 2 B -18 0 -8 -18 -20 C -2 8 0 -8 -14 D -8 18 8 0 -2 E -2 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 209: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) D A B E C (7) B A D C E (6) D B A C E (5) E C D A B (4) E C A B D (4) D E C A B (4) D B A E C (4) B A C D E (4) A B D C E (4) A B C E D (4) E C D B A (3) E C A D B (3) D B C E A (3) B A C E D (3) E D C A B (2) D C E B A (2) C E D B A (2) C E B D A (2) C E B A D (2) C E A B D (2) B D A C E (2) A B E C D (2) A B D E C (2) E A C D B (1) E A C B D (1) D E A C B (1) D C B E A (1) D B E C A (1) D A E B C (1) C B E D A (1) B D C A E (1) B C A E D (1) A E C B D (1) A D E B C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 -14 0 B 8 0 4 -12 6 C -2 -4 0 -16 -6 D 14 12 16 0 22 E 0 -6 6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -14 0 B 8 0 4 -12 6 C -2 -4 0 -16 -6 D 14 12 16 0 22 E 0 -6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=18 B=17 A=16 C=9 so C is eliminated. Round 2 votes counts: D=40 E=26 B=18 A=16 so A is eliminated. Round 3 votes counts: D=41 B=32 E=27 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:232 B:203 A:190 E:189 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -14 0 B 8 0 4 -12 6 C -2 -4 0 -16 -6 D 14 12 16 0 22 E 0 -6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -14 0 B 8 0 4 -12 6 C -2 -4 0 -16 -6 D 14 12 16 0 22 E 0 -6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -14 0 B 8 0 4 -12 6 C -2 -4 0 -16 -6 D 14 12 16 0 22 E 0 -6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 210: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (12) B C E A D (10) D A E B C (5) D E A B C (4) C B A E D (4) B D C E A (4) E B A D C (3) E A D C B (3) D B A C E (3) D A C E B (3) C A D E B (3) E A C D B (2) D B E A C (2) D A C B E (2) D A B E C (2) C B A D E (2) C A E D B (2) C A D B E (2) B E D A C (2) B E C A D (2) B E A C D (2) B D E C A (2) B C E D A (2) A E D C B (2) A D E C B (2) E D A B C (1) E B D A C (1) E A D B C (1) E A C B D (1) E A B D C (1) D C B A E (1) D B C A E (1) D B A E C (1) C E B A D (1) C D A E B (1) C B E A D (1) C B D A E (1) C A B D E (1) B E A D C (1) B D E A C (1) B C D A E (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 18 -4 6 B -4 0 4 -14 0 C -18 -4 0 -20 -8 D 4 14 20 0 14 E -6 0 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 -4 6 B -4 0 4 -14 0 C -18 -4 0 -20 -8 D 4 14 20 0 14 E -6 0 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=27 C=18 E=13 A=6 so A is eliminated. Round 2 votes counts: D=38 B=27 C=19 E=16 so E is eliminated. Round 3 votes counts: D=45 B=32 C=23 so C is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 A:212 E:194 B:193 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 18 -4 6 B -4 0 4 -14 0 C -18 -4 0 -20 -8 D 4 14 20 0 14 E -6 0 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 -4 6 B -4 0 4 -14 0 C -18 -4 0 -20 -8 D 4 14 20 0 14 E -6 0 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 -4 6 B -4 0 4 -14 0 C -18 -4 0 -20 -8 D 4 14 20 0 14 E -6 0 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 211: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (13) D B A C E (8) C E A D B (8) E C A B D (7) D B C A E (5) C E B D A (5) C E D A B (4) A E C D B (4) D B C E A (3) C E D B A (3) B A D E C (3) A C E D B (3) A B D E C (3) E C B A D (2) D C B E A (2) D B A E C (2) D A B C E (2) B D E C A (2) A E C B D (2) A E B C D (2) A D B E C (2) E C B D A (1) E C A D B (1) E A C B D (1) E A B C D (1) D C E B A (1) D C B A E (1) D A C B E (1) C D E B A (1) B E D A C (1) B E A D C (1) B D E A C (1) B D C A E (1) A D C E B (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 6 -14 10 B 12 0 4 -6 6 C -6 -4 0 -10 0 D 14 6 10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -14 10 B 12 0 4 -6 6 C -6 -4 0 -10 0 D 14 6 10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=22 C=21 A=19 E=13 so E is eliminated. Round 2 votes counts: C=32 D=25 B=22 A=21 so A is eliminated. Round 3 votes counts: C=42 D=29 B=29 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:218 B:208 A:195 C:190 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 6 -14 10 B 12 0 4 -6 6 C -6 -4 0 -10 0 D 14 6 10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -14 10 B 12 0 4 -6 6 C -6 -4 0 -10 0 D 14 6 10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -14 10 B 12 0 4 -6 6 C -6 -4 0 -10 0 D 14 6 10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 212: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A B E D (9) B E D A C (8) B E D C A (6) C A D E B (5) E D B C A (4) D B E A C (4) A D B E C (4) A C D E B (4) D A E B C (3) C A D B E (3) E B C D A (2) D E C B A (2) D E B C A (2) C B E A D (2) C A B D E (2) B E A D C (2) A D E C B (2) A D C E B (2) A C B E D (2) A C B D E (2) A B D E C (2) E D C B A (1) E D B A C (1) E B D C A (1) D E C A B (1) D E A B C (1) D C E B A (1) D A E C B (1) D A C E B (1) D A B E C (1) C D E B A (1) C D E A B (1) C D A E B (1) C B E D A (1) C B A E D (1) C A E B D (1) A D E B C (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 6 -6 0 B -4 0 4 -16 4 C -6 -4 0 -22 -18 D 6 16 22 0 16 E 0 -4 18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -6 0 B -4 0 4 -16 4 C -6 -4 0 -22 -18 D 6 16 22 0 16 E 0 -4 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 A=22 B=16 E=9 so E is eliminated. Round 2 votes counts: D=32 C=27 A=22 B=19 so B is eliminated. Round 3 votes counts: D=47 C=29 A=24 so A is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:202 E:199 B:194 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -6 0 B -4 0 4 -16 4 C -6 -4 0 -22 -18 D 6 16 22 0 16 E 0 -4 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -6 0 B -4 0 4 -16 4 C -6 -4 0 -22 -18 D 6 16 22 0 16 E 0 -4 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -6 0 B -4 0 4 -16 4 C -6 -4 0 -22 -18 D 6 16 22 0 16 E 0 -4 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 213: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (14) C B E A D (10) B C E D A (7) A D E C B (7) E A D C B (6) B C D A E (6) E D A B C (4) C E B A D (4) B D C A E (3) B D A E C (3) B C A D E (3) A E D C B (3) E C A D B (2) D E A B C (2) D B A E C (2) C B A E D (2) B C D E A (2) A E C D B (2) E D B A C (1) E C B D A (1) E C B A D (1) E B C D A (1) E A C D B (1) D B E A C (1) C E A D B (1) C E A B D (1) C B E D A (1) C B A D E (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C D A (1) B D E A C (1) B C E A D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -2 -2 0 B 6 0 6 0 -10 C 2 -6 0 2 -8 D 2 0 -2 0 -4 E 0 10 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.429211 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.570789 Sum of squares = 0.510022166558 Cumulative probabilities = A: 0.429211 B: 0.429211 C: 0.429211 D: 0.429211 E: 1.000000 A B C D E A 0 -6 -2 -2 0 B 6 0 6 0 -10 C 2 -6 0 2 -8 D 2 0 -2 0 -4 E 0 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=22 D=19 E=17 A=14 so A is eliminated. Round 2 votes counts: B=28 D=27 C=23 E=22 so E is eliminated. Round 3 votes counts: D=41 C=30 B=29 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 B:201 D:198 A:195 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -2 0 B 6 0 6 0 -10 C 2 -6 0 2 -8 D 2 0 -2 0 -4 E 0 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -2 0 B 6 0 6 0 -10 C 2 -6 0 2 -8 D 2 0 -2 0 -4 E 0 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -2 0 B 6 0 6 0 -10 C 2 -6 0 2 -8 D 2 0 -2 0 -4 E 0 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 214: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (15) A B E C D (15) D C E B A (13) C D E B A (11) A B E D C (8) E B A C D (6) C E D B A (5) D A B E C (3) B E A D C (3) B A E C D (3) E C B A D (2) D E B A C (2) D C E A B (2) D A B C E (2) E C D B A (1) E B A D C (1) D C A E B (1) D A C B E (1) C E B D A (1) C E B A D (1) C D E A B (1) C A B E D (1) B A E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -8 -16 2 B 0 0 -10 -14 6 C 8 10 0 -6 8 D 16 14 6 0 4 E -2 -6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -16 2 B 0 0 -10 -14 6 C 8 10 0 -6 8 D 16 14 6 0 4 E -2 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=24 C=20 E=10 B=7 so B is eliminated. Round 2 votes counts: D=39 A=28 C=20 E=13 so E is eliminated. Round 3 votes counts: D=39 A=38 C=23 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:210 B:191 E:190 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -16 2 B 0 0 -10 -14 6 C 8 10 0 -6 8 D 16 14 6 0 4 E -2 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -16 2 B 0 0 -10 -14 6 C 8 10 0 -6 8 D 16 14 6 0 4 E -2 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -16 2 B 0 0 -10 -14 6 C 8 10 0 -6 8 D 16 14 6 0 4 E -2 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 215: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) C A E D B (6) A C E D B (6) E C A D B (5) E B D C A (4) D A B E C (4) C E B D A (4) C E A D B (4) C E A B D (4) B C D E A (4) A E D C B (4) A D E B C (4) E C B D A (3) B D E A C (3) A E C D B (3) A D C E B (3) A D B E C (3) D A E B C (2) C E B A D (2) B D E C A (2) B D C A E (2) B D A E C (2) B D A C E (2) A D C B E (2) A D B C E (2) E D C B A (1) E D B A C (1) E D A B C (1) E B C D A (1) D E A B C (1) C B E D A (1) C B E A D (1) C A B E D (1) C A B D E (1) B E D C A (1) B D C E A (1) A D E C B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 18 4 8 12 B -18 0 -6 -20 -22 C -4 6 0 -4 -4 D -8 20 4 0 -6 E -12 22 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 8 12 B -18 0 -6 -20 -22 C -4 6 0 -4 -4 D -8 20 4 0 -6 E -12 22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=24 B=17 E=16 D=13 so D is eliminated. Round 2 votes counts: A=36 C=24 B=23 E=17 so E is eliminated. Round 3 votes counts: A=38 C=33 B=29 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:210 D:205 C:197 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 8 12 B -18 0 -6 -20 -22 C -4 6 0 -4 -4 D -8 20 4 0 -6 E -12 22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 8 12 B -18 0 -6 -20 -22 C -4 6 0 -4 -4 D -8 20 4 0 -6 E -12 22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 8 12 B -18 0 -6 -20 -22 C -4 6 0 -4 -4 D -8 20 4 0 -6 E -12 22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 216: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) C E D A B (8) B A E D C (8) A B D E C (6) D A B E C (5) C D B A E (5) C D B E A (4) C B D A E (4) C E B A D (3) C D A B E (3) C B E A D (3) B A D E C (3) E A D B C (2) E A B D C (2) D E A C B (2) D A E C B (2) D A E B C (2) D A C B E (2) C E B D A (2) B E A D C (2) B C E A D (2) B C A E D (2) B A C D E (2) E D A C B (1) E C B A D (1) E C A B D (1) D E A B C (1) D C E A B (1) D C A E B (1) D C A B E (1) D A B C E (1) C B E D A (1) B E C A D (1) B E A C D (1) B C A D E (1) B A D C E (1) A E D B C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 8 -8 -12 4 B -8 0 -10 -6 16 C 8 10 0 8 14 D 12 6 -8 0 16 E -4 -16 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -12 4 B -8 0 -10 -6 16 C 8 10 0 8 14 D 12 6 -8 0 16 E -4 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=43 B=23 D=18 A=9 E=7 so E is eliminated. Round 2 votes counts: C=45 B=23 D=19 A=13 so A is eliminated. Round 3 votes counts: C=45 B=32 D=23 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:213 A:196 B:196 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -12 4 B -8 0 -10 -6 16 C 8 10 0 8 14 D 12 6 -8 0 16 E -4 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -12 4 B -8 0 -10 -6 16 C 8 10 0 8 14 D 12 6 -8 0 16 E -4 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -12 4 B -8 0 -10 -6 16 C 8 10 0 8 14 D 12 6 -8 0 16 E -4 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 217: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A B C D E (7) E C A B D (6) E B A D C (6) C D A B E (5) B A E D C (5) B A D E C (5) A B D C E (5) E A B C D (4) C D E B A (4) A B E C D (4) E C D A B (3) D B A C E (3) C E D A B (3) B A D C E (3) E D C B A (2) E C D B A (2) C E D B A (2) C A E B D (2) C A D B E (2) A C B D E (2) A B E D C (2) A B C E D (2) E D B C A (1) E D B A C (1) E B D A C (1) E A C B D (1) E A B D C (1) D E B C A (1) D C E B A (1) D C B A E (1) C E A D B (1) C D E A B (1) C D A E B (1) C A D E B (1) A E B C D (1) Total count = 100 A B C D E A 0 24 6 36 18 B -24 0 4 30 8 C -6 -4 0 24 8 D -36 -30 -24 0 0 E -18 -8 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 6 36 18 B -24 0 4 30 8 C -6 -4 0 24 8 D -36 -30 -24 0 0 E -18 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997271 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=28 A=23 B=13 D=6 so D is eliminated. Round 2 votes counts: C=32 E=29 A=23 B=16 so B is eliminated. Round 3 votes counts: A=39 C=32 E=29 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:242 C:211 B:209 E:183 D:155 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 6 36 18 B -24 0 4 30 8 C -6 -4 0 24 8 D -36 -30 -24 0 0 E -18 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997271 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 6 36 18 B -24 0 4 30 8 C -6 -4 0 24 8 D -36 -30 -24 0 0 E -18 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997271 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 6 36 18 B -24 0 4 30 8 C -6 -4 0 24 8 D -36 -30 -24 0 0 E -18 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997271 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 218: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) D C B E A (9) D C A B E (6) E B A C D (5) C D B E A (5) D A C E B (4) D A B E C (4) C B E A D (4) B E D A C (4) B E C A D (4) A E B D C (4) A C E B D (4) D B C E A (3) B E C D A (3) D B E A C (2) D A E B C (2) D A C B E (2) C A E B D (2) C A D E B (2) A E C B D (2) A D E B C (2) A C E D B (2) A C D E B (2) E C B A D (1) E B A D C (1) E A B D C (1) E A B C D (1) D C A E B (1) D B E C A (1) C E B A D (1) C D A E B (1) C B D A E (1) B E A C D (1) B D E C A (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 8 10 2 6 B -8 0 0 0 0 C -10 0 0 2 -2 D -2 0 -2 0 -4 E -6 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 2 6 B -8 0 0 0 0 C -10 0 0 2 -2 D -2 0 -2 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=28 C=16 B=13 E=9 so E is eliminated. Round 2 votes counts: D=34 A=30 B=19 C=17 so C is eliminated. Round 3 votes counts: D=40 A=34 B=26 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:200 B:196 D:196 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 2 6 B -8 0 0 0 0 C -10 0 0 2 -2 D -2 0 -2 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 2 6 B -8 0 0 0 0 C -10 0 0 2 -2 D -2 0 -2 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 2 6 B -8 0 0 0 0 C -10 0 0 2 -2 D -2 0 -2 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 219: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) E D A C B (11) B C A D E (9) E D C A B (8) C B D E A (5) A B C D E (5) E D C B A (4) A E D B C (4) C D E B A (3) B C D E A (3) A E B D C (3) A B E D C (3) D E C B A (2) D C E B A (2) C D B E A (2) C B D A E (2) B C D A E (2) B A E C D (2) B A C E D (2) A C D E B (2) A B E C D (2) E D B C A (1) E D B A C (1) E B A D C (1) E A D C B (1) D E C A B (1) D E A C B (1) C D A E B (1) B E D C A (1) B E D A C (1) B A E D C (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 8 -2 4 B 12 0 8 10 6 C -8 -8 0 6 2 D 2 -10 -6 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 8 -2 4 B 12 0 8 10 6 C -8 -8 0 6 2 D 2 -10 -6 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=27 A=21 C=13 D=6 so D is eliminated. Round 2 votes counts: B=33 E=31 A=21 C=15 so C is eliminated. Round 3 votes counts: B=42 E=36 A=22 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:199 C:196 D:195 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 8 -2 4 B 12 0 8 10 6 C -8 -8 0 6 2 D 2 -10 -6 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 8 -2 4 B 12 0 8 10 6 C -8 -8 0 6 2 D 2 -10 -6 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 8 -2 4 B 12 0 8 10 6 C -8 -8 0 6 2 D 2 -10 -6 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 220: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) B C A D E (8) E D A C B (7) B E A D C (7) B A E D C (7) A D E C B (6) E D C A B (4) E D A B C (4) E A D B C (3) C D A B E (3) C B A D E (3) C A D E B (3) B C D A E (3) D A E C B (2) C D E A B (2) B E D C A (2) B C A E D (2) B A C E D (2) E B D A C (1) E B C D A (1) E B A D C (1) E A D C B (1) E A B D C (1) D E C A B (1) D C E A B (1) D C A E B (1) D A C E B (1) C E D A B (1) C D B A E (1) C B D E A (1) C B D A E (1) C A D B E (1) B E D A C (1) B E C D A (1) B E C A D (1) B C E A D (1) B A C D E (1) A D C E B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -4 2 16 B -10 0 -4 -8 -6 C 4 4 0 -4 -2 D -2 8 4 0 4 E -16 6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 2 16 B -10 0 -4 -8 -6 C 4 4 0 -4 -2 D -2 8 4 0 4 E -16 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000285 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=26 E=23 A=9 D=6 so D is eliminated. Round 2 votes counts: B=36 C=28 E=24 A=12 so A is eliminated. Round 3 votes counts: B=37 E=32 C=31 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 D:207 C:201 E:194 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 2 16 B -10 0 -4 -8 -6 C 4 4 0 -4 -2 D -2 8 4 0 4 E -16 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000285 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 2 16 B -10 0 -4 -8 -6 C 4 4 0 -4 -2 D -2 8 4 0 4 E -16 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000285 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 2 16 B -10 0 -4 -8 -6 C 4 4 0 -4 -2 D -2 8 4 0 4 E -16 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000285 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 221: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) E A C B D (9) D B C E A (7) A E D B C (5) A B C E D (5) A B C D E (5) E D C B A (4) E C A B D (3) D E C B A (3) D C B E A (3) D B A C E (3) B D C A E (3) A D E B C (3) E C D B A (2) E C B A D (2) E A D C B (2) D E A B C (2) D B C A E (2) D A E B C (2) D A B C E (2) C E B D A (2) C B E D A (2) B C D A E (2) E D A C B (1) E C B D A (1) E A C D B (1) D E B C A (1) D E B A C (1) D A B E C (1) C B E A D (1) C B D E A (1) C B A E D (1) B C D E A (1) B C A D E (1) B A D C E (1) B A C D E (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 16 8 2 B -10 0 -2 10 -12 C -16 2 0 8 -10 D -8 -10 -8 0 -10 E -2 12 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 8 2 B -10 0 -2 10 -12 C -16 2 0 8 -10 D -8 -10 -8 0 -10 E -2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998099 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=27 E=25 B=9 C=7 so C is eliminated. Round 2 votes counts: A=32 E=27 D=27 B=14 so B is eliminated. Round 3 votes counts: A=36 D=34 E=30 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:215 B:193 C:192 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 8 2 B -10 0 -2 10 -12 C -16 2 0 8 -10 D -8 -10 -8 0 -10 E -2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998099 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 8 2 B -10 0 -2 10 -12 C -16 2 0 8 -10 D -8 -10 -8 0 -10 E -2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998099 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 8 2 B -10 0 -2 10 -12 C -16 2 0 8 -10 D -8 -10 -8 0 -10 E -2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998099 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 222: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) A D E C B (7) E D C B A (5) E B D C A (5) B C E D A (5) E D B C A (4) E B C D A (4) C D B E A (4) A D C E B (4) D E C B A (3) C B D E A (3) A E D B C (3) A D C B E (3) A C D B E (3) E A D C B (2) D E A C B (2) C D B A E (2) C B D A E (2) B E C A D (2) B C D E A (2) B C A E D (2) A E D C B (2) A E B D C (2) A C B D E (2) A B C E D (2) A B C D E (2) E D A C B (1) E B C A D (1) E B A D C (1) D C E B A (1) D C A B E (1) D A E C B (1) D A C E B (1) C D A B E (1) C A B D E (1) B C E A D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -18 -18 -12 -10 B 18 0 -2 0 0 C 18 2 0 4 -12 D 12 0 -4 0 -6 E 10 0 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.280459 C: 0.000000 D: 0.000000 E: 0.719541 Sum of squares = 0.596396198361 Cumulative probabilities = A: 0.000000 B: 0.280459 C: 0.280459 D: 0.280459 E: 1.000000 A B C D E A 0 -18 -18 -12 -10 B 18 0 -2 0 0 C 18 2 0 4 -12 D 12 0 -4 0 -6 E 10 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=24 E=23 C=13 D=9 so D is eliminated. Round 2 votes counts: A=33 E=28 B=24 C=15 so C is eliminated. Round 3 votes counts: A=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:208 C:206 D:201 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -18 -12 -10 B 18 0 -2 0 0 C 18 2 0 4 -12 D 12 0 -4 0 -6 E 10 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -18 -12 -10 B 18 0 -2 0 0 C 18 2 0 4 -12 D 12 0 -4 0 -6 E 10 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -18 -12 -10 B 18 0 -2 0 0 C 18 2 0 4 -12 D 12 0 -4 0 -6 E 10 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 223: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (19) A B C D E (15) D E C B A (12) C B A D E (9) E D C B A (8) E D A B C (7) B A C D E (5) D E B A C (4) C D E B A (4) E D C A B (2) E D A C B (2) E C A B D (2) C E D B A (2) C D B A E (2) A E B D C (2) E C D A B (1) D E A B C (1) D C E B A (1) C E B A D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 12 8 6 B -4 0 8 8 2 C -12 -8 0 20 16 D -8 -8 -20 0 6 E -6 -2 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 8 6 B -4 0 8 8 2 C -12 -8 0 20 16 D -8 -8 -20 0 6 E -6 -2 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=22 D=18 C=18 B=5 so B is eliminated. Round 2 votes counts: A=42 E=22 D=18 C=18 so D is eliminated. Round 3 votes counts: A=42 E=39 C=19 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:208 B:207 D:185 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 8 6 B -4 0 8 8 2 C -12 -8 0 20 16 D -8 -8 -20 0 6 E -6 -2 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 8 6 B -4 0 8 8 2 C -12 -8 0 20 16 D -8 -8 -20 0 6 E -6 -2 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 8 6 B -4 0 8 8 2 C -12 -8 0 20 16 D -8 -8 -20 0 6 E -6 -2 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 224: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) C A E D B (8) C D A E B (6) D B E C A (5) B D E A C (5) A C E B D (5) D B C A E (4) C D E A B (4) B A E C D (4) A B E C D (4) D C E B A (3) D B C E A (3) B E D A C (3) B A C E D (3) E D C A B (2) E D B A C (2) E A C B D (2) D C B A E (2) D B E A C (2) C E A D B (2) C A D E B (2) C A D B E (2) C A B D E (2) B E A D C (2) B A E D C (2) E B A D C (1) D E C B A (1) D E C A B (1) C E D A B (1) C D A B E (1) C A E B D (1) B E A C D (1) B D A E C (1) B D A C E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -18 -12 2 B -10 0 -10 -20 0 C 18 10 0 2 24 D 12 20 -2 0 12 E -2 0 -24 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -18 -12 2 B -10 0 -10 -20 0 C 18 10 0 2 24 D 12 20 -2 0 12 E -2 0 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=29 B=22 A=12 E=7 so E is eliminated. Round 2 votes counts: D=34 C=29 B=23 A=14 so A is eliminated. Round 3 votes counts: C=37 D=34 B=29 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:221 A:191 E:181 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -18 -12 2 B -10 0 -10 -20 0 C 18 10 0 2 24 D 12 20 -2 0 12 E -2 0 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -18 -12 2 B -10 0 -10 -20 0 C 18 10 0 2 24 D 12 20 -2 0 12 E -2 0 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -18 -12 2 B -10 0 -10 -20 0 C 18 10 0 2 24 D 12 20 -2 0 12 E -2 0 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 225: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (11) C B A D E (9) A E D B C (9) E A D B C (8) C E D B A (6) E D A B C (5) D B E A C (4) C B D A E (4) C A E B D (4) C A B E D (3) B D A E C (3) E A D C B (2) C E B D A (2) C B E D A (2) C B A E D (2) B D C E A (2) B C D E A (2) B C D A E (2) A E C D B (2) E D C B A (1) E D A C B (1) E A C D B (1) D E A B C (1) C E D A B (1) C E A D B (1) C D E B A (1) C A E D B (1) B D C A E (1) B D A C E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E D C B (1) A D B E C (1) A C E D B (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -10 0 -2 B 12 0 -12 6 4 C 10 12 0 14 18 D 0 -6 -14 0 -8 E 2 -4 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 0 -2 B 12 0 -12 6 4 C 10 12 0 14 18 D 0 -6 -14 0 -8 E 2 -4 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=47 E=18 A=16 B=14 D=5 so D is eliminated. Round 2 votes counts: C=47 E=19 B=18 A=16 so A is eliminated. Round 3 votes counts: C=48 E=31 B=21 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:205 E:194 A:188 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 0 -2 B 12 0 -12 6 4 C 10 12 0 14 18 D 0 -6 -14 0 -8 E 2 -4 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 0 -2 B 12 0 -12 6 4 C 10 12 0 14 18 D 0 -6 -14 0 -8 E 2 -4 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 0 -2 B 12 0 -12 6 4 C 10 12 0 14 18 D 0 -6 -14 0 -8 E 2 -4 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 226: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (11) D E B A C (10) C A B E D (9) E D A C B (8) E D B C A (5) D E A C B (5) B C A E D (4) E D B A C (3) D A C B E (3) B A C D E (3) A C B D E (3) E D C B A (2) E C B A D (2) E C A B D (2) E B C A D (2) D B E C A (2) D B E A C (2) D B A C E (2) C A E B D (2) B D E C A (2) B D A C E (2) A C E B D (2) E C A D B (1) E B D C A (1) E A D C B (1) E A C D B (1) D E A B C (1) D B A E C (1) D A B C E (1) B E C A D (1) B D C A E (1) B C D A E (1) A D C E B (1) A C E D B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 4 -4 -2 B 14 0 10 -2 0 C -4 -10 0 -6 -4 D 4 2 6 0 4 E 2 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -4 -2 B 14 0 10 -2 0 C -4 -10 0 -6 -4 D 4 2 6 0 4 E 2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 B=25 C=11 A=9 so A is eliminated. Round 2 votes counts: E=28 D=28 B=25 C=19 so C is eliminated. Round 3 votes counts: B=38 E=33 D=29 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:208 E:201 A:192 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 4 -4 -2 B 14 0 10 -2 0 C -4 -10 0 -6 -4 D 4 2 6 0 4 E 2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -4 -2 B 14 0 10 -2 0 C -4 -10 0 -6 -4 D 4 2 6 0 4 E 2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -4 -2 B 14 0 10 -2 0 C -4 -10 0 -6 -4 D 4 2 6 0 4 E 2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 227: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) C B A E D (6) E A D B C (5) D E A C B (5) D E A B C (5) D C B E A (5) D A E B C (5) E A B C D (4) D E C A B (4) D C B A E (4) C B D A E (4) D C E B A (3) C D B E A (3) C B A D E (3) A E D B C (3) A E B D C (3) A B E C D (3) E C D A B (2) E A B D C (2) D C E A B (2) D B C A E (2) D B A E C (2) C D B A E (2) C B D E A (2) B A E C D (2) A E B C D (2) E D A B C (1) E C A D B (1) E A D C B (1) E A C B D (1) D A B E C (1) C E B A D (1) C B E D A (1) B C A E D (1) B A C E D (1) Total count = 100 A B C D E A 0 0 -8 -6 -12 B 0 0 -16 -12 0 C 8 16 0 -6 -4 D 6 12 6 0 4 E 12 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -6 -12 B 0 0 -16 -12 0 C 8 16 0 -6 -4 D 6 12 6 0 4 E 12 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=30 E=17 A=11 B=4 so B is eliminated. Round 2 votes counts: D=38 C=31 E=17 A=14 so A is eliminated. Round 3 votes counts: D=38 C=32 E=30 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:207 E:206 A:187 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -6 -12 B 0 0 -16 -12 0 C 8 16 0 -6 -4 D 6 12 6 0 4 E 12 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -6 -12 B 0 0 -16 -12 0 C 8 16 0 -6 -4 D 6 12 6 0 4 E 12 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -6 -12 B 0 0 -16 -12 0 C 8 16 0 -6 -4 D 6 12 6 0 4 E 12 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 228: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) B A C E D (9) B A C D E (9) E D C B A (5) E D B C A (4) E C D A B (4) E B D A C (4) D C E A B (3) C A D B E (3) A B C D E (3) E D B A C (2) E B A D C (2) D E B A C (2) D C A E B (2) D A C B E (2) C E D A B (2) C E B A D (2) C B A E D (2) C A B E D (2) C A B D E (2) B E A C D (2) B A E C D (2) B A D E C (2) B A D C E (2) A C B D E (2) E C D B A (1) E C B D A (1) E C B A D (1) E B C D A (1) E B C A D (1) E B A C D (1) D E C A B (1) D E A B C (1) D B A E C (1) D A B E C (1) C D E A B (1) C D A B E (1) B E A D C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -14 0 0 -6 B 14 0 2 2 -4 C 0 -2 0 6 -2 D 0 -2 -6 0 -20 E 6 4 2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 0 0 -6 B 14 0 2 2 -4 C 0 -2 0 6 -2 D 0 -2 -6 0 -20 E 6 4 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=27 C=15 D=13 A=7 so A is eliminated. Round 2 votes counts: E=38 B=30 C=18 D=14 so D is eliminated. Round 3 votes counts: E=42 B=33 C=25 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:207 C:201 A:190 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 0 0 -6 B 14 0 2 2 -4 C 0 -2 0 6 -2 D 0 -2 -6 0 -20 E 6 4 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 0 -6 B 14 0 2 2 -4 C 0 -2 0 6 -2 D 0 -2 -6 0 -20 E 6 4 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 0 -6 B 14 0 2 2 -4 C 0 -2 0 6 -2 D 0 -2 -6 0 -20 E 6 4 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 229: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (12) D B E C A (11) D A C E B (8) E B C A D (7) B E C A D (7) E C B A D (6) E C A B D (6) C A E B D (5) B D E C A (5) A C E D B (4) A C E B D (4) D C A E B (3) B D E A C (3) A D C E B (3) A C D E B (3) D B E A C (2) C E A B D (2) B E D C A (2) B E C D A (2) B E A C D (2) D E C B A (1) D A B C E (1) B D A E C (1) Total count = 100 A B C D E A 0 -22 -14 -2 -12 B 22 0 10 4 -4 C 14 -10 0 -4 -10 D 2 -4 4 0 6 E 12 4 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775539 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 A B C D E A 0 -22 -14 -2 -12 B 22 0 10 4 -4 C 14 -10 0 -4 -10 D 2 -4 4 0 6 E 12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775504 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=22 E=19 A=14 C=7 so C is eliminated. Round 2 votes counts: D=38 B=22 E=21 A=19 so A is eliminated. Round 3 votes counts: D=44 E=34 B=22 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:216 E:210 D:204 C:195 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -14 -2 -12 B 22 0 10 4 -4 C 14 -10 0 -4 -10 D 2 -4 4 0 6 E 12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775504 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 -2 -12 B 22 0 10 4 -4 C 14 -10 0 -4 -10 D 2 -4 4 0 6 E 12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775504 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 -2 -12 B 22 0 10 4 -4 C 14 -10 0 -4 -10 D 2 -4 4 0 6 E 12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775504 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 230: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) C A E B D (11) D B E A C (10) C A D E B (9) B E D C A (7) A C D B E (7) B E D A C (6) E B C A D (5) A C D E B (5) D A C B E (4) D A B C E (4) D B A E C (3) A D C B E (3) E C B A D (2) C E A B D (2) C A E D B (2) B E C A D (2) E B C D A (1) D E B A C (1) D A C E B (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -6 2 4 B -2 0 6 2 -2 C 6 -6 0 -2 2 D -2 -2 2 0 -4 E -4 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.38775510202 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 2 4 B -2 0 6 2 -2 C 6 -6 0 -2 2 D -2 -2 2 0 -4 E -4 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102044 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 E=20 A=18 B=15 so B is eliminated. Round 2 votes counts: E=35 C=24 D=23 A=18 so A is eliminated. Round 3 votes counts: C=39 E=35 D=26 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:202 A:201 C:200 E:200 D:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -6 2 4 B -2 0 6 2 -2 C 6 -6 0 -2 2 D -2 -2 2 0 -4 E -4 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102044 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 2 4 B -2 0 6 2 -2 C 6 -6 0 -2 2 D -2 -2 2 0 -4 E -4 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102044 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 2 4 B -2 0 6 2 -2 C 6 -6 0 -2 2 D -2 -2 2 0 -4 E -4 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102044 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 231: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (14) D A B C E (11) E C B D A (8) C E B D A (8) A D E B C (6) A D B E C (5) D B A C E (4) E C D A B (3) E C A B D (3) E A C D B (3) C B E D A (3) B D A C E (3) B A D C E (3) E A D C B (2) E A C B D (2) D A B E C (2) C E D B A (2) A D B C E (2) E D C A B (1) E C A D B (1) E A B C D (1) D E C A B (1) D E A C B (1) D C E A B (1) D C B E A (1) D B C A E (1) D A E C B (1) D A E B C (1) C E B A D (1) B D C A E (1) B A C D E (1) A E D B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 -6 -12 B 0 0 -12 0 -22 C -2 12 0 0 -14 D 6 0 0 0 -8 E 12 22 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 -6 -12 B 0 0 -12 0 -22 C -2 12 0 0 -14 D 6 0 0 0 -8 E 12 22 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=24 A=16 C=14 B=8 so B is eliminated. Round 2 votes counts: E=38 D=28 A=20 C=14 so C is eliminated. Round 3 votes counts: E=52 D=28 A=20 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:228 D:199 C:198 A:192 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -6 -12 B 0 0 -12 0 -22 C -2 12 0 0 -14 D 6 0 0 0 -8 E 12 22 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -6 -12 B 0 0 -12 0 -22 C -2 12 0 0 -14 D 6 0 0 0 -8 E 12 22 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -6 -12 B 0 0 -12 0 -22 C -2 12 0 0 -14 D 6 0 0 0 -8 E 12 22 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 232: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) E C A B D (7) B D A C E (7) D B A C E (5) E D C A B (4) D B E A C (4) B A C D E (4) E C A D B (3) D E A C B (3) D A B E C (3) C A B E D (3) B A D C E (3) B A C E D (3) A C B E D (3) E C D A B (2) D E B C A (2) D E B A C (2) D B E C A (2) D A C E B (2) D A B C E (2) C A E B D (2) B E C D A (2) B E C A D (2) B C E A D (2) A C E D B (2) E D C B A (1) E B D C A (1) E B C A D (1) D E A B C (1) D B A E C (1) D A E C B (1) C E A B D (1) C B E A D (1) B E D C A (1) B D E C A (1) B C A E D (1) A D B C E (1) A C E B D (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 8 6 0 B 6 0 8 14 12 C -8 -8 0 4 -2 D -6 -14 -4 0 -2 E 0 -12 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 6 0 B 6 0 8 14 12 C -8 -8 0 4 -2 D -6 -14 -4 0 -2 E 0 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=26 A=13 C=7 so C is eliminated. Round 2 votes counts: D=28 E=27 B=27 A=18 so A is eliminated. Round 3 votes counts: B=37 E=32 D=31 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:204 E:196 C:193 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 6 0 B 6 0 8 14 12 C -8 -8 0 4 -2 D -6 -14 -4 0 -2 E 0 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 6 0 B 6 0 8 14 12 C -8 -8 0 4 -2 D -6 -14 -4 0 -2 E 0 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 6 0 B 6 0 8 14 12 C -8 -8 0 4 -2 D -6 -14 -4 0 -2 E 0 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 233: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) E C A D B (5) E A C B D (5) E A B D C (5) B A E D C (5) A E B C D (5) A B D C E (5) D C B A E (4) C D E B A (4) C D B E A (4) B D C A E (4) E A B C D (3) B D A C E (3) B A D E C (3) A E B D C (3) A B E D C (3) E D C B A (2) E A C D B (2) C E A D B (2) C D B A E (2) B E D A C (2) B D E C A (2) B D A E C (2) A E C B D (2) A C D B E (2) A B E C D (2) E C D B A (1) E C D A B (1) E B D A C (1) D C E B A (1) D B C E A (1) D B C A E (1) C E D B A (1) C E D A B (1) C D E A B (1) C A D E B (1) B E A D C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 12 12 -2 B 0 0 4 14 6 C -12 -4 0 -10 -12 D -12 -14 10 0 -6 E 2 -6 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.251851 B: 0.748149 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.623155877081 Cumulative probabilities = A: 0.251851 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 12 -2 B 0 0 4 14 6 C -12 -4 0 -10 -12 D -12 -14 10 0 -6 E 2 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=24 B=22 C=16 D=13 so D is eliminated. Round 2 votes counts: C=27 E=25 B=24 A=24 so B is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 A:211 E:207 D:189 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 12 12 -2 B 0 0 4 14 6 C -12 -4 0 -10 -12 D -12 -14 10 0 -6 E 2 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 12 -2 B 0 0 4 14 6 C -12 -4 0 -10 -12 D -12 -14 10 0 -6 E 2 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 12 -2 B 0 0 4 14 6 C -12 -4 0 -10 -12 D -12 -14 10 0 -6 E 2 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 234: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) E A B D C (9) C D A B E (7) E B A C D (5) D C A B E (5) D A C B E (5) A D E B C (5) E B A D C (4) D A B C E (4) B E A C D (4) A E B D C (4) E B C A D (3) D A E B C (3) D A C E B (3) C D B E A (3) C D A E B (3) C B E D A (3) A E D B C (3) D C A E B (2) C E B A D (2) B E A D C (2) B C E A D (2) E A D B C (1) D A E C B (1) D A B E C (1) C D E B A (1) C B E A D (1) C B D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 14 10 -8 18 B -14 0 0 -20 0 C -10 0 0 -6 8 D 8 20 6 0 14 E -18 0 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 -8 18 B -14 0 0 -20 0 C -10 0 0 -6 8 D 8 20 6 0 14 E -18 0 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=24 E=22 A=13 B=8 so B is eliminated. Round 2 votes counts: C=35 E=28 D=24 A=13 so A is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:224 A:217 C:196 B:183 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 10 -8 18 B -14 0 0 -20 0 C -10 0 0 -6 8 D 8 20 6 0 14 E -18 0 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -8 18 B -14 0 0 -20 0 C -10 0 0 -6 8 D 8 20 6 0 14 E -18 0 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -8 18 B -14 0 0 -20 0 C -10 0 0 -6 8 D 8 20 6 0 14 E -18 0 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 235: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C E B D A (7) A B D C E (7) C B D E A (6) E A D C B (5) B D A C E (5) B A D C E (5) E C A D B (4) E A C D B (3) C E D B A (3) C B E D A (3) A E D B C (3) A E B D C (3) A D B E C (3) A B C E D (3) E C D A B (2) E C A B D (2) D E C B A (2) D A B E C (2) C D E B A (2) B D C A E (2) B C D A E (2) B C A D E (2) A D B C E (2) A B D E C (2) A B C D E (2) E C B A D (1) E A C B D (1) D E A B C (1) D C E B A (1) D B C E A (1) C E B A D (1) C B E A D (1) B C D E A (1) B A C D E (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 0 6 -8 B 6 0 -4 16 0 C 0 4 0 10 14 D -6 -16 -10 0 0 E 8 0 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.301730 B: 0.000000 C: 0.698270 D: 0.000000 E: 0.000000 Sum of squares = 0.578621984301 Cumulative probabilities = A: 0.301730 B: 0.301730 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 6 -8 B 6 0 -4 16 0 C 0 4 0 10 14 D -6 -16 -10 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000509 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 C=23 B=18 D=7 so D is eliminated. Round 2 votes counts: A=29 E=28 C=24 B=19 so B is eliminated. Round 3 votes counts: A=40 C=32 E=28 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:214 B:209 E:197 A:196 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 6 -8 B 6 0 -4 16 0 C 0 4 0 10 14 D -6 -16 -10 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000509 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 6 -8 B 6 0 -4 16 0 C 0 4 0 10 14 D -6 -16 -10 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000509 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 6 -8 B 6 0 -4 16 0 C 0 4 0 10 14 D -6 -16 -10 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000509 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 236: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) E C D A B (7) A B C D E (7) D B E A C (6) B D A E C (6) A C B D E (6) C E A B D (5) D E B C A (4) D B E C A (4) B D E A C (4) A C B E D (4) E D B C A (3) C E A D B (3) B A C D E (3) E D C B A (2) E C D B A (2) D E B A C (2) C E B D A (2) C E B A D (2) C A B D E (2) B D E C A (2) B A D C E (2) A B D C E (2) E D B A C (1) E D A B C (1) E A D C B (1) C E D B A (1) C B D E A (1) C B A E D (1) C A E D B (1) C A B E D (1) B D A C E (1) B C A D E (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -4 2 -6 B 0 0 0 24 8 C 4 0 0 16 8 D -2 -24 -16 0 8 E 6 -8 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.394453 C: 0.605547 D: 0.000000 E: 0.000000 Sum of squares = 0.522280406862 Cumulative probabilities = A: 0.000000 B: 0.394453 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 -6 B 0 0 0 24 8 C 4 0 0 16 8 D -2 -24 -16 0 8 E 6 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.500002 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997713 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=21 B=19 E=17 D=16 so D is eliminated. Round 2 votes counts: B=29 C=27 E=23 A=21 so A is eliminated. Round 3 votes counts: B=39 C=38 E=23 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:214 A:196 E:191 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 2 -6 B 0 0 0 24 8 C 4 0 0 16 8 D -2 -24 -16 0 8 E 6 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.500002 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997713 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 -6 B 0 0 0 24 8 C 4 0 0 16 8 D -2 -24 -16 0 8 E 6 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.500002 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997713 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 -6 B 0 0 0 24 8 C 4 0 0 16 8 D -2 -24 -16 0 8 E 6 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.500002 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997713 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 237: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) B D C E A (8) C E B D A (7) C B D E A (7) C A B D E (7) A E D B C (7) D B E A C (6) C E A B D (6) C A E B D (5) E D B A C (4) A C B D E (4) D B E C A (3) E D A B C (2) E C A D B (2) E A D B C (2) E A C D B (2) B D E C A (2) B D C A E (2) A E C D B (2) A D E B C (2) E D B C A (1) E C D B A (1) D B A E C (1) D B A C E (1) C A B E D (1) A E D C B (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -4 10 -6 B -14 0 -12 18 -8 C 4 12 0 10 22 D -10 -18 -10 0 -6 E 6 8 -22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 10 -6 B -14 0 -12 18 -8 C 4 12 0 10 22 D -10 -18 -10 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=30 E=14 B=12 D=11 so D is eliminated. Round 2 votes counts: C=33 A=30 B=23 E=14 so E is eliminated. Round 3 votes counts: C=36 A=36 B=28 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:207 E:199 B:192 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 10 -6 B -14 0 -12 18 -8 C 4 12 0 10 22 D -10 -18 -10 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 10 -6 B -14 0 -12 18 -8 C 4 12 0 10 22 D -10 -18 -10 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 10 -6 B -14 0 -12 18 -8 C 4 12 0 10 22 D -10 -18 -10 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 238: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (19) B A D E C (16) B D A E C (11) B A D C E (5) B A C D E (5) E D C A B (4) C E A D B (4) C A B E D (4) A C B E D (4) C A E D B (3) E C D A B (2) D E A B C (2) C E D B A (2) C E A B D (2) C A E B D (2) B D E A C (2) E D C B A (1) E C D B A (1) E B C D A (1) D E C B A (1) D E B C A (1) D E B A C (1) D B E C A (1) D B E A C (1) B C E D A (1) B A C E D (1) A C E D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 -2 8 B 2 0 0 12 6 C -2 0 0 8 10 D 2 -12 -8 0 -6 E -8 -6 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.815858 C: 0.184142 D: 0.000000 E: 0.000000 Sum of squares = 0.699533077161 Cumulative probabilities = A: 0.000000 B: 0.815858 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 8 B 2 0 0 12 6 C -2 0 0 8 10 D 2 -12 -8 0 -6 E -8 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500360 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258712 Cumulative probabilities = A: 0.000000 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=36 E=9 D=7 A=7 so D is eliminated. Round 2 votes counts: B=43 C=36 E=14 A=7 so A is eliminated. Round 3 votes counts: B=45 C=41 E=14 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:208 A:203 E:191 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 8 B 2 0 0 12 6 C -2 0 0 8 10 D 2 -12 -8 0 -6 E -8 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500360 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258712 Cumulative probabilities = A: 0.000000 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 8 B 2 0 0 12 6 C -2 0 0 8 10 D 2 -12 -8 0 -6 E -8 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500360 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258712 Cumulative probabilities = A: 0.000000 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 8 B 2 0 0 12 6 C -2 0 0 8 10 D 2 -12 -8 0 -6 E -8 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500360 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258712 Cumulative probabilities = A: 0.000000 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 239: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (13) A E C B D (8) D A E B C (6) A E D B C (6) D B E C A (5) C B E D A (5) A C B E D (5) D E B C A (4) D B C E A (4) E A B C D (3) C B D E A (3) C A B E D (3) A D E B C (3) E C B A D (2) E B C D A (2) E B A C D (2) D C B E A (2) D C B A E (2) D B C A E (2) B E D C A (2) B C E D A (2) A D C B E (2) E D B C A (1) E B D C A (1) D E B A C (1) D A C B E (1) C D A B E (1) C B A E D (1) C A B D E (1) B E C D A (1) B C D E A (1) A E B D C (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -16 8 -8 B 12 0 -6 14 14 C 16 6 0 14 4 D -8 -14 -14 0 -20 E 8 -14 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 8 -8 B 12 0 -6 14 14 C 16 6 0 14 4 D -8 -14 -14 0 -20 E 8 -14 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=27 C=27 E=11 B=6 so B is eliminated. Round 2 votes counts: C=30 A=29 D=27 E=14 so E is eliminated. Round 3 votes counts: C=35 A=34 D=31 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:217 E:205 A:186 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -16 8 -8 B 12 0 -6 14 14 C 16 6 0 14 4 D -8 -14 -14 0 -20 E 8 -14 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 8 -8 B 12 0 -6 14 14 C 16 6 0 14 4 D -8 -14 -14 0 -20 E 8 -14 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 8 -8 B 12 0 -6 14 14 C 16 6 0 14 4 D -8 -14 -14 0 -20 E 8 -14 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 240: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A C B D E (8) C A D E B (6) D E B C A (5) B E D A C (5) A B C D E (5) E B D A C (4) D C E A B (4) B A E D C (4) D E C B A (3) C A E D B (3) C A D B E (3) B A E C D (3) A C B E D (3) E D C A B (2) E C D A B (2) D B E C A (2) D B E A C (2) C D E A B (2) C D A E B (2) B E A D C (2) B D E A C (2) B D A C E (2) B A D E C (2) B A D C E (2) A B C E D (2) E C A D B (1) E B A D C (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A E B (1) D C A B E (1) C E D A B (1) C D A B E (1) B D A E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 4 -2 2 B 0 0 10 0 4 C -4 -10 0 -8 -4 D 2 0 8 0 10 E -2 -4 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.351260 C: 0.000000 D: 0.648739 E: 0.000000 Sum of squares = 0.544246836091 Cumulative probabilities = A: 0.000000 B: 0.351261 C: 0.351261 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -2 2 B 0 0 10 0 4 C -4 -10 0 -8 -4 D 2 0 8 0 10 E -2 -4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=20 A=20 D=19 C=18 so C is eliminated. Round 2 votes counts: A=32 D=24 B=23 E=21 so E is eliminated. Round 3 votes counts: D=37 A=34 B=29 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:207 A:202 E:194 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 -2 2 B 0 0 10 0 4 C -4 -10 0 -8 -4 D 2 0 8 0 10 E -2 -4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 2 B 0 0 10 0 4 C -4 -10 0 -8 -4 D 2 0 8 0 10 E -2 -4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 2 B 0 0 10 0 4 C -4 -10 0 -8 -4 D 2 0 8 0 10 E -2 -4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 241: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) C A D E B (6) B E D C A (6) A C D E B (6) A C D B E (6) E C D B A (5) C D E A B (4) E B A C D (3) C E D A B (3) B A E D C (3) A B D C E (3) E A B C D (2) D C E B A (2) D C A B E (2) C A E D B (2) B E D A C (2) B E A D C (2) B D E C A (2) B D A E C (2) B A D E C (2) A E C B D (2) A D B C E (2) A B E C D (2) A B D E C (2) A B C D E (2) E C B D A (1) E B C D A (1) E B C A D (1) E B A D C (1) E A C B D (1) D C B E A (1) D B C E A (1) D A C B E (1) C E D B A (1) C E A D B (1) C D E B A (1) C D A B E (1) C A D B E (1) B D E A C (1) A D C B E (1) A C E D B (1) A C B E D (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 2 12 2 B -10 0 -2 4 -2 C -2 2 0 12 2 D -12 -4 -12 0 0 E -2 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 12 2 B -10 0 -2 4 -2 C -2 2 0 12 2 D -12 -4 -12 0 0 E -2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=22 C=20 B=20 D=7 so D is eliminated. Round 2 votes counts: A=32 C=25 E=22 B=21 so B is eliminated. Round 3 votes counts: A=39 E=35 C=26 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:207 E:199 B:195 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 12 2 B -10 0 -2 4 -2 C -2 2 0 12 2 D -12 -4 -12 0 0 E -2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 12 2 B -10 0 -2 4 -2 C -2 2 0 12 2 D -12 -4 -12 0 0 E -2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 12 2 B -10 0 -2 4 -2 C -2 2 0 12 2 D -12 -4 -12 0 0 E -2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 242: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) A C D B E (10) E B D A C (8) A C B D E (7) E B D C A (6) E D B C A (5) D C B A E (5) D B C A E (5) A C E B D (5) E A C B D (4) B D E C A (4) A C E D B (4) E A B C D (3) D E B C A (3) D B E C A (2) C A D E B (2) B E D A C (2) A E C B D (2) E D C A B (1) E D B A C (1) E D A C B (1) E B A C D (1) E A C D B (1) D C A B E (1) C D A B E (1) C A E D B (1) B E D C A (1) B E A C D (1) B D C A E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 2 6 12 B -10 0 -14 -8 4 C -2 14 0 8 8 D -6 8 -8 0 4 E -12 -4 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 6 12 B -10 0 -14 -8 4 C -2 14 0 8 8 D -6 8 -8 0 4 E -12 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=29 D=16 C=15 B=9 so B is eliminated. Round 2 votes counts: E=35 A=29 D=21 C=15 so C is eliminated. Round 3 votes counts: A=43 E=35 D=22 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:214 D:199 B:186 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 6 12 B -10 0 -14 -8 4 C -2 14 0 8 8 D -6 8 -8 0 4 E -12 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 6 12 B -10 0 -14 -8 4 C -2 14 0 8 8 D -6 8 -8 0 4 E -12 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 6 12 B -10 0 -14 -8 4 C -2 14 0 8 8 D -6 8 -8 0 4 E -12 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 243: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (6) B D C E A (6) B D A E C (6) A C E D B (6) D B E C A (5) C E A D B (4) B D E A C (4) A B C E D (4) E D C A B (3) D B E A C (3) C E D A B (3) B A D E C (3) B A C D E (3) A B D E C (3) D E C A B (2) D E A C B (2) D B A E C (2) D A E B C (2) C E D B A (2) C B A E D (2) C A E D B (2) C A E B D (2) C A B E D (2) B D C A E (2) B D A C E (2) B A D C E (2) A D B E C (2) A C B E D (2) E C D A B (1) E C A D B (1) D E B C A (1) D C B E A (1) C E B A D (1) C E A B D (1) C B E D A (1) B C D E A (1) B C A E D (1) B C A D E (1) A E C D B (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -2 -10 4 B 10 0 20 12 28 C 2 -20 0 -16 4 D 10 -12 16 0 20 E -4 -28 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -10 4 B 10 0 20 12 28 C 2 -20 0 -16 4 D 10 -12 16 0 20 E -4 -28 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=20 A=20 D=18 E=5 so E is eliminated. Round 2 votes counts: B=37 C=22 D=21 A=20 so A is eliminated. Round 3 votes counts: B=44 C=32 D=24 so D is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:235 D:217 A:191 C:185 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -10 4 B 10 0 20 12 28 C 2 -20 0 -16 4 D 10 -12 16 0 20 E -4 -28 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -10 4 B 10 0 20 12 28 C 2 -20 0 -16 4 D 10 -12 16 0 20 E -4 -28 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -10 4 B 10 0 20 12 28 C 2 -20 0 -16 4 D 10 -12 16 0 20 E -4 -28 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 244: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (9) C B A E D (8) C B A D E (6) D E C A B (4) C E A B D (4) A E D B C (4) E D C A B (3) E D A B C (3) E A D C B (3) C B D E A (3) C A B E D (3) B C D A E (3) B C A D E (3) A B C E D (3) E D A C B (2) E A D B C (2) D E B A C (2) D B C E A (2) D B A E C (2) C E D B A (2) C E A D B (2) C D B E A (2) A C B E D (2) E C D A B (1) E A C D B (1) E A C B D (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) D C B E A (1) D B E A C (1) D B C A E (1) C E B D A (1) C D E B A (1) C B E A D (1) C B D A E (1) B D A C E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B C D (1) A C E B D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -10 2 -10 B -6 0 -12 -4 -2 C 10 12 0 6 10 D -2 4 -6 0 -2 E 10 2 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 2 -10 B -6 0 -12 -4 -2 C 10 12 0 6 10 D -2 4 -6 0 -2 E 10 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=26 E=16 A=14 B=10 so B is eliminated. Round 2 votes counts: C=40 D=27 A=17 E=16 so E is eliminated. Round 3 votes counts: C=41 D=35 A=24 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:202 D:197 A:194 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 2 -10 B -6 0 -12 -4 -2 C 10 12 0 6 10 D -2 4 -6 0 -2 E 10 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 2 -10 B -6 0 -12 -4 -2 C 10 12 0 6 10 D -2 4 -6 0 -2 E 10 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 2 -10 B -6 0 -12 -4 -2 C 10 12 0 6 10 D -2 4 -6 0 -2 E 10 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 245: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) D E C B A (5) D C E B A (5) C D E A B (5) B E A D C (5) C D E B A (4) A B E C D (4) E D C B A (3) E D B C A (3) E D B A C (3) D E B C A (3) C A D B E (3) B D E C A (3) A E B D C (3) A B E D C (3) A B C D E (3) E D A B C (2) D B E C A (2) C D B E A (2) C D B A E (2) C D A E B (2) C B D A E (2) A C E D B (2) A C B E D (2) A C B D E (2) A B C E D (2) E D C A B (1) E B A D C (1) E A D C B (1) E A D B C (1) D C B E A (1) C E A D B (1) C D A B E (1) C A E D B (1) C A B D E (1) B E D C A (1) B E D A C (1) B D C A E (1) B C A D E (1) B A E C D (1) B A C D E (1) A E D B C (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -6 -4 -6 B 14 0 4 -8 2 C 6 -4 0 -12 -10 D 4 8 12 0 -2 E 6 -2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.666667 Sum of squares = 0.500000000036 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.333333 E: 1.000000 A B C D E A 0 -14 -6 -4 -6 B 14 0 4 -8 2 C 6 -4 0 -12 -10 D 4 8 12 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.666667 Sum of squares = 0.500000001991 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 B=21 D=16 E=15 so E is eliminated. Round 2 votes counts: D=28 A=26 C=24 B=22 so B is eliminated. Round 3 votes counts: A=41 D=34 C=25 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:208 B:206 C:190 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 -6 B 14 0 4 -8 2 C 6 -4 0 -12 -10 D 4 8 12 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.666667 Sum of squares = 0.500000001991 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.333333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 -6 B 14 0 4 -8 2 C 6 -4 0 -12 -10 D 4 8 12 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.666667 Sum of squares = 0.500000001991 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 -6 B 14 0 4 -8 2 C 6 -4 0 -12 -10 D 4 8 12 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.666667 Sum of squares = 0.500000001991 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 246: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) D E B C A (11) E D C A B (6) D E C A B (6) A C B E D (6) E D B A C (5) E D A C B (5) C A D E B (5) C A B D E (4) B A C D E (4) A B C E D (3) E A C D B (2) C A D B E (2) C A B E D (2) B E D A C (2) B E A D C (2) B D E C A (2) B C A D E (2) E D C B A (1) E C D A B (1) E B D A C (1) E A C B D (1) D E C B A (1) D C E A B (1) D C A E B (1) D B E C A (1) D B C E A (1) C B D A E (1) C A E D B (1) B D E A C (1) B D C A E (1) B C D A E (1) A E C D B (1) A C E D B (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -2 2 0 B -2 0 0 -4 -2 C 2 0 0 4 0 D -2 4 -4 0 -8 E 0 2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.564981 D: 0.000000 E: 0.435019 Sum of squares = 0.508445156945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.564981 D: 0.564981 E: 1.000000 A B C D E A 0 2 -2 2 0 B -2 0 0 -4 -2 C 2 0 0 4 0 D -2 4 -4 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=22 D=22 C=15 A=14 so A is eliminated. Round 2 votes counts: B=31 C=24 E=23 D=22 so D is eliminated. Round 3 votes counts: E=41 B=33 C=26 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:205 C:203 A:201 B:196 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 2 0 B -2 0 0 -4 -2 C 2 0 0 4 0 D -2 4 -4 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 0 B -2 0 0 -4 -2 C 2 0 0 4 0 D -2 4 -4 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 0 B -2 0 0 -4 -2 C 2 0 0 4 0 D -2 4 -4 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 247: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) C B D E A (6) B C D A E (6) A E D B C (6) D C E B A (5) B C A D E (5) E A D C B (4) B A D C E (4) A B E C D (4) E D A C B (3) E C D A B (3) C D E B A (3) C D B E A (3) B A C E D (3) A E C B D (3) A B E D C (3) A B D E C (3) E C A D B (2) E A C D B (2) D C B E A (2) C E D B A (2) C E B A D (2) B A C D E (2) A E B D C (2) A E B C D (2) A D B E C (2) A B D C E (2) D E C A B (1) D E A C B (1) D B C E A (1) D B A C E (1) C B E D A (1) B D C A E (1) B C D E A (1) B C A E D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 8 4 B -2 0 0 4 4 C 2 0 0 6 4 D -8 -4 -6 0 -2 E -4 -4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.259479 C: 0.740521 D: 0.000000 E: 0.000000 Sum of squares = 0.615700763027 Cumulative probabilities = A: 0.000000 B: 0.259479 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 8 4 B -2 0 0 4 4 C 2 0 0 6 4 D -8 -4 -6 0 -2 E -4 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499878 C: 0.500122 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029628 Cumulative probabilities = A: 0.000000 B: 0.499878 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 E=20 C=17 D=11 so D is eliminated. Round 2 votes counts: A=29 B=25 C=24 E=22 so E is eliminated. Round 3 votes counts: A=39 C=36 B=25 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:206 C:206 B:203 E:195 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 8 4 B -2 0 0 4 4 C 2 0 0 6 4 D -8 -4 -6 0 -2 E -4 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499878 C: 0.500122 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029628 Cumulative probabilities = A: 0.000000 B: 0.499878 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 4 B -2 0 0 4 4 C 2 0 0 6 4 D -8 -4 -6 0 -2 E -4 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499878 C: 0.500122 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029628 Cumulative probabilities = A: 0.000000 B: 0.499878 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 4 B -2 0 0 4 4 C 2 0 0 6 4 D -8 -4 -6 0 -2 E -4 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499878 C: 0.500122 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029628 Cumulative probabilities = A: 0.000000 B: 0.499878 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 248: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) C A E D B (7) A C E D B (7) E A C B D (6) C E A D B (5) D B C A E (4) D B A C E (4) B D E A C (4) A E C B D (4) E C A B D (3) D C B A E (3) C D A B E (3) B D A E C (3) A E B D C (3) A E B C D (3) E C B D A (2) E B A D C (2) E A B D C (2) D B A E C (2) C E A B D (2) C D E A B (2) C D B A E (2) C A D E B (2) E B D A C (1) D C B E A (1) D B E C A (1) C E B D A (1) C D B E A (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E C A (1) B A E D C (1) A D C B E (1) A D B C E (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -2 6 6 B -8 0 -8 -16 -6 C 2 8 0 4 16 D -6 16 -4 0 -6 E -6 6 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 6 6 B -8 0 -8 -16 -6 C 2 8 0 4 16 D -6 16 -4 0 -6 E -6 6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 A=22 E=16 B=12 so B is eliminated. Round 2 votes counts: D=33 C=25 A=23 E=19 so E is eliminated. Round 3 votes counts: D=36 A=34 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:209 D:200 E:195 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 6 6 B -8 0 -8 -16 -6 C 2 8 0 4 16 D -6 16 -4 0 -6 E -6 6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 6 B -8 0 -8 -16 -6 C 2 8 0 4 16 D -6 16 -4 0 -6 E -6 6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 6 B -8 0 -8 -16 -6 C 2 8 0 4 16 D -6 16 -4 0 -6 E -6 6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 249: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) D C B E A (10) A E B C D (10) C D A E B (7) E B A C D (5) B E A D C (5) D B E C A (4) D B E A C (4) D B C E A (4) D A E B C (4) C D B E A (4) D C A E B (3) D C A B E (3) D A B E C (3) B E C A D (3) A E C B D (3) A E B D C (3) C B E A D (2) C A E B D (2) A C E B D (2) D B A E C (1) D A C E B (1) C E B A D (1) B E C D A (1) B D E A C (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 10 -2 -14 B 16 0 22 2 16 C -10 -22 0 6 -20 D 2 -2 -6 0 0 E 14 -16 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 10 -2 -14 B 16 0 22 2 16 C -10 -22 0 6 -20 D 2 -2 -6 0 0 E 14 -16 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999177 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=23 A=19 C=16 E=5 so E is eliminated. Round 2 votes counts: D=37 B=28 A=19 C=16 so C is eliminated. Round 3 votes counts: D=48 B=31 A=21 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:228 E:209 D:197 A:189 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 10 -2 -14 B 16 0 22 2 16 C -10 -22 0 6 -20 D 2 -2 -6 0 0 E 14 -16 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999177 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 -2 -14 B 16 0 22 2 16 C -10 -22 0 6 -20 D 2 -2 -6 0 0 E 14 -16 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999177 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 -2 -14 B 16 0 22 2 16 C -10 -22 0 6 -20 D 2 -2 -6 0 0 E 14 -16 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999177 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 250: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) B E D A C (7) C A E B D (6) B D E A C (6) E D B C A (5) A C B E D (5) A C B D E (5) E B D C A (4) C A D E B (4) B A D C E (4) A C D B E (4) D B E A C (3) D A C B E (3) C A E D B (3) B D A E C (3) B A C E D (3) A B C D E (3) E D C A B (2) E C A D B (2) D E B C A (2) A D C B E (2) A B C E D (2) E C D B A (1) E C B A D (1) D E C B A (1) D E B A C (1) D C E A B (1) D B A C E (1) C E D A B (1) C E A D B (1) C D A E B (1) B E A C D (1) B A E C D (1) B A D E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 14 0 6 B 6 0 -2 6 12 C -14 2 0 -12 2 D 0 -6 12 0 -6 E -6 -12 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305782 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 14 0 6 B 6 0 -2 6 12 C -14 2 0 -12 2 D 0 -6 12 0 -6 E -6 -12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305724 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 A=23 C=16 D=12 so D is eliminated. Round 2 votes counts: B=30 E=27 A=26 C=17 so C is eliminated. Round 3 votes counts: A=40 E=30 B=30 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:207 D:200 E:193 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 14 0 6 B 6 0 -2 6 12 C -14 2 0 -12 2 D 0 -6 12 0 -6 E -6 -12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305724 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 0 6 B 6 0 -2 6 12 C -14 2 0 -12 2 D 0 -6 12 0 -6 E -6 -12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305724 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 0 6 B 6 0 -2 6 12 C -14 2 0 -12 2 D 0 -6 12 0 -6 E -6 -12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.487603305724 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 251: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) D B C A E (6) A E B C D (6) A B D E C (6) E C A B D (5) E C A D B (4) E A C B D (4) D C B E A (4) A E B D C (4) E C D B A (3) D C B A E (3) B D A C E (3) A E C D B (3) A D B C E (3) A B E D C (3) E C B A D (2) D C A B E (2) D B A C E (2) C D E B A (2) C D B E A (2) C D A E B (2) A C D E B (2) A B D C E (2) E C B D A (1) E B A D C (1) E B A C D (1) E A C D B (1) E A B D C (1) D A C B E (1) D A B C E (1) C E B D A (1) C E A D B (1) C D B A E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D E A C (1) B D C A E (1) B C D E A (1) B A D C E (1) A D C B E (1) Total count = 100 A B C D E A 0 6 -4 4 8 B -6 0 -10 -8 -6 C 4 10 0 4 4 D -4 8 -4 0 -4 E -8 6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 4 8 B -6 0 -10 -8 -6 C 4 10 0 4 4 D -4 8 -4 0 -4 E -8 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 D=19 C=19 B=9 so B is eliminated. Round 2 votes counts: A=31 E=25 D=24 C=20 so C is eliminated. Round 3 votes counts: E=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:207 E:199 D:198 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 4 8 B -6 0 -10 -8 -6 C 4 10 0 4 4 D -4 8 -4 0 -4 E -8 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 4 8 B -6 0 -10 -8 -6 C 4 10 0 4 4 D -4 8 -4 0 -4 E -8 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 4 8 B -6 0 -10 -8 -6 C 4 10 0 4 4 D -4 8 -4 0 -4 E -8 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 252: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (7) D C A E B (6) B A E D C (6) E B C A D (5) D C E B A (5) C D A E B (5) B E A C D (5) B A E C D (5) A B D E C (5) D C E A B (4) C A D E B (4) E C D B A (3) E C B D A (3) D A B C E (3) C E D B A (3) C D E A B (3) A B C E D (3) E B D C A (2) D A C B E (2) A D C B E (2) A B E D C (2) A B D C E (2) E D C B A (1) E D B C A (1) E B C D A (1) E B A C D (1) D E C B A (1) D C A B E (1) C E D A B (1) C E A D B (1) C A D B E (1) B E D A C (1) B E A D C (1) B A D E C (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -2 8 16 B -10 0 6 4 0 C 2 -6 0 6 -2 D -8 -4 -6 0 -4 E -16 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.432098765431 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 8 16 B -10 0 6 4 0 C 2 -6 0 6 -2 D -8 -4 -6 0 -4 E -16 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.43209876542 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=22 B=19 C=18 E=17 so E is eliminated. Round 2 votes counts: B=28 D=24 C=24 A=24 so D is eliminated. Round 3 votes counts: C=42 B=29 A=29 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:216 B:200 C:200 E:195 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 8 16 B -10 0 6 4 0 C 2 -6 0 6 -2 D -8 -4 -6 0 -4 E -16 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.43209876542 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 8 16 B -10 0 6 4 0 C 2 -6 0 6 -2 D -8 -4 -6 0 -4 E -16 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.43209876542 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 8 16 B -10 0 6 4 0 C 2 -6 0 6 -2 D -8 -4 -6 0 -4 E -16 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.43209876542 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 253: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (12) E B C A D (9) D A C B E (8) E B D A C (6) E D C A B (5) E B A C D (5) B E A C D (5) E B A D C (4) D C A B E (4) C A D B E (4) C D A B E (3) B A C D E (3) E D B C A (2) D E A C B (2) D A B C E (2) C A B D E (2) B E C A D (2) B A C E D (2) A D C B E (2) E D B A C (1) E D A B C (1) E C D A B (1) E C B A D (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A B C (1) D C E A B (1) C D E A B (1) C D A E B (1) C B A E D (1) C A D E B (1) B E A D C (1) B C A E D (1) B A E D C (1) B A E C D (1) B A D C E (1) Total count = 100 A B C D E A 0 4 -8 -8 -2 B -4 0 0 -6 -14 C 8 0 0 -12 -2 D 8 6 12 0 -2 E 2 14 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -8 -8 -2 B -4 0 0 -6 -14 C 8 0 0 -12 -2 D 8 6 12 0 -2 E 2 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=31 B=17 C=13 A=2 so A is eliminated. Round 2 votes counts: E=37 D=33 B=17 C=13 so C is eliminated. Round 3 votes counts: D=43 E=37 B=20 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:210 C:197 A:193 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 -8 -2 B -4 0 0 -6 -14 C 8 0 0 -12 -2 D 8 6 12 0 -2 E 2 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -8 -2 B -4 0 0 -6 -14 C 8 0 0 -12 -2 D 8 6 12 0 -2 E 2 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -8 -2 B -4 0 0 -6 -14 C 8 0 0 -12 -2 D 8 6 12 0 -2 E 2 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 254: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) D B E C A (10) D A C E B (10) B E C A D (10) D B A E C (7) C E B A D (6) C E A B D (6) A D C E B (6) D A B C E (5) A C D E B (5) A C E D B (3) A C E B D (3) D B E A C (2) D A C B E (2) C A E B D (2) B E C D A (2) B D E C A (2) D C A E B (1) D B A C E (1) C E A D B (1) C A E D B (1) C A D E B (1) B E D C A (1) B D E A C (1) B D A E C (1) B A E C D (1) Total count = 100 A B C D E A 0 -8 -6 10 -2 B 8 0 -14 -10 -10 C 6 14 0 2 6 D -10 10 -2 0 8 E 2 10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 10 -2 B 8 0 -14 -10 -10 C 6 14 0 2 6 D -10 10 -2 0 8 E 2 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=18 C=17 A=17 E=10 so E is eliminated. Round 2 votes counts: D=38 C=27 B=18 A=17 so A is eliminated. Round 3 votes counts: D=44 C=38 B=18 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:203 E:199 A:197 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 10 -2 B 8 0 -14 -10 -10 C 6 14 0 2 6 D -10 10 -2 0 8 E 2 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 10 -2 B 8 0 -14 -10 -10 C 6 14 0 2 6 D -10 10 -2 0 8 E 2 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 10 -2 B 8 0 -14 -10 -10 C 6 14 0 2 6 D -10 10 -2 0 8 E 2 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 255: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) E A C D B (10) D B E A C (6) B C A E D (6) E C A B D (4) D A E C B (4) A E C D B (4) A E C B D (4) E C A D B (3) D B E C A (3) D B C A E (3) D B A C E (3) B C E A D (3) B C A D E (3) E A C B D (2) D E C A B (2) D B A E C (2) C E B A D (2) C E A B D (2) B D C E A (2) B D C A E (2) B D A C E (2) A C E B D (2) E D C A B (1) E D A C B (1) E C B A D (1) D E B A C (1) D E A B C (1) D A E B C (1) C B A E D (1) C A B E D (1) B C E D A (1) B C D A E (1) B A C D E (1) A E D C B (1) A D C E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 18 6 -12 B -14 0 -16 -16 -16 C -18 16 0 6 -24 D -6 16 -6 0 -2 E 12 16 24 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 18 6 -12 B -14 0 -16 -16 -16 C -18 16 0 6 -24 D -6 16 -6 0 -2 E 12 16 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=22 B=21 A=14 C=6 so C is eliminated. Round 2 votes counts: D=37 E=26 B=22 A=15 so A is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:213 D:201 C:190 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 18 6 -12 B -14 0 -16 -16 -16 C -18 16 0 6 -24 D -6 16 -6 0 -2 E 12 16 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 6 -12 B -14 0 -16 -16 -16 C -18 16 0 6 -24 D -6 16 -6 0 -2 E 12 16 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 6 -12 B -14 0 -16 -16 -16 C -18 16 0 6 -24 D -6 16 -6 0 -2 E 12 16 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 256: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) C A E D B (7) C D E A B (6) D E C B A (5) D E B A C (5) C A B E D (5) B D E A C (5) E D C A B (4) C A E B D (4) C A B D E (4) A C B E D (4) C D A E B (3) B D A E C (3) B A D E C (3) E A B D C (2) D C E B A (2) B E D A C (2) B E A D C (2) B A E D C (2) A C E B D (2) A B E C D (2) E D A C B (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D C B (1) E A B C D (1) D E B C A (1) D C E A B (1) D B E C A (1) D B E A C (1) C E D A B (1) C E A D B (1) C A D E B (1) B D C E A (1) B A C D E (1) A E C B D (1) A E B D C (1) A E B C D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 -4 -8 B -12 0 -8 -2 -26 C -6 8 0 -8 -10 D 4 2 8 0 -14 E 8 26 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 6 -4 -8 B -12 0 -8 -2 -26 C -6 8 0 -8 -10 D 4 2 8 0 -14 E 8 26 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=20 B=19 D=16 A=13 so A is eliminated. Round 2 votes counts: C=39 E=23 B=22 D=16 so D is eliminated. Round 3 votes counts: C=42 E=34 B=24 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:229 A:203 D:200 C:192 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 6 -4 -8 B -12 0 -8 -2 -26 C -6 8 0 -8 -10 D 4 2 8 0 -14 E 8 26 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 -8 B -12 0 -8 -2 -26 C -6 8 0 -8 -10 D 4 2 8 0 -14 E 8 26 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 -8 B -12 0 -8 -2 -26 C -6 8 0 -8 -10 D 4 2 8 0 -14 E 8 26 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 257: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) C A B D E (10) E D A B C (8) E D B A C (6) E C A D B (6) B C A D E (5) C B E A D (4) B D A C E (4) E D A C B (3) E A D C B (3) C E A B D (3) C A B E D (3) E C D B A (2) E C B A D (2) D E B A C (2) D B A E C (2) C B A E D (2) B E D C A (2) B D C A E (2) B D A E C (2) E D C A B (1) E D B C A (1) E C D A B (1) E B D C A (1) E B C D A (1) E A C D B (1) D B A C E (1) D A E B C (1) D A B E C (1) C A E D B (1) B E C D A (1) B D E A C (1) B C D E A (1) A E D C B (1) A D C B E (1) A C D E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -20 12 0 B 6 0 -14 12 10 C 20 14 0 14 2 D -12 -12 -14 0 -6 E 0 -10 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 12 0 B 6 0 -14 12 10 C 20 14 0 14 2 D -12 -12 -14 0 -6 E 0 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=34 B=18 D=7 A=5 so A is eliminated. Round 2 votes counts: E=37 C=36 B=19 D=8 so D is eliminated. Round 3 votes counts: E=40 C=37 B=23 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:207 E:197 A:193 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -20 12 0 B 6 0 -14 12 10 C 20 14 0 14 2 D -12 -12 -14 0 -6 E 0 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 12 0 B 6 0 -14 12 10 C 20 14 0 14 2 D -12 -12 -14 0 -6 E 0 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 12 0 B 6 0 -14 12 10 C 20 14 0 14 2 D -12 -12 -14 0 -6 E 0 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 258: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) E B C A D (9) D C A B E (8) E D C A B (7) B A C E D (5) B A C D E (5) E B A C D (4) D E C A B (4) C A B D E (4) E D C B A (3) E D B A C (3) E B D A C (3) C D A B E (3) C A D B E (3) A D C B E (3) E D B C A (2) E D A B C (2) E C B D A (2) D E A C B (2) D C A E B (2) B E A C D (2) A C D B E (2) A C B D E (2) A B C D E (2) E C B A D (1) E B A D C (1) D A B C E (1) C B A E D (1) B E C A D (1) B A E D C (1) B A E C D (1) A D B C E (1) Total count = 100 A B C D E A 0 12 0 -4 8 B -12 0 -14 -12 10 C 0 14 0 -6 4 D 4 12 6 0 4 E -8 -10 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 -4 8 B -12 0 -14 -12 10 C 0 14 0 -6 4 D 4 12 6 0 4 E -8 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=27 B=15 C=11 A=10 so A is eliminated. Round 2 votes counts: E=37 D=31 B=17 C=15 so C is eliminated. Round 3 votes counts: D=39 E=37 B=24 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:208 C:206 E:187 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 0 -4 8 B -12 0 -14 -12 10 C 0 14 0 -6 4 D 4 12 6 0 4 E -8 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -4 8 B -12 0 -14 -12 10 C 0 14 0 -6 4 D 4 12 6 0 4 E -8 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -4 8 B -12 0 -14 -12 10 C 0 14 0 -6 4 D 4 12 6 0 4 E -8 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 259: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B C A E D (8) B A C E D (7) D E A C B (5) D A E C B (5) B A C D E (5) C A E B D (4) B D E C A (4) B C E A D (4) E C D A B (3) C E B A D (3) B A D C E (3) D E A B C (2) D B E A C (2) D B A E C (2) B D E A C (2) B D C E A (2) B D A E C (2) B C E D A (2) B A D E C (2) A C E D B (2) A B D E C (2) E D C A B (1) E C A D B (1) E A D C B (1) D E C B A (1) D E B C A (1) D E B A C (1) D B E C A (1) D A B E C (1) C E D A B (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E A D (1) C B A E D (1) C A B E D (1) B D A C E (1) A E D C B (1) A E C D B (1) A D E C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 0 4 2 B 12 0 8 16 8 C 0 -8 0 -2 0 D -4 -16 2 0 6 E -2 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 4 2 B 12 0 8 16 8 C 0 -8 0 -2 0 D -4 -16 2 0 6 E -2 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 D=29 C=14 A=9 E=6 so E is eliminated. Round 2 votes counts: B=42 D=30 C=18 A=10 so A is eliminated. Round 3 votes counts: B=44 D=33 C=23 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:197 C:195 D:194 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 4 2 B 12 0 8 16 8 C 0 -8 0 -2 0 D -4 -16 2 0 6 E -2 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 4 2 B 12 0 8 16 8 C 0 -8 0 -2 0 D -4 -16 2 0 6 E -2 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 4 2 B 12 0 8 16 8 C 0 -8 0 -2 0 D -4 -16 2 0 6 E -2 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 260: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (5) D E C B A (5) C B A D E (5) C A E D B (4) E D A C B (3) D B E C A (3) C D E A B (3) C D B E A (3) C B D A E (3) B D C A E (3) B A E D C (3) A E B D C (3) A E B C D (3) A C E D B (3) A B E C D (3) A B C E D (3) E D B A C (2) E A D C B (2) D E C A B (2) C A B D E (2) B E A D C (2) B D E C A (2) B C D E A (2) B C D A E (2) A E C D B (2) A C E B D (2) E D C A B (1) E B D A C (1) E A B D C (1) D E B A C (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) C D E B A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D E A (1) C A D E B (1) C A B E D (1) B E D A C (1) B D E A C (1) B D A C E (1) B A E C D (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D C B (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -2 4 8 B 0 0 -2 0 0 C 2 2 0 0 0 D -4 0 0 0 0 E -8 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.790944 D: 0.121530 E: 0.087526 Sum of squares = 0.648022907416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.790944 D: 0.912474 E: 1.000000 A B C D E A 0 0 -2 4 8 B 0 0 -2 0 0 C 2 2 0 0 0 D -4 0 0 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.289474 E: 0.026316 Sum of squares = 0.552631689063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.973684 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=22 B=21 E=15 D=15 so E is eliminated. Round 2 votes counts: A=30 C=27 B=22 D=21 so D is eliminated. Round 3 votes counts: C=38 A=33 B=29 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:205 C:202 B:199 D:198 E:196 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 4 8 B 0 0 -2 0 0 C 2 2 0 0 0 D -4 0 0 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.289474 E: 0.026316 Sum of squares = 0.552631689063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.973684 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 8 B 0 0 -2 0 0 C 2 2 0 0 0 D -4 0 0 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.289474 E: 0.026316 Sum of squares = 0.552631689063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.973684 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 8 B 0 0 -2 0 0 C 2 2 0 0 0 D -4 0 0 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.289474 E: 0.026316 Sum of squares = 0.552631689063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.684211 D: 0.973684 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 261: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (13) B C E D A (9) A D E C B (9) A D C E B (6) A D E B C (5) E B D C A (4) D E C B A (4) A C D B E (4) C A B D E (3) B E D C A (3) B E C D A (3) B C E A D (3) A C D E B (3) A B C E D (3) E D B C A (2) E D B A C (2) E B C D A (2) D E A B C (2) D A E B C (2) B C A E D (2) A C B D E (2) E D C B A (1) D E C A B (1) D E A C B (1) D A C E B (1) C B E A D (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B E D (1) A D C B E (1) A C B E D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -10 0 -2 B 0 0 -10 8 8 C 10 10 0 8 16 D 0 -8 -8 0 -4 E 2 -8 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 0 -2 B 0 0 -10 8 8 C 10 10 0 8 16 D 0 -8 -8 0 -4 E 2 -8 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=21 B=20 E=11 D=11 so E is eliminated. Round 2 votes counts: A=37 B=26 C=21 D=16 so D is eliminated. Round 3 votes counts: A=43 B=30 C=27 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:222 B:203 A:194 E:191 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 0 -2 B 0 0 -10 8 8 C 10 10 0 8 16 D 0 -8 -8 0 -4 E 2 -8 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 0 -2 B 0 0 -10 8 8 C 10 10 0 8 16 D 0 -8 -8 0 -4 E 2 -8 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 0 -2 B 0 0 -10 8 8 C 10 10 0 8 16 D 0 -8 -8 0 -4 E 2 -8 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 262: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) D C A B E (8) E B C A D (7) E D C B A (6) D E C A B (6) D E C B A (5) D E A C B (5) C A B D E (5) A C B D E (5) E B D C A (4) B C A E D (4) B A C E D (4) E D A B C (3) C A D B E (3) E B D A C (2) E B A D C (2) D C A E B (2) D A C B E (2) C D B A E (2) C B A E D (2) A D C B E (2) A B C E D (2) A B C D E (2) E D B A C (1) E A D B C (1) B E C A D (1) B E A C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 8 -10 B 4 0 -6 8 -10 C 10 6 0 0 -12 D -8 -8 0 0 -4 E 10 10 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 8 -10 B 4 0 -6 8 -10 C 10 6 0 0 -12 D -8 -8 0 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=28 A=13 C=12 B=10 so B is eliminated. Round 2 votes counts: E=39 D=28 A=17 C=16 so C is eliminated. Round 3 votes counts: E=39 A=31 D=30 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:202 B:198 A:192 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 8 -10 B 4 0 -6 8 -10 C 10 6 0 0 -12 D -8 -8 0 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 8 -10 B 4 0 -6 8 -10 C 10 6 0 0 -12 D -8 -8 0 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 8 -10 B 4 0 -6 8 -10 C 10 6 0 0 -12 D -8 -8 0 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 263: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) E B D C A (8) D E B A C (8) A D C E B (7) A C B E D (7) D E B C A (6) C A B E D (6) B E C D A (5) D B E C A (4) A D C B E (4) A C D B E (4) E B C D A (3) D E A B C (3) C B E D A (3) A C D E B (3) C B E A D (2) B E D C A (2) B C E A D (2) A C E B D (2) E D B C A (1) E B C A D (1) D C B A E (1) D A E C B (1) C B A E D (1) C A D B E (1) B D E C A (1) B C E D A (1) A E B D C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 2 4 -14 0 B -2 0 14 -10 -12 C -4 -14 0 -18 -12 D 14 10 18 0 10 E 0 12 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -14 0 B -2 0 14 -10 -12 C -4 -14 0 -18 -12 D 14 10 18 0 10 E 0 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=30 E=13 C=13 B=11 so B is eliminated. Round 2 votes counts: D=34 A=30 E=20 C=16 so C is eliminated. Round 3 votes counts: A=38 D=34 E=28 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 E:207 A:196 B:195 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -14 0 B -2 0 14 -10 -12 C -4 -14 0 -18 -12 D 14 10 18 0 10 E 0 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -14 0 B -2 0 14 -10 -12 C -4 -14 0 -18 -12 D 14 10 18 0 10 E 0 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -14 0 B -2 0 14 -10 -12 C -4 -14 0 -18 -12 D 14 10 18 0 10 E 0 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 264: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) A C B E D (8) C A B D E (7) E D B C A (5) D E B C A (5) E B D A C (4) D E C B A (4) D E C A B (4) C A D E B (4) B C A E D (4) B A C E D (4) A B C E D (4) C D A E B (3) B E D A C (3) E D A B C (2) C B A D E (2) C A D B E (2) B E D C A (2) B A E C D (2) A E B D C (2) A C D E B (2) E D A C B (1) E A D B C (1) D E A B C (1) D C E B A (1) D C A E B (1) D B E C A (1) C D B E A (1) C B D E A (1) C A B E D (1) B E A D C (1) B C E A D (1) A E C D B (1) A D E C B (1) A C E D B (1) A C D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 2 2 4 B 2 0 6 -4 -6 C -2 -6 0 2 -2 D -2 4 -2 0 -16 E -4 6 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888884 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 -2 2 2 4 B 2 0 6 -4 -6 C -2 -6 0 2 -2 D -2 4 -2 0 -16 E -4 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=22 C=21 D=17 B=17 so D is eliminated. Round 2 votes counts: E=37 C=23 A=22 B=18 so B is eliminated. Round 3 votes counts: E=44 C=28 A=28 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:203 B:199 C:196 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 4 B 2 0 6 -4 -6 C -2 -6 0 2 -2 D -2 4 -2 0 -16 E -4 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 4 B 2 0 6 -4 -6 C -2 -6 0 2 -2 D -2 4 -2 0 -16 E -4 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 4 B 2 0 6 -4 -6 C -2 -6 0 2 -2 D -2 4 -2 0 -16 E -4 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 265: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) A E C D B (7) E A D B C (6) D B C A E (6) D A E B C (5) C B E A D (5) B D C E A (5) A E D C B (5) E A C B D (4) D E A B C (4) D A E C B (3) C A B E D (3) E D A B C (2) E A D C B (2) E A B D C (2) D C B A E (2) D B E A C (2) D B C E A (2) C D A B E (2) C B A E D (2) C B A D E (2) C A E B D (2) B C D E A (2) B C D A E (2) A C E D B (2) E C B A D (1) E A B C D (1) D B E C A (1) D A C E B (1) D A C B E (1) C D B A E (1) C B D E A (1) C A D E B (1) B E C A D (1) B C E A D (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 10 -2 -2 16 B -10 0 -16 -12 0 C 2 16 0 0 4 D 2 12 0 0 6 E -16 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.251497 D: 0.748503 E: 0.000000 Sum of squares = 0.623507798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.251497 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -2 16 B -10 0 -16 -12 0 C 2 16 0 0 4 D 2 12 0 0 6 E -16 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=27 E=18 A=16 B=11 so B is eliminated. Round 2 votes counts: C=33 D=32 E=19 A=16 so A is eliminated. Round 3 votes counts: C=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:211 D:210 E:187 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 -2 16 B -10 0 -16 -12 0 C 2 16 0 0 4 D 2 12 0 0 6 E -16 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -2 16 B -10 0 -16 -12 0 C 2 16 0 0 4 D 2 12 0 0 6 E -16 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -2 16 B -10 0 -16 -12 0 C 2 16 0 0 4 D 2 12 0 0 6 E -16 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 266: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) A E C B D (7) A B E D C (7) A C D E B (5) C E D B A (4) B D E C A (4) A B E C D (4) A B D E C (4) D C E B A (3) D C B E A (3) D B C E A (3) B E D C A (3) B D E A C (3) B D A E C (3) A C E D B (3) A B D C E (3) E D B C A (2) E C D B A (2) E A C B D (2) C E A D B (2) C A E D B (2) C A D E B (2) A C E B D (2) E D C B A (1) E B C A D (1) E B A D C (1) E A B C D (1) D E C B A (1) D E B C A (1) D B E C A (1) D A B C E (1) C E D A B (1) C D E A B (1) C D A E B (1) B E A D C (1) B A E D C (1) B A D E C (1) A E B C D (1) A D C B E (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 8 8 2 B -6 0 -6 2 -8 C -8 6 0 4 -4 D -8 -2 -4 0 0 E -2 8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 8 2 B -6 0 -6 2 -8 C -8 6 0 4 -4 D -8 -2 -4 0 0 E -2 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994401 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=21 B=16 D=13 E=10 so E is eliminated. Round 2 votes counts: A=43 C=23 B=18 D=16 so D is eliminated. Round 3 votes counts: A=44 C=31 B=25 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:205 C:199 D:193 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 8 2 B -6 0 -6 2 -8 C -8 6 0 4 -4 D -8 -2 -4 0 0 E -2 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994401 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 8 2 B -6 0 -6 2 -8 C -8 6 0 4 -4 D -8 -2 -4 0 0 E -2 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994401 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 8 2 B -6 0 -6 2 -8 C -8 6 0 4 -4 D -8 -2 -4 0 0 E -2 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994401 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 267: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) B D E A C (8) D B C A E (6) E D B C A (5) E B D A C (5) C A D B E (5) B D A C E (5) A C B D E (5) B E D A C (4) E C A D B (3) B D E C A (3) A C E B D (3) A C D B E (3) A B C D E (3) D C B A E (2) D C A B E (2) D B E C A (2) D B C E A (2) D B A C E (2) C D A E B (2) C A E D B (2) B A D C E (2) A E C B D (2) A C B E D (2) E D C B A (1) E C D A B (1) E B A C D (1) E A C B D (1) D E B C A (1) C A D E B (1) B E A D C (1) B D A E C (1) A E B C D (1) A D C B E (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 6 -22 6 B 20 0 26 16 22 C -6 -26 0 -26 2 D 22 -16 26 0 16 E -6 -22 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 -22 6 B 20 0 26 16 22 C -6 -26 0 -26 2 D 22 -16 26 0 16 E -6 -22 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 A=23 D=17 C=10 so C is eliminated. Round 2 votes counts: A=31 E=26 B=24 D=19 so D is eliminated. Round 3 votes counts: B=38 A=35 E=27 so E is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:242 D:224 A:185 E:177 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 6 -22 6 B 20 0 26 16 22 C -6 -26 0 -26 2 D 22 -16 26 0 16 E -6 -22 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -22 6 B 20 0 26 16 22 C -6 -26 0 -26 2 D 22 -16 26 0 16 E -6 -22 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -22 6 B 20 0 26 16 22 C -6 -26 0 -26 2 D 22 -16 26 0 16 E -6 -22 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 268: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (6) E B C D A (5) E A C D B (5) D A E B C (5) D A B C E (5) D B C A E (4) C B E A D (4) E C B A D (3) D E A B C (3) D A B E C (3) C E B A D (3) C B A E D (3) B D C A E (3) B C E D A (3) B C E A D (3) B C D E A (3) A E D C B (3) E C A B D (2) D B A E C (2) D B A C E (2) D A E C B (2) C B A D E (2) C A E B D (2) B D E C A (2) B C D A E (2) A E C D B (2) A C E D B (2) E D B C A (1) E D B A C (1) E A D C B (1) D E B A C (1) D B C E A (1) C A B E D (1) C A B D E (1) B E C D A (1) B D C E A (1) B C A D E (1) A D C E B (1) A D C B E (1) A D B C E (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -2 0 14 B 2 0 6 -6 2 C 2 -6 0 2 4 D 0 6 -2 0 8 E -14 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102038 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 0 14 B 2 0 6 -6 2 C 2 -6 0 2 4 D 0 6 -2 0 8 E -14 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102017 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=19 A=19 E=18 C=16 so C is eliminated. Round 2 votes counts: D=28 B=28 A=23 E=21 so E is eliminated. Round 3 votes counts: B=39 A=31 D=30 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:206 A:205 B:202 C:201 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 0 14 B 2 0 6 -6 2 C 2 -6 0 2 4 D 0 6 -2 0 8 E -14 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102017 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 0 14 B 2 0 6 -6 2 C 2 -6 0 2 4 D 0 6 -2 0 8 E -14 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102017 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 0 14 B 2 0 6 -6 2 C 2 -6 0 2 4 D 0 6 -2 0 8 E -14 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102017 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 269: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (16) D C B A E (7) E B A C D (6) E A C B D (6) E C D A B (5) C D A B E (5) E C A D B (4) E B A D C (4) D C A B E (4) D B C A E (4) C D E A B (3) B A E D C (3) B A E C D (3) B A C D E (3) A B C D E (3) E D C B A (2) E D C A B (2) D C E A B (2) B D A C E (2) B A D C E (2) A E B C D (2) E D B C A (1) E D B A C (1) D C E B A (1) D C B E A (1) D B E C A (1) C A E D B (1) C A D B E (1) B E A D C (1) B A C E D (1) A E C B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 12 18 -12 B -14 0 10 10 -14 C -12 -10 0 24 -18 D -18 -10 -24 0 -22 E 12 14 18 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 12 18 -12 B -14 0 10 10 -14 C -12 -10 0 24 -18 D -18 -10 -24 0 -22 E 12 14 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=47 D=20 B=15 C=10 A=8 so A is eliminated. Round 2 votes counts: E=50 D=20 B=20 C=10 so C is eliminated. Round 3 votes counts: E=51 D=29 B=20 so B is eliminated. Round 4 votes counts: E=61 D=39 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:233 A:216 B:196 C:192 D:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 12 18 -12 B -14 0 10 10 -14 C -12 -10 0 24 -18 D -18 -10 -24 0 -22 E 12 14 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 18 -12 B -14 0 10 10 -14 C -12 -10 0 24 -18 D -18 -10 -24 0 -22 E 12 14 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 18 -12 B -14 0 10 10 -14 C -12 -10 0 24 -18 D -18 -10 -24 0 -22 E 12 14 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 270: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) A B E D C (8) E A B C D (7) E C D A B (6) C E D B A (6) C D B E A (6) A E B D C (6) C D E B A (5) A B D E C (5) E C D B A (4) E A C D B (4) B C D A E (4) E C B D A (3) A D B C E (3) E B A C D (2) E A C B D (2) E A B D C (2) B D C A E (2) B A D C E (2) A D E C B (2) A B D C E (2) E B C A D (1) D C A E B (1) D C A B E (1) D B C A E (1) C B E D A (1) C B D E A (1) B D A C E (1) A E D C B (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -2 -2 0 B -4 0 -4 -2 -6 C 2 4 0 4 -10 D 2 2 -4 0 -8 E 0 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.467381 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.532619 Sum of squares = 0.502127989314 Cumulative probabilities = A: 0.467381 B: 0.467381 C: 0.467381 D: 0.467381 E: 1.000000 A B C D E A 0 4 -2 -2 0 B -4 0 -4 -2 -6 C 2 4 0 4 -10 D 2 2 -4 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=29 C=19 D=12 B=9 so B is eliminated. Round 2 votes counts: E=31 A=31 C=23 D=15 so D is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 A:200 C:200 D:196 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 -2 0 B -4 0 -4 -2 -6 C 2 4 0 4 -10 D 2 2 -4 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -2 0 B -4 0 -4 -2 -6 C 2 4 0 4 -10 D 2 2 -4 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -2 0 B -4 0 -4 -2 -6 C 2 4 0 4 -10 D 2 2 -4 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 271: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (6) E B A D C (5) C D B E A (5) A C D E B (5) C B E D A (4) C B D E A (4) C A E B D (4) B E C D A (4) E C B A D (3) E B C A D (3) D C B E A (3) D C B A E (3) D A B E C (3) C D A B E (3) B D C E A (3) B C E D A (3) A E D B C (3) A E B D C (3) A D E C B (3) E A B C D (2) D C A B E (2) D B E A C (2) D B C E A (2) D B A E C (2) D B A C E (2) C E B A D (2) B E D C A (2) A D B E C (2) A C E D B (2) E B A C D (1) E A B D C (1) D A C B E (1) D A B C E (1) C E B D A (1) C A D E B (1) B E D A C (1) B C D E A (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 -12 -6 -4 -4 B 12 0 -6 -4 6 C 6 6 0 -2 16 D 4 4 2 0 8 E 4 -6 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -4 -4 B 12 0 -6 -4 6 C 6 6 0 -2 16 D 4 4 2 0 8 E 4 -6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 D=21 E=15 B=14 so B is eliminated. Round 2 votes counts: C=28 A=26 D=24 E=22 so E is eliminated. Round 3 votes counts: C=38 A=35 D=27 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:209 B:204 A:187 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -4 -4 B 12 0 -6 -4 6 C 6 6 0 -2 16 D 4 4 2 0 8 E 4 -6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -4 -4 B 12 0 -6 -4 6 C 6 6 0 -2 16 D 4 4 2 0 8 E 4 -6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -4 -4 B 12 0 -6 -4 6 C 6 6 0 -2 16 D 4 4 2 0 8 E 4 -6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 272: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) B E C A D (7) D C E B A (6) D C B E A (4) A D C B E (4) A B E C D (4) E B A C D (3) E A B C D (3) D C B A E (3) D C A E B (3) D A C E B (3) D A C B E (3) A E B D C (3) A E B C D (3) A D E C B (3) A D B C E (3) E C D B A (2) D C E A B (2) D C A B E (2) C B D E A (2) B C E D A (2) B A E C D (2) B A C E D (2) A D C E B (2) A B E D C (2) A B C E D (2) A B C D E (2) E D C B A (1) E C B D A (1) E B C A D (1) E A D C B (1) E A D B C (1) D A E C B (1) C E B D A (1) C D E B A (1) C D B E A (1) C D B A E (1) B E C D A (1) B E A C D (1) B C D E A (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 -6 -2 2 0 B 6 0 6 6 2 C 2 -6 0 6 4 D -2 -6 -6 0 -2 E 0 -2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 2 0 B 6 0 6 6 2 C 2 -6 0 6 4 D -2 -6 -6 0 -2 E 0 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=27 E=21 B=18 C=6 so C is eliminated. Round 2 votes counts: D=30 A=28 E=22 B=20 so B is eliminated. Round 3 votes counts: A=34 E=33 D=33 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:210 C:203 E:198 A:197 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 2 0 B 6 0 6 6 2 C 2 -6 0 6 4 D -2 -6 -6 0 -2 E 0 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 0 B 6 0 6 6 2 C 2 -6 0 6 4 D -2 -6 -6 0 -2 E 0 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 0 B 6 0 6 6 2 C 2 -6 0 6 4 D -2 -6 -6 0 -2 E 0 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 273: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) A C D E B (8) E B A D C (7) C D A B E (6) C A D E B (6) B E D C A (5) C D B A E (4) E B A C D (3) E A C B D (3) D C B A E (3) D C A B E (3) B D E C A (3) A E C B D (3) A E B D C (3) E C B D A (2) E B C D A (2) D A C B E (2) C D B E A (2) B D C E A (2) A D C B E (2) A C D B E (2) E C B A D (1) E C A B D (1) E B D C A (1) E B D A C (1) E B C A D (1) E A B D C (1) D B C A E (1) D B A C E (1) C E D B A (1) C E B D A (1) C E A D B (1) C A E D B (1) C A D B E (1) B E A D C (1) B D E A C (1) B C D E A (1) B A D E C (1) A E D C B (1) A E C D B (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 2 16 2 B -10 0 -12 6 -16 C -2 12 0 18 -2 D -16 -6 -18 0 2 E -2 16 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 16 2 B -10 0 -12 6 -16 C -2 12 0 18 -2 D -16 -6 -18 0 2 E -2 16 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=23 A=22 B=14 D=10 so D is eliminated. Round 2 votes counts: E=31 C=29 A=24 B=16 so B is eliminated. Round 3 votes counts: E=41 C=33 A=26 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:215 C:213 E:207 B:184 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 16 2 B -10 0 -12 6 -16 C -2 12 0 18 -2 D -16 -6 -18 0 2 E -2 16 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 16 2 B -10 0 -12 6 -16 C -2 12 0 18 -2 D -16 -6 -18 0 2 E -2 16 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 16 2 B -10 0 -12 6 -16 C -2 12 0 18 -2 D -16 -6 -18 0 2 E -2 16 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 274: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (13) C E B D A (9) A D E B C (9) A D B E C (9) C B D E A (6) E D B C A (5) A C E D B (5) C A B E D (4) B D E C A (4) A E D B C (4) E D B A C (3) A C B D E (3) E B D C A (2) C E D B A (2) B E D C A (2) B C E D A (2) E D A B C (1) E B C D A (1) D E B A C (1) D E A B C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E D A B (1) C E A D B (1) C B E A D (1) C A E D B (1) C A B D E (1) B D E A C (1) B D C E A (1) B C D E A (1) B A D E C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -14 -18 -18 B 12 0 6 8 6 C 14 -6 0 4 4 D 18 -8 -4 0 -14 E 18 -6 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -18 -18 B 12 0 6 8 6 C 14 -6 0 4 4 D 18 -8 -4 0 -14 E 18 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=32 E=12 B=12 D=5 so D is eliminated. Round 2 votes counts: C=39 A=34 E=14 B=13 so B is eliminated. Round 3 votes counts: C=43 A=35 E=22 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:216 E:211 C:208 D:196 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 -18 -18 B 12 0 6 8 6 C 14 -6 0 4 4 D 18 -8 -4 0 -14 E 18 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -18 -18 B 12 0 6 8 6 C 14 -6 0 4 4 D 18 -8 -4 0 -14 E 18 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -18 -18 B 12 0 6 8 6 C 14 -6 0 4 4 D 18 -8 -4 0 -14 E 18 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 275: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) E C B A D (6) E C A B D (6) D C E B A (6) D C E A B (6) D A B C E (6) C E D B A (6) B D A C E (6) A B E C D (6) E C A D B (5) C E D A B (5) D C B E A (3) A E C B D (3) A E B C D (3) A D E C B (3) D B A C E (2) C E B D A (2) C D E B A (2) B A E C D (2) B A D C E (2) A D B E C (2) A D B C E (2) E A C B D (1) E A B C D (1) D B C A E (1) D A C E B (1) D A C B E (1) B C D E A (1) B A D E C (1) Total count = 100 A B C D E A 0 20 2 4 0 B -20 0 -12 -2 -12 C -2 12 0 -2 4 D -4 2 2 0 8 E 0 12 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.899512 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.100488 Sum of squares = 0.819218957424 Cumulative probabilities = A: 0.899512 B: 0.899512 C: 0.899512 D: 0.899512 E: 1.000000 A B C D E A 0 20 2 4 0 B -20 0 -12 -2 -12 C -2 12 0 -2 4 D -4 2 2 0 8 E 0 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556612944 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 E=19 C=15 B=12 so B is eliminated. Round 2 votes counts: A=33 D=32 E=19 C=16 so C is eliminated. Round 3 votes counts: D=35 A=33 E=32 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:206 D:204 E:200 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 2 4 0 B -20 0 -12 -2 -12 C -2 12 0 -2 4 D -4 2 2 0 8 E 0 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556612944 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 2 4 0 B -20 0 -12 -2 -12 C -2 12 0 -2 4 D -4 2 2 0 8 E 0 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556612944 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 2 4 0 B -20 0 -12 -2 -12 C -2 12 0 -2 4 D -4 2 2 0 8 E 0 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556612944 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 276: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) A B E C D (7) D E C B A (6) C A E B D (6) C D E B A (5) D B A E C (4) D A B E C (4) A D B E C (4) A C B E D (4) A B C E D (4) E B A C D (3) D B E A C (3) D A C B E (3) C E D B A (3) B E A C D (3) A B E D C (3) E C B A D (2) D E B C A (2) D C E A B (2) D C A B E (2) D A B C E (2) A B D E C (2) E D C B A (1) E D B C A (1) E C D B A (1) E B C D A (1) D E B A C (1) C E D A B (1) C E B D A (1) C E B A D (1) C D A E B (1) C A E D B (1) C A D E B (1) C A D B E (1) B E A D C (1) B A E C D (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 2 -8 4 B 0 0 -8 -20 -2 C -2 8 0 -2 0 D 8 20 2 0 8 E -4 2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -8 4 B 0 0 -8 -20 -2 C -2 8 0 -2 0 D 8 20 2 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999969615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=26 C=21 E=9 B=5 so B is eliminated. Round 2 votes counts: D=39 A=27 C=21 E=13 so E is eliminated. Round 3 votes counts: D=41 A=34 C=25 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:202 A:199 E:195 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -8 4 B 0 0 -8 -20 -2 C -2 8 0 -2 0 D 8 20 2 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999969615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -8 4 B 0 0 -8 -20 -2 C -2 8 0 -2 0 D 8 20 2 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999969615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -8 4 B 0 0 -8 -20 -2 C -2 8 0 -2 0 D 8 20 2 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999969615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 277: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) B C D A E (7) C B E A D (6) B E D A C (6) A D E C B (6) E A D B C (5) D A E B C (5) C B D A E (5) E B A D C (4) E A D C B (4) A D E B C (4) D A B E C (3) C A D E B (3) A E D B C (3) E B C A D (2) E A C D B (2) D A C B E (2) C E A B D (2) C A E D B (2) C A D B E (2) B E C D A (2) B C E D A (2) A E D C B (2) E C B A D (1) E C A D B (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A B C E (1) C D A B E (1) C B A D E (1) B E C A D (1) B D C A E (1) B D A E C (1) B C E A D (1) B C D E A (1) Total count = 100 A B C D E A 0 0 2 4 6 B 0 0 2 0 6 C -2 -2 0 -2 -6 D -4 0 2 0 -6 E -6 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.486652 B: 0.513348 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500356353954 Cumulative probabilities = A: 0.486652 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 6 B 0 0 2 0 6 C -2 -2 0 -2 -6 D -4 0 2 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=22 E=19 D=15 A=15 so D is eliminated. Round 2 votes counts: C=30 A=27 B=24 E=19 so E is eliminated. Round 3 votes counts: A=38 C=32 B=30 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:204 E:200 D:196 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 4 6 B 0 0 2 0 6 C -2 -2 0 -2 -6 D -4 0 2 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 6 B 0 0 2 0 6 C -2 -2 0 -2 -6 D -4 0 2 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 6 B 0 0 2 0 6 C -2 -2 0 -2 -6 D -4 0 2 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 278: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (7) E D B C A (6) D A E C B (6) D A C E B (6) E B D A C (5) D E A C B (5) C B A E D (5) C A B D E (5) E B D C A (4) D E A B C (4) C A D B E (4) B C E A D (4) E D B A C (3) E B C D A (3) D A E B C (3) B C A E D (3) D E C A B (2) C E B D A (2) C D E B A (2) C B E A D (2) B E C A D (2) A D C B E (2) A B E D C (2) E D C B A (1) E B A D C (1) D E B A C (1) D C A E B (1) C D E A B (1) C B E D A (1) C B A D E (1) C A B E D (1) B E C D A (1) B A C E D (1) A D E C B (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -2 -14 0 B -4 0 -12 -14 -16 C 2 12 0 -10 -2 D 14 14 10 0 6 E 0 16 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -14 0 B -4 0 -12 -14 -16 C 2 12 0 -10 -2 D 14 14 10 0 6 E 0 16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 E=23 A=14 B=11 so B is eliminated. Round 2 votes counts: C=31 D=28 E=26 A=15 so A is eliminated. Round 3 votes counts: C=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:206 C:201 A:194 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -14 0 B -4 0 -12 -14 -16 C 2 12 0 -10 -2 D 14 14 10 0 6 E 0 16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -14 0 B -4 0 -12 -14 -16 C 2 12 0 -10 -2 D 14 14 10 0 6 E 0 16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -14 0 B -4 0 -12 -14 -16 C 2 12 0 -10 -2 D 14 14 10 0 6 E 0 16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 279: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) D C A E B (6) B A C E D (6) C E D A B (5) C E B A D (5) E C B A D (4) D E C A B (4) B A E C D (4) A D B C E (4) A B D E C (4) E C B D A (3) E B C A D (3) E B A D C (3) D C E A B (3) D A B E C (3) B C E A D (3) A B D C E (3) E D A B C (2) E C D A B (2) D A E B C (2) D A C E B (2) C D E A B (2) C B A D E (2) B E C A D (2) E D C A B (1) E D A C B (1) E B A C D (1) D E A C B (1) D A E C B (1) D A C B E (1) D A B C E (1) C E D B A (1) C D A B E (1) C B E A D (1) B E A C D (1) B C A E D (1) B C A D E (1) B A D E C (1) B A C D E (1) Total count = 100 A B C D E A 0 -2 -16 0 -12 B 2 0 -8 -2 -20 C 16 8 0 14 -2 D 0 2 -14 0 -14 E 12 20 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -16 0 -12 B 2 0 -8 -2 -20 C 16 8 0 14 -2 D 0 2 -14 0 -14 E 12 20 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=24 B=20 C=17 A=11 so A is eliminated. Round 2 votes counts: E=28 D=28 B=27 C=17 so C is eliminated. Round 3 votes counts: E=39 D=31 B=30 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 C:218 D:187 B:186 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -16 0 -12 B 2 0 -8 -2 -20 C 16 8 0 14 -2 D 0 2 -14 0 -14 E 12 20 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 0 -12 B 2 0 -8 -2 -20 C 16 8 0 14 -2 D 0 2 -14 0 -14 E 12 20 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 0 -12 B 2 0 -8 -2 -20 C 16 8 0 14 -2 D 0 2 -14 0 -14 E 12 20 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 280: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) E A B C D (8) D C E B A (8) A B E C D (8) B A C D E (6) D C B A E (5) C D B A E (5) E C D A B (4) A B C E D (4) E D C A B (3) E A C B D (3) E A B D C (3) D E C B A (3) D C B E A (3) B D C A E (3) A B C D E (3) E A C D B (2) B A E D C (2) B A D E C (2) A E B C D (2) E D B A C (1) E D A B C (1) E B A D C (1) E A D C B (1) D B C A E (1) C B A D E (1) B D A E C (1) B D A C E (1) B C D A E (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 4 -2 -4 B 10 0 -2 4 -2 C -4 2 0 -2 -14 D 2 -4 2 0 -12 E 4 2 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 4 -2 -4 B 10 0 -2 4 -2 C -4 2 0 -2 -14 D 2 -4 2 0 -12 E 4 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=20 A=20 B=16 C=6 so C is eliminated. Round 2 votes counts: E=38 D=25 A=20 B=17 so B is eliminated. Round 3 votes counts: E=38 D=31 A=31 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:205 A:194 D:194 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 4 -2 -4 B 10 0 -2 4 -2 C -4 2 0 -2 -14 D 2 -4 2 0 -12 E 4 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -2 -4 B 10 0 -2 4 -2 C -4 2 0 -2 -14 D 2 -4 2 0 -12 E 4 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -2 -4 B 10 0 -2 4 -2 C -4 2 0 -2 -14 D 2 -4 2 0 -12 E 4 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 281: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (14) E A B C D (10) D C E B A (8) A B D C E (7) E A C B D (5) A B E C D (5) B A C D E (4) E D C A B (3) E A B D C (3) D C B E A (3) C D B A E (3) E C D A B (2) D E C B A (2) D B C A E (2) C D E B A (2) B D A C E (2) B C D A E (2) B C A D E (2) B A D C E (2) A E B D C (2) A E B C D (2) A B E D C (2) E D C B A (1) E C D B A (1) E C A D B (1) E C A B D (1) E B A C D (1) E A D B C (1) E A C D B (1) C D B E A (1) C B D A E (1) B A E C D (1) A D B E C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 2 6 8 B 4 0 2 8 8 C -2 -2 0 -8 8 D -6 -8 8 0 16 E -8 -8 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 6 8 B 4 0 2 8 8 C -2 -2 0 -8 8 D -6 -8 8 0 16 E -8 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=29 A=21 B=13 C=7 so C is eliminated. Round 2 votes counts: D=35 E=30 A=21 B=14 so B is eliminated. Round 3 votes counts: D=40 E=30 A=30 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:211 A:206 D:205 C:198 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 6 8 B 4 0 2 8 8 C -2 -2 0 -8 8 D -6 -8 8 0 16 E -8 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 8 B 4 0 2 8 8 C -2 -2 0 -8 8 D -6 -8 8 0 16 E -8 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 8 B 4 0 2 8 8 C -2 -2 0 -8 8 D -6 -8 8 0 16 E -8 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 282: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E C A D B (8) E C D A B (7) E A C B D (7) B D A E C (6) B A D C E (6) D B C E A (5) E A B C D (3) D E C B A (3) D E B C A (3) C D E A B (3) E D C B A (2) E B D A C (2) D C B E A (2) D C B A E (2) D B A C E (2) C E A D B (2) C A D B E (2) B A D E C (2) A E C B D (2) A C E B D (2) A B C D E (2) E D B C A (1) E D B A C (1) E C A B D (1) E B A D C (1) D E B A C (1) D C E B A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E A B D (1) C D A E B (1) C D A B E (1) C A E B D (1) B A E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 6 -14 -10 B 10 0 4 0 -6 C -6 -4 0 -10 -8 D 14 0 10 0 12 E 10 6 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.409650 C: 0.000000 D: 0.590350 E: 0.000000 Sum of squares = 0.516326375405 Cumulative probabilities = A: 0.000000 B: 0.409650 C: 0.409650 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -14 -10 B 10 0 4 0 -6 C -6 -4 0 -10 -8 D 14 0 10 0 12 E 10 6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=26 D=22 C=11 A=8 so A is eliminated. Round 2 votes counts: E=35 B=30 D=22 C=13 so C is eliminated. Round 3 votes counts: E=41 B=30 D=29 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:206 B:204 A:186 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 6 -14 -10 B 10 0 4 0 -6 C -6 -4 0 -10 -8 D 14 0 10 0 12 E 10 6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -14 -10 B 10 0 4 0 -6 C -6 -4 0 -10 -8 D 14 0 10 0 12 E 10 6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -14 -10 B 10 0 4 0 -6 C -6 -4 0 -10 -8 D 14 0 10 0 12 E 10 6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 283: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) A D B E C (8) C B E D A (6) A D E C B (6) E D B A C (5) C E B D A (5) B C E D A (5) E C B D A (4) D A E B C (4) E B D C A (3) D B E A C (3) C B E A D (3) C B A E D (3) B E C D A (3) A C D E B (3) A C D B E (3) E B C D A (2) D A B E C (2) C E A D B (2) C A D E B (2) B E D C A (2) A D C E B (2) A D B C E (2) E D C B A (1) E D B C A (1) E C D A B (1) E B D A C (1) D E B A C (1) D B A E C (1) C B A D E (1) C A E D B (1) C A D B E (1) C A B D E (1) B D E A C (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 6 -4 2 B 4 0 8 -16 -6 C -6 -8 0 -6 -18 D 4 16 6 0 4 E -2 6 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -4 2 B 4 0 8 -16 -6 C -6 -8 0 -6 -18 D 4 16 6 0 4 E -2 6 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=25 E=18 B=12 D=11 so D is eliminated. Round 2 votes counts: A=40 C=25 E=19 B=16 so B is eliminated. Round 3 votes counts: A=42 C=30 E=28 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 E:209 A:200 B:195 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -4 2 B 4 0 8 -16 -6 C -6 -8 0 -6 -18 D 4 16 6 0 4 E -2 6 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 2 B 4 0 8 -16 -6 C -6 -8 0 -6 -18 D 4 16 6 0 4 E -2 6 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 2 B 4 0 8 -16 -6 C -6 -8 0 -6 -18 D 4 16 6 0 4 E -2 6 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 284: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) A D E B C (8) C B E D A (7) C D E B A (6) B C A E D (6) D E C B A (5) D E C A B (5) E D C B A (4) A D E C B (4) D C E A B (3) C B D E A (3) B A C E D (3) A B E C D (3) E D A B C (2) D C A E B (2) C A B D E (2) B C E A D (2) B A E D C (2) A E D B C (2) A B D E C (2) A B C D E (2) E D B A C (1) E B D A C (1) E B C D A (1) E A D B C (1) D E A C B (1) D E A B C (1) D A E C B (1) C E D B A (1) C D E A B (1) C D B E A (1) C D A E B (1) C B A D E (1) B E C D A (1) B E A D C (1) B A E C D (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -4 4 4 B -6 0 2 -2 -4 C 4 -2 0 -14 -14 D -4 2 14 0 0 E -4 4 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.106381 E: 0.075437 Sum of squares = 0.455024197945 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.924563 E: 1.000000 A B C D E A 0 6 -4 4 4 B -6 0 2 -2 -4 C 4 -2 0 -14 -14 D -4 2 14 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.090909 E: 0.090909 Sum of squares = 0.454545454545 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.909091 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=23 D=18 B=16 E=10 so E is eliminated. Round 2 votes counts: A=34 D=25 C=23 B=18 so B is eliminated. Round 3 votes counts: A=41 C=33 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:207 D:206 A:205 B:195 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -4 4 4 B -6 0 2 -2 -4 C 4 -2 0 -14 -14 D -4 2 14 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.090909 E: 0.090909 Sum of squares = 0.454545454545 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.909091 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 4 4 B -6 0 2 -2 -4 C 4 -2 0 -14 -14 D -4 2 14 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.090909 E: 0.090909 Sum of squares = 0.454545454545 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.909091 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 4 4 B -6 0 2 -2 -4 C 4 -2 0 -14 -14 D -4 2 14 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.090909 E: 0.090909 Sum of squares = 0.454545454545 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.909091 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 285: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) B E C A D (9) C A D B E (8) E B A D C (7) D A E C B (6) D A C E B (4) C B A D E (4) B E A D C (4) A D C B E (4) E D A B C (3) E B D C A (3) E B D A C (3) D E A B C (3) E B C D A (2) D E C A B (2) D E A C B (2) D A E B C (2) C B E A D (2) C B A E D (2) E D C B A (1) E D B A C (1) E C D B A (1) E B C A D (1) E A B D C (1) D C E A B (1) D C A E B (1) D C A B E (1) D A C B E (1) C D E B A (1) C D A E B (1) C B E D A (1) B E A C D (1) B C E A D (1) B C A E D (1) B C A D E (1) B A C E D (1) A D E B C (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -6 2 0 B -6 0 -6 -8 6 C 6 6 0 -4 -8 D -2 8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 2 0 B -6 0 -6 -8 6 C 6 6 0 -4 -8 D -2 8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888896 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=23 D=23 B=18 A=8 so A is eliminated. Round 2 votes counts: C=29 D=28 E=23 B=20 so B is eliminated. Round 3 votes counts: E=38 C=34 D=28 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:209 A:201 C:200 E:197 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 2 0 B -6 0 -6 -8 6 C 6 6 0 -4 -8 D -2 8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888896 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 2 0 B -6 0 -6 -8 6 C 6 6 0 -4 -8 D -2 8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888896 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 2 0 B -6 0 -6 -8 6 C 6 6 0 -4 -8 D -2 8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888896 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 286: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) C E B D A (6) A D B C E (6) D C E A B (5) D E C B A (4) D C A E B (4) C D E A B (4) B E C A D (4) A D B E C (4) C B E A D (3) C A B E D (3) B E D A C (3) B A E C D (3) E D B C A (2) E C B D A (2) E B D C A (2) D E B A C (2) D E A C B (2) C E B A D (2) B E D C A (2) B E A D C (2) A C D E B (2) A C D B E (2) A B D E C (2) E D C B A (1) E C D B A (1) D E C A B (1) D E B C A (1) D B E A C (1) D B A E C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E D B A (1) C D E B A (1) C D A E B (1) C A E B D (1) B E C D A (1) B D A E C (1) B A E D C (1) B A C E D (1) A D C E B (1) A C B E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 -4 -6 B -4 0 -14 -16 4 C 4 14 0 -20 8 D 4 16 20 0 16 E 6 -4 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -4 -6 B -4 0 -14 -16 4 C 4 14 0 -20 8 D 4 16 20 0 16 E 6 -4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=24 C=22 B=18 E=8 so E is eliminated. Round 2 votes counts: A=28 D=27 C=25 B=20 so B is eliminated. Round 3 votes counts: D=35 A=35 C=30 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 C:203 A:195 E:189 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -4 -6 B -4 0 -14 -16 4 C 4 14 0 -20 8 D 4 16 20 0 16 E 6 -4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -4 -6 B -4 0 -14 -16 4 C 4 14 0 -20 8 D 4 16 20 0 16 E 6 -4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -4 -6 B -4 0 -14 -16 4 C 4 14 0 -20 8 D 4 16 20 0 16 E 6 -4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) B E D A C (9) D E B C A (7) D E B A C (6) B A C E D (6) A C D E B (6) D E C A B (5) A C B D E (5) C A D E B (4) A C B E D (4) C A B E D (3) C A B D E (3) B A E D C (3) A B C E D (3) A B C D E (3) D E A B C (2) B C A E D (2) A D E C B (2) A B D E C (2) E D C B A (1) E D B A C (1) D E C B A (1) D A E B C (1) C D E A B (1) C B A E D (1) B E D C A (1) B E C D A (1) B D E A C (1) B A E C D (1) B A D E C (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 16 2 4 B 8 0 26 -2 0 C -16 -26 0 -12 -16 D -2 2 12 0 4 E -4 0 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 -8 16 2 4 B 8 0 26 -2 0 C -16 -26 0 -12 -16 D -2 2 12 0 4 E -4 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000212 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 D=22 E=14 C=12 so C is eliminated. Round 2 votes counts: A=37 B=26 D=23 E=14 so E is eliminated. Round 3 votes counts: D=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:216 D:208 A:207 E:204 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 16 2 4 B 8 0 26 -2 0 C -16 -26 0 -12 -16 D -2 2 12 0 4 E -4 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000212 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 2 4 B 8 0 26 -2 0 C -16 -26 0 -12 -16 D -2 2 12 0 4 E -4 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000212 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 2 4 B 8 0 26 -2 0 C -16 -26 0 -12 -16 D -2 2 12 0 4 E -4 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000212 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 288: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (10) B C E A D (9) D A E C B (8) B E C D A (7) E B D A C (6) E D A C B (4) D E A C B (4) C A D B E (4) B C A D E (4) C B A D E (3) C A D E B (3) B E D A C (3) B D A E C (3) A D C B E (3) E C B D A (2) D A E B C (2) C E A D B (2) C B E A D (2) B A D C E (2) A D E C B (2) E D B A C (1) E D A B C (1) E B C D A (1) D E A B C (1) D A B E C (1) C E A B D (1) C B A E D (1) C A B D E (1) B E D C A (1) B D E A C (1) B C E D A (1) B C A E D (1) B A C D E (1) A D E B C (1) A D B C E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 14 6 6 B -2 0 -4 0 0 C -14 4 0 -10 2 D -6 0 10 0 14 E -6 0 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 6 6 B -2 0 -4 0 0 C -14 4 0 -10 2 D -6 0 10 0 14 E -6 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=19 C=17 D=16 E=15 so E is eliminated. Round 2 votes counts: B=40 D=22 C=19 A=19 so C is eliminated. Round 3 votes counts: B=48 A=30 D=22 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:209 B:197 C:191 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 6 6 B -2 0 -4 0 0 C -14 4 0 -10 2 D -6 0 10 0 14 E -6 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 6 6 B -2 0 -4 0 0 C -14 4 0 -10 2 D -6 0 10 0 14 E -6 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 6 6 B -2 0 -4 0 0 C -14 4 0 -10 2 D -6 0 10 0 14 E -6 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 289: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) D A C E B (8) B E D A C (8) E B A D C (7) C B E A D (6) A D E B C (6) C B E D A (5) C A D E B (5) B E C A D (5) D A E B C (4) A E D B C (4) C D A E B (3) C B D E A (3) B E C D A (3) A D C E B (3) D C A B E (2) D B E A C (2) C A E B D (2) C A D B E (2) A C D E B (2) E B A C D (1) E A D B C (1) E A B D C (1) E A B C D (1) D A C B E (1) C D B E A (1) C B A E D (1) B E D C A (1) B C E D A (1) B C E A D (1) A E B D C (1) Total count = 100 A B C D E A 0 10 0 -2 6 B -10 0 -6 -6 2 C 0 6 0 2 10 D 2 6 -2 0 2 E -6 -2 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.102798 B: 0.000000 C: 0.897202 D: 0.000000 E: 0.000000 Sum of squares = 0.815538797053 Cumulative probabilities = A: 0.102798 B: 0.102798 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -2 6 B -10 0 -6 -6 2 C 0 6 0 2 10 D 2 6 -2 0 2 E -6 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499653 B: 0.000000 C: 0.500347 D: 0.000000 E: 0.000000 Sum of squares = 0.500000240958 Cumulative probabilities = A: 0.499653 B: 0.499653 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=19 D=17 A=16 E=11 so E is eliminated. Round 2 votes counts: C=37 B=27 A=19 D=17 so D is eliminated. Round 3 votes counts: C=39 A=32 B=29 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:209 A:207 D:204 B:190 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 0 -2 6 B -10 0 -6 -6 2 C 0 6 0 2 10 D 2 6 -2 0 2 E -6 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499653 B: 0.000000 C: 0.500347 D: 0.000000 E: 0.000000 Sum of squares = 0.500000240958 Cumulative probabilities = A: 0.499653 B: 0.499653 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -2 6 B -10 0 -6 -6 2 C 0 6 0 2 10 D 2 6 -2 0 2 E -6 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499653 B: 0.000000 C: 0.500347 D: 0.000000 E: 0.000000 Sum of squares = 0.500000240958 Cumulative probabilities = A: 0.499653 B: 0.499653 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -2 6 B -10 0 -6 -6 2 C 0 6 0 2 10 D 2 6 -2 0 2 E -6 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499653 B: 0.000000 C: 0.500347 D: 0.000000 E: 0.000000 Sum of squares = 0.500000240958 Cumulative probabilities = A: 0.499653 B: 0.499653 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 290: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) C A B D E (8) D A B E C (6) E B A C D (5) C A D B E (5) C A B E D (5) E D B A C (4) D C E A B (4) E B A D C (3) C E A B D (3) C D A B E (3) E C B D A (2) E B D A C (2) D C A B E (2) D A C B E (2) D A B C E (2) C E D B A (2) C E D A B (2) C E B A D (2) C D E A B (2) C D A E B (2) B E A C D (2) B A C E D (2) E D B C A (1) E C D B A (1) E B C D A (1) D E C B A (1) D E C A B (1) D E A C B (1) D E A B C (1) D B E A C (1) D B A E C (1) C B E A D (1) C A E B D (1) B E A D C (1) B A E D C (1) B A D E C (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 -12 -10 B -8 0 -2 -12 -6 C -2 2 0 2 2 D 12 12 -2 0 18 E 10 6 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.750000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000156 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -12 -10 B -8 0 -2 -12 -6 C -2 2 0 2 2 D 12 12 -2 0 18 E 10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.750000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999947 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=34 E=19 B=7 A=4 so A is eliminated. Round 2 votes counts: C=37 D=34 E=19 B=10 so B is eliminated. Round 3 votes counts: C=40 D=37 E=23 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:202 E:198 A:194 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 8 2 -12 -10 B -8 0 -2 -12 -6 C -2 2 0 2 2 D 12 12 -2 0 18 E 10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.750000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999947 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -12 -10 B -8 0 -2 -12 -6 C -2 2 0 2 2 D 12 12 -2 0 18 E 10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.750000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999947 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -12 -10 B -8 0 -2 -12 -6 C -2 2 0 2 2 D 12 12 -2 0 18 E 10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.750000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999947 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 291: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) A B C D E (7) E D C B A (5) E A D B C (5) E A B D C (5) A C D B E (5) E B A C D (4) C B D A E (4) B A C E D (4) E B D C A (3) E B D A C (3) E B C A D (3) E B A D C (3) D E C A B (3) A D E C B (3) A D C B E (3) E D B C A (2) D A C E B (2) C D B E A (2) C D B A E (2) B C A E D (2) B C A D E (2) A E D B C (2) E D C A B (1) E D A C B (1) E D A B C (1) E C B D A (1) E B C D A (1) D C E A B (1) D C A E B (1) D A E C B (1) C D E B A (1) C A D B E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E A D (1) B C D A E (1) B A C D E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 6 10 2 B -4 0 10 0 -8 C -6 -10 0 -8 -4 D -10 0 8 0 -4 E -2 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 10 2 B -4 0 10 0 -8 C -6 -10 0 -8 -4 D -10 0 8 0 -4 E -2 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=22 D=16 B=14 C=10 so C is eliminated. Round 2 votes counts: E=38 A=23 D=21 B=18 so B is eliminated. Round 3 votes counts: E=42 A=32 D=26 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:207 B:199 D:197 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 10 2 B -4 0 10 0 -8 C -6 -10 0 -8 -4 D -10 0 8 0 -4 E -2 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 10 2 B -4 0 10 0 -8 C -6 -10 0 -8 -4 D -10 0 8 0 -4 E -2 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 10 2 B -4 0 10 0 -8 C -6 -10 0 -8 -4 D -10 0 8 0 -4 E -2 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 292: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) D B C E A (9) D A C B E (9) D C B A E (8) E B C A D (6) A E C B D (5) B C E D A (4) A E D B C (4) A D E C B (4) E B A C D (3) E A B C D (3) C B E A D (3) B E C D A (3) B C D E A (3) A D E B C (3) E C B A D (2) D A E B C (2) C B E D A (2) A D C B E (2) E B C D A (1) E A C B D (1) D C B E A (1) D C A B E (1) D B E C A (1) D B C A E (1) C D B E A (1) C B D E A (1) C B A D E (1) B E C A D (1) B D C E A (1) A E D C B (1) A E C D B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 0 4 8 B 4 0 10 2 4 C 0 -10 0 4 -2 D -4 -2 -4 0 -2 E -8 -4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 4 8 B 4 0 10 2 4 C 0 -10 0 4 -2 D -4 -2 -4 0 -2 E -8 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=32 A=32 E=16 B=12 C=8 so C is eliminated. Round 2 votes counts: D=33 A=32 B=19 E=16 so E is eliminated. Round 3 votes counts: A=36 D=33 B=31 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:204 C:196 E:196 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 4 8 B 4 0 10 2 4 C 0 -10 0 4 -2 D -4 -2 -4 0 -2 E -8 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 4 8 B 4 0 10 2 4 C 0 -10 0 4 -2 D -4 -2 -4 0 -2 E -8 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 4 8 B 4 0 10 2 4 C 0 -10 0 4 -2 D -4 -2 -4 0 -2 E -8 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 293: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) D E C B A (7) C B D E A (7) A E B C D (7) E A D C B (5) B D C A E (5) B A C D E (5) A B C E D (5) D C E B A (4) C D B E A (4) B C D A E (4) A E D B C (4) A B D C E (3) E D C B A (2) E D A C B (2) E C D B A (2) E A D B C (2) E A C D B (2) D B C E A (2) B C A D E (2) B A C E D (2) A D E B C (2) A B C D E (2) E A C B D (1) D B C A E (1) D B A C E (1) D A B E C (1) C E D B A (1) C D E B A (1) C B A E D (1) B C A E D (1) B A D C E (1) A E C B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -22 -4 -4 0 B 22 0 4 -2 14 C 4 -4 0 -2 20 D 4 2 2 0 20 E 0 -14 -20 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -4 -4 0 B 22 0 4 -2 14 C 4 -4 0 -2 20 D 4 2 2 0 20 E 0 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 B=20 E=16 C=14 so C is eliminated. Round 2 votes counts: D=29 B=28 A=26 E=17 so E is eliminated. Round 3 votes counts: D=36 A=36 B=28 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 D:214 C:209 A:185 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -4 -4 0 B 22 0 4 -2 14 C 4 -4 0 -2 20 D 4 2 2 0 20 E 0 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -4 -4 0 B 22 0 4 -2 14 C 4 -4 0 -2 20 D 4 2 2 0 20 E 0 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -4 -4 0 B 22 0 4 -2 14 C 4 -4 0 -2 20 D 4 2 2 0 20 E 0 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 294: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) D C B A E (8) A E C D B (8) E B A D C (7) B E D C A (6) D C A E B (5) B D C E A (5) E B A C D (4) E A C D B (4) B E A C D (4) B C D A E (4) D B C E A (3) B D C A E (3) A C E D B (3) E A C B D (2) D E A C B (2) D C A B E (2) C D A E B (2) C D A B E (2) B E D A C (2) B E A D C (2) A C D E B (2) E A B D C (1) D E B C A (1) D B C A E (1) C A D E B (1) B E C D A (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -10 4 0 -16 B 10 0 16 10 -6 C -4 -16 0 0 -14 D 0 -10 0 0 -12 E 16 6 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 4 0 -16 B 10 0 16 10 -6 C -4 -16 0 0 -14 D 0 -10 0 0 -12 E 16 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=28 D=22 A=14 C=5 so C is eliminated. Round 2 votes counts: B=31 E=28 D=26 A=15 so A is eliminated. Round 3 votes counts: E=39 B=31 D=30 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:215 A:189 D:189 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 4 0 -16 B 10 0 16 10 -6 C -4 -16 0 0 -14 D 0 -10 0 0 -12 E 16 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 0 -16 B 10 0 16 10 -6 C -4 -16 0 0 -14 D 0 -10 0 0 -12 E 16 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 0 -16 B 10 0 16 10 -6 C -4 -16 0 0 -14 D 0 -10 0 0 -12 E 16 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 295: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) E B D C A (5) E A C B D (5) B E D A C (5) E B D A C (4) C E A B D (4) C D A B E (4) C A D B E (4) B D A E C (4) E C B A D (3) D B A C E (3) D A B C E (3) C E B D A (3) C E B A D (3) C B E D A (3) E C B D A (2) E B C D A (2) E B A D C (2) E B A C D (2) E A B D C (2) C A E D B (2) B E D C A (2) B E A D C (2) B D E C A (2) B D E A C (2) B D C E A (2) A C D B E (2) E C A B D (1) E B C A D (1) E A B C D (1) D C B A E (1) D B A E C (1) C E D B A (1) C D E B A (1) C B D E A (1) C A D E B (1) A E D B C (1) A E C B D (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 4 -2 -22 B 14 0 16 30 4 C -4 -16 0 -6 -22 D 2 -30 6 0 -16 E 22 -4 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -2 -22 B 14 0 16 30 4 C -4 -16 0 -6 -22 D 2 -30 6 0 -16 E 22 -4 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=27 B=19 A=16 D=8 so D is eliminated. Round 2 votes counts: E=30 C=28 B=23 A=19 so A is eliminated. Round 3 votes counts: B=37 E=33 C=30 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:232 E:228 A:183 D:181 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 -2 -22 B 14 0 16 30 4 C -4 -16 0 -6 -22 D 2 -30 6 0 -16 E 22 -4 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -2 -22 B 14 0 16 30 4 C -4 -16 0 -6 -22 D 2 -30 6 0 -16 E 22 -4 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -2 -22 B 14 0 16 30 4 C -4 -16 0 -6 -22 D 2 -30 6 0 -16 E 22 -4 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 296: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) B A E C D (6) B A D E C (6) A B E D C (6) E C D A B (5) D C E A B (5) D C B E A (5) C E D A B (5) E A C D B (4) C D E A B (4) B D A C E (4) B A D C E (4) A B E C D (4) E C A D B (3) E C A B D (3) E A C B D (3) E D C A B (2) D B A C E (2) C D E B A (2) C D B E A (2) A E C B D (2) A E B C D (2) D E C A B (1) D E A C B (1) D C B A E (1) D C A B E (1) D B C A E (1) D A E C B (1) C E D B A (1) B C D E A (1) B C A E D (1) B A E D C (1) B A C E D (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -2 -4 -10 B -8 0 -20 -10 -6 C 2 20 0 -2 -4 D 4 10 2 0 0 E 10 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.615949 E: 0.384051 Sum of squares = 0.526888543648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.615949 E: 1.000000 A B C D E A 0 8 -2 -4 -10 B -8 0 -20 -10 -6 C 2 20 0 -2 -4 D 4 10 2 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=24 E=20 A=16 C=14 so C is eliminated. Round 2 votes counts: D=34 E=26 B=24 A=16 so A is eliminated. Round 3 votes counts: B=35 D=34 E=31 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:210 C:208 D:208 A:196 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -2 -4 -10 B -8 0 -20 -10 -6 C 2 20 0 -2 -4 D 4 10 2 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -4 -10 B -8 0 -20 -10 -6 C 2 20 0 -2 -4 D 4 10 2 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -4 -10 B -8 0 -20 -10 -6 C 2 20 0 -2 -4 D 4 10 2 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 297: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) A E D B C (9) D C B A E (8) C B E A D (8) B C E A D (8) C D B E A (7) A E B D C (7) A D E B C (7) E A B C D (5) C B E D A (5) C B D E A (5) E B C A D (4) D A C B E (4) D C B E A (3) D A C E B (3) D C A B E (2) D A E B C (2) B E C A D (2) E B A C D (1) B C A E D (1) Total count = 100 A B C D E A 0 -4 -6 4 4 B 4 0 -8 -8 6 C 6 8 0 -8 8 D -4 8 8 0 0 E -4 -6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.35802469132 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 4 4 B 4 0 -8 -8 6 C 6 8 0 -8 8 D -4 8 8 0 0 E -4 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691361 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=25 A=23 B=11 E=10 so E is eliminated. Round 2 votes counts: D=31 A=28 C=25 B=16 so B is eliminated. Round 3 votes counts: C=40 D=31 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:206 A:199 B:197 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 4 4 B 4 0 -8 -8 6 C 6 8 0 -8 8 D -4 8 8 0 0 E -4 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691361 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 4 B 4 0 -8 -8 6 C 6 8 0 -8 8 D -4 8 8 0 0 E -4 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691361 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 4 B 4 0 -8 -8 6 C 6 8 0 -8 8 D -4 8 8 0 0 E -4 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691361 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 298: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) E C B D A (5) E C A B D (5) B E C D A (5) A B D E C (5) E B C A D (4) A E C D B (4) A D E C B (4) A D B C E (4) E C A D B (3) D A C E B (3) D A C B E (3) D A B C E (3) C E B D A (3) A D C B E (3) E C B A D (2) E B C D A (2) E A B C D (2) C D B E A (2) B D C E A (2) B D A C E (2) B C E D A (2) B C D E A (2) B A E D C (2) A D B E C (2) E C D B A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B A C E (1) C E A D B (1) C D E B A (1) C D E A B (1) C B E D A (1) B E C A D (1) B E A D C (1) B D C A E (1) B D A E C (1) B A D C E (1) A E D C B (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 14 6 12 8 B -14 0 -10 0 -8 C -6 10 0 -4 -4 D -12 0 4 0 8 E -8 8 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 12 8 B -14 0 -10 0 -8 C -6 10 0 -4 -4 D -12 0 4 0 8 E -8 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=24 B=20 D=13 C=9 so C is eliminated. Round 2 votes counts: A=34 E=28 B=21 D=17 so D is eliminated. Round 3 votes counts: A=45 E=30 B=25 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:200 C:198 E:198 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 12 8 B -14 0 -10 0 -8 C -6 10 0 -4 -4 D -12 0 4 0 8 E -8 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 12 8 B -14 0 -10 0 -8 C -6 10 0 -4 -4 D -12 0 4 0 8 E -8 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 12 8 B -14 0 -10 0 -8 C -6 10 0 -4 -4 D -12 0 4 0 8 E -8 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 299: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) D E A B C (8) D A E B C (7) A D C E B (7) A C D B E (7) A C B D E (7) E D B C A (6) C B E A D (5) C A B E D (5) E B C D A (4) C B A E D (4) B E C D A (4) D E B A C (3) A D C B E (3) D A E C B (2) C E B A D (2) B C E D A (2) B C E A D (2) B C A D E (2) A D E C B (2) A C D E B (2) E D A B C (1) E C D B A (1) C E A D B (1) C A B D E (1) B E D C A (1) B D E C A (1) A D B C E (1) Total count = 100 A B C D E A 0 8 0 2 0 B -8 0 2 -2 -10 C 0 -2 0 -2 2 D -2 2 2 0 6 E 0 10 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.536068 B: 0.000000 C: 0.463932 D: 0.000000 E: 0.000000 Sum of squares = 0.502601735022 Cumulative probabilities = A: 0.536068 B: 0.536068 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 2 0 B -8 0 2 -2 -10 C 0 -2 0 -2 2 D -2 2 2 0 6 E 0 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.000000 C: 0.499863 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037274 Cumulative probabilities = A: 0.500137 B: 0.500137 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=21 D=20 C=18 B=12 so B is eliminated. Round 2 votes counts: A=29 E=26 C=24 D=21 so D is eliminated. Round 3 votes counts: E=38 A=38 C=24 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:205 D:204 E:201 C:199 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 2 0 B -8 0 2 -2 -10 C 0 -2 0 -2 2 D -2 2 2 0 6 E 0 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.000000 C: 0.499863 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037274 Cumulative probabilities = A: 0.500137 B: 0.500137 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 2 0 B -8 0 2 -2 -10 C 0 -2 0 -2 2 D -2 2 2 0 6 E 0 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.000000 C: 0.499863 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037274 Cumulative probabilities = A: 0.500137 B: 0.500137 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 2 0 B -8 0 2 -2 -10 C 0 -2 0 -2 2 D -2 2 2 0 6 E 0 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.000000 C: 0.499863 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037274 Cumulative probabilities = A: 0.500137 B: 0.500137 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 300: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) B D E C A (6) E A B C D (5) C D B A E (5) B E D C A (5) A C E D B (5) E B A C D (4) D B C E A (4) D B C A E (4) C A D B E (4) A C E B D (4) E A D B C (3) E A C B D (3) D B E C A (3) C B A D E (3) C A D E B (3) E B D A C (2) E B A D C (2) D C B A E (2) D C A B E (2) C D A B E (2) C B D A E (2) C A B E D (2) B E C A D (2) B D C E A (2) A E C B D (2) E A D C B (1) E A C D B (1) D E A B C (1) D A C E B (1) C B E A D (1) C A E B D (1) B C E A D (1) B C D E A (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -10 16 6 B -2 0 -10 -4 2 C 10 10 0 20 14 D -16 4 -20 0 8 E -6 -2 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 16 6 B -2 0 -10 -4 2 C 10 10 0 20 14 D -16 4 -20 0 8 E -6 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 A=22 E=21 D=17 B=17 so D is eliminated. Round 2 votes counts: B=28 C=27 A=23 E=22 so E is eliminated. Round 3 votes counts: A=37 B=36 C=27 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:227 A:207 B:193 D:188 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 16 6 B -2 0 -10 -4 2 C 10 10 0 20 14 D -16 4 -20 0 8 E -6 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 16 6 B -2 0 -10 -4 2 C 10 10 0 20 14 D -16 4 -20 0 8 E -6 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 16 6 B -2 0 -10 -4 2 C 10 10 0 20 14 D -16 4 -20 0 8 E -6 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 301: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) E A B D C (6) B E C D A (6) A E B D C (6) E B D C A (5) C B D E A (5) A C D B E (5) C D B E A (4) B C D E A (4) A E B C D (4) E B C D A (3) E A D B C (3) E A B C D (3) D C B E A (3) C D B A E (3) A D C E B (3) E D C B A (2) E B C A D (2) E B A D C (2) E B A C D (2) D C A B E (2) C A D B E (2) B C E D A (2) E D B C A (1) E D B A C (1) E D A C B (1) D C E B A (1) D C E A B (1) D C B A E (1) B E D C A (1) B A E C D (1) A E D B C (1) A D E C B (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 4 8 -16 B -2 0 6 6 -18 C -4 -6 0 -6 -24 D -8 -6 6 0 -26 E 16 18 24 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 8 -16 B -2 0 6 6 -18 C -4 -6 0 -6 -24 D -8 -6 6 0 -26 E 16 18 24 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=31 C=14 B=14 D=8 so D is eliminated. Round 2 votes counts: A=33 E=31 C=22 B=14 so B is eliminated. Round 3 votes counts: E=38 A=34 C=28 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:242 A:199 B:196 D:183 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 8 -16 B -2 0 6 6 -18 C -4 -6 0 -6 -24 D -8 -6 6 0 -26 E 16 18 24 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 -16 B -2 0 6 6 -18 C -4 -6 0 -6 -24 D -8 -6 6 0 -26 E 16 18 24 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 -16 B -2 0 6 6 -18 C -4 -6 0 -6 -24 D -8 -6 6 0 -26 E 16 18 24 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 302: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (19) A B D E C (11) A B E D C (9) B A D C E (8) C D E B A (6) A B D C E (6) E A C B D (5) D B A C E (5) E C A B D (4) A E B D C (4) E C D B A (3) E C D A B (3) E C A D B (3) D B C A E (3) C D B E A (3) B D A C E (3) A E B C D (2) E B A C D (1) D C B E A (1) D A B C E (1) Total count = 100 A B C D E A 0 -4 10 6 4 B 4 0 6 6 0 C -10 -6 0 -2 10 D -6 -6 2 0 -6 E -4 0 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.792383 C: 0.000000 D: 0.000000 E: 0.207617 Sum of squares = 0.670976125035 Cumulative probabilities = A: 0.000000 B: 0.792383 C: 0.792383 D: 0.792383 E: 1.000000 A B C D E A 0 -4 10 6 4 B 4 0 6 6 0 C -10 -6 0 -2 10 D -6 -6 2 0 -6 E -4 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.531250001984 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 E=19 B=11 D=10 so D is eliminated. Round 2 votes counts: A=33 C=29 E=19 B=19 so E is eliminated. Round 3 votes counts: C=42 A=38 B=20 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:208 C:196 E:196 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 6 4 B 4 0 6 6 0 C -10 -6 0 -2 10 D -6 -6 2 0 -6 E -4 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.531250001984 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 6 4 B 4 0 6 6 0 C -10 -6 0 -2 10 D -6 -6 2 0 -6 E -4 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.531250001984 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 6 4 B 4 0 6 6 0 C -10 -6 0 -2 10 D -6 -6 2 0 -6 E -4 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.531250001984 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 303: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (13) C A D B E (7) B E C A D (5) B C E A D (5) B C A D E (5) E D C A B (4) E D A B C (4) D A E C B (4) E D B A C (3) E C D A B (3) D E A C B (3) D A C E B (3) C E D A B (3) C E B A D (3) C B A D E (3) C A B D E (3) E B C D A (2) D A E B C (2) D A B E C (2) B A D E C (2) B A C D E (2) A D B C E (2) E C B D A (1) E B D C A (1) E B D A C (1) D E A B C (1) D C E A B (1) C E A D B (1) C B E A D (1) B E D A C (1) B E C D A (1) B D E A C (1) B C A E D (1) B A D C E (1) A D C B E (1) A D B E C (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 22 0 -8 -16 B -22 0 -10 -18 -6 C 0 10 0 -4 -10 D 8 18 4 0 -6 E 16 6 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 0 -8 -16 B -22 0 -10 -18 -6 C 0 10 0 -4 -10 D 8 18 4 0 -6 E 16 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=24 C=21 D=16 A=7 so A is eliminated. Round 2 votes counts: E=32 B=26 C=22 D=20 so D is eliminated. Round 3 votes counts: E=42 B=31 C=27 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:212 A:199 C:198 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 0 -8 -16 B -22 0 -10 -18 -6 C 0 10 0 -4 -10 D 8 18 4 0 -6 E 16 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 0 -8 -16 B -22 0 -10 -18 -6 C 0 10 0 -4 -10 D 8 18 4 0 -6 E 16 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 0 -8 -16 B -22 0 -10 -18 -6 C 0 10 0 -4 -10 D 8 18 4 0 -6 E 16 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 304: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) C D A E B (7) B E A D C (7) A E C D B (7) E B A C D (5) C D A B E (5) E C D A B (4) E B D C A (4) D C E B A (4) D C B A E (4) B E D C A (4) B A E D C (4) A C D E B (4) A B E C D (4) E B A D C (3) D C E A B (3) B D C E A (3) B D C A E (3) E D C B A (2) E A C D B (2) D C A B E (2) B A E C D (2) B A D C E (2) A E B C D (2) E A B C D (1) D C B E A (1) C A D E B (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 8 14 B -4 0 -10 -8 0 C -6 10 0 8 -4 D -8 8 -8 0 -4 E -14 0 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 8 14 B -4 0 -10 -8 0 C -6 10 0 8 -4 D -8 8 -8 0 -4 E -14 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 E=21 D=14 C=13 so C is eliminated. Round 2 votes counts: A=28 D=26 B=25 E=21 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:216 C:204 E:197 D:194 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 8 14 B -4 0 -10 -8 0 C -6 10 0 8 -4 D -8 8 -8 0 -4 E -14 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 8 14 B -4 0 -10 -8 0 C -6 10 0 8 -4 D -8 8 -8 0 -4 E -14 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 8 14 B -4 0 -10 -8 0 C -6 10 0 8 -4 D -8 8 -8 0 -4 E -14 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 305: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) E B A C D (12) D C A E B (10) E A B C D (7) B E D C A (6) B E A C D (6) A E C D B (6) A C D E B (6) E B D A C (4) A C E D B (4) E B A D C (3) D C B A E (2) D B C E A (2) D B C A E (2) C D A E B (2) B E A D C (2) B D E C A (2) B D C E A (2) A E C B D (2) E D A B C (1) D A C E B (1) C D A B E (1) B E D A C (1) B C D A E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 14 0 4 B -10 0 4 0 -18 C -14 -4 0 -2 -6 D 0 0 2 0 -12 E -4 18 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.811144 B: 0.000000 C: 0.000000 D: 0.188856 E: 0.000000 Sum of squares = 0.693620863221 Cumulative probabilities = A: 0.811144 B: 0.811144 C: 0.811144 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 0 4 B -10 0 4 0 -18 C -14 -4 0 -2 -6 D 0 0 2 0 -12 E -4 18 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000071349 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 B=20 A=20 C=3 so C is eliminated. Round 2 votes counts: D=33 E=27 B=20 A=20 so B is eliminated. Round 3 votes counts: E=42 D=38 A=20 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:214 D:195 B:188 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 0 4 B -10 0 4 0 -18 C -14 -4 0 -2 -6 D 0 0 2 0 -12 E -4 18 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000071349 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 0 4 B -10 0 4 0 -18 C -14 -4 0 -2 -6 D 0 0 2 0 -12 E -4 18 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000071349 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 0 4 B -10 0 4 0 -18 C -14 -4 0 -2 -6 D 0 0 2 0 -12 E -4 18 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000071349 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 306: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) C B A D E (7) D C E B A (6) C B D A E (6) B C A E D (6) A E B C D (6) A B E C D (6) E A D B C (5) E D A B C (4) D C B E A (4) E D B A C (3) D E C B A (3) D E C A B (3) C D B E A (3) E B D C A (2) D E A C B (2) D C E A B (2) C D B A E (2) C B D E A (2) C A B D E (2) B A E C D (2) B A C E D (2) A E D B C (2) E D B C A (1) E B C D A (1) E A B D C (1) D E B C A (1) D E A B C (1) D C A E B (1) B E A C D (1) A E B D C (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -4 6 10 B 4 0 8 12 10 C 4 -8 0 16 10 D -6 -12 -16 0 -6 E -10 -10 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 6 10 B 4 0 8 12 10 C 4 -8 0 16 10 D -6 -12 -16 0 -6 E -10 -10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 C=22 E=17 B=11 so B is eliminated. Round 2 votes counts: A=31 C=28 D=23 E=18 so E is eliminated. Round 3 votes counts: A=38 D=33 C=29 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:217 C:211 A:204 E:188 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 6 10 B 4 0 8 12 10 C 4 -8 0 16 10 D -6 -12 -16 0 -6 E -10 -10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 6 10 B 4 0 8 12 10 C 4 -8 0 16 10 D -6 -12 -16 0 -6 E -10 -10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 6 10 B 4 0 8 12 10 C 4 -8 0 16 10 D -6 -12 -16 0 -6 E -10 -10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 307: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) D C B E A (7) C B E A D (7) C B E D A (6) A D E B C (6) D C B A E (4) A E B C D (4) D B C E A (3) D A C E B (3) B C E D A (3) A E D C B (3) A E D B C (3) E A B C D (2) D B E A C (2) D A E B C (2) D A B C E (2) C E B A D (2) C A E B D (2) B D E C A (2) B C E A D (2) A E C B D (2) A D E C B (2) A C E B D (2) E C B A D (1) E B D A C (1) E B C A D (1) E B A C D (1) E A B D C (1) D E B A C (1) D E A B C (1) D B E C A (1) D B A E C (1) D A C B E (1) D A B E C (1) C E A B D (1) C D B E A (1) C D B A E (1) C B D A E (1) C A B E D (1) B E C D A (1) B E C A D (1) B E A C D (1) B C D E A (1) A E B D C (1) Total count = 100 A B C D E A 0 -20 -14 -8 -16 B 20 0 -10 10 16 C 14 10 0 4 16 D 8 -10 -4 0 2 E 16 -16 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -14 -8 -16 B 20 0 -10 10 16 C 14 10 0 4 16 D 8 -10 -4 0 2 E 16 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 A=23 B=11 E=7 so E is eliminated. Round 2 votes counts: C=31 D=29 A=26 B=14 so B is eliminated. Round 3 votes counts: C=40 D=32 A=28 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:218 D:198 E:191 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -14 -8 -16 B 20 0 -10 10 16 C 14 10 0 4 16 D 8 -10 -4 0 2 E 16 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 -8 -16 B 20 0 -10 10 16 C 14 10 0 4 16 D 8 -10 -4 0 2 E 16 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 -8 -16 B 20 0 -10 10 16 C 14 10 0 4 16 D 8 -10 -4 0 2 E 16 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 308: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) D E B A C (6) C A B E D (6) E A C B D (5) B D A C E (5) A C B E D (5) E D C B A (4) E C D B A (4) E C A D B (4) D E B C A (4) D B E A C (4) D B A E C (4) C A B D E (4) A B C D E (4) E D B C A (3) E C A B D (3) D B A C E (3) E D A C B (2) E A C D B (2) D B C A E (2) C E B A D (2) B D C A E (2) A C B D E (2) E C D A B (1) E A D B C (1) D B E C A (1) D A B E C (1) C D B E A (1) C B E D A (1) C B D A E (1) C B A D E (1) C A E B D (1) B C A D E (1) B A D C E (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -6 2 -10 B 0 0 -14 4 -2 C 6 14 0 12 0 D -2 -4 -12 0 -6 E 10 2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.479298 D: 0.000000 E: 0.520702 Sum of squares = 0.500857120616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.479298 D: 0.479298 E: 1.000000 A B C D E A 0 0 -6 2 -10 B 0 0 -14 4 -2 C 6 14 0 12 0 D -2 -4 -12 0 -6 E 10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=25 C=24 A=13 B=9 so B is eliminated. Round 2 votes counts: D=32 E=29 C=25 A=14 so A is eliminated. Round 3 votes counts: C=37 D=33 E=30 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:209 B:194 A:193 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 2 -10 B 0 0 -14 4 -2 C 6 14 0 12 0 D -2 -4 -12 0 -6 E 10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 -10 B 0 0 -14 4 -2 C 6 14 0 12 0 D -2 -4 -12 0 -6 E 10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 -10 B 0 0 -14 4 -2 C 6 14 0 12 0 D -2 -4 -12 0 -6 E 10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 309: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) D B C A E (6) B D E A C (6) E D B C A (5) E B D A C (5) D C A B E (5) C A D B E (5) B D A C E (5) E A C B D (4) D B E C A (4) A C E B D (4) A C B D E (4) E D C A B (3) E D B A C (3) D E B C A (3) A E C B D (3) A C B E D (3) E C D A B (2) E A C D B (2) C D A B E (2) C A E D B (2) B D C A E (2) B A E C D (2) A B C D E (2) E D C B A (1) E A B C D (1) D E C A B (1) D B C E A (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A C D (1) B D E C A (1) B C A D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 8 -12 12 B 8 0 14 8 20 C -8 -14 0 -4 4 D 12 -8 4 0 14 E -12 -20 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -12 12 B 8 0 14 8 20 C -8 -14 0 -4 4 D 12 -8 4 0 14 E -12 -20 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=26 B=26 D=20 A=17 C=11 so C is eliminated. Round 2 votes counts: E=26 B=26 A=26 D=22 so D is eliminated. Round 3 votes counts: B=37 A=33 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:211 A:200 C:189 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -12 12 B 8 0 14 8 20 C -8 -14 0 -4 4 D 12 -8 4 0 14 E -12 -20 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -12 12 B 8 0 14 8 20 C -8 -14 0 -4 4 D 12 -8 4 0 14 E -12 -20 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -12 12 B 8 0 14 8 20 C -8 -14 0 -4 4 D 12 -8 4 0 14 E -12 -20 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 310: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (14) E C A D B (10) C E A D B (8) C E B A D (7) B C E D A (7) D A B C E (4) B D A C E (4) E A C D B (3) D B A C E (3) D A B E C (3) A E D C B (3) A D E C B (3) A D C E B (3) E C B A D (2) E A D C B (2) D A C B E (2) C D A E B (2) C A D E B (2) B E C D A (2) E C A B D (1) E B A C D (1) D B A E C (1) C E D A B (1) C E A B D (1) C B E A D (1) C B D E A (1) C A E D B (1) B E C A D (1) B D E A C (1) B D C A E (1) B C D A E (1) A E C D B (1) A D E B C (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 6 2 6 2 B -6 0 -10 -10 -6 C -2 10 0 8 0 D -6 10 -8 0 -6 E -2 6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 6 2 B -6 0 -10 -10 -6 C -2 10 0 8 0 D -6 10 -8 0 -6 E -2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=24 E=19 D=13 A=13 so D is eliminated. Round 2 votes counts: B=35 C=24 A=22 E=19 so E is eliminated. Round 3 votes counts: C=37 B=36 A=27 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:208 C:208 E:205 D:195 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 6 2 B -6 0 -10 -10 -6 C -2 10 0 8 0 D -6 10 -8 0 -6 E -2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 6 2 B -6 0 -10 -10 -6 C -2 10 0 8 0 D -6 10 -8 0 -6 E -2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 6 2 B -6 0 -10 -10 -6 C -2 10 0 8 0 D -6 10 -8 0 -6 E -2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 311: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (12) B E A D C (9) C A D E B (6) B E C D A (6) C D A E B (5) B A E D C (5) A C D E B (5) B E D C A (4) B E C A D (4) A C D B E (4) E D C A B (3) C D E A B (3) B A C D E (3) E D C B A (2) E C B D A (2) E B D C A (2) B C E D A (2) A D C B E (2) A D B C E (2) E D B C A (1) E C D B A (1) E B C D A (1) E A B D C (1) D C E A B (1) D C A E B (1) D A C E B (1) C E D B A (1) C B E D A (1) C B A D E (1) C A D B E (1) B E D A C (1) B A D E C (1) A D E C B (1) A D E B C (1) A B E D C (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 4 24 10 B -6 0 -6 -6 0 C -4 6 0 -6 6 D -24 6 6 0 6 E -10 0 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 24 10 B -6 0 -6 -6 0 C -4 6 0 -6 6 D -24 6 6 0 6 E -10 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=31 C=18 E=13 D=3 so D is eliminated. Round 2 votes counts: B=35 A=32 C=20 E=13 so E is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:201 D:197 B:191 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 24 10 B -6 0 -6 -6 0 C -4 6 0 -6 6 D -24 6 6 0 6 E -10 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 24 10 B -6 0 -6 -6 0 C -4 6 0 -6 6 D -24 6 6 0 6 E -10 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 24 10 B -6 0 -6 -6 0 C -4 6 0 -6 6 D -24 6 6 0 6 E -10 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 312: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) B E D C A (8) A D C E B (8) A C E D B (8) D A C E B (5) B D E C A (5) B D A E C (5) E C A B D (4) E B C D A (4) D B E C A (4) D A B C E (4) D B A E C (3) C E A D B (3) B E C D A (3) E C B A D (2) D C E A B (2) B D E A C (2) A C D E B (2) A B D C E (2) A B C E D (2) E C D A B (1) E C B D A (1) D E B C A (1) D B A C E (1) D A C B E (1) C E A B D (1) B A E C D (1) B A D E C (1) B A D C E (1) A D C B E (1) A D B C E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 0 0 -4 B 6 0 18 10 16 C 0 -18 0 -10 -12 D 0 -10 10 0 0 E 4 -16 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 0 -4 B 6 0 18 10 16 C 0 -18 0 -10 -12 D 0 -10 10 0 0 E 4 -16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=26 D=21 E=12 C=4 so C is eliminated. Round 2 votes counts: B=37 A=26 D=21 E=16 so E is eliminated. Round 3 votes counts: B=44 A=34 D=22 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:200 E:200 A:195 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 0 -4 B 6 0 18 10 16 C 0 -18 0 -10 -12 D 0 -10 10 0 0 E 4 -16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 -4 B 6 0 18 10 16 C 0 -18 0 -10 -12 D 0 -10 10 0 0 E 4 -16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 -4 B 6 0 18 10 16 C 0 -18 0 -10 -12 D 0 -10 10 0 0 E 4 -16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 313: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) C D B A E (10) B A D C E (7) C E D B A (5) A B D E C (5) E C B A D (4) D C A B E (4) B A E D C (4) B A D E C (4) E C D A B (3) E A D B C (3) D A B E C (3) C E D A B (3) C E B A D (3) C D E A B (3) E D A B C (2) E B A C D (2) D E A B C (2) D A B C E (2) C D A B E (2) C B E A D (2) B D A C E (2) A B E D C (2) E A B C D (1) D E A C B (1) D B C A E (1) D A E B C (1) D A C B E (1) C D E B A (1) C D B E A (1) C B D E A (1) B E A D C (1) B A E C D (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 0 14 4 2 B 0 0 12 2 8 C -14 -12 0 -14 -2 D -4 -2 14 0 2 E -2 -8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.361651 B: 0.638349 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.538280856432 Cumulative probabilities = A: 0.361651 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 4 2 B 0 0 12 2 8 C -14 -12 0 -14 -2 D -4 -2 14 0 2 E -2 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 B=20 D=15 A=8 so A is eliminated. Round 2 votes counts: C=31 E=27 B=27 D=15 so D is eliminated. Round 3 votes counts: C=36 B=33 E=31 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:210 D:205 E:195 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 4 2 B 0 0 12 2 8 C -14 -12 0 -14 -2 D -4 -2 14 0 2 E -2 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 4 2 B 0 0 12 2 8 C -14 -12 0 -14 -2 D -4 -2 14 0 2 E -2 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 4 2 B 0 0 12 2 8 C -14 -12 0 -14 -2 D -4 -2 14 0 2 E -2 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 314: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (12) C B D A E (12) A E D B C (10) D B C A E (9) C B D E A (7) E A D C B (6) B D C A E (6) B C D A E (5) A D E B C (5) E A C B D (4) E A C D B (3) D A B C E (3) E C A B D (2) D B A C E (2) C B E D A (2) B D C E A (2) E C B D A (1) E C B A D (1) E B C D A (1) C E B D A (1) C E B A D (1) C A D B E (1) C A B D E (1) B C D E A (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -6 -4 12 B 2 0 14 -6 6 C 6 -14 0 -12 8 D 4 6 12 0 12 E -12 -6 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -4 12 B 2 0 14 -6 6 C 6 -14 0 -12 8 D 4 6 12 0 12 E -12 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999346 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=25 A=17 D=14 B=14 so D is eliminated. Round 2 votes counts: E=30 C=25 B=25 A=20 so A is eliminated. Round 3 votes counts: E=45 B=29 C=26 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:217 B:208 A:200 C:194 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 12 B 2 0 14 -6 6 C 6 -14 0 -12 8 D 4 6 12 0 12 E -12 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999346 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 12 B 2 0 14 -6 6 C 6 -14 0 -12 8 D 4 6 12 0 12 E -12 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999346 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 12 B 2 0 14 -6 6 C 6 -14 0 -12 8 D 4 6 12 0 12 E -12 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999346 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 315: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) B D E A C (7) B A C D E (6) C A E D B (5) E D B C A (4) D E A C B (4) B E D C A (4) A C D E B (4) A C B D E (4) D E B A C (3) D E A B C (3) D A E B C (3) C A B D E (3) B E C D A (3) A B D C E (3) E D C B A (2) E C B D A (2) D A E C B (2) C B A E D (2) C A E B D (2) B C E A D (2) B C A E D (2) B A D E C (2) A D C E B (2) E D B A C (1) E C D B A (1) E B D C A (1) D B E A C (1) D B A E C (1) D A B E C (1) C E D A B (1) C E A D B (1) C B E A D (1) C A B E D (1) B D A E C (1) B C A D E (1) B A D C E (1) A D E B C (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 -8 0 B -4 0 6 -2 -2 C -6 -6 0 -14 -12 D 8 2 14 0 12 E 0 2 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -8 0 B -4 0 6 -2 -2 C -6 -6 0 -14 -12 D 8 2 14 0 12 E 0 2 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=20 D=18 A=17 C=16 so C is eliminated. Round 2 votes counts: B=32 A=28 E=22 D=18 so D is eliminated. Round 3 votes counts: B=34 A=34 E=32 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:218 A:201 E:201 B:199 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -8 0 B -4 0 6 -2 -2 C -6 -6 0 -14 -12 D 8 2 14 0 12 E 0 2 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -8 0 B -4 0 6 -2 -2 C -6 -6 0 -14 -12 D 8 2 14 0 12 E 0 2 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -8 0 B -4 0 6 -2 -2 C -6 -6 0 -14 -12 D 8 2 14 0 12 E 0 2 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 316: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) E D B C A (8) D E B A C (8) C A B D E (7) A C B D E (6) D B E A C (5) C A E D B (5) B D E C A (5) E C A D B (4) C A E B D (4) A C D B E (4) E B D C A (3) E A C D B (3) B D E A C (3) A C D E B (3) C A B E D (2) B E D C A (2) B D A C E (2) B C A D E (2) A C E D B (2) E D A B C (1) E B C D A (1) E A D C B (1) D B A E C (1) D B A C E (1) D A E B C (1) C B A D E (1) B D C E A (1) B D C A E (1) B A C D E (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 8 -6 -10 B 10 0 14 -16 -10 C -8 -14 0 -6 -14 D 6 16 6 0 6 E 10 10 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -6 -10 B 10 0 14 -16 -10 C -8 -14 0 -6 -14 D 6 16 6 0 6 E 10 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=19 B=17 A=17 D=16 so D is eliminated. Round 2 votes counts: E=39 B=24 C=19 A=18 so A is eliminated. Round 3 votes counts: E=41 C=34 B=25 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:217 E:214 B:199 A:191 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 -6 -10 B 10 0 14 -16 -10 C -8 -14 0 -6 -14 D 6 16 6 0 6 E 10 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -6 -10 B 10 0 14 -16 -10 C -8 -14 0 -6 -14 D 6 16 6 0 6 E 10 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -6 -10 B 10 0 14 -16 -10 C -8 -14 0 -6 -14 D 6 16 6 0 6 E 10 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 317: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) D E B A C (8) C A B E D (8) B A C D E (8) B D A E C (7) D B E A C (6) A B C E D (6) C A E B D (5) B A D C E (5) C A B D E (4) E C D A B (3) D E C B A (3) A B C D E (3) D E C A B (2) D E B C A (2) D C E B A (2) C E D A B (2) A C B E D (2) E D C B A (1) E D B A C (1) E C A D B (1) E A C D B (1) E A C B D (1) E A B C D (1) D B A E C (1) C E A D B (1) C D E A B (1) C B A D E (1) C A E D B (1) B E A D C (1) B D E A C (1) B A E D C (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 2 10 4 8 B -2 0 4 12 12 C -10 -4 0 0 0 D -4 -12 0 0 12 E -8 -12 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 8 B -2 0 4 12 12 C -10 -4 0 0 0 D -4 -12 0 0 12 E -8 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=24 B=24 C=23 E=17 A=12 so A is eliminated. Round 2 votes counts: B=33 C=26 D=24 E=17 so E is eliminated. Round 3 votes counts: D=34 B=34 C=32 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:212 D:198 C:193 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 8 B -2 0 4 12 12 C -10 -4 0 0 0 D -4 -12 0 0 12 E -8 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 8 B -2 0 4 12 12 C -10 -4 0 0 0 D -4 -12 0 0 12 E -8 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 8 B -2 0 4 12 12 C -10 -4 0 0 0 D -4 -12 0 0 12 E -8 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 318: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) C B D A E (8) A E D B C (6) A E C D B (6) E D B C A (5) E A D C B (5) E A D B C (5) A E C B D (5) C B D E A (4) E D A B C (3) E C A B D (3) D E B A C (3) C B A D E (3) A D B C E (3) E C D B A (2) E C B D A (2) E C B A D (2) B D C E A (2) A E D C B (2) A D E B C (2) A D B E C (2) A C B E D (2) E D B A C (1) E C A D B (1) E A C D B (1) E A C B D (1) D E B C A (1) D B E A C (1) D B C A E (1) C B E D A (1) C B A E D (1) C A E B D (1) C A B D E (1) B D C A E (1) B C D E A (1) B C D A E (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 10 -4 B -4 0 -4 -16 -16 C -2 4 0 -4 -18 D -10 16 4 0 -12 E 4 16 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 10 -4 B -4 0 -4 -16 -16 C -2 4 0 -4 -18 D -10 16 4 0 -12 E 4 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=31 A=31 C=19 D=14 B=5 so B is eliminated. Round 2 votes counts: E=31 A=31 C=21 D=17 so D is eliminated. Round 3 votes counts: E=36 C=33 A=31 so A is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:206 D:199 C:190 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 10 -4 B -4 0 -4 -16 -16 C -2 4 0 -4 -18 D -10 16 4 0 -12 E 4 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 10 -4 B -4 0 -4 -16 -16 C -2 4 0 -4 -18 D -10 16 4 0 -12 E 4 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 10 -4 B -4 0 -4 -16 -16 C -2 4 0 -4 -18 D -10 16 4 0 -12 E 4 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 319: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) A C E D B (8) C A D E B (6) E D B A C (5) D B E C A (5) D B C A E (5) A E C D B (5) E D A C B (4) E A C D B (4) C A B D E (4) B E D A C (4) B D C E A (4) C A D B E (3) C A B E D (3) B D C A E (3) A E C B D (3) A C E B D (3) E D A B C (2) E B D A C (2) E A B C D (2) D E B A C (2) D C B A E (2) C A E B D (2) E B A D C (1) E A D C B (1) E A C B D (1) D E A C B (1) D B E A C (1) D B C E A (1) C D B A E (1) C B A D E (1) B E C D A (1) B D E A C (1) B C A D E (1) Total count = 100 A B C D E A 0 4 0 -4 0 B -4 0 -4 -12 -4 C 0 4 0 -4 -6 D 4 12 4 0 -2 E 0 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250863 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.749137 Sum of squares = 0.62413811516 Cumulative probabilities = A: 0.250863 B: 0.250863 C: 0.250863 D: 0.250863 E: 1.000000 A B C D E A 0 4 0 -4 0 B -4 0 -4 -12 -4 C 0 4 0 -4 -6 D 4 12 4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556377 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 C=20 A=19 D=17 so D is eliminated. Round 2 votes counts: B=34 E=25 C=22 A=19 so A is eliminated. Round 3 votes counts: B=34 E=33 C=33 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:209 E:206 A:200 C:197 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -4 0 B -4 0 -4 -12 -4 C 0 4 0 -4 -6 D 4 12 4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556377 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -4 0 B -4 0 -4 -12 -4 C 0 4 0 -4 -6 D 4 12 4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556377 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -4 0 B -4 0 -4 -12 -4 C 0 4 0 -4 -6 D 4 12 4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556377 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 320: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) D A B E C (7) C A B E D (6) C E A D B (5) A B D C E (5) E D C B A (4) E D C A B (4) B D A E C (4) D E A B C (3) D B A E C (3) C E D A B (3) C A E D B (3) A B C D E (3) E C D A B (2) D E C A B (2) D E A C B (2) C E B D A (2) C E A B D (2) C B E A D (2) B C E A D (2) B A D E C (2) B A D C E (2) B A C E D (2) B A C D E (2) A D B E C (2) A C B E D (2) E D B C A (1) E C B D A (1) E B D C A (1) D E B C A (1) D E B A C (1) D B E A C (1) D A E C B (1) D A E B C (1) C E D B A (1) C E B A D (1) C B A E D (1) C A E B D (1) B C A E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -10 -8 0 B -12 0 -10 -12 -2 C 10 10 0 6 -4 D 8 12 -6 0 -14 E 0 2 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.168368 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.831632 Sum of squares = 0.719960061977 Cumulative probabilities = A: 0.168368 B: 0.168368 C: 0.168368 D: 0.168368 E: 1.000000 A B C D E A 0 12 -10 -8 0 B -12 0 -10 -12 -2 C 10 10 0 6 -4 D 8 12 -6 0 -14 E 0 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.59183695587 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 D=22 B=15 A=14 so A is eliminated. Round 2 votes counts: C=30 D=24 B=24 E=22 so E is eliminated. Round 3 votes counts: C=42 D=33 B=25 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 E:210 D:200 A:197 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -10 -8 0 B -12 0 -10 -12 -2 C 10 10 0 6 -4 D 8 12 -6 0 -14 E 0 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.59183695587 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 -8 0 B -12 0 -10 -12 -2 C 10 10 0 6 -4 D 8 12 -6 0 -14 E 0 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.59183695587 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 -8 0 B -12 0 -10 -12 -2 C 10 10 0 6 -4 D 8 12 -6 0 -14 E 0 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.59183695587 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 321: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) B D A E C (10) D B E C A (9) E C A B D (6) E C D A B (5) B A D C E (5) A B D C E (5) E C D B A (4) D E B C A (4) B D E A C (4) A E C B D (4) E D C B A (3) D B A C E (3) C E A D B (3) D B E A C (2) C E D B A (2) C E A B D (2) B A D E C (2) E D B C A (1) E C A D B (1) E A B C D (1) D C E B A (1) D C B A E (1) D C A B E (1) C E D A B (1) C A E D B (1) C A E B D (1) C A D B E (1) B E D A C (1) A D C B E (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 6 -4 0 B 4 0 0 12 -2 C -6 0 0 -8 -18 D 4 -12 8 0 0 E 0 2 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.133317 E: 0.866683 Sum of squares = 0.768912607717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.133317 E: 1.000000 A B C D E A 0 -4 6 -4 0 B 4 0 0 12 -2 C -6 0 0 -8 -18 D 4 -12 8 0 0 E 0 2 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102184503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=22 E=21 D=21 C=11 so C is eliminated. Round 2 votes counts: E=29 A=28 B=22 D=21 so D is eliminated. Round 3 votes counts: B=37 E=34 A=29 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:207 D:200 A:199 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 -4 0 B 4 0 0 12 -2 C -6 0 0 -8 -18 D 4 -12 8 0 0 E 0 2 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102184503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 0 B 4 0 0 12 -2 C -6 0 0 -8 -18 D 4 -12 8 0 0 E 0 2 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102184503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 0 B 4 0 0 12 -2 C -6 0 0 -8 -18 D 4 -12 8 0 0 E 0 2 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102184503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 322: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (20) E D C B A (12) E C D B A (7) E D A C B (5) E D C A B (4) C B E D A (4) B C A D E (4) A D E B C (4) A B D C E (4) A B C E D (4) E B C A D (3) D E C A B (3) D E A C B (3) D C E B A (2) C E D B A (2) C D B E A (2) B C D A E (2) B C A E D (2) B A C D E (2) A D B E C (2) E C D A B (1) E C B D A (1) E A D C B (1) D E C B A (1) D A B C E (1) B C E A D (1) A E D C B (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -2 0 -4 B -10 0 2 -4 -2 C 2 -2 0 10 0 D 0 4 -10 0 2 E 4 2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.329998 D: 0.000000 E: 0.670002 Sum of squares = 0.557801575177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.329998 D: 0.329998 E: 1.000000 A B C D E A 0 10 -2 0 -4 B -10 0 2 -4 -2 C 2 -2 0 10 0 D 0 4 -10 0 2 E 4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=34 B=11 D=10 C=8 so C is eliminated. Round 2 votes counts: A=37 E=36 B=15 D=12 so D is eliminated. Round 3 votes counts: E=45 A=38 B=17 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:205 A:202 E:202 D:198 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 0 -4 B -10 0 2 -4 -2 C 2 -2 0 10 0 D 0 4 -10 0 2 E 4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 0 -4 B -10 0 2 -4 -2 C 2 -2 0 10 0 D 0 4 -10 0 2 E 4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 0 -4 B -10 0 2 -4 -2 C 2 -2 0 10 0 D 0 4 -10 0 2 E 4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 323: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D C A B E (6) C D B E A (6) E B C D A (5) B D A C E (5) E B A C D (4) D A B C E (4) D B C A E (3) D A C B E (3) C D B A E (3) A D B C E (3) A B D E C (3) E C B A D (2) E C A B D (2) E B A D C (2) E A C B D (2) E A B D C (2) E A B C D (2) D C B A E (2) C E D B A (2) C E D A B (2) C D A E B (2) C D A B E (2) B D A E C (2) B A E D C (2) B A D E C (2) A E C D B (2) E C D A B (1) E B C A D (1) E A C D B (1) D B C E A (1) D B A C E (1) C E B D A (1) C D E B A (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E C A (1) B D C A E (1) B C D E A (1) B A D C E (1) A E B D C (1) A D C E B (1) A D C B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -4 -26 4 B 16 0 4 6 16 C 4 -4 0 -2 6 D 26 -6 2 0 12 E -4 -16 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -26 4 B 16 0 4 6 16 C 4 -4 0 -2 6 D 26 -6 2 0 12 E -4 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=20 C=19 B=18 A=13 so A is eliminated. Round 2 votes counts: E=33 D=25 B=23 C=19 so C is eliminated. Round 3 votes counts: D=39 E=38 B=23 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:221 D:217 C:202 E:181 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 -26 4 B 16 0 4 6 16 C 4 -4 0 -2 6 D 26 -6 2 0 12 E -4 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -26 4 B 16 0 4 6 16 C 4 -4 0 -2 6 D 26 -6 2 0 12 E -4 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -26 4 B 16 0 4 6 16 C 4 -4 0 -2 6 D 26 -6 2 0 12 E -4 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 324: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D A C E B (7) D A C B E (7) C B E D A (6) A E D B C (6) E B C A D (5) E A B C D (5) B C E A D (5) E A B D C (4) D A B C E (4) A E B D C (4) A D E B C (3) E C B A D (2) D C B A E (2) D A E C B (2) C D B E A (2) C D B A E (2) C B E A D (2) B E C A D (2) B C E D A (2) B C D A E (2) A D E C B (2) A D B E C (2) D C E A B (1) D C A E B (1) D C A B E (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B D A (1) C D E A B (1) C B D A E (1) B E A C D (1) B D C A E (1) B A E C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 20 8 4 B -8 0 18 8 -8 C -20 -18 0 -2 -2 D -8 -8 2 0 -12 E -4 8 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 20 8 4 B -8 0 18 8 -8 C -20 -18 0 -2 -2 D -8 -8 2 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 A=19 C=16 B=14 so B is eliminated. Round 2 votes counts: D=28 E=27 C=25 A=20 so A is eliminated. Round 3 votes counts: E=39 D=36 C=25 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:220 E:209 B:205 D:187 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 20 8 4 B -8 0 18 8 -8 C -20 -18 0 -2 -2 D -8 -8 2 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 20 8 4 B -8 0 18 8 -8 C -20 -18 0 -2 -2 D -8 -8 2 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 20 8 4 B -8 0 18 8 -8 C -20 -18 0 -2 -2 D -8 -8 2 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 325: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) C B A D E (6) B C A D E (6) D E A C B (5) B E D C A (5) B E C D A (5) E A D C B (4) E D A B C (3) C B A E D (3) C A D B E (3) B C D A E (3) B C A E D (3) A D C E B (3) A C D E B (3) E D B A C (2) E B D C A (2) E B A C D (2) E A C B D (2) D E B A C (2) D C A B E (2) D A C E B (2) C A B E D (2) B D C E A (2) B C D E A (2) A E D C B (2) A D E C B (2) A C E B D (2) A C D B E (2) E D B C A (1) E A D B C (1) E A B D C (1) D E B C A (1) D E A B C (1) D C A E B (1) D B E C A (1) D B C A E (1) C A B D E (1) B E C A D (1) B D E C A (1) B C E D A (1) B C E A D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -8 2 -4 B 2 0 -2 4 4 C 8 2 0 -2 -2 D -2 -4 2 0 0 E 4 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.168992 E: 0.081008 Sum of squares = 0.347620566042 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.918992 E: 1.000000 A B C D E A 0 -2 -8 2 -4 B 2 0 -2 4 4 C 8 2 0 -2 -2 D -2 -4 2 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.125000 E: 0.125000 Sum of squares = 0.343750000022 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.875000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=24 D=16 C=15 A=15 so C is eliminated. Round 2 votes counts: B=39 E=24 A=21 D=16 so D is eliminated. Round 3 votes counts: B=41 E=33 A=26 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:204 C:203 E:201 D:198 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 2 -4 B 2 0 -2 4 4 C 8 2 0 -2 -2 D -2 -4 2 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.125000 E: 0.125000 Sum of squares = 0.343750000022 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 2 -4 B 2 0 -2 4 4 C 8 2 0 -2 -2 D -2 -4 2 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.125000 E: 0.125000 Sum of squares = 0.343750000022 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 2 -4 B 2 0 -2 4 4 C 8 2 0 -2 -2 D -2 -4 2 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.125000 E: 0.125000 Sum of squares = 0.343750000022 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 326: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) E D B A C (9) C D E A B (7) C A B D E (7) B A E C D (7) C D A E B (6) B A E D C (6) D E C B A (5) D E B C A (5) C A D B E (5) A C B E D (5) D E B A C (4) D E C A B (3) B E D A C (3) B E A D C (3) E B D A C (2) C A D E B (2) C A B E D (2) B A C D E (2) E D C B A (1) D C E B A (1) D C E A B (1) D B C E A (1) B D E A C (1) A E B C D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 8 2 8 B 0 0 8 0 6 C -8 -8 0 10 0 D -2 0 -10 0 2 E -8 -6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.561639 B: 0.438361 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.507598779436 Cumulative probabilities = A: 0.561639 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 8 B 0 0 8 0 6 C -8 -8 0 10 0 D -2 0 -10 0 2 E -8 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=22 D=20 A=17 E=12 so E is eliminated. Round 2 votes counts: D=30 C=29 B=24 A=17 so A is eliminated. Round 3 votes counts: C=35 B=35 D=30 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:207 C:197 D:195 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 8 B 0 0 8 0 6 C -8 -8 0 10 0 D -2 0 -10 0 2 E -8 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 8 B 0 0 8 0 6 C -8 -8 0 10 0 D -2 0 -10 0 2 E -8 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 8 B 0 0 8 0 6 C -8 -8 0 10 0 D -2 0 -10 0 2 E -8 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 327: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (13) C E D A B (6) C E A D B (6) C E A B D (6) D C E B A (5) E C B A D (4) D A B C E (4) A C E B D (4) E C D B A (3) D C E A B (3) D B E C A (3) D B E A C (3) B A D E C (3) A B D E C (3) E C B D A (2) E B C A D (2) D B A C E (2) D A C E B (2) C E B A D (2) B D E C A (2) A B E C D (2) A B D C E (2) E C A B D (1) E B A C D (1) E A C B D (1) D E C B A (1) D C B E A (1) D B C A E (1) C E D B A (1) C D E A B (1) C D A E B (1) C A E D B (1) B A E C D (1) A E C B D (1) A E B C D (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 -8 -8 B 0 0 -10 -22 -14 C 4 10 0 0 6 D 8 22 0 0 6 E 8 14 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.677736 D: 0.322264 E: 0.000000 Sum of squares = 0.563179877342 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.677736 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -8 -8 B 0 0 -10 -22 -14 C 4 10 0 0 6 D 8 22 0 0 6 E 8 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=24 A=18 E=14 B=6 so B is eliminated. Round 2 votes counts: D=40 C=24 A=22 E=14 so E is eliminated. Round 3 votes counts: D=40 C=36 A=24 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:210 E:205 A:190 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -8 -8 B 0 0 -10 -22 -14 C 4 10 0 0 6 D 8 22 0 0 6 E 8 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -8 -8 B 0 0 -10 -22 -14 C 4 10 0 0 6 D 8 22 0 0 6 E 8 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -8 -8 B 0 0 -10 -22 -14 C 4 10 0 0 6 D 8 22 0 0 6 E 8 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 328: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) E C A B D (6) E C D B A (5) C E D B A (5) B D A E C (5) D C E B A (4) D B A C E (4) C E D A B (4) C E A D B (4) B A D E C (4) E C A D B (3) D B C E A (3) C D E A B (3) B D E C A (3) A C E B D (3) E D B C A (2) E B D C A (2) D E C B A (2) D C B E A (2) B D A C E (2) B A D C E (2) A D B C E (2) A C E D B (2) A B E C D (2) A B D C E (2) E C B D A (1) E A B C D (1) D E B C A (1) C D E B A (1) C A E D B (1) C A D E B (1) B E A D C (1) B E A C D (1) B A E D C (1) A E C B D (1) A C D E B (1) A C D B E (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 -24 -16 -26 B 20 0 0 -20 -6 C 24 0 0 -4 -2 D 16 20 4 0 8 E 26 6 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -24 -16 -26 B 20 0 0 -20 -6 C 24 0 0 -4 -2 D 16 20 4 0 8 E 26 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=20 C=19 B=19 A=17 so A is eliminated. Round 2 votes counts: D=27 C=26 B=26 E=21 so E is eliminated. Round 3 votes counts: C=42 D=29 B=29 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:224 E:213 C:209 B:197 A:157 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -24 -16 -26 B 20 0 0 -20 -6 C 24 0 0 -4 -2 D 16 20 4 0 8 E 26 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -24 -16 -26 B 20 0 0 -20 -6 C 24 0 0 -4 -2 D 16 20 4 0 8 E 26 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -24 -16 -26 B 20 0 0 -20 -6 C 24 0 0 -4 -2 D 16 20 4 0 8 E 26 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 329: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (11) B C A D E (6) E A D C B (5) D C B E A (5) D B C E A (5) D B C A E (5) A E B C D (5) C D B E A (4) C B D E A (4) C B D A E (4) C B A E D (4) E D C A B (3) C B A D E (3) B D C A E (3) A B E C D (3) E C D A B (2) E C A D B (2) E A D B C (2) E A C D B (2) E A C B D (2) D C E B A (2) D B A E C (2) B C D A E (2) B A C D E (2) A E D B C (2) D E C B A (1) D E C A B (1) D E A C B (1) D B A C E (1) C A E B D (1) B A D C E (1) A E C B D (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -14 10 18 B 8 0 4 10 14 C 14 -4 0 -2 8 D -10 -10 2 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 10 18 B 8 0 4 10 14 C 14 -4 0 -2 8 D -10 -10 2 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 C=20 E=18 B=14 so B is eliminated. Round 2 votes counts: C=28 A=28 D=26 E=18 so E is eliminated. Round 3 votes counts: A=39 C=32 D=29 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:208 A:203 D:193 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -14 10 18 B 8 0 4 10 14 C 14 -4 0 -2 8 D -10 -10 2 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 10 18 B 8 0 4 10 14 C 14 -4 0 -2 8 D -10 -10 2 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 10 18 B 8 0 4 10 14 C 14 -4 0 -2 8 D -10 -10 2 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 330: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) C D A E B (12) B E A D C (11) A E B C D (10) C D A B E (6) E A B C D (5) E B A D C (4) C D B E A (4) A C E D B (4) D C B A E (3) B E D A C (3) A E B D C (3) E A B D C (2) D B C E A (2) C A D E B (2) B E C D A (2) B D E C A (2) E B A C D (1) E A C B D (1) D C A B E (1) C E A B D (1) C A E D B (1) B E C A D (1) B E A C D (1) A E D B C (1) A E C B D (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 8 2 6 -4 B -8 0 -2 -4 0 C -2 2 0 6 2 D -6 4 -6 0 -4 E 4 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 8 2 6 -4 B -8 0 -2 -4 0 C -2 2 0 6 2 D -6 4 -6 0 -4 E 4 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000107 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 B=20 D=18 E=13 so E is eliminated. Round 2 votes counts: A=31 C=26 B=25 D=18 so D is eliminated. Round 3 votes counts: C=42 A=31 B=27 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:206 C:204 E:203 D:194 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 2 6 -4 B -8 0 -2 -4 0 C -2 2 0 6 2 D -6 4 -6 0 -4 E 4 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000107 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 6 -4 B -8 0 -2 -4 0 C -2 2 0 6 2 D -6 4 -6 0 -4 E 4 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000107 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 6 -4 B -8 0 -2 -4 0 C -2 2 0 6 2 D -6 4 -6 0 -4 E 4 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000107 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 331: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D B A C E (9) D A B E C (5) A E D C B (5) C E B A D (4) B C D E A (4) A D E B C (4) A D B E C (4) E A C B D (3) C E A B D (3) A E C B D (3) E D C A B (2) E D A C B (2) E C A D B (2) E A C D B (2) D E C A B (2) D B C E A (2) D B C A E (2) C B D E A (2) B D C A E (2) B D A C E (2) B C D A E (2) B C A E D (2) B A D C E (2) A E B C D (2) A B C E D (2) E C D B A (1) E C D A B (1) D E C B A (1) D E B A C (1) D E A C B (1) D B A E C (1) D A E C B (1) C B E D A (1) C B E A D (1) B C E A D (1) B A C E D (1) A E D B C (1) A E C D B (1) A D E C B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 18 10 12 8 B -18 0 2 -2 -6 C -10 -2 0 -4 -14 D -12 2 4 0 0 E -8 6 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 10 12 8 B -18 0 2 -2 -6 C -10 -2 0 -4 -14 D -12 2 4 0 0 E -8 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 E=23 B=16 C=11 so C is eliminated. Round 2 votes counts: E=30 D=25 A=25 B=20 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:224 E:206 D:197 B:188 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 12 8 B -18 0 2 -2 -6 C -10 -2 0 -4 -14 D -12 2 4 0 0 E -8 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 12 8 B -18 0 2 -2 -6 C -10 -2 0 -4 -14 D -12 2 4 0 0 E -8 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 12 8 B -18 0 2 -2 -6 C -10 -2 0 -4 -14 D -12 2 4 0 0 E -8 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 332: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) D E C B A (7) A B C E D (7) B E D A C (6) E C D B A (5) C E D A B (5) C A D E B (5) B A E D C (5) B A D E C (5) D E B C A (4) C D A E B (4) B A E C D (4) E D C B A (3) C D E A B (3) E B C D A (2) C A E D B (2) B E A D C (2) A D B C E (2) A C D B E (2) A C B D E (2) A B D E C (2) A B C D E (2) E D B C A (1) E C B D A (1) E C B A D (1) D C E A B (1) D B E A C (1) D A C B E (1) D A B C E (1) C E D B A (1) C A E B D (1) B E A C D (1) B D A E C (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 0 8 6 12 B 0 0 -6 0 8 C -8 6 0 12 -4 D -6 0 -12 0 -10 E -12 -8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.775908 B: 0.224092 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.652250522341 Cumulative probabilities = A: 0.775908 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 6 12 B 0 0 -6 0 8 C -8 6 0 12 -4 D -6 0 -12 0 -10 E -12 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=24 C=21 D=15 E=13 so E is eliminated. Round 2 votes counts: C=28 A=27 B=26 D=19 so D is eliminated. Round 3 votes counts: C=39 B=32 A=29 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:203 B:201 E:197 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 6 12 B 0 0 -6 0 8 C -8 6 0 12 -4 D -6 0 -12 0 -10 E -12 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 6 12 B 0 0 -6 0 8 C -8 6 0 12 -4 D -6 0 -12 0 -10 E -12 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 6 12 B 0 0 -6 0 8 C -8 6 0 12 -4 D -6 0 -12 0 -10 E -12 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 333: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (18) E D B C A (14) B D C E A (6) C A B D E (5) B C D A E (5) D B E C A (4) C B A D E (4) A C B E D (4) D C B A E (3) C B D A E (3) B D E C A (3) A E C D B (3) A C E B D (3) E D B A C (2) E D A C B (2) E B D C A (2) D E B C A (2) D B C E A (2) A C E D B (2) A C D B E (2) A B C E D (2) E D C B A (1) E A D C B (1) E A D B C (1) E A C D B (1) D C A E B (1) C D B A E (1) C D A B E (1) B E D C A (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -16 -6 16 B 6 0 -10 14 30 C 16 10 0 10 26 D 6 -14 -10 0 22 E -16 -30 -26 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -6 16 B 6 0 -10 14 30 C 16 10 0 10 26 D 6 -14 -10 0 22 E -16 -30 -26 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=24 B=15 C=14 D=12 so D is eliminated. Round 2 votes counts: A=35 E=26 B=21 C=18 so C is eliminated. Round 3 votes counts: A=42 B=32 E=26 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:231 B:220 D:202 A:194 E:153 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -6 16 B 6 0 -10 14 30 C 16 10 0 10 26 D 6 -14 -10 0 22 E -16 -30 -26 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -6 16 B 6 0 -10 14 30 C 16 10 0 10 26 D 6 -14 -10 0 22 E -16 -30 -26 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -6 16 B 6 0 -10 14 30 C 16 10 0 10 26 D 6 -14 -10 0 22 E -16 -30 -26 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 334: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A B C E D (8) E B A C D (6) E B C A D (5) E A D B C (4) D C B E A (4) D C B A E (4) D C A B E (4) A D C B E (4) E D A B C (3) E A B D C (3) D E C B A (3) D E A C B (3) C B D A E (3) C B A D E (3) B C A E D (3) A E B C D (3) A C B D E (3) E D B A C (2) E B C D A (2) E A B C D (2) B C A D E (2) E D A C B (1) E B D A C (1) E B A D C (1) D E C A B (1) D C E B A (1) D A E C B (1) D A C B E (1) C D B A E (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E A D (1) B A C E D (1) A E D C B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 2 12 -8 B 8 0 18 6 -2 C -2 -18 0 0 -8 D -12 -6 0 0 -20 E 8 2 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 2 12 -8 B 8 0 18 6 -2 C -2 -18 0 0 -8 D -12 -6 0 0 -20 E 8 2 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=22 A=21 B=10 C=9 so C is eliminated. Round 2 votes counts: E=38 D=23 A=23 B=16 so B is eliminated. Round 3 votes counts: E=42 A=32 D=26 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:215 A:199 C:186 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 2 12 -8 B 8 0 18 6 -2 C -2 -18 0 0 -8 D -12 -6 0 0 -20 E 8 2 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 12 -8 B 8 0 18 6 -2 C -2 -18 0 0 -8 D -12 -6 0 0 -20 E 8 2 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 12 -8 B 8 0 18 6 -2 C -2 -18 0 0 -8 D -12 -6 0 0 -20 E 8 2 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 335: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (15) D B E A C (5) D B C E A (5) C B D A E (5) B D C A E (5) A E C B D (5) E A C D B (4) E A C B D (4) C D B A E (4) C B A D E (4) C A E B D (4) C A B E D (4) E D A B C (3) E C A D B (3) D E B A C (3) B D A C E (3) A C E B D (3) E A D B C (2) D E C A B (2) D B E C A (2) C E A B D (2) B D A E C (2) E C D A B (1) E A B D C (1) D E B C A (1) D C B E A (1) D B A C E (1) C E A D B (1) C D E B A (1) B A D E C (1) A E B D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -20 -18 18 B 16 0 4 -8 18 C 20 -4 0 -8 16 D 18 8 8 0 20 E -18 -18 -16 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -20 -18 18 B 16 0 4 -8 18 C 20 -4 0 -8 16 D 18 8 8 0 20 E -18 -18 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=25 E=18 B=11 A=11 so B is eliminated. Round 2 votes counts: D=45 C=25 E=18 A=12 so A is eliminated. Round 3 votes counts: D=46 C=28 E=26 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 B:215 C:212 A:182 E:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -20 -18 18 B 16 0 4 -8 18 C 20 -4 0 -8 16 D 18 8 8 0 20 E -18 -18 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -20 -18 18 B 16 0 4 -8 18 C 20 -4 0 -8 16 D 18 8 8 0 20 E -18 -18 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -20 -18 18 B 16 0 4 -8 18 C 20 -4 0 -8 16 D 18 8 8 0 20 E -18 -18 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 336: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) B E D C A (9) E B A D C (7) E A B D C (6) B E A C D (6) C D A B E (5) B E D A C (5) B C D E A (4) A E D C B (4) A E B C D (4) A C D E B (4) E B A C D (3) D C A E B (3) C A D B E (3) E D B C A (2) E A D B C (2) D C B E A (2) D C B A E (2) D C A B E (2) C D A E B (2) B E A D C (2) B C A D E (2) A C B D E (2) E D A C B (1) E B D A C (1) D E C A B (1) D E B C A (1) C B D A E (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C E A (1) B C A E D (1) Total count = 100 A B C D E A 0 -22 -6 -6 -12 B 22 0 18 14 18 C 6 -18 0 -4 -14 D 6 -14 4 0 -10 E 12 -18 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 -6 -12 B 22 0 18 14 18 C 6 -18 0 -4 -14 D 6 -14 4 0 -10 E 12 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=22 C=20 A=14 D=11 so D is eliminated. Round 2 votes counts: B=33 C=29 E=24 A=14 so A is eliminated. Round 3 votes counts: C=35 B=33 E=32 so E is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:236 E:209 D:193 C:185 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -6 -6 -12 B 22 0 18 14 18 C 6 -18 0 -4 -14 D 6 -14 4 0 -10 E 12 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -6 -12 B 22 0 18 14 18 C 6 -18 0 -4 -14 D 6 -14 4 0 -10 E 12 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -6 -12 B 22 0 18 14 18 C 6 -18 0 -4 -14 D 6 -14 4 0 -10 E 12 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 337: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (14) C B A E D (10) D E A C B (9) E A D B C (5) D E B C A (5) B C A E D (5) D E A B C (4) D C B E A (4) A E B C D (4) D C E B A (3) C B D E A (3) C B D A E (3) A E D B C (3) D E C B A (2) D E C A B (2) D E B A C (2) D A E C B (2) C B A D E (2) A D E C B (2) A B C E D (2) E B D C A (1) E B A C D (1) D C B A E (1) C D B E A (1) B E C A D (1) B C E D A (1) B C E A D (1) B C D E A (1) B A C E D (1) A E C D B (1) A D E B C (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 8 -16 -20 B -4 0 6 -22 -24 C -8 -6 0 -20 -20 D 16 22 20 0 -4 E 20 24 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 -16 -20 B -4 0 6 -22 -24 C -8 -6 0 -20 -20 D 16 22 20 0 -4 E 20 24 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=21 C=19 A=16 B=10 so B is eliminated. Round 2 votes counts: D=34 C=27 E=22 A=17 so A is eliminated. Round 3 votes counts: D=37 C=32 E=31 so E is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:234 D:227 A:188 B:178 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 -16 -20 B -4 0 6 -22 -24 C -8 -6 0 -20 -20 D 16 22 20 0 -4 E 20 24 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -16 -20 B -4 0 6 -22 -24 C -8 -6 0 -20 -20 D 16 22 20 0 -4 E 20 24 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -16 -20 B -4 0 6 -22 -24 C -8 -6 0 -20 -20 D 16 22 20 0 -4 E 20 24 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 338: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) A C D E B (10) E C A B D (6) B E D C A (6) D A C B E (5) C A D E B (5) B E C A D (5) B D E C A (4) B D E A C (4) A D C B E (4) D B A C E (3) D A B C E (3) C E A D B (3) C E A B D (3) B D A C E (3) E C B A D (2) E C A D B (2) E B C D A (2) D C A E B (2) D B A E C (2) B E D A C (2) B E C D A (2) E D B C A (1) E B D C A (1) E B C A D (1) D E C A B (1) D B E A C (1) C A E D B (1) B E A C D (1) B D A E C (1) B A E C D (1) A D C E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 16 6 -6 6 B -16 0 -12 -8 -2 C -6 12 0 -10 10 D 6 8 10 0 20 E -6 2 -10 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 -6 6 B -16 0 -12 -8 -2 C -6 12 0 -10 10 D 6 8 10 0 20 E -6 2 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 A=17 E=15 C=12 so C is eliminated. Round 2 votes counts: B=29 D=27 A=23 E=21 so E is eliminated. Round 3 votes counts: A=37 B=35 D=28 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:222 A:211 C:203 E:183 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 -6 6 B -16 0 -12 -8 -2 C -6 12 0 -10 10 D 6 8 10 0 20 E -6 2 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -6 6 B -16 0 -12 -8 -2 C -6 12 0 -10 10 D 6 8 10 0 20 E -6 2 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -6 6 B -16 0 -12 -8 -2 C -6 12 0 -10 10 D 6 8 10 0 20 E -6 2 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 339: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (13) E D B C A (9) D B C E A (9) A C B D E (8) A E C B D (7) E A D C B (6) E A D B C (6) B C D A E (6) C B A D E (4) E D B A C (3) E D A B C (3) D E B C A (3) A C E B D (3) E A C B D (2) E A B D C (2) D B E C A (2) B D C A E (2) A E D C B (2) A E C D B (2) E D A C B (1) E A B C D (1) D C B E A (1) D C B A E (1) C D B A E (1) C A B D E (1) B D C E A (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -6 -10 2 B 10 0 -6 2 0 C 6 6 0 -2 2 D 10 -2 2 0 4 E -2 0 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000006 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -10 2 B 10 0 -6 2 0 C 6 6 0 -2 2 D 10 -2 2 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999997 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=23 C=19 D=16 B=9 so B is eliminated. Round 2 votes counts: E=33 C=25 A=23 D=19 so D is eliminated. Round 3 votes counts: C=39 E=38 A=23 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:207 C:206 B:203 E:196 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 -10 2 B 10 0 -6 2 0 C 6 6 0 -2 2 D 10 -2 2 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999997 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -10 2 B 10 0 -6 2 0 C 6 6 0 -2 2 D 10 -2 2 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999997 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -10 2 B 10 0 -6 2 0 C 6 6 0 -2 2 D 10 -2 2 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999997 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 340: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (12) E D A C B (9) D E B C A (5) C B D E A (5) B C D A E (5) A B C E D (5) B C A D E (4) A E D B C (4) A D E B C (4) E D C B A (3) D E C B A (3) D B E C A (3) B C D E A (3) A E C B D (3) A B C D E (3) E A D C B (2) D E A B C (2) D B E A C (2) D B C E A (2) C B A E D (2) C B A D E (2) B D C E A (2) B A C D E (2) E D C A B (1) E A C D B (1) D C B E A (1) D B A E C (1) D A E B C (1) D A B E C (1) C E B D A (1) C B E D A (1) C B E A D (1) B D C A E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 10 -4 6 B -2 0 6 -14 -2 C -10 -6 0 -22 -18 D 4 14 22 0 6 E -6 2 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 -4 6 B -2 0 6 -14 -2 C -10 -6 0 -22 -18 D 4 14 22 0 6 E -6 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=21 B=17 E=16 C=12 so C is eliminated. Round 2 votes counts: A=34 B=28 D=21 E=17 so E is eliminated. Round 3 votes counts: A=37 D=34 B=29 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:207 E:204 B:194 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 10 -4 6 B -2 0 6 -14 -2 C -10 -6 0 -22 -18 D 4 14 22 0 6 E -6 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -4 6 B -2 0 6 -14 -2 C -10 -6 0 -22 -18 D 4 14 22 0 6 E -6 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -4 6 B -2 0 6 -14 -2 C -10 -6 0 -22 -18 D 4 14 22 0 6 E -6 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 341: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) E B A D C (5) C A D E B (5) A E C B D (5) A E B C D (5) A C E B D (5) D A E C B (4) E B D A C (3) D C A E B (3) D B E C A (3) C D B E A (3) C D B A E (3) B E A C D (3) B C D E A (3) A E D B C (3) A D C E B (3) E A D B C (2) D E B A C (2) D C B E A (2) D B C E A (2) C B D E A (2) C B D A E (2) C A D B E (2) C A B D E (2) B E D C A (2) B D E C A (2) A E D C B (2) A C D E B (2) E D A B C (1) C D A E B (1) C D A B E (1) C B A E D (1) C B A D E (1) B E C A D (1) B D C E A (1) B C E D A (1) B C E A D (1) B C A E D (1) A E C D B (1) A E B D C (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 12 10 18 10 B -12 0 -4 6 -22 C -10 4 0 4 -4 D -18 -6 -4 0 0 E -10 22 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 18 10 B -12 0 -4 6 -22 C -10 4 0 4 -4 D -18 -6 -4 0 0 E -10 22 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 E=17 D=16 B=15 so B is eliminated. Round 2 votes counts: C=29 A=29 E=23 D=19 so D is eliminated. Round 3 votes counts: C=37 A=33 E=30 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:208 C:197 D:186 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 18 10 B -12 0 -4 6 -22 C -10 4 0 4 -4 D -18 -6 -4 0 0 E -10 22 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 18 10 B -12 0 -4 6 -22 C -10 4 0 4 -4 D -18 -6 -4 0 0 E -10 22 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 18 10 B -12 0 -4 6 -22 C -10 4 0 4 -4 D -18 -6 -4 0 0 E -10 22 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 342: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) E D A B C (10) E D C B A (8) C B A E D (8) A B C E D (7) D E C B A (6) B A C E D (4) A B C D E (4) D E C A B (3) C D B E A (3) B C A E D (3) A B E C D (3) D E A C B (2) D C A E B (2) D A E B C (2) C E D B A (2) C E B D A (2) C B E A D (2) C B D E A (2) C B D A E (2) C B A D E (2) A E B D C (2) E D A C B (1) E C B D A (1) E B D C A (1) E B A C D (1) D A C B E (1) C D B A E (1) C B E D A (1) A E D B C (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 2 2 -22 -12 B -2 0 0 -8 -12 C -2 0 0 -2 -6 D 22 8 2 0 -14 E 12 12 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 -22 -12 B -2 0 0 -8 -12 C -2 0 0 -2 -6 D 22 8 2 0 -14 E 12 12 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 E=22 A=19 B=7 so B is eliminated. Round 2 votes counts: C=28 D=27 A=23 E=22 so E is eliminated. Round 3 votes counts: D=47 C=29 A=24 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:222 D:209 C:195 B:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 -22 -12 B -2 0 0 -8 -12 C -2 0 0 -2 -6 D 22 8 2 0 -14 E 12 12 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -22 -12 B -2 0 0 -8 -12 C -2 0 0 -2 -6 D 22 8 2 0 -14 E 12 12 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -22 -12 B -2 0 0 -8 -12 C -2 0 0 -2 -6 D 22 8 2 0 -14 E 12 12 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 343: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) E C B D A (9) B E C A D (9) C E B D A (7) A D B E C (7) A D B C E (6) E B C D A (4) D A C E B (4) C E B A D (4) E C D B A (3) D A C B E (3) C E D A B (3) A D C B E (3) E D C B A (2) E C B A D (2) E B C A D (2) D C A E B (2) C E D B A (2) C B E A D (2) B A E C D (2) A B C E D (2) D E C B A (1) D C E B A (1) D B A E C (1) D A B C E (1) C D E A B (1) C A B E D (1) B E C D A (1) B E A C D (1) B C E A D (1) B A D E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -14 -10 -10 B 10 0 -4 -4 4 C 14 4 0 16 -10 D 10 4 -16 0 -14 E 10 -4 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407408 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 -10 -14 -10 -10 B 10 0 -4 -4 4 C 14 4 0 16 -10 D 10 4 -16 0 -14 E 10 -4 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407405 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=22 C=20 A=20 B=15 so B is eliminated. Round 2 votes counts: E=33 D=23 A=23 C=21 so C is eliminated. Round 3 votes counts: E=52 D=24 A=24 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:212 B:203 D:192 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 -10 -10 B 10 0 -4 -4 4 C 14 4 0 16 -10 D 10 4 -16 0 -14 E 10 -4 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407405 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -10 -10 B 10 0 -4 -4 4 C 14 4 0 16 -10 D 10 4 -16 0 -14 E 10 -4 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407405 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -10 -10 B 10 0 -4 -4 4 C 14 4 0 16 -10 D 10 4 -16 0 -14 E 10 -4 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407405 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 344: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) D E C B A (7) E D C B A (6) C D E A B (5) D E B A C (4) C E D B A (4) E C D B A (3) D B A E C (3) C E A B D (3) C A B E D (3) B A E D C (3) B A E C D (3) A B D E C (3) E D B A C (2) E B A D C (2) D E B C A (2) D C A B E (2) D A B E C (2) D A B C E (2) C E D A B (2) C D A E B (2) B A D E C (2) A C B D E (2) A B E C D (2) A B D C E (2) A B C D E (2) E D B C A (1) E C B D A (1) E C B A D (1) E B D C A (1) E B A C D (1) D C E B A (1) D C E A B (1) C E B A D (1) C E A D B (1) C D E B A (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E A C (1) A D B C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -2 -10 -6 B 4 0 -2 -10 -4 C 2 2 0 0 -4 D 10 10 0 0 -2 E 6 4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 -10 -6 B 4 0 -2 -10 -4 C 2 2 0 0 -4 D 10 10 0 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 A=22 E=18 B=11 so B is eliminated. Round 2 votes counts: A=30 D=25 C=25 E=20 so E is eliminated. Round 3 votes counts: D=36 A=34 C=30 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:208 C:200 B:194 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 -10 -6 B 4 0 -2 -10 -4 C 2 2 0 0 -4 D 10 10 0 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -10 -6 B 4 0 -2 -10 -4 C 2 2 0 0 -4 D 10 10 0 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -10 -6 B 4 0 -2 -10 -4 C 2 2 0 0 -4 D 10 10 0 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 345: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) C A E B D (6) A B D E C (6) E C D A B (5) B D A E C (5) A C E B D (5) A B D C E (5) D B E C A (4) A C B E D (4) E C D B A (3) D B C E A (3) B D C A E (3) B D A C E (3) B A D E C (3) E D C B A (2) E C A D B (2) D B E A C (2) C E D B A (2) C E D A B (2) C B E D A (2) C A E D B (2) C A B E D (2) B A D C E (2) A E C D B (2) A B C D E (2) E D C A B (1) E D A C B (1) D E B C A (1) D E B A C (1) D E A B C (1) D C B E A (1) D A E B C (1) C E B D A (1) C E A D B (1) C B A D E (1) B C D A E (1) A E D C B (1) A E D B C (1) A E B D C (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -4 -4 16 B -6 0 8 22 14 C 4 -8 0 -10 12 D 4 -22 10 0 4 E -16 -14 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.687500 B: 0.125000 C: 0.000000 D: 0.187500 E: 0.000000 Sum of squares = 0.523437499122 Cumulative probabilities = A: 0.687500 B: 0.812500 C: 0.812500 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -4 16 B -6 0 8 22 14 C 4 -8 0 -10 12 D 4 -22 10 0 4 E -16 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.687500 B: 0.125000 C: 0.000000 D: 0.187500 E: 0.000000 Sum of squares = 0.523437498041 Cumulative probabilities = A: 0.687500 B: 0.812500 C: 0.812500 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 C=19 E=14 D=14 so E is eliminated. Round 2 votes counts: C=29 A=29 B=24 D=18 so D is eliminated. Round 3 votes counts: B=35 C=33 A=32 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:207 C:199 D:198 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -4 -4 16 B -6 0 8 22 14 C 4 -8 0 -10 12 D 4 -22 10 0 4 E -16 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.687500 B: 0.125000 C: 0.000000 D: 0.187500 E: 0.000000 Sum of squares = 0.523437498041 Cumulative probabilities = A: 0.687500 B: 0.812500 C: 0.812500 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 16 B -6 0 8 22 14 C 4 -8 0 -10 12 D 4 -22 10 0 4 E -16 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.687500 B: 0.125000 C: 0.000000 D: 0.187500 E: 0.000000 Sum of squares = 0.523437498041 Cumulative probabilities = A: 0.687500 B: 0.812500 C: 0.812500 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 16 B -6 0 8 22 14 C 4 -8 0 -10 12 D 4 -22 10 0 4 E -16 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.687500 B: 0.125000 C: 0.000000 D: 0.187500 E: 0.000000 Sum of squares = 0.523437498041 Cumulative probabilities = A: 0.687500 B: 0.812500 C: 0.812500 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 346: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) C E A D B (8) D B C A E (7) E C A D B (6) E A C B D (6) C A E D B (5) B D E A C (5) B D A E C (5) E A B C D (4) D C B E A (4) B A E D C (4) A E C D B (4) C D E A B (3) C D A E B (3) B E A D C (3) E A C D B (2) D C B A E (2) D B C E A (2) B D A C E (2) A E B C D (2) A C E D B (2) E A B D C (1) D C E A B (1) D B E C A (1) C E D A B (1) C D B A E (1) B D E C A (1) B D C E A (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -16 -2 2 B 0 0 0 -4 0 C 16 0 0 -2 10 D 2 4 2 0 0 E -2 0 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.880952 E: 0.119048 Sum of squares = 0.790249441177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.880952 E: 1.000000 A B C D E A 0 0 -16 -2 2 B 0 0 0 -4 0 C 16 0 0 -2 10 D 2 4 2 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222307834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=21 E=19 D=17 A=10 so A is eliminated. Round 2 votes counts: B=33 E=26 C=24 D=17 so D is eliminated. Round 3 votes counts: B=43 C=31 E=26 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 D:204 B:198 E:194 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -16 -2 2 B 0 0 0 -4 0 C 16 0 0 -2 10 D 2 4 2 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222307834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -2 2 B 0 0 0 -4 0 C 16 0 0 -2 10 D 2 4 2 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222307834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -2 2 B 0 0 0 -4 0 C 16 0 0 -2 10 D 2 4 2 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222307834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 347: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) D A C B E (12) D E A C B (6) D A C E B (6) B E C A D (6) B C E A D (6) E D A B C (5) B C A E D (5) A D C B E (5) E B D A C (4) D A E C B (4) E D B A C (3) D E A B C (3) C A B D E (3) A C D B E (3) E D B C A (2) E B A C D (2) C B A D E (2) B A C E D (2) E B D C A (1) E B C D A (1) C B D E A (1) C B D A E (1) C B A E D (1) C A D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 12 2 -8 B 2 0 10 0 -2 C -12 -10 0 -2 -2 D -2 0 2 0 -6 E 8 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999023 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 12 2 -8 B 2 0 10 0 -2 C -12 -10 0 -2 -2 D -2 0 2 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=31 B=19 C=9 A=9 so C is eliminated. Round 2 votes counts: E=32 D=31 B=24 A=13 so A is eliminated. Round 3 votes counts: D=40 E=32 B=28 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:205 A:202 D:197 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 12 2 -8 B 2 0 10 0 -2 C -12 -10 0 -2 -2 D -2 0 2 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 2 -8 B 2 0 10 0 -2 C -12 -10 0 -2 -2 D -2 0 2 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 2 -8 B 2 0 10 0 -2 C -12 -10 0 -2 -2 D -2 0 2 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 348: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) B C D A E (7) E A B C D (6) B C A D E (6) E A B D C (5) E B A D C (4) E A D C B (4) D C B E A (4) D C B A E (4) E D A B C (3) D E A C B (3) D C A B E (3) B D C E A (3) E D A C B (2) E B D A C (2) E A C B D (2) D C E B A (2) D C A E B (2) D B E C A (2) C D B A E (2) C B D A E (2) B D C A E (2) B A E C D (2) E D B A C (1) D E C B A (1) D E B C A (1) D C E A B (1) D B C E A (1) C D A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D C A (1) B E C A D (1) B E A C D (1) B C D E A (1) B C A E D (1) B A C E D (1) A E C D B (1) A E C B D (1) A E B C D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -4 -2 -16 B 20 0 22 20 0 C 4 -22 0 -2 -4 D 2 -20 2 0 4 E 16 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.513079 C: 0.000000 D: 0.000000 E: 0.486921 Sum of squares = 0.500342119173 Cumulative probabilities = A: 0.000000 B: 0.513079 C: 0.513079 D: 0.513079 E: 1.000000 A B C D E A 0 -20 -4 -2 -16 B 20 0 22 20 0 C 4 -22 0 -2 -4 D 2 -20 2 0 4 E 16 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=26 D=24 C=9 A=5 so A is eliminated. Round 2 votes counts: E=39 B=28 D=24 C=9 so C is eliminated. Round 3 votes counts: E=39 B=32 D=29 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:208 D:194 C:188 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -4 -2 -16 B 20 0 22 20 0 C 4 -22 0 -2 -4 D 2 -20 2 0 4 E 16 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -2 -16 B 20 0 22 20 0 C 4 -22 0 -2 -4 D 2 -20 2 0 4 E 16 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -2 -16 B 20 0 22 20 0 C 4 -22 0 -2 -4 D 2 -20 2 0 4 E 16 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 349: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (16) A B C E D (14) C B A D E (8) E D A B C (7) E D C B A (6) C B A E D (5) E D A C B (4) E D C A B (3) E A D B C (3) D E C A B (3) D C E B A (3) B C A D E (3) A B C D E (3) E A B C D (2) D E A B C (2) D C B E A (2) C B D A E (2) B C A E D (2) A B E C D (2) E C B A D (1) D C B A E (1) D A B C E (1) C D E B A (1) C D B E A (1) C B E D A (1) C B E A D (1) B A C E D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -8 -18 -6 -12 B 8 0 -16 -6 -6 C 18 16 0 -4 -2 D 6 6 4 0 -8 E 12 6 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -18 -6 -12 B 8 0 -16 -6 -6 C 18 16 0 -4 -2 D 6 6 4 0 -8 E 12 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 A=21 C=19 B=6 so B is eliminated. Round 2 votes counts: D=28 E=26 C=24 A=22 so A is eliminated. Round 3 votes counts: C=42 E=30 D=28 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:214 D:204 B:190 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -18 -6 -12 B 8 0 -16 -6 -6 C 18 16 0 -4 -2 D 6 6 4 0 -8 E 12 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -6 -12 B 8 0 -16 -6 -6 C 18 16 0 -4 -2 D 6 6 4 0 -8 E 12 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -6 -12 B 8 0 -16 -6 -6 C 18 16 0 -4 -2 D 6 6 4 0 -8 E 12 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 350: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) C D B A E (7) E A B D C (5) B E A D C (5) E A C D B (4) C D A E B (4) B D C A E (4) B C D E A (4) A E D B C (4) E C A B D (3) E A C B D (3) D B A C E (3) D A C B E (3) C E B D A (3) B A E D C (3) B A D E C (3) A E D C B (3) E B C A D (2) E A D B C (2) E A B C D (2) D C B A E (2) D B C A E (2) C D A B E (2) C B E D A (2) C B D E A (2) A E B D C (2) A D E C B (2) E B A C D (1) D C A E B (1) D C A B E (1) C E D B A (1) C E A D B (1) C D E B A (1) C A E D B (1) B E D A C (1) B E C A D (1) B D A E C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 12 12 0 B 10 0 6 8 -6 C -12 -6 0 -12 -10 D -12 -8 12 0 -12 E 0 6 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.239431 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.760569 Sum of squares = 0.635791966314 Cumulative probabilities = A: 0.239431 B: 0.239431 C: 0.239431 D: 0.239431 E: 1.000000 A B C D E A 0 -10 12 12 0 B 10 0 6 8 -6 C -12 -6 0 -12 -10 D -12 -8 12 0 -12 E 0 6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250111473 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=24 B=22 A=13 D=12 so D is eliminated. Round 2 votes counts: E=29 C=28 B=27 A=16 so A is eliminated. Round 3 votes counts: E=40 C=32 B=28 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:209 A:207 D:190 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 12 12 0 B 10 0 6 8 -6 C -12 -6 0 -12 -10 D -12 -8 12 0 -12 E 0 6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250111473 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 12 12 0 B 10 0 6 8 -6 C -12 -6 0 -12 -10 D -12 -8 12 0 -12 E 0 6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250111473 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 12 12 0 B 10 0 6 8 -6 C -12 -6 0 -12 -10 D -12 -8 12 0 -12 E 0 6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250111473 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 351: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) C E B A D (7) D B A E C (6) D E B A C (5) B D A E C (5) A D B E C (5) E D C B A (4) C A E B D (4) C A B E D (4) D A B E C (3) C E D A B (3) C E A B D (3) C B E A D (3) B A C D E (3) A D B C E (3) A B C D E (3) E D C A B (2) E C D A B (2) D E A B C (2) D B E A C (2) D A E B C (2) C E B D A (2) C E A D B (2) B A D E C (2) B A D C E (2) E D B C A (1) E B D C A (1) D E B C A (1) C E D B A (1) C B A E D (1) C A E D B (1) B D E A C (1) A D E C B (1) A D C E B (1) A D C B E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 0 -2 0 B 10 0 -2 -12 -6 C 0 2 0 -4 -8 D 2 12 4 0 2 E 0 6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -2 0 B 10 0 -2 -12 -6 C 0 2 0 -4 -8 D 2 12 4 0 2 E 0 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=21 E=18 A=17 B=13 so B is eliminated. Round 2 votes counts: C=31 D=27 A=24 E=18 so E is eliminated. Round 3 votes counts: C=41 D=35 A=24 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:206 B:195 C:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 0 -2 0 B 10 0 -2 -12 -6 C 0 2 0 -4 -8 D 2 12 4 0 2 E 0 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -2 0 B 10 0 -2 -12 -6 C 0 2 0 -4 -8 D 2 12 4 0 2 E 0 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -2 0 B 10 0 -2 -12 -6 C 0 2 0 -4 -8 D 2 12 4 0 2 E 0 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 352: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (7) C B D E A (6) B C D E A (6) D B E C A (5) B D C E A (5) A E D C B (5) A E D B C (5) C E D B A (3) C E A B D (3) C B E D A (3) B A C D E (3) A C E B D (3) A B D E C (3) E C D B A (2) E C D A B (2) E A D B C (2) D E A B C (2) D B A E C (2) C A E B D (2) B C D A E (2) B A D C E (2) A E C B D (2) A C B E D (2) A B E D C (2) A B C E D (2) A B C D E (2) E D A C B (1) E D A B C (1) E C A D B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E B C A (1) D C B E A (1) D B E A C (1) D B C E A (1) C E B D A (1) C E B A D (1) C E A D B (1) C B E A D (1) A E B D C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 6 4 8 -6 B -6 0 0 4 0 C -4 0 0 12 0 D -8 -4 -12 0 -10 E 6 0 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.326068 C: 0.058261 D: 0.000000 E: 0.615670 Sum of squares = 0.488764981348 Cumulative probabilities = A: 0.000000 B: 0.326068 C: 0.384330 D: 0.384330 E: 1.000000 A B C D E A 0 6 4 8 -6 B -6 0 0 4 0 C -4 0 0 12 0 D -8 -4 -12 0 -10 E 6 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.258065 C: 0.290323 D: 0.000000 E: 0.451613 Sum of squares = 0.354838718345 Cumulative probabilities = A: 0.000000 B: 0.258065 C: 0.548387 D: 0.548387 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=21 B=18 D=14 E=11 so E is eliminated. Round 2 votes counts: A=40 C=26 B=18 D=16 so D is eliminated. Round 3 votes counts: A=44 C=28 B=28 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:206 C:204 B:199 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 8 -6 B -6 0 0 4 0 C -4 0 0 12 0 D -8 -4 -12 0 -10 E 6 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.258065 C: 0.290323 D: 0.000000 E: 0.451613 Sum of squares = 0.354838718345 Cumulative probabilities = A: 0.000000 B: 0.258065 C: 0.548387 D: 0.548387 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 8 -6 B -6 0 0 4 0 C -4 0 0 12 0 D -8 -4 -12 0 -10 E 6 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.258065 C: 0.290323 D: 0.000000 E: 0.451613 Sum of squares = 0.354838718345 Cumulative probabilities = A: 0.000000 B: 0.258065 C: 0.548387 D: 0.548387 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 8 -6 B -6 0 0 4 0 C -4 0 0 12 0 D -8 -4 -12 0 -10 E 6 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.258065 C: 0.290323 D: 0.000000 E: 0.451613 Sum of squares = 0.354838718345 Cumulative probabilities = A: 0.000000 B: 0.258065 C: 0.548387 D: 0.548387 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 353: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) B C E A D (9) D A E C B (8) B A D C E (8) A D B E C (8) E C D A B (6) D A B C E (4) C E D A B (4) C B E D A (4) B A D E C (4) A D B C E (4) D E C A B (3) C E D B A (3) B A E D C (3) B A E C D (2) B A C E D (2) B A C D E (2) A D E B C (2) E D A C B (1) E C B D A (1) D E A C B (1) D C E A B (1) D C A E B (1) C E B A D (1) C D E A B (1) B E C A D (1) B E A C D (1) B D A C E (1) B C A E D (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 6 0 4 B 8 0 6 4 10 C -6 -6 0 -2 16 D 0 -4 2 0 0 E -4 -10 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 0 4 B 8 0 6 4 10 C -6 -6 0 -2 16 D 0 -4 2 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=24 D=18 A=16 E=8 so E is eliminated. Round 2 votes counts: B=34 C=31 D=19 A=16 so A is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:201 C:201 D:199 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 0 4 B 8 0 6 4 10 C -6 -6 0 -2 16 D 0 -4 2 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 0 4 B 8 0 6 4 10 C -6 -6 0 -2 16 D 0 -4 2 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 0 4 B 8 0 6 4 10 C -6 -6 0 -2 16 D 0 -4 2 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 354: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) A C D E B (10) E B D A C (9) A C D B E (7) E B D C A (6) C D A B E (6) B E D C A (6) B D E C A (5) A C E B D (5) E D B C A (4) D E B C A (4) D C B A E (3) D B E C A (3) E B A D C (2) D B C E A (2) C A B D E (2) A E C D B (2) A C B E D (2) A C B D E (2) E D A C B (1) E B A C D (1) E A B D C (1) E A B C D (1) D C E A B (1) D C B E A (1) B E A D C (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 6 -6 -2 4 B -6 0 -8 -10 0 C 6 8 0 2 4 D 2 10 -2 0 12 E -4 0 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -2 4 B -6 0 -8 -10 0 C 6 8 0 2 4 D 2 10 -2 0 12 E -4 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=25 C=18 D=14 B=12 so B is eliminated. Round 2 votes counts: E=32 A=31 D=19 C=18 so C is eliminated. Round 3 votes counts: A=43 E=32 D=25 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:211 C:210 A:201 E:190 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -2 4 B -6 0 -8 -10 0 C 6 8 0 2 4 D 2 10 -2 0 12 E -4 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -2 4 B -6 0 -8 -10 0 C 6 8 0 2 4 D 2 10 -2 0 12 E -4 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -2 4 B -6 0 -8 -10 0 C 6 8 0 2 4 D 2 10 -2 0 12 E -4 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 355: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) E C A D B (7) C E B D A (7) B D C A E (7) C E A B D (6) D B A E C (5) E C A B D (4) E A C D B (4) E A C B D (4) D B C A E (4) C B D E A (4) A B D E C (4) A B D C E (4) E C D B A (3) A E C B D (3) E A D C B (2) E A D B C (2) D B C E A (2) C E D B A (2) C E B A D (2) C B D A E (2) A D B E C (2) E D C B A (1) E C D A B (1) D B A C E (1) D A B E C (1) C B E D A (1) C B A D E (1) C A E B D (1) B C D A E (1) B A D C E (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 -12 -4 -4 B 8 0 -10 24 0 C 12 10 0 6 12 D 4 -24 -6 0 -2 E 4 0 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -4 -4 B 8 0 -10 24 0 C 12 10 0 6 12 D 4 -24 -6 0 -2 E 4 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 B=19 A=14 D=13 so D is eliminated. Round 2 votes counts: B=31 E=28 C=26 A=15 so A is eliminated. Round 3 votes counts: B=42 E=32 C=26 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:211 E:197 A:186 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -4 -4 B 8 0 -10 24 0 C 12 10 0 6 12 D 4 -24 -6 0 -2 E 4 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -4 -4 B 8 0 -10 24 0 C 12 10 0 6 12 D 4 -24 -6 0 -2 E 4 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -4 -4 B 8 0 -10 24 0 C 12 10 0 6 12 D 4 -24 -6 0 -2 E 4 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 356: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (19) E C D A B (8) C D A E B (7) D C A B E (6) A D B C E (6) E C D B A (5) B A D E C (5) D A C B E (4) B E A D C (4) E C B D A (3) E B C A D (3) C D E A B (3) E B C D A (2) E B A C D (2) D C A E B (2) C E D B A (2) C E D A B (2) B A E D C (2) A D C B E (2) A B D C E (2) E B A D C (1) E A D C B (1) E A C D B (1) D A C E B (1) C D A B E (1) B E A C D (1) B D C A E (1) B D A C E (1) B A C D E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 10 4 24 B 4 0 4 -2 12 C -10 -4 0 -18 20 D -4 2 18 0 22 E -24 -12 -20 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000006 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 4 24 B 4 0 4 -2 12 C -10 -4 0 -18 20 D -4 2 18 0 22 E -24 -12 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000136 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=26 C=15 D=13 A=12 so A is eliminated. Round 2 votes counts: B=37 E=27 D=21 C=15 so C is eliminated. Round 3 votes counts: B=37 D=32 E=31 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:217 B:209 C:194 E:161 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 4 24 B 4 0 4 -2 12 C -10 -4 0 -18 20 D -4 2 18 0 22 E -24 -12 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000136 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 4 24 B 4 0 4 -2 12 C -10 -4 0 -18 20 D -4 2 18 0 22 E -24 -12 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000136 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 4 24 B 4 0 4 -2 12 C -10 -4 0 -18 20 D -4 2 18 0 22 E -24 -12 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000136 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 357: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (6) E B C A D (5) B C D E A (5) A D E B C (5) E C B D A (4) D A C B E (4) A E B D C (4) E C D A B (3) E A D C B (3) E A D B C (3) E A C B D (3) E A B D C (3) D C B A E (3) D C A E B (3) D C A B E (3) C E D B A (3) C D B A E (3) B C E D A (3) B C D A E (3) A D B E C (3) E C A B D (2) E B A C D (2) E A B C D (2) C B D E A (2) B D C A E (2) A E D C B (2) A D B C E (2) A B E D C (2) E C B A D (1) E A C D B (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E A B (1) C D B E A (1) C B D A E (1) B E C D A (1) B A D E C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 18 0 4 2 B -18 0 6 -4 -20 C 0 -6 0 -6 -16 D -4 4 6 0 -8 E -2 20 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.956349 B: 0.000000 C: 0.043651 D: 0.000000 E: 0.000000 Sum of squares = 0.916508541743 Cumulative probabilities = A: 0.956349 B: 0.956349 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 4 2 B -18 0 6 -4 -20 C 0 -6 0 -6 -16 D -4 4 6 0 -8 E -2 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469143565 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=26 D=15 B=15 C=12 so C is eliminated. Round 2 votes counts: E=36 A=26 D=20 B=18 so B is eliminated. Round 3 votes counts: E=40 D=33 A=27 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:212 D:199 C:186 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 4 2 B -18 0 6 -4 -20 C 0 -6 0 -6 -16 D -4 4 6 0 -8 E -2 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469143565 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 4 2 B -18 0 6 -4 -20 C 0 -6 0 -6 -16 D -4 4 6 0 -8 E -2 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469143565 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 4 2 B -18 0 6 -4 -20 C 0 -6 0 -6 -16 D -4 4 6 0 -8 E -2 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469143565 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 358: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (5) A E C D B (5) E B C D A (4) D B C E A (4) D B A E C (4) B E C D A (4) B D E C A (4) A E C B D (4) A D C E B (4) A D B E C (4) A C E D B (4) E C A B D (3) E A C B D (3) D B A C E (3) C A E D B (3) A C D E B (3) E C B D A (2) E B C A D (2) D C B E A (2) D C B A E (2) D A B E C (2) C E B D A (2) C E A B D (2) C D B E A (2) B E D C A (2) B D C E A (2) A E B D C (2) A D B C E (2) A B D E C (2) E C B A D (1) E A B C D (1) D B C A E (1) D A C B E (1) C E D B A (1) C E A D B (1) C A D E B (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E D C (1) A E D B C (1) A D C B E (1) Total count = 100 A B C D E A 0 8 10 2 12 B -8 0 6 -12 2 C -10 -6 0 -4 -8 D -2 12 4 0 2 E -12 -2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 2 12 B -8 0 6 -12 2 C -10 -6 0 -4 -8 D -2 12 4 0 2 E -12 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=24 E=16 B=16 C=12 so C is eliminated. Round 2 votes counts: A=36 D=26 E=22 B=16 so B is eliminated. Round 3 votes counts: A=37 D=34 E=29 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:208 E:196 B:194 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 2 12 B -8 0 6 -12 2 C -10 -6 0 -4 -8 D -2 12 4 0 2 E -12 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 2 12 B -8 0 6 -12 2 C -10 -6 0 -4 -8 D -2 12 4 0 2 E -12 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 2 12 B -8 0 6 -12 2 C -10 -6 0 -4 -8 D -2 12 4 0 2 E -12 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 359: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) E D B C A (8) E C A B D (7) D E B A C (7) D B E A C (6) C A B E D (6) E D B A C (5) E B D C A (5) D E A C B (5) C A E B D (5) C A B D E (4) A C D E B (4) E C A D B (3) B D E A C (3) D B A C E (2) D A C B E (2) D A B C E (2) C A D B E (2) A D C B E (2) E D C A B (1) E C B A D (1) E B D A C (1) E B C A D (1) D B A E C (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E C A (1) B D A C E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 6 -4 -14 B -10 0 -8 -22 -8 C -6 8 0 -8 -16 D 4 22 8 0 8 E 14 8 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -4 -14 B -10 0 -8 -22 -8 C -6 8 0 -8 -16 D 4 22 8 0 8 E 14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=25 C=18 A=17 B=8 so B is eliminated. Round 2 votes counts: E=35 D=30 C=18 A=17 so A is eliminated. Round 3 votes counts: E=35 C=33 D=32 so D is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:215 A:199 C:189 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 -4 -14 B -10 0 -8 -22 -8 C -6 8 0 -8 -16 D 4 22 8 0 8 E 14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -4 -14 B -10 0 -8 -22 -8 C -6 8 0 -8 -16 D 4 22 8 0 8 E 14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -4 -14 B -10 0 -8 -22 -8 C -6 8 0 -8 -16 D 4 22 8 0 8 E 14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 360: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) D A B C E (9) C E A D B (8) E C B A D (7) D B A C E (7) A D C E B (7) B D A E C (6) E C A B D (5) B D E A C (5) B E C A D (4) B D A C E (4) E C A D B (3) D B A E C (3) C A E D B (3) B E D C A (3) D A C E B (2) A D C B E (2) E B C A D (1) D A E B C (1) D A C B E (1) D A B E C (1) C E A B D (1) B C E D A (1) B C E A D (1) B A D C E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 4 -8 0 B 8 0 16 0 20 C -4 -16 0 -4 0 D 8 0 4 0 2 E 0 -20 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.406797 C: 0.000000 D: 0.593203 E: 0.000000 Sum of squares = 0.517373729005 Cumulative probabilities = A: 0.000000 B: 0.406797 C: 0.406797 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -8 0 B 8 0 16 0 20 C -4 -16 0 -4 0 D 8 0 4 0 2 E 0 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=24 E=16 C=12 A=12 so C is eliminated. Round 2 votes counts: B=36 E=25 D=24 A=15 so A is eliminated. Round 3 votes counts: B=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:207 A:194 E:189 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -8 0 B 8 0 16 0 20 C -4 -16 0 -4 0 D 8 0 4 0 2 E 0 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -8 0 B 8 0 16 0 20 C -4 -16 0 -4 0 D 8 0 4 0 2 E 0 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -8 0 B 8 0 16 0 20 C -4 -16 0 -4 0 D 8 0 4 0 2 E 0 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 361: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) C E A B D (8) D B A C E (7) E C A B D (6) D A B C E (6) B D E C A (5) A D B C E (5) D B A E C (4) C A E D B (4) B D E A C (4) E C B A D (3) D B E A C (3) B E D C A (3) B D A C E (3) A C E D B (3) A C D B E (3) E B D A C (2) D A B E C (2) C E A D B (2) C A D B E (2) B D A E C (2) A C D E B (2) E D A B C (1) E C B D A (1) E C A D B (1) E A C D B (1) C E B A D (1) C B E D A (1) C B E A D (1) C A E B D (1) C A B D E (1) B D C A E (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 -2 -10 -6 B 2 0 16 6 8 C 2 -16 0 -16 4 D 10 -6 16 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -10 -6 B 2 0 16 6 8 C 2 -16 0 -16 4 D 10 -6 16 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989166 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 C=21 B=19 A=14 so A is eliminated. Round 2 votes counts: C=29 D=28 E=24 B=19 so B is eliminated. Round 3 votes counts: D=43 C=30 E=27 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:211 E:196 A:190 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 -6 B 2 0 16 6 8 C 2 -16 0 -16 4 D 10 -6 16 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989166 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 -6 B 2 0 16 6 8 C 2 -16 0 -16 4 D 10 -6 16 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989166 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 -6 B 2 0 16 6 8 C 2 -16 0 -16 4 D 10 -6 16 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989166 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 362: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) D A E B C (8) C B D A E (7) C B E A D (5) D B C A E (4) D A B E C (4) B D C A E (4) A E D C B (4) E D A B C (3) E A C B D (3) D B E A C (3) C B A E D (3) B D C E A (3) B C E D A (3) B C D E A (3) B C D A E (3) E A D C B (2) E A C D B (2) E A B D C (2) D E B A C (2) C E B A D (2) A D E C B (2) A D C B E (2) E C A B D (1) E B D A C (1) E B C D A (1) E B C A D (1) D E A B C (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C B E (1) D A B C E (1) C B E D A (1) C B D E A (1) C A B D E (1) B E C D A (1) B D E C A (1) B C E A D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 8 -18 -2 B 4 0 22 -4 8 C -8 -22 0 -22 -6 D 18 4 22 0 12 E 2 -8 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -18 -2 B 4 0 22 -4 8 C -8 -22 0 -22 -6 D 18 4 22 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 C=20 B=19 A=10 so A is eliminated. Round 2 votes counts: D=33 E=28 C=20 B=19 so B is eliminated. Round 3 votes counts: D=41 C=30 E=29 so E is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:215 E:194 A:192 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -18 -2 B 4 0 22 -4 8 C -8 -22 0 -22 -6 D 18 4 22 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -18 -2 B 4 0 22 -4 8 C -8 -22 0 -22 -6 D 18 4 22 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -18 -2 B 4 0 22 -4 8 C -8 -22 0 -22 -6 D 18 4 22 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 363: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) D A E B C (10) B C A E D (8) D B A C E (5) B C A D E (5) E A D C B (4) D C B E A (4) C B A E D (4) B C D A E (4) A B C E D (4) E D A C B (3) E A C B D (3) D E A B C (3) D B C A E (3) D A B E C (3) C B E A D (2) C B D E A (2) A E C B D (2) A E B C D (2) D E C A B (1) D C E B A (1) D B C E A (1) C E D B A (1) C E B A D (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E D A (1) B D A C E (1) B C D E A (1) B A D C E (1) B A C E D (1) A E D C B (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 16 -16 14 B -4 0 10 -12 6 C -16 -10 0 -12 6 D 16 12 12 0 22 E -14 -6 -6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 -16 14 B -4 0 10 -12 6 C -16 -10 0 -12 6 D 16 12 12 0 22 E -14 -6 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 B=21 C=14 A=12 E=10 so E is eliminated. Round 2 votes counts: D=46 B=21 A=19 C=14 so C is eliminated. Round 3 votes counts: D=49 B=31 A=20 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:231 A:209 B:200 C:184 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 16 -16 14 B -4 0 10 -12 6 C -16 -10 0 -12 6 D 16 12 12 0 22 E -14 -6 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 -16 14 B -4 0 10 -12 6 C -16 -10 0 -12 6 D 16 12 12 0 22 E -14 -6 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 -16 14 B -4 0 10 -12 6 C -16 -10 0 -12 6 D 16 12 12 0 22 E -14 -6 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 364: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) E A D B C (10) D C B A E (8) D E C B A (7) B C A E D (7) A E B C D (6) E D A B C (5) E A B C D (5) C B D A E (4) B C D E A (4) A B C E D (4) A E D C B (3) E D B C A (2) C B A D E (2) B C D A E (2) B C A D E (2) A E D B C (2) A D C B E (2) E B C A D (1) E B A C D (1) E A D C B (1) D E A C B (1) D C E B A (1) D C A B E (1) D A E C B (1) C D B A E (1) C D A B E (1) C A B D E (1) B E C A D (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -12 2 0 B 8 0 6 -16 8 C 12 -6 0 -10 6 D -2 16 10 0 0 E 0 -8 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.416667 B: 0.000000 C: 0.083333 D: 0.500000 E: 0.000000 Sum of squares = 0.430555555438 Cumulative probabilities = A: 0.416667 B: 0.416667 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 2 0 B 8 0 6 -16 8 C 12 -6 0 -10 6 D -2 16 10 0 0 E 0 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.000000 C: 0.083333 D: 0.500000 E: 0.000000 Sum of squares = 0.43055555233 Cumulative probabilities = A: 0.416667 B: 0.416667 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=25 A=20 B=16 C=9 so C is eliminated. Round 2 votes counts: D=32 E=25 B=22 A=21 so A is eliminated. Round 3 votes counts: E=36 D=35 B=29 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:203 C:201 E:193 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 2 0 B 8 0 6 -16 8 C 12 -6 0 -10 6 D -2 16 10 0 0 E 0 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.000000 C: 0.083333 D: 0.500000 E: 0.000000 Sum of squares = 0.43055555233 Cumulative probabilities = A: 0.416667 B: 0.416667 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 2 0 B 8 0 6 -16 8 C 12 -6 0 -10 6 D -2 16 10 0 0 E 0 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.000000 C: 0.083333 D: 0.500000 E: 0.000000 Sum of squares = 0.43055555233 Cumulative probabilities = A: 0.416667 B: 0.416667 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 2 0 B 8 0 6 -16 8 C 12 -6 0 -10 6 D -2 16 10 0 0 E 0 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.000000 C: 0.083333 D: 0.500000 E: 0.000000 Sum of squares = 0.43055555233 Cumulative probabilities = A: 0.416667 B: 0.416667 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 365: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (13) B A C E D (11) B C E D A (9) B A E C D (9) B A D E C (9) D C E A B (8) A E C D B (8) A D E C B (8) B D C E A (5) D C E B A (4) B C E A D (3) A B D E C (3) D A E C B (2) B D A E C (2) E C D A B (1) D B E C A (1) D A B E C (1) C E D B A (1) C E D A B (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 8 4 8 B 8 0 6 2 6 C -8 -6 0 -14 -16 D -4 -2 14 0 12 E -8 -6 16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 4 8 B 8 0 6 2 6 C -8 -6 0 -14 -16 D -4 -2 14 0 12 E -8 -6 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=48 D=29 A=20 C=2 E=1 so E is eliminated. Round 2 votes counts: B=48 D=29 A=20 C=3 so C is eliminated. Round 3 votes counts: B=48 D=32 A=20 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:210 A:206 E:195 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 4 8 B 8 0 6 2 6 C -8 -6 0 -14 -16 D -4 -2 14 0 12 E -8 -6 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 4 8 B 8 0 6 2 6 C -8 -6 0 -14 -16 D -4 -2 14 0 12 E -8 -6 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 4 8 B 8 0 6 2 6 C -8 -6 0 -14 -16 D -4 -2 14 0 12 E -8 -6 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 366: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (21) A C E B D (11) C A E B D (10) D B E C A (7) B D E A C (7) E A B C D (5) C A D B E (5) C A D E B (4) C A B D E (4) D B C A E (3) C A E D B (3) E B D A C (2) E A C B D (2) A C B D E (2) E D B A C (1) E C A D B (1) D E B C A (1) D E B A C (1) D B C E A (1) C E A D B (1) C E A B D (1) C D A B E (1) B D A C E (1) B A C D E (1) A E C B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 16 8 -2 B -8 0 6 0 12 C -16 -6 0 10 0 D -8 0 -10 0 18 E 2 -12 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.285714 Sum of squares = 0.499999999943 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.714286 E: 1.000000 A B C D E A 0 8 16 8 -2 B -8 0 6 0 12 C -16 -6 0 10 0 D -8 0 -10 0 18 E 2 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.285714 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=29 A=17 E=11 B=9 so B is eliminated. Round 2 votes counts: D=42 C=29 A=18 E=11 so E is eliminated. Round 3 votes counts: D=45 C=30 A=25 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:215 B:205 D:200 C:194 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 8 -2 B -8 0 6 0 12 C -16 -6 0 10 0 D -8 0 -10 0 18 E 2 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.285714 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.714286 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 8 -2 B -8 0 6 0 12 C -16 -6 0 10 0 D -8 0 -10 0 18 E 2 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.285714 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.714286 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 8 -2 B -8 0 6 0 12 C -16 -6 0 10 0 D -8 0 -10 0 18 E 2 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.285714 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.714286 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 367: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) B A C D E (10) E D A C B (8) E C D A B (6) C E D A B (6) B A D E C (6) C D E A B (5) B C A D E (5) C E D B A (4) B A E D C (4) A D E C B (3) E A D C B (2) C D A E B (2) B C A E D (2) B A D C E (2) A D E B C (2) A B D E C (2) E D C B A (1) E C D B A (1) D C E A B (1) D A E C B (1) C D B A E (1) C B A D E (1) B E C D A (1) B E A D C (1) B E A C D (1) B C E D A (1) B A C E D (1) A E D C B (1) A E D B C (1) A C D E B (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 16 0 -2 -2 B -16 0 -18 -20 -16 C 0 18 0 0 -10 D 2 20 0 0 -10 E 2 16 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 0 -2 -2 B -16 0 -18 -20 -16 C 0 18 0 0 -10 D 2 20 0 0 -10 E 2 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=31 C=19 A=14 D=2 so D is eliminated. Round 2 votes counts: B=34 E=31 C=20 A=15 so A is eliminated. Round 3 votes counts: E=39 B=38 C=23 so C is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:206 D:206 C:204 B:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 0 -2 -2 B -16 0 -18 -20 -16 C 0 18 0 0 -10 D 2 20 0 0 -10 E 2 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 -2 -2 B -16 0 -18 -20 -16 C 0 18 0 0 -10 D 2 20 0 0 -10 E 2 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 -2 -2 B -16 0 -18 -20 -16 C 0 18 0 0 -10 D 2 20 0 0 -10 E 2 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 368: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) D B E C A (8) D C B E A (7) A C B E D (6) A E C B D (5) A E B C D (5) A C E B D (5) C A D B E (4) C D B A E (3) C B A E D (3) C A B E D (3) A E D B C (3) A E B D C (3) E B A D C (2) E A B D C (2) E A B C D (2) D E B A C (2) D E A B C (2) C D B E A (2) C B D E A (2) C B A D E (2) A C E D B (2) A C D E B (2) E B D C A (1) E B C A D (1) E B A C D (1) D E C B A (1) D B C E A (1) D A E C B (1) C B E D A (1) C A B D E (1) B E D C A (1) B D C E A (1) A D E C B (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -8 10 2 B 2 0 -6 -6 -8 C 8 6 0 2 -6 D -10 6 -2 0 8 E -2 8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.270137 B: 0.000000 C: 0.229863 D: 0.104863 E: 0.395137 Sum of squares = 0.292940322923 Cumulative probabilities = A: 0.270137 B: 0.270137 C: 0.500000 D: 0.604863 E: 1.000000 A B C D E A 0 -2 -8 10 2 B 2 0 -6 -6 -8 C 8 6 0 2 -6 D -10 6 -2 0 8 E -2 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.312500 D: 0.187500 E: 0.312500 Sum of squares = 0.265625000003 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.500000 D: 0.687500 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=34 A=34 C=21 E=9 B=2 so B is eliminated. Round 2 votes counts: D=35 A=34 C=21 E=10 so E is eliminated. Round 3 votes counts: A=41 D=37 C=22 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:205 E:202 A:201 D:201 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 10 2 B 2 0 -6 -6 -8 C 8 6 0 2 -6 D -10 6 -2 0 8 E -2 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.312500 D: 0.187500 E: 0.312500 Sum of squares = 0.265625000003 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.500000 D: 0.687500 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 10 2 B 2 0 -6 -6 -8 C 8 6 0 2 -6 D -10 6 -2 0 8 E -2 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.312500 D: 0.187500 E: 0.312500 Sum of squares = 0.265625000003 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.500000 D: 0.687500 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 10 2 B 2 0 -6 -6 -8 C 8 6 0 2 -6 D -10 6 -2 0 8 E -2 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.312500 D: 0.187500 E: 0.312500 Sum of squares = 0.265625000003 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.500000 D: 0.687500 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 369: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) D B A C E (5) C E D B A (5) A C D E B (5) E C D B A (4) E B D C A (4) D B C A E (4) C D A E B (4) C A D E B (4) E C A B D (3) E B C D A (3) D A C B E (3) D A B C E (3) C D E A B (3) C A E D B (3) B E D C A (3) B E D A C (3) B D E A C (3) E B C A D (2) E B A C D (2) D C A B E (2) B A E D C (2) A E C B D (2) A B E D C (2) A B E C D (2) A B D C E (2) E C B A D (1) E C A D B (1) E B A D C (1) D C B E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D A B E (1) B E A D C (1) B D E C A (1) B D A E C (1) B A D E C (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -4 -10 12 B -4 0 -10 -20 -4 C 4 10 0 2 16 D 10 20 -2 0 4 E -12 4 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -10 12 B -4 0 -10 -20 -4 C 4 10 0 2 16 D 10 20 -2 0 4 E -12 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=23 A=23 E=21 D=18 B=15 so B is eliminated. Round 2 votes counts: E=28 A=26 D=23 C=23 so D is eliminated. Round 3 votes counts: A=38 E=32 C=30 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:216 D:216 A:201 E:186 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -10 12 B -4 0 -10 -20 -4 C 4 10 0 2 16 D 10 20 -2 0 4 E -12 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -10 12 B -4 0 -10 -20 -4 C 4 10 0 2 16 D 10 20 -2 0 4 E -12 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -10 12 B -4 0 -10 -20 -4 C 4 10 0 2 16 D 10 20 -2 0 4 E -12 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 370: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) C E B D A (5) C A D E B (5) B A D E C (5) D E C A B (4) C B E A D (4) B C A E D (4) A D B C E (4) A B D C E (4) E D C B A (3) E C D B A (3) D A E B C (3) C E D B A (3) C B A E D (3) E B D A C (2) E B C D A (2) D E A B C (2) D A E C B (2) D A C E B (2) C D E A B (2) B E D A C (2) B E C A D (2) B A C E D (2) A C D B E (2) A C B D E (2) A B C D E (2) E D C A B (1) E C B D A (1) E B D C A (1) D E A C B (1) D C E A B (1) D B A E C (1) C E D A B (1) C E B A D (1) C A B E D (1) B E C D A (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D C E (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 4 14 12 B -2 0 0 12 6 C -4 0 0 -4 4 D -14 -12 4 0 10 E -12 -6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 14 12 B -2 0 0 12 6 C -4 0 0 -4 4 D -14 -12 4 0 10 E -12 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 B=20 D=16 E=13 so E is eliminated. Round 2 votes counts: C=29 A=26 B=25 D=20 so D is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:208 C:198 D:194 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 14 12 B -2 0 0 12 6 C -4 0 0 -4 4 D -14 -12 4 0 10 E -12 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 14 12 B -2 0 0 12 6 C -4 0 0 -4 4 D -14 -12 4 0 10 E -12 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 14 12 B -2 0 0 12 6 C -4 0 0 -4 4 D -14 -12 4 0 10 E -12 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 371: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) A E D C B (6) A D C E B (6) B A C D E (5) A D E C B (5) E D C B A (4) C D E B A (4) C D B E A (4) C B D E A (4) B E C D A (4) A E B D C (4) A B E D C (4) E D A C B (3) D C E A B (3) B A E C D (3) A B E C D (3) E B D C A (2) C D A E B (2) C A B D E (2) B C E D A (2) A E D B C (2) A B C D E (2) E D C A B (1) E C D B A (1) E A B D C (1) D E C A B (1) D C E B A (1) D C A E B (1) C D E A B (1) C D B A E (1) C A D B E (1) B E A D C (1) B C A D E (1) B A C E D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -4 -2 2 B 0 0 -6 4 2 C 4 6 0 10 10 D 2 -4 -10 0 16 E -2 -2 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -2 2 B 0 0 -6 4 2 C 4 6 0 10 10 D 2 -4 -10 0 16 E -2 -2 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=29 C=19 E=12 D=6 so D is eliminated. Round 2 votes counts: A=34 B=29 C=24 E=13 so E is eliminated. Round 3 votes counts: A=38 C=31 B=31 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:215 D:202 B:200 A:198 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -2 2 B 0 0 -6 4 2 C 4 6 0 10 10 D 2 -4 -10 0 16 E -2 -2 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 2 B 0 0 -6 4 2 C 4 6 0 10 10 D 2 -4 -10 0 16 E -2 -2 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 2 B 0 0 -6 4 2 C 4 6 0 10 10 D 2 -4 -10 0 16 E -2 -2 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 372: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C A E (7) B C D A E (7) E A D C B (6) D A E B C (5) C B D E A (5) A E B C D (5) A E C B D (4) A E B D C (4) E A C D B (3) D E A C B (3) D B C E A (3) B D C A E (3) B A D E C (3) E D C A B (2) E D A C B (2) D C B E A (2) D B A C E (2) C E A B D (2) C B E A D (2) B D A C E (2) A E D B C (2) A B E C D (2) E C A D B (1) D E C B A (1) D C E B A (1) D B E A C (1) D A B E C (1) C E D B A (1) C D E B A (1) C B E D A (1) B D A E C (1) B C D E A (1) B C A E D (1) B C A D E (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 16 -4 4 B -8 0 6 10 -8 C -16 -6 0 -6 -16 D 4 -10 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900826378 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 -4 4 B -8 0 6 10 -8 C -16 -6 0 -6 -16 D 4 -10 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825905 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=24 B=19 A=19 C=12 so C is eliminated. Round 2 votes counts: E=27 D=27 B=27 A=19 so A is eliminated. Round 3 votes counts: E=42 B=30 D=28 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:208 D:202 B:200 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 -4 4 B -8 0 6 10 -8 C -16 -6 0 -6 -16 D 4 -10 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825905 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -4 4 B -8 0 6 10 -8 C -16 -6 0 -6 -16 D 4 -10 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825905 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -4 4 B -8 0 6 10 -8 C -16 -6 0 -6 -16 D 4 -10 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825905 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 373: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) E A C B D (8) E C B D A (7) E C D B A (6) D B A C E (6) A D B C E (6) A B D C E (6) A D B E C (5) E C B A D (4) E C A B D (4) D A B C E (4) A E B D C (4) C D B E A (3) A D E B C (3) E A C D B (2) C E D B A (2) B C D A E (2) B A D C E (2) A E D B C (2) E D A B C (1) E C A D B (1) E A D C B (1) D C B E A (1) D C B A E (1) D A E B C (1) D A B E C (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) C B A E D (1) A E B C D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 10 6 14 B -4 0 10 -10 6 C -10 -10 0 -8 -4 D -6 10 8 0 4 E -14 -6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 6 14 B -4 0 10 -10 6 C -10 -10 0 -8 -4 D -6 10 8 0 4 E -14 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=29 D=23 C=10 B=4 so B is eliminated. Round 2 votes counts: E=34 A=31 D=23 C=12 so C is eliminated. Round 3 votes counts: E=38 A=32 D=30 so D is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:208 B:201 E:190 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 6 14 B -4 0 10 -10 6 C -10 -10 0 -8 -4 D -6 10 8 0 4 E -14 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 6 14 B -4 0 10 -10 6 C -10 -10 0 -8 -4 D -6 10 8 0 4 E -14 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 6 14 B -4 0 10 -10 6 C -10 -10 0 -8 -4 D -6 10 8 0 4 E -14 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 374: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) D A E B C (7) B C E A D (7) B E C D A (6) D E B A C (5) D A E C B (5) C B E A D (5) E B C D A (4) D A C E B (4) A C B D E (4) D E C A B (3) C E B A D (3) B E C A D (3) E D B C A (2) E C B D A (2) E B D C A (2) D E A B C (2) D B E A C (2) C E B D A (2) C B A E D (2) B C A E D (2) B A E C D (2) A C D E B (2) E D C B A (1) E C D B A (1) D E C B A (1) D E B C A (1) D E A C B (1) D A B E C (1) C E A D B (1) B E D C A (1) B E D A C (1) B A D E C (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 2 -8 -12 B 12 0 2 -2 -16 C -2 -2 0 -2 -8 D 8 2 2 0 4 E 12 16 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -8 -12 B 12 0 2 -2 -16 C -2 -2 0 -2 -8 D 8 2 2 0 4 E 12 16 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=23 A=20 C=13 E=12 so E is eliminated. Round 2 votes counts: D=35 B=29 A=20 C=16 so C is eliminated. Round 3 votes counts: B=43 D=36 A=21 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:208 B:198 C:193 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -8 -12 B 12 0 2 -2 -16 C -2 -2 0 -2 -8 D 8 2 2 0 4 E 12 16 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -8 -12 B 12 0 2 -2 -16 C -2 -2 0 -2 -8 D 8 2 2 0 4 E 12 16 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -8 -12 B 12 0 2 -2 -16 C -2 -2 0 -2 -8 D 8 2 2 0 4 E 12 16 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 375: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) C D A B E (9) E B D A C (7) A B D C E (6) E D C B A (4) E C D B A (4) E C A B D (4) E A B C D (4) C E A D B (4) A C B D E (4) D B A C E (3) C D E B A (3) C A D B E (3) B A D E C (3) E C D A B (2) D B A E C (2) C D E A B (2) B D E A C (2) B A D C E (2) A B C D E (2) E D B C A (1) E B D C A (1) E B A C D (1) E A C B D (1) E A B D C (1) D C E B A (1) D C B E A (1) D C B A E (1) D B E A C (1) D B C A E (1) C E A B D (1) C A E D B (1) C A D E B (1) B E D A C (1) B D A E C (1) A E C B D (1) A C E B D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 12 6 -14 B 2 0 2 10 -12 C -12 -2 0 -2 -6 D -6 -10 2 0 -2 E 14 12 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 12 6 -14 B 2 0 2 10 -12 C -12 -2 0 -2 -6 D -6 -10 2 0 -2 E 14 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=24 A=16 D=10 B=9 so B is eliminated. Round 2 votes counts: E=42 C=24 A=21 D=13 so D is eliminated. Round 3 votes counts: E=45 C=28 A=27 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:201 B:201 D:192 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 12 6 -14 B 2 0 2 10 -12 C -12 -2 0 -2 -6 D -6 -10 2 0 -2 E 14 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 6 -14 B 2 0 2 10 -12 C -12 -2 0 -2 -6 D -6 -10 2 0 -2 E 14 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 6 -14 B 2 0 2 10 -12 C -12 -2 0 -2 -6 D -6 -10 2 0 -2 E 14 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 376: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) B D C E A (8) D B C E A (7) B E A D C (6) D C A E B (5) C D A E B (5) B E A C D (5) A E C D B (5) D C A B E (4) A E B C D (4) D B C A E (3) B E D C A (3) B D E C A (3) B A E D C (3) A E C B D (3) A C E D B (3) E B C A D (2) E A B C D (2) D C B A E (2) D B A C E (2) C D E B A (2) C D E A B (2) B E D A C (2) B D E A C (2) E B A C D (1) E A C B D (1) C A D E B (1) B E C D A (1) B C E D A (1) B A E C D (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -28 -16 -22 -14 B 28 0 16 0 28 C 16 -16 0 -22 8 D 22 0 22 0 12 E 14 -28 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.564081 C: 0.000000 D: 0.435919 E: 0.000000 Sum of squares = 0.508212630702 Cumulative probabilities = A: 0.000000 B: 0.564081 C: 0.564081 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -16 -22 -14 B 28 0 16 0 28 C 16 -16 0 -22 8 D 22 0 22 0 12 E 14 -28 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=32 A=16 C=10 E=6 so E is eliminated. Round 2 votes counts: B=39 D=32 A=19 C=10 so C is eliminated. Round 3 votes counts: D=41 B=39 A=20 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:236 D:228 C:193 E:183 A:160 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -16 -22 -14 B 28 0 16 0 28 C 16 -16 0 -22 8 D 22 0 22 0 12 E 14 -28 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -16 -22 -14 B 28 0 16 0 28 C 16 -16 0 -22 8 D 22 0 22 0 12 E 14 -28 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -16 -22 -14 B 28 0 16 0 28 C 16 -16 0 -22 8 D 22 0 22 0 12 E 14 -28 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 377: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (13) E B D C A (10) E B D A C (9) D C A E B (9) B E D A C (7) A C D B E (7) E D B C A (6) A C B D E (5) E D C A B (4) D E C A B (4) B A C E D (4) B A C D E (4) D A C B E (2) C A D B E (2) B D E A C (2) E D B A C (1) E B C A D (1) E B A C D (1) D E B C A (1) D C E A B (1) D C A B E (1) D A C E B (1) C D A E B (1) C A E D B (1) C A D E B (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 16 -18 -20 B 20 0 22 16 -2 C -16 -22 0 -16 -20 D 18 -16 16 0 -18 E 20 2 20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 16 -18 -20 B 20 0 22 16 -2 C -16 -22 0 -16 -20 D 18 -16 16 0 -18 E 20 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=31 D=19 A=13 C=5 so C is eliminated. Round 2 votes counts: E=32 B=31 D=20 A=17 so A is eliminated. Round 3 votes counts: B=37 E=33 D=30 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 B:228 D:200 A:179 C:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 16 -18 -20 B 20 0 22 16 -2 C -16 -22 0 -16 -20 D 18 -16 16 0 -18 E 20 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 16 -18 -20 B 20 0 22 16 -2 C -16 -22 0 -16 -20 D 18 -16 16 0 -18 E 20 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 16 -18 -20 B 20 0 22 16 -2 C -16 -22 0 -16 -20 D 18 -16 16 0 -18 E 20 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 378: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) B A E D C (9) A E B C D (8) E A C D B (7) E C A D B (6) B A E C D (6) C E D A B (4) C D E A B (4) C D B E A (4) B D A C E (4) E A C B D (3) D C E A B (3) D C B E A (3) B D C E A (3) A E C D B (3) D C B A E (2) D B C E A (2) D B C A E (2) C D E B A (2) E C D A B (1) E C B A D (1) E A B C D (1) D C E B A (1) D A E C B (1) D A C E B (1) D A B E C (1) C B D E A (1) B C D E A (1) B A D E C (1) B A D C E (1) A E D C B (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 0 6 B 6 0 4 4 6 C 0 -4 0 6 -4 D 0 -4 -6 0 -4 E -6 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 0 6 B 6 0 4 4 6 C 0 -4 0 6 -4 D 0 -4 -6 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=19 D=16 C=15 A=15 so C is eliminated. Round 2 votes counts: B=36 D=26 E=23 A=15 so A is eliminated. Round 3 votes counts: B=39 E=35 D=26 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:200 C:199 E:198 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 0 6 B 6 0 4 4 6 C 0 -4 0 6 -4 D 0 -4 -6 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 6 B 6 0 4 4 6 C 0 -4 0 6 -4 D 0 -4 -6 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 6 B 6 0 4 4 6 C 0 -4 0 6 -4 D 0 -4 -6 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 379: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) E B A C D (8) D C A E B (8) B E A C D (8) B E A D C (7) A E C B D (6) D C B E A (5) D C A B E (5) E A B C D (4) D B C E A (4) B E D C A (4) A C E D B (4) D C B A E (3) C D A E B (3) B E D A C (3) A C E B D (3) D B E C A (2) C D A B E (2) C A D E B (2) B D C E A (2) E B A D C (1) E A B D C (1) C B E A D (1) C A E B D (1) B E C D A (1) B D E C A (1) B D E A C (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 12 12 -6 B 2 0 12 22 -2 C -12 -12 0 6 -12 D -12 -22 -6 0 -22 E 6 2 12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999253 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 12 12 -6 B 2 0 12 22 -2 C -12 -12 0 6 -12 D -12 -22 -6 0 -22 E 6 2 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 A=23 E=14 C=9 so C is eliminated. Round 2 votes counts: D=32 B=28 A=26 E=14 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: B=61 D=39 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:221 B:217 A:208 C:185 D:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 12 12 -6 B 2 0 12 22 -2 C -12 -12 0 6 -12 D -12 -22 -6 0 -22 E 6 2 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 12 -6 B 2 0 12 22 -2 C -12 -12 0 6 -12 D -12 -22 -6 0 -22 E 6 2 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 12 -6 B 2 0 12 22 -2 C -12 -12 0 6 -12 D -12 -22 -6 0 -22 E 6 2 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 380: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) D B A C E (10) E C A B D (6) E C A D B (5) E B D A C (5) C A D E B (5) C A D B E (5) B D E A C (5) E B C D A (4) D A C B E (4) C A E D B (4) B D A C E (4) E C B A D (3) E B D C A (3) D A B C E (3) B D E C A (3) A C D E B (3) E B C A D (2) B D A E C (2) A D C B E (2) E A C D B (1) D C B A E (1) D B C A E (1) D B A E C (1) C D A B E (1) C B E A D (1) B E D C A (1) B E D A C (1) B D C E A (1) B D C A E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 -2 18 B -2 0 -6 -16 16 C -6 6 0 4 16 D 2 16 -4 0 26 E -18 -16 -16 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -2 18 B -2 0 -6 -16 16 C -6 6 0 4 16 D 2 16 -4 0 26 E -18 -16 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889564 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=20 B=18 A=17 C=16 so C is eliminated. Round 2 votes counts: A=31 E=29 D=21 B=19 so B is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: D=63 E=37 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:212 C:210 B:196 E:162 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 -2 18 B -2 0 -6 -16 16 C -6 6 0 4 16 D 2 16 -4 0 26 E -18 -16 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889564 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -2 18 B -2 0 -6 -16 16 C -6 6 0 4 16 D 2 16 -4 0 26 E -18 -16 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889564 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -2 18 B -2 0 -6 -16 16 C -6 6 0 4 16 D 2 16 -4 0 26 E -18 -16 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889564 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 381: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) A B D E C (7) E C D B A (5) E C B D A (5) D E C A B (5) C E D B A (4) C E B A D (4) C D A E B (4) D C A E B (3) D A E B C (3) D A C E B (3) D A B C E (3) C E B D A (3) A D B C E (3) E B C A D (2) D C E A B (2) D A E C B (2) C E D A B (2) C D A B E (2) C B E A D (2) C A B D E (2) B E C A D (2) B E A C D (2) B A E D C (2) B A C E D (2) B A C D E (2) A B D C E (2) E D C B A (1) E D B A C (1) E B A D C (1) E B A C D (1) D E A C B (1) D C A B E (1) D A C B E (1) C D E A B (1) C A D B E (1) B E A D C (1) B C E A D (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 14 -4 -22 8 B -14 0 -8 -14 -6 C 4 8 0 -4 -4 D 22 14 4 0 16 E -8 6 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 -22 8 B -14 0 -8 -14 -6 C 4 8 0 -4 -4 D 22 14 4 0 16 E -8 6 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=25 E=16 B=14 A=12 so A is eliminated. Round 2 votes counts: D=36 C=25 B=23 E=16 so E is eliminated. Round 3 votes counts: D=38 C=35 B=27 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:228 C:202 A:198 E:193 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -4 -22 8 B -14 0 -8 -14 -6 C 4 8 0 -4 -4 D 22 14 4 0 16 E -8 6 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -22 8 B -14 0 -8 -14 -6 C 4 8 0 -4 -4 D 22 14 4 0 16 E -8 6 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -22 8 B -14 0 -8 -14 -6 C 4 8 0 -4 -4 D 22 14 4 0 16 E -8 6 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 382: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) C A D E B (6) E B A C D (5) B E D C A (5) B E C D A (5) E B C A D (4) E B A D C (4) D B A C E (4) A D C B E (4) C B E D A (3) B E D A C (3) B D E A C (3) A C D E B (3) E C B A D (2) E C A B D (2) E A C D B (2) E A B D C (2) D C A B E (2) D B C A E (2) D A C B E (2) D A B C E (2) C E A D B (2) B D C A E (2) A D E B C (2) A C E D B (2) E C A D B (1) E A B C D (1) D C B A E (1) D B A E C (1) D A B E C (1) C D E A B (1) C D B E A (1) C B D E A (1) B E A D C (1) B D E C A (1) B D C E A (1) B D A E C (1) B A E D C (1) A E C D B (1) A D E C B (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -4 -4 0 B 2 0 4 -6 16 C 4 -4 0 4 2 D 4 6 -4 0 8 E 0 -16 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775498 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 0 B 2 0 4 -6 16 C 4 -4 0 4 2 D 4 6 -4 0 8 E 0 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=23 B=23 D=15 A=15 so D is eliminated. Round 2 votes counts: B=30 C=27 E=23 A=20 so A is eliminated. Round 3 votes counts: C=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:207 C:203 A:195 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 0 B 2 0 4 -6 16 C 4 -4 0 4 2 D 4 6 -4 0 8 E 0 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 0 B 2 0 4 -6 16 C 4 -4 0 4 2 D 4 6 -4 0 8 E 0 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 0 B 2 0 4 -6 16 C 4 -4 0 4 2 D 4 6 -4 0 8 E 0 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 383: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) D E A C B (7) A B C D E (7) B C A E D (5) E D C A B (4) D E A B C (4) D A E C B (4) A C D B E (4) E D B C A (3) E C B D A (3) E B D C A (3) E B C D A (3) D E C A B (3) D E B A C (3) B E C D A (3) B A C E D (3) B A C D E (3) A D B C E (3) A C B D E (3) E D C B A (2) D A E B C (2) C E B D A (2) A D E C B (2) A D C E B (2) A D C B E (2) A D B E C (2) E C D B A (1) D B E A C (1) C E B A D (1) C A E D B (1) C A E B D (1) C A B E D (1) B E D C A (1) B A E D C (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 10 2 -4 B -6 0 14 0 -2 C -10 -14 0 -2 -4 D -2 0 2 0 8 E 4 2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.42857142857 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 A B C D E A 0 6 10 2 -4 B -6 0 14 0 -2 C -10 -14 0 -2 -4 D -2 0 2 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428644 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 D=24 E=19 C=6 so C is eliminated. Round 2 votes counts: A=29 B=25 D=24 E=22 so E is eliminated. Round 3 votes counts: B=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:207 D:204 B:203 E:201 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 2 -4 B -6 0 14 0 -2 C -10 -14 0 -2 -4 D -2 0 2 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428644 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 2 -4 B -6 0 14 0 -2 C -10 -14 0 -2 -4 D -2 0 2 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428644 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 2 -4 B -6 0 14 0 -2 C -10 -14 0 -2 -4 D -2 0 2 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428644 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 384: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) E D C B A (7) A D E B C (7) A B D E C (7) E D C A B (6) D E A B C (6) C E D B A (6) C B E D A (6) C B A E D (5) B C E D A (5) A B C D E (5) D E A C B (4) C E D A B (4) B C A E D (3) E D B C A (2) E C D A B (2) D E B C A (2) A D E C B (2) A C E D B (2) E C D B A (1) D E B A C (1) D A E B C (1) C E B D A (1) C A E D B (1) C A B E D (1) B D E C A (1) B A D E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -6 -10 -8 B 2 0 2 -10 -12 C 6 -2 0 6 0 D 10 10 -6 0 -4 E 8 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.463220 D: 0.000000 E: 0.536780 Sum of squares = 0.502705594056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.463220 D: 0.463220 E: 1.000000 A B C D E A 0 -2 -6 -10 -8 B 2 0 2 -10 -12 C 6 -2 0 6 0 D 10 10 -6 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 B=20 E=18 D=14 so D is eliminated. Round 2 votes counts: E=31 A=25 C=24 B=20 so B is eliminated. Round 3 votes counts: A=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:205 D:205 B:191 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 -10 -8 B 2 0 2 -10 -12 C 6 -2 0 6 0 D 10 10 -6 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -10 -8 B 2 0 2 -10 -12 C 6 -2 0 6 0 D 10 10 -6 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -10 -8 B 2 0 2 -10 -12 C 6 -2 0 6 0 D 10 10 -6 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 385: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) E B D C A (7) C A D E B (7) C D E B A (5) E B C A D (4) C D A B E (4) A B D E C (4) E A B D C (3) D C B E A (3) D B E C A (3) C D B E A (3) A D C B E (3) A D B C E (3) E B A C D (2) E A B C D (2) D C B A E (2) D B E A C (2) D A C B E (2) C D A E B (2) C A E D B (2) C A D B E (2) B E D C A (2) B E A D C (2) A E B C D (2) A C D B E (2) A B E D C (2) E D C B A (1) E B C D A (1) E A C B D (1) D B A E C (1) D A B C E (1) C E D B A (1) C E B A D (1) C E A B D (1) C D B A E (1) B E D A C (1) B D E A C (1) B A E D C (1) B A D E C (1) A E C B D (1) A E B D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -4 14 -8 B 6 0 8 -2 -8 C 4 -8 0 -8 -6 D -14 2 8 0 4 E 8 8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.000000 D: 0.307692 E: 0.538462 Sum of squares = 0.408284023673 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.153846 D: 0.461538 E: 1.000000 A B C D E A 0 -6 -4 14 -8 B 6 0 8 -2 -8 C 4 -8 0 -8 -6 D -14 2 8 0 4 E 8 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.000000 D: 0.307692 E: 0.538462 Sum of squares = 0.40828402369 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.153846 D: 0.461538 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=29 C=29 A=20 D=14 B=8 so B is eliminated. Round 2 votes counts: E=34 C=29 A=22 D=15 so D is eliminated. Round 3 votes counts: E=40 C=34 A=26 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:202 D:200 A:198 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 14 -8 B 6 0 8 -2 -8 C 4 -8 0 -8 -6 D -14 2 8 0 4 E 8 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.000000 D: 0.307692 E: 0.538462 Sum of squares = 0.40828402369 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.153846 D: 0.461538 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 14 -8 B 6 0 8 -2 -8 C 4 -8 0 -8 -6 D -14 2 8 0 4 E 8 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.000000 D: 0.307692 E: 0.538462 Sum of squares = 0.40828402369 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.153846 D: 0.461538 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 14 -8 B 6 0 8 -2 -8 C 4 -8 0 -8 -6 D -14 2 8 0 4 E 8 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.000000 D: 0.307692 E: 0.538462 Sum of squares = 0.40828402369 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.153846 D: 0.461538 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 386: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) C D A B E (7) E C B A D (6) C B D A E (6) E C A D B (5) D A B E C (5) D A B C E (5) C D A E B (5) A D B E C (5) C E B A D (4) C D B A E (4) E B A D C (3) E C A B D (2) E A C D B (2) E A B D C (2) D C A B E (2) D A C E B (2) C E D A B (2) C E B D A (2) B E A D C (2) B C E A D (2) B C A D E (2) A D E B C (2) A B D E C (2) E D C A B (1) E D A C B (1) E B C A D (1) E B A C D (1) E A D C B (1) E A D B C (1) D A C B E (1) C D E A B (1) B E A C D (1) B A E D C (1) B A D C E (1) A E D B C (1) Total count = 100 A B C D E A 0 10 -4 12 20 B -10 0 -8 -6 10 C 4 8 0 6 -8 D -12 6 -6 0 18 E -20 -10 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000004 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 10 -4 12 20 B -10 0 -8 -6 10 C 4 8 0 6 -8 D -12 6 -6 0 18 E -20 -10 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999661 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 B=18 D=15 A=10 so A is eliminated. Round 2 votes counts: C=31 E=27 D=22 B=20 so B is eliminated. Round 3 votes counts: C=35 D=34 E=31 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:205 D:203 B:193 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -4 12 20 B -10 0 -8 -6 10 C 4 8 0 6 -8 D -12 6 -6 0 18 E -20 -10 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999661 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 12 20 B -10 0 -8 -6 10 C 4 8 0 6 -8 D -12 6 -6 0 18 E -20 -10 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999661 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 12 20 B -10 0 -8 -6 10 C 4 8 0 6 -8 D -12 6 -6 0 18 E -20 -10 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999661 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 387: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (11) A D B C E (10) E C B D A (8) A C D E B (8) C E A D B (6) B E D C A (6) A D C B E (6) B E D A C (5) E C B A D (4) D A B C E (4) C E A B D (4) D B A E C (3) B E C D A (3) B D A E C (3) E C A B D (2) E B D C A (2) B D E C A (2) A C E D B (2) E B C D A (1) E A C B D (1) D C A E B (1) D B A C E (1) D A C B E (1) C E D B A (1) C A E D B (1) B A D E C (1) A C E B D (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 18 -4 -12 B 2 0 6 10 16 C -18 -6 0 -14 -6 D 4 -10 14 0 6 E 12 -16 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 18 -4 -12 B 2 0 6 10 16 C -18 -6 0 -14 -6 D 4 -10 14 0 6 E 12 -16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=29 E=18 C=12 D=10 so D is eliminated. Round 2 votes counts: B=35 A=34 E=18 C=13 so C is eliminated. Round 3 votes counts: A=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:207 A:200 E:198 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 18 -4 -12 B 2 0 6 10 16 C -18 -6 0 -14 -6 D 4 -10 14 0 6 E 12 -16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 18 -4 -12 B 2 0 6 10 16 C -18 -6 0 -14 -6 D 4 -10 14 0 6 E 12 -16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 18 -4 -12 B 2 0 6 10 16 C -18 -6 0 -14 -6 D 4 -10 14 0 6 E 12 -16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 388: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) A E D C B (6) E A D C B (5) D C A E B (5) E B A D C (4) E A B D C (4) B C E D A (4) E D B C A (3) E B D C A (3) E A D B C (3) D C E B A (3) D C A B E (3) C D B A E (3) B E D C A (3) B C D E A (3) B A C E D (3) A E C D B (3) A C B D E (3) D E B C A (2) D C B E A (2) D A C E B (2) C A B D E (2) B E C D A (2) A E B C D (2) A D E C B (2) A C B E D (2) E D B A C (1) E B D A C (1) E B C A D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C B A E (1) D B E C A (1) D B C E A (1) C B D A E (1) C B A D E (1) B E C A D (1) B E A C D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -6 -6 -2 B -10 0 -8 -12 -6 C 6 8 0 -14 -4 D 6 12 14 0 -10 E 2 6 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -6 -6 -2 B -10 0 -8 -12 -6 C 6 8 0 -14 -4 D 6 12 14 0 -10 E 2 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=22 A=21 B=17 C=14 so C is eliminated. Round 2 votes counts: D=32 E=26 A=23 B=19 so B is eliminated. Round 3 votes counts: E=37 D=36 A=27 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:211 A:198 C:198 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -6 -6 -2 B -10 0 -8 -12 -6 C 6 8 0 -14 -4 D 6 12 14 0 -10 E 2 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -6 -2 B -10 0 -8 -12 -6 C 6 8 0 -14 -4 D 6 12 14 0 -10 E 2 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -6 -2 B -10 0 -8 -12 -6 C 6 8 0 -14 -4 D 6 12 14 0 -10 E 2 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 389: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) B D A E C (9) C E B A D (8) D B A E C (6) C E A B D (5) D A B E C (4) D A C E B (3) B D C E A (3) B C E A D (3) A D E C B (3) E C A B D (2) E A C B D (2) D B A C E (2) D A C B E (2) D A B C E (2) C D E A B (2) C A E D B (2) B E C D A (2) B E C A D (2) B D C A E (2) B C E D A (2) A E C D B (2) E C B A D (1) E B C A D (1) E A B C D (1) D C A E B (1) C E D B A (1) C E D A B (1) C E B D A (1) C B E D A (1) C B E A D (1) B E A C D (1) B D E C A (1) B D E A C (1) B C D E A (1) B A E D C (1) B A E C D (1) B A D E C (1) A E B D C (1) A E B C D (1) A D E B C (1) A D B E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -6 6 -6 B 4 0 2 12 0 C 6 -2 0 10 8 D -6 -12 -10 0 -8 E 6 0 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.887733 C: 0.000000 D: 0.000000 E: 0.112267 Sum of squares = 0.800673519446 Cumulative probabilities = A: 0.000000 B: 0.887733 C: 0.887733 D: 0.887733 E: 1.000000 A B C D E A 0 -4 -6 6 -6 B 4 0 2 12 0 C 6 -2 0 10 8 D -6 -12 -10 0 -8 E 6 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000051157 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=30 D=20 A=11 E=7 so E is eliminated. Round 2 votes counts: C=35 B=31 D=20 A=14 so A is eliminated. Round 3 votes counts: C=40 B=35 D=25 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:209 E:203 A:195 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 6 -6 B 4 0 2 12 0 C 6 -2 0 10 8 D -6 -12 -10 0 -8 E 6 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000051157 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 -6 B 4 0 2 12 0 C 6 -2 0 10 8 D -6 -12 -10 0 -8 E 6 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000051157 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 -6 B 4 0 2 12 0 C 6 -2 0 10 8 D -6 -12 -10 0 -8 E 6 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000051157 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 390: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (11) C A B E D (8) C E B D A (7) C A D E B (7) A B E D C (7) E B D C A (5) C D E B A (5) C B E D A (5) A D C E B (5) A D B E C (5) D E B A C (4) A C D E B (4) C B E A D (3) D E B C A (2) D A E B C (2) B E D C A (2) B E D A C (2) B E C D A (2) B C E D A (2) A D E C B (2) A C B E D (2) E D B C A (1) E B C D A (1) D E A B C (1) C E D B A (1) B A E D C (1) A C D B E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -2 16 14 B -14 0 0 -2 -16 C 2 0 0 -2 0 D -16 2 2 0 0 E -14 16 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.800000 D: 0.100000 E: 0.000000 Sum of squares = 0.659999999997 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 16 14 B -14 0 0 -2 -16 C 2 0 0 -2 0 D -16 2 2 0 0 E -14 16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.800000 D: 0.100000 E: 0.000000 Sum of squares = 0.659999999976 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=36 D=9 B=9 E=7 so E is eliminated. Round 2 votes counts: A=39 C=36 B=15 D=10 so D is eliminated. Round 3 votes counts: A=42 C=36 B=22 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:221 E:201 C:200 D:194 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 14 -2 16 14 B -14 0 0 -2 -16 C 2 0 0 -2 0 D -16 2 2 0 0 E -14 16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.800000 D: 0.100000 E: 0.000000 Sum of squares = 0.659999999976 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 16 14 B -14 0 0 -2 -16 C 2 0 0 -2 0 D -16 2 2 0 0 E -14 16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.800000 D: 0.100000 E: 0.000000 Sum of squares = 0.659999999976 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 16 14 B -14 0 0 -2 -16 C 2 0 0 -2 0 D -16 2 2 0 0 E -14 16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.800000 D: 0.100000 E: 0.000000 Sum of squares = 0.659999999976 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 391: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) C D B A E (7) E C B A D (6) C E B A D (6) C B E A D (5) E B A C D (4) D C A B E (4) E A B D C (3) C E D A B (3) C D E B A (3) B A D E C (3) A D B E C (3) E A B C D (2) D C B A E (2) D A B E C (2) D A B C E (2) C E B D A (2) C D E A B (2) C D B E A (2) B E A D C (2) B E A C D (2) B D A C E (2) B C E A D (2) B A E D C (2) A E D B C (2) A D E C B (2) A B E D C (2) E C A D B (1) E B C A D (1) E B A D C (1) E A D B C (1) E A C D B (1) D E C A B (1) D B A C E (1) D A E B C (1) D A C E B (1) D A C B E (1) C B E D A (1) C B D E A (1) C B D A E (1) B E C A D (1) B D A E C (1) Total count = 100 A B C D E A 0 -16 -18 4 -6 B 16 0 -20 0 -2 C 18 20 0 6 12 D -4 0 -6 0 0 E 6 2 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -18 4 -6 B 16 0 -20 0 -2 C 18 20 0 6 12 D -4 0 -6 0 0 E 6 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=23 E=20 B=15 A=9 so A is eliminated. Round 2 votes counts: C=33 D=28 E=22 B=17 so B is eliminated. Round 3 votes counts: C=35 D=34 E=31 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:198 B:197 D:195 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -18 4 -6 B 16 0 -20 0 -2 C 18 20 0 6 12 D -4 0 -6 0 0 E 6 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 4 -6 B 16 0 -20 0 -2 C 18 20 0 6 12 D -4 0 -6 0 0 E 6 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 4 -6 B 16 0 -20 0 -2 C 18 20 0 6 12 D -4 0 -6 0 0 E 6 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 392: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (16) A C E B D (13) E C D B A (12) E C A D B (12) D B E C A (11) C E A B D (9) A B D C E (7) D B A E C (5) D B E A C (4) A C B D E (4) E D B C A (3) C A E B D (3) D B A C E (1) Total count = 100 A B C D E A 0 -4 0 -4 -2 B 4 0 -6 4 -4 C 0 6 0 6 6 D 4 -4 -6 0 -4 E 2 4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.413564 B: 0.000000 C: 0.586436 D: 0.000000 E: 0.000000 Sum of squares = 0.514942489633 Cumulative probabilities = A: 0.413564 B: 0.413564 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -4 -2 B 4 0 -6 4 -4 C 0 6 0 6 6 D 4 -4 -6 0 -4 E 2 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=24 D=21 B=16 C=12 so C is eliminated. Round 2 votes counts: E=36 A=27 D=21 B=16 so B is eliminated. Round 3 votes counts: D=37 E=36 A=27 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:202 B:199 A:195 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 -4 -2 B 4 0 -6 4 -4 C 0 6 0 6 6 D 4 -4 -6 0 -4 E 2 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -4 -2 B 4 0 -6 4 -4 C 0 6 0 6 6 D 4 -4 -6 0 -4 E 2 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -4 -2 B 4 0 -6 4 -4 C 0 6 0 6 6 D 4 -4 -6 0 -4 E 2 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 393: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) B C A D E (8) D A E C B (7) C B D E A (7) A E D C B (7) A D E C B (6) A D E B C (6) D E A C B (5) C B E D A (5) E D C A B (3) E C D A B (3) B C E D A (3) B C D A E (3) A E D B C (3) D E C A B (2) C E D B A (2) C E B D A (2) B C A E D (2) A E B D C (2) A B D C E (2) E D A C B (1) E C D B A (1) E A C D B (1) D C E B A (1) D C E A B (1) C D E B A (1) B C E A D (1) B A C E D (1) B A C D E (1) A E B C D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -8 -12 4 B -6 0 -10 0 -10 C 8 10 0 6 0 D 12 0 -6 0 18 E -4 10 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.851951 D: 0.000000 E: 0.148049 Sum of squares = 0.747739575945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.851951 D: 0.851951 E: 1.000000 A B C D E A 0 6 -8 -12 4 B -6 0 -10 0 -10 C 8 10 0 6 0 D 12 0 -6 0 18 E -4 10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500005383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=28 C=17 D=16 E=9 so E is eliminated. Round 2 votes counts: A=31 B=28 C=21 D=20 so D is eliminated. Round 3 votes counts: A=44 C=28 B=28 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 D:212 A:195 E:194 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -12 4 B -6 0 -10 0 -10 C 8 10 0 6 0 D 12 0 -6 0 18 E -4 10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500005383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -12 4 B -6 0 -10 0 -10 C 8 10 0 6 0 D 12 0 -6 0 18 E -4 10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500005383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -12 4 B -6 0 -10 0 -10 C 8 10 0 6 0 D 12 0 -6 0 18 E -4 10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500005383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 394: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) C E B A D (6) C B E A D (5) A D E C B (5) E C A B D (4) E C A D B (3) E B D C A (3) E B C D A (3) D E A B C (3) D A E C B (3) D A E B C (3) C A E B D (3) B D E C A (3) E D C A B (2) E D B A C (2) E A D C B (2) E A C D B (2) D B E A C (2) B E C D A (2) B D C E A (2) B C E A D (2) B C D A E (2) A E C D B (2) A D C E B (2) A D C B E (2) A C E D B (2) A B C D E (2) E D C B A (1) E D A B C (1) E C B D A (1) E C B A D (1) D E B A C (1) D E A C B (1) D A B E C (1) D A B C E (1) C E A B D (1) C B A E D (1) C A B E D (1) B D C A E (1) B D A C E (1) B C E D A (1) B C D E A (1) B C A E D (1) B A C D E (1) A E D C B (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 0 6 -10 B -2 0 -8 -2 -18 C 0 8 0 0 -16 D -6 2 0 0 -6 E 10 18 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 6 -10 B -2 0 -8 -2 -18 C 0 8 0 0 -16 D -6 2 0 0 -6 E 10 18 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=21 A=20 C=17 B=17 so C is eliminated. Round 2 votes counts: E=32 A=24 B=23 D=21 so D is eliminated. Round 3 votes counts: E=37 A=32 B=31 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:199 C:196 D:195 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 6 -10 B -2 0 -8 -2 -18 C 0 8 0 0 -16 D -6 2 0 0 -6 E 10 18 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 -10 B -2 0 -8 -2 -18 C 0 8 0 0 -16 D -6 2 0 0 -6 E 10 18 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 -10 B -2 0 -8 -2 -18 C 0 8 0 0 -16 D -6 2 0 0 -6 E 10 18 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 395: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) B E C D A (8) A D C B E (8) D A E C B (5) C B A D E (5) B C A E D (5) A D C E B (5) D E A C B (4) E D A C B (3) E B D A C (3) E B C D A (3) D A C E B (3) B C E D A (3) B C A D E (3) A C D B E (3) A C B D E (3) E B D C A (2) E B A D C (2) E A D B C (2) D C A E B (2) C D E B A (2) C B E D A (2) C A D B E (2) C A B D E (2) B E A C D (2) E D C A B (1) D E C A B (1) D C E B A (1) C D A E B (1) C D A B E (1) B E C A D (1) B A C D E (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 10 2 B 4 0 -10 10 18 C 8 10 0 14 24 D -10 -10 -14 0 8 E -2 -18 -24 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 10 2 B 4 0 -10 10 18 C 8 10 0 14 24 D -10 -10 -14 0 8 E -2 -18 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=21 E=16 D=16 C=15 so C is eliminated. Round 2 votes counts: B=39 A=25 D=20 E=16 so E is eliminated. Round 3 votes counts: B=49 A=27 D=24 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:228 B:211 A:200 D:187 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 10 2 B 4 0 -10 10 18 C 8 10 0 14 24 D -10 -10 -14 0 8 E -2 -18 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 10 2 B 4 0 -10 10 18 C 8 10 0 14 24 D -10 -10 -14 0 8 E -2 -18 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 10 2 B 4 0 -10 10 18 C 8 10 0 14 24 D -10 -10 -14 0 8 E -2 -18 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 396: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) D E B C A (9) B C A E D (9) D E A C B (7) C B A D E (5) A C B E D (5) E D B A C (4) D E A B C (4) D E C A B (3) D E B A C (3) B E D A C (3) B D E C A (3) A C D E B (3) E D A C B (2) C A B E D (2) C A B D E (2) B C D E A (2) B A C E D (2) A C E D B (2) E B D A C (1) E A D C B (1) E A D B C (1) D E C B A (1) D C B E A (1) D A E C B (1) C D A E B (1) C B D E A (1) C B A E D (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B D C E A (1) B C E D A (1) B C D A E (1) B C A D E (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 4 -20 -20 B 4 0 14 -12 -14 C -4 -14 0 -16 -14 D 20 12 16 0 4 E 20 14 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -20 -20 B 4 0 14 -12 -14 C -4 -14 0 -16 -14 D 20 12 16 0 4 E 20 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 E=18 A=14 C=13 so C is eliminated. Round 2 votes counts: B=33 D=30 A=19 E=18 so E is eliminated. Round 3 votes counts: D=45 B=34 A=21 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 E:222 B:196 A:180 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -20 -20 B 4 0 14 -12 -14 C -4 -14 0 -16 -14 D 20 12 16 0 4 E 20 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -20 -20 B 4 0 14 -12 -14 C -4 -14 0 -16 -14 D 20 12 16 0 4 E 20 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -20 -20 B 4 0 14 -12 -14 C -4 -14 0 -16 -14 D 20 12 16 0 4 E 20 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 397: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) B D A E C (8) A B C D E (8) C A E B D (7) A C B E D (6) E D B C A (5) C E D A B (5) C E A D B (5) A B D C E (5) D E B C A (4) D B E A C (4) B D E A C (4) A C B D E (4) E C D A B (3) D E C B A (3) B A D C E (3) E C D B A (2) D B A E C (2) B A D E C (2) A B C E D (2) E B D C A (1) E B C D A (1) D E B A C (1) C A E D B (1) C A D E B (1) C A B E D (1) B A E C D (1) B A C E D (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 6 -2 8 B 2 0 6 10 6 C -6 -6 0 0 2 D 2 -10 0 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -2 8 B 2 0 6 10 6 C -6 -6 0 0 2 D 2 -10 0 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=20 C=20 B=20 D=14 so D is eliminated. Round 2 votes counts: E=28 B=26 A=26 C=20 so C is eliminated. Round 3 votes counts: E=38 A=36 B=26 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:212 A:205 D:197 C:195 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 -2 8 B 2 0 6 10 6 C -6 -6 0 0 2 D 2 -10 0 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 8 B 2 0 6 10 6 C -6 -6 0 0 2 D 2 -10 0 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 8 B 2 0 6 10 6 C -6 -6 0 0 2 D 2 -10 0 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 398: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) B A E D C (9) C E D B A (6) B E A D C (6) B A D E C (6) A D B E C (6) C D A E B (5) B E C A D (5) A B D E C (5) A B D C E (4) E D C A B (3) D C A E B (3) C E D A B (3) E B C D A (2) B C A D E (2) B A D C E (2) A D E B C (2) A D C B E (2) A D B C E (2) E D C B A (1) E D A C B (1) E D A B C (1) E C D B A (1) E C B D A (1) E B C A D (1) D C E A B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D A B E (1) C B E D A (1) C B D A E (1) C B A D E (1) C A D B E (1) B E A C D (1) A D E C B (1) Total count = 100 A B C D E A 0 4 -2 12 8 B -4 0 8 -4 10 C 2 -8 0 -12 -4 D -12 4 12 0 14 E -8 -10 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428606 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 12 8 B -4 0 8 -4 10 C 2 -8 0 -12 -4 D -12 4 12 0 14 E -8 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=31 B=31 A=22 E=11 D=5 so D is eliminated. Round 2 votes counts: C=35 B=31 A=23 E=11 so E is eliminated. Round 3 votes counts: C=41 B=34 A=25 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 D:209 B:205 C:189 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 12 8 B -4 0 8 -4 10 C 2 -8 0 -12 -4 D -12 4 12 0 14 E -8 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 12 8 B -4 0 8 -4 10 C 2 -8 0 -12 -4 D -12 4 12 0 14 E -8 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 12 8 B -4 0 8 -4 10 C 2 -8 0 -12 -4 D -12 4 12 0 14 E -8 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 399: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) C B E D A (6) B A C D E (5) C E B D A (4) C B E A D (4) A D B E C (4) E D C B A (3) E C D B A (3) E C A D B (3) E A D C B (3) D E B C A (3) D A B E C (3) C E A B D (3) B D C A E (3) B C D E A (3) A C B E D (3) E C D A B (2) E C B D A (2) E C A B D (2) D E A C B (2) C E B A D (2) B D A C E (2) B C E D A (2) B A D C E (2) A E D C B (2) A D E C B (2) A D E B C (2) A D B C E (2) A B D C E (2) E D A C B (1) E A C D B (1) D E C B A (1) D E A B C (1) D B E A C (1) D A E B C (1) C B A E D (1) C A E B D (1) B C D A E (1) B C A E D (1) B C A D E (1) A E C D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -2 -2 -4 B 14 0 -6 4 8 C 2 6 0 6 0 D 2 -4 -6 0 -2 E 4 -8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.693187 D: 0.000000 E: 0.306813 Sum of squares = 0.574642459679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.693187 D: 0.693187 E: 1.000000 A B C D E A 0 -14 -2 -2 -4 B 14 0 -6 4 8 C 2 6 0 6 0 D 2 -4 -6 0 -2 E 4 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204131457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=21 E=20 B=20 A=20 D=19 so D is eliminated. Round 2 votes counts: B=28 E=27 A=24 C=21 so C is eliminated. Round 3 votes counts: B=39 E=36 A=25 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:207 E:199 D:195 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -2 -2 -4 B 14 0 -6 4 8 C 2 6 0 6 0 D 2 -4 -6 0 -2 E 4 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204131457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -2 -4 B 14 0 -6 4 8 C 2 6 0 6 0 D 2 -4 -6 0 -2 E 4 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204131457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -2 -4 B 14 0 -6 4 8 C 2 6 0 6 0 D 2 -4 -6 0 -2 E 4 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204131457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 400: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C E B D A (5) C D A E B (5) B A D E C (5) E C B A D (4) E C A D B (4) D A B E C (4) C B E D A (4) A D E B C (4) E B A C D (3) D C A B E (3) C E D A B (3) C D B A E (3) B E A C D (3) B D C A E (3) B D A E C (3) E B C A D (2) E B A D C (2) E A C D B (2) D C A E B (2) D B A C E (2) D A C E B (2) D A C B E (2) D A B C E (2) B A E D C (2) E C A B D (1) E A D B C (1) E A B D C (1) E A B C D (1) D B A E C (1) C E D B A (1) C E B A D (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A B E (1) C B E A D (1) C B D E A (1) B C E D A (1) B C E A D (1) Total count = 100 A B C D E A 0 -18 2 2 -12 B 18 0 4 12 6 C -2 -4 0 0 -8 D -2 -12 0 0 -12 E 12 -6 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 2 2 -12 B 18 0 4 12 6 C -2 -4 0 0 -8 D -2 -12 0 0 -12 E 12 -6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=28 E=21 D=18 A=4 so A is eliminated. Round 2 votes counts: B=29 C=28 D=22 E=21 so E is eliminated. Round 3 votes counts: C=39 B=38 D=23 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:213 C:193 A:187 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 2 2 -12 B 18 0 4 12 6 C -2 -4 0 0 -8 D -2 -12 0 0 -12 E 12 -6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 2 2 -12 B 18 0 4 12 6 C -2 -4 0 0 -8 D -2 -12 0 0 -12 E 12 -6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 2 2 -12 B 18 0 4 12 6 C -2 -4 0 0 -8 D -2 -12 0 0 -12 E 12 -6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 401: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) E C B D A (7) D A B E C (7) C E D B A (6) C E B D A (6) A D B C E (5) E B C D A (4) C E B A D (4) B A D E C (4) A D B E C (4) D E C B A (3) B E C D A (3) B A E D C (3) D E B A C (2) D C E A B (2) D A C E B (2) C E D A B (2) B E C A D (2) B E A C D (2) B C E A D (2) A D C E B (2) A D C B E (2) A C D E B (2) A B D C E (2) E C B A D (1) E B C A D (1) D E C A B (1) D C A E B (1) C D A E B (1) C B E A D (1) C A B E D (1) B E D A C (1) B E A D C (1) B A E C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 4 4 -2 B 8 0 12 16 6 C -4 -12 0 -6 -18 D -4 -16 6 0 2 E 2 -6 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 4 -2 B 8 0 12 16 6 C -4 -12 0 -6 -18 D -4 -16 6 0 2 E 2 -6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=21 B=19 D=18 E=13 so E is eliminated. Round 2 votes counts: C=29 A=29 B=24 D=18 so D is eliminated. Round 3 votes counts: A=38 C=36 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:221 E:206 A:199 D:194 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 4 -2 B 8 0 12 16 6 C -4 -12 0 -6 -18 D -4 -16 6 0 2 E 2 -6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 4 -2 B 8 0 12 16 6 C -4 -12 0 -6 -18 D -4 -16 6 0 2 E 2 -6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 4 -2 B 8 0 12 16 6 C -4 -12 0 -6 -18 D -4 -16 6 0 2 E 2 -6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 402: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) D B A C E (8) E C A B D (6) D C E B A (5) D B C E A (5) C E D B A (5) C E B A D (5) A B E C D (5) C E B D A (4) B D C E A (4) A E C B D (4) D A B E C (3) A E B C D (3) A D B E C (3) E C B A D (2) E A C B D (2) D C B A E (2) B D C A E (2) B D A C E (2) B C E A D (2) B A D E C (2) A E B D C (2) A D E B C (2) D C E A B (1) D B C A E (1) D B A E C (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A B D (1) C D E B A (1) C D E A B (1) C B E A D (1) B C E D A (1) A E C D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 -14 -14 -8 B 22 0 -2 0 6 C 14 2 0 -12 22 D 14 0 12 0 10 E 8 -6 -22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.115177 C: 0.000000 D: 0.884823 E: 0.000000 Sum of squares = 0.796177911659 Cumulative probabilities = A: 0.000000 B: 0.115177 C: 0.115177 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -14 -14 -8 B 22 0 -2 0 6 C 14 2 0 -12 22 D 14 0 12 0 10 E 8 -6 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=22 C=18 B=13 E=10 so E is eliminated. Round 2 votes counts: D=37 C=26 A=24 B=13 so B is eliminated. Round 3 votes counts: D=45 C=29 A=26 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:213 C:213 E:185 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -14 -14 -8 B 22 0 -2 0 6 C 14 2 0 -12 22 D 14 0 12 0 10 E 8 -6 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 -14 -8 B 22 0 -2 0 6 C 14 2 0 -12 22 D 14 0 12 0 10 E 8 -6 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 -14 -8 B 22 0 -2 0 6 C 14 2 0 -12 22 D 14 0 12 0 10 E 8 -6 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 403: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (14) E A C B D (7) B D E A C (7) D B C E A (6) B E D A C (6) E A B C D (5) C E A D B (5) B D C A E (4) A E C B D (4) D C B A E (3) C E D A B (3) C D E A B (3) C D A E B (3) C A E D B (3) E B A D C (2) D C B E A (2) C D A B E (2) B D A E C (2) A E B C D (2) E D B C A (1) E C D A B (1) E B D A C (1) E A C D B (1) D C E B A (1) C E D B A (1) C D E B A (1) B E A D C (1) B D C E A (1) B D A C E (1) B A E D C (1) B A D E C (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -8 -26 -10 B 12 0 12 -2 8 C 8 -12 0 -10 14 D 26 2 10 0 6 E 10 -8 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -26 -10 B 12 0 12 -2 8 C 8 -12 0 -10 14 D 26 2 10 0 6 E 10 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=24 C=21 E=18 A=11 so A is eliminated. Round 2 votes counts: D=26 C=25 B=25 E=24 so E is eliminated. Round 3 votes counts: C=38 B=35 D=27 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:215 C:200 E:191 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -8 -26 -10 B 12 0 12 -2 8 C 8 -12 0 -10 14 D 26 2 10 0 6 E 10 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -26 -10 B 12 0 12 -2 8 C 8 -12 0 -10 14 D 26 2 10 0 6 E 10 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -26 -10 B 12 0 12 -2 8 C 8 -12 0 -10 14 D 26 2 10 0 6 E 10 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 404: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) D A C E B (7) C E B A D (7) B E C A D (7) B C E D A (7) A D E C B (7) A E C D B (4) A D C E B (4) D A E B C (3) C E A D B (3) B D C E A (3) B D A E C (3) A E D C B (3) E C B A D (2) E C A B D (2) D B C A E (2) D B A E C (2) D A C B E (2) C B E D A (2) B E A C D (2) B C D E A (2) A D E B C (2) E B C A D (1) D C B E A (1) D C A E B (1) D B A C E (1) D A E C B (1) D A B C E (1) C E D B A (1) C E A B D (1) C D A E B (1) C A E D B (1) B E C D A (1) B E A D C (1) B C E A D (1) B A E D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 8 2 12 B -6 0 0 -12 -2 C -8 0 0 -10 -4 D -2 12 10 0 4 E -12 2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 2 12 B -6 0 0 -12 -2 C -8 0 0 -10 -4 D -2 12 10 0 4 E -12 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=28 A=22 C=16 E=5 so E is eliminated. Round 2 votes counts: D=29 B=29 A=22 C=20 so C is eliminated. Round 3 votes counts: B=40 D=31 A=29 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:212 E:195 B:190 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 12 B -6 0 0 -12 -2 C -8 0 0 -10 -4 D -2 12 10 0 4 E -12 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 12 B -6 0 0 -12 -2 C -8 0 0 -10 -4 D -2 12 10 0 4 E -12 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 12 B -6 0 0 -12 -2 C -8 0 0 -10 -4 D -2 12 10 0 4 E -12 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 405: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) A D C E B (10) E A D C B (7) A D E C B (7) E B D C A (6) A E D C B (6) A D C B E (6) E A B D C (5) B C E D A (5) E B A C D (3) A E B C D (3) E D C B A (2) E B C D A (2) E B A D C (2) B E C D A (2) B C D A E (2) A E D B C (2) A C B D E (2) E D A C B (1) E B C A D (1) E A D B C (1) E A B C D (1) D E B C A (1) D C A B E (1) D A C E B (1) D A C B E (1) C D A B E (1) C B D E A (1) B E C A D (1) B E A C D (1) A C D B E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 24 22 -10 B -16 0 6 4 -22 C -24 -6 0 -18 -10 D -22 -4 18 0 -4 E 10 22 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 24 22 -10 B -16 0 6 4 -22 C -24 -6 0 -18 -10 D -22 -4 18 0 -4 E 10 22 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=31 B=24 D=4 C=2 so C is eliminated. Round 2 votes counts: A=39 E=31 B=25 D=5 so D is eliminated. Round 3 votes counts: A=43 E=32 B=25 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:226 E:223 D:194 B:186 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 24 22 -10 B -16 0 6 4 -22 C -24 -6 0 -18 -10 D -22 -4 18 0 -4 E 10 22 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 24 22 -10 B -16 0 6 4 -22 C -24 -6 0 -18 -10 D -22 -4 18 0 -4 E 10 22 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 24 22 -10 B -16 0 6 4 -22 C -24 -6 0 -18 -10 D -22 -4 18 0 -4 E 10 22 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 406: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) B A C D E (11) D E B A C (10) A C B E D (10) C A B E D (8) D E B C A (7) D B A C E (4) E D C B A (3) E C A B D (3) D E A C B (3) B D A C E (3) A B C E D (3) A B C D E (3) E C D A B (2) E C A D B (2) D E C B A (2) D E A B C (2) C B A E D (2) C A E B D (2) E D A C B (1) E C B A D (1) E A C D B (1) D B E A C (1) B C A D E (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 8 -2 -2 B -8 0 -6 -2 -4 C -8 6 0 -2 -2 D 2 2 2 0 -2 E 2 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 -2 -2 B -8 0 -6 -2 -4 C -8 6 0 -2 -2 D 2 2 2 0 -2 E 2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=26 A=17 B=16 C=12 so C is eliminated. Round 2 votes counts: D=29 A=27 E=26 B=18 so B is eliminated. Round 3 votes counts: A=42 D=32 E=26 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:206 E:205 D:202 C:197 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 -2 -2 B -8 0 -6 -2 -4 C -8 6 0 -2 -2 D 2 2 2 0 -2 E 2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -2 -2 B -8 0 -6 -2 -4 C -8 6 0 -2 -2 D 2 2 2 0 -2 E 2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -2 -2 B -8 0 -6 -2 -4 C -8 6 0 -2 -2 D 2 2 2 0 -2 E 2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 407: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) E B D A C (9) D B E A C (7) C D A B E (5) C E D B A (4) C D E B A (4) A E B D C (4) E D B C A (3) D B C E A (3) C A D B E (3) B D E A C (3) A C E B D (3) A C D B E (3) A C B D E (3) A B E D C (3) E D C B A (2) E C B A D (2) E A B D C (2) D B A E C (2) C E B D A (2) C A D E B (2) C A B D E (2) E B D C A (1) E B C A D (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E C A (1) C E D A B (1) C E A B D (1) C D B E A (1) C D A E B (1) C A B E D (1) B E D A C (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B E C (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -4 -6 -2 B -2 0 -10 6 -12 C 4 10 0 6 8 D 6 -6 -6 0 -8 E 2 12 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -6 -2 B -2 0 -10 6 -12 C 4 10 0 6 8 D 6 -6 -6 0 -8 E 2 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=23 E=20 D=16 B=4 so B is eliminated. Round 2 votes counts: C=37 A=23 E=21 D=19 so D is eliminated. Round 3 votes counts: C=42 E=33 A=25 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:207 A:195 D:193 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -6 -2 B -2 0 -10 6 -12 C 4 10 0 6 8 D 6 -6 -6 0 -8 E 2 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -6 -2 B -2 0 -10 6 -12 C 4 10 0 6 8 D 6 -6 -6 0 -8 E 2 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -6 -2 B -2 0 -10 6 -12 C 4 10 0 6 8 D 6 -6 -6 0 -8 E 2 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 408: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) E C B D A (7) E B C D A (7) B C E D A (7) A D B C E (7) E A D C B (6) A D C E B (5) B C D A E (4) E B C A D (3) B C D E A (3) B A D C E (3) E C D A B (2) E A C D B (2) D A C B E (2) C B E D A (2) C B D E A (2) C B D A E (2) B E C D A (2) B D C A E (2) A E D B C (2) E D A C B (1) E C D B A (1) E C B A D (1) E B A D C (1) E A D B C (1) D C A E B (1) D C A B E (1) D A C E B (1) C E D B A (1) C E B D A (1) C D B A E (1) C D A B E (1) B E C A D (1) B A D E C (1) A E D C B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 0 -14 C 4 8 0 0 -6 D 2 0 0 0 2 E 2 14 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.193408 D: 0.806592 E: 0.000000 Sum of squares = 0.687996890988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.193408 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 0 -14 C 4 8 0 0 -6 D 2 0 0 0 2 E 2 14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000324084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=30 B=23 C=10 D=5 so D is eliminated. Round 2 votes counts: A=33 E=32 B=23 C=12 so C is eliminated. Round 3 votes counts: A=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:203 D:202 A:194 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 0 -14 C 4 8 0 0 -6 D 2 0 0 0 2 E 2 14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000324084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 0 -14 C 4 8 0 0 -6 D 2 0 0 0 2 E 2 14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000324084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 0 -14 C 4 8 0 0 -6 D 2 0 0 0 2 E 2 14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000324084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 409: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (12) E C A B D (10) E A C B D (7) E C D A B (5) E A B C D (4) D B C A E (4) B D A C E (4) B C D A E (4) C E B D A (3) C D E B A (3) E D C B A (2) E A D C B (2) D C B E A (2) D C B A E (2) D B C E A (2) D A B E C (2) C E B A D (2) C D B E A (2) C B E D A (2) C B D E A (2) C B D A E (2) A E D B C (2) A E B D C (2) A D B E C (2) A B D E C (2) A B D C E (2) E C D B A (1) D E A B C (1) D B E C A (1) D B A E C (1) D A B C E (1) C E D B A (1) C E A B D (1) C B E A D (1) B D C A E (1) B A C D E (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -6 -20 -8 B 10 0 0 4 6 C 6 0 0 4 8 D 20 -4 -4 0 6 E 8 -6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.604316 C: 0.395684 D: 0.000000 E: 0.000000 Sum of squares = 0.52176356492 Cumulative probabilities = A: 0.000000 B: 0.604316 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -20 -8 B 10 0 0 4 6 C 6 0 0 4 8 D 20 -4 -4 0 6 E 8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999462 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=28 C=19 A=12 B=10 so B is eliminated. Round 2 votes counts: D=33 E=31 C=23 A=13 so A is eliminated. Round 3 votes counts: D=39 E=37 C=24 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 C:209 D:209 E:194 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -20 -8 B 10 0 0 4 6 C 6 0 0 4 8 D 20 -4 -4 0 6 E 8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999462 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -20 -8 B 10 0 0 4 6 C 6 0 0 4 8 D 20 -4 -4 0 6 E 8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999462 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -20 -8 B 10 0 0 4 6 C 6 0 0 4 8 D 20 -4 -4 0 6 E 8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999462 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 410: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (13) A B D C E (8) E C B D A (6) E C A D B (6) D A B E C (6) C E A B D (6) E D C B A (5) E C D B A (5) B D A C E (4) E C D A B (3) D E A B C (3) C E B A D (3) E C B A D (2) D E B C A (2) D B E A C (2) D A B C E (2) C E B D A (2) C A E B D (2) B A D C E (2) A D B C E (2) A C B E D (2) E D C A B (1) E D A C B (1) E C A B D (1) E A C D B (1) D E B A C (1) D B E C A (1) D A E B C (1) C B A E D (1) C A B E D (1) B C E D A (1) B A C D E (1) A E C D B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 -18 -4 B 2 0 0 -14 -4 C -4 0 0 -8 -22 D 18 14 8 0 0 E 4 4 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.503003 E: 0.496997 Sum of squares = 0.500018037514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.503003 E: 1.000000 A B C D E A 0 -2 4 -18 -4 B 2 0 0 -14 -4 C -4 0 0 -8 -22 D 18 14 8 0 0 E 4 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=31 D=31 C=15 A=15 B=8 so B is eliminated. Round 2 votes counts: D=35 E=31 A=18 C=16 so C is eliminated. Round 3 votes counts: E=43 D=35 A=22 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:215 B:192 A:190 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -18 -4 B 2 0 0 -14 -4 C -4 0 0 -8 -22 D 18 14 8 0 0 E 4 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -18 -4 B 2 0 0 -14 -4 C -4 0 0 -8 -22 D 18 14 8 0 0 E 4 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -18 -4 B 2 0 0 -14 -4 C -4 0 0 -8 -22 D 18 14 8 0 0 E 4 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 411: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) B C A D E (7) E A C B D (6) E D A C B (5) E B C A D (5) C A B D E (5) B C D A E (5) E D B A C (4) E B D C A (4) D B C A E (4) D A C B E (4) C B A D E (4) E A C D B (3) E C A B D (2) E A D C B (2) D B E C A (2) C E A B D (2) B D C E A (2) B C A E D (2) A D C E B (2) A C E B D (2) A C D B E (2) E D A B C (1) D E B A C (1) D E A C B (1) D E A B C (1) D C A B E (1) D B A C E (1) D A C E B (1) C A D B E (1) C A B E D (1) B E D C A (1) B E C A D (1) B D E C A (1) B C E A D (1) A D E C B (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -22 2 10 B 10 0 6 24 14 C 22 -6 0 0 18 D -2 -24 0 0 16 E -10 -14 -18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 2 10 B 10 0 6 24 14 C 22 -6 0 0 18 D -2 -24 0 0 16 E -10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=30 D=16 C=13 A=9 so A is eliminated. Round 2 votes counts: E=32 B=30 D=20 C=18 so C is eliminated. Round 3 votes counts: B=41 E=36 D=23 so D is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:217 D:195 A:190 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -22 2 10 B 10 0 6 24 14 C 22 -6 0 0 18 D -2 -24 0 0 16 E -10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 2 10 B 10 0 6 24 14 C 22 -6 0 0 18 D -2 -24 0 0 16 E -10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 2 10 B 10 0 6 24 14 C 22 -6 0 0 18 D -2 -24 0 0 16 E -10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 412: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (17) C D A E B (9) B E D A C (8) D C B E A (6) C D A B E (6) C A D E B (5) A E B C D (5) E B A D C (4) A C E B D (4) D C A E B (3) D B E C A (3) D B C E A (3) B D E C A (3) A B E C D (3) E B D A C (2) D E B C A (2) C D B E A (2) B E A C D (2) A C D E B (2) E A B D C (1) D C E B A (1) D C E A B (1) C D B A E (1) C B D E A (1) B E D C A (1) B A E C D (1) A E C B D (1) A E B D C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 6 -4 -14 B 14 0 12 8 14 C -6 -12 0 -14 -8 D 4 -8 14 0 -2 E 14 -14 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 -4 -14 B 14 0 12 8 14 C -6 -12 0 -14 -8 D 4 -8 14 0 -2 E 14 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=24 D=19 A=18 E=7 so E is eliminated. Round 2 votes counts: B=38 C=24 D=19 A=19 so D is eliminated. Round 3 votes counts: B=46 C=35 A=19 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:205 D:204 A:187 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 6 -4 -14 B 14 0 12 8 14 C -6 -12 0 -14 -8 D 4 -8 14 0 -2 E 14 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 -4 -14 B 14 0 12 8 14 C -6 -12 0 -14 -8 D 4 -8 14 0 -2 E 14 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 -4 -14 B 14 0 12 8 14 C -6 -12 0 -14 -8 D 4 -8 14 0 -2 E 14 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 413: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (12) C D B A E (11) C D E A B (10) D C B A E (6) B A D E C (6) C E B A D (5) E B A C D (4) E A B D C (4) E A B C D (4) C D E B A (4) D E A B C (3) C E D A B (3) E D A B C (2) D C A B E (2) D A B E C (2) C D A B E (2) C B D A E (2) B A E C D (2) E C D A B (1) E C A B D (1) E B A D C (1) D E A C B (1) D B C A E (1) D B A E C (1) D B A C E (1) C E A B D (1) C D B E A (1) C D A E B (1) C B A D E (1) B E A C D (1) B D A C E (1) B C A E D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -22 -6 -10 8 B 22 0 -4 -6 10 C 6 4 0 10 6 D 10 6 -10 0 14 E -8 -10 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 -10 8 B 22 0 -4 -6 10 C 6 4 0 10 6 D 10 6 -10 0 14 E -8 -10 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 B=23 E=17 D=17 A=2 so A is eliminated. Round 2 votes counts: C=41 B=24 D=18 E=17 so E is eliminated. Round 3 votes counts: C=43 B=37 D=20 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:211 D:210 A:185 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -6 -10 8 B 22 0 -4 -6 10 C 6 4 0 10 6 D 10 6 -10 0 14 E -8 -10 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -10 8 B 22 0 -4 -6 10 C 6 4 0 10 6 D 10 6 -10 0 14 E -8 -10 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -10 8 B 22 0 -4 -6 10 C 6 4 0 10 6 D 10 6 -10 0 14 E -8 -10 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 414: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (15) D C E A B (9) B A E C D (9) C E D B A (8) A B E C D (8) A B D E C (7) D A B C E (6) B A D E C (6) E C B A D (4) C E A B D (3) E C A B D (2) D E C B A (2) D B A C E (2) C E D A B (2) A C E B D (2) E C D B A (1) E B A C D (1) C E B D A (1) C E B A D (1) C D E A B (1) B E A C D (1) B D A E C (1) B A E D C (1) A E C B D (1) A D B C E (1) A C D E B (1) A B E D C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 2 4 -2 B 6 0 -6 4 -8 C -2 6 0 -4 10 D -4 -4 4 0 6 E 2 8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 -2 B 6 0 -6 4 -8 C -2 6 0 -4 10 D -4 -4 4 0 6 E 2 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102016 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=24 B=18 C=16 E=8 so E is eliminated. Round 2 votes counts: D=34 A=24 C=23 B=19 so B is eliminated. Round 3 votes counts: A=42 D=35 C=23 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:205 D:201 A:199 B:198 E:197 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 4 -2 B 6 0 -6 4 -8 C -2 6 0 -4 10 D -4 -4 4 0 6 E 2 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102016 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 -2 B 6 0 -6 4 -8 C -2 6 0 -4 10 D -4 -4 4 0 6 E 2 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102016 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 -2 B 6 0 -6 4 -8 C -2 6 0 -4 10 D -4 -4 4 0 6 E 2 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102016 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 415: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) E D A B C (9) E D A C B (8) C B A D E (6) D E A C B (5) B C A D E (5) B A C D E (5) C B A E D (4) E C D B A (3) C A B D E (3) B C A E D (3) A C B D E (3) A B C D E (3) E D C B A (2) E D B A C (2) E C D A B (2) E B C D A (2) D A E C B (2) C D E A B (2) C A D B E (2) B A D E C (2) A D B E C (2) A C D B E (2) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) D E A B C (1) D A E B C (1) C D A E B (1) C B E D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B A D C E (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 14 0 -8 -6 B -14 0 -12 -12 -8 C 0 12 0 0 -14 D 8 12 0 0 -4 E 6 8 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 0 -8 -6 B -14 0 -12 -12 -8 C 0 12 0 0 -14 D 8 12 0 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=19 B=19 A=12 D=9 so D is eliminated. Round 2 votes counts: E=47 C=19 B=19 A=15 so A is eliminated. Round 3 votes counts: E=51 B=25 C=24 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:208 A:200 C:199 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 -8 -6 B -14 0 -12 -12 -8 C 0 12 0 0 -14 D 8 12 0 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 -8 -6 B -14 0 -12 -12 -8 C 0 12 0 0 -14 D 8 12 0 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 -8 -6 B -14 0 -12 -12 -8 C 0 12 0 0 -14 D 8 12 0 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 416: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) C D E A B (6) D C E A B (5) A E B D C (5) E A D C B (4) C B A E D (4) B A E D C (4) B A C E D (4) A E C B D (4) D E C A B (3) D E A C B (3) D C B E A (3) D B E A C (3) C A E B D (3) B A E C D (3) A E C D B (3) E D A C B (2) E D A B C (2) D E B A C (2) D B E C A (2) C B D A E (2) C A E D B (2) B D C A E (2) B D A E C (2) B C A E D (2) A B E C D (2) E D C A B (1) E A C D B (1) C E A D B (1) C D E B A (1) C D B E A (1) C B D E A (1) B D E A C (1) B D C E A (1) B C D A E (1) B A D E C (1) B A C D E (1) A E B C D (1) A C E B D (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 18 18 -2 0 B -18 0 -4 -4 -14 C -18 4 0 -8 -14 D 2 4 8 0 -4 E 0 14 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.217159 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.782841 Sum of squares = 0.659998429205 Cumulative probabilities = A: 0.217159 B: 0.217159 C: 0.217159 D: 0.217159 E: 1.000000 A B C D E A 0 18 18 -2 0 B -18 0 -4 -4 -14 C -18 4 0 -8 -14 D 2 4 8 0 -4 E 0 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=22 C=21 A=19 E=10 so E is eliminated. Round 2 votes counts: D=33 A=24 B=22 C=21 so C is eliminated. Round 3 votes counts: D=41 A=30 B=29 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:217 E:216 D:205 C:182 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 18 -2 0 B -18 0 -4 -4 -14 C -18 4 0 -8 -14 D 2 4 8 0 -4 E 0 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 -2 0 B -18 0 -4 -4 -14 C -18 4 0 -8 -14 D 2 4 8 0 -4 E 0 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 -2 0 B -18 0 -4 -4 -14 C -18 4 0 -8 -14 D 2 4 8 0 -4 E 0 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 417: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) E D A B C (7) A B C D E (7) C A B D E (6) E D B C A (5) D E A B C (5) A D E B C (5) C B E D A (4) C B A E D (4) A C B D E (4) A B D C E (4) E D C B A (3) E D C A B (3) E D B A C (2) E C B D A (2) D E B A C (2) D B A E C (2) D A E B C (2) C E B D A (2) C E B A D (2) B C A D E (2) A D B E C (2) A D B C E (2) E D A C B (1) E A D C B (1) D A B E C (1) C B E A D (1) C A B E D (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A C D B E (1) Total count = 100 A B C D E A 0 8 2 8 16 B -8 0 8 8 12 C -2 -8 0 -4 10 D -8 -8 4 0 20 E -16 -12 -10 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 8 16 B -8 0 8 8 12 C -2 -8 0 -4 10 D -8 -8 4 0 20 E -16 -12 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=27 E=24 D=12 B=7 so B is eliminated. Round 2 votes counts: C=34 A=27 E=24 D=15 so D is eliminated. Round 3 votes counts: C=36 A=33 E=31 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:210 D:204 C:198 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 8 16 B -8 0 8 8 12 C -2 -8 0 -4 10 D -8 -8 4 0 20 E -16 -12 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 8 16 B -8 0 8 8 12 C -2 -8 0 -4 10 D -8 -8 4 0 20 E -16 -12 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 8 16 B -8 0 8 8 12 C -2 -8 0 -4 10 D -8 -8 4 0 20 E -16 -12 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 418: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) B E C A D (6) E C B D A (5) E B C D A (5) D A B E C (5) A D C E B (5) A B C E D (5) B E C D A (4) A D B C E (4) D E C B A (3) C E D A B (3) C E B A D (3) A D B E C (3) A C B E D (3) A B E C D (3) A B D E C (3) D C E A B (2) D B E A C (2) D A B C E (2) C E D B A (2) C D E A B (2) B A E D C (2) E C B A D (1) D E B C A (1) D C E B A (1) D C A E B (1) D B A E C (1) D A E B C (1) C E B D A (1) C E A D B (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D C A (1) B E A C D (1) B D E A C (1) B C A E D (1) B A D E C (1) A D C B E (1) A C D E B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 16 6 -2 10 B -16 0 6 -2 2 C -6 -6 0 4 0 D 2 2 -4 0 0 E -10 -2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888853 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 -2 10 B -16 0 6 -2 2 C -6 -6 0 4 0 D 2 2 -4 0 0 E -10 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888399 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=26 B=17 C=16 E=11 so E is eliminated. Round 2 votes counts: A=30 D=26 C=22 B=22 so C is eliminated. Round 3 votes counts: D=34 A=34 B=32 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:215 D:200 C:196 B:195 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 -2 10 B -16 0 6 -2 2 C -6 -6 0 4 0 D 2 2 -4 0 0 E -10 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888399 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -2 10 B -16 0 6 -2 2 C -6 -6 0 4 0 D 2 2 -4 0 0 E -10 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888399 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -2 10 B -16 0 6 -2 2 C -6 -6 0 4 0 D 2 2 -4 0 0 E -10 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888399 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 419: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (14) D B A C E (9) C A E B D (7) B A C D E (7) E D C A B (6) E C A D B (6) E C A B D (6) D B E A C (6) A C B D E (6) C A E D B (5) C A B E D (3) B D E A C (3) E D C B A (2) E D B C A (2) D E B A C (2) E C D B A (1) E C D A B (1) E C B A D (1) E B C A D (1) D E A C B (1) D E A B C (1) D B A E C (1) D A C E B (1) D A C B E (1) C E A D B (1) B E D C A (1) B D C A E (1) B D A E C (1) A D C B E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 12 -8 18 B 4 0 2 4 12 C -12 -2 0 -8 16 D 8 -4 8 0 14 E -18 -12 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 -8 18 B 4 0 2 4 12 C -12 -2 0 -8 16 D 8 -4 8 0 14 E -18 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=26 D=22 C=16 A=9 so A is eliminated. Round 2 votes counts: B=28 E=26 D=24 C=22 so C is eliminated. Round 3 votes counts: E=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:211 A:209 C:197 E:170 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 -8 18 B 4 0 2 4 12 C -12 -2 0 -8 16 D 8 -4 8 0 14 E -18 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -8 18 B 4 0 2 4 12 C -12 -2 0 -8 16 D 8 -4 8 0 14 E -18 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -8 18 B 4 0 2 4 12 C -12 -2 0 -8 16 D 8 -4 8 0 14 E -18 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 420: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) D A B E C (8) B A D C E (7) A B D C E (7) D E C B A (6) E D C B A (5) E D C A B (5) E C D B A (5) B A C D E (5) D A E B C (4) B C A E D (4) B A C E D (4) A B C D E (4) E C D A B (3) E C B D A (3) A D B C E (3) D E B C A (2) C B E A D (2) C B A E D (2) E C B A D (1) D E C A B (1) D E A C B (1) D E A B C (1) D B E A C (1) D B A E C (1) D A B C E (1) C E A B D (1) C A E B D (1) B C A D E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -2 6 8 B 16 0 8 6 4 C 2 -8 0 -6 6 D -6 -6 6 0 6 E -8 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 6 8 B 16 0 8 6 4 C 2 -8 0 -6 6 D -6 -6 6 0 6 E -8 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=22 B=21 A=16 C=15 so C is eliminated. Round 2 votes counts: E=32 D=26 B=25 A=17 so A is eliminated. Round 3 votes counts: B=38 E=33 D=29 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:200 A:198 C:197 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 6 8 B 16 0 8 6 4 C 2 -8 0 -6 6 D -6 -6 6 0 6 E -8 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 6 8 B 16 0 8 6 4 C 2 -8 0 -6 6 D -6 -6 6 0 6 E -8 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 6 8 B 16 0 8 6 4 C 2 -8 0 -6 6 D -6 -6 6 0 6 E -8 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 421: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) B D E C A (7) A C D E B (7) D E B A C (6) C A B D E (5) D E A B C (4) B E D C A (4) A E D C B (4) A E D B C (4) A C E D B (4) E B D A C (3) C A D B E (3) C A B E D (3) B C D E A (3) E A D B C (2) D B E C A (2) D A E B C (2) C B D E A (2) C B A E D (2) C B A D E (2) C A D E B (2) B E D A C (2) A E C D B (2) A C E B D (2) E D A B C (1) E B A D C (1) E B A C D (1) D A C E B (1) C D A B E (1) C B E A D (1) C B D A E (1) B E A C D (1) B D E A C (1) B C E D A (1) A E C B D (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 2 22 0 -2 B -2 0 10 -14 -18 C -22 -10 0 -12 -18 D 0 14 12 0 4 E 2 18 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.317169 B: 0.000000 C: 0.000000 D: 0.682831 E: 0.000000 Sum of squares = 0.566854381988 Cumulative probabilities = A: 0.317169 B: 0.317169 C: 0.317169 D: 1.000000 E: 1.000000 A B C D E A 0 2 22 0 -2 B -2 0 10 -14 -18 C -22 -10 0 -12 -18 D 0 14 12 0 4 E 2 18 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 B=19 E=17 D=15 so D is eliminated. Round 2 votes counts: A=30 E=27 C=22 B=21 so B is eliminated. Round 3 votes counts: E=44 A=30 C=26 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:215 A:211 B:188 C:169 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 22 0 -2 B -2 0 10 -14 -18 C -22 -10 0 -12 -18 D 0 14 12 0 4 E 2 18 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 22 0 -2 B -2 0 10 -14 -18 C -22 -10 0 -12 -18 D 0 14 12 0 4 E 2 18 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 22 0 -2 B -2 0 10 -14 -18 C -22 -10 0 -12 -18 D 0 14 12 0 4 E 2 18 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 422: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) E D A C B (6) B D E C A (6) E A D C B (5) C A B D E (5) A C E D B (5) E D B C A (4) E D A B C (4) D E A B C (4) D B E A C (4) D E B C A (3) C A B E D (3) B C D E A (3) A C D B E (3) E D C A B (2) E A C D B (2) D B E C A (2) B D C A E (2) B D A C E (2) B A C D E (2) A D E C B (2) A D B C E (2) A C E B D (2) A C B D E (2) E C B A D (1) E B D C A (1) D A E B C (1) D A B E C (1) D A B C E (1) C E A B D (1) C B A E D (1) C B A D E (1) C A E B D (1) B E D C A (1) B C E D A (1) B C A D E (1) A E D C B (1) A E C D B (1) A D C B E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 14 22 -12 -16 B -14 0 8 -26 -10 C -22 -8 0 -28 -18 D 12 26 28 0 16 E 16 10 18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 22 -12 -16 B -14 0 8 -26 -10 C -22 -8 0 -28 -18 D 12 26 28 0 16 E 16 10 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=24 A=21 B=18 C=12 so C is eliminated. Round 2 votes counts: A=30 E=26 D=24 B=20 so B is eliminated. Round 3 votes counts: D=37 A=35 E=28 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:241 E:214 A:204 B:179 C:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 22 -12 -16 B -14 0 8 -26 -10 C -22 -8 0 -28 -18 D 12 26 28 0 16 E 16 10 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 22 -12 -16 B -14 0 8 -26 -10 C -22 -8 0 -28 -18 D 12 26 28 0 16 E 16 10 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 22 -12 -16 B -14 0 8 -26 -10 C -22 -8 0 -28 -18 D 12 26 28 0 16 E 16 10 18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 423: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (14) C E A D B (12) A C D B E (7) D A B E C (6) C A D E B (6) B E D A C (5) B D A E C (5) E C B D A (4) C E B A D (4) A D C B E (4) E D B A C (3) C E A B D (3) C A E D B (3) E C D A B (2) E B D A C (2) E B C D A (2) D B A E C (2) C E D A B (2) B D E A C (2) A D C E B (2) E D C B A (1) E C D B A (1) E C B A D (1) E B D C A (1) D A C E B (1) C E B D A (1) C B A E D (1) C A D B E (1) C A B D E (1) B A D E C (1) Total count = 100 A B C D E A 0 28 8 20 8 B -28 0 -14 -34 -2 C -8 14 0 2 24 D -20 34 -2 0 4 E -8 2 -24 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 8 20 8 B -28 0 -14 -34 -2 C -8 14 0 2 24 D -20 34 -2 0 4 E -8 2 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=27 E=17 B=13 D=9 so D is eliminated. Round 2 votes counts: C=34 A=34 E=17 B=15 so B is eliminated. Round 3 votes counts: A=42 C=34 E=24 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:232 C:216 D:208 E:183 B:161 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 8 20 8 B -28 0 -14 -34 -2 C -8 14 0 2 24 D -20 34 -2 0 4 E -8 2 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 8 20 8 B -28 0 -14 -34 -2 C -8 14 0 2 24 D -20 34 -2 0 4 E -8 2 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 8 20 8 B -28 0 -14 -34 -2 C -8 14 0 2 24 D -20 34 -2 0 4 E -8 2 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 424: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) E B D A C (6) B E D C A (6) D B E C A (5) D B E A C (5) A D C B E (5) E B C A D (4) D A C B E (4) C D A B E (4) C A D B E (4) B E C D A (4) E A B C D (3) C E B A D (3) C A E B D (3) B D E C A (3) A D E B C (3) A C E B D (3) A C D E B (3) E B C D A (2) D C A B E (2) D B A E C (2) D A B E C (2) C B E A D (2) C A D E B (2) A D C E B (2) E A B D C (1) D E B A C (1) D C B E A (1) D B C E A (1) C D B A E (1) C B E D A (1) C A B E D (1) A E D B C (1) A E B D C (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -12 -14 -10 B 8 0 16 2 8 C 12 -16 0 -16 -14 D 14 -2 16 0 2 E 10 -8 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -14 -10 B 8 0 16 2 8 C 12 -16 0 -16 -14 D 14 -2 16 0 2 E 10 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 C=21 A=20 B=13 so B is eliminated. Round 2 votes counts: E=33 D=26 C=21 A=20 so A is eliminated. Round 3 votes counts: E=36 D=36 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:217 D:215 E:207 C:183 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -14 -10 B 8 0 16 2 8 C 12 -16 0 -16 -14 D 14 -2 16 0 2 E 10 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -14 -10 B 8 0 16 2 8 C 12 -16 0 -16 -14 D 14 -2 16 0 2 E 10 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -14 -10 B 8 0 16 2 8 C 12 -16 0 -16 -14 D 14 -2 16 0 2 E 10 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 425: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) B A E C D (9) B A C E D (9) B A C D E (8) D C E A B (7) E C D B A (6) A B D E C (6) A B D C E (6) D A B E C (4) C D E A B (4) E D C A B (3) D E C A B (3) D C A B E (3) B A E D C (3) A B C D E (3) E D C B A (2) E C B A D (2) C D E B A (2) C B A E D (2) C B A D E (2) E D B A C (1) E B A C D (1) D A E B C (1) D A C B E (1) C D B A E (1) C D A B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 6 4 20 B 14 0 4 4 18 C -6 -4 0 18 16 D -4 -4 -18 0 4 E -20 -18 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 4 20 B 14 0 4 4 18 C -6 -4 0 18 16 D -4 -4 -18 0 4 E -20 -18 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=21 D=19 A=16 E=15 so E is eliminated. Round 2 votes counts: B=30 C=29 D=25 A=16 so A is eliminated. Round 3 votes counts: B=46 C=29 D=25 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:212 A:208 D:189 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 6 4 20 B 14 0 4 4 18 C -6 -4 0 18 16 D -4 -4 -18 0 4 E -20 -18 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 4 20 B 14 0 4 4 18 C -6 -4 0 18 16 D -4 -4 -18 0 4 E -20 -18 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 4 20 B 14 0 4 4 18 C -6 -4 0 18 16 D -4 -4 -18 0 4 E -20 -18 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 426: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (11) C A E D B (7) D E B C A (6) B D E A C (5) C A B D E (4) B A D E C (4) E D C B A (3) E D B A C (3) C D E B A (3) C D B E A (3) C A B E D (3) B A E D C (3) A E B D C (3) A B E D C (3) A B C E D (3) E D C A B (2) D E C B A (2) C E D A B (2) C E A D B (2) B D E C A (2) B C D A E (2) B C A D E (2) A E D C B (2) A E C D B (2) A B E C D (2) E D A C B (1) E A D B C (1) D C E B A (1) D B C E A (1) C D B A E (1) C B D E A (1) C B A D E (1) C A D E B (1) B E A D C (1) B D C E A (1) B C D E A (1) A E D B C (1) A E B C D (1) A C E D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -2 20 18 B -8 0 -8 10 10 C 2 8 0 10 6 D -20 -10 -10 0 -16 E -18 -10 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 20 18 B -8 0 -8 10 10 C 2 8 0 10 6 D -20 -10 -10 0 -16 E -18 -10 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=28 B=21 E=10 D=10 so E is eliminated. Round 2 votes counts: A=32 C=28 B=21 D=19 so D is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:222 C:213 B:202 E:191 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 20 18 B -8 0 -8 10 10 C 2 8 0 10 6 D -20 -10 -10 0 -16 E -18 -10 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 20 18 B -8 0 -8 10 10 C 2 8 0 10 6 D -20 -10 -10 0 -16 E -18 -10 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 20 18 B -8 0 -8 10 10 C 2 8 0 10 6 D -20 -10 -10 0 -16 E -18 -10 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 427: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (13) C B E D A (7) D B C A E (6) C B E A D (6) A E D C B (6) E C B A D (5) A D E B C (5) E A D B C (4) D A B E C (4) D A B C E (4) B C D E A (4) D B A C E (3) C E B A D (3) B C E D A (3) A E C D B (3) E B C A D (2) E A C D B (2) E A C B D (2) C B A E D (2) A E D B C (2) A D E C B (2) E C A B D (1) E B D A C (1) E B C D A (1) E A B D C (1) D A C B E (1) C D B A E (1) C B D E A (1) C B D A E (1) B D C E A (1) B C D A E (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 4 10 -4 12 B -4 0 10 -14 -10 C -10 -10 0 -8 -10 D 4 14 8 0 -4 E -12 10 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.44000000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 A B C D E A 0 4 10 -4 12 B -4 0 10 -14 -10 C -10 -10 0 -8 -10 D 4 14 8 0 -4 E -12 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.4400000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=21 A=20 E=19 B=9 so B is eliminated. Round 2 votes counts: D=32 C=29 A=20 E=19 so E is eliminated. Round 3 votes counts: C=38 D=33 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:211 D:211 E:206 B:191 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 -4 12 B -4 0 10 -14 -10 C -10 -10 0 -8 -10 D 4 14 8 0 -4 E -12 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.4400000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -4 12 B -4 0 10 -14 -10 C -10 -10 0 -8 -10 D 4 14 8 0 -4 E -12 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.4400000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -4 12 B -4 0 10 -14 -10 C -10 -10 0 -8 -10 D 4 14 8 0 -4 E -12 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.4400000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 428: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (11) B C A D E (8) A B D C E (6) E D C A B (5) C B D A E (5) E A D B C (4) C D B A E (4) C B E D A (4) E C D B A (3) E A B D C (3) D C A B E (3) B C E A D (3) A D B C E (3) A B E D C (3) E C B D A (2) E B A C D (2) D A C E B (2) C D E B A (2) C B D E A (2) B A D C E (2) B A C E D (2) B A C D E (2) A E B D C (2) A D E C B (2) A D B E C (2) E C D A B (1) E B C A D (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C B E (1) C E D B A (1) C D B E A (1) B C A E D (1) B A E D C (1) A E D B C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 4 0 8 B -8 0 -4 0 8 C -4 4 0 -12 8 D 0 0 12 0 0 E -8 -8 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.611374 B: 0.000000 C: 0.000000 D: 0.388626 E: 0.000000 Sum of squares = 0.524808301546 Cumulative probabilities = A: 0.611374 B: 0.611374 C: 0.611374 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 0 8 B -8 0 -4 0 8 C -4 4 0 -12 8 D 0 0 12 0 0 E -8 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=21 C=19 B=19 D=9 so D is eliminated. Round 2 votes counts: E=32 A=25 C=24 B=19 so B is eliminated. Round 3 votes counts: C=36 E=32 A=32 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:206 B:198 C:198 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 0 8 B -8 0 -4 0 8 C -4 4 0 -12 8 D 0 0 12 0 0 E -8 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 0 8 B -8 0 -4 0 8 C -4 4 0 -12 8 D 0 0 12 0 0 E -8 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 0 8 B -8 0 -4 0 8 C -4 4 0 -12 8 D 0 0 12 0 0 E -8 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 429: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (14) A D B C E (8) E C B D A (7) D E A C B (6) B C E A D (6) B C A E D (6) A B C E D (6) A B C D E (6) D A E C B (5) A D C B E (5) D E C A B (3) E D A C B (2) E C D B A (2) E B C A D (2) D A C E B (2) B E C A D (2) A D E C B (2) A D E B C (2) E D C A B (1) E B C D A (1) D E C B A (1) D C A B E (1) C E D B A (1) C E B D A (1) C D B A E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B D E (1) B E C D A (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 0 -10 0 -2 B 0 0 -16 -12 -4 C 10 16 0 -4 -2 D 0 12 4 0 -6 E 2 4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -10 0 -2 B 0 0 -16 -12 -4 C 10 16 0 -4 -2 D 0 12 4 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=29 A=29 D=18 B=17 C=7 so C is eliminated. Round 2 votes counts: E=31 A=30 B=20 D=19 so D is eliminated. Round 3 votes counts: E=41 A=38 B=21 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:207 D:205 A:194 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 0 -2 B 0 0 -16 -12 -4 C 10 16 0 -4 -2 D 0 12 4 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 0 -2 B 0 0 -16 -12 -4 C 10 16 0 -4 -2 D 0 12 4 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 0 -2 B 0 0 -16 -12 -4 C 10 16 0 -4 -2 D 0 12 4 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 430: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E B C D A (8) E B C A D (7) B E C A D (7) D A E C B (6) C A D B E (5) C A B D E (5) B C E A D (5) D A E B C (4) D A C E B (4) A D C E B (4) C E B A D (3) C B E A D (3) C B A E D (3) A D C B E (3) E D A B C (2) E B D C A (2) D A B C E (2) C A D E B (2) B D A C E (2) A C D B E (2) E D B A C (1) E C B A D (1) E C A B D (1) E B D A C (1) D E A B C (1) D B A E C (1) D A B E C (1) C E A B D (1) B E D A C (1) B E C D A (1) B C E D A (1) B C A D E (1) Total count = 100 A B C D E A 0 4 -12 6 8 B -4 0 -4 6 4 C 12 4 0 12 10 D -6 -6 -12 0 4 E -8 -4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 6 8 B -4 0 -4 6 4 C 12 4 0 12 10 D -6 -6 -12 0 4 E -8 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 C=22 B=18 A=9 so A is eliminated. Round 2 votes counts: D=35 C=24 E=23 B=18 so B is eliminated. Round 3 votes counts: D=37 E=32 C=31 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:219 A:203 B:201 D:190 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 6 8 B -4 0 -4 6 4 C 12 4 0 12 10 D -6 -6 -12 0 4 E -8 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 6 8 B -4 0 -4 6 4 C 12 4 0 12 10 D -6 -6 -12 0 4 E -8 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 6 8 B -4 0 -4 6 4 C 12 4 0 12 10 D -6 -6 -12 0 4 E -8 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 431: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) E C D A B (7) A D B E C (7) E C A D B (6) C E B D A (6) E C A B D (5) B A D C E (5) D B A C E (4) D A B E C (4) B D C E A (4) B D A C E (4) C E B A D (3) C B E D A (3) B C E D A (3) A E D C B (3) A B D E C (3) D A E C B (2) D A B C E (2) B C D E A (2) B A C E D (2) E C B A D (1) E B C A D (1) D C E A B (1) D C B E A (1) D B C A E (1) D A E B C (1) C E D A B (1) B C E A D (1) B A D E C (1) B A C D E (1) A E C B D (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -10 -16 -16 -14 B 10 0 -6 -6 -4 C 16 6 0 12 12 D 16 6 -12 0 -12 E 14 4 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -16 -14 B 10 0 -6 -6 -4 C 16 6 0 12 12 D 16 6 -12 0 -12 E 14 4 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 E=20 D=16 A=16 so D is eliminated. Round 2 votes counts: B=28 C=27 A=25 E=20 so E is eliminated. Round 3 votes counts: C=46 B=29 A=25 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:223 E:209 D:199 B:197 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -16 -16 -14 B 10 0 -6 -6 -4 C 16 6 0 12 12 D 16 6 -12 0 -12 E 14 4 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -16 -14 B 10 0 -6 -6 -4 C 16 6 0 12 12 D 16 6 -12 0 -12 E 14 4 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -16 -14 B 10 0 -6 -6 -4 C 16 6 0 12 12 D 16 6 -12 0 -12 E 14 4 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 432: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) B A D E C (7) E C A D B (6) D C B E A (6) C D E B A (6) D B C E A (5) C D E A B (5) B D A E C (5) D B C A E (4) C E D A B (4) C D B E A (4) B D C A E (4) A B E D C (4) E A C D B (3) B A E D C (3) A E B C D (3) D C E B A (2) C E A D B (2) C A E B D (2) B D A C E (2) E D C A B (1) E D A C B (1) E D A B C (1) E C D A B (1) E A C B D (1) D B E C A (1) B D E A C (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B D C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -6 -6 0 B 6 0 -10 -4 0 C 6 10 0 2 0 D 6 4 -2 0 8 E 0 0 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.903936 D: 0.000000 E: 0.096064 Sum of squares = 0.826328163066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.903936 D: 0.903936 E: 1.000000 A B C D E A 0 -6 -6 -6 0 B 6 0 -10 -4 0 C 6 10 0 2 0 D 6 4 -2 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000017477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 A=20 D=18 E=14 so E is eliminated. Round 2 votes counts: C=30 B=25 A=24 D=21 so D is eliminated. Round 3 votes counts: C=39 B=35 A=26 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:208 B:196 E:196 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -6 0 B 6 0 -10 -4 0 C 6 10 0 2 0 D 6 4 -2 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000017477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -6 0 B 6 0 -10 -4 0 C 6 10 0 2 0 D 6 4 -2 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000017477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -6 0 B 6 0 -10 -4 0 C 6 10 0 2 0 D 6 4 -2 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000017477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 433: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (12) E C A B D (10) D A C E B (9) B D A C E (7) D B A C E (5) B A D C E (4) E B C A D (3) C E A D B (3) B E D C A (3) B E C A D (3) B C A E D (3) B A C E D (3) A C D B E (3) E C D A B (2) D E C A B (2) D A C B E (2) C E A B D (2) C A E D B (2) B D E A C (2) B D A E C (2) B A C D E (2) A D C E B (2) A C D E B (2) A C B E D (2) E D C A B (1) E C B A D (1) D E B C A (1) D B E C A (1) D B E A C (1) D B A E C (1) D A B C E (1) C A E B D (1) B D E C A (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -2 18 4 B -14 0 -12 0 -6 C 2 12 0 8 8 D -18 0 -8 0 -2 E -4 6 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 18 4 B -14 0 -12 0 -6 C 2 12 0 8 8 D -18 0 -8 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=29 D=23 A=10 C=8 so C is eliminated. Round 2 votes counts: E=34 B=30 D=23 A=13 so A is eliminated. Round 3 votes counts: E=37 B=33 D=30 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:217 C:215 E:198 D:186 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 18 4 B -14 0 -12 0 -6 C 2 12 0 8 8 D -18 0 -8 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 18 4 B -14 0 -12 0 -6 C 2 12 0 8 8 D -18 0 -8 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 18 4 B -14 0 -12 0 -6 C 2 12 0 8 8 D -18 0 -8 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 434: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (13) D B E A C (12) C A D B E (11) C A E D B (10) C A E B D (10) E B D A C (8) D B E C A (7) E A C B D (6) B D E A C (4) A E C B D (4) D B C A E (2) C D B A E (2) E A B D C (1) E A B C D (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A C E (1) C D A B E (1) C A D E B (1) C A B D E (1) A E B C D (1) A C B D E (1) Total count = 100 A B C D E A 0 22 4 20 18 B -22 0 -24 0 -10 C -4 24 0 24 12 D -20 0 -24 0 -8 E -18 10 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 4 20 18 B -22 0 -24 0 -10 C -4 24 0 24 12 D -20 0 -24 0 -8 E -18 10 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=25 A=19 E=16 B=4 so B is eliminated. Round 2 votes counts: C=36 D=29 A=19 E=16 so E is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=62 D=38 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:232 C:228 E:194 D:174 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 4 20 18 B -22 0 -24 0 -10 C -4 24 0 24 12 D -20 0 -24 0 -8 E -18 10 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 4 20 18 B -22 0 -24 0 -10 C -4 24 0 24 12 D -20 0 -24 0 -8 E -18 10 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 4 20 18 B -22 0 -24 0 -10 C -4 24 0 24 12 D -20 0 -24 0 -8 E -18 10 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 435: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (13) B E C A D (9) A C D E B (8) E B C A D (6) E B D A C (5) D E B A C (5) B E D C A (5) E D B A C (4) D E A C B (3) D A E C B (3) D A C B E (3) B E D A C (3) B D E A C (3) E D A C B (2) E C A D B (2) D E A B C (2) D B E A C (2) D B A C E (2) C A D E B (2) C A D B E (2) C A B E D (2) B C E A D (2) B C A E D (2) E C B A D (1) E C A B D (1) E B D C A (1) E A C D B (1) C B A E D (1) C A E D B (1) B E C D A (1) B D C A E (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 -10 18 -18 -16 B 10 0 10 -10 -20 C -18 -10 0 -14 -18 D 18 10 14 0 2 E 16 20 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 18 -18 -16 B 10 0 10 -10 -20 C -18 -10 0 -14 -18 D 18 10 14 0 2 E 16 20 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=28 E=23 C=8 A=8 so C is eliminated. Round 2 votes counts: D=33 B=29 E=23 A=15 so A is eliminated. Round 3 votes counts: D=45 B=31 E=24 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:226 D:222 B:195 A:187 C:170 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 18 -18 -16 B 10 0 10 -10 -20 C -18 -10 0 -14 -18 D 18 10 14 0 2 E 16 20 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 18 -18 -16 B 10 0 10 -10 -20 C -18 -10 0 -14 -18 D 18 10 14 0 2 E 16 20 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 18 -18 -16 B 10 0 10 -10 -20 C -18 -10 0 -14 -18 D 18 10 14 0 2 E 16 20 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 436: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) B D A E C (5) B A E C D (5) E C D B A (4) E C B D A (4) C E A D B (4) B E C A D (4) B A D E C (4) A D B C E (4) A C E D B (4) A C E B D (4) D E C B A (3) D B E C A (3) D B E A C (3) D A C E B (3) C A E D B (3) B D E C A (3) B A E D C (3) E C D A B (2) E B C A D (2) D E B C A (2) D B A E C (2) D A C B E (2) D A B C E (2) C E A B D (2) B D E A C (2) A D C E B (2) E C A B D (1) D C E B A (1) D C E A B (1) C E D A B (1) C D E A B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B A D C E (1) A D C B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -22 2 6 -6 B 22 0 0 4 0 C -2 0 0 2 -24 D -6 -4 -2 0 -10 E 6 0 24 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.514487 C: 0.000000 D: 0.000000 E: 0.485513 Sum of squares = 0.500419752863 Cumulative probabilities = A: 0.000000 B: 0.514487 C: 0.514487 D: 0.514487 E: 1.000000 A B C D E A 0 -22 2 6 -6 B 22 0 0 4 0 C -2 0 0 2 -24 D -6 -4 -2 0 -10 E 6 0 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=22 E=19 A=17 C=11 so C is eliminated. Round 2 votes counts: B=31 E=26 D=23 A=20 so A is eliminated. Round 3 votes counts: E=37 B=33 D=30 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:220 B:213 A:190 D:189 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 2 6 -6 B 22 0 0 4 0 C -2 0 0 2 -24 D -6 -4 -2 0 -10 E 6 0 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 2 6 -6 B 22 0 0 4 0 C -2 0 0 2 -24 D -6 -4 -2 0 -10 E 6 0 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 2 6 -6 B 22 0 0 4 0 C -2 0 0 2 -24 D -6 -4 -2 0 -10 E 6 0 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 437: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) B A D E C (7) E D A C B (6) E A D B C (6) C D E A B (6) C B E D A (5) B C A D E (5) C B D A E (4) B A E D C (4) D E A C B (3) D A E C B (3) C B D E A (3) B C D A E (3) B C A E D (3) E C D A B (2) E A D C B (2) E A B D C (2) D A E B C (2) C D A E B (2) C D A B E (2) B E A D C (2) B C E A D (2) E D C A B (1) E D A B C (1) E C A D B (1) E B A D C (1) D C E A B (1) C E B D A (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E A D (1) B D A C E (1) B A E C D (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 -8 -14 -16 B -4 0 -10 -2 -6 C 8 10 0 10 8 D 14 2 -10 0 -6 E 16 6 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -14 -16 B -4 0 -10 -2 -6 C 8 10 0 10 8 D 14 2 -10 0 -6 E 16 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=31 E=22 D=9 A=2 so A is eliminated. Round 2 votes counts: C=36 B=31 E=23 D=10 so D is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:210 D:200 B:189 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -14 -16 B -4 0 -10 -2 -6 C 8 10 0 10 8 D 14 2 -10 0 -6 E 16 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -14 -16 B -4 0 -10 -2 -6 C 8 10 0 10 8 D 14 2 -10 0 -6 E 16 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -14 -16 B -4 0 -10 -2 -6 C 8 10 0 10 8 D 14 2 -10 0 -6 E 16 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 438: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (7) B D A E C (7) A C E B D (7) E C A D B (5) D E B C A (5) D B E C A (5) D E C A B (4) B D A C E (4) B A C E D (4) A C B E D (4) C E D A B (3) C A E D B (3) B D C E A (3) B A D C E (3) A E D C B (3) A B D E C (3) E C D A B (2) D E C B A (2) D E B A C (2) C A E B D (2) C A B E D (2) B C E D A (2) B A D E C (2) B A C D E (2) A B C E D (2) E A C D B (1) D B E A C (1) C E A B D (1) B D E C A (1) B D E A C (1) B C E A D (1) B C D E A (1) B C A E D (1) A E C D B (1) A D E B C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 0 14 6 B -6 0 4 8 0 C 0 -4 0 6 6 D -14 -8 -6 0 -6 E -6 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.684870 B: 0.000000 C: 0.315130 D: 0.000000 E: 0.000000 Sum of squares = 0.568353525225 Cumulative probabilities = A: 0.684870 B: 0.684870 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 14 6 B -6 0 4 8 0 C 0 -4 0 6 6 D -14 -8 -6 0 -6 E -6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 D=19 C=18 E=8 so E is eliminated. Round 2 votes counts: B=32 C=25 A=24 D=19 so D is eliminated. Round 3 votes counts: B=45 C=31 A=24 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:213 C:204 B:203 E:197 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 14 6 B -6 0 4 8 0 C 0 -4 0 6 6 D -14 -8 -6 0 -6 E -6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 14 6 B -6 0 4 8 0 C 0 -4 0 6 6 D -14 -8 -6 0 -6 E -6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 14 6 B -6 0 4 8 0 C 0 -4 0 6 6 D -14 -8 -6 0 -6 E -6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 439: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) D E B C A (7) B A E C D (7) C D A B E (6) C A B E D (5) B E A D C (5) B E A C D (5) D C E B A (4) D C E A B (4) C A D B E (4) A E B D C (4) A C B E D (4) E B D A C (3) D C A E B (3) C D A E B (3) C A D E B (3) A B E C D (3) E D B A C (2) E B A D C (2) D E C B A (2) B C A E D (2) E A B D C (1) D E C A B (1) D E B A C (1) D C B E A (1) D A E B C (1) C D B E A (1) C B A D E (1) C A B D E (1) B E D A C (1) A C E D B (1) A C E B D (1) A C D E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -12 4 4 B 4 0 8 10 12 C 12 -8 0 -2 -8 D -4 -10 2 0 -12 E -4 -12 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 4 4 B 4 0 8 10 12 C 12 -8 0 -2 -8 D -4 -10 2 0 -12 E -4 -12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=24 C=24 A=16 E=8 so E is eliminated. Round 2 votes counts: B=33 D=26 C=24 A=17 so A is eliminated. Round 3 votes counts: B=43 C=31 D=26 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:202 C:197 A:196 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 4 4 B 4 0 8 10 12 C 12 -8 0 -2 -8 D -4 -10 2 0 -12 E -4 -12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 4 4 B 4 0 8 10 12 C 12 -8 0 -2 -8 D -4 -10 2 0 -12 E -4 -12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 4 4 B 4 0 8 10 12 C 12 -8 0 -2 -8 D -4 -10 2 0 -12 E -4 -12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 440: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) C E B A D (7) B C E D A (7) E A C D B (6) D B A C E (6) C E A B D (6) E C A B D (5) B D C A E (5) B C D E A (5) D A B C E (4) C B E A D (4) A D E C B (4) D B A E C (3) D A B E C (3) A C E D B (3) E A D C B (2) D A E C B (2) C B A E D (2) B E C D A (2) B D C E A (2) B C E A D (2) A E C D B (2) E C B A D (1) E B D C A (1) E B C D A (1) D B E C A (1) D B E A C (1) C B A D E (1) B D E C A (1) B C D A E (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -8 -6 -8 B 6 0 6 6 0 C 8 -6 0 12 14 D 6 -6 -12 0 -2 E 8 0 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.806229 C: 0.000000 D: 0.000000 E: 0.193771 Sum of squares = 0.68755197698 Cumulative probabilities = A: 0.000000 B: 0.806229 C: 0.806229 D: 0.806229 E: 1.000000 A B C D E A 0 -6 -8 -6 -8 B 6 0 6 6 0 C 8 -6 0 12 14 D 6 -6 -12 0 -2 E 8 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000224067 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 C=20 E=16 A=11 so A is eliminated. Round 2 votes counts: D=33 B=25 C=24 E=18 so E is eliminated. Round 3 votes counts: C=38 D=35 B=27 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:209 E:198 D:193 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -6 -8 B 6 0 6 6 0 C 8 -6 0 12 14 D 6 -6 -12 0 -2 E 8 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000224067 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -6 -8 B 6 0 6 6 0 C 8 -6 0 12 14 D 6 -6 -12 0 -2 E 8 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000224067 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -6 -8 B 6 0 6 6 0 C 8 -6 0 12 14 D 6 -6 -12 0 -2 E 8 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000224067 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 441: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (12) E D A B C (8) C B A D E (7) A D E C B (6) D E A C B (5) D E A B C (5) B C E A D (5) B C A E D (4) A C D B E (4) E D B C A (3) E D B A C (3) D A E C B (3) C B A E D (3) A D C E B (3) A C B E D (3) E A D B C (2) D E B A C (2) D B E C A (2) C A B D E (2) B E D C A (2) B E C D A (2) A E B C D (2) E B D C A (1) D E B C A (1) D B C E A (1) C D B E A (1) C D A B E (1) C B D E A (1) B D C E A (1) B C D E A (1) A E D C B (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 0 -10 -16 B 4 0 14 -4 6 C 0 -14 0 2 4 D 10 4 -2 0 -4 E 16 -6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 A B C D E A 0 -4 0 -10 -16 B 4 0 14 -4 6 C 0 -14 0 2 4 D 10 4 -2 0 -4 E 16 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775356 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=22 D=19 E=17 C=15 so C is eliminated. Round 2 votes counts: B=38 A=24 D=21 E=17 so E is eliminated. Round 3 votes counts: B=39 D=35 A=26 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:210 E:205 D:204 C:196 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 -10 -16 B 4 0 14 -4 6 C 0 -14 0 2 4 D 10 4 -2 0 -4 E 16 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775356 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -10 -16 B 4 0 14 -4 6 C 0 -14 0 2 4 D 10 4 -2 0 -4 E 16 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775356 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -10 -16 B 4 0 14 -4 6 C 0 -14 0 2 4 D 10 4 -2 0 -4 E 16 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775356 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 442: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (11) E B D C A (7) C E A B D (6) C A D E B (6) A D C B E (6) E C B A D (5) D B A C E (4) D A C B E (4) C E A D B (4) A C D E B (4) E B C D A (3) D B A E C (3) B E D A C (3) B D A E C (3) E C B D A (2) E C A D B (2) D E B C A (2) D A B C E (2) A C D B E (2) A B D C E (2) E D C B A (1) E C D B A (1) E C A B D (1) E B C A D (1) C A E B D (1) B E D C A (1) B E A C D (1) Total count = 100 A B C D E A 0 4 -10 6 -2 B -4 0 -14 -6 -16 C 10 14 0 2 6 D -6 6 -2 0 -2 E 2 16 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 6 -2 B -4 0 -14 -6 -16 C 10 14 0 2 6 D -6 6 -2 0 -2 E 2 16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=23 B=19 D=15 A=14 so A is eliminated. Round 2 votes counts: C=35 E=23 D=21 B=21 so D is eliminated. Round 3 votes counts: C=45 B=30 E=25 so E is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:207 A:199 D:198 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 6 -2 B -4 0 -14 -6 -16 C 10 14 0 2 6 D -6 6 -2 0 -2 E 2 16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 6 -2 B -4 0 -14 -6 -16 C 10 14 0 2 6 D -6 6 -2 0 -2 E 2 16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 6 -2 B -4 0 -14 -6 -16 C 10 14 0 2 6 D -6 6 -2 0 -2 E 2 16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 443: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) D C E B A (8) C D E A B (7) A B E C D (7) E B A C D (6) D E C B A (5) D B A C E (5) A B D C E (4) E C D B A (3) E C A B D (3) E B A D C (3) D B A E C (3) C E D A B (3) C D A E B (3) C D A B E (3) B A D E C (3) E D C B A (2) E C D A B (2) D C A B E (2) B E A D C (2) A B C E D (2) E D B C A (1) E B D A C (1) E B C A D (1) D E B C A (1) D E B A C (1) D C E A B (1) D B E C A (1) D B E A C (1) C E A B D (1) C D E B A (1) C A B D E (1) B E D A C (1) B A E C D (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 2 -10 -10 B 20 0 10 -6 -6 C -2 -10 0 -12 -16 D 10 6 12 0 4 E 10 6 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 2 -10 -10 B 20 0 10 -6 -6 C -2 -10 0 -12 -16 D 10 6 12 0 4 E 10 6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=22 C=19 B=17 A=14 so A is eliminated. Round 2 votes counts: B=31 D=28 E=22 C=19 so C is eliminated. Round 3 votes counts: D=42 B=32 E=26 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:214 B:209 A:181 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 2 -10 -10 B 20 0 10 -6 -6 C -2 -10 0 -12 -16 D 10 6 12 0 4 E 10 6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 2 -10 -10 B 20 0 10 -6 -6 C -2 -10 0 -12 -16 D 10 6 12 0 4 E 10 6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 2 -10 -10 B 20 0 10 -6 -6 C -2 -10 0 -12 -16 D 10 6 12 0 4 E 10 6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 444: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) C B A E D (8) B E A D C (8) C D A E B (6) D A E C B (5) B E A C D (5) D E A C B (4) C A D E B (4) A E D B C (4) A D E C B (4) D E A B C (3) D C A E B (3) D A E B C (3) C D B A E (3) E B D A C (2) E A D B C (2) D C E B A (2) D C B E A (2) C B D A E (2) C B A D E (2) B E D A C (2) B C E A D (2) A C D E B (2) E D B A C (1) E B A D C (1) E A B D C (1) D E B A C (1) C D B E A (1) C A B E D (1) C A B D E (1) B E C D A (1) B D E C A (1) B C E D A (1) B C A E D (1) A E D C B (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 2 -2 4 B 8 0 -20 -2 0 C -2 20 0 0 -2 D 2 2 0 0 14 E -4 0 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.329044 D: 0.670956 E: 0.000000 Sum of squares = 0.558451916111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.329044 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -2 4 B 8 0 -20 -2 0 C -2 20 0 0 -2 D 2 2 0 0 14 E -4 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=23 B=21 A=13 E=7 so E is eliminated. Round 2 votes counts: C=36 D=24 B=24 A=16 so A is eliminated. Round 3 votes counts: C=39 D=35 B=26 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:209 C:208 A:198 B:193 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -2 4 B 8 0 -20 -2 0 C -2 20 0 0 -2 D 2 2 0 0 14 E -4 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 4 B 8 0 -20 -2 0 C -2 20 0 0 -2 D 2 2 0 0 14 E -4 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 4 B 8 0 -20 -2 0 C -2 20 0 0 -2 D 2 2 0 0 14 E -4 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 445: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) B C D E A (8) E C A B D (6) E A C D B (6) E A C B D (5) D B A E C (5) D B A C E (5) D A B E C (5) E C D B A (3) C E A B D (3) B D C E A (3) B C E D A (3) A E D C B (3) E C D A B (2) E C A D B (2) D A E B C (2) C E B D A (2) C B E D A (2) C A E B D (2) B C D A E (2) B C A D E (2) A E C D B (2) A D B E C (2) A B D C E (2) E C B A D (1) D E C B A (1) D E A C B (1) D C E B A (1) D B C E A (1) C E D B A (1) C B E A D (1) C B A E D (1) B D C A E (1) B D A C E (1) B A C E D (1) A E C B D (1) A D E C B (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -12 2 -20 B 6 0 -10 12 -8 C 12 10 0 30 2 D -2 -12 -30 0 -12 E 20 8 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 2 -20 B 6 0 -10 12 -8 C 12 10 0 30 2 D -2 -12 -30 0 -12 E 20 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=21 B=21 C=20 A=13 so A is eliminated. Round 2 votes counts: E=31 D=25 B=24 C=20 so C is eliminated. Round 3 votes counts: E=47 B=28 D=25 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:227 E:219 B:200 A:182 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 2 -20 B 6 0 -10 12 -8 C 12 10 0 30 2 D -2 -12 -30 0 -12 E 20 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 2 -20 B 6 0 -10 12 -8 C 12 10 0 30 2 D -2 -12 -30 0 -12 E 20 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 2 -20 B 6 0 -10 12 -8 C 12 10 0 30 2 D -2 -12 -30 0 -12 E 20 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 446: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) A B D E C (7) B E A C D (6) B A E C D (6) A B D C E (6) D C E A B (5) C E D A B (5) A D B C E (5) E C D A B (4) E C B D A (4) E B C A D (4) C E B D A (4) D A C B E (3) D A B C E (3) C D E A B (3) B A D E C (3) E A D C B (2) D A C E B (2) B A D C E (2) B A C E D (2) B A C D E (2) E C D B A (1) E A B D C (1) D E C A B (1) D C A B E (1) D A E C B (1) D A B E C (1) C D E B A (1) C D B A E (1) C D A B E (1) B C E A D (1) B C A E D (1) B C A D E (1) B A E D C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 6 10 4 2 B -6 0 8 2 8 C -10 -8 0 8 12 D -4 -2 -8 0 2 E -2 -8 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 4 2 B -6 0 8 2 8 C -10 -8 0 8 12 D -4 -2 -8 0 2 E -2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=22 A=20 D=17 E=16 so E is eliminated. Round 2 votes counts: C=31 B=29 A=23 D=17 so D is eliminated. Round 3 votes counts: C=38 A=33 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:206 C:201 D:194 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 4 2 B -6 0 8 2 8 C -10 -8 0 8 12 D -4 -2 -8 0 2 E -2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 4 2 B -6 0 8 2 8 C -10 -8 0 8 12 D -4 -2 -8 0 2 E -2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 4 2 B -6 0 8 2 8 C -10 -8 0 8 12 D -4 -2 -8 0 2 E -2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 447: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) E A D B C (5) D E A B C (5) C A B D E (5) A D E C B (5) E D B A C (4) D A E C B (4) C B E A D (4) C B A D E (4) A D C E B (4) A D C B E (4) E D A B C (3) E B D A C (3) E A B C D (3) C B D A E (3) B E C D A (3) A E D C B (3) E B C D A (2) E B C A D (2) C B A E D (2) C A D B E (2) C A B E D (2) B D E C A (2) B C E D A (2) B C E A D (2) B C D E A (2) A C D B E (2) E B D C A (1) E A B D C (1) D E B A C (1) D C B A E (1) D A C E B (1) C D B A E (1) C A E B D (1) B E D C A (1) B E C A D (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 18 14 8 6 B -18 0 -12 -6 0 C -14 12 0 -12 -2 D -8 6 12 0 8 E -6 0 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 14 8 6 B -18 0 -12 -6 0 C -14 12 0 -12 -2 D -8 6 12 0 8 E -6 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 A=20 D=19 B=13 so B is eliminated. Round 2 votes counts: C=30 E=29 D=21 A=20 so A is eliminated. Round 3 votes counts: D=35 E=33 C=32 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:223 D:209 E:194 C:192 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 14 8 6 B -18 0 -12 -6 0 C -14 12 0 -12 -2 D -8 6 12 0 8 E -6 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 14 8 6 B -18 0 -12 -6 0 C -14 12 0 -12 -2 D -8 6 12 0 8 E -6 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 14 8 6 B -18 0 -12 -6 0 C -14 12 0 -12 -2 D -8 6 12 0 8 E -6 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 448: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) E D A C B (9) B C D A E (7) B C A D E (7) A E C D B (7) E A D C B (5) A C E D B (5) D E A C B (4) B E D A C (4) B D E C A (4) A C B E D (4) D E C B A (3) D E C A B (3) D E B C A (3) B D C E A (3) B A C E D (3) D E B A C (2) D B C E A (2) C A B E D (2) E D B A C (1) E D A B C (1) E A D B C (1) D B E C A (1) C D B A E (1) C D A E B (1) C B A E D (1) C A D E B (1) B E A D C (1) B E A C D (1) A E D C B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 2 2 4 B 8 0 4 -2 2 C -2 -4 0 4 -4 D -2 2 -4 0 -16 E -4 -2 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.660000000067 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 A B C D E A 0 -8 2 2 4 B 8 0 4 -2 2 C -2 -4 0 4 -4 D -2 2 -4 0 -16 E -4 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999999995 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=19 D=18 E=17 C=6 so C is eliminated. Round 2 votes counts: B=41 A=22 D=20 E=17 so E is eliminated. Round 3 votes counts: B=41 D=31 A=28 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:207 B:206 A:200 C:197 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 2 4 B 8 0 4 -2 2 C -2 -4 0 4 -4 D -2 2 -4 0 -16 E -4 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999999995 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 2 4 B 8 0 4 -2 2 C -2 -4 0 4 -4 D -2 2 -4 0 -16 E -4 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999999995 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 2 4 B 8 0 4 -2 2 C -2 -4 0 4 -4 D -2 2 -4 0 -16 E -4 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999999995 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 449: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) D C B E A (10) E A B D C (6) C D E A B (6) C D B A E (6) A E B C D (6) B A E C D (4) A E C B D (4) E A D B C (3) D E C A B (3) D B C E A (3) B E A D C (3) B D C E A (3) B C A E D (3) E A C D B (2) D C E B A (2) D C E A B (2) C A E B D (2) A E C D B (2) A E B D C (2) A B E C D (2) E D A C B (1) E A D C B (1) D E B A C (1) D E A C B (1) D B E C A (1) C D E B A (1) C D A E B (1) C B D A E (1) C A E D B (1) C A D E B (1) B D E C A (1) B D E A C (1) B A E D C (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 -6 -16 -10 -24 B 6 0 -16 -20 2 C 16 16 0 8 8 D 10 20 -8 0 10 E 24 -2 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -10 -24 B 6 0 -16 -20 2 C 16 16 0 8 8 D 10 20 -8 0 10 E 24 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=23 B=17 A=17 E=13 so E is eliminated. Round 2 votes counts: C=30 A=29 D=24 B=17 so B is eliminated. Round 3 votes counts: A=38 C=33 D=29 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 D:216 E:202 B:186 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -10 -24 B 6 0 -16 -20 2 C 16 16 0 8 8 D 10 20 -8 0 10 E 24 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -10 -24 B 6 0 -16 -20 2 C 16 16 0 8 8 D 10 20 -8 0 10 E 24 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -10 -24 B 6 0 -16 -20 2 C 16 16 0 8 8 D 10 20 -8 0 10 E 24 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 450: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) C E B D A (10) A D B E C (9) A D C B E (7) C D A B E (6) C A D E B (6) C E B A D (5) B E D A C (5) C E A D B (4) E B A D C (3) B E D C A (3) B E A D C (3) A D B C E (3) E C B D A (2) D A B E C (2) D A B C E (2) C A E D B (2) C A D B E (2) A D E B C (2) E C B A D (1) E B D A C (1) E A D C B (1) D B A E C (1) D A C B E (1) C E D B A (1) C E D A B (1) B E C D A (1) B D E A C (1) B D A E C (1) B D A C E (1) A D C E B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -8 2 -4 B -2 0 -4 -6 -2 C 8 4 0 6 8 D -2 6 -6 0 -8 E 4 2 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 2 -4 B -2 0 -4 -6 -2 C 8 4 0 6 8 D -2 6 -6 0 -8 E 4 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=24 E=18 B=15 D=6 so D is eliminated. Round 2 votes counts: C=37 A=29 E=18 B=16 so B is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:203 A:196 D:195 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 2 -4 B -2 0 -4 -6 -2 C 8 4 0 6 8 D -2 6 -6 0 -8 E 4 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 2 -4 B -2 0 -4 -6 -2 C 8 4 0 6 8 D -2 6 -6 0 -8 E 4 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 2 -4 B -2 0 -4 -6 -2 C 8 4 0 6 8 D -2 6 -6 0 -8 E 4 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 451: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) C B A D E (8) E D B A C (7) E D A B C (7) D E A C B (7) B E D A C (7) C A D B E (6) B C A E D (6) E B D A C (5) C D A E B (4) C A B D E (4) A C D E B (4) E A D B C (2) D E C A B (2) B E A C D (2) B C E D A (2) A E D B C (2) D E C B A (1) D E A B C (1) D C A E B (1) D A E C B (1) D A C E B (1) C D A B E (1) C B A E D (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B A E C D (1) B A C E D (1) A D E C B (1) Total count = 100 A B C D E A 0 8 0 2 4 B -8 0 -4 -16 -12 C 0 4 0 6 0 D -2 16 -6 0 4 E -4 12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.637378 B: 0.000000 C: 0.362622 D: 0.000000 E: 0.000000 Sum of squares = 0.537745432463 Cumulative probabilities = A: 0.637378 B: 0.637378 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 2 4 B -8 0 -4 -16 -12 C 0 4 0 6 0 D -2 16 -6 0 4 E -4 12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=24 E=21 D=14 A=7 so A is eliminated. Round 2 votes counts: C=38 B=24 E=23 D=15 so D is eliminated. Round 3 votes counts: C=40 E=36 B=24 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:207 D:206 C:205 E:202 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 2 4 B -8 0 -4 -16 -12 C 0 4 0 6 0 D -2 16 -6 0 4 E -4 12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 2 4 B -8 0 -4 -16 -12 C 0 4 0 6 0 D -2 16 -6 0 4 E -4 12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 2 4 B -8 0 -4 -16 -12 C 0 4 0 6 0 D -2 16 -6 0 4 E -4 12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 452: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D C A (10) C A B E D (7) C A B D E (7) C A E B D (6) A D C B E (4) E D B C A (3) D A B E C (3) C B E A D (3) B D E C A (3) A D E B C (3) A D B C E (3) A C E D B (3) E D B A C (2) E C B D A (2) E C B A D (2) D E B A C (2) D A E B C (2) C E B A D (2) C B E D A (2) C B A D E (2) B E D C A (2) A C B D E (2) E C A B D (1) E B C D A (1) D B E A C (1) D B A E C (1) D A B C E (1) C E B D A (1) C B D E A (1) C B A E D (1) B E C D A (1) B D C A E (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 -16 22 22 B -14 0 -22 14 14 C 16 22 0 14 20 D -22 -14 -14 0 2 E -22 -14 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -16 22 22 B -14 0 -22 14 14 C 16 22 0 14 20 D -22 -14 -14 0 2 E -22 -14 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=30 E=21 D=10 B=7 so B is eliminated. Round 2 votes counts: C=32 A=30 E=24 D=14 so D is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:236 A:221 B:196 D:176 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -16 22 22 B -14 0 -22 14 14 C 16 22 0 14 20 D -22 -14 -14 0 2 E -22 -14 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -16 22 22 B -14 0 -22 14 14 C 16 22 0 14 20 D -22 -14 -14 0 2 E -22 -14 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -16 22 22 B -14 0 -22 14 14 C 16 22 0 14 20 D -22 -14 -14 0 2 E -22 -14 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 453: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) E C B D A (8) C E B D A (6) C E B A D (5) A D E B C (5) A D B E C (5) E D A B C (4) C B E A D (4) E C B A D (3) E B C D A (3) D A E B C (3) D A B E C (3) C B E D A (3) B C E D A (3) B C D A E (3) A D C B E (3) E D A C B (2) E C D B A (2) D B A C E (2) D A B C E (2) C B A E D (2) C B A D E (2) B C D E A (2) A E D C B (2) A D E C B (2) A C B D E (2) E D C A B (1) E D B A C (1) D E B A C (1) D E A B C (1) D B A E C (1) C A E B D (1) B D C E A (1) B D A C E (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 2 -6 0 B 6 0 2 -2 -2 C -2 -2 0 0 4 D 6 2 0 0 -2 E 0 2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.430355 D: 0.569645 E: 0.000000 Sum of squares = 0.509700821031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.430355 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -6 0 B 6 0 2 -2 -2 C -2 -2 0 0 4 D 6 2 0 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499922 D: 0.500078 E: 0.000000 Sum of squares = 0.500000012272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499922 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=24 C=23 D=13 B=10 so B is eliminated. Round 2 votes counts: C=31 A=30 E=24 D=15 so D is eliminated. Round 3 votes counts: A=42 C=32 E=26 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:203 B:202 C:200 E:200 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -6 0 B 6 0 2 -2 -2 C -2 -2 0 0 4 D 6 2 0 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499922 D: 0.500078 E: 0.000000 Sum of squares = 0.500000012272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499922 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -6 0 B 6 0 2 -2 -2 C -2 -2 0 0 4 D 6 2 0 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499922 D: 0.500078 E: 0.000000 Sum of squares = 0.500000012272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499922 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -6 0 B 6 0 2 -2 -2 C -2 -2 0 0 4 D 6 2 0 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499922 D: 0.500078 E: 0.000000 Sum of squares = 0.500000012272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499922 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 454: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (16) E D C B A (12) D C B A E (9) A B C D E (8) D E C B A (7) A B E C D (6) E D C A B (5) E D A B C (5) B A C D E (5) E A B C D (4) C D B A E (4) D C E B A (3) C B A D E (3) E A B D C (2) D E C A B (2) D E A B C (2) D C B E A (1) D A B E C (1) C D E B A (1) C B A E D (1) B A C E D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -4 12 B -6 0 4 -4 14 C -4 -4 0 -2 6 D 4 4 2 0 -6 E -12 -14 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677553 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 A B C D E A 0 6 4 -4 12 B -6 0 4 -4 14 C -4 -4 0 -2 6 D 4 4 2 0 -6 E -12 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677533 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=28 D=25 C=9 B=6 so B is eliminated. Round 2 votes counts: A=38 E=28 D=25 C=9 so C is eliminated. Round 3 votes counts: A=42 D=30 E=28 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:209 B:204 D:202 C:198 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 -4 12 B -6 0 4 -4 14 C -4 -4 0 -2 6 D 4 4 2 0 -6 E -12 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677533 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 12 B -6 0 4 -4 14 C -4 -4 0 -2 6 D 4 4 2 0 -6 E -12 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677533 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 12 B -6 0 4 -4 14 C -4 -4 0 -2 6 D 4 4 2 0 -6 E -12 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677533 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 455: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (13) D E C A B (7) B A C E D (6) E D A C B (5) E D A B C (5) C B A D E (5) B C A D E (5) D E B A C (4) A C B E D (4) A B C E D (4) C D B E A (3) B D A E C (3) E D C A B (2) E C D A B (2) E A D B C (2) D E C B A (2) D E B C A (2) D E A B C (2) D C E B A (2) D B E A C (2) C E D A B (2) C B A E D (2) B C D A E (2) E C A D B (1) E A D C B (1) E A C D B (1) D C B E A (1) D B C E A (1) C E A D B (1) C D E B A (1) C D E A B (1) C A E D B (1) C A E B D (1) B A E D C (1) B A D E C (1) B A D C E (1) A E C B D (1) Total count = 100 A B C D E A 0 12 -14 2 0 B -12 0 -18 -2 8 C 14 18 0 12 12 D -2 2 -12 0 -10 E 0 -8 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -14 2 0 B -12 0 -18 -2 8 C 14 18 0 12 12 D -2 2 -12 0 -10 E 0 -8 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=23 E=19 B=19 A=9 so A is eliminated. Round 2 votes counts: C=34 D=23 B=23 E=20 so E is eliminated. Round 3 votes counts: C=39 D=38 B=23 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:200 E:195 D:189 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -14 2 0 B -12 0 -18 -2 8 C 14 18 0 12 12 D -2 2 -12 0 -10 E 0 -8 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -14 2 0 B -12 0 -18 -2 8 C 14 18 0 12 12 D -2 2 -12 0 -10 E 0 -8 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -14 2 0 B -12 0 -18 -2 8 C 14 18 0 12 12 D -2 2 -12 0 -10 E 0 -8 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 456: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) C D E A B (9) D E C A B (8) C D A E B (7) A B C E D (7) A B C D E (7) A B E D C (6) B A C E D (5) C D E B A (4) C A B D E (4) B E A D C (4) B A E D C (4) E D C B A (3) C A D E B (3) D C E A B (2) A C B D E (2) E D B C A (1) E D A B C (1) E A D B C (1) D E C B A (1) D A E C B (1) C D B E A (1) C B D E A (1) C B A D E (1) B E D A C (1) B E C D A (1) B C D E A (1) A E D B C (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 22 6 -6 0 B -22 0 4 -10 -8 C -6 -4 0 10 10 D 6 10 -10 0 6 E 0 8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.272727 D: 0.272727 E: 0.000000 Sum of squares = 0.355371900837 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.727273 D: 1.000000 E: 1.000000 A B C D E A 0 22 6 -6 0 B -22 0 4 -10 -8 C -6 -4 0 10 10 D 6 10 -10 0 6 E 0 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.272727 D: 0.272727 E: 0.000000 Sum of squares = 0.35537190082 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.727273 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=25 E=17 B=16 D=12 so D is eliminated. Round 2 votes counts: C=32 E=26 A=26 B=16 so B is eliminated. Round 3 votes counts: A=35 C=33 E=32 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:206 C:205 E:196 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 22 6 -6 0 B -22 0 4 -10 -8 C -6 -4 0 10 10 D 6 10 -10 0 6 E 0 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.272727 D: 0.272727 E: 0.000000 Sum of squares = 0.35537190082 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.727273 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 6 -6 0 B -22 0 4 -10 -8 C -6 -4 0 10 10 D 6 10 -10 0 6 E 0 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.272727 D: 0.272727 E: 0.000000 Sum of squares = 0.35537190082 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.727273 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 6 -6 0 B -22 0 4 -10 -8 C -6 -4 0 10 10 D 6 10 -10 0 6 E 0 8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.272727 D: 0.272727 E: 0.000000 Sum of squares = 0.35537190082 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.727273 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 457: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B D E A C (8) A B E D C (7) A B C E D (7) D E B A C (6) C D E B A (6) B D E C A (5) E D B A C (4) C B A D E (4) C A B D E (4) A C E D B (4) E D A B C (3) D E B C A (3) C E D A B (3) C B D E A (3) B A E D C (3) A E D B C (3) A C B E D (3) D E C B A (2) C A B E D (2) B A D E C (2) E D A C B (1) D B E C A (1) D B E A C (1) C D E A B (1) C D B E A (1) B E A D C (1) B D A E C (1) B C D A E (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 12 0 2 B 6 0 14 6 10 C -12 -14 0 -2 -2 D 0 -6 2 0 0 E -2 -10 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 0 2 B 6 0 14 6 10 C -12 -14 0 -2 -2 D 0 -6 2 0 0 E -2 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=25 B=22 D=13 E=8 so E is eliminated. Round 2 votes counts: C=32 A=25 B=22 D=21 so D is eliminated. Round 3 votes counts: B=37 C=34 A=29 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:204 D:198 E:195 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 0 2 B 6 0 14 6 10 C -12 -14 0 -2 -2 D 0 -6 2 0 0 E -2 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 0 2 B 6 0 14 6 10 C -12 -14 0 -2 -2 D 0 -6 2 0 0 E -2 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 0 2 B 6 0 14 6 10 C -12 -14 0 -2 -2 D 0 -6 2 0 0 E -2 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 458: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (17) A D E B C (12) A D C B E (9) D A C B E (6) E B C D A (4) D A E C B (4) C B E A D (4) B C E A D (4) E A B D C (3) D C A B E (3) D A E B C (3) B C E D A (3) E B A C D (2) E A D B C (2) D E A B C (2) C B D A E (2) B E C D A (2) B E C A D (2) E D B A C (1) E B D C A (1) E B C A D (1) D E C B A (1) D C E A B (1) D C B E A (1) D A C E B (1) C D B A E (1) C B D E A (1) C B A E D (1) C A D B E (1) A E D B C (1) A E B D C (1) A D E C B (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 0 -8 -4 B -4 0 -12 -2 16 C 0 12 0 -8 14 D 8 2 8 0 2 E 4 -16 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -8 -4 B -4 0 -12 -2 16 C 0 12 0 -8 14 D 8 2 8 0 2 E 4 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 D=22 E=14 B=11 so B is eliminated. Round 2 votes counts: C=34 A=26 D=22 E=18 so E is eliminated. Round 3 votes counts: C=43 A=33 D=24 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:210 C:209 B:199 A:196 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -8 -4 B -4 0 -12 -2 16 C 0 12 0 -8 14 D 8 2 8 0 2 E 4 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -8 -4 B -4 0 -12 -2 16 C 0 12 0 -8 14 D 8 2 8 0 2 E 4 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -8 -4 B -4 0 -12 -2 16 C 0 12 0 -8 14 D 8 2 8 0 2 E 4 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 459: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) D B A C E (5) E B A D C (4) C B D E A (4) A E D B C (4) E C B A D (3) E A C D B (3) D A B C E (3) C E A D B (3) C B E D A (3) C A D E B (3) B E D A C (3) B C D E A (3) A D B E C (3) E C A D B (2) E B D A C (2) E A D B C (2) E A B D C (2) D B C A E (2) C E B A D (2) C D B A E (2) C A E D B (2) B D E A C (2) B D C A E (2) B D A E C (2) B D A C E (2) A D B C E (2) A B D E C (2) E C B D A (1) E C A B D (1) E B C A D (1) D C A B E (1) D B A E C (1) D A B E C (1) C E D A B (1) C E A B D (1) C D E A B (1) C D A E B (1) C B D A E (1) C A D B E (1) B E D C A (1) B E A D C (1) A E D C B (1) A E C D B (1) A D E B C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -2 -6 -10 B 12 0 2 4 -4 C 2 -2 0 -2 12 D 6 -4 2 0 -6 E 10 4 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.222222 D: 0.000000 E: 0.111111 Sum of squares = 0.506172839576 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 -12 -2 -6 -10 B 12 0 2 4 -4 C 2 -2 0 -2 12 D 6 -4 2 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.222222 D: 0.000000 E: 0.111111 Sum of squares = 0.506172839517 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=21 B=16 A=16 D=13 so D is eliminated. Round 2 votes counts: C=35 B=24 E=21 A=20 so A is eliminated. Round 3 votes counts: C=37 B=35 E=28 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:205 E:204 D:199 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 -6 -10 B 12 0 2 4 -4 C 2 -2 0 -2 12 D 6 -4 2 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.222222 D: 0.000000 E: 0.111111 Sum of squares = 0.506172839517 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -6 -10 B 12 0 2 4 -4 C 2 -2 0 -2 12 D 6 -4 2 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.222222 D: 0.000000 E: 0.111111 Sum of squares = 0.506172839517 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -6 -10 B 12 0 2 4 -4 C 2 -2 0 -2 12 D 6 -4 2 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.222222 D: 0.000000 E: 0.111111 Sum of squares = 0.506172839517 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 460: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) B D C A E (8) A C E B D (8) B C D A E (7) E D A C B (6) E A C D B (6) D B E C A (5) E A D C B (4) D E B A C (4) D E A B C (4) C B A E D (4) C A E B D (4) B C A D E (4) A E C D B (4) B D C E A (3) D B A C E (2) B C A E D (2) A E C B D (2) E C B D A (1) D E A C B (1) D B C E A (1) D B C A E (1) D B A E C (1) D A E C B (1) D A E B C (1) C E A B D (1) C A B E D (1) B C D E A (1) A E D C B (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 14 -14 6 B 8 0 8 -6 2 C -14 -8 0 -8 -4 D 14 6 8 0 10 E -6 -2 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 14 -14 6 B 8 0 8 -6 2 C -14 -8 0 -8 -4 D 14 6 8 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=25 E=17 A=17 C=10 so C is eliminated. Round 2 votes counts: D=31 B=29 A=22 E=18 so E is eliminated. Round 3 votes counts: D=37 A=33 B=30 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:206 A:199 E:193 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 14 -14 6 B 8 0 8 -6 2 C -14 -8 0 -8 -4 D 14 6 8 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 -14 6 B 8 0 8 -6 2 C -14 -8 0 -8 -4 D 14 6 8 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 -14 6 B 8 0 8 -6 2 C -14 -8 0 -8 -4 D 14 6 8 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 461: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) C E A D B (7) E C A D B (6) C E D B A (6) B D A E C (6) D B A E C (5) C B D A E (5) B D A C E (5) C D B E A (4) A B E D C (4) E A C D B (2) E A C B D (2) D E A B C (2) D B E C A (2) D B C E A (2) D B C A E (2) C E D A B (2) C B D E A (2) B C D A E (2) A E C B D (2) A E B D C (2) A E B C D (2) E D A B C (1) E A D C B (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E A C (1) D A B E C (1) C B E D A (1) C B E A D (1) C A E B D (1) B D C A E (1) B A D E C (1) A E D B C (1) A D E B C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -16 -6 -14 B -2 0 -12 2 -4 C 16 12 0 16 10 D 6 -2 -16 0 -8 E 14 4 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 -6 -14 B -2 0 -12 2 -4 C 16 12 0 16 10 D 6 -2 -16 0 -8 E 14 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 D=18 B=15 A=14 E=12 so E is eliminated. Round 2 votes counts: C=47 D=19 A=19 B=15 so B is eliminated. Round 3 votes counts: C=49 D=31 A=20 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:208 B:192 D:190 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 -6 -14 B -2 0 -12 2 -4 C 16 12 0 16 10 D 6 -2 -16 0 -8 E 14 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -6 -14 B -2 0 -12 2 -4 C 16 12 0 16 10 D 6 -2 -16 0 -8 E 14 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -6 -14 B -2 0 -12 2 -4 C 16 12 0 16 10 D 6 -2 -16 0 -8 E 14 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 462: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) D C A E B (6) D A E C B (6) C A D B E (6) E B D A C (5) D E A B C (5) E D B A C (4) D A C E B (4) C B A E D (4) C A B E D (4) C A B D E (4) B E C A D (4) E B A D C (3) D E A C B (3) D C B E A (3) B E D C A (3) B C E A D (3) A E D B C (3) A C B E D (3) C D A B E (2) C B D A E (2) C A D E B (2) B E A C D (2) A C D E B (2) E D A B C (1) E A D B C (1) E A B D C (1) D C E B A (1) B E D A C (1) B E A D C (1) B C A E D (1) B A E C D (1) A E B C D (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 10 -4 6 B -12 0 -6 -12 -8 C -10 6 0 -14 -2 D 4 12 14 0 6 E -6 8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 -4 6 B -12 0 -6 -12 -8 C -10 6 0 -14 -2 D 4 12 14 0 6 E -6 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=24 B=16 E=15 A=11 so A is eliminated. Round 2 votes counts: D=34 C=30 E=19 B=17 so B is eliminated. Round 3 votes counts: C=35 D=34 E=31 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:212 E:199 C:190 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 10 -4 6 B -12 0 -6 -12 -8 C -10 6 0 -14 -2 D 4 12 14 0 6 E -6 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -4 6 B -12 0 -6 -12 -8 C -10 6 0 -14 -2 D 4 12 14 0 6 E -6 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -4 6 B -12 0 -6 -12 -8 C -10 6 0 -14 -2 D 4 12 14 0 6 E -6 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 463: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) A C D B E (8) B C A E D (7) D E A C B (6) C A B D E (6) B E C D A (6) C B A E D (5) B C E A D (5) D A E C B (4) C B A D E (4) B C A D E (3) A E C D B (3) A D E C B (3) E D B A C (2) E D A C B (2) E D A B C (2) E B C D A (2) E B C A D (2) E A D C B (2) D E B C A (2) D A C B E (2) B C E D A (2) A D C E B (2) D E B A C (1) D E A B C (1) D C A B E (1) D A E B C (1) D A C E B (1) B E D C A (1) B E C A D (1) B D E C A (1) A E C B D (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -14 8 6 B 6 0 -4 10 6 C 14 4 0 12 -4 D -8 -10 -12 0 -4 E -6 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.34693877553 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 -6 -14 8 6 B 6 0 -4 10 6 C 14 4 0 12 -4 D -8 -10 -12 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775513 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=21 D=19 A=19 C=15 so C is eliminated. Round 2 votes counts: B=35 A=25 E=21 D=19 so D is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:209 E:198 A:197 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 8 6 B 6 0 -4 10 6 C 14 4 0 12 -4 D -8 -10 -12 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775513 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 8 6 B 6 0 -4 10 6 C 14 4 0 12 -4 D -8 -10 -12 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775513 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 8 6 B 6 0 -4 10 6 C 14 4 0 12 -4 D -8 -10 -12 0 -4 E -6 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775513 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 464: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) B C A E D (10) A D E B C (7) E D A C B (6) D E C A B (5) C D E A B (5) A E D B C (5) C B A D E (4) B A E C D (4) A B E D C (4) E D C A B (3) E D A B C (3) C D E B A (3) C B D A E (3) B A C E D (3) A B D E C (3) E A B D C (2) D A E C B (2) B C E A D (2) E A D B C (1) D E A B C (1) D C E A B (1) C E D B A (1) C E B D A (1) C D B E A (1) C D A E B (1) C B E D A (1) C B D E A (1) B E A D C (1) B C A D E (1) B A E D C (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 24 14 0 0 B -24 0 0 -14 -18 C -14 0 0 -16 -20 D 0 14 16 0 4 E 0 18 20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333905 B: 0.000000 C: 0.000000 D: 0.666095 E: 0.000000 Sum of squares = 0.555175213112 Cumulative probabilities = A: 0.333905 B: 0.333905 C: 0.333905 D: 1.000000 E: 1.000000 A B C D E A 0 24 14 0 0 B -24 0 0 -14 -18 C -14 0 0 -16 -20 D 0 14 16 0 4 E 0 18 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.4999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 D=21 C=21 A=20 E=15 so E is eliminated. Round 2 votes counts: D=33 B=23 A=23 C=21 so C is eliminated. Round 3 votes counts: D=44 B=33 A=23 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:219 D:217 E:217 C:175 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 24 14 0 0 B -24 0 0 -14 -18 C -14 0 0 -16 -20 D 0 14 16 0 4 E 0 18 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.4999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 14 0 0 B -24 0 0 -14 -18 C -14 0 0 -16 -20 D 0 14 16 0 4 E 0 18 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.4999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 14 0 0 B -24 0 0 -14 -18 C -14 0 0 -16 -20 D 0 14 16 0 4 E 0 18 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.4999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 465: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) D C A B E (6) C D E A B (6) C D B E A (6) C D E B A (5) B E A D C (5) B A E D C (5) E B A D C (4) D C A E B (4) E B C A D (3) D A C E B (3) D A C B E (3) C E B D A (3) C D B A E (3) C D A E B (3) A D E C B (3) A B D E C (3) E C A D B (2) E C A B D (2) C B E D A (2) B E C A D (2) B E A C D (2) B D A C E (2) B A D E C (2) A E B D C (2) E C B A D (1) E B C D A (1) E A B D C (1) D A B C E (1) C E D B A (1) C D A B E (1) C B D E A (1) C B D A E (1) B D A E C (1) A E D B C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -14 -6 -6 -8 B 14 0 -12 0 -6 C 6 12 0 4 2 D 6 0 -4 0 12 E 8 6 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -6 -8 B 14 0 -12 0 -6 C 6 12 0 4 2 D 6 0 -4 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=21 B=19 D=17 A=11 so A is eliminated. Round 2 votes counts: C=32 E=24 D=22 B=22 so D is eliminated. Round 3 votes counts: C=48 E=28 B=24 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:207 E:200 B:198 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -6 -6 -8 B 14 0 -12 0 -6 C 6 12 0 4 2 D 6 0 -4 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -6 -8 B 14 0 -12 0 -6 C 6 12 0 4 2 D 6 0 -4 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -6 -8 B 14 0 -12 0 -6 C 6 12 0 4 2 D 6 0 -4 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 466: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) C A D E B (6) B E D A C (5) A D C E B (5) E C D A B (4) E B C D A (4) B E C D A (4) B E C A D (4) B C A E D (4) B A D C E (4) A C D B E (4) E D C B A (3) E C B D A (3) C E A D B (3) C B E A D (3) B D E A C (3) B D A E C (3) B A C D E (3) D E C A B (2) D E A C B (2) D A E C B (2) B C E A D (2) A D B C E (2) A C D E B (2) E D B C A (1) E B D C A (1) E B D A C (1) D C A E B (1) D B E A C (1) D B A E C (1) D A C E B (1) C E D A B (1) C E A B D (1) C A B D E (1) B E D C A (1) B A D E C (1) B A C E D (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -14 -4 -14 B 6 0 -6 0 -2 C 14 6 0 0 -8 D 4 0 0 0 -8 E 14 2 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -14 -4 -14 B 6 0 -6 0 -2 C 14 6 0 0 -8 D 4 0 0 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=25 C=15 A=15 D=10 so D is eliminated. Round 2 votes counts: B=37 E=29 A=18 C=16 so C is eliminated. Round 3 votes counts: B=40 E=34 A=26 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:206 B:199 D:198 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -14 -4 -14 B 6 0 -6 0 -2 C 14 6 0 0 -8 D 4 0 0 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -4 -14 B 6 0 -6 0 -2 C 14 6 0 0 -8 D 4 0 0 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -4 -14 B 6 0 -6 0 -2 C 14 6 0 0 -8 D 4 0 0 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 467: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (11) A E C B D (7) D B C E A (6) E D C B A (5) E D A C B (5) D E B C A (5) D E A B C (4) B C A D E (4) A C E B D (4) A B C D E (4) E A D C B (3) E D C A B (2) E C D B A (2) E C B D A (2) E A C B D (2) D E C B A (2) D B E C A (2) D B C A E (2) D A B C E (2) B C D E A (2) B C D A E (2) A E D C B (2) A E C D B (2) A B D C E (2) E D A B C (1) E C D A B (1) E C A B D (1) E A C D B (1) D E B A C (1) C E B D A (1) C E B A D (1) C B E D A (1) C B A E D (1) B D C E A (1) B D C A E (1) B A C D E (1) A E D B C (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 16 12 0 -2 B -16 0 -12 -4 -12 C -12 12 0 0 -6 D 0 4 0 0 -12 E 2 12 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 12 0 -2 B -16 0 -12 -4 -12 C -12 12 0 0 -6 D 0 4 0 0 -12 E 2 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=25 D=24 B=11 C=4 so C is eliminated. Round 2 votes counts: A=36 E=27 D=24 B=13 so B is eliminated. Round 3 votes counts: A=42 D=30 E=28 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:213 C:197 D:196 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 12 0 -2 B -16 0 -12 -4 -12 C -12 12 0 0 -6 D 0 4 0 0 -12 E 2 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 0 -2 B -16 0 -12 -4 -12 C -12 12 0 0 -6 D 0 4 0 0 -12 E 2 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 0 -2 B -16 0 -12 -4 -12 C -12 12 0 0 -6 D 0 4 0 0 -12 E 2 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 468: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) E D A C B (6) E C A D B (6) D A B C E (6) B C E A D (6) B E C D A (4) B D A C E (4) B C A D E (4) A D C E B (4) E A D C B (3) D A E B C (3) D A B E C (3) C E A D B (3) B C E D A (3) E A C D B (2) D B A E C (2) C E B A D (2) C B E A D (2) C A B E D (2) B D E C A (2) B D C A E (2) B D A E C (2) B C A E D (2) B A C D E (2) A C D E B (2) E C D A B (1) E C B A D (1) D E A C B (1) D B A C E (1) D A C E B (1) C E A B D (1) C B A E D (1) C A E D B (1) C A E B D (1) B E D C A (1) B E D A C (1) B D E A C (1) Total count = 100 A B C D E A 0 14 10 -10 8 B -14 0 -2 -12 2 C -10 2 0 -8 0 D 10 12 8 0 2 E -8 -2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 -10 8 B -14 0 -2 -12 2 C -10 2 0 -8 0 D 10 12 8 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=28 E=19 C=13 A=6 so A is eliminated. Round 2 votes counts: B=34 D=32 E=19 C=15 so C is eliminated. Round 3 votes counts: B=39 D=34 E=27 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:211 E:194 C:192 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 10 -10 8 B -14 0 -2 -12 2 C -10 2 0 -8 0 D 10 12 8 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -10 8 B -14 0 -2 -12 2 C -10 2 0 -8 0 D 10 12 8 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -10 8 B -14 0 -2 -12 2 C -10 2 0 -8 0 D 10 12 8 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 469: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) D A B E C (7) C E B A D (6) B C E D A (4) E A D C B (3) D E C A B (3) D C B E A (3) D B A E C (3) D B A C E (3) D A E B C (3) C B E A D (3) B C D A E (3) A E D C B (3) A B D E C (3) E C B A D (2) E C A D B (2) E C A B D (2) E A B C D (2) D C E A B (2) C E B D A (2) C B E D A (2) B D C A E (2) B D A C E (2) B C A E D (2) A E D B C (2) A D E C B (2) A D B E C (2) A B E C D (2) E B C A D (1) E A C B D (1) D E A C B (1) D B C A E (1) C E D B A (1) C E D A B (1) C D B E A (1) B E C A D (1) B C E A D (1) B C D E A (1) B A E C D (1) B A D C E (1) B A C E D (1) B A C D E (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 8 -6 10 B -4 0 2 -6 0 C -8 -2 0 -12 -14 D 6 6 12 0 6 E -10 0 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -6 10 B -4 0 2 -6 0 C -8 -2 0 -12 -14 D 6 6 12 0 6 E -10 0 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=20 A=17 C=16 E=13 so E is eliminated. Round 2 votes counts: D=34 A=23 C=22 B=21 so B is eliminated. Round 3 votes counts: D=38 C=35 A=27 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:208 E:199 B:196 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -6 10 B -4 0 2 -6 0 C -8 -2 0 -12 -14 D 6 6 12 0 6 E -10 0 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -6 10 B -4 0 2 -6 0 C -8 -2 0 -12 -14 D 6 6 12 0 6 E -10 0 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -6 10 B -4 0 2 -6 0 C -8 -2 0 -12 -14 D 6 6 12 0 6 E -10 0 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 470: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (11) B D A C E (11) E C A D B (6) D B E C A (5) E C D B A (4) E C D A B (4) E C A B D (4) D A B E C (4) A C E B D (4) E C B D A (3) D E A C B (3) C E B A D (3) B A D C E (2) A B D C E (2) A B C E D (2) A B C D E (2) E D C B A (1) E C B A D (1) D E C A B (1) D B E A C (1) D B A C E (1) D A E B C (1) D A B C E (1) C E B D A (1) C B A E D (1) C A E B D (1) C A B E D (1) B E D C A (1) B D C E A (1) B C E D A (1) B C D A E (1) A D E C B (1) A D B E C (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -2 -12 -2 B -2 0 -2 8 2 C 2 2 0 2 -4 D 12 -8 -2 0 0 E 2 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.157328 B: 0.421336 C: 0.132004 D: 0.000000 E: 0.289332 Sum of squares = 0.303414215902 Cumulative probabilities = A: 0.157328 B: 0.578664 C: 0.710668 D: 0.710668 E: 1.000000 A B C D E A 0 2 -2 -12 -2 B -2 0 -2 8 2 C 2 2 0 2 -4 D 12 -8 -2 0 0 E 2 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199998 B: 0.400001 C: 0.100002 D: 0.000000 E: 0.299999 Sum of squares = 0.30000000001 Cumulative probabilities = A: 0.199998 B: 0.599999 C: 0.700001 D: 0.700001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 C=18 B=17 A=14 so A is eliminated. Round 2 votes counts: D=31 E=23 C=23 B=23 so E is eliminated. Round 3 votes counts: C=45 D=32 B=23 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:203 E:202 C:201 D:201 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 -12 -2 B -2 0 -2 8 2 C 2 2 0 2 -4 D 12 -8 -2 0 0 E 2 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199998 B: 0.400001 C: 0.100002 D: 0.000000 E: 0.299999 Sum of squares = 0.30000000001 Cumulative probabilities = A: 0.199998 B: 0.599999 C: 0.700001 D: 0.700001 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -12 -2 B -2 0 -2 8 2 C 2 2 0 2 -4 D 12 -8 -2 0 0 E 2 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199998 B: 0.400001 C: 0.100002 D: 0.000000 E: 0.299999 Sum of squares = 0.30000000001 Cumulative probabilities = A: 0.199998 B: 0.599999 C: 0.700001 D: 0.700001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -12 -2 B -2 0 -2 8 2 C 2 2 0 2 -4 D 12 -8 -2 0 0 E 2 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199998 B: 0.400001 C: 0.100002 D: 0.000000 E: 0.299999 Sum of squares = 0.30000000001 Cumulative probabilities = A: 0.199998 B: 0.599999 C: 0.700001 D: 0.700001 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 471: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) A C D E B (8) E C D A B (7) D C A E B (7) D C A B E (5) B A D C E (5) E C A D B (4) B E D A C (4) B D E C A (4) B D A C E (4) E D C A B (3) E B D C A (3) E B C D A (2) E B C A D (2) D A C B E (2) B E D C A (2) B A E C D (2) B A D E C (2) B A C D E (2) A D C B E (2) E D B C A (1) E C D B A (1) E C B D A (1) E B A C D (1) E A C D B (1) E A C B D (1) D E C B A (1) D C E B A (1) D C E A B (1) C E A D B (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A D C (1) B D C A E (1) B D A E C (1) A C E D B (1) A C D B E (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -4 -2 B 0 0 -6 0 0 C 0 6 0 0 -6 D 4 0 0 0 2 E 2 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.185212 D: 0.814788 E: 0.000000 Sum of squares = 0.698182628609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.185212 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 -2 B 0 0 -6 0 0 C 0 6 0 0 -6 D 4 0 0 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000118391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=27 D=17 A=15 C=4 so C is eliminated. Round 2 votes counts: B=37 E=28 D=18 A=17 so A is eliminated. Round 3 votes counts: B=40 E=30 D=30 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:203 E:203 C:200 A:197 B:197 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -4 -2 B 0 0 -6 0 0 C 0 6 0 0 -6 D 4 0 0 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000118391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 -2 B 0 0 -6 0 0 C 0 6 0 0 -6 D 4 0 0 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000118391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 -2 B 0 0 -6 0 0 C 0 6 0 0 -6 D 4 0 0 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000118391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 472: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) A E B D C (7) C D B E A (6) B A C E D (6) D E A C B (5) D C E A B (5) C B D A E (5) B C A D E (5) B A E D C (5) A B E D C (5) E A D B C (4) C B A E D (4) E D A C B (3) E D A B C (3) C D B A E (3) B A E C D (3) A B E C D (3) D E A B C (2) C D E B A (2) C B A D E (2) A B C E D (2) E D C A B (1) E C D A B (1) E A B D C (1) D E C B A (1) D E B A C (1) D C B E A (1) D B E C A (1) C D E A B (1) C A B E D (1) B D C E A (1) B D A E C (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 6 6 0 8 B -6 0 2 6 10 C -6 -2 0 -8 -10 D 0 -6 8 0 -2 E -8 -10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625390 B: 0.000000 C: 0.000000 D: 0.374610 E: 0.000000 Sum of squares = 0.531445265189 Cumulative probabilities = A: 0.625390 B: 0.625390 C: 0.625390 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 0 8 B -6 0 2 6 10 C -6 -2 0 -8 -10 D 0 -6 8 0 -2 E -8 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500156 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.000000 Sum of squares = 0.500000048645 Cumulative probabilities = A: 0.500156 B: 0.500156 C: 0.500156 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=21 A=19 E=13 so E is eliminated. Round 2 votes counts: D=30 C=25 A=24 B=21 so B is eliminated. Round 3 votes counts: A=38 D=32 C=30 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:206 D:200 E:197 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 0 8 B -6 0 2 6 10 C -6 -2 0 -8 -10 D 0 -6 8 0 -2 E -8 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500156 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.000000 Sum of squares = 0.500000048645 Cumulative probabilities = A: 0.500156 B: 0.500156 C: 0.500156 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 0 8 B -6 0 2 6 10 C -6 -2 0 -8 -10 D 0 -6 8 0 -2 E -8 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500156 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.000000 Sum of squares = 0.500000048645 Cumulative probabilities = A: 0.500156 B: 0.500156 C: 0.500156 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 0 8 B -6 0 2 6 10 C -6 -2 0 -8 -10 D 0 -6 8 0 -2 E -8 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500156 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.000000 Sum of squares = 0.500000048645 Cumulative probabilities = A: 0.500156 B: 0.500156 C: 0.500156 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 473: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) C B D A E (6) B D C A E (6) E B D A C (5) E A C D B (5) E C A B D (4) E B D C A (4) D B A C E (4) B E D C A (4) E A B D C (3) C B E D A (3) A C D E B (3) E B C D A (2) E A D C B (2) C E A B D (2) C D B A E (2) C B D E A (2) C A D E B (2) B E C D A (2) B D E C A (2) B D E A C (2) A E D C B (2) A E C D B (2) A C D B E (2) E C B D A (1) E B A D C (1) E B A C D (1) E A D B C (1) E A C B D (1) E A B C D (1) D C A B E (1) D B A E C (1) D A C B E (1) C E A D B (1) C A E D B (1) C A E B D (1) B E D A C (1) B D C E A (1) B D A E C (1) B C E D A (1) B C D E A (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -16 -6 -6 B 6 0 -10 16 4 C 16 10 0 10 0 D 6 -16 -10 0 -2 E 6 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.453388 D: 0.000000 E: 0.546612 Sum of squares = 0.504345342642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.453388 D: 0.453388 E: 1.000000 A B C D E A 0 -6 -16 -6 -6 B 6 0 -10 16 4 C 16 10 0 10 0 D 6 -16 -10 0 -2 E 6 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=29 B=21 A=12 D=7 so D is eliminated. Round 2 votes counts: E=31 C=30 B=26 A=13 so A is eliminated. Round 3 votes counts: E=37 C=37 B=26 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:208 E:202 D:189 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -6 -6 B 6 0 -10 16 4 C 16 10 0 10 0 D 6 -16 -10 0 -2 E 6 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -6 -6 B 6 0 -10 16 4 C 16 10 0 10 0 D 6 -16 -10 0 -2 E 6 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -6 -6 B 6 0 -10 16 4 C 16 10 0 10 0 D 6 -16 -10 0 -2 E 6 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 474: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) E C A D B (9) E B D C A (7) C A D B E (5) B D A C E (5) E A C B D (4) D B C A E (4) C A E D B (4) B D E C A (4) A C D B E (4) E D B C A (3) E B A D C (3) E B A C D (3) C A D E B (3) D C B A E (2) D B E C A (2) C D A B E (2) B D E A C (2) B D A E C (2) B A D E C (2) A C E D B (2) A C E B D (2) E D C B A (1) E C D B A (1) E C A B D (1) E A C D B (1) E A B C D (1) D C B E A (1) D C A B E (1) D B C E A (1) C E A D B (1) B E D A C (1) B D C E A (1) B D C A E (1) A E C B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -8 -4 -16 B 14 0 8 6 -16 C 8 -8 0 -10 -18 D 4 -6 10 0 -12 E 16 16 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -8 -4 -16 B 14 0 8 6 -16 C 8 -8 0 -10 -18 D 4 -6 10 0 -12 E 16 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=45 B=18 C=15 D=11 A=11 so D is eliminated. Round 2 votes counts: E=45 B=25 C=19 A=11 so A is eliminated. Round 3 votes counts: E=46 C=28 B=26 so B is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:206 D:198 C:186 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -8 -4 -16 B 14 0 8 6 -16 C 8 -8 0 -10 -18 D 4 -6 10 0 -12 E 16 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -4 -16 B 14 0 8 6 -16 C 8 -8 0 -10 -18 D 4 -6 10 0 -12 E 16 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -4 -16 B 14 0 8 6 -16 C 8 -8 0 -10 -18 D 4 -6 10 0 -12 E 16 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 475: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (7) C A B D E (6) E D B A C (5) D B E C A (5) C A E D B (5) B D E A C (5) D E B C A (4) B A E D C (4) E A D C B (3) B E D A C (3) B D E C A (3) B D C E A (3) A C E D B (3) A C B E D (3) A B C E D (3) E A D B C (2) E A B D C (2) D E B A C (2) C D B E A (2) C B A D E (2) C A D B E (2) B E A D C (2) B C A D E (2) B A C D E (2) A B C D E (2) E D C A B (1) E D B C A (1) E D A C B (1) E D A B C (1) E B A D C (1) D E C B A (1) D C E B A (1) C E D A B (1) C D E B A (1) C D E A B (1) C D B A E (1) C A D E B (1) B D C A E (1) B C D A E (1) A C E B D (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 10 12 -2 B 4 0 12 -2 10 C -10 -12 0 -4 -10 D -12 2 4 0 -2 E 2 -10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.666667 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839644 Cumulative probabilities = A: 0.111111 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 12 -2 B 4 0 12 -2 10 C -10 -12 0 -4 -10 D -12 2 4 0 -2 E 2 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.666667 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839504 Cumulative probabilities = A: 0.111111 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=22 A=22 E=17 D=13 so D is eliminated. Round 2 votes counts: B=31 E=24 C=23 A=22 so A is eliminated. Round 3 votes counts: B=38 E=31 C=31 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:208 E:202 D:196 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 12 -2 B 4 0 12 -2 10 C -10 -12 0 -4 -10 D -12 2 4 0 -2 E 2 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.666667 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839504 Cumulative probabilities = A: 0.111111 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 12 -2 B 4 0 12 -2 10 C -10 -12 0 -4 -10 D -12 2 4 0 -2 E 2 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.666667 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839504 Cumulative probabilities = A: 0.111111 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 12 -2 B 4 0 12 -2 10 C -10 -12 0 -4 -10 D -12 2 4 0 -2 E 2 -10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.666667 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839504 Cumulative probabilities = A: 0.111111 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 476: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) D E C B A (7) B A C D E (7) E D B A C (6) A B C E D (5) E A D B C (4) D E B C A (4) A C B E D (4) A B E D C (4) E D A C B (3) B A E D C (3) E D C B A (2) E C D A B (2) E B D A C (2) D E C A B (2) D C E B A (2) C D E A B (2) C A E D B (2) C A D E B (2) B E D A C (2) B E A D C (2) B D C E A (2) B A C E D (2) A C E D B (2) A C B D E (2) A B C D E (2) E B A D C (1) D B C E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D A E B (1) C B D A E (1) B D E C A (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 14 -2 -14 B -6 0 4 -12 -20 C -14 -4 0 -18 -16 D 2 12 18 0 -24 E 14 20 16 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 14 -2 -14 B -6 0 4 -12 -20 C -14 -4 0 -18 -16 D 2 12 18 0 -24 E 14 20 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 B=20 D=16 C=11 so C is eliminated. Round 2 votes counts: E=30 A=29 B=21 D=20 so D is eliminated. Round 3 votes counts: E=48 A=30 B=22 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:237 D:204 A:202 B:183 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 -2 -14 B -6 0 4 -12 -20 C -14 -4 0 -18 -16 D 2 12 18 0 -24 E 14 20 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -2 -14 B -6 0 4 -12 -20 C -14 -4 0 -18 -16 D 2 12 18 0 -24 E 14 20 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -2 -14 B -6 0 4 -12 -20 C -14 -4 0 -18 -16 D 2 12 18 0 -24 E 14 20 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 477: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (13) B A C D E (13) A B E C D (9) E D C A B (8) B C A D E (8) E A D B C (7) D C E B A (6) C B D A E (6) A E B D C (5) E A D C B (4) D E C B A (4) A B C E D (4) E D A C B (3) B C D A E (3) E A B D C (2) C D B A E (2) C D E B A (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -2 8 4 B 12 0 6 4 18 C 2 -6 0 20 12 D -8 -4 -20 0 12 E -4 -18 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999078 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 8 4 B 12 0 6 4 18 C 2 -6 0 20 12 D -8 -4 -20 0 12 E -4 -18 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 C=22 A=20 D=10 so D is eliminated. Round 2 votes counts: E=28 C=28 B=24 A=20 so A is eliminated. Round 3 votes counts: B=38 E=34 C=28 so C is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:214 A:199 D:190 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 8 4 B 12 0 6 4 18 C 2 -6 0 20 12 D -8 -4 -20 0 12 E -4 -18 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 8 4 B 12 0 6 4 18 C 2 -6 0 20 12 D -8 -4 -20 0 12 E -4 -18 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 8 4 B 12 0 6 4 18 C 2 -6 0 20 12 D -8 -4 -20 0 12 E -4 -18 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 478: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) B A C E D (6) E A D B C (5) D E A C B (5) C B A E D (5) C B A D E (5) A D E B C (5) A B E C D (5) C B D A E (4) E B A C D (3) D E A B C (3) C D B E A (3) B C A E D (3) A B E D C (3) D A E B C (2) D A C B E (2) C D B A E (2) C B E A D (2) B C E A D (2) B C A D E (2) A E B D C (2) A D B C E (2) A B C E D (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A C B (1) E C B D A (1) D E C B A (1) D E C A B (1) D C E A B (1) D C B A E (1) D C A E B (1) D C A B E (1) D A E C B (1) C D E B A (1) C B E D A (1) C B D E A (1) B E A C D (1) B A E C D (1) A E D B C (1) A E B C D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 8 16 18 B 6 0 4 2 12 C -8 -4 0 2 6 D -16 -2 -2 0 4 E -18 -12 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 16 18 B 6 0 4 2 12 C -8 -4 0 2 6 D -16 -2 -2 0 4 E -18 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 A=23 B=15 E=13 so E is eliminated. Round 2 votes counts: D=29 A=28 C=25 B=18 so B is eliminated. Round 3 votes counts: A=39 C=32 D=29 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:212 C:198 D:192 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 16 18 B 6 0 4 2 12 C -8 -4 0 2 6 D -16 -2 -2 0 4 E -18 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 16 18 B 6 0 4 2 12 C -8 -4 0 2 6 D -16 -2 -2 0 4 E -18 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 16 18 B 6 0 4 2 12 C -8 -4 0 2 6 D -16 -2 -2 0 4 E -18 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 479: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) C E B D A (7) B C D E A (6) E C A B D (4) D E A C B (4) B D C A E (4) A E C B D (4) E A C D B (3) D C E B A (3) D A B E C (3) C E B A D (3) C B E D A (3) B D C E A (3) B C E A D (3) D E C A B (2) D B C E A (2) D B A C E (2) C B E A D (2) B C E D A (2) B A D C E (2) A E C D B (2) A E B C D (2) A B D C E (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B A D (1) E A D C B (1) D E C B A (1) D B C A E (1) D A E C B (1) D A E B C (1) D A B C E (1) B D A C E (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) A D E C B (1) A D B E C (1) A C B E D (1) A B E D C (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -6 -2 -6 B 2 0 -12 16 -6 C 6 12 0 2 6 D 2 -16 -2 0 -10 E 6 6 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -2 -6 B 2 0 -12 16 -6 C 6 12 0 2 6 D 2 -16 -2 0 -10 E 6 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 D=21 C=15 E=12 so E is eliminated. Round 2 votes counts: A=31 B=25 D=22 C=22 so D is eliminated. Round 3 votes counts: A=41 B=30 C=29 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 E:208 B:200 A:192 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -2 -6 B 2 0 -12 16 -6 C 6 12 0 2 6 D 2 -16 -2 0 -10 E 6 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -2 -6 B 2 0 -12 16 -6 C 6 12 0 2 6 D 2 -16 -2 0 -10 E 6 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -2 -6 B 2 0 -12 16 -6 C 6 12 0 2 6 D 2 -16 -2 0 -10 E 6 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 480: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) A B D C E (7) E D C A B (6) B A D C E (6) D C E B A (5) E C D B A (4) D C B E A (4) D B C A E (4) B A C D E (4) A E B C D (4) E A C B D (3) D E A C B (3) B D C A E (3) B C D A E (3) A B E C D (3) E D C B A (2) E C D A B (2) E C A D B (2) D E C A B (2) D B A C E (2) C E D B A (2) C E B A D (2) B D A C E (2) A B D E C (2) A B C D E (2) E D A C B (1) E C A B D (1) D A E B C (1) D A B C E (1) C E B D A (1) C D E B A (1) C D B E A (1) C B D E A (1) B C A D E (1) B A C E D (1) A E D B C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -14 -10 -18 -2 B 14 0 -2 -6 -4 C 10 2 0 -24 6 D 18 6 24 0 28 E 2 4 -6 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -18 -2 B 14 0 -2 -6 -4 C 10 2 0 -24 6 D 18 6 24 0 28 E 2 4 -6 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=21 A=21 B=20 C=8 so C is eliminated. Round 2 votes counts: D=32 E=26 B=21 A=21 so B is eliminated. Round 3 votes counts: D=41 A=33 E=26 so E is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:238 B:201 C:197 E:186 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 -18 -2 B 14 0 -2 -6 -4 C 10 2 0 -24 6 D 18 6 24 0 28 E 2 4 -6 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -18 -2 B 14 0 -2 -6 -4 C 10 2 0 -24 6 D 18 6 24 0 28 E 2 4 -6 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -18 -2 B 14 0 -2 -6 -4 C 10 2 0 -24 6 D 18 6 24 0 28 E 2 4 -6 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 481: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) D E B A C (8) C A E D B (6) A C B D E (6) E D C B A (5) C A B E D (5) B D E A C (5) E D C A B (4) D E B C A (4) B D E C A (4) B A D E C (4) A C B E D (4) E D B C A (3) E D B A C (3) D E A B C (3) C B E D A (3) C B A E D (2) B D A E C (2) B C A D E (2) A C E D B (2) A B D E C (2) E D A B C (1) E C D A B (1) C E A D B (1) C B E A D (1) C A E B D (1) C A B D E (1) B D A C E (1) B A D C E (1) B A C D E (1) A E D C B (1) A D B E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 14 6 8 B -2 0 10 10 12 C -14 -10 0 -4 -2 D -6 -10 4 0 14 E -8 -12 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 6 8 B -2 0 10 10 12 C -14 -10 0 -4 -2 D -6 -10 4 0 14 E -8 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=20 B=20 E=17 D=15 so D is eliminated. Round 2 votes counts: E=32 A=28 C=20 B=20 so C is eliminated. Round 3 votes counts: A=41 E=33 B=26 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:215 D:201 C:185 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 6 8 B -2 0 10 10 12 C -14 -10 0 -4 -2 D -6 -10 4 0 14 E -8 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 6 8 B -2 0 10 10 12 C -14 -10 0 -4 -2 D -6 -10 4 0 14 E -8 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 6 8 B -2 0 10 10 12 C -14 -10 0 -4 -2 D -6 -10 4 0 14 E -8 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 482: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) D E B C A (9) C A D E B (7) C A D B E (6) B E D A C (5) A C B E D (5) E B D A C (4) D E C B A (3) C D A E B (3) C D A B E (3) C A E D B (3) A E C B D (3) A C E D B (3) A C E B D (3) E D B C A (2) D B E C A (2) C A B D E (2) B D E C A (2) B D E A C (2) B A E C D (2) B A D C E (2) A C B D E (2) A B C E D (2) E C D A B (1) E C A D B (1) E A C D B (1) D E C A B (1) D E B A C (1) D C E A B (1) D C B E A (1) D B E A C (1) D B C E A (1) C D B A E (1) B E A D C (1) B D C A E (1) B D A E C (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 0 -10 2 B 4 0 0 -22 -12 C 0 0 0 0 -6 D 10 22 0 0 4 E -2 12 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.308271 D: 0.691729 E: 0.000000 Sum of squares = 0.573519887919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.308271 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -10 2 B 4 0 0 -22 -12 C 0 0 0 0 -6 D 10 22 0 0 4 E -2 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399999 D: 0.600001 E: 0.000000 Sum of squares = 0.520000408056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=20 E=19 A=19 B=17 so B is eliminated. Round 2 votes counts: D=26 E=25 C=25 A=24 so A is eliminated. Round 3 votes counts: C=41 E=31 D=28 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:206 C:197 A:194 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -10 2 B 4 0 0 -22 -12 C 0 0 0 0 -6 D 10 22 0 0 4 E -2 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399999 D: 0.600001 E: 0.000000 Sum of squares = 0.520000408056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399999 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -10 2 B 4 0 0 -22 -12 C 0 0 0 0 -6 D 10 22 0 0 4 E -2 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399999 D: 0.600001 E: 0.000000 Sum of squares = 0.520000408056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -10 2 B 4 0 0 -22 -12 C 0 0 0 0 -6 D 10 22 0 0 4 E -2 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399999 D: 0.600001 E: 0.000000 Sum of squares = 0.520000408056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 483: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) B A C D E (9) B E D A C (5) B C A E D (5) E D C B A (4) E D B A C (4) D E A C B (4) C A B D E (4) B A C E D (4) A B C D E (4) D E C A B (3) C A D E B (3) B C A D E (3) A D C E B (3) A C D E B (3) E D B C A (2) E D A C B (2) E B D C A (2) D A E B C (2) C E D B A (2) C B A E D (2) C B A D E (2) C A E D B (2) B E C D A (2) B A D E C (2) A C B D E (2) E C D A B (1) D A E C B (1) C E D A B (1) B A E D C (1) B A E C D (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 0 6 12 B 0 0 -4 -2 -2 C 0 4 0 2 2 D -6 2 -2 0 -4 E -12 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.351662 B: 0.000000 C: 0.648338 D: 0.000000 E: 0.000000 Sum of squares = 0.544008458858 Cumulative probabilities = A: 0.351662 B: 0.351662 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 6 12 B 0 0 -4 -2 -2 C 0 4 0 2 2 D -6 2 -2 0 -4 E -12 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=27 C=16 A=15 D=10 so D is eliminated. Round 2 votes counts: E=34 B=32 A=18 C=16 so C is eliminated. Round 3 votes counts: E=37 B=36 A=27 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:209 C:204 B:196 E:196 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 6 12 B 0 0 -4 -2 -2 C 0 4 0 2 2 D -6 2 -2 0 -4 E -12 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 6 12 B 0 0 -4 -2 -2 C 0 4 0 2 2 D -6 2 -2 0 -4 E -12 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 6 12 B 0 0 -4 -2 -2 C 0 4 0 2 2 D -6 2 -2 0 -4 E -12 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 484: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) D B C E A (8) A E C D B (8) E A B D C (6) B D C E A (6) A C E D B (6) E A B C D (5) C D B A E (5) E A D B C (4) E A C D B (3) B E D A C (3) A E C B D (3) A C D E B (3) D C A B E (2) D B E C A (2) D B C A E (2) C D A B E (2) C B D A E (2) C A D E B (2) B E A C D (2) B D E C A (2) E D A B C (1) E B D A C (1) E B A D C (1) E B A C D (1) E A D C B (1) E A C B D (1) D C B E A (1) D C A E B (1) C B A D E (1) C A D B E (1) B E D C A (1) B D C A E (1) B A E C D (1) A E D C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 6 4 2 B -4 0 -6 -22 2 C -6 6 0 -4 6 D -4 22 4 0 -2 E -2 -2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 2 B -4 0 -6 -22 2 C -6 6 0 -4 6 D -4 22 4 0 -2 E -2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 A=23 B=16 C=13 so C is eliminated. Round 2 votes counts: D=31 A=26 E=24 B=19 so B is eliminated. Round 3 votes counts: D=42 E=30 A=28 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:210 A:208 C:201 E:196 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 2 B -4 0 -6 -22 2 C -6 6 0 -4 6 D -4 22 4 0 -2 E -2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 2 B -4 0 -6 -22 2 C -6 6 0 -4 6 D -4 22 4 0 -2 E -2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 2 B -4 0 -6 -22 2 C -6 6 0 -4 6 D -4 22 4 0 -2 E -2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 485: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) E D C A B (8) C A E D B (8) C A B E D (8) D E B A C (7) D E B C A (5) B A C D E (5) C E D A B (4) C E A D B (4) A C B E D (4) B A D E C (3) E D A C B (2) E C D A B (2) D B E A C (2) C B A D E (2) B D A C E (2) B C A D E (2) A E D C B (2) A B D C E (2) A B C D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C A D B (1) E A C D B (1) D E A B C (1) D B E C A (1) C A E B D (1) C A B D E (1) B D C E A (1) B D A E C (1) B A D C E (1) A C E B D (1) A C B D E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 2 -4 B -10 0 -2 -2 0 C 0 2 0 -4 0 D -2 2 4 0 0 E 4 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.480726 E: 0.519274 Sum of squares = 0.500742981066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.480726 E: 1.000000 A B C D E A 0 10 0 2 -4 B -10 0 -2 -2 0 C 0 2 0 -4 0 D -2 2 4 0 0 E 4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 E=17 D=16 A=14 so A is eliminated. Round 2 votes counts: C=34 B=31 E=19 D=16 so D is eliminated. Round 3 votes counts: C=34 B=34 E=32 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:204 D:202 E:202 C:199 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 0 2 -4 B -10 0 -2 -2 0 C 0 2 0 -4 0 D -2 2 4 0 0 E 4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 2 -4 B -10 0 -2 -2 0 C 0 2 0 -4 0 D -2 2 4 0 0 E 4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 2 -4 B -10 0 -2 -2 0 C 0 2 0 -4 0 D -2 2 4 0 0 E 4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 486: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) E B D C A (9) D B E A C (9) E C A B D (7) B D E C A (7) A C E D B (7) E C B D A (6) C A E B D (6) C E A B D (5) C A B D E (5) A D B C E (4) E D B A C (3) C E B D A (3) A D C B E (2) E D B C A (1) E B C D A (1) E A D B C (1) E A C D B (1) D B A E C (1) D B A C E (1) C E B A D (1) C B A D E (1) C A E D B (1) B D C E A (1) B D A E C (1) B D A C E (1) B C D E A (1) A E D B C (1) A D B E C (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -10 10 -12 B -6 0 -14 12 -8 C 10 14 0 14 2 D -10 -12 -14 0 -6 E 12 8 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 10 -12 B -6 0 -14 12 -8 C 10 14 0 14 2 D -10 -12 -14 0 -6 E 12 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 C=22 D=11 B=11 so D is eliminated. Round 2 votes counts: E=29 A=27 C=22 B=22 so C is eliminated. Round 3 votes counts: A=39 E=38 B=23 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:220 E:212 A:197 B:192 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 10 -12 B -6 0 -14 12 -8 C 10 14 0 14 2 D -10 -12 -14 0 -6 E 12 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 10 -12 B -6 0 -14 12 -8 C 10 14 0 14 2 D -10 -12 -14 0 -6 E 12 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 10 -12 B -6 0 -14 12 -8 C 10 14 0 14 2 D -10 -12 -14 0 -6 E 12 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 487: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) D C E B A (7) B C E D A (7) A D E C B (6) E C A B D (5) A E C B D (5) E C B D A (4) C D E B A (4) E A C D B (3) D B C A E (3) C E B D A (3) B D C E A (3) A E D C B (3) A E B C D (3) A D B E C (3) A B E C D (3) A B D E C (3) E C A D B (2) D B C E A (2) D A C E B (2) C E D B A (2) C E D A B (2) C D B E A (2) B D C A E (2) B D A C E (2) B C D E A (2) B A D C E (2) E D A C B (1) E C B A D (1) E A C B D (1) D A B C E (1) C B E D A (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -4 0 -4 B -6 0 -24 -4 -26 C 4 24 0 16 -6 D 0 4 -16 0 -10 E 4 26 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 0 -4 B -6 0 -24 -4 -26 C 4 24 0 16 -6 D 0 4 -16 0 -10 E 4 26 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=18 E=17 D=15 C=14 so C is eliminated. Round 2 votes counts: A=36 E=24 D=21 B=19 so B is eliminated. Round 3 votes counts: A=38 E=32 D=30 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:219 A:199 D:189 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 0 -4 B -6 0 -24 -4 -26 C 4 24 0 16 -6 D 0 4 -16 0 -10 E 4 26 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 -4 B -6 0 -24 -4 -26 C 4 24 0 16 -6 D 0 4 -16 0 -10 E 4 26 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 -4 B -6 0 -24 -4 -26 C 4 24 0 16 -6 D 0 4 -16 0 -10 E 4 26 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 488: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E A B D C (8) A B E D C (7) C D E B A (6) E C D B A (5) E A C B D (5) D B A C E (4) C D B E A (4) A B D E C (4) E C A B D (3) E A B C D (3) D C B E A (3) D B C A E (3) B D A C E (3) B A D E C (3) A E B D C (3) E B D A C (2) D C B A E (2) D B E A C (2) C E D B A (2) C E D A B (2) C A B D E (2) B A D C E (2) E D C B A (1) E D B A C (1) E C A D B (1) E B A D C (1) D B A E C (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A E B (1) B D A E C (1) A E B C D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 6 -6 -4 B 10 0 0 2 4 C -6 0 0 -4 -4 D 6 -2 4 0 6 E 4 -4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.739779 C: 0.260221 D: 0.000000 E: 0.000000 Sum of squares = 0.614988185549 Cumulative probabilities = A: 0.000000 B: 0.739779 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -6 -4 B 10 0 0 2 4 C -6 0 0 -4 -4 D 6 -2 4 0 6 E 4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555561718 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=29 A=17 D=15 B=9 so B is eliminated. Round 2 votes counts: E=30 C=29 A=22 D=19 so D is eliminated. Round 3 votes counts: C=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:208 D:207 E:199 A:193 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 -6 -4 B 10 0 0 2 4 C -6 0 0 -4 -4 D 6 -2 4 0 6 E 4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555561718 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -6 -4 B 10 0 0 2 4 C -6 0 0 -4 -4 D 6 -2 4 0 6 E 4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555561718 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -6 -4 B 10 0 0 2 4 C -6 0 0 -4 -4 D 6 -2 4 0 6 E 4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555561718 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 489: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (16) D C E A B (12) B A E D C (11) A B E C D (6) B D E C A (5) D C B E A (4) E C A D B (3) D E C A B (3) D C E B A (3) B E A D C (3) B A C E D (3) E D C A B (2) E A C D B (2) D E B C A (2) D B C E A (2) C D E A B (2) C A E D B (2) B E D A C (2) B D C E A (2) A C E D B (2) D E C B A (1) D B E C A (1) C D A E B (1) C A D E B (1) B D E A C (1) B D C A E (1) B D A C E (1) B A D E C (1) B A D C E (1) A E C D B (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -20 6 10 0 B 20 0 16 12 24 C -6 -16 0 -16 -20 D -10 -12 16 0 -10 E 0 -24 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 10 0 B 20 0 16 12 24 C -6 -16 0 -16 -20 D -10 -12 16 0 -10 E 0 -24 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=47 D=28 A=12 E=7 C=6 so C is eliminated. Round 2 votes counts: B=47 D=31 A=15 E=7 so E is eliminated. Round 3 votes counts: B=47 D=33 A=20 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:236 E:203 A:198 D:192 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 6 10 0 B 20 0 16 12 24 C -6 -16 0 -16 -20 D -10 -12 16 0 -10 E 0 -24 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 10 0 B 20 0 16 12 24 C -6 -16 0 -16 -20 D -10 -12 16 0 -10 E 0 -24 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 10 0 B 20 0 16 12 24 C -6 -16 0 -16 -20 D -10 -12 16 0 -10 E 0 -24 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 490: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) E D C B A (7) A B D C E (6) C E D A B (5) B A D E C (5) D B E A C (4) E C B A D (3) E B C D A (3) D E C B A (3) D E C A B (3) D C E A B (3) B A E C D (3) E C D A B (2) E C B D A (2) D E B C A (2) D B A E C (2) D A C E B (2) C D E A B (2) C A E D B (2) B E D C A (2) B E C D A (2) B A E D C (2) B A D C E (2) B A C E D (2) A D C B E (2) A C B D E (2) A B C E D (2) E B D C A (1) D E A B C (1) D A B E C (1) D A B C E (1) C E D B A (1) C E A B D (1) C D A E B (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E A D (1) A C D E B (1) A C D B E (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -8 -20 -18 B 18 0 -4 -8 -6 C 8 4 0 -6 -22 D 20 8 6 0 -4 E 18 6 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -8 -20 -18 B 18 0 -4 -8 -6 C 8 4 0 -6 -22 D 20 8 6 0 -4 E 18 6 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=23 D=22 A=17 C=12 so C is eliminated. Round 2 votes counts: E=33 D=25 B=23 A=19 so A is eliminated. Round 3 votes counts: B=36 E=35 D=29 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:215 B:200 C:192 A:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -8 -20 -18 B 18 0 -4 -8 -6 C 8 4 0 -6 -22 D 20 8 6 0 -4 E 18 6 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -20 -18 B 18 0 -4 -8 -6 C 8 4 0 -6 -22 D 20 8 6 0 -4 E 18 6 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -20 -18 B 18 0 -4 -8 -6 C 8 4 0 -6 -22 D 20 8 6 0 -4 E 18 6 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 491: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) E B A C D (7) D C A B E (7) A C E B D (6) B E C D A (5) E B D A C (4) D A C B E (4) C A D B E (4) A E C D B (4) A E C B D (4) A D C E B (4) E A B C D (3) D C B A E (3) D A C E B (3) B E D C A (3) B D E C A (3) B C E D A (3) A C D E B (3) E B C A D (2) D A E B C (2) C E B A D (2) B C D E A (2) A D E B C (2) E D A B C (1) E A C B D (1) D E A B C (1) D B E A C (1) D B C E A (1) D B C A E (1) C B E A D (1) C B A E D (1) C A E B D (1) A E D B C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 4 -6 2 B -4 0 4 -2 -4 C -4 -4 0 0 -8 D 6 2 0 0 0 E -2 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.626360 E: 0.373640 Sum of squares = 0.531933470831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.626360 E: 1.000000 A B C D E A 0 4 4 -6 2 B -4 0 4 -2 -4 C -4 -4 0 0 -8 D 6 2 0 0 0 E -2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=25 E=18 B=16 C=9 so C is eliminated. Round 2 votes counts: D=32 A=30 E=20 B=18 so B is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:205 D:204 A:202 B:197 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -6 2 B -4 0 4 -2 -4 C -4 -4 0 0 -8 D 6 2 0 0 0 E -2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -6 2 B -4 0 4 -2 -4 C -4 -4 0 0 -8 D 6 2 0 0 0 E -2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -6 2 B -4 0 4 -2 -4 C -4 -4 0 0 -8 D 6 2 0 0 0 E -2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 492: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) D C A B E (7) A C E B D (6) D C B E A (5) B E A D C (5) E B A C D (4) C E D B A (4) E C B A D (3) E B C D A (3) D B E A C (3) C E B D A (3) C E A B D (3) C A D E B (3) B A E D C (3) A E C B D (3) A B E D C (3) E C B D A (2) E A B C D (2) C A E D B (2) B E D A C (2) B D E A C (2) A E B C D (2) A C B E D (2) A C B D E (2) A B E C D (2) E B D C A (1) E A C B D (1) D C E B A (1) D C B A E (1) D B E C A (1) D B A C E (1) D A B E C (1) D A B C E (1) C D E B A (1) C D E A B (1) C D A E B (1) B D E C A (1) A D B C E (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 14 0 4 B 8 0 -4 14 8 C -14 4 0 4 -4 D 0 -14 -4 0 -14 E -4 -8 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.538462 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.408284023666 Cumulative probabilities = A: 0.153846 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 14 0 4 B 8 0 -4 14 8 C -14 4 0 4 -4 D 0 -14 -4 0 -14 E -4 -8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.538462 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.408284023659 Cumulative probabilities = A: 0.153846 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 C=18 E=16 B=13 so B is eliminated. Round 2 votes counts: D=32 A=27 E=23 C=18 so C is eliminated. Round 3 votes counts: D=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:213 A:205 E:203 C:195 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 14 0 4 B 8 0 -4 14 8 C -14 4 0 4 -4 D 0 -14 -4 0 -14 E -4 -8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.538462 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.408284023659 Cumulative probabilities = A: 0.153846 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 0 4 B 8 0 -4 14 8 C -14 4 0 4 -4 D 0 -14 -4 0 -14 E -4 -8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.538462 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.408284023659 Cumulative probabilities = A: 0.153846 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 0 4 B 8 0 -4 14 8 C -14 4 0 4 -4 D 0 -14 -4 0 -14 E -4 -8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.538462 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.408284023659 Cumulative probabilities = A: 0.153846 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 493: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) B E D A C (7) C A E B D (6) C A D E B (6) C A B E D (5) C A E D B (4) B E C A D (4) B D E A C (4) E B D A C (3) D E B A C (3) D C B A E (3) D A C E B (3) B E A C D (3) E A C B D (2) C B A E D (2) C B A D E (2) C A D B E (2) C A B D E (2) B E D C A (2) B D E C A (2) B C A E D (2) A C E D B (2) A C E B D (2) A C D E B (2) E B A D C (1) E B A C D (1) E A D B C (1) E A B C D (1) D E A B C (1) D C A B E (1) D B E C A (1) D B C A E (1) D A E C B (1) C D A E B (1) C D A B E (1) B E C D A (1) B E A D C (1) B D C E A (1) B D C A E (1) B C E A D (1) B C D E A (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -4 10 4 B 8 0 0 14 14 C 4 0 0 8 2 D -10 -14 -8 0 -6 E -4 -14 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.433732 C: 0.566268 D: 0.000000 E: 0.000000 Sum of squares = 0.508782853552 Cumulative probabilities = A: 0.000000 B: 0.433732 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 10 4 B 8 0 0 14 14 C 4 0 0 8 2 D -10 -14 -8 0 -6 E -4 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=30 D=21 E=9 A=9 so E is eliminated. Round 2 votes counts: B=35 C=31 D=21 A=13 so A is eliminated. Round 3 votes counts: C=40 B=36 D=24 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:207 A:201 E:193 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 10 4 B 8 0 0 14 14 C 4 0 0 8 2 D -10 -14 -8 0 -6 E -4 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 10 4 B 8 0 0 14 14 C 4 0 0 8 2 D -10 -14 -8 0 -6 E -4 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 10 4 B 8 0 0 14 14 C 4 0 0 8 2 D -10 -14 -8 0 -6 E -4 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 494: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) A E C D B (9) D B E A C (8) D B C A E (8) B D C E A (6) D B C E A (5) B D E C A (5) A E C B D (5) A C E D B (5) E B D C A (4) E C A B D (3) D B A E C (3) D B A C E (3) E C B D A (2) E A D B C (2) E A B D C (2) C E B A D (2) C D B A E (2) C B D E A (2) C B D A E (2) A D B E C (2) D B E C A (1) C E B D A (1) C E A B D (1) C A E B D (1) C A D B E (1) C A B D E (1) B D C A E (1) A E D C B (1) A D E B C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 4 -6 -6 B 10 0 2 -4 2 C -4 -2 0 -4 -14 D 6 4 4 0 4 E 6 -2 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.140445 C: 0.000000 D: 0.859555 E: 0.000000 Sum of squares = 0.758559452827 Cumulative probabilities = A: 0.000000 B: 0.140445 C: 0.140445 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -6 -6 B 10 0 2 -4 2 C -4 -2 0 -4 -14 D 6 4 4 0 4 E 6 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=25 E=22 C=13 B=12 so B is eliminated. Round 2 votes counts: D=40 A=25 E=22 C=13 so C is eliminated. Round 3 votes counts: D=46 A=28 E=26 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:207 B:205 A:191 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 4 -6 -6 B 10 0 2 -4 2 C -4 -2 0 -4 -14 D 6 4 4 0 4 E 6 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -6 -6 B 10 0 2 -4 2 C -4 -2 0 -4 -14 D 6 4 4 0 4 E 6 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -6 -6 B 10 0 2 -4 2 C -4 -2 0 -4 -14 D 6 4 4 0 4 E 6 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 495: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (15) A B C E D (10) C E B D A (8) A D C B E (5) C B E A D (4) A B E C D (4) D E B C A (3) D C E B A (3) C B E D A (3) B C A E D (3) A D E B C (3) A D B E C (3) A D B C E (3) E C B D A (2) D A E C B (2) D A E B C (2) B C E A D (2) B A C E D (2) A C D B E (2) A C B E D (2) A B D C E (2) A B C D E (2) E D B C A (1) E C D B A (1) E B C D A (1) D E C A B (1) D E B A C (1) D E A C B (1) D A C E B (1) C D E B A (1) C B A E D (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B A E C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -4 4 -2 B 12 0 -6 2 6 C 4 6 0 6 10 D -4 -2 -6 0 2 E 2 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 4 -2 B 12 0 -6 2 6 C 4 6 0 6 10 D -4 -2 -6 0 2 E 2 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=29 C=17 B=12 E=5 so E is eliminated. Round 2 votes counts: A=37 D=30 C=20 B=13 so B is eliminated. Round 3 votes counts: A=41 D=30 C=29 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:213 B:207 D:195 A:193 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -4 4 -2 B 12 0 -6 2 6 C 4 6 0 6 10 D -4 -2 -6 0 2 E 2 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 4 -2 B 12 0 -6 2 6 C 4 6 0 6 10 D -4 -2 -6 0 2 E 2 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 4 -2 B 12 0 -6 2 6 C 4 6 0 6 10 D -4 -2 -6 0 2 E 2 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 496: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) E B D C A (8) E C D B A (7) D B E A C (7) A B D C E (6) C E D A B (5) D B A E C (4) C A E B D (4) A C B E D (4) C A E D B (3) C A B E D (3) B D E A C (3) A B C D E (3) E D C B A (2) D C A E B (2) C E D B A (2) C E A D B (2) C E A B D (2) B E D A C (2) B A E D C (2) A C D B E (2) A B D E C (2) E D B C A (1) E B D A C (1) D E C B A (1) D E B C A (1) D E B A C (1) C E B D A (1) C D E A B (1) B E A C D (1) B D A E C (1) B A D E C (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 8 10 0 4 B -8 0 -8 12 12 C -10 8 0 4 8 D 0 -12 -4 0 0 E -4 -12 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.705617 B: 0.000000 C: 0.000000 D: 0.294383 E: 0.000000 Sum of squares = 0.584557077517 Cumulative probabilities = A: 0.705617 B: 0.705617 C: 0.705617 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 0 4 B -8 0 -8 12 12 C -10 8 0 4 8 D 0 -12 -4 0 0 E -4 -12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000668 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=23 E=19 D=16 B=10 so B is eliminated. Round 2 votes counts: A=35 C=23 E=22 D=20 so D is eliminated. Round 3 votes counts: A=40 E=35 C=25 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:205 B:204 D:192 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 0 4 B -8 0 -8 12 12 C -10 8 0 4 8 D 0 -12 -4 0 0 E -4 -12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000668 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 4 B -8 0 -8 12 12 C -10 8 0 4 8 D 0 -12 -4 0 0 E -4 -12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000668 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 4 B -8 0 -8 12 12 C -10 8 0 4 8 D 0 -12 -4 0 0 E -4 -12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000668 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 497: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (11) A E D B C (9) C B A E D (8) B C D A E (8) D A E B C (5) E D A C B (4) E A D C B (4) D E A B C (3) C E D B A (3) A D B E C (3) A B C E D (3) E A D B C (2) E A C D B (2) C B E A D (2) B D C A E (2) B D A C E (2) B C A E D (2) B C A D E (2) A B E C D (2) E D C A B (1) E D A B C (1) E C D B A (1) E C B A D (1) E A C B D (1) D E C A B (1) D E A C B (1) D C E B A (1) D C B E A (1) D B C E A (1) C E B D A (1) C D B E A (1) C B E D A (1) C B D A E (1) C B A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D C B (1) A E C B D (1) A E B C D (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 0 12 B 6 0 0 10 12 C 0 0 0 12 10 D 0 -10 -12 0 -6 E -12 -12 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.482285 C: 0.517715 D: 0.000000 E: 0.000000 Sum of squares = 0.500627642931 Cumulative probabilities = A: 0.000000 B: 0.482285 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 0 12 B 6 0 0 10 12 C 0 0 0 12 10 D 0 -10 -12 0 -6 E -12 -12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=22 B=19 E=17 D=13 so D is eliminated. Round 2 votes counts: C=31 A=27 E=22 B=20 so B is eliminated. Round 3 votes counts: C=46 A=32 E=22 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:214 C:211 A:203 D:186 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 0 12 B 6 0 0 10 12 C 0 0 0 12 10 D 0 -10 -12 0 -6 E -12 -12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 12 B 6 0 0 10 12 C 0 0 0 12 10 D 0 -10 -12 0 -6 E -12 -12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 12 B 6 0 0 10 12 C 0 0 0 12 10 D 0 -10 -12 0 -6 E -12 -12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 498: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (9) E C B A D (7) D B A C E (7) D A B C E (7) A C D E B (7) E C A B D (6) D A C E B (3) C A E D B (3) B E D A C (3) B E C A D (3) B D E A C (3) D B A E C (2) C A D E B (2) B E D C A (2) B E C D A (2) B E A C D (2) B D E C A (2) A D C E B (2) A B E D C (2) E C B D A (1) E B C A D (1) E B A C D (1) E A C B D (1) D C A B E (1) D B C E A (1) D B C A E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D E A B (1) B A D E C (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 0 6 6 0 B 0 0 0 -6 0 C -6 0 0 4 4 D -6 6 -4 0 2 E 0 0 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.486736 B: 0.204653 C: 0.000000 D: 0.000000 E: 0.308611 Sum of squares = 0.37403540047 Cumulative probabilities = A: 0.486736 B: 0.691389 C: 0.691389 D: 0.691389 E: 1.000000 A B C D E A 0 0 6 6 0 B 0 0 0 -6 0 C -6 0 0 4 4 D -6 6 -4 0 2 E 0 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.392857 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.321429 Sum of squares = 0.339285718759 Cumulative probabilities = A: 0.392857 B: 0.678571 C: 0.678571 D: 0.678571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 C=20 E=17 A=14 so A is eliminated. Round 2 votes counts: B=29 C=28 D=26 E=17 so E is eliminated. Round 3 votes counts: C=43 B=31 D=26 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:206 C:201 D:199 B:197 E:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 6 0 B 0 0 0 -6 0 C -6 0 0 4 4 D -6 6 -4 0 2 E 0 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.392857 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.321429 Sum of squares = 0.339285718759 Cumulative probabilities = A: 0.392857 B: 0.678571 C: 0.678571 D: 0.678571 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 0 B 0 0 0 -6 0 C -6 0 0 4 4 D -6 6 -4 0 2 E 0 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.392857 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.321429 Sum of squares = 0.339285718759 Cumulative probabilities = A: 0.392857 B: 0.678571 C: 0.678571 D: 0.678571 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 0 B 0 0 0 -6 0 C -6 0 0 4 4 D -6 6 -4 0 2 E 0 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.392857 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.321429 Sum of squares = 0.339285718759 Cumulative probabilities = A: 0.392857 B: 0.678571 C: 0.678571 D: 0.678571 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 499: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (20) D E B C A (10) E D B C A (8) E D B A C (7) C A B D E (7) E D A C B (5) E D A B C (5) C B A D E (5) B C A D E (5) B C D E A (4) A E D C B (3) A C B D E (3) E A D C B (2) B D C E A (2) B C D A E (2) A C E D B (2) A C E B D (2) E A D B C (1) D E C A B (1) D B E C A (1) C B A E D (1) B E D C A (1) B D E C A (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 6 4 6 4 B -6 0 -6 8 4 C -4 6 0 6 6 D -6 -8 -6 0 -18 E -4 -4 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 6 4 B -6 0 -6 8 4 C -4 6 0 6 6 D -6 -8 -6 0 -18 E -4 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=28 B=15 C=13 D=12 so D is eliminated. Round 2 votes counts: E=39 A=32 B=16 C=13 so C is eliminated. Round 3 votes counts: E=39 A=39 B=22 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:207 E:202 B:200 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 6 4 B -6 0 -6 8 4 C -4 6 0 6 6 D -6 -8 -6 0 -18 E -4 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 6 4 B -6 0 -6 8 4 C -4 6 0 6 6 D -6 -8 -6 0 -18 E -4 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 6 4 B -6 0 -6 8 4 C -4 6 0 6 6 D -6 -8 -6 0 -18 E -4 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 500: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (11) D A C E B (10) E B A D C (5) E B A C D (5) D C A B E (5) C D A B E (5) B E C A D (5) B C E A D (5) E A B C D (3) D A E C B (3) B C A E D (3) B C A D E (3) A D C E B (3) E D A B C (2) E B C A D (2) E A D C B (2) E A D B C (2) E A C B D (2) E A B D C (2) C A D B E (2) B E C D A (2) B C E D A (2) A E D C B (2) E D A C B (1) E B D C A (1) D E B A C (1) D E A C B (1) D C B A E (1) D A C B E (1) C D B A E (1) C B D A E (1) C A B D E (1) B D C A E (1) B C D E A (1) A E C D B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -4 0 12 B 0 0 12 10 0 C 4 -12 0 14 16 D 0 -10 -14 0 4 E -12 0 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.446833 B: 0.553167 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.505653479514 Cumulative probabilities = A: 0.446833 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 0 12 B 0 0 12 10 0 C 4 -12 0 14 16 D 0 -10 -14 0 4 E -12 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=27 D=22 C=10 A=8 so A is eliminated. Round 2 votes counts: B=33 E=30 D=25 C=12 so C is eliminated. Round 3 votes counts: B=35 D=34 E=31 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:211 A:204 D:190 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 0 12 B 0 0 12 10 0 C 4 -12 0 14 16 D 0 -10 -14 0 4 E -12 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 0 12 B 0 0 12 10 0 C 4 -12 0 14 16 D 0 -10 -14 0 4 E -12 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 0 12 B 0 0 12 10 0 C 4 -12 0 14 16 D 0 -10 -14 0 4 E -12 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 501: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) A B E D C (9) E A C D B (7) E C D B A (6) C D E B A (6) C D B E A (6) A E B D C (6) E C D A B (4) D C B E A (4) B A D C E (4) A E C D B (4) A E B C D (4) A B D C E (4) E A B C D (3) D C B A E (3) E B A D C (2) B D C E A (2) A D C B E (2) A C D B E (2) E B D C A (1) D C E B A (1) D C A B E (1) D B C A E (1) C E D B A (1) C E D A B (1) C D E A B (1) C D B A E (1) C D A B E (1) B E D C A (1) B D A C E (1) B A E D C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 0 -2 8 B 0 0 -4 -6 4 C 0 4 0 -4 4 D 2 6 4 0 0 E -8 -4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.875674 E: 0.124326 Sum of squares = 0.78226256611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.875674 E: 1.000000 A B C D E A 0 0 0 -2 8 B 0 0 -4 -6 4 C 0 4 0 -4 4 D 2 6 4 0 0 E -8 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=23 B=18 C=17 D=10 so D is eliminated. Round 2 votes counts: A=32 C=26 E=23 B=19 so B is eliminated. Round 3 votes counts: C=38 A=38 E=24 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:206 A:203 C:202 B:197 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -2 8 B 0 0 -4 -6 4 C 0 4 0 -4 4 D 2 6 4 0 0 E -8 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 8 B 0 0 -4 -6 4 C 0 4 0 -4 4 D 2 6 4 0 0 E -8 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 8 B 0 0 -4 -6 4 C 0 4 0 -4 4 D 2 6 4 0 0 E -8 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 502: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) E D C A B (8) B A C D E (8) A E B C D (8) D C E B A (7) A B E C D (7) C D E B A (6) E A B C D (4) D C B A E (4) B C D A E (4) A E B D C (4) C D B E A (3) B A D C E (3) A B E D C (3) E C D A B (2) D B C A E (2) C D B A E (2) C B D A E (2) B A D E C (2) E D A C B (1) E C A B D (1) E A D C B (1) E A C B D (1) D E C A B (1) D B A C E (1) C E D A B (1) C D E A B (1) C B A E D (1) C B A D E (1) B D C A E (1) B A C E D (1) Total count = 100 A B C D E A 0 -14 -12 -10 8 B 14 0 -4 2 8 C 12 4 0 6 14 D 10 -2 -6 0 14 E -8 -8 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -10 8 B 14 0 -4 2 8 C 12 4 0 6 14 D 10 -2 -6 0 14 E -8 -8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 B=19 E=18 C=17 so C is eliminated. Round 2 votes counts: D=36 B=23 A=22 E=19 so E is eliminated. Round 3 votes counts: D=48 A=29 B=23 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:218 B:210 D:208 A:186 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -12 -10 8 B 14 0 -4 2 8 C 12 4 0 6 14 D 10 -2 -6 0 14 E -8 -8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -10 8 B 14 0 -4 2 8 C 12 4 0 6 14 D 10 -2 -6 0 14 E -8 -8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -10 8 B 14 0 -4 2 8 C 12 4 0 6 14 D 10 -2 -6 0 14 E -8 -8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 503: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (28) C D B A E (17) E A B C D (7) D C B A E (7) C E D A B (7) E C A B D (5) C E A B D (4) B A D E C (4) D B A E C (3) D B A C E (3) C D E B A (3) A E B D C (3) D B C A E (2) B A D C E (2) A B E D C (2) E C A D B (1) C D B E A (1) B D A E C (1) Total count = 100 A B C D E A 0 14 6 12 -12 B -14 0 10 12 -16 C -6 -10 0 -10 -8 D -12 -12 10 0 -14 E 12 16 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 6 12 -12 B -14 0 10 12 -16 C -6 -10 0 -10 -8 D -12 -12 10 0 -14 E 12 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=32 D=15 B=7 A=5 so A is eliminated. Round 2 votes counts: E=44 C=32 D=15 B=9 so B is eliminated. Round 3 votes counts: E=46 C=32 D=22 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:210 B:196 D:186 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 6 12 -12 B -14 0 10 12 -16 C -6 -10 0 -10 -8 D -12 -12 10 0 -14 E 12 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 12 -12 B -14 0 10 12 -16 C -6 -10 0 -10 -8 D -12 -12 10 0 -14 E 12 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 12 -12 B -14 0 10 12 -16 C -6 -10 0 -10 -8 D -12 -12 10 0 -14 E 12 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 504: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) A B C E D (11) A D E B C (6) A B E D C (6) B A C E D (5) D E C A B (4) A C D E B (4) E D C B A (3) E D B C A (3) D E A C B (3) D C E A B (3) C E D B A (3) C B E D A (3) B C E A D (3) A D E C B (3) D E B C A (2) D C E B A (2) C D E B A (2) C B A E D (2) B E D A C (2) B E C D A (2) B C E D A (2) A C B E D (2) A B D E C (2) E D B A C (1) E B D C A (1) E B C D A (1) D A E C B (1) D A C E B (1) C A B E D (1) C A B D E (1) B E A D C (1) B A E C D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 0 -4 B 0 0 0 -4 -8 C -2 0 0 -10 -8 D 0 4 10 0 -8 E 4 8 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 0 -4 B 0 0 0 -4 -8 C -2 0 0 -10 -8 D 0 4 10 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=27 B=16 C=12 E=9 so E is eliminated. Round 2 votes counts: A=36 D=34 B=18 C=12 so C is eliminated. Round 3 votes counts: D=39 A=38 B=23 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:214 D:203 A:199 B:194 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 0 -4 B 0 0 0 -4 -8 C -2 0 0 -10 -8 D 0 4 10 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 0 -4 B 0 0 0 -4 -8 C -2 0 0 -10 -8 D 0 4 10 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 0 -4 B 0 0 0 -4 -8 C -2 0 0 -10 -8 D 0 4 10 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 505: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (8) A C D E B (7) B E A C D (6) D C B E A (5) D C A B E (5) B E D C A (5) A E B C D (5) E B A C D (4) B E C D A (4) E A B D C (3) E A B C D (3) C D B A E (3) A C D B E (3) E B D A C (2) E A D B C (2) D C E B A (2) D C A E B (2) B C D E A (2) A E D B C (2) A E C B D (2) A D C E B (2) A B E C D (2) E D B A C (1) D E B C A (1) D A C E B (1) C D B E A (1) C B D E A (1) C A B D E (1) B C E A D (1) B C A E D (1) A E B D C (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 16 14 -4 B -4 0 8 10 0 C -16 -8 0 14 -4 D -14 -10 -14 0 -10 E 4 0 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.385432 C: 0.000000 D: 0.000000 E: 0.614568 Sum of squares = 0.526251726614 Cumulative probabilities = A: 0.000000 B: 0.385432 C: 0.385432 D: 0.385432 E: 1.000000 A B C D E A 0 4 16 14 -4 B -4 0 8 10 0 C -16 -8 0 14 -4 D -14 -10 -14 0 -10 E 4 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000024872 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=24 B=19 D=16 C=14 so C is eliminated. Round 2 votes counts: D=28 A=28 E=24 B=20 so B is eliminated. Round 3 votes counts: E=40 D=31 A=29 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:215 E:209 B:207 C:193 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 16 14 -4 B -4 0 8 10 0 C -16 -8 0 14 -4 D -14 -10 -14 0 -10 E 4 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000024872 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 14 -4 B -4 0 8 10 0 C -16 -8 0 14 -4 D -14 -10 -14 0 -10 E 4 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000024872 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 14 -4 B -4 0 8 10 0 C -16 -8 0 14 -4 D -14 -10 -14 0 -10 E 4 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000024872 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 506: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) D A B C E (7) C E B D A (7) E B A C D (6) C E B A D (6) B A E C D (6) A B E D C (5) A B D E C (5) D A B E C (4) A D B E C (4) A B E C D (4) E C B D A (3) E C B A D (3) D E C A B (3) D A E B C (3) E D A B C (2) E B C A D (2) D C E A B (2) D C A B E (2) D A C B E (2) C D B A E (2) C B E A D (2) D E A B C (1) D C E B A (1) D C A E B (1) D A C E B (1) C D E B A (1) C B D A E (1) B E C A D (1) B E A C D (1) B A C E D (1) A E D B C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 10 -2 2 B 2 0 10 10 -4 C -10 -10 0 10 -10 D 2 -10 -10 0 -20 E -2 4 10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 10 -2 2 B 2 0 10 10 -4 C -10 -10 0 10 -10 D 2 -10 -10 0 -20 E -2 4 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999498 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 A=21 E=16 B=9 so B is eliminated. Round 2 votes counts: A=28 D=27 C=27 E=18 so E is eliminated. Round 3 votes counts: C=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:216 B:209 A:204 C:190 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 10 -2 2 B 2 0 10 10 -4 C -10 -10 0 10 -10 D 2 -10 -10 0 -20 E -2 4 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999498 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -2 2 B 2 0 10 10 -4 C -10 -10 0 10 -10 D 2 -10 -10 0 -20 E -2 4 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999498 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -2 2 B 2 0 10 10 -4 C -10 -10 0 10 -10 D 2 -10 -10 0 -20 E -2 4 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999498 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 507: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (16) D E A C B (15) D E A B C (13) C B A E D (11) A D E C B (8) B C D A E (4) D E B A C (3) C B A D E (3) B C A D E (3) D B E C A (2) B D E C A (2) B D C E A (2) B C D E A (2) A C E D B (2) A C E B D (2) E D A C B (1) E A D C B (1) E A D B C (1) D B E A C (1) D B C E A (1) D A E C B (1) C A B E D (1) B D E A C (1) B C E D A (1) B C E A D (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 2 2 6 B 6 0 6 -2 2 C -2 -6 0 -6 -2 D -2 2 6 0 22 E -6 -2 2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000175 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 2 6 B 6 0 6 -2 2 C -2 -6 0 -6 -2 D -2 2 6 0 22 E -6 -2 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999996 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=32 C=15 A=14 E=3 so E is eliminated. Round 2 votes counts: D=37 B=32 A=16 C=15 so C is eliminated. Round 3 votes counts: B=46 D=37 A=17 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:206 A:202 C:192 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 2 6 B 6 0 6 -2 2 C -2 -6 0 -6 -2 D -2 2 6 0 22 E -6 -2 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999996 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 6 B 6 0 6 -2 2 C -2 -6 0 -6 -2 D -2 2 6 0 22 E -6 -2 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999996 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 6 B 6 0 6 -2 2 C -2 -6 0 -6 -2 D -2 2 6 0 22 E -6 -2 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999996 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 508: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (7) B D A C E (7) D B A C E (6) C E A D B (6) A D C E B (6) E C A D B (5) B E C D A (4) E B A C D (3) C E D A B (3) B E D A C (3) B E C A D (3) E C B A D (2) E B C A D (2) E A C B D (2) D B C A E (2) D A C B E (2) D A B C E (2) B E D C A (2) B D E C A (2) B D C E A (2) B D C A E (2) B D A E C (2) B C E D A (2) B C D E A (2) B A E D C (2) A E C D B (2) A D B C E (2) A C D E B (2) E A C D B (1) E A B C D (1) D C A E B (1) D C A B E (1) D A C E B (1) C D E A B (1) C D B E A (1) B E A C D (1) B D E A C (1) A E B D C (1) A D B E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 0 2 -12 B 2 0 12 8 6 C 0 -12 0 2 4 D -2 -8 -2 0 -6 E 12 -6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 -12 B 2 0 12 8 6 C 0 -12 0 2 4 D -2 -8 -2 0 -6 E 12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=23 A=16 D=15 C=11 so C is eliminated. Round 2 votes counts: B=35 E=32 D=17 A=16 so A is eliminated. Round 3 votes counts: E=36 B=36 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:204 C:197 A:194 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 2 -12 B 2 0 12 8 6 C 0 -12 0 2 4 D -2 -8 -2 0 -6 E 12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 -12 B 2 0 12 8 6 C 0 -12 0 2 4 D -2 -8 -2 0 -6 E 12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 -12 B 2 0 12 8 6 C 0 -12 0 2 4 D -2 -8 -2 0 -6 E 12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 509: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) B A D C E (7) E C A D B (6) A C E B D (6) E C A B D (5) C E B A D (5) D E C A B (4) D B A E C (4) B D A C E (4) A E C D B (4) E C D A B (3) D B E A C (3) B D C E A (3) A B D C E (3) E C D B A (2) E C B D A (2) D E C B A (2) D E B C A (2) D B A C E (2) D A E C B (2) D A B E C (2) C E B D A (2) B C E D A (2) B C E A D (2) B A C E D (2) A D B C E (2) E C B A D (1) D E B A C (1) D E A C B (1) D B E C A (1) C B E A D (1) B A C D E (1) A E C B D (1) A D E C B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -4 16 -14 B -2 0 -16 16 -18 C 4 16 0 12 6 D -16 -16 -12 0 -10 E 14 18 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 16 -14 B -2 0 -16 16 -18 C 4 16 0 12 6 D -16 -16 -12 0 -10 E 14 18 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=21 E=19 A=19 C=17 so C is eliminated. Round 2 votes counts: E=35 D=24 B=22 A=19 so A is eliminated. Round 3 votes counts: E=46 D=27 B=27 so D is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:218 A:200 B:190 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 16 -14 B -2 0 -16 16 -18 C 4 16 0 12 6 D -16 -16 -12 0 -10 E 14 18 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 16 -14 B -2 0 -16 16 -18 C 4 16 0 12 6 D -16 -16 -12 0 -10 E 14 18 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 16 -14 B -2 0 -16 16 -18 C 4 16 0 12 6 D -16 -16 -12 0 -10 E 14 18 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 510: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (7) B A E D C (7) D B A E C (6) B D A E C (6) A E B D C (6) D C B E A (5) D C A E B (5) C E A B D (5) E A C B D (4) D A B E C (4) A E B C D (4) E A B C D (3) B E A C D (3) B A E C D (3) B A D E C (3) E C A B D (2) D C E B A (2) D C A B E (2) C E B A D (2) C E A D B (2) C D B E A (2) B E C A D (2) A B E D C (2) E B C A D (1) D C E A B (1) D C B A E (1) D B C A E (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C B E (1) C E D A B (1) C D E B A (1) A E D C B (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 8 16 4 14 B -8 0 8 6 0 C -16 -8 0 -16 -22 D -4 -6 16 0 4 E -14 0 22 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 4 14 B -8 0 8 6 0 C -16 -8 0 -16 -22 D -4 -6 16 0 4 E -14 0 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=24 C=20 A=15 E=10 so E is eliminated. Round 2 votes counts: D=31 B=25 C=22 A=22 so C is eliminated. Round 3 votes counts: D=42 A=31 B=27 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:205 B:203 E:202 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 4 14 B -8 0 8 6 0 C -16 -8 0 -16 -22 D -4 -6 16 0 4 E -14 0 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 4 14 B -8 0 8 6 0 C -16 -8 0 -16 -22 D -4 -6 16 0 4 E -14 0 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 4 14 B -8 0 8 6 0 C -16 -8 0 -16 -22 D -4 -6 16 0 4 E -14 0 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 511: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) B A C E D (6) E D C B A (5) E C D B A (5) C D E A B (5) B A E C D (5) B E A C D (4) B A E D C (4) A B D C E (4) E D C A B (3) D E C A B (3) D E A B C (3) D A E B C (3) D A B E C (3) C B E A D (3) C B A D E (3) B C A E D (3) E D A B C (2) E B D A C (2) E B C A D (2) E B A D C (2) D E A C B (2) B C E A D (2) B A C D E (2) A B D E C (2) E B A C D (1) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C D A B E (1) C A B D E (1) B A D E C (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 6 -2 -8 B 18 0 8 14 0 C -6 -8 0 8 -18 D 2 -14 -8 0 -16 E 8 0 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.501086 C: 0.000000 D: 0.000000 E: 0.498914 Sum of squares = 0.500002360204 Cumulative probabilities = A: 0.000000 B: 0.501086 C: 0.501086 D: 0.501086 E: 1.000000 A B C D E A 0 -18 6 -2 -8 B 18 0 8 14 0 C -6 -8 0 8 -18 D 2 -14 -8 0 -16 E 8 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 D=20 C=16 A=9 so A is eliminated. Round 2 votes counts: B=34 E=28 D=21 C=17 so C is eliminated. Round 3 votes counts: B=42 E=31 D=27 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:221 B:220 A:189 C:188 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 -2 -8 B 18 0 8 14 0 C -6 -8 0 8 -18 D 2 -14 -8 0 -16 E 8 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 -2 -8 B 18 0 8 14 0 C -6 -8 0 8 -18 D 2 -14 -8 0 -16 E 8 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 -2 -8 B 18 0 8 14 0 C -6 -8 0 8 -18 D 2 -14 -8 0 -16 E 8 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 512: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (17) B D E C A (12) C A D B E (9) A C D B E (7) E B D A C (6) B D C A E (6) E A B C D (5) E B A D C (3) E A C B D (3) D C B A E (3) D B C A E (3) A E C D B (3) E D B C A (2) E B A C D (2) D B C E A (2) C D A B E (2) B E D C A (2) B E D A C (2) B D C E A (2) E C A D B (1) E A C D B (1) E A B D C (1) D C A B E (1) D B E C A (1) A E B C D (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -26 -26 -22 -24 B 26 0 34 26 4 C 26 -34 0 -26 -24 D 22 -26 26 0 -2 E 24 -4 24 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -26 -22 -24 B 26 0 34 26 4 C 26 -34 0 -26 -24 D 22 -26 26 0 -2 E 24 -4 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972145 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=24 A=14 C=11 D=10 so D is eliminated. Round 2 votes counts: E=41 B=30 C=15 A=14 so A is eliminated. Round 3 votes counts: E=45 B=30 C=25 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:245 E:223 D:210 C:171 A:151 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -26 -22 -24 B 26 0 34 26 4 C 26 -34 0 -26 -24 D 22 -26 26 0 -2 E 24 -4 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972145 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -26 -22 -24 B 26 0 34 26 4 C 26 -34 0 -26 -24 D 22 -26 26 0 -2 E 24 -4 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972145 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -26 -22 -24 B 26 0 34 26 4 C 26 -34 0 -26 -24 D 22 -26 26 0 -2 E 24 -4 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972145 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 513: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) C A B E D (7) D E A B C (6) B C E D A (6) E D B A C (5) C B A E D (5) B D E C A (5) A C E D B (5) D E B A C (4) D B E A C (4) B E D C A (4) B C D E A (4) A E D C B (4) E B D C A (3) C B E A D (3) C A E B D (3) A D C E B (3) E D A B C (2) E A D C B (2) C A B D E (2) A C D E B (2) E B C D A (1) D A E C B (1) D A E B C (1) D A B E C (1) C B E D A (1) C B A D E (1) B E C D A (1) B C D A E (1) B C A D E (1) B A C D E (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 4 0 -2 B 0 0 0 -2 -4 C -4 0 0 -8 -8 D 0 2 8 0 -6 E 2 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 0 -2 B 0 0 0 -2 -4 C -4 0 0 -8 -8 D 0 2 8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=23 C=22 D=17 E=13 so E is eliminated. Round 2 votes counts: B=27 A=27 D=24 C=22 so C is eliminated. Round 3 votes counts: A=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:210 D:202 A:201 B:197 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 0 -2 B 0 0 0 -2 -4 C -4 0 0 -8 -8 D 0 2 8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 -2 B 0 0 0 -2 -4 C -4 0 0 -8 -8 D 0 2 8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 -2 B 0 0 0 -2 -4 C -4 0 0 -8 -8 D 0 2 8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 514: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) E B A C D (9) D A C B E (8) E B C A D (7) E D B C A (6) A D C B E (6) E B D C A (5) D C E B A (5) A B C E D (5) E C B D A (3) D E C B A (3) A C D B E (3) E B D A C (2) D E B C A (2) D C A E B (2) D C A B E (2) B E A C D (2) B C E A D (2) A C B E D (2) A B E C D (2) D E C A B (1) D E B A C (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D A B E (1) C A D B E (1) C A B E D (1) B E C A D (1) A E B D C (1) A D B E C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -20 -8 -10 -22 B 20 0 14 8 -22 C 8 -14 0 4 -14 D 10 -8 -4 0 -18 E 22 22 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -8 -10 -22 B 20 0 14 8 -22 C 8 -14 0 4 -14 D 10 -8 -4 0 -18 E 22 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 D=26 A=22 C=5 B=5 so C is eliminated. Round 2 votes counts: E=43 D=28 A=24 B=5 so B is eliminated. Round 3 votes counts: E=48 D=28 A=24 so A is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:238 B:210 C:192 D:190 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -8 -10 -22 B 20 0 14 8 -22 C 8 -14 0 4 -14 D 10 -8 -4 0 -18 E 22 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -10 -22 B 20 0 14 8 -22 C 8 -14 0 4 -14 D 10 -8 -4 0 -18 E 22 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -10 -22 B 20 0 14 8 -22 C 8 -14 0 4 -14 D 10 -8 -4 0 -18 E 22 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 515: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (13) B E D C A (13) C D A B E (9) A C D E B (8) E A B C D (5) E B A C D (4) E A B D C (4) C D B A E (4) B D C E A (4) E A C D B (3) D C A B E (3) C D A E B (3) B D C A E (3) E B C D A (2) D C B A E (2) C D B E A (2) A D C E B (2) A C E D B (2) E C A D B (1) E B D C A (1) C E D A B (1) C D E A B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D A E C (1) B D A C E (1) B C E D A (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -2 -4 -14 B 6 0 12 10 -10 C 2 -12 0 -4 -6 D 4 -10 4 0 -12 E 14 10 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 -4 -14 B 6 0 12 10 -10 C 2 -12 0 -4 -6 D 4 -10 4 0 -12 E 14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=25 C=21 A=16 D=5 so D is eliminated. Round 2 votes counts: E=33 C=26 B=25 A=16 so A is eliminated. Round 3 votes counts: C=38 E=37 B=25 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:209 D:193 C:190 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 -14 B 6 0 12 10 -10 C 2 -12 0 -4 -6 D 4 -10 4 0 -12 E 14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 -14 B 6 0 12 10 -10 C 2 -12 0 -4 -6 D 4 -10 4 0 -12 E 14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 -14 B 6 0 12 10 -10 C 2 -12 0 -4 -6 D 4 -10 4 0 -12 E 14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 516: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) E A B D C (7) E B A D C (6) C E D B A (6) C A E D B (5) E C B D A (4) C D E B A (4) C D B E A (4) E B D A C (3) D C B E A (3) B E D A C (3) B D A E C (3) A D B C E (3) A C D B E (3) A B E D C (3) A B D E C (3) E B D C A (2) D B E C A (2) D B C E A (2) C D B A E (2) B A E D C (2) A E B C D (2) A D C B E (2) A B D C E (2) E D B C A (1) E C D B A (1) E B C A D (1) E A C B D (1) E A B C D (1) D E B C A (1) D C B A E (1) D B A E C (1) C E D A B (1) C D A E B (1) C A E B D (1) C A D B E (1) B E A D C (1) A E C B D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 14 10 -8 B 6 0 14 10 -18 C -14 -14 0 -18 -14 D -10 -10 18 0 -22 E 8 18 14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 14 10 -8 B 6 0 14 10 -18 C -14 -14 0 -18 -14 D -10 -10 18 0 -22 E 8 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 C=25 D=10 B=9 so B is eliminated. Round 2 votes counts: E=31 A=31 C=25 D=13 so D is eliminated. Round 3 votes counts: A=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:206 A:205 D:188 C:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 14 10 -8 B 6 0 14 10 -18 C -14 -14 0 -18 -14 D -10 -10 18 0 -22 E 8 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 10 -8 B 6 0 14 10 -18 C -14 -14 0 -18 -14 D -10 -10 18 0 -22 E 8 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 10 -8 B 6 0 14 10 -18 C -14 -14 0 -18 -14 D -10 -10 18 0 -22 E 8 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 517: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) A C E D B (7) D B C E A (6) D B A C E (6) C A E B D (6) D A B E C (5) C E A B D (4) A E C D B (4) A C E B D (4) A C D E B (4) E C A B D (3) E A C B D (3) D B C A E (3) D B A E C (3) B E C D A (3) B E C A D (3) B D E C A (3) A E C B D (3) A D C E B (3) D A E C B (2) D A C E B (2) C E B A D (2) B E D C A (2) B C E D A (2) A E D C B (2) E B C A D (1) E B A D C (1) D B E A C (1) D A C B E (1) D A B C E (1) C B E D A (1) B D C E A (1) B C E A D (1) Total count = 100 A B C D E A 0 8 4 2 12 B -8 0 -2 -14 -2 C -4 2 0 2 8 D -2 14 -2 0 -4 E -12 2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 2 12 B -8 0 -2 -14 -2 C -4 2 0 2 8 D -2 14 -2 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=27 B=15 C=13 E=8 so E is eliminated. Round 2 votes counts: D=37 A=30 B=17 C=16 so C is eliminated. Round 3 votes counts: A=43 D=37 B=20 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:204 D:203 E:193 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 2 12 B -8 0 -2 -14 -2 C -4 2 0 2 8 D -2 14 -2 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 12 B -8 0 -2 -14 -2 C -4 2 0 2 8 D -2 14 -2 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 12 B -8 0 -2 -14 -2 C -4 2 0 2 8 D -2 14 -2 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 518: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) C A D E B (8) B E D A C (7) B D E A C (5) E D B A C (4) E C A D B (4) E B D A C (4) D A C B E (4) A D C B E (4) A C D E B (4) E D A B C (3) E C A B D (3) E B D C A (3) D A C E B (3) C A E D B (3) A D C E B (3) E B C D A (2) E B C A D (2) C E A D B (2) C A B D E (2) B E C D A (2) B E C A D (2) B C A E D (2) A D E C B (2) E A D C B (1) E A C D B (1) D E B A C (1) D B A E C (1) C B A D E (1) B E D C A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 16 -2 12 2 B -16 0 -12 -18 -6 C 2 12 0 4 2 D -12 18 -4 0 4 E -2 6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 12 2 B -16 0 -12 -18 -6 C 2 12 0 4 2 D -12 18 -4 0 4 E -2 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=27 C=27 B=24 A=13 D=9 so D is eliminated. Round 2 votes counts: E=28 C=27 B=25 A=20 so A is eliminated. Round 3 votes counts: C=45 E=30 B=25 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:210 D:203 E:199 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -2 12 2 B -16 0 -12 -18 -6 C 2 12 0 4 2 D -12 18 -4 0 4 E -2 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 12 2 B -16 0 -12 -18 -6 C 2 12 0 4 2 D -12 18 -4 0 4 E -2 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 12 2 B -16 0 -12 -18 -6 C 2 12 0 4 2 D -12 18 -4 0 4 E -2 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 519: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) C D E A B (6) B A C D E (6) D E C B A (5) B A E C D (5) A B E C D (5) D C E B A (4) B E A D C (4) B C A D E (4) B A E D C (4) E D C A B (3) E D B C A (3) B E D A C (3) B D C E A (3) E D A C B (2) E B D A C (2) C D E B A (2) C D B E A (2) C D A B E (2) B E D C A (2) B C D E A (2) B C D A E (2) B A C E D (2) A E B D C (2) A C E D B (2) A C D E B (2) E D C B A (1) E D B A C (1) E D A B C (1) E B D C A (1) E B A D C (1) E A D C B (1) D C E A B (1) D C B E A (1) C D A E B (1) C B A D E (1) C A D B E (1) A E C D B (1) A E B C D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -6 -12 -14 B 22 0 12 4 2 C 6 -12 0 -4 -10 D 12 -4 4 0 4 E 14 -2 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 -12 -14 B 22 0 12 4 2 C 6 -12 0 -4 -10 D 12 -4 4 0 4 E 14 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=17 E=16 C=15 A=15 so C is eliminated. Round 2 votes counts: B=38 D=30 E=16 A=16 so E is eliminated. Round 3 votes counts: B=42 D=41 A=17 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:209 D:208 C:190 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -6 -12 -14 B 22 0 12 4 2 C 6 -12 0 -4 -10 D 12 -4 4 0 4 E 14 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -12 -14 B 22 0 12 4 2 C 6 -12 0 -4 -10 D 12 -4 4 0 4 E 14 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -12 -14 B 22 0 12 4 2 C 6 -12 0 -4 -10 D 12 -4 4 0 4 E 14 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 520: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (13) A E B C D (10) C D B A E (7) B A E D C (6) B A C E D (6) C D E A B (5) B A E C D (5) D C B E A (4) C D B E A (4) E A D B C (3) D E C A B (3) E A D C B (2) E A B D C (2) D C E B A (2) C E D A B (2) B C D A E (2) B C A D E (2) B A C D E (2) A E C B D (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A D B (1) E A C D B (1) E A C B D (1) E A B C D (1) D E A C B (1) D C B A E (1) D B E A C (1) D B C E A (1) D B A C E (1) C D E B A (1) C B A D E (1) B D A E C (1) B A D E C (1) A E B D C (1) Total count = 100 A B C D E A 0 4 0 -2 0 B -4 0 -6 -10 -6 C 0 6 0 12 8 D 2 10 -12 0 6 E 0 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.475982 B: 0.000000 C: 0.524018 D: 0.000000 E: 0.000000 Sum of squares = 0.501153761163 Cumulative probabilities = A: 0.475982 B: 0.475982 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -2 0 B -4 0 -6 -10 -6 C 0 6 0 12 8 D 2 10 -12 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 C=20 A=15 E=13 so E is eliminated. Round 2 votes counts: D=28 B=25 A=25 C=22 so C is eliminated. Round 3 votes counts: D=48 B=26 A=26 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:203 A:201 E:196 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -2 0 B -4 0 -6 -10 -6 C 0 6 0 12 8 D 2 10 -12 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -2 0 B -4 0 -6 -10 -6 C 0 6 0 12 8 D 2 10 -12 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -2 0 B -4 0 -6 -10 -6 C 0 6 0 12 8 D 2 10 -12 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 521: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) A B D C E (6) E C B D A (5) B D E C A (5) B A D E C (5) A E C D B (5) A B E D C (5) A B D E C (5) E B D C A (4) E C D A B (3) E B C D A (3) E A B C D (3) D C B E A (3) C D A B E (3) B D A C E (3) E C A D B (2) D C B A E (2) D B C E A (2) C E D B A (2) C E D A B (2) C D E B A (2) C D E A B (2) A E B D C (2) A C D B E (2) E C A B D (1) E A C D B (1) E A C B D (1) D C A B E (1) D B C A E (1) C D B E A (1) C D A E B (1) C A E D B (1) B E D A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B A E D C (1) B A D C E (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -12 -14 -4 B 4 0 2 8 2 C 12 -2 0 -2 -22 D 14 -8 2 0 0 E 4 -2 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -14 -4 B 4 0 2 8 2 C 12 -2 0 -2 -22 D 14 -8 2 0 0 E 4 -2 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=27 B=19 C=14 D=9 so D is eliminated. Round 2 votes counts: E=31 A=27 B=22 C=20 so C is eliminated. Round 3 votes counts: E=39 A=33 B=28 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:208 D:204 C:193 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 -14 -4 B 4 0 2 8 2 C 12 -2 0 -2 -22 D 14 -8 2 0 0 E 4 -2 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -14 -4 B 4 0 2 8 2 C 12 -2 0 -2 -22 D 14 -8 2 0 0 E 4 -2 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -14 -4 B 4 0 2 8 2 C 12 -2 0 -2 -22 D 14 -8 2 0 0 E 4 -2 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 522: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) B A E C D (10) D C E A B (7) E C D A B (5) D A C B E (5) D A B C E (5) B E A C D (5) B A D E C (5) B A D C E (4) E D C B A (3) E C D B A (3) E B C A D (3) B E D A C (3) B D A E C (3) A B C D E (3) E C B D A (2) D C A E B (2) D B A C E (2) B A C E D (2) A C D B E (2) A C B D E (2) A B C E D (2) E D B C A (1) E C A B D (1) E B C D A (1) D E C B A (1) D B E A C (1) C E D A B (1) C A E B D (1) B E A D C (1) A D C B E (1) A D B C E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 18 10 4 B 20 0 6 20 18 C -18 -6 0 8 -16 D -10 -20 -8 0 -10 E -4 -18 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999493 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 18 10 4 B 20 0 6 20 18 C -18 -6 0 8 -16 D -10 -20 -8 0 -10 E -4 -18 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=29 D=23 A=13 C=2 so C is eliminated. Round 2 votes counts: B=33 E=30 D=23 A=14 so A is eliminated. Round 3 votes counts: B=42 E=31 D=27 so D is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:232 A:206 E:202 C:184 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 18 10 4 B 20 0 6 20 18 C -18 -6 0 8 -16 D -10 -20 -8 0 -10 E -4 -18 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 18 10 4 B 20 0 6 20 18 C -18 -6 0 8 -16 D -10 -20 -8 0 -10 E -4 -18 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 18 10 4 B 20 0 6 20 18 C -18 -6 0 8 -16 D -10 -20 -8 0 -10 E -4 -18 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 523: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) E B C A D (8) E D A B C (7) E C B A D (7) D A C B E (7) D A B C E (6) D E A C B (4) E C B D A (3) D C A B E (3) C A B D E (3) B E C A D (3) A B C D E (3) E D C A B (2) E D A C B (2) D A E C B (2) D A E B C (2) D A C E B (2) C D A B E (2) C B A E D (2) B C A D E (2) B A C E D (2) A D B E C (2) A D B C E (2) E D C B A (1) E D B A C (1) E C D B A (1) E B A C D (1) E A B D C (1) D E C A B (1) C E B A D (1) C B E D A (1) C B E A D (1) B E A C D (1) B C E A D (1) B A E D C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 6 6 B -6 0 -12 4 8 C 4 12 0 4 0 D -6 -4 -4 0 6 E -6 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.763493 D: 0.000000 E: 0.236507 Sum of squares = 0.638856824677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.763493 D: 0.763493 E: 1.000000 A B C D E A 0 6 -4 6 6 B -6 0 -12 4 8 C 4 12 0 4 0 D -6 -4 -4 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000056672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=27 C=20 B=10 A=9 so A is eliminated. Round 2 votes counts: E=34 D=32 C=20 B=14 so B is eliminated. Round 3 votes counts: E=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 A:207 B:197 D:196 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 6 6 B -6 0 -12 4 8 C 4 12 0 4 0 D -6 -4 -4 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000056672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 6 6 B -6 0 -12 4 8 C 4 12 0 4 0 D -6 -4 -4 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000056672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 6 6 B -6 0 -12 4 8 C 4 12 0 4 0 D -6 -4 -4 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000056672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 524: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) D E C A B (9) E D A B C (8) E A D B C (8) E A B C D (8) C D B A E (8) E A B D C (7) D C E B A (7) C B D A E (6) C B A D E (5) D C B A E (4) D E C B A (3) A B E C D (3) D C E A B (2) B C A D E (2) B A E C D (2) A E B C D (2) A B C E D (2) E D A C B (1) D E A C B (1) D C B E A (1) B C A E D (1) Total count = 100 A B C D E A 0 2 4 0 -10 B -2 0 6 -4 -12 C -4 -6 0 -2 -4 D 0 4 2 0 -4 E 10 12 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 0 -10 B -2 0 6 -4 -12 C -4 -6 0 -2 -4 D 0 4 2 0 -4 E 10 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=27 C=19 B=15 A=7 so A is eliminated. Round 2 votes counts: E=34 D=27 B=20 C=19 so C is eliminated. Round 3 votes counts: D=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:201 A:198 B:194 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 0 -10 B -2 0 6 -4 -12 C -4 -6 0 -2 -4 D 0 4 2 0 -4 E 10 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 -10 B -2 0 6 -4 -12 C -4 -6 0 -2 -4 D 0 4 2 0 -4 E 10 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 -10 B -2 0 6 -4 -12 C -4 -6 0 -2 -4 D 0 4 2 0 -4 E 10 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 525: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) D E B C A (9) E D A C B (8) D E C A B (8) E A C D B (5) C A B D E (5) A C B E D (5) B D C A E (4) B C D A E (3) B A C E D (3) A E C B D (3) A C E B D (3) E D B A C (2) E A C B D (2) E A B C D (2) D C B A E (2) D C A E B (2) D B E C A (2) D B C A E (2) C D A B E (2) B D E C A (2) E D C A B (1) E D B C A (1) E D A B C (1) E B A C D (1) E A D C B (1) E A B D C (1) D C A B E (1) D B C E A (1) B A C D E (1) A E C D B (1) A E B C D (1) A C E D B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -14 -2 6 B -10 0 -2 0 -6 C 14 2 0 4 -2 D 2 0 -4 0 14 E -6 6 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.540000000397 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 A B C D E A 0 10 -14 -2 6 B -10 0 -2 0 -6 C 14 2 0 4 -2 D 2 0 -4 0 14 E -6 6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.540000000419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 B=25 A=16 C=7 so C is eliminated. Round 2 votes counts: D=29 E=25 B=25 A=21 so A is eliminated. Round 3 votes counts: B=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:209 D:206 A:200 E:194 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -14 -2 6 B -10 0 -2 0 -6 C 14 2 0 4 -2 D 2 0 -4 0 14 E -6 6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.540000000419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 -2 6 B -10 0 -2 0 -6 C 14 2 0 4 -2 D 2 0 -4 0 14 E -6 6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.540000000419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 -2 6 B -10 0 -2 0 -6 C 14 2 0 4 -2 D 2 0 -4 0 14 E -6 6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.540000000419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 526: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) A C D B E (9) A D C E B (6) D E B A C (5) B E C A D (5) E D B C A (4) D E A B C (4) D A C E B (4) C A D E B (4) B C E A D (4) A C D E B (4) D A E C B (3) C B E D A (3) C A B E D (3) B E D A C (3) D C E A B (2) D A E B C (2) C B A E D (2) C A D B E (2) B C A E D (2) A C B E D (2) A C B D E (2) E B C D A (1) D E C A B (1) D C A E B (1) C D E A B (1) C B E A D (1) B E D C A (1) B E C D A (1) B D E A C (1) B A E D C (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 14 6 8 8 B -14 0 -4 -14 -6 C -6 4 0 -6 10 D -8 14 6 0 14 E -8 6 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 8 8 B -14 0 -4 -14 -6 C -6 4 0 -6 10 D -8 14 6 0 14 E -8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 B=18 C=16 E=14 so E is eliminated. Round 2 votes counts: A=30 B=28 D=26 C=16 so C is eliminated. Round 3 votes counts: A=39 B=34 D=27 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:213 C:201 E:187 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 8 8 B -14 0 -4 -14 -6 C -6 4 0 -6 10 D -8 14 6 0 14 E -8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 8 8 B -14 0 -4 -14 -6 C -6 4 0 -6 10 D -8 14 6 0 14 E -8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 8 8 B -14 0 -4 -14 -6 C -6 4 0 -6 10 D -8 14 6 0 14 E -8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 527: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) E D A B C (10) C E B A D (8) D E A B C (6) D E B A C (5) B A C D E (5) A B C D E (5) E D C B A (4) E C A B D (3) B D A C E (3) E D B A C (2) E D A C B (2) E C A D B (2) E A D B C (2) D B C A E (2) D B A E C (2) C E B D A (2) B C A D E (2) B A D C E (2) A D E B C (2) A D B E C (2) A B D C E (2) E D B C A (1) E C D A B (1) E A D C B (1) D B E C A (1) D B A C E (1) D A B E C (1) C E D B A (1) C B D A E (1) C B A E D (1) C A E B D (1) C A B E D (1) B D C A E (1) A E D B C (1) A D B C E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 14 8 -2 B 10 0 20 0 -8 C -14 -20 0 -10 2 D -8 0 10 0 12 E 2 8 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.539052 C: 0.000000 D: 0.460948 E: 0.000000 Sum of squares = 0.503050130398 Cumulative probabilities = A: 0.000000 B: 0.539052 C: 0.539052 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 8 -2 B 10 0 20 0 -8 C -14 -20 0 -10 2 D -8 0 10 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999998473 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 D=18 A=15 B=13 so B is eliminated. Round 2 votes counts: E=28 C=28 D=22 A=22 so D is eliminated. Round 3 votes counts: E=40 C=31 A=29 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:211 D:207 A:205 E:198 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 8 -2 B 10 0 20 0 -8 C -14 -20 0 -10 2 D -8 0 10 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999998473 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 8 -2 B 10 0 20 0 -8 C -14 -20 0 -10 2 D -8 0 10 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999998473 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 8 -2 B 10 0 20 0 -8 C -14 -20 0 -10 2 D -8 0 10 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999998473 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 528: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) D C A B E (9) A E B C D (9) C D A B E (7) E B A D C (6) E B A C D (6) D C B E A (5) A C D E B (5) A B E D C (5) C D E B A (4) B E D C A (4) B E A D C (4) E B D C A (2) B E D A C (2) B D E C A (2) A D C B E (2) A C E D B (2) A C E B D (2) E C B D A (1) E B C D A (1) E A B C D (1) D C B A E (1) D B C A E (1) D B A E C (1) C E D B A (1) C E A D B (1) C D E A B (1) C A D E B (1) B D E A C (1) B A E D C (1) A C D B E (1) Total count = 100 A B C D E A 0 14 -4 -8 16 B -14 0 -8 -6 -8 C 4 8 0 8 8 D 8 6 -8 0 4 E -16 8 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 -8 16 B -14 0 -8 -6 -8 C 4 8 0 8 8 D 8 6 -8 0 4 E -16 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 E=17 D=17 B=14 so B is eliminated. Round 2 votes counts: E=27 A=27 C=26 D=20 so D is eliminated. Round 3 votes counts: C=42 E=30 A=28 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:209 D:205 E:190 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 -8 16 B -14 0 -8 -6 -8 C 4 8 0 8 8 D 8 6 -8 0 4 E -16 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -8 16 B -14 0 -8 -6 -8 C 4 8 0 8 8 D 8 6 -8 0 4 E -16 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -8 16 B -14 0 -8 -6 -8 C 4 8 0 8 8 D 8 6 -8 0 4 E -16 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 529: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C A B E D (6) B E D C A (5) B C A E D (5) E D B C A (4) D B E C A (4) B D E C A (4) A D C E B (4) A C D E B (4) E B D C A (3) D E B C A (3) D E A C B (3) D E A B C (3) C B A E D (3) C A E B D (3) B C E D A (3) A D E C B (3) A C D B E (3) A C B E D (3) D A B E C (2) C A E D B (2) B C A D E (2) A D E B C (2) A C E D B (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C A D B (1) D B E A C (1) D A E C B (1) D A E B C (1) C E A B D (1) C B E A D (1) B E C D A (1) B D A E C (1) B C D E A (1) A D C B E (1) Total count = 100 A B C D E A 0 0 -10 0 4 B 0 0 8 -8 0 C 10 -8 0 -8 -4 D 0 8 8 0 8 E -4 0 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.315488 B: 0.000000 C: 0.000000 D: 0.684512 E: 0.000000 Sum of squares = 0.568089707026 Cumulative probabilities = A: 0.315488 B: 0.315488 C: 0.315488 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 0 4 B 0 0 8 -8 0 C 10 -8 0 -8 -4 D 0 8 8 0 8 E -4 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.506172909588 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 B=22 C=16 E=11 so E is eliminated. Round 2 votes counts: D=31 A=26 B=25 C=18 so C is eliminated. Round 3 votes counts: A=39 D=32 B=29 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:212 B:200 A:197 E:196 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 0 4 B 0 0 8 -8 0 C 10 -8 0 -8 -4 D 0 8 8 0 8 E -4 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.506172909588 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 0 4 B 0 0 8 -8 0 C 10 -8 0 -8 -4 D 0 8 8 0 8 E -4 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.506172909588 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 0 4 B 0 0 8 -8 0 C 10 -8 0 -8 -4 D 0 8 8 0 8 E -4 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.506172909588 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 530: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (9) D A B C E (8) C B A E D (7) E D C B A (6) E D B C A (6) D E A B C (6) D A E B C (6) C B E A D (6) A B C D E (6) E C B A D (5) B C A E D (5) E C B D A (4) A C B D E (4) E D B A C (3) E B C A D (3) C A B D E (3) D A C B E (2) D A B E C (2) E B C D A (1) D E A C B (1) C E B A D (1) C B A D E (1) C A D B E (1) B A C E D (1) B A C D E (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 10 16 B 0 0 16 -2 16 C -2 -16 0 -2 14 D -10 2 2 0 4 E -16 -16 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.644148 B: 0.355852 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.541557450991 Cumulative probabilities = A: 0.644148 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 10 16 B 0 0 16 -2 16 C -2 -16 0 -2 14 D -10 2 2 0 4 E -16 -16 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=25 A=21 C=19 B=7 so B is eliminated. Round 2 votes counts: E=28 D=25 C=24 A=23 so A is eliminated. Round 3 votes counts: D=36 C=36 E=28 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:215 A:214 D:199 C:197 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 10 16 B 0 0 16 -2 16 C -2 -16 0 -2 14 D -10 2 2 0 4 E -16 -16 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 10 16 B 0 0 16 -2 16 C -2 -16 0 -2 14 D -10 2 2 0 4 E -16 -16 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 10 16 B 0 0 16 -2 16 C -2 -16 0 -2 14 D -10 2 2 0 4 E -16 -16 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 531: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) B E C D A (10) B A D C E (10) A D C E B (10) D C A E B (6) A B D C E (6) B A E D C (5) E C D A B (4) B E A C D (4) E C B D A (3) C D E A B (3) A D C B E (3) E B A C D (2) C D A E B (2) B C D A E (2) B A E C D (2) A D E C B (2) E D C A B (1) E C D B A (1) E A D C B (1) E A C D B (1) D C E A B (1) C E D A B (1) C D E B A (1) C D B A E (1) C D A B E (1) B E C A D (1) B C E D A (1) B C D E A (1) A E D C B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -2 0 6 B 10 0 14 18 -2 C 2 -14 0 4 0 D 0 -18 -4 0 2 E -6 2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765427 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.444444 E: 1.000000 A B C D E A 0 -10 -2 0 6 B 10 0 14 18 -2 C 2 -14 0 4 0 D 0 -18 -4 0 2 E -6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.555556 Sum of squares = 0.432098764899 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=24 A=24 C=9 D=7 so D is eliminated. Round 2 votes counts: B=36 E=24 A=24 C=16 so C is eliminated. Round 3 votes counts: B=37 A=33 E=30 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:197 E:197 C:196 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 0 6 B 10 0 14 18 -2 C 2 -14 0 4 0 D 0 -18 -4 0 2 E -6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.555556 Sum of squares = 0.432098764899 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.444444 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 0 6 B 10 0 14 18 -2 C 2 -14 0 4 0 D 0 -18 -4 0 2 E -6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.555556 Sum of squares = 0.432098764899 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 0 6 B 10 0 14 18 -2 C 2 -14 0 4 0 D 0 -18 -4 0 2 E -6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.555556 Sum of squares = 0.432098764899 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.444444 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 532: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) A D E C B (8) D E A B C (7) D B E A C (6) C B E A D (6) C A E D B (6) C A E B D (6) B D E A C (6) B C D E A (6) D A E B C (5) B C E D A (5) A D C E B (4) B E C D A (3) B D E C A (3) E B D A C (2) D E B A C (2) D A E C B (2) C A D E B (2) B C E A D (2) E D A B C (1) E A D B C (1) D A C E B (1) C E B A D (1) C E A B D (1) C A D B E (1) C A B E D (1) B E D A C (1) B C D A E (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -4 -2 -6 B 4 0 4 4 -2 C 4 -4 0 0 4 D 2 -4 0 0 12 E 6 2 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.222222 Sum of squares = 0.506172839506 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.777778 E: 1.000000 A B C D E A 0 -4 -4 -2 -6 B 4 0 4 4 -2 C 4 -4 0 0 4 D 2 -4 0 0 12 E 6 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.222222 Sum of squares = 0.506172839856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=27 D=23 A=14 E=4 so E is eliminated. Round 2 votes counts: C=32 B=29 D=24 A=15 so A is eliminated. Round 3 votes counts: D=38 C=33 B=29 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:205 D:205 C:202 E:196 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -6 B 4 0 4 4 -2 C 4 -4 0 0 4 D 2 -4 0 0 12 E 6 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.222222 Sum of squares = 0.506172839856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.777778 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -6 B 4 0 4 4 -2 C 4 -4 0 0 4 D 2 -4 0 0 12 E 6 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.222222 Sum of squares = 0.506172839856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.777778 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -6 B 4 0 4 4 -2 C 4 -4 0 0 4 D 2 -4 0 0 12 E 6 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.222222 Sum of squares = 0.506172839856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.777778 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 533: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) B A E D C (8) B E D A C (6) B A C E D (5) E D A C B (4) E D A B C (4) E A D C B (4) D C E B A (4) B E A D C (4) B C D E A (4) A E D C B (4) E D B A C (3) D E C A B (3) C D A E B (3) C B A D E (3) C A D E B (3) B C A D E (3) A C E D B (3) D E B C A (2) C D B E A (2) C A B D E (2) B A E C D (2) A E D B C (2) A C D E B (2) E B A D C (1) E A D B C (1) D E A C B (1) D C E A B (1) C D E B A (1) C B D E A (1) B E D C A (1) B D E C A (1) B C A E D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 12 0 -14 B 4 0 0 -12 -12 C -12 0 0 -12 -6 D 0 12 12 0 -10 E 14 12 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 12 0 -14 B 4 0 0 -12 -12 C -12 0 0 -12 -6 D 0 12 12 0 -10 E 14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=24 E=17 A=13 D=11 so D is eliminated. Round 2 votes counts: B=35 C=29 E=23 A=13 so A is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:221 D:207 A:197 B:190 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 12 0 -14 B 4 0 0 -12 -12 C -12 0 0 -12 -6 D 0 12 12 0 -10 E 14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 0 -14 B 4 0 0 -12 -12 C -12 0 0 -12 -6 D 0 12 12 0 -10 E 14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 0 -14 B 4 0 0 -12 -12 C -12 0 0 -12 -6 D 0 12 12 0 -10 E 14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 534: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C E B A D (7) C A E B D (7) E B C D A (6) B E D C A (6) E C B D A (5) E B D C A (5) D A B E C (5) C B E D A (5) A C E D B (5) D B A E C (4) A C D E B (4) D B E C A (3) A D C B E (3) A D B E C (3) C E A B D (2) B D E C A (2) A D E B C (2) A D B C E (2) A C E B D (2) A C D B E (2) E D B C A (1) E C A B D (1) E A C D B (1) D E B A C (1) D E A B C (1) C E B D A (1) C A B E D (1) B E C D A (1) B C E D A (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -14 -8 -12 -16 B 14 0 4 4 -6 C 8 -4 0 2 -14 D 12 -4 -2 0 -14 E 16 6 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -8 -12 -16 B 14 0 4 4 -6 C 8 -4 0 2 -14 D 12 -4 -2 0 -14 E 16 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 C=23 E=19 B=10 so B is eliminated. Round 2 votes counts: E=26 D=25 A=25 C=24 so C is eliminated. Round 3 votes counts: E=42 A=33 D=25 so D is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:208 C:196 D:196 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -8 -12 -16 B 14 0 4 4 -6 C 8 -4 0 2 -14 D 12 -4 -2 0 -14 E 16 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -12 -16 B 14 0 4 4 -6 C 8 -4 0 2 -14 D 12 -4 -2 0 -14 E 16 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -12 -16 B 14 0 4 4 -6 C 8 -4 0 2 -14 D 12 -4 -2 0 -14 E 16 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 535: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (16) C B A E D (12) D E A B C (11) B A C E D (7) E A D B C (5) D E A C B (5) C B A D E (5) D E C B A (4) C D B A E (4) C B D A E (4) A B E C D (4) D C E B A (3) D C B E A (3) D C B A E (3) C D B E A (3) B C A E D (2) A E B C D (2) E A B D C (1) E A B C D (1) D E C A B (1) D A E B C (1) D A C B E (1) A E D B C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 12 -18 -6 B 0 0 4 -22 -2 C -12 -4 0 -10 -4 D 18 22 10 0 -4 E 6 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 12 -18 -6 B 0 0 4 -22 -2 C -12 -4 0 -10 -4 D 18 22 10 0 -4 E 6 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=28 E=23 B=9 A=8 so A is eliminated. Round 2 votes counts: D=32 C=28 E=26 B=14 so B is eliminated. Round 3 votes counts: C=38 D=32 E=30 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:208 A:194 B:190 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 -18 -6 B 0 0 4 -22 -2 C -12 -4 0 -10 -4 D 18 22 10 0 -4 E 6 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -18 -6 B 0 0 4 -22 -2 C -12 -4 0 -10 -4 D 18 22 10 0 -4 E 6 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -18 -6 B 0 0 4 -22 -2 C -12 -4 0 -10 -4 D 18 22 10 0 -4 E 6 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 536: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) D E A C B (7) B C A E D (7) D E C A B (5) B C D E A (5) C B E A D (4) C B D E A (4) B A E C D (4) A B E C D (4) E A C D B (3) D C E B A (3) E D C A B (2) E A D C B (2) D E A B C (2) D C E A B (2) D B A E C (2) C E A B D (2) B D C E A (2) B D C A E (2) B C A D E (2) B A D C E (2) A E D C B (2) A E D B C (2) A E B C D (2) A D B E C (2) E C D A B (1) D B E C A (1) D B C E A (1) D A E B C (1) C E A D B (1) C D E B A (1) C D E A B (1) C D B E A (1) C A E B D (1) B C E D A (1) B A D E C (1) B A C D E (1) A E C B D (1) A E B D C (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 12 -2 B 8 0 12 14 14 C -2 -12 0 12 6 D -12 -14 -12 0 -2 E 2 -14 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 12 -2 B 8 0 12 14 14 C -2 -12 0 12 6 D -12 -14 -12 0 -2 E 2 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=24 A=16 C=15 E=8 so E is eliminated. Round 2 votes counts: B=37 D=26 A=21 C=16 so C is eliminated. Round 3 votes counts: B=45 D=30 A=25 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:202 C:202 E:192 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 12 -2 B 8 0 12 14 14 C -2 -12 0 12 6 D -12 -14 -12 0 -2 E 2 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 12 -2 B 8 0 12 14 14 C -2 -12 0 12 6 D -12 -14 -12 0 -2 E 2 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 12 -2 B 8 0 12 14 14 C -2 -12 0 12 6 D -12 -14 -12 0 -2 E 2 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 537: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) E A D B C (7) C B A D E (7) D E B C A (6) C B A E D (6) B C D E A (6) A E D B C (5) B D C E A (4) A E D C B (4) A C B E D (4) D E A B C (3) E D B C A (2) E D B A C (2) E D A B C (2) D B C E A (2) C B E D A (2) C B D A E (2) C A B E D (2) B E C D A (2) B D E C A (2) A E C D B (2) A D E C B (2) A C B D E (2) E B C A D (1) E A B C D (1) D E B A C (1) D C B E A (1) D C B A E (1) D B E C A (1) B E D C A (1) B E C A D (1) B C E D A (1) A E C B D (1) A E B C D (1) A D E B C (1) A D C E B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -18 -16 2 -12 B 18 0 4 10 12 C 16 -4 0 4 4 D -2 -10 -4 0 4 E 12 -12 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 2 -12 B 18 0 4 10 12 C 16 -4 0 4 4 D -2 -10 -4 0 4 E 12 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 B=17 E=15 D=15 so E is eliminated. Round 2 votes counts: A=34 C=27 D=21 B=18 so B is eliminated. Round 3 votes counts: C=38 A=34 D=28 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:222 C:210 E:196 D:194 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 2 -12 B 18 0 4 10 12 C 16 -4 0 4 4 D -2 -10 -4 0 4 E 12 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 2 -12 B 18 0 4 10 12 C 16 -4 0 4 4 D -2 -10 -4 0 4 E 12 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 2 -12 B 18 0 4 10 12 C 16 -4 0 4 4 D -2 -10 -4 0 4 E 12 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 538: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) D E C A B (8) C A E D B (7) B A C D E (7) A C B D E (6) A C B E D (5) D E B C A (4) B D E A C (4) B D A C E (4) B A D C E (4) D B E A C (3) C A E B D (3) B A C E D (3) A C E B D (3) A B C E D (3) E D C B A (2) E D B C A (2) E C D A B (2) D E C B A (2) D B A C E (2) C E A D B (2) E C A D B (1) D C E A B (1) D B E C A (1) D B A E C (1) D A C B E (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) B D A E C (1) B A D E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 12 2 -4 8 B -12 0 -14 -6 -2 C -2 14 0 -10 8 D 4 6 10 0 6 E -8 2 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 -4 8 B -12 0 -14 -6 -2 C -2 14 0 -10 8 D 4 6 10 0 6 E -8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 A=19 E=18 C=13 so C is eliminated. Round 2 votes counts: A=30 B=27 D=23 E=20 so E is eliminated. Round 3 votes counts: D=40 A=33 B=27 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:209 C:205 E:190 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 2 -4 8 B -12 0 -14 -6 -2 C -2 14 0 -10 8 D 4 6 10 0 6 E -8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -4 8 B -12 0 -14 -6 -2 C -2 14 0 -10 8 D 4 6 10 0 6 E -8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -4 8 B -12 0 -14 -6 -2 C -2 14 0 -10 8 D 4 6 10 0 6 E -8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 539: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) C D E A B (8) B A E C D (8) D C A E B (7) D C E A B (6) B A E D C (6) A E B C D (5) D C B E A (4) D C B A E (4) B E A C D (4) B D A E C (4) D B A C E (3) C E D A B (3) C D E B A (3) A B E C D (3) E A C B D (2) E A B C D (2) D B A E C (2) C E A D B (2) B D C E A (2) B D C A E (2) A E B D C (2) E C A B D (1) E B A C D (1) D C E B A (1) D C A B E (1) D B C E A (1) C D A E B (1) C B E D A (1) A E C D B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -10 -22 18 B 8 0 10 -12 8 C 10 -10 0 -10 14 D 22 12 10 0 16 E -18 -8 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -22 18 B 8 0 10 -12 8 C 10 -10 0 -10 14 D 22 12 10 0 16 E -18 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=26 C=18 A=13 E=6 so E is eliminated. Round 2 votes counts: D=37 B=27 C=19 A=17 so A is eliminated. Round 3 votes counts: B=40 D=38 C=22 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:230 B:207 C:202 A:189 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -22 18 B 8 0 10 -12 8 C 10 -10 0 -10 14 D 22 12 10 0 16 E -18 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -22 18 B 8 0 10 -12 8 C 10 -10 0 -10 14 D 22 12 10 0 16 E -18 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -22 18 B 8 0 10 -12 8 C 10 -10 0 -10 14 D 22 12 10 0 16 E -18 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 540: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D A E C B (6) C A D B E (6) B A D E C (6) B A D C E (6) E D A C B (5) B C E A D (5) E B D A C (4) B E C A D (4) A D B E C (4) E C D A B (3) D A E B C (3) C E B D A (3) B E C D A (3) B E A D C (3) A D C B E (3) A B D C E (3) E B C D A (2) C E D B A (2) C D A E B (2) C B E A D (2) B A E D C (2) B A C D E (2) A D E B C (2) A D B C E (2) E D A B C (1) D E A C B (1) C E D A B (1) C D E A B (1) C B E D A (1) C B A D E (1) C A B D E (1) B C E D A (1) B C A D E (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 10 10 4 B 10 0 10 16 16 C -10 -10 0 -2 -10 D -10 -16 2 0 2 E -4 -16 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 10 4 B 10 0 10 16 16 C -10 -10 0 -2 -10 D -10 -16 2 0 2 E -4 -16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=21 C=20 A=15 D=10 so D is eliminated. Round 2 votes counts: B=34 A=24 E=22 C=20 so C is eliminated. Round 3 votes counts: B=38 A=33 E=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:207 E:194 D:189 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 10 4 B 10 0 10 16 16 C -10 -10 0 -2 -10 D -10 -16 2 0 2 E -4 -16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 10 4 B 10 0 10 16 16 C -10 -10 0 -2 -10 D -10 -16 2 0 2 E -4 -16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 10 4 B 10 0 10 16 16 C -10 -10 0 -2 -10 D -10 -16 2 0 2 E -4 -16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 541: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) B E C D A (9) A D C B E (9) E B C D A (7) B A C D E (6) A D E C B (6) B C E D A (5) A B D C E (5) E C B D A (4) D A C E B (4) E C D B A (3) A D B C E (3) E D C A B (2) E A D C B (2) C B E D A (2) B C A D E (2) A B C D E (2) E D A C B (1) E B D A C (1) E B C A D (1) E B A C D (1) E A D B C (1) D E C A B (1) D E A C B (1) C E D B A (1) C D B A E (1) C D A B E (1) C B D A E (1) B E C A D (1) B E A C D (1) B C D A E (1) B C A E D (1) A D B E C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 14 10 12 B -4 0 -2 4 6 C -14 2 0 2 12 D -10 -4 -2 0 12 E -12 -6 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 10 12 B -4 0 -2 4 6 C -14 2 0 2 12 D -10 -4 -2 0 12 E -12 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=26 E=23 D=6 C=6 so D is eliminated. Round 2 votes counts: A=43 B=26 E=25 C=6 so C is eliminated. Round 3 votes counts: A=44 B=30 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 B:202 C:201 D:198 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 12 B -4 0 -2 4 6 C -14 2 0 2 12 D -10 -4 -2 0 12 E -12 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 12 B -4 0 -2 4 6 C -14 2 0 2 12 D -10 -4 -2 0 12 E -12 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 12 B -4 0 -2 4 6 C -14 2 0 2 12 D -10 -4 -2 0 12 E -12 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 542: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (14) B D E C A (7) B D C A E (7) C A E B D (6) E A C D B (5) B C A D E (5) A C E D B (5) E A C B D (4) D B E C A (4) D B C A E (4) C A B E D (4) B C A E D (4) D C A B E (3) C A D E B (3) C A B D E (3) A E C D B (3) E D A C B (2) D E B A C (2) D E A C B (2) C A E D B (2) B C D A E (2) E D B A C (1) E D A B C (1) E B D A C (1) D C A E B (1) D A E C B (1) D A C E B (1) C D A B E (1) B E D A C (1) B D E A C (1) Total count = 100 A B C D E A 0 -6 -12 -12 10 B 6 0 8 -10 20 C 12 -8 0 -6 2 D 12 10 6 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -12 10 B 6 0 8 -10 20 C 12 -8 0 -6 2 D 12 10 6 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=27 C=19 E=14 A=8 so A is eliminated. Round 2 votes counts: D=32 B=27 C=24 E=17 so E is eliminated. Round 3 votes counts: D=36 C=36 B=28 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 B:212 C:200 A:190 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 10 B 6 0 8 -10 20 C 12 -8 0 -6 2 D 12 10 6 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 10 B 6 0 8 -10 20 C 12 -8 0 -6 2 D 12 10 6 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 10 B 6 0 8 -10 20 C 12 -8 0 -6 2 D 12 10 6 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 543: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (14) E A C D B (10) A C E B D (10) C A B D E (9) B D C A E (7) A E C B D (7) E A D C B (5) E A D B C (5) B D C E A (5) A E C D B (5) E D A B C (4) C B A D E (4) B C D A E (4) E D B A C (2) D E B A C (2) C A B E D (2) D E B C A (1) D B C E A (1) C B D A E (1) C A E D B (1) C A E B D (1) Total count = 100 A B C D E A 0 18 0 18 2 B -18 0 -10 0 -6 C 0 10 0 8 -10 D -18 0 -8 0 -4 E -2 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.953041 B: 0.000000 C: 0.046959 D: 0.000000 E: 0.000000 Sum of squares = 0.910492512195 Cumulative probabilities = A: 0.953041 B: 0.953041 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 18 2 B -18 0 -10 0 -6 C 0 10 0 8 -10 D -18 0 -8 0 -4 E -2 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.72222222618 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 D=18 C=18 B=16 so B is eliminated. Round 2 votes counts: D=30 E=26 C=22 A=22 so C is eliminated. Round 3 votes counts: A=39 D=35 E=26 so E is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:209 C:204 D:185 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 18 2 B -18 0 -10 0 -6 C 0 10 0 8 -10 D -18 0 -8 0 -4 E -2 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.72222222618 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 18 2 B -18 0 -10 0 -6 C 0 10 0 8 -10 D -18 0 -8 0 -4 E -2 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.72222222618 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 18 2 B -18 0 -10 0 -6 C 0 10 0 8 -10 D -18 0 -8 0 -4 E -2 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.72222222618 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 544: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A C E (7) D A B C E (7) C A E D B (7) B D A E C (6) C E A D B (5) E C B A D (4) E B C D A (4) C E D A B (4) C E A B D (4) B A D E C (4) A C E B D (4) B D E A C (3) E B C A D (2) D B E C A (2) D B A E C (2) B E D C A (2) B E C A D (2) B A E D C (2) B A E C D (2) A D B C E (2) A B E C D (2) A B C E D (2) E A C B D (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A E B (1) D A C B E (1) C A E B D (1) B E D A C (1) B E A D C (1) B E A C D (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 14 6 B -6 0 8 12 0 C -2 -8 0 10 0 D -14 -12 -10 0 -20 E -6 0 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 14 6 B -6 0 8 12 0 C -2 -8 0 10 0 D -14 -12 -10 0 -20 E -6 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=23 C=21 E=19 A=13 so A is eliminated. Round 2 votes counts: B=28 C=27 D=26 E=19 so E is eliminated. Round 3 votes counts: C=40 B=34 D=26 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:207 E:207 C:200 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 14 6 B -6 0 8 12 0 C -2 -8 0 10 0 D -14 -12 -10 0 -20 E -6 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 14 6 B -6 0 8 12 0 C -2 -8 0 10 0 D -14 -12 -10 0 -20 E -6 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 14 6 B -6 0 8 12 0 C -2 -8 0 10 0 D -14 -12 -10 0 -20 E -6 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 545: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) B E A D C (6) D B A C E (5) B D A C E (5) A C D B E (5) E C A D B (3) E B D A C (3) E B C D A (3) E B A C D (3) D C B A E (3) C A E D B (3) C A D E B (3) B E D A C (3) A B C D E (3) E C D B A (2) E C D A B (2) E A C B D (2) D B C A E (2) C D A E B (2) B D E A C (2) B A E D C (2) B A D C E (2) A E C B D (2) A E B C D (2) A C E D B (2) A C E B D (2) E D B C A (1) E C A B D (1) E B D C A (1) E B A D C (1) E A C D B (1) E A B C D (1) D C B E A (1) D C A E B (1) D B E C A (1) D B C E A (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A B E (1) B D E C A (1) B D A E C (1) A D C B E (1) A D B C E (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 16 0 16 B -2 0 4 -4 10 C -16 -4 0 -4 12 D 0 4 4 0 2 E -16 -10 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.367053 B: 0.000000 C: 0.000000 D: 0.632947 E: 0.000000 Sum of squares = 0.535349865973 Cumulative probabilities = A: 0.367053 B: 0.367053 C: 0.367053 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 0 16 B -2 0 4 -4 10 C -16 -4 0 -4 12 D 0 4 4 0 2 E -16 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=22 D=21 A=21 C=12 so C is eliminated. Round 2 votes counts: A=27 E=26 D=25 B=22 so B is eliminated. Round 3 votes counts: E=35 D=34 A=31 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:217 D:205 B:204 C:194 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 0 16 B -2 0 4 -4 10 C -16 -4 0 -4 12 D 0 4 4 0 2 E -16 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 0 16 B -2 0 4 -4 10 C -16 -4 0 -4 12 D 0 4 4 0 2 E -16 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 0 16 B -2 0 4 -4 10 C -16 -4 0 -4 12 D 0 4 4 0 2 E -16 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 546: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) E B C A D (5) D C A E B (5) C E B A D (5) B E A C D (5) E C B A D (4) E B D C A (4) C E B D A (4) C D E B A (4) C A D E B (4) A D B E C (4) A B E C D (4) E B C D A (3) D A C B E (3) D A B E C (3) C D A E B (3) B A E D C (3) D E C B A (2) B E D A C (2) B E A D C (2) B A E C D (2) A D B C E (2) A C D B E (2) A C B E D (2) A B D E C (2) A B C E D (2) E C B D A (1) D C E A B (1) D B A E C (1) C E D B A (1) C E A B D (1) C D E A B (1) C A B E D (1) B D E A C (1) A D C B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -14 6 -8 B 14 0 -8 10 -12 C 14 8 0 10 2 D -6 -10 -10 0 -4 E 8 12 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 6 -8 B 14 0 -8 10 -12 C 14 8 0 10 2 D -6 -10 -10 0 -4 E 8 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 A=21 E=17 B=15 so B is eliminated. Round 2 votes counts: E=26 A=26 D=24 C=24 so D is eliminated. Round 3 votes counts: C=38 A=33 E=29 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:211 B:202 A:185 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 6 -8 B 14 0 -8 10 -12 C 14 8 0 10 2 D -6 -10 -10 0 -4 E 8 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 6 -8 B 14 0 -8 10 -12 C 14 8 0 10 2 D -6 -10 -10 0 -4 E 8 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 6 -8 B 14 0 -8 10 -12 C 14 8 0 10 2 D -6 -10 -10 0 -4 E 8 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 547: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) D C B A E (7) D B C E A (5) C D B A E (5) C D A B E (5) A E C B D (5) E B D A C (4) E A D B C (4) E A B D C (4) D C A B E (3) B E D C A (3) B D E C A (3) A E C D B (3) A C D E B (3) E A C D B (2) D E B C A (2) D E B A C (2) D E A C B (2) D B E C A (2) C B A D E (2) C A D B E (2) C A B D E (2) B E C A D (2) A D E C B (2) A D C E B (2) A C E B D (2) A C B E D (2) E D B A C (1) E D A C B (1) E D A B C (1) D C B E A (1) D C A E B (1) B E D A C (1) B E A C D (1) B D C E A (1) B C E A D (1) B C A E D (1) B C A D E (1) Total count = 100 A B C D E A 0 10 2 0 -4 B -10 0 -4 -12 0 C -2 4 0 -4 -8 D 0 12 4 0 6 E 4 0 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.260542 B: 0.000000 C: 0.000000 D: 0.739458 E: 0.000000 Sum of squares = 0.614680464611 Cumulative probabilities = A: 0.260542 B: 0.260542 C: 0.260542 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 0 -4 B -10 0 -4 -12 0 C -2 4 0 -4 -8 D 0 12 4 0 6 E 4 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=25 A=19 C=16 B=14 so B is eliminated. Round 2 votes counts: E=33 D=29 C=19 A=19 so C is eliminated. Round 3 votes counts: D=39 E=34 A=27 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:204 E:203 C:195 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 2 0 -4 B -10 0 -4 -12 0 C -2 4 0 -4 -8 D 0 12 4 0 6 E 4 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 0 -4 B -10 0 -4 -12 0 C -2 4 0 -4 -8 D 0 12 4 0 6 E 4 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 0 -4 B -10 0 -4 -12 0 C -2 4 0 -4 -8 D 0 12 4 0 6 E 4 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 548: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) C E D A B (9) E C D B A (6) C E D B A (5) D E C A B (4) C E B D A (4) E D C B A (3) E D B A C (3) E B D C A (3) D E B A C (3) C A B E D (3) B A E D C (3) A D B C E (3) A B C D E (3) E C D A B (2) E C B D A (2) E B C D A (2) D A B E C (2) C D E A B (2) B E C A D (2) B E A D C (2) A D C B E (2) E D C A B (1) E D B C A (1) D E A C B (1) D E A B C (1) D C E A B (1) D B E A C (1) D A E C B (1) D A C E B (1) C E A D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B E D A C (1) B D A E C (1) B C E A D (1) B A E C D (1) A D B E C (1) A C D B E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -10 -20 -22 B 10 0 -6 -12 -16 C 10 6 0 -4 -18 D 20 12 4 0 -12 E 22 16 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -10 -20 -22 B 10 0 -6 -12 -16 C 10 6 0 -4 -18 D 20 12 4 0 -12 E 22 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=23 B=22 D=15 A=13 so A is eliminated. Round 2 votes counts: C=29 B=27 E=23 D=21 so D is eliminated. Round 3 votes counts: B=34 E=33 C=33 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:234 D:212 C:197 B:188 A:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -10 -20 -22 B 10 0 -6 -12 -16 C 10 6 0 -4 -18 D 20 12 4 0 -12 E 22 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -20 -22 B 10 0 -6 -12 -16 C 10 6 0 -4 -18 D 20 12 4 0 -12 E 22 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -20 -22 B 10 0 -6 -12 -16 C 10 6 0 -4 -18 D 20 12 4 0 -12 E 22 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 549: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) C A D B E (7) E A D B C (5) C B D E A (5) C B D A E (5) B E D C A (4) A D E C B (4) D B C A E (3) B E C D A (3) A C E D B (3) A C D E B (3) E D B A C (2) E B A D C (2) E A B D C (2) D B E A C (2) D B C E A (2) D A E B C (2) C D A B E (2) C B E A D (2) B D C E A (2) B C E D A (2) A E D B C (2) A D C E B (2) E D A B C (1) E B D C A (1) E B C A D (1) E B A C D (1) D E A B C (1) D C B A E (1) D C A B E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C B E (1) C E B A D (1) C E A B D (1) C D B A E (1) C B E D A (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B E D (1) B E D A C (1) B D E A C (1) B C D E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -2 -10 -2 B 8 0 0 -6 4 C 2 0 0 -6 6 D 10 6 6 0 4 E 2 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -10 -2 B 8 0 0 -6 4 C 2 0 0 -6 6 D 10 6 6 0 4 E 2 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=22 A=18 D=16 B=14 so B is eliminated. Round 2 votes counts: C=33 E=30 D=19 A=18 so A is eliminated. Round 3 votes counts: C=40 E=35 D=25 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:213 B:203 C:201 E:194 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -10 -2 B 8 0 0 -6 4 C 2 0 0 -6 6 D 10 6 6 0 4 E 2 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -10 -2 B 8 0 0 -6 4 C 2 0 0 -6 6 D 10 6 6 0 4 E 2 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -10 -2 B 8 0 0 -6 4 C 2 0 0 -6 6 D 10 6 6 0 4 E 2 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 550: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D C A E B (6) E A B C D (4) D C A B E (4) B E D C A (4) A C E B D (4) E A D C B (3) D E A C B (3) D C B A E (3) D B E C A (3) D A C E B (3) C B A E D (3) B E C A D (3) B E A C D (3) B C A E D (3) A E C B D (3) E D B A C (2) E A B D C (2) D E B A C (2) D B C E A (2) C D A B E (2) C A D E B (2) C A B E D (2) B E D A C (2) B E C D A (2) B D C E A (2) A C E D B (2) E D A C B (1) E D A B C (1) E B A D C (1) E A D B C (1) E A C B D (1) D E B C A (1) D E A B C (1) D C B E A (1) D B C A E (1) D A E C B (1) C A B D E (1) B D C A E (1) B C A D E (1) A E C D B (1) A D E C B (1) A D C E B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 6 4 -6 B -4 0 0 2 -12 C -6 0 0 -6 -8 D -4 -2 6 0 -14 E 6 12 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 4 -6 B -4 0 0 2 -12 C -6 0 0 -6 -8 D -4 -2 6 0 -14 E 6 12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=24 B=21 A=14 C=10 so C is eliminated. Round 2 votes counts: D=33 E=24 B=24 A=19 so A is eliminated. Round 3 votes counts: D=38 E=34 B=28 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:204 B:193 D:193 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 4 -6 B -4 0 0 2 -12 C -6 0 0 -6 -8 D -4 -2 6 0 -14 E 6 12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 -6 B -4 0 0 2 -12 C -6 0 0 -6 -8 D -4 -2 6 0 -14 E 6 12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 -6 B -4 0 0 2 -12 C -6 0 0 -6 -8 D -4 -2 6 0 -14 E 6 12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 551: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) E D C A B (6) C D A B E (6) C A D B E (6) B A C E D (6) E B A D C (5) D C E A B (5) A B C D E (5) E D C B A (4) E D B A C (4) D E C A B (4) C D A E B (4) C A D E B (3) B E A D C (3) A C B D E (3) A B C E D (3) E B A C D (2) D E C B A (2) C D E A B (2) C A B D E (2) A B E C D (2) E D B C A (1) E C D A B (1) E B D A C (1) E A B C D (1) D E B C A (1) D E B A C (1) B E A C D (1) B A E C D (1) B A D C E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 10 6 16 12 B -10 0 0 0 4 C -6 0 0 24 18 D -16 0 -24 0 14 E -12 -4 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 16 12 B -10 0 0 0 4 C -6 0 0 24 18 D -16 0 -24 0 14 E -12 -4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=24 C=23 A=15 D=13 so D is eliminated. Round 2 votes counts: E=33 C=28 B=24 A=15 so A is eliminated. Round 3 votes counts: E=34 B=34 C=32 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:222 C:218 B:197 D:187 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 16 12 B -10 0 0 0 4 C -6 0 0 24 18 D -16 0 -24 0 14 E -12 -4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 16 12 B -10 0 0 0 4 C -6 0 0 24 18 D -16 0 -24 0 14 E -12 -4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 16 12 B -10 0 0 0 4 C -6 0 0 24 18 D -16 0 -24 0 14 E -12 -4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 552: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (12) E C B A D (8) C E A D B (8) D B A C E (5) D A B C E (5) A D C B E (5) E C A D B (4) C A E D B (4) A D B C E (4) E C A B D (3) D A C B E (3) B D A C E (3) E C B D A (2) E B C D A (2) D B A E C (2) C E B D A (2) C D B A E (2) C D A B E (2) B E D C A (2) B D E C A (2) B D E A C (2) A C E D B (2) A C D E B (2) A B D E C (2) E B D A C (1) D B C A E (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E D A (1) C A D E B (1) C A D B E (1) B E D A C (1) B E C D A (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 4 -4 18 B 0 0 -6 -8 18 C -4 6 0 -4 8 D 4 8 4 0 14 E -18 -18 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -4 18 B 0 0 -6 -8 18 C -4 6 0 -4 8 D 4 8 4 0 14 E -18 -18 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 E=20 D=17 A=17 so D is eliminated. Round 2 votes counts: B=31 A=26 C=23 E=20 so E is eliminated. Round 3 votes counts: C=40 B=34 A=26 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:215 A:209 C:203 B:202 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -4 18 B 0 0 -6 -8 18 C -4 6 0 -4 8 D 4 8 4 0 14 E -18 -18 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -4 18 B 0 0 -6 -8 18 C -4 6 0 -4 8 D 4 8 4 0 14 E -18 -18 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -4 18 B 0 0 -6 -8 18 C -4 6 0 -4 8 D 4 8 4 0 14 E -18 -18 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 553: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) C B D A E (9) E A D B C (7) C B E D A (7) B C D A E (6) E C A D B (4) D A C B E (4) A E D C B (4) E B A D C (3) D A C E B (3) C E A D B (3) C D B A E (3) C D A E B (3) B E A D C (3) E A B D C (2) D C B A E (2) D C A B E (2) C D A B E (2) C B E A D (2) B E C A D (2) B C D E A (2) A E D B C (2) A D E C B (2) A D E B C (2) E B A C D (1) D B A C E (1) D A E C B (1) C E B A D (1) C B D E A (1) B E A C D (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -2 -2 0 B -2 0 -26 -10 4 C 2 26 0 -4 10 D 2 10 4 0 -4 E 0 -4 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 0.222222 Sum of squares = 0.40740740737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.777778 E: 1.000000 A B C D E A 0 2 -2 -2 0 B -2 0 -26 -10 4 C 2 26 0 -4 10 D 2 10 4 0 -4 E 0 -4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 0.222222 Sum of squares = 0.407407404551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 B=19 D=13 A=11 so A is eliminated. Round 2 votes counts: E=32 C=31 B=19 D=18 so D is eliminated. Round 3 votes counts: C=43 E=37 B=20 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:206 A:199 E:195 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 -2 0 B -2 0 -26 -10 4 C 2 26 0 -4 10 D 2 10 4 0 -4 E 0 -4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 0.222222 Sum of squares = 0.407407404551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.777778 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -2 0 B -2 0 -26 -10 4 C 2 26 0 -4 10 D 2 10 4 0 -4 E 0 -4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 0.222222 Sum of squares = 0.407407404551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.777778 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -2 0 B -2 0 -26 -10 4 C 2 26 0 -4 10 D 2 10 4 0 -4 E 0 -4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 0.222222 Sum of squares = 0.407407404551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.777778 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 554: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) B D C E A (8) A D B E C (6) E A C B D (5) D B C E A (5) D B A E C (5) C E B D A (5) C E A B D (5) A E D B C (5) A D E B C (5) E C A B D (4) D B C A E (4) C B D E A (4) A C E B D (4) E C B D A (3) D B A C E (3) C E B A D (3) C B E D A (3) B C D E A (3) A E D C B (2) E C B A D (1) D B E C A (1) D B E A C (1) D A B E C (1) D A B C E (1) A E C B D (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 2 6 -2 B -2 0 -2 -2 -8 C -2 2 0 4 0 D -6 2 -4 0 -4 E 2 8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.280486 D: 0.000000 E: 0.719514 Sum of squares = 0.596372461163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.280486 D: 0.280486 E: 1.000000 A B C D E A 0 2 2 6 -2 B -2 0 -2 -2 -8 C -2 2 0 4 0 D -6 2 -4 0 -4 E 2 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.000000 E: 0.500240 Sum of squares = 0.500000114764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.499760 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=21 C=20 E=13 B=11 so B is eliminated. Round 2 votes counts: A=35 D=29 C=23 E=13 so E is eliminated. Round 3 votes counts: A=40 C=31 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:207 A:204 C:202 D:194 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 6 -2 B -2 0 -2 -2 -8 C -2 2 0 4 0 D -6 2 -4 0 -4 E 2 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.000000 E: 0.500240 Sum of squares = 0.500000114764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.499760 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 6 -2 B -2 0 -2 -2 -8 C -2 2 0 4 0 D -6 2 -4 0 -4 E 2 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.000000 E: 0.500240 Sum of squares = 0.500000114764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.499760 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 6 -2 B -2 0 -2 -2 -8 C -2 2 0 4 0 D -6 2 -4 0 -4 E 2 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.000000 E: 0.500240 Sum of squares = 0.500000114764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499760 D: 0.499760 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 555: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) C E B D A (8) C B E D A (7) E A D B C (6) B C D A E (6) A D B E C (6) E C B A D (5) B A D C E (5) A D E B C (5) D A B C E (4) E C B D A (3) D A E C B (3) D A B E C (3) C E B A D (3) E C D B A (2) D B A C E (2) D A E B C (2) B D A C E (2) E C D A B (1) E C A D B (1) E C A B D (1) E B A C D (1) E A C D B (1) D C A B E (1) C D E A B (1) C D B A E (1) C B E A D (1) C B D A E (1) B E C A D (1) B C E A D (1) B C A E D (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B D C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 8 6 -4 B 4 0 2 0 -8 C -8 -2 0 -4 -4 D -6 0 4 0 -8 E 4 8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 8 6 -4 B 4 0 2 0 -8 C -8 -2 0 -4 -4 D -6 0 4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=22 B=18 D=15 A=15 so D is eliminated. Round 2 votes counts: E=30 A=27 C=23 B=20 so B is eliminated. Round 3 votes counts: A=37 C=32 E=31 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:212 A:203 B:199 D:195 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 8 6 -4 B 4 0 2 0 -8 C -8 -2 0 -4 -4 D -6 0 4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 6 -4 B 4 0 2 0 -8 C -8 -2 0 -4 -4 D -6 0 4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 6 -4 B 4 0 2 0 -8 C -8 -2 0 -4 -4 D -6 0 4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 556: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) D E A C B (7) C B E A D (6) B A D E C (6) E D A C B (5) E D C A B (4) E C D A B (4) D E A B C (4) D A E B C (4) A E D C B (4) A D E B C (4) A D B E C (4) C E D B A (3) C E A D B (3) B D A E C (3) B A C D E (3) D A B E C (2) C E B D A (2) C E B A D (2) C B E D A (2) B C D E A (2) B C D A E (2) B C A E D (2) B A D C E (2) B A C E D (2) A B D E C (2) E A D C B (1) D E B A C (1) D B E C A (1) D B A E C (1) C E A B D (1) C B A E D (1) B C A D E (1) A E C D B (1) Total count = 100 A B C D E A 0 16 12 -10 -12 B -16 0 -8 -22 -16 C -12 8 0 -10 -16 D 10 22 10 0 -2 E 12 16 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999152 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 12 -10 -12 B -16 0 -8 -22 -16 C -12 8 0 -10 -16 D 10 22 10 0 -2 E 12 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=23 D=20 A=15 E=14 so E is eliminated. Round 2 votes counts: C=32 D=29 B=23 A=16 so A is eliminated. Round 3 votes counts: D=42 C=33 B=25 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:223 D:220 A:203 C:185 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 12 -10 -12 B -16 0 -8 -22 -16 C -12 8 0 -10 -16 D 10 22 10 0 -2 E 12 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 -10 -12 B -16 0 -8 -22 -16 C -12 8 0 -10 -16 D 10 22 10 0 -2 E 12 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 -10 -12 B -16 0 -8 -22 -16 C -12 8 0 -10 -16 D 10 22 10 0 -2 E 12 16 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 557: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (5) E A C B D (5) D A C E B (5) B C D A E (5) E B A C D (4) D C B A E (4) C B A D E (4) B C E A D (4) B C A D E (4) E D A B C (3) E B C A D (3) E A C D B (3) D E A B C (3) C B A E D (3) B C E D A (3) A C B E D (3) A C B D E (3) E D B C A (2) E D B A C (2) E B D C A (2) E A D B C (2) E A B C D (2) D E B C A (2) C B D A E (2) B C D E A (2) A D E C B (2) A D C B E (2) E B D A C (1) E B C D A (1) D E B A C (1) D E A C B (1) D C A B E (1) D B E C A (1) D A E C B (1) D A C B E (1) C A B E D (1) B E C D A (1) B D E C A (1) B C A E D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 6 10 -8 B 6 0 0 14 -8 C -6 0 0 14 -2 D -10 -14 -14 0 -10 E 8 8 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 10 -8 B 6 0 0 14 -8 C -6 0 0 14 -2 D -10 -14 -14 0 -10 E 8 8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=21 D=20 A=14 C=10 so C is eliminated. Round 2 votes counts: E=35 B=30 D=20 A=15 so A is eliminated. Round 3 votes counts: E=39 B=37 D=24 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:206 C:203 A:201 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 10 -8 B 6 0 0 14 -8 C -6 0 0 14 -2 D -10 -14 -14 0 -10 E 8 8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 10 -8 B 6 0 0 14 -8 C -6 0 0 14 -2 D -10 -14 -14 0 -10 E 8 8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 10 -8 B 6 0 0 14 -8 C -6 0 0 14 -2 D -10 -14 -14 0 -10 E 8 8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 558: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) B D C E A (8) A B D C E (8) E C D B A (7) A B D E C (7) C E D B A (6) A B C D E (6) A E C D B (5) A E C B D (5) E A D B C (3) E A C D B (3) C D B E A (3) C B D A E (3) B D C A E (3) B D A C E (3) A E D B C (3) A C E B D (3) E D C B A (2) E A D C B (2) D B E C A (2) B A D C E (2) E C D A B (1) E C A D B (1) D B C E A (1) C A E B D (1) A E D C B (1) A E B D C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 8 6 6 B -4 0 -4 20 12 C -8 4 0 6 12 D -6 -20 -6 0 10 E -6 -12 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 6 6 B -4 0 -4 20 12 C -8 4 0 6 12 D -6 -20 -6 0 10 E -6 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=21 E=19 B=16 D=3 so D is eliminated. Round 2 votes counts: A=41 C=21 E=19 B=19 so E is eliminated. Round 3 votes counts: A=49 C=32 B=19 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:212 C:207 D:189 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 6 B -4 0 -4 20 12 C -8 4 0 6 12 D -6 -20 -6 0 10 E -6 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 6 B -4 0 -4 20 12 C -8 4 0 6 12 D -6 -20 -6 0 10 E -6 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 6 B -4 0 -4 20 12 C -8 4 0 6 12 D -6 -20 -6 0 10 E -6 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 559: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) A C E B D (8) E A B C D (7) C A D E B (6) B E D A C (6) D E B C A (5) D B E C A (5) C A D B E (5) B D E C A (4) E D B A C (3) E A C B D (3) E B A D C (2) E B A C D (2) E A D C B (2) D C B A E (2) D C A B E (2) D B C E A (2) D B C A E (2) C D A B E (2) C A E D B (2) B D C A E (2) A E C D B (2) A E C B D (2) E A B D C (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A E B (1) C D A E B (1) C A B D E (1) B E A D C (1) B E A C D (1) B D E A C (1) B D C E A (1) B C A D E (1) B A C E D (1) A E B C D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 6 0 -14 B 2 0 14 8 -18 C -6 -14 0 -6 -12 D 0 -8 6 0 -8 E 14 18 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 0 -14 B 2 0 14 8 -18 C -6 -14 0 -6 -12 D 0 -8 6 0 -8 E 14 18 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=22 B=18 C=17 A=15 so A is eliminated. Round 2 votes counts: E=33 C=26 D=22 B=19 so B is eliminated. Round 3 votes counts: E=41 D=30 C=29 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:203 A:195 D:195 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 0 -14 B 2 0 14 8 -18 C -6 -14 0 -6 -12 D 0 -8 6 0 -8 E 14 18 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 -14 B 2 0 14 8 -18 C -6 -14 0 -6 -12 D 0 -8 6 0 -8 E 14 18 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 -14 B 2 0 14 8 -18 C -6 -14 0 -6 -12 D 0 -8 6 0 -8 E 14 18 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 560: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D C A (11) B D C E A (8) A E C D B (8) D B C E A (5) E B A D C (4) E A B D C (4) D C B A E (4) A E C B D (4) C A D B E (3) B E D C A (3) B D E C A (3) E B D A C (2) E A D B C (2) D C A B E (2) D B C A E (2) C D B A E (2) C D A B E (2) A C E D B (2) A C E B D (2) A C D E B (2) E B A C D (1) E A C D B (1) E A C B D (1) E A B C D (1) D E B A C (1) D B A C E (1) D A B C E (1) C B D A E (1) C A B D E (1) B E C D A (1) A E D C B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 2 4 2 4 B -2 0 0 -6 4 C -4 0 0 -12 4 D -2 6 12 0 4 E -4 -4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 2 4 B -2 0 0 -6 4 C -4 0 0 -12 4 D -2 6 12 0 4 E -4 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=27 D=16 B=15 C=9 so C is eliminated. Round 2 votes counts: A=37 E=27 D=20 B=16 so B is eliminated. Round 3 votes counts: A=37 D=32 E=31 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:206 B:198 C:194 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 2 4 B -2 0 0 -6 4 C -4 0 0 -12 4 D -2 6 12 0 4 E -4 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 4 B -2 0 0 -6 4 C -4 0 0 -12 4 D -2 6 12 0 4 E -4 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 4 B -2 0 0 -6 4 C -4 0 0 -12 4 D -2 6 12 0 4 E -4 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 561: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) A D E B C (6) A D C E B (6) C B D E A (5) D E A B C (4) D C E B A (4) C D A B E (4) C B E D A (4) E B D A C (3) E B C D A (3) D C A B E (3) C D B E A (3) A E B D C (3) E B D C A (2) D E C B A (2) D E B C A (2) D E B A C (2) D C A E B (2) D A C E B (2) D A C B E (2) C D B A E (2) B E C D A (2) B E C A D (2) B C E A D (2) A E B C D (2) A D C B E (2) A B E C D (2) E D B C A (1) E D B A C (1) E D A B C (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A C B (1) C B D A E (1) C B A E D (1) C A D B E (1) C A B D E (1) B D C E A (1) B C E D A (1) A E D B C (1) A D E C B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -12 -30 -12 B 4 0 -12 -22 -4 C 12 12 0 -24 12 D 30 22 24 0 30 E 12 4 -12 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -30 -12 B 4 0 -12 -22 -4 C 12 12 0 -24 12 D 30 22 24 0 30 E 12 4 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=25 C=22 E=14 B=8 so B is eliminated. Round 2 votes counts: D=32 C=25 A=25 E=18 so E is eliminated. Round 3 votes counts: D=40 C=32 A=28 so A is eliminated. Round 4 votes counts: D=62 C=38 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:253 C:206 E:187 B:183 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -12 -30 -12 B 4 0 -12 -22 -4 C 12 12 0 -24 12 D 30 22 24 0 30 E 12 4 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -30 -12 B 4 0 -12 -22 -4 C 12 12 0 -24 12 D 30 22 24 0 30 E 12 4 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -30 -12 B 4 0 -12 -22 -4 C 12 12 0 -24 12 D 30 22 24 0 30 E 12 4 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 562: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) A D E C B (6) A B D C E (6) D C E A B (5) B C E D A (5) A D C B E (4) A B E D C (4) E C B D A (3) C D E B A (3) C B E D A (3) B E C D A (3) B E C A D (3) B A E C D (3) E D C B A (2) E D C A B (2) E B C D A (2) E A B D C (2) D E C A B (2) D C A B E (2) D A E C B (2) D A C E B (2) C E B D A (2) B A C E D (2) A D C E B (2) A B C D E (2) E D A C B (1) D E A C B (1) D C A E B (1) C E D B A (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) B E A D C (1) B E A C D (1) B C E A D (1) B C A D E (1) B A E D C (1) B A C D E (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -8 -10 -8 B 2 0 -14 -2 -2 C 8 14 0 -2 -4 D 10 2 2 0 -6 E 8 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -8 -10 -8 B 2 0 -14 -2 -2 C 8 14 0 -2 -4 D 10 2 2 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=22 E=21 D=15 C=13 so C is eliminated. Round 2 votes counts: A=29 B=27 E=24 D=20 so D is eliminated. Round 3 votes counts: A=36 E=35 B=29 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:208 D:204 B:192 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 -10 -8 B 2 0 -14 -2 -2 C 8 14 0 -2 -4 D 10 2 2 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -10 -8 B 2 0 -14 -2 -2 C 8 14 0 -2 -4 D 10 2 2 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -10 -8 B 2 0 -14 -2 -2 C 8 14 0 -2 -4 D 10 2 2 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 563: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (9) B A C D E (7) A B E C D (6) E A B D C (5) D E C B A (4) E D C A B (3) D E C A B (3) D E B A C (3) D C E B A (3) D C B A E (3) D B E A C (3) C D E B A (3) C D E A B (3) C B D A E (3) C A B E D (3) E A D B C (2) E A B C D (2) D E A B C (2) D C E A B (2) C E D A B (2) B A C E D (2) A E B D C (2) A B E D C (2) A B C E D (2) E D B A C (1) D C B E A (1) D B C A E (1) C E A B D (1) C D B E A (1) C D B A E (1) B D C A E (1) B D A C E (1) B C A D E (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 0 4 -8 -8 B 0 0 10 -2 -2 C -4 -10 0 -8 2 D 8 2 8 0 12 E 8 2 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -8 -8 B 0 0 10 -2 -2 C -4 -10 0 -8 2 D 8 2 8 0 12 E 8 2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 E=23 B=14 A=12 so A is eliminated. Round 2 votes counts: C=26 E=25 D=25 B=24 so B is eliminated. Round 3 votes counts: C=38 E=34 D=28 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:215 B:203 E:198 A:194 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -8 -8 B 0 0 10 -2 -2 C -4 -10 0 -8 2 D 8 2 8 0 12 E 8 2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -8 -8 B 0 0 10 -2 -2 C -4 -10 0 -8 2 D 8 2 8 0 12 E 8 2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -8 -8 B 0 0 10 -2 -2 C -4 -10 0 -8 2 D 8 2 8 0 12 E 8 2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 564: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) A D E B C (9) A D E C B (8) D A E B C (7) C B E D A (7) E D A B C (5) E D A C B (4) E B D A C (4) E B C D A (4) C B A D E (4) A D B E C (4) E C B D A (3) E B D C A (3) C A D B E (3) C A B D E (3) B C E D A (3) E D C A B (2) C B A E D (2) B A D C E (2) A D C B E (2) E D B A C (1) E C D A B (1) C E B D A (1) C E B A D (1) C A D E B (1) B E D C A (1) B E C D A (1) B D A E C (1) B C A D E (1) B A D E C (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 0 4 0 B 0 0 -4 4 -8 C 0 4 0 -8 -18 D -4 -4 8 0 -4 E 0 8 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.671359 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.328641 Sum of squares = 0.558727862122 Cumulative probabilities = A: 0.671359 B: 0.671359 C: 0.671359 D: 0.671359 E: 1.000000 A B C D E A 0 0 0 4 0 B 0 0 -4 4 -8 C 0 4 0 -8 -18 D -4 -4 8 0 -4 E 0 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=27 A=24 B=11 D=7 so D is eliminated. Round 2 votes counts: C=31 A=31 E=27 B=11 so B is eliminated. Round 3 votes counts: A=36 C=35 E=29 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:202 D:198 B:196 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 0 B 0 0 -4 4 -8 C 0 4 0 -8 -18 D -4 -4 8 0 -4 E 0 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 0 B 0 0 -4 4 -8 C 0 4 0 -8 -18 D -4 -4 8 0 -4 E 0 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 0 B 0 0 -4 4 -8 C 0 4 0 -8 -18 D -4 -4 8 0 -4 E 0 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 565: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) A D C E B (8) C E B A D (7) E B C A D (6) D A B E C (6) C B E D A (6) A D E B C (6) D A E B C (4) E C B A D (3) D A C B E (3) D A B C E (3) C B E A D (3) C A D B E (3) B D E C A (3) A D E C B (3) E B C D A (2) E B A D C (2) C D A B E (2) C A E D B (2) A D C B E (2) E A D B C (1) E A B D C (1) E A B C D (1) D C A B E (1) C E B D A (1) C E A B D (1) C D B A E (1) C B D E A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B C E D A (1) B C D E A (1) A E D C B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -10 10 0 B -6 0 -6 0 -6 C 10 6 0 8 2 D -10 0 -8 0 0 E 0 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 10 0 B -6 0 -6 0 -6 C 10 6 0 8 2 D -10 0 -8 0 0 E 0 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=23 D=17 E=16 B=15 so B is eliminated. Round 2 votes counts: C=31 E=26 A=23 D=20 so D is eliminated. Round 3 votes counts: A=39 C=32 E=29 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:203 E:202 B:191 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 10 0 B -6 0 -6 0 -6 C 10 6 0 8 2 D -10 0 -8 0 0 E 0 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 10 0 B -6 0 -6 0 -6 C 10 6 0 8 2 D -10 0 -8 0 0 E 0 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 10 0 B -6 0 -6 0 -6 C 10 6 0 8 2 D -10 0 -8 0 0 E 0 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 566: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) B C D E A (7) E D A C B (5) E A D C B (4) C A D B E (4) B C D A E (4) B C A D E (4) A C D E B (4) E D A B C (3) E A D B C (3) E A B D C (3) C B D A E (3) B C A E D (3) A C D B E (3) E D B C A (2) E D B A C (2) E B D A C (2) D E C A B (2) D E A C B (2) D A E C B (2) C A B D E (2) B D E C A (2) B D C E A (2) A D E C B (2) A C E D B (2) E B D C A (1) E B A D C (1) E B A C D (1) D E B C A (1) D C E B A (1) D C B E A (1) D C A E B (1) D C A B E (1) D A C E B (1) C D B A E (1) C D A B E (1) C B A D E (1) B E D C A (1) B E C D A (1) B C E D A (1) B C E A D (1) A E B C D (1) A D C E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 4 0 2 B -14 0 -6 -14 -10 C -4 6 0 -8 0 D 0 14 8 0 6 E -2 10 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.434051 B: 0.000000 C: 0.000000 D: 0.565949 E: 0.000000 Sum of squares = 0.508698659644 Cumulative probabilities = A: 0.434051 B: 0.434051 C: 0.434051 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 0 2 B -14 0 -6 -14 -10 C -4 6 0 -8 0 D 0 14 8 0 6 E -2 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 A=23 D=12 C=12 so D is eliminated. Round 2 votes counts: E=32 B=26 A=26 C=16 so C is eliminated. Round 3 votes counts: A=35 E=33 B=32 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:210 E:201 C:197 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 0 2 B -14 0 -6 -14 -10 C -4 6 0 -8 0 D 0 14 8 0 6 E -2 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 0 2 B -14 0 -6 -14 -10 C -4 6 0 -8 0 D 0 14 8 0 6 E -2 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 0 2 B -14 0 -6 -14 -10 C -4 6 0 -8 0 D 0 14 8 0 6 E -2 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 567: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) C B D E A (7) A E D B C (7) E A B D C (4) E A B C D (4) D C A E B (4) C D B E A (4) B C E D A (4) B C D E A (4) E C A B D (3) D C B A E (3) D A B E C (3) B E A C D (3) B D C A E (3) A E D C B (3) E B A C D (2) E A C D B (2) C E B A D (2) C B E D A (2) B E C A D (2) B E A D C (2) B D A C E (2) B C E A D (2) B A E D C (2) E B C A D (1) E A D C B (1) E A C B D (1) D C A B E (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A D B (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A B E (1) C B E A D (1) C B D A E (1) A D E C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 2 8 -8 B 0 0 14 22 2 C -2 -14 0 0 -4 D -8 -22 0 0 -16 E 8 -2 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.167451 B: 0.832549 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.721177593975 Cumulative probabilities = A: 0.167451 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 8 -8 B 0 0 14 22 2 C -2 -14 0 0 -4 D -8 -22 0 0 -16 E 8 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199999 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000593288 Cumulative probabilities = A: 0.199999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=22 A=21 E=18 D=15 so D is eliminated. Round 2 votes counts: C=30 A=27 B=25 E=18 so E is eliminated. Round 3 votes counts: A=39 C=33 B=28 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:219 E:213 A:201 C:190 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 8 -8 B 0 0 14 22 2 C -2 -14 0 0 -4 D -8 -22 0 0 -16 E 8 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199999 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000593288 Cumulative probabilities = A: 0.199999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 8 -8 B 0 0 14 22 2 C -2 -14 0 0 -4 D -8 -22 0 0 -16 E 8 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199999 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000593288 Cumulative probabilities = A: 0.199999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 8 -8 B 0 0 14 22 2 C -2 -14 0 0 -4 D -8 -22 0 0 -16 E 8 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.199999 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000593288 Cumulative probabilities = A: 0.199999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (13) B E D A C (11) E B D A C (7) C A E B D (7) C A D E B (7) E B A D C (6) D B E C A (6) C D B E A (5) C A D B E (5) A C E B D (5) C D A B E (3) A C D E B (3) D C B E A (2) D B C E A (2) C A E D B (2) B D E A C (2) A E C B D (2) A C E D B (2) E B A C D (1) E A B D C (1) E A B C D (1) D C B A E (1) C E A B D (1) C B D E A (1) B E D C A (1) A E B D C (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -16 14 -8 -20 B 16 0 8 -4 4 C -14 -8 0 -8 -8 D 8 4 8 0 2 E 20 -4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 14 -8 -20 B 16 0 8 -4 4 C -14 -8 0 -8 -8 D 8 4 8 0 2 E 20 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=24 E=16 A=15 B=14 so B is eliminated. Round 2 votes counts: C=31 E=28 D=26 A=15 so A is eliminated. Round 3 votes counts: C=41 E=32 D=27 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:212 D:211 E:211 A:185 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 14 -8 -20 B 16 0 8 -4 4 C -14 -8 0 -8 -8 D 8 4 8 0 2 E 20 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 14 -8 -20 B 16 0 8 -4 4 C -14 -8 0 -8 -8 D 8 4 8 0 2 E 20 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 14 -8 -20 B 16 0 8 -4 4 C -14 -8 0 -8 -8 D 8 4 8 0 2 E 20 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 569: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) D C A B E (7) E D C A B (6) E D A B C (6) E B A C D (6) B C A D E (6) E A B D C (5) D C E A B (5) D C A E B (5) B E A C D (5) E D A C B (4) C D A B E (4) E A D B C (3) B A E C D (3) E D C B A (2) E D B A C (2) E B D A C (2) E B C D A (2) D C E B A (2) B C E D A (2) B A C D E (2) A B E C D (2) E B A D C (1) D E A C B (1) D A E C B (1) C D B E A (1) C D B A E (1) C B D A E (1) A E D B C (1) A E B D C (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 24 -4 -24 -26 B -24 0 2 -18 -26 C 4 -2 0 -24 -22 D 24 18 24 0 -8 E 26 26 22 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 -4 -24 -26 B -24 0 2 -18 -26 C 4 -2 0 -24 -22 D 24 18 24 0 -8 E 26 26 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=29 B=18 C=7 A=7 so C is eliminated. Round 2 votes counts: E=39 D=35 B=19 A=7 so A is eliminated. Round 3 votes counts: E=41 D=35 B=24 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:241 D:229 A:185 C:178 B:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 -4 -24 -26 B -24 0 2 -18 -26 C 4 -2 0 -24 -22 D 24 18 24 0 -8 E 26 26 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -4 -24 -26 B -24 0 2 -18 -26 C 4 -2 0 -24 -22 D 24 18 24 0 -8 E 26 26 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -4 -24 -26 B -24 0 2 -18 -26 C 4 -2 0 -24 -22 D 24 18 24 0 -8 E 26 26 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 570: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) C A D E B (6) B D C E A (6) C D B A E (5) E A B D C (4) D C A E B (4) D B C E A (4) B D E C A (4) A E C B D (4) A C E D B (4) E B A D C (3) D E B A C (3) B E D A C (3) A E C D B (3) E A B C D (2) D C B E A (2) D C B A E (2) D B E C A (2) C D A B E (2) C A E B D (2) B E A D C (2) B E A C D (2) B D E A C (2) B C D E A (2) B C A E D (2) A E D C B (2) A E B C D (2) A C D E B (2) E D A B C (1) E B A C D (1) E A D B C (1) D C A B E (1) C A E D B (1) C A D B E (1) B E D C A (1) B E C D A (1) B C E D A (1) B C E A D (1) B A E C D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 0 -12 -4 4 B 0 0 6 -6 -6 C 12 -6 0 2 8 D 4 6 -2 0 8 E -4 6 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102048 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -4 4 B 0 0 6 -6 -6 C 12 -6 0 2 8 D 4 6 -2 0 8 E -4 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102018 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 A=19 D=18 E=12 so E is eliminated. Round 2 votes counts: B=32 A=26 C=23 D=19 so D is eliminated. Round 3 votes counts: B=41 C=32 A=27 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:208 D:208 B:197 A:194 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -12 -4 4 B 0 0 6 -6 -6 C 12 -6 0 2 8 D 4 6 -2 0 8 E -4 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102018 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -4 4 B 0 0 6 -6 -6 C 12 -6 0 2 8 D 4 6 -2 0 8 E -4 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102018 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -4 4 B 0 0 6 -6 -6 C 12 -6 0 2 8 D 4 6 -2 0 8 E -4 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102018 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 571: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (8) E C D A B (7) E C A B D (7) D E C B A (6) D C E B A (6) A E C B D (5) A B E C D (5) E C A D B (4) E A C B D (4) D B C A E (4) C E A B D (4) B A D C E (4) A E B C D (4) C E D B A (3) A B D E C (3) A B C E D (3) D E B C A (2) D C B E A (2) D B C E A (2) C E D A B (2) E D C B A (1) E D C A B (1) E A B C D (1) D E B A C (1) D E A B C (1) D B E C A (1) D B A E C (1) D B A C E (1) D A B E C (1) C A E B D (1) C A B E D (1) A E B D C (1) A C B E D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -8 0 -10 B -16 0 -10 8 -22 C 8 10 0 4 -14 D 0 -8 -4 0 -12 E 10 22 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -8 0 -10 B -16 0 -10 8 -22 C 8 10 0 4 -14 D 0 -8 -4 0 -12 E 10 22 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=25 A=24 B=12 C=11 so C is eliminated. Round 2 votes counts: E=34 D=28 A=26 B=12 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:204 A:199 D:188 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -8 0 -10 B -16 0 -10 8 -22 C 8 10 0 4 -14 D 0 -8 -4 0 -12 E 10 22 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 0 -10 B -16 0 -10 8 -22 C 8 10 0 4 -14 D 0 -8 -4 0 -12 E 10 22 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 0 -10 B -16 0 -10 8 -22 C 8 10 0 4 -14 D 0 -8 -4 0 -12 E 10 22 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 572: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) A D C E B (7) A C D B E (7) E D B C A (6) D E B A C (6) E B D C A (5) D E A B C (5) B E D C A (5) D E B C A (4) C A B D E (4) A C D E B (4) A C B D E (4) C B A E D (3) B E C D A (3) B C E A D (3) B C A E D (3) D E A C B (2) D A E C B (2) B C E D A (2) E D B A C (1) E D A B C (1) E B D A C (1) E A D B C (1) D C A E B (1) D A C E B (1) C D A B E (1) C B D A E (1) C A D B E (1) C A B E D (1) B E C A D (1) B C D E A (1) A E B C D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 10 10 4 6 B -10 0 0 -22 -18 C -10 0 0 -16 -8 D -4 22 16 0 22 E -6 18 8 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999169 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 4 6 B -10 0 0 -22 -18 C -10 0 0 -16 -8 D -4 22 16 0 22 E -6 18 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=21 B=18 E=15 C=11 so C is eliminated. Round 2 votes counts: A=41 D=22 B=22 E=15 so E is eliminated. Round 3 votes counts: A=42 D=30 B=28 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:228 A:215 E:199 C:183 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 4 6 B -10 0 0 -22 -18 C -10 0 0 -16 -8 D -4 22 16 0 22 E -6 18 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 4 6 B -10 0 0 -22 -18 C -10 0 0 -16 -8 D -4 22 16 0 22 E -6 18 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 4 6 B -10 0 0 -22 -18 C -10 0 0 -16 -8 D -4 22 16 0 22 E -6 18 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 573: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) E B D A C (8) A C B D E (7) A B C E D (7) D E C A B (6) C D A B E (6) E D B A C (5) B A E C D (5) D C E A B (4) B E A C D (4) E D C B A (3) D E C B A (3) B A C E D (3) E B D C A (2) E B A D C (2) D C E B A (2) D C A E B (2) C D B E A (2) C D A E B (2) C A D B E (2) C A B D E (2) A C D B E (2) A C B E D (2) E D A C B (1) E A D B C (1) D C B E A (1) D A E C B (1) C D B A E (1) C B A D E (1) C A D E B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 0 -18 -8 B 4 0 -4 -12 -8 C 0 4 0 -2 -4 D 18 12 2 0 -10 E 8 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 -18 -8 B 4 0 -4 -12 -8 C 0 4 0 -2 -4 D 18 12 2 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=20 D=19 C=17 B=12 so B is eliminated. Round 2 votes counts: E=36 A=28 D=19 C=17 so C is eliminated. Round 3 votes counts: E=36 A=34 D=30 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:211 C:199 B:190 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -18 -8 B 4 0 -4 -12 -8 C 0 4 0 -2 -4 D 18 12 2 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -18 -8 B 4 0 -4 -12 -8 C 0 4 0 -2 -4 D 18 12 2 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -18 -8 B 4 0 -4 -12 -8 C 0 4 0 -2 -4 D 18 12 2 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 574: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E C D A B (8) A B D E C (8) B A D C E (5) A B C D E (5) E C A B D (4) E A D B C (4) C D B E A (4) A B E D C (4) E C D B A (3) E C A D B (3) D C B E A (3) D B C A E (3) C E D B A (3) B D A C E (3) E A D C B (2) D B C E A (2) C E B A D (2) C D E B A (2) A C B E D (2) A B C E D (2) E D A B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E B C A (1) D B E A C (1) D B A C E (1) D A E B C (1) C B D A E (1) C B A E D (1) C B A D E (1) C A B E D (1) B C A D E (1) B A C D E (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 24 14 26 8 B -24 0 14 14 24 C -14 -14 0 -4 10 D -26 -14 4 0 10 E -8 -24 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 14 26 8 B -24 0 14 14 24 C -14 -14 0 -4 10 D -26 -14 4 0 10 E -8 -24 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=28 C=15 D=12 B=10 so B is eliminated. Round 2 votes counts: A=41 E=28 C=16 D=15 so D is eliminated. Round 3 votes counts: A=46 E=30 C=24 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:236 B:214 C:189 D:187 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 14 26 8 B -24 0 14 14 24 C -14 -14 0 -4 10 D -26 -14 4 0 10 E -8 -24 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 14 26 8 B -24 0 14 14 24 C -14 -14 0 -4 10 D -26 -14 4 0 10 E -8 -24 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 14 26 8 B -24 0 14 14 24 C -14 -14 0 -4 10 D -26 -14 4 0 10 E -8 -24 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 575: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) C D A B E (8) B D E A C (6) B D E C A (5) B D C E A (5) E B A D C (4) E A B C D (4) B E D A C (4) B D C A E (4) A C D E B (4) E B D C A (3) E A C D B (3) D B C E A (3) A E C D B (3) A C D B E (3) E B A C D (2) E A C B D (2) D C B A E (2) D C A B E (2) C D E A B (2) C A E D B (2) C A D E B (2) B E A D C (2) A E B C D (2) A D C B E (2) A C E D B (2) E C A D B (1) E C A B D (1) E B C D A (1) E B C A D (1) D C B E A (1) C E A D B (1) B E D C A (1) B D A E C (1) B D A C E (1) A E C B D (1) Total count = 100 A B C D E A 0 -10 -8 -16 -4 B 10 0 16 0 18 C 8 -16 0 -10 6 D 16 0 10 0 20 E 4 -18 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.545008 C: 0.000000 D: 0.454992 E: 0.000000 Sum of squares = 0.504051421514 Cumulative probabilities = A: 0.000000 B: 0.545008 C: 0.545008 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -16 -4 B 10 0 16 0 18 C 8 -16 0 -10 6 D 16 0 10 0 20 E 4 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=22 D=17 A=17 C=15 so C is eliminated. Round 2 votes counts: B=29 D=27 E=23 A=21 so A is eliminated. Round 3 votes counts: D=38 E=33 B=29 so B is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:222 C:194 A:181 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -16 -4 B 10 0 16 0 18 C 8 -16 0 -10 6 D 16 0 10 0 20 E 4 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -16 -4 B 10 0 16 0 18 C 8 -16 0 -10 6 D 16 0 10 0 20 E 4 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -16 -4 B 10 0 16 0 18 C 8 -16 0 -10 6 D 16 0 10 0 20 E 4 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 576: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) E C A D B (8) E C D A B (7) D C B A E (5) B D A C E (5) E B A C D (4) E A C D B (4) C D A E B (4) B E D C A (4) B E A C D (4) B A D C E (4) B E A D C (3) B D C A E (3) B A E C D (3) A B D C E (3) E B D C A (2) E A B C D (2) D C A E B (2) B E D A C (2) B D E C A (2) B A E D C (2) A C E D B (2) A B E C D (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B D A (1) E B D A C (1) E B C D A (1) E B C A D (1) D C E A B (1) D A C B E (1) D A B C E (1) C D E A B (1) A C D E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -8 -10 -2 B 0 0 2 0 10 C 8 -2 0 -4 -12 D 10 0 4 0 -14 E 2 -10 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.661192 C: 0.000000 D: 0.338808 E: 0.000000 Sum of squares = 0.551965584708 Cumulative probabilities = A: 0.000000 B: 0.661192 C: 0.661192 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -10 -2 B 0 0 2 0 10 C 8 -2 0 -4 -12 D 10 0 4 0 -14 E 2 -10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.583333 C: 0.000000 D: 0.416667 E: 0.000000 Sum of squares = 0.513888904082 Cumulative probabilities = A: 0.000000 B: 0.583333 C: 0.583333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=32 D=19 A=10 C=5 so C is eliminated. Round 2 votes counts: E=34 B=32 D=24 A=10 so A is eliminated. Round 3 votes counts: B=38 E=36 D=26 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:209 B:206 D:200 C:195 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -10 -2 B 0 0 2 0 10 C 8 -2 0 -4 -12 D 10 0 4 0 -14 E 2 -10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.583333 C: 0.000000 D: 0.416667 E: 0.000000 Sum of squares = 0.513888904082 Cumulative probabilities = A: 0.000000 B: 0.583333 C: 0.583333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -10 -2 B 0 0 2 0 10 C 8 -2 0 -4 -12 D 10 0 4 0 -14 E 2 -10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.583333 C: 0.000000 D: 0.416667 E: 0.000000 Sum of squares = 0.513888904082 Cumulative probabilities = A: 0.000000 B: 0.583333 C: 0.583333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -10 -2 B 0 0 2 0 10 C 8 -2 0 -4 -12 D 10 0 4 0 -14 E 2 -10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.583333 C: 0.000000 D: 0.416667 E: 0.000000 Sum of squares = 0.513888904082 Cumulative probabilities = A: 0.000000 B: 0.583333 C: 0.583333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 577: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (17) E D C A B (8) C E D B A (8) B C E D A (5) B C D E A (5) E C D A B (4) C D E A B (4) C B E D A (4) B A D C E (4) A B D E C (4) C E D A B (3) D E C A B (2) D E A C B (2) C B D E A (2) B A E D C (2) B A E C D (2) B A C D E (2) A D E B C (2) A D B E C (2) A D B C E (2) E D A C B (1) E C D B A (1) E A D C B (1) D C E A B (1) D A C E B (1) C E B D A (1) C D E B A (1) B E C D A (1) B E C A D (1) B C E A D (1) B C A D E (1) B A D E C (1) A E D C B (1) A E D B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -6 -8 -12 B -16 0 -24 -24 -18 C 6 24 0 -8 -8 D 8 24 8 0 8 E 12 18 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -6 -8 -12 B -16 0 -24 -24 -18 C 6 24 0 -8 -8 D 8 24 8 0 8 E 12 18 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=25 C=23 E=15 D=6 so D is eliminated. Round 2 votes counts: A=32 B=25 C=24 E=19 so E is eliminated. Round 3 votes counts: C=39 A=36 B=25 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:224 E:215 C:207 A:195 B:159 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -6 -8 -12 B -16 0 -24 -24 -18 C 6 24 0 -8 -8 D 8 24 8 0 8 E 12 18 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 -8 -12 B -16 0 -24 -24 -18 C 6 24 0 -8 -8 D 8 24 8 0 8 E 12 18 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 -8 -12 B -16 0 -24 -24 -18 C 6 24 0 -8 -8 D 8 24 8 0 8 E 12 18 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 578: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) B D E A C (6) A C D B E (6) E B C D A (5) A D C B E (5) C A E B D (4) B E C A D (4) A C D E B (4) E C B D A (3) E B D C A (3) D E B A C (3) D B A E C (3) C E D A B (3) C E B A D (3) C A D E B (3) C A B E D (3) B C A E D (3) E B D A C (2) D E A C B (2) D A E C B (2) D A C E B (2) D A B C E (2) C A E D B (2) B E D C A (2) B E D A C (2) B D A E C (2) A D C E B (2) A C B D E (2) E D B A C (1) E C B A D (1) D E C A B (1) D B E A C (1) D A E B C (1) C E A D B (1) C E A B D (1) C D A E B (1) C B A E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 12 -6 12 B -10 0 -4 -2 0 C -12 4 0 2 0 D 6 2 -2 0 12 E -12 0 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.460000000129 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 -6 12 B -10 0 -4 -2 0 C -12 4 0 2 0 D 6 2 -2 0 12 E -12 0 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999905 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 A=21 B=19 E=15 so E is eliminated. Round 2 votes counts: B=29 C=26 D=24 A=21 so A is eliminated. Round 3 votes counts: C=38 D=31 B=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:214 D:209 C:197 B:192 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 12 -6 12 B -10 0 -4 -2 0 C -12 4 0 2 0 D 6 2 -2 0 12 E -12 0 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999905 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -6 12 B -10 0 -4 -2 0 C -12 4 0 2 0 D 6 2 -2 0 12 E -12 0 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999905 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -6 12 B -10 0 -4 -2 0 C -12 4 0 2 0 D 6 2 -2 0 12 E -12 0 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999905 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 579: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) E B D C A (10) A C D B E (10) E D C A B (9) E B A C D (9) B A C D E (6) C D A E B (5) D C E A B (4) D C A B E (4) D E C A B (3) D C A E B (3) E D C B A (2) E D B C A (2) E C D A B (2) C A D B E (2) B A D C E (2) B A C E D (2) A C B D E (2) E B D A C (1) E B C D A (1) E A C B D (1) D E C B A (1) C D A B E (1) C A E D B (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E C D (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 0 -16 B 2 0 0 0 -8 C 0 0 0 10 -12 D 0 0 -10 0 -6 E 16 8 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 0 -16 B 2 0 0 0 -8 C 0 0 0 10 -12 D 0 0 -10 0 -6 E 16 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=25 D=15 A=14 C=9 so C is eliminated. Round 2 votes counts: E=37 B=25 D=21 A=17 so A is eliminated. Round 3 votes counts: E=38 D=34 B=28 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:199 B:197 D:192 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 0 -16 B 2 0 0 0 -8 C 0 0 0 10 -12 D 0 0 -10 0 -6 E 16 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 -16 B 2 0 0 0 -8 C 0 0 0 10 -12 D 0 0 -10 0 -6 E 16 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 -16 B 2 0 0 0 -8 C 0 0 0 10 -12 D 0 0 -10 0 -6 E 16 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 580: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (6) E D C B A (5) B D E C A (5) B A D C E (5) A C E D B (5) E C D A B (4) E B D C A (4) C E D A B (4) B D E A C (4) B A D E C (4) E C D B A (3) E C A D B (3) D C E B A (3) B A E D C (3) A E C B D (3) A C D E B (3) A C B E D (3) A B D C E (3) A B C D E (3) E A C D B (2) D B E C A (2) D B C A E (2) C E A D B (2) B D C A E (2) B D A C E (2) A B C E D (2) E A B C D (1) D E C B A (1) D E B C A (1) D C B E A (1) D B C E A (1) C D E A B (1) C A E D B (1) C A D E B (1) B E A D C (1) B D A E C (1) A E C D B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 0 8 8 4 B 0 0 6 8 0 C -8 -6 0 0 -10 D -8 -8 0 0 -10 E -4 0 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.520406 B: 0.479594 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500832783301 Cumulative probabilities = A: 0.520406 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 8 4 B 0 0 6 8 0 C -8 -6 0 0 -10 D -8 -8 0 0 -10 E -4 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999715 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=27 E=22 D=11 C=9 so C is eliminated. Round 2 votes counts: A=33 E=28 B=27 D=12 so D is eliminated. Round 3 votes counts: E=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 E:208 B:207 C:188 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 8 4 B 0 0 6 8 0 C -8 -6 0 0 -10 D -8 -8 0 0 -10 E -4 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999715 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 8 4 B 0 0 6 8 0 C -8 -6 0 0 -10 D -8 -8 0 0 -10 E -4 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999715 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 8 4 B 0 0 6 8 0 C -8 -6 0 0 -10 D -8 -8 0 0 -10 E -4 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999715 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 581: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) E C A B D (6) E B C D A (6) E B C A D (6) D A C B E (6) D B A C E (5) D A B C E (5) C D A E B (5) C E D A B (4) C E A D B (4) C A D E B (4) B E D A C (4) E C B A D (3) D C A E B (3) B A D E C (3) A D B C E (3) E C D A B (2) D C E B A (2) B E D C A (2) B E A D C (2) B A D C E (2) A E C B D (2) E D C B A (1) E C B D A (1) E C A D B (1) D B A E C (1) D A C E B (1) C A E B D (1) B D E C A (1) B A E D C (1) B A E C D (1) A D C B E (1) A C E B D (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -4 -14 10 B -2 0 2 2 -8 C 4 -2 0 -4 -2 D 14 -2 4 0 4 E -10 8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428624 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 A B C D E A 0 2 -4 -14 10 B -2 0 2 2 -8 C 4 -2 0 -4 -2 D 14 -2 4 0 4 E -10 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428647 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 D=23 C=18 A=9 so A is eliminated. Round 2 votes counts: E=28 D=27 B=25 C=20 so C is eliminated. Round 3 votes counts: E=38 D=37 B=25 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:198 E:198 A:197 B:197 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -14 10 B -2 0 2 2 -8 C 4 -2 0 -4 -2 D 14 -2 4 0 4 E -10 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428647 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -14 10 B -2 0 2 2 -8 C 4 -2 0 -4 -2 D 14 -2 4 0 4 E -10 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428647 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -14 10 B -2 0 2 2 -8 C 4 -2 0 -4 -2 D 14 -2 4 0 4 E -10 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428647 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 582: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (15) C B A E D (10) D A E B C (9) A B C E D (7) D E C B A (5) D E B A C (5) C B E A D (5) A D B E C (4) D E B C A (3) C D E B A (3) C A B E D (3) E D B A C (2) D A E C B (2) B C E A D (2) A D E B C (2) A C B E D (2) E D C B A (1) E D B C A (1) E C D B A (1) E B C D A (1) D E C A B (1) D E A C B (1) D C E B A (1) D C A E B (1) D A C E B (1) D A B E C (1) C E D B A (1) C E B D A (1) C D E A B (1) C B E D A (1) C B D A E (1) C B A D E (1) C A D B E (1) A E D B C (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 -18 -2 B -10 0 10 -26 -18 C -10 -10 0 -14 -14 D 18 26 14 0 18 E 2 18 14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -18 -2 B -10 0 10 -26 -18 C -10 -10 0 -14 -14 D 18 26 14 0 18 E 2 18 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 C=28 A=19 E=6 B=2 so B is eliminated. Round 2 votes counts: D=45 C=30 A=19 E=6 so E is eliminated. Round 3 votes counts: D=49 C=32 A=19 so A is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:238 E:208 A:200 B:178 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -18 -2 B -10 0 10 -26 -18 C -10 -10 0 -14 -14 D 18 26 14 0 18 E 2 18 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -18 -2 B -10 0 10 -26 -18 C -10 -10 0 -14 -14 D 18 26 14 0 18 E 2 18 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -18 -2 B -10 0 10 -26 -18 C -10 -10 0 -14 -14 D 18 26 14 0 18 E 2 18 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 583: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) E A B C D (8) B C D E A (8) D C A B E (7) E A D B C (5) D C B A E (5) E A C B D (4) D B C A E (4) B E C A D (4) B D C E A (4) E B A C D (3) C B D A E (3) B C E D A (3) B C D A E (3) A E C B D (3) A D C E B (3) D B E C A (2) D B C E A (2) D A C E B (2) C D B A E (2) A D E C B (2) A C D E B (2) E B A D C (1) E A C D B (1) E A B D C (1) D B E A C (1) D A E C B (1) D A E B C (1) D A C B E (1) C A D B E (1) B C E A D (1) A C E D B (1) Total count = 100 A B C D E A 0 8 2 2 4 B -8 0 2 -8 2 C -2 -2 0 -6 4 D -2 8 6 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 2 4 B -8 0 2 -8 2 C -2 -2 0 -6 4 D -2 8 6 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999083 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 B=23 A=22 C=6 so C is eliminated. Round 2 votes counts: D=28 B=26 E=23 A=23 so E is eliminated. Round 3 votes counts: A=42 B=30 D=28 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:208 C:197 B:194 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 2 4 B -8 0 2 -8 2 C -2 -2 0 -6 4 D -2 8 6 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999083 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 4 B -8 0 2 -8 2 C -2 -2 0 -6 4 D -2 8 6 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999083 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 4 B -8 0 2 -8 2 C -2 -2 0 -6 4 D -2 8 6 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999083 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 584: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) C A B D E (6) E B D A C (5) C B A E D (5) D A E C B (4) C E B D A (4) C B E A D (4) B A D E C (4) B A D C E (4) C D E A B (3) C D A E B (3) B E C A D (3) A D B E C (3) A B D C E (3) E C D A B (2) D E A B C (2) D A C E B (2) C E D B A (2) C B A D E (2) C A D E B (2) B E A D C (2) B A E D C (2) B A C D E (2) A D B C E (2) A B D E C (2) E D B A C (1) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) C A D B E (1) B E D C A (1) B C E D A (1) B C E A D (1) B C A E D (1) B C A D E (1) B A C E D (1) A D C B E (1) Total count = 100 A B C D E A 0 0 -18 4 4 B 0 0 -14 14 4 C 18 14 0 14 26 D -4 -14 -14 0 2 E -4 -4 -26 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 4 4 B 0 0 -14 14 4 C 18 14 0 14 26 D -4 -14 -14 0 2 E -4 -4 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 B=23 E=12 D=12 A=11 so A is eliminated. Round 2 votes counts: C=42 B=28 D=18 E=12 so E is eliminated. Round 3 votes counts: C=46 B=35 D=19 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:236 B:202 A:195 D:185 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -18 4 4 B 0 0 -14 14 4 C 18 14 0 14 26 D -4 -14 -14 0 2 E -4 -4 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 4 4 B 0 0 -14 14 4 C 18 14 0 14 26 D -4 -14 -14 0 2 E -4 -4 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 4 4 B 0 0 -14 14 4 C 18 14 0 14 26 D -4 -14 -14 0 2 E -4 -4 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 585: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) B E C D A (6) A D E C B (6) E D B A C (5) E D A B C (5) C B A E D (5) C B A D E (5) D E B C A (4) D A E C B (4) C B D A E (4) A D C E B (4) E B D A C (3) C A B E D (3) A E D C B (3) A C D B E (3) E B D C A (2) C D B A E (2) C A D B E (2) B E D C A (2) B C E D A (2) A E D B C (2) A D C B E (2) A C D E B (2) E B A C D (1) D E B A C (1) D E A B C (1) D C B A E (1) D B E C A (1) D B C E A (1) D A C E B (1) D A C B E (1) B E C A D (1) B D E C A (1) B C E A D (1) B C D E A (1) A E B C D (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -4 4 24 B -2 0 -18 -4 8 C 4 18 0 0 2 D -4 4 0 0 12 E -24 -8 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.635682 D: 0.364318 E: 0.000000 Sum of squares = 0.536819334901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.635682 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 4 24 B -2 0 -18 -4 8 C 4 18 0 0 2 D -4 4 0 0 12 E -24 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.499997 E: 0.000000 Sum of squares = 0.50000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 E=16 D=15 B=14 so B is eliminated. Round 2 votes counts: C=33 A=26 E=25 D=16 so D is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:212 D:206 B:192 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 4 24 B -2 0 -18 -4 8 C 4 18 0 0 2 D -4 4 0 0 12 E -24 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.499997 E: 0.000000 Sum of squares = 0.50000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 24 B -2 0 -18 -4 8 C 4 18 0 0 2 D -4 4 0 0 12 E -24 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.499997 E: 0.000000 Sum of squares = 0.50000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 24 B -2 0 -18 -4 8 C 4 18 0 0 2 D -4 4 0 0 12 E -24 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.499997 E: 0.000000 Sum of squares = 0.50000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 586: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (6) D B A C E (5) C E B D A (5) A E C D B (5) D C B A E (4) C E B A D (4) A E C B D (4) A D E C B (4) E B C A D (3) E A C B D (3) D B C A E (3) C B E D A (3) B C D E A (3) A E B D C (3) E C B A D (2) E C A B D (2) D C A B E (2) D B A E C (2) D A B E C (2) C D A E B (2) C B D E A (2) B E C D A (2) B D C E A (2) B D A E C (2) B A D E C (2) A D E B C (2) A D C E B (2) E A B D C (1) E A B C D (1) D A C B E (1) C E D B A (1) C E D A B (1) C E A D B (1) C E A B D (1) C A E D B (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C A D (1) B A E D C (1) A E D C B (1) A E D B C (1) A E B C D (1) A C E D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 6 12 20 B -2 0 -6 4 -6 C -6 6 0 0 -10 D -12 -4 0 0 -4 E -20 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 12 20 B -2 0 -6 4 -6 C -6 6 0 0 -10 D -12 -4 0 0 -4 E -20 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=22 D=19 B=15 E=12 so E is eliminated. Round 2 votes counts: A=37 C=26 D=19 B=18 so B is eliminated. Round 3 votes counts: A=40 C=35 D=25 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:200 B:195 C:195 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 12 20 B -2 0 -6 4 -6 C -6 6 0 0 -10 D -12 -4 0 0 -4 E -20 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 12 20 B -2 0 -6 4 -6 C -6 6 0 0 -10 D -12 -4 0 0 -4 E -20 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 12 20 B -2 0 -6 4 -6 C -6 6 0 0 -10 D -12 -4 0 0 -4 E -20 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 587: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (15) A C B D E (12) A E D B C (8) C B D E A (6) E D B A C (5) E D A B C (5) D B E C A (5) A E D C B (5) A C B E D (5) E A D B C (4) C B D A E (4) C B A D E (4) D E B C A (3) C A B D E (3) A E C B D (3) D B C E A (2) B D C E A (2) B C D E A (2) A E C D B (2) E D C B A (1) C E B D A (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 -2 -2 B 0 0 8 -12 -4 C -4 -8 0 -14 -14 D 2 12 14 0 -8 E 2 4 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 -2 -2 B 0 0 8 -12 -4 C -4 -8 0 -14 -14 D 2 12 14 0 -8 E 2 4 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=30 C=18 D=10 B=4 so B is eliminated. Round 2 votes counts: A=38 E=30 C=20 D=12 so D is eliminated. Round 3 votes counts: E=38 A=38 C=24 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:210 A:200 B:196 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 -2 -2 B 0 0 8 -12 -4 C -4 -8 0 -14 -14 D 2 12 14 0 -8 E 2 4 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 -2 B 0 0 8 -12 -4 C -4 -8 0 -14 -14 D 2 12 14 0 -8 E 2 4 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 -2 B 0 0 8 -12 -4 C -4 -8 0 -14 -14 D 2 12 14 0 -8 E 2 4 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 588: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (14) E C D A B (9) C E D A B (8) C E D B A (7) B A D E C (7) D E C A B (5) D A B E C (5) C E B A D (5) B A C E D (4) E C A B D (3) D C E A B (3) D A E B C (3) C D E A B (3) B A E C D (3) B A D C E (3) D E A C B (2) D C B A E (2) D A B C E (2) B C A E D (2) B A E D C (2) B A C D E (2) D E A B C (1) C D E B A (1) C D B E A (1) C B E A D (1) B E C A D (1) A B E D C (1) Total count = 100 A B C D E A 0 18 -2 -4 0 B -18 0 0 -4 0 C 2 0 0 0 -12 D 4 4 0 0 8 E 0 0 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.274718 D: 0.725282 E: 0.000000 Sum of squares = 0.601504192909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.274718 D: 1.000000 E: 1.000000 A B C D E A 0 18 -2 -4 0 B -18 0 0 -4 0 C 2 0 0 0 -12 D 4 4 0 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399995 D: 0.600005 E: 0.000000 Sum of squares = 0.520001992181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399995 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 D=23 A=15 E=12 so E is eliminated. Round 2 votes counts: C=38 B=24 D=23 A=15 so A is eliminated. Round 3 votes counts: B=39 C=38 D=23 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:208 A:206 E:202 C:195 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -2 -4 0 B -18 0 0 -4 0 C 2 0 0 0 -12 D 4 4 0 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399995 D: 0.600005 E: 0.000000 Sum of squares = 0.520001992181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399995 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -2 -4 0 B -18 0 0 -4 0 C 2 0 0 0 -12 D 4 4 0 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399995 D: 0.600005 E: 0.000000 Sum of squares = 0.520001992181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399995 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -2 -4 0 B -18 0 0 -4 0 C 2 0 0 0 -12 D 4 4 0 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.399995 D: 0.600005 E: 0.000000 Sum of squares = 0.520001992181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.399995 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 589: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (14) D B A C E (12) C E A B D (11) E C A B D (9) E C D A B (8) D E B A C (6) E C A D B (5) E D B A C (3) E D A C B (3) C A B E D (3) B D A C E (3) E C D B A (2) E A D C B (2) D E B C A (2) C B A D E (2) B A D C E (2) B A C D E (2) E D C B A (1) E D C A B (1) E D A B C (1) E A C D B (1) D B E A C (1) D A B E C (1) D A B C E (1) C E B A D (1) C A B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 8 -18 -14 B 2 0 0 -28 -12 C -8 0 0 -8 -20 D 18 28 8 0 -2 E 14 12 20 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 -18 -14 B 2 0 0 -28 -12 C -8 0 0 -8 -20 D 18 28 8 0 -2 E 14 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=36 C=18 B=7 A=2 so A is eliminated. Round 2 votes counts: D=37 E=36 C=18 B=9 so B is eliminated. Round 3 votes counts: D=43 E=36 C=21 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:226 E:224 A:187 C:182 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 -18 -14 B 2 0 0 -28 -12 C -8 0 0 -8 -20 D 18 28 8 0 -2 E 14 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -18 -14 B 2 0 0 -28 -12 C -8 0 0 -8 -20 D 18 28 8 0 -2 E 14 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -18 -14 B 2 0 0 -28 -12 C -8 0 0 -8 -20 D 18 28 8 0 -2 E 14 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 590: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) D B E A C (7) A C D E B (7) D A B E C (6) A C B D E (6) C E A B D (5) D A E B C (4) C B E D A (4) B C E D A (4) A D E C B (4) C E B A D (3) C B E A D (3) B E C D A (3) A D B E C (3) E D B C A (2) E D A C B (2) E C B A D (2) E B D C A (2) E B C D A (2) D E A B C (2) B E D C A (2) A D B C E (2) A C E D B (2) E D C B A (1) E D C A B (1) E C D B A (1) E C D A B (1) D E B C A (1) D B E C A (1) C B A E D (1) C A E B D (1) C A B E D (1) B D E C A (1) B C D A E (1) B C A D E (1) A D E B C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -4 -12 -16 B 0 0 -10 0 -6 C 4 10 0 14 -14 D 12 0 -14 0 -2 E 16 6 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -4 -12 -16 B 0 0 -10 0 -6 C 4 10 0 14 -14 D 12 0 -14 0 -2 E 16 6 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 D=21 C=18 B=12 so B is eliminated. Round 2 votes counts: E=27 A=27 C=24 D=22 so D is eliminated. Round 3 votes counts: E=39 A=37 C=24 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:207 D:198 B:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 -12 -16 B 0 0 -10 0 -6 C 4 10 0 14 -14 D 12 0 -14 0 -2 E 16 6 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -12 -16 B 0 0 -10 0 -6 C 4 10 0 14 -14 D 12 0 -14 0 -2 E 16 6 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -12 -16 B 0 0 -10 0 -6 C 4 10 0 14 -14 D 12 0 -14 0 -2 E 16 6 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 591: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) E D A B C (6) C B E A D (6) E D A C B (5) C B A E D (5) B C A D E (5) A E D B C (5) A D B C E (5) E D C B A (4) C E B D A (4) C B A D E (4) B C D A E (4) A D E B C (4) E C B A D (3) E A D C B (3) D A E B C (3) D A B E C (3) E C B D A (2) E A D B C (2) D E B C A (2) D E A B C (2) D B C A E (2) C E B A D (2) B D A C E (2) A B C D E (2) E C D B A (1) E C A D B (1) E A C D B (1) D B A C E (1) C B E D A (1) C B D E A (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 6 0 8 B -2 0 12 -18 -2 C -6 -12 0 -16 2 D 0 18 16 0 -4 E -8 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.642485 B: 0.000000 C: 0.000000 D: 0.357515 E: 0.000000 Sum of squares = 0.540604209107 Cumulative probabilities = A: 0.642485 B: 0.642485 C: 0.642485 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 0 8 B -2 0 12 -18 -2 C -6 -12 0 -16 2 D 0 18 16 0 -4 E -8 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 D=20 A=18 B=11 so B is eliminated. Round 2 votes counts: C=32 E=28 D=22 A=18 so A is eliminated. Round 3 votes counts: E=34 C=34 D=32 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:215 A:208 E:198 B:195 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 0 8 B -2 0 12 -18 -2 C -6 -12 0 -16 2 D 0 18 16 0 -4 E -8 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 8 B -2 0 12 -18 -2 C -6 -12 0 -16 2 D 0 18 16 0 -4 E -8 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 8 B -2 0 12 -18 -2 C -6 -12 0 -16 2 D 0 18 16 0 -4 E -8 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 592: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) D B C E A (8) E A B C D (7) C A E D B (7) B D E A C (7) E A B D C (6) A E C B D (6) B E D A C (5) A E C D B (5) E A C B D (4) D B C A E (4) C D B A E (4) C D A E B (4) C A D E B (4) E B A D C (3) D E B A C (2) B E A D C (2) B D E C A (2) A C E D B (2) E B D A C (1) E B A C D (1) D E A B C (1) D B E C A (1) C D A B E (1) C B A D E (1) C A E B D (1) C A B D E (1) Total count = 100 A B C D E A 0 -2 4 0 0 B 2 0 0 -6 -8 C -4 0 0 -4 -6 D 0 6 4 0 0 E 0 8 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.378245 B: 0.000000 C: 0.000000 D: 0.379719 E: 0.242036 Sum of squares = 0.345837201319 Cumulative probabilities = A: 0.378245 B: 0.378245 C: 0.378245 D: 0.757964 E: 1.000000 A B C D E A 0 -2 4 0 0 B 2 0 0 -6 -8 C -4 0 0 -4 -6 D 0 6 4 0 0 E 0 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 E=22 B=16 A=13 so A is eliminated. Round 2 votes counts: E=33 D=26 C=25 B=16 so B is eliminated. Round 3 votes counts: E=40 D=35 C=25 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:207 D:205 A:201 B:194 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 0 0 B 2 0 0 -6 -8 C -4 0 0 -4 -6 D 0 6 4 0 0 E 0 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 0 0 B 2 0 0 -6 -8 C -4 0 0 -4 -6 D 0 6 4 0 0 E 0 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 0 0 B 2 0 0 -6 -8 C -4 0 0 -4 -6 D 0 6 4 0 0 E 0 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 593: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (14) C A D B E (13) C E B D A (5) C D A E B (5) B E D A C (5) B E A D C (5) A D B E C (5) A C D B E (5) E B D C A (4) E B C D A (4) D B E A C (4) C A D E B (4) A D C B E (4) D A B E C (3) C E B A D (3) C D E B A (3) E C B D A (2) E B C A D (2) E C B A D (1) E B A D C (1) D E B A C (1) D C E A B (1) C E A D B (1) C A E B D (1) B D E A C (1) B D A E C (1) B A D E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 2 -6 -14 B 14 0 2 2 -4 C -2 -2 0 0 -8 D 6 -2 0 0 2 E 14 4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 -14 2 -6 -14 B 14 0 2 2 -4 C -2 -2 0 0 -8 D 6 -2 0 0 2 E 14 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000135 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=28 A=15 B=13 D=9 so D is eliminated. Round 2 votes counts: C=36 E=29 A=18 B=17 so B is eliminated. Round 3 votes counts: E=44 C=36 A=20 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:207 D:203 C:194 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 2 -6 -14 B 14 0 2 2 -4 C -2 -2 0 0 -8 D 6 -2 0 0 2 E 14 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000135 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -6 -14 B 14 0 2 2 -4 C -2 -2 0 0 -8 D 6 -2 0 0 2 E 14 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000135 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -6 -14 B 14 0 2 2 -4 C -2 -2 0 0 -8 D 6 -2 0 0 2 E 14 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000135 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 594: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) D A B E C (10) C E B D A (9) A D C E B (9) A D B E C (8) C E B A D (6) E B C A D (4) D A B C E (4) C E A D B (4) D A C E B (3) D A C B E (3) E C B A D (2) C E A B D (2) C D A E B (2) C B E D A (2) B E C A D (2) B C E D A (2) B C D E A (2) A E B D C (2) A D C B E (2) E B A D C (1) D B A E C (1) D B A C E (1) C E D B A (1) C D E B A (1) B E D C A (1) B D E A C (1) B A E D C (1) A D E C B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -2 -8 -2 B -4 0 4 -4 2 C 2 -4 0 2 8 D 8 4 -2 0 0 E 2 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -8 -2 B -4 0 4 -4 2 C 2 -4 0 2 8 D 8 4 -2 0 0 E 2 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000025 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 D=22 B=20 E=7 so E is eliminated. Round 2 votes counts: C=29 B=25 A=24 D=22 so D is eliminated. Round 3 votes counts: A=44 C=29 B=27 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:205 C:204 B:199 A:196 E:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -8 -2 B -4 0 4 -4 2 C 2 -4 0 2 8 D 8 4 -2 0 0 E 2 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000025 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -8 -2 B -4 0 4 -4 2 C 2 -4 0 2 8 D 8 4 -2 0 0 E 2 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000025 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -8 -2 B -4 0 4 -4 2 C 2 -4 0 2 8 D 8 4 -2 0 0 E 2 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000025 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 595: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (12) A B D C E (8) A B C D E (8) D E B C A (7) E D C B A (6) D B A C E (5) E C A B D (4) A B C E D (4) C E A B D (3) A E C B D (3) A C E B D (3) E D C A B (2) D E C B A (2) D B C E A (2) D B C A E (2) D A B C E (2) A E D B C (2) A E C D B (2) A D B C E (2) A C B E D (2) E D A B C (1) E C B D A (1) E C A D B (1) E A D C B (1) E A C B D (1) D E A B C (1) D C B E A (1) D B A E C (1) D A B E C (1) C E B A D (1) C B E D A (1) C B D E A (1) C B A D E (1) C A E B D (1) B D C E A (1) B A D C E (1) A E B C D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 10 2 2 2 B -10 0 2 -10 -12 C -2 -2 0 0 -2 D -2 10 0 0 -4 E -2 12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 2 2 B -10 0 2 -10 -12 C -2 -2 0 0 -2 D -2 10 0 0 -4 E -2 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995736 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=29 D=24 C=8 B=2 so B is eliminated. Round 2 votes counts: A=38 E=29 D=25 C=8 so C is eliminated. Round 3 votes counts: A=40 E=34 D=26 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:208 D:202 C:197 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 2 2 B -10 0 2 -10 -12 C -2 -2 0 0 -2 D -2 10 0 0 -4 E -2 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995736 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 2 2 B -10 0 2 -10 -12 C -2 -2 0 0 -2 D -2 10 0 0 -4 E -2 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995736 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 2 2 B -10 0 2 -10 -12 C -2 -2 0 0 -2 D -2 10 0 0 -4 E -2 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995736 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 596: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) E B C A D (6) C D A B E (6) C E B D A (5) B E A C D (5) B A E D C (5) D A C E B (4) B C E A D (4) D A E B C (3) D A C B E (3) C D E A B (3) B E C A D (3) A D C B E (3) A D B C E (3) E C B D A (2) E B A C D (2) E A B D C (2) D A E C B (2) C D B A E (2) C D A E B (2) C B E D A (2) C B E A D (2) B E A D C (2) A D E B C (2) A D B E C (2) A B E D C (2) E D C A B (1) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) E B C D A (1) D E C A B (1) D E A B C (1) D C A B E (1) C E D B A (1) C D E B A (1) C D B E A (1) B A E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 10 8 -14 B 12 0 16 12 -2 C -10 -16 0 -2 -12 D -8 -12 2 0 -16 E 14 2 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 10 8 -14 B 12 0 16 12 -2 C -10 -16 0 -2 -12 D -8 -12 2 0 -16 E 14 2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 B=20 D=15 A=14 so A is eliminated. Round 2 votes counts: E=26 D=25 C=25 B=24 so B is eliminated. Round 3 votes counts: E=44 C=29 D=27 so D is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:219 A:196 D:183 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 10 8 -14 B 12 0 16 12 -2 C -10 -16 0 -2 -12 D -8 -12 2 0 -16 E 14 2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 8 -14 B 12 0 16 12 -2 C -10 -16 0 -2 -12 D -8 -12 2 0 -16 E 14 2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 8 -14 B 12 0 16 12 -2 C -10 -16 0 -2 -12 D -8 -12 2 0 -16 E 14 2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 597: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (15) D A C E B (14) C A D B E (9) D A C B E (7) A D C B E (7) C A B D E (6) B E C A D (6) E B D C A (4) B E D A C (4) D A E C B (3) C A D E B (3) E D B A C (2) E B C D A (2) E B C A D (2) D E A C B (2) C A B E D (2) A C D B E (2) E C A D B (1) D C A E B (1) D A B C E (1) C B A D E (1) B E A D C (1) B D A C E (1) B C E A D (1) B C A E D (1) B C A D E (1) A C B D E (1) Total count = 100 A B C D E A 0 18 20 -12 20 B -18 0 -18 -4 2 C -20 18 0 -24 16 D 12 4 24 0 18 E -20 -2 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 20 -12 20 B -18 0 -18 -4 2 C -20 18 0 -24 16 D 12 4 24 0 18 E -20 -2 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 C=21 B=15 A=10 so A is eliminated. Round 2 votes counts: D=35 E=26 C=24 B=15 so B is eliminated. Round 3 votes counts: E=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 A:223 C:195 B:181 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 20 -12 20 B -18 0 -18 -4 2 C -20 18 0 -24 16 D 12 4 24 0 18 E -20 -2 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 -12 20 B -18 0 -18 -4 2 C -20 18 0 -24 16 D 12 4 24 0 18 E -20 -2 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 -12 20 B -18 0 -18 -4 2 C -20 18 0 -24 16 D 12 4 24 0 18 E -20 -2 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 598: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) A B E D C (12) B A C D E (10) E D A C B (9) D E C A B (6) A E D B C (6) B C A D E (5) E D C A B (4) C D E A B (4) B A E D C (4) A B C E D (4) C B D E A (3) C B A D E (3) D E C B A (2) C B D A E (2) B A C E D (2) A B E C D (2) E D A B C (1) E A D B C (1) D C E B A (1) C E D A B (1) C D B E A (1) C B A E D (1) C A B E D (1) B C D E A (1) B A D E C (1) A E B D C (1) Total count = 100 A B C D E A 0 4 6 6 8 B -4 0 0 4 4 C -6 0 0 4 2 D -6 -4 -4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 6 8 B -4 0 0 4 4 C -6 0 0 4 2 D -6 -4 -4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=25 B=23 E=15 D=9 so D is eliminated. Round 2 votes counts: C=29 A=25 E=23 B=23 so E is eliminated. Round 3 votes counts: C=41 A=36 B=23 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:202 C:200 D:194 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 6 8 B -4 0 0 4 4 C -6 0 0 4 2 D -6 -4 -4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 6 8 B -4 0 0 4 4 C -6 0 0 4 2 D -6 -4 -4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 6 8 B -4 0 0 4 4 C -6 0 0 4 2 D -6 -4 -4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 599: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) C E D A B (7) E C D A B (6) E C A D B (6) D C E B A (6) D C B E A (6) E A C D B (5) A B E C D (5) C D E A B (4) B D A C E (4) A E B C D (4) E A C B D (3) D C B A E (3) E D C B A (2) E C D B A (2) E A B C D (2) D B C A E (2) C D E B A (2) B A E D C (2) A C B D E (2) A B E D C (2) A B D C E (2) D E C B A (1) D B C E A (1) C E A D B (1) C D A B E (1) C A E D B (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E A C (1) A E C B D (1) A C D B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -4 0 -16 B -10 0 -20 -14 -6 C 4 20 0 10 10 D 0 14 -10 0 -6 E 16 6 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 0 -16 B -10 0 -20 -14 -6 C 4 20 0 10 10 D 0 14 -10 0 -6 E 16 6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=20 D=19 A=19 C=16 so C is eliminated. Round 2 votes counts: E=34 D=26 B=20 A=20 so B is eliminated. Round 3 votes counts: E=37 A=32 D=31 so D is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:222 E:209 D:199 A:195 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 0 -16 B -10 0 -20 -14 -6 C 4 20 0 10 10 D 0 14 -10 0 -6 E 16 6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 0 -16 B -10 0 -20 -14 -6 C 4 20 0 10 10 D 0 14 -10 0 -6 E 16 6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 0 -16 B -10 0 -20 -14 -6 C 4 20 0 10 10 D 0 14 -10 0 -6 E 16 6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 600: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) E C D B A (7) B E C D A (6) A D C B E (6) E D C A B (5) C D E B A (5) A B E C D (5) D C E B A (4) B C D E A (4) B A E C D (4) E A B C D (3) D C E A B (3) C E D B A (3) C D B E A (3) B E A C D (3) A B D C E (3) E C B D A (2) E B A C D (2) D C B E A (2) D C A E B (2) D C A B E (2) B C E D A (2) B C D A E (2) B A D C E (2) A E D C B (2) A B E D C (2) E A D C B (1) E A C D B (1) D C B A E (1) C D B A E (1) A D C E B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -24 -26 -26 -30 B 24 0 -2 0 0 C 26 2 0 24 -6 D 26 0 -24 0 -14 E 30 0 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.489948 C: 0.000000 D: 0.000000 E: 0.510052 Sum of squares = 0.500202091789 Cumulative probabilities = A: 0.000000 B: 0.489948 C: 0.489948 D: 0.489948 E: 1.000000 A B C D E A 0 -24 -26 -26 -30 B 24 0 -2 0 0 C 26 2 0 24 -6 D 26 0 -24 0 -14 E 30 0 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=23 A=21 D=14 C=12 so C is eliminated. Round 2 votes counts: E=33 D=23 B=23 A=21 so A is eliminated. Round 3 votes counts: E=35 B=34 D=31 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:225 C:223 B:211 D:194 A:147 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 -26 -26 -30 B 24 0 -2 0 0 C 26 2 0 24 -6 D 26 0 -24 0 -14 E 30 0 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -26 -26 -30 B 24 0 -2 0 0 C 26 2 0 24 -6 D 26 0 -24 0 -14 E 30 0 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -26 -26 -30 B 24 0 -2 0 0 C 26 2 0 24 -6 D 26 0 -24 0 -14 E 30 0 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 601: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (13) A B E D C (12) B E D C A (7) C D E A B (6) B A E D C (6) A B C D E (6) E D C B A (5) C E D B A (5) B A D E C (5) A C B D E (4) D E C B A (3) C A D E B (3) E C D B A (2) B E D A C (2) B E A D C (2) A C D B E (2) A B E C D (2) A B D E C (2) A B C E D (2) E D B C A (1) E B D C A (1) E B C D A (1) D C E B A (1) C E D A B (1) C D A E B (1) B D E C A (1) A C E D B (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -2 0 -2 B 10 0 2 10 10 C 2 -2 0 2 -4 D 0 -10 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 0 -2 B 10 0 2 10 10 C 2 -2 0 2 -4 D 0 -10 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=29 B=23 E=10 D=4 so D is eliminated. Round 2 votes counts: A=34 C=30 B=23 E=13 so E is eliminated. Round 3 votes counts: C=40 A=34 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:216 C:199 E:199 A:193 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 0 -2 B 10 0 2 10 10 C 2 -2 0 2 -4 D 0 -10 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 0 -2 B 10 0 2 10 10 C 2 -2 0 2 -4 D 0 -10 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 0 -2 B 10 0 2 10 10 C 2 -2 0 2 -4 D 0 -10 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 602: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) E B D C A (6) E B C A D (6) D A C E B (6) A D C B E (6) E B C D A (5) B E C A D (5) A C D B E (5) D E C A B (4) D A E C B (4) C A B E D (4) A C B D E (4) E D B C A (3) E B D A C (3) B E A C D (3) B C E A D (3) D E B A C (2) D E A B C (2) D A E B C (2) C A D B E (2) A B C D E (2) E D C B A (1) E D C A B (1) E C D A B (1) E C B A D (1) D E A C B (1) D C A E B (1) D A B C E (1) C E D A B (1) C B A E D (1) C A D E B (1) C A B D E (1) B D A E C (1) B A D E C (1) B A D C E (1) A D B C E (1) Total count = 100 A B C D E A 0 16 6 -6 4 B -16 0 -6 -6 -2 C -6 6 0 -10 -4 D 6 6 10 0 12 E -4 2 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 -6 4 B -16 0 -6 -6 -2 C -6 6 0 -10 -4 D 6 6 10 0 12 E -4 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=27 A=18 B=14 C=10 so C is eliminated. Round 2 votes counts: D=31 E=28 A=26 B=15 so B is eliminated. Round 3 votes counts: E=39 D=32 A=29 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:210 E:195 C:193 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 -6 4 B -16 0 -6 -6 -2 C -6 6 0 -10 -4 D 6 6 10 0 12 E -4 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -6 4 B -16 0 -6 -6 -2 C -6 6 0 -10 -4 D 6 6 10 0 12 E -4 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -6 4 B -16 0 -6 -6 -2 C -6 6 0 -10 -4 D 6 6 10 0 12 E -4 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 603: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) D A E C B (7) D A C E B (6) B E C A D (6) E B A D C (5) A D E B C (5) C B E D A (4) C B D A E (4) A D E C B (4) A D C B E (4) E D A B C (3) E B C A D (3) C D A B E (3) C B D E A (3) B C E D A (3) E D A C B (2) E B A C D (2) E A B D C (2) D C A E B (2) D A C B E (2) C B A D E (2) B E A C D (2) A E D B C (2) A D C E B (2) E C B D A (1) E B D A C (1) E A D B C (1) C D E A B (1) C D B A E (1) C D A E B (1) C A D B E (1) B C E A D (1) B C A E D (1) B A E C D (1) B A C E D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 8 -6 2 B 0 0 -2 4 -18 C -8 2 0 2 -12 D 6 -4 -2 0 0 E -2 18 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.575555 E: 0.424445 Sum of squares = 0.51141707604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.575555 E: 1.000000 A B C D E A 0 0 8 -6 2 B 0 0 -2 4 -18 C -8 2 0 2 -12 D 6 -4 -2 0 0 E -2 18 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=20 A=19 D=17 B=15 so B is eliminated. Round 2 votes counts: E=37 C=25 A=21 D=17 so D is eliminated. Round 3 votes counts: E=37 A=36 C=27 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:214 A:202 D:200 B:192 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 -6 2 B 0 0 -2 4 -18 C -8 2 0 2 -12 D 6 -4 -2 0 0 E -2 18 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -6 2 B 0 0 -2 4 -18 C -8 2 0 2 -12 D 6 -4 -2 0 0 E -2 18 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -6 2 B 0 0 -2 4 -18 C -8 2 0 2 -12 D 6 -4 -2 0 0 E -2 18 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 604: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (13) B D A E C (13) C B E D A (11) D A B E C (8) E A D C B (7) C E A D B (7) B D A C E (7) C E B A D (6) B C D A E (6) A D E B C (5) E A D B C (4) C B D A E (4) D B A E C (2) B D C A E (2) C E B D A (1) C B D E A (1) A E D B C (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -2 -10 0 B 6 0 -2 2 10 C 2 2 0 -2 -10 D 10 -2 2 0 0 E 0 -10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.423732 C: 0.423732 D: 0.084738 E: 0.067799 Sum of squares = 0.370874167338 Cumulative probabilities = A: 0.000000 B: 0.423732 C: 0.847463 D: 0.932201 E: 1.000000 A B C D E A 0 -6 -2 -10 0 B 6 0 -2 2 10 C 2 2 0 -2 -10 D 10 -2 2 0 0 E 0 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.358025 C: 0.358025 D: 0.265432 E: 0.018519 Sum of squares = 0.327160493827 Cumulative probabilities = A: 0.000000 B: 0.358025 C: 0.716049 D: 0.981481 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=28 E=24 D=10 A=8 so A is eliminated. Round 2 votes counts: C=30 B=28 E=25 D=17 so D is eliminated. Round 3 votes counts: B=39 E=31 C=30 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:205 E:200 C:196 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -10 0 B 6 0 -2 2 10 C 2 2 0 -2 -10 D 10 -2 2 0 0 E 0 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.358025 C: 0.358025 D: 0.265432 E: 0.018519 Sum of squares = 0.327160493827 Cumulative probabilities = A: 0.000000 B: 0.358025 C: 0.716049 D: 0.981481 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -10 0 B 6 0 -2 2 10 C 2 2 0 -2 -10 D 10 -2 2 0 0 E 0 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.358025 C: 0.358025 D: 0.265432 E: 0.018519 Sum of squares = 0.327160493827 Cumulative probabilities = A: 0.000000 B: 0.358025 C: 0.716049 D: 0.981481 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -10 0 B 6 0 -2 2 10 C 2 2 0 -2 -10 D 10 -2 2 0 0 E 0 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.358025 C: 0.358025 D: 0.265432 E: 0.018519 Sum of squares = 0.327160493827 Cumulative probabilities = A: 0.000000 B: 0.358025 C: 0.716049 D: 0.981481 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 605: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) B A D E C (9) A D E B C (7) A E D C B (6) A D E C B (6) D A E B C (4) B D A E C (4) B C E D A (4) B C D E A (4) A B D E C (4) E D A C B (3) C E A D B (3) C B E A D (3) E C D A B (2) E C A D B (2) D E A B C (2) D B A E C (2) D A B E C (2) C E D B A (2) C E D A B (2) C E B D A (2) B C D A E (2) B C A D E (2) B A C D E (2) A E C D B (2) D E B C A (1) D A E C B (1) C E B A D (1) C B A E D (1) C A E D B (1) B D E C A (1) B D E A C (1) B D C A E (1) B A D C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 14 2 16 B 4 0 6 2 4 C -14 -6 0 -10 -18 D -2 -2 10 0 12 E -16 -4 18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 2 16 B 4 0 6 2 4 C -14 -6 0 -10 -18 D -2 -2 10 0 12 E -16 -4 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=26 C=24 D=12 E=7 so E is eliminated. Round 2 votes counts: B=31 C=28 A=26 D=15 so D is eliminated. Round 3 votes counts: A=38 B=34 C=28 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:214 D:209 B:208 E:193 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 2 16 B 4 0 6 2 4 C -14 -6 0 -10 -18 D -2 -2 10 0 12 E -16 -4 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 2 16 B 4 0 6 2 4 C -14 -6 0 -10 -18 D -2 -2 10 0 12 E -16 -4 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 2 16 B 4 0 6 2 4 C -14 -6 0 -10 -18 D -2 -2 10 0 12 E -16 -4 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 606: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) D B C E A (9) E A B C D (7) A C E D B (6) D C A B E (5) A E C B D (5) D C B A E (4) A C D E B (4) C B D E A (3) A E C D B (3) A D C E B (3) E B A D C (2) E B A C D (2) E A B D C (2) D B E C A (2) D B C A E (2) D B A C E (2) C D A B E (2) C A E D B (2) B E D C A (2) B E D A C (2) B E C D A (2) A E D C B (2) A E B D C (2) E C A B D (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) E A C B D (1) D A C B E (1) D A B E C (1) C D B E A (1) C D B A E (1) C B E D A (1) C A E B D (1) C A D E B (1) C A D B E (1) A E B C D (1) Total count = 100 A B C D E A 0 2 -6 -6 -2 B -2 0 4 -4 2 C 6 -4 0 -6 -2 D 6 4 6 0 4 E 2 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -6 -2 B -2 0 4 -4 2 C 6 -4 0 -6 -2 D 6 4 6 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 E=19 B=16 C=13 so C is eliminated. Round 2 votes counts: A=31 D=30 B=20 E=19 so E is eliminated. Round 3 votes counts: A=42 D=30 B=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:200 E:199 C:197 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -6 -2 B -2 0 4 -4 2 C 6 -4 0 -6 -2 D 6 4 6 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -6 -2 B -2 0 4 -4 2 C 6 -4 0 -6 -2 D 6 4 6 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -6 -2 B -2 0 4 -4 2 C 6 -4 0 -6 -2 D 6 4 6 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 607: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) C B A D E (6) A B D E C (6) E D A B C (5) A D E B C (5) D E C A B (4) D E A C B (4) D A E C B (4) C D E A B (4) C B E A D (4) B C A E D (4) C E D B A (3) B E C A D (3) A D B E C (3) E D C A B (2) E D B A C (2) E C D B A (2) D E A B C (2) D A E B C (2) C E B D A (2) B E A C D (2) B A E D C (2) B A C E D (2) B A C D E (2) A B D C E (2) E D C B A (1) E D A C B (1) E B D A C (1) E A D B C (1) E A B D C (1) D C E A B (1) C D E B A (1) C D A E B (1) C B A E D (1) B E A D C (1) B C A D E (1) B A E C D (1) A D C B E (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 2 -8 B -4 0 0 0 2 C -6 0 0 -4 -10 D -2 0 4 0 4 E 8 -2 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 A B C D E A 0 4 6 2 -8 B -4 0 0 0 2 C -6 0 0 -4 -10 D -2 0 4 0 4 E 8 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428519 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=20 B=18 D=17 E=16 so E is eliminated. Round 2 votes counts: C=31 D=28 A=22 B=19 so B is eliminated. Round 3 votes counts: C=39 A=32 D=29 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:203 A:202 B:199 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 2 -8 B -4 0 0 0 2 C -6 0 0 -4 -10 D -2 0 4 0 4 E 8 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428519 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 2 -8 B -4 0 0 0 2 C -6 0 0 -4 -10 D -2 0 4 0 4 E 8 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428519 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 2 -8 B -4 0 0 0 2 C -6 0 0 -4 -10 D -2 0 4 0 4 E 8 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428519 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 608: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E C A D B (9) E A D C B (7) B D A C E (7) B C E D A (6) A E D C B (6) C B E D A (4) B E C A D (4) B D C A E (4) E A C D B (3) D A C E B (3) D A B C E (3) B C D E A (3) A D E C B (3) E C A B D (2) D A C B E (2) C E B A D (2) C B D E A (2) B E C D A (2) A D B E C (2) E C B A D (1) E B A D C (1) E A C B D (1) D B C A E (1) D A B E C (1) C E D A B (1) C E B D A (1) C D E A B (1) C D B A E (1) C D A E B (1) B E A D C (1) B D E C A (1) B D A E C (1) B A D E C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 -12 -10 -4 B 6 0 0 10 12 C 12 0 0 8 4 D 10 -10 -8 0 -4 E 4 -12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.340630 C: 0.659370 D: 0.000000 E: 0.000000 Sum of squares = 0.550797474754 Cumulative probabilities = A: 0.000000 B: 0.340630 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -10 -4 B 6 0 0 10 12 C 12 0 0 8 4 D 10 -10 -8 0 -4 E 4 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=24 C=13 A=13 D=10 so D is eliminated. Round 2 votes counts: B=41 E=24 A=22 C=13 so C is eliminated. Round 3 votes counts: B=48 E=29 A=23 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:212 E:196 D:194 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -12 -10 -4 B 6 0 0 10 12 C 12 0 0 8 4 D 10 -10 -8 0 -4 E 4 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -10 -4 B 6 0 0 10 12 C 12 0 0 8 4 D 10 -10 -8 0 -4 E 4 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -10 -4 B 6 0 0 10 12 C 12 0 0 8 4 D 10 -10 -8 0 -4 E 4 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 609: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) D B A C E (8) A E B C D (8) E A C B D (6) D C E B A (6) D C B E A (5) D E C A B (4) C E A B D (4) A B E C D (4) E C A B D (3) D E A C B (3) B A C D E (3) D C B A E (2) D B C A E (2) C E B A D (2) C D E B A (2) C B A E D (2) C A E B D (2) B A D E C (2) B A C E D (2) A B E D C (2) E D C A B (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E A B C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E D A B (1) C D B E A (1) B D C A E (1) B D A C E (1) B C D A E (1) B C A E D (1) B A D C E (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 16 -6 12 B 8 0 4 -8 0 C -16 -4 0 -8 -4 D 6 8 8 0 12 E -12 0 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999294 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 16 -6 12 B 8 0 4 -8 0 C -16 -4 0 -8 -4 D 6 8 8 0 12 E -12 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 C=15 A=15 E=14 B=12 so B is eliminated. Round 2 votes counts: D=46 A=23 C=17 E=14 so E is eliminated. Round 3 votes counts: D=47 A=32 C=21 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:207 B:202 E:190 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 16 -6 12 B 8 0 4 -8 0 C -16 -4 0 -8 -4 D 6 8 8 0 12 E -12 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 -6 12 B 8 0 4 -8 0 C -16 -4 0 -8 -4 D 6 8 8 0 12 E -12 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 -6 12 B 8 0 4 -8 0 C -16 -4 0 -8 -4 D 6 8 8 0 12 E -12 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 610: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) E D B A C (10) E D B C A (8) A B C D E (7) E C D A B (5) C A B E D (5) C E A D B (4) C A E B D (4) B D A E C (4) E C A D B (3) D E B A C (3) D B E A C (3) B A D C E (3) B A C D E (3) E D C B A (2) E D C A B (2) C E D A B (2) B D E A C (2) B D A C E (2) B A D E C (2) A C B D E (2) E B D A C (1) E A C B D (1) D E C B A (1) D E B C A (1) D C B E A (1) D B A C E (1) C D B A E (1) C D A B E (1) B A E D C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -6 0 2 B -2 0 4 4 6 C 6 -4 0 4 0 D 0 -4 -4 0 0 E -2 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888978 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 0 2 B -2 0 4 4 6 C 6 -4 0 4 0 D 0 -4 -4 0 0 E -2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=30 B=17 A=11 D=10 so D is eliminated. Round 2 votes counts: E=37 C=31 B=21 A=11 so A is eliminated. Round 3 votes counts: E=37 C=34 B=29 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:206 C:203 A:199 D:196 E:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -6 0 2 B -2 0 4 4 6 C 6 -4 0 4 0 D 0 -4 -4 0 0 E -2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 2 B -2 0 4 4 6 C 6 -4 0 4 0 D 0 -4 -4 0 0 E -2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 2 B -2 0 4 4 6 C 6 -4 0 4 0 D 0 -4 -4 0 0 E -2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 611: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) A C E B D (8) A C D E B (8) C A D B E (7) B D E C A (7) E A C D B (6) E A C B D (5) A C E D B (5) E D B A C (4) C A B D E (4) B E D C A (4) E B D A C (3) E A D C B (3) A E C D B (3) D E A B C (2) D B C A E (2) C A D E B (2) C A B E D (2) B C D A E (2) E D A C B (1) E B A C D (1) D E B A C (1) D E A C B (1) D C B A E (1) D C A B E (1) D B E A C (1) D B C E A (1) D A C E B (1) C D B A E (1) B E D A C (1) B E C D A (1) B E C A D (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 22 12 14 14 B -22 0 -22 -4 -10 C -12 22 0 14 10 D -14 4 -14 0 2 E -14 10 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 14 14 B -22 0 -22 -4 -10 C -12 22 0 14 10 D -14 4 -14 0 2 E -14 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=24 E=23 C=16 D=11 so D is eliminated. Round 2 votes counts: B=28 E=27 A=27 C=18 so C is eliminated. Round 3 votes counts: A=43 B=30 E=27 so E is eliminated. Round 4 votes counts: A=61 B=39 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:231 C:217 E:192 D:189 B:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 12 14 14 B -22 0 -22 -4 -10 C -12 22 0 14 10 D -14 4 -14 0 2 E -14 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 14 14 B -22 0 -22 -4 -10 C -12 22 0 14 10 D -14 4 -14 0 2 E -14 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 14 14 B -22 0 -22 -4 -10 C -12 22 0 14 10 D -14 4 -14 0 2 E -14 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 612: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) B D C E A (9) A E D B C (6) D B E A C (5) D B A C E (5) C E B D A (5) A D B E C (5) D B E C A (4) E D B C A (3) E A D B C (3) C B D A E (3) C A B D E (3) A E C B D (3) A B D C E (3) E C B D A (2) E C A B D (2) E A C D B (2) D B A E C (2) B D A C E (2) B A D C E (2) A E C D B (2) A D B C E (2) A C B E D (2) A C B D E (2) E D B A C (1) E C D B A (1) E C A D B (1) E A C B D (1) D B C E A (1) D B C A E (1) C E B A D (1) C E A B D (1) C B A D E (1) B D C A E (1) A E D C B (1) Total count = 100 A B C D E A 0 -22 -2 -14 -8 B 22 0 10 10 30 C 2 -10 0 -12 12 D 14 -10 12 0 26 E 8 -30 -12 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -2 -14 -8 B 22 0 10 10 30 C 2 -10 0 -12 12 D 14 -10 12 0 26 E 8 -30 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998127 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 D=18 E=16 B=14 so B is eliminated. Round 2 votes counts: D=30 A=28 C=26 E=16 so E is eliminated. Round 3 votes counts: D=34 A=34 C=32 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:236 D:221 C:196 A:177 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -2 -14 -8 B 22 0 10 10 30 C 2 -10 0 -12 12 D 14 -10 12 0 26 E 8 -30 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998127 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -2 -14 -8 B 22 0 10 10 30 C 2 -10 0 -12 12 D 14 -10 12 0 26 E 8 -30 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998127 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -2 -14 -8 B 22 0 10 10 30 C 2 -10 0 -12 12 D 14 -10 12 0 26 E 8 -30 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998127 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 613: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (13) B C A D E (9) E B C D A (7) D A E C B (6) C B A D E (6) B C E D A (5) D A C E B (4) B E C D A (4) B E C A D (4) A D C E B (4) E D C B A (3) E D A B C (3) C A D B E (3) B C E A D (3) A D E C B (3) A C D B E (3) E D B A C (2) A D C B E (2) A C B D E (2) E B D A C (1) E B A D C (1) D E A C B (1) D C E A B (1) D C A E B (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D E A (1) C B D A E (1) C A B D E (1) B C A E D (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 0 -6 -12 2 B 0 0 -16 -6 -4 C 6 16 0 8 2 D 12 6 -8 0 4 E -2 4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -12 2 B 0 0 -16 -6 -4 C 6 16 0 8 2 D 12 6 -8 0 4 E -2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=27 C=15 A=15 D=13 so D is eliminated. Round 2 votes counts: E=31 B=27 A=25 C=17 so C is eliminated. Round 3 votes counts: B=36 E=32 A=32 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:216 D:207 E:198 A:192 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 -12 2 B 0 0 -16 -6 -4 C 6 16 0 8 2 D 12 6 -8 0 4 E -2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -12 2 B 0 0 -16 -6 -4 C 6 16 0 8 2 D 12 6 -8 0 4 E -2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -12 2 B 0 0 -16 -6 -4 C 6 16 0 8 2 D 12 6 -8 0 4 E -2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 614: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) C A E D B (7) B E D C A (7) B D A E C (5) A C E B D (5) A B D C E (4) E D B C A (3) E C D B A (3) E C D A B (3) E C A B D (3) D E B C A (3) D B A C E (3) C E A D B (3) C A D E B (3) B E D A C (3) B D E A C (3) A C D E B (3) E C B D A (2) E B C D A (2) D B E C A (2) B A D C E (2) A C E D B (2) A C D B E (2) A C B D E (2) A B C D E (2) E D C A B (1) D C A E B (1) D C A B E (1) D B E A C (1) D B C E A (1) D B A E C (1) C E D A B (1) C E A B D (1) B E A C D (1) B D E C A (1) B A E C D (1) B A D E C (1) A D C B E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -12 -10 -4 B 6 0 10 8 -8 C 12 -10 0 -6 -8 D 10 -8 6 0 -14 E 4 8 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -10 -4 B 6 0 10 8 -8 C 12 -10 0 -6 -8 D 10 -8 6 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=24 A=23 C=15 D=13 so D is eliminated. Round 2 votes counts: B=32 E=28 A=23 C=17 so C is eliminated. Round 3 votes counts: A=35 E=33 B=32 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:208 D:197 C:194 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -10 -4 B 6 0 10 8 -8 C 12 -10 0 -6 -8 D 10 -8 6 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -10 -4 B 6 0 10 8 -8 C 12 -10 0 -6 -8 D 10 -8 6 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -10 -4 B 6 0 10 8 -8 C 12 -10 0 -6 -8 D 10 -8 6 0 -14 E 4 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 615: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A D E B (10) B E C D A (8) A D C E B (7) C A B D E (4) B E C A D (4) A C D E B (4) E B D A C (3) C A D B E (3) B D E A C (3) B C E A D (3) E B C D A (2) D E A C B (2) C D A E B (2) C B E A D (2) C A E D B (2) B C E D A (2) B A C D E (2) A D C B E (2) A C D B E (2) A B D E C (2) E D C A B (1) E D B C A (1) E D B A C (1) E C B D A (1) E B D C A (1) D E C A B (1) D A E C B (1) D A B E C (1) C E D A B (1) C E B A D (1) C E A D B (1) C A B E D (1) B E D C A (1) B D A E C (1) B C A E D (1) B A D C E (1) A D B C E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -6 12 0 B -2 0 2 14 16 C 6 -2 0 16 10 D -12 -14 -16 0 4 E 0 -16 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000008 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 12 0 B -2 0 2 14 16 C 6 -2 0 16 10 D -12 -14 -16 0 4 E 0 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000012 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=27 A=21 E=10 D=5 so D is eliminated. Round 2 votes counts: B=37 C=27 A=23 E=13 so E is eliminated. Round 3 votes counts: B=45 C=30 A=25 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:215 A:204 E:185 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -6 12 0 B -2 0 2 14 16 C 6 -2 0 16 10 D -12 -14 -16 0 4 E 0 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000012 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 12 0 B -2 0 2 14 16 C 6 -2 0 16 10 D -12 -14 -16 0 4 E 0 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000012 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 12 0 B -2 0 2 14 16 C 6 -2 0 16 10 D -12 -14 -16 0 4 E 0 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000012 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 616: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) E A B C D (7) E B A C D (6) D C B A E (4) B D E C A (4) A E C D B (4) A C D E B (4) D C A E B (3) D A C B E (3) C D E B A (3) C D E A B (3) C D A E B (3) C A D E B (3) B E C D A (3) B E A D C (3) B E A C D (3) A B E D C (3) E C B D A (2) E A C B D (2) D B C A E (2) B A E D C (2) A E B D C (2) A D C E B (2) A B D E C (2) E C A D B (1) E C A B D (1) E B C D A (1) D C B E A (1) D B C E A (1) D B A C E (1) D A B C E (1) C E D A B (1) C E A D B (1) C D B E A (1) B E D C A (1) B E D A C (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 12 0 -4 2 B -12 0 0 -2 0 C 0 0 0 2 0 D 4 2 -2 0 10 E -2 0 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.269350 B: 0.000000 C: 0.730650 D: 0.000000 E: 0.000000 Sum of squares = 0.606399194818 Cumulative probabilities = A: 0.269350 B: 0.269350 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 -4 2 B -12 0 0 -2 0 C 0 0 0 2 0 D 4 2 -2 0 10 E -2 0 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555557853 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 E=20 A=19 C=15 so C is eliminated. Round 2 votes counts: D=34 E=22 B=22 A=22 so E is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:207 A:205 C:201 E:194 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 0 -4 2 B -12 0 0 -2 0 C 0 0 0 2 0 D 4 2 -2 0 10 E -2 0 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555557853 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -4 2 B -12 0 0 -2 0 C 0 0 0 2 0 D 4 2 -2 0 10 E -2 0 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555557853 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -4 2 B -12 0 0 -2 0 C 0 0 0 2 0 D 4 2 -2 0 10 E -2 0 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555557853 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 617: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) A C B D E (7) E D B C A (6) A D E B C (6) C B D E A (5) A E D B C (5) C B E D A (4) B C D E A (4) B C D A E (4) E D C B A (3) D B E C A (3) C B E A D (3) B D C A E (3) A D B C E (3) A C E B D (3) C E B D A (2) C E B A D (2) C B D A E (2) C A B E D (2) B C E D A (2) B C A D E (2) A E D C B (2) A E C D B (2) E D B A C (1) E B C D A (1) E A D C B (1) E A D B C (1) E A C D B (1) D E B C A (1) D E B A C (1) D E A B C (1) D B C E A (1) D A B C E (1) C E A B D (1) C B A D E (1) C A B D E (1) B D C E A (1) A D B E C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 -8 -6 B 4 0 14 4 4 C 8 -14 0 2 12 D 8 -4 -2 0 -2 E 6 -4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -8 -6 B 4 0 14 4 4 C 8 -14 0 2 12 D 8 -4 -2 0 -2 E 6 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=23 E=22 B=16 D=8 so D is eliminated. Round 2 votes counts: A=32 E=25 C=23 B=20 so B is eliminated. Round 3 votes counts: C=40 A=32 E=28 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:213 C:204 D:200 E:196 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -8 -6 B 4 0 14 4 4 C 8 -14 0 2 12 D 8 -4 -2 0 -2 E 6 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -8 -6 B 4 0 14 4 4 C 8 -14 0 2 12 D 8 -4 -2 0 -2 E 6 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -8 -6 B 4 0 14 4 4 C 8 -14 0 2 12 D 8 -4 -2 0 -2 E 6 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 618: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (6) C A E D B (5) B E D A C (5) A C B D E (5) E D B C A (4) D A C E B (4) C E A D B (4) C A D E B (4) B A C D E (4) C A B E D (3) B E D C A (3) B A D C E (3) A D C B E (3) A C D B E (3) A B C D E (3) E C D A B (2) E C B D A (2) E C B A D (2) E C A B D (2) E B D C A (2) D E C A B (2) D B E A C (2) D A E C B (2) B E C A D (2) B E A C D (2) B A D E C (2) A B D C E (2) E D C B A (1) E D C A B (1) E C D B A (1) E B C D A (1) E B C A D (1) D E C B A (1) D E B A C (1) D E A C B (1) D A B C E (1) C E D A B (1) C D E A B (1) C A E B D (1) B C A E D (1) B A E D C (1) B A C E D (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 4 6 12 12 B -4 0 -4 8 6 C -6 4 0 4 2 D -12 -8 -4 0 4 E -12 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 12 12 B -4 0 -4 8 6 C -6 4 0 4 2 D -12 -8 -4 0 4 E -12 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=19 C=19 A=18 D=14 so D is eliminated. Round 2 votes counts: B=32 A=25 E=24 C=19 so C is eliminated. Round 3 votes counts: A=38 B=32 E=30 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:203 C:202 D:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 12 12 B -4 0 -4 8 6 C -6 4 0 4 2 D -12 -8 -4 0 4 E -12 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 12 12 B -4 0 -4 8 6 C -6 4 0 4 2 D -12 -8 -4 0 4 E -12 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 12 12 B -4 0 -4 8 6 C -6 4 0 4 2 D -12 -8 -4 0 4 E -12 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 619: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) C B D A E (7) E D A B C (6) C D B E A (6) C B A D E (6) B C D A E (6) A E D B C (6) C B D E A (5) B C A D E (5) D E A B C (4) A E B D C (4) E A C D B (3) D B C E A (3) D C B E A (2) D B A E C (2) C B A E D (2) A C B E D (2) E D C B A (1) E D C A B (1) E C D B A (1) D E C B A (1) D E B A C (1) D C E B A (1) D A E B C (1) C E B A D (1) C E A B D (1) C A B D E (1) B D C A E (1) B D A C E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A E B C D (1) A D B E C (1) A C E B D (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 0 10 B 8 0 12 2 12 C 2 -12 0 6 12 D 0 -2 -6 0 18 E -10 -12 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 0 10 B 8 0 12 2 12 C 2 -12 0 6 12 D 0 -2 -6 0 18 E -10 -12 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=21 E=20 D=15 B=15 so D is eliminated. Round 2 votes counts: C=32 E=26 A=22 B=20 so B is eliminated. Round 3 votes counts: C=47 A=27 E=26 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:217 D:205 C:204 A:200 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 0 10 B 8 0 12 2 12 C 2 -12 0 6 12 D 0 -2 -6 0 18 E -10 -12 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 10 B 8 0 12 2 12 C 2 -12 0 6 12 D 0 -2 -6 0 18 E -10 -12 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 10 B 8 0 12 2 12 C 2 -12 0 6 12 D 0 -2 -6 0 18 E -10 -12 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 620: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) B A E D C (7) A E D C B (6) A E D B C (6) C B A E D (5) B C A E D (5) D E A C B (4) D E A B C (4) D B E A C (4) C E A D B (4) B D C E A (4) B C D E A (4) B C D A E (4) B D A E C (3) A E C B D (3) C B E A D (2) C A E D B (2) B C A D E (2) B A E C D (2) A E C D B (2) A E B C D (2) E A C D B (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) C E D A B (1) C E A B D (1) C D B E A (1) C B E D A (1) C A E B D (1) B D A C E (1) B A C E D (1) A E B D C (1) Total count = 100 A B C D E A 0 -22 -6 6 6 B 22 0 2 22 20 C 6 -2 0 12 10 D -6 -22 -12 0 -6 E -6 -20 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998504 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 6 6 B 22 0 2 22 20 C 6 -2 0 12 10 D -6 -22 -12 0 -6 E -6 -20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955351 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=30 A=20 D=16 E=1 so E is eliminated. Round 2 votes counts: B=33 C=30 A=21 D=16 so D is eliminated. Round 3 votes counts: B=38 C=33 A=29 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:233 C:213 A:192 E:185 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -6 6 6 B 22 0 2 22 20 C 6 -2 0 12 10 D -6 -22 -12 0 -6 E -6 -20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955351 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 6 6 B 22 0 2 22 20 C 6 -2 0 12 10 D -6 -22 -12 0 -6 E -6 -20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955351 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 6 6 B 22 0 2 22 20 C 6 -2 0 12 10 D -6 -22 -12 0 -6 E -6 -20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955351 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 621: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) D A B E C (7) C E B A D (6) E D C A B (5) E D A B C (5) C B A E D (5) B A D C E (5) B A C D E (5) E C D A B (4) E C B D A (4) C E B D A (4) A B D C E (4) E C B A D (3) C E D B A (3) C B E A D (3) C B A D E (3) B C A D E (3) A D B E C (3) E D A C B (2) E C D B A (2) D E C A B (2) D A B C E (2) B A C E D (2) E D C B A (1) E D B A C (1) D C A E B (1) D C A B E (1) D A E B C (1) D A C B E (1) C D E A B (1) B E A C D (1) B C A E D (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 -8 -8 B 6 0 -2 2 -4 C 4 2 0 0 0 D 8 -2 0 0 -6 E 8 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.511216 D: 0.000000 E: 0.488784 Sum of squares = 0.500251610267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.511216 D: 0.511216 E: 1.000000 A B C D E A 0 -6 -4 -8 -8 B 6 0 -2 2 -4 C 4 2 0 0 0 D 8 -2 0 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 D=22 B=18 A=8 so A is eliminated. Round 2 votes counts: E=27 D=25 C=25 B=23 so B is eliminated. Round 3 votes counts: C=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:203 B:201 D:200 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 -8 B 6 0 -2 2 -4 C 4 2 0 0 0 D 8 -2 0 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 -8 B 6 0 -2 2 -4 C 4 2 0 0 0 D 8 -2 0 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 -8 B 6 0 -2 2 -4 C 4 2 0 0 0 D 8 -2 0 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 622: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) D B E A C (8) C A E B D (7) C A B E D (6) C A E D B (5) B E D A C (5) E A B C D (4) D E B C A (4) C A D E B (4) B E A D C (4) D C B E A (3) B A E D C (3) B A E C D (3) A E C B D (3) E D A C B (2) D E C B A (2) D E C A B (2) C D A E B (2) C A B D E (2) B A C E D (2) E D C A B (1) E D B A C (1) E C A D B (1) E B A D C (1) E B A C D (1) E A C D B (1) E A C B D (1) D C E B A (1) D C A B E (1) D B E C A (1) D B C A E (1) D B A C E (1) C E A D B (1) C D A B E (1) C A D B E (1) B E A C D (1) B D E A C (1) B D C A E (1) B C A D E (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 4 8 -8 B 8 0 4 -4 -4 C -4 -4 0 -4 -16 D -8 4 4 0 -8 E 8 4 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 4 8 -8 B 8 0 4 -4 -4 C -4 -4 0 -4 -16 D -8 4 4 0 -8 E 8 4 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=29 B=22 E=13 A=4 so A is eliminated. Round 2 votes counts: D=32 C=30 B=22 E=16 so E is eliminated. Round 3 votes counts: D=36 C=36 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:218 B:202 A:198 D:196 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 8 -8 B 8 0 4 -4 -4 C -4 -4 0 -4 -16 D -8 4 4 0 -8 E 8 4 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 8 -8 B 8 0 4 -4 -4 C -4 -4 0 -4 -16 D -8 4 4 0 -8 E 8 4 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 8 -8 B 8 0 4 -4 -4 C -4 -4 0 -4 -16 D -8 4 4 0 -8 E 8 4 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 623: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) D C E B A (8) A B E C D (8) D E B C A (7) C D A B E (6) C A D B E (6) A B E D C (6) C D A E B (5) C A B E D (4) E B A C D (3) C D E B A (3) B E A C D (3) A D B E C (3) E B C D A (2) E B C A D (2) D E C B A (2) D C A E B (2) A D C B E (2) A C D B E (2) A B C E D (2) E D B C A (1) E D B A C (1) E B D A C (1) D E B A C (1) D E A B C (1) D C E A B (1) D C A B E (1) D A E B C (1) D A C B E (1) B C E A D (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -2 12 2 B -4 0 12 -8 -6 C 2 -12 0 -2 -10 D -12 8 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408124 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 4 -2 12 2 B -4 0 12 -8 -6 C 2 -12 0 -2 -10 D -12 8 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408162 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 A=24 E=22 B=5 so B is eliminated. Round 2 votes counts: E=25 D=25 C=25 A=25 so E is eliminated. Round 3 votes counts: A=43 C=29 D=28 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:208 E:205 D:201 B:197 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 12 2 B -4 0 12 -8 -6 C 2 -12 0 -2 -10 D -12 8 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408162 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 12 2 B -4 0 12 -8 -6 C 2 -12 0 -2 -10 D -12 8 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408162 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 12 2 B -4 0 12 -8 -6 C 2 -12 0 -2 -10 D -12 8 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408162 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 624: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) D A B E C (7) A C D B E (7) E C B D A (5) C A B E D (5) B E C A D (5) A C B D E (5) E B C D A (4) C A E B D (4) A D C B E (4) E D B C A (3) B E D A C (3) B E C D A (3) B C E A D (3) E B D C A (2) D E B C A (2) D E B A C (2) D A E C B (2) D A C E B (2) D A C B E (2) C E D B A (2) C E B D A (2) C E B A D (2) C B E A D (2) C A E D B (2) B D A E C (2) A D C E B (2) E D B A C (1) D E A C B (1) D B A E C (1) D A B C E (1) B E A D C (1) B E A C D (1) B A E C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 8 -8 12 B -6 0 0 4 10 C -8 0 0 10 -6 D 8 -4 -10 0 -6 E -12 -10 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.000000 Sum of squares = 0.337278106363 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.692308 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -8 12 B -6 0 0 4 10 C -8 0 0 10 -6 D 8 -4 -10 0 -6 E -12 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.000000 Sum of squares = 0.337278106451 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.692308 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=20 C=19 B=19 E=15 so E is eliminated. Round 2 votes counts: D=31 B=25 C=24 A=20 so A is eliminated. Round 3 votes counts: D=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:209 B:204 C:198 E:195 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 -8 12 B -6 0 0 4 10 C -8 0 0 10 -6 D 8 -4 -10 0 -6 E -12 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.000000 Sum of squares = 0.337278106451 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.692308 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 12 B -6 0 0 4 10 C -8 0 0 10 -6 D 8 -4 -10 0 -6 E -12 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.000000 Sum of squares = 0.337278106451 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.692308 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 12 B -6 0 0 4 10 C -8 0 0 10 -6 D 8 -4 -10 0 -6 E -12 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.000000 Sum of squares = 0.337278106451 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.692308 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 625: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) D B E A C (6) D B A C E (6) E C A B D (5) B D E C A (5) A C E D B (5) E A C B D (4) D B E C A (4) C A E B D (4) C A D B E (4) B E D C A (4) A E C D B (4) A C E B D (4) E B D C A (3) E B C D A (3) D A B E C (3) C E A B D (3) C A B D E (3) E D B A C (2) E A C D B (2) D A C B E (2) A E C B D (2) E C B A D (1) E B D A C (1) E A D B C (1) D C B A E (1) D B A E C (1) C E B A D (1) C B D A E (1) C B A D E (1) B D C A E (1) A E D C B (1) A D E C B (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 -2 10 B -2 0 -4 -8 4 C 4 4 0 -2 -6 D 2 8 2 0 0 E -10 -4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.891751 E: 0.108249 Sum of squares = 0.806938320445 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.891751 E: 1.000000 A B C D E A 0 2 -4 -2 10 B -2 0 -4 -8 4 C 4 4 0 -2 -6 D 2 8 2 0 0 E -10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222288399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 A=20 C=17 B=10 so B is eliminated. Round 2 votes counts: D=37 E=26 A=20 C=17 so C is eliminated. Round 3 votes counts: D=38 A=32 E=30 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:206 A:203 C:200 E:196 B:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -2 10 B -2 0 -4 -8 4 C 4 4 0 -2 -6 D 2 8 2 0 0 E -10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222288399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 10 B -2 0 -4 -8 4 C 4 4 0 -2 -6 D 2 8 2 0 0 E -10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222288399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 10 B -2 0 -4 -8 4 C 4 4 0 -2 -6 D 2 8 2 0 0 E -10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222288399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 626: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) C D B A E (6) E A C B D (5) E A B D C (4) D B A E C (4) C E D A B (4) C E A B D (4) C A B E D (4) A E B C D (4) E D B A C (3) E C A B D (3) E A D B C (3) D E B A C (3) D B E A C (3) D B A C E (3) C D E B A (3) C A E B D (3) D B C E A (2) C E A D B (2) C A B D E (2) B D A E C (2) B D A C E (2) B A D E C (2) E D C A B (1) E D A B C (1) E C D A B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E C B A (1) D E B C A (1) D C B A E (1) D B E C A (1) C D E A B (1) C D A E B (1) C B D A E (1) B D E A C (1) B D C A E (1) B A E D C (1) B A D C E (1) A E B D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 -12 0 B 2 0 10 -12 -4 C 4 -10 0 -4 2 D 12 12 4 0 2 E 0 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -12 0 B 2 0 10 -12 -4 C 4 -10 0 -4 2 D 12 12 4 0 2 E 0 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=28 E=24 B=10 A=7 so A is eliminated. Round 2 votes counts: C=31 E=29 D=28 B=12 so B is eliminated. Round 3 votes counts: D=37 C=32 E=31 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:200 B:198 C:196 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -12 0 B 2 0 10 -12 -4 C 4 -10 0 -4 2 D 12 12 4 0 2 E 0 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -12 0 B 2 0 10 -12 -4 C 4 -10 0 -4 2 D 12 12 4 0 2 E 0 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -12 0 B 2 0 10 -12 -4 C 4 -10 0 -4 2 D 12 12 4 0 2 E 0 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 627: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) E D B A C (7) B E D C A (6) B E C A D (6) E B D A C (5) D E A C B (5) E B A D C (4) D E A B C (4) D A C E B (4) B C A E D (4) A D C E B (4) D C A E B (3) D A E C B (3) C B A D E (3) C A B D E (3) B C E A D (3) E D A B C (2) D E C A B (2) D E B C A (2) D E B A C (2) C A B E D (2) B E C D A (2) B E A C D (2) B A C E D (2) A C B E D (2) E B D C A (1) E A D B C (1) E A B D C (1) D C A B E (1) C D A B E (1) C B A E D (1) B C D E A (1) A E D C B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 2 -2 -12 B 2 0 10 -2 -4 C -2 -10 0 -16 -12 D 2 2 16 0 -4 E 12 4 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 -2 -12 B 2 0 10 -2 -4 C -2 -10 0 -16 -12 D 2 2 16 0 -4 E 12 4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 E=21 C=18 A=9 so A is eliminated. Round 2 votes counts: D=30 B=26 E=22 C=22 so E is eliminated. Round 3 votes counts: D=41 B=37 C=22 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:208 B:203 A:193 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -2 -12 B 2 0 10 -2 -4 C -2 -10 0 -16 -12 D 2 2 16 0 -4 E 12 4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 -12 B 2 0 10 -2 -4 C -2 -10 0 -16 -12 D 2 2 16 0 -4 E 12 4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 -12 B 2 0 10 -2 -4 C -2 -10 0 -16 -12 D 2 2 16 0 -4 E 12 4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 628: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) E A D C B (6) B C D A E (5) A D E B C (5) D B A C E (4) C E D A B (4) C B D E A (4) A D B E C (4) E D A C B (3) E C A D B (3) C E B D A (3) B D A C E (3) B A E D C (3) E C A B D (2) E A D B C (2) D A B E C (2) C E B A D (2) C D E B A (2) C B D A E (2) B E C A D (2) B C E A D (2) B C A E D (2) A E B D C (2) A D E C B (2) E D C A B (1) E C D A B (1) E C B A D (1) E B C A D (1) E B A C D (1) E A C D B (1) E A B C D (1) D E A C B (1) D C E A B (1) D C B A E (1) D A E B C (1) D A C E B (1) C E D B A (1) C D E A B (1) C D B E A (1) C D B A E (1) B D C A E (1) B C D E A (1) B A D C E (1) A E D C B (1) A E D B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 10 -6 4 B -12 0 -8 -18 -18 C -10 8 0 -12 -14 D 6 18 12 0 6 E -4 18 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 -6 4 B -12 0 -8 -18 -18 C -10 8 0 -12 -14 D 6 18 12 0 6 E -4 18 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 C=21 B=20 D=19 A=17 so A is eliminated. Round 2 votes counts: D=30 E=27 B=22 C=21 so C is eliminated. Round 3 votes counts: E=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:211 A:210 C:186 B:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 10 -6 4 B -12 0 -8 -18 -18 C -10 8 0 -12 -14 D 6 18 12 0 6 E -4 18 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -6 4 B -12 0 -8 -18 -18 C -10 8 0 -12 -14 D 6 18 12 0 6 E -4 18 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -6 4 B -12 0 -8 -18 -18 C -10 8 0 -12 -14 D 6 18 12 0 6 E -4 18 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 629: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) A C B D E (10) C A E D B (9) A B C D E (7) E D B C A (6) D E B C A (6) E D C B A (5) E C D A B (4) B D A E C (4) B A D C E (4) A C B E D (4) E B D A C (3) C D E A B (3) B E D A C (3) A C E B D (3) D E C B A (2) D B E C A (2) C E A D B (2) C A D E B (2) E B D C A (1) D C E B A (1) C E D A B (1) C D A E B (1) C D A B E (1) C A E B D (1) B D E C A (1) B D A C E (1) B A E C D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 0 4 -10 0 B 0 0 0 8 -2 C -4 0 0 2 4 D 10 -8 -2 0 12 E 0 2 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.195652 B: 0.478261 C: 0.108696 D: 0.043478 E: 0.173913 Sum of squares = 0.310964083242 Cumulative probabilities = A: 0.195652 B: 0.673913 C: 0.782609 D: 0.826087 E: 1.000000 A B C D E A 0 0 4 -10 0 B 0 0 0 8 -2 C -4 0 0 2 4 D 10 -8 -2 0 12 E 0 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.478261 C: 0.108696 D: 0.043478 E: 0.173913 Sum of squares = 0.310964083183 Cumulative probabilities = A: 0.195652 B: 0.673913 C: 0.782609 D: 0.826087 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=24 C=20 E=19 D=11 so D is eliminated. Round 2 votes counts: E=27 B=26 A=26 C=21 so C is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:206 B:203 C:201 A:197 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 -10 0 B 0 0 0 8 -2 C -4 0 0 2 4 D 10 -8 -2 0 12 E 0 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.478261 C: 0.108696 D: 0.043478 E: 0.173913 Sum of squares = 0.310964083183 Cumulative probabilities = A: 0.195652 B: 0.673913 C: 0.782609 D: 0.826087 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -10 0 B 0 0 0 8 -2 C -4 0 0 2 4 D 10 -8 -2 0 12 E 0 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.478261 C: 0.108696 D: 0.043478 E: 0.173913 Sum of squares = 0.310964083183 Cumulative probabilities = A: 0.195652 B: 0.673913 C: 0.782609 D: 0.826087 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -10 0 B 0 0 0 8 -2 C -4 0 0 2 4 D 10 -8 -2 0 12 E 0 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.478261 C: 0.108696 D: 0.043478 E: 0.173913 Sum of squares = 0.310964083183 Cumulative probabilities = A: 0.195652 B: 0.673913 C: 0.782609 D: 0.826087 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 630: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) D A E B C (6) A C E B D (6) D E B A C (5) A C E D B (5) A D C E B (4) D B C E A (3) D B C A E (3) D A C B E (3) C B E A D (3) C B A E D (3) C A E B D (3) B C E D A (3) A C D E B (3) E C A B D (2) D B E A C (2) D B A E C (2) D B A C E (2) D A E C B (2) D A B C E (2) C A B E D (2) B E D C A (2) B E C D A (2) B C E A D (2) B C D E A (2) A E C D B (2) A D E C B (2) E D B A C (1) E D A B C (1) E C B A D (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) E A D B C (1) E A C B D (1) D E B C A (1) D E A B C (1) D A B E C (1) C E B A D (1) C B D A E (1) C A B D E (1) B D E C A (1) B D C E A (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 10 -10 8 B 2 0 8 -18 -4 C -10 -8 0 -6 8 D 10 18 6 0 10 E -8 4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -10 8 B 2 0 8 -18 -4 C -10 -8 0 -6 8 D 10 18 6 0 10 E -8 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=23 C=14 B=13 E=11 so E is eliminated. Round 2 votes counts: D=41 A=25 C=17 B=17 so C is eliminated. Round 3 votes counts: D=41 A=33 B=26 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:203 B:194 C:192 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -10 8 B 2 0 8 -18 -4 C -10 -8 0 -6 8 D 10 18 6 0 10 E -8 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -10 8 B 2 0 8 -18 -4 C -10 -8 0 -6 8 D 10 18 6 0 10 E -8 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -10 8 B 2 0 8 -18 -4 C -10 -8 0 -6 8 D 10 18 6 0 10 E -8 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 631: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (5) B E D A C (5) A E B D C (5) D C A E B (4) D C A B E (4) C D A B E (4) B E A C D (4) A E B C D (4) A C D E B (4) E B A C D (3) E A B D C (3) D A E B C (3) D A C E B (3) C D B A E (3) C D A E B (3) C A E B D (3) B E C A D (3) A C E D B (3) D B E A C (2) D A B E C (2) C B E D A (2) C B D E A (2) C A D E B (2) A E D B C (2) A D E C B (2) A D E B C (2) E B D A C (1) E B C A D (1) E A D B C (1) D E A B C (1) D C B A E (1) D B E C A (1) D A C B E (1) D A B C E (1) C E B A D (1) C D B E A (1) C B A E D (1) B E D C A (1) B E C D A (1) B E A D C (1) B D E C A (1) A E C B D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 20 22 6 20 B -20 0 6 -2 -18 C -22 -6 0 -6 -10 D -6 2 6 0 -4 E -20 18 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 22 6 20 B -20 0 6 -2 -18 C -22 -6 0 -6 -10 D -6 2 6 0 -4 E -20 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996268 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 C=22 B=16 E=14 so E is eliminated. Round 2 votes counts: A=29 B=26 D=23 C=22 so C is eliminated. Round 3 votes counts: D=34 A=34 B=32 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:206 D:199 B:183 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 22 6 20 B -20 0 6 -2 -18 C -22 -6 0 -6 -10 D -6 2 6 0 -4 E -20 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996268 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 22 6 20 B -20 0 6 -2 -18 C -22 -6 0 -6 -10 D -6 2 6 0 -4 E -20 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996268 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 22 6 20 B -20 0 6 -2 -18 C -22 -6 0 -6 -10 D -6 2 6 0 -4 E -20 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996268 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 632: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (7) D E C A B (6) B D C E A (6) B C D E A (6) A E D C B (6) A B E C D (6) A B C E D (6) A E C D B (5) D E C B A (4) D C E B A (4) B A D E C (4) B C D A E (3) B C A D E (3) B A C D E (3) A B E D C (3) E A D C B (2) D B E A C (2) B D C A E (2) A D E B C (2) A B D E C (2) E D C A B (1) E C D A B (1) E A C D B (1) D E B C A (1) D E A C B (1) D E A B C (1) D B C E A (1) C E D A B (1) C E A D B (1) C B D E A (1) C B A E D (1) B D A E C (1) B D A C E (1) B A D C E (1) A E D B C (1) A E C B D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 16 14 20 B 2 0 24 16 18 C -16 -24 0 -4 -2 D -14 -16 4 0 10 E -20 -18 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 14 20 B 2 0 24 16 18 C -16 -24 0 -4 -2 D -14 -16 4 0 10 E -20 -18 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989772 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=34 D=20 E=5 C=4 so C is eliminated. Round 2 votes counts: B=39 A=34 D=20 E=7 so E is eliminated. Round 3 votes counts: B=39 A=38 D=23 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:230 A:224 D:192 C:177 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 14 20 B 2 0 24 16 18 C -16 -24 0 -4 -2 D -14 -16 4 0 10 E -20 -18 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989772 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 14 20 B 2 0 24 16 18 C -16 -24 0 -4 -2 D -14 -16 4 0 10 E -20 -18 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989772 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 14 20 B 2 0 24 16 18 C -16 -24 0 -4 -2 D -14 -16 4 0 10 E -20 -18 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989772 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 633: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (16) B D C E A (10) E C A B D (9) D B C E A (6) C E B D A (6) A E C B D (6) B C D E A (5) A B D C E (5) B D A C E (4) A D B E C (4) E C D B A (3) E C D A B (3) D B A C E (3) C B E D A (3) E C A D B (2) C E B A D (2) A E D C B (2) E C B D A (1) E A C D B (1) D E C B A (1) D B A E C (1) D A B E C (1) C E A B D (1) B C E D A (1) A D E B C (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -6 4 -8 B -8 0 -14 12 -8 C 6 14 0 20 -6 D -4 -12 -20 0 -16 E 8 8 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -6 4 -8 B -8 0 -14 12 -8 C 6 14 0 20 -6 D -4 -12 -20 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=20 E=19 D=12 C=12 so D is eliminated. Round 2 votes counts: A=38 B=30 E=20 C=12 so C is eliminated. Round 3 votes counts: A=38 B=33 E=29 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 C:217 A:199 B:191 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 4 -8 B -8 0 -14 12 -8 C 6 14 0 20 -6 D -4 -12 -20 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 4 -8 B -8 0 -14 12 -8 C 6 14 0 20 -6 D -4 -12 -20 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 4 -8 B -8 0 -14 12 -8 C 6 14 0 20 -6 D -4 -12 -20 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 634: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) E B C D A (7) E B C A D (7) D A B C E (7) E C B A D (6) B E D A C (5) B A D E C (5) C E D A B (4) B D A E C (4) B A D C E (4) A D C B E (4) D C A E B (3) D A C B E (3) A B D C E (3) E C D B A (2) E C D A B (2) E C B D A (2) E B D C A (2) C E B A D (2) A C D B E (2) E D B A C (1) D E B A C (1) D A C E B (1) D A B E C (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A E B (1) C B A E D (1) C A E D B (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A D C (1) B E A C D (1) B D E A C (1) B A E D C (1) B A E C D (1) Total count = 100 A B C D E A 0 -10 8 4 4 B 10 0 22 12 10 C -8 -22 0 -10 -2 D -4 -12 10 0 2 E -4 -10 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 4 4 B 10 0 22 12 10 C -8 -22 0 -10 -2 D -4 -12 10 0 2 E -4 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 A=17 D=16 C=14 so C is eliminated. Round 2 votes counts: E=37 B=25 A=20 D=18 so D is eliminated. Round 3 votes counts: E=39 A=36 B=25 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:227 A:203 D:198 E:193 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 4 4 B 10 0 22 12 10 C -8 -22 0 -10 -2 D -4 -12 10 0 2 E -4 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 4 4 B 10 0 22 12 10 C -8 -22 0 -10 -2 D -4 -12 10 0 2 E -4 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 4 4 B 10 0 22 12 10 C -8 -22 0 -10 -2 D -4 -12 10 0 2 E -4 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 635: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) C B D E A (8) A D E B C (8) E A D B C (6) D A B C E (5) E A C B D (4) A E D C B (4) A E D B C (4) D B C E A (3) D B C A E (3) C E B A D (3) C B D A E (3) B D C E A (3) E C B A D (2) D B E C A (2) D B A C E (2) D A E B C (2) D A B E C (2) B C D E A (2) A E C D B (2) A D C B E (2) E D B A C (1) E C A B D (1) E B C D A (1) E A D C B (1) E A C D B (1) E A B D C (1) D C B A E (1) D B E A C (1) C E B D A (1) C A E B D (1) C A D B E (1) C A B D E (1) B E C D A (1) B D E C A (1) B D C A E (1) B C E D A (1) A E C B D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 4 -4 -4 B -4 0 0 -10 4 C -4 0 0 -10 6 D 4 10 10 0 8 E 4 -4 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -4 -4 B -4 0 0 -10 4 C -4 0 0 -10 6 D 4 10 10 0 8 E 4 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 D=21 E=18 B=9 so B is eliminated. Round 2 votes counts: C=29 D=26 A=26 E=19 so E is eliminated. Round 3 votes counts: A=39 C=34 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:200 C:196 B:195 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -4 -4 B -4 0 0 -10 4 C -4 0 0 -10 6 D 4 10 10 0 8 E 4 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -4 -4 B -4 0 0 -10 4 C -4 0 0 -10 6 D 4 10 10 0 8 E 4 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -4 -4 B -4 0 0 -10 4 C -4 0 0 -10 6 D 4 10 10 0 8 E 4 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 636: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) A C D E B (9) E D B A C (8) E B D A C (8) C A D B E (7) A D C E B (7) C B A E D (5) B E C D A (5) B C E A D (5) B E D C A (4) D A E C B (3) B E C A D (3) B C A E D (3) A D E C B (3) A C D B E (3) E D A B C (2) D E B A C (2) C B A D E (2) D E A C B (1) D E A B C (1) D C B A E (1) D A C E B (1) C A B E D (1) B D C E A (1) B C E D A (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 0 24 18 B -4 0 -14 2 6 C 0 14 0 16 20 D -24 -2 -16 0 6 E -18 -6 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.277392 B: 0.000000 C: 0.722608 D: 0.000000 E: 0.000000 Sum of squares = 0.599108284816 Cumulative probabilities = A: 0.277392 B: 0.277392 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 24 18 B -4 0 -14 2 6 C 0 14 0 16 20 D -24 -2 -16 0 6 E -18 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 B=22 E=18 D=9 so D is eliminated. Round 2 votes counts: C=28 A=28 E=22 B=22 so E is eliminated. Round 3 votes counts: B=40 A=32 C=28 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:225 A:223 B:195 D:182 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 24 18 B -4 0 -14 2 6 C 0 14 0 16 20 D -24 -2 -16 0 6 E -18 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 24 18 B -4 0 -14 2 6 C 0 14 0 16 20 D -24 -2 -16 0 6 E -18 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 24 18 B -4 0 -14 2 6 C 0 14 0 16 20 D -24 -2 -16 0 6 E -18 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 637: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) E D A B C (6) B C D A E (6) E A D B C (5) C B D A E (4) C A D B E (4) A D E B C (4) E C B A D (3) D A B E C (3) D A B C E (3) C E B A D (3) C B E D A (3) C B A D E (3) C A D E B (3) B D A C E (3) E C A D B (2) E B C D A (2) E A C D B (2) D B A E C (2) C E A D B (2) C B E A D (2) C B A E D (2) C A B D E (2) B E D A C (2) B E C D A (2) B C E D A (2) A E D C B (2) A E C D B (2) D E A B C (1) D B E A C (1) D A E B C (1) C E A B D (1) C A B E D (1) B D E A C (1) B D C A E (1) B C D E A (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 14 2 12 -4 B -14 0 -8 -12 -2 C -2 8 0 6 -4 D -12 12 -6 0 -10 E 4 2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 12 -4 B -14 0 -8 -12 -2 C -2 8 0 6 -4 D -12 12 -6 0 -10 E 4 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 B=18 D=11 A=10 so A is eliminated. Round 2 votes counts: E=35 C=31 B=18 D=16 so D is eliminated. Round 3 votes counts: E=41 C=32 B=27 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:210 C:204 D:192 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 12 -4 B -14 0 -8 -12 -2 C -2 8 0 6 -4 D -12 12 -6 0 -10 E 4 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 12 -4 B -14 0 -8 -12 -2 C -2 8 0 6 -4 D -12 12 -6 0 -10 E 4 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 12 -4 B -14 0 -8 -12 -2 C -2 8 0 6 -4 D -12 12 -6 0 -10 E 4 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 638: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (16) B C D E A (9) C B E D A (7) C B D E A (7) A B C D E (7) E D C B A (6) B C D A E (6) A D E B C (6) E D A C B (5) E A D C B (5) A E D B C (5) D E C B A (3) B C A D E (3) B A C D E (3) A D B C E (3) E D C A B (2) E C B D A (2) D E A C B (1) D B C E A (1) C E B D A (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 4 6 0 2 B -4 0 -12 -8 -8 C -6 12 0 -8 -6 D 0 8 8 0 0 E -2 8 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.644502 B: 0.000000 C: 0.000000 D: 0.355498 E: 0.000000 Sum of squares = 0.541761907987 Cumulative probabilities = A: 0.644502 B: 0.644502 C: 0.644502 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 0 2 B -4 0 -12 -8 -8 C -6 12 0 -8 -6 D 0 8 8 0 0 E -2 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=21 E=20 C=15 D=5 so D is eliminated. Round 2 votes counts: A=39 E=24 B=22 C=15 so C is eliminated. Round 3 votes counts: A=39 B=36 E=25 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:208 A:206 E:206 C:196 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 0 2 B -4 0 -12 -8 -8 C -6 12 0 -8 -6 D 0 8 8 0 0 E -2 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 0 2 B -4 0 -12 -8 -8 C -6 12 0 -8 -6 D 0 8 8 0 0 E -2 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 0 2 B -4 0 -12 -8 -8 C -6 12 0 -8 -6 D 0 8 8 0 0 E -2 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 639: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) D E A B C (10) C A B D E (8) A C B D E (8) E D B A C (7) B C E A D (6) B C A E D (6) E B D C A (5) D E A C B (4) D A E C B (3) D A E B C (3) B E C D A (3) B A C D E (3) A C D B E (3) D E B A C (2) C B A D E (2) B E D C A (2) A D E C B (2) A D C B E (2) E B D A C (1) C E B D A (1) C B E A D (1) B C E D A (1) B C A D E (1) A D C E B (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 16 14 B 6 0 4 22 20 C -4 -4 0 14 16 D -16 -22 -14 0 10 E -14 -20 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 16 14 B 6 0 4 22 20 C -4 -4 0 14 16 D -16 -22 -14 0 10 E -14 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=22 B=22 A=19 E=13 so E is eliminated. Round 2 votes counts: D=29 B=28 C=24 A=19 so A is eliminated. Round 3 votes counts: C=36 D=35 B=29 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:226 A:214 C:211 D:179 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 16 14 B 6 0 4 22 20 C -4 -4 0 14 16 D -16 -22 -14 0 10 E -14 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 16 14 B 6 0 4 22 20 C -4 -4 0 14 16 D -16 -22 -14 0 10 E -14 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 16 14 B 6 0 4 22 20 C -4 -4 0 14 16 D -16 -22 -14 0 10 E -14 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 640: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) B E D C A (7) A B D C E (7) E B C D A (5) D C E A B (5) B A D E C (5) A D C B E (5) A D C E B (4) A B C D E (4) E B D C A (3) C A D E B (3) E D C B A (2) E C D A B (2) E C B D A (2) D C A E B (2) C E D A B (2) C E A D B (2) B E C A D (2) B E A D C (2) B E A C D (2) B D E C A (2) B A E D C (2) B A D C E (2) A D B C E (2) A C E B D (2) A C D B E (2) E C D B A (1) E C A D B (1) E C A B D (1) E A B C D (1) D E C B A (1) D B C E A (1) D B A C E (1) C D A E B (1) B E D A C (1) B D A E C (1) B A E C D (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 10 22 10 B -14 0 4 8 2 C -10 -4 0 -10 10 D -22 -8 10 0 14 E -10 -2 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 22 10 B -14 0 4 8 2 C -10 -4 0 -10 10 D -22 -8 10 0 14 E -10 -2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=27 E=18 D=10 C=8 so C is eliminated. Round 2 votes counts: A=40 B=27 E=22 D=11 so D is eliminated. Round 3 votes counts: A=43 B=29 E=28 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:200 D:197 C:193 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 22 10 B -14 0 4 8 2 C -10 -4 0 -10 10 D -22 -8 10 0 14 E -10 -2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 22 10 B -14 0 4 8 2 C -10 -4 0 -10 10 D -22 -8 10 0 14 E -10 -2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 22 10 B -14 0 4 8 2 C -10 -4 0 -10 10 D -22 -8 10 0 14 E -10 -2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 641: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (11) E A C D B (8) D B A E C (6) C B A E D (6) D B E A C (5) E A D C B (4) C B D A E (4) B D A C E (4) A C E B D (4) E C A D B (3) E C A B D (3) D B E C A (3) D B C A E (3) C B E A D (3) C A B E D (3) B C D A E (3) A E C B D (3) D E B A C (2) D B A C E (2) C B E D A (2) E D A B C (1) E C D B A (1) E A C B D (1) D E B C A (1) D B C E A (1) C E D B A (1) C E B A D (1) C E A D B (1) C B D E A (1) C B A D E (1) C A E B D (1) B D C A E (1) B C A D E (1) B A D C E (1) A E D B C (1) A D E B C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -10 18 -6 B 6 0 -24 12 4 C 10 24 0 26 12 D -18 -12 -26 0 -20 E 6 -4 -12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 18 -6 B 6 0 -24 12 4 C 10 24 0 26 12 D -18 -12 -26 0 -20 E 6 -4 -12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=23 E=21 A=11 B=10 so B is eliminated. Round 2 votes counts: C=39 D=28 E=21 A=12 so A is eliminated. Round 3 votes counts: C=44 D=30 E=26 so E is eliminated. Round 4 votes counts: C=63 D=37 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:236 E:205 B:199 A:198 D:162 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 18 -6 B 6 0 -24 12 4 C 10 24 0 26 12 D -18 -12 -26 0 -20 E 6 -4 -12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 18 -6 B 6 0 -24 12 4 C 10 24 0 26 12 D -18 -12 -26 0 -20 E 6 -4 -12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 18 -6 B 6 0 -24 12 4 C 10 24 0 26 12 D -18 -12 -26 0 -20 E 6 -4 -12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 642: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) E D C B A (5) C B A D E (5) B D A C E (5) E C D A B (4) D B A E C (4) C E B D A (4) B C D A E (4) A E B D C (4) A B C D E (4) E D B A C (3) E D A B C (3) E A D B C (3) E A C D B (3) D B E A C (3) C B D E A (3) E D B C A (2) E C D B A (2) D E B C A (2) D B A C E (2) C E A B D (2) C B D A E (2) C A B D E (2) B D C A E (2) A C E B D (2) E D C A B (1) E C A D B (1) E A D C B (1) D E C B A (1) D E A B C (1) D C B E A (1) D B E C A (1) D B C E A (1) C D E B A (1) C B E A D (1) B A D C E (1) B A C D E (1) A D B E C (1) A C B E D (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -8 -14 2 B 12 0 0 8 10 C 8 0 0 2 8 D 14 -8 -2 0 0 E -2 -10 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.244450 C: 0.755550 D: 0.000000 E: 0.000000 Sum of squares = 0.630611482619 Cumulative probabilities = A: 0.000000 B: 0.244450 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -14 2 B 12 0 0 8 10 C 8 0 0 2 8 D 14 -8 -2 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 D=16 A=16 B=13 so B is eliminated. Round 2 votes counts: C=31 E=28 D=23 A=18 so A is eliminated. Round 3 votes counts: C=41 E=32 D=27 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:209 D:202 E:190 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -14 2 B 12 0 0 8 10 C 8 0 0 2 8 D 14 -8 -2 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -14 2 B 12 0 0 8 10 C 8 0 0 2 8 D 14 -8 -2 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -14 2 B 12 0 0 8 10 C 8 0 0 2 8 D 14 -8 -2 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 643: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) D C B A E (10) B C D E A (8) A E D B C (7) E A B C D (5) D C A E B (5) C D B E A (5) B E A C D (5) D A E C B (4) C B D E A (4) B C E D A (4) A E B D C (4) E B A C D (3) E A B D C (3) D B C A E (3) C D B A E (3) B C E A D (3) A D E C B (3) D C A B E (2) B E C A D (2) E A C B D (1) D A C E B (1) C D A E B (1) C B E A D (1) B D C E A (1) B A D E C (1) Total count = 100 A B C D E A 0 -6 -4 -2 10 B 6 0 -2 -10 4 C 4 2 0 -10 2 D 2 10 10 0 2 E -10 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -2 10 B 6 0 -2 -10 4 C 4 2 0 -10 2 D 2 10 10 0 2 E -10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 B=24 C=14 E=12 so E is eliminated. Round 2 votes counts: A=34 B=27 D=25 C=14 so C is eliminated. Round 3 votes counts: D=34 A=34 B=32 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:199 B:199 C:199 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 10 B 6 0 -2 -10 4 C 4 2 0 -10 2 D 2 10 10 0 2 E -10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 10 B 6 0 -2 -10 4 C 4 2 0 -10 2 D 2 10 10 0 2 E -10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 10 B 6 0 -2 -10 4 C 4 2 0 -10 2 D 2 10 10 0 2 E -10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 644: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (6) E A D B C (5) C D B A E (5) C B D E A (5) D C B A E (4) A D C B E (4) A C D B E (4) E D B A C (3) E A B C D (3) C D B E A (3) B C E D A (3) A E D B C (3) A D E B C (3) A D C E B (3) E B D C A (2) E B C D A (2) E B C A D (2) E B A D C (2) E A B D C (2) D C A B E (2) D A E B C (2) C D A B E (2) C B D A E (2) C B A E D (2) C A D B E (2) B E D C A (2) B E C D A (2) B D C E A (2) A E D C B (2) A E C B D (2) A D E C B (2) E D B C A (1) E B D A C (1) D A E C B (1) D A C B E (1) C A B D E (1) B D E C A (1) A E C D B (1) A E B C D (1) A C E B D (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -2 -4 6 B 0 0 -4 -8 10 C 2 4 0 4 10 D 4 8 -4 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -4 6 B 0 0 -4 -8 10 C 2 4 0 4 10 D 4 8 -4 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=23 C=22 B=16 D=10 so D is eliminated. Round 2 votes counts: A=33 C=28 E=23 B=16 so B is eliminated. Round 3 votes counts: C=39 A=33 E=28 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:211 C:210 A:200 B:199 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 -4 6 B 0 0 -4 -8 10 C 2 4 0 4 10 D 4 8 -4 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -4 6 B 0 0 -4 -8 10 C 2 4 0 4 10 D 4 8 -4 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -4 6 B 0 0 -4 -8 10 C 2 4 0 4 10 D 4 8 -4 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 645: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) D A C B E (5) B E C D A (5) B D C E A (5) A D C B E (5) D B C A E (4) B E D C A (4) B D E C A (4) A C E D B (4) E B C D A (3) E B A C D (3) E A C B D (3) D C A B E (3) B D A E C (3) A E C D B (3) E B C A D (2) C D E B A (2) C D A E B (2) B E A D C (2) B D A C E (2) B A E D C (2) A B D C E (2) E C B A D (1) E C A D B (1) E C A B D (1) E A C D B (1) E A B C D (1) D C B E A (1) D C A E B (1) D B C E A (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B E A (1) C A E D B (1) B E D A C (1) B E A C D (1) B D C A E (1) B C D E A (1) B A D E C (1) A E C B D (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 6 -2 6 B 2 0 -2 -2 10 C -6 2 0 2 14 D 2 2 -2 0 16 E -6 -10 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000125 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -2 6 B 2 0 -2 -2 10 C -6 2 0 2 14 D 2 2 -2 0 16 E -6 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999596 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=28 E=16 D=15 C=9 so C is eliminated. Round 2 votes counts: B=32 A=29 D=21 E=18 so E is eliminated. Round 3 votes counts: B=41 A=37 D=22 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:209 C:206 A:204 B:204 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 -2 6 B 2 0 -2 -2 10 C -6 2 0 2 14 D 2 2 -2 0 16 E -6 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999596 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 6 B 2 0 -2 -2 10 C -6 2 0 2 14 D 2 2 -2 0 16 E -6 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999596 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 6 B 2 0 -2 -2 10 C -6 2 0 2 14 D 2 2 -2 0 16 E -6 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999596 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 646: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (13) B D E A C (12) C D A E B (11) B E A D C (7) B A E D C (5) D C E A B (4) C D B E A (4) B A E C D (4) D B C E A (3) C D E A B (3) C D B A E (3) B D C E A (3) A E D B C (3) A E B C D (3) E A D B C (2) D C B E A (2) C A D E B (2) B C D E A (2) A E D C B (2) A E B D C (2) E A B D C (1) D C A E B (1) D B E C A (1) D B E A C (1) C D E B A (1) C B A E D (1) C A E B D (1) B A C E D (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -10 -2 8 B 0 0 0 -12 -2 C 10 0 0 2 12 D 2 12 -2 0 6 E -8 2 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.051885 C: 0.948115 D: 0.000000 E: 0.000000 Sum of squares = 0.901613262213 Cumulative probabilities = A: 0.000000 B: 0.051885 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -2 8 B 0 0 0 -12 -2 C 10 0 0 2 12 D 2 12 -2 0 6 E -8 2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041207 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=34 D=12 A=12 E=3 so E is eliminated. Round 2 votes counts: C=39 B=34 A=15 D=12 so D is eliminated. Round 3 votes counts: C=46 B=39 A=15 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 D:209 A:198 B:193 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -2 8 B 0 0 0 -12 -2 C 10 0 0 2 12 D 2 12 -2 0 6 E -8 2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041207 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -2 8 B 0 0 0 -12 -2 C 10 0 0 2 12 D 2 12 -2 0 6 E -8 2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041207 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -2 8 B 0 0 0 -12 -2 C 10 0 0 2 12 D 2 12 -2 0 6 E -8 2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041207 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 647: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (6) A D C B E (6) D A C E B (5) D A C B E (5) B E C A D (5) E C B D A (4) E B C D A (4) E B C A D (4) B E C D A (4) A D C E B (4) E C D A B (3) E B A D C (3) E A D B C (3) D C A B E (3) C D B A E (3) C D A B E (3) E D C A B (2) E B A C D (2) E A D C B (2) C E D A B (2) C B E D A (2) C B D A E (2) B C D A E (2) A E D C B (2) A D E C B (2) A D B C E (2) E D A C B (1) E A B D C (1) C D B E A (1) C D A E B (1) C B D E A (1) B E A D C (1) B E A C D (1) B C E A D (1) B C D E A (1) B C A D E (1) B A D C E (1) B A C D E (1) A E D B C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -10 -10 -8 B 0 0 -8 -4 6 C 10 8 0 8 8 D 10 4 -8 0 -8 E 8 -6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -10 -8 B 0 0 -8 -4 6 C 10 8 0 8 8 D 10 4 -8 0 -8 E 8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 A=19 C=15 D=13 so D is eliminated. Round 2 votes counts: E=29 A=29 B=24 C=18 so C is eliminated. Round 3 votes counts: A=36 B=33 E=31 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 E:201 D:199 B:197 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -10 -8 B 0 0 -8 -4 6 C 10 8 0 8 8 D 10 4 -8 0 -8 E 8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -10 -8 B 0 0 -8 -4 6 C 10 8 0 8 8 D 10 4 -8 0 -8 E 8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -10 -8 B 0 0 -8 -4 6 C 10 8 0 8 8 D 10 4 -8 0 -8 E 8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 648: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (9) E A C D B (8) D E B C A (8) D B C E A (8) A C E B D (8) E A D C B (7) D B E C A (7) A E C B D (6) C A B E D (5) E D B A C (4) E D A C B (4) E D A B C (4) C B A D E (4) A C B E D (4) D E B A C (3) B D C A E (3) B D C E A (2) E D C A B (1) C A B D E (1) B D A C E (1) B C A D E (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 -8 -12 B 0 0 2 -10 -8 C -2 -2 0 -4 -6 D 8 10 4 0 -4 E 12 8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 -8 -12 B 0 0 2 -10 -8 C -2 -2 0 -4 -6 D 8 10 4 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 A=20 B=16 C=10 so C is eliminated. Round 2 votes counts: E=28 D=26 A=26 B=20 so B is eliminated. Round 3 votes counts: D=41 A=31 E=28 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:215 D:209 C:193 B:192 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -8 -12 B 0 0 2 -10 -8 C -2 -2 0 -4 -6 D 8 10 4 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -8 -12 B 0 0 2 -10 -8 C -2 -2 0 -4 -6 D 8 10 4 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -8 -12 B 0 0 2 -10 -8 C -2 -2 0 -4 -6 D 8 10 4 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 649: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (14) E B C D A (11) C A E D B (10) B E D C A (9) A D C B E (8) C E A D B (7) D B A C E (6) D A B C E (6) E C A B D (5) D B A E C (5) B D E A C (5) C E A B D (3) C A D E B (3) A C D E B (3) B D A E C (2) B E D A C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -24 10 -10 B 6 0 -8 0 -12 C 24 8 0 14 -4 D -10 0 -14 0 -20 E 10 12 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -24 10 -10 B 6 0 -8 0 -12 C 24 8 0 14 -4 D -10 0 -14 0 -20 E 10 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=23 D=17 B=17 A=13 so A is eliminated. Round 2 votes counts: E=30 C=27 D=26 B=17 so B is eliminated. Round 3 votes counts: E=40 D=33 C=27 so C is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:221 B:193 A:185 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -24 10 -10 B 6 0 -8 0 -12 C 24 8 0 14 -4 D -10 0 -14 0 -20 E 10 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -24 10 -10 B 6 0 -8 0 -12 C 24 8 0 14 -4 D -10 0 -14 0 -20 E 10 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -24 10 -10 B 6 0 -8 0 -12 C 24 8 0 14 -4 D -10 0 -14 0 -20 E 10 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 650: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (6) D E B A C (5) B A C D E (5) E D C A B (4) C A E D B (4) C A B E D (4) B C E D A (4) A C D E B (4) E D C B A (3) E D B C A (3) D A E B C (3) D A B E C (3) E D A C B (2) E C D B A (2) E C B D A (2) D E A C B (2) D B E A C (2) C B E A D (2) C B A E D (2) B E D C A (2) B E C D A (2) B D A C E (2) B C E A D (2) B C A E D (2) A D E C B (2) A C B D E (2) A B D C E (2) A B C D E (2) E C D A B (1) E A D C B (1) D E B C A (1) D E A B C (1) D A E C B (1) C E D A B (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E D A (1) C A E B D (1) B D E A C (1) B D A E C (1) A D E B C (1) A D C E B (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -4 -12 -6 B 2 0 4 -4 2 C 4 -4 0 -2 -4 D 12 4 2 0 0 E 6 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.704337 E: 0.295663 Sum of squares = 0.583506837817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.704337 E: 1.000000 A B C D E A 0 -2 -4 -12 -6 B 2 0 4 -4 2 C 4 -4 0 -2 -4 D 12 4 2 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=19 E=18 D=18 C=18 so E is eliminated. Round 2 votes counts: D=30 B=27 C=23 A=20 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:204 B:202 C:197 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -12 -6 B 2 0 4 -4 2 C 4 -4 0 -2 -4 D 12 4 2 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -12 -6 B 2 0 4 -4 2 C 4 -4 0 -2 -4 D 12 4 2 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -12 -6 B 2 0 4 -4 2 C 4 -4 0 -2 -4 D 12 4 2 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 651: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (15) C D B A E (11) C D A E B (11) C D A B E (8) A E C D B (7) D C A E B (4) E B A D C (3) E A B D C (3) D C B A E (3) B E A C D (3) B D C E A (3) A E D C B (3) E B A C D (2) E A B C D (2) D C A B E (2) C A D E B (2) B E D C A (2) B E D A C (2) A C E D B (2) E A C B D (1) D B C E A (1) C B D E A (1) C B D A E (1) C A E D B (1) B E C D A (1) B D E C A (1) A E D B C (1) A E B D C (1) A E B C D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -4 -2 20 B -2 0 -18 -16 8 C 4 18 0 10 4 D 2 16 -10 0 0 E -20 -8 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -2 20 B -2 0 -18 -16 8 C 4 18 0 10 4 D 2 16 -10 0 0 E -20 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=27 A=17 E=11 D=10 so D is eliminated. Round 2 votes counts: C=44 B=28 A=17 E=11 so E is eliminated. Round 3 votes counts: C=44 B=33 A=23 so A is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:208 D:204 B:186 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -2 20 B -2 0 -18 -16 8 C 4 18 0 10 4 D 2 16 -10 0 0 E -20 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 20 B -2 0 -18 -16 8 C 4 18 0 10 4 D 2 16 -10 0 0 E -20 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 20 B -2 0 -18 -16 8 C 4 18 0 10 4 D 2 16 -10 0 0 E -20 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 652: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) B E D C A (5) B C D A E (5) D E B C A (4) D E A C B (4) C D A B E (4) A C D E B (4) E D A C B (3) E D A B C (3) E B D A C (3) E B A D C (3) D C B A E (3) D C A B E (3) C A B D E (3) B C A D E (3) E A D B C (2) E A B C D (2) D E C B A (2) D B C E A (2) C D B A E (2) C B A D E (2) B E A C D (2) B D E C A (2) B D C E A (2) A C D B E (2) E D B C A (1) E D B A C (1) E B D C A (1) E B A C D (1) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) D C B E A (1) D A C E B (1) C B D A E (1) C A D E B (1) B E C D A (1) B E C A D (1) B C E D A (1) B C E A D (1) B C D E A (1) B C A E D (1) A E C D B (1) A E C B D (1) A D C E B (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -18 -14 -6 B 6 0 0 -8 12 C 18 0 0 0 6 D 14 8 0 0 22 E 6 -12 -6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.529721 D: 0.470279 E: 0.000000 Sum of squares = 0.501766626941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.529721 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 -14 -6 B 6 0 0 -8 12 C 18 0 0 0 6 D 14 8 0 0 22 E 6 -12 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 D=21 C=19 A=12 so A is eliminated. Round 2 votes counts: C=28 E=25 B=25 D=22 so D is eliminated. Round 3 votes counts: C=37 E=36 B=27 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:222 C:212 B:205 E:183 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -18 -14 -6 B 6 0 0 -8 12 C 18 0 0 0 6 D 14 8 0 0 22 E 6 -12 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 -14 -6 B 6 0 0 -8 12 C 18 0 0 0 6 D 14 8 0 0 22 E 6 -12 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 -14 -6 B 6 0 0 -8 12 C 18 0 0 0 6 D 14 8 0 0 22 E 6 -12 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 653: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) B E D A C (7) A C D B E (7) E B C D A (5) E B D C A (4) B E D C A (4) A C D E B (4) A C B E D (4) E D B C A (3) E B C A D (3) D E C B A (3) D E C A B (3) D C E A B (3) B D E A C (3) A D C B E (3) E D C B A (2) D B E A C (2) B E C D A (2) B E A D C (2) B E A C D (2) A D C E B (2) A C B D E (2) E C B D A (1) D E B C A (1) D E B A C (1) D B A E C (1) D A C E B (1) D A C B E (1) C E D B A (1) C E D A B (1) C E A B D (1) C D E A B (1) C D A E B (1) C B E A D (1) C A E B D (1) C A D E B (1) B C A E D (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -4 -2 -30 B 24 0 12 16 14 C 4 -12 0 2 -20 D 2 -16 -2 0 -14 E 30 -14 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -4 -2 -30 B 24 0 12 16 14 C 4 -12 0 2 -20 D 2 -16 -2 0 -14 E 30 -14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=24 E=18 D=16 C=8 so C is eliminated. Round 2 votes counts: B=35 A=26 E=21 D=18 so D is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:225 C:187 D:185 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -4 -2 -30 B 24 0 12 16 14 C 4 -12 0 2 -20 D 2 -16 -2 0 -14 E 30 -14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -4 -2 -30 B 24 0 12 16 14 C 4 -12 0 2 -20 D 2 -16 -2 0 -14 E 30 -14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -4 -2 -30 B 24 0 12 16 14 C 4 -12 0 2 -20 D 2 -16 -2 0 -14 E 30 -14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 654: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) D C B E A (6) D B C E A (6) C E A D B (5) E B A C D (4) D C B A E (4) D B E C A (4) B E D C A (4) B E A D C (4) A B E D C (4) E C A B D (3) C E D B A (3) B D E C A (3) E B C D A (2) D B A E C (2) D A B C E (2) C E D A B (2) C E A B D (2) C D E B A (2) C D A E B (2) B E D A C (2) B D E A C (2) B D A E C (2) A E C B D (2) A E B D C (2) A D B E C (2) A B D E C (2) E C B D A (1) E A C B D (1) E A B C D (1) D B C A E (1) C E B D A (1) C D E A B (1) C D A B E (1) C B D E A (1) C A E D B (1) C A D E B (1) B A E D C (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -12 -8 -20 B 10 0 16 6 6 C 12 -16 0 -6 -12 D 8 -6 6 0 -10 E 20 -6 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 -20 B 10 0 16 6 6 C 12 -16 0 -6 -12 D 8 -6 6 0 -10 E 20 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999486 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=23 C=22 B=18 E=12 so E is eliminated. Round 2 votes counts: C=26 D=25 A=25 B=24 so B is eliminated. Round 3 votes counts: D=38 A=34 C=28 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 E:218 D:199 C:189 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 -20 B 10 0 16 6 6 C 12 -16 0 -6 -12 D 8 -6 6 0 -10 E 20 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999486 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 -20 B 10 0 16 6 6 C 12 -16 0 -6 -12 D 8 -6 6 0 -10 E 20 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999486 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 -20 B 10 0 16 6 6 C 12 -16 0 -6 -12 D 8 -6 6 0 -10 E 20 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999486 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 655: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E D C A B (8) E C D A B (7) B A E C D (6) B A D C E (6) E D C B A (5) E C D B A (5) C D E A B (5) B A C D E (5) D C A B E (4) E C B A D (3) E B C A D (3) E B A C D (3) D C A E B (3) B A E D C (3) A D C B E (3) E D B C A (2) B A C E D (2) A B D C E (2) A B C D E (2) E C B D A (1) E C A D B (1) E B D C A (1) E B C D A (1) D E C B A (1) D E C A B (1) D A C B E (1) C E D A B (1) C E A B D (1) C D A E B (1) C A E B D (1) C A B D E (1) B E D A C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -30 -14 -18 B -4 0 -26 -14 -26 C 30 26 0 0 -4 D 14 14 0 0 -12 E 18 26 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999047 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -30 -14 -18 B -4 0 -26 -14 -26 C 30 26 0 0 -4 D 14 14 0 0 -12 E 18 26 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=23 D=19 C=10 A=8 so A is eliminated. Round 2 votes counts: E=40 B=27 D=22 C=11 so C is eliminated. Round 3 votes counts: E=43 B=29 D=28 so D is eliminated. Round 4 votes counts: E=63 B=37 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 C:226 D:208 A:171 B:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -30 -14 -18 B -4 0 -26 -14 -26 C 30 26 0 0 -4 D 14 14 0 0 -12 E 18 26 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -30 -14 -18 B -4 0 -26 -14 -26 C 30 26 0 0 -4 D 14 14 0 0 -12 E 18 26 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -30 -14 -18 B -4 0 -26 -14 -26 C 30 26 0 0 -4 D 14 14 0 0 -12 E 18 26 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 656: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (6) A B C E D (6) D E B C A (5) D E A C B (4) D E A B C (4) D B E C A (4) B D C E A (4) A C B E D (4) E D C B A (3) E A C D B (3) D E C B A (3) D B E A C (3) C E B D A (3) C B E A D (3) A D B E C (3) E D A C B (2) E C D A B (2) E C A D B (2) D E C A B (2) D B A E C (2) D A B E C (2) C E A B D (2) C B A E D (2) C A B E D (2) B D A C E (2) B C A E D (2) B C A D E (2) A E C D B (2) A D E B C (2) A B D E C (2) A B D C E (2) E D C A B (1) E C D B A (1) E A D C B (1) C B E D A (1) B C E D A (1) B C D E A (1) B A D C E (1) B A C E D (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 8 0 -10 B -12 0 0 -4 2 C -8 0 0 -6 -8 D 0 4 6 0 -2 E 10 -2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999989 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 12 8 0 -10 B -12 0 0 -4 2 C -8 0 0 -6 -8 D 0 4 6 0 -2 E 10 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000133 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 E=15 B=14 C=13 so C is eliminated. Round 2 votes counts: A=31 D=29 E=20 B=20 so E is eliminated. Round 3 votes counts: A=39 D=38 B=23 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:205 D:204 B:193 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 0 -10 B -12 0 0 -4 2 C -8 0 0 -6 -8 D 0 4 6 0 -2 E 10 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000133 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 0 -10 B -12 0 0 -4 2 C -8 0 0 -6 -8 D 0 4 6 0 -2 E 10 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000133 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 0 -10 B -12 0 0 -4 2 C -8 0 0 -6 -8 D 0 4 6 0 -2 E 10 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000133 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 657: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) B A C D E (9) B D A C E (8) C A B E D (7) D E A C B (6) B C A E D (6) E C A D B (5) D E B A C (4) D B E A C (4) D B A C E (4) D A C E B (4) A C B D E (4) E C A B D (3) E B C A D (3) E B D C A (2) D E C A B (2) D A B C E (2) C A E B D (2) A C D B E (2) E D B C A (1) E D B A C (1) E C D A B (1) E C B A D (1) D A C B E (1) C A E D B (1) C A D E B (1) C A B D E (1) B E D A C (1) B E C A D (1) B D E A C (1) B A D C E (1) B A C E D (1) A C D E B (1) Total count = 100 A B C D E A 0 4 8 -2 10 B -4 0 -2 2 6 C -8 2 0 -2 10 D 2 -2 2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000048 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -2 10 B -4 0 -2 2 6 C -8 2 0 -2 10 D 2 -2 2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999711 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 E=26 C=12 A=7 so A is eliminated. Round 2 votes counts: B=28 D=27 E=26 C=19 so C is eliminated. Round 3 votes counts: B=40 D=31 E=29 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:210 D:206 B:201 C:201 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 -2 10 B -4 0 -2 2 6 C -8 2 0 -2 10 D 2 -2 2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999711 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 10 B -4 0 -2 2 6 C -8 2 0 -2 10 D 2 -2 2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999711 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 10 B -4 0 -2 2 6 C -8 2 0 -2 10 D 2 -2 2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999711 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 658: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) E C B D A (6) B D A C E (5) D B A E C (4) C E B A D (4) C E A B D (4) A B D C E (4) E C D A B (3) E B D C A (3) E B C D A (3) D A E B C (3) C E B D A (3) C A E D B (3) C A B E D (3) B E C D A (3) B D E A C (3) A D E B C (3) A C D B E (3) C E A D B (2) C A E B D (2) B D E C A (2) B C E D A (2) B C A D E (2) A D E C B (2) A D C B E (2) A D B E C (2) A D B C E (2) A C E D B (2) A C D E B (2) E D B A C (1) E D A B C (1) E C D B A (1) E C B A D (1) D A E C B (1) C B A E D (1) C A B D E (1) B E D C A (1) B D C E A (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -2 -8 12 B -6 0 6 12 0 C 2 -6 0 4 -2 D 8 -12 -4 0 2 E -12 0 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.440952 B: 0.204758 C: 0.267629 D: 0.086661 E: 0.000000 Sum of squares = 0.315499687238 Cumulative probabilities = A: 0.440952 B: 0.645710 C: 0.913339 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -8 12 B -6 0 6 12 0 C 2 -6 0 4 -2 D 8 -12 -4 0 2 E -12 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.446721 B: 0.233607 C: 0.192623 D: 0.127049 E: 0.000000 Sum of squares = 0.307377049234 Cumulative probabilities = A: 0.446721 B: 0.680328 C: 0.872951 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=23 A=23 B=20 E=19 D=15 so D is eliminated. Round 2 votes counts: A=34 B=24 C=23 E=19 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:206 A:204 C:199 D:197 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 6 -2 -8 12 B -6 0 6 12 0 C 2 -6 0 4 -2 D 8 -12 -4 0 2 E -12 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.446721 B: 0.233607 C: 0.192623 D: 0.127049 E: 0.000000 Sum of squares = 0.307377049234 Cumulative probabilities = A: 0.446721 B: 0.680328 C: 0.872951 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -8 12 B -6 0 6 12 0 C 2 -6 0 4 -2 D 8 -12 -4 0 2 E -12 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.446721 B: 0.233607 C: 0.192623 D: 0.127049 E: 0.000000 Sum of squares = 0.307377049234 Cumulative probabilities = A: 0.446721 B: 0.680328 C: 0.872951 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -8 12 B -6 0 6 12 0 C 2 -6 0 4 -2 D 8 -12 -4 0 2 E -12 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.446721 B: 0.233607 C: 0.192623 D: 0.127049 E: 0.000000 Sum of squares = 0.307377049234 Cumulative probabilities = A: 0.446721 B: 0.680328 C: 0.872951 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 659: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) B A E C D (7) E C D B A (5) C E D A B (5) B A E D C (5) C E D B A (4) D A C E B (3) C D E A B (3) B E D C A (3) B E A C D (3) B A D E C (3) A D C E B (3) A C D E B (3) A B D E C (3) A B D C E (3) A B C D E (3) E D C B A (2) E C B D A (2) E B D C A (2) D E C A B (2) D A E C B (2) C B E A D (2) B E D A C (2) B E C D A (2) B E A D C (2) B C E A D (2) B A C E D (2) A D C B E (2) E D B C A (1) D E C B A (1) D E B A C (1) D E A C B (1) C D A E B (1) C A E D B (1) C A D E B (1) B D E A C (1) B C A E D (1) A D B E C (1) A D B C E (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 6 0 -6 B 6 0 -2 -2 0 C -6 2 0 -2 -2 D 0 2 2 0 -8 E 6 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.384491 C: 0.000000 D: 0.000000 E: 0.615509 Sum of squares = 0.526684696356 Cumulative probabilities = A: 0.000000 B: 0.384491 C: 0.384491 D: 0.384491 E: 1.000000 A B C D E A 0 -6 6 0 -6 B 6 0 -2 -2 0 C -6 2 0 -2 -2 D 0 2 2 0 -8 E 6 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499872 C: 0.000000 D: 0.000000 E: 0.500128 Sum of squares = 0.500000032842 Cumulative probabilities = A: 0.000000 B: 0.499872 C: 0.499872 D: 0.499872 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=21 D=17 C=17 E=12 so E is eliminated. Round 2 votes counts: B=35 C=24 A=21 D=20 so D is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:208 B:201 D:198 A:197 C:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 0 -6 B 6 0 -2 -2 0 C -6 2 0 -2 -2 D 0 2 2 0 -8 E 6 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499872 C: 0.000000 D: 0.000000 E: 0.500128 Sum of squares = 0.500000032842 Cumulative probabilities = A: 0.000000 B: 0.499872 C: 0.499872 D: 0.499872 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 0 -6 B 6 0 -2 -2 0 C -6 2 0 -2 -2 D 0 2 2 0 -8 E 6 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499872 C: 0.000000 D: 0.000000 E: 0.500128 Sum of squares = 0.500000032842 Cumulative probabilities = A: 0.000000 B: 0.499872 C: 0.499872 D: 0.499872 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 0 -6 B 6 0 -2 -2 0 C -6 2 0 -2 -2 D 0 2 2 0 -8 E 6 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499872 C: 0.000000 D: 0.000000 E: 0.500128 Sum of squares = 0.500000032842 Cumulative probabilities = A: 0.000000 B: 0.499872 C: 0.499872 D: 0.499872 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 660: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) B A D E C (11) D E C B A (7) C A B D E (7) C E D A B (6) D E B A C (5) E D C B A (4) E D B A C (4) D B E A C (4) C D E A B (4) B A E D C (4) C A E D B (3) C A B E D (3) B D A E C (3) D B A E C (2) B E D A C (2) A B D E C (2) E C D B A (1) E B A C D (1) D E B C A (1) D C E B A (1) D C A B E (1) D A B C E (1) C E D B A (1) C E A D B (1) C E A B D (1) C A E B D (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 16 6 14 B 2 0 14 8 18 C -16 -14 0 -8 -6 D -6 -8 8 0 4 E -14 -18 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 6 14 B 2 0 14 8 18 C -16 -14 0 -8 -6 D -6 -8 8 0 4 E -14 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991813 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 A=21 B=20 E=10 so E is eliminated. Round 2 votes counts: D=30 C=28 B=21 A=21 so B is eliminated. Round 3 votes counts: A=37 D=35 C=28 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:221 A:217 D:199 E:185 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 6 14 B 2 0 14 8 18 C -16 -14 0 -8 -6 D -6 -8 8 0 4 E -14 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991813 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 6 14 B 2 0 14 8 18 C -16 -14 0 -8 -6 D -6 -8 8 0 4 E -14 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991813 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 6 14 B 2 0 14 8 18 C -16 -14 0 -8 -6 D -6 -8 8 0 4 E -14 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991813 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 661: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E A B D C (7) D A C E B (7) B E A D C (7) C D A E B (5) C B D E A (5) D C B A E (4) B C E D A (4) B C D E A (4) A E D C B (4) A E D B C (4) E A C D B (3) E A B C D (3) D C A B E (2) D B C A E (2) C E A D B (2) C D E A B (2) C D B E A (2) C B E D A (2) C B D A E (2) B E A C D (2) A E C D B (2) E C B A D (1) E A D B C (1) D C A E B (1) D B A E C (1) D A E C B (1) C E B D A (1) C A D E B (1) B E C A D (1) B D C E A (1) B D A E C (1) B D A C E (1) B C E A D (1) B C D A E (1) A E B D C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -6 -16 2 B 4 0 -14 -10 4 C 6 14 0 6 18 D 16 10 -6 0 8 E -2 -4 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -16 2 B 4 0 -14 -10 4 C 6 14 0 6 18 D 16 10 -6 0 8 E -2 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=23 D=18 E=15 A=13 so A is eliminated. Round 2 votes counts: C=31 E=26 B=23 D=20 so D is eliminated. Round 3 votes counts: C=45 E=29 B=26 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:214 B:192 A:188 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -16 2 B 4 0 -14 -10 4 C 6 14 0 6 18 D 16 10 -6 0 8 E -2 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -16 2 B 4 0 -14 -10 4 C 6 14 0 6 18 D 16 10 -6 0 8 E -2 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -16 2 B 4 0 -14 -10 4 C 6 14 0 6 18 D 16 10 -6 0 8 E -2 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 662: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) D A B C E (8) D B A E C (7) C E A B D (7) A C B E D (7) B A C E D (5) D B E A C (4) A C E B D (4) E B C A D (3) D B E C A (3) A D B C E (3) A B C E D (3) E D B C A (2) E C B D A (2) D E C B A (2) D E C A B (2) D B A C E (2) D A C B E (2) C A E B D (2) B E A C D (2) B A D C E (2) A C B D E (2) A B C D E (2) E D C B A (1) D E B C A (1) D E A B C (1) D A C E B (1) D A B E C (1) C E B A D (1) B E C D A (1) B E C A D (1) B D E A C (1) B A D E C (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 22 18 10 B 4 0 6 14 16 C -22 -6 0 12 8 D -18 -14 -12 0 -6 E -10 -16 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 22 18 10 B 4 0 6 14 16 C -22 -6 0 12 8 D -18 -14 -12 0 -6 E -10 -16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=24 E=19 B=13 C=10 so C is eliminated. Round 2 votes counts: D=34 E=27 A=26 B=13 so B is eliminated. Round 3 votes counts: D=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 B:220 C:196 E:186 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 22 18 10 B 4 0 6 14 16 C -22 -6 0 12 8 D -18 -14 -12 0 -6 E -10 -16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 18 10 B 4 0 6 14 16 C -22 -6 0 12 8 D -18 -14 -12 0 -6 E -10 -16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 18 10 B 4 0 6 14 16 C -22 -6 0 12 8 D -18 -14 -12 0 -6 E -10 -16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 663: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) C E D B A (8) B A C E D (8) D A B E C (7) C E B A D (7) D A E B C (6) E C D B A (5) D E C A B (5) C B E A D (4) B A E C D (4) D E A C B (3) D E A B C (3) D C A B E (3) B A E D C (3) E C B A D (2) D C A E B (2) D A E C B (2) C D E B A (2) C D E A B (2) B C A E D (2) E D C B A (1) E D C A B (1) E C D A B (1) D A C B E (1) C E B D A (1) C B D A E (1) C B A E D (1) B C E A D (1) B A C D E (1) A D B E C (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 -8 8 B 2 0 -4 -6 -2 C -2 4 0 2 -8 D 8 6 -2 0 0 E -8 2 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.601738 E: 0.398262 Sum of squares = 0.520701287324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.601738 E: 1.000000 A B C D E A 0 -2 2 -8 8 B 2 0 -4 -6 -2 C -2 4 0 2 -8 D 8 6 -2 0 0 E -8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 0.499932 Sum of squares = 0.500000009203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=26 B=19 A=13 E=10 so E is eliminated. Round 2 votes counts: D=34 C=34 B=19 A=13 so A is eliminated. Round 3 votes counts: D=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:206 E:201 A:200 C:198 B:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -8 8 B 2 0 -4 -6 -2 C -2 4 0 2 -8 D 8 6 -2 0 0 E -8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 0.499932 Sum of squares = 0.500000009203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -8 8 B 2 0 -4 -6 -2 C -2 4 0 2 -8 D 8 6 -2 0 0 E -8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 0.499932 Sum of squares = 0.500000009203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -8 8 B 2 0 -4 -6 -2 C -2 4 0 2 -8 D 8 6 -2 0 0 E -8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 0.499932 Sum of squares = 0.500000009203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500068 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 664: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) B E C D A (9) E A D C B (8) C D A B E (8) A E D C B (8) C D B A E (7) B E A C D (5) B C D E A (5) E A B D C (4) D C A E B (4) B E D C A (4) B C D A E (4) A D C E B (4) D C A B E (3) E B A C D (2) E A D B C (2) C B D A E (2) B E C A D (2) E D A C B (1) E D A B C (1) E A B C D (1) D A C E B (1) C A D E B (1) C A D B E (1) B E A D C (1) A E C B D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 0 2 -8 B 0 0 -2 -2 2 C 0 2 0 -2 -18 D -2 2 2 0 -18 E 8 -2 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.090909 E: 0.090909 Sum of squares = 0.685950412826 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.909091 E: 1.000000 A B C D E A 0 0 0 2 -8 B 0 0 -2 -2 2 C 0 2 0 -2 -18 D -2 2 2 0 -18 E 8 -2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.090909 E: 0.090909 Sum of squares = 0.685950413117 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 C=19 A=15 D=8 so D is eliminated. Round 2 votes counts: B=30 E=28 C=26 A=16 so A is eliminated. Round 3 votes counts: E=38 C=32 B=30 so B is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:199 A:197 D:192 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 2 -8 B 0 0 -2 -2 2 C 0 2 0 -2 -18 D -2 2 2 0 -18 E 8 -2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.090909 E: 0.090909 Sum of squares = 0.685950413117 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.909091 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -8 B 0 0 -2 -2 2 C 0 2 0 -2 -18 D -2 2 2 0 -18 E 8 -2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.090909 E: 0.090909 Sum of squares = 0.685950413117 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.909091 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -8 B 0 0 -2 -2 2 C 0 2 0 -2 -18 D -2 2 2 0 -18 E 8 -2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.090909 E: 0.090909 Sum of squares = 0.685950413117 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.909091 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 665: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) D B A C E (5) C E D A B (5) E C B D A (4) E C A D B (4) E B C D A (4) E B C A D (4) B D A C E (4) A B E D C (4) E C A B D (3) B A E D C (3) B A D E C (3) A D C B E (3) E C D A B (2) E C B A D (2) D C B E A (2) D C B A E (2) D C A E B (2) D B C A E (2) C E D B A (2) C A D E B (2) B E A C D (2) A E C D B (2) A D C E B (2) A C E D B (2) A B D C E (2) E C D B A (1) E A C B D (1) E A B C D (1) D C A B E (1) D B C E A (1) C E A D B (1) C D E B A (1) C D E A B (1) C D A E B (1) B E C D A (1) B E A D C (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -2 8 8 B 0 0 8 -8 6 C 2 -8 0 -8 8 D -8 8 8 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.626841 B: 0.373159 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.532177434673 Cumulative probabilities = A: 0.626841 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 8 8 B 0 0 8 -8 6 C 2 -8 0 -8 8 D -8 8 8 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035467 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 A=26 B=20 D=15 C=13 so C is eliminated. Round 2 votes counts: E=34 A=28 B=20 D=18 so D is eliminated. Round 3 votes counts: E=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 D:205 B:203 C:197 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 8 8 B 0 0 8 -8 6 C 2 -8 0 -8 8 D -8 8 8 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035467 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 8 8 B 0 0 8 -8 6 C 2 -8 0 -8 8 D -8 8 8 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035467 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 8 8 B 0 0 8 -8 6 C 2 -8 0 -8 8 D -8 8 8 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035467 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 666: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (6) E A B C D (5) D C E A B (5) D C B E A (5) C D A B E (5) A E C B D (5) A B E C D (5) E A C D B (4) D C E B A (4) E B A D C (3) D C B A E (3) B E D A C (3) E B D A C (2) D B C A E (2) C D E A B (2) C D B A E (2) C A D B E (2) B D C E A (2) B D C A E (2) B A E D C (2) B A D C E (2) A E C D B (2) A E B C D (2) A C E D B (2) A B C E D (2) E D C B A (1) E D C A B (1) E D B C A (1) E D A C B (1) E A D C B (1) E A B D C (1) D C A B E (1) D B C E A (1) C E D A B (1) C D A E B (1) C B D A E (1) C B A D E (1) C A E D B (1) B E D C A (1) B D E C A (1) B D A C E (1) B C D A E (1) B A E C D (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 0 -2 B -4 0 -6 2 8 C -6 6 0 -4 4 D 0 -2 4 0 -6 E 2 -8 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428568 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 4 6 0 -2 B -4 0 -6 2 8 C -6 6 0 -4 4 D 0 -2 4 0 -6 E 2 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428596 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 D=21 A=21 E=20 C=16 so C is eliminated. Round 2 votes counts: D=31 B=24 A=24 E=21 so E is eliminated. Round 3 votes counts: D=36 A=35 B=29 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:204 B:200 C:200 D:198 E:198 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 0 -2 B -4 0 -6 2 8 C -6 6 0 -4 4 D 0 -2 4 0 -6 E 2 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428596 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 0 -2 B -4 0 -6 2 8 C -6 6 0 -4 4 D 0 -2 4 0 -6 E 2 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428596 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 0 -2 B -4 0 -6 2 8 C -6 6 0 -4 4 D 0 -2 4 0 -6 E 2 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428596 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 667: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) D E C B A (7) E D C A B (5) C B E D A (5) A D E B C (5) E C D B A (4) D E B C A (4) D E A B C (4) A E D B C (4) A D B E C (4) A C B E D (4) A B D C E (4) A B C E D (4) E D A C B (3) C B E A D (3) A B D E C (3) A B C D E (3) D E B A C (2) C E B D A (2) C E B A D (2) B C D E A (2) B C A D E (2) B A C D E (2) E C D A B (1) E C A D B (1) D B E C A (1) C E A B D (1) C D B E A (1) C B D E A (1) B D C E A (1) B D C A E (1) B C D A E (1) B A D C E (1) A E D C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -10 -10 -20 B 4 0 -2 -14 -12 C 10 2 0 -22 -20 D 10 14 22 0 -2 E 20 12 20 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 -10 -20 B 4 0 -2 -14 -12 C 10 2 0 -22 -20 D 10 14 22 0 -2 E 20 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=24 D=18 C=15 B=10 so B is eliminated. Round 2 votes counts: A=36 E=24 D=20 C=20 so D is eliminated. Round 3 votes counts: E=42 A=36 C=22 so C is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:222 B:188 C:185 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 -10 -20 B 4 0 -2 -14 -12 C 10 2 0 -22 -20 D 10 14 22 0 -2 E 20 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -10 -20 B 4 0 -2 -14 -12 C 10 2 0 -22 -20 D 10 14 22 0 -2 E 20 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -10 -20 B 4 0 -2 -14 -12 C 10 2 0 -22 -20 D 10 14 22 0 -2 E 20 12 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 668: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (7) B D C E A (7) A E C D B (7) C D B E A (5) A E D B C (5) C A E D B (4) B D E C A (4) B D E A C (4) A C E D B (4) D B E C A (3) C D E B A (3) A E B D C (3) A B C D E (3) E D C B A (2) D E B C A (2) C E D B A (2) C E A D B (2) C B A D E (2) C A B D E (2) B D A E C (2) A E D C B (2) A C E B D (2) A C B E D (2) A B E D C (2) A B D E C (2) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) E C D A B (1) E A D C B (1) E A D B C (1) D E B A C (1) D C B E A (1) D B E A C (1) C B D A E (1) C A B E D (1) B C D E A (1) B C D A E (1) B A D E C (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -4 -4 -4 B 6 0 -6 -2 8 C 4 6 0 4 4 D 4 2 -4 0 10 E 4 -8 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -4 -4 B 6 0 -6 -2 8 C 4 6 0 4 4 D 4 2 -4 0 10 E 4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=29 B=21 E=9 D=8 so D is eliminated. Round 2 votes counts: A=33 C=30 B=25 E=12 so E is eliminated. Round 3 votes counts: A=37 C=33 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:206 B:203 A:191 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 -4 -4 B 6 0 -6 -2 8 C 4 6 0 4 4 D 4 2 -4 0 10 E 4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -4 -4 B 6 0 -6 -2 8 C 4 6 0 4 4 D 4 2 -4 0 10 E 4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -4 -4 B 6 0 -6 -2 8 C 4 6 0 4 4 D 4 2 -4 0 10 E 4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 669: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) E D A B C (7) D E A C B (6) C A B D E (6) A C D E B (5) D E C A B (4) C B A D E (4) E D B C A (3) E D B A C (3) E D A C B (3) D C E A B (3) B C A E D (3) A D E C B (3) A D C E B (3) A C B D E (3) A B E D C (3) A B C E D (3) D C E B A (2) C D A E B (2) C A D B E (2) B E D C A (2) B E D A C (2) B E A D C (2) B C A D E (2) B A E D C (2) A E D B C (2) E A D B C (1) D A E C B (1) C D A B E (1) C B D E A (1) B C E A D (1) B A E C D (1) B A C E D (1) A E D C B (1) A C D B E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 24 10 2 2 B -24 0 -2 -6 2 C -10 2 0 -6 6 D -2 6 6 0 0 E -2 -2 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 10 2 2 B -24 0 -2 -6 2 C -10 2 0 -6 6 D -2 6 6 0 0 E -2 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995286 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 E=17 D=16 C=16 so D is eliminated. Round 2 votes counts: E=27 A=27 B=25 C=21 so C is eliminated. Round 3 votes counts: A=38 E=32 B=30 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:205 C:196 E:195 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 10 2 2 B -24 0 -2 -6 2 C -10 2 0 -6 6 D -2 6 6 0 0 E -2 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995286 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 10 2 2 B -24 0 -2 -6 2 C -10 2 0 -6 6 D -2 6 6 0 0 E -2 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995286 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 10 2 2 B -24 0 -2 -6 2 C -10 2 0 -6 6 D -2 6 6 0 0 E -2 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995286 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 670: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (13) A D B C E (9) D A B C E (7) B C E A D (7) D A E C B (4) C E B D A (4) A D E C B (4) A D E B C (4) E C B A D (3) A B D C E (3) E D C B A (2) E D C A B (2) E C D B A (2) E A C B D (2) D E C B A (2) D B C E A (2) C B E D A (2) C B E A D (2) B D C A E (2) B C E D A (2) B C D E A (2) B A D C E (2) A E D C B (2) A D B E C (2) A B C D E (2) E C A D B (1) E C A B D (1) E A C D B (1) D C E B A (1) D B A C E (1) C E D B A (1) C B D E A (1) B C A D E (1) A B E D C (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 0 -6 B 4 0 0 6 2 C 6 0 0 -2 4 D 0 -6 2 0 0 E 6 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.524545 C: 0.475455 D: 0.000000 E: 0.000000 Sum of squares = 0.501204877323 Cumulative probabilities = A: 0.000000 B: 0.524545 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 0 -6 B 4 0 0 6 2 C 6 0 0 -2 4 D 0 -6 2 0 0 E 6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=27 D=17 B=16 C=10 so C is eliminated. Round 2 votes counts: E=32 A=30 B=21 D=17 so D is eliminated. Round 3 votes counts: A=41 E=35 B=24 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:206 C:204 E:200 D:198 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 0 -6 B 4 0 0 6 2 C 6 0 0 -2 4 D 0 -6 2 0 0 E 6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 -6 B 4 0 0 6 2 C 6 0 0 -2 4 D 0 -6 2 0 0 E 6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 -6 B 4 0 0 6 2 C 6 0 0 -2 4 D 0 -6 2 0 0 E 6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 671: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E A D C B (12) B C D A E (9) C B A D E (8) E C B A D (6) B C A D E (6) E D A C B (5) E B C D A (5) C B E A D (5) B D A C E (4) E D A B C (3) D A E B C (3) C A B D E (3) B D C A E (2) A D C B E (2) A D B C E (2) E D B C A (1) E D B A C (1) E C A D B (1) E C A B D (1) E A C D B (1) D E A B C (1) D A B E C (1) C E B A D (1) B C E D A (1) B C E A D (1) B A D C E (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 0 2 10 B 2 0 8 6 16 C 0 -8 0 -4 18 D -2 -6 4 0 12 E -10 -16 -18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 10 B 2 0 8 6 16 C 0 -8 0 -4 18 D -2 -6 4 0 12 E -10 -16 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997637 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=24 D=18 C=17 A=5 so A is eliminated. Round 2 votes counts: E=36 B=24 D=23 C=17 so C is eliminated. Round 3 votes counts: B=40 E=37 D=23 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:205 D:204 C:203 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 2 10 B 2 0 8 6 16 C 0 -8 0 -4 18 D -2 -6 4 0 12 E -10 -16 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997637 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 10 B 2 0 8 6 16 C 0 -8 0 -4 18 D -2 -6 4 0 12 E -10 -16 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997637 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 10 B 2 0 8 6 16 C 0 -8 0 -4 18 D -2 -6 4 0 12 E -10 -16 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997637 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 672: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) A C D B E (7) D B C A E (6) D B A C E (6) B D C E A (6) E C A B D (5) E A B D C (5) B D E C A (5) A E C D B (5) E A C B D (4) A D B C E (4) A E B D C (3) A B D E C (3) E C A D B (2) E B C D A (2) E A C D B (2) D C B A E (2) C E B D A (2) C D B E A (2) C A D B E (2) B D E A C (2) B D C A E (2) B D A C E (2) A C D E B (2) E B D A C (1) E B A D C (1) E A B C D (1) C E D B A (1) C E A D B (1) C A D E B (1) B D A E C (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 2 0 -4 B 2 0 4 10 2 C -2 -4 0 -2 -8 D 0 -10 2 0 8 E 4 -2 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 0 -4 B 2 0 4 10 2 C -2 -4 0 -2 -8 D 0 -10 2 0 8 E 4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=26 B=18 D=14 C=9 so C is eliminated. Round 2 votes counts: E=37 A=29 B=18 D=16 so D is eliminated. Round 3 votes counts: E=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:201 D:200 A:198 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 0 -4 B 2 0 4 10 2 C -2 -4 0 -2 -8 D 0 -10 2 0 8 E 4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 -4 B 2 0 4 10 2 C -2 -4 0 -2 -8 D 0 -10 2 0 8 E 4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 -4 B 2 0 4 10 2 C -2 -4 0 -2 -8 D 0 -10 2 0 8 E 4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 673: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) A C E D B (9) C E A D B (7) B D C E A (6) B C D E A (5) D B A E C (4) C E A B D (4) C A E D B (4) B D A E C (4) A E C D B (4) A D B E C (4) E C A D B (3) D B E A C (3) C A E B D (3) B D E C A (3) B D E A C (3) B D C A E (3) E A D B C (2) C E B D A (2) C B E D A (2) C B D E A (2) C A B D E (2) A E D C B (2) A D E B C (2) E D B A C (1) E D A B C (1) E C B D A (1) E A C D B (1) D A B E C (1) B C E D A (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 4 -2 6 B -2 0 4 2 6 C -4 -4 0 4 22 D 2 -2 -4 0 6 E -6 -6 -22 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333303 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -2 6 B -2 0 4 2 6 C -4 -4 0 4 22 D 2 -2 -4 0 6 E -6 -6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=26 A=23 E=9 D=8 so D is eliminated. Round 2 votes counts: B=41 C=26 A=24 E=9 so E is eliminated. Round 3 votes counts: B=42 C=30 A=28 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:209 A:205 B:205 D:201 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 -2 6 B -2 0 4 2 6 C -4 -4 0 4 22 D 2 -2 -4 0 6 E -6 -6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 6 B -2 0 4 2 6 C -4 -4 0 4 22 D 2 -2 -4 0 6 E -6 -6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 6 B -2 0 4 2 6 C -4 -4 0 4 22 D 2 -2 -4 0 6 E -6 -6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 674: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) E B C A D (6) C E A D B (6) B E C D A (6) E C B A D (5) D A C B E (5) D A B C E (5) B E C A D (5) B E D C A (4) B E A C D (4) A D B C E (4) E C A B D (3) D B A C E (3) D A C E B (3) C E D A B (3) B D A E C (3) E C B D A (2) C A E D B (2) B D E C A (2) B D E A C (2) B A E D C (2) A D C E B (2) A B D E C (2) E C A D B (1) E B C D A (1) D C E B A (1) D C E A B (1) D C A B E (1) D B C E A (1) C A D E B (1) B E A D C (1) B D A C E (1) A D C B E (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -18 -4 -8 B -2 0 6 0 6 C 18 -6 0 -6 2 D 4 0 6 0 -4 E 8 -6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.505780 C: 0.000000 D: 0.494220 E: 0.000000 Sum of squares = 0.500066778316 Cumulative probabilities = A: 0.000000 B: 0.505780 C: 0.505780 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 -4 -8 B -2 0 6 0 6 C 18 -6 0 -6 2 D 4 0 6 0 -4 E 8 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999803 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=28 E=18 C=12 A=12 so C is eliminated. Round 2 votes counts: B=30 D=28 E=27 A=15 so A is eliminated. Round 3 votes counts: D=37 B=33 E=30 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:205 C:204 D:203 E:202 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -18 -4 -8 B -2 0 6 0 6 C 18 -6 0 -6 2 D 4 0 6 0 -4 E 8 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999803 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -4 -8 B -2 0 6 0 6 C 18 -6 0 -6 2 D 4 0 6 0 -4 E 8 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999803 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -4 -8 B -2 0 6 0 6 C 18 -6 0 -6 2 D 4 0 6 0 -4 E 8 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999803 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 675: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (16) E D A C B (10) B C A E D (7) D E A C B (6) B D E A C (5) B C E A D (5) C B A E D (4) E D A B C (3) D E B A C (3) D E A B C (3) D A E C B (3) C B A D E (3) C A D E B (3) C A D B E (3) B E D A C (3) A D E C B (3) E C A D B (2) B E C D A (2) A D C E B (2) A C D E B (2) E D B A C (1) E A D C B (1) D A B C E (1) C A E D B (1) C A B D E (1) B D C E A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 -10 -2 8 6 B 10 0 10 4 12 C 2 -10 0 4 6 D -8 -4 -4 0 18 E -6 -12 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 8 6 B 10 0 10 4 12 C 2 -10 0 4 6 D -8 -4 -4 0 18 E -6 -12 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 E=17 D=16 C=15 A=8 so A is eliminated. Round 2 votes counts: B=44 D=21 E=18 C=17 so C is eliminated. Round 3 votes counts: B=52 D=29 E=19 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:201 C:201 D:201 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 8 6 B 10 0 10 4 12 C 2 -10 0 4 6 D -8 -4 -4 0 18 E -6 -12 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 8 6 B 10 0 10 4 12 C 2 -10 0 4 6 D -8 -4 -4 0 18 E -6 -12 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 8 6 B 10 0 10 4 12 C 2 -10 0 4 6 D -8 -4 -4 0 18 E -6 -12 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 676: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) B A C D E (11) A B C E D (10) D E C B A (7) D E C A B (4) E B A D C (3) E A C B D (3) D C E A B (3) C A D B E (3) C A B D E (3) A B E C D (3) E D C B A (2) E D B A C (2) E D A C B (2) E C D A B (2) E A C D B (2) E A B C D (2) D B C A E (2) C E A D B (2) C B A D E (2) C A E D B (2) B A E D C (2) B A C E D (2) A C E B D (2) E D A B C (1) E C A D B (1) E A B D C (1) D C E B A (1) D B E C A (1) D B C E A (1) C A B E D (1) B E A D C (1) B C A D E (1) B A D C E (1) A E C B D (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 22 2 22 -4 B -22 0 -8 2 -10 C -2 8 0 10 -4 D -22 -2 -10 0 -18 E 4 10 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 2 22 -4 B -22 0 -8 2 -10 C -2 8 0 10 -4 D -22 -2 -10 0 -18 E 4 10 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=19 B=18 A=18 C=13 so C is eliminated. Round 2 votes counts: E=34 A=27 B=20 D=19 so D is eliminated. Round 3 votes counts: E=49 A=27 B=24 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:221 E:218 C:206 B:181 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 2 22 -4 B -22 0 -8 2 -10 C -2 8 0 10 -4 D -22 -2 -10 0 -18 E 4 10 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 2 22 -4 B -22 0 -8 2 -10 C -2 8 0 10 -4 D -22 -2 -10 0 -18 E 4 10 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 2 22 -4 B -22 0 -8 2 -10 C -2 8 0 10 -4 D -22 -2 -10 0 -18 E 4 10 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 677: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) E D A B C (7) A D C B E (7) E D A C B (6) C B E D A (6) C B A D E (6) C B E A D (5) A D E B C (5) E D C B A (4) E C B D A (4) E B C D A (4) B C A E D (4) B C E D A (3) A B C D E (3) E C D B A (2) E A B D C (2) A C D B E (2) E D C A B (1) E B D C A (1) D C A E B (1) D A C E B (1) C E D B A (1) C E B D A (1) C D E A B (1) C D A B E (1) C B D E A (1) C B D A E (1) C A D B E (1) C A B D E (1) B E C A D (1) B C E A D (1) B C A D E (1) B A E C D (1) B A C D E (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 -10 -2 B -4 0 -24 -2 0 C 4 24 0 4 0 D 10 2 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.390622 D: 0.000000 E: 0.609378 Sum of squares = 0.523927261281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.390622 D: 0.390622 E: 1.000000 A B C D E A 0 4 -4 -10 -2 B -4 0 -24 -2 0 C 4 24 0 4 0 D 10 2 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=25 A=21 B=12 D=11 so D is eliminated. Round 2 votes counts: E=31 A=31 C=26 B=12 so B is eliminated. Round 3 votes counts: C=35 A=33 E=32 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:206 D:199 A:194 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -10 -2 B -4 0 -24 -2 0 C 4 24 0 4 0 D 10 2 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -10 -2 B -4 0 -24 -2 0 C 4 24 0 4 0 D 10 2 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -10 -2 B -4 0 -24 -2 0 C 4 24 0 4 0 D 10 2 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 678: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) C D B E A (8) C B D E A (8) C B D A E (8) A E B D C (8) B C A E D (7) E A D B C (5) E A B D C (5) A E D B C (5) D C A E B (4) B E A C D (3) B A E C D (3) B A C E D (3) E A B C D (2) D C B E A (2) D A C E B (2) C D B A E (2) B C E A D (2) A E D C B (2) A E B C D (2) A B E C D (2) E D A B C (1) E A D C B (1) D E C A B (1) D E A C B (1) C D A E B (1) B E C D A (1) B C D E A (1) B C A D E (1) Total count = 100 A B C D E A 0 2 -10 2 0 B -2 0 2 12 2 C 10 -2 0 8 16 D -2 -12 -8 0 -4 E 0 -2 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.5510204082 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 2 0 B -2 0 2 12 2 C 10 -2 0 8 16 D -2 -12 -8 0 -4 E 0 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020407359 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=21 D=19 A=19 E=14 so E is eliminated. Round 2 votes counts: A=32 C=27 B=21 D=20 so D is eliminated. Round 3 votes counts: C=43 A=36 B=21 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:207 A:197 E:193 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -10 2 0 B -2 0 2 12 2 C 10 -2 0 8 16 D -2 -12 -8 0 -4 E 0 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020407359 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 2 0 B -2 0 2 12 2 C 10 -2 0 8 16 D -2 -12 -8 0 -4 E 0 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020407359 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 2 0 B -2 0 2 12 2 C 10 -2 0 8 16 D -2 -12 -8 0 -4 E 0 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020407359 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 679: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (14) E A D B C (5) C B D A E (5) A B E C D (5) E D A C B (4) E A D C B (4) C B D E A (4) B C A E D (4) B C A D E (4) E D C A B (3) E A B C D (3) D E C A B (3) B A C E D (3) A E B C D (3) D E A B C (2) D C E B A (2) D C B E A (2) D C B A E (2) D A E C B (2) C B E D A (2) B C E A D (2) B C D A E (2) A E B D C (2) A D B E C (2) A B D C E (2) A B C E D (2) E C B D A (1) E B C A D (1) E A C B D (1) D C A E B (1) D C A B E (1) D A E B C (1) D A C B E (1) C E D B A (1) C E B D A (1) C D B E A (1) C B E A D (1) B A C D E (1) Total count = 100 A B C D E A 0 22 14 -10 -14 B -22 0 -12 -2 -8 C -14 12 0 -6 -12 D 10 2 6 0 4 E 14 8 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 14 -10 -14 B -22 0 -12 -2 -8 C -14 12 0 -6 -12 D 10 2 6 0 4 E 14 8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 B=16 A=16 C=15 so C is eliminated. Round 2 votes counts: D=32 B=28 E=24 A=16 so A is eliminated. Round 3 votes counts: B=37 D=34 E=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:215 D:211 A:206 C:190 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 14 -10 -14 B -22 0 -12 -2 -8 C -14 12 0 -6 -12 D 10 2 6 0 4 E 14 8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 14 -10 -14 B -22 0 -12 -2 -8 C -14 12 0 -6 -12 D 10 2 6 0 4 E 14 8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 14 -10 -14 B -22 0 -12 -2 -8 C -14 12 0 -6 -12 D 10 2 6 0 4 E 14 8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 680: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) D C E A B (7) A B D E C (7) C D E A B (5) D A B C E (4) B A E C D (4) E C D B A (3) E B C A D (3) B A D E C (3) A B C E D (3) E D B C A (2) D E B C A (2) D C A E B (2) D C A B E (2) D B E A C (2) D B A E C (2) D A C B E (2) C E A B D (2) B E D A C (2) B E A C D (2) B D A E C (2) A D B C E (2) A B E D C (2) A B D C E (2) E D C B A (1) E C B A D (1) E B D C A (1) E B A C D (1) D E C B A (1) D B A C E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A D B (1) C D E B A (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E A C (1) B A E D C (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -4 -18 -8 B 2 0 4 -18 -2 C 4 -4 0 -30 8 D 18 18 30 0 28 E 8 2 -8 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -18 -8 B 2 0 4 -18 -2 C 4 -4 0 -30 8 D 18 18 30 0 28 E 8 2 -8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=19 C=16 B=16 E=12 so E is eliminated. Round 2 votes counts: D=40 B=21 C=20 A=19 so A is eliminated. Round 3 votes counts: D=43 B=36 C=21 so C is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:247 B:193 C:189 E:187 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -18 -8 B 2 0 4 -18 -2 C 4 -4 0 -30 8 D 18 18 30 0 28 E 8 2 -8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -18 -8 B 2 0 4 -18 -2 C 4 -4 0 -30 8 D 18 18 30 0 28 E 8 2 -8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -18 -8 B 2 0 4 -18 -2 C 4 -4 0 -30 8 D 18 18 30 0 28 E 8 2 -8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 681: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) A E C D B (8) D E B A C (7) E D A B C (6) C B A D E (6) A C E B D (6) A C E D B (5) D B E C A (4) C A E D B (4) C A E B D (4) B C D A E (4) E A D C B (3) E A C D B (3) C A B E D (3) B D E C A (3) D E B C A (2) D B E A C (2) C B D A E (2) B D C A E (2) B C A D E (2) E D C A B (1) E D B A C (1) E A D B C (1) D E C B A (1) C E D B A (1) C E D A B (1) C A B D E (1) B D E A C (1) B C D E A (1) A E D B C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -8 -2 0 B 2 0 0 -2 -10 C 8 0 0 6 12 D 2 2 -6 0 4 E 0 10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.390808 C: 0.609192 D: 0.000000 E: 0.000000 Sum of squares = 0.523845789644 Cumulative probabilities = A: 0.000000 B: 0.390808 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -2 0 B 2 0 0 -2 -10 C 8 0 0 6 12 D 2 2 -6 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=22 A=22 D=16 E=15 so E is eliminated. Round 2 votes counts: A=29 B=25 D=24 C=22 so C is eliminated. Round 3 votes counts: A=41 B=33 D=26 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:201 E:197 B:195 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -2 0 B 2 0 0 -2 -10 C 8 0 0 6 12 D 2 2 -6 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -2 0 B 2 0 0 -2 -10 C 8 0 0 6 12 D 2 2 -6 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -2 0 B 2 0 0 -2 -10 C 8 0 0 6 12 D 2 2 -6 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 682: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) B C A E D (7) C B A E D (6) E A D C B (5) C B E A D (5) E A D B C (4) E A C B D (4) C E A D B (4) C E A B D (4) C B D A E (4) B A E C D (4) D E A C B (3) D B A E C (3) B C D A E (3) A E D B C (3) A E B C D (3) D B E A C (2) D B C A E (2) D A E B C (2) C E D A B (2) C D B E A (2) C B D E A (2) B D C A E (2) B C A D E (2) B A E D C (2) A E B D C (2) E D A C B (1) E D A B C (1) E C A D B (1) E C A B D (1) E A C D B (1) D C E A B (1) D C B E A (1) D C B A E (1) C E B A D (1) C B A D E (1) B D A E C (1) Total count = 100 A B C D E A 0 -2 -4 20 -4 B 2 0 0 8 0 C 4 0 0 14 0 D -20 -8 -14 0 -22 E 4 0 0 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.431683 C: 0.376422 D: 0.000000 E: 0.191896 Sum of squares = 0.36486718786 Cumulative probabilities = A: 0.000000 B: 0.431683 C: 0.808104 D: 0.808104 E: 1.000000 A B C D E A 0 -2 -4 20 -4 B 2 0 0 8 0 C 4 0 0 14 0 D -20 -8 -14 0 -22 E 4 0 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.333333333288 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=22 B=21 E=18 A=8 so A is eliminated. Round 2 votes counts: C=31 E=26 D=22 B=21 so B is eliminated. Round 3 votes counts: C=43 E=32 D=25 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:209 A:205 B:205 D:168 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 20 -4 B 2 0 0 8 0 C 4 0 0 14 0 D -20 -8 -14 0 -22 E 4 0 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.333333333288 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 20 -4 B 2 0 0 8 0 C 4 0 0 14 0 D -20 -8 -14 0 -22 E 4 0 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.333333333288 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 20 -4 B 2 0 0 8 0 C 4 0 0 14 0 D -20 -8 -14 0 -22 E 4 0 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.333333333288 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 683: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (5) A E C B D (5) E A C D B (4) D E A B C (4) D A E B C (4) B D C A E (4) B C D A E (4) E D A C B (3) E A D C B (3) D E C B A (3) D B C A E (3) D A B E C (3) C E D B A (3) C E A B D (3) C B A E D (3) C B A D E (3) E C A D B (2) E C A B D (2) E A D B C (2) D E B A C (2) D B C E A (2) C D E B A (2) C D B E A (2) C B E D A (2) A D E B C (2) A D B E C (2) A B C E D (2) E D C A B (1) E D A B C (1) E C D A B (1) D E A C B (1) D C E B A (1) D C B E A (1) D B E C A (1) D B A E C (1) D B A C E (1) C E D A B (1) C E B D A (1) C B D A E (1) C A E B D (1) C A B E D (1) B D A C E (1) B C A E D (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -8 -18 -6 B -2 0 -10 -14 -8 C 8 10 0 2 0 D 18 14 -2 0 10 E 6 8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.887033 D: 0.000000 E: 0.112967 Sum of squares = 0.799588351332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.887033 D: 0.887033 E: 1.000000 A B C D E A 0 2 -8 -18 -6 B -2 0 -10 -14 -8 C 8 10 0 2 0 D 18 14 -2 0 10 E 6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.72222263991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=27 E=19 A=14 B=12 so B is eliminated. Round 2 votes counts: C=33 D=32 E=19 A=16 so A is eliminated. Round 3 votes counts: D=37 C=37 E=26 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:210 E:202 A:185 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -18 -6 B -2 0 -10 -14 -8 C 8 10 0 2 0 D 18 14 -2 0 10 E 6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.72222263991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -18 -6 B -2 0 -10 -14 -8 C 8 10 0 2 0 D 18 14 -2 0 10 E 6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.72222263991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -18 -6 B -2 0 -10 -14 -8 C 8 10 0 2 0 D 18 14 -2 0 10 E 6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.72222263991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 684: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) A B D E C (9) B A D C E (8) B A D E C (7) E C D A B (6) E D C A B (4) D E C B A (4) D B A E C (4) B D A E C (4) E C D B A (3) C E D B A (3) B A C D E (3) A B D C E (3) D E C A B (2) D E B C A (2) D A B E C (2) C E A D B (2) C A E B D (2) B C E D A (2) B C A E D (2) B A C E D (2) A C B E D (2) A B C D E (2) E D C B A (1) D E A C B (1) D B E A C (1) D A E B C (1) C B E A D (1) C A E D B (1) C A B E D (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 6 -2 14 B -6 0 6 -4 8 C -6 -6 0 -10 -4 D 2 4 10 0 12 E -14 -8 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -2 14 B -6 0 6 -4 8 C -6 -6 0 -10 -4 D 2 4 10 0 12 E -14 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=21 A=20 D=17 E=14 so E is eliminated. Round 2 votes counts: C=30 B=28 D=22 A=20 so A is eliminated. Round 3 votes counts: B=42 C=34 D=24 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 A:212 B:202 C:187 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -2 14 B -6 0 6 -4 8 C -6 -6 0 -10 -4 D 2 4 10 0 12 E -14 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -2 14 B -6 0 6 -4 8 C -6 -6 0 -10 -4 D 2 4 10 0 12 E -14 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -2 14 B -6 0 6 -4 8 C -6 -6 0 -10 -4 D 2 4 10 0 12 E -14 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 685: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (15) E C A D B (13) B D A C E (11) D B A C E (7) D B C A E (4) B A E C D (4) E C D A B (3) B A D C E (3) B A C E D (3) E D C B A (2) E A C B D (2) D E C B A (2) D E C A B (2) D C A E B (2) D B E C A (2) D B C E A (2) C E A D B (2) B E A C D (2) A E C B D (2) A C E B D (2) A B C E D (2) E D C A B (1) E C D B A (1) D C E A B (1) D C B E A (1) D B A E C (1) C D E A B (1) B D A E C (1) B A E D C (1) B A D E C (1) B A C D E (1) A E B C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -8 12 -4 B -2 0 -6 4 -4 C 8 6 0 12 -14 D -12 -4 -12 0 -14 E 4 4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -8 12 -4 B -2 0 -6 4 -4 C 8 6 0 12 -14 D -12 -4 -12 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=27 D=24 A=9 C=3 so C is eliminated. Round 2 votes counts: E=39 B=27 D=25 A=9 so A is eliminated. Round 3 votes counts: E=44 B=30 D=26 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:206 A:201 B:196 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 12 -4 B -2 0 -6 4 -4 C 8 6 0 12 -14 D -12 -4 -12 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 12 -4 B -2 0 -6 4 -4 C 8 6 0 12 -14 D -12 -4 -12 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 12 -4 B -2 0 -6 4 -4 C 8 6 0 12 -14 D -12 -4 -12 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 686: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) A B E C D (7) E C D B A (6) D C E B A (5) D A C B E (5) C B D E A (5) E C B D A (4) A D E C B (4) A D B C E (4) E D A C B (3) D E A C B (3) C B E D A (3) B C D A E (3) B C A E D (3) B A E C D (3) A E D B C (3) A B D C E (3) E A D C B (2) E A B C D (2) D E C B A (2) D C B E A (2) B A C D E (2) A D E B C (2) A D B E C (2) A B C D E (2) E D C A B (1) E B C D A (1) E B C A D (1) D E C A B (1) D C A B E (1) D A E C B (1) C D E B A (1) B E A C D (1) B C E A D (1) B C D E A (1) B A C E D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 4 -10 -4 B 4 0 -12 -10 -2 C -4 12 0 -4 -16 D 10 10 4 0 -2 E 4 2 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 -10 -4 B 4 0 -12 -10 -2 C -4 12 0 -4 -16 D 10 10 4 0 -2 E 4 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 D=20 B=15 C=9 so C is eliminated. Round 2 votes counts: A=29 E=27 B=23 D=21 so D is eliminated. Round 3 votes counts: E=39 A=36 B=25 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:211 C:194 A:193 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 -10 -4 B 4 0 -12 -10 -2 C -4 12 0 -4 -16 D 10 10 4 0 -2 E 4 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -10 -4 B 4 0 -12 -10 -2 C -4 12 0 -4 -16 D 10 10 4 0 -2 E 4 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -10 -4 B 4 0 -12 -10 -2 C -4 12 0 -4 -16 D 10 10 4 0 -2 E 4 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 687: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B E D A (9) D A E B C (7) C B E A D (7) D A C B E (5) D A B E C (5) E C B A D (4) E B C A D (4) C E B A D (4) C B D A E (4) A D E B C (4) E A B D C (3) D A E C B (3) D A C E B (3) B C D A E (3) E A B C D (2) D C B A E (2) D C A B E (2) C B D E A (2) B C E A D (2) A E D B C (2) A D E C B (2) E B A D C (1) E B A C D (1) E A D C B (1) E A C D B (1) D A B C E (1) C E B D A (1) C D A E B (1) B E A D C (1) B E A C D (1) B D C A E (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 4 6 0 -8 B -4 0 -4 2 -8 C -6 4 0 -6 -4 D 0 -2 6 0 -10 E 8 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 0 -8 B -4 0 -4 2 -8 C -6 4 0 -6 -4 D 0 -2 6 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 E=26 B=9 A=9 so B is eliminated. Round 2 votes counts: C=34 D=29 E=28 A=9 so A is eliminated. Round 3 votes counts: D=35 C=34 E=31 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:215 A:201 D:197 C:194 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 0 -8 B -4 0 -4 2 -8 C -6 4 0 -6 -4 D 0 -2 6 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 0 -8 B -4 0 -4 2 -8 C -6 4 0 -6 -4 D 0 -2 6 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 0 -8 B -4 0 -4 2 -8 C -6 4 0 -6 -4 D 0 -2 6 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 688: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (18) E B D A C (11) C A B D E (7) B E D C A (7) C B A D E (4) A C E D B (4) A C D E B (4) E D A C B (3) D E A C B (3) B D E C A (3) A C D B E (3) E A C D B (2) E A C B D (2) D E B A C (2) D C A B E (2) D A C E B (2) C A B E D (2) B E D A C (2) B D C A E (2) B C D A E (2) E D B A C (1) E D A B C (1) E B D C A (1) E B C A D (1) E B A D C (1) E B A C D (1) E A D C B (1) D B E A C (1) D B A C E (1) C B A E D (1) C A E B D (1) C A D E B (1) B E C D A (1) B D C E A (1) B C A E D (1) Total count = 100 A B C D E A 0 12 -10 8 10 B -12 0 -20 2 16 C 10 20 0 10 12 D -8 -2 -10 0 12 E -10 -16 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 8 10 B -12 0 -20 2 16 C 10 20 0 10 12 D -8 -2 -10 0 12 E -10 -16 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=25 B=19 D=11 A=11 so D is eliminated. Round 2 votes counts: C=36 E=30 B=21 A=13 so A is eliminated. Round 3 votes counts: C=49 E=30 B=21 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:210 D:196 B:193 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 8 10 B -12 0 -20 2 16 C 10 20 0 10 12 D -8 -2 -10 0 12 E -10 -16 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 8 10 B -12 0 -20 2 16 C 10 20 0 10 12 D -8 -2 -10 0 12 E -10 -16 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 8 10 B -12 0 -20 2 16 C 10 20 0 10 12 D -8 -2 -10 0 12 E -10 -16 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 689: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (6) E C B A D (5) D A E C B (5) B C E A D (4) A B E C D (4) E D C A B (3) E C D B A (3) D C E A B (3) D C B A E (3) D A B C E (3) B A C E D (3) B A C D E (3) A D E B C (3) A D B E C (3) A D B C E (3) E D C B A (2) E C B D A (2) E B C A D (2) D E C A B (2) D E A C B (2) D C A E B (2) D C A B E (2) D A C E B (2) D A C B E (2) D A B E C (2) C E D B A (2) C B E D A (2) B A E C D (2) A B E D C (2) E C D A B (1) E B A C D (1) E A D C B (1) E A B C D (1) D E C B A (1) D C E B A (1) D C B E A (1) D B A C E (1) D A E B C (1) C E B A D (1) C D B E A (1) B E C A D (1) B C E D A (1) B A D C E (1) A E D B C (1) A E B D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 -12 2 B -2 0 -10 -12 -8 C 2 10 0 -8 -2 D 12 12 8 0 -4 E -2 8 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839504 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 A B C D E A 0 2 -2 -12 2 B -2 0 -10 -12 -8 C 2 10 0 -8 -2 D 12 12 8 0 -4 E -2 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839349 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=21 A=19 B=15 C=12 so C is eliminated. Round 2 votes counts: D=34 E=30 A=19 B=17 so B is eliminated. Round 3 votes counts: E=38 D=34 A=28 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:214 E:206 C:201 A:195 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 -12 2 B -2 0 -10 -12 -8 C 2 10 0 -8 -2 D 12 12 8 0 -4 E -2 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839349 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -12 2 B -2 0 -10 -12 -8 C 2 10 0 -8 -2 D 12 12 8 0 -4 E -2 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839349 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -12 2 B -2 0 -10 -12 -8 C 2 10 0 -8 -2 D 12 12 8 0 -4 E -2 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839349 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 690: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) C A D B E (8) B E A C D (8) E B D A C (7) D E B C A (7) D C A E B (7) D A C E B (6) E D B C A (5) B E D C A (5) D C A B E (3) B E C A D (3) B C A E D (3) B A C E D (3) A C D B E (3) A C B E D (3) E D B A C (2) E B A C D (2) D E C B A (2) D E C A B (2) D E B A C (2) C A B D E (2) A C D E B (2) E D A B C (1) D E A C B (1) D A E C B (1) C B A D E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -14 -20 -12 B 18 0 16 -4 -12 C 14 -16 0 -20 -14 D 20 4 20 0 -4 E 12 12 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -14 -20 -12 B 18 0 16 -4 -12 C 14 -16 0 -20 -14 D 20 4 20 0 -4 E 12 12 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=26 B=22 C=11 A=10 so A is eliminated. Round 2 votes counts: D=31 E=26 B=23 C=20 so C is eliminated. Round 3 votes counts: D=44 B=30 E=26 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:221 D:220 B:209 C:182 A:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -14 -20 -12 B 18 0 16 -4 -12 C 14 -16 0 -20 -14 D 20 4 20 0 -4 E 12 12 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 -20 -12 B 18 0 16 -4 -12 C 14 -16 0 -20 -14 D 20 4 20 0 -4 E 12 12 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 -20 -12 B 18 0 16 -4 -12 C 14 -16 0 -20 -14 D 20 4 20 0 -4 E 12 12 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 691: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) E C B D A (7) B D A E C (5) A B C E D (5) E C D B A (4) A C B E D (4) E D C A B (3) D E C B A (3) D E B C A (3) D B A E C (3) D A E B C (3) C E A B D (3) C A E B D (3) B A D E C (3) B A C E D (3) E C D A B (2) D A B E C (2) C E A D B (2) B E C D A (2) B D E C A (2) B D E A C (2) A D B E C (2) A D B C E (2) A B D C E (2) A B C D E (2) E D C B A (1) E D B C A (1) D E C A B (1) D E B A C (1) D E A C B (1) D B E C A (1) D B E A C (1) C E D B A (1) C E D A B (1) C E B D A (1) C B A E D (1) B E D C A (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C D E (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -16 -6 -4 -6 B 16 0 0 18 0 C 6 0 0 8 -8 D 4 -18 -8 0 -10 E 6 0 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.425144 C: 0.000000 D: 0.000000 E: 0.574856 Sum of squares = 0.511206861184 Cumulative probabilities = A: 0.000000 B: 0.425144 C: 0.425144 D: 0.425144 E: 1.000000 A B C D E A 0 -16 -6 -4 -6 B 16 0 0 18 0 C 6 0 0 8 -8 D 4 -18 -8 0 -10 E 6 0 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 A=21 C=20 D=19 E=18 so E is eliminated. Round 2 votes counts: C=33 D=24 B=22 A=21 so A is eliminated. Round 3 votes counts: C=39 B=31 D=30 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:212 C:203 A:184 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -4 -6 B 16 0 0 18 0 C 6 0 0 8 -8 D 4 -18 -8 0 -10 E 6 0 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -4 -6 B 16 0 0 18 0 C 6 0 0 8 -8 D 4 -18 -8 0 -10 E 6 0 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -4 -6 B 16 0 0 18 0 C 6 0 0 8 -8 D 4 -18 -8 0 -10 E 6 0 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 692: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) A D B E C (7) A B D E C (7) E B D C A (6) E B D A C (5) C D E A B (5) E D C B A (4) B A E D C (4) E D B A C (3) E B C D A (3) C B E A D (3) C B A E D (3) C A D E B (3) D C E A B (2) D A E C B (2) D A E B C (2) D A C E B (2) C E B D A (2) C A D B E (2) B E D A C (2) B C A E D (2) A D C B E (2) A C D B E (2) E D B C A (1) E C D B A (1) D E C A B (1) D E B A C (1) D E A C B (1) D E A B C (1) D C A E B (1) C E D A B (1) C D A E B (1) B E C D A (1) B E A C D (1) B C E A D (1) B A E C D (1) B A C E D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -8 -18 -12 B 12 0 0 -14 -18 C 8 0 0 -10 -8 D 18 14 10 0 -12 E 12 18 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -8 -18 -12 B 12 0 0 -14 -18 C 8 0 0 -10 -8 D 18 14 10 0 -12 E 12 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 A=20 D=13 B=13 so D is eliminated. Round 2 votes counts: C=34 E=27 A=26 B=13 so B is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:225 D:215 C:195 B:190 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 -18 -12 B 12 0 0 -14 -18 C 8 0 0 -10 -8 D 18 14 10 0 -12 E 12 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -18 -12 B 12 0 0 -14 -18 C 8 0 0 -10 -8 D 18 14 10 0 -12 E 12 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -18 -12 B 12 0 0 -14 -18 C 8 0 0 -10 -8 D 18 14 10 0 -12 E 12 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 693: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) E C A D B (7) D B A C E (7) E C A B D (6) B D A C E (5) B C D A E (5) B D E A C (4) A D C B E (4) A C E D B (4) E C B A D (3) C A E D B (3) A E C D B (3) E B D C A (2) E B D A C (2) E B C D A (2) E A D B C (2) E A C D B (2) D B A E C (2) D A B C E (2) C E B A D (2) C E A D B (2) C E A B D (2) C A D E B (2) C A D B E (2) C A B D E (2) B D C E A (2) B C E D A (2) E D B A C (1) E D A B C (1) E A C B D (1) C B A E D (1) C B A D E (1) B E C D A (1) B D A E C (1) B C D E A (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -12 4 10 B 4 0 0 6 2 C 12 0 0 12 20 D -4 -6 -12 0 2 E -10 -2 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.682858 C: 0.317142 D: 0.000000 E: 0.000000 Sum of squares = 0.56687401526 Cumulative probabilities = A: 0.000000 B: 0.682858 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 4 10 B 4 0 0 6 2 C 12 0 0 12 20 D -4 -6 -12 0 2 E -10 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=29 B=29 C=17 A=14 D=11 so D is eliminated. Round 2 votes counts: B=38 E=29 C=17 A=16 so A is eliminated. Round 3 votes counts: B=40 E=32 C=28 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:222 B:206 A:199 D:190 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 4 10 B 4 0 0 6 2 C 12 0 0 12 20 D -4 -6 -12 0 2 E -10 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 4 10 B 4 0 0 6 2 C 12 0 0 12 20 D -4 -6 -12 0 2 E -10 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 4 10 B 4 0 0 6 2 C 12 0 0 12 20 D -4 -6 -12 0 2 E -10 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 694: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) E B C D A (8) C E A B D (8) A C E D B (8) A D C B E (7) E C B A D (6) E C A B D (6) D B A E C (6) D A B C E (5) B D E C A (5) A C E B D (4) D B E A C (3) D A C B E (3) C A E D B (3) B D E A C (3) A C D E B (3) D B E C A (2) B E C D A (2) E C D B A (1) E C B D A (1) E B D C A (1) D B A C E (1) C A E B D (1) B D A E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 -4 -12 B 0 0 -4 12 0 C 8 4 0 4 -10 D 4 -12 -4 0 -18 E 12 0 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.284495 C: 0.000000 D: 0.000000 E: 0.715505 Sum of squares = 0.59288508489 Cumulative probabilities = A: 0.000000 B: 0.284495 C: 0.284495 D: 0.284495 E: 1.000000 A B C D E A 0 0 -8 -4 -12 B 0 0 -4 12 0 C 8 4 0 4 -10 D 4 -12 -4 0 -18 E 12 0 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 B=21 D=20 C=12 so C is eliminated. Round 2 votes counts: E=31 A=28 B=21 D=20 so D is eliminated. Round 3 votes counts: A=36 B=33 E=31 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:220 B:204 C:203 A:188 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -4 -12 B 0 0 -4 12 0 C 8 4 0 4 -10 D 4 -12 -4 0 -18 E 12 0 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 -12 B 0 0 -4 12 0 C 8 4 0 4 -10 D 4 -12 -4 0 -18 E 12 0 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 -12 B 0 0 -4 12 0 C 8 4 0 4 -10 D 4 -12 -4 0 -18 E 12 0 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 695: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) B D A E C (6) A D B C E (5) A C E D B (5) A C D E B (5) E B D C A (4) D B E C A (4) B D E C A (4) A D C B E (4) A C E B D (4) A C B D E (4) E C B D A (3) E B C D A (3) C E A D B (3) C A E D B (3) B E D C A (3) A B D C E (3) E D C B A (2) E C D B A (2) D A C E B (2) C E D A B (2) C E B D A (2) B A D E C (2) A D C E B (2) A C D B E (2) A B D E C (2) D E B C A (1) D C E A B (1) D B A E C (1) C E B A D (1) C E A B D (1) C A E B D (1) B E C D A (1) B A E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 18 0 8 B -2 0 2 12 6 C -18 -2 0 -10 4 D 0 -12 10 0 16 E -8 -6 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.870097 B: 0.000000 C: 0.000000 D: 0.129903 E: 0.000000 Sum of squares = 0.773944175013 Cumulative probabilities = A: 0.870097 B: 0.870097 C: 0.870097 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 0 8 B -2 0 2 12 6 C -18 -2 0 -10 4 D 0 -12 10 0 16 E -8 -6 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.755102291318 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=26 E=14 C=13 D=9 so D is eliminated. Round 2 votes counts: A=40 B=31 E=15 C=14 so C is eliminated. Round 3 votes counts: A=44 B=31 E=25 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:209 D:207 C:187 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 18 0 8 B -2 0 2 12 6 C -18 -2 0 -10 4 D 0 -12 10 0 16 E -8 -6 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.755102291318 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 0 8 B -2 0 2 12 6 C -18 -2 0 -10 4 D 0 -12 10 0 16 E -8 -6 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.755102291318 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 0 8 B -2 0 2 12 6 C -18 -2 0 -10 4 D 0 -12 10 0 16 E -8 -6 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.755102291318 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 696: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) B D C A E (10) E A C D B (8) C A D E B (8) B D C E A (7) A E C D B (6) E A B C D (5) B E D A C (5) A C E D B (5) D C B A E (4) C A E D B (4) C D A E B (3) B E A D C (3) E B A D C (2) E A C B D (2) E A B D C (2) D C A B E (2) C D A B E (2) B D E C A (2) A E C B D (2) D C A E B (1) D B C E A (1) C A D B E (1) B E A C D (1) B D E A C (1) B A E C D (1) Total count = 100 A B C D E A 0 2 -14 0 22 B -2 0 4 -14 4 C 14 -4 0 -4 20 D 0 14 4 0 8 E -22 -4 -20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.067082 B: 0.000000 C: 0.000000 D: 0.932918 E: 0.000000 Sum of squares = 0.874835389192 Cumulative probabilities = A: 0.067082 B: 0.067082 C: 0.067082 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 0 22 B -2 0 4 -14 4 C 14 -4 0 -4 20 D 0 14 4 0 8 E -22 -4 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321045353 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=20 E=19 C=18 A=13 so A is eliminated. Round 2 votes counts: B=30 E=27 C=23 D=20 so D is eliminated. Round 3 votes counts: B=43 C=30 E=27 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:213 A:205 B:196 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -14 0 22 B -2 0 4 -14 4 C 14 -4 0 -4 20 D 0 14 4 0 8 E -22 -4 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321045353 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 0 22 B -2 0 4 -14 4 C 14 -4 0 -4 20 D 0 14 4 0 8 E -22 -4 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321045353 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 0 22 B -2 0 4 -14 4 C 14 -4 0 -4 20 D 0 14 4 0 8 E -22 -4 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321045353 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 697: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) A B D E C (14) D B A C E (9) E C D B A (7) C E D B A (7) B D A E C (7) D B C A E (6) C E A D B (5) A B D C E (5) A E C B D (4) E A C B D (3) C D B E A (2) B D A C E (2) E C B D A (1) E B D A C (1) E A B D C (1) D B E C A (1) D B E A C (1) D B C E A (1) D A B C E (1) C E A B D (1) C D B A E (1) B A D E C (1) B A D C E (1) A E B D C (1) A D B C E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 8 6 10 B -4 0 6 16 8 C -8 -6 0 -6 -12 D -6 -16 6 0 8 E -10 -8 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 6 10 B -4 0 6 16 8 C -8 -6 0 -6 -12 D -6 -16 6 0 8 E -10 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 D=19 C=16 B=11 so B is eliminated. Round 2 votes counts: A=29 D=28 E=27 C=16 so C is eliminated. Round 3 votes counts: E=40 D=31 A=29 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:214 B:213 D:196 E:193 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 10 B -4 0 6 16 8 C -8 -6 0 -6 -12 D -6 -16 6 0 8 E -10 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 10 B -4 0 6 16 8 C -8 -6 0 -6 -12 D -6 -16 6 0 8 E -10 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 10 B -4 0 6 16 8 C -8 -6 0 -6 -12 D -6 -16 6 0 8 E -10 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 698: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) B A C D E (10) A C B D E (6) E A C D B (5) D B E C A (4) C A D E B (4) B D C A E (4) E B A C D (3) E A C B D (3) D E B C A (3) D C E A B (3) B A C E D (3) E D C B A (2) E D B C A (2) D E C B A (2) D E C A B (2) D C A E B (2) C A D B E (2) B E D C A (2) B D E C A (2) A C E D B (2) A C E B D (2) E D B A C (1) E D A C B (1) E C A D B (1) E B D A C (1) E B A D C (1) E A B C D (1) D B C E A (1) D B C A E (1) C D A B E (1) C A E D B (1) C A B D E (1) B E D A C (1) B E A C D (1) B D A C E (1) B C A D E (1) B A E D C (1) B A E C D (1) A E C B D (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -4 6 -6 B -4 0 -8 -4 -10 C 4 8 0 4 -4 D -6 4 -4 0 2 E 6 10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 4 -4 6 -6 B -4 0 -8 -4 -10 C 4 8 0 4 -4 D -6 4 -4 0 2 E 6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=27 D=18 A=14 C=9 so C is eliminated. Round 2 votes counts: E=32 B=27 A=22 D=19 so D is eliminated. Round 3 votes counts: E=42 B=33 A=25 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:206 A:200 D:198 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 6 -6 B -4 0 -8 -4 -10 C 4 8 0 4 -4 D -6 4 -4 0 2 E 6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 6 -6 B -4 0 -8 -4 -10 C 4 8 0 4 -4 D -6 4 -4 0 2 E 6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 6 -6 B -4 0 -8 -4 -10 C 4 8 0 4 -4 D -6 4 -4 0 2 E 6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 699: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (5) E A C B D (4) D C E B A (4) C D A E B (4) B E A D C (4) B E A C D (4) B D E C A (4) A E C B D (4) E C D B A (3) E C A D B (3) E B A C D (3) D C A E B (3) D C A B E (3) C A E D B (3) C A D E B (3) A B E C D (3) E C A B D (2) E B C A D (2) D C B E A (2) D C B A E (2) D B C E A (2) D B C A E (2) D B A C E (2) D A B C E (2) C E D A B (2) B E D C A (2) B E D A C (2) B D C E A (2) B D A E C (2) B A D E C (2) A D C B E (2) E D C B A (1) E B C D A (1) E A B C D (1) C E D B A (1) C E A D B (1) B D E A C (1) B A E C D (1) B A D C E (1) A E B C D (1) A C E D B (1) A C E B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 6 -2 B 10 0 0 6 4 C 4 0 0 -2 -10 D -6 -6 2 0 -10 E 2 -4 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.886404 C: 0.113596 D: 0.000000 E: 0.000000 Sum of squares = 0.798616040109 Cumulative probabilities = A: 0.000000 B: 0.886404 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 6 -2 B 10 0 0 6 4 C 4 0 0 -2 -10 D -6 -6 2 0 -10 E 2 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836736754 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=22 E=20 C=14 A=14 so C is eliminated. Round 2 votes counts: B=30 D=26 E=24 A=20 so A is eliminated. Round 3 votes counts: E=34 B=34 D=32 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 E:209 C:196 A:195 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 6 -2 B 10 0 0 6 4 C 4 0 0 -2 -10 D -6 -6 2 0 -10 E 2 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836736754 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 6 -2 B 10 0 0 6 4 C 4 0 0 -2 -10 D -6 -6 2 0 -10 E 2 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836736754 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 6 -2 B 10 0 0 6 4 C 4 0 0 -2 -10 D -6 -6 2 0 -10 E 2 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836736754 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 700: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) C B A D E (9) B C A D E (8) A B C E D (8) D E B C A (6) D C B E A (4) B A C D E (4) A C B E D (4) A C B D E (4) A B C D E (4) E D C A B (3) E D A C B (3) C D B E A (3) C B D A E (3) C A B D E (3) B A C E D (3) A E B D C (3) E D A B C (2) E A D B C (2) B D E A C (2) A B E C D (2) D E C A B (1) D C E B A (1) D B E C A (1) C D B A E (1) B C D A E (1) A E D B C (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -10 16 22 B 14 0 -6 18 28 C 10 6 0 20 24 D -16 -18 -20 0 32 E -22 -28 -24 -32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 16 22 B 14 0 -6 18 28 C 10 6 0 20 24 D -16 -18 -20 0 32 E -22 -28 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=24 C=19 B=18 E=10 so E is eliminated. Round 2 votes counts: D=32 A=31 C=19 B=18 so B is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:230 B:227 A:207 D:189 E:147 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -10 16 22 B 14 0 -6 18 28 C 10 6 0 20 24 D -16 -18 -20 0 32 E -22 -28 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 16 22 B 14 0 -6 18 28 C 10 6 0 20 24 D -16 -18 -20 0 32 E -22 -28 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 16 22 B 14 0 -6 18 28 C 10 6 0 20 24 D -16 -18 -20 0 32 E -22 -28 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 701: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) B E A C D (8) E A C B D (7) C A D E B (7) B D C A E (6) D C B A E (5) E B A C D (4) B E D A C (4) A C E D B (4) A C E B D (4) E B A D C (3) D C A B E (3) B E A D C (3) A E C D B (3) A C D E B (3) E A B C D (2) D B C A E (2) C A D B E (2) B E D C A (2) B D E C A (2) B D C E A (2) B C D A E (2) B C A E D (2) A E C B D (2) E A C D B (1) D E B C A (1) D E B A C (1) D E A C B (1) D A C E B (1) C D A E B (1) C D A B E (1) B E C A D (1) B D E A C (1) B C A D E (1) Total count = 100 A B C D E A 0 0 4 14 14 B 0 0 -6 12 -6 C -4 6 0 10 8 D -14 -12 -10 0 0 E -14 6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.775697 B: 0.224303 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.652017423715 Cumulative probabilities = A: 0.775697 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 14 14 B 0 0 -6 12 -6 C -4 6 0 10 8 D -14 -12 -10 0 0 E -14 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011121 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=22 E=17 A=16 C=11 so C is eliminated. Round 2 votes counts: B=34 A=25 D=24 E=17 so E is eliminated. Round 3 votes counts: B=41 A=35 D=24 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:210 B:200 E:192 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 14 14 B 0 0 -6 12 -6 C -4 6 0 10 8 D -14 -12 -10 0 0 E -14 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011121 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 14 14 B 0 0 -6 12 -6 C -4 6 0 10 8 D -14 -12 -10 0 0 E -14 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011121 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 14 14 B 0 0 -6 12 -6 C -4 6 0 10 8 D -14 -12 -10 0 0 E -14 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011121 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 702: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (11) E A B D C (8) C B D E A (7) A E B D C (7) E B A D C (5) A D B E C (4) E C A B D (3) E A C B D (3) C E B D A (3) C D B E A (3) C D A B E (3) B D E A C (3) E C B A D (2) D C B A E (2) D B C A E (2) C A E D B (2) B E D C A (2) B D C E A (2) B C E D A (2) A E D B C (2) A E C D B (2) A D E B C (2) A D C E B (2) E C B D A (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) D B C E A (1) D B A E C (1) D A C B E (1) C E D A B (1) C E A B D (1) C D A E B (1) C B E D A (1) C A D B E (1) B E D A C (1) B E C D A (1) B D E C A (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -12 -6 -12 B 12 0 -2 16 0 C 12 2 0 4 -6 D 6 -16 -4 0 -2 E 12 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.395920 C: 0.000000 D: 0.000000 E: 0.604080 Sum of squares = 0.521665222337 Cumulative probabilities = A: 0.000000 B: 0.395920 C: 0.395920 D: 0.395920 E: 1.000000 A B C D E A 0 -12 -12 -6 -12 B 12 0 -2 16 0 C 12 2 0 4 -6 D 6 -16 -4 0 -2 E 12 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=26 A=20 B=13 D=7 so D is eliminated. Round 2 votes counts: C=36 E=26 A=21 B=17 so B is eliminated. Round 3 votes counts: C=44 E=34 A=22 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:210 C:206 D:192 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -12 -6 -12 B 12 0 -2 16 0 C 12 2 0 4 -6 D 6 -16 -4 0 -2 E 12 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -6 -12 B 12 0 -2 16 0 C 12 2 0 4 -6 D 6 -16 -4 0 -2 E 12 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -6 -12 B 12 0 -2 16 0 C 12 2 0 4 -6 D 6 -16 -4 0 -2 E 12 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 703: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) B E C A D (6) A E B D C (6) D A B C E (5) A E D B C (5) A D E B C (5) E B A C D (4) D C A E B (4) D A C E B (4) C E B D A (4) B E A C D (4) B A E D C (4) D C A B E (3) D A B E C (3) C D E B A (3) C D B E A (3) C B E D A (3) A D B E C (3) E A D C B (2) C E D B A (2) C B D E A (2) A B E D C (2) A B D E C (2) E C B A D (1) E C A D B (1) E B C A D (1) E A B D C (1) E A B C D (1) D A E C B (1) C E D A B (1) C E B A D (1) C D E A B (1) C D B A E (1) C D A E B (1) B D C A E (1) B D A E C (1) B A D E C (1) Total count = 100 A B C D E A 0 16 22 0 18 B -16 0 10 -10 2 C -22 -10 0 -20 -8 D 0 10 20 0 2 E -18 -2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.513375 B: 0.000000 C: 0.000000 D: 0.486625 E: 0.000000 Sum of squares = 0.500357764567 Cumulative probabilities = A: 0.513375 B: 0.513375 C: 0.513375 D: 1.000000 E: 1.000000 A B C D E A 0 16 22 0 18 B -16 0 10 -10 2 C -22 -10 0 -20 -8 D 0 10 20 0 2 E -18 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 C=22 B=17 E=11 so E is eliminated. Round 2 votes counts: D=27 A=27 C=24 B=22 so B is eliminated. Round 3 votes counts: A=40 C=31 D=29 so D is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:228 D:216 B:193 E:193 C:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 22 0 18 B -16 0 10 -10 2 C -22 -10 0 -20 -8 D 0 10 20 0 2 E -18 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 22 0 18 B -16 0 10 -10 2 C -22 -10 0 -20 -8 D 0 10 20 0 2 E -18 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 22 0 18 B -16 0 10 -10 2 C -22 -10 0 -20 -8 D 0 10 20 0 2 E -18 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 704: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) A D E C B (6) D E B A C (5) B D E C A (5) D A E B C (4) C B A E D (4) A D B E C (4) A C B E D (4) A B C D E (4) E D A C B (3) D E B C A (3) D E A C B (3) D E A B C (3) A C E B D (3) E D C A B (2) E D B C A (2) D B E A C (2) D B A E C (2) C E A B D (2) C A E D B (2) C A E B D (2) C A B E D (2) B E D C A (2) B C E A D (2) B C D A E (2) B C A D E (2) A C E D B (2) A C B D E (2) E D C B A (1) E C A D B (1) E B D C A (1) C E B D A (1) C E B A D (1) C E A D B (1) C B E A D (1) B D C E A (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 -6 -6 B -4 0 12 6 4 C 2 -12 0 -2 2 D 6 -6 2 0 0 E 6 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775474 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 4 -2 -6 -6 B -4 0 12 6 4 C 2 -12 0 -2 2 D 6 -6 2 0 0 E 6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 D=22 C=16 E=10 so E is eliminated. Round 2 votes counts: D=30 A=27 B=26 C=17 so C is eliminated. Round 3 votes counts: A=37 B=33 D=30 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:209 D:201 E:200 A:195 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -2 -6 -6 B -4 0 12 6 4 C 2 -12 0 -2 2 D 6 -6 2 0 0 E 6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -6 -6 B -4 0 12 6 4 C 2 -12 0 -2 2 D 6 -6 2 0 0 E 6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -6 -6 B -4 0 12 6 4 C 2 -12 0 -2 2 D 6 -6 2 0 0 E 6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 705: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) A C B D E (7) D B A E C (5) C E B A D (5) A C E D B (5) E C D B A (4) C E A B D (4) A D B E C (4) A D B C E (4) A B D C E (4) E C B D A (3) C B E D A (3) B C E D A (3) B A D C E (3) A B C D E (3) E C A D B (2) E A C D B (2) D B E A C (2) D A B E C (2) C B A E D (2) B D C E A (2) B D A C E (2) B C D E A (2) A C E B D (2) E D C B A (1) E D C A B (1) E D B C A (1) E D A C B (1) E B D C A (1) E B C D A (1) E A D C B (1) D E B C A (1) D E B A C (1) D B E C A (1) C B A D E (1) C A E B D (1) B E C D A (1) A E C D B (1) A D E C B (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 4 8 -2 B 6 0 -12 18 4 C -4 12 0 22 24 D -8 -18 -22 0 -8 E 2 -4 -24 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545455 B: 0.181818 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.404958677663 Cumulative probabilities = A: 0.545455 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 8 -2 B 6 0 -12 18 4 C -4 12 0 22 24 D -8 -18 -22 0 -8 E 2 -4 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.181818 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.545455 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 E=18 B=13 D=12 so D is eliminated. Round 2 votes counts: A=35 C=24 B=21 E=20 so E is eliminated. Round 3 votes counts: A=39 C=35 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:227 B:208 A:202 E:191 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 4 8 -2 B 6 0 -12 18 4 C -4 12 0 22 24 D -8 -18 -22 0 -8 E 2 -4 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.181818 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.545455 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 8 -2 B 6 0 -12 18 4 C -4 12 0 22 24 D -8 -18 -22 0 -8 E 2 -4 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.181818 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.545455 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 8 -2 B 6 0 -12 18 4 C -4 12 0 22 24 D -8 -18 -22 0 -8 E 2 -4 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.181818 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.545455 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 706: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) E D A B C (7) C B A E D (6) E D A C B (5) E A B C D (5) D E C A B (5) D E A B C (5) E A B D C (4) C B A D E (4) A E B C D (4) E D C A B (3) E A D B C (3) D E A C B (3) A E B D C (3) A B E D C (3) A B C E D (3) D C B A E (2) D A B E C (2) C D B A E (2) C B E A D (2) C B D A E (2) B C A E D (2) B A C D E (2) A B E C D (2) E D C B A (1) E C B A D (1) E A D C B (1) D E C B A (1) D B A C E (1) D A B C E (1) C E D B A (1) C E B D A (1) C B D E A (1) B A C E D (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 24 16 -2 -14 B -24 0 2 -6 -24 C -16 -2 0 -22 -20 D 2 6 22 0 -16 E 14 24 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 16 -2 -14 B -24 0 2 -6 -24 C -16 -2 0 -22 -20 D 2 6 22 0 -16 E 14 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=28 C=19 A=18 B=5 so B is eliminated. Round 2 votes counts: E=30 D=28 C=21 A=21 so C is eliminated. Round 3 votes counts: E=34 D=33 A=33 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:237 A:212 D:207 B:174 C:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 16 -2 -14 B -24 0 2 -6 -24 C -16 -2 0 -22 -20 D 2 6 22 0 -16 E 14 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 16 -2 -14 B -24 0 2 -6 -24 C -16 -2 0 -22 -20 D 2 6 22 0 -16 E 14 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 16 -2 -14 B -24 0 2 -6 -24 C -16 -2 0 -22 -20 D 2 6 22 0 -16 E 14 24 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 707: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) C E D A B (6) B D A E C (6) B C D E A (5) A E C D B (5) D E C A B (4) D E A C B (4) B D C E A (4) B A E D C (4) A E D C B (4) C E A D B (3) B C A E D (3) B C A D E (3) D A E B C (2) C E B D A (2) C E A B D (2) C D E B A (2) C D E A B (2) C B E A D (2) B A E C D (2) B A C E D (2) A E C B D (2) A B E D C (2) A B E C D (2) E D A C B (1) E C A D B (1) E A D C B (1) E A C D B (1) D E B A C (1) D C E B A (1) D B E A C (1) D B C E A (1) D A E C B (1) C E D B A (1) C B E D A (1) C B D E A (1) B D E A C (1) B D C A E (1) B D A C E (1) B A D C E (1) A E D B C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 10 2 4 B 8 0 6 14 4 C -10 -6 0 -4 -12 D -2 -14 4 0 2 E -4 -4 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 2 4 B 8 0 6 14 4 C -10 -6 0 -4 -12 D -2 -14 4 0 2 E -4 -4 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=22 A=18 D=15 E=4 so E is eliminated. Round 2 votes counts: B=41 C=23 A=20 D=16 so D is eliminated. Round 3 votes counts: B=44 C=28 A=28 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:204 E:201 D:195 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 2 4 B 8 0 6 14 4 C -10 -6 0 -4 -12 D -2 -14 4 0 2 E -4 -4 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 2 4 B 8 0 6 14 4 C -10 -6 0 -4 -12 D -2 -14 4 0 2 E -4 -4 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 2 4 B 8 0 6 14 4 C -10 -6 0 -4 -12 D -2 -14 4 0 2 E -4 -4 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 708: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (5) E D A B C (5) D A C E B (5) C D E B A (5) B C E A D (5) B E C A D (4) A D E B C (4) A B E D C (4) A B C D E (4) D E A C B (3) C B E D A (3) C B A E D (3) B E C D A (3) B E A C D (3) B C A E D (3) E D B A C (2) E B D C A (2) E B A D C (2) E A D B C (2) D E C B A (2) D E C A B (2) D C A E B (2) C D B A E (2) C D A B E (2) B A E C D (2) A E B D C (2) A D B C E (2) A C D B E (2) E D B C A (1) E C B D A (1) E B C D A (1) E B C A D (1) D E A B C (1) C D A E B (1) C B E A D (1) C B A D E (1) C A D B E (1) C A B D E (1) B C E D A (1) B A C E D (1) A D B E C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 2 -10 B 8 0 14 0 2 C 6 -14 0 6 -6 D -2 0 -6 0 -16 E 10 -2 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.954601 C: 0.000000 D: 0.045399 E: 0.000000 Sum of squares = 0.913324899965 Cumulative probabilities = A: 0.000000 B: 0.954601 C: 0.954601 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 2 -10 B 8 0 14 0 2 C 6 -14 0 6 -6 D -2 0 -6 0 -16 E 10 -2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.888889 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469519648 Cumulative probabilities = A: 0.000000 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 A=21 C=20 D=15 so D is eliminated. Round 2 votes counts: E=30 A=26 C=22 B=22 so C is eliminated. Round 3 votes counts: E=35 A=33 B=32 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:212 C:196 A:189 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 2 -10 B 8 0 14 0 2 C 6 -14 0 6 -6 D -2 0 -6 0 -16 E 10 -2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.888889 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469519648 Cumulative probabilities = A: 0.000000 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 2 -10 B 8 0 14 0 2 C 6 -14 0 6 -6 D -2 0 -6 0 -16 E 10 -2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.888889 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469519648 Cumulative probabilities = A: 0.000000 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 2 -10 B 8 0 14 0 2 C 6 -14 0 6 -6 D -2 0 -6 0 -16 E 10 -2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.888889 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469519648 Cumulative probabilities = A: 0.000000 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 709: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) C D B E A (7) C A B D E (7) E D B C A (6) D E C B A (6) E D B A C (4) E D A B C (4) E D A C B (3) C D E B A (3) A E C D B (3) A E B D C (3) A B E D C (3) E B D A C (2) E A D B C (2) D E B C A (2) D C E B A (2) C D E A B (2) C B D E A (2) C B A D E (2) B D E C A (2) B A E D C (2) A E D C B (2) A E D B C (2) A C E D B (2) A C E B D (2) A C B E D (2) A C B D E (2) D E C A B (1) D B E C A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D A E (1) C A D E B (1) B E D A C (1) B C D E A (1) B C D A E (1) B A C E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 0 -8 -2 B -4 0 -6 -12 -6 C 0 6 0 4 0 D 8 12 -4 0 -6 E 2 6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.330129 D: 0.000000 E: 0.669871 Sum of squares = 0.557712137665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.330129 D: 0.330129 E: 1.000000 A B C D E A 0 4 0 -8 -2 B -4 0 -6 -12 -6 C 0 6 0 4 0 D 8 12 -4 0 -6 E 2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=28 E=21 D=12 B=9 so B is eliminated. Round 2 votes counts: A=34 C=30 E=22 D=14 so D is eliminated. Round 3 votes counts: E=34 A=34 C=32 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:207 C:205 D:205 A:197 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -8 -2 B -4 0 -6 -12 -6 C 0 6 0 4 0 D 8 12 -4 0 -6 E 2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -8 -2 B -4 0 -6 -12 -6 C 0 6 0 4 0 D 8 12 -4 0 -6 E 2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -8 -2 B -4 0 -6 -12 -6 C 0 6 0 4 0 D 8 12 -4 0 -6 E 2 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 710: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B C A D E (7) B A C D E (7) C E D B A (6) C B D E A (6) B A D C E (6) A B D E C (6) B C D E A (5) B C A E D (5) E D A C B (4) D C E B A (4) A B E C D (4) E A D C B (3) A D B E C (3) E C B A D (2) D E A C B (2) C D E B A (2) B C D A E (2) B A C E D (2) A E D B C (2) A D E B C (2) A B E D C (2) E C D B A (1) E C D A B (1) D E C A B (1) D C B E A (1) D A C E B (1) C E B D A (1) C B E D A (1) B D C E A (1) B D C A E (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -20 -8 6 4 B 20 0 10 16 18 C 8 -10 0 4 16 D -6 -16 -4 0 16 E -4 -18 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 6 4 B 20 0 10 16 18 C 8 -10 0 4 16 D -6 -16 -4 0 16 E -4 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=21 E=18 C=16 D=9 so D is eliminated. Round 2 votes counts: B=36 A=22 E=21 C=21 so E is eliminated. Round 3 votes counts: B=36 C=33 A=31 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:209 D:195 A:191 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -8 6 4 B 20 0 10 16 18 C 8 -10 0 4 16 D -6 -16 -4 0 16 E -4 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 6 4 B 20 0 10 16 18 C 8 -10 0 4 16 D -6 -16 -4 0 16 E -4 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 6 4 B 20 0 10 16 18 C 8 -10 0 4 16 D -6 -16 -4 0 16 E -4 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 711: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (14) A B D C E (8) E C D B A (7) D B E A C (7) C E D B A (6) C E A D B (6) D B A E C (5) A B D E C (5) E C A B D (4) B A D E C (4) E D C B A (3) D B E C A (3) D B C E A (3) B D A E C (3) D B A C E (2) D A B C E (2) A E C B D (2) A C E B D (2) E B A D C (1) E A C B D (1) E A B C D (1) D E B C A (1) D C E B A (1) C E D A B (1) C D E B A (1) C A E B D (1) B D E A C (1) A E B D C (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 8 -22 B -4 0 0 4 -6 C 2 0 0 -2 -2 D -8 -4 2 0 -4 E 22 6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -2 8 -22 B -4 0 0 4 -6 C 2 0 0 -2 -2 D -8 -4 2 0 -4 E 22 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 A=22 E=17 B=8 so B is eliminated. Round 2 votes counts: C=29 D=28 A=26 E=17 so E is eliminated. Round 3 votes counts: C=40 D=31 A=29 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:217 C:199 B:197 A:194 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 8 -22 B -4 0 0 4 -6 C 2 0 0 -2 -2 D -8 -4 2 0 -4 E 22 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 8 -22 B -4 0 0 4 -6 C 2 0 0 -2 -2 D -8 -4 2 0 -4 E 22 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 8 -22 B -4 0 0 4 -6 C 2 0 0 -2 -2 D -8 -4 2 0 -4 E 22 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 712: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) E D B C A (6) B A C D E (6) E D C A B (4) E B D C A (4) E A D C B (4) D E C B A (4) B D C A E (4) B A C E D (4) A C B D E (4) E D A C B (3) E B A C D (3) D C A B E (3) B E D C A (3) A C D E B (3) A C B E D (3) E B D A C (2) E A C B D (2) D E C A B (2) D C E A B (2) D C A E B (2) C D A B E (2) C A D B E (2) B D E C A (2) B A E C D (2) A C E D B (2) A C D B E (2) A B C D E (2) E D C B A (1) E D B A C (1) E B A D C (1) E A C D B (1) D B E C A (1) C A D E B (1) B E D A C (1) B E A D C (1) B C D A E (1) B C A D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 10 4 -10 B 10 0 6 8 4 C -10 -6 0 -2 -10 D -4 -8 2 0 -12 E 10 -4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 4 -10 B 10 0 6 8 4 C -10 -6 0 -2 -10 D -4 -8 2 0 -12 E 10 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=32 B=32 A=17 D=14 C=5 so C is eliminated. Round 2 votes counts: E=32 B=32 A=20 D=16 so D is eliminated. Round 3 votes counts: E=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:214 A:197 D:189 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 4 -10 B 10 0 6 8 4 C -10 -6 0 -2 -10 D -4 -8 2 0 -12 E 10 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 4 -10 B 10 0 6 8 4 C -10 -6 0 -2 -10 D -4 -8 2 0 -12 E 10 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 4 -10 B 10 0 6 8 4 C -10 -6 0 -2 -10 D -4 -8 2 0 -12 E 10 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 713: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (14) B C D A E (8) C E A D B (6) B D A E C (6) E C A D B (5) E A D C B (5) E A D B C (5) C B D A E (5) C A D E B (5) B D E A C (5) C A E D B (4) E D A B C (3) C B E D A (3) B E D A C (3) E B D A C (2) C B E A D (2) C A D B E (2) B C D E A (2) A D C B E (2) E D B A C (1) E A C D B (1) D E A B C (1) D A B E C (1) C E B A D (1) C B A E D (1) B E C D A (1) B D C A E (1) B D A C E (1) B C E D A (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 -22 12 6 B 14 0 -16 12 16 C 22 16 0 24 16 D -12 -12 -24 0 10 E -6 -16 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -22 12 6 B 14 0 -16 12 16 C 22 16 0 24 16 D -12 -12 -24 0 10 E -6 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=43 B=28 E=22 A=5 D=2 so D is eliminated. Round 2 votes counts: C=43 B=28 E=23 A=6 so A is eliminated. Round 3 votes counts: C=45 B=29 E=26 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:239 B:213 A:191 D:181 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -22 12 6 B 14 0 -16 12 16 C 22 16 0 24 16 D -12 -12 -24 0 10 E -6 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -22 12 6 B 14 0 -16 12 16 C 22 16 0 24 16 D -12 -12 -24 0 10 E -6 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -22 12 6 B 14 0 -16 12 16 C 22 16 0 24 16 D -12 -12 -24 0 10 E -6 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 714: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) B C D A E (7) E A B D C (6) C D B A E (6) E D A C B (5) E A D C B (5) C D B E A (5) C B D A E (5) B A C D E (5) A E B C D (5) B C A D E (4) E D C B A (3) E A D B C (3) D C B E A (3) A B C E D (3) E D C A B (2) E A B C D (2) D C E B A (2) D C B A E (2) B A D C E (2) A B E D C (2) A B E C D (2) A B C D E (2) E D A B C (1) E A C D B (1) D E C B A (1) D E B C A (1) D B E C A (1) C E D A B (1) C D E B A (1) C A E D B (1) C A B D E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 8 8 14 B -4 0 10 12 2 C -8 -10 0 4 2 D -8 -12 -4 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 14 B -4 0 10 12 2 C -8 -10 0 4 2 D -8 -12 -4 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=24 C=20 B=18 D=10 so D is eliminated. Round 2 votes counts: E=30 C=27 A=24 B=19 so B is eliminated. Round 3 votes counts: C=38 E=31 A=31 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:210 C:194 E:193 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 14 B -4 0 10 12 2 C -8 -10 0 4 2 D -8 -12 -4 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 14 B -4 0 10 12 2 C -8 -10 0 4 2 D -8 -12 -4 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 14 B -4 0 10 12 2 C -8 -10 0 4 2 D -8 -12 -4 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 715: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) E D C A B (8) E C D B A (7) A B D E C (7) C E D B A (6) A B D C E (6) B A C E D (5) A B C D E (5) B C A E D (4) B A E D C (4) D A E B C (3) C E B D A (3) C B A E D (3) B A E C D (3) A D B C E (3) C B E A D (2) C B A D E (2) B A C D E (2) A D B E C (2) E D A B C (1) E C B D A (1) E B C D A (1) D C E A B (1) D A E C B (1) D A C E B (1) C D E B A (1) C D A B E (1) C B D A E (1) C A B D E (1) B E C A D (1) B C A D E (1) B A D C E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -8 8 16 B -4 0 -2 6 12 C 8 2 0 2 2 D -8 -6 -2 0 2 E -16 -12 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 8 16 B -4 0 -2 6 12 C 8 2 0 2 2 D -8 -6 -2 0 2 E -16 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=21 C=20 E=18 D=16 so D is eliminated. Round 2 votes counts: A=30 E=28 C=21 B=21 so C is eliminated. Round 3 votes counts: E=39 A=32 B=29 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:207 B:206 D:193 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 8 16 B -4 0 -2 6 12 C 8 2 0 2 2 D -8 -6 -2 0 2 E -16 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 8 16 B -4 0 -2 6 12 C 8 2 0 2 2 D -8 -6 -2 0 2 E -16 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 8 16 B -4 0 -2 6 12 C 8 2 0 2 2 D -8 -6 -2 0 2 E -16 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 716: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) A B E D C (6) B E C D A (5) E B D C A (4) C A B E D (4) B E D A C (4) B E A D C (4) B A E D C (4) A D B E C (4) A C D B E (4) A C B E D (4) E B C D A (3) D E C B A (3) D E B C A (3) D E B A C (3) C D A E B (3) C A D E B (3) A D C E B (3) A B C E D (3) D A E B C (2) C E B D A (2) C A D B E (2) C A B D E (2) B E C A D (2) E D B C A (1) E B D A C (1) D E C A B (1) D C E B A (1) D C A E B (1) D B E A C (1) D A C E B (1) C D E B A (1) C B E D A (1) B E A C D (1) B D E A C (1) B C E A D (1) B A C E D (1) A D E B C (1) A D C B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 2 4 2 B 6 0 24 22 26 C -2 -24 0 -12 -22 D -4 -22 12 0 -16 E -2 -26 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 2 B 6 0 24 22 26 C -2 -24 0 -12 -22 D -4 -22 12 0 -16 E -2 -26 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995261 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=28 C=18 D=16 E=9 so E is eliminated. Round 2 votes counts: B=37 A=28 C=18 D=17 so D is eliminated. Round 3 votes counts: B=45 A=31 C=24 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:239 E:205 A:201 D:185 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 2 B 6 0 24 22 26 C -2 -24 0 -12 -22 D -4 -22 12 0 -16 E -2 -26 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995261 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 2 B 6 0 24 22 26 C -2 -24 0 -12 -22 D -4 -22 12 0 -16 E -2 -26 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995261 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 2 B 6 0 24 22 26 C -2 -24 0 -12 -22 D -4 -22 12 0 -16 E -2 -26 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995261 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 717: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) B A E D C (9) C D E B A (8) C D E A B (7) C D B E A (7) B A C E D (6) D C E B A (5) D E C A B (4) D C E A B (3) C D A E B (3) B E A D C (3) E D A B C (2) E B A D C (2) B E C A D (2) B C D E A (2) A E D C B (2) A E D B C (2) A E B D C (2) A B C E D (2) E D A C B (1) E B D A C (1) E A D B C (1) E A B D C (1) D C A E B (1) D A C E B (1) C D A B E (1) C B D E A (1) C B A D E (1) B E D C A (1) B E C D A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 0 0 -4 B 8 0 8 4 8 C 0 -8 0 -4 8 D 0 -4 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 0 -4 B 8 0 8 4 8 C 0 -8 0 -4 8 D 0 -4 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=28 A=21 D=14 E=8 so E is eliminated. Round 2 votes counts: B=32 C=28 A=23 D=17 so D is eliminated. Round 3 votes counts: C=41 B=32 A=27 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:199 C:198 E:195 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 0 -4 B 8 0 8 4 8 C 0 -8 0 -4 8 D 0 -4 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 0 -4 B 8 0 8 4 8 C 0 -8 0 -4 8 D 0 -4 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 0 -4 B 8 0 8 4 8 C 0 -8 0 -4 8 D 0 -4 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 718: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) A B E C D (9) D C E B A (8) A B E D C (7) E D A B C (5) C D E B A (5) B A C E D (5) A E D B C (5) E D A C B (4) D E A C B (4) B A C D E (4) A B C E D (4) D E C A B (3) C D B E A (3) C B A D E (3) A E B C D (3) E D C B A (2) B C A E D (2) A E B D C (2) A B D E C (2) E D C A B (1) E A D B C (1) E A B D C (1) C B E D A (1) C B D E A (1) C B D A E (1) B C E A D (1) B C A D E (1) A D E B C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 18 6 2 B -8 0 10 -4 -8 C -18 -10 0 -12 -18 D -6 4 12 0 -6 E -2 8 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999244 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 6 2 B -8 0 10 -4 -8 C -18 -10 0 -12 -18 D -6 4 12 0 -6 E -2 8 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=24 E=14 C=14 B=13 so B is eliminated. Round 2 votes counts: A=44 D=24 C=18 E=14 so E is eliminated. Round 3 votes counts: A=46 D=36 C=18 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:215 D:202 B:195 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 6 2 B -8 0 10 -4 -8 C -18 -10 0 -12 -18 D -6 4 12 0 -6 E -2 8 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 6 2 B -8 0 10 -4 -8 C -18 -10 0 -12 -18 D -6 4 12 0 -6 E -2 8 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 6 2 B -8 0 10 -4 -8 C -18 -10 0 -12 -18 D -6 4 12 0 -6 E -2 8 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 719: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) B A D C E (6) D E A C B (5) A D B E C (5) D B A E C (4) C E A D B (4) B A C D E (4) D E C A B (3) D E A B C (3) C E D A B (3) C E B A D (3) C E A B D (3) C B E A D (3) B D A E C (3) B A D E C (3) A D E C B (3) E D C B A (2) E D C A B (2) E D A C B (2) D B E A C (2) D A E B C (2) C E D B A (2) C E B D A (2) C A E B D (2) B D E C A (2) B D A C E (2) B C E A D (2) A E C D B (2) A D E B C (2) A B C E D (2) A B C D E (2) E C D B A (1) D E B A C (1) D A B E C (1) C B E D A (1) B D C E A (1) B A C E D (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 14 -2 -8 B -10 0 -2 -12 -10 C -14 2 0 -10 -12 D 2 12 10 0 10 E 8 10 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 -2 -8 B -10 0 -2 -12 -10 C -14 2 0 -10 -12 D 2 12 10 0 10 E 8 10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 D=21 A=18 E=14 so E is eliminated. Round 2 votes counts: C=31 D=27 B=24 A=18 so A is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:210 A:207 B:183 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 14 -2 -8 B -10 0 -2 -12 -10 C -14 2 0 -10 -12 D 2 12 10 0 10 E 8 10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -2 -8 B -10 0 -2 -12 -10 C -14 2 0 -10 -12 D 2 12 10 0 10 E 8 10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -2 -8 B -10 0 -2 -12 -10 C -14 2 0 -10 -12 D 2 12 10 0 10 E 8 10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 720: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) D E A C B (5) E D C B A (4) E C D B A (4) E A C B D (4) D A B C E (4) A D B E C (4) A D B C E (4) E D A C B (3) E C B D A (3) D C E B A (3) D A E C B (3) B A C D E (3) A B D C E (3) E C B A D (2) D C B A E (2) D B C A E (2) D A E B C (2) C E B D A (2) C B E A D (2) C B D E A (2) B C A E D (2) B C A D E (2) B A D C E (2) A E B C D (2) A D E B C (2) A B C D E (2) E D C A B (1) E C D A B (1) E A D C B (1) E A D B C (1) E A B C D (1) D E C A B (1) D C B E A (1) D A C B E (1) D A B E C (1) C E D B A (1) C E B A D (1) B D C A E (1) B C E A D (1) B C D A E (1) B A C E D (1) A E D B C (1) A E B D C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 6 -12 -4 B 0 0 -10 -4 2 C -6 10 0 -8 4 D 12 4 8 0 2 E 4 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -12 -4 B 0 0 -10 -4 2 C -6 10 0 -8 4 D 12 4 8 0 2 E 4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 A=21 C=16 B=13 so B is eliminated. Round 2 votes counts: A=27 D=26 E=25 C=22 so C is eliminated. Round 3 votes counts: E=40 A=31 D=29 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:213 C:200 E:198 A:195 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 -12 -4 B 0 0 -10 -4 2 C -6 10 0 -8 4 D 12 4 8 0 2 E 4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -12 -4 B 0 0 -10 -4 2 C -6 10 0 -8 4 D 12 4 8 0 2 E 4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -12 -4 B 0 0 -10 -4 2 C -6 10 0 -8 4 D 12 4 8 0 2 E 4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 721: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) B D E C A (8) D B A E C (7) A D C E B (7) E C B A D (6) D A B C E (6) C E A B D (5) D B E A C (4) D A C E B (4) D A B E C (4) B E C A D (4) B C E A D (4) A C E D B (4) E B C A D (3) B E D C A (3) A C E B D (3) E C A D B (2) D B E C A (2) D B A C E (2) D A C B E (2) C E B A D (2) C A E B D (2) A D B C E (2) A C D E B (2) E D B C A (1) D A E C B (1) B D A C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 0 -6 -4 B 10 0 18 0 16 C 0 -18 0 -8 -6 D 6 0 8 0 4 E 4 -16 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.305934 C: 0.000000 D: 0.694066 E: 0.000000 Sum of squares = 0.575323542446 Cumulative probabilities = A: 0.000000 B: 0.305934 C: 0.305934 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -6 -4 B 10 0 18 0 16 C 0 -18 0 -8 -6 D 6 0 8 0 4 E 4 -16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=19 E=12 C=9 so C is eliminated. Round 2 votes counts: D=32 B=28 A=21 E=19 so E is eliminated. Round 3 votes counts: B=39 D=33 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:209 E:195 A:190 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -6 -4 B 10 0 18 0 16 C 0 -18 0 -8 -6 D 6 0 8 0 4 E 4 -16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -6 -4 B 10 0 18 0 16 C 0 -18 0 -8 -6 D 6 0 8 0 4 E 4 -16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -6 -4 B 10 0 18 0 16 C 0 -18 0 -8 -6 D 6 0 8 0 4 E 4 -16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 722: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (18) C A D E B (14) A C D E B (6) E D B A C (5) B E C D A (5) E D A B C (4) E B D A C (4) C A B D E (4) A D C E B (4) E D A C B (3) C A D B E (3) B E D C A (3) B E C A D (3) E C D A B (2) D A E B C (2) D A C E B (2) B C A E D (2) B A D C E (2) E D C A B (1) E C D B A (1) E C B D A (1) E B D C A (1) D E A C B (1) C D A E B (1) C B A E D (1) C B A D E (1) B E A D C (1) B C E A D (1) B C A D E (1) A D C B E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 10 -8 -8 B 0 0 6 -2 -4 C -10 -6 0 -4 -10 D 8 2 4 0 -12 E 8 4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 10 -8 -8 B 0 0 6 -2 -4 C -10 -6 0 -4 -10 D 8 2 4 0 -12 E 8 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=24 E=22 A=13 D=5 so D is eliminated. Round 2 votes counts: B=36 C=24 E=23 A=17 so A is eliminated. Round 3 votes counts: C=38 B=37 E=25 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:217 D:201 B:200 A:197 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 -8 -8 B 0 0 6 -2 -4 C -10 -6 0 -4 -10 D 8 2 4 0 -12 E 8 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -8 -8 B 0 0 6 -2 -4 C -10 -6 0 -4 -10 D 8 2 4 0 -12 E 8 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -8 -8 B 0 0 6 -2 -4 C -10 -6 0 -4 -10 D 8 2 4 0 -12 E 8 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 723: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (14) E D C B A (11) A B E D C (7) A B C D E (7) C D E A B (6) D E C A B (5) E C D B A (4) D C E A B (4) D E A C B (3) C E D B A (3) C D E B A (3) C A B D E (3) A B C E D (3) D E C B A (2) C B E D A (2) B E C D A (2) B A E D C (2) B A E C D (2) A D E C B (2) A D E B C (2) A B D E C (2) A B D C E (2) E D B A C (1) D A E C B (1) C D A E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B C E D A (1) B C E A D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 0 -2 0 B 0 0 -4 0 2 C 0 4 0 8 4 D 2 0 -8 0 -10 E 0 -2 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.521670 B: 0.000000 C: 0.478330 D: 0.000000 E: 0.000000 Sum of squares = 0.500939159274 Cumulative probabilities = A: 0.521670 B: 0.521670 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -2 0 B 0 0 -4 0 2 C 0 4 0 8 4 D 2 0 -8 0 -10 E 0 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=24 C=19 E=16 D=15 so D is eliminated. Round 2 votes counts: A=27 E=26 B=24 C=23 so C is eliminated. Round 3 votes counts: E=42 A=32 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:208 E:202 A:199 B:199 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -2 0 B 0 0 -4 0 2 C 0 4 0 8 4 D 2 0 -8 0 -10 E 0 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 0 B 0 0 -4 0 2 C 0 4 0 8 4 D 2 0 -8 0 -10 E 0 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 0 B 0 0 -4 0 2 C 0 4 0 8 4 D 2 0 -8 0 -10 E 0 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 724: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) B E A D C (12) E B A C D (8) D C A B E (7) E A B C D (5) A D C E B (4) D A C B E (3) B E D A C (3) B E A C D (3) B A E D C (3) A C D E B (3) E C B A D (2) E B C A D (2) C E A D B (2) C D E B A (2) B D A C E (2) A B E D C (2) E C B D A (1) E C A B D (1) E B C D A (1) E B A D C (1) E A B D C (1) D C B A E (1) D C A E B (1) D B C E A (1) D B C A E (1) D A C E B (1) C E D A B (1) C D B E A (1) C D A B E (1) C A E D B (1) C A D E B (1) B E C D A (1) B D E C A (1) B D C E A (1) B D A E C (1) B A D E C (1) A E B C D (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 12 10 0 B -2 0 6 6 -6 C -12 -6 0 2 -2 D -10 -6 -2 0 -2 E 0 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.462799 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.537201 Sum of squares = 0.502767844472 Cumulative probabilities = A: 0.462799 B: 0.462799 C: 0.462799 D: 0.462799 E: 1.000000 A B C D E A 0 2 12 10 0 B -2 0 6 6 -6 C -12 -6 0 2 -2 D -10 -6 -2 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 E=22 D=15 A=12 so A is eliminated. Round 2 votes counts: B=30 C=26 E=23 D=21 so D is eliminated. Round 3 votes counts: C=43 B=34 E=23 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:212 E:205 B:202 C:191 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 10 0 B -2 0 6 6 -6 C -12 -6 0 2 -2 D -10 -6 -2 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 10 0 B -2 0 6 6 -6 C -12 -6 0 2 -2 D -10 -6 -2 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 10 0 B -2 0 6 6 -6 C -12 -6 0 2 -2 D -10 -6 -2 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 725: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E C B A D (10) C E A D B (9) B D A E C (9) B E D A C (7) B E C D A (7) A D C E B (6) C A D E B (5) D A C B E (4) E B C A D (3) B E C A D (3) E C A D B (2) D A C E B (2) C E D A B (2) B D A C E (2) B A D E C (2) A D C B E (2) E B C D A (1) D C A E B (1) D A B E C (1) C E B D A (1) C D A E B (1) C A E D B (1) B E A C D (1) B D E A C (1) B C E D A (1) A D E C B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 6 -6 4 B -4 0 6 -4 10 C -6 -6 0 -6 2 D 6 4 6 0 4 E -4 -10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -6 4 B -4 0 6 -4 10 C -6 -6 0 -6 2 D 6 4 6 0 4 E -4 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=21 C=19 E=16 A=11 so A is eliminated. Round 2 votes counts: B=33 D=32 C=19 E=16 so E is eliminated. Round 3 votes counts: B=37 D=32 C=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:204 B:204 C:192 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -6 4 B -4 0 6 -4 10 C -6 -6 0 -6 2 D 6 4 6 0 4 E -4 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -6 4 B -4 0 6 -4 10 C -6 -6 0 -6 2 D 6 4 6 0 4 E -4 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -6 4 B -4 0 6 -4 10 C -6 -6 0 -6 2 D 6 4 6 0 4 E -4 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 726: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) C B D E A (7) C A D E B (6) B C E D A (6) A E D B C (6) A C D E B (6) C B A E D (4) C B A D E (4) B E D C A (4) B E D A C (4) D E A B C (3) C B E D A (3) A E B D C (3) E D B A C (2) D E B C A (2) C D E B A (2) C D E A B (2) C D A E B (2) C A D B E (2) C A B D E (2) B C E A D (2) A E D C B (2) A B E D C (2) E D A B C (1) E B D A C (1) E A D B C (1) D E C A B (1) D E B A C (1) D E A C B (1) D A E C B (1) B D E C A (1) B C D E A (1) B A E D C (1) B A C E D (1) A D C E B (1) A C E B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -2 10 10 B -8 0 -12 0 -6 C 2 12 0 8 8 D -10 0 -8 0 6 E -10 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 10 10 B -8 0 -12 0 -6 C 2 12 0 8 8 D -10 0 -8 0 6 E -10 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=32 B=20 D=9 E=5 so E is eliminated. Round 2 votes counts: C=34 A=33 B=21 D=12 so D is eliminated. Round 3 votes counts: A=39 C=35 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:213 D:194 E:191 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 10 10 B -8 0 -12 0 -6 C 2 12 0 8 8 D -10 0 -8 0 6 E -10 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 10 10 B -8 0 -12 0 -6 C 2 12 0 8 8 D -10 0 -8 0 6 E -10 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 10 10 B -8 0 -12 0 -6 C 2 12 0 8 8 D -10 0 -8 0 6 E -10 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 727: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) C B E D A (8) D A E B C (6) A D E B C (6) B E A C D (5) B C E A D (5) C D B A E (4) C B E A D (4) A D B E C (4) A B E D C (4) E A D B C (3) C B D E A (3) B E C A D (3) E B A C D (2) E A B D C (2) D C E A B (2) D A E C B (2) C E D B A (2) C E B D A (2) C D E A B (2) C D B E A (2) C D A E B (2) C D A B E (2) B E A D C (2) A E D B C (2) A E B D C (2) E C B A D (1) E B C A D (1) E B A D C (1) E A D C B (1) D C A B E (1) D A C E B (1) D A C B E (1) C D E B A (1) B C A E D (1) B A E D C (1) Total count = 100 A B C D E A 0 4 -10 0 -4 B -4 0 0 -6 0 C 10 0 0 0 4 D 0 6 0 0 -4 E 4 0 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.616254 D: 0.383746 E: 0.000000 Sum of squares = 0.527030146163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.616254 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 0 -4 B -4 0 0 -6 0 C 10 0 0 0 4 D 0 6 0 0 -4 E 4 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500388 D: 0.499612 E: 0.000000 Sum of squares = 0.500000300596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500388 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=22 A=18 B=17 E=11 so E is eliminated. Round 2 votes counts: C=33 A=24 D=22 B=21 so B is eliminated. Round 3 votes counts: C=43 A=35 D=22 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 E:202 D:201 A:195 B:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 0 -4 B -4 0 0 -6 0 C 10 0 0 0 4 D 0 6 0 0 -4 E 4 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500388 D: 0.499612 E: 0.000000 Sum of squares = 0.500000300596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500388 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 0 -4 B -4 0 0 -6 0 C 10 0 0 0 4 D 0 6 0 0 -4 E 4 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500388 D: 0.499612 E: 0.000000 Sum of squares = 0.500000300596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500388 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 0 -4 B -4 0 0 -6 0 C 10 0 0 0 4 D 0 6 0 0 -4 E 4 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500388 D: 0.499612 E: 0.000000 Sum of squares = 0.500000300596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500388 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 728: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (13) A D B E C (13) C D A E B (11) B E A D C (10) E B C D A (6) D A C B E (6) E B C A D (5) D C A E B (5) C B E A D (4) B E C A D (4) D A C E B (3) C D E B A (3) A D C B E (3) A B E D C (3) E C B D A (2) C A D B E (2) E B D C A (1) E B D A C (1) D C A B E (1) D A E B C (1) D A B E C (1) C E B A D (1) B E A C D (1) Total count = 100 A B C D E A 0 -2 -16 -8 -2 B 2 0 -8 2 -4 C 16 8 0 4 4 D 8 -2 -4 0 -2 E 2 4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -8 -2 B 2 0 -8 2 -4 C 16 8 0 4 4 D 8 -2 -4 0 -2 E 2 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=19 D=17 E=15 B=15 so E is eliminated. Round 2 votes counts: C=36 B=28 A=19 D=17 so D is eliminated. Round 3 votes counts: C=42 A=30 B=28 so B is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:202 D:200 B:196 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 -8 -2 B 2 0 -8 2 -4 C 16 8 0 4 4 D 8 -2 -4 0 -2 E 2 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -8 -2 B 2 0 -8 2 -4 C 16 8 0 4 4 D 8 -2 -4 0 -2 E 2 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -8 -2 B 2 0 -8 2 -4 C 16 8 0 4 4 D 8 -2 -4 0 -2 E 2 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 729: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) C E A D B (7) E C A D B (6) B D C E A (6) E A C B D (4) D C B E A (4) A E C D B (4) A E B C D (4) E C A B D (3) E A B C D (3) D B A C E (3) D A B C E (3) C E D A B (3) B A D E C (3) A E C B D (3) A D B E C (3) A B D E C (3) D B C A E (2) C D E A B (2) B D A E C (2) A C E D B (2) A B E D C (2) A B E C D (2) E C B D A (1) E B C D A (1) E B A C D (1) D C E B A (1) D C E A B (1) D C A E B (1) C E D B A (1) C D E B A (1) B E D C A (1) B E A D C (1) B E A C D (1) B D E C A (1) B D E A C (1) B D A C E (1) B A E C D (1) A D B C E (1) Total count = 100 A B C D E A 0 14 -4 8 -20 B -14 0 12 -10 2 C 4 -12 0 0 -2 D -8 10 0 0 -2 E 20 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408174 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 A B C D E A 0 14 -4 8 -20 B -14 0 12 -10 2 C 4 -12 0 0 -2 D -8 10 0 0 -2 E 20 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408154 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=24 E=19 B=18 C=14 so C is eliminated. Round 2 votes counts: E=30 D=28 A=24 B=18 so B is eliminated. Round 3 votes counts: D=39 E=33 A=28 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:200 A:199 B:195 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 8 -20 B -14 0 12 -10 2 C 4 -12 0 0 -2 D -8 10 0 0 -2 E 20 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408154 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 8 -20 B -14 0 12 -10 2 C 4 -12 0 0 -2 D -8 10 0 0 -2 E 20 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408154 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 8 -20 B -14 0 12 -10 2 C 4 -12 0 0 -2 D -8 10 0 0 -2 E 20 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408154 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 730: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) E C B D A (6) E C B A D (6) E B C A D (5) C B E D A (5) A D B C E (5) E C D B A (4) D C B A E (4) A B D C E (4) E C D A B (3) D A E C B (3) C E D B A (3) B A C E D (3) E D C A B (2) E A B C D (2) D C E B A (2) D B C A E (2) D A C B E (2) C D E B A (2) B E C A D (2) B E A C D (2) B D A C E (2) B C E A D (2) A D B E C (2) A B D E C (2) E B A C D (1) E A C B D (1) D E C A B (1) D C E A B (1) D C B E A (1) D C A B E (1) D B A C E (1) D A E B C (1) D A C E B (1) C E B D A (1) C B D E A (1) B D C A E (1) B C D E A (1) B C A E D (1) B A E C D (1) B A C D E (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -20 -14 -16 -8 B 20 0 0 2 8 C 14 0 0 6 8 D 16 -2 -6 0 -4 E 8 -8 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.276996 C: 0.723004 D: 0.000000 E: 0.000000 Sum of squares = 0.599461131436 Cumulative probabilities = A: 0.000000 B: 0.276996 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -14 -16 -8 B 20 0 0 2 8 C 14 0 0 6 8 D 16 -2 -6 0 -4 E 8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999923 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 B=16 A=15 C=12 so C is eliminated. Round 2 votes counts: E=34 D=29 B=22 A=15 so A is eliminated. Round 3 votes counts: D=36 E=35 B=29 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:215 C:214 D:202 E:198 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -14 -16 -8 B 20 0 0 2 8 C 14 0 0 6 8 D 16 -2 -6 0 -4 E 8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999923 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 -16 -8 B 20 0 0 2 8 C 14 0 0 6 8 D 16 -2 -6 0 -4 E 8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999923 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 -16 -8 B 20 0 0 2 8 C 14 0 0 6 8 D 16 -2 -6 0 -4 E 8 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999923 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 731: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) A E D B C (7) E A B D C (6) C D B E A (5) C D B A E (5) D B E A C (4) D B A E C (4) A E C B D (4) E A C B D (3) C D A E B (3) B D E A C (3) B C D E A (3) E B A D C (2) E A B C D (2) D C B A E (2) D B C E A (2) D B C A E (2) D A E B C (2) C E A B D (2) C B E D A (2) C B D A E (2) C A E B D (2) C A D E B (2) B E D A C (2) B D C E A (2) E D B A C (1) D E B A C (1) D C B E A (1) D C A B E (1) D A C E B (1) D A B E C (1) C D A B E (1) C B E A D (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E C A (1) B C E A D (1) A E D C B (1) A E C D B (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 -2 -22 -12 B 16 0 -2 2 14 C 2 2 0 6 4 D 22 -2 -6 0 16 E 12 -14 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 -22 -12 B 16 0 -2 2 14 C 2 2 0 6 4 D 22 -2 -6 0 16 E 12 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=21 A=15 E=14 B=14 so E is eliminated. Round 2 votes counts: C=36 A=26 D=22 B=16 so B is eliminated. Round 3 votes counts: C=40 D=30 A=30 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:215 D:215 C:207 E:189 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -2 -22 -12 B 16 0 -2 2 14 C 2 2 0 6 4 D 22 -2 -6 0 16 E 12 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -22 -12 B 16 0 -2 2 14 C 2 2 0 6 4 D 22 -2 -6 0 16 E 12 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -22 -12 B 16 0 -2 2 14 C 2 2 0 6 4 D 22 -2 -6 0 16 E 12 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 732: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C B D E A (8) C B A D E (8) A E D B C (8) E D B C A (5) A B C D E (5) E D C B A (4) D E B C A (4) E A D C B (3) D B C E A (3) C A B E D (3) A C E B D (3) A C B D E (3) A B D C E (3) E C D B A (2) E A D B C (2) E A C D B (2) C B D A E (2) B D C E A (2) B C D E A (2) B C D A E (2) A E C D B (2) A E C B D (2) A B D E C (2) E D B A C (1) E A C B D (1) D B E C A (1) C E D B A (1) C E B D A (1) C E A B D (1) C A B D E (1) B D C A E (1) B C A D E (1) A D E B C (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -4 6 -2 B -4 0 4 4 -2 C 4 -4 0 2 2 D -6 -4 -2 0 0 E 2 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 -4 6 -2 B -4 0 4 4 -2 C 4 -4 0 2 2 D -6 -4 -2 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=28 C=25 D=8 B=8 so D is eliminated. Round 2 votes counts: E=32 A=31 C=25 B=12 so B is eliminated. Round 3 votes counts: C=36 E=33 A=31 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:202 C:202 B:201 E:201 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 6 -2 B -4 0 4 4 -2 C 4 -4 0 2 2 D -6 -4 -2 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 6 -2 B -4 0 4 4 -2 C 4 -4 0 2 2 D -6 -4 -2 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 6 -2 B -4 0 4 4 -2 C 4 -4 0 2 2 D -6 -4 -2 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 733: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (6) A B C D E (6) D E B C A (5) A C D B E (5) A C B E D (5) E D B C A (4) B E D C A (4) A C E D B (4) D B A E C (3) B E C D A (3) A D C E B (3) A C B D E (3) E C D A B (2) E C B D A (2) E B C D A (2) D E B A C (2) D B E C A (2) D B E A C (2) D A B E C (2) C E B A D (2) C E A D B (2) C E A B D (2) C B E A D (2) C A E B D (2) B D E A C (2) B D A E C (2) B A D C E (2) A D B E C (2) A C E B D (2) A B D C E (2) E D C A B (1) E C D B A (1) E B D C A (1) D E A C B (1) C E D A B (1) C E B D A (1) C A B E D (1) B D E C A (1) B C A E D (1) B C A D E (1) B A C D E (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 8 14 12 10 B -8 0 2 0 8 C -14 -2 0 14 12 D -12 0 -14 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 12 10 B -8 0 2 0 8 C -14 -2 0 14 12 D -12 0 -14 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 D=17 B=17 E=13 C=13 so E is eliminated. Round 2 votes counts: A=40 D=22 B=20 C=18 so C is eliminated. Round 3 votes counts: A=47 B=27 D=26 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:205 B:201 D:192 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 12 10 B -8 0 2 0 8 C -14 -2 0 14 12 D -12 0 -14 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 12 10 B -8 0 2 0 8 C -14 -2 0 14 12 D -12 0 -14 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 12 10 B -8 0 2 0 8 C -14 -2 0 14 12 D -12 0 -14 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 734: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) C E B D A (8) B C E D A (8) A B D C E (7) E C D B A (6) D E C B A (6) A D E C B (6) D E A C B (4) D A E C B (4) C B E D A (4) B C A E D (4) A D B E C (4) D A E B C (3) E D C B A (2) D E C A B (2) D E B C A (2) D B E C A (2) A E D C B (2) A E C D B (2) A D E B C (2) A C E B D (2) A B C D E (2) E C D A B (1) C A B E D (1) B D C E A (1) B D A C E (1) B C E A D (1) B C D E A (1) B A D C E (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 2 -10 4 B -4 0 -2 4 -4 C -2 2 0 2 4 D 10 -4 -2 0 -2 E -4 4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408227 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -10 4 B -4 0 -2 4 -4 C -2 2 0 2 4 D 10 -4 -2 0 -2 E -4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408633 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=23 B=18 C=13 E=9 so E is eliminated. Round 2 votes counts: A=37 D=25 C=20 B=18 so B is eliminated. Round 3 votes counts: A=39 C=34 D=27 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:203 D:201 A:200 E:199 B:197 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 2 -10 4 B -4 0 -2 4 -4 C -2 2 0 2 4 D 10 -4 -2 0 -2 E -4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408633 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -10 4 B -4 0 -2 4 -4 C -2 2 0 2 4 D 10 -4 -2 0 -2 E -4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408633 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -10 4 B -4 0 -2 4 -4 C -2 2 0 2 4 D 10 -4 -2 0 -2 E -4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408633 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 735: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) A D B C E (6) D E A B C (5) C B A E D (5) C A B D E (5) A D B E C (5) A C B D E (5) E D A B C (4) E B C D A (4) D A E B C (4) C B E D A (4) B C E D A (4) A D C B E (4) C B A D E (3) A D E B C (3) E D B A C (2) E D A C B (2) E C B D A (2) D E B A C (2) C A D B E (2) B C E A D (2) A D E C B (2) A C D B E (2) E C D A B (1) E C A D B (1) E B D C A (1) E A D C B (1) D A B E C (1) C E B D A (1) C E B A D (1) C E A B D (1) C A B E D (1) B E D C A (1) B D A E C (1) B A D C E (1) B A C D E (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 14 6 22 6 B -14 0 -4 4 24 C -6 4 0 6 16 D -22 -4 -6 0 8 E -6 -24 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 22 6 B -14 0 -4 4 24 C -6 4 0 6 16 D -22 -4 -6 0 8 E -6 -24 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=29 E=18 D=12 B=10 so B is eliminated. Round 2 votes counts: C=37 A=31 E=19 D=13 so D is eliminated. Round 3 votes counts: C=37 A=37 E=26 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:210 B:205 D:188 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 22 6 B -14 0 -4 4 24 C -6 4 0 6 16 D -22 -4 -6 0 8 E -6 -24 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 22 6 B -14 0 -4 4 24 C -6 4 0 6 16 D -22 -4 -6 0 8 E -6 -24 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 22 6 B -14 0 -4 4 24 C -6 4 0 6 16 D -22 -4 -6 0 8 E -6 -24 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 736: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) D C E B A (9) B A E C D (8) A E B D C (8) D C A E B (6) C D B A E (6) B E A C D (5) E A B D C (4) D C B E A (4) A E D B C (4) A B E C D (4) E B A D C (3) D E A B C (3) D C E A B (3) C B D A E (3) A B C E D (3) C B D E A (2) C A B D E (2) B C E D A (2) E D A B C (1) D E C B A (1) D E B A C (1) D E A C B (1) C D A B E (1) C B E A D (1) C B A E D (1) C B A D E (1) B E C D A (1) B C A E D (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -8 -8 -2 B 18 0 -2 -2 10 C 8 2 0 2 10 D 8 2 -2 0 8 E 2 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 -8 -2 B 18 0 -2 -2 10 C 8 2 0 2 10 D 8 2 -2 0 8 E 2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=27 A=20 B=17 E=8 so E is eliminated. Round 2 votes counts: D=29 C=27 A=24 B=20 so B is eliminated. Round 3 votes counts: A=40 C=31 D=29 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:212 C:211 D:208 E:187 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -8 -8 -2 B 18 0 -2 -2 10 C 8 2 0 2 10 D 8 2 -2 0 8 E 2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -8 -2 B 18 0 -2 -2 10 C 8 2 0 2 10 D 8 2 -2 0 8 E 2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -8 -2 B 18 0 -2 -2 10 C 8 2 0 2 10 D 8 2 -2 0 8 E 2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 737: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E D B A C (5) E C D B A (4) A E D B C (4) E C B D A (3) E A C B D (3) D B E C A (3) D B C E A (3) D B A E C (3) D B A C E (3) C E B D A (3) A E C B D (3) A D B E C (3) A D B C E (3) E D C A B (2) E D B C A (2) E D A B C (2) E C A D B (2) D B C A E (2) C B D E A (2) C B D A E (2) C A E B D (2) B D C A E (2) B D A C E (2) B C A D E (2) A C B E D (2) A C B D E (2) E D C B A (1) E D A C B (1) E C A B D (1) E A D B C (1) E A C D B (1) D E B C A (1) D E B A C (1) D B E A C (1) D A B E C (1) C E D B A (1) C E A B D (1) C A B E D (1) B C D A E (1) B A D C E (1) A E B D C (1) A D E B C (1) A C E B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 18 -2 12 B -4 0 24 -2 6 C -18 -24 0 -24 -2 D 2 2 24 0 6 E -12 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 -2 12 B -4 0 24 -2 6 C -18 -24 0 -24 -2 D 2 2 24 0 6 E -12 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=28 D=18 C=12 B=8 so B is eliminated. Round 2 votes counts: A=35 E=28 D=22 C=15 so C is eliminated. Round 3 votes counts: A=40 E=33 D=27 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:216 B:212 E:189 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 18 -2 12 B -4 0 24 -2 6 C -18 -24 0 -24 -2 D 2 2 24 0 6 E -12 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 -2 12 B -4 0 24 -2 6 C -18 -24 0 -24 -2 D 2 2 24 0 6 E -12 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 -2 12 B -4 0 24 -2 6 C -18 -24 0 -24 -2 D 2 2 24 0 6 E -12 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 738: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) D C E B A (7) A C D B E (7) D E C B A (6) B E A C D (6) E B D C A (5) D C E A B (5) C D A E B (5) A C B D E (5) B A E C D (4) A B E C D (4) E D B C A (3) E B C D A (3) E B A C D (3) D E B C A (3) A B C E D (3) A B C D E (3) E B A D C (2) D C A B E (2) B E D A C (2) A B D C E (2) E D C B A (1) E B D A C (1) D A C B E (1) C E B A D (1) C D E B A (1) C A E B D (1) C A D E B (1) C A D B E (1) B E A D C (1) A D C B E (1) Total count = 100 A B C D E A 0 2 -10 -10 0 B -2 0 -10 -8 -16 C 10 10 0 -4 12 D 10 8 4 0 20 E 0 16 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -10 0 B -2 0 -10 -8 -16 C 10 10 0 -4 12 D 10 8 4 0 20 E 0 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=25 E=18 B=13 C=10 so C is eliminated. Round 2 votes counts: D=40 A=28 E=19 B=13 so B is eliminated. Round 3 votes counts: D=40 A=32 E=28 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:214 E:192 A:191 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -10 -10 0 B -2 0 -10 -8 -16 C 10 10 0 -4 12 D 10 8 4 0 20 E 0 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -10 0 B -2 0 -10 -8 -16 C 10 10 0 -4 12 D 10 8 4 0 20 E 0 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -10 0 B -2 0 -10 -8 -16 C 10 10 0 -4 12 D 10 8 4 0 20 E 0 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 739: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) E B A C D (7) D C A E B (7) D C E B A (5) B E A D C (5) B A E C D (5) E C B D A (4) D C A B E (4) A B E C D (4) D A B C E (3) C E D B A (3) C D A E B (3) B A E D C (3) A D B C E (3) A B D E C (3) A B D C E (3) E B C A D (2) C E D A B (2) C D E A B (2) B A D E C (2) A C E D B (2) E D C B A (1) E C D B A (1) E C B A D (1) E C A B D (1) E B C D A (1) E A C B D (1) D C B E A (1) D B E C A (1) D B A C E (1) D A B E C (1) C E A D B (1) C E A B D (1) C A E D B (1) B E A C D (1) B D E A C (1) A E B C D (1) A D C B E (1) A C D E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 16 2 16 B -10 0 -2 -4 4 C -16 2 0 -8 6 D -2 4 8 0 0 E -16 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 2 16 B -10 0 -2 -4 4 C -16 2 0 -8 6 D -2 4 8 0 0 E -16 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980011 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=20 E=19 B=17 C=13 so C is eliminated. Round 2 votes counts: D=36 E=26 A=21 B=17 so B is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:222 D:205 B:194 C:192 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 2 16 B -10 0 -2 -4 4 C -16 2 0 -8 6 D -2 4 8 0 0 E -16 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980011 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 2 16 B -10 0 -2 -4 4 C -16 2 0 -8 6 D -2 4 8 0 0 E -16 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980011 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 2 16 B -10 0 -2 -4 4 C -16 2 0 -8 6 D -2 4 8 0 0 E -16 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980011 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 740: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) D C A E B (6) D A C B E (6) D A B C E (5) C E A D B (5) A E B D C (5) C D B E A (4) C D A E B (4) D B C A E (3) D A B E C (3) C A E D B (3) B E C D A (3) E C B A D (2) E C A B D (2) D C A B E (2) D B A C E (2) C E D A B (2) C E B D A (2) C D A B E (2) B D E C A (2) B C E D A (2) B A E D C (2) A D E C B (2) E B A D C (1) E B A C D (1) E A C B D (1) E A B C D (1) D B C E A (1) D B A E C (1) C D E A B (1) C B E D A (1) C A D E B (1) B E D A C (1) B E C A D (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B C D (1) A D C E B (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 18 -2 -14 14 B -18 0 2 -16 12 C 2 -2 0 -20 10 D 14 16 20 0 4 E -14 -12 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999088 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -2 -14 14 B -18 0 2 -16 12 C 2 -2 0 -20 10 D 14 16 20 0 4 E -14 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 B=23 A=15 E=8 so E is eliminated. Round 2 votes counts: D=29 C=29 B=25 A=17 so A is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:208 C:195 B:190 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -2 -14 14 B -18 0 2 -16 12 C 2 -2 0 -20 10 D 14 16 20 0 4 E -14 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -2 -14 14 B -18 0 2 -16 12 C 2 -2 0 -20 10 D 14 16 20 0 4 E -14 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -2 -14 14 B -18 0 2 -16 12 C 2 -2 0 -20 10 D 14 16 20 0 4 E -14 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 741: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) C D B E A (7) E A C D B (6) B D C A E (6) E C D A B (5) D C B A E (5) D B C A E (5) A E B D C (5) D B A C E (4) C E D B A (4) A B E D C (4) E C D B A (3) E A C B D (3) C D E B A (3) C B D E A (3) E C A D B (2) D C B E A (2) D A B C E (2) B C D E A (2) A E D B C (2) A B D E C (2) E D C A B (1) E C B D A (1) E C A B D (1) E B A C D (1) E A D C B (1) E A B C D (1) D E C A B (1) D C A B E (1) D A E C B (1) C E B D A (1) C D E A B (1) B D C E A (1) B D A C E (1) B C E D A (1) B C D A E (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -18 -14 -24 -2 B 18 0 -4 -14 12 C 14 4 0 -6 16 D 24 14 6 0 14 E 2 -12 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 -24 -2 B 18 0 -4 -14 12 C 14 4 0 -6 16 D 24 14 6 0 14 E 2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=21 B=20 C=19 A=15 so A is eliminated. Round 2 votes counts: E=33 B=26 D=22 C=19 so C is eliminated. Round 3 votes counts: E=38 D=33 B=29 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 C:214 B:206 E:180 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -14 -24 -2 B 18 0 -4 -14 12 C 14 4 0 -6 16 D 24 14 6 0 14 E 2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 -24 -2 B 18 0 -4 -14 12 C 14 4 0 -6 16 D 24 14 6 0 14 E 2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 -24 -2 B 18 0 -4 -14 12 C 14 4 0 -6 16 D 24 14 6 0 14 E 2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 742: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) D C A B E (10) C E B A D (9) C D E B A (7) C D A B E (6) E B A D C (4) E B A C D (4) C E D B A (4) C D E A B (4) A B E C D (4) A B D E C (4) D A B C E (3) B E A D C (3) B A E D C (3) A B E D C (3) E D C B A (2) C E B D A (2) A B D C E (2) E D B C A (1) E D B A C (1) E C B D A (1) E C B A D (1) E B C A D (1) D C E A B (1) D A C B E (1) C D A E B (1) C A D B E (1) C A B D E (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 14 -4 -14 10 B -14 0 -4 -16 14 C 4 4 0 -8 8 D 14 16 8 0 14 E -10 -14 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 -14 10 B -14 0 -4 -16 14 C 4 4 0 -8 8 D 14 16 8 0 14 E -10 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=28 A=16 E=15 B=6 so B is eliminated. Round 2 votes counts: C=35 D=28 A=19 E=18 so E is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:204 A:203 B:190 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -4 -14 10 B -14 0 -4 -16 14 C 4 4 0 -8 8 D 14 16 8 0 14 E -10 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -14 10 B -14 0 -4 -16 14 C 4 4 0 -8 8 D 14 16 8 0 14 E -10 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -14 10 B -14 0 -4 -16 14 C 4 4 0 -8 8 D 14 16 8 0 14 E -10 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 743: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) D E B C A (9) E B D C A (7) D A C B E (7) E B C A D (5) B C A D E (5) A C B E D (5) A C B D E (5) E D A C B (4) B C A E D (3) A E C B D (3) A C E B D (3) E D A B C (2) E A D C B (2) E A C B D (2) D E A C B (2) D E A B C (2) D B C E A (2) D B C A E (2) C A B E D (2) B D C A E (2) B C E A D (2) E D B A C (1) E C B A D (1) E C A B D (1) D E B A C (1) D B E C A (1) D B A C E (1) D A C E B (1) C B A E D (1) C A E B D (1) A E C D B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -8 -8 -8 B 6 0 10 -4 -20 C 8 -10 0 -14 -8 D 8 4 14 0 -14 E 8 20 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -8 -8 B 6 0 10 -4 -20 C 8 -10 0 -14 -8 D 8 4 14 0 -14 E 8 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=28 A=21 B=12 C=4 so C is eliminated. Round 2 votes counts: E=35 D=28 A=24 B=13 so B is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:206 B:196 C:188 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -8 -8 B 6 0 10 -4 -20 C 8 -10 0 -14 -8 D 8 4 14 0 -14 E 8 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -8 -8 B 6 0 10 -4 -20 C 8 -10 0 -14 -8 D 8 4 14 0 -14 E 8 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -8 -8 B 6 0 10 -4 -20 C 8 -10 0 -14 -8 D 8 4 14 0 -14 E 8 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 744: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) B E A D C (8) B A D C E (8) C A D E B (5) B A E C D (5) A C D B E (5) D C A B E (4) C D A E B (4) C A D B E (4) E C D B A (3) E C D A B (3) E B A C D (3) C E D A B (3) B A D E C (3) A B D C E (3) E C B D A (2) E B C D A (2) E B A D C (2) D A C B E (2) B E D A C (2) B D A C E (2) B A E D C (2) E C B A D (1) E C A D B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C B A E (1) D B C E A (1) C D E A B (1) C D A B E (1) B E A C D (1) B D E A C (1) B D A E C (1) B A C D E (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 8 14 8 B 16 0 14 18 14 C -8 -14 0 -2 -2 D -14 -18 2 0 2 E -8 -14 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 8 14 8 B 16 0 14 18 14 C -8 -14 0 -2 -2 D -14 -18 2 0 2 E -8 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=28 C=18 A=11 D=9 so D is eliminated. Round 2 votes counts: B=35 E=28 C=24 A=13 so A is eliminated. Round 3 votes counts: B=40 C=32 E=28 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:207 E:189 C:187 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 8 14 8 B 16 0 14 18 14 C -8 -14 0 -2 -2 D -14 -18 2 0 2 E -8 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 8 14 8 B 16 0 14 18 14 C -8 -14 0 -2 -2 D -14 -18 2 0 2 E -8 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 8 14 8 B 16 0 14 18 14 C -8 -14 0 -2 -2 D -14 -18 2 0 2 E -8 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 745: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) D A E C B (7) B E A D C (7) D C A E B (6) C D A B E (6) C B A E D (6) B E A C D (6) B E C A D (5) E B A D C (4) D A C E B (4) D E A C B (3) D E A B C (3) C D A E B (3) B E D A C (3) A E D B C (3) E B D A C (2) C B D E A (2) C B D A E (2) C B A D E (2) C A D B E (2) C A B E D (2) B E C D A (2) B C A E D (2) E D A B C (1) D E B A C (1) D C B E A (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -2 4 0 B 10 0 -2 10 16 C 2 2 0 4 2 D -4 -10 -4 0 -8 E 0 -16 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 4 0 B 10 0 -2 10 16 C 2 2 0 4 2 D -4 -10 -4 0 -8 E 0 -16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=25 C=25 A=8 E=7 so E is eliminated. Round 2 votes counts: B=41 D=26 C=25 A=8 so A is eliminated. Round 3 votes counts: B=41 D=32 C=27 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:205 A:196 E:195 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -2 4 0 B 10 0 -2 10 16 C 2 2 0 4 2 D -4 -10 -4 0 -8 E 0 -16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 4 0 B 10 0 -2 10 16 C 2 2 0 4 2 D -4 -10 -4 0 -8 E 0 -16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 4 0 B 10 0 -2 10 16 C 2 2 0 4 2 D -4 -10 -4 0 -8 E 0 -16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 746: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (16) E A B D C (15) A E C D B (14) C D B A E (12) A C E D B (8) B D E C A (7) A E C B D (4) D B C E A (3) C D B E A (3) E A D B C (2) D C B E A (2) C D A B E (2) B E D A C (2) E B A D C (1) E A D C B (1) D B E C A (1) C D A E B (1) C B D A E (1) B D C A E (1) B C D A E (1) A E B D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 0 -4 -6 B 0 0 4 2 6 C 0 -4 0 -6 0 D 4 -2 6 0 0 E 6 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.248355 B: 0.751645 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.626650382411 Cumulative probabilities = A: 0.248355 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 -6 B 0 0 4 2 6 C 0 -4 0 -6 0 D 4 -2 6 0 0 E 6 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556179 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 E=19 C=19 D=6 so D is eliminated. Round 2 votes counts: B=31 A=29 C=21 E=19 so E is eliminated. Round 3 votes counts: A=47 B=32 C=21 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:206 D:204 E:200 A:195 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -4 -6 B 0 0 4 2 6 C 0 -4 0 -6 0 D 4 -2 6 0 0 E 6 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556179 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 -6 B 0 0 4 2 6 C 0 -4 0 -6 0 D 4 -2 6 0 0 E 6 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556179 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 -6 B 0 0 4 2 6 C 0 -4 0 -6 0 D 4 -2 6 0 0 E 6 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556179 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 747: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) C D E A B (6) A B C E D (6) B A E D C (5) B A C E D (5) C B A E D (4) C A E B D (4) A E B D C (4) E D C A B (3) E D A C B (3) D E A B C (3) A B E C D (3) E C D A B (2) E A D C B (2) E A C D B (2) D C E B A (2) D B E A C (2) C E D A B (2) C E A D B (2) C D E B A (2) C D B E A (2) C A E D B (2) C A B E D (2) B D A E C (2) B C D A E (2) B A D E C (2) A E D B C (2) A B E D C (2) E D A B C (1) E C A D B (1) D E B C A (1) D E B A C (1) D E A C B (1) D C B E A (1) D B E C A (1) D B C E A (1) C B D A E (1) B D C A E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 22 0 8 6 B -22 0 -4 -2 -8 C 0 4 0 8 -2 D -8 2 -8 0 -22 E -6 8 2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.463823 B: 0.000000 C: 0.536177 D: 0.000000 E: 0.000000 Sum of squares = 0.502617549603 Cumulative probabilities = A: 0.463823 B: 0.463823 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 0 8 6 B -22 0 -4 -2 -8 C 0 4 0 8 -2 D -8 2 -8 0 -22 E -6 8 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=21 D=19 A=19 E=14 so E is eliminated. Round 2 votes counts: C=30 D=26 A=23 B=21 so B is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:213 C:205 B:182 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 0 8 6 B -22 0 -4 -2 -8 C 0 4 0 8 -2 D -8 2 -8 0 -22 E -6 8 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 0 8 6 B -22 0 -4 -2 -8 C 0 4 0 8 -2 D -8 2 -8 0 -22 E -6 8 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 0 8 6 B -22 0 -4 -2 -8 C 0 4 0 8 -2 D -8 2 -8 0 -22 E -6 8 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 748: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (7) E D C A B (6) B A C D E (5) E D A C B (4) E A D B C (4) C D E B A (4) C D B E A (4) C B D E A (4) C B D A E (4) E A B C D (3) D E C B A (3) D E C A B (3) D E A C B (3) C B E A D (3) C B A D E (3) B A C E D (3) A E D B C (3) A E B D C (3) A B E C D (3) A B C E D (3) E D A B C (2) D C B E A (2) D C B A E (2) D A E B C (2) C B A E D (2) B C A D E (2) A B D C E (2) E A C B D (1) D A B C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C B E D A (1) B D A C E (1) B C D A E (1) A E B C D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -8 2 0 B 8 0 -4 10 10 C 8 4 0 14 14 D -2 -10 -14 0 -6 E 0 -10 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 2 0 B 8 0 -4 10 10 C 8 4 0 14 14 D -2 -10 -14 0 -6 E 0 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=20 B=19 A=17 D=16 so D is eliminated. Round 2 votes counts: C=32 E=29 A=20 B=19 so B is eliminated. Round 3 votes counts: C=42 E=29 A=29 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:212 A:193 E:191 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 2 0 B 8 0 -4 10 10 C 8 4 0 14 14 D -2 -10 -14 0 -6 E 0 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 2 0 B 8 0 -4 10 10 C 8 4 0 14 14 D -2 -10 -14 0 -6 E 0 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 2 0 B 8 0 -4 10 10 C 8 4 0 14 14 D -2 -10 -14 0 -6 E 0 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 749: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) B D A E C (8) C D E B A (5) D B E C A (4) C D B E A (4) C A E D B (4) E A C D B (3) D B C E A (3) C E D B A (3) C A B D E (3) A E C D B (3) A E B D C (3) A C E D B (3) A B E D C (3) E D B A C (2) E D A B C (2) E A D B C (2) C E D A B (2) B D C E A (2) B D C A E (2) B A D E C (2) B A D C E (2) A C E B D (2) A C B E D (2) A C B D E (2) A B C D E (2) E D B C A (1) E D A C B (1) D E C B A (1) D E B C A (1) D C B E A (1) D B E A C (1) C E A D B (1) C D E A B (1) C B D A E (1) C A E B D (1) B D E C A (1) B D A C E (1) B C A D E (1) A E D B C (1) A E C B D (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 16 -12 0 B 10 0 12 2 14 C -16 -12 0 -10 -2 D 12 -2 10 0 18 E 0 -14 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 16 -12 0 B 10 0 12 2 14 C -16 -12 0 -10 -2 D 12 -2 10 0 18 E 0 -14 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 A=25 E=11 D=11 so E is eliminated. Round 2 votes counts: A=30 B=28 C=25 D=17 so D is eliminated. Round 3 votes counts: B=40 A=33 C=27 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:219 A:197 E:185 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 16 -12 0 B 10 0 12 2 14 C -16 -12 0 -10 -2 D 12 -2 10 0 18 E 0 -14 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 -12 0 B 10 0 12 2 14 C -16 -12 0 -10 -2 D 12 -2 10 0 18 E 0 -14 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 -12 0 B 10 0 12 2 14 C -16 -12 0 -10 -2 D 12 -2 10 0 18 E 0 -14 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 750: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) E A B C D (9) D B E A C (7) D C B A E (6) C D A B E (6) C A E B D (6) D C E B A (5) B E A D C (5) E B A C D (4) D C E A B (4) C A B E D (4) D C A B E (3) D B A E C (3) C E A B D (3) E D B A C (2) D E C B A (2) D E B A C (2) C D A E B (2) B E A C D (2) E B D A C (1) D E C A B (1) D C B E A (1) D C A E B (1) D B C E A (1) D B C A E (1) C E D A B (1) C A D B E (1) B E D A C (1) B D E A C (1) B A E D C (1) B A E C D (1) A E B C D (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 4 -2 -22 B 10 0 6 2 -8 C -4 -6 0 -14 -6 D 2 -2 14 0 -6 E 22 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 4 -2 -22 B 10 0 6 2 -8 C -4 -6 0 -14 -6 D 2 -2 14 0 -6 E 22 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=25 C=23 B=11 A=4 so A is eliminated. Round 2 votes counts: D=37 E=26 C=24 B=13 so B is eliminated. Round 3 votes counts: D=38 E=37 C=25 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:205 D:204 A:185 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 4 -2 -22 B 10 0 6 2 -8 C -4 -6 0 -14 -6 D 2 -2 14 0 -6 E 22 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -2 -22 B 10 0 6 2 -8 C -4 -6 0 -14 -6 D 2 -2 14 0 -6 E 22 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -2 -22 B 10 0 6 2 -8 C -4 -6 0 -14 -6 D 2 -2 14 0 -6 E 22 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 751: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (5) E B A D C (5) B E C A D (5) D E B C A (4) D E B A C (4) D A E B C (4) E B D A C (3) D E C B A (3) D E A B C (3) D C E B A (3) D C E A B (3) D A E C B (3) D A C E B (3) C D A B E (3) C B E D A (3) C B A E D (3) C A B E D (3) B E A C D (3) E D B A C (2) D C A E B (2) D A C B E (2) C B D E A (2) B C E A D (2) A D E B C (2) A D C B E (2) A D B E C (2) A C D B E (2) A C B D E (2) A B C E D (2) E C B D A (1) E B C A D (1) D E A C B (1) D C A B E (1) C E D B A (1) C D E B A (1) C D B E A (1) C D B A E (1) C B E A D (1) C B A D E (1) C A D B E (1) B E A D C (1) B C A E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -4 -18 -16 B 14 0 2 -8 -8 C 4 -2 0 -8 -6 D 18 8 8 0 16 E 16 8 6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -18 -16 B 14 0 2 -8 -8 C 4 -2 0 -8 -6 D 18 8 8 0 16 E 16 8 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=21 E=17 A=14 B=12 so B is eliminated. Round 2 votes counts: D=36 E=26 C=24 A=14 so A is eliminated. Round 3 votes counts: D=43 C=31 E=26 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:207 B:200 C:194 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -4 -18 -16 B 14 0 2 -8 -8 C 4 -2 0 -8 -6 D 18 8 8 0 16 E 16 8 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -18 -16 B 14 0 2 -8 -8 C 4 -2 0 -8 -6 D 18 8 8 0 16 E 16 8 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -18 -16 B 14 0 2 -8 -8 C 4 -2 0 -8 -6 D 18 8 8 0 16 E 16 8 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 752: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) B C A E D (11) D E A C B (10) D B C A E (9) E A C D B (6) D E C A B (5) E A C B D (4) C A E B D (4) E D A C B (3) D E A B C (3) D B C E A (3) C B A E D (3) B C D A E (3) B C A D E (3) E A D C B (2) D E B A C (2) D B E C A (2) D B E A C (2) C A B E D (2) A E C B D (2) E C A B D (1) D E C B A (1) D B A E C (1) D A B E C (1) C E A B D (1) B A C E D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -22 -16 10 B 10 0 10 0 10 C 22 -10 0 -14 10 D 16 0 14 0 18 E -10 -10 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.874930 C: 0.000000 D: 0.125070 E: 0.000000 Sum of squares = 0.781145479069 Cumulative probabilities = A: 0.000000 B: 0.874930 C: 0.874930 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 -16 10 B 10 0 10 0 10 C 22 -10 0 -14 10 D 16 0 14 0 18 E -10 -10 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=32 E=16 C=10 A=3 so A is eliminated. Round 2 votes counts: D=39 B=32 E=18 C=11 so C is eliminated. Round 3 votes counts: D=39 B=37 E=24 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:224 B:215 C:204 A:181 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -22 -16 10 B 10 0 10 0 10 C 22 -10 0 -14 10 D 16 0 14 0 18 E -10 -10 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 -16 10 B 10 0 10 0 10 C 22 -10 0 -14 10 D 16 0 14 0 18 E -10 -10 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 -16 10 B 10 0 10 0 10 C 22 -10 0 -14 10 D 16 0 14 0 18 E -10 -10 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 753: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (14) E C A D B (8) E A C D B (7) B D C A E (7) B E D A C (6) C D A B E (5) C A D E B (5) C A D B E (5) C E A D B (4) E C A B D (3) E B C D A (3) E A C B D (3) E A B D C (3) D B C A E (3) D A C B E (3) E C B D A (2) E B D C A (2) E B A D C (2) D C A B E (2) B D E A C (2) E B C A D (1) D B A C E (1) C D B A E (1) C D A E B (1) C A E D B (1) B E A D C (1) B D C E A (1) B D A E C (1) A D B C E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -8 -8 4 B -6 0 -4 2 8 C 8 4 0 2 12 D 8 -2 -2 0 8 E -4 -8 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -8 4 B -6 0 -4 2 8 C 8 4 0 2 12 D 8 -2 -2 0 8 E -4 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=32 C=22 D=9 A=3 so A is eliminated. Round 2 votes counts: E=34 B=32 C=24 D=10 so D is eliminated. Round 3 votes counts: B=37 E=34 C=29 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:206 B:200 A:197 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -8 4 B -6 0 -4 2 8 C 8 4 0 2 12 D 8 -2 -2 0 8 E -4 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -8 4 B -6 0 -4 2 8 C 8 4 0 2 12 D 8 -2 -2 0 8 E -4 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -8 4 B -6 0 -4 2 8 C 8 4 0 2 12 D 8 -2 -2 0 8 E -4 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 754: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) C E A D B (7) C D B A E (6) B D A E C (6) B A D E C (6) E C A B D (5) D A B E C (5) C E D A B (4) A E D B C (4) A D E B C (4) C E B D A (3) C B E D A (3) B D A C E (3) E C B A D (2) E C A D B (2) E B A C D (2) E A C D B (2) E A C B D (2) D B C A E (2) D B A C E (2) C D A E B (2) C B D E A (2) B D C A E (2) A B D E C (2) E A B D C (1) D C B A E (1) D A B C E (1) C E D B A (1) C E B A D (1) C E A B D (1) C D E A B (1) B E C A D (1) B E A D C (1) B E A C D (1) B D E C A (1) B C D E A (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 4 8 12 B -6 0 8 -6 10 C -4 -8 0 -2 -12 D -8 6 2 0 12 E -12 -10 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 8 12 B -6 0 8 -6 10 C -4 -8 0 -2 -12 D -8 6 2 0 12 E -12 -10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=22 A=20 E=16 D=11 so D is eliminated. Round 2 votes counts: C=32 B=26 A=26 E=16 so E is eliminated. Round 3 votes counts: C=41 A=31 B=28 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:206 B:203 E:189 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 8 12 B -6 0 8 -6 10 C -4 -8 0 -2 -12 D -8 6 2 0 12 E -12 -10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 8 12 B -6 0 8 -6 10 C -4 -8 0 -2 -12 D -8 6 2 0 12 E -12 -10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 8 12 B -6 0 8 -6 10 C -4 -8 0 -2 -12 D -8 6 2 0 12 E -12 -10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 755: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (15) E C D A B (14) C D A B E (9) B A D C E (9) E B A C D (6) B E A D C (6) E C D B A (4) E B C A D (4) C A D B E (4) E C A D B (3) D A B C E (3) E C B D A (2) E C A B D (2) D A C B E (2) C E D A B (2) C D E A B (2) B A D E C (2) A D B C E (2) E D C B A (1) E D C A B (1) E D B A C (1) E C B A D (1) E B D A C (1) C D A E B (1) B E A C D (1) B D A C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 0 12 -32 B 8 0 2 2 -20 C 0 -2 0 12 -28 D -12 -2 -12 0 -28 E 32 20 28 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 12 -32 B 8 0 2 2 -20 C 0 -2 0 12 -28 D -12 -2 -12 0 -28 E 32 20 28 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=55 B=19 C=18 D=5 A=3 so A is eliminated. Round 2 votes counts: E=55 C=19 B=19 D=7 so D is eliminated. Round 3 votes counts: E=55 B=24 C=21 so C is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:254 B:196 C:191 A:186 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 12 -32 B 8 0 2 2 -20 C 0 -2 0 12 -28 D -12 -2 -12 0 -28 E 32 20 28 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 12 -32 B 8 0 2 2 -20 C 0 -2 0 12 -28 D -12 -2 -12 0 -28 E 32 20 28 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 12 -32 B 8 0 2 2 -20 C 0 -2 0 12 -28 D -12 -2 -12 0 -28 E 32 20 28 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 756: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) B C D E A (8) A E D C B (8) C A B E D (7) E A D C B (6) D B E C A (6) B D C E A (6) B C D A E (6) A C B E D (6) D E B C A (4) A E C D B (4) A C E B D (4) D E B A C (3) D B C E A (3) C B D A E (3) C B A E D (3) C B A D E (3) A C E D B (3) E D B A C (2) A E C B D (2) E B A D C (1) D E A B C (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 0 -2 2 B -2 0 0 0 4 C 0 0 0 2 8 D 2 0 -2 0 -12 E -2 -4 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363880 B: 0.000000 C: 0.636120 D: 0.000000 E: 0.000000 Sum of squares = 0.537057247245 Cumulative probabilities = A: 0.363880 B: 0.363880 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 2 B -2 0 0 0 4 C 0 0 0 2 8 D 2 0 -2 0 -12 E -2 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499796 B: 0.000000 C: 0.500204 D: 0.000000 E: 0.000000 Sum of squares = 0.500000083157 Cumulative probabilities = A: 0.499796 B: 0.499796 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=21 E=18 D=17 C=16 so C is eliminated. Round 2 votes counts: A=35 B=30 E=18 D=17 so D is eliminated. Round 3 votes counts: B=39 A=35 E=26 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:205 A:201 B:201 E:199 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -2 2 B -2 0 0 0 4 C 0 0 0 2 8 D 2 0 -2 0 -12 E -2 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499796 B: 0.000000 C: 0.500204 D: 0.000000 E: 0.000000 Sum of squares = 0.500000083157 Cumulative probabilities = A: 0.499796 B: 0.499796 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 2 B -2 0 0 0 4 C 0 0 0 2 8 D 2 0 -2 0 -12 E -2 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499796 B: 0.000000 C: 0.500204 D: 0.000000 E: 0.000000 Sum of squares = 0.500000083157 Cumulative probabilities = A: 0.499796 B: 0.499796 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 2 B -2 0 0 0 4 C 0 0 0 2 8 D 2 0 -2 0 -12 E -2 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499796 B: 0.000000 C: 0.500204 D: 0.000000 E: 0.000000 Sum of squares = 0.500000083157 Cumulative probabilities = A: 0.499796 B: 0.499796 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 757: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) C D B E A (8) A E B D C (8) B C D A E (6) A E D C B (6) E A C B D (5) D C B A E (5) A E D B C (4) E A C D B (3) C B D E A (3) B D C A E (3) B A E C D (3) A D E B C (3) E C D A B (2) E A B C D (2) D C B E A (2) B C D E A (2) A D E C B (2) A D B C E (2) A B E D C (2) A B D E C (2) A B D C E (2) E D C A B (1) E C D B A (1) E C B A D (1) E B C A D (1) E A B D C (1) D E C A B (1) D C E B A (1) D C E A B (1) D A C E B (1) C E D B A (1) C D E B A (1) C B E D A (1) C B D A E (1) B A D C E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 18 16 18 6 B -18 0 -12 -8 -12 C -16 12 0 -14 -18 D -18 8 14 0 -6 E -6 12 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 18 6 B -18 0 -12 -8 -12 C -16 12 0 -14 -18 D -18 8 14 0 -6 E -6 12 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=26 C=15 B=15 D=11 so D is eliminated. Round 2 votes counts: A=34 E=27 C=24 B=15 so B is eliminated. Round 3 votes counts: A=38 C=35 E=27 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:215 D:199 C:182 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 18 6 B -18 0 -12 -8 -12 C -16 12 0 -14 -18 D -18 8 14 0 -6 E -6 12 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 18 6 B -18 0 -12 -8 -12 C -16 12 0 -14 -18 D -18 8 14 0 -6 E -6 12 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 18 6 B -18 0 -12 -8 -12 C -16 12 0 -14 -18 D -18 8 14 0 -6 E -6 12 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 758: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (10) B D E C A (6) A C E B D (6) C A E D B (5) E B C A D (4) E B A C D (4) B E D C A (4) B E A C D (4) B D E A C (4) A C E D B (4) E B C D A (3) D A C B E (3) C D A E B (3) B E C D A (3) A E C B D (3) D C B E A (2) D C A E B (2) D B E C A (2) C E A B D (2) B E D A C (2) A E B C D (2) A D C B E (2) A B D E C (2) E C B D A (1) E C A B D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A B E (1) D B C E A (1) D B C A E (1) D B A C E (1) D A C E B (1) D A B C E (1) C D E B A (1) C D E A B (1) C A D E B (1) B E C A D (1) B A E D C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 8 6 10 2 B -8 0 -4 8 -16 C -6 4 0 20 4 D -10 -8 -20 0 -2 E -2 16 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 10 2 B -8 0 -4 8 -16 C -6 4 0 20 4 D -10 -8 -20 0 -2 E -2 16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=25 D=17 E=14 C=13 so C is eliminated. Round 2 votes counts: A=37 B=25 D=22 E=16 so E is eliminated. Round 3 votes counts: A=41 B=37 D=22 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:211 E:206 B:190 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 10 2 B -8 0 -4 8 -16 C -6 4 0 20 4 D -10 -8 -20 0 -2 E -2 16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 10 2 B -8 0 -4 8 -16 C -6 4 0 20 4 D -10 -8 -20 0 -2 E -2 16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 10 2 B -8 0 -4 8 -16 C -6 4 0 20 4 D -10 -8 -20 0 -2 E -2 16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 759: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (14) A E C D B (11) B D C A E (10) E A C D B (6) E C D B A (5) B D E C A (5) B D A C E (4) A E B D C (4) A C D B E (4) E B D C A (3) E B A D C (3) C D B A E (3) A E C B D (3) A B D C E (3) E C D A B (2) A C D E B (2) E D C B A (1) E D B C A (1) E C A D B (1) E B D A C (1) E A B D C (1) D E B C A (1) D B C E A (1) D B C A E (1) C E D A B (1) C D B E A (1) C D A B E (1) C A D B E (1) B E D A C (1) B E A D C (1) B D E A C (1) A C E D B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -4 -14 0 B 14 0 14 12 6 C 4 -14 0 -14 -4 D 14 -12 14 0 6 E 0 -6 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999134 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -14 0 B 14 0 14 12 6 C 4 -14 0 -14 -4 D 14 -12 14 0 6 E 0 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=30 E=24 C=7 D=3 so D is eliminated. Round 2 votes counts: B=38 A=30 E=25 C=7 so C is eliminated. Round 3 votes counts: B=42 A=32 E=26 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:211 E:196 C:186 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 -14 0 B 14 0 14 12 6 C 4 -14 0 -14 -4 D 14 -12 14 0 6 E 0 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -14 0 B 14 0 14 12 6 C 4 -14 0 -14 -4 D 14 -12 14 0 6 E 0 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -14 0 B 14 0 14 12 6 C 4 -14 0 -14 -4 D 14 -12 14 0 6 E 0 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 760: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) B D A E C (9) E B D A C (7) C E A D B (6) C A D E B (6) B E D A C (6) A D B C E (6) C E A B D (5) A D C B E (5) E B C D A (4) D A B E C (4) C A E B D (4) E C B D A (3) E C D B A (2) E B D C A (2) D B A E C (2) D A B C E (2) C E D A B (2) B D E A C (2) A C D B E (2) E C B A D (1) E B C A D (1) D E A C B (1) D C E A B (1) D A C B E (1) C E B A D (1) C D E A B (1) C D A E B (1) B E A D C (1) B A E D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 2 0 8 B -16 0 -2 -4 4 C -2 2 0 -4 4 D 0 4 4 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363870 B: 0.000000 C: 0.000000 D: 0.636130 E: 0.000000 Sum of squares = 0.537062494276 Cumulative probabilities = A: 0.363870 B: 0.363870 C: 0.363870 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 0 8 B -16 0 -2 -4 4 C -2 2 0 -4 4 D 0 4 4 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=20 B=19 A=15 D=11 so D is eliminated. Round 2 votes counts: C=36 A=22 E=21 B=21 so E is eliminated. Round 3 votes counts: C=42 B=35 A=23 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 D:208 C:200 B:191 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 0 8 B -16 0 -2 -4 4 C -2 2 0 -4 4 D 0 4 4 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 0 8 B -16 0 -2 -4 4 C -2 2 0 -4 4 D 0 4 4 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 0 8 B -16 0 -2 -4 4 C -2 2 0 -4 4 D 0 4 4 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 761: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (15) A B C E D (12) B C A D E (7) A E B D C (6) D C E B A (5) C B A D E (5) E D A B C (4) D E C A B (4) D C B E A (4) A B E C D (4) E D C A B (3) E A D B C (3) D E A C B (3) C D B E A (3) A E B C D (3) E D A C B (2) D E A B C (2) C D B A E (2) C B A E D (2) C A B E D (2) B A C D E (2) E A C B D (1) D E B A C (1) C B D A E (1) B A C E D (1) A E D B C (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -6 2 0 B -4 0 -6 -4 -6 C 6 6 0 -8 -6 D -2 4 8 0 10 E 0 6 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 2 0 B -4 0 -6 -4 -6 C 6 6 0 -8 -6 D -2 4 8 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=28 C=15 E=13 B=10 so B is eliminated. Round 2 votes counts: D=34 A=31 C=22 E=13 so E is eliminated. Round 3 votes counts: D=43 A=35 C=22 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:210 E:201 A:200 C:199 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 2 0 B -4 0 -6 -4 -6 C 6 6 0 -8 -6 D -2 4 8 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 2 0 B -4 0 -6 -4 -6 C 6 6 0 -8 -6 D -2 4 8 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 2 0 B -4 0 -6 -4 -6 C 6 6 0 -8 -6 D -2 4 8 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 762: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) C A E B D (11) D E B C A (7) E D B A C (6) E A C B D (5) D E B A C (5) C A B E D (4) E A B D C (3) D B C A E (3) C A E D B (3) C A B D E (3) E C A D B (2) E C A B D (2) D C B A E (2) C D A E B (2) C D A B E (2) B D E A C (2) B D A C E (2) A C E B D (2) A C B E D (2) A B E C D (2) E D C A B (1) E D B C A (1) E D A B C (1) E C D A B (1) D E C B A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) D B A E C (1) B E A D C (1) B D A E C (1) B C A D E (1) B A C D E (1) A E C B D (1) A E B C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -8 -8 B -2 0 6 -8 -14 C 0 -6 0 -6 -12 D 8 8 6 0 0 E 8 14 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.570629 E: 0.429371 Sum of squares = 0.509976779571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.570629 E: 1.000000 A B C D E A 0 2 0 -8 -8 B -2 0 6 -8 -14 C 0 -6 0 -6 -12 D 8 8 6 0 0 E 8 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=25 E=22 A=10 B=8 so B is eliminated. Round 2 votes counts: D=40 C=26 E=23 A=11 so A is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:217 D:211 A:193 B:191 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -8 -8 B -2 0 6 -8 -14 C 0 -6 0 -6 -12 D 8 8 6 0 0 E 8 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -8 -8 B -2 0 6 -8 -14 C 0 -6 0 -6 -12 D 8 8 6 0 0 E 8 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -8 -8 B -2 0 6 -8 -14 C 0 -6 0 -6 -12 D 8 8 6 0 0 E 8 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 763: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) C E B D A (10) E C B D A (7) E D C A B (4) D A E C B (4) D A E B C (4) B A D E C (4) A D B E C (4) E C D B A (3) E C D A B (3) E B D C A (3) B E C A D (3) B C E A D (3) A D C E B (3) A B D C E (3) E B C D A (2) D E A C B (2) D A C E B (2) C B E A D (2) B E A D C (2) B A D C E (2) B A C D E (2) A D C B E (2) A D B C E (2) A B D E C (2) E D C B A (1) E B D A C (1) D E C A B (1) D E A B C (1) C E D B A (1) C D E A B (1) C D A E B (1) C B A E D (1) C A D B E (1) C A B D E (1) B C A E D (1) B A E D C (1) Total count = 100 A B C D E A 0 2 -18 -22 -20 B -2 0 -20 0 -28 C 18 20 0 4 -4 D 22 0 -4 0 -16 E 20 28 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -18 -22 -20 B -2 0 -20 0 -28 C 18 20 0 4 -4 D 22 0 -4 0 -16 E 20 28 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991423 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=24 B=18 A=16 D=14 so D is eliminated. Round 2 votes counts: E=28 C=28 A=26 B=18 so B is eliminated. Round 3 votes counts: A=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:234 C:219 D:201 B:175 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -18 -22 -20 B -2 0 -20 0 -28 C 18 20 0 4 -4 D 22 0 -4 0 -16 E 20 28 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991423 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -22 -20 B -2 0 -20 0 -28 C 18 20 0 4 -4 D 22 0 -4 0 -16 E 20 28 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991423 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -22 -20 B -2 0 -20 0 -28 C 18 20 0 4 -4 D 22 0 -4 0 -16 E 20 28 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991423 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 764: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (15) A B C E D (11) E C A B D (7) D E A B C (5) E D C A B (4) E A C B D (4) D C B E A (4) D B A C E (4) B A D C E (4) B A C D E (4) E D A C B (3) D E C A B (3) D B C A E (3) A E B C D (3) A C B E D (3) D E B C A (2) D E B A C (2) D C B A E (2) C A B E D (2) A B C D E (2) E D C B A (1) E C D A B (1) E C A D B (1) D E A C B (1) D C E B A (1) C E B A D (1) C E A B D (1) C D B E A (1) C B E D A (1) C B D A E (1) B C A D E (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -4 -8 -16 B -4 0 -14 -6 -10 C 4 14 0 -10 -4 D 8 6 10 0 12 E 16 10 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -8 -16 B -4 0 -14 -6 -10 C 4 14 0 -10 -4 D 8 6 10 0 12 E 16 10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 E=21 A=20 B=10 C=7 so C is eliminated. Round 2 votes counts: D=43 E=23 A=22 B=12 so B is eliminated. Round 3 votes counts: D=44 A=32 E=24 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:209 C:202 A:188 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -8 -16 B -4 0 -14 -6 -10 C 4 14 0 -10 -4 D 8 6 10 0 12 E 16 10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 -16 B -4 0 -14 -6 -10 C 4 14 0 -10 -4 D 8 6 10 0 12 E 16 10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 -16 B -4 0 -14 -6 -10 C 4 14 0 -10 -4 D 8 6 10 0 12 E 16 10 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 765: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (16) C E B D A (11) D B E C A (7) E C B D A (4) C E B A D (4) C B E D A (4) C B D E A (4) B D E C A (4) A D E B C (4) A C E D B (4) D B E A C (3) D B A E C (3) C A E B D (3) A E D B C (3) A D B C E (3) A C E B D (3) E D B C A (2) E B D C A (2) C A B E D (2) A C D B E (2) E C D B A (1) E C A D B (1) E B C D A (1) D E B A C (1) C E A B D (1) C B A E D (1) C B A D E (1) B E D C A (1) B D C E A (1) A D E C B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -10 2 -4 B 10 0 4 -2 8 C 10 -4 0 -4 -8 D -2 2 4 0 4 E 4 -8 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408187 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 2 -4 B 10 0 4 -2 8 C 10 -4 0 -4 -8 D -2 2 4 0 4 E 4 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.55102040808 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=31 D=14 E=11 B=6 so B is eliminated. Round 2 votes counts: A=38 C=31 D=19 E=12 so E is eliminated. Round 3 votes counts: C=38 A=38 D=24 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:210 D:204 E:200 C:197 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 2 -4 B 10 0 4 -2 8 C 10 -4 0 -4 -8 D -2 2 4 0 4 E 4 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.55102040808 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 2 -4 B 10 0 4 -2 8 C 10 -4 0 -4 -8 D -2 2 4 0 4 E 4 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.55102040808 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 2 -4 B 10 0 4 -2 8 C 10 -4 0 -4 -8 D -2 2 4 0 4 E 4 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.55102040808 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 766: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) D E C B A (7) A B C E D (7) E D B A C (6) E B D A C (3) D E C A B (3) D C E B A (3) C D B E A (3) C D A E B (3) C A B E D (3) A D E C B (3) A D C E B (3) E B A D C (2) D E B A C (2) D C E A B (2) C B D E A (2) C B D A E (2) C B A D E (2) C A D B E (2) B E D A C (2) B C A E D (2) A D E B C (2) A C D E B (2) A C B E D (2) E D B C A (1) E D A B C (1) E B D C A (1) E A D B C (1) E A B D C (1) D E B C A (1) D E A C B (1) D E A B C (1) D A E C B (1) C D E B A (1) C D E A B (1) C D B A E (1) C B E D A (1) C B A E D (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B A E D C (1) B A E C D (1) A E B D C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -2 -2 B 0 0 -16 -2 -6 C 8 16 0 -2 6 D 2 2 2 0 14 E 2 6 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -2 -2 B 0 0 -16 -2 -6 C 8 16 0 -2 6 D 2 2 2 0 14 E 2 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=22 D=21 E=16 B=11 so B is eliminated. Round 2 votes counts: C=33 A=24 E=22 D=21 so D is eliminated. Round 3 votes counts: C=38 E=37 A=25 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:210 A:194 E:194 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -2 -2 B 0 0 -16 -2 -6 C 8 16 0 -2 6 D 2 2 2 0 14 E 2 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 -2 B 0 0 -16 -2 -6 C 8 16 0 -2 6 D 2 2 2 0 14 E 2 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 -2 B 0 0 -16 -2 -6 C 8 16 0 -2 6 D 2 2 2 0 14 E 2 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 767: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) D E B A C (6) B C A E D (6) D E B C A (5) E D B C A (4) C A B E D (4) B E C D A (4) B A C E D (4) D E A C B (3) D E A B C (3) C B E D A (3) B E D C A (3) B C E A D (3) A D C E B (3) A C D E B (3) A C B E D (3) A B C E D (3) D A E B C (2) C E B D A (2) C A D E B (2) B E D A C (2) B C E D A (2) A D E C B (2) A C D B E (2) A C B D E (2) A B D E C (2) A B C D E (2) E C D B A (1) E C B D A (1) E B D C A (1) D E C A B (1) D A E C B (1) C D E A B (1) C B A E D (1) C A E B D (1) B A E C D (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -6 -6 -6 B 14 0 12 2 -2 C 6 -12 0 2 -6 D 6 -2 -2 0 0 E 6 2 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.261315 E: 0.738685 Sum of squares = 0.613941235201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.261315 E: 1.000000 A B C D E A 0 -14 -6 -6 -6 B 14 0 12 2 -2 C 6 -12 0 2 -6 D 6 -2 -2 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 0.500356 Sum of squares = 0.500000253038 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=25 A=25 C=14 E=7 so E is eliminated. Round 2 votes counts: D=33 B=26 A=25 C=16 so C is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:207 D:201 C:195 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -6 -6 -6 B 14 0 12 2 -2 C 6 -12 0 2 -6 D 6 -2 -2 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 0.500356 Sum of squares = 0.500000253038 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -6 -6 B 14 0 12 2 -2 C 6 -12 0 2 -6 D 6 -2 -2 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 0.500356 Sum of squares = 0.500000253038 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -6 -6 B 14 0 12 2 -2 C 6 -12 0 2 -6 D 6 -2 -2 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 0.500356 Sum of squares = 0.500000253038 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499644 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 768: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) C E D B A (8) A E C B D (7) A D B E C (7) D C B E A (5) C D E B A (5) A E B C D (5) D B C E A (4) C E A B D (4) A B D E C (4) E C A B D (3) D B C A E (3) D B A E C (3) A E B D C (3) E A B C D (2) D B A C E (2) D A B E C (2) C E B A D (2) B A E D C (2) B A D E C (2) A D E B C (2) E C B A D (1) E B A C D (1) E A C B D (1) D C A E B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E A B (1) C D B E A (1) B E D A C (1) B D E A C (1) B D C E A (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 14 18 14 14 B -14 0 12 0 0 C -18 -12 0 -12 -12 D -14 0 12 0 -4 E -14 0 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 14 14 B -14 0 12 0 0 C -18 -12 0 -12 -12 D -14 0 12 0 -4 E -14 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 D=23 C=23 E=8 B=7 so B is eliminated. Round 2 votes counts: A=43 D=25 C=23 E=9 so E is eliminated. Round 3 votes counts: A=47 C=27 D=26 so D is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:201 B:199 D:197 C:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 14 14 B -14 0 12 0 0 C -18 -12 0 -12 -12 D -14 0 12 0 -4 E -14 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 14 14 B -14 0 12 0 0 C -18 -12 0 -12 -12 D -14 0 12 0 -4 E -14 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 14 14 B -14 0 12 0 0 C -18 -12 0 -12 -12 D -14 0 12 0 -4 E -14 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 769: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D A C E B (7) B E C A D (6) B C E D A (6) C A E D B (5) D A E B C (4) C B E A D (4) B D E A C (4) D A C B E (3) C A D E B (3) A E D B C (3) A D E C B (3) E B A D C (2) E A C B D (2) D C B A E (2) D C A E B (2) D B A C E (2) C B E D A (2) B E D A C (2) B E C D A (2) B E A D C (2) B D C E A (2) B C D E A (2) A E D C B (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A C D (1) E A C D B (1) E A B D C (1) D C A B E (1) D B A E C (1) D A B E C (1) D A B C E (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D A E (1) B E A C D (1) B D A E C (1) B C E A D (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -6 2 -4 B 10 0 -8 6 -2 C 6 8 0 6 16 D -2 -6 -6 0 -10 E 4 2 -16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 2 -4 B 10 0 -8 6 -2 C 6 8 0 6 16 D -2 -6 -6 0 -10 E 4 2 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=26 D=24 A=11 E=10 so E is eliminated. Round 2 votes counts: B=33 C=28 D=24 A=15 so A is eliminated. Round 3 votes counts: B=34 D=33 C=33 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:203 E:200 A:191 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 2 -4 B 10 0 -8 6 -2 C 6 8 0 6 16 D -2 -6 -6 0 -10 E 4 2 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 2 -4 B 10 0 -8 6 -2 C 6 8 0 6 16 D -2 -6 -6 0 -10 E 4 2 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 2 -4 B 10 0 -8 6 -2 C 6 8 0 6 16 D -2 -6 -6 0 -10 E 4 2 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 770: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) A E B D C (8) D C E B A (6) B C D E A (6) B C D A E (6) E A D C B (5) D C E A B (5) C B D E A (4) A E D C B (4) A E D B C (4) D C B E A (3) B A E C D (3) B A C E D (3) E D A C B (2) E A D B C (2) D E A C B (2) D C A E B (2) C B D A E (2) B C E D A (2) B C A E D (2) A B E C D (2) A B C D E (2) E D C B A (1) E D C A B (1) E A B D C (1) E A B C D (1) D E C A B (1) D C A B E (1) D A E C B (1) B E C A D (1) B E A C D (1) B C E A D (1) B C A D E (1) B A C D E (1) A E B C D (1) A D E C B (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -8 -8 -8 B 4 0 -2 -2 2 C 8 2 0 -4 14 D 8 2 4 0 10 E 8 -2 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -8 -8 B 4 0 -2 -2 2 C 8 2 0 -4 14 D 8 2 4 0 10 E 8 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 D=21 C=15 E=13 so E is eliminated. Round 2 votes counts: A=33 B=27 D=25 C=15 so C is eliminated. Round 3 votes counts: D=34 B=33 A=33 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:210 B:201 E:191 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -8 -8 B 4 0 -2 -2 2 C 8 2 0 -4 14 D 8 2 4 0 10 E 8 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -8 -8 B 4 0 -2 -2 2 C 8 2 0 -4 14 D 8 2 4 0 10 E 8 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -8 -8 B 4 0 -2 -2 2 C 8 2 0 -4 14 D 8 2 4 0 10 E 8 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 771: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) E B C D A (8) A C D B E (8) B E D C A (7) E D B C A (5) E B D C A (5) C A E B D (5) D B A E C (4) E C B D A (3) E C A B D (3) D E B A C (3) D A B C E (3) B D E C A (3) A D B C E (3) A C B D E (3) E D C B A (2) E C B A D (2) D A C E B (2) C E A B D (2) C A E D B (2) B D E A C (2) E C D A B (1) E C A D B (1) E B C A D (1) D B A C E (1) D A E B C (1) C A B E D (1) B E C A D (1) B D A E C (1) B D A C E (1) B A D C E (1) B A C E D (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -4 -22 -16 B 20 0 22 4 4 C 4 -22 0 -10 -24 D 22 -4 10 0 -4 E 16 -4 24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998495 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -22 -16 B 20 0 22 4 4 C 4 -22 0 -10 -24 D 22 -4 10 0 -4 E 16 -4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995531 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=23 A=19 B=17 C=10 so C is eliminated. Round 2 votes counts: E=33 A=27 D=23 B=17 so B is eliminated. Round 3 votes counts: E=41 D=30 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:225 E:220 D:212 C:174 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -4 -22 -16 B 20 0 22 4 4 C 4 -22 0 -10 -24 D 22 -4 10 0 -4 E 16 -4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995531 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -22 -16 B 20 0 22 4 4 C 4 -22 0 -10 -24 D 22 -4 10 0 -4 E 16 -4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995531 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -22 -16 B 20 0 22 4 4 C 4 -22 0 -10 -24 D 22 -4 10 0 -4 E 16 -4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995531 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 772: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) B C E A D (8) E A C D B (5) A E D B C (5) A E B D C (5) E A B C D (4) D B A C E (4) C B E D A (4) C B E A D (4) E C A D B (3) E A C B D (3) D A B E C (3) C E B A D (3) C D E A B (3) C D B E A (3) A D E B C (3) E C B A D (2) D C A E B (2) D A E C B (2) D A B C E (2) C E B D A (2) B E A C D (2) B C D E A (2) B C D A E (2) E B C A D (1) D E A C B (1) D C B A E (1) D C A B E (1) D B A E C (1) D A C E B (1) C E A D B (1) C E A B D (1) C D B A E (1) B D A C E (1) B A E D C (1) B A D E C (1) A D E C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -6 10 -22 B 4 0 -6 14 4 C 6 6 0 26 10 D -10 -14 -26 0 -10 E 22 -4 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 10 -22 B 4 0 -6 14 4 C 6 6 0 26 10 D -10 -14 -26 0 -10 E 22 -4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=18 D=18 B=17 A=16 so A is eliminated. Round 2 votes counts: C=31 E=28 D=22 B=19 so B is eliminated. Round 3 votes counts: C=43 E=32 D=25 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:209 B:208 A:189 D:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 10 -22 B 4 0 -6 14 4 C 6 6 0 26 10 D -10 -14 -26 0 -10 E 22 -4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 10 -22 B 4 0 -6 14 4 C 6 6 0 26 10 D -10 -14 -26 0 -10 E 22 -4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 10 -22 B 4 0 -6 14 4 C 6 6 0 26 10 D -10 -14 -26 0 -10 E 22 -4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 773: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (15) C A E B D (9) D B E A C (8) A B D E C (7) D E B C A (6) A B D C E (6) E D B C A (5) D B E C A (5) C E A D B (5) C A E D B (5) B D E A C (5) A C B D E (4) D B A E C (3) A C E B D (3) E C D B A (2) B E D A C (2) A D B E C (2) E D C B A (1) D E C B A (1) C E B D A (1) B D A E C (1) A C D B E (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -10 -10 -12 B 10 0 4 -18 -6 C 10 -4 0 -6 2 D 10 18 6 0 0 E 12 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.890093 E: 0.109907 Sum of squares = 0.804344556101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.890093 E: 1.000000 A B C D E A 0 -10 -10 -10 -12 B 10 0 4 -18 -6 C 10 -4 0 -6 2 D 10 18 6 0 0 E 12 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=26 D=23 E=8 B=8 so E is eliminated. Round 2 votes counts: C=37 D=29 A=26 B=8 so B is eliminated. Round 3 votes counts: D=37 C=37 A=26 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:208 C:201 B:195 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 -10 -12 B 10 0 4 -18 -6 C 10 -4 0 -6 2 D 10 18 6 0 0 E 12 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -10 -12 B 10 0 4 -18 -6 C 10 -4 0 -6 2 D 10 18 6 0 0 E 12 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -10 -12 B 10 0 4 -18 -6 C 10 -4 0 -6 2 D 10 18 6 0 0 E 12 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 774: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (10) D B E C A (6) A D C E B (6) D E B C A (5) D B E A C (5) D A C E B (5) D A B E C (5) B E C A D (5) E B D C A (4) B E D C A (4) A C B E D (4) C E B A D (3) C A E B D (3) B E C D A (3) E C B D A (2) E C B A D (2) E B C D A (2) B D E C A (2) B C E A D (2) A C E D B (2) A C E B D (2) E D C B A (1) E C D A B (1) D E C B A (1) D E B A C (1) D E A C B (1) D B A E C (1) D A E C B (1) D A B C E (1) C E A B D (1) C B A E D (1) C A D E B (1) B D E A C (1) B A E C D (1) B A D E C (1) B A C E D (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 -4 -4 B 8 0 6 -8 -8 C -2 -6 0 -4 -10 D 4 8 4 0 10 E 4 8 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -4 -4 B 8 0 6 -8 -8 C -2 -6 0 -4 -10 D 4 8 4 0 10 E 4 8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=27 B=20 E=12 C=9 so C is eliminated. Round 2 votes counts: D=32 A=31 B=21 E=16 so E is eliminated. Round 3 votes counts: D=34 B=34 A=32 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:206 B:199 A:193 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -4 -4 B 8 0 6 -8 -8 C -2 -6 0 -4 -10 D 4 8 4 0 10 E 4 8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -4 -4 B 8 0 6 -8 -8 C -2 -6 0 -4 -10 D 4 8 4 0 10 E 4 8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -4 -4 B 8 0 6 -8 -8 C -2 -6 0 -4 -10 D 4 8 4 0 10 E 4 8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 775: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (13) E B C D A (8) D C A E B (8) A B D C E (7) C D E A B (6) B E A C D (6) E B A C D (5) D C A B E (5) D A C B E (5) B A E D C (5) C D A B E (3) B E A D C (3) E C D A B (2) E C B D A (2) E B C A D (2) E A B D C (2) D C E A B (2) D A C E B (2) C E D B A (2) B A E C D (2) B A D C E (2) B A C D E (2) E B A D C (1) C D E B A (1) C D B E A (1) C D B A E (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -12 -22 -12 B 12 0 -8 -4 -12 C 12 8 0 12 -4 D 22 4 -12 0 -8 E 12 12 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -12 -22 -12 B 12 0 -8 -4 -12 C 12 8 0 12 -4 D 22 4 -12 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=22 B=20 C=14 A=9 so A is eliminated. Round 2 votes counts: E=35 B=28 D=23 C=14 so C is eliminated. Round 3 votes counts: E=37 D=35 B=28 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:214 D:203 B:194 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -12 -22 -12 B 12 0 -8 -4 -12 C 12 8 0 12 -4 D 22 4 -12 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -22 -12 B 12 0 -8 -4 -12 C 12 8 0 12 -4 D 22 4 -12 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -22 -12 B 12 0 -8 -4 -12 C 12 8 0 12 -4 D 22 4 -12 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 776: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) A C E B D (7) C E A B D (6) A D B E C (6) E B C D A (5) D B E C A (5) D B A E C (5) C E B A D (5) E C B D A (4) B D E C A (4) A D B C E (4) A C E D B (4) A C D E B (4) D B E A C (3) D A B E C (3) C B E D A (3) C A E B D (3) B E D C A (3) E C A B D (2) D B C E A (2) D B A C E (2) D A B C E (2) C A D E B (2) B D C E A (2) A D C E B (2) E C B A D (1) D B C A E (1) A E C B D (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -10 -2 -4 B 4 0 -2 8 -10 C 10 2 0 8 12 D 2 -8 -8 0 -4 E 4 10 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -2 -4 B 4 0 -2 8 -10 C 10 2 0 8 12 D 2 -8 -8 0 -4 E 4 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 D=23 E=12 B=9 so B is eliminated. Round 2 votes counts: A=30 D=29 C=26 E=15 so E is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:203 B:200 D:191 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -2 -4 B 4 0 -2 8 -10 C 10 2 0 8 12 D 2 -8 -8 0 -4 E 4 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -2 -4 B 4 0 -2 8 -10 C 10 2 0 8 12 D 2 -8 -8 0 -4 E 4 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -2 -4 B 4 0 -2 8 -10 C 10 2 0 8 12 D 2 -8 -8 0 -4 E 4 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 777: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) A B C E D (7) A E B C D (6) E D A B C (5) B C A D E (5) E A D C B (4) E A D B C (4) D B C E A (4) C B D E A (4) B C D A E (4) E D A C B (3) C B D A E (3) C B A D E (3) B D C E A (3) B C D E A (3) A E D C B (3) A E C B D (3) A B E C D (3) E D C B A (2) E D C A B (2) D E C A B (2) C D B E A (2) C A B D E (2) B A C E D (2) A E B D C (2) E D B C A (1) D E B C A (1) D C E B A (1) D C B E A (1) C D E B A (1) C D A E B (1) A E D B C (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -10 -6 -6 B 0 0 2 4 -4 C 10 -2 0 2 -4 D 6 -4 -2 0 2 E 6 4 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999984 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 0 -10 -6 -6 B 0 0 2 4 -4 C 10 -2 0 2 -4 D 6 -4 -2 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=21 D=19 B=17 C=16 so C is eliminated. Round 2 votes counts: A=29 B=27 D=23 E=21 so E is eliminated. Round 3 votes counts: A=37 D=36 B=27 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:206 C:203 B:201 D:201 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 -6 -6 B 0 0 2 4 -4 C 10 -2 0 2 -4 D 6 -4 -2 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -6 -6 B 0 0 2 4 -4 C 10 -2 0 2 -4 D 6 -4 -2 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -6 -6 B 0 0 2 4 -4 C 10 -2 0 2 -4 D 6 -4 -2 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 778: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) E C B D A (7) E B C D A (7) D A C B E (7) B E C D A (6) A D E B C (5) D A C E B (4) A D C E B (4) A D B C E (4) C E B D A (3) B A E D C (3) A D E C B (3) E C D B A (2) E C D A B (2) E B A C D (2) E A D C B (2) E A B D C (2) C D B E A (2) B C E D A (2) A E D C B (2) A E D B C (2) A E B D C (2) E D A C B (1) E C B A D (1) E B C A D (1) E A B C D (1) C E D B A (1) C D E B A (1) C D E A B (1) C B E D A (1) C B D E A (1) B E A D C (1) B E A C D (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 16 0 2 B -6 0 -6 -4 -12 C -16 6 0 -6 -10 D 0 4 6 0 -8 E -2 12 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.961073 B: 0.000000 C: 0.000000 D: 0.038927 E: 0.000000 Sum of squares = 0.925176455745 Cumulative probabilities = A: 0.961073 B: 0.961073 C: 0.961073 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 0 2 B -6 0 -6 -4 -12 C -16 6 0 -6 -10 D 0 4 6 0 -8 E -2 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000469197 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=28 B=18 D=11 C=10 so C is eliminated. Round 2 votes counts: A=33 E=32 B=20 D=15 so D is eliminated. Round 3 votes counts: A=44 E=34 B=22 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:214 A:212 D:201 C:187 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 0 2 B -6 0 -6 -4 -12 C -16 6 0 -6 -10 D 0 4 6 0 -8 E -2 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000469197 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 0 2 B -6 0 -6 -4 -12 C -16 6 0 -6 -10 D 0 4 6 0 -8 E -2 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000469197 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 0 2 B -6 0 -6 -4 -12 C -16 6 0 -6 -10 D 0 4 6 0 -8 E -2 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000469197 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 779: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) B D A C E (7) B A E D C (6) B A D E C (6) E A B C D (5) D C B A E (5) D B C E A (5) E A C B D (4) D C E A B (4) D B C A E (4) A B E C D (4) E C A D B (3) D C B E A (3) B D A E C (3) B A E C D (3) A E C B D (3) A B D C E (3) D C A E B (2) D C A B E (2) D B A C E (2) C E D A B (2) C D E A B (2) B E A C D (2) E C D B A (1) E C B D A (1) D A B C E (1) C E A D B (1) C D A E B (1) C A E D B (1) B E A D C (1) B D E C A (1) A E B C D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -18 6 -10 12 B 18 0 8 0 18 C -6 -8 0 -28 12 D 10 0 28 0 24 E -12 -18 -12 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.498482 C: 0.000000 D: 0.501518 E: 0.000000 Sum of squares = 0.500004609323 Cumulative probabilities = A: 0.000000 B: 0.498482 C: 0.498482 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 -10 12 B 18 0 8 0 18 C -6 -8 0 -28 12 D 10 0 28 0 24 E -12 -18 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=29 E=14 A=13 C=7 so C is eliminated. Round 2 votes counts: D=40 B=29 E=17 A=14 so A is eliminated. Round 3 votes counts: D=42 B=36 E=22 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:231 B:222 A:195 C:185 E:167 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 -10 12 B 18 0 8 0 18 C -6 -8 0 -28 12 D 10 0 28 0 24 E -12 -18 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 -10 12 B 18 0 8 0 18 C -6 -8 0 -28 12 D 10 0 28 0 24 E -12 -18 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 -10 12 B 18 0 8 0 18 C -6 -8 0 -28 12 D 10 0 28 0 24 E -12 -18 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 780: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (14) C B D A E (8) E D A C B (5) E A D B C (5) D E C A B (5) B C A E D (5) B C A D E (5) C D B A E (4) B A C E D (4) A B E C D (4) D E A C B (3) D C E A B (3) D C A B E (3) B A E C D (3) A E B D C (3) A E B C D (3) E D C B A (2) E D C A B (2) E A B D C (2) E A B C D (2) D C B A E (2) C D B E A (2) B C E A D (2) B A C D E (2) A E D B C (2) E D A B C (1) E C B D A (1) E B A C D (1) D A C E B (1) B E A C D (1) Total count = 100 A B C D E A 0 -12 -16 -12 -2 B 12 0 -10 -8 -10 C 16 10 0 -6 10 D 12 8 6 0 4 E 2 10 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -12 -2 B 12 0 -10 -8 -10 C 16 10 0 -6 10 D 12 8 6 0 4 E 2 10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=22 E=21 C=14 A=12 so A is eliminated. Round 2 votes counts: D=31 E=29 B=26 C=14 so C is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:215 E:199 B:192 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -16 -12 -2 B 12 0 -10 -8 -10 C 16 10 0 -6 10 D 12 8 6 0 4 E 2 10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -12 -2 B 12 0 -10 -8 -10 C 16 10 0 -6 10 D 12 8 6 0 4 E 2 10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -12 -2 B 12 0 -10 -8 -10 C 16 10 0 -6 10 D 12 8 6 0 4 E 2 10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 781: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) D C A E B (6) E B C A D (5) D B A E C (5) B E A C D (5) A D C E B (5) D A C E B (4) D A C B E (4) B E D C A (4) B E C D A (4) A B D E C (4) E B A C D (3) C A E D B (3) B E C A D (3) A B E C D (3) E B C D A (2) E A B C D (2) D A B C E (2) C E B D A (2) C E B A D (2) C D E B A (2) B E A D C (2) A D C B E (2) A C D E B (2) E C B A D (1) D C B E A (1) D C B A E (1) D B E A C (1) D B A C E (1) C E A D B (1) C E A B D (1) C D E A B (1) C A E B D (1) C A D E B (1) B D E A C (1) B D A E C (1) A E C B D (1) A D B C E (1) A C E D B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 4 2 0 B 6 0 0 -2 -8 C -4 0 0 -6 4 D -2 2 6 0 4 E 0 8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000038 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 2 0 B 6 0 0 -2 -8 C -4 0 0 -6 4 D -2 2 6 0 4 E 0 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=21 B=20 C=14 E=13 so E is eliminated. Round 2 votes counts: D=32 B=30 A=23 C=15 so C is eliminated. Round 3 votes counts: D=35 B=35 A=30 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:205 A:200 E:200 B:198 C:197 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 2 0 B 6 0 0 -2 -8 C -4 0 0 -6 4 D -2 2 6 0 4 E 0 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 2 0 B 6 0 0 -2 -8 C -4 0 0 -6 4 D -2 2 6 0 4 E 0 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 2 0 B 6 0 0 -2 -8 C -4 0 0 -6 4 D -2 2 6 0 4 E 0 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 782: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) C B E A D (9) B C D A E (9) E C A B D (6) A E D C B (6) E A C D B (5) B C D E A (5) E A C B D (4) B D C A E (4) D E A C B (3) D A B E C (3) B C E D A (3) A D E C B (3) E A D C B (2) D B C E A (2) D B C A E (2) C E B A D (2) C B E D A (2) B D C E A (2) B D A C E (2) A D B E C (2) E C A D B (1) D C E B A (1) D B E C A (1) D B E A C (1) D B A E C (1) D B A C E (1) D A E C B (1) C E A B D (1) B C E A D (1) B C A D E (1) B A C D E (1) A E C B D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -4 -8 -2 B 0 0 6 8 6 C 4 -6 0 2 -2 D 8 -8 -2 0 14 E 2 -6 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.331953 B: 0.668047 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.556479799026 Cumulative probabilities = A: 0.331953 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -8 -2 B 0 0 6 8 6 C 4 -6 0 2 -2 D 8 -8 -2 0 14 E 2 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499989 B: 0.500011 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000228 Cumulative probabilities = A: 0.499989 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 E=18 C=14 A=13 so A is eliminated. Round 2 votes counts: D=32 B=29 E=25 C=14 so C is eliminated. Round 3 votes counts: B=40 D=32 E=28 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:206 C:199 A:193 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -8 -2 B 0 0 6 8 6 C 4 -6 0 2 -2 D 8 -8 -2 0 14 E 2 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499989 B: 0.500011 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000228 Cumulative probabilities = A: 0.499989 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -8 -2 B 0 0 6 8 6 C 4 -6 0 2 -2 D 8 -8 -2 0 14 E 2 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499989 B: 0.500011 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000228 Cumulative probabilities = A: 0.499989 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -8 -2 B 0 0 6 8 6 C 4 -6 0 2 -2 D 8 -8 -2 0 14 E 2 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499989 B: 0.500011 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000228 Cumulative probabilities = A: 0.499989 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 783: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D B C A E (7) B E A D C (6) B E A C D (6) B D E A C (5) A E C B D (5) A C E D B (5) D B A C E (4) C A D E B (4) E B D C A (3) E A C B D (3) E A B C D (3) D C A B E (3) C D A E B (3) B E D A C (3) E C A D B (2) E C A B D (2) D E B C A (2) D C B A E (2) D C A E B (2) D B C E A (2) C D E A B (2) C D A B E (2) C A E D B (2) B D A E C (2) B D A C E (2) E B A D C (1) D C E B A (1) D C E A B (1) D C B E A (1) C E A D B (1) B A E D C (1) B A D E C (1) A E C D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 16 6 -4 B 14 0 14 6 -4 C -16 -14 0 2 -8 D -6 -6 -2 0 -8 E 4 4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 16 6 -4 B 14 0 14 6 -4 C -16 -14 0 2 -8 D -6 -6 -2 0 -8 E 4 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 E=22 C=14 A=13 so A is eliminated. Round 2 votes counts: E=28 B=27 D=25 C=20 so C is eliminated. Round 3 votes counts: E=37 D=36 B=27 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:212 A:202 D:189 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 16 6 -4 B 14 0 14 6 -4 C -16 -14 0 2 -8 D -6 -6 -2 0 -8 E 4 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 16 6 -4 B 14 0 14 6 -4 C -16 -14 0 2 -8 D -6 -6 -2 0 -8 E 4 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 16 6 -4 B 14 0 14 6 -4 C -16 -14 0 2 -8 D -6 -6 -2 0 -8 E 4 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 784: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (6) B D C A E (5) B C D E A (5) A E C D B (5) C B E D A (4) B C A E D (4) E D A C B (3) D E C B A (3) D E A C B (3) D B E A C (3) D B C E A (3) C E A B D (3) C B E A D (3) B D C E A (3) B D A C E (3) A E D C B (3) E C D A B (2) E A D C B (2) E A C D B (2) C E D B A (2) C E B D A (2) C E B A D (2) C B D E A (2) C B A E D (2) B A D C E (2) B A C E D (2) A E D B C (2) A B C E D (2) E D C A B (1) D E C A B (1) D E A B C (1) D B A E C (1) D B A C E (1) C E D A B (1) C A E B D (1) B D A E C (1) B C D A E (1) B A C D E (1) A E B C D (1) A D E C B (1) A D E B C (1) A D B E C (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 0 -2 -2 B 10 0 -10 16 2 C 0 10 0 10 10 D 2 -16 -10 0 -14 E 2 -2 -10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.299664 B: 0.000000 C: 0.700336 D: 0.000000 E: 0.000000 Sum of squares = 0.580268835987 Cumulative probabilities = A: 0.299664 B: 0.299664 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -2 -2 B 10 0 -10 16 2 C 0 10 0 10 10 D 2 -16 -10 0 -14 E 2 -2 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499538 B: 0.000000 C: 0.500462 D: 0.000000 E: 0.000000 Sum of squares = 0.500000426464 Cumulative probabilities = A: 0.499538 B: 0.499538 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=25 C=22 D=16 E=10 so E is eliminated. Round 2 votes counts: A=29 B=27 C=24 D=20 so D is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:209 E:202 A:193 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 -2 -2 B 10 0 -10 16 2 C 0 10 0 10 10 D 2 -16 -10 0 -14 E 2 -2 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499538 B: 0.000000 C: 0.500462 D: 0.000000 E: 0.000000 Sum of squares = 0.500000426464 Cumulative probabilities = A: 0.499538 B: 0.499538 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -2 -2 B 10 0 -10 16 2 C 0 10 0 10 10 D 2 -16 -10 0 -14 E 2 -2 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499538 B: 0.000000 C: 0.500462 D: 0.000000 E: 0.000000 Sum of squares = 0.500000426464 Cumulative probabilities = A: 0.499538 B: 0.499538 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -2 -2 B 10 0 -10 16 2 C 0 10 0 10 10 D 2 -16 -10 0 -14 E 2 -2 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499538 B: 0.000000 C: 0.500462 D: 0.000000 E: 0.000000 Sum of squares = 0.500000426464 Cumulative probabilities = A: 0.499538 B: 0.499538 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 785: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) A B C E D (8) C B A E D (7) E A B C D (6) E D C A B (5) A B C D E (5) E D A B C (4) D B A C E (4) E D A C B (3) E A D B C (3) D C E B A (3) D C B A E (3) C D B A E (3) B C A D E (3) B A C D E (3) A B E C D (3) E D C B A (2) E C D B A (2) E C B A D (2) E C A B D (2) E A B D C (2) D E A C B (2) D E A B C (2) D A B E C (2) C E B A D (2) C B A D E (2) E C A D B (1) D E C A B (1) D B C A E (1) D A E B C (1) D A B C E (1) C B E A D (1) B C A E D (1) B A C E D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 2 6 -2 B -4 0 2 -2 -2 C -2 -2 0 4 -4 D -6 2 -4 0 -10 E 2 2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 6 -2 B -4 0 2 -2 -2 C -2 -2 0 4 -4 D -6 2 -4 0 -10 E 2 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=28 A=17 C=15 B=8 so B is eliminated. Round 2 votes counts: E=32 D=28 A=21 C=19 so C is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:209 A:205 C:198 B:197 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 6 -2 B -4 0 2 -2 -2 C -2 -2 0 4 -4 D -6 2 -4 0 -10 E 2 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 6 -2 B -4 0 2 -2 -2 C -2 -2 0 4 -4 D -6 2 -4 0 -10 E 2 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 6 -2 B -4 0 2 -2 -2 C -2 -2 0 4 -4 D -6 2 -4 0 -10 E 2 2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 786: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (6) D E A C B (5) D C A E B (5) D A E C B (5) C B D E A (5) D C E A B (4) A E D B C (4) A D E B C (4) C D B E A (3) C D A E B (3) C B E D A (3) C B A E D (3) B E C A D (3) B E A C D (3) B C A E D (3) B A E C D (3) B A C E D (3) A B E D C (3) E D A B C (2) E B A D C (2) E A D B C (2) E A B D C (2) D E C A B (2) D A C E B (2) C D E A B (2) C D A B E (2) C B A D E (2) B E A D C (2) A E B D C (2) E D B A C (1) E B D C A (1) E B D A C (1) D E A B C (1) D A E B C (1) C D E B A (1) C A B D E (1) B C E D A (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 0 0 -4 B -8 0 0 0 -6 C 0 0 0 -4 0 D 0 0 4 0 0 E 4 6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.475056 E: 0.524944 Sum of squares = 0.501244401947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.475056 E: 1.000000 A B C D E A 0 8 0 0 -4 B -8 0 0 0 -6 C 0 0 0 -4 0 D 0 0 4 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 B=24 A=15 E=11 so E is eliminated. Round 2 votes counts: D=28 B=28 C=25 A=19 so A is eliminated. Round 3 votes counts: D=39 B=35 C=26 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:205 A:202 D:202 C:198 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 0 -4 B -8 0 0 0 -6 C 0 0 0 -4 0 D 0 0 4 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 0 -4 B -8 0 0 0 -6 C 0 0 0 -4 0 D 0 0 4 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 0 -4 B -8 0 0 0 -6 C 0 0 0 -4 0 D 0 0 4 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 787: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (14) E C A B D (12) B D E A C (11) D B A C E (9) E C B A D (6) E A C D B (4) C E A D B (4) C A D E B (4) C A D B E (4) A C D E B (3) A C D B E (3) E B C D A (2) E A D C B (2) C D A B E (2) B E D C A (2) B E D A C (2) B D E C A (2) A E C D B (2) A D C B E (2) E C A D B (1) E A C B D (1) D B A E C (1) D A C B E (1) C E A B D (1) C D B A E (1) C A B D E (1) B D A E C (1) B D A C E (1) A C E D B (1) Total count = 100 A B C D E A 0 24 -12 30 0 B -24 0 -38 -16 -14 C 12 38 0 32 2 D -30 16 -32 0 -8 E 0 14 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 -12 30 0 B -24 0 -38 -16 -14 C 12 38 0 32 2 D -30 16 -32 0 -8 E 0 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999911428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=28 B=19 D=11 A=11 so D is eliminated. Round 2 votes counts: C=31 B=29 E=28 A=12 so A is eliminated. Round 3 votes counts: C=41 E=30 B=29 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:242 A:221 E:210 D:173 B:154 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -12 30 0 B -24 0 -38 -16 -14 C 12 38 0 32 2 D -30 16 -32 0 -8 E 0 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999911428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -12 30 0 B -24 0 -38 -16 -14 C 12 38 0 32 2 D -30 16 -32 0 -8 E 0 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999911428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -12 30 0 B -24 0 -38 -16 -14 C 12 38 0 32 2 D -30 16 -32 0 -8 E 0 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999911428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 788: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) E A C D B (7) B D E C A (7) A C E D B (6) A C D E B (6) E B D A C (5) D B C A E (5) C D A B E (5) B D C E A (5) B D C A E (5) A E C D B (4) A C D B E (4) E B A C D (3) D C B A E (3) D C A B E (3) E A C B D (2) C A D E B (2) C A D B E (2) B E D C A (2) E D B A C (1) E D A B C (1) E B A D C (1) E A B D C (1) E A B C D (1) D C E B A (1) D B C E A (1) C D A E B (1) C B D A E (1) C A B D E (1) B A E C D (1) B A C D E (1) A E C B D (1) A E B C D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -4 -8 8 B 0 0 -2 -4 -4 C 4 2 0 2 8 D 8 4 -2 0 8 E -8 4 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -8 8 B 0 0 -2 -4 -4 C 4 2 0 2 8 D 8 4 -2 0 8 E -8 4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=24 B=21 D=13 C=12 so C is eliminated. Round 2 votes counts: E=30 A=29 B=22 D=19 so D is eliminated. Round 3 votes counts: A=38 E=31 B=31 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:209 C:208 A:198 B:195 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -8 8 B 0 0 -2 -4 -4 C 4 2 0 2 8 D 8 4 -2 0 8 E -8 4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -8 8 B 0 0 -2 -4 -4 C 4 2 0 2 8 D 8 4 -2 0 8 E -8 4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -8 8 B 0 0 -2 -4 -4 C 4 2 0 2 8 D 8 4 -2 0 8 E -8 4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 789: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (5) C A E B D (5) C A D E B (5) E C B D A (4) D B A E C (4) D A B E C (4) D A B C E (4) C E D B A (4) C E B A D (4) A D B C E (4) A B D E C (4) A B D C E (4) E C D B A (3) D B E A C (3) C E A B D (3) B D E A C (3) A D C B E (3) A C D B E (3) E D C B A (2) E D B C A (2) E B C D A (2) B E D A C (2) B A D E C (2) A D B E C (2) A C B D E (2) A B C E D (2) E B C A D (1) D C E B A (1) D C A E B (1) D B E C A (1) D A C B E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E B A (1) C A E D B (1) B E D C A (1) B A E D C (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 6 2 10 10 B -6 0 -6 -2 4 C -2 6 0 0 6 D -10 2 0 0 4 E -10 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 10 10 B -6 0 -6 -2 4 C -2 6 0 0 6 D -10 2 0 0 4 E -10 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 E=19 D=19 B=9 so B is eliminated. Round 2 votes counts: A=30 C=26 E=22 D=22 so E is eliminated. Round 3 votes counts: C=41 A=30 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:205 D:198 B:195 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 10 10 B -6 0 -6 -2 4 C -2 6 0 0 6 D -10 2 0 0 4 E -10 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 10 10 B -6 0 -6 -2 4 C -2 6 0 0 6 D -10 2 0 0 4 E -10 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 10 10 B -6 0 -6 -2 4 C -2 6 0 0 6 D -10 2 0 0 4 E -10 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 790: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E D C B A (5) E A C B D (5) B D A C E (4) A B C D E (4) E C D A B (3) D B C E A (3) C D E B A (3) C B A D E (3) C A E B D (3) A E B D C (3) A C B D E (3) A B D C E (3) E D B A C (2) E C A D B (2) E C A B D (2) E A D C B (2) E A B D C (2) D B E A C (2) D B C A E (2) D B A E C (2) C E A D B (2) B A D E C (2) B A C D E (2) A E C B D (2) A C B E D (2) A B E C D (2) A B D E C (2) A B C E D (2) E D B C A (1) E D A B C (1) E C D B A (1) E A D B C (1) E A B C D (1) D E B C A (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E C A (1) C E D A B (1) C E A B D (1) C A B E D (1) C A B D E (1) B D C A E (1) B D A E C (1) B A D C E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 8 12 10 B -6 0 -4 24 4 C -8 4 0 10 4 D -12 -24 -10 0 2 E -10 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 12 10 B -6 0 -4 24 4 C -8 4 0 10 4 D -12 -24 -10 0 2 E -10 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 C=22 D=14 B=11 so B is eliminated. Round 2 votes counts: A=30 E=28 C=22 D=20 so D is eliminated. Round 3 votes counts: A=37 E=33 C=30 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:209 C:205 E:190 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 12 10 B -6 0 -4 24 4 C -8 4 0 10 4 D -12 -24 -10 0 2 E -10 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 12 10 B -6 0 -4 24 4 C -8 4 0 10 4 D -12 -24 -10 0 2 E -10 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 12 10 B -6 0 -4 24 4 C -8 4 0 10 4 D -12 -24 -10 0 2 E -10 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 791: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B A E D (6) D E A B C (5) C B D A E (5) B A D E C (5) D E A C B (4) C E A D B (4) A E B D C (4) A B E D C (4) D C E A B (3) C E A B D (3) C D B E A (3) B C A D E (3) B A E C D (3) A E B C D (3) D C E B A (2) C E D A B (2) C D E B A (2) C D E A B (2) B C A E D (2) B A E D C (2) B A C D E (2) A E D B C (2) A E C B D (2) A B E C D (2) E D C A B (1) E D A C B (1) E D A B C (1) E A D C B (1) E A C D B (1) E A C B D (1) D C B E A (1) D B C E A (1) D B A C E (1) D A E B C (1) C B E D A (1) B D A E C (1) B C D A E (1) B A D C E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 16 24 4 B -16 0 10 4 -8 C -16 -10 0 -4 -10 D -24 -4 4 0 -10 E -4 8 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 24 4 B -16 0 10 4 -8 C -16 -10 0 -4 -10 D -24 -4 4 0 -10 E -4 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=20 A=19 D=18 E=15 so E is eliminated. Round 2 votes counts: A=31 C=28 D=21 B=20 so B is eliminated. Round 3 votes counts: A=44 C=34 D=22 so D is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:212 B:195 D:183 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 24 4 B -16 0 10 4 -8 C -16 -10 0 -4 -10 D -24 -4 4 0 -10 E -4 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 24 4 B -16 0 10 4 -8 C -16 -10 0 -4 -10 D -24 -4 4 0 -10 E -4 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 24 4 B -16 0 10 4 -8 C -16 -10 0 -4 -10 D -24 -4 4 0 -10 E -4 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 792: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) D E A B C (8) A B C D E (8) E D C B A (6) E D A C B (6) E C A B D (4) D E B C A (4) D B A C E (4) C B A E D (4) E C B D A (3) E A C B D (3) D A B C E (3) B C A E D (3) B C A D E (3) A B D C E (3) A B C E D (3) E D C A B (2) E C D B A (2) E C B A D (2) D E B A C (2) D B A E C (2) D A B E C (2) C A B E D (2) A C B E D (2) D E C A B (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) D A E B C (1) C E B A D (1) C E A B D (1) C B E A D (1) B A C D E (1) A E C B D (1) Total count = 100 A B C D E A 0 16 16 -16 -14 B -16 0 2 -10 -10 C -16 -2 0 -12 -18 D 16 10 12 0 8 E 14 10 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 -16 -14 B -16 0 2 -10 -10 C -16 -2 0 -12 -18 D 16 10 12 0 8 E 14 10 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=28 A=17 C=9 B=7 so B is eliminated. Round 2 votes counts: D=39 E=28 A=18 C=15 so C is eliminated. Round 3 votes counts: D=39 E=31 A=30 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:217 A:201 B:183 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 16 -16 -14 B -16 0 2 -10 -10 C -16 -2 0 -12 -18 D 16 10 12 0 8 E 14 10 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 -16 -14 B -16 0 2 -10 -10 C -16 -2 0 -12 -18 D 16 10 12 0 8 E 14 10 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 -16 -14 B -16 0 2 -10 -10 C -16 -2 0 -12 -18 D 16 10 12 0 8 E 14 10 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 793: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D A C E B (9) D A C B E (5) D A B E C (5) C B E D A (5) B E C A D (5) A D E C B (5) A D E B C (5) D A B C E (4) C E B A D (4) A D B E C (4) E C B A D (3) E C A B D (3) C B D E A (3) A E B C D (3) E A C B D (2) E A B C D (2) D C A E B (2) D B C E A (2) D A E B C (2) C D E B A (2) E B C A D (1) D C B A E (1) D B C A E (1) D B A C E (1) C E B D A (1) C D B E A (1) C B E A D (1) B E A C D (1) B D C E A (1) B D A E C (1) B C E D A (1) A E D C B (1) A E C B D (1) A E B D C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 12 8 -12 6 B -12 0 -18 -20 4 C -8 18 0 -22 8 D 12 20 22 0 30 E -6 -4 -8 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -12 6 B -12 0 -18 -20 4 C -8 18 0 -22 8 D 12 20 22 0 30 E -6 -4 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=22 C=17 E=11 B=9 so B is eliminated. Round 2 votes counts: D=43 A=22 C=18 E=17 so E is eliminated. Round 3 votes counts: D=43 C=30 A=27 so A is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:242 A:207 C:198 B:177 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -12 6 B -12 0 -18 -20 4 C -8 18 0 -22 8 D 12 20 22 0 30 E -6 -4 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -12 6 B -12 0 -18 -20 4 C -8 18 0 -22 8 D 12 20 22 0 30 E -6 -4 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -12 6 B -12 0 -18 -20 4 C -8 18 0 -22 8 D 12 20 22 0 30 E -6 -4 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 794: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) D E B C A (7) C A D B E (7) D B E C A (5) B E D A C (5) D C E B A (4) C D A B E (4) C A D E B (4) A C B E D (4) E B A D C (3) D C B E A (3) B E A D C (3) B D E A C (3) A C D B E (3) E D C B A (2) E B D C A (2) E B D A C (2) D B C E A (2) C D A E B (2) B E D C A (2) B D E C A (2) A E B C D (2) A C E D B (2) A C E B D (2) A B E D C (2) A B D C E (2) A B C E D (2) E C D B A (1) E A C B D (1) E A B D C (1) D E C B A (1) C E D B A (1) C E D A B (1) C D E A B (1) C A E D B (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -20 -16 -20 B 12 0 12 -18 4 C 20 -12 0 -20 -8 D 16 18 20 0 4 E 20 -4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -20 -16 -20 B 12 0 12 -18 4 C 20 -12 0 -20 -8 D 16 18 20 0 4 E 20 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=22 A=22 C=21 E=20 B=15 so B is eliminated. Round 2 votes counts: E=30 D=27 A=22 C=21 so C is eliminated. Round 3 votes counts: D=34 A=34 E=32 so E is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:210 B:205 C:190 A:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -20 -16 -20 B 12 0 12 -18 4 C 20 -12 0 -20 -8 D 16 18 20 0 4 E 20 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -16 -20 B 12 0 12 -18 4 C 20 -12 0 -20 -8 D 16 18 20 0 4 E 20 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -16 -20 B 12 0 12 -18 4 C 20 -12 0 -20 -8 D 16 18 20 0 4 E 20 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 795: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D B C A E (11) C B D A E (9) E A C D B (8) B D C A E (7) D B A C E (5) C A B D E (5) E D B C A (4) E A D B C (4) D B E C A (4) C B A D E (4) E C A B D (3) E D B A C (2) A E C B D (2) A C E B D (2) A C B D E (2) E D A B C (1) E C B D A (1) D E B A C (1) D B E A C (1) D B A E C (1) D A B C E (1) C E B D A (1) C E A B D (1) C B D E A (1) C A B E D (1) B D C E A (1) B C D A E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -8 -4 8 B 8 0 -8 10 14 C 8 8 0 10 10 D 4 -10 -10 0 14 E -8 -14 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -4 8 B 8 0 -8 10 14 C 8 8 0 10 10 D 4 -10 -10 0 14 E -8 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=24 C=22 B=9 A=9 so B is eliminated. Round 2 votes counts: E=36 D=32 C=23 A=9 so A is eliminated. Round 3 votes counts: E=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:218 B:212 D:199 A:194 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 8 B 8 0 -8 10 14 C 8 8 0 10 10 D 4 -10 -10 0 14 E -8 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 8 B 8 0 -8 10 14 C 8 8 0 10 10 D 4 -10 -10 0 14 E -8 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 8 B 8 0 -8 10 14 C 8 8 0 10 10 D 4 -10 -10 0 14 E -8 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 796: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) B A C E D (7) D B C A E (5) C E A B D (5) E C A B D (4) D E C A B (4) D B A E C (4) B A E C D (4) A B E C D (4) E A C B D (3) D E A C B (3) D E A B C (3) E D A C B (2) E A B D C (2) E A B C D (2) D E B C A (2) D E B A C (2) D C E B A (2) D C E A B (2) D C B A E (2) D B E A C (2) C E D A B (2) C E A D B (2) C D B A E (2) C A E B D (2) B C A E D (2) B C A D E (2) B A D E C (2) E D C A B (1) E C A D B (1) E A D C B (1) D B A C E (1) C D E B A (1) C D E A B (1) C B D A E (1) C A B E D (1) B D A E C (1) B D A C E (1) B A E D C (1) A E C B D (1) A E B C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 12 6 B 2 0 -2 10 2 C 2 2 0 14 0 D -12 -10 -14 0 -14 E -6 -2 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.869304 D: 0.000000 E: 0.130696 Sum of squares = 0.7727712272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.869304 D: 0.869304 E: 1.000000 A B C D E A 0 -2 -2 12 6 B 2 0 -2 10 2 C 2 2 0 14 0 D -12 -10 -14 0 -14 E -6 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000045083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=24 B=20 E=16 A=8 so A is eliminated. Round 2 votes counts: D=32 C=25 B=25 E=18 so E is eliminated. Round 3 votes counts: D=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:207 B:206 E:203 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 12 6 B 2 0 -2 10 2 C 2 2 0 14 0 D -12 -10 -14 0 -14 E -6 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000045083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 12 6 B 2 0 -2 10 2 C 2 2 0 14 0 D -12 -10 -14 0 -14 E -6 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000045083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 12 6 B 2 0 -2 10 2 C 2 2 0 14 0 D -12 -10 -14 0 -14 E -6 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000045083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 797: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) B D E C A (9) A D E B C (8) E C A B D (5) B D C E A (5) E B C D A (4) E A C D B (4) D B A E C (3) C A B D E (3) B D C A E (3) B C E D A (3) B C D E A (3) E D B A C (2) E B D C A (2) E A D B C (2) D B E A C (2) D A B E C (2) D A B C E (2) C E A B D (2) C B D A E (2) C A D B E (2) A C E D B (2) E D A B C (1) E C B D A (1) E B D A C (1) D E A B C (1) D B C A E (1) C E B D A (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D E A (1) C A E D B (1) C A E B D (1) B C D A E (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E C B (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -6 -20 -4 B 12 0 34 0 14 C 6 -34 0 -18 -4 D 20 0 18 0 24 E 4 -14 4 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.516371 C: 0.000000 D: 0.483629 E: 0.000000 Sum of squares = 0.500536018005 Cumulative probabilities = A: 0.000000 B: 0.516371 C: 0.516371 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -20 -4 B 12 0 34 0 14 C 6 -34 0 -18 -4 D 20 0 18 0 24 E 4 -14 4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=22 D=20 A=18 C=16 so C is eliminated. Round 2 votes counts: B=29 E=26 A=25 D=20 so D is eliminated. Round 3 votes counts: B=44 A=29 E=27 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:231 B:230 E:185 A:179 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -20 -4 B 12 0 34 0 14 C 6 -34 0 -18 -4 D 20 0 18 0 24 E 4 -14 4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -20 -4 B 12 0 34 0 14 C 6 -34 0 -18 -4 D 20 0 18 0 24 E 4 -14 4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -20 -4 B 12 0 34 0 14 C 6 -34 0 -18 -4 D 20 0 18 0 24 E 4 -14 4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 798: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (14) B D E A C (13) C A E D B (8) A C E D B (8) C D E B A (5) A E B D C (5) A C B E D (5) D E B C A (4) C A B D E (4) C D B E A (3) E D B C A (2) D B E C A (2) C E D A B (2) C D E A B (2) B C D E A (2) A C E B D (2) A B E D C (2) E D C B A (1) E D B A C (1) D E C B A (1) D B C E A (1) C E A D B (1) C B D E A (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D C E A (1) B D C A E (1) B C D A E (1) B A D E C (1) A E D C B (1) A E D B C (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -18 -16 -14 B 12 0 4 10 10 C 18 -4 0 -4 2 D 16 -10 4 0 20 E 14 -10 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -16 -14 B 12 0 4 10 10 C 18 -4 0 -4 2 D 16 -10 4 0 20 E 14 -10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=29 A=25 D=8 E=4 so E is eliminated. Round 2 votes counts: B=34 C=29 A=25 D=12 so D is eliminated. Round 3 votes counts: B=44 C=31 A=25 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:215 C:206 E:191 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -18 -16 -14 B 12 0 4 10 10 C 18 -4 0 -4 2 D 16 -10 4 0 20 E 14 -10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -16 -14 B 12 0 4 10 10 C 18 -4 0 -4 2 D 16 -10 4 0 20 E 14 -10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -16 -14 B 12 0 4 10 10 C 18 -4 0 -4 2 D 16 -10 4 0 20 E 14 -10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 799: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (6) E B D A C (5) A E B D C (5) E C A D B (4) E A B D C (4) D C B E A (4) D B E C A (4) D B C A E (4) C D B A E (4) E D B C A (3) E D B A C (3) E A C B D (3) C E D B A (3) C D B E A (3) B D A C E (3) B A D C E (3) A B E D C (3) E C D A B (2) E B A D C (2) D B A C E (2) C E A D B (2) B E D A C (2) B D E A C (2) A E C B D (2) A E B C D (2) A C E B D (2) E C A B D (1) E A D B C (1) E A C D B (1) E A B C D (1) D E B C A (1) D C E B A (1) D B C E A (1) C D E B A (1) C D A B E (1) C B D A E (1) C A E D B (1) C A D B E (1) C A B D E (1) B D C A E (1) B D A E C (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 12 -4 -8 B 8 0 22 6 0 C -12 -22 0 -24 -6 D 4 -6 24 0 -4 E 8 0 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.415917 C: 0.000000 D: 0.000000 E: 0.584083 Sum of squares = 0.514139747237 Cumulative probabilities = A: 0.000000 B: 0.415917 C: 0.415917 D: 0.415917 E: 1.000000 A B C D E A 0 -8 12 -4 -8 B 8 0 22 6 0 C -12 -22 0 -24 -6 D 4 -6 24 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=23 C=18 D=17 B=12 so B is eliminated. Round 2 votes counts: E=32 A=26 D=24 C=18 so C is eliminated. Round 3 votes counts: E=37 D=34 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:218 D:209 E:209 A:196 C:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 -4 -8 B 8 0 22 6 0 C -12 -22 0 -24 -6 D 4 -6 24 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 -4 -8 B 8 0 22 6 0 C -12 -22 0 -24 -6 D 4 -6 24 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 -4 -8 B 8 0 22 6 0 C -12 -22 0 -24 -6 D 4 -6 24 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 800: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) D B A C E (8) B D A C E (7) E C A B D (6) D A C E B (6) B D E C A (6) E B C A D (5) D A C B E (5) B E C A D (5) A C D E B (5) C A E D B (4) A C E D B (4) E C B A D (3) E C A D B (3) C E A D B (3) A C D B E (3) E B D C A (2) D B E A C (2) C A E B D (2) B D E A C (2) E D C B A (1) E D C A B (1) E B C D A (1) D E C A B (1) D C A E B (1) C E A B D (1) B E C D A (1) Total count = 100 A B C D E A 0 -10 -16 -12 -10 B 10 0 2 6 2 C 16 -2 0 -8 -2 D 12 -6 8 0 -8 E 10 -2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -12 -10 B 10 0 2 6 2 C 16 -2 0 -8 -2 D 12 -6 8 0 -8 E 10 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998538 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=23 E=22 A=12 C=10 so C is eliminated. Round 2 votes counts: B=33 E=26 D=23 A=18 so A is eliminated. Round 3 votes counts: E=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 E:209 D:203 C:202 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -16 -12 -10 B 10 0 2 6 2 C 16 -2 0 -8 -2 D 12 -6 8 0 -8 E 10 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998538 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -12 -10 B 10 0 2 6 2 C 16 -2 0 -8 -2 D 12 -6 8 0 -8 E 10 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998538 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -12 -10 B 10 0 2 6 2 C 16 -2 0 -8 -2 D 12 -6 8 0 -8 E 10 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998538 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 801: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (16) D A B C E (9) A C E D B (7) E C B A D (6) C E A B D (6) E B C A D (5) D B A E C (5) D B A C E (5) B D E C A (4) D A E C B (3) C E B A D (3) B E D C A (3) B D A C E (3) D B E A C (2) D A C B E (2) C A E B D (2) A D C E B (2) A C D E B (2) E C A D B (1) E C A B D (1) D E B C A (1) D A E B C (1) D A C E B (1) C A E D B (1) C A B E D (1) B E C A D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A C D E (1) A E C D B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -6 -12 0 B 16 0 18 12 12 C 6 -18 0 14 -2 D 12 -12 -14 0 -14 E 0 -12 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -12 0 B 16 0 18 12 12 C 6 -18 0 14 -2 D 12 -12 -14 0 -14 E 0 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 A=14 E=13 C=13 so E is eliminated. Round 2 votes counts: B=36 D=29 C=21 A=14 so A is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 E:202 C:200 D:186 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -12 0 B 16 0 18 12 12 C 6 -18 0 14 -2 D 12 -12 -14 0 -14 E 0 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -12 0 B 16 0 18 12 12 C 6 -18 0 14 -2 D 12 -12 -14 0 -14 E 0 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -12 0 B 16 0 18 12 12 C 6 -18 0 14 -2 D 12 -12 -14 0 -14 E 0 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 802: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) C E D A B (7) C D B A E (7) A B E D C (7) E A B D C (5) B A D E C (5) C E A D B (4) B D A E C (4) D C E B A (3) D B A E C (3) C D E A B (3) E D B A C (2) E A C B D (2) D B E A C (2) D B C E A (2) D B C A E (2) C E A B D (2) C D B E A (2) C B A D E (2) B A E D C (2) A B E C D (2) A B C E D (2) E D C B A (1) E C D A B (1) E C A D B (1) E C A B D (1) E A B C D (1) D E B C A (1) D C B E A (1) D C B A E (1) D B E C A (1) C B D A E (1) C A E B D (1) C A D E B (1) B D E A C (1) B A D C E (1) A E C B D (1) A E B D C (1) A E B C D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -12 -12 -8 B 10 0 -10 -12 -2 C 12 10 0 10 10 D 12 12 -10 0 8 E 8 2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -12 -8 B 10 0 -10 -12 -2 C 12 10 0 10 10 D 12 12 -10 0 8 E 8 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 D=16 A=16 E=14 B=13 so B is eliminated. Round 2 votes counts: C=41 A=24 D=21 E=14 so E is eliminated. Round 3 votes counts: C=44 A=32 D=24 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:211 E:196 B:193 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -12 -8 B 10 0 -10 -12 -2 C 12 10 0 10 10 D 12 12 -10 0 8 E 8 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -12 -8 B 10 0 -10 -12 -2 C 12 10 0 10 10 D 12 12 -10 0 8 E 8 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -12 -8 B 10 0 -10 -12 -2 C 12 10 0 10 10 D 12 12 -10 0 8 E 8 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 803: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) E D C A B (5) B A C D E (5) E D B C A (4) E C D A B (4) C E A D B (4) B A D E C (4) A C B D E (4) E D C B A (3) C A B D E (3) B E A D C (3) B C A E D (3) A D B C E (3) A B C D E (3) E B D C A (2) E B C D A (2) D E C A B (2) D C A E B (2) D A C E B (2) C E D A B (2) C E B A D (2) C A D E B (2) C A D B E (2) B A C E D (2) A D C B E (2) A C D B E (2) E D B A C (1) E C D B A (1) E C B D A (1) E B C A D (1) D E A C B (1) D C E A B (1) D B A E C (1) D A E C B (1) D A E B C (1) C D E A B (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) C A B E D (1) B E C A D (1) B D A E C (1) B A E C D (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -6 8 0 B -2 0 -4 -12 -10 C 6 4 0 2 4 D -8 12 -2 0 6 E 0 10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 8 0 B -2 0 -4 -12 -10 C 6 4 0 2 4 D -8 12 -2 0 6 E 0 10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=21 B=21 D=19 A=15 so A is eliminated. Round 2 votes counts: C=27 B=25 E=24 D=24 so E is eliminated. Round 3 votes counts: D=37 C=33 B=30 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:208 D:204 A:202 E:200 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 8 0 B -2 0 -4 -12 -10 C 6 4 0 2 4 D -8 12 -2 0 6 E 0 10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 8 0 B -2 0 -4 -12 -10 C 6 4 0 2 4 D -8 12 -2 0 6 E 0 10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 8 0 B -2 0 -4 -12 -10 C 6 4 0 2 4 D -8 12 -2 0 6 E 0 10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 804: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) B E C A D (9) B E A D C (9) D A C E B (7) E A D B C (5) C B E D A (5) C B D A E (5) A D E C B (5) A D E B C (5) E B A D C (4) C E D A B (4) B C E D A (4) D A E C B (3) C D A B E (3) B A D E C (3) E B C A D (2) B C A D E (2) E D A C B (1) E C D B A (1) E C D A B (1) E C B D A (1) E A D C B (1) D E A C B (1) C E D B A (1) C E B D A (1) C D E A B (1) B E A C D (1) B A D C E (1) A E D B C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 0 -4 B -2 0 -6 -4 -12 C 2 6 0 4 -6 D 0 4 -4 0 -2 E 4 12 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 0 -4 B -2 0 -6 -4 -12 C 2 6 0 4 -6 D 0 4 -4 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=29 E=16 A=13 D=11 so D is eliminated. Round 2 votes counts: C=31 B=29 A=23 E=17 so E is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:203 D:199 A:198 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 0 -4 B -2 0 -6 -4 -12 C 2 6 0 4 -6 D 0 4 -4 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -4 B -2 0 -6 -4 -12 C 2 6 0 4 -6 D 0 4 -4 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -4 B -2 0 -6 -4 -12 C 2 6 0 4 -6 D 0 4 -4 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 805: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (20) A B D E C (11) C E D B A (9) D E C A B (8) D A E C B (6) B C E D A (5) A D B E C (5) D E C B A (4) C E B D A (4) B C E A D (4) C B E A D (3) A B C E D (3) E C D B A (2) D A E B C (2) C E D A B (2) B A D C E (2) B A C D E (2) A D E C B (2) A B D C E (2) E D C A B (1) B D A E C (1) B C A E D (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 14 12 16 B 14 0 18 16 18 C -14 -18 0 10 14 D -12 -16 -10 0 -8 E -16 -18 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 14 12 16 B 14 0 18 16 18 C -14 -18 0 10 14 D -12 -16 -10 0 -8 E -16 -18 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=24 D=20 C=18 E=3 so E is eliminated. Round 2 votes counts: B=35 A=24 D=21 C=20 so C is eliminated. Round 3 votes counts: B=42 D=34 A=24 so A is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:233 A:214 C:196 E:180 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 14 12 16 B 14 0 18 16 18 C -14 -18 0 10 14 D -12 -16 -10 0 -8 E -16 -18 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 14 12 16 B 14 0 18 16 18 C -14 -18 0 10 14 D -12 -16 -10 0 -8 E -16 -18 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 14 12 16 B 14 0 18 16 18 C -14 -18 0 10 14 D -12 -16 -10 0 -8 E -16 -18 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 806: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) A D E B C (10) A E D B C (7) E A D B C (4) C E B A D (4) C B D E A (4) B D E A C (4) E B D A C (3) D B A E C (3) C E A B D (3) C B D A E (3) C A D B E (3) B E D C A (3) E A B D C (2) D B C A E (2) C A E D B (2) C A E B D (2) C A D E B (2) B D E C A (2) B C D E A (2) E C B D A (1) E C B A D (1) E C A B D (1) E A C D B (1) E A C B D (1) E A B C D (1) D B E A C (1) D B A C E (1) D A E B C (1) D A B E C (1) C E A D B (1) C D A B E (1) B E D A C (1) B E C D A (1) B D C E A (1) B D C A E (1) B C E D A (1) A E D C B (1) A E C D B (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -6 2 -10 B 2 0 6 8 -2 C 6 -6 0 -2 -4 D -2 -8 2 0 -8 E 10 2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 2 -10 B 2 0 6 8 -2 C 6 -6 0 -2 -4 D -2 -8 2 0 -8 E 10 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=23 B=16 E=15 D=9 so D is eliminated. Round 2 votes counts: C=37 A=25 B=23 E=15 so E is eliminated. Round 3 votes counts: C=40 A=34 B=26 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 B:207 C:197 A:192 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 2 -10 B 2 0 6 8 -2 C 6 -6 0 -2 -4 D -2 -8 2 0 -8 E 10 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 -10 B 2 0 6 8 -2 C 6 -6 0 -2 -4 D -2 -8 2 0 -8 E 10 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 -10 B 2 0 6 8 -2 C 6 -6 0 -2 -4 D -2 -8 2 0 -8 E 10 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 807: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (18) E D C A B (16) C A B E D (14) D E B A C (9) E C A D B (5) B A C E D (5) B C A E D (4) A C B E D (4) D B E A C (3) C B A E D (3) D E C A B (2) D E A C B (2) D E A B C (2) C E A B D (2) C A E B D (2) B D A C E (2) B A D C E (2) E D C B A (1) C A E D B (1) A C E D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 0 26 16 B -6 0 -6 16 14 C 0 6 0 22 20 D -26 -16 -22 0 -18 E -16 -14 -20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.651034 B: 0.000000 C: 0.348966 D: 0.000000 E: 0.000000 Sum of squares = 0.545622818952 Cumulative probabilities = A: 0.651034 B: 0.651034 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 26 16 B -6 0 -6 16 14 C 0 6 0 22 20 D -26 -16 -22 0 -18 E -16 -14 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999515 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=22 C=22 D=18 A=7 so A is eliminated. Round 2 votes counts: B=33 C=27 E=22 D=18 so D is eliminated. Round 3 votes counts: E=37 B=36 C=27 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:224 C:224 B:209 E:184 D:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 26 16 B -6 0 -6 16 14 C 0 6 0 22 20 D -26 -16 -22 0 -18 E -16 -14 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999515 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 26 16 B -6 0 -6 16 14 C 0 6 0 22 20 D -26 -16 -22 0 -18 E -16 -14 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999515 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 26 16 B -6 0 -6 16 14 C 0 6 0 22 20 D -26 -16 -22 0 -18 E -16 -14 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999515 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 808: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) C B A D E (8) E A D B C (7) A E C B D (7) D B E C A (6) A E C D B (5) C A B E D (4) A C E B D (4) E D B A C (3) E A D C B (3) D E B A C (3) C A E D B (3) C A B D E (3) B C D A E (3) E D A C B (2) E D A B C (2) D B C E A (2) C B D A E (2) B D C E A (2) B D C A E (2) B C A D E (2) A E D B C (2) A E B D C (2) E D C A B (1) E A C D B (1) D E B C A (1) D C B E A (1) C D E B A (1) C D B E A (1) C A E B D (1) C A D B E (1) B D E C A (1) B D E A C (1) B D A C E (1) B C D E A (1) B A D E C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 8 12 6 B 0 0 2 -8 2 C -8 -2 0 -4 -14 D -12 8 4 0 2 E -6 -2 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.570618 B: 0.429382 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.509973832269 Cumulative probabilities = A: 0.570618 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 12 6 B 0 0 2 -8 2 C -8 -2 0 -4 -14 D -12 8 4 0 2 E -6 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=22 D=21 E=19 B=14 so B is eliminated. Round 2 votes counts: C=30 D=28 A=23 E=19 so E is eliminated. Round 3 votes counts: D=36 A=34 C=30 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:202 D:201 B:198 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 12 6 B 0 0 2 -8 2 C -8 -2 0 -4 -14 D -12 8 4 0 2 E -6 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 12 6 B 0 0 2 -8 2 C -8 -2 0 -4 -14 D -12 8 4 0 2 E -6 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 12 6 B 0 0 2 -8 2 C -8 -2 0 -4 -14 D -12 8 4 0 2 E -6 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 809: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (7) A E D B C (7) E D A B C (5) C B E A D (4) A C E B D (4) D E B A C (3) D A E B C (3) D A B C E (3) B D E C A (3) B D C E A (3) A E C D B (3) A E C B D (3) A D E B C (3) A D C B E (3) E D B A C (2) E B D C A (2) E A D B C (2) E A C B D (2) D B E C A (2) D B C E A (2) D B C A E (2) C D B A E (2) C B D A E (2) C B A D E (2) C A B E D (2) C A B D E (2) B E C D A (2) B C E D A (2) A C D B E (2) E D B C A (1) E C B A D (1) E C A B D (1) E B C A D (1) E B A D C (1) D E B C A (1) D B E A C (1) D A C B E (1) D A B E C (1) C E B A D (1) C A E B D (1) B E D C A (1) B C D E A (1) A E D C B (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 4 -4 -2 B -2 0 8 -4 0 C -4 -8 0 -10 -6 D 4 4 10 0 2 E 2 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -4 -2 B -2 0 8 -4 0 C -4 -8 0 -10 -6 D 4 4 10 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=23 D=19 E=18 B=12 so B is eliminated. Round 2 votes counts: A=28 C=26 D=25 E=21 so E is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:203 B:201 A:200 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -4 -2 B -2 0 8 -4 0 C -4 -8 0 -10 -6 D 4 4 10 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -4 -2 B -2 0 8 -4 0 C -4 -8 0 -10 -6 D 4 4 10 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -4 -2 B -2 0 8 -4 0 C -4 -8 0 -10 -6 D 4 4 10 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 810: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) A B C E D (8) D C A E B (6) B E A C D (6) E B D C A (5) C A D E B (4) E C D B A (3) E C A B D (3) D B E A C (3) D A B C E (3) C D A E B (3) C A E B D (3) B A D C E (3) E D C B A (2) E C B A D (2) E B C A D (2) D E C B A (2) D E B C A (2) D C A B E (2) C E A D B (2) C E A B D (2) B E D A C (2) B E A D C (2) B D E A C (2) B A E D C (2) B A D E C (2) A C B D E (2) A B D C E (2) E D C A B (1) E D B C A (1) E B A C D (1) D E C A B (1) D C E A B (1) D A C B E (1) C A E D B (1) B D A E C (1) B A C E D (1) A D C B E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 4 18 10 B 4 0 16 20 6 C -4 -16 0 4 -6 D -18 -20 -4 0 -14 E -10 -6 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999371 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 18 10 B 4 0 16 20 6 C -4 -16 0 4 -6 D -18 -20 -4 0 -14 E -10 -6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=21 E=20 C=15 A=15 so C is eliminated. Round 2 votes counts: B=29 E=24 D=24 A=23 so A is eliminated. Round 3 votes counts: B=42 D=30 E=28 so E is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:214 E:202 C:189 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 18 10 B 4 0 16 20 6 C -4 -16 0 4 -6 D -18 -20 -4 0 -14 E -10 -6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 18 10 B 4 0 16 20 6 C -4 -16 0 4 -6 D -18 -20 -4 0 -14 E -10 -6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 18 10 B 4 0 16 20 6 C -4 -16 0 4 -6 D -18 -20 -4 0 -14 E -10 -6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 811: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) D A C B E (5) C D A B E (5) B D A C E (5) E C A D B (4) E B A D C (4) E B A C D (4) E A C D B (4) C A D E B (4) B E D A C (4) B E C D A (4) A D C E B (4) E B C A D (3) B D C A E (3) B C D E A (3) B A E D C (3) D C A B E (2) D B C A E (2) C D A E B (2) A E C D B (2) A D E C B (2) E C D A B (1) E C B D A (1) E B C D A (1) E A C B D (1) E A B D C (1) D C B A E (1) D C A E B (1) C E D B A (1) C E D A B (1) C E A D B (1) C D B A E (1) B E D C A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E D A (1) B A D E C (1) A E B D C (1) A D C B E (1) A D B C E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 10 4 -4 B 10 0 10 8 12 C -10 -10 0 -10 -8 D -4 -8 10 0 -6 E 4 -12 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 4 -4 B 10 0 10 8 12 C -10 -10 0 -10 -8 D -4 -8 10 0 -6 E 4 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=24 C=15 A=13 D=11 so D is eliminated. Round 2 votes counts: B=39 E=24 C=19 A=18 so A is eliminated. Round 3 votes counts: B=41 C=30 E=29 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:203 A:200 D:196 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 4 -4 B 10 0 10 8 12 C -10 -10 0 -10 -8 D -4 -8 10 0 -6 E 4 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 4 -4 B 10 0 10 8 12 C -10 -10 0 -10 -8 D -4 -8 10 0 -6 E 4 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 4 -4 B 10 0 10 8 12 C -10 -10 0 -10 -8 D -4 -8 10 0 -6 E 4 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 812: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) D B C E A (8) A B D E C (8) C E A B D (7) B D C A E (7) C E D B A (6) A E D B C (6) E A C D B (5) E C A D B (4) A E B D C (4) C E A D B (3) C B D E A (3) A E C B D (3) D E B C A (2) D B E C A (2) D B A E C (2) B D C E A (2) B D A E C (2) A E C D B (2) E D B A C (1) E C D B A (1) E A D C B (1) E A D B C (1) E A C B D (1) D E B A C (1) D C B E A (1) D B C A E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D B E A (1) C A E B D (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -4 -4 -6 B 2 0 14 -2 -6 C 4 -14 0 -18 6 D 4 2 18 0 2 E 6 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -6 B 2 0 14 -2 -6 C 4 -14 0 -18 6 D 4 2 18 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=24 B=20 D=17 E=14 so E is eliminated. Round 2 votes counts: A=33 C=29 B=20 D=18 so D is eliminated. Round 3 votes counts: B=37 A=33 C=30 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:204 E:202 A:192 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -6 B 2 0 14 -2 -6 C 4 -14 0 -18 6 D 4 2 18 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -6 B 2 0 14 -2 -6 C 4 -14 0 -18 6 D 4 2 18 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -6 B 2 0 14 -2 -6 C 4 -14 0 -18 6 D 4 2 18 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 813: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) E A B C D (6) B C D A E (6) A E C D B (6) A C D B E (6) D C B A E (5) A E B C D (5) C D A B E (4) B D C E A (4) E B D C A (3) E A B D C (3) D C B E A (3) D B C E A (3) C D B A E (3) A C E B D (3) E D C A B (2) E D A C B (2) E B D A C (2) E A D B C (2) D B C A E (2) C A D B E (2) B E A C D (2) B D C A E (2) A C B D E (2) E D B C A (1) E A C D B (1) E A C B D (1) D E B C A (1) D C A E B (1) C B D A E (1) C B A D E (1) B E D C A (1) B E C D A (1) B D E C A (1) B C D E A (1) B A C E D (1) A E C B D (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 4 2 8 B -12 0 0 0 8 C -4 0 0 12 4 D -2 0 -12 0 -2 E -8 -8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 2 8 B -12 0 0 0 8 C -4 0 0 12 4 D -2 0 -12 0 -2 E -8 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=26 B=19 D=15 C=11 so C is eliminated. Round 2 votes counts: E=29 A=28 D=22 B=21 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:213 C:206 B:198 D:192 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 2 8 B -12 0 0 0 8 C -4 0 0 12 4 D -2 0 -12 0 -2 E -8 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 2 8 B -12 0 0 0 8 C -4 0 0 12 4 D -2 0 -12 0 -2 E -8 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 2 8 B -12 0 0 0 8 C -4 0 0 12 4 D -2 0 -12 0 -2 E -8 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 814: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) B E C D A (5) B E C A D (5) C D E A B (4) A D C E B (4) E B C D A (3) D C E A B (3) D A E B C (3) D A C E B (3) C E D A B (3) C E B D A (3) C B E A D (3) C B A E D (3) B E D C A (3) B E D A C (3) A D C B E (3) E D C B A (2) E D B C A (2) E D B A C (2) D E C A B (2) D E B A C (2) D A E C B (2) C E B A D (2) C A E B D (2) B E A D C (2) B E A C D (2) B C E A D (2) B C A E D (2) A D B E C (2) A C B D E (2) E C D B A (1) E B D C A (1) D E C B A (1) D E A B C (1) D B A E C (1) C E A B D (1) C A D B E (1) C A B E D (1) B A E C D (1) B A D E C (1) A C D E B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -26 0 -16 B 4 0 -12 -4 -12 C 26 12 0 12 4 D 0 4 -12 0 -8 E 16 12 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -26 0 -16 B 4 0 -12 -4 -12 C 26 12 0 12 4 D 0 4 -12 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 D=18 A=14 E=11 so E is eliminated. Round 2 votes counts: C=32 B=30 D=24 A=14 so A is eliminated. Round 3 votes counts: C=36 D=33 B=31 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:216 D:192 B:188 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -26 0 -16 B 4 0 -12 -4 -12 C 26 12 0 12 4 D 0 4 -12 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -26 0 -16 B 4 0 -12 -4 -12 C 26 12 0 12 4 D 0 4 -12 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -26 0 -16 B 4 0 -12 -4 -12 C 26 12 0 12 4 D 0 4 -12 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 815: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) D A B E C (6) E C A B D (5) D B A E C (5) D A E B C (5) B A E C D (5) D C B A E (4) D B A C E (4) D C E B A (3) D C E A B (3) C E D A B (3) C D E B A (3) C B E A D (3) B C A E D (3) B A C E D (3) A B E C D (3) A B D E C (3) E A C B D (2) E A B C D (2) D E C A B (2) C E B D A (2) C E A B D (2) B D A C E (2) B A D E C (2) A E B C D (2) A B E D C (2) E C D A B (1) E C A D B (1) E A B D C (1) D E A C B (1) D E A B C (1) D C B E A (1) D B C A E (1) C E D B A (1) C B A E D (1) B E C A D (1) B E A C D (1) B D A E C (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 -10 6 2 8 B 10 0 10 10 6 C -6 -10 0 2 -6 D -2 -10 -2 0 -4 E -8 -6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 2 8 B 10 0 10 10 6 C -6 -10 0 2 -6 D -2 -10 -2 0 -4 E -8 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=22 B=20 E=12 A=10 so A is eliminated. Round 2 votes counts: D=36 B=28 C=22 E=14 so E is eliminated. Round 3 votes counts: D=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:203 E:198 D:191 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 2 8 B 10 0 10 10 6 C -6 -10 0 2 -6 D -2 -10 -2 0 -4 E -8 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 2 8 B 10 0 10 10 6 C -6 -10 0 2 -6 D -2 -10 -2 0 -4 E -8 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 2 8 B 10 0 10 10 6 C -6 -10 0 2 -6 D -2 -10 -2 0 -4 E -8 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 816: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) A D C E B (7) E B A C D (6) A E B D C (6) E A B C D (5) D C A B E (5) D A C B E (5) B E C D A (5) B C D E A (5) A E D C B (5) D C B A E (4) B C E D A (4) A D E C B (4) A D C B E (4) E C B D A (3) E A B D C (3) C B D E A (3) E B C D A (2) E B C A D (2) C D B E A (2) C D B A E (2) A D B E C (2) E C A D B (1) E A C D B (1) D A C E B (1) C B E D A (1) C B D A E (1) B E D C A (1) B E C A D (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 16 16 12 10 B -16 0 0 -4 -10 C -16 0 0 -10 -10 D -12 4 10 0 -6 E -10 10 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 12 10 B -16 0 0 -4 -10 C -16 0 0 -10 -10 D -12 4 10 0 -6 E -10 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=23 B=16 D=15 C=9 so C is eliminated. Round 2 votes counts: A=37 E=23 B=21 D=19 so D is eliminated. Round 3 votes counts: A=48 B=29 E=23 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:208 D:198 B:185 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 12 10 B -16 0 0 -4 -10 C -16 0 0 -10 -10 D -12 4 10 0 -6 E -10 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 12 10 B -16 0 0 -4 -10 C -16 0 0 -10 -10 D -12 4 10 0 -6 E -10 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 12 10 B -16 0 0 -4 -10 C -16 0 0 -10 -10 D -12 4 10 0 -6 E -10 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 817: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (13) B E A C D (12) C A D E B (11) B E D A C (11) B A E C D (7) E B D C A (5) D E B C A (5) A C D B E (5) A C B E D (5) D B E C A (4) C D A E B (3) A C E B D (3) D E C B A (2) D C E A B (2) C A D B E (2) B E D C A (2) E B C A D (1) E B A D C (1) D C E B A (1) C D A B E (1) C A E D B (1) B E A D C (1) B A D C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -6 2 6 B 6 0 0 -2 2 C 6 0 0 4 -2 D -2 2 -4 0 2 E -6 -2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.553986 C: 0.446014 D: 0.000000 E: 0.000000 Sum of squares = 0.505829009197 Cumulative probabilities = A: 0.000000 B: 0.553986 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 2 6 B 6 0 0 -2 2 C 6 0 0 4 -2 D -2 2 -4 0 2 E -6 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500185 C: 0.499815 D: 0.000000 E: 0.000000 Sum of squares = 0.500000068242 Cumulative probabilities = A: 0.000000 B: 0.500185 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=27 C=18 A=14 E=7 so E is eliminated. Round 2 votes counts: B=41 D=27 C=18 A=14 so A is eliminated. Round 3 votes counts: B=41 C=32 D=27 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:204 B:203 D:199 A:198 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 2 6 B 6 0 0 -2 2 C 6 0 0 4 -2 D -2 2 -4 0 2 E -6 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500185 C: 0.499815 D: 0.000000 E: 0.000000 Sum of squares = 0.500000068242 Cumulative probabilities = A: 0.000000 B: 0.500185 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 2 6 B 6 0 0 -2 2 C 6 0 0 4 -2 D -2 2 -4 0 2 E -6 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500185 C: 0.499815 D: 0.000000 E: 0.000000 Sum of squares = 0.500000068242 Cumulative probabilities = A: 0.000000 B: 0.500185 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 2 6 B 6 0 0 -2 2 C 6 0 0 4 -2 D -2 2 -4 0 2 E -6 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500185 C: 0.499815 D: 0.000000 E: 0.000000 Sum of squares = 0.500000068242 Cumulative probabilities = A: 0.000000 B: 0.500185 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 818: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) A B E D C (7) D E A B C (6) D E A C B (5) C E D A B (5) C D E B A (5) C B D E A (5) E D A C B (4) D E C A B (4) B C D A E (4) B A C E D (4) E A D C B (3) D E C B A (3) D B E A C (3) B C A D E (3) A E D B C (3) D C E B A (2) C B A D E (2) C A B E D (2) B A D E C (2) E D C A B (1) E A D B C (1) E A C D B (1) D B A E C (1) C E A D B (1) C D B E A (1) C B D A E (1) C A E B D (1) B D C E A (1) B D A E C (1) B C A E D (1) B A E D C (1) B A E C D (1) A E D C B (1) A E C B D (1) A E B D C (1) A D E B C (1) A D B E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 0 -4 -2 B -2 0 -14 -4 0 C 0 14 0 -4 -6 D 4 4 4 0 2 E 2 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -4 -2 B -2 0 -14 -4 0 C 0 14 0 -4 -6 D 4 4 4 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=24 B=18 A=17 E=10 so E is eliminated. Round 2 votes counts: C=31 D=29 A=22 B=18 so B is eliminated. Round 3 votes counts: C=39 D=31 A=30 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:207 E:203 C:202 A:198 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -4 -2 B -2 0 -14 -4 0 C 0 14 0 -4 -6 D 4 4 4 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 -2 B -2 0 -14 -4 0 C 0 14 0 -4 -6 D 4 4 4 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 -2 B -2 0 -14 -4 0 C 0 14 0 -4 -6 D 4 4 4 0 2 E 2 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 819: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (9) D C B A E (7) B D C E A (6) A E C D B (6) E A B C D (5) E D A C B (4) C D B A E (4) E B A C D (3) E A C D B (3) B A C E D (3) A E D C B (3) E A B D C (2) D C B E A (2) D B E C A (2) D B C E A (2) C D A E B (2) C D A B E (2) A C E B D (2) E B D A C (1) E B A D C (1) E A D B C (1) E A C B D (1) D E A C B (1) D C E A B (1) D C A E B (1) C A D E B (1) C A D B E (1) B E D C A (1) B E C A D (1) B E A D C (1) B E A C D (1) B D E C A (1) B D C A E (1) B C E A D (1) B C D E A (1) B C A E D (1) B C A D E (1) B A E C D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 4 4 -6 B 2 0 -6 -8 0 C -4 6 0 2 -2 D -4 8 -2 0 -10 E 6 0 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.185329 C: 0.000000 D: 0.000000 E: 0.814671 Sum of squares = 0.698035356321 Cumulative probabilities = A: 0.000000 B: 0.185329 C: 0.185329 D: 0.185329 E: 1.000000 A B C D E A 0 -2 4 4 -6 B 2 0 -6 -8 0 C -4 6 0 2 -2 D -4 8 -2 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000678 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=29 D=16 A=13 C=10 so C is eliminated. Round 2 votes counts: E=32 B=29 D=24 A=15 so A is eliminated. Round 3 votes counts: E=44 B=30 D=26 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:209 C:201 A:200 D:196 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 4 -6 B 2 0 -6 -8 0 C -4 6 0 2 -2 D -4 8 -2 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000678 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 4 -6 B 2 0 -6 -8 0 C -4 6 0 2 -2 D -4 8 -2 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000678 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 4 -6 B 2 0 -6 -8 0 C -4 6 0 2 -2 D -4 8 -2 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000678 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 820: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) A D C E B (10) A C E D B (7) A C D E B (7) B E D C A (6) C E D A B (5) E C B D A (4) B D E C A (4) A D B C E (4) E C D B A (3) E C A B D (3) C A E D B (3) B A D E C (3) A D C B E (3) E C B A D (2) E B C A D (2) D E C B A (2) D C E B A (2) D B E C A (2) C D A E B (2) B A E C D (2) A B D C E (2) E C A D B (1) E B C D A (1) D C A E B (1) D B A C E (1) D A C B E (1) C E D B A (1) C E A D B (1) C E A B D (1) C A D E B (1) B E A C D (1) B D E A C (1) B A E D C (1) Total count = 100 A B C D E A 0 4 -14 8 -4 B -4 0 -20 -14 -18 C 14 20 0 14 4 D -8 14 -14 0 -8 E 4 18 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 8 -4 B -4 0 -20 -14 -18 C 14 20 0 14 4 D -8 14 -14 0 -8 E 4 18 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=28 E=16 C=14 D=9 so D is eliminated. Round 2 votes counts: A=34 B=31 E=18 C=17 so C is eliminated. Round 3 votes counts: A=41 B=31 E=28 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:226 E:213 A:197 D:192 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 8 -4 B -4 0 -20 -14 -18 C 14 20 0 14 4 D -8 14 -14 0 -8 E 4 18 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 8 -4 B -4 0 -20 -14 -18 C 14 20 0 14 4 D -8 14 -14 0 -8 E 4 18 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 8 -4 B -4 0 -20 -14 -18 C 14 20 0 14 4 D -8 14 -14 0 -8 E 4 18 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 821: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) C B A D E (8) A D B C E (8) E D A B C (7) E C B D A (7) E C B A D (7) E D A C B (5) D A E B C (4) D A B E C (4) E D C B A (3) E C D B A (3) D A B C E (3) C E B D A (3) A B D C E (3) A B C D E (3) E A D B C (2) D A E C B (2) C E D B A (2) C E B A D (2) C A D B E (2) B A C D E (2) E D B C A (1) E D B A C (1) E B C A D (1) D E A B C (1) D A C B E (1) C D A B E (1) C B E D A (1) C B A E D (1) C A B D E (1) B C A E D (1) B C A D E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -6 2 -8 B 4 0 -16 0 -2 C 6 16 0 10 4 D -2 0 -10 0 -10 E 8 2 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 -8 B 4 0 -16 0 -2 C 6 16 0 10 4 D -2 0 -10 0 -10 E 8 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=29 D=15 A=15 B=4 so B is eliminated. Round 2 votes counts: E=37 C=31 A=17 D=15 so D is eliminated. Round 3 votes counts: E=38 C=31 A=31 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:218 E:208 B:193 A:192 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 2 -8 B 4 0 -16 0 -2 C 6 16 0 10 4 D -2 0 -10 0 -10 E 8 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 -8 B 4 0 -16 0 -2 C 6 16 0 10 4 D -2 0 -10 0 -10 E 8 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 -8 B 4 0 -16 0 -2 C 6 16 0 10 4 D -2 0 -10 0 -10 E 8 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 822: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) A C E B D (7) A C B D E (7) E A C B D (6) A E C B D (6) E A C D B (5) D B C A E (5) E A D C B (4) D B E C A (4) C B A D E (4) E D A B C (3) D E B C A (3) C A B E D (3) A E C D B (3) E C A B D (2) D B C E A (2) C B D A E (2) C A B D E (2) B D C A E (2) A E D B C (2) A C B E D (2) E D B A C (1) E D A C B (1) E C B D A (1) E A D B C (1) D E B A C (1) D B A C E (1) C B E D A (1) C B A E D (1) B D C E A (1) B C D E A (1) B C D A E (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 6 16 2 B -14 0 -14 0 -18 C -6 14 0 8 -14 D -16 0 -8 0 -22 E -2 18 14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 16 2 B -14 0 -14 0 -18 C -6 14 0 8 -14 D -16 0 -8 0 -22 E -2 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=30 D=16 C=13 B=5 so B is eliminated. Round 2 votes counts: E=36 A=30 D=19 C=15 so C is eliminated. Round 3 votes counts: A=40 E=37 D=23 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:226 A:219 C:201 B:177 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 16 2 B -14 0 -14 0 -18 C -6 14 0 8 -14 D -16 0 -8 0 -22 E -2 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 16 2 B -14 0 -14 0 -18 C -6 14 0 8 -14 D -16 0 -8 0 -22 E -2 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 16 2 B -14 0 -14 0 -18 C -6 14 0 8 -14 D -16 0 -8 0 -22 E -2 18 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 823: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) A E C D B (9) B C D E A (6) A E D C B (6) A E C B D (6) D B E C A (5) D B C E A (4) B C A E D (4) A D E B C (4) A C E B D (4) A B C E D (4) E A C D B (3) E C A D B (2) C E A D B (2) C E A B D (2) C B E D A (2) B C E D A (2) A D E C B (2) A C B E D (2) E C D A B (1) E A D C B (1) E A C B D (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) D B A E C (1) D A E C B (1) D A B E C (1) C E B A D (1) C D E B A (1) C B E A D (1) B D C A E (1) B C D A E (1) B C A D E (1) B A C E D (1) A E D B C (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 2 20 4 B -12 0 0 2 -2 C -2 0 0 14 4 D -20 -2 -14 0 -10 E -4 2 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999726 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 20 4 B -12 0 0 2 -2 C -2 0 0 14 4 D -20 -2 -14 0 -10 E -4 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 B=26 D=16 C=9 E=8 so E is eliminated. Round 2 votes counts: A=46 B=26 D=16 C=12 so C is eliminated. Round 3 votes counts: A=52 B=30 D=18 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:208 E:202 B:194 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 20 4 B -12 0 0 2 -2 C -2 0 0 14 4 D -20 -2 -14 0 -10 E -4 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 20 4 B -12 0 0 2 -2 C -2 0 0 14 4 D -20 -2 -14 0 -10 E -4 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 20 4 B -12 0 0 2 -2 C -2 0 0 14 4 D -20 -2 -14 0 -10 E -4 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980656 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 824: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (16) A C B D E (14) D E A C B (12) B C A E D (10) E B D C A (6) E D B A C (5) A C D E B (5) B C E A D (4) A C D B E (4) C A B D E (3) D E A B C (2) D A E C B (2) C B A E D (2) C A B E D (2) B E D C A (2) B C E D A (2) B A C E D (2) A D C E B (2) D A C E B (1) C B A D E (1) B E C D A (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 2 2 0 B 2 0 0 0 -4 C -2 0 0 2 6 D -2 0 -2 0 -6 E 0 4 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.528712 C: 0.471288 D: 0.000000 E: 0.000000 Sum of squares = 0.501648733048 Cumulative probabilities = A: 0.000000 B: 0.528712 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 2 0 B 2 0 0 0 -4 C -2 0 0 2 6 D -2 0 -2 0 -6 E 0 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500175 C: 0.499825 D: 0.000000 E: 0.000000 Sum of squares = 0.500000061584 Cumulative probabilities = A: 0.000000 B: 0.500175 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 B=21 D=17 C=8 so C is eliminated. Round 2 votes counts: A=32 E=27 B=24 D=17 so D is eliminated. Round 3 votes counts: E=41 A=35 B=24 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:203 E:202 A:201 B:199 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 0 B 2 0 0 0 -4 C -2 0 0 2 6 D -2 0 -2 0 -6 E 0 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500175 C: 0.499825 D: 0.000000 E: 0.000000 Sum of squares = 0.500000061584 Cumulative probabilities = A: 0.000000 B: 0.500175 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 0 B 2 0 0 0 -4 C -2 0 0 2 6 D -2 0 -2 0 -6 E 0 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500175 C: 0.499825 D: 0.000000 E: 0.000000 Sum of squares = 0.500000061584 Cumulative probabilities = A: 0.000000 B: 0.500175 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 0 B 2 0 0 0 -4 C -2 0 0 2 6 D -2 0 -2 0 -6 E 0 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500175 C: 0.499825 D: 0.000000 E: 0.000000 Sum of squares = 0.500000061584 Cumulative probabilities = A: 0.000000 B: 0.500175 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 825: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) E A B D C (7) C D B A E (7) C B D E A (7) A E C B D (7) E B D C A (5) C D B E A (5) B D E C A (5) B D C E A (5) A E D B C (4) A E C D B (4) A E B D C (4) A C E B D (4) E A B C D (3) E C B D A (2) E B A D C (2) C D A B E (2) A E B C D (2) A C E D B (2) A C D B E (2) E B D A C (1) E A C B D (1) D C B E A (1) D B E C A (1) D B C A E (1) D A B C E (1) C E B D A (1) C B D A E (1) C A D B E (1) B E D C A (1) B E C D A (1) B C D E A (1) A D C B E (1) Total count = 100 A B C D E A 0 -10 -10 -12 -14 B 10 0 4 20 2 C 10 -4 0 6 0 D 12 -20 -6 0 -2 E 14 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -12 -14 B 10 0 4 20 2 C 10 -4 0 6 0 D 12 -20 -6 0 -2 E 14 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=24 E=21 B=13 D=12 so D is eliminated. Round 2 votes counts: A=31 C=25 B=23 E=21 so E is eliminated. Round 3 votes counts: A=42 B=31 C=27 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:207 C:206 D:192 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 -12 -14 B 10 0 4 20 2 C 10 -4 0 6 0 D 12 -20 -6 0 -2 E 14 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -12 -14 B 10 0 4 20 2 C 10 -4 0 6 0 D 12 -20 -6 0 -2 E 14 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -12 -14 B 10 0 4 20 2 C 10 -4 0 6 0 D 12 -20 -6 0 -2 E 14 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 826: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) D C E A B (12) B A E C D (11) B A E D C (9) E C A B D (8) B A D E C (8) E A B C D (7) D C B A E (5) E A C B D (4) B D A E C (4) D B C A E (3) C E D A B (3) A B E C D (3) D B A C E (2) C E A D B (2) C E A B D (2) E B A C D (1) D C E B A (1) A E B C D (1) Total count = 100 A B C D E A 0 12 0 12 -8 B -12 0 -2 16 -10 C 0 2 0 12 -12 D -12 -16 -12 0 -2 E 8 10 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 0 12 -8 B -12 0 -2 16 -10 C 0 2 0 12 -12 D -12 -16 -12 0 -2 E 8 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=23 C=21 E=20 A=4 so A is eliminated. Round 2 votes counts: B=35 D=23 E=21 C=21 so E is eliminated. Round 3 votes counts: B=44 C=33 D=23 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:216 A:208 C:201 B:196 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 0 12 -8 B -12 0 -2 16 -10 C 0 2 0 12 -12 D -12 -16 -12 0 -2 E 8 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 12 -8 B -12 0 -2 16 -10 C 0 2 0 12 -12 D -12 -16 -12 0 -2 E 8 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 12 -8 B -12 0 -2 16 -10 C 0 2 0 12 -12 D -12 -16 -12 0 -2 E 8 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 827: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) B A D C E (7) D A B C E (6) B D A E C (6) E C D A B (5) D B A E C (5) A C D E B (4) E C A D B (3) D E A C B (3) C E A D B (3) C A D E B (3) B A C D E (3) D B A C E (2) D A E C B (2) D A C B E (2) C E B A D (2) C A E B D (2) B E D C A (2) B C A E D (2) A D B C E (2) A C D B E (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E C B A D (1) E C A B D (1) E B D C A (1) D E B A C (1) D C A E B (1) D B E A C (1) D A C E B (1) D A B E C (1) C D E A B (1) C B E A D (1) C A E D B (1) C A B E D (1) B E C A D (1) B E A D C (1) B D E A C (1) B D A C E (1) B C E A D (1) B A D E C (1) B A C E D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 2 18 -2 24 B -2 0 -4 -20 10 C -18 4 0 -2 10 D 2 20 2 0 24 E -24 -10 -10 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 -2 24 B -2 0 -4 -20 10 C -18 4 0 -2 10 D 2 20 2 0 24 E -24 -10 -10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 E=20 C=14 A=14 so C is eliminated. Round 2 votes counts: B=28 D=26 E=25 A=21 so A is eliminated. Round 3 votes counts: D=39 B=33 E=28 so E is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 A:221 C:197 B:192 E:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 18 -2 24 B -2 0 -4 -20 10 C -18 4 0 -2 10 D 2 20 2 0 24 E -24 -10 -10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 -2 24 B -2 0 -4 -20 10 C -18 4 0 -2 10 D 2 20 2 0 24 E -24 -10 -10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 -2 24 B -2 0 -4 -20 10 C -18 4 0 -2 10 D 2 20 2 0 24 E -24 -10 -10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 828: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B A E C D (7) E A B D C (6) D E A B C (6) A B E C D (6) C D B E A (5) B C A E D (5) A E B D C (5) D C E A B (4) C D B A E (4) D E A C B (3) D C E B A (3) C B D E A (3) B A C E D (3) A E B C D (3) E B A D C (2) E A D B C (2) D E C A B (2) D A E C B (2) C D A B E (2) C B A D E (2) B E A C D (2) B C E D A (2) E D B A C (1) E D A B C (1) E B D A C (1) D E C B A (1) D C B E A (1) D A C E B (1) C B D A E (1) C B A E D (1) A E D C B (1) A E D B C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 10 -4 10 B -12 0 8 2 -8 C -10 -8 0 -4 -4 D 4 -2 4 0 0 E -10 8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.741789 E: 0.258211 Sum of squares = 0.616924006676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.741789 E: 1.000000 A B C D E A 0 12 10 -4 10 B -12 0 8 2 -8 C -10 -8 0 -4 -4 D 4 -2 4 0 0 E -10 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836737214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=19 C=18 A=18 E=13 so E is eliminated. Round 2 votes counts: D=34 A=26 B=22 C=18 so C is eliminated. Round 3 votes counts: D=45 B=29 A=26 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:214 D:203 E:201 B:195 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 10 -4 10 B -12 0 8 2 -8 C -10 -8 0 -4 -4 D 4 -2 4 0 0 E -10 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836737214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -4 10 B -12 0 8 2 -8 C -10 -8 0 -4 -4 D 4 -2 4 0 0 E -10 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836737214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -4 10 B -12 0 8 2 -8 C -10 -8 0 -4 -4 D 4 -2 4 0 0 E -10 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836737214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 829: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) D A B C E (6) E C B D A (5) A D B E C (4) E C D A B (3) E C A B D (3) D C E B A (3) C D E B A (3) B A C D E (3) A E B C D (3) A D B C E (3) A B E D C (3) E C B A D (2) E A D C B (2) E A C B D (2) E A B C D (2) D A E C B (2) C E D B A (2) C D B E A (2) C B E D A (2) B C E A D (2) B C D A E (2) A E D B C (2) A E B D C (2) A D E B C (2) E D C A B (1) E D A C B (1) E B C A D (1) E B A C D (1) E A D B C (1) E A C D B (1) D E C B A (1) D E C A B (1) D E A C B (1) D C B E A (1) D B C A E (1) D A E B C (1) D A C B E (1) C E B D A (1) C E B A D (1) C B E A D (1) B C E D A (1) B C D E A (1) B C A E D (1) B C A D E (1) B A D C E (1) B A C E D (1) A E D C B (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 0 0 -12 B -4 0 -2 -8 -18 C 0 2 0 14 -14 D 0 8 -14 0 -14 E 12 18 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 0 -12 B -4 0 -2 -8 -18 C 0 2 0 14 -14 D 0 8 -14 0 -14 E 12 18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=24 D=18 B=13 C=12 so C is eliminated. Round 2 votes counts: E=37 A=24 D=23 B=16 so B is eliminated. Round 3 votes counts: E=43 A=31 D=26 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:201 A:196 D:190 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 0 -12 B -4 0 -2 -8 -18 C 0 2 0 14 -14 D 0 8 -14 0 -14 E 12 18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 0 -12 B -4 0 -2 -8 -18 C 0 2 0 14 -14 D 0 8 -14 0 -14 E 12 18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 0 -12 B -4 0 -2 -8 -18 C 0 2 0 14 -14 D 0 8 -14 0 -14 E 12 18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 830: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) C E D A B (7) B A D E C (6) E C D B A (5) D C B E A (4) D B C E A (4) D B C A E (4) C A D B E (4) B D A C E (4) A B D C E (4) E D C B A (3) E A C B D (3) D C E B A (3) D C B A E (3) C D E B A (3) C D E A B (3) B A E D C (3) B A D C E (3) E C D A B (2) E B A D C (2) D B A C E (2) C E A D B (2) A C E B D (2) A B D E C (2) A B C E D (2) E D B C A (1) E B D A C (1) E A B D C (1) E A B C D (1) D B E C A (1) C D A E B (1) B E D A C (1) B D E A C (1) B D A E C (1) A E C B D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -16 -8 -12 B 10 0 -8 -20 2 C 16 8 0 -10 12 D 8 20 10 0 8 E 12 -2 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -8 -12 B 10 0 -8 -20 2 C 16 8 0 -10 12 D 8 20 10 0 8 E 12 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=21 C=20 B=19 A=13 so A is eliminated. Round 2 votes counts: B=29 E=28 C=22 D=21 so D is eliminated. Round 3 votes counts: B=40 C=32 E=28 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:223 C:213 E:195 B:192 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -16 -8 -12 B 10 0 -8 -20 2 C 16 8 0 -10 12 D 8 20 10 0 8 E 12 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -8 -12 B 10 0 -8 -20 2 C 16 8 0 -10 12 D 8 20 10 0 8 E 12 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -8 -12 B 10 0 -8 -20 2 C 16 8 0 -10 12 D 8 20 10 0 8 E 12 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 831: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) E B A C D (8) B E A C D (8) D C A B E (6) D A C E B (6) C A D B E (6) B C A E D (6) E B D C A (5) E D B A C (4) E B D A C (4) D C A E B (4) D E C A B (3) D E A C B (3) A C D B E (3) E D A C B (2) D A C B E (2) C A B D E (2) B E D C A (2) A C B D E (2) E D B C A (1) E A D C B (1) E A C D B (1) D E B C A (1) D C E A B (1) D C B A E (1) D B C E A (1) D B C A E (1) C D A B E (1) B C E D A (1) B C A D E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -8 2 -14 B 10 0 8 4 10 C 8 -8 0 4 -8 D -2 -4 -4 0 -12 E 14 -10 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 2 -14 B 10 0 8 4 10 C 8 -8 0 4 -8 D -2 -4 -4 0 -12 E 14 -10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=29 B=29 E=26 C=9 A=7 so A is eliminated. Round 2 votes counts: D=29 B=29 E=26 C=16 so C is eliminated. Round 3 votes counts: D=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:212 C:198 D:189 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 2 -14 B 10 0 8 4 10 C 8 -8 0 4 -8 D -2 -4 -4 0 -12 E 14 -10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 2 -14 B 10 0 8 4 10 C 8 -8 0 4 -8 D -2 -4 -4 0 -12 E 14 -10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 2 -14 B 10 0 8 4 10 C 8 -8 0 4 -8 D -2 -4 -4 0 -12 E 14 -10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 832: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) B D A E C (5) A D E B C (5) E D B A C (4) E D A B C (4) E C B D A (4) C B A D E (4) C A E D B (4) C E B D A (3) C E A D B (3) C A E B D (3) B E D C A (3) B D E A C (3) B C D E A (3) A D B E C (3) A D B C E (3) A C E D B (3) A C D E B (3) A C B D E (3) E D A C B (2) E C D A B (2) E B D C A (2) C B E D A (2) C A B D E (2) B D E C A (2) A E D B C (2) A C D B E (2) A B D C E (2) E C D B A (1) D E A B C (1) D B E A C (1) D B A E C (1) C E B A D (1) C B A E D (1) C A B E D (1) B E C D A (1) B D A C E (1) B A D C E (1) B A C D E (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 -2 4 B 0 0 10 -2 -8 C -4 -10 0 -4 -6 D 2 2 4 0 -6 E -4 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 0 4 -2 4 B 0 0 10 -2 -8 C -4 -10 0 -4 -6 D 2 2 4 0 -6 E -4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=25 C=24 B=20 D=3 so D is eliminated. Round 2 votes counts: A=28 E=26 C=24 B=22 so B is eliminated. Round 3 votes counts: A=37 E=36 C=27 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:203 D:201 B:200 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 -2 4 B 0 0 10 -2 -8 C -4 -10 0 -4 -6 D 2 2 4 0 -6 E -4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 4 B 0 0 10 -2 -8 C -4 -10 0 -4 -6 D 2 2 4 0 -6 E -4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 4 B 0 0 10 -2 -8 C -4 -10 0 -4 -6 D 2 2 4 0 -6 E -4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 833: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) E C A D B (7) C E D B A (7) A C B E D (7) A B D C E (7) D B A E C (5) B A D C E (5) A E C B D (5) E C D A B (4) E C D B A (3) E A C D B (3) B D A C E (3) E D A C B (2) E D A B C (2) E C A B D (2) E A D B C (2) D E B C A (2) D B C E A (2) C E B D A (2) A C E B D (2) E D C B A (1) E D C A B (1) E D B C A (1) D E C B A (1) D E B A C (1) D B E A C (1) D B C A E (1) D A B E C (1) C D E B A (1) C B D E A (1) C B A D E (1) C A E B D (1) C A B E D (1) B D C A E (1) B D A E C (1) A E C D B (1) A C B D E (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 20 4 12 -8 B -20 0 -24 2 -18 C -4 24 0 18 6 D -12 -2 -18 0 -26 E 8 18 -6 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691379 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 20 4 12 -8 B -20 0 -24 2 -18 C -4 24 0 18 6 D -12 -2 -18 0 -26 E 8 18 -6 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.358024690898 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 C=22 D=14 B=10 so B is eliminated. Round 2 votes counts: A=31 E=28 C=22 D=19 so D is eliminated. Round 3 votes counts: A=41 E=33 C=26 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:222 A:214 D:171 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 4 12 -8 B -20 0 -24 2 -18 C -4 24 0 18 6 D -12 -2 -18 0 -26 E 8 18 -6 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.358024690898 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 4 12 -8 B -20 0 -24 2 -18 C -4 24 0 18 6 D -12 -2 -18 0 -26 E 8 18 -6 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.358024690898 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 4 12 -8 B -20 0 -24 2 -18 C -4 24 0 18 6 D -12 -2 -18 0 -26 E 8 18 -6 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.358024690898 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 834: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) B A D E C (10) B C E A D (8) E C D A B (6) D E A C B (6) D A E C B (6) D A E B C (6) C E A D B (6) D A B E C (5) B D A E C (3) B A C E D (3) A D B E C (3) D E C A B (2) D B A E C (2) C E B A D (2) C B E D A (2) B C A D E (2) A B D E C (2) E D C A B (1) E C A D B (1) E A D C B (1) D C E A B (1) C E A B D (1) C B E A D (1) B D A C E (1) B C E D A (1) B C D E A (1) B C A E D (1) B A D C E (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 24 2 -12 -6 B -24 0 0 -22 -8 C -2 0 0 -4 -12 D 12 22 4 0 6 E 6 8 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 2 -12 -6 B -24 0 0 -22 -8 C -2 0 0 -4 -12 D 12 22 4 0 6 E 6 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=28 C=25 E=9 A=7 so A is eliminated. Round 2 votes counts: D=33 B=33 C=25 E=9 so E is eliminated. Round 3 votes counts: D=35 B=33 C=32 so C is eliminated. Round 4 votes counts: D=61 B=39 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:210 A:204 C:191 B:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 2 -12 -6 B -24 0 0 -22 -8 C -2 0 0 -4 -12 D 12 22 4 0 6 E 6 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 2 -12 -6 B -24 0 0 -22 -8 C -2 0 0 -4 -12 D 12 22 4 0 6 E 6 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 2 -12 -6 B -24 0 0 -22 -8 C -2 0 0 -4 -12 D 12 22 4 0 6 E 6 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 835: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (18) B A D E C (7) E C A D B (6) E A C B D (5) A E B C D (5) D C B A E (4) D B C A E (4) D C B E A (3) C D B A E (3) C B D A E (3) A E B D C (3) E D C A B (2) E C D A B (2) D B A E C (2) C D E B A (2) C D E A B (2) C D B E A (2) B D A C E (2) B A E D C (2) E D A B C (1) E A D B C (1) E A C D B (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D B C E A (1) D B A C E (1) C E D B A (1) C E A D B (1) C E A B D (1) C B A E D (1) C A E B D (1) C A B E D (1) B C D A E (1) B C A D E (1) B A C E D (1) B A C D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -26 -16 -10 B -12 0 -24 -22 -16 C 26 24 0 22 12 D 16 22 -22 0 -14 E 10 16 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -26 -16 -10 B -12 0 -24 -22 -16 C 26 24 0 22 12 D 16 22 -22 0 -14 E 10 16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=20 D=19 B=15 A=10 so A is eliminated. Round 2 votes counts: C=36 E=28 D=19 B=17 so B is eliminated. Round 3 votes counts: C=40 E=32 D=28 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:242 E:214 D:201 A:180 B:163 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -26 -16 -10 B -12 0 -24 -22 -16 C 26 24 0 22 12 D 16 22 -22 0 -14 E 10 16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -26 -16 -10 B -12 0 -24 -22 -16 C 26 24 0 22 12 D 16 22 -22 0 -14 E 10 16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -26 -16 -10 B -12 0 -24 -22 -16 C 26 24 0 22 12 D 16 22 -22 0 -14 E 10 16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 836: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) C A D B E (7) A E B C D (7) D B C E A (6) C D A B E (6) D C E B A (5) D C B A E (5) E B D A C (4) C A D E B (4) A C E B D (4) E B D C A (3) E A B C D (3) B D E C A (3) A E C B D (3) E C A D B (2) D E B C A (2) D C B E A (2) D B E C A (2) C A E D B (2) B E D A C (2) B D E A C (2) B A E D C (2) A C B D E (2) E D C B A (1) E D B C A (1) E B A C D (1) E A C B D (1) D B C A E (1) B E A D C (1) A C E D B (1) A C D B E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 10 -6 B 10 0 8 4 -12 C 4 -8 0 -8 -6 D -10 -4 8 0 -2 E 6 12 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 10 -6 B 10 0 8 4 -12 C 4 -8 0 -8 -6 D -10 -4 8 0 -2 E 6 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 A=20 C=19 B=10 so B is eliminated. Round 2 votes counts: E=31 D=28 A=22 C=19 so C is eliminated. Round 3 votes counts: A=35 D=34 E=31 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:213 B:205 D:196 A:195 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 10 -6 B 10 0 8 4 -12 C 4 -8 0 -8 -6 D -10 -4 8 0 -2 E 6 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 10 -6 B 10 0 8 4 -12 C 4 -8 0 -8 -6 D -10 -4 8 0 -2 E 6 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 10 -6 B 10 0 8 4 -12 C 4 -8 0 -8 -6 D -10 -4 8 0 -2 E 6 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 837: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) E D A C B (8) B C A D E (8) E D B C A (6) D E B A C (6) D E A B C (6) D A B C E (5) E C A B D (4) D E B C A (4) D E A C B (3) A B C D E (3) E D C A B (2) E C B A D (2) E C A D B (2) D B A C E (2) C B E A D (2) C A E B D (2) B D C A E (2) B C A E D (2) A C E B D (2) E D C B A (1) E C D B A (1) E C B D A (1) E A D C B (1) E A C B D (1) D B C A E (1) D A E C B (1) D A B E C (1) C A B E D (1) B D A C E (1) B C E D A (1) B C E A D (1) B A C D E (1) A D E C B (1) A D C B E (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -8 -4 -4 B 6 0 2 -4 -8 C 8 -2 0 -6 -2 D 4 4 6 0 -4 E 4 8 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -4 -4 B 6 0 2 -4 -8 C 8 -2 0 -6 -2 D 4 4 6 0 -4 E 4 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=29 D=29 C=16 B=16 A=10 so A is eliminated. Round 2 votes counts: D=31 E=29 C=20 B=20 so C is eliminated. Round 3 votes counts: B=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:209 D:205 C:199 B:198 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -4 -4 B 6 0 2 -4 -8 C 8 -2 0 -6 -2 D 4 4 6 0 -4 E 4 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -4 -4 B 6 0 2 -4 -8 C 8 -2 0 -6 -2 D 4 4 6 0 -4 E 4 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -4 -4 B 6 0 2 -4 -8 C 8 -2 0 -6 -2 D 4 4 6 0 -4 E 4 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 838: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) E C D A B (8) E C B A D (6) E B C A D (6) D A C B E (6) B E A C D (6) D C E A B (4) D C A E B (4) B E C A D (4) E C A B D (3) B A D E C (3) A B D C E (3) E D C A B (2) E C A D B (2) E B C D A (2) D A B C E (2) C E D A B (2) C E A D B (2) C D A E B (2) B E D A C (2) B D E A C (2) B D A E C (2) B D A C E (2) B A C E D (2) A D C E B (2) A D C B E (2) A D B C E (2) E C D B A (1) D B A E C (1) D B A C E (1) C A E D B (1) B E C D A (1) B A E C D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 0 12 -6 B 2 0 2 10 4 C 0 -2 0 2 -4 D -12 -10 -2 0 -4 E 6 -4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 12 -6 B 2 0 2 10 4 C 0 -2 0 2 -4 D -12 -10 -2 0 -4 E 6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=30 D=18 A=11 C=7 so C is eliminated. Round 2 votes counts: E=34 B=34 D=20 A=12 so A is eliminated. Round 3 votes counts: B=38 E=35 D=27 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:205 A:202 C:198 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 12 -6 B 2 0 2 10 4 C 0 -2 0 2 -4 D -12 -10 -2 0 -4 E 6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 12 -6 B 2 0 2 10 4 C 0 -2 0 2 -4 D -12 -10 -2 0 -4 E 6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 12 -6 B 2 0 2 10 4 C 0 -2 0 2 -4 D -12 -10 -2 0 -4 E 6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 839: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) A E C D B (9) B A E C D (7) E A C D B (5) D C B E A (4) B D A E C (4) B C E D A (4) B C E A D (4) B A D E C (4) D C E A B (3) C E D A B (3) B D A C E (3) B C D E A (3) A E D C B (3) E C A D B (2) E C A B D (2) E A C B D (2) D A C E B (2) D A B E C (2) C E B D A (2) C E A D B (2) C D E A B (2) A E B D C (2) E C B A D (1) D C E B A (1) D B C E A (1) D B C A E (1) D B A E C (1) D B A C E (1) D A E C B (1) C E B A D (1) C D B E A (1) C B D E A (1) B D C A E (1) A E C B D (1) A E B C D (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 2 -2 -8 B 10 0 4 10 8 C -2 -4 0 6 0 D 2 -10 -6 0 -4 E 8 -8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -2 -8 B 10 0 4 10 8 C -2 -4 0 6 0 D 2 -10 -6 0 -4 E 8 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=19 D=17 E=12 C=12 so E is eliminated. Round 2 votes counts: B=40 A=26 D=17 C=17 so D is eliminated. Round 3 votes counts: B=44 A=31 C=25 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:202 C:200 A:191 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 -2 -8 B 10 0 4 10 8 C -2 -4 0 6 0 D 2 -10 -6 0 -4 E 8 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -2 -8 B 10 0 4 10 8 C -2 -4 0 6 0 D 2 -10 -6 0 -4 E 8 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -2 -8 B 10 0 4 10 8 C -2 -4 0 6 0 D 2 -10 -6 0 -4 E 8 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 840: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) B D E A C (8) C E D A B (6) D E B A C (5) C E D B A (5) C B A E D (5) B C A D E (5) B A C D E (5) C B E D A (4) B D A E C (4) A D E B C (4) E C D A B (3) B C D E A (3) B A D E C (3) A E D C B (3) E D C A B (2) E D A B C (2) D E A B C (2) D A E B C (2) A B D E C (2) E A D C B (1) E A D B C (1) D B E C A (1) D B E A C (1) C B E A D (1) C B D E A (1) C B A D E (1) C A E D B (1) C A E B D (1) C A B E D (1) B D C A E (1) B C A E D (1) B A D C E (1) B A C E D (1) A E D B C (1) A E B D C (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 16 -16 -8 B 12 0 12 4 4 C -16 -12 0 -8 -10 D 16 -4 8 0 -2 E 8 -4 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 16 -16 -8 B 12 0 12 4 4 C -16 -12 0 -8 -10 D 16 -4 8 0 -2 E 8 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=26 E=17 A=14 D=11 so D is eliminated. Round 2 votes counts: B=34 C=26 E=24 A=16 so A is eliminated. Round 3 votes counts: B=38 E=35 C=27 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:209 E:208 A:190 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 16 -16 -8 B 12 0 12 4 4 C -16 -12 0 -8 -10 D 16 -4 8 0 -2 E 8 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 16 -16 -8 B 12 0 12 4 4 C -16 -12 0 -8 -10 D 16 -4 8 0 -2 E 8 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 16 -16 -8 B 12 0 12 4 4 C -16 -12 0 -8 -10 D 16 -4 8 0 -2 E 8 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 841: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (18) A D E C B (11) B C E D A (9) C B E D A (7) B E D A C (5) C B D E A (4) C A B D E (4) A C D E B (4) E D A B C (3) D E B A C (3) C B A E D (3) C A D B E (3) C A B E D (3) B E D C A (3) E B D A C (2) D E A B C (2) D A E B C (2) C B E A D (2) C A D E B (2) B E C D A (2) A E D B C (2) A D C E B (2) E D B A C (1) D A E C B (1) D A B E C (1) C B A D E (1) Total count = 100 A B C D E A 0 16 14 10 14 B -16 0 6 -10 -6 C -14 -6 0 -12 -12 D -10 10 12 0 16 E -14 6 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 10 14 B -16 0 6 -10 -6 C -14 -6 0 -12 -12 D -10 10 12 0 16 E -14 6 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=29 B=19 D=9 E=6 so E is eliminated. Round 2 votes counts: A=37 C=29 B=21 D=13 so D is eliminated. Round 3 votes counts: A=46 C=29 B=25 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:227 D:214 E:194 B:187 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 10 14 B -16 0 6 -10 -6 C -14 -6 0 -12 -12 D -10 10 12 0 16 E -14 6 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 10 14 B -16 0 6 -10 -6 C -14 -6 0 -12 -12 D -10 10 12 0 16 E -14 6 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 10 14 B -16 0 6 -10 -6 C -14 -6 0 -12 -12 D -10 10 12 0 16 E -14 6 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 842: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (19) A C E B D (11) D A E C B (6) A D C E B (6) D B A C E (5) D A B C E (5) E C B A D (4) E C A B D (4) B C A E D (4) D E C B A (3) D B A E C (3) B E C D A (3) B D E C A (3) E B C A D (2) D B C A E (2) B E D C A (2) B E C A D (2) B A D C E (2) A C E D B (2) A C D B E (2) E D B C A (1) E B D C A (1) D E B C A (1) D A C B E (1) D A B E C (1) B D C A E (1) B C E A D (1) A C D E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -6 -14 8 B 18 0 18 -16 16 C 6 -18 0 -24 -10 D 14 16 24 0 24 E -8 -16 10 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -14 8 B 18 0 18 -16 16 C 6 -18 0 -24 -10 D 14 16 24 0 24 E -8 -16 10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=46 A=24 B=18 E=12 so C is eliminated. Round 2 votes counts: D=46 A=24 B=18 E=12 so E is eliminated. Round 3 votes counts: D=47 A=28 B=25 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:239 B:218 A:185 E:181 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -6 -14 8 B 18 0 18 -16 16 C 6 -18 0 -24 -10 D 14 16 24 0 24 E -8 -16 10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -14 8 B 18 0 18 -16 16 C 6 -18 0 -24 -10 D 14 16 24 0 24 E -8 -16 10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -14 8 B 18 0 18 -16 16 C 6 -18 0 -24 -10 D 14 16 24 0 24 E -8 -16 10 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 843: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) D E A C B (8) D A E B C (8) E D C A B (7) B A C D E (7) A D B C E (7) D A C B E (4) C B A E D (4) B C E A D (4) B C A E D (4) A D B E C (4) E C B D A (3) C E B D A (3) B A C E D (3) E C D B A (2) D A B E C (2) C E B A D (2) B E A C D (2) A B D C E (2) A B C D E (2) E D A C B (1) E C D A B (1) E C B A D (1) E B D A C (1) E B C D A (1) E B C A D (1) D E C A B (1) D E A B C (1) D C A E B (1) D A B C E (1) B A E D C (1) A C D B E (1) Total count = 100 A B C D E A 0 2 10 10 2 B -2 0 2 2 16 C -10 -2 0 2 10 D -10 -2 -2 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 10 2 B -2 0 2 2 16 C -10 -2 0 2 10 D -10 -2 -2 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=21 C=19 E=18 A=16 so A is eliminated. Round 2 votes counts: D=37 B=25 C=20 E=18 so E is eliminated. Round 3 votes counts: D=45 B=28 C=27 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:212 B:209 C:200 D:192 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 10 2 B -2 0 2 2 16 C -10 -2 0 2 10 D -10 -2 -2 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 10 2 B -2 0 2 2 16 C -10 -2 0 2 10 D -10 -2 -2 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 10 2 B -2 0 2 2 16 C -10 -2 0 2 10 D -10 -2 -2 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 844: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) B C E A D (9) C E D B A (6) B C E D A (6) A B D E C (6) E C D A B (5) B A D E C (5) A D E B C (5) A D B E C (5) E D A C B (4) A D E C B (4) E D C A B (3) C E D A B (3) B C D E A (3) B C D A E (3) D E A C B (2) D C E A B (2) D A E C B (2) C E B D A (2) C B E D A (2) B C A E D (2) E C B A D (1) E C A D B (1) D A E B C (1) D A B E C (1) C D E A B (1) B D C A E (1) B D A C E (1) B A E C D (1) B A C D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 0 4 0 B 6 0 24 10 16 C 0 -24 0 -6 4 D -4 -10 6 0 6 E 0 -16 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 4 0 B 6 0 24 10 16 C 0 -24 0 -6 4 D -4 -10 6 0 6 E 0 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=22 E=14 C=14 D=8 so D is eliminated. Round 2 votes counts: B=42 A=26 E=16 C=16 so E is eliminated. Round 3 votes counts: B=42 A=32 C=26 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:199 D:199 C:187 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 4 0 B 6 0 24 10 16 C 0 -24 0 -6 4 D -4 -10 6 0 6 E 0 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 4 0 B 6 0 24 10 16 C 0 -24 0 -6 4 D -4 -10 6 0 6 E 0 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 4 0 B 6 0 24 10 16 C 0 -24 0 -6 4 D -4 -10 6 0 6 E 0 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 845: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (11) E C B A D (5) D B A E C (5) A D B C E (5) E B D C A (4) E B A C D (4) D B A C E (4) D A C B E (4) D A B C E (4) B E A D C (4) E C A B D (3) C E D B A (3) C D A B E (3) B E D A C (3) A B E D C (3) E C B D A (2) D C A B E (2) D B C A E (2) C E D A B (2) C A E D B (2) B D A E C (2) B A D E C (2) E C D B A (1) E B C D A (1) E B C A D (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B E A C (1) D A B E C (1) C D E B A (1) C A E B D (1) C A D E B (1) C A D B E (1) A E B C D (1) A C E B D (1) A C D B E (1) A C B E D (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 6 0 B -4 0 4 10 6 C -4 -4 0 0 6 D -6 -10 0 0 -14 E 0 -6 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.787269 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.212731 Sum of squares = 0.665046878353 Cumulative probabilities = A: 0.787269 B: 0.787269 C: 0.787269 D: 0.787269 E: 1.000000 A B C D E A 0 4 4 6 0 B -4 0 4 10 6 C -4 -4 0 0 6 D -6 -10 0 0 -14 E 0 -6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000026652 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 E=24 A=15 B=11 so B is eliminated. Round 2 votes counts: E=31 D=27 C=25 A=17 so A is eliminated. Round 3 votes counts: E=35 D=35 C=30 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:208 A:207 E:201 C:199 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 6 0 B -4 0 4 10 6 C -4 -4 0 0 6 D -6 -10 0 0 -14 E 0 -6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000026652 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 6 0 B -4 0 4 10 6 C -4 -4 0 0 6 D -6 -10 0 0 -14 E 0 -6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000026652 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 6 0 B -4 0 4 10 6 C -4 -4 0 0 6 D -6 -10 0 0 -14 E 0 -6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000026652 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 846: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) A D E B C (8) C B D E A (6) B C E A D (6) A E D B C (6) B C D A E (5) A D E C B (5) E C B D A (4) E B C A D (4) D E A C B (4) C B D A E (4) E A D C B (3) E A D B C (3) E A B C D (3) B C E D A (3) B C A D E (3) A E B D C (3) E D A C B (2) D C B A E (2) C D B A E (2) C B E D A (2) B A E C D (2) B A C E D (2) E C D A B (1) D C A E B (1) C E B D A (1) C D E B A (1) B E C A D (1) B C D E A (1) B C A E D (1) B A D C E (1) B A C D E (1) A E D C B (1) Total count = 100 A B C D E A 0 -4 4 6 10 B 4 0 6 6 -16 C -4 -6 0 6 -16 D -6 -6 -6 0 4 E -10 16 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.533333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.133333 Sum of squares = 0.413333333339 Cumulative probabilities = A: 0.533333 B: 0.866667 C: 0.866667 D: 0.866667 E: 1.000000 A B C D E A 0 -4 4 6 10 B 4 0 6 6 -16 C -4 -6 0 6 -16 D -6 -6 -6 0 4 E -10 16 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.533333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.133333 Sum of squares = 0.413333333337 Cumulative probabilities = A: 0.533333 B: 0.866667 C: 0.866667 D: 0.866667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=23 E=20 C=16 D=15 so D is eliminated. Round 2 votes counts: A=31 B=26 E=24 C=19 so C is eliminated. Round 3 votes counts: B=42 A=32 E=26 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:209 A:208 B:200 D:193 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 4 6 10 B 4 0 6 6 -16 C -4 -6 0 6 -16 D -6 -6 -6 0 4 E -10 16 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.533333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.133333 Sum of squares = 0.413333333337 Cumulative probabilities = A: 0.533333 B: 0.866667 C: 0.866667 D: 0.866667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 6 10 B 4 0 6 6 -16 C -4 -6 0 6 -16 D -6 -6 -6 0 4 E -10 16 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.533333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.133333 Sum of squares = 0.413333333337 Cumulative probabilities = A: 0.533333 B: 0.866667 C: 0.866667 D: 0.866667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 6 10 B 4 0 6 6 -16 C -4 -6 0 6 -16 D -6 -6 -6 0 4 E -10 16 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.533333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.133333 Sum of squares = 0.413333333337 Cumulative probabilities = A: 0.533333 B: 0.866667 C: 0.866667 D: 0.866667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 847: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) D B E C A (7) B D E C A (7) E B D A C (6) C D A B E (6) C A D E B (6) C A D B E (6) D C B A E (5) B E D A C (5) A C E D B (5) D B C E A (4) A E C B D (4) A C E B D (4) C D B A E (3) B E A D C (3) B D C E A (3) A E B C D (3) E D B A C (2) E B A C D (2) D C E B A (2) E A B D C (1) E A B C D (1) D E A C B (1) D E A B C (1) D C A E B (1) C D A E B (1) C A B E D (1) B E A C D (1) B C A E D (1) A E C D B (1) Total count = 100 A B C D E A 0 -16 -6 -8 -6 B 16 0 8 -2 4 C 6 -8 0 -10 -4 D 8 2 10 0 6 E 6 -4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -8 -6 B 16 0 8 -2 4 C 6 -8 0 -10 -4 D 8 2 10 0 6 E 6 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 D=21 B=20 E=19 A=17 so A is eliminated. Round 2 votes counts: C=32 E=27 D=21 B=20 so B is eliminated. Round 3 votes counts: E=36 C=33 D=31 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:213 D:213 E:200 C:192 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -6 -8 -6 B 16 0 8 -2 4 C 6 -8 0 -10 -4 D 8 2 10 0 6 E 6 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -8 -6 B 16 0 8 -2 4 C 6 -8 0 -10 -4 D 8 2 10 0 6 E 6 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -8 -6 B 16 0 8 -2 4 C 6 -8 0 -10 -4 D 8 2 10 0 6 E 6 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 848: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E C A D B (7) B E D A C (7) B D E A C (7) E B D C A (6) D B A C E (5) C A E D B (5) C A D B E (5) B D A C E (5) E D B C A (4) D B E A C (4) E C D A B (3) E C A B D (3) E B C A D (3) A D C B E (3) E D C A B (2) C A E B D (2) C A D E B (2) A D B C E (2) E C D B A (1) E C B D A (1) E C B A D (1) E B D A C (1) E B C D A (1) D E C A B (1) D E B C A (1) D B A E C (1) D A C B E (1) C E A D B (1) C E A B D (1) B D A E C (1) B A E D C (1) B A D C E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 -4 -10 B 2 0 4 -12 8 C 0 -4 0 -8 -12 D 4 12 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.631226 E: 0.368774 Sum of squares = 0.534440703675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.631226 E: 1.000000 A B C D E A 0 -2 0 -4 -10 B 2 0 4 -12 8 C 0 -4 0 -8 -12 D 4 12 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=22 C=16 A=16 D=13 so D is eliminated. Round 2 votes counts: E=35 B=32 A=17 C=16 so C is eliminated. Round 3 votes counts: E=37 B=32 A=31 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:212 E:207 B:201 A:192 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -4 -10 B 2 0 4 -12 8 C 0 -4 0 -8 -12 D 4 12 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 -10 B 2 0 4 -12 8 C 0 -4 0 -8 -12 D 4 12 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 -10 B 2 0 4 -12 8 C 0 -4 0 -8 -12 D 4 12 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 849: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) B D A C E (6) C D A B E (5) A D E C B (5) E C B D A (4) E C B A D (4) A D C E B (4) E B A D C (3) E A D C B (3) E A B D C (3) C D B A E (3) B E A D C (3) B A D C E (3) A D B C E (3) E C A B D (2) E B C D A (2) E B C A D (2) E A D B C (2) D C A B E (2) D A C B E (2) C D E B A (2) C D E A B (2) C D A E B (2) B E D A C (2) B C D A E (2) A D C B E (2) A D B E C (2) E C D B A (1) E B A C D (1) E A C D B (1) D B A C E (1) D A B C E (1) C E D B A (1) C B E D A (1) C B D E A (1) B D A E C (1) B C D E A (1) B A E D C (1) B A D E C (1) A E D B C (1) A C E D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 8 16 0 B -8 0 -14 -10 -10 C -8 14 0 -6 -10 D -16 10 6 0 4 E 0 10 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500379 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499621 Sum of squares = 0.500000283813 Cumulative probabilities = A: 0.500379 B: 0.500379 C: 0.500379 D: 0.500379 E: 1.000000 A B C D E A 0 8 8 16 0 B -8 0 -14 -10 -10 C -8 14 0 -6 -10 D -16 10 6 0 4 E 0 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=20 A=20 C=17 D=6 so D is eliminated. Round 2 votes counts: E=37 A=23 B=21 C=19 so C is eliminated. Round 3 votes counts: E=42 A=32 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:208 D:202 C:195 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 16 0 B -8 0 -14 -10 -10 C -8 14 0 -6 -10 D -16 10 6 0 4 E 0 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 16 0 B -8 0 -14 -10 -10 C -8 14 0 -6 -10 D -16 10 6 0 4 E 0 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 16 0 B -8 0 -14 -10 -10 C -8 14 0 -6 -10 D -16 10 6 0 4 E 0 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 850: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (6) C D E A B (6) B A E D C (6) E A C B D (5) D C B E A (5) C E A D B (5) B A D E C (5) D C E B A (4) D C B A E (4) C D B A E (4) A E B C D (4) E A B D C (3) D B A E C (3) C D B E A (3) A E B D C (3) A B E D C (3) E C A D B (2) E C A B D (2) E A C D B (2) E A B C D (2) D B A C E (2) C D E B A (2) C B A D E (2) B D A E C (2) B D A C E (2) A B E C D (2) E A D C B (1) D C E A B (1) D B E A C (1) D B C E A (1) C E D A B (1) C E A B D (1) C D A E B (1) C D A B E (1) C A E B D (1) B A E C D (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 -4 2 6 B 6 0 -8 -10 6 C 4 8 0 -4 4 D -2 10 4 0 10 E -6 -6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 2 6 B 6 0 -8 -10 6 C 4 8 0 -4 4 D -2 10 4 0 10 E -6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000013 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 E=17 B=16 A=13 so A is eliminated. Round 2 votes counts: D=27 C=27 E=25 B=21 so B is eliminated. Round 3 votes counts: E=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:206 A:199 B:197 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 2 6 B 6 0 -8 -10 6 C 4 8 0 -4 4 D -2 10 4 0 10 E -6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000013 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 2 6 B 6 0 -8 -10 6 C 4 8 0 -4 4 D -2 10 4 0 10 E -6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000013 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 2 6 B 6 0 -8 -10 6 C 4 8 0 -4 4 D -2 10 4 0 10 E -6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000013 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 851: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) C A D E B (7) C A D B E (7) D B E C A (6) D E B C A (5) C D A E B (5) B E A D C (5) A E B C D (5) E B A D C (4) E A B D C (4) C D A B E (4) E D B A C (3) C D B A E (3) B A E D C (3) A C B E D (3) E D B C A (2) E B D A C (2) D C E B A (2) D B C E A (2) C D E A B (2) C D B E A (2) B D C A E (2) A E C B D (2) A B E C D (2) D E C B A (1) D C B E A (1) D B C A E (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B E D (1) C A B D E (1) B E D A C (1) B D A E C (1) Total count = 100 A B C D E A 0 6 -14 8 16 B -6 0 -4 -8 -8 C 14 4 0 10 8 D -8 8 -10 0 6 E -16 8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 8 16 B -6 0 -4 -8 -8 C 14 4 0 10 8 D -8 8 -10 0 6 E -16 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=20 D=18 E=15 B=12 so B is eliminated. Round 2 votes counts: C=35 A=23 E=21 D=21 so E is eliminated. Round 3 votes counts: A=36 C=35 D=29 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:208 D:198 E:189 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 8 16 B -6 0 -4 -8 -8 C 14 4 0 10 8 D -8 8 -10 0 6 E -16 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 8 16 B -6 0 -4 -8 -8 C 14 4 0 10 8 D -8 8 -10 0 6 E -16 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 8 16 B -6 0 -4 -8 -8 C 14 4 0 10 8 D -8 8 -10 0 6 E -16 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 852: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) D B A E C (8) B A D C E (8) C E A D B (5) C B A E D (5) B A C E D (5) D A E B C (4) B D A E C (4) B A D E C (4) E A D B C (3) D E C A B (3) D E A B C (3) C B E A D (3) B D A C E (3) E C D A B (2) E C A D B (2) D A B E C (2) C E D A B (2) C E B A D (2) B C A E D (2) B C A D E (2) A B D E C (2) E D C A B (1) E A C D B (1) E A C B D (1) D E B A C (1) D E A C B (1) D C E B A (1) D B E A C (1) D B A C E (1) C E D B A (1) C D E B A (1) B D C A E (1) B A E D C (1) B A C D E (1) A E D B C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 14 20 12 B 10 0 16 12 6 C -14 -16 0 -6 6 D -20 -12 6 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 20 12 B 10 0 16 12 6 C -14 -16 0 -6 6 D -20 -12 6 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999182 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=29 D=25 E=10 A=5 so A is eliminated. Round 2 votes counts: B=33 C=29 D=25 E=13 so E is eliminated. Round 3 votes counts: C=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:218 D:188 E:187 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 20 12 B 10 0 16 12 6 C -14 -16 0 -6 6 D -20 -12 6 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999182 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 20 12 B 10 0 16 12 6 C -14 -16 0 -6 6 D -20 -12 6 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999182 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 20 12 B 10 0 16 12 6 C -14 -16 0 -6 6 D -20 -12 6 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999182 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 853: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) D C B E A (7) B D C A E (7) A E C B D (7) E C A B D (6) C B D E A (6) A B C E D (6) E A C D B (5) D B C A E (5) A E D B C (4) A E B D C (4) E C A D B (3) E A D C B (3) C D B E A (3) A E B C D (3) D B A C E (2) C E D B A (2) C B D A E (2) A B D E C (2) E C D B A (1) E C D A B (1) E A D B C (1) D E A B C (1) D B E A C (1) D B C E A (1) D B A E C (1) C E A B D (1) C B E A D (1) C B A D E (1) B C D A E (1) B A C D E (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 4 18 -4 B -16 0 -16 16 -4 C -4 16 0 20 -8 D -18 -16 -20 0 -16 E 4 4 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 4 18 -4 B -16 0 -16 16 -4 C -4 16 0 20 -8 D -18 -16 -20 0 -16 E 4 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=28 D=18 C=16 B=9 so B is eliminated. Round 2 votes counts: E=29 A=29 D=25 C=17 so C is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:217 E:216 C:212 B:190 D:165 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 4 18 -4 B -16 0 -16 16 -4 C -4 16 0 20 -8 D -18 -16 -20 0 -16 E 4 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 18 -4 B -16 0 -16 16 -4 C -4 16 0 20 -8 D -18 -16 -20 0 -16 E 4 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 18 -4 B -16 0 -16 16 -4 C -4 16 0 20 -8 D -18 -16 -20 0 -16 E 4 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 854: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) C D E A B (8) E C D B A (7) B A E D C (7) E C B D A (6) E B C A D (5) E B A C D (5) D C A B E (5) D A C B E (5) C E D B A (4) D C A E B (3) B E A D C (3) B E A C D (3) A D B C E (3) E B D C A (2) E B D A C (2) E B C D A (2) E B A D C (2) C E D A B (2) C D A B E (2) A B D C E (2) E D B A C (1) E C B A D (1) D C E A B (1) D B E A C (1) D A B C E (1) C E B A D (1) C D E B A (1) C A D E B (1) B D E A C (1) A D C B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -22 -28 -16 B 8 0 -16 -12 -28 C 22 16 0 18 2 D 28 12 -18 0 -8 E 16 28 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -22 -28 -16 B 8 0 -16 -12 -28 C 22 16 0 18 2 D 28 12 -18 0 -8 E 16 28 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=29 D=16 B=14 A=8 so A is eliminated. Round 2 votes counts: E=33 C=29 D=20 B=18 so B is eliminated. Round 3 votes counts: E=46 C=30 D=24 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:229 E:225 D:207 B:176 A:163 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -22 -28 -16 B 8 0 -16 -12 -28 C 22 16 0 18 2 D 28 12 -18 0 -8 E 16 28 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -22 -28 -16 B 8 0 -16 -12 -28 C 22 16 0 18 2 D 28 12 -18 0 -8 E 16 28 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -22 -28 -16 B 8 0 -16 -12 -28 C 22 16 0 18 2 D 28 12 -18 0 -8 E 16 28 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 855: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) A B D C E (5) E A B C D (4) B A C E D (4) A B E C D (4) E D C A B (3) D E C A B (3) D C E B A (3) D C E A B (3) D C B A E (3) D A E B C (3) D A B C E (3) C D E B A (3) B C A D E (3) B A C D E (3) A E B D C (3) A B E D C (3) E C D B A (2) E B A C D (2) C E D B A (2) C D B A E (2) C B A E D (2) A D B C E (2) A B D E C (2) E D A C B (1) E C D A B (1) E C B D A (1) E C B A D (1) E C A B D (1) E B C A D (1) E A D B C (1) D E C B A (1) D E A C B (1) D E A B C (1) D C B E A (1) D B C A E (1) D B A C E (1) D A C B E (1) C E B D A (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) B E C A D (1) B C D A E (1) B C A E D (1) B A E C D (1) A E D B C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -4 -2 4 B 4 0 6 0 10 C 4 -6 0 6 12 D 2 0 -6 0 10 E -4 -10 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.728490 C: 0.000000 D: 0.271510 E: 0.000000 Sum of squares = 0.604415495317 Cumulative probabilities = A: 0.000000 B: 0.728490 C: 0.728490 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 4 B 4 0 6 0 10 C 4 -6 0 6 12 D 2 0 -6 0 10 E -4 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500297 C: 0.000000 D: 0.499703 E: 0.000000 Sum of squares = 0.500000176412 Cumulative probabilities = A: 0.000000 B: 0.500297 C: 0.500297 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=22 C=21 E=18 B=14 so B is eliminated. Round 2 votes counts: A=30 C=26 D=25 E=19 so E is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:210 C:208 D:203 A:197 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 4 B 4 0 6 0 10 C 4 -6 0 6 12 D 2 0 -6 0 10 E -4 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500297 C: 0.000000 D: 0.499703 E: 0.000000 Sum of squares = 0.500000176412 Cumulative probabilities = A: 0.000000 B: 0.500297 C: 0.500297 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 4 B 4 0 6 0 10 C 4 -6 0 6 12 D 2 0 -6 0 10 E -4 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500297 C: 0.000000 D: 0.499703 E: 0.000000 Sum of squares = 0.500000176412 Cumulative probabilities = A: 0.000000 B: 0.500297 C: 0.500297 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 4 B 4 0 6 0 10 C 4 -6 0 6 12 D 2 0 -6 0 10 E -4 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500297 C: 0.000000 D: 0.499703 E: 0.000000 Sum of squares = 0.500000176412 Cumulative probabilities = A: 0.000000 B: 0.500297 C: 0.500297 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 856: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (15) A E D C B (13) B C D A E (10) D A E C B (8) B E A C D (8) B C E A D (7) B C D E A (7) C D B A E (4) C B D A E (4) B D C A E (4) E A C D B (3) D C B A E (3) B E A D C (3) A E D B C (3) E A B C D (1) D C A E B (1) D B C A E (1) D A E B C (1) C D E A B (1) C D A E B (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 14 10 10 B 2 0 -8 -10 2 C -14 8 0 -8 -14 D -10 10 8 0 -8 E -10 -2 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.421487603302 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 10 10 B 2 0 -8 -10 2 C -14 8 0 -8 -14 D -10 10 8 0 -8 E -10 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.421487603281 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=19 A=18 D=14 C=10 so C is eliminated. Round 2 votes counts: B=43 D=20 E=19 A=18 so A is eliminated. Round 3 votes counts: B=43 E=36 D=21 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 E:205 D:200 B:193 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 14 10 10 B 2 0 -8 -10 2 C -14 8 0 -8 -14 D -10 10 8 0 -8 E -10 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.421487603281 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 10 10 B 2 0 -8 -10 2 C -14 8 0 -8 -14 D -10 10 8 0 -8 E -10 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.421487603281 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 10 10 B 2 0 -8 -10 2 C -14 8 0 -8 -14 D -10 10 8 0 -8 E -10 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.421487603281 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 857: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) C A E B D (10) E C A D B (7) D B E A C (7) E C A B D (4) D E B C A (4) B A C E D (4) D B E C A (3) B A C D E (3) A C B E D (3) E D C B A (2) E C D B A (2) E C B A D (2) E B D C A (2) D E C A B (2) D B A E C (2) D A B C E (2) C A E D B (2) B E D C A (2) B D E A C (2) B D A C E (2) A C E D B (2) A B C E D (2) A B C D E (2) E D B C A (1) E C D A B (1) D E C B A (1) D C A E B (1) D A C E B (1) D A C B E (1) C E D A B (1) C E A B D (1) B E C A D (1) B E A C D (1) B D E C A (1) B D A E C (1) B A D C E (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 0 -4 6 B 10 0 8 -8 4 C 0 -8 0 2 4 D 4 8 -2 0 -2 E -6 -4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407406 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -4 6 B 10 0 8 -8 4 C 0 -8 0 2 4 D 4 8 -2 0 -2 E -6 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407446 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=21 B=18 C=14 A=12 so A is eliminated. Round 2 votes counts: D=35 C=22 B=22 E=21 so E is eliminated. Round 3 votes counts: D=38 C=38 B=24 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:207 D:204 C:199 A:196 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 0 -4 6 B 10 0 8 -8 4 C 0 -8 0 2 4 D 4 8 -2 0 -2 E -6 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407446 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -4 6 B 10 0 8 -8 4 C 0 -8 0 2 4 D 4 8 -2 0 -2 E -6 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407446 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -4 6 B 10 0 8 -8 4 C 0 -8 0 2 4 D 4 8 -2 0 -2 E -6 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407446 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 858: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) E B D C A (8) D C A B E (7) E C D B A (5) D B C A E (5) C A E D B (5) B D E A C (4) B D A C E (4) E C A B D (3) E B C D A (3) D B A C E (3) C D A E B (3) B D A E C (3) A B D C E (3) E C D A B (2) E A C B D (2) E A B C D (2) D E B C A (2) B D E C A (2) A D C B E (2) A D B C E (2) A C E D B (2) E C B D A (1) E B A D C (1) E B A C D (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C E A (1) D A B C E (1) C E D A B (1) C E A D B (1) C D A B E (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A D C (1) B A D C E (1) A E C B D (1) A C E B D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -26 -22 -4 B 0 0 2 -12 -8 C 26 -2 0 -10 -6 D 22 12 10 0 -2 E 4 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -26 -22 -4 B 0 0 2 -12 -8 C 26 -2 0 -10 -6 D 22 12 10 0 -2 E 4 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=22 B=17 A=13 C=12 so C is eliminated. Round 2 votes counts: E=38 D=26 A=19 B=17 so B is eliminated. Round 3 votes counts: E=41 D=39 A=20 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:210 C:204 B:191 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -26 -22 -4 B 0 0 2 -12 -8 C 26 -2 0 -10 -6 D 22 12 10 0 -2 E 4 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -26 -22 -4 B 0 0 2 -12 -8 C 26 -2 0 -10 -6 D 22 12 10 0 -2 E 4 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -26 -22 -4 B 0 0 2 -12 -8 C 26 -2 0 -10 -6 D 22 12 10 0 -2 E 4 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 859: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) E B D C A (6) A E B D C (6) E B D A C (5) C D B A E (5) B D E C A (5) A D B E C (5) A C D B E (5) D B E A C (4) A E D B C (4) A C E D B (4) E B C D A (3) D B E C A (3) C E B D A (3) C A D B E (3) A E C B D (3) E B A D C (2) D B C E A (2) D B A E C (2) C E A B D (2) C A E D B (2) C A E B D (2) B E D C A (2) A C E B D (2) E D B A C (1) E C B D A (1) E A B D C (1) E A B C D (1) D A B C E (1) C D B E A (1) B E D A C (1) B D C E A (1) B C D E A (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 0 -10 -6 B 12 0 14 10 2 C 0 -14 0 -8 -10 D 10 -10 8 0 -2 E 6 -2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -10 -6 B 12 0 14 10 2 C 0 -14 0 -8 -10 D 10 -10 8 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=26 E=20 D=12 B=10 so B is eliminated. Round 2 votes counts: A=32 C=27 E=23 D=18 so D is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:208 D:203 A:186 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -10 -6 B 12 0 14 10 2 C 0 -14 0 -8 -10 D 10 -10 8 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -10 -6 B 12 0 14 10 2 C 0 -14 0 -8 -10 D 10 -10 8 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -10 -6 B 12 0 14 10 2 C 0 -14 0 -8 -10 D 10 -10 8 0 -2 E 6 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 860: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) D E B A C (7) B D E A C (6) A D C B E (5) A C D B E (5) E B C D A (4) D B E A C (4) A C B D E (4) D B A E C (3) D A E C B (3) D A B E C (3) C E B A D (3) B E D C A (3) B E C A D (3) A D B C E (3) A C D E B (3) E D B C A (2) E C B D A (2) E B D C A (2) D E A B C (2) C B E A D (2) C A E D B (2) C A D E B (2) B E C D A (2) B C E A D (2) A D C E B (2) E D C B A (1) E D B A C (1) D E B C A (1) D A C E B (1) D A B C E (1) C E D A B (1) C B A E D (1) B E D A C (1) B C A E D (1) B A C E D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 14 0 2 B 4 0 6 -4 22 C -14 -6 0 -4 0 D 0 4 4 0 12 E -2 -22 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.191925 B: 0.000000 C: 0.000000 D: 0.808075 E: 0.000000 Sum of squares = 0.689820092825 Cumulative probabilities = A: 0.191925 B: 0.191925 C: 0.191925 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 0 2 B 4 0 6 -4 22 C -14 -6 0 -4 0 D 0 4 4 0 12 E -2 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499299 B: 0.000000 C: 0.000000 D: 0.500701 E: 0.000000 Sum of squares = 0.500000981444 Cumulative probabilities = A: 0.499299 B: 0.499299 C: 0.499299 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=24 C=20 B=19 E=12 so E is eliminated. Round 2 votes counts: D=29 B=25 A=24 C=22 so C is eliminated. Round 3 votes counts: A=37 B=33 D=30 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:210 A:206 C:188 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 14 0 2 B 4 0 6 -4 22 C -14 -6 0 -4 0 D 0 4 4 0 12 E -2 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499299 B: 0.000000 C: 0.000000 D: 0.500701 E: 0.000000 Sum of squares = 0.500000981444 Cumulative probabilities = A: 0.499299 B: 0.499299 C: 0.499299 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 0 2 B 4 0 6 -4 22 C -14 -6 0 -4 0 D 0 4 4 0 12 E -2 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499299 B: 0.000000 C: 0.000000 D: 0.500701 E: 0.000000 Sum of squares = 0.500000981444 Cumulative probabilities = A: 0.499299 B: 0.499299 C: 0.499299 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 0 2 B 4 0 6 -4 22 C -14 -6 0 -4 0 D 0 4 4 0 12 E -2 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499299 B: 0.000000 C: 0.000000 D: 0.500701 E: 0.000000 Sum of squares = 0.500000981444 Cumulative probabilities = A: 0.499299 B: 0.499299 C: 0.499299 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 861: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (14) D B E C A (8) E C D B A (6) A C E D B (6) A C E B D (6) D E B C A (5) A B C E D (5) B A D C E (4) E C D A B (3) B D E A C (3) B C A E D (3) E D C B A (2) D B E A C (2) D B A E C (2) D A E C B (2) C E A D B (2) B E C D A (2) B D A E C (2) B A C E D (2) A D E C B (2) A D C B E (2) A D B C E (2) A C B E D (2) A B D C E (2) D E C B A (1) D E A C B (1) C E D B A (1) C E A B D (1) C A E D B (1) C A E B D (1) B E D C A (1) B D C A E (1) B A D E C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -20 -4 -12 -4 B 20 0 20 0 18 C 4 -20 0 -16 -16 D 12 0 16 0 12 E 4 -18 16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.096322 C: 0.000000 D: 0.903678 E: 0.000000 Sum of squares = 0.82591126815 Cumulative probabilities = A: 0.000000 B: 0.096322 C: 0.096322 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -12 -4 B 20 0 20 0 18 C 4 -20 0 -16 -16 D 12 0 16 0 12 E 4 -18 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=29 D=21 E=11 C=6 so C is eliminated. Round 2 votes counts: B=33 A=31 D=21 E=15 so E is eliminated. Round 3 votes counts: A=34 D=33 B=33 so D is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:220 E:195 A:180 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -4 -12 -4 B 20 0 20 0 18 C 4 -20 0 -16 -16 D 12 0 16 0 12 E 4 -18 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -12 -4 B 20 0 20 0 18 C 4 -20 0 -16 -16 D 12 0 16 0 12 E 4 -18 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -12 -4 B 20 0 20 0 18 C 4 -20 0 -16 -16 D 12 0 16 0 12 E 4 -18 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 862: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) D E A C B (6) D A B C E (6) A C E B D (5) E C B A D (4) A E C D B (4) E C A D B (3) E B D C A (3) D E C A B (3) D E A B C (3) D B C E A (3) D B A C E (3) D A E C B (3) B C A E D (3) A D B C E (3) E D C B A (2) E C D B A (2) E C B D A (2) E C A B D (2) D E C B A (2) D E B C A (2) D B E C A (2) D B A E C (2) D A B E C (2) B A C E D (2) B A C D E (2) A E D C B (2) A C B E D (2) A B C E D (2) E D C A B (1) E D A C B (1) C E A B D (1) C B E A D (1) B E D C A (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D A E (1) B A D C E (1) A E C B D (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 4 4 -6 -6 B -4 0 2 -14 -6 C -4 -2 0 -8 -6 D 6 14 8 0 -4 E 6 6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 -6 -6 B -4 0 2 -14 -6 C -4 -2 0 -8 -6 D 6 14 8 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=21 E=20 B=20 C=2 so C is eliminated. Round 2 votes counts: D=37 E=21 B=21 A=21 so E is eliminated. Round 3 votes counts: D=43 B=30 A=27 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:211 A:198 C:190 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 -6 -6 B -4 0 2 -14 -6 C -4 -2 0 -8 -6 D 6 14 8 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -6 -6 B -4 0 2 -14 -6 C -4 -2 0 -8 -6 D 6 14 8 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -6 -6 B -4 0 2 -14 -6 C -4 -2 0 -8 -6 D 6 14 8 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 863: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (6) B A C D E (6) E C A D B (5) D E A C B (5) C E A B D (5) D A E C B (4) A C B D E (4) E D C A B (3) D B A E C (3) C E B A D (3) B D A C E (3) B C A E D (3) A D B C E (3) E D C B A (2) E D B C A (2) E C D B A (2) E C D A B (2) E C B A D (2) D E B C A (2) D E B A C (2) D A E B C (2) D A B C E (2) C B A E D (2) B E C D A (2) B D E C A (2) B D A E C (2) B A D C E (2) A B D C E (2) E C A B D (1) E B C A D (1) D E A B C (1) D B E A C (1) D A B E C (1) C E A D B (1) C B E A D (1) C A E D B (1) C A B E D (1) B C D A E (1) B C A D E (1) B A C E D (1) A D C B E (1) A C E D B (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 12 -2 B 4 0 6 8 6 C 2 -6 0 10 6 D -12 -8 -10 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 12 -2 B 4 0 6 8 6 C 2 -6 0 10 6 D -12 -8 -10 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 E=20 C=14 A=14 so C is eliminated. Round 2 votes counts: B=32 E=29 D=23 A=16 so A is eliminated. Round 3 votes counts: B=42 E=31 D=27 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:206 A:202 E:194 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 12 -2 B 4 0 6 8 6 C 2 -6 0 10 6 D -12 -8 -10 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 12 -2 B 4 0 6 8 6 C 2 -6 0 10 6 D -12 -8 -10 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 12 -2 B 4 0 6 8 6 C 2 -6 0 10 6 D -12 -8 -10 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 864: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) E B D A C (6) C D A B E (6) B A E D C (6) C A D B E (5) A D C B E (5) A B E D C (5) E B A C D (4) D C A B E (4) C E D B A (4) A B D C E (4) E C B D A (3) E B D C A (3) E B C A D (3) C D E A B (3) A B D E C (3) E D B C A (2) E B C D A (2) C D E B A (2) C D A E B (2) B A D E C (2) A C D B E (2) A C B E D (2) E D C B A (1) E C D B A (1) E C A B D (1) D B E A C (1) D B C A E (1) D B A E C (1) D B A C E (1) D A C B E (1) B E A D C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 14 12 4 B 10 0 14 14 2 C -14 -14 0 -18 -12 D -12 -14 18 0 -12 E -4 -2 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 12 4 B 10 0 14 14 2 C -14 -14 0 -18 -12 D -12 -14 18 0 -12 E -4 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=23 C=22 D=9 B=9 so D is eliminated. Round 2 votes counts: E=37 C=26 A=24 B=13 so B is eliminated. Round 3 votes counts: E=39 A=34 C=27 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:210 E:209 D:190 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 12 4 B 10 0 14 14 2 C -14 -14 0 -18 -12 D -12 -14 18 0 -12 E -4 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 12 4 B 10 0 14 14 2 C -14 -14 0 -18 -12 D -12 -14 18 0 -12 E -4 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 12 4 B 10 0 14 14 2 C -14 -14 0 -18 -12 D -12 -14 18 0 -12 E -4 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 865: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) E A B D C (6) D C E B A (5) E D A B C (4) E C A D B (4) E A B C D (4) D B C E A (4) B D C A E (4) B C A D E (4) A E B C D (4) E A D C B (3) D E A C B (3) D B C A E (3) C B D A E (3) B A E C D (3) B A C E D (3) A E B D C (3) E D A C B (2) E A C D B (2) E A C B D (2) D C E A B (2) C D E A B (2) C D B A E (2) C B A D E (2) B C D A E (2) B A C D E (2) A B E C D (2) E A D B C (1) D E C A B (1) D E A B C (1) D B E A C (1) C D B E A (1) C A B E D (1) B D A E C (1) B C A E D (1) B A E D C (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 2 2 -12 B 0 0 10 2 0 C -2 -10 0 -6 2 D -2 -2 6 0 2 E 12 0 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.717922 C: 0.000000 D: 0.000000 E: 0.282078 Sum of squares = 0.59497987673 Cumulative probabilities = A: 0.000000 B: 0.717922 C: 0.717922 D: 0.717922 E: 1.000000 A B C D E A 0 0 2 2 -12 B 0 0 10 2 0 C -2 -10 0 -6 2 D -2 -2 6 0 2 E 12 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500322 C: 0.000000 D: 0.000000 E: 0.499678 Sum of squares = 0.500000207249 Cumulative probabilities = A: 0.000000 B: 0.500322 C: 0.500322 D: 0.500322 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 B=21 A=12 C=11 so C is eliminated. Round 2 votes counts: D=33 E=28 B=26 A=13 so A is eliminated. Round 3 votes counts: E=36 D=33 B=31 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:206 E:204 D:202 A:196 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 2 -12 B 0 0 10 2 0 C -2 -10 0 -6 2 D -2 -2 6 0 2 E 12 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500322 C: 0.000000 D: 0.000000 E: 0.499678 Sum of squares = 0.500000207249 Cumulative probabilities = A: 0.000000 B: 0.500322 C: 0.500322 D: 0.500322 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 -12 B 0 0 10 2 0 C -2 -10 0 -6 2 D -2 -2 6 0 2 E 12 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500322 C: 0.000000 D: 0.000000 E: 0.499678 Sum of squares = 0.500000207249 Cumulative probabilities = A: 0.000000 B: 0.500322 C: 0.500322 D: 0.500322 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 -12 B 0 0 10 2 0 C -2 -10 0 -6 2 D -2 -2 6 0 2 E 12 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500322 C: 0.000000 D: 0.000000 E: 0.499678 Sum of squares = 0.500000207249 Cumulative probabilities = A: 0.000000 B: 0.500322 C: 0.500322 D: 0.500322 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 866: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (15) E A C D B (10) B D C A E (10) A E B C D (7) A B E D C (7) C D B E A (6) A E C D B (6) E C D A B (5) B A D C E (5) E C D B A (4) D C B E A (4) C D E B A (4) B D C E A (3) E A B C D (2) A B D E C (2) E C A D B (1) D C B A E (1) D B C A E (1) C D B A E (1) B E D C A (1) B E C D A (1) B D A C E (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 16 16 16 18 B -16 0 12 10 -10 C -16 -12 0 -2 -22 D -16 -10 2 0 -20 E -18 10 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 16 18 B -16 0 12 10 -10 C -16 -12 0 -2 -22 D -16 -10 2 0 -20 E -18 10 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 E=22 B=21 C=11 D=6 so D is eliminated. Round 2 votes counts: A=40 E=22 B=22 C=16 so C is eliminated. Round 3 votes counts: A=40 B=34 E=26 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 E:217 B:198 D:178 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 16 18 B -16 0 12 10 -10 C -16 -12 0 -2 -22 D -16 -10 2 0 -20 E -18 10 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 16 18 B -16 0 12 10 -10 C -16 -12 0 -2 -22 D -16 -10 2 0 -20 E -18 10 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 16 18 B -16 0 12 10 -10 C -16 -12 0 -2 -22 D -16 -10 2 0 -20 E -18 10 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 867: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) E C D B A (6) E D B C A (5) E D A B C (5) C D B E A (5) B D C E A (5) E A D B C (4) C E D B A (4) C B D E A (4) A B D E C (4) E D C B A (3) E D B A C (3) C B D A E (3) C A B D E (3) A C B D E (3) A B C D E (3) D E B C A (2) D B E A C (2) C A E B D (2) B D C A E (2) A E C D B (2) A E C B D (2) A E B D C (2) A C E B D (2) E C A D B (1) E A D C B (1) D E B A C (1) D C B E A (1) D B C E A (1) D B A E C (1) C E D A B (1) C E B D A (1) C D E B A (1) C B A E D (1) C B A D E (1) B D A E C (1) A E D B C (1) A D B E C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -22 -20 -30 -28 B 22 0 4 -18 2 C 20 -4 0 -8 -12 D 30 18 8 0 4 E 28 -2 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -20 -30 -28 B 22 0 4 -18 2 C 20 -4 0 -8 -12 D 30 18 8 0 4 E 28 -2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 A=22 D=16 B=8 so B is eliminated. Round 2 votes counts: E=28 C=26 D=24 A=22 so A is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:230 E:217 B:205 C:198 A:150 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -20 -30 -28 B 22 0 4 -18 2 C 20 -4 0 -8 -12 D 30 18 8 0 4 E 28 -2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -20 -30 -28 B 22 0 4 -18 2 C 20 -4 0 -8 -12 D 30 18 8 0 4 E 28 -2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -20 -30 -28 B 22 0 4 -18 2 C 20 -4 0 -8 -12 D 30 18 8 0 4 E 28 -2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 868: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) A B C D E (7) E D B A C (6) D E C B A (6) C D E B A (6) A B D E C (6) C E D A B (4) B A D E C (4) B A D C E (4) A B C E D (4) E D B C A (3) E D A B C (3) E C D A B (3) D E B C A (3) B D E A C (3) A B E D C (3) E A D C B (2) C E D B A (2) C A B D E (2) B D C E A (2) B C A D E (2) E D C A B (1) E D A C B (1) E C A D B (1) C D B E A (1) C B D A E (1) C B A D E (1) C A E D B (1) C A B E D (1) B D E C A (1) B D A E C (1) B A C D E (1) A C E D B (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 0 -12 -14 B 12 0 10 -6 -4 C 0 -10 0 -18 -14 D 12 6 18 0 6 E 14 4 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -12 -14 B 12 0 10 -6 -4 C 0 -10 0 -18 -14 D 12 6 18 0 6 E 14 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=25 C=19 B=18 D=9 so D is eliminated. Round 2 votes counts: E=38 A=25 C=19 B=18 so B is eliminated. Round 3 votes counts: E=42 A=35 C=23 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:213 B:206 A:181 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 0 -12 -14 B 12 0 10 -6 -4 C 0 -10 0 -18 -14 D 12 6 18 0 6 E 14 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -12 -14 B 12 0 10 -6 -4 C 0 -10 0 -18 -14 D 12 6 18 0 6 E 14 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -12 -14 B 12 0 10 -6 -4 C 0 -10 0 -18 -14 D 12 6 18 0 6 E 14 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 869: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (8) E C B D A (7) A B E C D (7) A B D C E (7) C E D B A (6) E B C A D (5) D C E B A (5) E C D B A (4) E C B A D (3) D E C A B (3) D C A B E (3) C E B D A (3) B A C E D (3) D C B A E (2) D A C B E (2) B A E C D (2) A D B E C (2) A D B C E (2) A B E D C (2) A B D E C (2) E D C A B (1) E D A C B (1) E C A D B (1) E B A C D (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A E B (1) D A E C B (1) C D E B A (1) C D B E A (1) C B E D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C E A D (1) B C D A E (1) B A D C E (1) B A C D E (1) A E D B C (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -6 0 B 2 0 2 6 4 C 4 -2 0 8 4 D 6 -6 -8 0 -8 E 0 -4 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -6 0 B 2 0 2 6 4 C 4 -2 0 8 4 D 6 -6 -8 0 -8 E 0 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 A=25 C=12 B=12 so C is eliminated. Round 2 votes counts: E=34 D=28 A=25 B=13 so B is eliminated. Round 3 votes counts: E=39 A=32 D=29 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:207 C:207 E:200 A:194 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 0 B 2 0 2 6 4 C 4 -2 0 8 4 D 6 -6 -8 0 -8 E 0 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 0 B 2 0 2 6 4 C 4 -2 0 8 4 D 6 -6 -8 0 -8 E 0 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 0 B 2 0 2 6 4 C 4 -2 0 8 4 D 6 -6 -8 0 -8 E 0 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 870: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (14) E D B A C (7) D E B A C (5) A C B D E (5) E D C A B (4) E D B C A (4) D E C A B (4) C A E B D (4) C A B E D (4) B A C E D (4) E B D A C (3) C A D E B (3) B D E A C (3) B A D C E (3) E C A D B (2) D E C B A (2) D E B C A (2) D B A C E (2) C A D B E (2) B E D A C (2) B A E C D (2) B A C D E (2) A C B E D (2) A B C E D (2) A B C D E (2) E B A D C (1) D C A B E (1) D B A E C (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E D B (1) B E A C D (1) B D A C E (1) B A D E C (1) Total count = 100 A B C D E A 0 8 -2 12 14 B -8 0 -4 12 8 C 2 4 0 8 12 D -12 -12 -8 0 10 E -14 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 12 14 B -8 0 -4 12 8 C 2 4 0 8 12 D -12 -12 -8 0 10 E -14 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=21 B=19 D=17 A=11 so A is eliminated. Round 2 votes counts: C=39 B=23 E=21 D=17 so D is eliminated. Round 3 votes counts: C=40 E=34 B=26 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:216 C:213 B:204 D:189 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 12 14 B -8 0 -4 12 8 C 2 4 0 8 12 D -12 -12 -8 0 10 E -14 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 12 14 B -8 0 -4 12 8 C 2 4 0 8 12 D -12 -12 -8 0 10 E -14 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 12 14 B -8 0 -4 12 8 C 2 4 0 8 12 D -12 -12 -8 0 10 E -14 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 871: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) B E A D C (7) E B C A D (5) D C A B E (5) D A C B E (5) A B E D C (5) E C B A D (4) C D A E B (4) A D B C E (4) D C B E A (3) C D E B A (3) D C B A E (2) D A B C E (2) C E B D A (2) C D E A B (2) B E C D A (2) B D A E C (2) B A E D C (2) A E C B D (2) A E B C D (2) A D C B E (2) A D B E C (2) A B D E C (2) E C B D A (1) E C A B D (1) E A B C D (1) D B C A E (1) D B A E C (1) D A B E C (1) C E D B A (1) C E B A D (1) C E A D B (1) C E A B D (1) C A E D B (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A C D (1) B D E C A (1) B D E A C (1) B A D E C (1) A E D B C (1) A E B D C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 14 18 2 B 4 0 12 12 8 C -14 -12 0 -8 -14 D -18 -12 8 0 -8 E -2 -8 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 18 2 B 4 0 12 12 8 C -14 -12 0 -8 -14 D -18 -12 8 0 -8 E -2 -8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=21 D=20 B=19 C=17 so C is eliminated. Round 2 votes counts: D=29 E=27 A=25 B=19 so B is eliminated. Round 3 votes counts: E=39 D=33 A=28 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:218 A:215 E:206 D:185 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 18 2 B 4 0 12 12 8 C -14 -12 0 -8 -14 D -18 -12 8 0 -8 E -2 -8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 18 2 B 4 0 12 12 8 C -14 -12 0 -8 -14 D -18 -12 8 0 -8 E -2 -8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 18 2 B 4 0 12 12 8 C -14 -12 0 -8 -14 D -18 -12 8 0 -8 E -2 -8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 872: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (14) D B C A E (7) C B A E D (7) E A D B C (5) D A B C E (5) C B E A D (5) E D A C B (4) E D A B C (4) E C B A D (4) D E A B C (4) C B D A E (4) E C A B D (3) E A B C D (3) D C B E A (3) D A E B C (3) D A B E C (3) C B A D E (3) D E C B A (2) C E B A D (2) B D C A E (2) E D C B A (1) E A D C B (1) E A C D B (1) C E B D A (1) C B E D A (1) C B D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) A E B C D (1) A D E B C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 8 -18 B -8 0 -14 12 -8 C -2 14 0 8 -8 D -8 -12 -8 0 -18 E 18 8 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 8 -18 B -8 0 -14 12 -8 C -2 14 0 8 -8 D -8 -12 -8 0 -18 E 18 8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=27 C=24 B=5 A=4 so A is eliminated. Round 2 votes counts: E=41 D=28 C=24 B=7 so B is eliminated. Round 3 votes counts: E=41 D=31 C=28 so C is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:206 A:200 B:191 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 8 -18 B -8 0 -14 12 -8 C -2 14 0 8 -8 D -8 -12 -8 0 -18 E 18 8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 8 -18 B -8 0 -14 12 -8 C -2 14 0 8 -8 D -8 -12 -8 0 -18 E 18 8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 8 -18 B -8 0 -14 12 -8 C -2 14 0 8 -8 D -8 -12 -8 0 -18 E 18 8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 873: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (7) D E B A C (5) B E D A C (5) B D E C A (5) D C B E A (4) D C A E B (4) C A B D E (4) B C A E D (4) A E C D B (4) A E C B D (4) E A B D C (3) D C E A B (3) C A E D B (3) B E A D C (3) B A E C D (3) B A C E D (3) E B D A C (2) D E C A B (2) D E A C B (2) D E A B C (2) C B A D E (2) B C E A D (2) B C D A E (2) E D B A C (1) E D A C B (1) E D A B C (1) E A B C D (1) D E B C A (1) D C E B A (1) D B E C A (1) D B E A C (1) D B C E A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B A E D (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B E D (1) B E A C D (1) B D E A C (1) A E B C D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 6 4 2 B 0 0 -2 12 -6 C -6 2 0 0 -2 D -4 -12 0 0 -6 E -2 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.860021 B: 0.139979 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.759230591277 Cumulative probabilities = A: 0.860021 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 2 B 0 0 -2 12 -6 C -6 2 0 0 -2 D -4 -12 0 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000064538 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 A=18 C=17 E=9 so E is eliminated. Round 2 votes counts: B=31 D=30 A=22 C=17 so C is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:206 B:202 C:197 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 2 B 0 0 -2 12 -6 C -6 2 0 0 -2 D -4 -12 0 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000064538 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 2 B 0 0 -2 12 -6 C -6 2 0 0 -2 D -4 -12 0 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000064538 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 2 B 0 0 -2 12 -6 C -6 2 0 0 -2 D -4 -12 0 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000064538 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 874: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) B A C D E (8) E D C A B (7) D A E B C (6) A D B E C (6) A D B C E (6) C E B D A (5) B C E A D (5) E C D B A (4) D E A C B (4) E C B D A (3) D A E C B (3) C E D B A (3) C B E A D (3) B C A E D (3) A B D E C (3) D E C A B (2) B E C A D (2) B A D C E (2) E D A B C (1) E B D C A (1) D E A B C (1) D C A E B (1) C D E A B (1) C B A E D (1) C A B D E (1) B E A D C (1) B A E D C (1) B A E C D (1) B A D E C (1) A D C B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 16 16 14 B 8 0 20 8 16 C -16 -20 0 4 6 D -16 -8 -4 0 -2 E -14 -16 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 16 16 14 B 8 0 20 8 16 C -16 -20 0 4 6 D -16 -8 -4 0 -2 E -14 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=19 D=17 E=16 C=14 so C is eliminated. Round 2 votes counts: B=38 E=24 A=20 D=18 so D is eliminated. Round 3 votes counts: B=38 E=32 A=30 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:219 C:187 D:185 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 16 16 14 B 8 0 20 8 16 C -16 -20 0 4 6 D -16 -8 -4 0 -2 E -14 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 16 14 B 8 0 20 8 16 C -16 -20 0 4 6 D -16 -8 -4 0 -2 E -14 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 16 14 B 8 0 20 8 16 C -16 -20 0 4 6 D -16 -8 -4 0 -2 E -14 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 875: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) E C A B D (8) B D C E A (8) E C A D B (7) A E C B D (7) B C E A D (6) D B A C E (5) B D A C E (5) B C D E A (5) D B C E A (4) D A B E C (4) A D E C B (4) C E D A B (3) C E A D B (3) B C E D A (3) D B A E C (2) D A E C B (2) B A D E C (2) D E C A B (1) D C E B A (1) D C E A B (1) D A C E B (1) D A C B E (1) C E D B A (1) C E B A D (1) C D E A B (1) C B E D A (1) B A E C D (1) A E D C B (1) Total count = 100 A B C D E A 0 10 -8 2 -8 B -10 0 -10 -6 -6 C 8 10 0 16 0 D -2 6 -16 0 -6 E 8 6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.348829 D: 0.000000 E: 0.651171 Sum of squares = 0.545705280925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.348829 D: 0.348829 E: 1.000000 A B C D E A 0 10 -8 2 -8 B -10 0 -10 -6 -6 C 8 10 0 16 0 D -2 6 -16 0 -6 E 8 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=23 D=22 E=15 C=10 so C is eliminated. Round 2 votes counts: B=31 E=23 D=23 A=23 so E is eliminated. Round 3 votes counts: A=41 B=32 D=27 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 E:210 A:198 D:191 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 2 -8 B -10 0 -10 -6 -6 C 8 10 0 16 0 D -2 6 -16 0 -6 E 8 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 2 -8 B -10 0 -10 -6 -6 C 8 10 0 16 0 D -2 6 -16 0 -6 E 8 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 2 -8 B -10 0 -10 -6 -6 C 8 10 0 16 0 D -2 6 -16 0 -6 E 8 6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 876: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) B C D E A (7) B C D A E (7) E B C A D (5) D A C B E (5) C B A E D (5) E B C D A (4) E A C B D (4) D E A B C (4) D B C A E (4) D A E C B (4) D A E B C (4) E C B A D (3) C B D A E (3) C B A D E (3) B C E A D (3) E A D C B (2) E A D B C (2) D B C E A (2) B D C E A (2) B C E D A (2) A E D C B (2) A D E C B (2) E B D C A (1) E A B C D (1) D B E C A (1) D B E A C (1) D B A C E (1) D A B C E (1) C E B A D (1) C A B D E (1) A E C D B (1) A E C B D (1) A D C B E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -26 -24 -6 -6 B 26 0 4 26 16 C 24 -4 0 22 16 D 6 -26 -22 0 8 E 6 -16 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -24 -6 -6 B 26 0 4 26 16 C 24 -4 0 22 16 D 6 -26 -22 0 8 E 6 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=22 C=21 B=21 A=9 so A is eliminated. Round 2 votes counts: D=30 E=26 C=23 B=21 so B is eliminated. Round 3 votes counts: C=42 D=32 E=26 so E is eliminated. Round 4 votes counts: C=61 D=39 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:236 C:229 D:183 E:183 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -24 -6 -6 B 26 0 4 26 16 C 24 -4 0 22 16 D 6 -26 -22 0 8 E 6 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -24 -6 -6 B 26 0 4 26 16 C 24 -4 0 22 16 D 6 -26 -22 0 8 E 6 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -24 -6 -6 B 26 0 4 26 16 C 24 -4 0 22 16 D 6 -26 -22 0 8 E 6 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 877: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) E A C D B (6) A E C D B (6) E D A C B (4) C A E D B (4) A E B D C (4) E D C A B (3) E A D C B (3) D B C E A (3) C D B E A (3) B D C A E (3) A E C B D (3) A C B E D (3) A B E D C (3) E D A B C (2) E C A D B (2) E A D B C (2) D E C B A (2) D E B C A (2) D C B E A (2) D B E C A (2) C B D A E (2) B D E A C (2) B D A E C (2) B C D A E (2) B C A D E (2) B A C D E (2) A E B C D (2) A C E B D (2) E B D A C (1) E A B D C (1) D B E A C (1) C E D A B (1) C E A D B (1) C A B E D (1) C A B D E (1) B D A C E (1) B A D E C (1) B A C E D (1) A E D C B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 14 10 6 -4 B -14 0 -2 -2 -6 C -10 2 0 -8 -12 D -6 2 8 0 -16 E 4 6 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 10 6 -4 B -14 0 -2 -2 -6 C -10 2 0 -8 -12 D -6 2 8 0 -16 E 4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 E=24 C=13 D=12 so D is eliminated. Round 2 votes counts: B=31 E=28 A=26 C=15 so C is eliminated. Round 3 votes counts: B=38 A=32 E=30 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 A:213 D:194 B:188 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 10 6 -4 B -14 0 -2 -2 -6 C -10 2 0 -8 -12 D -6 2 8 0 -16 E 4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 6 -4 B -14 0 -2 -2 -6 C -10 2 0 -8 -12 D -6 2 8 0 -16 E 4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 6 -4 B -14 0 -2 -2 -6 C -10 2 0 -8 -12 D -6 2 8 0 -16 E 4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 878: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) D B A E C (6) E D C B A (5) C E A B D (5) C A E B D (5) E D B C A (4) D E B C A (4) C A B E D (4) A B D C E (4) E C B A D (3) D A B C E (3) C A E D B (3) A D B C E (3) A B C E D (3) A B C D E (3) E D C A B (2) E C D B A (2) E C D A B (2) E C B D A (2) E B D C A (2) D E C B A (2) D B E A C (2) D A C B E (2) C E A D B (2) B E A C D (2) B A C E D (2) A C B D E (2) E B C D A (1) D E A C B (1) D E A B C (1) D A C E B (1) C E D A B (1) C B A E D (1) B D A E C (1) B D A C E (1) B A D E C (1) B A D C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 0 -8 -4 B 2 0 6 -12 -14 C 0 -6 0 -10 -4 D 8 12 10 0 -4 E 4 14 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 -8 -4 B 2 0 6 -12 -14 C 0 -6 0 -10 -4 D 8 12 10 0 -4 E 4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=23 C=21 A=17 B=8 so B is eliminated. Round 2 votes counts: D=33 E=25 C=21 A=21 so C is eliminated. Round 3 votes counts: A=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:213 A:193 B:191 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 -8 -4 B 2 0 6 -12 -14 C 0 -6 0 -10 -4 D 8 12 10 0 -4 E 4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -8 -4 B 2 0 6 -12 -14 C 0 -6 0 -10 -4 D 8 12 10 0 -4 E 4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -8 -4 B 2 0 6 -12 -14 C 0 -6 0 -10 -4 D 8 12 10 0 -4 E 4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 879: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (23) B C A D E (12) D A C B E (11) E D A C B (6) D E A C B (6) A C B D E (6) E D A B C (5) C B A D E (5) D A B C E (4) E B C D A (3) E D B C A (2) C B A E D (2) A C D B E (2) E D B A C (1) E C B A D (1) E B D C A (1) D E B A C (1) D E A B C (1) D B C A E (1) D B A C E (1) D A E C B (1) D A E B C (1) C A B D E (1) B D C A E (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 -12 -8 8 -2 B 12 0 18 14 -4 C 8 -18 0 14 -4 D -8 -14 -14 0 8 E 2 4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.000000 D: 0.153846 E: 0.538462 Sum of squares = 0.408284023664 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.307692 D: 0.461538 E: 1.000000 A B C D E A 0 -12 -8 8 -2 B 12 0 18 14 -4 C 8 -18 0 14 -4 D -8 -14 -14 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.000000 D: 0.153846 E: 0.538462 Sum of squares = 0.408284023655 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.307692 D: 0.461538 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 D=27 B=15 C=8 A=8 so C is eliminated. Round 2 votes counts: E=42 D=27 B=22 A=9 so A is eliminated. Round 3 votes counts: E=42 D=29 B=29 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:201 C:200 A:193 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 8 -2 B 12 0 18 14 -4 C 8 -18 0 14 -4 D -8 -14 -14 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.000000 D: 0.153846 E: 0.538462 Sum of squares = 0.408284023655 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.307692 D: 0.461538 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 8 -2 B 12 0 18 14 -4 C 8 -18 0 14 -4 D -8 -14 -14 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.000000 D: 0.153846 E: 0.538462 Sum of squares = 0.408284023655 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.307692 D: 0.461538 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 8 -2 B 12 0 18 14 -4 C 8 -18 0 14 -4 D -8 -14 -14 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.000000 D: 0.153846 E: 0.538462 Sum of squares = 0.408284023655 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.307692 D: 0.461538 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 880: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) C D E A B (7) A B E C D (7) D E C B A (6) C D A E B (6) D E B C A (5) C A B E D (5) E D B C A (4) C D A B E (4) A B C E D (4) E B D A C (3) C A D B E (3) B E A D C (3) E B D C A (2) D E B A C (2) D C E B A (2) D C A E B (2) D C A B E (2) C A B D E (2) B E A C D (2) B A E D C (2) A C B D E (2) A B D E C (2) E D C B A (1) E C B D A (1) E B C D A (1) E B A D C (1) E B A C D (1) D C E A B (1) C E A B D (1) C D E B A (1) A D B C E (1) A C B E D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -12 0 12 B -6 0 6 6 6 C 12 -6 0 18 -8 D 0 -6 -18 0 0 E -12 -6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.37499999996 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 0 12 B -6 0 6 6 6 C 12 -6 0 18 -8 D 0 -6 -18 0 0 E -12 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=20 A=20 B=17 E=14 so E is eliminated. Round 2 votes counts: C=30 D=25 B=25 A=20 so A is eliminated. Round 3 votes counts: B=41 C=33 D=26 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:206 A:203 E:195 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 6 -12 0 12 B -6 0 6 6 6 C 12 -6 0 18 -8 D 0 -6 -18 0 0 E -12 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 0 12 B -6 0 6 6 6 C 12 -6 0 18 -8 D 0 -6 -18 0 0 E -12 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 0 12 B -6 0 6 6 6 C 12 -6 0 18 -8 D 0 -6 -18 0 0 E -12 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 881: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) D A B C E (6) A D C E B (6) E B C D A (5) C E A B D (5) A D C B E (5) D B A C E (4) D A C E B (4) D A C B E (4) B C E A D (4) D A E C B (3) C A E D B (3) B E D C A (3) E D B A C (2) E C B A D (2) E C A D B (2) E B D A C (2) E B C A D (2) D B E A C (2) D B A E C (2) C E B A D (2) C B A E D (2) C A E B D (2) C A B E D (2) B C D A E (2) D E B A C (1) D E A B C (1) D A E B C (1) D A B E C (1) C B E A D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C D A (1) B D E A C (1) B D A E C (1) A E D C B (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 20 18 6 26 B -20 0 -16 -22 -10 C -18 16 0 -4 30 D -6 22 4 0 16 E -26 10 -30 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 6 26 B -20 0 -16 -22 -10 C -18 16 0 -4 30 D -6 22 4 0 16 E -26 10 -30 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998446 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=23 C=20 E=15 B=13 so B is eliminated. Round 2 votes counts: D=31 C=26 A=23 E=20 so E is eliminated. Round 3 votes counts: D=39 C=38 A=23 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:235 D:218 C:212 E:169 B:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 6 26 B -20 0 -16 -22 -10 C -18 16 0 -4 30 D -6 22 4 0 16 E -26 10 -30 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998446 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 6 26 B -20 0 -16 -22 -10 C -18 16 0 -4 30 D -6 22 4 0 16 E -26 10 -30 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998446 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 6 26 B -20 0 -16 -22 -10 C -18 16 0 -4 30 D -6 22 4 0 16 E -26 10 -30 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998446 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 882: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) E B D A C (6) E A B D C (6) A C E D B (6) E A C B D (5) B E D C A (5) E B D C A (4) E A C D B (4) B D E C A (4) B D C E A (4) A C D E B (4) A C D B E (4) D C B A E (3) C D A B E (3) A E D C B (3) E B A D C (2) E B A C D (2) E A D B C (2) D C A B E (2) B E D A C (2) B E C D A (2) B D C A E (2) A E C D B (2) A D C B E (2) E D B A C (1) E C A B D (1) E B C D A (1) E A D C B (1) E A B C D (1) D B C A E (1) D B A C E (1) D A E B C (1) C D B A E (1) C B A D E (1) C A E D B (1) C A E B D (1) Total count = 100 A B C D E A 0 16 10 14 -6 B -16 0 -6 -2 -8 C -10 6 0 -4 -10 D -14 2 4 0 -16 E 6 8 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 10 14 -6 B -16 0 -6 -2 -8 C -10 6 0 -4 -10 D -14 2 4 0 -16 E 6 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=21 B=19 C=16 D=8 so D is eliminated. Round 2 votes counts: E=36 A=22 C=21 B=21 so C is eliminated. Round 3 votes counts: A=38 E=36 B=26 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:217 C:191 D:188 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 10 14 -6 B -16 0 -6 -2 -8 C -10 6 0 -4 -10 D -14 2 4 0 -16 E 6 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 14 -6 B -16 0 -6 -2 -8 C -10 6 0 -4 -10 D -14 2 4 0 -16 E 6 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 14 -6 B -16 0 -6 -2 -8 C -10 6 0 -4 -10 D -14 2 4 0 -16 E 6 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 883: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) E C A D B (9) B A C E D (7) C E B A D (5) C E A B D (5) A C E D B (5) D A B C E (4) B C A E D (4) B D E C A (3) B D A C E (3) B A D C E (3) E C D A B (2) E C B D A (2) D E C A B (2) D E A C B (2) D B E C A (2) D B A C E (2) D A E B C (2) B E C D A (2) B D E A C (2) B D A E C (2) B A C D E (2) A D C B E (2) E C D B A (1) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) D E C B A (1) D E B C A (1) D E A B C (1) D A C E B (1) C E A D B (1) C A E D B (1) C A E B D (1) B C E A D (1) A D C E B (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 6 6 8 B -8 0 -8 -4 -16 C -6 8 0 8 2 D -6 4 -8 0 -2 E -8 16 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 6 8 B -8 0 -8 -4 -16 C -6 8 0 8 2 D -6 4 -8 0 -2 E -8 16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=29 B=29 E=18 C=13 A=11 so A is eliminated. Round 2 votes counts: D=33 B=31 E=18 C=18 so E is eliminated. Round 3 votes counts: C=34 D=33 B=33 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:206 E:204 D:194 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 6 8 B -8 0 -8 -4 -16 C -6 8 0 8 2 D -6 4 -8 0 -2 E -8 16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 6 8 B -8 0 -8 -4 -16 C -6 8 0 8 2 D -6 4 -8 0 -2 E -8 16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 6 8 B -8 0 -8 -4 -16 C -6 8 0 8 2 D -6 4 -8 0 -2 E -8 16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 884: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) E C A D B (8) E C B D A (7) B D A E C (7) C E A D B (5) E B C D A (4) E B C A D (4) D A B C E (4) A D C B E (4) E C B A D (3) E C A B D (3) D B A C E (3) C A D E B (3) A D B C E (3) E A C D B (2) C E D A B (2) B E D C A (2) B E C D A (2) B D C E A (2) A B D E C (2) E C D A B (1) E B A C D (1) E A B D C (1) E A B C D (1) D B C A E (1) D A C B E (1) C E D B A (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B E D A C (1) B D E A C (1) B C E D A (1) B A D E C (1) A E C D B (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -6 -6 -6 B 2 0 4 6 -6 C 6 -4 0 10 -6 D 6 -6 -10 0 -4 E 6 6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 -6 B 2 0 4 6 -6 C 6 -4 0 10 -6 D 6 -6 -10 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=27 C=15 A=14 D=9 so D is eliminated. Round 2 votes counts: E=35 B=31 A=19 C=15 so C is eliminated. Round 3 votes counts: E=44 B=31 A=25 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:203 C:203 D:193 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -6 B 2 0 4 6 -6 C 6 -4 0 10 -6 D 6 -6 -10 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -6 B 2 0 4 6 -6 C 6 -4 0 10 -6 D 6 -6 -10 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -6 B 2 0 4 6 -6 C 6 -4 0 10 -6 D 6 -6 -10 0 -4 E 6 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 885: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) E B D A C (8) D B E A C (8) A E C B D (8) C D B A E (7) C A D B E (7) A E D B C (5) C A E B D (4) A E C D B (4) A C E D B (4) D B C E A (3) C A E D B (3) B D E C A (3) A C E B D (3) E A C B D (2) D B E C A (2) C D B E A (2) C B D A E (2) C A D E B (2) B D E A C (2) E B A D C (1) D B C A E (1) D B A E C (1) C E B D A (1) C E A B D (1) C B D E A (1) B E D A C (1) B D C E A (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 12 20 14 6 B -12 0 -4 0 -16 C -20 4 0 4 -14 D -14 0 -4 0 -14 E -6 16 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 20 14 6 B -12 0 -4 0 -16 C -20 4 0 4 -14 D -14 0 -4 0 -14 E -6 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=26 E=22 D=15 B=7 so B is eliminated. Round 2 votes counts: C=30 A=26 E=23 D=21 so D is eliminated. Round 3 votes counts: E=38 C=35 A=27 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:226 E:219 C:187 B:184 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 20 14 6 B -12 0 -4 0 -16 C -20 4 0 4 -14 D -14 0 -4 0 -14 E -6 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 20 14 6 B -12 0 -4 0 -16 C -20 4 0 4 -14 D -14 0 -4 0 -14 E -6 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 20 14 6 B -12 0 -4 0 -16 C -20 4 0 4 -14 D -14 0 -4 0 -14 E -6 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 886: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) A E C D B (8) A D C E B (8) D B A C E (7) B D C E A (7) A C E D B (7) D A C E B (6) B E C D A (6) D A B C E (4) B E C A D (4) B D E C A (4) C E A B D (3) C A E D B (3) E C B A D (2) E B C A D (2) D B C A E (2) D A C B E (2) D A B E C (2) C E B A D (2) C E A D B (2) B C E D A (2) A D E C B (2) E C A D B (1) D B C E A (1) D B A E C (1) C E D B A (1) B D E A C (1) B D A C E (1) B C E A D (1) Total count = 100 A B C D E A 0 12 -2 6 6 B -12 0 -10 -14 -10 C 2 10 0 4 18 D -6 14 -4 0 -4 E -6 10 -18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 6 6 B -12 0 -10 -14 -10 C 2 10 0 4 18 D -6 14 -4 0 -4 E -6 10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 A=25 E=13 C=11 so C is eliminated. Round 2 votes counts: A=28 B=26 D=25 E=21 so E is eliminated. Round 3 votes counts: A=42 B=32 D=26 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 A:211 D:200 E:195 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 6 6 B -12 0 -10 -14 -10 C 2 10 0 4 18 D -6 14 -4 0 -4 E -6 10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 6 6 B -12 0 -10 -14 -10 C 2 10 0 4 18 D -6 14 -4 0 -4 E -6 10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 6 6 B -12 0 -10 -14 -10 C 2 10 0 4 18 D -6 14 -4 0 -4 E -6 10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 887: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) E C D B A (8) D E A C B (5) A D B C E (5) A B D C E (5) E D C A B (4) E C D A B (4) E C B D A (4) C E B D A (4) C E B A D (4) B C E D A (4) A D E C B (4) A B C D E (4) D A E C B (3) D A E B C (3) C B E A D (3) B C A E D (3) E D A C B (2) B D A C E (2) B C E A D (2) B C A D E (2) B A C D E (2) A D E B C (2) E D C B A (1) E D B C A (1) E A C D B (1) D A B E C (1) C E D A B (1) C E A D B (1) C E A B D (1) C B E D A (1) B D E C A (1) B A D C E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 2 2 -4 B -14 0 -4 -10 -6 C -2 4 0 0 -8 D -2 10 0 0 2 E 4 6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 14 2 2 -4 B -14 0 -4 -10 -6 C -2 4 0 0 -8 D -2 10 0 0 2 E 4 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000072 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=25 B=17 C=15 D=12 so D is eliminated. Round 2 votes counts: A=38 E=30 B=17 C=15 so C is eliminated. Round 3 votes counts: E=41 A=38 B=21 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 A:207 D:205 C:197 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 14 2 2 -4 B -14 0 -4 -10 -6 C -2 4 0 0 -8 D -2 10 0 0 2 E 4 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000072 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 2 -4 B -14 0 -4 -10 -6 C -2 4 0 0 -8 D -2 10 0 0 2 E 4 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000072 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 2 -4 B -14 0 -4 -10 -6 C -2 4 0 0 -8 D -2 10 0 0 2 E 4 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000072 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 888: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) E A B C D (7) D C B A E (6) C D B E A (5) B D A C E (5) A E B D C (5) E C D A B (4) D C A B E (4) C D E B A (4) B D C A E (4) B C D E A (4) B C D A E (4) E A D C B (3) E A C B D (3) C D E A B (3) B A D C E (3) A B E D C (3) E A B D C (2) B E A C D (2) A E D C B (2) E C A D B (1) E C A B D (1) E B C D A (1) E B A C D (1) D C E A B (1) D C A E B (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C E B D A (1) C D B A E (1) B E C A D (1) B A E C D (1) B A D E C (1) A D E C B (1) A D C E B (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 4 -4 10 B 8 0 12 16 14 C -4 -12 0 -10 2 D 4 -16 10 0 6 E -10 -14 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -4 10 B 8 0 12 16 14 C -4 -12 0 -10 2 D 4 -16 10 0 6 E -10 -14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=23 D=15 C=15 A=14 so A is eliminated. Round 2 votes counts: B=37 E=30 D=18 C=15 so C is eliminated. Round 3 votes counts: B=37 E=32 D=31 so D is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:202 A:201 C:188 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -4 10 B 8 0 12 16 14 C -4 -12 0 -10 2 D 4 -16 10 0 6 E -10 -14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -4 10 B 8 0 12 16 14 C -4 -12 0 -10 2 D 4 -16 10 0 6 E -10 -14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -4 10 B 8 0 12 16 14 C -4 -12 0 -10 2 D 4 -16 10 0 6 E -10 -14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 889: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (14) C A D E B (12) B D A E C (6) E B C A D (5) B D E A C (5) E C A B D (4) D B A C E (4) D A C B E (4) D A B C E (4) B E D C A (4) E C B A D (3) E B D C A (3) C A D B E (3) A D B C E (3) A C D E B (3) E C B D A (2) E B C D A (2) E B A D C (2) D B A E C (2) B D C E A (2) A D C B E (2) A C D B E (2) E B D A C (1) D C A B E (1) C E A D B (1) C E A B D (1) C D B E A (1) C A E D B (1) B D E C A (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -14 8 -12 -2 B 14 0 16 12 16 C -8 -16 0 -16 -12 D 12 -12 16 0 10 E 2 -16 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 -12 -2 B 14 0 16 12 16 C -8 -16 0 -16 -12 D 12 -12 16 0 10 E 2 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=22 C=19 D=15 A=12 so A is eliminated. Round 2 votes counts: B=32 E=24 C=24 D=20 so D is eliminated. Round 3 votes counts: B=45 C=31 E=24 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:213 E:194 A:190 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 -12 -2 B 14 0 16 12 16 C -8 -16 0 -16 -12 D 12 -12 16 0 10 E 2 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -12 -2 B 14 0 16 12 16 C -8 -16 0 -16 -12 D 12 -12 16 0 10 E 2 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -12 -2 B 14 0 16 12 16 C -8 -16 0 -16 -12 D 12 -12 16 0 10 E 2 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 890: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) E D B C A (11) A C B D E (10) C A B E D (8) D E A B C (7) B E C D A (6) B C A E D (5) E B D C A (4) A C D B E (4) D E A C B (3) C B E A D (3) B C E A D (3) A D C E B (3) E D A C B (2) C B A E D (2) B E D C A (2) B D E C A (2) E D C B A (1) E D C A B (1) E A D C B (1) D E B C A (1) D B E A C (1) C A E B D (1) B C E D A (1) B C A D E (1) B A C D E (1) A D E C B (1) A D C B E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -4 -8 -22 B 12 0 14 0 2 C 4 -14 0 -6 -10 D 8 0 6 0 -6 E 22 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.804528 C: 0.000000 D: 0.195472 E: 0.000000 Sum of squares = 0.685474741811 Cumulative probabilities = A: 0.000000 B: 0.804528 C: 0.804528 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -8 -22 B 12 0 14 0 2 C 4 -14 0 -6 -10 D 8 0 6 0 -6 E 22 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000049027 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=21 A=21 E=20 C=14 so C is eliminated. Round 2 votes counts: A=30 B=26 D=24 E=20 so E is eliminated. Round 3 votes counts: D=39 A=31 B=30 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:218 B:214 D:204 C:187 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -8 -22 B 12 0 14 0 2 C 4 -14 0 -6 -10 D 8 0 6 0 -6 E 22 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000049027 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -8 -22 B 12 0 14 0 2 C 4 -14 0 -6 -10 D 8 0 6 0 -6 E 22 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000049027 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -8 -22 B 12 0 14 0 2 C 4 -14 0 -6 -10 D 8 0 6 0 -6 E 22 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000049027 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 891: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) E B D A C (7) C D A B E (6) D B E C A (5) C D B E A (5) A E B D C (5) A B E D C (5) A B D E C (5) C A D E B (4) A E B C D (4) C E D B A (3) C A E D B (3) B D E A C (3) A C B D E (3) E B A D C (2) E A B D C (2) D E B C A (2) D C B E A (2) D B C E A (2) C D A E B (2) B D A E C (2) B A D E C (2) A C E B D (2) A C D B E (2) E D B C A (1) E B D C A (1) E B C A D (1) E A C B D (1) E A B C D (1) D C E B A (1) D B C A E (1) C D B A E (1) C A E B D (1) C A D B E (1) B E A D C (1) B D E C A (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -2 -6 2 B 2 0 8 0 -2 C 2 -8 0 -2 -2 D 6 0 2 0 18 E -2 2 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.596879 C: 0.000000 D: 0.403121 E: 0.000000 Sum of squares = 0.518771114886 Cumulative probabilities = A: 0.000000 B: 0.596879 C: 0.596879 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 2 B 2 0 8 0 -2 C 2 -8 0 -2 -2 D 6 0 2 0 18 E -2 2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=28 E=16 D=13 B=9 so B is eliminated. Round 2 votes counts: C=34 A=30 D=19 E=17 so E is eliminated. Round 3 votes counts: A=37 C=35 D=28 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:213 B:204 A:196 C:195 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 2 B 2 0 8 0 -2 C 2 -8 0 -2 -2 D 6 0 2 0 18 E -2 2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 2 B 2 0 8 0 -2 C 2 -8 0 -2 -2 D 6 0 2 0 18 E -2 2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 2 B 2 0 8 0 -2 C 2 -8 0 -2 -2 D 6 0 2 0 18 E -2 2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 892: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (19) E A D C B (11) C D B A E (10) B E A C D (6) B C D E A (6) E A B D C (5) E B A D C (4) B C A D E (4) A D C E B (3) E D C A B (2) E A B C D (2) D C E A B (2) C D A B E (2) B A E C D (2) A B E C D (2) E D C B A (1) E A D B C (1) D C E B A (1) D C A E B (1) D A E C B (1) D A C E B (1) C D B E A (1) C D A E B (1) C B D A E (1) C A D B E (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B A C D E (1) A E D C B (1) A E B C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -22 -12 -2 6 B 22 0 18 20 24 C 12 -18 0 28 14 D 2 -20 -28 0 12 E -6 -24 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -12 -2 6 B 22 0 18 20 24 C 12 -18 0 28 14 D 2 -20 -28 0 12 E -6 -24 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 E=26 C=16 A=9 D=6 so D is eliminated. Round 2 votes counts: B=43 E=26 C=20 A=11 so A is eliminated. Round 3 votes counts: B=46 E=29 C=25 so C is eliminated. Round 4 votes counts: B=62 E=38 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:242 C:218 A:185 D:183 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -12 -2 6 B 22 0 18 20 24 C 12 -18 0 28 14 D 2 -20 -28 0 12 E -6 -24 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -12 -2 6 B 22 0 18 20 24 C 12 -18 0 28 14 D 2 -20 -28 0 12 E -6 -24 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -12 -2 6 B 22 0 18 20 24 C 12 -18 0 28 14 D 2 -20 -28 0 12 E -6 -24 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 893: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) E D A C B (7) D A E B C (7) B A C D E (7) A D B E C (6) D A B C E (5) A D E B C (5) E C D B A (4) C E D B A (4) C B E D A (4) A E D B C (4) D E A B C (3) D A B E C (3) C B E A D (3) B D A C E (3) B A D C E (3) E C D A B (2) D E A C B (2) C E B A D (2) B C A E D (2) E D C A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E C A B D (1) D E C A B (1) D C E B A (1) D B C A E (1) C E B D A (1) C B D A E (1) A D B C E (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 18 -2 22 B -4 0 26 -16 4 C -18 -26 0 -10 -2 D 2 16 10 0 22 E -22 -4 2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 -2 22 B -4 0 26 -16 4 C -18 -26 0 -10 -2 D 2 16 10 0 22 E -22 -4 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 A=19 E=18 C=15 so C is eliminated. Round 2 votes counts: B=33 E=25 D=23 A=19 so A is eliminated. Round 3 votes counts: B=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:221 B:205 E:177 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 18 -2 22 B -4 0 26 -16 4 C -18 -26 0 -10 -2 D 2 16 10 0 22 E -22 -4 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 -2 22 B -4 0 26 -16 4 C -18 -26 0 -10 -2 D 2 16 10 0 22 E -22 -4 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 -2 22 B -4 0 26 -16 4 C -18 -26 0 -10 -2 D 2 16 10 0 22 E -22 -4 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 894: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B A C D E (8) D E C B A (7) E D C A B (6) E D A C B (6) B C A D E (5) D B E C A (4) D E B C A (3) D E B A C (3) B D E C A (3) B D E A C (3) B A C E D (3) A C E D B (3) A B C E D (3) D C B E A (2) C A E B D (2) C A B E D (2) B D A C E (2) B C D E A (2) B C D A E (2) B A D E C (2) A C E B D (2) A C B E D (2) A B E C D (2) E D A B C (1) E A D C B (1) E A C D B (1) D C E B A (1) D C E A B (1) C E D A B (1) C D E B A (1) C B A E D (1) C B A D E (1) C A E D B (1) B D C E A (1) B D C A E (1) B C A E D (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 -14 -12 -16 -10 B 14 0 2 0 2 C 12 -2 0 -14 -4 D 16 0 14 0 20 E 10 -2 4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.574489 C: 0.000000 D: 0.425511 E: 0.000000 Sum of squares = 0.511097312171 Cumulative probabilities = A: 0.000000 B: 0.574489 C: 0.574489 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -16 -10 B 14 0 2 0 2 C 12 -2 0 -14 -4 D 16 0 14 0 20 E 10 -2 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=29 E=15 A=13 C=9 so C is eliminated. Round 2 votes counts: B=36 D=30 A=18 E=16 so E is eliminated. Round 3 votes counts: D=44 B=36 A=20 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:225 B:209 C:196 E:196 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -16 -10 B 14 0 2 0 2 C 12 -2 0 -14 -4 D 16 0 14 0 20 E 10 -2 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -16 -10 B 14 0 2 0 2 C 12 -2 0 -14 -4 D 16 0 14 0 20 E 10 -2 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -16 -10 B 14 0 2 0 2 C 12 -2 0 -14 -4 D 16 0 14 0 20 E 10 -2 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 895: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (16) E D B C A (7) B C E D A (7) D E A B C (6) B E D C A (5) A C B E D (5) D E A C B (4) C A D E B (4) A D E C B (4) D E C A B (3) C D E A B (3) B E C D A (3) A D C E B (3) A C D B E (3) A C B D E (3) E B D C A (2) D A E C B (2) C A B E D (2) B C E A D (2) B C A E D (2) B A C E D (2) D E C B A (1) D E B C A (1) D C E A B (1) C E B D A (1) B E D A C (1) B E C A D (1) B E A D C (1) B A E C D (1) B A D E C (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 24 10 6 2 B -24 0 -10 -16 -16 C -10 10 0 12 12 D -6 16 -12 0 16 E -2 16 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 10 6 2 B -24 0 -10 -16 -16 C -10 10 0 12 12 D -6 16 -12 0 16 E -2 16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=26 D=18 C=10 E=9 so E is eliminated. Round 2 votes counts: A=37 B=28 D=25 C=10 so C is eliminated. Round 3 votes counts: A=43 B=29 D=28 so D is eliminated. Round 4 votes counts: A=62 B=38 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:212 D:207 E:193 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 10 6 2 B -24 0 -10 -16 -16 C -10 10 0 12 12 D -6 16 -12 0 16 E -2 16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 10 6 2 B -24 0 -10 -16 -16 C -10 10 0 12 12 D -6 16 -12 0 16 E -2 16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 10 6 2 B -24 0 -10 -16 -16 C -10 10 0 12 12 D -6 16 -12 0 16 E -2 16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 896: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (14) C B A E D (11) E D C B A (9) D E A B C (7) D A B C E (7) E C B A D (6) D E C A B (6) B A C E D (6) A B D C E (6) C E B A D (4) E C D B A (3) E D A B C (2) D E C B A (2) C E D B A (2) A B E D C (2) A B D E C (2) A B C E D (2) E C B D A (1) E B C A D (1) D E A C B (1) D C A B E (1) D A E B C (1) D A B E C (1) C B A D E (1) B C A E D (1) A D B C E (1) Total count = 100 A B C D E A 0 6 4 14 12 B -6 0 6 14 10 C -4 -6 0 4 12 D -14 -14 -4 0 0 E -12 -10 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 14 12 B -6 0 6 14 10 C -4 -6 0 4 12 D -14 -14 -4 0 0 E -12 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 E=22 C=18 B=7 so B is eliminated. Round 2 votes counts: A=33 D=26 E=22 C=19 so C is eliminated. Round 3 votes counts: A=46 E=28 D=26 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:212 C:203 D:184 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 14 12 B -6 0 6 14 10 C -4 -6 0 4 12 D -14 -14 -4 0 0 E -12 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 14 12 B -6 0 6 14 10 C -4 -6 0 4 12 D -14 -14 -4 0 0 E -12 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 14 12 B -6 0 6 14 10 C -4 -6 0 4 12 D -14 -14 -4 0 0 E -12 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 897: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (13) D C B A E (9) E A D B C (8) C B D A E (8) E D C B A (6) D A B C E (5) C D B A E (5) D C B E A (4) B C A D E (4) E D A B C (3) E C B D A (3) D A C B E (3) E C B A D (2) E A D C B (2) B A C D E (2) A E D B C (2) A E B D C (2) A D B C E (2) A B C D E (2) E D A C B (1) E B A C D (1) E A C B D (1) D E A C B (1) D C E B A (1) D A E B C (1) D A C E B (1) C E B D A (1) C D B E A (1) C B D E A (1) C B A D E (1) A E B C D (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 8 -8 2 B -2 0 -2 -10 0 C -8 2 0 -6 4 D 8 10 6 0 4 E -2 0 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -8 2 B -2 0 -2 -10 0 C -8 2 0 -6 4 D 8 10 6 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=25 C=17 A=12 B=6 so B is eliminated. Round 2 votes counts: E=40 D=25 C=21 A=14 so A is eliminated. Round 3 votes counts: E=46 D=28 C=26 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:202 C:196 E:195 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -8 2 B -2 0 -2 -10 0 C -8 2 0 -6 4 D 8 10 6 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -8 2 B -2 0 -2 -10 0 C -8 2 0 -6 4 D 8 10 6 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -8 2 B -2 0 -2 -10 0 C -8 2 0 -6 4 D 8 10 6 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 898: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) A E D B C (10) D A C B E (9) E B C A D (7) E A D C B (6) D C B A E (6) B C D A E (6) C B D A E (5) B C D E A (5) A D C B E (5) E C B D A (3) E B C D A (3) E A B C D (3) C B D E A (3) E B A C D (2) C B E D A (2) B C E D A (2) A D B C E (2) E D C A B (1) E C B A D (1) E A B D C (1) D C A B E (1) C D B E A (1) C D B A E (1) B E C D A (1) B C E A D (1) B C A D E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 0 0 2 -4 B 0 0 8 -8 2 C 0 -8 0 -6 2 D -2 8 6 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 A B C D E A 0 0 0 2 -4 B 0 0 8 -8 2 C 0 -8 0 -6 2 D -2 8 6 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=19 D=16 B=16 C=12 so C is eliminated. Round 2 votes counts: E=37 B=26 A=19 D=18 so D is eliminated. Round 3 votes counts: E=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:203 E:203 B:201 A:199 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 2 -4 B 0 0 8 -8 2 C 0 -8 0 -6 2 D -2 8 6 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -4 B 0 0 8 -8 2 C 0 -8 0 -6 2 D -2 8 6 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -4 B 0 0 8 -8 2 C 0 -8 0 -6 2 D -2 8 6 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 899: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) C A D B E (7) C D A B E (6) E B D C A (5) E B A C D (5) A C E B D (5) E B D A C (4) D C B E A (4) A C D E B (4) A C B E D (4) E D B A C (3) D C A B E (3) D B E C A (3) B E D C A (3) B D E C A (3) A E B C D (3) E B A D C (2) D E B C A (2) D C B A E (2) D A C E B (2) C D B A E (2) C A B E D (2) B E C D A (2) A E C B D (2) E A B D C (1) E A B C D (1) D E B A C (1) D B C E A (1) D A E C B (1) C A B D E (1) B C E D A (1) A E D C B (1) A E D B C (1) A D C E B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 14 6 4 18 B -14 0 -18 -10 10 C -6 18 0 14 14 D -4 10 -14 0 8 E -18 -10 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 4 18 B -14 0 -18 -10 10 C -6 18 0 14 14 D -4 10 -14 0 8 E -18 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997685 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=21 D=19 C=18 B=9 so B is eliminated. Round 2 votes counts: A=33 E=26 D=22 C=19 so C is eliminated. Round 3 votes counts: A=43 D=30 E=27 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:220 D:200 B:184 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 4 18 B -14 0 -18 -10 10 C -6 18 0 14 14 D -4 10 -14 0 8 E -18 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997685 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 4 18 B -14 0 -18 -10 10 C -6 18 0 14 14 D -4 10 -14 0 8 E -18 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997685 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 4 18 B -14 0 -18 -10 10 C -6 18 0 14 14 D -4 10 -14 0 8 E -18 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997685 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 900: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (15) D C E B A (12) B A E C D (11) C E D B A (7) A B D E C (7) D A B C E (5) E C A B D (4) D C E A B (4) D B A C E (4) A B E D C (3) E C D A B (2) E C B A D (2) E B A C D (2) E A B C D (2) D A E B C (2) C E B A D (2) A E B C D (2) E C D B A (1) E B C A D (1) D C B E A (1) D C A B E (1) D B C A E (1) D A C B E (1) D A B E C (1) C E D A B (1) C D E B A (1) C B E D A (1) B A D C E (1) B A C E D (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 18 10 14 B -4 0 20 10 8 C -18 -20 0 10 -14 D -10 -10 -10 0 -16 E -14 -8 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 10 14 B -4 0 20 10 8 C -18 -20 0 10 -14 D -10 -10 -10 0 -16 E -14 -8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=29 E=14 B=13 C=12 so C is eliminated. Round 2 votes counts: D=33 A=29 E=24 B=14 so B is eliminated. Round 3 votes counts: A=42 D=33 E=25 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 B:217 E:204 C:179 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 18 10 14 B -4 0 20 10 8 C -18 -20 0 10 -14 D -10 -10 -10 0 -16 E -14 -8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 10 14 B -4 0 20 10 8 C -18 -20 0 10 -14 D -10 -10 -10 0 -16 E -14 -8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 10 14 B -4 0 20 10 8 C -18 -20 0 10 -14 D -10 -10 -10 0 -16 E -14 -8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 901: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) D C B A E (7) D C A B E (7) A E B D C (7) E C D A B (6) B A D C E (6) C D E B A (5) E C B D A (4) B D C A E (4) A B D C E (4) E A C D B (3) E A D C B (2) E A B C D (2) D C A E B (2) D B C A E (2) C D B E A (2) C D B A E (2) B E C D A (2) B E A C D (2) B D A C E (2) B C D E A (2) A D E C B (2) A D C E B (2) A B E D C (2) A B D E C (2) E C A D B (1) E B A C D (1) E A C B D (1) D A C B E (1) C E D B A (1) C B E D A (1) C B D E A (1) A E D B C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -16 -20 10 B 6 0 -20 -14 2 C 16 20 0 -10 6 D 20 14 10 0 10 E -10 -2 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -20 10 B 6 0 -20 -14 2 C 16 20 0 -10 6 D 20 14 10 0 10 E -10 -2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 D=19 B=18 C=12 so C is eliminated. Round 2 votes counts: E=30 D=28 A=22 B=20 so B is eliminated. Round 3 votes counts: D=37 E=35 A=28 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:227 C:216 B:187 E:186 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -16 -20 10 B 6 0 -20 -14 2 C 16 20 0 -10 6 D 20 14 10 0 10 E -10 -2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -20 10 B 6 0 -20 -14 2 C 16 20 0 -10 6 D 20 14 10 0 10 E -10 -2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -20 10 B 6 0 -20 -14 2 C 16 20 0 -10 6 D 20 14 10 0 10 E -10 -2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 902: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (11) C A B D E (8) D A C B E (7) C A D B E (7) B C A D E (7) B E D C A (5) E B D C A (4) E B D A C (4) E B C D A (4) E A C D B (4) A D C E B (4) A C D B E (4) D A E C B (3) D A C E B (3) A C D E B (3) E B C A D (2) C B A D E (2) C A E B D (2) B E C A D (2) B D C A E (2) E D B A C (1) E A D C B (1) D E A B C (1) C B A E D (1) C A E D B (1) B E D A C (1) B D E C A (1) B D A C E (1) B C E D A (1) B C D A E (1) B C A E D (1) A E C D B (1) Total count = 100 A B C D E A 0 20 -2 0 16 B -20 0 -24 -2 2 C 2 24 0 2 10 D 0 2 -2 0 8 E -16 -2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -2 0 16 B -20 0 -24 -2 2 C 2 24 0 2 10 D 0 2 -2 0 8 E -16 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=22 C=21 D=14 A=12 so A is eliminated. Round 2 votes counts: E=32 C=28 B=22 D=18 so D is eliminated. Round 3 votes counts: C=42 E=36 B=22 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:217 D:204 E:182 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -2 0 16 B -20 0 -24 -2 2 C 2 24 0 2 10 D 0 2 -2 0 8 E -16 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -2 0 16 B -20 0 -24 -2 2 C 2 24 0 2 10 D 0 2 -2 0 8 E -16 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -2 0 16 B -20 0 -24 -2 2 C 2 24 0 2 10 D 0 2 -2 0 8 E -16 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 903: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) A D C E B (7) E B C A D (5) A D E C B (5) A D E B C (5) D A C E B (4) D A B C E (4) B E D A C (4) B E C A D (4) B C E D A (4) E C B A D (3) C E B A D (3) B E C D A (3) B D C A E (3) E B A D C (2) E A D B C (2) D B C A E (2) D B A C E (2) D A B E C (2) C D A B E (2) C A D E B (2) B C D E A (2) A E D C B (2) E C A D B (1) E C A B D (1) E B A C D (1) E A D C B (1) E A C D B (1) D C A B E (1) D A E C B (1) D A E B C (1) C E A B D (1) C B D E A (1) C B D A E (1) C A E D B (1) B D E A C (1) B D C E A (1) B C D A E (1) Total count = 100 A B C D E A 0 14 16 -6 18 B -14 0 -2 -18 2 C -16 2 0 -26 10 D 6 18 26 0 22 E -18 -2 -10 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 -6 18 B -14 0 -2 -18 2 C -16 2 0 -26 10 D 6 18 26 0 22 E -18 -2 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=23 A=19 E=17 C=11 so C is eliminated. Round 2 votes counts: D=32 B=25 A=22 E=21 so E is eliminated. Round 3 votes counts: B=39 D=32 A=29 so A is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:236 A:221 C:185 B:184 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 16 -6 18 B -14 0 -2 -18 2 C -16 2 0 -26 10 D 6 18 26 0 22 E -18 -2 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 -6 18 B -14 0 -2 -18 2 C -16 2 0 -26 10 D 6 18 26 0 22 E -18 -2 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 -6 18 B -14 0 -2 -18 2 C -16 2 0 -26 10 D 6 18 26 0 22 E -18 -2 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 904: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) E A D C B (6) E A C D B (6) D A E B C (6) D B C E A (5) D B A E C (5) C B D E A (5) E C A D B (4) B D C A E (4) A E C D B (4) D E A B C (3) C E A B D (3) C A B E D (3) E D A B C (2) E A C B D (2) D B C A E (2) C D E B A (2) C B E D A (2) B C D A E (2) A E C B D (2) A E B C D (2) A D E B C (2) A D B E C (2) A C B E D (2) E D C A B (1) E D A C B (1) E C A B D (1) E A D B C (1) D E C B A (1) D B E A C (1) C E B A D (1) C D B E A (1) C B E A D (1) C B A D E (1) C A E B D (1) B D C E A (1) B D A C E (1) B C D E A (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 24 14 8 -2 B -24 0 0 -26 -18 C -14 0 0 -6 -20 D -8 26 6 0 -6 E 2 18 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 14 8 -2 B -24 0 0 -26 -18 C -14 0 0 -6 -20 D -8 26 6 0 -6 E 2 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=23 A=22 C=20 B=11 so B is eliminated. Round 2 votes counts: D=29 E=24 C=24 A=23 so A is eliminated. Round 3 votes counts: E=40 D=34 C=26 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:222 D:209 C:180 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 14 8 -2 B -24 0 0 -26 -18 C -14 0 0 -6 -20 D -8 26 6 0 -6 E 2 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 14 8 -2 B -24 0 0 -26 -18 C -14 0 0 -6 -20 D -8 26 6 0 -6 E 2 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 14 8 -2 B -24 0 0 -26 -18 C -14 0 0 -6 -20 D -8 26 6 0 -6 E 2 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 905: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) B E C A D (9) D A C E B (7) A D E C B (6) E B A C D (4) E A B C D (4) D A E C B (4) C D B A E (4) C B E A D (4) C B D E A (4) E B A D C (3) E A B D C (3) D C A B E (3) D A E B C (3) B C E A D (3) D C B E A (2) D C A E B (2) C A B E D (2) B E D A C (2) B E C D A (2) B C D E A (2) A E D B C (2) A E C D B (2) A E C B D (2) E C A B D (1) D E A B C (1) D B C E A (1) D A C B E (1) C D A E B (1) C B E D A (1) C B D A E (1) C A E B D (1) B E A C D (1) B D C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 -8 -4 4 B 8 0 -18 0 6 C 8 18 0 -4 0 D 4 0 4 0 6 E -4 -6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.124483 C: 0.000000 D: 0.875517 E: 0.000000 Sum of squares = 0.782025472611 Cumulative probabilities = A: 0.000000 B: 0.124483 C: 0.124483 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -4 4 B 8 0 -18 0 6 C 8 18 0 -4 0 D 4 0 4 0 6 E -4 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.818182 E: 0.000000 Sum of squares = 0.702479528306 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=20 C=18 E=15 A=13 so A is eliminated. Round 2 votes counts: D=40 E=22 B=20 C=18 so C is eliminated. Round 3 votes counts: D=45 B=32 E=23 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 D:207 B:198 A:192 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 4 B 8 0 -18 0 6 C 8 18 0 -4 0 D 4 0 4 0 6 E -4 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.818182 E: 0.000000 Sum of squares = 0.702479528306 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 4 B 8 0 -18 0 6 C 8 18 0 -4 0 D 4 0 4 0 6 E -4 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.818182 E: 0.000000 Sum of squares = 0.702479528306 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 4 B 8 0 -18 0 6 C 8 18 0 -4 0 D 4 0 4 0 6 E -4 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.818182 E: 0.000000 Sum of squares = 0.702479528306 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 906: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) E A C B D (6) E C D A B (5) E C A D B (5) E C D B A (4) D C B A E (4) C E D A B (4) B A D C E (4) A C D B E (4) E D C B A (3) E B D C A (3) E B A D C (3) D C E B A (3) D C B E A (3) D B C A E (3) C E A D B (3) C D E A B (3) B A E D C (3) A E B C D (3) E D B C A (2) D E B C A (2) C A D E B (2) B A D E C (2) A B E D C (2) A B E C D (2) A B D C E (2) E C A B D (1) E B A C D (1) E A B C D (1) D B C E A (1) C A E D B (1) C A D B E (1) B E A D C (1) B D E C A (1) B D E A C (1) A E C B D (1) Total count = 100 A B C D E A 0 -8 -8 -4 -12 B 8 0 -6 -6 -12 C 8 6 0 -6 -4 D 4 6 6 0 -8 E 12 12 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -8 -4 -12 B 8 0 -6 -6 -12 C 8 6 0 -6 -4 D 4 6 6 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=22 D=16 C=14 A=14 so C is eliminated. Round 2 votes counts: E=41 B=22 D=19 A=18 so A is eliminated. Round 3 votes counts: E=46 B=28 D=26 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:204 C:202 B:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 -12 B 8 0 -6 -6 -12 C 8 6 0 -6 -4 D 4 6 6 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 -12 B 8 0 -6 -6 -12 C 8 6 0 -6 -4 D 4 6 6 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 -12 B 8 0 -6 -6 -12 C 8 6 0 -6 -4 D 4 6 6 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 907: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) E B D C A (8) C A D E B (8) B E D A C (8) A C D B E (7) D A C E B (5) E D B C A (4) D C E A B (3) D C A E B (3) C D A E B (3) B E A D C (3) A C D E B (3) D E B A C (2) D A B E C (2) C A E B D (2) B E D C A (2) B E C D A (2) B E A C D (2) B D E A C (2) B A D E C (2) A D C B E (2) A B C E D (2) E D B A C (1) E C D B A (1) E B C D A (1) D E C B A (1) D E B C A (1) D B E A C (1) D A E C B (1) C E B A D (1) C E A D B (1) C E A B D (1) C A E D B (1) C A B D E (1) B E C A D (1) B A E D C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 12 -8 -2 8 B -12 0 -8 -2 -2 C 8 8 0 -6 6 D 2 2 6 0 -2 E -8 2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000068 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 A B C D E A 0 12 -8 -2 8 B -12 0 -8 -2 -2 C 8 8 0 -6 6 D 2 2 6 0 -2 E -8 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000026 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=23 D=19 A=16 E=15 so E is eliminated. Round 2 votes counts: B=32 C=28 D=24 A=16 so A is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:208 A:205 D:204 E:195 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -8 -2 8 B -12 0 -8 -2 -2 C 8 8 0 -6 6 D 2 2 6 0 -2 E -8 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000026 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 -2 8 B -12 0 -8 -2 -2 C 8 8 0 -6 6 D 2 2 6 0 -2 E -8 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000026 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 -2 8 B -12 0 -8 -2 -2 C 8 8 0 -6 6 D 2 2 6 0 -2 E -8 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000026 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 908: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) D E B C A (6) D E A B C (6) C A B D E (5) A C E B D (5) A C B D E (5) C B A D E (4) C A B E D (4) B C E D A (4) B C D E A (4) A E D C B (4) A D E C B (4) E D A B C (3) D C B A E (3) B D C E A (3) A E C D B (3) A C E D B (3) E D B C A (2) E D B A C (2) E A D B C (2) D E B A C (2) D B E C A (2) D B C E A (2) A C D B E (2) E C A B D (1) E B D C A (1) E B C D A (1) E B C A D (1) D A E C B (1) D A E B C (1) C B D A E (1) C B A E D (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 22 10 12 16 B -22 0 -16 2 0 C -10 16 0 10 14 D -12 -2 -10 0 4 E -16 0 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 10 12 16 B -22 0 -16 2 0 C -10 16 0 10 14 D -12 -2 -10 0 4 E -16 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=23 C=15 E=13 B=11 so B is eliminated. Round 2 votes counts: A=38 D=26 C=23 E=13 so E is eliminated. Round 3 votes counts: A=40 D=34 C=26 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:215 D:190 E:183 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 10 12 16 B -22 0 -16 2 0 C -10 16 0 10 14 D -12 -2 -10 0 4 E -16 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 10 12 16 B -22 0 -16 2 0 C -10 16 0 10 14 D -12 -2 -10 0 4 E -16 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 10 12 16 B -22 0 -16 2 0 C -10 16 0 10 14 D -12 -2 -10 0 4 E -16 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 909: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (9) E A D B C (7) D E A B C (6) B A C E D (5) E D A C B (4) B C A D E (4) B A E C D (4) A B C E D (4) E D A B C (3) C D B E A (3) C B D A E (3) C B A D E (3) B D C E A (3) A C B E D (3) E A D C B (2) D E C A B (2) D E B A C (2) D C E B A (2) D C B E A (2) C A E D B (2) B C D A E (2) A E D B C (2) A C E B D (2) E D C A B (1) D E C B A (1) D E B C A (1) D B E A C (1) C D A B E (1) C B A E D (1) C A B E D (1) B A E D C (1) A E D C B (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 6 18 6 0 B -6 0 12 -6 0 C -18 -12 0 -4 0 D -6 6 4 0 -8 E 0 0 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.432970 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.567030 Sum of squares = 0.50898600879 Cumulative probabilities = A: 0.432970 B: 0.432970 C: 0.432970 D: 0.432970 E: 1.000000 A B C D E A 0 6 18 6 0 B -6 0 12 -6 0 C -18 -12 0 -4 0 D -6 6 4 0 -8 E 0 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 E=17 C=14 A=14 so C is eliminated. Round 2 votes counts: B=35 D=31 E=17 A=17 so E is eliminated. Round 3 votes counts: D=39 B=35 A=26 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:215 E:204 B:200 D:198 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 6 0 B -6 0 12 -6 0 C -18 -12 0 -4 0 D -6 6 4 0 -8 E 0 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 6 0 B -6 0 12 -6 0 C -18 -12 0 -4 0 D -6 6 4 0 -8 E 0 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 6 0 B -6 0 12 -6 0 C -18 -12 0 -4 0 D -6 6 4 0 -8 E 0 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 910: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) D C A E B (7) D A B C E (7) C E D A B (7) C E A B D (6) D A C B E (5) C E D B A (5) B E C A D (5) B A E C D (5) E B C A D (4) C D E A B (4) D B A E C (3) B A D E C (3) A D B C E (3) E C B D A (2) D A C E B (2) B E A C D (2) B A E D C (2) A B E C D (2) E C D B A (1) D E C B A (1) D E B C A (1) D C E A B (1) D B C A E (1) C E B D A (1) C E A D B (1) C A E D B (1) C A D E B (1) B E C D A (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) A D C E B (1) A C E B D (1) A C D E B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -16 -4 -4 B -4 0 -12 -6 -10 C 16 12 0 16 12 D 4 6 -16 0 -10 E 4 10 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -4 -4 B -4 0 -12 -6 -10 C 16 12 0 16 12 D 4 6 -16 0 -10 E 4 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 B=22 E=14 A=10 so A is eliminated. Round 2 votes counts: D=32 C=29 B=25 E=14 so E is eliminated. Round 3 votes counts: C=39 D=32 B=29 so B is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:206 D:192 A:190 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 -4 -4 B -4 0 -12 -6 -10 C 16 12 0 16 12 D 4 6 -16 0 -10 E 4 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -4 -4 B -4 0 -12 -6 -10 C 16 12 0 16 12 D 4 6 -16 0 -10 E 4 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -4 -4 B -4 0 -12 -6 -10 C 16 12 0 16 12 D 4 6 -16 0 -10 E 4 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 911: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) C B A E D (9) D E A B C (8) D A E B C (8) B C D E A (8) A D E C B (7) C B E A D (5) E D A B C (4) D E B A C (4) C B A D E (4) E A D B C (3) C B E D A (3) C B D E A (3) B C E D A (3) A E D B C (3) D B E A C (2) B D E C A (2) B D C E A (2) A D E B C (2) A C E D B (2) E A D C B (1) D A E C B (1) D A C B E (1) C B D A E (1) C A E B D (1) C A B D E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 16 0 4 B -8 0 -2 -16 -12 C -16 2 0 -18 -12 D 0 16 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.425299 B: 0.000000 C: 0.000000 D: 0.574701 E: 0.000000 Sum of squares = 0.511160430625 Cumulative probabilities = A: 0.425299 B: 0.425299 C: 0.425299 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 0 4 B -8 0 -2 -16 -12 C -16 2 0 -18 -12 D 0 16 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 D=24 B=15 E=8 so E is eliminated. Round 2 votes counts: A=30 D=28 C=27 B=15 so B is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:214 E:205 B:181 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 0 4 B -8 0 -2 -16 -12 C -16 2 0 -18 -12 D 0 16 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 0 4 B -8 0 -2 -16 -12 C -16 2 0 -18 -12 D 0 16 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 0 4 B -8 0 -2 -16 -12 C -16 2 0 -18 -12 D 0 16 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 912: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) B E C D A (8) B D E C A (6) D B E A C (5) D A B E C (5) D A B C E (5) A C E D B (5) E B C A D (4) C E A B D (4) B E D C A (4) A C D E B (4) D B C E A (3) D B C A E (3) D B A C E (3) C A E B D (3) A E C B D (3) E C A B D (2) C A E D B (2) B E D A C (2) A E D C B (2) A D C E B (2) E C B A D (1) E B C D A (1) E B A D C (1) E A C B D (1) E A B C D (1) D B E C A (1) D A C B E (1) C E B A D (1) C A D E B (1) B D E A C (1) B D C E A (1) A E D B C (1) A E C D B (1) A D E C B (1) A D E B C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 10 -14 6 B 6 0 28 -10 14 C -10 -28 0 -14 -20 D 14 10 14 0 4 E -6 -14 20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 -14 6 B 6 0 28 -10 14 C -10 -28 0 -14 -20 D 14 10 14 0 4 E -6 -14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=22 A=22 E=11 C=11 so E is eliminated. Round 2 votes counts: D=34 B=28 A=24 C=14 so C is eliminated. Round 3 votes counts: A=36 D=34 B=30 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:219 A:198 E:198 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 10 -14 6 B 6 0 28 -10 14 C -10 -28 0 -14 -20 D 14 10 14 0 4 E -6 -14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 -14 6 B 6 0 28 -10 14 C -10 -28 0 -14 -20 D 14 10 14 0 4 E -6 -14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 -14 6 B 6 0 28 -10 14 C -10 -28 0 -14 -20 D 14 10 14 0 4 E -6 -14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 913: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (12) B E D C A (10) E B C A D (7) B E A C D (5) E B A C D (4) D C A B E (4) C A E D B (4) E B C D A (3) D A C B E (3) B D E C A (3) B D E A C (3) A D C B E (3) A C D E B (3) D B E C A (2) C D A E B (2) C A D E B (2) B E C D A (2) B E A D C (2) B D A E C (2) B A E C D (2) A C E B D (2) E C D A B (1) E C B D A (1) E C A B D (1) E B D C A (1) D C E A B (1) D C B A E (1) D B A C E (1) D A C E B (1) C E D A B (1) C E A D B (1) C D E A B (1) B E D A C (1) B D A C E (1) B A D E C (1) B A D C E (1) A E C B D (1) A D B C E (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -20 -14 0 B 6 0 8 10 2 C 20 -8 0 -10 -6 D 14 -10 10 0 0 E 0 -2 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -14 0 B 6 0 8 10 2 C 20 -8 0 -10 -6 D 14 -10 10 0 0 E 0 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=25 E=18 A=13 C=11 so C is eliminated. Round 2 votes counts: B=33 D=28 E=20 A=19 so A is eliminated. Round 3 votes counts: D=38 B=35 E=27 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:207 E:202 C:198 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -20 -14 0 B 6 0 8 10 2 C 20 -8 0 -10 -6 D 14 -10 10 0 0 E 0 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -14 0 B 6 0 8 10 2 C 20 -8 0 -10 -6 D 14 -10 10 0 0 E 0 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -14 0 B 6 0 8 10 2 C 20 -8 0 -10 -6 D 14 -10 10 0 0 E 0 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 914: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (15) E A B D C (10) C D B A E (8) D C E A B (6) B A E D C (6) A B E D C (6) C D B E A (5) E B A D C (4) B A C E D (4) E A D B C (3) D E C A B (3) C D A B E (3) C B A D E (3) E D C A B (2) E D A B C (2) D E A C B (2) C D E A B (2) B E A C D (2) E D C B A (1) E B A C D (1) D C A E B (1) D C A B E (1) D A E B C (1) D A C B E (1) C D E B A (1) C B D A E (1) B C A D E (1) B A C D E (1) A E B D C (1) A D E B C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 24 20 12 B 6 0 18 14 18 C -24 -18 0 -2 -20 D -20 -14 2 0 -14 E -12 -18 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 24 20 12 B 6 0 18 14 18 C -24 -18 0 -2 -20 D -20 -14 2 0 -14 E -12 -18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 C=23 D=15 A=10 so A is eliminated. Round 2 votes counts: B=36 E=24 C=24 D=16 so D is eliminated. Round 3 votes counts: B=36 C=33 E=31 so E is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:225 E:202 D:177 C:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 24 20 12 B 6 0 18 14 18 C -24 -18 0 -2 -20 D -20 -14 2 0 -14 E -12 -18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 24 20 12 B 6 0 18 14 18 C -24 -18 0 -2 -20 D -20 -14 2 0 -14 E -12 -18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 24 20 12 B 6 0 18 14 18 C -24 -18 0 -2 -20 D -20 -14 2 0 -14 E -12 -18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 915: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) A E C D B (7) E A B D C (5) B E D C A (5) B D E C A (5) C D B A E (4) C B D A E (4) B E D A C (4) B D C E A (4) A E B D C (4) A C E B D (4) E B A D C (3) C B D E A (3) C A D E B (3) A C D E B (3) E D B A C (2) E B D C A (2) E A D B C (2) D E C B A (2) D C B E A (2) C D B E A (2) C D A E B (2) C A D B E (2) A E C B D (2) E B D A C (1) D E B C A (1) D E A C B (1) D B C E A (1) D A E C B (1) C A B D E (1) B E A D C (1) B C D E A (1) B C A D E (1) B A E C D (1) A E D C B (1) A E D B C (1) A E B C D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 10 6 6 B -2 0 -8 10 -12 C -10 8 0 4 -4 D -6 -10 -4 0 -12 E -6 12 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 6 6 B -2 0 -8 10 -12 C -10 8 0 4 -4 D -6 -10 -4 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=22 C=21 E=15 D=8 so D is eliminated. Round 2 votes counts: A=35 C=23 B=23 E=19 so E is eliminated. Round 3 votes counts: A=43 B=32 C=25 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:211 C:199 B:194 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 6 6 B -2 0 -8 10 -12 C -10 8 0 4 -4 D -6 -10 -4 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 6 6 B -2 0 -8 10 -12 C -10 8 0 4 -4 D -6 -10 -4 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 6 6 B -2 0 -8 10 -12 C -10 8 0 4 -4 D -6 -10 -4 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 916: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) E D B C A (7) D C E B A (7) C A D E B (7) A B E C D (7) A C D B E (6) D B E C A (5) C D E B A (4) D E C B A (3) C A D B E (3) B E D A C (3) A C E B D (3) A C B E D (3) A C B D E (3) A B E D C (3) E B D C A (2) E B A D C (2) D C B E A (2) C D E A B (2) C D A E B (2) C D A B E (2) B D E C A (2) B D E A C (2) B A E D C (2) E C B A D (1) D E B C A (1) D C B A E (1) D C A B E (1) D B C E A (1) A E B C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -6 6 -6 B 8 0 0 -8 16 C 6 0 0 -8 0 D -6 8 8 0 12 E 6 -16 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.338842975046 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 6 -6 B 8 0 0 -8 16 C 6 0 0 -8 0 D -6 8 8 0 12 E 6 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.33884297505 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=21 C=20 B=18 E=12 so E is eliminated. Round 2 votes counts: A=29 D=28 B=22 C=21 so C is eliminated. Round 3 votes counts: A=39 D=38 B=23 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:211 B:208 C:199 A:193 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 6 -6 B 8 0 0 -8 16 C 6 0 0 -8 0 D -6 8 8 0 12 E 6 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.33884297505 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 6 -6 B 8 0 0 -8 16 C 6 0 0 -8 0 D -6 8 8 0 12 E 6 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.33884297505 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 6 -6 B 8 0 0 -8 16 C 6 0 0 -8 0 D -6 8 8 0 12 E 6 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.33884297505 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 917: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) A D B E C (9) A D B C E (9) B C E A D (8) E C D B A (7) C E B D A (7) B A C E D (7) E C B D A (5) A B D C E (5) D A B E C (4) E D C B A (3) B C A E D (3) E C D A B (2) D E C B A (2) D E C A B (2) C E B A D (2) B C E D A (2) D E B C A (1) D E A C B (1) D B E C A (1) C E D B A (1) C E D A B (1) C B E A D (1) B E C D A (1) B D A E C (1) B A D E C (1) B A D C E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 -4 6 B 8 0 8 -10 8 C -2 -8 0 -2 -2 D 4 10 2 0 -4 E -6 -8 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.37190082644 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 A B C D E A 0 -8 2 -4 6 B 8 0 8 -10 8 C -2 -8 0 -2 -2 D 4 10 2 0 -4 E -6 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826371 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=24 D=22 E=17 C=12 so C is eliminated. Round 2 votes counts: E=28 B=25 A=25 D=22 so D is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:207 D:206 A:198 E:196 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -4 6 B 8 0 8 -10 8 C -2 -8 0 -2 -2 D 4 10 2 0 -4 E -6 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826371 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -4 6 B 8 0 8 -10 8 C -2 -8 0 -2 -2 D 4 10 2 0 -4 E -6 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826371 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -4 6 B 8 0 8 -10 8 C -2 -8 0 -2 -2 D 4 10 2 0 -4 E -6 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826371 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 918: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) D A E B C (8) A E C B D (7) D E B C A (6) D B E C A (6) E B C A D (4) C B E A D (4) D C B A E (3) D A C B E (3) C B D E A (3) A E D C B (3) A E D B C (3) E B C D A (2) E A C B D (2) D E A B C (2) D C B E A (2) D A B C E (2) B C E D A (2) B C D E A (2) A D C E B (2) E C B A D (1) E B D C A (1) E B A D C (1) E B A C D (1) E A B C D (1) D C A B E (1) D B A E C (1) D A B E C (1) C D B A E (1) C B E D A (1) C B D A E (1) C B A E D (1) C A B D E (1) B C E A D (1) A E B D C (1) A E B C D (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 -12 -22 -10 B 14 0 20 -22 4 C 12 -20 0 -24 -8 D 22 22 24 0 24 E 10 -4 8 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -22 -10 B 14 0 20 -22 4 C 12 -20 0 -24 -8 D 22 22 24 0 24 E 10 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=48 A=22 E=13 C=12 B=5 so B is eliminated. Round 2 votes counts: D=48 A=22 C=17 E=13 so E is eliminated. Round 3 votes counts: D=49 A=27 C=24 so C is eliminated. Round 4 votes counts: D=61 A=39 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:246 B:208 E:195 C:180 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -12 -22 -10 B 14 0 20 -22 4 C 12 -20 0 -24 -8 D 22 22 24 0 24 E 10 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -22 -10 B 14 0 20 -22 4 C 12 -20 0 -24 -8 D 22 22 24 0 24 E 10 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -22 -10 B 14 0 20 -22 4 C 12 -20 0 -24 -8 D 22 22 24 0 24 E 10 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 919: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) E D B C A (5) D B E C A (5) C A E D B (5) A C E B D (5) A B E D C (4) A B E C D (4) A B C D E (4) E C D A B (3) D E C B A (3) C E D A B (3) C D B E A (3) C A D E B (3) B A D E C (3) E D C A B (2) E D B A C (2) E D A B C (2) E B D A C (2) D E B C A (2) D C E B A (2) C D E B A (2) B D E A C (2) B D A E C (2) B C A D E (2) B A E D C (2) A E B C D (2) A C B D E (2) E D C B A (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B D C (1) D C B E A (1) C E A D B (1) C D B A E (1) C B D A E (1) C A E B D (1) C A B D E (1) B E A D C (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A C E (1) A E B D C (1) Total count = 100 A B C D E A 0 8 -2 4 4 B -8 0 0 2 -2 C 2 0 0 4 -6 D -4 -2 -4 0 -14 E -4 2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 8 -2 4 4 B -8 0 0 2 -2 C 2 0 0 4 -6 D -4 -2 -4 0 -14 E -4 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=21 C=21 B=16 D=13 so D is eliminated. Round 2 votes counts: A=29 E=26 C=24 B=21 so B is eliminated. Round 3 votes counts: A=37 E=35 C=28 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:207 C:200 B:196 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 4 4 B -8 0 0 2 -2 C 2 0 0 4 -6 D -4 -2 -4 0 -14 E -4 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 4 4 B -8 0 0 2 -2 C 2 0 0 4 -6 D -4 -2 -4 0 -14 E -4 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 4 4 B -8 0 0 2 -2 C 2 0 0 4 -6 D -4 -2 -4 0 -14 E -4 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 920: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) C D A B E (6) E D C B A (5) E D B C A (4) E B D C A (4) E B A D C (4) E A B D C (4) A B E D C (4) E C D B A (3) D C E B A (3) D C B A E (3) C D E B A (3) C A D E B (3) B D A E C (3) A E C B D (3) A C B D E (3) E B A C D (2) D C B E A (2) C E A D B (2) C D A E B (2) C A D B E (2) B E D A C (2) B A E D C (2) A E B D C (2) A E B C D (2) A C D B E (2) A B E C D (2) A B D E C (2) E C D A B (1) E B D A C (1) E A B C D (1) D B C E A (1) C D E A B (1) C A E D B (1) B E A D C (1) B D E A C (1) B A D E C (1) A C E D B (1) A C E B D (1) A C B E D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -2 -2 B 2 0 -8 -2 -18 C 4 8 0 -12 -20 D 2 2 12 0 -8 E 2 18 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -2 -2 B 2 0 -8 -2 -18 C 4 8 0 -12 -20 D 2 2 12 0 -8 E 2 18 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=26 C=20 D=15 B=10 so B is eliminated. Round 2 votes counts: E=32 A=29 C=20 D=19 so D is eliminated. Round 3 votes counts: E=39 A=32 C=29 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:204 A:195 C:190 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 -2 B 2 0 -8 -2 -18 C 4 8 0 -12 -20 D 2 2 12 0 -8 E 2 18 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 -2 B 2 0 -8 -2 -18 C 4 8 0 -12 -20 D 2 2 12 0 -8 E 2 18 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 -2 B 2 0 -8 -2 -18 C 4 8 0 -12 -20 D 2 2 12 0 -8 E 2 18 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 921: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (11) E B D A C (9) B E D C A (9) A C E D B (9) A C D E B (9) E B D C A (8) E A C D B (5) A C D B E (5) C A D B E (4) E D B C A (3) B E D A C (3) A C B D E (3) E D C A B (2) E B A C D (2) D B C A E (2) C A B D E (2) B A C D E (2) E D A C B (1) E B A D C (1) E A D C B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A B E (1) D B E C A (1) B D C A E (1) B C A D E (1) B A C E D (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 8 -4 -18 B 8 0 10 10 -8 C -8 -10 0 -8 -18 D 4 -10 8 0 -14 E 18 8 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 8 -4 -18 B 8 0 10 10 -8 C -8 -10 0 -8 -18 D 4 -10 8 0 -14 E 18 8 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=28 A=27 C=6 D=5 so D is eliminated. Round 2 votes counts: E=34 B=31 A=27 C=8 so C is eliminated. Round 3 votes counts: E=35 A=34 B=31 so B is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 B:210 D:194 A:189 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 8 -4 -18 B 8 0 10 10 -8 C -8 -10 0 -8 -18 D 4 -10 8 0 -14 E 18 8 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -4 -18 B 8 0 10 10 -8 C -8 -10 0 -8 -18 D 4 -10 8 0 -14 E 18 8 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -4 -18 B 8 0 10 10 -8 C -8 -10 0 -8 -18 D 4 -10 8 0 -14 E 18 8 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 922: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) B D A E C (10) A E D B C (9) C E B D A (8) C E A B D (5) C B E D A (5) A D E B C (5) E A D B C (4) C E A D B (4) B D A C E (4) A E D C B (4) E A C D B (3) D A B E C (3) C B D E A (3) D B A E C (2) C A E D B (2) B D E C A (2) A E C D B (2) E D A B C (1) E C D A B (1) E C B D A (1) E B D A C (1) E B C D A (1) E A D C B (1) C E B A D (1) C B D A E (1) B D C E A (1) B D C A E (1) B C D E A (1) B C D A E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 14 24 6 14 B -14 0 16 -8 -8 C -24 -16 0 -20 -22 D -6 8 20 0 -8 E -14 8 22 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 24 6 14 B -14 0 16 -8 -8 C -24 -16 0 -20 -22 D -6 8 20 0 -8 E -14 8 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=29 B=20 E=13 D=5 so D is eliminated. Round 2 votes counts: A=36 C=29 B=22 E=13 so E is eliminated. Round 3 votes counts: A=45 C=31 B=24 so B is eliminated. Round 4 votes counts: A=62 C=38 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:212 D:207 B:193 C:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 24 6 14 B -14 0 16 -8 -8 C -24 -16 0 -20 -22 D -6 8 20 0 -8 E -14 8 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 24 6 14 B -14 0 16 -8 -8 C -24 -16 0 -20 -22 D -6 8 20 0 -8 E -14 8 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 24 6 14 B -14 0 16 -8 -8 C -24 -16 0 -20 -22 D -6 8 20 0 -8 E -14 8 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 923: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (16) A B E D C (16) D B C A E (9) E A B C D (8) B D A C E (8) E C A D B (7) E A C B D (7) B A D E C (6) C E D A B (5) C D B E A (5) D C B A E (4) A E B C D (3) E C D A B (2) B A D C E (2) A E B D C (2) Total count = 100 A B C D E A 0 0 4 2 0 B 0 0 8 4 0 C -4 -8 0 6 -2 D -2 -4 -6 0 0 E 0 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.335991 B: 0.208836 C: 0.000000 D: 0.000000 E: 0.455173 Sum of squares = 0.363685096856 Cumulative probabilities = A: 0.335991 B: 0.544827 C: 0.544827 D: 0.544827 E: 1.000000 A B C D E A 0 0 4 2 0 B 0 0 8 4 0 C -4 -8 0 6 -2 D -2 -4 -6 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333334 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333305 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 A=21 B=16 D=13 so D is eliminated. Round 2 votes counts: C=30 B=25 E=24 A=21 so A is eliminated. Round 3 votes counts: B=41 C=30 E=29 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:206 A:203 E:201 C:196 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 2 0 B 0 0 8 4 0 C -4 -8 0 6 -2 D -2 -4 -6 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333334 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333305 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 2 0 B 0 0 8 4 0 C -4 -8 0 6 -2 D -2 -4 -6 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333334 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333305 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 2 0 B 0 0 8 4 0 C -4 -8 0 6 -2 D -2 -4 -6 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333334 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333305 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 924: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (15) A E B D C (11) C E A D B (8) C D B E A (8) C D B A E (6) C A E B D (6) E A C B D (5) C E A B D (5) E A B D C (4) D B C A E (4) D C B E A (3) C D E A B (3) A E C B D (3) E A C D B (2) D B E A C (2) D B C E A (2) B D E A C (2) B D C A E (2) D C B A E (1) D B A E C (1) C E D A B (1) C B D A E (1) C A E D B (1) B D A C E (1) B C A D E (1) B A E D C (1) A E B C D (1) Total count = 100 A B C D E A 0 0 -4 -4 10 B 0 0 -6 16 0 C 4 6 0 2 6 D 4 -16 -2 0 4 E -10 0 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -4 10 B 0 0 -6 16 0 C 4 6 0 2 6 D 4 -16 -2 0 4 E -10 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=22 A=15 D=13 E=11 so E is eliminated. Round 2 votes counts: C=39 A=26 B=22 D=13 so D is eliminated. Round 3 votes counts: C=43 B=31 A=26 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:205 A:201 D:195 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -4 10 B 0 0 -6 16 0 C 4 6 0 2 6 D 4 -16 -2 0 4 E -10 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 10 B 0 0 -6 16 0 C 4 6 0 2 6 D 4 -16 -2 0 4 E -10 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 10 B 0 0 -6 16 0 C 4 6 0 2 6 D 4 -16 -2 0 4 E -10 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 925: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (16) E D B C A (6) E D B A C (6) C B A D E (6) A C D E B (6) C A B D E (5) B E C D A (5) A E D C B (5) A C D B E (5) E D A B C (4) B C D A E (4) E B D C A (3) C B D A E (3) A C B D E (3) E B D A C (2) E A D B C (2) C A D B E (2) B D E C A (2) A E C D B (2) A C B E D (2) D E A C B (1) D C E B A (1) D C B E A (1) D B E C A (1) C D B A E (1) B D C E A (1) B C D E A (1) B C A E D (1) B C A D E (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -22 -20 -16 -4 B 22 0 10 10 20 C 20 -10 0 -4 -12 D 16 -10 4 0 -10 E 4 -20 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -20 -16 -4 B 22 0 10 10 20 C 20 -10 0 -4 -12 D 16 -10 4 0 -10 E 4 -20 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=25 E=23 C=17 D=4 so D is eliminated. Round 2 votes counts: B=32 A=25 E=24 C=19 so C is eliminated. Round 3 votes counts: B=43 A=32 E=25 so E is eliminated. Round 4 votes counts: B=61 A=39 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:203 D:200 C:197 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -20 -16 -4 B 22 0 10 10 20 C 20 -10 0 -4 -12 D 16 -10 4 0 -10 E 4 -20 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -20 -16 -4 B 22 0 10 10 20 C 20 -10 0 -4 -12 D 16 -10 4 0 -10 E 4 -20 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -20 -16 -4 B 22 0 10 10 20 C 20 -10 0 -4 -12 D 16 -10 4 0 -10 E 4 -20 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 926: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) D B A C E (11) C E B A D (10) D A B E C (8) C E B D A (6) C E D B A (4) B A E C D (4) E A B C D (3) D B C A E (3) D A B C E (3) C E D A B (3) E C B A D (2) D C E B A (2) C D E B A (2) B A D E C (2) A E B D C (2) A D B E C (2) A B E D C (2) A B D E C (2) E D C A B (1) E C D A B (1) E A C B D (1) D C E A B (1) D C B E A (1) D C B A E (1) D B A E C (1) D A E C B (1) C E A B D (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C A D (1) B D A E C (1) B D A C E (1) B C E A D (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -8 -8 -8 B 12 0 -2 6 -4 C 8 2 0 8 6 D 8 -6 -8 0 -12 E 8 4 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -8 -8 B 12 0 -2 6 -4 C 8 2 0 8 6 D 8 -6 -8 0 -12 E 8 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=29 E=19 B=10 A=10 so B is eliminated. Round 2 votes counts: D=34 C=30 E=20 A=16 so A is eliminated. Round 3 votes counts: D=40 E=30 C=30 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:209 B:206 D:191 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 -8 -8 B 12 0 -2 6 -4 C 8 2 0 8 6 D 8 -6 -8 0 -12 E 8 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -8 -8 B 12 0 -2 6 -4 C 8 2 0 8 6 D 8 -6 -8 0 -12 E 8 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -8 -8 B 12 0 -2 6 -4 C 8 2 0 8 6 D 8 -6 -8 0 -12 E 8 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 927: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (15) B D A C E (10) B A C E D (9) E D C A B (6) B A C D E (6) E C A D B (5) D E A C B (5) C A E B D (4) E B C A D (3) D E B C A (3) C A B E D (3) E C D A B (2) D B E C A (2) D B A E C (2) D A E C B (2) C E A B D (2) C B A E D (2) B C E A D (2) B C A E D (2) B A D C E (2) A C E B D (2) A C B E D (2) A B C E D (2) E C A B D (1) D E A B C (1) D B A C E (1) D A B C E (1) C E A D B (1) B D E C A (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -8 -2 2 B -10 0 -6 6 -4 C 8 6 0 -2 4 D 2 -6 2 0 4 E -2 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 -2 2 B -10 0 -6 6 -4 C 8 6 0 -2 4 D 2 -6 2 0 4 E -2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510205 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=32 B=32 E=17 C=12 A=7 so A is eliminated. Round 2 votes counts: B=34 D=32 E=17 C=17 so E is eliminated. Round 3 votes counts: D=38 B=37 C=25 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 A:201 D:201 E:197 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 -2 2 B -10 0 -6 6 -4 C 8 6 0 -2 4 D 2 -6 2 0 4 E -2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510205 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -2 2 B -10 0 -6 6 -4 C 8 6 0 -2 4 D 2 -6 2 0 4 E -2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510205 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -2 2 B -10 0 -6 6 -4 C 8 6 0 -2 4 D 2 -6 2 0 4 E -2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510205 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 928: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E C B A (7) C B D E A (7) D E A B C (6) A D E B C (6) A B C D E (5) E D A B C (4) A D B E C (4) A B C E D (4) E D A C B (3) C B E D A (3) C B A E D (3) C B A D E (3) A E D B C (3) A E B C D (3) E D C A B (2) E C D B A (2) E A D B C (2) D E C A B (2) D E A C B (2) D A E B C (2) C B E A D (2) C B D A E (2) B C A D E (2) B A C D E (2) E D C B A (1) E C B D A (1) E A C B D (1) D C B E A (1) D A B C E (1) C D E B A (1) B A C E D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 2 6 6 B -4 0 14 0 4 C -2 -14 0 4 -4 D -6 0 -4 0 8 E -6 -4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 6 6 B -4 0 14 0 4 C -2 -14 0 4 -4 D -6 0 -4 0 8 E -6 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=21 C=21 E=16 B=15 so B is eliminated. Round 2 votes counts: C=33 A=30 D=21 E=16 so E is eliminated. Round 3 votes counts: C=36 A=33 D=31 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:207 D:199 E:193 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 6 6 B -4 0 14 0 4 C -2 -14 0 4 -4 D -6 0 -4 0 8 E -6 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 6 6 B -4 0 14 0 4 C -2 -14 0 4 -4 D -6 0 -4 0 8 E -6 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 6 6 B -4 0 14 0 4 C -2 -14 0 4 -4 D -6 0 -4 0 8 E -6 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 929: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) C B E A D (10) A D E B C (10) C A B E D (9) D E B A C (6) A C B E D (6) D E B C A (5) E B D C A (4) D E A B C (4) D A E B C (4) B E C D A (4) A D C E B (4) A C D E B (3) E B D A C (2) C D A B E (2) C B A E D (2) A E D B C (2) A C D B E (2) E D B C A (1) D C B A E (1) D A C E B (1) C D B E A (1) B E D C A (1) B C E A D (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -2 8 2 B -4 0 -8 6 -2 C 2 8 0 6 6 D -8 -6 -6 0 -12 E -2 2 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 8 2 B -4 0 -8 6 -2 C 2 8 0 6 6 D -8 -6 -6 0 -12 E -2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=32 D=21 E=7 B=6 so B is eliminated. Round 2 votes counts: C=35 A=32 D=21 E=12 so E is eliminated. Round 3 votes counts: C=39 A=32 D=29 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:206 E:203 B:196 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 8 2 B -4 0 -8 6 -2 C 2 8 0 6 6 D -8 -6 -6 0 -12 E -2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 8 2 B -4 0 -8 6 -2 C 2 8 0 6 6 D -8 -6 -6 0 -12 E -2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 8 2 B -4 0 -8 6 -2 C 2 8 0 6 6 D -8 -6 -6 0 -12 E -2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 930: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) D B C E A (8) A E C B D (8) D B A E C (7) C E A D B (7) C E A B D (7) E A C B D (5) D B C A E (5) E C A B D (3) E B A C D (3) C D B E A (3) B D E A C (3) A E B D C (3) E A B C D (2) D C B A E (2) C D E B A (2) C A E D B (2) B D C E A (2) B A E D C (2) A B E D C (2) A B D E C (2) E C B A D (1) D C B E A (1) D B A C E (1) D A C B E (1) C E D A B (1) C B E D A (1) B E D A C (1) B E A D C (1) B C E D A (1) B C D E A (1) B A D E C (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 6 2 -6 B 10 0 10 18 8 C -6 -10 0 -2 -8 D -2 -18 2 0 -4 E 6 -8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 2 -6 B 10 0 10 18 8 C -6 -10 0 -2 -8 D -2 -18 2 0 -4 E 6 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 B=21 A=17 E=14 so E is eliminated. Round 2 votes counts: C=27 D=25 B=24 A=24 so B is eliminated. Round 3 votes counts: D=40 A=31 C=29 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:223 E:205 A:196 D:189 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 2 -6 B 10 0 10 18 8 C -6 -10 0 -2 -8 D -2 -18 2 0 -4 E 6 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 2 -6 B 10 0 10 18 8 C -6 -10 0 -2 -8 D -2 -18 2 0 -4 E 6 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 2 -6 B 10 0 10 18 8 C -6 -10 0 -2 -8 D -2 -18 2 0 -4 E 6 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 931: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) E B A C D (7) A E C D B (7) B E D C A (6) A C D E B (6) B E A D C (5) E C A B D (4) B E D A C (4) A C E D B (4) D C A B E (3) C D A E B (3) B D E C A (3) B A D E C (3) E C A D B (2) E A C D B (2) D C B A E (2) D B C E A (2) D B C A E (2) D A C E B (2) C E D A B (2) C A D E B (2) B E C D A (2) B E C A D (2) B E A C D (2) B D C A E (2) A D C E B (2) E C D A B (1) E A C B D (1) D B A C E (1) D A C B E (1) C E A D B (1) C D E B A (1) C D E A B (1) C A E D B (1) B D A E C (1) B D A C E (1) B A E D C (1) Total count = 100 A B C D E A 0 6 0 4 4 B -6 0 -12 -12 -14 C 0 12 0 2 -6 D -4 12 -2 0 -8 E -4 14 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.738714 B: 0.000000 C: 0.261286 D: 0.000000 E: 0.000000 Sum of squares = 0.613968848104 Cumulative probabilities = A: 0.738714 B: 0.738714 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 4 4 B -6 0 -12 -12 -14 C 0 12 0 2 -6 D -4 12 -2 0 -8 E -4 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000208683 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=21 A=19 E=17 C=11 so C is eliminated. Round 2 votes counts: B=32 D=26 A=22 E=20 so E is eliminated. Round 3 votes counts: B=39 A=32 D=29 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:212 A:207 C:204 D:199 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 4 4 B -6 0 -12 -12 -14 C 0 12 0 2 -6 D -4 12 -2 0 -8 E -4 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000208683 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 4 4 B -6 0 -12 -12 -14 C 0 12 0 2 -6 D -4 12 -2 0 -8 E -4 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000208683 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 4 4 B -6 0 -12 -12 -14 C 0 12 0 2 -6 D -4 12 -2 0 -8 E -4 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000208683 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 932: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) B A D C E (8) E B C A D (6) D A C B E (6) E C D A B (5) C A D B E (5) A B D C E (5) E B A D C (4) C D A E B (4) B E A D C (4) E B A C D (3) D C A B E (3) C A B D E (3) B A D E C (3) E C B A D (2) D E C A B (2) D E A B C (2) D C E A B (2) D A B C E (2) B D A E C (2) B D A C E (2) E D C B A (1) E D C A B (1) E D B A C (1) E C B D A (1) E B D C A (1) D B A E C (1) C E D A B (1) C E A D B (1) C E A B D (1) C D A B E (1) C A E D B (1) B E D A C (1) B E A C D (1) B D E A C (1) B A C D E (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 18 2 0 B 4 0 14 18 4 C -18 -14 0 -24 -2 D -2 -18 24 0 12 E 0 -4 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 18 2 0 B 4 0 14 18 4 C -18 -14 0 -24 -2 D -2 -18 24 0 12 E 0 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989829 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=23 D=18 C=17 A=8 so A is eliminated. Round 2 votes counts: E=34 B=28 D=19 C=19 so D is eliminated. Round 3 votes counts: E=38 C=31 B=31 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:208 D:208 E:193 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 18 2 0 B 4 0 14 18 4 C -18 -14 0 -24 -2 D -2 -18 24 0 12 E 0 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989829 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 18 2 0 B 4 0 14 18 4 C -18 -14 0 -24 -2 D -2 -18 24 0 12 E 0 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989829 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 18 2 0 B 4 0 14 18 4 C -18 -14 0 -24 -2 D -2 -18 24 0 12 E 0 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989829 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 933: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) B A E D C (11) C D E B A (9) D C E A B (6) B A C D E (6) E D A C B (5) B A E C D (5) A E B D C (5) E D C A B (4) C D B A E (4) A B E D C (4) E A B D C (3) B C A D E (3) E C D B A (2) E C D A B (2) E A D B C (2) D E C A B (2) B C D A E (2) E B A D C (1) E A D C B (1) D E A C B (1) D A C E B (1) C E D A B (1) C D A B E (1) C B D A E (1) B E C A D (1) B C E A D (1) B C A E D (1) B A C E D (1) Total count = 100 A B C D E A 0 4 -8 -10 -10 B -4 0 -8 -10 -18 C 8 8 0 8 2 D 10 10 -8 0 0 E 10 18 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -10 -10 B -4 0 -8 -10 -18 C 8 8 0 8 2 D 10 10 -8 0 0 E 10 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=30 E=20 D=10 A=9 so A is eliminated. Round 2 votes counts: B=35 C=30 E=25 D=10 so D is eliminated. Round 3 votes counts: C=37 B=35 E=28 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:213 D:206 A:188 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -10 B -4 0 -8 -10 -18 C 8 8 0 8 2 D 10 10 -8 0 0 E 10 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -10 B -4 0 -8 -10 -18 C 8 8 0 8 2 D 10 10 -8 0 0 E 10 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -10 B -4 0 -8 -10 -18 C 8 8 0 8 2 D 10 10 -8 0 0 E 10 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 934: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) E A B C D (7) B A E D C (5) D C B A E (4) D C A B E (4) C E A B D (4) C D A B E (4) B E A D C (4) A B E C D (4) E C A B D (3) D E B C A (3) C D E B A (3) C D A E B (3) C A E B D (3) C A B E D (3) A B C D E (3) E C D B A (2) E B C A D (2) E B A C D (2) D B A C E (2) C E D B A (2) C A D B E (2) B A D E C (2) A D B C E (2) A C B D E (2) A B D C E (2) E D B A C (1) E C B A D (1) E B D A C (1) E A C B D (1) D C E B A (1) D C B E A (1) D B A E C (1) D A C B E (1) C E A D B (1) C D E A B (1) C A E D B (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 4 32 2 B -8 0 6 22 -4 C -4 -6 0 10 -2 D -32 -22 -10 0 -16 E -2 4 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 32 2 B -8 0 6 22 -4 C -4 -6 0 10 -2 D -32 -22 -10 0 -16 E -2 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=27 D=17 A=16 B=11 so B is eliminated. Round 2 votes counts: E=33 C=27 A=23 D=17 so D is eliminated. Round 3 votes counts: C=37 E=36 A=27 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:223 E:210 B:208 C:199 D:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 32 2 B -8 0 6 22 -4 C -4 -6 0 10 -2 D -32 -22 -10 0 -16 E -2 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 32 2 B -8 0 6 22 -4 C -4 -6 0 10 -2 D -32 -22 -10 0 -16 E -2 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 32 2 B -8 0 6 22 -4 C -4 -6 0 10 -2 D -32 -22 -10 0 -16 E -2 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 935: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) A E C B D (8) E A B D C (7) C D B E A (7) C D A E B (7) C D B A E (5) B E A D C (5) D C B E A (4) D B E A C (4) D B C E A (4) C A E B D (4) B D E C A (4) A E C D B (4) C D A B E (3) B D E A C (3) A E B D C (3) A E B C D (3) E B C A D (2) E A B C D (2) D C B A E (2) B C D E A (2) A E D B C (2) D A E B C (1) C E A B D (1) C B E A D (1) C A D E B (1) B E D A C (1) B D C E A (1) B C E D A (1) Total count = 100 A B C D E A 0 8 -14 2 2 B -8 0 -10 -4 -6 C 14 10 0 18 2 D -2 4 -18 0 -4 E -2 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -14 2 2 B -8 0 -10 -4 -6 C 14 10 0 18 2 D -2 4 -18 0 -4 E -2 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=20 B=17 D=15 E=11 so E is eliminated. Round 2 votes counts: C=37 A=29 B=19 D=15 so D is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:203 A:199 D:190 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -14 2 2 B -8 0 -10 -4 -6 C 14 10 0 18 2 D -2 4 -18 0 -4 E -2 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -14 2 2 B -8 0 -10 -4 -6 C 14 10 0 18 2 D -2 4 -18 0 -4 E -2 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -14 2 2 B -8 0 -10 -4 -6 C 14 10 0 18 2 D -2 4 -18 0 -4 E -2 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 936: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) E B D A C (9) E D B A C (6) C A B D E (6) D A C B E (5) C E B A D (5) B E D A C (4) B A D C E (4) C E D A B (3) C E A D B (3) C D A E B (3) A D B C E (3) E D A B C (2) E C D B A (2) E C D A B (2) E B C D A (2) D A E B C (2) D A B E C (2) C A D E B (2) B E A D C (2) B E A C D (2) B D A E C (2) B C E A D (2) B A D E C (2) A C D B E (2) E D A C B (1) E C B D A (1) E C B A D (1) D E A C B (1) D E A B C (1) C E B D A (1) C D E A B (1) C B A E D (1) C B A D E (1) A D C B E (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 0 -2 B -6 0 -6 -4 4 C -8 6 0 4 10 D 0 4 -4 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.418467 B: 0.000000 C: 0.000000 D: 0.581533 E: 0.000000 Sum of squares = 0.513295166388 Cumulative probabilities = A: 0.418467 B: 0.418467 C: 0.418467 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 0 -2 B -6 0 -6 -4 4 C -8 6 0 4 10 D 0 4 -4 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.500220 E: 0.000000 Sum of squares = 0.500000096646 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=26 B=18 D=11 A=9 so A is eliminated. Round 2 votes counts: C=39 E=26 B=20 D=15 so D is eliminated. Round 3 votes counts: C=45 E=30 B=25 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:206 C:206 D:201 B:194 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 0 -2 B -6 0 -6 -4 4 C -8 6 0 4 10 D 0 4 -4 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.500220 E: 0.000000 Sum of squares = 0.500000096646 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 0 -2 B -6 0 -6 -4 4 C -8 6 0 4 10 D 0 4 -4 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.500220 E: 0.000000 Sum of squares = 0.500000096646 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 0 -2 B -6 0 -6 -4 4 C -8 6 0 4 10 D 0 4 -4 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.500220 E: 0.000000 Sum of squares = 0.500000096646 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 937: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) E A B D C (7) D E C B A (7) C D B A E (6) C B D A E (6) C A B D E (5) A C B D E (5) E A D B C (4) E A C B D (4) E D B A C (3) E C D B A (3) E A B C D (3) D C B A E (3) D B C A E (3) C B A D E (3) A B D C E (3) E D B C A (2) E D A B C (2) E C D A B (2) E C A D B (2) D C B E A (2) D B E A C (2) C D B E A (2) A E B C D (2) A B E C D (2) A B C D E (2) E D C A B (1) D B C E A (1) C E D B A (1) C E D A B (1) C D E B A (1) B D C A E (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -18 -12 -14 B 6 0 -24 -10 -6 C 18 24 0 2 -10 D 12 10 -2 0 4 E 14 6 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468750000171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 A B C D E A 0 -6 -18 -12 -14 B 6 0 -24 -10 -6 C 18 24 0 2 -10 D 12 10 -2 0 4 E 14 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=25 D=18 A=16 B=1 so B is eliminated. Round 2 votes counts: E=40 C=25 D=19 A=16 so A is eliminated. Round 3 votes counts: E=46 C=32 D=22 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:217 E:213 D:212 B:183 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -18 -12 -14 B 6 0 -24 -10 -6 C 18 24 0 2 -10 D 12 10 -2 0 4 E 14 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 -12 -14 B 6 0 -24 -10 -6 C 18 24 0 2 -10 D 12 10 -2 0 4 E 14 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 -12 -14 B 6 0 -24 -10 -6 C 18 24 0 2 -10 D 12 10 -2 0 4 E 14 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 938: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) B D A C E (10) E C A D B (9) C E B D A (7) A D E B C (6) E C A B D (5) D A B C E (5) B D C A E (5) A E D B C (5) E A D C B (4) D B A C E (4) C E D A B (3) E C B A D (2) E A C D B (2) C E D B A (2) C E A D B (2) C B E D A (2) C B D E A (2) B E A D C (2) B D A E C (2) E B A C D (1) E A D B C (1) D A B E C (1) C E B A D (1) C E A B D (1) C D A B E (1) C B D A E (1) B C D A E (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 12 8 8 B -16 0 12 -12 -4 C -12 -12 0 -16 -6 D -8 12 16 0 0 E -8 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 8 8 B -16 0 12 -12 -4 C -12 -12 0 -16 -6 D -8 12 16 0 0 E -8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 C=22 B=20 D=10 so D is eliminated. Round 2 votes counts: A=30 E=24 B=24 C=22 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:210 E:201 B:190 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 8 8 B -16 0 12 -12 -4 C -12 -12 0 -16 -6 D -8 12 16 0 0 E -8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 8 8 B -16 0 12 -12 -4 C -12 -12 0 -16 -6 D -8 12 16 0 0 E -8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 8 8 B -16 0 12 -12 -4 C -12 -12 0 -16 -6 D -8 12 16 0 0 E -8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 939: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) B D C E A (6) B D C A E (6) E A D C B (5) D C B E A (5) A E C D B (5) A E B C D (5) E A B D C (4) E D B A C (3) D C E A B (3) D B C E A (3) C D A E B (3) C A D E B (3) B A E C D (3) A C E D B (3) E D B C A (2) E B A D C (2) E A D B C (2) E A B C D (2) D E B C A (2) C B D A E (2) C B A D E (2) C A D B E (2) B E D A C (2) B D E C A (2) A C E B D (2) A C B D E (2) A B E C D (2) E C D A B (1) E B D A C (1) E A C B D (1) D B E C A (1) D B C A E (1) C D A B E (1) C A B D E (1) B C D A E (1) B C A D E (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 4 10 -6 B -10 0 2 -4 -12 C -4 -2 0 0 -2 D -10 4 0 0 -4 E 6 12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 4 10 -6 B -10 0 2 -4 -12 C -4 -2 0 0 -2 D -10 4 0 0 -4 E 6 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=21 A=21 D=15 C=14 so C is eliminated. Round 2 votes counts: E=29 A=27 B=25 D=19 so D is eliminated. Round 3 votes counts: B=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:209 C:196 D:195 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 4 10 -6 B -10 0 2 -4 -12 C -4 -2 0 0 -2 D -10 4 0 0 -4 E 6 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 10 -6 B -10 0 2 -4 -12 C -4 -2 0 0 -2 D -10 4 0 0 -4 E 6 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 10 -6 B -10 0 2 -4 -12 C -4 -2 0 0 -2 D -10 4 0 0 -4 E 6 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 940: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) C E B D A (11) E A B D C (8) D B A E C (6) C B D E A (6) A D B E C (5) C B E D A (4) A E D B C (4) D B A C E (3) D A B E C (3) C E A D B (3) C D A B E (3) B D A E C (3) A E C D B (3) A D E B C (3) E C A D B (2) E A C B D (2) C E A B D (2) B D C A E (2) E B C D A (1) E A D C B (1) E A C D B (1) E A B C D (1) D C B A E (1) D C A B E (1) C E B A D (1) C D B E A (1) B E C D A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D E A (1) B C D A E (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 -8 -2 -20 B -10 0 -10 16 -12 C 8 10 0 12 -16 D 2 -16 -12 0 -14 E 20 12 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -8 -2 -20 B -10 0 -10 16 -12 C 8 10 0 12 -16 D 2 -16 -12 0 -14 E 20 12 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=27 A=17 D=14 B=11 so B is eliminated. Round 2 votes counts: C=33 E=28 D=22 A=17 so A is eliminated. Round 3 votes counts: E=36 C=34 D=30 so D is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 C:207 B:192 A:190 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -8 -2 -20 B -10 0 -10 16 -12 C 8 10 0 12 -16 D 2 -16 -12 0 -14 E 20 12 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -2 -20 B -10 0 -10 16 -12 C 8 10 0 12 -16 D 2 -16 -12 0 -14 E 20 12 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -2 -20 B -10 0 -10 16 -12 C 8 10 0 12 -16 D 2 -16 -12 0 -14 E 20 12 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 941: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C A B (7) A C B E D (7) C A E D B (6) A C B D E (6) D E B A C (5) B D E C A (5) B D E A C (5) A C E D B (5) D E B C A (4) D E A C B (4) B E D C A (4) B C A E D (4) D E C B A (3) D E C A B (3) B A C E D (3) A C D E B (3) E D C B A (2) C E D A B (2) C A B E D (2) B A D E C (2) A D E C B (2) E D B C A (1) E C D B A (1) D E A B C (1) C E A D B (1) B C E D A (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 4 2 B -2 0 -8 0 0 C -8 8 0 4 2 D -4 0 -4 0 6 E -2 0 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 4 2 B -2 0 -8 0 0 C -8 8 0 4 2 D -4 0 -4 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=25 D=20 E=11 C=11 so E is eliminated. Round 2 votes counts: B=33 D=30 A=25 C=12 so C is eliminated. Round 3 votes counts: A=34 D=33 B=33 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:203 D:199 B:195 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 4 2 B -2 0 -8 0 0 C -8 8 0 4 2 D -4 0 -4 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 4 2 B -2 0 -8 0 0 C -8 8 0 4 2 D -4 0 -4 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 4 2 B -2 0 -8 0 0 C -8 8 0 4 2 D -4 0 -4 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 942: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) C A E D B (10) C D A E B (6) A E C B D (6) A E B C D (5) E A B D C (4) D C B E A (4) D B E C A (4) D B E A C (4) B E A D C (4) E A D C B (3) E A C D B (3) D B C E A (3) C A E B D (3) B E D A C (3) E A C B D (2) D E B A C (2) C D B A E (2) B C D A E (2) E B A D C (1) E A B C D (1) D C E B A (1) D C E A B (1) D B C A E (1) C D E A B (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D E B (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A E C (1) B A C E D (1) A E C D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 10 -4 -10 B 0 0 4 4 -4 C -10 -4 0 -2 -14 D 4 -4 2 0 2 E 10 4 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 0 10 -4 -10 B 0 0 4 4 -4 C -10 -4 0 -2 -14 D 4 -4 2 0 2 E 10 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=26 B=26 D=20 E=14 A=14 so E is eliminated. Round 2 votes counts: B=27 A=27 C=26 D=20 so D is eliminated. Round 3 votes counts: B=41 C=32 A=27 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:213 B:202 D:202 A:198 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 -4 -10 B 0 0 4 4 -4 C -10 -4 0 -2 -14 D 4 -4 2 0 2 E 10 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -4 -10 B 0 0 4 4 -4 C -10 -4 0 -2 -14 D 4 -4 2 0 2 E 10 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -4 -10 B 0 0 4 4 -4 C -10 -4 0 -2 -14 D 4 -4 2 0 2 E 10 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 943: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) C D E B A (11) E A C D B (10) B D C A E (9) A E B C D (7) A B E D C (7) D B C A E (5) B A D C E (5) E C D A B (4) E C A D B (4) E A C B D (4) C E D A B (3) B D A C E (3) B A D E C (3) A E B D C (3) D C E B A (2) C D B E A (2) A B E C D (2) C E D B A (1) C D E A B (1) C B D E A (1) B C E A D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -10 -6 -10 B 8 0 -8 -8 0 C 10 8 0 2 10 D 6 8 -2 0 8 E 10 0 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 -10 B 8 0 -8 -8 0 C 10 8 0 2 10 D 6 8 -2 0 8 E 10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=22 B=21 A=20 C=19 D=18 so D is eliminated. Round 2 votes counts: C=32 B=26 E=22 A=20 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:210 B:196 E:196 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 -10 B 8 0 -8 -8 0 C 10 8 0 2 10 D 6 8 -2 0 8 E 10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 -10 B 8 0 -8 -8 0 C 10 8 0 2 10 D 6 8 -2 0 8 E 10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 -10 B 8 0 -8 -8 0 C 10 8 0 2 10 D 6 8 -2 0 8 E 10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 944: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) E C D A B (7) C A B E D (7) B A C D E (7) E D C A B (6) A C B E D (6) C E A B D (5) E C A B D (4) D B A E C (4) C A B D E (4) E D A B C (3) E C A D B (3) E A B D C (3) D E B C A (3) B D A C E (3) A B C E D (3) E A C B D (2) D B E C A (2) C A E B D (2) B A D C E (2) A B C D E (2) E D C B A (1) E D B C A (1) E A D B C (1) D E C B A (1) D C B A E (1) D B E A C (1) D B A C E (1) C D B A E (1) C B A D E (1) B D A E C (1) B C A D E (1) A B E C D (1) Total count = 100 A B C D E A 0 18 0 8 -6 B -18 0 -2 8 -4 C 0 2 0 12 -8 D -8 -8 -12 0 -10 E 6 4 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 0 8 -6 B -18 0 -2 8 -4 C 0 2 0 12 -8 D -8 -8 -12 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=23 C=20 B=14 A=12 so A is eliminated. Round 2 votes counts: E=31 C=26 D=23 B=20 so B is eliminated. Round 3 votes counts: C=39 E=32 D=29 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:210 C:203 B:192 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 0 8 -6 B -18 0 -2 8 -4 C 0 2 0 12 -8 D -8 -8 -12 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 8 -6 B -18 0 -2 8 -4 C 0 2 0 12 -8 D -8 -8 -12 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 8 -6 B -18 0 -2 8 -4 C 0 2 0 12 -8 D -8 -8 -12 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 945: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) C E B A D (7) D A B E C (6) C E A D B (6) B C E A D (6) C E D A B (5) B A E C D (5) D A C E B (4) B D A E C (4) B A D E C (4) D B A E C (3) C B E A D (3) B E C A D (3) A D E B C (3) E C A D B (2) D C E A B (2) D C A E B (2) C D E A B (2) C B E D A (2) B C D A E (2) E C A B D (1) E A C B D (1) E A B C D (1) D C A B E (1) D B A C E (1) D A E B C (1) D A B C E (1) C E A B D (1) B E A C D (1) B D A C E (1) B C E D A (1) B C D E A (1) B C A E D (1) B A C E D (1) A E D C B (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 4 0 10 B -8 0 -2 -4 -2 C -4 2 0 4 0 D 0 4 -4 0 2 E -10 2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.747431 B: 0.000000 C: 0.000000 D: 0.252569 E: 0.000000 Sum of squares = 0.622444499019 Cumulative probabilities = A: 0.747431 B: 0.747431 C: 0.747431 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 0 10 B -8 0 -2 -4 -2 C -4 2 0 4 0 D 0 4 -4 0 2 E -10 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500116 B: 0.000000 C: 0.000000 D: 0.499884 E: 0.000000 Sum of squares = 0.500000026717 Cumulative probabilities = A: 0.500116 B: 0.500116 C: 0.500116 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=30 C=26 A=7 E=5 so E is eliminated. Round 2 votes counts: D=32 B=30 C=29 A=9 so A is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:211 C:201 D:201 E:195 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 0 10 B -8 0 -2 -4 -2 C -4 2 0 4 0 D 0 4 -4 0 2 E -10 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500116 B: 0.000000 C: 0.000000 D: 0.499884 E: 0.000000 Sum of squares = 0.500000026717 Cumulative probabilities = A: 0.500116 B: 0.500116 C: 0.500116 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 0 10 B -8 0 -2 -4 -2 C -4 2 0 4 0 D 0 4 -4 0 2 E -10 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500116 B: 0.000000 C: 0.000000 D: 0.499884 E: 0.000000 Sum of squares = 0.500000026717 Cumulative probabilities = A: 0.500116 B: 0.500116 C: 0.500116 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 0 10 B -8 0 -2 -4 -2 C -4 2 0 4 0 D 0 4 -4 0 2 E -10 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500116 B: 0.000000 C: 0.000000 D: 0.499884 E: 0.000000 Sum of squares = 0.500000026717 Cumulative probabilities = A: 0.500116 B: 0.500116 C: 0.500116 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 946: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) C B E A D (8) A E C B D (8) D A E B C (7) D B A E C (6) D A E C B (6) B C E A D (6) D C B E A (5) A E B C D (4) D C A E B (3) B E A C D (3) B C D E A (3) D B C E A (2) D A C E B (2) C E B A D (2) C A E B D (2) B C E D A (2) B A E C D (2) D C E A B (1) D B E A C (1) D B A C E (1) D A B E C (1) C E A D B (1) C D E B A (1) C B D E A (1) B E D C A (1) B E C A D (1) B E A D C (1) B D E A C (1) B D C E A (1) B A E D C (1) A E C D B (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -4 10 -6 B -2 0 -10 18 -6 C 4 10 0 16 8 D -10 -18 -16 0 -12 E 6 6 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 10 -6 B -2 0 -10 18 -6 C 4 10 0 16 8 D -10 -18 -16 0 -12 E 6 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=27 B=22 A=16 so E is eliminated. Round 2 votes counts: D=35 C=27 B=22 A=16 so A is eliminated. Round 3 votes counts: D=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:208 A:201 B:200 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 10 -6 B -2 0 -10 18 -6 C 4 10 0 16 8 D -10 -18 -16 0 -12 E 6 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 10 -6 B -2 0 -10 18 -6 C 4 10 0 16 8 D -10 -18 -16 0 -12 E 6 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 10 -6 B -2 0 -10 18 -6 C 4 10 0 16 8 D -10 -18 -16 0 -12 E 6 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 947: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) A B E C D (7) D A C B E (5) E C B D A (4) D A C E B (4) D A B C E (4) C E B D A (4) C D E B A (4) A D B E C (4) A B D E C (4) E C B A D (3) E B C A D (3) D C B E A (3) C E B A D (3) B E C A D (3) B E A C D (3) B A E D C (3) A B E D C (3) E B C D A (2) D C E B A (2) D C A E B (2) C E A B D (2) B D A E C (2) A D B C E (2) A C D E B (2) E B A C D (1) E A B C D (1) D C E A B (1) D B E A C (1) D B A C E (1) C D E A B (1) C D A E B (1) C A E B D (1) C A D E B (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E C A (1) B D E A C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 2 -4 -6 B 8 0 -2 10 0 C -2 2 0 10 2 D 4 -10 -10 0 -8 E 6 0 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999998482 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -4 -6 B 8 0 -2 10 0 C -2 2 0 10 2 D 4 -10 -10 0 -8 E 6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 A=23 B=16 E=14 so E is eliminated. Round 2 votes counts: C=31 A=24 D=23 B=22 so B is eliminated. Round 3 votes counts: C=40 A=32 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:208 C:206 E:206 A:192 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -4 -6 B 8 0 -2 10 0 C -2 2 0 10 2 D 4 -10 -10 0 -8 E 6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -4 -6 B 8 0 -2 10 0 C -2 2 0 10 2 D 4 -10 -10 0 -8 E 6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -4 -6 B 8 0 -2 10 0 C -2 2 0 10 2 D 4 -10 -10 0 -8 E 6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 948: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) E D A C B (9) B C A D E (6) A B D E C (6) E A D B C (5) C B E D A (5) C B A D E (4) B A D E C (4) B A D C E (4) E C D A B (3) D A B E C (3) C E D B A (3) C E D A B (3) B A C D E (3) E B C A D (2) D E A B C (2) C B E A D (2) C B D E A (2) C B A E D (2) A E D B C (2) A D B E C (2) A B E D C (2) E D C A B (1) E D A B C (1) E C A D B (1) E A D C B (1) D E A C B (1) D C E A B (1) C E B A D (1) C D E B A (1) C B D A E (1) B D A C E (1) B C E A D (1) B C D A E (1) B A E D C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -2 2 -12 B 10 0 -4 20 0 C 2 4 0 4 4 D -2 -20 -4 0 -14 E 12 0 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 2 -12 B 10 0 -4 20 0 C 2 4 0 4 4 D -2 -20 -4 0 -14 E 12 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=23 B=21 A=14 D=7 so D is eliminated. Round 2 votes counts: C=36 E=26 B=21 A=17 so A is eliminated. Round 3 votes counts: C=36 B=34 E=30 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:213 E:211 C:207 A:189 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -2 2 -12 B 10 0 -4 20 0 C 2 4 0 4 4 D -2 -20 -4 0 -14 E 12 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 2 -12 B 10 0 -4 20 0 C 2 4 0 4 4 D -2 -20 -4 0 -14 E 12 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 2 -12 B 10 0 -4 20 0 C 2 4 0 4 4 D -2 -20 -4 0 -14 E 12 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 949: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (6) E A B D C (5) C D B A E (5) E B A D C (4) E B A C D (4) C D A E B (4) B C D A E (4) A D C E B (4) E C D A B (3) E A D B C (3) E A C D B (3) C E B D A (3) C B D E A (3) C B D A E (3) B C D E A (3) A E D C B (3) A D C B E (3) E C A D B (2) D C A E B (2) D A C B E (2) C D B E A (2) B E A D C (2) B E A C D (2) B A E D C (2) B A D C E (2) A D B C E (2) E C D B A (1) E C B D A (1) E C B A D (1) E A D C B (1) E A C B D (1) E A B C D (1) C E D B A (1) C E D A B (1) C D E B A (1) C D E A B (1) C D A B E (1) B E C A D (1) B A D E C (1) A E D B C (1) A D E C B (1) A D E B C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 6 0 B 4 0 -6 4 -6 C -2 6 0 4 8 D -6 -4 -4 0 6 E 0 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 6 0 B 4 0 -6 4 -6 C -2 6 0 4 8 D -6 -4 -4 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888794 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=25 B=23 A=18 D=4 so D is eliminated. Round 2 votes counts: E=30 C=27 B=23 A=20 so A is eliminated. Round 3 votes counts: E=36 C=36 B=28 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 A:202 B:198 D:196 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 6 0 B 4 0 -6 4 -6 C -2 6 0 4 8 D -6 -4 -4 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888794 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 0 B 4 0 -6 4 -6 C -2 6 0 4 8 D -6 -4 -4 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888794 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 0 B 4 0 -6 4 -6 C -2 6 0 4 8 D -6 -4 -4 0 6 E 0 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888794 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 950: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) D C A E B (9) C B D E A (7) A E D C B (7) B E A C D (5) A E B D C (5) D C B A E (4) C D B E A (4) B A E D C (4) E B A C D (3) E A C D B (3) C D E A B (3) B E C A D (3) B C D E A (3) A D E B C (3) E A C B D (2) D A E C B (2) B E A D C (2) B C E D A (2) A D E C B (2) E A B D C (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C E B A D (1) C E A D B (1) C D E B A (1) C D B A E (1) C D A E B (1) C B E D A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D A E (1) B A D E C (1) A E D B C (1) Total count = 100 A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -12 C -8 2 0 6 -8 D -8 -8 -6 0 -4 E 8 12 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -12 C -8 2 0 6 -8 D -8 -8 -6 0 -4 E 8 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=21 E=19 D=18 A=18 so D is eliminated. Round 2 votes counts: C=34 B=26 A=21 E=19 so E is eliminated. Round 3 votes counts: A=37 C=34 B=29 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:206 C:196 B:195 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -12 C -8 2 0 6 -8 D -8 -8 -6 0 -4 E 8 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -12 C -8 2 0 6 -8 D -8 -8 -6 0 -4 E 8 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -12 C -8 2 0 6 -8 D -8 -8 -6 0 -4 E 8 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 951: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C A D E B (8) C A D B E (6) E B D A C (5) C A E D B (5) A C D E B (5) E B A D C (4) D B A E C (4) B E C D A (4) E C A B D (3) D A C B E (3) D A B C E (3) A D C E B (3) E B A C D (2) C E A D B (2) C E A B D (2) C D A B E (2) B D E A C (2) B D C A E (2) B D A E C (2) B C E D A (2) A D E C B (2) A C E D B (2) E C B A D (1) E B D C A (1) E B C A D (1) E A D B C (1) E A C D B (1) E A B D C (1) D C A B E (1) D A B E C (1) C D B A E (1) C B E A D (1) C B D A E (1) C B A E D (1) B E D C A (1) B D A C E (1) B C D E A (1) B C D A E (1) A D E B C (1) Total count = 100 A B C D E A 0 4 6 4 10 B -4 0 0 -2 0 C -6 0 0 4 6 D -4 2 -4 0 0 E -10 0 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 10 B -4 0 0 -2 0 C -6 0 0 4 6 D -4 2 -4 0 0 E -10 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=26 E=20 A=13 D=12 so D is eliminated. Round 2 votes counts: C=30 B=30 E=20 A=20 so E is eliminated. Round 3 votes counts: B=43 C=34 A=23 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:212 C:202 B:197 D:197 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 10 B -4 0 0 -2 0 C -6 0 0 4 6 D -4 2 -4 0 0 E -10 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 10 B -4 0 0 -2 0 C -6 0 0 4 6 D -4 2 -4 0 0 E -10 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 10 B -4 0 0 -2 0 C -6 0 0 4 6 D -4 2 -4 0 0 E -10 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 952: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) D A C B E (4) C E D B A (4) C E A B D (4) C D E A B (4) B E A D C (4) A C D B E (4) E C D B A (3) E B D A C (3) D E B C A (3) D C E B A (3) D B A E C (3) C E D A B (3) C D A E B (3) C A E B D (3) B A E D C (3) A C E B D (3) E D C B A (2) E C B A D (2) E B A D C (2) D C A E B (2) D C A B E (2) D B E A C (2) B D A E C (2) A C B E D (2) A B E D C (2) A B E C D (2) A B D E C (2) A B D C E (2) A B C D E (2) E D B C A (1) E C A B D (1) E B C D A (1) E B C A D (1) D C E A B (1) D A B C E (1) C D E B A (1) C A E D B (1) B E A C D (1) B D E A C (1) B A D E C (1) A D C B E (1) A D B C E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 10 2 -4 B -4 0 -8 2 -12 C -10 8 0 4 6 D -2 -2 -4 0 -8 E 4 12 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000057 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 10 2 -4 B -4 0 -8 2 -12 C -10 8 0 4 6 D -2 -2 -4 0 -8 E 4 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000042 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=23 A=23 E=21 D=21 B=12 so B is eliminated. Round 2 votes counts: A=27 E=26 D=24 C=23 so C is eliminated. Round 3 votes counts: E=37 D=32 A=31 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:209 A:206 C:204 D:192 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 2 -4 B -4 0 -8 2 -12 C -10 8 0 4 6 D -2 -2 -4 0 -8 E 4 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000042 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 2 -4 B -4 0 -8 2 -12 C -10 8 0 4 6 D -2 -2 -4 0 -8 E 4 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000042 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 2 -4 B -4 0 -8 2 -12 C -10 8 0 4 6 D -2 -2 -4 0 -8 E 4 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000042 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 953: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) A E D B C (9) E A D C B (7) C B D E A (7) C B E A D (5) E A D B C (4) D E A C B (4) D B C A E (4) D A E B C (4) E A B C D (3) D C E A B (3) D A E C B (3) B C E A D (3) E D A C B (2) E C A D B (2) E A C D B (2) E A C B D (2) D A B E C (2) C E B A D (2) C B D A E (2) B D C A E (2) B C D A E (2) B A E C D (2) A E B C D (2) A D E B C (2) E C A B D (1) E B A C D (1) D A C B E (1) C D E B A (1) C D B E A (1) C D B A E (1) B E C A D (1) B C A E D (1) B C A D E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 10 4 4 -2 B -10 0 -10 -22 -10 C -4 10 0 -16 -10 D -4 22 16 0 -2 E 2 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 4 4 -2 B -10 0 -10 -22 -10 C -4 10 0 -16 -10 D -4 22 16 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=24 C=19 A=15 B=12 so B is eliminated. Round 2 votes counts: D=32 C=26 E=25 A=17 so A is eliminated. Round 3 votes counts: E=40 D=34 C=26 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:212 A:208 C:190 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 4 4 -2 B -10 0 -10 -22 -10 C -4 10 0 -16 -10 D -4 22 16 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 4 -2 B -10 0 -10 -22 -10 C -4 10 0 -16 -10 D -4 22 16 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 4 -2 B -10 0 -10 -22 -10 C -4 10 0 -16 -10 D -4 22 16 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 954: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (5) E C A D B (4) A E C B D (4) A E B D C (4) E C D A B (3) D E C A B (3) D C E B A (3) D C B E A (3) D A E B C (3) C B D E A (3) B D C A E (3) B D A C E (3) B A C D E (3) A B E D C (3) E D C A B (2) E D A C B (2) E C A B D (2) E A D C B (2) E A D B C (2) D B A C E (2) C E D B A (2) C E D A B (2) C E B D A (2) C E A B D (2) C D B E A (2) C B E A D (2) C A B E D (2) B C A D E (2) B A D C E (2) B A C E D (2) A B E C D (2) A B D E C (2) E A C B D (1) D E C B A (1) D E A B C (1) D B C E A (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A D B (1) C D E B A (1) C B E D A (1) B D A E C (1) B C D A E (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 16 2 -4 -8 B -16 0 -16 0 -14 C -2 16 0 -4 -6 D 4 0 4 0 -6 E 8 14 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 2 -4 -8 B -16 0 -16 0 -14 C -2 16 0 -4 -6 D 4 0 4 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=21 A=20 E=18 B=17 so B is eliminated. Round 2 votes counts: D=31 A=27 C=24 E=18 so E is eliminated. Round 3 votes counts: D=35 C=33 A=32 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:217 A:203 C:202 D:201 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 2 -4 -8 B -16 0 -16 0 -14 C -2 16 0 -4 -6 D 4 0 4 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 -4 -8 B -16 0 -16 0 -14 C -2 16 0 -4 -6 D 4 0 4 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 -4 -8 B -16 0 -16 0 -14 C -2 16 0 -4 -6 D 4 0 4 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 955: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) C D E A B (7) A B C D E (6) D C E B A (5) A B E C D (5) E D B C A (4) E B A D C (4) C D A B E (4) B E A D C (4) E C D A B (3) C A D B E (3) A E C B D (3) E D C B A (2) E C A D B (2) E B D A C (2) E A B D C (2) E A B C D (2) D E C B A (2) D C B E A (2) C D A E B (2) B A D C E (2) B A C D E (2) A E B C D (2) A C B D E (2) E D C A B (1) E D B A C (1) E D A B C (1) E A C D B (1) D E B C A (1) D B E C A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D B A E (1) C A D E B (1) B D E A C (1) B D C A E (1) B D A C E (1) B A D E C (1) A C E D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 10 12 -2 B -4 0 8 4 -4 C -10 -8 0 4 -10 D -12 -4 -4 0 -8 E 2 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 12 -2 B -4 0 8 4 -4 C -10 -8 0 4 -10 D -12 -4 -4 0 -8 E 2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=22 C=21 A=21 D=11 so D is eliminated. Round 2 votes counts: E=28 C=28 B=23 A=21 so A is eliminated. Round 3 votes counts: B=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:212 B:202 C:188 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 12 -2 B -4 0 8 4 -4 C -10 -8 0 4 -10 D -12 -4 -4 0 -8 E 2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 12 -2 B -4 0 8 4 -4 C -10 -8 0 4 -10 D -12 -4 -4 0 -8 E 2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 12 -2 B -4 0 8 4 -4 C -10 -8 0 4 -10 D -12 -4 -4 0 -8 E 2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 956: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) B E C A D (8) D A E C B (6) D A C E B (6) C B A E D (6) D A E B C (5) C B E D A (5) C B D A E (5) E D A B C (4) D E A B C (4) A D E C B (4) C B D E A (3) C A D B E (3) A D E B C (3) E B D A C (2) E A D B C (2) D E B A C (2) D C A E B (2) D C A B E (2) C D A B E (2) C B E A D (2) B E C D A (2) B C E D A (2) A C B E D (2) E B A D C (1) E B A C D (1) E A B D C (1) D E A C B (1) D C B E A (1) C A B D E (1) B E D C A (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 0 -8 -10 -2 B 0 0 -2 6 8 C 8 2 0 4 2 D 10 -6 -4 0 0 E 2 -8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -10 -2 B 0 0 -2 6 8 C 8 2 0 4 2 D 10 -6 -4 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 B=22 E=11 A=11 so E is eliminated. Round 2 votes counts: D=33 C=27 B=26 A=14 so A is eliminated. Round 3 votes counts: D=42 C=29 B=29 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:206 D:200 E:196 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -10 -2 B 0 0 -2 6 8 C 8 2 0 4 2 D 10 -6 -4 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -10 -2 B 0 0 -2 6 8 C 8 2 0 4 2 D 10 -6 -4 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -10 -2 B 0 0 -2 6 8 C 8 2 0 4 2 D 10 -6 -4 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 957: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) A D B C E (9) B D E A C (8) B D A E C (8) C E B A D (7) B E C D A (7) D A B E C (6) A D C E B (6) A C D E B (6) E C B D A (4) C E A B D (4) E C D A B (3) A D C B E (3) D B A E C (2) B E D C A (2) B D A C E (2) B A D C E (2) A D E C B (2) E D C A B (1) E C D B A (1) E C B A D (1) E C A D B (1) E B D C A (1) C E B D A (1) C B E A D (1) C A E D B (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 4 12 8 -2 B -4 0 -2 -2 4 C -12 2 0 -6 4 D -8 2 6 0 12 E 2 -4 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016528764 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 A B C D E A 0 4 12 8 -2 B -4 0 -2 -2 4 C -12 2 0 -6 4 D -8 2 6 0 12 E 2 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016528882 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 C=23 E=12 D=8 so D is eliminated. Round 2 votes counts: A=33 B=32 C=23 E=12 so E is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:206 B:198 C:194 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 8 -2 B -4 0 -2 -2 4 C -12 2 0 -6 4 D -8 2 6 0 12 E 2 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016528882 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 8 -2 B -4 0 -2 -2 4 C -12 2 0 -6 4 D -8 2 6 0 12 E 2 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016528882 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 8 -2 B -4 0 -2 -2 4 C -12 2 0 -6 4 D -8 2 6 0 12 E 2 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016528882 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 958: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) B C E A D (7) D A E C B (6) B C A E D (6) A D B E C (6) E D C B A (5) D A E B C (5) B A C E D (5) E C B D A (4) C E B D A (4) E C D B A (3) D E C A B (3) D E A C B (3) C B A E D (3) A D B C E (3) E D C A B (2) E B C A D (2) D E A B C (2) B A C D E (2) A D C B E (2) A B D C E (2) E D B A C (1) E D A C B (1) E D A B C (1) D C E A B (1) D A C E B (1) D A C B E (1) D A B E C (1) C D A E B (1) C B E D A (1) C B A D E (1) B E C A D (1) B E A C D (1) B A E C D (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -6 8 -2 B 12 0 -2 4 10 C 6 2 0 4 0 D -8 -4 -4 0 -16 E 2 -10 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.887540 D: 0.000000 E: 0.112460 Sum of squares = 0.800374439528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.887540 D: 0.887540 E: 1.000000 A B C D E A 0 -12 -6 8 -2 B 12 0 -2 4 10 C 6 2 0 4 0 D -8 -4 -4 0 -16 E 2 -10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.722222669114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 E=19 C=19 A=16 so A is eliminated. Round 2 votes counts: D=34 B=28 E=19 C=19 so E is eliminated. Round 3 votes counts: D=44 B=30 C=26 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:206 E:204 A:194 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -6 8 -2 B 12 0 -2 4 10 C 6 2 0 4 0 D -8 -4 -4 0 -16 E 2 -10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.722222669114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 8 -2 B 12 0 -2 4 10 C 6 2 0 4 0 D -8 -4 -4 0 -16 E 2 -10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.722222669114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 8 -2 B 12 0 -2 4 10 C 6 2 0 4 0 D -8 -4 -4 0 -16 E 2 -10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.000000 E: 0.166666 Sum of squares = 0.722222669114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833334 D: 0.833334 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 959: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) A E B C D (8) B A E C D (5) D E C A B (4) D C E A B (4) D C B E A (4) C B E D A (4) B C D E A (4) B A C E D (4) E C D A B (3) D B C A E (3) D A E C B (3) A E D C B (3) E C D B A (2) E C B D A (2) E A D C B (2) C B D E A (2) B D A C E (2) B C A E D (2) B A C D E (2) A E B D C (2) A D E C B (2) A B D E C (2) E D A C B (1) E C B A D (1) E B A C D (1) E A C B D (1) D E C B A (1) D E A C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D B E A (1) B E C A D (1) B E A C D (1) B D C A E (1) B C E D A (1) B A D C E (1) A D B E C (1) A D B C E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -6 -14 -8 B 16 0 -10 2 -10 C 6 10 0 -4 2 D 14 -2 4 0 6 E 8 10 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000001 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -14 -8 B 16 0 -10 2 -10 C 6 10 0 -4 2 D 14 -2 4 0 6 E 8 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000138 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=24 A=22 E=13 C=9 so C is eliminated. Round 2 votes counts: D=34 B=30 A=22 E=14 so E is eliminated. Round 3 votes counts: D=40 B=35 A=25 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:211 C:207 E:205 B:199 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -6 -14 -8 B 16 0 -10 2 -10 C 6 10 0 -4 2 D 14 -2 4 0 6 E 8 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000138 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -14 -8 B 16 0 -10 2 -10 C 6 10 0 -4 2 D 14 -2 4 0 6 E 8 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000138 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -14 -8 B 16 0 -10 2 -10 C 6 10 0 -4 2 D 14 -2 4 0 6 E 8 10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000138 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 960: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) C A D E B (7) E D B C A (6) E B D C A (5) C D E B A (5) C A B E D (5) B E D A C (5) A B E D C (5) D E B A C (4) C E D B A (4) A C B E D (4) A C B D E (4) C E B D A (3) C D E A B (3) D E B C A (2) D B E A C (2) C A B D E (2) B E D C A (2) B A E C D (2) A D E B C (2) A D B E C (2) E C D B A (1) E C B D A (1) E B D A C (1) D E C B A (1) C B E D A (1) C B A E D (1) C A E B D (1) B E C D A (1) B E A D C (1) B C E D A (1) B C E A D (1) B A E D C (1) B A C E D (1) A C D E B (1) A C D B E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 4 0 B 4 0 10 18 6 C 6 -10 0 6 -6 D -4 -18 -6 0 -10 E 0 -6 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 4 0 B 4 0 10 18 6 C 6 -10 0 6 -6 D -4 -18 -6 0 -10 E 0 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=30 B=15 E=14 D=9 so D is eliminated. Round 2 votes counts: C=32 A=30 E=21 B=17 so B is eliminated. Round 3 votes counts: C=34 A=34 E=32 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:219 E:205 C:198 A:197 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 4 0 B 4 0 10 18 6 C 6 -10 0 6 -6 D -4 -18 -6 0 -10 E 0 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 0 B 4 0 10 18 6 C 6 -10 0 6 -6 D -4 -18 -6 0 -10 E 0 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 0 B 4 0 10 18 6 C 6 -10 0 6 -6 D -4 -18 -6 0 -10 E 0 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 961: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) C A D E B (6) B D A E C (6) A D E B C (5) D A B E C (4) D A B C E (4) C E B A D (4) C E A D B (4) B E C D A (4) B D E A C (4) B C D A E (4) E B D A C (3) E A D B C (3) A D C E B (3) E D A B C (2) E C B A D (2) E C A D B (2) E B C D A (2) E A D C B (2) D B A E C (2) C E A B D (2) C A E D B (2) C A D B E (2) B D A C E (2) B C E D A (2) A E D C B (2) A D E C B (2) A D C B E (2) D A E B C (1) C E B D A (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A D E (1) B E D C A (1) A E D B C (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 16 -4 4 B -2 0 18 -2 0 C -16 -18 0 -16 -14 D 4 2 16 0 0 E -4 0 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.707447 E: 0.292553 Sum of squares = 0.58606865968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.707447 E: 1.000000 A B C D E A 0 2 16 -4 4 B -2 0 18 -2 0 C -16 -18 0 -16 -14 D 4 2 16 0 0 E -4 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 0.498952 Sum of squares = 0.500002195399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=25 A=17 E=16 D=11 so D is eliminated. Round 2 votes counts: B=33 A=26 C=25 E=16 so E is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:209 B:207 E:205 C:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 16 -4 4 B -2 0 18 -2 0 C -16 -18 0 -16 -14 D 4 2 16 0 0 E -4 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 0.498952 Sum of squares = 0.500002195399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 -4 4 B -2 0 18 -2 0 C -16 -18 0 -16 -14 D 4 2 16 0 0 E -4 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 0.498952 Sum of squares = 0.500002195399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 -4 4 B -2 0 18 -2 0 C -16 -18 0 -16 -14 D 4 2 16 0 0 E -4 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 0.498952 Sum of squares = 0.500002195399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501048 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 962: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (17) A B D C E (12) C E A B D (9) E C D B A (7) E C B D A (6) A D B C E (6) E C A B D (5) D A B C E (5) E C B A D (4) B A D E C (3) A C E B D (3) A B D E C (3) D E C B A (2) D B E A C (2) D B A C E (2) C E D A B (2) C E A D B (2) B D A E C (2) E B A C D (1) D C E B A (1) D B E C A (1) C E D B A (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 20 4 14 B -2 0 12 4 14 C -20 -12 0 -12 -6 D -4 -4 12 0 16 E -14 -14 6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999281 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 20 4 14 B -2 0 12 4 14 C -20 -12 0 -12 -6 D -4 -4 12 0 16 E -14 -14 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=28 E=23 C=14 B=5 so B is eliminated. Round 2 votes counts: D=32 A=31 E=23 C=14 so C is eliminated. Round 3 votes counts: E=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:220 B:214 D:210 E:181 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 20 4 14 B -2 0 12 4 14 C -20 -12 0 -12 -6 D -4 -4 12 0 16 E -14 -14 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 20 4 14 B -2 0 12 4 14 C -20 -12 0 -12 -6 D -4 -4 12 0 16 E -14 -14 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 20 4 14 B -2 0 12 4 14 C -20 -12 0 -12 -6 D -4 -4 12 0 16 E -14 -14 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 963: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) A C B E D (6) C B D E A (5) B E A D C (5) A E D B C (5) A D E C B (5) E D B A C (4) E B D A C (4) D C E B A (4) C A D B E (4) C A B D E (4) A E D C B (4) A C D E B (4) D E B C A (3) C D A E B (3) B E D A C (3) A E B D C (3) A B E C D (3) C B A E D (2) B C E D A (2) A C E B D (2) E B D C A (1) E A D B C (1) D E C B A (1) D E B A C (1) D E A C B (1) D C E A B (1) D B E C A (1) D A E C B (1) C D A B E (1) C A D E B (1) B E A C D (1) B D E C A (1) B C D E A (1) B C A E D (1) B A C E D (1) A D E B C (1) A C E D B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 16 12 8 B -6 0 -2 4 -2 C -16 2 0 -12 -10 D -12 -4 12 0 -12 E -8 2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 12 8 B -6 0 -2 4 -2 C -16 2 0 -12 -10 D -12 -4 12 0 -12 E -8 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=21 C=20 D=13 E=10 so E is eliminated. Round 2 votes counts: A=37 B=26 C=20 D=17 so D is eliminated. Round 3 votes counts: A=39 B=35 C=26 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:208 B:197 D:192 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 12 8 B -6 0 -2 4 -2 C -16 2 0 -12 -10 D -12 -4 12 0 -12 E -8 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 12 8 B -6 0 -2 4 -2 C -16 2 0 -12 -10 D -12 -4 12 0 -12 E -8 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 12 8 B -6 0 -2 4 -2 C -16 2 0 -12 -10 D -12 -4 12 0 -12 E -8 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 964: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (16) C A B E D (11) B C A D E (8) D E A B C (7) D E B C A (6) D E B A C (6) E A C D B (5) D B E C A (5) E A D C B (4) C B A E D (4) B D C E A (4) B D C A E (4) A C B E D (4) B C D A E (2) E D C A B (1) E C A D B (1) D E C B A (1) C E D B A (1) C E D A B (1) C B A D E (1) B D E A C (1) B D A C E (1) B A C D E (1) A E D B C (1) A E C D B (1) A C E D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 0 -12 -18 B -10 0 -4 -14 -4 C 0 4 0 -14 -10 D 12 14 14 0 -4 E 18 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 0 -12 -18 B -10 0 -4 -14 -4 C 0 4 0 -14 -10 D 12 14 14 0 -4 E 18 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 B=21 C=18 A=9 so A is eliminated. Round 2 votes counts: E=29 D=25 C=23 B=23 so C is eliminated. Round 3 votes counts: B=43 E=32 D=25 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:218 A:190 C:190 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 0 -12 -18 B -10 0 -4 -14 -4 C 0 4 0 -14 -10 D 12 14 14 0 -4 E 18 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -12 -18 B -10 0 -4 -14 -4 C 0 4 0 -14 -10 D 12 14 14 0 -4 E 18 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -12 -18 B -10 0 -4 -14 -4 C 0 4 0 -14 -10 D 12 14 14 0 -4 E 18 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 965: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) A E C B D (7) D C B A E (6) D C A B E (6) B E C A D (4) B D E C A (4) B C D E A (4) A E B C D (4) E B C A D (3) E B A C D (3) E A B D C (3) A C E B D (3) A C D B E (3) E B D A C (2) E B C D A (2) E A D B C (2) D B E C A (2) D B C E A (2) C D B A E (2) C D A B E (2) C B D A E (2) C B A E D (2) C A B E D (2) B C E D A (2) A E D C B (2) A C B E D (2) E D B A C (1) E B D C A (1) E B A D C (1) D E B A C (1) D E A B C (1) D C B E A (1) D A E C B (1) D A C B E (1) C B E D A (1) C A D B E (1) B E D C A (1) B E C D A (1) B D C E A (1) A E D B C (1) A E C D B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -4 6 0 B -2 0 6 24 4 C 4 -6 0 18 -10 D -6 -24 -18 0 -16 E 0 -4 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.679930 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.320070 Sum of squares = 0.564749882058 Cumulative probabilities = A: 0.679930 B: 0.679930 C: 0.679930 D: 0.679930 E: 1.000000 A B C D E A 0 2 -4 6 0 B -2 0 6 24 4 C 4 -6 0 18 -10 D -6 -24 -18 0 -16 E 0 -4 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555561117 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 D=21 B=17 C=12 so C is eliminated. Round 2 votes counts: A=28 E=25 D=25 B=22 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:216 E:211 C:203 A:202 D:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -4 6 0 B -2 0 6 24 4 C 4 -6 0 18 -10 D -6 -24 -18 0 -16 E 0 -4 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555561117 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 6 0 B -2 0 6 24 4 C 4 -6 0 18 -10 D -6 -24 -18 0 -16 E 0 -4 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555561117 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 6 0 B -2 0 6 24 4 C 4 -6 0 18 -10 D -6 -24 -18 0 -16 E 0 -4 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555561117 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 966: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (12) E B C D A (9) D A C B E (9) E B A C D (6) A D C B E (6) A D C E B (5) D C A B E (4) C D B A E (4) C B E D A (4) A D E C B (4) A D E B C (4) E A B D C (3) C B D E A (3) C B D A E (3) A E D B C (3) E A D B C (2) B E C A D (2) B C E D A (2) A E B C D (2) E D B C A (1) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) E A B C D (1) D E A C B (1) D A C E B (1) C E B D A (1) C D A B E (1) A E B D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 2 6 -2 B 2 0 4 6 -20 C -2 -4 0 6 -12 D -6 -6 -6 0 -8 E 2 20 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 6 -2 B 2 0 4 6 -20 C -2 -4 0 6 -12 D -6 -6 -6 0 -8 E 2 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=27 C=16 D=15 B=4 so B is eliminated. Round 2 votes counts: E=40 A=27 C=18 D=15 so D is eliminated. Round 3 votes counts: E=41 A=37 C=22 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:202 B:196 C:194 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 6 -2 B 2 0 4 6 -20 C -2 -4 0 6 -12 D -6 -6 -6 0 -8 E 2 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 6 -2 B 2 0 4 6 -20 C -2 -4 0 6 -12 D -6 -6 -6 0 -8 E 2 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 6 -2 B 2 0 4 6 -20 C -2 -4 0 6 -12 D -6 -6 -6 0 -8 E 2 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 967: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) A D E B C (6) E A D C B (5) D A E C B (5) D A B C E (5) E A D B C (4) C B D E A (4) B E C A D (4) A E D B C (4) E B C A D (3) E B A D C (3) D A C B E (3) C E D B A (3) C E B A D (3) B C D A E (3) E C D A B (2) E C B A D (2) E B A C D (2) D E A C B (2) C E B D A (2) C D A E B (2) C B D A E (2) B E A D C (2) B A E D C (2) A D B E C (2) A D B C E (2) E C B D A (1) E C A B D (1) E A B D C (1) D C A E B (1) D C A B E (1) C D E A B (1) C D A B E (1) C B E A D (1) B E A C D (1) B C E A D (1) B C A E D (1) B C A D E (1) B A D E C (1) B A D C E (1) B A C D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 8 12 -8 B 0 0 4 2 -6 C -8 -4 0 -4 -10 D -12 -2 4 0 -10 E 8 6 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 12 -8 B 0 0 4 2 -6 C -8 -4 0 -4 -10 D -12 -2 4 0 -10 E 8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 B=18 D=17 A=16 so A is eliminated. Round 2 votes counts: E=28 D=27 C=25 B=20 so B is eliminated. Round 3 votes counts: E=38 C=32 D=30 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:206 B:200 D:190 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 12 -8 B 0 0 4 2 -6 C -8 -4 0 -4 -10 D -12 -2 4 0 -10 E 8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 12 -8 B 0 0 4 2 -6 C -8 -4 0 -4 -10 D -12 -2 4 0 -10 E 8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 12 -8 B 0 0 4 2 -6 C -8 -4 0 -4 -10 D -12 -2 4 0 -10 E 8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 968: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (5) D A B C E (5) E A C B D (4) D C A E B (4) C D E A B (4) A E C D B (4) A D C E B (4) E C B D A (3) E C B A D (3) C E D B A (3) B E A C D (3) B D E C A (3) B D A E C (3) B A D E C (3) E C A B D (2) E B C A D (2) E B A C D (2) D C A B E (2) D A C B E (2) C E D A B (2) C D E B A (2) B E C D A (2) B E C A D (2) B D C A E (2) B D A C E (2) B A E D C (2) A E D C B (2) A E B D C (2) A E B C D (2) A D B E C (2) A B D E C (2) E A C D B (1) E A B C D (1) D C B A E (1) D B C A E (1) D B A C E (1) B E A D C (1) B D C E A (1) B C E D A (1) B C D E A (1) A E C B D (1) A D E C B (1) A D C B E (1) A D B C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 12 8 10 4 B -12 0 -4 2 -10 C -8 4 0 2 -18 D -10 -2 -2 0 -2 E -4 10 18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 10 4 B -12 0 -4 2 -10 C -8 4 0 2 -18 D -10 -2 -2 0 -2 E -4 10 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 E=23 D=16 C=11 so C is eliminated. Round 2 votes counts: E=28 B=26 A=24 D=22 so D is eliminated. Round 3 votes counts: A=37 E=34 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:213 D:192 C:190 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 10 4 B -12 0 -4 2 -10 C -8 4 0 2 -18 D -10 -2 -2 0 -2 E -4 10 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 10 4 B -12 0 -4 2 -10 C -8 4 0 2 -18 D -10 -2 -2 0 -2 E -4 10 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 10 4 B -12 0 -4 2 -10 C -8 4 0 2 -18 D -10 -2 -2 0 -2 E -4 10 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 969: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) B D C E A (10) B D A C E (10) E C A D B (6) A E B C D (5) B D E C A (4) A E C B D (4) E A C D B (3) D B C E A (3) D B A C E (3) C E A D B (3) C D E B A (3) B D C A E (3) E A C B D (2) D B C A E (2) C E D A B (2) C D E A B (2) B E D C A (2) B D A E C (2) B A E C D (2) B A D E C (2) E C D B A (1) E C A B D (1) D C E B A (1) D C E A B (1) D C B E A (1) C E D B A (1) C D A E B (1) C A D E B (1) B E D A C (1) B E C D A (1) B E A C D (1) B A D C E (1) A D C E B (1) A C E D B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 -8 2 B 8 0 6 4 -2 C -2 -6 0 4 2 D 8 -4 -4 0 4 E -2 2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999984 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 A B C D E A 0 -8 2 -8 2 B 8 0 6 4 -2 C -2 -6 0 4 2 D 8 -4 -4 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999872 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=24 E=13 C=13 D=11 so D is eliminated. Round 2 votes counts: B=47 A=24 C=16 E=13 so E is eliminated. Round 3 votes counts: B=47 A=29 C=24 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:202 C:199 E:197 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -8 2 B 8 0 6 4 -2 C -2 -6 0 4 2 D 8 -4 -4 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999872 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -8 2 B 8 0 6 4 -2 C -2 -6 0 4 2 D 8 -4 -4 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999872 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -8 2 B 8 0 6 4 -2 C -2 -6 0 4 2 D 8 -4 -4 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999872 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 970: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) E D C A B (6) B A E C D (6) B A C D E (6) E B A D C (5) C B A D E (5) D C E A B (4) C D B A E (4) C D A B E (4) C B D A E (4) B E A C D (4) B C A D E (4) D E C A B (3) C D E B A (3) B A E D C (3) B A C E D (3) E D A C B (2) E C B D A (2) D E A C B (2) D A E B C (2) C E D B A (2) C D B E A (2) A B D C E (2) E D A B C (1) E C D B A (1) E C B A D (1) E B A C D (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B A D (1) C D E A B (1) C B A E D (1) B C E A D (1) B C A E D (1) B A D C E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -22 -2 8 16 B 22 0 -2 18 18 C 2 2 0 14 2 D -8 -18 -14 0 0 E -16 -18 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -2 8 16 B 22 0 -2 18 18 C 2 2 0 14 2 D -8 -18 -14 0 0 E -16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=27 E=19 D=14 A=11 so A is eliminated. Round 2 votes counts: B=38 C=27 E=21 D=14 so D is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:228 C:210 A:200 E:182 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -2 8 16 B 22 0 -2 18 18 C 2 2 0 14 2 D -8 -18 -14 0 0 E -16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -2 8 16 B 22 0 -2 18 18 C 2 2 0 14 2 D -8 -18 -14 0 0 E -16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -2 8 16 B 22 0 -2 18 18 C 2 2 0 14 2 D -8 -18 -14 0 0 E -16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 971: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) E C A D B (10) A E C B D (10) A B D E C (10) D B C E A (7) C D B E A (6) A E B D C (6) B D A C E (5) E A C D B (4) D C B E A (3) D B C A E (3) B D C A E (3) A E C D B (3) A E B C D (3) E C D B A (2) E A C B D (2) B D C E A (2) A B D C E (2) E C B D A (1) D B A C E (1) C E B D A (1) C D E B A (1) B D E C A (1) B A D C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -2 6 -2 B -4 0 -10 -4 -10 C 2 10 0 10 -6 D -6 4 -10 0 -8 E 2 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -2 6 -2 B -4 0 -10 -4 -10 C 2 10 0 10 -6 D -6 4 -10 0 -8 E 2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=19 C=19 D=14 B=12 so B is eliminated. Round 2 votes counts: A=37 D=25 E=19 C=19 so E is eliminated. Round 3 votes counts: A=43 C=32 D=25 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:208 A:203 D:190 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 6 -2 B -4 0 -10 -4 -10 C 2 10 0 10 -6 D -6 4 -10 0 -8 E 2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 -2 B -4 0 -10 -4 -10 C 2 10 0 10 -6 D -6 4 -10 0 -8 E 2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 -2 B -4 0 -10 -4 -10 C 2 10 0 10 -6 D -6 4 -10 0 -8 E 2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 972: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (16) D C A B E (14) B A E C D (13) A B C E D (6) D C A E B (5) E B C A D (4) D C E A B (4) A B C D E (4) E C B A D (3) D E C B A (3) D B A E C (3) D A C B E (3) D A B C E (3) C A B E D (3) E B D A C (2) C E A B D (2) C A B D E (2) E D C B A (1) E D B A C (1) E C D B A (1) E C B D A (1) D E B A C (1) C E D A B (1) C D E A B (1) B E A C D (1) B A E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 10 12 16 B 2 0 12 18 8 C -10 -12 0 16 -2 D -12 -18 -16 0 -12 E -16 -8 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 12 16 B 2 0 12 18 8 C -10 -12 0 16 -2 D -12 -18 -16 0 -12 E -16 -8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=29 B=15 A=11 C=9 so C is eliminated. Round 2 votes counts: D=37 E=32 A=16 B=15 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:220 A:218 C:196 E:195 D:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 12 16 B 2 0 12 18 8 C -10 -12 0 16 -2 D -12 -18 -16 0 -12 E -16 -8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 12 16 B 2 0 12 18 8 C -10 -12 0 16 -2 D -12 -18 -16 0 -12 E -16 -8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 12 16 B 2 0 12 18 8 C -10 -12 0 16 -2 D -12 -18 -16 0 -12 E -16 -8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 973: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) E B C A D (7) D C E B A (7) C D E B A (7) A E B D C (7) A D B E C (5) E A B C D (4) D A B E C (4) C D B E A (4) A B E D C (4) D C A B E (3) D A C B E (3) C E B D A (3) C E B A D (3) A D E B C (3) A B E C D (3) D C A E B (2) D A E B C (2) D A B C E (2) C B E D A (2) C B E A D (2) B E A C D (2) E D C A B (1) E A B D C (1) D C E A B (1) D C B A E (1) D A E C B (1) D A C E B (1) C E D B A (1) C B D E A (1) C B A D E (1) B C E A D (1) B A E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 6 8 -12 B 4 0 12 4 -20 C -6 -12 0 2 -10 D -8 -4 -2 0 -2 E 12 20 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 6 8 -12 B 4 0 12 4 -20 C -6 -12 0 2 -10 D -8 -4 -2 0 -2 E 12 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 A=23 E=22 B=4 so B is eliminated. Round 2 votes counts: D=27 C=25 E=24 A=24 so E is eliminated. Round 3 votes counts: A=40 C=32 D=28 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:222 B:200 A:199 D:192 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 8 -12 B 4 0 12 4 -20 C -6 -12 0 2 -10 D -8 -4 -2 0 -2 E 12 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 8 -12 B 4 0 12 4 -20 C -6 -12 0 2 -10 D -8 -4 -2 0 -2 E 12 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 8 -12 B 4 0 12 4 -20 C -6 -12 0 2 -10 D -8 -4 -2 0 -2 E 12 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 974: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C D B A E (7) C B A D E (7) E D B A C (6) D E C B A (6) D C E B A (5) D C B E A (5) E A B D C (4) A B E C D (4) E D C A B (3) E D A C B (3) D E C A B (3) D E B C A (3) D B C E A (3) B A C E D (3) A B C E D (3) E B D A C (2) E A C D B (2) D C E A B (2) D C B A E (2) A C B E D (2) E B A D C (1) E A D B C (1) E A C B D (1) D E B A C (1) D B E C A (1) C D E A B (1) C D A E B (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D A C (1) B D C E A (1) B D A E C (1) B C D A E (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -14 -8 -34 -28 B 14 0 -6 -30 -10 C 8 6 0 -26 -6 D 34 30 26 0 6 E 28 10 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -34 -28 B 14 0 -6 -30 -10 C 8 6 0 -26 -6 D 34 30 26 0 6 E 28 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=31 C=19 A=10 B=8 so B is eliminated. Round 2 votes counts: E=33 D=33 C=20 A=14 so A is eliminated. Round 3 votes counts: E=38 D=33 C=29 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:248 E:219 C:191 B:184 A:158 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -8 -34 -28 B 14 0 -6 -30 -10 C 8 6 0 -26 -6 D 34 30 26 0 6 E 28 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -34 -28 B 14 0 -6 -30 -10 C 8 6 0 -26 -6 D 34 30 26 0 6 E 28 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -34 -28 B 14 0 -6 -30 -10 C 8 6 0 -26 -6 D 34 30 26 0 6 E 28 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 975: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) D B A C E (9) E C A B D (8) B D A C E (8) D B C A E (5) B A D C E (5) E C D A B (4) E A C B D (4) C A E B D (4) E D C A B (3) E B A C D (3) B A C D E (3) A C B E D (3) A B C E D (3) E D B C A (2) D E B C A (2) D C A B E (2) D B E C A (2) D B E A C (2) C E A D B (2) B E D A C (2) A C E B D (2) A C D B E (2) A C B D E (2) D C E A B (1) D B C E A (1) C E D A B (1) C A E D B (1) C A D E B (1) B D A E C (1) B A C E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 10 8 B -8 0 2 2 6 C -4 -2 0 8 16 D -10 -2 -8 0 -4 E -8 -6 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 10 8 B -8 0 2 2 6 C -4 -2 0 8 16 D -10 -2 -8 0 -4 E -8 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=24 B=20 A=14 C=9 so C is eliminated. Round 2 votes counts: E=36 D=24 B=20 A=20 so B is eliminated. Round 3 votes counts: E=38 D=33 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:215 C:209 B:201 D:188 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 10 8 B -8 0 2 2 6 C -4 -2 0 8 16 D -10 -2 -8 0 -4 E -8 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 10 8 B -8 0 2 2 6 C -4 -2 0 8 16 D -10 -2 -8 0 -4 E -8 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 10 8 B -8 0 2 2 6 C -4 -2 0 8 16 D -10 -2 -8 0 -4 E -8 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 976: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (15) B E D A C (11) A C B E D (9) C D A E B (8) E B D A C (7) C A D E B (6) D E C B A (4) C A D B E (4) A B C E D (4) E D B A C (3) D C E A B (3) B E A D C (3) B A E D C (3) A B E C D (3) D E C A B (2) D B E C A (2) C A E D B (2) C A B D E (2) E A C D B (1) D E B A C (1) D C E B A (1) C A B E D (1) B C A D E (1) B A E C D (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -2 -14 -6 B 4 0 10 -6 -12 C 2 -10 0 -10 -14 D 14 6 10 0 -2 E 6 12 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 -14 -6 B 4 0 10 -6 -12 C 2 -10 0 -10 -14 D 14 6 10 0 -2 E 6 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 B=19 A=19 E=11 so E is eliminated. Round 2 votes counts: D=31 B=26 C=23 A=20 so A is eliminated. Round 3 votes counts: C=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:217 D:214 B:198 A:187 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 -14 -6 B 4 0 10 -6 -12 C 2 -10 0 -10 -14 D 14 6 10 0 -2 E 6 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -14 -6 B 4 0 10 -6 -12 C 2 -10 0 -10 -14 D 14 6 10 0 -2 E 6 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -14 -6 B 4 0 10 -6 -12 C 2 -10 0 -10 -14 D 14 6 10 0 -2 E 6 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 977: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) E D A B C (7) C B A D E (7) B C A E D (6) D E A B C (5) A D E B C (5) E B C D A (4) C B E D A (4) C B D A E (4) E D B A C (3) D A C B E (3) C B A E D (3) E D C B A (2) E C D B A (2) C D B E A (2) C B E A D (2) B A C D E (2) A E D B C (2) A E B D C (2) A D C B E (2) A D B C E (2) A B C D E (2) E D B C A (1) E C B D A (1) E B C A D (1) E A D B C (1) D E C A B (1) D C E B A (1) D C A B E (1) D A E B C (1) D A C E B (1) C D E B A (1) C D B A E (1) C B D E A (1) C A B D E (1) B E A C D (1) A D B E C (1) A C B D E (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 8 -10 2 B -2 0 0 -8 0 C -8 0 0 -4 -2 D 10 8 4 0 12 E -2 0 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -10 2 B -2 0 0 -8 0 C -8 0 0 -4 -2 D 10 8 4 0 12 E -2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=22 D=22 A=21 B=9 so B is eliminated. Round 2 votes counts: C=32 E=23 A=23 D=22 so D is eliminated. Round 3 votes counts: E=38 C=34 A=28 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:217 A:201 B:195 E:194 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -10 2 B -2 0 0 -8 0 C -8 0 0 -4 -2 D 10 8 4 0 12 E -2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -10 2 B -2 0 0 -8 0 C -8 0 0 -4 -2 D 10 8 4 0 12 E -2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -10 2 B -2 0 0 -8 0 C -8 0 0 -4 -2 D 10 8 4 0 12 E -2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 978: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (15) A E C D B (11) E A C B D (9) B D C E A (6) C B D E A (5) A E B D C (4) E A B C D (3) D B A C E (3) B E C D A (3) A E D C B (3) A E D B C (3) E C B D A (2) D C B A E (2) D B C E A (2) C E B D A (2) A D E C B (2) A D E B C (2) A D C E B (2) A D B E C (2) E C B A D (1) E C A D B (1) E C A B D (1) E B C A D (1) E A B D C (1) D A C B E (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A E B (1) C B E D A (1) C A E D B (1) B E D C A (1) B E D A C (1) B D A E C (1) B C D E A (1) B A E D C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 4 2 4 12 B -4 0 2 -6 -10 C -2 -2 0 -4 -10 D -4 6 4 0 -8 E -12 10 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 4 12 B -4 0 2 -6 -10 C -2 -2 0 -4 -10 D -4 6 4 0 -8 E -12 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=23 E=19 B=14 C=13 so C is eliminated. Round 2 votes counts: A=32 D=25 E=23 B=20 so B is eliminated. Round 3 votes counts: D=38 A=33 E=29 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:208 D:199 B:191 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 4 12 B -4 0 2 -6 -10 C -2 -2 0 -4 -10 D -4 6 4 0 -8 E -12 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 12 B -4 0 2 -6 -10 C -2 -2 0 -4 -10 D -4 6 4 0 -8 E -12 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 12 B -4 0 2 -6 -10 C -2 -2 0 -4 -10 D -4 6 4 0 -8 E -12 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 979: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) D E A C B (7) A D E C B (5) A C B D E (5) E D A B C (4) E A D B C (4) C D B E A (4) A E D B C (4) E D A C B (3) E B D A C (3) D E C B A (3) D A E C B (3) B E A C D (3) B C A E D (3) A D C E B (3) E D B C A (2) E B D C A (2) E B A D C (2) E A B D C (2) D E C A B (2) C D A B E (2) C B D E A (2) B E C D A (2) B C E D A (2) B C E A D (2) A E D C B (2) A E B D C (2) A C D B E (2) A B E C D (2) A B C E D (2) D C E B A (1) D A C E B (1) C D B A E (1) C B D A E (1) C A D B E (1) B C A D E (1) A E B C D (1) Total count = 100 A B C D E A 0 14 20 10 0 B -14 0 -14 -8 -12 C -20 14 0 -10 -16 D -10 8 10 0 6 E 0 12 16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.612513 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.387487 Sum of squares = 0.52531812548 Cumulative probabilities = A: 0.612513 B: 0.612513 C: 0.612513 D: 0.612513 E: 1.000000 A B C D E A 0 14 20 10 0 B -14 0 -14 -8 -12 C -20 14 0 -10 -16 D -10 8 10 0 6 E 0 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 C=20 D=17 B=13 so B is eliminated. Round 2 votes counts: C=28 A=28 E=27 D=17 so D is eliminated. Round 3 votes counts: E=39 A=32 C=29 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:211 D:207 C:184 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 10 0 B -14 0 -14 -8 -12 C -20 14 0 -10 -16 D -10 8 10 0 6 E 0 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 10 0 B -14 0 -14 -8 -12 C -20 14 0 -10 -16 D -10 8 10 0 6 E 0 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 10 0 B -14 0 -14 -8 -12 C -20 14 0 -10 -16 D -10 8 10 0 6 E 0 12 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 980: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) B D C A E (8) A B E C D (8) D C E B A (7) A E B C D (7) E C D A B (6) E A C D B (6) D B C E A (5) B D A C E (5) B A D C E (5) A B E D C (5) D C B E A (4) C D E A B (4) A E C B D (4) E A C B D (3) D B C A E (3) C E D A B (2) B A D E C (2) E C A D B (1) D E B C A (1) D C B A E (1) C D B A E (1) C A B D E (1) B A E D C (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -10 6 B 2 0 2 0 0 C 4 -2 0 6 12 D 10 0 -6 0 12 E -6 0 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.822675 C: 0.000000 D: 0.177325 E: 0.000000 Sum of squares = 0.708238313651 Cumulative probabilities = A: 0.000000 B: 0.822675 C: 0.822675 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -10 6 B 2 0 2 0 0 C 4 -2 0 6 12 D 10 0 -6 0 12 E -6 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000024358 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=21 B=21 E=16 C=16 so E is eliminated. Round 2 votes counts: A=35 C=23 D=21 B=21 so D is eliminated. Round 3 votes counts: C=35 A=35 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:208 B:202 A:195 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -10 6 B 2 0 2 0 0 C 4 -2 0 6 12 D 10 0 -6 0 12 E -6 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000024358 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -10 6 B 2 0 2 0 0 C 4 -2 0 6 12 D 10 0 -6 0 12 E -6 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000024358 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -10 6 B 2 0 2 0 0 C 4 -2 0 6 12 D 10 0 -6 0 12 E -6 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000024358 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 981: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) E B C D A (9) E B D C A (8) C A E B D (8) C A D E B (7) E B C A D (6) A D C B E (6) D A C B E (5) D A B E C (5) C A D B E (4) B E D A C (4) C E B A D (3) A C D B E (3) E B D A C (2) D E B A C (2) D B A E C (2) D A C E B (2) D A B C E (2) B E D C A (2) B E C A D (2) B D E A C (2) A C D E B (2) E D C A B (1) C E A B D (1) C A E D B (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -4 -12 -4 B 4 0 12 -6 -6 C 4 -12 0 -8 -10 D 12 6 8 0 6 E 4 6 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -12 -4 B 4 0 12 -6 -6 C 4 -12 0 -8 -10 D 12 6 8 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 C=24 A=12 B=10 so B is eliminated. Round 2 votes counts: E=34 D=30 C=24 A=12 so A is eliminated. Round 3 votes counts: D=37 E=34 C=29 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:207 B:202 A:188 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -12 -4 B 4 0 12 -6 -6 C 4 -12 0 -8 -10 D 12 6 8 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -12 -4 B 4 0 12 -6 -6 C 4 -12 0 -8 -10 D 12 6 8 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -12 -4 B 4 0 12 -6 -6 C 4 -12 0 -8 -10 D 12 6 8 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 982: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) B C A E D (7) A D E C B (7) E D B C A (5) D E A C B (5) A B C D E (5) E D C B A (4) E D B A C (4) D E A B C (4) C B A E D (4) A C D E B (4) D E C A B (3) C A B D E (3) B A C D E (3) A C B D E (3) A B D E C (3) C E D B A (2) C B E D A (2) C A E D B (2) C A D E B (2) B E D C A (2) B A D E C (2) A D E B C (2) E D C A B (1) E C D B A (1) D E B A C (1) D A E B C (1) C E B D A (1) C B A D E (1) B E D A C (1) B E C D A (1) B D E A C (1) B C E A D (1) B A E D C (1) A D C E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 0 6 6 B 4 0 6 0 0 C 0 -6 0 2 2 D -6 0 -2 0 6 E -6 0 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.706615 C: 0.000000 D: 0.293385 E: 0.000000 Sum of squares = 0.585379245682 Cumulative probabilities = A: 0.000000 B: 0.706615 C: 0.706615 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 6 6 B 4 0 6 0 0 C 0 -6 0 2 2 D -6 0 -2 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000013062 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 C=17 E=15 D=14 so D is eliminated. Round 2 votes counts: E=28 A=28 B=27 C=17 so C is eliminated. Round 3 votes counts: A=35 B=34 E=31 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:205 A:204 C:199 D:199 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 6 6 B 4 0 6 0 0 C 0 -6 0 2 2 D -6 0 -2 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000013062 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 6 6 B 4 0 6 0 0 C 0 -6 0 2 2 D -6 0 -2 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000013062 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 6 6 B 4 0 6 0 0 C 0 -6 0 2 2 D -6 0 -2 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000013062 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 983: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) E A B D C (6) D A B C E (5) C B A D E (5) B A C E D (5) B A C D E (5) E C B A D (4) E B A C D (4) D E A B C (4) D C A B E (4) D A B E C (4) C B D A E (4) B C A E D (4) A D B C E (4) E C D B A (3) A B D E C (3) E A D B C (2) D A E B C (2) D A C B E (2) C E B A D (2) C B A E D (2) B C A D E (2) A B E D C (2) E D C A B (1) E D A C B (1) E C D A B (1) E B C A D (1) E B A D C (1) D E C A B (1) C D E B A (1) C D E A B (1) C D A B E (1) C B E A D (1) B A E C D (1) B A D C E (1) Total count = 100 A B C D E A 0 8 24 30 32 B -8 0 32 26 30 C -24 -32 0 -6 18 D -30 -26 6 0 18 E -32 -30 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 24 30 32 B -8 0 32 26 30 C -24 -32 0 -6 18 D -30 -26 6 0 18 E -32 -30 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 A=19 B=18 C=17 so C is eliminated. Round 2 votes counts: B=30 E=26 D=25 A=19 so A is eliminated. Round 3 votes counts: B=45 D=29 E=26 so E is eliminated. Round 4 votes counts: B=63 D=37 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:247 B:240 D:184 C:178 E:151 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 24 30 32 B -8 0 32 26 30 C -24 -32 0 -6 18 D -30 -26 6 0 18 E -32 -30 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 24 30 32 B -8 0 32 26 30 C -24 -32 0 -6 18 D -30 -26 6 0 18 E -32 -30 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 24 30 32 B -8 0 32 26 30 C -24 -32 0 -6 18 D -30 -26 6 0 18 E -32 -30 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 984: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) A E B D C (7) A B E C D (7) C D B A E (5) C D A B E (5) A E D C B (5) E D C A B (4) E A D C B (4) B E C D A (4) E A B D C (3) C D B E A (3) B C A D E (3) B A C D E (3) A D E C B (3) A B E D C (3) E B A D C (2) D C E A B (2) D C A E B (2) C D E B A (2) C B D A E (2) B C E D A (2) B C D E A (2) B C D A E (2) B A E C D (2) A E D B C (2) A C D B E (2) A C B D E (2) E B D C A (1) E A D B C (1) D E C A B (1) D C E B A (1) C B D E A (1) C A D B E (1) B E A C D (1) B A C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 26 14 22 32 B -26 0 -6 -2 4 C -14 6 0 2 0 D -22 2 -2 0 2 E -32 -4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 14 22 32 B -26 0 -6 -2 4 C -14 6 0 2 0 D -22 2 -2 0 2 E -32 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=20 C=19 E=15 D=6 so D is eliminated. Round 2 votes counts: A=40 C=24 B=20 E=16 so E is eliminated. Round 3 votes counts: A=48 C=29 B=23 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:247 C:197 D:190 B:185 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 14 22 32 B -26 0 -6 -2 4 C -14 6 0 2 0 D -22 2 -2 0 2 E -32 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 14 22 32 B -26 0 -6 -2 4 C -14 6 0 2 0 D -22 2 -2 0 2 E -32 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 14 22 32 B -26 0 -6 -2 4 C -14 6 0 2 0 D -22 2 -2 0 2 E -32 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 985: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) C D E B A (6) A B E D C (6) C D B E A (5) C D B A E (5) B D A E C (4) A E B C D (4) E D A B C (3) D B E A C (3) C D E A B (3) B D C A E (3) B A E D C (3) B A D E C (3) A E C B D (3) E D A C B (2) E A D B C (2) D E C A B (2) D C E B A (2) D B C E A (2) C E D A B (2) B C D A E (2) B C A D E (2) B A C E D (2) A C E B D (2) A B E C D (2) E D C A B (1) E C A D B (1) E A C B D (1) D E C B A (1) D E B A C (1) D C E A B (1) D C B E A (1) D B E C A (1) C E A D B (1) C B A E D (1) C B A D E (1) C A E B D (1) C A D E B (1) C A B E D (1) C A B D E (1) B D E A C (1) B D A C E (1) B A D C E (1) Total count = 100 A B C D E A 0 -2 6 -4 16 B 2 0 10 8 2 C -6 -10 0 -6 -6 D 4 -8 6 0 6 E -16 -2 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -4 16 B 2 0 10 8 2 C -6 -10 0 -6 -6 D 4 -8 6 0 6 E -16 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=26 B=22 D=14 E=10 so E is eliminated. Round 2 votes counts: C=29 A=29 B=22 D=20 so D is eliminated. Round 3 votes counts: C=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:211 A:208 D:204 E:191 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 -4 16 B 2 0 10 8 2 C -6 -10 0 -6 -6 D 4 -8 6 0 6 E -16 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -4 16 B 2 0 10 8 2 C -6 -10 0 -6 -6 D 4 -8 6 0 6 E -16 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -4 16 B 2 0 10 8 2 C -6 -10 0 -6 -6 D 4 -8 6 0 6 E -16 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 986: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (14) C E A B D (9) E C A B D (6) E A C B D (5) D B E A C (5) D B A E C (5) C A E B D (5) E C A D B (4) E D B C A (3) E C D A B (3) D E C B A (3) A E C B D (3) A C B E D (3) D E B A C (2) D B E C A (2) C E A D B (2) B D E A C (2) B D A E C (2) B D A C E (2) B A D C E (2) A C E B D (2) E D C B A (1) E D B A C (1) E C B A D (1) E B D A C (1) E A B C D (1) D E B C A (1) D C E B A (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A D C (1) A D C B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 12 0 -10 B 0 0 -6 -2 -14 C -12 6 0 -2 -6 D 0 2 2 0 -6 E 10 14 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 12 0 -10 B 0 0 -6 -2 -14 C -12 6 0 -2 -6 D 0 2 2 0 -6 E 10 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=26 C=19 A=12 B=10 so B is eliminated. Round 2 votes counts: D=39 E=28 C=19 A=14 so A is eliminated. Round 3 votes counts: D=43 E=31 C=26 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:201 D:199 C:193 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 0 -10 B 0 0 -6 -2 -14 C -12 6 0 -2 -6 D 0 2 2 0 -6 E 10 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 0 -10 B 0 0 -6 -2 -14 C -12 6 0 -2 -6 D 0 2 2 0 -6 E 10 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 0 -10 B 0 0 -6 -2 -14 C -12 6 0 -2 -6 D 0 2 2 0 -6 E 10 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 987: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (10) B D E A C (9) E C B A D (8) A D C B E (6) E B C D A (5) C E B A D (4) B E D C A (4) A C D E B (4) D A B E C (3) C A D E B (3) B D A E C (3) A D C E B (3) A D B E C (3) A D B C E (3) E C B D A (2) E B A D C (2) D A B C E (2) C E B D A (2) C A E D B (2) A D E B C (2) A C E D B (2) E C A B D (1) E B D C A (1) E B C A D (1) C E A D B (1) C A D B E (1) B E C D A (1) B D E C A (1) Total count = 100 A B C D E A 0 -8 6 12 -4 B 8 0 2 8 -6 C -6 -2 0 -6 -14 D -12 -8 6 0 8 E 4 6 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.272727 E: 0.363636 Sum of squares = 0.338842975171 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.636364 E: 1.000000 A B C D E A 0 -8 6 12 -4 B 8 0 2 8 -6 C -6 -2 0 -6 -14 D -12 -8 6 0 8 E 4 6 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.272727 E: 0.363636 Sum of squares = 0.338842975185 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=23 A=23 E=20 B=18 D=16 so D is eliminated. Round 2 votes counts: B=29 A=28 C=23 E=20 so E is eliminated. Round 3 votes counts: B=38 C=34 A=28 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:208 B:206 A:203 D:197 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 12 -4 B 8 0 2 8 -6 C -6 -2 0 -6 -14 D -12 -8 6 0 8 E 4 6 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.272727 E: 0.363636 Sum of squares = 0.338842975185 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.636364 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 12 -4 B 8 0 2 8 -6 C -6 -2 0 -6 -14 D -12 -8 6 0 8 E 4 6 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.272727 E: 0.363636 Sum of squares = 0.338842975185 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.636364 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 12 -4 B 8 0 2 8 -6 C -6 -2 0 -6 -14 D -12 -8 6 0 8 E 4 6 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.272727 E: 0.363636 Sum of squares = 0.338842975185 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.636364 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 988: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) D B C A E (8) E C A B D (7) D C E B A (7) D B A C E (7) B D A C E (7) A B E C D (6) E C D A B (4) D E C B A (4) E C A D B (3) E A C B D (3) D C B E A (3) D B C E A (3) B A C E D (3) A E C B D (3) E D C A B (2) D E C A B (2) C E D B A (2) C D E B A (2) A E B D C (2) E D A C B (1) E A D B C (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B A D (1) C E A B D (1) C B D A E (1) C A B E D (1) A E B C D (1) A D B E C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -4 -12 6 B 16 0 4 -6 6 C 4 -4 0 -20 14 D 12 6 20 0 14 E -6 -6 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -12 6 B 16 0 4 -6 6 C 4 -4 0 -20 14 D 12 6 20 0 14 E -6 -6 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=21 B=19 A=15 C=9 so C is eliminated. Round 2 votes counts: D=38 E=26 B=20 A=16 so A is eliminated. Round 3 votes counts: D=39 E=32 B=29 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:210 C:197 A:187 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -4 -12 6 B 16 0 4 -6 6 C 4 -4 0 -20 14 D 12 6 20 0 14 E -6 -6 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -12 6 B 16 0 4 -6 6 C 4 -4 0 -20 14 D 12 6 20 0 14 E -6 -6 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -12 6 B 16 0 4 -6 6 C 4 -4 0 -20 14 D 12 6 20 0 14 E -6 -6 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 989: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E A B D C (9) B C D E A (8) A E D B C (7) C B D E A (6) B E A D C (5) A E B D C (5) C D A E B (4) A E D C B (4) A E C B D (4) E A D B C (3) D B C E A (3) C D A B E (3) B D E A C (3) A E C D B (3) D C B E A (2) D C A E B (2) D A E C B (2) C A D E B (2) B D C E A (2) B C E A D (2) E A B C D (1) D C B A E (1) D C A B E (1) C D B E A (1) C A E D B (1) B E D A C (1) B E A C D (1) B D E C A (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 2 0 4 B -8 0 2 -2 0 C -2 -2 0 -4 2 D 0 2 4 0 4 E -4 0 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.431739 B: 0.000000 C: 0.000000 D: 0.568261 E: 0.000000 Sum of squares = 0.50931912755 Cumulative probabilities = A: 0.431739 B: 0.431739 C: 0.431739 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 0 4 B -8 0 2 -2 0 C -2 -2 0 -4 2 D 0 2 4 0 4 E -4 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 B=23 E=13 D=11 so D is eliminated. Round 2 votes counts: C=33 A=28 B=26 E=13 so E is eliminated. Round 3 votes counts: A=41 C=33 B=26 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:207 D:205 C:197 B:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 0 4 B -8 0 2 -2 0 C -2 -2 0 -4 2 D 0 2 4 0 4 E -4 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 0 4 B -8 0 2 -2 0 C -2 -2 0 -4 2 D 0 2 4 0 4 E -4 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 0 4 B -8 0 2 -2 0 C -2 -2 0 -4 2 D 0 2 4 0 4 E -4 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 990: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) E C D B A (6) E B C A D (5) B A C D E (5) E D C A B (4) E A B D C (4) D A C B E (4) B A E C D (4) A D B E C (4) A B E D C (4) A B D C E (4) E C B D A (3) D C A E B (3) D A B C E (3) C D E B A (3) B A C E D (3) D E C A B (2) D A E B C (2) C E D B A (2) C E B D A (2) C D E A B (2) C D B E A (2) C D B A E (2) C B E D A (2) A D B C E (2) E D A C B (1) E D A B C (1) E B A D C (1) E B A C D (1) D E A C B (1) D A B E C (1) C E D A B (1) C D A B E (1) C B E A D (1) C B D A E (1) C B A E D (1) C B A D E (1) B C A E D (1) B A E D C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -4 -12 -2 B -6 0 -4 -10 -4 C 4 4 0 -2 6 D 12 10 2 0 4 E 2 4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -12 -2 B -6 0 -4 -10 -4 C 4 4 0 -2 6 D 12 10 2 0 4 E 2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 C=21 A=16 B=14 so B is eliminated. Round 2 votes counts: A=29 E=26 D=23 C=22 so C is eliminated. Round 3 votes counts: E=34 D=34 A=32 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:206 E:198 A:194 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -12 -2 B -6 0 -4 -10 -4 C 4 4 0 -2 6 D 12 10 2 0 4 E 2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -12 -2 B -6 0 -4 -10 -4 C 4 4 0 -2 6 D 12 10 2 0 4 E 2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -12 -2 B -6 0 -4 -10 -4 C 4 4 0 -2 6 D 12 10 2 0 4 E 2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 991: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) E D A C B (7) E B C A D (6) B C E A D (5) B C A E D (5) D A C E B (4) C B A D E (4) B A C D E (4) E B D C A (3) E B D A C (3) E B C D A (3) D E A C B (3) D A C B E (3) C A D B E (3) C A B D E (3) B C A D E (3) A D C B E (3) E D B C A (2) E D B A C (2) E C B D A (2) E B A C D (2) D A E C B (2) B E C A D (2) B E A D C (2) B E A C D (2) E D C A B (1) E C D B A (1) E B A D C (1) D E C A B (1) D E A B C (1) D C A B E (1) B A E C D (1) A E B D C (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 10 2 -18 B 6 0 20 10 -10 C -10 -20 0 -2 -16 D -2 -10 2 0 -22 E 18 10 16 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 10 2 -18 B 6 0 20 10 -10 C -10 -20 0 -2 -16 D -2 -10 2 0 -22 E 18 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 B=24 D=15 C=10 A=8 so A is eliminated. Round 2 votes counts: E=44 B=26 D=18 C=12 so C is eliminated. Round 3 votes counts: E=44 B=34 D=22 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:233 B:213 A:194 D:184 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 10 2 -18 B 6 0 20 10 -10 C -10 -20 0 -2 -16 D -2 -10 2 0 -22 E 18 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 2 -18 B 6 0 20 10 -10 C -10 -20 0 -2 -16 D -2 -10 2 0 -22 E 18 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 2 -18 B 6 0 20 10 -10 C -10 -20 0 -2 -16 D -2 -10 2 0 -22 E 18 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 992: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (12) E B D A C (9) A E C B D (7) C A D B E (6) A C E D B (6) E A B D C (5) B D E C A (5) A C E B D (5) E D B A C (4) D B C E A (3) D B C A E (3) C B A D E (3) C A E B D (3) B E D C A (3) A C D E B (3) E A C B D (2) E A B C D (2) C D B A E (2) C D A B E (2) C A B D E (2) E B D C A (1) E B A D C (1) D E A B C (1) D C B A E (1) D B E A C (1) D B A E C (1) D A B E C (1) C B E D A (1) C B D A E (1) C A D E B (1) B D E A C (1) B D C E A (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 0 -6 -4 B 6 0 8 4 -2 C 0 -8 0 -8 -12 D 6 -4 8 0 2 E 4 2 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000013 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -6 0 -6 -4 B 6 0 8 4 -2 C 0 -8 0 -8 -12 D 6 -4 8 0 2 E 4 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999924 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=23 A=22 C=21 B=10 so B is eliminated. Round 2 votes counts: D=30 E=27 A=22 C=21 so C is eliminated. Round 3 votes counts: A=37 D=35 E=28 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:208 E:208 D:206 A:192 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -6 -4 B 6 0 8 4 -2 C 0 -8 0 -8 -12 D 6 -4 8 0 2 E 4 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999924 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 -4 B 6 0 8 4 -2 C 0 -8 0 -8 -12 D 6 -4 8 0 2 E 4 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999924 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 -4 B 6 0 8 4 -2 C 0 -8 0 -8 -12 D 6 -4 8 0 2 E 4 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999924 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 993: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) C D B A E (6) B C D A E (6) E A D C B (5) E A B D C (5) B D C E A (5) E A B C D (4) D C B A E (4) B D C A E (4) A E C D B (4) E B A D C (3) E A C D B (3) A E B C D (3) A C E D B (3) A B C E D (3) E D A C B (2) E A C B D (2) D C A E B (2) C D A E B (2) C B D A E (2) C A D B E (2) C A B D E (2) B A E C D (2) A E C B D (2) E D C A B (1) E B D A C (1) E A D B C (1) D E C B A (1) D E C A B (1) D B C E A (1) C D A B E (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E C A (1) B A C E D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -2 0 4 B -2 0 -12 2 4 C 2 12 0 4 10 D 0 -2 -4 0 0 E -4 -4 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 0 4 B -2 0 -12 2 4 C 2 12 0 4 10 D 0 -2 -4 0 0 E -4 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 D=17 A=17 C=16 so C is eliminated. Round 2 votes counts: E=27 D=26 B=25 A=22 so A is eliminated. Round 3 votes counts: E=40 D=30 B=30 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:214 A:202 D:197 B:196 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 0 4 B -2 0 -12 2 4 C 2 12 0 4 10 D 0 -2 -4 0 0 E -4 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 4 B -2 0 -12 2 4 C 2 12 0 4 10 D 0 -2 -4 0 0 E -4 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 4 B -2 0 -12 2 4 C 2 12 0 4 10 D 0 -2 -4 0 0 E -4 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 994: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) C B E D A (6) B C A E D (6) E D C B A (5) C B D E A (5) C B D A E (5) E B C D A (4) D A E C B (4) A D E C B (4) E B A C D (3) E A B D C (3) D E A C B (3) C E B D A (3) B A C E D (3) A D E B C (3) E D A B C (2) E C B D A (2) E B C A D (2) D C E B A (2) B C E D A (2) B A E C D (2) A E D B C (2) A E B D C (2) A B C D E (2) E D A C B (1) E A D B C (1) D E C B A (1) D C E A B (1) D C B A E (1) D A C B E (1) C D B E A (1) C D B A E (1) C A D B E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C A D E (1) A E B C D (1) A D C E B (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -30 -16 -2 -14 B 30 0 2 24 0 C 16 -2 0 20 2 D 2 -24 -20 0 -20 E 14 0 -2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.654331 C: 0.000000 D: 0.000000 E: 0.345668 Sum of squares = 0.54763631294 Cumulative probabilities = A: 0.000000 B: 0.654331 C: 0.654332 D: 0.654332 E: 1.000000 A B C D E A 0 -30 -16 -2 -14 B 30 0 2 24 0 C 16 -2 0 20 2 D 2 -24 -20 0 -20 E 14 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500554 C: 0.000000 D: 0.000000 E: 0.499446 Sum of squares = 0.500000614355 Cumulative probabilities = A: 0.000000 B: 0.500554 C: 0.500554 D: 0.500554 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 C=22 A=18 D=13 so D is eliminated. Round 2 votes counts: E=27 C=26 B=24 A=23 so A is eliminated. Round 3 votes counts: E=43 C=29 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:228 C:218 E:216 A:169 D:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -16 -2 -14 B 30 0 2 24 0 C 16 -2 0 20 2 D 2 -24 -20 0 -20 E 14 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500554 C: 0.000000 D: 0.000000 E: 0.499446 Sum of squares = 0.500000614355 Cumulative probabilities = A: 0.000000 B: 0.500554 C: 0.500554 D: 0.500554 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -16 -2 -14 B 30 0 2 24 0 C 16 -2 0 20 2 D 2 -24 -20 0 -20 E 14 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500554 C: 0.000000 D: 0.000000 E: 0.499446 Sum of squares = 0.500000614355 Cumulative probabilities = A: 0.000000 B: 0.500554 C: 0.500554 D: 0.500554 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -16 -2 -14 B 30 0 2 24 0 C 16 -2 0 20 2 D 2 -24 -20 0 -20 E 14 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500554 C: 0.000000 D: 0.000000 E: 0.499446 Sum of squares = 0.500000614355 Cumulative probabilities = A: 0.000000 B: 0.500554 C: 0.500554 D: 0.500554 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 995: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) A C E B D (8) E D C B A (7) D B E C A (7) D E B C A (6) B D C E A (6) A C B E D (6) A B C D E (4) E A D B C (3) C B D E A (3) C A B E D (3) B C D E A (3) B C A D E (3) A E D B C (3) E D B C A (2) E D A C B (2) D E B A C (2) D B E A C (2) C E D B A (2) C A E B D (2) C A B D E (2) B D E C A (2) E D A B C (1) E A D C B (1) C E D A B (1) C B A E D (1) C B A D E (1) C A E D B (1) B D A C E (1) B C D A E (1) B A D E C (1) B A C D E (1) A E D C B (1) A E C B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -6 4 0 B 2 0 -2 2 -2 C 6 2 0 4 2 D -4 -2 -4 0 -6 E 0 2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 4 0 B 2 0 -2 2 -2 C 6 2 0 4 2 D -4 -2 -4 0 -6 E 0 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=18 D=17 E=16 C=16 so E is eliminated. Round 2 votes counts: A=37 D=29 B=18 C=16 so C is eliminated. Round 3 votes counts: A=45 D=32 B=23 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:207 E:203 B:200 A:198 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 4 0 B 2 0 -2 2 -2 C 6 2 0 4 2 D -4 -2 -4 0 -6 E 0 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 4 0 B 2 0 -2 2 -2 C 6 2 0 4 2 D -4 -2 -4 0 -6 E 0 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 4 0 B 2 0 -2 2 -2 C 6 2 0 4 2 D -4 -2 -4 0 -6 E 0 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 996: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (11) D B E C A (10) E A D C B (7) E D B A C (6) A E C B D (6) C B D A E (5) E A C D B (4) E A B D C (4) B D C A E (4) A C E B D (4) E A D B C (3) D B C E A (3) D B C A E (3) C B A D E (3) E A C B D (2) E A B C D (2) D E B C A (2) D E B A C (2) C D B A E (2) C A D B E (2) C A B D E (2) B D C E A (2) B C D A E (2) A C E D B (2) E D A C B (1) E B A D C (1) D C E B A (1) C A B E D (1) B D E C A (1) B D E A C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 14 10 -4 B -4 0 -6 4 4 C -14 6 0 -2 -6 D -10 -4 2 0 -10 E 4 -4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 14 10 -4 B -4 0 -6 4 4 C -14 6 0 -2 -6 D -10 -4 2 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=24 D=21 C=15 B=10 so B is eliminated. Round 2 votes counts: E=30 D=29 A=24 C=17 so C is eliminated. Round 3 votes counts: D=38 A=32 E=30 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:208 B:199 C:192 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 -4 B -4 0 -6 4 4 C -14 6 0 -2 -6 D -10 -4 2 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 -4 B -4 0 -6 4 4 C -14 6 0 -2 -6 D -10 -4 2 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 -4 B -4 0 -6 4 4 C -14 6 0 -2 -6 D -10 -4 2 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 997: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) A B D E C (7) E C B D A (5) C D E B A (5) B E A D C (5) E B C D A (4) C E D B A (4) E B A D C (3) E B A C D (3) D C A B E (3) D B C A E (3) D A C B E (3) C D E A B (3) C D A E B (3) B A D E C (3) E A C B D (2) E A B C D (2) D B A C E (2) C E B D A (2) C E A D B (2) C D A B E (2) C A D E B (2) B D A E C (2) A E B C D (2) A D B C E (2) A C D B E (2) A C B D E (2) E C A B D (1) E B D C A (1) E B D A C (1) D C B E A (1) D C B A E (1) D B A E C (1) C E D A B (1) C E A B D (1) C A E D B (1) B E D A C (1) B A E D C (1) A E C B D (1) A E B D C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 10 4 6 B -4 0 4 18 -2 C -10 -4 0 4 -8 D -4 -18 -4 0 -6 E -6 2 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 4 6 B -4 0 4 18 -2 C -10 -4 0 4 -8 D -4 -18 -4 0 -6 E -6 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 E=22 D=14 B=12 so B is eliminated. Round 2 votes counts: A=30 E=28 C=26 D=16 so D is eliminated. Round 3 votes counts: A=38 C=34 E=28 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:208 E:205 C:191 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 4 6 B -4 0 4 18 -2 C -10 -4 0 4 -8 D -4 -18 -4 0 -6 E -6 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 4 6 B -4 0 4 18 -2 C -10 -4 0 4 -8 D -4 -18 -4 0 -6 E -6 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 4 6 B -4 0 4 18 -2 C -10 -4 0 4 -8 D -4 -18 -4 0 -6 E -6 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 998: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (5) C A B D E (5) B C A D E (5) E D C A B (4) E C B A D (4) C A B E D (4) A D C B E (4) A C D B E (4) E D B A C (3) E D A C B (3) E B C D A (3) D E B A C (3) D B E A C (3) C B A E D (3) C A E D B (3) C A E B D (3) B E D C A (3) A D C E B (3) A C B D E (3) D B A C E (2) D A E B C (2) D A C B E (2) B A C D E (2) A D B C E (2) A C D E B (2) E D C B A (1) E C B D A (1) E B C A D (1) D E A C B (1) D E A B C (1) D B A E C (1) D A E C B (1) D A C E B (1) D A B E C (1) D A B C E (1) C B E A D (1) C B A D E (1) B E D A C (1) B E C D A (1) B D E A C (1) B D C E A (1) B D A C E (1) B C A E D (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 0 16 18 B -12 0 -18 -6 10 C 0 18 0 4 12 D -16 6 -4 0 10 E -18 -10 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.450149 B: 0.000000 C: 0.549851 D: 0.000000 E: 0.000000 Sum of squares = 0.504970177015 Cumulative probabilities = A: 0.450149 B: 0.450149 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 16 18 B -12 0 -18 -6 10 C 0 18 0 4 12 D -16 6 -4 0 10 E -18 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=20 D=19 A=19 B=17 so B is eliminated. Round 2 votes counts: E=30 C=26 D=22 A=22 so D is eliminated. Round 3 votes counts: E=39 A=34 C=27 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:217 D:198 B:187 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 16 18 B -12 0 -18 -6 10 C 0 18 0 4 12 D -16 6 -4 0 10 E -18 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 16 18 B -12 0 -18 -6 10 C 0 18 0 4 12 D -16 6 -4 0 10 E -18 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 16 18 B -12 0 -18 -6 10 C 0 18 0 4 12 D -16 6 -4 0 10 E -18 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 999: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) D E A B C (9) C E D B A (9) D A E B C (7) C B A D E (5) E D C B A (4) D E C A B (4) D A B E C (4) C D E B A (4) C B E A D (4) C E B D A (3) B C A E D (3) B A C E D (3) A B D C E (3) E D A B C (2) E C D B A (2) D E A C B (2) C D E A B (2) A D B E C (2) A B C D E (2) E D C A B (1) E D B C A (1) E D B A C (1) E B C A D (1) E A D B C (1) C E B A D (1) C D B A E (1) C D A B E (1) C A B D E (1) B E A D C (1) B C E A D (1) B A E D C (1) B A C D E (1) A E D B C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -16 -14 -6 B 12 0 -8 -18 -12 C 16 8 0 8 8 D 14 18 -8 0 -2 E 6 12 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -14 -6 B 12 0 -8 -18 -12 C 16 8 0 8 8 D 14 18 -8 0 -2 E 6 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 D=26 E=13 B=10 A=10 so B is eliminated. Round 2 votes counts: C=45 D=26 A=15 E=14 so E is eliminated. Round 3 votes counts: C=48 D=35 A=17 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:211 E:206 B:187 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -16 -14 -6 B 12 0 -8 -18 -12 C 16 8 0 8 8 D 14 18 -8 0 -2 E 6 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -14 -6 B 12 0 -8 -18 -12 C 16 8 0 8 8 D 14 18 -8 0 -2 E 6 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -14 -6 B 12 0 -8 -18 -12 C 16 8 0 8 8 D 14 18 -8 0 -2 E 6 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1000: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) C E A D B (6) E C B A D (5) D A C E B (5) B E C A D (5) D A B E C (4) B E C D A (4) E C A D B (3) E B C A D (3) C D A E B (3) B E A D C (3) A D C E B (3) A D B E C (3) E C A B D (2) C E B D A (2) C E B A D (2) C E A B D (2) B E D C A (2) B C E D A (2) B A D E C (2) A E D C B (2) A D E C B (2) A C D E B (2) E B A C D (1) E A D C B (1) E A C D B (1) E A B D C (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C B E (1) D A B C E (1) C E D B A (1) C E D A B (1) C D E A B (1) C D B A E (1) C D A B E (1) C B D E A (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A C D (1) B D E A C (1) B D A C E (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 4 -4 8 -4 B -4 0 -8 -2 -10 C 4 8 0 6 -16 D -8 2 -6 0 -6 E 4 10 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -4 8 -4 B -4 0 -8 -2 -10 C 4 8 0 6 -16 D -8 2 -6 0 -6 E 4 10 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=23 E=17 D=16 A=14 so A is eliminated. Round 2 votes counts: B=30 D=25 C=25 E=20 so E is eliminated. Round 3 votes counts: C=37 B=35 D=28 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:218 A:202 C:201 D:191 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 8 -4 B -4 0 -8 -2 -10 C 4 8 0 6 -16 D -8 2 -6 0 -6 E 4 10 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 8 -4 B -4 0 -8 -2 -10 C 4 8 0 6 -16 D -8 2 -6 0 -6 E 4 10 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 8 -4 B -4 0 -8 -2 -10 C 4 8 0 6 -16 D -8 2 -6 0 -6 E 4 10 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1001: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) D B C A E (5) C A B E D (5) D B A C E (4) B D C A E (4) B C D A E (4) D E A B C (3) D B E C A (3) D A E C B (3) C A E B D (3) B D E C A (3) B D C E A (3) B C E A D (3) B C A E D (3) A E C D B (3) E D A C B (2) E D A B C (2) E C A B D (2) E A D C B (2) D E B A C (2) D E A C B (2) D B E A C (2) D A E B C (2) D A C E B (2) D A B C E (2) C B A D E (2) B E C D A (2) B E C A D (2) B C A D E (2) A D E C B (2) E B C A D (1) E A C B D (1) D B A E C (1) D A C B E (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) C A B D E (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -4 -4 4 B -2 0 6 -8 8 C 4 -6 0 -2 -2 D 4 8 2 0 6 E -4 -8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -4 4 B -2 0 6 -8 8 C 4 -6 0 -2 -2 D 4 8 2 0 6 E -4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=26 E=20 C=15 A=7 so A is eliminated. Round 2 votes counts: D=34 B=26 E=24 C=16 so C is eliminated. Round 3 votes counts: B=36 D=34 E=30 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:202 A:199 C:197 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -4 4 B -2 0 6 -8 8 C 4 -6 0 -2 -2 D 4 8 2 0 6 E -4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -4 4 B -2 0 6 -8 8 C 4 -6 0 -2 -2 D 4 8 2 0 6 E -4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -4 4 B -2 0 6 -8 8 C 4 -6 0 -2 -2 D 4 8 2 0 6 E -4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1002: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (10) E D B A C (9) D E B A C (6) E D C A B (5) D E C B A (5) C A B E D (5) D B E A C (4) A C B D E (4) A B C D E (4) E B D A C (3) C E A B D (3) B A D C E (3) E D C B A (2) E D B C A (2) E C A B D (2) E B A D C (2) C E A D B (2) C D E A B (2) C D B A E (2) C D A B E (2) C A D B E (2) B A C D E (2) A C B E D (2) E C D A B (1) E C A D B (1) E A B D C (1) E A B C D (1) D E C A B (1) D C E B A (1) D C B E A (1) D B A E C (1) D B A C E (1) C E D A B (1) C A E B D (1) C A D E B (1) B E A D C (1) B D A C E (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -4 0 -12 B -8 0 -14 -4 -6 C 4 14 0 2 4 D 0 4 -2 0 6 E 12 6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 0 -12 B -8 0 -14 -4 -6 C 4 14 0 2 4 D 0 4 -2 0 6 E 12 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=29 D=20 A=13 B=7 so B is eliminated. Round 2 votes counts: C=31 E=30 D=21 A=18 so A is eliminated. Round 3 votes counts: C=45 E=31 D=24 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:204 E:204 A:196 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 0 -12 B -8 0 -14 -4 -6 C 4 14 0 2 4 D 0 4 -2 0 6 E 12 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 0 -12 B -8 0 -14 -4 -6 C 4 14 0 2 4 D 0 4 -2 0 6 E 12 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 0 -12 B -8 0 -14 -4 -6 C 4 14 0 2 4 D 0 4 -2 0 6 E 12 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1003: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) C E D B A (9) B A C E D (7) A B D E C (7) D E C A B (6) B C E A D (6) A B C E D (5) E D C B A (3) C E B D A (3) C D E B A (3) B E C A D (3) E D B C A (2) E C B D A (2) D E A C B (2) D E A B C (2) D C E A B (2) D A E B C (2) C B A E D (2) A D E B C (2) A D B C E (2) A B C D E (2) E C D B A (1) D E C B A (1) D C A E B (1) D A E C B (1) D A C E B (1) C D E A B (1) C D A E B (1) C B E D A (1) C B E A D (1) B E D A C (1) B E C D A (1) B C A E D (1) B A E D C (1) A D B E C (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 0 8 0 B 16 0 14 14 10 C 0 -14 0 24 0 D -8 -14 -24 0 -24 E 0 -10 0 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 8 0 B 16 0 14 14 10 C 0 -14 0 24 0 D -8 -14 -24 0 -24 E 0 -10 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=23 C=21 D=18 E=8 so E is eliminated. Round 2 votes counts: B=30 C=24 D=23 A=23 so D is eliminated. Round 3 votes counts: C=37 B=32 A=31 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:207 C:205 A:196 D:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 8 0 B 16 0 14 14 10 C 0 -14 0 24 0 D -8 -14 -24 0 -24 E 0 -10 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 8 0 B 16 0 14 14 10 C 0 -14 0 24 0 D -8 -14 -24 0 -24 E 0 -10 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 8 0 B 16 0 14 14 10 C 0 -14 0 24 0 D -8 -14 -24 0 -24 E 0 -10 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1004: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) D E B A C (8) C A D B E (8) B E A C D (8) E B D A C (6) C A B E D (6) A B C E D (5) D E C B A (4) D B E A C (4) A C B E D (4) D C A B E (3) D A C B E (3) D A B C E (3) C A B D E (3) E D B A C (2) E B C D A (2) E B C A D (2) C E B A D (2) C D A E B (2) A C D B E (2) A C B D E (2) E D C B A (1) E B D C A (1) E B A C D (1) D B A E C (1) C E D A B (1) C D A B E (1) C A E B D (1) B E D A C (1) B D E A C (1) B A E D C (1) B A E C D (1) Total count = 100 A B C D E A 0 -12 6 -8 -8 B 12 0 14 -6 14 C -6 -14 0 2 -8 D 8 6 -2 0 10 E 8 -14 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.636364 E: 0.000000 Sum of squares = 0.487603305825 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -8 -8 B 12 0 14 -6 14 C -6 -14 0 2 -8 D 8 6 -2 0 10 E 8 -14 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.636364 E: 0.000000 Sum of squares = 0.48760330578 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=24 E=15 A=13 B=12 so B is eliminated. Round 2 votes counts: D=37 E=24 C=24 A=15 so A is eliminated. Round 3 votes counts: D=37 C=37 E=26 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:217 D:211 E:196 A:189 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 6 -8 -8 B 12 0 14 -6 14 C -6 -14 0 2 -8 D 8 6 -2 0 10 E 8 -14 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.636364 E: 0.000000 Sum of squares = 0.48760330578 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -8 -8 B 12 0 14 -6 14 C -6 -14 0 2 -8 D 8 6 -2 0 10 E 8 -14 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.636364 E: 0.000000 Sum of squares = 0.48760330578 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -8 -8 B 12 0 14 -6 14 C -6 -14 0 2 -8 D 8 6 -2 0 10 E 8 -14 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.636364 E: 0.000000 Sum of squares = 0.48760330578 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1005: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) D A E B C (7) A D C B E (6) D A C E B (5) D E C A B (4) D C A E B (4) D C A B E (4) B E C A D (4) E B C A D (3) E B A D C (3) D A E C B (3) B E A C D (3) A D B E C (3) A D B C E (3) E D B C A (2) E D A B C (2) D E A B C (2) D A C B E (2) C E B D A (2) C D A B E (2) C B E A D (2) C B A D E (2) C A B D E (2) B C A E D (2) B A E C D (2) A C B D E (2) A B D C E (2) E D C B A (1) E B D A C (1) E B A C D (1) D E C B A (1) D C E A B (1) C D E B A (1) C D E A B (1) C B E D A (1) C A D B E (1) B E A D C (1) B C E A D (1) B A C E D (1) A E B D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 18 4 -6 12 B -18 0 6 -10 -4 C -4 -6 0 -16 -2 D 6 10 16 0 18 E -12 4 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 -6 12 B -18 0 6 -10 -4 C -4 -6 0 -16 -2 D 6 10 16 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=20 A=19 C=14 B=14 so C is eliminated. Round 2 votes counts: D=37 E=22 A=22 B=19 so B is eliminated. Round 3 votes counts: D=37 E=34 A=29 so A is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:214 E:188 B:187 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 4 -6 12 B -18 0 6 -10 -4 C -4 -6 0 -16 -2 D 6 10 16 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 -6 12 B -18 0 6 -10 -4 C -4 -6 0 -16 -2 D 6 10 16 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 -6 12 B -18 0 6 -10 -4 C -4 -6 0 -16 -2 D 6 10 16 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1006: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) E A C B D (5) C E A B D (5) C D B E A (5) D C B E A (4) C E B A D (4) C D B A E (4) C B E A D (4) C B D E A (4) E A B C D (3) D C B A E (3) B C D E A (3) A E B D C (3) A D E C B (3) A D E B C (3) E C B A D (2) D C A B E (2) D B C E A (2) D A B E C (2) C D A E B (2) C B E D A (2) B E A D C (2) B E A C D (2) B D E C A (2) A E D C B (2) E B A D C (1) E B A C D (1) D B E A C (1) D B C A E (1) D A E B C (1) D A C E B (1) C E D B A (1) C E A D B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E C A D (1) B D E A C (1) B D C E A (1) B C E A D (1) B A E D C (1) A E C D B (1) A E C B D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -20 -14 2 -16 B 20 0 -22 0 10 C 14 22 0 12 10 D -2 0 -12 0 8 E 16 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -14 2 -16 B 20 0 -22 0 10 C 14 22 0 12 10 D -2 0 -12 0 8 E 16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=24 A=15 B=14 E=12 so E is eliminated. Round 2 votes counts: C=37 D=24 A=23 B=16 so B is eliminated. Round 3 votes counts: C=42 A=30 D=28 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:204 D:197 E:194 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -14 2 -16 B 20 0 -22 0 10 C 14 22 0 12 10 D -2 0 -12 0 8 E 16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 2 -16 B 20 0 -22 0 10 C 14 22 0 12 10 D -2 0 -12 0 8 E 16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 2 -16 B 20 0 -22 0 10 C 14 22 0 12 10 D -2 0 -12 0 8 E 16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1007: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (13) D A C B E (10) A C D E B (6) E B C A D (5) D C B E A (5) C E B A D (5) E B A C D (4) C B E D A (4) E B A D C (3) D C A B E (3) D A B E C (3) C A D E B (3) B D E C A (3) A D C E B (3) A D B E C (3) D C B A E (2) D B E C A (2) D B E A C (2) C D E B A (2) C D A B E (2) C A E B D (2) B E D C A (2) A D C B E (2) E C B A D (1) E B C D A (1) D B A E C (1) C E A B D (1) C D A E B (1) C A E D B (1) B E C D A (1) B E A D C (1) A E B D C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 8 -14 -10 B 14 0 -8 -2 18 C -8 8 0 -18 6 D 14 2 18 0 6 E 10 -18 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 -14 -10 B 14 0 -8 -2 18 C -8 8 0 -18 6 D 14 2 18 0 6 E 10 -18 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=21 B=20 A=17 E=14 so E is eliminated. Round 2 votes counts: B=33 D=28 C=22 A=17 so A is eliminated. Round 3 votes counts: D=36 B=35 C=29 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:211 C:194 E:190 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 8 -14 -10 B 14 0 -8 -2 18 C -8 8 0 -18 6 D 14 2 18 0 6 E 10 -18 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -14 -10 B 14 0 -8 -2 18 C -8 8 0 -18 6 D 14 2 18 0 6 E 10 -18 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -14 -10 B 14 0 -8 -2 18 C -8 8 0 -18 6 D 14 2 18 0 6 E 10 -18 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1008: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (14) B A D C E (12) E C A D B (6) B A E C D (5) D C E A B (4) D A C E B (4) A E D C B (4) C D E B A (3) B D C A E (3) A D B C E (3) A B D C E (3) E C D B A (2) E C B D A (2) E A C D B (2) C E D B A (2) C E B D A (2) C D B E A (2) B D A C E (2) B C E D A (2) B C D E A (2) A D E C B (2) A B D E C (2) E D C A B (1) E C B A D (1) E B A C D (1) E A D C B (1) E A B C D (1) D E A C B (1) D A E C B (1) C E D A B (1) C B D E A (1) B E C A D (1) B C E A D (1) B C D A E (1) B C A E D (1) B A D E C (1) B A C E D (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -4 0 -6 B -4 0 -12 -10 -12 C 4 12 0 8 0 D 0 10 -8 0 -4 E 6 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.379095 D: 0.000000 E: 0.620905 Sum of squares = 0.529235967673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.379095 D: 0.379095 E: 1.000000 A B C D E A 0 4 -4 0 -6 B -4 0 -12 -10 -12 C 4 12 0 8 0 D 0 10 -8 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=31 A=16 C=11 D=10 so D is eliminated. Round 2 votes counts: E=32 B=32 A=21 C=15 so C is eliminated. Round 3 votes counts: E=44 B=35 A=21 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:211 D:199 A:197 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 0 -6 B -4 0 -12 -10 -12 C 4 12 0 8 0 D 0 10 -8 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 -6 B -4 0 -12 -10 -12 C 4 12 0 8 0 D 0 10 -8 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 -6 B -4 0 -12 -10 -12 C 4 12 0 8 0 D 0 10 -8 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1009: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) D E B A C (6) D B E A C (6) B D E A C (6) E D B A C (5) C E A D B (5) C A B E D (5) E C D B A (4) C E D B A (4) A C B D E (4) A B D E C (4) A B C D E (4) E B D C A (3) C E A B D (3) C D E A B (3) C A D B E (3) B A D E C (3) E C B D A (2) E B C D A (2) C A B D E (2) B E D A C (2) A D C B E (2) A D B C E (2) A C D B E (2) E D C B A (1) D E B C A (1) D C A E B (1) D A B E C (1) C E D A B (1) C E B D A (1) C A E B D (1) C A D E B (1) B D A E C (1) B A E D C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 -12 -22 B 8 0 10 -10 0 C -2 -10 0 -4 -10 D 12 10 4 0 6 E 22 0 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -12 -22 B 8 0 10 -10 0 C -2 -10 0 -4 -10 D 12 10 4 0 6 E 22 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=23 A=20 D=15 B=13 so B is eliminated. Round 2 votes counts: C=29 E=25 A=24 D=22 so D is eliminated. Round 3 votes counts: E=44 C=30 A=26 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:213 B:204 C:187 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -12 -22 B 8 0 10 -10 0 C -2 -10 0 -4 -10 D 12 10 4 0 6 E 22 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -12 -22 B 8 0 10 -10 0 C -2 -10 0 -4 -10 D 12 10 4 0 6 E 22 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -12 -22 B 8 0 10 -10 0 C -2 -10 0 -4 -10 D 12 10 4 0 6 E 22 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1010: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) E C A D B (6) D C B E A (6) A E C D B (6) C D B E A (5) B D C E A (5) B A D E C (5) A E B D C (5) A E B C D (5) E A B C D (4) B D C A E (4) A E C B D (4) A B D E C (4) C D E B A (3) B D E C A (3) A B D C E (3) E A C B D (2) B E A D C (2) B D A E C (2) B A E D C (2) A C E D B (2) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D B C E A (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D B A (1) C E D A B (1) C D E A B (1) C D B A E (1) C A D E B (1) B E D C A (1) A C D E B (1) Total count = 100 A B C D E A 0 6 12 18 -4 B -6 0 0 8 -6 C -12 0 0 6 -24 D -18 -8 -6 0 -4 E 4 6 24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 12 18 -4 B -6 0 0 8 -6 C -12 0 0 6 -24 D -18 -8 -6 0 -4 E 4 6 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 E=23 C=13 D=10 so D is eliminated. Round 2 votes counts: A=31 B=27 E=23 C=19 so C is eliminated. Round 3 votes counts: B=39 A=32 E=29 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 A:216 B:198 C:185 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 18 -4 B -6 0 0 8 -6 C -12 0 0 6 -24 D -18 -8 -6 0 -4 E 4 6 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 18 -4 B -6 0 0 8 -6 C -12 0 0 6 -24 D -18 -8 -6 0 -4 E 4 6 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 18 -4 B -6 0 0 8 -6 C -12 0 0 6 -24 D -18 -8 -6 0 -4 E 4 6 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1011: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (19) A E C B D (7) A E B C D (7) D C B E A (6) E C A D B (5) E C A B D (5) C E D B A (5) B D C E A (4) B C D E A (3) A E C D B (3) A B E C D (3) A B D E C (3) D C E B A (2) D C E A B (2) D B A C E (2) C E D A B (2) C E B A D (2) B C E D A (2) A E D B C (2) E D C A B (1) E A C D B (1) E A C B D (1) D E C A B (1) D A E C B (1) D A B C E (1) C E B D A (1) C E A B D (1) C D B E A (1) B C E A D (1) B A E C D (1) B A D E C (1) B A C E D (1) A E D C B (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -26 -6 -30 B 2 0 2 -12 -2 C 26 -2 0 4 10 D 6 12 -4 0 -6 E 30 2 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839467 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -26 -6 -30 B 2 0 2 -12 -2 C 26 -2 0 4 10 D 6 12 -4 0 -6 E 30 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839554 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=28 E=13 B=13 C=12 so C is eliminated. Round 2 votes counts: D=35 A=28 E=24 B=13 so B is eliminated. Round 3 votes counts: D=42 A=31 E=27 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:219 E:214 D:204 B:195 A:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -26 -6 -30 B 2 0 2 -12 -2 C 26 -2 0 4 10 D 6 12 -4 0 -6 E 30 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839554 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -26 -6 -30 B 2 0 2 -12 -2 C 26 -2 0 4 10 D 6 12 -4 0 -6 E 30 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839554 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -26 -6 -30 B 2 0 2 -12 -2 C 26 -2 0 4 10 D 6 12 -4 0 -6 E 30 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839554 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1012: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) C D B E A (8) D C A B E (6) D C B E A (5) B E C D A (5) B E A D C (5) A E B C D (5) D C A E B (4) A D C E B (4) B E C A D (3) B D C E A (3) A E C D B (3) E B C A D (2) E A B C D (2) D A C B E (2) C E D A B (2) C D E B A (2) B C E D A (2) A E D B C (2) A E B D C (2) A C D E B (2) E A C D B (1) E A B D C (1) D C B A E (1) D B C E A (1) D A C E B (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D E A (1) C A D E B (1) B E D C A (1) B E A C D (1) B D E C A (1) B A E D C (1) A E D C B (1) A E C B D (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -8 0 -18 B 8 0 -6 -8 0 C 8 6 0 12 4 D 0 8 -12 0 -4 E 18 0 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 0 -18 B 8 0 -6 -8 0 C 8 6 0 12 4 D 0 8 -12 0 -4 E 18 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 B=22 D=20 C=20 E=15 so E is eliminated. Round 2 votes counts: B=33 A=27 D=20 C=20 so D is eliminated. Round 3 votes counts: C=36 B=34 A=30 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:209 B:197 D:196 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 0 -18 B 8 0 -6 -8 0 C 8 6 0 12 4 D 0 8 -12 0 -4 E 18 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 0 -18 B 8 0 -6 -8 0 C 8 6 0 12 4 D 0 8 -12 0 -4 E 18 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 0 -18 B 8 0 -6 -8 0 C 8 6 0 12 4 D 0 8 -12 0 -4 E 18 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1013: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C B A D E (6) C A B E D (6) E D A B C (5) B D E C A (5) A C B E D (5) D E B C A (4) A E C D B (4) A C E B D (4) E D A C B (3) B D C E A (3) B C D A E (3) B C A D E (3) B A D C E (3) B A C D E (3) A E D B C (3) A C E D B (3) E D C B A (2) E D B C A (2) D B E A C (2) C B E D A (2) C A E D B (2) C A E B D (2) B D E A C (2) B D A C E (2) A E D C B (2) E D B A C (1) E C D B A (1) E C A D B (1) C E D A B (1) C B E A D (1) C B A E D (1) C A B D E (1) B D C A E (1) A D E B C (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 6 8 16 B 8 0 4 12 4 C -6 -4 0 2 8 D -8 -12 -2 0 -2 E -16 -4 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 8 16 B 8 0 4 12 4 C -6 -4 0 2 8 D -8 -12 -2 0 -2 E -16 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999093 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=25 A=25 C=22 E=15 D=13 so D is eliminated. Round 2 votes counts: B=27 E=26 A=25 C=22 so C is eliminated. Round 3 votes counts: B=37 A=36 E=27 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:211 C:200 D:188 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 8 16 B 8 0 4 12 4 C -6 -4 0 2 8 D -8 -12 -2 0 -2 E -16 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999093 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 8 16 B 8 0 4 12 4 C -6 -4 0 2 8 D -8 -12 -2 0 -2 E -16 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999093 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 8 16 B 8 0 4 12 4 C -6 -4 0 2 8 D -8 -12 -2 0 -2 E -16 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999093 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1014: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (13) A E C D B (12) B D C E A (11) E A B D C (10) C D B A E (9) C A D B E (5) A E C B D (5) E B D A C (4) D B C A E (4) B D E C A (4) D B C E A (3) B D E A C (3) E B A D C (2) E A C B D (2) E A B C D (2) C A E D B (2) A E B D C (2) D C B A E (1) D B A E C (1) C A D E B (1) B E D A C (1) A E D B C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 14 20 18 16 B -14 0 -2 -6 -16 C -20 2 0 6 0 D -18 6 -6 0 -14 E -16 16 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 18 16 B -14 0 -2 -6 -16 C -20 2 0 6 0 D -18 6 -6 0 -14 E -16 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=20 B=19 C=17 D=9 so D is eliminated. Round 2 votes counts: A=35 B=27 E=20 C=18 so C is eliminated. Round 3 votes counts: A=43 B=37 E=20 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:207 C:194 D:184 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 18 16 B -14 0 -2 -6 -16 C -20 2 0 6 0 D -18 6 -6 0 -14 E -16 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 18 16 B -14 0 -2 -6 -16 C -20 2 0 6 0 D -18 6 -6 0 -14 E -16 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 18 16 B -14 0 -2 -6 -16 C -20 2 0 6 0 D -18 6 -6 0 -14 E -16 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1015: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) E D B A C (9) E A C D B (7) B C A D E (7) A C E B D (7) D E B C A (6) C A B E D (5) B D C E A (5) B C D A E (5) A E C D B (5) E D A B C (4) C A B D E (4) A C B D E (4) E A D C B (3) D E B A C (3) E D A C B (2) C B A D E (2) B D C A E (2) A C E D B (2) A C B E D (2) E D B C A (1) E C B D A (1) E C A D B (1) E B D C A (1) B D E C A (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 -2 0 -8 B 8 0 10 -6 -4 C 2 -10 0 6 -8 D 0 6 -6 0 0 E 8 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.402673 E: 0.597327 Sum of squares = 0.51894508494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.402673 E: 1.000000 A B C D E A 0 -8 -2 0 -8 B 8 0 10 -6 -4 C 2 -10 0 6 -8 D 0 6 -6 0 0 E 8 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=21 A=20 D=19 C=11 so C is eliminated. Round 2 votes counts: E=29 A=29 B=23 D=19 so D is eliminated. Round 3 votes counts: E=38 B=33 A=29 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:204 D:200 C:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 0 -8 B 8 0 10 -6 -4 C 2 -10 0 6 -8 D 0 6 -6 0 0 E 8 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 -8 B 8 0 10 -6 -4 C 2 -10 0 6 -8 D 0 6 -6 0 0 E 8 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 -8 B 8 0 10 -6 -4 C 2 -10 0 6 -8 D 0 6 -6 0 0 E 8 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1016: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (12) D C E B A (11) A B E C D (10) E A D C B (9) B C D E A (9) B A C D E (8) A E B C D (7) A E D C B (5) E D A C B (3) C D B E A (3) B C A D E (3) A E B D C (3) A B C E D (3) E D C A B (2) D C B E A (2) E D C B A (1) D E C B A (1) D E C A B (1) D C E A B (1) C D A E B (1) B D C E A (1) B A E D C (1) B A E C D (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 4 12 B 6 0 18 18 8 C -4 -18 0 18 12 D -4 -18 -18 0 8 E -12 -8 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 4 12 B 6 0 18 18 8 C -4 -18 0 18 12 D -4 -18 -18 0 8 E -12 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=30 D=16 E=15 C=4 so C is eliminated. Round 2 votes counts: B=35 A=30 D=20 E=15 so E is eliminated. Round 3 votes counts: A=39 B=35 D=26 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:207 C:204 D:184 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 4 12 B 6 0 18 18 8 C -4 -18 0 18 12 D -4 -18 -18 0 8 E -12 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 4 12 B 6 0 18 18 8 C -4 -18 0 18 12 D -4 -18 -18 0 8 E -12 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 4 12 B 6 0 18 18 8 C -4 -18 0 18 12 D -4 -18 -18 0 8 E -12 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1017: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) D E A B C (7) E D A B C (6) E C D B A (5) C B A D E (5) B A C D E (5) D E C B A (3) C E B A D (3) C B D E A (3) A E B C D (3) A B D C E (3) A B C E D (3) E D C B A (2) E D A C B (2) E A B C D (2) D C E B A (2) D A B E C (2) D A B C E (2) C B E A D (2) B C A D E (2) B A D C E (2) A B D E C (2) E D C A B (1) E C D A B (1) E C B A D (1) E C A D B (1) E C A B D (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D E C A B (1) D B C A E (1) D A E B C (1) C E D B A (1) C D E B A (1) C D B E A (1) B D C A E (1) B C A E D (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 14 0 B 2 0 4 10 -2 C -4 -4 0 10 0 D -14 -10 -10 0 -8 E 0 2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.335554 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.664446 Sum of squares = 0.554084925692 Cumulative probabilities = A: 0.335554 B: 0.335554 C: 0.335554 D: 0.335554 E: 1.000000 A B C D E A 0 -2 4 14 0 B 2 0 4 10 -2 C -4 -4 0 10 0 D -14 -10 -10 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499893 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500107 Sum of squares = 0.500000022895 Cumulative probabilities = A: 0.499893 B: 0.499893 C: 0.499893 D: 0.499893 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 D=19 A=18 B=12 so B is eliminated. Round 2 votes counts: C=28 E=26 A=26 D=20 so D is eliminated. Round 3 votes counts: E=37 C=32 A=31 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:208 B:207 E:205 C:201 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 14 0 B 2 0 4 10 -2 C -4 -4 0 10 0 D -14 -10 -10 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499893 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500107 Sum of squares = 0.500000022895 Cumulative probabilities = A: 0.499893 B: 0.499893 C: 0.499893 D: 0.499893 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 14 0 B 2 0 4 10 -2 C -4 -4 0 10 0 D -14 -10 -10 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499893 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500107 Sum of squares = 0.500000022895 Cumulative probabilities = A: 0.499893 B: 0.499893 C: 0.499893 D: 0.499893 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 14 0 B 2 0 4 10 -2 C -4 -4 0 10 0 D -14 -10 -10 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499893 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500107 Sum of squares = 0.500000022895 Cumulative probabilities = A: 0.499893 B: 0.499893 C: 0.499893 D: 0.499893 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1018: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (20) B C E A D (14) E A D C B (7) E C B A D (6) B C D A E (6) A E D C B (5) D A B C E (4) C B E A D (4) B C E D A (4) B C D E A (4) A D E C B (4) D B A C E (3) D A E B C (3) B D C A E (3) E D A C B (1) E B C A D (1) E A C D B (1) E A C B D (1) D A C B E (1) D A B E C (1) C E B A D (1) C B A E D (1) C B A D E (1) C A B E D (1) B E C A D (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 6 0 10 B -2 0 -12 -4 -4 C -6 12 0 -6 -4 D 0 4 6 0 2 E -10 4 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.334051 B: 0.000000 C: 0.000000 D: 0.665949 E: 0.000000 Sum of squares = 0.555078304237 Cumulative probabilities = A: 0.334051 B: 0.334051 C: 0.334051 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 0 10 B -2 0 -12 -4 -4 C -6 12 0 -6 -4 D 0 4 6 0 2 E -10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=32 B=32 E=17 A=11 C=8 so C is eliminated. Round 2 votes counts: B=38 D=32 E=18 A=12 so A is eliminated. Round 3 votes counts: B=39 D=37 E=24 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:206 C:198 E:198 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 0 10 B -2 0 -12 -4 -4 C -6 12 0 -6 -4 D 0 4 6 0 2 E -10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 10 B -2 0 -12 -4 -4 C -6 12 0 -6 -4 D 0 4 6 0 2 E -10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 10 B -2 0 -12 -4 -4 C -6 12 0 -6 -4 D 0 4 6 0 2 E -10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1019: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) D B E A C (6) E D A B C (5) C D B E A (5) C B A E D (5) B A C E D (5) E A D B C (4) D E C A B (4) D E B A C (3) C D E A B (3) C B D A E (3) C A E B D (3) B A D E C (3) A B E C D (3) E D A C B (2) E A D C B (2) D C B E A (2) C B A D E (2) B A E D C (2) A E C B D (2) E D C A B (1) E A C D B (1) E A C B D (1) E A B D C (1) E A B C D (1) D E C B A (1) D E A C B (1) D B E C A (1) D B C E A (1) D B C A E (1) D B A E C (1) C E D A B (1) C E A B D (1) C A B E D (1) B D C A E (1) B D A E C (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D C E (1) A E B D C (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 22 -10 -16 B -4 0 16 -12 2 C -22 -16 0 -14 -20 D 10 12 14 0 6 E 16 -2 20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 22 -10 -16 B -4 0 16 -12 2 C -22 -16 0 -14 -20 D 10 12 14 0 6 E 16 -2 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=24 E=18 B=17 A=9 so A is eliminated. Round 2 votes counts: D=32 C=25 B=22 E=21 so E is eliminated. Round 3 votes counts: D=46 C=29 B=25 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:214 B:201 A:200 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 22 -10 -16 B -4 0 16 -12 2 C -22 -16 0 -14 -20 D 10 12 14 0 6 E 16 -2 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 22 -10 -16 B -4 0 16 -12 2 C -22 -16 0 -14 -20 D 10 12 14 0 6 E 16 -2 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 22 -10 -16 B -4 0 16 -12 2 C -22 -16 0 -14 -20 D 10 12 14 0 6 E 16 -2 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1020: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (6) C A D B E (5) B E D C A (5) D B A C E (4) C D B A E (4) C B D A E (4) B E D A C (4) A C D B E (4) E B C D A (3) E B A D C (3) D B C A E (3) C B A D E (3) C A E D B (3) C A D E B (3) B C E D A (3) B C D E A (3) A E C D B (3) A D E B C (3) E C B A D (2) E A D C B (2) D A B C E (2) C E B A D (2) B D E C A (2) B D E A C (2) B D C A E (2) E D A B C (1) E B D C A (1) E B A C D (1) E A C B D (1) E A B D C (1) D B E A C (1) D A E B C (1) D A C B E (1) C E A D B (1) C E A B D (1) C A E B D (1) C A B D E (1) B E C D A (1) B D A E C (1) B C D A E (1) A E D C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -22 -8 -10 8 B 22 0 8 8 14 C 8 -8 0 4 10 D 10 -8 -4 0 6 E -8 -14 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -8 -10 8 B 22 0 8 8 14 C 8 -8 0 4 10 D 10 -8 -4 0 6 E -8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=24 E=21 A=15 D=12 so D is eliminated. Round 2 votes counts: B=32 C=28 E=21 A=19 so A is eliminated. Round 3 votes counts: C=37 B=34 E=29 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:207 D:202 A:184 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -8 -10 8 B 22 0 8 8 14 C 8 -8 0 4 10 D 10 -8 -4 0 6 E -8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -8 -10 8 B 22 0 8 8 14 C 8 -8 0 4 10 D 10 -8 -4 0 6 E -8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -8 -10 8 B 22 0 8 8 14 C 8 -8 0 4 10 D 10 -8 -4 0 6 E -8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1021: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) A B D E C (6) D C E A B (5) D A E B C (5) C D E B A (5) C B E D A (4) C B E A D (4) A D E B C (4) E D C B A (3) C E D B A (3) C E B D A (3) B E C D A (3) B E A D C (3) B A E C D (3) A C D E B (3) A C B D E (3) A B E D C (3) E D B C A (2) D E B C A (2) C D E A B (2) C A B E D (2) B E A C D (2) A D C E B (2) A D B C E (2) A B C E D (2) E D B A C (1) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C A E B (1) C B A E D (1) C A D E B (1) C A D B E (1) B C E A D (1) B A E D C (1) B A C E D (1) A D B E C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -8 -4 -12 B 4 0 -8 -6 -10 C 8 8 0 -2 -6 D 4 6 2 0 4 E 12 10 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -4 -12 B 4 0 -8 -6 -10 C 8 8 0 -2 -6 D 4 6 2 0 4 E 12 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=26 D=22 B=14 E=10 so E is eliminated. Round 2 votes counts: D=28 C=28 A=28 B=16 so B is eliminated. Round 3 votes counts: A=38 C=33 D=29 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 D:208 C:204 B:190 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 -12 B 4 0 -8 -6 -10 C 8 8 0 -2 -6 D 4 6 2 0 4 E 12 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 -12 B 4 0 -8 -6 -10 C 8 8 0 -2 -6 D 4 6 2 0 4 E 12 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 -12 B 4 0 -8 -6 -10 C 8 8 0 -2 -6 D 4 6 2 0 4 E 12 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1022: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) A D B C E (9) A C D B E (9) E C B D A (7) D B E A C (6) C A E B D (6) C E A B D (5) D B E C A (4) D B A E C (4) A D C B E (4) A C B D E (4) E B D C A (3) E B C D A (3) B D E C A (3) A C E B D (3) D E B A C (2) D A B E C (2) C E B D A (2) C A B E D (2) A C E D B (2) A C D E B (2) E C B A D (1) D E B C A (1) D E A B C (1) D A B C E (1) C B E A D (1) B E D C A (1) A D B E C (1) Total count = 100 A B C D E A 0 2 0 20 -2 B -2 0 -18 4 2 C 0 18 0 16 22 D -20 -4 -16 0 6 E 2 -2 -22 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.505951 B: 0.000000 C: 0.494049 D: 0.000000 E: 0.000000 Sum of squares = 0.500070837872 Cumulative probabilities = A: 0.505951 B: 0.505951 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 20 -2 B -2 0 -18 4 2 C 0 18 0 16 22 D -20 -4 -16 0 6 E 2 -2 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=27 D=21 E=14 B=4 so B is eliminated. Round 2 votes counts: A=34 C=27 D=24 E=15 so E is eliminated. Round 3 votes counts: C=38 A=34 D=28 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:228 A:210 B:193 E:186 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 20 -2 B -2 0 -18 4 2 C 0 18 0 16 22 D -20 -4 -16 0 6 E 2 -2 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 20 -2 B -2 0 -18 4 2 C 0 18 0 16 22 D -20 -4 -16 0 6 E 2 -2 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 20 -2 B -2 0 -18 4 2 C 0 18 0 16 22 D -20 -4 -16 0 6 E 2 -2 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1023: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (13) C D E A B (6) B D E A C (6) D E C A B (4) B A C E D (4) E A D B C (3) D C E B A (3) C D B E A (3) C A E D B (3) B E A D C (3) B D C E A (3) B D A E C (3) B C A D E (3) A E B D C (3) A C B E D (3) E D A C B (2) D E B C A (2) D C E A B (2) C B D A E (2) C A B E D (2) B A E C D (2) A E D B C (2) A B E C D (2) A B C E D (2) E D A B C (1) E A D C B (1) E A C D B (1) D E C B A (1) D E B A C (1) D B E C A (1) D B C E A (1) C D E B A (1) C D A E B (1) C B A D E (1) B D E C A (1) B D C A E (1) B C D A E (1) B A D E C (1) B A C D E (1) A E C D B (1) A E C B D (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 16 8 8 B 16 0 22 18 20 C -16 -22 0 -18 -12 D -8 -18 18 0 -2 E -8 -20 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 16 8 8 B 16 0 22 18 20 C -16 -22 0 -18 -12 D -8 -18 18 0 -2 E -8 -20 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 C=19 A=16 D=15 E=8 so E is eliminated. Round 2 votes counts: B=42 A=21 C=19 D=18 so D is eliminated. Round 3 votes counts: B=47 C=29 A=24 so A is eliminated. Round 4 votes counts: B=61 C=39 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:238 A:208 D:195 E:193 C:166 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 16 8 8 B 16 0 22 18 20 C -16 -22 0 -18 -12 D -8 -18 18 0 -2 E -8 -20 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 16 8 8 B 16 0 22 18 20 C -16 -22 0 -18 -12 D -8 -18 18 0 -2 E -8 -20 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 16 8 8 B 16 0 22 18 20 C -16 -22 0 -18 -12 D -8 -18 18 0 -2 E -8 -20 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1024: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) D C A B E (9) D B A C E (7) C D A E B (7) E C A B D (6) C E A D B (6) C A E D B (4) B E A D C (4) B D A E C (4) E A B C D (3) D C B E A (3) C D E A B (3) E A C B D (2) D C B A E (2) D A B C E (2) C D E B A (2) B D E A C (2) B A D E C (2) A E C B D (2) A B E D C (2) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A D C (1) D B A E C (1) C E D A B (1) C A D E B (1) B E D C A (1) B E D A C (1) B A E D C (1) A E B C D (1) A D C B E (1) A D B C E (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 2 2 2 B -10 0 -8 -6 -12 C -2 8 0 8 6 D -2 6 -8 0 -2 E -2 12 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999446 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 2 2 B -10 0 -8 -6 -12 C -2 8 0 8 6 D -2 6 -8 0 -2 E -2 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 C=24 B=15 A=11 so A is eliminated. Round 2 votes counts: E=29 C=28 D=26 B=17 so B is eliminated. Round 3 votes counts: E=38 D=34 C=28 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:210 A:208 E:203 D:197 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 2 2 B -10 0 -8 -6 -12 C -2 8 0 8 6 D -2 6 -8 0 -2 E -2 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 2 2 B -10 0 -8 -6 -12 C -2 8 0 8 6 D -2 6 -8 0 -2 E -2 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 2 2 B -10 0 -8 -6 -12 C -2 8 0 8 6 D -2 6 -8 0 -2 E -2 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996388 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1025: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) B C A D E (7) D E A C B (6) E D C A B (5) E D A C B (5) D E C B A (3) D E C A B (3) D E B A C (3) C B E A D (3) C B D E A (3) C B A E D (3) A B C E D (3) E D C B A (2) E C D B A (2) E C D A B (2) E A D C B (2) D B C E A (2) C E A B D (2) C A B E D (2) B C A E D (2) B A C E D (2) B A C D E (2) A E B D C (2) A B E D C (2) A B D E C (2) A B C D E (2) E D A B C (1) E C A D B (1) E A C D B (1) E A C B D (1) D C E B A (1) D B A C E (1) D A E B C (1) C E B A D (1) C B E D A (1) B D C E A (1) B D C A E (1) B C D A E (1) A E D C B (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 18 4 -12 -26 B -18 0 0 -12 -16 C -4 0 0 -18 -18 D 12 12 18 0 8 E 26 16 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 -12 -26 B -18 0 0 -12 -16 C -4 0 0 -18 -18 D 12 12 18 0 8 E 26 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=22 B=16 C=15 A=15 so C is eliminated. Round 2 votes counts: D=32 B=26 E=25 A=17 so A is eliminated. Round 3 votes counts: B=38 D=34 E=28 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:226 D:225 A:192 C:180 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 4 -12 -26 B -18 0 0 -12 -16 C -4 0 0 -18 -18 D 12 12 18 0 8 E 26 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 -12 -26 B -18 0 0 -12 -16 C -4 0 0 -18 -18 D 12 12 18 0 8 E 26 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 -12 -26 B -18 0 0 -12 -16 C -4 0 0 -18 -18 D 12 12 18 0 8 E 26 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1026: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (14) C A E B D (14) B D E A C (9) D B C A E (6) D E C A B (3) D C B A E (3) D B E C A (3) C A E D B (3) B E A D C (3) B A E C D (3) B A C E D (3) E C A D B (2) D E B C A (2) D B C E A (2) C A D B E (2) C A B E D (2) B E D A C (2) B C A D E (2) E D A C B (1) E C A B D (1) E B A C D (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A E B (1) C E A D B (1) C E A B D (1) C A D E B (1) C A B D E (1) B E A C D (1) B D A E C (1) B D A C E (1) B A E D C (1) B A D E C (1) B A C D E (1) A E C B D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -22 -6 -2 -2 B 22 0 16 6 24 C 6 -16 0 -12 -6 D 2 -6 12 0 12 E 2 -24 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 -2 -2 B 22 0 16 6 24 C 6 -16 0 -12 -6 D 2 -6 12 0 12 E 2 -24 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998461 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=28 C=25 E=7 A=3 so A is eliminated. Round 2 votes counts: D=37 B=29 C=26 E=8 so E is eliminated. Round 3 votes counts: D=38 C=31 B=31 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:234 D:210 C:186 E:186 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -6 -2 -2 B 22 0 16 6 24 C 6 -16 0 -12 -6 D 2 -6 12 0 12 E 2 -24 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998461 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -2 -2 B 22 0 16 6 24 C 6 -16 0 -12 -6 D 2 -6 12 0 12 E 2 -24 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998461 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -2 -2 B 22 0 16 6 24 C 6 -16 0 -12 -6 D 2 -6 12 0 12 E 2 -24 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998461 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1027: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) D B A C E (7) E D B A C (6) E C B A D (6) E C D B A (4) E C A D B (4) D A B C E (4) E C B D A (3) C A E B D (3) C A B D E (3) B A D C E (3) B A C D E (3) A C B D E (3) E D C A B (2) E D B C A (2) E C A B D (2) E B C D A (2) E B C A D (2) D E A C B (2) D B A E C (2) C E A D B (2) C E A B D (2) C A D E B (2) B E D A C (2) B D E A C (2) B D A E C (2) B D A C E (2) B A C E D (2) A D B C E (2) A B D C E (2) E B D A C (1) D E C A B (1) D E B A C (1) D B E A C (1) D A C B E (1) C D A E B (1) C A D B E (1) Total count = 100 A B C D E A 0 -6 -4 -16 -14 B 6 0 -4 -10 -16 C 4 4 0 10 -14 D 16 10 -10 0 -10 E 14 16 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -16 -14 B 6 0 -4 -10 -16 C 4 4 0 10 -14 D 16 10 -10 0 -10 E 14 16 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 D=19 B=16 C=14 A=7 so A is eliminated. Round 2 votes counts: E=44 D=21 B=18 C=17 so C is eliminated. Round 3 votes counts: E=51 D=25 B=24 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:203 C:202 B:188 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -16 -14 B 6 0 -4 -10 -16 C 4 4 0 10 -14 D 16 10 -10 0 -10 E 14 16 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -16 -14 B 6 0 -4 -10 -16 C 4 4 0 10 -14 D 16 10 -10 0 -10 E 14 16 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -16 -14 B 6 0 -4 -10 -16 C 4 4 0 10 -14 D 16 10 -10 0 -10 E 14 16 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1028: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (9) E D A C B (7) C B A E D (6) C B D E A (5) C B D A E (5) A B C D E (5) D E C B A (4) D E A B C (4) B C A D E (4) E D C B A (3) E A D B C (3) C E B A D (3) C D E B A (3) A E B D C (3) A B C E D (3) E D C A B (2) C E D B A (2) C D B E A (2) B A C E D (2) B A C D E (2) A E D B C (2) A B E C D (2) E C D B A (1) D E C A B (1) D E A C B (1) D B C A E (1) D A E B C (1) C B E D A (1) B C A E D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -6 -6 -4 B 8 0 -10 4 -2 C 6 10 0 12 8 D 6 -4 -12 0 -2 E 4 2 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -6 -4 B 8 0 -10 4 -2 C 6 10 0 12 8 D 6 -4 -12 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=26 A=17 D=12 B=9 so B is eliminated. Round 2 votes counts: C=41 E=26 A=21 D=12 so D is eliminated. Round 3 votes counts: C=42 E=36 A=22 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:200 E:200 D:194 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -6 -4 B 8 0 -10 4 -2 C 6 10 0 12 8 D 6 -4 -12 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -6 -4 B 8 0 -10 4 -2 C 6 10 0 12 8 D 6 -4 -12 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -6 -4 B 8 0 -10 4 -2 C 6 10 0 12 8 D 6 -4 -12 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1029: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) B D C E A (6) E A D B C (4) D B E A C (4) C A E B D (4) B D E C A (4) B C D E A (4) A C D E B (4) D B C A E (3) D A C B E (3) C B E A D (3) C A D B E (3) A D E C B (3) A D C E B (3) A C E B D (3) E C B A D (2) E B D A C (2) E B C A D (2) E B A C D (2) E A B C D (2) D C B A E (2) D C A B E (2) D B C E A (2) C D B A E (2) C B D E A (2) B C E D A (2) A E D C B (2) A E D B C (2) A E C D B (2) A E C B D (2) A D C B E (2) A C E D B (2) E C A B D (1) E A C B D (1) D E B A C (1) D B E C A (1) D A B C E (1) C E B A D (1) C B D A E (1) B E C D A (1) B D E A C (1) Total count = 100 A B C D E A 0 -8 -8 0 -8 B 8 0 0 4 10 C 8 0 0 -8 10 D 0 -4 8 0 8 E 8 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.730133 C: 0.269867 D: 0.000000 E: 0.000000 Sum of squares = 0.605922720541 Cumulative probabilities = A: 0.000000 B: 0.730133 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 0 -8 B 8 0 0 4 10 C 8 0 0 -8 10 D 0 -4 8 0 8 E 8 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555644404 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=24 D=19 E=16 C=16 so E is eliminated. Round 2 votes counts: A=32 B=30 D=19 C=19 so D is eliminated. Round 3 votes counts: B=41 A=36 C=23 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:206 C:205 E:190 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 0 -8 B 8 0 0 4 10 C 8 0 0 -8 10 D 0 -4 8 0 8 E 8 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555644404 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 0 -8 B 8 0 0 4 10 C 8 0 0 -8 10 D 0 -4 8 0 8 E 8 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555644404 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 0 -8 B 8 0 0 4 10 C 8 0 0 -8 10 D 0 -4 8 0 8 E 8 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555644404 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1030: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) D B E A C (6) D B A C E (6) E C A B D (5) E A C B D (5) C D E A B (4) C A B E D (4) E D C A B (3) E A B C D (3) D C B A E (3) C E D A B (3) C E A B D (3) C A E B D (3) B A E D C (3) B A C E D (3) A C B E D (3) E C D A B (2) E C A D B (2) E B A C D (2) D E C B A (2) D E C A B (2) D C B E A (2) C E A D B (2) B D A E C (2) B A E C D (2) A E B C D (2) A B C E D (2) E B A D C (1) D E B C A (1) D E B A C (1) D C E B A (1) D C E A B (1) D B A E C (1) C D A B E (1) C A D B E (1) C A B D E (1) B C A D E (1) B A D C E (1) B A C D E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -8 4 -2 B -8 0 -10 -2 0 C 8 10 0 14 8 D -4 2 -14 0 -10 E 2 0 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 4 -2 B -8 0 -10 -2 0 C 8 10 0 14 8 D -4 2 -14 0 -10 E 2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=23 C=22 B=13 A=9 so A is eliminated. Round 2 votes counts: D=33 E=26 C=26 B=15 so B is eliminated. Round 3 votes counts: D=36 C=33 E=31 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:202 A:201 B:190 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 4 -2 B -8 0 -10 -2 0 C 8 10 0 14 8 D -4 2 -14 0 -10 E 2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 4 -2 B -8 0 -10 -2 0 C 8 10 0 14 8 D -4 2 -14 0 -10 E 2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 4 -2 B -8 0 -10 -2 0 C 8 10 0 14 8 D -4 2 -14 0 -10 E 2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1031: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) D C E A B (7) D E C A B (5) C D E B A (5) B A E C D (5) A B E D C (5) E C D B A (4) B E C D A (4) B C E D A (4) B A E D C (4) C D A E B (3) B E A C D (3) A B D C E (3) E D C B A (2) E D C A B (2) E B D C A (2) D C A E B (2) C E D B A (2) C D B E A (2) B A C D E (2) A D E C B (2) A D C E B (2) A B D E C (2) E D B C A (1) E D A C B (1) E C B D A (1) E A D C B (1) D E A C B (1) D A C E B (1) C D B A E (1) C B E D A (1) C B D E A (1) C A B D E (1) B E D A C (1) B E A D C (1) B C D E A (1) B C D A E (1) B A C E D (1) A E D C B (1) A E B D C (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -20 -26 -20 B -2 0 -16 -8 -8 C 20 16 0 6 2 D 26 8 -6 0 6 E 20 8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -20 -26 -20 B -2 0 -16 -8 -8 C 20 16 0 6 2 D 26 8 -6 0 6 E 20 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999970927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 A=19 D=16 E=14 so E is eliminated. Round 2 votes counts: C=29 B=29 D=22 A=20 so A is eliminated. Round 3 votes counts: B=41 C=31 D=28 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:217 E:210 B:183 A:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -20 -26 -20 B -2 0 -16 -8 -8 C 20 16 0 6 2 D 26 8 -6 0 6 E 20 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999970927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -20 -26 -20 B -2 0 -16 -8 -8 C 20 16 0 6 2 D 26 8 -6 0 6 E 20 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999970927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -20 -26 -20 B -2 0 -16 -8 -8 C 20 16 0 6 2 D 26 8 -6 0 6 E 20 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999970927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1032: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) C D A B E (7) E B A D C (6) C E A D B (6) D C A B E (5) B E A D C (5) B D A E C (5) E C A B D (4) E B A C D (3) D A B C E (3) C E D B A (3) C D A E B (3) C A D E B (3) B A D E C (3) A D C B E (3) E C B D A (2) E B D A C (2) E B C D A (2) D B A C E (2) C E D A B (2) A E B D C (2) A D B C E (2) A C E D B (2) A C D E B (2) A B E D C (2) E A B C D (1) D C B A E (1) D A C B E (1) C D B E A (1) C A E D B (1) B E D A C (1) B D E A C (1) B A E D C (1) A E B C D (1) A D B E C (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 2 18 4 B -6 0 -12 2 -8 C -2 12 0 6 -4 D -18 -2 -6 0 -10 E -4 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 18 4 B -6 0 -12 2 -8 C -2 12 0 6 -4 D -18 -2 -6 0 -10 E -4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999691 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 A=17 B=16 D=12 so D is eliminated. Round 2 votes counts: C=32 E=29 A=21 B=18 so B is eliminated. Round 3 votes counts: E=36 C=32 A=32 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:209 C:206 B:188 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 18 4 B -6 0 -12 2 -8 C -2 12 0 6 -4 D -18 -2 -6 0 -10 E -4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999691 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 18 4 B -6 0 -12 2 -8 C -2 12 0 6 -4 D -18 -2 -6 0 -10 E -4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999691 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 18 4 B -6 0 -12 2 -8 C -2 12 0 6 -4 D -18 -2 -6 0 -10 E -4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999691 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1033: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (12) C A D B E (10) E B C D A (9) E B D C A (7) E B D A C (7) A D C B E (7) A D B E C (7) E C B D A (6) C E A B D (4) A C D B E (4) B D A E C (3) E A D B C (2) D B A E C (2) D B A C E (2) D A B E C (2) C E B D A (2) B D E C A (2) E D B A C (1) E C A D B (1) C E B A D (1) C B D E A (1) C B D A E (1) C A E D B (1) C A D E B (1) C A B D E (1) B E D A C (1) B D E A C (1) B C D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 6 4 4 8 B -6 0 20 -6 14 C -4 -20 0 -14 -4 D -4 6 14 0 14 E -8 -14 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 4 8 B -6 0 20 -6 14 C -4 -20 0 -14 -4 D -4 6 14 0 14 E -8 -14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=31 C=22 B=8 D=6 so D is eliminated. Round 2 votes counts: E=33 A=33 C=22 B=12 so B is eliminated. Round 3 votes counts: A=40 E=37 C=23 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:211 B:211 E:184 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 4 8 B -6 0 20 -6 14 C -4 -20 0 -14 -4 D -4 6 14 0 14 E -8 -14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 4 8 B -6 0 20 -6 14 C -4 -20 0 -14 -4 D -4 6 14 0 14 E -8 -14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 4 8 B -6 0 20 -6 14 C -4 -20 0 -14 -4 D -4 6 14 0 14 E -8 -14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1034: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) B E D C A (8) C A B E D (6) D E B A C (5) C A E D B (4) A D B E C (4) A C D E B (4) D A E C B (3) D A E B C (3) C A E B D (3) C A D E B (3) A D C E B (3) A C D B E (3) A C B D E (3) E D B C A (2) E B D C A (2) D E A B C (2) D B E A C (2) C E B D A (2) C E A B D (2) B C E A D (2) B A D E C (2) A D B C E (2) E D C B A (1) D E C B A (1) D E C A B (1) D E A C B (1) D B A E C (1) D A C E B (1) C E D B A (1) C E D A B (1) C E B A D (1) C E A D B (1) C D E A B (1) C B E A D (1) C B A E D (1) B E D A C (1) B C A E D (1) B A E D C (1) A D C B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -12 0 2 B -8 0 2 -2 4 C 12 -2 0 4 -4 D 0 2 -4 0 -6 E -2 -4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.545455 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.438016528868 Cumulative probabilities = A: 0.090909 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 0 2 B -8 0 2 -2 4 C 12 -2 0 4 -4 D 0 2 -4 0 -6 E -2 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.545455 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.438016528414 Cumulative probabilities = A: 0.090909 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=26 A=22 D=20 E=5 so E is eliminated. Round 2 votes counts: B=28 C=27 D=23 A=22 so A is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:205 E:202 A:199 B:198 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 0 2 B -8 0 2 -2 4 C 12 -2 0 4 -4 D 0 2 -4 0 -6 E -2 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.545455 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.438016528414 Cumulative probabilities = A: 0.090909 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 0 2 B -8 0 2 -2 4 C 12 -2 0 4 -4 D 0 2 -4 0 -6 E -2 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.545455 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.438016528414 Cumulative probabilities = A: 0.090909 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 0 2 B -8 0 2 -2 4 C 12 -2 0 4 -4 D 0 2 -4 0 -6 E -2 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.545455 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.438016528414 Cumulative probabilities = A: 0.090909 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1035: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) D A B E C (7) A B E D C (6) A E B C D (5) C E D B A (4) B E A C D (4) B A E C D (4) A B D E C (4) E C B A D (3) D C B A E (3) D C A E B (3) D B A C E (3) D A C E B (3) C E D A B (3) C D E B A (3) B A E D C (3) A D B E C (3) E C A B D (2) E A C B D (2) D C E B A (2) D C B E A (2) D B A E C (2) D A B C E (2) C E A B D (2) B D A E C (2) B A D E C (2) E B C A D (1) E B A C D (1) E A B C D (1) D C E A B (1) D C A B E (1) D B C A E (1) D A E B C (1) C E A D B (1) C D E A B (1) C B D E A (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 14 10 14 B 0 0 8 8 2 C -14 -8 0 -4 -10 D -10 -8 4 0 -6 E -14 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.406214 B: 0.593786 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.517591731439 Cumulative probabilities = A: 0.406214 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 10 14 B 0 0 8 8 2 C -14 -8 0 -4 -10 D -10 -8 4 0 -6 E -14 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=24 A=20 B=15 E=10 so E is eliminated. Round 2 votes counts: D=31 C=29 A=23 B=17 so B is eliminated. Round 3 votes counts: A=37 D=33 C=30 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:209 E:200 D:190 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 10 14 B 0 0 8 8 2 C -14 -8 0 -4 -10 D -10 -8 4 0 -6 E -14 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 10 14 B 0 0 8 8 2 C -14 -8 0 -4 -10 D -10 -8 4 0 -6 E -14 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 10 14 B 0 0 8 8 2 C -14 -8 0 -4 -10 D -10 -8 4 0 -6 E -14 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1036: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) E B D C A (9) B E D A C (9) B E D C A (6) C D A E B (5) B E A D C (5) A C D E B (5) A C D B E (5) E D B C A (4) D E C B A (4) A B C E D (4) E C D B A (3) D E B C A (3) E C B D A (2) D C A E B (2) C D E A B (2) B E C A D (2) B E A C D (2) B A E D C (2) A C B D E (2) E B C D A (1) D C E A B (1) D A C E B (1) D A B E C (1) C E D B A (1) C A E D B (1) B A E C D (1) A D C E B (1) A D C B E (1) A C B E D (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -12 -8 -8 B 8 0 6 0 -10 C 12 -6 0 -2 -12 D 8 0 2 0 -8 E 8 10 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 -8 -8 B 8 0 6 0 -10 C 12 -6 0 -2 -12 D 8 0 2 0 -8 E 8 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=23 E=19 C=19 D=12 so D is eliminated. Round 2 votes counts: B=27 E=26 A=25 C=22 so C is eliminated. Round 3 votes counts: A=43 E=30 B=27 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:202 D:201 C:196 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 -8 -8 B 8 0 6 0 -10 C 12 -6 0 -2 -12 D 8 0 2 0 -8 E 8 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -8 -8 B 8 0 6 0 -10 C 12 -6 0 -2 -12 D 8 0 2 0 -8 E 8 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -8 -8 B 8 0 6 0 -10 C 12 -6 0 -2 -12 D 8 0 2 0 -8 E 8 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1037: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E B A C D (8) E B D A C (7) E B A D C (6) C D A B E (6) D E C A B (5) D C A E B (5) B E A C D (5) B C A D E (5) E D A B C (4) C A D B E (4) B E C A D (4) E A D C B (3) D C A B E (3) E D B A C (2) D E A C B (2) C B A D E (2) C A B D E (2) B A E C D (2) E D C A B (1) E A C B D (1) E A B D C (1) D B E C A (1) D A C E B (1) C B D A E (1) B E C D A (1) B C E A D (1) B C D A E (1) B C A E D (1) B A C E D (1) A E B C D (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 14 0 -24 B -4 0 2 0 -18 C -14 -2 0 -4 -28 D 0 0 4 0 -18 E 24 18 28 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 14 0 -24 B -4 0 2 0 -18 C -14 -2 0 -4 -28 D 0 0 4 0 -18 E 24 18 28 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 B=21 D=17 C=15 A=4 so A is eliminated. Round 2 votes counts: E=44 B=21 D=18 C=17 so C is eliminated. Round 3 votes counts: E=44 D=30 B=26 so B is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:244 A:197 D:193 B:190 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 14 0 -24 B -4 0 2 0 -18 C -14 -2 0 -4 -28 D 0 0 4 0 -18 E 24 18 28 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 0 -24 B -4 0 2 0 -18 C -14 -2 0 -4 -28 D 0 0 4 0 -18 E 24 18 28 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 0 -24 B -4 0 2 0 -18 C -14 -2 0 -4 -28 D 0 0 4 0 -18 E 24 18 28 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1038: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) C E D A B (9) A B D C E (7) E C D B A (4) D E C B A (4) D B E C A (4) C A D E B (4) B A D E C (4) C E A D B (3) C D E A B (3) B D E C A (3) B D A E C (3) A B C E D (3) E D C B A (2) E D B C A (2) E B D C A (2) D E B C A (2) D C E A B (2) C A E D B (2) B D E A C (2) B A E D C (2) A C E D B (2) A C B E D (2) E C D A B (1) E C B A D (1) E C A D B (1) E B C D A (1) E A C B D (1) E A B C D (1) D C E B A (1) D B E A C (1) D A C B E (1) B E D C A (1) B E D A C (1) B E A D C (1) B E A C D (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 16 -4 2 -8 B -16 0 -14 -2 -18 C 4 14 0 6 8 D -2 2 -6 0 -10 E 8 18 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 2 -8 B -16 0 -14 -2 -18 C 4 14 0 6 8 D -2 2 -6 0 -10 E 8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=21 B=18 E=16 D=15 so D is eliminated. Round 2 votes counts: A=31 C=24 B=23 E=22 so E is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:214 A:203 D:192 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 2 -8 B -16 0 -14 -2 -18 C 4 14 0 6 8 D -2 2 -6 0 -10 E 8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 2 -8 B -16 0 -14 -2 -18 C 4 14 0 6 8 D -2 2 -6 0 -10 E 8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 2 -8 B -16 0 -14 -2 -18 C 4 14 0 6 8 D -2 2 -6 0 -10 E 8 18 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1039: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) A B D E C (10) E C D B A (9) B D A C E (9) E C A D B (8) E A C B D (7) A E B D C (7) A B D C E (7) A E C B D (4) E A C D B (3) D B C E A (3) E C D A B (2) D C B E A (2) D B A C E (2) C D B E A (2) B A D C E (2) A C E B D (2) E A B D C (1) D B C A E (1) C E D A B (1) C D B A E (1) B D C A E (1) B D A E C (1) A E B C D (1) Total count = 100 A B C D E A 0 6 12 4 -4 B -6 0 -10 4 -18 C -12 10 0 8 -6 D -4 -4 -8 0 -18 E 4 18 6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 12 4 -4 B -6 0 -10 4 -18 C -12 10 0 8 -6 D -4 -4 -8 0 -18 E 4 18 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=30 C=18 B=13 D=8 so D is eliminated. Round 2 votes counts: A=31 E=30 C=20 B=19 so B is eliminated. Round 3 votes counts: A=45 E=30 C=25 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:209 C:200 B:185 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 4 -4 B -6 0 -10 4 -18 C -12 10 0 8 -6 D -4 -4 -8 0 -18 E 4 18 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 4 -4 B -6 0 -10 4 -18 C -12 10 0 8 -6 D -4 -4 -8 0 -18 E 4 18 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 4 -4 B -6 0 -10 4 -18 C -12 10 0 8 -6 D -4 -4 -8 0 -18 E 4 18 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1040: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (17) E D C A B (14) B E A D C (5) B A C E D (5) A B C D E (5) E D C B A (3) E B D A C (3) D C E A B (3) C D A E B (3) B D C A E (3) A E B C D (3) A B C E D (3) E C D A B (2) E A C B D (2) E A B C D (2) C D E A B (2) C D A B E (2) B A E D C (2) B A E C D (2) B A D E C (2) A C D B E (2) A C B D E (2) E D B C A (1) E A C D B (1) D E C B A (1) D E C A B (1) D C E B A (1) D C B A E (1) D B C A E (1) C B D A E (1) C A D B E (1) B E A C D (1) B A D C E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 20 16 16 B 0 0 12 20 12 C -20 -12 0 16 10 D -16 -20 -16 0 -2 E -16 -12 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.467137 B: 0.532863 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502159993114 Cumulative probabilities = A: 0.467137 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 20 16 16 B 0 0 12 20 12 C -20 -12 0 16 10 D -16 -20 -16 0 -2 E -16 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=28 A=17 C=9 D=8 so D is eliminated. Round 2 votes counts: B=39 E=30 A=17 C=14 so C is eliminated. Round 3 votes counts: B=41 E=36 A=23 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:226 B:222 C:197 E:182 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 20 16 16 B 0 0 12 20 12 C -20 -12 0 16 10 D -16 -20 -16 0 -2 E -16 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 20 16 16 B 0 0 12 20 12 C -20 -12 0 16 10 D -16 -20 -16 0 -2 E -16 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 20 16 16 B 0 0 12 20 12 C -20 -12 0 16 10 D -16 -20 -16 0 -2 E -16 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1041: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) C B A D E (7) A C B E D (7) E D A B C (6) D E B C A (5) D B C E A (5) E A B C D (4) D B E C A (4) C A B D E (4) B C A D E (4) A E D C B (4) E A D B C (3) B C D E A (3) E D B A C (2) E D A C B (2) E B D C A (2) E A D C B (2) E A B D C (2) D E A B C (2) D C B A E (2) C B D A E (2) B C D A E (2) A E C B D (2) A C B D E (2) E B A C D (1) D E C A B (1) D E A C B (1) D C E B A (1) D C B E A (1) C D A B E (1) C A B E D (1) B E A C D (1) B C A E D (1) B A E C D (1) B A C E D (1) A E C D B (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -8 0 -12 B 6 0 18 14 16 C 8 -18 0 -6 4 D 0 -14 6 0 10 E 12 -16 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 0 -12 B 6 0 18 14 16 C 8 -18 0 -6 4 D 0 -14 6 0 10 E 12 -16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 B=21 A=18 C=15 so C is eliminated. Round 2 votes counts: B=30 E=24 D=23 A=23 so D is eliminated. Round 3 votes counts: B=42 E=34 A=24 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 D:201 C:194 E:191 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 0 -12 B 6 0 18 14 16 C 8 -18 0 -6 4 D 0 -14 6 0 10 E 12 -16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 -12 B 6 0 18 14 16 C 8 -18 0 -6 4 D 0 -14 6 0 10 E 12 -16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 -12 B 6 0 18 14 16 C 8 -18 0 -6 4 D 0 -14 6 0 10 E 12 -16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1042: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (6) D A B C E (6) D B A E C (5) C A E B D (5) B E A D C (5) E C B A D (4) C E B A D (4) C E A B D (4) B E D A C (4) E B C A D (3) D C A E B (3) C A E D B (3) C A D E B (3) B E A C D (3) B D A E C (3) A C D E B (3) A B E D C (3) E B A C D (2) D B E C A (2) D B E A C (2) D A C B E (2) D A B E C (2) B D E A C (2) A E C B D (2) E C A B D (1) E B C D A (1) E A B C D (1) D C E B A (1) D C B A E (1) D B C A E (1) C E B D A (1) C E A D B (1) C D E A B (1) B E C D A (1) B E C A D (1) B D E C A (1) B A E D C (1) A E B C D (1) A D B E C (1) A C E D B (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 4 10 10 B -4 0 6 12 8 C -4 -6 0 -4 -6 D -10 -12 4 0 -8 E -10 -8 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 10 10 B -4 0 6 12 8 C -4 -6 0 -4 -6 D -10 -12 4 0 -8 E -10 -8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=22 B=21 A=14 E=12 so E is eliminated. Round 2 votes counts: D=31 C=27 B=27 A=15 so A is eliminated. Round 3 votes counts: C=34 B=34 D=32 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:211 E:198 C:190 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 10 10 B -4 0 6 12 8 C -4 -6 0 -4 -6 D -10 -12 4 0 -8 E -10 -8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 10 10 B -4 0 6 12 8 C -4 -6 0 -4 -6 D -10 -12 4 0 -8 E -10 -8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 10 10 B -4 0 6 12 8 C -4 -6 0 -4 -6 D -10 -12 4 0 -8 E -10 -8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1043: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) B E C A D (10) E C A D B (9) B D A E C (8) D A C E B (7) C E A D B (7) B A D E C (5) D C A E B (3) D B A C E (3) D A B C E (3) B D C E A (3) B C E D A (3) A E C D B (3) A D E C B (3) A D B E C (3) E C B A D (2) E C A B D (2) E A C D B (2) C E D A B (2) B D C A E (2) A B D E C (2) C E D B A (1) C E B D A (1) B D E C A (1) B A E C D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 8 2 14 B 4 0 12 4 12 C -8 -12 0 -12 -2 D -2 -4 12 0 12 E -14 -12 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 2 14 B 4 0 12 4 12 C -8 -12 0 -12 -2 D -2 -4 12 0 12 E -14 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=45 D=16 E=15 A=13 C=11 so C is eliminated. Round 2 votes counts: B=45 E=26 D=16 A=13 so A is eliminated. Round 3 votes counts: B=47 E=30 D=23 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:210 D:209 C:183 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 2 14 B 4 0 12 4 12 C -8 -12 0 -12 -2 D -2 -4 12 0 12 E -14 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 2 14 B 4 0 12 4 12 C -8 -12 0 -12 -2 D -2 -4 12 0 12 E -14 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 2 14 B 4 0 12 4 12 C -8 -12 0 -12 -2 D -2 -4 12 0 12 E -14 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1044: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (16) B C E D A (14) E B C D A (6) B C A E D (6) A D C E B (4) E D C B A (3) D E A C B (3) C B E D A (3) B E C D A (3) A D C B E (3) A C D B E (3) A C B D E (3) D C A E B (2) D A E C B (2) C D B E A (2) B C E A D (2) A D E B C (2) A B C D E (2) E D B C A (1) E D A C B (1) E D A B C (1) E C B D A (1) E B D C A (1) E A D B C (1) E A B D C (1) D E C A B (1) D A E B C (1) C B A E D (1) C A D B E (1) C A B D E (1) B E C A D (1) B A E C D (1) A E D B C (1) A E B D C (1) A D B E C (1) A B E D C (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 2 10 10 B -10 0 0 2 2 C -2 0 0 2 -2 D -10 -2 -2 0 -2 E -10 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 10 10 B -10 0 0 2 2 C -2 0 0 2 -2 D -10 -2 -2 0 -2 E -10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=27 E=16 D=9 C=8 so C is eliminated. Round 2 votes counts: A=42 B=31 E=16 D=11 so D is eliminated. Round 3 votes counts: A=47 B=33 E=20 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:199 B:197 E:196 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 10 10 B -10 0 0 2 2 C -2 0 0 2 -2 D -10 -2 -2 0 -2 E -10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 10 10 B -10 0 0 2 2 C -2 0 0 2 -2 D -10 -2 -2 0 -2 E -10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 10 10 B -10 0 0 2 2 C -2 0 0 2 -2 D -10 -2 -2 0 -2 E -10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1045: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (13) B D C A E (10) B D A C E (9) D A B E C (8) C E A B D (5) A D E B C (5) E C A D B (4) D B A C E (4) C E B D A (4) C B E A D (4) A D B E C (4) E C D A B (3) D E A C B (3) B C A D E (3) E A C D B (2) D B A E C (2) D A E B C (2) C B E D A (2) B C D E A (2) B C D A E (2) A B D C E (2) E D C A B (1) E D A C B (1) E C A B D (1) B A D C E (1) A E D C B (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -12 -8 -6 10 B 12 0 8 18 6 C 8 -8 0 -6 22 D 6 -18 6 0 14 E -10 -6 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -6 10 B 12 0 8 18 6 C 8 -8 0 -6 22 D 6 -18 6 0 14 E -10 -6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=27 D=19 A=14 E=12 so E is eliminated. Round 2 votes counts: C=36 B=27 D=21 A=16 so A is eliminated. Round 3 votes counts: C=40 D=31 B=29 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:222 C:208 D:204 A:192 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -6 10 B 12 0 8 18 6 C 8 -8 0 -6 22 D 6 -18 6 0 14 E -10 -6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -6 10 B 12 0 8 18 6 C 8 -8 0 -6 22 D 6 -18 6 0 14 E -10 -6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -6 10 B 12 0 8 18 6 C 8 -8 0 -6 22 D 6 -18 6 0 14 E -10 -6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1046: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (13) A C D E B (11) A C D B E (9) E D B C A (5) C D A E B (5) B C A D E (4) A C B D E (4) E D C A B (3) E B D C A (3) D E C A B (3) B E C D A (3) B C D E A (3) B A C E D (3) B A C D E (3) A E B D C (3) E D C B A (2) E D A C B (2) D C E A B (2) C B D E A (2) A B C D E (2) E A D C B (1) E A B D C (1) D E C B A (1) D C E B A (1) C D B A E (1) C B D A E (1) C A D B E (1) C A B D E (1) B E A D C (1) B C E D A (1) B A E C D (1) A E C B D (1) A D C E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -10 -2 6 B -4 0 -4 4 10 C 10 4 0 16 12 D 2 -4 -16 0 10 E -6 -10 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -2 6 B -4 0 -4 4 10 C 10 4 0 16 12 D 2 -4 -16 0 10 E -6 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=32 E=17 C=11 D=7 so D is eliminated. Round 2 votes counts: A=33 B=32 E=21 C=14 so C is eliminated. Round 3 votes counts: A=40 B=36 E=24 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:221 B:203 A:199 D:196 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 -2 6 B -4 0 -4 4 10 C 10 4 0 16 12 D 2 -4 -16 0 10 E -6 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -2 6 B -4 0 -4 4 10 C 10 4 0 16 12 D 2 -4 -16 0 10 E -6 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -2 6 B -4 0 -4 4 10 C 10 4 0 16 12 D 2 -4 -16 0 10 E -6 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1047: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) E C B A D (10) C E D A B (10) E C D A B (7) E B C A D (6) D C A E B (6) B E A C D (6) E C B D A (4) B A D C E (4) A D B C E (4) D A B C E (3) C D E A B (3) C D A E B (3) B E C D A (3) B A E D C (3) E A B C D (2) D A C B E (2) E C D B A (1) E C A D B (1) E B A C D (1) D A C E B (1) C E D B A (1) C D E B A (1) B E C A D (1) B E A D C (1) B D A C E (1) A E B D C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -14 8 -16 B 10 0 -2 12 -18 C 14 2 0 20 -20 D -8 -12 -20 0 -16 E 16 18 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -14 8 -16 B 10 0 -2 12 -18 C 14 2 0 20 -20 D -8 -12 -20 0 -16 E 16 18 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=31 C=18 D=12 A=7 so A is eliminated. Round 2 votes counts: E=33 B=32 C=18 D=17 so D is eliminated. Round 3 votes counts: B=39 E=33 C=28 so C is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:235 C:208 B:201 A:184 D:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -14 8 -16 B 10 0 -2 12 -18 C 14 2 0 20 -20 D -8 -12 -20 0 -16 E 16 18 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 8 -16 B 10 0 -2 12 -18 C 14 2 0 20 -20 D -8 -12 -20 0 -16 E 16 18 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 8 -16 B 10 0 -2 12 -18 C 14 2 0 20 -20 D -8 -12 -20 0 -16 E 16 18 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1048: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (14) E B D C A (8) B D E A C (7) B D A C E (6) C E A D B (5) C A B D E (4) B E D A C (4) A D C B E (4) E C B D A (3) E C A D B (3) E B C D A (3) C A E B D (3) E D B A C (2) E C B A D (2) E C A B D (2) D B A E C (2) B E D C A (2) B D A E C (2) B C A D E (2) A D C E B (2) A C D B E (2) E D C A B (1) E D A C B (1) D E A B C (1) D B E A C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) C A D E B (1) C A B E D (1) B C A E D (1) B A C D E (1) A E C D B (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -18 10 4 B -2 0 -10 10 -12 C 18 10 0 8 8 D -10 -10 -8 0 -22 E -4 12 -8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 10 4 B -2 0 -10 10 -12 C 18 10 0 8 8 D -10 -10 -8 0 -22 E -4 12 -8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=25 B=25 A=12 D=6 so D is eliminated. Round 2 votes counts: C=32 B=28 E=26 A=14 so A is eliminated. Round 3 votes counts: C=42 B=31 E=27 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:211 A:199 B:193 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -18 10 4 B -2 0 -10 10 -12 C 18 10 0 8 8 D -10 -10 -8 0 -22 E -4 12 -8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 10 4 B -2 0 -10 10 -12 C 18 10 0 8 8 D -10 -10 -8 0 -22 E -4 12 -8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 10 4 B -2 0 -10 10 -12 C 18 10 0 8 8 D -10 -10 -8 0 -22 E -4 12 -8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1049: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (14) B A D C E (10) E C B D A (6) E C B A D (6) D A B E C (5) A D B C E (5) D A E C B (4) D A B C E (4) B C A D E (4) E B D A C (3) D A C B E (3) B A C D E (3) E D A C B (2) E D A B C (2) E C D B A (2) D A E B C (2) D A C E B (2) C E D A B (2) C B E A D (2) C A B D E (2) B E C A D (2) B A E C D (2) B A D E C (2) E D B C A (1) E D B A C (1) D E A C B (1) D E A B C (1) C E B A D (1) C E A D B (1) C D A E B (1) C B A D E (1) B C E A D (1) B C A E D (1) B A E D C (1) Total count = 100 A B C D E A 0 2 6 -12 4 B -2 0 0 -6 -4 C -6 0 0 2 -14 D 12 6 -2 0 0 E -4 4 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.522146 E: 0.477854 Sum of squares = 0.500980890897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.522146 E: 1.000000 A B C D E A 0 2 6 -12 4 B -2 0 0 -6 -4 C -6 0 0 2 -14 D 12 6 -2 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=26 D=22 C=10 A=5 so A is eliminated. Round 2 votes counts: E=37 D=27 B=26 C=10 so C is eliminated. Round 3 votes counts: E=41 B=31 D=28 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:208 E:207 A:200 B:194 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -12 4 B -2 0 0 -6 -4 C -6 0 0 2 -14 D 12 6 -2 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -12 4 B -2 0 0 -6 -4 C -6 0 0 2 -14 D 12 6 -2 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -12 4 B -2 0 0 -6 -4 C -6 0 0 2 -14 D 12 6 -2 0 0 E -4 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1050: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) C B A E D (7) D E B A C (6) D E A B C (6) B C E D A (5) B C A D E (5) A E D C B (5) A C B D E (5) B D E C A (4) E C B D A (3) C B A D E (3) C A B E D (3) B C D A E (3) A D E C B (3) A C E D B (3) A C D E B (3) E D B A C (2) E D A B C (2) E A D C B (2) D A E C B (2) C E A B D (2) B C D E A (2) E D B C A (1) D E B C A (1) D E A C B (1) D A E B C (1) C A E D B (1) C A E B D (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A C E (1) B C A E D (1) B A C D E (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 12 -4 4 B -4 0 -8 0 -8 C -12 8 0 0 2 D 4 0 0 0 6 E -4 8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.194527 D: 0.805473 E: 0.000000 Sum of squares = 0.68662689507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.194527 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 -4 4 B -4 0 -8 0 -8 C -12 8 0 0 2 D 4 0 0 0 6 E -4 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000019366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=22 E=19 D=17 C=17 so D is eliminated. Round 2 votes counts: E=33 B=25 A=25 C=17 so C is eliminated. Round 3 votes counts: E=35 B=35 A=30 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:208 D:205 C:199 E:198 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 12 -4 4 B -4 0 -8 0 -8 C -12 8 0 0 2 D 4 0 0 0 6 E -4 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000019366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 -4 4 B -4 0 -8 0 -8 C -12 8 0 0 2 D 4 0 0 0 6 E -4 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000019366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 -4 4 B -4 0 -8 0 -8 C -12 8 0 0 2 D 4 0 0 0 6 E -4 8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000019366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1051: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) A B D E C (8) D C E B A (7) C E D B A (7) D A B C E (6) B A D E C (6) A B E C D (6) D B A E C (5) E C D B A (4) E C A B D (4) B A E C D (4) A B D C E (4) E C B A D (3) D E C B A (3) B E C D A (3) E C B D A (2) D C E A B (2) D B E C A (2) D A C E B (2) B D A E C (2) A E C B D (2) A B C E D (2) C E B D A (1) C E A D B (1) C D E A B (1) B E D C A (1) B D E C A (1) A D B C E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 0 -14 0 B 2 0 2 0 2 C 0 -2 0 0 -12 D 14 0 0 0 0 E 0 -2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.808663 C: 0.000000 D: 0.191337 E: 0.000000 Sum of squares = 0.690545969721 Cumulative probabilities = A: 0.000000 B: 0.808663 C: 0.808663 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -14 0 B 2 0 2 0 2 C 0 -2 0 0 -12 D 14 0 0 0 0 E 0 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999678 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 C=18 B=17 E=13 so E is eliminated. Round 2 votes counts: C=31 D=27 A=25 B=17 so B is eliminated. Round 3 votes counts: A=35 C=34 D=31 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:207 E:205 B:203 C:193 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -14 0 B 2 0 2 0 2 C 0 -2 0 0 -12 D 14 0 0 0 0 E 0 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999678 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -14 0 B 2 0 2 0 2 C 0 -2 0 0 -12 D 14 0 0 0 0 E 0 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999678 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -14 0 B 2 0 2 0 2 C 0 -2 0 0 -12 D 14 0 0 0 0 E 0 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999678 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1052: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) C B E A D (5) A D B E C (5) D A E C B (4) D A C B E (4) C D B A E (4) C D A B E (4) E D A C B (3) E C D B A (3) E C B A D (3) E B A D C (3) E A B D C (3) D A E B C (3) B A C D E (3) E C B D A (2) E B C A D (2) D E A C B (2) D C A E B (2) D A B C E (2) C E B D A (2) C D E A B (2) C B E D A (2) C B A D E (2) B A E D C (2) B A D C E (2) A D E B C (2) A B D E C (2) A B D C E (2) E D C B A (1) E C D A B (1) E B D A C (1) E B C D A (1) E B A C D (1) E A D B C (1) D E A B C (1) D C E A B (1) D A C E B (1) D A B E C (1) C E D B A (1) C E B A D (1) C D B E A (1) C D A E B (1) C B D A E (1) B E C A D (1) B E A D C (1) B E A C D (1) B C A D E (1) Total count = 100 A B C D E A 0 6 12 -14 -4 B -6 0 -6 -12 -8 C -12 6 0 -10 -12 D 14 12 10 0 6 E 4 8 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 -14 -4 B -6 0 -6 -12 -8 C -12 6 0 -10 -12 D 14 12 10 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=26 D=21 B=11 A=11 so B is eliminated. Round 2 votes counts: E=34 C=27 D=21 A=18 so A is eliminated. Round 3 votes counts: E=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:209 A:200 C:186 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 12 -14 -4 B -6 0 -6 -12 -8 C -12 6 0 -10 -12 D 14 12 10 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 -14 -4 B -6 0 -6 -12 -8 C -12 6 0 -10 -12 D 14 12 10 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 -14 -4 B -6 0 -6 -12 -8 C -12 6 0 -10 -12 D 14 12 10 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1053: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) A E C D B (8) E A B D C (7) D C B A E (7) E A B C D (6) D B C E A (5) A E D C B (5) E A C B D (4) C A D B E (4) B D C E A (4) A C D E B (4) D B C A E (3) C D B A E (3) B D E C A (3) B C D E A (3) A E D B C (3) E B A D C (2) E A D B C (2) D C A B E (2) D B E A C (2) D A C B E (2) B E C D A (2) D E B A C (1) D E A B C (1) D B E C A (1) C B D A E (1) C A B E D (1) B E D C A (1) B C E A D (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 22 18 18 10 B -22 0 -6 -10 -10 C -18 6 0 -4 -14 D -18 10 4 0 -6 E -10 10 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 18 18 10 B -22 0 -6 -10 -10 C -18 6 0 -4 -14 D -18 10 4 0 -6 E -10 10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=24 E=21 B=14 C=9 so C is eliminated. Round 2 votes counts: A=37 D=27 E=21 B=15 so B is eliminated. Round 3 votes counts: D=38 A=37 E=25 so E is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:210 D:195 C:185 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 18 18 10 B -22 0 -6 -10 -10 C -18 6 0 -4 -14 D -18 10 4 0 -6 E -10 10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 18 10 B -22 0 -6 -10 -10 C -18 6 0 -4 -14 D -18 10 4 0 -6 E -10 10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 18 10 B -22 0 -6 -10 -10 C -18 6 0 -4 -14 D -18 10 4 0 -6 E -10 10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1054: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (16) A D C B E (13) B E C D A (7) E A B D C (6) A E D C B (6) C D B E A (5) C D B A E (4) B C E D A (4) E B A D C (3) E B A C D (3) D C A B E (3) C D A B E (3) B C D E A (3) A D E C B (3) A D C E B (3) E B D C A (2) D C E B A (2) D C A E B (2) D A C E B (2) A B E C D (2) A B D C E (2) E D C A B (1) E B C A D (1) E A D C B (1) D E C A B (1) D C E A B (1) B A E C D (1) Total count = 100 A B C D E A 0 -2 -10 -12 -12 B 2 0 0 0 -6 C 10 0 0 -2 -6 D 12 0 2 0 -6 E 12 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -10 -12 -12 B 2 0 0 0 -6 C 10 0 0 -2 -6 D 12 0 2 0 -6 E 12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=29 B=15 C=12 D=11 so D is eliminated. Round 2 votes counts: E=34 A=31 C=20 B=15 so B is eliminated. Round 3 votes counts: E=41 A=32 C=27 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:204 C:201 B:198 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -10 -12 -12 B 2 0 0 0 -6 C 10 0 0 -2 -6 D 12 0 2 0 -6 E 12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -12 -12 B 2 0 0 0 -6 C 10 0 0 -2 -6 D 12 0 2 0 -6 E 12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -12 -12 B 2 0 0 0 -6 C 10 0 0 -2 -6 D 12 0 2 0 -6 E 12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1055: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) B E A D C (6) E A D B C (5) C A E B D (5) B D E A C (5) B C D E A (5) E A B D C (4) D B E A C (4) D A E C B (4) C B D A E (4) B E D A C (4) B D C E A (4) D A E B C (3) A E C D B (3) E A D C B (2) E A C D B (2) E A C B D (2) D E B A C (2) D E A B C (2) D C A E B (2) C B E A D (2) C B A E D (2) C A D E B (2) B E A C D (2) B C E D A (2) A E D C B (2) D B C A E (1) D B A E C (1) D A C E B (1) C B A D E (1) B D E C A (1) B D C A E (1) B C D A E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 12 6 -8 B -4 0 6 2 -8 C -12 -6 0 -8 -8 D -6 -2 8 0 -10 E 8 8 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 12 6 -8 B -4 0 6 2 -8 C -12 -6 0 -8 -8 D -6 -2 8 0 -10 E 8 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=27 D=20 E=15 A=7 so A is eliminated. Round 2 votes counts: B=31 C=29 E=20 D=20 so E is eliminated. Round 3 votes counts: C=36 B=35 D=29 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:217 A:207 B:198 D:195 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 12 6 -8 B -4 0 6 2 -8 C -12 -6 0 -8 -8 D -6 -2 8 0 -10 E 8 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 6 -8 B -4 0 6 2 -8 C -12 -6 0 -8 -8 D -6 -2 8 0 -10 E 8 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 6 -8 B -4 0 6 2 -8 C -12 -6 0 -8 -8 D -6 -2 8 0 -10 E 8 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1056: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) E C B A D (6) D A B C E (6) C B E A D (6) E C D B A (5) D A E B C (4) B A C D E (4) E D A B C (3) D E C A B (3) D A B E C (3) B C A D E (3) A D B E C (3) E C D A B (2) E A D B C (2) E A C D B (2) E A B C D (2) D C E B A (2) D C B A E (2) C E B D A (2) C E B A D (2) C B D E A (2) B D C A E (2) B C A E D (2) A E D B C (2) A B D C E (2) E D C B A (1) E D C A B (1) E C A B D (1) E B A C D (1) D E C B A (1) D E A C B (1) D E A B C (1) D C A B E (1) D B C A E (1) D B A C E (1) C E D B A (1) C D B E A (1) C B E D A (1) C B A E D (1) C B A D E (1) B C D A E (1) B A C E D (1) A E B C D (1) A D E B C (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -2 -8 -10 B 0 0 -2 -12 -6 C 2 2 0 0 -8 D 8 12 0 0 -6 E 10 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 -8 -10 B 0 0 -2 -12 -6 C 2 2 0 0 -8 D 8 12 0 0 -6 E 10 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=26 C=17 B=13 A=12 so A is eliminated. Round 2 votes counts: E=35 D=30 B=18 C=17 so C is eliminated. Round 3 votes counts: E=40 D=31 B=29 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:207 C:198 A:190 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 -8 -10 B 0 0 -2 -12 -6 C 2 2 0 0 -8 D 8 12 0 0 -6 E 10 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -8 -10 B 0 0 -2 -12 -6 C 2 2 0 0 -8 D 8 12 0 0 -6 E 10 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -8 -10 B 0 0 -2 -12 -6 C 2 2 0 0 -8 D 8 12 0 0 -6 E 10 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1057: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) E B D A C (9) C A D B E (8) D B E C A (6) D B E A C (5) A C B D E (5) B E D A C (4) D C B A E (3) C D A B E (3) C A E D B (3) E C A D B (2) E C A B D (2) E A C B D (2) D C E B A (2) C E D A B (2) C D E A B (2) B D A E C (2) B A E D C (2) B A D E C (2) A C B E D (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B A C (1) E C D A B (1) E A B C D (1) D E B A C (1) D C B E A (1) D B C E A (1) D B A C E (1) D A B C E (1) C E A D B (1) C A E B D (1) B D E A C (1) B D A C E (1) A E C B D (1) A E B C D (1) A C E B D (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 0 -18 -12 B 8 0 12 -14 10 C 0 -12 0 -12 -12 D 18 14 12 0 -2 E 12 -10 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.000000 D: 0.384615 E: 0.538462 Sum of squares = 0.443786982247 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.076923 D: 0.461538 E: 1.000000 A B C D E A 0 -8 0 -18 -12 B 8 0 12 -14 10 C 0 -12 0 -12 -12 D 18 14 12 0 -2 E 12 -10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.000000 D: 0.384615 E: 0.538462 Sum of squares = 0.443786982107 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.076923 D: 0.461538 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=21 C=20 A=17 B=12 so B is eliminated. Round 2 votes counts: E=34 D=25 A=21 C=20 so C is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:221 B:208 E:208 C:182 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -18 -12 B 8 0 12 -14 10 C 0 -12 0 -12 -12 D 18 14 12 0 -2 E 12 -10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.000000 D: 0.384615 E: 0.538462 Sum of squares = 0.443786982107 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.076923 D: 0.461538 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -18 -12 B 8 0 12 -14 10 C 0 -12 0 -12 -12 D 18 14 12 0 -2 E 12 -10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.000000 D: 0.384615 E: 0.538462 Sum of squares = 0.443786982107 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.076923 D: 0.461538 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -18 -12 B 8 0 12 -14 10 C 0 -12 0 -12 -12 D 18 14 12 0 -2 E 12 -10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.000000 D: 0.384615 E: 0.538462 Sum of squares = 0.443786982107 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.076923 D: 0.461538 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1058: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (22) D A E B C (14) D C A B E (10) E B A C D (7) A E B D C (7) A D E B C (7) D A C E B (6) B E A C D (6) C B E D A (5) B E C A D (4) D A E C B (2) C D B E A (2) B C E A D (2) E A B C D (1) D C E B A (1) D C B E A (1) D C A E B (1) C B D E A (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 2 14 -4 B 2 0 -2 12 6 C -2 2 0 2 2 D -14 -12 -2 0 -10 E 4 -6 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 14 -4 B 2 0 -2 12 6 C -2 2 0 2 2 D -14 -12 -2 0 -10 E 4 -6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=30 A=15 B=12 E=8 so E is eliminated. Round 2 votes counts: D=35 C=30 B=19 A=16 so A is eliminated. Round 3 votes counts: D=42 C=30 B=28 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:209 A:205 E:203 C:202 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 14 -4 B 2 0 -2 12 6 C -2 2 0 2 2 D -14 -12 -2 0 -10 E 4 -6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 14 -4 B 2 0 -2 12 6 C -2 2 0 2 2 D -14 -12 -2 0 -10 E 4 -6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 14 -4 B 2 0 -2 12 6 C -2 2 0 2 2 D -14 -12 -2 0 -10 E 4 -6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1059: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) D C E A B (6) C D E A B (6) D E C A B (5) C B E A D (5) B A C E D (5) E D C B A (4) E C D B A (4) C B A D E (4) B A E C D (4) A B D E C (4) E D C A B (3) E B C A D (3) D A B E C (3) D A B C E (3) C E D B A (3) C D A B E (3) B E A C D (3) A B D C E (3) A B C D E (3) D E A B C (2) C E B A D (2) C D E B A (2) C A B D E (2) E C B A D (1) E B D C A (1) E B A D C (1) E B A C D (1) D C A E B (1) D A C B E (1) C B A E D (1) B A D C E (1) B A C D E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -12 6 -4 B 6 0 -6 6 10 C 12 6 0 6 4 D -6 -6 -6 0 2 E 4 -10 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 6 -4 B 6 0 -6 6 10 C 12 6 0 6 4 D -6 -6 -6 0 2 E 4 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=21 B=21 E=18 A=12 so A is eliminated. Round 2 votes counts: B=32 C=28 D=22 E=18 so E is eliminated. Round 3 votes counts: B=38 C=33 D=29 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:208 E:194 A:192 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 6 -4 B 6 0 -6 6 10 C 12 6 0 6 4 D -6 -6 -6 0 2 E 4 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 6 -4 B 6 0 -6 6 10 C 12 6 0 6 4 D -6 -6 -6 0 2 E 4 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 6 -4 B 6 0 -6 6 10 C 12 6 0 6 4 D -6 -6 -6 0 2 E 4 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1060: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) D E A C B (6) E D C B A (5) E D A B C (5) C B A D E (5) E D B A C (4) D A E C B (4) B C A E D (4) A D C B E (4) A B C E D (4) E C B D A (3) E B A C D (3) D E A B C (3) D A E B C (3) D A C B E (3) C B A E D (3) C D B A E (2) C B E A D (2) B C E A D (2) A D B C E (2) A C B D E (2) E D B C A (1) E B D C A (1) E A D B C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A E B (1) D C A B E (1) D A C E B (1) C E B D A (1) C D A B E (1) C B D A E (1) C A D B E (1) B E C A D (1) B E A C D (1) B A C E D (1) A E D B C (1) A D E B C (1) A D B E C (1) A C D B E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 8 0 2 B 0 0 -2 -12 -8 C -8 2 0 -6 -8 D 0 12 6 0 -4 E -2 8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.785110 B: 0.000000 C: 0.000000 D: 0.214890 E: 0.000000 Sum of squares = 0.662575439661 Cumulative probabilities = A: 0.785110 B: 0.785110 C: 0.785110 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 0 2 B 0 0 -2 -12 -8 C -8 2 0 -6 -8 D 0 12 6 0 -4 E -2 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555761532 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 A=19 C=16 B=9 so B is eliminated. Round 2 votes counts: E=32 D=26 C=22 A=20 so A is eliminated. Round 3 votes counts: E=35 D=34 C=31 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:209 D:207 A:205 C:190 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 0 2 B 0 0 -2 -12 -8 C -8 2 0 -6 -8 D 0 12 6 0 -4 E -2 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555761532 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 0 2 B 0 0 -2 -12 -8 C -8 2 0 -6 -8 D 0 12 6 0 -4 E -2 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555761532 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 0 2 B 0 0 -2 -12 -8 C -8 2 0 -6 -8 D 0 12 6 0 -4 E -2 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555761532 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1061: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (6) D C E A B (5) C A B D E (4) B D A C E (4) B A E C D (4) A C B E D (4) E D A C B (3) E A C D B (3) D C B A E (3) C D A E B (3) C B A D E (3) B E A C D (3) B D E A C (3) B D C A E (3) A B C E D (3) E D B A C (2) E A C B D (2) D E C A B (2) D E B C A (2) D B E C A (2) D B C E A (2) C E A D B (2) C A E D B (2) C A D E B (2) B E A D C (2) B D E C A (2) A C E D B (2) A C E B D (2) E D A B C (1) E C D A B (1) E C A D B (1) E A D C B (1) D E C B A (1) D C E B A (1) D C B E A (1) D C A E B (1) D C A B E (1) C B D A E (1) C A D B E (1) B E D A C (1) B D A E C (1) B C D A E (1) B A D E C (1) B A C D E (1) A E C D B (1) A E C B D (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 6 14 B 0 0 -8 8 16 C -6 8 0 10 18 D -6 -8 -10 0 4 E -14 -16 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.763961 B: 0.236039 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.639351237211 Cumulative probabilities = A: 0.763961 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 6 14 B 0 0 -8 8 16 C -6 8 0 10 18 D -6 -8 -10 0 4 E -14 -16 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571431 B: 0.428569 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204786348 Cumulative probabilities = A: 0.571431 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=21 C=18 A=15 E=14 so E is eliminated. Round 2 votes counts: B=32 D=27 A=21 C=20 so C is eliminated. Round 3 votes counts: B=36 A=33 D=31 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:213 B:208 D:190 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 6 14 B 0 0 -8 8 16 C -6 8 0 10 18 D -6 -8 -10 0 4 E -14 -16 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571431 B: 0.428569 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204786348 Cumulative probabilities = A: 0.571431 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 14 B 0 0 -8 8 16 C -6 8 0 10 18 D -6 -8 -10 0 4 E -14 -16 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571431 B: 0.428569 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204786348 Cumulative probabilities = A: 0.571431 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 14 B 0 0 -8 8 16 C -6 8 0 10 18 D -6 -8 -10 0 4 E -14 -16 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571431 B: 0.428569 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204786348 Cumulative probabilities = A: 0.571431 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1062: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) D C E B A (7) D A C B E (5) B E A C D (5) A B D E C (5) A B C E D (5) B E C A D (4) A B E C D (4) E C B D A (3) D C E A B (3) C D E B A (3) C B E A D (3) A D C B E (3) E D B C A (2) E C D B A (2) E B C A D (2) E B A C D (2) D C A B E (2) C E D B A (2) C E B D A (2) C D B E A (2) C B E D A (2) B C A E D (2) E D B A C (1) E D A B C (1) E B D C A (1) E B A D C (1) D E C B A (1) D E B C A (1) D E A B C (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B C E (1) C D B A E (1) C A B D E (1) B A E C D (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 6 -2 B 0 0 10 10 16 C -6 -10 0 -6 -4 D -6 -10 6 0 -12 E 2 -16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.649915 B: 0.350085 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.54494904152 Cumulative probabilities = A: 0.649915 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 6 -2 B 0 0 10 10 16 C -6 -10 0 -6 -4 D -6 -10 6 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=25 C=16 E=15 B=12 so B is eliminated. Round 2 votes counts: A=33 D=25 E=24 C=18 so C is eliminated. Round 3 votes counts: A=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:218 A:205 E:201 D:189 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 6 -2 B 0 0 10 10 16 C -6 -10 0 -6 -4 D -6 -10 6 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 -2 B 0 0 10 10 16 C -6 -10 0 -6 -4 D -6 -10 6 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 -2 B 0 0 10 10 16 C -6 -10 0 -6 -4 D -6 -10 6 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1063: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (6) C D A E B (5) A E C B D (5) D C A B E (4) D B A C E (4) C E D A B (4) B E D C A (4) B D E A C (4) B A E D C (4) B A D E C (4) A C D E B (4) E C A B D (3) D C E B A (3) D C B E A (3) D C B A E (3) D B C A E (3) C A D E B (3) B E A C D (3) E A C B D (2) E A B C D (2) D B C E A (2) D A B C E (2) C A E D B (2) B E A D C (2) A E B C D (2) A C E D B (2) A B E D C (2) E C D B A (1) E B C D A (1) E B A C D (1) D C A E B (1) D B E C A (1) D B A E C (1) D A C E B (1) C E A D B (1) C D E A B (1) B D E C A (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 8 -10 22 B 2 0 4 -4 12 C -8 -4 0 -16 0 D 10 4 16 0 18 E -22 -12 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -10 22 B 2 0 4 -4 12 C -8 -4 0 -16 0 D 10 4 16 0 18 E -22 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=28 B=28 A=18 C=16 E=10 so E is eliminated. Round 2 votes counts: B=30 D=28 A=22 C=20 so C is eliminated. Round 3 votes counts: D=39 A=31 B=30 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 A:209 B:207 C:186 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 -10 22 B 2 0 4 -4 12 C -8 -4 0 -16 0 D 10 4 16 0 18 E -22 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -10 22 B 2 0 4 -4 12 C -8 -4 0 -16 0 D 10 4 16 0 18 E -22 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -10 22 B 2 0 4 -4 12 C -8 -4 0 -16 0 D 10 4 16 0 18 E -22 -12 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1064: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E C A D B (9) B D E A C (8) D B E A C (6) E B D C A (5) C A E D B (5) E C A B D (4) B D E C A (4) A C E D B (4) A C D B E (4) A C B D E (4) D A B C E (3) C E A D B (3) C A E B D (3) B D A E C (3) A C D E B (3) E D C A B (2) E D B C A (2) E C D B A (2) D E B A C (2) D E A B C (2) C E A B D (2) E C B A D (1) E A C D B (1) D B A C E (1) D A C E B (1) C A B E D (1) C A B D E (1) B E D C A (1) B C E D A (1) B A D C E (1) Total count = 100 A B C D E A 0 4 8 -8 -10 B -4 0 0 0 -2 C -8 0 0 -4 -4 D 8 0 4 0 8 E 10 2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.474959 C: 0.000000 D: 0.525041 E: 0.000000 Sum of squares = 0.501254100897 Cumulative probabilities = A: 0.000000 B: 0.474959 C: 0.474959 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -8 -10 B -4 0 0 0 -2 C -8 0 0 -4 -4 D 8 0 4 0 8 E 10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=26 D=15 C=15 A=15 so D is eliminated. Round 2 votes counts: B=36 E=30 A=19 C=15 so C is eliminated. Round 3 votes counts: B=36 E=35 A=29 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:204 A:197 B:197 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -8 -10 B -4 0 0 0 -2 C -8 0 0 -4 -4 D 8 0 4 0 8 E 10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -8 -10 B -4 0 0 0 -2 C -8 0 0 -4 -4 D 8 0 4 0 8 E 10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -8 -10 B -4 0 0 0 -2 C -8 0 0 -4 -4 D 8 0 4 0 8 E 10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1065: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (11) E D B A C (8) C A B D E (7) D B C A E (6) E A C B D (4) C A D B E (4) A E B C D (4) E D B C A (3) E A B D C (3) D C B E A (3) D C B A E (3) B A D C E (3) E C A D B (2) E B D A C (2) E A D B C (2) D E B C A (2) D E B A C (2) C D E A B (2) C A B E D (2) A C E B D (2) A C B E D (2) A C B D E (2) A B E D C (2) A B C D E (2) E D C B A (1) E D C A B (1) E D A B C (1) E A D C B (1) E A B C D (1) D E C B A (1) D B E C A (1) C D B E A (1) C B D A E (1) C A E B D (1) B E D A C (1) B D A E C (1) B D A C E (1) B C D A E (1) B C A D E (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -8 -6 16 B 6 0 -2 -10 12 C 8 2 0 2 12 D 6 10 -2 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999035 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -6 16 B 6 0 -2 -10 12 C 8 2 0 2 12 D 6 10 -2 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998427 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=29 C=29 D=18 A=16 B=8 so B is eliminated. Round 2 votes counts: C=31 E=30 D=20 A=19 so A is eliminated. Round 3 votes counts: C=40 E=37 D=23 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:212 B:203 A:198 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -6 16 B 6 0 -2 -10 12 C 8 2 0 2 12 D 6 10 -2 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998427 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -6 16 B 6 0 -2 -10 12 C 8 2 0 2 12 D 6 10 -2 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998427 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -6 16 B 6 0 -2 -10 12 C 8 2 0 2 12 D 6 10 -2 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998427 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1066: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) C D E A B (6) C A D B E (6) B A E C D (6) E D B C A (5) E D C B A (4) E B D A C (4) C E D A B (4) C A B D E (4) B E A D C (4) E D B A C (3) E B A D C (3) D C E A B (3) D B A E C (3) B A E D C (3) A C B D E (3) A B C E D (3) E C D B A (2) D E C B A (2) D C A E B (2) D B E A C (2) D A C B E (2) C D A B E (2) B A D C E (2) E D C A B (1) E C D A B (1) E C B A D (1) E B A C D (1) D E B C A (1) D E B A C (1) D C A B E (1) D A B C E (1) C D A E B (1) B E D A C (1) B D A E C (1) B A C E D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 8 -6 2 B 2 0 8 -6 10 C -8 -8 0 2 2 D 6 6 -2 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -6 2 B 2 0 8 -6 10 C -8 -8 0 2 2 D 6 6 -2 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999357 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 B=19 D=18 A=15 so A is eliminated. Round 2 votes counts: B=30 C=27 E=25 D=18 so D is eliminated. Round 3 votes counts: B=36 C=35 E=29 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:207 D:207 A:201 C:194 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 -6 2 B 2 0 8 -6 10 C -8 -8 0 2 2 D 6 6 -2 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999357 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -6 2 B 2 0 8 -6 10 C -8 -8 0 2 2 D 6 6 -2 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999357 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -6 2 B 2 0 8 -6 10 C -8 -8 0 2 2 D 6 6 -2 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999357 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1067: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) D E C B A (9) D E B A C (8) E D C A B (5) C A B E D (5) A C B E D (5) A B C E D (5) E C D A B (4) D E B C A (4) C E D A B (4) A B D E C (4) D B E A C (3) C B A D E (3) B D C A E (3) B C A D E (3) E D A C B (2) E D A B C (2) E C A D B (2) C E A B D (2) B D C E A (2) A C E B D (2) E D C B A (1) D B E C A (1) C E A D B (1) C B D E A (1) C B D A E (1) C B A E D (1) B D E A C (1) B D A E C (1) B D A C E (1) B A D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -4 -6 -4 B 12 0 4 8 8 C 4 -4 0 4 4 D 6 -8 -4 0 18 E 4 -8 -4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -6 -4 B 12 0 4 8 8 C 4 -4 0 4 4 D 6 -8 -4 0 18 E 4 -8 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=24 C=18 A=17 E=16 so E is eliminated. Round 2 votes counts: D=35 C=24 B=24 A=17 so A is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:206 C:204 A:187 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 -4 B 12 0 4 8 8 C 4 -4 0 4 4 D 6 -8 -4 0 18 E 4 -8 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 -4 B 12 0 4 8 8 C 4 -4 0 4 4 D 6 -8 -4 0 18 E 4 -8 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 -4 B 12 0 4 8 8 C 4 -4 0 4 4 D 6 -8 -4 0 18 E 4 -8 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1068: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) C B D E A (8) C B E D A (6) A E D B C (6) D A C B E (5) A E D C B (5) D B C A E (4) D A B C E (4) E B C A D (3) E B A C D (3) E A C B D (3) E A B C D (3) D C A B E (3) D B A C E (3) D A C E B (3) C E B A D (3) E C B A D (2) E A C D B (2) E A B D C (2) D A E B C (2) D A B E C (2) B C E A D (2) B C D E A (2) A D E C B (2) E C A D B (1) E C A B D (1) E A D C B (1) D C B A E (1) C E A B D (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E A D (1) C B D A E (1) B E A C D (1) B D C A E (1) B C D A E (1) Total count = 100 A B C D E A 0 12 12 2 8 B -12 0 -4 -12 -4 C -12 4 0 -6 4 D -2 12 6 0 8 E -8 4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 2 8 B -12 0 -4 -12 -4 C -12 4 0 -6 4 D -2 12 6 0 8 E -8 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=23 A=22 E=21 B=7 so B is eliminated. Round 2 votes counts: D=28 C=28 E=22 A=22 so E is eliminated. Round 3 votes counts: A=37 C=35 D=28 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:212 C:195 E:192 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 2 8 B -12 0 -4 -12 -4 C -12 4 0 -6 4 D -2 12 6 0 8 E -8 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 2 8 B -12 0 -4 -12 -4 C -12 4 0 -6 4 D -2 12 6 0 8 E -8 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 2 8 B -12 0 -4 -12 -4 C -12 4 0 -6 4 D -2 12 6 0 8 E -8 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1069: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E A B D C (8) B D C E A (7) B D C A E (7) E B A D C (6) C D A B E (5) B E A D C (5) E A C B D (4) C D A E B (4) E A B C D (3) D B C A E (3) C A D E B (3) A E D C B (3) A E C D B (3) D C B A E (2) B E D A C (2) B D E C A (2) B D E A C (2) A E B D C (2) A C D E B (2) E C B D A (1) E B A C D (1) E A D C B (1) E A C D B (1) D C A B E (1) D B A C E (1) D A C B E (1) C E B D A (1) C E A D B (1) C A E D B (1) B D A E C (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D B C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 0 -6 6 B 8 0 12 14 6 C 0 -12 0 -16 6 D 6 -14 16 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -6 6 B 8 0 12 14 6 C 0 -12 0 -16 6 D 6 -14 16 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999092 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=25 C=24 A=13 D=8 so D is eliminated. Round 2 votes counts: B=34 C=27 E=25 A=14 so A is eliminated. Round 3 votes counts: B=35 E=34 C=31 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:208 A:196 C:189 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -6 6 B 8 0 12 14 6 C 0 -12 0 -16 6 D 6 -14 16 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999092 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -6 6 B 8 0 12 14 6 C 0 -12 0 -16 6 D 6 -14 16 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999092 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -6 6 B 8 0 12 14 6 C 0 -12 0 -16 6 D 6 -14 16 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999092 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1070: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B C D A E (8) B D C E A (7) B C A D E (7) A E C D B (7) E A C D B (5) D B C E A (4) C A D E B (4) B D C A E (4) A C B E D (4) E A D B C (3) D C B A E (3) C A B D E (3) A C E D B (3) E D A B C (2) D E C A B (2) C D B A E (2) C A D B E (2) B E D A C (2) B D E C A (2) A E C B D (2) E A B D C (1) E A B C D (1) D E B C A (1) D C A E B (1) D C A B E (1) D B C A E (1) C D A B E (1) C B D A E (1) C B A D E (1) B E D C A (1) B E A D C (1) B E A C D (1) B A E C D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -12 14 14 B -6 0 -6 -6 14 C 12 6 0 8 16 D -14 6 -8 0 12 E -14 -14 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 14 14 B -6 0 -6 -6 14 C 12 6 0 8 16 D -14 6 -8 0 12 E -14 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=22 A=17 C=14 D=13 so D is eliminated. Round 2 votes counts: B=39 E=25 C=19 A=17 so A is eliminated. Round 3 votes counts: B=39 E=34 C=27 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:221 A:211 B:198 D:198 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 14 14 B -6 0 -6 -6 14 C 12 6 0 8 16 D -14 6 -8 0 12 E -14 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 14 14 B -6 0 -6 -6 14 C 12 6 0 8 16 D -14 6 -8 0 12 E -14 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 14 14 B -6 0 -6 -6 14 C 12 6 0 8 16 D -14 6 -8 0 12 E -14 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1071: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (14) C B A D E (8) B D A C E (7) C B E D A (6) A D B C E (6) E A D C B (5) E C B D A (4) C E B A D (4) C B D A E (4) A D E B C (4) E C A D B (3) E A D B C (3) D A E B C (3) A D B E C (3) D A B E C (2) C B E A D (2) C A D B E (2) B E C D A (2) B C D A E (2) B A D C E (2) A D C B E (2) E D B C A (1) E C D A B (1) E B D C A (1) E B D A C (1) E B C D A (1) D A B C E (1) C E B D A (1) C B D E A (1) C A E D B (1) B D E A C (1) B C E D A (1) A E D B C (1) Total count = 100 A B C D E A 0 2 10 -8 -4 B -2 0 12 -4 4 C -10 -12 0 -14 0 D 8 4 14 0 -4 E 4 -4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 2 10 -8 -4 B -2 0 12 -4 4 C -10 -12 0 -14 0 D 8 4 14 0 -4 E 4 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=29 A=16 B=15 D=6 so D is eliminated. Round 2 votes counts: E=34 C=29 A=22 B=15 so B is eliminated. Round 3 votes counts: E=37 C=32 A=31 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:211 B:205 E:202 A:200 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 10 -8 -4 B -2 0 12 -4 4 C -10 -12 0 -14 0 D 8 4 14 0 -4 E 4 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -8 -4 B -2 0 12 -4 4 C -10 -12 0 -14 0 D 8 4 14 0 -4 E 4 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -8 -4 B -2 0 12 -4 4 C -10 -12 0 -14 0 D 8 4 14 0 -4 E 4 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1072: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (7) B D E C A (6) D B C E A (5) C E B A D (5) D B A E C (4) C E A B D (4) E A B C D (3) D C B A E (3) D B E A C (3) D B C A E (3) D A C E B (3) B E D A C (3) B E C A D (3) B D E A C (3) E B C A D (2) E B A C D (2) D C A B E (2) D B A C E (2) D A B E C (2) D A B C E (2) C D B E A (2) C D A E B (2) C B E D A (2) C B E A D (2) C A E D B (2) B E A D C (2) A E C D B (2) A D E C B (2) A C E D B (2) E A C B D (1) E A B D C (1) D B E C A (1) D A E C B (1) D A E B C (1) D A C B E (1) C A E B D (1) C A D E B (1) B E C D A (1) B E A C D (1) B C E D A (1) A E B D C (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 4 -6 -6 B 12 0 4 4 8 C -4 -4 0 -4 -4 D 6 -4 4 0 2 E 6 -8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -6 -6 B 12 0 4 4 8 C -4 -4 0 -4 -4 D 6 -4 4 0 2 E 6 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=21 B=20 A=17 E=9 so E is eliminated. Round 2 votes counts: D=33 B=24 A=22 C=21 so C is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:204 E:200 C:192 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 -6 -6 B 12 0 4 4 8 C -4 -4 0 -4 -4 D 6 -4 4 0 2 E 6 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -6 -6 B 12 0 4 4 8 C -4 -4 0 -4 -4 D 6 -4 4 0 2 E 6 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -6 -6 B 12 0 4 4 8 C -4 -4 0 -4 -4 D 6 -4 4 0 2 E 6 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1073: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) D B A C E (6) D C B A E (5) B E A C D (5) B A D E C (5) D C A E B (4) C E D A B (4) C D E B A (4) A E B C D (4) D C E A B (3) C E D B A (3) C E B D A (3) C D B E A (3) B D C E A (3) A D B E C (3) E C A D B (2) E C A B D (2) D C E B A (2) D A B C E (2) C E A D B (2) C D E A B (2) B C E A D (2) B A E C D (2) A B E D C (2) A B D E C (2) E C B A D (1) E B C A D (1) E A C D B (1) E A C B D (1) D B C A E (1) D A C E B (1) D A C B E (1) D A B E C (1) C B E D A (1) C B D E A (1) B D A E C (1) B D A C E (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -18 0 -6 6 B 18 0 2 -8 12 C 0 -2 0 -8 10 D 6 8 8 0 6 E -6 -12 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 0 -6 6 B 18 0 2 -8 12 C 0 -2 0 -8 10 D 6 8 8 0 6 E -6 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=26 C=23 A=15 E=8 so E is eliminated. Round 2 votes counts: B=29 C=28 D=26 A=17 so A is eliminated. Round 3 votes counts: B=37 D=32 C=31 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:212 C:200 A:191 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 0 -6 6 B 18 0 2 -8 12 C 0 -2 0 -8 10 D 6 8 8 0 6 E -6 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -6 6 B 18 0 2 -8 12 C 0 -2 0 -8 10 D 6 8 8 0 6 E -6 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -6 6 B 18 0 2 -8 12 C 0 -2 0 -8 10 D 6 8 8 0 6 E -6 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1074: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) B E A C D (7) D A C E B (6) A C D E B (5) E D B C A (4) E D A C B (4) C A D B E (4) B E C A D (4) A C D B E (4) E B A D C (3) E B A C D (3) D C A E B (3) D C A B E (3) C B A D E (3) B E C D A (3) E D B A C (2) E D A B C (2) E B D C A (2) E B D A C (2) E A D B C (2) D E C A B (2) D E B C A (2) D C B A E (2) C D B A E (2) B E D C A (2) B C E D A (2) B C E A D (2) B C D E A (2) B C A E D (2) B C A D E (2) E A D C B (1) D E A C B (1) D C B E A (1) C B D A E (1) B A C E D (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 -10 -10 -6 B 8 0 -2 -18 8 C 10 2 0 10 4 D 10 18 -10 0 0 E 6 -8 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -10 -6 B 8 0 -2 -18 8 C 10 2 0 10 4 D 10 18 -10 0 0 E 6 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 D=20 C=17 A=11 so A is eliminated. Round 2 votes counts: E=27 B=27 C=26 D=20 so D is eliminated. Round 3 votes counts: C=41 E=32 B=27 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:209 B:198 E:197 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 -10 -6 B 8 0 -2 -18 8 C 10 2 0 10 4 D 10 18 -10 0 0 E 6 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -10 -6 B 8 0 -2 -18 8 C 10 2 0 10 4 D 10 18 -10 0 0 E 6 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -10 -6 B 8 0 -2 -18 8 C 10 2 0 10 4 D 10 18 -10 0 0 E 6 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1075: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (13) E A D C B (6) A E D B C (5) E B C A D (4) D A B C E (4) B D A C E (4) B C E D A (4) A D E B C (4) E A D B C (3) D A E C B (3) D A C E B (3) B C E A D (3) C E D B A (2) C E B D A (2) C B E A D (2) C B D E A (2) B E C A D (2) B E A C D (2) B C D A E (2) A D E C B (2) A D B C E (2) A B D E C (2) E D A C B (1) E C D A B (1) E C A D B (1) E B A C D (1) E A C B D (1) E A B C D (1) D E A C B (1) D C A E B (1) D B A C E (1) D A E B C (1) D A C B E (1) D A B E C (1) C E D A B (1) C E B A D (1) C D B A E (1) C D A B E (1) C B D A E (1) B E A D C (1) B D C A E (1) B A D E C (1) B A D C E (1) B A C E D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 10 -4 -10 B 4 0 6 6 10 C -10 -6 0 -2 8 D 4 -6 2 0 -18 E 10 -10 -8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -4 -10 B 4 0 6 6 10 C -10 -6 0 -2 8 D 4 -6 2 0 -18 E 10 -10 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=22 E=19 A=17 D=16 so D is eliminated. Round 2 votes counts: A=30 C=27 B=23 E=20 so E is eliminated. Round 3 votes counts: A=43 C=29 B=28 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:213 E:205 A:196 C:195 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 -4 -10 B 4 0 6 6 10 C -10 -6 0 -2 8 D 4 -6 2 0 -18 E 10 -10 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -4 -10 B 4 0 6 6 10 C -10 -6 0 -2 8 D 4 -6 2 0 -18 E 10 -10 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -4 -10 B 4 0 6 6 10 C -10 -6 0 -2 8 D 4 -6 2 0 -18 E 10 -10 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1076: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) E C A D B (7) E A C B D (6) B D C E A (5) B D A E C (5) A E B C D (5) D C A E B (4) D B C E A (4) D C E A B (3) D C B E A (3) C E B D A (3) C D E A B (3) A E C D B (3) E B C A D (2) E A C D B (2) D C B A E (2) D B C A E (2) D B A C E (2) D A C E B (2) C E D A B (2) C D A E B (2) B D A C E (2) A E D C B (2) E C A B D (1) E A B C D (1) D A B C E (1) C A E D B (1) B E C D A (1) B E A C D (1) B D E C A (1) B D E A C (1) B D C A E (1) B C D E A (1) B A E D C (1) B A E C D (1) B A D E C (1) A E C B D (1) A E B D C (1) A D E C B (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 22 -12 0 -10 B -22 0 -18 -16 -26 C 12 18 0 6 10 D 0 16 -6 0 -2 E 10 26 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -12 0 -10 B -22 0 -18 -16 -26 C 12 18 0 6 10 D 0 16 -6 0 -2 E 10 26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 B=21 E=19 C=19 A=18 so A is eliminated. Round 2 votes counts: E=31 D=26 B=22 C=21 so C is eliminated. Round 3 votes counts: E=46 D=32 B=22 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:223 E:214 D:204 A:200 B:159 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -12 0 -10 B -22 0 -18 -16 -26 C 12 18 0 6 10 D 0 16 -6 0 -2 E 10 26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -12 0 -10 B -22 0 -18 -16 -26 C 12 18 0 6 10 D 0 16 -6 0 -2 E 10 26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -12 0 -10 B -22 0 -18 -16 -26 C 12 18 0 6 10 D 0 16 -6 0 -2 E 10 26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1077: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (16) A D E C B (13) E C B A D (8) D A B C E (8) B C D E A (7) B C E D A (6) E A C B D (5) A D E B C (5) A E C B D (4) E C B D A (3) E C A B D (3) D B C A E (3) D A E B C (3) C B E D A (3) C B E A D (3) E A C D B (2) D B A C E (2) C E B A D (2) D B C E A (1) D A B E C (1) C E B D A (1) A C E B D (1) Total count = 100 A B C D E A 0 22 20 24 12 B -22 0 -28 -8 -32 C -20 28 0 -4 -26 D -24 8 4 0 -14 E -12 32 26 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 20 24 12 B -22 0 -28 -8 -32 C -20 28 0 -4 -26 D -24 8 4 0 -14 E -12 32 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=21 D=18 B=13 C=9 so C is eliminated. Round 2 votes counts: A=39 E=24 B=19 D=18 so D is eliminated. Round 3 votes counts: A=51 B=25 E=24 so E is eliminated. Round 4 votes counts: A=61 B=39 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:239 E:230 C:189 D:187 B:155 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 20 24 12 B -22 0 -28 -8 -32 C -20 28 0 -4 -26 D -24 8 4 0 -14 E -12 32 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 20 24 12 B -22 0 -28 -8 -32 C -20 28 0 -4 -26 D -24 8 4 0 -14 E -12 32 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 20 24 12 B -22 0 -28 -8 -32 C -20 28 0 -4 -26 D -24 8 4 0 -14 E -12 32 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1078: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) D B E A C (7) C B D E A (7) A E D B C (6) D B C E A (5) D A E B C (5) D C B E A (4) A E C B D (4) C D B E A (3) C D B A E (3) C D A E B (3) C B E A D (3) C B A E D (3) C A E D B (3) C A E B D (3) B D E A C (3) A E C D B (3) A C E B D (3) E A D B C (2) D E A B C (2) B E A C D (2) B C E A D (2) B C D E A (2) E A B D C (1) D E B A C (1) D C B A E (1) D A E C B (1) D A C E B (1) D A B E C (1) C B D A E (1) C B A D E (1) B E D A C (1) B E C A D (1) B E A D C (1) A E D C B (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -18 -6 -18 -10 B 18 0 2 -6 18 C 6 -2 0 -2 14 D 18 6 2 0 18 E 10 -18 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -18 -10 B 18 0 2 -6 18 C 6 -2 0 -2 14 D 18 6 2 0 18 E 10 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=28 B=20 A=19 E=3 so E is eliminated. Round 2 votes counts: C=30 D=28 A=22 B=20 so B is eliminated. Round 3 votes counts: D=40 C=35 A=25 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:216 C:208 E:180 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -6 -18 -10 B 18 0 2 -6 18 C 6 -2 0 -2 14 D 18 6 2 0 18 E 10 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -18 -10 B 18 0 2 -6 18 C 6 -2 0 -2 14 D 18 6 2 0 18 E 10 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -18 -10 B 18 0 2 -6 18 C 6 -2 0 -2 14 D 18 6 2 0 18 E 10 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1079: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (5) D C B A E (5) A D E C B (5) E B C D A (4) E B A C D (4) E A D B C (4) B C E D A (4) A E D C B (4) E A B D C (3) E A B C D (3) D E A C B (3) D A E C B (3) D A C B E (3) C D B A E (3) B E C D A (3) B C D E A (3) A E D B C (3) A D C B E (3) A B C E D (3) E D A C B (2) E D A B C (2) D C B E A (2) D C A B E (2) C B D A E (2) B C A E D (2) A D C E B (2) A C B E D (2) A C B D E (2) E D B C A (1) E D B A C (1) E B D C A (1) E A D C B (1) D E C B A (1) D E B C A (1) D A C E B (1) C D B E A (1) B E C A D (1) B C E A D (1) B C D A E (1) B C A D E (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 12 2 -2 B -6 0 2 -10 -10 C -12 -2 0 -6 -12 D -2 10 6 0 -10 E 2 10 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 12 2 -2 B -6 0 2 -10 -10 C -12 -2 0 -6 -12 D -2 10 6 0 -10 E 2 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=26 D=21 B=16 C=6 so C is eliminated. Round 2 votes counts: E=31 A=26 D=25 B=18 so B is eliminated. Round 3 votes counts: E=40 D=31 A=29 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:209 D:202 B:188 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 2 -2 B -6 0 2 -10 -10 C -12 -2 0 -6 -12 D -2 10 6 0 -10 E 2 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 2 -2 B -6 0 2 -10 -10 C -12 -2 0 -6 -12 D -2 10 6 0 -10 E 2 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 2 -2 B -6 0 2 -10 -10 C -12 -2 0 -6 -12 D -2 10 6 0 -10 E 2 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1080: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) D C B A E (8) E A C B D (6) B D C E A (5) E A B C D (4) B C E A D (4) A E C D B (4) E B D A C (3) E B C A D (3) E A D C B (3) E A B D C (3) D C A B E (3) D B E C A (3) D B C E A (3) C D B A E (3) C B D A E (3) A D E C B (3) A C D E B (3) E A C D B (2) D E A B C (2) B E D C A (2) B E C D A (2) B D E C A (2) B C E D A (2) A E C B D (2) A D C E B (2) A C E B D (2) E B A D C (1) E B A C D (1) D E B A C (1) D A E C B (1) D A C E B (1) C B A E D (1) C A E B D (1) B E C A D (1) Total count = 100 A B C D E A 0 -16 -12 -8 -6 B 16 0 4 -4 4 C 12 -4 0 -12 2 D 8 4 12 0 6 E 6 -4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 -8 -6 B 16 0 4 -4 4 C 12 -4 0 -12 2 D 8 4 12 0 6 E 6 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=26 B=18 A=16 C=8 so C is eliminated. Round 2 votes counts: D=35 E=26 B=22 A=17 so A is eliminated. Round 3 votes counts: D=43 E=35 B=22 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:210 C:199 E:197 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -12 -8 -6 B 16 0 4 -4 4 C 12 -4 0 -12 2 D 8 4 12 0 6 E 6 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -8 -6 B 16 0 4 -4 4 C 12 -4 0 -12 2 D 8 4 12 0 6 E 6 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -8 -6 B 16 0 4 -4 4 C 12 -4 0 -12 2 D 8 4 12 0 6 E 6 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1081: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) E B C A D (6) D A B C E (6) C A D E B (6) B E C D A (6) E B C D A (5) A D C E B (5) D A C B E (4) C A E D B (4) B D E C A (4) E C A B D (3) C E B A D (3) B E D A C (3) B D E A C (3) B D A E C (3) A C D E B (3) E C A D B (2) D C A B E (2) D B A E C (2) D A B E C (2) C E A D B (2) B E D C A (2) A E D C B (2) A D C B E (2) A D B E C (2) E C B D A (1) E A B D C (1) D B A C E (1) C E D B A (1) C B E D A (1) C B D A E (1) B D C E A (1) B D A C E (1) B C E D A (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -14 0 -2 B 2 0 0 2 -6 C 14 0 0 6 -10 D 0 -2 -6 0 -2 E 2 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -14 0 -2 B 2 0 0 2 -6 C 14 0 0 6 -10 D 0 -2 -6 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 C=18 D=17 A=17 so D is eliminated. Round 2 votes counts: A=29 B=27 E=24 C=20 so C is eliminated. Round 3 votes counts: A=41 E=30 B=29 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:205 B:199 D:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -14 0 -2 B 2 0 0 2 -6 C 14 0 0 6 -10 D 0 -2 -6 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 0 -2 B 2 0 0 2 -6 C 14 0 0 6 -10 D 0 -2 -6 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 0 -2 B 2 0 0 2 -6 C 14 0 0 6 -10 D 0 -2 -6 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1082: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E A C D B (7) A B C D E (6) E D C B A (5) E B A D C (5) D C B E A (5) B A E D C (5) A C D B E (5) D C E B A (4) C D E A B (4) C D B A E (4) B D C A E (4) A B E D C (3) A B E C D (3) B E A D C (2) B D C E A (2) A E C B D (2) A B C E D (2) E D C A B (1) E D A B C (1) E B D C A (1) E A D C B (1) E A B D C (1) D E C B A (1) D B C E A (1) C E D A B (1) C D E B A (1) C D A E B (1) C D A B E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D E C A (1) B C D A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E B C D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 0 -2 -12 B -8 0 -14 -12 0 C 0 14 0 8 -6 D 2 12 -8 0 -10 E 12 0 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.210684 C: 0.000000 D: 0.000000 E: 0.789316 Sum of squares = 0.667407454081 Cumulative probabilities = A: 0.000000 B: 0.210684 C: 0.210684 D: 0.210684 E: 1.000000 A B C D E A 0 8 0 -2 -12 B -8 0 -14 -12 0 C 0 14 0 8 -6 D 2 12 -8 0 -10 E 12 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000039262 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=25 B=19 C=13 D=11 so D is eliminated. Round 2 votes counts: E=33 A=25 C=22 B=20 so B is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:208 D:198 A:197 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 -2 -12 B -8 0 -14 -12 0 C 0 14 0 8 -6 D 2 12 -8 0 -10 E 12 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000039262 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -2 -12 B -8 0 -14 -12 0 C 0 14 0 8 -6 D 2 12 -8 0 -10 E 12 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000039262 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -2 -12 B -8 0 -14 -12 0 C 0 14 0 8 -6 D 2 12 -8 0 -10 E 12 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000039262 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1083: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (18) B C E D A (13) A D E B C (9) C B E D A (8) D A E C B (7) B C E A D (7) B C A E D (4) A D B E C (4) D E A C B (3) A B D C E (3) E C D B A (2) E C B D A (2) D E C B A (2) B A C E D (2) A D B C E (2) A B C E D (2) E D C B A (1) E B C D A (1) D E C A B (1) D A C E B (1) C E D B A (1) B E C D A (1) B A C D E (1) A E D B C (1) A D C E B (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 14 14 16 B -10 0 6 -6 0 C -14 -6 0 -10 -8 D -14 6 10 0 8 E -16 0 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 14 16 B -10 0 6 -6 0 C -14 -6 0 -10 -8 D -14 6 10 0 8 E -16 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 B=28 D=14 C=9 E=6 so E is eliminated. Round 2 votes counts: A=43 B=29 D=15 C=13 so C is eliminated. Round 3 votes counts: A=43 B=39 D=18 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 D:205 B:195 E:192 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 14 16 B -10 0 6 -6 0 C -14 -6 0 -10 -8 D -14 6 10 0 8 E -16 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 14 16 B -10 0 6 -6 0 C -14 -6 0 -10 -8 D -14 6 10 0 8 E -16 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 14 16 B -10 0 6 -6 0 C -14 -6 0 -10 -8 D -14 6 10 0 8 E -16 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1084: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) D E B C A (7) C E A B D (6) A C B E D (6) D B A E C (5) C A E B D (5) A C E D B (5) A C E B D (5) E C D B A (4) E C A D B (4) B A D C E (4) A B D C E (4) E C D A B (3) E C B A D (3) C E A D B (3) C E B A D (2) B A C D E (2) A D B C E (2) A B C D E (2) E D C B A (1) E D B C A (1) E B C D A (1) D E C B A (1) D E C A B (1) D E A C B (1) D B E C A (1) D A B E C (1) D A B C E (1) C B E A D (1) C A B E D (1) B D E C A (1) B D A E C (1) B D A C E (1) B C E A D (1) B A C E D (1) A E C D B (1) A C D E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 22 0 B -8 0 -10 -2 -10 C -6 10 0 18 10 D -22 2 -18 0 -10 E 0 10 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.807376 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.192624 Sum of squares = 0.68895987271 Cumulative probabilities = A: 0.807376 B: 0.807376 C: 0.807376 D: 0.807376 E: 1.000000 A B C D E A 0 8 6 22 0 B -8 0 -10 -2 -10 C -6 10 0 18 10 D -22 2 -18 0 -10 E 0 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250483037 Cumulative probabilities = A: 0.625001 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 C=18 E=17 B=11 so B is eliminated. Round 2 votes counts: A=35 D=29 C=19 E=17 so E is eliminated. Round 3 votes counts: A=35 C=34 D=31 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:216 E:205 B:185 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 22 0 B -8 0 -10 -2 -10 C -6 10 0 18 10 D -22 2 -18 0 -10 E 0 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250483037 Cumulative probabilities = A: 0.625001 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 22 0 B -8 0 -10 -2 -10 C -6 10 0 18 10 D -22 2 -18 0 -10 E 0 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250483037 Cumulative probabilities = A: 0.625001 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 22 0 B -8 0 -10 -2 -10 C -6 10 0 18 10 D -22 2 -18 0 -10 E 0 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250483037 Cumulative probabilities = A: 0.625001 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1085: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) B E D A C (7) D E B C A (6) C A D E B (6) B A E D C (6) C D A E B (5) A B C E D (5) E B A D C (4) D E C B A (4) A C B E D (4) D E C A B (3) D C E B A (3) D C E A B (3) D B E C A (3) C D A B E (3) C A D B E (3) C A B D E (3) A C E D B (3) E B D A C (2) B A E C D (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D A B (1) E A C D B (1) D C B A E (1) D B C E A (1) D B C A E (1) C D E A B (1) C A E D B (1) B D E C A (1) B D E A C (1) B D A C E (1) B A C D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -4 0 -6 B 10 0 4 -4 6 C 4 -4 0 -18 -8 D 0 4 18 0 0 E 6 -6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.704024 E: 0.295976 Sum of squares = 0.583251369593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.704024 E: 1.000000 A B C D E A 0 -10 -4 0 -6 B 10 0 4 -4 6 C 4 -4 0 -18 -8 D 0 4 18 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000029813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=25 C=22 A=14 E=11 so E is eliminated. Round 2 votes counts: B=34 D=28 C=23 A=15 so A is eliminated. Round 3 votes counts: B=40 C=32 D=28 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:208 E:204 A:190 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 0 -6 B 10 0 4 -4 6 C 4 -4 0 -18 -8 D 0 4 18 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000029813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 0 -6 B 10 0 4 -4 6 C 4 -4 0 -18 -8 D 0 4 18 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000029813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 0 -6 B 10 0 4 -4 6 C 4 -4 0 -18 -8 D 0 4 18 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000029813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1086: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) A E C B D (8) B D A E C (7) D B C A E (6) C B D A E (6) A E B D C (6) E A B D C (5) E A C D B (4) C E A D B (4) B D C A E (4) E A D C B (3) E A D B C (3) C D B E A (3) E D C B A (2) E A C B D (2) D B E A C (2) C E D A B (2) C D B A E (2) C A E B D (2) B A D E C (2) A C E B D (2) E D A C B (1) E A B C D (1) D E B C A (1) D C B A E (1) D B E C A (1) C D E B A (1) C B A D E (1) C A B D E (1) B D E A C (1) B D A C E (1) A E B C D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 2 -6 4 B 6 0 10 4 4 C -2 -10 0 -18 -2 D 6 -4 18 0 6 E -4 -4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -6 4 B 6 0 10 4 4 C -2 -10 0 -18 -2 D 6 -4 18 0 6 E -4 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 E=21 A=19 B=15 so B is eliminated. Round 2 votes counts: D=36 C=22 E=21 A=21 so E is eliminated. Round 3 votes counts: D=39 A=39 C=22 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:212 A:197 E:194 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -6 4 B 6 0 10 4 4 C -2 -10 0 -18 -2 D 6 -4 18 0 6 E -4 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -6 4 B 6 0 10 4 4 C -2 -10 0 -18 -2 D 6 -4 18 0 6 E -4 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -6 4 B 6 0 10 4 4 C -2 -10 0 -18 -2 D 6 -4 18 0 6 E -4 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1087: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) A C E B D (6) C D B A E (5) C A D B E (5) B E D C A (5) A E C D B (5) D B E C A (4) D B C E A (4) E A C B D (3) C B E D A (3) B C D E A (3) A E D B C (3) A E B C D (3) E B D C A (2) E A B D C (2) E A B C D (2) D C B E A (2) D C B A E (2) D B E A C (2) C D A B E (2) A E B D C (2) A D E B C (2) A C E D B (2) A C D E B (2) E D B A C (1) E C B A D (1) E C A B D (1) E B A D C (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B C E (1) C E A B D (1) C B E A D (1) C B D E A (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B E D (1) B E C D A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 10 -2 8 12 B -10 0 -10 6 -2 C 2 10 0 20 -4 D -8 -6 -20 0 -16 E -12 2 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.111111 Sum of squares = 0.506172839522 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 10 -2 8 12 B -10 0 -10 6 -2 C 2 10 0 20 -4 D -8 -6 -20 0 -16 E -12 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.111111 Sum of squares = 0.50617283977 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=22 D=18 E=13 B=13 so E is eliminated. Round 2 votes counts: A=41 C=24 D=19 B=16 so B is eliminated. Round 3 votes counts: A=42 D=29 C=29 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:214 E:205 B:192 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 8 12 B -10 0 -10 6 -2 C 2 10 0 20 -4 D -8 -6 -20 0 -16 E -12 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.111111 Sum of squares = 0.50617283977 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 8 12 B -10 0 -10 6 -2 C 2 10 0 20 -4 D -8 -6 -20 0 -16 E -12 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.111111 Sum of squares = 0.50617283977 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 8 12 B -10 0 -10 6 -2 C 2 10 0 20 -4 D -8 -6 -20 0 -16 E -12 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.111111 Sum of squares = 0.50617283977 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1088: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E B D A (7) C A E D B (7) B D E A C (7) C E A B D (6) E C B D A (5) E B D C A (5) B D E C A (5) E C A B D (4) A D B E C (4) A D B C E (4) E B C D A (3) C A D B E (3) A C E D B (3) A C D B E (3) D B E A C (2) D A B E C (2) D A B C E (2) C B D E A (2) B E D C A (2) B D C E A (2) A E C D B (2) A C D E B (2) E C B A D (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B D C (1) D B A C E (1) C A E B D (1) Total count = 100 A B C D E A 0 -10 -6 -14 -10 B 10 0 6 8 0 C 6 -6 0 0 -14 D 14 -8 0 0 0 E 10 0 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.772367 C: 0.000000 D: 0.000000 E: 0.227633 Sum of squares = 0.648367057465 Cumulative probabilities = A: 0.000000 B: 0.772367 C: 0.772367 D: 0.772367 E: 1.000000 A B C D E A 0 -10 -6 -14 -10 B 10 0 6 8 0 C 6 -6 0 0 -14 D 14 -8 0 0 0 E 10 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=22 D=18 A=18 B=16 so B is eliminated. Round 2 votes counts: D=32 C=26 E=24 A=18 so A is eliminated. Round 3 votes counts: D=40 C=34 E=26 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:212 E:212 D:203 C:193 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -14 -10 B 10 0 6 8 0 C 6 -6 0 0 -14 D 14 -8 0 0 0 E 10 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -14 -10 B 10 0 6 8 0 C 6 -6 0 0 -14 D 14 -8 0 0 0 E 10 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -14 -10 B 10 0 6 8 0 C 6 -6 0 0 -14 D 14 -8 0 0 0 E 10 0 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1089: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (15) C A D E B (10) C A E D B (9) B D E A C (8) D E A B C (6) C B D E A (6) A E D B C (5) B A E D C (4) C D A E B (3) C B E D A (3) C A B E D (3) B E D C A (3) B E A D C (3) D E B A C (2) C B A E D (2) A E B D C (2) A D E C B (2) E D B A C (1) D E C B A (1) D C E B A (1) C D E A B (1) C B E A D (1) C B A D E (1) C A E B D (1) B E C D A (1) A D E B C (1) A C E D B (1) A C D E B (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -2 -4 B 4 0 6 12 6 C -8 -6 0 -10 -10 D 2 -12 10 0 -14 E 4 -6 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -2 -4 B 4 0 6 12 6 C -8 -6 0 -10 -10 D 2 -12 10 0 -14 E 4 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=34 A=15 D=10 E=1 so E is eliminated. Round 2 votes counts: C=40 B=34 A=15 D=11 so D is eliminated. Round 3 votes counts: C=42 B=37 A=21 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:211 A:199 D:193 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 -2 -4 B 4 0 6 12 6 C -8 -6 0 -10 -10 D 2 -12 10 0 -14 E 4 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -2 -4 B 4 0 6 12 6 C -8 -6 0 -10 -10 D 2 -12 10 0 -14 E 4 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -2 -4 B 4 0 6 12 6 C -8 -6 0 -10 -10 D 2 -12 10 0 -14 E 4 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1090: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) A C E D B (9) E D B C A (6) B D C A E (6) B C A D E (6) E A C D B (5) B D C E A (5) A C B D E (5) E A D C B (4) C A D B E (4) B E D C A (4) A E C D B (4) E D A C B (3) B C D A E (3) B A C D E (3) E D C A B (2) E D B A C (2) E B A D C (2) E A B D C (2) D E B C A (2) A C D B E (2) A C B E D (2) E C A D B (1) E B D C A (1) E B D A C (1) D E C B A (1) D C E A B (1) D B E C A (1) B E A D C (1) A C E B D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -4 6 -6 B 6 0 10 4 4 C 4 -10 0 -6 -2 D -6 -4 6 0 0 E 6 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 6 -6 B 6 0 10 4 4 C 4 -10 0 -6 -2 D -6 -4 6 0 0 E 6 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=29 A=25 D=5 C=4 so C is eliminated. Round 2 votes counts: B=37 E=29 A=29 D=5 so D is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:202 D:198 A:195 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 6 -6 B 6 0 10 4 4 C 4 -10 0 -6 -2 D -6 -4 6 0 0 E 6 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 6 -6 B 6 0 10 4 4 C 4 -10 0 -6 -2 D -6 -4 6 0 0 E 6 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 6 -6 B 6 0 10 4 4 C 4 -10 0 -6 -2 D -6 -4 6 0 0 E 6 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1091: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) E A C D B (7) D B E C A (6) D B E A C (6) B D C A E (6) E C A D B (5) C B D A E (4) B D E A C (4) B D A E C (4) E A D C B (3) E A D B C (3) D B C E A (3) C A E D B (3) C A B D E (3) B D C E A (3) A E C B D (3) E D B A C (2) E D A C B (2) E C D B A (2) D E B C A (2) B D A C E (2) A E C D B (2) A C E B D (2) A C B E D (2) E B D A C (1) E A B D C (1) D C B E A (1) C E D B A (1) C E A D B (1) C D E B A (1) C D B E A (1) C B A D E (1) C A B E D (1) B A D C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 -2 0 C 6 10 0 0 -6 D 2 2 0 0 0 E 10 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.745215 E: 0.254785 Sum of squares = 0.620260945725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.745215 E: 1.000000 A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 -2 0 C 6 10 0 0 -6 D 2 2 0 0 0 E 10 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 B=20 D=18 A=11 so A is eliminated. Round 2 votes counts: E=31 C=30 B=21 D=18 so D is eliminated. Round 3 votes counts: B=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:208 C:205 D:202 B:195 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 -2 0 C 6 10 0 0 -6 D 2 2 0 0 0 E 10 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 -2 0 C 6 10 0 0 -6 D 2 2 0 0 0 E 10 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 -2 0 C 6 10 0 0 -6 D 2 2 0 0 0 E 10 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1092: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) D B A E C (7) C E A B D (7) C B A D E (7) C A E B D (7) C A B E D (7) B D C A E (6) E A D C B (5) B D A E C (4) E D A B C (3) E A C D B (3) B D A C E (3) B C D A E (3) A D B E C (3) E C A D B (2) D E B A C (2) C E B D A (2) C E A D B (2) C B D A E (2) C A B D E (2) B D C E A (2) E D C A B (1) E D A C B (1) E C D B A (1) E A D B C (1) D E A B C (1) C E B A D (1) C B A E D (1) C A E D B (1) B C D E A (1) B C A D E (1) A D E B C (1) A C E D B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -12 8 16 B 0 0 -10 16 14 C 12 10 0 6 16 D -8 -16 -6 0 6 E -16 -14 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 8 16 B 0 0 -10 16 14 C 12 10 0 6 16 D -8 -16 -6 0 6 E -16 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=20 E=17 D=17 A=7 so A is eliminated. Round 2 votes counts: C=42 D=21 B=20 E=17 so E is eliminated. Round 3 votes counts: C=48 D=32 B=20 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:210 A:206 D:188 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 8 16 B 0 0 -10 16 14 C 12 10 0 6 16 D -8 -16 -6 0 6 E -16 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 8 16 B 0 0 -10 16 14 C 12 10 0 6 16 D -8 -16 -6 0 6 E -16 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 8 16 B 0 0 -10 16 14 C 12 10 0 6 16 D -8 -16 -6 0 6 E -16 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1093: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) E A C B D (6) B A E D C (6) C D E A B (5) B D C E A (5) B C E A D (5) A E B C D (5) D C A E B (4) A E D C B (4) D C E A B (3) D A E C B (3) C E D A B (3) B D A C E (3) A E C D B (3) E C A D B (2) D C E B A (2) D B C E A (2) D A C E B (2) D A B E C (2) B C D E A (2) B A E C D (2) A E C B D (2) A B E D C (2) E C B A D (1) E A D C B (1) D C B E A (1) D B C A E (1) D A E B C (1) C E B D A (1) C E B A D (1) C E A B D (1) C D E B A (1) C D B E A (1) C B D E A (1) B E A C D (1) B D C A E (1) B D A E C (1) B C E D A (1) B A D E C (1) A D E C B (1) A D E B C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 20 12 8 -6 B -20 0 -12 -2 -20 C -12 12 0 4 -8 D -8 2 -4 0 -10 E 6 20 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 12 8 -6 B -20 0 -12 -2 -20 C -12 12 0 4 -8 D -8 2 -4 0 -10 E 6 20 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=21 A=20 E=17 C=14 so C is eliminated. Round 2 votes counts: B=29 D=28 E=23 A=20 so A is eliminated. Round 3 votes counts: E=37 B=32 D=31 so D is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:217 C:198 D:190 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 12 8 -6 B -20 0 -12 -2 -20 C -12 12 0 4 -8 D -8 2 -4 0 -10 E 6 20 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 12 8 -6 B -20 0 -12 -2 -20 C -12 12 0 4 -8 D -8 2 -4 0 -10 E 6 20 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 12 8 -6 B -20 0 -12 -2 -20 C -12 12 0 4 -8 D -8 2 -4 0 -10 E 6 20 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1094: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (5) A C E D B (5) B D E C A (4) E C A D B (3) E B D A C (3) E A B C D (3) D E C B A (3) D C B A E (3) C A E D B (3) C A D E B (3) B D E A C (3) A C D E B (3) A B C D E (3) E D B C A (2) E C A B D (2) E B D C A (2) E B C D A (2) E B A D C (2) E A C B D (2) D C E A B (2) D C B E A (2) D B E C A (2) D B C A E (2) C E A D B (2) C D E A B (2) C D A E B (2) C A D B E (2) B E A D C (2) B D A E C (2) A E B C D (2) A C E B D (2) A C D B E (2) E D C B A (1) E C D B A (1) E C B A D (1) E B A C D (1) D C A E B (1) D B C E A (1) C E D A B (1) C D A B E (1) B E D C A (1) B E D A C (1) B D A C E (1) B A E D C (1) B A E C D (1) B A D E C (1) B A D C E (1) A C B E D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -12 0 -2 B -8 0 -10 -8 -12 C 12 10 0 2 2 D 0 8 -2 0 4 E 2 12 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 0 -2 B -8 0 -10 -8 -12 C 12 10 0 2 2 D 0 8 -2 0 4 E 2 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=21 A=20 B=18 C=16 so C is eliminated. Round 2 votes counts: E=28 A=28 D=26 B=18 so B is eliminated. Round 3 votes counts: D=36 E=32 A=32 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:213 D:205 E:204 A:197 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 0 -2 B -8 0 -10 -8 -12 C 12 10 0 2 2 D 0 8 -2 0 4 E 2 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 0 -2 B -8 0 -10 -8 -12 C 12 10 0 2 2 D 0 8 -2 0 4 E 2 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 0 -2 B -8 0 -10 -8 -12 C 12 10 0 2 2 D 0 8 -2 0 4 E 2 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1095: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) A E B D C (7) D C A E B (6) A C D B E (6) D C E B A (4) C D B E A (4) B E A C D (4) B C E D A (4) A B E C D (4) E B A D C (3) C D E B A (3) C B D E A (3) E B D C A (2) E B D A C (2) E A D B C (2) D E C B A (2) C D B A E (2) C D A B E (2) C A D B E (2) B E C D A (2) B E C A D (2) B E A D C (2) B C A E D (2) A D E C B (2) A B E D C (2) A B C E D (2) E D A C B (1) D E A C B (1) D C E A B (1) D A C E B (1) C D A E B (1) C B D A E (1) C A B D E (1) B C D E A (1) B C A D E (1) B A E C D (1) A E D C B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 8 14 14 B -10 0 -12 -4 0 C -8 12 0 0 20 D -14 4 0 0 14 E -14 0 -20 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 14 14 B -10 0 -12 -4 0 C -8 12 0 0 20 D -14 4 0 0 14 E -14 0 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=19 B=19 D=15 E=10 so E is eliminated. Round 2 votes counts: A=39 B=26 C=19 D=16 so D is eliminated. Round 3 votes counts: A=42 C=32 B=26 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:212 D:202 B:187 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 14 14 B -10 0 -12 -4 0 C -8 12 0 0 20 D -14 4 0 0 14 E -14 0 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 14 14 B -10 0 -12 -4 0 C -8 12 0 0 20 D -14 4 0 0 14 E -14 0 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 14 14 B -10 0 -12 -4 0 C -8 12 0 0 20 D -14 4 0 0 14 E -14 0 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1096: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (13) B D E C A (6) B D C E A (6) A E D B C (6) C E D B A (5) C E B D A (5) C B D E A (5) A C E D B (5) C E A D B (4) A D B E C (4) A B D E C (4) E C D B A (3) D B E A C (3) C A E D B (3) B D E A C (3) B D A C E (3) A B D C E (3) E D B C A (2) E D B A C (2) E A D C B (2) C E A B D (2) C A E B D (2) B C D E A (2) E A D B C (1) E A C D B (1) D B A E C (1) C A B D E (1) B D C A E (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -20 6 -20 -4 B 20 0 20 14 10 C -6 -20 0 -20 -4 D 20 -14 20 0 10 E 4 -10 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 -20 -4 B 20 0 20 14 10 C -6 -20 0 -20 -4 D 20 -14 20 0 10 E 4 -10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998448 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=27 A=24 E=11 D=4 so D is eliminated. Round 2 votes counts: B=38 C=27 A=24 E=11 so E is eliminated. Round 3 votes counts: B=42 C=30 A=28 so A is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 D:218 E:194 A:181 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 6 -20 -4 B 20 0 20 14 10 C -6 -20 0 -20 -4 D 20 -14 20 0 10 E 4 -10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998448 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -20 -4 B 20 0 20 14 10 C -6 -20 0 -20 -4 D 20 -14 20 0 10 E 4 -10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998448 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -20 -4 B 20 0 20 14 10 C -6 -20 0 -20 -4 D 20 -14 20 0 10 E 4 -10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998448 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1097: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (14) E B A C D (11) D C A E B (5) C D E B A (5) A E B D C (5) D A C E B (4) A D B E C (4) E B C A D (3) D C E B A (3) D C E A B (3) D A C B E (3) B E C A D (3) B E A C D (3) B A E C D (3) A D E B C (3) E C B D A (2) E C B A D (2) E A B D C (2) D A E C B (2) C D B E A (2) C B E D A (2) A B E D C (2) E D A B C (1) E B A D C (1) E A D B C (1) D E C B A (1) D E C A B (1) D C B E A (1) D A E B C (1) C E B D A (1) C D B A E (1) B C E A D (1) B C A E D (1) A E B C D (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -2 -4 2 B -8 0 -4 -12 -16 C 2 4 0 -16 -6 D 4 12 16 0 8 E -2 16 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -4 2 B -8 0 -4 -12 -16 C 2 4 0 -16 -6 D 4 12 16 0 8 E -2 16 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=23 A=17 C=11 B=11 so C is eliminated. Round 2 votes counts: D=46 E=24 A=17 B=13 so B is eliminated. Round 3 votes counts: D=46 E=33 A=21 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:206 A:202 C:192 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -2 -4 2 B -8 0 -4 -12 -16 C 2 4 0 -16 -6 D 4 12 16 0 8 E -2 16 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -4 2 B -8 0 -4 -12 -16 C 2 4 0 -16 -6 D 4 12 16 0 8 E -2 16 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -4 2 B -8 0 -4 -12 -16 C 2 4 0 -16 -6 D 4 12 16 0 8 E -2 16 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1098: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B D E A (9) C E A D B (8) B D A E C (7) A E D B C (6) C B D A E (5) C B A E D (4) C A E B D (4) C A B E D (4) E D A B C (3) D E A B C (3) D B E A C (3) C E D A B (3) B D E A C (3) A E B D C (3) C D B E A (2) C B A D E (2) C A E D B (2) B A D E C (2) A E C D B (2) A E C B D (2) E A D C B (1) D E B A C (1) D C B E A (1) D B E C A (1) D B C E A (1) C E B D A (1) C D E B A (1) C D E A B (1) B D C A E (1) B D A C E (1) B C D A E (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -2 4 -4 B -6 0 -4 2 -2 C 2 4 0 4 4 D -4 -2 -4 0 -8 E 4 2 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 4 -4 B -6 0 -4 2 -2 C 2 4 0 4 4 D -4 -2 -4 0 -8 E 4 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 B=16 E=14 A=14 D=10 so D is eliminated. Round 2 votes counts: C=47 B=21 E=18 A=14 so A is eliminated. Round 3 votes counts: C=47 E=31 B=22 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:207 E:205 A:202 B:195 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 4 -4 B -6 0 -4 2 -2 C 2 4 0 4 4 D -4 -2 -4 0 -8 E 4 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 4 -4 B -6 0 -4 2 -2 C 2 4 0 4 4 D -4 -2 -4 0 -8 E 4 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 4 -4 B -6 0 -4 2 -2 C 2 4 0 4 4 D -4 -2 -4 0 -8 E 4 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1099: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) E D C A B (7) C A B D E (7) B A D C E (6) E C D A B (4) E C B A D (4) E B D A C (4) C B A E D (4) B D E A C (4) A B C D E (4) E D B C A (3) D E A B C (3) D B A E C (3) C A D B E (3) C A B E D (3) B E D A C (3) B E A D C (3) B A C D E (3) D E B A C (2) D B E A C (2) D A E B C (2) C E A D B (2) B E A C D (2) A B D C E (2) E D C B A (1) E C D B A (1) E C A B D (1) D A B C E (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B D A E C (1) B D A C E (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 12 -2 -8 B 12 0 16 10 8 C -12 -16 0 -14 -18 D 2 -10 14 0 -4 E 8 -8 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 12 -2 -8 B 12 0 16 10 8 C -12 -16 0 -14 -18 D 2 -10 14 0 -4 E 8 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=24 C=23 D=13 A=7 so A is eliminated. Round 2 votes counts: E=33 B=30 C=24 D=13 so D is eliminated. Round 3 votes counts: E=40 B=36 C=24 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:211 D:201 A:195 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 12 -2 -8 B 12 0 16 10 8 C -12 -16 0 -14 -18 D 2 -10 14 0 -4 E 8 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 12 -2 -8 B 12 0 16 10 8 C -12 -16 0 -14 -18 D 2 -10 14 0 -4 E 8 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 12 -2 -8 B 12 0 16 10 8 C -12 -16 0 -14 -18 D 2 -10 14 0 -4 E 8 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1100: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) D E A B C (7) C B E A D (7) E C B D A (6) D A E B C (6) E C B A D (5) A D B C E (5) E D A B C (4) D E A C B (4) C B A E D (4) C B A D E (4) A B C D E (4) E B C A D (3) B C A E D (3) A D B E C (3) E D A C B (2) E C D B A (2) D A C B E (2) C E B A D (2) B C E A D (2) E D C A B (1) E D B C A (1) E D B A C (1) E B C D A (1) D C A B E (1) D A E C B (1) D A B E C (1) C B D A E (1) B C A D E (1) B A E C D (1) B A C E D (1) B A C D E (1) A D E B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 12 -2 4 B -8 0 16 -4 6 C -12 -16 0 -4 -2 D 2 4 4 0 6 E -4 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -2 4 B -8 0 16 -4 6 C -12 -16 0 -4 -2 D 2 4 4 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=26 C=18 A=15 B=9 so B is eliminated. Round 2 votes counts: D=32 E=26 C=24 A=18 so A is eliminated. Round 3 votes counts: D=42 C=30 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:211 D:208 B:205 E:193 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -2 4 B -8 0 16 -4 6 C -12 -16 0 -4 -2 D 2 4 4 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -2 4 B -8 0 16 -4 6 C -12 -16 0 -4 -2 D 2 4 4 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -2 4 B -8 0 16 -4 6 C -12 -16 0 -4 -2 D 2 4 4 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1101: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) D C A E B (8) C D A E B (7) B E C A D (6) C D E A B (5) A E B C D (4) E C A B D (3) E A B C D (3) D C B E A (3) D C B A E (3) B A E D C (3) D B C E A (2) D B C A E (2) D B A E C (2) D A C B E (2) D A B E C (2) C E B A D (2) B D E C A (2) B D A E C (2) A E D B C (2) A E B D C (2) A D E B C (2) A B E D C (2) E B A C D (1) E A C B D (1) D A C E B (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E A D (1) C B D E A (1) C A E D B (1) B E D C A (1) B E A D C (1) B C D E A (1) B A D E C (1) A E C D B (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -8 2 -2 B 0 0 8 2 2 C 8 -8 0 10 -10 D -2 -2 -10 0 0 E 2 -2 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.335665 B: 0.664335 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55401201558 Cumulative probabilities = A: 0.335665 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 2 -2 B 0 0 8 2 2 C 8 -8 0 10 -10 D -2 -2 -10 0 0 E 2 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499620 B: 0.500380 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000288994 Cumulative probabilities = A: 0.499620 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 C=23 A=15 E=8 so E is eliminated. Round 2 votes counts: B=30 C=26 D=25 A=19 so A is eliminated. Round 3 votes counts: B=41 D=31 C=28 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:206 E:205 C:200 A:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 2 -2 B 0 0 8 2 2 C 8 -8 0 10 -10 D -2 -2 -10 0 0 E 2 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499620 B: 0.500380 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000288994 Cumulative probabilities = A: 0.499620 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 -2 B 0 0 8 2 2 C 8 -8 0 10 -10 D -2 -2 -10 0 0 E 2 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499620 B: 0.500380 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000288994 Cumulative probabilities = A: 0.499620 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 -2 B 0 0 8 2 2 C 8 -8 0 10 -10 D -2 -2 -10 0 0 E 2 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499620 B: 0.500380 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000288994 Cumulative probabilities = A: 0.499620 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1102: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D B C E A (8) A C E B D (7) D B E C A (6) E A C B D (5) D E B C A (5) D B C A E (5) E B C A D (4) E A C D B (4) C A B E D (4) E A D C B (3) D E A B C (3) C B A D E (3) B D C E A (3) B C D A E (3) A C B E D (3) E B C D A (2) B C D E A (2) A D E C B (2) A C D B E (2) E D B A C (1) E C A B D (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B C D (1) D E B A C (1) D B E A C (1) D A B C E (1) C B D A E (1) C A B D E (1) B E C D A (1) B D C A E (1) B C E A D (1) B C A D E (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -4 10 -10 B 2 0 6 12 -4 C 4 -6 0 14 -6 D -10 -12 -14 0 0 E 10 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.155598 E: 0.844402 Sum of squares = 0.7372249847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.155598 E: 1.000000 A B C D E A 0 -2 -4 10 -10 B 2 0 6 12 -4 C 4 -6 0 14 -6 D -10 -12 -14 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000016022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=25 E=24 B=12 C=9 so C is eliminated. Round 2 votes counts: D=30 A=30 E=24 B=16 so B is eliminated. Round 3 votes counts: D=40 A=34 E=26 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:210 B:208 C:203 A:197 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 10 -10 B 2 0 6 12 -4 C 4 -6 0 14 -6 D -10 -12 -14 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000016022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 -10 B 2 0 6 12 -4 C 4 -6 0 14 -6 D -10 -12 -14 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000016022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 -10 B 2 0 6 12 -4 C 4 -6 0 14 -6 D -10 -12 -14 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000016022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1103: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) E C A D B (5) E A D C B (5) C A E D B (5) D A B E C (4) C A D B E (4) B E D A C (4) B D A C E (4) A C D B E (4) E B D A C (3) D B A E C (3) D B A C E (3) C E A B D (3) B D A E C (3) A B D C E (3) E D A B C (2) E C B D A (2) E C A B D (2) E B D C A (2) E B C D A (2) E A C D B (2) D A B C E (2) C B A D E (2) C A D E B (2) B D C A E (2) A C D E B (2) E D B A C (1) E C B A D (1) C E B A D (1) C E A D B (1) C B E D A (1) C B E A D (1) C A E B D (1) C A B E D (1) C A B D E (1) B D E C A (1) B C D A E (1) A E C D B (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 18 2 4 B -8 0 -2 0 8 C -18 2 0 -8 -6 D -2 0 8 0 6 E -4 -8 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 2 4 B -8 0 -2 0 8 C -18 2 0 -8 -6 D -2 0 8 0 6 E -4 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985127 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=24 C=23 A=14 D=12 so D is eliminated. Round 2 votes counts: B=30 E=27 C=23 A=20 so A is eliminated. Round 3 votes counts: B=39 C=32 E=29 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:216 D:206 B:199 E:194 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 2 4 B -8 0 -2 0 8 C -18 2 0 -8 -6 D -2 0 8 0 6 E -4 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985127 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 2 4 B -8 0 -2 0 8 C -18 2 0 -8 -6 D -2 0 8 0 6 E -4 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985127 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 2 4 B -8 0 -2 0 8 C -18 2 0 -8 -6 D -2 0 8 0 6 E -4 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985127 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1104: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (13) A C D E B (9) E B D A C (8) C A D B E (8) D C A B E (6) B E D A C (5) E B A C D (4) E A C B D (4) C D A B E (4) C A D E B (4) A C E D B (4) E B A D C (3) D B C E A (3) B D E C A (3) A E C B D (3) E B C A D (2) D B E C A (2) D B C A E (2) C A E B D (2) A C E B D (2) E C A B D (1) E A B C D (1) D C B A E (1) D B E A C (1) C B A E D (1) C A E D B (1) B D E A C (1) B D C E A (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -8 0 -4 B 0 0 -2 8 2 C 8 2 0 0 -4 D 0 -8 0 0 -8 E 4 -2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000028 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 0 -8 0 -4 B 0 0 -2 8 2 C 8 2 0 0 -4 D 0 -8 0 0 -8 E 4 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=23 B=23 C=20 A=19 D=15 so D is eliminated. Round 2 votes counts: B=31 C=27 E=23 A=19 so A is eliminated. Round 3 votes counts: C=42 B=31 E=27 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:207 B:204 C:203 A:194 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 0 -4 B 0 0 -2 8 2 C 8 2 0 0 -4 D 0 -8 0 0 -8 E 4 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 0 -4 B 0 0 -2 8 2 C 8 2 0 0 -4 D 0 -8 0 0 -8 E 4 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 0 -4 B 0 0 -2 8 2 C 8 2 0 0 -4 D 0 -8 0 0 -8 E 4 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1105: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (17) C B D A E (10) E D C A B (6) E A B D C (5) C D E A B (5) B C A D E (5) B A E D C (5) E A D B C (4) C D E B A (4) A B E D C (4) E A B C D (3) C D B E A (3) B A C E D (3) E D A C B (2) E C D A B (2) E A D C B (2) D E C A B (2) C E D A B (2) C B A D E (2) E C A D B (1) E C A B D (1) E A C B D (1) D E A B C (1) D C E B A (1) D C B A E (1) D B A C E (1) C B A E D (1) B A E C D (1) B A D E C (1) B A D C E (1) B A C D E (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -26 -14 10 B 14 0 -26 -8 14 C 26 26 0 26 14 D 14 8 -26 0 10 E -10 -14 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -26 -14 10 B 14 0 -26 -8 14 C 26 26 0 26 14 D 14 8 -26 0 10 E -10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 E=27 B=17 D=6 A=6 so D is eliminated. Round 2 votes counts: C=46 E=30 B=18 A=6 so A is eliminated. Round 3 votes counts: C=46 E=31 B=23 so B is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:246 D:203 B:197 A:178 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -26 -14 10 B 14 0 -26 -8 14 C 26 26 0 26 14 D 14 8 -26 0 10 E -10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -26 -14 10 B 14 0 -26 -8 14 C 26 26 0 26 14 D 14 8 -26 0 10 E -10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -26 -14 10 B 14 0 -26 -8 14 C 26 26 0 26 14 D 14 8 -26 0 10 E -10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1106: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (11) E C D A B (10) C E B D A (9) A D B E C (8) D A B E C (6) C B E A D (5) E C D B A (4) D B A C E (4) C E B A D (4) E C A D B (3) A D E B C (3) A D B C E (3) A B D C E (3) E D A C B (2) E C B A D (2) E C A B D (2) D A E B C (2) B D A C E (2) A E D B C (2) E C B D A (1) E A D C B (1) D E C A B (1) D A E C B (1) C E D B A (1) C D B E A (1) C B D E A (1) B D C A E (1) B C E D A (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D C B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 4 4 4 B 0 0 2 -6 2 C -4 -2 0 -4 -2 D -4 6 4 0 2 E -4 -2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.744066 B: 0.255934 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.619136002668 Cumulative probabilities = A: 0.744066 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 4 4 B 0 0 2 -6 2 C -4 -2 0 -4 -2 D -4 6 4 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001348 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=22 C=21 B=18 D=14 so D is eliminated. Round 2 votes counts: A=31 E=26 B=22 C=21 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 D:204 B:199 E:197 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 4 4 B 0 0 2 -6 2 C -4 -2 0 -4 -2 D -4 6 4 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001348 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 4 B 0 0 2 -6 2 C -4 -2 0 -4 -2 D -4 6 4 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001348 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 4 B 0 0 2 -6 2 C -4 -2 0 -4 -2 D -4 6 4 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001348 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1107: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) D E A B C (9) A E D C B (9) E D A C B (8) B C A D E (6) D E B A C (5) A D E B C (5) E D C A B (4) C B E D A (4) D E A C B (3) B C D E A (3) E D C B A (2) D A E B C (2) C E D B A (2) C B D E A (2) C A B E D (2) B D E C A (2) B D C E A (2) B A D E C (2) A C E D B (2) A C B E D (2) A B D E C (2) E C D A B (1) D E C B A (1) D E B C A (1) D B E A C (1) C E A D B (1) C A E B D (1) B C A E D (1) B A C E D (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 4 6 -4 -2 B -4 0 -12 -14 -14 C -6 12 0 -18 -16 D 4 14 18 0 -6 E 2 14 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 -4 -2 B -4 0 -12 -14 -14 C -6 12 0 -18 -16 D 4 14 18 0 -6 E 2 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=22 A=21 B=18 E=15 so E is eliminated. Round 2 votes counts: D=36 C=25 A=21 B=18 so B is eliminated. Round 3 votes counts: D=40 C=35 A=25 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:219 D:215 A:202 C:186 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 -4 -2 B -4 0 -12 -14 -14 C -6 12 0 -18 -16 D 4 14 18 0 -6 E 2 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -4 -2 B -4 0 -12 -14 -14 C -6 12 0 -18 -16 D 4 14 18 0 -6 E 2 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -4 -2 B -4 0 -12 -14 -14 C -6 12 0 -18 -16 D 4 14 18 0 -6 E 2 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1108: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) C A B D E (10) B A E D C (10) D E B A C (6) B A D E C (6) A B C D E (5) E D B A C (4) E C D A B (4) A B C E D (4) D B E A C (3) C D E A B (3) C A B E D (3) B A C D E (3) A B E C D (3) E D C B A (2) D C E B A (2) C D B A E (2) C D A B E (2) E D C A B (1) E D B C A (1) E C A D B (1) E B A D C (1) E A C B D (1) E A B D C (1) E A B C D (1) D E B C A (1) D C B A E (1) D B C E A (1) C E D A B (1) C A E B D (1) C A D B E (1) B D A E C (1) B A D C E (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 6 10 12 B 10 0 6 8 18 C -6 -6 0 -6 -16 D -10 -8 6 0 18 E -12 -18 16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 10 12 B 10 0 6 8 18 C -6 -6 0 -6 -16 D -10 -8 6 0 18 E -12 -18 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 B=21 E=17 A=15 so A is eliminated. Round 2 votes counts: B=34 C=25 D=24 E=17 so E is eliminated. Round 3 votes counts: B=37 D=32 C=31 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:209 D:203 E:184 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 10 12 B 10 0 6 8 18 C -6 -6 0 -6 -16 D -10 -8 6 0 18 E -12 -18 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 10 12 B 10 0 6 8 18 C -6 -6 0 -6 -16 D -10 -8 6 0 18 E -12 -18 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 10 12 B 10 0 6 8 18 C -6 -6 0 -6 -16 D -10 -8 6 0 18 E -12 -18 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1109: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (12) E B C D A (6) A B E C D (6) D C A E B (5) B A E C D (5) E C B D A (4) E B A C D (4) D C B A E (4) D C A B E (4) B E A C D (4) A D C B E (4) A D B C E (4) A B D E C (4) A B D C E (4) D C E A B (3) D A C B E (3) C D E B A (3) B C D E A (3) E A C D B (2) B E C D A (2) B A E D C (2) A E B C D (2) E D C A B (1) E C D A B (1) E C B A D (1) E B C A D (1) E A B C D (1) B C E D A (1) A E D C B (1) A E D B C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -2 -4 2 B 4 0 2 2 2 C 2 -2 0 16 -22 D 4 -2 -16 0 -16 E -2 -2 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 2 B 4 0 2 2 2 C 2 -2 0 16 -22 D 4 -2 -16 0 -16 E -2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998089 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=28 D=19 B=17 C=3 so C is eliminated. Round 2 votes counts: E=33 A=28 D=22 B=17 so B is eliminated. Round 3 votes counts: E=40 A=35 D=25 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:217 B:205 C:197 A:196 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 2 B 4 0 2 2 2 C 2 -2 0 16 -22 D 4 -2 -16 0 -16 E -2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998089 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 2 B 4 0 2 2 2 C 2 -2 0 16 -22 D 4 -2 -16 0 -16 E -2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998089 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 2 B 4 0 2 2 2 C 2 -2 0 16 -22 D 4 -2 -16 0 -16 E -2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998089 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1110: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) D E C A B (7) B A C D E (7) D E B C A (6) C A B E D (6) E D C A B (5) D E A C B (4) B D E C A (4) B A C E D (4) E D A C B (3) E C A D B (3) D E B A C (3) D B E A C (3) B D A C E (3) B C A E D (3) A C E B D (3) D E C B A (2) D B E C A (2) C A E B D (2) B D E A C (2) B D A E C (2) B C D E A (2) B A D C E (2) A B C E D (2) E D C B A (1) E A D C B (1) D A E B C (1) C E B A D (1) C B A E D (1) C A E D B (1) B D C E A (1) B C A D E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 0 4 -2 0 B 0 0 -2 14 12 C -4 2 0 -6 2 D 2 -14 6 0 6 E 0 -12 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.640550 B: 0.359450 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.53950860344 Cumulative probabilities = A: 0.640550 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -2 0 B 0 0 -2 14 12 C -4 2 0 -6 2 D 2 -14 6 0 6 E 0 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=28 A=17 E=13 C=11 so C is eliminated. Round 2 votes counts: B=32 D=28 A=26 E=14 so E is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:201 D:200 C:197 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 -2 0 B 0 0 -2 14 12 C -4 2 0 -6 2 D 2 -14 6 0 6 E 0 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 0 B 0 0 -2 14 12 C -4 2 0 -6 2 D 2 -14 6 0 6 E 0 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 0 B 0 0 -2 14 12 C -4 2 0 -6 2 D 2 -14 6 0 6 E 0 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1111: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) C A B E D (8) A C B D E (7) E D C B A (6) E D B C A (6) C E B D A (5) E D B A C (4) E B D C A (4) C B E D A (4) C A B D E (4) B E D A C (4) A C D B E (4) A B D E C (4) D A E B C (3) C E D B A (3) A B C D E (3) D E A B C (2) C B A E D (2) C A D E B (2) C A D B E (2) A D C E B (2) A D B E C (2) E B D A C (1) D E A C B (1) D A E C B (1) C E B A D (1) C E A D B (1) C A E B D (1) B A E D C (1) B A D E C (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 2 -8 -4 B 4 0 -8 0 -8 C -2 8 0 -6 -2 D 8 0 6 0 -2 E 4 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 -8 -4 B 4 0 -8 0 -8 C -2 8 0 -6 -2 D 8 0 6 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=23 E=21 D=17 B=6 so B is eliminated. Round 2 votes counts: C=33 E=25 A=25 D=17 so D is eliminated. Round 3 votes counts: E=38 C=33 A=29 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:206 C:199 B:194 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -8 -4 B 4 0 -8 0 -8 C -2 8 0 -6 -2 D 8 0 6 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -8 -4 B 4 0 -8 0 -8 C -2 8 0 -6 -2 D 8 0 6 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -8 -4 B 4 0 -8 0 -8 C -2 8 0 -6 -2 D 8 0 6 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1112: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (6) E D A C B (5) E C A B D (5) E A C D B (5) D A B C E (5) E D B A C (4) D A E C B (4) E B C A D (3) D B C A E (3) D A C B E (3) C A D B E (3) B E C D A (3) B D E C A (3) B D C A E (3) B C D A E (3) B C A D E (3) E B C D A (2) E A D C B (2) E A C B D (2) C B E A D (2) C A E B D (2) C A B D E (2) B C E D A (2) B C E A D (2) A E D C B (2) A D C E B (2) A C D E B (2) E D A B C (1) E C B A D (1) E C A D B (1) E B D A C (1) D E B A C (1) D E A B C (1) D C B A E (1) D B E A C (1) D A C E B (1) C B D A E (1) C A B E D (1) B E D C A (1) B E C A D (1) A E C D B (1) A D E C B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 4 -10 2 B -6 0 -2 -14 0 C -4 2 0 -2 -2 D 10 14 2 0 0 E -2 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.404826 E: 0.595174 Sum of squares = 0.518116336478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.404826 E: 1.000000 A B C D E A 0 6 4 -10 2 B -6 0 -2 -14 0 C -4 2 0 -2 -2 D 10 14 2 0 0 E -2 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=26 B=21 C=11 A=10 so A is eliminated. Round 2 votes counts: E=35 D=29 B=21 C=15 so C is eliminated. Round 3 votes counts: E=38 D=35 B=27 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:201 E:200 C:197 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -10 2 B -6 0 -2 -14 0 C -4 2 0 -2 -2 D 10 14 2 0 0 E -2 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -10 2 B -6 0 -2 -14 0 C -4 2 0 -2 -2 D 10 14 2 0 0 E -2 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -10 2 B -6 0 -2 -14 0 C -4 2 0 -2 -2 D 10 14 2 0 0 E -2 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1113: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (15) D C A B E (14) B A C D E (10) E D C A B (9) A B C D E (7) C D A B E (6) E B A D C (5) D C E A B (5) E D C B A (4) D C B A E (4) B A E C D (4) D C A E B (3) E A B C D (2) C A D B E (2) B E A C D (2) E D B C A (1) E B D C A (1) E A D C B (1) D E C A B (1) B C D A E (1) B C A D E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -4 2 8 B -2 0 2 0 6 C 4 -2 0 4 8 D -2 0 -4 0 8 E -8 -6 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 2 8 B -2 0 2 0 6 C 4 -2 0 4 8 D -2 0 -4 0 8 E -8 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999952 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=27 B=19 C=8 A=8 so C is eliminated. Round 2 votes counts: E=38 D=33 B=19 A=10 so A is eliminated. Round 3 votes counts: E=38 D=35 B=27 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:207 A:204 B:203 D:201 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -4 2 8 B -2 0 2 0 6 C 4 -2 0 4 8 D -2 0 -4 0 8 E -8 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999952 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 2 8 B -2 0 2 0 6 C 4 -2 0 4 8 D -2 0 -4 0 8 E -8 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999952 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 2 8 B -2 0 2 0 6 C 4 -2 0 4 8 D -2 0 -4 0 8 E -8 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999952 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1114: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) B C D E A (6) E D A B C (5) B D C E A (5) A E D C B (5) A C B E D (5) E A D C B (4) E A D B C (4) D E C B A (4) C B D E A (4) B D E C A (4) B C A D E (4) A C E D B (4) A B E D C (4) E D B A C (3) D C E B A (3) D C B E A (3) C B D A E (3) A E D B C (3) C D B E A (2) B A E D C (2) A E C D B (2) A B E C D (2) E B A D C (1) D E B C A (1) D C E A B (1) D B E C A (1) C D E A B (1) C B A D E (1) C A D E B (1) B C D A E (1) B A C E D (1) B A C D E (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 10 -6 -16 B 0 0 0 -6 0 C -10 0 0 -18 -4 D 6 6 18 0 -8 E 16 0 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.320051 C: 0.000000 D: 0.000000 E: 0.679949 Sum of squares = 0.5647633648 Cumulative probabilities = A: 0.000000 B: 0.320051 C: 0.320051 D: 0.320051 E: 1.000000 A B C D E A 0 0 10 -6 -16 B 0 0 0 -6 0 C -10 0 0 -18 -4 D 6 6 18 0 -8 E 16 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=24 E=23 D=13 C=12 so C is eliminated. Round 2 votes counts: B=32 A=29 E=23 D=16 so D is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:214 D:211 B:197 A:194 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 -6 -16 B 0 0 0 -6 0 C -10 0 0 -18 -4 D 6 6 18 0 -8 E 16 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -6 -16 B 0 0 0 -6 0 C -10 0 0 -18 -4 D 6 6 18 0 -8 E 16 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -6 -16 B 0 0 0 -6 0 C -10 0 0 -18 -4 D 6 6 18 0 -8 E 16 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1115: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) A E C B D (8) C D B A E (7) E A B D C (6) A E C D B (6) E A D B C (5) A C E D B (4) D B E A C (3) C A E D B (3) C A E B D (3) B E A D C (3) B D E A C (3) B D C E A (3) B C A E D (3) E D A B C (2) E A D C B (2) E A B C D (2) D E B A C (2) D C B A E (2) D B E C A (2) C D A B E (2) C B D A E (2) C B A E D (2) C A D E B (2) B D E C A (2) A E B C D (2) E B D A C (1) D E A B C (1) D C E A B (1) D C A E B (1) C D A E B (1) C B A D E (1) C A D B E (1) C A B E D (1) B E D A C (1) B C D E A (1) Total count = 100 A B C D E A 0 6 2 8 2 B -6 0 2 -12 -4 C -2 -2 0 2 -2 D -8 12 -2 0 -8 E -2 4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 8 2 B -6 0 2 -12 -4 C -2 -2 0 2 -2 D -8 12 -2 0 -8 E -2 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=21 A=20 E=18 B=16 so B is eliminated. Round 2 votes counts: D=29 C=29 E=22 A=20 so A is eliminated. Round 3 votes counts: E=38 C=33 D=29 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:209 E:206 C:198 D:197 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 8 2 B -6 0 2 -12 -4 C -2 -2 0 2 -2 D -8 12 -2 0 -8 E -2 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 8 2 B -6 0 2 -12 -4 C -2 -2 0 2 -2 D -8 12 -2 0 -8 E -2 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 8 2 B -6 0 2 -12 -4 C -2 -2 0 2 -2 D -8 12 -2 0 -8 E -2 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1116: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) C A E D B (6) B D E C A (6) E D C B A (4) D B E A C (4) C A E B D (4) B D A E C (4) E C B D A (3) E C A D B (3) E B C D A (3) D B A E C (3) C E A B D (3) C B A E D (3) B E C D A (3) B D A C E (3) A C D E B (3) E D B C A (2) E B D C A (2) D A B E C (2) C A B E D (2) B D E A C (2) B C D E A (2) A D E C B (2) A D C E B (2) A D B C E (2) E D A C B (1) E C D A B (1) E C A B D (1) D E A B C (1) D B E C A (1) D A E C B (1) D A E B C (1) C E B D A (1) C E B A D (1) C B E A D (1) B E D C A (1) B C E D A (1) B C D A E (1) B C A E D (1) B C A D E (1) A E C D B (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -14 -6 6 B 6 0 -8 2 -10 C 14 8 0 12 -4 D 6 -2 -12 0 -14 E -6 10 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.430555555583 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 A B C D E A 0 -6 -14 -6 6 B 6 0 -8 2 -10 C 14 8 0 12 -4 D 6 -2 -12 0 -14 E -6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.430555555544 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=21 A=21 E=20 D=13 so D is eliminated. Round 2 votes counts: B=33 A=25 E=21 C=21 so E is eliminated. Round 3 votes counts: B=40 C=33 A=27 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:211 B:195 A:190 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -6 6 B 6 0 -8 2 -10 C 14 8 0 12 -4 D 6 -2 -12 0 -14 E -6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.430555555544 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -6 6 B 6 0 -8 2 -10 C 14 8 0 12 -4 D 6 -2 -12 0 -14 E -6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.430555555544 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -6 6 B 6 0 -8 2 -10 C 14 8 0 12 -4 D 6 -2 -12 0 -14 E -6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.430555555544 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1117: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) A E B D C (9) E A D B C (8) E D A B C (7) C D E B A (7) C A B E D (5) D E C B A (3) D B E C A (3) C B D E A (3) C B D A E (3) C B A D E (3) B A E D C (3) A E C D B (3) E D B A C (2) C D E A B (2) B D C A E (2) A E D C B (2) A E D B C (2) A E B C D (2) A B E D C (2) E D A C B (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B D C (1) D E B C A (1) D E B A C (1) D C E B A (1) C E A D B (1) C D B A E (1) C A E B D (1) B D E A C (1) B D C E A (1) B C D A E (1) B C A D E (1) B A D E C (1) B A D C E (1) A E C B D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 2 -10 B -4 0 0 -14 -18 C -2 0 0 -6 -12 D -2 14 6 0 -10 E 10 18 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 2 -10 B -4 0 0 -14 -18 C -2 0 0 -6 -12 D -2 14 6 0 -10 E 10 18 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=23 E=22 B=11 D=9 so D is eliminated. Round 2 votes counts: C=36 E=27 A=23 B=14 so B is eliminated. Round 3 votes counts: C=41 E=31 A=28 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:204 A:199 C:190 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 2 -10 B -4 0 0 -14 -18 C -2 0 0 -6 -12 D -2 14 6 0 -10 E 10 18 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 -10 B -4 0 0 -14 -18 C -2 0 0 -6 -12 D -2 14 6 0 -10 E 10 18 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 -10 B -4 0 0 -14 -18 C -2 0 0 -6 -12 D -2 14 6 0 -10 E 10 18 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1118: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) A E B C D (8) E A B C D (6) B D E C A (6) D C B E A (5) A E C B D (5) E B A D C (4) D C B A E (4) C D B E A (4) B E D C A (4) D B C E A (3) C D E B A (3) B E A D C (3) E A C B D (2) D C A B E (2) D B A C E (2) C A E D B (2) B D E A C (2) B D A E C (2) A D C B E (2) A C E D B (2) A C D E B (2) E C B A D (1) E C A B D (1) E A B D C (1) D B C A E (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A D B (1) C E A B D (1) C D A E B (1) C D A B E (1) C B E D A (1) B E C D A (1) B D C E A (1) A E C D B (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 8 6 -6 B 0 0 10 18 -6 C -8 -10 0 -6 -16 D -6 -18 6 0 -12 E 6 6 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 6 -6 B 0 0 10 18 -6 C -8 -10 0 -6 -16 D -6 -18 6 0 -12 E 6 6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=19 D=18 C=17 E=15 so E is eliminated. Round 2 votes counts: A=40 B=23 C=19 D=18 so D is eliminated. Round 3 votes counts: A=40 C=30 B=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:220 B:211 A:204 D:185 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 6 -6 B 0 0 10 18 -6 C -8 -10 0 -6 -16 D -6 -18 6 0 -12 E 6 6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 6 -6 B 0 0 10 18 -6 C -8 -10 0 -6 -16 D -6 -18 6 0 -12 E 6 6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 6 -6 B 0 0 10 18 -6 C -8 -10 0 -6 -16 D -6 -18 6 0 -12 E 6 6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1119: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) B A C D E (10) E D C A B (7) E D B A C (7) C A B E D (7) E D C B A (5) A C B D E (5) C A B D E (4) B D E A C (4) E D B C A (3) D E A C B (3) C A E D B (3) B A D C E (3) B A C E D (3) D E C A B (2) D B E A C (2) C A D E B (2) B D A E C (2) B A D E C (2) E C D A B (1) E C B D A (1) E B C D A (1) D A B E C (1) C E D A B (1) C E B A D (1) C B A E D (1) C A E B D (1) B E C A D (1) B E A C D (1) B C A E D (1) B A E D C (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 16 -2 -2 B 20 0 10 2 2 C -16 -10 0 -8 -10 D 2 -2 8 0 8 E 2 -2 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 16 -2 -2 B 20 0 10 2 2 C -16 -10 0 -8 -10 D 2 -2 8 0 8 E 2 -2 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 C=20 D=19 A=8 so A is eliminated. Round 2 votes counts: B=30 C=26 E=25 D=19 so D is eliminated. Round 3 votes counts: E=41 B=33 C=26 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:208 E:201 A:196 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 16 -2 -2 B 20 0 10 2 2 C -16 -10 0 -8 -10 D 2 -2 8 0 8 E 2 -2 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 16 -2 -2 B 20 0 10 2 2 C -16 -10 0 -8 -10 D 2 -2 8 0 8 E 2 -2 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 16 -2 -2 B 20 0 10 2 2 C -16 -10 0 -8 -10 D 2 -2 8 0 8 E 2 -2 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1120: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) B A E C D (7) C D B E A (5) A B E C D (5) A B C D E (5) E B A C D (4) D C E A B (4) D A E C B (4) B A C E D (4) A B E D C (4) E D A C B (3) D C E B A (3) B C A E D (3) B C A D E (3) E D C A B (2) E C D B A (2) E C B D A (2) D E C A B (2) D E A C B (2) C D B A E (2) C B E A D (2) C B D E A (2) C B A D E (2) A E B D C (2) A D E B C (2) E D C B A (1) E D A B C (1) E B C A D (1) E A D B C (1) E A B D C (1) D C B A E (1) D A E B C (1) D A C B E (1) C E D B A (1) C E B D A (1) C B E D A (1) C B D A E (1) C B A E D (1) B C E A D (1) B A C D E (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 0 2 2 B 16 0 -4 8 4 C 0 4 0 26 2 D -2 -8 -26 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166802 B: 0.000000 C: 0.833198 D: 0.000000 E: 0.000000 Sum of squares = 0.722041972275 Cumulative probabilities = A: 0.166802 B: 0.166802 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 2 2 B 16 0 -4 8 4 C 0 4 0 26 2 D -2 -8 -26 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000050877 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=20 B=19 E=18 D=18 so E is eliminated. Round 2 votes counts: C=29 D=25 B=24 A=22 so A is eliminated. Round 3 votes counts: B=42 D=29 C=29 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:212 E:197 A:194 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 0 2 2 B 16 0 -4 8 4 C 0 4 0 26 2 D -2 -8 -26 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000050877 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 2 2 B 16 0 -4 8 4 C 0 4 0 26 2 D -2 -8 -26 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000050877 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 2 2 B 16 0 -4 8 4 C 0 4 0 26 2 D -2 -8 -26 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000050877 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1121: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (5) D B E A C (5) D B A E C (4) B D E A C (4) B C A E D (4) B A C D E (4) E D B C A (3) E C D A B (3) C A E B D (3) C A B E D (3) B D A C E (3) A D C E B (3) A D B C E (3) A C B D E (3) E C A D B (2) E B D C A (2) D E B A C (2) D E A C B (2) D E A B C (2) D A E C B (2) D A C E B (2) D A B C E (2) C E A B D (2) C B A E D (2) B E D C A (2) B D A E C (2) B C E A D (2) B A C E D (2) A D C B E (2) A B D C E (2) E C D B A (1) E C B D A (1) E C B A D (1) E C A B D (1) E B C D A (1) D E C B A (1) D E C A B (1) D E B C A (1) D A B E C (1) C E B A D (1) C E A D B (1) C B E A D (1) B E C D A (1) B E C A D (1) B D E C A (1) B A D C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 6 -8 0 B 6 0 10 2 14 C -6 -10 0 -16 -4 D 8 -2 16 0 8 E 0 -14 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -8 0 B 6 0 10 2 14 C -6 -10 0 -16 -4 D 8 -2 16 0 8 E 0 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 E=20 A=15 C=13 so C is eliminated. Round 2 votes counts: B=30 D=25 E=24 A=21 so A is eliminated. Round 3 votes counts: B=39 D=34 E=27 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:215 A:196 E:191 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -8 0 B 6 0 10 2 14 C -6 -10 0 -16 -4 D 8 -2 16 0 8 E 0 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -8 0 B 6 0 10 2 14 C -6 -10 0 -16 -4 D 8 -2 16 0 8 E 0 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -8 0 B 6 0 10 2 14 C -6 -10 0 -16 -4 D 8 -2 16 0 8 E 0 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1122: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) D C E A B (10) A C B D E (8) E D B C A (6) E B A D C (6) D E C B A (6) A B E C D (6) B A E D C (4) B A E C D (4) A B C E D (4) E D C B A (3) E D B A C (3) C D A E B (3) C A D B E (3) A B E D C (3) E B D C A (2) E B D A C (2) C D E A B (2) C D A B E (2) A D C E B (2) A C D B E (2) E A D B C (1) D E A C B (1) D C A E B (1) C B A D E (1) C A B D E (1) B E A D C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 -4 -8 B -2 0 -14 -14 -20 C 2 14 0 -24 4 D 4 14 24 0 10 E 8 20 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -4 -8 B -2 0 -14 -14 -20 C 2 14 0 -24 4 D 4 14 24 0 10 E 8 20 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 E=23 C=12 B=9 so B is eliminated. Round 2 votes counts: A=35 D=29 E=24 C=12 so C is eliminated. Round 3 votes counts: A=40 D=36 E=24 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 E:207 C:198 A:194 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -4 -8 B -2 0 -14 -14 -20 C 2 14 0 -24 4 D 4 14 24 0 10 E 8 20 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 -8 B -2 0 -14 -14 -20 C 2 14 0 -24 4 D 4 14 24 0 10 E 8 20 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 -8 B -2 0 -14 -14 -20 C 2 14 0 -24 4 D 4 14 24 0 10 E 8 20 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1123: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) D A B C E (7) C E A D B (7) B E D C A (7) B D E A C (7) B D A E C (7) A C D E B (7) E C B D A (5) C E A B D (5) E B C D A (4) D B A E C (4) D B A C E (4) A D B C E (4) E B C A D (3) A D C E B (3) A D C B E (3) D A C B E (2) B D E C A (2) A C E D B (2) E C B A D (1) E C A B D (1) D A C E B (1) B E D A C (1) B E C D A (1) B E C A D (1) B A E C D (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 10 -4 10 B -2 0 10 -4 6 C -10 -10 0 -8 10 D 4 4 8 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 -4 10 B -2 0 10 -4 6 C -10 -10 0 -8 10 D 4 4 8 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=20 A=20 D=18 E=14 so E is eliminated. Round 2 votes counts: B=35 C=27 A=20 D=18 so D is eliminated. Round 3 votes counts: B=43 A=30 C=27 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:209 B:205 C:191 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 10 -4 10 B -2 0 10 -4 6 C -10 -10 0 -8 10 D 4 4 8 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -4 10 B -2 0 10 -4 6 C -10 -10 0 -8 10 D 4 4 8 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -4 10 B -2 0 10 -4 6 C -10 -10 0 -8 10 D 4 4 8 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1124: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) B D E C A (8) C A E D B (7) A C E B D (7) D B E C A (6) A C B E D (6) E D C A B (5) D E B C A (5) C E A D B (5) B D E A C (5) B D A C E (4) B A C D E (4) E C A D B (3) E A C B D (3) B A C E D (3) D E C B A (2) D E C A B (2) B D A E C (2) A B C E D (2) E C D A B (1) E B A D C (1) D C E A B (1) D C B A E (1) C E D A B (1) C D E A B (1) C D A E B (1) B E D A C (1) B A E C D (1) B A D E C (1) B A D C E (1) A E C B D (1) Total count = 100 A B C D E A 0 10 2 8 0 B -10 0 -12 0 -10 C -2 12 0 10 6 D -8 0 -10 0 -12 E 0 10 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.838760 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.161240 Sum of squares = 0.729516156617 Cumulative probabilities = A: 0.838760 B: 0.838760 C: 0.838760 D: 0.838760 E: 1.000000 A B C D E A 0 10 2 8 0 B -10 0 -12 0 -10 C -2 12 0 10 6 D -8 0 -10 0 -12 E 0 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000024626 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=25 D=17 C=15 E=13 so E is eliminated. Round 2 votes counts: B=31 A=28 D=22 C=19 so C is eliminated. Round 3 votes counts: A=43 B=31 D=26 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:210 E:208 D:185 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 8 0 B -10 0 -12 0 -10 C -2 12 0 10 6 D -8 0 -10 0 -12 E 0 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000024626 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 8 0 B -10 0 -12 0 -10 C -2 12 0 10 6 D -8 0 -10 0 -12 E 0 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000024626 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 8 0 B -10 0 -12 0 -10 C -2 12 0 10 6 D -8 0 -10 0 -12 E 0 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000024626 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1125: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) E B C A D (8) D B E A C (8) E C A B D (7) D A C E B (6) D A C B E (6) C A E B D (5) B E A C D (4) B D E A C (4) D A B C E (3) C E A B D (3) B E D A C (3) E B D C A (2) D C A E B (2) D B C A E (2) D B A E C (2) C A E D B (2) C A D E B (2) B E D C A (2) A C E D B (2) A C D E B (2) E C D B A (1) E C A D B (1) E B A C D (1) D E C A B (1) D C E A B (1) D B E C A (1) B E C D A (1) B E A D C (1) B D E C A (1) B A D C E (1) A C E B D (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 16 -12 0 B 4 0 12 -4 6 C -16 -12 0 -12 4 D 12 4 12 0 8 E 0 -6 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 -12 0 B 4 0 12 -4 6 C -16 -12 0 -12 4 D 12 4 12 0 8 E 0 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 E=20 B=17 C=12 A=9 so A is eliminated. Round 2 votes counts: D=42 E=20 C=19 B=19 so C is eliminated. Round 3 votes counts: D=46 E=33 B=21 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:209 A:200 E:191 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 16 -12 0 B 4 0 12 -4 6 C -16 -12 0 -12 4 D 12 4 12 0 8 E 0 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 -12 0 B 4 0 12 -4 6 C -16 -12 0 -12 4 D 12 4 12 0 8 E 0 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 -12 0 B 4 0 12 -4 6 C -16 -12 0 -12 4 D 12 4 12 0 8 E 0 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1126: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (13) C A D E B (9) B E C A D (9) D A C B E (6) B E D A C (6) E B C A D (5) D A E C B (5) C A D B E (4) B E D C A (4) B E C D A (4) C B E A D (3) A C D E B (3) E D B A C (2) E C B A D (2) E B D A C (2) D E A B C (2) D B A E C (2) C E B A D (2) C A B D E (2) A D C E B (2) E B A C D (1) E A C B D (1) D A E B C (1) D A B E C (1) D A B C E (1) C E A B D (1) C D A B E (1) C B D A E (1) C B A E D (1) C A E D B (1) B C E D A (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 6 -4 -6 8 B -6 0 -14 -6 -4 C 4 14 0 6 6 D 6 6 -6 0 8 E -8 4 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -6 8 B -6 0 -14 -6 -4 C 4 14 0 6 6 D 6 6 -6 0 8 E -8 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 C=25 E=13 A=5 so A is eliminated. Round 2 votes counts: D=33 C=28 B=26 E=13 so E is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:207 A:202 E:191 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -6 8 B -6 0 -14 -6 -4 C 4 14 0 6 6 D 6 6 -6 0 8 E -8 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -6 8 B -6 0 -14 -6 -4 C 4 14 0 6 6 D 6 6 -6 0 8 E -8 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -6 8 B -6 0 -14 -6 -4 C 4 14 0 6 6 D 6 6 -6 0 8 E -8 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1127: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) B D A E C (8) E D B C A (7) E C D B A (7) C E A D B (7) C E A B D (7) A B D C E (7) D B E A C (6) B A D C E (6) A C B D E (6) D E B A C (4) D B A E C (4) C A B E D (4) E C D A B (3) C A B D E (3) E D C B A (2) A D B C E (2) A B C D E (2) C E B A D (1) C B A D E (1) C A E D B (1) C A E B D (1) B A D E C (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 12 2 -4 B 12 0 14 -4 4 C -12 -14 0 -12 -2 D -2 4 12 0 4 E 4 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.111111 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839511 Cumulative probabilities = A: 0.222222 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 -12 12 2 -4 B 12 0 14 -4 4 C -12 -14 0 -12 -2 D -2 4 12 0 4 E 4 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.111111 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839548 Cumulative probabilities = A: 0.222222 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 A=18 B=16 D=14 so D is eliminated. Round 2 votes counts: E=31 B=26 C=25 A=18 so A is eliminated. Round 3 votes counts: B=38 E=31 C=31 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:209 A:199 E:199 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 12 2 -4 B 12 0 14 -4 4 C -12 -14 0 -12 -2 D -2 4 12 0 4 E 4 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.111111 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839548 Cumulative probabilities = A: 0.222222 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 12 2 -4 B 12 0 14 -4 4 C -12 -14 0 -12 -2 D -2 4 12 0 4 E 4 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.111111 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839548 Cumulative probabilities = A: 0.222222 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 12 2 -4 B 12 0 14 -4 4 C -12 -14 0 -12 -2 D -2 4 12 0 4 E 4 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.111111 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839548 Cumulative probabilities = A: 0.222222 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1128: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (7) A E D C B (6) E D B C A (5) B C D E A (5) A E D B C (5) D E C B A (4) C B A D E (4) E D B A C (3) E D A C B (3) E A D B C (3) D E A C B (3) C A B D E (3) B A C E D (3) A D E C B (3) A C D E B (3) E D A B C (2) E B D A C (2) D C E A B (2) C D E B A (2) C D B E A (2) C B D E A (2) C B D A E (2) B E D C A (2) B E C D A (2) B C A D E (2) B A E C D (2) A C D B E (2) A B C D E (2) E D C B A (1) E A B D C (1) D C A E B (1) D A E C B (1) C D A E B (1) C D A B E (1) C A D E B (1) C A D B E (1) B C E D A (1) B C A E D (1) A E B D C (1) A D C E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 10 6 10 B -10 0 -12 -12 -8 C -10 12 0 2 0 D -6 12 -2 0 10 E -10 8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 6 10 B -10 0 -12 -12 -8 C -10 12 0 2 0 D -6 12 -2 0 10 E -10 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=20 C=19 B=18 D=11 so D is eliminated. Round 2 votes counts: A=33 E=27 C=22 B=18 so B is eliminated. Round 3 votes counts: A=38 E=31 C=31 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:207 C:202 E:194 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 6 10 B -10 0 -12 -12 -8 C -10 12 0 2 0 D -6 12 -2 0 10 E -10 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 6 10 B -10 0 -12 -12 -8 C -10 12 0 2 0 D -6 12 -2 0 10 E -10 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 6 10 B -10 0 -12 -12 -8 C -10 12 0 2 0 D -6 12 -2 0 10 E -10 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1129: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) A D B C E (9) E C B D A (8) E B C A D (7) B E A D C (7) B A E D C (7) C E D B A (6) C D A E B (5) E B A C D (4) D A C B E (4) B E A C D (4) A D C B E (4) C E B D A (3) C D E B A (3) C D E A B (3) B E C A D (3) A D B E C (3) D C A E B (2) D C A B E (2) B A D E C (2) C D B E A (1) C D A B E (1) B E C D A (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -30 -8 4 -18 B 30 0 14 12 -2 C 8 -14 0 18 -12 D -4 -12 -18 0 -18 E 18 2 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -30 -8 4 -18 B 30 0 14 12 -2 C 8 -14 0 18 -12 D -4 -12 -18 0 -18 E 18 2 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=24 C=22 A=18 D=8 so D is eliminated. Round 2 votes counts: E=28 C=26 B=24 A=22 so A is eliminated. Round 3 votes counts: B=37 C=35 E=28 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:225 C:200 A:174 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -30 -8 4 -18 B 30 0 14 12 -2 C 8 -14 0 18 -12 D -4 -12 -18 0 -18 E 18 2 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -8 4 -18 B 30 0 14 12 -2 C 8 -14 0 18 -12 D -4 -12 -18 0 -18 E 18 2 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -8 4 -18 B 30 0 14 12 -2 C 8 -14 0 18 -12 D -4 -12 -18 0 -18 E 18 2 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1130: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (13) E C B D A (9) E B C D A (9) C E B D A (9) C E A D B (7) A D B E C (7) A D C B E (6) E C B A D (4) D A B C E (3) C E B A D (3) C A D E B (3) B D A E C (3) B E D A C (2) B E C D A (2) B D A C E (2) B A D E C (2) A D C E B (2) A C D E B (2) E C A D B (1) E A B D C (1) D B A E C (1) D A B E C (1) C E D A B (1) C B E D A (1) C A D B E (1) B E D C A (1) B C E D A (1) A E D B C (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -4 10 -2 B -2 0 0 -2 -8 C 4 0 0 8 10 D -10 2 -8 0 -6 E 2 8 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.387756 C: 0.612244 D: 0.000000 E: 0.000000 Sum of squares = 0.525197322014 Cumulative probabilities = A: 0.000000 B: 0.387756 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 10 -2 B -2 0 0 -2 -8 C 4 0 0 8 10 D -10 2 -8 0 -6 E 2 8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=25 E=24 B=13 D=5 so D is eliminated. Round 2 votes counts: A=37 C=25 E=24 B=14 so B is eliminated. Round 3 votes counts: A=45 E=29 C=26 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:211 A:203 E:203 B:194 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 10 -2 B -2 0 0 -2 -8 C 4 0 0 8 10 D -10 2 -8 0 -6 E 2 8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 10 -2 B -2 0 0 -2 -8 C 4 0 0 8 10 D -10 2 -8 0 -6 E 2 8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 10 -2 B -2 0 0 -2 -8 C 4 0 0 8 10 D -10 2 -8 0 -6 E 2 8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1131: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) C A E D B (7) E A C B D (6) B E A D C (6) E A B C D (5) D B E C A (5) C D A E B (4) C A E B D (4) B D E A C (4) D C B A E (3) D B C A E (3) C D A B E (3) C A B E D (3) B E A C D (3) B A E C D (3) E B A D C (2) E A C D B (2) D E A C B (2) D C E A B (2) D B E A C (2) B E D A C (2) B D C A E (2) A E C B D (2) E D A C B (1) E C A D B (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) D C A E B (1) C D B A E (1) C B D A E (1) C A D E B (1) B D C E A (1) B C D A E (1) B C A E D (1) B A C E D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -6 4 -14 B 4 0 12 4 10 C 6 -12 0 4 -4 D -4 -4 -4 0 -10 E 14 -10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 4 -14 B 4 0 12 4 10 C 6 -12 0 4 -4 D -4 -4 -4 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 B=24 E=21 A=4 so A is eliminated. Round 2 votes counts: D=27 B=25 E=24 C=24 so E is eliminated. Round 3 votes counts: B=36 C=35 D=29 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:209 C:197 A:190 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 4 -14 B 4 0 12 4 10 C 6 -12 0 4 -4 D -4 -4 -4 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 -14 B 4 0 12 4 10 C 6 -12 0 4 -4 D -4 -4 -4 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 -14 B 4 0 12 4 10 C 6 -12 0 4 -4 D -4 -4 -4 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1132: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) B D C E A (9) D A E C B (8) B D A C E (7) A E C D B (7) D B A E C (6) B C E A D (5) E C A D B (4) E C A B D (4) E A C D B (4) E C D A B (3) D E A C B (3) D B E C A (3) D B A C E (3) C A E B D (3) D E C A B (2) D A B E C (2) C B E A D (2) B D C A E (2) B C D E A (2) A C E B D (2) D E C B A (1) D A B C E (1) C E B A D (1) B C E D A (1) B C A E D (1) B A C E D (1) A E D C B (1) A D E C B (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -4 -6 -6 B -12 0 -12 0 -8 C 4 12 0 -2 2 D 6 0 2 0 4 E 6 8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.082253 C: 0.000000 D: 0.917747 E: 0.000000 Sum of squares = 0.849025395619 Cumulative probabilities = A: 0.000000 B: 0.082253 C: 0.082253 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 -6 -6 B -12 0 -12 0 -8 C 4 12 0 -2 2 D 6 0 2 0 4 E 6 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.75510205217 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=28 E=15 C=15 A=13 so A is eliminated. Round 2 votes counts: D=31 B=29 E=23 C=17 so C is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:208 D:206 E:204 A:198 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -4 -6 -6 B -12 0 -12 0 -8 C 4 12 0 -2 2 D 6 0 2 0 4 E 6 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.75510205217 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 -6 -6 B -12 0 -12 0 -8 C 4 12 0 -2 2 D 6 0 2 0 4 E 6 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.75510205217 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 -6 -6 B -12 0 -12 0 -8 C 4 12 0 -2 2 D 6 0 2 0 4 E 6 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.75510205217 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1133: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) C B A D E (10) D E B A C (6) E D A C B (5) D E B C A (5) C E A B D (5) C A E B D (4) C A B E D (4) E D C B A (3) E C A B D (3) C A B D E (3) B A C D E (3) E D C A B (2) E A D B C (2) D B E A C (2) D B C A E (2) D B A E C (2) C E D B A (2) B D A C E (2) B C A D E (2) B A D C E (2) A B C D E (2) E D B C A (1) E D B A C (1) E C D A B (1) E C A D B (1) E A C B D (1) D B E C A (1) C E B A D (1) C D B A E (1) C B A E D (1) B D C A E (1) B C D A E (1) A E B D C (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -8 2 -6 B -2 0 0 4 -12 C 8 0 0 -2 0 D -2 -4 2 0 -6 E 6 12 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.410948 D: 0.000000 E: 0.589052 Sum of squares = 0.515860457566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.410948 D: 0.410948 E: 1.000000 A B C D E A 0 2 -8 2 -6 B -2 0 0 4 -12 C 8 0 0 -2 0 D -2 -4 2 0 -6 E 6 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=31 C=31 D=18 B=11 A=9 so A is eliminated. Round 2 votes counts: C=34 E=32 D=18 B=16 so B is eliminated. Round 3 votes counts: C=43 E=33 D=24 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:203 A:195 B:195 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 2 -6 B -2 0 0 4 -12 C 8 0 0 -2 0 D -2 -4 2 0 -6 E 6 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 2 -6 B -2 0 0 4 -12 C 8 0 0 -2 0 D -2 -4 2 0 -6 E 6 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 2 -6 B -2 0 0 4 -12 C 8 0 0 -2 0 D -2 -4 2 0 -6 E 6 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1134: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) C B D A E (11) C B D E A (9) E C A D B (7) C E A B D (7) E A D B C (6) C A E B D (5) B D C A E (5) D B E C A (4) E A C D B (3) D B A E C (3) E D A B C (2) E C D A B (2) D B E A C (2) C E D B A (2) C A B D E (2) B D A C E (2) A E C D B (2) A E C B D (2) A B D C E (2) E D C B A (1) E D B C A (1) E C D B A (1) C E B D A (1) C E B A D (1) B D C E A (1) B C D A E (1) A D E B C (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -22 4 0 B -10 0 -14 2 -14 C 22 14 0 16 2 D -4 -2 -16 0 -12 E 0 14 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -22 4 0 B -10 0 -14 2 -14 C 22 14 0 16 2 D -4 -2 -16 0 -12 E 0 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=23 A=21 D=9 B=9 so D is eliminated. Round 2 votes counts: C=38 E=23 A=21 B=18 so B is eliminated. Round 3 votes counts: C=45 E=29 A=26 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:212 A:196 D:183 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -22 4 0 B -10 0 -14 2 -14 C 22 14 0 16 2 D -4 -2 -16 0 -12 E 0 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -22 4 0 B -10 0 -14 2 -14 C 22 14 0 16 2 D -4 -2 -16 0 -12 E 0 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -22 4 0 B -10 0 -14 2 -14 C 22 14 0 16 2 D -4 -2 -16 0 -12 E 0 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1135: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (12) B D C A E (11) C B E A D (8) D B A C E (6) D A E C B (6) E C A B D (5) E A C D B (5) D A B E C (5) C E B A D (5) C E A B D (5) D A E B C (4) B D C E A (4) D B C A E (3) D B A E C (3) E A D C B (2) E A C B D (2) B C D E A (2) A D E C B (2) E C A D B (1) D B E C A (1) D A B C E (1) C A E B D (1) C A B E D (1) B C D A E (1) B C A D E (1) A E D C B (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -22 6 -4 B 14 0 8 20 18 C 22 -8 0 2 24 D -6 -20 -2 0 0 E 4 -18 -24 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -22 6 -4 B 14 0 8 20 18 C 22 -8 0 2 24 D -6 -20 -2 0 0 E 4 -18 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 C=20 E=15 A=5 so A is eliminated. Round 2 votes counts: D=31 B=31 C=21 E=17 so E is eliminated. Round 3 votes counts: C=35 D=34 B=31 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:230 C:220 D:186 A:183 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -22 6 -4 B 14 0 8 20 18 C 22 -8 0 2 24 D -6 -20 -2 0 0 E 4 -18 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -22 6 -4 B 14 0 8 20 18 C 22 -8 0 2 24 D -6 -20 -2 0 0 E 4 -18 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -22 6 -4 B 14 0 8 20 18 C 22 -8 0 2 24 D -6 -20 -2 0 0 E 4 -18 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1136: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) D C B A E (9) D A B E C (6) C D B E A (5) C D B A E (5) C B E A D (4) E C B A D (3) E C A D B (3) D A C E B (3) C D E B A (3) B A E D C (3) A E B D C (3) E B A C D (2) E A D B C (2) E A C B D (2) E A B D C (2) D C A E B (2) D B C A E (2) D A E B C (2) C E D A B (2) C D E A B (2) C B A E D (2) B E A C D (2) B A E C D (2) A D E B C (2) E C D A B (1) E B C A D (1) E A B C D (1) D E A C B (1) D C E A B (1) D B A E C (1) D A E C B (1) D A B C E (1) C E D B A (1) C B E D A (1) C B D E A (1) C B D A E (1) B D A C E (1) B C A E D (1) B A D E C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 -20 -4 0 B 22 0 -26 -12 0 C 20 26 0 10 14 D 4 12 -10 0 4 E 0 0 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -20 -4 0 B 22 0 -26 -12 0 C 20 26 0 10 14 D 4 12 -10 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=29 E=17 B=10 A=7 so A is eliminated. Round 2 votes counts: C=37 D=32 E=20 B=11 so B is eliminated. Round 3 votes counts: C=38 D=35 E=27 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:235 D:205 B:192 E:191 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -20 -4 0 B 22 0 -26 -12 0 C 20 26 0 10 14 D 4 12 -10 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -20 -4 0 B 22 0 -26 -12 0 C 20 26 0 10 14 D 4 12 -10 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -20 -4 0 B 22 0 -26 -12 0 C 20 26 0 10 14 D 4 12 -10 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1137: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (23) A C D E B (19) E B D C A (7) D C A E B (7) C A D E B (6) A C D B E (5) B E D A C (4) B E C A D (4) C A E D B (3) E D B C A (2) E C A B D (2) B E A C D (2) B D E C A (2) B D A C E (2) B A C E D (2) A C B D E (2) E D C A B (1) D E C A B (1) D E B C A (1) D B A C E (1) D A C E B (1) D A C B E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -18 -6 2 B 0 0 0 4 -2 C 18 0 0 -6 2 D 6 -4 6 0 -4 E -2 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 A B C D E A 0 0 -18 -6 2 B 0 0 0 4 -2 C 18 0 0 -6 2 D 6 -4 6 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.38888888864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=28 E=12 D=12 C=9 so C is eliminated. Round 2 votes counts: B=39 A=37 E=12 D=12 so E is eliminated. Round 3 votes counts: B=46 A=39 D=15 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:207 D:202 B:201 E:201 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -18 -6 2 B 0 0 0 4 -2 C 18 0 0 -6 2 D 6 -4 6 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.38888888864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 -6 2 B 0 0 0 4 -2 C 18 0 0 -6 2 D 6 -4 6 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.38888888864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 -6 2 B 0 0 0 4 -2 C 18 0 0 -6 2 D 6 -4 6 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.38888888864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1138: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) C B E A D (10) B C A E D (10) A B C D E (9) E D C B A (6) A D B C E (6) D E C B A (5) D E C A B (5) D E A C B (5) C E B D A (5) B A C E D (4) A B D C E (3) A B C E D (3) E C D B A (2) E C B D A (2) E C B A D (2) D A B E C (2) C B A E D (2) D E A B C (1) C E D B A (1) C E B A D (1) C B D E A (1) B C D A E (1) B A C D E (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -6 6 8 B 6 0 6 10 8 C 6 -6 0 10 14 D -6 -10 -10 0 2 E -8 -8 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 6 8 B 6 0 6 10 8 C 6 -6 0 10 14 D -6 -10 -10 0 2 E -8 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=23 C=20 B=16 E=12 so E is eliminated. Round 2 votes counts: D=35 C=26 A=23 B=16 so B is eliminated. Round 3 votes counts: C=37 D=35 A=28 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:212 A:201 D:188 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 6 8 B 6 0 6 10 8 C 6 -6 0 10 14 D -6 -10 -10 0 2 E -8 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 6 8 B 6 0 6 10 8 C 6 -6 0 10 14 D -6 -10 -10 0 2 E -8 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 6 8 B 6 0 6 10 8 C 6 -6 0 10 14 D -6 -10 -10 0 2 E -8 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1139: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) D B A E C (6) C E A B D (5) C A E B D (5) B D E C A (5) E C A D B (4) C B A D E (4) B D C A E (4) E D B A C (3) C B E D A (3) B D A C E (3) A E D C B (3) A C E D B (3) E C B D A (2) E C A B D (2) D E B A C (2) D E A B C (2) D B A C E (2) D A B C E (2) C E B A D (2) B C D A E (2) A D E C B (2) A D E B C (2) A D C B E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E C B A D (1) E B D C A (1) E B C D A (1) E A D B C (1) E A C D B (1) D A E B C (1) D A B E C (1) C E B D A (1) C B E A D (1) C A B E D (1) C A B D E (1) B E C D A (1) B D E A C (1) B C D E A (1) B A C D E (1) A D B E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 0 6 14 B -2 0 6 0 4 C 0 -6 0 -8 8 D -6 0 8 0 12 E -14 -4 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.827089 B: 0.000000 C: 0.172911 D: 0.000000 E: 0.000000 Sum of squares = 0.713974867606 Cumulative probabilities = A: 0.827089 B: 0.827089 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 6 14 B -2 0 6 0 4 C 0 -6 0 -8 8 D -6 0 8 0 12 E -14 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000059947 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 E=19 B=18 D=16 so D is eliminated. Round 2 votes counts: A=28 B=26 E=23 C=23 so E is eliminated. Round 3 votes counts: C=34 B=34 A=32 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 D:207 B:204 C:197 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 6 14 B -2 0 6 0 4 C 0 -6 0 -8 8 D -6 0 8 0 12 E -14 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000059947 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 14 B -2 0 6 0 4 C 0 -6 0 -8 8 D -6 0 8 0 12 E -14 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000059947 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 14 B -2 0 6 0 4 C 0 -6 0 -8 8 D -6 0 8 0 12 E -14 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000059947 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1140: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (12) D B E A C (11) A C E B D (9) B E D C A (6) E B D C A (5) D A C B E (5) A C D B E (5) E B C A D (4) D B E C A (4) C A E B D (4) E C B A D (3) E B C D A (3) D B A E C (3) A D C B E (3) D C A E B (2) D A B C E (2) C E A B D (2) B E D A C (2) B E A C D (2) B D E C A (2) B D E A C (2) A C E D B (2) E C A B D (1) D B A C E (1) D A C E B (1) C A E D B (1) B E A D C (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 26 2 4 B -2 0 -2 -6 0 C -26 2 0 -2 2 D -2 6 2 0 8 E -4 0 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 26 2 4 B -2 0 -2 -6 0 C -26 2 0 -2 2 D -2 6 2 0 8 E -4 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=29 E=16 B=15 C=7 so C is eliminated. Round 2 votes counts: A=38 D=29 E=18 B=15 so B is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:207 B:195 E:193 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 26 2 4 B -2 0 -2 -6 0 C -26 2 0 -2 2 D -2 6 2 0 8 E -4 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 26 2 4 B -2 0 -2 -6 0 C -26 2 0 -2 2 D -2 6 2 0 8 E -4 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 26 2 4 B -2 0 -2 -6 0 C -26 2 0 -2 2 D -2 6 2 0 8 E -4 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1141: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) A D E B C (7) B C E A D (6) C E D A B (5) E C D A B (4) C E B D A (4) B A D E C (4) E D A C B (3) D E A C B (3) D A E C B (3) D A E B C (3) D A B E C (3) B C A D E (3) B A D C E (3) A D B E C (3) A B D E C (3) E B C A D (2) D A C B E (2) D A B C E (2) C E D B A (2) C D E A B (2) C B D A E (2) B A E D C (2) E C B A D (1) E B A C D (1) E A D B C (1) E A B D C (1) E A B C D (1) D C A E B (1) D A C E B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C B E A D (1) C B A D E (1) B E C A D (1) B E A D C (1) B D C A E (1) B D A C E (1) B C A E D (1) B A C E D (1) B A C D E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 4 8 4 B -2 0 8 4 -6 C -4 -8 0 0 2 D -8 -4 0 0 4 E -4 6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 4 B -2 0 8 4 -6 C -4 -8 0 0 2 D -8 -4 0 0 4 E -4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995269 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 D=18 A=15 E=14 so E is eliminated. Round 2 votes counts: C=33 B=28 D=21 A=18 so A is eliminated. Round 3 votes counts: B=35 C=33 D=32 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:202 E:198 D:196 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 4 B -2 0 8 4 -6 C -4 -8 0 0 2 D -8 -4 0 0 4 E -4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995269 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 4 B -2 0 8 4 -6 C -4 -8 0 0 2 D -8 -4 0 0 4 E -4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995269 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 4 B -2 0 8 4 -6 C -4 -8 0 0 2 D -8 -4 0 0 4 E -4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995269 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1142: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) C B E A D (7) A D B E C (7) D E A B C (6) C E B D A (5) C A D B E (5) E B D C A (4) E D B A C (3) E C B D A (3) E B D A C (3) D A E C B (3) C B A E D (3) C B A D E (3) C A B D E (3) A D C B E (3) E D A B C (2) D E A C B (2) C E D A B (2) C E B A D (2) C D E A B (2) B E C D A (2) B C E A D (2) B C A E D (2) A D E C B (2) A C B D E (2) E D C B A (1) E D C A B (1) E D A C B (1) E B C D A (1) C D A E B (1) C A D E B (1) C A B E D (1) B E C A D (1) B C E D A (1) B A C E D (1) B A C D E (1) A D C E B (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -6 0 -2 B -10 0 -10 -4 -6 C 6 10 0 4 0 D 0 4 -4 0 4 E 2 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.747015 D: 0.000000 E: 0.252985 Sum of squares = 0.622032380006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.747015 D: 0.747015 E: 1.000000 A B C D E A 0 10 -6 0 -2 B -10 0 -10 -4 -6 C 6 10 0 4 0 D 0 4 -4 0 4 E 2 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.000000 E: 0.499870 Sum of squares = 0.500000033874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.500130 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=19 D=18 A=18 B=10 so B is eliminated. Round 2 votes counts: C=40 E=22 A=20 D=18 so D is eliminated. Round 3 votes counts: C=40 E=30 A=30 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:202 E:202 A:201 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 0 -2 B -10 0 -10 -4 -6 C 6 10 0 4 0 D 0 4 -4 0 4 E 2 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.000000 E: 0.499870 Sum of squares = 0.500000033874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.500130 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 0 -2 B -10 0 -10 -4 -6 C 6 10 0 4 0 D 0 4 -4 0 4 E 2 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.000000 E: 0.499870 Sum of squares = 0.500000033874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.500130 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 0 -2 B -10 0 -10 -4 -6 C 6 10 0 4 0 D 0 4 -4 0 4 E 2 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.000000 E: 0.499870 Sum of squares = 0.500000033874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500130 D: 0.500130 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1143: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (6) C B E A D (5) C B A E D (5) B A E C D (5) D E C B A (4) C B E D A (4) C B A D E (4) A B E C D (4) D E C A B (3) D E A B C (3) D A E C B (3) C D A B E (3) B E C A D (3) A D E B C (3) E D B C A (2) E C B D A (2) E B A D C (2) E A B D C (2) C E B D A (2) B E A C D (2) B C E A D (2) B C A E D (2) A D B E C (2) A B C E D (2) A B C D E (2) E D C B A (1) E D A B C (1) E C D B A (1) E B D A C (1) E B C D A (1) E B A C D (1) D E A C B (1) D C E B A (1) D C E A B (1) D C A E B (1) D A E B C (1) D A C B E (1) D A B E C (1) C D B E A (1) C D B A E (1) C B D A E (1) C A D B E (1) C A B D E (1) B A C E D (1) A E D B C (1) A E B D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -16 6 -4 B 20 0 -6 14 10 C 16 6 0 24 -4 D -6 -14 -24 0 -8 E 4 -10 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.380000000064 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 -20 -16 6 -4 B 20 0 -6 14 10 C 16 6 0 24 -4 D -6 -14 -24 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999953 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=20 A=17 B=15 E=14 so E is eliminated. Round 2 votes counts: C=37 D=24 B=20 A=19 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:219 E:203 A:183 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -16 6 -4 B 20 0 -6 14 10 C 16 6 0 24 -4 D -6 -14 -24 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999953 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 6 -4 B 20 0 -6 14 10 C 16 6 0 24 -4 D -6 -14 -24 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999953 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 6 -4 B 20 0 -6 14 10 C 16 6 0 24 -4 D -6 -14 -24 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999953 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1144: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) D C E A B (7) A B E D C (7) A E B D C (6) A B E C D (6) B A C E D (5) E D A C B (4) D C E B A (4) C D E B A (4) E C D A B (3) E A D C B (3) B A D C E (3) B A C D E (3) A E D B C (3) E C B A D (2) E A B C D (2) D C A B E (2) D B A C E (2) D A C B E (2) C E D B A (2) B E A C D (2) B C A E D (2) A D B E C (2) E C A B D (1) E A C B D (1) E A B D C (1) D E C A B (1) D C B A E (1) D B C A E (1) C E B D A (1) C D B E A (1) C B E D A (1) C B D A E (1) B C E A D (1) B C D A E (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 28 24 18 B -8 0 18 12 8 C -28 -18 0 -4 -8 D -24 -12 4 0 -24 E -18 -8 8 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 28 24 18 B -8 0 18 12 8 C -28 -18 0 -4 -8 D -24 -12 4 0 -24 E -18 -8 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 D=20 E=17 C=10 so C is eliminated. Round 2 votes counts: B=28 A=27 D=25 E=20 so E is eliminated. Round 3 votes counts: A=35 D=34 B=31 so B is eliminated. Round 4 votes counts: A=62 D=38 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:239 B:215 E:203 D:172 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 28 24 18 B -8 0 18 12 8 C -28 -18 0 -4 -8 D -24 -12 4 0 -24 E -18 -8 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 28 24 18 B -8 0 18 12 8 C -28 -18 0 -4 -8 D -24 -12 4 0 -24 E -18 -8 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 28 24 18 B -8 0 18 12 8 C -28 -18 0 -4 -8 D -24 -12 4 0 -24 E -18 -8 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1145: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) C D A E B (8) E C B A D (7) D C A B E (5) B A D E C (5) E C A B D (4) E B A C D (4) D A C B E (4) C E D A B (4) A D C B E (4) E B C A D (3) D B A C E (3) C E A D B (3) B E A D C (3) B D A E C (3) B A E D C (3) A D B C E (3) A B D E C (3) E C B D A (2) E B C D A (2) C E D B A (2) C D E A B (2) C A D E B (2) A C D E B (2) A B E D C (2) E A C B D (1) D C B E A (1) D C B A E (1) C D E B A (1) C A E D B (1) B E D C A (1) A B E C D (1) Total count = 100 A B C D E A 0 18 2 2 20 B -18 0 -8 -12 4 C -2 8 0 -2 12 D -2 12 2 0 14 E -20 -4 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999243 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 2 20 B -18 0 -8 -12 4 C -2 8 0 -2 12 D -2 12 2 0 14 E -20 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 C=23 B=15 A=15 so B is eliminated. Round 2 votes counts: E=27 D=27 C=23 A=23 so C is eliminated. Round 3 votes counts: D=38 E=36 A=26 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:221 D:213 C:208 B:183 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 2 20 B -18 0 -8 -12 4 C -2 8 0 -2 12 D -2 12 2 0 14 E -20 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 2 20 B -18 0 -8 -12 4 C -2 8 0 -2 12 D -2 12 2 0 14 E -20 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 2 20 B -18 0 -8 -12 4 C -2 8 0 -2 12 D -2 12 2 0 14 E -20 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1146: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (11) C D A E B (9) E A B D C (8) B E A D C (8) E B A C D (6) D C A E B (5) E A D B C (4) B C D E A (4) E A D C B (3) E A C D B (3) D C A B E (3) B E A C D (3) B D C A E (3) B C D A E (3) A E D C B (3) E B A D C (2) D B A C E (2) C B D A E (2) B E C A D (2) B D C E A (2) B D A E C (2) E C A D B (1) E A B C D (1) D A C E B (1) C E D B A (1) C D E A B (1) C D B E A (1) C B D E A (1) C A D E B (1) A E D B C (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 0 -2 -2 B 6 0 4 -6 -6 C 0 -4 0 2 2 D 2 6 -2 0 6 E 2 6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888774 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 -2 B 6 0 4 -6 -6 C 0 -4 0 2 2 D 2 6 -2 0 6 E 2 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888881 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 B=27 D=11 A=7 so A is eliminated. Round 2 votes counts: E=32 C=28 B=27 D=13 so D is eliminated. Round 3 votes counts: C=37 E=34 B=29 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:206 C:200 E:200 B:199 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -2 -2 B 6 0 4 -6 -6 C 0 -4 0 2 2 D 2 6 -2 0 6 E 2 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888881 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 -2 B 6 0 4 -6 -6 C 0 -4 0 2 2 D 2 6 -2 0 6 E 2 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888881 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 -2 B 6 0 4 -6 -6 C 0 -4 0 2 2 D 2 6 -2 0 6 E 2 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888881 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1147: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (13) A C D B E (11) E B D A C (6) C A D B E (5) B E D C A (5) B D E C A (5) C D A B E (4) B E D A C (4) E A B C D (3) C A D E B (3) A E C B D (3) A C E D B (3) E C A D B (2) E A C D B (2) D C A B E (2) D B E C A (2) C A E D B (2) B E A D C (2) B A E D C (2) A C D E B (2) E D C B A (1) E D B C A (1) E C D B A (1) E B A D C (1) E A C B D (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B E A (1) D C B A E (1) D B C E A (1) D B C A E (1) B D C A E (1) B A D C E (1) A E C D B (1) A E B C D (1) A C E B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -8 -4 -8 B 4 0 2 2 0 C 8 -2 0 -8 -16 D 4 -2 8 0 -10 E 8 0 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.708707 C: 0.000000 D: 0.000000 E: 0.291293 Sum of squares = 0.587116979568 Cumulative probabilities = A: 0.000000 B: 0.708707 C: 0.708707 D: 0.708707 E: 1.000000 A B C D E A 0 -4 -8 -4 -8 B 4 0 2 2 0 C 8 -2 0 -8 -16 D 4 -2 8 0 -10 E 8 0 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=24 B=20 C=14 D=11 so D is eliminated. Round 2 votes counts: E=33 B=24 A=24 C=19 so C is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:204 D:200 C:191 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 -8 B 4 0 2 2 0 C 8 -2 0 -8 -16 D 4 -2 8 0 -10 E 8 0 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 -8 B 4 0 2 2 0 C 8 -2 0 -8 -16 D 4 -2 8 0 -10 E 8 0 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 -8 B 4 0 2 2 0 C 8 -2 0 -8 -16 D 4 -2 8 0 -10 E 8 0 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1148: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) C B E D A (9) D B E A C (5) A D E B C (5) A D C B E (5) E B D C A (4) E B C D A (4) D A E B C (4) D A B E C (4) C E B A D (4) A D B C E (4) A C D B E (4) B E D C A (3) A D B E C (3) E D B A C (2) E C B D A (2) D B A C E (2) C A E B D (2) B D E C A (2) B D C A E (2) B C D E A (2) A C E D B (2) E D B C A (1) E A C D B (1) D E B A C (1) D B A E C (1) D A B C E (1) C B E A D (1) C A B E D (1) B E C D A (1) B D C E A (1) B D A C E (1) B C E D A (1) A E D C B (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -20 -2 -28 -10 B 20 0 8 6 8 C 2 -8 0 -4 12 D 28 -6 4 0 -2 E 10 -8 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -2 -28 -10 B 20 0 8 6 8 C 2 -8 0 -4 12 D 28 -6 4 0 -2 E 10 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 D=18 E=14 B=13 so B is eliminated. Round 2 votes counts: C=31 A=27 D=24 E=18 so E is eliminated. Round 3 votes counts: C=38 D=34 A=28 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:221 D:212 C:201 E:196 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -2 -28 -10 B 20 0 8 6 8 C 2 -8 0 -4 12 D 28 -6 4 0 -2 E 10 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -2 -28 -10 B 20 0 8 6 8 C 2 -8 0 -4 12 D 28 -6 4 0 -2 E 10 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -2 -28 -10 B 20 0 8 6 8 C 2 -8 0 -4 12 D 28 -6 4 0 -2 E 10 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1149: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (15) C D A B E (15) D C E A B (10) B A C E D (8) E B A D C (7) D E C B A (7) B A E C D (7) C A B D E (5) A B C E D (4) D E C A B (2) D E B C A (2) D E B A C (2) D E A B C (2) C D B A E (2) A B E C D (2) E D B C A (1) E B A C D (1) D C E B A (1) D C A E B (1) D C A B E (1) C D A E B (1) C B A D E (1) C A D B E (1) C A B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -2 -24 0 B 8 0 4 -26 -4 C 2 -4 0 -4 4 D 24 26 4 0 8 E 0 4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -24 0 B 8 0 4 -26 -4 C 2 -4 0 -4 4 D 24 26 4 0 8 E 0 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 E=24 B=15 A=7 so A is eliminated. Round 2 votes counts: D=28 C=26 E=24 B=22 so B is eliminated. Round 3 votes counts: C=38 E=33 D=29 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:231 C:199 E:196 B:191 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -24 0 B 8 0 4 -26 -4 C 2 -4 0 -4 4 D 24 26 4 0 8 E 0 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -24 0 B 8 0 4 -26 -4 C 2 -4 0 -4 4 D 24 26 4 0 8 E 0 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -24 0 B 8 0 4 -26 -4 C 2 -4 0 -4 4 D 24 26 4 0 8 E 0 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1150: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (5) D B C E A (5) E A B C D (4) D E B A C (4) E B A D C (3) D E B C A (3) D C B A E (3) D B E A C (3) C D A B E (3) C B D A E (3) C A D E B (3) C A B E D (3) B A C E D (3) A C B E D (3) A B E C D (3) A B C E D (3) E D C A B (2) E D B A C (2) E D A B C (2) E B D A C (2) E A C B D (2) D E C B A (2) D C B E A (2) D C A E B (2) C B A D E (2) C A D B E (2) B E D A C (2) B C A D E (2) B A E D C (2) B A E C D (2) A C E B D (2) E D A C B (1) E C A D B (1) E B A C D (1) E A D C B (1) E A D B C (1) D C E A B (1) D C A B E (1) D B E C A (1) D B C A E (1) C D A E B (1) C A E B D (1) C A B D E (1) B D A E C (1) B C D A E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 0 -2 -6 0 B 0 0 4 -4 4 C 2 -4 0 -4 -4 D 6 4 4 0 4 E 0 -4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -6 0 B 0 0 4 -4 4 C 2 -4 0 -4 -4 D 6 4 4 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=22 C=19 B=13 A=13 so B is eliminated. Round 2 votes counts: D=34 E=24 C=22 A=20 so A is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:202 E:198 A:196 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -2 -6 0 B 0 0 4 -4 4 C 2 -4 0 -4 -4 D 6 4 4 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -6 0 B 0 0 4 -4 4 C 2 -4 0 -4 -4 D 6 4 4 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -6 0 B 0 0 4 -4 4 C 2 -4 0 -4 -4 D 6 4 4 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1151: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) A C B D E (8) C A B D E (7) E D B C A (6) A D E C B (6) E D A B C (5) E B D C A (5) B C E D A (5) D E B C A (4) C B A E D (4) B E D C A (4) A C D E B (4) A C D B E (4) D E B A C (3) D E A B C (3) C A B E D (3) C B A D E (2) B E C D A (2) A E D B C (2) A E C B D (2) E A D B C (1) D B A E C (1) D A E B C (1) C B E A D (1) C B D E A (1) B D C E A (1) A E D C B (1) A E C D B (1) A D E B C (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 10 -2 -2 B -2 0 8 -8 -12 C -10 -8 0 -10 -16 D 2 8 10 0 -6 E 2 12 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 10 -2 -2 B -2 0 8 -8 -12 C -10 -8 0 -10 -16 D 2 8 10 0 -6 E 2 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=27 C=18 D=12 B=12 so D is eliminated. Round 2 votes counts: E=37 A=32 C=18 B=13 so B is eliminated. Round 3 votes counts: E=43 A=33 C=24 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:207 A:204 B:193 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 10 -2 -2 B -2 0 8 -8 -12 C -10 -8 0 -10 -16 D 2 8 10 0 -6 E 2 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -2 -2 B -2 0 8 -8 -12 C -10 -8 0 -10 -16 D 2 8 10 0 -6 E 2 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -2 -2 B -2 0 8 -8 -12 C -10 -8 0 -10 -16 D 2 8 10 0 -6 E 2 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1152: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (11) D B E C A (10) A C E D B (9) A C D E B (7) A C B E D (5) D E B C A (4) B E C A D (4) B A E C D (4) A C E B D (4) B E A D C (3) B D E A C (3) A D C E B (3) E C B D A (2) D E C B A (2) D A C E B (2) B E D C A (2) B E D A C (2) A D C B E (2) A D B C E (2) A B C D E (2) E D C B A (1) E C D B A (1) D C A E B (1) D B E A C (1) D B A E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A B D (1) C A D E B (1) B E C D A (1) B D A E C (1) B C A E D (1) B A D E C (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 10 6 -2 B 14 0 10 2 18 C -10 -10 0 -6 -8 D -6 -2 6 0 12 E 2 -18 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 6 -2 B 14 0 10 2 18 C -10 -10 0 -6 -8 D -6 -2 6 0 12 E 2 -18 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998797 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=33 D=21 C=5 E=4 so E is eliminated. Round 2 votes counts: A=37 B=33 D=22 C=8 so C is eliminated. Round 3 votes counts: A=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:205 A:200 E:190 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 6 -2 B 14 0 10 2 18 C -10 -10 0 -6 -8 D -6 -2 6 0 12 E 2 -18 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998797 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 6 -2 B 14 0 10 2 18 C -10 -10 0 -6 -8 D -6 -2 6 0 12 E 2 -18 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998797 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 6 -2 B 14 0 10 2 18 C -10 -10 0 -6 -8 D -6 -2 6 0 12 E 2 -18 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998797 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1153: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) A D E B C (8) B C E A D (6) D E A C B (5) D A E C B (4) C B D A E (4) C B A D E (4) A D B C E (4) E D C B A (3) E C B D A (3) E B C A D (3) E A D B C (3) A E D B C (3) E D C A B (2) E D A B C (2) D C B A E (2) D C A B E (2) D A C E B (2) D A C B E (2) C E D B A (2) C B E A D (2) B E C A D (2) B C A E D (2) B C A D E (2) B A C E D (2) A B E C D (2) A B C D E (2) E B A C D (1) E A B D C (1) D E C A B (1) D C B E A (1) D C A E B (1) D A E B C (1) C E B D A (1) C D B E A (1) C D B A E (1) C B D E A (1) A E B C D (1) A D C B E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -8 2 4 B 2 0 -6 -4 6 C 8 6 0 0 6 D -2 4 0 0 0 E -4 -6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.523570 D: 0.476430 E: 0.000000 Sum of squares = 0.501111117659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.523570 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 2 4 B 2 0 -6 -4 6 C 8 6 0 0 6 D -2 4 0 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=23 D=21 E=18 B=14 so B is eliminated. Round 2 votes counts: C=34 A=25 D=21 E=20 so E is eliminated. Round 3 votes counts: C=42 A=30 D=28 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:201 B:199 A:198 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 2 4 B 2 0 -6 -4 6 C 8 6 0 0 6 D -2 4 0 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 2 4 B 2 0 -6 -4 6 C 8 6 0 0 6 D -2 4 0 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 2 4 B 2 0 -6 -4 6 C 8 6 0 0 6 D -2 4 0 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1154: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) E B C D A (5) D C B A E (5) D C A B E (5) B D C E A (5) B A E D C (5) A E C D B (5) A D C B E (5) E C D B A (4) C D E B A (4) B E C D A (4) B E A D C (4) B D C A E (4) A E B D C (4) A B D C E (4) C D E A B (3) C D B E A (3) E C D A B (2) E B A C D (2) C D A E B (2) B E D C A (2) A E D C B (2) A D B C E (2) A B E D C (2) E B A D C (1) E A C D B (1) E A C B D (1) E A B C D (1) D B C E A (1) C A D E B (1) B D A C E (1) B A D C E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 0 0 14 B 2 0 -2 -6 8 C 0 2 0 -8 8 D 0 6 8 0 8 E -14 -8 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.298435 B: 0.000000 C: 0.000000 D: 0.701565 E: 0.000000 Sum of squares = 0.581256994104 Cumulative probabilities = A: 0.298435 B: 0.298435 C: 0.298435 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 0 14 B 2 0 -2 -6 8 C 0 2 0 -8 8 D 0 6 8 0 8 E -14 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=26 E=17 C=13 D=11 so D is eliminated. Round 2 votes counts: A=33 B=27 C=23 E=17 so E is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 A:206 B:201 C:201 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 0 14 B 2 0 -2 -6 8 C 0 2 0 -8 8 D 0 6 8 0 8 E -14 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 14 B 2 0 -2 -6 8 C 0 2 0 -8 8 D 0 6 8 0 8 E -14 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 14 B 2 0 -2 -6 8 C 0 2 0 -8 8 D 0 6 8 0 8 E -14 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1155: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) A D C E B (9) B C D E A (8) B E C D A (7) D A C B E (6) E B A C D (5) D C B A E (3) D B C A E (3) D A C E B (3) C D B A E (3) C B D E A (3) B E C A D (3) B E A D C (3) A E D B C (3) A E C D B (3) E B C A D (2) E A B D C (2) C D A B E (2) B C E D A (2) A E D C B (2) A D E C B (2) A C D E B (2) E C B A D (1) E C A B D (1) E B A D C (1) E A C B D (1) E A B C D (1) D B A C E (1) D A B C E (1) B E D C A (1) B E D A C (1) B D C E A (1) A E B D C (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -2 -12 14 B -4 0 -6 -12 18 C 2 6 0 -10 18 D 12 12 10 0 18 E -14 -18 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -12 14 B -4 0 -6 -12 18 C 2 6 0 -10 18 D 12 12 10 0 18 E -14 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=26 A=24 E=14 C=8 so C is eliminated. Round 2 votes counts: D=33 B=29 A=24 E=14 so E is eliminated. Round 3 votes counts: B=38 D=33 A=29 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:208 A:202 B:198 E:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -12 14 B -4 0 -6 -12 18 C 2 6 0 -10 18 D 12 12 10 0 18 E -14 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -12 14 B -4 0 -6 -12 18 C 2 6 0 -10 18 D 12 12 10 0 18 E -14 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -12 14 B -4 0 -6 -12 18 C 2 6 0 -10 18 D 12 12 10 0 18 E -14 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1156: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (14) B A D C E (10) D C E B A (8) C E D B A (6) E C D B A (5) D C E A B (5) A B D E C (5) A D B C E (4) A B D C E (4) E C A D B (3) D A B C E (3) C D E B A (3) C D E A B (3) D C B E A (2) D C B A E (2) D B A C E (2) D A C B E (2) C E D A B (2) E C B D A (1) E C B A D (1) E A C D B (1) E A B C D (1) D C A E B (1) D B C E A (1) D B C A E (1) D A C E B (1) B E C A D (1) B E A C D (1) B D A C E (1) B C D E A (1) B A E C D (1) B A D E C (1) A E D C B (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -20 -28 -18 B -4 0 -24 -42 -16 C 20 24 0 -8 24 D 28 42 8 0 20 E 18 16 -24 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -20 -28 -18 B -4 0 -24 -42 -16 C 20 24 0 -8 24 D 28 42 8 0 20 E 18 16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=16 A=16 C=14 so C is eliminated. Round 2 votes counts: E=34 D=34 B=16 A=16 so B is eliminated. Round 3 votes counts: E=36 D=36 A=28 so A is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:249 C:230 E:195 A:169 B:157 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -20 -28 -18 B -4 0 -24 -42 -16 C 20 24 0 -8 24 D 28 42 8 0 20 E 18 16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -28 -18 B -4 0 -24 -42 -16 C 20 24 0 -8 24 D 28 42 8 0 20 E 18 16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -28 -18 B -4 0 -24 -42 -16 C 20 24 0 -8 24 D 28 42 8 0 20 E 18 16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1157: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) B C E D A (9) C D B E A (6) E D C B A (5) A D C E B (5) D C E A B (4) C D E B A (4) B E D C A (4) B E C D A (4) A E D B C (4) E B D C A (3) D E C A B (3) A C D B E (3) A B E D C (3) D C E B A (2) C D E A B (2) B A E C D (2) B A C E D (2) A C D E B (2) A B C E D (2) A B C D E (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A B C (1) E B D A C (1) E B C D A (1) D E C B A (1) D E A C B (1) D C A E B (1) C E B D A (1) C D A E B (1) C B D E A (1) C B A D E (1) B E D A C (1) B E A C D (1) B C D E A (1) B C A E D (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -14 -20 -18 B 6 0 -12 -18 -12 C 14 12 0 -6 2 D 20 18 6 0 2 E 18 12 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997449 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -20 -18 B 6 0 -12 -18 -12 C 14 12 0 -6 2 D 20 18 6 0 2 E 18 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=25 C=16 E=14 D=12 so D is eliminated. Round 2 votes counts: A=33 B=25 C=23 E=19 so E is eliminated. Round 3 votes counts: A=35 C=33 B=32 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:223 E:213 C:211 B:182 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -14 -20 -18 B 6 0 -12 -18 -12 C 14 12 0 -6 2 D 20 18 6 0 2 E 18 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -20 -18 B 6 0 -12 -18 -12 C 14 12 0 -6 2 D 20 18 6 0 2 E 18 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -20 -18 B 6 0 -12 -18 -12 C 14 12 0 -6 2 D 20 18 6 0 2 E 18 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1158: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) E A B D C (6) E D B A C (5) C D E A B (5) A B E D C (5) A B E C D (5) E D C B A (4) C D B A E (4) B A E D C (4) A E B D C (4) A B C E D (4) A B C D E (4) E A B C D (3) C D E B A (3) C D A E B (3) C A B D E (3) B A C D E (3) E C D A B (2) D C B A E (2) D B C A E (2) C E D A B (2) C D B E A (2) C A B E D (2) A C B E D (2) E D C A B (1) E D A B C (1) E B D A C (1) E B A D C (1) E A C B D (1) D C B E A (1) D B E A C (1) D B A E C (1) C D A B E (1) B D C A E (1) B C D A E (1) B A D E C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 12 8 0 8 B -12 0 6 6 0 C -8 -6 0 4 8 D 0 -6 -4 0 -8 E -8 0 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.639208 B: 0.000000 C: 0.000000 D: 0.360792 E: 0.000000 Sum of squares = 0.538757594266 Cumulative probabilities = A: 0.639208 B: 0.639208 C: 0.639208 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 0 8 B -12 0 6 6 0 C -8 -6 0 4 8 D 0 -6 -4 0 -8 E -8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500244 B: 0.000000 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000118705 Cumulative probabilities = A: 0.500244 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 C=25 D=14 B=10 so B is eliminated. Round 2 votes counts: A=34 C=26 E=25 D=15 so D is eliminated. Round 3 votes counts: C=39 A=35 E=26 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:200 C:199 E:196 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 0 8 B -12 0 6 6 0 C -8 -6 0 4 8 D 0 -6 -4 0 -8 E -8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500244 B: 0.000000 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000118705 Cumulative probabilities = A: 0.500244 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 0 8 B -12 0 6 6 0 C -8 -6 0 4 8 D 0 -6 -4 0 -8 E -8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500244 B: 0.000000 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000118705 Cumulative probabilities = A: 0.500244 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 0 8 B -12 0 6 6 0 C -8 -6 0 4 8 D 0 -6 -4 0 -8 E -8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500244 B: 0.000000 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000118705 Cumulative probabilities = A: 0.500244 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1159: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (21) A D E B C (11) A E D B C (7) C B D E A (5) E D A B C (4) E A D B C (4) B C E D A (4) E B D A C (3) C B E A D (3) C B D A E (3) A D C E B (3) A D C B E (3) E B D C A (2) E B C D A (2) D A E B C (2) C D B A E (2) B E C D A (2) A D E C B (2) A C E D B (2) A C D E B (2) A C D B E (2) E D B A C (1) E B A D C (1) D E A B C (1) D B E A C (1) D A C B E (1) D A B E C (1) C E B D A (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D B E (1) Total count = 100 A B C D E A 0 -4 2 -12 -10 B 4 0 -8 -2 0 C -2 8 0 6 12 D 12 2 -6 0 -18 E 10 0 -12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.460000000011 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -12 -10 B 4 0 -8 -2 0 C -2 8 0 6 12 D 12 2 -6 0 -18 E 10 0 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=32 E=17 D=6 B=6 so D is eliminated. Round 2 votes counts: C=39 A=36 E=18 B=7 so B is eliminated. Round 3 votes counts: C=43 A=36 E=21 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:212 E:208 B:197 D:195 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 -12 -10 B 4 0 -8 -2 0 C -2 8 0 6 12 D 12 2 -6 0 -18 E 10 0 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -12 -10 B 4 0 -8 -2 0 C -2 8 0 6 12 D 12 2 -6 0 -18 E 10 0 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -12 -10 B 4 0 -8 -2 0 C -2 8 0 6 12 D 12 2 -6 0 -18 E 10 0 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1160: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A D C E B (9) D A B C E (5) C E B A D (5) A D E C B (5) E B C A D (4) D B A E C (4) D A B E C (4) E C B A D (3) D A C B E (3) B E C A D (3) B D C E A (3) E C A B D (2) E B D A C (2) E A C B D (2) E A B C D (2) D B E A C (2) D A E C B (2) D A C E B (2) C B E A D (2) C A E D B (2) B E D C A (2) B D C A E (2) B C D E A (2) A E C D B (2) A C E D B (2) A C D E B (2) E A D C B (1) E A D B C (1) E A C D B (1) D C A B E (1) D B E C A (1) D B C A E (1) C D A B E (1) C A E B D (1) B D E C A (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 2 4 0 0 B -2 0 2 -4 -2 C -4 -2 0 -4 -12 D 0 4 4 0 2 E 0 2 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.402383 B: 0.000000 C: 0.000000 D: 0.597617 E: 0.000000 Sum of squares = 0.519058207662 Cumulative probabilities = A: 0.402383 B: 0.402383 C: 0.402383 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 0 0 B -2 0 2 -4 -2 C -4 -2 0 -4 -12 D 0 4 4 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 A=21 E=18 C=11 so C is eliminated. Round 2 votes counts: B=27 D=26 A=24 E=23 so E is eliminated. Round 3 votes counts: B=41 A=33 D=26 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:205 A:203 B:197 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 0 B -2 0 2 -4 -2 C -4 -2 0 -4 -12 D 0 4 4 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 0 B -2 0 2 -4 -2 C -4 -2 0 -4 -12 D 0 4 4 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 0 B -2 0 2 -4 -2 C -4 -2 0 -4 -12 D 0 4 4 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1161: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) E C B A D (6) D B A E C (6) D A B C E (6) C E B A D (6) C E A B D (6) E C B D A (5) C E A D B (5) C E D A B (4) B D A E C (4) A D C B E (4) E B A D C (3) D A B E C (3) A D B E C (3) E C A B D (2) E B C D A (2) D B A C E (2) C D A E B (2) C A D B E (2) B D E A C (2) A B D E C (2) E B D C A (1) E B D A C (1) E A B C D (1) D C B A E (1) D C A B E (1) D A C B E (1) C E D B A (1) C E B D A (1) C D E B A (1) C D B A E (1) C A E B D (1) C A D E B (1) B E D A C (1) B A E D C (1) B A D E C (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 2 8 4 B -8 0 -4 -6 2 C -2 4 0 -4 10 D -8 6 4 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 8 4 B -8 0 -4 -6 2 C -2 4 0 -4 10 D -8 6 4 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=21 D=20 A=19 B=9 so B is eliminated. Round 2 votes counts: C=31 D=26 E=22 A=21 so A is eliminated. Round 3 votes counts: D=44 C=32 E=24 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:211 C:204 D:203 B:192 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 8 4 B -8 0 -4 -6 2 C -2 4 0 -4 10 D -8 6 4 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 8 4 B -8 0 -4 -6 2 C -2 4 0 -4 10 D -8 6 4 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 8 4 B -8 0 -4 -6 2 C -2 4 0 -4 10 D -8 6 4 0 4 E -4 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1162: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) E A B D C (7) C D B E A (6) A E B D C (6) A B E C D (6) D E C A B (5) C B D A E (5) B A C E D (5) D C E A B (4) C D B A E (4) B A E C D (4) E A C B D (3) C B A E D (3) B C A E D (3) E D A C B (2) E A D C B (2) E A C D B (2) D E A B C (2) D C E B A (2) D C B E A (2) D C B A E (2) C E A B D (2) C D E B A (2) B C A D E (2) A E B C D (2) E D A B C (1) E C D A B (1) E C A B D (1) E A B C D (1) D E A C B (1) D B C A E (1) B D C A E (1) B C D A E (1) B A E D C (1) A B E D C (1) Total count = 100 A B C D E A 0 12 6 16 -6 B -12 0 2 8 -6 C -6 -2 0 6 -10 D -16 -8 -6 0 -20 E 6 6 10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 6 16 -6 B -12 0 2 8 -6 C -6 -2 0 6 -10 D -16 -8 -6 0 -20 E 6 6 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=22 D=19 B=17 A=15 so A is eliminated. Round 2 votes counts: E=35 B=24 C=22 D=19 so D is eliminated. Round 3 votes counts: E=43 C=32 B=25 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:214 B:196 C:194 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 6 16 -6 B -12 0 2 8 -6 C -6 -2 0 6 -10 D -16 -8 -6 0 -20 E 6 6 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 16 -6 B -12 0 2 8 -6 C -6 -2 0 6 -10 D -16 -8 -6 0 -20 E 6 6 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 16 -6 B -12 0 2 8 -6 C -6 -2 0 6 -10 D -16 -8 -6 0 -20 E 6 6 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1163: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) C B D E A (7) B A C E D (7) D E C A B (6) E D C B A (5) D E A C B (4) D A C E B (4) E D A C B (3) B C E D A (3) B A E C D (3) A B E C D (3) A B C E D (3) E A D B C (2) D C E A B (2) C E D B A (2) C E B D A (2) C D B E A (2) C B E D A (2) C B D A E (2) B E A C D (2) B C A D E (2) A E D B C (2) A E B D C (2) A B E D C (2) A B C D E (2) E D B C A (1) E B D C A (1) E B C D A (1) E B C A D (1) E B A D C (1) E B A C D (1) D E C B A (1) D C B E A (1) D C A E B (1) D A E C B (1) B C E A D (1) B C A E D (1) B A C D E (1) A D E C B (1) A D E B C (1) A D B C E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -4 -18 -18 B 16 0 -10 4 -6 C 4 10 0 -2 10 D 18 -4 2 0 -2 E 18 6 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999994 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -18 -18 B 16 0 -10 4 -6 C 4 10 0 -2 10 D 18 -4 2 0 -2 E 18 6 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999604 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=20 A=19 C=17 E=16 so E is eliminated. Round 2 votes counts: D=37 B=25 A=21 C=17 so C is eliminated. Round 3 votes counts: D=41 B=38 A=21 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 E:208 D:207 B:202 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -4 -18 -18 B 16 0 -10 4 -6 C 4 10 0 -2 10 D 18 -4 2 0 -2 E 18 6 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999604 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -18 -18 B 16 0 -10 4 -6 C 4 10 0 -2 10 D 18 -4 2 0 -2 E 18 6 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999604 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -18 -18 B 16 0 -10 4 -6 C 4 10 0 -2 10 D 18 -4 2 0 -2 E 18 6 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999604 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1164: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C A E B D (8) A E C D B (8) A C E D B (8) E D A B C (7) D E B A C (6) C A B E D (6) B D E C A (6) E A D C B (4) E A C D B (4) B D C A E (4) B C D A E (4) B C A D E (4) C B A E D (3) B D C E A (3) D E A B C (2) D B A E C (2) C B A D E (2) C A B D E (2) E D B A C (1) E D A C B (1) E B D C A (1) C B E A D (1) B E D C A (1) B C E D A (1) A E D C B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 8 4 6 B -4 0 2 -8 -2 C -8 -2 0 2 -6 D -4 8 -2 0 -10 E -6 2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 4 6 B -4 0 2 -8 -2 C -8 -2 0 2 -6 D -4 8 -2 0 -10 E -6 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 C=22 D=19 E=18 A=18 so E is eliminated. Round 2 votes counts: D=28 A=26 B=24 C=22 so C is eliminated. Round 3 votes counts: A=42 B=30 D=28 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:206 D:196 B:194 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 6 B -4 0 2 -8 -2 C -8 -2 0 2 -6 D -4 8 -2 0 -10 E -6 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 6 B -4 0 2 -8 -2 C -8 -2 0 2 -6 D -4 8 -2 0 -10 E -6 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 6 B -4 0 2 -8 -2 C -8 -2 0 2 -6 D -4 8 -2 0 -10 E -6 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1165: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D E A C B (6) E C A D B (5) D E C B A (5) A B C E D (5) E D C A B (4) B D C A E (4) B A D C E (4) E D C B A (3) E D A C B (3) E C D A B (3) D A B E C (3) B A C E D (3) E C A B D (2) D E C A B (2) D E B C A (2) D B C E A (2) D A E B C (2) C E B A D (2) C B A E D (2) C A E B D (2) B D A C E (2) B C D E A (2) B C A D E (2) B A C D E (2) A E C B D (2) A D E C B (2) A D B E C (2) E C D B A (1) D E B A C (1) D B E C A (1) C E B D A (1) C E A B D (1) C B E A D (1) B C E D A (1) B C D A E (1) A E D C B (1) A D E B C (1) A C E B D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -16 2 4 B 2 0 0 2 -4 C 16 0 0 -2 -2 D -2 -2 2 0 -6 E -4 4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.727273 Sum of squares = 0.570247933903 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.272727 D: 0.272727 E: 1.000000 A B C D E A 0 -2 -16 2 4 B 2 0 0 2 -4 C 16 0 0 -2 -2 D -2 -2 2 0 -6 E -4 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.727273 Sum of squares = 0.570247933824 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.272727 D: 0.272727 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=24 E=21 A=16 C=9 so C is eliminated. Round 2 votes counts: B=33 E=25 D=24 A=18 so A is eliminated. Round 3 votes counts: B=40 E=31 D=29 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:206 E:204 B:200 D:196 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 2 4 B 2 0 0 2 -4 C 16 0 0 -2 -2 D -2 -2 2 0 -6 E -4 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.727273 Sum of squares = 0.570247933824 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.272727 D: 0.272727 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 2 4 B 2 0 0 2 -4 C 16 0 0 -2 -2 D -2 -2 2 0 -6 E -4 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.727273 Sum of squares = 0.570247933824 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.272727 D: 0.272727 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 2 4 B 2 0 0 2 -4 C 16 0 0 -2 -2 D -2 -2 2 0 -6 E -4 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.727273 Sum of squares = 0.570247933824 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.272727 D: 0.272727 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1166: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) A C B E D (8) E D C B A (7) B A D C E (7) D B E C A (6) C A E D B (6) B D A C E (6) A C E B D (6) B D A E C (5) E C A D B (4) C E A D B (4) D B A C E (3) E C D B A (2) D E B C A (2) B D E A C (2) B A C D E (2) A C B D E (2) A B C D E (2) E C B D A (1) E C A B D (1) E B C A D (1) D E C A B (1) D C A B E (1) D B E A C (1) D B A E C (1) C E A B D (1) C A D E B (1) B E A C D (1) B D E C A (1) A D B C E (1) A C E D B (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 2 10 B -4 0 -14 -4 4 C -2 14 0 10 8 D -2 4 -10 0 -8 E -10 -4 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 10 B -4 0 -14 -4 4 C -2 14 0 10 8 D -2 4 -10 0 -8 E -10 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 A=23 D=15 C=12 so C is eliminated. Round 2 votes counts: E=31 A=30 B=24 D=15 so D is eliminated. Round 3 votes counts: B=35 E=34 A=31 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:215 A:209 E:193 D:192 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 10 B -4 0 -14 -4 4 C -2 14 0 10 8 D -2 4 -10 0 -8 E -10 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 10 B -4 0 -14 -4 4 C -2 14 0 10 8 D -2 4 -10 0 -8 E -10 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 10 B -4 0 -14 -4 4 C -2 14 0 10 8 D -2 4 -10 0 -8 E -10 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1167: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A D C B (8) C B A D E (7) C B A E D (6) C B E A D (5) B C A D E (4) A E D C B (4) E D A C B (3) E D A B C (3) E C A B D (3) D E A B C (3) D B A C E (3) B C E D A (3) E D B C A (2) E D B A C (2) E B D C A (2) E B C D A (2) E A C D B (2) D A E B C (2) D A B C E (2) B D A C E (2) B C D E A (2) A D E C B (2) A C D B E (2) E D C B A (1) E C B D A (1) E C B A D (1) E C A D B (1) D B E A C (1) D A E C B (1) C A B D E (1) B E C D A (1) B D C A E (1) B C E A D (1) B A D C E (1) A E C D B (1) A D C E B (1) A D C B E (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -8 6 6 B 16 0 -4 8 8 C 8 4 0 8 8 D -6 -8 -8 0 -6 E -6 -8 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 6 6 B 16 0 -4 8 8 C 8 4 0 8 8 D -6 -8 -8 0 -6 E -6 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=25 C=19 A=13 D=12 so D is eliminated. Round 2 votes counts: E=34 B=29 C=19 A=18 so A is eliminated. Round 3 votes counts: E=44 B=32 C=24 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:214 A:194 E:192 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -8 6 6 B 16 0 -4 8 8 C 8 4 0 8 8 D -6 -8 -8 0 -6 E -6 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 6 6 B 16 0 -4 8 8 C 8 4 0 8 8 D -6 -8 -8 0 -6 E -6 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 6 6 B 16 0 -4 8 8 C 8 4 0 8 8 D -6 -8 -8 0 -6 E -6 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1168: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (14) B E D A C (10) C D E A B (8) D C E A B (5) C A D E B (5) B A C E D (5) B E A D C (4) B A E C D (4) E D B C A (3) E D B A C (3) E D A C B (3) D E C A B (3) C D E B A (3) C D A E B (3) C A B D E (3) A C B D E (3) A B E D C (3) A B C E D (3) E D A B C (2) B E D C A (2) A E D B C (2) E B D A C (1) D E C B A (1) D E A C B (1) D C E B A (1) C B D E A (1) C A D B E (1) B A C D E (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 22 0 -2 B 6 0 16 10 10 C -22 -16 0 -18 -14 D 0 -10 18 0 -18 E 2 -10 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 22 0 -2 B 6 0 16 10 10 C -22 -16 0 -18 -14 D 0 -10 18 0 -18 E 2 -10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 C=24 A=13 E=12 D=11 so D is eliminated. Round 2 votes counts: B=40 C=30 E=17 A=13 so A is eliminated. Round 3 votes counts: B=47 C=33 E=20 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:212 A:207 D:195 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 22 0 -2 B 6 0 16 10 10 C -22 -16 0 -18 -14 D 0 -10 18 0 -18 E 2 -10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 22 0 -2 B 6 0 16 10 10 C -22 -16 0 -18 -14 D 0 -10 18 0 -18 E 2 -10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 22 0 -2 B 6 0 16 10 10 C -22 -16 0 -18 -14 D 0 -10 18 0 -18 E 2 -10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1169: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) E B A C D (6) D C A B E (6) C A D B E (5) B D A E C (5) E C A B D (4) E B D A C (4) C A E B D (4) A C B E D (4) D E B C A (3) D B A E C (3) E D C B A (2) E D B C A (2) E D B A C (2) E C D B A (2) E C B A D (2) E B A D C (2) D E C B A (2) D B A C E (2) C E A D B (2) C E A B D (2) C D A B E (2) C A E D B (2) C A D E B (2) C A B D E (2) B A E C D (2) A B E C D (2) E C B D A (1) D E B A C (1) D C E A B (1) D C B A E (1) D C A E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E A B (1) C A B E D (1) B E D A C (1) B E A D C (1) B D E A C (1) B A D E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -2 -8 -2 B 8 0 -6 -8 2 C 2 6 0 -2 -14 D 8 8 2 0 2 E 2 -2 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -8 -2 B 8 0 -6 -8 2 C 2 6 0 -2 -14 D 8 8 2 0 2 E 2 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 C=24 B=11 A=8 so A is eliminated. Round 2 votes counts: D=30 C=30 E=27 B=13 so B is eliminated. Round 3 votes counts: D=37 E=33 C=30 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:206 B:198 C:196 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -8 -2 B 8 0 -6 -8 2 C 2 6 0 -2 -14 D 8 8 2 0 2 E 2 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -8 -2 B 8 0 -6 -8 2 C 2 6 0 -2 -14 D 8 8 2 0 2 E 2 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -8 -2 B 8 0 -6 -8 2 C 2 6 0 -2 -14 D 8 8 2 0 2 E 2 -2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1170: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) A D B E C (6) B C E D A (5) A E D B C (5) D A E C B (4) D A B C E (4) C B E D A (4) B C D E A (4) E A D C B (3) D C A E B (3) D A C E B (3) B E C A D (3) B C E A D (3) B A E D C (3) A E D C B (3) A B D E C (3) E C D A B (2) E C B A D (2) E C A B D (2) D C A B E (2) C E D A B (2) C E B D A (2) C B D E A (2) B D A C E (2) B A E C D (2) B A D E C (2) A D E B C (2) A B E D C (2) E B A C D (1) E A D B C (1) E A B D C (1) D E C A B (1) D B A C E (1) D A C B E (1) C E D B A (1) C D E B A (1) C D E A B (1) C D B E A (1) B E A C D (1) B D C A E (1) B A D C E (1) Total count = 100 A B C D E A 0 16 16 6 14 B -16 0 6 -8 6 C -16 -6 0 -22 -12 D -6 8 22 0 4 E -14 -6 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 6 14 B -16 0 6 -8 6 C -16 -6 0 -22 -12 D -6 8 22 0 4 E -14 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 D=19 C=14 E=12 so E is eliminated. Round 2 votes counts: A=33 B=28 C=20 D=19 so D is eliminated. Round 3 votes counts: A=45 B=29 C=26 so C is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:214 B:194 E:194 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 6 14 B -16 0 6 -8 6 C -16 -6 0 -22 -12 D -6 8 22 0 4 E -14 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 6 14 B -16 0 6 -8 6 C -16 -6 0 -22 -12 D -6 8 22 0 4 E -14 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 6 14 B -16 0 6 -8 6 C -16 -6 0 -22 -12 D -6 8 22 0 4 E -14 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1171: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) A C D E B (7) B E D C A (6) A C B E D (6) A B E D C (6) D E B C A (5) C A D E B (5) B E D A C (5) A B C E D (5) C D A E B (4) C A B D E (4) E B D C A (3) C A B E D (3) B E C D A (3) B E A C D (3) A B E C D (3) E B D A C (2) D E B A C (2) D C A E B (2) B E A D C (2) A D E B C (2) A C D B E (2) E D B C A (1) D E A C B (1) D E A B C (1) D C E B A (1) C E D B A (1) C D E B A (1) C D E A B (1) C B E A D (1) C B A E D (1) B E C A D (1) B C E D A (1) B A E D C (1) A D C E B (1) Total count = 100 A B C D E A 0 20 -2 20 18 B -20 0 4 12 20 C 2 -4 0 18 6 D -20 -12 -18 0 -8 E -18 -20 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.076923 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.621301775152 Cumulative probabilities = A: 0.153846 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -2 20 18 B -20 0 4 12 20 C 2 -4 0 18 6 D -20 -12 -18 0 -8 E -18 -20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.076923 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.62130177481 Cumulative probabilities = A: 0.153846 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 B=22 D=12 E=6 so E is eliminated. Round 2 votes counts: A=32 C=28 B=27 D=13 so D is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:211 B:208 E:182 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 -2 20 18 B -20 0 4 12 20 C 2 -4 0 18 6 D -20 -12 -18 0 -8 E -18 -20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.076923 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.62130177481 Cumulative probabilities = A: 0.153846 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -2 20 18 B -20 0 4 12 20 C 2 -4 0 18 6 D -20 -12 -18 0 -8 E -18 -20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.076923 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.62130177481 Cumulative probabilities = A: 0.153846 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -2 20 18 B -20 0 4 12 20 C 2 -4 0 18 6 D -20 -12 -18 0 -8 E -18 -20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.076923 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.62130177481 Cumulative probabilities = A: 0.153846 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1172: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) C E A D B (8) D B E C A (7) A B D C E (7) E C D B A (6) B D E C A (6) D B E A C (5) C E B D A (5) A C E D B (5) B D C E A (4) A D B E C (4) A C B E D (4) D E B C A (3) B D A E C (3) A B C D E (3) E D B C A (2) E C D A B (2) C E D B A (2) C E A B D (2) B A D E C (2) E D C B A (1) E D C A B (1) E D A C B (1) E C B D A (1) C A E D B (1) B D E A C (1) B A D C E (1) A E D B C (1) A E C D B (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -2 0 -14 B -2 0 0 0 -2 C 2 0 0 0 4 D 0 0 0 0 -4 E 14 2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.170530 C: 0.599690 D: 0.229780 E: 0.000000 Sum of squares = 0.4415072471 Cumulative probabilities = A: 0.000000 B: 0.170530 C: 0.770220 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 0 -14 B -2 0 0 0 -2 C 2 0 0 0 4 D 0 0 0 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.423077 D: 0.269231 E: 0.000000 Sum of squares = 0.346153847308 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.730769 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=18 B=17 D=15 E=14 so E is eliminated. Round 2 votes counts: A=36 C=27 D=20 B=17 so B is eliminated. Round 3 votes counts: A=39 D=34 C=27 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:208 C:203 B:198 D:198 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 0 -14 B -2 0 0 0 -2 C 2 0 0 0 4 D 0 0 0 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.423077 D: 0.269231 E: 0.000000 Sum of squares = 0.346153847308 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.730769 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -14 B -2 0 0 0 -2 C 2 0 0 0 4 D 0 0 0 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.423077 D: 0.269231 E: 0.000000 Sum of squares = 0.346153847308 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.730769 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -14 B -2 0 0 0 -2 C 2 0 0 0 4 D 0 0 0 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.307692 C: 0.423077 D: 0.269231 E: 0.000000 Sum of squares = 0.346153847308 Cumulative probabilities = A: 0.000000 B: 0.307692 C: 0.730769 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1173: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) D C E A B (9) C D E A B (9) B A E C D (9) E A C B D (7) B D A E C (7) D B C A E (6) D C B E A (5) C E D A B (5) C E A D B (5) A E B C D (5) B D A C E (4) A B E C D (4) D C E B A (3) B A D E C (3) E C A D B (2) E D A C B (1) E C A B D (1) E A B C D (1) D B E A C (1) D B C E A (1) D B A E C (1) C D B A E (1) Total count = 100 A B C D E A 0 -2 6 -6 0 B 2 0 4 2 4 C -6 -4 0 -2 -4 D 6 -2 2 0 0 E 0 -4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -6 0 B 2 0 4 2 4 C -6 -4 0 -2 -4 D 6 -2 2 0 0 E 0 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=26 C=20 E=12 A=9 so A is eliminated. Round 2 votes counts: B=37 D=26 C=20 E=17 so E is eliminated. Round 3 votes counts: B=43 C=30 D=27 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:206 D:203 E:200 A:199 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 -6 0 B 2 0 4 2 4 C -6 -4 0 -2 -4 D 6 -2 2 0 0 E 0 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -6 0 B 2 0 4 2 4 C -6 -4 0 -2 -4 D 6 -2 2 0 0 E 0 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -6 0 B 2 0 4 2 4 C -6 -4 0 -2 -4 D 6 -2 2 0 0 E 0 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1174: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) D B E A C (8) A E D C B (8) C A E B D (7) B D C E A (7) B C D E A (5) A C E D B (5) E A D C B (4) C B A D E (4) B D E C A (4) A E C D B (4) A D E C B (4) E D A B C (3) D A E B C (3) C E A B D (3) C B E A D (3) D E B A C (2) D E A B C (2) D B A E C (2) C A B E D (2) B C D A E (2) D B A C E (1) D A B E C (1) C E B A D (1) C B E D A (1) C B D A E (1) C A D B E (1) B C E D A (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -2 14 12 B 2 0 -16 0 4 C 2 16 0 0 8 D -14 0 0 0 -2 E -12 -4 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.928195 D: 0.071805 E: 0.000000 Sum of squares = 0.866702438592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.928195 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 14 12 B 2 0 -16 0 4 C 2 16 0 0 8 D -14 0 0 0 -2 E -12 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.875000 D: 0.125000 E: 0.000000 Sum of squares = 0.781250012588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=23 D=19 B=19 E=7 so E is eliminated. Round 2 votes counts: C=32 A=27 D=22 B=19 so B is eliminated. Round 3 votes counts: C=40 D=33 A=27 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:211 B:195 D:192 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 14 12 B 2 0 -16 0 4 C 2 16 0 0 8 D -14 0 0 0 -2 E -12 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.875000 D: 0.125000 E: 0.000000 Sum of squares = 0.781250012588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 14 12 B 2 0 -16 0 4 C 2 16 0 0 8 D -14 0 0 0 -2 E -12 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.875000 D: 0.125000 E: 0.000000 Sum of squares = 0.781250012588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 14 12 B 2 0 -16 0 4 C 2 16 0 0 8 D -14 0 0 0 -2 E -12 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.875000 D: 0.125000 E: 0.000000 Sum of squares = 0.781250012588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1175: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (13) A E D C B (6) B C A E D (5) C B A E D (4) B A C D E (4) E D A C B (3) E C A D B (3) D E C A B (3) D B C E A (3) C D E B A (3) C D B E A (3) C B D E A (3) B C D A E (3) B C A D E (3) A E B D C (3) A B E D C (3) E A C D B (2) D E A C B (2) D E A B C (2) D B A E C (2) C B E D A (2) B D C E A (2) B D C A E (2) B D A C E (2) B C D E A (2) A E B C D (2) A B D E C (2) E D C A B (1) E A D C B (1) D E C B A (1) D C E B A (1) D B E C A (1) D A E B C (1) C E D B A (1) C E B D A (1) C E A D B (1) C B E A D (1) C A E B D (1) B A C E D (1) A D E B C (1) Total count = 100 A B C D E A 0 0 0 12 16 B 0 0 14 -8 -4 C 0 -14 0 -10 -4 D -12 8 10 0 -8 E -16 4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.656426 B: 0.343574 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.548938070673 Cumulative probabilities = A: 0.656426 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 12 16 B 0 0 14 -8 -4 C 0 -14 0 -10 -4 D -12 8 10 0 -8 E -16 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 C=20 D=16 E=10 so E is eliminated. Round 2 votes counts: A=33 B=24 C=23 D=20 so D is eliminated. Round 3 votes counts: A=41 B=30 C=29 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:201 E:200 D:199 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 12 16 B 0 0 14 -8 -4 C 0 -14 0 -10 -4 D -12 8 10 0 -8 E -16 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 12 16 B 0 0 14 -8 -4 C 0 -14 0 -10 -4 D -12 8 10 0 -8 E -16 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 12 16 B 0 0 14 -8 -4 C 0 -14 0 -10 -4 D -12 8 10 0 -8 E -16 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1176: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) D B C A E (6) C D A B E (6) E C D B A (5) D B A C E (5) D C E B A (4) C D E B A (4) B E A D C (4) B A D E C (4) E B D A C (3) E A B C D (3) D C B A E (3) D C A B E (3) C D A E B (3) B A E D C (3) A C B D E (3) A B E C D (3) A B D C E (3) E B A D C (2) D C B E A (2) C D E A B (2) C A E D B (2) A C D B E (2) E D C B A (1) E D B C A (1) E C A B D (1) E B D C A (1) E B C A D (1) E A C B D (1) D B E C A (1) D B C E A (1) C E D A B (1) C E A D B (1) B D A C E (1) B A D C E (1) A E B C D (1) A D B C E (1) A C B E D (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -12 -6 22 B 6 0 -4 -22 26 C 12 4 0 -2 28 D 6 22 2 0 26 E -22 -26 -28 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -6 22 B 6 0 -4 -22 26 C 12 4 0 -2 28 D 6 22 2 0 26 E -22 -26 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 E=19 A=17 B=13 so B is eliminated. Round 2 votes counts: D=26 C=26 A=25 E=23 so E is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:228 C:221 B:203 A:199 E:149 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -6 22 B 6 0 -4 -22 26 C 12 4 0 -2 28 D 6 22 2 0 26 E -22 -26 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -6 22 B 6 0 -4 -22 26 C 12 4 0 -2 28 D 6 22 2 0 26 E -22 -26 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -6 22 B 6 0 -4 -22 26 C 12 4 0 -2 28 D 6 22 2 0 26 E -22 -26 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999199 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1177: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (13) C B D E A (8) A C B D E (7) E D B C A (6) D B E C A (6) E A D B C (4) B D C E A (4) A D E B C (4) A D B E C (4) E C D B A (3) E C B D A (3) D B C E A (3) C B D A E (3) A E D C B (3) D E B C A (2) A E C D B (2) A E C B D (2) A C E B D (2) A C B E D (2) E D C A B (1) E D B A C (1) E C A B D (1) E A D C B (1) D E B A C (1) D E A B C (1) D B E A C (1) D B C A E (1) D B A C E (1) C E B D A (1) C E B A D (1) C B E D A (1) C B E A D (1) C B A E D (1) C A B D E (1) B C D E A (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 4 4 -2 B -2 0 12 -18 -4 C -4 -12 0 -18 -18 D -4 18 18 0 2 E 2 4 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000016 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 2 4 4 -2 B -2 0 12 -18 -4 C -4 -12 0 -18 -18 D -4 18 18 0 2 E 2 4 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000057 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=20 C=17 D=16 B=5 so B is eliminated. Round 2 votes counts: A=42 E=20 D=20 C=18 so C is eliminated. Round 3 votes counts: A=44 D=32 E=24 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:217 E:211 A:204 B:194 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 4 -2 B -2 0 12 -18 -4 C -4 -12 0 -18 -18 D -4 18 18 0 2 E 2 4 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000057 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 4 -2 B -2 0 12 -18 -4 C -4 -12 0 -18 -18 D -4 18 18 0 2 E 2 4 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000057 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 4 -2 B -2 0 12 -18 -4 C -4 -12 0 -18 -18 D -4 18 18 0 2 E 2 4 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000057 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1178: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (7) C D E B A (6) B A E C D (6) C D A B E (5) E D C A B (4) D E C A B (4) B C A D E (4) A B E D C (4) E D C B A (3) E B A D C (3) D C E A B (3) B C E D A (3) B C D E A (3) A E D C B (3) A E D B C (3) A D C E B (3) E D A C B (2) D C A E B (2) C D B E A (2) C D B A E (2) C D A E B (2) C B D E A (2) B E A C D (2) B C D A E (2) B A C D E (2) A E B D C (2) A D C B E (2) E D B C A (1) E D B A C (1) E B D C A (1) E B D A C (1) E B C D A (1) E A D B C (1) E A B D C (1) D C E B A (1) C B D A E (1) C B A D E (1) C A B D E (1) B A E D C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -22 -18 -4 B 12 0 0 -2 4 C 22 0 0 8 -2 D 18 2 -8 0 0 E 4 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.541660 C: 0.458340 D: 0.000000 E: 0.000000 Sum of squares = 0.503471062569 Cumulative probabilities = A: 0.000000 B: 0.541660 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -22 -18 -4 B 12 0 0 -2 4 C 22 0 0 8 -2 D 18 2 -8 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999876 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=22 E=19 A=19 D=10 so D is eliminated. Round 2 votes counts: B=30 C=28 E=23 A=19 so A is eliminated. Round 3 votes counts: B=35 C=34 E=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:214 B:207 D:206 E:201 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -22 -18 -4 B 12 0 0 -2 4 C 22 0 0 8 -2 D 18 2 -8 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999876 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -22 -18 -4 B 12 0 0 -2 4 C 22 0 0 8 -2 D 18 2 -8 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999876 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -22 -18 -4 B 12 0 0 -2 4 C 22 0 0 8 -2 D 18 2 -8 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999876 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1179: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) A E D B C (7) E A C D B (5) C B D E A (5) E A D C B (4) D B C A E (4) C B E A D (4) A E B C D (4) D A B E C (3) C E D A B (3) B C E A D (3) B A E C D (3) A E B D C (3) D A E B C (2) C E B A D (2) C D B E A (2) C B E D A (2) B D A C E (2) B C E D A (2) B C D E A (2) B C D A E (2) B A D E C (2) B A C E D (2) A E D C B (2) A D E B C (2) E D A C B (1) E C D A B (1) E C B A D (1) E C A D B (1) E C A B D (1) E B C A D (1) D E C A B (1) D C B E A (1) D C B A E (1) D B A E C (1) D B A C E (1) D A E C B (1) C E D B A (1) C D E A B (1) B E A C D (1) B D C A E (1) B C A E D (1) B C A D E (1) A D E C B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 12 20 -4 B -4 0 4 8 -4 C -12 -4 0 18 -14 D -20 -8 -18 0 -28 E 4 4 14 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 12 20 -4 B -4 0 4 8 -4 C -12 -4 0 18 -14 D -20 -8 -18 0 -28 E 4 4 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 A=21 C=20 D=15 so D is eliminated. Round 2 votes counts: B=28 A=27 E=23 C=22 so C is eliminated. Round 3 votes counts: B=43 E=30 A=27 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:216 B:202 C:194 D:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 12 20 -4 B -4 0 4 8 -4 C -12 -4 0 18 -14 D -20 -8 -18 0 -28 E 4 4 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 20 -4 B -4 0 4 8 -4 C -12 -4 0 18 -14 D -20 -8 -18 0 -28 E 4 4 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 20 -4 B -4 0 4 8 -4 C -12 -4 0 18 -14 D -20 -8 -18 0 -28 E 4 4 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1180: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) C E D A B (8) E C A B D (7) E C A D B (5) A B E C D (5) A B D C E (5) A C E B D (4) A B E D C (4) E C D B A (3) C E D B A (3) C D E B A (3) E C B D A (2) E C B A D (2) E A C B D (2) D C E B A (2) D C B A E (2) D B C A E (2) D A C B E (2) C E A D B (2) C D E A B (2) B D A E C (2) B D A C E (2) B A D C E (2) A E C B D (2) A D C B E (2) E B C A D (1) E B A C D (1) D C A E B (1) D B E C A (1) D B C E A (1) D B A C E (1) D A B C E (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D E A C (1) B A E D C (1) A D B C E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 12 0 18 2 B -12 0 -12 12 -4 C 0 12 0 8 -4 D -18 -12 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.906573 B: 0.000000 C: 0.093427 D: 0.000000 E: 0.000000 Sum of squares = 0.830603064844 Cumulative probabilities = A: 0.906573 B: 0.906573 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 18 2 B -12 0 -12 12 -4 C 0 12 0 8 -4 D -18 -12 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555618862 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 B=20 C=19 D=13 so D is eliminated. Round 2 votes counts: A=28 B=25 C=24 E=23 so E is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:208 E:208 B:192 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 18 2 B -12 0 -12 12 -4 C 0 12 0 8 -4 D -18 -12 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555618862 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 18 2 B -12 0 -12 12 -4 C 0 12 0 8 -4 D -18 -12 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555618862 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 18 2 B -12 0 -12 12 -4 C 0 12 0 8 -4 D -18 -12 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555618862 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1181: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) D E A C B (9) D B C A E (9) E A C B D (7) B D C A E (6) B C D A E (5) B C A E D (5) D A C E B (4) B E C A D (4) B C E A D (4) E A D C B (3) D E B A C (3) D B E A C (3) E D A C B (2) E B A C D (2) D C B A E (2) D C A B E (2) D A E C B (2) C B A E D (2) E A B D C (1) E A B C D (1) D E A B C (1) D C A E B (1) D A C B E (1) C E A B D (1) C A E B D (1) C A D B E (1) C A B E D (1) B E D C A (1) B E D A C (1) B D E C A (1) B C D E A (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 -8 -14 B -2 0 -4 -12 -2 C -6 4 0 -4 -6 D 8 12 4 0 2 E 14 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -8 -14 B -2 0 -4 -12 -2 C -6 4 0 -4 -6 D 8 12 4 0 2 E 14 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=28 B=28 C=6 A=1 so A is eliminated. Round 2 votes counts: D=37 E=28 B=28 C=7 so C is eliminated. Round 3 votes counts: D=38 E=31 B=31 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:210 C:194 A:193 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -8 -14 B -2 0 -4 -12 -2 C -6 4 0 -4 -6 D 8 12 4 0 2 E 14 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -8 -14 B -2 0 -4 -12 -2 C -6 4 0 -4 -6 D 8 12 4 0 2 E 14 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -8 -14 B -2 0 -4 -12 -2 C -6 4 0 -4 -6 D 8 12 4 0 2 E 14 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1182: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (6) E D C B A (5) E C D A B (5) A E C B D (5) D C B E A (4) C E D A B (4) C D E A B (4) C A E D B (4) B A E D C (4) A B E C D (4) D C E B A (3) D B E C A (3) B D E A C (3) B D A C E (3) B A D E C (3) A C B E D (3) E D C A B (2) E C A D B (2) E A C D B (2) E A B D C (2) D E B C A (2) C D A E B (2) B D C A E (2) B D A E C (2) B A D C E (2) A C E B D (2) E D B C A (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D B C E A (1) C E A D B (1) C D E B A (1) C D B A E (1) C B D A E (1) C A E B D (1) B E D A C (1) B E A D C (1) B A C D E (1) A E B C D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 2 -2 -2 B -8 0 -8 6 -6 C -2 8 0 6 -6 D 2 -6 -6 0 -22 E 2 6 6 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 -2 -2 B -8 0 -8 6 -6 C -2 8 0 6 -6 D 2 -6 -6 0 -22 E 2 6 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 A=23 B=22 C=19 D=13 so D is eliminated. Round 2 votes counts: C=26 B=26 E=25 A=23 so A is eliminated. Round 3 votes counts: B=37 C=32 E=31 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:218 A:203 C:203 B:192 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 -2 -2 B -8 0 -8 6 -6 C -2 8 0 6 -6 D 2 -6 -6 0 -22 E 2 6 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 -2 B -8 0 -8 6 -6 C -2 8 0 6 -6 D 2 -6 -6 0 -22 E 2 6 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 -2 B -8 0 -8 6 -6 C -2 8 0 6 -6 D 2 -6 -6 0 -22 E 2 6 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1183: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) E C D A B (9) B A D E C (7) D C A E B (5) E D C B A (4) E B D C A (4) B D E A C (4) A B C D E (4) C E A D B (3) B A E D C (3) B A D C E (3) E D C A B (2) D E C B A (2) D E B C A (2) D C E A B (2) D A C B E (2) D A B C E (2) C E D A B (2) C D E A B (2) C A E D B (2) B E D A C (2) B E A D C (2) B E A C D (2) B D A E C (2) B A C D E (2) A C D B E (2) A B D C E (2) A B C E D (2) E D B C A (1) E C A B D (1) E B C A D (1) C D A E B (1) C A E B D (1) B E D C A (1) B D A C E (1) B A E C D (1) B A C E D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -8 -18 -10 B 8 0 0 -6 -6 C 8 0 0 -8 -18 D 18 6 8 0 -6 E 10 6 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -8 -18 -10 B 8 0 0 -6 -6 C 8 0 0 -8 -18 D 18 6 8 0 -6 E 10 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=31 B=31 D=15 A=12 C=11 so C is eliminated. Round 2 votes counts: E=36 B=31 D=18 A=15 so A is eliminated. Round 3 votes counts: B=40 E=39 D=21 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:213 B:198 C:191 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -18 -10 B 8 0 0 -6 -6 C 8 0 0 -8 -18 D 18 6 8 0 -6 E 10 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -18 -10 B 8 0 0 -6 -6 C 8 0 0 -8 -18 D 18 6 8 0 -6 E 10 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -18 -10 B 8 0 0 -6 -6 C 8 0 0 -8 -18 D 18 6 8 0 -6 E 10 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1184: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) D C B E A (11) A E B C D (11) D E C B A (5) D B C A E (5) B C A D E (5) E D C B A (4) A E D C B (4) A B C E D (4) B D C A E (3) B C D A E (3) E D C A B (2) E C D B A (2) D C E B A (2) C D B E A (2) C B D E A (2) B C D E A (2) B A C E D (2) B A C D E (2) A E C B D (2) A E B D C (2) A B E C D (2) E D A C B (1) E C A B D (1) E A C D B (1) D E C A B (1) D A E B C (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E B A (1) B C A E D (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -6 0 2 B 6 0 -8 -14 -6 C 6 8 0 -12 -2 D 0 14 12 0 -4 E -2 6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.399442 B: 0.000000 C: 0.114525 D: 0.142459 E: 0.343574 Sum of squares = 0.311007787099 Cumulative probabilities = A: 0.399442 B: 0.399442 C: 0.513967 D: 0.656426 E: 1.000000 A B C D E A 0 -6 -6 0 2 B 6 0 -8 -14 -6 C 6 8 0 -12 -2 D 0 14 12 0 -4 E -2 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.396341 B: 0.000000 C: 0.115854 D: 0.140244 E: 0.347561 Sum of squares = 0.310975609742 Cumulative probabilities = A: 0.396341 B: 0.396341 C: 0.512195 D: 0.652439 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 E=22 B=18 C=6 so C is eliminated. Round 2 votes counts: D=30 A=27 E=23 B=20 so B is eliminated. Round 3 votes counts: D=40 A=37 E=23 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:211 E:205 C:200 A:195 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -6 0 2 B 6 0 -8 -14 -6 C 6 8 0 -12 -2 D 0 14 12 0 -4 E -2 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.396341 B: 0.000000 C: 0.115854 D: 0.140244 E: 0.347561 Sum of squares = 0.310975609742 Cumulative probabilities = A: 0.396341 B: 0.396341 C: 0.512195 D: 0.652439 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 0 2 B 6 0 -8 -14 -6 C 6 8 0 -12 -2 D 0 14 12 0 -4 E -2 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.396341 B: 0.000000 C: 0.115854 D: 0.140244 E: 0.347561 Sum of squares = 0.310975609742 Cumulative probabilities = A: 0.396341 B: 0.396341 C: 0.512195 D: 0.652439 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 0 2 B 6 0 -8 -14 -6 C 6 8 0 -12 -2 D 0 14 12 0 -4 E -2 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.396341 B: 0.000000 C: 0.115854 D: 0.140244 E: 0.347561 Sum of squares = 0.310975609742 Cumulative probabilities = A: 0.396341 B: 0.396341 C: 0.512195 D: 0.652439 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1185: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) A E C D B (9) E A D B C (7) E A B D C (7) C B D A E (7) D B C E A (6) A E C B D (6) D B E C A (4) B D C E A (4) C B A D E (3) B D E C A (3) A E B C D (3) A C E D B (3) E D B A C (2) C D E A B (2) C D B A E (2) C A E B D (2) C A D E B (2) C A B E D (2) C A B D E (2) B C D A E (2) E D C B A (1) E D A B C (1) E B A D C (1) E A D C B (1) D E C B A (1) D E B A C (1) C D B E A (1) C A D B E (1) B D E A C (1) B D A E C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 20 10 22 14 B -20 0 -12 12 -18 C -10 12 0 18 0 D -22 -12 -18 0 -12 E -14 18 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 22 14 B -20 0 -12 12 -18 C -10 12 0 18 0 D -22 -12 -18 0 -12 E -14 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 E=20 D=12 B=11 so B is eliminated. Round 2 votes counts: A=33 C=26 D=21 E=20 so E is eliminated. Round 3 votes counts: A=49 C=26 D=25 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:233 C:210 E:208 B:181 D:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 22 14 B -20 0 -12 12 -18 C -10 12 0 18 0 D -22 -12 -18 0 -12 E -14 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 22 14 B -20 0 -12 12 -18 C -10 12 0 18 0 D -22 -12 -18 0 -12 E -14 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 22 14 B -20 0 -12 12 -18 C -10 12 0 18 0 D -22 -12 -18 0 -12 E -14 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1186: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) D A E C B (10) B C E A D (9) B C E D A (8) C B A E D (7) C B A D E (7) B C D E A (4) D E A C B (3) D B C E A (3) C A B D E (3) B E C D A (3) B E C A D (3) E D A B C (2) E B A C D (2) E A D C B (2) D A C E B (2) C A B E D (2) B C D A E (2) A D C E B (2) A C D E B (2) E D B A C (1) E B D C A (1) E B A D C (1) E A B C D (1) D E B A C (1) D C B A E (1) C D B A E (1) C B D A E (1) C A D B E (1) B D E C A (1) B C A E D (1) A E D C B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -16 -8 -10 B 14 0 6 12 14 C 16 -6 0 16 14 D 8 -12 -16 0 10 E 10 -14 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -8 -10 B 14 0 6 12 14 C 16 -6 0 16 14 D 8 -12 -16 0 10 E 10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=30 C=22 E=10 A=7 so A is eliminated. Round 2 votes counts: D=33 B=31 C=25 E=11 so E is eliminated. Round 3 votes counts: D=39 B=36 C=25 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:220 D:195 E:186 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -8 -10 B 14 0 6 12 14 C 16 -6 0 16 14 D 8 -12 -16 0 10 E 10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -8 -10 B 14 0 6 12 14 C 16 -6 0 16 14 D 8 -12 -16 0 10 E 10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -8 -10 B 14 0 6 12 14 C 16 -6 0 16 14 D 8 -12 -16 0 10 E 10 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1187: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) C D A E B (6) E B A C D (5) B D C E A (5) E D C A B (4) E B D C A (4) C D A B E (4) C A D E B (4) B E D C A (4) B D E C A (4) B D C A E (4) A E C D B (4) A C E D B (4) E B A D C (3) E A B C D (3) A B C D E (3) E D B C A (2) E C D A B (2) E B D A C (2) D C E A B (2) D C B A E (2) D C A B E (2) B E A D C (2) B A D C E (2) A E C B D (2) A E B C D (2) A C D E B (2) D E C B A (1) D C E B A (1) D C A E B (1) D B C E A (1) C A E D B (1) C A D B E (1) B D A C E (1) B A C E D (1) B A C D E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -10 -4 -2 B -10 0 -2 -2 -22 C 10 2 0 6 0 D 4 2 -6 0 -2 E 2 22 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.575796 D: 0.000000 E: 0.424204 Sum of squares = 0.511490036132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.575796 D: 0.575796 E: 1.000000 A B C D E A 0 10 -10 -4 -2 B -10 0 -2 -2 -22 C 10 2 0 6 0 D 4 2 -6 0 -2 E 2 22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=24 A=19 C=16 D=10 so D is eliminated. Round 2 votes counts: E=32 B=25 C=24 A=19 so A is eliminated. Round 3 votes counts: E=40 C=32 B=28 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:209 D:199 A:197 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 -4 -2 B -10 0 -2 -2 -22 C 10 2 0 6 0 D 4 2 -6 0 -2 E 2 22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 -4 -2 B -10 0 -2 -2 -22 C 10 2 0 6 0 D 4 2 -6 0 -2 E 2 22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 -4 -2 B -10 0 -2 -2 -22 C 10 2 0 6 0 D 4 2 -6 0 -2 E 2 22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1188: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) E B D A C (7) D C B A E (7) D B C A E (7) E A C B D (6) C D A B E (6) C A D B E (6) E B D C A (5) D B A C E (5) A C E D B (5) C D B A E (4) C A E D B (4) B E D C A (4) B E D A C (4) E C B D A (2) E C B A D (2) E B A D C (2) E A B D C (2) D B C E A (2) A E C D B (2) A E C B D (2) A C D E B (2) E C A B D (1) C A D E B (1) B D E A C (1) B D A C E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -22 -18 -26 8 B 22 0 -2 -6 14 C 18 2 0 -12 4 D 26 6 12 0 4 E -8 -14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -18 -26 8 B 22 0 -2 -6 14 C 18 2 0 -12 4 D 26 6 12 0 4 E -8 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=21 C=21 B=18 A=13 so A is eliminated. Round 2 votes counts: E=31 C=29 D=22 B=18 so B is eliminated. Round 3 votes counts: E=39 D=32 C=29 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:214 C:206 E:185 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -18 -26 8 B 22 0 -2 -6 14 C 18 2 0 -12 4 D 26 6 12 0 4 E -8 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -18 -26 8 B 22 0 -2 -6 14 C 18 2 0 -12 4 D 26 6 12 0 4 E -8 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -18 -26 8 B 22 0 -2 -6 14 C 18 2 0 -12 4 D 26 6 12 0 4 E -8 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1189: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) E A C D B (6) C E A D B (6) C E A B D (5) E C A D B (4) D B A E C (4) C E B A D (4) B E D A C (4) A E D C B (4) A D C E B (4) A D E B C (3) E C B A D (2) E C A B D (2) D B A C E (2) D A B E C (2) D A B C E (2) C E B D A (2) B E A D C (2) B D E C A (2) B D C E A (2) B D C A E (2) B C E D A (2) B A D E C (2) A E D B C (2) A E C D B (2) A D B E C (2) A C E D B (2) E B C A D (1) E A B D C (1) E A B C D (1) D C B A E (1) D B C A E (1) D A C E B (1) C E D A B (1) C D B E A (1) C D B A E (1) C B E D A (1) C B D E A (1) C A D E B (1) B C D E A (1) A E B D C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 14 18 -2 B -8 0 -4 -6 -12 C -14 4 0 -8 -14 D -18 6 8 0 -10 E 2 12 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 14 18 -2 B -8 0 -4 -6 -12 C -14 4 0 -8 -14 D -18 6 8 0 -10 E 2 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 A=22 E=17 D=13 so D is eliminated. Round 2 votes counts: B=32 A=27 C=24 E=17 so E is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:219 D:193 B:185 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 14 18 -2 B -8 0 -4 -6 -12 C -14 4 0 -8 -14 D -18 6 8 0 -10 E 2 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 18 -2 B -8 0 -4 -6 -12 C -14 4 0 -8 -14 D -18 6 8 0 -10 E 2 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 18 -2 B -8 0 -4 -6 -12 C -14 4 0 -8 -14 D -18 6 8 0 -10 E 2 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1190: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) C A E B D (12) D B A E C (9) C E A B D (7) B E A D C (5) E A C B D (4) D C E A B (4) E C A B D (3) C E A D B (3) B D A E C (3) A C E B D (3) E A B C D (2) D E B C A (2) D C B A E (2) C D E A B (2) C D A E B (2) C A E D B (2) B E D A C (2) B A E C D (2) A C B E D (2) E D B A C (1) E C D A B (1) E C A D B (1) E B A C D (1) D E B A C (1) D C E B A (1) D B E C A (1) D B C A E (1) C D A B E (1) C A B D E (1) B E A C D (1) B D A C E (1) B A E D C (1) B A D E C (1) B A C D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -4 C -8 2 0 6 -8 D -8 -8 -6 0 -10 E 8 4 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -4 C -8 2 0 6 -8 D -8 -8 -6 0 -10 E 8 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=30 B=17 E=13 A=7 so A is eliminated. Round 2 votes counts: C=35 D=33 B=19 E=13 so E is eliminated. Round 3 votes counts: C=44 D=34 B=22 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:215 A:206 B:199 C:196 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -4 C -8 2 0 6 -8 D -8 -8 -6 0 -10 E 8 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -4 C -8 2 0 6 -8 D -8 -8 -6 0 -10 E 8 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 -8 B -4 0 -2 8 -4 C -8 2 0 6 -8 D -8 -8 -6 0 -10 E 8 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1191: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) D E A C B (7) B C A D E (6) C B D A E (5) B A E C D (5) C B A D E (4) B C A E D (4) B A C E D (4) E A D B C (3) D E C A B (3) D C A E B (3) B E A D C (3) B A E D C (3) B A C D E (3) A E B D C (3) A D E C B (3) E D A B C (2) E B D A C (2) E B A D C (2) C D E B A (2) C D B A E (2) C D A B E (2) B E A C D (2) A E D B C (2) A B D C E (2) E C D B A (1) D C E A B (1) D A E C B (1) D A C E B (1) C D E A B (1) C D B E A (1) C D A E B (1) C B D E A (1) C A D B E (1) C A B D E (1) B C E D A (1) B C E A D (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 18 8 16 B 4 0 -2 6 6 C -18 2 0 -2 -4 D -8 -6 2 0 6 E -16 -6 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.083333 B: 0.750000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.597222222321 Cumulative probabilities = A: 0.083333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 18 8 16 B 4 0 -2 6 6 C -18 2 0 -2 -4 D -8 -6 2 0 6 E -16 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.750000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.597222222235 Cumulative probabilities = A: 0.083333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=21 E=19 D=16 A=12 so A is eliminated. Round 2 votes counts: B=35 E=24 C=22 D=19 so D is eliminated. Round 3 votes counts: E=38 B=35 C=27 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:207 D:197 C:189 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 18 8 16 B 4 0 -2 6 6 C -18 2 0 -2 -4 D -8 -6 2 0 6 E -16 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.750000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.597222222235 Cumulative probabilities = A: 0.083333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 18 8 16 B 4 0 -2 6 6 C -18 2 0 -2 -4 D -8 -6 2 0 6 E -16 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.750000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.597222222235 Cumulative probabilities = A: 0.083333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 18 8 16 B 4 0 -2 6 6 C -18 2 0 -2 -4 D -8 -6 2 0 6 E -16 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.750000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.597222222235 Cumulative probabilities = A: 0.083333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1192: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) D E B C A (6) B E D A C (6) A C B E D (6) A B E C D (6) D C E B A (5) C D A E B (5) C A E B D (5) D B E A C (4) B A E D C (4) E B D C A (3) D B E C A (3) C D E A B (3) E B D A C (2) E B A D C (2) D B A E C (2) D A B C E (2) C E A B D (2) B E A D C (2) B D E A C (2) A C D B E (2) E D B C A (1) E C B D A (1) E C B A D (1) E B C A D (1) E B A C D (1) E A B C D (1) D E C B A (1) D E B A C (1) D C E A B (1) D C B E A (1) D C A E B (1) D C A B E (1) D A C B E (1) C E D B A (1) C A D B E (1) B A E C D (1) A C E B D (1) A C B D E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 -6 -2 B 2 0 4 0 -6 C 2 -4 0 -4 -4 D 6 0 4 0 4 E 2 6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.236438 C: 0.000000 D: 0.763562 E: 0.000000 Sum of squares = 0.638930149909 Cumulative probabilities = A: 0.000000 B: 0.236438 C: 0.236438 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 -2 B 2 0 4 0 -6 C 2 -4 0 -4 -4 D 6 0 4 0 4 E 2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000001229 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 A=18 B=15 E=13 so E is eliminated. Round 2 votes counts: D=30 C=27 B=24 A=19 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:207 E:204 B:200 C:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 -2 B 2 0 4 0 -6 C 2 -4 0 -4 -4 D 6 0 4 0 4 E 2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000001229 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 -2 B 2 0 4 0 -6 C 2 -4 0 -4 -4 D 6 0 4 0 4 E 2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000001229 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 -2 B 2 0 4 0 -6 C 2 -4 0 -4 -4 D 6 0 4 0 4 E 2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000001229 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1193: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) B E C D A (8) A D C B E (8) E B C A D (7) A E C D B (7) B D C E A (6) B C D E A (6) A D E C B (4) E C B A D (3) E B C D A (3) E B A C D (3) D B C A E (3) A E D C B (3) E C A D B (2) E A C D B (2) D C B A E (2) D B A C E (2) D A C B E (2) B C E D A (2) A E B C D (2) A D B C E (2) E C B D A (1) E C A B D (1) E A C B D (1) E A B C D (1) D C E A B (1) D C B E A (1) D B C E A (1) D A C E B (1) B E A C D (1) B D C A E (1) B D A C E (1) B A E D C (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 4 18 0 B 6 0 4 0 -2 C -4 -4 0 0 -4 D -18 0 0 0 2 E 0 2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.235522 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.764478 Sum of squares = 0.639897349007 Cumulative probabilities = A: 0.235522 B: 0.235522 C: 0.235522 D: 0.235522 E: 1.000000 A B C D E A 0 -6 4 18 0 B 6 0 4 0 -2 C -4 -4 0 0 -4 D -18 0 0 0 2 E 0 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000035463 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=27 E=24 D=13 so C is eliminated. Round 2 votes counts: A=36 B=27 E=24 D=13 so D is eliminated. Round 3 votes counts: A=39 B=36 E=25 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:208 B:204 E:202 C:194 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 18 0 B 6 0 4 0 -2 C -4 -4 0 0 -4 D -18 0 0 0 2 E 0 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000035463 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 18 0 B 6 0 4 0 -2 C -4 -4 0 0 -4 D -18 0 0 0 2 E 0 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000035463 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 18 0 B 6 0 4 0 -2 C -4 -4 0 0 -4 D -18 0 0 0 2 E 0 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000035463 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1194: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (11) C B D E A (9) E D B C A (7) E C B D A (7) B C D E A (6) D B C A E (5) A E C B D (5) A D E B C (5) A C B D E (5) E A D B C (4) C B E D A (4) A C B E D (4) A B C D E (4) A E D B C (3) E A C B D (2) D E B C A (2) D B C E A (2) C B D A E (2) B C D A E (2) E D A B C (1) E C D B A (1) E C B A D (1) E A D C B (1) D A E B C (1) D A B E C (1) B D C A E (1) B A C D E (1) A E D C B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 -2 6 B 0 0 16 8 18 C -2 -16 0 6 14 D 2 -8 -6 0 18 E -6 -18 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.561375 B: 0.438625 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.507533716821 Cumulative probabilities = A: 0.561375 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -2 6 B 0 0 16 8 18 C -2 -16 0 6 14 D 2 -8 -6 0 18 E -6 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 E=24 C=15 D=11 B=10 so B is eliminated. Round 2 votes counts: A=41 E=24 C=23 D=12 so D is eliminated. Round 3 votes counts: A=43 C=31 E=26 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:221 A:203 D:203 C:201 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -2 6 B 0 0 16 8 18 C -2 -16 0 6 14 D 2 -8 -6 0 18 E -6 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 6 B 0 0 16 8 18 C -2 -16 0 6 14 D 2 -8 -6 0 18 E -6 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 6 B 0 0 16 8 18 C -2 -16 0 6 14 D 2 -8 -6 0 18 E -6 -18 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1195: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) E C A D B (7) E C D B A (6) C E B A D (6) C E B D A (5) C B E D A (5) A D B E C (5) E C D A B (4) B D A C E (4) B C D A E (4) D B A E C (3) C E A D B (3) A B D C E (3) E D C B A (2) E D B C A (2) E D A B C (2) E C B D A (2) D E A B C (2) C E A B D (2) B D C A E (2) B D A E C (2) A E C D B (2) A D E C B (2) A D E B C (2) E D B A C (1) E D A C B (1) E A D C B (1) E A C D B (1) D B E A C (1) D A E B C (1) C B E A D (1) C B D A E (1) C B A E D (1) C B A D E (1) C A E B D (1) B D E C A (1) A E D C B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -12 -18 -10 B 0 0 -12 -14 -12 C 12 12 0 8 -18 D 18 14 -8 0 -12 E 10 12 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -12 -18 -10 B 0 0 -12 -14 -12 C 12 12 0 8 -18 D 18 14 -8 0 -12 E 10 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 A=17 D=15 B=13 so B is eliminated. Round 2 votes counts: C=30 E=29 D=24 A=17 so A is eliminated. Round 3 votes counts: D=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:226 C:207 D:206 B:181 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -12 -18 -10 B 0 0 -12 -14 -12 C 12 12 0 8 -18 D 18 14 -8 0 -12 E 10 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -18 -10 B 0 0 -12 -14 -12 C 12 12 0 8 -18 D 18 14 -8 0 -12 E 10 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -18 -10 B 0 0 -12 -14 -12 C 12 12 0 8 -18 D 18 14 -8 0 -12 E 10 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1196: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (7) C E D A B (6) C D E B A (6) A E C D B (5) A E C B D (5) D C E A B (4) B D C E A (4) B A D C E (4) B A C E D (4) E C D A B (3) D C E B A (3) C E D B A (3) B D C A E (3) B C D E A (3) B A E C D (3) E C A D B (2) D C B E A (2) D B C A E (2) D B A C E (2) D A E B C (2) B C D A E (2) B C A E D (2) B A E D C (2) A E D C B (2) A E B C D (2) E D C A B (1) E C A B D (1) E A D C B (1) E A C B D (1) D E C A B (1) D B C E A (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E D A (1) B C E D A (1) B C E A D (1) A E D B C (1) A E B D C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -16 -8 -4 6 B 16 0 0 -2 -8 C 8 0 0 6 12 D 4 2 -6 0 0 E -6 8 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.294541 C: 0.705459 D: 0.000000 E: 0.000000 Sum of squares = 0.584426695787 Cumulative probabilities = A: 0.000000 B: 0.294541 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -4 6 B 16 0 0 -2 -8 C 8 0 0 6 12 D 4 2 -6 0 0 E -6 8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=19 C=18 A=18 E=9 so E is eliminated. Round 2 votes counts: B=36 C=24 D=20 A=20 so D is eliminated. Round 3 votes counts: B=42 C=35 A=23 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:203 D:200 E:195 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -8 -4 6 B 16 0 0 -2 -8 C 8 0 0 6 12 D 4 2 -6 0 0 E -6 8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -4 6 B 16 0 0 -2 -8 C 8 0 0 6 12 D 4 2 -6 0 0 E -6 8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -4 6 B 16 0 0 -2 -8 C 8 0 0 6 12 D 4 2 -6 0 0 E -6 8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1197: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) B D A E C (8) D B C A E (7) D B A E C (5) C D E B A (5) E C A B D (4) C E A B D (4) C D B E A (4) A E C D B (4) A E B D C (4) A E B C D (4) D B C E A (3) D B A C E (3) B D C A E (3) E A C D B (2) C E D B A (2) B D C E A (2) B A D E C (2) A E C B D (2) A B E D C (2) A B D E C (2) E C A D B (1) E B A C D (1) E A B C D (1) D C B E A (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A D B (1) C D E A B (1) C D A E B (1) C B E D A (1) C A E D B (1) B E A D C (1) B D E A C (1) B D A C E (1) B A E D C (1) A E D B C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 12 0 6 B 6 0 8 10 -4 C -12 -8 0 2 -14 D 0 -10 -2 0 2 E -6 4 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999999 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 12 0 6 B 6 0 8 10 -4 C -12 -8 0 2 -14 D 0 -10 -2 0 2 E -6 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000012 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 A=21 D=19 B=19 E=18 so E is eliminated. Round 2 votes counts: A=33 C=28 B=20 D=19 so D is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:206 E:205 D:195 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 0 6 B 6 0 8 10 -4 C -12 -8 0 2 -14 D 0 -10 -2 0 2 E -6 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000012 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 0 6 B 6 0 8 10 -4 C -12 -8 0 2 -14 D 0 -10 -2 0 2 E -6 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000012 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 0 6 B 6 0 8 10 -4 C -12 -8 0 2 -14 D 0 -10 -2 0 2 E -6 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000012 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1198: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A B D E (9) D E B C A (8) D E B A C (6) A C E D B (6) E D B A C (5) B D E C A (5) E D A B C (4) C A B E D (4) B E D C A (4) A C B E D (4) D E C B A (3) C D E A B (3) C A D E B (3) B A E D C (3) A B E D C (3) D E A C B (2) D E A B C (2) C B D E A (2) C B A E D (2) B C D E A (2) A E D B C (2) A B C E D (2) D B E C A (1) C D E B A (1) B C E D A (1) B A C E D (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 4 -20 -20 B 10 0 20 6 8 C -4 -20 0 -18 -18 D 20 -6 18 0 -4 E 20 -8 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -20 -20 B 10 0 20 6 8 C -4 -20 0 -18 -18 D 20 -6 18 0 -4 E 20 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 D=22 A=18 E=9 so E is eliminated. Round 2 votes counts: D=31 B=27 C=24 A=18 so A is eliminated. Round 3 votes counts: C=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:222 E:217 D:214 A:177 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 -20 -20 B 10 0 20 6 8 C -4 -20 0 -18 -18 D 20 -6 18 0 -4 E 20 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -20 -20 B 10 0 20 6 8 C -4 -20 0 -18 -18 D 20 -6 18 0 -4 E 20 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -20 -20 B 10 0 20 6 8 C -4 -20 0 -18 -18 D 20 -6 18 0 -4 E 20 -8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1199: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) D C A E B (5) C D E B A (5) A C D B E (5) D E C A B (4) D C E B A (4) B E A D C (4) B A C E D (4) A B C D E (4) E D C B A (3) E D A C B (3) E B A D C (3) C D B A E (3) C A D B E (3) B E A C D (3) B A E D C (3) B A E C D (3) A B C E D (3) E D A B C (2) D E C B A (2) D C E A B (2) C D A E B (2) C B D E A (2) C B A D E (2) A E B D C (2) A C B D E (2) A B E D C (2) A B E C D (2) E D B C A (1) E B D A C (1) E B C D A (1) D E A C B (1) C D E A B (1) B E D C A (1) B E C D A (1) B A C D E (1) A E D B C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 6 2 -2 12 B -6 0 -16 -12 10 C -2 16 0 8 12 D 2 12 -8 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999794 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -2 12 B -6 0 -16 -12 10 C -2 16 0 8 12 D 2 12 -8 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999784 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 B=20 D=18 E=14 so E is eliminated. Round 2 votes counts: D=27 C=25 B=25 A=23 so A is eliminated. Round 3 votes counts: B=38 C=32 D=30 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:210 A:209 B:188 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 -2 12 B -6 0 -16 -12 10 C -2 16 0 8 12 D 2 12 -8 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999784 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -2 12 B -6 0 -16 -12 10 C -2 16 0 8 12 D 2 12 -8 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999784 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -2 12 B -6 0 -16 -12 10 C -2 16 0 8 12 D 2 12 -8 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999784 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1200: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (13) D C E A B (10) B A E C D (8) D E B A C (6) C D A B E (6) C A B D E (5) A B E C D (5) E D B A C (4) E B A D C (4) D E C B A (4) D C E B A (4) B A E D C (4) A B C E D (4) E D A B C (3) C D E A B (3) C D B A E (3) C B A E D (2) C A D B E (2) E B D A C (1) E A B D C (1) D E C A B (1) D C B E A (1) C D A E B (1) C B D A E (1) C B A D E (1) B C A D E (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 -16 4 16 B -12 0 -14 4 14 C 16 14 0 12 14 D -4 -4 -12 0 -2 E -16 -14 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -16 4 16 B -12 0 -14 4 14 C 16 14 0 12 14 D -4 -4 -12 0 -2 E -16 -14 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=26 E=13 B=13 A=11 so A is eliminated. Round 2 votes counts: C=37 D=26 B=23 E=14 so E is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:208 B:196 D:189 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -16 4 16 B -12 0 -14 4 14 C 16 14 0 12 14 D -4 -4 -12 0 -2 E -16 -14 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -16 4 16 B -12 0 -14 4 14 C 16 14 0 12 14 D -4 -4 -12 0 -2 E -16 -14 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -16 4 16 B -12 0 -14 4 14 C 16 14 0 12 14 D -4 -4 -12 0 -2 E -16 -14 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1201: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D E A C B (5) C B A D E (5) B C E A D (5) B C A E D (5) E B A D C (4) D A C E B (4) E D C B A (3) E B C D A (3) E A D B C (3) D A E C B (3) C D A B E (3) C A B D E (3) B A E C D (3) B A C E D (3) A D E B C (3) E D B C A (2) E B D C A (2) E B C A D (2) E A B D C (2) D E C B A (2) D E C A B (2) D E A B C (2) D C A E B (2) D A C B E (2) C B A E D (2) C A D B E (2) B A C D E (2) A D B C E (2) A C B D E (2) A B C D E (2) E D C A B (1) D A E B C (1) C D E B A (1) C D E A B (1) C D B A E (1) C B D A E (1) B E C A D (1) B E A C D (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 2 8 6 B -4 0 8 -2 -8 C -2 -8 0 -2 -2 D -8 2 2 0 4 E -6 8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 8 6 B -4 0 8 -2 -8 C -2 -8 0 -2 -2 D -8 2 2 0 4 E -6 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 B=20 C=19 A=11 so A is eliminated. Round 2 votes counts: E=28 D=28 C=22 B=22 so C is eliminated. Round 3 votes counts: D=37 B=35 E=28 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:200 E:200 B:197 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 8 6 B -4 0 8 -2 -8 C -2 -8 0 -2 -2 D -8 2 2 0 4 E -6 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 8 6 B -4 0 8 -2 -8 C -2 -8 0 -2 -2 D -8 2 2 0 4 E -6 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 8 6 B -4 0 8 -2 -8 C -2 -8 0 -2 -2 D -8 2 2 0 4 E -6 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1202: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) E C B A D (8) E C A B D (6) A B C D E (6) C E B A D (5) C B A E D (5) B C A D E (5) E D C B A (4) E D A B C (4) E C D B A (4) D A B E C (4) B A D C E (4) B A C D E (4) E D C A B (3) C B E A D (3) A D B C E (3) A B D C E (3) E C D A B (2) D E B A C (2) C B A D E (2) B D A C E (2) E D B C A (1) E D B A C (1) D E A C B (1) D E A B C (1) D A E B C (1) D A C B E (1) C E A B D (1) B D E C A (1) B C E A D (1) B C D A E (1) B C A E D (1) Total count = 100 A B C D E A 0 -8 -6 14 4 B 8 0 10 16 12 C 6 -10 0 8 14 D -14 -16 -8 0 2 E -4 -12 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 14 4 B 8 0 10 16 12 C 6 -10 0 8 14 D -14 -16 -8 0 2 E -4 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=20 B=19 C=16 A=12 so A is eliminated. Round 2 votes counts: E=33 B=28 D=23 C=16 so C is eliminated. Round 3 votes counts: E=39 B=38 D=23 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:209 A:202 E:184 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 14 4 B 8 0 10 16 12 C 6 -10 0 8 14 D -14 -16 -8 0 2 E -4 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 14 4 B 8 0 10 16 12 C 6 -10 0 8 14 D -14 -16 -8 0 2 E -4 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 14 4 B 8 0 10 16 12 C 6 -10 0 8 14 D -14 -16 -8 0 2 E -4 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1203: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) E D B A C (7) C A B D E (6) E A C D B (5) B D C A E (5) A E C B D (5) D E B C A (4) B D E C A (4) A B C D E (4) A C E B D (3) A C B E D (3) A C B D E (3) E D C B A (2) E D A C B (2) E A D C B (2) D B E C A (2) D B C E A (2) C D B A E (2) C D A B E (2) C A E D B (2) B D C E A (2) B C D A E (2) B A D C E (2) B A C D E (2) E D C A B (1) E D A B C (1) E C A D B (1) E B D A C (1) E A D B C (1) E A C B D (1) E A B D C (1) D E C B A (1) D C E B A (1) D C E A B (1) D B C A E (1) C D A E B (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D E A C (1) B C A D E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -6 -6 -2 B 2 0 6 0 -4 C 6 -6 0 -4 -4 D 6 0 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.348557 C: 0.000000 D: 0.651443 E: 0.000000 Sum of squares = 0.545869886397 Cumulative probabilities = A: 0.000000 B: 0.348557 C: 0.348557 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 -2 B 2 0 6 0 -4 C 6 -6 0 -4 -4 D 6 0 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499890 C: 0.000000 D: 0.500110 E: 0.000000 Sum of squares = 0.500000024407 Cumulative probabilities = A: 0.000000 B: 0.499890 C: 0.499890 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=20 A=20 C=16 D=12 so D is eliminated. Round 2 votes counts: E=37 B=25 A=20 C=18 so C is eliminated. Round 3 votes counts: E=39 A=33 B=28 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:207 E:203 B:202 C:196 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -2 B 2 0 6 0 -4 C 6 -6 0 -4 -4 D 6 0 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499890 C: 0.000000 D: 0.500110 E: 0.000000 Sum of squares = 0.500000024407 Cumulative probabilities = A: 0.000000 B: 0.499890 C: 0.499890 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -2 B 2 0 6 0 -4 C 6 -6 0 -4 -4 D 6 0 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499890 C: 0.000000 D: 0.500110 E: 0.000000 Sum of squares = 0.500000024407 Cumulative probabilities = A: 0.000000 B: 0.499890 C: 0.499890 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -2 B 2 0 6 0 -4 C 6 -6 0 -4 -4 D 6 0 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499890 C: 0.000000 D: 0.500110 E: 0.000000 Sum of squares = 0.500000024407 Cumulative probabilities = A: 0.000000 B: 0.499890 C: 0.499890 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1204: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) C E B A D (10) A D B E C (7) D B E A C (6) D A B E C (6) C A E B D (5) E B D C A (4) D B A E C (4) C A D E B (4) E B D A C (3) D B E C A (3) C E A B D (3) C D E B A (3) B E D A C (3) E B C D A (2) E B C A D (2) E B A D C (2) E B A C D (2) A D C B E (2) A C D B E (2) E D C B A (1) E C B A D (1) D E B C A (1) D C A B E (1) D A B C E (1) C E D B A (1) C A D B E (1) B D E A C (1) B A D E C (1) A E B D C (1) A E B C D (1) A D B C E (1) A C E B D (1) A C B D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 -6 -2 -18 B 22 0 6 12 -16 C 6 -6 0 0 -6 D 2 -12 0 0 -8 E 18 16 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -22 -6 -2 -18 B 22 0 6 12 -16 C 6 -6 0 0 -6 D 2 -12 0 0 -8 E 18 16 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=22 A=18 E=17 B=5 so B is eliminated. Round 2 votes counts: C=38 D=23 E=20 A=19 so A is eliminated. Round 3 votes counts: C=42 D=35 E=23 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:224 B:212 C:197 D:191 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -22 -6 -2 -18 B 22 0 6 12 -16 C 6 -6 0 0 -6 D 2 -12 0 0 -8 E 18 16 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -2 -18 B 22 0 6 12 -16 C 6 -6 0 0 -6 D 2 -12 0 0 -8 E 18 16 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -2 -18 B 22 0 6 12 -16 C 6 -6 0 0 -6 D 2 -12 0 0 -8 E 18 16 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1205: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) C A D E B (8) C A D B E (8) B E A D C (6) E B D A C (5) B E C A D (4) E D B A C (3) E B C D A (3) D A E C B (3) D A B E C (3) C A B D E (3) B E A C D (3) A C D B E (3) E B D C A (2) D E B A C (2) D C A E B (2) D A C B E (2) C B E A D (2) C B A E D (2) B E D A C (2) B C A E D (2) A D B C E (2) A C B D E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B D A (1) E C B A D (1) D E C A B (1) D E A C B (1) D C E A B (1) D A E B C (1) C E D A B (1) C E B A D (1) C D A E B (1) B A E D C (1) B A E C D (1) B A C E D (1) A D C B E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 10 8 6 16 B -10 0 -14 -14 -2 C -8 14 0 -2 8 D -6 14 2 0 12 E -16 2 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 6 16 B -10 0 -14 -14 -2 C -8 14 0 -2 8 D -6 14 2 0 12 E -16 2 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 B=20 E=18 A=10 so A is eliminated. Round 2 votes counts: C=32 D=29 B=21 E=18 so E is eliminated. Round 3 votes counts: C=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:220 D:211 C:206 E:183 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 6 16 B -10 0 -14 -14 -2 C -8 14 0 -2 8 D -6 14 2 0 12 E -16 2 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 6 16 B -10 0 -14 -14 -2 C -8 14 0 -2 8 D -6 14 2 0 12 E -16 2 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 6 16 B -10 0 -14 -14 -2 C -8 14 0 -2 8 D -6 14 2 0 12 E -16 2 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1206: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) E D A C B (6) D E A B C (6) A D B C E (6) E D C B A (5) C B E D A (5) C B D E A (5) A B C E D (5) D E C B A (4) C B A E D (4) A B C D E (4) D B C A E (3) C E B D A (3) B C D A E (3) A E D B C (3) E A D B C (2) E A C B D (2) C B E A D (2) C B A D E (2) B A C D E (2) A E C B D (2) A E B C D (2) A D E B C (2) E D C A B (1) E D A B C (1) E C B A D (1) E A D C B (1) D E C A B (1) D E A C B (1) D C E B A (1) D A E B C (1) D A B C E (1) C D E B A (1) C B D A E (1) B C A E D (1) B A C E D (1) Total count = 100 A B C D E A 0 -6 -4 2 4 B 6 0 4 8 8 C 4 -4 0 10 18 D -2 -8 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 2 4 B 6 0 4 8 8 C 4 -4 0 10 18 D -2 -8 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 E=19 D=18 B=16 so B is eliminated. Round 2 votes counts: C=36 A=27 E=19 D=18 so D is eliminated. Round 3 votes counts: C=40 E=31 A=29 so A is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:213 A:198 D:193 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 2 4 B 6 0 4 8 8 C 4 -4 0 10 18 D -2 -8 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 2 4 B 6 0 4 8 8 C 4 -4 0 10 18 D -2 -8 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 2 4 B 6 0 4 8 8 C 4 -4 0 10 18 D -2 -8 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1207: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) E A D C B (6) E A C D B (6) D C E B A (6) C D B E A (6) A E B C D (6) D C B E A (5) B C D A E (5) B A C D E (5) A E B D C (5) E C D A B (4) E A D B C (3) A E C D B (3) A E C B D (3) A B E C D (3) D E C B A (2) C D E B A (2) B D C E A (2) B D C A E (2) B A E D C (2) A B E D C (2) E D B A C (1) E D A C B (1) E C A D B (1) E A B D C (1) D E C A B (1) D B E C A (1) D B C E A (1) C D E A B (1) C D B A E (1) C A E D B (1) B D A C E (1) B A D E C (1) B A D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 2 0 -16 B -12 0 -14 -20 -22 C -2 14 0 -4 -20 D 0 20 4 0 -14 E 16 22 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 0 -16 B -12 0 -14 -20 -22 C -2 14 0 -4 -20 D 0 20 4 0 -14 E 16 22 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=23 B=19 D=16 C=11 so C is eliminated. Round 2 votes counts: E=31 D=26 A=24 B=19 so B is eliminated. Round 3 votes counts: D=36 A=33 E=31 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:236 D:205 A:199 C:194 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 0 -16 B -12 0 -14 -20 -22 C -2 14 0 -4 -20 D 0 20 4 0 -14 E 16 22 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 0 -16 B -12 0 -14 -20 -22 C -2 14 0 -4 -20 D 0 20 4 0 -14 E 16 22 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 0 -16 B -12 0 -14 -20 -22 C -2 14 0 -4 -20 D 0 20 4 0 -14 E 16 22 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1208: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) C E A D B (8) A B D E C (6) E C B D A (5) E B C D A (5) A D C B E (5) D B A E C (4) D A C B E (4) B D E A C (4) A D B C E (4) E C B A D (3) E B C A D (3) C E A B D (3) B E D C A (3) E C A B D (2) D A B C E (2) C E B D A (2) C A D E B (2) B E D A C (2) B E A D C (2) B A D E C (2) A C D B E (2) A B E D C (2) E C D B A (1) E B D C A (1) E B A D C (1) E B A C D (1) D B E C A (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E B A (1) C A E D B (1) B A E D C (1) A E C B D (1) A D B E C (1) A C E D B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 14 8 0 B 6 0 12 22 14 C -14 -12 0 -12 -22 D -8 -22 12 0 -2 E 0 -14 22 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 14 8 0 B 6 0 12 22 14 C -14 -12 0 -12 -22 D -8 -22 12 0 -2 E 0 -14 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 E=22 C=19 D=12 so D is eliminated. Round 2 votes counts: A=31 B=28 E=22 C=19 so C is eliminated. Round 3 votes counts: E=38 A=34 B=28 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:227 A:208 E:205 D:190 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 14 8 0 B 6 0 12 22 14 C -14 -12 0 -12 -22 D -8 -22 12 0 -2 E 0 -14 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 8 0 B 6 0 12 22 14 C -14 -12 0 -12 -22 D -8 -22 12 0 -2 E 0 -14 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 8 0 B 6 0 12 22 14 C -14 -12 0 -12 -22 D -8 -22 12 0 -2 E 0 -14 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1209: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (14) D B E C A (6) E A C B D (5) D E B C A (5) C A B E D (5) C A B D E (5) E D B A C (4) E B C A D (4) E A C D B (4) D E B A C (4) B C A E D (4) B C A D E (4) A C D B E (4) A C E D B (3) E C A B D (2) E B D C A (2) D B C A E (2) D A C E B (2) D A C B E (2) B E C D A (2) B D E C A (2) B D C A E (2) E D A C B (1) E D A B C (1) E C B A D (1) E A D C B (1) D E A B C (1) D B A C E (1) C A E B D (1) B E C A D (1) B C E A D (1) A D C E B (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 2 26 6 B -10 0 -8 14 -14 C -2 8 0 26 8 D -26 -14 -26 0 -14 E -6 14 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 26 6 B -10 0 -8 14 -14 C -2 8 0 26 8 D -26 -14 -26 0 -14 E -6 14 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 D=23 B=16 C=11 so C is eliminated. Round 2 votes counts: A=36 E=25 D=23 B=16 so B is eliminated. Round 3 votes counts: A=44 E=29 D=27 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:220 E:207 B:191 D:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 26 6 B -10 0 -8 14 -14 C -2 8 0 26 8 D -26 -14 -26 0 -14 E -6 14 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 26 6 B -10 0 -8 14 -14 C -2 8 0 26 8 D -26 -14 -26 0 -14 E -6 14 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 26 6 B -10 0 -8 14 -14 C -2 8 0 26 8 D -26 -14 -26 0 -14 E -6 14 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1210: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) C A E B D (10) D B A E C (7) D E B C A (6) B A D C E (6) E C D B A (5) D B E A C (5) D B A C E (5) A C B D E (5) E C A B D (4) E D C B A (3) E C D A B (3) A C B E D (3) A B C D E (3) E D B A C (2) E C A D B (2) C E A D B (2) B D A C E (2) B A D E C (2) A B C E D (2) E D B C A (1) E C B A D (1) D E C B A (1) D E B A C (1) D C E B A (1) D C E A B (1) D C A B E (1) C E D A B (1) C D A E B (1) C A B E D (1) B D A E C (1) B A E D C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 6 2 B 0 0 -12 2 -10 C 8 12 0 8 10 D -6 -2 -8 0 -2 E -2 10 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 6 2 B 0 0 -12 2 -10 C 8 12 0 8 10 D -6 -2 -8 0 -2 E -2 10 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 E=21 A=14 B=12 so B is eliminated. Round 2 votes counts: D=31 C=25 A=23 E=21 so E is eliminated. Round 3 votes counts: C=40 D=37 A=23 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:200 E:200 D:191 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 6 2 B 0 0 -12 2 -10 C 8 12 0 8 10 D -6 -2 -8 0 -2 E -2 10 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 6 2 B 0 0 -12 2 -10 C 8 12 0 8 10 D -6 -2 -8 0 -2 E -2 10 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 6 2 B 0 0 -12 2 -10 C 8 12 0 8 10 D -6 -2 -8 0 -2 E -2 10 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1211: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) D E C B A (6) D B A E C (6) D A B E C (6) C E B A D (6) A B C D E (6) A B C E D (5) E C B D A (4) C A E B D (4) A D B C E (4) A C B E D (4) E C B A D (3) D A B C E (3) B E C D A (3) A C E B D (3) E D C B A (2) E C D B A (2) D E C A B (2) D E B C A (2) D B E C A (2) C E A B D (2) B D E C A (2) B A D C E (2) A C B D E (2) E C D A B (1) D E B A C (1) D A E B C (1) C B E A D (1) B D E A C (1) B D A E C (1) B C A E D (1) B A C E D (1) B A C D E (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 14 -4 6 B 8 0 10 4 18 C -14 -10 0 0 -4 D 4 -4 0 0 16 E -6 -18 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 14 -4 6 B 8 0 10 4 18 C -14 -10 0 0 -4 D 4 -4 0 0 16 E -6 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996599 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=27 C=13 E=12 B=12 so E is eliminated. Round 2 votes counts: D=38 A=27 C=23 B=12 so B is eliminated. Round 3 votes counts: D=42 A=31 C=27 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:220 D:208 A:204 C:186 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 14 -4 6 B 8 0 10 4 18 C -14 -10 0 0 -4 D 4 -4 0 0 16 E -6 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996599 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 -4 6 B 8 0 10 4 18 C -14 -10 0 0 -4 D 4 -4 0 0 16 E -6 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996599 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 -4 6 B 8 0 10 4 18 C -14 -10 0 0 -4 D 4 -4 0 0 16 E -6 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996599 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1212: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) C A E D B (7) A C B E D (7) B A D E C (5) B A C E D (5) A B C E D (5) E C D A B (4) D B E A C (4) B A C D E (4) A C E D B (4) A C E B D (4) E D C B A (3) B D E A C (3) B D A E C (3) B A D C E (3) A B C D E (3) D E C B A (2) D E C A B (2) D E B C A (2) D B E C A (2) C E A D B (2) C A E B D (2) B E D C A (2) B D E C A (2) B C E A D (2) E D B C A (1) D E B A C (1) D E A C B (1) D A B E C (1) C E D A B (1) C E B A D (1) C E A B D (1) A D E C B (1) A C D E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 14 18 14 B -10 0 0 8 6 C -14 0 0 6 6 D -18 -8 -6 0 -16 E -14 -6 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 18 14 B -10 0 0 8 6 C -14 0 0 6 6 D -18 -8 -6 0 -16 E -14 -6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=27 E=15 D=15 C=14 so C is eliminated. Round 2 votes counts: A=36 B=29 E=20 D=15 so D is eliminated. Round 3 votes counts: A=37 B=35 E=28 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:202 C:199 E:195 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 18 14 B -10 0 0 8 6 C -14 0 0 6 6 D -18 -8 -6 0 -16 E -14 -6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 18 14 B -10 0 0 8 6 C -14 0 0 6 6 D -18 -8 -6 0 -16 E -14 -6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 18 14 B -10 0 0 8 6 C -14 0 0 6 6 D -18 -8 -6 0 -16 E -14 -6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1213: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (5) C D B A E (5) C D A B E (5) A B E C D (5) E A C B D (4) E A B D C (4) D C E B A (4) D B C E A (4) B A E D C (4) A E B C D (4) A B C D E (4) E D C B A (3) D C B E A (3) C D E B A (3) C D E A B (3) E D B A C (2) E C D A B (2) E B A D C (2) D E C B A (2) D B C A E (2) C E D A B (2) C B D A E (2) B D E A C (2) B A D E C (2) B A C D E (2) E D C A B (1) E D B C A (1) E C A D B (1) E A C D B (1) E A B C D (1) D C B A E (1) D B E C A (1) C E A D B (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D B E (1) B E A D C (1) B D A E C (1) B D A C E (1) B A D C E (1) A E C B D (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -12 -6 B 8 0 2 0 0 C 0 -2 0 4 -2 D 12 0 -4 0 2 E 6 0 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.768007 C: 0.000000 D: 0.231993 E: 0.000000 Sum of squares = 0.643655488614 Cumulative probabilities = A: 0.000000 B: 0.768007 C: 0.768007 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -12 -6 B 8 0 2 0 0 C 0 -2 0 4 -2 D 12 0 -4 0 2 E 6 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666668 C: 0.000000 D: 0.333332 E: 0.000000 Sum of squares = 0.555556278421 Cumulative probabilities = A: 0.000000 B: 0.666668 C: 0.666668 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 D=17 A=17 B=14 so B is eliminated. Round 2 votes counts: E=28 A=26 C=25 D=21 so D is eliminated. Round 3 votes counts: C=39 E=33 A=28 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:205 D:205 E:203 C:200 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -12 -6 B 8 0 2 0 0 C 0 -2 0 4 -2 D 12 0 -4 0 2 E 6 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666668 C: 0.000000 D: 0.333332 E: 0.000000 Sum of squares = 0.555556278421 Cumulative probabilities = A: 0.000000 B: 0.666668 C: 0.666668 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -12 -6 B 8 0 2 0 0 C 0 -2 0 4 -2 D 12 0 -4 0 2 E 6 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666668 C: 0.000000 D: 0.333332 E: 0.000000 Sum of squares = 0.555556278421 Cumulative probabilities = A: 0.000000 B: 0.666668 C: 0.666668 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -12 -6 B 8 0 2 0 0 C 0 -2 0 4 -2 D 12 0 -4 0 2 E 6 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666668 C: 0.000000 D: 0.333332 E: 0.000000 Sum of squares = 0.555556278421 Cumulative probabilities = A: 0.000000 B: 0.666668 C: 0.666668 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1214: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (24) A C D E B (17) A B C D E (7) E B D C A (5) D C E A B (5) B E A D C (5) A B C E D (5) E D B C A (4) C A D E B (4) A C B D E (4) D E C B A (3) C D E A B (3) B E D A C (3) B A E C D (3) B A C E D (2) A B E C D (2) E D C B A (1) B A E D C (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 4 -6 B 2 0 24 24 16 C -2 -24 0 -4 -4 D -4 -24 4 0 -12 E 6 -16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998346 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 4 -6 B 2 0 24 24 16 C -2 -24 0 -4 -4 D -4 -24 4 0 -12 E 6 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998272 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=37 E=10 D=8 C=7 so C is eliminated. Round 2 votes counts: A=41 B=38 D=11 E=10 so E is eliminated. Round 3 votes counts: B=43 A=41 D=16 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:203 A:199 C:183 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 4 -6 B 2 0 24 24 16 C -2 -24 0 -4 -4 D -4 -24 4 0 -12 E 6 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998272 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -6 B 2 0 24 24 16 C -2 -24 0 -4 -4 D -4 -24 4 0 -12 E 6 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998272 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -6 B 2 0 24 24 16 C -2 -24 0 -4 -4 D -4 -24 4 0 -12 E 6 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998272 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1215: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) B A C E D (6) D A E C B (5) B C A E D (5) E D A C B (4) C B A D E (4) A D E B C (4) E B A D C (3) D E A C B (3) D C A E B (3) C B D E A (3) B C E A D (3) A B D C E (3) E D A B C (2) E B C D A (2) E A D B C (2) D E A B C (2) D C E A B (2) D A C E B (2) C E D B A (2) C D E A B (2) B E A C D (2) B C A D E (2) A D B E C (2) A D B C E (2) E D C A B (1) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E A B D C (1) D E C A B (1) D C A B E (1) D A E B C (1) D A C B E (1) C D B E A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E A D (1) C A B D E (1) B E C A D (1) B E A D C (1) B A E D C (1) B A E C D (1) B A C D E (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 4 -4 0 B 2 0 0 0 6 C -4 0 0 -2 12 D 4 0 2 0 0 E 0 -6 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.501557 C: 0.000000 D: 0.498443 E: 0.000000 Sum of squares = 0.500004849286 Cumulative probabilities = A: 0.000000 B: 0.501557 C: 0.501557 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -4 0 B 2 0 0 0 6 C -4 0 0 -2 12 D 4 0 2 0 0 E 0 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=23 D=21 E=19 A=13 so A is eliminated. Round 2 votes counts: D=30 B=27 C=24 E=19 so E is eliminated. Round 3 votes counts: D=41 B=33 C=26 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:204 C:203 D:203 A:199 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 -4 0 B 2 0 0 0 6 C -4 0 0 -2 12 D 4 0 2 0 0 E 0 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -4 0 B 2 0 0 0 6 C -4 0 0 -2 12 D 4 0 2 0 0 E 0 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -4 0 B 2 0 0 0 6 C -4 0 0 -2 12 D 4 0 2 0 0 E 0 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1216: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) E D A C B (7) B D E A C (7) B D E C A (6) C A B E D (5) B E D A C (5) B C A E D (5) C B A D E (4) C A E D B (4) B C A D E (4) E A D C B (3) D E A C B (3) C B A E D (3) C A D E B (3) A E C D B (3) A C D E B (3) E D B A C (2) E A C D B (2) D E B A C (2) D B E A C (2) C A E B D (2) B E C A D (2) B D C A E (2) B C D E A (2) B C D A E (2) A E D C B (2) E D A B C (1) E B A D C (1) E A D B C (1) D A C E B (1) B E C D A (1) B C E A D (1) Total count = 100 A B C D E A 0 -2 8 14 4 B 2 0 -8 4 2 C -8 8 0 10 0 D -14 -4 -10 0 -18 E -4 -2 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407405 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 14 4 B 2 0 -8 4 2 C -8 8 0 10 0 D -14 -4 -10 0 -18 E -4 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407406 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=21 E=17 A=17 D=8 so D is eliminated. Round 2 votes counts: B=39 E=22 C=21 A=18 so A is eliminated. Round 3 votes counts: B=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:212 E:206 C:205 B:200 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 14 4 B 2 0 -8 4 2 C -8 8 0 10 0 D -14 -4 -10 0 -18 E -4 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407406 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 14 4 B 2 0 -8 4 2 C -8 8 0 10 0 D -14 -4 -10 0 -18 E -4 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407406 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 14 4 B 2 0 -8 4 2 C -8 8 0 10 0 D -14 -4 -10 0 -18 E -4 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407406 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1217: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (14) E B D C A (11) D B E A C (9) C A E B D (8) C E A B D (6) A D C B E (5) E B C D A (4) C E B A D (4) B E D C A (4) B D E A C (4) D A B E C (3) C A E D B (3) C A D B E (3) A C E D B (3) E C B D A (2) D B A E C (2) C A D E B (2) A D B E C (2) A D B C E (2) E C B A D (1) E B D A C (1) D B A C E (1) D A B C E (1) C B E D A (1) C B E A D (1) B E D A C (1) B E C D A (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -2 10 0 B -6 0 -8 -2 8 C 2 8 0 8 10 D -10 2 -8 0 -2 E 0 -8 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 10 0 B -6 0 -8 -2 8 C 2 8 0 8 10 D -10 2 -8 0 -2 E 0 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 E=19 D=16 B=10 so B is eliminated. Round 2 votes counts: C=28 A=27 E=25 D=20 so D is eliminated. Round 3 votes counts: E=38 A=34 C=28 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:214 A:207 B:196 E:192 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 10 0 B -6 0 -8 -2 8 C 2 8 0 8 10 D -10 2 -8 0 -2 E 0 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 10 0 B -6 0 -8 -2 8 C 2 8 0 8 10 D -10 2 -8 0 -2 E 0 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 10 0 B -6 0 -8 -2 8 C 2 8 0 8 10 D -10 2 -8 0 -2 E 0 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1218: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) B C A E D (11) D E A C B (8) E D A C B (6) D B A C E (6) A C E B D (6) E A C D B (5) E A C B D (4) A C B E D (4) E D C A B (3) D E B A C (3) D B E A C (3) D B C A E (3) C A B E D (3) B C A D E (3) D A B C E (2) C B A E D (2) C A E B D (2) B D A C E (2) B A C D E (2) E D C B A (1) E C A D B (1) E B C A D (1) D E C A B (1) D E A B C (1) D B A E C (1) B A C E D (1) A E C D B (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 22 14 20 2 B -22 0 -20 8 -10 C -14 20 0 18 -2 D -20 -8 -18 0 -28 E -2 10 2 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 14 20 2 B -22 0 -20 8 -10 C -14 20 0 18 -2 D -20 -8 -18 0 -28 E -2 10 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962368 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=28 B=19 A=13 C=7 so C is eliminated. Round 2 votes counts: E=33 D=28 B=21 A=18 so A is eliminated. Round 3 votes counts: E=42 D=29 B=29 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:229 E:219 C:211 B:178 D:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 14 20 2 B -22 0 -20 8 -10 C -14 20 0 18 -2 D -20 -8 -18 0 -28 E -2 10 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962368 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 14 20 2 B -22 0 -20 8 -10 C -14 20 0 18 -2 D -20 -8 -18 0 -28 E -2 10 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962368 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 14 20 2 B -22 0 -20 8 -10 C -14 20 0 18 -2 D -20 -8 -18 0 -28 E -2 10 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962368 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1219: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (13) D C A E B (7) A E B C D (7) E B A C D (5) E A B C D (5) D A C E B (5) C D A E B (5) C A E B D (5) D C A B E (4) C A D E B (4) B E D A C (4) D C B A E (3) D B C E A (3) A C E D B (3) A C E B D (3) D C B E A (2) D A C B E (2) B E D C A (2) B E C A D (2) B D E A C (2) A E D C B (2) A E C B D (2) E A C B D (1) D B E C A (1) D B E A C (1) D A B E C (1) C D B E A (1) C B E D A (1) C A E D B (1) B E A D C (1) B D E C A (1) A D E B C (1) Total count = 100 A B C D E A 0 16 16 10 10 B -16 0 -2 8 -12 C -16 2 0 16 -2 D -10 -8 -16 0 -14 E -10 12 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 10 10 B -16 0 -2 8 -12 C -16 2 0 16 -2 D -10 -8 -16 0 -14 E -10 12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=25 A=18 C=17 E=11 so E is eliminated. Round 2 votes counts: B=30 D=29 A=24 C=17 so C is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:209 C:200 B:189 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 10 10 B -16 0 -2 8 -12 C -16 2 0 16 -2 D -10 -8 -16 0 -14 E -10 12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 10 10 B -16 0 -2 8 -12 C -16 2 0 16 -2 D -10 -8 -16 0 -14 E -10 12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 10 10 B -16 0 -2 8 -12 C -16 2 0 16 -2 D -10 -8 -16 0 -14 E -10 12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1220: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (13) D A B E C (8) E C A B D (5) C B D A E (5) B D A C E (5) D A E B C (4) C B E D A (4) E C A D B (3) E B A C D (3) E A D C B (3) D B A C E (3) B A D E C (3) A D B E C (3) E A D B C (2) D C B A E (2) D C A E B (2) D B C A E (2) D A C E B (2) D A C B E (2) D A B C E (2) C E A B D (2) C D B A E (2) C B E A D (2) B C E A D (2) A E B D C (2) E C B A D (1) E A B D C (1) D A E C B (1) C E D A B (1) C D E B A (1) C B D E A (1) B D A E C (1) B C E D A (1) B A E D C (1) B A D C E (1) A E D B C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 2 2 10 B 6 0 -4 10 4 C -2 4 0 -8 10 D -2 -10 8 0 4 E -10 -4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.454545 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826439 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.818182 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 2 10 B 6 0 -4 10 4 C -2 4 0 -8 10 D -2 -10 8 0 4 E -10 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.454545 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826434 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.818182 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=28 E=18 B=14 A=9 so A is eliminated. Round 2 votes counts: D=32 C=31 E=21 B=16 so B is eliminated. Round 3 votes counts: D=43 C=34 E=23 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:208 A:204 C:202 D:200 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 2 10 B 6 0 -4 10 4 C -2 4 0 -8 10 D -2 -10 8 0 4 E -10 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.454545 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826434 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.818182 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 10 B 6 0 -4 10 4 C -2 4 0 -8 10 D -2 -10 8 0 4 E -10 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.454545 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826434 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.818182 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 10 B 6 0 -4 10 4 C -2 4 0 -8 10 D -2 -10 8 0 4 E -10 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.454545 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826434 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.818182 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1221: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (12) D E C B A (9) A B C E D (9) E D B C A (7) A E B C D (6) A C B D E (5) C B D E A (4) B C A D E (4) E D A C B (3) B C D E A (3) B A C D E (3) A B C D E (3) E D C A B (2) E B A D C (2) E A D C B (2) D C E B A (2) D C B E A (2) C D B E A (2) C B D A E (2) E D A B C (1) E A D B C (1) D E B C A (1) C D B A E (1) C D A B E (1) C A B D E (1) B E C D A (1) B E A C D (1) B C E D A (1) B C D A E (1) B A C E D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A D E C B (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -12 -10 -12 B 18 0 -4 -2 -8 C 12 4 0 2 -8 D 10 2 -2 0 -8 E 12 8 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -12 -10 -12 B 18 0 -4 -2 -8 C 12 4 0 2 -8 D 10 2 -2 0 -8 E 12 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=30 A=30 B=15 D=14 C=11 so C is eliminated. Round 2 votes counts: A=31 E=30 B=21 D=18 so D is eliminated. Round 3 votes counts: E=42 A=32 B=26 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:205 B:202 D:201 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -12 -10 -12 B 18 0 -4 -2 -8 C 12 4 0 2 -8 D 10 2 -2 0 -8 E 12 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -10 -12 B 18 0 -4 -2 -8 C 12 4 0 2 -8 D 10 2 -2 0 -8 E 12 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -10 -12 B 18 0 -4 -2 -8 C 12 4 0 2 -8 D 10 2 -2 0 -8 E 12 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1222: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (6) B D A C E (6) B A E D C (6) E C A D B (5) B D A E C (5) A B E C D (5) E C A B D (4) D C E B A (4) D C B E A (4) D B C E A (4) A E C B D (4) C E D A B (3) C D E A B (3) C A E B D (3) A B C E D (3) E D C A B (2) E C D A B (2) D E C A B (2) D E B C A (2) D B E C A (2) D B E A C (2) D B C A E (2) B A E C D (2) B A D E C (2) A E B C D (2) A C E B D (2) A C B E D (2) E A C B D (1) E A B C D (1) D E C B A (1) D E B A C (1) D C E A B (1) D B A E C (1) C D A E B (1) B E D A C (1) B D E A C (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 4 -2 0 -4 B -4 0 0 4 0 C 2 0 0 0 -8 D 0 -4 0 0 -8 E 4 0 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.334709 C: 0.000000 D: 0.000000 E: 0.665291 Sum of squares = 0.554642501144 Cumulative probabilities = A: 0.000000 B: 0.334709 C: 0.334709 D: 0.334709 E: 1.000000 A B C D E A 0 4 -2 0 -4 B -4 0 0 4 0 C 2 0 0 0 -8 D 0 -4 0 0 -8 E 4 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499910 C: 0.000000 D: 0.000000 E: 0.500090 Sum of squares = 0.500000016266 Cumulative probabilities = A: 0.000000 B: 0.499910 C: 0.499910 D: 0.499910 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 A=18 C=16 E=15 so E is eliminated. Round 2 votes counts: D=28 C=27 B=25 A=20 so A is eliminated. Round 3 votes counts: C=36 B=36 D=28 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:210 B:200 A:199 C:197 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 0 -4 B -4 0 0 4 0 C 2 0 0 0 -8 D 0 -4 0 0 -8 E 4 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499910 C: 0.000000 D: 0.000000 E: 0.500090 Sum of squares = 0.500000016266 Cumulative probabilities = A: 0.000000 B: 0.499910 C: 0.499910 D: 0.499910 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 0 -4 B -4 0 0 4 0 C 2 0 0 0 -8 D 0 -4 0 0 -8 E 4 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499910 C: 0.000000 D: 0.000000 E: 0.500090 Sum of squares = 0.500000016266 Cumulative probabilities = A: 0.000000 B: 0.499910 C: 0.499910 D: 0.499910 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 0 -4 B -4 0 0 4 0 C 2 0 0 0 -8 D 0 -4 0 0 -8 E 4 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499910 C: 0.000000 D: 0.000000 E: 0.500090 Sum of squares = 0.500000016266 Cumulative probabilities = A: 0.000000 B: 0.499910 C: 0.499910 D: 0.499910 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1223: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) D A C E B (8) B C E A D (8) C D A B E (6) E D A B C (5) E A B D C (5) B E C A D (5) E B A D C (4) E B A C D (4) D E A B C (4) C B D A E (4) E A D B C (3) D A E B C (3) C B D E A (3) C B A E D (3) B C E D A (3) A D E C B (3) E B D A C (2) D C A B E (2) E D B A C (1) D E B A C (1) D C A E B (1) D B E C A (1) C B E D A (1) C B E A D (1) C B A D E (1) C A B E D (1) C A B D E (1) B E C D A (1) A E D C B (1) A E D B C (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 14 16 -12 -4 B -14 0 2 -6 -16 C -16 -2 0 -12 -8 D 12 6 12 0 2 E 4 16 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 -12 -4 B -14 0 2 -6 -16 C -16 -2 0 -12 -8 D 12 6 12 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=24 C=21 B=17 A=8 so A is eliminated. Round 2 votes counts: D=34 E=26 C=23 B=17 so B is eliminated. Round 3 votes counts: D=34 C=34 E=32 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:213 A:207 B:183 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 16 -12 -4 B -14 0 2 -6 -16 C -16 -2 0 -12 -8 D 12 6 12 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 -12 -4 B -14 0 2 -6 -16 C -16 -2 0 -12 -8 D 12 6 12 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 -12 -4 B -14 0 2 -6 -16 C -16 -2 0 -12 -8 D 12 6 12 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1224: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) B A C E D (5) E A C D B (4) C D E A B (4) C A B E D (4) B E A D C (4) B A E D C (4) A E B C D (4) A B E C D (4) E D B A C (3) E A D C B (3) E A D B C (3) E A B D C (3) D E B C A (3) D C B E A (3) C D A E B (3) B A E C D (3) E D A B C (2) E A C B D (2) D C E A B (2) C B A D E (2) C A E D B (2) C A E B D (2) B E D A C (2) A E C B D (2) E B A D C (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C E A (1) D B C A E (1) C E A D B (1) C D A B E (1) C B D A E (1) C A D B E (1) B D E A C (1) B D C A E (1) B C D A E (1) B C A D E (1) B A C D E (1) A C E D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 8 18 12 B 0 0 0 0 6 C -8 0 0 20 2 D -18 0 -20 0 -22 E -12 -6 -2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.518317 B: 0.481683 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50067104282 Cumulative probabilities = A: 0.518317 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 18 12 B 0 0 0 0 6 C -8 0 0 20 2 D -18 0 -20 0 -22 E -12 -6 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=23 E=21 D=13 A=13 so D is eliminated. Round 2 votes counts: C=37 B=26 E=24 A=13 so A is eliminated. Round 3 votes counts: C=39 B=31 E=30 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:219 C:207 B:203 E:201 D:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 18 12 B 0 0 0 0 6 C -8 0 0 20 2 D -18 0 -20 0 -22 E -12 -6 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 18 12 B 0 0 0 0 6 C -8 0 0 20 2 D -18 0 -20 0 -22 E -12 -6 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 18 12 B 0 0 0 0 6 C -8 0 0 20 2 D -18 0 -20 0 -22 E -12 -6 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1225: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) D B E C A (8) A C B E D (7) C D A E B (6) A C E D B (6) E B D A C (5) B E D A C (5) A C D B E (5) C A D E B (4) B E A D C (4) B D E C A (4) C A D B E (3) B E D C A (3) A B C E D (3) E D B C A (2) E A B D C (2) D C E B A (2) A E B C D (2) A B C D E (2) E D C B A (1) E B D C A (1) E B A D C (1) D E C B A (1) D E B C A (1) D C B E A (1) C D E B A (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B A E D C (1) A E C D B (1) A E C B D (1) A E B D C (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 22 10 14 B -10 0 2 16 6 C -22 -2 0 4 4 D -10 -16 -4 0 -14 E -14 -6 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 22 10 14 B -10 0 2 16 6 C -22 -2 0 4 4 D -10 -16 -4 0 -14 E -14 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=21 C=14 D=13 E=12 so E is eliminated. Round 2 votes counts: A=42 B=28 D=16 C=14 so C is eliminated. Round 3 votes counts: A=49 B=28 D=23 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:207 E:195 C:192 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 22 10 14 B -10 0 2 16 6 C -22 -2 0 4 4 D -10 -16 -4 0 -14 E -14 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 22 10 14 B -10 0 2 16 6 C -22 -2 0 4 4 D -10 -16 -4 0 -14 E -14 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 22 10 14 B -10 0 2 16 6 C -22 -2 0 4 4 D -10 -16 -4 0 -14 E -14 -6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1226: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) B D E C A (8) B D A E C (7) B A D E C (6) C D E B A (5) A B E C D (5) C E A D B (4) B A D C E (4) E C A D B (3) D C B E A (3) D B C E A (3) C E D A B (3) A B E D C (3) A B C D E (3) E D C B A (2) D C E B A (2) C D E A B (2) B D E A C (2) B D C E A (2) B D A C E (2) B A E D C (2) A E C D B (2) A E B C D (2) A B C E D (2) E C D B A (1) E C D A B (1) E A C D B (1) D E C B A (1) D B E C A (1) C A E D B (1) B D C A E (1) A E C B D (1) A C D B E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 14 8 12 B 4 0 10 6 16 C -14 -10 0 -2 2 D -8 -6 2 0 12 E -12 -16 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 8 12 B 4 0 10 6 16 C -14 -10 0 -2 2 D -8 -6 2 0 12 E -12 -16 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=33 C=15 D=10 E=8 so E is eliminated. Round 2 votes counts: B=34 A=34 C=20 D=12 so D is eliminated. Round 3 votes counts: B=38 A=34 C=28 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:215 D:200 C:188 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 8 12 B 4 0 10 6 16 C -14 -10 0 -2 2 D -8 -6 2 0 12 E -12 -16 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 8 12 B 4 0 10 6 16 C -14 -10 0 -2 2 D -8 -6 2 0 12 E -12 -16 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 8 12 B 4 0 10 6 16 C -14 -10 0 -2 2 D -8 -6 2 0 12 E -12 -16 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1227: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) E D C A B (7) A E D B C (5) A B D E C (5) E D C B A (4) E D A C B (4) C E D B A (4) C D E B A (4) C B D A E (4) C A B E D (4) B C A D E (4) A B C D E (4) D E C B A (3) D E B C A (3) B A D C E (3) E C D A B (2) C E A D B (2) C B D E A (2) C B A E D (2) C B A D E (2) B A D E C (2) A C B E D (2) A B D C E (2) A B C E D (2) E C A D B (1) E A D B C (1) D C E B A (1) D B E A C (1) D B C E A (1) C E B D A (1) C E A B D (1) B D A C E (1) B C D E A (1) B C D A E (1) B A C D E (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -8 -8 -6 B -10 0 0 -8 -6 C 8 0 0 -10 -2 D 8 8 10 0 -8 E 6 6 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -8 -8 -6 B -10 0 0 -8 -6 C 8 0 0 -10 -2 D 8 8 10 0 -8 E 6 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 A=23 B=13 D=9 so D is eliminated. Round 2 votes counts: E=35 C=27 A=23 B=15 so B is eliminated. Round 3 votes counts: E=36 C=34 A=30 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:209 C:198 A:194 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -8 -8 -6 B -10 0 0 -8 -6 C 8 0 0 -10 -2 D 8 8 10 0 -8 E 6 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -8 -6 B -10 0 0 -8 -6 C 8 0 0 -10 -2 D 8 8 10 0 -8 E 6 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -8 -6 B -10 0 0 -8 -6 C 8 0 0 -10 -2 D 8 8 10 0 -8 E 6 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1228: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B E D A C (7) B E C D A (7) B A C D E (6) E C D B A (5) B E D C A (5) B A D E C (5) C A B D E (4) E D A C B (3) E B D C A (3) C E D A B (3) C B E D A (3) B C E A D (3) B A E D C (3) A D C E B (3) A C D E B (3) E B C D A (2) C D E A B (2) C A D E B (2) B E C A D (2) B C A D E (2) B A D C E (2) B A C E D (2) A B C D E (2) E D B A C (1) E D A B C (1) D E C A B (1) C D A E B (1) C B E A D (1) C A D B E (1) B E A D C (1) B C E D A (1) B C A E D (1) A D E C B (1) A D C B E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 -12 -4 -16 B 24 0 16 26 24 C 12 -16 0 8 -8 D 4 -26 -8 0 -22 E 16 -24 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -12 -4 -16 B 24 0 16 26 24 C 12 -16 0 8 -8 D 4 -26 -8 0 -22 E 16 -24 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=47 E=22 C=17 A=13 D=1 so D is eliminated. Round 2 votes counts: B=47 E=23 C=17 A=13 so A is eliminated. Round 3 votes counts: B=51 C=25 E=24 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:245 E:211 C:198 D:174 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -12 -4 -16 B 24 0 16 26 24 C 12 -16 0 8 -8 D 4 -26 -8 0 -22 E 16 -24 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -12 -4 -16 B 24 0 16 26 24 C 12 -16 0 8 -8 D 4 -26 -8 0 -22 E 16 -24 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -12 -4 -16 B 24 0 16 26 24 C 12 -16 0 8 -8 D 4 -26 -8 0 -22 E 16 -24 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1229: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (7) E B A C D (5) D C A B E (5) B E A D C (5) B D A E C (4) E C D B A (3) D C E A B (3) D C A E B (3) D B E C A (3) D B A C E (3) C E A D B (3) B E A C D (3) B D E A C (3) B A E C D (3) A E B C D (3) E C B A D (2) D B C E A (2) D A C B E (2) C D A E B (2) C A D E B (2) B E D A C (2) B D E C A (2) A E C B D (2) A C E B D (2) A B C E D (2) E C D A B (1) E C A B D (1) E B C A D (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B E A (1) D C B A E (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B C E (1) C E D A B (1) C D E A B (1) C A E D B (1) B E D C A (1) B D A C E (1) B A E D C (1) A D C B E (1) A D B C E (1) A C E D B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 14 -2 4 B 18 0 20 6 18 C -14 -20 0 -18 -16 D 2 -6 18 0 10 E -4 -18 16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 14 -2 4 B 18 0 20 6 18 C -14 -20 0 -18 -16 D 2 -6 18 0 10 E -4 -18 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=30 E=14 A=14 C=10 so C is eliminated. Round 2 votes counts: D=33 B=32 E=18 A=17 so A is eliminated. Round 3 votes counts: D=37 B=36 E=27 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 D:212 A:199 E:192 C:166 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 14 -2 4 B 18 0 20 6 18 C -14 -20 0 -18 -16 D 2 -6 18 0 10 E -4 -18 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 14 -2 4 B 18 0 20 6 18 C -14 -20 0 -18 -16 D 2 -6 18 0 10 E -4 -18 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 14 -2 4 B 18 0 20 6 18 C -14 -20 0 -18 -16 D 2 -6 18 0 10 E -4 -18 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1230: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (13) D A C E B (11) B C E A D (9) C A D E B (8) C A D B E (7) E D A B C (6) D A E C B (6) C B A D E (6) B C A D E (6) B E C D A (4) D E A C B (3) B E C A D (3) B C D A E (3) E D A C B (2) E A D C B (2) D E A B C (2) C D A E B (2) E C A D B (1) E B D A C (1) E B A D C (1) D A C B E (1) C A B D E (1) B D A C E (1) B C E D A (1) Total count = 100 A B C D E A 0 4 -2 -12 4 B -4 0 0 -2 10 C 2 0 0 2 12 D 12 2 -2 0 14 E -4 -10 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.146390 C: 0.853610 D: 0.000000 E: 0.000000 Sum of squares = 0.750080512461 Cumulative probabilities = A: 0.000000 B: 0.146390 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -12 4 B -4 0 0 -2 10 C 2 0 0 2 12 D 12 2 -2 0 14 E -4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555603987 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 C=24 D=23 E=13 so A is eliminated. Round 2 votes counts: B=40 C=24 D=23 E=13 so E is eliminated. Round 3 votes counts: B=42 D=33 C=25 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 C:208 B:202 A:197 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -12 4 B -4 0 0 -2 10 C 2 0 0 2 12 D 12 2 -2 0 14 E -4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555603987 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -12 4 B -4 0 0 -2 10 C 2 0 0 2 12 D 12 2 -2 0 14 E -4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555603987 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -12 4 B -4 0 0 -2 10 C 2 0 0 2 12 D 12 2 -2 0 14 E -4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555603987 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1231: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) B D A C E (7) D B C A E (6) E B C D A (5) E B A C D (5) B D A E C (5) E C B D A (4) A C E B D (4) A C D B E (4) E C B A D (3) D B A C E (3) C E A D B (3) C A E D B (3) B D E A C (3) A E C B D (3) E C D B A (2) D C A B E (2) D A B C E (2) C E D A B (2) B A D C E (2) A D B C E (2) A C D E B (2) A B D C E (2) E C D A B (1) E C A D B (1) E B D C A (1) E B C A D (1) E A C B D (1) D B E C A (1) D A C B E (1) C E A B D (1) B E D A C (1) B D E C A (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 6 2 B 0 0 -4 28 -12 C 0 4 0 22 -2 D -6 -28 -22 0 -10 E -2 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.548867 B: 0.000000 C: 0.451133 D: 0.000000 E: 0.000000 Sum of squares = 0.504775851466 Cumulative probabilities = A: 0.548867 B: 0.548867 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 6 2 B 0 0 -4 28 -12 C 0 4 0 22 -2 D -6 -28 -22 0 -10 E -2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500465 B: 0.000000 C: 0.499535 D: 0.000000 E: 0.000000 Sum of squares = 0.500000431611 Cumulative probabilities = A: 0.500465 B: 0.500465 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=20 B=19 D=15 C=9 so C is eliminated. Round 2 votes counts: E=43 A=23 B=19 D=15 so D is eliminated. Round 3 votes counts: E=43 B=29 A=28 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:211 B:206 A:204 D:167 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 6 2 B 0 0 -4 28 -12 C 0 4 0 22 -2 D -6 -28 -22 0 -10 E -2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500465 B: 0.000000 C: 0.499535 D: 0.000000 E: 0.000000 Sum of squares = 0.500000431611 Cumulative probabilities = A: 0.500465 B: 0.500465 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 6 2 B 0 0 -4 28 -12 C 0 4 0 22 -2 D -6 -28 -22 0 -10 E -2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500465 B: 0.000000 C: 0.499535 D: 0.000000 E: 0.000000 Sum of squares = 0.500000431611 Cumulative probabilities = A: 0.500465 B: 0.500465 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 6 2 B 0 0 -4 28 -12 C 0 4 0 22 -2 D -6 -28 -22 0 -10 E -2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500465 B: 0.000000 C: 0.499535 D: 0.000000 E: 0.000000 Sum of squares = 0.500000431611 Cumulative probabilities = A: 0.500465 B: 0.500465 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1232: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) D B E A C (5) D B A E C (5) C A E D B (5) C A E B D (5) A D C B E (5) E C B D A (4) D A B E C (4) A C D E B (4) E B C D A (3) D E B C A (3) C E B A D (3) C E A B D (3) B D E C A (3) A C E D B (3) A C E B D (3) E C B A D (2) E B D C A (2) D A C E B (2) B E D C A (2) B E C D A (2) B D A E C (2) A D C E B (2) A D B C E (2) A B C E D (2) E C D B A (1) E C D A B (1) D E C B A (1) D E C A B (1) D A E C B (1) D A E B C (1) D A B C E (1) C E A D B (1) C A D E B (1) C A B E D (1) B E C A D (1) B A E C D (1) B A C E D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -8 -6 6 B 0 0 -2 -18 -4 C 8 2 0 -2 -8 D 6 18 2 0 8 E -6 4 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -6 6 B 0 0 -2 -18 -4 C 8 2 0 -2 -8 D 6 18 2 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=23 C=19 E=13 B=12 so B is eliminated. Round 2 votes counts: D=38 A=25 C=19 E=18 so E is eliminated. Round 3 votes counts: D=42 C=33 A=25 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:200 E:199 A:196 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -6 6 B 0 0 -2 -18 -4 C 8 2 0 -2 -8 D 6 18 2 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -6 6 B 0 0 -2 -18 -4 C 8 2 0 -2 -8 D 6 18 2 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -6 6 B 0 0 -2 -18 -4 C 8 2 0 -2 -8 D 6 18 2 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1233: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) A B D E C (6) E D B C A (5) C A E D B (4) B D E A C (4) A C D E B (4) A C B D E (4) E D B A C (3) C E D A B (3) C A E B D (3) C A B E D (3) B E D A C (3) A D B E C (3) A C D B E (3) A C B E D (3) E C D B A (2) E C B D A (2) D E A C B (2) C E B D A (2) C B E A D (2) C B A E D (2) C A D B E (2) B A E D C (2) B A D E C (2) A B D C E (2) E D C B A (1) E B D C A (1) D E B C A (1) D E B A C (1) D E A B C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E A D B (1) C D E A B (1) C B E D A (1) C A B D E (1) B E D C A (1) B D A E C (1) B C E D A (1) B A E C D (1) B A C E D (1) A D C E B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 18 4 20 20 B -18 0 -12 -2 4 C -4 12 0 12 10 D -20 2 -12 0 4 E -20 -4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 20 20 B -18 0 -12 -2 4 C -4 12 0 12 10 D -20 2 -12 0 4 E -20 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=28 B=16 E=14 D=8 so D is eliminated. Round 2 votes counts: C=34 A=30 E=19 B=17 so B is eliminated. Round 3 votes counts: A=37 C=35 E=28 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:231 C:215 D:187 B:186 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 20 20 B -18 0 -12 -2 4 C -4 12 0 12 10 D -20 2 -12 0 4 E -20 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 20 20 B -18 0 -12 -2 4 C -4 12 0 12 10 D -20 2 -12 0 4 E -20 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 20 20 B -18 0 -12 -2 4 C -4 12 0 12 10 D -20 2 -12 0 4 E -20 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1234: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) D B A E C (6) D A B C E (5) C E A D B (5) B E C A D (5) B D A E C (5) B A E C D (4) A C E B D (4) E C D B A (3) D B E A C (3) D A C E B (3) D A C B E (3) B E D C A (3) A B C E D (3) E C D A B (2) E B D C A (2) D E C B A (2) D B E C A (2) C E A B D (2) C A E D B (2) C A E B D (2) B E C D A (2) B E A C D (2) B D E A C (2) B A D E C (2) A D B C E (2) A C D E B (2) A B D C E (2) E D B C A (1) E C A D B (1) E C A B D (1) E B C A D (1) D E C A B (1) D C E A B (1) D C A E B (1) D B A C E (1) C E D A B (1) C D E A B (1) B E A D C (1) B D E C A (1) B A D C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 4 -2 -2 B 10 0 12 2 12 C -4 -12 0 0 -16 D 2 -2 0 0 -6 E 2 -12 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -2 -2 B 10 0 12 2 12 C -4 -12 0 0 -16 D 2 -2 0 0 -6 E 2 -12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=28 B=28 E=17 A=14 C=13 so C is eliminated. Round 2 votes counts: D=29 B=28 E=25 A=18 so A is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:206 D:197 A:195 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 -2 -2 B 10 0 12 2 12 C -4 -12 0 0 -16 D 2 -2 0 0 -6 E 2 -12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -2 -2 B 10 0 12 2 12 C -4 -12 0 0 -16 D 2 -2 0 0 -6 E 2 -12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -2 -2 B 10 0 12 2 12 C -4 -12 0 0 -16 D 2 -2 0 0 -6 E 2 -12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1235: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (15) B D C A E (11) B C A D E (11) B A C E D (10) E A C D B (8) D E C A B (8) E A D C B (5) A C B E D (5) D E C B A (4) D E B C A (3) D B E C A (3) E D C A B (2) B C D A E (2) B A C D E (2) A E C B D (2) E D B A C (1) E A C B D (1) D C E B A (1) D C A E B (1) D B C A E (1) C D A B E (1) B E A C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 4 -6 -4 B 0 0 -8 -6 -4 C -4 8 0 -10 -6 D 6 6 10 0 -4 E 4 4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 -6 -4 B 0 0 -8 -6 -4 C -4 8 0 -10 -6 D 6 6 10 0 -4 E 4 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=32 D=21 A=9 C=1 so C is eliminated. Round 2 votes counts: B=37 E=32 D=22 A=9 so A is eliminated. Round 3 votes counts: B=43 E=35 D=22 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:209 E:209 A:197 C:194 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 -6 -4 B 0 0 -8 -6 -4 C -4 8 0 -10 -6 D 6 6 10 0 -4 E 4 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -6 -4 B 0 0 -8 -6 -4 C -4 8 0 -10 -6 D 6 6 10 0 -4 E 4 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -6 -4 B 0 0 -8 -6 -4 C -4 8 0 -10 -6 D 6 6 10 0 -4 E 4 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1236: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (5) A B E C D (5) E D C A B (4) C D E B A (4) C D B E A (4) B D C E A (4) B D A E C (4) B A D E C (4) A E C D B (4) A C E B D (4) D E C B A (3) D E C A B (3) D C E B A (3) D C E A B (3) C E A D B (3) C A E D B (3) B D E A C (3) B A E C D (3) A B C E D (3) D B C E A (2) C E D A B (2) C D E A B (2) C A E B D (2) B D E C A (2) B C D A E (2) B A D C E (2) A E C B D (2) A B E D C (2) E A D C B (1) E A D B C (1) E A C D B (1) D C B E A (1) D B E A C (1) B D C A E (1) B C A D E (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 2 4 6 B 2 0 -2 6 4 C -2 2 0 -2 0 D -4 -6 2 0 0 E -6 -4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 4 6 B 2 0 -2 6 4 C -2 2 0 -2 0 D -4 -6 2 0 0 E -6 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=24 C=20 D=16 E=7 so E is eliminated. Round 2 votes counts: B=33 A=27 D=20 C=20 so D is eliminated. Round 3 votes counts: C=37 B=36 A=27 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:205 B:205 C:199 D:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 4 6 B 2 0 -2 6 4 C -2 2 0 -2 0 D -4 -6 2 0 0 E -6 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 6 B 2 0 -2 6 4 C -2 2 0 -2 0 D -4 -6 2 0 0 E -6 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 6 B 2 0 -2 6 4 C -2 2 0 -2 0 D -4 -6 2 0 0 E -6 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1237: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (15) C B D A E (8) E A B C D (7) E A B D C (6) B C A D E (6) E D A C B (4) A E B D C (4) E A D C B (3) D C B A E (3) D A B E C (3) C D E B A (3) C D B A E (3) C B E A D (3) B C D A E (3) E C D A B (2) C E D B A (2) C D B E A (2) B A C D E (2) A B E C D (2) E C D B A (1) E C A D B (1) E C A B D (1) E B C A D (1) E A C B D (1) D C E B A (1) D C B E A (1) D B C A E (1) D B A C E (1) D A E B C (1) C E B D A (1) C B D E A (1) C B A D E (1) B A D C E (1) B A C E D (1) A D E B C (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 10 18 -12 B -8 0 16 2 -10 C -10 -16 0 4 -10 D -18 -2 -4 0 -10 E 12 10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 10 18 -12 B -8 0 16 2 -10 C -10 -16 0 4 -10 D -18 -2 -4 0 -10 E 12 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=24 B=13 D=11 A=10 so A is eliminated. Round 2 votes counts: E=46 C=24 B=17 D=13 so D is eliminated. Round 3 votes counts: E=48 C=29 B=23 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:212 B:200 C:184 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 18 -12 B -8 0 16 2 -10 C -10 -16 0 4 -10 D -18 -2 -4 0 -10 E 12 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 18 -12 B -8 0 16 2 -10 C -10 -16 0 4 -10 D -18 -2 -4 0 -10 E 12 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 18 -12 B -8 0 16 2 -10 C -10 -16 0 4 -10 D -18 -2 -4 0 -10 E 12 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1238: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) C D A B E (8) B E C D A (7) E B A C D (6) D C A E B (6) D A C E B (6) E B A D C (5) C D B A E (4) B E D C A (4) B E A C D (4) D C A B E (3) C B D E A (3) A E C B D (3) E B D A C (2) E A B D C (2) D A E C B (2) C B D A E (2) A E B D C (2) A E B C D (2) A D E C B (2) A D C E B (2) A C D E B (2) D E B C A (1) D C B E A (1) D A E B C (1) C D B E A (1) C D A E B (1) C B A D E (1) C A D B E (1) B D C E A (1) B C E A D (1) B C D E A (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -14 -8 0 B 10 0 4 14 6 C 14 -4 0 16 -12 D 8 -14 -16 0 0 E 0 -6 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -8 0 B 10 0 4 14 6 C 14 -4 0 16 -12 D 8 -14 -16 0 0 E 0 -6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=21 D=20 E=15 A=15 so E is eliminated. Round 2 votes counts: B=42 C=21 D=20 A=17 so A is eliminated. Round 3 votes counts: B=48 D=26 C=26 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:207 E:203 D:189 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 -8 0 B 10 0 4 14 6 C 14 -4 0 16 -12 D 8 -14 -16 0 0 E 0 -6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -8 0 B 10 0 4 14 6 C 14 -4 0 16 -12 D 8 -14 -16 0 0 E 0 -6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -8 0 B 10 0 4 14 6 C 14 -4 0 16 -12 D 8 -14 -16 0 0 E 0 -6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1239: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) C B E A D (6) E C B D A (5) E A D B C (5) B C E A D (5) D A C E B (4) A D E B C (4) A D B E C (4) D A E C B (3) D A C B E (3) C D A B E (3) C B E D A (3) C B D A E (3) B C A E D (3) B A E D C (3) B A D E C (3) A B E D C (3) E D A B C (2) E C D A B (2) E B A D C (2) D C A B E (2) C E B D A (2) C D E B A (2) C D E A B (2) B E C A D (2) B C A D E (2) E D A C B (1) E C B A D (1) E B C A D (1) D E C A B (1) D E A C B (1) D A E B C (1) C E D B A (1) C E D A B (1) C E B A D (1) C D B A E (1) C B D E A (1) B A E C D (1) B A C D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 12 6 B 12 0 -12 14 16 C 14 12 0 12 8 D -12 -14 -12 0 0 E -6 -16 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 12 6 B 12 0 -12 14 16 C 14 12 0 12 8 D -12 -14 -12 0 0 E -6 -16 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=20 E=19 D=15 A=13 so A is eliminated. Round 2 votes counts: C=33 B=25 D=23 E=19 so E is eliminated. Round 3 votes counts: C=41 D=31 B=28 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 B:215 A:196 E:185 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 12 6 B 12 0 -12 14 16 C 14 12 0 12 8 D -12 -14 -12 0 0 E -6 -16 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 12 6 B 12 0 -12 14 16 C 14 12 0 12 8 D -12 -14 -12 0 0 E -6 -16 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 12 6 B 12 0 -12 14 16 C 14 12 0 12 8 D -12 -14 -12 0 0 E -6 -16 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1240: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) B E D C A (9) E B D C A (6) B E D A C (6) A B C D E (6) C A D E B (5) A C D E B (5) B A D C E (4) A C E D B (4) A C B D E (4) E D B C A (3) E C D A B (3) D C E A B (3) B A E D C (3) B A C D E (3) D E C B A (2) B D E C A (2) B D C A E (2) B A E C D (2) B A C E D (2) A C E B D (2) A C D B E (2) E D C A B (1) E C A D B (1) E B A C D (1) D C B A E (1) D C A E B (1) D C A B E (1) D B E C A (1) C E D A B (1) C D E A B (1) B E A D C (1) B D A C E (1) B A D E C (1) Total count = 100 A B C D E A 0 -20 -6 -8 -2 B 20 0 6 10 2 C 6 -6 0 -16 -4 D 8 -10 16 0 -10 E 2 -2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -8 -2 B 20 0 6 10 2 C 6 -6 0 -16 -4 D 8 -10 16 0 -10 E 2 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=25 A=23 D=9 C=7 so C is eliminated. Round 2 votes counts: B=36 A=28 E=26 D=10 so D is eliminated. Round 3 votes counts: B=38 E=32 A=30 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:207 D:202 C:190 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -6 -8 -2 B 20 0 6 10 2 C 6 -6 0 -16 -4 D 8 -10 16 0 -10 E 2 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -8 -2 B 20 0 6 10 2 C 6 -6 0 -16 -4 D 8 -10 16 0 -10 E 2 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -8 -2 B 20 0 6 10 2 C 6 -6 0 -16 -4 D 8 -10 16 0 -10 E 2 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1241: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) D E C B A (5) B C D E A (5) A E C B D (5) D B C E A (4) C B E D A (4) B C A E D (4) A E D C B (4) A D E C B (4) A D E B C (4) D A B E C (3) C B E A D (3) A E C D B (3) A D B C E (3) E D C A B (2) E C D A B (2) D E B C A (2) D E A C B (2) D C B E A (2) D B E C A (2) C E B D A (2) B D C E A (2) B D A C E (2) B A C D E (2) A C E B D (2) A B D C E (2) E D C B A (1) E C D B A (1) E C A D B (1) E C A B D (1) E A C D B (1) D E C A B (1) D C E B A (1) D B C A E (1) D B A C E (1) D A E C B (1) C E D B A (1) B C D A E (1) B C A D E (1) B A D C E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 4 10 B -4 0 2 -6 8 C -2 -2 0 -2 8 D -4 6 2 0 8 E -10 -8 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 4 10 B -4 0 2 -6 8 C -2 -2 0 -2 8 D -4 6 2 0 8 E -10 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=25 B=18 C=10 E=9 so E is eliminated. Round 2 votes counts: A=39 D=28 B=18 C=15 so C is eliminated. Round 3 votes counts: A=41 D=32 B=27 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:206 C:201 B:200 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 4 10 B -4 0 2 -6 8 C -2 -2 0 -2 8 D -4 6 2 0 8 E -10 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 10 B -4 0 2 -6 8 C -2 -2 0 -2 8 D -4 6 2 0 8 E -10 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 10 B -4 0 2 -6 8 C -2 -2 0 -2 8 D -4 6 2 0 8 E -10 -8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1242: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) B A C E D (9) A B C D E (9) B A E D C (8) E D C B A (6) C E D A B (6) A C B D E (6) D E C A B (5) B A C D E (5) A B D C E (4) E D C A B (3) E C D B A (3) E C D A B (3) C D E A B (3) D E B A C (2) D C E A B (2) C A D E B (2) B E D A C (2) B A D C E (2) E D B C A (1) E B D A C (1) D E C B A (1) C E D B A (1) C B A E D (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C A D (1) B A E C D (1) Total count = 100 A B C D E A 0 -10 18 20 18 B 10 0 12 24 22 C -18 -12 0 4 4 D -20 -24 -4 0 4 E -18 -22 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 18 20 18 B 10 0 12 24 22 C -18 -12 0 4 4 D -20 -24 -4 0 4 E -18 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=19 E=17 C=15 D=10 so D is eliminated. Round 2 votes counts: B=39 E=25 A=19 C=17 so C is eliminated. Round 3 votes counts: B=40 E=37 A=23 so A is eliminated. Round 4 votes counts: B=61 E=39 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:234 A:223 C:189 D:178 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 18 20 18 B 10 0 12 24 22 C -18 -12 0 4 4 D -20 -24 -4 0 4 E -18 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 18 20 18 B 10 0 12 24 22 C -18 -12 0 4 4 D -20 -24 -4 0 4 E -18 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 18 20 18 B 10 0 12 24 22 C -18 -12 0 4 4 D -20 -24 -4 0 4 E -18 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1243: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) C D B E A (7) E A B D C (5) A E B D C (5) E B C A D (4) E B A D C (4) A D E B C (4) E B C D A (3) D B C E A (3) C D B A E (3) C B E D A (3) B C D E A (3) A E C B D (3) E C B A D (2) E B D C A (2) E B D A C (2) E B A C D (2) E A B C D (2) D C B A E (2) D A B E C (2) C E B A D (2) C B D E A (2) C A D B E (2) A E D C B (2) A E D B C (2) A D B E C (2) A C E D B (2) D C B E A (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A E C (1) D A C B E (1) C E B D A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E C D A (1) B D E C A (1) B C E D A (1) A E C D B (1) A D C E B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -12 -2 -8 B 8 0 6 2 -4 C 12 -6 0 10 -6 D 2 -2 -10 0 -4 E 8 4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 -2 -8 B 8 0 6 2 -4 C 12 -6 0 10 -6 D 2 -2 -10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=26 A=24 D=13 B=7 so B is eliminated. Round 2 votes counts: C=34 E=28 A=24 D=14 so D is eliminated. Round 3 votes counts: C=41 E=31 A=28 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:206 C:205 D:193 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 -2 -8 B 8 0 6 2 -4 C 12 -6 0 10 -6 D 2 -2 -10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -2 -8 B 8 0 6 2 -4 C 12 -6 0 10 -6 D 2 -2 -10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -2 -8 B 8 0 6 2 -4 C 12 -6 0 10 -6 D 2 -2 -10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1244: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) C E A D B (5) E A D C B (4) C E D A B (4) C E B A D (4) C B E D A (4) C B D E A (4) B D C A E (4) B D A C E (4) A E D C B (4) A D B E C (4) D B A E C (3) C E D B A (3) C D B E A (3) C B E A D (3) B C D A E (3) A D E B C (3) A B E D C (3) E A C D B (2) E A C B D (2) D E A C B (2) C E B D A (2) C E A B D (2) B D A E C (2) B C A E D (2) B A E C D (2) B A D C E (2) A B D E C (2) E D A C B (1) D B C A E (1) D A B E C (1) C D E B A (1) B C D E A (1) B C A D E (1) B A D E C (1) B A C D E (1) A E B D C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 6 14 6 B 2 0 0 4 4 C -6 0 0 2 8 D -14 -4 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.856097 C: 0.143903 D: 0.000000 E: 0.000000 Sum of squares = 0.753610626513 Cumulative probabilities = A: 0.000000 B: 0.856097 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 14 6 B 2 0 0 4 4 C -6 0 0 2 8 D -14 -4 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000306564 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=26 B=23 E=9 D=7 so D is eliminated. Round 2 votes counts: C=35 B=27 A=27 E=11 so E is eliminated. Round 3 votes counts: A=38 C=35 B=27 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:205 C:202 E:198 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 14 6 B 2 0 0 4 4 C -6 0 0 2 8 D -14 -4 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000306564 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 14 6 B 2 0 0 4 4 C -6 0 0 2 8 D -14 -4 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000306564 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 14 6 B 2 0 0 4 4 C -6 0 0 2 8 D -14 -4 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000306564 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1245: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) D A B E C (5) C E B D A (5) C E B A D (5) E C B A D (4) E B C A D (4) D C B E A (4) D A C B E (4) D A B C E (4) C D E B A (4) A B E D C (4) D C A B E (3) A E B C D (3) E A C B D (2) D C B A E (2) D C A E B (2) D B A E C (2) C E A B D (2) C D B E A (2) C D A E B (2) C B E D A (2) B E C D A (2) B E A D C (2) B D E C A (2) B A E D C (2) A D E B C (2) E B A C D (1) D B C E A (1) D B C A E (1) D B A C E (1) C E D A B (1) C B D E A (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E A C (1) B D A E C (1) B C E D A (1) B A D E C (1) A E D C B (1) A E C B D (1) A E B D C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -4 -10 4 B 6 0 2 0 16 C 4 -2 0 -12 -2 D 10 0 12 0 8 E -4 -16 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.417236 C: 0.000000 D: 0.582764 E: 0.000000 Sum of squares = 0.513699790531 Cumulative probabilities = A: 0.000000 B: 0.417236 C: 0.417236 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -10 4 B 6 0 2 0 16 C 4 -2 0 -12 -2 D 10 0 12 0 8 E -4 -16 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 A=21 B=14 E=11 so E is eliminated. Round 2 votes counts: D=29 C=29 A=23 B=19 so B is eliminated. Round 3 votes counts: C=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:212 C:194 A:192 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 4 B 6 0 2 0 16 C 4 -2 0 -12 -2 D 10 0 12 0 8 E -4 -16 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 4 B 6 0 2 0 16 C 4 -2 0 -12 -2 D 10 0 12 0 8 E -4 -16 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 4 B 6 0 2 0 16 C 4 -2 0 -12 -2 D 10 0 12 0 8 E -4 -16 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1246: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) E D C A B (7) C E D B A (7) B A D E C (7) D E C A B (6) A B C D E (6) E D C B A (5) B A C E D (5) C E D A B (4) C B E D A (4) E C D B A (3) D E B A C (3) B A E D C (3) A D E C B (3) A D C E B (3) A D B E C (3) D E A C B (2) C E B D A (2) B C E D A (2) B C A E D (2) D E A B C (1) D A E C B (1) C D E A B (1) C A D E B (1) B E D C A (1) B E C D A (1) B E A D C (1) B A C D E (1) A D E B C (1) A D B C E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 8 0 0 B -6 0 -2 -6 -2 C -8 2 0 -18 -18 D 0 6 18 0 6 E 0 2 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.530364 B: 0.000000 C: 0.000000 D: 0.469636 E: 0.000000 Sum of squares = 0.501843887361 Cumulative probabilities = A: 0.530364 B: 0.530364 C: 0.530364 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 0 0 B -6 0 -2 -6 -2 C -8 2 0 -18 -18 D 0 6 18 0 6 E 0 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999874 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=23 C=19 E=15 D=13 so D is eliminated. Round 2 votes counts: A=31 E=27 B=23 C=19 so C is eliminated. Round 3 votes counts: E=41 A=32 B=27 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:207 E:207 B:192 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 0 0 B -6 0 -2 -6 -2 C -8 2 0 -18 -18 D 0 6 18 0 6 E 0 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999874 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 0 0 B -6 0 -2 -6 -2 C -8 2 0 -18 -18 D 0 6 18 0 6 E 0 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999874 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 0 0 B -6 0 -2 -6 -2 C -8 2 0 -18 -18 D 0 6 18 0 6 E 0 2 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999874 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1247: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) D C E B A (8) D B E A C (7) A B E C D (7) D C A E B (5) D E B C A (4) D A B E C (4) C A E B D (4) C A B E D (4) E B D C A (3) E B D A C (3) A C B E D (3) E D B C A (2) E B C A D (2) D E C B A (2) D A C B E (2) C E D B A (2) C E A B D (2) C D A E B (2) B E A D C (2) B E A C D (2) A C D B E (2) E C B D A (1) E C B A D (1) D E B A C (1) D C E A B (1) D A B C E (1) C E B D A (1) C D E B A (1) C D E A B (1) C A D E B (1) B E D A C (1) B D E A C (1) B A D E C (1) A D B C E (1) A C B D E (1) A B E D C (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -14 -6 -16 B 10 0 -8 6 -14 C 14 8 0 -4 8 D 6 -6 4 0 -2 E 16 14 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 A B C D E A 0 -10 -14 -6 -16 B 10 0 -8 6 -14 C 14 8 0 -4 8 D 6 -6 4 0 -2 E 16 14 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=28 A=18 E=12 B=7 so B is eliminated. Round 2 votes counts: D=36 C=28 A=19 E=17 so E is eliminated. Round 3 votes counts: D=45 C=32 A=23 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:213 E:212 D:201 B:197 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -6 -16 B 10 0 -8 6 -14 C 14 8 0 -4 8 D 6 -6 4 0 -2 E 16 14 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -6 -16 B 10 0 -8 6 -14 C 14 8 0 -4 8 D 6 -6 4 0 -2 E 16 14 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -6 -16 B 10 0 -8 6 -14 C 14 8 0 -4 8 D 6 -6 4 0 -2 E 16 14 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1248: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) C A D E B (6) C B E A D (5) C A E B D (5) D B E A C (4) A D E B C (4) A C E D B (4) E D B A C (3) E A B D C (3) C B D E A (3) B E D A C (3) B C E D A (3) A E D C B (3) D E B A C (2) D E A B C (2) C D B A E (2) C D A E B (2) C D A B E (2) B E D C A (2) B E A D C (2) B D E C A (2) B D C E A (2) B C D E A (2) A E D B C (2) A D C E B (2) A C D E B (2) E B D A C (1) E B C A D (1) E B A D C (1) E A C B D (1) E A B C D (1) D B E C A (1) D A C E B (1) D A B E C (1) C B E D A (1) C B D A E (1) C A E D B (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C D A (1) A E C D B (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 10 0 -10 B 2 0 6 6 -2 C -10 -6 0 -2 -4 D 0 -6 2 0 6 E 10 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.142857 E: 0.428571 Sum of squares = 0.387755102074 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.571429 E: 1.000000 A B C D E A 0 -2 10 0 -10 B 2 0 6 6 -2 C -10 -6 0 -2 -4 D 0 -6 2 0 6 E 10 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.142857 E: 0.428571 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 A=21 E=11 D=11 so E is eliminated. Round 2 votes counts: C=31 B=29 A=26 D=14 so D is eliminated. Round 3 votes counts: B=39 C=31 A=30 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:206 E:205 D:201 A:199 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 0 -10 B 2 0 6 6 -2 C -10 -6 0 -2 -4 D 0 -6 2 0 6 E 10 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.142857 E: 0.428571 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.571429 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 0 -10 B 2 0 6 6 -2 C -10 -6 0 -2 -4 D 0 -6 2 0 6 E 10 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.142857 E: 0.428571 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.571429 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 0 -10 B 2 0 6 6 -2 C -10 -6 0 -2 -4 D 0 -6 2 0 6 E 10 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.142857 E: 0.428571 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.571429 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1249: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E B A C D (9) A B C E D (7) C A D B E (6) E D B C A (5) C D A E B (5) E B D A C (4) D E C A B (4) B A E C D (4) A C B E D (4) C D A B E (3) B E A D C (3) B E A C D (3) A E C B D (3) A B C D E (3) E C A B D (2) D E C B A (2) D C E A B (2) D C B A E (2) D C A E B (2) C E D A B (2) C A D E B (2) B A C E D (2) E D C B A (1) E B D C A (1) E B A D C (1) E A B C D (1) D E B C A (1) D C E B A (1) B E D A C (1) B D A C E (1) B A E D C (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 16 -2 6 14 B -16 0 -4 2 2 C 2 4 0 16 6 D -6 -2 -16 0 -10 E -14 -2 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 6 14 B -16 0 -4 2 2 C 2 4 0 16 6 D -6 -2 -16 0 -10 E -14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 A=19 C=18 B=15 so B is eliminated. Round 2 votes counts: E=31 A=26 D=25 C=18 so C is eliminated. Round 3 votes counts: A=34 E=33 D=33 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:214 E:194 B:192 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -2 6 14 B -16 0 -4 2 2 C 2 4 0 16 6 D -6 -2 -16 0 -10 E -14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 6 14 B -16 0 -4 2 2 C 2 4 0 16 6 D -6 -2 -16 0 -10 E -14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 6 14 B -16 0 -4 2 2 C 2 4 0 16 6 D -6 -2 -16 0 -10 E -14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1250: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D C A E B (8) B E A C D (7) D A C B E (5) B A E C D (5) E B D C A (4) C D A E B (4) C D A B E (4) C A D B E (4) D A B C E (3) C D E A B (3) B A C E D (3) E D C B A (2) E D B C A (2) E D B A C (2) E C D B A (2) E B A C D (2) D C E A B (2) D B A E C (2) D A C E B (2) C E A B D (2) C B A E D (2) B E A D C (2) B A E D C (2) A D C B E (2) A B C E D (2) E C B A D (1) E B D A C (1) E B C D A (1) E B A D C (1) D E A B C (1) C E D B A (1) C B E A D (1) C A B E D (1) B E D A C (1) B D E A C (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -4 -2 6 B 6 0 4 -2 2 C 4 -4 0 10 6 D 2 2 -10 0 -10 E -6 -2 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408019 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 A B C D E A 0 -6 -4 -2 6 B 6 0 4 -2 2 C 4 -4 0 10 6 D 2 2 -10 0 -10 E -6 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408138 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 C=22 B=21 A=8 so A is eliminated. Round 2 votes counts: E=26 D=26 C=24 B=24 so C is eliminated. Round 3 votes counts: D=42 E=29 B=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:208 B:205 E:198 A:197 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 6 B 6 0 4 -2 2 C 4 -4 0 10 6 D 2 2 -10 0 -10 E -6 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408138 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 6 B 6 0 4 -2 2 C 4 -4 0 10 6 D 2 2 -10 0 -10 E -6 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408138 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 6 B 6 0 4 -2 2 C 4 -4 0 10 6 D 2 2 -10 0 -10 E -6 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408138 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1251: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) D A E C B (7) D B A E C (6) E C D A B (5) B A D C E (5) A B D C E (5) D A B E C (4) B C A E D (4) E D C A B (3) D E B C A (3) D E A C B (3) C E A D B (3) C E A B D (3) C A E B D (3) B D A E C (3) B C E A D (3) B A C D E (3) A D B C E (3) A B C D E (3) E D C B A (2) E C D B A (2) C E B A D (2) B E C D A (2) A B C E D (2) E C B A D (1) E B C D A (1) D E C B A (1) D E C A B (1) D E A B C (1) D B E A C (1) D A E B C (1) D A B C E (1) C B A E D (1) C A E D B (1) C A B E D (1) B A C E D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 2 -8 12 B -4 0 4 0 -2 C -2 -4 0 -2 -8 D 8 0 2 0 6 E -12 2 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.462837 C: 0.000000 D: 0.537163 E: 0.000000 Sum of squares = 0.502762180728 Cumulative probabilities = A: 0.000000 B: 0.462837 C: 0.462837 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -8 12 B -4 0 4 0 -2 C -2 -4 0 -2 -8 D 8 0 2 0 6 E -12 2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 B=21 A=15 C=14 so C is eliminated. Round 2 votes counts: E=29 D=29 B=22 A=20 so A is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:208 A:205 B:199 E:196 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -8 12 B -4 0 4 0 -2 C -2 -4 0 -2 -8 D 8 0 2 0 6 E -12 2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -8 12 B -4 0 4 0 -2 C -2 -4 0 -2 -8 D 8 0 2 0 6 E -12 2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -8 12 B -4 0 4 0 -2 C -2 -4 0 -2 -8 D 8 0 2 0 6 E -12 2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1252: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) A D E B C (11) D A C E B (5) B E C D A (5) B C E D A (5) E B D A C (4) D A E C B (4) A D E C B (4) D E A C B (3) C E B D A (3) C D E B A (3) C A D B E (3) B C E A D (3) A D C E B (3) A B C D E (3) E D B A C (2) C E D B A (2) C D A E B (2) C D A B E (2) C B E A D (2) C B A E D (2) C B A D E (2) A D B E C (2) A C D E B (2) E D B C A (1) E D A B C (1) E C D B A (1) E B D C A (1) C D E A B (1) C A B D E (1) B E D A C (1) B E A D C (1) B A C E D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -2 -16 -2 B 2 0 -16 -6 -6 C 2 16 0 12 16 D 16 6 -12 0 4 E 2 6 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -16 -2 B 2 0 -16 -6 -6 C 2 16 0 12 16 D 16 6 -12 0 4 E 2 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=27 B=16 D=12 E=10 so E is eliminated. Round 2 votes counts: C=36 A=27 B=21 D=16 so D is eliminated. Round 3 votes counts: A=40 C=36 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:207 E:194 A:189 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -16 -2 B 2 0 -16 -6 -6 C 2 16 0 12 16 D 16 6 -12 0 4 E 2 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -16 -2 B 2 0 -16 -6 -6 C 2 16 0 12 16 D 16 6 -12 0 4 E 2 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -16 -2 B 2 0 -16 -6 -6 C 2 16 0 12 16 D 16 6 -12 0 4 E 2 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1253: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (18) C D B A E (12) A E B C D (10) E A D C B (7) D C E A B (7) D C B E A (7) E D A C B (5) B C A D E (5) D C E B A (4) D C B A E (4) B C D A E (4) B A E C D (4) B A E D C (3) E A B C D (2) D E C A B (2) E A D B C (1) E A C D B (1) C B D A E (1) B A C E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 8 8 -8 B -10 0 0 0 -14 C -8 0 0 -16 -8 D -8 0 16 0 -8 E 8 14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 8 8 -8 B -10 0 0 0 -14 C -8 0 0 -16 -8 D -8 0 16 0 -8 E 8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=24 B=17 C=13 A=12 so A is eliminated. Round 2 votes counts: E=44 D=24 B=19 C=13 so C is eliminated. Round 3 votes counts: E=44 D=36 B=20 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:209 D:200 B:188 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 8 8 -8 B -10 0 0 0 -14 C -8 0 0 -16 -8 D -8 0 16 0 -8 E 8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 8 -8 B -10 0 0 0 -14 C -8 0 0 -16 -8 D -8 0 16 0 -8 E 8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 8 -8 B -10 0 0 0 -14 C -8 0 0 -16 -8 D -8 0 16 0 -8 E 8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1254: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) B A D C E (7) B A E D C (6) D C B E A (5) A B E C D (5) D C A E B (4) C E D A B (4) B D A C E (4) A E C D B (4) A E B C D (4) E C A D B (3) E A C B D (3) D C E A B (3) B D C A E (3) E C D A B (2) E B C A D (2) E A C D B (2) D B C E A (2) D B C A E (2) C D E A B (2) B E A C D (2) B D C E A (2) B A E C D (2) A B D C E (2) E C D B A (1) E C B D A (1) E C A B D (1) E B A C D (1) E A B C D (1) D C A B E (1) D A C B E (1) D A B C E (1) C E D B A (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E C A (1) B A D E C (1) A D C E B (1) A D C B E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 2 4 2 B 6 0 4 4 4 C -2 -4 0 -12 8 D -4 -4 12 0 4 E -2 -4 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 2 B 6 0 4 4 4 C -2 -4 0 -12 8 D -4 -4 12 0 4 E -2 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=26 A=19 E=17 C=7 so C is eliminated. Round 2 votes counts: B=31 D=28 E=22 A=19 so A is eliminated. Round 3 votes counts: B=39 D=31 E=30 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 D:204 A:201 C:195 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 2 B 6 0 4 4 4 C -2 -4 0 -12 8 D -4 -4 12 0 4 E -2 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 2 B 6 0 4 4 4 C -2 -4 0 -12 8 D -4 -4 12 0 4 E -2 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 2 B 6 0 4 4 4 C -2 -4 0 -12 8 D -4 -4 12 0 4 E -2 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1255: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (14) D C A B E (12) A C D B E (7) E B A D C (4) D C A E B (4) B E A C D (4) E D A C B (3) D E C A B (3) B E D C A (3) B E C D A (3) B C D A E (3) E D B C A (2) E D B A C (2) E B D C A (2) E A B C D (2) D E B C A (2) D B C E A (2) D A C E B (2) B D E C A (2) B D C A E (2) B A C E D (2) A C B D E (2) E D C A B (1) E B D A C (1) E B C A D (1) E A D C B (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E A B (1) D C B A E (1) D B E C A (1) D B C A E (1) C A B D E (1) B D C E A (1) B C A E D (1) B C A D E (1) B A E C D (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 0 -10 -16 B 12 0 18 4 0 C 0 -18 0 -12 -12 D 10 -4 12 0 -2 E 16 0 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.480965 C: 0.000000 D: 0.000000 E: 0.519034 Sum of squares = 0.500724613658 Cumulative probabilities = A: 0.000000 B: 0.480965 C: 0.480965 D: 0.480966 E: 1.000000 A B C D E A 0 -12 0 -10 -16 B 12 0 18 4 0 C 0 -18 0 -12 -12 D 10 -4 12 0 -2 E 16 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=30 B=23 A=11 C=1 so C is eliminated. Round 2 votes counts: E=35 D=30 B=23 A=12 so A is eliminated. Round 3 votes counts: E=37 D=37 B=26 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:215 D:208 A:181 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -10 -16 B 12 0 18 4 0 C 0 -18 0 -12 -12 D 10 -4 12 0 -2 E 16 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -10 -16 B 12 0 18 4 0 C 0 -18 0 -12 -12 D 10 -4 12 0 -2 E 16 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -10 -16 B 12 0 18 4 0 C 0 -18 0 -12 -12 D 10 -4 12 0 -2 E 16 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1256: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (6) D A B C E (6) A D C E B (6) C E A D B (5) B E C D A (5) B D A C E (5) B C A D E (5) B E D C A (4) E C B A D (3) E C A B D (3) E B C D A (3) E B C A D (3) C A D E B (3) B D E C A (3) A C D E B (3) E D A B C (2) D B A E C (2) D B A C E (2) C E A B D (2) C B A D E (2) C A E D B (2) C A D B E (2) B D C E A (2) B D A E C (2) A D C B E (2) E D A C B (1) E B D C A (1) E B D A C (1) E A D C B (1) E A C D B (1) D A E C B (1) D A E B C (1) C A E B D (1) C A B E D (1) C A B D E (1) B E C A D (1) B D E A C (1) B D C A E (1) B C E D A (1) B C E A D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -22 12 0 B -4 0 4 4 0 C 22 -4 0 12 10 D -12 -4 -12 0 2 E 0 0 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.133333 B: 0.733333 C: 0.133333 D: 0.000000 E: 0.000000 Sum of squares = 0.573333333074 Cumulative probabilities = A: 0.133333 B: 0.866667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -22 12 0 B -4 0 4 4 0 C 22 -4 0 12 10 D -12 -4 -12 0 2 E 0 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.133333 B: 0.733333 C: 0.133333 D: 0.000000 E: 0.000000 Sum of squares = 0.573333332635 Cumulative probabilities = A: 0.133333 B: 0.866667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 C=19 A=13 D=12 so D is eliminated. Round 2 votes counts: B=35 E=25 A=21 C=19 so C is eliminated. Round 3 votes counts: B=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:202 A:197 E:194 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -22 12 0 B -4 0 4 4 0 C 22 -4 0 12 10 D -12 -4 -12 0 2 E 0 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.133333 B: 0.733333 C: 0.133333 D: 0.000000 E: 0.000000 Sum of squares = 0.573333332635 Cumulative probabilities = A: 0.133333 B: 0.866667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -22 12 0 B -4 0 4 4 0 C 22 -4 0 12 10 D -12 -4 -12 0 2 E 0 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.133333 B: 0.733333 C: 0.133333 D: 0.000000 E: 0.000000 Sum of squares = 0.573333332635 Cumulative probabilities = A: 0.133333 B: 0.866667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -22 12 0 B -4 0 4 4 0 C 22 -4 0 12 10 D -12 -4 -12 0 2 E 0 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.133333 B: 0.733333 C: 0.133333 D: 0.000000 E: 0.000000 Sum of squares = 0.573333332635 Cumulative probabilities = A: 0.133333 B: 0.866667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1257: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (5) C B E D A (5) A B D E C (5) C B A D E (4) B A E D C (4) A B D C E (4) E D A C B (3) E D A B C (3) D A C E B (3) C B D A E (3) B C E A D (3) A D E C B (3) E D C B A (2) E B C D A (2) D E A C B (2) D C E A B (2) D C A E B (2) C E D B A (2) C E B D A (2) C D E B A (2) C D E A B (2) C D A E B (2) C D A B E (2) B E C D A (2) B E A D C (2) B C A E D (2) B C A D E (2) B A C E D (2) A D C E B (2) E C D B A (1) E C B D A (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A B C (1) D A E C B (1) D A E B C (1) C E D A B (1) C B D E A (1) C A D B E (1) B E A C D (1) B A E C D (1) A E B D C (1) A D E B C (1) A D B E C (1) A D B C E (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -4 -8 2 B -10 0 -12 2 -2 C 4 12 0 -8 4 D 8 -2 8 0 2 E -2 2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.43801652872 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -8 2 B -10 0 -12 2 -2 C 4 12 0 -8 4 D 8 -2 8 0 2 E -2 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528912 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=21 E=20 B=19 D=13 so D is eliminated. Round 2 votes counts: C=31 A=26 E=24 B=19 so B is eliminated. Round 3 votes counts: C=38 A=33 E=29 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:206 A:200 E:197 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 -8 2 B -10 0 -12 2 -2 C 4 12 0 -8 4 D 8 -2 8 0 2 E -2 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528912 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -8 2 B -10 0 -12 2 -2 C 4 12 0 -8 4 D 8 -2 8 0 2 E -2 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528912 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -8 2 B -10 0 -12 2 -2 C 4 12 0 -8 4 D 8 -2 8 0 2 E -2 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528912 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1258: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) C B A E D (8) E D C B A (7) A B C E D (6) D E A B C (5) A B C D E (5) E D B A C (4) D E C A B (4) B A C E D (4) A B D E C (4) E D B C A (3) E D A B C (3) C E D B A (3) C D A B E (3) E C D B A (2) D E A C B (2) D C A E B (2) D A B C E (2) C E B D A (2) C A D B E (2) C A B D E (2) B A E C D (2) D E B C A (1) D E B A C (1) D C A B E (1) D A E B C (1) D A B E C (1) C D B E A (1) B E C A D (1) B E A C D (1) B C A E D (1) B A E D C (1) A D E B C (1) A D B E C (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -6 -16 0 B 4 0 0 -20 -2 C 6 0 0 -12 -12 D 16 20 12 0 2 E 0 2 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -16 0 B 4 0 0 -20 -2 C 6 0 0 -12 -12 D 16 20 12 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999961937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=21 A=20 E=19 B=10 so B is eliminated. Round 2 votes counts: D=30 A=27 C=22 E=21 so E is eliminated. Round 3 votes counts: D=47 A=28 C=25 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:206 B:191 C:191 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -16 0 B 4 0 0 -20 -2 C 6 0 0 -12 -12 D 16 20 12 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999961937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -16 0 B 4 0 0 -20 -2 C 6 0 0 -12 -12 D 16 20 12 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999961937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -16 0 B 4 0 0 -20 -2 C 6 0 0 -12 -12 D 16 20 12 0 2 E 0 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999961937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1259: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (15) D E A C B (12) C B A D E (9) B C A E D (6) B C A D E (6) E D A B C (5) D A E C B (4) B E C A D (4) B C E A D (4) B C D A E (3) E D B C A (2) D E C A B (2) C B D A E (2) C A B D E (2) B A C E D (2) A E D C B (2) E B D C A (1) E B C D A (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E B C A (1) D C B A E (1) D C A B E (1) C D B A E (1) C A D B E (1) B E D C A (1) B C E D A (1) B C D E A (1) A E C D B (1) A D C E B (1) A C D E B (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -2 -8 -8 B -4 0 -16 -4 -4 C 2 16 0 -2 -10 D 8 4 2 0 0 E 8 4 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.386756 E: 0.613244 Sum of squares = 0.525648499309 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.386756 E: 1.000000 A B C D E A 0 4 -2 -8 -8 B -4 0 -16 -4 -4 C 2 16 0 -2 -10 D 8 4 2 0 0 E 8 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=28 B=28 D=21 C=15 A=8 so A is eliminated. Round 2 votes counts: E=31 B=28 D=22 C=19 so C is eliminated. Round 3 votes counts: B=43 E=31 D=26 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:207 C:203 A:193 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -8 -8 B -4 0 -16 -4 -4 C 2 16 0 -2 -10 D 8 4 2 0 0 E 8 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -8 -8 B -4 0 -16 -4 -4 C 2 16 0 -2 -10 D 8 4 2 0 0 E 8 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -8 -8 B -4 0 -16 -4 -4 C 2 16 0 -2 -10 D 8 4 2 0 0 E 8 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1260: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) E A C D B (9) B D C A E (9) A E C D B (9) E A D C B (5) D B C A E (5) B D E A C (5) D B A E C (4) C B D A E (4) B D C E A (4) A E D B C (4) E A D B C (3) C B E A D (3) C A E B D (3) D B E A C (2) C B D E A (2) A E D C B (2) E C A B D (1) E B A D C (1) E A C B D (1) E A B C D (1) D E B A C (1) D E A B C (1) D B A C E (1) D A E B C (1) C D B A E (1) C B E D A (1) C B A D E (1) B E D C A (1) B E D A C (1) B D E C A (1) A C E D B (1) Total count = 100 A B C D E A 0 6 4 12 -10 B -6 0 -10 2 -10 C -4 10 0 -2 -6 D -12 -2 2 0 -16 E 10 10 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 4 12 -10 B -6 0 -10 2 -10 C -4 10 0 -2 -6 D -12 -2 2 0 -16 E 10 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=21 B=21 A=16 D=15 so D is eliminated. Round 2 votes counts: B=33 C=27 E=23 A=17 so A is eliminated. Round 3 votes counts: E=39 B=33 C=28 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:206 C:199 B:188 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 12 -10 B -6 0 -10 2 -10 C -4 10 0 -2 -6 D -12 -2 2 0 -16 E 10 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 12 -10 B -6 0 -10 2 -10 C -4 10 0 -2 -6 D -12 -2 2 0 -16 E 10 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 12 -10 B -6 0 -10 2 -10 C -4 10 0 -2 -6 D -12 -2 2 0 -16 E 10 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1261: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (9) A D E C B (9) D E A C B (8) B A D E C (8) B A C D E (8) C E D A B (5) B C A D E (5) B C E D A (4) B C A E D (4) E D C A B (3) E D A C B (3) D E B A C (3) C B A E D (3) A D E B C (3) B E D C A (2) A C D E B (2) E D C B A (1) E C D B A (1) E C D A B (1) D E A B C (1) D B E A C (1) C A B E D (1) B D E C A (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 14 14 14 B 0 0 0 -8 -6 C -14 0 0 -6 -10 D -14 8 6 0 18 E -14 6 10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.552707 B: 0.447293 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.505556058735 Cumulative probabilities = A: 0.552707 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 14 14 B 0 0 0 -8 -6 C -14 0 0 -6 -10 D -14 8 6 0 18 E -14 6 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=21 A=16 D=13 E=9 so E is eliminated. Round 2 votes counts: B=41 C=23 D=20 A=16 so A is eliminated. Round 3 votes counts: B=42 D=32 C=26 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:221 D:209 B:193 E:192 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 14 14 B 0 0 0 -8 -6 C -14 0 0 -6 -10 D -14 8 6 0 18 E -14 6 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 14 14 B 0 0 0 -8 -6 C -14 0 0 -6 -10 D -14 8 6 0 18 E -14 6 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 14 14 B 0 0 0 -8 -6 C -14 0 0 -6 -10 D -14 8 6 0 18 E -14 6 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1262: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) C D E B A (11) A B E D C (11) E A B C D (10) D B A C E (8) E C A B D (7) D C B A E (7) B A D E C (5) A E B C D (4) E A C B D (3) C E D B A (3) C E A D B (3) B A E D C (3) E C A D B (2) D B C A E (2) C D E A B (2) B D A E C (2) D C A B E (1) D A B E C (1) C E B A D (1) C D B E A (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -2 2 -8 B -14 0 -4 -4 -16 C 2 4 0 16 0 D -2 4 -16 0 -18 E 8 16 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600623 D: 0.000000 E: 0.399377 Sum of squares = 0.520249955397 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600623 D: 0.600623 E: 1.000000 A B C D E A 0 14 -2 2 -8 B -14 0 -4 -4 -16 C 2 4 0 16 0 D -2 4 -16 0 -18 E 8 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=22 D=19 A=17 B=10 so B is eliminated. Round 2 votes counts: C=32 A=25 E=22 D=21 so D is eliminated. Round 3 votes counts: C=42 A=36 E=22 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:211 A:203 D:184 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 2 -8 B -14 0 -4 -4 -16 C 2 4 0 16 0 D -2 4 -16 0 -18 E 8 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 2 -8 B -14 0 -4 -4 -16 C 2 4 0 16 0 D -2 4 -16 0 -18 E 8 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 2 -8 B -14 0 -4 -4 -16 C 2 4 0 16 0 D -2 4 -16 0 -18 E 8 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1263: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) D A B E C (9) C E B A D (8) B E A C D (8) D C A E B (7) E B C A D (5) C D A E B (5) E B A C D (4) C E B D A (4) B E A D C (4) A B E D C (4) E C B A D (3) C E D B A (3) C D E B A (3) C D E A B (3) D C B E A (2) D C A B E (2) D A B C E (2) B A E D C (2) A D B E C (2) D B A E C (1) D A C E B (1) C E D A B (1) B E D A C (1) B E C D A (1) B E C A D (1) B D E A C (1) B A D E C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 4 -12 -4 B 4 0 -4 -2 4 C -4 4 0 0 2 D 12 2 0 0 0 E 4 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.564121 D: 0.435879 E: 0.000000 Sum of squares = 0.50822303512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.564121 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -12 -4 B 4 0 -4 -2 4 C -4 4 0 0 2 D 12 2 0 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=27 B=19 E=12 A=8 so A is eliminated. Round 2 votes counts: D=36 C=27 B=24 E=13 so E is eliminated. Round 3 votes counts: D=36 B=34 C=30 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:207 B:201 C:201 E:199 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -12 -4 B 4 0 -4 -2 4 C -4 4 0 0 2 D 12 2 0 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -12 -4 B 4 0 -4 -2 4 C -4 4 0 0 2 D 12 2 0 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -12 -4 B 4 0 -4 -2 4 C -4 4 0 0 2 D 12 2 0 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1264: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) C B D E A (8) C B D A E (8) D B E C A (7) E D A B C (6) B C D E A (6) A E D C B (6) E A D B C (5) B D C E A (5) D E A B C (4) A E C D B (4) D B C E A (3) C B A D E (3) C A B E D (3) A E C B D (3) E D B A C (2) C A B D E (2) A C E B D (2) A C B E D (2) E B D C A (1) E B D A C (1) E A B D C (1) E A B C D (1) D E B C A (1) D E B A C (1) D C B E A (1) D A E C B (1) D A C E B (1) C B A E D (1) A E B D C (1) A D C E B (1) Total count = 100 A B C D E A 0 4 2 -12 -6 B -4 0 8 -4 -2 C -2 -8 0 -14 -8 D 12 4 14 0 4 E 6 2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -12 -6 B -4 0 8 -4 -2 C -2 -8 0 -14 -8 D 12 4 14 0 4 E 6 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=25 D=19 E=17 B=11 so B is eliminated. Round 2 votes counts: C=31 A=28 D=24 E=17 so E is eliminated. Round 3 votes counts: A=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:206 B:199 A:194 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -12 -6 B -4 0 8 -4 -2 C -2 -8 0 -14 -8 D 12 4 14 0 4 E 6 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -12 -6 B -4 0 8 -4 -2 C -2 -8 0 -14 -8 D 12 4 14 0 4 E 6 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -12 -6 B -4 0 8 -4 -2 C -2 -8 0 -14 -8 D 12 4 14 0 4 E 6 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1265: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) C B A E D (5) B C E D A (5) A D E B C (5) D E A B C (4) C E D B A (4) A D E C B (4) A C B D E (4) E D B C A (3) D E A C B (3) D A E B C (3) C E B D A (3) C B E D A (3) C A B E D (3) B C A D E (3) A C E D B (3) A B C D E (3) E D C B A (2) E D C A B (2) D E C B A (2) D E B C A (2) D E B A C (2) B C D E A (2) B A D C E (2) A C D E B (2) E B C D A (1) E A D C B (1) D B E C A (1) D B E A C (1) D B A E C (1) D A E C B (1) C E D A B (1) C E A B D (1) C A E D B (1) B E C D A (1) B D E C A (1) B D C E A (1) B D A E C (1) B C E A D (1) B C A E D (1) B A C D E (1) A E C D B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 2 0 6 B -2 0 -8 0 -2 C -2 8 0 16 16 D 0 0 -16 0 0 E -6 2 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.923724 B: 0.000000 C: 0.000000 D: 0.076276 E: 0.000000 Sum of squares = 0.859083882592 Cumulative probabilities = A: 0.923724 B: 0.923724 C: 0.923724 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 6 B -2 0 -8 0 -2 C -2 8 0 16 16 D 0 0 -16 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469142948 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=21 D=20 B=19 E=9 so E is eliminated. Round 2 votes counts: A=32 D=27 C=21 B=20 so B is eliminated. Round 3 votes counts: C=35 A=35 D=30 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:219 A:205 B:194 D:192 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 6 B -2 0 -8 0 -2 C -2 8 0 16 16 D 0 0 -16 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469142948 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 6 B -2 0 -8 0 -2 C -2 8 0 16 16 D 0 0 -16 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469142948 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 6 B -2 0 -8 0 -2 C -2 8 0 16 16 D 0 0 -16 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.802469142948 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1266: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) E C A B D (6) D B A C E (6) A E D C B (6) E C B D A (5) D B C E A (5) C E B A D (5) A E C B D (5) E A C B D (4) D B C A E (4) D A B E C (4) A D E B C (4) A D B C E (4) C B E D A (3) C B E A D (3) B C E D A (3) A D B E C (3) E C B A D (2) E A C D B (2) D A B C E (2) B D C A E (2) A E C D B (2) A C E B D (2) E C D B A (1) D B E C A (1) D B A E C (1) D A E B C (1) C E B D A (1) C E A B D (1) C A E B D (1) B D C E A (1) B D A C E (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -4 2 -2 B 4 0 0 6 2 C 4 0 0 8 4 D -2 -6 -8 0 -4 E 2 -2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.570949 C: 0.429051 D: 0.000000 E: 0.000000 Sum of squares = 0.51006758519 Cumulative probabilities = A: 0.000000 B: 0.570949 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 2 -2 B 4 0 0 6 2 C 4 0 0 8 4 D -2 -6 -8 0 -4 E 2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 E=20 B=15 C=14 so C is eliminated. Round 2 votes counts: A=28 E=27 D=24 B=21 so B is eliminated. Round 3 votes counts: E=36 D=36 A=28 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:208 B:206 E:200 A:196 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 2 -2 B 4 0 0 6 2 C 4 0 0 8 4 D -2 -6 -8 0 -4 E 2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 2 -2 B 4 0 0 6 2 C 4 0 0 8 4 D -2 -6 -8 0 -4 E 2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 2 -2 B 4 0 0 6 2 C 4 0 0 8 4 D -2 -6 -8 0 -4 E 2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1267: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) E D A C B (6) D C B A E (6) E C B A D (5) B C D A E (5) E A B C D (4) B C A E D (4) B C A D E (4) A E B C D (4) A B C E D (4) E D C A B (3) E A C B D (3) D E C B A (3) D E A B C (3) D C B E A (3) D A B C E (3) C B E A D (3) E C D B A (2) D E C A B (2) D C E B A (2) D A E B C (2) C E B A D (2) C D B E A (2) C B D E A (2) C B D A E (2) A E D B C (2) A E B D C (2) E C D A B (1) E C B D A (1) E A D C B (1) E A D B C (1) D E A C B (1) D B C E A (1) D B A C E (1) C D E B A (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -22 -18 -4 B 12 0 -2 -6 -2 C 22 2 0 0 4 D 18 6 0 0 2 E 4 2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.641379 D: 0.358621 E: 0.000000 Sum of squares = 0.539975803867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.641379 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -22 -18 -4 B 12 0 -2 -6 -2 C 22 2 0 0 4 D 18 6 0 0 2 E 4 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999995815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=27 A=14 B=13 C=12 so C is eliminated. Round 2 votes counts: D=37 E=29 B=20 A=14 so A is eliminated. Round 3 votes counts: E=37 D=37 B=26 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:213 B:201 E:200 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -22 -18 -4 B 12 0 -2 -6 -2 C 22 2 0 0 4 D 18 6 0 0 2 E 4 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999995815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -22 -18 -4 B 12 0 -2 -6 -2 C 22 2 0 0 4 D 18 6 0 0 2 E 4 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999995815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -22 -18 -4 B 12 0 -2 -6 -2 C 22 2 0 0 4 D 18 6 0 0 2 E 4 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999995815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1268: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (14) B D E A C (6) B D C A E (6) E D B A C (5) E C A D B (4) E A D C B (4) C E A D B (4) E D A B C (3) E A C D B (3) D B E A C (3) C A B E D (3) C A B D E (3) B D C E A (3) B D A E C (3) D B A E C (2) D A E B C (2) D A B E C (2) C B A D E (2) B E D C A (2) B D A C E (2) B C D A E (2) B C A D E (2) A D E C B (2) A C E D B (2) E D C B A (1) E D B C A (1) E D A C B (1) D A B C E (1) C E A B D (1) C B E D A (1) C B A E D (1) C A E B D (1) B E D A C (1) B D E C A (1) B A D C E (1) B A C D E (1) A E D C B (1) A E C D B (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 8 -4 4 12 B -8 0 0 -16 -2 C 4 0 0 -10 2 D -4 16 10 0 -8 E -12 2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407416 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 4 12 B -8 0 0 -16 -2 C 4 0 0 -10 2 D -4 16 10 0 -8 E -12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407408 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=30 B=30 E=22 D=10 A=8 so A is eliminated. Round 2 votes counts: C=32 B=30 E=24 D=14 so D is eliminated. Round 3 votes counts: B=39 C=33 E=28 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:210 D:207 C:198 E:198 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -4 4 12 B -8 0 0 -16 -2 C 4 0 0 -10 2 D -4 16 10 0 -8 E -12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407408 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 4 12 B -8 0 0 -16 -2 C 4 0 0 -10 2 D -4 16 10 0 -8 E -12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407408 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 4 12 B -8 0 0 -16 -2 C 4 0 0 -10 2 D -4 16 10 0 -8 E -12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407408 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1269: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) C A E B D (8) D B A C E (7) B D A C E (7) E C A B D (6) D B E A C (5) E A C D B (4) A C E D B (4) A C B D E (4) E D C A B (3) D E B A C (3) C E A B D (3) E D B C A (2) E D B A C (2) E D A C B (2) E C D A B (2) D E A C B (2) D B E C A (2) D B A E C (2) B E D C A (2) B A C D E (2) A C D B E (2) E D A B C (1) E C D B A (1) E B D C A (1) E B C A D (1) D A B C E (1) C B A E D (1) C A E D B (1) C A B E D (1) B D E C A (1) B D E A C (1) B D C A E (1) B C E A D (1) B C A E D (1) B C A D E (1) A C E B D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 4 4 -8 B -12 0 -10 -12 -14 C -4 10 0 10 -4 D -4 12 -10 0 -14 E 8 14 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 4 4 -8 B -12 0 -10 -12 -14 C -4 10 0 10 -4 D -4 12 -10 0 -14 E 8 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=22 B=17 C=14 A=13 so A is eliminated. Round 2 votes counts: E=34 C=26 D=22 B=18 so B is eliminated. Round 3 votes counts: E=36 D=32 C=32 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:206 C:206 D:192 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 4 4 -8 B -12 0 -10 -12 -14 C -4 10 0 10 -4 D -4 12 -10 0 -14 E 8 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 4 -8 B -12 0 -10 -12 -14 C -4 10 0 10 -4 D -4 12 -10 0 -14 E 8 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 4 -8 B -12 0 -10 -12 -14 C -4 10 0 10 -4 D -4 12 -10 0 -14 E 8 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1270: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) C B E D A (9) C D E B A (8) A D E B C (8) D C E B A (6) A B E D C (6) D A E B C (5) A C B E D (4) E D B A C (3) D E C B A (3) D E A B C (3) C D A E B (3) C B E A D (3) C A B E D (3) A C D E B (3) E B D A C (2) D E B C A (2) C B A E D (2) B E D C A (2) B E A D C (2) E D A B C (1) C D E A B (1) C D B E A (1) C D B A E (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C D A (1) B A E C D (1) A E D B C (1) A D C E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 6 -24 -16 B 14 0 0 -22 -20 C -6 0 0 -12 -4 D 24 22 12 0 14 E 16 20 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 -24 -16 B 14 0 0 -22 -20 C -6 0 0 -12 -4 D 24 22 12 0 14 E 16 20 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=29 A=25 B=7 E=6 so E is eliminated. Round 2 votes counts: D=33 C=33 A=25 B=9 so B is eliminated. Round 3 votes counts: D=38 C=34 A=28 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:236 E:213 C:189 B:186 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 6 -24 -16 B 14 0 0 -22 -20 C -6 0 0 -12 -4 D 24 22 12 0 14 E 16 20 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 -24 -16 B 14 0 0 -22 -20 C -6 0 0 -12 -4 D 24 22 12 0 14 E 16 20 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 -24 -16 B 14 0 0 -22 -20 C -6 0 0 -12 -4 D 24 22 12 0 14 E 16 20 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1271: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (6) B A E D C (6) A D C B E (6) E D C A B (5) C D E B A (4) C B D A E (4) B C A D E (4) A B D C E (4) E C D B A (3) E C D A B (3) E B C D A (3) E B A D C (3) B A E C D (3) B A C D E (3) E D A C B (2) D C E A B (2) D C A E B (2) D A C E B (2) B E C D A (2) B C E D A (2) A D E C B (2) A C D B E (2) A B E D C (2) A B D E C (2) A B C D E (2) E D A B C (1) E C B D A (1) E B D C A (1) E A D B C (1) D A E C B (1) C E D B A (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A D B E (1) B E A D C (1) B E A C D (1) B C D E A (1) B C A E D (1) B A D C E (1) B A C E D (1) A E D C B (1) A E B D C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 2 0 0 10 B -2 0 -6 0 8 C 0 6 0 4 10 D 0 0 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.416167 B: 0.000000 C: 0.583833 D: 0.000000 E: 0.000000 Sum of squares = 0.51405579353 Cumulative probabilities = A: 0.416167 B: 0.416167 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 10 B -2 0 -6 0 8 C 0 6 0 4 10 D 0 0 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 E=23 C=20 D=7 so D is eliminated. Round 2 votes counts: A=27 B=26 C=24 E=23 so E is eliminated. Round 3 votes counts: C=36 B=33 A=31 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:206 D:202 B:200 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 0 10 B -2 0 -6 0 8 C 0 6 0 4 10 D 0 0 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 10 B -2 0 -6 0 8 C 0 6 0 4 10 D 0 0 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 10 B -2 0 -6 0 8 C 0 6 0 4 10 D 0 0 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1272: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (18) A C B D E (13) C A B D E (10) C E D B A (6) E D C B A (5) E D B A C (5) C D E B A (4) A B C D E (4) C B D E A (3) C A E D B (3) B D E C A (3) B D E A C (3) A C E D B (3) A B D E C (3) C D B E A (2) E D A C B (1) E D A B C (1) E C D B A (1) E A C D B (1) D E B C A (1) C B D A E (1) C A D B E (1) B E D A C (1) B D C E A (1) B D A E C (1) B D A C E (1) A E C D B (1) A E B D C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -18 -16 -12 B 12 0 -12 -6 -2 C 18 12 0 8 8 D 16 6 -8 0 4 E 12 2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -16 -12 B 12 0 -12 -6 -2 C 18 12 0 8 8 D 16 6 -8 0 4 E 12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=30 A=27 B=10 D=1 so D is eliminated. Round 2 votes counts: E=33 C=30 A=27 B=10 so B is eliminated. Round 3 votes counts: E=40 C=31 A=29 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:209 E:201 B:196 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -18 -16 -12 B 12 0 -12 -6 -2 C 18 12 0 8 8 D 16 6 -8 0 4 E 12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -16 -12 B 12 0 -12 -6 -2 C 18 12 0 8 8 D 16 6 -8 0 4 E 12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -16 -12 B 12 0 -12 -6 -2 C 18 12 0 8 8 D 16 6 -8 0 4 E 12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1273: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) C B E D A (6) C B A E D (6) A D E B C (6) A E D B C (5) A E B D C (5) A C D E B (5) C B D E A (4) C A D B E (4) C A B E D (4) E B D A C (3) D E A B C (3) D B E C A (3) D A E B C (3) C D B E A (3) A D E C B (3) E D B A C (2) C D A B E (2) C B E A D (2) B E D A C (2) B E C D A (2) B D E C A (2) B C E A D (2) A C E D B (2) E D A B C (1) E B A D C (1) D E B C A (1) D A E C B (1) C D A E B (1) C A E B D (1) B E A D C (1) B E A C D (1) B C E D A (1) B A E C D (1) A E D C B (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 12 4 4 B 2 0 8 -10 -6 C -12 -8 0 -4 -12 D -4 10 4 0 0 E -4 6 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.468749999903 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 4 4 B 2 0 8 -10 -6 C -12 -8 0 -4 -12 D -4 10 4 0 0 E -4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999999 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=29 D=19 B=12 E=7 so E is eliminated. Round 2 votes counts: C=33 A=29 D=22 B=16 so B is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:207 D:205 B:197 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 12 4 4 B 2 0 8 -10 -6 C -12 -8 0 -4 -12 D -4 10 4 0 0 E -4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999999 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 4 4 B 2 0 8 -10 -6 C -12 -8 0 -4 -12 D -4 10 4 0 0 E -4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999999 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 4 4 B 2 0 8 -10 -6 C -12 -8 0 -4 -12 D -4 10 4 0 0 E -4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999999 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1274: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C A E B D (8) E C D A B (7) E C A D B (6) B D A E C (5) B D A C E (5) B A D C E (5) E D B A C (4) D E B C A (4) A C B E D (4) A C B D E (4) A B C D E (4) B D E A C (3) E D C A B (2) E D B C A (2) D B E C A (2) C E A D B (2) C D A B E (2) C A B E D (2) C A B D E (2) A E C B D (2) A C E B D (2) E D C B A (1) E B D A C (1) D E B A C (1) D C E B A (1) D B C A E (1) C E D A B (1) C D E A B (1) C B D A E (1) C A E D B (1) B E D A C (1) B A D E C (1) B A C D E (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 8 -8 4 B -4 0 2 6 6 C -8 -2 0 4 -4 D 8 -6 -4 0 4 E -4 -6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.35802469136 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -8 4 B -4 0 2 6 6 C -8 -2 0 4 -4 D 8 -6 -4 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 B=21 C=20 D=18 A=18 so D is eliminated. Round 2 votes counts: B=33 E=28 C=21 A=18 so A is eliminated. Round 3 votes counts: B=38 E=31 C=31 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:205 A:204 D:201 C:195 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 8 -8 4 B -4 0 2 6 6 C -8 -2 0 4 -4 D 8 -6 -4 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -8 4 B -4 0 2 6 6 C -8 -2 0 4 -4 D 8 -6 -4 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -8 4 B -4 0 2 6 6 C -8 -2 0 4 -4 D 8 -6 -4 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1275: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) C E B A D (7) C E A B D (7) C A E D B (6) E C B A D (5) D A B C E (5) B D E C A (5) A E C D B (5) A D C E B (5) A C E D B (5) C B E A D (4) A C D E B (4) D B A C E (3) D A B E C (3) E C A D B (2) E C A B D (2) D A E C B (2) D A E B C (2) D A C E B (2) D A C B E (2) B E C D A (2) B D C E A (2) E D A B C (1) E B C D A (1) E B C A D (1) E A D C B (1) D B E A C (1) C E A D B (1) B E D C A (1) B E C A D (1) B D E A C (1) B D A E C (1) B D A C E (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 12 4 16 8 B -12 0 -22 -16 -20 C -4 22 0 8 10 D -16 16 -8 0 -6 E -8 20 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 16 8 B -12 0 -22 -16 -20 C -4 22 0 8 10 D -16 16 -8 0 -6 E -8 20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 A=20 B=15 E=13 so E is eliminated. Round 2 votes counts: C=34 D=28 A=21 B=17 so B is eliminated. Round 3 votes counts: C=40 D=39 A=21 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:220 C:218 E:204 D:193 B:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 16 8 B -12 0 -22 -16 -20 C -4 22 0 8 10 D -16 16 -8 0 -6 E -8 20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 16 8 B -12 0 -22 -16 -20 C -4 22 0 8 10 D -16 16 -8 0 -6 E -8 20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 16 8 B -12 0 -22 -16 -20 C -4 22 0 8 10 D -16 16 -8 0 -6 E -8 20 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1276: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (12) B D A E C (11) C E A D B (8) E A D C B (7) B D C A E (5) E A D B C (4) E A B D C (4) B E C A D (4) D A E C B (3) C E B A D (3) C E A B D (3) C D A E B (3) C B E D A (3) C B D A E (3) A E D B C (3) D A E B C (2) D A B E C (2) C D E A B (2) B E A D C (2) B D A C E (2) E C A D B (1) E A C D B (1) E A C B D (1) D C A E B (1) C E B D A (1) C D A B E (1) C B E A D (1) C B D E A (1) B C E D A (1) B C E A D (1) B C D E A (1) A E D C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -10 -8 2 B 2 0 12 20 2 C 10 -12 0 2 4 D 8 -20 -2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -8 2 B 2 0 12 20 2 C 10 -12 0 2 4 D 8 -20 -2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=29 E=18 D=8 A=6 so A is eliminated. Round 2 votes counts: B=40 C=29 E=22 D=9 so D is eliminated. Round 3 votes counts: B=42 C=30 E=28 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:202 E:195 D:194 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -10 -8 2 B 2 0 12 20 2 C 10 -12 0 2 4 D 8 -20 -2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -8 2 B 2 0 12 20 2 C 10 -12 0 2 4 D 8 -20 -2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -8 2 B 2 0 12 20 2 C 10 -12 0 2 4 D 8 -20 -2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1277: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) E D B C A (11) E D B A C (10) D E B A C (9) C A B E D (8) C A B D E (6) B A C D E (6) A C B D E (6) B A C E D (5) C A D E B (3) D E C A B (2) D B E A C (2) B E D A C (2) B E C A D (2) B C A E D (2) E C A D B (1) E B D C A (1) D E A C B (1) D A C E B (1) C E A B D (1) C B A E D (1) C A E D B (1) C A E B D (1) B D E A C (1) B C E A D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 -6 -14 B 6 0 8 -8 -10 C 8 -8 0 -6 -10 D 6 8 6 0 -24 E 14 10 10 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -6 -14 B 6 0 8 -8 -10 C 8 -8 0 -6 -10 D 6 8 6 0 -24 E 14 10 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=21 B=19 D=15 A=9 so A is eliminated. Round 2 votes counts: E=36 C=28 B=21 D=15 so D is eliminated. Round 3 votes counts: E=48 C=29 B=23 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:229 B:198 D:198 C:192 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -6 -14 B 6 0 8 -8 -10 C 8 -8 0 -6 -10 D 6 8 6 0 -24 E 14 10 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -6 -14 B 6 0 8 -8 -10 C 8 -8 0 -6 -10 D 6 8 6 0 -24 E 14 10 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -6 -14 B 6 0 8 -8 -10 C 8 -8 0 -6 -10 D 6 8 6 0 -24 E 14 10 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1278: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) A C E D B (7) D B A E C (5) D B A C E (5) D A E C B (5) D A B E C (5) A D E C B (5) E C B A D (4) D A B C E (4) A D C E B (4) E C A D B (3) E C A B D (3) C A E B D (3) B E C D A (3) B C E D A (3) E D A C B (2) E B D C A (2) D A E B C (2) C E A B D (2) C B E A D (2) C B A E D (2) B E D C A (2) B D E C A (2) B D C A E (2) B D A C E (2) A E D C B (2) A D C B E (2) A D B C E (2) E B C D A (1) E A C D B (1) D B E A C (1) B D E A C (1) B D A E C (1) Total count = 100 A B C D E A 0 4 12 4 16 B -4 0 -14 -10 -12 C -12 14 0 -12 0 D -4 10 12 0 -4 E -16 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 4 16 B -4 0 -14 -10 -12 C -12 14 0 -12 0 D -4 10 12 0 -4 E -16 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=22 C=19 E=16 B=16 so E is eliminated. Round 2 votes counts: D=29 C=29 A=23 B=19 so B is eliminated. Round 3 votes counts: D=41 C=36 A=23 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:207 E:200 C:195 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 4 16 B -4 0 -14 -10 -12 C -12 14 0 -12 0 D -4 10 12 0 -4 E -16 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 4 16 B -4 0 -14 -10 -12 C -12 14 0 -12 0 D -4 10 12 0 -4 E -16 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 4 16 B -4 0 -14 -10 -12 C -12 14 0 -12 0 D -4 10 12 0 -4 E -16 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1279: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) A D E B C (12) C B E A D (7) C B A D E (7) B E D A C (7) C A D E B (5) B C E D A (5) C A D B E (4) E D A B C (3) C E B D A (3) C B A E D (3) B A D E C (3) A D B E C (3) E B D A C (2) D E A B C (2) D A E C B (2) D A E B C (2) C E D B A (2) C E D A B (2) B E A D C (2) E D C A B (1) E C D B A (1) E B D C A (1) E B C D A (1) D A C E B (1) C D A E B (1) B E D C A (1) B C A D E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -20 -18 -2 -8 B 20 0 -10 14 14 C 18 10 0 12 12 D 2 -14 -12 0 -10 E 8 -14 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -18 -2 -8 B 20 0 -10 14 14 C 18 10 0 12 12 D 2 -14 -12 0 -10 E 8 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=48 B=19 A=17 E=9 D=7 so D is eliminated. Round 2 votes counts: C=48 A=22 B=19 E=11 so E is eliminated. Round 3 votes counts: C=50 A=27 B=23 so B is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:219 E:196 D:183 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -18 -2 -8 B 20 0 -10 14 14 C 18 10 0 12 12 D 2 -14 -12 0 -10 E 8 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 -2 -8 B 20 0 -10 14 14 C 18 10 0 12 12 D 2 -14 -12 0 -10 E 8 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 -2 -8 B 20 0 -10 14 14 C 18 10 0 12 12 D 2 -14 -12 0 -10 E 8 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1280: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (10) B D C E A (8) B C D E A (8) B D C A E (6) D A B E C (5) C E B D A (5) B D A C E (5) A E C D B (5) E C A D B (4) C E B A D (4) C B E D A (4) A E D B C (4) A D E B C (4) A E D C B (3) A D B E C (3) E A C D B (2) D B A C E (2) C B D E A (2) E C A B D (1) E A D C B (1) D B E A C (1) B D E C A (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -6 -16 -2 B 14 0 18 10 14 C 6 -18 0 -8 10 D 16 -10 8 0 14 E 2 -14 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -16 -2 B 14 0 18 10 14 C 6 -18 0 -8 10 D 16 -10 8 0 14 E 2 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 A=20 D=19 E=8 so E is eliminated. Round 2 votes counts: C=30 B=28 A=23 D=19 so D is eliminated. Round 3 votes counts: B=42 C=30 A=28 so A is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:228 D:214 C:195 E:182 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -16 -2 B 14 0 18 10 14 C 6 -18 0 -8 10 D 16 -10 8 0 14 E 2 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -16 -2 B 14 0 18 10 14 C 6 -18 0 -8 10 D 16 -10 8 0 14 E 2 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -16 -2 B 14 0 18 10 14 C 6 -18 0 -8 10 D 16 -10 8 0 14 E 2 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1281: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) C B D E A (9) D A C B E (8) D C A B E (7) A D E C B (7) E A B D C (6) A E D C B (6) C D B A E (5) A D C E B (5) E B A C D (4) E A B C D (4) B E C A D (4) E B C A D (3) D C B A E (3) B C D E A (3) E A D B C (2) D C A E B (2) D A C E B (2) A E D B C (2) A D E B C (2) E C B A D (1) C E B D A (1) C B E D A (1) C B D A E (1) B C D A E (1) Total count = 100 A B C D E A 0 6 -4 -8 2 B -6 0 -16 -2 6 C 4 16 0 -4 18 D 8 2 4 0 10 E -2 -6 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -8 2 B -6 0 -16 -2 6 C 4 16 0 -4 18 D 8 2 4 0 10 E -2 -6 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=22 A=22 E=20 B=19 C=17 so C is eliminated. Round 2 votes counts: B=30 D=27 A=22 E=21 so E is eliminated. Round 3 votes counts: B=39 A=34 D=27 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 D:212 A:198 B:191 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -8 2 B -6 0 -16 -2 6 C 4 16 0 -4 18 D 8 2 4 0 10 E -2 -6 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -8 2 B -6 0 -16 -2 6 C 4 16 0 -4 18 D 8 2 4 0 10 E -2 -6 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -8 2 B -6 0 -16 -2 6 C 4 16 0 -4 18 D 8 2 4 0 10 E -2 -6 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1282: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (19) E C D B A (15) E C D A B (10) E B C D A (8) A D C B E (7) A B D C E (7) B E A C D (4) B A E D C (4) E C B D A (3) E A B C D (2) D C A B E (2) C D E A B (2) B E D C A (2) B E C D A (2) B D A C E (2) A B E D C (2) E A C D B (1) D C A E B (1) C D E B A (1) C D A E B (1) B A E C D (1) B A D E C (1) A D C E B (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -24 6 2 0 B 24 0 10 14 8 C -6 -10 0 2 -10 D -2 -14 -2 0 -8 E 0 -8 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 6 2 0 B 24 0 10 14 8 C -6 -10 0 2 -10 D -2 -14 -2 0 -8 E 0 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=35 A=19 C=4 D=3 so D is eliminated. Round 2 votes counts: E=39 B=35 A=19 C=7 so C is eliminated. Round 3 votes counts: E=42 B=35 A=23 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:228 E:205 A:192 C:188 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 6 2 0 B 24 0 10 14 8 C -6 -10 0 2 -10 D -2 -14 -2 0 -8 E 0 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 6 2 0 B 24 0 10 14 8 C -6 -10 0 2 -10 D -2 -14 -2 0 -8 E 0 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 6 2 0 B 24 0 10 14 8 C -6 -10 0 2 -10 D -2 -14 -2 0 -8 E 0 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1283: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) C A D B E (7) C D A B E (6) A C D B E (6) E B D C A (5) E B A D C (5) D B E C A (5) B D E C A (5) C A E D B (4) E D B C A (3) E A B C D (3) D B C A E (3) C A D E B (3) B D E A C (3) A C E D B (3) A C E B D (3) E D C B A (2) D C B A E (2) D B C E A (2) B E D A C (2) A C D E B (2) A B E C D (2) E C A D B (1) E A C D B (1) E A C B D (1) D C E B A (1) D C B E A (1) C D E B A (1) C D E A B (1) B A E D C (1) B A D E C (1) A E C D B (1) A E C B D (1) A E B C D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -4 -6 -6 B 6 0 4 -10 -6 C 4 -4 0 -4 -8 D 6 10 4 0 0 E 6 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.397810 E: 0.602190 Sum of squares = 0.520885703844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.397810 E: 1.000000 A B C D E A 0 -6 -4 -6 -6 B 6 0 4 -10 -6 C 4 -4 0 -4 -8 D 6 10 4 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=22 A=20 D=14 B=12 so B is eliminated. Round 2 votes counts: E=34 D=22 C=22 A=22 so D is eliminated. Round 3 votes counts: E=47 C=31 A=22 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:210 B:197 C:194 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -6 -6 B 6 0 4 -10 -6 C 4 -4 0 -4 -8 D 6 10 4 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -6 -6 B 6 0 4 -10 -6 C 4 -4 0 -4 -8 D 6 10 4 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -6 -6 B 6 0 4 -10 -6 C 4 -4 0 -4 -8 D 6 10 4 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1284: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (14) A C D B E (14) C A B D E (9) D B A C E (6) E C A B D (5) B D C A E (5) E D B A C (4) E C B D A (4) E A C B D (4) C A E B D (4) B D E C A (4) E A C D B (3) A C E D B (3) E D B C A (2) E A D B C (2) D B C A E (2) D B A E C (2) C B D A E (2) B D C E A (2) A C B D E (2) E A D C B (1) D B E A C (1) D A B C E (1) C E B A D (1) B C D A E (1) A E C D B (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -10 0 6 B 0 0 -6 14 4 C 10 6 0 6 6 D 0 -14 -6 0 4 E -6 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 0 6 B 0 0 -6 14 4 C 10 6 0 6 6 D 0 -14 -6 0 4 E -6 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=21 C=16 D=12 B=12 so D is eliminated. Round 2 votes counts: E=39 B=23 A=22 C=16 so C is eliminated. Round 3 votes counts: E=40 A=35 B=25 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:214 B:206 A:198 D:192 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 0 6 B 0 0 -6 14 4 C 10 6 0 6 6 D 0 -14 -6 0 4 E -6 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 0 6 B 0 0 -6 14 4 C 10 6 0 6 6 D 0 -14 -6 0 4 E -6 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 0 6 B 0 0 -6 14 4 C 10 6 0 6 6 D 0 -14 -6 0 4 E -6 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1285: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (12) B A E C D (8) A B E D C (7) B E A C D (5) B C E D A (5) C E D B A (4) C D E A B (4) E C D A B (3) D A C E B (3) C E D A B (3) C B E D A (3) A E D C B (3) E B A C D (2) D C B A E (2) C D E B A (2) B A D E C (2) A E D B C (2) A E B C D (2) A D B E C (2) A B E C D (2) A B D E C (2) E D C A B (1) E D A C B (1) E C D B A (1) E C A D B (1) E B C A D (1) E A C B D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C A E B (1) D A E C B (1) C E B D A (1) C B D E A (1) B E C A D (1) B D A C E (1) B C E A D (1) B C D A E (1) B A E D C (1) B A C E D (1) B A C D E (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 12 2 -4 -12 B -12 0 -2 0 -8 C -2 2 0 10 -8 D 4 0 -10 0 -22 E 12 8 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 -4 -12 B -12 0 -2 0 -8 C -2 2 0 10 -8 D 4 0 -10 0 -22 E 12 8 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=22 D=21 C=18 E=12 so E is eliminated. Round 2 votes counts: B=30 A=24 D=23 C=23 so D is eliminated. Round 3 votes counts: C=40 B=30 A=30 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:225 C:201 A:199 B:189 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 -4 -12 B -12 0 -2 0 -8 C -2 2 0 10 -8 D 4 0 -10 0 -22 E 12 8 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -4 -12 B -12 0 -2 0 -8 C -2 2 0 10 -8 D 4 0 -10 0 -22 E 12 8 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -4 -12 B -12 0 -2 0 -8 C -2 2 0 10 -8 D 4 0 -10 0 -22 E 12 8 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1286: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) A D B C E (7) E D B C A (6) C B E A D (5) B C E A D (5) B C A E D (5) A B C D E (5) E C B D A (4) B E C D A (4) B D A E C (4) E B C D A (3) D E A B C (3) D B E A C (3) D A B E C (3) A B D C E (3) E D C A B (2) E C D B A (2) D A E C B (2) C B A E D (2) A C D B E (2) A C B D E (2) E D A C B (1) E C D A B (1) D B A E C (1) D A E B C (1) C E B D A (1) C E B A D (1) C E A D B (1) C A B E D (1) B D E C A (1) B D C A E (1) B C E D A (1) B A D C E (1) B A C D E (1) A D E C B (1) A D B E C (1) A C E D B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 8 -8 -8 B 2 0 20 2 18 C -8 -20 0 -2 -6 D 8 -2 2 0 6 E 8 -18 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -8 -8 B 2 0 20 2 18 C -8 -20 0 -2 -6 D 8 -2 2 0 6 E 8 -18 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998529 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 B=23 E=19 C=11 so C is eliminated. Round 2 votes counts: B=30 A=25 D=23 E=22 so E is eliminated. Round 3 votes counts: B=39 D=35 A=26 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:207 A:195 E:195 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 -8 -8 B 2 0 20 2 18 C -8 -20 0 -2 -6 D 8 -2 2 0 6 E 8 -18 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998529 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -8 -8 B 2 0 20 2 18 C -8 -20 0 -2 -6 D 8 -2 2 0 6 E 8 -18 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998529 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -8 -8 B 2 0 20 2 18 C -8 -20 0 -2 -6 D 8 -2 2 0 6 E 8 -18 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998529 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1287: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) C B A E D (9) C A B E D (7) A B E D C (7) D C A B E (6) E B A D C (5) D E B A C (5) C D E B A (4) C D A B E (4) D C E B A (3) D A E B C (3) C B E A D (3) C A B D E (3) A B E C D (3) D E C B A (2) D C E A B (2) D C A E B (2) D A B E C (2) C E B A D (2) B E A C D (2) B A E C D (2) A B C E D (2) E A D B C (1) E A B D C (1) D A C E B (1) D A C B E (1) C E B D A (1) C D B A E (1) C B A D E (1) B C A E D (1) B A E D C (1) B A C E D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -2 6 18 B -14 0 -4 4 14 C 2 4 0 -8 8 D -6 -4 8 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000055 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 6 18 B -14 0 -4 4 14 C 2 4 0 -8 8 D -6 -4 8 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=35 A=14 E=7 B=7 so E is eliminated. Round 2 votes counts: D=37 C=35 A=16 B=12 so B is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:218 C:203 D:201 B:200 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 -2 6 18 B -14 0 -4 4 14 C 2 4 0 -8 8 D -6 -4 8 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 6 18 B -14 0 -4 4 14 C 2 4 0 -8 8 D -6 -4 8 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 6 18 B -14 0 -4 4 14 C 2 4 0 -8 8 D -6 -4 8 0 4 E -18 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1288: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) B E D A C (8) E D B C A (7) D E C B A (7) E D C B A (6) C D A E B (6) D E B C A (5) D C E A B (5) B A E C D (4) A B C E D (4) E D C A B (3) E B D C A (3) A C D E B (3) A C B E D (3) C D E A B (2) C A E D B (2) C A D E B (2) C A D B E (2) B E D C A (2) A E B C D (2) A C E B D (2) E B D A C (1) D C E B A (1) D B E C A (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) B A C E D (1) B A C D E (1) A C E D B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -10 -20 -8 B 2 0 -12 -8 -16 C 10 12 0 -6 -6 D 20 8 6 0 0 E 8 16 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.435589 E: 0.564411 Sum of squares = 0.50829753657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.435589 E: 1.000000 A B C D E A 0 -2 -10 -20 -8 B 2 0 -12 -8 -16 C 10 12 0 -6 -6 D 20 8 6 0 0 E 8 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=20 B=20 D=19 C=14 so C is eliminated. Round 2 votes counts: A=33 D=27 E=20 B=20 so E is eliminated. Round 3 votes counts: D=43 A=33 B=24 so B is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:215 C:205 B:183 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -10 -20 -8 B 2 0 -12 -8 -16 C 10 12 0 -6 -6 D 20 8 6 0 0 E 8 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -20 -8 B 2 0 -12 -8 -16 C 10 12 0 -6 -6 D 20 8 6 0 0 E 8 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -20 -8 B 2 0 -12 -8 -16 C 10 12 0 -6 -6 D 20 8 6 0 0 E 8 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1289: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) A B E D C (9) B A C D E (7) C D E B A (6) C D E A B (6) E D C B A (5) B A E D C (5) A B C D E (5) D E C A B (4) C B D E A (4) A E D B C (4) A B D E C (4) B C A D E (3) B A C E D (3) A D E B C (3) E D B C A (2) E D A C B (2) E D A B C (2) C E D B A (2) C B E D A (2) B C E D A (2) B C A E D (2) E D B A C (1) E A D B C (1) D E A C B (1) C B A D E (1) C A B D E (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 0 2 0 B -10 0 6 -4 -4 C 0 -6 0 -10 -8 D -2 4 10 0 -4 E 0 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.405682 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.594318 Sum of squares = 0.517791789929 Cumulative probabilities = A: 0.405682 B: 0.405682 C: 0.405682 D: 0.405682 E: 1.000000 A B C D E A 0 10 0 2 0 B -10 0 6 -4 -4 C 0 -6 0 -10 -8 D -2 4 10 0 -4 E 0 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 C=22 B=22 D=5 so D is eliminated. Round 2 votes counts: E=28 A=28 C=22 B=22 so C is eliminated. Round 3 votes counts: E=42 B=29 A=29 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:206 D:204 B:194 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 2 0 B -10 0 6 -4 -4 C 0 -6 0 -10 -8 D -2 4 10 0 -4 E 0 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 2 0 B -10 0 6 -4 -4 C 0 -6 0 -10 -8 D -2 4 10 0 -4 E 0 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 2 0 B -10 0 6 -4 -4 C 0 -6 0 -10 -8 D -2 4 10 0 -4 E 0 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1290: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (5) E B A D C (5) A B E C D (5) A B D C E (5) D B C A E (4) D B A E C (4) D B A C E (4) C D A B E (4) C A D B E (4) B D A E C (4) E A B D C (3) D C B A E (3) C E D A B (3) A B E D C (3) E C D B A (2) E C A B D (2) E A C B D (2) D C B E A (2) D B E C A (2) D B E A C (2) C E D B A (2) C E A D B (2) C D A E B (2) C A E B D (2) B E A D C (2) A E B C D (2) A C B E D (2) A C B D E (2) A B C D E (2) E D B C A (1) E D B A C (1) E C A D B (1) E A B C D (1) D B C E A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D B A E (1) C A E D B (1) C A B E D (1) B D E A C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 14 0 14 B -6 0 16 2 20 C -14 -16 0 -4 4 D 0 -2 4 0 0 E -14 -20 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.604163 B: 0.000000 C: 0.000000 D: 0.395837 E: 0.000000 Sum of squares = 0.521699924991 Cumulative probabilities = A: 0.604163 B: 0.604163 C: 0.604163 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 0 14 B -6 0 16 2 20 C -14 -16 0 -4 4 D 0 -2 4 0 0 E -14 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.499999999041 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=23 A=23 D=22 B=7 so B is eliminated. Round 2 votes counts: D=27 E=25 C=25 A=23 so A is eliminated. Round 3 votes counts: E=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:217 B:216 D:201 C:185 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 0 14 B -6 0 16 2 20 C -14 -16 0 -4 4 D 0 -2 4 0 0 E -14 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.499999999041 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 0 14 B -6 0 16 2 20 C -14 -16 0 -4 4 D 0 -2 4 0 0 E -14 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.499999999041 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 0 14 B -6 0 16 2 20 C -14 -16 0 -4 4 D 0 -2 4 0 0 E -14 -20 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.000000 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.499999999041 Cumulative probabilities = A: 0.500002 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1291: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) E B A D C (6) C D B E A (6) A E B D C (6) D C B E A (5) E B C A D (4) C E B D A (4) A D C E B (4) E B C D A (3) E A B C D (3) D C B A E (3) D B A E C (3) D A B E C (3) C E B A D (3) C E A B D (3) C D A B E (3) B E A D C (3) E B A C D (2) D B E A C (2) D A B C E (2) C D B A E (2) C D A E B (2) B E D A C (2) B D E A C (2) A C D E B (2) E C B A D (1) E A B D C (1) D C A B E (1) D B E C A (1) C B E D A (1) C A E D B (1) B D E C A (1) B C E D A (1) A E D B C (1) A D E C B (1) A D B E C (1) A C E D B (1) A C E B D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -4 6 -8 B 10 0 -2 -2 -12 C 4 2 0 0 4 D -6 2 0 0 4 E 8 12 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.746600 D: 0.253400 E: 0.000000 Sum of squares = 0.621623400354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.746600 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 6 -8 B 10 0 -2 -2 -12 C 4 2 0 0 4 D -6 2 0 0 4 E 8 12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000025455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=20 D=20 A=19 B=9 so B is eliminated. Round 2 votes counts: C=33 E=25 D=23 A=19 so A is eliminated. Round 3 votes counts: C=37 E=33 D=30 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:206 C:205 D:200 B:197 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 6 -8 B 10 0 -2 -2 -12 C 4 2 0 0 4 D -6 2 0 0 4 E 8 12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000025455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 6 -8 B 10 0 -2 -2 -12 C 4 2 0 0 4 D -6 2 0 0 4 E 8 12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000025455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 6 -8 B 10 0 -2 -2 -12 C 4 2 0 0 4 D -6 2 0 0 4 E 8 12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000025455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1292: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) E A C B D (8) E A D C B (7) A E D C B (5) E D A B C (4) E A D B C (4) D B C A E (4) C B A E D (4) B D C E A (4) B C D A E (4) E A C D B (3) B C D E A (3) E D B C A (2) D E B C A (2) D C A B E (2) D A E C B (2) D A C B E (2) B E C A D (2) B D C A E (2) B C E D A (2) A E C D B (2) A D E C B (2) A C E B D (2) A C B E D (2) E B D C A (1) E B C A D (1) D E B A C (1) D E A B C (1) D C B A E (1) D B E C A (1) D B C E A (1) D A B C E (1) C E B A D (1) C D B A E (1) C B E A D (1) C B D A E (1) C B A D E (1) C A B E D (1) B E C D A (1) B C E A D (1) Total count = 100 A B C D E A 0 16 12 14 -2 B -16 0 -16 4 -16 C -12 16 0 2 -18 D -14 -4 -2 0 -28 E 2 16 18 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 12 14 -2 B -16 0 -16 4 -16 C -12 16 0 2 -18 D -14 -4 -2 0 -28 E 2 16 18 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=23 B=19 D=18 C=10 so C is eliminated. Round 2 votes counts: E=31 B=26 A=24 D=19 so D is eliminated. Round 3 votes counts: E=35 B=34 A=31 so A is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:220 C:194 B:178 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 12 14 -2 B -16 0 -16 4 -16 C -12 16 0 2 -18 D -14 -4 -2 0 -28 E 2 16 18 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 14 -2 B -16 0 -16 4 -16 C -12 16 0 2 -18 D -14 -4 -2 0 -28 E 2 16 18 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 14 -2 B -16 0 -16 4 -16 C -12 16 0 2 -18 D -14 -4 -2 0 -28 E 2 16 18 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1293: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) C E D B A (9) D B A C E (7) D B E C A (6) C E A D B (6) A E B C D (6) A B D E C (6) A B D C E (5) D B C E A (4) C E D A B (4) B D A E C (4) A C E B D (4) C A E D B (3) B A D E C (3) E C D B A (2) D C E B A (2) D B A E C (2) C E A B D (2) C D E B A (2) A B E C D (2) E C A D B (1) E B C D A (1) E A C B D (1) D E B C A (1) D B E A C (1) C D A E B (1) C A E B D (1) B A D C E (1) A D B C E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 8 16 B -10 0 2 -6 -14 C -10 -2 0 14 8 D -8 6 -14 0 -6 E -16 14 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 8 16 B -10 0 2 -6 -14 C -10 -2 0 14 8 D -8 6 -14 0 -6 E -16 14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=28 D=23 B=8 E=5 so E is eliminated. Round 2 votes counts: A=37 C=31 D=23 B=9 so B is eliminated. Round 3 votes counts: A=41 C=32 D=27 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:205 E:198 D:189 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 8 16 B -10 0 2 -6 -14 C -10 -2 0 14 8 D -8 6 -14 0 -6 E -16 14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 8 16 B -10 0 2 -6 -14 C -10 -2 0 14 8 D -8 6 -14 0 -6 E -16 14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 8 16 B -10 0 2 -6 -14 C -10 -2 0 14 8 D -8 6 -14 0 -6 E -16 14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1294: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (6) E D C B A (5) D C E B A (5) D A C B E (5) B E C A D (5) A D C B E (5) E B C D A (4) A B C E D (4) E C B D A (3) E B C A D (3) D C A B E (3) C D B A E (3) A E D B C (3) A C D B E (3) A B C D E (3) E D A C B (2) D E A C B (2) D C E A B (2) D C B E A (2) D A C E B (2) C B D A E (2) B C E A D (2) A E B C D (2) A D E C B (2) E D B C A (1) E C D B A (1) E B D C A (1) E B A C D (1) E A D B C (1) E A B C D (1) D C B A E (1) D C A E B (1) D A E C B (1) C E B D A (1) C B E D A (1) C B D E A (1) C A B D E (1) B E A C D (1) B C E D A (1) B C A D E (1) B A E C D (1) B A C D E (1) A D E B C (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 0 0 8 B 6 0 -10 -2 10 C 0 10 0 8 16 D 0 2 -8 0 -2 E -8 -10 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.435037 B: 0.000000 C: 0.564963 D: 0.000000 E: 0.000000 Sum of squares = 0.508440410382 Cumulative probabilities = A: 0.435037 B: 0.435037 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 0 8 B 6 0 -10 -2 10 C 0 10 0 8 16 D 0 2 -8 0 -2 E -8 -10 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 E=23 B=18 C=9 so C is eliminated. Round 2 votes counts: D=27 A=27 E=24 B=22 so B is eliminated. Round 3 votes counts: A=36 E=34 D=30 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:217 B:202 A:201 D:196 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 0 8 B 6 0 -10 -2 10 C 0 10 0 8 16 D 0 2 -8 0 -2 E -8 -10 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 8 B 6 0 -10 -2 10 C 0 10 0 8 16 D 0 2 -8 0 -2 E -8 -10 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 8 B 6 0 -10 -2 10 C 0 10 0 8 16 D 0 2 -8 0 -2 E -8 -10 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1295: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) D E B A C (8) C A B E D (8) D E A B C (7) B C A E D (6) A C D E B (6) D E A C B (5) C A B D E (5) B E D C A (5) B C E A D (4) A D E C B (4) E B D C A (3) C B A D E (3) A D C E B (3) E D B C A (2) E D A B C (2) B E C D A (2) A C E D B (2) A C D B E (2) A C B E D (2) E D B A C (1) E B D A C (1) E B C D A (1) D B E C A (1) D A E C B (1) C A D B E (1) B E C A D (1) B C E D A (1) B C D E A (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -10 18 10 B 2 0 -8 10 8 C 10 8 0 14 12 D -18 -10 -14 0 -4 E -10 -8 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 18 10 B 2 0 -8 10 8 C 10 8 0 14 12 D -18 -10 -14 0 -4 E -10 -8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=22 B=20 A=20 E=10 so E is eliminated. Round 2 votes counts: C=28 D=27 B=25 A=20 so A is eliminated. Round 3 votes counts: C=41 D=34 B=25 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:208 B:206 E:187 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 18 10 B 2 0 -8 10 8 C 10 8 0 14 12 D -18 -10 -14 0 -4 E -10 -8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 18 10 B 2 0 -8 10 8 C 10 8 0 14 12 D -18 -10 -14 0 -4 E -10 -8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 18 10 B 2 0 -8 10 8 C 10 8 0 14 12 D -18 -10 -14 0 -4 E -10 -8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1296: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (7) B E D C A (5) E B C A D (4) D A E C B (4) B E C D A (4) B C E D A (4) A D C E B (4) A C E D B (4) A C E B D (4) E B C D A (3) D E B A C (3) D B E C A (3) D B E A C (3) D B C A E (3) D B A C E (3) C B E A D (3) C A E B D (3) E D A B C (2) E C A B D (2) E B D A C (2) E A C B D (2) D A C B E (2) C D A B E (2) C B D A E (2) B D C E A (2) A E C D B (2) A E C B D (2) A C D E B (2) E B D C A (1) E A D B C (1) E A B C D (1) D E A B C (1) D B C E A (1) D A E B C (1) C E B A D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C A D (1) B C E A D (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -10 -16 -14 B 12 0 12 12 -2 C 10 -12 0 0 -12 D 16 -12 0 0 -8 E 14 2 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -10 -16 -14 B 12 0 12 12 -2 C 10 -12 0 0 -12 D 16 -12 0 0 -8 E 14 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=24 B=24 A=20 E=18 C=14 so C is eliminated. Round 2 votes counts: B=29 D=26 A=26 E=19 so E is eliminated. Round 3 votes counts: B=40 A=32 D=28 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:217 D:198 C:193 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -10 -16 -14 B 12 0 12 12 -2 C 10 -12 0 0 -12 D 16 -12 0 0 -8 E 14 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -16 -14 B 12 0 12 12 -2 C 10 -12 0 0 -12 D 16 -12 0 0 -8 E 14 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -16 -14 B 12 0 12 12 -2 C 10 -12 0 0 -12 D 16 -12 0 0 -8 E 14 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1297: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) B E C A D (8) D C A E B (5) D C A B E (5) C D B A E (5) A D E C B (5) E B C D A (4) D A E C B (4) E B A C D (3) E A B D C (3) D E C A B (3) D C E A B (3) B C E D A (3) A E D B C (3) A D E B C (3) E D A C B (2) E C B D A (2) E B C A D (2) E A D B C (2) D C E B A (2) C D A B E (2) C B D A E (2) B C E A D (2) A D C B E (2) E D C A B (1) E D A B C (1) D E A C B (1) C D E B A (1) C D B E A (1) C B D E A (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A C D (1) B C A E D (1) B A E C D (1) A E B D C (1) A B E D C (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 20 -12 -12 8 B -20 0 -16 -20 -18 C 12 16 0 -14 -6 D 12 20 14 0 16 E -8 18 6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -12 -12 8 B -20 0 -16 -20 -18 C 12 16 0 -14 -6 D 12 20 14 0 16 E -8 18 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=20 A=18 B=16 C=15 so C is eliminated. Round 2 votes counts: D=40 E=20 B=20 A=20 so E is eliminated. Round 3 votes counts: D=44 B=31 A=25 so A is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:231 C:204 A:202 E:200 B:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 -12 -12 8 B -20 0 -16 -20 -18 C 12 16 0 -14 -6 D 12 20 14 0 16 E -8 18 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -12 -12 8 B -20 0 -16 -20 -18 C 12 16 0 -14 -6 D 12 20 14 0 16 E -8 18 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -12 -12 8 B -20 0 -16 -20 -18 C 12 16 0 -14 -6 D 12 20 14 0 16 E -8 18 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1298: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (15) A E D C B (11) E A D C B (5) C B E D A (5) B D C A E (5) A E B C D (5) C B D E A (4) E D C A B (3) E D A C B (3) B D A C E (3) E C D A B (2) E A C D B (2) D C E B A (2) D B C A E (2) D A E C B (2) C B E A D (2) A E D B C (2) A E C D B (2) A E B D C (2) A D E B C (2) A D B E C (2) E D C B A (1) E C B A D (1) E C A D B (1) E C A B D (1) D E C B A (1) D C B E A (1) D B C E A (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B D A (1) B C E A D (1) B C D A E (1) B C A E D (1) B A D C E (1) B A C D E (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 -10 -4 B -2 0 0 4 0 C 2 0 0 -6 -2 D 10 -4 6 0 -6 E 4 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.374850 C: 0.000000 D: 0.000000 E: 0.625150 Sum of squares = 0.531325014426 Cumulative probabilities = A: 0.000000 B: 0.374850 C: 0.374850 D: 0.374850 E: 1.000000 A B C D E A 0 2 -2 -10 -4 B -2 0 0 4 0 C 2 0 0 -6 -2 D 10 -4 6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 E=19 D=12 C=12 so D is eliminated. Round 2 votes counts: A=34 B=31 E=20 C=15 so C is eliminated. Round 3 votes counts: B=43 A=34 E=23 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:203 B:201 C:197 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 -10 -4 B -2 0 0 4 0 C 2 0 0 -6 -2 D 10 -4 6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -10 -4 B -2 0 0 4 0 C 2 0 0 -6 -2 D 10 -4 6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -10 -4 B -2 0 0 4 0 C 2 0 0 -6 -2 D 10 -4 6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1299: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (15) B C A E D (11) C B A D E (5) C A D E B (5) B C D A E (5) A E D C B (5) E D A C B (4) B E D A C (4) B E A D C (4) A E C D B (4) E A D C B (3) C D A E B (3) C B D A E (3) B C D E A (3) B C A D E (3) E A D B C (2) D E A B C (2) D C A E B (2) D A E C B (2) B D C E A (2) E D B A C (1) E D A B C (1) D E B A C (1) D C E A B (1) C B A E D (1) C A E D B (1) B E D C A (1) B E C A D (1) B D E C A (1) B C E A D (1) B A E C D (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 4 2 -2 6 B -4 0 -10 -6 -8 C -2 10 0 -4 -6 D 2 6 4 0 8 E -6 8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -2 6 B -4 0 -10 -6 -8 C -2 10 0 -4 -6 D 2 6 4 0 8 E -6 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=23 C=18 E=11 A=11 so E is eliminated. Round 2 votes counts: B=37 D=29 C=18 A=16 so A is eliminated. Round 3 votes counts: D=39 B=38 C=23 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:205 E:200 C:199 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -2 6 B -4 0 -10 -6 -8 C -2 10 0 -4 -6 D 2 6 4 0 8 E -6 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -2 6 B -4 0 -10 -6 -8 C -2 10 0 -4 -6 D 2 6 4 0 8 E -6 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -2 6 B -4 0 -10 -6 -8 C -2 10 0 -4 -6 D 2 6 4 0 8 E -6 8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1300: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (6) A C D B E (6) E B C A D (5) C D A E B (5) E B D A C (4) C E D B A (4) C E D A B (4) B E D A C (4) B E A D C (4) A B D C E (4) E B A C D (3) D C A E B (3) D A B C E (3) C E A B D (3) C D E A B (3) C A E D B (3) A D B C E (3) A B E C D (3) A B D E C (3) E C B A D (2) E B C D A (2) D B E A C (2) D A B E C (2) C E A D B (2) B D A E C (2) E C B D A (1) E B D C A (1) E B A D C (1) D C E B A (1) D C E A B (1) D C A B E (1) D B A E C (1) D A C B E (1) C E B D A (1) C A E B D (1) C A D E B (1) B E A C D (1) B A E C D (1) B A D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 18 12 8 2 B -18 0 2 -4 -2 C -12 -2 0 4 14 D -8 4 -4 0 0 E -2 2 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 12 8 2 B -18 0 2 -4 -2 C -12 -2 0 4 14 D -8 4 -4 0 0 E -2 2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 E=19 D=15 B=13 so B is eliminated. Round 2 votes counts: E=28 A=28 C=27 D=17 so D is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:202 D:196 E:193 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 12 8 2 B -18 0 2 -4 -2 C -12 -2 0 4 14 D -8 4 -4 0 0 E -2 2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 8 2 B -18 0 2 -4 -2 C -12 -2 0 4 14 D -8 4 -4 0 0 E -2 2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 8 2 B -18 0 2 -4 -2 C -12 -2 0 4 14 D -8 4 -4 0 0 E -2 2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1301: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) E A C B D (8) B A E D C (8) C D E A B (7) B D A E C (5) D C B E A (4) B D A C E (4) E A C D B (3) D B C E A (3) C E A D B (3) C D B A E (3) B E A D C (3) A E C B D (3) E A B C D (2) D C E B A (2) D C B A E (2) C D B E A (2) C D A E B (2) C A E D B (2) B A D E C (2) B A C E D (2) A E B D C (2) A C E B D (2) E D A C B (1) E C D A B (1) E A D B C (1) D B E A C (1) C E D A B (1) C D A B E (1) C B A D E (1) C A E B D (1) B E D A C (1) B D C A E (1) B C D A E (1) B A E C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 26 16 14 B -8 0 2 22 -6 C -26 -2 0 18 -12 D -16 -22 -18 0 -18 E -14 6 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 26 16 14 B -8 0 2 22 -6 C -26 -2 0 18 -12 D -16 -22 -18 0 -18 E -14 6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 A=21 E=16 D=12 so D is eliminated. Round 2 votes counts: B=32 C=31 A=21 E=16 so E is eliminated. Round 3 votes counts: A=36 C=32 B=32 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:232 E:211 B:205 C:189 D:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 26 16 14 B -8 0 2 22 -6 C -26 -2 0 18 -12 D -16 -22 -18 0 -18 E -14 6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 26 16 14 B -8 0 2 22 -6 C -26 -2 0 18 -12 D -16 -22 -18 0 -18 E -14 6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 26 16 14 B -8 0 2 22 -6 C -26 -2 0 18 -12 D -16 -22 -18 0 -18 E -14 6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1302: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (16) D B A C E (15) C E A D B (12) A B D E C (11) B D A E C (8) C E D B A (6) C D B E A (5) B D A C E (3) A D B E C (3) E C B A D (2) D B C A E (2) C E B D A (2) B A D E C (2) E C B D A (1) E C A D B (1) E A C B D (1) E A B D C (1) D C B A E (1) D B A E C (1) C E B A D (1) C E A B D (1) C D E B A (1) B D C E A (1) A E C B D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -4 8 -2 B 2 0 -2 6 6 C 4 2 0 0 0 D -8 -6 0 0 6 E 2 -6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.819082 D: 0.180918 E: 0.000000 Sum of squares = 0.703627022273 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.819082 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 8 -2 B 2 0 -2 6 6 C 4 2 0 0 0 D -8 -6 0 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000080947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=22 D=19 A=17 B=14 so B is eliminated. Round 2 votes counts: D=31 C=28 E=22 A=19 so A is eliminated. Round 3 votes counts: D=47 C=28 E=25 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:206 C:203 A:200 D:196 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 8 -2 B 2 0 -2 6 6 C 4 2 0 0 0 D -8 -6 0 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000080947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 -2 B 2 0 -2 6 6 C 4 2 0 0 0 D -8 -6 0 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000080947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 -2 B 2 0 -2 6 6 C 4 2 0 0 0 D -8 -6 0 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000080947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1303: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) E C A B D (9) D B A C E (8) D B E A C (7) B D A C E (6) A C B D E (5) E D B C A (4) E C A D B (4) E B D A C (4) C E A D B (4) D B C A E (3) D B A E C (3) C A E D B (3) B A D C E (3) E C D A B (2) E B A D C (2) E A C B D (2) D B E C A (2) A C B E D (2) A B C D E (2) E D C B A (1) E C D B A (1) E B D C A (1) D E B C A (1) D E B A C (1) D C B A E (1) D B C E A (1) C E A B D (1) C D A B E (1) C A D B E (1) B D A E C (1) A D B C E (1) A C E B D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 4 6 B 0 0 2 0 -2 C 0 -2 0 -2 8 D -4 0 2 0 -4 E -6 2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.464363 B: 0.535637 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50254003398 Cumulative probabilities = A: 0.464363 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 6 B 0 0 2 0 -2 C 0 -2 0 -2 8 D -4 0 2 0 -4 E -6 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 C=20 A=13 B=10 so B is eliminated. Round 2 votes counts: D=34 E=30 C=20 A=16 so A is eliminated. Round 3 votes counts: D=38 E=31 C=31 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:205 C:202 B:200 D:197 E:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 6 B 0 0 2 0 -2 C 0 -2 0 -2 8 D -4 0 2 0 -4 E -6 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 6 B 0 0 2 0 -2 C 0 -2 0 -2 8 D -4 0 2 0 -4 E -6 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 6 B 0 0 2 0 -2 C 0 -2 0 -2 8 D -4 0 2 0 -4 E -6 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1304: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) E B C D A (9) B E A D C (8) D C A B E (7) E B A C D (6) C D E A B (6) A D C B E (6) A D B C E (6) E C B D A (4) E B C A D (4) A B D E C (4) D B A E C (3) B A D E C (3) A C D B E (3) D A C B E (2) C D E B A (2) B E D C A (2) A E B C D (2) A B E C D (2) E C D B A (1) E B D C A (1) E B A D C (1) D C B E A (1) C A D E B (1) B E A C D (1) B A E D C (1) Total count = 100 A B C D E A 0 6 -4 -4 8 B -6 0 6 -4 -2 C 4 -6 0 10 -4 D 4 4 -10 0 16 E -8 2 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999854 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -4 8 B -6 0 6 -4 -2 C 4 -6 0 10 -4 D 4 4 -10 0 16 E -8 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=23 A=23 B=15 D=13 so D is eliminated. Round 2 votes counts: C=31 E=26 A=25 B=18 so B is eliminated. Round 3 votes counts: E=37 A=32 C=31 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:207 A:203 C:202 B:197 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -4 -4 8 B -6 0 6 -4 -2 C 4 -6 0 10 -4 D 4 4 -10 0 16 E -8 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 8 B -6 0 6 -4 -2 C 4 -6 0 10 -4 D 4 4 -10 0 16 E -8 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 8 B -6 0 6 -4 -2 C 4 -6 0 10 -4 D 4 4 -10 0 16 E -8 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1305: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) E C A B D (8) D B A C E (8) C E A D B (8) B D A E C (6) D B A E C (5) C E B D A (5) C E A B D (5) A B D E C (4) D B C A E (3) C E D B A (3) C D B E A (3) C D B A E (3) E A C D B (2) E A B D C (2) C D A B E (2) B D C E A (2) B D A C E (2) A E D B C (2) E B D C A (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B A E (1) D C A B E (1) D A B E C (1) D A B C E (1) C D E B A (1) B E D A C (1) B D E A C (1) A E C D B (1) A E B D C (1) A D C B E (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 8 -2 8 B -6 0 8 -14 14 C -8 -8 0 -12 0 D 2 14 12 0 12 E -8 -14 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -2 8 B -6 0 8 -14 14 C -8 -8 0 -12 0 D 2 14 12 0 12 E -8 -14 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=21 D=20 E=17 B=12 so B is eliminated. Round 2 votes counts: D=31 C=30 A=21 E=18 so E is eliminated. Round 3 votes counts: C=38 D=34 A=28 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:210 B:201 C:186 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -2 8 B -6 0 8 -14 14 C -8 -8 0 -12 0 D 2 14 12 0 12 E -8 -14 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -2 8 B -6 0 8 -14 14 C -8 -8 0 -12 0 D 2 14 12 0 12 E -8 -14 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -2 8 B -6 0 8 -14 14 C -8 -8 0 -12 0 D 2 14 12 0 12 E -8 -14 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1306: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (14) E A D B C (10) C B D A E (10) C B D E A (9) C E A D B (7) C E B D A (6) A E D B C (6) C E A B D (5) B D C A E (5) B D A C E (5) E C A D B (4) B D A E C (4) A D B E C (4) E A C D B (2) A D E B C (2) E C A B D (1) E B D A C (1) E A C B D (1) D B A C E (1) D A B E C (1) C D B A E (1) C B E D A (1) Total count = 100 A B C D E A 0 -14 2 -16 6 B 14 0 6 -4 10 C -2 -6 0 -6 0 D 16 4 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -16 6 B 14 0 6 -4 10 C -2 -6 0 -6 0 D 16 4 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=19 D=16 B=14 A=12 so A is eliminated. Round 2 votes counts: C=39 E=25 D=22 B=14 so B is eliminated. Round 3 votes counts: C=39 D=36 E=25 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:213 C:193 A:189 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 2 -16 6 B 14 0 6 -4 10 C -2 -6 0 -6 0 D 16 4 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -16 6 B 14 0 6 -4 10 C -2 -6 0 -6 0 D 16 4 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -16 6 B 14 0 6 -4 10 C -2 -6 0 -6 0 D 16 4 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1307: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) B C E A D (8) E B C A D (7) D A E C B (7) A D E B C (7) E C B D A (6) B E C A D (6) E A D B C (5) D A C E B (5) A D B E C (4) A D B C E (4) E D A C B (3) D A C B E (3) C E D A B (3) C B D A E (3) C E B D A (2) B E A D C (2) B C E D A (2) B A D E C (2) E B A D C (1) E A D C B (1) D E A C B (1) D C A B E (1) C E D B A (1) C D A B E (1) B C D A E (1) B C A E D (1) B C A D E (1) B A D C E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -4 4 -14 B 6 0 4 4 -2 C 4 -4 0 2 -6 D -4 -4 -2 0 -14 E 14 2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 4 -14 B 6 0 4 4 -2 C 4 -4 0 2 -6 D -4 -4 -2 0 -14 E 14 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977212 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 C=19 D=17 A=17 so D is eliminated. Round 2 votes counts: A=32 E=24 B=24 C=20 so C is eliminated. Round 3 votes counts: B=36 A=34 E=30 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:206 C:198 A:190 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 4 -14 B 6 0 4 4 -2 C 4 -4 0 2 -6 D -4 -4 -2 0 -14 E 14 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977212 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 4 -14 B 6 0 4 4 -2 C 4 -4 0 2 -6 D -4 -4 -2 0 -14 E 14 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977212 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 4 -14 B 6 0 4 4 -2 C 4 -4 0 2 -6 D -4 -4 -2 0 -14 E 14 2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977212 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1308: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (14) B D C A E (11) A C E D B (9) D B E A C (8) E A C B D (7) D B C A E (5) C A B D E (5) E A D B C (4) B D E A C (4) A E C D B (4) E B D A C (3) C A E B D (3) B E D A C (3) B D E C A (3) E D B A C (2) B C D A E (2) E D A B C (1) E A B D C (1) D E B A C (1) D B A E C (1) C E A B D (1) C B D A E (1) C B A D E (1) C A E D B (1) C A D E B (1) C A B E D (1) B D C E A (1) B C D E A (1) A D C E B (1) Total count = 100 A B C D E A 0 6 26 6 -8 B -6 0 2 -4 -6 C -26 -2 0 2 -12 D -6 4 -2 0 -8 E 8 6 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 26 6 -8 B -6 0 2 -4 -6 C -26 -2 0 2 -12 D -6 4 -2 0 -8 E 8 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=25 D=15 C=14 A=14 so C is eliminated. Round 2 votes counts: E=33 B=27 A=25 D=15 so D is eliminated. Round 3 votes counts: B=41 E=34 A=25 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:215 D:194 B:193 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 26 6 -8 B -6 0 2 -4 -6 C -26 -2 0 2 -12 D -6 4 -2 0 -8 E 8 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 26 6 -8 B -6 0 2 -4 -6 C -26 -2 0 2 -12 D -6 4 -2 0 -8 E 8 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 26 6 -8 B -6 0 2 -4 -6 C -26 -2 0 2 -12 D -6 4 -2 0 -8 E 8 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1309: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) C D B E A (8) C B D E A (7) D C B A E (6) E A B D C (5) B C D E A (5) D A E C B (4) C D B A E (4) C B E A D (4) A E D B C (4) E B A C D (3) D C A B E (3) B E A D C (3) A E D C B (3) E A C B D (2) D C A E B (2) D B E A C (2) D B C E A (2) C E A B D (2) C A E D B (2) B E A C D (2) B C E A D (2) A E C D B (2) A E B D C (2) A E B C D (2) A D E C B (2) D B E C A (1) D B C A E (1) D B A E C (1) C E B A D (1) C D A E B (1) C D A B E (1) B D C E A (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 -6 2 -18 B 6 0 -10 2 6 C 6 10 0 16 4 D -2 -2 -16 0 2 E 18 -6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 2 -18 B 6 0 -10 2 6 C 6 10 0 16 4 D -2 -2 -16 0 2 E 18 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 E=19 A=16 B=13 so B is eliminated. Round 2 votes counts: C=37 E=24 D=23 A=16 so A is eliminated. Round 3 votes counts: E=38 C=37 D=25 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:203 B:202 D:191 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 2 -18 B 6 0 -10 2 6 C 6 10 0 16 4 D -2 -2 -16 0 2 E 18 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 2 -18 B 6 0 -10 2 6 C 6 10 0 16 4 D -2 -2 -16 0 2 E 18 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 2 -18 B 6 0 -10 2 6 C 6 10 0 16 4 D -2 -2 -16 0 2 E 18 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1310: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) D B C E A (5) C D E B A (5) C A E D B (5) E D B C A (4) D E B C A (4) A C E B D (4) A B E D C (4) E D B A C (3) E A D B C (3) C E D B A (3) C E A D B (3) C A D B E (3) B D E A C (3) B A D E C (3) A E C B D (3) A E B D C (3) A C B E D (3) E C A D B (2) D B E C A (2) C A D E B (2) C A B D E (2) B D C E A (2) B D C A E (2) B D A E C (2) B D A C E (2) A C E D B (2) A C B D E (2) E D C A B (1) E A B D C (1) D C B E A (1) C E D A B (1) C D B A E (1) B E D A C (1) B D E C A (1) B A D C E (1) A E B C D (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -10 2 -2 B 2 0 2 -12 0 C 10 -2 0 0 14 D -2 12 0 0 4 E 2 0 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.489844 D: 0.510156 E: 0.000000 Sum of squares = 0.500206281657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.489844 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 2 -2 B 2 0 2 -12 0 C 10 -2 0 0 14 D -2 12 0 0 4 E 2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=26 B=17 E=14 D=12 so D is eliminated. Round 2 votes counts: C=32 A=26 B=24 E=18 so E is eliminated. Round 3 votes counts: C=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:211 D:207 B:196 A:194 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 2 -2 B 2 0 2 -12 0 C 10 -2 0 0 14 D -2 12 0 0 4 E 2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 2 -2 B 2 0 2 -12 0 C 10 -2 0 0 14 D -2 12 0 0 4 E 2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 2 -2 B 2 0 2 -12 0 C 10 -2 0 0 14 D -2 12 0 0 4 E 2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1311: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (11) B D A E C (9) A D B C E (9) C E A D B (8) E C D A B (5) B E D A C (5) E C B D A (4) D A B E C (4) C E A B D (4) E C B A D (3) E B C D A (3) C A B D E (3) E C D B A (2) E B D A C (2) D A B C E (2) B E C A D (2) B E A D C (2) B A C D E (2) E D B A C (1) E D A C B (1) E B D C A (1) D B E A C (1) D B A E C (1) D A E C B (1) D A C E B (1) C D E A B (1) C D A E B (1) C B E A D (1) C B A E D (1) C B A D E (1) C A E D B (1) C A D E B (1) B D E A C (1) B C A D E (1) B A D E C (1) A D C B E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 14 8 6 B 10 0 18 16 18 C -14 -18 0 -10 2 D -8 -16 10 0 8 E -6 -18 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 8 6 B 10 0 18 16 18 C -14 -18 0 -10 2 D -8 -16 10 0 8 E -6 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=22 C=22 A=12 D=10 so D is eliminated. Round 2 votes counts: B=36 E=22 C=22 A=20 so A is eliminated. Round 3 votes counts: B=52 C=25 E=23 so E is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:209 D:197 E:183 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 8 6 B 10 0 18 16 18 C -14 -18 0 -10 2 D -8 -16 10 0 8 E -6 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 8 6 B 10 0 18 16 18 C -14 -18 0 -10 2 D -8 -16 10 0 8 E -6 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 8 6 B 10 0 18 16 18 C -14 -18 0 -10 2 D -8 -16 10 0 8 E -6 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1312: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) B A C D E (9) E D C A B (7) E D B C A (6) D B E A C (6) D E B C A (5) D E B A C (5) C A E D B (5) C A E B D (5) E D C B A (3) E C A D B (3) C A B E D (3) E C D A B (2) E B D C A (2) D E A B C (2) B E D C A (2) B E C D A (2) B D E C A (2) B D A E C (2) B C A E D (2) B A D C E (2) E C B D A (1) E C A B D (1) D E A C B (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A B D (1) B D E A C (1) A C E D B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -4 -8 -4 B 4 0 2 -4 -6 C 4 -2 0 -2 -10 D 8 4 2 0 6 E 4 6 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -8 -4 B 4 0 2 -4 -6 C 4 -2 0 -2 -10 D 8 4 2 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=24 B=22 A=15 C=14 so C is eliminated. Round 2 votes counts: A=28 E=26 D=24 B=22 so B is eliminated. Round 3 votes counts: A=41 E=30 D=29 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:207 B:198 C:195 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -8 -4 B 4 0 2 -4 -6 C 4 -2 0 -2 -10 D 8 4 2 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -8 -4 B 4 0 2 -4 -6 C 4 -2 0 -2 -10 D 8 4 2 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -8 -4 B 4 0 2 -4 -6 C 4 -2 0 -2 -10 D 8 4 2 0 6 E 4 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1313: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) A D C E B (9) C E A B D (8) D B A E C (7) C A E B D (7) B D E C A (7) B E D C A (6) C E B A D (5) B E C D A (4) B D E A C (4) A C D E B (4) D B E A C (3) C E B D A (3) A D C B E (3) A C E D B (3) D A B C E (2) C B E D A (2) C A E D B (2) A D E B C (2) E C B D A (1) E B C D A (1) D B C A E (1) D A E B C (1) D A C B E (1) C B E A D (1) C A D B E (1) C A B E D (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 8 -2 -6 8 B -8 0 -4 2 6 C 2 4 0 -12 8 D 6 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -6 8 B -8 0 -4 2 6 C 2 4 0 -12 8 D 6 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.40625000007 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=24 B=22 A=22 E=2 so E is eliminated. Round 2 votes counts: C=31 D=24 B=23 A=22 so A is eliminated. Round 3 votes counts: D=39 C=38 B=23 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:204 C:201 B:198 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -2 -6 8 B -8 0 -4 2 6 C 2 4 0 -12 8 D 6 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.40625000007 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -6 8 B -8 0 -4 2 6 C 2 4 0 -12 8 D 6 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.40625000007 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -6 8 B -8 0 -4 2 6 C 2 4 0 -12 8 D 6 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.40625000007 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1314: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) D A E C B (6) D A C E B (5) C B E A D (5) C B D E A (5) E B A C D (4) C B E D A (4) C B D A E (4) B E C A D (4) A D B E C (4) D E A C B (3) D A C B E (3) B C A D E (3) A D E B C (3) A B E D C (3) E B C A D (2) E B A D C (2) D A E B C (2) C D B A E (2) C B A D E (2) B E A C D (2) B C A E D (2) B A E C D (2) A E D B C (2) A B D E C (2) E D C A B (1) E B C D A (1) E A D B C (1) E A B D C (1) D E A B C (1) D C A E B (1) C E D B A (1) C E B D A (1) C D E B A (1) C D A B E (1) B A C E D (1) A E B D C (1) A D C B E (1) A D B C E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 4 16 6 B 12 0 8 22 22 C -4 -8 0 14 4 D -16 -22 -14 0 2 E -6 -22 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 16 6 B 12 0 8 22 22 C -4 -8 0 14 4 D -16 -22 -14 0 2 E -6 -22 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=22 D=21 A=19 E=12 so E is eliminated. Round 2 votes counts: B=31 C=26 D=22 A=21 so A is eliminated. Round 3 votes counts: B=40 D=34 C=26 so C is eliminated. Round 4 votes counts: B=61 D=39 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:232 A:207 C:203 E:183 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 16 6 B 12 0 8 22 22 C -4 -8 0 14 4 D -16 -22 -14 0 2 E -6 -22 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 16 6 B 12 0 8 22 22 C -4 -8 0 14 4 D -16 -22 -14 0 2 E -6 -22 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 16 6 B 12 0 8 22 22 C -4 -8 0 14 4 D -16 -22 -14 0 2 E -6 -22 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1315: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D A C B E (7) E B C A D (6) B E C D A (6) B E C A D (5) A D E C B (5) A C D E B (5) D C A B E (4) B C D E A (4) D A C E B (3) C A D E B (3) B D C E A (3) B C E D A (3) A E C D B (3) A D C E B (3) E C A B D (2) E A B D C (2) E A B C D (2) D C B A E (2) D B A E C (2) C D A B E (2) B E D A C (2) B E A D C (2) A E D C B (2) A D E B C (2) D B E A C (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B C E (1) C D B E A (1) C D B A E (1) C B E A D (1) B E D C A (1) B D E A C (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 10 6 0 B 2 0 10 -4 2 C -10 -10 0 6 -8 D -6 4 -6 0 6 E 0 -2 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 6 0 B 2 0 10 -4 2 C -10 -10 0 6 -8 D -6 4 -6 0 6 E 0 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888874 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 A=22 E=20 C=8 so C is eliminated. Round 2 votes counts: B=28 D=27 A=25 E=20 so E is eliminated. Round 3 votes counts: B=42 A=31 D=27 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:207 B:205 E:200 D:199 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 10 6 0 B 2 0 10 -4 2 C -10 -10 0 6 -8 D -6 4 -6 0 6 E 0 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888874 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 6 0 B 2 0 10 -4 2 C -10 -10 0 6 -8 D -6 4 -6 0 6 E 0 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888874 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 6 0 B 2 0 10 -4 2 C -10 -10 0 6 -8 D -6 4 -6 0 6 E 0 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888874 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1316: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) C A E B D (9) D B E A C (8) B D C A E (7) E D B A C (5) E D A B C (5) D B C E A (5) C A E D B (5) E A D C B (4) B D E A C (4) A C E B D (4) D B E C A (3) B D A E C (3) E B D A C (2) E A C B D (2) C D B A E (2) C B D A E (2) C B A D E (2) C A B E D (2) E D A C B (1) E C D A B (1) E C A D B (1) E B A D C (1) E A D B C (1) D E C B A (1) D E B A C (1) C E A D B (1) C A B D E (1) B D A C E (1) B C D A E (1) B C A D E (1) B A C D E (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 12 -4 -12 B 0 0 0 -10 -14 C -12 0 0 -4 -10 D 4 10 4 0 -14 E 12 14 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 12 -4 -12 B 0 0 0 -10 -14 C -12 0 0 -4 -10 D 4 10 4 0 -14 E 12 14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=24 D=18 B=18 A=7 so A is eliminated. Round 2 votes counts: E=35 C=29 D=18 B=18 so D is eliminated. Round 3 votes counts: E=37 B=34 C=29 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:202 A:198 B:188 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 -4 -12 B 0 0 0 -10 -14 C -12 0 0 -4 -10 D 4 10 4 0 -14 E 12 14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -4 -12 B 0 0 0 -10 -14 C -12 0 0 -4 -10 D 4 10 4 0 -14 E 12 14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -4 -12 B 0 0 0 -10 -14 C -12 0 0 -4 -10 D 4 10 4 0 -14 E 12 14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1317: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (14) D B C E A (13) B C E A D (9) A B C E D (8) D A B C E (7) D E C B A (5) A E C B D (5) A E B C D (5) D A B E C (4) B C E D A (4) D E C A B (3) C E B D A (2) C E B A D (2) C B E A D (2) B C A E D (2) A D E C B (2) A B E C D (2) E C B D A (1) E C B A D (1) E C A B D (1) E A D C B (1) E A C B D (1) D C E B A (1) D C B E A (1) D A E B C (1) B C D E A (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 4 -14 4 B -12 0 16 -6 10 C -4 -16 0 -8 6 D 14 6 8 0 8 E -4 -10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -14 4 B -12 0 16 -6 10 C -4 -16 0 -8 6 D 14 6 8 0 8 E -4 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=49 A=24 B=16 C=6 E=5 so E is eliminated. Round 2 votes counts: D=49 A=26 B=16 C=9 so C is eliminated. Round 3 votes counts: D=49 A=27 B=24 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:204 A:203 C:189 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 4 -14 4 B -12 0 16 -6 10 C -4 -16 0 -8 6 D 14 6 8 0 8 E -4 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -14 4 B -12 0 16 -6 10 C -4 -16 0 -8 6 D 14 6 8 0 8 E -4 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -14 4 B -12 0 16 -6 10 C -4 -16 0 -8 6 D 14 6 8 0 8 E -4 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1318: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) D A C B E (10) C E A D B (10) B D A E C (9) B E D A C (8) E C B A D (7) E B C D A (6) C A D E B (6) B E C D A (6) C E B A D (4) B D E A C (4) A D C E B (4) E B C A D (3) D A C E B (2) A D B E C (2) E C B D A (1) D B A E C (1) D A B E C (1) C E A B D (1) C D A E B (1) C A E D B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 2 8 -20 0 B -2 0 4 -2 8 C -8 -4 0 -8 4 D 20 2 8 0 6 E 0 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -20 0 B -2 0 4 -2 8 C -8 -4 0 -8 4 D 20 2 8 0 6 E 0 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 C=23 E=17 A=8 so A is eliminated. Round 2 votes counts: D=33 B=27 C=23 E=17 so E is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:204 A:195 C:192 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -20 0 B -2 0 4 -2 8 C -8 -4 0 -8 4 D 20 2 8 0 6 E 0 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -20 0 B -2 0 4 -2 8 C -8 -4 0 -8 4 D 20 2 8 0 6 E 0 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -20 0 B -2 0 4 -2 8 C -8 -4 0 -8 4 D 20 2 8 0 6 E 0 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1319: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) B E D A C (10) C A D E B (7) E A D B C (6) B C E D A (6) B C D A E (6) E D A B C (5) C A D B E (5) B E C D A (5) C B D A E (4) A D E C B (4) E A D C B (3) E D B A C (2) E B C A D (2) E B A D C (2) C B A D E (2) B D E A C (2) A D C E B (2) E C A D B (1) E A C D B (1) E A B C D (1) D E A B C (1) D A E B C (1) D A C E B (1) D A B E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C D A B E (1) C B E A D (1) C A E D B (1) C A B D E (1) B C E A D (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -8 8 -10 -24 B 8 0 24 10 -10 C -8 -24 0 -4 -18 D 10 -10 4 0 -22 E 24 10 18 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 8 -10 -24 B 8 0 24 10 -10 C -8 -24 0 -4 -18 D 10 -10 4 0 -22 E 24 10 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=30 C=25 A=8 D=4 so D is eliminated. Round 2 votes counts: E=34 B=30 C=25 A=11 so A is eliminated. Round 3 votes counts: E=41 B=31 C=28 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:237 B:216 D:191 A:183 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 8 -10 -24 B 8 0 24 10 -10 C -8 -24 0 -4 -18 D 10 -10 4 0 -22 E 24 10 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -10 -24 B 8 0 24 10 -10 C -8 -24 0 -4 -18 D 10 -10 4 0 -22 E 24 10 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -10 -24 B 8 0 24 10 -10 C -8 -24 0 -4 -18 D 10 -10 4 0 -22 E 24 10 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1320: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) E B C A D (8) A D C B E (8) C A D B E (7) A D C E B (7) E B D A C (6) D A C B E (5) E B C D A (4) D B E A C (4) C A B E D (4) B E D C A (4) B E C D A (4) D A E B C (3) C D A B E (3) E A B D C (2) D A B C E (2) C A E B D (2) A C D E B (2) A C D B E (2) E C B A D (1) E B A D C (1) E A B C D (1) D A E C B (1) D A B E C (1) C E A B D (1) C B E D A (1) C B E A D (1) C B D E A (1) C A D E B (1) C A B D E (1) B E D A C (1) B C E D A (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 -4 2 2 B -10 0 0 4 0 C 4 0 0 -8 0 D -2 -4 8 0 -2 E -2 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.42857142858 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 2 2 B -10 0 0 4 0 C 4 0 0 -8 0 D -2 -4 8 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428524 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 A=21 D=16 B=10 so B is eliminated. Round 2 votes counts: E=40 C=23 A=21 D=16 so D is eliminated. Round 3 votes counts: E=44 A=33 C=23 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:205 D:200 E:200 C:198 B:197 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -4 2 2 B -10 0 0 4 0 C 4 0 0 -8 0 D -2 -4 8 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428524 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 2 2 B -10 0 0 4 0 C 4 0 0 -8 0 D -2 -4 8 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428524 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 2 2 B -10 0 0 4 0 C 4 0 0 -8 0 D -2 -4 8 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428524 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1321: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (16) B E D C A (10) A D B E C (10) C E B D A (7) C A D E B (7) B E C D A (5) E B D C A (4) C A E D B (4) A D E B C (4) C A B E D (3) B D E A C (3) A C D B E (3) C E D A B (2) C E A B D (2) C A B D E (2) A D C E B (2) E D C B A (1) E B C D A (1) D E A B C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B A D (1) B C E A D (1) B C A E D (1) B A D E C (1) B A C D E (1) A D C B E (1) A D B C E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -4 -8 -12 B 8 0 24 18 22 C 4 -24 0 -14 -18 D 8 -18 14 0 -18 E 12 -22 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -8 -12 B 8 0 24 18 22 C 4 -24 0 -14 -18 D 8 -18 14 0 -18 E 12 -22 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 C=29 A=23 E=6 D=4 so D is eliminated. Round 2 votes counts: B=39 C=29 A=25 E=7 so E is eliminated. Round 3 votes counts: B=44 C=30 A=26 so A is eliminated. Round 4 votes counts: B=62 C=38 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:236 E:213 D:193 A:184 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -8 -12 B 8 0 24 18 22 C 4 -24 0 -14 -18 D 8 -18 14 0 -18 E 12 -22 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -8 -12 B 8 0 24 18 22 C 4 -24 0 -14 -18 D 8 -18 14 0 -18 E 12 -22 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -8 -12 B 8 0 24 18 22 C 4 -24 0 -14 -18 D 8 -18 14 0 -18 E 12 -22 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1322: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) E B A D C (8) E B C A D (7) C D A B E (7) A D C B E (7) A D B E C (7) D A C B E (6) A D E B C (6) E C B D A (4) C E D B A (4) C E B D A (4) C B E D A (4) C D A E B (3) C B D A E (3) E D A C B (2) E C D A B (2) D A C E B (2) C B D E A (2) B E C D A (2) B E A D C (2) A D C E B (2) E B A C D (1) E A B D C (1) B D A C E (1) B C E D A (1) B A D E C (1) A D E C B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -2 -10 -4 B 4 0 -6 0 -10 C 2 6 0 4 -6 D 10 0 -4 0 0 E 4 10 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.357189 E: 0.642811 Sum of squares = 0.540789707166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.357189 E: 1.000000 A B C D E A 0 -4 -2 -10 -4 B 4 0 -6 0 -10 C 2 6 0 4 -6 D 10 0 -4 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=27 A=25 D=8 B=7 so B is eliminated. Round 2 votes counts: E=37 C=28 A=26 D=9 so D is eliminated. Round 3 votes counts: E=37 A=35 C=28 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:203 D:203 B:194 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 -10 -4 B 4 0 -6 0 -10 C 2 6 0 4 -6 D 10 0 -4 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -10 -4 B 4 0 -6 0 -10 C 2 6 0 4 -6 D 10 0 -4 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -10 -4 B 4 0 -6 0 -10 C 2 6 0 4 -6 D 10 0 -4 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1323: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B D E A C (8) B E A D C (5) D A E B C (4) C E A D B (4) C E A B D (4) C B E A D (4) B D C A E (4) B D A E C (4) E C A B D (3) E A B C D (3) D C A E B (3) D A E C B (3) B C E A D (3) A D E C B (3) E A D C B (2) E A C D B (2) D B A E C (2) C D B A E (2) C D A E B (2) C A D E B (2) B C E D A (2) B C D E A (2) A E D C B (2) E C B A D (1) E C A D B (1) E B A C D (1) E A D B C (1) D B C A E (1) D B A C E (1) D A C E B (1) D A B E C (1) C E B A D (1) C D A B E (1) C B D A E (1) C A E B D (1) B E D A C (1) B E C D A (1) B E C A D (1) B D C E A (1) B C D A E (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 6 -8 8 -2 B -6 0 -4 4 -8 C 8 4 0 4 -2 D -8 -4 -4 0 -6 E 2 8 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 8 -2 B -6 0 -4 4 -8 C 8 4 0 4 -2 D -8 -4 -4 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=30 D=16 E=14 A=7 so A is eliminated. Round 2 votes counts: B=33 C=30 D=19 E=18 so E is eliminated. Round 3 votes counts: C=38 B=37 D=25 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:207 A:202 B:193 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 8 -2 B -6 0 -4 4 -8 C 8 4 0 4 -2 D -8 -4 -4 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 8 -2 B -6 0 -4 4 -8 C 8 4 0 4 -2 D -8 -4 -4 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 8 -2 B -6 0 -4 4 -8 C 8 4 0 4 -2 D -8 -4 -4 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1324: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) D B C E A (5) C B D A E (5) B D C A E (5) A E C B D (5) A C E B D (5) D B C A E (4) B C D A E (4) A C B E D (4) A B C D E (4) E D B C A (3) E A C D B (3) E A C B D (3) D E B C A (3) D C B E A (3) C B A D E (3) A B D C E (3) E D B A C (2) E D A B C (2) E A B D C (2) D B E C A (2) C D B E A (2) B D A C E (2) A E B D C (2) A C B D E (2) E D C A B (1) E D A C B (1) E C D B A (1) E C A D B (1) E A D C B (1) D E C B A (1) D E B A C (1) D C E B A (1) D C B A E (1) C E A D B (1) C D B A E (1) C A E B D (1) C A B D E (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 0 2 0 8 B 0 0 6 6 4 C -2 -6 0 -6 16 D 0 -6 6 0 8 E -8 -4 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.518488 B: 0.481512 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500683599699 Cumulative probabilities = A: 0.518488 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 0 8 B 0 0 6 6 4 C -2 -6 0 -6 16 D 0 -6 6 0 8 E -8 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 D=21 C=14 B=12 so B is eliminated. Round 2 votes counts: D=28 E=27 A=27 C=18 so C is eliminated. Round 3 votes counts: D=40 A=32 E=28 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:208 A:205 D:204 C:201 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 0 8 B 0 0 6 6 4 C -2 -6 0 -6 16 D 0 -6 6 0 8 E -8 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 0 8 B 0 0 6 6 4 C -2 -6 0 -6 16 D 0 -6 6 0 8 E -8 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 0 8 B 0 0 6 6 4 C -2 -6 0 -6 16 D 0 -6 6 0 8 E -8 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1325: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (12) E D B A C (9) E C D A B (6) D E C B A (6) E D C B A (5) E D C A B (4) E C A D B (4) D C E B A (4) B A D C E (4) A C B D E (4) E D A B C (3) C E A D B (3) A C B E D (3) A B E C D (3) E A C D B (2) D B E A C (2) D B C A E (2) C A E D B (2) B D C A E (2) B A C D E (2) A E B C D (2) A C E B D (2) E D B C A (1) E A D B C (1) E A C B D (1) E A B C D (1) D E B C A (1) D C B E A (1) C E D A B (1) C A E B D (1) C A B D E (1) B D E A C (1) B D A E C (1) B A D E C (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 16 12 2 -12 B -16 0 -2 -14 -20 C -12 2 0 2 -12 D -2 14 -2 0 -12 E 12 20 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 12 2 -12 B -16 0 -2 -14 -20 C -12 2 0 2 -12 D -2 14 -2 0 -12 E 12 20 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=28 D=16 B=11 C=8 so C is eliminated. Round 2 votes counts: E=41 A=32 D=16 B=11 so B is eliminated. Round 3 votes counts: E=41 A=39 D=20 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:209 D:199 C:190 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 12 2 -12 B -16 0 -2 -14 -20 C -12 2 0 2 -12 D -2 14 -2 0 -12 E 12 20 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 2 -12 B -16 0 -2 -14 -20 C -12 2 0 2 -12 D -2 14 -2 0 -12 E 12 20 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 2 -12 B -16 0 -2 -14 -20 C -12 2 0 2 -12 D -2 14 -2 0 -12 E 12 20 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1326: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) D B A E C (10) C E A B D (8) D B E C A (5) C E B A D (5) C A E B D (5) B D E C A (5) A E C B D (5) D B C E A (4) A D B E C (4) C D B E A (3) A E B D C (3) A D E B C (3) E B C A D (2) C A E D B (2) B E D C A (2) B D C E A (2) A E B C D (2) A D E C B (2) A C E D B (2) E C B A D (1) E C A B D (1) E A C B D (1) E A B C D (1) D C B E A (1) D B A C E (1) D A B E C (1) D A B C E (1) C E B D A (1) C D E A B (1) C B D E A (1) C A D E B (1) B E C D A (1) B D E A C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -2 0 -12 B 10 0 16 -6 4 C 2 -16 0 -10 -22 D 0 6 10 0 14 E 12 -4 22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.215284 B: 0.000000 C: 0.000000 D: 0.784716 E: 0.000000 Sum of squares = 0.662126028885 Cumulative probabilities = A: 0.215284 B: 0.215284 C: 0.215284 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 0 -12 B 10 0 16 -6 4 C 2 -16 0 -10 -22 D 0 6 10 0 14 E 12 -4 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250004739 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=27 A=23 B=11 E=6 so E is eliminated. Round 2 votes counts: D=33 C=29 A=25 B=13 so B is eliminated. Round 3 votes counts: D=43 C=32 A=25 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:212 E:208 A:188 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 0 -12 B 10 0 16 -6 4 C 2 -16 0 -10 -22 D 0 6 10 0 14 E 12 -4 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250004739 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 0 -12 B 10 0 16 -6 4 C 2 -16 0 -10 -22 D 0 6 10 0 14 E 12 -4 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250004739 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 0 -12 B 10 0 16 -6 4 C 2 -16 0 -10 -22 D 0 6 10 0 14 E 12 -4 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250004739 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1327: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) E A C B D (7) D C E B A (7) C E D A B (6) D B C A E (5) B A D E C (5) A E B C D (5) A B E C D (5) E C A D B (4) D C B E A (4) D C B A E (4) D B C E A (3) B D A E C (3) B D A C E (3) B A E D C (3) E C D A B (2) E B A D C (2) D E C B A (2) A E C B D (2) E A B C D (1) D E C A B (1) D C E A B (1) D B E C A (1) D B E A C (1) D B A C E (1) C D E B A (1) C D B A E (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D B E (1) B D E A C (1) B A D C E (1) B A C E D (1) A C E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -12 -18 -8 B -2 0 -16 -16 -8 C 12 16 0 4 10 D 18 16 -4 0 18 E 8 8 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -18 -8 B -2 0 -16 -16 -8 C 12 16 0 4 10 D 18 16 -4 0 18 E 8 8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=22 B=17 E=16 A=15 so A is eliminated. Round 2 votes counts: D=30 C=24 E=23 B=23 so E is eliminated. Round 3 votes counts: C=39 B=31 D=30 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:224 C:221 E:194 A:182 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -18 -8 B -2 0 -16 -16 -8 C 12 16 0 4 10 D 18 16 -4 0 18 E 8 8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -18 -8 B -2 0 -16 -16 -8 C 12 16 0 4 10 D 18 16 -4 0 18 E 8 8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -18 -8 B -2 0 -16 -16 -8 C 12 16 0 4 10 D 18 16 -4 0 18 E 8 8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1328: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (13) D C E B A (11) D C B E A (9) A B C E D (9) E D C A B (6) D E C B A (6) C D B E A (6) E A D C B (5) E D C B A (4) E A B D C (4) C D B A E (4) C B D A E (4) B A C D E (4) E D A C B (3) A E B C D (3) A B C D E (3) E A D B C (2) B C A D E (2) B C D A E (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 -6 -8 -12 B 2 0 -16 -12 10 C 6 16 0 -2 6 D 8 12 2 0 0 E 12 -10 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.842695 E: 0.157305 Sum of squares = 0.734880259121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.842695 E: 1.000000 A B C D E A 0 -2 -6 -8 -12 B 2 0 -16 -12 10 C 6 16 0 -2 6 D 8 12 2 0 0 E 12 -10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000119935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=26 E=24 C=14 B=7 so B is eliminated. Round 2 votes counts: A=33 D=26 E=24 C=17 so C is eliminated. Round 3 votes counts: D=41 A=35 E=24 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:211 E:198 B:192 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 -12 B 2 0 -16 -12 10 C 6 16 0 -2 6 D 8 12 2 0 0 E 12 -10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000119935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 -12 B 2 0 -16 -12 10 C 6 16 0 -2 6 D 8 12 2 0 0 E 12 -10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000119935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 -12 B 2 0 -16 -12 10 C 6 16 0 -2 6 D 8 12 2 0 0 E 12 -10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000119935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1329: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) E A D C B (9) B C D A E (9) B E D A C (8) B E A C D (7) B C A D E (5) D C A E B (4) B D C E A (4) D A C E B (3) C A D B E (3) B E A D C (3) E B A D C (2) E A C D B (2) D A E C B (2) C D B A E (2) C D A E B (2) C D A B E (2) B E D C A (2) B D C A E (2) B C A E D (2) A C D E B (2) E D B A C (1) E D A C B (1) E B D A C (1) D E A C B (1) D C B A E (1) D C A B E (1) C B D A E (1) B E C D A (1) B E C A D (1) B D E C A (1) B C D E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -8 0 10 B 8 0 0 2 12 C 8 0 0 6 10 D 0 -2 -6 0 12 E -10 -12 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.322252 C: 0.677748 D: 0.000000 E: 0.000000 Sum of squares = 0.563188941206 Cumulative probabilities = A: 0.000000 B: 0.322252 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 0 10 B 8 0 0 2 12 C 8 0 0 6 10 D 0 -2 -6 0 12 E -10 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=46 C=20 E=16 D=12 A=6 so A is eliminated. Round 2 votes counts: B=46 C=23 E=19 D=12 so D is eliminated. Round 3 votes counts: B=46 C=32 E=22 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 B:211 D:202 A:197 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 0 10 B 8 0 0 2 12 C 8 0 0 6 10 D 0 -2 -6 0 12 E -10 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 0 10 B 8 0 0 2 12 C 8 0 0 6 10 D 0 -2 -6 0 12 E -10 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 0 10 B 8 0 0 2 12 C 8 0 0 6 10 D 0 -2 -6 0 12 E -10 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1330: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) B C D A E (9) E A D C B (8) E A D B C (7) B C A D E (7) A E D B C (5) D A E C B (4) C B E D A (4) C B D E A (4) E D A C B (3) D E A C B (3) C E D A B (3) C D B A E (3) D C A E B (2) C B E A D (2) B C E A D (2) B A E D C (2) A E B D C (2) E C D A B (1) E C A B D (1) D C E A B (1) D C B A E (1) D A E B C (1) D A C B E (1) C E B D A (1) C E B A D (1) C D E A B (1) C D B E A (1) B E C A D (1) B E A C D (1) B A E C D (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -12 -8 10 B 10 0 -12 8 10 C 12 12 0 12 14 D 8 -8 -12 0 6 E -10 -10 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 10 B 10 0 -12 8 10 C 12 12 0 12 14 D 8 -8 -12 0 6 E -10 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 E=20 D=13 A=9 so A is eliminated. Round 2 votes counts: C=31 B=28 E=27 D=14 so D is eliminated. Round 3 votes counts: E=36 C=36 B=28 so B is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:208 D:197 A:190 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 10 B 10 0 -12 8 10 C 12 12 0 12 14 D 8 -8 -12 0 6 E -10 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 10 B 10 0 -12 8 10 C 12 12 0 12 14 D 8 -8 -12 0 6 E -10 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 10 B 10 0 -12 8 10 C 12 12 0 12 14 D 8 -8 -12 0 6 E -10 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1331: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) B D E A C (7) D B C A E (6) C A E D B (6) B D A C E (6) E C A B D (5) E B D C A (5) E B C A D (5) E A C D B (5) D C A B E (5) E B A C D (4) C A D E B (4) E C A D B (3) A C E D B (3) E B D A C (2) E B C D A (2) E A B C D (2) D A C B E (2) C D A B E (2) C A D B E (2) B E D A C (2) A E C D B (2) E C B A D (1) E B A D C (1) E A C B D (1) D C B A E (1) D A B C E (1) C E A D B (1) B D E C A (1) B D A E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 2 -6 6 B 8 0 10 -10 -6 C -2 -10 0 0 2 D 6 10 0 0 0 E -6 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.394870 D: 0.605130 E: 0.000000 Sum of squares = 0.522104547531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.394870 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -6 6 B 8 0 10 -10 -6 C -2 -10 0 0 2 D 6 10 0 0 0 E -6 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499574 D: 0.500426 E: 0.000000 Sum of squares = 0.500000363407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499574 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=25 B=17 C=15 A=7 so A is eliminated. Round 2 votes counts: E=38 D=25 C=20 B=17 so B is eliminated. Round 3 votes counts: E=40 D=40 C=20 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:201 E:199 A:197 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -6 6 B 8 0 10 -10 -6 C -2 -10 0 0 2 D 6 10 0 0 0 E -6 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499574 D: 0.500426 E: 0.000000 Sum of squares = 0.500000363407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499574 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -6 6 B 8 0 10 -10 -6 C -2 -10 0 0 2 D 6 10 0 0 0 E -6 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499574 D: 0.500426 E: 0.000000 Sum of squares = 0.500000363407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499574 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -6 6 B 8 0 10 -10 -6 C -2 -10 0 0 2 D 6 10 0 0 0 E -6 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499574 D: 0.500426 E: 0.000000 Sum of squares = 0.500000363407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499574 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1332: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) D A E B C (9) B E C D A (9) D A E C B (6) A D E C B (6) A D C E B (6) E B C D A (5) D A B E C (4) C B A D E (3) C A E D B (3) B E D A C (3) B C E A D (3) A D C B E (3) E C B A D (2) E B D C A (2) D A B C E (2) C E B A D (2) B E D C A (2) B D A E C (2) B C E D A (2) E D A B C (1) E B D A C (1) E A D C B (1) E A C D B (1) D E A B C (1) D B A E C (1) C E A D B (1) C E A B D (1) C A D B E (1) C A B D E (1) B D A C E (1) B C D A E (1) B C A D E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 -4 2 B 4 0 0 6 2 C 2 0 0 -2 -12 D 4 -6 2 0 -2 E -2 -2 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.901682 C: 0.098318 D: 0.000000 E: 0.000000 Sum of squares = 0.82269707573 Cumulative probabilities = A: 0.000000 B: 0.901682 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 2 B 4 0 0 6 2 C 2 0 0 -2 -12 D 4 -6 2 0 -2 E -2 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.75510207467 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 D=23 A=16 E=13 so E is eliminated. Round 2 votes counts: B=32 C=26 D=24 A=18 so A is eliminated. Round 3 votes counts: D=40 B=32 C=28 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:206 E:205 D:199 A:196 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 2 B 4 0 0 6 2 C 2 0 0 -2 -12 D 4 -6 2 0 -2 E -2 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.75510207467 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 2 B 4 0 0 6 2 C 2 0 0 -2 -12 D 4 -6 2 0 -2 E -2 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.75510207467 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 2 B 4 0 0 6 2 C 2 0 0 -2 -12 D 4 -6 2 0 -2 E -2 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.75510207467 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1333: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) B C E A D (7) B E C D A (6) A D C E B (6) C B E A D (5) E B C D A (4) D E B A C (4) D A C B E (4) B E C A D (4) A D C B E (4) E D A C B (3) E C B A D (3) E B C A D (3) D E A B C (3) D A E B C (3) C B A E D (3) A C D B E (3) E D B A C (2) E C A B D (2) E B D C A (2) E B D A C (2) D B E A C (2) B D E C A (2) E C D B A (1) E A D C B (1) E A C D B (1) D E A C B (1) D B A E C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C A E B D (1) C A B E D (1) B E D C A (1) B D A C E (1) B C A E D (1) A D E C B (1) Total count = 100 A B C D E A 0 -10 4 -4 -22 B 10 0 0 0 -6 C -4 0 0 -6 -20 D 4 0 6 0 -10 E 22 6 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 4 -4 -22 B 10 0 0 0 -6 C -4 0 0 -6 -20 D 4 0 6 0 -10 E 22 6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 B=22 A=14 C=12 so C is eliminated. Round 2 votes counts: B=30 D=28 E=26 A=16 so A is eliminated. Round 3 votes counts: D=42 B=31 E=27 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:229 B:202 D:200 C:185 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 4 -4 -22 B 10 0 0 0 -6 C -4 0 0 -6 -20 D 4 0 6 0 -10 E 22 6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -4 -22 B 10 0 0 0 -6 C -4 0 0 -6 -20 D 4 0 6 0 -10 E 22 6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -4 -22 B 10 0 0 0 -6 C -4 0 0 -6 -20 D 4 0 6 0 -10 E 22 6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1334: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) A D E B C (6) B D A C E (5) B C D A E (5) E C B A D (4) E B C A D (4) E A B D C (4) A D E C B (4) E A D C B (3) E A D B C (3) D A B C E (3) C E B D A (3) C D B A E (3) C B E D A (3) C B D E A (3) B D A E C (3) B C E D A (3) B A D E C (3) A D B E C (3) E A C D B (2) D C A B E (2) D A C B E (2) C E D A B (2) C D A E B (2) C D A B E (2) B E A D C (2) B D C A E (2) E C A B D (1) E B A D C (1) D A C E B (1) D A B E C (1) C E B A D (1) C E A D B (1) B E C A D (1) B E A C D (1) B C D E A (1) A E D C B (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 0 -6 14 B 10 0 6 16 12 C 0 -6 0 -2 2 D 6 -16 2 0 18 E -14 -12 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -6 14 B 10 0 6 16 12 C 0 -6 0 -2 2 D 6 -16 2 0 18 E -14 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=26 E=22 A=16 D=9 so D is eliminated. Round 2 votes counts: C=29 B=26 A=23 E=22 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:222 D:205 A:199 C:197 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -6 14 B 10 0 6 16 12 C 0 -6 0 -2 2 D 6 -16 2 0 18 E -14 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -6 14 B 10 0 6 16 12 C 0 -6 0 -2 2 D 6 -16 2 0 18 E -14 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -6 14 B 10 0 6 16 12 C 0 -6 0 -2 2 D 6 -16 2 0 18 E -14 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1335: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) D A E B C (6) C E B D A (6) A D B C E (6) D A C E B (5) E D C A B (4) D A C B E (4) D A B E C (4) E D A B C (3) E C D B A (3) E C D A B (3) E C B A D (3) D A E C B (3) D A B C E (3) C B E A D (3) B C A E D (3) A D B E C (3) E C B D A (2) D E C A B (2) D E A C B (2) B C E A D (2) B A D E C (2) B A D C E (2) B A C D E (2) A B D C E (2) E D A C B (1) E B C D A (1) E B C A D (1) E B A D C (1) D C A E B (1) C E D B A (1) C E D A B (1) C D A E B (1) C B A E D (1) C B A D E (1) B E C A D (1) B C A D E (1) B A E D C (1) B A E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 4 -12 6 B -10 0 -8 -12 -14 C -4 8 0 -12 4 D 12 12 12 0 2 E -6 14 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -12 6 B -10 0 -8 -12 -14 C -4 8 0 -12 4 D 12 12 12 0 2 E -6 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=22 C=21 B=15 A=12 so A is eliminated. Round 2 votes counts: D=39 E=22 C=21 B=18 so B is eliminated. Round 3 votes counts: D=46 C=29 E=25 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:204 E:201 C:198 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -12 6 B -10 0 -8 -12 -14 C -4 8 0 -12 4 D 12 12 12 0 2 E -6 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -12 6 B -10 0 -8 -12 -14 C -4 8 0 -12 4 D 12 12 12 0 2 E -6 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -12 6 B -10 0 -8 -12 -14 C -4 8 0 -12 4 D 12 12 12 0 2 E -6 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1336: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) A D C E B (9) B D C E A (7) A E C D B (7) A E B D C (6) E A B C D (5) A E D C B (5) B C E D A (4) A B E D C (4) D B C A E (3) B C D E A (3) E B C A D (2) E B A C D (2) D C B E A (2) D C A B E (2) C D B E A (2) B E A D C (2) B A E D C (2) A E B C D (2) A D B C E (2) E C B D A (1) E C B A D (1) E B C D A (1) E A C D B (1) E A C B D (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D B A (1) C D A E B (1) C B E D A (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C A E (1) A E D B C (1) A D E C B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 12 14 22 B -6 0 4 -4 0 C -12 -4 0 -22 4 D -14 4 22 0 -4 E -22 0 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 14 22 B -6 0 4 -4 0 C -12 -4 0 -22 4 D -14 4 22 0 -4 E -22 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=22 D=19 E=14 C=5 so C is eliminated. Round 2 votes counts: A=40 B=23 D=22 E=15 so E is eliminated. Round 3 votes counts: A=47 B=30 D=23 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 D:204 B:197 E:189 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 14 22 B -6 0 4 -4 0 C -12 -4 0 -22 4 D -14 4 22 0 -4 E -22 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 14 22 B -6 0 4 -4 0 C -12 -4 0 -22 4 D -14 4 22 0 -4 E -22 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 14 22 B -6 0 4 -4 0 C -12 -4 0 -22 4 D -14 4 22 0 -4 E -22 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1337: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E C B A D (8) C E B A D (8) B A E D C (7) D A B E C (6) A B D E C (6) C E D B A (5) D C A E B (4) C D E A B (4) A B E C D (4) E B A C D (3) D C E A B (3) D C A B E (3) C E A B D (3) C D E B A (3) B E A C D (3) A D B E C (3) A B E D C (3) E B C A D (2) C E B D A (2) B A E C D (2) E D B A C (1) E B A D C (1) D E C B A (1) D C E B A (1) D B E A C (1) C E D A B (1) C D A B E (1) B E A D C (1) B A D E C (1) Total count = 100 A B C D E A 0 0 2 10 -2 B 0 0 6 8 0 C -2 -6 0 -2 -6 D -10 -8 2 0 -8 E 2 0 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.397380 C: 0.000000 D: 0.000000 E: 0.602620 Sum of squares = 0.521061686397 Cumulative probabilities = A: 0.000000 B: 0.397380 C: 0.397380 D: 0.397380 E: 1.000000 A B C D E A 0 0 2 10 -2 B 0 0 6 8 0 C -2 -6 0 -2 -6 D -10 -8 2 0 -8 E 2 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=27 A=16 E=15 B=14 so B is eliminated. Round 2 votes counts: D=28 C=27 A=26 E=19 so E is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:208 B:207 A:205 C:192 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 10 -2 B 0 0 6 8 0 C -2 -6 0 -2 -6 D -10 -8 2 0 -8 E 2 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 10 -2 B 0 0 6 8 0 C -2 -6 0 -2 -6 D -10 -8 2 0 -8 E 2 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 10 -2 B 0 0 6 8 0 C -2 -6 0 -2 -6 D -10 -8 2 0 -8 E 2 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1338: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B E A C D (9) B E A D C (8) E B A C D (7) E A B C D (6) D C B A E (6) D C A B E (5) C D A E B (5) A E B C D (5) D C E A B (4) D C B E A (3) C D E B A (3) B D C E A (3) B A E D C (3) E A C B D (2) C D E A B (2) C A D E B (2) B D C A E (2) A C E D B (2) E C D A B (1) E B C A D (1) E A C D B (1) D C E B A (1) D B C A E (1) B E D A C (1) B E C D A (1) B D A E C (1) A E C B D (1) A D C E B (1) A D B C E (1) A C D E B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 4 -6 B 0 0 2 4 -8 C -2 -2 0 0 2 D -4 -4 0 0 0 E 6 8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999696 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 0 2 4 -6 B 0 0 2 4 -8 C -2 -2 0 0 2 D -4 -4 0 0 0 E 6 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=28 E=18 A=13 C=12 so C is eliminated. Round 2 votes counts: D=39 B=28 E=18 A=15 so A is eliminated. Round 3 votes counts: D=44 B=30 E=26 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:206 A:200 B:199 C:199 D:196 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 2 4 -6 B 0 0 2 4 -8 C -2 -2 0 0 2 D -4 -4 0 0 0 E 6 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 -6 B 0 0 2 4 -8 C -2 -2 0 0 2 D -4 -4 0 0 0 E 6 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 -6 B 0 0 2 4 -8 C -2 -2 0 0 2 D -4 -4 0 0 0 E 6 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1339: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) A B D E C (8) C E D B A (7) E D C B A (5) B A D E C (5) A B C D E (5) D E C A B (4) D E A B C (4) C E B D A (4) C A B E D (4) B A C E D (4) D E B A C (3) A D B E C (3) E D B C A (2) E C D B A (2) D E A C B (2) D A E B C (2) C B E A D (2) C A E D B (2) C A D E B (2) B C A E D (2) A D C B E (2) A B D C E (2) E D C A B (1) E C D A B (1) E B D C A (1) D A E C B (1) D A B E C (1) C E A D B (1) C D E A B (1) C D A E B (1) C B A E D (1) B E C D A (1) B D A E C (1) B A E D C (1) A D E C B (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 18 -4 -4 2 B -18 0 -10 -18 -14 C 4 10 0 0 2 D 4 18 0 0 2 E -2 14 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.809445 D: 0.190555 E: 0.000000 Sum of squares = 0.691512592151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.809445 D: 1.000000 E: 1.000000 A B C D E A 0 18 -4 -4 2 B -18 0 -10 -18 -14 C 4 10 0 0 2 D 4 18 0 0 2 E -2 14 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=24 D=17 B=14 E=12 so E is eliminated. Round 2 votes counts: C=36 D=25 A=24 B=15 so B is eliminated. Round 3 votes counts: C=39 A=34 D=27 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:208 A:206 E:204 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -4 -4 2 B -18 0 -10 -18 -14 C 4 10 0 0 2 D 4 18 0 0 2 E -2 14 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 -4 2 B -18 0 -10 -18 -14 C 4 10 0 0 2 D 4 18 0 0 2 E -2 14 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 -4 2 B -18 0 -10 -18 -14 C 4 10 0 0 2 D 4 18 0 0 2 E -2 14 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1340: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) E C A D B (7) B D A C E (7) D B C E A (5) C A E D B (5) A E C B D (5) E C D B A (3) D C E B A (3) C E A D B (3) B D E C A (3) A B E C D (3) A B D C E (3) E C D A B (2) E C A B D (2) D A B C E (2) B D E A C (2) B D C E A (2) B D C A E (2) B D A E C (2) B A D E C (2) B A D C E (2) A D B C E (2) A C D E B (2) A B C E D (2) A B C D E (2) E D B C A (1) E C B D A (1) E B D C A (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C A E (1) D B A C E (1) C E D A B (1) C D E A B (1) B E D C A (1) B A E C D (1) A E C D B (1) A E B C D (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 14 6 12 20 B -14 0 -2 -10 -2 C -6 2 0 8 22 D -12 10 -8 0 -4 E -20 2 -22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 12 20 B -14 0 -2 -10 -2 C -6 2 0 8 22 D -12 10 -8 0 -4 E -20 2 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999404 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=24 E=17 D=15 C=10 so C is eliminated. Round 2 votes counts: A=39 B=24 E=21 D=16 so D is eliminated. Round 3 votes counts: A=41 B=34 E=25 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 C:213 D:193 B:186 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 12 20 B -14 0 -2 -10 -2 C -6 2 0 8 22 D -12 10 -8 0 -4 E -20 2 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999404 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 12 20 B -14 0 -2 -10 -2 C -6 2 0 8 22 D -12 10 -8 0 -4 E -20 2 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999404 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 12 20 B -14 0 -2 -10 -2 C -6 2 0 8 22 D -12 10 -8 0 -4 E -20 2 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999404 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1341: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) D C B E A (5) D B E C A (4) C D B A E (4) B D E C A (4) A E C B D (4) A C E D B (4) E B A D C (3) E A B D C (3) D B C E A (3) C D A E B (3) B E D A C (3) B E A D C (3) B D E A C (3) B D C E A (3) A E B D C (3) A C E B D (3) D C B A E (2) C A E D B (2) C A D E B (2) C A D B E (2) B D A E C (2) B A E D C (2) A E C D B (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E B D A C (1) E A D B C (1) E A C D B (1) D E C B A (1) D C E B A (1) C D E A B (1) C D B E A (1) C D A B E (1) C B D A E (1) C B A D E (1) B D C A E (1) B C A D E (1) B A D E C (1) A C D E B (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 8 6 10 B 0 0 4 10 4 C -8 -4 0 -4 -10 D -6 -10 4 0 0 E -10 -4 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.671849 B: 0.328151 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.559064299435 Cumulative probabilities = A: 0.671849 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 6 10 B 0 0 4 10 4 C -8 -4 0 -4 -10 D -6 -10 4 0 0 E -10 -4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=23 C=18 D=16 E=13 so E is eliminated. Round 2 votes counts: A=35 B=27 C=20 D=18 so D is eliminated. Round 3 votes counts: A=36 B=34 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:209 E:198 D:194 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 6 10 B 0 0 4 10 4 C -8 -4 0 -4 -10 D -6 -10 4 0 0 E -10 -4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 6 10 B 0 0 4 10 4 C -8 -4 0 -4 -10 D -6 -10 4 0 0 E -10 -4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 6 10 B 0 0 4 10 4 C -8 -4 0 -4 -10 D -6 -10 4 0 0 E -10 -4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1342: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) E A D C B (9) D E C A B (9) D C B E A (9) C D B E A (7) B C D A E (7) B C A D E (7) A E D B C (7) E D A C B (5) C B D A E (5) A B E C D (5) B A C E D (4) E A D B C (3) D E C B A (3) D C E B A (3) D E A C B (1) D C B A E (1) B C A E D (1) B A C D E (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -4 0 2 B -4 0 -4 -14 -4 C 4 4 0 -2 -8 D 0 14 2 0 6 E -2 4 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.218548 B: 0.000000 C: 0.000000 D: 0.781452 E: 0.000000 Sum of squares = 0.658430606817 Cumulative probabilities = A: 0.218548 B: 0.218548 C: 0.218548 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 0 2 B -4 0 -4 -14 -4 C 4 4 0 -2 -8 D 0 14 2 0 6 E -2 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555871341 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 B=20 E=17 C=12 so C is eliminated. Round 2 votes counts: D=33 B=25 A=25 E=17 so E is eliminated. Round 3 votes counts: D=38 A=37 B=25 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:211 E:202 A:201 C:199 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 0 2 B -4 0 -4 -14 -4 C 4 4 0 -2 -8 D 0 14 2 0 6 E -2 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555871341 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 2 B -4 0 -4 -14 -4 C 4 4 0 -2 -8 D 0 14 2 0 6 E -2 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555871341 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 2 B -4 0 -4 -14 -4 C 4 4 0 -2 -8 D 0 14 2 0 6 E -2 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555871341 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1343: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A B E C D (11) A D C E B (9) E C D B A (7) B A E C D (6) A B D C E (6) D C E B A (5) A D C B E (5) A E C D B (4) C D E B A (3) B E A C D (3) A E B C D (3) E B C D A (2) E A B C D (2) D C B E A (2) D C A E B (2) B D C E A (2) B A D C E (2) A B D E C (2) E C A D B (1) E A C D B (1) D C E A B (1) D C A B E (1) D B C E A (1) C D E A B (1) B E D C A (1) B D C A E (1) B D A C E (1) A E D C B (1) A E C B D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 18 18 14 B -6 0 10 10 12 C -18 -10 0 14 -14 D -18 -10 -14 0 -10 E -14 -12 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 18 14 B -6 0 10 10 12 C -18 -10 0 14 -14 D -18 -10 -14 0 -10 E -14 -12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 B=27 E=13 D=12 C=4 so C is eliminated. Round 2 votes counts: A=44 B=27 D=16 E=13 so E is eliminated. Round 3 votes counts: A=48 B=29 D=23 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:213 E:199 C:186 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 18 14 B -6 0 10 10 12 C -18 -10 0 14 -14 D -18 -10 -14 0 -10 E -14 -12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 18 14 B -6 0 10 10 12 C -18 -10 0 14 -14 D -18 -10 -14 0 -10 E -14 -12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 18 14 B -6 0 10 10 12 C -18 -10 0 14 -14 D -18 -10 -14 0 -10 E -14 -12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1344: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) E C A D B (6) D B A E C (5) D A E B C (5) C E B A D (5) C B A D E (4) A D B C E (4) E D A C B (3) E C B D A (3) D A E C B (3) B D A C E (3) A D E C B (3) E D C A B (2) E C D B A (2) D E A B C (2) D B E A C (2) D A B C E (2) C E B D A (2) C E A D B (2) C B E A D (2) C A E B D (2) B E C D A (2) B D E C A (2) B C A D E (2) B A C D E (2) A B C D E (2) E D C B A (1) E D B C A (1) E C B A D (1) E C A B D (1) E A C D B (1) D E B A C (1) D B A C E (1) C E A B D (1) C B E D A (1) C B A E D (1) C A B E D (1) B D C E A (1) B C E D A (1) B C D E A (1) B C D A E (1) A E C D B (1) A D E B C (1) A D C E B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -10 8 B -6 0 2 -16 0 C -4 -2 0 -6 -14 D 10 16 6 0 16 E -8 0 14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -10 8 B -6 0 2 -16 0 C -4 -2 0 -6 -14 D 10 16 6 0 16 E -8 0 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 C=21 B=15 A=14 so A is eliminated. Round 2 votes counts: D=39 E=22 C=21 B=18 so B is eliminated. Round 3 votes counts: D=46 C=30 E=24 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 A:204 E:195 B:190 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -10 8 B -6 0 2 -16 0 C -4 -2 0 -6 -14 D 10 16 6 0 16 E -8 0 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -10 8 B -6 0 2 -16 0 C -4 -2 0 -6 -14 D 10 16 6 0 16 E -8 0 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -10 8 B -6 0 2 -16 0 C -4 -2 0 -6 -14 D 10 16 6 0 16 E -8 0 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1345: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) D E B A C (7) B D C A E (7) E A C B D (6) E A C D B (5) D E C A B (5) B C A D E (5) D C B A E (4) C A B E D (4) A C B E D (4) E D A C B (3) D E B C A (3) C A E D B (3) C A D B E (3) A C E B D (3) E D B A C (2) E B A D C (2) E A D C B (2) D B E A C (2) D B C A E (2) B E D A C (2) B E A C D (2) B A C E D (2) A E C D B (2) E A D B C (1) E A B C D (1) D E C B A (1) D C B E A (1) C E A D B (1) C D A B E (1) C A E B D (1) C A D E B (1) B D E C A (1) B D E A C (1) B C A E D (1) B A E C D (1) Total count = 100 A B C D E A 0 -8 -4 0 -12 B 8 0 0 -14 2 C 4 0 0 -8 -14 D 0 14 8 0 4 E 12 -2 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.186581 B: 0.000000 C: 0.000000 D: 0.813419 E: 0.000000 Sum of squares = 0.696462623114 Cumulative probabilities = A: 0.186581 B: 0.186581 C: 0.186581 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 0 -12 B 8 0 0 -14 2 C 4 0 0 -8 -14 D 0 14 8 0 4 E 12 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000054513 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=22 B=22 C=14 A=9 so A is eliminated. Round 2 votes counts: D=33 E=24 B=22 C=21 so C is eliminated. Round 3 votes counts: D=38 E=32 B=30 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:210 B:198 C:191 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 0 -12 B 8 0 0 -14 2 C 4 0 0 -8 -14 D 0 14 8 0 4 E 12 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000054513 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 0 -12 B 8 0 0 -14 2 C 4 0 0 -8 -14 D 0 14 8 0 4 E 12 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000054513 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 0 -12 B 8 0 0 -14 2 C 4 0 0 -8 -14 D 0 14 8 0 4 E 12 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000054513 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1346: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) D A B E C (7) C E B A D (7) C A D E B (7) D A C B E (6) D B E A C (5) C D A B E (5) B E C D A (5) A D E B C (5) A D C E B (5) E B C A D (4) E B A D C (4) E B A C D (4) D C A B E (4) C E A B D (3) D A B C E (2) C B E D A (2) C A E D B (2) C A E B D (2) B E A D C (2) B D E A C (2) A E B D C (2) D B A E C (1) C D B A E (1) C D A E B (1) C B D E A (1) B E D C A (1) B D E C A (1) Total count = 100 A B C D E A 0 2 8 -6 0 B -2 0 8 -2 8 C -8 -8 0 -12 -4 D 6 2 12 0 6 E 0 -8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -6 0 B -2 0 8 -2 8 C -8 -8 0 -12 -4 D 6 2 12 0 6 E 0 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 B=20 E=12 A=12 so E is eliminated. Round 2 votes counts: B=32 C=31 D=25 A=12 so A is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:206 A:202 E:195 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -6 0 B -2 0 8 -2 8 C -8 -8 0 -12 -4 D 6 2 12 0 6 E 0 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -6 0 B -2 0 8 -2 8 C -8 -8 0 -12 -4 D 6 2 12 0 6 E 0 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -6 0 B -2 0 8 -2 8 C -8 -8 0 -12 -4 D 6 2 12 0 6 E 0 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1347: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) E D B A C (8) E C B D A (8) C A B D E (8) C B A D E (6) E B D C A (5) D A B E C (4) A D B C E (4) E C A D B (3) E B C D A (3) D E B A C (3) D A E B C (3) C B A E D (3) A C D B E (3) A C B D E (3) E D A B C (2) C B E D A (2) A D C B E (2) E D B C A (1) E C D B A (1) E A C D B (1) D E A B C (1) D B A E C (1) D B A C E (1) C E B D A (1) C E A B D (1) C B E A D (1) C A E B D (1) B E D C A (1) B E D A C (1) B D E A C (1) B C E D A (1) B C A D E (1) B A C D E (1) A E D C B (1) A D E B C (1) A D B E C (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -12 4 -8 B 18 0 -10 16 -8 C 12 10 0 18 0 D -4 -16 -18 0 -10 E 8 8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.196718 D: 0.000000 E: 0.803282 Sum of squares = 0.6839600329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.196718 D: 0.196718 E: 1.000000 A B C D E A 0 -18 -12 4 -8 B 18 0 -10 16 -8 C 12 10 0 18 0 D -4 -16 -18 0 -10 E 8 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=32 C=32 A=17 D=13 B=6 so B is eliminated. Round 2 votes counts: E=34 C=34 A=18 D=14 so D is eliminated. Round 3 votes counts: E=39 C=34 A=27 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:213 B:208 A:183 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -12 4 -8 B 18 0 -10 16 -8 C 12 10 0 18 0 D -4 -16 -18 0 -10 E 8 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 4 -8 B 18 0 -10 16 -8 C 12 10 0 18 0 D -4 -16 -18 0 -10 E 8 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 4 -8 B 18 0 -10 16 -8 C 12 10 0 18 0 D -4 -16 -18 0 -10 E 8 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1348: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) B A D E C (7) A B D C E (5) E D B A C (4) E C D B A (4) E C D A B (4) D C A E B (4) C D A E B (4) B A E D C (4) D A C B E (3) C E D A B (3) C D E A B (3) B E A D C (3) B E A C D (3) B A D C E (3) B A C E D (3) E D C A B (2) E D A C B (2) E B D A C (2) E B A D C (2) D E A B C (2) C B E A D (2) C A D B E (2) B C A E D (2) E D A B C (1) E C B D A (1) E C B A D (1) E B C A D (1) E B A C D (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A B E (1) D A B E C (1) D A B C E (1) C E D B A (1) B C A D E (1) B A E C D (1) A D C B E (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 24 8 10 B 10 0 18 4 10 C -24 -18 0 -6 2 D -8 -4 6 0 6 E -10 -10 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 24 8 10 B 10 0 18 4 10 C -24 -18 0 -6 2 D -8 -4 6 0 6 E -10 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=25 D=15 C=15 A=9 so A is eliminated. Round 2 votes counts: B=42 E=25 D=18 C=15 so C is eliminated. Round 3 votes counts: B=44 E=29 D=27 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:216 D:200 E:186 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 24 8 10 B 10 0 18 4 10 C -24 -18 0 -6 2 D -8 -4 6 0 6 E -10 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 24 8 10 B 10 0 18 4 10 C -24 -18 0 -6 2 D -8 -4 6 0 6 E -10 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 24 8 10 B 10 0 18 4 10 C -24 -18 0 -6 2 D -8 -4 6 0 6 E -10 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1349: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) C E B A D (7) E C B D A (5) E B C A D (5) A D C B E (5) D B E A C (4) D A B E C (4) A C E B D (4) D B A E C (3) C E D B A (3) C E B D A (3) C D A E B (3) B E A D C (3) B D A E C (3) A D B E C (3) D C E B A (2) D C B E A (2) D B E C A (2) D A C B E (2) C E A B D (2) C A E D B (2) B E A C D (2) A B E D C (2) A B E C D (2) E B A C D (1) E A B C D (1) D C A B E (1) D B A C E (1) D A C E B (1) D A B C E (1) C D E B A (1) C D E A B (1) C A E B D (1) C A D E B (1) B E D A C (1) B A E D C (1) B A D E C (1) A E C B D (1) A C D E B (1) A C D B E (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 2 14 -6 B 16 0 -16 12 -6 C -2 16 0 14 -6 D -14 -12 -14 0 -12 E 6 6 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 2 14 -6 B 16 0 -16 12 -6 C -2 16 0 14 -6 D -14 -12 -14 0 -12 E 6 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 A=22 E=20 B=11 so B is eliminated. Round 2 votes counts: E=26 D=26 C=24 A=24 so C is eliminated. Round 3 votes counts: E=41 D=31 A=28 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:211 B:203 A:197 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 2 14 -6 B 16 0 -16 12 -6 C -2 16 0 14 -6 D -14 -12 -14 0 -12 E 6 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 14 -6 B 16 0 -16 12 -6 C -2 16 0 14 -6 D -14 -12 -14 0 -12 E 6 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 14 -6 B 16 0 -16 12 -6 C -2 16 0 14 -6 D -14 -12 -14 0 -12 E 6 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1350: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) C B E A D (7) B A C E D (7) D A B E C (6) A B D E C (6) E D C A B (4) E D A C B (4) D E C A B (4) D A E B C (4) B C A E D (4) E C B A D (3) E C A B D (3) D E A C B (3) D A B C E (3) A B E D C (3) D E A B C (2) D C B A E (2) C E D B A (2) C E B D A (2) C B D E A (2) B D A C E (2) B A D C E (2) B A C D E (2) A D B E C (2) A B E C D (2) E C D B A (1) E A D C B (1) E A D B C (1) E A B D C (1) E A B C D (1) D C E B A (1) C D E B A (1) C B D A E (1) C B A E D (1) B A E C D (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 10 12 0 B -4 0 2 14 8 C -10 -2 0 -6 -6 D -12 -14 6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.712275 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.287725 Sum of squares = 0.590121560488 Cumulative probabilities = A: 0.712275 B: 0.712275 C: 0.712275 D: 0.712275 E: 1.000000 A B C D E A 0 4 10 12 0 B -4 0 2 14 8 C -10 -2 0 -6 -6 D -12 -14 6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555688798 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 E=19 B=18 A=15 so A is eliminated. Round 2 votes counts: D=29 B=29 C=23 E=19 so E is eliminated. Round 3 votes counts: D=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:210 E:204 C:188 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 12 0 B -4 0 2 14 8 C -10 -2 0 -6 -6 D -12 -14 6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555688798 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 12 0 B -4 0 2 14 8 C -10 -2 0 -6 -6 D -12 -14 6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555688798 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 12 0 B -4 0 2 14 8 C -10 -2 0 -6 -6 D -12 -14 6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555688798 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1351: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) C D A B E (8) D C E A B (7) B E A D C (5) B A E C D (5) E D C B A (4) E A B D C (4) E B A C D (3) D A C B E (3) C D E A B (3) B A E D C (3) A B D E C (3) E D A B C (2) E C B D A (2) E B C D A (2) D E C A B (2) C D B A E (2) C D A E B (2) B E A C D (2) B A C D E (2) A D C B E (2) A C B D E (2) A B E D C (2) A B D C E (2) A B C D E (2) E D C A B (1) E C D B A (1) E B D C A (1) E B D A C (1) E B C A D (1) E A D B C (1) D C A E B (1) D C A B E (1) D A C E B (1) C E B D A (1) C D B E A (1) C B E A D (1) C B A D E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 16 8 -12 B -2 0 6 12 -2 C -16 -6 0 -16 -12 D -8 -12 16 0 -6 E 12 2 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 16 8 -12 B -2 0 6 12 -2 C -16 -6 0 -16 -12 D -8 -12 16 0 -6 E 12 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=19 B=17 D=15 A=15 so D is eliminated. Round 2 votes counts: E=36 C=28 A=19 B=17 so B is eliminated. Round 3 votes counts: E=43 A=29 C=28 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:207 B:207 D:195 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 16 8 -12 B -2 0 6 12 -2 C -16 -6 0 -16 -12 D -8 -12 16 0 -6 E 12 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 8 -12 B -2 0 6 12 -2 C -16 -6 0 -16 -12 D -8 -12 16 0 -6 E 12 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 8 -12 B -2 0 6 12 -2 C -16 -6 0 -16 -12 D -8 -12 16 0 -6 E 12 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1352: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (6) D E A C B (5) C A E D B (5) B C E A D (5) E D B C A (4) E C B A D (4) B D E A C (4) B D A E C (4) A D C B E (4) E D C A B (3) E B D C A (3) C B A E D (3) C A E B D (3) B E C D A (3) E D A C B (2) E C D B A (2) E C A D B (2) E B C D A (2) D E A B C (2) D B A E C (2) C B E A D (2) B C A E D (2) B A D C E (2) A D C E B (2) A C D E B (2) A C D B E (2) E D B A C (1) E C D A B (1) E C B D A (1) E C A B D (1) E B C A D (1) D E B A C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C A D E B (1) C A B D E (1) B E D C A (1) B D A C E (1) B A C D E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -16 6 -4 B 14 0 0 8 -4 C 16 0 0 8 -4 D -6 -8 -8 0 -6 E 4 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -16 6 -4 B 14 0 0 8 -4 C 16 0 0 8 -4 D -6 -8 -8 0 -6 E 4 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=27 C=17 D=15 A=12 so A is eliminated. Round 2 votes counts: B=30 E=27 C=22 D=21 so D is eliminated. Round 3 votes counts: E=37 B=35 C=28 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:210 B:209 E:209 A:186 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -16 6 -4 B 14 0 0 8 -4 C 16 0 0 8 -4 D -6 -8 -8 0 -6 E 4 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 6 -4 B 14 0 0 8 -4 C 16 0 0 8 -4 D -6 -8 -8 0 -6 E 4 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 6 -4 B 14 0 0 8 -4 C 16 0 0 8 -4 D -6 -8 -8 0 -6 E 4 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1353: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) D C B E A (7) E D C A B (5) D E C B A (5) D E C A B (5) D C E B A (5) A E B C D (5) A B E C D (5) B D C A E (4) B A E C D (4) B A C D E (4) E A B D C (3) D B C E A (3) B A E D C (3) A C B E D (3) E C D A B (2) D B E A C (2) C E D A B (2) C A E D B (2) B D A C E (2) B A C E D (2) E D A C B (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E B A C (1) D C E A B (1) D C B A E (1) C E A D B (1) C D B E A (1) C D A E B (1) C D A B E (1) C A B D E (1) B E D A C (1) B C A D E (1) A C E B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -16 -18 -16 B -8 0 -14 -16 -6 C 16 14 0 -2 6 D 18 16 2 0 8 E 16 6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999092 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -16 -18 -16 B -8 0 -14 -16 -6 C 16 14 0 -2 6 D 18 16 2 0 8 E 16 6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=21 C=18 A=16 E=15 so E is eliminated. Round 2 votes counts: D=36 A=22 C=21 B=21 so C is eliminated. Round 3 votes counts: D=52 A=27 B=21 so B is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:217 E:204 A:179 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -16 -18 -16 B -8 0 -14 -16 -6 C 16 14 0 -2 6 D 18 16 2 0 8 E 16 6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 -18 -16 B -8 0 -14 -16 -6 C 16 14 0 -2 6 D 18 16 2 0 8 E 16 6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 -18 -16 B -8 0 -14 -16 -6 C 16 14 0 -2 6 D 18 16 2 0 8 E 16 6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1354: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) C D B E A (7) D C B E A (5) B D E A C (5) E B A D C (4) E A B C D (4) D B C E A (4) C E A D B (4) E B D C A (3) E A B D C (3) D B C A E (3) C E D B A (3) C D B A E (3) C A E D B (3) A E B D C (3) A E B C D (3) A C E B D (3) E C D A B (2) E C A D B (2) E B D A C (2) E A C B D (2) C E D A B (2) C D A B E (2) C A D E B (2) C A D B E (2) B A D E C (2) E D B C A (1) E C D B A (1) E C A B D (1) D B E C A (1) D B A C E (1) C D E B A (1) B E D C A (1) B D E C A (1) B D A E C (1) A D B C E (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -8 2 -18 B -2 0 -10 -2 -16 C 8 10 0 14 -4 D -2 2 -14 0 -12 E 18 16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -8 2 -18 B -2 0 -10 -2 -16 C 8 10 0 14 -4 D -2 2 -14 0 -12 E 18 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=25 A=22 D=14 B=10 so B is eliminated. Round 2 votes counts: C=29 E=26 A=24 D=21 so D is eliminated. Round 3 votes counts: C=41 E=33 A=26 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:214 A:189 D:187 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 2 -18 B -2 0 -10 -2 -16 C 8 10 0 14 -4 D -2 2 -14 0 -12 E 18 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 2 -18 B -2 0 -10 -2 -16 C 8 10 0 14 -4 D -2 2 -14 0 -12 E 18 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 2 -18 B -2 0 -10 -2 -16 C 8 10 0 14 -4 D -2 2 -14 0 -12 E 18 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1355: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) D A E C B (7) B E D A C (6) B E C D A (6) E B C D A (5) C E B D A (5) C B E A D (5) E D B C A (4) D E B A C (4) A D C B E (4) E B D C A (3) C A E D B (3) B E D C A (3) B C E A D (3) A D B E C (3) D B E A C (2) D A E B C (2) C E B A D (2) C A B E D (2) B E C A D (2) B E A D C (2) B A D E C (2) A C B D E (2) E D B A C (1) E C B D A (1) D B A E C (1) D A B E C (1) C E A B D (1) C B A E D (1) C A D E B (1) B D A E C (1) B A E D C (1) A D B C E (1) A C D E B (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 6 -4 -10 B 20 0 14 14 4 C -6 -14 0 -16 -16 D 4 -14 16 0 -14 E 10 -4 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 -4 -10 B 20 0 14 14 4 C -6 -14 0 -16 -16 D 4 -14 16 0 -14 E 10 -4 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=23 C=20 D=17 E=14 so E is eliminated. Round 2 votes counts: B=34 A=23 D=22 C=21 so C is eliminated. Round 3 votes counts: B=48 A=30 D=22 so D is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:218 D:196 A:186 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 6 -4 -10 B 20 0 14 14 4 C -6 -14 0 -16 -16 D 4 -14 16 0 -14 E 10 -4 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -4 -10 B 20 0 14 14 4 C -6 -14 0 -16 -16 D 4 -14 16 0 -14 E 10 -4 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -4 -10 B 20 0 14 14 4 C -6 -14 0 -16 -16 D 4 -14 16 0 -14 E 10 -4 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1356: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) D E A C B (8) C B A D E (7) A E D B C (7) E D A C B (6) C B D E A (6) B C A E D (6) A E D C B (4) D E B A C (3) C B D A E (3) C B A E D (3) B A C E D (3) A C E D B (3) E D A B C (2) D E C B A (2) D E B C A (2) D B E C A (2) C D E A B (2) B D E C A (2) A B E C D (2) A B C E D (2) D E C A B (1) D C E B A (1) D C E A B (1) C A B E D (1) C A B D E (1) B E D A C (1) B D E A C (1) B C D E A (1) B C D A E (1) B A E C D (1) A E C D B (1) A E B D C (1) A C D E B (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -4 10 18 B 10 0 -4 8 10 C 4 4 0 10 6 D -10 -8 -10 0 10 E -18 -10 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 10 18 B 10 0 -4 8 10 C 4 4 0 10 6 D -10 -8 -10 0 10 E -18 -10 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=23 A=23 D=20 E=8 so E is eliminated. Round 2 votes counts: D=28 B=26 C=23 A=23 so C is eliminated. Round 3 votes counts: B=45 D=30 A=25 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:212 A:207 D:191 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 10 18 B 10 0 -4 8 10 C 4 4 0 10 6 D -10 -8 -10 0 10 E -18 -10 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 10 18 B 10 0 -4 8 10 C 4 4 0 10 6 D -10 -8 -10 0 10 E -18 -10 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 10 18 B 10 0 -4 8 10 C 4 4 0 10 6 D -10 -8 -10 0 10 E -18 -10 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1357: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (15) B E C A D (10) D C A E B (9) B A E D C (8) C D E A B (7) B E A C D (7) E B C D A (6) C E D B A (6) A D C B E (6) E C B D A (5) C E B D A (5) B A E C D (4) B E C D A (3) A B D E C (3) A D B E C (2) E D C B A (1) A D C E B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 -14 0 B 10 0 -10 4 -10 C 4 10 0 6 2 D 14 -4 -6 0 -10 E 0 10 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -14 0 B 10 0 -10 4 -10 C 4 10 0 6 2 D 14 -4 -6 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=24 C=18 A=14 E=12 so E is eliminated. Round 2 votes counts: B=38 D=25 C=23 A=14 so A is eliminated. Round 3 votes counts: B=42 D=35 C=23 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 E:209 B:197 D:197 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -14 0 B 10 0 -10 4 -10 C 4 10 0 6 2 D 14 -4 -6 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -14 0 B 10 0 -10 4 -10 C 4 10 0 6 2 D 14 -4 -6 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -14 0 B 10 0 -10 4 -10 C 4 10 0 6 2 D 14 -4 -6 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1358: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) D E A C B (6) D E B C A (5) D B A E C (4) C E B A D (4) B C E D A (4) B A D C E (4) E C D A B (3) E C A D B (3) E A C D B (3) C A E B D (3) B C D E A (3) B C A D E (3) B A C E D (3) B A C D E (3) A C B E D (3) A B C E D (3) D E C A B (2) D A E C B (2) D A E B C (2) D A B E C (2) C E B D A (2) C E A D B (2) C E A B D (2) B D C A E (2) B D A C E (2) B C A E D (2) A D E C B (2) A B D C E (2) E D C B A (1) E C D B A (1) C B E A D (1) C B A E D (1) B C D A E (1) A E D C B (1) A E C D B (1) A D B E C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -8 -2 -2 B -8 0 -4 0 -8 C 8 4 0 4 2 D 2 0 -4 0 -4 E 2 8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -2 -2 B -8 0 -4 0 -8 C 8 4 0 4 2 D 2 0 -4 0 -4 E 2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 E=20 C=15 A=15 so C is eliminated. Round 2 votes counts: E=30 B=29 D=23 A=18 so A is eliminated. Round 3 votes counts: B=39 E=35 D=26 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:206 A:198 D:197 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -2 -2 B -8 0 -4 0 -8 C 8 4 0 4 2 D 2 0 -4 0 -4 E 2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -2 -2 B -8 0 -4 0 -8 C 8 4 0 4 2 D 2 0 -4 0 -4 E 2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -2 -2 B -8 0 -4 0 -8 C 8 4 0 4 2 D 2 0 -4 0 -4 E 2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1359: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E A D (9) D B A C E (7) C E B A D (6) B D C A E (6) A D E C B (6) D A B C E (5) A E D C B (5) E C B A D (4) D B C A E (4) B C E D A (4) B C D E A (3) A E C D B (3) E A D C B (2) E A C D B (2) D B E A C (2) D B A E C (2) D A B E C (2) C E A B D (2) B E C D A (2) B D C E A (2) E D B C A (1) E D A B C (1) E C B D A (1) E C A B D (1) E B C D A (1) D E A B C (1) D A E C B (1) C B A E D (1) C A B D E (1) B C E A D (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 -14 0 -12 10 B 14 0 10 -10 6 C 0 -10 0 -16 6 D 12 10 16 0 8 E -10 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -12 10 B 14 0 10 -10 6 C 0 -10 0 -16 6 D 12 10 16 0 8 E -10 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=19 B=19 A=15 E=13 so E is eliminated. Round 2 votes counts: D=36 C=25 B=20 A=19 so A is eliminated. Round 3 votes counts: D=50 C=30 B=20 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:210 A:192 C:190 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 0 -12 10 B 14 0 10 -10 6 C 0 -10 0 -16 6 D 12 10 16 0 8 E -10 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -12 10 B 14 0 10 -10 6 C 0 -10 0 -16 6 D 12 10 16 0 8 E -10 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -12 10 B 14 0 10 -10 6 C 0 -10 0 -16 6 D 12 10 16 0 8 E -10 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1360: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (14) C D E B A (13) C D E A B (9) B A E D C (7) D E B A C (5) C A E B D (5) C A B E D (5) A E B D C (5) D C E B A (4) D B E A C (3) C E D A B (3) C D B E A (3) E B A D C (2) C E A D B (2) C D B A E (2) C B A D E (2) A C E B D (2) A B E C D (2) E D B A C (1) E B D A C (1) D E B C A (1) D B E C A (1) D B C E A (1) B E A D C (1) B D E A C (1) B D C A E (1) B D A E C (1) B A D E C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -4 0 -2 B 4 0 0 4 -6 C 4 0 0 0 8 D 0 -4 0 0 -4 E 2 6 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.416174 C: 0.583826 D: 0.000000 E: 0.000000 Sum of squares = 0.514053662531 Cumulative probabilities = A: 0.000000 B: 0.416174 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 0 -2 B 4 0 0 4 -6 C 4 0 0 0 8 D 0 -4 0 0 -4 E 2 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=24 D=15 B=13 E=4 so E is eliminated. Round 2 votes counts: C=44 A=24 D=16 B=16 so D is eliminated. Round 3 votes counts: C=48 B=28 A=24 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:206 E:202 B:201 D:196 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 0 -2 B 4 0 0 4 -6 C 4 0 0 0 8 D 0 -4 0 0 -4 E 2 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 -2 B 4 0 0 4 -6 C 4 0 0 0 8 D 0 -4 0 0 -4 E 2 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 -2 B 4 0 0 4 -6 C 4 0 0 0 8 D 0 -4 0 0 -4 E 2 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1361: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) A E C B D (8) D B E A C (7) D B C E A (7) D B C A E (7) C D A E B (6) C A E D B (6) E A B C D (5) D B A E C (4) C E A B D (4) C A E B D (4) D C B A E (3) C A D E B (3) B E A D C (3) D C B E A (2) B E D A C (2) A E B C D (2) A C E B D (2) A B E D C (2) E C A B D (1) E B A C D (1) E A C B D (1) D C A E B (1) D C A B E (1) D B E C A (1) D A C B E (1) C E A D B (1) B E A C D (1) B D E C A (1) B D A E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 4 -8 6 B 0 0 10 -2 8 C -4 -10 0 -8 -2 D 8 2 8 0 12 E -6 -8 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -8 6 B 0 0 10 -2 8 C -4 -10 0 -8 -2 D 8 2 8 0 12 E -6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=24 B=18 A=16 E=8 so E is eliminated. Round 2 votes counts: D=34 C=25 A=22 B=19 so B is eliminated. Round 3 votes counts: D=48 A=27 C=25 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:208 A:201 C:188 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -8 6 B 0 0 10 -2 8 C -4 -10 0 -8 -2 D 8 2 8 0 12 E -6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -8 6 B 0 0 10 -2 8 C -4 -10 0 -8 -2 D 8 2 8 0 12 E -6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -8 6 B 0 0 10 -2 8 C -4 -10 0 -8 -2 D 8 2 8 0 12 E -6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1362: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) B C D A E (7) E A D B C (6) E A C D B (6) B C E A D (6) D A E C B (5) C D A E B (5) C B D A E (5) D C A E B (3) D A E B C (3) D A C E B (3) C E D A B (3) C B D E A (3) B E C A D (3) B E A D C (3) B E A C D (3) B C D E A (3) A E D C B (3) C D B A E (2) B D A E C (2) B C E D A (2) B A D E C (2) A D E B C (2) E C A D B (1) E A B D C (1) D B A E C (1) D A B E C (1) D A B C E (1) C D A B E (1) C B E D A (1) C B E A D (1) B D C A E (1) B A E D C (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 8 6 -4 0 B -8 0 -2 -12 -2 C -6 2 0 4 -6 D 4 12 -4 0 2 E 0 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 A B C D E A 0 8 6 -4 0 B -8 0 -2 -12 -2 C -6 2 0 4 -6 D 4 12 -4 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=22 C=21 D=17 A=7 so A is eliminated. Round 2 votes counts: B=33 E=26 C=21 D=20 so D is eliminated. Round 3 votes counts: E=37 B=36 C=27 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:207 A:205 E:203 C:197 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 -4 0 B -8 0 -2 -12 -2 C -6 2 0 4 -6 D 4 12 -4 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -4 0 B -8 0 -2 -12 -2 C -6 2 0 4 -6 D 4 12 -4 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -4 0 B -8 0 -2 -12 -2 C -6 2 0 4 -6 D 4 12 -4 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1363: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) A E D C B (8) D A E B C (7) B C D E A (7) C E A B D (6) C B E D A (6) C B E A D (6) E A C D B (5) B D C E A (5) B D C A E (5) A E C D B (5) D B A C E (4) E C A B D (3) C E B A D (3) B D A C E (3) E C A D B (2) E A C B D (2) D B C A E (2) D A B E C (2) A D E B C (2) E A D C B (1) C E B D A (1) C B D E A (1) B C E D A (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 4 -12 2 B 12 0 2 -2 10 C -4 -2 0 -4 0 D 12 2 4 0 2 E -2 -10 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -12 2 B 12 0 2 -2 10 C -4 -2 0 -4 0 D 12 2 4 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=23 B=21 A=16 E=13 so E is eliminated. Round 2 votes counts: C=28 D=27 A=24 B=21 so B is eliminated. Round 3 votes counts: D=40 C=36 A=24 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:211 D:210 C:195 E:193 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -12 2 B 12 0 2 -2 10 C -4 -2 0 -4 0 D 12 2 4 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -12 2 B 12 0 2 -2 10 C -4 -2 0 -4 0 D 12 2 4 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -12 2 B 12 0 2 -2 10 C -4 -2 0 -4 0 D 12 2 4 0 2 E -2 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1364: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) C D E A B (8) B A E D C (8) D A B E C (5) B A D E C (5) D E A B C (4) D A E B C (4) C E D B A (4) C E A B D (4) C D B A E (4) C B D A E (4) E C A D B (3) E A B D C (3) C E B A D (3) E C D A B (2) C D E B A (2) C B A E D (2) C B A D E (2) A E B D C (2) E D C A B (1) E D A C B (1) E D A B C (1) E A D B C (1) D E C A B (1) D E A C B (1) D B A E C (1) C D B E A (1) C D A E B (1) C D A B E (1) B E A D C (1) B D A E C (1) B D A C E (1) B C A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -10 -18 -4 B -14 0 -12 -16 -16 C 10 12 0 10 4 D 18 16 -10 0 0 E 4 16 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 -18 -4 B -14 0 -12 -16 -16 C 10 12 0 10 4 D 18 16 -10 0 0 E 4 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=47 B=20 D=16 E=12 A=5 so A is eliminated. Round 2 votes counts: C=47 B=22 D=16 E=15 so E is eliminated. Round 3 votes counts: C=52 B=27 D=21 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:212 E:208 A:191 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 -18 -4 B -14 0 -12 -16 -16 C 10 12 0 10 4 D 18 16 -10 0 0 E 4 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 -18 -4 B -14 0 -12 -16 -16 C 10 12 0 10 4 D 18 16 -10 0 0 E 4 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 -18 -4 B -14 0 -12 -16 -16 C 10 12 0 10 4 D 18 16 -10 0 0 E 4 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1365: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) B A C E D (11) D E C B A (8) C E D A B (8) E C D B A (6) E C D A B (4) D B E C A (4) D A B E C (4) A B D C E (4) D E C A B (3) B A D E C (3) B A D C E (3) D C E A B (2) D B A E C (2) C E D B A (2) C E A D B (2) B D E C A (2) B D A E C (2) B A C D E (2) A D B E C (2) E D C A B (1) E C B D A (1) D A E C B (1) C E A B D (1) C B E A D (1) C A E B D (1) B E C D A (1) B C E A D (1) B A E C D (1) A D C E B (1) A D B C E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 4 -2 6 B 0 0 12 -4 14 C -4 -12 0 14 10 D 2 4 -14 0 -12 E -6 -14 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.466667 C: 0.133333 D: 0.400000 E: 0.000000 Sum of squares = 0.395555555482 Cumulative probabilities = A: 0.000000 B: 0.466667 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -2 6 B 0 0 12 -4 14 C -4 -12 0 14 10 D 2 4 -14 0 -12 E -6 -14 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.466667 C: 0.133333 D: 0.400000 E: 0.000000 Sum of squares = 0.395555554949 Cumulative probabilities = A: 0.000000 B: 0.466667 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=24 A=23 C=15 E=12 so E is eliminated. Round 2 votes counts: C=26 B=26 D=25 A=23 so A is eliminated. Round 3 votes counts: B=43 D=29 C=28 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:211 A:204 C:204 E:191 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 -2 6 B 0 0 12 -4 14 C -4 -12 0 14 10 D 2 4 -14 0 -12 E -6 -14 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.466667 C: 0.133333 D: 0.400000 E: 0.000000 Sum of squares = 0.395555554949 Cumulative probabilities = A: 0.000000 B: 0.466667 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 6 B 0 0 12 -4 14 C -4 -12 0 14 10 D 2 4 -14 0 -12 E -6 -14 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.466667 C: 0.133333 D: 0.400000 E: 0.000000 Sum of squares = 0.395555554949 Cumulative probabilities = A: 0.000000 B: 0.466667 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 6 B 0 0 12 -4 14 C -4 -12 0 14 10 D 2 4 -14 0 -12 E -6 -14 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.466667 C: 0.133333 D: 0.400000 E: 0.000000 Sum of squares = 0.395555554949 Cumulative probabilities = A: 0.000000 B: 0.466667 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1366: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) A D E C B (11) E D B C A (10) C B A E D (9) A C B D E (7) C A B D E (5) B C E A D (5) A C D B E (5) E D B A C (4) B E C D A (4) A D C E B (4) E B D C A (3) D A E C B (3) C B E D A (2) C B A D E (2) B C E D A (2) B C A E D (2) E D A B C (1) E A D B C (1) E A B D C (1) D E C B A (1) D E A C B (1) D C A E B (1) D A E B C (1) B E C A D (1) A D E B C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 6 12 6 B -10 0 -4 -10 -8 C -6 4 0 -8 -10 D -12 10 8 0 6 E -6 8 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 12 6 B -10 0 -4 -10 -8 C -6 4 0 -8 -10 D -12 10 8 0 6 E -6 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=20 D=18 C=18 B=14 so B is eliminated. Round 2 votes counts: A=30 C=27 E=25 D=18 so D is eliminated. Round 3 votes counts: E=38 A=34 C=28 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:206 E:203 C:190 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 12 6 B -10 0 -4 -10 -8 C -6 4 0 -8 -10 D -12 10 8 0 6 E -6 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 12 6 B -10 0 -4 -10 -8 C -6 4 0 -8 -10 D -12 10 8 0 6 E -6 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 12 6 B -10 0 -4 -10 -8 C -6 4 0 -8 -10 D -12 10 8 0 6 E -6 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1367: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) E A D C B (6) A E C D B (6) A C E D B (5) A C D E B (5) C A D E B (4) B D C E A (4) B C A E D (4) E B D A C (3) C B D A E (3) C A D B E (3) B D E C A (3) B C D A E (3) E A D B C (2) D E B A C (2) D B E C A (2) C B A D E (2) C A B D E (2) B E D A C (2) B E C D A (2) B C E A D (2) B C D E A (2) E D B A C (1) E D A C B (1) E D A B C (1) E B A C D (1) E A C D B (1) E A B D C (1) D E C A B (1) D C E B A (1) D C B A E (1) D B C E A (1) C D B A E (1) C B A E D (1) C A B E D (1) B E C A D (1) B E A D C (1) B C E D A (1) B C A D E (1) A E D C B (1) A E C B D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -16 6 -8 B 14 0 4 8 10 C 16 -4 0 6 -2 D -6 -8 -6 0 -16 E 8 -10 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 6 -8 B 14 0 4 8 10 C 16 -4 0 6 -2 D -6 -8 -6 0 -16 E 8 -10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=20 E=17 C=17 D=8 so D is eliminated. Round 2 votes counts: B=41 E=20 A=20 C=19 so C is eliminated. Round 3 votes counts: B=49 A=30 E=21 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:208 E:208 A:184 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 6 -8 B 14 0 4 8 10 C 16 -4 0 6 -2 D -6 -8 -6 0 -16 E 8 -10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 6 -8 B 14 0 4 8 10 C 16 -4 0 6 -2 D -6 -8 -6 0 -16 E 8 -10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 6 -8 B 14 0 4 8 10 C 16 -4 0 6 -2 D -6 -8 -6 0 -16 E 8 -10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1368: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) E A B C D (6) D C B A E (6) E B A C D (5) E A C B D (4) D C A B E (4) D B C E A (4) D B C A E (4) C D B A E (4) C D A B E (4) E B C A D (3) C B D E A (3) B E C D A (3) B E C A D (3) B D E A C (3) B D C E A (3) A E D B C (3) A E C B D (3) D A E B C (2) C B E D A (2) B E D C A (2) B E A C D (2) B D E C A (2) A E D C B (2) E C B A D (1) E B A D C (1) E A D B C (1) D E A B C (1) D C A E B (1) D B E C A (1) D B E A C (1) D B A E C (1) C E B A D (1) C A E D B (1) C A E B D (1) B C E D A (1) B C E A D (1) B C D E A (1) A E C D B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -12 -6 -22 B 16 0 20 14 10 C 12 -20 0 4 -14 D 6 -14 -4 0 -8 E 22 -10 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 -6 -22 B 16 0 20 14 10 C 12 -20 0 4 -14 D 6 -14 -4 0 -8 E 22 -10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 B=21 C=16 A=11 so A is eliminated. Round 2 votes counts: E=36 D=25 B=21 C=18 so C is eliminated. Round 3 votes counts: E=40 D=34 B=26 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:230 E:217 C:191 D:190 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 -6 -22 B 16 0 20 14 10 C 12 -20 0 4 -14 D 6 -14 -4 0 -8 E 22 -10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -6 -22 B 16 0 20 14 10 C 12 -20 0 4 -14 D 6 -14 -4 0 -8 E 22 -10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -6 -22 B 16 0 20 14 10 C 12 -20 0 4 -14 D 6 -14 -4 0 -8 E 22 -10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1369: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) A E D C B (9) A E C D B (7) C B A E D (6) C A E B D (6) D E A B C (5) C A E D B (5) E A D C B (4) D B A E C (4) C B E A D (4) D A E C B (3) B D E C A (3) A C E D B (3) D A E B C (2) C E A B D (2) C B D A E (2) C A B E D (2) B E D A C (2) B D E A C (2) B D C A E (2) B D A C E (2) B C D A E (2) A D C E B (2) E A D B C (1) E A C D B (1) D B E A C (1) C E B A D (1) C E A D B (1) C B A D E (1) C A B D E (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 10 -2 14 20 B -10 0 -22 2 -6 C 2 22 0 10 6 D -14 -2 -10 0 -14 E -20 6 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 14 20 B -10 0 -22 2 -6 C 2 22 0 10 6 D -14 -2 -10 0 -14 E -20 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 A=22 D=15 E=6 so E is eliminated. Round 2 votes counts: C=31 A=28 B=26 D=15 so D is eliminated. Round 3 votes counts: A=38 C=31 B=31 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:220 E:197 B:182 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 14 20 B -10 0 -22 2 -6 C 2 22 0 10 6 D -14 -2 -10 0 -14 E -20 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 14 20 B -10 0 -22 2 -6 C 2 22 0 10 6 D -14 -2 -10 0 -14 E -20 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 14 20 B -10 0 -22 2 -6 C 2 22 0 10 6 D -14 -2 -10 0 -14 E -20 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1370: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) C D E B A (7) C D E A B (7) A B D C E (7) B A E D C (6) E B A C D (5) D C B A E (5) D C A B E (4) A D C B E (4) A D B C E (4) E C D B A (3) E C D A B (3) E B D C A (3) E B A D C (3) E A B C D (3) B E A D C (3) A E B C D (3) E C A D B (2) E B C D A (2) E A B D C (2) C D A E B (2) C D A B E (2) A E B D C (2) E B C A D (1) E A C B D (1) D B A C E (1) C D B A E (1) B E D C A (1) B A D E C (1) A C D E B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 16 14 18 8 B -16 0 16 8 0 C -14 -16 0 -12 -8 D -18 -8 12 0 -4 E -8 0 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 18 8 B -16 0 16 8 0 C -14 -16 0 -12 -8 D -18 -8 12 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=28 C=19 B=11 D=10 so D is eliminated. Round 2 votes counts: A=32 E=28 C=28 B=12 so B is eliminated. Round 3 votes counts: A=40 E=32 C=28 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:204 E:202 D:191 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 18 8 B -16 0 16 8 0 C -14 -16 0 -12 -8 D -18 -8 12 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 18 8 B -16 0 16 8 0 C -14 -16 0 -12 -8 D -18 -8 12 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 18 8 B -16 0 16 8 0 C -14 -16 0 -12 -8 D -18 -8 12 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1371: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) A C E B D (8) D B E C A (7) D B C E A (7) C A D B E (6) E A B D C (5) D C B A E (5) A E C B D (5) E B D A C (4) D C B E A (4) C D A B E (4) B D E C A (4) E D B A C (3) E A D B C (3) C A B D E (3) A E C D B (3) A C E D B (3) C B D E A (2) B E D C A (2) B D C E A (2) A E B C D (2) E A B C D (1) D E B A C (1) D E A B C (1) D A C E B (1) C B D A E (1) C B A E D (1) B E D A C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -14 -16 6 B 6 0 -14 -18 16 C 14 14 0 -2 16 D 16 18 2 0 18 E -6 -16 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -16 6 B 6 0 -14 -18 16 C 14 14 0 -2 16 D 16 18 2 0 18 E -6 -16 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 A=23 E=16 B=9 so B is eliminated. Round 2 votes counts: D=32 C=26 A=23 E=19 so E is eliminated. Round 3 votes counts: D=42 A=32 C=26 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 C:221 B:195 A:185 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -14 -16 6 B 6 0 -14 -18 16 C 14 14 0 -2 16 D 16 18 2 0 18 E -6 -16 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -16 6 B 6 0 -14 -18 16 C 14 14 0 -2 16 D 16 18 2 0 18 E -6 -16 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -16 6 B 6 0 -14 -18 16 C 14 14 0 -2 16 D 16 18 2 0 18 E -6 -16 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1372: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B C A E D (9) D C E A B (6) C D B A E (6) B A E C D (6) C D B E A (5) B E A C D (5) E A B D C (4) D A C E B (4) C B D E A (4) A B E C D (4) E A D B C (3) B C E A D (3) E D B A C (2) E B A D C (2) D E C A B (2) D E A C B (2) D C E B A (2) D A E C B (2) C D A B E (2) C B D A E (2) C B A D E (2) A E B D C (2) E D B C A (1) E B D C A (1) E B C D A (1) E B A C D (1) D E C B A (1) C A D B E (1) B E C D A (1) B E C A D (1) B A C E D (1) A D E B C (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -18 -6 6 B 12 0 -2 0 8 C 18 2 0 10 16 D 6 0 -10 0 6 E -6 -8 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -6 6 B 12 0 -2 0 8 C 18 2 0 10 16 D 6 0 -10 0 6 E -6 -8 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=26 C=22 E=15 A=9 so A is eliminated. Round 2 votes counts: B=31 D=30 C=22 E=17 so E is eliminated. Round 3 votes counts: B=42 D=36 C=22 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:223 B:209 D:201 A:185 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -18 -6 6 B 12 0 -2 0 8 C 18 2 0 10 16 D 6 0 -10 0 6 E -6 -8 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -6 6 B 12 0 -2 0 8 C 18 2 0 10 16 D 6 0 -10 0 6 E -6 -8 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -6 6 B 12 0 -2 0 8 C 18 2 0 10 16 D 6 0 -10 0 6 E -6 -8 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1373: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) C A E B D (6) E A C B D (5) D B C E A (5) D A E C B (5) A E C D B (4) A C D E B (4) E A B C D (3) B E D C A (3) B D E C A (3) B D C E A (3) A E C B D (3) E B C A D (2) D C B A E (2) D B E C A (2) D B E A C (2) D A C E B (2) C B D A E (2) C A D B E (2) B D C A E (2) B C E D A (2) B C D E A (2) B C D A E (2) A E D C B (2) A C E D B (2) A C E B D (2) E C A B D (1) E B D A C (1) E B C D A (1) E B A D C (1) E A D C B (1) E A D B C (1) E A B D C (1) D E B A C (1) D C A B E (1) D A E B C (1) D A B C E (1) C E B A D (1) C D B A E (1) C B E A D (1) C B A E D (1) C A B E D (1) C A B D E (1) B E D A C (1) B E C D A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -12 -6 10 B 0 0 -2 6 -2 C 12 2 0 2 8 D 6 -6 -2 0 4 E -10 2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -6 10 B 0 0 -2 6 -2 C 12 2 0 2 8 D 6 -6 -2 0 4 E -10 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=20 A=18 E=17 C=16 so C is eliminated. Round 2 votes counts: D=30 A=28 B=24 E=18 so E is eliminated. Round 3 votes counts: A=40 D=30 B=30 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 B:201 D:201 A:196 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 -6 10 B 0 0 -2 6 -2 C 12 2 0 2 8 D 6 -6 -2 0 4 E -10 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -6 10 B 0 0 -2 6 -2 C 12 2 0 2 8 D 6 -6 -2 0 4 E -10 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -6 10 B 0 0 -2 6 -2 C 12 2 0 2 8 D 6 -6 -2 0 4 E -10 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1374: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) C B D E A (7) A D E C B (5) D E A B C (4) C A B D E (4) A D C E B (4) E A D B C (3) C D B A E (3) C B D A E (3) B C D E A (3) A C B E D (3) A C B D E (3) E D A B C (2) E B D C A (2) E B C A D (2) D E B C A (2) C B A D E (2) B E D C A (2) B E C D A (2) B D E C A (2) B C E D A (2) A E D C B (2) A E C B D (2) A E B C D (2) A D E B C (2) A C D B E (2) E D B A C (1) E B A C D (1) E A B D C (1) E A B C D (1) D C B E A (1) D C B A E (1) D B E C A (1) D A E B C (1) D A C E B (1) D A C B E (1) C D B E A (1) C D A B E (1) C B A E D (1) C A D B E (1) C A B E D (1) B D C E A (1) A E B D C (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 20 10 12 18 B -20 0 -4 -2 -4 C -10 4 0 0 -2 D -12 2 0 0 14 E -18 4 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 12 18 B -20 0 -4 -2 -4 C -10 4 0 0 -2 D -12 2 0 0 14 E -18 4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=24 E=13 D=12 B=12 so D is eliminated. Round 2 votes counts: A=42 C=26 E=19 B=13 so B is eliminated. Round 3 votes counts: A=42 C=32 E=26 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 D:202 C:196 E:187 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 12 18 B -20 0 -4 -2 -4 C -10 4 0 0 -2 D -12 2 0 0 14 E -18 4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 12 18 B -20 0 -4 -2 -4 C -10 4 0 0 -2 D -12 2 0 0 14 E -18 4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 12 18 B -20 0 -4 -2 -4 C -10 4 0 0 -2 D -12 2 0 0 14 E -18 4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1375: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) E C D B A (6) C E A B D (6) A B D E C (6) A B D C E (6) E C A D B (5) A E C B D (5) A E B D C (5) A B C D E (4) D B E C A (3) D B C A E (3) C E D B A (3) C A E B D (3) A C E B D (3) A B E D C (3) E D B C A (2) E D B A C (2) E A C D B (2) D B E A C (2) D B C E A (2) D B A C E (2) C D B E A (2) B D A E C (2) E D C B A (1) E C A B D (1) E A D B C (1) E A C B D (1) D C B E A (1) C E B A D (1) C E A D B (1) C D B A E (1) C B D A E (1) C B A D E (1) C A B D E (1) B D C A E (1) B A D E C (1) B A D C E (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 10 10 18 16 B -10 0 8 20 4 C -10 -8 0 -4 2 D -18 -20 4 0 -4 E -16 -4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 18 16 B -10 0 8 20 4 C -10 -8 0 -4 2 D -18 -20 4 0 -4 E -16 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=21 C=20 D=13 B=12 so B is eliminated. Round 2 votes counts: A=36 D=23 E=21 C=20 so C is eliminated. Round 3 votes counts: A=41 E=32 D=27 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 B:211 E:191 C:190 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 18 16 B -10 0 8 20 4 C -10 -8 0 -4 2 D -18 -20 4 0 -4 E -16 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 18 16 B -10 0 8 20 4 C -10 -8 0 -4 2 D -18 -20 4 0 -4 E -16 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 18 16 B -10 0 8 20 4 C -10 -8 0 -4 2 D -18 -20 4 0 -4 E -16 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1376: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) E B C D A (8) D A C E B (8) A C B D E (7) C B A E D (6) E B D C A (5) D A E C B (5) D E B A C (4) D A E B C (4) C A B E D (4) E D B C A (3) D E A B C (3) C B E A D (3) E D C B A (2) E D B A C (2) D E C B A (2) D E A C B (2) B C A E D (2) A D C B E (2) A D B C E (2) A C D B E (2) A C B E D (2) A B E D C (2) A B C E D (2) E B C A D (1) D E B C A (1) D C A E B (1) D A C B E (1) C A D B E (1) C A B D E (1) B C E A D (1) B A E C D (1) Total count = 100 A B C D E A 0 -2 -2 -2 6 B 2 0 2 10 -2 C 2 -2 0 2 -10 D 2 -10 -2 0 -8 E -6 2 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999947 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -2 -2 -2 6 B 2 0 2 10 -2 C 2 -2 0 2 -10 D 2 -10 -2 0 -8 E -6 2 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000104 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=21 A=19 C=15 B=14 so B is eliminated. Round 2 votes counts: E=31 D=31 A=20 C=18 so C is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:207 B:206 A:200 C:196 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 -2 6 B 2 0 2 10 -2 C 2 -2 0 2 -10 D 2 -10 -2 0 -8 E -6 2 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000104 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -2 6 B 2 0 2 10 -2 C 2 -2 0 2 -10 D 2 -10 -2 0 -8 E -6 2 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000104 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -2 6 B 2 0 2 10 -2 C 2 -2 0 2 -10 D 2 -10 -2 0 -8 E -6 2 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000104 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1377: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (22) B A E C D (12) D C A E B (5) B A E D C (5) A B E C D (5) E B C A D (4) C D E B A (4) C D E A B (4) D C E B A (3) D B A C E (3) C E D A B (3) B A D E C (3) E C B A D (2) E A B C D (2) D B C A E (2) D A B E C (2) D A B C E (2) C E B D A (2) C E B A D (2) B D A C E (2) E C D A B (1) E C A D B (1) E C A B D (1) E B A C D (1) E A D C B (1) E A C B D (1) D A E C B (1) C E D B A (1) B C A E D (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -16 -14 -10 B -4 0 -8 -10 -22 C 16 8 0 -4 14 D 14 10 4 0 6 E 10 22 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -14 -10 B -4 0 -8 -10 -22 C 16 8 0 -4 14 D 14 10 4 0 6 E 10 22 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=24 C=16 E=14 A=6 so A is eliminated. Round 2 votes counts: D=40 B=30 C=16 E=14 so E is eliminated. Round 3 votes counts: D=41 B=37 C=22 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:217 D:217 E:206 A:182 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -16 -14 -10 B -4 0 -8 -10 -22 C 16 8 0 -4 14 D 14 10 4 0 6 E 10 22 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -14 -10 B -4 0 -8 -10 -22 C 16 8 0 -4 14 D 14 10 4 0 6 E 10 22 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -14 -10 B -4 0 -8 -10 -22 C 16 8 0 -4 14 D 14 10 4 0 6 E 10 22 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1378: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C A D B E (8) B E A D C (8) E B D C A (7) D C A E B (7) C D A E B (7) E D B C A (5) D C E A B (4) A C D B E (4) D E C B A (3) C D A B E (3) B E A C D (3) A C B E D (3) E B A D C (2) D E C A B (2) D A C E B (2) C B A D E (2) C A D E B (2) C A B D E (2) B E D C A (2) B E D A C (2) B E C D A (2) A C D E B (2) A C B D E (2) E A D B C (1) E A B D C (1) D E A C B (1) D C E B A (1) B C E A D (1) B A E C D (1) B A C E D (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -16 -12 -6 B -4 0 -12 -6 -12 C 16 12 0 -14 4 D 12 6 14 0 6 E 6 12 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -12 -6 B -4 0 -12 -6 -12 C 16 12 0 -14 4 D 12 6 14 0 6 E 6 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 D=20 B=20 A=12 so A is eliminated. Round 2 votes counts: C=35 E=24 D=21 B=20 so B is eliminated. Round 3 votes counts: E=42 C=37 D=21 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:209 E:204 A:185 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -16 -12 -6 B -4 0 -12 -6 -12 C 16 12 0 -14 4 D 12 6 14 0 6 E 6 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -12 -6 B -4 0 -12 -6 -12 C 16 12 0 -14 4 D 12 6 14 0 6 E 6 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -12 -6 B -4 0 -12 -6 -12 C 16 12 0 -14 4 D 12 6 14 0 6 E 6 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1379: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (5) D A E B C (5) D A B E C (5) B C D A E (5) E C A B D (4) D B A C E (4) A E D C B (4) A E C B D (4) E D C B A (3) E D A C B (3) E A C B D (3) D E A B C (3) D B C A E (3) D A B C E (3) C B A E D (3) B C D E A (3) B C A D E (3) A D E B C (3) E D C A B (2) E C B D A (2) E A D C B (2) E A C D B (2) D B C E A (2) D B A E C (2) B D C A E (2) A D E C B (2) A D B C E (2) E C D B A (1) E C A D B (1) E B D C A (1) D E B C A (1) D E B A C (1) D E A C B (1) D B E C A (1) C E B D A (1) C B E D A (1) C B E A D (1) B D C E A (1) B A C D E (1) A E C D B (1) A D B E C (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 -12 10 B -6 0 6 -16 -12 C -8 -6 0 -14 -26 D 12 16 14 0 10 E -10 12 26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -12 10 B -6 0 6 -16 -12 C -8 -6 0 -14 -26 D 12 16 14 0 10 E -10 12 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=29 A=19 B=15 C=6 so C is eliminated. Round 2 votes counts: D=31 E=30 B=20 A=19 so A is eliminated. Round 3 votes counts: E=40 D=39 B=21 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 E:209 A:206 B:186 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -12 10 B -6 0 6 -16 -12 C -8 -6 0 -14 -26 D 12 16 14 0 10 E -10 12 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -12 10 B -6 0 6 -16 -12 C -8 -6 0 -14 -26 D 12 16 14 0 10 E -10 12 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -12 10 B -6 0 6 -16 -12 C -8 -6 0 -14 -26 D 12 16 14 0 10 E -10 12 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1380: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B D A C (5) E B A D C (5) E B A C D (5) E D A B C (4) B E C D A (4) B E C A D (4) B C D E A (4) A D C E B (4) E B D C A (3) D C A B E (3) A D E C B (3) A C B E D (3) A C B D E (3) E D B A C (2) E A D B C (2) E A B D C (2) D C B E A (2) C D A B E (2) C B D A E (2) C A B D E (2) B C E D A (2) A E D B C (2) A E B C D (2) A C D E B (2) E B C A D (1) E A B C D (1) D E B C A (1) D E A C B (1) D E A B C (1) D C E B A (1) D C A E B (1) D B C E A (1) D A E C B (1) D A C E B (1) D A C B E (1) C D B A E (1) C B A E D (1) C A D B E (1) B E D C A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 24 10 -4 B -8 0 6 2 -6 C -24 -6 0 2 -4 D -10 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 24 10 -4 B -8 0 6 2 -6 C -24 -6 0 2 -4 D -10 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=30 B=16 D=14 C=9 so C is eliminated. Round 2 votes counts: A=34 E=30 B=19 D=17 so D is eliminated. Round 3 votes counts: A=43 E=34 B=23 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:219 E:209 B:197 D:191 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 24 10 -4 B -8 0 6 2 -6 C -24 -6 0 2 -4 D -10 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 24 10 -4 B -8 0 6 2 -6 C -24 -6 0 2 -4 D -10 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 24 10 -4 B -8 0 6 2 -6 C -24 -6 0 2 -4 D -10 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1381: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) D E C A B (10) B C A D E (9) E D A C B (6) E A C D B (5) C A D E B (4) B D E C A (4) A C B E D (4) D E B C A (3) C D A E B (3) B E D A C (3) B D C A E (3) E D B A C (2) E A C B D (2) D E B A C (2) D C E A B (2) B E A D C (2) B C D A E (2) B A E C D (2) A B C E D (2) E B A D C (1) E A D C B (1) D E A C B (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C E A (1) C D A B E (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B E D (1) B C A E D (1) B A C D E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -2 8 6 B 6 0 4 6 6 C 2 -4 0 12 8 D -8 -6 -12 0 2 E -6 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 8 6 B 6 0 4 6 6 C 2 -4 0 12 8 D -8 -6 -12 0 2 E -6 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 D=22 E=17 C=13 A=8 so A is eliminated. Round 2 votes counts: B=42 D=22 E=18 C=18 so E is eliminated. Round 3 votes counts: B=43 D=31 C=26 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:209 A:203 E:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 8 6 B 6 0 4 6 6 C 2 -4 0 12 8 D -8 -6 -12 0 2 E -6 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 8 6 B 6 0 4 6 6 C 2 -4 0 12 8 D -8 -6 -12 0 2 E -6 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 8 6 B 6 0 4 6 6 C 2 -4 0 12 8 D -8 -6 -12 0 2 E -6 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1382: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (5) C B E D A (5) A D C E B (5) E C B A D (4) D C A B E (4) D A C B E (4) B C E D A (4) A D E B C (4) E B C A D (3) E B A D C (3) E A B D C (3) D A C E B (3) C E B A D (3) C B D E A (3) B D A E C (3) A D E C B (3) E A C D B (2) D C B A E (2) D A B E C (2) D A B C E (2) C E A D B (2) B E D C A (2) B E C D A (2) B E C A D (2) B E A D C (2) B D E A C (2) A E D C B (2) A E B D C (2) E C A B D (1) E B A C D (1) E A D C B (1) E A D B C (1) E A C B D (1) E A B C D (1) D C A E B (1) C E D A B (1) C E B D A (1) C D A E B (1) C D A B E (1) C A D E B (1) B D C A E (1) B A E D C (1) A E D B C (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 12 2 0 B -2 0 -4 0 -4 C -12 4 0 -20 -2 D -2 0 20 0 -4 E 0 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.503048 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.496952 Sum of squares = 0.500018538213 Cumulative probabilities = A: 0.503048 B: 0.503048 C: 0.503048 D: 0.503048 E: 1.000000 A B C D E A 0 2 12 2 0 B -2 0 -4 0 -4 C -12 4 0 -20 -2 D -2 0 20 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999768 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=21 B=19 A=19 C=18 so C is eliminated. Round 2 votes counts: E=28 B=27 D=25 A=20 so A is eliminated. Round 3 votes counts: D=39 E=34 B=27 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:208 D:207 E:205 B:195 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 2 0 B -2 0 -4 0 -4 C -12 4 0 -20 -2 D -2 0 20 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999768 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 2 0 B -2 0 -4 0 -4 C -12 4 0 -20 -2 D -2 0 20 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999768 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 2 0 B -2 0 -4 0 -4 C -12 4 0 -20 -2 D -2 0 20 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999768 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1383: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) B E D C A (6) A C D E B (6) D E B C A (5) A C E B D (5) D E C B A (4) C D B E A (4) C A D E B (4) B E D A C (4) A E B D C (4) A B E C D (4) D E B A C (3) C D A E B (3) C A D B E (3) C A B E D (3) B D E C A (3) E D B A C (2) E B A D C (2) D B E C A (2) C D A B E (2) B E C D A (2) B E A D C (2) A D E C B (2) A B E D C (2) E B D A C (1) E A B D C (1) D C E A B (1) D C B E A (1) D C A E B (1) D B C E A (1) C D E B A (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B C A E D (1) A E D C B (1) A E C B D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -2 2 6 B -4 0 -6 4 4 C 2 6 0 4 -4 D -2 -4 -4 0 -2 E -6 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888868 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 4 -2 2 6 B -4 0 -6 4 4 C 2 6 0 4 -4 D -2 -4 -4 0 -2 E -6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=23 B=19 D=18 E=6 so E is eliminated. Round 2 votes counts: A=35 C=23 B=22 D=20 so D is eliminated. Round 3 votes counts: B=35 A=35 C=30 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:205 C:204 B:199 E:198 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 2 6 B -4 0 -6 4 4 C 2 6 0 4 -4 D -2 -4 -4 0 -2 E -6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 6 B -4 0 -6 4 4 C 2 6 0 4 -4 D -2 -4 -4 0 -2 E -6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 6 B -4 0 -6 4 4 C 2 6 0 4 -4 D -2 -4 -4 0 -2 E -6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1384: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (5) B E D C A (5) A B C D E (5) D E B C A (4) D E B A C (4) D A B E C (4) B D E C A (4) A D E C B (4) E D C B A (3) E C B D A (3) C E D A B (3) C A E B D (3) B C E D A (3) A B D C E (3) E D B C A (2) E C D B A (2) E C D A B (2) D E C A B (2) D B A E C (2) C E B A D (2) C E A D B (2) C E A B D (2) C B E A D (2) C A E D B (2) B E C D A (2) B D E A C (2) B A D C E (2) A D B E C (2) A D B C E (2) A C E D B (2) D E A B C (1) D B E A C (1) D A E C B (1) C A B E D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A D E C (1) B A C D E (1) A D E B C (1) A C E B D (1) A C D B E (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -12 -12 -14 B 6 0 12 6 -2 C 12 -12 0 -4 -8 D 12 -6 4 0 -2 E 14 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998273 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -12 -14 B 6 0 12 6 -2 C 12 -12 0 -4 -8 D 12 -6 4 0 -2 E 14 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 C=22 D=19 E=12 so E is eliminated. Round 2 votes counts: C=29 D=24 A=24 B=23 so B is eliminated. Round 3 votes counts: D=36 C=36 A=28 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 B:211 D:204 C:194 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 -14 B 6 0 12 6 -2 C 12 -12 0 -4 -8 D 12 -6 4 0 -2 E 14 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 -14 B 6 0 12 6 -2 C 12 -12 0 -4 -8 D 12 -6 4 0 -2 E 14 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 -14 B 6 0 12 6 -2 C 12 -12 0 -4 -8 D 12 -6 4 0 -2 E 14 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1385: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (12) C E D B A (9) A C E D B (9) A B D E C (9) E C B D A (8) C E A D B (8) B D A E C (5) A C D E B (4) E C D B A (3) D C E B A (3) C E D A B (3) B D E A C (3) A D B C E (3) E D C B A (2) C E A B D (2) C D E A B (2) B E D C A (2) A D C B E (2) A B D C E (2) E B C D A (1) D B A E C (1) C A E D B (1) B E D A C (1) B E A C D (1) B A E D C (1) B A D E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -12 -10 -20 B 6 0 -14 0 -12 C 12 14 0 6 -2 D 10 0 -6 0 -6 E 20 12 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -10 -20 B 6 0 -14 0 -12 C 12 14 0 6 -2 D 10 0 -6 0 -6 E 20 12 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=26 C=25 E=14 D=4 so D is eliminated. Round 2 votes counts: A=31 C=28 B=27 E=14 so E is eliminated. Round 3 votes counts: C=41 A=31 B=28 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:220 C:215 D:199 B:190 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -10 -20 B 6 0 -14 0 -12 C 12 14 0 6 -2 D 10 0 -6 0 -6 E 20 12 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -10 -20 B 6 0 -14 0 -12 C 12 14 0 6 -2 D 10 0 -6 0 -6 E 20 12 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -10 -20 B 6 0 -14 0 -12 C 12 14 0 6 -2 D 10 0 -6 0 -6 E 20 12 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1386: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) A E D C B (8) B D C A E (7) B C D E A (7) E C A D B (4) E A D C B (4) B A D C E (4) A E B D C (4) E A C D B (3) B A E C D (3) E C B D A (2) E C B A D (2) E B C A D (2) E A B C D (2) D C E A B (2) D A C E B (2) C E D B A (2) C E B D A (2) C D B E A (2) C B D E A (2) B E C A D (2) B D A C E (2) B C E D A (2) B C E A D (2) A E D B C (2) A D E C B (2) A B E D C (2) D C B E A (1) D C B A E (1) D B C A E (1) D B A C E (1) D A E C B (1) C D E A B (1) C B E D A (1) B E A C D (1) B C D A E (1) B A D E C (1) B A C D E (1) A E B C D (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -8 4 -8 B 4 0 0 6 -10 C 8 0 0 6 -10 D -4 -6 -6 0 -18 E 8 10 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 4 -8 B 4 0 0 6 -10 C 8 0 0 6 -10 D -4 -6 -6 0 -18 E 8 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=27 A=21 C=10 D=9 so D is eliminated. Round 2 votes counts: B=35 E=27 A=24 C=14 so C is eliminated. Round 3 votes counts: B=42 E=34 A=24 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:202 B:200 A:192 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 4 -8 B 4 0 0 6 -10 C 8 0 0 6 -10 D -4 -6 -6 0 -18 E 8 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 4 -8 B 4 0 0 6 -10 C 8 0 0 6 -10 D -4 -6 -6 0 -18 E 8 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 4 -8 B 4 0 0 6 -10 C 8 0 0 6 -10 D -4 -6 -6 0 -18 E 8 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1387: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) E C D A B (6) B A C D E (6) E D C A B (5) D C A B E (5) C D E A B (5) B E A D C (5) D C A E B (4) B A D E C (4) E B D A C (3) E B C A D (3) C E D A B (3) B A E D C (3) B A E C D (3) B A D C E (3) A D C B E (3) A D B C E (3) E C D B A (2) E B A D C (2) D C E A B (2) C B A E D (2) C B A D E (2) C A D B E (2) B C E A D (2) E D A C B (1) E D A B C (1) E C B D A (1) E B D C A (1) E B C D A (1) D E A C B (1) D A E B C (1) C D A B E (1) C A B D E (1) B E C A D (1) B E A C D (1) B C A D E (1) B A C E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 0 0 8 B -6 0 -6 -4 16 C 0 6 0 -10 10 D 0 4 10 0 6 E -8 -16 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.599893 B: 0.000000 C: 0.000000 D: 0.400107 E: 0.000000 Sum of squares = 0.51995736019 Cumulative probabilities = A: 0.599893 B: 0.599893 C: 0.599893 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 8 B -6 0 -6 -4 16 C 0 6 0 -10 10 D 0 4 10 0 6 E -8 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 D=20 C=16 A=8 so A is eliminated. Round 2 votes counts: B=32 E=26 D=26 C=16 so C is eliminated. Round 3 votes counts: B=37 D=34 E=29 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:207 C:203 B:200 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 0 8 B -6 0 -6 -4 16 C 0 6 0 -10 10 D 0 4 10 0 6 E -8 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 8 B -6 0 -6 -4 16 C 0 6 0 -10 10 D 0 4 10 0 6 E -8 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 8 B -6 0 -6 -4 16 C 0 6 0 -10 10 D 0 4 10 0 6 E -8 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1388: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (13) E C B A D (7) E D C A B (5) B A C E D (5) A B C D E (5) D E C B A (4) D E C A B (4) D A B C E (4) C D B A E (4) B A C D E (4) A B C E D (4) E D A B C (3) E A B C D (3) D C B A E (3) C B A D E (3) B A E C D (3) A B E C D (3) D C E B A (2) D C A E B (2) C D E B A (2) A B D E C (2) A B D C E (2) E D C B A (1) E C A B D (1) E B C A D (1) E B A C D (1) D E A B C (1) D A E B C (1) C E D B A (1) C E B A D (1) C D B E A (1) C B A E D (1) A E B D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -12 -2 -2 B 14 0 -10 -4 -8 C 12 10 0 26 -10 D 2 4 -26 0 -10 E 2 8 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -12 -2 -2 B 14 0 -10 -4 -8 C 12 10 0 26 -10 D 2 4 -26 0 -10 E 2 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=21 A=19 C=13 B=12 so B is eliminated. Round 2 votes counts: E=35 A=31 D=21 C=13 so C is eliminated. Round 3 votes counts: E=37 A=35 D=28 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:215 B:196 A:185 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -12 -2 -2 B 14 0 -10 -4 -8 C 12 10 0 26 -10 D 2 4 -26 0 -10 E 2 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -2 -2 B 14 0 -10 -4 -8 C 12 10 0 26 -10 D 2 4 -26 0 -10 E 2 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -2 -2 B 14 0 -10 -4 -8 C 12 10 0 26 -10 D 2 4 -26 0 -10 E 2 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1389: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) C E B D A (9) E C B D A (8) E C B A D (7) D B C A E (6) B C D E A (6) E C D B A (4) D B C E A (4) C B E D A (4) E A C B D (3) B C E D A (3) A E C D B (3) A E C B D (3) A D E B C (3) A D B E C (3) A D B C E (3) E C A B D (2) D E C B A (2) B D C E A (2) B D C A E (2) A D E C B (2) D E B C A (1) D A E B C (1) D A B E C (1) C E A B D (1) C B D E A (1) C A B E D (1) B C A D E (1) A E B D C (1) A C E B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 -28 -8 -14 B 20 0 -22 14 -22 C 28 22 0 18 -8 D 8 -14 -18 0 -22 E 14 22 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -28 -8 -14 B 20 0 -22 14 -22 C 28 22 0 18 -8 D 8 -14 -18 0 -22 E 14 22 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 C=16 D=15 B=14 so B is eliminated. Round 2 votes counts: A=31 C=26 E=24 D=19 so D is eliminated. Round 3 votes counts: C=40 A=33 E=27 so E is eliminated. Round 4 votes counts: C=64 A=36 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:233 C:230 B:195 D:177 A:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -28 -8 -14 B 20 0 -22 14 -22 C 28 22 0 18 -8 D 8 -14 -18 0 -22 E 14 22 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -28 -8 -14 B 20 0 -22 14 -22 C 28 22 0 18 -8 D 8 -14 -18 0 -22 E 14 22 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -28 -8 -14 B 20 0 -22 14 -22 C 28 22 0 18 -8 D 8 -14 -18 0 -22 E 14 22 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1390: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) B C D E A (8) A D E C B (7) C E A D B (6) B D A C E (6) B D A E C (5) B C E D A (5) E A C D B (4) D A B E C (4) C E B A D (4) A D B E C (4) E C A D B (3) E A D C B (3) C E D A B (3) B D C A E (3) B A D E C (3) A E D C B (3) D A E C B (2) D A E B C (2) C E B D A (2) B C D A E (2) B C A D E (2) B A D C E (2) E C D A B (1) D E A C B (1) D B C E A (1) C E A B D (1) C B E A D (1) B C A E D (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -4 -8 -4 B 8 0 0 10 14 C 4 0 0 4 12 D 8 -10 -4 0 6 E 4 -14 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.588577 C: 0.411423 D: 0.000000 E: 0.000000 Sum of squares = 0.515691694748 Cumulative probabilities = A: 0.000000 B: 0.588577 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -8 -4 B 8 0 0 10 14 C 4 0 0 4 12 D 8 -10 -4 0 6 E 4 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=26 A=16 E=11 D=10 so D is eliminated. Round 2 votes counts: B=38 C=26 A=24 E=12 so E is eliminated. Round 3 votes counts: B=38 A=32 C=30 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:210 D:200 A:188 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -8 -4 B 8 0 0 10 14 C 4 0 0 4 12 D 8 -10 -4 0 6 E 4 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -8 -4 B 8 0 0 10 14 C 4 0 0 4 12 D 8 -10 -4 0 6 E 4 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -8 -4 B 8 0 0 10 14 C 4 0 0 4 12 D 8 -10 -4 0 6 E 4 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1391: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (16) A B D C E (9) B D C A E (8) E C D A B (5) E A B C D (5) C D E B A (5) B A D C E (5) E A C D B (4) D C B A E (4) B C D E A (4) A B E D C (4) D C E A B (3) A E D C B (3) E B C D A (2) D C B E A (2) C D B E A (2) A E B D C (2) A E B C D (2) A D C B E (2) A D B C E (2) E D C A B (1) E B A C D (1) E A D C B (1) D C E B A (1) D C A E B (1) C B D E A (1) B E C D A (1) B A E C D (1) A E C B D (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -12 -12 -8 B 6 0 -6 -6 -8 C 12 6 0 2 0 D 12 6 -2 0 0 E 8 8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.594319 D: 0.000000 E: 0.405681 Sum of squares = 0.517791979005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.594319 D: 0.594319 E: 1.000000 A B C D E A 0 -6 -12 -12 -8 B 6 0 -6 -6 -8 C 12 6 0 2 0 D 12 6 -2 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.4999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=27 B=19 D=11 C=8 so C is eliminated. Round 2 votes counts: E=35 A=27 B=20 D=18 so D is eliminated. Round 3 votes counts: E=44 B=28 A=28 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:210 D:208 E:208 B:193 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 -8 B 6 0 -6 -6 -8 C 12 6 0 2 0 D 12 6 -2 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.4999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 -8 B 6 0 -6 -6 -8 C 12 6 0 2 0 D 12 6 -2 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.4999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 -8 B 6 0 -6 -6 -8 C 12 6 0 2 0 D 12 6 -2 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.4999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1392: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) E B D C A (8) B A C E D (7) D E C A B (6) D A C E B (6) A C D B E (6) D C A E B (5) A C D E B (5) B E D C A (4) B E D A C (4) A C B D E (4) A B C D E (4) B A C D E (3) A D C E B (3) E D C B A (2) E D B C A (2) D E A C B (2) B E C D A (2) B E A C D (2) E D C A B (1) E B C D A (1) E B C A D (1) D E C B A (1) D E B C A (1) D E B A C (1) D A C B E (1) C A D E B (1) C A B E D (1) B E A D C (1) B D A E C (1) B A E C D (1) B A D C E (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 8 4 2 B 6 0 10 10 8 C -8 -10 0 -4 -2 D -4 -10 4 0 6 E -2 -8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 4 2 B 6 0 10 10 8 C -8 -10 0 -4 -2 D -4 -10 4 0 6 E -2 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=24 D=23 E=15 C=2 so C is eliminated. Round 2 votes counts: B=36 A=26 D=23 E=15 so E is eliminated. Round 3 votes counts: B=46 D=28 A=26 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:204 D:198 E:193 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 4 2 B 6 0 10 10 8 C -8 -10 0 -4 -2 D -4 -10 4 0 6 E -2 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 4 2 B 6 0 10 10 8 C -8 -10 0 -4 -2 D -4 -10 4 0 6 E -2 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 4 2 B 6 0 10 10 8 C -8 -10 0 -4 -2 D -4 -10 4 0 6 E -2 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1393: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) B A E D C (7) B A D E C (7) E B A C D (6) D A C B E (5) C D A E B (5) B E A D C (5) A B D E C (5) E B C A D (4) C E D B A (4) C D E A B (4) A D C B E (4) E C B D A (3) C D A B E (3) A D B C E (3) E C B A D (2) E C A D B (2) D C B A E (2) D B A C E (2) C E D A B (2) C D E B A (2) B D A E C (2) A E B D C (2) A B E D C (2) A B D C E (2) E C D B A (1) E C D A B (1) E C A B D (1) E A B D C (1) D B C A E (1) D A B C E (1) B D E C A (1) A E B C D (1) Total count = 100 A B C D E A 0 2 10 8 22 B -2 0 4 2 18 C -10 -4 0 -18 -6 D -8 -2 18 0 12 E -22 -18 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 8 22 B -2 0 4 2 18 C -10 -4 0 -18 -6 D -8 -2 18 0 12 E -22 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 E=21 C=20 A=19 D=18 so D is eliminated. Round 2 votes counts: C=29 B=25 A=25 E=21 so E is eliminated. Round 3 votes counts: C=39 B=35 A=26 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:221 B:211 D:210 C:181 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 22 B -2 0 4 2 18 C -10 -4 0 -18 -6 D -8 -2 18 0 12 E -22 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 22 B -2 0 4 2 18 C -10 -4 0 -18 -6 D -8 -2 18 0 12 E -22 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 22 B -2 0 4 2 18 C -10 -4 0 -18 -6 D -8 -2 18 0 12 E -22 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1394: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (21) E A C D B (20) B C D A E (6) E A D C B (4) B C A D E (4) E A C B D (3) D B C E A (3) B C A E D (3) A E C D B (3) D E A C B (2) D C B A E (2) D B C A E (2) C A E D B (2) C A E B D (2) A E C B D (2) A C E B D (2) E D A C B (1) E D A B C (1) E A D B C (1) E A B D C (1) E A B C D (1) D E A B C (1) D C A E B (1) D B E C A (1) C D B A E (1) C D A E B (1) C D A B E (1) C A B E D (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E A C (1) B D C E A (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -4 6 12 B -4 0 2 4 0 C 4 -2 0 10 8 D -6 -4 -10 0 -4 E -12 0 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 6 12 B -4 0 2 4 0 C 4 -2 0 10 8 D -6 -4 -10 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999916 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=32 D=12 A=9 C=8 so C is eliminated. Round 2 votes counts: B=39 E=32 D=15 A=14 so A is eliminated. Round 3 votes counts: E=45 B=40 D=15 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:210 A:209 B:201 E:192 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 6 12 B -4 0 2 4 0 C 4 -2 0 10 8 D -6 -4 -10 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999916 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 6 12 B -4 0 2 4 0 C 4 -2 0 10 8 D -6 -4 -10 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999916 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 6 12 B -4 0 2 4 0 C 4 -2 0 10 8 D -6 -4 -10 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999916 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1395: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) C A E D B (7) C A D B E (6) E B A C D (5) E A C D B (5) E A C B D (5) D C B A E (5) D B C E A (4) B D C A E (4) D C A E B (3) D C A B E (3) D B E C A (3) C A B D E (3) A E C B D (3) E B D A C (2) E A D C B (2) D B C A E (2) C D A E B (2) C A D E B (2) B E D A C (2) B E A C D (2) B D E C A (2) B D E A C (2) A C E D B (2) E D B A C (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D B E A C (1) C D A B E (1) C B A D E (1) C A B E D (1) B D A E C (1) B C D A E (1) B C A D E (1) B A E C D (1) B A C E D (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 14 -8 16 20 B -14 0 -26 -6 -4 C 8 26 0 18 16 D -16 6 -18 0 0 E -20 4 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 16 20 B -14 0 -26 -6 -4 C 8 26 0 18 16 D -16 6 -18 0 0 E -20 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 E=21 B=17 A=15 so A is eliminated. Round 2 votes counts: C=34 E=25 D=24 B=17 so B is eliminated. Round 3 votes counts: C=37 D=33 E=30 so E is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:234 A:221 D:186 E:184 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -8 16 20 B -14 0 -26 -6 -4 C 8 26 0 18 16 D -16 6 -18 0 0 E -20 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 16 20 B -14 0 -26 -6 -4 C 8 26 0 18 16 D -16 6 -18 0 0 E -20 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 16 20 B -14 0 -26 -6 -4 C 8 26 0 18 16 D -16 6 -18 0 0 E -20 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1396: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) E C A B D (9) C E D B A (9) B A D E C (9) A B E C D (9) D C E B A (7) E C D A B (4) E A C B D (4) A B D E C (4) E C A D B (3) C E D A B (3) C D E B A (3) B A D C E (3) A B E D C (3) E C B A D (2) D C E A B (2) D C B E A (2) D B C A E (2) D A B C E (2) B A E C D (2) E A B C D (1) D C B A E (1) B D A C E (1) B A E D C (1) A E B C D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 6 6 2 B 6 0 2 0 4 C -6 -2 0 0 -4 D -6 0 0 0 -2 E -2 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.641242 C: 0.000000 D: 0.358758 E: 0.000000 Sum of squares = 0.539898676486 Cumulative probabilities = A: 0.000000 B: 0.641242 C: 0.641242 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 6 2 B 6 0 2 0 4 C -6 -2 0 0 -4 D -6 0 0 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500310 C: 0.000000 D: 0.499690 E: 0.000000 Sum of squares = 0.500000192005 Cumulative probabilities = A: 0.000000 B: 0.500310 C: 0.500310 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=23 A=19 B=16 C=15 so C is eliminated. Round 2 votes counts: E=35 D=30 A=19 B=16 so B is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:206 A:204 E:200 D:196 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 6 2 B 6 0 2 0 4 C -6 -2 0 0 -4 D -6 0 0 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500310 C: 0.000000 D: 0.499690 E: 0.000000 Sum of squares = 0.500000192005 Cumulative probabilities = A: 0.000000 B: 0.500310 C: 0.500310 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 6 2 B 6 0 2 0 4 C -6 -2 0 0 -4 D -6 0 0 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500310 C: 0.000000 D: 0.499690 E: 0.000000 Sum of squares = 0.500000192005 Cumulative probabilities = A: 0.000000 B: 0.500310 C: 0.500310 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 6 2 B 6 0 2 0 4 C -6 -2 0 0 -4 D -6 0 0 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500310 C: 0.000000 D: 0.499690 E: 0.000000 Sum of squares = 0.500000192005 Cumulative probabilities = A: 0.000000 B: 0.500310 C: 0.500310 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1397: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (13) A E B D C (8) A D E B C (8) C B D E A (5) C D A B E (4) A E C B D (4) D C B A E (3) D C A B E (3) D B E C A (3) D B C E A (3) D B A E C (3) C B E D A (3) E B C A D (2) E B A C D (2) E A C B D (2) D C B E A (2) D A B E C (2) D A B C E (2) C E B A D (2) C A E B D (2) A E B C D (2) A C E B D (2) E C B A D (1) E B C D A (1) E B A D C (1) E A B D C (1) D B E A C (1) D B C A E (1) C E B D A (1) C E A B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B D E C A (1) A E D B C (1) A E C D B (1) A D E C B (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -8 -8 4 B 2 0 -8 -12 6 C 8 8 0 4 2 D 8 12 -4 0 18 E -4 -6 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -8 4 B 2 0 -8 -12 6 C 8 8 0 4 2 D 8 12 -4 0 18 E -4 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=31 D=23 E=10 B=4 so B is eliminated. Round 2 votes counts: C=32 A=31 D=24 E=13 so E is eliminated. Round 3 votes counts: C=37 A=37 D=26 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:211 B:194 A:193 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -8 4 B 2 0 -8 -12 6 C 8 8 0 4 2 D 8 12 -4 0 18 E -4 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -8 4 B 2 0 -8 -12 6 C 8 8 0 4 2 D 8 12 -4 0 18 E -4 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -8 4 B 2 0 -8 -12 6 C 8 8 0 4 2 D 8 12 -4 0 18 E -4 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1398: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (14) C D A E B (13) B D E A C (6) D C A E B (5) B E A C D (5) B C E A D (5) D A C E B (4) C D B A E (4) C A E D B (4) A E D C B (4) D C B A E (3) D B A E C (3) B E D A C (3) B C E D A (3) E B A D C (2) E A D B C (2) E A B D C (2) C A D E B (2) B C D E A (2) A E C D B (2) E B A C D (1) E A C D B (1) D B E A C (1) D B C A E (1) D A E C B (1) D A E B C (1) D A B E C (1) C E A D B (1) C D A B E (1) C B D E A (1) B D C E A (1) B D C A E (1) Total count = 100 A B C D E A 0 -12 6 -10 0 B 12 0 8 -8 10 C -6 -8 0 -10 2 D 10 8 10 0 2 E 0 -10 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -10 0 B 12 0 8 -8 10 C -6 -8 0 -10 2 D 10 8 10 0 2 E 0 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 C=26 D=20 E=8 A=6 so A is eliminated. Round 2 votes counts: B=40 C=26 D=20 E=14 so E is eliminated. Round 3 votes counts: B=45 C=29 D=26 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:215 B:211 E:193 A:192 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 6 -10 0 B 12 0 8 -8 10 C -6 -8 0 -10 2 D 10 8 10 0 2 E 0 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -10 0 B 12 0 8 -8 10 C -6 -8 0 -10 2 D 10 8 10 0 2 E 0 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -10 0 B 12 0 8 -8 10 C -6 -8 0 -10 2 D 10 8 10 0 2 E 0 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1399: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) C E D A B (8) B C A D E (7) E C D A B (6) B A D C E (6) E D A C B (5) C E D B A (5) B A D E C (5) C E B D A (4) C B E A D (4) E D A B C (3) E B D A C (3) C B A D E (3) B A C D E (3) E D C A B (2) E C B D A (2) D A E B C (2) C B E D A (2) C B A E D (2) B E D A C (2) A C D B E (2) A B D C E (2) E D B C A (1) D E A C B (1) D A E C B (1) C D A E B (1) C A D E B (1) C A B D E (1) B E C D A (1) B E A D C (1) B D A E C (1) B A C E D (1) A D E B C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 0 0 B 6 0 -2 2 8 C 0 2 0 6 6 D 0 -2 -6 0 -4 E 0 -8 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.189790 B: 0.000000 C: 0.810210 D: 0.000000 E: 0.000000 Sum of squares = 0.692460073128 Cumulative probabilities = A: 0.189790 B: 0.189790 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 0 0 B 6 0 -2 2 8 C 0 2 0 6 6 D 0 -2 -6 0 -4 E 0 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000150688 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 E=22 A=16 D=4 so D is eliminated. Round 2 votes counts: C=31 B=27 E=23 A=19 so A is eliminated. Round 3 votes counts: B=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:207 C:207 A:197 E:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 0 0 B 6 0 -2 2 8 C 0 2 0 6 6 D 0 -2 -6 0 -4 E 0 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000150688 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 0 B 6 0 -2 2 8 C 0 2 0 6 6 D 0 -2 -6 0 -4 E 0 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000150688 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 0 B 6 0 -2 2 8 C 0 2 0 6 6 D 0 -2 -6 0 -4 E 0 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000150688 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1400: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) A B D E C (9) C E D B A (8) D E B A C (7) D A B E C (6) A D B E C (6) C A B E D (5) E D B A C (4) E B D A C (4) E B A D C (4) C E B D A (4) E C D B A (2) E C B A D (2) D E B C A (2) D B A E C (2) D A B C E (2) C D E B A (2) C D A B E (2) C A D B E (2) C A B D E (2) A B E C D (2) A B D C E (2) E D C B A (1) E D B C A (1) E B A C D (1) E A B C D (1) D B E A C (1) C E A B D (1) C A E D B (1) A D B C E (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 10 4 -10 B 10 0 14 -2 -10 C -10 -14 0 -6 -12 D -4 2 6 0 -4 E 10 10 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 10 4 -10 B 10 0 14 -2 -10 C -10 -14 0 -6 -12 D -4 2 6 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=23 E=20 D=20 so B is eliminated. Round 2 votes counts: C=37 A=23 E=20 D=20 so E is eliminated. Round 3 votes counts: C=41 D=30 A=29 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:218 B:206 D:200 A:197 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 10 4 -10 B 10 0 14 -2 -10 C -10 -14 0 -6 -12 D -4 2 6 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 4 -10 B 10 0 14 -2 -10 C -10 -14 0 -6 -12 D -4 2 6 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 4 -10 B 10 0 14 -2 -10 C -10 -14 0 -6 -12 D -4 2 6 0 -4 E 10 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1401: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) D A E C B (6) C B D E A (6) E A D C B (5) C B E D A (5) B C E D A (5) B C D E A (5) B C D A E (5) A D E B C (5) B E C A D (4) B C E A D (4) A E D B C (4) E D A C B (3) A D B E C (3) E B C A D (2) E A C D B (2) D E C B A (2) D E A C B (2) D C E B A (2) C D B E A (2) B C A E D (2) A D B C E (2) E C D B A (1) E C B D A (1) E B A C D (1) E A D B C (1) E A C B D (1) E A B C D (1) D E C A B (1) D C B A E (1) D C A B E (1) D B C A E (1) B A E C D (1) B A C E D (1) A E D C B (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 2 -12 B 2 0 -2 -8 -2 C 0 2 0 -2 -14 D -2 8 2 0 8 E 12 2 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.090909 Sum of squares = 0.438016528925 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.909091 E: 1.000000 A B C D E A 0 -2 0 2 -12 B 2 0 -2 -8 -2 C 0 2 0 -2 -14 D -2 8 2 0 8 E 12 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.090909 Sum of squares = 0.438016528918 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 E=18 D=16 C=13 so C is eliminated. Round 2 votes counts: B=38 A=26 E=18 D=18 so E is eliminated. Round 3 votes counts: B=42 A=36 D=22 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:210 D:208 B:195 A:194 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 2 -12 B 2 0 -2 -8 -2 C 0 2 0 -2 -14 D -2 8 2 0 8 E 12 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.090909 Sum of squares = 0.438016528918 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.909091 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 -12 B 2 0 -2 -8 -2 C 0 2 0 -2 -14 D -2 8 2 0 8 E 12 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.090909 Sum of squares = 0.438016528918 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.909091 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 -12 B 2 0 -2 -8 -2 C 0 2 0 -2 -14 D -2 8 2 0 8 E 12 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.090909 Sum of squares = 0.438016528918 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.909091 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1402: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) C B E A D (6) D E A B C (5) C D E B A (5) B C E A D (5) E C B D A (4) D A C B E (4) C E B D A (4) A D E B C (4) A D B E C (4) E D B A C (3) D E C B A (3) D A E C B (3) D A E B C (3) A B C D E (3) E D A B C (2) E B C D A (2) E B C A D (2) D E A C B (2) D A B E C (2) C E D B A (2) C A B D E (2) B E A C D (2) A D B C E (2) A B E D C (2) E D B C A (1) E C D B A (1) E B A C D (1) E A D B C (1) D E C A B (1) D C E A B (1) D C A B E (1) C E B A D (1) C D B E A (1) C D B A E (1) C B A E D (1) B A E C D (1) B A C E D (1) A E D B C (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 0 -14 -22 B 6 0 2 -6 -4 C 0 -2 0 6 -4 D 14 6 -6 0 -2 E 22 4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 0 -14 -22 B 6 0 2 -6 -4 C 0 -2 0 6 -4 D 14 6 -6 0 -2 E 22 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 A=20 E=17 B=9 so B is eliminated. Round 2 votes counts: C=34 D=25 A=22 E=19 so E is eliminated. Round 3 votes counts: C=43 D=31 A=26 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:216 D:206 C:200 B:199 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 -14 -22 B 6 0 2 -6 -4 C 0 -2 0 6 -4 D 14 6 -6 0 -2 E 22 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -14 -22 B 6 0 2 -6 -4 C 0 -2 0 6 -4 D 14 6 -6 0 -2 E 22 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -14 -22 B 6 0 2 -6 -4 C 0 -2 0 6 -4 D 14 6 -6 0 -2 E 22 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1403: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) A E D B C (8) D E C B A (7) A B C D E (7) E D A C B (6) D E A C B (6) C B D E A (6) C B D A E (5) E D C B A (4) B C A E D (4) D A C E B (3) C B E D A (3) A D E B C (3) A B C E D (3) E D A B C (2) E A D B C (2) D E C A B (2) D A E C B (2) B A C D E (2) A E B D C (2) A D E C B (2) A B E C D (2) E B A C D (1) E A B D C (1) D C E B A (1) D C B E A (1) C D B A E (1) C B A D E (1) B C E D A (1) B C E A D (1) A D B C E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 10 0 12 B -8 0 -2 -2 -4 C -10 2 0 -8 -2 D 0 2 8 0 18 E -12 4 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.557783 B: 0.000000 C: 0.000000 D: 0.442217 E: 0.000000 Sum of squares = 0.506677731081 Cumulative probabilities = A: 0.557783 B: 0.557783 C: 0.557783 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 0 12 B -8 0 -2 -2 -4 C -10 2 0 -8 -2 D 0 2 8 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 E=16 C=16 B=16 so E is eliminated. Round 2 votes counts: D=34 A=33 B=17 C=16 so C is eliminated. Round 3 votes counts: D=35 A=33 B=32 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:214 B:192 C:191 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 0 12 B -8 0 -2 -2 -4 C -10 2 0 -8 -2 D 0 2 8 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 12 B -8 0 -2 -2 -4 C -10 2 0 -8 -2 D 0 2 8 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 12 B -8 0 -2 -2 -4 C -10 2 0 -8 -2 D 0 2 8 0 18 E -12 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1404: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (13) A D C E B (11) B E C D A (8) A C E D B (8) B A D E C (6) D E C B A (5) D C E A B (4) B E D C A (4) A C D E B (4) A C B E D (4) A B C E D (4) D A C E B (3) D A B E C (3) A C E B D (3) E C B D A (2) D B E C A (2) B E C A D (2) A B D C E (2) E D B C A (1) D E C A B (1) D E B C A (1) D B A E C (1) C E D B A (1) C E A D B (1) B D E A C (1) B A E C D (1) B A C E D (1) A C D B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 10 0 8 B -2 0 4 6 10 C -10 -4 0 -18 -4 D 0 -6 18 0 20 E -8 -10 4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.911086 B: 0.000000 C: 0.000000 D: 0.088914 E: 0.000000 Sum of squares = 0.837982760113 Cumulative probabilities = A: 0.911086 B: 0.911086 C: 0.911086 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 0 8 B -2 0 4 6 10 C -10 -4 0 -18 -4 D 0 -6 18 0 20 E -8 -10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000023998 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=36 D=20 E=3 C=2 so C is eliminated. Round 2 votes counts: A=39 B=36 D=20 E=5 so E is eliminated. Round 3 votes counts: A=40 B=38 D=22 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:210 B:209 E:183 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 0 8 B -2 0 4 6 10 C -10 -4 0 -18 -4 D 0 -6 18 0 20 E -8 -10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000023998 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 0 8 B -2 0 4 6 10 C -10 -4 0 -18 -4 D 0 -6 18 0 20 E -8 -10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000023998 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 0 8 B -2 0 4 6 10 C -10 -4 0 -18 -4 D 0 -6 18 0 20 E -8 -10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000023998 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1405: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (5) D E A C B (4) C A B E D (4) B E D A C (4) B C A E D (4) E A C D B (3) D E A B C (3) D C A E B (3) D B E C A (3) D B C E A (3) C B A D E (3) C A E D B (3) C A D E B (3) B E A C D (3) B D C A E (3) A C E D B (3) A C E B D (3) E D B A C (2) E B A D C (2) E A B C D (2) D C B E A (2) D C B A E (2) D B E A C (2) C B D A E (2) B E A D C (2) B D E A C (2) B C A D E (2) A E C D B (2) E D A C B (1) E D A B C (1) E A B D C (1) D E C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) C D B A E (1) C D A E B (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B D E (1) B D C E A (1) B C D A E (1) B A E C D (1) B A C E D (1) A E B C D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 0 0 B 8 0 2 -4 4 C -2 -2 0 0 6 D 0 4 0 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.253354 B: 0.000000 C: 0.000000 D: 0.746646 E: 0.000000 Sum of squares = 0.621668379978 Cumulative probabilities = A: 0.253354 B: 0.253354 C: 0.253354 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 0 0 B 8 0 2 -4 4 C -2 -2 0 0 6 D 0 4 0 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555571866 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=24 C=21 E=12 A=12 so E is eliminated. Round 2 votes counts: D=35 B=26 C=21 A=18 so A is eliminated. Round 3 votes counts: D=35 C=33 B=32 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:205 D:204 C:201 A:197 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 0 0 B 8 0 2 -4 4 C -2 -2 0 0 6 D 0 4 0 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555571866 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 0 0 B 8 0 2 -4 4 C -2 -2 0 0 6 D 0 4 0 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555571866 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 0 0 B 8 0 2 -4 4 C -2 -2 0 0 6 D 0 4 0 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555571866 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1406: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) B E A C D (12) D E B C A (8) D C A E B (8) A C B D E (6) B E D C A (5) A C B E D (5) C A D E B (4) B A E C D (4) D C A B E (3) B E D A C (3) B A C E D (3) E D B A C (2) E B D C A (2) D C E B A (2) D C E A B (2) C D A B E (2) C A D B E (2) A C E B D (2) A C D B E (2) E D B C A (1) E B A C D (1) D E C B A (1) D B E C A (1) D B C A E (1) C D A E B (1) B D E C A (1) A E B C D (1) A C E D B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 12 -10 -6 B 18 0 16 16 2 C -12 -16 0 -4 -8 D 10 -16 4 0 -10 E 6 -2 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 12 -10 -6 B 18 0 16 16 2 C -12 -16 0 -4 -8 D 10 -16 4 0 -10 E 6 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985825 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=26 A=19 E=18 C=9 so C is eliminated. Round 2 votes counts: D=29 B=28 A=25 E=18 so E is eliminated. Round 3 votes counts: B=43 D=32 A=25 so A is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:211 D:194 A:189 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 12 -10 -6 B 18 0 16 16 2 C -12 -16 0 -4 -8 D 10 -16 4 0 -10 E 6 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985825 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 12 -10 -6 B 18 0 16 16 2 C -12 -16 0 -4 -8 D 10 -16 4 0 -10 E 6 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985825 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 12 -10 -6 B 18 0 16 16 2 C -12 -16 0 -4 -8 D 10 -16 4 0 -10 E 6 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985825 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1407: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) C B E A D (6) D E A C B (5) D A E B C (5) C E A B D (5) B A D C E (5) E D A C B (4) E C D A B (4) C E A D B (4) B D A C E (4) B A C D E (4) E C D B A (3) D E A B C (3) D B A E C (3) D A B E C (3) B D A E C (3) B C A E D (3) E D B C A (2) E C A D B (2) D B E A C (2) C B A E D (2) C A E D B (2) B C D A E (2) A D E C B (2) E D C B A (1) E A D C B (1) D E B A C (1) D A E C B (1) C E D A B (1) C E B D A (1) C B A D E (1) C A B E D (1) B E D C A (1) B D C A E (1) B C E D A (1) B C A D E (1) A E C D B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -2 -2 -8 B 8 0 -10 0 -10 C 2 10 0 4 6 D 2 0 -4 0 -4 E 8 10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -2 -8 B 8 0 -10 0 -10 C 2 10 0 4 6 D 2 0 -4 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=25 D=23 E=17 A=5 so A is eliminated. Round 2 votes counts: C=31 B=26 D=25 E=18 so E is eliminated. Round 3 votes counts: C=41 D=33 B=26 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 E:208 D:197 B:194 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 -2 -8 B 8 0 -10 0 -10 C 2 10 0 4 6 D 2 0 -4 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -2 -8 B 8 0 -10 0 -10 C 2 10 0 4 6 D 2 0 -4 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -2 -8 B 8 0 -10 0 -10 C 2 10 0 4 6 D 2 0 -4 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1408: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (14) A C E D B (12) D E B A C (7) B C A E D (7) C A E B D (5) A C E B D (5) D E A C B (4) D B E A C (4) B C E A D (4) D A E C B (3) B C E D A (3) E D C A B (2) D E A B C (2) C A B E D (2) B E D C A (2) B D E A C (2) B D C E A (2) A D E C B (2) E D C B A (1) E D A C B (1) E A C D B (1) D E B C A (1) D B E C A (1) D B A E C (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E D B (1) B E C D A (1) B D A C E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D C B (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 2 -4 -8 B 12 0 10 6 -2 C -2 -10 0 -4 -2 D 4 -6 4 0 -6 E 8 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 2 -4 -8 B 12 0 10 6 -2 C -2 -10 0 -4 -2 D 4 -6 4 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=23 A=22 C=11 E=5 so E is eliminated. Round 2 votes counts: B=39 D=27 A=23 C=11 so C is eliminated. Round 3 votes counts: B=41 A=32 D=27 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:209 D:198 C:191 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 2 -4 -8 B 12 0 10 6 -2 C -2 -10 0 -4 -2 D 4 -6 4 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -4 -8 B 12 0 10 6 -2 C -2 -10 0 -4 -2 D 4 -6 4 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -4 -8 B 12 0 10 6 -2 C -2 -10 0 -4 -2 D 4 -6 4 0 -6 E 8 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1409: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B C A E D (8) A E D B C (8) A E B D C (8) B A E C D (6) D E A C B (5) C D B E A (5) C B D E A (5) E A D B C (4) D E C A B (4) C B D A E (4) C D E B A (3) C B A D E (3) B A C E D (3) A D E C B (3) E D A B C (2) C D B A E (2) B C E D A (2) B C D E A (2) A B E C D (2) E D A C B (1) E B A D C (1) D C E B A (1) D A E C B (1) C D E A B (1) B E D C A (1) B E A D C (1) B E A C D (1) B C E A D (1) B C A D E (1) A E D C B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -2 6 4 B 0 0 2 0 -2 C 2 -2 0 2 2 D -6 0 -2 0 -2 E -4 2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.431019 B: 0.568981 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.509516742723 Cumulative probabilities = A: 0.431019 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 6 4 B 0 0 2 0 -2 C 2 -2 0 2 2 D -6 0 -2 0 -2 E -4 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 C=23 D=19 E=8 so E is eliminated. Round 2 votes counts: A=28 B=27 C=23 D=22 so D is eliminated. Round 3 votes counts: A=37 C=36 B=27 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:204 C:202 B:200 E:199 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 6 4 B 0 0 2 0 -2 C 2 -2 0 2 2 D -6 0 -2 0 -2 E -4 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 6 4 B 0 0 2 0 -2 C 2 -2 0 2 2 D -6 0 -2 0 -2 E -4 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 6 4 B 0 0 2 0 -2 C 2 -2 0 2 2 D -6 0 -2 0 -2 E -4 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1410: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) A E C B D (8) D B E C A (7) C A E B D (5) D E B A C (4) D B E A C (4) B E D C A (4) A E C D B (4) E A B C D (3) D B C E A (3) D A E B C (3) B C E D A (3) A C D E B (3) E A C B D (2) D A E C B (2) C B E A D (2) C A D B E (2) B E C D A (2) B D C E A (2) B C D E A (2) A D E C B (2) A D C E B (2) E D B A C (1) E C B A D (1) E C A B D (1) E B D A C (1) E B C A D (1) E B A D C (1) D C B A E (1) D C A B E (1) D B C A E (1) D A B E C (1) D A B C E (1) C E B A D (1) C D A B E (1) C B D A E (1) C B A D E (1) C A B D E (1) B E C A D (1) B D E C A (1) B D E A C (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 8 10 6 4 B -8 0 -4 10 -14 C -10 4 0 10 -14 D -6 -10 -10 0 -6 E -4 14 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 6 4 B -8 0 -4 10 -14 C -10 4 0 10 -14 D -6 -10 -10 0 -6 E -4 14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=28 B=16 C=14 E=11 so E is eliminated. Round 2 votes counts: A=36 D=29 B=19 C=16 so C is eliminated. Round 3 votes counts: A=45 D=30 B=25 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:214 C:195 B:192 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 6 4 B -8 0 -4 10 -14 C -10 4 0 10 -14 D -6 -10 -10 0 -6 E -4 14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 6 4 B -8 0 -4 10 -14 C -10 4 0 10 -14 D -6 -10 -10 0 -6 E -4 14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 6 4 B -8 0 -4 10 -14 C -10 4 0 10 -14 D -6 -10 -10 0 -6 E -4 14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1411: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (14) B C E A D (11) C B E D A (7) D A C E B (6) D A E B C (5) D A C B E (5) C E B A D (4) B E C A D (4) E B C A D (3) D A B E C (3) C B E A D (3) B D A C E (3) A D E C B (3) E A B D C (2) D A B C E (2) C E D A B (2) C B D A E (2) B A E D C (2) B A D E C (2) A E D B C (2) A D E B C (2) E C B A D (1) E B A C D (1) E A D C B (1) E A C B D (1) D C A E B (1) C D E A B (1) C D B A E (1) C D A B E (1) B E A D C (1) B E A C D (1) B D C A E (1) B C E D A (1) A D B E C (1) Total count = 100 A B C D E A 0 4 14 -10 12 B -4 0 -6 0 2 C -14 6 0 -12 2 D 10 0 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.413595 C: 0.000000 D: 0.586405 E: 0.000000 Sum of squares = 0.514931615684 Cumulative probabilities = A: 0.000000 B: 0.413595 C: 0.413595 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 -10 12 B -4 0 -6 0 2 C -14 6 0 -12 2 D 10 0 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=26 C=21 E=9 A=8 so A is eliminated. Round 2 votes counts: D=42 B=26 C=21 E=11 so E is eliminated. Round 3 votes counts: D=45 B=32 C=23 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:214 A:210 B:196 C:191 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 14 -10 12 B -4 0 -6 0 2 C -14 6 0 -12 2 D 10 0 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -10 12 B -4 0 -6 0 2 C -14 6 0 -12 2 D 10 0 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -10 12 B -4 0 -6 0 2 C -14 6 0 -12 2 D 10 0 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1412: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) A E C D B (6) D E B A C (5) D E A B C (5) D B E A C (5) B C D A E (5) E D A B C (4) E A D C B (4) D E A C B (4) C A E D B (4) B E D A C (4) D C A E B (3) C B D A E (3) C A E B D (3) C A D E B (3) B D E C A (3) E A C D B (2) D B E C A (2) C B A E D (2) C A B E D (2) B E A D C (2) B D E A C (2) B D C A E (2) B C A E D (2) A C E D B (2) E D B A C (1) E B D A C (1) E B A D C (1) E B A C D (1) E A C B D (1) E A B C D (1) D A E C B (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A C D (1) B C E D A (1) B C A D E (1) Total count = 100 A B C D E A 0 -6 6 -18 -16 B 6 0 14 -4 -4 C -6 -14 0 -14 -12 D 18 4 14 0 10 E 16 4 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -18 -16 B 6 0 14 -4 -4 C -6 -14 0 -14 -12 D 18 4 14 0 10 E 16 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=25 C=20 E=16 A=8 so A is eliminated. Round 2 votes counts: B=31 D=25 E=22 C=22 so E is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:211 B:206 A:183 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 6 -18 -16 B 6 0 14 -4 -4 C -6 -14 0 -14 -12 D 18 4 14 0 10 E 16 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -18 -16 B 6 0 14 -4 -4 C -6 -14 0 -14 -12 D 18 4 14 0 10 E 16 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -18 -16 B 6 0 14 -4 -4 C -6 -14 0 -14 -12 D 18 4 14 0 10 E 16 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1413: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (16) E D C A B (14) B C A D E (7) D E A C B (6) E B D C A (5) B C A E D (5) B E D A C (4) A D C E B (3) E D C B A (2) E D B A C (2) E C D A B (2) E B C D A (2) C E D A B (2) C A D E B (2) C A B D E (2) B E D C A (2) B A D C E (2) A D E C B (2) A C D B E (2) E D A B C (1) D E C A B (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D B A (1) C B E A D (1) C B A D E (1) C A D B E (1) B E A D C (1) B E A C D (1) B C E D A (1) B C E A D (1) B A E D C (1) B A D E C (1) A D C B E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 20 -4 -28 -30 B -20 0 -24 -24 -30 C 4 24 0 -36 -30 D 28 24 36 0 -28 E 30 30 30 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 -4 -28 -30 B -20 0 -24 -24 -30 C 4 24 0 -36 -30 D 28 24 36 0 -28 E 30 30 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 B=26 D=10 C=10 A=10 so D is eliminated. Round 2 votes counts: E=51 B=26 A=13 C=10 so C is eliminated. Round 3 votes counts: E=54 B=28 A=18 so A is eliminated. Round 4 votes counts: E=65 B=35 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:259 D:230 C:181 A:179 B:151 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 -4 -28 -30 B -20 0 -24 -24 -30 C 4 24 0 -36 -30 D 28 24 36 0 -28 E 30 30 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -4 -28 -30 B -20 0 -24 -24 -30 C 4 24 0 -36 -30 D 28 24 36 0 -28 E 30 30 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -4 -28 -30 B -20 0 -24 -24 -30 C 4 24 0 -36 -30 D 28 24 36 0 -28 E 30 30 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1414: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) A C E B D (8) D B E C A (7) D B C E A (7) C D B E A (6) C B D E A (5) E D B C A (3) E B D C A (3) C D A B E (3) C B D A E (3) A E B C D (3) E D A B C (2) E B D A C (2) E A B D C (2) D C B A E (2) D C A B E (2) C D B A E (2) C A B E D (2) B E D C A (2) B D C E A (2) A E C D B (2) A D E C B (2) A C D B E (2) E B C D A (1) E B C A D (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B C D (1) D E B A C (1) D A B C E (1) C E B A D (1) C B E D A (1) C B A D E (1) C A D B E (1) B E C D A (1) A D E B C (1) A D C E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -12 -12 -2 B 6 0 -14 8 4 C 12 14 0 16 4 D 12 -8 -16 0 0 E 2 -4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -12 -2 B 6 0 -14 8 4 C 12 14 0 16 4 D 12 -8 -16 0 0 E 2 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=25 D=20 E=18 B=5 so B is eliminated. Round 2 votes counts: A=32 C=25 D=22 E=21 so E is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:223 B:202 E:197 D:194 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 -2 B 6 0 -14 8 4 C 12 14 0 16 4 D 12 -8 -16 0 0 E 2 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 -2 B 6 0 -14 8 4 C 12 14 0 16 4 D 12 -8 -16 0 0 E 2 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 -2 B 6 0 -14 8 4 C 12 14 0 16 4 D 12 -8 -16 0 0 E 2 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1415: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) E B A D C (7) C D A B E (5) C B E A D (5) D E C A B (4) D E A C B (4) C E B D A (4) C D E B A (4) A B E D C (4) D A C E B (3) D A C B E (3) C D E A B (3) C B A D E (3) B E C A D (3) A E B D C (3) A B D C E (3) E D C B A (2) E B A C D (2) D E A B C (2) D C E A B (2) D A E B C (2) C D B E A (2) C D B A E (2) A D E B C (2) E D B A C (1) E C B D A (1) E B C D A (1) E B C A D (1) E A B D C (1) D C E B A (1) D C A E B (1) D C A B E (1) D A E C B (1) C D A E B (1) C B D A E (1) B E A D C (1) B A C E D (1) A D B E C (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 6 -2 -20 B 2 0 -8 4 -6 C -6 8 0 0 -4 D 2 -4 0 0 8 E 20 6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.117647 B: 0.098039 C: 0.264706 D: 0.500000 E: 0.019608 Sum of squares = 0.343906191334 Cumulative probabilities = A: 0.117647 B: 0.215686 C: 0.480392 D: 0.980392 E: 1.000000 A B C D E A 0 -2 6 -2 -20 B 2 0 -8 4 -6 C -6 8 0 0 -4 D 2 -4 0 0 8 E 20 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.098039 C: 0.264706 D: 0.500000 E: 0.019608 Sum of squares = 0.343906190132 Cumulative probabilities = A: 0.117647 B: 0.215686 C: 0.480392 D: 0.980392 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=24 E=16 A=16 B=14 so B is eliminated. Round 2 votes counts: C=30 E=29 D=24 A=17 so A is eliminated. Round 3 votes counts: E=36 C=33 D=31 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:203 C:199 B:196 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -2 -20 B 2 0 -8 4 -6 C -6 8 0 0 -4 D 2 -4 0 0 8 E 20 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.098039 C: 0.264706 D: 0.500000 E: 0.019608 Sum of squares = 0.343906190132 Cumulative probabilities = A: 0.117647 B: 0.215686 C: 0.480392 D: 0.980392 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 -20 B 2 0 -8 4 -6 C -6 8 0 0 -4 D 2 -4 0 0 8 E 20 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.098039 C: 0.264706 D: 0.500000 E: 0.019608 Sum of squares = 0.343906190132 Cumulative probabilities = A: 0.117647 B: 0.215686 C: 0.480392 D: 0.980392 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 -20 B 2 0 -8 4 -6 C -6 8 0 0 -4 D 2 -4 0 0 8 E 20 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.098039 C: 0.264706 D: 0.500000 E: 0.019608 Sum of squares = 0.343906190132 Cumulative probabilities = A: 0.117647 B: 0.215686 C: 0.480392 D: 0.980392 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1416: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) A B D E C (9) A D B C E (6) E C B D A (5) C E D B A (5) D C E B A (4) D C A E B (4) B A E C D (4) B A D E C (4) E C D B A (3) C E B D A (3) C E A D B (3) C E A B D (3) B E D C A (3) B E C D A (3) D C E A B (2) D B A E C (2) D A C E B (2) C D E A B (2) C A D E B (2) B E A C D (2) B D A E C (2) A C E B D (2) A B D C E (2) E C A B D (1) E B C A D (1) D E C B A (1) D B E A C (1) D A B E C (1) C E B A D (1) C A E D B (1) B E C A D (1) B D E C A (1) A D C B E (1) A D B E C (1) A C D E B (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 2 12 10 B -8 0 6 16 8 C -2 -6 0 6 -10 D -12 -16 -6 0 -4 E -10 -8 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 12 10 B -8 0 6 16 8 C -2 -6 0 6 -10 D -12 -16 -6 0 -4 E -10 -8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=20 B=20 D=17 E=10 so E is eliminated. Round 2 votes counts: A=33 C=29 B=21 D=17 so D is eliminated. Round 3 votes counts: C=40 A=36 B=24 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:211 E:198 C:194 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 12 10 B -8 0 6 16 8 C -2 -6 0 6 -10 D -12 -16 -6 0 -4 E -10 -8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 12 10 B -8 0 6 16 8 C -2 -6 0 6 -10 D -12 -16 -6 0 -4 E -10 -8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 12 10 B -8 0 6 16 8 C -2 -6 0 6 -10 D -12 -16 -6 0 -4 E -10 -8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1417: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) E C D B A (6) C B A E D (6) C B E D A (4) A D B E C (4) A B D E C (4) E C B D A (3) D A E B C (3) C E A D B (3) C A B D E (3) B E D C A (3) B D A E C (3) A D C E B (3) E D A C B (2) E D A B C (2) D E B A C (2) D E A C B (2) D E A B C (2) D A B E C (2) C E D A B (2) C E A B D (2) B C A D E (2) B A D E C (2) B A D C E (2) A D E C B (2) A D E B C (2) A D C B E (2) E D C A B (1) E D B A C (1) E C D A B (1) E B D C A (1) D B E A C (1) C E D B A (1) C E B A D (1) C B E A D (1) C A E D B (1) C A D E B (1) C A D B E (1) B D E A C (1) B C E D A (1) B C D A E (1) B A C D E (1) A E D C B (1) A C D E B (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -4 -4 0 B 0 0 -18 0 -6 C 4 18 0 4 -2 D 4 0 -4 0 0 E 0 6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.246644 E: 0.753356 Sum of squares = 0.628378048769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.246644 E: 1.000000 A B C D E A 0 0 -4 -4 0 B 0 0 -18 0 -6 C 4 18 0 4 -2 D 4 0 -4 0 0 E 0 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555690112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=22 E=17 B=16 D=12 so D is eliminated. Round 2 votes counts: C=33 A=27 E=23 B=17 so B is eliminated. Round 3 votes counts: C=37 A=35 E=28 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:204 D:200 A:196 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 -4 0 B 0 0 -18 0 -6 C 4 18 0 4 -2 D 4 0 -4 0 0 E 0 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555690112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 0 B 0 0 -18 0 -6 C 4 18 0 4 -2 D 4 0 -4 0 0 E 0 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555690112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 0 B 0 0 -18 0 -6 C 4 18 0 4 -2 D 4 0 -4 0 0 E 0 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555690112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1418: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) E C D B A (10) A E D C B (10) E D C A B (9) B C E D A (6) A D E C B (5) B A C D E (4) A B E D C (4) A B D C E (4) E D A C B (3) C E D B A (3) C D E B A (3) E A D C B (2) D E C A B (2) D E A C B (2) C D B E A (2) A D C B E (2) E D C B A (1) D C E A B (1) D C B E A (1) D C A E B (1) D A E C B (1) B E A C D (1) B C D A E (1) B C A D E (1) B A E C D (1) A E D B C (1) A E B D C (1) A E B C D (1) A D C E B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -10 -20 -20 B -4 0 -18 -20 -14 C 10 18 0 -4 -12 D 20 20 4 0 -8 E 20 14 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -10 -20 -20 B -4 0 -18 -20 -14 C 10 18 0 -4 -12 D 20 20 4 0 -8 E 20 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=28 E=25 D=8 C=8 so D is eliminated. Round 2 votes counts: A=32 E=29 B=28 C=11 so C is eliminated. Round 3 votes counts: E=36 A=33 B=31 so B is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:218 C:206 A:177 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -10 -20 -20 B -4 0 -18 -20 -14 C 10 18 0 -4 -12 D 20 20 4 0 -8 E 20 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -20 -20 B -4 0 -18 -20 -14 C 10 18 0 -4 -12 D 20 20 4 0 -8 E 20 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -20 -20 B -4 0 -18 -20 -14 C 10 18 0 -4 -12 D 20 20 4 0 -8 E 20 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1419: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) E B D A C (6) C D A B E (6) A C D E B (6) E A B C D (5) E B A D C (4) C A D B E (4) B D E C A (4) A E C B D (4) A D C E B (4) E A B D C (3) B D C E A (3) B C E D A (3) E B C A D (2) D C B A E (2) D C A B E (2) D B E A C (2) D B C E A (2) C B E A D (2) C B D A E (2) C A E B D (2) B E C A D (2) B C D E A (2) A E D C B (2) A C E D B (2) A C D B E (2) E B C D A (1) E A D B C (1) D E A B C (1) D B E C A (1) D A E C B (1) D A E B C (1) C D B E A (1) C D B A E (1) C B D E A (1) B E D C A (1) B E C D A (1) A E D B C (1) A E C D B (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 0 10 12 -10 B 0 0 8 12 -12 C -10 -8 0 16 -6 D -12 -12 -16 0 -2 E 10 12 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 10 12 -10 B 0 0 8 12 -12 C -10 -8 0 16 -6 D -12 -12 -16 0 -2 E 10 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=24 C=19 B=16 D=12 so D is eliminated. Round 2 votes counts: E=30 A=26 C=23 B=21 so B is eliminated. Round 3 votes counts: E=41 C=33 A=26 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:206 B:204 C:196 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 12 -10 B 0 0 8 12 -12 C -10 -8 0 16 -6 D -12 -12 -16 0 -2 E 10 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 12 -10 B 0 0 8 12 -12 C -10 -8 0 16 -6 D -12 -12 -16 0 -2 E 10 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 12 -10 B 0 0 8 12 -12 C -10 -8 0 16 -6 D -12 -12 -16 0 -2 E 10 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1420: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) D C A B E (8) E C D A B (6) E B A C D (6) B E A D C (6) B A D C E (6) A D C B E (5) E C D B A (4) D A C B E (4) C D E A B (4) B A E D C (4) A B D C E (4) E C B D A (3) C E D A B (3) B E D C A (3) B E A C D (3) E C A D B (2) E C A B D (2) D C A E B (2) C A D E B (2) B E D A C (2) B D A C E (2) E B D C A (1) E B C D A (1) E A B C D (1) D B A C E (1) C D E B A (1) B A D E C (1) A D B C E (1) A C D E B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -2 -8 4 B -12 0 -12 -6 4 C 2 12 0 -2 6 D 8 6 2 0 4 E -4 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 -8 4 B -12 0 -12 -6 4 C 2 12 0 -2 6 D 8 6 2 0 4 E -4 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=26 C=19 D=15 A=13 so A is eliminated. Round 2 votes counts: B=33 E=26 D=21 C=20 so C is eliminated. Round 3 votes counts: D=38 B=33 E=29 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:209 A:203 E:191 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -2 -8 4 B -12 0 -12 -6 4 C 2 12 0 -2 6 D 8 6 2 0 4 E -4 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -8 4 B -12 0 -12 -6 4 C 2 12 0 -2 6 D 8 6 2 0 4 E -4 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -8 4 B -12 0 -12 -6 4 C 2 12 0 -2 6 D 8 6 2 0 4 E -4 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1421: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (12) D B C A E (9) B D C E A (7) C D B A E (6) C A D E B (6) B D E C A (6) E B A D C (5) E A B C D (5) B D E A C (5) E A B D C (4) C D A B E (4) A E C D B (4) A C E D B (4) E B D A C (2) E A C D B (2) D C B A E (2) C A E D B (2) B E D A C (2) A E B D C (2) E B A C D (1) D B C E A (1) C E D B A (1) C D B E A (1) C B D E A (1) C A D B E (1) B E D C A (1) B E A D C (1) B D C A E (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 2 2 -14 B 4 0 6 10 -2 C -2 -6 0 2 -4 D -2 -10 -2 0 4 E 14 2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000012 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 A B C D E A 0 -4 2 2 -14 B 4 0 6 10 -2 C -2 -6 0 2 -4 D -2 -10 -2 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=23 C=22 D=12 A=12 so D is eliminated. Round 2 votes counts: B=33 E=31 C=24 A=12 so A is eliminated. Round 3 votes counts: E=37 B=34 C=29 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:209 E:208 C:195 D:195 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 2 -14 B 4 0 6 10 -2 C -2 -6 0 2 -4 D -2 -10 -2 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 2 -14 B 4 0 6 10 -2 C -2 -6 0 2 -4 D -2 -10 -2 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 2 -14 B 4 0 6 10 -2 C -2 -6 0 2 -4 D -2 -10 -2 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1422: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) C D B A E (11) E A C B D (5) D B C A E (5) C E A D B (4) B D A E C (4) A E B D C (4) E C A B D (3) E A C D B (3) D C B A E (3) C E D A B (3) C D B E A (3) C B E D A (3) A E D B C (3) A D B E C (3) E A D B C (2) E A B C D (2) D B A E C (2) C E D B A (2) C E B D A (2) C D E A B (2) C D A E B (2) C D A B E (2) C B D E A (2) A B D E C (2) E C A D B (1) E B A D C (1) E A D C B (1) D B A C E (1) C E B A D (1) C B D A E (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C A E (1) B A E D C (1) A D C E B (1) Total count = 100 A B C D E A 0 10 -4 0 -6 B -10 0 -12 -10 -8 C 4 12 0 6 0 D 0 10 -6 0 -8 E 6 8 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.479929 D: 0.000000 E: 0.520071 Sum of squares = 0.500805648681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.479929 D: 0.479929 E: 1.000000 A B C D E A 0 10 -4 0 -6 B -10 0 -12 -10 -8 C 4 12 0 6 0 D 0 10 -6 0 -8 E 6 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=29 A=13 D=11 B=8 so B is eliminated. Round 2 votes counts: C=39 E=31 D=16 A=14 so A is eliminated. Round 3 votes counts: E=39 C=39 D=22 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 E:211 A:200 D:198 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 0 -6 B -10 0 -12 -10 -8 C 4 12 0 6 0 D 0 10 -6 0 -8 E 6 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 0 -6 B -10 0 -12 -10 -8 C 4 12 0 6 0 D 0 10 -6 0 -8 E 6 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 0 -6 B -10 0 -12 -10 -8 C 4 12 0 6 0 D 0 10 -6 0 -8 E 6 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1423: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) D C B E A (5) C E A D B (5) A C E D B (5) A C D B E (5) E B D C A (4) B D A E C (4) A C E B D (4) A C D E B (4) D B C E A (3) C A E D B (3) B E D C A (3) B E D A C (3) B D E C A (3) B A D E C (3) A D C B E (3) E C B D A (2) E C A B D (2) E B D A C (2) E B C A D (2) E B A C D (2) E A B C D (2) D B A C E (2) C E D A B (2) C D A B E (2) C A D E B (2) A D B C E (2) A B E D C (2) E D B C A (1) E B C D A (1) E B A D C (1) E A C B D (1) D E B C A (1) D C E B A (1) D C B A E (1) C E D B A (1) C A D B E (1) B E A D C (1) B D E A C (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -2 4 -10 B 6 0 -2 -10 2 C 2 2 0 -4 4 D -4 10 4 0 -2 E 10 -2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 -6 -2 4 -10 B 6 0 -2 -10 2 C 2 2 0 -4 4 D -4 10 4 0 -2 E 10 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.360000000082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=20 D=19 B=18 C=16 so C is eliminated. Round 2 votes counts: A=33 E=28 D=21 B=18 so B is eliminated. Round 3 votes counts: A=36 E=35 D=29 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:204 E:203 C:202 B:198 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 4 -10 B 6 0 -2 -10 2 C 2 2 0 -4 4 D -4 10 4 0 -2 E 10 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.360000000082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 4 -10 B 6 0 -2 -10 2 C 2 2 0 -4 4 D -4 10 4 0 -2 E 10 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.360000000082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 4 -10 B 6 0 -2 -10 2 C 2 2 0 -4 4 D -4 10 4 0 -2 E 10 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.360000000082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1424: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (16) D C A B E (14) D C A E B (5) C D A B E (5) C A D B E (5) B E A C D (5) D C B A E (4) B E D A C (4) E A C B D (3) D E C B A (3) D E B C A (3) D B C A E (3) E B D A C (2) E B A D C (2) E A B C D (2) C A D E B (2) A E C B D (2) A C D B E (2) A C B D E (2) A B C E D (2) E B D C A (1) D E C A B (1) D C E A B (1) D C B E A (1) D B E C A (1) D B C E A (1) C D A E B (1) C A E D B (1) B E A D C (1) B A E C D (1) B A D C E (1) B A C D E (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -4 0 6 B 0 0 -6 -8 6 C 4 6 0 4 4 D 0 8 -4 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 0 6 B 0 0 -6 -8 6 C 4 6 0 4 4 D 0 8 -4 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=26 C=14 B=13 A=10 so A is eliminated. Round 2 votes counts: D=37 E=29 C=19 B=15 so B is eliminated. Round 3 votes counts: E=40 D=38 C=22 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:209 D:208 A:201 B:196 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 0 6 B 0 0 -6 -8 6 C 4 6 0 4 4 D 0 8 -4 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 0 6 B 0 0 -6 -8 6 C 4 6 0 4 4 D 0 8 -4 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 0 6 B 0 0 -6 -8 6 C 4 6 0 4 4 D 0 8 -4 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1425: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C B D A E (8) B E C A D (8) E A D B C (7) B C E A D (7) B C E D A (6) C D A B E (5) A D E C B (5) E B A C D (4) D A C E B (4) B C D E A (4) E B A D C (3) E A B D C (3) D C A E B (3) E D A B C (2) D C A B E (2) C D B A E (2) A E D C B (2) D E B A C (1) D E A B C (1) D C B A E (1) D B A E C (1) C D A E B (1) C B E D A (1) C B E A D (1) C B D E A (1) C B A D E (1) B E D C A (1) B E C D A (1) B E A C D (1) B D C E A (1) A E D B C (1) A E C D B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -8 -8 -6 B 6 0 4 2 4 C 8 -4 0 6 0 D 8 -2 -6 0 2 E 6 -4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999581 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -8 -6 B 6 0 4 2 4 C 8 -4 0 6 0 D 8 -2 -6 0 2 E 6 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=21 C=20 E=19 A=11 so A is eliminated. Round 2 votes counts: B=29 D=27 E=23 C=21 so C is eliminated. Round 3 votes counts: B=41 D=36 E=23 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 C:205 D:201 E:200 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -8 -6 B 6 0 4 2 4 C 8 -4 0 6 0 D 8 -2 -6 0 2 E 6 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -8 -6 B 6 0 4 2 4 C 8 -4 0 6 0 D 8 -2 -6 0 2 E 6 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -8 -6 B 6 0 4 2 4 C 8 -4 0 6 0 D 8 -2 -6 0 2 E 6 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1426: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (19) D B E C A (12) E B D A C (11) C A D B E (10) C A E B D (7) B E D A C (4) E B A D C (3) C D A B E (3) C A D E B (3) A C D E B (3) E B A C D (2) D C B A E (2) D C A B E (2) D B C E A (2) C A E D B (2) B D E C A (2) A E B C D (2) A C E D B (2) E A B C D (1) D E B A C (1) D B E A C (1) C D B A E (1) C A B E D (1) C A B D E (1) B D E A C (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 16 4 16 20 B -16 0 -16 10 -16 C -4 16 0 18 16 D -16 -10 -18 0 -12 E -20 16 -16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 16 20 B -16 0 -16 10 -16 C -4 16 0 18 16 D -16 -10 -18 0 -12 E -20 16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=28 A=28 D=20 E=17 B=7 so B is eliminated. Round 2 votes counts: C=28 A=28 D=23 E=21 so E is eliminated. Round 3 votes counts: D=38 A=34 C=28 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:223 E:196 B:181 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 4 16 20 B -16 0 -16 10 -16 C -4 16 0 18 16 D -16 -10 -18 0 -12 E -20 16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 16 20 B -16 0 -16 10 -16 C -4 16 0 18 16 D -16 -10 -18 0 -12 E -20 16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 16 20 B -16 0 -16 10 -16 C -4 16 0 18 16 D -16 -10 -18 0 -12 E -20 16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1427: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) E B D C A (8) B D E C A (8) A C E D B (8) C D E A B (5) A C E B D (4) A C D B E (4) E C D B A (3) E C A D B (3) E B C D A (3) D E B C A (3) C E D A B (3) B D E A C (3) A C D E B (3) D E C B A (2) D C E B A (2) D B E C A (2) C A E D B (2) A B E C D (2) A B D C E (2) E B A C D (1) E A C B D (1) D C A B E (1) D B C E A (1) D B C A E (1) D A C B E (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B A E D C (1) B A D E C (1) B A D C E (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B C E (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -18 -16 -20 B 4 0 4 4 -10 C 18 -4 0 -2 -10 D 16 -4 2 0 -10 E 20 10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -18 -16 -20 B 4 0 4 4 -10 C 18 -4 0 -2 -10 D 16 -4 2 0 -10 E 20 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=26 E=19 D=13 C=12 so C is eliminated. Round 2 votes counts: A=33 B=26 E=22 D=19 so D is eliminated. Round 3 votes counts: A=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:202 B:201 C:201 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -18 -16 -20 B 4 0 4 4 -10 C 18 -4 0 -2 -10 D 16 -4 2 0 -10 E 20 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -16 -20 B 4 0 4 4 -10 C 18 -4 0 -2 -10 D 16 -4 2 0 -10 E 20 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -16 -20 B 4 0 4 4 -10 C 18 -4 0 -2 -10 D 16 -4 2 0 -10 E 20 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1428: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (13) A D E B C (7) B D A E C (5) B C A D E (5) C E A D B (4) B D E A C (4) B C D E A (4) A E D C B (4) A E C D B (4) D A B E C (3) B D E C A (3) E D B A C (2) E D A B C (2) D E B A C (2) D E A B C (2) D B A E C (2) C B A D E (2) C A E D B (2) C A E B D (2) B C E D A (2) A D E C B (2) A C E D B (2) A C D E B (2) A C D B E (2) A C B D E (2) E D C B A (1) E D A C B (1) E C D B A (1) E B D C A (1) D A E B C (1) C E D A B (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) C A B E D (1) B C D A E (1) B A C D E (1) A D C E B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 6 -2 8 B 4 0 -2 0 8 C -6 2 0 10 4 D 2 0 -10 0 6 E -8 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888939 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -2 8 B 4 0 -2 0 8 C -6 2 0 10 4 D 2 0 -10 0 6 E -8 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=28 B=25 D=10 E=8 so E is eliminated. Round 2 votes counts: C=30 A=28 B=26 D=16 so D is eliminated. Round 3 votes counts: A=37 B=32 C=31 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:205 C:205 A:204 D:199 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 -2 8 B 4 0 -2 0 8 C -6 2 0 10 4 D 2 0 -10 0 6 E -8 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -2 8 B 4 0 -2 0 8 C -6 2 0 10 4 D 2 0 -10 0 6 E -8 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -2 8 B 4 0 -2 0 8 C -6 2 0 10 4 D 2 0 -10 0 6 E -8 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1429: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C D E B A (8) D E B A C (7) C A D E B (7) D E B C A (6) A B E D C (6) C D E A B (5) A B C E D (5) D E C B A (4) E D B C A (3) E D B A C (3) C A B E D (3) C A B D E (3) B A E D C (3) A C B D E (3) D C E B A (2) D C E A B (2) C E D B A (2) B E A D C (2) A C D E B (2) A B D E C (2) E D C B A (1) E B D C A (1) C D A E B (1) C B E D A (1) C B E A D (1) C A D B E (1) A D E C B (1) A D E B C (1) A C D B E (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -12 -16 B 8 0 2 -14 -12 C 2 -2 0 -8 -2 D 12 14 8 0 14 E 16 12 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -12 -16 B 8 0 2 -14 -12 C 2 -2 0 -8 -2 D 12 14 8 0 14 E 16 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=24 D=21 B=15 E=8 so E is eliminated. Round 2 votes counts: C=32 D=28 A=24 B=16 so B is eliminated. Round 3 votes counts: D=39 C=32 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:208 C:195 B:192 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -12 -16 B 8 0 2 -14 -12 C 2 -2 0 -8 -2 D 12 14 8 0 14 E 16 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -12 -16 B 8 0 2 -14 -12 C 2 -2 0 -8 -2 D 12 14 8 0 14 E 16 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -12 -16 B 8 0 2 -14 -12 C 2 -2 0 -8 -2 D 12 14 8 0 14 E 16 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1430: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) D E A C B (6) C B E A D (6) E B C A D (5) D A B C E (5) E D A B C (4) E B A C D (4) C B A D E (4) B A C E D (4) A B D C E (4) E C B D A (3) E C B A D (3) D C A B E (3) D A E B C (3) C E B A D (3) C A B D E (3) B C E A D (3) A B C D E (3) E C D B A (2) D E C A B (2) D E A B C (2) D C E A B (2) D A B E C (2) C D E A B (2) C B A E D (2) E D C B A (1) E B D A C (1) E B A D C (1) E A D B C (1) D C A E B (1) D A E C B (1) D A C E B (1) C D A B E (1) C A D B E (1) B E A C D (1) B C A E D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 12 4 2 -4 B -12 0 -8 4 4 C -4 8 0 2 12 D -2 -4 -2 0 8 E 4 -4 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.579932 B: 0.000000 C: 0.059521 D: 0.200684 E: 0.159863 Sum of squares = 0.4056937791 Cumulative probabilities = A: 0.579932 B: 0.579932 C: 0.639453 D: 0.840137 E: 1.000000 A B C D E A 0 12 4 2 -4 B -12 0 -8 4 4 C -4 8 0 2 12 D -2 -4 -2 0 8 E 4 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.584416 B: 0.000000 C: 0.090910 D: 0.155843 E: 0.168831 Sum of squares = 0.402597402585 Cumulative probabilities = A: 0.584416 B: 0.584416 C: 0.675326 D: 0.831169 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=25 C=22 B=9 A=9 so B is eliminated. Round 2 votes counts: D=35 E=26 C=26 A=13 so A is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:207 D:200 B:194 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 2 -4 B -12 0 -8 4 4 C -4 8 0 2 12 D -2 -4 -2 0 8 E 4 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.584416 B: 0.000000 C: 0.090910 D: 0.155843 E: 0.168831 Sum of squares = 0.402597402585 Cumulative probabilities = A: 0.584416 B: 0.584416 C: 0.675326 D: 0.831169 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 2 -4 B -12 0 -8 4 4 C -4 8 0 2 12 D -2 -4 -2 0 8 E 4 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.584416 B: 0.000000 C: 0.090910 D: 0.155843 E: 0.168831 Sum of squares = 0.402597402585 Cumulative probabilities = A: 0.584416 B: 0.584416 C: 0.675326 D: 0.831169 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 2 -4 B -12 0 -8 4 4 C -4 8 0 2 12 D -2 -4 -2 0 8 E 4 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.584416 B: 0.000000 C: 0.090910 D: 0.155843 E: 0.168831 Sum of squares = 0.402597402585 Cumulative probabilities = A: 0.584416 B: 0.584416 C: 0.675326 D: 0.831169 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1431: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (16) A D B E C (11) D C E B A (6) A D C E B (6) D C A E B (5) D A C E B (5) B E C A D (5) B E C D A (4) A B E D C (4) D B E A C (3) C D E B A (3) A B D E C (3) E B C D A (2) D E B C A (2) D A E B C (2) C E D B A (2) C A E B D (2) C A B E D (2) B E A C D (2) A C B E D (2) D E C A B (1) D B E C A (1) D A B E C (1) C E D A B (1) C E B A D (1) C D E A B (1) C A D E B (1) B E A D C (1) B D E A C (1) A D C B E (1) A C E D B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -10 -12 -4 B -2 0 -14 -8 -16 C 10 14 0 -6 14 D 12 8 6 0 8 E 4 16 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -12 -4 B -2 0 -14 -8 -16 C 10 14 0 -6 14 D 12 8 6 0 8 E 4 16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=29 D=26 B=13 E=2 so E is eliminated. Round 2 votes counts: A=30 C=29 D=26 B=15 so B is eliminated. Round 3 votes counts: C=40 A=33 D=27 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:216 E:199 A:188 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -10 -12 -4 B -2 0 -14 -8 -16 C 10 14 0 -6 14 D 12 8 6 0 8 E 4 16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -12 -4 B -2 0 -14 -8 -16 C 10 14 0 -6 14 D 12 8 6 0 8 E 4 16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -12 -4 B -2 0 -14 -8 -16 C 10 14 0 -6 14 D 12 8 6 0 8 E 4 16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1432: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) A E B D C (6) A E B C D (6) E A D B C (5) D E C B A (5) A B C E D (5) E D A C B (4) E D A B C (4) D E C A B (4) D C B E A (4) C B A D E (4) B C A D E (4) A B E C D (4) D C E B A (3) C D B E A (3) C D B A E (3) C B D E A (3) B C D A E (3) A E D B C (3) A C B E D (3) E D B A C (2) D E B C A (2) C A B D E (2) B C D E A (2) E A D C B (1) D B E C A (1) C D E B A (1) B A C E D (1) A E D C B (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -6 -6 12 B 0 0 -2 8 4 C 6 2 0 8 0 D 6 -8 -8 0 6 E -12 -4 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.803758 D: 0.000000 E: 0.196242 Sum of squares = 0.684538438508 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.803758 D: 0.803758 E: 1.000000 A B C D E A 0 0 -6 -6 12 B 0 0 -2 8 4 C 6 2 0 8 0 D 6 -8 -8 0 6 E -12 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555562097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=25 D=19 E=16 B=10 so B is eliminated. Round 2 votes counts: C=34 A=31 D=19 E=16 so E is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:205 A:200 D:198 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 -6 12 B 0 0 -2 8 4 C 6 2 0 8 0 D 6 -8 -8 0 6 E -12 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555562097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -6 12 B 0 0 -2 8 4 C 6 2 0 8 0 D 6 -8 -8 0 6 E -12 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555562097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -6 12 B 0 0 -2 8 4 C 6 2 0 8 0 D 6 -8 -8 0 6 E -12 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555562097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1433: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (14) A D C E B (14) B A E C D (9) B E C A D (7) A B D C E (6) D A C E B (5) A D C B E (5) E C B D A (4) E B C D A (4) D C E A B (4) C D E B A (4) E C D B A (3) D C A E B (3) C E D B A (2) B E A C D (2) A D B C E (2) A B E C D (2) A B D E C (2) E B A C D (1) D C E B A (1) C B E D A (1) B C E D A (1) B C D E A (1) B C A D E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 0 6 2 B 10 0 8 12 8 C 0 -8 0 12 0 D -6 -12 -12 0 -2 E -2 -8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 6 2 B 10 0 8 12 8 C 0 -8 0 12 0 D -6 -12 -12 0 -2 E -2 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=33 D=13 E=12 C=7 so C is eliminated. Round 2 votes counts: B=36 A=33 D=17 E=14 so E is eliminated. Round 3 votes counts: B=45 A=33 D=22 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:202 A:199 E:196 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 6 2 B 10 0 8 12 8 C 0 -8 0 12 0 D -6 -12 -12 0 -2 E -2 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 2 B 10 0 8 12 8 C 0 -8 0 12 0 D -6 -12 -12 0 -2 E -2 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 2 B 10 0 8 12 8 C 0 -8 0 12 0 D -6 -12 -12 0 -2 E -2 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1434: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (7) C D B E A (6) E A D B C (5) C B D E A (5) D E C A B (4) B A C E D (4) A B E C D (4) E D C A B (3) E D A C B (3) D E B A C (3) D C E B A (3) C D E B A (3) C B A E D (3) C A B E D (3) B A E D C (3) B A D E C (3) A B E D C (3) D E C B A (2) D B E A C (2) C B D A E (2) C B A D E (2) B D A E C (2) B C D A E (2) B C A E D (2) A E D B C (2) A E B D C (2) A B C E D (2) E D A B C (1) E A D C B (1) D E B C A (1) D E A B C (1) D C E A B (1) D C B E A (1) D B C E A (1) C A E D B (1) C A E B D (1) B D E C A (1) B A E C D (1) A E C D B (1) A E C B D (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 -8 6 6 B 18 0 6 10 18 C 8 -6 0 4 0 D -6 -10 -4 0 4 E -6 -18 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 6 6 B 18 0 6 10 18 C 8 -6 0 4 0 D -6 -10 -4 0 4 E -6 -18 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 D=19 A=17 E=13 so E is eliminated. Round 2 votes counts: D=26 C=26 B=25 A=23 so A is eliminated. Round 3 votes counts: B=37 D=34 C=29 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:203 A:193 D:192 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -8 6 6 B 18 0 6 10 18 C 8 -6 0 4 0 D -6 -10 -4 0 4 E -6 -18 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 6 6 B 18 0 6 10 18 C 8 -6 0 4 0 D -6 -10 -4 0 4 E -6 -18 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 6 6 B 18 0 6 10 18 C 8 -6 0 4 0 D -6 -10 -4 0 4 E -6 -18 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1435: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) C D E A B (7) B A E D C (7) D C E B A (6) D C B E A (6) C E D A B (6) A E C B D (6) D B C E A (4) D B C A E (4) C E A D B (4) B A D E C (4) A E B C D (4) A C E B D (4) E A C B D (3) C D E B A (3) B D A C E (3) A B E D C (3) D C B A E (2) D B A C E (2) C A E D B (2) B D A E C (2) E C B A D (1) E B D A C (1) E B A D C (1) E A B C D (1) D C A B E (1) D A B C E (1) B E A D C (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 8 4 12 B -6 0 -4 2 2 C -8 4 0 2 16 D -4 -2 -2 0 -6 E -12 -2 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 4 12 B -6 0 -4 2 2 C -8 4 0 2 16 D -4 -2 -2 0 -6 E -12 -2 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 C=22 B=17 E=7 so E is eliminated. Round 2 votes counts: A=32 D=26 C=23 B=19 so B is eliminated. Round 3 votes counts: A=45 D=32 C=23 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:207 B:197 D:193 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 4 12 B -6 0 -4 2 2 C -8 4 0 2 16 D -4 -2 -2 0 -6 E -12 -2 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 4 12 B -6 0 -4 2 2 C -8 4 0 2 16 D -4 -2 -2 0 -6 E -12 -2 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 4 12 B -6 0 -4 2 2 C -8 4 0 2 16 D -4 -2 -2 0 -6 E -12 -2 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1436: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) E A C D B (8) E A C B D (7) B D E A C (7) B D C A E (7) B D E C A (6) E A B C D (5) B E D A C (5) A C E B D (5) D B C E A (4) B D C E A (4) A E C B D (4) A C E D B (4) E D B A C (2) E B A C D (2) D C B A E (2) C A E D B (2) C A D E B (2) B E A D C (2) A E C D B (2) E D A C B (1) E B A D C (1) D C A E B (1) D C A B E (1) D B E C A (1) D B E A C (1) C D A B E (1) C B D A E (1) C A E B D (1) B C D A E (1) B A E C D (1) Total count = 100 A B C D E A 0 -12 14 -8 -12 B 12 0 16 18 6 C -14 -16 0 -8 -10 D 8 -18 8 0 -4 E 12 -6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 14 -8 -12 B 12 0 16 18 6 C -14 -16 0 -8 -10 D 8 -18 8 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=26 D=19 A=15 C=7 so C is eliminated. Round 2 votes counts: B=34 E=26 D=20 A=20 so D is eliminated. Round 3 votes counts: B=51 E=26 A=23 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:210 D:197 A:191 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 14 -8 -12 B 12 0 16 18 6 C -14 -16 0 -8 -10 D 8 -18 8 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 14 -8 -12 B 12 0 16 18 6 C -14 -16 0 -8 -10 D 8 -18 8 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 14 -8 -12 B 12 0 16 18 6 C -14 -16 0 -8 -10 D 8 -18 8 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1437: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) C B E D A (8) C D E B A (7) B E C D A (7) A C D E B (6) A D E C B (5) D E C B A (3) C E D B A (3) C B A E D (3) C A D E B (3) B E D A C (3) B A E C D (3) A B C E D (3) A B C D E (3) C D E A B (2) C D A E B (2) C A B D E (2) B A E D C (2) A D E B C (2) A D C E B (2) A C B D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D C A (1) E B D A C (1) D E B C A (1) D E B A C (1) D E A B C (1) D C E A B (1) C A D B E (1) C A B E D (1) B E A D C (1) B C E D A (1) B C A E D (1) B A C E D (1) A E D B C (1) A E B D C (1) A C D B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 -16 -8 -6 B 18 0 -6 10 8 C 16 6 0 20 8 D 8 -10 -20 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 -8 -6 B 18 0 -6 10 8 C 16 6 0 20 8 D 8 -10 -20 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=28 A=28 D=7 E=5 so E is eliminated. Round 2 votes counts: C=32 B=30 A=28 D=10 so D is eliminated. Round 3 votes counts: C=37 B=34 A=29 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:215 E:199 D:185 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -16 -8 -6 B 18 0 -6 10 8 C 16 6 0 20 8 D 8 -10 -20 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -8 -6 B 18 0 -6 10 8 C 16 6 0 20 8 D 8 -10 -20 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -8 -6 B 18 0 -6 10 8 C 16 6 0 20 8 D 8 -10 -20 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1438: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (5) D C E A B (5) C A B E D (5) E C B A D (4) D E C A B (4) D A B C E (4) B E C A D (4) B A D C E (4) E C B D A (3) E B C A D (3) C E A D B (3) C E A B D (3) C A D E B (3) B E A C D (3) E D C B A (2) E D C A B (2) E D B C A (2) E D B A C (2) E C D B A (2) D B A C E (2) C A E B D (2) B D A C E (2) B A C E D (2) A D C B E (2) A C D E B (2) A C B D E (2) A B D C E (2) A B C D E (2) E B D C A (1) E B D A C (1) E B C D A (1) D E B C A (1) D E B A C (1) D C A E B (1) D B E A C (1) D B A E C (1) D A C E B (1) D A C B E (1) C E D A B (1) C D A E B (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E A C (1) B A D E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -18 0 -16 B -6 0 -10 -2 -12 C 18 10 0 6 4 D 0 2 -6 0 -8 E 16 12 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -18 0 -16 B -6 0 -10 -2 -12 C 18 10 0 6 4 D 0 2 -6 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=22 B=20 C=18 A=12 so A is eliminated. Round 2 votes counts: E=28 D=25 B=24 C=23 so C is eliminated. Round 3 votes counts: E=37 D=32 B=31 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:216 D:194 A:186 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -18 0 -16 B -6 0 -10 -2 -12 C 18 10 0 6 4 D 0 2 -6 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 0 -16 B -6 0 -10 -2 -12 C 18 10 0 6 4 D 0 2 -6 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 0 -16 B -6 0 -10 -2 -12 C 18 10 0 6 4 D 0 2 -6 0 -8 E 16 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1439: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (6) E A D C B (5) C D A B E (5) B E A D C (5) B C D E A (5) A E D C B (5) E D A C B (4) B E D C A (4) E D B A C (3) C D B E A (3) C D B A E (3) C A D E B (3) B E C D A (3) E A B D C (2) D C E B A (2) C D A E B (2) C B A D E (2) C A D B E (2) C A B D E (2) B E D A C (2) B E A C D (2) B D C E A (2) B C D A E (2) B C A D E (2) B A E C D (2) B A C E D (2) A C D E B (2) A B E C D (2) E D A B C (1) E B D A C (1) E A D B C (1) D E C A B (1) D E A C B (1) D C E A B (1) C B D E A (1) B D E C A (1) B C E A D (1) B A E D C (1) A E D B C (1) A E B D C (1) A E B C D (1) A D E C B (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -6 -6 -2 B 10 0 -4 6 22 C 6 4 0 10 0 D 6 -6 -10 0 -2 E 2 -22 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.901539 D: 0.000000 E: 0.098461 Sum of squares = 0.822467361903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.901539 D: 0.901539 E: 1.000000 A B C D E A 0 -10 -6 -6 -2 B 10 0 -4 6 22 C 6 4 0 10 0 D 6 -6 -10 0 -2 E 2 -22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.000000 E: 0.153846 Sum of squares = 0.739645016051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.846154 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=29 E=17 A=15 D=5 so D is eliminated. Round 2 votes counts: B=34 C=32 E=19 A=15 so A is eliminated. Round 3 votes counts: B=37 C=35 E=28 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:217 C:210 D:194 E:191 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 -6 -2 B 10 0 -4 6 22 C 6 4 0 10 0 D 6 -6 -10 0 -2 E 2 -22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.000000 E: 0.153846 Sum of squares = 0.739645016051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.846154 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -6 -2 B 10 0 -4 6 22 C 6 4 0 10 0 D 6 -6 -10 0 -2 E 2 -22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.000000 E: 0.153846 Sum of squares = 0.739645016051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.846154 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -6 -2 B 10 0 -4 6 22 C 6 4 0 10 0 D 6 -6 -10 0 -2 E 2 -22 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.000000 E: 0.153846 Sum of squares = 0.739645016051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.846154 D: 0.846154 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1440: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) E B C A D (5) B E C A D (5) A D C B E (5) A D B C E (5) E B C D A (4) E B A C D (4) D C A B E (4) D A C B E (4) B C D A E (4) E A D C B (3) C D B A E (3) C B E D A (3) C B D A E (3) B E C D A (3) A D E B C (3) E D A C B (2) E C D A B (2) E A D B C (2) E A B D C (2) D E C A B (2) D A E C B (2) D A C E B (2) C D A B E (2) B C A D E (2) B A C D E (2) A D E C B (2) E C D B A (1) E B A D C (1) C E D B A (1) C E D A B (1) C E B D A (1) C D E A B (1) C B D E A (1) B C E A D (1) B A E D C (1) B A D C E (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -14 -8 -6 B 8 0 -8 2 -2 C 14 8 0 14 -6 D 8 -2 -14 0 -2 E 6 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -14 -8 -6 B 8 0 -8 2 -2 C 14 8 0 14 -6 D 8 -2 -14 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=19 A=17 C=16 D=14 so D is eliminated. Round 2 votes counts: E=36 A=25 C=20 B=19 so B is eliminated. Round 3 votes counts: E=44 A=29 C=27 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:208 B:200 D:195 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 -8 -6 B 8 0 -8 2 -2 C 14 8 0 14 -6 D 8 -2 -14 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -8 -6 B 8 0 -8 2 -2 C 14 8 0 14 -6 D 8 -2 -14 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -8 -6 B 8 0 -8 2 -2 C 14 8 0 14 -6 D 8 -2 -14 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1441: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) C A D E B (8) E D C A B (5) D E C A B (5) B D E A C (5) E B D C A (4) A C B E D (4) A C B D E (4) C E A D B (3) B E D A C (3) B A E C D (3) B A C E D (3) A C D B E (3) A B C E D (3) E D C B A (2) E D B C A (2) E C A D B (2) D E C B A (2) D E B C A (2) D C E A B (2) D C A E B (2) B D A C E (2) B A D C E (2) E C D A B (1) E B D A C (1) E B A C D (1) D C E B A (1) D B E C A (1) D B C E A (1) D B C A E (1) D A C B E (1) C E D A B (1) C D E A B (1) C D A E B (1) C A E D B (1) B E A D C (1) B D E C A (1) B D A E C (1) B A D E C (1) A C E D B (1) A C E B D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 2 4 6 B -2 0 -2 2 2 C -2 2 0 4 12 D -4 -2 -4 0 14 E -6 -2 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 4 6 B -2 0 -2 2 2 C -2 2 0 4 12 D -4 -2 -4 0 14 E -6 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=18 D=18 A=18 C=15 so C is eliminated. Round 2 votes counts: B=31 A=27 E=22 D=20 so D is eliminated. Round 3 votes counts: E=35 B=34 A=31 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:208 A:207 D:202 B:200 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 6 B -2 0 -2 2 2 C -2 2 0 4 12 D -4 -2 -4 0 14 E -6 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 6 B -2 0 -2 2 2 C -2 2 0 4 12 D -4 -2 -4 0 14 E -6 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 6 B -2 0 -2 2 2 C -2 2 0 4 12 D -4 -2 -4 0 14 E -6 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1442: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) C E D A B (8) A D E B C (5) D A E B C (4) D A B E C (4) B A D C E (4) E C B A D (3) E A D B C (3) E A B D C (3) D A E C B (3) C E D B A (3) C E B A D (3) C B E A D (3) B E A C D (3) B C A D E (3) A D B E C (3) D C A E B (2) D A C B E (2) C D E A B (2) C B E D A (2) C B D A E (2) B E C A D (2) B C D A E (2) B A D E C (2) E D A C B (1) E C A D B (1) E B C A D (1) E A C B D (1) D E A C B (1) D C B A E (1) D C A B E (1) D A C E B (1) D A B C E (1) C E B D A (1) C D B E A (1) C D A E B (1) C D A B E (1) C B D E A (1) B C E D A (1) B C A E D (1) B A E D C (1) B A E C D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -10 8 -6 B 0 0 10 2 4 C 10 -10 0 12 12 D -8 -2 -12 0 -6 E 6 -4 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.241427 B: 0.758573 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.633720336975 Cumulative probabilities = A: 0.241427 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 8 -6 B 0 0 10 2 4 C 10 -10 0 12 12 D -8 -2 -12 0 -6 E 6 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000003192 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=28 D=20 E=13 A=10 so A is eliminated. Round 2 votes counts: B=30 D=28 C=28 E=14 so E is eliminated. Round 3 votes counts: B=35 C=33 D=32 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 B:208 E:198 A:196 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -10 8 -6 B 0 0 10 2 4 C 10 -10 0 12 12 D -8 -2 -12 0 -6 E 6 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000003192 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 8 -6 B 0 0 10 2 4 C 10 -10 0 12 12 D -8 -2 -12 0 -6 E 6 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000003192 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 8 -6 B 0 0 10 2 4 C 10 -10 0 12 12 D -8 -2 -12 0 -6 E 6 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000003192 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1443: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) D A C B E (12) E B A C D (6) D C A B E (6) D C A E B (5) D E A B C (4) B E A C D (4) E D B C A (3) E B C D A (3) D C E A B (3) C D A B E (3) C A B D E (3) A C D B E (3) E D B A C (2) E B D A C (2) E B A D C (2) D E C A B (2) D A E C B (2) B C A E D (2) A B C E D (2) A B C D E (2) E D A B C (1) E C B A D (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A C B (1) D C E B A (1) D A B E C (1) D A B C E (1) C E B D A (1) C E B A D (1) C D E B A (1) C B A E D (1) C A B E D (1) B E C A D (1) Total count = 100 A B C D E A 0 4 -8 -14 -12 B -4 0 4 -8 -16 C 8 -4 0 -2 -4 D 14 8 2 0 6 E 12 16 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -14 -12 B -4 0 4 -8 -16 C 8 -4 0 -2 -4 D 14 8 2 0 6 E 12 16 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994351 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=34 C=11 B=7 A=7 so B is eliminated. Round 2 votes counts: D=41 E=39 C=13 A=7 so A is eliminated. Round 3 votes counts: D=41 E=39 C=20 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:213 C:199 B:188 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -14 -12 B -4 0 4 -8 -16 C 8 -4 0 -2 -4 D 14 8 2 0 6 E 12 16 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994351 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -14 -12 B -4 0 4 -8 -16 C 8 -4 0 -2 -4 D 14 8 2 0 6 E 12 16 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994351 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -14 -12 B -4 0 4 -8 -16 C 8 -4 0 -2 -4 D 14 8 2 0 6 E 12 16 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994351 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1444: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) D A E B C (10) B A D C E (9) D E A C B (8) E C D A B (7) D A B E C (7) C E B A D (6) B C A E D (5) E C B A D (4) D A E C B (4) E D C A B (3) B C D A E (3) E C A B D (2) D E C A B (2) C E B D A (2) C B E D A (2) B C E A D (2) B C A D E (2) A D B E C (2) E C A D B (1) E A D C B (1) E A B C D (1) D A B C E (1) C B D E A (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 2 -6 0 -6 B -2 0 -8 6 -6 C 6 8 0 2 -8 D 0 -6 -2 0 0 E 6 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.344538 E: 0.655462 Sum of squares = 0.548336605267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.344538 E: 1.000000 A B C D E A 0 2 -6 0 -6 B -2 0 -8 6 -6 C 6 8 0 2 -8 D 0 -6 -2 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 0.500338 Sum of squares = 0.500000228815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=23 C=22 E=19 A=4 so A is eliminated. Round 2 votes counts: D=34 B=23 C=22 E=21 so E is eliminated. Round 3 votes counts: D=39 C=36 B=25 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:210 C:204 D:196 A:195 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 0 -6 B -2 0 -8 6 -6 C 6 8 0 2 -8 D 0 -6 -2 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 0.500338 Sum of squares = 0.500000228815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 -6 B -2 0 -8 6 -6 C 6 8 0 2 -8 D 0 -6 -2 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 0.500338 Sum of squares = 0.500000228815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 -6 B -2 0 -8 6 -6 C 6 8 0 2 -8 D 0 -6 -2 0 0 E 6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 0.500338 Sum of squares = 0.500000228815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499662 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1445: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (12) A E D B C (12) C B D E A (9) C B E D A (6) B C D E A (6) A D E B C (5) E D A C B (4) D E A C B (4) B C A E D (4) E A D C B (3) D A E B C (3) A E D C B (3) D B C A E (2) D A E C B (2) C E B D A (2) C D B E A (2) C B E A D (2) A E B C D (2) E D C A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E A C D B (1) E A C B D (1) D E C B A (1) D C B E A (1) D B C E A (1) D B A C E (1) B D A C E (1) B C A D E (1) B A C E D (1) B A C D E (1) A D E C B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -6 -18 6 B 10 0 8 4 4 C 6 -8 0 8 8 D 18 -4 -8 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -18 6 B 10 0 8 4 4 C 6 -8 0 8 8 D 18 -4 -8 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=25 C=21 D=15 E=13 so E is eliminated. Round 2 votes counts: A=30 B=26 C=24 D=20 so D is eliminated. Round 3 votes counts: A=43 B=30 C=27 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:207 D:207 E:187 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -18 6 B 10 0 8 4 4 C 6 -8 0 8 8 D 18 -4 -8 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -18 6 B 10 0 8 4 4 C 6 -8 0 8 8 D 18 -4 -8 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -18 6 B 10 0 8 4 4 C 6 -8 0 8 8 D 18 -4 -8 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1446: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) D A B E C (5) A C B D E (5) A B E C D (5) D C A B E (4) D A B C E (4) B E A D C (4) E C B A D (3) E B C D A (3) D C E B A (3) D B E A C (3) D B A E C (3) C A E B D (3) B A E D C (3) B A E C D (3) E B D A C (2) E B C A D (2) E B A C D (2) C E D A B (2) C E A B D (2) C D E A B (2) B E D A C (2) B D A E C (2) A D B E C (2) A D B C E (2) A C D B E (2) A B D E C (2) A B C E D (2) E D B C A (1) E C D B A (1) E B D C A (1) D E C B A (1) D A C B E (1) C E B D A (1) C D E B A (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E A C D (1) B A D E C (1) A D C B E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 18 12 14 B -2 0 12 14 22 C -18 -12 0 4 -6 D -12 -14 -4 0 -4 E -14 -22 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999239 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 12 14 B -2 0 12 14 22 C -18 -12 0 4 -6 D -12 -14 -4 0 -4 E -14 -22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=23 C=22 B=16 E=15 so E is eliminated. Round 2 votes counts: C=26 B=26 D=25 A=23 so A is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:223 B:223 E:187 C:184 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 18 12 14 B -2 0 12 14 22 C -18 -12 0 4 -6 D -12 -14 -4 0 -4 E -14 -22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 12 14 B -2 0 12 14 22 C -18 -12 0 4 -6 D -12 -14 -4 0 -4 E -14 -22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 12 14 B -2 0 12 14 22 C -18 -12 0 4 -6 D -12 -14 -4 0 -4 E -14 -22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1447: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) D A E B C (6) C B E A D (6) D A B E C (5) E C A B D (4) E A B C D (4) B A E C D (4) B A D C E (4) D E A C B (3) D C E A B (3) D C B A E (3) C E D B A (3) C E B A D (3) E D C A B (2) E D A C B (2) E C D A B (2) D E C A B (2) C B E D A (2) C B A E D (2) B D A C E (2) B C A E D (2) B A D E C (2) B A C E D (2) A E D B C (2) A D E B C (2) A B D E C (2) E C B A D (1) E A D C B (1) E A C B D (1) E A B D C (1) D E A B C (1) D C E B A (1) D B C A E (1) D B A C E (1) D A E C B (1) D A B C E (1) C E B D A (1) C D E B A (1) C D B A E (1) C B D E A (1) B E C A D (1) B A E D C (1) B A C D E (1) A E B D C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 16 10 -4 B -8 0 8 0 -10 C -16 -8 0 -14 -18 D -10 0 14 0 -12 E 4 10 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 16 10 -4 B -8 0 8 0 -10 C -16 -8 0 -14 -18 D -10 0 14 0 -12 E 4 10 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 C=20 B=19 A=9 so A is eliminated. Round 2 votes counts: D=30 E=28 B=22 C=20 so C is eliminated. Round 3 votes counts: E=35 B=33 D=32 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:215 D:196 B:195 C:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 10 -4 B -8 0 8 0 -10 C -16 -8 0 -14 -18 D -10 0 14 0 -12 E 4 10 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 10 -4 B -8 0 8 0 -10 C -16 -8 0 -14 -18 D -10 0 14 0 -12 E 4 10 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 10 -4 B -8 0 8 0 -10 C -16 -8 0 -14 -18 D -10 0 14 0 -12 E 4 10 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1448: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) E C A D B (7) C E D A B (6) B D A C E (6) E A C B D (5) E A C D B (4) D B A C E (4) C E D B A (4) C E A D B (4) C D E B A (4) B D C E A (4) B A D E C (3) A E C D B (3) E C B A D (2) D C A E B (2) D A B E C (2) D A B C E (2) C E B D A (2) B C E D A (2) A D B E C (2) A B E D C (2) E B A C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A B E (1) D A C B E (1) C D B E A (1) C B D E A (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E A C (1) B D C A E (1) B D A E C (1) B C D E A (1) B A E D C (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 12 -8 -2 -24 B -12 0 -22 -6 -18 C 8 22 0 20 0 D 2 6 -20 0 -16 E 24 18 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.427945 D: 0.000000 E: 0.572055 Sum of squares = 0.51038395485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.427945 D: 0.427945 E: 1.000000 A B C D E A 0 12 -8 -2 -24 B -12 0 -22 -6 -18 C 8 22 0 20 0 D 2 6 -20 0 -16 E 24 18 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 C=22 D=15 A=13 so A is eliminated. Round 2 votes counts: E=33 B=25 C=23 D=19 so D is eliminated. Round 3 votes counts: E=35 B=35 C=30 so C is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:225 A:189 D:186 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -8 -2 -24 B -12 0 -22 -6 -18 C 8 22 0 20 0 D 2 6 -20 0 -16 E 24 18 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 -2 -24 B -12 0 -22 -6 -18 C 8 22 0 20 0 D 2 6 -20 0 -16 E 24 18 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 -2 -24 B -12 0 -22 -6 -18 C 8 22 0 20 0 D 2 6 -20 0 -16 E 24 18 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1449: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (10) D A B E C (7) A D E B C (7) C B E D A (6) C B D E A (5) C E B A D (4) A E D B C (4) E C B A D (3) E A C B D (3) D B C A E (3) D A C B E (3) C D A B E (3) C A E D B (3) A E D C B (3) A D E C B (3) A D C E B (3) E C A B D (2) D C A B E (2) D B A E C (2) B D A E C (2) B C D E A (2) A E C D B (2) E B A D C (1) E B A C D (1) E A B D C (1) E A B C D (1) D A B C E (1) C E B D A (1) C E A D B (1) C E A B D (1) C D B E A (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E C A (1) B D E A C (1) B D C A E (1) B C D A E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 -6 4 B -4 0 0 2 10 C 4 0 0 6 4 D 6 -2 -6 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.240992 C: 0.759008 D: 0.000000 E: 0.000000 Sum of squares = 0.634170690379 Cumulative probabilities = A: 0.000000 B: 0.240992 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -6 4 B -4 0 0 2 10 C 4 0 0 6 4 D 6 -2 -6 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499583 C: 0.500417 D: 0.000000 E: 0.000000 Sum of squares = 0.500000347071 Cumulative probabilities = A: 0.000000 B: 0.499583 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=24 B=21 D=18 E=12 so E is eliminated. Round 2 votes counts: C=30 A=29 B=23 D=18 so D is eliminated. Round 3 votes counts: A=40 C=32 B=28 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:204 A:199 D:198 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -6 4 B -4 0 0 2 10 C 4 0 0 6 4 D 6 -2 -6 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499583 C: 0.500417 D: 0.000000 E: 0.000000 Sum of squares = 0.500000347071 Cumulative probabilities = A: 0.000000 B: 0.499583 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 4 B -4 0 0 2 10 C 4 0 0 6 4 D 6 -2 -6 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499583 C: 0.500417 D: 0.000000 E: 0.000000 Sum of squares = 0.500000347071 Cumulative probabilities = A: 0.000000 B: 0.499583 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 4 B -4 0 0 2 10 C 4 0 0 6 4 D 6 -2 -6 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499583 C: 0.500417 D: 0.000000 E: 0.000000 Sum of squares = 0.500000347071 Cumulative probabilities = A: 0.000000 B: 0.499583 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1450: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (18) B C E D A (12) B A E C D (6) C E D B A (5) C D E B A (5) D C E A B (4) D A C E B (4) C E B D A (4) A D C E B (4) A B D E C (4) C B E D A (3) B A D C E (3) A D B E C (3) A B E D C (3) A B D C E (3) D E A C B (2) C B D E A (2) B E C A D (2) A D E B C (2) E B C D A (1) E A C D B (1) D E C A B (1) D C A E B (1) B E C D A (1) B C E A D (1) B C A E D (1) B A C E D (1) A E D C B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 6 14 10 12 B -6 0 -12 -6 -6 C -14 12 0 -10 10 D -10 6 10 0 16 E -12 6 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 10 12 B -6 0 -12 -6 -6 C -14 12 0 -10 10 D -10 6 10 0 16 E -12 6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=27 C=19 D=12 E=2 so E is eliminated. Round 2 votes counts: A=41 B=28 C=19 D=12 so D is eliminated. Round 3 votes counts: A=47 B=28 C=25 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:211 C:199 B:185 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 10 12 B -6 0 -12 -6 -6 C -14 12 0 -10 10 D -10 6 10 0 16 E -12 6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 10 12 B -6 0 -12 -6 -6 C -14 12 0 -10 10 D -10 6 10 0 16 E -12 6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 10 12 B -6 0 -12 -6 -6 C -14 12 0 -10 10 D -10 6 10 0 16 E -12 6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1451: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) C A E D B (8) A C E B D (8) B D C A E (7) B D A E C (6) E A D B C (4) D B E A C (4) D B C E A (4) B D E A C (4) E D C B A (3) E C A D B (3) A E C B D (3) A C E D B (3) A C B E D (3) E C D A B (2) D C B E A (2) C E D A B (2) C E A D B (2) C A B D E (2) B D E C A (2) B D A C E (2) B C A D E (2) E D B A C (1) E A D C B (1) E A C D B (1) D E C B A (1) D E B A C (1) C D E A B (1) C B D A E (1) C A E B D (1) C A B E D (1) B C D A E (1) B A D E C (1) B A D C E (1) A E B C D (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 -4 8 B 2 0 4 -2 8 C 6 -4 0 -4 6 D 4 2 4 0 2 E -8 -8 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -4 8 B 2 0 4 -2 8 C 6 -4 0 -4 6 D 4 2 4 0 2 E -8 -8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=21 D=20 C=18 E=15 so E is eliminated. Round 2 votes counts: A=27 B=26 D=24 C=23 so C is eliminated. Round 3 votes counts: A=44 D=29 B=27 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:206 D:206 C:202 A:198 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 8 B 2 0 4 -2 8 C 6 -4 0 -4 6 D 4 2 4 0 2 E -8 -8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 8 B 2 0 4 -2 8 C 6 -4 0 -4 6 D 4 2 4 0 2 E -8 -8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 8 B 2 0 4 -2 8 C 6 -4 0 -4 6 D 4 2 4 0 2 E -8 -8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1452: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (13) C B A D E (12) B C E A D (11) D A E C B (10) E B D A C (9) B E C D A (8) B C A D E (8) C A D B E (6) E D A C B (4) B C E D A (3) C A B D E (2) B E D C A (2) B C A E D (2) A D C E B (2) E D B A C (1) E A D C B (1) D E A C B (1) C B A E D (1) C A D E B (1) B C D A E (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -16 -14 -4 -6 B 16 0 16 18 14 C 14 -16 0 10 0 D 4 -18 -10 0 -10 E 6 -14 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -4 -6 B 16 0 16 18 14 C 14 -16 0 10 0 D 4 -18 -10 0 -10 E 6 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=28 C=22 D=11 A=4 so A is eliminated. Round 2 votes counts: B=35 E=28 C=22 D=15 so D is eliminated. Round 3 votes counts: E=40 B=35 C=25 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:204 E:201 D:183 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -14 -4 -6 B 16 0 16 18 14 C 14 -16 0 10 0 D 4 -18 -10 0 -10 E 6 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -4 -6 B 16 0 16 18 14 C 14 -16 0 10 0 D 4 -18 -10 0 -10 E 6 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -4 -6 B 16 0 16 18 14 C 14 -16 0 10 0 D 4 -18 -10 0 -10 E 6 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1453: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) C D A E B (9) E D C A B (8) B A C D E (7) E D A C B (6) B E A D C (6) C A D E B (4) B A E D C (4) E D B A C (3) E B D C A (3) C D A B E (3) C A D B E (3) E A D B C (2) D E A C B (2) D A C E B (2) C D E A B (2) B E D C A (2) B E D A C (2) B C E D A (2) A D C E B (2) A C D B E (2) E D A B C (1) E C D A B (1) E B D A C (1) D E C A B (1) D C E A B (1) D C A E B (1) C B D A E (1) C A B D E (1) B E A C D (1) B A E C D (1) B A C E D (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -4 -2 12 B -12 0 -4 -14 -4 C 4 4 0 0 8 D 2 14 0 0 12 E -12 4 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.619053 D: 0.380947 E: 0.000000 Sum of squares = 0.528347044252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.619053 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 -2 12 B -12 0 -4 -14 -4 C 4 4 0 0 8 D 2 14 0 0 12 E -12 4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=25 C=23 A=9 D=7 so D is eliminated. Round 2 votes counts: B=36 E=28 C=25 A=11 so A is eliminated. Round 3 votes counts: B=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:214 A:209 C:208 E:186 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 -2 12 B -12 0 -4 -14 -4 C 4 4 0 0 8 D 2 14 0 0 12 E -12 4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 -2 12 B -12 0 -4 -14 -4 C 4 4 0 0 8 D 2 14 0 0 12 E -12 4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 -2 12 B -12 0 -4 -14 -4 C 4 4 0 0 8 D 2 14 0 0 12 E -12 4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1454: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) A C D E B (9) B E D C A (8) C A D E B (7) A D E C B (6) B E D A C (5) B D E A C (5) E D B A C (4) C B E D A (4) E B D A C (3) C B E A D (3) C A B D E (3) B E C D A (3) A D E B C (3) A D C E B (3) D E B A C (2) D E A B C (2) D A E B C (2) C B A E D (2) C B A D E (2) C A B E D (2) B C D E A (2) E D A B C (1) E C D A B (1) D B E A C (1) C A E B D (1) C A D B E (1) B D E C A (1) B D C A E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -14 -4 -12 -12 B 14 0 8 12 10 C 4 -8 0 6 6 D 12 -12 -6 0 2 E 12 -10 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -12 -12 B 14 0 8 12 10 C 4 -8 0 6 6 D 12 -12 -6 0 2 E 12 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=25 A=23 E=9 D=7 so D is eliminated. Round 2 votes counts: B=37 C=25 A=25 E=13 so E is eliminated. Round 3 votes counts: B=46 A=28 C=26 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:204 D:198 E:197 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 -12 -12 B 14 0 8 12 10 C 4 -8 0 6 6 D 12 -12 -6 0 2 E 12 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -12 -12 B 14 0 8 12 10 C 4 -8 0 6 6 D 12 -12 -6 0 2 E 12 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -12 -12 B 14 0 8 12 10 C 4 -8 0 6 6 D 12 -12 -6 0 2 E 12 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1455: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (6) A B E C D (6) C A B D E (5) B A E C D (5) E D B A C (4) D C E A B (4) B A E D C (4) E D A B C (3) E A B D C (3) C D E B A (3) C A B E D (3) B E D C A (3) B C E A D (3) A E B D C (3) A B C E D (3) E B A D C (2) E A D B C (2) D E C A B (2) D E B A C (2) D E A C B (2) D E A B C (2) D C E B A (2) C D E A B (2) C D B E A (2) C D A E B (2) C D A B E (2) C B D E A (2) B E A C D (2) A E D B C (2) E B D A C (1) D E B C A (1) D C B E A (1) C D B A E (1) C B D A E (1) C B A E D (1) C B A D E (1) C A D E B (1) B C A E D (1) B A C E D (1) A E B C D (1) A D E B C (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 14 14 16 4 B -14 0 22 16 8 C -14 -22 0 -2 -14 D -16 -16 2 0 -18 E -4 -8 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 16 4 B -14 0 22 16 8 C -14 -22 0 -2 -14 D -16 -16 2 0 -18 E -4 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 B=19 D=16 E=15 so E is eliminated. Round 2 votes counts: A=29 C=26 D=23 B=22 so B is eliminated. Round 3 votes counts: A=43 C=30 D=27 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:216 E:210 D:176 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 16 4 B -14 0 22 16 8 C -14 -22 0 -2 -14 D -16 -16 2 0 -18 E -4 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 16 4 B -14 0 22 16 8 C -14 -22 0 -2 -14 D -16 -16 2 0 -18 E -4 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 16 4 B -14 0 22 16 8 C -14 -22 0 -2 -14 D -16 -16 2 0 -18 E -4 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1456: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (19) A D E B C (11) C A B E D (8) B E D C A (6) D E B A C (5) A C D E B (5) D E A B C (4) C B E A D (4) A C E D B (4) A C B D E (4) E D B C A (3) C B A E D (3) C A E B D (3) E B D C A (2) D E B C A (2) C A B D E (2) B E C D A (2) B C E D A (2) A D E C B (2) E D B A C (1) E D A B C (1) E C B D A (1) E B C D A (1) C E B D A (1) C B A D E (1) B D A E C (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 -20 -2 -8 B 8 0 -16 20 4 C 20 16 0 20 12 D 2 -20 -20 0 -26 E 8 -4 -12 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -20 -2 -8 B 8 0 -16 20 4 C 20 16 0 20 12 D 2 -20 -20 0 -26 E 8 -4 -12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=28 D=11 B=11 E=9 so E is eliminated. Round 2 votes counts: C=42 A=28 D=16 B=14 so B is eliminated. Round 3 votes counts: C=47 A=28 D=25 so D is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:234 E:209 B:208 A:181 D:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -20 -2 -8 B 8 0 -16 20 4 C 20 16 0 20 12 D 2 -20 -20 0 -26 E 8 -4 -12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 -2 -8 B 8 0 -16 20 4 C 20 16 0 20 12 D 2 -20 -20 0 -26 E 8 -4 -12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 -2 -8 B 8 0 -16 20 4 C 20 16 0 20 12 D 2 -20 -20 0 -26 E 8 -4 -12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1457: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) E A C B D (5) D B A C E (5) B D E A C (5) A E D C B (5) E C A B D (4) D B A E C (4) D A E B C (4) C E A B D (4) A E C D B (4) E A D B C (3) E A C D B (3) D A B E C (3) C E B A D (3) C A E D B (3) E B A D C (2) E A B D C (2) D A C E B (2) B E C D A (2) B C E D A (2) B C E A D (2) B C D E A (2) A D E C B (2) A D C E B (2) E C B A D (1) E C A D B (1) E B A C D (1) E A D C B (1) D E B A C (1) D C B A E (1) D B C A E (1) D A B C E (1) C B E D A (1) C B D E A (1) C B D A E (1) B E C A D (1) B D C E A (1) B D C A E (1) A E D B C (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 16 24 -16 B -4 0 -10 2 -12 C -16 10 0 4 -12 D -24 -2 -4 0 -24 E 16 12 12 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 16 24 -16 B -4 0 -10 2 -12 C -16 10 0 4 -12 D -24 -2 -4 0 -24 E 16 12 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=23 C=23 D=22 B=16 A=16 so B is eliminated. Round 2 votes counts: D=29 C=29 E=26 A=16 so A is eliminated. Round 3 votes counts: E=36 D=34 C=30 so C is eliminated. Round 4 votes counts: E=62 D=38 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:214 C:193 B:188 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 16 24 -16 B -4 0 -10 2 -12 C -16 10 0 4 -12 D -24 -2 -4 0 -24 E 16 12 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 24 -16 B -4 0 -10 2 -12 C -16 10 0 4 -12 D -24 -2 -4 0 -24 E 16 12 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 24 -16 B -4 0 -10 2 -12 C -16 10 0 4 -12 D -24 -2 -4 0 -24 E 16 12 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1458: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (6) E A D C B (5) A D B E C (5) E C A D B (4) D A E B C (4) C E D B A (4) C B E D A (4) B C A D E (4) B A D C E (4) E D A C B (3) E C D A B (3) D E A C B (3) B D C A E (3) B C D E A (3) B C D A E (3) A D E B C (3) E D C A B (2) D E C B A (2) D E A B C (2) D B C E A (2) C E B D A (2) C E B A D (2) C D B E A (2) C B D E A (2) B D A C E (2) B C A E D (2) A E C B D (2) A B D E C (2) E D C B A (1) E C A B D (1) D E C A B (1) D C E B A (1) D B C A E (1) D B A E C (1) D A B E C (1) C E A B D (1) C D E B A (1) C B A E D (1) C A B E D (1) B A C E D (1) B A C D E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 -6 -2 B -2 0 2 -16 -8 C 6 -2 0 -10 -4 D 6 16 10 0 6 E 2 8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -6 -2 B -2 0 2 -16 -8 C 6 -2 0 -10 -4 D 6 16 10 0 6 E 2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 C=20 A=20 E=19 D=18 so D is eliminated. Round 2 votes counts: E=27 B=27 A=25 C=21 so C is eliminated. Round 3 votes counts: E=38 B=36 A=26 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:219 E:204 C:195 A:194 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -6 -2 B -2 0 2 -16 -8 C 6 -2 0 -10 -4 D 6 16 10 0 6 E 2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -6 -2 B -2 0 2 -16 -8 C 6 -2 0 -10 -4 D 6 16 10 0 6 E 2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -6 -2 B -2 0 2 -16 -8 C 6 -2 0 -10 -4 D 6 16 10 0 6 E 2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1459: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) A D E B C (7) E A D C B (6) B C D A E (6) D A B C E (5) A E D B C (5) E C B A D (4) D B A C E (4) C B E A D (4) C B D E A (4) B D A C E (4) B A D C E (4) C E D B A (3) C B E D A (3) A D B E C (3) E C D A B (2) E C A D B (2) B C E A D (2) B C A D E (2) A E B D C (2) A B D C E (2) E C D B A (1) E C A B D (1) E A C D B (1) D C B E A (1) D C B A E (1) D C A B E (1) D A E B C (1) D A B E C (1) C E D A B (1) C E B A D (1) C D E B A (1) B E A C D (1) B D C A E (1) B C A E D (1) B A C E D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -2 2 6 B 18 0 8 8 6 C 2 -8 0 2 24 D -2 -8 -2 0 0 E -6 -6 -24 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -2 2 6 B 18 0 8 8 6 C 2 -8 0 2 24 D -2 -8 -2 0 0 E -6 -6 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=23 A=20 E=17 D=14 so D is eliminated. Round 2 votes counts: C=29 B=27 A=27 E=17 so E is eliminated. Round 3 votes counts: C=39 A=34 B=27 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:220 C:210 A:194 D:194 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -2 2 6 B 18 0 8 8 6 C 2 -8 0 2 24 D -2 -8 -2 0 0 E -6 -6 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 2 6 B 18 0 8 8 6 C 2 -8 0 2 24 D -2 -8 -2 0 0 E -6 -6 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 2 6 B 18 0 8 8 6 C 2 -8 0 2 24 D -2 -8 -2 0 0 E -6 -6 -24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1460: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) E C D A B (5) D A B E C (5) B D C A E (5) D B A C E (4) B C A E D (4) A B C E D (4) E C A B D (3) D E C B A (3) C E B D A (3) C E B A D (3) B D A C E (3) B C D E A (3) B C D A E (3) B A D C E (3) D B A E C (2) C E A B D (2) B C A D E (2) B A C E D (2) A E D C B (2) A E C D B (2) A E B C D (2) A B D C E (2) E D C A B (1) E D A C B (1) E C D B A (1) E C A D B (1) E A C D B (1) D E A C B (1) D C E B A (1) D B E C A (1) D B C E A (1) D B C A E (1) D A E C B (1) D A E B C (1) C E D B A (1) C B E A D (1) C B A E D (1) B D C E A (1) B A C D E (1) A D E C B (1) A D E B C (1) A D B E C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 6 4 32 B 0 0 6 22 2 C -6 -6 0 14 4 D -4 -22 -14 0 -4 E -32 -2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.420621 B: 0.579379 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.512602161928 Cumulative probabilities = A: 0.420621 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 32 B 0 0 6 22 2 C -6 -6 0 14 4 D -4 -22 -14 0 -4 E -32 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 D=21 E=13 C=11 so C is eliminated. Round 2 votes counts: B=29 A=28 E=22 D=21 so D is eliminated. Round 3 votes counts: B=38 A=35 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:215 C:203 E:183 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 32 B 0 0 6 22 2 C -6 -6 0 14 4 D -4 -22 -14 0 -4 E -32 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 32 B 0 0 6 22 2 C -6 -6 0 14 4 D -4 -22 -14 0 -4 E -32 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 32 B 0 0 6 22 2 C -6 -6 0 14 4 D -4 -22 -14 0 -4 E -32 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1461: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A E B C D (8) A D E B C (8) D A C E B (6) B E C A D (6) C B E D A (5) D C B A E (4) D A C B E (4) D A B C E (4) E C A B D (3) D B C E A (3) A D E C B (3) A D B E C (3) E C B A D (2) E B A C D (2) D A B E C (2) C E B D A (2) C E B A D (2) C D B E A (2) B E A C D (2) B D C E A (2) B C E D A (2) A E C B D (2) A B E C D (2) E A C B D (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) C D E A B (1) C B E A D (1) B C E A D (1) B C D E A (1) B A E D C (1) A E D C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 4 0 8 B -4 0 2 -6 16 C -4 -2 0 -10 4 D 0 6 10 0 12 E -8 -16 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.760930 B: 0.000000 C: 0.000000 D: 0.239070 E: 0.000000 Sum of squares = 0.636169288054 Cumulative probabilities = A: 0.760930 B: 0.760930 C: 0.760930 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 8 B -4 0 2 -6 16 C -4 -2 0 -10 4 D 0 6 10 0 12 E -8 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=29 B=15 C=13 E=8 so E is eliminated. Round 2 votes counts: D=35 A=30 C=18 B=17 so B is eliminated. Round 3 votes counts: D=37 A=35 C=28 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:208 B:204 C:194 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 8 B -4 0 2 -6 16 C -4 -2 0 -10 4 D 0 6 10 0 12 E -8 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 8 B -4 0 2 -6 16 C -4 -2 0 -10 4 D 0 6 10 0 12 E -8 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 8 B -4 0 2 -6 16 C -4 -2 0 -10 4 D 0 6 10 0 12 E -8 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1462: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) D E B C A (7) B D E C A (7) A C B E D (7) A C E D B (6) E D C A B (5) B A D C E (5) D E C B A (4) B D E A C (4) E C D A B (3) D E C A B (3) C B A E D (3) C A E D B (3) E D C B A (2) E D B C A (2) E A D C B (2) D E B A C (2) C E A D B (2) B D A E C (2) B C E D A (2) B C A E D (2) A C E B D (2) E D A C B (1) D E A B C (1) D B E A C (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A B D (1) C A E B D (1) C A B E D (1) B D C E A (1) B C D E A (1) B A D E C (1) A E C D B (1) A D E C B (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -6 -4 -8 B 14 0 -2 0 -6 C 6 2 0 -6 -2 D 4 0 6 0 0 E 8 6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.338022 E: 0.661978 Sum of squares = 0.55247358602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.338022 E: 1.000000 A B C D E A 0 -14 -6 -4 -8 B 14 0 -2 0 -6 C 6 2 0 -6 -2 D 4 0 6 0 0 E 8 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=19 A=19 E=15 C=14 so C is eliminated. Round 2 votes counts: B=36 A=24 E=21 D=19 so D is eliminated. Round 3 votes counts: E=38 B=38 A=24 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:205 B:203 C:200 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 -8 B 14 0 -2 0 -6 C 6 2 0 -6 -2 D 4 0 6 0 0 E 8 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 -8 B 14 0 -2 0 -6 C 6 2 0 -6 -2 D 4 0 6 0 0 E 8 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 -8 B 14 0 -2 0 -6 C 6 2 0 -6 -2 D 4 0 6 0 0 E 8 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1463: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) B D E C A (9) A E D B C (9) E D A B C (8) D E B A C (8) B D E A C (8) B D C E A (5) C B D E A (4) A C E D B (4) C B A D E (3) C A B E D (3) B C D A E (3) A E D C B (3) C E D B A (2) C E D A B (2) B C D E A (2) B C A D E (2) A E C D B (2) E D A C B (1) D E B C A (1) D B E A C (1) D B C E A (1) C D E B A (1) C B D A E (1) C B A E D (1) B D A E C (1) B A D E C (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -16 -6 B 8 0 22 -8 -4 C 2 -22 0 -14 -6 D 16 8 14 0 6 E 6 4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -16 -6 B 8 0 22 -8 -4 C 2 -22 0 -14 -6 D 16 8 14 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=28 A=21 D=11 E=9 so E is eliminated. Round 2 votes counts: B=31 C=28 A=21 D=20 so D is eliminated. Round 3 votes counts: B=42 A=30 C=28 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:209 E:205 A:184 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -16 -6 B 8 0 22 -8 -4 C 2 -22 0 -14 -6 D 16 8 14 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -16 -6 B 8 0 22 -8 -4 C 2 -22 0 -14 -6 D 16 8 14 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -16 -6 B 8 0 22 -8 -4 C 2 -22 0 -14 -6 D 16 8 14 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1464: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C B A D E (8) D A E C B (7) B C E A D (6) B C A D E (6) E D B A C (5) D E A B C (5) A D C B E (5) C B E A D (4) A D C E B (4) E D C A B (3) E C D B A (3) E B D A C (3) B C A E D (3) E D A C B (2) E B D C A (2) D E A C B (2) D A C E B (2) C B E D A (2) C A D B E (2) C A B D E (2) B E A D C (2) B A D E C (2) A D E C B (2) A C D B E (2) A C B D E (2) E B C D A (1) D A E B C (1) C E D B A (1) C B A E D (1) A E D B C (1) Total count = 100 A B C D E A 0 2 12 4 0 B -2 0 -8 -12 -6 C -12 8 0 -14 0 D -4 12 14 0 4 E 0 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.655027 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.344973 Sum of squares = 0.548066872408 Cumulative probabilities = A: 0.655027 B: 0.655027 C: 0.655027 D: 0.655027 E: 1.000000 A B C D E A 0 2 12 4 0 B -2 0 -8 -12 -6 C -12 8 0 -14 0 D -4 12 14 0 4 E 0 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500321 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499679 Sum of squares = 0.500000205478 Cumulative probabilities = A: 0.500321 B: 0.500321 C: 0.500321 D: 0.500321 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=20 B=19 D=17 A=16 so A is eliminated. Round 2 votes counts: E=29 D=28 C=24 B=19 so B is eliminated. Round 3 votes counts: C=39 E=31 D=30 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:213 A:209 E:201 C:191 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 4 0 B -2 0 -8 -12 -6 C -12 8 0 -14 0 D -4 12 14 0 4 E 0 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500321 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499679 Sum of squares = 0.500000205478 Cumulative probabilities = A: 0.500321 B: 0.500321 C: 0.500321 D: 0.500321 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 4 0 B -2 0 -8 -12 -6 C -12 8 0 -14 0 D -4 12 14 0 4 E 0 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500321 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499679 Sum of squares = 0.500000205478 Cumulative probabilities = A: 0.500321 B: 0.500321 C: 0.500321 D: 0.500321 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 4 0 B -2 0 -8 -12 -6 C -12 8 0 -14 0 D -4 12 14 0 4 E 0 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500321 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499679 Sum of squares = 0.500000205478 Cumulative probabilities = A: 0.500321 B: 0.500321 C: 0.500321 D: 0.500321 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1465: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (20) C D E A B (18) B C D E A (8) A E D C B (8) A B E D C (7) A E B D C (6) C D E B A (5) B A E C D (5) E D C A B (3) D C E A B (3) C D A E B (3) B C D A E (3) B A C D E (3) E A D C B (2) A C D E B (2) E D A C B (1) C D B E A (1) B E D C A (1) B D C E A (1) Total count = 100 A B C D E A 0 6 8 6 14 B -6 0 8 8 -2 C -8 -8 0 -4 -6 D -6 -8 4 0 -6 E -14 2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 6 14 B -6 0 8 8 -2 C -8 -8 0 -4 -6 D -6 -8 4 0 -6 E -14 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=27 A=23 E=6 D=3 so D is eliminated. Round 2 votes counts: B=41 C=30 A=23 E=6 so E is eliminated. Round 3 votes counts: B=41 C=33 A=26 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:204 E:200 D:192 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 6 14 B -6 0 8 8 -2 C -8 -8 0 -4 -6 D -6 -8 4 0 -6 E -14 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 6 14 B -6 0 8 8 -2 C -8 -8 0 -4 -6 D -6 -8 4 0 -6 E -14 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 6 14 B -6 0 8 8 -2 C -8 -8 0 -4 -6 D -6 -8 4 0 -6 E -14 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1466: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) B E A C D (8) D C A B E (7) E A B C D (6) D B C E A (5) D A C E B (5) C D A E B (5) A E B D C (5) A E B C D (5) B E C A D (4) A E C B D (4) D C B A E (3) D C A E B (3) C D B E A (3) B D C E A (3) D C B E A (2) C E B A D (2) C A D E B (2) B E D C A (2) B E C D A (2) B C E D A (2) A C E B D (2) E B A C D (1) E A B D C (1) D A E B C (1) C B E A D (1) B E D A C (1) B D E C A (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 6 10 -8 B 0 0 16 20 8 C -6 -16 0 -4 -6 D -10 -20 4 0 -12 E 8 -8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.294215 B: 0.705785 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.584694811696 Cumulative probabilities = A: 0.294215 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 10 -8 B 0 0 16 20 8 C -6 -16 0 -4 -6 D -10 -20 4 0 -12 E 8 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499527 B: 0.500473 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000447955 Cumulative probabilities = A: 0.499527 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=26 A=20 C=13 E=8 so E is eliminated. Round 2 votes counts: B=34 A=27 D=26 C=13 so C is eliminated. Round 3 votes counts: B=37 D=34 A=29 so A is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:209 A:204 C:184 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 10 -8 B 0 0 16 20 8 C -6 -16 0 -4 -6 D -10 -20 4 0 -12 E 8 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499527 B: 0.500473 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000447955 Cumulative probabilities = A: 0.499527 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 10 -8 B 0 0 16 20 8 C -6 -16 0 -4 -6 D -10 -20 4 0 -12 E 8 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499527 B: 0.500473 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000447955 Cumulative probabilities = A: 0.499527 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 10 -8 B 0 0 16 20 8 C -6 -16 0 -4 -6 D -10 -20 4 0 -12 E 8 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499527 B: 0.500473 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000447955 Cumulative probabilities = A: 0.499527 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1467: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) A B E D C (8) A B E C D (8) E B D C A (7) E D C B A (6) C D E B A (6) C D A B E (6) B E A D C (6) A C D B E (6) A B C D E (5) B A E C D (4) A D C B E (3) E B A D C (2) D C A E B (2) C D A E B (2) A E B D C (2) E B D A C (1) E B C D A (1) E A B D C (1) D E C B A (1) D C E A B (1) C D E A B (1) C D B E A (1) C D B A E (1) B E C D A (1) B E A C D (1) B C E D A (1) B C A D E (1) B A E D C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 0 2 B 8 0 4 4 10 C 0 -4 0 -6 0 D 0 -4 6 0 -4 E -2 -10 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 0 2 B 8 0 4 4 10 C 0 -4 0 -6 0 D 0 -4 6 0 -4 E -2 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=18 C=17 D=16 B=16 so D is eliminated. Round 2 votes counts: A=33 C=32 E=19 B=16 so B is eliminated. Round 3 votes counts: A=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:213 D:199 A:197 E:196 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 0 2 B 8 0 4 4 10 C 0 -4 0 -6 0 D 0 -4 6 0 -4 E -2 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 0 2 B 8 0 4 4 10 C 0 -4 0 -6 0 D 0 -4 6 0 -4 E -2 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 0 2 B 8 0 4 4 10 C 0 -4 0 -6 0 D 0 -4 6 0 -4 E -2 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1468: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (21) D A C E B (8) D A C B E (7) A C D E B (7) A D C E B (5) E B C A D (4) D A B E C (4) C A E B D (4) B E D C A (4) D A E B C (3) C E B A D (3) C A D E B (3) B E D A C (3) B E C D A (3) B C E A D (3) D B E A C (2) D B A E C (2) C B E A D (2) E B D A C (1) D C A E B (1) D C A B E (1) D B A C E (1) C E A B D (1) C A E D B (1) C A B D E (1) B D E A C (1) A E D B C (1) A E C D B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -4 18 4 B 0 0 6 2 10 C 4 -6 0 10 -2 D -18 -2 -10 0 -6 E -4 -10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.277150 B: 0.722850 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.599324111085 Cumulative probabilities = A: 0.277150 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 18 4 B 0 0 6 2 10 C 4 -6 0 10 -2 D -18 -2 -10 0 -6 E -4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=29 A=16 C=15 E=5 so E is eliminated. Round 2 votes counts: B=40 D=29 A=16 C=15 so C is eliminated. Round 3 votes counts: B=45 D=29 A=26 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:209 C:203 E:197 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 18 4 B 0 0 6 2 10 C 4 -6 0 10 -2 D -18 -2 -10 0 -6 E -4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 18 4 B 0 0 6 2 10 C 4 -6 0 10 -2 D -18 -2 -10 0 -6 E -4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 18 4 B 0 0 6 2 10 C 4 -6 0 10 -2 D -18 -2 -10 0 -6 E -4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1469: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) B E A D C (8) B E D A C (6) A B E C D (6) E B D A C (5) D C E B A (5) B A E C D (5) A C B E D (5) D E B C A (4) D E B A C (4) C A D E B (4) C A D B E (4) A C E B D (4) A B C E D (4) E D B A C (3) C A B D E (3) D E C A B (2) D C E A B (2) C D E A B (2) C D A B E (2) A E B C D (2) E D A C B (1) E A B D C (1) D E C B A (1) C A E D B (1) C A E B D (1) C A B E D (1) B D E A C (1) B A E D C (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 16 4 10 B -12 0 4 10 -4 C -16 -4 0 10 -2 D -4 -10 -10 0 -12 E -10 4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 4 10 B -12 0 4 10 -4 C -16 -4 0 10 -2 D -4 -10 -10 0 -12 E -10 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994015 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=22 A=22 D=18 E=10 so E is eliminated. Round 2 votes counts: C=28 B=27 A=23 D=22 so D is eliminated. Round 3 votes counts: C=38 B=38 A=24 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:221 E:204 B:199 C:194 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 4 10 B -12 0 4 10 -4 C -16 -4 0 10 -2 D -4 -10 -10 0 -12 E -10 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994015 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 4 10 B -12 0 4 10 -4 C -16 -4 0 10 -2 D -4 -10 -10 0 -12 E -10 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994015 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 4 10 B -12 0 4 10 -4 C -16 -4 0 10 -2 D -4 -10 -10 0 -12 E -10 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994015 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1470: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (16) C A D B E (14) E B D A C (10) E C B D A (9) C E B D A (6) C B D A E (5) B D A E C (5) E C A D B (4) E B C D A (4) A D B C E (4) E A D B C (3) C E B A D (3) C E A D B (3) C B E D A (3) E C A B D (2) B E D A C (2) E D B A C (1) E A D C B (1) D A B E C (1) C A E D B (1) A E D C B (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -8 8 -2 B -4 0 -6 -2 2 C 8 6 0 10 -20 D -8 2 -10 0 -6 E 2 -2 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 -8 8 -2 B -4 0 -6 -2 2 C 8 6 0 10 -20 D -8 2 -10 0 -6 E 2 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=34 A=23 B=7 D=1 so D is eliminated. Round 2 votes counts: C=35 E=34 A=24 B=7 so B is eliminated. Round 3 votes counts: E=36 C=35 A=29 so A is eliminated. Round 4 votes counts: E=60 C=40 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 C:202 A:201 B:195 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 8 -2 B -4 0 -6 -2 2 C 8 6 0 10 -20 D -8 2 -10 0 -6 E 2 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 8 -2 B -4 0 -6 -2 2 C 8 6 0 10 -20 D -8 2 -10 0 -6 E 2 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 8 -2 B -4 0 -6 -2 2 C 8 6 0 10 -20 D -8 2 -10 0 -6 E 2 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999946 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1471: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (7) E C D B A (5) C D E B A (5) A B E D C (5) E C D A B (4) E A C B D (4) E A B C D (4) C D B E A (4) A E D B C (4) E C A D B (3) E A C D B (3) D C B A E (3) C D B A E (3) A B D E C (3) E D C A B (2) E C B A D (2) E A D C B (2) D C E B A (2) D B C A E (2) D B A C E (2) C E D B A (2) C E B D A (2) B D C A E (2) B C A D E (2) A E B D C (2) A B E C D (2) E D A C B (1) E C B D A (1) E B C A D (1) D C E A B (1) D C B E A (1) D A B C E (1) C B E D A (1) C B D E A (1) C B D A E (1) B D A C E (1) B C D A E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B C D (1) A D E B C (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -2 4 -2 B -6 0 -6 -6 -4 C 2 6 0 10 -2 D -4 6 -10 0 -4 E 2 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -2 4 -2 B -6 0 -6 -6 -4 C 2 6 0 10 -2 D -4 6 -10 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=28 C=19 D=12 B=9 so B is eliminated. Round 2 votes counts: E=32 A=31 C=22 D=15 so D is eliminated. Round 3 votes counts: A=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:208 E:206 A:203 D:194 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 4 -2 B -6 0 -6 -6 -4 C 2 6 0 10 -2 D -4 6 -10 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 4 -2 B -6 0 -6 -6 -4 C 2 6 0 10 -2 D -4 6 -10 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 4 -2 B -6 0 -6 -6 -4 C 2 6 0 10 -2 D -4 6 -10 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1472: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (10) A C B D E (10) E D A B C (8) B C D E A (7) C B A D E (6) A D B C E (5) C B E A D (4) E D A C B (3) D E A B C (3) D B C E A (3) C B E D A (3) A E D C B (3) A D E B C (3) E C B D A (2) E A C B D (2) D A E B C (2) E D B C A (1) E C B A D (1) E A D B C (1) D B C A E (1) D B A C E (1) C E B A D (1) C B D E A (1) C B D A E (1) C B A E D (1) C A B D E (1) B C E D A (1) B C A D E (1) A E D B C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 10 10 6 -4 B -10 0 -4 8 14 C -10 4 0 8 18 D -6 -8 -8 0 10 E 4 -14 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.562500 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.312500 Sum of squares = 0.429687499777 Cumulative probabilities = A: 0.562500 B: 0.562500 C: 0.687500 D: 0.687500 E: 1.000000 A B C D E A 0 10 10 6 -4 B -10 0 -4 8 14 C -10 4 0 8 18 D -6 -8 -8 0 10 E 4 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.562500 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.312500 Sum of squares = 0.429687499995 Cumulative probabilities = A: 0.562500 B: 0.562500 C: 0.687500 D: 0.687500 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=24 B=19 C=18 D=10 so D is eliminated. Round 2 votes counts: E=32 A=26 B=24 C=18 so C is eliminated. Round 3 votes counts: B=40 E=33 A=27 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:211 C:210 B:204 D:194 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 6 -4 B -10 0 -4 8 14 C -10 4 0 8 18 D -6 -8 -8 0 10 E 4 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.562500 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.312500 Sum of squares = 0.429687499995 Cumulative probabilities = A: 0.562500 B: 0.562500 C: 0.687500 D: 0.687500 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 6 -4 B -10 0 -4 8 14 C -10 4 0 8 18 D -6 -8 -8 0 10 E 4 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.562500 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.312500 Sum of squares = 0.429687499995 Cumulative probabilities = A: 0.562500 B: 0.562500 C: 0.687500 D: 0.687500 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 6 -4 B -10 0 -4 8 14 C -10 4 0 8 18 D -6 -8 -8 0 10 E 4 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.562500 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.312500 Sum of squares = 0.429687499995 Cumulative probabilities = A: 0.562500 B: 0.562500 C: 0.687500 D: 0.687500 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1473: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) D C B E A (9) C E A B D (8) C D B E A (7) E A C B D (6) A E B D C (5) E A B C D (4) D C B A E (4) C E D A B (4) C D E B A (4) A E B C D (4) C B D E A (3) E A C D B (2) D C E A B (2) D B A C E (2) C B E D A (2) B E A C D (2) B A D E C (2) A E C D B (2) A D B E C (2) E C A B D (1) D C A E B (1) D B A E C (1) D A E C B (1) D A E B C (1) C E B A D (1) C E A D B (1) C B E A D (1) B D C A E (1) B D A E C (1) B D A C E (1) B C E A D (1) B A E D C (1) A E D C B (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -20 -8 -16 B 6 0 -22 -8 2 C 20 22 0 8 24 D 8 8 -8 0 6 E 16 -2 -24 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -8 -16 B 6 0 -22 -8 2 C 20 22 0 8 24 D 8 8 -8 0 6 E 16 -2 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=31 C=31 A=16 E=13 B=9 so B is eliminated. Round 2 votes counts: D=34 C=32 A=19 E=15 so E is eliminated. Round 3 votes counts: D=34 C=33 A=33 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:237 D:207 E:192 B:189 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -20 -8 -16 B 6 0 -22 -8 2 C 20 22 0 8 24 D 8 8 -8 0 6 E 16 -2 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -8 -16 B 6 0 -22 -8 2 C 20 22 0 8 24 D 8 8 -8 0 6 E 16 -2 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -8 -16 B 6 0 -22 -8 2 C 20 22 0 8 24 D 8 8 -8 0 6 E 16 -2 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1474: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) B E C A D (8) D A B E C (7) D C A E B (6) C E B A D (6) D A C E B (5) C E B D A (5) C A E B D (5) B E C D A (4) B E A C D (4) A C E B D (3) A C D E B (3) E C B D A (2) E B C A D (2) D B E C A (2) D B E A C (2) C D A E B (2) B E D A C (2) A D C E B (2) A D B E C (2) A B E D C (2) A B E C D (2) E C B A D (1) E B C D A (1) D C E B A (1) D B A E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C D E A B (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E C A (1) B A E C D (1) A E B C D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -2 -2 10 B -12 0 -8 6 0 C 2 8 0 10 6 D 2 -6 -10 0 -6 E -10 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 -2 10 B -12 0 -8 6 0 C 2 8 0 10 6 D 2 -6 -10 0 -6 E -10 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 B=21 A=17 E=6 so E is eliminated. Round 2 votes counts: D=33 C=26 B=24 A=17 so A is eliminated. Round 3 votes counts: D=38 C=32 B=30 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:209 E:195 B:193 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 -2 10 B -12 0 -8 6 0 C 2 8 0 10 6 D 2 -6 -10 0 -6 E -10 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -2 10 B -12 0 -8 6 0 C 2 8 0 10 6 D 2 -6 -10 0 -6 E -10 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -2 10 B -12 0 -8 6 0 C 2 8 0 10 6 D 2 -6 -10 0 -6 E -10 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1475: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) C B A D E (8) E A D B C (7) A E D C B (5) E D B A C (4) D E A C B (4) B C E A D (4) E D A C B (3) E B D C A (3) B C A E D (3) B C A D E (3) A D E C B (3) A C B D E (3) A B C E D (3) E D A B C (2) E B D A C (2) E A D C B (2) D C B E A (2) D C A E B (2) C D B E A (2) C B D A E (2) C A D B E (2) B E C D A (2) B E A C D (2) B C E D A (2) A C B E D (2) E B A D C (1) E A B D C (1) D E C A B (1) D E B C A (1) D C B A E (1) D B E C A (1) B E D C A (1) B E C A D (1) A E C D B (1) A E B C D (1) A D C E B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -2 10 -16 B 10 0 8 8 10 C 2 -8 0 6 4 D -10 -8 -6 0 -6 E 16 -10 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 10 -16 B 10 0 8 8 10 C 2 -8 0 6 4 D -10 -8 -6 0 -6 E 16 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 A=21 C=14 D=12 so D is eliminated. Round 2 votes counts: E=31 B=29 A=21 C=19 so C is eliminated. Round 3 votes counts: B=44 E=31 A=25 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:204 C:202 A:191 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 10 -16 B 10 0 8 8 10 C 2 -8 0 6 4 D -10 -8 -6 0 -6 E 16 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 10 -16 B 10 0 8 8 10 C 2 -8 0 6 4 D -10 -8 -6 0 -6 E 16 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 10 -16 B 10 0 8 8 10 C 2 -8 0 6 4 D -10 -8 -6 0 -6 E 16 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1476: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (17) B C A E D (16) D A E B C (7) E A D C B (6) D C B E A (6) C B E A D (6) C B D E A (6) A E B C D (6) A E D B C (5) B D C A E (3) B C D A E (3) D E A B C (2) D C E A B (2) D B C E A (2) C B A E D (2) E A C B D (1) D E C A B (1) D C E B A (1) D B C A E (1) D A B E C (1) B D A E C (1) B A E D C (1) B A E C D (1) A E D C B (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 2 -6 0 B -2 0 2 -6 0 C -2 -2 0 -16 -4 D 6 6 16 0 8 E 0 0 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -6 0 B -2 0 2 -6 0 C -2 -2 0 -16 -4 D 6 6 16 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=25 C=14 A=14 E=7 so E is eliminated. Round 2 votes counts: D=40 B=25 A=21 C=14 so C is eliminated. Round 3 votes counts: D=40 B=39 A=21 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:199 E:198 B:197 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 2 -6 0 B -2 0 2 -6 0 C -2 -2 0 -16 -4 D 6 6 16 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -6 0 B -2 0 2 -6 0 C -2 -2 0 -16 -4 D 6 6 16 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -6 0 B -2 0 2 -6 0 C -2 -2 0 -16 -4 D 6 6 16 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1477: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) C A B E D (7) C B E A D (6) A C D B E (6) A D C B E (5) E B D C A (4) E B C D A (4) C B A E D (4) C A D B E (4) C A B D E (4) E D B C A (3) D B E A C (3) B E C A D (3) A C B D E (3) E D B A C (2) D A C B E (2) D A B E C (2) C E B D A (2) C E B A D (2) C A D E B (2) B D A E C (2) B A C E D (2) A B C D E (2) E C B D A (1) E B D A C (1) D E A C B (1) D E A B C (1) D C A E B (1) D B A E C (1) D A E C B (1) D A B C E (1) C A E D B (1) B E D A C (1) B E A C D (1) A D B E C (1) A D B C E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 4 12 8 B 6 0 -6 2 26 C -4 6 0 12 14 D -12 -2 -12 0 8 E -8 -26 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 12 8 B 6 0 -6 2 26 C -4 6 0 12 14 D -12 -2 -12 0 8 E -8 -26 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 A=20 E=15 B=9 so B is eliminated. Round 2 votes counts: C=32 D=26 A=22 E=20 so E is eliminated. Round 3 votes counts: C=40 D=37 A=23 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:214 A:209 D:191 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 4 12 8 B 6 0 -6 2 26 C -4 6 0 12 14 D -12 -2 -12 0 8 E -8 -26 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 12 8 B 6 0 -6 2 26 C -4 6 0 12 14 D -12 -2 -12 0 8 E -8 -26 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 12 8 B 6 0 -6 2 26 C -4 6 0 12 14 D -12 -2 -12 0 8 E -8 -26 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1478: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) A B D C E (9) E C D B A (8) E C D A B (8) A B D E C (8) B D A C E (7) E C A B D (5) D B A C E (5) B A D E C (5) D C B A E (4) C D E B A (4) B A D C E (4) A E B D C (4) E C A D B (3) D B C A E (3) E A B C D (2) C E D A B (2) A D B C E (2) A B E D C (2) E C B D A (1) E B C D A (1) E A C B D (1) D C A B E (1) C E A D B (1) C A D E B (1) Total count = 100 A B C D E A 0 -2 -2 -6 10 B 2 0 4 -2 0 C 2 -4 0 -8 4 D 6 2 8 0 6 E -10 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 10 B 2 0 4 -2 0 C 2 -4 0 -8 4 D 6 2 8 0 6 E -10 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=25 C=17 B=16 D=13 so D is eliminated. Round 2 votes counts: E=29 A=25 B=24 C=22 so C is eliminated. Round 3 votes counts: E=45 B=28 A=27 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:202 A:200 C:197 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 10 B 2 0 4 -2 0 C 2 -4 0 -8 4 D 6 2 8 0 6 E -10 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 10 B 2 0 4 -2 0 C 2 -4 0 -8 4 D 6 2 8 0 6 E -10 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 10 B 2 0 4 -2 0 C 2 -4 0 -8 4 D 6 2 8 0 6 E -10 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1479: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (5) B C A D E (5) E D C A B (4) E D A C B (4) E C D A B (4) D A E C B (4) B D A E C (4) A D E C B (4) A C D E B (4) D E A B C (3) C B E D A (3) C B E A D (3) B D E A C (3) B C E A D (3) B A C D E (3) E D B C A (2) E C A D B (2) E B C D A (2) D A E B C (2) D A B E C (2) C A E D B (2) C A B E D (2) B D E C A (2) B C A E D (2) B A D C E (2) A D B E C (2) D E B A C (1) D E A C B (1) D B E A C (1) D B A E C (1) C E D B A (1) C E B D A (1) C E A D B (1) C E A B D (1) C B A E D (1) C A E B D (1) C A B D E (1) B E D C A (1) B E C D A (1) B D C E A (1) B C D A E (1) B A D E C (1) A E D C B (1) A D C B E (1) A D B C E (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -4 -8 0 B 0 0 4 2 8 C 4 -4 0 4 -4 D 8 -2 -4 0 4 E 0 -8 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.141092 B: 0.858908 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.757629697082 Cumulative probabilities = A: 0.141092 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -8 0 B 0 0 4 2 8 C 4 -4 0 4 -4 D 8 -2 -4 0 4 E 0 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017089 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=18 C=17 A=16 D=15 so D is eliminated. Round 2 votes counts: B=36 A=24 E=23 C=17 so C is eliminated. Round 3 votes counts: B=43 A=30 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:207 D:203 C:200 E:196 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -8 0 B 0 0 4 2 8 C 4 -4 0 4 -4 D 8 -2 -4 0 4 E 0 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017089 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -8 0 B 0 0 4 2 8 C 4 -4 0 4 -4 D 8 -2 -4 0 4 E 0 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017089 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -8 0 B 0 0 4 2 8 C 4 -4 0 4 -4 D 8 -2 -4 0 4 E 0 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017089 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1480: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) D A B E C (9) E C A D B (7) C E B A D (7) D A B C E (6) E C B A D (5) D A E C B (5) A B D E C (5) C E D A B (4) C E B D A (4) B A D E C (4) B A D C E (4) C B E A D (3) A D B E C (3) E C A B D (2) E A D C B (2) E A D B C (2) E A B D C (2) D E A C B (2) C D B A E (2) B D A C E (2) E D A C B (1) D C A E B (1) D B C A E (1) D B A C E (1) D A E B C (1) C E D B A (1) C D E A B (1) C D A B E (1) B E C A D (1) B D C A E (1) B A C D E (1) Total count = 100 A B C D E A 0 26 0 -4 -6 B -26 0 -14 -18 -12 C 0 14 0 -4 -20 D 4 18 4 0 0 E 6 12 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.648453 E: 0.351547 Sum of squares = 0.544076299766 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.648453 E: 1.000000 A B C D E A 0 26 0 -4 -6 B -26 0 -14 -18 -12 C 0 14 0 -4 -20 D 4 18 4 0 0 E 6 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 C=23 B=13 A=8 so A is eliminated. Round 2 votes counts: E=30 D=29 C=23 B=18 so B is eliminated. Round 3 votes counts: D=45 E=31 C=24 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:219 D:213 A:208 C:195 B:165 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 26 0 -4 -6 B -26 0 -14 -18 -12 C 0 14 0 -4 -20 D 4 18 4 0 0 E 6 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 0 -4 -6 B -26 0 -14 -18 -12 C 0 14 0 -4 -20 D 4 18 4 0 0 E 6 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 0 -4 -6 B -26 0 -14 -18 -12 C 0 14 0 -4 -20 D 4 18 4 0 0 E 6 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1481: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) A D C E B (7) C D A E B (5) C B E D A (5) B E A D C (5) B C E A D (4) A C D E B (4) E D B A C (3) E B D A C (3) C D E B A (3) C A D E B (3) C A D B E (3) B E D A C (3) B E C D A (3) B C E D A (3) E A D B C (2) D C E A B (2) D A E C B (2) D A E B C (2) C D B E A (2) B A E D C (2) A D E C B (2) A B E D C (2) A B C E D (2) E B A D C (1) D E C B A (1) D E A C B (1) D C A E B (1) D A C E B (1) C E D B A (1) C D E A B (1) C B E A D (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B D E (1) B C A E D (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 12 14 14 10 B -12 0 -4 -18 -10 C -14 4 0 -6 14 D -14 18 6 0 16 E -10 10 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 14 10 B -12 0 -4 -18 -10 C -14 4 0 -6 14 D -14 18 6 0 16 E -10 10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 B=21 D=10 E=9 so E is eliminated. Round 2 votes counts: A=34 C=28 B=25 D=13 so D is eliminated. Round 3 votes counts: A=40 C=32 B=28 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:213 C:199 E:185 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 14 10 B -12 0 -4 -18 -10 C -14 4 0 -6 14 D -14 18 6 0 16 E -10 10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 14 10 B -12 0 -4 -18 -10 C -14 4 0 -6 14 D -14 18 6 0 16 E -10 10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 14 10 B -12 0 -4 -18 -10 C -14 4 0 -6 14 D -14 18 6 0 16 E -10 10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1482: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) E C B A D (8) D B A E C (6) B A E D C (6) E C A B D (5) E B C A D (5) D C A E B (5) C E A D B (5) B E A C D (5) B D A E C (5) C E D A B (4) D A C B E (3) C E A B D (3) A B C E D (3) E B A C D (2) D C E A B (2) D C A B E (2) C D E A B (2) C D A E B (2) B D E A C (2) B A D E C (2) A B E C D (2) E D B C A (1) E C B D A (1) D B E A C (1) D B A C E (1) B E A D C (1) B A E C D (1) A D C B E (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 10 4 6 B -6 0 14 6 10 C -10 -14 0 -4 -8 D -4 -6 4 0 -6 E -6 -10 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 4 6 B -6 0 14 6 10 C -10 -14 0 -4 -8 D -4 -6 4 0 -6 E -6 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 B=22 C=16 A=9 so A is eliminated. Round 2 votes counts: D=33 B=29 E=22 C=16 so C is eliminated. Round 3 votes counts: D=37 E=34 B=29 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:213 B:212 E:199 D:194 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 4 6 B -6 0 14 6 10 C -10 -14 0 -4 -8 D -4 -6 4 0 -6 E -6 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 4 6 B -6 0 14 6 10 C -10 -14 0 -4 -8 D -4 -6 4 0 -6 E -6 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 4 6 B -6 0 14 6 10 C -10 -14 0 -4 -8 D -4 -6 4 0 -6 E -6 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1483: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) C E B A D (11) B E C D A (11) A D C E B (9) B D E C A (7) C E A B D (6) B E D C A (6) E C B A D (3) E B D C A (3) D A E B C (3) C B E A D (3) A D C B E (3) D B E A C (2) D B A E C (2) C A E B D (2) A D E C B (2) A D B E C (2) A D B C E (2) A C D E B (2) E C B D A (1) D A B C E (1) C B E D A (1) B D A C E (1) B C E D A (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -10 -4 -10 B 4 0 10 12 10 C 10 -10 0 -14 -14 D 4 -12 14 0 0 E 10 -10 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -4 -10 B 4 0 10 12 10 C 10 -10 0 -14 -14 D 4 -12 14 0 0 E 10 -10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=23 A=23 D=21 E=7 so E is eliminated. Round 2 votes counts: B=29 C=27 A=23 D=21 so D is eliminated. Round 3 votes counts: A=40 B=33 C=27 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:207 D:203 A:186 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -4 -10 B 4 0 10 12 10 C 10 -10 0 -14 -14 D 4 -12 14 0 0 E 10 -10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -4 -10 B 4 0 10 12 10 C 10 -10 0 -14 -14 D 4 -12 14 0 0 E 10 -10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -4 -10 B 4 0 10 12 10 C 10 -10 0 -14 -14 D 4 -12 14 0 0 E 10 -10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1484: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) E D C B A (9) C A E D B (8) B E D A C (8) E D B C A (6) B A D E C (6) B D E A C (5) A C B D E (5) B E D C A (4) B A C D E (4) A B C D E (4) E D C A B (3) C A D E B (3) A C D E B (3) E C D A B (2) D E B C A (2) C E A D B (2) C D A E B (2) B E A D C (2) B D A E C (2) A C D B E (2) D E C B A (1) D E C A B (1) D B E A C (1) C D E A B (1) C A E B D (1) B D E C A (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 -6 -12 -16 -16 B 6 0 -6 -12 -8 C 12 6 0 -6 -8 D 16 12 6 0 -12 E 16 8 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -16 -16 B 6 0 -6 -12 -8 C 12 6 0 -6 -8 D 16 12 6 0 -12 E 16 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=27 E=20 A=14 D=5 so D is eliminated. Round 2 votes counts: B=35 C=27 E=24 A=14 so A is eliminated. Round 3 votes counts: B=39 C=37 E=24 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:222 D:211 C:202 B:190 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -16 -16 B 6 0 -6 -12 -8 C 12 6 0 -6 -8 D 16 12 6 0 -12 E 16 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -16 -16 B 6 0 -6 -12 -8 C 12 6 0 -6 -8 D 16 12 6 0 -12 E 16 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -16 -16 B 6 0 -6 -12 -8 C 12 6 0 -6 -8 D 16 12 6 0 -12 E 16 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1485: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (15) E A B D C (8) C D B A E (8) E B A C D (7) D A C E B (7) B C E D A (7) C B D E A (6) A E D B C (5) E B C A D (4) C D B E A (4) B E C A D (4) A D E C B (4) E A B C D (3) D C B A E (3) D A C B E (3) A E B D C (3) D A E C B (2) C B E D A (2) A D E B C (2) B E C D A (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 8 -8 -16 8 B -8 0 -12 -10 8 C 8 12 0 -8 14 D 16 10 8 0 12 E -8 -8 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -16 8 B -8 0 -12 -10 8 C 8 12 0 -8 14 D 16 10 8 0 12 E -8 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=22 C=20 A=16 B=12 so B is eliminated. Round 2 votes counts: D=30 E=27 C=27 A=16 so A is eliminated. Round 3 votes counts: D=38 E=35 C=27 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:213 A:196 B:189 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -16 8 B -8 0 -12 -10 8 C 8 12 0 -8 14 D 16 10 8 0 12 E -8 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -16 8 B -8 0 -12 -10 8 C 8 12 0 -8 14 D 16 10 8 0 12 E -8 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -16 8 B -8 0 -12 -10 8 C 8 12 0 -8 14 D 16 10 8 0 12 E -8 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1486: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (15) A C D E B (12) B E C D A (11) B C E D A (7) A D C E B (7) D E B A C (6) C A B E D (6) A C B D E (5) C B A E D (3) B D E A C (3) D A E C B (2) D A B E C (2) C B E A D (2) C A E B D (2) B C E A D (2) A D E C B (2) A D C B E (2) A C D B E (2) E D B C A (1) E B D C A (1) D E A C B (1) D E A B C (1) D B E A C (1) D A E B C (1) C E A B D (1) B D E C A (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -4 -6 -6 B 6 0 6 20 26 C 4 -6 0 8 4 D 6 -20 -8 0 -2 E 6 -26 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -6 -6 B 6 0 6 20 26 C 4 -6 0 8 4 D 6 -20 -8 0 -2 E 6 -26 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=31 D=14 C=14 E=2 so E is eliminated. Round 2 votes counts: B=40 A=31 D=15 C=14 so C is eliminated. Round 3 votes counts: B=45 A=40 D=15 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 C:205 A:189 E:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -6 -6 B 6 0 6 20 26 C 4 -6 0 8 4 D 6 -20 -8 0 -2 E 6 -26 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -6 -6 B 6 0 6 20 26 C 4 -6 0 8 4 D 6 -20 -8 0 -2 E 6 -26 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -6 -6 B 6 0 6 20 26 C 4 -6 0 8 4 D 6 -20 -8 0 -2 E 6 -26 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996323 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1487: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) C D B A E (6) A E C B D (6) E A B D C (5) D C B E A (5) D B C E A (5) C A E B D (5) B D E A C (5) B D A E C (5) A E B C D (5) D E A B C (4) C E A D B (4) C D E A B (4) D C E A B (3) E A C B D (2) D C B A E (2) C D A E B (2) C A E D B (2) B D C A E (2) A E B D C (2) E B A D C (1) E A D B C (1) E A C D B (1) D C E B A (1) D B E C A (1) C E D A B (1) C D E B A (1) C D B E A (1) C B A D E (1) C A B E D (1) B E A D C (1) B C A D E (1) B A E D C (1) B A D E C (1) B A C E D (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 4 -12 -8 B -2 0 0 -4 -2 C -4 0 0 -6 0 D 12 4 6 0 16 E 8 2 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -12 -8 B -2 0 0 -4 -2 C -4 0 0 -6 0 D 12 4 6 0 16 E 8 2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 B=17 A=16 E=10 so E is eliminated. Round 2 votes counts: D=29 C=28 A=25 B=18 so B is eliminated. Round 3 votes counts: D=41 A=30 C=29 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:197 B:196 C:195 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -12 -8 B -2 0 0 -4 -2 C -4 0 0 -6 0 D 12 4 6 0 16 E 8 2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -12 -8 B -2 0 0 -4 -2 C -4 0 0 -6 0 D 12 4 6 0 16 E 8 2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -12 -8 B -2 0 0 -4 -2 C -4 0 0 -6 0 D 12 4 6 0 16 E 8 2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1488: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) A D B E C (7) C E B D A (6) C E D A B (5) B E C A D (5) A D B C E (5) E C B A D (4) E B C A D (4) D C A E B (4) A B E D C (4) D A C E B (3) C E D B A (3) B A E C D (3) A E C D B (3) A B D E C (3) E C B D A (2) D A B C E (2) C D E B A (2) B E A C D (2) E C A B D (1) E B C D A (1) E B A C D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A B E (1) D B E C A (1) D B C E A (1) C E A D B (1) C D E A B (1) B E C D A (1) B E A D C (1) B D E C A (1) B A E D C (1) B A D E C (1) A E D C B (1) A E B C D (1) A D E C B (1) A D E B C (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 4 6 6 B -14 0 -4 -12 4 C -4 4 0 -4 -4 D -6 12 4 0 -4 E -6 -4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 6 6 B -14 0 -4 -12 4 C -4 4 0 -4 -4 D -6 12 4 0 -4 E -6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 C=18 B=15 E=14 so E is eliminated. Round 2 votes counts: A=29 D=25 C=25 B=21 so B is eliminated. Round 3 votes counts: A=38 C=36 D=26 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:203 E:199 C:196 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 6 6 B -14 0 -4 -12 4 C -4 4 0 -4 -4 D -6 12 4 0 -4 E -6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 6 6 B -14 0 -4 -12 4 C -4 4 0 -4 -4 D -6 12 4 0 -4 E -6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 6 6 B -14 0 -4 -12 4 C -4 4 0 -4 -4 D -6 12 4 0 -4 E -6 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1489: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) E B C A D (8) E B A C D (6) D C A B E (6) C E D B A (6) A E B D C (6) D C E A B (5) A D B C E (4) A B D E C (4) E C D A B (3) D A C B E (3) C E B D A (3) C D B A E (3) B E A C D (3) A B E D C (3) E D C A B (2) E C B A D (2) E A D B C (2) D C A E B (2) C D E B A (2) B A E D C (2) B A E C D (2) A D E B C (2) E C D B A (1) E C A D B (1) E A B C D (1) D E C A B (1) D A C E B (1) D A B C E (1) C D E A B (1) C B D E A (1) C B D A E (1) B A D E C (1) B A C E D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -12 0 -12 B 0 0 -4 6 -26 C 12 4 0 6 -18 D 0 -6 -6 0 -20 E 12 26 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -12 0 -12 B 0 0 -4 6 -26 C 12 4 0 6 -18 D 0 -6 -6 0 -20 E 12 26 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=21 D=19 C=17 B=9 so B is eliminated. Round 2 votes counts: E=37 A=27 D=19 C=17 so C is eliminated. Round 3 votes counts: E=46 D=27 A=27 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:238 C:202 A:188 B:188 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -12 0 -12 B 0 0 -4 6 -26 C 12 4 0 6 -18 D 0 -6 -6 0 -20 E 12 26 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 0 -12 B 0 0 -4 6 -26 C 12 4 0 6 -18 D 0 -6 -6 0 -20 E 12 26 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 0 -12 B 0 0 -4 6 -26 C 12 4 0 6 -18 D 0 -6 -6 0 -20 E 12 26 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1490: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) D C B A E (6) A E B C D (6) E D A C B (5) E A D C B (5) E A B C D (5) A E B D C (5) C D B E A (4) B C D A E (4) A D E B C (4) E A D B C (3) D C E B A (3) D A E C B (3) B C A E D (3) B C A D E (3) E D C B A (2) D E A C B (2) D C A E B (2) D A C B E (2) D A B C E (2) C D E B A (2) C B E D A (2) C B D E A (2) A E D C B (2) A B D C E (2) E C B D A (1) E C B A D (1) E A B D C (1) D E C B A (1) D C E A B (1) D B C A E (1) C B D A E (1) B E C A D (1) B E A C D (1) B C E D A (1) B C E A D (1) B A C E D (1) A E D B C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 4 -6 0 B -6 0 -6 -14 -10 C -4 6 0 -18 0 D 6 14 18 0 4 E 0 10 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -6 0 B -6 0 -6 -14 -10 C -4 6 0 -18 0 D 6 14 18 0 4 E 0 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=23 A=22 B=15 C=11 so C is eliminated. Round 2 votes counts: D=35 E=23 A=22 B=20 so B is eliminated. Round 3 votes counts: D=42 E=29 A=29 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:203 A:202 C:192 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -6 0 B -6 0 -6 -14 -10 C -4 6 0 -18 0 D 6 14 18 0 4 E 0 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -6 0 B -6 0 -6 -14 -10 C -4 6 0 -18 0 D 6 14 18 0 4 E 0 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -6 0 B -6 0 -6 -14 -10 C -4 6 0 -18 0 D 6 14 18 0 4 E 0 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1491: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) D C B A E (7) A D C E B (7) D C A B E (6) E A B C D (5) C D B A E (5) C B D E A (5) C D B E A (4) A E B D C (4) A D E C B (4) E B C A D (3) D C A E B (3) D A C B E (3) C D A E B (3) B E C A D (3) B C E D A (3) A E C D B (3) E B A D C (2) E A B D C (2) D B C A E (2) D A C E B (2) B D E C A (2) B D C E A (2) A E D B C (2) A C E D B (2) D B C E A (1) C B E D A (1) C A D E B (1) B E D C A (1) B E C D A (1) B E A C D (1) B C D E A (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -8 -4 12 B 2 0 -14 -14 -4 C 8 14 0 -4 16 D 4 14 4 0 18 E -12 4 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -4 12 B 2 0 -14 -14 -4 C 8 14 0 -4 16 D 4 14 4 0 18 E -12 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=19 C=19 B=14 so B is eliminated. Round 2 votes counts: D=28 E=25 A=24 C=23 so C is eliminated. Round 3 votes counts: D=46 E=29 A=25 so A is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:217 A:199 B:185 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 12 B 2 0 -14 -14 -4 C 8 14 0 -4 16 D 4 14 4 0 18 E -12 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 12 B 2 0 -14 -14 -4 C 8 14 0 -4 16 D 4 14 4 0 18 E -12 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 12 B 2 0 -14 -14 -4 C 8 14 0 -4 16 D 4 14 4 0 18 E -12 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1492: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) B A D C E (8) A D E B C (8) C E B D A (7) D E A C B (6) B A D E C (6) D A E C B (5) A D B E C (5) C E D A B (4) B C A E D (4) B A C D E (4) A B D E C (4) C B E D A (3) C B E A D (3) B C E A D (3) E C D B A (2) E C B D A (2) D A E B C (2) B A C E D (2) E D A C B (1) E D A B C (1) E C D A B (1) E A D B C (1) D E C A B (1) C E D B A (1) C B D A E (1) B E C A D (1) B E A C D (1) B A E D C (1) B A E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 14 26 24 B 20 0 26 24 16 C -14 -26 0 0 2 D -26 -24 0 0 22 E -24 -16 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 14 26 24 B 20 0 26 24 16 C -14 -26 0 0 2 D -26 -24 0 0 22 E -24 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=19 A=18 D=14 E=8 so E is eliminated. Round 2 votes counts: B=41 C=24 A=19 D=16 so D is eliminated. Round 3 votes counts: B=41 A=34 C=25 so C is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:243 A:222 D:186 C:181 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 14 26 24 B 20 0 26 24 16 C -14 -26 0 0 2 D -26 -24 0 0 22 E -24 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 14 26 24 B 20 0 26 24 16 C -14 -26 0 0 2 D -26 -24 0 0 22 E -24 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 14 26 24 B 20 0 26 24 16 C -14 -26 0 0 2 D -26 -24 0 0 22 E -24 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1493: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) E A D B C (7) C B D E A (7) E A C B D (6) D B C E A (6) E C B D A (5) A E C B D (5) D B A C E (4) C B D A E (4) A B C D E (4) A E D B C (3) A D E B C (3) A C B E D (3) E D A C B (2) E A D C B (2) D E B C A (2) D B E C A (2) D A B C E (2) C B A D E (2) A E C D B (2) A E B C D (2) A D B E C (2) A C E B D (2) E D C B A (1) E C D B A (1) E C B A D (1) E A C D B (1) D E B A C (1) D C E B A (1) D A E B C (1) C E B A D (1) C E A B D (1) C D B E A (1) C B E A D (1) B D C A E (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 10 2 2 B -2 0 0 -6 0 C -10 0 0 0 2 D -2 6 0 0 8 E -2 0 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999881 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 2 2 B -2 0 0 -6 0 C -10 0 0 0 2 D -2 6 0 0 8 E -2 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=27 E=26 C=17 B=1 so B is eliminated. Round 2 votes counts: A=29 D=28 E=26 C=17 so C is eliminated. Round 3 votes counts: D=40 A=31 E=29 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:206 B:196 C:196 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 2 2 B -2 0 0 -6 0 C -10 0 0 0 2 D -2 6 0 0 8 E -2 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 2 2 B -2 0 0 -6 0 C -10 0 0 0 2 D -2 6 0 0 8 E -2 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 2 2 B -2 0 0 -6 0 C -10 0 0 0 2 D -2 6 0 0 8 E -2 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1494: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (14) D A E B C (10) C B A E D (9) D C E A B (7) C D E B A (5) B A E D C (4) A B E D C (4) D E A B C (3) D C A E B (3) D A B E C (3) C D B A E (3) B A E C D (3) E B A C D (2) E A B C D (2) D E A C B (2) D A E C B (2) D A C B E (2) C E B A D (2) C D E A B (2) C D B E A (2) B E A C D (2) A E B D C (2) E D A B C (1) E C A B D (1) E A B D C (1) D E C A B (1) D C B E A (1) D C B A E (1) C E A B D (1) C B E D A (1) C B D A E (1) B E C A D (1) B C E A D (1) B A C E D (1) Total count = 100 A B C D E A 0 -6 -12 0 -4 B 6 0 -20 4 6 C 12 20 0 6 12 D 0 -4 -6 0 -4 E 4 -6 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 0 -4 B 6 0 -20 4 6 C 12 20 0 6 12 D 0 -4 -6 0 -4 E 4 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 D=35 B=12 E=7 A=6 so A is eliminated. Round 2 votes counts: C=40 D=35 B=16 E=9 so E is eliminated. Round 3 votes counts: C=41 D=36 B=23 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:198 E:195 D:193 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 0 -4 B 6 0 -20 4 6 C 12 20 0 6 12 D 0 -4 -6 0 -4 E 4 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 0 -4 B 6 0 -20 4 6 C 12 20 0 6 12 D 0 -4 -6 0 -4 E 4 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 0 -4 B 6 0 -20 4 6 C 12 20 0 6 12 D 0 -4 -6 0 -4 E 4 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1495: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) E A B D C (6) C D B A E (6) B E C D A (6) C D A B E (5) C B D E A (5) A E D B C (5) B C E D A (4) A D E C B (4) E B D C A (3) D E A C B (3) C D B E A (3) B E A C D (3) A D C E B (3) A C D E B (3) E B D A C (2) E A D B C (2) D C E A B (2) D A C E B (2) C A D B E (2) B E D C A (2) B C E A D (2) A E B D C (2) A E B C D (2) E D B C A (1) E D A B C (1) E B A C D (1) D E B A C (1) D C A E B (1) D A E C B (1) C D A E B (1) C B A E D (1) C B A D E (1) B E C A D (1) B C D E A (1) A E D C B (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 6 0 -14 B 2 0 8 4 -10 C -6 -8 0 0 -10 D 0 -4 0 0 -10 E 14 10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 0 -14 B 2 0 8 4 -10 C -6 -8 0 0 -10 D 0 -4 0 0 -10 E 14 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 A=23 B=19 D=10 so D is eliminated. Round 2 votes counts: E=28 C=27 A=26 B=19 so B is eliminated. Round 3 votes counts: E=40 C=34 A=26 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:202 A:195 D:193 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 0 -14 B 2 0 8 4 -10 C -6 -8 0 0 -10 D 0 -4 0 0 -10 E 14 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 -14 B 2 0 8 4 -10 C -6 -8 0 0 -10 D 0 -4 0 0 -10 E 14 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 -14 B 2 0 8 4 -10 C -6 -8 0 0 -10 D 0 -4 0 0 -10 E 14 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1496: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) B D E A C (9) C A E D B (7) C D E B A (6) A C B E D (5) C E A D B (4) B A D E C (4) A C E D B (4) C D B E A (3) C B D E A (3) B A C D E (3) D B E C A (2) D B C E A (2) C E D B A (2) C A E B D (2) C A B D E (2) B D A E C (2) A E C D B (2) A E B D C (2) A C E B D (2) A B E D C (2) E D C B A (1) E D B A C (1) E D A B C (1) E C A D B (1) E A D C B (1) E A D B C (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E A C (1) C B D A E (1) C B A D E (1) C A D E B (1) B D E C A (1) B D C E A (1) B D A C E (1) A E D B C (1) A E C B D (1) A E B C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 0 -6 B -4 0 -22 -8 -6 C 6 22 0 24 28 D 0 8 -24 0 -2 E 6 6 -28 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 0 -6 B -4 0 -22 -8 -6 C 6 22 0 24 28 D 0 8 -24 0 -2 E 6 6 -28 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=23 B=21 D=9 E=6 so E is eliminated. Round 2 votes counts: C=42 A=25 B=21 D=12 so D is eliminated. Round 3 votes counts: C=46 B=28 A=26 so A is eliminated. Round 4 votes counts: C=61 B=39 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:240 A:196 E:193 D:191 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 0 -6 B -4 0 -22 -8 -6 C 6 22 0 24 28 D 0 8 -24 0 -2 E 6 6 -28 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 0 -6 B -4 0 -22 -8 -6 C 6 22 0 24 28 D 0 8 -24 0 -2 E 6 6 -28 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 0 -6 B -4 0 -22 -8 -6 C 6 22 0 24 28 D 0 8 -24 0 -2 E 6 6 -28 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1497: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) E A D B C (8) D A E B C (6) C B D A E (6) D A B C E (4) B C D A E (4) E C B A D (3) D A E C B (3) D A B E C (3) C D B A E (3) C D A E B (3) C B E D A (3) C B E A D (3) B E C A D (3) A D E B C (3) A D B E C (3) E C A B D (2) E B C A D (2) D E A C B (2) C E D A B (2) C E B A D (2) C D A B E (2) B D A C E (2) B C E A D (2) B A E D C (2) B A D E C (2) A E D B C (2) E C D A B (1) E A D C B (1) D C B A E (1) D C A B E (1) D A C B E (1) C E B D A (1) B E A D C (1) B C A D E (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 16 10 4 8 B -16 0 20 2 -4 C -10 -20 0 -14 -16 D -4 -2 14 0 2 E -8 4 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 4 8 B -16 0 20 2 -4 C -10 -20 0 -14 -16 D -4 -2 14 0 2 E -8 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 D=21 B=18 A=9 so A is eliminated. Round 2 votes counts: E=30 D=27 C=25 B=18 so B is eliminated. Round 3 votes counts: E=36 D=32 C=32 so D is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:219 D:205 E:205 B:201 C:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 4 8 B -16 0 20 2 -4 C -10 -20 0 -14 -16 D -4 -2 14 0 2 E -8 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 4 8 B -16 0 20 2 -4 C -10 -20 0 -14 -16 D -4 -2 14 0 2 E -8 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 4 8 B -16 0 20 2 -4 C -10 -20 0 -14 -16 D -4 -2 14 0 2 E -8 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1498: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) E B A D C (8) C A E D B (8) A C E B D (8) D B C E A (7) A E B C D (7) D C B E A (6) A E C B D (6) E A B D C (5) D B E C A (5) B E D A C (5) C D B A E (4) C D A B E (4) B D E A C (4) C A E B D (3) C A D E B (2) A E B D C (2) E B D C A (1) E B D A C (1) E B A C D (1) E A B C D (1) C D B E A (1) C A D B E (1) B E A D C (1) Total count = 100 A B C D E A 0 -6 16 6 -10 B 6 0 14 6 -6 C -16 -14 0 -8 -12 D -6 -6 8 0 -14 E 10 6 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 16 6 -10 B 6 0 14 6 -6 C -16 -14 0 -8 -12 D -6 -6 8 0 -14 E 10 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=23 A=23 E=17 B=10 so B is eliminated. Round 2 votes counts: D=31 E=23 C=23 A=23 so E is eliminated. Round 3 votes counts: A=39 D=38 C=23 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:221 B:210 A:203 D:191 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 16 6 -10 B 6 0 14 6 -6 C -16 -14 0 -8 -12 D -6 -6 8 0 -14 E 10 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 6 -10 B 6 0 14 6 -6 C -16 -14 0 -8 -12 D -6 -6 8 0 -14 E 10 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 6 -10 B 6 0 14 6 -6 C -16 -14 0 -8 -12 D -6 -6 8 0 -14 E 10 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1499: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) C B E D A (7) D A E B C (6) D C A E B (4) D A C E B (4) B E C A D (4) B E A C D (4) B C E D A (4) A E D B C (4) A E B D C (4) A D E B C (4) E B A D C (3) D A E C B (3) C D A B E (3) C B E A D (3) E A B D C (2) D E A B C (2) D C E B A (2) D C E A B (2) D C B E A (2) C B D E A (2) C A D B E (2) B C E A D (2) A C D E B (2) A C B E D (2) E B D A C (1) E B A C D (1) D E B C A (1) D C B A E (1) D C A B E (1) D B E C A (1) C D B A E (1) C D A E B (1) C B A D E (1) B E C D A (1) A E B C D (1) A D C E B (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 -14 -4 B -2 0 -6 -10 2 C 6 6 0 4 14 D 14 10 -4 0 8 E 4 -2 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -14 -4 B -2 0 -6 -10 2 C 6 6 0 4 14 D 14 10 -4 0 8 E 4 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 A=21 B=15 E=7 so E is eliminated. Round 2 votes counts: D=29 C=28 A=23 B=20 so B is eliminated. Round 3 votes counts: C=39 A=31 D=30 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:214 B:192 E:190 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -14 -4 B -2 0 -6 -10 2 C 6 6 0 4 14 D 14 10 -4 0 8 E 4 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -14 -4 B -2 0 -6 -10 2 C 6 6 0 4 14 D 14 10 -4 0 8 E 4 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -14 -4 B -2 0 -6 -10 2 C 6 6 0 4 14 D 14 10 -4 0 8 E 4 -2 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1500: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) A C D E B (6) E D B A C (5) D E A B C (5) C B A E D (5) C A B E D (5) A D C E B (5) E B D C A (4) C B E A D (4) B E C D A (4) B C E D A (4) A D E C B (4) A C D B E (4) E B C D A (3) B E D C A (3) B D E A C (3) E D B C A (2) D B E A C (2) D B A E C (2) C E B D A (2) C E A D B (2) C B E D A (2) B D A E C (2) A D B E C (2) A C B D E (2) E D A B C (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) B A C D E (1) A D E B C (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 16 -10 -14 B 16 0 12 -6 -6 C -16 -12 0 -4 -8 D 10 6 4 0 2 E 14 6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 16 -10 -14 B 16 0 12 -6 -6 C -16 -12 0 -4 -8 D 10 6 4 0 2 E 14 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=22 D=20 B=17 E=15 so E is eliminated. Round 2 votes counts: D=28 A=26 B=24 C=22 so C is eliminated. Round 3 votes counts: B=38 A=34 D=28 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:213 D:211 B:208 A:188 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 16 -10 -14 B 16 0 12 -6 -6 C -16 -12 0 -4 -8 D 10 6 4 0 2 E 14 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 16 -10 -14 B 16 0 12 -6 -6 C -16 -12 0 -4 -8 D 10 6 4 0 2 E 14 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 16 -10 -14 B 16 0 12 -6 -6 C -16 -12 0 -4 -8 D 10 6 4 0 2 E 14 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1501: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) E C A B D (7) B A D E C (6) E A C B D (5) B D A C E (5) D C B A E (4) D A B E C (4) C E D B A (4) B A D C E (4) A B D E C (4) E A B C D (3) C E B D A (3) C D E B A (3) A B E D C (3) E C D A B (2) E C B A D (2) D C E B A (2) D C E A B (2) D A E B C (2) C E D A B (2) C E B A D (2) B A E C D (2) B A C E D (2) A E B D C (2) A D B E C (2) A B E C D (2) E A D C B (1) D C B E A (1) D B C A E (1) D B A E C (1) C E A D B (1) C B E A D (1) B D C A E (1) B C D A E (1) B C A D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -14 20 2 18 B 14 0 16 14 12 C -20 -16 0 -12 2 D -2 -14 12 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 20 2 18 B 14 0 16 14 12 C -20 -16 0 -12 2 D -2 -14 12 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=22 E=20 C=16 A=14 so A is eliminated. Round 2 votes counts: B=31 D=30 E=23 C=16 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:228 A:213 D:203 E:179 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 20 2 18 B 14 0 16 14 12 C -20 -16 0 -12 2 D -2 -14 12 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 20 2 18 B 14 0 16 14 12 C -20 -16 0 -12 2 D -2 -14 12 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 20 2 18 B 14 0 16 14 12 C -20 -16 0 -12 2 D -2 -14 12 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1502: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) C D A E B (6) B D E C A (6) D C E B A (5) D C E A B (5) D C B E A (5) A C E D B (5) A C D E B (5) C D E A B (4) B D C A E (4) A E B C D (4) A B E C D (4) E D C B A (3) E A C D B (3) C D A B E (3) B E D C A (3) A E C B D (3) E B A D C (2) B E D A C (2) B A E C D (2) B A D C E (2) A E C D B (2) A C B D E (2) E D B C A (1) E C D A B (1) E C A D B (1) E B D C A (1) E A B C D (1) D C A B E (1) C E D A B (1) C A D E B (1) B D E A C (1) B D C E A (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 -6 -8 B -8 0 -12 -4 -8 C 4 12 0 0 4 D 6 4 0 0 4 E 8 8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.265743 D: 0.734257 E: 0.000000 Sum of squares = 0.609752616697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.265743 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -6 -8 B -8 0 -12 -4 -8 C 4 12 0 0 4 D 6 4 0 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=27 D=16 C=15 E=13 so E is eliminated. Round 2 votes counts: B=32 A=31 D=20 C=17 so C is eliminated. Round 3 votes counts: D=35 A=33 B=32 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:210 D:207 E:204 A:195 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -6 -8 B -8 0 -12 -4 -8 C 4 12 0 0 4 D 6 4 0 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -6 -8 B -8 0 -12 -4 -8 C 4 12 0 0 4 D 6 4 0 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -6 -8 B -8 0 -12 -4 -8 C 4 12 0 0 4 D 6 4 0 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1503: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) B E A C D (8) D C A E B (6) B D C E A (6) E A B C D (5) C A E D B (5) D E A C B (4) D B E A C (4) B E A D C (4) A E C D B (4) A E C B D (4) E A B D C (3) C D B A E (3) C B A E D (3) B C D A E (3) B C A E D (3) A E D C B (3) D E A B C (2) D B C E A (2) D A E C B (2) C A D E B (2) B C D E A (2) E D A B C (1) E A D C B (1) E A D B C (1) D E B A C (1) D C B E A (1) D B C A E (1) C D A E B (1) C B D A E (1) C A B E D (1) B D E A C (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 0 18 8 B -14 0 -4 10 -14 C 0 4 0 14 2 D -18 -10 -14 0 -16 E -8 14 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.422644 B: 0.000000 C: 0.577356 D: 0.000000 E: 0.000000 Sum of squares = 0.511967929019 Cumulative probabilities = A: 0.422644 B: 0.422644 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 18 8 B -14 0 -4 10 -14 C 0 4 0 14 2 D -18 -10 -14 0 -16 E -8 14 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 D=23 A=13 E=11 so E is eliminated. Round 2 votes counts: B=27 C=26 D=24 A=23 so A is eliminated. Round 3 votes counts: B=36 C=35 D=29 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:220 C:210 E:210 B:189 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 18 8 B -14 0 -4 10 -14 C 0 4 0 14 2 D -18 -10 -14 0 -16 E -8 14 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 18 8 B -14 0 -4 10 -14 C 0 4 0 14 2 D -18 -10 -14 0 -16 E -8 14 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 18 8 B -14 0 -4 10 -14 C 0 4 0 14 2 D -18 -10 -14 0 -16 E -8 14 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1504: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (6) B E A D C (5) B C D E A (5) B A E C D (5) E A D C B (4) D E C A B (4) C A D E B (4) A E C D B (4) D C E A B (3) C D E A B (3) C D A E B (3) C B D A E (3) B D E C A (3) B D C E A (3) B C A D E (3) A E D C B (3) E D A C B (2) E A D B C (2) E A B D C (2) D E A C B (2) D C B E A (2) D B E C A (2) C B A D E (2) C A D B E (2) B E D A C (2) B A C E D (2) A E B D C (2) A C B E D (2) A B E C D (2) E D B A C (1) E D A B C (1) E B A D C (1) D E B C A (1) D C E B A (1) D B C E A (1) C D B E A (1) C D A B E (1) C A E D B (1) C A B D E (1) A E B C D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -10 0 -2 B -2 0 2 2 8 C 10 -2 0 6 2 D 0 -2 -6 0 12 E 2 -8 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408173 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 0 -2 B -2 0 2 2 8 C 10 -2 0 6 2 D 0 -2 -6 0 12 E 2 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040804 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=21 D=16 A=16 E=13 so E is eliminated. Round 2 votes counts: B=35 A=24 C=21 D=20 so D is eliminated. Round 3 votes counts: B=40 C=31 A=29 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:205 D:202 A:195 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -10 0 -2 B -2 0 2 2 8 C 10 -2 0 6 2 D 0 -2 -6 0 12 E 2 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040804 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 0 -2 B -2 0 2 2 8 C 10 -2 0 6 2 D 0 -2 -6 0 12 E 2 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040804 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 0 -2 B -2 0 2 2 8 C 10 -2 0 6 2 D 0 -2 -6 0 12 E 2 -8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040804 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1505: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) C E A B D (8) A B D C E (7) B D A C E (6) A D B E C (6) E C B D A (5) E C A B D (5) E A D B C (4) C B D E A (4) A E C D B (4) E D B C A (3) E C D B A (3) E A C D B (3) D B E C A (3) B D C A E (3) A E D B C (3) A D B C E (3) E C A D B (2) D B A E C (2) D B A C E (2) B A D C E (2) A C B D E (2) E D B A C (1) E C D A B (1) E A D C B (1) D B E A C (1) D B C A E (1) C B D A E (1) C A B E D (1) B D C E A (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -4 4 -12 B -2 0 -4 14 -10 C 4 4 0 2 6 D -4 -14 -2 0 -12 E 12 10 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 4 -12 B -2 0 -4 14 -10 C 4 4 0 2 6 D -4 -14 -2 0 -12 E 12 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 C=25 B=12 D=9 so D is eliminated. Round 2 votes counts: E=28 A=26 C=25 B=21 so B is eliminated. Round 3 votes counts: A=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:208 B:199 A:195 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 4 -12 B -2 0 -4 14 -10 C 4 4 0 2 6 D -4 -14 -2 0 -12 E 12 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 -12 B -2 0 -4 14 -10 C 4 4 0 2 6 D -4 -14 -2 0 -12 E 12 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 -12 B -2 0 -4 14 -10 C 4 4 0 2 6 D -4 -14 -2 0 -12 E 12 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1506: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) E A D C B (8) D C B E A (8) D C B A E (8) C D B E A (8) A E D C B (8) B C D E A (7) E A C D B (4) E A B C D (4) B A C D E (4) C D E B A (3) B D C A E (3) E D C A B (2) E C D B A (2) D C E A B (2) B C D A E (2) A B E C D (2) E C D A B (1) E C B D A (1) D C A B E (1) D B C A E (1) C B D E A (1) B D C E A (1) B A D C E (1) B A C E D (1) A E C B D (1) A E B D C (1) A D C E B (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -2 -2 -4 B 2 0 -18 -16 0 C 2 18 0 6 6 D 2 16 -6 0 4 E 4 0 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -2 -4 B 2 0 -18 -16 0 C 2 18 0 6 6 D 2 16 -6 0 4 E 4 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997257 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 D=20 B=19 C=12 so C is eliminated. Round 2 votes counts: D=31 A=27 E=22 B=20 so B is eliminated. Round 3 votes counts: D=45 A=33 E=22 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:208 E:197 A:195 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -2 -4 B 2 0 -18 -16 0 C 2 18 0 6 6 D 2 16 -6 0 4 E 4 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997257 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -2 -4 B 2 0 -18 -16 0 C 2 18 0 6 6 D 2 16 -6 0 4 E 4 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997257 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -2 -4 B 2 0 -18 -16 0 C 2 18 0 6 6 D 2 16 -6 0 4 E 4 0 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997257 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1507: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (12) C E B D A (6) C B D A E (6) E A B D C (5) C A D B E (5) B D C A E (5) E B D A C (4) C B E D A (4) E C B D A (3) C D B A E (3) C B D E A (3) B E D A C (3) A E D B C (3) E C A B D (2) E B D C A (2) E B C D A (2) E A C D B (2) E A C B D (2) D B C A E (2) C E B A D (2) B D C E A (2) A D B E C (2) E A D C B (1) E A B C D (1) D C B A E (1) D B A E C (1) D B A C E (1) D A C B E (1) D A B E C (1) C E A B D (1) C D A B E (1) C A E D B (1) C A B D E (1) B D E C A (1) B D E A C (1) B C D E A (1) B C D A E (1) A E D C B (1) A D E B C (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -10 -10 -20 B 8 0 2 16 -6 C 10 -2 0 0 0 D 10 -16 0 0 -16 E 20 6 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.523080 D: 0.000000 E: 0.476920 Sum of squares = 0.50106537128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.523080 D: 0.523080 E: 1.000000 A B C D E A 0 -8 -10 -10 -20 B 8 0 2 16 -6 C 10 -2 0 0 0 D 10 -16 0 0 -16 E 20 6 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=33 B=14 A=10 D=7 so D is eliminated. Round 2 votes counts: E=36 C=34 B=18 A=12 so A is eliminated. Round 3 votes counts: E=41 C=38 B=21 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:221 B:210 C:204 D:189 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -10 -20 B 8 0 2 16 -6 C 10 -2 0 0 0 D 10 -16 0 0 -16 E 20 6 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -10 -20 B 8 0 2 16 -6 C 10 -2 0 0 0 D 10 -16 0 0 -16 E 20 6 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -10 -20 B 8 0 2 16 -6 C 10 -2 0 0 0 D 10 -16 0 0 -16 E 20 6 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1508: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) B A E C D (9) A B E D C (7) C D B E A (6) B A C E D (6) E A B D C (4) D E C A B (4) D C E A B (4) C D A B E (4) C B D A E (4) E B A D C (3) D C E B A (3) C D B A E (3) C B D E A (3) B C A D E (3) A B E C D (3) E D A C B (2) E A D B C (2) D E C B A (2) D C A E B (2) D A C E B (2) B C A E D (2) B A E D C (2) B A C D E (2) A E B D C (2) E A D C B (1) C D E A B (1) B E A D C (1) A C D B E (1) Total count = 100 A B C D E A 0 -22 -6 -4 4 B 22 0 -8 2 12 C 6 8 0 18 16 D 4 -2 -18 0 12 E -4 -12 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 -4 4 B 22 0 -8 2 12 C 6 8 0 18 16 D 4 -2 -18 0 12 E -4 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=25 D=17 A=13 E=12 so E is eliminated. Round 2 votes counts: C=33 B=28 A=20 D=19 so D is eliminated. Round 3 votes counts: C=48 B=28 A=24 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:214 D:198 A:186 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -6 -4 4 B 22 0 -8 2 12 C 6 8 0 18 16 D 4 -2 -18 0 12 E -4 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -4 4 B 22 0 -8 2 12 C 6 8 0 18 16 D 4 -2 -18 0 12 E -4 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -4 4 B 22 0 -8 2 12 C 6 8 0 18 16 D 4 -2 -18 0 12 E -4 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1509: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (6) E B A C D (5) C D E B A (5) B D A E C (5) B A E D C (5) E C B A D (4) D C A B E (4) A E C B D (4) A E B C D (4) A D C E B (4) A C E D B (4) E C A B D (3) E A B C D (3) D C B A E (3) C D E A B (3) B E D C A (3) A D B C E (3) E B C A D (2) D C B E A (2) C E D A B (2) C D A E B (2) B D C E A (2) A D C B E (2) A B E D C (2) E C B D A (1) E C A D B (1) E B A D C (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D B A (1) C E A D B (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E A C (1) B D C A E (1) B A D E C (1) A E C D B (1) A E B D C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 14 20 4 B 4 0 0 14 -8 C -14 0 0 -2 -12 D -20 -14 2 0 -12 E -4 8 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000041 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 14 20 4 B 4 0 0 14 -8 C -14 0 0 -2 -12 D -20 -14 2 0 -12 E -4 8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 E=20 C=15 D=12 so D is eliminated. Round 2 votes counts: B=28 A=28 C=24 E=20 so E is eliminated. Round 3 votes counts: B=36 C=33 A=31 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:217 E:214 B:205 C:186 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 14 20 4 B 4 0 0 14 -8 C -14 0 0 -2 -12 D -20 -14 2 0 -12 E -4 8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 20 4 B 4 0 0 14 -8 C -14 0 0 -2 -12 D -20 -14 2 0 -12 E -4 8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 20 4 B 4 0 0 14 -8 C -14 0 0 -2 -12 D -20 -14 2 0 -12 E -4 8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1510: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) A B C E D (10) D E C B A (8) B C A E D (8) A B C D E (7) D E A C B (6) A B D C E (6) C B E A D (5) E D C B A (4) D A E B C (4) E C D B A (3) D A B E C (3) C E B A D (3) A D B C E (3) E C B D A (2) C E B D A (2) B C E A D (2) B A C E D (2) A E C D B (2) E D C A B (1) E D A C B (1) D E A B C (1) D A E C B (1) C A E B D (1) B A D C E (1) B A C D E (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 18 0 6 2 B -18 0 -2 0 -2 C 0 2 0 -4 2 D -6 0 4 0 6 E -2 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636129 B: 0.000000 C: 0.363871 D: 0.000000 E: 0.000000 Sum of squares = 0.537062307787 Cumulative probabilities = A: 0.636129 B: 0.636129 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 6 2 B -18 0 -2 0 -2 C 0 2 0 -4 2 D -6 0 4 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=30 B=14 E=11 C=11 so E is eliminated. Round 2 votes counts: D=40 A=30 C=16 B=14 so B is eliminated. Round 3 votes counts: D=40 A=34 C=26 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:202 C:200 E:196 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 6 2 B -18 0 -2 0 -2 C 0 2 0 -4 2 D -6 0 4 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 6 2 B -18 0 -2 0 -2 C 0 2 0 -4 2 D -6 0 4 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 6 2 B -18 0 -2 0 -2 C 0 2 0 -4 2 D -6 0 4 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1511: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (5) B A C E D (5) B A C D E (5) E D C A B (4) E C D A B (4) B C A E D (4) B A D C E (4) A C E D B (4) E C A D B (3) D E C A B (3) D B A E C (3) D A E C B (3) C E A B D (3) C B E A D (3) C A E D B (3) B D A E C (3) B D A C E (3) B C A D E (3) A C E B D (3) E D C B A (2) E D A C B (2) E C B D A (2) D B E A C (2) D A B E C (2) C A E B D (2) B E C D A (2) B C E A D (2) A D C E B (2) A C B E D (2) E C D B A (1) D E B A C (1) D E A B C (1) D B E C A (1) C E B A D (1) B E D C A (1) B D E C A (1) A D E C B (1) A D C B E (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 10 8 12 B -2 0 -10 2 -2 C -10 10 0 8 6 D -8 -2 -8 0 -8 E -12 2 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 8 12 B -2 0 -10 2 -2 C -10 10 0 8 6 D -8 -2 -8 0 -8 E -12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997245 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=21 E=18 A=16 C=12 so C is eliminated. Round 2 votes counts: B=36 E=22 D=21 A=21 so D is eliminated. Round 3 votes counts: B=42 E=32 A=26 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:216 C:207 E:196 B:194 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 12 B -2 0 -10 2 -2 C -10 10 0 8 6 D -8 -2 -8 0 -8 E -12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997245 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 12 B -2 0 -10 2 -2 C -10 10 0 8 6 D -8 -2 -8 0 -8 E -12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997245 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 12 B -2 0 -10 2 -2 C -10 10 0 8 6 D -8 -2 -8 0 -8 E -12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997245 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1512: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) B A D C E (9) B D C E A (8) E C D A B (7) E C A D B (6) E A C D B (5) D C E B A (3) B E C D A (3) B D C A E (3) B A D E C (3) A E C D B (3) A B E C D (3) A B D C E (3) E C B D A (2) E B C D A (2) D C B E A (2) D B C E A (2) B E A C D (2) B D A C E (2) B A E C D (2) A E C B D (2) A E B C D (2) A D C E B (2) A D B C E (2) A C E D B (2) E B A C D (1) D C E A B (1) D C A E B (1) D B E C A (1) C E D A B (1) C D E A B (1) B E D C A (1) B D E C A (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -12 -4 -20 B 16 0 0 -2 -6 C 12 0 0 12 -14 D 4 2 -12 0 -10 E 20 6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -12 -4 -20 B 16 0 0 -2 -6 C 12 0 0 12 -14 D 4 2 -12 0 -10 E 20 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=34 B=34 A=20 D=10 C=2 so C is eliminated. Round 2 votes counts: E=35 B=34 A=20 D=11 so D is eliminated. Round 3 votes counts: E=40 B=39 A=21 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:205 B:204 D:192 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -12 -4 -20 B 16 0 0 -2 -6 C 12 0 0 12 -14 D 4 2 -12 0 -10 E 20 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -4 -20 B 16 0 0 -2 -6 C 12 0 0 12 -14 D 4 2 -12 0 -10 E 20 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -4 -20 B 16 0 0 -2 -6 C 12 0 0 12 -14 D 4 2 -12 0 -10 E 20 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1513: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) D E C A B (8) D E A C B (5) B C A D E (5) B A C E D (5) D E C B A (4) D E A B C (4) C B D A E (4) B C D A E (4) D C E B A (3) C D E A B (3) C B A D E (3) C A B E D (3) B A D C E (3) A B E C D (3) A B C E D (3) E D A C B (2) E D A B C (2) D E B C A (2) C D B E A (2) B C A E D (2) A E B D C (2) A B E D C (2) E D C A B (1) E A D C B (1) D E B A C (1) D C B E A (1) D B E A C (1) D B C E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C A E B D (1) B D A C E (1) B A D E C (1) B A C D E (1) A E D B C (1) A E C D B (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -18 -2 12 B 8 0 -10 6 10 C 18 10 0 6 16 D 2 -6 -6 0 18 E -12 -10 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 -2 12 B 8 0 -10 6 10 C 18 10 0 6 16 D 2 -6 -6 0 18 E -12 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=28 B=22 A=14 E=6 so E is eliminated. Round 2 votes counts: D=35 C=28 B=22 A=15 so A is eliminated. Round 3 votes counts: D=38 B=32 C=30 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:225 B:207 D:204 A:192 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -18 -2 12 B 8 0 -10 6 10 C 18 10 0 6 16 D 2 -6 -6 0 18 E -12 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -2 12 B 8 0 -10 6 10 C 18 10 0 6 16 D 2 -6 -6 0 18 E -12 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -2 12 B 8 0 -10 6 10 C 18 10 0 6 16 D 2 -6 -6 0 18 E -12 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1514: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (12) A B D C E (8) D A E C B (7) B A D E C (6) E C B D A (5) C E D A B (4) C E B A D (4) B C E D A (4) B A C E D (4) A D B E C (4) D E C A B (3) D A C E B (3) B A D C E (3) A D C E B (3) A D B C E (3) E C D A B (2) C E D B A (2) B E C D A (2) B E C A D (2) A D E C B (2) A B D E C (2) E D C B A (1) E D C A B (1) E C D B A (1) E B C D A (1) D E A C B (1) D C A E B (1) D A E B C (1) C D E A B (1) C B E D A (1) C A D E B (1) B E A C D (1) B C E A D (1) B A C D E (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 -6 2 B 2 0 -12 16 -14 C -2 12 0 2 16 D 6 -16 -2 0 2 E -2 14 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -6 2 B 2 0 -12 16 -14 C -2 12 0 2 16 D 6 -16 -2 0 2 E -2 14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000024 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=24 A=24 D=16 E=11 so E is eliminated. Round 2 votes counts: C=33 B=25 A=24 D=18 so D is eliminated. Round 3 votes counts: C=39 A=36 B=25 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:214 A:198 E:197 B:196 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 2 -6 2 B 2 0 -12 16 -14 C -2 12 0 2 16 D 6 -16 -2 0 2 E -2 14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000024 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -6 2 B 2 0 -12 16 -14 C -2 12 0 2 16 D 6 -16 -2 0 2 E -2 14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000024 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -6 2 B 2 0 -12 16 -14 C -2 12 0 2 16 D 6 -16 -2 0 2 E -2 14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000024 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1515: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) E A B C D (7) B A D C E (6) A B E D C (6) D B A C E (5) C D B A E (5) E C D A B (4) E A B D C (4) D B A E C (4) C E A B D (4) B D A C E (4) E C A B D (3) D E B A C (3) C B A D E (3) E D C B A (2) E D A B C (2) C E D B A (2) C E D A B (2) B D A E C (2) B A D E C (2) A B E C D (2) A B D E C (2) E D C A B (1) E C A D B (1) E A C B D (1) D C B E A (1) D C B A E (1) D B C A E (1) C E A D B (1) C D B E A (1) C B D A E (1) C A E B D (1) C A B E D (1) A E B D C (1) A E B C D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 8 -4 0 B 8 0 6 6 -2 C -8 -6 0 6 4 D 4 -6 -6 0 6 E 0 2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.374713 C: 0.094253 D: 0.062069 E: 0.468966 Sum of squares = 0.373074384902 Cumulative probabilities = A: 0.000000 B: 0.374713 C: 0.468966 D: 0.531034 E: 1.000000 A B C D E A 0 -8 8 -4 0 B 8 0 6 6 -2 C -8 -6 0 6 4 D 4 -6 -6 0 6 E 0 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.381818 C: 0.081818 D: 0.072727 E: 0.463636 Sum of squares = 0.372727272724 Cumulative probabilities = A: 0.000000 B: 0.381818 C: 0.463636 D: 0.536364 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=25 D=15 B=14 A=14 so B is eliminated. Round 2 votes counts: C=32 E=25 A=22 D=21 so D is eliminated. Round 3 votes counts: A=37 C=35 E=28 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:209 D:199 A:198 C:198 E:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -4 0 B 8 0 6 6 -2 C -8 -6 0 6 4 D 4 -6 -6 0 6 E 0 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.381818 C: 0.081818 D: 0.072727 E: 0.463636 Sum of squares = 0.372727272724 Cumulative probabilities = A: 0.000000 B: 0.381818 C: 0.463636 D: 0.536364 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -4 0 B 8 0 6 6 -2 C -8 -6 0 6 4 D 4 -6 -6 0 6 E 0 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.381818 C: 0.081818 D: 0.072727 E: 0.463636 Sum of squares = 0.372727272724 Cumulative probabilities = A: 0.000000 B: 0.381818 C: 0.463636 D: 0.536364 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -4 0 B 8 0 6 6 -2 C -8 -6 0 6 4 D 4 -6 -6 0 6 E 0 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.381818 C: 0.081818 D: 0.072727 E: 0.463636 Sum of squares = 0.372727272724 Cumulative probabilities = A: 0.000000 B: 0.381818 C: 0.463636 D: 0.536364 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1516: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (7) C B D E A (6) D B C E A (5) A E D B C (5) C B E D A (4) C A D E B (4) B C D E A (4) A E D C B (4) E B C A D (3) E A B D C (3) E A B C D (3) C E B A D (3) C D B A E (3) A E C B D (3) A D E B C (3) E B A C D (2) E A D B C (2) D C B A E (2) D C A E B (2) D C A B E (2) D A C B E (2) D A B E C (2) C A E B D (2) B C E A D (2) E C B A D (1) E B A D C (1) E A C B D (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A E C (1) D A E B C (1) D A C E B (1) C E A B D (1) C D B E A (1) C A E D B (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B D C E A (1) B C E D A (1) A E B C D (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -18 14 -14 B 8 0 -6 6 0 C 18 6 0 16 14 D -14 -6 -16 0 -10 E 14 0 -14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 14 -14 B 8 0 -6 6 0 C 18 6 0 16 14 D -14 -6 -16 0 -10 E 14 0 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=21 A=19 E=16 B=12 so B is eliminated. Round 2 votes counts: C=39 D=22 E=20 A=19 so A is eliminated. Round 3 votes counts: C=41 E=33 D=26 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:205 B:204 A:187 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -18 14 -14 B 8 0 -6 6 0 C 18 6 0 16 14 D -14 -6 -16 0 -10 E 14 0 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 14 -14 B 8 0 -6 6 0 C 18 6 0 16 14 D -14 -6 -16 0 -10 E 14 0 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 14 -14 B 8 0 -6 6 0 C 18 6 0 16 14 D -14 -6 -16 0 -10 E 14 0 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1517: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) C E A B D (6) E C A D B (5) D B A E C (5) C B D A E (5) A B D E C (5) E A D B C (4) C D B E A (4) C A E B D (4) A B C D E (4) E D A B C (3) E A C D B (3) E A C B D (3) C A B E D (3) B D A C E (3) B C D A E (3) E D A C B (2) E C D A B (2) E C A B D (2) D B E A C (2) C B A D E (2) B D C A E (2) B A D C E (2) A B E D C (2) E D C A B (1) E C D B A (1) D B C E A (1) D B A C E (1) C E B D A (1) C E B A D (1) C E A D B (1) C D E B A (1) C A B D E (1) B D A E C (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -10 6 -4 B -10 0 -18 8 -4 C 10 18 0 26 10 D -6 -8 -26 0 -12 E 4 4 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -10 6 -4 B -10 0 -18 8 -4 C 10 18 0 26 10 D -6 -8 -26 0 -12 E 4 4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=26 A=15 B=12 D=9 so D is eliminated. Round 2 votes counts: C=38 E=26 B=21 A=15 so A is eliminated. Round 3 votes counts: C=39 B=32 E=29 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 E:205 A:201 B:188 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 6 -4 B -10 0 -18 8 -4 C 10 18 0 26 10 D -6 -8 -26 0 -12 E 4 4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 6 -4 B -10 0 -18 8 -4 C 10 18 0 26 10 D -6 -8 -26 0 -12 E 4 4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 6 -4 B -10 0 -18 8 -4 C 10 18 0 26 10 D -6 -8 -26 0 -12 E 4 4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1518: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (16) E B D C A (13) A C D B E (7) A E B C D (6) A E C B D (5) E B A D C (4) E B A C D (4) E A B C D (4) D C B A E (4) C D B A E (3) C D A B E (3) A E D C B (3) A C B D E (3) E D B C A (2) E B D A C (2) E A B D C (2) C B D A E (2) E D C B A (1) E B C D A (1) D E C B A (1) D E B C A (1) D C A E B (1) D C A B E (1) D B E C A (1) B E D C A (1) B D E C A (1) B D C E A (1) B C D E A (1) A E C D B (1) A E B D C (1) A D C B E (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -8 -12 -12 B 18 0 -8 6 -6 C 8 8 0 -14 -8 D 12 -6 14 0 -4 E 12 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -8 -12 -12 B 18 0 -8 6 -6 C 8 8 0 -14 -8 D 12 -6 14 0 -4 E 12 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=30 D=25 C=8 B=4 so B is eliminated. Round 2 votes counts: E=34 A=30 D=27 C=9 so C is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:208 B:205 C:197 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -8 -12 -12 B 18 0 -8 6 -6 C 8 8 0 -14 -8 D 12 -6 14 0 -4 E 12 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -12 -12 B 18 0 -8 6 -6 C 8 8 0 -14 -8 D 12 -6 14 0 -4 E 12 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -12 -12 B 18 0 -8 6 -6 C 8 8 0 -14 -8 D 12 -6 14 0 -4 E 12 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1519: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) E C A D B (8) A B E D C (8) C D E B A (6) B D A C E (6) A E C B D (6) D C B E A (5) C E D A B (5) E C D A B (4) D B C E A (3) C E D B A (3) C D B E A (3) A E B C D (3) A B E C D (3) E A C D B (2) D E C B A (2) D C E B A (2) C E A D B (2) B D A E C (2) B A D E C (2) B A D C E (2) B A C D E (2) A B D E C (2) D B E A C (1) C B D E A (1) C B A D E (1) B C A D E (1) A E B D C (1) A C E B D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -12 -6 6 B 4 0 -4 8 10 C 12 4 0 8 12 D 6 -8 -8 0 4 E -6 -10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -6 6 B 4 0 -4 8 10 C 12 4 0 8 12 D 6 -8 -8 0 4 E -6 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 C=21 E=14 D=13 so D is eliminated. Round 2 votes counts: B=29 C=28 A=27 E=16 so E is eliminated. Round 3 votes counts: C=42 B=29 A=29 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:209 D:197 A:192 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -6 6 B 4 0 -4 8 10 C 12 4 0 8 12 D 6 -8 -8 0 4 E -6 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -6 6 B 4 0 -4 8 10 C 12 4 0 8 12 D 6 -8 -8 0 4 E -6 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -6 6 B 4 0 -4 8 10 C 12 4 0 8 12 D 6 -8 -8 0 4 E -6 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1520: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (9) D E A B C (8) A B D C E (7) C B A E D (6) C E B A D (5) E D C B A (4) E C D B A (4) E C D A B (4) D A B E C (4) E D C A B (3) E D A C B (3) E C B A D (3) E C A B D (3) C B E A D (3) B A C D E (3) A B C D E (3) D E B A C (2) D B A C E (2) C B D E A (2) B A D C E (2) E A B C D (1) D E C B A (1) D B C A E (1) D A E B C (1) C E B D A (1) C B D A E (1) B C A D E (1) A E B D C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 -2 -14 B 0 0 -4 4 -8 C 2 4 0 0 -8 D 2 -4 0 0 -4 E 14 8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 -2 -14 B 0 0 -4 4 -8 C 2 4 0 0 -8 D 2 -4 0 0 -4 E 14 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=27 D=19 A=13 B=6 so B is eliminated. Round 2 votes counts: E=35 C=28 D=19 A=18 so A is eliminated. Round 3 votes counts: E=37 C=34 D=29 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:199 D:197 B:196 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 -2 -14 B 0 0 -4 4 -8 C 2 4 0 0 -8 D 2 -4 0 0 -4 E 14 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 -14 B 0 0 -4 4 -8 C 2 4 0 0 -8 D 2 -4 0 0 -4 E 14 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 -14 B 0 0 -4 4 -8 C 2 4 0 0 -8 D 2 -4 0 0 -4 E 14 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1521: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) E A C B D (9) E A C D B (8) D A E B C (8) B C D A E (8) E C A B D (7) A E D B C (7) C B E A D (6) C B D E A (5) D A B E C (4) C B E D A (4) D B C A E (3) D B A C E (3) A D E B C (3) C B A E D (2) A E D C B (2) A E C B D (2) E D C A B (1) D E C A B (1) D E A B C (1) D B C E A (1) C E B D A (1) C E B A D (1) B D C A E (1) B C D E A (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 28 16 16 -10 B -28 0 -16 -2 -22 C -16 16 0 8 -28 D -16 2 -8 0 -20 E 10 22 28 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 28 16 16 -10 B -28 0 -16 -2 -22 C -16 16 0 8 -28 D -16 2 -8 0 -20 E 10 22 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=21 C=19 A=16 B=10 so B is eliminated. Round 2 votes counts: E=34 C=28 D=22 A=16 so A is eliminated. Round 3 votes counts: E=46 C=28 D=26 so D is eliminated. Round 4 votes counts: E=64 C=36 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:240 A:225 C:190 D:179 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 28 16 16 -10 B -28 0 -16 -2 -22 C -16 16 0 8 -28 D -16 2 -8 0 -20 E 10 22 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 16 16 -10 B -28 0 -16 -2 -22 C -16 16 0 8 -28 D -16 2 -8 0 -20 E 10 22 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 16 16 -10 B -28 0 -16 -2 -22 C -16 16 0 8 -28 D -16 2 -8 0 -20 E 10 22 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1522: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) D A B E C (9) C E B A D (8) E C B A D (7) E C D B A (5) E D A B C (4) D E A B C (4) C B E A D (4) B C A D E (4) E D A C B (3) E C A B D (3) E A D B C (3) D A E B C (3) C B D A E (3) C B A E D (3) B A C E D (3) E A B C D (2) D C B A E (2) A D B E C (2) A B D C E (2) E D C A B (1) E C D A B (1) E A D C B (1) E A B D C (1) D E C B A (1) D E C A B (1) D E A C B (1) D C E B A (1) D B C A E (1) D B A C E (1) C E D B A (1) C D E B A (1) C D B A E (1) C B A D E (1) B A E C D (1) B A C D E (1) Total count = 100 A B C D E A 0 2 2 -8 -6 B -2 0 2 -14 -4 C -2 -2 0 -2 -6 D 8 14 2 0 -2 E 6 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 -8 -6 B -2 0 2 -14 -4 C -2 -2 0 -2 -6 D 8 14 2 0 -2 E 6 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=31 C=22 B=9 A=4 so A is eliminated. Round 2 votes counts: D=36 E=31 C=22 B=11 so B is eliminated. Round 3 votes counts: D=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:209 A:195 C:194 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 -8 -6 B -2 0 2 -14 -4 C -2 -2 0 -2 -6 D 8 14 2 0 -2 E 6 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -8 -6 B -2 0 2 -14 -4 C -2 -2 0 -2 -6 D 8 14 2 0 -2 E 6 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -8 -6 B -2 0 2 -14 -4 C -2 -2 0 -2 -6 D 8 14 2 0 -2 E 6 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1523: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (12) E D B C A (9) A D B E C (9) A C B D E (9) C B D E A (6) C B E D A (5) A B D C E (5) C A B E D (3) C A B D E (3) A E D B C (3) A D E B C (3) A C E D B (3) E C D B A (2) E C B D A (2) D B E C A (2) C A E B D (2) B D A C E (2) A D B C E (2) A C E B D (2) E D B A C (1) E D A B C (1) E B D C A (1) D E B C A (1) D B E A C (1) C E B A D (1) C B D A E (1) C B A D E (1) B D E C A (1) B D E A C (1) B D C E A (1) B A D C E (1) A E D C B (1) A E C D B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -6 2 6 B 2 0 -10 20 10 C 6 10 0 8 20 D -2 -20 -8 0 2 E -6 -10 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 2 6 B 2 0 -10 20 10 C 6 10 0 8 20 D -2 -20 -8 0 2 E -6 -10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=34 E=16 B=6 D=4 so D is eliminated. Round 2 votes counts: A=40 C=34 E=17 B=9 so B is eliminated. Round 3 votes counts: A=43 C=35 E=22 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:211 A:200 D:186 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 2 6 B 2 0 -10 20 10 C 6 10 0 8 20 D -2 -20 -8 0 2 E -6 -10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 6 B 2 0 -10 20 10 C 6 10 0 8 20 D -2 -20 -8 0 2 E -6 -10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 6 B 2 0 -10 20 10 C 6 10 0 8 20 D -2 -20 -8 0 2 E -6 -10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1524: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E B C A D (8) E C A B D (6) A D C E B (6) E B D A C (5) D B A C E (5) B E D C A (5) B E C A D (5) A C D E B (5) C B A D E (4) E C B A D (3) D A C E B (3) B E C D A (3) B C E A D (3) E A C D B (2) D E B A C (2) D A B E C (2) C A E D B (2) C A D E B (2) C A D B E (2) B D E A C (2) B D A C E (2) E D B A C (1) E B D C A (1) E A D C B (1) D E A C B (1) D E A B C (1) D A E C B (1) D A B C E (1) C E A D B (1) C A B D E (1) B E D A C (1) B D C A E (1) B C E D A (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 -8 0 4 -4 B 8 0 2 6 -2 C 0 -2 0 0 0 D -4 -6 0 0 4 E 4 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.38888888879 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 A B C D E A 0 -8 0 4 -4 B 8 0 2 6 -2 C 0 -2 0 0 0 D -4 -6 0 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 B=25 C=12 A=11 so A is eliminated. Round 2 votes counts: D=31 E=27 B=25 C=17 so C is eliminated. Round 3 votes counts: D=40 E=30 B=30 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:207 E:201 C:199 D:197 A:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 4 -4 B 8 0 2 6 -2 C 0 -2 0 0 0 D -4 -6 0 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 4 -4 B 8 0 2 6 -2 C 0 -2 0 0 0 D -4 -6 0 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 4 -4 B 8 0 2 6 -2 C 0 -2 0 0 0 D -4 -6 0 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1525: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) C D A B E (7) E B D A C (6) D C A B E (6) B E A C D (6) E B A C D (5) E B C A D (4) C D E A B (4) A D C B E (4) D A E C B (3) D A C B E (3) C A D B E (3) B A E C D (3) E C B D A (2) D C A E B (2) D A C E B (2) C E B D A (2) C B E A D (2) B E C A D (2) B A E D C (2) A D B E C (2) E D C B A (1) E D B A C (1) E D A B C (1) E C D B A (1) E B C D A (1) E A D B C (1) D E C A B (1) D E A B C (1) C D A E B (1) C B A D E (1) C A B D E (1) B E A D C (1) B C E A D (1) B C A E D (1) B A C D E (1) A E B D C (1) A D E B C (1) A C D B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 14 10 -6 B 6 0 6 8 -2 C -14 -6 0 0 -14 D -10 -8 0 0 -8 E 6 2 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 14 10 -6 B 6 0 6 8 -2 C -14 -6 0 0 -14 D -10 -8 0 0 -8 E 6 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=21 D=18 B=17 A=11 so A is eliminated. Round 2 votes counts: E=34 D=25 C=22 B=19 so B is eliminated. Round 3 votes counts: E=48 D=26 C=26 so D is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:209 A:206 D:187 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 14 10 -6 B 6 0 6 8 -2 C -14 -6 0 0 -14 D -10 -8 0 0 -8 E 6 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 10 -6 B 6 0 6 8 -2 C -14 -6 0 0 -14 D -10 -8 0 0 -8 E 6 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 10 -6 B 6 0 6 8 -2 C -14 -6 0 0 -14 D -10 -8 0 0 -8 E 6 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1526: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) B C E D A (7) E A D C B (6) C B E A D (6) D A B C E (5) E B C D A (4) E D B A C (3) E D A B C (3) D A B E C (3) C E B A D (3) C A B D E (3) B D A C E (3) A D B C E (3) E D A C B (2) E C B D A (2) E C A B D (2) E B D C A (2) D E B A C (2) D E A B C (2) D A E B C (2) C B A D E (2) B D C A E (2) B C D E A (2) B C D A E (2) A E C D B (2) A C E D B (2) A C D B E (2) E C B A D (1) E C A D B (1) E A C D B (1) D B E A C (1) C E A D B (1) C E A B D (1) C A E D B (1) B E D C A (1) B D C E A (1) A D E B C (1) A D C E B (1) A D C B E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 12 12 2 -8 B -12 0 -2 -10 -10 C -12 2 0 -6 0 D -2 10 6 0 -2 E 8 10 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.157601 D: 0.000000 E: 0.842399 Sum of squares = 0.734473903302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.157601 D: 0.157601 E: 1.000000 A B C D E A 0 12 12 2 -8 B -12 0 -2 -10 -10 C -12 2 0 -6 0 D -2 10 6 0 -2 E 8 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500000021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=23 B=18 C=17 D=15 so D is eliminated. Round 2 votes counts: A=33 E=31 B=19 C=17 so C is eliminated. Round 3 votes counts: A=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:209 D:206 C:192 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 12 2 -8 B -12 0 -2 -10 -10 C -12 2 0 -6 0 D -2 10 6 0 -2 E 8 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500000021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 2 -8 B -12 0 -2 -10 -10 C -12 2 0 -6 0 D -2 10 6 0 -2 E 8 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500000021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 2 -8 B -12 0 -2 -10 -10 C -12 2 0 -6 0 D -2 10 6 0 -2 E 8 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500000021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1527: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) E B D C A (9) C A D E B (7) A C D E B (7) A C B E D (6) D B E A C (5) C A E B D (5) D E B C A (4) C A B E D (4) B E C D A (4) A D C E B (4) A C D B E (4) D E B A C (3) D A C E B (3) C E B A D (3) A D C B E (3) D C A E B (2) C B E A D (2) B E C A D (2) E B C D A (1) E B C A D (1) D E A B C (1) D A E B C (1) D A B E C (1) C D E B A (1) C D A E B (1) C B A E D (1) B E A D C (1) B D E A C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -16 4 2 B -2 0 -6 6 -6 C 16 6 0 0 8 D -4 -6 0 0 -2 E -2 6 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.637174 D: 0.362826 E: 0.000000 Sum of squares = 0.537633570718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.637174 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 4 2 B -2 0 -6 6 -6 C 16 6 0 0 8 D -4 -6 0 0 -2 E -2 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500318 D: 0.499682 E: 0.000000 Sum of squares = 0.500000202716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500318 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 D=20 B=19 E=11 so E is eliminated. Round 2 votes counts: B=30 A=26 C=24 D=20 so D is eliminated. Round 3 votes counts: B=42 A=32 C=26 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:215 E:199 A:196 B:196 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 4 2 B -2 0 -6 6 -6 C 16 6 0 0 8 D -4 -6 0 0 -2 E -2 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500318 D: 0.499682 E: 0.000000 Sum of squares = 0.500000202716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500318 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 4 2 B -2 0 -6 6 -6 C 16 6 0 0 8 D -4 -6 0 0 -2 E -2 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500318 D: 0.499682 E: 0.000000 Sum of squares = 0.500000202716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500318 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 4 2 B -2 0 -6 6 -6 C 16 6 0 0 8 D -4 -6 0 0 -2 E -2 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500318 D: 0.499682 E: 0.000000 Sum of squares = 0.500000202716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500318 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1528: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (16) C A E B D (11) A E C D B (8) E A D B C (7) E A C D B (5) B D E C A (5) B D E A C (5) C B D A E (4) B D C A E (4) A C E D B (4) D B C A E (3) C B A D E (3) C A B D E (3) B D C E A (3) E B D A C (2) E D B A C (1) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) D B C E A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D B A E (1) C B E A D (1) C B A E D (1) C A E D B (1) C A D B E (1) C A B E D (1) B E D C A (1) B E C D A (1) Total count = 100 A B C D E A 0 -10 8 0 -6 B 10 0 6 -4 12 C -8 -6 0 -6 -12 D 0 4 6 0 4 E 6 -12 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.211065 B: 0.000000 C: 0.000000 D: 0.788935 E: 0.000000 Sum of squares = 0.666967297197 Cumulative probabilities = A: 0.211065 B: 0.211065 C: 0.211065 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 0 -6 B 10 0 6 -4 12 C -8 -6 0 -6 -12 D 0 4 6 0 4 E 6 -12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836737034 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 B=19 E=18 A=12 so A is eliminated. Round 2 votes counts: C=32 E=26 D=23 B=19 so B is eliminated. Round 3 votes counts: D=40 C=32 E=28 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:207 E:201 A:196 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 0 -6 B 10 0 6 -4 12 C -8 -6 0 -6 -12 D 0 4 6 0 4 E 6 -12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836737034 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 0 -6 B 10 0 6 -4 12 C -8 -6 0 -6 -12 D 0 4 6 0 4 E 6 -12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836737034 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 0 -6 B 10 0 6 -4 12 C -8 -6 0 -6 -12 D 0 4 6 0 4 E 6 -12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836737034 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1529: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (17) A C B E D (12) C A B E D (7) D E B A C (6) A C D B E (5) E B D C A (4) D E C A B (4) A C B D E (4) E B C A D (3) D C E A B (3) D C A E B (3) D B E A C (3) D A C E B (3) B E C A D (3) B E A C D (3) E D B C A (2) C A E B D (2) C A D E B (2) E C B D A (1) D E C B A (1) D E A B C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C B E (1) C E A B D (1) C D E A B (1) C A E D B (1) B E D A C (1) B D E A C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -10 -10 -10 B -6 0 -4 -12 -14 C 10 4 0 -6 -4 D 10 12 6 0 16 E 10 14 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -10 -10 B -6 0 -4 -12 -14 C 10 4 0 -6 -4 D 10 12 6 0 16 E 10 14 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 A=22 C=14 E=10 B=9 so B is eliminated. Round 2 votes counts: D=46 A=23 E=17 C=14 so C is eliminated. Round 3 votes counts: D=47 A=35 E=18 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:206 C:202 A:188 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -10 -10 -10 B -6 0 -4 -12 -14 C 10 4 0 -6 -4 D 10 12 6 0 16 E 10 14 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -10 -10 B -6 0 -4 -12 -14 C 10 4 0 -6 -4 D 10 12 6 0 16 E 10 14 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -10 -10 B -6 0 -4 -12 -14 C 10 4 0 -6 -4 D 10 12 6 0 16 E 10 14 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1530: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) A D B C E (6) D A B E C (5) B E C D A (5) A D C B E (5) A B D C E (5) C E A B D (4) B C E A D (4) E C B D A (3) C E A D B (3) B D E C A (3) B D A E C (3) A D C E B (3) A C E D B (3) A C E B D (3) A C D E B (3) D E C B A (2) D E B C A (2) D B A E C (2) C A E D B (2) B E D C A (2) B A D E C (2) A D B E C (2) E D C B A (1) E D B C A (1) E C D B A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E B A C (1) D B E C A (1) D B E A C (1) D A E C B (1) D A E B C (1) C E D A B (1) C E B D A (1) C B E A D (1) C A E B D (1) B D E A C (1) B A C E D (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 4 18 4 B -6 0 4 4 6 C -4 -4 0 -4 14 D -18 -4 4 0 4 E -4 -6 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 18 4 B -6 0 4 4 6 C -4 -4 0 -4 14 D -18 -4 4 0 4 E -4 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=21 C=20 D=17 E=8 so E is eliminated. Round 2 votes counts: A=34 C=24 B=23 D=19 so D is eliminated. Round 3 votes counts: A=41 B=31 C=28 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:204 C:201 D:193 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 18 4 B -6 0 4 4 6 C -4 -4 0 -4 14 D -18 -4 4 0 4 E -4 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 18 4 B -6 0 4 4 6 C -4 -4 0 -4 14 D -18 -4 4 0 4 E -4 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 18 4 B -6 0 4 4 6 C -4 -4 0 -4 14 D -18 -4 4 0 4 E -4 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1531: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (9) D B A C E (5) A E B C D (5) A B D C E (5) E C B A D (4) E C A B D (4) E A C B D (4) D E C A B (4) D C B E A (4) D A B C E (4) D B C A E (3) E D C A B (2) E C D B A (2) E C D A B (2) C B E A D (2) B D A C E (2) B C D A E (2) B C A E D (2) B A C D E (2) A B D E C (2) A B C E D (2) E A D C B (1) E A C D B (1) D E C B A (1) D E A C B (1) D B C E A (1) D A E C B (1) D A E B C (1) D A B E C (1) C E B D A (1) C D B E A (1) B C E A D (1) B C D E A (1) B C A D E (1) B A D C E (1) A E D B C (1) A E C B D (1) A E B D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 10 6 4 8 B -10 0 10 8 8 C -6 -10 0 -4 0 D -4 -8 4 0 6 E -8 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 4 8 B -10 0 10 8 8 C -6 -10 0 -4 0 D -4 -8 4 0 6 E -8 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=29 E=20 B=12 C=4 so C is eliminated. Round 2 votes counts: D=36 A=29 E=21 B=14 so B is eliminated. Round 3 votes counts: D=41 A=35 E=24 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:208 D:199 C:190 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 4 8 B -10 0 10 8 8 C -6 -10 0 -4 0 D -4 -8 4 0 6 E -8 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 8 B -10 0 10 8 8 C -6 -10 0 -4 0 D -4 -8 4 0 6 E -8 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 8 B -10 0 10 8 8 C -6 -10 0 -4 0 D -4 -8 4 0 6 E -8 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1532: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (14) A D C B E (11) D A C E B (10) B E C A D (6) E B C A D (5) B C E A D (5) B C A D E (5) E B D C A (4) E B D A C (4) D A E C B (4) A C B D E (4) E D B C A (3) B E C D A (3) B C A E D (3) E D A C B (2) D E A C B (2) D C A E B (2) D A C B E (2) B E A C D (2) A D C E B (2) E D B A C (1) E D A B C (1) D C A B E (1) C D A B E (1) C A D B E (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -6 -8 -4 B 10 0 12 10 -10 C 6 -12 0 0 -4 D 8 -10 0 0 -6 E 4 10 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 -8 -4 B 10 0 12 10 -10 C 6 -12 0 0 -4 D 8 -10 0 0 -6 E 4 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=24 D=21 A=19 C=2 so C is eliminated. Round 2 votes counts: E=34 B=24 D=22 A=20 so A is eliminated. Round 3 votes counts: D=38 E=34 B=28 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:211 D:196 C:195 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 -8 -4 B 10 0 12 10 -10 C 6 -12 0 0 -4 D 8 -10 0 0 -6 E 4 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -8 -4 B 10 0 12 10 -10 C 6 -12 0 0 -4 D 8 -10 0 0 -6 E 4 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -8 -4 B 10 0 12 10 -10 C 6 -12 0 0 -4 D 8 -10 0 0 -6 E 4 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1533: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) B D C A E (7) A D B E C (7) E C A D B (6) C B E D A (6) B D A C E (6) B C D E A (6) D B A E C (5) C E B A D (5) E A D C B (4) E A C D B (4) A E D B C (4) C E A B D (3) A D E B C (3) E C B A D (2) E C A B D (2) D A B E C (2) B C D A E (2) B A D C E (2) E D B C A (1) E C B D A (1) D E B A C (1) D E A B C (1) D B A C E (1) C B E A D (1) C B D E A (1) B D C E A (1) B C A D E (1) A E D C B (1) A E C D B (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -10 -2 -10 B 18 0 4 16 2 C 10 -4 0 6 10 D 2 -16 -6 0 -2 E 10 -2 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997404 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -2 -10 B 18 0 4 16 2 C 10 -4 0 6 10 D 2 -16 -6 0 -2 E 10 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 E=20 A=19 D=10 so D is eliminated. Round 2 votes counts: B=31 C=26 E=22 A=21 so A is eliminated. Round 3 votes counts: B=42 E=31 C=27 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:211 E:200 D:189 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 -2 -10 B 18 0 4 16 2 C 10 -4 0 6 10 D 2 -16 -6 0 -2 E 10 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -2 -10 B 18 0 4 16 2 C 10 -4 0 6 10 D 2 -16 -6 0 -2 E 10 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -2 -10 B 18 0 4 16 2 C 10 -4 0 6 10 D 2 -16 -6 0 -2 E 10 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1534: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) C D B A E (8) A E D B C (8) E A C B D (6) E A B D C (6) B D C E A (6) A E C D B (6) D B C A E (5) D B A C E (5) C B D A E (4) E C A B D (3) E A D B C (3) B D C A E (3) A D B E C (3) E C B D A (2) E A C D B (2) C B E D A (2) A D B C E (2) E B D A C (1) E B C D A (1) E B A D C (1) E A B C D (1) D C B A E (1) D A B C E (1) C E B D A (1) C D B E A (1) C A E D B (1) B D A E C (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 0 -8 4 B 8 0 -2 0 10 C 0 2 0 4 8 D 8 0 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.109751 B: 0.000000 C: 0.890249 D: 0.000000 E: 0.000000 Sum of squares = 0.804588805302 Cumulative probabilities = A: 0.109751 B: 0.109751 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -8 4 B 8 0 -2 0 10 C 0 2 0 4 8 D 8 0 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000351 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 A=23 D=12 B=12 so D is eliminated. Round 2 votes counts: C=28 E=26 A=24 B=22 so B is eliminated. Round 3 votes counts: C=44 A=30 E=26 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:208 C:207 D:206 A:194 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 0 -8 4 B 8 0 -2 0 10 C 0 2 0 4 8 D 8 0 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000351 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -8 4 B 8 0 -2 0 10 C 0 2 0 4 8 D 8 0 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000351 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -8 4 B 8 0 -2 0 10 C 0 2 0 4 8 D 8 0 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000351 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1535: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) C E D B A (8) C E A B D (8) D E C B A (7) B A D C E (5) A B D C E (5) A B C E D (5) C A B E D (4) E D C B A (3) E C D A B (3) D E B C A (3) D B E A C (3) C E D A B (3) B D A E C (3) B D A C E (3) A B C D E (3) D E B A C (2) C E A D B (2) C D B A E (2) B A D E C (2) A C B E D (2) A B D E C (2) E C D B A (1) E C A D B (1) E A D B C (1) D C E B A (1) D B A C E (1) C D E B A (1) C B D A E (1) C A B D E (1) B A C D E (1) A C E B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 2 -10 6 B 14 0 2 -4 10 C -2 -2 0 -4 14 D 10 4 4 0 12 E -6 -10 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999372 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -10 6 B 14 0 2 -4 10 C -2 -2 0 -4 14 D 10 4 4 0 12 E -6 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=27 A=20 B=14 E=9 so E is eliminated. Round 2 votes counts: C=35 D=30 A=21 B=14 so B is eliminated. Round 3 votes counts: D=36 C=35 A=29 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:211 C:203 A:192 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 2 -10 6 B 14 0 2 -4 10 C -2 -2 0 -4 14 D 10 4 4 0 12 E -6 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -10 6 B 14 0 2 -4 10 C -2 -2 0 -4 14 D 10 4 4 0 12 E -6 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -10 6 B 14 0 2 -4 10 C -2 -2 0 -4 14 D 10 4 4 0 12 E -6 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1536: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C B A D E (7) B C D A E (7) E A D C B (6) E D B A C (5) C B A E D (5) B E C D A (5) B E D C A (4) E C B A D (3) D A E B C (3) C B E A D (3) B C E D A (3) B C E A D (3) B C D E A (3) A D E C B (3) A C D B E (3) E B D A C (2) D E A B C (2) D A B C E (2) C B D A E (2) C A B E D (2) A E C D B (2) A D C E B (2) E D A C B (1) E B C D A (1) E A C D B (1) E A C B D (1) D E B A C (1) D B E A C (1) D B A E C (1) D B A C E (1) D A E C B (1) D A B E C (1) C A E B D (1) C A D B E (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 0 -10 -8 B 14 0 8 4 10 C 0 -8 0 8 -6 D 10 -4 -8 0 -14 E 8 -10 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -10 -8 B 14 0 8 4 10 C 0 -8 0 8 -6 D 10 -4 -8 0 -14 E 8 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=25 C=21 D=13 A=12 so A is eliminated. Round 2 votes counts: E=31 C=25 B=25 D=19 so D is eliminated. Round 3 votes counts: E=41 B=31 C=28 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:209 C:197 D:192 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -10 -8 B 14 0 8 4 10 C 0 -8 0 8 -6 D 10 -4 -8 0 -14 E 8 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -10 -8 B 14 0 8 4 10 C 0 -8 0 8 -6 D 10 -4 -8 0 -14 E 8 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -10 -8 B 14 0 8 4 10 C 0 -8 0 8 -6 D 10 -4 -8 0 -14 E 8 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1537: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (19) E C D B A (18) B A D C E (13) E C D A B (11) B C D E A (6) D C E A B (5) B A E C D (5) D C E B A (3) D C A E B (2) C D E B A (2) A D C E B (2) A B E C D (2) E C B D A (1) E B C D A (1) D A C E B (1) C E D B A (1) B E C D A (1) B C E D A (1) B A E D C (1) B A C D E (1) A E D C B (1) A E C D B (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -4 -6 0 B 8 0 2 4 -2 C 4 -2 0 4 12 D 6 -4 -4 0 10 E 0 2 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000063 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -8 -4 -6 0 B 8 0 2 4 -2 C 4 -2 0 4 12 D 6 -4 -4 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999993 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=28 A=27 D=11 C=3 so C is eliminated. Round 2 votes counts: E=32 B=28 A=27 D=13 so D is eliminated. Round 3 votes counts: E=42 A=30 B=28 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:209 B:206 D:204 A:191 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -6 0 B 8 0 2 4 -2 C 4 -2 0 4 12 D 6 -4 -4 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999993 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -6 0 B 8 0 2 4 -2 C 4 -2 0 4 12 D 6 -4 -4 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999993 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -6 0 B 8 0 2 4 -2 C 4 -2 0 4 12 D 6 -4 -4 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999993 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1538: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) E D C A B (7) D C B E A (6) D E C B A (5) B A C D E (5) E A C B D (4) E A B C D (4) D C E B A (4) A B C E D (4) E C A B D (3) D B C A E (3) C D B E A (3) C B A D E (3) B C D A E (3) A E B C D (3) A B E D C (3) E D C B A (2) E D A C B (2) E A D C B (2) E A D B C (2) C D E B A (2) C B D A E (2) B C A D E (2) A E B D C (2) A B C D E (2) E C D A B (1) E A C D B (1) D E C A B (1) D E B C A (1) D C B A E (1) D B A E C (1) D B A C E (1) C B E D A (1) C B A E D (1) B D C A E (1) B D A C E (1) B A D C E (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -4 4 -2 B 0 0 -2 10 8 C 4 2 0 6 -4 D -4 -10 -6 0 -2 E 2 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428582 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 0 -4 4 -2 B 0 0 -2 10 8 C 4 2 0 6 -4 D -4 -10 -6 0 -2 E 2 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 A=23 B=14 C=12 so C is eliminated. Round 2 votes counts: E=28 D=28 A=23 B=21 so B is eliminated. Round 3 votes counts: A=36 D=35 E=29 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:208 C:204 E:200 A:199 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 4 -2 B 0 0 -2 10 8 C 4 2 0 6 -4 D -4 -10 -6 0 -2 E 2 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 -2 B 0 0 -2 10 8 C 4 2 0 6 -4 D -4 -10 -6 0 -2 E 2 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 -2 B 0 0 -2 10 8 C 4 2 0 6 -4 D -4 -10 -6 0 -2 E 2 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1539: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (14) E A C B D (8) D B C E A (6) A E C B D (6) B D C E A (5) B C E D A (5) A E D C B (5) D A C B E (4) B C D E A (4) E A D B C (3) E A B C D (3) D C B A E (3) B C E A D (3) E D A B C (2) E C B A D (2) E B C A D (2) E A D C B (2) D E A B C (2) D B A C E (2) D A E B C (2) C B E A D (2) C A B D E (2) B C D A E (2) A C E B D (2) E B C D A (1) C E B A D (1) C B D A E (1) C B A E D (1) C B A D E (1) B E C D A (1) A D E C B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -12 -8 -4 B 12 0 14 6 16 C 12 -14 0 -4 20 D 8 -6 4 0 0 E 4 -16 -20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -8 -4 B 12 0 14 6 16 C 12 -14 0 -4 20 D 8 -6 4 0 0 E 4 -16 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=23 B=20 A=16 C=8 so C is eliminated. Round 2 votes counts: D=33 B=25 E=24 A=18 so A is eliminated. Round 3 votes counts: E=37 D=35 B=28 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:224 C:207 D:203 E:184 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -8 -4 B 12 0 14 6 16 C 12 -14 0 -4 20 D 8 -6 4 0 0 E 4 -16 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -8 -4 B 12 0 14 6 16 C 12 -14 0 -4 20 D 8 -6 4 0 0 E 4 -16 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -8 -4 B 12 0 14 6 16 C 12 -14 0 -4 20 D 8 -6 4 0 0 E 4 -16 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1540: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) E B A C D (8) A B D C E (8) E C D B A (7) C D E A B (7) B A E D C (7) B A E C D (5) E B C A D (4) B A D E C (4) E C D A B (3) D A C B E (3) D A B C E (3) A D B C E (3) E D C B A (2) E C B D A (2) D E C A B (2) D C E A B (2) D C A E B (2) D A B E C (2) C E D A B (2) C D A B E (2) B E A C D (2) E D B C A (1) E C B A D (1) E B D A C (1) C D A E B (1) B E A D C (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -2 -10 10 B -8 0 2 -10 10 C 2 -2 0 -10 -6 D 10 10 10 0 8 E -10 -10 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -10 10 B -8 0 2 -10 10 C 2 -2 0 -10 -6 D 10 10 10 0 8 E -10 -10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=27 B=20 C=12 A=12 so C is eliminated. Round 2 votes counts: D=37 E=31 B=20 A=12 so A is eliminated. Round 3 votes counts: D=40 E=31 B=29 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:203 B:197 C:192 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -2 -10 10 B -8 0 2 -10 10 C 2 -2 0 -10 -6 D 10 10 10 0 8 E -10 -10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -10 10 B -8 0 2 -10 10 C 2 -2 0 -10 -6 D 10 10 10 0 8 E -10 -10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -10 10 B -8 0 2 -10 10 C 2 -2 0 -10 -6 D 10 10 10 0 8 E -10 -10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1541: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) E A B D C (6) E A C B D (5) E C A B D (4) B A D E C (4) E A B C D (3) D C B E A (3) D C B A E (3) D B A C E (3) C E A D B (3) C D E A B (3) C D A E B (3) B D A E C (3) B D A C E (3) A B E D C (3) A B D E C (3) A B D C E (3) E C D B A (2) E C B A D (2) E B D C A (2) D B C A E (2) C E D B A (2) C E D A B (2) C D B E A (2) B E A D C (2) A E B D C (2) E D C B A (1) E C D A B (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) D C A B E (1) D B C E A (1) D A C B E (1) C A E D B (1) C A D E B (1) C A D B E (1) B D E C A (1) B D E A C (1) A D C B E (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 18 -4 -2 0 B -18 0 -8 4 2 C 4 8 0 0 2 D 2 -4 0 0 6 E 0 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.691595 D: 0.308405 E: 0.000000 Sum of squares = 0.573417456477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.691595 D: 1.000000 E: 1.000000 A B C D E A 0 18 -4 -2 0 B -18 0 -8 4 2 C 4 8 0 0 2 D 2 -4 0 0 6 E 0 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=27 A=15 D=14 B=14 so D is eliminated. Round 2 votes counts: C=34 E=30 B=20 A=16 so A is eliminated. Round 3 votes counts: C=39 E=32 B=29 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:207 A:206 D:202 E:195 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -4 -2 0 B -18 0 -8 4 2 C 4 8 0 0 2 D 2 -4 0 0 6 E 0 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 -2 0 B -18 0 -8 4 2 C 4 8 0 0 2 D 2 -4 0 0 6 E 0 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 -2 0 B -18 0 -8 4 2 C 4 8 0 0 2 D 2 -4 0 0 6 E 0 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1542: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (13) A B E C D (10) A B E D C (9) A D C E B (8) C D E B A (7) B E C D A (7) D C A E B (6) B E A C D (6) D A C E B (3) B E C A D (3) B A E C D (3) A D C B E (3) A C D B E (3) E B C D A (2) D E C B A (2) D C E A B (2) A D E B C (2) A B C E D (2) E D C B A (1) C E D B A (1) C D A E B (1) C B E D A (1) B E D C A (1) B C E A D (1) A E B D C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 6 6 B -4 0 -2 -6 2 C -4 2 0 -4 4 D -6 6 4 0 4 E -6 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 6 6 B -4 0 -2 -6 2 C -4 2 0 -4 4 D -6 6 4 0 4 E -6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 D=26 B=21 C=10 E=3 so E is eliminated. Round 2 votes counts: A=40 D=27 B=23 C=10 so C is eliminated. Round 3 votes counts: A=40 D=36 B=24 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:204 C:199 B:195 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 6 6 B -4 0 -2 -6 2 C -4 2 0 -4 4 D -6 6 4 0 4 E -6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 6 6 B -4 0 -2 -6 2 C -4 2 0 -4 4 D -6 6 4 0 4 E -6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 6 6 B -4 0 -2 -6 2 C -4 2 0 -4 4 D -6 6 4 0 4 E -6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1543: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) A D C B E (9) B E C A D (8) D E C B A (5) D E B A C (5) D A C E B (5) B E A C D (4) D B E A C (3) D A C B E (3) D A B E C (3) C B E A D (3) A D B C E (3) A C B D E (3) E C B D A (2) E C B A D (2) D E B C A (2) D C A E B (2) C E D B A (2) C D E A B (2) C A D E B (2) C A B E D (2) B E D A C (2) B E A D C (2) E D C B A (1) E B D C A (1) E B C D A (1) D E A B C (1) D C E A B (1) D A E B C (1) D A B C E (1) C D E B A (1) C D A E B (1) C B A E D (1) B A D E C (1) B A C E D (1) A C D E B (1) A C D B E (1) A C B E D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 4 10 -14 B 12 0 -18 -10 6 C -4 18 0 -6 10 D -10 10 6 0 16 E 14 -6 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.085714 C: 0.200000 D: 0.192857 E: 0.121429 Sum of squares = 0.259285714291 Cumulative probabilities = A: 0.400000 B: 0.485714 C: 0.685714 D: 0.878571 E: 1.000000 A B C D E A 0 -12 4 10 -14 B 12 0 -18 -10 6 C -4 18 0 -6 10 D -10 10 6 0 16 E 14 -6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.085714 C: 0.200000 D: 0.192857 E: 0.121429 Sum of squares = 0.259285714286 Cumulative probabilities = A: 0.400000 B: 0.485714 C: 0.685714 D: 0.878571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=23 A=20 B=18 E=7 so E is eliminated. Round 2 votes counts: D=33 C=27 B=20 A=20 so B is eliminated. Round 3 votes counts: D=36 C=36 A=28 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:209 B:195 A:194 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 4 10 -14 B 12 0 -18 -10 6 C -4 18 0 -6 10 D -10 10 6 0 16 E 14 -6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.085714 C: 0.200000 D: 0.192857 E: 0.121429 Sum of squares = 0.259285714286 Cumulative probabilities = A: 0.400000 B: 0.485714 C: 0.685714 D: 0.878571 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 10 -14 B 12 0 -18 -10 6 C -4 18 0 -6 10 D -10 10 6 0 16 E 14 -6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.085714 C: 0.200000 D: 0.192857 E: 0.121429 Sum of squares = 0.259285714286 Cumulative probabilities = A: 0.400000 B: 0.485714 C: 0.685714 D: 0.878571 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 10 -14 B 12 0 -18 -10 6 C -4 18 0 -6 10 D -10 10 6 0 16 E 14 -6 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.085714 C: 0.200000 D: 0.192857 E: 0.121429 Sum of squares = 0.259285714286 Cumulative probabilities = A: 0.400000 B: 0.485714 C: 0.685714 D: 0.878571 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1544: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) D A B E C (8) B D A C E (8) E C A D B (6) D B A C E (6) C E B A D (5) A D B E C (5) D E A C B (4) C B E D A (4) C B E A D (4) B C A E D (4) E A D C B (3) B C A D E (3) A D E C B (3) E D A C B (2) E C A B D (2) D A E B C (2) C E B D A (2) C B D E A (2) B C D E A (2) B C D A E (2) B A D C E (2) A D E B C (2) A B D E C (2) E D C A B (1) E C D B A (1) E A C D B (1) D E C B A (1) D C B E A (1) D B A E C (1) C E A B D (1) C D E B A (1) Total count = 100 A B C D E A 0 2 -2 -14 -4 B -2 0 -6 -14 8 C 2 6 0 -2 -6 D 14 14 2 0 10 E 4 -8 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -14 -4 B -2 0 -6 -14 8 C 2 6 0 -2 -6 D 14 14 2 0 10 E 4 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 B=21 C=19 A=12 so A is eliminated. Round 2 votes counts: D=33 E=25 B=23 C=19 so C is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:200 E:196 B:193 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -14 -4 B -2 0 -6 -14 8 C 2 6 0 -2 -6 D 14 14 2 0 10 E 4 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -14 -4 B -2 0 -6 -14 8 C 2 6 0 -2 -6 D 14 14 2 0 10 E 4 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -14 -4 B -2 0 -6 -14 8 C 2 6 0 -2 -6 D 14 14 2 0 10 E 4 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1545: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (8) E D B C A (7) E C B A D (7) D B A E C (7) E B D C A (6) D E B A C (5) D A C B E (5) A C B D E (5) E C A B D (4) D B A C E (4) A B C D E (4) E D C B A (3) D B E A C (3) D A C E B (3) C A E B D (3) C A B E D (3) A B D C E (3) D E A C B (2) C B E A D (2) B D A C E (2) A D C E B (2) A C D B E (2) D A E B C (1) D A B E C (1) C E A D B (1) C E A B D (1) B E D C A (1) B E C A D (1) B C A E D (1) B A C D E (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 0 20 -16 14 B 0 0 12 -12 8 C -20 -12 0 -28 4 D 16 12 28 0 20 E -14 -8 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 20 -16 14 B 0 0 12 -12 8 C -20 -12 0 -28 4 D 16 12 28 0 20 E -14 -8 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=27 A=18 C=10 B=6 so B is eliminated. Round 2 votes counts: D=41 E=29 A=19 C=11 so C is eliminated. Round 3 votes counts: D=41 E=33 A=26 so A is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:238 A:209 B:204 E:177 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 20 -16 14 B 0 0 12 -12 8 C -20 -12 0 -28 4 D 16 12 28 0 20 E -14 -8 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 20 -16 14 B 0 0 12 -12 8 C -20 -12 0 -28 4 D 16 12 28 0 20 E -14 -8 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 20 -16 14 B 0 0 12 -12 8 C -20 -12 0 -28 4 D 16 12 28 0 20 E -14 -8 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1546: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) E B A C D (5) C D A B E (5) C A D E B (5) B D A C E (5) E C D B A (4) E B A D C (4) C A E D B (4) B E D A C (4) A C D B E (4) E A C B D (3) D A C B E (3) C E D A B (3) C A D B E (3) B D E A C (3) E C D A B (2) E C B D A (2) E B D A C (2) D C B A E (2) D B C A E (2) D B A C E (2) D A B C E (2) C E A D B (2) B D E C A (2) A D C B E (2) E D C B A (1) E C A D B (1) E B C A D (1) E A C D B (1) D C E B A (1) D B C E A (1) C D E A B (1) B E D C A (1) B D A E C (1) A E C D B (1) A E B C D (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -8 -20 -10 B 8 0 -4 -6 -12 C 8 4 0 2 -2 D 20 6 -2 0 -10 E 10 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -8 -20 -10 B 8 0 -4 -6 -12 C 8 4 0 2 -2 D 20 6 -2 0 -10 E 10 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=23 B=16 D=13 A=11 so A is eliminated. Round 2 votes counts: E=39 C=29 B=17 D=15 so D is eliminated. Round 3 votes counts: E=39 C=37 B=24 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:207 C:206 B:193 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -20 -10 B 8 0 -4 -6 -12 C 8 4 0 2 -2 D 20 6 -2 0 -10 E 10 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -20 -10 B 8 0 -4 -6 -12 C 8 4 0 2 -2 D 20 6 -2 0 -10 E 10 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -20 -10 B 8 0 -4 -6 -12 C 8 4 0 2 -2 D 20 6 -2 0 -10 E 10 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1547: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) A C B E D (10) D E B C A (9) D A E C B (6) C B E A D (6) C B A E D (5) A D C B E (4) D E B A C (3) D E A B C (3) B C E A D (3) A D C E B (3) E D B C A (2) E B D C A (2) E B C D A (2) E B A C D (2) E A B C D (2) D C A B E (2) D B C E A (2) C B A D E (2) C A B E D (2) A E C B D (2) A C E B D (2) A C D B E (2) E D B A C (1) E B D A C (1) E B C A D (1) E A D B C (1) D A C E B (1) D A C B E (1) B E D C A (1) A E D B C (1) A E C D B (1) A E B C D (1) A D E C B (1) A D E B C (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 2 8 -2 B 4 0 -4 -10 2 C -2 4 0 -10 -6 D -8 10 10 0 4 E 2 -2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.363636 C: 0.000000 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826381 Cumulative probabilities = A: 0.454545 B: 0.818182 C: 0.818182 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 8 -2 B 4 0 -4 -10 2 C -2 4 0 -10 -6 D -8 10 10 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.363636 C: 0.000000 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826443 Cumulative probabilities = A: 0.454545 B: 0.818182 C: 0.818182 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=30 C=15 E=14 B=4 so B is eliminated. Round 2 votes counts: D=37 A=30 C=18 E=15 so E is eliminated. Round 3 votes counts: D=44 A=35 C=21 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:208 A:202 E:201 B:196 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 2 8 -2 B 4 0 -4 -10 2 C -2 4 0 -10 -6 D -8 10 10 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.363636 C: 0.000000 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826443 Cumulative probabilities = A: 0.454545 B: 0.818182 C: 0.818182 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 -2 B 4 0 -4 -10 2 C -2 4 0 -10 -6 D -8 10 10 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.363636 C: 0.000000 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826443 Cumulative probabilities = A: 0.454545 B: 0.818182 C: 0.818182 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 -2 B 4 0 -4 -10 2 C -2 4 0 -10 -6 D -8 10 10 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.363636 C: 0.000000 D: 0.181818 E: 0.000000 Sum of squares = 0.371900826443 Cumulative probabilities = A: 0.454545 B: 0.818182 C: 0.818182 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1548: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) D C A E B (7) E A B D C (6) A D C B E (6) E B C D A (5) A E B D C (5) E B A C D (4) D C A B E (4) C D A B E (4) A E D C B (4) A C D B E (4) E B A D C (3) C D B E A (3) C D B A E (3) C B D E A (3) B A E C D (3) E D C B A (2) B C E D A (2) B C D E A (2) B C D A E (2) A D E C B (2) A D C E B (2) E D C A B (1) E B D C A (1) E B C A D (1) D C E A B (1) D A C E B (1) C D E B A (1) C B D A E (1) C B A D E (1) B E C D A (1) B A C D E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 10 12 6 B 0 0 0 10 8 C -10 0 0 8 0 D -12 -10 -8 0 0 E -6 -8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.439909 B: 0.560091 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.507221762303 Cumulative probabilities = A: 0.439909 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 12 6 B 0 0 0 10 8 C -10 0 0 8 0 D -12 -10 -8 0 0 E -6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 B=22 C=16 D=13 so D is eliminated. Round 2 votes counts: C=28 A=27 E=23 B=22 so B is eliminated. Round 3 votes counts: E=35 C=34 A=31 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 B:209 C:199 E:193 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 12 6 B 0 0 0 10 8 C -10 0 0 8 0 D -12 -10 -8 0 0 E -6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 12 6 B 0 0 0 10 8 C -10 0 0 8 0 D -12 -10 -8 0 0 E -6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 12 6 B 0 0 0 10 8 C -10 0 0 8 0 D -12 -10 -8 0 0 E -6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1549: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) E B A D C (8) D C E B A (6) D C A E B (6) C A D B E (6) B A E C D (6) D C A B E (5) E D B C A (4) E B D A C (4) E B A C D (4) D C E A B (4) A B E C D (4) D E B C A (3) B E A C D (3) A C B E D (3) A B C E D (3) E B D C A (2) D E B A C (2) C A B E D (2) A C D B E (2) A C B D E (2) E D B A C (1) D E C B A (1) D A E B C (1) D A C E B (1) C E D B A (1) C D E B A (1) C B E A D (1) B E A D C (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -6 -6 8 B -4 0 -6 -12 2 C 6 6 0 -2 10 D 6 12 2 0 4 E -8 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -6 8 B -4 0 -6 -12 2 C 6 6 0 -2 10 D 6 12 2 0 4 E -8 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=23 C=22 A=16 B=10 so B is eliminated. Round 2 votes counts: D=29 E=27 C=22 A=22 so C is eliminated. Round 3 votes counts: D=41 A=30 E=29 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:210 A:200 B:190 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 -6 8 B -4 0 -6 -12 2 C 6 6 0 -2 10 D 6 12 2 0 4 E -8 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 8 B -4 0 -6 -12 2 C 6 6 0 -2 10 D 6 12 2 0 4 E -8 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 8 B -4 0 -6 -12 2 C 6 6 0 -2 10 D 6 12 2 0 4 E -8 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1550: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) B E D C A (8) D E C B A (7) C A D E B (7) A C D E B (7) B E D A C (5) D E C A B (4) D C E A B (4) B E A C D (3) B A E C D (3) A D C E B (3) A B D E C (3) A B C D E (3) E D B C A (2) E B D C A (2) D E B C A (2) C E D B A (2) C D E A B (2) C A B E D (2) B E A D C (2) B C E A D (2) A C D B E (2) A B E D C (2) E C D B A (1) E C B D A (1) D E A C B (1) D A C E B (1) C D A E B (1) C A E B D (1) B E C D A (1) B E C A D (1) B C E D A (1) B A E D C (1) B A C E D (1) A D E C B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -2 10 -2 B -10 0 2 6 2 C 2 -2 0 4 0 D -10 -6 -4 0 -2 E 2 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.285881 D: 0.000000 E: 0.428405 Sum of squares = 0.306074933968 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.571595 D: 0.571595 E: 1.000000 A B C D E A 0 10 -2 10 -2 B -10 0 2 6 2 C 2 -2 0 4 0 D -10 -6 -4 0 -2 E 2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.357143 D: 0.000000 E: 0.357143 Sum of squares = 0.295918367349 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.642857 D: 0.642857 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=28 D=19 C=15 E=6 so E is eliminated. Round 2 votes counts: A=32 B=30 D=21 C=17 so C is eliminated. Round 3 votes counts: A=42 B=31 D=27 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:202 E:201 B:200 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 10 -2 B -10 0 2 6 2 C 2 -2 0 4 0 D -10 -6 -4 0 -2 E 2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.357143 D: 0.000000 E: 0.357143 Sum of squares = 0.295918367349 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.642857 D: 0.642857 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 10 -2 B -10 0 2 6 2 C 2 -2 0 4 0 D -10 -6 -4 0 -2 E 2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.357143 D: 0.000000 E: 0.357143 Sum of squares = 0.295918367349 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.642857 D: 0.642857 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 10 -2 B -10 0 2 6 2 C 2 -2 0 4 0 D -10 -6 -4 0 -2 E 2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.357143 D: 0.000000 E: 0.357143 Sum of squares = 0.295918367349 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.642857 D: 0.642857 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1551: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) A D E C B (9) D C A E B (5) A E B D C (4) E B D A C (3) E B A D C (3) E A D B C (3) E A B D C (3) D A E C B (3) C D B E A (3) C D B A E (3) C B D A E (3) B E C A D (3) B C E D A (3) A E D C B (3) A C D E B (3) E D B A C (2) D A C E B (2) C B E D A (2) C A D B E (2) B C D E A (2) A E B C D (2) A D C E B (2) A C B E D (2) E D A B C (1) E B A C D (1) D E C B A (1) D E B C A (1) D E A C B (1) D E A B C (1) D C B E A (1) D A E B C (1) C B D E A (1) C B A E D (1) C B A D E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B E A C D (1) B D C E A (1) B C E A D (1) A C E D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 18 28 14 16 B -18 0 2 -14 -28 C -28 -2 0 -24 -20 D -14 14 24 0 -8 E -16 28 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 28 14 16 B -18 0 2 -14 -28 C -28 -2 0 -24 -20 D -14 14 24 0 -8 E -16 28 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=16 D=16 C=16 B=15 so B is eliminated. Round 2 votes counts: A=37 E=24 C=22 D=17 so D is eliminated. Round 3 votes counts: A=43 C=29 E=28 so E is eliminated. Round 4 votes counts: A=64 C=36 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:238 E:220 D:208 B:171 C:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 28 14 16 B -18 0 2 -14 -28 C -28 -2 0 -24 -20 D -14 14 24 0 -8 E -16 28 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 28 14 16 B -18 0 2 -14 -28 C -28 -2 0 -24 -20 D -14 14 24 0 -8 E -16 28 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 28 14 16 B -18 0 2 -14 -28 C -28 -2 0 -24 -20 D -14 14 24 0 -8 E -16 28 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1552: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) E A B D C (9) C B D A E (8) D A C E B (7) A D E C B (7) B E C A D (5) C B E D A (4) B C E D A (4) B C D E A (4) A D E B C (4) E A D C B (3) D C A B E (3) C B D E A (3) A E D C B (3) E B C A D (2) E B A D C (2) E A B C D (2) D A E B C (2) D A C B E (2) C D B A E (2) B C E A D (2) E C A B D (1) E B A C D (1) D C B A E (1) D B C A E (1) D B A C E (1) D A B E C (1) C E B A D (1) C D A B E (1) C B E A D (1) C A E D B (1) B E C D A (1) B C D A E (1) Total count = 100 A B C D E A 0 12 8 8 -10 B -12 0 4 2 -10 C -8 -4 0 -12 -6 D -8 -2 12 0 -4 E 10 10 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 8 8 -10 B -12 0 4 2 -10 C -8 -4 0 -12 -6 D -8 -2 12 0 -4 E 10 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=21 D=18 B=17 A=14 so A is eliminated. Round 2 votes counts: E=33 D=29 C=21 B=17 so B is eliminated. Round 3 votes counts: E=39 C=32 D=29 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:209 D:199 B:192 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 8 8 -10 B -12 0 4 2 -10 C -8 -4 0 -12 -6 D -8 -2 12 0 -4 E 10 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 8 -10 B -12 0 4 2 -10 C -8 -4 0 -12 -6 D -8 -2 12 0 -4 E 10 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 8 -10 B -12 0 4 2 -10 C -8 -4 0 -12 -6 D -8 -2 12 0 -4 E 10 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1553: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (14) E D B C A (12) B D E A C (10) B D E C A (9) A C E D B (9) C A E D B (8) E C D B A (4) C A E B D (4) A B D C E (3) E D B A C (2) D B E A C (2) C E B D A (2) C A B E D (2) B D A E C (2) B D A C E (2) E C D A B (1) E C A D B (1) E B D C A (1) D B E C A (1) C E D B A (1) C E A D B (1) C B E D A (1) B C D E A (1) A E C D B (1) A D B E C (1) A D B C E (1) A C E B D (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 0 2 -2 2 B 0 0 -6 6 2 C -2 6 0 8 6 D 2 -6 -8 0 -4 E -2 -2 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.697600 B: 0.092800 C: 0.104800 D: 0.104800 E: 0.000000 Sum of squares = 0.51722358762 Cumulative probabilities = A: 0.697600 B: 0.790400 C: 0.895200 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -2 2 B 0 0 -6 6 2 C -2 6 0 8 6 D 2 -6 -8 0 -4 E -2 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.666750 B: 0.000251 C: 0.166499 D: 0.166499 E: 0.000000 Sum of squares = 0.500000125932 Cumulative probabilities = A: 0.666750 B: 0.667001 C: 0.833501 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=24 E=21 C=19 D=3 so D is eliminated. Round 2 votes counts: A=33 B=27 E=21 C=19 so C is eliminated. Round 3 votes counts: A=47 B=28 E=25 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:209 A:201 B:201 E:197 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 -2 2 B 0 0 -6 6 2 C -2 6 0 8 6 D 2 -6 -8 0 -4 E -2 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.666750 B: 0.000251 C: 0.166499 D: 0.166499 E: 0.000000 Sum of squares = 0.500000125932 Cumulative probabilities = A: 0.666750 B: 0.667001 C: 0.833501 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 2 B 0 0 -6 6 2 C -2 6 0 8 6 D 2 -6 -8 0 -4 E -2 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.666750 B: 0.000251 C: 0.166499 D: 0.166499 E: 0.000000 Sum of squares = 0.500000125932 Cumulative probabilities = A: 0.666750 B: 0.667001 C: 0.833501 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 2 B 0 0 -6 6 2 C -2 6 0 8 6 D 2 -6 -8 0 -4 E -2 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.666750 B: 0.000251 C: 0.166499 D: 0.166499 E: 0.000000 Sum of squares = 0.500000125932 Cumulative probabilities = A: 0.666750 B: 0.667001 C: 0.833501 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1554: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) D C A B E (7) D B A E C (7) D A C B E (5) E C B A D (4) E B A D C (4) C E B A D (4) A E B D C (4) E B C A D (3) C E A B D (3) A E C B D (3) E C B D A (2) D A B C E (2) C E B D A (2) C A D E B (2) B E C D A (2) B E A D C (2) B D E C A (2) B D E A C (2) A D B E C (2) A C D E B (2) A B D E C (2) E C A B D (1) E B C D A (1) E A B C D (1) D C B A E (1) D B C E A (1) D B A C E (1) D A B E C (1) C E D B A (1) C E D A B (1) C D E A B (1) C D B E A (1) C D A E B (1) C D A B E (1) C B D E A (1) C A E D B (1) B E D C A (1) B E D A C (1) B D C E A (1) B D A E C (1) B A E D C (1) B A D E C (1) A E C D B (1) A E B C D (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 10 6 2 B 8 0 2 14 -6 C -10 -2 0 -4 -16 D -6 -14 4 0 -4 E -2 6 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406250000081 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -8 10 6 2 B 8 0 2 14 -6 C -10 -2 0 -4 -16 D -6 -14 4 0 -4 E -2 6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=23 C=19 A=19 B=14 so B is eliminated. Round 2 votes counts: D=31 E=29 A=21 C=19 so C is eliminated. Round 3 votes counts: E=40 D=36 A=24 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:209 A:205 D:190 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 10 6 2 B 8 0 2 14 -6 C -10 -2 0 -4 -16 D -6 -14 4 0 -4 E -2 6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 6 2 B 8 0 2 14 -6 C -10 -2 0 -4 -16 D -6 -14 4 0 -4 E -2 6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 6 2 B 8 0 2 14 -6 C -10 -2 0 -4 -16 D -6 -14 4 0 -4 E -2 6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1555: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (14) C D A B E (11) D C B E A (8) A C B E D (8) D E B C A (6) D C E B A (5) D C A E B (5) E B D A C (4) C A D B E (4) E B A C D (3) C D B A E (3) B E A D C (3) A E B D C (3) A B E C D (3) E B D C A (2) C A D E B (2) B E D C A (2) A E B C D (2) E D B A C (1) E A B D C (1) D E B A C (1) D B E C A (1) D B C E A (1) D A E B C (1) D A C E B (1) B E C D A (1) B A E C D (1) A E D B C (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -2 -6 -6 B 12 0 2 -6 -6 C 2 -2 0 -22 -2 D 6 6 22 0 2 E 6 6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -6 -6 B 12 0 2 -6 -6 C 2 -2 0 -22 -2 D 6 6 22 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 C=20 A=19 B=7 so B is eliminated. Round 2 votes counts: E=31 D=29 C=20 A=20 so C is eliminated. Round 3 votes counts: D=43 E=31 A=26 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:206 B:201 C:188 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -6 -6 B 12 0 2 -6 -6 C 2 -2 0 -22 -2 D 6 6 22 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -6 -6 B 12 0 2 -6 -6 C 2 -2 0 -22 -2 D 6 6 22 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -6 -6 B 12 0 2 -6 -6 C 2 -2 0 -22 -2 D 6 6 22 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1556: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) E B A C D (6) D B C A E (6) D C B A E (5) B D C A E (5) A E C B D (5) E A B C D (4) D B E C A (4) C A D E B (4) D C A E B (3) D C A B E (3) B E A C D (3) B A E C D (3) A C E B D (3) E A C B D (2) D E B C A (2) D C B E A (2) D B C E A (2) C D A E B (2) C D A B E (2) C A D B E (2) C A B D E (2) B E D A C (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) D C E A B (1) C D B A E (1) C B A D E (1) C A E D B (1) B E A D C (1) B D E C A (1) B D C E A (1) B D A C E (1) B C A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -6 4 10 B 8 0 2 -4 6 C 6 -2 0 8 4 D -4 4 -8 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 4 10 B 8 0 2 -4 6 C 6 -2 0 8 4 D -4 4 -8 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428495 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=21 C=15 A=10 so A is eliminated. Round 2 votes counts: E=31 D=28 B=22 C=19 so C is eliminated. Round 3 votes counts: D=39 E=35 B=26 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:208 B:206 A:200 D:199 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 4 10 B 8 0 2 -4 6 C 6 -2 0 8 4 D -4 4 -8 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428495 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 4 10 B 8 0 2 -4 6 C 6 -2 0 8 4 D -4 4 -8 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428495 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 4 10 B 8 0 2 -4 6 C 6 -2 0 8 4 D -4 4 -8 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428495 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1557: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (5) A D E B C (5) E A C D B (4) C D E B A (4) B E A C D (4) B C E A D (4) A E D C B (4) A E B C D (4) E C A B D (3) E A C B D (3) D B C A E (3) C D B E A (3) C B D E A (3) B C D E A (3) A E B D C (3) E B C A D (2) E A D C B (2) E A B C D (2) D B A C E (2) C E D B A (2) C E D A B (2) C D E A B (2) B A E D C (2) A E C D B (2) A D E C B (2) E D A C B (1) E C D A B (1) E C B A D (1) E C A D B (1) D C E B A (1) D C B E A (1) D C B A E (1) D B A E C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) C B E D A (1) C B E A D (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D A E (1) B C A D E (1) B A E C D (1) B A D E C (1) A E D B C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 2 12 -16 B -4 0 -6 -6 -22 C -2 6 0 16 -12 D -12 6 -16 0 -10 E 16 22 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 12 -16 B -4 0 -6 -6 -22 C -2 6 0 16 -12 D -12 6 -16 0 -10 E 16 22 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=20 C=20 B=20 D=17 so D is eliminated. Round 2 votes counts: C=28 B=26 A=26 E=20 so E is eliminated. Round 3 votes counts: A=38 C=34 B=28 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:230 C:204 A:201 D:184 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 12 -16 B -4 0 -6 -6 -22 C -2 6 0 16 -12 D -12 6 -16 0 -10 E 16 22 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 12 -16 B -4 0 -6 -6 -22 C -2 6 0 16 -12 D -12 6 -16 0 -10 E 16 22 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 12 -16 B -4 0 -6 -6 -22 C -2 6 0 16 -12 D -12 6 -16 0 -10 E 16 22 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1558: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) B E D A C (7) D B E A C (6) C D A B E (6) A C E B D (6) E B D C A (4) E B D A C (4) D C B E A (4) D B E C A (4) A B E D C (4) E B A D C (3) D E B C A (3) D C A B E (3) C A D E B (3) A E C B D (3) D B C E A (2) C E A B D (2) C D E B A (2) C A D B E (2) A E B D C (2) A D B E C (2) A C D B E (2) E C B D A (1) E C B A D (1) E B C D A (1) D C E B A (1) D A C B E (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E D B (1) B D E A C (1) A E B C D (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -10 -8 2 B -10 0 -8 4 -6 C 10 8 0 -8 4 D 8 -4 8 0 -6 E -2 6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.555556 E: 1.000000 A B C D E A 0 10 -10 -8 2 B -10 0 -8 4 -6 C 10 8 0 -8 4 D 8 -4 8 0 -6 E -2 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.222222 E: 0.444444 Sum of squares = 0.35802469136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 A=22 E=14 B=8 so B is eliminated. Round 2 votes counts: C=31 D=26 A=22 E=21 so E is eliminated. Round 3 votes counts: D=41 C=34 A=25 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:203 E:203 A:197 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -10 -8 2 B -10 0 -8 4 -6 C 10 8 0 -8 4 D 8 -4 8 0 -6 E -2 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.222222 E: 0.444444 Sum of squares = 0.35802469136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.555556 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 -8 2 B -10 0 -8 4 -6 C 10 8 0 -8 4 D 8 -4 8 0 -6 E -2 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.222222 E: 0.444444 Sum of squares = 0.35802469136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 -8 2 B -10 0 -8 4 -6 C 10 8 0 -8 4 D 8 -4 8 0 -6 E -2 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.222222 E: 0.444444 Sum of squares = 0.35802469136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.555556 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1559: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) E B D C A (9) E D C A B (8) B A C E D (8) A C D B E (8) B E D C A (7) B E D A C (7) A C D E B (7) E D B C A (6) C A D E B (6) B E A C D (6) D E C A B (5) B A C D E (5) A C B D E (4) D A C E B (1) C A D B E (1) B A E C D (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -4 -6 4 B -2 0 -2 -6 -6 C 4 2 0 -8 2 D 6 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775273 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 A B C D E A 0 2 -4 -6 4 B -2 0 -2 -6 -6 C 4 2 0 -8 2 D 6 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775412 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=23 A=20 D=16 C=7 so C is eliminated. Round 2 votes counts: B=34 A=27 E=23 D=16 so D is eliminated. Round 3 votes counts: A=38 B=34 E=28 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:208 E:202 C:200 A:198 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -6 4 B -2 0 -2 -6 -6 C 4 2 0 -8 2 D 6 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775412 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -6 4 B -2 0 -2 -6 -6 C 4 2 0 -8 2 D 6 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775412 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -6 4 B -2 0 -2 -6 -6 C 4 2 0 -8 2 D 6 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775412 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1560: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) C A B D E (9) B A C E D (7) E D C A B (6) D C A B E (6) B A C D E (6) E B A C D (5) D E C A B (5) D C E A B (3) D C B A E (3) D C A E B (3) C A D B E (3) B A E C D (3) A B C E D (3) E B D A C (2) E B A D C (2) D E C B A (2) C D A B E (2) C B A D E (2) C A E D B (2) E D B C A (1) E D A C B (1) E D A B C (1) E A D C B (1) D E B C A (1) D B E A C (1) D B C E A (1) D B C A E (1) C A D E B (1) C A B E D (1) B E A C D (1) B D C A E (1) B C A D E (1) A E C B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -8 0 14 B 0 0 -6 -8 6 C 8 6 0 -2 14 D 0 8 2 0 4 E -14 -6 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.117253 B: 0.000000 C: 0.000000 D: 0.882747 E: 0.000000 Sum of squares = 0.792990208491 Cumulative probabilities = A: 0.117253 B: 0.117253 C: 0.117253 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 0 14 B 0 0 -6 -8 6 C 8 6 0 -2 14 D 0 8 2 0 4 E -14 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000023282 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 C=20 B=19 A=6 so A is eliminated. Round 2 votes counts: E=30 D=26 C=22 B=22 so C is eliminated. Round 3 votes counts: B=36 E=32 D=32 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:207 A:203 B:196 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 0 14 B 0 0 -6 -8 6 C 8 6 0 -2 14 D 0 8 2 0 4 E -14 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000023282 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 0 14 B 0 0 -6 -8 6 C 8 6 0 -2 14 D 0 8 2 0 4 E -14 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000023282 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 0 14 B 0 0 -6 -8 6 C 8 6 0 -2 14 D 0 8 2 0 4 E -14 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000023282 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1561: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E A B D (7) C A E D B (7) B D E A C (7) E C B D A (5) D B A C E (5) C A D B E (5) A D B C E (5) E C B A D (4) E B C D A (4) D A B C E (4) C E A D B (4) B E D A C (4) E C A B D (3) C A D E B (3) B E D C A (3) A D B E C (3) A C D B E (3) E B D A C (2) D A C B E (2) C A E B D (2) A D C B E (2) A C E D B (2) E B C A D (1) E B A C D (1) D C B A E (1) D B C A E (1) C E B D A (1) B D E C A (1) Total count = 100 A B C D E A 0 4 -4 4 6 B -4 0 -2 -10 8 C 4 2 0 4 8 D -4 10 -4 0 0 E -6 -8 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 4 6 B -4 0 -2 -10 8 C 4 2 0 4 8 D -4 10 -4 0 0 E -6 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 E=20 B=15 A=15 so B is eliminated. Round 2 votes counts: D=29 C=29 E=27 A=15 so A is eliminated. Round 3 votes counts: D=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:205 D:201 B:196 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 6 B -4 0 -2 -10 8 C 4 2 0 4 8 D -4 10 -4 0 0 E -6 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 6 B -4 0 -2 -10 8 C 4 2 0 4 8 D -4 10 -4 0 0 E -6 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 6 B -4 0 -2 -10 8 C 4 2 0 4 8 D -4 10 -4 0 0 E -6 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1562: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) C B D E A (8) B C D E A (8) A E D C B (8) C D E A B (6) A E D B C (6) A C E D B (6) E D C A B (5) B A E D C (5) A B E D C (5) C A E D B (3) B D E A C (3) B A C D E (3) E D A C B (2) D E C B A (2) D E C A B (2) B D E C A (2) B D C E A (2) B A D E C (2) B A C E D (2) E D B A C (1) E D A B C (1) D E B C A (1) D E B A C (1) C B A D E (1) C A D E B (1) C A B E D (1) C A B D E (1) B C A D E (1) A E B D C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 -4 -4 B 0 0 -10 -6 -8 C 4 10 0 2 6 D 4 6 -2 0 4 E 4 8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -4 -4 B 0 0 -10 -6 -8 C 4 10 0 2 6 D 4 6 -2 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 A=28 E=9 D=6 so D is eliminated. Round 2 votes counts: C=29 B=28 A=28 E=15 so E is eliminated. Round 3 votes counts: C=38 B=31 A=31 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:206 E:201 A:194 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -4 -4 B 0 0 -10 -6 -8 C 4 10 0 2 6 D 4 6 -2 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 -4 B 0 0 -10 -6 -8 C 4 10 0 2 6 D 4 6 -2 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 -4 B 0 0 -10 -6 -8 C 4 10 0 2 6 D 4 6 -2 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1563: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) C B D E A (8) A E D B C (8) D B C A E (6) B C D E A (6) D C B A E (5) D A B C E (5) C B E D A (5) B D C E A (4) A D B C E (4) E A B C D (3) D B A C E (3) A E D C B (3) A E C D B (3) A D E C B (3) E C B D A (2) E C B A D (2) E B C A D (2) D A C B E (2) C E B D A (2) C D B A E (2) A E B D C (2) A D B E C (2) E C A B D (1) E B C D A (1) E B A C D (1) E A C D B (1) D C A B E (1) C E D A B (1) C D B E A (1) A D C E B (1) Total count = 100 A B C D E A 0 0 2 -8 0 B 0 0 -6 -2 8 C -2 6 0 2 12 D 8 2 -2 0 6 E 0 -8 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000000124 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -8 0 B 0 0 -6 -2 8 C -2 6 0 2 12 D 8 2 -2 0 6 E 0 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999646 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 D=22 C=19 B=10 so B is eliminated. Round 2 votes counts: D=26 A=26 C=25 E=23 so E is eliminated. Round 3 votes counts: A=41 C=33 D=26 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:209 D:207 B:200 A:197 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 2 -8 0 B 0 0 -6 -2 8 C -2 6 0 2 12 D 8 2 -2 0 6 E 0 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999646 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -8 0 B 0 0 -6 -2 8 C -2 6 0 2 12 D 8 2 -2 0 6 E 0 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999646 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -8 0 B 0 0 -6 -2 8 C -2 6 0 2 12 D 8 2 -2 0 6 E 0 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999646 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1564: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (7) A C E D B (6) E D C B A (5) E D B C A (5) A B E C D (5) B D E C A (4) B D E A C (4) E C A D B (3) C A D E B (3) B E D A C (3) A E C B D (3) A E B C D (3) A B C D E (3) E C D A B (2) E B D C A (2) E A B D C (2) D C B E A (2) D B C E A (2) C E D A B (2) C D E A B (2) C D A B E (2) B D C E A (2) B A C D E (2) A C D E B (2) A C D B E (2) A B C E D (2) E D C A B (1) E B D A C (1) E B A D C (1) E A C D B (1) D E B C A (1) D C E B A (1) D C B A E (1) D B E C A (1) C E A D B (1) C D E B A (1) C D B A E (1) C A E D B (1) C A D B E (1) B E A D C (1) B D C A E (1) B A E D C (1) B A D C E (1) A E B D C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 14 6 8 0 B -14 0 -2 -12 -14 C -6 2 0 12 -16 D -8 12 -12 0 -22 E 0 14 16 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.863967 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.136033 Sum of squares = 0.764943253026 Cumulative probabilities = A: 0.863967 B: 0.863967 C: 0.863967 D: 0.863967 E: 1.000000 A B C D E A 0 14 6 8 0 B -14 0 -2 -12 -14 C -6 2 0 12 -16 D -8 12 -12 0 -22 E 0 14 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=23 B=19 C=14 D=8 so D is eliminated. Round 2 votes counts: A=36 E=24 B=22 C=18 so C is eliminated. Round 3 votes counts: A=43 E=31 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:226 A:214 C:196 D:185 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 8 0 B -14 0 -2 -12 -14 C -6 2 0 12 -16 D -8 12 -12 0 -22 E 0 14 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 8 0 B -14 0 -2 -12 -14 C -6 2 0 12 -16 D -8 12 -12 0 -22 E 0 14 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 8 0 B -14 0 -2 -12 -14 C -6 2 0 12 -16 D -8 12 -12 0 -22 E 0 14 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1565: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) B D E A C (8) D B C E A (7) B E A C D (6) A E C D B (6) E A C B D (5) D A E C B (5) D B C A E (4) C E A B D (4) A E C B D (4) E A B C D (3) D A E B C (3) D A C B E (3) C E B A D (3) C A E D B (3) E C A B D (2) D C A E B (2) C B E D A (2) B D C E A (2) B D A E C (2) D C B E A (1) D C B A E (1) D B A C E (1) D A B E C (1) C D B E A (1) C B E A D (1) C B D E A (1) B E D A C (1) B E A D C (1) B D E C A (1) B C E A D (1) B A E D C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -14 28 -16 0 B 14 0 10 -2 16 C -28 -10 0 -16 -26 D 16 2 16 0 12 E 0 -16 26 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 28 -16 0 B 14 0 10 -2 16 C -28 -10 0 -16 -26 D 16 2 16 0 12 E 0 -16 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=23 C=15 A=12 E=10 so E is eliminated. Round 2 votes counts: D=40 B=23 A=20 C=17 so C is eliminated. Round 3 votes counts: D=41 B=30 A=29 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:219 A:199 E:199 C:160 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 28 -16 0 B 14 0 10 -2 16 C -28 -10 0 -16 -26 D 16 2 16 0 12 E 0 -16 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 28 -16 0 B 14 0 10 -2 16 C -28 -10 0 -16 -26 D 16 2 16 0 12 E 0 -16 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 28 -16 0 B 14 0 10 -2 16 C -28 -10 0 -16 -26 D 16 2 16 0 12 E 0 -16 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1566: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) C E A D B (11) E C B A D (8) E B C A D (8) C A D E B (8) D A B C E (7) A D C B E (7) E B C D A (6) D A B E C (4) C A E D B (4) E B D C A (3) D A C B E (3) E C A D B (2) E B D A C (2) B E C D A (2) B D E A C (2) B D A E C (2) E C A B D (1) D B A E C (1) C E B A D (1) C E A B D (1) C A D B E (1) B E D C A (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -14 10 -20 B -4 0 0 -2 -14 C 14 0 0 10 -8 D -10 2 -10 0 -24 E 20 14 8 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -14 10 -20 B -4 0 0 -2 -14 C 14 0 0 10 -8 D -10 2 -10 0 -24 E 20 14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 B=19 D=15 A=10 so A is eliminated. Round 2 votes counts: E=30 C=28 D=23 B=19 so B is eliminated. Round 3 votes counts: E=45 C=28 D=27 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:233 C:208 A:190 B:190 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -14 10 -20 B -4 0 0 -2 -14 C 14 0 0 10 -8 D -10 2 -10 0 -24 E 20 14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 10 -20 B -4 0 0 -2 -14 C 14 0 0 10 -8 D -10 2 -10 0 -24 E 20 14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 10 -20 B -4 0 0 -2 -14 C 14 0 0 10 -8 D -10 2 -10 0 -24 E 20 14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1567: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) D C B A E (6) A E C D B (6) E A B C D (4) D A B E C (4) C E A B D (4) B D E A C (4) A E D B C (4) A E B D C (4) E B A D C (3) E B A C D (3) E A C B D (3) C D B E A (3) C B E D A (3) B E A C D (3) D B A E C (2) D A E B C (2) C E B A D (2) C D A B E (2) C A E D B (2) C A D E B (2) B E D A C (2) B D C E A (2) B C D E A (2) A E D C B (2) A E B C D (2) E C A B D (1) D C B E A (1) D C A E B (1) D C A B E (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A C E (1) C E B D A (1) C D B A E (1) C D A E B (1) B E A D C (1) A E C B D (1) A D E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 10 0 -4 B 2 0 -6 6 -2 C -10 6 0 10 -6 D 0 -6 -10 0 -4 E 4 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 10 0 -4 B 2 0 -6 6 -2 C -10 6 0 10 -6 D 0 -6 -10 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=22 D=21 E=14 B=14 so E is eliminated. Round 2 votes counts: C=30 A=29 D=21 B=20 so B is eliminated. Round 3 votes counts: A=39 C=32 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:202 B:200 C:200 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 10 0 -4 B 2 0 -6 6 -2 C -10 6 0 10 -6 D 0 -6 -10 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 0 -4 B 2 0 -6 6 -2 C -10 6 0 10 -6 D 0 -6 -10 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 0 -4 B 2 0 -6 6 -2 C -10 6 0 10 -6 D 0 -6 -10 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) C D B A E (5) C B E A D (5) A E C D B (5) E A B C D (4) D B C E A (4) C B D E A (4) B C E D A (4) B C D E A (4) A E D C B (4) A E D B C (4) E B C A D (3) D B C A E (3) C B E D A (3) E B A D C (2) D B A C E (2) D A E C B (2) C E B A D (2) C B A E D (2) B E D C A (2) B E C D A (2) B E C A D (2) A E C B D (2) A D E C B (2) E D A B C (1) E C A B D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B A E C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E A B D (1) C D B E A (1) C D A B E (1) C A E D B (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 -14 -6 2 B 10 0 0 -4 8 C 14 0 0 16 4 D 6 4 -16 0 -10 E -2 -8 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.598199 C: 0.401801 D: 0.000000 E: 0.000000 Sum of squares = 0.51928619288 Cumulative probabilities = A: 0.000000 B: 0.598199 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -6 2 B 10 0 0 -4 8 C 14 0 0 16 4 D 6 4 -16 0 -10 E -2 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 A=18 B=16 E=12 so E is eliminated. Round 2 votes counts: C=30 D=26 A=23 B=21 so B is eliminated. Round 3 votes counts: C=46 D=29 A=25 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:207 E:198 D:192 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -6 2 B 10 0 0 -4 8 C 14 0 0 16 4 D 6 4 -16 0 -10 E -2 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -6 2 B 10 0 0 -4 8 C 14 0 0 16 4 D 6 4 -16 0 -10 E -2 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -6 2 B 10 0 0 -4 8 C 14 0 0 16 4 D 6 4 -16 0 -10 E -2 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1569: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) E D C A B (7) B A D E C (6) D E A B C (5) B D A E C (5) E D C B A (4) D E A C B (4) D A E B C (4) C E D A B (4) C E B A D (4) C B A E D (4) B A C D E (4) A B D E C (4) E D B C A (3) E C D B A (3) E C D A B (3) D A B E C (3) C E A D B (3) C B E A D (3) A B D C E (3) D B E A C (2) C A E D B (2) C B E D A (1) C A E B D (1) C A B E D (1) B E D C A (1) B D E A C (1) B C A D E (1) B A D C E (1) A D B E C (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 -18 -14 B 4 0 -8 -18 -14 C 8 8 0 -10 -12 D 18 18 10 0 -6 E 14 14 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 -18 -14 B 4 0 -8 -18 -14 C 8 8 0 -10 -12 D 18 18 10 0 -6 E 14 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=20 B=19 D=18 A=11 so A is eliminated. Round 2 votes counts: C=33 B=27 E=20 D=20 so E is eliminated. Round 3 votes counts: C=39 D=34 B=27 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:223 D:220 C:197 B:182 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 -18 -14 B 4 0 -8 -18 -14 C 8 8 0 -10 -12 D 18 18 10 0 -6 E 14 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -18 -14 B 4 0 -8 -18 -14 C 8 8 0 -10 -12 D 18 18 10 0 -6 E 14 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -18 -14 B 4 0 -8 -18 -14 C 8 8 0 -10 -12 D 18 18 10 0 -6 E 14 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1570: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) D C B E A (8) B A D E C (8) C D E B A (7) E A C D B (5) D B C E A (5) A E B C D (5) C E A D B (4) C D B E A (4) B D A C E (4) E A C B D (3) A E C B D (3) A B E D C (3) E D C B A (2) E C D A B (2) E C A D B (2) E A B C D (2) C D B A E (2) B D A E C (2) E C D B A (1) E A B D C (1) D E C B A (1) D C E B A (1) D B E C A (1) D B C A E (1) C E D B A (1) C A E B D (1) B D E A C (1) B D C A E (1) B A E D C (1) B A D C E (1) B A C D E (1) A E B D C (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -14 -14 -28 B 6 0 -22 -20 -12 C 14 22 0 16 12 D 14 20 -16 0 -2 E 28 12 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -14 -28 B 6 0 -22 -20 -12 C 14 22 0 16 12 D 14 20 -16 0 -2 E 28 12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=19 E=18 D=17 A=14 so A is eliminated. Round 2 votes counts: C=33 E=27 B=23 D=17 so D is eliminated. Round 3 votes counts: C=42 B=30 E=28 so E is eliminated. Round 4 votes counts: C=61 B=39 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 E:215 D:208 B:176 A:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -14 -28 B 6 0 -22 -20 -12 C 14 22 0 16 12 D 14 20 -16 0 -2 E 28 12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -14 -28 B 6 0 -22 -20 -12 C 14 22 0 16 12 D 14 20 -16 0 -2 E 28 12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -14 -28 B 6 0 -22 -20 -12 C 14 22 0 16 12 D 14 20 -16 0 -2 E 28 12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1571: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) C D B E A (9) A E C D B (9) B E A D C (7) A C D E B (5) E B A D C (4) D C B E A (4) C D B A E (3) C A D E B (3) B E D C A (3) A E B D C (3) A E B C D (3) A C E D B (3) A B E C D (3) E D C A B (2) E B D C A (2) C D E A B (2) C D A B E (2) B E D A C (2) B C D A E (2) A B E D C (2) E D B C A (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D C E B A (1) D B E C A (1) D B C E A (1) C D E B A (1) C D A E B (1) B D E C A (1) B C D E A (1) B A E D C (1) A C D B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -2 0 -14 B 10 0 4 -4 12 C 2 -4 0 2 4 D 0 4 -2 0 0 E 14 -12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 0 -14 B 10 0 4 -4 12 C 2 -4 0 2 4 D 0 4 -2 0 0 E 14 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=28 C=21 E=13 D=7 so D is eliminated. Round 2 votes counts: A=31 B=30 C=26 E=13 so E is eliminated. Round 3 votes counts: B=37 A=35 C=28 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:202 D:201 E:199 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 0 -14 B 10 0 4 -4 12 C 2 -4 0 2 4 D 0 4 -2 0 0 E 14 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 0 -14 B 10 0 4 -4 12 C 2 -4 0 2 4 D 0 4 -2 0 0 E 14 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 0 -14 B 10 0 4 -4 12 C 2 -4 0 2 4 D 0 4 -2 0 0 E 14 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1572: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) A B E D C (8) B A E C D (7) D E C A B (5) C D E A B (5) B E C D A (4) A E B D C (4) E D C B A (3) E D A C B (3) D C E A B (3) D C A E B (3) B C E D A (3) B C A E D (3) E A D B C (2) D A C E B (2) C D E B A (2) C D B E A (2) C D B A E (2) C D A E B (2) B E A C D (2) B C D A E (2) B A E D C (2) B A C E D (2) B A C D E (2) E B D C A (1) E B D A C (1) E A D C B (1) E A B D C (1) D E C B A (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A D C (1) B C E A D (1) A E D C B (1) A E D B C (1) A D C E B (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -12 -8 0 B 4 0 0 18 16 C 12 0 0 10 2 D 8 -18 -10 0 -4 E 0 -16 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333054 C: 0.666946 D: 0.000000 E: 0.000000 Sum of squares = 0.555741671417 Cumulative probabilities = A: 0.000000 B: 0.333054 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -8 0 B 4 0 0 18 16 C 12 0 0 10 2 D 8 -18 -10 0 -4 E 0 -16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=26 A=18 D=14 E=12 so E is eliminated. Round 2 votes counts: B=32 C=26 A=22 D=20 so D is eliminated. Round 3 votes counts: C=41 B=32 A=27 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:212 E:193 A:188 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 -8 0 B 4 0 0 18 16 C 12 0 0 10 2 D 8 -18 -10 0 -4 E 0 -16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -8 0 B 4 0 0 18 16 C 12 0 0 10 2 D 8 -18 -10 0 -4 E 0 -16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -8 0 B 4 0 0 18 16 C 12 0 0 10 2 D 8 -18 -10 0 -4 E 0 -16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1573: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (11) B A C E D (9) A B D E C (9) C E D B A (8) E C D B A (6) D C E A B (6) D A C E B (6) E D C A B (4) D E C A B (4) C E B D A (3) C B E D A (3) B A E D C (3) A D E C B (3) D A E C B (2) C D E A B (2) B E C D A (2) B A D E C (2) B A C D E (2) A B D C E (2) E D C B A (1) E C B D A (1) E B D C A (1) C E D A B (1) B E C A D (1) B C E D A (1) B C E A D (1) B C A E D (1) B A D C E (1) A D E B C (1) A D C E B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 8 -2 10 B 14 0 -2 6 0 C -8 2 0 4 -4 D 2 -6 -4 0 -14 E -10 0 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.521342 C: 0.000000 D: 0.000000 E: 0.478658 Sum of squares = 0.500910960042 Cumulative probabilities = A: 0.000000 B: 0.521342 C: 0.521342 D: 0.521342 E: 1.000000 A B C D E A 0 -14 8 -2 10 B 14 0 -2 6 0 C -8 2 0 4 -4 D 2 -6 -4 0 -14 E -10 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999288 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=18 A=18 C=17 E=13 so E is eliminated. Round 2 votes counts: B=35 C=24 D=23 A=18 so A is eliminated. Round 3 votes counts: B=46 D=30 C=24 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:204 A:201 C:197 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 -2 10 B 14 0 -2 6 0 C -8 2 0 4 -4 D 2 -6 -4 0 -14 E -10 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999288 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -2 10 B 14 0 -2 6 0 C -8 2 0 4 -4 D 2 -6 -4 0 -14 E -10 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999288 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -2 10 B 14 0 -2 6 0 C -8 2 0 4 -4 D 2 -6 -4 0 -14 E -10 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.499999999288 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1574: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (15) B C D E A (13) A E B C D (9) D C B E A (8) B A C E D (7) D C E B A (5) D E C A B (4) B C A D E (4) D E C B A (3) B C A E D (3) A E D B C (3) A B C E D (3) D A E C B (2) C E D B A (2) B C D A E (2) A E C D B (2) A E B D C (2) A B E C D (2) E D C A B (1) E D A C B (1) E A D C B (1) D E A C B (1) C D B E A (1) C B D E A (1) B C E D A (1) A E C B D (1) A D E C B (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 10 16 B 0 0 2 -2 -6 C -4 -2 0 4 2 D -10 2 -4 0 -6 E -16 6 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.499503 B: 0.500497 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000477046 Cumulative probabilities = A: 0.499503 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 10 16 B 0 0 2 -2 -6 C -4 -2 0 4 2 D -10 2 -4 0 -6 E -16 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=30 D=23 C=4 E=3 so E is eliminated. Round 2 votes counts: A=41 B=30 D=25 C=4 so C is eliminated. Round 3 votes counts: A=41 B=31 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:200 B:197 E:197 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 10 16 B 0 0 2 -2 -6 C -4 -2 0 4 2 D -10 2 -4 0 -6 E -16 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 16 B 0 0 2 -2 -6 C -4 -2 0 4 2 D -10 2 -4 0 -6 E -16 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 16 B 0 0 2 -2 -6 C -4 -2 0 4 2 D -10 2 -4 0 -6 E -16 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1575: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (17) E B A D C (12) A B E C D (11) D E C B A (8) C D A B E (8) E D C B A (7) A B C D E (5) E B D C A (3) D C E A B (3) B E A C D (3) A B C E D (3) E D B C A (2) D C A B E (2) B C D E A (2) B A E C D (2) A C B D E (2) E D C A B (1) E B D A C (1) E A B D C (1) D C B E A (1) D C A E B (1) C D B E A (1) C B A D E (1) C A D B E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -20 -16 -14 -24 B 20 0 -6 -4 -16 C 16 6 0 -20 -6 D 14 4 20 0 4 E 24 16 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999195 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -16 -14 -24 B 20 0 -6 -4 -16 C 16 6 0 -20 -6 D 14 4 20 0 4 E 24 16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=27 A=23 C=11 B=7 so B is eliminated. Round 2 votes counts: D=32 E=30 A=25 C=13 so C is eliminated. Round 3 votes counts: D=43 E=30 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:221 C:198 B:197 A:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -16 -14 -24 B 20 0 -6 -4 -16 C 16 6 0 -20 -6 D 14 4 20 0 4 E 24 16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 -14 -24 B 20 0 -6 -4 -16 C 16 6 0 -20 -6 D 14 4 20 0 4 E 24 16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 -14 -24 B 20 0 -6 -4 -16 C 16 6 0 -20 -6 D 14 4 20 0 4 E 24 16 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1576: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (11) C E B A D (10) C E B D A (8) B E C D A (8) B E C A D (6) D C E B A (5) D A C E B (5) B E D C A (5) A C D E B (5) D B E A C (4) C E A B D (4) B E A C D (4) A D C B E (4) A D C E B (3) E C B A D (2) C A E B D (2) B D E A C (2) A D B E C (2) E B C A D (1) D E B C A (1) D B E C A (1) D A C B E (1) B E D A C (1) B E A D C (1) B D E C A (1) B A E C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -22 -8 -6 -28 B 22 0 0 16 6 C 8 0 0 4 -4 D 6 -16 -4 0 -10 E 28 -6 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.597750 C: 0.402250 D: 0.000000 E: 0.000000 Sum of squares = 0.519109987661 Cumulative probabilities = A: 0.000000 B: 0.597750 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -8 -6 -28 B 22 0 0 16 6 C 8 0 0 4 -4 D 6 -16 -4 0 -10 E 28 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=28 C=24 A=16 E=3 so E is eliminated. Round 2 votes counts: B=30 D=28 C=26 A=16 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:222 E:218 C:204 D:188 A:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -8 -6 -28 B 22 0 0 16 6 C 8 0 0 4 -4 D 6 -16 -4 0 -10 E 28 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -8 -6 -28 B 22 0 0 16 6 C 8 0 0 4 -4 D 6 -16 -4 0 -10 E 28 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -8 -6 -28 B 22 0 0 16 6 C 8 0 0 4 -4 D 6 -16 -4 0 -10 E 28 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1577: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (12) A D E C B (9) B C D E A (8) A D B E C (8) B C E D A (7) E C B D A (5) C E D B A (4) B D A C E (4) E C A D B (3) E A C D B (3) C E B D A (3) A E D C B (3) A B D C E (3) E C D A B (2) D A B C E (2) B E C D A (2) B D C E A (2) B D C A E (2) A B D E C (2) E D C A B (1) E C D B A (1) D E C A B (1) D E A C B (1) D C A E B (1) D B C A E (1) D A E C B (1) D A C E B (1) C D E B A (1) C D E A B (1) C B E D A (1) B C D A E (1) B A E C D (1) B A D E C (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 6 -6 8 B 12 0 14 12 16 C -6 -14 0 -12 8 D 6 -12 12 0 24 E -8 -16 -8 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -6 8 B 12 0 14 12 16 C -6 -14 0 -12 8 D 6 -12 12 0 24 E -8 -16 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=27 E=15 C=10 D=8 so D is eliminated. Round 2 votes counts: B=41 A=31 E=17 C=11 so C is eliminated. Round 3 votes counts: B=42 A=32 E=26 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:227 D:215 A:198 C:188 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 -6 8 B 12 0 14 12 16 C -6 -14 0 -12 8 D 6 -12 12 0 24 E -8 -16 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -6 8 B 12 0 14 12 16 C -6 -14 0 -12 8 D 6 -12 12 0 24 E -8 -16 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -6 8 B 12 0 14 12 16 C -6 -14 0 -12 8 D 6 -12 12 0 24 E -8 -16 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1578: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) B E A D C (9) C D A E B (7) E A D B C (5) B C E A D (5) C B E A D (4) C B D E A (4) C B D A E (4) B A E D C (4) B A D E C (4) E A D C B (3) D A E C B (3) D A E B C (3) C B E D A (3) B E C A D (3) E B A D C (2) E A B D C (2) C E B D A (2) B D A C E (2) B C D A E (2) A E D B C (2) A D B E C (2) E D A C B (1) E C B A D (1) E C A D B (1) D E C A B (1) D C A E B (1) D C A B E (1) D B A C E (1) D A C E B (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A B E (1) B D A E C (1) B C A D E (1) Total count = 100 A B C D E A 0 -6 -10 -2 -18 B 6 0 -4 6 4 C 10 4 0 4 6 D 2 -6 -4 0 -18 E 18 -4 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -2 -18 B 6 0 -4 6 4 C 10 4 0 4 6 D 2 -6 -4 0 -18 E 18 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=31 E=15 D=11 A=4 so A is eliminated. Round 2 votes counts: C=39 B=31 E=17 D=13 so D is eliminated. Round 3 votes counts: C=42 B=34 E=24 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:212 B:206 D:187 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -2 -18 B 6 0 -4 6 4 C 10 4 0 4 6 D 2 -6 -4 0 -18 E 18 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -2 -18 B 6 0 -4 6 4 C 10 4 0 4 6 D 2 -6 -4 0 -18 E 18 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -2 -18 B 6 0 -4 6 4 C 10 4 0 4 6 D 2 -6 -4 0 -18 E 18 -4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1579: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) B C E D A (6) B A D C E (6) A D E C B (6) E C D B A (5) D C E A B (5) C E B D A (5) A D B C E (5) E D C A B (4) A B D E C (4) A B D C E (4) D E C A B (3) D A E C B (3) C E D B A (3) A D E B C (3) A B E D C (3) E C B D A (2) B E C A D (2) B C D E A (2) B A E C D (2) B A C E D (2) B A C D E (2) A E D C B (2) E C B A D (1) E A D C B (1) D E A C B (1) D A C B E (1) C B E D A (1) B E A C D (1) B C E A D (1) B A E D C (1) A E C D B (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 0 -2 -6 B -16 0 -8 -8 -10 C 0 8 0 -6 -14 D 2 8 6 0 -8 E 6 10 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 0 -2 -6 B -16 0 -8 -8 -10 C 0 8 0 -6 -14 D 2 8 6 0 -8 E 6 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=25 E=23 D=13 C=9 so C is eliminated. Round 2 votes counts: E=31 A=30 B=26 D=13 so D is eliminated. Round 3 votes counts: E=40 A=34 B=26 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:204 D:204 C:194 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 0 -2 -6 B -16 0 -8 -8 -10 C 0 8 0 -6 -14 D 2 8 6 0 -8 E 6 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 -2 -6 B -16 0 -8 -8 -10 C 0 8 0 -6 -14 D 2 8 6 0 -8 E 6 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 -2 -6 B -16 0 -8 -8 -10 C 0 8 0 -6 -14 D 2 8 6 0 -8 E 6 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1580: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) C D B E A (7) A E B D C (7) A B C D E (7) E D C B A (4) C E D B A (4) A B E D C (4) E A B D C (3) C E D A B (3) A B D C E (3) A B C E D (3) E C D A B (2) E B A D C (2) E A D C B (2) D C B E A (2) C D A B E (2) B E D A C (2) B E A D C (2) B D C E A (2) B D A E C (2) A E B C D (2) A C E D B (2) A C B D E (2) E D A C B (1) E C D B A (1) E B D A C (1) E A D B C (1) E A C D B (1) D C E B A (1) D B C E A (1) C D B A E (1) C B D A E (1) C A E D B (1) B D E C A (1) B D C A E (1) B D A C E (1) B A E D C (1) B A D E C (1) B A D C E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 12 0 -6 B -2 0 -2 6 -2 C -12 2 0 8 14 D 0 -6 -8 0 -8 E 6 2 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.437500 B: 0.000000 C: 0.187500 D: 0.000000 E: 0.375000 Sum of squares = 0.367187499944 Cumulative probabilities = A: 0.437500 B: 0.437500 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 2 12 0 -6 B -2 0 -2 6 -2 C -12 2 0 8 14 D 0 -6 -8 0 -8 E 6 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.000000 C: 0.187500 D: 0.000000 E: 0.375000 Sum of squares = 0.3671875 Cumulative probabilities = A: 0.437500 B: 0.437500 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=29 E=18 B=14 D=4 so D is eliminated. Round 2 votes counts: A=35 C=32 E=18 B=15 so B is eliminated. Round 3 votes counts: A=41 C=36 E=23 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:206 A:204 E:201 B:200 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 0 -6 B -2 0 -2 6 -2 C -12 2 0 8 14 D 0 -6 -8 0 -8 E 6 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.000000 C: 0.187500 D: 0.000000 E: 0.375000 Sum of squares = 0.3671875 Cumulative probabilities = A: 0.437500 B: 0.437500 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 0 -6 B -2 0 -2 6 -2 C -12 2 0 8 14 D 0 -6 -8 0 -8 E 6 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.000000 C: 0.187500 D: 0.000000 E: 0.375000 Sum of squares = 0.3671875 Cumulative probabilities = A: 0.437500 B: 0.437500 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 0 -6 B -2 0 -2 6 -2 C -12 2 0 8 14 D 0 -6 -8 0 -8 E 6 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.000000 C: 0.187500 D: 0.000000 E: 0.375000 Sum of squares = 0.3671875 Cumulative probabilities = A: 0.437500 B: 0.437500 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1581: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) A C B D E (7) C B A D E (6) A D C B E (6) D A E C B (5) A D E C B (5) E B C D A (4) D A C B E (4) C B E D A (4) C B E A D (4) E D B A C (3) E D B C A (2) E D A B C (2) E B D C A (2) E B C A D (2) D E A B C (2) C D B A E (2) C B D A E (2) C B A E D (2) C A D B E (2) C A B D E (2) B E C A D (2) B C E D A (2) A E D B C (2) A D C E B (2) A C D B E (2) E B D A C (1) E B A C D (1) E A B C D (1) D E C B A (1) D E C A B (1) D E A C B (1) D C B A E (1) D C A B E (1) B C A E D (1) A E B D C (1) A E B C D (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -4 20 12 B 2 0 -22 10 20 C 4 22 0 14 20 D -20 -10 -14 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 20 12 B 2 0 -22 10 20 C 4 22 0 14 20 D -20 -10 -14 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=24 E=18 D=16 B=14 so B is eliminated. Round 2 votes counts: C=36 A=28 E=20 D=16 so D is eliminated. Round 3 votes counts: C=38 A=37 E=25 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:230 A:213 B:205 D:181 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 20 12 B 2 0 -22 10 20 C 4 22 0 14 20 D -20 -10 -14 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 20 12 B 2 0 -22 10 20 C 4 22 0 14 20 D -20 -10 -14 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 20 12 B 2 0 -22 10 20 C 4 22 0 14 20 D -20 -10 -14 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1582: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) E D B A C (8) C B A D E (7) C A D E B (5) B C A D E (5) B A C E D (5) A C B E D (5) D E C A B (4) B C D A E (4) E A C D B (3) D E C B A (3) D E B A C (3) B E D A C (3) B D E C A (3) B D C E A (3) E D A B C (2) E A D C B (2) D B E C A (2) C A D B E (2) B A E D C (2) B A E C D (2) A E C D B (2) A C E B D (2) A B C E D (2) E D A C B (1) D E B C A (1) D C E A B (1) D C B E A (1) C D E A B (1) C A E D B (1) B D E A C (1) B D C A E (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -8 16 16 B 8 0 -4 14 16 C 8 4 0 18 12 D -16 -14 -18 0 -6 E -16 -16 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 16 16 B 8 0 -4 14 16 C 8 4 0 18 12 D -16 -14 -18 0 -6 E -16 -16 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=26 E=16 D=15 A=14 so A is eliminated. Round 2 votes counts: C=34 B=31 E=20 D=15 so D is eliminated. Round 3 votes counts: C=36 B=33 E=31 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:217 A:208 E:181 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 16 16 B 8 0 -4 14 16 C 8 4 0 18 12 D -16 -14 -18 0 -6 E -16 -16 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 16 16 B 8 0 -4 14 16 C 8 4 0 18 12 D -16 -14 -18 0 -6 E -16 -16 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 16 16 B 8 0 -4 14 16 C 8 4 0 18 12 D -16 -14 -18 0 -6 E -16 -16 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1583: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (14) B A D C E (13) E C B D A (8) C E D A B (8) B A D E C (8) D A C E B (6) E B C A D (5) D A E C B (4) B E C A D (4) C D A B E (3) B C E A D (3) A D B E C (3) D A C B E (2) C D E A B (2) C D A E B (2) A D B C E (2) E C D B A (1) E A D B C (1) D C A E B (1) C B E D A (1) C B D A E (1) B E A D C (1) B C E D A (1) B C D A E (1) B C A D E (1) A D E C B (1) A D E B C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -12 -10 2 B -4 0 -8 -2 -8 C 12 8 0 10 -4 D 10 2 -10 0 6 E -2 8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.200000 E: 0.500000 Sum of squares = 0.38 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.500000 E: 1.000000 A B C D E A 0 4 -12 -10 2 B -4 0 -8 -2 -8 C 12 8 0 10 -4 D 10 2 -10 0 6 E -2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=29 C=17 D=13 A=9 so A is eliminated. Round 2 votes counts: B=34 E=29 D=20 C=17 so C is eliminated. Round 3 votes counts: E=37 B=36 D=27 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:213 D:204 E:202 A:192 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -10 2 B -4 0 -8 -2 -8 C 12 8 0 10 -4 D 10 2 -10 0 6 E -2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -10 2 B -4 0 -8 -2 -8 C 12 8 0 10 -4 D 10 2 -10 0 6 E -2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -10 2 B -4 0 -8 -2 -8 C 12 8 0 10 -4 D 10 2 -10 0 6 E -2 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1584: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) B A E C D (11) C D B A E (8) D C E A B (6) B C A D E (6) A E B D C (5) D C A B E (4) C D E B A (4) C D B E A (4) C B D A E (4) A B E D C (4) D C E B A (3) E B A C D (2) E A D B C (2) D C A E B (2) A E D B C (2) A B D C E (2) E D C A B (1) E D A C B (1) E C D B A (1) E B C D A (1) D E C A B (1) D E A C B (1) D A E C B (1) C B D E A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C E A D (1) B A C E D (1) B A C D E (1) A D B C E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 12 10 B 2 0 16 16 8 C -2 -16 0 0 0 D -12 -16 0 0 2 E -10 -8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 12 10 B 2 0 16 16 8 C -2 -16 0 0 0 D -12 -16 0 0 2 E -10 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993065 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=21 C=21 D=18 A=17 so A is eliminated. Round 2 votes counts: B=32 E=28 C=21 D=19 so D is eliminated. Round 3 votes counts: C=36 B=33 E=31 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:211 C:191 E:190 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 12 10 B 2 0 16 16 8 C -2 -16 0 0 0 D -12 -16 0 0 2 E -10 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993065 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 12 10 B 2 0 16 16 8 C -2 -16 0 0 0 D -12 -16 0 0 2 E -10 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993065 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 12 10 B 2 0 16 16 8 C -2 -16 0 0 0 D -12 -16 0 0 2 E -10 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993065 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1585: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (12) C E A B D (10) D B C E A (9) C E B D A (7) B D C E A (7) A E C B D (6) D B A C E (5) B C E D A (5) D B A E C (4) D A B E C (4) B D E C A (4) A D B E C (4) A D E C B (3) E C A B D (2) E A C B D (2) A E D C B (2) A D E B C (2) A D B C E (2) A C E B D (2) E C B D A (1) E C B A D (1) D A B C E (1) C E B A D (1) C D B E A (1) C B D E A (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 8 2 2 -2 B -8 0 -6 -2 -6 C -2 6 0 4 6 D -2 2 -4 0 -4 E 2 6 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000053 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 8 2 2 -2 B -8 0 -6 -2 -6 C -2 6 0 4 6 D -2 2 -4 0 -4 E 2 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=23 C=20 B=16 E=6 so E is eliminated. Round 2 votes counts: A=37 C=24 D=23 B=16 so B is eliminated. Round 3 votes counts: A=37 D=34 C=29 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:205 E:203 D:196 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 2 -2 B -8 0 -6 -2 -6 C -2 6 0 4 6 D -2 2 -4 0 -4 E 2 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 -2 B -8 0 -6 -2 -6 C -2 6 0 4 6 D -2 2 -4 0 -4 E 2 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 -2 B -8 0 -6 -2 -6 C -2 6 0 4 6 D -2 2 -4 0 -4 E 2 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1586: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (13) C A B E D (8) A C B E D (8) B D E C A (6) E D A C B (5) D E A B C (5) D E B C A (4) B C A E D (4) A C E B D (4) E D B C A (3) C A B D E (3) B C D A E (3) B C A D E (3) A E C D B (3) A C E D B (3) E B D C A (2) D B E C A (2) D A E C B (2) C B A E D (2) A D C E B (2) A C D E B (2) A C B D E (2) E D A B C (1) E C B A D (1) D E A C B (1) D B E A C (1) C B A D E (1) B E D C A (1) B E C D A (1) B C E A D (1) B C D E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 2 8 -2 4 B -2 0 2 2 -6 C -8 -2 0 0 -6 D 2 -2 0 0 4 E -4 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -2 4 B -2 0 2 2 -6 C -8 -2 0 0 -6 D 2 -2 0 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333317 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=26 B=20 C=14 E=12 so E is eliminated. Round 2 votes counts: D=37 A=26 B=22 C=15 so C is eliminated. Round 3 votes counts: D=37 A=37 B=26 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:206 D:202 E:202 B:198 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 -2 4 B -2 0 2 2 -6 C -8 -2 0 0 -6 D 2 -2 0 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333317 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -2 4 B -2 0 2 2 -6 C -8 -2 0 0 -6 D 2 -2 0 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333317 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -2 4 B -2 0 2 2 -6 C -8 -2 0 0 -6 D 2 -2 0 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333317 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1587: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) B E D C A (8) D E B A C (7) B E D A C (7) A C D E B (7) C B A E D (5) E B D A C (4) D A E C B (4) C A B E D (4) B C E D A (4) E D B A C (3) D B E C A (3) C A B D E (3) A E D C B (3) E B A D C (2) E A D B C (2) C A D E B (2) B E C A D (2) A D E C B (2) A D C E B (2) D C E B A (1) D C A E B (1) D B E A C (1) C D B E A (1) C D A E B (1) C D A B E (1) C B E A D (1) C B D A E (1) C B A D E (1) C A D B E (1) B E C D A (1) B D C E A (1) B C E A D (1) B C D E A (1) A E C D B (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 12 -16 -16 B 10 0 12 -4 -6 C -12 -12 0 -18 -18 D 16 4 18 0 -2 E 16 6 18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 12 -16 -16 B 10 0 12 -4 -6 C -12 -12 0 -18 -18 D 16 4 18 0 -2 E 16 6 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 C=21 A=18 E=11 so E is eliminated. Round 2 votes counts: B=31 D=28 C=21 A=20 so A is eliminated. Round 3 votes counts: D=37 B=32 C=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:221 D:218 B:206 A:185 C:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 12 -16 -16 B 10 0 12 -4 -6 C -12 -12 0 -18 -18 D 16 4 18 0 -2 E 16 6 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 12 -16 -16 B 10 0 12 -4 -6 C -12 -12 0 -18 -18 D 16 4 18 0 -2 E 16 6 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 12 -16 -16 B 10 0 12 -4 -6 C -12 -12 0 -18 -18 D 16 4 18 0 -2 E 16 6 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1588: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) A C B E D (9) A C E B D (7) B D C A E (6) E A D C B (5) D B E C A (5) D B C E A (5) C A B D E (5) B C D A E (5) E D B A C (4) E D A C B (4) C B A D E (4) B C A D E (4) A E C B D (4) C A B E D (3) E D C A B (2) E D A B C (2) B D C E A (2) B A C D E (2) E A D B C (1) E A C D B (1) D E B A C (1) D B E A C (1) C D B E A (1) C D B A E (1) C B D A E (1) B D E C A (1) B D E A C (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -12 -6 6 B 10 0 4 10 12 C 12 -4 0 -2 10 D 6 -10 2 0 12 E -6 -12 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -6 6 B 10 0 4 10 12 C 12 -4 0 -2 10 D 6 -10 2 0 12 E -6 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=22 B=21 E=19 C=15 so C is eliminated. Round 2 votes counts: A=30 B=26 D=25 E=19 so E is eliminated. Round 3 votes counts: D=37 A=37 B=26 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:218 C:208 D:205 A:189 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -6 6 B 10 0 4 10 12 C 12 -4 0 -2 10 D 6 -10 2 0 12 E -6 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -6 6 B 10 0 4 10 12 C 12 -4 0 -2 10 D 6 -10 2 0 12 E -6 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -6 6 B 10 0 4 10 12 C 12 -4 0 -2 10 D 6 -10 2 0 12 E -6 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1589: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) B E A D C (10) B C D E A (5) E D C A B (4) D C E A B (4) C D E A B (4) B C A D E (4) E D C B A (3) E B D A C (3) E A D B C (3) C A D E B (3) B E D C A (3) B E D A C (3) B A C D E (3) A C D E B (3) E D A C B (2) E B A D C (2) D E C A B (2) C A D B E (2) C A B D E (2) B E A C D (2) B C D A E (2) A D C E B (2) A B E D C (2) E B D C A (1) E A B D C (1) D C A E B (1) C D B E A (1) C D B A E (1) C D A B E (1) B E C D A (1) B C A E D (1) B A E D C (1) B A E C D (1) B A C E D (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E C B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -10 -2 -8 B -4 0 4 2 -4 C 10 -4 0 -2 4 D 2 -2 2 0 6 E 8 4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.109574 B: 0.390426 C: 0.000000 D: 0.406382 E: 0.093618 Sum of squares = 0.338350070407 Cumulative probabilities = A: 0.109574 B: 0.500000 C: 0.500000 D: 0.906382 E: 1.000000 A B C D E A 0 4 -10 -2 -8 B -4 0 4 2 -4 C 10 -4 0 -2 4 D 2 -2 2 0 6 E 8 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.134615 B: 0.365385 C: 0.000000 D: 0.423077 E: 0.076923 Sum of squares = 0.336538461538 Cumulative probabilities = A: 0.134615 B: 0.500000 C: 0.500000 D: 0.923077 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=24 E=19 A=13 D=7 so D is eliminated. Round 2 votes counts: B=37 C=29 E=21 A=13 so A is eliminated. Round 3 votes counts: B=40 C=35 E=25 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:204 D:204 E:201 B:199 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -10 -2 -8 B -4 0 4 2 -4 C 10 -4 0 -2 4 D 2 -2 2 0 6 E 8 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.134615 B: 0.365385 C: 0.000000 D: 0.423077 E: 0.076923 Sum of squares = 0.336538461538 Cumulative probabilities = A: 0.134615 B: 0.500000 C: 0.500000 D: 0.923077 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -2 -8 B -4 0 4 2 -4 C 10 -4 0 -2 4 D 2 -2 2 0 6 E 8 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.134615 B: 0.365385 C: 0.000000 D: 0.423077 E: 0.076923 Sum of squares = 0.336538461538 Cumulative probabilities = A: 0.134615 B: 0.500000 C: 0.500000 D: 0.923077 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -2 -8 B -4 0 4 2 -4 C 10 -4 0 -2 4 D 2 -2 2 0 6 E 8 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.134615 B: 0.365385 C: 0.000000 D: 0.423077 E: 0.076923 Sum of squares = 0.336538461538 Cumulative probabilities = A: 0.134615 B: 0.500000 C: 0.500000 D: 0.923077 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1590: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) A D C B E (7) E A D B C (6) C B E D A (6) B C E D A (6) A E D C B (6) C B D E A (5) A D E C B (5) C B D A E (4) E C B A D (3) E B D A C (3) E B C D A (3) E B C A D (3) B C D E A (3) E C A B D (2) E A D C B (2) D A E B C (2) D A C B E (2) D A B E C (2) D A B C E (2) C E B A D (2) C E A B D (2) B E C D A (2) B D C E A (2) E B A C D (1) E A B C D (1) D E B A C (1) D B E A C (1) D B C A E (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B E D (1) A E D B C (1) A E C D B (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 8 4 10 -8 B -8 0 -4 0 -6 C -4 4 0 -2 -6 D -10 0 2 0 -6 E 8 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 4 10 -8 B -8 0 -4 0 -6 C -4 4 0 -2 -6 D -10 0 2 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 C=23 B=13 D=11 so D is eliminated. Round 2 votes counts: A=37 E=25 C=23 B=15 so B is eliminated. Round 3 votes counts: A=37 C=35 E=28 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:213 A:207 C:196 D:193 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 4 10 -8 B -8 0 -4 0 -6 C -4 4 0 -2 -6 D -10 0 2 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 10 -8 B -8 0 -4 0 -6 C -4 4 0 -2 -6 D -10 0 2 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 10 -8 B -8 0 -4 0 -6 C -4 4 0 -2 -6 D -10 0 2 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1591: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) A E C B D (9) D B C E A (8) E B D A C (7) C D B A E (7) C A E D B (7) B D E C A (7) E A C B D (5) B D E A C (5) D B C A E (4) B D C E A (4) A E C D B (4) D B E A C (3) C B D A E (3) C A D B E (3) E A D B C (2) C B D E A (2) A C E D B (2) A C E B D (2) E B D C A (1) D C B A E (1) B E D A C (1) Total count = 100 A B C D E A 0 -6 6 -6 -16 B 6 0 10 18 -4 C -6 -10 0 -12 -14 D 6 -18 12 0 -6 E 16 4 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 -6 -16 B 6 0 10 18 -4 C -6 -10 0 -12 -14 D 6 -18 12 0 -6 E 16 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 B=17 A=17 D=16 so D is eliminated. Round 2 votes counts: B=32 E=28 C=23 A=17 so A is eliminated. Round 3 votes counts: E=41 B=32 C=27 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:215 D:197 A:189 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 -6 -16 B 6 0 10 18 -4 C -6 -10 0 -12 -14 D 6 -18 12 0 -6 E 16 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -6 -16 B 6 0 10 18 -4 C -6 -10 0 -12 -14 D 6 -18 12 0 -6 E 16 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -6 -16 B 6 0 10 18 -4 C -6 -10 0 -12 -14 D 6 -18 12 0 -6 E 16 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1592: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) E B D A C (8) A C E B D (8) D B E C A (6) C A D B E (6) D B A C E (4) A C D B E (4) E D B C A (3) C A E D B (3) C A E B D (3) B E D C A (3) B D E A C (3) A E C B D (3) E C B A D (2) D B C E A (2) D B C A E (2) C E A D B (2) C D E B A (2) C D A B E (2) C A D E B (2) B E D A C (2) E C D B A (1) E C D A B (1) E C B D A (1) E C A B D (1) E B A D C (1) E B A C D (1) E A C B D (1) D C B A E (1) D B A E C (1) D A C B E (1) D A B C E (1) C E D A B (1) C D B E A (1) B D E C A (1) A E B D C (1) A D B C E (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 -16 -8 B 12 0 6 6 -12 C 14 -6 0 -6 -4 D 16 -6 6 0 -16 E 8 12 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -14 -16 -8 B 12 0 6 6 -12 C 14 -6 0 -6 -4 D 16 -6 6 0 -16 E 8 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 A=20 D=18 B=9 so B is eliminated. Round 2 votes counts: E=36 D=22 C=22 A=20 so A is eliminated. Round 3 votes counts: E=41 C=35 D=24 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:206 D:200 C:199 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -14 -16 -8 B 12 0 6 6 -12 C 14 -6 0 -6 -4 D 16 -6 6 0 -16 E 8 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -16 -8 B 12 0 6 6 -12 C 14 -6 0 -6 -4 D 16 -6 6 0 -16 E 8 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -16 -8 B 12 0 6 6 -12 C 14 -6 0 -6 -4 D 16 -6 6 0 -16 E 8 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1593: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) D A C E B (6) B E C A D (5) B E A C D (5) B C E A D (5) D A E C B (4) C D A B E (4) C B D E A (4) A D E B C (4) E B A C D (3) E A B D C (3) D A E B C (3) C D B A E (3) C B E D A (3) C B E A D (3) B E D A C (3) B C E D A (3) B C D E A (3) A E D B C (3) E B A D C (2) E A B C D (2) D C A E B (2) D C A B E (2) D B C A E (2) D A C B E (2) B E A D C (2) A D C E B (2) E B C A D (1) E A D B C (1) D C B A E (1) D B A E C (1) D B A C E (1) D A B C E (1) C E A D B (1) C E A B D (1) B D A E C (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 0 -12 0 B 2 0 8 -2 8 C 0 -8 0 6 12 D 12 2 -6 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999993 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -12 0 B 2 0 8 -2 8 C 0 -8 0 6 12 D 12 2 -6 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000075 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 D=25 E=12 A=10 so A is eliminated. Round 2 votes counts: D=32 B=27 C=26 E=15 so E is eliminated. Round 3 votes counts: B=38 D=36 C=26 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:208 D:208 C:205 A:193 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -12 0 B 2 0 8 -2 8 C 0 -8 0 6 12 D 12 2 -6 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000075 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -12 0 B 2 0 8 -2 8 C 0 -8 0 6 12 D 12 2 -6 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000075 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -12 0 B 2 0 8 -2 8 C 0 -8 0 6 12 D 12 2 -6 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000075 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1594: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (24) E A D C B (23) E A D B C (8) D A C B E (6) A E D C B (6) C B D A E (5) B C D E A (5) B C E D A (4) E B C A D (2) E A B D C (2) D C B A E (2) A D E C B (2) E C A B D (1) E A C D B (1) D C A B E (1) D A C E B (1) C B E A D (1) C B A D E (1) B E C D A (1) B D C A E (1) B C E A D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 0 2 B -6 0 -4 -4 4 C -2 4 0 -6 8 D 0 4 6 0 0 E -2 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.554752 B: 0.000000 C: 0.000000 D: 0.445248 E: 0.000000 Sum of squares = 0.505995587765 Cumulative probabilities = A: 0.554752 B: 0.554752 C: 0.554752 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 0 2 B -6 0 -4 -4 4 C -2 4 0 -6 8 D 0 4 6 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=36 D=10 A=10 C=7 so C is eliminated. Round 2 votes counts: B=43 E=37 D=10 A=10 so D is eliminated. Round 3 votes counts: B=45 E=37 A=18 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:205 D:205 C:202 B:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 0 2 B -6 0 -4 -4 4 C -2 4 0 -6 8 D 0 4 6 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 2 B -6 0 -4 -4 4 C -2 4 0 -6 8 D 0 4 6 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 2 B -6 0 -4 -4 4 C -2 4 0 -6 8 D 0 4 6 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1595: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (5) C B A E D (5) E C A D B (4) E A C D B (4) D E B A C (4) D E A B C (4) B D A C E (4) B C A D E (4) A C B E D (4) E D C A B (3) E D A B C (3) C A E B D (3) B C D A E (3) B A C D E (3) A D B C E (3) A C E B D (3) E D C B A (2) E C D A B (2) E A D C B (2) D B E C A (2) D A E B C (2) C E B D A (2) C E A B D (2) C B E A D (2) C B A D E (2) B D C E A (2) B D C A E (2) A B C D E (2) E D B C A (1) E C D B A (1) D E B C A (1) D B E A C (1) D B A E C (1) D A B E C (1) C B E D A (1) B E C D A (1) B D E C A (1) B C D E A (1) B A D C E (1) A E D C B (1) A E C D B (1) A D E B C (1) A D B E C (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 6 6 0 -2 B -6 0 0 -2 -4 C -6 0 0 2 2 D 0 2 -2 0 -6 E 2 4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000007 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 6 6 0 -2 B -6 0 0 -2 -4 C -6 0 0 2 2 D 0 2 -2 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000013 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=22 A=18 C=17 D=16 so D is eliminated. Round 2 votes counts: E=36 B=26 A=21 C=17 so C is eliminated. Round 3 votes counts: E=40 B=36 A=24 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:205 E:205 C:199 D:197 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 0 -2 B -6 0 0 -2 -4 C -6 0 0 2 2 D 0 2 -2 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000013 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 0 -2 B -6 0 0 -2 -4 C -6 0 0 2 2 D 0 2 -2 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000013 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 0 -2 B -6 0 0 -2 -4 C -6 0 0 2 2 D 0 2 -2 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000013 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1596: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) B D A C E (10) E C A D B (8) A C B D E (8) D B E C A (7) D B A E C (7) A B D C E (7) E D B C A (5) D B E A C (5) A C E B D (5) E C A B D (4) E C D B A (3) E A C D B (3) A C B E D (3) D E B C A (2) C A E B D (2) E D C B A (1) E D B A C (1) E A D B C (1) D B C E A (1) D B A C E (1) C B D E A (1) C B D A E (1) B D E C A (1) B D C A E (1) A E D B C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 6 -6 B -6 0 2 8 8 C -6 -2 0 -2 2 D -6 -8 2 0 6 E 6 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999982 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 6 6 6 -6 B -6 0 2 8 8 C -6 -2 0 -2 2 D -6 -8 2 0 6 E 6 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999998 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=25 D=23 C=14 B=12 so B is eliminated. Round 2 votes counts: D=35 E=26 A=25 C=14 so C is eliminated. Round 3 votes counts: D=37 E=36 A=27 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:206 B:206 D:197 C:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 -6 B -6 0 2 8 8 C -6 -2 0 -2 2 D -6 -8 2 0 6 E 6 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999998 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 -6 B -6 0 2 8 8 C -6 -2 0 -2 2 D -6 -8 2 0 6 E 6 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999998 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 -6 B -6 0 2 8 8 C -6 -2 0 -2 2 D -6 -8 2 0 6 E 6 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999998 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1597: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) C B A E D (7) A D E B C (7) D E C B A (5) B C D E A (5) A D B E C (5) E D C A B (4) A B D E C (4) D E B C A (3) C E B D A (3) C B E D A (3) B A D C E (3) B A C D E (3) A B C E D (3) E D C B A (2) E D A C B (2) E C D B A (2) D E B A C (2) D E A B C (2) D C E B A (2) C E D B A (2) C E B A D (2) C B E A D (2) B D C E A (2) B D A C E (2) E C D A B (1) E A D C B (1) D E C A B (1) D E A C B (1) D B C E A (1) D B A E C (1) C E A B D (1) C D B E A (1) B D C A E (1) B C D A E (1) B C A E D (1) B C A D E (1) A E D B C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -6 2 0 B 14 0 -2 -8 -4 C 6 2 0 -20 -4 D -2 8 20 0 8 E 0 4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.083333 C: 0.000000 D: 0.583333 E: 0.000000 Sum of squares = 0.458333333434 Cumulative probabilities = A: 0.333333 B: 0.416667 C: 0.416667 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 2 0 B 14 0 -2 -8 -4 C 6 2 0 -20 -4 D -2 8 20 0 8 E 0 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.083333 C: 0.000000 D: 0.583333 E: 0.000000 Sum of squares = 0.458333333332 Cumulative probabilities = A: 0.333333 B: 0.416667 C: 0.416667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=21 B=19 D=18 E=12 so E is eliminated. Round 2 votes counts: A=31 D=26 C=24 B=19 so B is eliminated. Round 3 votes counts: A=37 C=32 D=31 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 B:200 E:200 C:192 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 2 0 B 14 0 -2 -8 -4 C 6 2 0 -20 -4 D -2 8 20 0 8 E 0 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.083333 C: 0.000000 D: 0.583333 E: 0.000000 Sum of squares = 0.458333333332 Cumulative probabilities = A: 0.333333 B: 0.416667 C: 0.416667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 2 0 B 14 0 -2 -8 -4 C 6 2 0 -20 -4 D -2 8 20 0 8 E 0 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.083333 C: 0.000000 D: 0.583333 E: 0.000000 Sum of squares = 0.458333333332 Cumulative probabilities = A: 0.333333 B: 0.416667 C: 0.416667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 2 0 B 14 0 -2 -8 -4 C 6 2 0 -20 -4 D -2 8 20 0 8 E 0 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.083333 C: 0.000000 D: 0.583333 E: 0.000000 Sum of squares = 0.458333333332 Cumulative probabilities = A: 0.333333 B: 0.416667 C: 0.416667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1598: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) C E D A B (6) D B A E C (5) D A B E C (5) C E A B D (5) C D E A B (4) B A D E C (4) E A B C D (3) D E A C B (3) D C E B A (3) D C B A E (3) C E B A D (3) C B E A D (3) C B A D E (3) B A E D C (3) B A E C D (3) B A D C E (3) E D A B C (2) D E C A B (2) D E A B C (2) D C E A B (2) D B C A E (2) C D E B A (2) C D B E A (2) C D B A E (2) A B E D C (2) E C D A B (1) E A D B C (1) D C B E A (1) D B A C E (1) D A E B C (1) C E B D A (1) C E A D B (1) C B D E A (1) C B A E D (1) B D A C E (1) B C A E D (1) B C A D E (1) B A C E D (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 0 -4 -8 B 0 0 2 -4 0 C 0 -2 0 -8 6 D 4 4 8 0 8 E 8 0 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 -8 B 0 0 2 -4 0 C 0 -2 0 -8 6 D 4 4 8 0 8 E 8 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=30 B=17 E=13 A=6 so A is eliminated. Round 2 votes counts: C=34 D=31 B=20 E=15 so E is eliminated. Round 3 votes counts: C=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:199 C:198 E:197 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -4 -8 B 0 0 2 -4 0 C 0 -2 0 -8 6 D 4 4 8 0 8 E 8 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 -8 B 0 0 2 -4 0 C 0 -2 0 -8 6 D 4 4 8 0 8 E 8 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 -8 B 0 0 2 -4 0 C 0 -2 0 -8 6 D 4 4 8 0 8 E 8 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1599: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) D A E C B (6) B C E D A (6) A D B E C (6) D E A C B (5) E D C A B (4) E C D A B (4) D E C A B (4) C E D B A (4) E C D B A (3) C E B D A (3) C B E D A (3) B D C E A (3) B A D C E (3) B A C E D (3) A D E B C (3) E C A D B (2) E A C D B (2) D E C B A (2) C E A B D (2) B C A E D (2) B A D E C (2) B A C D E (2) A B D E C (2) E D C B A (1) D E B C A (1) D B E C A (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B A D (1) C D B E A (1) C B E A D (1) B D A C E (1) B C E A D (1) A E D C B (1) A E C D B (1) A C E B D (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 -10 -10 B -10 0 -20 -22 -16 C 0 20 0 -10 -18 D 10 22 10 0 4 E 10 16 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -10 -10 B -10 0 -20 -22 -16 C 0 20 0 -10 -18 D 10 22 10 0 4 E 10 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 D=21 E=16 C=16 so E is eliminated. Round 2 votes counts: D=26 A=26 C=25 B=23 so B is eliminated. Round 3 votes counts: A=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:223 E:220 C:196 A:195 B:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -10 -10 B -10 0 -20 -22 -16 C 0 20 0 -10 -18 D 10 22 10 0 4 E 10 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -10 -10 B -10 0 -20 -22 -16 C 0 20 0 -10 -18 D 10 22 10 0 4 E 10 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -10 -10 B -10 0 -20 -22 -16 C 0 20 0 -10 -18 D 10 22 10 0 4 E 10 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1600: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) E B D C A (7) A D B E C (7) C A E D B (6) C A E B D (6) D B A E C (5) C A D B E (5) E B C D A (4) C D A B E (4) E C B D A (3) E B D A C (3) C E A B D (3) C D B E A (3) A C E D B (3) E B A D C (2) E A B D C (2) D A B E C (2) B D E A C (2) A D C B E (2) A D B C E (2) A C D B E (2) E C A B D (1) E A B C D (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B C E (1) C D E B A (1) C D B A E (1) C B E D A (1) C A D E B (1) B D E C A (1) A E D B C (1) A E C B D (1) A E B D C (1) A E B C D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -18 -2 8 B -8 0 -8 -2 -16 C 18 8 0 16 8 D 2 2 -16 0 -12 E -8 16 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 -2 8 B -8 0 -8 -2 -16 C 18 8 0 16 8 D 2 2 -16 0 -12 E -8 16 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=23 A=22 D=12 B=3 so B is eliminated. Round 2 votes counts: C=40 E=23 A=22 D=15 so D is eliminated. Round 3 votes counts: C=42 A=30 E=28 so E is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:206 A:198 D:188 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 -2 8 B -8 0 -8 -2 -16 C 18 8 0 16 8 D 2 2 -16 0 -12 E -8 16 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -2 8 B -8 0 -8 -2 -16 C 18 8 0 16 8 D 2 2 -16 0 -12 E -8 16 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -2 8 B -8 0 -8 -2 -16 C 18 8 0 16 8 D 2 2 -16 0 -12 E -8 16 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1601: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) B C A D E (8) C A B E D (7) D B E A C (6) C A E B D (6) E A C B D (5) E D A C B (4) E C A D B (4) E A C D B (4) B D A C E (4) E C A B D (3) D E A B C (3) C B A E D (3) E D A B C (2) E A D B C (2) D E B A C (2) D B A E C (2) C E A D B (2) B C D A E (2) A E B C D (2) A C E B D (2) A B C E D (2) E D B A C (1) E A B C D (1) D E C A B (1) D E B C A (1) D C B E A (1) D C B A E (1) D B E C A (1) C E A B D (1) C B D A E (1) C B A D E (1) B D C A E (1) B A E D C (1) B A E C D (1) B A C D E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -6 16 12 B -6 0 4 8 6 C 6 -4 0 16 6 D -16 -8 -16 0 -10 E -12 -6 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 16 12 B -6 0 4 8 6 C 6 -4 0 16 6 D -16 -8 -16 0 -10 E -12 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=26 C=21 B=18 A=8 so A is eliminated. Round 2 votes counts: E=29 D=27 C=24 B=20 so B is eliminated. Round 3 votes counts: C=37 D=32 E=31 so E is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:212 B:206 E:193 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 16 12 B -6 0 4 8 6 C 6 -4 0 16 6 D -16 -8 -16 0 -10 E -12 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 16 12 B -6 0 4 8 6 C 6 -4 0 16 6 D -16 -8 -16 0 -10 E -12 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 16 12 B -6 0 4 8 6 C 6 -4 0 16 6 D -16 -8 -16 0 -10 E -12 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1602: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (14) B D A E C (10) C E B A D (6) D A B E C (5) E C A D B (4) D A B C E (4) E C A B D (3) D B A C E (3) C E B D A (3) B E C D A (3) B D E C A (3) B D A C E (3) A D E C B (3) A D C E B (3) A C D E B (3) E B C D A (2) D B A E C (2) C E A B D (2) C A E D B (2) B D E A C (2) A D B E C (2) A C E D B (2) E C B D A (1) E C B A D (1) E B A D C (1) E A D C B (1) E A C D B (1) C B E D A (1) C A D E B (1) C A D B E (1) B E D C A (1) B D C E A (1) B D C A E (1) B C D E A (1) A E D C B (1) A E C D B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -2 8 -2 B -10 0 -10 -10 -10 C 2 10 0 4 6 D -8 10 -4 0 0 E 2 10 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 8 -2 B -10 0 -10 -10 -10 C 2 10 0 4 6 D -8 10 -4 0 0 E 2 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998336 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=25 A=17 E=14 D=14 so E is eliminated. Round 2 votes counts: C=39 B=28 A=19 D=14 so D is eliminated. Round 3 votes counts: C=39 B=33 A=28 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:207 E:203 D:199 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 8 -2 B -10 0 -10 -10 -10 C 2 10 0 4 6 D -8 10 -4 0 0 E 2 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998336 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 8 -2 B -10 0 -10 -10 -10 C 2 10 0 4 6 D -8 10 -4 0 0 E 2 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998336 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 8 -2 B -10 0 -10 -10 -10 C 2 10 0 4 6 D -8 10 -4 0 0 E 2 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998336 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1603: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) C D A E B (9) C A E D B (6) D E A B C (5) C B A E D (5) B E A D C (5) C B D E A (4) B C E A D (4) D E A C B (3) D A E C B (3) C B D A E (3) C A D E B (3) B D E A C (3) E A D B C (2) D E B A C (2) D B E C A (2) C D B A E (2) C B A D E (2) B C D E A (2) B C A E D (2) A E C D B (2) E D A B C (1) E B A D C (1) E A B D C (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A E B (1) D B C E A (1) D A C E B (1) C D E A B (1) C D A B E (1) B E D A C (1) B E C A D (1) B E A C D (1) B D C E A (1) B A E D C (1) B A C E D (1) A E B D C (1) A D E C B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -6 2 12 B -8 0 -22 -20 -12 C 6 22 0 2 6 D -2 20 -2 0 6 E -12 12 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 2 12 B -8 0 -22 -20 -12 C 6 22 0 2 6 D -2 20 -2 0 6 E -12 12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=22 D=21 A=16 E=5 so E is eliminated. Round 2 votes counts: C=36 B=23 D=22 A=19 so A is eliminated. Round 3 votes counts: C=39 D=35 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:211 A:208 E:194 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 2 12 B -8 0 -22 -20 -12 C 6 22 0 2 6 D -2 20 -2 0 6 E -12 12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 2 12 B -8 0 -22 -20 -12 C 6 22 0 2 6 D -2 20 -2 0 6 E -12 12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 2 12 B -8 0 -22 -20 -12 C 6 22 0 2 6 D -2 20 -2 0 6 E -12 12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1604: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (7) C B D E A (6) E A B C D (5) D E A C B (5) D C E B A (5) D A E C B (4) B C E A D (4) A D E B C (4) A B E C D (4) E C D B A (3) E B A C D (3) D C B A E (3) D A C B E (3) A E B D C (3) A D B E C (3) E C B D A (2) E A D C B (2) E A D B C (2) E A B D C (2) D E C A B (2) D C B E A (2) C D E B A (2) C D B E A (2) C D B A E (2) B C D A E (2) A E D B C (2) A E B C D (2) E D C B A (1) E D C A B (1) E C B A D (1) E B C A D (1) C B D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) A E D C B (1) A D E C B (1) A D B C E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 16 8 -2 B 0 0 2 -2 -8 C -16 -2 0 4 -8 D -8 2 -4 0 4 E 2 8 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428567 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.428571 E: 1.000000 A B C D E A 0 0 16 8 -2 B 0 0 2 -2 -8 C -16 -2 0 4 -8 D -8 2 -4 0 4 E 2 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=23 B=16 C=13 so C is eliminated. Round 2 votes counts: D=30 A=24 E=23 B=23 so E is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:207 D:197 B:196 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 16 8 -2 B 0 0 2 -2 -8 C -16 -2 0 4 -8 D -8 2 -4 0 4 E 2 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.428571 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 8 -2 B 0 0 2 -2 -8 C -16 -2 0 4 -8 D -8 2 -4 0 4 E 2 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 8 -2 B 0 0 2 -2 -8 C -16 -2 0 4 -8 D -8 2 -4 0 4 E 2 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.428571 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1605: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (13) D B C A E (11) D B A E C (11) C E A D B (10) D C B E A (7) C D E B A (5) E A C B D (4) C E D A B (4) B D A E C (4) A E B D C (3) A E B C D (3) E C A B D (2) E A B C D (2) D C B A E (2) D B C E A (2) C D E A B (2) C D B E A (2) B A E D C (2) B A D E C (2) A B E D C (2) E B A D C (1) E B A C D (1) E A B D C (1) D B E C A (1) D B A C E (1) C E B D A (1) C D A E B (1) Total count = 100 A B C D E A 0 -6 -26 -8 -16 B 6 0 -6 -18 -6 C 26 6 0 0 22 D 8 18 0 0 2 E 16 6 -22 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.465640 D: 0.534360 E: 0.000000 Sum of squares = 0.502361161921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.465640 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -26 -8 -16 B 6 0 -6 -18 -6 C 26 6 0 0 22 D 8 18 0 0 2 E 16 6 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=35 E=11 B=8 A=8 so B is eliminated. Round 2 votes counts: D=39 C=38 A=12 E=11 so E is eliminated. Round 3 votes counts: C=40 D=39 A=21 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:214 E:199 B:188 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -26 -8 -16 B 6 0 -6 -18 -6 C 26 6 0 0 22 D 8 18 0 0 2 E 16 6 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -26 -8 -16 B 6 0 -6 -18 -6 C 26 6 0 0 22 D 8 18 0 0 2 E 16 6 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -26 -8 -16 B 6 0 -6 -18 -6 C 26 6 0 0 22 D 8 18 0 0 2 E 16 6 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1606: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (14) E B A C D (8) B E A C D (7) A C D B E (7) D C A E B (6) C D A B E (4) A C B D E (4) E B D C A (3) D C E A B (3) B A C D E (3) A C D E B (3) A B C E D (3) A B C D E (3) E D C B A (2) D C B E A (2) B E C D A (2) E D B C A (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A D C (1) E A D C B (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C E A (1) D B C A E (1) D A C E B (1) C A D B E (1) B E D C A (1) B D E C A (1) B D C E A (1) B A E C D (1) B A C E D (1) A E B C D (1) A D C E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 0 0 14 B -14 0 -8 -6 22 C 0 8 0 6 24 D 0 6 -6 0 22 E -14 -22 -24 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.406212 B: 0.000000 C: 0.593788 D: 0.000000 E: 0.000000 Sum of squares = 0.517592255682 Cumulative probabilities = A: 0.406212 B: 0.406212 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 0 14 B -14 0 -8 -6 22 C 0 8 0 6 24 D 0 6 -6 0 22 E -14 -22 -24 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=24 E=21 B=17 C=5 so C is eliminated. Round 2 votes counts: D=37 A=25 E=21 B=17 so B is eliminated. Round 3 votes counts: D=39 E=31 A=30 so A is eliminated. Round 4 votes counts: D=61 E=39 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:219 A:214 D:211 B:197 E:159 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 0 14 B -14 0 -8 -6 22 C 0 8 0 6 24 D 0 6 -6 0 22 E -14 -22 -24 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 0 14 B -14 0 -8 -6 22 C 0 8 0 6 24 D 0 6 -6 0 22 E -14 -22 -24 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 0 14 B -14 0 -8 -6 22 C 0 8 0 6 24 D 0 6 -6 0 22 E -14 -22 -24 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1607: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) B D E C A (7) A C B D E (7) E C D B A (6) A C E D B (6) D B E C A (5) A D B E C (5) E D B C A (4) E D B A C (3) C E B D A (3) A D B C E (3) A B C D E (3) D E B A C (2) D B E A C (2) D B A E C (2) C A E D B (2) C A E B D (2) C A B D E (2) B D C E A (2) B D C A E (2) B D A C E (2) B C D E A (2) A E D B C (2) A B D C E (2) E C B D A (1) E B D C A (1) E A D B C (1) D E B C A (1) D E A B C (1) D A B E C (1) C E A B D (1) C B E A D (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B E D (1) B D E A C (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 0 8 2 10 B 0 0 8 2 6 C -8 -8 0 -2 4 D -2 -2 2 0 10 E -10 -6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.411316 B: 0.588684 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.515729572151 Cumulative probabilities = A: 0.411316 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 10 B 0 0 8 2 6 C -8 -8 0 -2 4 D -2 -2 2 0 10 E -10 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=16 B=16 C=15 D=14 so D is eliminated. Round 2 votes counts: A=40 B=25 E=20 C=15 so C is eliminated. Round 3 votes counts: A=47 B=29 E=24 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:208 D:204 C:193 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 10 B 0 0 8 2 6 C -8 -8 0 -2 4 D -2 -2 2 0 10 E -10 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 10 B 0 0 8 2 6 C -8 -8 0 -2 4 D -2 -2 2 0 10 E -10 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 10 B 0 0 8 2 6 C -8 -8 0 -2 4 D -2 -2 2 0 10 E -10 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1608: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (13) A B D E C (11) C E B A D (7) A B E C D (7) C E D B A (6) D C E B A (5) D C E A B (5) D A C E B (5) E C B D A (4) A D B C E (4) D A B E C (3) D A B C E (3) C E B D A (3) C D E B A (3) E C B A D (2) D E C A B (2) D A C B E (2) B E C A D (2) B A C E D (2) A B E D C (2) A B C E D (2) E D C B A (1) E C D B A (1) E B C A D (1) D E C B A (1) B A E D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 14 12 14 B 4 0 6 16 8 C -14 -6 0 6 -4 D -12 -16 -6 0 -8 E -14 -8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 12 14 B 4 0 6 16 8 C -14 -6 0 6 -4 D -12 -16 -6 0 -8 E -14 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 C=19 B=18 E=9 so E is eliminated. Round 2 votes counts: A=28 D=27 C=26 B=19 so B is eliminated. Round 3 votes counts: A=44 C=29 D=27 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:217 E:195 C:191 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 12 14 B 4 0 6 16 8 C -14 -6 0 6 -4 D -12 -16 -6 0 -8 E -14 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 12 14 B 4 0 6 16 8 C -14 -6 0 6 -4 D -12 -16 -6 0 -8 E -14 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 12 14 B 4 0 6 16 8 C -14 -6 0 6 -4 D -12 -16 -6 0 -8 E -14 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1609: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (16) A E C B D (8) D A B C E (7) E C B A D (6) D B C A E (6) A D E B C (6) D B C E A (5) C B D E A (5) B C D E A (5) A D B C E (4) D B A C E (3) C B E D A (3) C B D A E (2) B D C E A (2) A E D C B (2) A D C B E (2) E D A B C (1) E C B D A (1) E C A B D (1) E B D C A (1) E B C D A (1) E A D C B (1) D B E C A (1) D A E B C (1) D A B E C (1) C B E A D (1) C B A D E (1) B E C D A (1) B C E D A (1) B C D A E (1) A D E C B (1) A D B E C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 12 4 -4 B -8 0 -4 16 6 C -12 4 0 10 0 D -4 -16 -10 0 10 E 4 -6 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.444444 Sum of squares = 0.358024690712 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 0.555556 D: 0.555556 E: 1.000000 A B C D E A 0 8 12 4 -4 B -8 0 -4 16 6 C -12 4 0 10 0 D -4 -16 -10 0 10 E 4 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691206 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 D=24 C=12 B=10 so B is eliminated. Round 2 votes counts: E=29 D=26 A=26 C=19 so C is eliminated. Round 3 votes counts: D=39 E=34 A=27 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:210 B:205 C:201 E:194 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 4 -4 B -8 0 -4 16 6 C -12 4 0 10 0 D -4 -16 -10 0 10 E 4 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691206 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 4 -4 B -8 0 -4 16 6 C -12 4 0 10 0 D -4 -16 -10 0 10 E 4 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691206 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 4 -4 B -8 0 -4 16 6 C -12 4 0 10 0 D -4 -16 -10 0 10 E 4 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691206 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1610: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) E A B C D (10) B A D C E (7) E C A D B (6) D B A C E (6) C D A B E (6) E C D A B (5) E C D B A (4) C E D A B (4) B A E D C (4) A B E D C (4) A B D C E (3) E B A C D (2) D C A B E (2) C E A D B (2) C D E A B (2) B D A C E (2) B A D E C (2) E C B A D (1) E C A B D (1) E B D A C (1) E B A D C (1) E A B D C (1) D E C B A (1) D C B E A (1) D A B C E (1) C D E B A (1) B E D A C (1) A E C B D (1) A D B C E (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 6 4 12 B -10 0 -2 -6 12 C -6 2 0 2 8 D -4 6 -2 0 -4 E -12 -12 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 4 12 B -10 0 -2 -6 12 C -6 2 0 2 8 D -4 6 -2 0 -4 E -12 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=22 B=16 C=15 A=15 so C is eliminated. Round 2 votes counts: E=38 D=31 B=16 A=15 so A is eliminated. Round 3 votes counts: E=40 D=32 B=28 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:216 C:203 D:198 B:197 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 4 12 B -10 0 -2 -6 12 C -6 2 0 2 8 D -4 6 -2 0 -4 E -12 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 12 B -10 0 -2 -6 12 C -6 2 0 2 8 D -4 6 -2 0 -4 E -12 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 12 B -10 0 -2 -6 12 C -6 2 0 2 8 D -4 6 -2 0 -4 E -12 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998211 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1611: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) E A B C D (7) D C B A E (6) A B E C D (6) D E C B A (5) D C E B A (5) A E B C D (5) E B A C D (4) D C A B E (4) C D B E A (4) C B D E A (4) E A B D C (3) B E A C D (3) A B C E D (3) E B C A D (2) D A E C B (2) D A C E B (2) B E C A D (2) A E B D C (2) A C D B E (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) E B C D A (1) E B A D C (1) D C A E B (1) D A C B E (1) C D B A E (1) C D A B E (1) C B D A E (1) B C E A D (1) B C A E D (1) A E D B C (1) A D E B C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -6 -8 -16 B 12 0 -8 -6 4 C 6 8 0 -2 0 D 8 6 2 0 6 E 16 -4 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -8 -16 B 12 0 -8 -6 4 C 6 8 0 -2 0 D 8 6 2 0 6 E 16 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=23 A=22 C=11 B=7 so B is eliminated. Round 2 votes counts: D=37 E=28 A=22 C=13 so C is eliminated. Round 3 votes counts: D=48 E=29 A=23 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:206 E:203 B:201 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -8 -16 B 12 0 -8 -6 4 C 6 8 0 -2 0 D 8 6 2 0 6 E 16 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -8 -16 B 12 0 -8 -6 4 C 6 8 0 -2 0 D 8 6 2 0 6 E 16 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -8 -16 B 12 0 -8 -6 4 C 6 8 0 -2 0 D 8 6 2 0 6 E 16 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1612: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) D A B C E (8) E C B D A (7) C E B A D (7) A D B E C (6) A D E C B (5) D A B E C (4) B D E C A (4) B D C E A (4) A E C D B (4) E C A D B (3) E C A B D (3) C E A B D (3) C B E D A (3) B C E D A (3) A E D C B (3) A C D B E (3) C E B D A (2) C A E D B (2) B C D E A (2) A D C E B (2) A D B C E (2) A C D E B (2) E C B A D (1) E A C D B (1) D B A E C (1) B E D C A (1) B E C D A (1) B D E A C (1) B D C A E (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 6 6 -2 8 B -6 0 -4 -14 6 C -6 4 0 -4 8 D 2 14 4 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -2 8 B -6 0 -4 -14 6 C -6 4 0 -4 8 D 2 14 4 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 C=17 B=17 E=15 so E is eliminated. Round 2 votes counts: C=31 A=30 D=22 B=17 so B is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:209 C:201 B:191 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -2 8 B -6 0 -4 -14 6 C -6 4 0 -4 8 D 2 14 4 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -2 8 B -6 0 -4 -14 6 C -6 4 0 -4 8 D 2 14 4 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -2 8 B -6 0 -4 -14 6 C -6 4 0 -4 8 D 2 14 4 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1613: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A C D E B (8) C A D B E (7) E B D A C (6) C A B E D (6) B E C D A (5) A D E C B (5) C A E B D (4) C A D E B (4) B E D A C (4) E D B A C (3) D E B A C (3) D E A B C (3) C E B A D (3) C B E A D (3) B D E A C (3) E C B A D (2) D A C E B (2) C B A D E (2) C A E D B (2) B C E A D (2) A E D C B (2) A D C E B (2) E B C A D (1) E A D B C (1) D B E A C (1) D A C B E (1) C D A B E (1) B E C A D (1) B D E C A (1) B D C E A (1) B C D E A (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 14 -4 B 0 0 -12 6 -4 C 8 12 0 8 0 D -14 -6 -8 0 -8 E 4 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285770 D: 0.000000 E: 0.714229 Sum of squares = 0.591788556947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285771 D: 0.285771 E: 1.000000 A B C D E A 0 0 -8 14 -4 B 0 0 -12 6 -4 C 8 12 0 8 0 D -14 -6 -8 0 -8 E 4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=26 A=19 E=13 D=10 so D is eliminated. Round 2 votes counts: C=32 B=27 A=22 E=19 so E is eliminated. Round 3 votes counts: B=40 C=34 A=26 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:208 A:201 B:195 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 14 -4 B 0 0 -12 6 -4 C 8 12 0 8 0 D -14 -6 -8 0 -8 E 4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 14 -4 B 0 0 -12 6 -4 C 8 12 0 8 0 D -14 -6 -8 0 -8 E 4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 14 -4 B 0 0 -12 6 -4 C 8 12 0 8 0 D -14 -6 -8 0 -8 E 4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1614: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) C E A D B (6) E C A D B (5) C E D A B (5) A E B C D (5) E A C B D (4) C D E A B (4) B A E D C (4) E C A B D (3) E A B C D (3) D C B E A (3) D C B A E (3) C D E B A (3) C A B D E (3) B D A E C (3) E D B A C (2) D E B A C (2) D B E A C (2) D B A E C (2) C D B A E (2) C A B E D (2) B D A C E (2) B A C D E (2) A C E B D (2) A B E D C (2) A B E C D (2) E D A C B (1) E D A B C (1) E C D A B (1) E B A D C (1) D E C B A (1) D E B C A (1) D B C E A (1) D B A C E (1) C D B E A (1) C D A E B (1) C A D B E (1) B A D E C (1) A E C B D (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -10 -2 0 B -8 0 -6 -16 -6 C 10 6 0 14 4 D 2 16 -14 0 -4 E 0 6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 -2 0 B -8 0 -6 -16 -6 C 10 6 0 14 4 D 2 16 -14 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=25 E=21 A=14 B=12 so B is eliminated. Round 2 votes counts: D=30 C=28 E=21 A=21 so E is eliminated. Round 3 votes counts: C=37 D=34 A=29 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:203 D:200 A:198 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 -2 0 B -8 0 -6 -16 -6 C 10 6 0 14 4 D 2 16 -14 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 -2 0 B -8 0 -6 -16 -6 C 10 6 0 14 4 D 2 16 -14 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 -2 0 B -8 0 -6 -16 -6 C 10 6 0 14 4 D 2 16 -14 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1615: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) B D C A E (10) E A C D B (7) B C D E A (7) E C A B D (6) D B A E C (5) D A B E C (5) C B E A D (5) A E D C B (5) E A C B D (4) D B A C E (4) C A E D B (3) A D E B C (3) E A D C B (2) D B C A E (2) C B E D A (2) C B D E A (2) B D E A C (2) A E C D B (2) A D E C B (2) E A D B C (1) D A E B C (1) D A B C E (1) C E B A D (1) B D C E A (1) B C E D A (1) B C E A D (1) B C D A E (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 12 -8 12 -8 B -12 0 -8 10 -2 C 8 8 0 8 8 D -12 -10 -8 0 -6 E 8 2 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -8 12 -8 B -12 0 -8 10 -2 C 8 8 0 8 8 D -12 -10 -8 0 -6 E 8 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 E=20 D=18 A=14 so A is eliminated. Round 2 votes counts: E=28 C=25 D=24 B=23 so B is eliminated. Round 3 votes counts: D=37 C=35 E=28 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:204 E:204 B:194 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -8 12 -8 B -12 0 -8 10 -2 C 8 8 0 8 8 D -12 -10 -8 0 -6 E 8 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 12 -8 B -12 0 -8 10 -2 C 8 8 0 8 8 D -12 -10 -8 0 -6 E 8 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 12 -8 B -12 0 -8 10 -2 C 8 8 0 8 8 D -12 -10 -8 0 -6 E 8 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1616: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) B A C E D (9) D B E C A (7) B D A C E (6) E C A D B (5) A C E B D (5) D B A C E (4) B C A E D (4) C E A B D (3) A B C E D (3) E D C A B (2) E C B A D (2) E C A B D (2) D E C B A (2) D E A C B (2) D B E A C (2) D B A E C (2) D A B E C (2) C E B A D (2) C B E A D (2) B D E C A (2) A E C D B (2) A D C E B (2) E C D A B (1) D E B C A (1) D B C E A (1) D A E C B (1) D A E B C (1) D A B C E (1) B E D C A (1) B E C A D (1) B D C E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B A C D E (1) A E C B D (1) A D E C B (1) A D B C E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -4 -2 -2 B 6 0 8 -2 8 C 4 -8 0 -8 0 D 2 2 8 0 4 E 2 -8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -2 -2 B 6 0 8 -2 8 C 4 -8 0 -8 0 D 2 2 8 0 4 E 2 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=28 A=17 E=12 C=7 so C is eliminated. Round 2 votes counts: D=36 B=30 E=17 A=17 so E is eliminated. Round 3 votes counts: D=39 B=34 A=27 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:208 E:195 C:194 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 -2 B 6 0 8 -2 8 C 4 -8 0 -8 0 D 2 2 8 0 4 E 2 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 -2 B 6 0 8 -2 8 C 4 -8 0 -8 0 D 2 2 8 0 4 E 2 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 -2 B 6 0 8 -2 8 C 4 -8 0 -8 0 D 2 2 8 0 4 E 2 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1617: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) D E C A B (5) C E A D B (5) B A C E D (5) E C D A B (4) E B C A D (4) E B A C D (4) A C B E D (4) A B C E D (4) D E C B A (3) D E B C A (3) D C E A B (3) C A D E B (3) B E D A C (3) E D C A B (2) E C B D A (2) E C A D B (2) E C A B D (2) E B D C A (2) D E B A C (2) D B E A C (2) D A B C E (2) C E D A B (2) C D A E B (2) A C D B E (2) A B C D E (2) E D C B A (1) E D B C A (1) E C D B A (1) E B C D A (1) D C A E B (1) D B A C E (1) D A C B E (1) C E A B D (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B D E (1) B E A D C (1) B D E A C (1) B D A C E (1) B A E C D (1) B A D C E (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -8 8 -28 B -4 0 -2 0 -16 C 8 2 0 28 -8 D -8 0 -28 0 -22 E 28 16 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999207 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -8 8 -28 B -4 0 -2 0 -16 C 8 2 0 28 -8 D -8 0 -28 0 -22 E 28 16 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 B=21 C=17 A=13 so A is eliminated. Round 2 votes counts: B=27 E=26 C=24 D=23 so D is eliminated. Round 3 votes counts: E=39 B=32 C=29 so C is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:237 C:215 B:189 A:188 D:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 8 -28 B -4 0 -2 0 -16 C 8 2 0 28 -8 D -8 0 -28 0 -22 E 28 16 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 8 -28 B -4 0 -2 0 -16 C 8 2 0 28 -8 D -8 0 -28 0 -22 E 28 16 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 8 -28 B -4 0 -2 0 -16 C 8 2 0 28 -8 D -8 0 -28 0 -22 E 28 16 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1618: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) A D B E C (8) E C B D A (6) E B C D A (6) A C D E B (5) C E B D A (4) E C A B D (3) D B A E C (3) C E D B A (3) C D A B E (3) C A D E B (3) B D E C A (3) A B E D C (3) D B C A E (2) C E A D B (2) C D E B A (2) C D B E A (2) B E D A C (2) B E C D A (2) B D E A C (2) A E C B D (2) A E B C D (2) E C B A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A C E (1) D A C B E (1) C E A B D (1) C D A E B (1) C A E D B (1) B E D C A (1) B E A D C (1) B D A E C (1) B C D E A (1) B A E D C (1) B A D E C (1) A E D B C (1) A E B D C (1) A D C E B (1) A D B C E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 2 8 B -2 0 -6 -6 4 C 0 6 0 6 -6 D -2 6 -6 0 8 E -8 -4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.599986 B: 0.000000 C: 0.400014 D: 0.000000 E: 0.000000 Sum of squares = 0.519994275153 Cumulative probabilities = A: 0.599986 B: 0.599986 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 8 B -2 0 -6 -6 4 C 0 6 0 6 -6 D -2 6 -6 0 8 E -8 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=22 E=18 B=15 D=10 so D is eliminated. Round 2 votes counts: A=36 C=24 B=22 E=18 so E is eliminated. Round 3 votes counts: A=37 C=34 B=29 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:206 C:203 D:203 B:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 2 8 B -2 0 -6 -6 4 C 0 6 0 6 -6 D -2 6 -6 0 8 E -8 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 8 B -2 0 -6 -6 4 C 0 6 0 6 -6 D -2 6 -6 0 8 E -8 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 8 B -2 0 -6 -6 4 C 0 6 0 6 -6 D -2 6 -6 0 8 E -8 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1619: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (6) B A D C E (6) B D A C E (5) A B D C E (5) E C D B A (4) E C B D A (4) E B A C D (4) D B A C E (4) B E A D C (4) E C A B D (3) D C E B A (3) D A B C E (3) C D A E B (3) A C D E B (3) E C D A B (2) E C A D B (2) D C E A B (2) D C B A E (2) D B E C A (2) C D E A B (2) B D A E C (2) B A E D C (2) B A D E C (2) A D C B E (2) A B C D E (2) E B D C A (1) E B C D A (1) E B C A D (1) E A C D B (1) E A C B D (1) E A B C D (1) D C B E A (1) D C A E B (1) C E A D B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B D C E A (1) A E C B D (1) A D C E B (1) A D B C E (1) A C E D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 10 -4 -4 B 8 0 6 6 0 C -10 -6 0 -4 12 D 4 -6 4 0 6 E 4 0 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.776054 C: 0.000000 D: 0.000000 E: 0.223946 Sum of squares = 0.652411596789 Cumulative probabilities = A: 0.000000 B: 0.776054 C: 0.776054 D: 0.776054 E: 1.000000 A B C D E A 0 -8 10 -4 -4 B 8 0 6 6 0 C -10 -6 0 -4 12 D 4 -6 4 0 6 E 4 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555568648 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 D=18 A=18 C=12 so C is eliminated. Round 2 votes counts: E=32 B=27 D=23 A=18 so A is eliminated. Round 3 votes counts: E=35 B=35 D=30 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:204 A:197 C:196 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 -4 -4 B 8 0 6 6 0 C -10 -6 0 -4 12 D 4 -6 4 0 6 E 4 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555568648 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 -4 -4 B 8 0 6 6 0 C -10 -6 0 -4 12 D 4 -6 4 0 6 E 4 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555568648 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 -4 -4 B 8 0 6 6 0 C -10 -6 0 -4 12 D 4 -6 4 0 6 E 4 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555568648 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1620: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) A D C B E (7) D C A B E (6) E B A D C (5) E B C D A (4) E B A C D (4) B E D C A (4) A D C E B (4) E A B C D (3) C D B E A (3) B D C E A (3) B D A E C (3) A C E D B (3) A C D E B (3) E C B A D (2) D B A E C (2) D A B C E (2) C E A B D (2) C D A B E (2) C A E D B (2) C A D E B (2) B E D A C (2) B E C D A (2) B D E A C (2) B D C A E (2) A E C D B (2) A D E C B (2) E C A B D (1) E B D A C (1) E B C A D (1) E A C D B (1) E A C B D (1) E A B D C (1) D B C A E (1) C E B D A (1) C D B A E (1) C D A E B (1) B C E D A (1) A E D C B (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 12 18 -2 12 B -12 0 -10 -8 2 C -18 10 0 -16 8 D 2 8 16 0 8 E -12 -2 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 18 -2 12 B -12 0 -10 -8 2 C -18 10 0 -16 8 D 2 8 16 0 8 E -12 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 D=19 B=19 C=14 so C is eliminated. Round 2 votes counts: A=28 E=27 D=26 B=19 so B is eliminated. Round 3 votes counts: E=36 D=36 A=28 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:220 D:217 C:192 B:186 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 18 -2 12 B -12 0 -10 -8 2 C -18 10 0 -16 8 D 2 8 16 0 8 E -12 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 18 -2 12 B -12 0 -10 -8 2 C -18 10 0 -16 8 D 2 8 16 0 8 E -12 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 18 -2 12 B -12 0 -10 -8 2 C -18 10 0 -16 8 D 2 8 16 0 8 E -12 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1621: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) C B D E A (6) E A B D C (4) D A C E B (4) C D A B E (4) C B E A D (4) C A E B D (4) C A D E B (4) B C D E A (4) A D C E B (4) C D B A E (3) B E C A D (3) B C E D A (3) A E D B C (3) A C D E B (3) E B A D C (2) E A B C D (2) D C B A E (2) D C A B E (2) D B E A C (2) C D A E B (2) C B E D A (2) C B D A E (2) B E D A C (2) B D E A C (2) B C E A D (2) A D E B C (2) E B A C D (1) E A D B C (1) D E B A C (1) D E A B C (1) D B C E A (1) D B A E C (1) D A E C B (1) C E A B D (1) C A D B E (1) B E D C A (1) B E A D C (1) B D C E A (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 8 -4 -12 6 B -8 0 -2 -6 -2 C 4 2 0 4 20 D 12 6 -4 0 24 E -6 2 -20 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -12 6 B -8 0 -2 -6 -2 C 4 2 0 4 20 D 12 6 -4 0 24 E -6 2 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=24 B=19 A=14 E=10 so E is eliminated. Round 2 votes counts: C=33 D=24 B=22 A=21 so A is eliminated. Round 3 votes counts: C=37 D=35 B=28 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:215 A:199 B:191 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -12 6 B -8 0 -2 -6 -2 C 4 2 0 4 20 D 12 6 -4 0 24 E -6 2 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -12 6 B -8 0 -2 -6 -2 C 4 2 0 4 20 D 12 6 -4 0 24 E -6 2 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -12 6 B -8 0 -2 -6 -2 C 4 2 0 4 20 D 12 6 -4 0 24 E -6 2 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1622: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) E A D B C (10) A D E C B (10) B C E D A (7) C B D A E (6) E C B D A (5) B C D E A (5) D C A B E (4) D A C B E (4) B C D A E (4) E B A C D (3) C B E D A (3) B E C A D (3) A E D C B (3) A E D B C (3) E D A C B (2) E A B D C (2) D C B A E (2) D A E C B (2) C D B A E (2) B C E A D (2) A D E B C (2) E A B C D (1) D A C E B (1) C B D E A (1) B E C D A (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -12 2 -12 B 10 0 10 8 -10 C 12 -10 0 8 -16 D -2 -8 -8 0 -12 E 12 10 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 2 -12 B 10 0 10 8 -10 C 12 -10 0 8 -16 D -2 -8 -8 0 -12 E 12 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=22 A=19 D=13 C=12 so C is eliminated. Round 2 votes counts: E=34 B=32 A=19 D=15 so D is eliminated. Round 3 votes counts: B=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:209 C:197 D:185 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 2 -12 B 10 0 10 8 -10 C 12 -10 0 8 -16 D -2 -8 -8 0 -12 E 12 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 2 -12 B 10 0 10 8 -10 C 12 -10 0 8 -16 D -2 -8 -8 0 -12 E 12 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 2 -12 B 10 0 10 8 -10 C 12 -10 0 8 -16 D -2 -8 -8 0 -12 E 12 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1623: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) C E B D A (8) A C E B D (7) D B E C A (6) D B A E C (6) D A B E C (6) B D E C A (5) D B C E A (4) C E B A D (4) B E C D A (4) A D C E B (4) E C B A D (3) D B E A C (2) D B A C E (2) C E A B D (2) C D B E A (2) B E D C A (2) A D C B E (2) A D B C E (2) A C E D B (2) A B E D C (2) E C B D A (1) E B C D A (1) E B C A D (1) D C A E B (1) D A C B E (1) C E A D B (1) C D E B A (1) C D E A B (1) C A E D B (1) C A E B D (1) B D A E C (1) A E C B D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 2 -8 4 B 6 0 14 -10 18 C -2 -14 0 -14 -8 D 8 10 14 0 14 E -4 -18 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -8 4 B 6 0 14 -10 18 C -2 -14 0 -14 -8 D 8 10 14 0 14 E -4 -18 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=28 C=21 B=12 E=6 so E is eliminated. Round 2 votes counts: A=33 D=28 C=25 B=14 so B is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:214 A:196 E:186 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -8 4 B 6 0 14 -10 18 C -2 -14 0 -14 -8 D 8 10 14 0 14 E -4 -18 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -8 4 B 6 0 14 -10 18 C -2 -14 0 -14 -8 D 8 10 14 0 14 E -4 -18 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -8 4 B 6 0 14 -10 18 C -2 -14 0 -14 -8 D 8 10 14 0 14 E -4 -18 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1624: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (10) B E C D A (8) E B C D A (4) E B A C D (4) E A B D C (4) E A B C D (4) D C B A E (4) C B D E A (4) A D C B E (4) A C D B E (4) E A D B C (3) C D B E A (3) C B D A E (3) B D C E A (3) A E B C D (3) E D B C A (2) E B D C A (2) D B E C A (2) D A C B E (2) C D B A E (2) A E C D B (2) E B A D C (1) E A D C B (1) D E B C A (1) D C B E A (1) D C A B E (1) C D A B E (1) B E D C A (1) B D E C A (1) B C E D A (1) A E C B D (1) A D E C B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -8 -12 -20 B 14 0 10 10 12 C 8 -10 0 12 -12 D 12 -10 -12 0 -4 E 20 -12 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -12 -20 B 14 0 10 10 12 C 8 -10 0 12 -12 D 12 -10 -12 0 -4 E 20 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 B=24 C=13 D=11 so D is eliminated. Round 2 votes counts: A=29 E=26 B=26 C=19 so C is eliminated. Round 3 votes counts: B=43 A=31 E=26 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:212 C:199 D:193 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 -12 -20 B 14 0 10 10 12 C 8 -10 0 12 -12 D 12 -10 -12 0 -4 E 20 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -12 -20 B 14 0 10 10 12 C 8 -10 0 12 -12 D 12 -10 -12 0 -4 E 20 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -12 -20 B 14 0 10 10 12 C 8 -10 0 12 -12 D 12 -10 -12 0 -4 E 20 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1625: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) C D B E A (6) C D A B E (6) B D E A C (6) B E D A C (5) E A B D C (4) D C B E A (4) C D B A E (4) C A E B D (4) E B D A C (3) D B C E A (3) C A E D B (3) C A D B E (3) D B E A C (2) D B A E C (2) C E B D A (2) C E A B D (2) B D E C A (2) A E D B C (2) A E C B D (2) A E B C D (2) A D E B C (2) A C E B D (2) E C A B D (1) E B D C A (1) E B A D C (1) E A C B D (1) E A B C D (1) D B A C E (1) D A B E C (1) C E B A D (1) C D E B A (1) C B E D A (1) C B D E A (1) C A D E B (1) B E A D C (1) A D B E C (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 8 -2 2 B -6 0 6 12 0 C -8 -6 0 -8 -6 D 2 -12 8 0 -4 E -2 0 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.542833 B: 0.042833 C: 0.000000 D: 0.271417 E: 0.142917 Sum of squares = 0.390594858456 Cumulative probabilities = A: 0.542833 B: 0.585667 C: 0.585667 D: 0.857083 E: 1.000000 A B C D E A 0 6 8 -2 2 B -6 0 6 12 0 C -8 -6 0 -8 -6 D 2 -12 8 0 -4 E -2 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500053 B: 0.000053 C: 0.000000 D: 0.250026 E: 0.249868 Sum of squares = 0.3750000236 Cumulative probabilities = A: 0.500053 B: 0.500105 C: 0.500105 D: 0.750132 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=26 B=14 D=13 E=12 so E is eliminated. Round 2 votes counts: C=36 A=32 B=19 D=13 so D is eliminated. Round 3 votes counts: C=40 A=33 B=27 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:207 B:206 E:204 D:197 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 -2 2 B -6 0 6 12 0 C -8 -6 0 -8 -6 D 2 -12 8 0 -4 E -2 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500053 B: 0.000053 C: 0.000000 D: 0.250026 E: 0.249868 Sum of squares = 0.3750000236 Cumulative probabilities = A: 0.500053 B: 0.500105 C: 0.500105 D: 0.750132 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -2 2 B -6 0 6 12 0 C -8 -6 0 -8 -6 D 2 -12 8 0 -4 E -2 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500053 B: 0.000053 C: 0.000000 D: 0.250026 E: 0.249868 Sum of squares = 0.3750000236 Cumulative probabilities = A: 0.500053 B: 0.500105 C: 0.500105 D: 0.750132 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -2 2 B -6 0 6 12 0 C -8 -6 0 -8 -6 D 2 -12 8 0 -4 E -2 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500053 B: 0.000053 C: 0.000000 D: 0.250026 E: 0.249868 Sum of squares = 0.3750000236 Cumulative probabilities = A: 0.500053 B: 0.500105 C: 0.500105 D: 0.750132 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1626: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (17) A B C D E (12) C B A E D (10) D E A B C (8) C B A D E (8) E C D B A (6) B A C D E (5) E D A B C (4) E C B A D (3) D A B C E (3) E A B D C (2) D E C B A (2) D A B E C (2) C E B A D (2) A B D C E (2) A B C E D (2) E D C A B (1) E C B D A (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) C E B D A (1) C D E B A (1) C B D A E (1) C A B E D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 -12 2 -2 B 18 0 -14 6 0 C 12 14 0 10 2 D -2 -6 -10 0 -6 E 2 0 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -12 2 -2 B 18 0 -14 6 0 C 12 14 0 10 2 D -2 -6 -10 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=24 A=18 D=17 B=5 so B is eliminated. Round 2 votes counts: E=36 C=24 A=23 D=17 so D is eliminated. Round 3 votes counts: E=46 A=28 C=26 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:219 B:205 E:203 D:188 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -12 2 -2 B 18 0 -14 6 0 C 12 14 0 10 2 D -2 -6 -10 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 2 -2 B 18 0 -14 6 0 C 12 14 0 10 2 D -2 -6 -10 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 2 -2 B 18 0 -14 6 0 C 12 14 0 10 2 D -2 -6 -10 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1627: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) E A C D B (6) C D E A B (6) B D A C E (6) B A D C E (5) E C D A B (4) D C B A E (4) B E A D C (4) B D C A E (4) A B E C D (4) D B E C A (3) D B C E A (3) B D E C A (3) B A E C D (3) A E B C D (3) E D B C A (2) E C A D B (2) D C E B A (2) C E D A B (2) C D A E B (2) C A D E B (2) B A E D C (2) A E C D B (2) A C E D B (2) A C D E B (2) A B C E D (2) E D C A B (1) E B D A C (1) E B A C D (1) E A B C D (1) D C B E A (1) D B C A E (1) C D A B E (1) B E D A C (1) B D C E A (1) B A D E C (1) A E C B D (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 0 -8 0 B 8 0 4 -10 4 C 0 -4 0 -2 -2 D 8 10 2 0 0 E 0 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.488662 E: 0.511338 Sum of squares = 0.500257085885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.488662 E: 1.000000 A B C D E A 0 -8 0 -8 0 B 8 0 4 -10 4 C 0 -4 0 -2 -2 D 8 10 2 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=24 A=19 D=14 C=13 so C is eliminated. Round 2 votes counts: B=30 E=26 D=23 A=21 so A is eliminated. Round 3 votes counts: B=38 E=34 D=28 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:210 B:203 E:199 C:196 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -8 0 B 8 0 4 -10 4 C 0 -4 0 -2 -2 D 8 10 2 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -8 0 B 8 0 4 -10 4 C 0 -4 0 -2 -2 D 8 10 2 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -8 0 B 8 0 4 -10 4 C 0 -4 0 -2 -2 D 8 10 2 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1628: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) C E B A D (8) D A B E C (7) A D E B C (7) D A B C E (6) A D C E B (5) A D C B E (5) C B D E A (4) C A D B E (4) A D E C B (4) C B D A E (3) B C E D A (3) E B D A C (2) E B C A D (2) D A E B C (2) C D A B E (2) C A E D B (2) B E C D A (2) B D C A E (2) B D A E C (2) A C D E B (2) E C B A D (1) E B C D A (1) D C A B E (1) D B A E C (1) D B A C E (1) C E A B D (1) C D B A E (1) C B E A D (1) B D E C A (1) B D E A C (1) B D C E A (1) A E D C B (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -4 -10 20 B -2 0 -14 -6 22 C 4 14 0 -2 28 D 10 6 2 0 26 E -20 -22 -28 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -10 20 B -2 0 -14 -6 22 C 4 14 0 -2 28 D 10 6 2 0 26 E -20 -22 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=26 D=18 B=12 E=6 so E is eliminated. Round 2 votes counts: C=39 A=26 D=18 B=17 so B is eliminated. Round 3 votes counts: C=47 D=27 A=26 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:222 D:222 A:204 B:200 E:152 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -10 20 B -2 0 -14 -6 22 C 4 14 0 -2 28 D 10 6 2 0 26 E -20 -22 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -10 20 B -2 0 -14 -6 22 C 4 14 0 -2 28 D 10 6 2 0 26 E -20 -22 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -10 20 B -2 0 -14 -6 22 C 4 14 0 -2 28 D 10 6 2 0 26 E -20 -22 -28 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1629: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (14) D E A C B (8) B C E D A (6) A C B E D (6) A B C E D (6) E C B D A (5) E D C B A (4) D B C E A (4) C B E A D (4) B C D E A (4) B C A E D (4) D E B C A (3) A D B C E (3) A C E B D (3) E A D C B (2) E A C B D (2) B C D A E (2) B A C E D (2) A E D C B (2) A E C B D (2) A B C D E (2) E C D B A (1) E C B A D (1) D E B A C (1) D E A B C (1) D B C A E (1) D A E C B (1) D A E B C (1) D A B E C (1) C B E D A (1) B D C E A (1) B C A D E (1) A D E C B (1) Total count = 100 A B C D E A 0 -18 -12 -18 -24 B 18 0 -14 4 -4 C 12 14 0 4 0 D 18 -4 -4 0 -2 E 24 4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.823135 D: 0.000000 E: 0.176865 Sum of squares = 0.708832559623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.823135 D: 0.823135 E: 1.000000 A B C D E A 0 -18 -12 -18 -24 B 18 0 -14 4 -4 C 12 14 0 4 0 D 18 -4 -4 0 -2 E 24 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=25 B=20 E=15 C=5 so C is eliminated. Round 2 votes counts: D=35 B=25 A=25 E=15 so E is eliminated. Round 3 votes counts: D=40 B=31 A=29 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:215 E:215 D:204 B:202 A:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -12 -18 -24 B 18 0 -14 4 -4 C 12 14 0 4 0 D 18 -4 -4 0 -2 E 24 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -18 -24 B 18 0 -14 4 -4 C 12 14 0 4 0 D 18 -4 -4 0 -2 E 24 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -18 -24 B 18 0 -14 4 -4 C 12 14 0 4 0 D 18 -4 -4 0 -2 E 24 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1630: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) E B A D C (7) C A D E B (6) A C E B D (6) D C B E A (5) D B E A C (5) C D B E A (5) C D A B E (5) B E D A C (5) B D E A C (5) A E B D C (5) C D B A E (4) A E C B D (4) D B C E A (3) C A E B D (3) B D E C A (3) A C E D B (3) A E B C D (2) E B D A C (1) E A C B D (1) E A B C D (1) D E B A C (1) D C A B E (1) D B E C A (1) C B E A D (1) C B D E A (1) C A E D B (1) C A D B E (1) B E D C A (1) B E A D C (1) B D C E A (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 0 16 6 -14 B 0 0 6 14 -4 C -16 -6 0 -12 -8 D -6 -14 12 0 -4 E 14 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 16 6 -14 B 0 0 6 14 -4 C -16 -6 0 -12 -8 D -6 -14 12 0 -4 E 14 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=22 E=19 D=16 B=16 so D is eliminated. Round 2 votes counts: C=33 B=25 A=22 E=20 so E is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:215 B:208 A:204 D:194 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 16 6 -14 B 0 0 6 14 -4 C -16 -6 0 -12 -8 D -6 -14 12 0 -4 E 14 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 6 -14 B 0 0 6 14 -4 C -16 -6 0 -12 -8 D -6 -14 12 0 -4 E 14 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 6 -14 B 0 0 6 14 -4 C -16 -6 0 -12 -8 D -6 -14 12 0 -4 E 14 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1631: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (16) D E C B A (10) E D A B C (8) D E B C A (8) A B C E D (8) E D C B A (6) E D A C B (5) B C D A E (4) B C A D E (4) E A D B C (3) D B C E A (3) A E B C D (3) A C B E D (3) E D C A B (2) E A D C B (2) D C E B A (2) A E C B D (2) A C E B D (2) A B E C D (2) A B C D E (2) D C B E A (1) C B D A E (1) B D C E A (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -12 -16 -2 -2 B 12 0 -4 -2 -10 C 16 4 0 -6 -6 D 2 2 6 0 4 E 2 10 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -2 -2 B 12 0 -4 -2 -10 C 16 4 0 -6 -6 D 2 2 6 0 4 E 2 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 A=24 C=17 B=9 so B is eliminated. Round 2 votes counts: E=26 D=25 C=25 A=24 so A is eliminated. Round 3 votes counts: C=40 E=35 D=25 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:207 E:207 C:204 B:198 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -16 -2 -2 B 12 0 -4 -2 -10 C 16 4 0 -6 -6 D 2 2 6 0 4 E 2 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -2 -2 B 12 0 -4 -2 -10 C 16 4 0 -6 -6 D 2 2 6 0 4 E 2 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -2 -2 B 12 0 -4 -2 -10 C 16 4 0 -6 -6 D 2 2 6 0 4 E 2 10 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1632: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) C E B D A (10) C E D B A (8) E C D A B (5) E C A D B (5) D A B C E (5) A D B E C (5) E C B A D (4) D B A C E (4) B D C E A (3) A D B C E (3) A B D E C (3) E A C D B (2) D C B E A (2) D B C A E (2) B D C A E (2) B A D C E (2) A E D B C (2) A D E B C (2) A B E D C (2) E C D B A (1) E C B D A (1) E C A B D (1) E A C B D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A B E (1) D A C B E (1) C E D A B (1) C D B E A (1) C B E D A (1) B A E C D (1) B A D E C (1) A E D C B (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 0 -24 2 B 12 0 4 -6 4 C 0 -4 0 -14 22 D 24 6 14 0 4 E -2 -4 -22 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -24 2 B 12 0 4 -6 4 C 0 -4 0 -14 22 D 24 6 14 0 4 E -2 -4 -22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=21 C=21 B=21 A=20 D=17 so D is eliminated. Round 2 votes counts: B=27 C=26 A=26 E=21 so E is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:224 B:207 C:202 E:184 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 0 -24 2 B 12 0 4 -6 4 C 0 -4 0 -14 22 D 24 6 14 0 4 E -2 -4 -22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -24 2 B 12 0 4 -6 4 C 0 -4 0 -14 22 D 24 6 14 0 4 E -2 -4 -22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -24 2 B 12 0 4 -6 4 C 0 -4 0 -14 22 D 24 6 14 0 4 E -2 -4 -22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1633: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) E C A D B (6) B D C A E (5) B D A C E (5) B C D E A (5) E A B C D (4) A D B E C (4) E C B A D (3) E C A B D (3) C E B D A (3) C B E D A (3) C B D E A (3) A E D C B (3) A E D B C (3) A E C D B (3) E C B D A (2) E B C D A (2) D B C A E (2) C D B E A (2) B E C D A (2) B A E D C (2) A E B D C (2) A D E C B (2) A D B C E (2) A B E D C (2) A B D E C (2) E B C A D (1) E A B D C (1) D C B A E (1) D A B C E (1) C E D B A (1) C E D A B (1) C D E A B (1) C D A B E (1) C B D A E (1) B E A D C (1) B C E D A (1) B A D E C (1) B A D C E (1) A D E B C (1) Total count = 100 A B C D E A 0 6 2 16 -12 B -6 0 0 10 -6 C -2 0 0 18 -22 D -16 -10 -18 0 -20 E 12 6 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 2 16 -12 B -6 0 0 10 -6 C -2 0 0 18 -22 D -16 -10 -18 0 -20 E 12 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=24 B=23 C=16 D=4 so D is eliminated. Round 2 votes counts: E=33 B=25 A=25 C=17 so C is eliminated. Round 3 votes counts: E=39 B=35 A=26 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 A:206 B:199 C:197 D:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 16 -12 B -6 0 0 10 -6 C -2 0 0 18 -22 D -16 -10 -18 0 -20 E 12 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 16 -12 B -6 0 0 10 -6 C -2 0 0 18 -22 D -16 -10 -18 0 -20 E 12 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 16 -12 B -6 0 0 10 -6 C -2 0 0 18 -22 D -16 -10 -18 0 -20 E 12 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1634: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) D E A B C (8) D B A E C (7) D E B A C (6) B A C D E (6) E D C A B (5) E C D A B (5) C A B E D (5) E D C B A (4) C E A B D (4) C B A D E (4) A B D E C (4) E C D B A (3) D A B E C (3) C B A E D (3) A B D C E (3) E C A D B (2) C E D B A (2) C E B A D (2) B D A E C (2) A B C D E (2) E D A B C (1) D E B C A (1) D A E B C (1) C E D A B (1) C E B D A (1) C A E B D (1) B D A C E (1) B C A D E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 12 -2 10 B 6 0 16 -2 4 C -12 -16 0 -16 -8 D 2 2 16 0 20 E -10 -4 8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -2 10 B 6 0 16 -2 4 C -12 -16 0 -16 -8 D 2 2 16 0 20 E -10 -4 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 E=20 B=20 A=11 so A is eliminated. Round 2 votes counts: B=29 D=27 C=23 E=21 so E is eliminated. Round 3 votes counts: D=38 C=33 B=29 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:212 A:207 E:187 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 12 -2 10 B 6 0 16 -2 4 C -12 -16 0 -16 -8 D 2 2 16 0 20 E -10 -4 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -2 10 B 6 0 16 -2 4 C -12 -16 0 -16 -8 D 2 2 16 0 20 E -10 -4 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -2 10 B 6 0 16 -2 4 C -12 -16 0 -16 -8 D 2 2 16 0 20 E -10 -4 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1635: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) D E B A C (7) D A E C B (7) D B E C A (6) A C E B D (6) C A B E D (5) B C D A E (5) D E A B C (4) C B A E D (4) C B A D E (4) B C E A D (4) A E D C B (4) E A D C B (3) D B C E A (3) A E C B D (3) E D A B C (2) E A C B D (2) D B C A E (2) D A C B E (2) B C E D A (2) B C D E A (2) A D E C B (2) E D B A C (1) E C B A D (1) E B C D A (1) E B C A D (1) D B E A C (1) D A E B C (1) C E B A D (1) C B E A D (1) C A B D E (1) B E C D A (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -6 -12 -4 B 12 0 4 6 6 C 6 -4 0 -8 4 D 12 -6 8 0 14 E 4 -6 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -12 -4 B 12 0 4 6 6 C 6 -4 0 -8 4 D 12 -6 8 0 14 E 4 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=23 A=17 C=16 E=11 so E is eliminated. Round 2 votes counts: D=36 B=25 A=22 C=17 so C is eliminated. Round 3 votes counts: D=36 B=36 A=28 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:214 C:199 E:190 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -12 -4 B 12 0 4 6 6 C 6 -4 0 -8 4 D 12 -6 8 0 14 E 4 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -12 -4 B 12 0 4 6 6 C 6 -4 0 -8 4 D 12 -6 8 0 14 E 4 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -12 -4 B 12 0 4 6 6 C 6 -4 0 -8 4 D 12 -6 8 0 14 E 4 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1636: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) B A D C E (6) A B E C D (6) C D E B A (5) A E B C D (5) E D C B A (4) D C B E A (4) C D B A E (4) E C A D B (3) E A D C B (3) E A C D B (3) E A B D C (3) D C B A E (3) D B C A E (3) B D C A E (3) B A C D E (3) E D C A B (2) E C D A B (2) E A B C D (2) D C E B A (2) D B C E A (2) C D A B E (2) B D A C E (2) A E C B D (2) A C B D E (2) E D B C A (1) E D B A C (1) E A D B C (1) C E D A B (1) C D E A B (1) C B D A E (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E A C (1) B D A E C (1) B C D A E (1) B A E D C (1) A E B D C (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 6 2 B -2 0 -6 4 0 C -8 6 0 8 -2 D -6 -4 -8 0 -4 E -2 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 6 2 B -2 0 -6 4 0 C -8 6 0 8 -2 D -6 -4 -8 0 -4 E -2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=19 A=19 C=16 D=14 so D is eliminated. Round 2 votes counts: E=32 C=25 B=24 A=19 so A is eliminated. Round 3 votes counts: E=40 B=33 C=27 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:209 C:202 E:202 B:198 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 6 2 B -2 0 -6 4 0 C -8 6 0 8 -2 D -6 -4 -8 0 -4 E -2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 6 2 B -2 0 -6 4 0 C -8 6 0 8 -2 D -6 -4 -8 0 -4 E -2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 6 2 B -2 0 -6 4 0 C -8 6 0 8 -2 D -6 -4 -8 0 -4 E -2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1637: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) A C B D E (10) A C B E D (9) D E B C A (8) A B E D C (8) C A D B E (7) C D E B A (6) C A D E B (6) D E C B A (5) C D E A B (5) E D B C A (4) E B D A C (4) D C E B A (4) B A E D C (4) A B C E D (3) C D A E B (2) A B E C D (2) C E D A B (1) B E A D C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 2 4 B -8 0 -10 4 10 C -4 10 0 4 8 D -2 -4 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 2 4 B -8 0 -10 4 10 C -4 10 0 4 8 D -2 -4 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=27 D=17 B=15 E=8 so E is eliminated. Round 2 votes counts: A=33 C=27 D=21 B=19 so B is eliminated. Round 3 votes counts: A=38 D=35 C=27 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:209 D:199 B:198 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 2 4 B -8 0 -10 4 10 C -4 10 0 4 8 D -2 -4 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 4 B -8 0 -10 4 10 C -4 10 0 4 8 D -2 -4 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 4 B -8 0 -10 4 10 C -4 10 0 4 8 D -2 -4 -4 0 8 E -4 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1638: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) D A C E B (7) B E C A D (7) A D C B E (7) E B C D A (6) D C E B A (6) D C A E B (5) B A E C D (5) C E D B A (4) A B D E C (4) E C B D A (3) C E B D A (3) C D A E B (3) B E A C D (3) A D B C E (3) D A C B E (2) B E D A C (2) B E C D A (2) B A E D C (2) A D C E B (2) A C D B E (2) A B E D C (2) A B E C D (2) E B C A D (1) D A B C E (1) C E D A B (1) C E A D B (1) C D E B A (1) C D E A B (1) C A E D B (1) B E D C A (1) B E A D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -6 -10 0 B -6 0 -12 -10 -4 C 6 12 0 -8 16 D 10 10 8 0 6 E 0 4 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -10 0 B -6 0 -12 -10 -4 C 6 12 0 -8 16 D 10 10 8 0 6 E 0 4 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=24 B=23 C=15 E=10 so E is eliminated. Round 2 votes counts: B=30 D=28 A=24 C=18 so C is eliminated. Round 3 votes counts: D=38 B=36 A=26 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:213 A:195 E:191 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 -10 0 B -6 0 -12 -10 -4 C 6 12 0 -8 16 D 10 10 8 0 6 E 0 4 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -10 0 B -6 0 -12 -10 -4 C 6 12 0 -8 16 D 10 10 8 0 6 E 0 4 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -10 0 B -6 0 -12 -10 -4 C 6 12 0 -8 16 D 10 10 8 0 6 E 0 4 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1639: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) B E A C D (8) E B A C D (7) D C A B E (6) A B C E D (6) E D B A C (5) E D A B C (5) D C E A B (5) D C A E B (5) B A E C D (5) E B A D C (3) E A B C D (3) D E C A B (3) E B D A C (2) E A B D C (2) D E C B A (2) D E A C B (2) C D A B E (2) C B A D E (2) C A B D E (2) B C A E D (2) B C A D E (2) A C B E D (2) E D B C A (1) D E B C A (1) B E D C A (1) B D C A E (1) A E C B D (1) A E B C D (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 30 18 0 B 4 0 32 24 2 C -30 -32 0 12 -6 D -18 -24 -12 0 -30 E 0 -2 6 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 30 18 0 B 4 0 32 24 2 C -30 -32 0 12 -6 D -18 -24 -12 0 -30 E 0 -2 6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=28 D=24 A=13 C=6 so C is eliminated. Round 2 votes counts: B=31 E=28 D=26 A=15 so A is eliminated. Round 3 votes counts: B=43 E=30 D=27 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:222 E:217 C:172 D:158 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 30 18 0 B 4 0 32 24 2 C -30 -32 0 12 -6 D -18 -24 -12 0 -30 E 0 -2 6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 30 18 0 B 4 0 32 24 2 C -30 -32 0 12 -6 D -18 -24 -12 0 -30 E 0 -2 6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 30 18 0 B 4 0 32 24 2 C -30 -32 0 12 -6 D -18 -24 -12 0 -30 E 0 -2 6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1640: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E D A C B (5) E A B D C (5) D C A B E (5) E C B D A (4) B E C A D (4) E D C B A (3) E B C A D (3) E A D B C (3) D E A C B (3) D A E C B (3) D A C E B (3) B C A D E (3) E D C A B (2) E A D C B (2) E A B C D (2) D E C A B (2) D C E B A (2) D C B E A (2) D C B A E (2) D A C B E (2) C D B E A (2) C B E D A (2) B C E A D (2) B C D A E (2) B A C D E (2) A E D B C (2) A E B D C (2) A D E C B (2) A D B C E (2) A B E C D (2) A B D E C (2) E C D B A (1) E B D C A (1) E B A C D (1) D E C B A (1) D C A E B (1) C B D E A (1) B C E D A (1) B A C E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -6 -14 -8 B -6 0 -16 0 -6 C 6 16 0 -14 -10 D 14 0 14 0 2 E 8 6 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.233021 C: 0.000000 D: 0.766979 E: 0.000000 Sum of squares = 0.642555215933 Cumulative probabilities = A: 0.000000 B: 0.233021 C: 0.233021 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -14 -8 B -6 0 -16 0 -6 C 6 16 0 -14 -10 D 14 0 14 0 2 E 8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000012201 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=26 B=15 A=15 C=12 so C is eliminated. Round 2 votes counts: E=32 D=28 B=25 A=15 so A is eliminated. Round 3 votes counts: E=36 D=32 B=32 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:211 C:199 A:189 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 -14 -8 B -6 0 -16 0 -6 C 6 16 0 -14 -10 D 14 0 14 0 2 E 8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000012201 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -14 -8 B -6 0 -16 0 -6 C 6 16 0 -14 -10 D 14 0 14 0 2 E 8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000012201 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -14 -8 B -6 0 -16 0 -6 C 6 16 0 -14 -10 D 14 0 14 0 2 E 8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000012201 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1641: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (13) D A C E B (11) A D B E C (11) C D E B A (10) C E B A D (9) B E C A D (8) D C A E B (7) D C E A B (4) D A B E C (4) D A B C E (4) B E A C D (4) A B E D C (3) E B C A D (2) D A C B E (2) B A E C D (2) A B D E C (2) E C B A D (1) D C A B E (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -10 -12 -2 B -2 0 -16 -8 -16 C 10 16 0 0 22 D 12 8 0 0 12 E 2 16 -22 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.723023 D: 0.276977 E: 0.000000 Sum of squares = 0.599478876702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.723023 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -12 -2 B -2 0 -16 -8 -16 C 10 16 0 0 22 D 12 8 0 0 12 E 2 16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=32 A=18 B=14 E=3 so E is eliminated. Round 2 votes counts: D=33 C=33 A=18 B=16 so B is eliminated. Round 3 votes counts: C=43 D=33 A=24 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 D:216 E:192 A:189 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -12 -2 B -2 0 -16 -8 -16 C 10 16 0 0 22 D 12 8 0 0 12 E 2 16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -12 -2 B -2 0 -16 -8 -16 C 10 16 0 0 22 D 12 8 0 0 12 E 2 16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -12 -2 B -2 0 -16 -8 -16 C 10 16 0 0 22 D 12 8 0 0 12 E 2 16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1642: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (5) E C A D B (4) E B C D A (4) E A D C B (4) E A C D B (4) D A E B C (4) C B E A D (4) A D C E B (4) E D A B C (3) E C B A D (3) D B A E C (3) D A E C B (3) D A B C E (3) C E A B D (3) C B A D E (3) C A B E D (3) B C D A E (3) B C A D E (3) A E D C B (3) A D E C B (3) E D A C B (2) C E B A D (2) C B A E D (2) B D E C A (2) B D E A C (2) A D B C E (2) E D C A B (1) E D B A C (1) E C B D A (1) E B D C A (1) D E A B C (1) D B E A C (1) D B A C E (1) D A B E C (1) C A E B D (1) C A D B E (1) C A B D E (1) B E D C A (1) B D C E A (1) B D C A E (1) B D A E C (1) B C E A D (1) A E C D B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 8 8 8 B -8 0 -8 -2 -8 C -8 8 0 -8 -8 D -8 2 8 0 2 E -8 8 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 8 B -8 0 -8 -2 -8 C -8 8 0 -8 -8 D -8 2 8 0 2 E -8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=20 B=20 D=17 A=15 so A is eliminated. Round 2 votes counts: E=32 D=26 C=21 B=21 so C is eliminated. Round 3 votes counts: E=38 B=34 D=28 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:216 E:203 D:202 C:192 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 8 B -8 0 -8 -2 -8 C -8 8 0 -8 -8 D -8 2 8 0 2 E -8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 8 B -8 0 -8 -2 -8 C -8 8 0 -8 -8 D -8 2 8 0 2 E -8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 8 B -8 0 -8 -2 -8 C -8 8 0 -8 -8 D -8 2 8 0 2 E -8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1643: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D C B A (9) D E C B A (7) E D C A B (5) B D C E A (5) A B C E D (5) D B E C A (4) B C A D E (4) A C B E D (4) D E B C A (3) C B D E A (3) A C E B D (3) E D A B C (2) E A D C B (2) D B C E A (2) B C D A E (2) B A C D E (2) A E D C B (2) A E C B D (2) A B D C E (2) E D B C A (1) E D A C B (1) E C D B A (1) E C D A B (1) E A D B C (1) D C E B A (1) D C B E A (1) C E D B A (1) C E A D B (1) C D E B A (1) C B D A E (1) C B A D E (1) B D E C A (1) B D E A C (1) B D C A E (1) B D A E C (1) B C D E A (1) B A D C E (1) A E C D B (1) A E B D C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -14 -12 -10 B 10 0 2 6 8 C 14 -2 0 -10 6 D 12 -6 10 0 14 E 10 -8 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -12 -10 B 10 0 2 6 8 C 14 -2 0 -10 6 D 12 -6 10 0 14 E 10 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=23 B=19 D=18 C=8 so C is eliminated. Round 2 votes counts: A=32 E=25 B=24 D=19 so D is eliminated. Round 3 votes counts: E=37 A=32 B=31 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 B:213 C:204 E:191 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 -12 -10 B 10 0 2 6 8 C 14 -2 0 -10 6 D 12 -6 10 0 14 E 10 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -12 -10 B 10 0 2 6 8 C 14 -2 0 -10 6 D 12 -6 10 0 14 E 10 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -12 -10 B 10 0 2 6 8 C 14 -2 0 -10 6 D 12 -6 10 0 14 E 10 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1644: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (15) E D C B A (7) B A E D C (6) A B C D E (6) D C E A B (5) C A D B E (4) E D B C A (3) E B C D A (3) E B A D C (3) D E C A B (3) B A C E D (3) A D B C E (3) E D A B C (2) E C D B A (2) E B D A C (2) D C A E B (2) C E D A B (2) C D E A B (2) C D A B E (2) C B A D E (2) B E A C D (2) B C A E D (2) B A E C D (2) B A C D E (2) A D B E C (2) E D B A C (1) E C D A B (1) E B C A D (1) E B A C D (1) D E A B C (1) C D A E B (1) C B E D A (1) B E C A D (1) B E A D C (1) B A D E C (1) B A D C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -18 -10 -18 B -6 0 0 -18 -14 C 18 0 0 -18 -20 D 10 18 18 0 -22 E 18 14 20 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -18 -10 -18 B -6 0 0 -18 -14 C 18 0 0 -18 -20 D 10 18 18 0 -22 E 18 14 20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=21 C=14 A=13 D=11 so D is eliminated. Round 2 votes counts: E=45 C=21 B=21 A=13 so A is eliminated. Round 3 votes counts: E=45 B=33 C=22 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:237 D:212 C:190 B:181 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -18 -10 -18 B -6 0 0 -18 -14 C 18 0 0 -18 -20 D 10 18 18 0 -22 E 18 14 20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 -10 -18 B -6 0 0 -18 -14 C 18 0 0 -18 -20 D 10 18 18 0 -22 E 18 14 20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 -10 -18 B -6 0 0 -18 -14 C 18 0 0 -18 -20 D 10 18 18 0 -22 E 18 14 20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1645: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) D E A B C (7) D A E C B (6) A D E C B (6) C B A E D (5) B C E A D (5) A C D B E (5) E B D C A (4) C B A D E (4) C A B D E (4) B C E D A (4) E D B C A (3) B C A E D (3) E D B A C (2) E B A D C (2) D A E B C (2) D A C E B (2) C B E D A (2) C B D A E (2) A E D B C (2) A D E B C (2) E D A B C (1) E B D A C (1) D E B C A (1) D E B A C (1) D E A C B (1) D C E B A (1) D C E A B (1) C D B E A (1) C B E A D (1) B E D C A (1) B E C D A (1) B E A C D (1) B A C E D (1) A E B D C (1) A E B C D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 14 12 18 B -8 0 -6 -10 -16 C -14 6 0 -16 6 D -12 10 16 0 16 E -18 16 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 12 18 B -8 0 -6 -10 -16 C -14 6 0 -16 6 D -12 10 16 0 16 E -18 16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 C=19 B=16 E=13 so E is eliminated. Round 2 votes counts: A=30 D=28 B=23 C=19 so C is eliminated. Round 3 votes counts: B=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:215 C:191 E:188 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 12 18 B -8 0 -6 -10 -16 C -14 6 0 -16 6 D -12 10 16 0 16 E -18 16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 12 18 B -8 0 -6 -10 -16 C -14 6 0 -16 6 D -12 10 16 0 16 E -18 16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 12 18 B -8 0 -6 -10 -16 C -14 6 0 -16 6 D -12 10 16 0 16 E -18 16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1646: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (17) D C A E B (9) B E C A D (7) D A C E B (6) B E D C A (6) E D C A B (5) D A C B E (5) E B C D A (4) B A C E D (4) E D B C A (3) E B D C A (3) D C E A B (3) B A C D E (3) A D C B E (3) A C D B E (3) E C D A B (2) C A D E B (2) E D C B A (1) E C B D A (1) E C A D B (1) E B C A D (1) E B A C D (1) D E C A B (1) D E B C A (1) D B A C E (1) B E D A C (1) B E C D A (1) B A E C D (1) B A D C E (1) A D C E B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -2 -6 -18 B 14 0 12 4 8 C 2 -12 0 -2 -14 D 6 -4 2 0 -18 E 18 -8 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -6 -18 B 14 0 12 4 8 C 2 -12 0 -2 -14 D 6 -4 2 0 -18 E 18 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=26 E=22 A=9 C=2 so C is eliminated. Round 2 votes counts: B=41 D=26 E=22 A=11 so A is eliminated. Round 3 votes counts: B=42 D=36 E=22 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:221 B:219 D:193 C:187 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -6 -18 B 14 0 12 4 8 C 2 -12 0 -2 -14 D 6 -4 2 0 -18 E 18 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -6 -18 B 14 0 12 4 8 C 2 -12 0 -2 -14 D 6 -4 2 0 -18 E 18 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -6 -18 B 14 0 12 4 8 C 2 -12 0 -2 -14 D 6 -4 2 0 -18 E 18 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1647: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (15) D B A E C (12) B D A E C (11) C E A B D (9) D A E C B (6) C E A D B (6) E C A D B (5) B D C E A (5) B D A C E (5) A E C D B (4) E A C D B (3) D A B E C (3) B C D E A (3) A E D C B (3) C E B A D (2) B D C A E (2) A D E C B (2) D C E A B (1) D B C A E (1) C B E A D (1) B C E D A (1) Total count = 100 A B C D E A 0 -16 -2 0 -2 B 16 0 16 8 18 C 2 -16 0 -2 2 D 0 -8 2 0 2 E 2 -18 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 0 -2 B 16 0 16 8 18 C 2 -16 0 -2 2 D 0 -8 2 0 2 E 2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 D=23 C=18 A=9 E=8 so E is eliminated. Round 2 votes counts: B=42 D=23 C=23 A=12 so A is eliminated. Round 3 votes counts: B=42 C=30 D=28 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:198 C:193 A:190 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 0 -2 B 16 0 16 8 18 C 2 -16 0 -2 2 D 0 -8 2 0 2 E 2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 0 -2 B 16 0 16 8 18 C 2 -16 0 -2 2 D 0 -8 2 0 2 E 2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 0 -2 B 16 0 16 8 18 C 2 -16 0 -2 2 D 0 -8 2 0 2 E 2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1648: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) E D A B C (5) E B D A C (5) E B A D C (4) D A E C B (4) C D E A B (4) C B E D A (4) C B A E D (4) C B E A D (3) C A D B E (3) B E C A D (3) B E A D C (3) B E A C D (3) A D C E B (3) A B D E C (3) E C D B A (2) E B D C A (2) D C A E B (2) D A C E B (2) C E B D A (2) C D E B A (2) C B A D E (2) B C E A D (2) B C A E D (2) B A E C D (2) A D E C B (2) A D E B C (2) A C B D E (2) E D C B A (1) E D B A C (1) E B C D A (1) E A D B C (1) D E C A B (1) D E A C B (1) D E A B C (1) D A E B C (1) C D A B E (1) C A B D E (1) B A C E D (1) A E B D C (1) A D C B E (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 0 0 -2 B -2 0 -12 2 -16 C 0 12 0 6 0 D 0 -2 -6 0 -6 E 2 16 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.696222 D: 0.000000 E: 0.303778 Sum of squares = 0.577006178509 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.696222 D: 0.696222 E: 1.000000 A B C D E A 0 2 0 0 -2 B -2 0 -12 2 -16 C 0 12 0 6 0 D 0 -2 -6 0 -6 E 2 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=22 B=16 A=16 D=12 so D is eliminated. Round 2 votes counts: C=36 E=25 A=23 B=16 so B is eliminated. Round 3 votes counts: C=40 E=34 A=26 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:209 A:200 D:193 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 0 -2 B -2 0 -12 2 -16 C 0 12 0 6 0 D 0 -2 -6 0 -6 E 2 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 -2 B -2 0 -12 2 -16 C 0 12 0 6 0 D 0 -2 -6 0 -6 E 2 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 -2 B -2 0 -12 2 -16 C 0 12 0 6 0 D 0 -2 -6 0 -6 E 2 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1649: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) E A B D C (7) D A E C B (7) E B A C D (6) C B E D A (6) C B D E A (6) B C E A D (6) E A B C D (4) D C A E B (4) E A D C B (3) D C B A E (3) A E D B C (3) A E B D C (3) D A E B C (2) D A C E B (2) D A B E C (2) B E A C D (2) B A D C E (2) E C D A B (1) E C B A D (1) E C A B D (1) E B C A D (1) E A D B C (1) E A C B D (1) D E C A B (1) D C E A B (1) D C A B E (1) D B C A E (1) D B A C E (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E B A (1) C D E A B (1) C D B A E (1) B E C A D (1) B D C A E (1) B A E D C (1) A E D C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 0 -4 -26 B 2 0 -8 2 -12 C 0 8 0 2 -2 D 4 -2 -2 0 -4 E 26 12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 -4 -26 B 2 0 -8 2 -12 C 0 8 0 2 -2 D 4 -2 -2 0 -4 E 26 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 D=25 B=13 A=9 so A is eliminated. Round 2 votes counts: E=33 C=27 D=26 B=14 so B is eliminated. Round 3 votes counts: E=37 C=33 D=30 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:204 D:198 B:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 -4 -26 B 2 0 -8 2 -12 C 0 8 0 2 -2 D 4 -2 -2 0 -4 E 26 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 -26 B 2 0 -8 2 -12 C 0 8 0 2 -2 D 4 -2 -2 0 -4 E 26 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 -26 B 2 0 -8 2 -12 C 0 8 0 2 -2 D 4 -2 -2 0 -4 E 26 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1650: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) E C D B A (6) E B A D C (6) E C B D A (5) B A E D C (5) D A C B E (4) C E A D B (4) C A D B E (4) A B D C E (4) E C B A D (3) E B D A C (3) C E D A B (3) C D E A B (3) C D A E B (3) B A D C E (3) E B C D A (2) E B A C D (2) B E A D C (2) B D A E C (2) B A D E C (2) A D C B E (2) A D B C E (2) A C D B E (2) A B E C D (2) A B C D E (2) E D B C A (1) E C A B D (1) E B D C A (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A B E (1) D B A C E (1) D A B C E (1) C A B D E (1) B E D A C (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -6 -4 10 B -6 0 -16 -4 10 C 6 16 0 12 12 D 4 4 -12 0 6 E -10 -10 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -4 10 B -6 0 -16 -4 10 C 6 16 0 12 12 D 4 4 -12 0 6 E -10 -10 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=30 C=30 B=15 A=15 D=10 so D is eliminated. Round 2 votes counts: C=34 E=30 A=20 B=16 so B is eliminated. Round 3 votes counts: C=34 E=33 A=33 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:203 D:201 B:192 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -4 10 B -6 0 -16 -4 10 C 6 16 0 12 12 D 4 4 -12 0 6 E -10 -10 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -4 10 B -6 0 -16 -4 10 C 6 16 0 12 12 D 4 4 -12 0 6 E -10 -10 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -4 10 B -6 0 -16 -4 10 C 6 16 0 12 12 D 4 4 -12 0 6 E -10 -10 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1651: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (13) E D B A C (11) D E B A C (7) C A B E D (6) D E A B C (5) C B A E D (5) C B A D E (5) C A B D E (4) B D E C A (4) B C D E A (4) A E C D B (4) E D A B C (3) B E D C A (3) E D B C A (2) D B E A C (2) B D C E A (2) B C D A E (2) A E D C B (2) A C D B E (2) E C B A D (1) E B C D A (1) D A E B C (1) D A B C E (1) C A E B D (1) B E C D A (1) B D C A E (1) B C E D A (1) B C A D E (1) A D E C B (1) A C E B D (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 12 -2 6 B 6 0 4 -10 -8 C -12 -4 0 10 4 D 2 10 -10 0 -12 E -6 8 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.100000 Sum of squares = 0.459999999942 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.900000 E: 1.000000 A B C D E A 0 -6 12 -2 6 B 6 0 4 -10 -8 C -12 -4 0 10 4 D 2 10 -10 0 -12 E -6 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.100000 Sum of squares = 0.459999999768 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=21 B=19 E=18 D=16 so D is eliminated. Round 2 votes counts: E=30 A=28 C=21 B=21 so C is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:205 E:205 C:199 B:196 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -6 12 -2 6 B 6 0 4 -10 -8 C -12 -4 0 10 4 D 2 10 -10 0 -12 E -6 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.100000 Sum of squares = 0.459999999768 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -2 6 B 6 0 4 -10 -8 C -12 -4 0 10 4 D 2 10 -10 0 -12 E -6 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.100000 Sum of squares = 0.459999999768 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.900000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -2 6 B 6 0 4 -10 -8 C -12 -4 0 10 4 D 2 10 -10 0 -12 E -6 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.100000 Sum of squares = 0.459999999768 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.900000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1652: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) B E C A D (7) E B C A D (6) C B E A D (5) A D E C B (5) A D E B C (5) D A E C B (4) D A B E C (4) E B A C D (3) D A E B C (3) C D A B E (3) C B D E A (3) C B D A E (3) B E D A C (3) B E C D A (3) B C E D A (3) E A D B C (2) E A B D C (2) D C A B E (2) D A B C E (2) B C E A D (2) A E D B C (2) A D C E B (2) E C B A D (1) E C A B D (1) E B A D C (1) E A C B D (1) D E A B C (1) D A C B E (1) C E B A D (1) C E A B D (1) C D B A E (1) C D A E B (1) C B E D A (1) C B A E D (1) C A D E B (1) B D E C A (1) B D C E A (1) B D C A E (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 0 2 B -6 0 4 2 -6 C -2 -4 0 -2 -10 D 0 -2 2 0 8 E -2 6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.577988 B: 0.000000 C: 0.000000 D: 0.422012 E: 0.000000 Sum of squares = 0.512164230029 Cumulative probabilities = A: 0.577988 B: 0.577988 C: 0.577988 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 0 2 B -6 0 4 2 -6 C -2 -4 0 -2 -10 D 0 -2 2 0 8 E -2 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=21 B=21 E=17 A=15 so A is eliminated. Round 2 votes counts: D=38 C=22 B=21 E=19 so E is eliminated. Round 3 votes counts: D=42 B=33 C=25 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:205 D:204 E:203 B:197 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 0 2 B -6 0 4 2 -6 C -2 -4 0 -2 -10 D 0 -2 2 0 8 E -2 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 2 B -6 0 4 2 -6 C -2 -4 0 -2 -10 D 0 -2 2 0 8 E -2 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 2 B -6 0 4 2 -6 C -2 -4 0 -2 -10 D 0 -2 2 0 8 E -2 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1653: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) B E A C D (6) E D B A C (5) D C E A B (5) D C A B E (5) E B D A C (4) E A B C D (4) D B E C A (4) C D A E B (4) B A E C D (4) E B A D C (3) D E C A B (3) C A E B D (3) C A D B E (3) B E D A C (3) B E A D C (3) D E C B A (2) D E B C A (2) D B E A C (2) C A B E D (2) C A B D E (2) B A C E D (2) A C E B D (2) A B C E D (2) E D C B A (1) E D C A B (1) E C D A B (1) E C A D B (1) E A C D B (1) E A C B D (1) D E B A C (1) D C E B A (1) D C B E A (1) D C B A E (1) D C A E B (1) C E A D B (1) C A E D B (1) C A D E B (1) B D E A C (1) B D A C E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 6 8 2 -14 B -6 0 -4 4 -2 C -8 4 0 0 -10 D -2 -4 0 0 -20 E 14 2 10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 2 -14 B -6 0 -4 4 -2 C -8 4 0 0 -10 D -2 -4 0 0 -20 E 14 2 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=22 B=20 C=17 A=13 so A is eliminated. Round 2 votes counts: D=28 C=26 E=24 B=22 so B is eliminated. Round 3 votes counts: E=40 D=30 C=30 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:201 B:196 C:193 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 2 -14 B -6 0 -4 4 -2 C -8 4 0 0 -10 D -2 -4 0 0 -20 E 14 2 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 -14 B -6 0 -4 4 -2 C -8 4 0 0 -10 D -2 -4 0 0 -20 E 14 2 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 -14 B -6 0 -4 4 -2 C -8 4 0 0 -10 D -2 -4 0 0 -20 E 14 2 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1654: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B A E C D (7) E A D C B (5) C D E A B (5) C D B E A (5) B C D A E (5) A E B D C (5) E D C A B (4) B E A D C (4) B A E D C (4) E D A C B (3) D E C A B (3) C D B A E (3) C D A E B (3) B C D E A (3) B A C D E (3) E A B D C (2) C D E B A (2) C D A B E (2) C B D A E (2) B E D C A (2) A E C D B (2) A B E C D (2) E B A D C (1) E A D B C (1) D C E B A (1) D C B E A (1) D C A E B (1) D B E C A (1) C A D E B (1) C A D B E (1) B E D A C (1) B D C E A (1) B C A D E (1) A E D C B (1) A C D B E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -12 -14 -8 B -6 0 -10 -10 2 C 12 10 0 -2 2 D 14 10 2 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -14 -8 B -6 0 -10 -10 2 C 12 10 0 -2 2 D 14 10 2 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=24 E=16 D=16 A=13 so A is eliminated. Round 2 votes counts: B=35 C=25 E=24 D=16 so D is eliminated. Round 3 votes counts: C=37 B=36 E=27 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:211 E:198 B:188 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -12 -14 -8 B -6 0 -10 -10 2 C 12 10 0 -2 2 D 14 10 2 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -14 -8 B -6 0 -10 -10 2 C 12 10 0 -2 2 D 14 10 2 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -14 -8 B -6 0 -10 -10 2 C 12 10 0 -2 2 D 14 10 2 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1655: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) D E A B C (7) C A B D E (7) C B E A D (6) D A E C B (5) E D B A C (4) D A E B C (4) A D B E C (4) A B D E C (4) E B D C A (3) C E D B A (3) C E B D A (3) C B A E D (3) C A D E B (3) C A D B E (3) E C D B A (2) E B C D A (2) C D E A B (2) C D A E B (2) C A B E D (2) B E C D A (2) A D C B E (2) A D B C E (2) E D C B A (1) E D B C A (1) E C B D A (1) D E A C B (1) D C A E B (1) D A C E B (1) C B E D A (1) B E D A C (1) B E C A D (1) B E A D C (1) B C A E D (1) B A E D C (1) A D E B C (1) A D C E B (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 0 -8 2 B -12 0 -4 -2 -10 C 0 4 0 -6 -6 D 8 2 6 0 8 E -2 10 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 -8 2 B -12 0 -4 -2 -10 C 0 4 0 -6 -6 D 8 2 6 0 8 E -2 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=21 D=19 A=18 B=7 so B is eliminated. Round 2 votes counts: C=36 E=26 D=19 A=19 so D is eliminated. Round 3 votes counts: C=37 E=34 A=29 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:212 A:203 E:203 C:196 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 0 -8 2 B -12 0 -4 -2 -10 C 0 4 0 -6 -6 D 8 2 6 0 8 E -2 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -8 2 B -12 0 -4 -2 -10 C 0 4 0 -6 -6 D 8 2 6 0 8 E -2 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -8 2 B -12 0 -4 -2 -10 C 0 4 0 -6 -6 D 8 2 6 0 8 E -2 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1656: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (7) D A C B E (6) A D B C E (6) C E B A D (5) E B C A D (4) D A E B C (4) C B A E D (4) C B A D E (4) B C E A D (4) E B D A C (3) D E A B C (3) C E D B A (3) C E B D A (3) C D A B E (3) C B E A D (3) A D B E C (3) A B D C E (3) E D B C A (2) E C B D A (2) E C B A D (2) E B C D A (2) E B A D C (2) D E C A B (2) D C A E B (2) C E D A B (2) B A C E D (2) B A C D E (2) A C D B E (2) E D A B C (1) E C D A B (1) D C A B E (1) D A C E B (1) C D E A B (1) C A D B E (1) B C A E D (1) B A E D C (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 0 -4 16 10 B 0 0 2 10 10 C 4 -2 0 4 20 D -16 -10 -4 0 4 E -10 -10 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.063557 B: 0.936443 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.880964598038 Cumulative probabilities = A: 0.063557 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 16 10 B 0 0 2 10 10 C 4 -2 0 4 20 D -16 -10 -4 0 4 E -10 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556233134 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=22 E=19 D=19 B=11 so B is eliminated. Round 2 votes counts: C=34 A=28 E=19 D=19 so E is eliminated. Round 3 votes counts: C=45 A=30 D=25 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:211 B:211 D:187 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 16 10 B 0 0 2 10 10 C 4 -2 0 4 20 D -16 -10 -4 0 4 E -10 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556233134 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 16 10 B 0 0 2 10 10 C 4 -2 0 4 20 D -16 -10 -4 0 4 E -10 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556233134 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 16 10 B 0 0 2 10 10 C 4 -2 0 4 20 D -16 -10 -4 0 4 E -10 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556233134 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1657: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) C E B D A (9) A D B E C (9) E D C A B (6) E C D B A (6) E C D A B (5) D E A C B (5) C E B A D (5) D A E C B (4) B C A E D (4) D A B E C (3) C E D B A (3) C B E A D (3) B C E A D (3) B A C E D (3) D E C A B (2) B C E D A (2) B C D A E (2) B A C D E (2) A B E C D (2) A B D E C (2) E D A C B (1) E C B A D (1) E C A D B (1) D B A C E (1) C B E D A (1) B C A D E (1) A D B C E (1) Total count = 100 A B C D E A 0 -18 -8 0 -6 B 18 0 -4 6 4 C 8 4 0 6 6 D 0 -6 -6 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 0 -6 B 18 0 -4 6 4 C 8 4 0 6 6 D 0 -6 -6 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=21 E=20 D=15 A=14 so A is eliminated. Round 2 votes counts: B=34 D=25 C=21 E=20 so E is eliminated. Round 3 votes counts: C=34 B=34 D=32 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:212 C:212 E:203 D:189 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -8 0 -6 B 18 0 -4 6 4 C 8 4 0 6 6 D 0 -6 -6 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 0 -6 B 18 0 -4 6 4 C 8 4 0 6 6 D 0 -6 -6 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 0 -6 B 18 0 -4 6 4 C 8 4 0 6 6 D 0 -6 -6 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1658: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) E B C D A (7) A C D E B (7) B E D C A (6) E B A C D (5) D C A B E (5) B D C A E (5) A E C D B (5) A D C B E (5) E A C D B (4) E A B D C (4) C D A B E (4) B A D C E (4) E B D C A (3) C D B A E (3) B D C E A (3) B A E D C (3) E A B C D (2) B E A D C (2) B C D E A (2) A B D C E (2) E C D B A (1) E C D A B (1) E C A D B (1) D C B A E (1) C D E A B (1) C D B E A (1) C D A E B (1) B E C D A (1) A E B D C (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 8 10 -4 B 10 0 18 18 -2 C -8 -18 0 -6 -10 D -10 -18 6 0 -10 E 4 2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 8 10 -4 B 10 0 18 18 -2 C -8 -18 0 -6 -10 D -10 -18 6 0 -10 E 4 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=26 A=22 C=10 D=6 so D is eliminated. Round 2 votes counts: E=36 B=26 A=22 C=16 so C is eliminated. Round 3 votes counts: E=37 A=32 B=31 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:213 A:202 D:184 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 8 10 -4 B 10 0 18 18 -2 C -8 -18 0 -6 -10 D -10 -18 6 0 -10 E 4 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 10 -4 B 10 0 18 18 -2 C -8 -18 0 -6 -10 D -10 -18 6 0 -10 E 4 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 10 -4 B 10 0 18 18 -2 C -8 -18 0 -6 -10 D -10 -18 6 0 -10 E 4 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1659: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A C B E (10) B E C D A (10) A D C E B (10) E A C B D (6) B C E D A (5) A E D C B (5) D B C A E (4) A D E C B (4) D B A C E (3) D A B E C (3) B D C E A (3) E C B A D (2) C B E D A (2) B E D C A (2) B C D E A (2) A E C D B (2) E C A B D (1) E B C D A (1) E A B C D (1) D C B A E (1) D B A E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C D B A E (1) C D A B E (1) C A E D B (1) C A D E B (1) B D E A C (1) A D E B C (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 2 -2 4 B -2 0 -2 -4 2 C -2 2 0 0 -2 D 2 4 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999953 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 2 2 -2 4 B -2 0 -2 -4 2 C -2 2 0 0 -2 D 2 4 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 B=23 E=21 C=8 so C is eliminated. Round 2 votes counts: A=27 D=25 B=25 E=23 so E is eliminated. Round 3 votes counts: B=39 A=36 D=25 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:203 D:202 C:199 E:199 B:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 -2 4 B -2 0 -2 -4 2 C -2 2 0 0 -2 D 2 4 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -2 4 B -2 0 -2 -4 2 C -2 2 0 0 -2 D 2 4 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -2 4 B -2 0 -2 -4 2 C -2 2 0 0 -2 D 2 4 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1660: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C E D A B (8) E C A B D (7) A B D E C (7) C D B A E (6) B D A C E (6) E A B C D (5) D B A C E (5) B A D E C (5) E C B A D (4) D B C A E (4) A E B D C (4) C E D B A (3) C D B E A (3) C D A E B (3) E A C B D (2) D C B A E (2) D C A B E (2) C E A D B (2) B E A D C (2) E C A D B (1) E B A C D (1) C D E B A (1) C D A B E (1) C B E D A (1) B D A E C (1) B A E D C (1) A E D C B (1) A D B E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 10 4 8 2 B -10 0 6 12 -2 C -4 -6 0 -4 -4 D -8 -12 4 0 -4 E -2 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 8 2 B -10 0 6 12 -2 C -4 -6 0 -4 -4 D -8 -12 4 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=28 B=15 A=15 D=13 so D is eliminated. Round 2 votes counts: C=32 E=29 B=24 A=15 so A is eliminated. Round 3 votes counts: E=34 B=34 C=32 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:204 B:203 C:191 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 8 2 B -10 0 6 12 -2 C -4 -6 0 -4 -4 D -8 -12 4 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 8 2 B -10 0 6 12 -2 C -4 -6 0 -4 -4 D -8 -12 4 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 8 2 B -10 0 6 12 -2 C -4 -6 0 -4 -4 D -8 -12 4 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1661: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) A B D C E (9) D B E A C (4) C E A B D (4) C A E B D (4) B A D E C (4) E D C B A (3) D E C B A (3) D E B C A (3) D E B A C (3) C E A D B (3) A B E C D (3) A B C E D (3) E D B C A (2) E C D A B (2) D C B E A (2) D B A E C (2) C A D B E (2) C A B E D (2) B D A E C (2) B D A C E (2) A C B D E (2) A B C D E (2) E D B A C (1) E C D B A (1) E C B A D (1) E C A D B (1) E C A B D (1) E B D C A (1) E B A D C (1) E B A C D (1) E A B C D (1) D C E B A (1) D B E C A (1) C E D B A (1) C D E B A (1) C A E D B (1) C A B D E (1) B E D A C (1) B A E D C (1) A E C B D (1) A E B C D (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 16 -8 2 -12 B -16 0 -2 0 -10 C 8 2 0 6 8 D -2 0 -6 0 -10 E 12 10 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -8 2 -12 B -16 0 -2 0 -10 C 8 2 0 6 8 D -2 0 -6 0 -10 E 12 10 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=23 D=19 E=16 B=10 so B is eliminated. Round 2 votes counts: C=32 A=28 D=23 E=17 so E is eliminated. Round 3 votes counts: C=38 D=31 A=31 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:212 A:199 D:191 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -8 2 -12 B -16 0 -2 0 -10 C 8 2 0 6 8 D -2 0 -6 0 -10 E 12 10 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 2 -12 B -16 0 -2 0 -10 C 8 2 0 6 8 D -2 0 -6 0 -10 E 12 10 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 2 -12 B -16 0 -2 0 -10 C 8 2 0 6 8 D -2 0 -6 0 -10 E 12 10 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1662: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) B A D C E (7) A D B E C (7) C B E A D (6) E C D A B (5) D A E B C (5) D A B E C (5) C E D B A (5) B C A E D (5) E C B A D (4) C E B A D (4) B C A D E (4) B A D E C (4) D E A C B (3) C D E A B (3) E D C A B (2) E A B C D (2) C D E B A (2) C D B E A (2) A E B D C (2) A B D E C (2) E A D B C (1) E A B D C (1) D B A C E (1) D A E C B (1) D A B C E (1) C E D A B (1) C E B D A (1) C D B A E (1) C B E D A (1) B E C A D (1) B D C A E (1) B C D A E (1) B A C D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 2 4 -2 B 2 0 4 -6 0 C -2 -4 0 -2 -6 D -4 6 2 0 4 E 2 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999985 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 A B C D E A 0 -2 2 4 -2 B 2 0 4 -6 0 C -2 -4 0 -2 -6 D -4 6 2 0 4 E 2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999281 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 E=22 D=16 A=12 so A is eliminated. Round 2 votes counts: C=26 B=26 E=24 D=24 so E is eliminated. Round 3 votes counts: C=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:204 E:202 A:201 B:200 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 4 -2 B 2 0 4 -6 0 C -2 -4 0 -2 -6 D -4 6 2 0 4 E 2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999281 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -2 B 2 0 4 -6 0 C -2 -4 0 -2 -6 D -4 6 2 0 4 E 2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999281 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -2 B 2 0 4 -6 0 C -2 -4 0 -2 -6 D -4 6 2 0 4 E 2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999281 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1663: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (6) A E C D B (6) A D C E B (6) E A C B D (5) D C B A E (5) B E D A C (5) B D C E A (5) D B C A E (4) B E C D A (4) A D C B E (4) A C D E B (4) E C A D B (3) E B C A D (3) C D A E B (3) B D E C A (3) B D C A E (3) E B C D A (2) E B A D C (2) E B A C D (2) E A B D C (2) D A C B E (2) C A D E B (2) B E A D C (2) B D A C E (2) A C E D B (2) E C A B D (1) E A B C D (1) C E A D B (1) C D B E A (1) C D B A E (1) C B E D A (1) C A E D B (1) B E D C A (1) A E D B C (1) A E B D C (1) A E B C D (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 8 0 4 12 B -8 0 -10 -8 0 C 0 10 0 -12 8 D -4 8 12 0 6 E -12 0 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.876066 B: 0.000000 C: 0.123934 D: 0.000000 E: 0.000000 Sum of squares = 0.782851378056 Cumulative probabilities = A: 0.876066 B: 0.876066 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 4 12 B -8 0 -10 -8 0 C 0 10 0 -12 8 D -4 8 12 0 6 E -12 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000871 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 E=21 D=17 C=10 so C is eliminated. Round 2 votes counts: A=30 B=26 E=22 D=22 so E is eliminated. Round 3 votes counts: A=43 B=35 D=22 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:211 C:203 B:187 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 4 12 B -8 0 -10 -8 0 C 0 10 0 -12 8 D -4 8 12 0 6 E -12 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000871 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 4 12 B -8 0 -10 -8 0 C 0 10 0 -12 8 D -4 8 12 0 6 E -12 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000871 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 4 12 B -8 0 -10 -8 0 C 0 10 0 -12 8 D -4 8 12 0 6 E -12 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000871 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1664: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (6) B D A C E (6) B C D E A (6) A E D B C (6) A B D E C (6) E A C B D (5) A E B C D (5) E C A B D (4) E A C D B (4) D B C A E (4) C E D B A (4) C E B D A (4) C E B A D (3) B D C A E (3) A E D C B (3) A E B D C (3) A D E B C (3) A D B E C (3) E C D A B (2) E C B A D (2) E A D C B (2) C B D E A (2) B C D A E (2) A B D C E (2) E C A D B (1) D C E B A (1) D C B E A (1) D B C E A (1) D A B E C (1) D A B C E (1) C D E B A (1) C B E A D (1) B A C D E (1) A D E C B (1) Total count = 100 A B C D E A 0 4 16 10 12 B -4 0 18 10 -8 C -16 -18 0 -6 -2 D -10 -10 6 0 2 E -12 8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 10 12 B -4 0 18 10 -8 C -16 -18 0 -6 -2 D -10 -10 6 0 2 E -12 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=20 B=18 D=15 C=15 so D is eliminated. Round 2 votes counts: A=34 B=29 E=20 C=17 so C is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:208 E:198 D:194 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 10 12 B -4 0 18 10 -8 C -16 -18 0 -6 -2 D -10 -10 6 0 2 E -12 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 10 12 B -4 0 18 10 -8 C -16 -18 0 -6 -2 D -10 -10 6 0 2 E -12 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 10 12 B -4 0 18 10 -8 C -16 -18 0 -6 -2 D -10 -10 6 0 2 E -12 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1665: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) C B A D E (6) E D B C A (5) D C E A B (5) D E C A B (4) D E A C B (4) B E A D C (4) A D E C B (4) A C B D E (4) A B E C D (4) A B C E D (4) E D A C B (3) D E C B A (3) C D B E A (3) B C A E D (3) B A E C D (3) A E D B C (3) E A B D C (2) C A D B E (2) C A B D E (2) B E C A D (2) B C A D E (2) A C D B E (2) A B E D C (2) E D C B A (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D B C (1) D C E B A (1) D C A E B (1) D A C E B (1) C D A B E (1) C A D E B (1) B E D A C (1) B E C D A (1) B E A C D (1) B C E D A (1) B C D E A (1) B A E D C (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -2 12 -4 B -6 0 -10 6 12 C 2 10 0 0 -6 D -12 -6 0 0 8 E 4 -12 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.123596 B: 0.157303 C: 0.337079 D: 0.078652 E: 0.303371 Sum of squares = 0.251862138622 Cumulative probabilities = A: 0.123596 B: 0.280899 C: 0.617978 D: 0.696629 E: 1.000000 A B C D E A 0 6 -2 12 -4 B -6 0 -10 6 12 C 2 10 0 0 -6 D -12 -6 0 0 8 E 4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.123596 B: 0.157303 C: 0.337079 D: 0.078652 E: 0.303371 Sum of squares = 0.251862138619 Cumulative probabilities = A: 0.123596 B: 0.280899 C: 0.617978 D: 0.696629 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=21 B=20 D=19 E=15 so E is eliminated. Round 2 votes counts: D=29 A=28 B=22 C=21 so C is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 C:203 B:201 D:195 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 12 -4 B -6 0 -10 6 12 C 2 10 0 0 -6 D -12 -6 0 0 8 E 4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.123596 B: 0.157303 C: 0.337079 D: 0.078652 E: 0.303371 Sum of squares = 0.251862138619 Cumulative probabilities = A: 0.123596 B: 0.280899 C: 0.617978 D: 0.696629 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 12 -4 B -6 0 -10 6 12 C 2 10 0 0 -6 D -12 -6 0 0 8 E 4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.123596 B: 0.157303 C: 0.337079 D: 0.078652 E: 0.303371 Sum of squares = 0.251862138619 Cumulative probabilities = A: 0.123596 B: 0.280899 C: 0.617978 D: 0.696629 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 12 -4 B -6 0 -10 6 12 C 2 10 0 0 -6 D -12 -6 0 0 8 E 4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.123596 B: 0.157303 C: 0.337079 D: 0.078652 E: 0.303371 Sum of squares = 0.251862138619 Cumulative probabilities = A: 0.123596 B: 0.280899 C: 0.617978 D: 0.696629 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1666: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) C A E D B (7) B D E C A (7) A C B E D (7) B E D A C (6) A C E D B (6) E D A C B (5) D E C A B (4) C A B D E (4) E B D A C (3) D B E C A (3) C A D E B (3) B C A D E (3) E A C D B (2) D C B E A (2) D C B A E (2) D C A E B (2) C A D B E (2) B E A D C (2) B D E A C (2) A E C D B (2) E D A B C (1) D E C B A (1) D E B A C (1) D C E B A (1) D C E A B (1) D B C E A (1) C E A D B (1) C D B A E (1) C D A E B (1) C D A B E (1) C B A D E (1) C A B E D (1) B E A C D (1) B D C A E (1) B A E C D (1) B A C E D (1) A C E B D (1) Total count = 100 A B C D E A 0 2 0 -10 -6 B -2 0 -16 -18 -2 C 0 16 0 -8 0 D 10 18 8 0 -12 E 6 2 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.365473 D: 0.000000 E: 0.634527 Sum of squares = 0.536194950171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.365473 D: 0.365473 E: 1.000000 A B C D E A 0 2 0 -10 -6 B -2 0 -16 -18 -2 C 0 16 0 -8 0 D 10 18 8 0 -12 E 6 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=22 E=20 D=18 A=16 so A is eliminated. Round 2 votes counts: C=36 B=24 E=22 D=18 so D is eliminated. Round 3 votes counts: C=44 E=28 B=28 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:212 E:210 C:204 A:193 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -10 -6 B -2 0 -16 -18 -2 C 0 16 0 -8 0 D 10 18 8 0 -12 E 6 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -10 -6 B -2 0 -16 -18 -2 C 0 16 0 -8 0 D 10 18 8 0 -12 E 6 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -10 -6 B -2 0 -16 -18 -2 C 0 16 0 -8 0 D 10 18 8 0 -12 E 6 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1667: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) D C A E B (7) B E A D C (6) D B C E A (5) C D A E B (5) B E A C D (5) B D E C A (5) B D E A C (5) A E C B D (5) E A B C D (4) E B A C D (3) E A C B D (3) D C B A E (3) D B C A E (3) A C E B D (3) D B A C E (2) C E A B D (2) B E D C A (2) B D C E A (2) A E B C D (2) A C E D B (2) E C B A D (1) E C A B D (1) D C A B E (1) D B E A C (1) D B A E C (1) D A B C E (1) C E A D B (1) C D E B A (1) C D B E A (1) C A D E B (1) B E D A C (1) B E C D A (1) B E C A D (1) B D A E C (1) B C E A D (1) B A E D C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -4 4 -4 B 4 0 8 12 -2 C 4 -8 0 2 -2 D -4 -12 -2 0 -8 E 4 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 4 -4 B 4 0 8 12 -2 C 4 -8 0 2 -2 D -4 -12 -2 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=24 C=19 A=14 E=12 so E is eliminated. Round 2 votes counts: B=34 D=24 C=21 A=21 so C is eliminated. Round 3 votes counts: B=35 A=34 D=31 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:208 C:198 A:196 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 4 -4 B 4 0 8 12 -2 C 4 -8 0 2 -2 D -4 -12 -2 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 4 -4 B 4 0 8 12 -2 C 4 -8 0 2 -2 D -4 -12 -2 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 4 -4 B 4 0 8 12 -2 C 4 -8 0 2 -2 D -4 -12 -2 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1668: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) A C B E D (11) A C D B E (7) D E B C A (5) D A C E B (5) A D E B C (4) D E C B A (3) D E A B C (3) D C E B A (3) C D A B E (3) C B A E D (3) B E C D A (3) B E A C D (3) A E B D C (3) A D C E B (3) A C B D E (3) E D B C A (2) E D B A C (2) D E B A C (2) D C E A B (2) D C A E B (2) C A D B E (2) C A B D E (2) D A E B C (1) C D E A B (1) C D B A E (1) C B E D A (1) C B E A D (1) C B D E A (1) C A B E D (1) B E C A D (1) B C E D A (1) A D E C B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 0 -6 8 B -12 0 -12 -4 -8 C 0 12 0 -6 8 D 6 4 6 0 8 E -8 8 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 -6 8 B -12 0 -12 -4 -8 C 0 12 0 -6 8 D 6 4 6 0 8 E -8 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=26 E=16 C=16 B=8 so B is eliminated. Round 2 votes counts: A=34 D=26 E=23 C=17 so C is eliminated. Round 3 votes counts: A=42 D=32 E=26 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 C:207 E:192 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 0 -6 8 B -12 0 -12 -4 -8 C 0 12 0 -6 8 D 6 4 6 0 8 E -8 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -6 8 B -12 0 -12 -4 -8 C 0 12 0 -6 8 D 6 4 6 0 8 E -8 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -6 8 B -12 0 -12 -4 -8 C 0 12 0 -6 8 D 6 4 6 0 8 E -8 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1669: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) D E B C A (6) D B E C A (6) A C E B D (6) B C A E D (5) E D C B A (4) D A E B C (4) D A B E C (4) B C E A D (4) D E C B A (3) D B A E C (3) C E A B D (3) C B E A D (3) C B A E D (3) C A B E D (3) A C B D E (3) E D C A B (2) E D B C A (2) E C D B A (2) C E B A D (2) B D E C A (2) B D A C E (2) B C E D A (2) A B C E D (2) A B C D E (2) E C D A B (1) E C A D B (1) D E C A B (1) D B E A C (1) D A E C B (1) C E B D A (1) B A D C E (1) B A C D E (1) A D B C E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -12 6 8 B 6 0 -4 10 18 C 12 4 0 14 14 D -6 -10 -14 0 -14 E -8 -18 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 6 8 B 6 0 -4 10 18 C 12 4 0 14 14 D -6 -10 -14 0 -14 E -8 -18 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 B=17 C=15 E=12 so E is eliminated. Round 2 votes counts: D=37 A=27 C=19 B=17 so B is eliminated. Round 3 votes counts: D=41 C=30 A=29 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:215 A:198 E:187 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 6 8 B 6 0 -4 10 18 C 12 4 0 14 14 D -6 -10 -14 0 -14 E -8 -18 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 6 8 B 6 0 -4 10 18 C 12 4 0 14 14 D -6 -10 -14 0 -14 E -8 -18 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 6 8 B 6 0 -4 10 18 C 12 4 0 14 14 D -6 -10 -14 0 -14 E -8 -18 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1670: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B E D A C (8) D B E C A (7) C A D E B (7) A C D B E (7) B D E A C (6) E B D C A (5) E A C B D (5) E C A B D (4) C A D B E (4) A C B D E (4) E B D A C (3) D E B C A (3) D C B A E (3) D B C A E (3) A C E B D (3) E D B C A (2) E B A D C (2) E A B C D (2) D A B C E (2) B D A E C (2) B D A C E (2) E B A C D (1) D B E A C (1) D B C E A (1) D A C B E (1) C E A D B (1) B A E C D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 2 -2 B 0 0 4 0 8 C -4 -4 0 -2 -4 D -2 0 2 0 8 E 2 -8 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.483213 B: 0.516787 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500563564558 Cumulative probabilities = A: 0.483213 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 2 -2 B 0 0 4 0 8 C -4 -4 0 -2 -4 D -2 0 2 0 8 E 2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=21 C=20 B=19 A=16 so A is eliminated. Round 2 votes counts: C=35 E=24 D=21 B=20 so B is eliminated. Round 3 votes counts: C=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:206 D:204 A:202 E:195 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 2 -2 B 0 0 4 0 8 C -4 -4 0 -2 -4 D -2 0 2 0 8 E 2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 2 -2 B 0 0 4 0 8 C -4 -4 0 -2 -4 D -2 0 2 0 8 E 2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 2 -2 B 0 0 4 0 8 C -4 -4 0 -2 -4 D -2 0 2 0 8 E 2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1671: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) C E B D A (7) E C A D B (5) C E D A B (5) B A D E C (5) A B D E C (5) E A C D B (4) C E D B A (4) B D C A E (4) A D E B C (4) E C B A D (3) D B C A E (3) D A B C E (3) C B E D A (3) B D A C E (3) B C E D A (3) E C A B D (2) E A D C B (2) C E B A D (2) C D B E A (2) B C D E A (2) A E D C B (2) A E D B C (2) A E B D C (2) E B C A D (1) E B A C D (1) E A D B C (1) E A C B D (1) D C B A E (1) D C A E B (1) D B A C E (1) D A E C B (1) C D B A E (1) C B D E A (1) B C D A E (1) B A E D C (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -2 8 2 B -2 0 4 -6 -2 C 2 -4 0 -4 -6 D -8 6 4 0 -2 E -2 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999975 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 2 -2 8 2 B -2 0 4 -6 -2 C 2 -4 0 -4 -6 D -8 6 4 0 -2 E -2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999878 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 E=20 B=19 D=10 so D is eliminated. Round 2 votes counts: A=30 C=27 B=23 E=20 so E is eliminated. Round 3 votes counts: A=38 C=37 B=25 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:205 E:204 D:200 B:197 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 8 2 B -2 0 4 -6 -2 C 2 -4 0 -4 -6 D -8 6 4 0 -2 E -2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999878 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 2 B -2 0 4 -6 -2 C 2 -4 0 -4 -6 D -8 6 4 0 -2 E -2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999878 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 2 B -2 0 4 -6 -2 C 2 -4 0 -4 -6 D -8 6 4 0 -2 E -2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999878 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1672: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) D E C A B (6) B E C D A (6) E B D C A (5) D E B C A (5) B E D C A (5) B C E A D (5) E D B C A (4) D E C B A (4) D A C E B (3) B E C A D (3) A D C E B (3) A C B D E (3) D E A C B (2) D A E C B (2) D A E B C (2) C E D B A (2) C E D A B (2) C B A E D (2) B E D A C (2) B A E D C (2) A D B E C (2) A C D B E (2) E C D B A (1) D E B A C (1) D C E A B (1) C E B A D (1) C D E A B (1) C B E D A (1) C B E A D (1) C A E D B (1) C A E B D (1) C A D E B (1) B D E A C (1) B C A E D (1) B A E C D (1) B A C E D (1) A D E C B (1) A D C B E (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -18 -12 -18 B 8 0 -4 -12 -16 C 18 4 0 -4 -10 D 12 12 4 0 2 E 18 16 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 -12 -18 B 8 0 -4 -12 -16 C 18 4 0 -4 -10 D 12 12 4 0 2 E 18 16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 A=24 C=13 E=10 so E is eliminated. Round 2 votes counts: B=32 D=30 A=24 C=14 so C is eliminated. Round 3 votes counts: B=37 D=36 A=27 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:221 D:215 C:204 B:188 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -18 -12 -18 B 8 0 -4 -12 -16 C 18 4 0 -4 -10 D 12 12 4 0 2 E 18 16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -12 -18 B 8 0 -4 -12 -16 C 18 4 0 -4 -10 D 12 12 4 0 2 E 18 16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -12 -18 B 8 0 -4 -12 -16 C 18 4 0 -4 -10 D 12 12 4 0 2 E 18 16 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1673: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) E A D B C (7) B C D E A (7) D C B A E (5) D C A B E (5) D A C B E (5) E B C D A (4) C D B E A (4) B C E D A (4) A E D C B (4) E B A C D (3) E A B D C (3) D C B E A (3) C B D E A (3) A E D B C (3) E D C B A (2) D E A C B (2) C D B A E (2) B C E A D (2) B C D A E (2) A E B C D (2) A D E C B (2) A D C E B (2) A B C D E (2) E D B C A (1) E C B D A (1) E B C A D (1) E A D C B (1) E A B C D (1) D E C B A (1) D A E C B (1) D A C E B (1) C B D A E (1) B E C A D (1) B C A E D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 0 -8 -2 B -4 0 -10 -22 16 C 0 10 0 -16 20 D 8 22 16 0 18 E 2 -16 -20 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -8 -2 B -4 0 -10 -22 16 C 0 10 0 -16 20 D 8 22 16 0 18 E 2 -16 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 D=23 B=17 C=10 so C is eliminated. Round 2 votes counts: D=29 A=26 E=24 B=21 so B is eliminated. Round 3 votes counts: D=42 E=31 A=27 so A is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:232 C:207 A:197 B:190 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -8 -2 B -4 0 -10 -22 16 C 0 10 0 -16 20 D 8 22 16 0 18 E 2 -16 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -8 -2 B -4 0 -10 -22 16 C 0 10 0 -16 20 D 8 22 16 0 18 E 2 -16 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -8 -2 B -4 0 -10 -22 16 C 0 10 0 -16 20 D 8 22 16 0 18 E 2 -16 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1674: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (13) B A D E C (11) D E B A C (10) C E D A B (8) C D E B A (7) A B E D C (6) C D B A E (5) D E C B A (4) C B A D E (4) B A C D E (4) A B C E D (4) C D E A B (3) C A B D E (3) B A D C E (3) E D A B C (2) D B A E C (2) B A E D C (2) E D C A B (1) E C D A B (1) E A B D C (1) D C E B A (1) D C B A E (1) C E A B D (1) C B D A E (1) B C A D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -8 8 22 B 12 0 -6 10 22 C 8 6 0 12 18 D -8 -10 -12 0 20 E -22 -22 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 8 22 B 12 0 -6 10 22 C 8 6 0 12 18 D -8 -10 -12 0 20 E -22 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=45 B=21 D=18 A=11 E=5 so E is eliminated. Round 2 votes counts: C=46 D=21 B=21 A=12 so A is eliminated. Round 3 votes counts: C=46 B=33 D=21 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:219 A:205 D:195 E:159 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 8 22 B 12 0 -6 10 22 C 8 6 0 12 18 D -8 -10 -12 0 20 E -22 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 8 22 B 12 0 -6 10 22 C 8 6 0 12 18 D -8 -10 -12 0 20 E -22 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 8 22 B 12 0 -6 10 22 C 8 6 0 12 18 D -8 -10 -12 0 20 E -22 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1675: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) D B A C E (7) E C A B D (6) D B A E C (6) D A B E C (6) E D C A B (4) D E C B A (4) D E C A B (4) D B C A E (4) C E D B A (4) C E B A D (4) B A D C E (4) E C A D B (3) A E B C D (3) E C D B A (2) E A C B D (2) D E A C B (2) D E A B C (2) D C B A E (2) C B E A D (2) B D A C E (2) B C A D E (2) B A C D E (2) A B E C D (2) A B D E C (2) E D A C B (1) E A B C D (1) D C E B A (1) D A E B C (1) C E A B D (1) C B E D A (1) C B D A E (1) C B A E D (1) C A E B D (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -10 -24 -4 B 0 0 -8 -24 -8 C 10 8 0 -6 -20 D 24 24 6 0 6 E 4 8 20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -24 -4 B 0 0 -8 -24 -8 C 10 8 0 -6 -20 D 24 24 6 0 6 E 4 8 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=27 C=15 B=11 A=8 so A is eliminated. Round 2 votes counts: D=40 E=30 C=15 B=15 so C is eliminated. Round 3 votes counts: E=40 D=40 B=20 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:230 E:213 C:196 A:181 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -24 -4 B 0 0 -8 -24 -8 C 10 8 0 -6 -20 D 24 24 6 0 6 E 4 8 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -24 -4 B 0 0 -8 -24 -8 C 10 8 0 -6 -20 D 24 24 6 0 6 E 4 8 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -24 -4 B 0 0 -8 -24 -8 C 10 8 0 -6 -20 D 24 24 6 0 6 E 4 8 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1676: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) C B D E A (11) B C E D A (8) A E B C D (6) D A E C B (5) E A B D C (4) C D B A E (4) A E D C B (4) A B C E D (4) E D A B C (3) E A D B C (3) C B D A E (3) B C E A D (3) B C D E A (3) B C A E D (3) A D E C B (3) D E A C B (2) D C E A B (2) D C B E A (2) D C A E B (2) B E A C D (2) E D B A C (1) E A B C D (1) D E C A B (1) D C B A E (1) D B C E A (1) C B A E D (1) B E C A D (1) A E B D C (1) A D E B C (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 12 8 2 4 B -12 0 16 4 -4 C -8 -16 0 2 -2 D -2 -4 -2 0 -16 E -4 4 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 2 4 B -12 0 16 4 -4 C -8 -16 0 2 -2 D -2 -4 -2 0 -16 E -4 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=20 C=19 D=16 E=12 so E is eliminated. Round 2 votes counts: A=41 D=20 B=20 C=19 so C is eliminated. Round 3 votes counts: A=41 B=35 D=24 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:209 B:202 C:188 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 2 4 B -12 0 16 4 -4 C -8 -16 0 2 -2 D -2 -4 -2 0 -16 E -4 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 2 4 B -12 0 16 4 -4 C -8 -16 0 2 -2 D -2 -4 -2 0 -16 E -4 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 2 4 B -12 0 16 4 -4 C -8 -16 0 2 -2 D -2 -4 -2 0 -16 E -4 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1677: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) A D C B E (7) A E B D C (6) C D B E A (5) A D E B C (5) E D C B A (4) E B C D A (4) D C E B A (4) D A E C B (4) C E B D A (3) B E C A D (3) B E A C D (3) A E D B C (3) E C B D A (2) E B A C D (2) D A E B C (2) D A C E B (2) C B D E A (2) C B D A E (2) B E C D A (2) B A C E D (2) A C B D E (2) E D B C A (1) E D B A C (1) E B D C A (1) E B A D C (1) E A B D C (1) D E C B A (1) D E C A B (1) D E A B C (1) D C B E A (1) D C A E B (1) C E D B A (1) C B A E D (1) C B A D E (1) C A B D E (1) B C E A D (1) B A E C D (1) A D E C B (1) A D C E B (1) A D B E C (1) A C D B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -2 -8 -8 B 16 0 -14 4 -6 C 2 14 0 -4 -6 D 8 -4 4 0 -6 E 8 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -2 -8 -8 B 16 0 -14 4 -6 C 2 14 0 -4 -6 D 8 -4 4 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 E=17 D=17 B=12 so B is eliminated. Round 2 votes counts: A=32 C=26 E=25 D=17 so D is eliminated. Round 3 votes counts: A=40 C=32 E=28 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:203 D:201 B:200 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -2 -8 -8 B 16 0 -14 4 -6 C 2 14 0 -4 -6 D 8 -4 4 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -8 -8 B 16 0 -14 4 -6 C 2 14 0 -4 -6 D 8 -4 4 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -8 -8 B 16 0 -14 4 -6 C 2 14 0 -4 -6 D 8 -4 4 0 -6 E 8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1678: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) A B C E D (8) E D C B A (7) B A C D E (7) D E C B A (6) A B C D E (5) E C D B A (4) D C E B A (4) C D E B A (4) A B E D C (4) A B D E C (4) C E D B A (3) C B A D E (3) B A D C E (3) E D A B C (2) E A D B C (2) C D B E A (2) B C D A E (2) B C A D E (2) A E B D C (2) A B E C D (2) E C D A B (1) E A D C B (1) E A B D C (1) D E A B C (1) D C B A E (1) D B A E C (1) C E B D A (1) C B E A D (1) C B D A E (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 -6 -6 B 4 0 -4 -6 -6 C 10 4 0 -8 -4 D 6 6 8 0 -6 E 6 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -6 B 4 0 -4 -6 -6 C 10 4 0 -8 -4 D 6 6 8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=27 C=15 B=14 D=13 so D is eliminated. Round 2 votes counts: E=38 A=27 C=20 B=15 so B is eliminated. Round 3 votes counts: E=38 A=38 C=24 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:207 C:201 B:194 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -6 B 4 0 -4 -6 -6 C 10 4 0 -8 -4 D 6 6 8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -6 B 4 0 -4 -6 -6 C 10 4 0 -8 -4 D 6 6 8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -6 B 4 0 -4 -6 -6 C 10 4 0 -8 -4 D 6 6 8 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1679: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) B E A D C (6) B D A E C (6) D A E C B (5) C D A E B (5) D C A E B (4) D B A C E (4) C B E A D (4) C B D E A (4) A E D B C (4) D B A E C (3) B D C E A (3) B D C A E (3) B C E D A (3) B C D E A (3) E B A C D (2) E A C D B (2) D C B A E (2) D A E B C (2) D A C E B (2) D A B E C (2) C E A B D (2) C D B A E (2) B E D A C (2) A E D C B (2) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) E A B C D (1) D C A B E (1) D B C A E (1) C E A D B (1) C D A B E (1) C B D A E (1) B E A C D (1) B D E C A (1) B D E A C (1) B C E A D (1) A E C D B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 14 -22 6 B 10 0 6 6 10 C -14 -6 0 -16 -4 D 22 -6 16 0 14 E -6 -10 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 -22 6 B 10 0 6 6 10 C -14 -6 0 -16 -4 D 22 -6 16 0 14 E -6 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=26 C=20 E=15 A=9 so A is eliminated. Round 2 votes counts: B=30 D=27 E=22 C=21 so C is eliminated. Round 3 votes counts: B=39 D=35 E=26 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:223 B:216 A:194 E:187 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 -22 6 B 10 0 6 6 10 C -14 -6 0 -16 -4 D 22 -6 16 0 14 E -6 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 -22 6 B 10 0 6 6 10 C -14 -6 0 -16 -4 D 22 -6 16 0 14 E -6 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 -22 6 B 10 0 6 6 10 C -14 -6 0 -16 -4 D 22 -6 16 0 14 E -6 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1680: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) A E C D B (7) D B A E C (6) B D C E A (6) C E B D A (5) C E A D B (5) E C A D B (4) B D C A E (4) B D A C E (4) B C D E A (4) A D B E C (4) D B E A C (3) D A B E C (3) C A E B D (3) A E D B C (3) E D C B A (2) E C D B A (2) D B E C A (2) C E B A D (2) B D A E C (2) B A D C E (2) E D A C B (1) E C D A B (1) E A C D B (1) D E C B A (1) D E B A C (1) D E A B C (1) D A E B C (1) C E D B A (1) C B E D A (1) C B E A D (1) C B D E A (1) C B A D E (1) C A B E D (1) B C D A E (1) A E D C B (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 -6 -8 B 4 0 -2 0 -4 C 14 2 0 2 6 D 6 0 -2 0 -2 E 8 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -6 -8 B 4 0 -2 0 -4 C 14 2 0 2 6 D 6 0 -2 0 -2 E 8 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=23 D=18 A=18 E=11 so E is eliminated. Round 2 votes counts: C=37 B=23 D=21 A=19 so A is eliminated. Round 3 votes counts: C=46 D=29 B=25 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:204 D:201 B:199 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -6 -8 B 4 0 -2 0 -4 C 14 2 0 2 6 D 6 0 -2 0 -2 E 8 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -6 -8 B 4 0 -2 0 -4 C 14 2 0 2 6 D 6 0 -2 0 -2 E 8 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -6 -8 B 4 0 -2 0 -4 C 14 2 0 2 6 D 6 0 -2 0 -2 E 8 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1681: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) E A D C B (6) E D A B C (5) D A E C B (5) C A D B E (5) B E C A D (5) D A C E B (4) C D B A E (4) C A D E B (4) C A B D E (4) A C D E B (4) E B D A C (3) E A B C D (3) D E A B C (3) C B A E D (3) C B A D E (3) B C D E A (3) B C A E D (3) E A D B C (2) C B D A E (2) C A B E D (2) B E D C A (2) B C E D A (2) B C D A E (2) E D B A C (1) E D A C B (1) E B A D C (1) D C A E B (1) D C A B E (1) D B E A C (1) C D A B E (1) B C A D E (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E C B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 14 -10 18 10 B -14 0 -12 -6 2 C 10 12 0 22 16 D -18 6 -22 0 0 E -10 -2 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 18 10 B -14 0 -12 -6 2 C 10 12 0 22 16 D -18 6 -22 0 0 E -10 -2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 E=22 D=15 A=10 so A is eliminated. Round 2 votes counts: C=33 E=25 B=25 D=17 so D is eliminated. Round 3 votes counts: C=40 E=34 B=26 so B is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:230 A:216 E:186 B:185 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 18 10 B -14 0 -12 -6 2 C 10 12 0 22 16 D -18 6 -22 0 0 E -10 -2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 18 10 B -14 0 -12 -6 2 C 10 12 0 22 16 D -18 6 -22 0 0 E -10 -2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 18 10 B -14 0 -12 -6 2 C 10 12 0 22 16 D -18 6 -22 0 0 E -10 -2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1682: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) B C D A E (8) A E D C B (7) A E B D C (6) E A D C B (5) C B D E A (5) C D B E A (4) A E B C D (4) D C A E B (3) B C E D A (3) B A E C D (3) B A C D E (3) A E D B C (3) A D E C B (3) E D C A B (2) E B C D A (2) E A B C D (2) D E C A B (2) D C E B A (2) D C E A B (2) D C B E A (2) D C B A E (2) D A C B E (2) C D B A E (2) A B E C D (2) E C D B A (1) E A B D C (1) D C A B E (1) D A C E B (1) C B D A E (1) B E C A D (1) B E A C D (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -10 -14 6 B 4 0 4 10 6 C 10 -4 0 8 8 D 14 -10 -8 0 14 E -6 -6 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -14 6 B 4 0 4 10 6 C 10 -4 0 8 8 D 14 -10 -8 0 14 E -6 -6 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=27 D=17 E=13 C=12 so C is eliminated. Round 2 votes counts: B=37 A=27 D=23 E=13 so E is eliminated. Round 3 votes counts: B=39 A=35 D=26 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:211 D:205 A:189 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -14 6 B 4 0 4 10 6 C 10 -4 0 8 8 D 14 -10 -8 0 14 E -6 -6 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -14 6 B 4 0 4 10 6 C 10 -4 0 8 8 D 14 -10 -8 0 14 E -6 -6 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -14 6 B 4 0 4 10 6 C 10 -4 0 8 8 D 14 -10 -8 0 14 E -6 -6 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1683: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) B C D E A (7) B C E D A (6) E D C B A (5) D C B A E (5) A D C B E (5) E B A C D (4) B C D A E (4) E D C A B (3) E A D C B (3) D A C B E (3) A E D C B (3) A D E C B (3) A D C E B (3) A B C D E (3) E C B D A (2) E B C A D (2) E A B D C (2) E A B C D (2) C B D E A (2) B E C D A (2) B C A D E (2) B A C D E (2) A E B D C (2) A E B C D (2) A B D C E (2) E D A C B (1) E A D B C (1) D E C A B (1) D C E B A (1) D C B E A (1) D C A E B (1) D C A B E (1) D A C E B (1) B E A C D (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -10 -12 -12 B 14 0 12 18 -4 C 10 -12 0 6 0 D 12 -18 -6 0 -6 E 12 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.163393 D: 0.000000 E: 0.836607 Sum of squares = 0.726608577602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.163393 D: 0.163393 E: 1.000000 A B C D E A 0 -14 -10 -12 -12 B 14 0 12 18 -4 C 10 -12 0 6 0 D 12 -18 -6 0 -6 E 12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000493001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=25 A=24 D=14 C=2 so C is eliminated. Round 2 votes counts: E=35 B=27 A=24 D=14 so D is eliminated. Round 3 votes counts: E=37 B=33 A=30 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:211 C:202 D:191 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -10 -12 -12 B 14 0 12 18 -4 C 10 -12 0 6 0 D 12 -18 -6 0 -6 E 12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000493001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -12 -12 B 14 0 12 18 -4 C 10 -12 0 6 0 D 12 -18 -6 0 -6 E 12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000493001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -12 -12 B 14 0 12 18 -4 C 10 -12 0 6 0 D 12 -18 -6 0 -6 E 12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000493001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1684: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (14) A C B E D (11) E D B A C (8) C A D B E (8) C A B D E (8) E B D A C (5) D E B A C (4) D C E B A (4) C A D E B (4) B E A D C (4) D E C B A (3) B E D A C (3) B A E D C (3) A B E C D (3) C D E A B (2) C D A E B (2) A B C E D (2) E D C A B (1) E B A D C (1) D E C A B (1) D B E C A (1) C D A B E (1) C A E B D (1) C A B E D (1) B D E C A (1) B D E A C (1) A E D B C (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -4 -2 -6 B 4 0 2 -10 -6 C 4 -2 0 -10 -10 D 2 10 10 0 8 E 6 6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 -6 B 4 0 2 -10 -6 C 4 -2 0 -10 -10 D 2 10 10 0 8 E 6 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 A=19 E=15 B=12 so B is eliminated. Round 2 votes counts: D=29 C=27 E=22 A=22 so E is eliminated. Round 3 votes counts: D=46 C=27 A=27 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:207 B:195 A:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -6 B 4 0 2 -10 -6 C 4 -2 0 -10 -10 D 2 10 10 0 8 E 6 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -6 B 4 0 2 -10 -6 C 4 -2 0 -10 -10 D 2 10 10 0 8 E 6 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -6 B 4 0 2 -10 -6 C 4 -2 0 -10 -10 D 2 10 10 0 8 E 6 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1685: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (14) A E B C D (10) D C E B A (9) B E C D A (8) A D C E B (8) A B E C D (5) B E A C D (4) B C E D A (4) E B C A D (3) D A C E B (3) D A C B E (3) B E C A D (3) A D E B C (3) E B C D A (2) D C A B E (2) C D E B A (2) A D B E C (2) A B E D C (2) E C B D A (1) E C B A D (1) E B A C D (1) E A B C D (1) D C A E B (1) D B C E A (1) B D E C A (1) B D C E A (1) B C D E A (1) B A E C D (1) A E C B D (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 -8 -6 -14 B 14 0 8 0 8 C 8 -8 0 -4 2 D 6 0 4 0 6 E 14 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.305758 C: 0.000000 D: 0.694242 E: 0.000000 Sum of squares = 0.575459690081 Cumulative probabilities = A: 0.000000 B: 0.305758 C: 0.305758 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -6 -14 B 14 0 8 0 8 C 8 -8 0 -4 2 D 6 0 4 0 6 E 14 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=33 A=33 B=23 E=9 C=2 so C is eliminated. Round 2 votes counts: D=35 A=33 B=23 E=9 so E is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:215 D:208 C:199 E:199 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 -6 -14 B 14 0 8 0 8 C 8 -8 0 -4 2 D 6 0 4 0 6 E 14 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -6 -14 B 14 0 8 0 8 C 8 -8 0 -4 2 D 6 0 4 0 6 E 14 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -6 -14 B 14 0 8 0 8 C 8 -8 0 -4 2 D 6 0 4 0 6 E 14 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1686: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) E A C D B (11) A E B D C (8) A E B C D (7) B D C E A (6) A E C D B (6) A E C B D (5) D B C E A (4) B A D E C (4) A E D B C (4) C B D E A (3) B C D E A (3) A B D E C (3) E C D A B (2) E C A D B (2) C D E B A (2) C D B E A (2) B D A C E (2) A E D C B (2) E D C A B (1) E A D B C (1) D C B E A (1) D B E C A (1) D B C A E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C B E D A (1) C B A E D (1) Total count = 100 A B C D E A 0 8 8 8 14 B -8 0 16 14 -8 C -8 -16 0 -4 -16 D -8 -14 4 0 -8 E -14 8 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 14 B -8 0 16 14 -8 C -8 -16 0 -4 -16 D -8 -14 4 0 -8 E -14 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=28 E=17 C=12 D=8 so D is eliminated. Round 2 votes counts: A=36 B=34 E=17 C=13 so C is eliminated. Round 3 votes counts: B=42 A=36 E=22 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:209 B:207 D:187 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 14 B -8 0 16 14 -8 C -8 -16 0 -4 -16 D -8 -14 4 0 -8 E -14 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 14 B -8 0 16 14 -8 C -8 -16 0 -4 -16 D -8 -14 4 0 -8 E -14 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 14 B -8 0 16 14 -8 C -8 -16 0 -4 -16 D -8 -14 4 0 -8 E -14 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1687: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (6) E B C D A (5) D E A C B (5) B C E A D (5) B C A E D (5) B C A D E (5) E D A B C (4) E C B D A (4) D A E C B (4) C B A D E (4) B A C D E (4) A D B C E (4) E D C B A (3) A D C E B (3) E D C A B (2) E D B A C (2) E D A C B (2) E B D C A (2) E B A D C (2) C E B D A (2) C B E D A (2) C A D B E (2) C A B D E (2) B E C A D (2) A C B D E (2) A B D C E (2) E D B C A (1) E C D B A (1) E C D A B (1) E B D A C (1) D C A E B (1) D A C E B (1) C D E A B (1) C B E A D (1) C B D E A (1) B E A C D (1) B A D E C (1) B A C E D (1) A D E B C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 10 0 B 10 0 0 10 4 C 4 0 0 4 10 D -10 -10 -4 0 2 E 0 -4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.618318 C: 0.381682 D: 0.000000 E: 0.000000 Sum of squares = 0.527998161673 Cumulative probabilities = A: 0.000000 B: 0.618318 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 10 0 B 10 0 0 10 4 C 4 0 0 4 10 D -10 -10 -4 0 2 E 0 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=24 A=20 C=15 D=11 so D is eliminated. Round 2 votes counts: E=35 A=25 B=24 C=16 so C is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:209 A:198 E:192 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 10 0 B 10 0 0 10 4 C 4 0 0 4 10 D -10 -10 -4 0 2 E 0 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 10 0 B 10 0 0 10 4 C 4 0 0 4 10 D -10 -10 -4 0 2 E 0 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 10 0 B 10 0 0 10 4 C 4 0 0 4 10 D -10 -10 -4 0 2 E 0 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1688: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) D C E B A (7) B A E D C (5) A B E C D (5) D E C B A (4) C D E A B (4) B A D C E (4) A B C D E (4) E D C B A (3) E D A B C (3) E C D A B (3) C D A B E (3) B A D E C (3) A B E D C (3) E A B C D (2) D C B E A (2) D C B A E (2) D B C A E (2) C D B A E (2) C B D A E (2) C A D B E (2) C A B D E (2) B D A E C (2) B D A C E (2) A B C E D (2) E D B C A (1) E C A D B (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B D C (1) D C E A B (1) D B E C A (1) D B C E A (1) C E D A B (1) C D E B A (1) C D A E B (1) C A E D B (1) B E D A C (1) B D E A C (1) B C A D E (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 4 -2 18 B 14 0 10 8 22 C -4 -10 0 -2 12 D 2 -8 2 0 24 E -18 -22 -12 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -2 18 B 14 0 10 8 22 C -4 -10 0 -2 12 D 2 -8 2 0 24 E -18 -22 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=20 C=19 E=17 A=17 so E is eliminated. Round 2 votes counts: B=29 D=27 C=23 A=21 so A is eliminated. Round 3 votes counts: B=47 D=27 C=26 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 D:210 A:203 C:198 E:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 -2 18 B 14 0 10 8 22 C -4 -10 0 -2 12 D 2 -8 2 0 24 E -18 -22 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -2 18 B 14 0 10 8 22 C -4 -10 0 -2 12 D 2 -8 2 0 24 E -18 -22 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -2 18 B 14 0 10 8 22 C -4 -10 0 -2 12 D 2 -8 2 0 24 E -18 -22 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1689: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) B C E D A (8) B C A E D (7) A C D B E (7) E D B C A (5) C B A D E (5) B C E A D (5) A D E C B (5) A C B D E (5) D E C B A (4) D E A C B (4) A D C B E (4) E B C D A (3) D A E C B (3) D A C E B (3) B E C D A (3) A B C E D (3) E B D C A (2) E B C A D (2) E A D B C (2) D C E B A (1) D C A B E (1) C D A B E (1) C B D A E (1) C A B D E (1) B E C A D (1) B E A C D (1) B C D E A (1) B C A D E (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 0 -4 2 -2 B 0 0 8 0 10 C 4 -8 0 12 10 D -2 0 -12 0 -4 E 2 -10 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.512768 B: 0.487232 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500326016596 Cumulative probabilities = A: 0.512768 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 -2 B 0 0 8 0 10 C 4 -8 0 12 10 D -2 0 -12 0 -4 E 2 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 E=23 D=16 C=8 so C is eliminated. Round 2 votes counts: B=33 A=27 E=23 D=17 so D is eliminated. Round 3 votes counts: A=35 B=33 E=32 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:209 C:209 A:198 E:193 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 2 -2 B 0 0 8 0 10 C 4 -8 0 12 10 D -2 0 -12 0 -4 E 2 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 -2 B 0 0 8 0 10 C 4 -8 0 12 10 D -2 0 -12 0 -4 E 2 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 -2 B 0 0 8 0 10 C 4 -8 0 12 10 D -2 0 -12 0 -4 E 2 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1690: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (10) B E C D A (7) B C D A E (5) A C D B E (5) E D A B C (4) E B D C A (4) E D A C B (3) E B C D A (3) E A D C B (3) E A B C D (3) D C A B E (3) B C A D E (3) A E D C B (3) A D E C B (3) A C B D E (3) E D B C A (2) D A C E B (2) C D B A E (2) C D A B E (2) C B D A E (2) C A D B E (2) C A B D E (2) B E C A D (2) B D C E A (2) B C E D A (2) A C D E B (2) E D B A C (1) E B C A D (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E A B (1) D C B E A (1) D C A E B (1) D A E C B (1) D A C B E (1) C B A D E (1) B C D E A (1) B C A E D (1) A D C B E (1) Total count = 100 A B C D E A 0 16 0 -2 10 B -16 0 -10 -10 -4 C 0 10 0 0 10 D 2 10 0 0 14 E -10 4 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.528879 D: 0.471121 E: 0.000000 Sum of squares = 0.501668002419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.528879 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 -2 10 B -16 0 -10 -10 -4 C 0 10 0 0 10 D 2 10 0 0 14 E -10 4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 B=23 D=11 C=11 so D is eliminated. Round 2 votes counts: A=31 E=29 B=23 C=17 so C is eliminated. Round 3 votes counts: A=41 E=30 B=29 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:212 C:210 E:185 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 16 0 -2 10 B -16 0 -10 -10 -4 C 0 10 0 0 10 D 2 10 0 0 14 E -10 4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 -2 10 B -16 0 -10 -10 -4 C 0 10 0 0 10 D 2 10 0 0 14 E -10 4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 -2 10 B -16 0 -10 -10 -4 C 0 10 0 0 10 D 2 10 0 0 14 E -10 4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1691: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E B D A C (8) C A D B E (8) E B A C D (5) D E B C A (5) D C A E B (5) E D B A C (4) B E D C A (4) B D E C A (4) A C D E B (4) C A B D E (3) B E A C D (3) A C E D B (3) A C E B D (3) A C B E D (3) A C B D E (3) E D B C A (2) E B D C A (2) D E C A B (2) C A D E B (2) B E D A C (2) B D C E A (2) B C D A E (2) E D A C B (1) E B A D C (1) E A C B D (1) E A B C D (1) D B E C A (1) D B C E A (1) D B C A E (1) C D A E B (1) C D A B E (1) C B A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 4 -14 -16 2 B -4 0 -4 -4 -2 C 14 4 0 -12 6 D 16 4 12 0 12 E -2 2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 -16 2 B -4 0 -4 -4 -2 C 14 4 0 -12 6 D 16 4 12 0 12 E -2 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 B=17 A=17 C=16 so C is eliminated. Round 2 votes counts: A=30 D=27 E=25 B=18 so B is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:206 B:193 E:191 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -14 -16 2 B -4 0 -4 -4 -2 C 14 4 0 -12 6 D 16 4 12 0 12 E -2 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 -16 2 B -4 0 -4 -4 -2 C 14 4 0 -12 6 D 16 4 12 0 12 E -2 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 -16 2 B -4 0 -4 -4 -2 C 14 4 0 -12 6 D 16 4 12 0 12 E -2 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1692: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) C E D B A (8) A D B C E (8) C D E B A (7) E C B D A (6) E C B A D (6) D A B C E (5) A B E D C (4) E C D A B (3) E B A C D (3) D C E A B (3) D A C B E (3) C E B D A (3) A B D E C (3) E C D B A (2) E B C A D (2) E A B C D (2) D C B A E (2) D C A E B (2) D A C E B (2) C D E A B (2) B E C A D (2) A E D B C (2) E A D B C (1) D A E C B (1) C D B E A (1) C B D E A (1) B E A C D (1) B C E D A (1) B A E D C (1) B A E C D (1) B A C E D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -20 -22 -8 B -4 0 -22 -24 -10 C 20 22 0 4 18 D 22 24 -4 0 2 E 8 10 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999297 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -20 -22 -8 B -4 0 -22 -24 -10 C 20 22 0 4 18 D 22 24 -4 0 2 E 8 10 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 C=22 A=19 B=7 so B is eliminated. Round 2 votes counts: E=28 D=27 C=23 A=22 so A is eliminated. Round 3 votes counts: D=40 E=36 C=24 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:232 D:222 E:199 A:177 B:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -20 -22 -8 B -4 0 -22 -24 -10 C 20 22 0 4 18 D 22 24 -4 0 2 E 8 10 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -22 -8 B -4 0 -22 -24 -10 C 20 22 0 4 18 D 22 24 -4 0 2 E 8 10 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -22 -8 B -4 0 -22 -24 -10 C 20 22 0 4 18 D 22 24 -4 0 2 E 8 10 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1693: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) E C B A D (8) C D E B A (8) D C E A B (7) D A B C E (5) B E A C D (5) A B D E C (5) D C A E B (4) D C A B E (4) D A C B E (4) C D E A B (4) E B C A D (3) E B A C D (3) D A B E C (3) C E D B A (3) A D B C E (3) A B E D C (3) E C D B A (2) E C B D A (2) C E B D A (2) C E B A D (2) A B C E D (2) E D B A C (1) E B D C A (1) E B D A C (1) D E C A B (1) D C E B A (1) C A D B E (1) B A E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 -6 -8 B 4 0 -6 -2 -6 C 6 6 0 12 2 D 6 2 -12 0 2 E 8 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -6 -8 B 4 0 -6 -2 -6 C 6 6 0 12 2 D 6 2 -12 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 C=20 B=15 A=15 so B is eliminated. Round 2 votes counts: D=29 E=26 A=25 C=20 so C is eliminated. Round 3 votes counts: D=41 E=33 A=26 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:213 E:205 D:199 B:195 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -6 -8 B 4 0 -6 -2 -6 C 6 6 0 12 2 D 6 2 -12 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -6 -8 B 4 0 -6 -2 -6 C 6 6 0 12 2 D 6 2 -12 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -6 -8 B 4 0 -6 -2 -6 C 6 6 0 12 2 D 6 2 -12 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1694: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (25) D A C E B (13) D A B E C (9) C E B A D (7) C E B D A (4) B E C D A (4) A D C E B (4) D B E C A (3) D A B C E (3) A C E B D (3) D C A E B (2) D B E A C (2) C E A B D (2) C D E B A (2) C B E D A (2) B A E C D (2) E C B A D (1) D C E B A (1) D C E A B (1) D B C E A (1) D A C B E (1) C D E A B (1) C A E B D (1) B E A C D (1) B D E C A (1) B C E D A (1) A E C D B (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 -14 -18 -2 -18 B 14 0 6 10 12 C 18 -6 0 16 -2 D 2 -10 -16 0 -10 E 18 -12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -18 -2 -18 B 14 0 6 10 12 C 18 -6 0 16 -2 D 2 -10 -16 0 -10 E 18 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=34 C=19 A=10 E=1 so E is eliminated. Round 2 votes counts: D=36 B=34 C=20 A=10 so A is eliminated. Round 3 votes counts: D=41 B=34 C=25 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:213 E:209 D:183 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -18 -2 -18 B 14 0 6 10 12 C 18 -6 0 16 -2 D 2 -10 -16 0 -10 E 18 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -18 -2 -18 B 14 0 6 10 12 C 18 -6 0 16 -2 D 2 -10 -16 0 -10 E 18 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -18 -2 -18 B 14 0 6 10 12 C 18 -6 0 16 -2 D 2 -10 -16 0 -10 E 18 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1695: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (12) C D B A E (9) D C E A B (7) B C A E D (7) A E B C D (6) E A D B C (5) D E A C B (5) E A D C B (4) E A B C D (4) D E C A B (4) D C B E A (4) C B D A E (4) E A B D C (3) D E A B C (3) C E A D B (3) C D E A B (3) C E D A B (2) B C A D E (2) B A E D C (2) A E B D C (2) E D A C B (1) E A C D B (1) D C B A E (1) C D B E A (1) C A B E D (1) B C D A E (1) B A C E D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 12 2 10 0 B -12 0 -2 -6 -8 C -2 2 0 18 -8 D -10 6 -18 0 -12 E 0 8 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.546891 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.453109 Sum of squares = 0.504397480338 Cumulative probabilities = A: 0.546891 B: 0.546891 C: 0.546891 D: 0.546891 E: 1.000000 A B C D E A 0 12 2 10 0 B -12 0 -2 -6 -8 C -2 2 0 18 -8 D -10 6 -18 0 -12 E 0 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 C=23 E=18 A=10 so A is eliminated. Round 2 votes counts: E=27 B=26 D=24 C=23 so C is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:212 C:205 B:186 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 10 0 B -12 0 -2 -6 -8 C -2 2 0 18 -8 D -10 6 -18 0 -12 E 0 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 10 0 B -12 0 -2 -6 -8 C -2 2 0 18 -8 D -10 6 -18 0 -12 E 0 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 10 0 B -12 0 -2 -6 -8 C -2 2 0 18 -8 D -10 6 -18 0 -12 E 0 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1696: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) E B C A D (7) A D C B E (7) E B C D A (5) D A C E B (5) E C B D A (4) D A C B E (4) B E C A D (4) B C E A D (4) D A E C B (3) C E B D A (3) C D A B E (3) C B D A E (3) B C A D E (3) A D E B C (3) E D C A B (2) E D A C B (2) E D A B C (2) E B A D C (2) C D E B A (2) C B E D A (2) C B A D E (2) B C E D A (2) B A E D C (2) A D B C E (2) E C D A B (1) E A B D C (1) D E C A B (1) C D B A E (1) C D A E B (1) C B D E A (1) B E C D A (1) B C D A E (1) B A C D E (1) A E D B C (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -6 2 8 B 0 0 -12 0 -8 C 6 12 0 4 -2 D -2 0 -4 0 10 E -8 8 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.125000 E: 0.250000 Sum of squares = 0.468749999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.750000 E: 1.000000 A B C D E A 0 0 -6 2 8 B 0 0 -12 0 -8 C 6 12 0 4 -2 D -2 0 -4 0 10 E -8 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.125000 E: 0.250000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=25 C=18 B=18 D=13 so D is eliminated. Round 2 votes counts: A=37 E=27 C=18 B=18 so C is eliminated. Round 3 votes counts: A=41 E=32 B=27 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:210 A:202 D:202 E:196 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 2 8 B 0 0 -12 0 -8 C 6 12 0 4 -2 D -2 0 -4 0 10 E -8 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.125000 E: 0.250000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 8 B 0 0 -12 0 -8 C 6 12 0 4 -2 D -2 0 -4 0 10 E -8 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.125000 E: 0.250000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 8 B 0 0 -12 0 -8 C 6 12 0 4 -2 D -2 0 -4 0 10 E -8 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.125000 E: 0.250000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1697: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) A E C B D (7) B D C A E (6) D B C E A (5) D A E B C (5) A E B C D (5) E A C B D (4) D C E B A (4) A B C E D (4) D E A C B (3) D B C A E (3) B C D E A (3) A E D C B (3) E C B A D (2) E C A B D (2) E A C D B (2) D A B E C (2) B C D A E (2) B C A E D (2) B C A D E (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) D E C B A (1) D E C A B (1) D E A B C (1) D B E C A (1) D B E A C (1) D A E C B (1) D A B C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C B D E A (1) C A B E D (1) B D A C E (1) B C E D A (1) B C E A D (1) B A C D E (1) A E D B C (1) A E B D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 12 -18 20 B 2 0 22 -2 0 C -12 -22 0 -8 6 D 18 2 8 0 14 E -20 0 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999517 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 -18 20 B 2 0 22 -2 0 C -12 -22 0 -8 6 D 18 2 8 0 14 E -20 0 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=22 B=19 E=14 C=5 so C is eliminated. Round 2 votes counts: D=40 A=23 B=20 E=17 so E is eliminated. Round 3 votes counts: D=44 A=32 B=24 so B is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:211 A:206 C:182 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 -18 20 B 2 0 22 -2 0 C -12 -22 0 -8 6 D 18 2 8 0 14 E -20 0 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -18 20 B 2 0 22 -2 0 C -12 -22 0 -8 6 D 18 2 8 0 14 E -20 0 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -18 20 B 2 0 22 -2 0 C -12 -22 0 -8 6 D 18 2 8 0 14 E -20 0 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (13) D A E B C (11) E C B D A (10) E D C B A (8) A D B C E (7) A B C D E (7) B C A E D (6) B A C D E (6) D E A C B (5) E C D B A (4) D A E C B (3) C E B D A (3) B C E A D (3) A D B E C (3) A B D C E (3) D E A B C (2) A D E B C (2) E C D A B (1) E B C D A (1) B C A D E (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 0 4 0 B 10 0 4 6 0 C 0 -4 0 12 0 D -4 -6 -12 0 2 E 0 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.516528 C: 0.000000 D: 0.000000 E: 0.483472 Sum of squares = 0.500546319401 Cumulative probabilities = A: 0.000000 B: 0.516528 C: 0.516528 D: 0.516528 E: 1.000000 A B C D E A 0 -10 0 4 0 B 10 0 4 6 0 C 0 -4 0 12 0 D -4 -6 -12 0 2 E 0 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=23 D=21 C=16 B=16 so C is eliminated. Round 2 votes counts: B=29 E=27 A=23 D=21 so D is eliminated. Round 3 votes counts: A=37 E=34 B=29 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:210 C:204 E:199 A:197 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 4 0 B 10 0 4 6 0 C 0 -4 0 12 0 D -4 -6 -12 0 2 E 0 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 4 0 B 10 0 4 6 0 C 0 -4 0 12 0 D -4 -6 -12 0 2 E 0 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 4 0 B 10 0 4 6 0 C 0 -4 0 12 0 D -4 -6 -12 0 2 E 0 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1699: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (18) B E D C A (17) C A B E D (11) A C D B E (8) D E B A C (7) C A B D E (6) E B D C A (5) E D B A C (3) C A D B E (3) E B D A C (2) B E C D A (2) B C E D A (2) E D A B C (1) D B E C A (1) D B E A C (1) D A E B C (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A E D (1) C B A D E (1) B C E A D (1) B C D E A (1) A D E C B (1) A D E B C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -10 10 8 B -4 0 -10 8 16 C 10 10 0 20 16 D -10 -8 -20 0 0 E -8 -16 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 10 8 B -4 0 -10 8 16 C 10 10 0 20 16 D -10 -8 -20 0 0 E -8 -16 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 B=23 E=11 D=10 so D is eliminated. Round 2 votes counts: A=31 C=26 B=25 E=18 so E is eliminated. Round 3 votes counts: B=42 A=32 C=26 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:228 A:206 B:205 D:181 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 10 8 B -4 0 -10 8 16 C 10 10 0 20 16 D -10 -8 -20 0 0 E -8 -16 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 10 8 B -4 0 -10 8 16 C 10 10 0 20 16 D -10 -8 -20 0 0 E -8 -16 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 10 8 B -4 0 -10 8 16 C 10 10 0 20 16 D -10 -8 -20 0 0 E -8 -16 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1700: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D C E B A (8) E C B D A (7) E B C A D (6) A D B E C (6) D A C B E (4) D A B C E (4) C E D B A (4) B E C A D (4) A B E C D (4) E C D B A (3) B A E C D (3) A D B C E (3) D E C A B (2) D C B E A (2) D A C E B (2) C E B D A (2) C D B E A (2) A E B C D (2) A D E B C (2) A B D E C (2) E C A D B (1) E C A B D (1) E A C D B (1) E A C B D (1) E A B C D (1) D E A C B (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) C D E B A (1) C B E D A (1) C B D E A (1) B D C A E (1) B C E D A (1) B C A E D (1) B A C E D (1) A E D C B (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -18 0 -18 B 12 0 -14 -2 -16 C 18 14 0 12 -16 D 0 2 -12 0 -8 E 18 16 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -18 0 -18 B 12 0 -14 -2 -16 C 18 14 0 12 -16 D 0 2 -12 0 -8 E 18 16 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=27 A=22 C=11 B=11 so C is eliminated. Round 2 votes counts: E=35 D=30 A=22 B=13 so B is eliminated. Round 3 votes counts: E=41 D=32 A=27 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:214 D:191 B:190 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -18 0 -18 B 12 0 -14 -2 -16 C 18 14 0 12 -16 D 0 2 -12 0 -8 E 18 16 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 0 -18 B 12 0 -14 -2 -16 C 18 14 0 12 -16 D 0 2 -12 0 -8 E 18 16 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 0 -18 B 12 0 -14 -2 -16 C 18 14 0 12 -16 D 0 2 -12 0 -8 E 18 16 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1701: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) E D C A B (10) D E C A B (7) C B D E A (7) C D E B A (6) B A C E D (6) A E D C B (6) E D A C B (5) C E D B A (4) A E D B C (4) C D B E A (3) B C A D E (3) B A C D E (3) E A D C B (2) B C E D A (2) A E B D C (2) A D E B C (2) A B D E C (2) E D C B A (1) E C D B A (1) E C B D A (1) D E C B A (1) D E A C B (1) D C E B A (1) D C B A E (1) B E C A D (1) B C D E A (1) B C A E D (1) B A E C D (1) A D B E C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -2 -4 -8 B -12 0 -14 -14 -8 C 2 14 0 -18 -22 D 4 14 18 0 -20 E 8 8 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -2 -4 -8 B -12 0 -14 -14 -8 C 2 14 0 -18 -22 D 4 14 18 0 -20 E 8 8 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=20 C=20 B=18 D=11 so D is eliminated. Round 2 votes counts: A=31 E=29 C=22 B=18 so B is eliminated. Round 3 votes counts: A=41 E=30 C=29 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 D:208 A:199 C:188 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -2 -4 -8 B -12 0 -14 -14 -8 C 2 14 0 -18 -22 D 4 14 18 0 -20 E 8 8 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -4 -8 B -12 0 -14 -14 -8 C 2 14 0 -18 -22 D 4 14 18 0 -20 E 8 8 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -4 -8 B -12 0 -14 -14 -8 C 2 14 0 -18 -22 D 4 14 18 0 -20 E 8 8 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1702: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (7) E B A C D (6) C D E A B (6) E C D B A (5) B A D E C (5) D E C B A (4) D B A C E (4) B A E D C (4) E B D C A (3) D C E B A (3) C A D B E (3) A D B C E (3) E D C B A (2) E D B C A (2) E C B D A (2) D C E A B (2) D C A B E (2) D A B C E (2) C E D B A (2) C E D A B (2) C E A B D (2) C D A E B (2) C A E D B (2) C A E B D (2) B E A C D (2) B D A E C (2) A C B E D (2) A B D C E (2) A B C D E (2) E C B A D (1) E C A B D (1) E B C A D (1) D E B C A (1) D B E C A (1) D B E A C (1) D B C E A (1) D A C B E (1) B E A D C (1) A C E B D (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -4 0 -2 B 6 0 4 -4 -4 C 4 -4 0 6 -4 D 0 4 -6 0 -2 E 2 4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 0 -2 B 6 0 4 -4 -4 C 4 -4 0 6 -4 D 0 4 -6 0 -2 E 2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 D=22 C=21 A=20 B=14 so B is eliminated. Round 2 votes counts: A=29 E=26 D=24 C=21 so C is eliminated. Round 3 votes counts: A=36 E=32 D=32 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 B:201 C:201 D:198 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 0 -2 B 6 0 4 -4 -4 C 4 -4 0 6 -4 D 0 4 -6 0 -2 E 2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 0 -2 B 6 0 4 -4 -4 C 4 -4 0 6 -4 D 0 4 -6 0 -2 E 2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 0 -2 B 6 0 4 -4 -4 C 4 -4 0 6 -4 D 0 4 -6 0 -2 E 2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1703: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) E C A B D (7) B E D C A (7) B E C A D (7) D B A E C (5) B E C D A (5) D B E C A (4) D B E A C (4) D A C B E (4) C A E B D (4) A C E D B (4) D A E C B (3) D A C E B (3) D A B C E (3) B A C E D (3) A C D E B (3) D B A C E (2) D A B E C (2) B D E A C (2) B D A E C (2) A C E B D (2) E D B C A (1) E C B D A (1) E C B A D (1) E C A D B (1) E B C D A (1) E B C A D (1) D E C A B (1) D E B C A (1) C E B A D (1) C E A B D (1) C A E D B (1) C A B E D (1) B D A C E (1) B C E A D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -16 -10 -20 -10 B 16 0 20 12 24 C 10 -20 0 -10 -28 D 20 -12 10 0 0 E 10 -24 28 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -20 -10 B 16 0 20 12 24 C 10 -20 0 -10 -28 D 20 -12 10 0 0 E 10 -24 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=32 E=13 A=11 C=8 so C is eliminated. Round 2 votes counts: B=36 D=32 A=17 E=15 so E is eliminated. Round 3 votes counts: B=41 D=33 A=26 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:236 D:209 E:207 C:176 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -20 -10 B 16 0 20 12 24 C 10 -20 0 -10 -28 D 20 -12 10 0 0 E 10 -24 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -20 -10 B 16 0 20 12 24 C 10 -20 0 -10 -28 D 20 -12 10 0 0 E 10 -24 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -20 -10 B 16 0 20 12 24 C 10 -20 0 -10 -28 D 20 -12 10 0 0 E 10 -24 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1704: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (14) C D B A E (14) A E B D C (14) A E B C D (8) E A B D C (6) D B C E A (5) B D E C A (5) E B D A C (4) A E C D B (4) A E C B D (4) A C D B E (4) E B A D C (3) D C B E A (3) C A D B E (3) B D C E A (3) A C E D B (2) E B D C A (1) C B D A E (1) B E D C A (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 0 -2 10 B 8 0 0 0 6 C 0 0 0 10 0 D 2 0 -10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.561773 C: 0.438227 D: 0.000000 E: 0.000000 Sum of squares = 0.507631676072 Cumulative probabilities = A: 0.000000 B: 0.561773 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -2 10 B 8 0 0 0 6 C 0 0 0 10 0 D 2 0 -10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=32 E=14 B=9 D=8 so D is eliminated. Round 2 votes counts: A=37 C=35 E=14 B=14 so E is eliminated. Round 3 votes counts: A=43 C=35 B=22 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:207 C:205 A:200 D:199 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -2 10 B 8 0 0 0 6 C 0 0 0 10 0 D 2 0 -10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -2 10 B 8 0 0 0 6 C 0 0 0 10 0 D 2 0 -10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -2 10 B 8 0 0 0 6 C 0 0 0 10 0 D 2 0 -10 0 6 E -10 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1705: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) B C E D A (9) E B C A D (7) D A C B E (7) E A D C B (6) C B D A E (5) E B C D A (4) E A D B C (4) A E D C B (4) A D E B C (4) E B A C D (3) C D B A E (3) C B E D A (3) C B D E A (3) B E C A D (3) E C B D A (2) E C B A D (2) E A B D C (2) D C A B E (2) D A E C B (2) D A C E B (2) B E C D A (2) A E D B C (2) A D C E B (2) E D A C B (1) C D B E A (1) B C D A E (1) B C A D E (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 0 4 6 -4 B 0 0 -14 -4 -16 C -4 14 0 0 -18 D -6 4 0 0 -8 E 4 16 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 6 -4 B 0 0 -14 -4 -16 C -4 14 0 0 -18 D -6 4 0 0 -8 E 4 16 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=24 B=17 C=15 D=13 so D is eliminated. Round 2 votes counts: A=35 E=31 C=17 B=17 so C is eliminated. Round 3 votes counts: A=37 B=32 E=31 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:223 A:203 C:196 D:195 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 6 -4 B 0 0 -14 -4 -16 C -4 14 0 0 -18 D -6 4 0 0 -8 E 4 16 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 -4 B 0 0 -14 -4 -16 C -4 14 0 0 -18 D -6 4 0 0 -8 E 4 16 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 -4 B 0 0 -14 -4 -16 C -4 14 0 0 -18 D -6 4 0 0 -8 E 4 16 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1706: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) D A B E C (7) E B A C D (6) D C A B E (6) B E A D C (6) C E A B D (5) C A E B D (5) D C A E B (4) D B A E C (4) C A E D B (4) C E B A D (3) C B E A D (3) B E D A C (3) B E A C D (3) B D E A C (3) A D E B C (3) E B A D C (2) D B A C E (2) B E C A D (2) A E B D C (2) E B C A D (1) D C B A E (1) D B E A C (1) D A C E B (1) C E B D A (1) C E A D B (1) C D B E A (1) C D B A E (1) C B E D A (1) C A D E B (1) B E C D A (1) B C E D A (1) A E D B C (1) A E C B D (1) A E B C D (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 0 8 12 B -8 0 0 0 -6 C 0 0 0 4 4 D -8 0 -4 0 -8 E -12 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.424735 B: 0.000000 C: 0.575265 D: 0.000000 E: 0.000000 Sum of squares = 0.511329595377 Cumulative probabilities = A: 0.424735 B: 0.424735 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 8 12 B -8 0 0 0 -6 C 0 0 0 4 4 D -8 0 -4 0 -8 E -12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=26 B=19 A=12 E=9 so E is eliminated. Round 2 votes counts: C=34 B=28 D=26 A=12 so A is eliminated. Round 3 votes counts: C=37 D=32 B=31 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:204 E:199 B:193 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 8 12 B -8 0 0 0 -6 C 0 0 0 4 4 D -8 0 -4 0 -8 E -12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 8 12 B -8 0 0 0 -6 C 0 0 0 4 4 D -8 0 -4 0 -8 E -12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 8 12 B -8 0 0 0 -6 C 0 0 0 4 4 D -8 0 -4 0 -8 E -12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1707: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (16) E D B C A (14) A C B D E (11) E C B A D (8) D E B A C (7) D B A C E (6) E C A B D (4) E D C B A (3) C A B E D (3) D E A B C (2) D B E C A (2) D B E A C (2) D A E B C (2) D A C B E (2) C E A B D (2) A C B E D (2) E C D A B (1) E C B D A (1) E B C D A (1) E B C A D (1) D E B C A (1) D B A E C (1) C B A D E (1) B D C A E (1) B D A C E (1) B C E A D (1) B C A E D (1) A D C B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 10 -26 0 B 4 0 22 -20 6 C -10 -22 0 -24 0 D 26 20 24 0 16 E 0 -6 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -26 0 B 4 0 22 -20 6 C -10 -22 0 -24 0 D 26 20 24 0 16 E 0 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=33 A=16 C=6 B=4 so B is eliminated. Round 2 votes counts: D=43 E=33 A=16 C=8 so C is eliminated. Round 3 votes counts: D=43 E=36 A=21 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:243 B:206 A:190 E:189 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 10 -26 0 B 4 0 22 -20 6 C -10 -22 0 -24 0 D 26 20 24 0 16 E 0 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -26 0 B 4 0 22 -20 6 C -10 -22 0 -24 0 D 26 20 24 0 16 E 0 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -26 0 B 4 0 22 -20 6 C -10 -22 0 -24 0 D 26 20 24 0 16 E 0 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1708: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) A E C D B (10) A E C B D (8) E C A D B (7) C E D B A (6) C E B D A (6) E C D A B (4) D B A C E (4) B D C E A (4) A D B E C (4) A B E C D (4) A B D E C (4) E C A B D (3) D A B C E (3) B D A C E (3) C E B A D (2) B C D E A (2) A E B C D (2) A D E B C (2) E C D B A (1) E A C D B (1) E A C B D (1) D C E B A (1) D C B E A (1) D A E C B (1) C B E D A (1) B C E D A (1) B A C E D (1) A E D C B (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 14 2 4 -2 B -14 0 -10 -14 -14 C -2 10 0 20 -8 D -4 14 -20 0 -18 E 2 14 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 4 -2 B -14 0 -10 -14 -14 C -2 10 0 20 -8 D -4 14 -20 0 -18 E 2 14 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=20 E=17 C=15 B=11 so B is eliminated. Round 2 votes counts: A=38 D=27 C=18 E=17 so E is eliminated. Round 3 votes counts: A=40 C=33 D=27 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:221 C:210 A:209 D:186 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 4 -2 B -14 0 -10 -14 -14 C -2 10 0 20 -8 D -4 14 -20 0 -18 E 2 14 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 4 -2 B -14 0 -10 -14 -14 C -2 10 0 20 -8 D -4 14 -20 0 -18 E 2 14 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 4 -2 B -14 0 -10 -14 -14 C -2 10 0 20 -8 D -4 14 -20 0 -18 E 2 14 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1709: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C D B A E (8) D A E C B (7) B C E A D (7) E A B D C (5) D C A E B (5) B E C A D (5) A E D B C (5) D A C E B (4) C B E D A (4) C B D E A (4) C B D A E (4) D C A B E (3) C D B E A (3) A D E B C (3) E D A C B (2) E A B C D (2) D E A C B (2) B E A C D (2) B C E D A (2) B C A E D (2) A B E D C (2) E D C A B (1) E C B A D (1) E B C A D (1) E B A C D (1) D C B A E (1) D A C B E (1) A E B D C (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -2 -2 -4 B -10 0 0 -12 -2 C 2 0 0 -6 -2 D 2 12 6 0 -8 E 4 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -2 -2 -4 B -10 0 0 -12 -2 C 2 0 0 -6 -2 D 2 12 6 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 C=23 B=18 A=13 so A is eliminated. Round 2 votes counts: E=30 D=27 C=23 B=20 so B is eliminated. Round 3 votes counts: E=39 C=34 D=27 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:206 A:201 C:197 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -2 -2 -4 B -10 0 0 -12 -2 C 2 0 0 -6 -2 D 2 12 6 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -2 -4 B -10 0 0 -12 -2 C 2 0 0 -6 -2 D 2 12 6 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -2 -4 B -10 0 0 -12 -2 C 2 0 0 -6 -2 D 2 12 6 0 -8 E 4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1710: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) D A C E B (8) E B C D A (7) B E C A D (6) E D A B C (5) C A D B E (5) A D C E B (5) C B E A D (4) C B A D E (4) B C E A D (4) A D C B E (4) E D A C B (3) E B D A C (3) D E A C B (3) D A E B C (3) B E D A C (3) B E C D A (3) E C D A B (2) E C B D A (2) C E D A B (2) B A D C E (2) E D C A B (1) C D A E B (1) C B A E D (1) C A D E B (1) C A B D E (1) B E A C D (1) B C A D E (1) B A D E C (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 16 10 -14 2 B -16 0 -16 -10 -14 C -10 16 0 -8 -10 D 14 10 8 0 6 E -2 14 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 -14 2 B -16 0 -16 -10 -14 C -10 16 0 -8 -10 D 14 10 8 0 6 E -2 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=23 B=21 C=19 A=12 so A is eliminated. Round 2 votes counts: D=35 E=23 B=23 C=19 so C is eliminated. Round 3 votes counts: D=42 B=33 E=25 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:208 A:207 C:194 B:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 10 -14 2 B -16 0 -16 -10 -14 C -10 16 0 -8 -10 D 14 10 8 0 6 E -2 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 -14 2 B -16 0 -16 -10 -14 C -10 16 0 -8 -10 D 14 10 8 0 6 E -2 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 -14 2 B -16 0 -16 -10 -14 C -10 16 0 -8 -10 D 14 10 8 0 6 E -2 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1711: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A B E D C (7) C D E A B (6) D C E A B (5) C D A E B (5) B A E D C (5) A D C E B (5) A D C B E (5) A B D C E (5) E B C D A (4) D C A E B (4) B E A C D (4) E C D B A (3) B A E C D (3) E D C A B (2) E C D A B (2) E A D C B (2) E A B D C (2) C E D B A (2) B C D E A (2) B C D A E (2) B A C D E (2) E D A C B (1) E B A D C (1) C D E B A (1) C D B A E (1) B E C A D (1) B C E D A (1) B C A D E (1) B A D C E (1) A E D C B (1) A D E C B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -6 -4 0 B -10 0 6 6 6 C 6 -6 0 4 -2 D 4 -6 -4 0 -4 E 0 -6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.35537190082 Cumulative probabilities = A: 0.272727 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 -4 0 B -10 0 6 6 6 C 6 -6 0 4 -2 D 4 -6 -4 0 -4 E 0 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.355371900821 Cumulative probabilities = A: 0.272727 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=26 E=17 C=15 D=9 so D is eliminated. Round 2 votes counts: B=33 A=26 C=24 E=17 so E is eliminated. Round 3 votes counts: B=38 C=31 A=31 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:204 C:201 A:200 E:200 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -6 -4 0 B -10 0 6 6 6 C 6 -6 0 4 -2 D 4 -6 -4 0 -4 E 0 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.355371900821 Cumulative probabilities = A: 0.272727 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -4 0 B -10 0 6 6 6 C 6 -6 0 4 -2 D 4 -6 -4 0 -4 E 0 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.355371900821 Cumulative probabilities = A: 0.272727 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -4 0 B -10 0 6 6 6 C 6 -6 0 4 -2 D 4 -6 -4 0 -4 E 0 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.355371900821 Cumulative probabilities = A: 0.272727 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1712: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) C B E A D (10) C B A E D (9) B C E D A (8) A D C B E (6) E D B C A (5) E B C D A (5) D A E B C (5) A D E C B (5) D E B A C (3) D A B C E (3) B E C D A (3) A D C E B (3) A C B E D (3) A C B D E (3) E D A B C (2) E C B A D (2) E B D C A (2) D E A B C (2) C E B A D (2) C A B E D (2) D B E C A (1) D A B E C (1) C B E D A (1) C A E B D (1) C A B D E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -22 -24 -2 -12 B 22 0 0 6 4 C 24 0 0 4 8 D 2 -6 -4 0 -12 E 12 -4 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.412409 C: 0.587591 D: 0.000000 E: 0.000000 Sum of squares = 0.515344516951 Cumulative probabilities = A: 0.000000 B: 0.412409 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -24 -2 -12 B 22 0 0 6 4 C 24 0 0 4 8 D 2 -6 -4 0 -12 E 12 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999814 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 A=22 E=16 B=11 so B is eliminated. Round 2 votes counts: C=34 D=25 A=22 E=19 so E is eliminated. Round 3 votes counts: C=44 D=34 A=22 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:216 E:206 D:190 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -24 -2 -12 B 22 0 0 6 4 C 24 0 0 4 8 D 2 -6 -4 0 -12 E 12 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999814 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -24 -2 -12 B 22 0 0 6 4 C 24 0 0 4 8 D 2 -6 -4 0 -12 E 12 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999814 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -24 -2 -12 B 22 0 0 6 4 C 24 0 0 4 8 D 2 -6 -4 0 -12 E 12 -4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999814 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1713: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) B A D E C (8) B C E D A (7) B A D C E (7) A D E C B (5) A B D E C (5) D A C E B (3) B D A C E (3) B C D E A (3) B A E C D (3) A D C E B (3) A D B C E (3) A B E D C (3) D C E A B (2) C E D A B (2) C D E A B (2) B E C D A (2) B E C A D (2) B E A C D (2) B D C E A (2) B D C A E (2) A B E C D (2) E C B D A (1) E B C A D (1) E A C D B (1) D C B E A (1) D C A E B (1) D B C A E (1) D B A C E (1) D A B C E (1) C E B D A (1) C D B E A (1) B A E D C (1) B A C E D (1) B A C D E (1) A E B C D (1) A D C B E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 14 4 16 B 2 0 28 18 30 C -14 -28 0 -10 0 D -4 -18 10 0 16 E -16 -30 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 4 16 B 2 0 28 18 30 C -14 -28 0 -10 0 D -4 -18 10 0 16 E -16 -30 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999951717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 A=25 E=15 D=10 C=6 so C is eliminated. Round 2 votes counts: B=44 A=25 E=18 D=13 so D is eliminated. Round 3 votes counts: B=48 A=30 E=22 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:239 A:216 D:202 C:174 E:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 14 4 16 B 2 0 28 18 30 C -14 -28 0 -10 0 D -4 -18 10 0 16 E -16 -30 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999951717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 4 16 B 2 0 28 18 30 C -14 -28 0 -10 0 D -4 -18 10 0 16 E -16 -30 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999951717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 4 16 B 2 0 28 18 30 C -14 -28 0 -10 0 D -4 -18 10 0 16 E -16 -30 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999951717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1714: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) D E A C B (8) B C A E D (8) B A E C D (6) E D A B C (5) C B D E A (5) C B D A E (5) A E D C B (5) A E B D C (5) E A D B C (4) D E C A B (3) D C E A B (3) D C B E A (3) C D B A E (3) B C A D E (3) B A C E D (3) A B E C D (3) D E C B A (2) D C E B A (2) B C D E A (2) A E B C D (2) E A D C B (1) D E B A C (1) D B E C A (1) C D E A B (1) C D B E A (1) C B A D E (1) B E A C D (1) B C E D A (1) B C E A D (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 2 10 8 10 B -2 0 10 -8 -6 C -10 -10 0 -6 -16 D -8 8 6 0 -10 E -10 6 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 8 10 B -2 0 10 -8 -6 C -10 -10 0 -6 -16 D -8 8 6 0 -10 E -10 6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983129 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 D=23 C=16 E=10 so E is eliminated. Round 2 votes counts: A=31 D=28 B=25 C=16 so C is eliminated. Round 3 votes counts: B=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:215 E:211 D:198 B:197 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 10 B -2 0 10 -8 -6 C -10 -10 0 -6 -16 D -8 8 6 0 -10 E -10 6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983129 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 10 B -2 0 10 -8 -6 C -10 -10 0 -6 -16 D -8 8 6 0 -10 E -10 6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983129 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 10 B -2 0 10 -8 -6 C -10 -10 0 -6 -16 D -8 8 6 0 -10 E -10 6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983129 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1715: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (5) C D E A B (5) C A D B E (5) E B A C D (4) D E B C A (4) D C E B A (4) B E A D C (4) A C E B D (4) E B D A C (3) D C E A B (3) C A E B D (3) C A D E B (3) A B C E D (3) A B C D E (3) E D C B A (2) E B A D C (2) D E C B A (2) D C A E B (2) D B C E A (2) D B A E C (2) D B A C E (2) C E A D B (2) C D A E B (2) C A E D B (2) B D E A C (2) A C B E D (2) A B E C D (2) E C B A D (1) E B D C A (1) E B C D A (1) D C B E A (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C A E (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E B A (1) C D A B E (1) B E D A C (1) B E A C D (1) B A E D C (1) B A D C E (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -12 -6 -8 B 2 0 0 -16 -14 C 12 0 0 -4 18 D 6 16 4 0 8 E 8 14 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -6 -8 B 2 0 0 -16 -14 C 12 0 0 -4 18 D 6 16 4 0 8 E 8 14 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 E=19 A=18 B=10 so B is eliminated. Round 2 votes counts: D=30 E=25 C=25 A=20 so A is eliminated. Round 3 votes counts: C=39 D=32 E=29 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:213 E:198 A:186 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 -6 -8 B 2 0 0 -16 -14 C 12 0 0 -4 18 D 6 16 4 0 8 E 8 14 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -6 -8 B 2 0 0 -16 -14 C 12 0 0 -4 18 D 6 16 4 0 8 E 8 14 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -6 -8 B 2 0 0 -16 -14 C 12 0 0 -4 18 D 6 16 4 0 8 E 8 14 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1716: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) E C D A B (12) D A E B C (12) D A B C E (9) E C B A D (8) C E B A D (8) C B E A D (6) A D B C E (4) E C D B A (3) D A E C B (3) B C E A D (3) E D C A B (2) E C B D A (2) C E D B A (2) B C A D E (2) A B D C E (2) E D A C B (1) E A D B C (1) D E C A B (1) C E B D A (1) C D B A E (1) B A E C D (1) B A D C E (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 22 -2 -24 0 B -22 0 0 -30 -12 C 2 0 0 0 -20 D 24 30 0 0 0 E 0 12 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.401377 E: 0.598623 Sum of squares = 0.519453135145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.401377 E: 1.000000 A B C D E A 0 22 -2 -24 0 B -22 0 0 -30 -12 C 2 0 0 0 -20 D 24 30 0 0 0 E 0 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=29 C=18 B=8 A=7 so A is eliminated. Round 2 votes counts: D=43 E=29 C=18 B=10 so B is eliminated. Round 3 votes counts: D=46 E=30 C=24 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:216 A:198 C:191 B:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 22 -2 -24 0 B -22 0 0 -30 -12 C 2 0 0 0 -20 D 24 30 0 0 0 E 0 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -2 -24 0 B -22 0 0 -30 -12 C 2 0 0 0 -20 D 24 30 0 0 0 E 0 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -2 -24 0 B -22 0 0 -30 -12 C 2 0 0 0 -20 D 24 30 0 0 0 E 0 12 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1717: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (12) B A D E C (6) A E B C D (6) E C A D B (5) C E A D B (5) D C B E A (4) B D A E C (4) B D A C E (4) E D C A B (3) D C E B A (3) D B C E A (3) C E D A B (3) C D B E A (3) C A E B D (3) B C D A E (3) A E C B D (3) A E B D C (3) E A D C B (2) C D E A B (2) C B A E D (2) C A E D B (2) B D C A E (2) B A E C D (2) A B E C D (2) E D A B C (1) E A C D B (1) E A B D C (1) D E C A B (1) D E B A C (1) D E A B C (1) D C E A B (1) D B E C A (1) D B E A C (1) D B C A E (1) C D E B A (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 4 14 14 B 8 0 12 10 4 C -4 -12 0 -12 -14 D -14 -10 12 0 -16 E -14 -4 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 14 14 B 8 0 12 10 4 C -4 -12 0 -12 -14 D -14 -10 12 0 -16 E -14 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999159 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=21 D=17 A=15 E=13 so E is eliminated. Round 2 votes counts: B=34 C=26 D=21 A=19 so A is eliminated. Round 3 votes counts: B=47 C=30 D=23 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:212 E:206 D:186 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 14 14 B 8 0 12 10 4 C -4 -12 0 -12 -14 D -14 -10 12 0 -16 E -14 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999159 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 14 14 B 8 0 12 10 4 C -4 -12 0 -12 -14 D -14 -10 12 0 -16 E -14 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999159 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 14 14 B 8 0 12 10 4 C -4 -12 0 -12 -14 D -14 -10 12 0 -16 E -14 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999159 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1718: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) D A E B C (7) D A B E C (7) A B D C E (7) E C B D A (6) E D C B A (5) E C D B A (5) D E A C B (5) B C A E D (5) B A C D E (5) E C B A D (4) D E A B C (4) C E B A D (4) A D B C E (3) A B C D E (3) E D C A B (2) D B A C E (2) C B E D A (2) B D A C E (2) B A D C E (2) B A C E D (2) E D A C B (1) E C D A B (1) E A C D B (1) D A E C B (1) C E B D A (1) B D C A E (1) B C E A D (1) A D E B C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 10 -4 0 B 10 0 6 10 4 C -10 -6 0 0 0 D 4 -10 0 0 0 E 0 -4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -4 0 B 10 0 6 10 4 C -10 -6 0 0 0 D 4 -10 0 0 0 E 0 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 B=18 A=16 C=15 so C is eliminated. Round 2 votes counts: E=30 B=28 D=26 A=16 so A is eliminated. Round 3 votes counts: B=40 E=30 D=30 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:198 E:198 D:197 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 -4 0 B 10 0 6 10 4 C -10 -6 0 0 0 D 4 -10 0 0 0 E 0 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -4 0 B 10 0 6 10 4 C -10 -6 0 0 0 D 4 -10 0 0 0 E 0 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -4 0 B 10 0 6 10 4 C -10 -6 0 0 0 D 4 -10 0 0 0 E 0 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1719: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D E A C B (6) D A B C E (6) E B C A D (5) D A C E B (5) B C A E D (5) B D A C E (4) A D C E B (4) E D A C B (3) E C B A D (3) E C A D B (3) E B C D A (3) B E D C A (3) B D E A C (3) A C D E B (3) E D C A B (2) E C A B D (2) D B A C E (2) D A C B E (2) C E A B D (2) B C E A D (2) A C D B E (2) A C B D E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C D A B (1) E C B D A (1) D E B A C (1) D B E A C (1) D B A E C (1) D A E C B (1) D A E B C (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E B D (1) B D E C A (1) B A D C E (1) B A C D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 0 -12 B 2 0 4 4 -2 C -2 -4 0 0 -6 D 0 -4 0 0 -2 E 12 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 0 -12 B 2 0 4 4 -2 C -2 -4 0 0 -6 D 0 -4 0 0 -2 E 12 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=26 D=26 A=13 C=6 so C is eliminated. Round 2 votes counts: B=31 E=29 D=26 A=14 so A is eliminated. Round 3 votes counts: D=35 B=35 E=30 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:211 B:204 D:197 A:194 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 0 -12 B 2 0 4 4 -2 C -2 -4 0 0 -6 D 0 -4 0 0 -2 E 12 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 -12 B 2 0 4 4 -2 C -2 -4 0 0 -6 D 0 -4 0 0 -2 E 12 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 -12 B 2 0 4 4 -2 C -2 -4 0 0 -6 D 0 -4 0 0 -2 E 12 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1720: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) D A B E C (8) E C A D B (5) D B A E C (5) B D A C E (5) B D C E A (4) B D A E C (4) B C A E D (4) A D E B C (4) E C A B D (3) D B C E A (3) D B A C E (3) C E B D A (3) A D B E C (3) E A C D B (2) D A E C B (2) D A E B C (2) C E B A D (2) C E A D B (2) C B E D A (2) B D C A E (2) B C D E A (2) B C D A E (2) A E D C B (2) A E C D B (2) A E C B D (2) A B E C D (2) E A C B D (1) D E C A B (1) D E A C B (1) D C E B A (1) D C B E A (1) C E D A B (1) B C E A D (1) B A D C E (1) A E B D C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 4 -4 12 B -10 0 14 2 6 C -4 -14 0 -10 -4 D 4 -2 10 0 12 E -12 -6 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000176 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -4 12 B -10 0 14 2 6 C -4 -14 0 -10 -4 D 4 -2 10 0 12 E -12 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999738 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 C=19 A=18 E=11 so E is eliminated. Round 2 votes counts: D=27 C=27 B=25 A=21 so A is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:211 B:206 E:187 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -4 12 B -10 0 14 2 6 C -4 -14 0 -10 -4 D 4 -2 10 0 12 E -12 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999738 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -4 12 B -10 0 14 2 6 C -4 -14 0 -10 -4 D 4 -2 10 0 12 E -12 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999738 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -4 12 B -10 0 14 2 6 C -4 -14 0 -10 -4 D 4 -2 10 0 12 E -12 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999738 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1721: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) D E B C A (9) A C B E D (9) C A B E D (5) D E B A C (4) D A B E C (4) C E B A D (4) C E A B D (4) A D B C E (4) A C B D E (4) E D C B A (3) E C D B A (3) D A E B C (3) A B C D E (3) E C B D A (2) D B E A C (2) D B A E C (2) C A E B D (2) A D B E C (2) A C E B D (2) A C D B E (2) E B D C A (1) E B C D A (1) D E C B A (1) D E A B C (1) D B E C A (1) C E B D A (1) C E A D B (1) C B A E D (1) B E D C A (1) B E C D A (1) B D E A C (1) B A C E D (1) A D C E B (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -6 -6 -6 B 2 0 8 -12 -12 C 6 -8 0 -6 -8 D 6 12 6 0 -10 E 6 12 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 -6 B 2 0 8 -12 -12 C 6 -8 0 -6 -8 D 6 12 6 0 -10 E 6 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=27 E=22 C=18 B=4 so B is eliminated. Round 2 votes counts: A=30 D=28 E=24 C=18 so C is eliminated. Round 3 votes counts: A=38 E=34 D=28 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:207 B:193 C:192 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -6 B 2 0 8 -12 -12 C 6 -8 0 -6 -8 D 6 12 6 0 -10 E 6 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -6 B 2 0 8 -12 -12 C 6 -8 0 -6 -8 D 6 12 6 0 -10 E 6 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -6 B 2 0 8 -12 -12 C 6 -8 0 -6 -8 D 6 12 6 0 -10 E 6 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1722: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) C B D A E (7) E A B D C (5) C D B A E (5) B C D E A (5) A C E D B (5) D C B E A (4) C B A E D (4) A E D C B (4) D C A E B (3) C B A D E (3) C A D B E (3) C A B E D (3) B D C E A (3) A E B C D (3) E D B A C (2) E D A B C (2) D B E C A (2) C B D E A (2) B C E A D (2) A E C B D (2) A C E B D (2) E B D A C (1) E B A D C (1) E A D C B (1) D E B C A (1) D E B A C (1) D E A C B (1) D C E A B (1) D B C E A (1) D A C E B (1) C D B E A (1) C D A E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E C A (1) B C E D A (1) B C D A E (1) B C A E D (1) A E D B C (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -14 0 2 B 6 0 -10 4 8 C 14 10 0 10 20 D 0 -4 -10 0 -6 E -2 -8 -20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 0 2 B 6 0 -10 4 8 C 14 10 0 10 20 D 0 -4 -10 0 -6 E -2 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=19 A=19 B=18 D=15 so D is eliminated. Round 2 votes counts: C=37 E=22 B=21 A=20 so A is eliminated. Round 3 votes counts: C=46 E=33 B=21 so B is eliminated. Round 4 votes counts: C=60 E=40 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:204 A:191 D:190 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 0 2 B 6 0 -10 4 8 C 14 10 0 10 20 D 0 -4 -10 0 -6 E -2 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 0 2 B 6 0 -10 4 8 C 14 10 0 10 20 D 0 -4 -10 0 -6 E -2 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 0 2 B 6 0 -10 4 8 C 14 10 0 10 20 D 0 -4 -10 0 -6 E -2 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1723: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (7) E D A B C (6) D C A B E (6) B C A E D (6) E D C B A (5) C B D A E (5) C B A D E (5) A B C D E (5) E B A C D (4) E A B D C (4) B A C D E (4) E D A C B (3) E C D B A (3) E A D B C (3) D C E B A (3) D A C B E (3) C B E A D (3) E B C A D (2) D E C A B (2) D C E A B (2) D C B A E (2) B C A D E (2) B A C E D (2) E D C A B (1) E C B D A (1) D E A C B (1) D A B C E (1) C D A B E (1) B E C A D (1) B E A C D (1) A E B D C (1) A E B C D (1) A D C B E (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -14 -4 10 B 12 0 -8 -2 16 C 14 8 0 8 20 D 4 2 -8 0 4 E -10 -16 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -4 10 B 12 0 -8 -2 16 C 14 8 0 8 20 D 4 2 -8 0 4 E -10 -16 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=21 D=20 B=16 A=11 so A is eliminated. Round 2 votes counts: E=34 B=23 D=22 C=21 so C is eliminated. Round 3 votes counts: B=36 E=34 D=30 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:225 B:209 D:201 A:190 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -4 10 B 12 0 -8 -2 16 C 14 8 0 8 20 D 4 2 -8 0 4 E -10 -16 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -4 10 B 12 0 -8 -2 16 C 14 8 0 8 20 D 4 2 -8 0 4 E -10 -16 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -4 10 B 12 0 -8 -2 16 C 14 8 0 8 20 D 4 2 -8 0 4 E -10 -16 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1724: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) B E C A D (11) D A C E B (7) B E D C A (7) D B E A C (6) C E B A D (6) C A E B D (6) A C D E B (6) E B C A D (4) D A E B C (4) B E C D A (4) D C A B E (3) B D E A C (3) E B A D C (2) E B A C D (2) C A D E B (2) A D C E B (2) D E B A C (1) D C B E A (1) D B A C E (1) D A C B E (1) D A B E C (1) C D A B E (1) C B E A D (1) C A E D B (1) C A D B E (1) B D E C A (1) B D C E A (1) A E D B C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -24 0 -6 -22 B 24 0 20 20 8 C 0 -20 0 -6 -16 D 6 -20 6 0 -16 E 22 -8 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 0 -6 -22 B 24 0 20 20 8 C 0 -20 0 -6 -16 D 6 -20 6 0 -16 E 22 -8 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=25 C=18 A=11 E=8 so E is eliminated. Round 2 votes counts: B=46 D=25 C=18 A=11 so A is eliminated. Round 3 votes counts: B=46 D=28 C=26 so C is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:236 E:223 D:188 C:179 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 0 -6 -22 B 24 0 20 20 8 C 0 -20 0 -6 -16 D 6 -20 6 0 -16 E 22 -8 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 0 -6 -22 B 24 0 20 20 8 C 0 -20 0 -6 -16 D 6 -20 6 0 -16 E 22 -8 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 0 -6 -22 B 24 0 20 20 8 C 0 -20 0 -6 -16 D 6 -20 6 0 -16 E 22 -8 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1725: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) A C D B E (7) D E B A C (6) D A C E B (5) E B D C A (4) A C B D E (4) E D B C A (3) D E B C A (3) D E A B C (3) D A E B C (3) C B E A D (3) C B A E D (3) B A C E D (3) A D C B E (3) A C B E D (3) A B D E C (3) A B C E D (3) D E C B A (2) D A B E C (2) C D A E B (2) B E C A D (2) B C E A D (2) B A E C D (2) A B C D E (2) E D C B A (1) E D B A C (1) E C D B A (1) E C B D A (1) E B D A C (1) E B C D A (1) E B C A D (1) D E C A B (1) D E A C B (1) D C A E B (1) C E B A D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E A D C (1) B C A E D (1) B A E D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 12 16 22 B -12 0 0 4 14 C -12 0 0 8 10 D -16 -4 -8 0 4 E -22 -14 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 16 22 B -12 0 0 4 14 C -12 0 0 8 10 D -16 -4 -8 0 4 E -22 -14 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 C=20 E=14 B=12 so B is eliminated. Round 2 votes counts: A=33 D=27 C=23 E=17 so E is eliminated. Round 3 votes counts: D=37 A=34 C=29 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:231 B:203 C:203 D:188 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 16 22 B -12 0 0 4 14 C -12 0 0 8 10 D -16 -4 -8 0 4 E -22 -14 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 16 22 B -12 0 0 4 14 C -12 0 0 8 10 D -16 -4 -8 0 4 E -22 -14 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 16 22 B -12 0 0 4 14 C -12 0 0 8 10 D -16 -4 -8 0 4 E -22 -14 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1726: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) C A D B E (5) A D E C B (5) E B D C A (4) E B D A C (4) D A E C B (4) D A C E B (4) C A B D E (4) B E C A D (4) B E A D C (4) B E A C D (4) A D C E B (4) A C D B E (4) D E A C B (3) D E A B C (3) C B A D E (3) B C E A D (3) E D B C A (2) E B C D A (2) B C A E D (2) A D C B E (2) A B E D C (2) E D C B A (1) E D B A C (1) E A D B C (1) E A B D C (1) D C E A B (1) D A E B C (1) C D E B A (1) C D A B E (1) C B E D A (1) C B D A E (1) C A D E B (1) B A E C D (1) A E B D C (1) A D E B C (1) A D B E C (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 6 10 -2 B -2 0 8 6 8 C -6 -8 0 0 -24 D -10 -6 0 0 0 E 2 -8 24 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000133 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 2 6 10 -2 B -2 0 8 6 8 C -6 -8 0 0 -24 D -10 -6 0 0 0 E 2 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=22 C=17 E=16 D=16 so E is eliminated. Round 2 votes counts: B=39 A=24 D=20 C=17 so C is eliminated. Round 3 votes counts: B=44 A=34 D=22 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:210 E:209 A:208 D:192 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 10 -2 B -2 0 8 6 8 C -6 -8 0 0 -24 D -10 -6 0 0 0 E 2 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 10 -2 B -2 0 8 6 8 C -6 -8 0 0 -24 D -10 -6 0 0 0 E 2 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 10 -2 B -2 0 8 6 8 C -6 -8 0 0 -24 D -10 -6 0 0 0 E 2 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1727: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) E B C D A (7) B E A D C (6) D A C B E (5) C E B D A (5) A D B C E (5) E B A C D (4) D C A B E (4) B E C D A (4) A D B E C (4) A C D E B (4) E C B A D (3) E B C A D (3) C E D B A (3) C D E A B (3) C D A B E (3) B E D A C (3) A D C B E (3) E C B D A (2) E B A D C (2) D A B C E (2) C D E B A (2) C A D E B (2) A E B D C (2) A B E D C (2) A B D E C (2) D B C E A (1) D A B E C (1) C E B A D (1) C D B E A (1) B A E D C (1) B A D E C (1) A E B C D (1) Total count = 100 A B C D E A 0 2 -4 -8 0 B -2 0 2 -2 -4 C 4 -2 0 12 4 D 8 2 -12 0 2 E 0 4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999897 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 2 -4 -8 0 B -2 0 2 -2 -4 C 4 -2 0 12 4 D 8 2 -12 0 2 E 0 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 E=21 B=15 D=13 so D is eliminated. Round 2 votes counts: C=32 A=31 E=21 B=16 so B is eliminated. Round 3 votes counts: E=34 C=33 A=33 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:209 D:200 E:199 B:197 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -8 0 B -2 0 2 -2 -4 C 4 -2 0 12 4 D 8 2 -12 0 2 E 0 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -8 0 B -2 0 2 -2 -4 C 4 -2 0 12 4 D 8 2 -12 0 2 E 0 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -8 0 B -2 0 2 -2 -4 C 4 -2 0 12 4 D 8 2 -12 0 2 E 0 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1728: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) D E C B A (9) E B A D C (8) D C E B A (8) E D B A C (5) B A E D C (5) E D C B A (4) C D E B A (4) C A D B E (4) B E A D C (4) E B D A C (3) C A B D E (3) A B E C D (3) E D B C A (2) D C B A E (2) D C A B E (2) C E A B D (2) C D E A B (2) C D A E B (2) C A B E D (2) A C B E D (2) A B E D C (2) A B C E D (2) E C D B A (1) E C B D A (1) E B A C D (1) D E B A C (1) D C E A B (1) D C B E A (1) D B E A C (1) C E D A B (1) C A E B D (1) B A D E C (1) Total count = 100 A B C D E A 0 -22 -24 -20 -18 B 22 0 -24 -20 -12 C 24 24 0 -18 -2 D 20 20 18 0 2 E 18 12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -24 -20 -18 B 22 0 -24 -20 -12 C 24 24 0 -18 -2 D 20 20 18 0 2 E 18 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999949929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=25 D=25 B=10 A=9 so A is eliminated. Round 2 votes counts: C=33 E=25 D=25 B=17 so B is eliminated. Round 3 votes counts: E=39 C=35 D=26 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:230 E:215 C:214 B:183 A:158 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -24 -20 -18 B 22 0 -24 -20 -12 C 24 24 0 -18 -2 D 20 20 18 0 2 E 18 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999949929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -24 -20 -18 B 22 0 -24 -20 -12 C 24 24 0 -18 -2 D 20 20 18 0 2 E 18 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999949929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -24 -20 -18 B 22 0 -24 -20 -12 C 24 24 0 -18 -2 D 20 20 18 0 2 E 18 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999949929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1729: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (17) B C D A E (17) E A D B C (6) A D E C B (6) E B C A D (4) B D C A E (4) B C E D A (4) A D C E B (4) E C A D B (3) E A C D B (3) D A C B E (3) C D A E B (3) B E C D A (3) B C D E A (3) D A C E B (2) C B D A E (2) B E C A D (2) E C B A D (1) E B A C D (1) E A C B D (1) E A B D C (1) D C A E B (1) D C A B E (1) C E D A B (1) C E B D A (1) C D B A E (1) C B D E A (1) C A D E B (1) B E D A C (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 10 -6 4 -6 B -10 0 -6 -8 -16 C 6 6 0 4 -2 D -4 8 -4 0 -2 E 6 16 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -6 4 -6 B -10 0 -6 -8 -16 C 6 6 0 4 -2 D -4 8 -4 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=34 A=12 C=10 D=7 so D is eliminated. Round 2 votes counts: E=37 B=34 A=17 C=12 so C is eliminated. Round 3 votes counts: E=39 B=38 A=23 so A is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 C:207 A:201 D:199 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -6 4 -6 B -10 0 -6 -8 -16 C 6 6 0 4 -2 D -4 8 -4 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 4 -6 B -10 0 -6 -8 -16 C 6 6 0 4 -2 D -4 8 -4 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 4 -6 B -10 0 -6 -8 -16 C 6 6 0 4 -2 D -4 8 -4 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1730: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) C B A D E (8) A D B C E (7) A B C D E (7) E D C B A (6) E C B D A (5) D A C B E (5) C B A E D (5) C B E A D (4) E C D B A (3) E C B A D (3) D E A C B (3) D A E B C (3) B C A E D (3) B A C E D (3) E D C A B (2) D E A B C (2) D A B C E (2) E D A C B (1) E B C A D (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E A B (1) D C A B E (1) D A C E B (1) C E B D A (1) C E B A D (1) C B D A E (1) B E C A D (1) B C E A D (1) B A C D E (1) A D E B C (1) A D B E C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 4 2 2 B -6 0 -6 -4 4 C -4 6 0 0 6 D -2 4 0 0 -8 E -2 -4 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 2 B -6 0 -6 -4 4 C -4 6 0 0 6 D -2 4 0 0 -8 E -2 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=20 D=19 A=18 B=9 so B is eliminated. Round 2 votes counts: E=35 C=24 A=22 D=19 so D is eliminated. Round 3 votes counts: E=41 A=33 C=26 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 C:204 E:198 D:197 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 2 B -6 0 -6 -4 4 C -4 6 0 0 6 D -2 4 0 0 -8 E -2 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 2 B -6 0 -6 -4 4 C -4 6 0 0 6 D -2 4 0 0 -8 E -2 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 2 B -6 0 -6 -4 4 C -4 6 0 0 6 D -2 4 0 0 -8 E -2 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1731: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) D C A B E (5) D B C A E (5) A E C B D (5) A C E B D (5) E D B C A (4) E B D C A (4) E A C B D (4) D E B C A (4) D B E C A (4) D A C B E (4) A C B E D (4) E B A C D (3) D A C E B (3) C A B D E (3) B D E C A (3) B D C E A (3) E B C D A (2) E B C A D (2) E A B C D (2) C B A D E (2) B C D A E (2) B C A D E (2) A C B D E (2) E D A B C (1) E B D A C (1) E A D B C (1) E A B D C (1) D E B A C (1) D C B A E (1) C B A E D (1) C A D B E (1) B E C A D (1) B D C A E (1) B C E A D (1) B C A E D (1) A E D C B (1) A E C D B (1) A D C B E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -16 -8 4 B 8 0 10 10 8 C 16 -10 0 -8 10 D 8 -10 8 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -8 4 B 8 0 10 10 8 C 16 -10 0 -8 10 D 8 -10 8 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=25 A=21 B=14 C=7 so C is eliminated. Round 2 votes counts: D=33 E=25 A=25 B=17 so B is eliminated. Round 3 votes counts: D=42 A=31 E=27 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:207 C:204 A:186 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 4 B 8 0 10 10 8 C 16 -10 0 -8 10 D 8 -10 8 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 4 B 8 0 10 10 8 C 16 -10 0 -8 10 D 8 -10 8 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 4 B 8 0 10 10 8 C 16 -10 0 -8 10 D 8 -10 8 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1732: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D A E C B (9) D B A C E (8) D A C E B (7) B C E A D (7) D B E A C (4) B E C A D (4) B D E C A (4) B D C A E (4) A C E D B (4) D B A E C (3) D A B C E (3) A E C D B (3) E C B A D (2) E A C D B (2) E A C B D (2) C E A B D (2) C A E D B (2) B E C D A (2) B D E A C (2) A C D E B (2) E D B A C (1) E B D A C (1) E B C A D (1) E B A C D (1) D E B A C (1) D A E B C (1) D A C B E (1) D A B E C (1) C A E B D (1) B E D C A (1) B C E D A (1) B C A E D (1) B C A D E (1) A D E C B (1) Total count = 100 A B C D E A 0 2 14 -8 4 B -2 0 4 -6 -6 C -14 -4 0 -4 -12 D 8 6 4 0 4 E -4 6 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 -8 4 B -2 0 4 -6 -6 C -14 -4 0 -4 -12 D 8 6 4 0 4 E -4 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=27 E=20 A=10 C=5 so C is eliminated. Round 2 votes counts: D=38 B=27 E=22 A=13 so A is eliminated. Round 3 votes counts: D=41 E=32 B=27 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:206 E:205 B:195 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 14 -8 4 B -2 0 4 -6 -6 C -14 -4 0 -4 -12 D 8 6 4 0 4 E -4 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -8 4 B -2 0 4 -6 -6 C -14 -4 0 -4 -12 D 8 6 4 0 4 E -4 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -8 4 B -2 0 4 -6 -6 C -14 -4 0 -4 -12 D 8 6 4 0 4 E -4 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1733: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) C E D A B (9) D E B A C (8) A B C E D (6) C A B E D (5) B D E A C (5) B A D E C (5) E D C B A (4) C A E D B (4) C A E B D (4) A C B D E (4) A B C D E (4) D E C B A (3) D E C A B (3) C D E A B (3) E D C A B (2) E D B C A (2) D B E A C (2) C A D E B (2) C A B D E (2) B D A E C (2) B A C D E (2) A C B E D (2) E D B A C (1) E C D B A (1) E C B D A (1) D C E A B (1) C E A D B (1) B A E D C (1) B A D C E (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -12 -12 -10 B -6 0 -4 -12 -16 C 12 4 0 2 2 D 12 12 -2 0 14 E 10 16 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -12 -10 B -6 0 -4 -12 -16 C 12 4 0 2 2 D 12 12 -2 0 14 E 10 16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=26 A=17 B=16 E=11 so E is eliminated. Round 2 votes counts: D=35 C=32 A=17 B=16 so B is eliminated. Round 3 votes counts: D=42 C=32 A=26 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:210 E:205 A:186 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -12 -10 B -6 0 -4 -12 -16 C 12 4 0 2 2 D 12 12 -2 0 14 E 10 16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -12 -10 B -6 0 -4 -12 -16 C 12 4 0 2 2 D 12 12 -2 0 14 E 10 16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -12 -10 B -6 0 -4 -12 -16 C 12 4 0 2 2 D 12 12 -2 0 14 E 10 16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1734: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (6) C D E A B (5) B D C E A (5) A E B D C (5) E A D B C (4) B A E D C (4) A C E B D (4) A B E D C (4) E D A B C (3) D E C B A (3) D B E A C (3) C D E B A (3) C B A E D (3) C A E D B (3) B D C A E (3) B C D A E (3) B A C E D (3) A E D C B (3) A B E C D (3) E D A C B (2) D E C A B (2) D E B A C (2) D B E C A (2) C A E B D (2) B D A E C (2) A E D B C (2) E D C A B (1) E A D C B (1) D E B C A (1) D C E B A (1) D B C E A (1) C E A D B (1) C D B E A (1) C B D E A (1) C A D E B (1) B D E A C (1) B D A C E (1) B C A D E (1) B A C D E (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 10 14 8 14 B -10 0 10 -2 -14 C -14 -10 0 -12 -12 D -8 2 12 0 -14 E -14 14 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 8 14 B -10 0 10 -2 -14 C -14 -10 0 -12 -12 D -8 2 12 0 -14 E -14 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 C=20 D=15 E=11 so E is eliminated. Round 2 votes counts: A=35 B=24 D=21 C=20 so C is eliminated. Round 3 votes counts: A=42 D=30 B=28 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:213 D:196 B:192 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 8 14 B -10 0 10 -2 -14 C -14 -10 0 -12 -12 D -8 2 12 0 -14 E -14 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 8 14 B -10 0 10 -2 -14 C -14 -10 0 -12 -12 D -8 2 12 0 -14 E -14 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 8 14 B -10 0 10 -2 -14 C -14 -10 0 -12 -12 D -8 2 12 0 -14 E -14 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1735: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) C E D A B (7) C A B E D (7) A B C D E (7) B A D E C (6) D E C B A (5) E C D A B (4) D E C A B (4) A B D C E (4) E D C A B (3) E D B C A (3) D E B A C (3) D C E A B (3) C E A B D (3) C A E B D (3) B A D C E (3) B A C D E (3) A C B D E (3) E C D B A (2) C E A D B (2) B A E D C (2) B A E C D (2) A B C E D (2) E D B A C (1) D A B E C (1) C E D B A (1) C D A E B (1) C A D E B (1) C A B D E (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 14 -22 2 -4 B -14 0 -22 -6 -14 C 22 22 0 0 6 D -2 6 0 0 -8 E 4 14 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.825773 D: 0.174227 E: 0.000000 Sum of squares = 0.712256305523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.825773 D: 1.000000 E: 1.000000 A B C D E A 0 14 -22 2 -4 B -14 0 -22 -6 -14 C 22 22 0 0 6 D -2 6 0 0 -8 E 4 14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.428571 E: 0.000000 Sum of squares = 0.510204182799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 B=17 A=17 D=16 so D is eliminated. Round 2 votes counts: E=36 C=29 A=18 B=17 so B is eliminated. Round 3 votes counts: E=36 A=35 C=29 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:225 E:210 D:198 A:195 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -22 2 -4 B -14 0 -22 -6 -14 C 22 22 0 0 6 D -2 6 0 0 -8 E 4 14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.428571 E: 0.000000 Sum of squares = 0.510204182799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -22 2 -4 B -14 0 -22 -6 -14 C 22 22 0 0 6 D -2 6 0 0 -8 E 4 14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.428571 E: 0.000000 Sum of squares = 0.510204182799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -22 2 -4 B -14 0 -22 -6 -14 C 22 22 0 0 6 D -2 6 0 0 -8 E 4 14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.428571 E: 0.000000 Sum of squares = 0.510204182799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1736: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) B E A C D (9) A D C E B (8) C D B E A (7) A E B D C (7) C D B A E (6) C D A B E (6) D C A B E (5) A B E D C (4) E B A D C (3) E A B D C (3) D C E A B (3) A E D C B (3) E B A C D (2) C D E B A (2) B E C D A (2) B E C A D (2) B C D A E (2) A D E C B (2) A D C B E (2) A B D C E (2) E B C D A (1) E A D C B (1) D C E B A (1) C E D B A (1) B C E D A (1) B C D E A (1) B A E C D (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 18 -2 2 22 B -18 0 -16 -18 4 C 2 16 0 -12 18 D -2 18 12 0 18 E -22 -4 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.125000 D: 0.125000 E: 0.000000 Sum of squares = 0.59375000004 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -2 2 22 B -18 0 -16 -18 4 C 2 16 0 -12 18 D -2 18 12 0 18 E -22 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.125000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749998447 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=22 D=20 B=18 E=10 so E is eliminated. Round 2 votes counts: A=34 B=24 C=22 D=20 so D is eliminated. Round 3 votes counts: C=42 A=34 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:223 A:220 C:212 B:176 E:169 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 18 -2 2 22 B -18 0 -16 -18 4 C 2 16 0 -12 18 D -2 18 12 0 18 E -22 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.125000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749998447 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -2 2 22 B -18 0 -16 -18 4 C 2 16 0 -12 18 D -2 18 12 0 18 E -22 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.125000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749998447 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -2 2 22 B -18 0 -16 -18 4 C 2 16 0 -12 18 D -2 18 12 0 18 E -22 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.125000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749998447 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1737: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) C D E A B (6) A B C E D (5) D E C B A (4) D E B C A (4) C A E D B (4) B D E A C (4) A C E D B (4) E D C B A (3) E B D A C (3) B D E C A (3) B A C E D (3) A E C D B (3) A C B E D (3) A B C D E (3) E C D A B (2) E A B D C (2) D C E B A (2) D C B E A (2) D B E C A (2) B E D A C (2) B C D A E (2) B A D E C (2) B A D C E (2) B A C D E (2) A E C B D (2) A C E B D (2) A C B D E (2) A B E C D (2) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) E B A D C (1) D C E A B (1) C E D A B (1) C D A E B (1) C A D E B (1) B E A D C (1) B D C A E (1) B D A E C (1) B A E C D (1) Total count = 100 A B C D E A 0 6 6 -10 -8 B -6 0 -2 -2 -14 C -6 2 0 -2 -6 D 10 2 2 0 -10 E 8 14 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 -10 -8 B -6 0 -2 -2 -14 C -6 2 0 -2 -6 D 10 2 2 0 -10 E 8 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=24 E=22 D=15 C=13 so C is eliminated. Round 2 votes counts: A=31 B=24 E=23 D=22 so D is eliminated. Round 3 votes counts: E=40 A=32 B=28 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:202 A:197 C:194 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -10 -8 B -6 0 -2 -2 -14 C -6 2 0 -2 -6 D 10 2 2 0 -10 E 8 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -10 -8 B -6 0 -2 -2 -14 C -6 2 0 -2 -6 D 10 2 2 0 -10 E 8 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -10 -8 B -6 0 -2 -2 -14 C -6 2 0 -2 -6 D 10 2 2 0 -10 E 8 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1738: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) E B A D C (6) A E D C B (6) A E B D C (6) A C D B E (6) C D B A E (5) E B D C A (4) C D A B E (4) B C D E A (4) A E B C D (4) A D C E B (4) D C A B E (3) C B D E A (3) B E C D A (3) A C B D E (3) E B D A C (2) E B A C D (2) D C E A B (2) C A B D E (2) B D C E A (2) E B C D A (1) E B C A D (1) E A B D C (1) D E C B A (1) D E B C A (1) D C B A E (1) D A E C B (1) D A C E B (1) C D B E A (1) C B D A E (1) C A D B E (1) B E D C A (1) B D E C A (1) A E D B C (1) A E C D B (1) A E C B D (1) A D E C B (1) A D C B E (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -2 -2 10 B -2 0 -18 -2 4 C 2 18 0 -10 8 D 2 2 10 0 18 E -10 -4 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -2 10 B -2 0 -18 -2 4 C 2 18 0 -10 8 D 2 2 10 0 18 E -10 -4 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=19 E=17 C=17 B=11 so B is eliminated. Round 2 votes counts: A=36 D=22 E=21 C=21 so E is eliminated. Round 3 votes counts: A=45 D=29 C=26 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:209 A:204 B:191 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -2 10 B -2 0 -18 -2 4 C 2 18 0 -10 8 D 2 2 10 0 18 E -10 -4 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -2 10 B -2 0 -18 -2 4 C 2 18 0 -10 8 D 2 2 10 0 18 E -10 -4 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -2 10 B -2 0 -18 -2 4 C 2 18 0 -10 8 D 2 2 10 0 18 E -10 -4 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1739: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) A C E B D (5) A C D B E (5) E B D A C (4) E B C A D (4) D B E C A (4) B E D C A (4) A C D E B (4) E B D C A (3) E B C D A (3) D B E A C (3) D B C E A (3) D A C B E (3) B D E C A (3) A E D B C (3) A E B C D (3) A D C E B (3) E A B C D (2) D E B A C (2) D C B A E (2) D C A B E (2) C A B E D (2) B E C D A (2) B D C E A (2) A D E C B (2) A D C B E (2) A C B D E (2) E B A D C (1) E A C B D (1) D C B E A (1) D B C A E (1) D A E C B (1) C D B A E (1) C D A B E (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) C A E B D (1) C A D B E (1) A E C D B (1) A E C B D (1) A D E B C (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 12 4 0 B 4 0 6 6 -2 C -12 -6 0 0 -6 D -4 -6 0 0 2 E 0 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.080340 B: 0.151796 C: 0.000000 D: 0.151796 E: 0.616068 Sum of squares = 0.432078326092 Cumulative probabilities = A: 0.080340 B: 0.232136 C: 0.232136 D: 0.383932 E: 1.000000 A B C D E A 0 -4 12 4 0 B 4 0 6 6 -2 C -12 -6 0 0 -6 D -4 -6 0 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068182 B: 0.159091 C: 0.000000 D: 0.159091 E: 0.613636 Sum of squares = 0.431818181817 Cumulative probabilities = A: 0.068182 B: 0.227273 C: 0.227273 D: 0.386364 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=23 D=22 B=11 C=10 so C is eliminated. Round 2 votes counts: A=38 D=24 E=23 B=15 so B is eliminated. Round 3 votes counts: A=38 E=31 D=31 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:207 A:206 E:203 D:196 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 4 0 B 4 0 6 6 -2 C -12 -6 0 0 -6 D -4 -6 0 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068182 B: 0.159091 C: 0.000000 D: 0.159091 E: 0.613636 Sum of squares = 0.431818181817 Cumulative probabilities = A: 0.068182 B: 0.227273 C: 0.227273 D: 0.386364 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 4 0 B 4 0 6 6 -2 C -12 -6 0 0 -6 D -4 -6 0 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068182 B: 0.159091 C: 0.000000 D: 0.159091 E: 0.613636 Sum of squares = 0.431818181817 Cumulative probabilities = A: 0.068182 B: 0.227273 C: 0.227273 D: 0.386364 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 4 0 B 4 0 6 6 -2 C -12 -6 0 0 -6 D -4 -6 0 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068182 B: 0.159091 C: 0.000000 D: 0.159091 E: 0.613636 Sum of squares = 0.431818181817 Cumulative probabilities = A: 0.068182 B: 0.227273 C: 0.227273 D: 0.386364 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1740: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) B E A D C (9) A B E D C (8) D C E B A (7) A C D B E (7) A B E C D (6) B E D C A (5) E B D A C (4) B E D A C (4) E B C A D (3) D B E A C (3) C D E B A (3) C A D E B (3) A C E B D (3) E B D C A (2) E B A C D (2) D E B C A (2) A C B E D (2) E D B C A (1) E C A B D (1) E B A D C (1) D C B E A (1) D C A B E (1) D B E C A (1) D B A C E (1) C E B D A (1) C E A D B (1) C D E A B (1) C D A B E (1) C A D B E (1) B A E D C (1) A E B C D (1) A D C B E (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 12 6 -4 B 2 0 14 8 10 C -12 -14 0 -10 -12 D -6 -8 10 0 -10 E 4 -10 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 6 -4 B 2 0 14 8 10 C -12 -14 0 -10 -12 D -6 -8 10 0 -10 E 4 -10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=20 B=19 D=16 E=14 so E is eliminated. Round 2 votes counts: B=31 A=31 C=21 D=17 so D is eliminated. Round 3 votes counts: B=39 A=31 C=30 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:208 A:206 D:193 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 12 6 -4 B 2 0 14 8 10 C -12 -14 0 -10 -12 D -6 -8 10 0 -10 E 4 -10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 6 -4 B 2 0 14 8 10 C -12 -14 0 -10 -12 D -6 -8 10 0 -10 E 4 -10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 6 -4 B 2 0 14 8 10 C -12 -14 0 -10 -12 D -6 -8 10 0 -10 E 4 -10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1741: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (7) E C A B D (5) B A D C E (5) E D B C A (4) E B D C A (4) E B C A D (4) D C A E B (4) C A E B D (4) A C B D E (4) A B C D E (4) E B C D A (3) D A C B E (3) D A B C E (3) B E A C D (3) B D A E C (3) B A D E C (3) E D C B A (2) E D C A B (2) E C D A B (2) D E B C A (2) D B E A C (2) D B A C E (2) C D A E B (2) C A E D B (2) C A D E B (2) B E D A C (2) B D A C E (2) A C D B E (2) E C D B A (1) D E C A B (1) D C E A B (1) D B A E C (1) C D E A B (1) B E A D C (1) B D E A C (1) B A C E D (1) B A C D E (1) A D C B E (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 4 4 B -6 0 6 6 -6 C 6 -6 0 0 8 D -4 -6 0 0 2 E -4 6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 4 4 B -6 0 6 6 -6 C 6 -6 0 0 8 D -4 -6 0 0 2 E -4 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=22 D=19 C=18 A=14 so A is eliminated. Round 2 votes counts: B=28 E=27 C=25 D=20 so D is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:204 C:204 B:200 D:196 E:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -6 4 4 B -6 0 6 6 -6 C 6 -6 0 0 8 D -4 -6 0 0 2 E -4 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 4 4 B -6 0 6 6 -6 C 6 -6 0 0 8 D -4 -6 0 0 2 E -4 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 4 4 B -6 0 6 6 -6 C 6 -6 0 0 8 D -4 -6 0 0 2 E -4 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1742: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (12) A D B C E (11) A E D B C (9) C B D E A (8) C B D A E (7) E A D B C (5) A D E B C (5) C E B D A (4) C B E D A (4) A D B E C (4) E C B A D (3) E D B C A (2) E A C D B (2) D B C A E (2) D B A C E (2) C E A B D (2) B D C A E (2) A E C D B (2) E C A B D (1) E A D C B (1) E A C B D (1) D B E C A (1) D B A E C (1) C E B A D (1) C B A D E (1) C A B E D (1) C A B D E (1) B D C E A (1) B C D E A (1) A D C B E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -8 6 2 B 4 0 -8 2 -4 C 8 8 0 6 2 D -6 -2 -6 0 -4 E -2 4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 6 2 B 4 0 -8 2 -4 C 8 8 0 6 2 D -6 -2 -6 0 -4 E -2 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=29 E=27 D=6 B=4 so B is eliminated. Round 2 votes counts: A=34 C=30 E=27 D=9 so D is eliminated. Round 3 votes counts: A=37 C=35 E=28 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:202 A:198 B:197 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 6 2 B 4 0 -8 2 -4 C 8 8 0 6 2 D -6 -2 -6 0 -4 E -2 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 6 2 B 4 0 -8 2 -4 C 8 8 0 6 2 D -6 -2 -6 0 -4 E -2 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 6 2 B 4 0 -8 2 -4 C 8 8 0 6 2 D -6 -2 -6 0 -4 E -2 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1743: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) B E C D A (9) E B A D C (6) C D A E B (5) B E C A D (5) A C D E B (5) E B D A C (4) B C E D A (4) D C A E B (3) D A C E B (3) C D A B E (3) C A D E B (3) B E D C A (3) A D E C B (3) A D C E B (3) E D A B C (2) E B D C A (2) E A B D C (2) D E A C B (2) C B A D E (2) B E D A C (2) B E A D C (2) A E D C B (2) A B C E D (2) E D B A C (1) E A D B C (1) D E B C A (1) D C E A B (1) C D B E A (1) C B D E A (1) C B D A E (1) C A B D E (1) B E A C D (1) B C D E A (1) B C A E D (1) A C E D B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -12 2 -2 B -6 0 0 0 0 C 12 0 0 14 2 D -2 0 -14 0 -2 E 2 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.475489 C: 0.524511 D: 0.000000 E: 0.000000 Sum of squares = 0.501201606543 Cumulative probabilities = A: 0.000000 B: 0.475489 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 2 -2 B -6 0 0 0 0 C 12 0 0 14 2 D -2 0 -14 0 -2 E 2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=26 E=18 A=18 D=10 so D is eliminated. Round 2 votes counts: C=30 B=28 E=21 A=21 so E is eliminated. Round 3 votes counts: B=42 C=30 A=28 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:214 E:201 A:197 B:197 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 2 -2 B -6 0 0 0 0 C 12 0 0 14 2 D -2 0 -14 0 -2 E 2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 2 -2 B -6 0 0 0 0 C 12 0 0 14 2 D -2 0 -14 0 -2 E 2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 2 -2 B -6 0 0 0 0 C 12 0 0 14 2 D -2 0 -14 0 -2 E 2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1744: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (7) E A C B D (7) D B A E C (7) C E A D B (7) D B E A C (5) C E A B D (5) B D A E C (4) B D A C E (4) D E C A B (3) D C E A B (3) D C B E A (3) D B C A E (3) D B A C E (3) C E D A B (3) B A E C D (3) A E C B D (3) A E B C D (3) E C A D B (2) D E C B A (2) C D B A E (2) C A E B D (2) B A E D C (2) B A D E C (2) A B C E D (2) E C D A B (1) D E B C A (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C E A (1) C D E A B (1) C B A E D (1) C A B E D (1) B E D A C (1) B A C E D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -2 0 -8 B -2 0 -12 0 -4 C 2 12 0 6 -8 D 0 0 -6 0 -4 E 8 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 0 -8 B -2 0 -12 0 -4 C 2 12 0 6 -8 D 0 0 -6 0 -4 E 8 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=22 B=18 E=17 A=9 so A is eliminated. Round 2 votes counts: D=34 E=23 C=23 B=20 so B is eliminated. Round 3 votes counts: D=44 E=29 C=27 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:206 A:196 D:195 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 0 -8 B -2 0 -12 0 -4 C 2 12 0 6 -8 D 0 0 -6 0 -4 E 8 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -8 B -2 0 -12 0 -4 C 2 12 0 6 -8 D 0 0 -6 0 -4 E 8 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -8 B -2 0 -12 0 -4 C 2 12 0 6 -8 D 0 0 -6 0 -4 E 8 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1745: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (12) A D B E C (11) A D B C E (7) E C D A B (5) C E B A D (5) B A D C E (5) D A E C B (4) D A B E C (4) B D A C E (4) B C E D A (4) A D E C B (4) A B D C E (4) D A B C E (3) C B E D A (3) B C E A D (3) A D E B C (3) E C D B A (2) E C B A D (2) E C A D B (2) E C A B D (2) E A D C B (2) D E A C B (2) D B A C E (2) E C B D A (1) D E C B A (1) C B E A D (1) B C A E D (1) B A C D E (1) Total count = 100 A B C D E A 0 6 12 6 6 B -6 0 4 -4 6 C -12 -4 0 -12 10 D -6 4 12 0 10 E -6 -6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 6 6 B -6 0 4 -4 6 C -12 -4 0 -12 10 D -6 4 12 0 10 E -6 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=21 B=18 E=16 D=16 so E is eliminated. Round 2 votes counts: C=35 A=31 B=18 D=16 so D is eliminated. Round 3 votes counts: A=44 C=36 B=20 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:210 B:200 C:191 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 6 6 B -6 0 4 -4 6 C -12 -4 0 -12 10 D -6 4 12 0 10 E -6 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 6 6 B -6 0 4 -4 6 C -12 -4 0 -12 10 D -6 4 12 0 10 E -6 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 6 6 B -6 0 4 -4 6 C -12 -4 0 -12 10 D -6 4 12 0 10 E -6 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1746: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) D E A B C (10) C B A E D (6) B D E C A (6) B D C E A (6) B C A D E (6) A E D C B (6) A E C D B (6) D E B A C (5) B D E A C (5) A C E D B (5) E A D C B (4) D B E A C (4) C A E D B (3) C A E B D (3) C A B E D (2) A D E B C (2) E D A C B (1) E D A B C (1) E A D B C (1) C B E D A (1) C B E A D (1) C B A D E (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 2 -6 -16 B 10 0 20 4 4 C -2 -20 0 -2 -2 D 6 -4 2 0 18 E 16 -4 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -6 -16 B 10 0 20 4 4 C -2 -20 0 -2 -2 D 6 -4 2 0 18 E 16 -4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=20 D=19 C=17 E=7 so E is eliminated. Round 2 votes counts: B=37 A=25 D=21 C=17 so C is eliminated. Round 3 votes counts: B=46 A=33 D=21 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:211 E:198 C:187 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 -6 -16 B 10 0 20 4 4 C -2 -20 0 -2 -2 D 6 -4 2 0 18 E 16 -4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -6 -16 B 10 0 20 4 4 C -2 -20 0 -2 -2 D 6 -4 2 0 18 E 16 -4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -6 -16 B 10 0 20 4 4 C -2 -20 0 -2 -2 D 6 -4 2 0 18 E 16 -4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1747: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) C E A B D (8) B A D C E (6) C E B A D (5) D B A E C (4) D B A C E (4) C B E A D (4) E C A D B (3) E A C D B (3) E A C B D (3) D B C A E (3) D A B E C (3) B C A E D (3) B A C D E (3) A E D C B (3) A E B C D (3) E C D A B (2) D E A C B (2) D A E B C (2) C E D B A (2) C B A E D (2) B A C E D (2) A E D B C (2) A E C B D (2) E D A C B (1) E A D C B (1) D E C B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D C B E A (1) D B C E A (1) C E D A B (1) C E B D A (1) B D C A E (1) B C D A E (1) B A D E C (1) A E C D B (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 0 34 -2 B -8 0 -14 12 -16 C 0 14 0 18 -2 D -34 -12 -18 0 -24 E 2 16 2 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 34 -2 B -8 0 -14 12 -16 C 0 14 0 18 -2 D -34 -12 -18 0 -24 E 2 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999936956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 E=22 B=17 A=14 so A is eliminated. Round 2 votes counts: E=33 D=25 C=23 B=19 so B is eliminated. Round 3 votes counts: E=34 D=34 C=32 so C is eliminated. Round 4 votes counts: E=62 D=38 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:220 C:215 B:187 D:156 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 34 -2 B -8 0 -14 12 -16 C 0 14 0 18 -2 D -34 -12 -18 0 -24 E 2 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999936956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 34 -2 B -8 0 -14 12 -16 C 0 14 0 18 -2 D -34 -12 -18 0 -24 E 2 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999936956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 34 -2 B -8 0 -14 12 -16 C 0 14 0 18 -2 D -34 -12 -18 0 -24 E 2 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999936956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1748: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) D B E A C (10) B D E C A (10) B E D C A (9) C A E B D (6) E C B D A (3) D E B C A (3) C E A B D (3) C A E D B (3) A D C B E (3) A D B C E (3) A C E B D (3) A B C D E (3) D B A E C (2) D A B E C (2) C B E A D (2) B E C D A (2) B D E A C (2) A D C E B (2) A D B E C (2) A C D E B (2) A C D B E (2) A C B D E (2) E B D C A (1) E B C D A (1) C E B D A (1) C E B A D (1) C B E D A (1) B D A E C (1) B C E D A (1) B C D E A (1) A E D C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 4 0 -2 B 2 0 6 8 18 C -4 -6 0 -2 2 D 0 -8 2 0 0 E 2 -18 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 0 -2 B 2 0 6 8 18 C -4 -6 0 -2 2 D 0 -8 2 0 0 E 2 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=26 D=17 C=17 E=5 so E is eliminated. Round 2 votes counts: A=35 B=28 C=20 D=17 so D is eliminated. Round 3 votes counts: B=43 A=37 C=20 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:200 D:197 C:195 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 0 -2 B 2 0 6 8 18 C -4 -6 0 -2 2 D 0 -8 2 0 0 E 2 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 0 -2 B 2 0 6 8 18 C -4 -6 0 -2 2 D 0 -8 2 0 0 E 2 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 0 -2 B 2 0 6 8 18 C -4 -6 0 -2 2 D 0 -8 2 0 0 E 2 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1749: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) E B C D A (6) C E B D A (6) B E C A D (6) D A C E B (5) B E C D A (5) A D B E C (5) A B E D C (5) E C B D A (4) D C A E B (4) A D C B E (4) D A E C B (3) D A E B C (3) C E D B A (3) C D E B A (3) A D C E B (3) A B E C D (3) E D B A C (2) C D A E B (2) C B E A D (2) B E A D C (2) B A E C D (2) A C D B E (2) D E C B A (1) D E B A C (1) D E A B C (1) D C E A B (1) C B A E D (1) C A D B E (1) B C E A D (1) B A E D C (1) A D E B C (1) A D B C E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 8 0 -4 B 8 0 10 8 2 C -8 -10 0 12 -20 D 0 -8 -12 0 -14 E 4 -2 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 0 -4 B 8 0 10 8 2 C -8 -10 0 12 -20 D 0 -8 -12 0 -14 E 4 -2 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 D=19 C=18 E=12 so E is eliminated. Round 2 votes counts: B=31 A=26 C=22 D=21 so D is eliminated. Round 3 votes counts: A=38 B=34 C=28 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:214 A:198 C:187 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 0 -4 B 8 0 10 8 2 C -8 -10 0 12 -20 D 0 -8 -12 0 -14 E 4 -2 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 0 -4 B 8 0 10 8 2 C -8 -10 0 12 -20 D 0 -8 -12 0 -14 E 4 -2 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 0 -4 B 8 0 10 8 2 C -8 -10 0 12 -20 D 0 -8 -12 0 -14 E 4 -2 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1750: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) E A B C D (6) C E A D B (6) C D E A B (6) E C A B D (5) E B C A D (5) D C A E B (5) D C B A E (4) D C A B E (4) D B C A E (4) C E D A B (4) B E A C D (4) C D E B A (3) C D A E B (3) B E A D C (3) B D A E C (3) A E B D C (3) E B A C D (2) D B A E C (2) D B A C E (2) D A C E B (2) D A C B E (2) B A D E C (2) E A C D B (1) E A C B D (1) D B C E A (1) D A E B C (1) D A B E C (1) D A B C E (1) C E D B A (1) C E B A D (1) C D B E A (1) C A E D B (1) B D C E A (1) Total count = 100 A B C D E A 0 4 -10 -2 -2 B -4 0 0 -10 -12 C 10 0 0 0 4 D 2 10 0 0 -4 E 2 12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.706317 D: 0.293683 E: 0.000000 Sum of squares = 0.585133086588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.706317 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -2 -2 B -4 0 0 -10 -12 C 10 0 0 0 4 D 2 10 0 0 -4 E 2 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500189 D: 0.499811 E: 0.000000 Sum of squares = 0.500000071577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500189 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 B=22 E=20 A=3 so A is eliminated. Round 2 votes counts: D=29 C=26 E=23 B=22 so B is eliminated. Round 3 votes counts: E=39 D=35 C=26 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:207 E:207 D:204 A:195 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 -2 -2 B -4 0 0 -10 -12 C 10 0 0 0 4 D 2 10 0 0 -4 E 2 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500189 D: 0.499811 E: 0.000000 Sum of squares = 0.500000071577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500189 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -2 -2 B -4 0 0 -10 -12 C 10 0 0 0 4 D 2 10 0 0 -4 E 2 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500189 D: 0.499811 E: 0.000000 Sum of squares = 0.500000071577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500189 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -2 -2 B -4 0 0 -10 -12 C 10 0 0 0 4 D 2 10 0 0 -4 E 2 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500189 D: 0.499811 E: 0.000000 Sum of squares = 0.500000071577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500189 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1751: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (10) A D E C B (9) E C A B D (8) D A E C B (8) B D C E A (8) D A E B C (7) B C E D A (7) D B A C E (6) E A C B D (5) D A B C E (5) C E B A D (5) C B E A D (5) B C D E A (4) A E D C B (3) E C B A D (2) E A C D B (2) D B C A E (2) D A B E C (2) A E C D B (2) Total count = 100 A B C D E A 0 2 -2 2 -12 B -2 0 2 8 -2 C 2 -2 0 0 4 D -2 -8 0 0 2 E 12 2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 2 -2 2 -12 B -2 0 2 8 -2 C 2 -2 0 0 4 D -2 -8 0 0 2 E 12 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000063 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=29 E=17 A=14 C=10 so C is eliminated. Round 2 votes counts: B=34 D=30 E=22 A=14 so A is eliminated. Round 3 votes counts: D=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:204 B:203 C:202 D:196 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 2 -12 B -2 0 2 8 -2 C 2 -2 0 0 4 D -2 -8 0 0 2 E 12 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000063 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 -12 B -2 0 2 8 -2 C 2 -2 0 0 4 D -2 -8 0 0 2 E 12 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000063 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 -12 B -2 0 2 8 -2 C 2 -2 0 0 4 D -2 -8 0 0 2 E 12 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000063 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1752: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (15) D A B E C (9) C E B A D (5) A D E B C (5) C E B D A (4) C E A B D (4) A D E C B (4) E A C D B (3) D B E A C (3) A E D C B (3) A D B E C (3) E C B A D (2) E C A D B (2) D B A E C (2) D A E B C (2) C B E A D (2) C B D E A (2) C B D A E (2) C B A E D (2) B D E C A (2) B D A E C (2) B D A C E (2) B C D E A (2) B C D A E (2) A E D B C (2) E D A B C (1) E C A B D (1) E B C D A (1) E A D C B (1) E A D B C (1) D E A B C (1) C B A D E (1) C A E D B (1) C A E B D (1) C A B E D (1) C A B D E (1) B C E D A (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -8 -6 -6 B 4 0 -18 10 8 C 8 18 0 14 0 D 6 -10 -14 0 -8 E 6 -8 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.508277 D: 0.000000 E: 0.491723 Sum of squares = 0.500137001683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.508277 D: 0.508277 E: 1.000000 A B C D E A 0 -4 -8 -6 -6 B 4 0 -18 10 8 C 8 18 0 14 0 D 6 -10 -14 0 -8 E 6 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=19 D=17 E=12 B=11 so B is eliminated. Round 2 votes counts: C=46 D=23 A=19 E=12 so E is eliminated. Round 3 votes counts: C=52 D=24 A=24 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:203 B:202 A:188 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 -6 B 4 0 -18 10 8 C 8 18 0 14 0 D 6 -10 -14 0 -8 E 6 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 -6 B 4 0 -18 10 8 C 8 18 0 14 0 D 6 -10 -14 0 -8 E 6 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 -6 B 4 0 -18 10 8 C 8 18 0 14 0 D 6 -10 -14 0 -8 E 6 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1753: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) E C B A D (9) D B C E A (8) B D C E A (8) E C A B D (7) A D B E C (6) A E C B D (5) E C B D A (3) D B C A E (3) D B A C E (3) B D C A E (3) A D E C B (3) E C D A B (2) E C A D B (2) D E C B A (2) C E B D A (2) C B E D A (2) B D A C E (2) B C E D A (2) B C D E A (2) B C A E D (2) E D C B A (1) E C D B A (1) E A D C B (1) E A C B D (1) D C B E A (1) D A B E C (1) C E B A D (1) B C D A E (1) B A C E D (1) A E D C B (1) A E C D B (1) A E B C D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -24 -14 -10 B 14 0 10 6 14 C 24 -10 0 -8 6 D 14 -6 8 0 8 E 10 -14 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -24 -14 -10 B 14 0 10 6 14 C 24 -10 0 -8 6 D 14 -6 8 0 8 E 10 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 B=21 A=19 C=5 so C is eliminated. Round 2 votes counts: E=30 D=28 B=23 A=19 so A is eliminated. Round 3 votes counts: E=38 D=38 B=24 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:222 D:212 C:206 E:191 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -24 -14 -10 B 14 0 10 6 14 C 24 -10 0 -8 6 D 14 -6 8 0 8 E 10 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -24 -14 -10 B 14 0 10 6 14 C 24 -10 0 -8 6 D 14 -6 8 0 8 E 10 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -24 -14 -10 B 14 0 10 6 14 C 24 -10 0 -8 6 D 14 -6 8 0 8 E 10 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1754: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) B C A E D (10) D E A B C (9) A D E B C (7) A B C D E (7) C B E D A (5) C B A E D (5) A D E C B (5) A C B D E (5) E D C B A (4) E D C A B (3) E D B C A (3) E C D B A (3) C A B D E (3) C B E A D (2) B E D A C (2) B C E D A (2) B C A D E (2) E D B A C (1) E D A C B (1) E B D C A (1) D A E C B (1) C E D A B (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C D A (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 16 4 0 -2 B -16 0 -6 -8 -8 C -4 6 0 -2 -10 D 0 8 2 0 8 E 2 8 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333715 B: 0.000000 C: 0.000000 D: 0.666285 E: 0.000000 Sum of squares = 0.555301354841 Cumulative probabilities = A: 0.333715 B: 0.333715 C: 0.333715 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 0 -2 B -16 0 -6 -8 -8 C -4 6 0 -2 -10 D 0 8 2 0 8 E 2 8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=22 C=18 B=18 E=16 so E is eliminated. Round 2 votes counts: D=34 A=26 C=21 B=19 so B is eliminated. Round 3 votes counts: D=38 C=36 A=26 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:209 E:206 C:195 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 4 0 -2 B -16 0 -6 -8 -8 C -4 6 0 -2 -10 D 0 8 2 0 8 E 2 8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 0 -2 B -16 0 -6 -8 -8 C -4 6 0 -2 -10 D 0 8 2 0 8 E 2 8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 0 -2 B -16 0 -6 -8 -8 C -4 6 0 -2 -10 D 0 8 2 0 8 E 2 8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1755: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) E B A D C (6) C D A B E (6) E C D A B (5) E A B D C (5) C D B A E (5) C B D A E (5) D C A B E (4) B A D C E (4) A D B C E (4) E D A C B (3) E A D C B (3) D A C B E (3) B A E D C (3) E C B D A (2) D E C A B (2) D C E A B (2) D C A E B (2) D A E C B (2) C E D A B (2) C D A E B (2) B E A D C (2) A D E B C (2) E D C A B (1) E C D B A (1) E B C A D (1) D A C E B (1) C D E B A (1) C D E A B (1) C B E D A (1) B E C A D (1) B C E A D (1) B C D A E (1) B C A D E (1) B A D E C (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E C B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 26 6 -4 4 B -26 0 -10 -26 -10 C -6 10 0 -26 -4 D 4 26 26 0 4 E -4 10 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 6 -4 4 B -26 0 -10 -26 -10 C -6 10 0 -26 -4 D 4 26 26 0 4 E -4 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=23 D=16 B=15 A=11 so A is eliminated. Round 2 votes counts: E=37 D=24 C=23 B=16 so B is eliminated. Round 3 votes counts: E=43 D=30 C=27 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:216 E:203 C:187 B:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 26 6 -4 4 B -26 0 -10 -26 -10 C -6 10 0 -26 -4 D 4 26 26 0 4 E -4 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 6 -4 4 B -26 0 -10 -26 -10 C -6 10 0 -26 -4 D 4 26 26 0 4 E -4 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 6 -4 4 B -26 0 -10 -26 -10 C -6 10 0 -26 -4 D 4 26 26 0 4 E -4 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1756: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) A C B D E (8) E B D C A (6) C B E D A (6) C B E A D (6) C B A E D (6) C A B E D (6) D E A B C (5) A D E C B (5) D E B C A (4) D E B A C (4) A D E B C (4) B E C D A (3) B C E D A (3) A D C E B (3) A C D E B (3) D C E A B (2) D A E C B (2) A C D B E (2) D E A C B (1) C E D A B (1) C D E A B (1) C A B D E (1) B C A E D (1) B A C E D (1) A D B E C (1) A C B E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -16 0 -8 B 4 0 -8 0 -6 C 16 8 0 0 6 D 0 0 0 0 -4 E 8 6 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.595551 D: 0.404449 E: 0.000000 Sum of squares = 0.51826009834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.595551 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 0 -8 B 4 0 -8 0 -6 C 16 8 0 0 6 D 0 0 0 0 -4 E 8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 E=18 D=18 B=8 so B is eliminated. Round 2 votes counts: C=31 A=30 E=21 D=18 so D is eliminated. Round 3 votes counts: E=35 C=33 A=32 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:206 D:198 B:195 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 0 -8 B 4 0 -8 0 -6 C 16 8 0 0 6 D 0 0 0 0 -4 E 8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 0 -8 B 4 0 -8 0 -6 C 16 8 0 0 6 D 0 0 0 0 -4 E 8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 0 -8 B 4 0 -8 0 -6 C 16 8 0 0 6 D 0 0 0 0 -4 E 8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1757: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (13) A C E D B (8) C A E D B (7) B C D E A (7) B D E A C (6) A E C D B (6) D E B A C (5) C B D A E (5) C A B D E (5) D B E A C (4) E D A B C (3) C E D A B (3) C B A D E (3) C A E B D (3) C A B E D (3) A E D C B (3) E D B A C (2) C B D E A (2) B D C E A (2) A E D B C (2) A C E B D (2) A B E D C (2) E A C D B (1) D B E C A (1) B C A D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -10 -6 2 B 2 0 -2 10 10 C 10 2 0 14 4 D 6 -10 -14 0 8 E -2 -10 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -6 2 B 2 0 -2 10 10 C 10 2 0 14 4 D 6 -10 -14 0 8 E -2 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=29 A=24 D=10 E=6 so E is eliminated. Round 2 votes counts: C=31 B=29 A=25 D=15 so D is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:210 D:195 A:192 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 -6 2 B 2 0 -2 10 10 C 10 2 0 14 4 D 6 -10 -14 0 8 E -2 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -6 2 B 2 0 -2 10 10 C 10 2 0 14 4 D 6 -10 -14 0 8 E -2 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -6 2 B 2 0 -2 10 10 C 10 2 0 14 4 D 6 -10 -14 0 8 E -2 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1758: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) D E B A C (5) A C B D E (5) E D B A C (4) D A E B C (4) D A C B E (4) C A B E D (4) B C E D A (4) A C D B E (4) E D B C A (3) D B C E A (3) D B C A E (3) B E D C A (3) B E C D A (3) A D E C B (3) A D C E B (3) A D C B E (3) E D A B C (2) E B C D A (2) D B E C A (2) D A E C B (2) C B E A D (2) C B A E D (2) C A B D E (2) B D E C A (2) A C B E D (2) E B C A D (1) E A D B C (1) E A C B D (1) D E B C A (1) D E A B C (1) D B A E C (1) C B D A E (1) C B A D E (1) C A E B D (1) B D C E A (1) B C E A D (1) B C D A E (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 2 -16 4 B 4 0 8 -6 8 C -2 -8 0 -16 0 D 16 6 16 0 6 E -4 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -16 4 B 4 0 8 -6 8 C -2 -8 0 -16 0 D 16 6 16 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 E=20 B=15 C=13 so C is eliminated. Round 2 votes counts: A=33 D=26 B=21 E=20 so E is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:207 A:193 E:191 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -16 4 B 4 0 8 -6 8 C -2 -8 0 -16 0 D 16 6 16 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -16 4 B 4 0 8 -6 8 C -2 -8 0 -16 0 D 16 6 16 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -16 4 B 4 0 8 -6 8 C -2 -8 0 -16 0 D 16 6 16 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1759: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D C A B E (8) A E B C D (6) A D C E B (5) E B A C D (4) C D B E A (4) C B D E A (4) B E C A D (4) A D E B C (4) E B D C A (3) E B C D A (3) E B A D C (3) B E C D A (3) A C B E D (3) E B D A C (2) E A B D C (2) D C B A E (2) D A E C B (2) C D B A E (2) C B E D A (2) B E D C A (2) B E A C D (2) A E B D C (2) A D E C B (2) A D C B E (2) A C E B D (2) E B C A D (1) E A B C D (1) D E B C A (1) D C E B A (1) D A C E B (1) D A C B E (1) C D A B E (1) B C E D A (1) A E D C B (1) A E C B D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -2 -4 -4 B 6 0 -8 6 6 C 2 8 0 -6 0 D 4 -6 6 0 0 E 4 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.400000 E: 0.000000 Sum of squares = 0.340000000003 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -4 -4 B 6 0 -8 6 6 C 2 8 0 -6 0 D 4 -6 6 0 0 E 4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.400000 E: 0.000000 Sum of squares = 0.340000000002 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=25 E=19 C=13 B=12 so B is eliminated. Round 2 votes counts: A=31 E=30 D=25 C=14 so C is eliminated. Round 3 votes counts: D=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:205 C:202 D:202 E:199 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 -4 B 6 0 -8 6 6 C 2 8 0 -6 0 D 4 -6 6 0 0 E 4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.400000 E: 0.000000 Sum of squares = 0.340000000002 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 -4 B 6 0 -8 6 6 C 2 8 0 -6 0 D 4 -6 6 0 0 E 4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.400000 E: 0.000000 Sum of squares = 0.340000000002 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 -4 B 6 0 -8 6 6 C 2 8 0 -6 0 D 4 -6 6 0 0 E 4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.400000 E: 0.000000 Sum of squares = 0.340000000002 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1760: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (11) E D B A C (10) B D C A E (6) D E B A C (5) C A B D E (5) A E C D B (5) E D B C A (4) E D A B C (4) E A C D B (4) B C D A E (4) D B E C A (3) E C A D B (2) E C A B D (2) E A D C B (2) D B E A C (2) D A B E C (2) C E B A D (2) C A E B D (2) C A B E D (2) B D C E A (2) B C A D E (2) A C E D B (2) A C E B D (2) A C B E D (2) E D C A B (1) E D A C B (1) E C B A D (1) E B D C A (1) D E A B C (1) C B A D E (1) B E D C A (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 -8 -20 -22 B 14 0 20 -2 -8 C 8 -20 0 -16 -28 D 20 2 16 0 -4 E 22 8 28 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -8 -20 -22 B 14 0 20 -2 -8 C 8 -20 0 -16 -28 D 20 2 16 0 -4 E 22 8 28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=28 A=15 D=13 C=12 so C is eliminated. Round 2 votes counts: E=34 B=29 A=24 D=13 so D is eliminated. Round 3 votes counts: E=40 B=34 A=26 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:231 D:217 B:212 C:172 A:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -8 -20 -22 B 14 0 20 -2 -8 C 8 -20 0 -16 -28 D 20 2 16 0 -4 E 22 8 28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -20 -22 B 14 0 20 -2 -8 C 8 -20 0 -16 -28 D 20 2 16 0 -4 E 22 8 28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -20 -22 B 14 0 20 -2 -8 C 8 -20 0 -16 -28 D 20 2 16 0 -4 E 22 8 28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1761: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) B C D E A (9) A D E B C (9) E A C B D (8) D B C A E (6) B D C A E (6) A E D B C (6) E C A B D (5) E A D C B (5) C E B A D (5) C B E A D (5) D C B E A (4) D B A C E (4) C B E D A (3) C B D E A (3) E C A D B (2) B C D A E (2) B A C E D (2) A B E C D (2) D A B C E (1) B D A C E (1) B C E D A (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -2 2 0 B 2 0 18 4 18 C 2 -18 0 -6 4 D -2 -4 6 0 10 E 0 -18 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 2 0 B 2 0 18 4 18 C 2 -18 0 -6 4 D -2 -4 6 0 10 E 0 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=21 E=20 A=19 C=16 so C is eliminated. Round 2 votes counts: B=32 E=25 D=24 A=19 so A is eliminated. Round 3 votes counts: D=34 B=34 E=32 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:205 A:199 C:191 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 2 0 B 2 0 18 4 18 C 2 -18 0 -6 4 D -2 -4 6 0 10 E 0 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 0 B 2 0 18 4 18 C 2 -18 0 -6 4 D -2 -4 6 0 10 E 0 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 0 B 2 0 18 4 18 C 2 -18 0 -6 4 D -2 -4 6 0 10 E 0 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1762: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (18) E D C A B (8) A C D E B (7) A C B D E (7) E B D C A (6) D E C A B (5) C A D E B (5) B A C D E (5) A C D B E (5) E D B C A (4) B A E C D (4) B A C E D (4) A B C D E (4) B E D A C (3) E D C B A (2) D C E A B (2) C D A E B (2) A C E D B (2) E B A D C (1) E A D C B (1) D E C B A (1) D C E B A (1) C A D B E (1) B E A D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -10 -4 -6 B 0 0 2 8 6 C 10 -2 0 -6 -8 D 4 -8 6 0 -10 E 6 -6 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.099683 B: 0.900317 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.820507666104 Cumulative probabilities = A: 0.099683 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -4 -6 B 0 0 2 8 6 C 10 -2 0 -6 -8 D 4 -8 6 0 -10 E 6 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222984 Cumulative probabilities = A: 0.166667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=26 E=22 D=9 C=8 so C is eliminated. Round 2 votes counts: B=35 A=32 E=22 D=11 so D is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:209 B:208 C:197 D:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -10 -4 -6 B 0 0 2 8 6 C 10 -2 0 -6 -8 D 4 -8 6 0 -10 E 6 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222984 Cumulative probabilities = A: 0.166667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 -6 B 0 0 2 8 6 C 10 -2 0 -6 -8 D 4 -8 6 0 -10 E 6 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222984 Cumulative probabilities = A: 0.166667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 -6 B 0 0 2 8 6 C 10 -2 0 -6 -8 D 4 -8 6 0 -10 E 6 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222984 Cumulative probabilities = A: 0.166667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1763: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) E D B C A (6) C A B D E (5) E D C B A (4) D B A E C (4) C D B E A (4) E D A B C (3) E A D B C (3) E A B D C (3) D E C B A (3) C E D A B (3) C A E B D (3) C A B E D (3) A E B D C (3) A C B D E (3) E D B A C (2) E C D B A (2) E A C B D (2) D C E B A (2) C E D B A (2) C E A B D (2) C D B A E (2) C B A D E (2) B A D C E (2) B A C D E (2) A C B E D (2) A B E C D (2) A B C E D (2) E D C A B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E B C A (1) D E B A C (1) D C B A E (1) D B E A C (1) D B A C E (1) C D E B A (1) B D A E C (1) B C D A E (1) B A D E C (1) A E C B D (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 4 8 2 B -8 0 -2 2 -4 C -4 2 0 10 4 D -8 -2 -10 0 -6 E -2 4 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 8 2 B -8 0 -2 2 -4 C -4 2 0 10 4 D -8 -2 -10 0 -6 E -2 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=27 A=23 D=14 B=7 so B is eliminated. Round 2 votes counts: E=29 C=28 A=28 D=15 so D is eliminated. Round 3 votes counts: E=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:206 E:202 B:194 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 8 2 B -8 0 -2 2 -4 C -4 2 0 10 4 D -8 -2 -10 0 -6 E -2 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 8 2 B -8 0 -2 2 -4 C -4 2 0 10 4 D -8 -2 -10 0 -6 E -2 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 8 2 B -8 0 -2 2 -4 C -4 2 0 10 4 D -8 -2 -10 0 -6 E -2 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1764: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) A B E D C (12) B A E D C (8) E B D C A (6) C D E A B (6) C D A E B (6) A B C D E (6) E D C B A (5) A C D B E (5) A C D E B (4) A B E C D (4) D E C B A (3) D C E B A (3) B E A D C (3) A C B D E (3) E D B C A (2) E B D A C (2) B E D A C (2) B E C D A (2) E D B A C (1) E B A D C (1) D C E A B (1) C A D B E (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 6 -2 2 B 2 0 2 2 -4 C -6 -2 0 2 -4 D 2 -2 -2 0 0 E -2 4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.551234 E: 0.448766 Sum of squares = 0.505249850175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.551234 E: 1.000000 A B C D E A 0 -2 6 -2 2 B 2 0 2 2 -4 C -6 -2 0 2 -4 D 2 -2 -2 0 0 E -2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 0.499762 Sum of squares = 0.50000011363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=25 E=17 B=16 D=7 so D is eliminated. Round 2 votes counts: A=35 C=29 E=20 B=16 so B is eliminated. Round 3 votes counts: A=44 C=29 E=27 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:203 A:202 B:201 D:199 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 -2 2 B 2 0 2 2 -4 C -6 -2 0 2 -4 D 2 -2 -2 0 0 E -2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 0.499762 Sum of squares = 0.50000011363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 2 B 2 0 2 2 -4 C -6 -2 0 2 -4 D 2 -2 -2 0 0 E -2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 0.499762 Sum of squares = 0.50000011363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 2 B 2 0 2 2 -4 C -6 -2 0 2 -4 D 2 -2 -2 0 0 E -2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 0.499762 Sum of squares = 0.50000011363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500238 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1765: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) A C E B D (6) E B A C D (5) E B D C A (4) E B C D A (4) C D E B A (4) B E A D C (4) A B E C D (4) E B C A D (3) D B E A C (3) C D E A B (3) C A E B D (3) B E D A C (3) A B D E C (3) E B D A C (2) D E B C A (2) D C E B A (2) C E D B A (2) C E B A D (2) C D A E B (2) C D A B E (2) C A E D B (2) C A D E B (2) C A D B E (2) B D E A C (2) B A E D C (2) A D C B E (2) E D C B A (1) E C D B A (1) E C A B D (1) E B A D C (1) D E C B A (1) D C B A E (1) D C A B E (1) D B E C A (1) D B A E C (1) D A C B E (1) D A B E C (1) D A B C E (1) A E B C D (1) A D B C E (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -4 -2 -14 B 14 0 12 8 -20 C 4 -12 0 6 -14 D 2 -8 -6 0 -18 E 14 20 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -4 -2 -14 B 14 0 12 8 -20 C 4 -12 0 6 -14 D 2 -8 -6 0 -18 E 14 20 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=24 A=22 D=15 B=11 so B is eliminated. Round 2 votes counts: E=35 C=24 A=24 D=17 so D is eliminated. Round 3 votes counts: E=44 C=28 A=28 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 B:207 C:192 D:185 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -4 -2 -14 B 14 0 12 8 -20 C 4 -12 0 6 -14 D 2 -8 -6 0 -18 E 14 20 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -2 -14 B 14 0 12 8 -20 C 4 -12 0 6 -14 D 2 -8 -6 0 -18 E 14 20 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -2 -14 B 14 0 12 8 -20 C 4 -12 0 6 -14 D 2 -8 -6 0 -18 E 14 20 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1766: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (14) C A E B D (11) A C D E B (8) B D E C A (7) B E D C A (6) C E B A D (4) C A E D B (4) E D B A C (3) E B D A C (3) D B A E C (3) E B D C A (2) E A B D C (2) D E A B C (2) D B C E A (2) C B D A E (2) C A D E B (2) A C D B E (2) E D A B C (1) E C B D A (1) E C B A D (1) E B C D A (1) E B A D C (1) D E B A C (1) D C B A E (1) C E B D A (1) C E A B D (1) C D B A E (1) C B E A D (1) C B D E A (1) C B A E D (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C D A (1) A E D C B (1) A E C B D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 -6 -8 -14 B 18 0 0 2 -10 C 6 0 0 -4 -6 D 8 -2 4 0 0 E 14 10 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.477192 E: 0.522808 Sum of squares = 0.501040383364 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.477192 E: 1.000000 A B C D E A 0 -18 -6 -8 -14 B 18 0 0 2 -10 C 6 0 0 -4 -6 D 8 -2 4 0 0 E 14 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=23 A=16 E=15 B=15 so E is eliminated. Round 2 votes counts: C=33 D=27 B=22 A=18 so A is eliminated. Round 3 votes counts: C=46 D=30 B=24 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:215 B:205 D:205 C:198 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -6 -8 -14 B 18 0 0 2 -10 C 6 0 0 -4 -6 D 8 -2 4 0 0 E 14 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -8 -14 B 18 0 0 2 -10 C 6 0 0 -4 -6 D 8 -2 4 0 0 E 14 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -8 -14 B 18 0 0 2 -10 C 6 0 0 -4 -6 D 8 -2 4 0 0 E 14 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1767: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) D B E C A (9) B D E A C (9) B D A C E (9) A C E B D (9) D B C A E (5) E D B C A (4) E C D A B (4) E A C B D (4) C A E D B (4) B D A E C (4) E B D A C (3) B A D C E (3) E D C B A (2) E B A C D (2) D E B C A (2) D B C E A (2) C A D B E (2) B A C D E (2) A E C B D (2) E D C A B (1) E A B C D (1) D E C B A (1) C E A D B (1) B D C A E (1) B A E D C (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 6 -12 -8 B 18 0 18 8 2 C -6 -18 0 -12 -18 D 12 -8 12 0 2 E 8 -2 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 -12 -8 B 18 0 18 8 2 C -6 -18 0 -12 -18 D 12 -8 12 0 2 E 8 -2 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985286 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=29 D=19 A=15 C=7 so C is eliminated. Round 2 votes counts: E=31 B=29 A=21 D=19 so D is eliminated. Round 3 votes counts: B=45 E=34 A=21 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:211 D:209 A:184 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 -12 -8 B 18 0 18 8 2 C -6 -18 0 -12 -18 D 12 -8 12 0 2 E 8 -2 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985286 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 -12 -8 B 18 0 18 8 2 C -6 -18 0 -12 -18 D 12 -8 12 0 2 E 8 -2 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985286 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 -12 -8 B 18 0 18 8 2 C -6 -18 0 -12 -18 D 12 -8 12 0 2 E 8 -2 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985286 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1768: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (6) D A C E B (5) E C A B D (4) C E B D A (4) B E C D A (4) B E C A D (4) A D E C B (4) A D B E C (4) E C B D A (3) E C B A D (3) E B C A D (3) E B A C D (3) D B C A E (3) B D C E A (3) B D A C E (3) A E D C B (3) A E C D B (3) A D C E B (3) D C B E A (2) D B C E A (2) D A C B E (2) D A B C E (2) C D E A B (2) B E A C D (2) B C E D A (2) B A D E C (2) B A D C E (2) A D B C E (2) A B D E C (2) E C D A B (1) E C A D B (1) E A C B D (1) D C E B A (1) D B A C E (1) C E D A B (1) C E A D B (1) C D B E A (1) C B D E A (1) B C D E A (1) B A E D C (1) B A E C D (1) A E C B D (1) Total count = 100 A B C D E A 0 -16 -6 0 -12 B 16 0 -6 0 -6 C 6 6 0 6 0 D 0 0 -6 0 -4 E 12 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.440142 D: 0.000000 E: 0.559858 Sum of squares = 0.507165899266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.440142 D: 0.440142 E: 1.000000 A B C D E A 0 -16 -6 0 -12 B 16 0 -6 0 -6 C 6 6 0 6 0 D 0 0 -6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=22 E=19 D=18 C=16 so C is eliminated. Round 2 votes counts: E=31 B=26 A=22 D=21 so D is eliminated. Round 3 votes counts: B=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:209 B:202 D:195 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -6 0 -12 B 16 0 -6 0 -6 C 6 6 0 6 0 D 0 0 -6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 0 -12 B 16 0 -6 0 -6 C 6 6 0 6 0 D 0 0 -6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 0 -12 B 16 0 -6 0 -6 C 6 6 0 6 0 D 0 0 -6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1769: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) A D C B E (11) D C B A E (10) B C D A E (9) E B C D A (8) E A B C D (8) A E D C B (8) E A D B C (4) C B D A E (4) E B C A D (3) E A D C B (3) E A B D C (3) D A C B E (3) B C E D A (3) A D E C B (3) E B A C D (2) D B C A E (1) C D B A E (1) B E C D A (1) B D C A E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -6 -6 6 B 10 0 10 10 16 C 6 -10 0 6 14 D 6 -10 -6 0 14 E -6 -16 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -6 6 B 10 0 10 10 16 C 6 -10 0 6 14 D 6 -10 -6 0 14 E -6 -16 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=26 A=24 D=14 C=5 so C is eliminated. Round 2 votes counts: E=31 B=30 A=24 D=15 so D is eliminated. Round 3 votes counts: B=42 E=31 A=27 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:208 D:202 A:192 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -6 6 B 10 0 10 10 16 C 6 -10 0 6 14 D 6 -10 -6 0 14 E -6 -16 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -6 6 B 10 0 10 10 16 C 6 -10 0 6 14 D 6 -10 -6 0 14 E -6 -16 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -6 6 B 10 0 10 10 16 C 6 -10 0 6 14 D 6 -10 -6 0 14 E -6 -16 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1770: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) E D C B A (8) D E A C B (8) B A C E D (7) A B C D E (7) D A E B C (6) E B C A D (5) E C B D A (4) D A C B E (4) D E A B C (3) D A E C B (3) C B E A D (3) C B A E D (3) B C E A D (3) A B D C E (3) E C D B A (2) E C B A D (2) C A B D E (2) A D C B E (2) A C B D E (2) E D B C A (1) E D A B C (1) E B D C A (1) E B C D A (1) E B A C D (1) D E C B A (1) D E C A B (1) D A B C E (1) C D E B A (1) B E C A D (1) A D B E C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 8 6 B 8 0 8 12 2 C -2 -8 0 10 0 D -8 -12 -10 0 -8 E -6 -2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 8 6 B 8 0 8 12 2 C -2 -8 0 10 0 D -8 -12 -10 0 -8 E -6 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=26 B=21 A=17 C=9 so C is eliminated. Round 2 votes counts: D=28 B=27 E=26 A=19 so A is eliminated. Round 3 votes counts: B=42 D=32 E=26 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:204 C:200 E:200 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 8 6 B 8 0 8 12 2 C -2 -8 0 10 0 D -8 -12 -10 0 -8 E -6 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 8 6 B 8 0 8 12 2 C -2 -8 0 10 0 D -8 -12 -10 0 -8 E -6 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 8 6 B 8 0 8 12 2 C -2 -8 0 10 0 D -8 -12 -10 0 -8 E -6 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1771: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) B D C A E (7) E D A C B (6) D C E A B (5) D C B A E (4) C D A E B (4) D E C A B (3) C D E A B (3) B E D A C (3) B E A C D (3) B D E A C (3) B A E C D (3) A E B C D (3) A C E B D (3) E A D C B (2) E A C B D (2) E A B C D (2) D B C E A (2) C A B E D (2) B E A D C (2) B D E C A (2) B C D A E (2) B A C E D (2) B A C D E (2) E D A B C (1) E C D A B (1) E C A D B (1) E B D A C (1) E B A D C (1) E A B D C (1) D E A C B (1) D C E B A (1) D C B E A (1) D B E C A (1) D B E A C (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) B D C E A (1) B C A E D (1) B C A D E (1) B A D C E (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 4 6 -12 -14 B -4 0 -8 2 -2 C -6 8 0 0 -4 D 12 -2 0 0 0 E 14 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.276214 E: 0.723786 Sum of squares = 0.600160458961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.276214 E: 1.000000 A B C D E A 0 4 6 -12 -14 B -4 0 -8 2 -2 C -6 8 0 0 -4 D 12 -2 0 0 0 E 14 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 0.500387 Sum of squares = 0.500000299345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=25 D=19 C=14 A=9 so A is eliminated. Round 2 votes counts: B=33 E=30 D=19 C=18 so C is eliminated. Round 3 votes counts: B=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 D:205 C:199 B:194 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 -12 -14 B -4 0 -8 2 -2 C -6 8 0 0 -4 D 12 -2 0 0 0 E 14 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 0.500387 Sum of squares = 0.500000299345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -12 -14 B -4 0 -8 2 -2 C -6 8 0 0 -4 D 12 -2 0 0 0 E 14 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 0.500387 Sum of squares = 0.500000299345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -12 -14 B -4 0 -8 2 -2 C -6 8 0 0 -4 D 12 -2 0 0 0 E 14 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 0.500387 Sum of squares = 0.500000299345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499613 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1772: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) E C D A B (9) B A E C D (7) A B E C D (6) A B D C E (6) D C E B A (5) B D C E A (5) B A D C E (5) E C A D B (4) B A E D C (4) A E B C D (4) B D A C E (3) A E C D B (3) A D C E B (3) E B C D A (2) D C A E B (2) D B C E A (2) B D E C A (2) A E C B D (2) A D B C E (2) A B E D C (2) E B A C D (1) E A C D B (1) E A B C D (1) D C E A B (1) D C B E A (1) C D E B A (1) C D E A B (1) B E D C A (1) B E C D A (1) B D C A E (1) B A D E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 6 8 6 B 2 0 14 10 0 C -6 -14 0 6 -22 D -8 -10 -6 0 -16 E -6 0 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.815430 C: 0.000000 D: 0.000000 E: 0.184570 Sum of squares = 0.698991654982 Cumulative probabilities = A: 0.000000 B: 0.815430 C: 0.815430 D: 0.815430 E: 1.000000 A B C D E A 0 -2 6 8 6 B 2 0 14 10 0 C -6 -14 0 6 -22 D -8 -10 -6 0 -16 E -6 0 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000141421 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=30 A=30 E=27 D=11 C=2 so C is eliminated. Round 2 votes counts: B=30 A=30 E=27 D=13 so D is eliminated. Round 3 votes counts: E=35 B=33 A=32 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:213 A:209 C:182 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 8 6 B 2 0 14 10 0 C -6 -14 0 6 -22 D -8 -10 -6 0 -16 E -6 0 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000141421 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 8 6 B 2 0 14 10 0 C -6 -14 0 6 -22 D -8 -10 -6 0 -16 E -6 0 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000141421 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 8 6 B 2 0 14 10 0 C -6 -14 0 6 -22 D -8 -10 -6 0 -16 E -6 0 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000141421 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1773: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) A B C D E (8) B C A D E (6) E D C B A (5) D E C B A (5) A E D B C (5) E C D B A (4) C D B E A (4) A B E C D (4) A B C E D (4) E D C A B (3) E D A C B (3) D E C A B (3) D C B A E (3) C B D E A (3) A E B D C (3) A E B C D (3) E A D B C (2) E A B C D (2) D C E B A (2) D C B E A (2) C D B A E (2) B A C E D (2) E C B D A (1) E A D C B (1) E A B D C (1) D C A B E (1) D A B C E (1) C E D B A (1) C B D A E (1) B C E A D (1) B C D A E (1) B C A E D (1) A D E B C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 2 10 14 B 6 0 12 2 10 C -2 -12 0 14 6 D -10 -2 -14 0 6 E -14 -10 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 10 14 B 6 0 12 2 10 C -2 -12 0 14 6 D -10 -2 -14 0 6 E -14 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=22 B=20 D=17 C=11 so C is eliminated. Round 2 votes counts: A=30 B=24 E=23 D=23 so E is eliminated. Round 3 votes counts: D=39 A=36 B=25 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:210 C:203 D:190 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 10 14 B 6 0 12 2 10 C -2 -12 0 14 6 D -10 -2 -14 0 6 E -14 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 10 14 B 6 0 12 2 10 C -2 -12 0 14 6 D -10 -2 -14 0 6 E -14 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 10 14 B 6 0 12 2 10 C -2 -12 0 14 6 D -10 -2 -14 0 6 E -14 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1774: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) B C A E D (7) C B A E D (5) E D B A C (4) D E B A C (4) D A E C B (4) C A B E D (4) B C E A D (4) A C B E D (4) E D B C A (3) E B D C A (3) E B C A D (3) D E A C B (3) D B C A E (3) D A C B E (3) C A B D E (3) A C D B E (3) A C B D E (3) E B A C D (2) E A C B D (2) D E B C A (2) D B E C A (2) D A C E B (2) B E C D A (2) B E C A D (2) B C E D A (2) A D C E B (2) A C E B D (2) E B D A C (1) E A B C D (1) D B C E A (1) C A D B E (1) B E D C A (1) B D C E A (1) A E C D B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 2 2 -2 B 4 0 12 4 2 C -2 -12 0 6 4 D -2 -4 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 2 -2 B 4 0 12 4 2 C -2 -12 0 6 4 D -2 -4 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=19 B=19 A=17 C=13 so C is eliminated. Round 2 votes counts: D=32 A=25 B=24 E=19 so E is eliminated. Round 3 votes counts: D=39 B=33 A=28 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:202 A:199 C:198 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 2 -2 B 4 0 12 4 2 C -2 -12 0 6 4 D -2 -4 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 2 -2 B 4 0 12 4 2 C -2 -12 0 6 4 D -2 -4 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 2 -2 B 4 0 12 4 2 C -2 -12 0 6 4 D -2 -4 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1775: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) D C A E B (6) C A B E D (6) A C B D E (6) E D C B A (5) E B C D A (5) E B C A D (5) C A D B E (5) B E A C D (4) E D B A C (3) D E B A C (3) D A C E B (3) C A B D E (3) B C E A D (3) B A C E D (3) A C D B E (3) E D B C A (2) D E C A B (2) D E A C B (2) D A E C B (2) D A B C E (2) C B E A D (2) B C A E D (2) A D C B E (2) E D C A B (1) E C B A D (1) E B D C A (1) E B D A C (1) E B A D C (1) D C A B E (1) D A B E C (1) C E D A B (1) C D A E B (1) C D A B E (1) C B A E D (1) B A E C D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -8 0 16 B -14 0 -22 -8 10 C 8 22 0 6 20 D 0 8 -6 0 4 E -16 -10 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 0 16 B -14 0 -22 -8 10 C 8 22 0 6 20 D 0 8 -6 0 4 E -16 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 C=20 B=13 A=13 so B is eliminated. Round 2 votes counts: E=29 D=29 C=25 A=17 so A is eliminated. Round 3 votes counts: C=37 D=33 E=30 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:211 D:203 B:183 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -8 0 16 B -14 0 -22 -8 10 C 8 22 0 6 20 D 0 8 -6 0 4 E -16 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 0 16 B -14 0 -22 -8 10 C 8 22 0 6 20 D 0 8 -6 0 4 E -16 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 0 16 B -14 0 -22 -8 10 C 8 22 0 6 20 D 0 8 -6 0 4 E -16 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1776: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (12) E D A C B (9) E B C A D (8) E A D C B (8) D C B A E (8) B C A D E (6) A D C B E (6) E B C D A (5) E A B C D (5) A B C D E (4) E D B C A (3) D C A B E (3) B C D E A (3) E D C B A (2) D A C B E (2) A C B D E (2) E B D C A (1) E A D B C (1) E A B D C (1) D C B E A (1) D B C E A (1) D A E C B (1) C B D A E (1) B C E D A (1) B C A E D (1) B A C D E (1) A E D C B (1) A E C B D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -12 -6 2 B 8 0 6 4 4 C 12 -6 0 0 6 D 6 -4 0 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -6 2 B 8 0 6 4 4 C 12 -6 0 0 6 D 6 -4 0 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 B=24 D=16 A=16 C=1 so C is eliminated. Round 2 votes counts: E=43 B=25 D=16 A=16 so D is eliminated. Round 3 votes counts: E=43 B=35 A=22 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:206 D:204 E:191 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -6 2 B 8 0 6 4 4 C 12 -6 0 0 6 D 6 -4 0 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -6 2 B 8 0 6 4 4 C 12 -6 0 0 6 D 6 -4 0 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -6 2 B 8 0 6 4 4 C 12 -6 0 0 6 D 6 -4 0 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1777: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (12) B A E D C (12) E A B C D (10) C D E A B (10) D C B E A (8) B D C A E (8) E A C D B (5) E A C B D (4) B E A D C (4) D B C A E (3) C E D A B (3) C D A E B (3) B D A C E (3) E C A D B (2) D C A B E (2) C E A D B (2) B E D C A (2) B E A C D (2) B D A E C (1) B A E C D (1) B A D E C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -10 -10 -4 B 14 0 -2 0 20 C 10 2 0 -12 8 D 10 0 12 0 2 E 4 -20 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.464815 C: 0.000000 D: 0.535185 E: 0.000000 Sum of squares = 0.50247592481 Cumulative probabilities = A: 0.000000 B: 0.464815 C: 0.464815 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -10 -4 B 14 0 -2 0 20 C 10 2 0 -12 8 D 10 0 12 0 2 E 4 -20 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=25 E=21 C=18 A=2 so A is eliminated. Round 2 votes counts: B=35 D=25 E=22 C=18 so C is eliminated. Round 3 votes counts: D=38 B=35 E=27 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:212 C:204 E:187 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 -10 -4 B 14 0 -2 0 20 C 10 2 0 -12 8 D 10 0 12 0 2 E 4 -20 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -10 -4 B 14 0 -2 0 20 C 10 2 0 -12 8 D 10 0 12 0 2 E 4 -20 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -10 -4 B 14 0 -2 0 20 C 10 2 0 -12 8 D 10 0 12 0 2 E 4 -20 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1778: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) B A C E D (7) B C A D E (6) E D A C B (5) B D E C A (4) B D C A E (4) A E C B D (4) A C E D B (4) E A D C B (3) E A C D B (3) D E C B A (3) D E B C A (3) D B E C A (3) C A E D B (3) B D C E A (3) B C D A E (3) E A D B C (2) D E A C B (2) D C B E A (2) C A D E B (2) C A B E D (2) B D E A C (2) B A E C D (2) A E B C D (2) E D C A B (1) E D B A C (1) E D A B C (1) E B A D C (1) E A C B D (1) D E B A C (1) D C E A B (1) D C B A E (1) D C A E B (1) C D B A E (1) C D A E B (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A D C (1) B C A E D (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 -4 -12 -2 0 B 4 0 -4 -6 -6 C 12 4 0 -4 -6 D 2 6 4 0 10 E 0 6 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 0 B 4 0 -4 -6 -6 C 12 4 0 -4 -6 D 2 6 4 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=24 E=18 C=13 A=11 so A is eliminated. Round 2 votes counts: B=34 E=25 D=24 C=17 so C is eliminated. Round 3 votes counts: B=39 E=32 D=29 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:211 C:203 E:201 B:194 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 0 B 4 0 -4 -6 -6 C 12 4 0 -4 -6 D 2 6 4 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 0 B 4 0 -4 -6 -6 C 12 4 0 -4 -6 D 2 6 4 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 0 B 4 0 -4 -6 -6 C 12 4 0 -4 -6 D 2 6 4 0 10 E 0 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1779: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) A E D C B (8) D C E B A (7) A E D B C (7) A C D B E (5) E B D C A (4) E B A D C (4) E D B C A (3) C D B A E (3) B E D C A (3) B E C D A (3) A E B C D (3) A D C E B (3) E D C B A (2) E D A C B (2) D E C B A (2) D E C A B (2) B C E D A (2) B C D E A (2) A E B D C (2) A D E C B (2) E D B A C (1) E D A B C (1) E B D A C (1) E A D B C (1) D E B C A (1) D E A C B (1) D C E A B (1) D C B E A (1) C D B E A (1) C B D A E (1) C A B D E (1) B E C A D (1) B E A C D (1) B D C E A (1) B C D A E (1) B A E C D (1) B A C D E (1) A C D E B (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 0 -8 -12 B 10 0 -6 -10 -18 C 0 6 0 -20 -14 D 8 10 20 0 -6 E 12 18 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 0 -8 -12 B 10 0 -6 -10 -18 C 0 6 0 -20 -14 D 8 10 20 0 -6 E 12 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=19 B=16 D=15 C=14 so C is eliminated. Round 2 votes counts: A=37 B=25 E=19 D=19 so E is eliminated. Round 3 votes counts: A=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:225 D:216 B:188 C:186 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 0 -8 -12 B 10 0 -6 -10 -18 C 0 6 0 -20 -14 D 8 10 20 0 -6 E 12 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -8 -12 B 10 0 -6 -10 -18 C 0 6 0 -20 -14 D 8 10 20 0 -6 E 12 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -8 -12 B 10 0 -6 -10 -18 C 0 6 0 -20 -14 D 8 10 20 0 -6 E 12 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1780: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) C E D A B (8) C A E D B (8) B D E A C (8) E D C A B (6) C A B D E (6) E D A B C (5) D E B A C (5) C E A D B (5) B D A E C (5) B A C D E (5) C A B E D (4) B A D E C (4) C B A D E (3) E C D B A (2) D E A B C (2) C B E A D (2) E D A C B (1) E C D A B (1) D B E A C (1) C E D B A (1) C E B D A (1) C B A E D (1) C A E B D (1) B C A D E (1) B A D C E (1) A D B E C (1) A C D B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -10 -14 B -2 0 -2 -12 -10 C 0 2 0 2 -2 D 10 12 -2 0 -10 E 14 10 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 -10 -14 B -2 0 -2 -12 -10 C 0 2 0 2 -2 D 10 12 -2 0 -10 E 14 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=24 B=24 D=8 A=4 so A is eliminated. Round 2 votes counts: C=41 B=26 E=24 D=9 so D is eliminated. Round 3 votes counts: C=41 E=31 B=28 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:205 C:201 A:189 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -10 -14 B -2 0 -2 -12 -10 C 0 2 0 2 -2 D 10 12 -2 0 -10 E 14 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -10 -14 B -2 0 -2 -12 -10 C 0 2 0 2 -2 D 10 12 -2 0 -10 E 14 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -10 -14 B -2 0 -2 -12 -10 C 0 2 0 2 -2 D 10 12 -2 0 -10 E 14 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1781: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (11) D B C A E (5) C B D E A (5) A E D B C (5) E A C D B (4) E A C B D (4) C E D A B (4) C B E D A (4) A E B D C (4) D C B E A (3) D A E C B (3) D A B E C (3) C E B A D (3) B D A C E (3) B C D A E (3) A E D C B (3) E C A D B (2) C E B D A (2) C D E B A (2) C D B E A (2) B E A C D (2) B D C A E (2) B A C E D (2) A E B C D (2) A D E B C (2) E C A B D (1) E A D C B (1) E A B C D (1) D E C A B (1) D C E A B (1) D C A E B (1) D A B C E (1) C E D B A (1) B D C E A (1) B C E D A (1) B A E C D (1) B A D E C (1) A E C D B (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -10 -18 -12 B 8 0 4 8 4 C 10 -4 0 16 14 D 18 -8 -16 0 4 E 12 -4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -18 -12 B 8 0 4 8 4 C 10 -4 0 16 14 D 18 -8 -16 0 4 E 12 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 A=19 D=18 E=13 so E is eliminated. Round 2 votes counts: A=29 B=27 C=26 D=18 so D is eliminated. Round 3 votes counts: A=36 C=32 B=32 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:218 B:212 D:199 E:195 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -18 -12 B 8 0 4 8 4 C 10 -4 0 16 14 D 18 -8 -16 0 4 E 12 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -18 -12 B 8 0 4 8 4 C 10 -4 0 16 14 D 18 -8 -16 0 4 E 12 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -18 -12 B 8 0 4 8 4 C 10 -4 0 16 14 D 18 -8 -16 0 4 E 12 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1782: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (17) E D C B A (10) A D C E B (10) B E D C A (8) A C D E B (8) A B C D E (4) D E C A B (3) D C E A B (3) B A C D E (3) D E A C B (2) C D E A B (2) B C E D A (2) B C A E D (2) B A E D C (2) B A E C D (2) B A C E D (2) A D E C B (2) A D C B E (2) A C D B E (2) E D C A B (1) E D B C A (1) E C D B A (1) E B D C A (1) E B C D A (1) D C A E B (1) C E B D A (1) C B D E A (1) C A D B E (1) B E D A C (1) B C A D E (1) A C B D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -14 -12 -10 B 12 0 -2 2 6 C 14 2 0 2 -8 D 12 -2 -2 0 -6 E 10 -6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -12 -14 -12 -10 B 12 0 -2 2 6 C 14 2 0 2 -8 D 12 -2 -2 0 -6 E 10 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250000049 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=31 E=15 D=9 C=5 so C is eliminated. Round 2 votes counts: B=41 A=32 E=16 D=11 so D is eliminated. Round 3 votes counts: B=41 A=33 E=26 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:209 C:205 D:201 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 -12 -10 B 12 0 -2 2 6 C 14 2 0 2 -8 D 12 -2 -2 0 -6 E 10 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250000049 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -12 -10 B 12 0 -2 2 6 C 14 2 0 2 -8 D 12 -2 -2 0 -6 E 10 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250000049 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -12 -10 B 12 0 -2 2 6 C 14 2 0 2 -8 D 12 -2 -2 0 -6 E 10 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250000049 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1783: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) A B C D E (5) D C B E A (4) D B E C A (4) D B C E A (4) A C E D B (4) A B D C E (4) E C D B A (3) E C A D B (3) E B A D C (3) C E D B A (3) C A E D B (3) C A D B E (3) B E A D C (3) B A D C E (3) A B E C D (3) E D B C A (2) E B D C A (2) E A B D C (2) D E B C A (2) C D E B A (2) C D B A E (2) C D A E B (2) C D A B E (2) C A D E B (2) B A E D C (2) B A D E C (2) A E C B D (2) A E B C D (2) A C B D E (2) A B E D C (2) E C D A B (1) E A B C D (1) D E C B A (1) D C E B A (1) C E D A B (1) C E A D B (1) C D E A B (1) B D A E C (1) B D A C E (1) A E C D B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -10 10 0 B 2 0 -2 -18 -2 C 10 2 0 2 4 D -10 18 -2 0 -2 E 0 2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 10 0 B 2 0 -2 -18 -2 C 10 2 0 2 4 D -10 18 -2 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=23 C=22 D=16 B=12 so B is eliminated. Round 2 votes counts: A=34 E=26 C=22 D=18 so D is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:209 D:202 E:200 A:199 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 10 0 B 2 0 -2 -18 -2 C 10 2 0 2 4 D -10 18 -2 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 10 0 B 2 0 -2 -18 -2 C 10 2 0 2 4 D -10 18 -2 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 10 0 B 2 0 -2 -18 -2 C 10 2 0 2 4 D -10 18 -2 0 -2 E 0 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1784: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) B A E C D (8) D C A E B (7) D A C E B (6) B D C E A (5) B A E D C (5) B E A C D (4) D C E B A (3) D A B C E (3) C E D B A (3) B E C A D (3) B D A E C (3) A E C B D (3) A B E D C (3) E A B C D (2) D C E A B (2) D C B E A (2) D B C E A (2) C D E B A (2) B C D E A (2) E B C A D (1) E A C D B (1) E A C B D (1) D C B A E (1) D B C A E (1) D A C B E (1) D A B E C (1) C E B D A (1) C E A D B (1) C E A B D (1) C D E A B (1) B E C D A (1) B C E D A (1) B C E A D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B C D (1) A D E C B (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 18 4 20 B -2 0 22 14 18 C -18 -22 0 -2 -8 D -4 -14 2 0 -8 E -20 -18 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 4 20 B -2 0 22 14 18 C -18 -22 0 -2 -8 D -4 -14 2 0 -8 E -20 -18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=29 A=24 C=9 E=5 so E is eliminated. Round 2 votes counts: B=34 D=29 A=28 C=9 so C is eliminated. Round 3 votes counts: D=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:222 E:189 D:188 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 18 4 20 B -2 0 22 14 18 C -18 -22 0 -2 -8 D -4 -14 2 0 -8 E -20 -18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 4 20 B -2 0 22 14 18 C -18 -22 0 -2 -8 D -4 -14 2 0 -8 E -20 -18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 4 20 B -2 0 22 14 18 C -18 -22 0 -2 -8 D -4 -14 2 0 -8 E -20 -18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1785: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) B E D A C (10) B E C D A (8) E B D A C (6) C D A E B (6) C A D E B (5) D A E C B (4) A D C E B (4) A C D E B (4) E D B A C (3) E B D C A (3) D A C E B (3) C A B E D (3) D A E B C (2) C B A E D (2) C A B D E (2) B E C A D (2) B C E A D (2) B A E C D (2) A B C E D (2) E D C B A (1) E C B D A (1) E B C D A (1) D E C A B (1) D E A C B (1) D E A B C (1) C E B D A (1) C D E A B (1) C B E D A (1) C B E A D (1) B E D C A (1) B E A D C (1) B A C E D (1) A D E B C (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -6 -10 8 B -6 0 -6 0 2 C 6 6 0 14 0 D 10 0 -14 0 -4 E -8 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.702786 D: 0.000000 E: 0.297214 Sum of squares = 0.582244026684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.702786 D: 0.702786 E: 1.000000 A B C D E A 0 6 -6 -10 8 B -6 0 -6 0 2 C 6 6 0 14 0 D 10 0 -14 0 -4 E -8 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204106053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=27 E=15 A=13 D=12 so D is eliminated. Round 2 votes counts: C=33 B=27 A=22 E=18 so E is eliminated. Round 3 votes counts: B=40 C=36 A=24 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:199 E:197 D:196 B:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -10 8 B -6 0 -6 0 2 C 6 6 0 14 0 D 10 0 -14 0 -4 E -8 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204106053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -10 8 B -6 0 -6 0 2 C 6 6 0 14 0 D 10 0 -14 0 -4 E -8 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204106053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -10 8 B -6 0 -6 0 2 C 6 6 0 14 0 D 10 0 -14 0 -4 E -8 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204106053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1786: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) E D B A C (8) D E B A C (8) D B E A C (7) D B C A E (6) E A D B C (5) C B A D E (5) C A B D E (5) E D A B C (4) E A D C B (4) E A B D C (3) D E B C A (3) C A E B D (3) A E C B D (3) E A C D B (2) D C B E A (2) C B D A E (2) B D A E C (2) B C D A E (2) B C A D E (2) E D A C B (1) E C D A B (1) E C A D B (1) E A B C D (1) D E C A B (1) D E A B C (1) D B C E A (1) D B A E C (1) C E D A B (1) C A E D B (1) B A D C E (1) B A C E D (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -2 -8 B 4 0 16 -16 -2 C -8 -16 0 -16 -12 D 2 16 16 0 0 E 8 2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.381564 E: 0.618436 Sum of squares = 0.528054173549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.381564 E: 1.000000 A B C D E A 0 -4 8 -2 -8 B 4 0 16 -16 -2 C -8 -16 0 -16 -12 D 2 16 16 0 0 E 8 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=30 D=30 C=27 B=9 A=4 so A is eliminated. Round 2 votes counts: E=33 D=30 C=27 B=10 so B is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:211 B:201 A:197 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -2 -8 B 4 0 16 -16 -2 C -8 -16 0 -16 -12 D 2 16 16 0 0 E 8 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -2 -8 B 4 0 16 -16 -2 C -8 -16 0 -16 -12 D 2 16 16 0 0 E 8 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -2 -8 B 4 0 16 -16 -2 C -8 -16 0 -16 -12 D 2 16 16 0 0 E 8 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1787: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A B C D (8) C D B E A (8) E C D A B (4) E A D C B (4) D C B A E (4) A E B D C (4) E B C D A (3) C D E B A (3) B C D E A (3) A E D B C (3) A D B C E (3) E B A C D (2) E A C D B (2) E A C B D (2) E A B D C (2) D B C A E (2) D A C E B (2) C B D E A (2) B C E D A (2) B A E D C (2) B A D C E (2) A D E C B (2) A B D C E (2) E D C A B (1) E C D B A (1) E B C A D (1) D C E A B (1) D C A E B (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C B E D A (1) B E C D A (1) B D C A E (1) B C A D E (1) B A D E C (1) A E D C B (1) A E B C D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -4 -10 -6 B 6 0 16 6 0 C 4 -16 0 14 6 D 10 -6 -14 0 6 E 6 0 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.672165 C: 0.000000 D: 0.000000 E: 0.327835 Sum of squares = 0.55928188807 Cumulative probabilities = A: 0.000000 B: 0.672165 C: 0.672165 D: 0.672165 E: 1.000000 A B C D E A 0 -6 -4 -10 -6 B 6 0 16 6 0 C 4 -16 0 14 6 D 10 -6 -14 0 6 E 6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500284 C: 0.000000 D: 0.000000 E: 0.499716 Sum of squares = 0.50000016187 Cumulative probabilities = A: 0.000000 B: 0.500284 C: 0.500284 D: 0.500284 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=23 A=18 C=16 D=13 so D is eliminated. Round 2 votes counts: E=30 B=26 C=22 A=22 so C is eliminated. Round 3 votes counts: B=41 E=36 A=23 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:204 D:198 E:197 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 -6 B 6 0 16 6 0 C 4 -16 0 14 6 D 10 -6 -14 0 6 E 6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500284 C: 0.000000 D: 0.000000 E: 0.499716 Sum of squares = 0.50000016187 Cumulative probabilities = A: 0.000000 B: 0.500284 C: 0.500284 D: 0.500284 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 -6 B 6 0 16 6 0 C 4 -16 0 14 6 D 10 -6 -14 0 6 E 6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500284 C: 0.000000 D: 0.000000 E: 0.499716 Sum of squares = 0.50000016187 Cumulative probabilities = A: 0.000000 B: 0.500284 C: 0.500284 D: 0.500284 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 -6 B 6 0 16 6 0 C 4 -16 0 14 6 D 10 -6 -14 0 6 E 6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500284 C: 0.000000 D: 0.000000 E: 0.499716 Sum of squares = 0.50000016187 Cumulative probabilities = A: 0.000000 B: 0.500284 C: 0.500284 D: 0.500284 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1788: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) C A D E B (9) E B A C D (8) E A C B D (8) B E D C A (8) B D E C A (7) A C D E B (7) E A B C D (6) D C A B E (6) C D A B E (5) E B A D C (4) B E D A C (4) B E A D C (4) A C E D B (4) C D A E B (3) A E C D B (3) D B C A E (2) B D C E A (2) E A C D B (1) Total count = 100 A B C D E A 0 4 -2 8 -4 B -4 0 -10 2 -6 C 2 10 0 8 -6 D -8 -2 -8 0 0 E 4 6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.203827 E: 0.796173 Sum of squares = 0.675436806793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.203827 E: 1.000000 A B C D E A 0 4 -2 8 -4 B -4 0 -10 2 -6 C 2 10 0 8 -6 D -8 -2 -8 0 0 E 4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 D=17 C=17 A=14 so A is eliminated. Round 2 votes counts: E=30 C=28 B=25 D=17 so D is eliminated. Round 3 votes counts: C=43 E=30 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:208 C:207 A:203 B:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 8 -4 B -4 0 -10 2 -6 C 2 10 0 8 -6 D -8 -2 -8 0 0 E 4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 8 -4 B -4 0 -10 2 -6 C 2 10 0 8 -6 D -8 -2 -8 0 0 E 4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 8 -4 B -4 0 -10 2 -6 C 2 10 0 8 -6 D -8 -2 -8 0 0 E 4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1789: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) A E B C D (8) D C B E A (6) D B C A E (6) A E D C B (6) D E C B A (5) D C E B A (5) A E C B D (5) D A B C E (4) E C D B A (3) E C B D A (3) E C B A D (3) D B C E A (3) A B D C E (3) D E A C B (2) B D C A E (2) B C E A D (2) A D E C B (2) A D B C E (2) A B E C D (2) A B C E D (2) E D C B A (1) E D A C B (1) E C D A B (1) E A D C B (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E D A (1) C B E A D (1) C B D E A (1) B C D E A (1) B C A E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 2 4 B -10 0 -14 -10 -20 C -10 14 0 -14 -14 D -2 10 14 0 -2 E -4 20 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 2 4 B -10 0 -14 -10 -20 C -10 14 0 -14 -14 D -2 10 14 0 -2 E -4 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=35 A=35 E=21 B=6 C=3 so C is eliminated. Round 2 votes counts: D=35 A=35 E=21 B=9 so B is eliminated. Round 3 votes counts: D=39 A=36 E=25 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:213 D:210 C:188 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 2 4 B -10 0 -14 -10 -20 C -10 14 0 -14 -14 D -2 10 14 0 -2 E -4 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 2 4 B -10 0 -14 -10 -20 C -10 14 0 -14 -14 D -2 10 14 0 -2 E -4 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 2 4 B -10 0 -14 -10 -20 C -10 14 0 -14 -14 D -2 10 14 0 -2 E -4 20 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1790: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) A B E D C (7) E C A B D (6) C D E B A (6) C D B A E (6) A B D E C (6) C E D B A (5) C E D A B (5) E C D A B (4) E A B C D (4) D C B A E (4) D B A C E (4) B D A C E (4) A E B D C (4) D B C A E (3) C E A B D (3) B A D E C (3) E C A D B (2) D C B E A (2) C E B A D (2) B A D C E (2) A E B C D (2) E D A C B (1) E A C B D (1) E A B D C (1) D A B E C (1) C D B E A (1) B D A E C (1) Total count = 100 A B C D E A 0 -6 2 -14 14 B 6 0 4 -8 8 C -2 -4 0 -6 -6 D 14 8 6 0 6 E -14 -8 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -14 14 B 6 0 4 -8 8 C -2 -4 0 -6 -6 D 14 8 6 0 6 E -14 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 E=19 A=19 B=10 so B is eliminated. Round 2 votes counts: D=29 C=28 A=24 E=19 so E is eliminated. Round 3 votes counts: C=40 D=30 A=30 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:217 B:205 A:198 C:191 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -14 14 B 6 0 4 -8 8 C -2 -4 0 -6 -6 D 14 8 6 0 6 E -14 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -14 14 B 6 0 4 -8 8 C -2 -4 0 -6 -6 D 14 8 6 0 6 E -14 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -14 14 B 6 0 4 -8 8 C -2 -4 0 -6 -6 D 14 8 6 0 6 E -14 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1791: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) B A C E D (10) A B E C D (9) A E D C B (7) E D C A B (6) D C E B A (6) B C D E A (5) B A D C E (4) A E B C D (4) D E C B A (3) D C B E A (3) C D B E A (3) B A C D E (3) E C A D B (2) D E A C B (2) C D E B A (2) C B E D A (2) B C D A E (2) B C A E D (2) A E C D B (2) A B E D C (2) E D A C B (1) C E D A B (1) C B D E A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C E D A (1) A E D B C (1) A D E B C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -2 -2 2 B 0 0 -2 0 4 C 2 2 0 0 -2 D 2 0 0 0 -2 E -2 -4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999996 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 0 -2 -2 2 B 0 0 -2 0 4 C 2 2 0 0 -2 D 2 0 0 0 -2 E -2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999927 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=28 D=24 E=9 C=9 so E is eliminated. Round 2 votes counts: D=31 B=30 A=28 C=11 so C is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:201 C:201 D:200 A:199 E:199 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 -2 2 B 0 0 -2 0 4 C 2 2 0 0 -2 D 2 0 0 0 -2 E -2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999927 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 2 B 0 0 -2 0 4 C 2 2 0 0 -2 D 2 0 0 0 -2 E -2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999927 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 2 B 0 0 -2 0 4 C 2 2 0 0 -2 D 2 0 0 0 -2 E -2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999927 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1792: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (17) C E B D A (15) A C E B D (8) D B A E C (7) C A E B D (7) D B E C A (6) C E A B D (6) D B E A C (4) D A B E C (3) A C D B E (3) E B D C A (2) E B D A C (2) D B C E A (2) C E B A D (2) B D E C A (2) A E B D C (2) A D C B E (2) A D B C E (2) A C E D B (2) E C B A D (1) E B C D A (1) C D B E A (1) C A D E B (1) B E D A C (1) A E C B D (1) Total count = 100 A B C D E A 0 8 8 8 10 B -8 0 2 0 0 C -8 -2 0 -4 2 D -8 0 4 0 0 E -10 0 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 10 B -8 0 2 0 0 C -8 -2 0 -4 2 D -8 0 4 0 0 E -10 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=32 D=22 E=6 B=3 so B is eliminated. Round 2 votes counts: A=37 C=32 D=24 E=7 so E is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:198 B:197 C:194 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 10 B -8 0 2 0 0 C -8 -2 0 -4 2 D -8 0 4 0 0 E -10 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 10 B -8 0 2 0 0 C -8 -2 0 -4 2 D -8 0 4 0 0 E -10 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 10 B -8 0 2 0 0 C -8 -2 0 -4 2 D -8 0 4 0 0 E -10 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1793: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (13) D E A B C (9) D E A C B (5) A E D B C (5) D E B A C (4) C A D E B (4) C A B E D (4) B E D A C (3) B E A D C (3) B C A E D (3) B A E D C (3) B A E C D (3) A E D C B (3) A C E B D (3) E D A B C (2) D C E A B (2) D C A E B (2) D A E C B (2) C D B E A (2) C B D A E (2) B D E A C (2) B C E D A (2) B C D E A (2) E B A D C (1) E A D B C (1) D C E B A (1) D B E A C (1) D B C E A (1) D A C E B (1) C D A E B (1) C D A B E (1) C B A D E (1) C A E B D (1) B E A C D (1) B C E A D (1) B A C E D (1) A E C D B (1) A E B D C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 14 10 14 B 0 0 -2 0 -2 C -14 2 0 -4 0 D -10 0 4 0 -12 E -14 2 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.420109 B: 0.579891 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.51276507845 Cumulative probabilities = A: 0.420109 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 10 14 B 0 0 -2 0 -2 C -14 2 0 -4 0 D -10 0 4 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=28 B=24 A=15 E=4 so E is eliminated. Round 2 votes counts: D=30 C=29 B=25 A=16 so A is eliminated. Round 3 votes counts: D=39 C=35 B=26 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:219 E:200 B:198 C:192 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 10 14 B 0 0 -2 0 -2 C -14 2 0 -4 0 D -10 0 4 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 10 14 B 0 0 -2 0 -2 C -14 2 0 -4 0 D -10 0 4 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 10 14 B 0 0 -2 0 -2 C -14 2 0 -4 0 D -10 0 4 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1794: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) E D C A B (7) D E B A C (7) D E C B A (5) A B E D C (5) D B A E C (4) B A D E C (4) E D A C B (3) E A B D C (3) D C E B A (3) C E A B D (3) C D B A E (3) C B D A E (3) C B A E D (3) C B A D E (3) C A B E D (3) B D A E C (3) B A C D E (3) A B C E D (3) E A D B C (2) D E C A B (2) D E B C A (2) C D E B A (2) B C D A E (2) B A C E D (2) A E B C D (2) E C D A B (1) D E A B C (1) D C B E A (1) D B E C A (1) D B A C E (1) C E D A B (1) C A E B D (1) A E B D C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 8 -20 -4 B 4 0 8 -8 -10 C -8 -8 0 -26 -22 D 20 8 26 0 0 E 4 10 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.362872 E: 0.637128 Sum of squares = 0.537608355635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.362872 E: 1.000000 A B C D E A 0 -4 8 -20 -4 B 4 0 8 -8 -10 C -8 -8 0 -26 -22 D 20 8 26 0 0 E 4 10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 C=22 B=14 A=13 so A is eliminated. Round 2 votes counts: E=27 D=27 C=24 B=22 so B is eliminated. Round 3 votes counts: D=34 C=34 E=32 so E is eliminated. Round 4 votes counts: D=63 C=37 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:218 B:197 A:190 C:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -20 -4 B 4 0 8 -8 -10 C -8 -8 0 -26 -22 D 20 8 26 0 0 E 4 10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -20 -4 B 4 0 8 -8 -10 C -8 -8 0 -26 -22 D 20 8 26 0 0 E 4 10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -20 -4 B 4 0 8 -8 -10 C -8 -8 0 -26 -22 D 20 8 26 0 0 E 4 10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1795: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) C A D B E (9) A C D E B (8) E B D A C (7) E A C B D (7) B E D C A (7) A C E D B (7) E B A C D (6) B D E C A (6) E B A D C (5) D C A B E (4) E B D C A (3) B D C E A (3) E A C D B (2) D C B A E (2) D B C A E (2) C D A B E (2) A E C B D (2) A C D B E (2) E D A B C (1) D C A E B (1) D B E C A (1) D B C E A (1) B E A C D (1) B C D A E (1) A E C D B (1) Total count = 100 A B C D E A 0 10 16 18 -18 B -10 0 6 14 -18 C -16 -6 0 14 -16 D -18 -14 -14 0 -16 E 18 18 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 16 18 -18 B -10 0 6 14 -18 C -16 -6 0 14 -16 D -18 -14 -14 0 -16 E 18 18 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=20 B=18 D=11 C=11 so D is eliminated. Round 2 votes counts: E=40 B=22 A=20 C=18 so C is eliminated. Round 3 votes counts: E=40 A=36 B=24 so B is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:234 A:213 B:196 C:188 D:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 16 18 -18 B -10 0 6 14 -18 C -16 -6 0 14 -16 D -18 -14 -14 0 -16 E 18 18 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 18 -18 B -10 0 6 14 -18 C -16 -6 0 14 -16 D -18 -14 -14 0 -16 E 18 18 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 18 -18 B -10 0 6 14 -18 C -16 -6 0 14 -16 D -18 -14 -14 0 -16 E 18 18 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1796: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) E B C A D (5) D C B E A (5) A D B C E (5) E B A C D (4) D A C B E (4) C D A E B (4) B D E C A (4) A E B C D (4) A B E D C (4) E C A B D (3) E B C D A (3) D C E B A (3) D B C E A (3) B E D C A (3) B E A D C (3) B A E D C (3) A D C E B (3) A C D E B (3) A B E C D (3) E C B D A (2) C E D A B (2) C D E A B (2) B E A C D (2) B D E A C (2) A C E D B (2) E A B C D (1) D C B A E (1) D C A E B (1) D B A C E (1) D A C E B (1) C E D B A (1) B E C D A (1) B D A E C (1) B A E C D (1) A E C B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 14 12 2 B -4 0 10 4 10 C -14 -10 0 -12 -2 D -12 -4 12 0 4 E -2 -10 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 12 2 B -4 0 10 4 10 C -14 -10 0 -12 -2 D -12 -4 12 0 4 E -2 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=20 D=19 E=18 C=9 so C is eliminated. Round 2 votes counts: A=34 D=25 E=21 B=20 so B is eliminated. Round 3 votes counts: A=38 D=32 E=30 so E is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:210 D:200 E:193 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 12 2 B -4 0 10 4 10 C -14 -10 0 -12 -2 D -12 -4 12 0 4 E -2 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 12 2 B -4 0 10 4 10 C -14 -10 0 -12 -2 D -12 -4 12 0 4 E -2 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 12 2 B -4 0 10 4 10 C -14 -10 0 -12 -2 D -12 -4 12 0 4 E -2 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1797: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) D B E A C (7) D E B A C (5) A E C B D (5) D E A C B (4) C B A E D (4) C A E B D (4) C A B D E (4) B E D A C (4) B A E C D (4) D C A E B (3) C A D E B (3) B D E A C (3) B A C E D (3) E A B C D (2) D C A B E (2) D B C E A (2) C A D B E (2) B D E C A (2) B C D A E (2) B C A E D (2) E D A B C (1) E B D A C (1) E A D C B (1) E A D B C (1) D E A B C (1) D C E B A (1) D C B E A (1) D C B A E (1) D B E C A (1) D A E C B (1) C D B A E (1) C D A E B (1) C D A B E (1) C A E D B (1) B E C D A (1) B E A D C (1) B E A C D (1) B D C E A (1) B C A D E (1) A E B C D (1) A C E D B (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 6 18 B -2 0 -2 16 24 C 0 2 0 12 4 D -6 -16 -12 0 0 E -18 -24 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.536705 B: 0.000000 C: 0.463295 D: 0.000000 E: 0.000000 Sum of squares = 0.502694533048 Cumulative probabilities = A: 0.536705 B: 0.536705 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 6 18 B -2 0 -2 16 24 C 0 2 0 12 4 D -6 -16 -12 0 0 E -18 -24 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 B=25 A=10 E=6 so E is eliminated. Round 2 votes counts: D=30 C=30 B=26 A=14 so A is eliminated. Round 3 votes counts: C=37 D=32 B=31 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:218 A:213 C:209 D:183 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 6 18 B -2 0 -2 16 24 C 0 2 0 12 4 D -6 -16 -12 0 0 E -18 -24 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 18 B -2 0 -2 16 24 C 0 2 0 12 4 D -6 -16 -12 0 0 E -18 -24 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 18 B -2 0 -2 16 24 C 0 2 0 12 4 D -6 -16 -12 0 0 E -18 -24 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1798: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (14) B D C E A (11) B D C A E (11) A E C D B (8) D C B A E (7) E A B C D (6) D B C A E (6) C D B A E (6) E A B D C (5) C D A E B (5) B E A D C (5) E A C B D (3) C A E D B (3) B E D A C (3) C A D E B (2) A C E D B (2) C B D A E (1) B D E A C (1) B C D E A (1) Total count = 100 A B C D E A 0 -4 -6 -4 2 B 4 0 -2 -6 4 C 6 2 0 2 10 D 4 6 -2 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 2 B 4 0 -2 -6 4 C 6 2 0 2 10 D 4 6 -2 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=28 C=17 D=13 A=10 so A is eliminated. Round 2 votes counts: E=36 B=32 C=19 D=13 so D is eliminated. Round 3 votes counts: B=38 E=36 C=26 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:210 D:205 B:200 A:194 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 2 B 4 0 -2 -6 4 C 6 2 0 2 10 D 4 6 -2 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 2 B 4 0 -2 -6 4 C 6 2 0 2 10 D 4 6 -2 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 2 B 4 0 -2 -6 4 C 6 2 0 2 10 D 4 6 -2 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1799: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) D A E C B (8) E D A B C (6) D A C B E (6) C B A E D (5) A D B C E (5) E C B D A (4) D A E B C (4) D A B C E (4) B C E A D (4) B C A D E (4) E D C B A (3) E D A C B (3) D E A C B (3) D E A B C (3) C B A D E (3) B C A E D (3) A D C B E (3) A B C D E (3) E D B A C (2) E C D B A (2) D A C E B (2) E D C A B (1) E D B C A (1) E C B A D (1) E B C A D (1) E A D B C (1) E A B D C (1) C E B D A (1) C B E D A (1) C B E A D (1) C A B D E (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 12 -24 6 B -10 0 2 -14 -10 C -12 -2 0 -12 -4 D 24 14 12 0 2 E -6 10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 -24 6 B -10 0 2 -14 -10 C -12 -2 0 -12 -4 D 24 14 12 0 2 E -6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=30 C=12 B=12 A=12 so C is eliminated. Round 2 votes counts: E=35 D=30 B=22 A=13 so A is eliminated. Round 3 votes counts: D=38 E=35 B=27 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 E:203 A:202 C:185 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 12 -24 6 B -10 0 2 -14 -10 C -12 -2 0 -12 -4 D 24 14 12 0 2 E -6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -24 6 B -10 0 2 -14 -10 C -12 -2 0 -12 -4 D 24 14 12 0 2 E -6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -24 6 B -10 0 2 -14 -10 C -12 -2 0 -12 -4 D 24 14 12 0 2 E -6 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1800: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) C B A D E (8) E D C A B (7) E D A C B (6) D A E C B (6) C B E D A (6) A D E B C (5) D E A C B (4) A D E C B (4) E D A B C (3) B A D C E (3) A B D E C (3) E D C B A (2) E B C D A (2) C D E A B (2) C D A E B (2) B E A D C (2) B A C D E (2) A D C E B (2) A D B E C (2) A B D C E (2) E D B A C (1) E C D B A (1) E C D A B (1) E C B D A (1) E B A D C (1) E A D B C (1) D A E B C (1) D A C E B (1) C E D B A (1) C B D A E (1) B E C A D (1) B C E D A (1) B C E A D (1) B C A E D (1) B A C E D (1) A D B C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 6 2 12 B -10 0 -14 -8 -8 C -6 14 0 -14 -8 D -2 8 14 0 20 E -12 8 8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 2 12 B -10 0 -14 -8 -8 C -6 14 0 -14 -8 D -2 8 14 0 20 E -12 8 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998458 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=21 A=21 C=20 D=12 so D is eliminated. Round 2 votes counts: E=30 A=29 B=21 C=20 so C is eliminated. Round 3 votes counts: B=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:220 A:215 C:193 E:192 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 2 12 B -10 0 -14 -8 -8 C -6 14 0 -14 -8 D -2 8 14 0 20 E -12 8 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998458 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 12 B -10 0 -14 -8 -8 C -6 14 0 -14 -8 D -2 8 14 0 20 E -12 8 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998458 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 12 B -10 0 -14 -8 -8 C -6 14 0 -14 -8 D -2 8 14 0 20 E -12 8 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998458 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1801: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) C A B D E (9) E D B C A (8) D B A C E (6) D B E C A (5) D B C A E (5) D E B C A (4) D B E A C (4) E D A C B (3) E C A D B (3) D E B A C (3) B D C A E (3) A C E B D (3) A C B D E (3) E D C A B (2) E D A B C (2) E C A B D (2) E A C D B (2) E A B C D (2) B C A D E (2) A C B E D (2) E D B A C (1) E B D A C (1) E A B D C (1) D E C B A (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A E C (1) C E A B D (1) C D E A B (1) C B A D E (1) C A E B D (1) C A D B E (1) B D A C E (1) B C D A E (1) B A C D E (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 -10 -14 B 2 0 6 -10 -2 C 6 -6 0 -8 -10 D 10 10 8 0 12 E 14 2 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -10 -14 B 2 0 6 -10 -2 C 6 -6 0 -8 -10 D 10 10 8 0 12 E 14 2 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=32 C=14 A=10 B=8 so B is eliminated. Round 2 votes counts: E=36 D=36 C=17 A=11 so A is eliminated. Round 3 votes counts: E=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:207 B:198 C:191 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -10 -14 B 2 0 6 -10 -2 C 6 -6 0 -8 -10 D 10 10 8 0 12 E 14 2 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -10 -14 B 2 0 6 -10 -2 C 6 -6 0 -8 -10 D 10 10 8 0 12 E 14 2 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -10 -14 B 2 0 6 -10 -2 C 6 -6 0 -8 -10 D 10 10 8 0 12 E 14 2 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1802: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) C D A E B (7) C D E A B (6) B E A D C (6) E B A C D (5) D C B A E (5) D A C B E (5) D A B C E (5) C D A B E (5) B A E D C (5) C E B A D (4) C D B E A (4) E C B A D (3) E B A D C (3) E C A B D (2) E A B D C (2) E A B C D (2) D A B E C (2) C E D B A (2) C E D A B (2) C E B D A (2) C D E B A (2) C D B A E (2) C E A D B (1) C E A B D (1) B E A C D (1) B D E C A (1) B D A E C (1) B A D E C (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -18 -22 2 B -6 0 -26 -18 10 C 18 26 0 2 26 D 22 18 -2 0 16 E -2 -10 -26 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -18 -22 2 B -6 0 -26 -18 10 C 18 26 0 2 26 D 22 18 -2 0 16 E -2 -10 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=27 E=17 B=15 A=3 so A is eliminated. Round 2 votes counts: C=38 D=28 E=18 B=16 so B is eliminated. Round 3 votes counts: C=38 D=32 E=30 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:236 D:227 A:184 B:180 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -18 -22 2 B -6 0 -26 -18 10 C 18 26 0 2 26 D 22 18 -2 0 16 E -2 -10 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 -22 2 B -6 0 -26 -18 10 C 18 26 0 2 26 D 22 18 -2 0 16 E -2 -10 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 -22 2 B -6 0 -26 -18 10 C 18 26 0 2 26 D 22 18 -2 0 16 E -2 -10 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1803: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) E B D C A (8) E B C A D (7) B E A D C (7) C E D A B (4) A D C B E (4) D C A B E (3) D A C B E (3) D A B C E (3) C D A E B (3) C A D E B (3) B E A C D (3) A C D B E (3) A C B E D (3) E D B C A (2) E C B D A (2) E C B A D (2) E B C D A (2) D E C B A (2) C D E A B (2) C A E D B (2) C A D B E (2) B A E D C (2) B A E C D (2) B A D E C (2) A B D C E (2) A B C E D (2) A B C D E (2) E C D B A (1) E B A C D (1) D E B A C (1) D C E A B (1) D C A E B (1) C E A D B (1) B D E A C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 4 4 -12 B 8 0 14 16 10 C -4 -14 0 -4 -8 D -4 -16 4 0 -20 E 12 -10 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 4 -12 B 8 0 14 16 10 C -4 -14 0 -4 -8 D -4 -16 4 0 -20 E 12 -10 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 A=18 C=17 D=14 so D is eliminated. Round 2 votes counts: E=28 B=26 A=24 C=22 so C is eliminated. Round 3 votes counts: A=38 E=36 B=26 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:224 E:215 A:194 C:185 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 4 -12 B 8 0 14 16 10 C -4 -14 0 -4 -8 D -4 -16 4 0 -20 E 12 -10 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 4 -12 B 8 0 14 16 10 C -4 -14 0 -4 -8 D -4 -16 4 0 -20 E 12 -10 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 4 -12 B 8 0 14 16 10 C -4 -14 0 -4 -8 D -4 -16 4 0 -20 E 12 -10 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1804: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) A D B C E (9) A B D C E (9) E C D B A (7) B C A E D (7) D A E B C (6) D A B E C (6) E D C A B (5) C E B A D (5) C B E A D (5) D A B C E (4) B A C E D (4) E C B D A (3) B A C D E (3) A D B E C (3) E C B A D (2) D E C A B (2) C B A E D (2) E D C B A (1) E D A C B (1) D E A B C (1) B C A D E (1) A D E B C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 20 20 6 14 B -20 0 12 -14 10 C -20 -12 0 -20 0 D -6 14 20 0 16 E -14 -10 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 20 6 14 B -20 0 12 -14 10 C -20 -12 0 -20 0 D -6 14 20 0 16 E -14 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=24 E=19 B=15 C=12 so C is eliminated. Round 2 votes counts: D=30 E=24 A=24 B=22 so B is eliminated. Round 3 votes counts: A=41 D=30 E=29 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:230 D:222 B:194 E:180 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 20 6 14 B -20 0 12 -14 10 C -20 -12 0 -20 0 D -6 14 20 0 16 E -14 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 20 6 14 B -20 0 12 -14 10 C -20 -12 0 -20 0 D -6 14 20 0 16 E -14 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 20 6 14 B -20 0 12 -14 10 C -20 -12 0 -20 0 D -6 14 20 0 16 E -14 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1805: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) D E A C B (11) C B D A E (7) C B A D E (7) A E B C D (7) E D A B C (6) A E D B C (5) D E C B A (4) C B D E A (4) B C A E D (4) A B E C D (4) D E C A B (3) D C E B A (3) C D B E A (3) C B A E D (3) A E B D C (3) C B E D A (2) A D E C B (2) E D C A B (1) E A B D C (1) E A B C D (1) D E A B C (1) D C B E A (1) D C A E B (1) D A E C B (1) C A B D E (1) B A C E D (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 22 12 4 -4 B -22 0 -10 -8 -22 C -12 10 0 -10 -22 D -4 8 10 0 0 E 4 22 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.293121 E: 0.706879 Sum of squares = 0.585598126522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.293121 E: 1.000000 A B C D E A 0 22 12 4 -4 B -22 0 -10 -8 -22 C -12 10 0 -10 -22 D -4 8 10 0 0 E 4 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.500156 Sum of squares = 0.500000048698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 A=23 E=20 B=5 so B is eliminated. Round 2 votes counts: C=31 D=25 A=24 E=20 so E is eliminated. Round 3 votes counts: A=37 D=32 C=31 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:224 A:217 D:207 C:183 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 12 4 -4 B -22 0 -10 -8 -22 C -12 10 0 -10 -22 D -4 8 10 0 0 E 4 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.500156 Sum of squares = 0.500000048698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 4 -4 B -22 0 -10 -8 -22 C -12 10 0 -10 -22 D -4 8 10 0 0 E 4 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.500156 Sum of squares = 0.500000048698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 4 -4 B -22 0 -10 -8 -22 C -12 10 0 -10 -22 D -4 8 10 0 0 E 4 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 0.500156 Sum of squares = 0.500000048698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499844 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1806: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) E A B D C (7) C D E A B (7) C D B E A (5) C D B A E (5) E A D C B (4) D E A B C (4) D C B E A (4) B A E D C (4) B A E C D (4) A E B C D (4) E A D B C (3) D C B A E (3) C B D A E (3) B C D A E (3) A E B D C (3) E D A C B (2) E A C B D (2) D C E B A (2) D B C A E (2) C D E B A (2) B D C A E (2) A B E D C (2) E C D A B (1) E C A B D (1) E A C D B (1) D E A C B (1) C E A D B (1) C E A B D (1) B D A E C (1) B D A C E (1) B C A E D (1) B A D E C (1) B A C E D (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -6 -16 -16 B -12 0 -10 -16 -14 C 6 10 0 -14 6 D 16 16 14 0 12 E 16 14 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 -16 -16 B -12 0 -10 -16 -14 C 6 10 0 -14 6 D 16 16 14 0 12 E 16 14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=24 E=21 B=18 A=11 so A is eliminated. Round 2 votes counts: E=29 D=26 C=24 B=21 so B is eliminated. Round 3 votes counts: E=40 D=31 C=29 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:206 C:204 A:187 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -6 -16 -16 B -12 0 -10 -16 -14 C 6 10 0 -14 6 D 16 16 14 0 12 E 16 14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -16 -16 B -12 0 -10 -16 -14 C 6 10 0 -14 6 D 16 16 14 0 12 E 16 14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -16 -16 B -12 0 -10 -16 -14 C 6 10 0 -14 6 D 16 16 14 0 12 E 16 14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1807: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) D A E B C (6) C B A E D (6) A B E C D (6) A B C E D (6) D E A C B (5) B C A E D (5) A E B D C (4) E D C B A (3) D E C B A (3) D E C A B (3) C D B E A (3) C B E A D (3) C B D E A (3) B A C E D (3) D E A B C (2) D C E B A (2) C E D B A (2) C B E D A (2) C B A D E (2) A E D B C (2) A E B C D (2) A D E B C (2) A D B E C (2) E D B C A (1) E B D A C (1) E B C A D (1) E B A C D (1) E A B D C (1) E A B C D (1) D C B E A (1) D A E C B (1) D A C B E (1) C D B A E (1) C B D A E (1) B E A C D (1) B C E A D (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 14 2 4 B -8 0 16 2 -2 C -14 -16 0 2 -12 D -2 -2 -2 0 -20 E -4 2 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998854 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 2 4 B -8 0 16 2 -2 C -14 -16 0 2 -12 D -2 -2 -2 0 -20 E -4 2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 C=23 E=17 B=10 so B is eliminated. Round 2 votes counts: C=29 A=29 D=24 E=18 so E is eliminated. Round 3 votes counts: D=37 A=33 C=30 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:214 B:204 D:187 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 2 4 B -8 0 16 2 -2 C -14 -16 0 2 -12 D -2 -2 -2 0 -20 E -4 2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 2 4 B -8 0 16 2 -2 C -14 -16 0 2 -12 D -2 -2 -2 0 -20 E -4 2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 2 4 B -8 0 16 2 -2 C -14 -16 0 2 -12 D -2 -2 -2 0 -20 E -4 2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1808: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) C A E B D (9) D B E A C (7) E A C B D (5) D B C A E (5) C A E D B (5) C A D E B (4) C A D B E (4) E B D A C (3) E B A D C (3) E A B C D (3) B D E A C (3) A C E B D (3) D C B A E (2) D A C B E (2) C D A B E (2) C A B D E (2) B E D C A (2) B D E C A (2) A E D C B (2) A E C D B (2) A C D E B (2) E C A B D (1) E B C A D (1) E B A C D (1) E A C D B (1) E A B D C (1) D B E C A (1) D B A E C (1) D B A C E (1) D A E B C (1) D A B E C (1) C E A B D (1) C D B A E (1) C B E A D (1) C A B E D (1) B C E A D (1) B C D E A (1) A E C B D (1) Total count = 100 A B C D E A 0 28 8 30 24 B -28 0 -24 -10 -20 C -8 24 0 26 16 D -30 10 -26 0 -16 E -24 20 -16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 8 30 24 B -28 0 -24 -10 -20 C -8 24 0 26 16 D -30 10 -26 0 -16 E -24 20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=21 A=21 E=19 B=9 so B is eliminated. Round 2 votes counts: C=32 D=26 E=21 A=21 so E is eliminated. Round 3 votes counts: A=35 C=34 D=31 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:245 C:229 E:198 D:169 B:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 8 30 24 B -28 0 -24 -10 -20 C -8 24 0 26 16 D -30 10 -26 0 -16 E -24 20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 8 30 24 B -28 0 -24 -10 -20 C -8 24 0 26 16 D -30 10 -26 0 -16 E -24 20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 8 30 24 B -28 0 -24 -10 -20 C -8 24 0 26 16 D -30 10 -26 0 -16 E -24 20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1809: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (16) C E D B A (11) E A B C D (8) A E B D C (7) E A B D C (5) C E D A B (5) E C B A D (4) D C B A E (4) A B D E C (4) D C A B E (3) D B A C E (3) B A D C E (3) A D B C E (3) E C A B D (2) E A C D B (2) E A C B D (2) D B C A E (2) C E B D A (2) C D A B E (2) A B E D C (2) E C A D B (1) E B C D A (1) E B C A D (1) C D E B A (1) C D B E A (1) B D C A E (1) B C D A E (1) B A E D C (1) B A D E C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -16 -6 8 B 6 0 -12 -10 -4 C 16 12 0 20 16 D 6 10 -20 0 -8 E -8 4 -16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -6 8 B 6 0 -12 -10 -4 C 16 12 0 20 16 D 6 10 -20 0 -8 E -8 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=26 A=17 D=12 B=7 so B is eliminated. Round 2 votes counts: C=39 E=26 A=22 D=13 so D is eliminated. Round 3 votes counts: C=49 E=26 A=25 so A is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:232 D:194 E:194 A:190 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -6 8 B 6 0 -12 -10 -4 C 16 12 0 20 16 D 6 10 -20 0 -8 E -8 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -6 8 B 6 0 -12 -10 -4 C 16 12 0 20 16 D 6 10 -20 0 -8 E -8 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -6 8 B 6 0 -12 -10 -4 C 16 12 0 20 16 D 6 10 -20 0 -8 E -8 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1810: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) E C B A D (6) D A B C E (6) E C A B D (5) C B E D A (5) B C D E A (4) A E D B C (4) A D B C E (4) E D B A C (3) E A C D B (3) D B E C A (3) C E A B D (3) A E D C B (3) D B C A E (2) D B A C E (2) C E B D A (2) C E B A D (2) C B E A D (2) C B A D E (2) B D C E A (2) B C E D A (2) B C D A E (2) A D E B C (2) A D C E B (2) A D B E C (2) A C B D E (2) E D A B C (1) E B D C A (1) E A D C B (1) E A C B D (1) D E A B C (1) D A E B C (1) D A B E C (1) C B D A E (1) B E C D A (1) B D E C A (1) B D C A E (1) A E C D B (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -14 -4 -18 B 8 0 -6 12 -6 C 14 6 0 12 -2 D 4 -12 -12 0 -14 E 18 6 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -14 -4 -18 B 8 0 -6 12 -6 C 14 6 0 12 -2 D 4 -12 -12 0 -14 E 18 6 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=23 C=17 D=16 B=13 so B is eliminated. Round 2 votes counts: E=32 C=25 A=23 D=20 so D is eliminated. Round 3 votes counts: E=37 A=33 C=30 so C is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:215 B:204 D:183 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 -4 -18 B 8 0 -6 12 -6 C 14 6 0 12 -2 D 4 -12 -12 0 -14 E 18 6 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -4 -18 B 8 0 -6 12 -6 C 14 6 0 12 -2 D 4 -12 -12 0 -14 E 18 6 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -4 -18 B 8 0 -6 12 -6 C 14 6 0 12 -2 D 4 -12 -12 0 -14 E 18 6 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1811: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C A D E B (8) E B A D C (7) E A B D C (5) A C E D B (5) C E B A D (4) C B D A E (4) C A E D B (4) B D E A C (4) E B A C D (3) D A B C E (3) C D A B E (3) B E C D A (3) D B C A E (2) D B A E C (2) D A C B E (2) D A B E C (2) C E A B D (2) C A E B D (2) B E C A D (2) B C D E A (2) A D C E B (2) A C D E B (2) E C B A D (1) E B C A D (1) E A C B D (1) E A B C D (1) D C B A E (1) D A C E B (1) C D B A E (1) C B E A D (1) C A D B E (1) B E D C A (1) B E A D C (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 8 12 -4 B 6 0 6 16 -4 C -8 -6 0 6 4 D -12 -16 -6 0 -12 E 4 4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775501 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 -6 8 12 -4 B 6 0 6 16 -4 C -8 -6 0 6 4 D -12 -16 -6 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775413 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=26 E=19 D=13 A=12 so A is eliminated. Round 2 votes counts: C=37 B=26 E=20 D=17 so D is eliminated. Round 3 votes counts: C=43 B=35 E=22 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:208 A:205 C:198 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 12 -4 B 6 0 6 16 -4 C -8 -6 0 6 4 D -12 -16 -6 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775413 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 12 -4 B 6 0 6 16 -4 C -8 -6 0 6 4 D -12 -16 -6 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775413 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 12 -4 B 6 0 6 16 -4 C -8 -6 0 6 4 D -12 -16 -6 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775413 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1812: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) B C E A D (7) D A B C E (6) E B C A D (5) D A B E C (5) C E B D A (5) D A C B E (4) A E D B C (4) A D B E C (4) D A E C B (3) C B E D A (3) B C D A E (3) E C B A D (2) E A B D C (2) D C A B E (2) C E D A B (2) C D B E A (2) C D B A E (2) C D A E B (2) B E C A D (2) B A E D C (2) B A D E C (2) A D E B C (2) A B E D C (2) A B D E C (2) E B A D C (1) E B A C D (1) E A C D B (1) D C B A E (1) D B A C E (1) D A E B C (1) D A C E B (1) C E D B A (1) C E A D B (1) C D E B A (1) C D A B E (1) B E A D C (1) B D C A E (1) B D A C E (1) B C D E A (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -4 2 8 B 6 0 14 4 10 C 4 -14 0 0 12 D -2 -4 0 0 -4 E -8 -10 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 2 8 B 6 0 14 4 10 C 4 -14 0 0 12 D -2 -4 0 0 -4 E -8 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 B=20 A=16 E=12 so E is eliminated. Round 2 votes counts: C=30 B=27 D=24 A=19 so A is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:201 A:200 D:195 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 2 8 B 6 0 14 4 10 C 4 -14 0 0 12 D -2 -4 0 0 -4 E -8 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 2 8 B 6 0 14 4 10 C 4 -14 0 0 12 D -2 -4 0 0 -4 E -8 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 2 8 B 6 0 14 4 10 C 4 -14 0 0 12 D -2 -4 0 0 -4 E -8 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1813: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (14) D E A C B (11) E A D B C (8) B A C E D (7) D C E B A (6) B C A E D (6) E D A B C (5) C B A E D (5) C B A D E (4) C B D A E (3) B A E D C (3) A E B D C (3) A B E D C (3) A B C E D (3) D E C A B (2) C D B E A (2) A E D B C (2) D C B E A (1) D B C E A (1) C D E B A (1) C D E A B (1) C A E D B (1) C A D E B (1) C A B E D (1) B D C A E (1) B C D E A (1) B C D A E (1) B A E C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 24 0 -6 B -14 0 20 -12 -10 C -24 -20 0 -20 -6 D 0 12 20 0 0 E 6 10 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.331910 E: 0.668090 Sum of squares = 0.556508447535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.331910 E: 1.000000 A B C D E A 0 14 24 0 -6 B -14 0 20 -12 -10 C -24 -20 0 -20 -6 D 0 12 20 0 0 E 6 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=20 C=19 E=13 A=13 so E is eliminated. Round 2 votes counts: D=40 A=21 B=20 C=19 so C is eliminated. Round 3 votes counts: D=44 B=32 A=24 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:216 E:211 B:192 C:165 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 14 24 0 -6 B -14 0 20 -12 -10 C -24 -20 0 -20 -6 D 0 12 20 0 0 E 6 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 24 0 -6 B -14 0 20 -12 -10 C -24 -20 0 -20 -6 D 0 12 20 0 0 E 6 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 24 0 -6 B -14 0 20 -12 -10 C -24 -20 0 -20 -6 D 0 12 20 0 0 E 6 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1814: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) A C B E D (10) E D B C A (9) A C B D E (9) B E D C A (5) B E C D A (5) A C D B E (5) E B D C A (4) C A B E D (4) D E B A C (3) C B A E D (3) B C E D A (3) B C E A D (3) B A E C D (3) A D C E B (3) A C D E B (3) D E A C B (2) D E A B C (2) C B E D A (2) C A D B E (2) E D B A C (1) D C A E B (1) D A E C B (1) C D E B A (1) C A D E B (1) B E C A D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -10 -12 -2 -6 B 10 0 4 6 10 C 12 -4 0 10 0 D 2 -6 -10 0 -8 E 6 -10 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -2 -6 B 10 0 4 6 10 C 12 -4 0 10 0 D 2 -6 -10 0 -8 E 6 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=21 B=20 E=14 C=13 so C is eliminated. Round 2 votes counts: A=39 B=25 D=22 E=14 so E is eliminated. Round 3 votes counts: A=39 D=32 B=29 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:215 C:209 E:202 D:189 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -2 -6 B 10 0 4 6 10 C 12 -4 0 10 0 D 2 -6 -10 0 -8 E 6 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -2 -6 B 10 0 4 6 10 C 12 -4 0 10 0 D 2 -6 -10 0 -8 E 6 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -2 -6 B 10 0 4 6 10 C 12 -4 0 10 0 D 2 -6 -10 0 -8 E 6 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1815: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (5) C B E D A (5) E A D C B (4) E A B C D (4) D B A C E (4) B D C A E (4) B C E A D (4) A D B E C (4) E C D A B (3) E C A B D (3) D A E C B (3) D A B C E (3) C D B E A (3) B A E C D (3) A E D B C (3) A B D E C (3) E D C A B (2) E C B A D (2) E C A D B (2) D A C B E (2) C B D A E (2) B D A C E (2) B C D A E (2) B C A E D (2) A B E D C (2) E D A C B (1) E C B D A (1) E A C D B (1) E A C B D (1) E A B D C (1) D E C A B (1) D E A C B (1) D C B A E (1) D C A B E (1) D B C A E (1) C E D B A (1) C E B A D (1) C D E B A (1) C D B A E (1) C B D E A (1) B C D E A (1) B C A D E (1) B A D C E (1) B A C E D (1) A E B D C (1) A D E B C (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 -4 2 B 0 0 2 10 14 C 2 -2 0 4 4 D 4 -10 -4 0 -8 E -2 -14 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.388729 B: 0.611271 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.524762594501 Cumulative probabilities = A: 0.388729 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -4 2 B 0 0 2 10 14 C 2 -2 0 4 4 D 4 -10 -4 0 -8 E -2 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499792 B: 0.500208 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000086432 Cumulative probabilities = A: 0.499792 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=21 C=20 D=17 A=17 so D is eliminated. Round 2 votes counts: E=27 B=26 A=25 C=22 so C is eliminated. Round 3 votes counts: B=39 E=35 A=26 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:204 A:198 E:194 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -4 2 B 0 0 2 10 14 C 2 -2 0 4 4 D 4 -10 -4 0 -8 E -2 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499792 B: 0.500208 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000086432 Cumulative probabilities = A: 0.499792 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -4 2 B 0 0 2 10 14 C 2 -2 0 4 4 D 4 -10 -4 0 -8 E -2 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499792 B: 0.500208 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000086432 Cumulative probabilities = A: 0.499792 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -4 2 B 0 0 2 10 14 C 2 -2 0 4 4 D 4 -10 -4 0 -8 E -2 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499792 B: 0.500208 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000086432 Cumulative probabilities = A: 0.499792 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1816: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E C A D B (9) E B D A C (7) E C D A B (5) D A B C E (5) B E C A D (5) B D A C E (5) B C A D E (4) E D A B C (3) E B C D A (3) D A E C B (3) B A D C E (3) A D C E B (3) E D C A B (2) E D B A C (2) E B C A D (2) D A E B C (2) D A C E B (2) C E B A D (2) C E A D B (2) C B A D E (2) B E D A C (2) B D A E C (2) B C E A D (2) B C A E D (2) A D C B E (2) E C B D A (1) E C B A D (1) D E A C B (1) D A C B E (1) D A B E C (1) C E A B D (1) C A D E B (1) B E D C A (1) A B D C E (1) Total count = 100 A B C D E A 0 8 10 -16 -22 B -8 0 4 -8 -24 C -10 -4 0 -16 -24 D 16 8 16 0 -24 E 22 24 24 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 10 -16 -22 B -8 0 4 -8 -24 C -10 -4 0 -16 -24 D 16 8 16 0 -24 E 22 24 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=45 B=26 D=15 C=8 A=6 so A is eliminated. Round 2 votes counts: E=45 B=27 D=20 C=8 so C is eliminated. Round 3 votes counts: E=50 B=29 D=21 so D is eliminated. Round 4 votes counts: E=62 B=38 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:247 D:208 A:190 B:182 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 -16 -22 B -8 0 4 -8 -24 C -10 -4 0 -16 -24 D 16 8 16 0 -24 E 22 24 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 -16 -22 B -8 0 4 -8 -24 C -10 -4 0 -16 -24 D 16 8 16 0 -24 E 22 24 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 -16 -22 B -8 0 4 -8 -24 C -10 -4 0 -16 -24 D 16 8 16 0 -24 E 22 24 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1817: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) E B A D C (5) D B C A E (5) C E A D B (4) C D A B E (4) B E D A C (4) A E B D C (4) A B D E C (4) E C B A D (3) E A C B D (3) C D B A E (3) C A D E B (3) B D C E A (3) A E D B C (3) A D B C E (3) E C B D A (2) E C A D B (2) D B A C E (2) D A C B E (2) C D E B A (2) C D B E A (2) A D C B E (2) A D B E C (2) A C E D B (2) A B E D C (2) E C A B D (1) E B D A C (1) E B C D A (1) E B A C D (1) E A C D B (1) E A B C D (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B D A (1) C B E D A (1) C B D E A (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) B C D E A (1) B A E D C (1) B A D E C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 12 16 -4 B -8 0 12 4 -2 C -12 -12 0 -12 -8 D -16 -4 12 0 -10 E 4 2 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 12 16 -4 B -8 0 12 4 -2 C -12 -12 0 -12 -8 D -16 -4 12 0 -10 E 4 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=24 A=24 B=15 D=10 so D is eliminated. Round 2 votes counts: E=27 A=27 C=24 B=22 so B is eliminated. Round 3 votes counts: E=35 C=33 A=32 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:216 E:212 B:203 D:191 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 16 -4 B -8 0 12 4 -2 C -12 -12 0 -12 -8 D -16 -4 12 0 -10 E 4 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 16 -4 B -8 0 12 4 -2 C -12 -12 0 -12 -8 D -16 -4 12 0 -10 E 4 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 16 -4 B -8 0 12 4 -2 C -12 -12 0 -12 -8 D -16 -4 12 0 -10 E 4 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1818: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) C D E A B (7) B D A E C (7) D B A E C (6) E A C B D (5) C E D A B (5) C E A B D (5) D C B E A (4) D C B A E (4) D B A C E (4) B A E D C (4) A B E C D (4) E C A B D (3) D C E B A (3) B E A C D (3) A E B C D (3) A B E D C (3) E B A C D (2) D C E A B (2) D C A E B (2) C E D B A (2) C E B A D (2) C E A D B (2) C D E B A (2) B A E C D (2) D C A B E (1) D B C A E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 10 0 -14 B -8 0 -2 8 -14 C -10 2 0 18 -4 D 0 -8 -18 0 -14 E 14 14 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 10 0 -14 B -8 0 -2 8 -14 C -10 2 0 18 -4 D 0 -8 -18 0 -14 E 14 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 E=20 B=16 A=12 so A is eliminated. Round 2 votes counts: D=27 C=27 E=23 B=23 so E is eliminated. Round 3 votes counts: B=38 C=35 D=27 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:203 A:202 B:192 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 0 -14 B -8 0 -2 8 -14 C -10 2 0 18 -4 D 0 -8 -18 0 -14 E 14 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 -14 B -8 0 -2 8 -14 C -10 2 0 18 -4 D 0 -8 -18 0 -14 E 14 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 -14 B -8 0 -2 8 -14 C -10 2 0 18 -4 D 0 -8 -18 0 -14 E 14 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1819: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) D B A E C (10) E C D A B (8) D B E A C (8) E C D B A (7) A C E B D (6) E D C B A (5) E C A D B (5) E C A B D (5) D E B C A (5) B A D C E (5) B D A C E (4) A C B E D (4) A B C D E (4) D B E C A (3) C A E B D (3) B D A E C (2) A B D C E (2) E A B C D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -6 -4 -18 B -2 0 -12 -2 -14 C 6 12 0 12 -18 D 4 2 -12 0 -12 E 18 14 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -6 -4 -18 B -2 0 -12 -2 -14 C 6 12 0 12 -18 D 4 2 -12 0 -12 E 18 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=26 A=17 C=15 B=11 so B is eliminated. Round 2 votes counts: D=32 E=31 A=22 C=15 so C is eliminated. Round 3 votes counts: E=43 D=32 A=25 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:231 C:206 D:191 A:187 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -18 B -2 0 -12 -2 -14 C 6 12 0 12 -18 D 4 2 -12 0 -12 E 18 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -18 B -2 0 -12 -2 -14 C 6 12 0 12 -18 D 4 2 -12 0 -12 E 18 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -18 B -2 0 -12 -2 -14 C 6 12 0 12 -18 D 4 2 -12 0 -12 E 18 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1820: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) A E D C B (10) B C D E A (8) E D A B C (7) C B D A E (6) B D E A C (6) B D E C A (5) C B A E D (4) C B A D E (4) A E D B C (4) E A D B C (3) D E B A C (3) C A E B D (3) C A B E D (3) B D C E A (3) A C E D B (3) D B E A C (2) D A E B C (2) C A D E B (2) B E D C A (2) B E D A C (2) E D B A C (1) D B A E C (1) C A B D E (1) B C E D A (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 4 -8 2 14 B -4 0 0 -4 -4 C 8 0 0 -4 0 D -2 4 4 0 -12 E -14 4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.674393 D: 0.000000 E: 0.325607 Sum of squares = 0.560825546537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.674393 D: 0.674393 E: 1.000000 A B C D E A 0 4 -8 2 14 B -4 0 0 -4 -4 C 8 0 0 -4 0 D -2 4 4 0 -12 E -14 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.000000 E: 0.363636 Sum of squares = 0.537190092368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.636364 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=27 A=19 E=11 D=8 so D is eliminated. Round 2 votes counts: C=35 B=30 A=21 E=14 so E is eliminated. Round 3 votes counts: C=35 B=34 A=31 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:206 C:202 E:201 D:197 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 2 14 B -4 0 0 -4 -4 C 8 0 0 -4 0 D -2 4 4 0 -12 E -14 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.000000 E: 0.363636 Sum of squares = 0.537190092368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.636364 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 2 14 B -4 0 0 -4 -4 C 8 0 0 -4 0 D -2 4 4 0 -12 E -14 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.000000 E: 0.363636 Sum of squares = 0.537190092368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.636364 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 2 14 B -4 0 0 -4 -4 C 8 0 0 -4 0 D -2 4 4 0 -12 E -14 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.000000 E: 0.363636 Sum of squares = 0.537190092368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636364 D: 0.636364 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1821: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) C B E D A (6) E C A B D (5) E A D C B (5) E A C B D (5) C B D A E (4) A D B C E (4) E A C D B (3) D B A C E (3) C E B D A (3) C B A E D (3) E D C B A (2) D E B C A (2) D B C E A (2) D B C A E (2) D A E B C (2) C E B A D (2) C B E A D (2) B D C E A (2) B D C A E (2) B C D E A (2) B C D A E (2) A E D B C (2) A E C B D (2) A D E B C (2) A D B E C (2) A C E B D (2) A C B D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B A D (1) E A D B C (1) D E A B C (1) D B E C A (1) D B E A C (1) D A B E C (1) D A B C E (1) C B D E A (1) C B A D E (1) C A B E D (1) C A B D E (1) B D A C E (1) B C E D A (1) A E D C B (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -10 -2 -14 B 8 0 -20 16 -2 C 10 20 0 12 2 D 2 -16 -12 0 -14 E 14 2 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -2 -14 B 8 0 -20 16 -2 C 10 20 0 12 2 D 2 -16 -12 0 -14 E 14 2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=24 A=19 D=16 B=10 so B is eliminated. Round 2 votes counts: E=31 C=29 D=21 A=19 so A is eliminated. Round 3 votes counts: E=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:214 B:201 A:183 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 -2 -14 B 8 0 -20 16 -2 C 10 20 0 12 2 D 2 -16 -12 0 -14 E 14 2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -2 -14 B 8 0 -20 16 -2 C 10 20 0 12 2 D 2 -16 -12 0 -14 E 14 2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -2 -14 B 8 0 -20 16 -2 C 10 20 0 12 2 D 2 -16 -12 0 -14 E 14 2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1822: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (12) D C A B E (10) A B E C D (7) D C E B A (6) C D A B E (6) D B E A C (5) C A B E D (5) E B A D C (4) C D E B A (4) C A E B D (4) A B E D C (4) E A B C D (3) C E B A D (3) C D A E B (3) B E A D C (3) E C B A D (2) E B C A D (2) D C B A E (2) C D E A B (2) E D B C A (1) E D B A C (1) D E C B A (1) D E B C A (1) D E B A C (1) D C B E A (1) D A C B E (1) C E A B D (1) C A D B E (1) B D E A C (1) B A E D C (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -10 8 -8 B 2 0 -6 6 -4 C 10 6 0 12 0 D -8 -6 -12 0 -8 E 8 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.391702 D: 0.000000 E: 0.608298 Sum of squares = 0.523456892656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.391702 D: 0.391702 E: 1.000000 A B C D E A 0 -2 -10 8 -8 B 2 0 -6 6 -4 C 10 6 0 12 0 D -8 -6 -12 0 -8 E 8 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=28 E=25 A=13 B=5 so B is eliminated. Round 2 votes counts: D=29 C=29 E=28 A=14 so A is eliminated. Round 3 votes counts: E=40 D=30 C=30 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:210 B:199 A:194 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 8 -8 B 2 0 -6 6 -4 C 10 6 0 12 0 D -8 -6 -12 0 -8 E 8 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 8 -8 B 2 0 -6 6 -4 C 10 6 0 12 0 D -8 -6 -12 0 -8 E 8 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 8 -8 B 2 0 -6 6 -4 C 10 6 0 12 0 D -8 -6 -12 0 -8 E 8 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1823: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) D E A B C (9) C B A E D (7) B D C A E (6) B C D A E (6) C B E A D (4) B C A E D (4) B C A D E (4) E C A B D (3) E A C D B (3) D E A C B (3) D A E B C (3) C E A B D (3) C A B E D (3) E C B A D (2) E A D C B (2) D B C A E (2) D B A E C (2) D A B E C (2) B C E D A (2) B C E A D (2) B C D E A (2) A E D C B (2) E C D A B (1) E C B D A (1) E B C D A (1) E A C B D (1) D E B C A (1) D B E A C (1) D B C E A (1) D B A C E (1) D A E C B (1) C E B A D (1) C A E B D (1) A E C D B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -14 -10 -6 B 0 0 2 8 2 C 14 -2 0 4 2 D 10 -8 -4 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.068065 B: 0.931935 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.873136059873 Cumulative probabilities = A: 0.068065 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -10 -6 B 0 0 2 8 2 C 14 -2 0 4 2 D 10 -8 -4 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250006506 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 E=24 C=19 A=5 so A is eliminated. Round 2 votes counts: E=27 D=27 B=27 C=19 so C is eliminated. Round 3 votes counts: B=41 E=32 D=27 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:206 E:205 D:195 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -14 -10 -6 B 0 0 2 8 2 C 14 -2 0 4 2 D 10 -8 -4 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250006506 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -10 -6 B 0 0 2 8 2 C 14 -2 0 4 2 D 10 -8 -4 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250006506 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -10 -6 B 0 0 2 8 2 C 14 -2 0 4 2 D 10 -8 -4 0 -8 E 6 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250006506 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1824: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (15) C D A E B (14) D A E C B (9) E A B D C (8) D C A E B (7) C B E A D (6) D A E B C (5) B C E A D (5) D A C E B (4) C B D E A (3) B E A C D (3) C A E D B (2) B E C A D (2) A D E B C (2) E B A D C (1) E A C B D (1) D C B A E (1) D B C A E (1) D A C B E (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A B E (1) C A D E B (1) B E D A C (1) B D E A C (1) B C E D A (1) B C D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 16 4 -4 0 B -16 0 -6 -4 -14 C -4 6 0 -14 2 D 4 4 14 0 6 E 0 14 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 -4 0 B -16 0 -6 -4 -14 C -4 6 0 -14 2 D 4 4 14 0 6 E 0 14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=29 D=28 E=10 A=3 so A is eliminated. Round 2 votes counts: D=30 C=30 B=29 E=11 so E is eliminated. Round 3 votes counts: B=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:214 A:208 E:203 C:195 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 4 -4 0 B -16 0 -6 -4 -14 C -4 6 0 -14 2 D 4 4 14 0 6 E 0 14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 -4 0 B -16 0 -6 -4 -14 C -4 6 0 -14 2 D 4 4 14 0 6 E 0 14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 -4 0 B -16 0 -6 -4 -14 C -4 6 0 -14 2 D 4 4 14 0 6 E 0 14 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1825: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C A D (8) E C B D A (5) E B D A C (5) E B C A D (5) E B D C A (4) E B C D A (4) D A E C B (4) A D C B E (4) C D E A B (3) C D A E B (3) C A D E B (3) B E A C D (3) B A C D E (3) E D B A C (2) D A E B C (2) D A B E C (2) C E B D A (2) C E B A D (2) C A D B E (2) C A B D E (2) B E D A C (2) B E A D C (2) B A E D C (2) A D B E C (2) A D B C E (2) A C D B E (2) E D C A B (1) E D A B C (1) D C A E B (1) D A C B E (1) C E D A B (1) B C E A D (1) B A D C E (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 -8 -2 B 4 0 6 6 -18 C -6 -6 0 0 -8 D 8 -6 0 0 -2 E 2 18 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 6 -8 -2 B 4 0 6 6 -18 C -6 -6 0 0 -8 D 8 -6 0 0 -2 E 2 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 D=21 C=18 A=11 so A is eliminated. Round 2 votes counts: D=29 E=27 B=24 C=20 so C is eliminated. Round 3 votes counts: D=42 E=32 B=26 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:200 B:199 A:196 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 -8 -2 B 4 0 6 6 -18 C -6 -6 0 0 -8 D 8 -6 0 0 -2 E 2 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -8 -2 B 4 0 6 6 -18 C -6 -6 0 0 -8 D 8 -6 0 0 -2 E 2 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -8 -2 B 4 0 6 6 -18 C -6 -6 0 0 -8 D 8 -6 0 0 -2 E 2 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1826: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D C E A B (8) D B A E C (8) B A E C D (8) C E B A D (7) D A C E B (5) B E A C D (5) D B C E A (4) C E A B D (4) B D A E C (4) D C E B A (3) B E C A D (3) D C B A E (2) D C A B E (2) C E A D B (2) C D E A B (2) B D E C A (2) B A E D C (2) B A D E C (2) A E B C D (2) A D E C B (2) A D E B C (2) A B E C D (2) E C A B D (1) D B C A E (1) D A E B C (1) D A C B E (1) C B E A D (1) B D E A C (1) A E C D B (1) A D B E C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 16 12 14 B 6 0 -2 8 0 C -16 2 0 -4 -14 D -12 -8 4 0 2 E -14 0 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.756540 C: 0.000000 D: 0.000000 E: 0.243460 Sum of squares = 0.631625373202 Cumulative probabilities = A: 0.000000 B: 0.756540 C: 0.756540 D: 0.756540 E: 1.000000 A B C D E A 0 -6 16 12 14 B 6 0 -2 8 0 C -16 2 0 -4 -14 D -12 -8 4 0 2 E -14 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000011028 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=27 A=21 C=16 E=1 so E is eliminated. Round 2 votes counts: D=35 B=27 A=21 C=17 so C is eliminated. Round 3 votes counts: D=37 B=35 A=28 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:218 B:206 E:199 D:193 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 16 12 14 B 6 0 -2 8 0 C -16 2 0 -4 -14 D -12 -8 4 0 2 E -14 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000011028 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 12 14 B 6 0 -2 8 0 C -16 2 0 -4 -14 D -12 -8 4 0 2 E -14 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000011028 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 12 14 B 6 0 -2 8 0 C -16 2 0 -4 -14 D -12 -8 4 0 2 E -14 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.580000011028 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1827: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (14) E D C A B (10) D E A C B (8) D E A B C (7) C B A E D (7) E D C B A (5) C B E A D (5) B A C D E (4) C E B D A (3) C A B E D (3) A D E C B (3) A B C E D (3) A B C D E (3) E C D B A (2) C E B A D (2) C B E D A (2) B C E D A (2) A D B E C (2) A C B E D (2) E C B D A (1) D E C B A (1) D E C A B (1) D E B A C (1) D B A E C (1) D A E B C (1) D A B E C (1) C E A D B (1) B E D C A (1) B D A E C (1) B C E A D (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -10 -22 4 -6 B 10 0 -12 12 8 C 22 12 0 12 8 D -4 -12 -12 0 -30 E 6 -8 -8 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 4 -6 B 10 0 -12 12 8 C 22 12 0 12 8 D -4 -12 -12 0 -30 E 6 -8 -8 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 D=21 E=18 A=13 so A is eliminated. Round 2 votes counts: B=31 D=26 C=25 E=18 so E is eliminated. Round 3 votes counts: D=41 B=31 C=28 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:227 E:210 B:209 A:183 D:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -22 4 -6 B 10 0 -12 12 8 C 22 12 0 12 8 D -4 -12 -12 0 -30 E 6 -8 -8 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 4 -6 B 10 0 -12 12 8 C 22 12 0 12 8 D -4 -12 -12 0 -30 E 6 -8 -8 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 4 -6 B 10 0 -12 12 8 C 22 12 0 12 8 D -4 -12 -12 0 -30 E 6 -8 -8 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1828: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) B E D A C (9) E D A B C (8) B E C D A (7) E B D A C (6) A D C E B (6) E D B A C (5) C A B D E (5) D A E C B (4) C B A E D (4) C B A D E (4) B C E A D (4) A C D E B (3) A C D B E (3) D E A C B (2) D A C E B (2) C B E D A (2) C A D B E (2) B E D C A (2) B C A E D (2) A D E C B (2) D E C A B (1) D E A B C (1) D A E B C (1) C D E A B (1) B E A D C (1) B C E D A (1) B A E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 12 -4 0 B -4 0 0 0 -2 C -12 0 0 -4 -2 D 4 0 4 0 -4 E 0 2 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.264603 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.735397 Sum of squares = 0.610823006562 Cumulative probabilities = A: 0.264603 B: 0.264603 C: 0.264603 D: 0.264603 E: 1.000000 A B C D E A 0 4 12 -4 0 B -4 0 0 0 -2 C -12 0 0 -4 -2 D 4 0 4 0 -4 E 0 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500220 Sum of squares = 0.500000097212 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 0.499780 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=27 B=27 E=19 A=16 D=11 so D is eliminated. Round 2 votes counts: C=27 B=27 E=23 A=23 so E is eliminated. Round 3 votes counts: B=38 A=34 C=28 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:204 D:202 B:197 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 12 -4 0 B -4 0 0 0 -2 C -12 0 0 -4 -2 D 4 0 4 0 -4 E 0 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500220 Sum of squares = 0.500000097212 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 0.499780 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 -4 0 B -4 0 0 0 -2 C -12 0 0 -4 -2 D 4 0 4 0 -4 E 0 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500220 Sum of squares = 0.500000097212 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 0.499780 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 -4 0 B -4 0 0 0 -2 C -12 0 0 -4 -2 D 4 0 4 0 -4 E 0 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499780 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500220 Sum of squares = 0.500000097212 Cumulative probabilities = A: 0.499780 B: 0.499780 C: 0.499780 D: 0.499780 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1829: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) C B A E D (8) B C D A E (7) B D C A E (6) A E C D B (5) A C E B D (5) E A D C B (4) C B E A D (4) B D C E A (4) D E A B C (3) D B A C E (3) C A E B D (3) B C D E A (3) A E D C B (3) A E C B D (3) E C A D B (2) E A C B D (2) D B E A C (2) D B A E C (2) D A E B C (2) B C A D E (2) E D C A B (1) E D A B C (1) E C A B D (1) D E B C A (1) D E B A C (1) D B C E A (1) D B C A E (1) C E B D A (1) C E B A D (1) C E A B D (1) C A B E D (1) B D A C E (1) B C E D A (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 2 18 8 B -2 0 -14 8 -6 C -2 14 0 24 8 D -18 -8 -24 0 -18 E -8 6 -8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 18 8 B -2 0 -14 8 -6 C -2 14 0 24 8 D -18 -8 -24 0 -18 E -8 6 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 C=19 A=18 D=16 so D is eliminated. Round 2 votes counts: B=33 E=28 A=20 C=19 so C is eliminated. Round 3 votes counts: B=45 E=31 A=24 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:222 A:215 E:204 B:193 D:166 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 18 8 B -2 0 -14 8 -6 C -2 14 0 24 8 D -18 -8 -24 0 -18 E -8 6 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 18 8 B -2 0 -14 8 -6 C -2 14 0 24 8 D -18 -8 -24 0 -18 E -8 6 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 18 8 B -2 0 -14 8 -6 C -2 14 0 24 8 D -18 -8 -24 0 -18 E -8 6 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1830: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) E A D C B (8) A B C D E (8) D C B A E (6) B C D A E (6) D C B E A (5) C D B E A (5) A E B C D (5) E D C A B (4) E A B D C (4) D C E B A (4) C B D A E (4) C D B A E (3) A E B D C (3) A B E C D (3) A B C E D (3) E C D B A (2) E A D B C (2) C D E B A (2) B D C A E (2) B A C D E (2) A D B C E (2) A B D C E (2) E D A C B (1) E A C D B (1) E A C B D (1) C E D B A (1) B C E A D (1) B C A D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -8 -6 2 B 4 0 -10 -8 8 C 8 10 0 -4 14 D 6 8 4 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -6 2 B 4 0 -10 -8 8 C 8 10 0 -4 14 D 6 8 4 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=27 D=15 C=15 B=12 so B is eliminated. Round 2 votes counts: E=31 A=29 C=23 D=17 so D is eliminated. Round 3 votes counts: C=40 E=31 A=29 so A is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:212 B:197 A:192 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 2 B 4 0 -10 -8 8 C 8 10 0 -4 14 D 6 8 4 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 2 B 4 0 -10 -8 8 C 8 10 0 -4 14 D 6 8 4 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 2 B 4 0 -10 -8 8 C 8 10 0 -4 14 D 6 8 4 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1831: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) D E A C B (6) A B C E D (6) E B A C D (5) E A D B C (5) E A B D C (5) D E C B A (5) C B D A E (5) E D C B A (4) E D A B C (4) B A C E D (4) A B E C D (4) D C E B A (3) C E B D A (3) C D E B A (3) C D B A E (3) C B A D E (3) B C A E D (3) B A C D E (3) A B D E C (3) A D B E C (2) E D C A B (1) E D A C B (1) E C D B A (1) E C B D A (1) E A B C D (1) D E A B C (1) D C E A B (1) D C A B E (1) C D B E A (1) C B E A D (1) C B D E A (1) C B A E D (1) C A D B E (1) B C A D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 -2 -18 B 2 0 -4 2 -12 C -2 4 0 4 -10 D 2 -2 -4 0 0 E 18 12 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.491004 E: 0.508996 Sum of squares = 0.500161831284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.491004 E: 1.000000 A B C D E A 0 -2 2 -2 -18 B 2 0 -4 2 -12 C -2 4 0 4 -10 D 2 -2 -4 0 0 E 18 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 C=22 A=16 B=11 so B is eliminated. Round 2 votes counts: E=28 C=26 D=23 A=23 so D is eliminated. Round 3 votes counts: E=46 C=31 A=23 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:198 D:198 B:194 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -2 -18 B 2 0 -4 2 -12 C -2 4 0 4 -10 D 2 -2 -4 0 0 E 18 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 -18 B 2 0 -4 2 -12 C -2 4 0 4 -10 D 2 -2 -4 0 0 E 18 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 -18 B 2 0 -4 2 -12 C -2 4 0 4 -10 D 2 -2 -4 0 0 E 18 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1832: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) D E A B C (6) D B A E C (5) C B A E D (5) C E D A B (4) B C A E D (4) B A C E D (4) E D A C B (3) E C A D B (3) C A E B D (3) B C D A E (3) E D C A B (2) E D A B C (2) E A C D B (2) E A C B D (2) D E C A B (2) D C B E A (2) D B C A E (2) D B A C E (2) C D B E A (2) C B D E A (2) B D C A E (2) B D A E C (2) B C A D E (2) B A E C D (2) A E C B D (2) A E B D C (2) A C E B D (2) A B E C D (2) E A D C B (1) E A D B C (1) D E C B A (1) D C E A B (1) D B E C A (1) D B C E A (1) D A E B C (1) C E B A D (1) C B E A D (1) C A B E D (1) B D A C E (1) B A D E C (1) B A D C E (1) A E B C D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 6 4 B -6 0 0 12 0 C 6 0 0 14 10 D -6 -12 -14 0 -20 E -4 0 -10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363013 C: 0.636987 D: 0.000000 E: 0.000000 Sum of squares = 0.53753069607 Cumulative probabilities = A: 0.000000 B: 0.363013 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 6 4 B -6 0 0 12 0 C 6 0 0 14 10 D -6 -12 -14 0 -20 E -4 0 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499613 C: 0.500387 D: 0.000000 E: 0.000000 Sum of squares = 0.500000299119 Cumulative probabilities = A: 0.000000 B: 0.499613 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=24 B=22 E=16 A=11 so A is eliminated. Round 2 votes counts: C=29 B=26 D=24 E=21 so E is eliminated. Round 3 votes counts: C=38 D=33 B=29 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:205 B:203 E:203 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 6 4 B -6 0 0 12 0 C 6 0 0 14 10 D -6 -12 -14 0 -20 E -4 0 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499613 C: 0.500387 D: 0.000000 E: 0.000000 Sum of squares = 0.500000299119 Cumulative probabilities = A: 0.000000 B: 0.499613 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 6 4 B -6 0 0 12 0 C 6 0 0 14 10 D -6 -12 -14 0 -20 E -4 0 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499613 C: 0.500387 D: 0.000000 E: 0.000000 Sum of squares = 0.500000299119 Cumulative probabilities = A: 0.000000 B: 0.499613 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 6 4 B -6 0 0 12 0 C 6 0 0 14 10 D -6 -12 -14 0 -20 E -4 0 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499613 C: 0.500387 D: 0.000000 E: 0.000000 Sum of squares = 0.500000299119 Cumulative probabilities = A: 0.000000 B: 0.499613 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1833: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (12) E B C A D (9) C D A B E (9) E B A D C (6) D A E B C (6) B E C A D (6) D A C E B (5) D C A B E (4) A D E B C (4) E A B D C (3) C D B A E (3) C B D E A (3) E C B D A (2) E B A C D (2) C E B D A (2) C B E D A (2) C B E A D (2) C B D A E (2) C A D B E (2) B C E A D (2) A D B E C (2) E D A C B (1) E A D B C (1) D E A C B (1) D A B E C (1) C D E B A (1) C D E A B (1) C B A E D (1) C B A D E (1) B E A C D (1) B C A E D (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 8 -6 -10 10 B -8 0 -8 -8 10 C 6 8 0 4 8 D 10 8 -4 0 16 E -10 -10 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 -10 10 B -8 0 -8 -8 10 C 6 8 0 4 8 D 10 8 -4 0 16 E -10 -10 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=29 C=29 E=24 B=10 A=8 so A is eliminated. Round 2 votes counts: D=36 C=29 E=25 B=10 so B is eliminated. Round 3 votes counts: D=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:215 C:213 A:201 B:193 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 -10 10 B -8 0 -8 -8 10 C 6 8 0 4 8 D 10 8 -4 0 16 E -10 -10 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -10 10 B -8 0 -8 -8 10 C 6 8 0 4 8 D 10 8 -4 0 16 E -10 -10 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -10 10 B -8 0 -8 -8 10 C 6 8 0 4 8 D 10 8 -4 0 16 E -10 -10 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1834: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) D C A B E (7) E B A C D (6) D B A C E (6) B D A E C (6) D B C A E (4) C E D B A (4) B A E D C (4) A D C B E (4) E C A B D (3) E B A D C (3) D A C B E (3) D A B C E (3) C E D A B (3) C D E B A (3) B A D E C (3) A B E D C (3) E B D C A (2) E B C D A (2) C D E A B (2) B D E C A (2) A B D E C (2) E C B D A (1) E C A D B (1) E B C A D (1) E A B D C (1) C E A D B (1) C D B E A (1) C D B A E (1) C D A E B (1) C D A B E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D A C E (1) A E C D B (1) A D C E B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 16 -8 10 B 18 0 20 0 26 C -16 -20 0 -32 0 D 8 0 32 0 10 E -10 -26 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.449541 C: 0.000000 D: 0.550459 E: 0.000000 Sum of squares = 0.505092223159 Cumulative probabilities = A: 0.000000 B: 0.449541 C: 0.449541 D: 1.000000 E: 1.000000 A B C D E A 0 -18 16 -8 10 B 18 0 20 0 26 C -16 -20 0 -32 0 D 8 0 32 0 10 E -10 -26 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 E=20 C=19 A=13 so A is eliminated. Round 2 votes counts: B=31 D=29 E=21 C=19 so C is eliminated. Round 3 votes counts: D=40 B=31 E=29 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:232 D:225 A:200 E:177 C:166 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 16 -8 10 B 18 0 20 0 26 C -16 -20 0 -32 0 D 8 0 32 0 10 E -10 -26 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 16 -8 10 B 18 0 20 0 26 C -16 -20 0 -32 0 D 8 0 32 0 10 E -10 -26 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 16 -8 10 B 18 0 20 0 26 C -16 -20 0 -32 0 D 8 0 32 0 10 E -10 -26 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1835: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (14) E B D C A (12) A D C E B (7) B E A C D (6) B E D C A (5) A C D E B (5) D E C B A (4) C A D B E (4) C D A B E (3) A E D C B (3) A C B D E (3) A B C E D (3) E A D B C (2) E A B D C (2) B E C D A (2) A E B C D (2) A D E C B (2) E D B C A (1) E B A C D (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E B A (1) D C B E A (1) D C A E B (1) D A C E B (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 10 18 10 B -12 0 -10 -8 4 C -10 10 0 8 4 D -18 8 -8 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 18 10 B -12 0 -10 -8 4 C -10 10 0 8 4 D -18 8 -8 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 B=19 E=18 D=11 C=11 so D is eliminated. Round 2 votes counts: A=42 E=25 B=19 C=14 so C is eliminated. Round 3 votes counts: A=50 E=26 B=24 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:206 D:197 B:187 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 18 10 B -12 0 -10 -8 4 C -10 10 0 8 4 D -18 8 -8 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 18 10 B -12 0 -10 -8 4 C -10 10 0 8 4 D -18 8 -8 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 18 10 B -12 0 -10 -8 4 C -10 10 0 8 4 D -18 8 -8 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1836: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (17) B A C E D (11) D E C B A (10) D C E B A (5) D C E A B (5) E D A B C (4) B A C D E (4) B A E D C (3) A B E C D (3) A B C E D (3) E D C A B (2) D E B C A (2) C B A D E (2) B A E C D (2) B A D E C (2) E D A C B (1) E C D A B (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E B A C (1) D E A C B (1) D B E A C (1) D B C E A (1) D B C A E (1) C E D A B (1) C D E B A (1) C D B E A (1) C D B A E (1) C B D A E (1) C A E B D (1) C A B E D (1) B D C A E (1) B D A C E (1) B C A D E (1) A E B C D (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -10 -18 -16 B 4 0 -10 -16 -16 C 10 10 0 -22 -12 D 18 16 22 0 18 E 16 16 12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -18 -16 B 4 0 -10 -16 -16 C 10 10 0 -22 -12 D 18 16 22 0 18 E 16 16 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 B=25 E=12 A=10 C=9 so C is eliminated. Round 2 votes counts: D=47 B=28 E=13 A=12 so A is eliminated. Round 3 votes counts: D=47 B=37 E=16 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:237 E:213 C:193 B:181 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -18 -16 B 4 0 -10 -16 -16 C 10 10 0 -22 -12 D 18 16 22 0 18 E 16 16 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -18 -16 B 4 0 -10 -16 -16 C 10 10 0 -22 -12 D 18 16 22 0 18 E 16 16 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -18 -16 B 4 0 -10 -16 -16 C 10 10 0 -22 -12 D 18 16 22 0 18 E 16 16 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1837: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (15) B C A D E (9) E B A C D (6) B C A E D (6) E D B A C (4) E A D C B (4) D E A C B (4) D A C B E (4) D C A E B (3) D C A B E (3) D A C E B (3) A C D B E (3) E D A B C (2) E B D C A (2) E A C D B (2) D E B C A (2) D B C A E (2) B E C A D (2) B D C A E (2) B C E A D (2) A E C D B (2) A C D E B (2) E B D A C (1) E B C A D (1) E B A D C (1) E A C B D (1) D C B A E (1) D B E C A (1) D A E C B (1) C B A D E (1) C A B D E (1) B E D C A (1) B E C D A (1) B D E C A (1) B C E D A (1) A D E C B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 16 -8 -8 B -6 0 -6 -20 -16 C -16 6 0 -18 -10 D 8 20 18 0 -10 E 8 16 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 16 -8 -8 B -6 0 -6 -20 -16 C -16 6 0 -18 -10 D 8 20 18 0 -10 E 8 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=25 D=24 A=10 C=2 so C is eliminated. Round 2 votes counts: E=39 B=26 D=24 A=11 so A is eliminated. Round 3 votes counts: E=42 D=31 B=27 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:218 A:203 C:181 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 16 -8 -8 B -6 0 -6 -20 -16 C -16 6 0 -18 -10 D 8 20 18 0 -10 E 8 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 -8 -8 B -6 0 -6 -20 -16 C -16 6 0 -18 -10 D 8 20 18 0 -10 E 8 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 -8 -8 B -6 0 -6 -20 -16 C -16 6 0 -18 -10 D 8 20 18 0 -10 E 8 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1838: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) E D C A B (6) C B E D A (6) A E D C B (5) D E C A B (4) D A B E C (4) C E B D A (4) B C D E A (4) B C A E D (4) A E D B C (4) E C D B A (3) C B D E A (3) A B D E C (3) E D A C B (2) D E A C B (2) D C E B A (2) D B C E A (2) C E D B A (2) C E B A D (2) C D E B A (2) B C E A D (2) B C D A E (2) B C A D E (2) B A C D E (2) E D C B A (1) E C D A B (1) E C A D B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E A B C (1) D B A C E (1) D A E C B (1) D A E B C (1) C B E A D (1) B D C A E (1) B D A C E (1) B C E D A (1) A E C B D (1) A E B C D (1) A D B E C (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -14 -16 -10 B -2 0 -4 -14 -14 C 14 4 0 -8 -8 D 16 14 8 0 -2 E 10 14 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -14 -16 -10 B -2 0 -4 -14 -14 C 14 4 0 -8 -8 D 16 14 8 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=20 D=19 B=19 E=16 so E is eliminated. Round 2 votes counts: D=28 A=28 C=25 B=19 so B is eliminated. Round 3 votes counts: C=40 D=30 A=30 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:218 E:217 C:201 B:183 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -14 -16 -10 B -2 0 -4 -14 -14 C 14 4 0 -8 -8 D 16 14 8 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -16 -10 B -2 0 -4 -14 -14 C 14 4 0 -8 -8 D 16 14 8 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -16 -10 B -2 0 -4 -14 -14 C 14 4 0 -8 -8 D 16 14 8 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1839: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (7) E B A D C (6) C D A B E (6) B E D C A (6) A C E D B (6) E B D C A (5) B E D A C (5) B D C E A (5) D C B E A (4) A E B C D (4) A B E C D (4) D C B A E (3) D B C E A (3) C D A E B (3) A E C B D (3) A C E B D (3) A C D E B (3) A B C E D (3) E D C B A (2) E C D A B (2) D C E B A (2) D C A B E (2) C D E A B (2) C A D E B (2) C A D B E (2) A C B D E (2) E C A D B (1) E B D A C (1) E A C B D (1) C E D A B (1) A E C D B (1) Total count = 100 A B C D E A 0 2 -2 -4 -6 B -2 0 -2 10 4 C 2 2 0 -2 4 D 4 -10 2 0 -22 E 6 -4 -4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408605 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -4 -6 B -2 0 -2 10 4 C 2 2 0 -2 4 D 4 -10 2 0 -22 E 6 -4 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040822 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 E=18 C=16 D=14 so D is eliminated. Round 2 votes counts: A=29 C=27 B=26 E=18 so E is eliminated. Round 3 votes counts: B=38 C=32 A=30 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:210 B:205 C:203 A:195 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 -4 -6 B -2 0 -2 10 4 C 2 2 0 -2 4 D 4 -10 2 0 -22 E 6 -4 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040822 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 -6 B -2 0 -2 10 4 C 2 2 0 -2 4 D 4 -10 2 0 -22 E 6 -4 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040822 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 -6 B -2 0 -2 10 4 C 2 2 0 -2 4 D 4 -10 2 0 -22 E 6 -4 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040822 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1840: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) D B A E C (7) D B A C E (6) E C A B D (5) D B E C A (5) E C A D B (4) D A E B C (4) B D C A E (4) A C E D B (4) D B E A C (3) C E B A D (3) C E A B D (3) B D E C A (3) B C E A D (3) A D C B E (3) E C B D A (2) D A E C B (2) D A B E C (2) C B E A D (2) C A E B D (2) A E C D B (2) A B C D E (2) E D B C A (1) E C B A D (1) E A C D B (1) D E C B A (1) D E A C B (1) D A B C E (1) C B A E D (1) B E D C A (1) B E C D A (1) B D C E A (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) A D E C B (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -22 6 -18 16 B 22 0 22 2 24 C -6 -22 0 -18 6 D 18 -2 18 0 24 E -16 -24 -6 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 6 -18 16 B 22 0 22 2 24 C -6 -22 0 -18 6 D 18 -2 18 0 24 E -16 -24 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995307 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 E=14 A=14 C=11 so C is eliminated. Round 2 votes counts: D=32 B=32 E=20 A=16 so A is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:235 D:229 A:191 C:180 E:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 6 -18 16 B 22 0 22 2 24 C -6 -22 0 -18 6 D 18 -2 18 0 24 E -16 -24 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995307 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 6 -18 16 B 22 0 22 2 24 C -6 -22 0 -18 6 D 18 -2 18 0 24 E -16 -24 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995307 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 6 -18 16 B 22 0 22 2 24 C -6 -22 0 -18 6 D 18 -2 18 0 24 E -16 -24 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995307 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1841: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) D C E A B (8) D C A B E (6) D A C B E (6) E B C A D (5) E B A C D (5) D C A E B (5) B A E C D (5) E C B A D (3) C D A E B (3) B E D A C (3) B E A D C (3) E D B C A (2) E B D C A (2) D B E C A (2) D A B C E (2) C E A D B (2) C A E D B (2) C A D E B (2) B D A E C (2) A C D B E (2) E C D B A (1) E C A B D (1) E B D A C (1) E B C D A (1) D E B C A (1) D C E B A (1) D C B A E (1) D B E A C (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E A B (1) C A E B D (1) B A E D C (1) B A D E C (1) A D C B E (1) A D B C E (1) A C E B D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 -2 -6 B 4 0 2 -4 2 C 2 -2 0 -4 0 D 2 4 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999973 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -4 -2 -2 -6 B 4 0 2 -4 2 C 2 -2 0 -4 0 D 2 4 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=24 E=21 C=12 A=8 so A is eliminated. Round 2 votes counts: D=37 B=26 E=21 C=16 so C is eliminated. Round 3 votes counts: D=45 E=28 B=27 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:204 E:203 B:202 C:198 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 -6 B 4 0 2 -4 2 C 2 -2 0 -4 0 D 2 4 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 -6 B 4 0 2 -4 2 C 2 -2 0 -4 0 D 2 4 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 -6 B 4 0 2 -4 2 C 2 -2 0 -4 0 D 2 4 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1842: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (12) A D C B E (11) D A C B E (8) E B C A D (7) B E C D A (6) A D E C B (6) E B A C D (5) C B E D A (4) A D C E B (4) D C A B E (3) C B D E A (3) A E D B C (3) E B D C A (2) E A D B C (2) C D A B E (2) C A B D E (2) B C E D A (2) A D E B C (2) A C D B E (2) E D B C A (1) E D B A C (1) E C B A D (1) E B D A C (1) E B A D C (1) D B E C A (1) D A E B C (1) C D B A E (1) C B E A D (1) C A D B E (1) C A B E D (1) B C D E A (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -2 2 -2 B 0 0 -4 -2 -2 C 2 4 0 6 -4 D -2 2 -6 0 -2 E 2 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 2 -2 B 0 0 -4 -2 -2 C 2 4 0 6 -4 D -2 2 -6 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=30 C=15 D=13 B=9 so B is eliminated. Round 2 votes counts: E=39 A=30 C=18 D=13 so D is eliminated. Round 3 votes counts: E=40 A=39 C=21 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:205 C:204 A:199 B:196 D:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 2 -2 B 0 0 -4 -2 -2 C 2 4 0 6 -4 D -2 2 -6 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 -2 B 0 0 -4 -2 -2 C 2 4 0 6 -4 D -2 2 -6 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 -2 B 0 0 -4 -2 -2 C 2 4 0 6 -4 D -2 2 -6 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1843: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) B E A C D (6) A C D B E (6) A B C D E (6) C A D E B (5) B A E C D (5) E D C B A (4) D E C A B (4) B A E D C (4) E D B C A (3) E B D C A (3) D E C B A (3) D C E A B (3) D A C E B (3) C D E A B (3) B E D A C (3) B E C A D (3) B E A D C (3) B A D E C (3) B A C E D (3) E B C D A (2) C A E B D (2) A D C E B (2) A D B C E (2) A C B D E (2) E C D B A (1) E C B D A (1) D E B A C (1) C E D A B (1) C D A E B (1) B C E A D (1) A D B E C (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 6 14 10 B -2 0 0 -2 0 C -6 0 0 2 0 D -14 2 -2 0 6 E -10 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 14 10 B -2 0 0 -2 0 C -6 0 0 2 0 D -14 2 -2 0 6 E -10 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=22 D=21 E=14 C=12 so C is eliminated. Round 2 votes counts: B=31 A=29 D=25 E=15 so E is eliminated. Round 3 votes counts: B=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:216 B:198 C:198 D:196 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 14 10 B -2 0 0 -2 0 C -6 0 0 2 0 D -14 2 -2 0 6 E -10 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 14 10 B -2 0 0 -2 0 C -6 0 0 2 0 D -14 2 -2 0 6 E -10 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 14 10 B -2 0 0 -2 0 C -6 0 0 2 0 D -14 2 -2 0 6 E -10 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1844: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) D A C E B (8) A E D B C (6) B C E D A (5) E B C A D (4) D C A B E (4) C B E D A (4) C B D E A (4) E B A C D (3) D C A E B (3) C D A B E (3) B E A C D (3) A D E C B (3) A D E B C (3) E C B A D (2) E A B D C (2) D A B C E (2) C E B D A (2) C D E A B (2) C D B E A (2) C D A E B (2) B E A D C (2) B D A C E (2) B C E A D (2) B C D E A (2) B A E D C (2) A E B D C (2) A B E D C (2) E B A D C (1) E A D C B (1) D C B A E (1) D B A C E (1) D A C B E (1) C D E B A (1) C D B A E (1) C B D A E (1) B A D E C (1) Total count = 100 A B C D E A 0 -12 -10 -2 -4 B 12 0 10 12 10 C 10 -10 0 6 6 D 2 -12 -6 0 -6 E 4 -10 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -2 -4 B 12 0 10 12 10 C 10 -10 0 6 6 D 2 -12 -6 0 -6 E 4 -10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 D=20 A=16 E=13 so E is eliminated. Round 2 votes counts: B=37 C=24 D=20 A=19 so A is eliminated. Round 3 votes counts: B=43 D=33 C=24 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:206 E:197 D:189 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 -2 -4 B 12 0 10 12 10 C 10 -10 0 6 6 D 2 -12 -6 0 -6 E 4 -10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -2 -4 B 12 0 10 12 10 C 10 -10 0 6 6 D 2 -12 -6 0 -6 E 4 -10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -2 -4 B 12 0 10 12 10 C 10 -10 0 6 6 D 2 -12 -6 0 -6 E 4 -10 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1845: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) A B C E D (11) D E B A C (10) C A B E D (9) E D B A C (6) D E C B A (6) C D E A B (6) B E D A C (5) A B E D C (5) D E B C A (4) C D A E B (4) B A E D C (4) A B E C D (4) D C E B A (3) B E A D C (3) A C B E D (3) E D B C A (2) C A B D E (2) D C E A B (1) Total count = 100 A B C D E A 0 -10 2 -18 -16 B 10 0 8 -8 -8 C -2 -8 0 2 2 D 18 8 -2 0 -4 E 16 8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.444444 Sum of squares = 0.407407407402 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 0.555556 E: 1.000000 A B C D E A 0 -10 2 -18 -16 B 10 0 8 -8 -8 C -2 -8 0 2 2 D 18 8 -2 0 -4 E 16 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.444444 Sum of squares = 0.407407407379 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=24 A=23 B=12 E=8 so E is eliminated. Round 2 votes counts: C=33 D=32 A=23 B=12 so B is eliminated. Round 3 votes counts: D=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:213 D:210 B:201 C:197 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 -18 -16 B 10 0 8 -8 -8 C -2 -8 0 2 2 D 18 8 -2 0 -4 E 16 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.444444 Sum of squares = 0.407407407379 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -18 -16 B 10 0 8 -8 -8 C -2 -8 0 2 2 D 18 8 -2 0 -4 E 16 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.444444 Sum of squares = 0.407407407379 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -18 -16 B 10 0 8 -8 -8 C -2 -8 0 2 2 D 18 8 -2 0 -4 E 16 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.444444 Sum of squares = 0.407407407379 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1846: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) A D B E C (6) D A B E C (5) C D B E A (5) B A D E C (5) E B C A D (4) D A B C E (4) C E B D A (4) A B E D C (4) E A B C D (3) D C A B E (3) C E D B A (3) C E D A B (3) C E B A D (3) A D E B C (3) E C B A D (2) E B A C D (2) E A D B C (2) E A C D B (2) E A B D C (2) D B A C E (2) D A E C B (2) C E A B D (2) C D E B A (2) C D B A E (2) C B D E A (2) B A E D C (2) E C A D B (1) E B A D C (1) D A E B C (1) D A C B E (1) C D E A B (1) C D A E B (1) C B D A E (1) B E C A D (1) B E A D C (1) B D A C E (1) B C E D A (1) B C E A D (1) A E D B C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 4 0 -8 B 2 0 6 -2 8 C -4 -6 0 4 -4 D 0 2 -4 0 -2 E 8 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.044722 B: 0.303518 C: 0.151759 D: 0.500000 E: 0.000000 Sum of squares = 0.367154407036 Cumulative probabilities = A: 0.044722 B: 0.348241 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 0 -8 B 2 0 6 -2 8 C -4 -6 0 4 -4 D 0 2 -4 0 -2 E 8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.277778 C: 0.138889 D: 0.500000 E: 0.000000 Sum of squares = 0.353395102906 Cumulative probabilities = A: 0.083333 B: 0.361111 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=19 D=18 A=16 B=12 so B is eliminated. Round 2 votes counts: C=37 A=23 E=21 D=19 so D is eliminated. Round 3 votes counts: C=40 A=39 E=21 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:207 E:203 D:198 A:197 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 0 -8 B 2 0 6 -2 8 C -4 -6 0 4 -4 D 0 2 -4 0 -2 E 8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.277778 C: 0.138889 D: 0.500000 E: 0.000000 Sum of squares = 0.353395102906 Cumulative probabilities = A: 0.083333 B: 0.361111 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 0 -8 B 2 0 6 -2 8 C -4 -6 0 4 -4 D 0 2 -4 0 -2 E 8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.277778 C: 0.138889 D: 0.500000 E: 0.000000 Sum of squares = 0.353395102906 Cumulative probabilities = A: 0.083333 B: 0.361111 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 0 -8 B 2 0 6 -2 8 C -4 -6 0 4 -4 D 0 2 -4 0 -2 E 8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.277778 C: 0.138889 D: 0.500000 E: 0.000000 Sum of squares = 0.353395102906 Cumulative probabilities = A: 0.083333 B: 0.361111 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1847: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (18) A D E B C (8) D E A C B (7) D A E C B (6) A B D E C (5) D E C A B (4) D A E B C (3) C E D B A (3) C B E A D (3) C B A E D (3) A B C E D (3) D E A B C (2) C E B D A (2) C D E B A (2) C B A D E (2) B C E A D (2) B C A E D (2) B A C E D (2) A D E C B (2) A D C E B (2) A D B E C (2) A B E D C (2) E D C B A (1) E D B C A (1) E B C D A (1) D E C B A (1) D C E B A (1) C A B D E (1) B E C A D (1) B E A D C (1) B A E C D (1) A E D B C (1) A E B D C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -4 0 B -6 0 -20 2 4 C -4 20 0 -6 0 D 4 -2 6 0 4 E 0 -4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.071429 D: 0.714286 E: 0.000000 Sum of squares = 0.561224489802 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.285714 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 0 B -6 0 -20 2 4 C -4 20 0 -6 0 D 4 -2 6 0 4 E 0 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.071429 D: 0.714286 E: 0.000000 Sum of squares = 0.561224490308 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=30 D=24 B=9 E=3 so E is eliminated. Round 2 votes counts: C=34 A=30 D=26 B=10 so B is eliminated. Round 3 votes counts: C=40 A=34 D=26 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:206 C:205 A:203 E:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 0 B -6 0 -20 2 4 C -4 20 0 -6 0 D 4 -2 6 0 4 E 0 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.071429 D: 0.714286 E: 0.000000 Sum of squares = 0.561224490308 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 0 B -6 0 -20 2 4 C -4 20 0 -6 0 D 4 -2 6 0 4 E 0 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.071429 D: 0.714286 E: 0.000000 Sum of squares = 0.561224490308 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 0 B -6 0 -20 2 4 C -4 20 0 -6 0 D 4 -2 6 0 4 E 0 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.071429 D: 0.714286 E: 0.000000 Sum of squares = 0.561224490308 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1848: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) B E D A C (7) E A C D B (6) C A E D B (6) D A C E B (5) C A D E B (5) D E A C B (4) C B A E D (4) B C E A D (4) B C A E D (4) B C A D E (4) E D A B C (3) D A E C B (3) C D A B E (3) B E C A D (3) B C D A E (3) E B A D C (2) D C B A E (2) B D E A C (2) B D C A E (2) A E D C B (2) E D A C B (1) E B D A C (1) E A C B D (1) D E A B C (1) D C A E B (1) D B E A C (1) D B A C E (1) C E B A D (1) C E A B D (1) C D A E B (1) C A B E D (1) C A B D E (1) B E C D A (1) B E A D C (1) B D E C A (1) B C D E A (1) A E C D B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 2 12 2 B -10 0 -16 -10 -8 C -2 16 0 4 4 D -12 10 -4 0 -16 E -2 8 -4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 12 2 B -10 0 -16 -10 -8 C -2 16 0 4 4 D -12 10 -4 0 -16 E -2 8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=23 E=21 D=18 A=5 so A is eliminated. Round 2 votes counts: B=33 E=24 C=24 D=19 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:211 E:209 D:189 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 12 2 B -10 0 -16 -10 -8 C -2 16 0 4 4 D -12 10 -4 0 -16 E -2 8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 12 2 B -10 0 -16 -10 -8 C -2 16 0 4 4 D -12 10 -4 0 -16 E -2 8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 12 2 B -10 0 -16 -10 -8 C -2 16 0 4 4 D -12 10 -4 0 -16 E -2 8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1849: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) E B C A D (8) B E D A C (8) D A C B E (6) D B A C E (5) C A D E B (5) B D E A C (5) B D A E C (5) E C A B D (4) B E C D A (4) D C A B E (3) C D A B E (3) B E D C A (3) B D A C E (3) E C B A D (2) E B A C D (2) D A B C E (2) C E B A D (2) C A E D B (2) B D C A E (2) B C E D A (2) E A D C B (1) E A C D B (1) E A B C D (1) D A C E B (1) D A B E C (1) B E C A D (1) B D E C A (1) B C D E A (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -8 -10 -14 B 8 0 8 8 10 C 8 -8 0 2 -20 D 10 -8 -2 0 -8 E 14 -10 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -10 -14 B 8 0 8 8 10 C 8 -8 0 2 -20 D 10 -8 -2 0 -8 E 14 -10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=30 D=18 C=12 A=5 so A is eliminated. Round 2 votes counts: B=35 E=31 D=20 C=14 so C is eliminated. Round 3 votes counts: E=36 B=35 D=29 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:216 D:196 C:191 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -10 -14 B 8 0 8 8 10 C 8 -8 0 2 -20 D 10 -8 -2 0 -8 E 14 -10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -10 -14 B 8 0 8 8 10 C 8 -8 0 2 -20 D 10 -8 -2 0 -8 E 14 -10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -10 -14 B 8 0 8 8 10 C 8 -8 0 2 -20 D 10 -8 -2 0 -8 E 14 -10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1850: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (6) B C D E A (6) D E A B C (5) A E D C B (5) A C E D B (5) D B E A C (4) C A E D B (4) B D C E A (4) B C D A E (4) E A D C B (3) C B E D A (3) C B E A D (3) C B A D E (3) C A E B D (3) B D E A C (3) B D A E C (3) A D E B C (3) D E B A C (2) C B D E A (2) C B A E D (2) C A B D E (2) A E C D B (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D B C (1) D B A E C (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A D B (1) C A B E D (1) B E D C A (1) B D C A E (1) B C A D E (1) A E D B C (1) A D E C B (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -6 -8 -6 B 8 0 8 2 8 C 6 -8 0 -6 -2 D 8 -2 6 0 12 E 6 -8 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -8 -6 B 8 0 8 2 8 C 6 -8 0 -6 -2 D 8 -2 6 0 12 E 6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=25 A=20 D=14 E=12 so E is eliminated. Round 2 votes counts: B=29 C=28 A=24 D=19 so D is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:212 C:195 E:194 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -8 -6 B 8 0 8 2 8 C 6 -8 0 -6 -2 D 8 -2 6 0 12 E 6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -8 -6 B 8 0 8 2 8 C 6 -8 0 -6 -2 D 8 -2 6 0 12 E 6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -8 -6 B 8 0 8 2 8 C 6 -8 0 -6 -2 D 8 -2 6 0 12 E 6 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1851: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (10) E A D C B (8) B C D A E (6) A C D B E (6) B E D C A (5) B E A C D (5) A D C E B (5) E D C A B (4) E D B C A (4) E B A D C (4) D C A E B (4) C D A B E (4) B D C E A (4) E B A C D (3) B E C D A (3) B A C D E (3) E A B D C (2) E A B C D (2) B E C A D (2) B C D E A (2) A E D C B (2) A E C D B (2) A C D E B (2) E B D A C (1) D C B E A (1) C D B A E (1) B C A D E (1) B A E C D (1) A E B C D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -2 2 -20 B 12 0 20 14 -8 C 2 -20 0 -8 -18 D -2 -14 8 0 -18 E 20 8 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -2 2 -20 B 12 0 20 14 -8 C 2 -20 0 -8 -18 D -2 -14 8 0 -18 E 20 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=32 A=20 D=5 C=5 so D is eliminated. Round 2 votes counts: E=38 B=32 A=20 C=10 so C is eliminated. Round 3 votes counts: E=38 B=34 A=28 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:232 B:219 D:187 A:184 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -2 2 -20 B 12 0 20 14 -8 C 2 -20 0 -8 -18 D -2 -14 8 0 -18 E 20 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 2 -20 B 12 0 20 14 -8 C 2 -20 0 -8 -18 D -2 -14 8 0 -18 E 20 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 2 -20 B 12 0 20 14 -8 C 2 -20 0 -8 -18 D -2 -14 8 0 -18 E 20 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1852: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) D E C B A (7) E D C A B (6) A B E D C (6) B A D E C (5) A E D C B (5) A E B D C (5) C E D A B (4) C D B E A (4) B A E D C (4) E D A C B (3) E D A B C (3) C D E A B (3) A E D B C (3) A B E C D (3) A B C E D (3) E A D C B (2) E A D B C (2) C B D E A (2) C B A D E (2) B D E C A (2) A C B E D (2) E D B A C (1) D E C A B (1) D E B A C (1) D B E C A (1) C E A D B (1) C B D A E (1) C A E D B (1) C A B E D (1) C A B D E (1) B C D E A (1) B C A D E (1) B A C E D (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 12 2 0 -8 B -12 0 -14 -18 -18 C -2 14 0 -14 -20 D 0 18 14 0 -14 E 8 18 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 0 -8 B -12 0 -14 -18 -18 C -2 14 0 -14 -20 D 0 18 14 0 -14 E 8 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=28 E=17 B=15 D=10 so D is eliminated. Round 2 votes counts: C=30 A=28 E=26 B=16 so B is eliminated. Round 3 votes counts: A=39 C=32 E=29 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:230 D:209 A:203 C:189 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 0 -8 B -12 0 -14 -18 -18 C -2 14 0 -14 -20 D 0 18 14 0 -14 E 8 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 0 -8 B -12 0 -14 -18 -18 C -2 14 0 -14 -20 D 0 18 14 0 -14 E 8 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 0 -8 B -12 0 -14 -18 -18 C -2 14 0 -14 -20 D 0 18 14 0 -14 E 8 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1853: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) C A B E D (6) A C E D B (6) D E B A C (4) C A E D B (4) C A E B D (4) B D E A C (4) B D A E C (4) E C D B A (3) E C A D B (3) B D C A E (3) B A D C E (3) B A C D E (3) A E C D B (3) A B C D E (3) E D A C B (2) E D A B C (2) E C D A B (2) C E B D A (2) C E A D B (2) C B E D A (2) C B A D E (2) C A B D E (2) B D E C A (2) B D A C E (2) B C D E A (2) A C E B D (2) A C B E D (2) A C B D E (2) E D C B A (1) E D C A B (1) E C B D A (1) E A D C B (1) D B A E C (1) D A E B C (1) C E A B D (1) B C D A E (1) B A D E C (1) A E D B C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 12 6 16 B -4 0 -8 8 6 C -12 8 0 16 8 D -6 -8 -16 0 -2 E -16 -6 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 6 16 B -4 0 -8 8 6 C -12 8 0 16 8 D -6 -8 -16 0 -2 E -16 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=25 B=25 A=21 E=16 D=13 so D is eliminated. Round 2 votes counts: B=33 C=25 A=22 E=20 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:210 B:201 E:186 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 6 16 B -4 0 -8 8 6 C -12 8 0 16 8 D -6 -8 -16 0 -2 E -16 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 6 16 B -4 0 -8 8 6 C -12 8 0 16 8 D -6 -8 -16 0 -2 E -16 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 6 16 B -4 0 -8 8 6 C -12 8 0 16 8 D -6 -8 -16 0 -2 E -16 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1854: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) C D A B E (6) B E A D C (6) E B A D C (5) C E B A D (5) C B D E A (5) C D B A E (4) B D A E C (4) A E D B C (4) E C B A D (3) D A B E C (3) C B E A D (3) C B D A E (3) E B A C D (2) E A D C B (2) D B C A E (2) D B A E C (2) D A C E B (2) C D B E A (2) B A D E C (2) A D E B C (2) A D B E C (2) E C A D B (1) E C A B D (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B C D (1) D C A B E (1) D A E C B (1) D A E B C (1) C E D B A (1) C E B D A (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A E B (1) C B E D A (1) C A E D B (1) B E A C D (1) B D C A E (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D C B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 6 12 -8 B 12 0 -4 10 2 C -6 4 0 0 -10 D -12 -10 0 0 -4 E 8 -2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999993 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -12 6 12 -8 B 12 0 -4 10 2 C -6 4 0 0 -10 D -12 -10 0 0 -4 E 8 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999856 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=24 B=17 D=12 A=11 so A is eliminated. Round 2 votes counts: C=37 E=29 D=17 B=17 so D is eliminated. Round 3 votes counts: C=40 E=34 B=26 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:210 A:199 C:194 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 6 12 -8 B 12 0 -4 10 2 C -6 4 0 0 -10 D -12 -10 0 0 -4 E 8 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999856 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 12 -8 B 12 0 -4 10 2 C -6 4 0 0 -10 D -12 -10 0 0 -4 E 8 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999856 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 12 -8 B 12 0 -4 10 2 C -6 4 0 0 -10 D -12 -10 0 0 -4 E 8 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999856 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1855: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) D B A C E (8) D A B C E (8) E C B A D (7) B D C E A (5) B C E D A (5) A D E C B (5) E C A D B (4) C E B A D (4) B E C D A (4) B D A C E (4) D B A E C (3) A E D C B (3) A E C D B (3) A D C E B (3) A C E D B (3) E C A B D (2) E A C D B (2) B D E C A (2) B D A E C (2) E B C A D (1) D A C B E (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E D A (1) C A E D B (1) B E D C A (1) B E C A D (1) B D C A E (1) B C D E A (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 14 -12 14 B 2 0 16 -14 16 C -14 -16 0 -16 -2 D 12 14 16 0 10 E -14 -16 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 -12 14 B 2 0 16 -14 16 C -14 -16 0 -16 -2 D 12 14 16 0 10 E -14 -16 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 A=20 E=16 C=9 so C is eliminated. Round 2 votes counts: D=29 B=27 E=23 A=21 so A is eliminated. Round 3 votes counts: D=40 E=33 B=27 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:210 A:207 E:181 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 14 -12 14 B 2 0 16 -14 16 C -14 -16 0 -16 -2 D 12 14 16 0 10 E -14 -16 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 -12 14 B 2 0 16 -14 16 C -14 -16 0 -16 -2 D 12 14 16 0 10 E -14 -16 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 -12 14 B 2 0 16 -14 16 C -14 -16 0 -16 -2 D 12 14 16 0 10 E -14 -16 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1856: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) C D A B E (9) D C A E B (8) C A B E D (8) D C E B A (7) C B A E D (5) A C B E D (5) E B A D C (4) A E B D C (4) A B E C D (4) D C E A B (3) C D B A E (3) C A D B E (3) C A B D E (3) B E A C D (3) E B D A C (2) E A B D C (2) D E B C A (2) D E A B C (2) B A E C D (2) A E B C D (2) A B C E D (2) E D B A C (1) D E C B A (1) C D A E B (1) C B D E A (1) B C A E D (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 16 -10 0 24 B -16 0 -16 -2 -2 C 10 16 0 6 18 D 0 2 -6 0 8 E -24 2 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -10 0 24 B -16 0 -16 -2 -2 C 10 16 0 6 18 D 0 2 -6 0 8 E -24 2 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=33 C=33 A=19 E=9 B=6 so B is eliminated. Round 2 votes counts: C=34 D=33 A=21 E=12 so E is eliminated. Round 3 votes counts: D=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 A:215 D:202 B:182 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -10 0 24 B -16 0 -16 -2 -2 C 10 16 0 6 18 D 0 2 -6 0 8 E -24 2 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -10 0 24 B -16 0 -16 -2 -2 C 10 16 0 6 18 D 0 2 -6 0 8 E -24 2 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -10 0 24 B -16 0 -16 -2 -2 C 10 16 0 6 18 D 0 2 -6 0 8 E -24 2 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1857: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) A B C D E (8) D E C B A (6) D E C A B (6) E D C B A (5) A B C E D (5) E B C D A (4) D A C E B (4) D A C B E (4) A D B C E (4) D E B C A (3) D E A C B (3) B E C A D (3) B C A E D (3) B A E C D (3) A C B D E (3) A B D C E (3) E D B C A (2) E C B D A (2) D C E A B (2) D A E C B (2) C D E A B (2) C B E A D (2) B E A C D (2) A D C B E (2) E C D B A (1) E C B A D (1) E B D C A (1) E B C A D (1) D A E B C (1) C E B A D (1) C A B E D (1) B C E A D (1) B A E D C (1) Total count = 100 A B C D E A 0 0 6 4 4 B 0 0 6 6 6 C -6 -6 0 2 6 D -4 -6 -2 0 6 E -4 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.585043 B: 0.414957 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.514464589261 Cumulative probabilities = A: 0.585043 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 4 B 0 0 6 6 6 C -6 -6 0 2 6 D -4 -6 -2 0 6 E -4 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=25 B=21 E=17 C=6 so C is eliminated. Round 2 votes counts: D=33 A=26 B=23 E=18 so E is eliminated. Round 3 votes counts: D=41 B=33 A=26 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:207 C:198 D:197 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 4 B 0 0 6 6 6 C -6 -6 0 2 6 D -4 -6 -2 0 6 E -4 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 4 B 0 0 6 6 6 C -6 -6 0 2 6 D -4 -6 -2 0 6 E -4 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 4 B 0 0 6 6 6 C -6 -6 0 2 6 D -4 -6 -2 0 6 E -4 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1858: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) B A D C E (8) C B D A E (7) A B D E C (7) D E A B C (6) C B A D E (6) E C D A B (5) B A D E C (5) E D A B C (4) D B A E C (4) C E D B A (4) C B A E D (4) E C A D B (3) C E A D B (3) B D A C E (3) A D B E C (3) E A D B C (2) C E B D A (2) B C D A E (2) B A C D E (2) E A C D B (1) D E C B A (1) D A B E C (1) C E B A D (1) C E A B D (1) C D E B A (1) B D C A E (1) B D A E C (1) B C A D E (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -2 -2 14 B 6 0 4 6 12 C 2 -4 0 4 12 D 2 -6 -4 0 20 E -14 -12 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -2 14 B 6 0 4 6 12 C 2 -4 0 4 12 D 2 -6 -4 0 20 E -14 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=23 E=15 D=12 A=12 so D is eliminated. Round 2 votes counts: C=38 B=27 E=22 A=13 so A is eliminated. Round 3 votes counts: B=39 C=38 E=23 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:207 D:206 A:202 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -2 14 B 6 0 4 6 12 C 2 -4 0 4 12 D 2 -6 -4 0 20 E -14 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -2 14 B 6 0 4 6 12 C 2 -4 0 4 12 D 2 -6 -4 0 20 E -14 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -2 14 B 6 0 4 6 12 C 2 -4 0 4 12 D 2 -6 -4 0 20 E -14 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1859: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) E D A B C (9) D E C A B (5) C D B A E (5) B A C E D (5) D C E A B (4) C B D A E (4) C B A D E (4) E A D B C (3) C D A B E (3) B C A D E (3) B A C D E (3) A E B D C (3) A D E B C (3) E D A C B (2) E B D A C (2) E B A D C (2) E A B D C (2) D E A C B (2) D A E B C (2) C D E B A (2) C B D E A (2) B E C A D (2) A E D B C (2) A B E D C (2) A B C D E (2) E D B C A (1) E C B D A (1) C E B D A (1) C D B E A (1) C B E D A (1) C B A E D (1) B C E A D (1) B A E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 6 14 B 10 0 24 12 8 C 8 -24 0 10 12 D -6 -12 -10 0 -8 E -14 -8 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 6 14 B 10 0 24 12 8 C 8 -24 0 10 12 D -6 -12 -10 0 -8 E -14 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=24 E=22 D=13 A=13 so D is eliminated. Round 2 votes counts: E=29 C=28 B=28 A=15 so A is eliminated. Round 3 votes counts: E=39 B=33 C=28 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:203 A:201 E:187 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 6 14 B 10 0 24 12 8 C 8 -24 0 10 12 D -6 -12 -10 0 -8 E -14 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 6 14 B 10 0 24 12 8 C 8 -24 0 10 12 D -6 -12 -10 0 -8 E -14 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 6 14 B 10 0 24 12 8 C 8 -24 0 10 12 D -6 -12 -10 0 -8 E -14 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1860: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) E A C D B (6) B D A C E (6) A E C D B (6) A D C B E (6) D B C A E (5) B E C D A (4) E C B A D (3) B D E C A (3) B D C E A (3) B D C A E (3) B D A E C (3) B C D E A (3) A D C E B (3) A D B C E (3) E C A D B (2) E B C D A (2) E B C A D (2) E B A C D (2) E A B C D (2) D A C B E (2) D A B C E (2) C D A E B (2) C A D E B (2) B E A D C (2) A C D E B (2) E C A B D (1) E B A D C (1) D C A B E (1) D B A C E (1) C E D A B (1) C E A D B (1) C B D E A (1) B E D A C (1) B D E A C (1) B A D E C (1) B A D C E (1) A E D B C (1) A C E D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 10 0 6 B 8 0 8 6 8 C -10 -8 0 -2 -2 D 0 -6 2 0 10 E -6 -8 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 0 6 B 8 0 8 6 8 C -10 -8 0 -2 -2 D 0 -6 2 0 10 E -6 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=27 A=24 D=11 C=7 so C is eliminated. Round 2 votes counts: B=32 E=29 A=26 D=13 so D is eliminated. Round 3 votes counts: B=38 A=33 E=29 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:204 D:203 C:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 0 6 B 8 0 8 6 8 C -10 -8 0 -2 -2 D 0 -6 2 0 10 E -6 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 0 6 B 8 0 8 6 8 C -10 -8 0 -2 -2 D 0 -6 2 0 10 E -6 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 0 6 B 8 0 8 6 8 C -10 -8 0 -2 -2 D 0 -6 2 0 10 E -6 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1861: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (14) A E D B C (12) D B C A E (7) C E B D A (7) E A C B D (6) E A B C D (6) E A D B C (5) A D B E C (4) E C B A D (3) E C A B D (3) C B E D A (3) B C D A E (3) A D E B C (3) A D B C E (3) E A D C B (2) C D B A E (2) C B D A E (2) B D C A E (2) A E B D C (2) E C D A B (1) E C B D A (1) E B A C D (1) E A C D B (1) E A B D C (1) D C B A E (1) D B A C E (1) D A C B E (1) C D B E A (1) B C A D E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -2 8 -10 B -2 0 4 12 -8 C 2 -4 0 10 -4 D -8 -12 -10 0 -10 E 10 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 8 -10 B -2 0 4 12 -8 C 2 -4 0 10 -4 D -8 -12 -10 0 -10 E 10 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=29 A=25 D=10 B=6 so B is eliminated. Round 2 votes counts: C=33 E=30 A=25 D=12 so D is eliminated. Round 3 votes counts: C=43 E=30 A=27 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:203 C:202 A:199 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 8 -10 B -2 0 4 12 -8 C 2 -4 0 10 -4 D -8 -12 -10 0 -10 E 10 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 -10 B -2 0 4 12 -8 C 2 -4 0 10 -4 D -8 -12 -10 0 -10 E 10 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 -10 B -2 0 4 12 -8 C 2 -4 0 10 -4 D -8 -12 -10 0 -10 E 10 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1862: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (6) C E B D A (6) B E C A D (6) E B D A C (4) E B C D A (4) D C A E B (4) D A C E B (4) A C D B E (4) A C B E D (4) A B E D C (4) A B E C D (4) C D E B A (3) C D A B E (3) B E A D C (3) B E A C D (3) A D C B E (3) A D B E C (3) E D B C A (2) E D B A C (2) E B D C A (2) D C E B A (2) C B E A D (2) C B A E D (2) C A B E D (2) B C E A D (2) E C B D A (1) E B A D C (1) D E B C A (1) D E B A C (1) D E A B C (1) D C E A B (1) C E D B A (1) C B E D A (1) C A D B E (1) C A B D E (1) B A E C D (1) A D E B C (1) A D C E B (1) A D B C E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 6 2 2 B 0 0 6 10 4 C -6 -6 0 4 -2 D -2 -10 -4 0 -14 E -2 -4 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.522705 B: 0.477295 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501031071484 Cumulative probabilities = A: 0.522705 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 2 2 B 0 0 6 10 4 C -6 -6 0 4 -2 D -2 -10 -4 0 -14 E -2 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 D=20 E=16 B=15 so B is eliminated. Round 2 votes counts: E=28 A=28 C=24 D=20 so D is eliminated. Round 3 votes counts: A=38 E=31 C=31 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:205 E:205 C:195 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 2 2 B 0 0 6 10 4 C -6 -6 0 4 -2 D -2 -10 -4 0 -14 E -2 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 2 2 B 0 0 6 10 4 C -6 -6 0 4 -2 D -2 -10 -4 0 -14 E -2 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 2 2 B 0 0 6 10 4 C -6 -6 0 4 -2 D -2 -10 -4 0 -14 E -2 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1863: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) D B C A E (9) C E A B D (9) B D C A E (8) E A C D B (7) A E D B C (7) D B A E C (6) E A C B D (5) A E C D B (5) C E B A D (4) C E A D B (4) C E D A B (3) C D B E A (3) B D A E C (3) A E D C B (3) E C A B D (2) A D E B C (2) E A B C D (1) D C B A E (1) D B A C E (1) D A E B C (1) C B E D A (1) C A E D B (1) B D C E A (1) B C D E A (1) B A D E C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -14 4 -2 B -2 0 -16 -8 -8 C 14 16 0 12 12 D -4 8 -12 0 -4 E 2 8 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 4 -2 B -2 0 -16 -8 -8 C 14 16 0 12 12 D -4 8 -12 0 -4 E 2 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=18 A=18 E=15 B=14 so B is eliminated. Round 2 votes counts: C=36 D=30 A=19 E=15 so E is eliminated. Round 3 votes counts: C=38 A=32 D=30 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:201 A:195 D:194 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 4 -2 B -2 0 -16 -8 -8 C 14 16 0 12 12 D -4 8 -12 0 -4 E 2 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 4 -2 B -2 0 -16 -8 -8 C 14 16 0 12 12 D -4 8 -12 0 -4 E 2 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 4 -2 B -2 0 -16 -8 -8 C 14 16 0 12 12 D -4 8 -12 0 -4 E 2 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1864: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (18) B D A C E (13) E A C D B (6) D A C B E (5) E C A B D (4) B E D C A (4) B D C A E (4) B D E A C (3) B C D A E (3) A D C B E (3) E D B A C (2) E D A C B (2) E C B A D (2) E B D C A (2) E B C D A (2) C E A D B (2) C A D E B (2) B E D A C (2) B D A E C (2) A D C E B (2) E A D C B (1) D E A B C (1) D B A C E (1) D A E C B (1) D A C E B (1) C E A B D (1) C B A D E (1) C A E D B (1) C A D B E (1) C A B D E (1) B E C D A (1) B C E A D (1) B C A D E (1) A E D C B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -2 2 -8 B -12 0 -16 -6 -6 C 2 16 0 0 -8 D -2 6 0 0 -6 E 8 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -2 2 -8 B -12 0 -16 -6 -6 C 2 16 0 0 -8 D -2 6 0 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=34 D=9 C=9 A=9 so D is eliminated. Round 2 votes counts: E=40 B=35 A=16 C=9 so C is eliminated. Round 3 votes counts: E=43 B=36 A=21 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:205 A:202 D:199 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -2 2 -8 B -12 0 -16 -6 -6 C 2 16 0 0 -8 D -2 6 0 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 2 -8 B -12 0 -16 -6 -6 C 2 16 0 0 -8 D -2 6 0 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 2 -8 B -12 0 -16 -6 -6 C 2 16 0 0 -8 D -2 6 0 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1865: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B A C E (6) A E B C D (6) A B E D C (6) E C A B D (5) D C B E A (5) C E D A B (5) C D E B A (5) C D B E A (5) A E B D C (5) C D E A B (4) B A D E C (4) A B D E C (4) D C B A E (3) D B C A E (3) C E D B A (3) C E A D B (3) E A B C D (2) B D A E C (2) B D A C E (2) B A E D C (2) E C A D B (1) E B C A D (1) E B A D C (1) D B C E A (1) C A E D B (1) B E A D C (1) B D E C A (1) B D E A C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 8 8 8 -8 B -8 0 -2 6 -4 C -8 2 0 2 -4 D -8 -6 -2 0 -4 E 8 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 8 -8 B -8 0 -2 6 -4 C -8 2 0 2 -4 D -8 -6 -2 0 -4 E 8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 E=20 D=18 B=13 so B is eliminated. Round 2 votes counts: A=29 C=26 D=24 E=21 so E is eliminated. Round 3 votes counts: A=43 C=33 D=24 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:208 B:196 C:196 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 8 -8 B -8 0 -2 6 -4 C -8 2 0 2 -4 D -8 -6 -2 0 -4 E 8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 -8 B -8 0 -2 6 -4 C -8 2 0 2 -4 D -8 -6 -2 0 -4 E 8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 -8 B -8 0 -2 6 -4 C -8 2 0 2 -4 D -8 -6 -2 0 -4 E 8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1866: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) D B E C A (6) A E D C B (6) B C D E A (5) A E C D B (5) A E C B D (5) D B C E A (4) C B D E A (4) C A B E D (4) A C B E D (4) E A D B C (3) D E B C A (3) D E B A C (3) D E A B C (3) C B A D E (3) B D C E A (3) B C D A E (3) B C A D E (3) A C E B D (3) E D C B A (2) E C A D B (2) D B A E C (2) C B D A E (2) C B A E D (2) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) D B A C E (1) D A B E C (1) C E B A D (1) C B E A D (1) B D C A E (1) B D A C E (1) A E D B C (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 0 4 0 B 4 0 -4 -6 6 C 0 4 0 -4 -6 D -4 6 4 0 2 E 0 -6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.448167 B: 0.207333 C: 0.137168 D: 0.207333 E: 0.000000 Sum of squares = 0.305642178999 Cumulative probabilities = A: 0.448167 B: 0.655499 C: 0.792667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 4 0 B 4 0 -4 -6 6 C 0 4 0 -4 -6 D -4 6 4 0 2 E 0 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.451219 B: 0.195122 C: 0.158536 D: 0.195122 E: 0.000000 Sum of squares = 0.304878048781 Cumulative probabilities = A: 0.451219 B: 0.646342 C: 0.804878 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 E=17 C=17 B=16 so B is eliminated. Round 2 votes counts: D=28 C=28 A=27 E=17 so E is eliminated. Round 3 votes counts: A=36 D=34 C=30 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:204 A:200 B:200 E:199 C:197 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 0 4 0 B 4 0 -4 -6 6 C 0 4 0 -4 -6 D -4 6 4 0 2 E 0 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.451219 B: 0.195122 C: 0.158536 D: 0.195122 E: 0.000000 Sum of squares = 0.304878048781 Cumulative probabilities = A: 0.451219 B: 0.646342 C: 0.804878 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 4 0 B 4 0 -4 -6 6 C 0 4 0 -4 -6 D -4 6 4 0 2 E 0 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.451219 B: 0.195122 C: 0.158536 D: 0.195122 E: 0.000000 Sum of squares = 0.304878048781 Cumulative probabilities = A: 0.451219 B: 0.646342 C: 0.804878 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 4 0 B 4 0 -4 -6 6 C 0 4 0 -4 -6 D -4 6 4 0 2 E 0 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.451219 B: 0.195122 C: 0.158536 D: 0.195122 E: 0.000000 Sum of squares = 0.304878048781 Cumulative probabilities = A: 0.451219 B: 0.646342 C: 0.804878 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1867: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) B D A C E (8) C D E B A (7) A E B C D (7) E A C B D (6) A B E D C (6) E C A D B (4) E C A B D (4) D C E B A (4) C E A D B (4) B D A E C (4) B A D E C (4) A B D E C (4) D B C A E (3) C E D B A (3) C E D A B (3) C D B E A (3) B A D C E (3) A E B D C (3) D B C E A (2) D B A C E (2) A E C B D (2) E C D A B (1) C A B E D (1) B D C A E (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 2 0 0 B 8 0 -2 10 4 C -2 2 0 -6 8 D 0 -10 6 0 10 E 0 -4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765425 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 0 0 B 8 0 -2 10 4 C -2 2 0 -6 8 D 0 -10 6 0 10 E 0 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765401 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 C=21 B=21 D=20 E=15 so E is eliminated. Round 2 votes counts: C=30 A=29 B=21 D=20 so D is eliminated. Round 3 votes counts: C=43 A=29 B=28 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:210 D:203 C:201 A:197 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 0 0 B 8 0 -2 10 4 C -2 2 0 -6 8 D 0 -10 6 0 10 E 0 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765401 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 0 0 B 8 0 -2 10 4 C -2 2 0 -6 8 D 0 -10 6 0 10 E 0 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765401 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 0 0 B 8 0 -2 10 4 C -2 2 0 -6 8 D 0 -10 6 0 10 E 0 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765401 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1868: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) C D B E A (6) C D B A E (6) D C A E B (5) C B A E D (5) B E A C D (5) B C A E D (5) A E B D C (5) E D A B C (4) E A D B C (4) E A B D C (4) D E A C B (4) D A E C B (4) A E D B C (4) D C E A B (3) D A E B C (3) C B D E A (3) B A E C D (3) E B A D C (2) D E A B C (2) C B E A D (2) B C E A D (2) A B E C D (2) D E C A B (1) D C E B A (1) C D E B A (1) C D A B E (1) C B E D A (1) C B A D E (1) C A D B E (1) B A C E D (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -2 -4 10 B 2 0 -4 -2 4 C 2 4 0 4 2 D 4 2 -4 0 0 E -10 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -4 10 B 2 0 -4 -2 4 C 2 4 0 4 2 D 4 2 -4 0 0 E -10 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=23 B=16 E=14 A=13 so A is eliminated. Round 2 votes counts: C=34 D=24 E=23 B=19 so B is eliminated. Round 3 votes counts: C=42 E=34 D=24 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 A:201 D:201 B:200 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -4 10 B 2 0 -4 -2 4 C 2 4 0 4 2 D 4 2 -4 0 0 E -10 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -4 10 B 2 0 -4 -2 4 C 2 4 0 4 2 D 4 2 -4 0 0 E -10 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -4 10 B 2 0 -4 -2 4 C 2 4 0 4 2 D 4 2 -4 0 0 E -10 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1869: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) D C A E B (5) D A E B C (5) B E C A D (5) B C E A D (5) C D B E A (4) C B E A D (4) A B E D C (4) E D B A C (3) E B A D C (3) A E B D C (3) A D E B C (3) E D A B C (2) E A B D C (2) D E A B C (2) D C E B A (2) D A E C B (2) D A C E B (2) C D E B A (2) C D A B E (2) C B A D E (2) C A D B E (2) C A B E D (2) C A B D E (2) B E A C D (2) B A C E D (2) A E D B C (2) A D C E B (2) A D B E C (2) E D C B A (1) E B D C A (1) D E C B A (1) C E B D A (1) C B D E A (1) B E A D C (1) B C E D A (1) B A E D C (1) B A E C D (1) A D C B E (1) A C B D E (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 10 -2 B 2 0 6 10 12 C 2 -6 0 -2 4 D -10 -10 2 0 -12 E 2 -12 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 10 -2 B 2 0 6 10 12 C 2 -6 0 -2 4 D -10 -10 2 0 -12 E 2 -12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992033 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=21 D=19 B=18 E=12 so E is eliminated. Round 2 votes counts: C=30 D=25 A=23 B=22 so B is eliminated. Round 3 votes counts: C=41 A=33 D=26 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:215 A:202 C:199 E:199 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 10 -2 B 2 0 6 10 12 C 2 -6 0 -2 4 D -10 -10 2 0 -12 E 2 -12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992033 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 10 -2 B 2 0 6 10 12 C 2 -6 0 -2 4 D -10 -10 2 0 -12 E 2 -12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992033 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 10 -2 B 2 0 6 10 12 C 2 -6 0 -2 4 D -10 -10 2 0 -12 E 2 -12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992033 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1870: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) A D C E B (7) B E C D A (6) D A E C B (5) C E B A D (5) C E A D B (5) A B D C E (5) E C D B A (4) D E C A B (4) E C D A B (3) B D E C A (3) B D A E C (3) B C E A D (3) A C D E B (3) E B C D A (2) D A E B C (2) B E D C A (2) B C A E D (2) B A C D E (2) A D C B E (2) A C B E D (2) E D C B A (1) E D B C A (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E A B (1) D B E C A (1) D A C E B (1) D A B E C (1) D A B C E (1) C E D A B (1) C E A B D (1) C B E A D (1) C A E D B (1) B D E A C (1) B A D C E (1) B A C E D (1) A D B C E (1) A C E D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -16 -10 -14 B 0 0 -20 2 -22 C 16 20 0 8 -2 D 10 -2 -8 0 -4 E 14 22 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -16 -10 -14 B 0 0 -20 2 -22 C 16 20 0 8 -2 D 10 -2 -8 0 -4 E 14 22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 E=20 D=19 C=14 so C is eliminated. Round 2 votes counts: E=32 B=25 A=24 D=19 so D is eliminated. Round 3 votes counts: E=40 A=34 B=26 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:221 E:221 D:198 A:180 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -16 -10 -14 B 0 0 -20 2 -22 C 16 20 0 8 -2 D 10 -2 -8 0 -4 E 14 22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -10 -14 B 0 0 -20 2 -22 C 16 20 0 8 -2 D 10 -2 -8 0 -4 E 14 22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -10 -14 B 0 0 -20 2 -22 C 16 20 0 8 -2 D 10 -2 -8 0 -4 E 14 22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1871: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) E C D B A (8) E D C B A (6) A B C D E (6) B E C D A (5) A D E C B (5) D E C A B (4) B C E D A (4) B A E D C (4) E D C A B (3) E B D C A (3) D E A C B (3) B C E A D (3) A B D C E (3) E C B D A (2) D A E C B (2) D A C E B (2) C B E D A (2) B E A C D (2) B A E C D (2) B A C D E (2) A D C E B (2) A C D E B (2) A C B D E (2) A B D E C (2) E D A B C (1) E B C D A (1) D E A B C (1) D C E A B (1) C D E A B (1) C A D E B (1) C A B D E (1) B E A D C (1) B C A E D (1) B C A D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 6 2 -2 B 14 0 6 14 2 C -6 -6 0 12 -12 D -2 -14 -12 0 -16 E 2 -2 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 2 -2 B 14 0 6 14 2 C -6 -6 0 12 -12 D -2 -14 -12 0 -16 E 2 -2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=24 A=23 D=13 C=5 so C is eliminated. Round 2 votes counts: B=37 A=25 E=24 D=14 so D is eliminated. Round 3 votes counts: B=37 E=34 A=29 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:214 A:196 C:194 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 6 2 -2 B 14 0 6 14 2 C -6 -6 0 12 -12 D -2 -14 -12 0 -16 E 2 -2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 2 -2 B 14 0 6 14 2 C -6 -6 0 12 -12 D -2 -14 -12 0 -16 E 2 -2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 2 -2 B 14 0 6 14 2 C -6 -6 0 12 -12 D -2 -14 -12 0 -16 E 2 -2 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1872: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) A C E B D (9) D A B E C (8) C E B A D (8) D E B C A (6) C A E B D (5) A D B C E (5) E C B D A (4) D B A E C (4) E B C D A (3) C E A B D (3) A C E D B (3) A C B E D (3) E C B A D (2) D B E A C (2) D A C E B (2) D A B C E (2) C B E A D (2) B E D C A (2) B E C D A (2) B E C A D (2) B C E A D (2) E D B C A (1) E B D C A (1) C E B D A (1) C E A D B (1) C A E D B (1) C A D E B (1) B D E C A (1) B D E A C (1) A D C B E (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -14 2 -6 B 6 0 4 6 -4 C 14 -4 0 10 4 D -2 -6 -10 0 -10 E 6 4 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -6 -14 2 -6 B 6 0 4 6 -4 C 14 -4 0 10 4 D -2 -6 -10 0 -10 E 6 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=24 C=22 E=11 B=10 so B is eliminated. Round 2 votes counts: D=35 C=24 A=24 E=17 so E is eliminated. Round 3 votes counts: D=39 C=37 A=24 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:208 B:206 A:188 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 2 -6 B 6 0 4 6 -4 C 14 -4 0 10 4 D -2 -6 -10 0 -10 E 6 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 2 -6 B 6 0 4 6 -4 C 14 -4 0 10 4 D -2 -6 -10 0 -10 E 6 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 2 -6 B 6 0 4 6 -4 C 14 -4 0 10 4 D -2 -6 -10 0 -10 E 6 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1873: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) A E D C B (8) B C D E A (7) E A D C B (6) B C E D A (6) B C D A E (5) A C D E B (5) E A B D C (4) B E C D A (4) A E C D B (4) A D E C B (4) E B A D C (3) E B A C D (3) E A B C D (3) C D A B E (3) B E C A D (3) D A C E B (2) C D A E B (2) C B D A E (2) B C E A D (2) E D A B C (1) D E A C B (1) D E A B C (1) D C B A E (1) D C A B E (1) D B E C A (1) D B E A C (1) D B C A E (1) C D B A E (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A C D (1) B C A E D (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 14 16 18 -12 B -14 0 12 -8 -16 C -16 -12 0 10 -16 D -18 8 -10 0 -18 E 12 16 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 16 18 -12 B -14 0 12 -8 -16 C -16 -12 0 10 -16 D -18 8 -10 0 -18 E 12 16 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 A=23 C=10 D=9 so D is eliminated. Round 2 votes counts: B=33 E=30 A=25 C=12 so C is eliminated. Round 3 votes counts: B=37 A=33 E=30 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:231 A:218 B:187 C:183 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 16 18 -12 B -14 0 12 -8 -16 C -16 -12 0 10 -16 D -18 8 -10 0 -18 E 12 16 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 18 -12 B -14 0 12 -8 -16 C -16 -12 0 10 -16 D -18 8 -10 0 -18 E 12 16 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 18 -12 B -14 0 12 -8 -16 C -16 -12 0 10 -16 D -18 8 -10 0 -18 E 12 16 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1874: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) B C E A D (8) A D E B C (8) B E C A D (6) E B A D C (5) E A D B C (5) C D A E B (5) D C A E B (4) D A C E B (4) C D E A B (4) C D A B E (4) C B E D A (4) C B D A E (4) E A B D C (3) B E A D C (3) D C E A B (2) C D E B A (2) C D B A E (2) B C E D A (2) B A E D C (2) A D E C B (2) E C B D A (1) E B C A D (1) E A D C B (1) D C A B E (1) D A C B E (1) C E B D A (1) C B D E A (1) B E A C D (1) B C A E D (1) B C A D E (1) A E D C B (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -8 -2 0 B -10 0 -6 -12 -16 C 8 6 0 -4 4 D 2 12 4 0 10 E 0 16 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 -2 0 B -10 0 -6 -12 -16 C 8 6 0 -4 4 D 2 12 4 0 10 E 0 16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996047 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 D=21 E=16 A=12 so A is eliminated. Round 2 votes counts: D=32 C=27 B=24 E=17 so E is eliminated. Round 3 votes counts: D=39 B=33 C=28 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:207 E:201 A:200 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -8 -2 0 B -10 0 -6 -12 -16 C 8 6 0 -4 4 D 2 12 4 0 10 E 0 16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996047 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -2 0 B -10 0 -6 -12 -16 C 8 6 0 -4 4 D 2 12 4 0 10 E 0 16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996047 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -2 0 B -10 0 -6 -12 -16 C 8 6 0 -4 4 D 2 12 4 0 10 E 0 16 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996047 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1875: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) E A C D B (7) D B E A C (6) C A B E D (6) E D B C A (5) E C A D B (5) B D E C A (5) B D C A E (5) A C E D B (5) E D B A C (4) E D A C B (4) D E B A C (4) A C D B E (4) B C A D E (3) A C B E D (3) A C B D E (3) D B E C A (2) D B A C E (2) B D A C E (2) B A C D E (2) E D A B C (1) E C A B D (1) D E B C A (1) D E A C B (1) D A C E B (1) D A B C E (1) C B A E D (1) C A E B D (1) B E D C A (1) B C A E D (1) A E C D B (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 12 26 10 6 B -12 0 -8 -12 -4 C -26 8 0 8 4 D -10 12 -8 0 -12 E -6 4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 26 10 6 B -12 0 -8 -12 -4 C -26 8 0 8 4 D -10 12 -8 0 -12 E -6 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 B=19 D=18 C=8 so C is eliminated. Round 2 votes counts: A=35 E=27 B=20 D=18 so D is eliminated. Round 3 votes counts: A=37 E=33 B=30 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:203 C:197 D:191 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 26 10 6 B -12 0 -8 -12 -4 C -26 8 0 8 4 D -10 12 -8 0 -12 E -6 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 26 10 6 B -12 0 -8 -12 -4 C -26 8 0 8 4 D -10 12 -8 0 -12 E -6 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 26 10 6 B -12 0 -8 -12 -4 C -26 8 0 8 4 D -10 12 -8 0 -12 E -6 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1876: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D B A C E (5) A D E B C (5) E C B D A (4) D E C B A (4) D A B C E (4) C E B A D (4) B C E A D (4) A B D C E (4) E C D B A (3) E C D A B (3) D A E C B (3) B C D A E (3) B C A E D (3) B A D C E (3) A E B C D (3) A D B E C (3) A D B C E (3) E D C A B (2) D B C A E (2) C E B D A (2) C B D E A (2) B D C A E (2) A E D C B (2) A D E C B (2) E D A C B (1) E C A B D (1) E A C D B (1) D E C A B (1) D E A C B (1) D C B E A (1) D B C E A (1) C E D B A (1) C D E B A (1) C B E D A (1) B D A C E (1) B C A D E (1) A E D B C (1) A B E C D (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -8 4 8 B 12 0 4 0 -6 C 8 -4 0 -4 0 D -4 0 4 0 8 E -8 6 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.445191 C: 0.000000 D: 0.554809 E: 0.000000 Sum of squares = 0.506008050759 Cumulative probabilities = A: 0.000000 B: 0.445191 C: 0.445191 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 4 8 B 12 0 4 0 -6 C 8 -4 0 -4 0 D -4 0 4 0 8 E -8 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999807 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=23 D=22 B=17 C=11 so C is eliminated. Round 2 votes counts: E=30 A=27 D=23 B=20 so B is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:205 D:204 C:200 A:196 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 4 8 B 12 0 4 0 -6 C 8 -4 0 -4 0 D -4 0 4 0 8 E -8 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999807 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 4 8 B 12 0 4 0 -6 C 8 -4 0 -4 0 D -4 0 4 0 8 E -8 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999807 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 4 8 B 12 0 4 0 -6 C 8 -4 0 -4 0 D -4 0 4 0 8 E -8 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999807 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1877: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) D E B A C (7) D B E C A (5) D B A E C (5) B D C E A (5) A E C D B (5) E A C D B (4) C A E B D (4) B D C A E (4) B C D A E (4) E D B C A (3) C B A D E (3) C A B E D (3) A E D C B (3) E D A B C (2) E A D C B (2) D B E A C (2) D A B E C (2) C B A E D (2) B D E C A (2) B C A D E (2) A D E B C (2) A C E D B (2) A C B E D (2) E D B A C (1) E D A C B (1) E C B A D (1) E C A B D (1) D E B C A (1) D B C E A (1) D B A C E (1) C B E D A (1) C B E A D (1) C A B D E (1) B C D E A (1) A D B C E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 10 4 18 B 4 0 6 0 2 C -10 -6 0 -2 2 D -4 0 2 0 4 E -18 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.629076 C: 0.000000 D: 0.370924 E: 0.000000 Sum of squares = 0.533321435866 Cumulative probabilities = A: 0.000000 B: 0.629076 C: 0.629076 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 4 18 B 4 0 6 0 2 C -10 -6 0 -2 2 D -4 0 2 0 4 E -18 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500318 C: 0.000000 D: 0.499682 E: 0.000000 Sum of squares = 0.500000201738 Cumulative probabilities = A: 0.000000 B: 0.500318 C: 0.500318 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=24 B=18 E=15 C=15 so E is eliminated. Round 2 votes counts: A=34 D=31 B=18 C=17 so C is eliminated. Round 3 votes counts: A=43 D=31 B=26 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:206 D:201 C:192 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 4 18 B 4 0 6 0 2 C -10 -6 0 -2 2 D -4 0 2 0 4 E -18 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500318 C: 0.000000 D: 0.499682 E: 0.000000 Sum of squares = 0.500000201738 Cumulative probabilities = A: 0.000000 B: 0.500318 C: 0.500318 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 4 18 B 4 0 6 0 2 C -10 -6 0 -2 2 D -4 0 2 0 4 E -18 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500318 C: 0.000000 D: 0.499682 E: 0.000000 Sum of squares = 0.500000201738 Cumulative probabilities = A: 0.000000 B: 0.500318 C: 0.500318 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 4 18 B 4 0 6 0 2 C -10 -6 0 -2 2 D -4 0 2 0 4 E -18 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500318 C: 0.000000 D: 0.499682 E: 0.000000 Sum of squares = 0.500000201738 Cumulative probabilities = A: 0.000000 B: 0.500318 C: 0.500318 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1878: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) D C A B E (9) E B A D C (8) E B D C A (6) C D A B E (6) C D A E B (5) E A B C D (4) B A E D C (4) E D C B A (3) E C D A B (3) D C E B A (3) C D E A B (3) B E D C A (3) A B D C E (3) E B C D A (2) E A C D B (2) C A D E B (2) B E A D C (2) B D C A E (2) B D A C E (2) A D C B E (2) A C D B E (2) A B C D E (2) E C D B A (1) E B D A C (1) E A C B D (1) D C B A E (1) D C A E B (1) B E D A C (1) B D E A C (1) B D C E A (1) B A D E C (1) A E C D B (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 -8 -8 B 2 0 6 10 -16 C 2 -6 0 -8 -10 D 8 -10 8 0 -8 E 8 16 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -2 -8 -8 B 2 0 6 10 -16 C 2 -6 0 -8 -10 D 8 -10 8 0 -8 E 8 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=17 C=16 D=14 A=13 so A is eliminated. Round 2 votes counts: E=43 B=22 C=19 D=16 so D is eliminated. Round 3 votes counts: E=43 C=35 B=22 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:201 D:199 A:190 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 -8 -8 B 2 0 6 10 -16 C 2 -6 0 -8 -10 D 8 -10 8 0 -8 E 8 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -8 -8 B 2 0 6 10 -16 C 2 -6 0 -8 -10 D 8 -10 8 0 -8 E 8 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -8 -8 B 2 0 6 10 -16 C 2 -6 0 -8 -10 D 8 -10 8 0 -8 E 8 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1879: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (13) A C E B D (11) E B D A C (8) C B D A E (8) A E C B D (7) C A D B E (6) C A B D E (5) E D B A C (4) D B C E A (4) A E D B C (4) A C E D B (4) D B E A C (3) E B A D C (2) D A B E C (2) C A E B D (2) B D E C A (2) E A D B C (1) E A C B D (1) E A B D C (1) D E B A C (1) D E A B C (1) D C B E A (1) C D B E A (1) C B E D A (1) C B D E A (1) C A B E D (1) B E D C A (1) A E C D B (1) A E B D C (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 8 -2 8 B 0 0 -2 6 0 C -8 2 0 2 -8 D 2 -6 -2 0 -2 E -8 0 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.485688 B: 0.514312 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500409635742 Cumulative probabilities = A: 0.485688 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -2 8 B 0 0 -2 6 0 C -8 2 0 2 -8 D 2 -6 -2 0 -2 E -8 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=25 C=25 E=17 B=3 so B is eliminated. Round 2 votes counts: A=30 D=27 C=25 E=18 so E is eliminated. Round 3 votes counts: D=40 A=35 C=25 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:207 B:202 E:201 D:196 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 -2 8 B 0 0 -2 6 0 C -8 2 0 2 -8 D 2 -6 -2 0 -2 E -8 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -2 8 B 0 0 -2 6 0 C -8 2 0 2 -8 D 2 -6 -2 0 -2 E -8 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -2 8 B 0 0 -2 6 0 C -8 2 0 2 -8 D 2 -6 -2 0 -2 E -8 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1880: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) A B C E D (7) E B C D A (5) D E A B C (5) D C E B A (5) A C B D E (5) A B E C D (5) D E B C A (4) D A C B E (4) B E C A D (4) A D C B E (4) E D B C A (3) D E C B A (3) C A B E D (3) B E A C D (3) A C D B E (3) A C B E D (3) D E C A B (2) D A E B C (2) B A E C D (2) E D C B A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C A D (1) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D A B E (1) C B E A D (1) C A D B E (1) C A B D E (1) B E A D C (1) B C E A D (1) B A E D C (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 18 0 -4 10 B -18 0 -4 -4 6 C 0 4 0 0 8 D 4 4 0 0 8 E -10 -6 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.405957 D: 0.594043 E: 0.000000 Sum of squares = 0.517688027971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.405957 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 -4 10 B -18 0 -4 -4 6 C 0 4 0 0 8 D 4 4 0 0 8 E -10 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=28 E=13 B=13 C=9 so C is eliminated. Round 2 votes counts: D=39 A=33 E=14 B=14 so E is eliminated. Round 3 votes counts: D=43 A=33 B=24 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:212 D:208 C:206 B:190 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 18 0 -4 10 B -18 0 -4 -4 6 C 0 4 0 0 8 D 4 4 0 0 8 E -10 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 -4 10 B -18 0 -4 -4 6 C 0 4 0 0 8 D 4 4 0 0 8 E -10 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 -4 10 B -18 0 -4 -4 6 C 0 4 0 0 8 D 4 4 0 0 8 E -10 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1881: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) A C B E D (6) A C B D E (6) A E D C B (5) E D A B C (4) E B D C A (4) E A D C B (4) C B D A E (4) A E B C D (4) E D B A C (3) E B C D A (3) E A B C D (3) D E B C A (3) C B A D E (3) C A B D E (3) A E C B D (3) E A D B C (2) D C B A E (2) D B E C A (2) D B C E A (2) B C E D A (2) B C A E D (2) A C E B D (2) E D A C B (1) E B D A C (1) E B C A D (1) D E C B A (1) D E A C B (1) D C B E A (1) D A C B E (1) C D B A E (1) C D A B E (1) B E C A D (1) B D C E A (1) B C D E A (1) B C D A E (1) B C A D E (1) A E C D B (1) A D E C B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 0 0 -2 B 0 0 4 6 -14 C 0 -4 0 2 -16 D 0 -6 -2 0 -26 E 2 14 16 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 0 -2 B 0 0 4 6 -14 C 0 -4 0 2 -16 D 0 -6 -2 0 -26 E 2 14 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=30 D=13 C=12 B=9 so B is eliminated. Round 2 votes counts: E=37 A=30 C=19 D=14 so D is eliminated. Round 3 votes counts: E=44 A=31 C=25 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 A:199 B:198 C:191 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 0 -2 B 0 0 4 6 -14 C 0 -4 0 2 -16 D 0 -6 -2 0 -26 E 2 14 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 -2 B 0 0 4 6 -14 C 0 -4 0 2 -16 D 0 -6 -2 0 -26 E 2 14 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 -2 B 0 0 4 6 -14 C 0 -4 0 2 -16 D 0 -6 -2 0 -26 E 2 14 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1882: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (7) C D E A B (6) B A E C D (6) E B D A C (5) D C E A B (5) A B E C D (5) C D A E B (4) C D A B E (4) C A D E B (4) B E D A C (4) A B C E D (4) E D C A B (3) E B A D C (3) D E C B A (3) D E B C A (3) D C B E A (3) B D E C A (3) C A D B E (2) B A C D E (2) A E C B D (2) A C D E B (2) A C B D E (2) E D C B A (1) E D B C A (1) E C A D B (1) E B D C A (1) E A B D C (1) D E C A B (1) D C E B A (1) D C B A E (1) D B E C A (1) C A E D B (1) B C D A E (1) B A E D C (1) B A D C E (1) B A C E D (1) A C E D B (1) A C E B D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -2 -6 B -2 0 2 2 0 C 0 -2 0 2 -4 D 2 -2 -2 0 2 E 6 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.615561 C: 0.000000 D: 0.000000 E: 0.384439 Sum of squares = 0.526708581746 Cumulative probabilities = A: 0.000000 B: 0.615561 C: 0.615561 D: 0.615561 E: 1.000000 A B C D E A 0 2 0 -2 -6 B -2 0 2 2 0 C 0 -2 0 2 -4 D 2 -2 -2 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500206 C: 0.000000 D: 0.000000 E: 0.499794 Sum of squares = 0.500000084827 Cumulative probabilities = A: 0.000000 B: 0.500206 C: 0.500206 D: 0.500206 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=21 A=19 D=18 E=16 so E is eliminated. Round 2 votes counts: B=35 D=23 C=22 A=20 so A is eliminated. Round 3 votes counts: B=46 C=31 D=23 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:204 B:201 D:200 C:198 A:197 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 0 -2 -6 B -2 0 2 2 0 C 0 -2 0 2 -4 D 2 -2 -2 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500206 C: 0.000000 D: 0.000000 E: 0.499794 Sum of squares = 0.500000084827 Cumulative probabilities = A: 0.000000 B: 0.500206 C: 0.500206 D: 0.500206 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 -6 B -2 0 2 2 0 C 0 -2 0 2 -4 D 2 -2 -2 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500206 C: 0.000000 D: 0.000000 E: 0.499794 Sum of squares = 0.500000084827 Cumulative probabilities = A: 0.000000 B: 0.500206 C: 0.500206 D: 0.500206 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 -6 B -2 0 2 2 0 C 0 -2 0 2 -4 D 2 -2 -2 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500206 C: 0.000000 D: 0.000000 E: 0.499794 Sum of squares = 0.500000084827 Cumulative probabilities = A: 0.000000 B: 0.500206 C: 0.500206 D: 0.500206 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1883: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (10) A D C E B (8) B C E D A (6) A D C B E (6) E B C D A (5) D A E C B (5) B C E A D (5) B C A E D (5) A D E B C (4) A C D B E (4) C B E D A (3) C B A E D (3) C A B D E (3) B E C D A (3) E D B C A (2) E D A C B (2) E B D C A (2) D E A C B (2) D E A B C (2) A D E C B (2) E D B A C (1) E B D A C (1) D A E B C (1) C D A B E (1) C B E A D (1) B E D A C (1) B C A D E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 0 6 8 B -2 0 -2 0 6 C 0 2 0 2 14 D -6 0 -2 0 0 E -8 -6 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.782380 B: 0.000000 C: 0.217620 D: 0.000000 E: 0.000000 Sum of squares = 0.659476387555 Cumulative probabilities = A: 0.782380 B: 0.782380 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 6 8 B -2 0 -2 0 6 C 0 2 0 2 14 D -6 0 -2 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 C=21 B=21 D=10 so D is eliminated. Round 2 votes counts: A=31 E=27 C=21 B=21 so C is eliminated. Round 3 votes counts: B=38 A=35 E=27 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:209 A:208 B:201 D:196 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 6 8 B -2 0 -2 0 6 C 0 2 0 2 14 D -6 0 -2 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 8 B -2 0 -2 0 6 C 0 2 0 2 14 D -6 0 -2 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 8 B -2 0 -2 0 6 C 0 2 0 2 14 D -6 0 -2 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1884: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) D B C E A (6) B C D A E (6) B C A D E (6) E A D C B (5) D E A B C (5) C B A E D (5) E D A C B (4) E A C D B (4) A E C B D (4) D E C B A (3) D B E C A (3) D B C A E (3) C B D A E (3) B C A E D (3) A C E B D (3) E A C B D (2) D C E B A (2) B D C A E (2) B A C E D (2) A E B C D (2) E C D A B (1) E C A B D (1) D E C A B (1) D E B C A (1) D C B E A (1) D A B E C (1) C E A B D (1) C D B E A (1) C B E D A (1) C B D E A (1) C A E B D (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B D C (1) A D B E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 -8 -4 B 0 0 -10 -6 -4 C 4 10 0 0 0 D 8 6 0 0 12 E 4 4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.475108 D: 0.524892 E: 0.000000 Sum of squares = 0.501239193952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.475108 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -8 -4 B 0 0 -10 -6 -4 C 4 10 0 0 0 D 8 6 0 0 12 E 4 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=20 E=17 A=15 C=13 so C is eliminated. Round 2 votes counts: D=36 B=30 E=18 A=16 so A is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 C:207 E:198 A:192 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -8 -4 B 0 0 -10 -6 -4 C 4 10 0 0 0 D 8 6 0 0 12 E 4 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -8 -4 B 0 0 -10 -6 -4 C 4 10 0 0 0 D 8 6 0 0 12 E 4 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -8 -4 B 0 0 -10 -6 -4 C 4 10 0 0 0 D 8 6 0 0 12 E 4 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1885: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) A C D B E (7) E B A C D (6) B E C A D (6) E B A D C (5) D C A E B (5) C D A B E (5) E B D C A (4) B E C D A (4) B E A C D (4) E D B C A (3) D A C E B (3) B E D C A (3) A D C E B (3) A C D E B (3) E B D A C (2) E A D C B (2) E A B C D (2) D C E B A (2) A E D C B (2) A E C D B (2) A E B C D (2) A B C D E (2) E D C A B (1) E A D B C (1) E A B D C (1) D E C B A (1) D C E A B (1) D C B E A (1) D A E C B (1) C A D B E (1) B D E C A (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A D E C B (1) Total count = 100 A B C D E A 0 6 0 4 -4 B -6 0 2 -6 -6 C 0 -2 0 -2 -8 D -4 6 2 0 -4 E 4 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 0 4 -4 B -6 0 2 -6 -6 C 0 -2 0 -2 -8 D -4 6 2 0 -4 E 4 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 D=22 A=22 C=6 so C is eliminated. Round 2 votes counts: E=27 D=27 B=23 A=23 so B is eliminated. Round 3 votes counts: E=45 D=30 A=25 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:203 D:200 C:194 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 4 -4 B -6 0 2 -6 -6 C 0 -2 0 -2 -8 D -4 6 2 0 -4 E 4 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 4 -4 B -6 0 2 -6 -6 C 0 -2 0 -2 -8 D -4 6 2 0 -4 E 4 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 4 -4 B -6 0 2 -6 -6 C 0 -2 0 -2 -8 D -4 6 2 0 -4 E 4 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1886: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) B C D A E (6) C B A D E (5) A D E C B (5) E A D C B (4) E A D B C (4) B D E C A (4) B C E D A (4) A D C E B (4) E D A B C (3) E B D C A (3) E B D A C (3) E A C D B (3) D A B C E (3) C A D B E (3) B E D C A (3) B D C A E (3) B C D E A (3) D B C A E (2) D A E C B (2) D A C B E (2) C A B D E (2) B E C D A (2) B D C E A (2) A E D C B (2) E D B A C (1) E C B A D (1) E B C D A (1) D E A C B (1) D E A B C (1) D C A B E (1) D A C E B (1) C E A B D (1) C D B A E (1) C D A B E (1) C B E A D (1) C B D A E (1) C B A E D (1) B C E A D (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -16 -8 -4 B 8 0 10 6 4 C 16 -10 0 -10 2 D 8 -6 10 0 10 E 4 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -8 -4 B 8 0 10 6 4 C 16 -10 0 -10 2 D 8 -6 10 0 10 E 4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=28 C=16 A=14 D=13 so D is eliminated. Round 2 votes counts: E=31 B=30 A=22 C=17 so C is eliminated. Round 3 votes counts: B=39 E=32 A=29 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:211 C:199 E:194 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 -4 B 8 0 10 6 4 C 16 -10 0 -10 2 D 8 -6 10 0 10 E 4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 -4 B 8 0 10 6 4 C 16 -10 0 -10 2 D 8 -6 10 0 10 E 4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 -4 B 8 0 10 6 4 C 16 -10 0 -10 2 D 8 -6 10 0 10 E 4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1887: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) A C E B D (7) D C A B E (6) D B A C E (5) C A D E B (5) E B D A C (4) C A D B E (4) B D E A C (4) B D A E C (4) E B D C A (3) C D A E B (3) C A E B D (3) B E D A C (3) B A D C E (3) E D C B A (2) E C A B D (2) D E C B A (2) D C E B A (2) D B E C A (2) D B C A E (2) D A B C E (2) C E A D B (2) C D A B E (2) B A E D C (2) A C B E D (2) A C B D E (2) E D B C A (1) E C B D A (1) E B C A D (1) E B A C D (1) E A C B D (1) E A B C D (1) D B A E C (1) D A C B E (1) C E D A B (1) C A E D B (1) B E A D C (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 10 -20 14 B 8 0 2 -6 14 C -10 -2 0 -20 8 D 20 6 20 0 20 E -14 -14 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 -20 14 B 8 0 2 -6 14 C -10 -2 0 -20 8 D 20 6 20 0 20 E -14 -14 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=21 B=18 E=17 A=12 so A is eliminated. Round 2 votes counts: D=32 C=32 B=19 E=17 so E is eliminated. Round 3 votes counts: C=36 D=35 B=29 so B is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:233 B:209 A:198 C:188 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 10 -20 14 B 8 0 2 -6 14 C -10 -2 0 -20 8 D 20 6 20 0 20 E -14 -14 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 -20 14 B 8 0 2 -6 14 C -10 -2 0 -20 8 D 20 6 20 0 20 E -14 -14 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 -20 14 B 8 0 2 -6 14 C -10 -2 0 -20 8 D 20 6 20 0 20 E -14 -14 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1888: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) D E C B A (7) E C D B A (6) B A D E C (6) D C E A B (5) A B D C E (5) B A C E D (4) D E C A B (3) D A C E B (3) D A B E C (3) A D C B E (3) A C B D E (3) A B C E D (3) D E B C A (2) D B E C A (2) C E D A B (2) C D E A B (2) C A E B D (2) B E C D A (2) B E C A D (2) B E A C D (2) B D E A C (2) B A E D C (2) B A E C D (2) A D B E C (2) A D B C E (2) A C D E B (2) A B C D E (2) E D C B A (1) E C B A D (1) E B C D A (1) D B E A C (1) C E D B A (1) C E A B D (1) C D A E B (1) C A D E B (1) B E D C A (1) A C E D B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 0 -4 -2 B 4 0 -6 0 2 C 0 6 0 -2 -12 D 4 0 2 0 16 E 2 -2 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166020 C: 0.000000 D: 0.833980 E: 0.000000 Sum of squares = 0.723084803052 Cumulative probabilities = A: 0.000000 B: 0.166020 C: 0.166020 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -4 -2 B 4 0 -6 0 2 C 0 6 0 -2 -12 D 4 0 2 0 16 E 2 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.62500004612 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 B=23 E=16 C=10 so C is eliminated. Round 2 votes counts: D=29 A=28 B=23 E=20 so E is eliminated. Round 3 votes counts: D=39 B=32 A=29 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:200 E:198 C:196 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -4 -2 B 4 0 -6 0 2 C 0 6 0 -2 -12 D 4 0 2 0 16 E 2 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.62500004612 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -4 -2 B 4 0 -6 0 2 C 0 6 0 -2 -12 D 4 0 2 0 16 E 2 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.62500004612 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -4 -2 B 4 0 -6 0 2 C 0 6 0 -2 -12 D 4 0 2 0 16 E 2 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.62500004612 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1889: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) B C D A E (9) C B E D A (8) C B D A E (7) E A D B C (6) A D E B C (6) E C A B D (5) E A D C B (4) D A B C E (4) C E B A D (4) C B E A D (4) D A B E C (3) B D C A E (3) A E D C B (3) E C A D B (2) E B C A D (2) E A B D C (2) D A E C B (2) B C D E A (2) A E D B C (2) E C B A D (1) E B A D C (1) E B A C D (1) E A C D B (1) D C B A E (1) D A C B E (1) C E B D A (1) B E C A D (1) B E A C D (1) B C E A D (1) B A E D C (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -2 0 6 B -4 0 10 8 -6 C 2 -10 0 0 -10 D 0 -8 0 0 -2 E -6 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999886 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 0 6 B -4 0 10 8 -6 C 2 -10 0 0 -10 D 0 -8 0 0 -2 E -6 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.46874999975 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 D=20 B=18 A=13 so A is eliminated. Round 2 votes counts: E=30 D=28 C=24 B=18 so B is eliminated. Round 3 votes counts: C=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:206 A:204 B:204 D:195 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 0 6 B -4 0 10 8 -6 C 2 -10 0 0 -10 D 0 -8 0 0 -2 E -6 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.46874999975 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 0 6 B -4 0 10 8 -6 C 2 -10 0 0 -10 D 0 -8 0 0 -2 E -6 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.46874999975 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 0 6 B -4 0 10 8 -6 C 2 -10 0 0 -10 D 0 -8 0 0 -2 E -6 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.46874999975 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1890: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) D C A B E (6) D C B E A (5) C D B A E (5) C D A B E (5) E B A C D (4) B E C A D (4) B C E A D (4) E A B D C (3) D C B A E (3) D A E C B (3) C B D E A (3) B E A C D (3) B C A E D (3) A D E C B (3) E B A D C (2) E A D B C (2) E A B C D (2) D E B C A (2) D E B A C (2) D E A C B (2) C D B E A (2) C A D B E (2) C A B D E (2) B E C D A (2) A E B C D (2) E D A B C (1) E B D C A (1) D E A B C (1) D B E C A (1) D A C E B (1) C B A E D (1) C B A D E (1) C A B E D (1) B C E D A (1) B A E C D (1) A E D B C (1) A E B D C (1) A D C E B (1) A C E B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -22 -6 6 B 0 0 -10 -10 16 C 22 10 0 4 12 D 6 10 -4 0 14 E -6 -16 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -22 -6 6 B 0 0 -10 -10 16 C 22 10 0 4 12 D 6 10 -4 0 14 E -6 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=22 B=18 E=15 A=12 so A is eliminated. Round 2 votes counts: D=37 C=24 B=20 E=19 so E is eliminated. Round 3 votes counts: D=41 B=35 C=24 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:224 D:213 B:198 A:189 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -22 -6 6 B 0 0 -10 -10 16 C 22 10 0 4 12 D 6 10 -4 0 14 E -6 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -22 -6 6 B 0 0 -10 -10 16 C 22 10 0 4 12 D 6 10 -4 0 14 E -6 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -22 -6 6 B 0 0 -10 -10 16 C 22 10 0 4 12 D 6 10 -4 0 14 E -6 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1891: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (16) A C B D E (12) D B E A C (9) C A E B D (8) C A E D B (6) B D A C E (6) B D E A C (5) E C A D B (4) E C A B D (4) A C E B D (4) D B E C A (3) C E A D B (3) A C D B E (3) A B C D E (3) E C D B A (2) E C D A B (2) E B D C A (2) B D E C A (2) E C B D A (1) D B A E C (1) D B A C E (1) B E D C A (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -8 -2 -8 B 0 0 2 0 -4 C 8 -2 0 4 -4 D 2 0 -4 0 -6 E 8 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -8 -2 -8 B 0 0 2 0 -4 C 8 -2 0 4 -4 D 2 0 -4 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=23 C=17 B=15 D=14 so D is eliminated. Round 2 votes counts: E=31 B=29 A=23 C=17 so C is eliminated. Round 3 votes counts: A=37 E=34 B=29 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:203 B:199 D:196 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -2 -8 B 0 0 2 0 -4 C 8 -2 0 4 -4 D 2 0 -4 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 -8 B 0 0 2 0 -4 C 8 -2 0 4 -4 D 2 0 -4 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 -8 B 0 0 2 0 -4 C 8 -2 0 4 -4 D 2 0 -4 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1892: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (7) E C D A B (5) D C B E A (5) D B C A E (5) A B C E D (5) E D C A B (4) C D B A E (4) E D B A C (3) D C E B A (3) D C B A E (3) C D E B A (3) B D A C E (3) B A C D E (3) E C A D B (2) E A C B D (2) E A B C D (2) D B E C A (2) D B E A C (2) D B C E A (2) D B A C E (2) C E D A B (2) C E A D B (2) C A E D B (2) C A E B D (2) B A D E C (2) A E B C D (2) A B E C D (2) E D A B C (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E B C A (1) D C E A B (1) C D A E B (1) C B A D E (1) C A D B E (1) C A B E D (1) B D C A E (1) B D A E C (1) B C A D E (1) B A E D C (1) B A D C E (1) A E C B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -22 -22 -6 B 0 0 -10 -32 0 C 22 10 0 8 26 D 22 32 -8 0 12 E 6 0 -26 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -22 -22 -6 B 0 0 -10 -32 0 C 22 10 0 8 26 D 22 32 -8 0 12 E 6 0 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 E=23 B=13 A=12 so A is eliminated. Round 2 votes counts: C=27 E=26 D=26 B=21 so B is eliminated. Round 3 votes counts: C=37 D=34 E=29 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:233 D:229 E:184 B:179 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -22 -22 -6 B 0 0 -10 -32 0 C 22 10 0 8 26 D 22 32 -8 0 12 E 6 0 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -22 -22 -6 B 0 0 -10 -32 0 C 22 10 0 8 26 D 22 32 -8 0 12 E 6 0 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -22 -22 -6 B 0 0 -10 -32 0 C 22 10 0 8 26 D 22 32 -8 0 12 E 6 0 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1893: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (15) B A C D E (9) C D A E B (7) B E D C A (6) D E C A B (5) B E A D C (5) E D C B A (3) E D A C B (3) D C E A B (3) B A C E D (3) A E C D B (3) A C B D E (3) E D B C A (2) E B D C A (2) E A D C B (2) E A C D B (2) D C E B A (2) D C A E B (2) C D A B E (2) B E D A C (2) B E A C D (2) B D E C A (2) B C D A E (2) E A B D C (1) D C B A E (1) C B D A E (1) C A D E B (1) C A D B E (1) B D C A E (1) B C A D E (1) B A E C D (1) A E C B D (1) A C D E B (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -18 -22 -14 B -10 0 -18 -10 -10 C 18 18 0 -14 -14 D 22 10 14 0 -10 E 14 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -18 -22 -14 B -10 0 -18 -10 -10 C 18 18 0 -14 -14 D 22 10 14 0 -10 E 14 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=30 D=13 C=12 A=11 so A is eliminated. Round 2 votes counts: B=36 E=34 C=17 D=13 so D is eliminated. Round 3 votes counts: E=39 B=36 C=25 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:218 C:204 A:178 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -18 -22 -14 B -10 0 -18 -10 -10 C 18 18 0 -14 -14 D 22 10 14 0 -10 E 14 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -18 -22 -14 B -10 0 -18 -10 -10 C 18 18 0 -14 -14 D 22 10 14 0 -10 E 14 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -18 -22 -14 B -10 0 -18 -10 -10 C 18 18 0 -14 -14 D 22 10 14 0 -10 E 14 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1894: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (5) C B E D A (5) B E D C A (5) E A D B C (4) C B D E A (4) C B D A E (4) A D E B C (4) E B D A C (3) E A B D C (3) E A B C D (3) D B A C E (3) C E B A D (3) C D B A E (3) C D A B E (3) B D E C A (3) B C E D A (3) A D E C B (3) A D C E B (3) A C D E B (3) E C B A D (2) E B A D C (2) E B A C D (2) D C A B E (2) D A E B C (2) D A C B E (2) D A B E C (2) C B E A D (2) B E D A C (2) B C D E A (2) E A D C B (1) E A C B D (1) D B C A E (1) D A B C E (1) C A E D B (1) C A E B D (1) C A D E B (1) C A D B E (1) B E C D A (1) B D E A C (1) A E D C B (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 -4 -4 -14 B 12 0 4 14 0 C 4 -4 0 4 -2 D 4 -14 -4 0 -4 E 14 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.432002 C: 0.000000 D: 0.000000 E: 0.567998 Sum of squares = 0.509247340723 Cumulative probabilities = A: 0.000000 B: 0.432002 C: 0.432002 D: 0.432002 E: 1.000000 A B C D E A 0 -12 -4 -4 -14 B 12 0 4 14 0 C 4 -4 0 4 -2 D 4 -14 -4 0 -4 E 14 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=17 A=16 D=13 so D is eliminated. Round 2 votes counts: C=30 E=26 A=23 B=21 so B is eliminated. Round 3 votes counts: E=38 C=36 A=26 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:210 C:201 D:191 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -4 -14 B 12 0 4 14 0 C 4 -4 0 4 -2 D 4 -14 -4 0 -4 E 14 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -4 -14 B 12 0 4 14 0 C 4 -4 0 4 -2 D 4 -14 -4 0 -4 E 14 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -4 -14 B 12 0 4 14 0 C 4 -4 0 4 -2 D 4 -14 -4 0 -4 E 14 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1895: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D B C E A (8) D B E A C (6) A C E D B (6) A C E B D (5) C D B A E (4) C A E B D (4) E B D A C (3) E B A D C (3) E A B D C (3) D C B A E (3) D B E C A (3) C A D E B (3) C A D B E (3) E B A C D (2) D C B E A (2) D A E B C (2) C D B E A (2) C B E A D (2) C B D E A (2) C A E D B (2) B D E C A (2) B D E A C (2) B C D E A (2) E B C A D (1) E A B C D (1) D E B A C (1) D E A B C (1) D B C A E (1) D B A E C (1) C B E D A (1) C B D A E (1) C B A E D (1) B E C A D (1) B E A C D (1) B D C E A (1) B C E D A (1) B C E A D (1) A E B C D (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -2 2 -4 B 16 0 -4 0 2 C 2 4 0 14 12 D -2 0 -14 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 2 -4 B 16 0 -4 0 2 C 2 4 0 14 12 D -2 0 -14 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 A=23 E=13 B=11 so B is eliminated. Round 2 votes counts: D=33 C=29 A=23 E=15 so E is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:216 B:207 D:194 E:193 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -2 2 -4 B 16 0 -4 0 2 C 2 4 0 14 12 D -2 0 -14 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 2 -4 B 16 0 -4 0 2 C 2 4 0 14 12 D -2 0 -14 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 2 -4 B 16 0 -4 0 2 C 2 4 0 14 12 D -2 0 -14 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1896: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) B E A D C (9) C D A E B (6) C D A B E (6) A D B E C (6) E B C A D (5) C D B A E (5) E C A D B (4) B A D E C (4) E B A C D (3) C D E A B (3) A B D E C (3) E B A D C (2) D C A E B (2) D A C E B (2) D A C B E (2) B D A C E (2) B C E D A (2) A D E C B (2) A D C E B (2) E C D A B (1) E C B D A (1) E C B A D (1) E A D C B (1) E A D B C (1) E A B D C (1) D C A B E (1) D B A C E (1) C E D B A (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C A D (1) B E A C D (1) B C D E A (1) B C D A E (1) B A E D C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 -10 -2 -4 B -12 0 -8 -20 0 C 10 8 0 12 4 D 2 20 -12 0 6 E 4 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 -2 -4 B -12 0 -8 -20 0 C 10 8 0 12 4 D 2 20 -12 0 6 E 4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=22 E=20 A=15 D=8 so D is eliminated. Round 2 votes counts: C=38 B=23 E=20 A=19 so A is eliminated. Round 3 votes counts: C=44 B=33 E=23 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:208 A:198 E:197 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 -2 -4 B -12 0 -8 -20 0 C 10 8 0 12 4 D 2 20 -12 0 6 E 4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 -2 -4 B -12 0 -8 -20 0 C 10 8 0 12 4 D 2 20 -12 0 6 E 4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 -2 -4 B -12 0 -8 -20 0 C 10 8 0 12 4 D 2 20 -12 0 6 E 4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1897: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) A B D E C (10) B A D E C (7) C E A D B (5) B D A E C (5) D E B C A (4) C E D A B (4) C A E B D (4) B A D C E (4) E D C A B (3) E C D A B (3) D E C B A (3) D B A E C (3) E C D B A (2) E C A D B (2) C E A B D (2) C A B E D (2) B D A C E (2) A B E D C (2) A B C E D (2) E C A B D (1) E A D B C (1) D E B A C (1) D E A B C (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C E A (1) C D E B A (1) B D C A E (1) B A C D E (1) A E C B D (1) A E B D C (1) A E B C D (1) A C E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 -2 -2 B 4 0 0 -2 -10 C 8 0 0 -4 -6 D 2 2 4 0 -4 E 2 10 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 -2 -2 B 4 0 0 -2 -10 C 8 0 0 -4 -6 D 2 2 4 0 -4 E 2 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=20 A=20 D=16 E=12 so E is eliminated. Round 2 votes counts: C=40 A=21 B=20 D=19 so D is eliminated. Round 3 votes counts: C=47 B=31 A=22 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:211 D:202 C:199 B:196 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 -2 -2 B 4 0 0 -2 -10 C 8 0 0 -4 -6 D 2 2 4 0 -4 E 2 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -2 -2 B 4 0 0 -2 -10 C 8 0 0 -4 -6 D 2 2 4 0 -4 E 2 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -2 -2 B 4 0 0 -2 -10 C 8 0 0 -4 -6 D 2 2 4 0 -4 E 2 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1898: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) C A B E D (11) C D E A B (8) D E B A C (6) E D B A C (4) B A E D C (4) A B C E D (4) E C D A B (3) E A B D C (3) D B A E C (3) C D A B E (3) C B A D E (3) A C B E D (3) E D C A B (2) E C A B D (2) E B A D C (2) E A B C D (2) D E C B A (2) D C B A E (2) D B E A C (2) D B C A E (2) D B A C E (2) C D B A E (2) B D A E C (2) B A E C D (2) B A C D E (2) D E C A B (1) C E D A B (1) C E A B D (1) C A E D B (1) C A E B D (1) C A D B E (1) B A C E D (1) Total count = 100 A B C D E A 0 18 -16 10 22 B -18 0 -18 10 22 C 16 18 0 26 20 D -10 -10 -26 0 6 E -22 -22 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -16 10 22 B -18 0 -18 10 22 C 16 18 0 26 20 D -10 -10 -26 0 6 E -22 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 D=20 E=18 B=11 A=7 so A is eliminated. Round 2 votes counts: C=47 D=20 E=18 B=15 so B is eliminated. Round 3 votes counts: C=54 E=24 D=22 so D is eliminated. Round 4 votes counts: C=60 E=40 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:240 A:217 B:198 D:180 E:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -16 10 22 B -18 0 -18 10 22 C 16 18 0 26 20 D -10 -10 -26 0 6 E -22 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -16 10 22 B -18 0 -18 10 22 C 16 18 0 26 20 D -10 -10 -26 0 6 E -22 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -16 10 22 B -18 0 -18 10 22 C 16 18 0 26 20 D -10 -10 -26 0 6 E -22 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1899: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) E A B D C (7) D C A B E (7) E B C A D (6) E A D B C (6) D A C E B (6) A D E B C (6) D A B C E (5) C B D E A (5) C B D A E (5) C D B A E (4) C B E D A (4) E A D C B (3) B E C A D (3) B C E A D (3) E B A C D (2) C E B D A (2) A E D B C (2) A D E C B (2) A D B E C (2) E C D A B (1) E B A D C (1) E A B C D (1) D C A E B (1) D A E C B (1) C D A E B (1) C D A B E (1) C B E A D (1) B C E D A (1) B A D E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 24 10 -4 8 B -24 0 -4 -14 4 C -10 4 0 -20 10 D 4 14 20 0 12 E -8 -4 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 10 -4 8 B -24 0 -4 -14 4 C -10 4 0 -20 10 D 4 14 20 0 12 E -8 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 C=23 A=14 B=8 so B is eliminated. Round 2 votes counts: E=30 D=28 C=27 A=15 so A is eliminated. Round 3 votes counts: D=40 E=33 C=27 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:219 C:192 E:183 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 10 -4 8 B -24 0 -4 -14 4 C -10 4 0 -20 10 D 4 14 20 0 12 E -8 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 10 -4 8 B -24 0 -4 -14 4 C -10 4 0 -20 10 D 4 14 20 0 12 E -8 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 10 -4 8 B -24 0 -4 -14 4 C -10 4 0 -20 10 D 4 14 20 0 12 E -8 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1900: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (13) B E A D C (13) C D A E B (12) E D C B A (10) A B C D E (7) E B D C A (6) A C B D E (6) B E D A C (5) B A E C D (5) A C D B E (4) C A D E B (3) B A C E D (3) D E C A B (2) C A D B E (2) B A E D C (2) E D B C A (1) E B D A C (1) E B A D C (1) C D E A B (1) B E D C A (1) B A C D E (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -2 -4 -8 B -2 0 -8 2 -2 C 2 8 0 -10 6 D 4 -2 10 0 4 E 8 2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.100000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999852 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -4 -8 B -2 0 -8 2 -2 C 2 8 0 -10 6 D 4 -2 10 0 4 E 8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.100000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999837 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=19 C=18 A=18 D=15 so D is eliminated. Round 2 votes counts: C=31 B=30 E=21 A=18 so A is eliminated. Round 3 votes counts: C=42 B=37 E=21 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:203 E:200 B:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -4 -8 B -2 0 -8 2 -2 C 2 8 0 -10 6 D 4 -2 10 0 4 E 8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.100000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999837 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 -8 B -2 0 -8 2 -2 C 2 8 0 -10 6 D 4 -2 10 0 4 E 8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.100000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999837 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 -8 B -2 0 -8 2 -2 C 2 8 0 -10 6 D 4 -2 10 0 4 E 8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.100000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999837 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1901: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (7) E C D B A (6) A C B E D (6) C E B D A (5) B D A E C (5) D E B C A (4) D B E A C (4) A D E B C (4) A B C D E (4) E D C B A (3) D B E C A (3) D B A E C (3) C E D A B (3) C E A D B (3) C E A B D (3) B D E C A (3) A D B E C (3) A C E B D (3) A B D C E (3) E D A B C (2) E A C D B (2) C E D B A (2) C A E D B (2) C A E B D (2) A E D C B (2) E C D A B (1) D E B A C (1) D A E B C (1) C B E D A (1) C A B D E (1) B D C E A (1) B D A C E (1) B C D E A (1) B A D E C (1) B A C D E (1) A E D B C (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 12 0 4 B -10 0 6 -4 -4 C -12 -6 0 -4 -14 D 0 4 4 0 2 E -4 4 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.301230 B: 0.000000 C: 0.000000 D: 0.698770 E: 0.000000 Sum of squares = 0.579018791364 Cumulative probabilities = A: 0.301230 B: 0.301230 C: 0.301230 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 0 4 B -10 0 6 -4 -4 C -12 -6 0 -4 -14 D 0 4 4 0 2 E -4 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=22 D=16 E=14 B=13 so B is eliminated. Round 2 votes counts: A=37 D=26 C=23 E=14 so E is eliminated. Round 3 votes counts: A=39 D=31 C=30 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:206 D:205 B:194 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 0 4 B -10 0 6 -4 -4 C -12 -6 0 -4 -14 D 0 4 4 0 2 E -4 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 0 4 B -10 0 6 -4 -4 C -12 -6 0 -4 -14 D 0 4 4 0 2 E -4 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 0 4 B -10 0 6 -4 -4 C -12 -6 0 -4 -14 D 0 4 4 0 2 E -4 4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1902: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) D E B A C (7) C A E D B (6) A C B D E (6) D A E C B (5) D B E A C (4) B E D C A (4) B D E A C (4) A C D E B (4) E D C A B (3) E D B C A (3) E B C D A (3) D E A C B (3) C A B E D (3) B A D C E (3) E D B A C (2) D A B E C (2) C E A D B (2) C A E B D (2) C A B D E (2) B E D A C (2) B C E A D (2) B A C D E (2) A D C E B (2) E C D A B (1) E C B D A (1) E C B A D (1) D E A B C (1) D B A E C (1) D A C E B (1) D A C B E (1) C E B A D (1) C D E A B (1) C B E A D (1) C B A E D (1) B E C D A (1) B E C A D (1) B D A C E (1) A D C B E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 8 -16 -10 B 4 0 2 -2 -12 C -8 -2 0 -14 -12 D 16 2 14 0 6 E 10 12 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -16 -10 B 4 0 2 -2 -12 C -8 -2 0 -14 -12 D 16 2 14 0 6 E 10 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=21 B=20 C=19 A=15 so A is eliminated. Round 2 votes counts: C=30 D=28 E=21 B=21 so E is eliminated. Round 3 votes counts: D=36 C=33 B=31 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:214 B:196 A:189 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -16 -10 B 4 0 2 -2 -12 C -8 -2 0 -14 -12 D 16 2 14 0 6 E 10 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -16 -10 B 4 0 2 -2 -12 C -8 -2 0 -14 -12 D 16 2 14 0 6 E 10 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -16 -10 B 4 0 2 -2 -12 C -8 -2 0 -14 -12 D 16 2 14 0 6 E 10 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) C E A D B (9) C A E D B (7) B D E A C (7) A D E C B (7) E D A B C (6) C B A D E (6) B D A E C (6) C B E A D (5) D A B E C (4) E B D A C (3) D A E B C (3) B C E D A (3) B C D A E (3) D E A B C (2) C E B A D (2) B E D A C (2) B D C A E (2) B D A C E (2) A D C E B (2) E D B A C (1) E C A D B (1) C E A B D (1) C B E D A (1) C A D B E (1) B C D E A (1) B C A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 10 14 8 -10 B -10 0 -8 -10 -12 C -14 8 0 -18 -8 D -8 10 18 0 -6 E 10 12 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 8 -10 B -10 0 -8 -10 -12 C -14 8 0 -18 -8 D -8 10 18 0 -6 E 10 12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=27 E=22 A=10 D=9 so D is eliminated. Round 2 votes counts: C=32 B=27 E=24 A=17 so A is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:211 D:207 C:184 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 8 -10 B -10 0 -8 -10 -12 C -14 8 0 -18 -8 D -8 10 18 0 -6 E 10 12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 8 -10 B -10 0 -8 -10 -12 C -14 8 0 -18 -8 D -8 10 18 0 -6 E 10 12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 8 -10 B -10 0 -8 -10 -12 C -14 8 0 -18 -8 D -8 10 18 0 -6 E 10 12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1904: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (16) B A D E C (6) A D B C E (6) C A D B E (5) D A B E C (4) C E B A D (4) C D A E B (4) E C D A B (3) E C B D A (3) D C A E B (3) D A E B C (3) D A C B E (3) C B E A D (3) B E A D C (3) E C D B A (2) E B A D C (2) C E A D B (2) C A B D E (2) B E A C D (2) A D C B E (2) A D B E C (2) A B D C E (2) E D C A B (1) E D B C A (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) D E A C B (1) D A E C B (1) D A C E B (1) D A B C E (1) C E B D A (1) C B A D E (1) C A E D B (1) B E C A D (1) B C A D E (1) B A E D C (1) B A D C E (1) A C B D E (1) Total count = 100 A B C D E A 0 28 -14 -2 2 B -28 0 -20 -24 -6 C 14 20 0 8 20 D 2 24 -8 0 0 E -2 6 -20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 -14 -2 2 B -28 0 -20 -24 -6 C 14 20 0 8 20 D 2 24 -8 0 0 E -2 6 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=17 E=16 B=15 A=13 so A is eliminated. Round 2 votes counts: C=40 D=27 B=17 E=16 so E is eliminated. Round 3 votes counts: C=48 D=29 B=23 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:231 D:209 A:207 E:192 B:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 28 -14 -2 2 B -28 0 -20 -24 -6 C 14 20 0 8 20 D 2 24 -8 0 0 E -2 6 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 -14 -2 2 B -28 0 -20 -24 -6 C 14 20 0 8 20 D 2 24 -8 0 0 E -2 6 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 -14 -2 2 B -28 0 -20 -24 -6 C 14 20 0 8 20 D 2 24 -8 0 0 E -2 6 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1905: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (16) C B E D A (12) A B E D C (9) B E A D C (6) A D E B C (6) D C E B A (3) C D A E B (3) C A B E D (3) B E C D A (3) A E B D C (3) A D B E C (3) A C B E D (3) D E A B C (2) D A E B C (2) D A C E B (2) C B E A D (2) C B A E D (2) C A B D E (2) A B E C D (2) E C B D A (1) E B D C A (1) E B D A C (1) E B A D C (1) D E C B A (1) D E B A C (1) D C A E B (1) D A E C B (1) C E B D A (1) C D E A B (1) C D A B E (1) C A D B E (1) B C A E D (1) A E D B C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -10 -6 -4 B 4 0 -16 6 2 C 10 16 0 10 12 D 6 -6 -10 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -4 B 4 0 -16 6 2 C 10 16 0 10 12 D 6 -6 -10 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=29 D=13 B=10 E=4 so E is eliminated. Round 2 votes counts: C=45 A=29 D=13 B=13 so D is eliminated. Round 3 votes counts: C=50 A=36 B=14 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:198 E:197 D:193 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -4 B 4 0 -16 6 2 C 10 16 0 10 12 D 6 -6 -10 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 -16 6 2 C 10 16 0 10 12 D 6 -6 -10 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 -16 6 2 C 10 16 0 10 12 D 6 -6 -10 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1906: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) B D E A C (11) E B D A C (5) C A D B E (5) B E D C A (5) D B E A C (4) D B A E C (4) B E C D A (4) E A D B C (3) D A B E C (3) B C D E A (3) A E D C B (3) A C D E B (3) E C B A D (2) E B C A D (2) C E B A D (2) C B D A E (2) C B A D E (2) C A D E B (2) C A B E D (2) C A B D E (2) B D E C A (2) B C E D A (2) E D B A C (1) E A C D B (1) E A B D C (1) E A B C D (1) D E B A C (1) D B C A E (1) D B A C E (1) D A E B C (1) C B A E D (1) C A E B D (1) B E D A C (1) B D C E A (1) B D C A E (1) B D A C E (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -6 -8 -4 B 18 0 20 8 16 C 6 -20 0 -2 -14 D 8 -8 2 0 2 E 4 -16 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -8 -4 B 18 0 20 8 16 C 6 -20 0 -2 -14 D 8 -8 2 0 2 E 4 -16 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=30 E=16 D=15 A=8 so A is eliminated. Round 2 votes counts: C=33 B=31 E=20 D=16 so D is eliminated. Round 3 votes counts: B=44 C=33 E=23 so E is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:231 D:202 E:200 C:185 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -6 -8 -4 B 18 0 20 8 16 C 6 -20 0 -2 -14 D 8 -8 2 0 2 E 4 -16 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -8 -4 B 18 0 20 8 16 C 6 -20 0 -2 -14 D 8 -8 2 0 2 E 4 -16 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -8 -4 B 18 0 20 8 16 C 6 -20 0 -2 -14 D 8 -8 2 0 2 E 4 -16 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1907: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) A C B D E (8) D E B C A (7) D B A C E (6) E C B D A (5) E C A B D (5) A D B C E (4) A B D C E (4) E D B C A (3) D E B A C (3) D B A E C (3) D A B E C (3) C E A B D (3) C B A D E (3) C A E B D (3) C A B E D (3) E D C A B (2) E A D C B (2) A B C D E (2) E D A C B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C B A D (1) E A C D B (1) E A C B D (1) D E A B C (1) D B E C A (1) D B C E A (1) D B C A E (1) D A B C E (1) C B E D A (1) C B E A D (1) C B A E D (1) B D C E A (1) B C D A E (1) B C A D E (1) A E C B D (1) A D E B C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -10 -6 -4 B 0 0 -10 -6 -6 C 10 10 0 -10 -6 D 6 6 10 0 4 E 4 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -6 -4 B 0 0 -10 -6 -6 C 10 10 0 -10 -6 D 6 6 10 0 4 E 4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=27 A=22 C=15 B=3 so B is eliminated. Round 2 votes counts: E=33 D=28 A=22 C=17 so C is eliminated. Round 3 votes counts: E=38 A=33 D=29 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:206 C:202 A:190 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -6 -4 B 0 0 -10 -6 -6 C 10 10 0 -10 -6 D 6 6 10 0 4 E 4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -6 -4 B 0 0 -10 -6 -6 C 10 10 0 -10 -6 D 6 6 10 0 4 E 4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -6 -4 B 0 0 -10 -6 -6 C 10 10 0 -10 -6 D 6 6 10 0 4 E 4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1908: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) E D B C A (8) A C B E D (8) D E B C A (7) C A E B D (5) B E D A C (5) E B D A C (4) C A E D B (4) C A D E B (4) B D A E C (4) B A E C D (4) A B C E D (3) E D C B A (2) E C B D A (2) E B D C A (2) C D A E B (2) B D E A C (2) A B C D E (2) E C D B A (1) E C B A D (1) E C A D B (1) E C A B D (1) E B C A D (1) D E C B A (1) D E C A B (1) D C A E B (1) D B A E C (1) D B A C E (1) D A C B E (1) C E D A B (1) C E A B D (1) C A D B E (1) C A B E D (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 -24 4 -10 -4 B 24 0 20 8 0 C -4 -20 0 -6 -22 D 10 -8 6 0 -18 E 4 0 22 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.504628 C: 0.000000 D: 0.000000 E: 0.495372 Sum of squares = 0.500042843896 Cumulative probabilities = A: 0.000000 B: 0.504628 C: 0.504628 D: 0.504628 E: 1.000000 A B C D E A 0 -24 4 -10 -4 B 24 0 20 8 0 C -4 -20 0 -6 -22 D 10 -8 6 0 -18 E 4 0 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 D=22 B=22 C=19 A=14 so A is eliminated. Round 2 votes counts: C=28 B=27 E=23 D=22 so D is eliminated. Round 3 votes counts: B=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:222 D:195 A:183 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 4 -10 -4 B 24 0 20 8 0 C -4 -20 0 -6 -22 D 10 -8 6 0 -18 E 4 0 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 4 -10 -4 B 24 0 20 8 0 C -4 -20 0 -6 -22 D 10 -8 6 0 -18 E 4 0 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 4 -10 -4 B 24 0 20 8 0 C -4 -20 0 -6 -22 D 10 -8 6 0 -18 E 4 0 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1909: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) A D E B C (10) B C D E A (8) B D C E A (6) C B E D A (5) A E C D B (5) E A D C B (4) B C A D E (4) D E A B C (3) C B D E A (3) C B A E D (3) B C A E D (3) E C D A B (2) E C A D B (2) D E B C A (2) D E B A C (2) D E A C B (2) D B C E A (2) D A E B C (2) C E B A D (2) C B E A D (2) B C D A E (2) A E B D C (2) A D B E C (2) E D C A B (1) E C D B A (1) E A C D B (1) E A C B D (1) D E C B A (1) D E C A B (1) D B C A E (1) D B A E C (1) C E D B A (1) C A B E D (1) B A D C E (1) A E C B D (1) Total count = 100 A B C D E A 0 0 -6 8 -4 B 0 0 2 -12 -12 C 6 -2 0 -6 -12 D -8 12 6 0 6 E 4 12 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.35802469136 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 A B C D E A 0 0 -6 8 -4 B 0 0 2 -12 -12 C 6 -2 0 -6 -12 D -8 12 6 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691369 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 D=17 C=17 E=12 so E is eliminated. Round 2 votes counts: A=36 B=24 C=22 D=18 so D is eliminated. Round 3 votes counts: A=43 B=32 C=25 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:211 D:208 A:199 C:193 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -6 8 -4 B 0 0 2 -12 -12 C 6 -2 0 -6 -12 D -8 12 6 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691369 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 8 -4 B 0 0 2 -12 -12 C 6 -2 0 -6 -12 D -8 12 6 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691369 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 8 -4 B 0 0 2 -12 -12 C 6 -2 0 -6 -12 D -8 12 6 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691369 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1910: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) A D E C B (8) B C E D A (7) A E D C B (7) B D C A E (6) B C D E A (5) A D E B C (5) E C A B D (4) E A C D B (4) E A C B D (4) D B A C E (4) D A B E C (3) D A B C E (3) C E B A D (3) B E C A D (3) B D C E A (3) E C B A D (2) D B C A E (2) D A E C B (2) C D E B A (2) B D A C E (2) E C A D B (1) E B A C D (1) D C E A B (1) D C B E A (1) D C A B E (1) D A C B E (1) C E B D A (1) C D B E A (1) C B E D A (1) C B E A D (1) B C D A E (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 6 -6 B 8 0 10 8 8 C 8 -10 0 -2 8 D -6 -8 2 0 4 E 6 -8 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 6 -6 B 8 0 10 8 8 C 8 -10 0 -2 8 D -6 -8 2 0 4 E 6 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=22 D=18 E=16 C=9 so C is eliminated. Round 2 votes counts: B=37 A=22 D=21 E=20 so E is eliminated. Round 3 votes counts: B=44 A=35 D=21 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:202 D:196 E:193 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 6 -6 B 8 0 10 8 8 C 8 -10 0 -2 8 D -6 -8 2 0 4 E 6 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 6 -6 B 8 0 10 8 8 C 8 -10 0 -2 8 D -6 -8 2 0 4 E 6 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 6 -6 B 8 0 10 8 8 C 8 -10 0 -2 8 D -6 -8 2 0 4 E 6 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1911: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) E B D C A (10) E A D C B (5) E A D B C (5) A E C D B (5) E D B A C (4) D A C B E (3) C B D A E (3) B E C D A (3) B C D A E (3) A E D C B (3) A C E B D (3) E C A B D (2) E B C A D (2) E A B D C (2) E A B C D (2) D C B A E (2) D B E C A (2) D B C A E (2) D A E B C (2) D A B C E (2) B E D C A (2) B D E C A (2) B D C E A (2) E D B C A (1) E D A B C (1) E C B A D (1) E B D A C (1) E B C D A (1) D E A B C (1) D B C E A (1) C B E D A (1) C A E B D (1) C A D B E (1) C A B D E (1) B C D E A (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 12 12 0 -4 B -12 0 4 -12 -12 C -12 -4 0 -12 -16 D 0 12 12 0 -12 E 4 12 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 12 0 -4 B -12 0 4 -12 -12 C -12 -4 0 -12 -16 D 0 12 12 0 -12 E 4 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=28 D=15 B=13 C=7 so C is eliminated. Round 2 votes counts: E=37 A=31 B=17 D=15 so D is eliminated. Round 3 votes counts: E=38 A=38 B=24 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:210 D:206 B:184 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 12 0 -4 B -12 0 4 -12 -12 C -12 -4 0 -12 -16 D 0 12 12 0 -12 E 4 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 0 -4 B -12 0 4 -12 -12 C -12 -4 0 -12 -16 D 0 12 12 0 -12 E 4 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 0 -4 B -12 0 4 -12 -12 C -12 -4 0 -12 -16 D 0 12 12 0 -12 E 4 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1912: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) C B E D A (10) B C E D A (9) A D E C B (8) E C B A D (7) A D E B C (7) D A B E C (5) E A D C B (4) C E B A D (3) C B E A D (3) B C D A E (3) E A B D C (2) D B A C E (2) D A E B C (2) C B D A E (2) B E C A D (2) B D C A E (2) B C D E A (2) A E D C B (2) A E D B C (2) E C A D B (1) E C A B D (1) E B C A D (1) E A D B C (1) E A C D B (1) E A C B D (1) D C B A E (1) D B C A E (1) D A C B E (1) C D B A E (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 2 -4 4 B -2 0 4 -4 10 C -2 -4 0 -4 4 D 4 4 4 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999991 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 A B C D E A 0 2 2 -4 4 B -2 0 4 -4 10 C -2 -4 0 -4 4 D 4 4 4 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=21 E=19 C=19 B=18 so B is eliminated. Round 2 votes counts: C=33 D=25 E=21 A=21 so E is eliminated. Round 3 votes counts: C=45 A=30 D=25 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:205 B:204 A:202 C:197 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 2 -4 4 B -2 0 4 -4 10 C -2 -4 0 -4 4 D 4 4 4 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -4 4 B -2 0 4 -4 10 C -2 -4 0 -4 4 D 4 4 4 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -4 4 B -2 0 4 -4 10 C -2 -4 0 -4 4 D 4 4 4 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1913: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) A B E C D (8) D C E A B (6) C D E B A (6) C E B D A (5) C B A E D (4) C A B D E (4) D E A B C (3) D C E B A (3) C A B E D (3) B A E C D (3) A B C E D (3) E B D A C (2) E B C D A (2) E B A D C (2) D E C A B (2) C D A B E (2) C B E D A (2) B E A C D (2) B A C E D (2) A B D E C (2) E D B C A (1) E D A B C (1) E C D B A (1) E B D C A (1) D E C B A (1) D E B C A (1) D E A C B (1) D C A B E (1) D A E B C (1) D A C B E (1) C E D B A (1) C D E A B (1) C D A E B (1) B E C A D (1) B C E A D (1) B A E D C (1) A D E B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 16 0 8 8 B -16 0 4 22 14 C 0 -4 0 10 4 D -8 -22 -10 0 -12 E -8 -14 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500556 B: 0.000000 C: 0.499444 D: 0.000000 E: 0.000000 Sum of squares = 0.500000615847 Cumulative probabilities = A: 0.500556 B: 0.500556 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 8 8 B -16 0 4 22 14 C 0 -4 0 10 4 D -8 -22 -10 0 -12 E -8 -14 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=29 D=20 E=10 B=10 so E is eliminated. Round 2 votes counts: A=31 C=30 D=22 B=17 so B is eliminated. Round 3 votes counts: A=41 C=34 D=25 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:212 C:205 E:193 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 8 8 B -16 0 4 22 14 C 0 -4 0 10 4 D -8 -22 -10 0 -12 E -8 -14 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 8 8 B -16 0 4 22 14 C 0 -4 0 10 4 D -8 -22 -10 0 -12 E -8 -14 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 8 8 B -16 0 4 22 14 C 0 -4 0 10 4 D -8 -22 -10 0 -12 E -8 -14 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1914: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) A E B D C (10) E B A C D (8) D A C B E (7) D C B E A (6) D C B A E (6) E B C A D (5) C D B E A (5) A E D B C (5) A D C E B (5) B E C D A (4) A D E C B (4) D C A B E (3) D A C E B (3) C B E D A (2) C B D E A (2) A D E B C (2) E C B A D (1) E C A B D (1) E A B C D (1) C E B D A (1) C D E B A (1) B E C A D (1) B C E D A (1) B C D E A (1) A D C B E (1) A D B C E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 20 16 20 B -12 0 4 2 -16 C -20 -4 0 -8 -10 D -16 -2 8 0 -6 E -20 16 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 20 16 20 B -12 0 4 2 -16 C -20 -4 0 -8 -10 D -16 -2 8 0 -6 E -20 16 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 D=25 E=16 C=11 B=7 so B is eliminated. Round 2 votes counts: A=41 D=25 E=21 C=13 so C is eliminated. Round 3 votes counts: A=41 D=34 E=25 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:206 D:192 B:189 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 20 16 20 B -12 0 4 2 -16 C -20 -4 0 -8 -10 D -16 -2 8 0 -6 E -20 16 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 20 16 20 B -12 0 4 2 -16 C -20 -4 0 -8 -10 D -16 -2 8 0 -6 E -20 16 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 20 16 20 B -12 0 4 2 -16 C -20 -4 0 -8 -10 D -16 -2 8 0 -6 E -20 16 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1915: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) D C A E B (7) E B C A D (6) C D A B E (6) B E A D C (6) E B C D A (5) C B E A D (5) D A E B C (4) B E C A D (4) A D C B E (4) E B A D C (3) D A C E B (3) D A C B E (3) C D E A B (3) E B D A C (2) D E B A C (2) C E B D A (2) B E A C D (2) B C A E D (2) B A E D C (2) E D C B A (1) E D B C A (1) E D B A C (1) E C B D A (1) D E A B C (1) D C E A B (1) D A E C B (1) C E D B A (1) C D E B A (1) C B A E D (1) C A D B E (1) C A B D E (1) B A E C D (1) A D B E C (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -18 -12 4 B -2 0 -8 -6 -12 C 18 8 0 8 8 D 12 6 -8 0 6 E -4 12 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 -12 4 B -2 0 -8 -6 -12 C 18 8 0 8 8 D 12 6 -8 0 6 E -4 12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=22 E=20 B=17 A=10 so A is eliminated. Round 2 votes counts: C=33 D=27 E=20 B=20 so E is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:208 E:197 A:188 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -18 -12 4 B -2 0 -8 -6 -12 C 18 8 0 8 8 D 12 6 -8 0 6 E -4 12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -12 4 B -2 0 -8 -6 -12 C 18 8 0 8 8 D 12 6 -8 0 6 E -4 12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -12 4 B -2 0 -8 -6 -12 C 18 8 0 8 8 D 12 6 -8 0 6 E -4 12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1916: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (12) A B C E D (9) C E D A B (8) D E B C A (6) B A E D C (6) E D C B A (5) D C E B A (5) B A D E C (5) C E D B A (4) C D E B A (4) C A E B D (4) A B D C E (4) A C B E D (3) E C D B A (2) D B E A C (2) C D E A B (2) B D E A C (2) A B E C D (2) E C A B D (1) E B D A C (1) E B A C D (1) D A B C E (1) C E A B D (1) C D A E B (1) C A E D B (1) C A B E D (1) C A B D E (1) B D A E C (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -16 -12 -12 B 12 0 -14 -8 -16 C 16 14 0 -4 4 D 12 8 4 0 0 E 12 16 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.717950 E: 0.282050 Sum of squares = 0.595004832905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.717950 E: 1.000000 A B C D E A 0 -12 -16 -12 -12 B 12 0 -14 -8 -16 C 16 14 0 -4 4 D 12 8 4 0 0 E 12 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 0.499371 Sum of squares = 0.500000790368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 A=23 B=14 E=10 so E is eliminated. Round 2 votes counts: D=31 C=30 A=23 B=16 so B is eliminated. Round 3 votes counts: D=35 A=35 C=30 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:212 E:212 B:187 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -16 -12 -12 B 12 0 -14 -8 -16 C 16 14 0 -4 4 D 12 8 4 0 0 E 12 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 0.499371 Sum of squares = 0.500000790368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -12 -12 B 12 0 -14 -8 -16 C 16 14 0 -4 4 D 12 8 4 0 0 E 12 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 0.499371 Sum of squares = 0.500000790368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -12 -12 B 12 0 -14 -8 -16 C 16 14 0 -4 4 D 12 8 4 0 0 E 12 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 0.499371 Sum of squares = 0.500000790368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500629 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1917: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) C E A D B (6) E A C D B (5) E A C B D (5) D C B E A (5) E A B C D (4) D C B A E (4) D B E A C (4) C D B A E (4) B D A E C (4) B D A C E (4) E D A C B (3) C A E B D (3) B A C E D (3) A E C B D (3) A E B C D (3) E D A B C (2) E C A D B (2) E C A B D (2) D E A C B (2) D B C E A (2) D B A E C (2) C B D A E (2) B D C A E (2) B A E C D (2) E A D C B (1) E A D B C (1) D E C A B (1) D E B A C (1) D C E A B (1) D B A C E (1) C E A B D (1) C D E A B (1) C A E D B (1) C A B E D (1) B C D A E (1) B C A E D (1) B A E D C (1) B A D E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 6 -6 2 B 2 0 -6 -12 4 C -6 6 0 2 0 D 6 12 -2 0 -2 E -2 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.116279 B: 0.046512 C: 0.023256 D: 0.209302 E: 0.604651 Sum of squares = 0.425635477914 Cumulative probabilities = A: 0.116279 B: 0.162791 C: 0.186047 D: 0.395349 E: 1.000000 A B C D E A 0 -2 6 -6 2 B 2 0 -6 -12 4 C -6 6 0 2 0 D 6 12 -2 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.116279 B: 0.046512 C: 0.023256 D: 0.209302 E: 0.604651 Sum of squares = 0.425635478598 Cumulative probabilities = A: 0.116279 B: 0.162791 C: 0.186047 D: 0.395349 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=25 C=19 B=19 A=7 so A is eliminated. Round 2 votes counts: E=31 D=30 B=20 C=19 so C is eliminated. Round 3 votes counts: E=42 D=35 B=23 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:207 C:201 A:200 E:198 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -6 2 B 2 0 -6 -12 4 C -6 6 0 2 0 D 6 12 -2 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.116279 B: 0.046512 C: 0.023256 D: 0.209302 E: 0.604651 Sum of squares = 0.425635478598 Cumulative probabilities = A: 0.116279 B: 0.162791 C: 0.186047 D: 0.395349 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -6 2 B 2 0 -6 -12 4 C -6 6 0 2 0 D 6 12 -2 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.116279 B: 0.046512 C: 0.023256 D: 0.209302 E: 0.604651 Sum of squares = 0.425635478598 Cumulative probabilities = A: 0.116279 B: 0.162791 C: 0.186047 D: 0.395349 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -6 2 B 2 0 -6 -12 4 C -6 6 0 2 0 D 6 12 -2 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.116279 B: 0.046512 C: 0.023256 D: 0.209302 E: 0.604651 Sum of squares = 0.425635478598 Cumulative probabilities = A: 0.116279 B: 0.162791 C: 0.186047 D: 0.395349 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1918: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) A E C B D (8) E A C B D (6) D B C A E (6) C E A D B (6) E A B C D (5) D B C E A (5) E C A D B (4) E A B D C (4) D B E C A (4) D B E A C (4) C B A D E (4) B D C A E (4) E A C D B (3) C A B D E (3) B D A E C (3) A E B C D (3) E A D B C (2) C D B A E (2) C B D A E (2) A C E B D (2) E D B A C (1) E D A B C (1) D B A E C (1) C A B E D (1) B E A D C (1) B D E A C (1) B C D A E (1) B A E D C (1) A C B E D (1) Total count = 100 A B C D E A 0 20 -6 30 6 B -20 0 -6 22 -12 C 6 6 0 24 -4 D -30 -22 -24 0 -20 E -6 12 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999986 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 20 -6 30 6 B -20 0 -6 22 -12 C 6 6 0 24 -4 D -30 -22 -24 0 -20 E -6 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=26 D=20 A=14 B=11 so B is eliminated. Round 2 votes counts: C=30 D=28 E=27 A=15 so A is eliminated. Round 3 votes counts: E=39 C=33 D=28 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:225 C:216 E:215 B:192 D:152 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -6 30 6 B -20 0 -6 22 -12 C 6 6 0 24 -4 D -30 -22 -24 0 -20 E -6 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -6 30 6 B -20 0 -6 22 -12 C 6 6 0 24 -4 D -30 -22 -24 0 -20 E -6 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -6 30 6 B -20 0 -6 22 -12 C 6 6 0 24 -4 D -30 -22 -24 0 -20 E -6 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1919: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) B D E C A (6) B C E A D (6) B C D E A (5) A C B E D (5) D E A B C (4) D A E C B (4) A C E D B (4) E D B A C (3) E D A B C (3) E B D A C (3) E A D B C (3) D E B A C (3) D B E C A (3) C B A E D (3) C B A D E (3) B E D C A (3) A E C D B (3) A C E B D (3) D E B C A (2) D E A C B (2) D C B E A (2) C D B A E (2) C A B D E (2) B C A E D (2) E B A D C (1) E B A C D (1) D C E A B (1) D C A E B (1) D A C E B (1) C A D E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B C E D A (1) A E D C B (1) A E C B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 -2 -8 B 2 0 4 10 8 C 4 -4 0 6 4 D 2 -10 -6 0 -12 E 8 -8 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -2 -8 B 2 0 4 10 8 C 4 -4 0 6 4 D 2 -10 -6 0 -12 E 8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999118 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 C=19 A=19 E=14 so E is eliminated. Round 2 votes counts: B=30 D=29 A=22 C=19 so C is eliminated. Round 3 votes counts: B=36 A=33 D=31 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:205 E:204 A:192 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 -8 B 2 0 4 10 8 C 4 -4 0 6 4 D 2 -10 -6 0 -12 E 8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999118 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 -8 B 2 0 4 10 8 C 4 -4 0 6 4 D 2 -10 -6 0 -12 E 8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999118 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 -8 B 2 0 4 10 8 C 4 -4 0 6 4 D 2 -10 -6 0 -12 E 8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999118 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1920: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (14) A C E B D (13) D E B C A (12) A C E D B (7) B D E C A (4) D E C A B (3) D B A E C (3) D A C E B (3) E D C B A (2) E C B A D (2) E B D C A (2) D B E A C (2) D A B C E (2) C E A B D (2) C A E B D (2) B E D C A (2) B A D C E (2) B A C E D (2) A C D E B (2) A B C E D (2) A B C D E (2) E C A D B (1) E C A B D (1) E B C D A (1) E B C A D (1) D E C B A (1) D E A C B (1) D B A C E (1) D A C B E (1) B E C D A (1) B D A C E (1) B C A E D (1) B A C D E (1) A D C E B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -4 -12 -4 B 10 0 12 -14 -14 C 4 -12 0 -16 -6 D 12 14 16 0 14 E 4 14 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -12 -4 B 10 0 12 -14 -14 C 4 -12 0 -16 -6 D 12 14 16 0 14 E 4 14 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 A=29 B=14 E=10 C=4 so C is eliminated. Round 2 votes counts: D=43 A=31 B=14 E=12 so E is eliminated. Round 3 votes counts: D=45 A=35 B=20 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:205 B:197 A:185 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 -12 -4 B 10 0 12 -14 -14 C 4 -12 0 -16 -6 D 12 14 16 0 14 E 4 14 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -12 -4 B 10 0 12 -14 -14 C 4 -12 0 -16 -6 D 12 14 16 0 14 E 4 14 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -12 -4 B 10 0 12 -14 -14 C 4 -12 0 -16 -6 D 12 14 16 0 14 E 4 14 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1921: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) D C B E A (6) E C D A B (5) B A D C E (5) E C D B A (4) B E C D A (4) A E C D B (4) A D C E B (4) A D B C E (4) E A B C D (3) C E D B A (3) C D E A B (3) B D C A E (3) B A D E C (3) A E C B D (3) E C A D B (2) D C B A E (2) D A C B E (2) B E A D C (2) B E A C D (2) A D C B E (2) E C B A D (1) E B C D A (1) E B A C D (1) E A C D B (1) E A C B D (1) D C E A B (1) D C A E B (1) D B C E A (1) D B C A E (1) D B A C E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E B A (1) C D A E B (1) B E D C A (1) B E D A C (1) B D A C E (1) A E B D C (1) A E B C D (1) A B E D C (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 -10 -14 B 10 0 0 -2 10 C 8 0 0 -10 12 D 10 2 10 0 8 E 14 -10 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -10 -14 B 10 0 0 -2 10 C 8 0 0 -10 12 D 10 2 10 0 8 E 14 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 E=19 D=15 C=11 so C is eliminated. Round 2 votes counts: B=32 E=25 A=23 D=20 so D is eliminated. Round 3 votes counts: B=43 E=30 A=27 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:215 B:209 C:205 E:192 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -8 -10 -14 B 10 0 0 -2 10 C 8 0 0 -10 12 D 10 2 10 0 8 E 14 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -10 -14 B 10 0 0 -2 10 C 8 0 0 -10 12 D 10 2 10 0 8 E 14 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -10 -14 B 10 0 0 -2 10 C 8 0 0 -10 12 D 10 2 10 0 8 E 14 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1922: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) D A B E C (8) E B C D A (7) B E D C A (7) A D C B E (6) E B C A D (5) D A C B E (5) C B E D A (5) B E D A C (5) A C D E B (5) E B A D C (4) B E C D A (4) C E B A D (3) E C B A D (2) D A B C E (2) C E B D A (2) B D E A C (2) B C D E A (2) A D E B C (2) A D C E B (2) A D B E C (2) E B D A C (1) E A B C D (1) D B A E C (1) C E A B D (1) C A E D B (1) B D C E A (1) B C E D A (1) A E D B C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -2 -26 -6 B 4 0 14 8 22 C 2 -14 0 -2 -8 D 26 -8 2 0 -2 E 6 -22 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -26 -6 B 4 0 14 8 22 C 2 -14 0 -2 -8 D 26 -8 2 0 -2 E 6 -22 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996353 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=22 B=22 E=20 A=20 D=16 so D is eliminated. Round 2 votes counts: A=35 B=23 C=22 E=20 so E is eliminated. Round 3 votes counts: B=40 A=36 C=24 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:224 D:209 E:197 C:189 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -26 -6 B 4 0 14 8 22 C 2 -14 0 -2 -8 D 26 -8 2 0 -2 E 6 -22 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996353 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -26 -6 B 4 0 14 8 22 C 2 -14 0 -2 -8 D 26 -8 2 0 -2 E 6 -22 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996353 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -26 -6 B 4 0 14 8 22 C 2 -14 0 -2 -8 D 26 -8 2 0 -2 E 6 -22 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996353 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1923: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) D C A E B (10) B A E D C (10) C D E A B (9) B E A C D (6) D A C E B (5) D C A B E (4) C E D B A (4) A D C E B (4) A D C B E (4) A E D C B (3) A D B C E (3) E C B D A (2) E B C A D (2) E B A C D (2) C E D A B (2) C D A E B (2) B E C A D (2) B A D C E (2) A B E D C (2) E D C A B (1) E C D A B (1) E A D C B (1) D C E A B (1) C B E D A (1) C B D E A (1) B E A D C (1) B C A D E (1) A E B D C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -8 -8 6 B -10 0 -12 -10 -2 C 8 12 0 -8 6 D 8 10 8 0 -4 E -6 2 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.444444 Sum of squares = 0.358024691351 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 1.000000 A B C D E A 0 10 -8 -8 6 B -10 0 -12 -10 -2 C 8 12 0 -8 6 D 8 10 8 0 -4 E -6 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.444444 Sum of squares = 0.358024691221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=20 C=19 A=19 E=9 so E is eliminated. Round 2 votes counts: B=37 C=22 D=21 A=20 so A is eliminated. Round 3 votes counts: B=41 D=37 C=22 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:209 A:200 E:197 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -8 -8 6 B -10 0 -12 -10 -2 C 8 12 0 -8 6 D 8 10 8 0 -4 E -6 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.444444 Sum of squares = 0.358024691221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -8 6 B -10 0 -12 -10 -2 C 8 12 0 -8 6 D 8 10 8 0 -4 E -6 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.444444 Sum of squares = 0.358024691221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -8 6 B -10 0 -12 -10 -2 C 8 12 0 -8 6 D 8 10 8 0 -4 E -6 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 0.444444 Sum of squares = 0.358024691221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.555556 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1924: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) E C A B D (7) A B D E C (7) E A C B D (5) C E D B A (5) B D A C E (5) B A D E C (5) E C B A D (4) B A E C D (4) A E C D B (4) A E B C D (4) E C A D B (3) D C B E A (3) C E D A B (3) C D E B A (3) A D B E C (3) D C E A B (2) D B C E A (2) B C E A D (2) A E C B D (2) A B E D C (2) E C D A B (1) E B C A D (1) E A C D B (1) D C B A E (1) D B C A E (1) D A C E B (1) D A B C E (1) C E B A D (1) C D E A B (1) B E A C D (1) B D C E A (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D C B (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 0 14 20 4 B 0 0 2 8 -2 C -14 -2 0 8 -16 D -20 -8 -8 0 -8 E -4 2 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.479220 B: 0.520780 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500863541017 Cumulative probabilities = A: 0.479220 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 20 4 B 0 0 2 8 -2 C -14 -2 0 8 -16 D -20 -8 -8 0 -8 E -4 2 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=22 B=21 D=19 C=13 so C is eliminated. Round 2 votes counts: E=31 A=25 D=23 B=21 so B is eliminated. Round 3 votes counts: E=35 A=35 D=30 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:211 B:204 C:188 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 20 4 B 0 0 2 8 -2 C -14 -2 0 8 -16 D -20 -8 -8 0 -8 E -4 2 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 20 4 B 0 0 2 8 -2 C -14 -2 0 8 -16 D -20 -8 -8 0 -8 E -4 2 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 20 4 B 0 0 2 8 -2 C -14 -2 0 8 -16 D -20 -8 -8 0 -8 E -4 2 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1925: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) C A D E B (8) A B D E C (7) B E A D C (6) C D A E B (5) C B E D A (5) B E C D A (5) D E A B C (4) C E D B A (4) B E D A C (4) C E B D A (3) B E C A D (3) B A E D C (3) A D E B C (3) A D C E B (3) A D B E C (3) E B C D A (2) C A D B E (2) C A B D E (2) B C E D A (2) A D E C B (2) A C D E B (2) E D B A C (1) E C B D A (1) E B D C A (1) E B D A C (1) C D E B A (1) C B E A D (1) C B A E D (1) B E D C A (1) B C A E D (1) A D C B E (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -12 4 -6 B -8 0 -2 4 2 C 12 2 0 18 4 D -4 -4 -18 0 8 E 6 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 4 -6 B -8 0 -2 4 2 C 12 2 0 18 4 D -4 -4 -18 0 8 E 6 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=25 A=25 E=6 D=4 so D is eliminated. Round 2 votes counts: C=40 B=25 A=25 E=10 so E is eliminated. Round 3 votes counts: C=41 B=30 A=29 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:198 A:197 E:196 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 4 -6 B -8 0 -2 4 2 C 12 2 0 18 4 D -4 -4 -18 0 8 E 6 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 4 -6 B -8 0 -2 4 2 C 12 2 0 18 4 D -4 -4 -18 0 8 E 6 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 4 -6 B -8 0 -2 4 2 C 12 2 0 18 4 D -4 -4 -18 0 8 E 6 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1926: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) C D A B E (6) B A E D C (6) D C A B E (5) B A D E C (5) E B A D C (4) D B A C E (4) A B E D C (4) D B A E C (3) D A B C E (3) C E D B A (3) C E D A B (3) C E A B D (3) A B D C E (3) E C B A D (2) E B D A C (2) E B A C D (2) E A C B D (2) E A B C D (2) D C B E A (2) D C B A E (2) D A C B E (2) C D E A B (2) C A D E B (2) C A D B E (2) A C B E D (2) A B D E C (2) E C D B A (1) E C A B D (1) E B D C A (1) E B C D A (1) D B E C A (1) D B C E A (1) C E A D B (1) C A E D B (1) C A E B D (1) B E D A C (1) A E B C D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 18 4 0 30 B -18 0 -4 -4 16 C -4 4 0 -2 16 D 0 4 2 0 10 E -30 -16 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.359735 B: 0.000000 C: 0.000000 D: 0.640265 E: 0.000000 Sum of squares = 0.539348190394 Cumulative probabilities = A: 0.359735 B: 0.359735 C: 0.359735 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 0 30 B -18 0 -4 -4 16 C -4 4 0 -2 16 D 0 4 2 0 10 E -30 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=23 E=18 A=16 B=12 so B is eliminated. Round 2 votes counts: C=31 A=27 D=23 E=19 so E is eliminated. Round 3 votes counts: A=37 C=36 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:208 C:207 B:195 E:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 0 30 B -18 0 -4 -4 16 C -4 4 0 -2 16 D 0 4 2 0 10 E -30 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 0 30 B -18 0 -4 -4 16 C -4 4 0 -2 16 D 0 4 2 0 10 E -30 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 0 30 B -18 0 -4 -4 16 C -4 4 0 -2 16 D 0 4 2 0 10 E -30 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1927: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) D A E B C (7) D B E A C (6) C B E A D (6) E A B C D (5) D C B E A (5) D B C E A (5) A E B C D (5) E B A C D (4) D A C E B (4) B C E D A (4) C D B A E (3) C A E B D (3) B E D A C (3) A E B D C (3) C D B E A (2) C B E D A (2) A E D B C (2) A D E C B (2) A C E D B (2) D C B A E (1) D C A B E (1) D B C A E (1) D A E C B (1) D A C B E (1) C E B A D (1) C B D E A (1) C A D E B (1) C A B E D (1) B E C D A (1) B E A C D (1) B D E C A (1) B D C E A (1) B C E A D (1) A E D C B (1) A E C B D (1) A D E B C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 2 0 -14 B 14 0 18 6 10 C -2 -18 0 6 -4 D 0 -6 -6 0 -10 E 14 -10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 0 -14 B 14 0 18 6 10 C -2 -18 0 6 -4 D 0 -6 -6 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=20 B=20 A=19 E=9 so E is eliminated. Round 2 votes counts: D=32 B=24 A=24 C=20 so C is eliminated. Round 3 votes counts: D=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:209 C:191 D:189 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 0 -14 B 14 0 18 6 10 C -2 -18 0 6 -4 D 0 -6 -6 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 0 -14 B 14 0 18 6 10 C -2 -18 0 6 -4 D 0 -6 -6 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 0 -14 B 14 0 18 6 10 C -2 -18 0 6 -4 D 0 -6 -6 0 -10 E 14 -10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1928: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) E D C A B (9) D B E A C (9) C A E B D (8) C A B E D (8) E C D A B (7) D B A E C (7) B D A C E (7) E C A D B (6) A B C D E (6) E D C B A (5) A C B E D (5) C E A B D (4) D E B A C (2) B A D C E (2) E D B C A (1) D B A C E (1) B D E A C (1) B A C D E (1) Total count = 100 A B C D E A 0 6 -18 -20 -10 B -6 0 -4 -16 -6 C 18 4 0 -10 -16 D 20 16 10 0 -6 E 10 6 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -18 -20 -10 B -6 0 -4 -16 -6 C 18 4 0 -10 -16 D 20 16 10 0 -6 E 10 6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=28 C=20 B=11 A=11 so B is eliminated. Round 2 votes counts: D=38 E=28 C=20 A=14 so A is eliminated. Round 3 votes counts: D=40 C=32 E=28 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:219 C:198 B:184 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -18 -20 -10 B -6 0 -4 -16 -6 C 18 4 0 -10 -16 D 20 16 10 0 -6 E 10 6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 -20 -10 B -6 0 -4 -16 -6 C 18 4 0 -10 -16 D 20 16 10 0 -6 E 10 6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 -20 -10 B -6 0 -4 -16 -6 C 18 4 0 -10 -16 D 20 16 10 0 -6 E 10 6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1929: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) E C D B A (5) E B C D A (5) E B A D C (5) A D C B E (5) D E C A B (4) D C E A B (4) D C A E B (4) E D C A B (3) D A C B E (3) C D A B E (3) B C A E D (3) B A E C D (3) A D B C E (3) A B C D E (3) E D B A C (2) E D A B C (2) E A D B C (2) D A C E B (2) C D E B A (2) A D B E C (2) A C D B E (2) A C B D E (2) A B E D C (2) A B D C E (2) E D A C B (1) E C B D A (1) E B D C A (1) E B C A D (1) E B A C D (1) E A B D C (1) D C E B A (1) D A E C B (1) C E D B A (1) C D B E A (1) C D B A E (1) C B D A E (1) B E C D A (1) B E C A D (1) B C E D A (1) B C A D E (1) B A C E D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -6 -16 -8 B -6 0 -10 -26 -16 C 6 10 0 -20 -8 D 16 26 20 0 -2 E 8 16 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999338 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -6 -16 -8 B -6 0 -10 -26 -16 C 6 10 0 -20 -8 D 16 26 20 0 -2 E 8 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=23 D=19 B=11 C=9 so C is eliminated. Round 2 votes counts: E=39 D=26 A=23 B=12 so B is eliminated. Round 3 votes counts: E=42 A=31 D=27 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:230 E:217 C:194 A:188 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 -16 -8 B -6 0 -10 -26 -16 C 6 10 0 -20 -8 D 16 26 20 0 -2 E 8 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -16 -8 B -6 0 -10 -26 -16 C 6 10 0 -20 -8 D 16 26 20 0 -2 E 8 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -16 -8 B -6 0 -10 -26 -16 C 6 10 0 -20 -8 D 16 26 20 0 -2 E 8 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1930: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) A E B D C (8) E D B C A (7) E B D C A (7) A C B D E (6) A C E B D (5) A C D B E (5) E B D A C (4) B D C E A (4) C A D B E (3) A E D B C (3) A E C B D (3) A B E C D (3) E D B A C (2) D B C E A (2) C B D A E (2) B D E C A (2) B A E D C (2) A E B C D (2) A C E D B (2) A C D E B (2) A C B E D (2) E B A D C (1) E A B D C (1) D C E B A (1) D C B E A (1) D B E C A (1) C D E B A (1) C D B A E (1) C D A B E (1) C A D E B (1) C A B D E (1) B E D C A (1) B C D A E (1) B C A D E (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 6 12 B 0 0 6 14 -2 C -6 -6 0 4 4 D -6 -14 -4 0 -8 E -12 2 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.556875 B: 0.443125 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.506469557796 Cumulative probabilities = A: 0.556875 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 6 12 B 0 0 6 14 -2 C -6 -6 0 4 4 D -6 -14 -4 0 -8 E -12 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 E=22 C=19 B=11 D=5 so D is eliminated. Round 2 votes counts: A=43 E=22 C=21 B=14 so B is eliminated. Round 3 votes counts: A=45 C=29 E=26 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:209 C:198 E:197 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 6 12 B 0 0 6 14 -2 C -6 -6 0 4 4 D -6 -14 -4 0 -8 E -12 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 12 B 0 0 6 14 -2 C -6 -6 0 4 4 D -6 -14 -4 0 -8 E -12 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 12 B 0 0 6 14 -2 C -6 -6 0 4 4 D -6 -14 -4 0 -8 E -12 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1931: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) A C B D E (8) E D C B A (5) A C D B E (5) A B C D E (5) C E D B A (4) C D E A B (4) C A D E B (4) B A E D C (4) A B C E D (4) E C D B A (3) D E B A C (3) D C E A B (3) C D A E B (3) B A D E C (3) A B D E C (3) D E C A B (2) C A E D B (2) B E C D A (2) B A E C D (2) B A C E D (2) A D B E C (2) E D B A C (1) E B D A C (1) E B C D A (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) D A E B C (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B D E (1) B E D C A (1) B E A C D (1) B C E A D (1) A D C E B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 6 16 6 10 B -6 0 -4 2 12 C -16 4 0 12 6 D -6 -2 -12 0 10 E -10 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 6 10 B -6 0 -4 2 12 C -16 4 0 12 6 D -6 -2 -12 0 10 E -10 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=25 C=21 D=13 E=11 so E is eliminated. Round 2 votes counts: A=30 B=27 C=24 D=19 so D is eliminated. Round 3 votes counts: C=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:203 B:202 D:195 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 6 10 B -6 0 -4 2 12 C -16 4 0 12 6 D -6 -2 -12 0 10 E -10 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 6 10 B -6 0 -4 2 12 C -16 4 0 12 6 D -6 -2 -12 0 10 E -10 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 6 10 B -6 0 -4 2 12 C -16 4 0 12 6 D -6 -2 -12 0 10 E -10 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1932: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) B D C A E (8) A C D B E (8) D B C A E (6) B E D C A (6) B D E C A (6) A E C D B (6) E B D C A (5) B D C E A (5) E A C D B (4) E A C B D (4) E A B C D (4) A C E D B (4) E B D A C (3) E A B D C (3) D C B A E (3) A C D E B (3) E B A D C (2) C D B A E (2) C D A B E (2) C B A D E (2) E B A C D (1) E A D B C (1) D B E C A (1) A E D C B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -8 6 10 B 0 0 2 0 14 C 8 -2 0 -2 6 D -6 0 2 0 10 E -10 -14 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.128874 B: 0.871126 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.775468748817 Cumulative probabilities = A: 0.128874 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 6 10 B 0 0 2 0 14 C 8 -2 0 -2 6 D -6 0 2 0 10 E -10 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000156306 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 A=24 C=14 D=10 so D is eliminated. Round 2 votes counts: B=32 E=27 A=24 C=17 so C is eliminated. Round 3 votes counts: B=39 A=34 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:208 C:205 A:204 D:203 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 6 10 B 0 0 2 0 14 C 8 -2 0 -2 6 D -6 0 2 0 10 E -10 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000156306 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 6 10 B 0 0 2 0 14 C 8 -2 0 -2 6 D -6 0 2 0 10 E -10 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000156306 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 6 10 B 0 0 2 0 14 C 8 -2 0 -2 6 D -6 0 2 0 10 E -10 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000156306 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1933: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) D C A E B (8) C D A E B (7) B E A C D (6) A B C D E (6) E C D B A (5) A C D B E (5) E D C B A (4) D C E A B (4) B A E D C (4) B A E C D (4) E B D C A (3) E B D A C (3) E B C D A (3) D E C A B (3) C E D B A (3) B A C E D (3) E D B C A (2) C E B D A (2) C D E A B (2) E B A D C (1) D E C B A (1) D A C E B (1) C D E B A (1) C D A B E (1) C B A E D (1) C A D B E (1) B E C A D (1) B E A D C (1) B C A E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -6 -6 10 B -2 0 -22 -18 -6 C 6 22 0 6 18 D 6 18 -6 0 6 E -10 6 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -6 10 B -2 0 -22 -18 -6 C 6 22 0 6 18 D 6 18 -6 0 6 E -10 6 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=21 B=20 C=18 D=17 so D is eliminated. Round 2 votes counts: C=30 E=25 A=25 B=20 so B is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:226 D:212 A:200 E:186 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -6 10 B -2 0 -22 -18 -6 C 6 22 0 6 18 D 6 18 -6 0 6 E -10 6 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -6 10 B -2 0 -22 -18 -6 C 6 22 0 6 18 D 6 18 -6 0 6 E -10 6 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -6 10 B -2 0 -22 -18 -6 C 6 22 0 6 18 D 6 18 -6 0 6 E -10 6 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1934: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (16) D A C B E (12) C A D E B (9) B E D A C (9) A C D E B (7) E B D A C (5) C A E D B (5) E C A B D (4) B E D C A (4) E C B A D (3) E B D C A (3) D A B C E (3) C E A B D (3) B D E A C (3) A C D B E (3) B D A E C (2) E C A D B (1) D B E A C (1) D B A E C (1) D B A C E (1) C A D B E (1) B E C D A (1) B C E A D (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -4 8 -8 B 2 0 2 10 -12 C 4 -2 0 10 -6 D -8 -10 -10 0 -10 E 8 12 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 8 -8 B 2 0 2 10 -12 C 4 -2 0 10 -6 D -8 -10 -10 0 -10 E 8 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=21 D=18 C=18 A=11 so A is eliminated. Round 2 votes counts: E=32 C=28 B=21 D=19 so D is eliminated. Round 3 votes counts: C=41 E=32 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:203 B:201 A:197 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 8 -8 B 2 0 2 10 -12 C 4 -2 0 10 -6 D -8 -10 -10 0 -10 E 8 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 -8 B 2 0 2 10 -12 C 4 -2 0 10 -6 D -8 -10 -10 0 -10 E 8 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 -8 B 2 0 2 10 -12 C 4 -2 0 10 -6 D -8 -10 -10 0 -10 E 8 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1935: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (17) B D A E C (6) D E A C B (4) D B A E C (4) D A B E C (4) B D A C E (4) B C E A D (4) A E C D B (4) E C A D B (3) D A E C B (3) C E A B D (3) C B E A D (3) B C D E A (3) B C A E D (3) A D E C B (3) E A C D B (2) D A E B C (2) C E B A D (2) B D E A C (2) B D C E A (2) B D C A E (2) B C E D A (2) A E D C B (2) A C E D B (2) E D A C B (1) D E C A B (1) D E A B C (1) D B E C A (1) C E D B A (1) C E B D A (1) C D B E A (1) C B E D A (1) C A E D B (1) C A B E D (1) B D E C A (1) B C A D E (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -8 6 -12 B -10 0 -12 -14 -6 C 8 12 0 12 10 D -6 14 -12 0 -8 E 12 6 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 6 -12 B -10 0 -12 -14 -6 C 8 12 0 12 10 D -6 14 -12 0 -8 E 12 6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=31 B=31 D=20 A=12 E=6 so E is eliminated. Round 2 votes counts: C=34 B=31 D=21 A=14 so A is eliminated. Round 3 votes counts: C=42 B=32 D=26 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 E:208 A:198 D:194 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 6 -12 B -10 0 -12 -14 -6 C 8 12 0 12 10 D -6 14 -12 0 -8 E 12 6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 6 -12 B -10 0 -12 -14 -6 C 8 12 0 12 10 D -6 14 -12 0 -8 E 12 6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 6 -12 B -10 0 -12 -14 -6 C 8 12 0 12 10 D -6 14 -12 0 -8 E 12 6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1936: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (12) B E D C A (11) B D E C A (8) A C E D B (8) C A D E B (6) B E D A C (6) B A E D C (5) C D E A B (4) E D A C B (3) B D C E A (3) B A C D E (3) A C B D E (3) E D C A B (2) E A D C B (2) D E C A B (2) D E B C A (2) B C D A E (2) A C D B E (2) E D B C A (1) E D B A C (1) E B D C A (1) E A D B C (1) E A B D C (1) D E C B A (1) D C E B A (1) B D C A E (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C E D (1) A E B C D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 8 2 0 B 0 0 8 4 2 C -8 -8 0 -8 -4 D -2 -4 8 0 4 E 0 -2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.305627 B: 0.694373 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.575562070593 Cumulative probabilities = A: 0.305627 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 0 B 0 0 8 4 2 C -8 -8 0 -8 -4 D -2 -4 8 0 4 E 0 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 A=29 E=12 C=10 D=6 so D is eliminated. Round 2 votes counts: B=43 A=29 E=17 C=11 so C is eliminated. Round 3 votes counts: B=43 A=35 E=22 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:207 A:205 D:203 E:199 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 0 B 0 0 8 4 2 C -8 -8 0 -8 -4 D -2 -4 8 0 4 E 0 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 0 B 0 0 8 4 2 C -8 -8 0 -8 -4 D -2 -4 8 0 4 E 0 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 0 B 0 0 8 4 2 C -8 -8 0 -8 -4 D -2 -4 8 0 4 E 0 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1937: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) C E D A B (6) C D E B A (6) D E C B A (5) B A D E C (5) B A D C E (5) A E C B D (5) A B C E D (5) E D C A B (4) E A D C B (4) E A C D B (4) D B C E A (4) D C E B A (3) D C B E A (3) C B A D E (3) B D C A E (3) B D A E C (3) B D A C E (3) B C D A E (3) E D A C B (2) E A D B C (2) C B D A E (2) E C D A B (1) E A C B D (1) D E B A C (1) D B E C A (1) D B C A E (1) C E A D B (1) C D B E A (1) C A E B D (1) A E B D C (1) A E B C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -20 4 -4 2 B 20 0 -6 2 2 C -4 6 0 -2 18 D 4 -2 2 0 20 E -2 -2 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000085 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 4 -4 2 B 20 0 -6 2 2 C -4 6 0 -2 18 D 4 -2 2 0 20 E -2 -2 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=20 E=18 D=18 A=14 so A is eliminated. Round 2 votes counts: B=36 E=25 C=21 D=18 so D is eliminated. Round 3 votes counts: B=42 E=31 C=27 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:212 B:209 C:209 A:191 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 4 -4 2 B 20 0 -6 2 2 C -4 6 0 -2 18 D 4 -2 2 0 20 E -2 -2 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 4 -4 2 B 20 0 -6 2 2 C -4 6 0 -2 18 D 4 -2 2 0 20 E -2 -2 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 4 -4 2 B 20 0 -6 2 2 C -4 6 0 -2 18 D 4 -2 2 0 20 E -2 -2 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1938: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) A C D B E (6) D C B E A (5) C D A B E (5) C D B E A (4) B E C D A (4) B C D E A (4) A E D C B (4) A E B C D (4) A D E C B (4) A D C E B (4) E B D A C (3) E B A D C (3) C A D B E (3) B C E D A (3) A E D B C (3) A C B D E (3) E D B C A (2) E B A C D (2) E A B C D (2) C D B A E (2) C B D A E (2) B E D C A (2) A C D E B (2) A C B E D (2) E A D B C (1) E A B D C (1) D C E B A (1) D C B A E (1) D C A E B (1) D B C E A (1) D A C E B (1) C B D E A (1) C B A D E (1) B C D A E (1) A E C B D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -4 -4 4 B 2 0 -8 -2 2 C 4 8 0 8 8 D 4 2 -8 0 4 E -4 -2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 4 B 2 0 -8 -2 2 C 4 8 0 8 8 D 4 2 -8 0 4 E -4 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=23 C=18 B=14 D=10 so D is eliminated. Round 2 votes counts: A=36 C=26 E=23 B=15 so B is eliminated. Round 3 votes counts: A=36 C=35 E=29 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:201 A:197 B:197 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 4 B 2 0 -8 -2 2 C 4 8 0 8 8 D 4 2 -8 0 4 E -4 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 4 B 2 0 -8 -2 2 C 4 8 0 8 8 D 4 2 -8 0 4 E -4 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 4 B 2 0 -8 -2 2 C 4 8 0 8 8 D 4 2 -8 0 4 E -4 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1939: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A B C D E (9) A C B E D (6) E D C B A (5) B D E C A (5) C A E B D (4) A C E B D (4) A B D C E (4) E C D B A (3) C A E D B (3) C A B E D (3) A D B E C (3) E D C A B (2) E C D A B (2) D E B A C (2) D A E C B (2) C E D B A (2) C E D A B (2) C B E D A (2) C B A E D (2) B C E D A (2) B A D E C (2) A D E C B (2) A C E D B (2) E D B C A (1) D E C A B (1) D E A C B (1) D E A B C (1) D B E A C (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E A D (1) B E D C A (1) B E C D A (1) B D E A C (1) A D E B C (1) A D C E B (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 18 -8 6 4 B -18 0 -12 2 -8 C 8 12 0 8 6 D -6 -2 -8 0 -2 E -4 8 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -8 6 4 B -18 0 -12 2 -8 C 8 12 0 8 6 D -6 -2 -8 0 -2 E -4 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=22 D=17 E=13 B=12 so B is eliminated. Round 2 votes counts: A=38 C=24 D=23 E=15 so E is eliminated. Round 3 votes counts: A=38 D=32 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:217 A:210 E:200 D:191 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -8 6 4 B -18 0 -12 2 -8 C 8 12 0 8 6 D -6 -2 -8 0 -2 E -4 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 6 4 B -18 0 -12 2 -8 C 8 12 0 8 6 D -6 -2 -8 0 -2 E -4 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 6 4 B -18 0 -12 2 -8 C 8 12 0 8 6 D -6 -2 -8 0 -2 E -4 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1940: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) D C B A E (8) C D B A E (7) A E B C D (7) E B A D C (6) E A B D C (6) B D C E A (5) A C D E B (5) E A B C D (4) C D A B E (4) C A D E B (4) D C B E A (3) D B C E A (3) C D A E B (3) B E D C A (3) B E A D C (3) B E A C D (3) E B A C D (2) D C A E B (2) C D B E A (2) B E D A C (2) B C D E A (2) A C E D B (2) D C A B E (1) B E C A D (1) B D E C A (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 2 8 8 B 2 0 -4 -8 -4 C -2 4 0 14 2 D -8 8 -14 0 0 E -8 4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 8 8 B 2 0 -4 -8 -4 C -2 4 0 14 2 D -8 8 -14 0 0 E -8 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.37499999991 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=20 B=20 E=18 D=17 so D is eliminated. Round 2 votes counts: C=34 A=25 B=23 E=18 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:209 A:208 E:197 B:193 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 8 8 B 2 0 -4 -8 -4 C -2 4 0 14 2 D -8 8 -14 0 0 E -8 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.37499999991 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 8 8 B 2 0 -4 -8 -4 C -2 4 0 14 2 D -8 8 -14 0 0 E -8 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.37499999991 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 8 8 B 2 0 -4 -8 -4 C -2 4 0 14 2 D -8 8 -14 0 0 E -8 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.37499999991 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1941: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (10) A E C D B (10) B D C E A (9) A C D B E (6) B E D C A (5) C D B A E (4) E D A C B (3) E A B D C (3) D C B A E (3) E D C A B (2) E A D B C (2) E A C D B (2) B C D E A (2) B C A D E (2) B A C D E (2) A E B C D (2) A B C D E (2) E D C B A (1) E D B C A (1) E B D C A (1) E B D A C (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A B E (1) D A C E B (1) C B D A E (1) C A D B E (1) B E C D A (1) B E A C D (1) B A E C D (1) A E C B D (1) A D C E B (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 4 0 2 B -6 0 -10 -8 8 C -4 10 0 2 2 D 0 8 -2 0 2 E -2 -8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.587003 B: 0.000000 C: 0.000000 D: 0.412997 E: 0.000000 Sum of squares = 0.515139063951 Cumulative probabilities = A: 0.587003 B: 0.587003 C: 0.587003 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 2 B -6 0 -10 -8 8 C -4 10 0 2 2 D 0 8 -2 0 2 E -2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=27 A=25 D=9 C=6 so C is eliminated. Round 2 votes counts: B=34 E=27 A=26 D=13 so D is eliminated. Round 3 votes counts: B=42 E=30 A=28 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:206 C:205 D:204 E:193 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 2 B -6 0 -10 -8 8 C -4 10 0 2 2 D 0 8 -2 0 2 E -2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 2 B -6 0 -10 -8 8 C -4 10 0 2 2 D 0 8 -2 0 2 E -2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 2 B -6 0 -10 -8 8 C -4 10 0 2 2 D 0 8 -2 0 2 E -2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1942: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) D A E C B (7) B E C A D (7) A D E B C (7) D A C E B (6) B E A C D (5) D C E B A (4) C D E B A (4) C D B E A (4) A D B E C (4) A D B C E (4) E C D B A (3) C E B D A (3) C B D E A (3) D C E A B (2) C B A D E (2) B E C D A (2) A D E C B (2) A D C B E (2) A B E D C (2) A B E C D (2) A B D C E (2) E D B A C (1) E B D C A (1) E B C D A (1) E B A C D (1) E A D B C (1) E A B D C (1) D C A B E (1) D A C B E (1) C D B A E (1) C B D A E (1) B C E A D (1) B A E C D (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 -8 -6 B 8 0 -10 -8 10 C -2 10 0 2 2 D 8 8 -2 0 14 E 6 -10 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000000104 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -8 -6 B 8 0 -10 -8 10 C -2 10 0 2 2 D 8 8 -2 0 14 E 6 -10 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=27 A=27 D=21 B=16 E=9 so E is eliminated. Round 2 votes counts: C=30 A=29 D=22 B=19 so B is eliminated. Round 3 votes counts: C=41 A=36 D=23 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:214 C:206 B:200 A:190 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 2 -8 -6 B 8 0 -10 -8 10 C -2 10 0 2 2 D 8 8 -2 0 14 E 6 -10 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -8 -6 B 8 0 -10 -8 10 C -2 10 0 2 2 D 8 8 -2 0 14 E 6 -10 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -8 -6 B 8 0 -10 -8 10 C -2 10 0 2 2 D 8 8 -2 0 14 E 6 -10 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1943: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) A B C D E (7) E B C D A (6) E A D C B (5) B E C D A (5) A D C B E (5) E D C A B (4) B C D A E (4) A D C E B (4) E B D C A (3) E B A C D (3) D A C B E (3) C B D A E (3) E D A C B (2) E A D B C (2) E A B C D (2) D C B A E (2) D C A B E (2) B E C A D (2) B C A D E (2) B A E C D (2) B A C D E (2) A E B C D (2) A C D B E (2) A C B D E (2) A B E C D (2) E D A B C (1) E B A D C (1) E A B D C (1) D E C B A (1) D C E B A (1) D C B E A (1) D A C E B (1) C D B A E (1) C D A B E (1) C A D B E (1) B E A C D (1) A E D C B (1) Total count = 100 A B C D E A 0 0 2 -2 -2 B 0 0 -4 0 0 C -2 4 0 0 -12 D 2 0 0 0 -10 E 2 0 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.539890 C: 0.000000 D: 0.000000 E: 0.460110 Sum of squares = 0.503182408949 Cumulative probabilities = A: 0.000000 B: 0.539890 C: 0.539890 D: 0.539890 E: 1.000000 A B C D E A 0 0 2 -2 -2 B 0 0 -4 0 0 C -2 4 0 0 -12 D 2 0 0 0 -10 E 2 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=25 B=18 D=11 C=6 so C is eliminated. Round 2 votes counts: E=40 A=26 B=21 D=13 so D is eliminated. Round 3 votes counts: E=42 A=33 B=25 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:199 B:198 D:196 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -2 -2 B 0 0 -4 0 0 C -2 4 0 0 -12 D 2 0 0 0 -10 E 2 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -2 B 0 0 -4 0 0 C -2 4 0 0 -12 D 2 0 0 0 -10 E 2 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -2 B 0 0 -4 0 0 C -2 4 0 0 -12 D 2 0 0 0 -10 E 2 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1944: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) D E C A B (11) A B D E C (11) B A C E D (10) C B E D A (8) B C A E D (8) B A D E C (7) A D E B C (7) C E D A B (6) D A E C B (4) B A C D E (4) D E A C B (3) E D C A B (1) E C D A B (1) E A D B C (1) B A E C D (1) B A D C E (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 2 4 10 B 6 0 2 0 2 C -2 -2 0 4 2 D -4 0 -4 0 0 E -10 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.784401 C: 0.000000 D: 0.215599 E: 0.000000 Sum of squares = 0.661767772745 Cumulative probabilities = A: 0.000000 B: 0.784401 C: 0.784401 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 10 B 6 0 2 0 2 C -2 -2 0 4 2 D -4 0 -4 0 0 E -10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556289 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=28 A=20 D=18 E=3 so E is eliminated. Round 2 votes counts: B=31 C=29 A=21 D=19 so D is eliminated. Round 3 votes counts: C=41 B=31 A=28 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:205 B:205 C:201 D:196 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 10 B 6 0 2 0 2 C -2 -2 0 4 2 D -4 0 -4 0 0 E -10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556289 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 10 B 6 0 2 0 2 C -2 -2 0 4 2 D -4 0 -4 0 0 E -10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556289 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 10 B 6 0 2 0 2 C -2 -2 0 4 2 D -4 0 -4 0 0 E -10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556289 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1945: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) B D A E C (10) A D B E C (10) C E B D A (8) E C A B D (6) C E B A D (6) C E A B D (6) C E A D B (5) D B A E C (4) D A B E C (3) B E C A D (3) B A D E C (3) D B A C E (2) D A C E B (2) D A B C E (2) C E D A B (2) C A E D B (2) B E C D A (2) A C D E B (2) A B D E C (2) D C E A B (1) D A C B E (1) C B E D A (1) C A D E B (1) B D E C A (1) B C E D A (1) B A E C D (1) A E C B D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -10 20 -4 B 4 0 -12 22 -8 C 10 12 0 16 -12 D -20 -22 -16 0 -10 E 4 8 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 20 -4 B 4 0 -12 22 -8 C 10 12 0 16 -12 D -20 -22 -16 0 -10 E 4 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=21 A=17 E=16 D=15 so D is eliminated. Round 2 votes counts: C=32 B=27 A=25 E=16 so E is eliminated. Round 3 votes counts: C=48 B=27 A=25 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:217 C:213 B:203 A:201 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 20 -4 B 4 0 -12 22 -8 C 10 12 0 16 -12 D -20 -22 -16 0 -10 E 4 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 20 -4 B 4 0 -12 22 -8 C 10 12 0 16 -12 D -20 -22 -16 0 -10 E 4 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 20 -4 B 4 0 -12 22 -8 C 10 12 0 16 -12 D -20 -22 -16 0 -10 E 4 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1946: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) B D C A E (7) E A B D C (6) C D B A E (6) B E A D C (6) E A C B D (5) E B A D C (4) E A B C D (4) C E A D B (4) B A E D C (4) E A C D B (3) C D E A B (3) C B D E A (3) D C B A E (2) D B C A E (2) C D A E B (2) C B E D A (2) B D A E C (2) B D A C E (2) A E C D B (2) A E B D C (2) E C B A D (1) E C A B D (1) E B A C D (1) D C A B E (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C D E B A (1) C B E A D (1) C A E D B (1) B E C A D (1) B D C E A (1) B C E D A (1) B C D E A (1) B A D E C (1) A E D C B (1) A E D B C (1) A D E C B (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 0 8 -16 B 14 0 -2 18 8 C 0 2 0 4 4 D -8 -18 -4 0 -8 E 16 -8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.076832 B: 0.000000 C: 0.923168 D: 0.000000 E: 0.000000 Sum of squares = 0.858142283779 Cumulative probabilities = A: 0.076832 B: 0.076832 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 8 -16 B 14 0 -2 18 8 C 0 2 0 4 4 D -8 -18 -4 0 -8 E 16 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250026467 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=26 E=25 A=9 D=8 so D is eliminated. Round 2 votes counts: C=35 B=28 E=25 A=12 so A is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:219 E:206 C:205 A:189 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 0 8 -16 B 14 0 -2 18 8 C 0 2 0 4 4 D -8 -18 -4 0 -8 E 16 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250026467 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 8 -16 B 14 0 -2 18 8 C 0 2 0 4 4 D -8 -18 -4 0 -8 E 16 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250026467 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 8 -16 B 14 0 -2 18 8 C 0 2 0 4 4 D -8 -18 -4 0 -8 E 16 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250026467 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1947: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) C E B A D (9) D A B E C (8) C D E A B (6) A D B E C (6) E C B A D (5) C E B D A (5) B A E D C (5) B A D E C (5) A B D E C (5) D A B C E (4) B E A C D (4) E B C A D (3) B E C A D (3) D A C E B (2) D A C B E (2) B A E C D (2) E C A B D (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) C E D B A (1) C E A D B (1) C E A B D (1) C D E B A (1) C D A E B (1) C B E A D (1) B E A D C (1) B C A E D (1) Total count = 100 A B C D E A 0 8 -10 6 -12 B -8 0 -6 2 -4 C 10 6 0 16 2 D -6 -2 -16 0 -12 E 12 4 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 6 -12 B -8 0 -6 2 -4 C 10 6 0 16 2 D -6 -2 -16 0 -12 E 12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=21 D=20 A=11 E=9 so E is eliminated. Round 2 votes counts: C=45 B=24 D=20 A=11 so A is eliminated. Round 3 votes counts: C=45 B=29 D=26 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:213 A:196 B:192 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 6 -12 B -8 0 -6 2 -4 C 10 6 0 16 2 D -6 -2 -16 0 -12 E 12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 6 -12 B -8 0 -6 2 -4 C 10 6 0 16 2 D -6 -2 -16 0 -12 E 12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 6 -12 B -8 0 -6 2 -4 C 10 6 0 16 2 D -6 -2 -16 0 -12 E 12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1948: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (14) E C B A D (8) A D E C B (8) B C E A D (7) C E B D A (6) D A B C E (5) B D A C E (4) B A D C E (4) A E C D B (4) A D B E C (4) D B C E A (3) D B A C E (3) D A B E C (3) C E B A D (3) B D C E A (3) E C D B A (2) E C A B D (2) D B C A E (2) D A E B C (2) A B D E C (2) E C D A B (1) E C B D A (1) E C A D B (1) D A E C B (1) C E D B A (1) C B E A D (1) B E C A D (1) B C D E A (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -28 -14 -4 -10 B 28 0 22 16 18 C 14 -22 0 6 18 D 4 -16 -6 0 -4 E 10 -18 -18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -14 -4 -10 B 28 0 22 16 18 C 14 -22 0 6 18 D 4 -16 -6 0 -4 E 10 -18 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=21 D=19 E=15 C=11 so C is eliminated. Round 2 votes counts: B=35 E=25 A=21 D=19 so D is eliminated. Round 3 votes counts: B=43 A=32 E=25 so E is eliminated. Round 4 votes counts: B=64 A=36 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:242 C:208 D:189 E:189 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -14 -4 -10 B 28 0 22 16 18 C 14 -22 0 6 18 D 4 -16 -6 0 -4 E 10 -18 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -14 -4 -10 B 28 0 22 16 18 C 14 -22 0 6 18 D 4 -16 -6 0 -4 E 10 -18 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -14 -4 -10 B 28 0 22 16 18 C 14 -22 0 6 18 D 4 -16 -6 0 -4 E 10 -18 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1949: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) E A C B D (8) C A E D B (7) C A E B D (7) A C E B D (7) B D E A C (6) D E B A C (5) D B E C A (5) C A B E D (5) E A C D B (4) D B C A E (4) B E A C D (3) B C A E D (3) C D A E B (2) C A D E B (2) B E D A C (2) B D C A E (2) E D B A C (1) E D A C B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C E A (1) C D A B E (1) C B A D E (1) C A B D E (1) B D E C A (1) B C A D E (1) B A C E D (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 8 -4 B -2 0 -4 4 -4 C -6 4 0 12 -2 D -8 -4 -12 0 -8 E 4 4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 6 8 -4 B -2 0 -4 4 -4 C -6 4 0 12 -2 D -8 -4 -12 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 B=19 E=18 A=8 so A is eliminated. Round 2 votes counts: C=34 D=29 B=19 E=18 so E is eliminated. Round 3 votes counts: C=46 D=31 B=23 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:209 A:206 C:204 B:197 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 8 -4 B -2 0 -4 4 -4 C -6 4 0 12 -2 D -8 -4 -12 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 8 -4 B -2 0 -4 4 -4 C -6 4 0 12 -2 D -8 -4 -12 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 8 -4 B -2 0 -4 4 -4 C -6 4 0 12 -2 D -8 -4 -12 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1950: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) E B A C D (7) C D A B E (7) D C A B E (6) C D A E B (6) B E D A C (6) C A E D B (5) A E C B D (5) D C B A E (4) D B E A C (4) C A D E B (4) E B A D C (3) E A B C D (3) D B C E A (3) C E A B D (3) E A B D C (2) D A C E B (2) D A C B E (2) C D B E A (2) B D E A C (2) A E D C B (2) E B C A D (1) D C B E A (1) D B E C A (1) D B A C E (1) D A B E C (1) C E A D B (1) C B E D A (1) C B D E A (1) A E C D B (1) A E B C D (1) A D E C B (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 10 8 0 2 B -10 0 -14 -12 0 C -8 14 0 2 4 D 0 12 -2 0 -2 E -2 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.686164 B: 0.000000 C: 0.000000 D: 0.313836 E: 0.000000 Sum of squares = 0.569314327262 Cumulative probabilities = A: 0.686164 B: 0.686164 C: 0.686164 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 0 2 B -10 0 -14 -12 0 C -8 14 0 2 4 D 0 12 -2 0 -2 E -2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500173 B: 0.000000 C: 0.000000 D: 0.499827 E: 0.000000 Sum of squares = 0.500000060173 Cumulative probabilities = A: 0.500173 B: 0.500173 C: 0.500173 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 E=16 B=16 A=13 so A is eliminated. Round 2 votes counts: C=33 D=26 E=25 B=16 so B is eliminated. Round 3 votes counts: E=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:210 C:206 D:204 E:198 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 0 2 B -10 0 -14 -12 0 C -8 14 0 2 4 D 0 12 -2 0 -2 E -2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500173 B: 0.000000 C: 0.000000 D: 0.499827 E: 0.000000 Sum of squares = 0.500000060173 Cumulative probabilities = A: 0.500173 B: 0.500173 C: 0.500173 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 0 2 B -10 0 -14 -12 0 C -8 14 0 2 4 D 0 12 -2 0 -2 E -2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500173 B: 0.000000 C: 0.000000 D: 0.499827 E: 0.000000 Sum of squares = 0.500000060173 Cumulative probabilities = A: 0.500173 B: 0.500173 C: 0.500173 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 0 2 B -10 0 -14 -12 0 C -8 14 0 2 4 D 0 12 -2 0 -2 E -2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500173 B: 0.000000 C: 0.000000 D: 0.499827 E: 0.000000 Sum of squares = 0.500000060173 Cumulative probabilities = A: 0.500173 B: 0.500173 C: 0.500173 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1951: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) C A D B E (8) E B A D C (5) A B E C D (5) E B D A C (4) E A B D C (4) D E C B A (4) C D A B E (4) E B D C A (3) D E B C A (3) D C E B A (3) D C E A B (3) D C B E A (3) C A B D E (3) B E D A C (3) B D E C A (3) B A E C D (3) A E B C D (3) E D B A C (2) C D A E B (2) B E D C A (2) A C E B D (2) A C B E D (2) A C B D E (2) A B C E D (2) E D C B A (1) E A D C B (1) E A B C D (1) D C B A E (1) D B C E A (1) C D E A B (1) C D B A E (1) C A D E B (1) B E A D C (1) B C D A E (1) B A C D E (1) A E C B D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -12 -6 -12 B 6 0 10 2 -6 C 12 -10 0 -10 -14 D 6 -2 10 0 -8 E 12 6 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -6 -12 B 6 0 10 2 -6 C 12 -10 0 -10 -14 D 6 -2 10 0 -8 E 12 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=20 A=19 D=18 B=14 so B is eliminated. Round 2 votes counts: E=35 A=23 D=21 C=21 so D is eliminated. Round 3 votes counts: E=45 C=32 A=23 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:206 D:203 C:189 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -6 -12 B 6 0 10 2 -6 C 12 -10 0 -10 -14 D 6 -2 10 0 -8 E 12 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -6 -12 B 6 0 10 2 -6 C 12 -10 0 -10 -14 D 6 -2 10 0 -8 E 12 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -6 -12 B 6 0 10 2 -6 C 12 -10 0 -10 -14 D 6 -2 10 0 -8 E 12 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1952: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D B C A E (8) C B D A E (8) E A C B D (7) A E C B D (6) E A D C B (5) E A B C D (5) D B C E A (4) B D C A E (4) D E A C B (3) D E A B C (3) D C B A E (3) D B E C A (3) C A E B D (3) C A B E D (3) B C A E D (3) E A D B C (2) D E B A C (2) D C E A B (2) D C B E A (2) D B E A C (2) C B A E D (2) C B A D E (2) A C E B D (2) E D A B C (1) E A B D C (1) D C E B A (1) B E A D C (1) B A E C D (1) B A D E C (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 8 -4 B -4 0 -14 2 -2 C -4 14 0 4 -4 D -8 -2 -4 0 -4 E 4 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 8 -4 B -4 0 -14 2 -2 C -4 14 0 4 -4 D -8 -2 -4 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=29 C=18 B=11 A=9 so A is eliminated. Round 2 votes counts: E=35 D=33 C=20 B=12 so B is eliminated. Round 3 votes counts: E=38 D=38 C=24 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:206 C:205 B:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 8 -4 B -4 0 -14 2 -2 C -4 14 0 4 -4 D -8 -2 -4 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 8 -4 B -4 0 -14 2 -2 C -4 14 0 4 -4 D -8 -2 -4 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 8 -4 B -4 0 -14 2 -2 C -4 14 0 4 -4 D -8 -2 -4 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1953: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B E A C D (8) B A E C D (7) D C E A B (6) E D C B A (5) C D A B E (5) B A C E D (5) B A C D E (5) A B C D E (5) E B A D C (4) E B A C D (4) D E C A B (4) D C A B E (4) E D C A B (3) E D B C A (2) E D B A C (2) E B D C A (2) E B D A C (2) D A C E B (2) B C A E D (2) A D C B E (2) A C B D E (2) E D A B C (1) E B C D A (1) E B C A D (1) D E A C B (1) D A C B E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E C A D (1) B C A D E (1) Total count = 100 A B C D E A 0 -6 2 0 6 B 6 0 6 4 2 C -2 -6 0 0 4 D 0 -4 0 0 0 E -6 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 0 6 B 6 0 6 4 2 C -2 -6 0 0 4 D 0 -4 0 0 0 E -6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=27 D=27 A=9 C=8 so C is eliminated. Round 2 votes counts: D=32 B=30 E=27 A=11 so A is eliminated. Round 3 votes counts: B=38 D=35 E=27 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:201 C:198 D:198 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 0 6 B 6 0 6 4 2 C -2 -6 0 0 4 D 0 -4 0 0 0 E -6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 6 B 6 0 6 4 2 C -2 -6 0 0 4 D 0 -4 0 0 0 E -6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 6 B 6 0 6 4 2 C -2 -6 0 0 4 D 0 -4 0 0 0 E -6 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1954: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) C B D E A (6) D B E A C (5) D A E B C (5) B E D C A (5) A E D B C (5) A D E B C (5) E A B C D (3) D A C B E (3) C D B A E (3) C B E D A (3) C B E A D (3) A E B C D (3) E B A D C (2) D C B E A (2) D C B A E (2) D C A B E (2) C A E B D (2) C A D B E (2) B C D E A (2) A E D C B (2) A E B D C (2) A D C E B (2) E D B A C (1) E B D A C (1) E B C A D (1) E B A C D (1) E A C B D (1) D E B A C (1) D B E C A (1) D B C E A (1) D A C E B (1) D A B E C (1) D A B C E (1) C D A B E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E D A (1) A E C D B (1) A E C B D (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 4 -18 -2 B 4 0 4 -14 10 C -4 -4 0 -10 -4 D 18 14 10 0 14 E 2 -10 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -18 -2 B 4 0 4 -14 10 C -4 -4 0 -10 -4 D 18 14 10 0 14 E 2 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=25 A=24 B=13 E=10 so E is eliminated. Round 2 votes counts: C=28 A=28 D=26 B=18 so B is eliminated. Round 3 votes counts: D=36 C=33 A=31 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:202 E:191 A:190 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -18 -2 B 4 0 4 -14 10 C -4 -4 0 -10 -4 D 18 14 10 0 14 E 2 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -18 -2 B 4 0 4 -14 10 C -4 -4 0 -10 -4 D 18 14 10 0 14 E 2 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -18 -2 B 4 0 4 -14 10 C -4 -4 0 -10 -4 D 18 14 10 0 14 E 2 -10 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1955: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) D C E A B (9) E B D A C (6) C A B D E (6) E D B A C (5) E B A D C (5) D C E B A (5) A B E C D (5) D E B A C (4) D E B C A (3) C D B A E (3) C B A D E (3) B E A C D (3) B C A E D (3) B A C E D (3) E D A B C (2) D E C B A (2) D E A B C (2) D C A E B (2) B E A D C (2) A C D B E (2) A B E D C (2) A B C E D (2) E A B D C (1) D E C A B (1) C D B E A (1) C A D B E (1) B E C A D (1) B A E C D (1) A E D B C (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -4 -14 -4 B 0 0 2 -12 4 C 4 -2 0 -6 8 D 14 12 6 0 14 E 4 -4 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -14 -4 B 0 0 2 -12 4 C 4 -2 0 -6 8 D 14 12 6 0 14 E 4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 E=19 A=14 B=13 so B is eliminated. Round 2 votes counts: C=29 D=28 E=25 A=18 so A is eliminated. Round 3 votes counts: C=37 E=34 D=29 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:223 C:202 B:197 A:189 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -14 -4 B 0 0 2 -12 4 C 4 -2 0 -6 8 D 14 12 6 0 14 E 4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -14 -4 B 0 0 2 -12 4 C 4 -2 0 -6 8 D 14 12 6 0 14 E 4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -14 -4 B 0 0 2 -12 4 C 4 -2 0 -6 8 D 14 12 6 0 14 E 4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1956: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) B E A D C (9) B E A C D (7) B C D E A (7) A E D C B (7) C B D A E (4) C A D E B (4) A C D E B (4) E B A D C (3) D C A E B (3) C D A B E (3) C B A D E (3) A C E D B (3) E D A B C (2) B E D A C (2) B E C D A (2) B C E A D (2) B C D A E (2) A E C D B (2) A C E B D (2) E A D B C (1) E A B D C (1) D E B C A (1) D E A C B (1) D C B E A (1) D B C E A (1) D A C E B (1) C B D E A (1) C B A E D (1) C A E D B (1) C A D B E (1) C A B E D (1) B E D C A (1) B D C E A (1) B C E D A (1) B C A E D (1) B A E C D (1) B A C E D (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -2 14 12 B 4 0 -6 8 6 C 2 6 0 28 16 D -14 -8 -28 0 -4 E -12 -6 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 14 12 B 4 0 -6 8 6 C 2 6 0 28 16 D -14 -8 -28 0 -4 E -12 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=28 A=20 D=8 E=7 so E is eliminated. Round 2 votes counts: B=40 C=28 A=22 D=10 so D is eliminated. Round 3 votes counts: B=42 C=32 A=26 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:210 B:206 E:185 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 14 12 B 4 0 -6 8 6 C 2 6 0 28 16 D -14 -8 -28 0 -4 E -12 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 14 12 B 4 0 -6 8 6 C 2 6 0 28 16 D -14 -8 -28 0 -4 E -12 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 14 12 B 4 0 -6 8 6 C 2 6 0 28 16 D -14 -8 -28 0 -4 E -12 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1957: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) B E A D C (8) E B D A C (7) A B C E D (6) E B A D C (5) D E C B A (4) C D A E B (4) B A E D C (4) A C B D E (4) E B A C D (3) D E B C A (3) D C E B A (3) C A D E B (3) C A D B E (3) B D E A C (3) E B D C A (2) E A B C D (2) D C E A B (2) D B E A C (2) C A E D B (2) A E B C D (2) A C B E D (2) A B D C E (2) A B C D E (2) D E B A C (1) D C A E B (1) D C A B E (1) D B E C A (1) D B C A E (1) D B A C E (1) C D E A B (1) C D A B E (1) C A E B D (1) C A B E D (1) B E D A C (1) B D A E C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 16 4 -12 B 16 0 30 14 -10 C -16 -30 0 -22 -14 D -4 -14 22 0 -10 E 12 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 16 4 -12 B 16 0 30 14 -10 C -16 -30 0 -22 -14 D -4 -14 22 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=20 A=20 B=17 C=16 so C is eliminated. Round 2 votes counts: A=30 E=27 D=26 B=17 so B is eliminated. Round 3 votes counts: E=36 A=34 D=30 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:225 E:223 D:197 A:196 C:159 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 16 4 -12 B 16 0 30 14 -10 C -16 -30 0 -22 -14 D -4 -14 22 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 16 4 -12 B 16 0 30 14 -10 C -16 -30 0 -22 -14 D -4 -14 22 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 16 4 -12 B 16 0 30 14 -10 C -16 -30 0 -22 -14 D -4 -14 22 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1958: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) B A C D E (8) E D C A B (7) B C A E D (5) C B A D E (4) B A C E D (4) E B A D C (3) B C A D E (3) A D C B E (3) A B C D E (3) E D A C B (2) E D A B C (2) E C D B A (2) E B C D A (2) D E C A B (2) D E A C B (2) D C E A B (2) D C A E B (2) D A E C B (2) C D A E B (2) B E C A D (2) B A E C D (2) A E D B C (2) A D E B C (2) A C D B E (2) A C B D E (2) A B D E C (2) E D B A C (1) E B D A C (1) E A D B C (1) D A C E B (1) D A C B E (1) C E D B A (1) C D E B A (1) C D A B E (1) C B E D A (1) C B D E A (1) C B D A E (1) B E A D C (1) B E A C D (1) B C E A D (1) B A E D C (1) A D E C B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 4 10 12 B 8 0 -2 -2 2 C -4 2 0 -2 -2 D -10 2 2 0 -2 E -12 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.500000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.41999999961 Cumulative probabilities = A: 0.100000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 10 12 B 8 0 -2 -2 2 C -4 2 0 -2 -2 D -10 2 2 0 -2 E -12 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.500000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999123 Cumulative probabilities = A: 0.100000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=28 A=19 D=12 C=12 so D is eliminated. Round 2 votes counts: E=33 B=28 A=23 C=16 so C is eliminated. Round 3 votes counts: E=37 B=35 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:203 C:197 D:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 10 12 B 8 0 -2 -2 2 C -4 2 0 -2 -2 D -10 2 2 0 -2 E -12 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.500000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999123 Cumulative probabilities = A: 0.100000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 10 12 B 8 0 -2 -2 2 C -4 2 0 -2 -2 D -10 2 2 0 -2 E -12 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.500000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999123 Cumulative probabilities = A: 0.100000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 10 12 B 8 0 -2 -2 2 C -4 2 0 -2 -2 D -10 2 2 0 -2 E -12 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.500000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999123 Cumulative probabilities = A: 0.100000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1959: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) B E D C A (7) D A C B E (6) A C D E B (5) E C A B D (4) B A E D C (4) E C D B A (3) E B C D A (3) D C A E B (3) C E A D B (3) C A D E B (3) B E D A C (3) B D A E C (3) E C B A D (2) E A C B D (2) D C E B A (2) D C A B E (2) D B C E A (2) D A B C E (2) C E D A B (2) B A D E C (2) A E C B D (2) A D C E B (2) A D C B E (2) A C D B E (2) A B E C D (2) E C B D A (1) E B D C A (1) E B C A D (1) E A B C D (1) D E B C A (1) D C E A B (1) D B A C E (1) C D E A B (1) C D A E B (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E C A (1) B D E A C (1) B A E C D (1) A C E D B (1) A C E B D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 8 4 -6 B 0 0 -2 10 8 C -8 2 0 4 -12 D -4 -10 -4 0 -12 E 6 -8 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.457242 B: 0.542758 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.503656482175 Cumulative probabilities = A: 0.457242 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 4 -6 B 0 0 -2 10 8 C -8 2 0 4 -12 D -4 -10 -4 0 -12 E 6 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=20 A=19 E=18 C=10 so C is eliminated. Round 2 votes counts: B=33 E=23 D=22 A=22 so D is eliminated. Round 3 votes counts: B=36 A=36 E=28 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:211 B:208 A:203 C:193 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 8 4 -6 B 0 0 -2 10 8 C -8 2 0 4 -12 D -4 -10 -4 0 -12 E 6 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 4 -6 B 0 0 -2 10 8 C -8 2 0 4 -12 D -4 -10 -4 0 -12 E 6 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 4 -6 B 0 0 -2 10 8 C -8 2 0 4 -12 D -4 -10 -4 0 -12 E 6 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1960: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E C B A D (6) D A B E C (6) C B E D A (6) C E B A D (5) A D C E B (5) E C B D A (4) C A E D B (4) A D E B C (4) E B D C A (3) E A D B C (3) D A C B E (3) D A B C E (3) C B D A E (3) B C D E A (3) A D E C B (3) A D C B E (3) E B D A C (2) E A C D B (2) C E B D A (2) C B E A D (2) C A D B E (2) B E D A C (2) A D B E C (2) E D B A C (1) E D A B C (1) E B C A D (1) E B A D C (1) E A D C B (1) E A B D C (1) D B C A E (1) D B A E C (1) D A E B C (1) C D B A E (1) C B D E A (1) C B A D E (1) C A B D E (1) B D E A C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -6 -4 -10 B 8 0 -10 4 -14 C 6 10 0 2 -6 D 4 -4 -2 0 -8 E 10 14 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 -4 -10 B 8 0 -10 4 -14 C 6 10 0 2 -6 D 4 -4 -2 0 -8 E 10 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=28 A=18 D=15 B=6 so B is eliminated. Round 2 votes counts: E=35 C=31 A=18 D=16 so D is eliminated. Round 3 votes counts: E=36 C=32 A=32 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:206 D:195 B:194 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -10 B 8 0 -10 4 -14 C 6 10 0 2 -6 D 4 -4 -2 0 -8 E 10 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -10 B 8 0 -10 4 -14 C 6 10 0 2 -6 D 4 -4 -2 0 -8 E 10 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -10 B 8 0 -10 4 -14 C 6 10 0 2 -6 D 4 -4 -2 0 -8 E 10 14 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1961: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) D A C E B (8) D A E B C (5) C B E A D (5) E A B C D (4) C B D E A (4) C A D E B (4) E A B D C (3) C D A B E (3) C B E D A (3) B E D C A (3) B E D A C (3) B C E D A (3) A E C B D (3) E B A C D (2) D C B A E (2) D C A B E (2) D B E A C (2) D A C B E (2) D A B E C (2) C A E B D (2) B E C D A (2) B D E C A (2) B C E A D (2) A E D C B (2) A D E B C (2) A D C E B (2) E D A B C (1) E B C A D (1) E A D B C (1) D E A B C (1) D A E C B (1) C E B A D (1) C D B E A (1) C D B A E (1) C D A E B (1) C B D A E (1) C A D B E (1) B E C A D (1) B C D E A (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 4 8 -8 -8 B -4 0 0 8 -8 C -8 0 0 -8 -2 D 8 -8 8 0 -2 E 8 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 -8 -8 B -4 0 0 8 -8 C -8 0 0 -8 -2 D 8 -8 8 0 -2 E 8 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 E=20 B=17 A=11 so A is eliminated. Round 2 votes counts: D=30 C=27 E=26 B=17 so B is eliminated. Round 3 votes counts: E=35 C=33 D=32 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:210 D:203 A:198 B:198 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 -8 -8 B -4 0 0 8 -8 C -8 0 0 -8 -2 D 8 -8 8 0 -2 E 8 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -8 -8 B -4 0 0 8 -8 C -8 0 0 -8 -2 D 8 -8 8 0 -2 E 8 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -8 -8 B -4 0 0 8 -8 C -8 0 0 -8 -2 D 8 -8 8 0 -2 E 8 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1962: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) A C E D B (9) C A B E D (8) C B D E A (7) B C D E A (7) E D A B C (6) B D E C A (6) B D E A C (6) A E D C B (5) D E A B C (4) A E D B C (4) A E C D B (4) C B A D E (3) E D B A C (2) D E C A B (2) D E A C B (2) C B D A E (2) C A D E B (2) B C A E D (2) E D A C B (1) E A D B C (1) D C E A B (1) D B E C A (1) C D E A B (1) C B A E D (1) C A E D B (1) C A E B D (1) B C A D E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 8 -14 -12 B -6 0 -2 -10 -12 C -8 2 0 0 -6 D 14 10 0 0 8 E 12 12 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.355011 D: 0.644989 E: 0.000000 Sum of squares = 0.542043477075 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.355011 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -14 -12 B -6 0 -2 -10 -12 C -8 2 0 0 -6 D 14 10 0 0 8 E 12 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 B=22 D=19 E=10 so E is eliminated. Round 2 votes counts: D=28 C=26 A=24 B=22 so B is eliminated. Round 3 votes counts: D=40 C=36 A=24 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:216 E:211 A:194 C:194 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -14 -12 B -6 0 -2 -10 -12 C -8 2 0 0 -6 D 14 10 0 0 8 E 12 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -14 -12 B -6 0 -2 -10 -12 C -8 2 0 0 -6 D 14 10 0 0 8 E 12 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -14 -12 B -6 0 -2 -10 -12 C -8 2 0 0 -6 D 14 10 0 0 8 E 12 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1963: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) E B D C A (8) A E B D C (7) A D B C E (7) A B D E C (7) C D B E A (6) A E C D B (6) E C B D A (4) E C D B A (3) C A D B E (3) B D E C A (3) A C E D B (3) E A C B D (2) D C B A E (2) D B C A E (2) C E D B A (2) A E C B D (2) A C D E B (2) A B E D C (2) A B D C E (2) E C A D B (1) E C A B D (1) E B D A C (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B E A (1) D B C E A (1) C E D A B (1) C D E B A (1) C D B A E (1) B E D C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 16 10 18 18 B -16 0 -4 -6 6 C -10 4 0 2 -6 D -18 6 -2 0 6 E -18 -6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 18 18 B -16 0 -4 -6 6 C -10 4 0 2 -6 D -18 6 -2 0 6 E -18 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=49 E=23 C=14 B=8 D=6 so D is eliminated. Round 2 votes counts: A=49 E=23 C=17 B=11 so B is eliminated. Round 3 votes counts: A=50 E=28 C=22 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 D:196 C:195 B:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 18 18 B -16 0 -4 -6 6 C -10 4 0 2 -6 D -18 6 -2 0 6 E -18 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 18 18 B -16 0 -4 -6 6 C -10 4 0 2 -6 D -18 6 -2 0 6 E -18 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 18 18 B -16 0 -4 -6 6 C -10 4 0 2 -6 D -18 6 -2 0 6 E -18 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1964: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) D B E A C (10) E A C B D (8) A E C B D (6) D C B A E (5) C A B E D (5) E A C D B (4) D B E C A (4) B D C A E (4) D E A C B (3) C A E B D (3) B E A C D (3) A C E B D (3) D E B A C (2) D B C E A (2) C D A E B (2) C A E D B (2) B E A D C (2) B D E A C (2) B C A E D (2) B C A D E (2) E D A B C (1) E B A D C (1) E B A C D (1) E A D C B (1) E A B C D (1) D C A E B (1) C D B A E (1) C D A B E (1) C B D A E (1) C A D E B (1) B D C E A (1) B C D A E (1) B A E C D (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -12 2 -4 8 B 12 0 4 -4 16 C -2 -4 0 0 -4 D 4 4 0 0 8 E -8 -16 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.289227 D: 0.710773 E: 0.000000 Sum of squares = 0.588850396868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.289227 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -4 8 B 12 0 4 -4 16 C -2 -4 0 0 -4 D 4 4 0 0 8 E -8 -16 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499112 D: 0.500888 E: 0.000000 Sum of squares = 0.500001576977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499112 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=18 E=17 C=16 A=11 so A is eliminated. Round 2 votes counts: D=38 E=25 C=19 B=18 so B is eliminated. Round 3 votes counts: D=45 E=31 C=24 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:214 D:208 A:197 C:195 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -4 8 B 12 0 4 -4 16 C -2 -4 0 0 -4 D 4 4 0 0 8 E -8 -16 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499112 D: 0.500888 E: 0.000000 Sum of squares = 0.500001576977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499112 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -4 8 B 12 0 4 -4 16 C -2 -4 0 0 -4 D 4 4 0 0 8 E -8 -16 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499112 D: 0.500888 E: 0.000000 Sum of squares = 0.500001576977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499112 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -4 8 B 12 0 4 -4 16 C -2 -4 0 0 -4 D 4 4 0 0 8 E -8 -16 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499112 D: 0.500888 E: 0.000000 Sum of squares = 0.500001576977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499112 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1965: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (6) C B E D A (5) C B A D E (5) A D E B C (5) A B C D E (5) E D C B A (4) E B C A D (4) D A E C B (4) C B D E A (4) B C A E D (4) A D E C B (4) E B C D A (3) D E C B A (3) D E A C B (3) D E A B C (3) B C E D A (3) E D A B C (2) E B A C D (2) E A D B C (2) E A B D C (2) A E B D C (2) A E B C D (2) A D C B E (2) A C B D E (2) E D B C A (1) E B D C A (1) E A B C D (1) D E C A B (1) D C E B A (1) D C E A B (1) D A C E B (1) D A C B E (1) C D B A E (1) C B A E D (1) B E C D A (1) B E C A D (1) B E A C D (1) B A E C D (1) A E D B C (1) A D C E B (1) A D B E C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 0 14 -10 B 4 0 12 16 -8 C 0 -12 0 6 -12 D -14 -16 -6 0 -2 E 10 8 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 14 -10 B 4 0 12 16 -8 C 0 -12 0 6 -12 D -14 -16 -6 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999978021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 D=18 B=17 C=16 so C is eliminated. Round 2 votes counts: B=32 A=27 E=22 D=19 so D is eliminated. Round 3 votes counts: E=34 B=33 A=33 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:212 A:200 C:191 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 14 -10 B 4 0 12 16 -8 C 0 -12 0 6 -12 D -14 -16 -6 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999978021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 14 -10 B 4 0 12 16 -8 C 0 -12 0 6 -12 D -14 -16 -6 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999978021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 14 -10 B 4 0 12 16 -8 C 0 -12 0 6 -12 D -14 -16 -6 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999978021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1966: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) E C A D B (7) D B C E A (6) B D A C E (6) D B A E C (5) C B D E A (4) E A C D B (3) C E D B A (3) C E D A B (3) C D E B A (3) C B A E D (3) B D C A E (3) B D A E C (3) A E D B C (3) A B E C D (3) D C E B A (2) C E A D B (2) C E A B D (2) B A D E C (2) A E C D B (2) A E B D C (2) A C E B D (2) A B D E C (2) A B C E D (2) E D A C B (1) E C D A B (1) E C A B D (1) D E C B A (1) D E B A C (1) D B E C A (1) D B E A C (1) D B A C E (1) C D B E A (1) C B E A D (1) B D C E A (1) B C D A E (1) B A C D E (1) A E D C B (1) A E B C D (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 8 4 10 B 0 0 -4 0 -2 C -8 4 0 10 -6 D -4 0 -10 0 -6 E -10 2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.513798 B: 0.486202 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500380699628 Cumulative probabilities = A: 0.513798 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 4 10 B 0 0 -4 0 -2 C -8 4 0 10 -6 D -4 0 -10 0 -6 E -10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=22 D=18 B=17 E=13 so E is eliminated. Round 2 votes counts: A=33 C=31 D=19 B=17 so B is eliminated. Round 3 votes counts: A=36 D=32 C=32 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:202 C:200 B:197 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 4 10 B 0 0 -4 0 -2 C -8 4 0 10 -6 D -4 0 -10 0 -6 E -10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 4 10 B 0 0 -4 0 -2 C -8 4 0 10 -6 D -4 0 -10 0 -6 E -10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 4 10 B 0 0 -4 0 -2 C -8 4 0 10 -6 D -4 0 -10 0 -6 E -10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1967: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (13) E D C B A (12) C D E A B (8) A B C D E (8) E B A D C (7) B E A D C (6) B A E C D (5) A B D C E (5) E C D B A (4) D C E A B (4) C D A E B (4) A B D E C (3) E B A C D (2) D E C A B (2) C E D B A (2) B E A C D (2) E C B D A (1) E B C A D (1) D C E B A (1) D C A E B (1) C E B A D (1) C D E B A (1) C D A B E (1) C A D B E (1) C A B D E (1) A D B E C (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 10 18 -8 B 16 0 10 16 -2 C -10 -10 0 -12 -20 D -18 -16 12 0 -16 E 8 2 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 10 18 -8 B 16 0 10 16 -2 C -10 -10 0 -12 -20 D -18 -16 12 0 -16 E 8 2 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 A=20 C=19 D=8 so D is eliminated. Round 2 votes counts: E=29 B=26 C=25 A=20 so A is eliminated. Round 3 votes counts: B=45 E=29 C=26 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:220 A:202 D:181 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 10 18 -8 B 16 0 10 16 -2 C -10 -10 0 -12 -20 D -18 -16 12 0 -16 E 8 2 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 18 -8 B 16 0 10 16 -2 C -10 -10 0 -12 -20 D -18 -16 12 0 -16 E 8 2 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 18 -8 B 16 0 10 16 -2 C -10 -10 0 -12 -20 D -18 -16 12 0 -16 E 8 2 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1968: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) B E C A D (6) E A D B C (4) D C B A E (4) E D B A C (3) E B A D C (3) D C A B E (3) D B C A E (3) D A C E B (3) D A C B E (3) C D A B E (3) C A B E D (3) B C D A E (3) A D C E B (3) A C E B D (3) A C D E B (3) A C B E D (3) E D B C A (2) E B D A C (2) E A D C B (2) D E B C A (2) C D B A E (2) C B A E D (2) C A D B E (2) B E D C A (2) B E C D A (2) B D C E A (2) E D A B C (1) E B D C A (1) E B C A D (1) E A C B D (1) E A B C D (1) D E B A C (1) D E A C B (1) C B D A E (1) C B A D E (1) C A B D E (1) B D E C A (1) B C E D A (1) B C E A D (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D C B (1) A E C B D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 -4 6 4 B 12 0 4 4 4 C 4 -4 0 4 8 D -6 -4 -4 0 -10 E -4 -4 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 6 4 B 12 0 4 4 4 C 4 -4 0 4 8 D -6 -4 -4 0 -10 E -4 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=21 D=20 A=16 C=15 so C is eliminated. Round 2 votes counts: E=28 D=25 B=25 A=22 so A is eliminated. Round 3 votes counts: E=34 D=34 B=32 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:212 C:206 A:197 E:197 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 6 4 B 12 0 4 4 4 C 4 -4 0 4 8 D -6 -4 -4 0 -10 E -4 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 6 4 B 12 0 4 4 4 C 4 -4 0 4 8 D -6 -4 -4 0 -10 E -4 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 6 4 B 12 0 4 4 4 C 4 -4 0 4 8 D -6 -4 -4 0 -10 E -4 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1969: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) B C E D A (7) A D B E C (7) C B E D A (6) C E B D A (5) C E A D B (5) A D E C B (5) E D A C B (4) C B E A D (4) B C A D E (4) B A D C E (4) D E A B C (3) C E D B A (3) C B A E D (3) B E D A C (3) B A C D E (3) E C D A B (2) D E A C B (2) C E D A B (2) A D C E B (2) E D B A C (1) E C D B A (1) D A E C B (1) D A E B C (1) D A B E C (1) C B A D E (1) C A E D B (1) C A B D E (1) B E D C A (1) B D E A C (1) B D A E C (1) B C E A D (1) B C A E D (1) B A D E C (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 10 -2 B 2 0 2 -2 6 C -4 -2 0 4 12 D -10 2 -4 0 0 E 2 -6 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408185 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 10 -2 B 2 0 2 -2 6 C -4 -2 0 4 12 D -10 2 -4 0 0 E 2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408163 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 A=26 E=8 D=8 so E is eliminated. Round 2 votes counts: C=34 B=27 A=26 D=13 so D is eliminated. Round 3 votes counts: A=38 C=34 B=28 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:205 C:205 B:204 D:194 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 10 -2 B 2 0 2 -2 6 C -4 -2 0 4 12 D -10 2 -4 0 0 E 2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408163 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 -2 B 2 0 2 -2 6 C -4 -2 0 4 12 D -10 2 -4 0 0 E 2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408163 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 -2 B 2 0 2 -2 6 C -4 -2 0 4 12 D -10 2 -4 0 0 E 2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408163 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1970: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) E D B A C (7) B E D A C (7) B A C E D (7) D E B C A (6) B A E C D (6) D C A E B (5) B E A C D (5) E B D A C (4) C D A E B (4) C A D E B (4) C A D B E (4) E D A B C (3) C A B D E (3) A C E B D (3) E D A C B (2) E B A D C (2) D C E A B (2) D C B A E (2) D C A B E (2) B C A D E (2) A C E D B (2) E A D C B (1) D E C B A (1) D B C E A (1) C B A D E (1) B E A D C (1) B D C A E (1) B C D A E (1) A E B C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 6 -12 0 B 8 0 10 -8 -10 C -6 -10 0 -10 -8 D 12 8 10 0 -6 E 0 10 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.192173 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.807827 Sum of squares = 0.689515415632 Cumulative probabilities = A: 0.192173 B: 0.192173 C: 0.192173 D: 0.192173 E: 1.000000 A B C D E A 0 -8 6 -12 0 B 8 0 10 -8 -10 C -6 -10 0 -10 -8 D 12 8 10 0 -6 E 0 10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555557055 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=27 E=19 C=16 A=8 so A is eliminated. Round 2 votes counts: B=31 D=27 C=22 E=20 so E is eliminated. Round 3 votes counts: D=40 B=38 C=22 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:212 B:200 A:193 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 -12 0 B 8 0 10 -8 -10 C -6 -10 0 -10 -8 D 12 8 10 0 -6 E 0 10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555557055 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -12 0 B 8 0 10 -8 -10 C -6 -10 0 -10 -8 D 12 8 10 0 -6 E 0 10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555557055 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -12 0 B 8 0 10 -8 -10 C -6 -10 0 -10 -8 D 12 8 10 0 -6 E 0 10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555557055 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1971: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (12) D C B E A (7) D B C E A (7) A C E B D (7) A E D B C (5) A E C B D (5) E A B D C (4) B E D C A (4) A C D E B (4) D B E A C (3) C A B E D (3) B D E C A (3) E B D C A (2) E B D A C (2) E B C D A (2) E B A C D (2) D C B A E (2) D A B E C (2) C B D E A (2) E C B A D (1) E C A B D (1) E B A D C (1) E A B C D (1) D E B A C (1) D B C A E (1) D B A E C (1) C E B D A (1) C E B A D (1) C D B E A (1) C D B A E (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B D E (1) B D C E A (1) A E C D B (1) A E B C D (1) A D E B C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -14 -10 -20 B 20 0 12 -4 12 C 14 -12 0 -20 -10 D 10 4 20 0 6 E 20 -12 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -14 -10 -20 B 20 0 12 -4 12 C 14 -12 0 -20 -10 D 10 4 20 0 6 E 20 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=26 E=16 C=14 B=8 so B is eliminated. Round 2 votes counts: D=40 A=26 E=20 C=14 so C is eliminated. Round 3 votes counts: D=44 A=33 E=23 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:220 D:220 E:206 C:186 A:168 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -14 -10 -20 B 20 0 12 -4 12 C 14 -12 0 -20 -10 D 10 4 20 0 6 E 20 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 -10 -20 B 20 0 12 -4 12 C 14 -12 0 -20 -10 D 10 4 20 0 6 E 20 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 -10 -20 B 20 0 12 -4 12 C 14 -12 0 -20 -10 D 10 4 20 0 6 E 20 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1972: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (14) C A E B D (13) A C E B D (11) D B E A C (9) E B C D A (4) C E B D A (4) A C D E B (4) D B A E C (3) D A B C E (3) B D E C A (3) A C E D B (3) A C D B E (3) E B D C A (2) D A B E C (2) C E B A D (2) C E A B D (2) B E D C A (2) A D C B E (2) A D B E C (2) A D B C E (2) E C B D A (1) E A B C D (1) D B C E A (1) D B C A E (1) C D A B E (1) C A E D B (1) C A D E B (1) B E D A C (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 6 -4 -2 8 B -6 0 0 -6 -2 C 4 0 0 4 8 D 2 6 -4 0 2 E -8 2 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.238249 C: 0.761751 D: 0.000000 E: 0.000000 Sum of squares = 0.637027338048 Cumulative probabilities = A: 0.000000 B: 0.238249 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -2 8 B -6 0 0 -6 -2 C 4 0 0 4 8 D 2 6 -4 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000013493 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=29 C=24 E=8 B=6 so B is eliminated. Round 2 votes counts: D=36 A=29 C=24 E=11 so E is eliminated. Round 3 votes counts: D=41 A=30 C=29 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:208 A:204 D:203 B:193 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -2 8 B -6 0 0 -6 -2 C 4 0 0 4 8 D 2 6 -4 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000013493 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -2 8 B -6 0 0 -6 -2 C 4 0 0 4 8 D 2 6 -4 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000013493 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -2 8 B -6 0 0 -6 -2 C 4 0 0 4 8 D 2 6 -4 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000013493 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1973: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) E D B A C (6) D A E C B (6) C A D B E (6) B E C A D (6) B E C D A (5) B C E A D (5) A D C E B (5) E D A B C (4) D E A C B (4) C B A E D (4) A D E C B (4) A C D B E (4) E D B C A (3) C A B D E (3) D E C B A (2) D A C E B (2) B E A C D (2) B C E D A (2) A C D E B (2) E B D C A (1) D E C A B (1) D E A B C (1) D C E B A (1) D C A E B (1) D A E B C (1) C D A E B (1) C D A B E (1) C A D E B (1) C A B E D (1) B E D C A (1) B E A D C (1) B C A E D (1) B A E C D (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 2 -6 14 10 B -2 0 -12 -16 2 C 6 12 0 6 -4 D -14 16 -6 0 8 E -10 -2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999994 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 2 -6 14 10 B -2 0 -12 -16 2 C 6 12 0 6 -4 D -14 16 -6 0 8 E -10 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=24 D=19 A=18 E=14 so E is eliminated. Round 2 votes counts: D=32 B=26 C=24 A=18 so A is eliminated. Round 3 votes counts: D=43 C=30 B=27 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:210 C:210 D:202 E:192 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 14 10 B -2 0 -12 -16 2 C 6 12 0 6 -4 D -14 16 -6 0 8 E -10 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 14 10 B -2 0 -12 -16 2 C 6 12 0 6 -4 D -14 16 -6 0 8 E -10 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 14 10 B -2 0 -12 -16 2 C 6 12 0 6 -4 D -14 16 -6 0 8 E -10 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.300000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1974: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) C E D A B (11) E D C A B (6) E C D B A (5) A B D E C (5) E D A B C (4) D E A C B (4) D A E B C (4) B A E D C (4) B A D E C (4) A D B E C (4) E C D A B (3) D E C A B (3) C B A D E (3) B A D C E (3) E D B A C (2) C E D B A (2) C B A E D (2) B C E A D (2) A D C B E (2) E D B C A (1) E C B D A (1) E B A D C (1) D E A B C (1) D C A E B (1) C E B D A (1) C D A E B (1) C B E D A (1) C A B D E (1) B E A D C (1) B C A E D (1) B C A D E (1) B A C E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 8 -2 2 B -4 0 6 -8 -2 C -8 -6 0 -2 -6 D 2 8 2 0 2 E -2 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -2 2 B -4 0 6 -8 -2 C -8 -6 0 -2 -6 D 2 8 2 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 C=22 D=13 A=13 so D is eliminated. Round 2 votes counts: E=31 B=29 C=23 A=17 so A is eliminated. Round 3 votes counts: B=40 E=35 C=25 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:207 A:206 E:202 B:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -2 2 B -4 0 6 -8 -2 C -8 -6 0 -2 -6 D 2 8 2 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 2 B -4 0 6 -8 -2 C -8 -6 0 -2 -6 D 2 8 2 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 2 B -4 0 6 -8 -2 C -8 -6 0 -2 -6 D 2 8 2 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1975: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) D C B A E (6) A D C E B (6) B E C D A (4) B C E D A (4) A E D B C (4) A E B D C (4) A D B C E (4) E C B D A (3) E C B A D (3) E B A C D (3) E A C D B (3) C E B D A (3) C D E B A (3) C D B E A (3) A E D C B (3) E C A B D (2) E B C D A (2) E B C A D (2) D C A E B (2) D C A B E (2) D A C B E (2) D A B C E (2) C E D B A (2) C B E D A (2) B E A C D (2) A E B C D (2) A D E B C (2) E C D B A (1) E A C B D (1) D C B E A (1) D B A C E (1) D A C E B (1) C B D E A (1) B D C A E (1) B C D E A (1) A D C B E (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 6 -8 B -4 0 -2 2 -20 C -4 2 0 10 -2 D -6 -2 -10 0 -18 E 8 20 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 6 -8 B -4 0 -2 2 -20 C -4 2 0 10 -2 D -6 -2 -10 0 -18 E 8 20 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=28 D=17 C=14 B=12 so B is eliminated. Round 2 votes counts: E=34 A=29 C=19 D=18 so D is eliminated. Round 3 votes counts: A=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:203 C:203 B:188 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 6 -8 B -4 0 -2 2 -20 C -4 2 0 10 -2 D -6 -2 -10 0 -18 E 8 20 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 6 -8 B -4 0 -2 2 -20 C -4 2 0 10 -2 D -6 -2 -10 0 -18 E 8 20 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 6 -8 B -4 0 -2 2 -20 C -4 2 0 10 -2 D -6 -2 -10 0 -18 E 8 20 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1976: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) E D A B C (7) D E A C B (7) C B D A E (6) B C A E D (5) D C E A B (4) C B A D E (4) B E A D C (4) B A E C D (4) C D B E A (3) C D B A E (3) C D A E B (3) B E A C D (3) B C E D A (3) B C E A D (3) A E D B C (3) A E B D C (3) E B A D C (2) E A B D C (2) C D E B A (2) C D E A B (2) C D A B E (2) C B D E A (2) A D E C B (2) A B E D C (2) E D A C B (1) D E C A B (1) D C A E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B C D E A (1) B C A D E (1) B A C E D (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 4 0 -14 B -2 0 8 -4 0 C -4 -8 0 -2 -4 D 0 4 2 0 -8 E 14 0 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.423139 C: 0.000000 D: 0.000000 E: 0.576861 Sum of squares = 0.511815290175 Cumulative probabilities = A: 0.000000 B: 0.423139 C: 0.423139 D: 0.423139 E: 1.000000 A B C D E A 0 2 4 0 -14 B -2 0 8 -4 0 C -4 -8 0 -2 -4 D 0 4 2 0 -8 E 14 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=27 E=20 D=13 A=12 so A is eliminated. Round 2 votes counts: B=29 C=28 E=27 D=16 so D is eliminated. Round 3 votes counts: E=37 C=34 B=29 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 B:201 D:199 A:196 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 0 -14 B -2 0 8 -4 0 C -4 -8 0 -2 -4 D 0 4 2 0 -8 E 14 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 -14 B -2 0 8 -4 0 C -4 -8 0 -2 -4 D 0 4 2 0 -8 E 14 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 -14 B -2 0 8 -4 0 C -4 -8 0 -2 -4 D 0 4 2 0 -8 E 14 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1977: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) B C A D E (10) D E C B A (7) A C B D E (6) E B C D A (4) E A D C B (4) D A C B E (4) B C D A E (4) D E A C B (3) D C B E A (3) D C B A E (3) D A E C B (3) A D C B E (3) E D A C B (2) E D A B C (2) E B C A D (2) E A B C D (2) C B D A E (2) C B A D E (2) B C D E A (2) B C A E D (2) A D E C B (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B A C (1) E A D B C (1) D B C E A (1) B E C D A (1) B E C A D (1) B C E A D (1) B A C E D (1) A E D C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -22 -18 -12 -2 B 22 0 4 -8 2 C 18 -4 0 -8 0 D 12 8 8 0 14 E 2 -2 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -18 -12 -2 B 22 0 4 -8 2 C 18 -4 0 -8 0 D 12 8 8 0 14 E 2 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=24 B=22 A=18 C=4 so C is eliminated. Round 2 votes counts: E=32 B=26 D=24 A=18 so A is eliminated. Round 3 votes counts: B=37 E=34 D=29 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:221 B:210 C:203 E:193 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -18 -12 -2 B 22 0 4 -8 2 C 18 -4 0 -8 0 D 12 8 8 0 14 E 2 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -18 -12 -2 B 22 0 4 -8 2 C 18 -4 0 -8 0 D 12 8 8 0 14 E 2 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -18 -12 -2 B 22 0 4 -8 2 C 18 -4 0 -8 0 D 12 8 8 0 14 E 2 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1978: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) E A D C B (6) E A C D B (6) A E D B C (6) D B A C E (5) C E B D A (5) C B E D A (5) B D C A E (5) A D E B C (5) B C D A E (4) A E D C B (4) A D B E C (4) E C D A B (3) B C D E A (3) A B E D C (3) E C A B D (2) D B C A E (2) D A B C E (2) C E D B A (2) B D A C E (2) B C E A D (2) A B D C E (2) E D A C B (1) E C B D A (1) E C B A D (1) E B C A D (1) E A C B D (1) D C A E B (1) D A B E C (1) C D E B A (1) C D E A B (1) C D B E A (1) C B D E A (1) B E C A D (1) A E B D C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 2 8 -2 B -16 0 2 -18 -10 C -2 -2 0 -4 -10 D -8 18 4 0 -16 E 2 10 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 2 8 -2 B -16 0 2 -18 -10 C -2 -2 0 -4 -10 D -8 18 4 0 -16 E 2 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 B=17 C=16 D=11 so D is eliminated. Round 2 votes counts: A=30 E=29 B=24 C=17 so C is eliminated. Round 3 votes counts: E=38 B=31 A=31 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:212 D:199 C:191 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 2 8 -2 B -16 0 2 -18 -10 C -2 -2 0 -4 -10 D -8 18 4 0 -16 E 2 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 8 -2 B -16 0 2 -18 -10 C -2 -2 0 -4 -10 D -8 18 4 0 -16 E 2 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 8 -2 B -16 0 2 -18 -10 C -2 -2 0 -4 -10 D -8 18 4 0 -16 E 2 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1979: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (6) E C D A B (5) A E B C D (5) E D A C B (4) E C A D B (4) E A D C B (4) E A C D B (4) B D C A E (4) A E B D C (4) A B E D C (4) D C B E A (3) C D E B A (3) C B D E A (3) B C D A E (3) B A C E D (3) B A C D E (3) A B C E D (3) D C E B A (2) C E D A B (2) C E A B D (2) C D B E A (2) B D A C E (2) A B D E C (2) D E C B A (1) D E C A B (1) D E B C A (1) D E A C B (1) D B E A C (1) D B C A E (1) D B A E C (1) D B A C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C B E A D (1) C B A E D (1) C A E B D (1) B D A E C (1) B C D E A (1) B C A E D (1) B C A D E (1) B A D E C (1) B A D C E (1) A E D C B (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 10 8 10 4 B -10 0 0 0 -12 C -8 0 0 6 -4 D -10 0 -6 0 -20 E -4 12 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 10 4 B -10 0 0 0 -12 C -8 0 0 6 -4 D -10 0 -6 0 -20 E -4 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=21 B=21 C=18 D=13 so D is eliminated. Round 2 votes counts: A=27 E=25 B=25 C=23 so C is eliminated. Round 3 votes counts: E=37 B=35 A=28 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:216 E:216 C:197 B:189 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 10 4 B -10 0 0 0 -12 C -8 0 0 6 -4 D -10 0 -6 0 -20 E -4 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 10 4 B -10 0 0 0 -12 C -8 0 0 6 -4 D -10 0 -6 0 -20 E -4 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 10 4 B -10 0 0 0 -12 C -8 0 0 6 -4 D -10 0 -6 0 -20 E -4 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1980: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) E D A C B (5) D E C A B (5) D E C B A (4) D E B C A (4) D C E A B (4) C D A E B (4) B C A D E (4) B A E C D (4) A C B E D (4) E A B C D (3) D B C E A (3) C B A D E (3) C A D E B (3) B D C A E (3) B A C D E (3) E D B A C (2) C A E D B (2) C A D B E (2) C A B D E (2) B D E A C (2) A E C D B (2) E D C A B (1) E D A B C (1) E C D A B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A D B C (1) E A C D B (1) D E B A C (1) D C E B A (1) D C A B E (1) D B E C A (1) D B E A C (1) C E D A B (1) C E A D B (1) C A E B D (1) B E D A C (1) B E A D C (1) B D E C A (1) B C D A E (1) B A D E C (1) A C E D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 2 2 B 4 0 2 -6 -4 C 6 -2 0 8 6 D -2 6 -8 0 8 E -2 4 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406249999972 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 2 B 4 0 2 -6 -4 C 6 -2 0 8 6 D -2 6 -8 0 8 E -2 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406249999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 C=19 E=18 A=9 so A is eliminated. Round 2 votes counts: B=31 D=25 C=24 E=20 so E is eliminated. Round 3 votes counts: B=37 D=35 C=28 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:209 D:202 B:198 A:197 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 2 2 B 4 0 2 -6 -4 C 6 -2 0 8 6 D -2 6 -8 0 8 E -2 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406249999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 2 B 4 0 2 -6 -4 C 6 -2 0 8 6 D -2 6 -8 0 8 E -2 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406249999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 2 B 4 0 2 -6 -4 C 6 -2 0 8 6 D -2 6 -8 0 8 E -2 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406249999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1981: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) B C A D E (9) E D A C B (8) D A C E B (7) D A E C B (6) D A C B E (6) E B D C A (4) E B C A D (4) B C A E D (4) E D B A C (3) E B D A C (3) D E A C B (3) C B A D E (3) B C E A D (3) E D B C A (2) C B D A E (2) B E C D A (2) B E C A D (2) B C D A E (2) A D C B E (2) A C D B E (2) E B C D A (1) E B A D C (1) E B A C D (1) E A D C B (1) E A D B C (1) D C B A E (1) C D A B E (1) C A B D E (1) B A C E D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 18 -22 2 B -4 0 6 -12 -16 C -18 -6 0 -20 -6 D 22 12 20 0 -4 E -2 16 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.785714 Sum of squares = 0.642857142839 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.214286 E: 1.000000 A B C D E A 0 4 18 -22 2 B -4 0 6 -12 -16 C -18 -6 0 -20 -6 D 22 12 20 0 -4 E -2 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.785714 Sum of squares = 0.642857142894 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.214286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=23 B=23 A=8 C=7 so C is eliminated. Round 2 votes counts: E=39 B=28 D=24 A=9 so A is eliminated. Round 3 votes counts: E=40 D=30 B=30 so D is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:225 E:212 A:201 B:187 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 18 -22 2 B -4 0 6 -12 -16 C -18 -6 0 -20 -6 D 22 12 20 0 -4 E -2 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.785714 Sum of squares = 0.642857142894 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.214286 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 -22 2 B -4 0 6 -12 -16 C -18 -6 0 -20 -6 D 22 12 20 0 -4 E -2 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.785714 Sum of squares = 0.642857142894 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.214286 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 -22 2 B -4 0 6 -12 -16 C -18 -6 0 -20 -6 D 22 12 20 0 -4 E -2 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.071429 E: 0.785714 Sum of squares = 0.642857142894 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.214286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1982: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (13) B E C D A (8) D E B A C (7) A D C E B (7) C B E A D (6) E B D C A (4) C A B E D (4) A D E C B (4) A D E B C (4) A D C B E (4) E B C D A (3) C A D B E (3) A C D E B (3) A C D B E (3) E B D A C (2) D E A B C (2) C E B D A (2) C B E D A (2) C B A E D (2) C A D E B (2) B E D A C (2) B C E D A (2) E C B D A (1) D A E C B (1) C E A B D (1) C A B D E (1) B E D C A (1) B E C A D (1) B D E A C (1) A D B E C (1) A D B C E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 12 14 -2 10 B -12 0 6 -10 -12 C -14 -6 0 -10 -12 D 2 10 10 0 18 E -10 12 12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 -2 10 B -12 0 6 -10 -12 C -14 -6 0 -10 -12 D 2 10 10 0 18 E -10 12 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=23 C=23 B=15 E=10 so E is eliminated. Round 2 votes counts: A=29 C=24 B=24 D=23 so D is eliminated. Round 3 votes counts: A=45 B=31 C=24 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:220 A:217 E:198 B:186 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 14 -2 10 B -12 0 6 -10 -12 C -14 -6 0 -10 -12 D 2 10 10 0 18 E -10 12 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 -2 10 B -12 0 6 -10 -12 C -14 -6 0 -10 -12 D 2 10 10 0 18 E -10 12 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 -2 10 B -12 0 6 -10 -12 C -14 -6 0 -10 -12 D 2 10 10 0 18 E -10 12 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1983: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (25) B D A C E (19) E B D A C (9) E B C A D (5) C A D B E (5) B E D A C (5) A C D B E (5) E B D C A (4) C A E D B (4) D B A C E (3) D A C B E (3) D A B C E (2) C A D E B (2) B D A E C (2) A D C B E (2) E D C B A (1) E C B A D (1) E C A B D (1) C E A B D (1) B D E A C (1) Total count = 100 A B C D E A 0 0 2 2 -6 B 0 0 0 -4 -6 C -2 0 0 -2 -8 D -2 4 2 0 -12 E 6 6 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 2 -6 B 0 0 0 -4 -6 C -2 0 0 -2 -8 D -2 4 2 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=46 B=27 C=12 D=8 A=7 so A is eliminated. Round 2 votes counts: E=46 B=27 C=17 D=10 so D is eliminated. Round 3 votes counts: E=46 B=32 C=22 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:199 D:196 B:195 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 2 -6 B 0 0 0 -4 -6 C -2 0 0 -2 -8 D -2 4 2 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 -6 B 0 0 0 -4 -6 C -2 0 0 -2 -8 D -2 4 2 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 -6 B 0 0 0 -4 -6 C -2 0 0 -2 -8 D -2 4 2 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1984: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) A B C E D (7) D C E B A (6) C B A D E (6) E A B D C (5) C B A E D (5) B A C E D (5) A B E D C (5) D E C A B (4) B C A E D (4) A E B D C (4) E D B A C (3) D E C B A (3) D E A B C (3) D E A C B (2) C D B E A (2) C D B A E (2) C A B D E (2) B E A C D (2) B A E C D (2) A B E C D (2) E D B C A (1) E B C D A (1) E A D B C (1) D E B A C (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D B A (1) C D E B A (1) C B E A D (1) C B D A E (1) B C E A D (1) A D C B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 16 10 4 B -4 0 18 10 0 C -16 -18 0 -6 -2 D -10 -10 6 0 -20 E -4 0 2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 10 4 B -4 0 18 10 0 C -16 -18 0 -6 -2 D -10 -10 6 0 -20 E -4 0 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=21 C=21 A=21 B=14 so B is eliminated. Round 2 votes counts: A=28 C=26 E=23 D=23 so E is eliminated. Round 3 votes counts: D=37 A=36 C=27 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:212 E:209 D:183 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 10 4 B -4 0 18 10 0 C -16 -18 0 -6 -2 D -10 -10 6 0 -20 E -4 0 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 10 4 B -4 0 18 10 0 C -16 -18 0 -6 -2 D -10 -10 6 0 -20 E -4 0 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 10 4 B -4 0 18 10 0 C -16 -18 0 -6 -2 D -10 -10 6 0 -20 E -4 0 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1985: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) A B C E D (10) A C B E D (7) D E C B A (6) B E D C A (6) A C D E B (6) E D B C A (5) D E C A B (4) D C E A B (4) C A D E B (4) A C B D E (4) E B D C A (3) C A E D B (3) D E B A C (2) D C A E B (2) C D E A B (2) C D A E B (2) C A B E D (2) B E A D C (2) B C E A D (2) B A C E D (2) A B D E C (2) E C D B A (1) D E A B C (1) D B E A C (1) C E D B A (1) B E D A C (1) B E C D A (1) B A E D C (1) B A E C D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -16 -4 -4 B -10 0 0 -12 -14 C 16 0 0 -2 6 D 4 12 2 0 4 E 4 14 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -16 -4 -4 B -10 0 0 -12 -14 C 16 0 0 -2 6 D 4 12 2 0 4 E 4 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=30 B=16 C=14 E=9 so E is eliminated. Round 2 votes counts: D=35 A=31 B=19 C=15 so C is eliminated. Round 3 votes counts: D=41 A=40 B=19 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:210 E:204 A:193 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -16 -4 -4 B -10 0 0 -12 -14 C 16 0 0 -2 6 D 4 12 2 0 4 E 4 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -16 -4 -4 B -10 0 0 -12 -14 C 16 0 0 -2 6 D 4 12 2 0 4 E 4 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -16 -4 -4 B -10 0 0 -12 -14 C 16 0 0 -2 6 D 4 12 2 0 4 E 4 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1986: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) D C B E A (7) E A C D B (6) A E B D C (6) A E B C D (5) E A B D C (4) D C E B A (4) D C E A B (4) E A C B D (3) D E C A B (3) C D E A B (3) C D B A E (3) B D C A E (3) B D A E C (3) B C D A E (3) B A D E C (3) A B E C D (3) E D A C B (2) E C A D B (2) E A D C B (2) C D E B A (2) C A E B D (2) B A E C D (2) B A C E D (2) A B E D C (2) E C D A B (1) E A D B C (1) E A B C D (1) D E A C B (1) D B C A E (1) C E D A B (1) C E A B D (1) C B D A E (1) C B A D E (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 6 -4 -4 -16 B -6 0 -18 -6 -8 C 4 18 0 4 -2 D 4 6 -4 0 6 E 16 8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.666667 E: 1.000000 A B C D E A 0 6 -4 -4 -16 B -6 0 -18 -6 -8 C 4 18 0 4 -2 D 4 6 -4 0 6 E 16 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888889386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=22 D=20 B=18 A=16 so A is eliminated. Round 2 votes counts: E=33 C=24 B=23 D=20 so D is eliminated. Round 3 votes counts: C=39 E=37 B=24 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:210 D:206 A:191 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -4 -16 B -6 0 -18 -6 -8 C 4 18 0 4 -2 D 4 6 -4 0 6 E 16 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888889386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 -16 B -6 0 -18 -6 -8 C 4 18 0 4 -2 D 4 6 -4 0 6 E 16 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888889386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 -16 B -6 0 -18 -6 -8 C 4 18 0 4 -2 D 4 6 -4 0 6 E 16 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888889386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1987: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) D C B A E (7) D C A B E (6) A D B C E (6) A B E D C (6) E C B D A (5) D C E A B (5) A B D C E (5) E C D B A (4) B A E C D (4) A B E C D (4) E B C A D (3) E B A C D (3) E A B C D (3) D C E B A (3) D B A C E (3) D A C B E (3) A D B E C (3) E C B A D (2) B E A C D (2) A B D E C (2) E C A D B (1) D C B E A (1) D C A E B (1) D A B C E (1) C E B D A (1) C D B E A (1) C B E D A (1) B E C A D (1) B D A C E (1) B A C D E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -2 -4 10 B 4 0 0 -8 16 C 2 0 0 -8 10 D 4 8 8 0 16 E -10 -16 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 10 B 4 0 0 -8 16 C 2 0 0 -8 10 D 4 8 8 0 16 E -10 -16 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=28 E=21 C=12 B=9 so B is eliminated. Round 2 votes counts: A=33 D=31 E=24 C=12 so C is eliminated. Round 3 votes counts: D=41 A=33 E=26 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:206 C:202 A:200 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 10 B 4 0 0 -8 16 C 2 0 0 -8 10 D 4 8 8 0 16 E -10 -16 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 10 B 4 0 0 -8 16 C 2 0 0 -8 10 D 4 8 8 0 16 E -10 -16 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 10 B 4 0 0 -8 16 C 2 0 0 -8 10 D 4 8 8 0 16 E -10 -16 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 1988: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) E C B D A (8) A D B C E (8) E C A D B (5) E C D B A (4) D B A C E (4) A D C B E (4) A B D C E (4) E B C D A (3) E A C D B (3) C E A D B (3) B E D C A (3) B D E A C (3) E C B A D (2) E C A B D (2) E B D A C (2) D B C A E (2) C E D A B (2) B D E C A (2) B D C E A (2) B D A E C (2) B A D C E (2) A E C D B (2) A E B D C (2) A C E D B (2) A B D E C (2) E C D A B (1) E A C B D (1) D C A B E (1) D A B C E (1) C E D B A (1) C A E D B (1) C A D E B (1) B E C D A (1) B D C A E (1) B A D E C (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 10 -4 4 B 4 0 10 6 8 C -10 -10 0 -12 0 D 4 -6 12 0 2 E -4 -8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -4 4 B 4 0 10 6 8 C -10 -10 0 -12 0 D 4 -6 12 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=27 B=26 D=8 C=8 so D is eliminated. Round 2 votes counts: B=32 E=31 A=28 C=9 so C is eliminated. Round 3 votes counts: E=37 B=32 A=31 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:206 A:203 E:193 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 -4 4 B 4 0 10 6 8 C -10 -10 0 -12 0 D 4 -6 12 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -4 4 B 4 0 10 6 8 C -10 -10 0 -12 0 D 4 -6 12 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -4 4 B 4 0 10 6 8 C -10 -10 0 -12 0 D 4 -6 12 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1989: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) D E C B A (7) B A E C D (6) B A C E D (6) D B E A C (5) E C A B D (4) D E C A B (4) D E B C A (4) D C E A B (4) D C A B E (4) B E A C D (4) A B C E D (4) E C D A B (3) B A E D C (3) E D C A B (2) E D B C A (2) C D E A B (2) C A E B D (2) C A D E B (2) B E D A C (2) B D E A C (2) A C D B E (2) A B C D E (2) E D B A C (1) E B D A C (1) E B C A D (1) E B A C D (1) D C A E B (1) D B E C A (1) D B A C E (1) C D A B E (1) C A E D B (1) C A D B E (1) C A B E D (1) B E A D C (1) B D A E C (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 6 4 -2 B 0 0 -2 4 14 C -6 2 0 6 -12 D -4 -4 -6 0 -10 E 2 -14 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.641628 B: 0.358372 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.540117227072 Cumulative probabilities = A: 0.641628 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 -2 B 0 0 -2 4 14 C -6 2 0 6 -12 D -4 -4 -6 0 -10 E 2 -14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 A=18 E=15 C=10 so C is eliminated. Round 2 votes counts: D=34 B=26 A=25 E=15 so E is eliminated. Round 3 votes counts: D=42 B=29 A=29 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:208 E:205 A:204 C:195 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 -2 B 0 0 -2 4 14 C -6 2 0 6 -12 D -4 -4 -6 0 -10 E 2 -14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 -2 B 0 0 -2 4 14 C -6 2 0 6 -12 D -4 -4 -6 0 -10 E 2 -14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 -2 B 0 0 -2 4 14 C -6 2 0 6 -12 D -4 -4 -6 0 -10 E 2 -14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1990: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) E C D A B (7) E B C D A (6) B E A D C (6) B E A C D (6) E B A D C (5) C A D B E (5) B A C D E (5) D C A B E (3) B A D C E (3) A C D B E (3) A B D E C (3) A B D C E (3) E D A C B (2) E B D C A (2) E B D A C (2) D E C A B (2) D C A E B (2) D A E C B (2) D A C B E (2) C D E A B (2) C D A B E (2) C B A D E (2) B A C E D (2) A D C B E (2) A C B D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E B C A D (1) E B A C D (1) D C E A B (1) D A E B C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 6 2 -12 B -10 0 0 0 2 C -6 0 0 -8 -20 D -2 0 8 0 -6 E 12 -2 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.43055555544 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 A B C D E A 0 10 6 2 -12 B -10 0 0 0 2 C -6 0 0 -8 -20 D -2 0 8 0 -6 E 12 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555488 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=22 A=15 D=13 C=11 so C is eliminated. Round 2 votes counts: E=39 B=24 A=20 D=17 so D is eliminated. Round 3 votes counts: E=44 A=32 B=24 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:203 D:200 B:196 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 2 -12 B -10 0 0 0 2 C -6 0 0 -8 -20 D -2 0 8 0 -6 E 12 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555488 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 -12 B -10 0 0 0 2 C -6 0 0 -8 -20 D -2 0 8 0 -6 E 12 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555488 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 -12 B -10 0 0 0 2 C -6 0 0 -8 -20 D -2 0 8 0 -6 E 12 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555488 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 1991: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) C D E B A (7) E A C B D (6) B D A C E (6) E C A B D (5) E A C D B (5) D B C A E (5) C D B E A (5) A E B C D (5) D C B E A (4) D C B A E (4) C E A D B (4) E C A D B (3) E A B C D (3) D B A E C (3) B A E D C (3) A E B D C (3) A B E D C (3) A B E C D (2) E A B D C (1) D C E B A (1) D C E A B (1) D B A C E (1) D A E B C (1) D A B E C (1) C E D B A (1) C E D A B (1) C D E A B (1) C B D E A (1) B D C A E (1) B C D A E (1) B A E C D (1) B A D E C (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -6 10 -14 B -10 0 -16 4 -14 C 6 16 0 20 4 D -10 -4 -20 0 -10 E 14 14 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 10 -14 B -10 0 -16 4 -14 C 6 16 0 20 4 D -10 -4 -20 0 -10 E 14 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=23 D=21 A=15 B=13 so B is eliminated. Round 2 votes counts: C=29 D=28 E=23 A=20 so A is eliminated. Round 3 votes counts: E=41 D=30 C=29 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:223 E:217 A:200 B:182 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 10 -14 B -10 0 -16 4 -14 C 6 16 0 20 4 D -10 -4 -20 0 -10 E 14 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 10 -14 B -10 0 -16 4 -14 C 6 16 0 20 4 D -10 -4 -20 0 -10 E 14 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 10 -14 B -10 0 -16 4 -14 C 6 16 0 20 4 D -10 -4 -20 0 -10 E 14 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1992: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) E D B A C (6) E D A B C (5) C B D A E (5) B C D A E (5) A C E B D (5) E A D B C (4) C D B E A (4) A E C B D (4) A E B D C (4) E A D C B (3) D E B C A (3) D E B A C (3) D B C E A (3) A C B E D (3) D B E C A (2) C B A D E (2) C A B E D (2) C A B D E (2) B D E C A (2) B D E A C (2) B D C E A (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E B D A C (1) E A C D B (1) D C E B A (1) D B E A C (1) C E A D B (1) C D E B A (1) C A E D B (1) C A E B D (1) B D C A E (1) B C A D E (1) B A E D C (1) B A D C E (1) A E D C B (1) A E D B C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 16 0 2 B -6 0 -2 -10 -20 C -16 2 0 2 -16 D 0 10 -2 0 -18 E -2 20 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.934284 B: 0.000000 C: 0.000000 D: 0.065716 E: 0.000000 Sum of squares = 0.877205009798 Cumulative probabilities = A: 0.934284 B: 0.934284 C: 0.934284 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 0 2 B -6 0 -2 -10 -20 C -16 2 0 2 -16 D 0 10 -2 0 -18 E -2 20 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.900000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.820000027383 Cumulative probabilities = A: 0.900000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 C=19 B=15 D=13 so D is eliminated. Round 2 votes counts: E=30 A=29 B=21 C=20 so C is eliminated. Round 3 votes counts: A=35 E=33 B=32 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:226 A:212 D:195 C:186 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 0 2 B -6 0 -2 -10 -20 C -16 2 0 2 -16 D 0 10 -2 0 -18 E -2 20 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.900000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.820000027383 Cumulative probabilities = A: 0.900000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 0 2 B -6 0 -2 -10 -20 C -16 2 0 2 -16 D 0 10 -2 0 -18 E -2 20 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.900000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.820000027383 Cumulative probabilities = A: 0.900000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 0 2 B -6 0 -2 -10 -20 C -16 2 0 2 -16 D 0 10 -2 0 -18 E -2 20 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.900000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.820000027383 Cumulative probabilities = A: 0.900000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1993: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C B D A E (6) C D E B A (5) C D B E A (5) E D C A B (4) B A D C E (4) A E D B C (4) E D A C B (3) E A D C B (3) E A D B C (3) E A B C D (3) C E D B A (3) C E B D A (3) C D B A E (3) B C D A E (3) B A C D E (3) A B D E C (3) E C D B A (2) E C D A B (2) E B C A D (2) E A C B D (2) D C B A E (2) D A B C E (2) C B D E A (2) A E B D C (2) A D B C E (2) A B E D C (2) E C B A D (1) E C A D B (1) E A C D B (1) D C E B A (1) D C B E A (1) D C A B E (1) D B C A E (1) B D C A E (1) B D A C E (1) B C A E D (1) B C A D E (1) B A C E D (1) A D E B C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -2 -2 -8 B 4 0 -2 0 -6 C 2 2 0 2 6 D 2 0 -2 0 -2 E 8 6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -2 -8 B 4 0 -2 0 -6 C 2 2 0 2 6 D 2 0 -2 0 -2 E 8 6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=27 A=16 B=15 D=8 so D is eliminated. Round 2 votes counts: E=34 C=32 A=18 B=16 so B is eliminated. Round 3 votes counts: C=39 E=34 A=27 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 E:205 D:199 B:198 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 -8 B 4 0 -2 0 -6 C 2 2 0 2 6 D 2 0 -2 0 -2 E 8 6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 -8 B 4 0 -2 0 -6 C 2 2 0 2 6 D 2 0 -2 0 -2 E 8 6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 -8 B 4 0 -2 0 -6 C 2 2 0 2 6 D 2 0 -2 0 -2 E 8 6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1994: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) A B E C D (7) D C E B A (6) C D B E A (5) A C B E D (5) E B C D A (4) E B C A D (4) D C A E B (4) D A E B C (4) C A B E D (4) A E B D C (4) A D B E C (4) C B E D A (3) A D C B E (3) E B D C A (2) D E C B A (2) D E A B C (2) D C B E A (2) D A C B E (2) C A D B E (2) B E C A D (2) B E A C D (2) A D E B C (2) E C B A D (1) E B A D C (1) E B A C D (1) E A B D C (1) D E B C A (1) D C E A B (1) D A C E B (1) C D E B A (1) C D B A E (1) C D A B E (1) C B D E A (1) C A B D E (1) A E D B C (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -14 14 -2 B -4 0 -12 10 14 C 14 12 0 12 8 D -14 -10 -12 0 -6 E 2 -14 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 14 -2 B -4 0 -12 10 14 C 14 12 0 12 8 D -14 -10 -12 0 -6 E 2 -14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=28 D=25 E=14 B=4 so B is eliminated. Round 2 votes counts: A=29 C=28 D=25 E=18 so E is eliminated. Round 3 votes counts: C=39 A=34 D=27 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 B:204 A:201 E:193 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 14 -2 B -4 0 -12 10 14 C 14 12 0 12 8 D -14 -10 -12 0 -6 E 2 -14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 14 -2 B -4 0 -12 10 14 C 14 12 0 12 8 D -14 -10 -12 0 -6 E 2 -14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 14 -2 B -4 0 -12 10 14 C 14 12 0 12 8 D -14 -10 -12 0 -6 E 2 -14 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1995: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (16) C D E A B (13) B A E D C (13) D C B A E (9) D B C A E (9) B A D E C (6) B D A C E (5) E A C B D (4) D C B E A (3) C E D A B (3) A E B C D (3) E C A D B (2) D C E B A (2) D B A C E (2) C E A D B (2) A B E C D (2) D C E A B (1) B E A D C (1) B E A C D (1) B D A E C (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 12 4 4 B 6 0 22 8 8 C -12 -22 0 -6 -2 D -4 -8 6 0 2 E -4 -8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 4 4 B 6 0 22 8 8 C -12 -22 0 -6 -2 D -4 -8 6 0 2 E -4 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=26 E=22 C=18 A=6 so A is eliminated. Round 2 votes counts: B=31 D=26 E=25 C=18 so C is eliminated. Round 3 votes counts: D=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:207 D:198 E:194 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 4 4 B 6 0 22 8 8 C -12 -22 0 -6 -2 D -4 -8 6 0 2 E -4 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 4 4 B 6 0 22 8 8 C -12 -22 0 -6 -2 D -4 -8 6 0 2 E -4 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 4 4 B 6 0 22 8 8 C -12 -22 0 -6 -2 D -4 -8 6 0 2 E -4 -8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1996: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) D B E A C (7) D B A E C (7) D E B C A (6) E C B A D (5) A D B C E (5) D C A E B (4) D A B E C (4) C A E D B (4) A C B E D (4) E B D C A (3) B A D E C (3) A B D C E (3) A B C E D (3) E C D B A (2) D A B C E (2) C E A D B (2) C A E B D (2) C A D E B (2) A C D E B (2) A C D B E (2) E D C B A (1) E D B C A (1) E B C D A (1) E B C A D (1) D E C B A (1) D C E B A (1) D C E A B (1) D B E C A (1) C E D A B (1) C D E A B (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) A C E D B (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 2 6 4 B -10 0 8 -16 -2 C -2 -8 0 -10 2 D -6 16 10 0 14 E -4 2 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 6 4 B -10 0 8 -16 -2 C -2 -8 0 -10 2 D -6 16 10 0 14 E -4 2 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=23 C=21 E=14 B=8 so B is eliminated. Round 2 votes counts: D=36 A=26 C=21 E=17 so E is eliminated. Round 3 votes counts: D=42 C=30 A=28 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:211 C:191 E:191 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 6 4 B -10 0 8 -16 -2 C -2 -8 0 -10 2 D -6 16 10 0 14 E -4 2 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 6 4 B -10 0 8 -16 -2 C -2 -8 0 -10 2 D -6 16 10 0 14 E -4 2 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 6 4 B -10 0 8 -16 -2 C -2 -8 0 -10 2 D -6 16 10 0 14 E -4 2 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 1997: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) E C D A B (6) C E B D A (6) A B D C E (5) E A C D B (4) C D B E A (4) E C D B A (3) E C A B D (3) D C E B A (3) D B C A E (3) C E D B A (3) B C D A E (3) A D E B C (3) A D B E C (3) E D C A B (2) E C A D B (2) E A D C B (2) D C B E A (2) D B A C E (2) C E B A D (2) C D E B A (2) C B E D A (2) C B D E A (2) B D C A E (2) B A C E D (2) A E D C B (2) A E D B C (2) A E B D C (2) A E B C D (2) A B D E C (2) E A C B D (1) D E C A B (1) D A B C E (1) C B E A D (1) B D A C E (1) B C D E A (1) B C A D E (1) B A E C D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -8 2 -4 B 10 0 -6 -2 -2 C 8 6 0 2 16 D -2 2 -2 0 2 E 4 2 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 2 -4 B 10 0 -6 -2 -2 C 8 6 0 2 16 D -2 2 -2 0 2 E 4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 A=23 C=22 B=20 D=12 so D is eliminated. Round 2 votes counts: C=27 B=25 E=24 A=24 so E is eliminated. Round 3 votes counts: C=44 A=31 B=25 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:200 D:200 E:194 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 2 -4 B 10 0 -6 -2 -2 C 8 6 0 2 16 D -2 2 -2 0 2 E 4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 2 -4 B 10 0 -6 -2 -2 C 8 6 0 2 16 D -2 2 -2 0 2 E 4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 2 -4 B 10 0 -6 -2 -2 C 8 6 0 2 16 D -2 2 -2 0 2 E 4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 1998: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) B E A D C (8) C D A E B (7) A C B E D (7) B E A C D (6) E B D A C (5) E B A D C (5) C A D B E (5) A B E D C (5) D E B A C (4) A B C E D (4) C B E A D (3) B A E C D (3) E D B A C (2) D E C A B (2) D E B C A (2) D C E A B (2) D C A E B (2) C D E B A (2) C B A E D (2) A C D B E (2) A C B D E (2) A B E C D (2) E D B C A (1) C D E A B (1) C D A B E (1) C B E D A (1) C A B E D (1) C A B D E (1) A E B D C (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 18 20 -4 B 4 0 0 12 8 C -18 0 0 0 4 D -20 -12 0 0 -12 E 4 -8 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.896037 C: 0.103963 D: 0.000000 E: 0.000000 Sum of squares = 0.813690783016 Cumulative probabilities = A: 0.000000 B: 0.896037 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 18 20 -4 B 4 0 0 12 8 C -18 0 0 0 4 D -20 -12 0 0 -12 E 4 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.702479367998 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 D=20 B=17 E=13 so E is eliminated. Round 2 votes counts: B=27 A=26 C=24 D=23 so D is eliminated. Round 3 votes counts: C=38 B=36 A=26 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:212 E:202 C:193 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 18 20 -4 B 4 0 0 12 8 C -18 0 0 0 4 D -20 -12 0 0 -12 E 4 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.702479367998 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 18 20 -4 B 4 0 0 12 8 C -18 0 0 0 4 D -20 -12 0 0 -12 E 4 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.702479367998 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 18 20 -4 B 4 0 0 12 8 C -18 0 0 0 4 D -20 -12 0 0 -12 E 4 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.702479367998 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 1999: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) E B D C A (7) C B A D E (7) C A B D E (6) E D A B C (4) E B D A C (4) C B E D A (4) C A D B E (4) A D E B C (4) E B C D A (3) E A D B C (3) B D E C A (3) A D E C B (3) A D C B E (3) E C B D A (2) E A D C B (2) E A C D B (2) D B E A C (2) C B D E A (2) C B D A E (2) C B A E D (2) C A E B D (2) C A B E D (2) B C D E A (2) A D B E C (2) A C D B E (2) D B A E C (1) D A E B C (1) D A B E C (1) C E B D A (1) C B E A D (1) C A E D B (1) B E D C A (1) B D C A E (1) B D A C E (1) B C E D A (1) A E D C B (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -8 -2 -4 B 10 0 0 8 2 C 8 0 0 -8 -8 D 2 -8 8 0 -2 E 4 -2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.921455 C: 0.078545 D: 0.000000 E: 0.000000 Sum of squares = 0.855248949071 Cumulative probabilities = A: 0.000000 B: 0.921455 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -2 -4 B 10 0 0 8 2 C 8 0 0 -8 -8 D 2 -8 8 0 -2 E 4 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000534 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=34 A=17 B=9 D=5 so D is eliminated. Round 2 votes counts: E=35 C=34 A=19 B=12 so B is eliminated. Round 3 votes counts: E=41 C=38 A=21 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:206 D:200 C:196 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -2 -4 B 10 0 0 8 2 C 8 0 0 -8 -8 D 2 -8 8 0 -2 E 4 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000534 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -2 -4 B 10 0 0 8 2 C 8 0 0 -8 -8 D 2 -8 8 0 -2 E 4 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000534 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -2 -4 B 10 0 0 8 2 C 8 0 0 -8 -8 D 2 -8 8 0 -2 E 4 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000534 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2000: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) B C D A E (10) C D A E B (8) E A D C B (6) B C E A D (6) E A B D C (5) C B D A E (5) E A D B C (4) D A E C B (4) B E D A C (4) D C A E B (3) B E A C D (3) B D C A E (3) E A C D B (2) E A B C D (2) C D B A E (2) B D E A C (2) B D C E A (2) B D A E C (2) A E D C B (2) E B A D C (1) E B A C D (1) D E A B C (1) D B C A E (1) D B A E C (1) D A E B C (1) C E A D B (1) C D A B E (1) C B E A D (1) C A E D B (1) B E C A D (1) B C E D A (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 -16 6 -4 -10 B 16 0 28 22 14 C -6 -28 0 -8 -8 D 4 -22 8 0 -4 E 10 -14 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 -4 -10 B 16 0 28 22 14 C -6 -28 0 -8 -8 D 4 -22 8 0 -4 E 10 -14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=46 E=21 C=19 D=11 A=3 so A is eliminated. Round 2 votes counts: B=46 E=23 C=19 D=12 so D is eliminated. Round 3 votes counts: B=48 E=30 C=22 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:240 E:204 D:193 A:188 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 -4 -10 B 16 0 28 22 14 C -6 -28 0 -8 -8 D 4 -22 8 0 -4 E 10 -14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -4 -10 B 16 0 28 22 14 C -6 -28 0 -8 -8 D 4 -22 8 0 -4 E 10 -14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -4 -10 B 16 0 28 22 14 C -6 -28 0 -8 -8 D 4 -22 8 0 -4 E 10 -14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2001: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) B D C A E (8) E A D C B (7) D C B E A (6) E D A C B (5) B C D A E (5) B C A E D (5) A E C B D (5) E A C D B (4) D E A C B (4) D B A E C (4) C B A E D (4) B C A D E (3) A E D C B (3) A E C D B (3) D E C A B (2) D C E B A (2) D B E C A (2) D B C E A (2) C E A D B (2) C E A B D (2) B D C E A (2) A E B D C (2) E C D A B (1) D A E B C (1) C D B E A (1) C B E A D (1) C B D E A (1) C A E B D (1) C A B E D (1) B A C E D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -2 -8 -4 B -6 0 -12 -14 -6 C 2 12 0 -16 -4 D 8 14 16 0 2 E 4 6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -8 -4 B -6 0 -12 -14 -6 C 2 12 0 -16 -4 D 8 14 16 0 2 E 4 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999968556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=24 E=17 A=15 C=13 so C is eliminated. Round 2 votes counts: D=32 B=30 E=21 A=17 so A is eliminated. Round 3 votes counts: E=36 D=32 B=32 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:220 E:206 C:197 A:196 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 -8 -4 B -6 0 -12 -14 -6 C 2 12 0 -16 -4 D 8 14 16 0 2 E 4 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999968556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -8 -4 B -6 0 -12 -14 -6 C 2 12 0 -16 -4 D 8 14 16 0 2 E 4 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999968556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -8 -4 B -6 0 -12 -14 -6 C 2 12 0 -16 -4 D 8 14 16 0 2 E 4 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999968556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2002: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) E C D B A (5) E C B D A (5) D A B E C (4) C E A B D (4) A D E C B (4) A D B C E (4) A C B E D (4) E D C A B (3) D E B C A (3) D A E C B (3) C E B D A (3) C E B A D (3) C A E B D (3) A C E B D (3) A B C D E (3) E D C B A (2) E C D A B (2) D E A C B (2) D B E C A (2) D B A E C (2) C B E A D (2) B E C D A (2) B D E C A (2) B D A C E (2) B C E D A (2) B C D E A (2) B A C E D (2) A E C D B (2) A B C E D (2) E C A D B (1) D E C A B (1) D A E B C (1) C B E D A (1) B D C E A (1) B D A E C (1) B C A E D (1) B C A D E (1) B A C D E (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -16 -2 -10 B 4 0 -8 14 -4 C 16 8 0 24 4 D 2 -14 -24 0 -20 E 10 4 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -2 -10 B 4 0 -8 14 -4 C 16 8 0 24 4 D 2 -14 -24 0 -20 E 10 4 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 E=18 D=18 C=16 so C is eliminated. Round 2 votes counts: E=28 B=27 A=27 D=18 so D is eliminated. Round 3 votes counts: A=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:226 E:215 B:203 A:184 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 -2 -10 B 4 0 -8 14 -4 C 16 8 0 24 4 D 2 -14 -24 0 -20 E 10 4 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -2 -10 B 4 0 -8 14 -4 C 16 8 0 24 4 D 2 -14 -24 0 -20 E 10 4 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -2 -10 B 4 0 -8 14 -4 C 16 8 0 24 4 D 2 -14 -24 0 -20 E 10 4 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2003: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) A C E D B (10) B E D C A (9) B D E C A (8) E B C A D (6) D B A E C (5) C A E B D (5) A C D E B (5) D A C B E (3) C A E D B (3) B D E A C (3) E C A B D (2) E B A C D (2) E A C B D (2) D B C E A (2) D B A C E (2) B E D A C (2) A D C E B (2) E C B A D (1) E B D A C (1) E B A D C (1) E A C D B (1) D E B A C (1) D A E C B (1) D A C E B (1) D A B C E (1) C B A E D (1) C A D E B (1) B E C D A (1) B E C A D (1) B C E A D (1) B C D E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 16 -4 -12 B 18 0 16 -2 2 C -16 -16 0 -8 -22 D 4 2 8 0 -6 E 12 -2 22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999761 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 -18 16 -4 -12 B 18 0 16 -2 2 C -16 -16 0 -8 -22 D 4 2 8 0 -6 E 12 -2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 A=21 E=16 C=10 so C is eliminated. Round 2 votes counts: A=30 D=27 B=27 E=16 so E is eliminated. Round 3 votes counts: B=38 A=35 D=27 so D is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:219 B:217 D:204 A:191 C:169 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 16 -4 -12 B 18 0 16 -2 2 C -16 -16 0 -8 -22 D 4 2 8 0 -6 E 12 -2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 16 -4 -12 B 18 0 16 -2 2 C -16 -16 0 -8 -22 D 4 2 8 0 -6 E 12 -2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 16 -4 -12 B 18 0 16 -2 2 C -16 -16 0 -8 -22 D 4 2 8 0 -6 E 12 -2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2004: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) E B C D A (8) C E B A D (8) E B D C A (7) D B A E C (6) D A B E C (6) A D B C E (6) E C B D A (5) E B D A C (5) A C D E B (4) E C B A D (3) B D E A C (3) A C D B E (3) D A B C E (2) C E A B D (2) C A E D B (2) B E C D A (2) A D B E C (2) E B A C D (1) D B E A C (1) D B A C E (1) C E B D A (1) C E A D B (1) C B E D A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B C E D A (1) A E D C B (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 12 -2 -2 B 10 0 8 0 -6 C -12 -8 0 -8 -8 D 2 0 8 0 -6 E 2 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 12 -2 -2 B 10 0 8 0 -6 C -12 -8 0 -8 -8 D 2 0 8 0 -6 E 2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=29 C=17 D=16 B=8 so B is eliminated. Round 2 votes counts: E=33 A=30 D=19 C=18 so C is eliminated. Round 3 votes counts: E=47 A=34 D=19 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:206 D:202 A:199 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 12 -2 -2 B 10 0 8 0 -6 C -12 -8 0 -8 -8 D 2 0 8 0 -6 E 2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 12 -2 -2 B 10 0 8 0 -6 C -12 -8 0 -8 -8 D 2 0 8 0 -6 E 2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 12 -2 -2 B 10 0 8 0 -6 C -12 -8 0 -8 -8 D 2 0 8 0 -6 E 2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2005: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) E B D A C (7) D E C B A (6) D C E B A (5) C D A B E (5) A C D B E (5) E B D C A (4) C D A E B (4) B E A D C (4) D C A E B (3) C D B E A (3) C A B E D (3) C A B D E (3) B C A E D (3) B A E C D (3) A D C E B (3) A C D E B (3) E B A D C (2) D E B C A (2) D E A B C (2) D A E C B (2) D A C E B (2) C A D B E (2) B E D C A (2) A C B D E (2) E D B A C (1) C D E B A (1) C B E A D (1) C B A E D (1) C A D E B (1) B E D A C (1) B E C D A (1) B E A C D (1) B A C E D (1) A D E B C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 -2 14 B -2 0 -26 -2 2 C 0 26 0 6 20 D 2 2 -6 0 10 E -14 -2 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.324553 B: 0.000000 C: 0.675447 D: 0.000000 E: 0.000000 Sum of squares = 0.561563041349 Cumulative probabilities = A: 0.324553 B: 0.324553 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 14 B -2 0 -26 -2 2 C 0 26 0 6 20 D 2 2 -6 0 10 E -14 -2 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 D=22 B=16 E=14 so E is eliminated. Round 2 votes counts: B=29 C=24 A=24 D=23 so D is eliminated. Round 3 votes counts: C=38 B=32 A=30 so A is eliminated. Round 4 votes counts: C=63 B=37 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:207 D:204 B:186 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -2 14 B -2 0 -26 -2 2 C 0 26 0 6 20 D 2 2 -6 0 10 E -14 -2 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 14 B -2 0 -26 -2 2 C 0 26 0 6 20 D 2 2 -6 0 10 E -14 -2 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 14 B -2 0 -26 -2 2 C 0 26 0 6 20 D 2 2 -6 0 10 E -14 -2 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2006: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) E B C D A (6) A E D B C (6) A D B E C (6) E B D C A (5) C E A B D (5) E A B D C (4) D B E A C (4) A E C B D (4) A D E B C (4) E D B A C (3) E B A D C (3) C B E D A (3) C A E B D (3) C A D B E (3) A D C B E (3) E C B A D (2) D B A E C (2) D A B E C (2) C E B D A (2) B D C E A (2) A D B C E (2) A C D B E (2) E A C B D (1) E A B C D (1) D E B A C (1) D B E C A (1) D B C E A (1) D B A C E (1) D A B C E (1) C D A B E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B E D (1) B E D C A (1) B D E C A (1) B C D E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 2 4 22 -10 B -2 0 16 10 -18 C -4 -16 0 -10 -18 D -22 -10 10 0 -18 E 10 18 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 22 -10 B -2 0 16 10 -18 C -4 -16 0 -10 -18 D -22 -10 10 0 -18 E 10 18 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=28 E=25 D=13 B=5 so B is eliminated. Round 2 votes counts: C=29 A=29 E=26 D=16 so D is eliminated. Round 3 votes counts: A=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:209 B:203 D:180 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 22 -10 B -2 0 16 10 -18 C -4 -16 0 -10 -18 D -22 -10 10 0 -18 E 10 18 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 22 -10 B -2 0 16 10 -18 C -4 -16 0 -10 -18 D -22 -10 10 0 -18 E 10 18 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 22 -10 B -2 0 16 10 -18 C -4 -16 0 -10 -18 D -22 -10 10 0 -18 E 10 18 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2007: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (7) C D A E B (6) B E D A C (6) B A E C D (6) C A D B E (5) B E A D C (5) B E A C D (5) E B D A C (4) E B A D C (4) C A B D E (4) D E C A B (3) D C E B A (3) D C A E B (3) A E B C D (3) E A B D C (2) D B E C A (2) C B D A E (2) C A D E B (2) B E D C A (2) B A C E D (2) A C E D B (2) A C B E D (2) E D B A C (1) E D A B C (1) D E B C A (1) D E B A C (1) D E A C B (1) D E A B C (1) D A C E B (1) C D A B E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B E D (1) B D E C A (1) B C E A D (1) B C A E D (1) A E D B C (1) A E B D C (1) A D E C B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 18 18 10 B 0 0 16 26 12 C -18 -16 0 10 -18 D -18 -26 -10 0 -18 E -10 -12 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.731215 B: 0.268785 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.60692103794 Cumulative probabilities = A: 0.731215 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 18 10 B 0 0 16 26 12 C -18 -16 0 10 -18 D -18 -26 -10 0 -18 E -10 -12 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=24 A=19 D=16 E=12 so E is eliminated. Round 2 votes counts: B=37 C=24 A=21 D=18 so D is eliminated. Round 3 votes counts: B=42 C=33 A=25 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:223 E:207 C:179 D:164 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 18 10 B 0 0 16 26 12 C -18 -16 0 10 -18 D -18 -26 -10 0 -18 E -10 -12 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 18 10 B 0 0 16 26 12 C -18 -16 0 10 -18 D -18 -26 -10 0 -18 E -10 -12 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 18 10 B 0 0 16 26 12 C -18 -16 0 10 -18 D -18 -26 -10 0 -18 E -10 -12 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2008: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) E A C B D (9) A C E D B (7) E A B C D (6) A E C D B (6) E B D A C (4) D C A B E (4) C A D E B (4) B E C A D (4) B D E A C (4) E B A C D (3) E A D C B (3) C D A B E (3) B D E C A (3) B D C A E (3) E B C A D (2) E B A D C (2) C A E B D (2) C A B E D (2) B E D A C (2) B D C E A (2) B C D A E (2) A E C B D (2) E A C D B (1) D B C E A (1) D B A C E (1) D A E C B (1) D A C E B (1) C A E D B (1) C A D B E (1) B E D C A (1) A C D E B (1) Total count = 100 A B C D E A 0 8 6 12 6 B -8 0 4 6 -10 C -6 -4 0 12 -6 D -12 -6 -12 0 -14 E -6 10 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 12 6 B -8 0 4 6 -10 C -6 -4 0 12 -6 D -12 -6 -12 0 -14 E -6 10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=21 D=20 A=16 C=13 so C is eliminated. Round 2 votes counts: E=30 A=26 D=23 B=21 so B is eliminated. Round 3 votes counts: E=37 D=37 A=26 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:216 E:212 C:198 B:196 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 12 6 B -8 0 4 6 -10 C -6 -4 0 12 -6 D -12 -6 -12 0 -14 E -6 10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 12 6 B -8 0 4 6 -10 C -6 -4 0 12 -6 D -12 -6 -12 0 -14 E -6 10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 12 6 B -8 0 4 6 -10 C -6 -4 0 12 -6 D -12 -6 -12 0 -14 E -6 10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2009: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A C E (8) B D A E C (8) A D C B E (8) B D A C E (7) C E A D B (5) E C B A D (4) E A B C D (4) C A D E B (4) E A C B D (3) D A B C E (3) B E D C A (3) B D E C A (3) A E C D B (3) A C E D B (3) A C D E B (3) E C A D B (2) D A C B E (2) C D A B E (2) C A E D B (2) A D B C E (2) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C D B (1) D B C A E (1) C D A E B (1) C A D B E (1) B D E A C (1) B D C A E (1) B A D E C (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 18 22 16 24 B -18 0 -6 -6 2 C -22 6 0 -2 6 D -16 6 2 0 14 E -24 -2 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 22 16 24 B -18 0 -6 -6 2 C -22 6 0 -2 6 D -16 6 2 0 14 E -24 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 A=21 C=15 D=14 so D is eliminated. Round 2 votes counts: B=33 E=26 A=26 C=15 so C is eliminated. Round 3 votes counts: A=36 B=33 E=31 so E is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:240 D:203 C:194 B:186 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 22 16 24 B -18 0 -6 -6 2 C -22 6 0 -2 6 D -16 6 2 0 14 E -24 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 22 16 24 B -18 0 -6 -6 2 C -22 6 0 -2 6 D -16 6 2 0 14 E -24 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 22 16 24 B -18 0 -6 -6 2 C -22 6 0 -2 6 D -16 6 2 0 14 E -24 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2010: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (9) C E A D B (9) B D E C A (8) A C E D B (8) B D A E C (6) C A E D B (5) E C A B D (4) B D E A C (4) E C A D B (3) D B C E A (3) B A D E C (3) A E C B D (3) E C D B A (2) E C B D A (2) D E B C A (2) A C D E B (2) A B D E C (2) E D C B A (1) E C D A B (1) D C E B A (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D A B (1) B E D C A (1) B D A C E (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -6 10 -2 B -10 0 -14 -2 -10 C 6 14 0 10 -4 D -10 2 -10 0 -6 E 2 10 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -6 10 -2 B -10 0 -14 -2 -10 C 6 14 0 10 -4 D -10 2 -10 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=23 D=18 C=15 E=13 so E is eliminated. Round 2 votes counts: A=31 C=27 B=23 D=19 so D is eliminated. Round 3 votes counts: B=39 A=32 C=29 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 E:211 A:206 D:188 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -6 10 -2 B -10 0 -14 -2 -10 C 6 14 0 10 -4 D -10 2 -10 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 10 -2 B -10 0 -14 -2 -10 C 6 14 0 10 -4 D -10 2 -10 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 10 -2 B -10 0 -14 -2 -10 C 6 14 0 10 -4 D -10 2 -10 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2011: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (19) C A D E B (8) B E D A C (8) D A B E C (6) C A D B E (6) A D B E C (6) C E B A D (5) E B D C A (4) E B C D A (4) D A E B C (4) C E B D A (4) A D B C E (4) B D E A C (3) D B E A C (2) C E A B D (2) C B A E D (2) A D C E B (2) A D C B E (2) A C D B E (2) E C B D A (1) D E B A C (1) D B A E C (1) C E A D B (1) B E C D A (1) B E A D C (1) B A D C E (1) Total count = 100 A B C D E A 0 -14 24 -16 -12 B 14 0 30 10 -10 C -24 -30 0 -28 -22 D 16 -10 28 0 -4 E 12 10 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 24 -16 -12 B 14 0 30 10 -10 C -24 -30 0 -28 -22 D 16 -10 28 0 -4 E 12 10 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 A=16 D=14 B=14 so D is eliminated. Round 2 votes counts: E=29 C=28 A=26 B=17 so B is eliminated. Round 3 votes counts: E=44 C=28 A=28 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:222 D:215 A:191 C:148 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 24 -16 -12 B 14 0 30 10 -10 C -24 -30 0 -28 -22 D 16 -10 28 0 -4 E 12 10 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 24 -16 -12 B 14 0 30 10 -10 C -24 -30 0 -28 -22 D 16 -10 28 0 -4 E 12 10 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 24 -16 -12 B 14 0 30 10 -10 C -24 -30 0 -28 -22 D 16 -10 28 0 -4 E 12 10 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2012: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (13) C D A B E (12) C D A E B (11) E C D A B (9) E B A D C (9) B A D C E (8) E C B D A (5) B A D E C (4) A D C B E (4) E B D A C (3) D A C B E (3) C E D A B (3) A D B C E (3) E B C D A (2) E B C A D (2) C A D B E (2) E D C A B (1) D A C E B (1) D A B C E (1) C B A D E (1) C A D E B (1) B E D A C (1) B C A D E (1) Total count = 100 A B C D E A 0 2 0 -4 4 B -2 0 -6 -2 6 C 0 6 0 -2 2 D 4 2 2 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -4 4 B -2 0 -6 -2 6 C 0 6 0 -2 2 D 4 2 2 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 B=27 A=7 D=5 so D is eliminated. Round 2 votes counts: E=31 C=30 B=27 A=12 so A is eliminated. Round 3 votes counts: C=38 E=31 B=31 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:206 C:203 A:201 B:198 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -4 4 B -2 0 -6 -2 6 C 0 6 0 -2 2 D 4 2 2 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 4 B -2 0 -6 -2 6 C 0 6 0 -2 2 D 4 2 2 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 4 B -2 0 -6 -2 6 C 0 6 0 -2 2 D 4 2 2 0 4 E -4 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2013: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) D C A E B (8) C D A B E (8) E A B D C (4) C D A E B (4) E B A D C (3) E A B C D (3) D E A C B (3) C A D E B (3) C A B E D (3) A C D E B (3) D E B A C (2) D E A B C (2) D B E A C (2) D A C E B (2) C D B A E (2) C B A D E (2) C A E D B (2) B E D C A (2) B E C A D (2) B C E A D (2) B C D E A (2) B C A E D (2) A E C B D (2) E D B A C (1) E D A C B (1) E B A C D (1) E A D B C (1) D C B E A (1) D C B A E (1) D A E C B (1) C B A E D (1) C A E B D (1) C A B D E (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E A C (1) B D C E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 12 0 6 2 B -12 0 -2 2 0 C 0 2 0 22 0 D -6 -2 -22 0 0 E -2 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.614631 B: 0.000000 C: 0.385369 D: 0.000000 E: 0.000000 Sum of squares = 0.526280345453 Cumulative probabilities = A: 0.614631 B: 0.614631 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 6 2 B -12 0 -2 2 0 C 0 2 0 22 0 D -6 -2 -22 0 0 E -2 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999278 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=27 D=22 E=14 A=9 so A is eliminated. Round 2 votes counts: C=30 B=29 D=22 E=19 so E is eliminated. Round 3 votes counts: B=41 C=33 D=26 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:210 E:199 B:194 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 6 2 B -12 0 -2 2 0 C 0 2 0 22 0 D -6 -2 -22 0 0 E -2 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999278 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 6 2 B -12 0 -2 2 0 C 0 2 0 22 0 D -6 -2 -22 0 0 E -2 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999278 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 6 2 B -12 0 -2 2 0 C 0 2 0 22 0 D -6 -2 -22 0 0 E -2 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999278 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2014: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) B D A C E (9) E B D A C (7) C E A D B (7) B D E A C (6) E C A D B (5) C A D B E (5) B D A E C (4) E C B D A (3) E C A B D (3) D B C A E (3) C A D E B (3) B E D A C (3) A C E D B (3) E C B A D (2) E B D C A (2) E B A D C (2) E A B D C (2) C A E D B (2) A C D B E (2) E B C A D (1) E B A C D (1) E A C D B (1) E A C B D (1) E A B C D (1) D C A B E (1) D A B C E (1) C D B E A (1) C D B A E (1) C D A B E (1) B E D C A (1) B D E C A (1) B D C A E (1) A E B D C (1) A D C B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 14 -10 0 B 16 0 16 6 6 C -14 -16 0 -14 4 D 10 -6 14 0 4 E 0 -6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 14 -10 0 B 16 0 16 6 6 C -14 -16 0 -14 4 D 10 -6 14 0 4 E 0 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=25 C=20 D=15 A=9 so A is eliminated. Round 2 votes counts: E=32 B=27 C=25 D=16 so D is eliminated. Round 3 votes counts: B=41 E=32 C=27 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:211 A:194 E:193 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 14 -10 0 B 16 0 16 6 6 C -14 -16 0 -14 4 D 10 -6 14 0 4 E 0 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 14 -10 0 B 16 0 16 6 6 C -14 -16 0 -14 4 D 10 -6 14 0 4 E 0 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 14 -10 0 B 16 0 16 6 6 C -14 -16 0 -14 4 D 10 -6 14 0 4 E 0 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2015: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (14) C D A E B (10) E B A C D (9) B E A D C (7) C D E A B (5) E C B A D (4) E B A D C (4) D A C B E (4) B A E D C (4) B A D E C (4) E C D B A (3) D A B C E (3) C D A B E (3) E C B D A (2) E B D C A (2) E B C D A (2) E B C A D (2) C E D A B (2) A C D B E (2) A B D E C (2) E A C B D (1) E A B C D (1) D B E A C (1) D B A C E (1) C E D B A (1) C E A D B (1) C D E B A (1) C A E D B (1) A E C B D (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -6 -8 4 B -6 0 -12 -8 -4 C 6 12 0 2 0 D 8 8 -2 0 4 E -4 4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.758892 D: 0.000000 E: 0.241108 Sum of squares = 0.634050285226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.758892 D: 0.758892 E: 1.000000 A B C D E A 0 6 -6 -8 4 B -6 0 -12 -8 -4 C 6 12 0 2 0 D 8 8 -2 0 4 E -4 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555556227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=24 D=23 B=15 A=8 so A is eliminated. Round 2 votes counts: E=31 C=26 D=25 B=18 so B is eliminated. Round 3 votes counts: E=43 D=31 C=26 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 D:209 A:198 E:198 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -8 4 B -6 0 -12 -8 -4 C 6 12 0 2 0 D 8 8 -2 0 4 E -4 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555556227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -8 4 B -6 0 -12 -8 -4 C 6 12 0 2 0 D 8 8 -2 0 4 E -4 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555556227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -8 4 B -6 0 -12 -8 -4 C 6 12 0 2 0 D 8 8 -2 0 4 E -4 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555556227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2016: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) C E A D B (7) C A E B D (7) E C D A B (6) C E D A B (6) B A D C E (6) D B E C A (5) A B C E D (5) E D C A B (4) B A C D E (4) A C E B D (4) A B D E C (4) D E B C A (2) D B E A C (2) C B D E A (2) C B A E D (2) C A B E D (2) B D A E C (2) A C B E D (2) A B E D C (2) E D C B A (1) E C D B A (1) D E C B A (1) D E B A C (1) D E A B C (1) D C E B A (1) D B C E A (1) C E D B A (1) C A E D B (1) B D E A C (1) B D C A E (1) B D A C E (1) B C D E A (1) B A C E D (1) A E C D B (1) A D B E C (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -4 18 12 B -10 0 0 12 8 C 4 0 0 8 14 D -18 -12 -8 0 -8 E -12 -8 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100610 C: 0.899390 D: 0.000000 E: 0.000000 Sum of squares = 0.819024754782 Cumulative probabilities = A: 0.000000 B: 0.100610 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 18 12 B -10 0 0 12 8 C 4 0 0 8 14 D -18 -12 -8 0 -8 E -12 -8 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836781023 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 A=21 D=14 E=12 so E is eliminated. Round 2 votes counts: C=35 B=25 A=21 D=19 so D is eliminated. Round 3 votes counts: C=42 B=36 A=22 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:218 C:213 B:205 E:187 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 18 12 B -10 0 0 12 8 C 4 0 0 8 14 D -18 -12 -8 0 -8 E -12 -8 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836781023 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 18 12 B -10 0 0 12 8 C 4 0 0 8 14 D -18 -12 -8 0 -8 E -12 -8 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836781023 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 18 12 B -10 0 0 12 8 C 4 0 0 8 14 D -18 -12 -8 0 -8 E -12 -8 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836781023 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2017: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) A D E B C (9) C B A E D (7) B C E D A (6) B C A E D (5) C B E D A (4) B A C D E (4) A B D C E (4) E D C B A (3) E D B A C (3) D A E C B (3) D A E B C (3) C E D A B (3) C E B D A (3) A D E C B (3) A D B E C (3) E D B C A (2) D E A B C (2) C B E A D (2) C A B D E (2) B E D A C (2) A D C E B (2) A C D E B (2) A C B D E (2) A B D E C (2) E D C A B (1) E D A C B (1) E C B D A (1) D B E A C (1) C E D B A (1) C E B A D (1) B E D C A (1) B A D E C (1) B A C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 16 2 8 B -4 0 0 -2 -4 C -16 0 0 -10 0 D -2 2 10 0 6 E -8 4 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 2 8 B -4 0 0 -2 -4 C -16 0 0 -10 0 D -2 2 10 0 6 E -8 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987678 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=23 B=20 D=18 E=11 so E is eliminated. Round 2 votes counts: D=28 A=28 C=24 B=20 so B is eliminated. Round 3 votes counts: C=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:208 B:195 E:195 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 2 8 B -4 0 0 -2 -4 C -16 0 0 -10 0 D -2 2 10 0 6 E -8 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987678 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 2 8 B -4 0 0 -2 -4 C -16 0 0 -10 0 D -2 2 10 0 6 E -8 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987678 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 2 8 B -4 0 0 -2 -4 C -16 0 0 -10 0 D -2 2 10 0 6 E -8 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987678 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2018: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) E C A D B (6) D B E A C (5) C A B E D (5) A C B D E (5) E C D B A (4) D B A E C (4) C A E B D (4) A C E B D (4) A C B E D (4) A B C D E (4) E D C B A (3) E C D A B (3) B A D C E (3) E D B C A (2) D E B C A (2) D E B A C (2) C E B A D (2) C E A B D (2) C B A E D (2) C B A D E (2) B D E C A (2) B D C A E (2) B D A E C (2) B A C D E (2) E D C A B (1) E D A C B (1) E A D C B (1) E A C D B (1) D E A B C (1) D B E C A (1) D A B E C (1) C E B D A (1) C E A D B (1) B C D E A (1) B C D A E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 6 2 16 B 6 0 -6 18 18 C -6 6 0 10 16 D -2 -18 -10 0 6 E -16 -18 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 2 16 B 6 0 -6 18 18 C -6 6 0 10 16 D -2 -18 -10 0 6 E -16 -18 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=22 A=20 C=19 D=16 so D is eliminated. Round 2 votes counts: B=33 E=27 A=21 C=19 so C is eliminated. Round 3 votes counts: B=37 E=33 A=30 so A is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:213 A:209 D:188 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -6 6 2 16 B 6 0 -6 18 18 C -6 6 0 10 16 D -2 -18 -10 0 6 E -16 -18 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 2 16 B 6 0 -6 18 18 C -6 6 0 10 16 D -2 -18 -10 0 6 E -16 -18 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 2 16 B 6 0 -6 18 18 C -6 6 0 10 16 D -2 -18 -10 0 6 E -16 -18 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2019: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (24) D E A B C (20) B C A E D (6) E A B D C (5) C D B A E (5) B A E C D (5) E D A B C (4) D C E A B (4) D C E B A (3) D C B A E (3) C D B E A (2) C B A D E (2) B A E D C (2) B A C E D (2) A E B C D (2) A B E D C (2) E A D B C (1) D E C A B (1) D B A E C (1) D A E B C (1) C D E B A (1) C B D A E (1) A E D B C (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -4 8 18 B 14 0 8 6 12 C 4 -8 0 2 6 D -8 -6 -2 0 -12 E -18 -12 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 8 18 B 14 0 8 6 12 C 4 -8 0 2 6 D -8 -6 -2 0 -12 E -18 -12 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=33 B=15 E=10 A=7 so A is eliminated. Round 2 votes counts: C=35 D=33 B=18 E=14 so E is eliminated. Round 3 votes counts: D=39 C=35 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:220 A:204 C:202 E:188 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 8 18 B 14 0 8 6 12 C 4 -8 0 2 6 D -8 -6 -2 0 -12 E -18 -12 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 8 18 B 14 0 8 6 12 C 4 -8 0 2 6 D -8 -6 -2 0 -12 E -18 -12 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 8 18 B 14 0 8 6 12 C 4 -8 0 2 6 D -8 -6 -2 0 -12 E -18 -12 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2020: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (16) D A E B C (15) B E A C D (8) C D B A E (6) E A B C D (5) D C A E B (5) D A E C B (5) A E D B C (5) C D B E A (4) D C B A E (3) C B E D A (3) E A D B C (2) E A B D C (2) D A B E C (2) C E A B D (2) B C E A D (2) B C A E D (2) B A E D C (2) A E B D C (2) E A D C B (1) D E A C B (1) D B A E C (1) C E B A D (1) C B D E A (1) B E C A D (1) B D C A E (1) B A D E C (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 6 6 2 B 4 0 4 -2 6 C -6 -4 0 2 -8 D -6 2 -2 0 -8 E -2 -6 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 6 2 B 4 0 4 -2 6 C -6 -4 0 2 -8 D -6 2 -2 0 -8 E -2 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=32 B=17 E=10 A=8 so A is eliminated. Round 2 votes counts: D=33 C=33 E=17 B=17 so E is eliminated. Round 3 votes counts: D=41 C=33 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:206 A:205 E:204 D:193 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 6 2 B 4 0 4 -2 6 C -6 -4 0 2 -8 D -6 2 -2 0 -8 E -2 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 6 2 B 4 0 4 -2 6 C -6 -4 0 2 -8 D -6 2 -2 0 -8 E -2 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 6 2 B 4 0 4 -2 6 C -6 -4 0 2 -8 D -6 2 -2 0 -8 E -2 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2021: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) B D E A C (9) E A D C B (5) B D C A E (5) E B A D C (4) C B D A E (4) C A E D B (4) C A D E B (4) E A D B C (3) C E A D B (3) B D A E C (3) B C D E A (3) B C D A E (3) E B A C D (2) E A C D B (2) C E B A D (2) C D B A E (2) C B A D E (2) B E C D A (2) B E C A D (2) A D E C B (2) E C A D B (1) E C A B D (1) E B D A C (1) E B C A D (1) E A B D C (1) D C A B E (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C E B (1) C D A B E (1) C B E A D (1) C A E B D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E A C D (1) B D E C A (1) B D A C E (1) A E D C B (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -24 6 -4 -14 B 24 0 12 28 14 C -6 -12 0 -6 -14 D 4 -28 6 0 -6 E 14 -14 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 6 -4 -14 B 24 0 12 28 14 C -6 -12 0 -6 -14 D 4 -28 6 0 -6 E 14 -14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 C=27 E=21 D=5 A=5 so D is eliminated. Round 2 votes counts: B=43 C=28 E=21 A=8 so A is eliminated. Round 3 votes counts: B=43 C=30 E=27 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:239 E:210 D:188 A:182 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 6 -4 -14 B 24 0 12 28 14 C -6 -12 0 -6 -14 D 4 -28 6 0 -6 E 14 -14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 6 -4 -14 B 24 0 12 28 14 C -6 -12 0 -6 -14 D 4 -28 6 0 -6 E 14 -14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 6 -4 -14 B 24 0 12 28 14 C -6 -12 0 -6 -14 D 4 -28 6 0 -6 E 14 -14 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2022: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) E A C D B (6) D C E A B (6) D C B E A (6) B D C A E (6) A B E C D (6) D B C E A (5) B A C E D (5) A E C B D (5) E C D A B (4) B A E D C (4) B A D C E (4) A B E D C (4) E D A C B (3) E A D C B (3) D E C A B (3) C D E A B (3) E D C A B (2) E C A D B (2) B D A C E (2) B C D A E (2) B A D E C (2) A E B D C (2) D C E B A (1) C E D A B (1) C D E B A (1) C B D E A (1) C A E B D (1) B D A E C (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 18 12 6 6 B -18 0 4 6 -2 C -12 -4 0 -10 -10 D -6 -6 10 0 -12 E -6 2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 12 6 6 B -18 0 4 6 -2 C -12 -4 0 -10 -10 D -6 -6 10 0 -12 E -6 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=25 D=21 E=20 C=7 so C is eliminated. Round 2 votes counts: B=28 A=26 D=25 E=21 so E is eliminated. Round 3 votes counts: A=37 D=35 B=28 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:209 B:195 D:193 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 12 6 6 B -18 0 4 6 -2 C -12 -4 0 -10 -10 D -6 -6 10 0 -12 E -6 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 6 6 B -18 0 4 6 -2 C -12 -4 0 -10 -10 D -6 -6 10 0 -12 E -6 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 6 6 B -18 0 4 6 -2 C -12 -4 0 -10 -10 D -6 -6 10 0 -12 E -6 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2023: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) C B E D A (6) D B E A C (5) D A B E C (5) C A E B D (5) B E D C A (5) A D C E B (4) E C A B D (3) E B C A D (3) C B D E A (3) C B D A E (3) B E D A C (3) B D E C A (3) A C D E B (3) E B D A C (2) E B C D A (2) D B A E C (2) D A C B E (2) D A B C E (2) C E A B D (2) C B E A D (2) B E C D A (2) A E D B C (2) A E C D B (2) A D C B E (2) E D B A C (1) E B D C A (1) E B A C D (1) E A B D C (1) E A B C D (1) D C A B E (1) D B E C A (1) D B A C E (1) C E B A D (1) C D B A E (1) C D A B E (1) C A E D B (1) C A D E B (1) B D E A C (1) B D C E A (1) A D E C B (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -14 -8 0 B 0 0 -8 4 24 C 14 8 0 6 4 D 8 -4 -6 0 6 E 0 -24 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -8 0 B 0 0 -8 4 24 C 14 8 0 6 4 D 8 -4 -6 0 6 E 0 -24 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=19 A=16 E=15 B=15 so E is eliminated. Round 2 votes counts: C=38 B=24 D=20 A=18 so A is eliminated. Round 3 votes counts: C=44 D=30 B=26 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:210 D:202 A:189 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 -8 0 B 0 0 -8 4 24 C 14 8 0 6 4 D 8 -4 -6 0 6 E 0 -24 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -8 0 B 0 0 -8 4 24 C 14 8 0 6 4 D 8 -4 -6 0 6 E 0 -24 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -8 0 B 0 0 -8 4 24 C 14 8 0 6 4 D 8 -4 -6 0 6 E 0 -24 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2024: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) A C E D B (8) D C A E B (7) E B A C D (6) C A D E B (6) D C A B E (5) B E D A C (5) A C D E B (5) B E D C A (4) E B D C A (3) D B E C A (3) A C E B D (3) A C B E D (3) E D C A B (2) E A C D B (2) D C E A B (2) D C B A E (2) D B C A E (2) C D A B E (2) B D E C A (2) B D E A C (2) B D C A E (2) B A C D E (2) A C B D E (2) E C A D B (1) E B D A C (1) E B A D C (1) E A B C D (1) D E C A B (1) D E B C A (1) C D A E B (1) C A E D B (1) C A D B E (1) B D A C E (1) A C D B E (1) Total count = 100 A B C D E A 0 26 4 4 8 B -26 0 -28 -6 -22 C -4 28 0 8 12 D -4 6 -8 0 0 E -8 22 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 4 4 8 B -26 0 -28 -6 -22 C -4 28 0 8 12 D -4 6 -8 0 0 E -8 22 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999669 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 A=22 B=18 C=11 so C is eliminated. Round 2 votes counts: A=30 E=26 D=26 B=18 so B is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:222 A:221 E:201 D:197 B:159 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 4 4 8 B -26 0 -28 -6 -22 C -4 28 0 8 12 D -4 6 -8 0 0 E -8 22 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999669 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 4 4 8 B -26 0 -28 -6 -22 C -4 28 0 8 12 D -4 6 -8 0 0 E -8 22 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999669 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 4 4 8 B -26 0 -28 -6 -22 C -4 28 0 8 12 D -4 6 -8 0 0 E -8 22 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999669 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2025: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (6) D C E A B (5) A D E C B (5) E C D A B (4) D A E C B (4) B A D C E (4) E B A C D (3) D C B A E (3) D A C B E (3) C B D E A (3) B C E A D (3) B C D A E (3) B A E D C (3) A E D B C (3) A D B E C (3) E B C A D (2) E A D C B (2) E A B C D (2) D A C E B (2) C E B D A (2) C D B E A (2) B E A C D (2) B D A C E (2) B C E D A (2) B A E C D (2) A E D C B (2) A D E B C (2) A D B C E (2) A B E D C (2) A B D E C (2) A B D C E (2) E D C A B (1) E C B D A (1) E C B A D (1) E C A B D (1) D E C A B (1) D C A E B (1) D B A C E (1) C E D A B (1) C D E A B (1) C B E D A (1) B D C A E (1) B C A E D (1) A E B D C (1) Total count = 100 A B C D E A 0 4 8 0 8 B -4 0 -4 -8 -6 C -8 4 0 -14 2 D 0 8 14 0 4 E -8 6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.705753 B: 0.000000 C: 0.000000 D: 0.294247 E: 0.000000 Sum of squares = 0.584668362972 Cumulative probabilities = A: 0.705753 B: 0.705753 C: 0.705753 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 0 8 B -4 0 -4 -8 -6 C -8 4 0 -14 2 D 0 8 14 0 4 E -8 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 D=20 E=17 C=16 so C is eliminated. Round 2 votes counts: B=27 E=26 A=24 D=23 so D is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:210 E:196 C:192 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 0 8 B -4 0 -4 -8 -6 C -8 4 0 -14 2 D 0 8 14 0 4 E -8 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 0 8 B -4 0 -4 -8 -6 C -8 4 0 -14 2 D 0 8 14 0 4 E -8 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 0 8 B -4 0 -4 -8 -6 C -8 4 0 -14 2 D 0 8 14 0 4 E -8 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2026: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) E D B C A (9) D C E B A (6) B E A D C (6) C A D B E (5) B A E D C (5) A B C E D (5) C D E A B (4) E B D C A (3) D C A E B (3) C D E B A (3) B E D A C (3) A C B E D (3) A C B D E (3) A B E C D (3) E B D A C (2) D E B C A (2) D E B A C (2) D C E A B (2) C D A E B (2) C B A E D (2) C A D E B (2) C A B E D (2) C A B D E (2) B A E C D (2) E D B A C (1) D A C E B (1) B E C D A (1) B E C A D (1) B E A C D (1) A C D B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -20 -10 -14 B 20 0 -4 -8 -6 C 20 4 0 -16 -8 D 10 8 16 0 0 E 14 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.361785 E: 0.638215 Sum of squares = 0.538207002447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.361785 E: 1.000000 A B C D E A 0 -20 -20 -10 -14 B 20 0 -4 -8 -6 C 20 4 0 -16 -8 D 10 8 16 0 0 E 14 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=22 B=19 A=17 E=15 so E is eliminated. Round 2 votes counts: D=37 B=24 C=22 A=17 so A is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:214 B:201 C:200 A:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -20 -10 -14 B 20 0 -4 -8 -6 C 20 4 0 -16 -8 D 10 8 16 0 0 E 14 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -20 -10 -14 B 20 0 -4 -8 -6 C 20 4 0 -16 -8 D 10 8 16 0 0 E 14 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -20 -10 -14 B 20 0 -4 -8 -6 C 20 4 0 -16 -8 D 10 8 16 0 0 E 14 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2027: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) C B D A E (11) A E C B D (9) D B C A E (7) E D B C A (6) E D B A C (5) E D A B C (5) E A C B D (4) A C B E D (4) E A D B C (3) E A C D B (3) E A B D C (3) D C B E A (3) D B E C A (3) A C B D E (3) D E B C A (2) C A B D E (2) A E B C D (2) A C E B D (2) E C D A B (1) C E D B A (1) C B A D E (1) B D C A E (1) B D A C E (1) B C D A E (1) B A D C E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -4 -20 -4 B 12 0 12 -4 8 C 4 -12 0 -8 6 D 20 4 8 0 0 E 4 -8 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.824486 E: 0.175514 Sum of squares = 0.71058259052 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.824486 E: 1.000000 A B C D E A 0 -12 -4 -20 -4 B 12 0 12 -4 8 C 4 -12 0 -8 6 D 20 4 8 0 0 E 4 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555856854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=28 A=23 C=15 B=4 so B is eliminated. Round 2 votes counts: E=30 D=30 A=24 C=16 so C is eliminated. Round 3 votes counts: D=42 E=31 A=27 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:214 C:195 E:195 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -4 -20 -4 B 12 0 12 -4 8 C 4 -12 0 -8 6 D 20 4 8 0 0 E 4 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555856854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -20 -4 B 12 0 12 -4 8 C 4 -12 0 -8 6 D 20 4 8 0 0 E 4 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555856854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -20 -4 B 12 0 12 -4 8 C 4 -12 0 -8 6 D 20 4 8 0 0 E 4 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555856854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2028: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (8) A D E C B (7) D E A C B (6) B C E D A (6) B A C D E (6) D E B C A (5) D A E B C (5) E D A C B (4) D E A B C (4) C B A E D (4) E D C A B (2) E C D B A (2) D B E A C (2) C B E D A (2) C B E A D (2) B E D C A (2) B D E C A (2) B D A E C (2) A D E B C (2) A D B C E (2) A C E D B (2) A B C D E (2) E D C B A (1) E D B C A (1) E C D A B (1) E C B D A (1) E B D C A (1) E A C D B (1) D E B A C (1) D B A E C (1) C E B D A (1) C E B A D (1) C A E D B (1) C A B E D (1) B C E A D (1) B C A D E (1) B A D C E (1) A D C B E (1) A D B E C (1) A C E B D (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 8 -4 2 B 8 0 14 -4 0 C -8 -14 0 -8 -10 D 4 4 8 0 6 E -2 0 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -4 2 B 8 0 14 -4 0 C -8 -14 0 -8 -10 D 4 4 8 0 6 E -2 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=24 A=21 E=14 C=12 so C is eliminated. Round 2 votes counts: B=37 D=24 A=23 E=16 so E is eliminated. Round 3 votes counts: B=41 D=35 A=24 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:209 E:201 A:199 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -4 2 B 8 0 14 -4 0 C -8 -14 0 -8 -10 D 4 4 8 0 6 E -2 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -4 2 B 8 0 14 -4 0 C -8 -14 0 -8 -10 D 4 4 8 0 6 E -2 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -4 2 B 8 0 14 -4 0 C -8 -14 0 -8 -10 D 4 4 8 0 6 E -2 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2029: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) D C A B E (6) D A C B E (6) A E D B C (6) E B C A D (5) E A D B C (5) B C E A D (5) E A B D C (4) E A B C D (4) D C B A E (4) A D E C B (4) A D C B E (4) E B A C D (3) D A E C B (3) D C B E A (2) D A E B C (2) D A C E B (2) C B D A E (2) C B A D E (2) B E C D A (2) B E C A D (2) A E B C D (2) A C D B E (2) A C B E D (2) E D B C A (1) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) E B C D A (1) C D B A E (1) C B E D A (1) C B D E A (1) C B A E D (1) C A B D E (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 12 8 8 4 B -12 0 8 -2 6 C -8 -8 0 -8 2 D -8 2 8 0 -14 E -4 -6 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 8 4 B -12 0 8 -2 6 C -8 -8 0 -8 2 D -8 2 8 0 -14 E -4 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 A=22 B=17 C=9 so C is eliminated. Round 2 votes counts: E=27 D=26 B=24 A=23 so A is eliminated. Round 3 votes counts: E=36 D=36 B=28 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:216 E:201 B:200 D:194 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 8 4 B -12 0 8 -2 6 C -8 -8 0 -8 2 D -8 2 8 0 -14 E -4 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 8 4 B -12 0 8 -2 6 C -8 -8 0 -8 2 D -8 2 8 0 -14 E -4 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 8 4 B -12 0 8 -2 6 C -8 -8 0 -8 2 D -8 2 8 0 -14 E -4 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2030: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) B A D E C (7) D A B E C (6) C E D A B (5) D E C A B (4) B A E C D (4) A D B C E (4) A B C D E (4) E D C B A (3) E C D B A (3) C E B A D (3) C B A E D (3) B C A E D (3) A C B D E (3) A B D E C (3) A B D C E (3) E D C A B (2) E C D A B (2) E C B D A (2) D C A E B (2) D A E B C (2) C E D B A (2) C B E A D (2) B A C D E (2) E D B A C (1) D E B A C (1) D E A C B (1) D E A B C (1) D C E A B (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A D B (1) C D A B E (1) C A D E B (1) C A B E D (1) C A B D E (1) B E C A D (1) B C E A D (1) B A E D C (1) B A D C E (1) A D B E C (1) Total count = 100 A B C D E A 0 4 12 16 28 B -4 0 10 6 22 C -12 -10 0 6 8 D -16 -6 -6 0 4 E -28 -22 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 16 28 B -4 0 10 6 22 C -12 -10 0 6 8 D -16 -6 -6 0 4 E -28 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995512 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=21 C=20 A=18 E=13 so E is eliminated. Round 2 votes counts: B=28 D=27 C=27 A=18 so A is eliminated. Round 3 votes counts: B=38 D=32 C=30 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:230 B:217 C:196 D:188 E:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 16 28 B -4 0 10 6 22 C -12 -10 0 6 8 D -16 -6 -6 0 4 E -28 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995512 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 16 28 B -4 0 10 6 22 C -12 -10 0 6 8 D -16 -6 -6 0 4 E -28 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995512 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 16 28 B -4 0 10 6 22 C -12 -10 0 6 8 D -16 -6 -6 0 4 E -28 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995512 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2031: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) B E C A D (7) B E A C D (7) D A C E B (5) C D A E B (5) D C A E B (4) D A E B C (4) C D E A B (4) A E D B C (4) E B A C D (3) C D B E A (3) B E A D C (3) B C E A D (3) B C D A E (3) A D E B C (3) E C B A D (2) E B A D C (2) E A D C B (2) E A B D C (2) D A B E C (2) C E D A B (2) C D B A E (2) C B E D A (2) C B D A E (2) A E B D C (2) E A D B C (1) E A B C D (1) D C B A E (1) D C A B E (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C D E B A (1) C B D E A (1) B C E D A (1) B A E D C (1) A E D C B (1) Total count = 100 A B C D E A 0 4 6 -10 0 B -4 0 2 -12 -16 C -6 -2 0 4 -8 D 10 12 -4 0 2 E 0 16 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 A B C D E A 0 4 6 -10 0 B -4 0 2 -12 -16 C -6 -2 0 4 -8 D 10 12 -4 0 2 E 0 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 B=25 E=13 A=10 so A is eliminated. Round 2 votes counts: D=30 C=25 B=25 E=20 so E is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:210 A:200 C:194 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 -10 0 B -4 0 2 -12 -16 C -6 -2 0 4 -8 D 10 12 -4 0 2 E 0 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -10 0 B -4 0 2 -12 -16 C -6 -2 0 4 -8 D 10 12 -4 0 2 E 0 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -10 0 B -4 0 2 -12 -16 C -6 -2 0 4 -8 D 10 12 -4 0 2 E 0 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2032: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) A C D E B (10) B E D C A (9) D A C E B (7) D A C B E (7) B E C A D (6) B E C D A (5) D B E A C (4) C A E B D (4) A D C E B (4) B D E C A (3) A C E D B (3) E C B A D (2) D A B C E (2) C E B A D (2) C D A B E (2) C A D B E (2) B E D A C (2) A C E B D (2) E B A D C (1) E A C B D (1) D C A B E (1) D B C E A (1) D B C A E (1) D B A E C (1) D A E B C (1) D A B E C (1) C B E A D (1) C B A E D (1) C A D E B (1) C A B E D (1) C A B D E (1) B D E A C (1) Total count = 100 A B C D E A 0 0 -6 4 4 B 0 0 -4 4 4 C 6 4 0 8 6 D -4 -4 -8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 4 4 B 0 0 -4 4 4 C 6 4 0 8 6 D -4 -4 -8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 A=19 C=15 E=14 so E is eliminated. Round 2 votes counts: B=37 D=26 A=20 C=17 so C is eliminated. Round 3 votes counts: B=43 A=29 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 B:202 A:201 E:193 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 4 4 B 0 0 -4 4 4 C 6 4 0 8 6 D -4 -4 -8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 4 4 B 0 0 -4 4 4 C 6 4 0 8 6 D -4 -4 -8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 4 4 B 0 0 -4 4 4 C 6 4 0 8 6 D -4 -4 -8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2033: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) D A C B E (9) A D B C E (7) C D A E B (6) B E A D C (6) C D A B E (5) B E C A D (5) E C B D A (4) C E D B A (4) C D E A B (4) B E A C D (4) B A E D C (4) E B A C D (3) A D C B E (3) E B C D A (2) E B A D C (2) D C A B E (2) A B D C E (2) E D C B A (1) E D C A B (1) E B D C A (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D A B (1) C E B D A (1) C D B E A (1) C B E D A (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -2 6 -4 B 0 0 8 0 6 C 2 -8 0 6 -2 D -6 0 -6 0 -6 E 4 -6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.406697 B: 0.593303 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.517410995289 Cumulative probabilities = A: 0.406697 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 6 -4 B 0 0 8 0 6 C 2 -8 0 6 -2 D -6 0 -6 0 -6 E 4 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 B=19 A=18 D=15 so D is eliminated. Round 2 votes counts: A=30 C=26 E=25 B=19 so B is eliminated. Round 3 votes counts: E=40 A=34 C=26 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:207 E:203 A:200 C:199 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 6 -4 B 0 0 8 0 6 C 2 -8 0 6 -2 D -6 0 -6 0 -6 E 4 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 6 -4 B 0 0 8 0 6 C 2 -8 0 6 -2 D -6 0 -6 0 -6 E 4 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 6 -4 B 0 0 8 0 6 C 2 -8 0 6 -2 D -6 0 -6 0 -6 E 4 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2034: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) B A D E C (8) A B D C E (8) E D B C A (7) E D C B A (6) C A E D B (6) A C B D E (6) C A D E B (4) A C D E B (4) A C D B E (4) A B C D E (4) A B C E D (3) E C D B A (2) D E B C A (2) D C E A B (2) C D A E B (2) B D A E C (2) B A E C D (2) A D C B E (2) A C B E D (2) E C D A B (1) E C B A D (1) E B D C A (1) D E B A C (1) D C A E B (1) D B A E C (1) C E A D B (1) C E A B D (1) C A E B D (1) B E D A C (1) B E C A D (1) B E A D C (1) B A E D C (1) B A C E D (1) Total count = 100 A B C D E A 0 24 2 22 24 B -24 0 -12 -12 -6 C -2 12 0 12 24 D -22 12 -12 0 2 E -24 6 -24 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 2 22 24 B -24 0 -12 -12 -6 C -2 12 0 12 24 D -22 12 -12 0 2 E -24 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973597 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=25 E=18 B=17 D=7 so D is eliminated. Round 2 votes counts: A=33 C=28 E=21 B=18 so B is eliminated. Round 3 votes counts: A=48 C=28 E=24 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:236 C:223 D:190 E:178 B:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 2 22 24 B -24 0 -12 -12 -6 C -2 12 0 12 24 D -22 12 -12 0 2 E -24 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973597 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 2 22 24 B -24 0 -12 -12 -6 C -2 12 0 12 24 D -22 12 -12 0 2 E -24 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973597 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 2 22 24 B -24 0 -12 -12 -6 C -2 12 0 12 24 D -22 12 -12 0 2 E -24 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973597 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2035: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) E B D C A (9) A C B D E (9) B E C D A (6) D E A B C (5) D A E C B (5) D E B C A (4) D E A C B (4) C A B E D (4) E D B A C (3) D C A E B (3) C B A E D (3) B C E D A (3) B C A E D (3) D E C A B (2) D A C E B (2) C D A E B (2) C B E D A (2) C A B D E (2) A D C E B (2) A C D B E (2) E B D A C (1) D E C B A (1) D E B A C (1) D A E B C (1) C B E A D (1) C B A D E (1) C A D B E (1) B E D C A (1) B E C A D (1) B E A D C (1) B C E A D (1) A D C B E (1) A D B E C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -20 -30 -12 B 4 0 6 0 -10 C 20 -6 0 -16 -14 D 30 0 16 0 0 E 12 10 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.471042 E: 0.528958 Sum of squares = 0.501677091501 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.471042 E: 1.000000 A B C D E A 0 -4 -20 -30 -12 B 4 0 6 0 -10 C 20 -6 0 -16 -14 D 30 0 16 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 A=17 C=16 B=16 so C is eliminated. Round 2 votes counts: D=30 A=24 E=23 B=23 so E is eliminated. Round 3 votes counts: D=43 B=33 A=24 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:223 E:218 B:200 C:192 A:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -20 -30 -12 B 4 0 6 0 -10 C 20 -6 0 -16 -14 D 30 0 16 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -20 -30 -12 B 4 0 6 0 -10 C 20 -6 0 -16 -14 D 30 0 16 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -20 -30 -12 B 4 0 6 0 -10 C 20 -6 0 -16 -14 D 30 0 16 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2036: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) A E C B D (9) D B E C A (7) B C D A E (7) A C B D E (7) E D B C A (6) E D A B C (6) E A C D B (6) D E B C A (6) E A D C B (5) E D B A C (4) C A B D E (4) D B C A E (3) B D C A E (3) A C E B D (3) A C B E D (3) E A D B C (2) E D C B A (1) E D A C B (1) E A C B D (1) C B D A E (1) B D C E A (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 -6 -4 -16 -16 B 6 0 18 -18 0 C 4 -18 0 -14 -8 D 16 18 14 0 6 E 16 0 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -16 -16 B 6 0 18 -18 0 C 4 -18 0 -14 -8 D 16 18 14 0 6 E 16 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=28 A=22 B=13 C=5 so C is eliminated. Round 2 votes counts: E=32 D=28 A=26 B=14 so B is eliminated. Round 3 votes counts: D=40 E=32 A=28 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:209 B:203 C:182 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -16 -16 B 6 0 18 -18 0 C 4 -18 0 -14 -8 D 16 18 14 0 6 E 16 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -16 -16 B 6 0 18 -18 0 C 4 -18 0 -14 -8 D 16 18 14 0 6 E 16 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -16 -16 B 6 0 18 -18 0 C 4 -18 0 -14 -8 D 16 18 14 0 6 E 16 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2037: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (9) D B E A C (6) D B A C E (6) D A B E C (6) C E A B D (5) B D A C E (5) E D B C A (4) B D C A E (4) E D B A C (3) E C A D B (3) E B D C A (3) C E B D A (3) C B D A E (3) E D A B C (2) E C A B D (2) C A E B D (2) C A B D E (2) B D C E A (2) B C D A E (2) A E D B C (2) A D E B C (2) A C E D B (2) A C D B E (2) A B D C E (2) E C D B A (1) E C B D A (1) E C B A D (1) E B C D A (1) E A D B C (1) E A C D B (1) D B A E C (1) D A B C E (1) C B D E A (1) C B A D E (1) C A B E D (1) B D E C A (1) B C A D E (1) A E D C B (1) A E C D B (1) A D B E C (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 12 -12 18 B 0 0 30 -12 16 C -12 -30 0 -24 12 D 12 12 24 0 20 E -18 -16 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 -12 18 B 0 0 30 -12 16 C -12 -30 0 -24 12 D 12 12 24 0 20 E -18 -16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 D=20 C=18 B=15 so B is eliminated. Round 2 votes counts: D=32 A=24 E=23 C=21 so C is eliminated. Round 3 votes counts: D=38 E=31 A=31 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:234 B:217 A:209 C:173 E:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 12 -12 18 B 0 0 30 -12 16 C -12 -30 0 -24 12 D 12 12 24 0 20 E -18 -16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -12 18 B 0 0 30 -12 16 C -12 -30 0 -24 12 D 12 12 24 0 20 E -18 -16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -12 18 B 0 0 30 -12 16 C -12 -30 0 -24 12 D 12 12 24 0 20 E -18 -16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2038: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) C B D A E (10) E A B C D (7) D C B E A (7) E A B D C (5) D C B A E (5) C B A D E (5) A B C D E (5) E D C B A (4) E D A B C (4) D E B C A (4) D B C A E (4) E A C B D (3) D E C B A (3) E D A C B (2) D B C E A (2) C D B E A (2) A E B C D (2) A C B E D (2) A B E C D (2) E D C A B (1) E C A B D (1) E A D C B (1) D E B A C (1) C D B A E (1) B D A C E (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 -2 -16 B 2 0 6 -6 -2 C 0 -6 0 -14 -6 D 2 6 14 0 4 E 16 2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -2 -16 B 2 0 6 -6 -2 C 0 -6 0 -14 -6 D 2 6 14 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=26 C=18 A=14 B=3 so B is eliminated. Round 2 votes counts: E=39 D=27 C=19 A=15 so A is eliminated. Round 3 votes counts: E=45 C=28 D=27 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:210 B:200 A:190 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -2 -16 B 2 0 6 -6 -2 C 0 -6 0 -14 -6 D 2 6 14 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -2 -16 B 2 0 6 -6 -2 C 0 -6 0 -14 -6 D 2 6 14 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -2 -16 B 2 0 6 -6 -2 C 0 -6 0 -14 -6 D 2 6 14 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2039: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) E D B A C (9) E D B C A (7) D B E A C (6) A C B D E (6) C E B D A (5) E D C B A (4) E C D B A (4) E D A B C (3) D E B A C (3) C E A B D (3) C A E B D (3) C A B E D (3) B D A C E (3) E C A D B (2) D A B E C (2) C E D B A (2) C B E D A (2) C B D A E (2) C B A D E (2) B C A D E (2) B A C D E (2) A D B E C (2) A B D C E (2) A B C D E (2) E C D A B (1) D B A E C (1) C E B A D (1) C B D E A (1) B D E A C (1) B C D A E (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -14 -14 -8 B 16 0 -6 8 6 C 14 6 0 12 8 D 14 -8 -12 0 2 E 8 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -14 -8 B 16 0 -6 8 6 C 14 6 0 12 8 D 14 -8 -12 0 2 E 8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=30 A=13 D=12 B=9 so B is eliminated. Round 2 votes counts: C=39 E=30 D=16 A=15 so A is eliminated. Round 3 votes counts: C=49 E=30 D=21 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:212 D:198 E:196 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -14 -14 -8 B 16 0 -6 8 6 C 14 6 0 12 8 D 14 -8 -12 0 2 E 8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -14 -8 B 16 0 -6 8 6 C 14 6 0 12 8 D 14 -8 -12 0 2 E 8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -14 -8 B 16 0 -6 8 6 C 14 6 0 12 8 D 14 -8 -12 0 2 E 8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2040: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) A C B D E (8) E D B C A (7) C B A E D (6) B C E D A (6) A D E C B (6) E D A B C (4) D B E C A (4) D E B C A (3) D E A B C (3) D B C E A (3) D A E B C (3) C B E D A (3) C B A D E (3) A C B E D (3) E A D C B (2) D B C A E (2) C B E A D (2) C B D A E (2) B D C E A (2) B C D E A (2) A E C D B (2) A D E B C (2) E D B A C (1) E B D C A (1) E A D B C (1) D E B A C (1) D A B E C (1) C E B A D (1) B D E C A (1) B C D A E (1) A E C B D (1) A D C E B (1) A D C B E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 2 0 6 B 2 0 -4 -14 2 C -2 4 0 -16 -4 D 0 14 16 0 0 E -6 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.627952 B: 0.000000 C: 0.000000 D: 0.372048 E: 0.000000 Sum of squares = 0.532743586008 Cumulative probabilities = A: 0.627952 B: 0.627952 C: 0.627952 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 0 6 B 2 0 -4 -14 2 C -2 4 0 -16 -4 D 0 14 16 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=20 C=17 E=16 B=12 so B is eliminated. Round 2 votes counts: A=35 C=26 D=23 E=16 so E is eliminated. Round 3 votes counts: A=38 D=36 C=26 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:203 E:198 B:193 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 0 6 B 2 0 -4 -14 2 C -2 4 0 -16 -4 D 0 14 16 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 6 B 2 0 -4 -14 2 C -2 4 0 -16 -4 D 0 14 16 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 6 B 2 0 -4 -14 2 C -2 4 0 -16 -4 D 0 14 16 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2041: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) D A E C B (8) B C E D A (7) B C E A D (7) A D E B C (7) A B D E C (7) E C D A B (6) A D E C B (6) D E A C B (5) C E B D A (5) E D C A B (4) B A C E D (3) B A C D E (3) A D B E C (3) B C D E A (2) B C A D E (2) B A E C D (2) A E D C B (2) A B D C E (2) E D A C B (1) E C D B A (1) D E C A B (1) D A E B C (1) D A B C E (1) C E D B A (1) C B E D A (1) B C A E D (1) A E D B C (1) Total count = 100 A B C D E A 0 10 24 12 18 B -10 0 18 4 2 C -24 -18 0 -18 -10 D -12 -4 18 0 16 E -18 -2 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 24 12 18 B -10 0 18 4 2 C -24 -18 0 -18 -10 D -12 -4 18 0 16 E -18 -2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=28 D=16 E=12 C=7 so C is eliminated. Round 2 votes counts: B=38 A=28 E=18 D=16 so D is eliminated. Round 3 votes counts: B=38 A=38 E=24 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:232 D:209 B:207 E:187 C:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 24 12 18 B -10 0 18 4 2 C -24 -18 0 -18 -10 D -12 -4 18 0 16 E -18 -2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 24 12 18 B -10 0 18 4 2 C -24 -18 0 -18 -10 D -12 -4 18 0 16 E -18 -2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 24 12 18 B -10 0 18 4 2 C -24 -18 0 -18 -10 D -12 -4 18 0 16 E -18 -2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2042: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (15) C A E D B (14) B D E A C (14) C A D E B (9) B E A D C (6) C D E A B (4) C B D E A (4) A E D B C (4) D E A B C (3) C B A E D (3) B A E D C (3) A E D C B (3) E D A B C (2) D A E B C (2) C D A E B (2) A D E C B (2) A D E B C (2) E A D B C (1) D E B A C (1) C D A B E (1) C B E A D (1) C B A D E (1) C A B E D (1) B C E D A (1) B C A E D (1) Total count = 100 A B C D E A 0 0 16 2 -4 B 0 0 10 0 2 C -16 -10 0 -16 -16 D -2 0 16 0 -10 E 4 -2 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.287147 B: 0.712853 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.590612984092 Cumulative probabilities = A: 0.287147 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 2 -4 B 0 0 10 0 2 C -16 -10 0 -16 -16 D -2 0 16 0 -10 E 4 -2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555743506 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=40 B=40 A=11 D=6 E=3 so E is eliminated. Round 2 votes counts: C=40 B=40 A=12 D=8 so D is eliminated. Round 3 votes counts: B=41 C=40 A=19 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:214 A:207 B:206 D:202 C:171 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 16 2 -4 B 0 0 10 0 2 C -16 -10 0 -16 -16 D -2 0 16 0 -10 E 4 -2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555743506 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 2 -4 B 0 0 10 0 2 C -16 -10 0 -16 -16 D -2 0 16 0 -10 E 4 -2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555743506 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 2 -4 B 0 0 10 0 2 C -16 -10 0 -16 -16 D -2 0 16 0 -10 E 4 -2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555743506 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2043: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (24) D E A B C (16) C B E A D (10) E D A B C (6) C B A D E (6) E D C A B (4) B C A D E (4) E D A C B (3) D A E B C (3) C E B D A (3) B A C D E (3) A B D E C (3) E C D A B (2) C E D B A (2) B A D E C (2) B A D C E (2) A B D C E (2) E B A D C (1) C E B A D (1) C B E D A (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -18 -14 20 2 B 18 0 -12 26 16 C 14 12 0 12 16 D -20 -26 -12 0 -16 E -2 -16 -16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 20 2 B 18 0 -12 26 16 C 14 12 0 12 16 D -20 -26 -12 0 -16 E -2 -16 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=47 D=19 E=16 B=11 A=7 so A is eliminated. Round 2 votes counts: C=47 D=20 E=17 B=16 so B is eliminated. Round 3 votes counts: C=54 D=29 E=17 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:224 A:195 E:191 D:163 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -14 20 2 B 18 0 -12 26 16 C 14 12 0 12 16 D -20 -26 -12 0 -16 E -2 -16 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 20 2 B 18 0 -12 26 16 C 14 12 0 12 16 D -20 -26 -12 0 -16 E -2 -16 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 20 2 B 18 0 -12 26 16 C 14 12 0 12 16 D -20 -26 -12 0 -16 E -2 -16 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2044: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) E A D B C (8) E A D C B (5) C D A B E (5) E B A D C (4) E A B D C (4) B D C A E (4) D B A E C (3) C E B A D (3) C A E D B (3) C A D E B (3) B E D A C (3) B D A E C (3) B C E D A (3) A D E C B (3) E C A B D (2) E B A C D (2) E A B C D (2) C D B A E (2) C D A E B (2) C B E A D (2) C B D E A (2) B E A D C (2) B D E A C (2) E C A D B (1) E B C A D (1) D C B A E (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E D A (1) B E D C A (1) B E C D A (1) B E C A D (1) B D E C A (1) B C D A E (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -2 2 -2 B 6 0 -2 10 -2 C 2 2 0 -2 -10 D -2 -10 2 0 -4 E 2 2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 2 -2 B 6 0 -2 10 -2 C 2 2 0 -2 -10 D -2 -10 2 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=29 B=22 D=8 A=8 so D is eliminated. Round 2 votes counts: C=35 E=29 B=25 A=11 so A is eliminated. Round 3 votes counts: E=38 C=37 B=25 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:206 A:196 C:196 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 2 -2 B 6 0 -2 10 -2 C 2 2 0 -2 -10 D -2 -10 2 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 -2 B 6 0 -2 10 -2 C 2 2 0 -2 -10 D -2 -10 2 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 -2 B 6 0 -2 10 -2 C 2 2 0 -2 -10 D -2 -10 2 0 -4 E 2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2045: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) E D A B C (7) C B E A D (7) A C D E B (6) D A E B C (5) B C E D A (5) B C A D E (5) A D C E B (5) E D B A C (4) E A D C B (4) C B A D E (4) C A D E B (4) C A D B E (4) E B D A C (3) D E A B C (3) C B A E D (3) B E D A C (3) D E B A C (2) D A E C B (2) C A B D E (2) B E C D A (2) A E D C B (2) E D A C B (1) E C B D A (1) C A B E D (1) B E D C A (1) B D E A C (1) B D A C E (1) B C E A D (1) B C D E A (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 10 16 14 8 B -10 0 -10 -16 -14 C -16 10 0 -6 2 D -14 16 6 0 10 E -8 14 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 14 8 B -10 0 -10 -16 -14 C -16 10 0 -6 2 D -14 16 6 0 10 E -8 14 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=22 B=21 E=20 D=12 so D is eliminated. Round 2 votes counts: A=29 E=25 C=25 B=21 so B is eliminated. Round 3 votes counts: C=38 E=32 A=30 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:224 D:209 E:197 C:195 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 14 8 B -10 0 -10 -16 -14 C -16 10 0 -6 2 D -14 16 6 0 10 E -8 14 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 14 8 B -10 0 -10 -16 -14 C -16 10 0 -6 2 D -14 16 6 0 10 E -8 14 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 14 8 B -10 0 -10 -16 -14 C -16 10 0 -6 2 D -14 16 6 0 10 E -8 14 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2046: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) A E B D C (8) E A D C B (7) D C E B A (7) E D C B A (6) B A C D E (6) A B E C D (6) C D B E A (4) B C D A E (4) B C A D E (4) A B C D E (4) D B C E A (3) A B E D C (3) E D A C B (2) E C D B A (2) E A C D B (2) C E D B A (2) C D E B A (2) B D C E A (2) B D C A E (2) B C D E A (2) A E B C D (2) A B D C E (2) A B C E D (2) D E C B A (1) D C B E A (1) C B D E A (1) B D A C E (1) B A D E C (1) B A D C E (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -4 -2 0 -4 B 4 0 8 2 -2 C 2 -8 0 -12 0 D 0 -2 12 0 -4 E 4 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.064348 D: 0.000000 E: 0.935652 Sum of squares = 0.879585776039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.064348 D: 0.064348 E: 1.000000 A B C D E A 0 -4 -2 0 -4 B 4 0 8 2 -2 C 2 -8 0 -12 0 D 0 -2 12 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000002871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 B=23 D=12 C=9 so C is eliminated. Round 2 votes counts: E=29 A=29 B=24 D=18 so D is eliminated. Round 3 votes counts: E=39 B=32 A=29 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:206 E:205 D:203 A:195 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 0 -4 B 4 0 8 2 -2 C 2 -8 0 -12 0 D 0 -2 12 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000002871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 0 -4 B 4 0 8 2 -2 C 2 -8 0 -12 0 D 0 -2 12 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000002871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 0 -4 B 4 0 8 2 -2 C 2 -8 0 -12 0 D 0 -2 12 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000002871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2047: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) D B A E C (7) B A D C E (7) E C D A B (6) C A E B D (5) B A C D E (5) E D C A B (4) E C A B D (4) C A B E D (4) D E B C A (3) D B C A E (3) D B A C E (3) C B A D E (3) B D A C E (3) A E C B D (3) E D A B C (2) E C A D B (2) E A C D B (2) E A C B D (2) D E B A C (2) C B A E D (2) A E B C D (2) A B C E D (2) E D A C B (1) E A D B C (1) D E C A B (1) D C B E A (1) D B E A C (1) D B C E A (1) C E D B A (1) C E A B D (1) C B D E A (1) C B D A E (1) B D C A E (1) B D A E C (1) B A C E D (1) A C E B D (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 0 14 B 8 0 -6 4 0 C 2 6 0 0 -4 D 0 -4 0 0 6 E -14 0 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.465756 D: 0.534244 E: 0.000000 Sum of squares = 0.50234525049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.465756 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 0 14 B 8 0 -6 4 0 C 2 6 0 0 -4 D 0 -4 0 0 6 E -14 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 C=18 B=18 A=11 so A is eliminated. Round 2 votes counts: E=29 D=29 B=22 C=20 so C is eliminated. Round 3 votes counts: E=37 B=34 D=29 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:203 A:202 C:202 D:201 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 0 14 B 8 0 -6 4 0 C 2 6 0 0 -4 D 0 -4 0 0 6 E -14 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 14 B 8 0 -6 4 0 C 2 6 0 0 -4 D 0 -4 0 0 6 E -14 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 14 B 8 0 -6 4 0 C 2 6 0 0 -4 D 0 -4 0 0 6 E -14 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2048: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (17) B E D A C (11) A E D C B (7) E D A B C (5) E B D A C (4) D E A B C (4) C D A E B (4) B C D E A (4) B E A D C (3) A C D E B (3) E B A D C (2) C B D E A (2) C B D A E (2) C B A D E (2) C A D B E (2) C A B E D (2) B E D C A (2) B C E A D (2) A E D B C (2) E A D B C (1) E A B D C (1) D E A C B (1) D C E B A (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C E B (1) C D E A B (1) C D A B E (1) C A E D B (1) B E C D A (1) B E A C D (1) B D E A C (1) B C A D E (1) A E C D B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 22 6 4 4 B -22 0 -10 -16 -26 C -6 10 0 -2 2 D -4 16 2 0 4 E -4 26 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 6 4 4 B -22 0 -10 -16 -26 C -6 10 0 -2 2 D -4 16 2 0 4 E -4 26 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=26 A=17 E=13 D=10 so D is eliminated. Round 2 votes counts: C=37 B=26 A=19 E=18 so E is eliminated. Round 3 votes counts: C=37 B=32 A=31 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:218 D:209 E:208 C:202 B:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 6 4 4 B -22 0 -10 -16 -26 C -6 10 0 -2 2 D -4 16 2 0 4 E -4 26 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 6 4 4 B -22 0 -10 -16 -26 C -6 10 0 -2 2 D -4 16 2 0 4 E -4 26 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 6 4 4 B -22 0 -10 -16 -26 C -6 10 0 -2 2 D -4 16 2 0 4 E -4 26 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2049: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) C B D A E (7) E A D C B (6) B E D A C (6) B C D E A (6) B C D A E (6) A C E D B (6) E A D B C (5) A E D C B (5) E D B A C (3) D E A B C (3) C B A E D (3) C B A D E (3) B D E C A (3) A E C D B (3) E A B D C (2) D B E C A (2) C D A B E (2) C A E D B (2) C A E B D (2) C A D E B (2) C A B E D (2) E D A B C (1) E B A C D (1) E A C B D (1) D E B A C (1) D C B A E (1) D B E A C (1) D B C E A (1) C D B A E (1) C A B D E (1) B E C D A (1) B D C A E (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -10 -10 -4 B 12 0 4 8 12 C 10 -4 0 -2 10 D 10 -8 2 0 2 E 4 -12 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -10 -4 B 12 0 4 8 12 C 10 -4 0 -2 10 D 10 -8 2 0 2 E 4 -12 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=25 E=19 A=15 D=9 so D is eliminated. Round 2 votes counts: B=36 C=26 E=23 A=15 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:207 D:203 E:190 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 -10 -4 B 12 0 4 8 12 C 10 -4 0 -2 10 D 10 -8 2 0 2 E 4 -12 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -10 -4 B 12 0 4 8 12 C 10 -4 0 -2 10 D 10 -8 2 0 2 E 4 -12 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -10 -4 B 12 0 4 8 12 C 10 -4 0 -2 10 D 10 -8 2 0 2 E 4 -12 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2050: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (13) C B E D A (10) A D E B C (9) D B A C E (4) D A B E C (4) D A B C E (4) B C E D A (4) A E D C B (4) A D B C E (4) E C A B D (3) E A D B C (3) B E C D A (3) E A C D B (2) C E A B D (2) C B E A D (2) C B A D E (2) B D C A E (2) B C D E A (2) B C D A E (2) A C E D B (2) E D B A C (1) E D A B C (1) E C B D A (1) E C B A D (1) D E B A C (1) D B E A C (1) D B A E C (1) D A E B C (1) C E B D A (1) C B D E A (1) C B D A E (1) C A E D B (1) C A E B D (1) B D E C A (1) B D C E A (1) A D E C B (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -8 6 -8 B 10 0 0 6 2 C 8 0 0 10 22 D -6 -6 -10 0 -10 E 8 -2 -22 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.591869 C: 0.408131 D: 0.000000 E: 0.000000 Sum of squares = 0.516879811529 Cumulative probabilities = A: 0.000000 B: 0.591869 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 6 -8 B 10 0 0 6 2 C 8 0 0 10 22 D -6 -6 -10 0 -10 E 8 -2 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=23 D=16 B=15 E=12 so E is eliminated. Round 2 votes counts: C=39 A=28 D=18 B=15 so B is eliminated. Round 3 votes counts: C=50 A=28 D=22 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:209 E:197 A:190 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 6 -8 B 10 0 0 6 2 C 8 0 0 10 22 D -6 -6 -10 0 -10 E 8 -2 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 6 -8 B 10 0 0 6 2 C 8 0 0 10 22 D -6 -6 -10 0 -10 E 8 -2 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 6 -8 B 10 0 0 6 2 C 8 0 0 10 22 D -6 -6 -10 0 -10 E 8 -2 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2051: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) C E D A B (9) A D B C E (6) D C A E B (5) C D E A B (5) E C B D A (4) E B C A D (4) D C B A E (4) B E A C D (4) B A E D C (4) B A D E C (4) E B C D A (3) C E D B A (3) C D E B A (3) B A D C E (3) E C A D B (2) D A C E B (2) D A C B E (2) C D B E A (2) A D B E C (2) A B E D C (2) A B D E C (2) E C B A D (1) E C A B D (1) E B A C D (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A B E (1) D A B C E (1) B E C D A (1) B E C A D (1) B D C E A (1) B C E D A (1) B C D E A (1) B A E C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -24 -16 -18 B -8 0 -10 -14 -10 C 24 10 0 18 2 D 16 14 -18 0 -8 E 18 10 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -24 -16 -18 B -8 0 -10 -14 -10 C 24 10 0 18 2 D 16 14 -18 0 -8 E 18 10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=22 B=21 D=16 A=14 so A is eliminated. Round 2 votes counts: E=27 B=27 D=24 C=22 so C is eliminated. Round 3 votes counts: E=39 D=34 B=27 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:227 E:217 D:202 B:179 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -24 -16 -18 B -8 0 -10 -14 -10 C 24 10 0 18 2 D 16 14 -18 0 -8 E 18 10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -24 -16 -18 B -8 0 -10 -14 -10 C 24 10 0 18 2 D 16 14 -18 0 -8 E 18 10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -24 -16 -18 B -8 0 -10 -14 -10 C 24 10 0 18 2 D 16 14 -18 0 -8 E 18 10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2052: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) D C E B A (5) A B E C D (5) D C B E A (4) B C E A D (4) A E B C D (4) A B D C E (4) E D C A B (3) E C B A D (3) E A D C B (3) D E C A B (3) B C D A E (3) B A C E D (3) A E D C B (3) A E B D C (3) A B E D C (3) E C A D B (2) D B C A E (2) C E D B A (2) C D B E A (2) C B E D A (2) B D C A E (2) B C A D E (2) B A C D E (2) A E D B C (2) E B C A D (1) D E C B A (1) D E A C B (1) D C E A B (1) D C B A E (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E B A (1) C B D E A (1) B E C A D (1) B E A C D (1) B D A C E (1) B C A E D (1) B A D C E (1) A D E C B (1) A D E B C (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -10 6 2 B 8 0 0 -2 0 C 10 0 0 -4 -2 D -6 2 4 0 -8 E -2 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.534643 C: 0.000000 D: 0.000000 E: 0.465357 Sum of squares = 0.502400299033 Cumulative probabilities = A: 0.000000 B: 0.534643 C: 0.534643 D: 0.534643 E: 1.000000 A B C D E A 0 -8 -10 6 2 B 8 0 0 -2 0 C 10 0 0 -4 -2 D -6 2 4 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 B=21 E=19 C=9 so C is eliminated. Round 2 votes counts: A=29 D=25 B=24 E=22 so E is eliminated. Round 3 votes counts: D=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:204 B:203 C:202 D:196 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 6 2 B 8 0 0 -2 0 C 10 0 0 -4 -2 D -6 2 4 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 6 2 B 8 0 0 -2 0 C 10 0 0 -4 -2 D -6 2 4 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 6 2 B 8 0 0 -2 0 C 10 0 0 -4 -2 D -6 2 4 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2053: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) E D A B C (7) D E A B C (5) B E D C A (5) B E C D A (5) B C E A D (5) C B A D E (4) C A D B E (4) C A B D E (4) E D A C B (3) E B D C A (3) D E A C B (3) D A E C B (3) B C A D E (3) A C D B E (3) E D B C A (2) E D B A C (2) E C D A B (2) E B D A C (2) D A E B C (2) B E D A C (2) B C A E D (2) A D E C B (2) A D C E B (2) A D C B E (2) E D C B A (1) E D C A B (1) E C D B A (1) E B C D A (1) D E C A B (1) C E B D A (1) C B E D A (1) C A D E B (1) B E C A D (1) B D A E C (1) B A C E D (1) B A C D E (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -12 -8 -8 B 4 0 4 2 8 C 12 -4 0 0 -10 D 8 -2 0 0 -12 E 8 -8 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -8 -8 B 4 0 4 2 8 C 12 -4 0 0 -10 D 8 -2 0 0 -12 E 8 -8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 C=23 D=14 A=12 so A is eliminated. Round 2 votes counts: C=27 B=27 E=25 D=21 so D is eliminated. Round 3 votes counts: E=41 C=31 B=28 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:209 C:199 D:197 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 -8 -8 B 4 0 4 2 8 C 12 -4 0 0 -10 D 8 -2 0 0 -12 E 8 -8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -8 -8 B 4 0 4 2 8 C 12 -4 0 0 -10 D 8 -2 0 0 -12 E 8 -8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -8 -8 B 4 0 4 2 8 C 12 -4 0 0 -10 D 8 -2 0 0 -12 E 8 -8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997711 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2054: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) A C B E D (7) D E B C A (5) D B A C E (5) A C E D B (5) D B E C A (4) C A E B D (4) B E D C A (4) E D B C A (3) D B E A C (3) D A E C B (3) D A C E B (3) D A C B E (3) A C D E B (3) E C A B D (2) E B D C A (2) D E A C B (2) D B A E C (2) C E A B D (2) B E C D A (2) B D E C A (2) B D C A E (2) B D A C E (2) A C B D E (2) E D C B A (1) E D C A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E B C A D (1) E A C D B (1) D E B A C (1) D A B C E (1) C E B A D (1) C B E A D (1) C B A E D (1) B E C A D (1) B C E D A (1) B C E A D (1) B A D C E (1) B A C D E (1) A D C E B (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 12 -6 12 B -2 0 -10 2 -4 C -12 10 0 -6 14 D 6 -2 6 0 -4 E -12 4 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958676797 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 A B C D E A 0 2 12 -6 12 B -2 0 -10 2 -4 C -12 10 0 -6 14 D 6 -2 6 0 -4 E -12 4 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=28 B=17 E=14 C=9 so C is eliminated. Round 2 votes counts: D=32 A=32 B=19 E=17 so E is eliminated. Round 3 votes counts: A=38 D=37 B=25 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:210 C:203 D:203 B:193 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 12 -6 12 B -2 0 -10 2 -4 C -12 10 0 -6 14 D 6 -2 6 0 -4 E -12 4 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 -6 12 B -2 0 -10 2 -4 C -12 10 0 -6 14 D 6 -2 6 0 -4 E -12 4 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 -6 12 B -2 0 -10 2 -4 C -12 10 0 -6 14 D 6 -2 6 0 -4 E -12 4 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677681 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2055: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) B D A E C (7) B D A C E (7) E A C B D (6) E C A D B (5) E A B D C (5) D B C A E (5) C D B E A (5) A E C B D (5) A B D C E (5) C E D B A (4) C D B A E (4) B D E C A (4) A E B D C (4) E C A B D (3) C A E D B (3) B D E A C (3) A B D E C (3) B A D E C (2) B A D C E (2) A B E D C (2) E B D A C (1) C E D A B (1) C A D B E (1) B E D A C (1) B D C A E (1) A E B C D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 12 14 8 B -8 0 6 26 8 C -12 -6 0 -4 -4 D -14 -26 4 0 2 E -8 -8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 14 8 B -8 0 6 26 8 C -12 -6 0 -4 -4 D -14 -26 4 0 2 E -8 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 A=22 E=20 D=5 so D is eliminated. Round 2 votes counts: B=32 C=26 A=22 E=20 so E is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:216 E:193 C:187 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 14 8 B -8 0 6 26 8 C -12 -6 0 -4 -4 D -14 -26 4 0 2 E -8 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 14 8 B -8 0 6 26 8 C -12 -6 0 -4 -4 D -14 -26 4 0 2 E -8 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 14 8 B -8 0 6 26 8 C -12 -6 0 -4 -4 D -14 -26 4 0 2 E -8 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2056: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (15) D B C E A (14) D B C A E (7) E A C B D (6) C B D E A (6) D A B C E (5) C B D A E (5) A E D B C (5) D A E B C (4) B C D E A (4) E C B A D (3) B D C E A (3) E C A B D (2) D C B A E (2) A E D C B (2) A D E B C (2) A C B D E (2) E B D C A (1) E A D B C (1) E A B D C (1) D E B A C (1) D C B E A (1) C D B E A (1) C D B A E (1) C B E D A (1) C B E A D (1) C B A D E (1) B E C D A (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -8 -14 6 B 6 0 0 4 14 C 8 0 0 -2 12 D 14 -4 2 0 22 E -6 -14 -12 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.583076 C: 0.416924 D: 0.000000 E: 0.000000 Sum of squares = 0.513803120176 Cumulative probabilities = A: 0.000000 B: 0.583076 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -14 6 B 6 0 0 4 14 C 8 0 0 -2 12 D 14 -4 2 0 22 E -6 -14 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=28 C=16 E=14 B=8 so B is eliminated. Round 2 votes counts: D=37 A=28 C=20 E=15 so E is eliminated. Round 3 votes counts: D=38 A=36 C=26 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 B:212 C:209 A:189 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -14 6 B 6 0 0 4 14 C 8 0 0 -2 12 D 14 -4 2 0 22 E -6 -14 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -14 6 B 6 0 0 4 14 C 8 0 0 -2 12 D 14 -4 2 0 22 E -6 -14 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -14 6 B 6 0 0 4 14 C 8 0 0 -2 12 D 14 -4 2 0 22 E -6 -14 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2057: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E A B D C (7) D B E A C (6) B D E A C (6) C A E B D (5) E B D A C (4) D E B A C (4) D B E C A (4) D B C E A (4) E D B A C (3) D C B E A (3) C D B E A (3) C A E D B (3) C A D B E (3) C A B E D (3) B E A D C (3) A E B C D (3) D B C A E (2) C D E A B (2) C D A B E (2) B C D A E (2) A E C B D (2) A C E B D (2) E D C A B (1) E B A D C (1) E A D B C (1) D E C B A (1) D C B A E (1) C E A D B (1) C D A E B (1) C B A D E (1) C A D E B (1) C A B D E (1) B A C E D (1) B A C D E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -20 -8 -18 -8 B 20 0 6 -12 16 C 8 -6 0 -4 6 D 18 12 4 0 16 E 8 -16 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 -18 -8 B 20 0 6 -12 16 C 8 -6 0 -4 6 D 18 12 4 0 16 E 8 -16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=25 E=17 B=13 A=9 so A is eliminated. Round 2 votes counts: C=39 D=25 E=22 B=14 so B is eliminated. Round 3 votes counts: C=43 D=31 E=26 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 B:215 C:202 E:185 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -8 -18 -8 B 20 0 6 -12 16 C 8 -6 0 -4 6 D 18 12 4 0 16 E 8 -16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -18 -8 B 20 0 6 -12 16 C 8 -6 0 -4 6 D 18 12 4 0 16 E 8 -16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -18 -8 B 20 0 6 -12 16 C 8 -6 0 -4 6 D 18 12 4 0 16 E 8 -16 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2058: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) E B A C D (11) D C A B E (8) D C A E B (6) C A D E B (6) B E D A C (6) A C E B D (4) D B E C A (3) D B C E A (3) C A E D B (3) B E C A D (3) B D E C A (3) A E C B D (3) A C D E B (3) E A C B D (2) E A B C D (2) D C B A E (2) D B E A C (2) C D A E B (2) C A E B D (2) B E D C A (2) A D C E B (2) E B D A C (1) E B C A D (1) D E A B C (1) D A E B C (1) D A C E B (1) C B D A E (1) C A B D E (1) B D E A C (1) B C E D A (1) A E D B C (1) Total count = 100 A B C D E A 0 -4 6 12 -8 B 4 0 8 12 -4 C -6 -8 0 14 -10 D -12 -12 -14 0 -8 E 8 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 6 12 -8 B 4 0 8 12 -4 C -6 -8 0 14 -10 D -12 -12 -14 0 -8 E 8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 E=17 C=15 A=13 so A is eliminated. Round 2 votes counts: D=29 B=28 C=22 E=21 so E is eliminated. Round 3 votes counts: B=43 D=30 C=27 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:210 A:203 C:195 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 12 -8 B 4 0 8 12 -4 C -6 -8 0 14 -10 D -12 -12 -14 0 -8 E 8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 12 -8 B 4 0 8 12 -4 C -6 -8 0 14 -10 D -12 -12 -14 0 -8 E 8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 12 -8 B 4 0 8 12 -4 C -6 -8 0 14 -10 D -12 -12 -14 0 -8 E 8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2059: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (23) B E D C A (18) B E D A C (6) B A E D C (5) B A D E C (4) E D C B A (3) E D C A B (3) C E D A B (3) C D E A B (3) C A D E B (3) B E C D A (3) E D B C A (2) D E C A B (2) C E D B A (2) B C A E D (2) B A C E D (2) A B D C E (2) A B C D E (2) E D B A C (1) E B D C A (1) D E A B C (1) D A E B C (1) C D A E B (1) C B E D A (1) C A B D E (1) B E A D C (1) B C E D A (1) A D C E B (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 2 -4 -2 B 4 0 6 2 0 C -2 -6 0 -4 -4 D 4 -2 4 0 -8 E 2 0 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.457604 C: 0.000000 D: 0.000000 E: 0.542396 Sum of squares = 0.503594904111 Cumulative probabilities = A: 0.000000 B: 0.457604 C: 0.457604 D: 0.457604 E: 1.000000 A B C D E A 0 -4 2 -4 -2 B 4 0 6 2 0 C -2 -6 0 -4 -4 D 4 -2 4 0 -8 E 2 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=30 C=14 E=10 D=4 so D is eliminated. Round 2 votes counts: B=42 A=31 C=14 E=13 so E is eliminated. Round 3 votes counts: B=46 A=32 C=22 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:207 B:206 D:199 A:196 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -4 -2 B 4 0 6 2 0 C -2 -6 0 -4 -4 D 4 -2 4 0 -8 E 2 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -4 -2 B 4 0 6 2 0 C -2 -6 0 -4 -4 D 4 -2 4 0 -8 E 2 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -4 -2 B 4 0 6 2 0 C -2 -6 0 -4 -4 D 4 -2 4 0 -8 E 2 0 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2060: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) D B C A E (10) E C A D B (6) E A B D C (6) A C B D E (6) E B D A C (5) E A C B D (5) C A D B E (5) B D A C E (4) E B A D C (3) C D B A E (3) C D A B E (3) C A E D B (3) C A B D E (3) B D E A C (3) B A D C E (3) A C E B D (3) E A B C D (2) D C B A E (2) A B C D E (2) E D C B A (1) E C D B A (1) E A C D B (1) D B E C A (1) C E A D B (1) C D E B A (1) B E D A C (1) B E A D C (1) B D C A E (1) A E B C D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -4 6 2 B 2 0 10 2 0 C 4 -10 0 -6 4 D -6 -2 6 0 -4 E -2 0 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.601779 C: 0.000000 D: 0.000000 E: 0.398221 Sum of squares = 0.520717829286 Cumulative probabilities = A: 0.000000 B: 0.601779 C: 0.601779 D: 0.601779 E: 1.000000 A B C D E A 0 -2 -4 6 2 B 2 0 10 2 0 C 4 -10 0 -6 4 D -6 -2 6 0 -4 E -2 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091469 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=19 A=14 D=13 B=13 so D is eliminated. Round 2 votes counts: E=41 B=24 C=21 A=14 so A is eliminated. Round 3 votes counts: E=42 C=31 B=27 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:207 A:201 E:199 D:197 C:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 6 2 B 2 0 10 2 0 C 4 -10 0 -6 4 D -6 -2 6 0 -4 E -2 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091469 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 6 2 B 2 0 10 2 0 C 4 -10 0 -6 4 D -6 -2 6 0 -4 E -2 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091469 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 6 2 B 2 0 10 2 0 C 4 -10 0 -6 4 D -6 -2 6 0 -4 E -2 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091469 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2061: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) A C D E B (7) A D C B E (6) E B C D A (4) E B C A D (4) B E D C A (4) B E A D C (4) E C B A D (3) E B A C D (3) D C B E A (3) C D A E B (3) C A D E B (3) B E D A C (3) B D E A C (3) A E B C D (3) A B E D C (3) E C B D A (2) D B A C E (2) D A B C E (2) C E B D A (2) C E A D B (2) C D E B A (2) A E C B D (2) A D B E C (2) A C E B D (2) A C D B E (2) E C A B D (1) E B D C A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B E C A (1) D B C A E (1) D A C B E (1) C E A B D (1) C A E D B (1) C A E B D (1) B A E D C (1) A D B C E (1) A C E D B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 0 10 -2 B 4 0 4 14 2 C 0 -4 0 0 -8 D -10 -14 0 0 2 E 2 -2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 10 -2 B 4 0 4 14 2 C 0 -4 0 0 -8 D -10 -14 0 0 2 E 2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=23 E=18 C=15 D=13 so D is eliminated. Round 2 votes counts: A=34 B=27 C=21 E=18 so E is eliminated. Round 3 votes counts: B=39 A=34 C=27 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:203 A:202 C:194 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 10 -2 B 4 0 4 14 2 C 0 -4 0 0 -8 D -10 -14 0 0 2 E 2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 10 -2 B 4 0 4 14 2 C 0 -4 0 0 -8 D -10 -14 0 0 2 E 2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 10 -2 B 4 0 4 14 2 C 0 -4 0 0 -8 D -10 -14 0 0 2 E 2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2062: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (13) D E B C A (9) D E A C B (6) A C D E B (6) B C A E D (5) E D B A C (4) D E B A C (4) D E A B C (4) C A B E D (4) A C B D E (4) E D A B C (3) D E C B A (3) C B D E A (3) C B A E D (3) C A B D E (3) E B D A C (2) B E D C A (2) B C E D A (2) B A C E D (2) A E D C B (2) A D E C B (2) E D B C A (1) D E C A B (1) D C E B A (1) D C E A B (1) D A E C B (1) C B D A E (1) C B A D E (1) C A D E B (1) C A D B E (1) B D E C A (1) A E D B C (1) A E B D C (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 14 2 6 B -12 0 -14 -2 -8 C -14 14 0 2 4 D -2 2 -2 0 6 E -6 8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 2 6 B -12 0 -14 -2 -8 C -14 14 0 2 4 D -2 2 -2 0 6 E -6 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=30 C=17 B=12 E=10 so E is eliminated. Round 2 votes counts: D=38 A=31 C=17 B=14 so B is eliminated. Round 3 votes counts: D=43 A=33 C=24 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:203 D:202 E:196 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 2 6 B -12 0 -14 -2 -8 C -14 14 0 2 4 D -2 2 -2 0 6 E -6 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 2 6 B -12 0 -14 -2 -8 C -14 14 0 2 4 D -2 2 -2 0 6 E -6 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 2 6 B -12 0 -14 -2 -8 C -14 14 0 2 4 D -2 2 -2 0 6 E -6 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2063: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) D C A B E (8) D C A E B (6) D B C A E (6) B D E C A (6) A C E D B (6) E B A C D (5) B E D A C (5) B D C A E (4) E A C B D (3) D C E A B (3) C A D E B (3) B E A C D (3) E B D A C (2) E A B C D (2) D C B E A (2) C A D B E (2) B E D C A (2) B E A D C (2) B A E C D (2) B A C D E (2) A E C D B (2) A E C B D (2) A C D E B (2) E D C A B (1) E B D C A (1) E B A D C (1) D C B A E (1) D B C E A (1) C D B A E (1) C A B D E (1) B D A C E (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 4 0 4 B -6 0 -8 -6 2 C -4 8 0 -4 2 D 0 6 4 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.561600 B: 0.000000 C: 0.000000 D: 0.438400 E: 0.000000 Sum of squares = 0.507589035822 Cumulative probabilities = A: 0.561600 B: 0.561600 C: 0.561600 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 4 B -6 0 -8 -6 2 C -4 8 0 -4 2 D 0 6 4 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 E=24 A=15 C=7 so C is eliminated. Round 2 votes counts: D=28 B=27 E=24 A=21 so A is eliminated. Round 3 votes counts: E=35 D=35 B=30 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:207 D:204 C:201 E:197 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 4 B -6 0 -8 -6 2 C -4 8 0 -4 2 D 0 6 4 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 4 B -6 0 -8 -6 2 C -4 8 0 -4 2 D 0 6 4 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 4 B -6 0 -8 -6 2 C -4 8 0 -4 2 D 0 6 4 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2064: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) E B C D A (7) A D B C E (5) D B E A C (4) C E A B D (4) B E D A C (4) E C B D A (3) D B A E C (3) C E B A D (3) A E B D C (3) A D B E C (3) A C D E B (3) A B D E C (3) E B D C A (2) D C A B E (2) D A C B E (2) D A B E C (2) D A B C E (2) C D A B E (2) C A E B D (2) C A D E B (2) B E A D C (2) B D E A C (2) B A E D C (2) A D C B E (2) E C B A D (1) E B D A C (1) E B A C D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B A C E (1) C E D B A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D A E B (1) C B E D A (1) C A E D B (1) C A D B E (1) B E D C A (1) B D E C A (1) B D A E C (1) B A D E C (1) A E C B D (1) A E B C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 4 -10 -2 B 6 0 6 16 0 C -4 -6 0 -4 -2 D 10 -16 4 0 -8 E 2 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.237201 C: 0.000000 D: 0.000000 E: 0.762799 Sum of squares = 0.638126464387 Cumulative probabilities = A: 0.000000 B: 0.237201 C: 0.237201 D: 0.237201 E: 1.000000 A B C D E A 0 -6 4 -10 -2 B 6 0 6 16 0 C -4 -6 0 -4 -2 D 10 -16 4 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=23 D=18 E=16 B=14 so B is eliminated. Round 2 votes counts: C=29 A=26 E=23 D=22 so D is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:214 E:206 D:195 A:193 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -10 -2 B 6 0 6 16 0 C -4 -6 0 -4 -2 D 10 -16 4 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -10 -2 B 6 0 6 16 0 C -4 -6 0 -4 -2 D 10 -16 4 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -10 -2 B 6 0 6 16 0 C -4 -6 0 -4 -2 D 10 -16 4 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2065: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) E A B C D (8) C D B A E (8) B C D A E (5) A E D C B (5) E A D C B (4) A E B C D (4) A D C E B (4) A C D B E (4) E D A C B (3) B E C D A (3) B E A C D (3) B A C D E (3) E B D C A (2) E B A D C (2) E A D B C (2) E A B D C (2) D E C B A (2) D C E B A (2) D C E A B (2) D C A E B (2) A E C D B (2) A C B D E (2) A B C D E (2) E D B C A (1) E B A C D (1) D E C A B (1) D C B A E (1) D C A B E (1) D B C E A (1) C D A B E (1) C B D A E (1) B E C A D (1) B C D E A (1) B A E C D (1) A E C B D (1) A D E C B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 12 8 0 B -6 0 -12 -12 -2 C -12 12 0 6 0 D -8 12 -6 0 6 E 0 2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.673240 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.326760 Sum of squares = 0.560023899193 Cumulative probabilities = A: 0.673240 B: 0.673240 C: 0.673240 D: 0.673240 E: 1.000000 A B C D E A 0 6 12 8 0 B -6 0 -12 -12 -2 C -12 12 0 6 0 D -8 12 -6 0 6 E 0 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 D=21 B=17 C=10 so C is eliminated. Round 2 votes counts: D=30 A=27 E=25 B=18 so B is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:213 C:203 D:202 E:198 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 8 0 B -6 0 -12 -12 -2 C -12 12 0 6 0 D -8 12 -6 0 6 E 0 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 8 0 B -6 0 -12 -12 -2 C -12 12 0 6 0 D -8 12 -6 0 6 E 0 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 8 0 B -6 0 -12 -12 -2 C -12 12 0 6 0 D -8 12 -6 0 6 E 0 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2066: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (15) D E A B C (8) D A E B C (8) C B D A E (8) C B A E D (6) E A B C D (5) E A D B C (4) C A E D B (4) D C A E B (3) D B E A C (3) D B C E A (3) C D B A E (3) C B E A D (3) E A B D C (2) D C A B E (2) C B D E A (2) B E A C D (2) B D C E A (2) B C E A D (2) A E D C B (2) A E C B D (2) A D E C B (2) E D A B C (1) D C B A E (1) D B E C A (1) D B C A E (1) C A E B D (1) B C E D A (1) B C D E A (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 22 12 -26 20 B -22 0 -12 -24 -18 C -12 12 0 -16 -12 D 26 24 16 0 26 E -20 18 12 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 -26 20 B -22 0 -12 -24 -18 C -12 12 0 -16 -12 D 26 24 16 0 26 E -20 18 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 C=27 E=12 B=8 A=8 so B is eliminated. Round 2 votes counts: D=47 C=31 E=14 A=8 so A is eliminated. Round 3 votes counts: D=49 C=32 E=19 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:246 A:214 E:192 C:186 B:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 12 -26 20 B -22 0 -12 -24 -18 C -12 12 0 -16 -12 D 26 24 16 0 26 E -20 18 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 -26 20 B -22 0 -12 -24 -18 C -12 12 0 -16 -12 D 26 24 16 0 26 E -20 18 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 -26 20 B -22 0 -12 -24 -18 C -12 12 0 -16 -12 D 26 24 16 0 26 E -20 18 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2067: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) E C D A B (7) D E B A C (6) E D B A C (5) B A D C E (5) A B C D E (5) E D C B A (4) E D B C A (4) E B D A C (4) E C B A D (3) E C A D B (3) E C A B D (3) D B E A C (3) C A D B E (3) C A B E D (3) B A C D E (3) A B D C E (3) E D C A B (2) D E A B C (2) D B A C E (2) C A E D B (2) B E D A C (2) B D A E C (2) E C B D A (1) E B D C A (1) E B C A D (1) D E C A B (1) D E A C B (1) D B A E C (1) D A B C E (1) C E B A D (1) C E A D B (1) C A E B D (1) B D E A C (1) B A E C D (1) B A C E D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 0 0 -12 B 2 0 8 2 -6 C 0 -8 0 -2 -16 D 0 -2 2 0 0 E 12 6 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.517893 E: 0.482107 Sum of squares = 0.500640278709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.517893 E: 1.000000 A B C D E A 0 -2 0 0 -12 B 2 0 8 2 -6 C 0 -8 0 -2 -16 D 0 -2 2 0 0 E 12 6 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=20 D=17 B=15 A=10 so A is eliminated. Round 2 votes counts: E=38 B=23 C=21 D=18 so D is eliminated. Round 3 votes counts: E=48 B=31 C=21 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:203 D:200 A:193 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 0 -12 B 2 0 8 2 -6 C 0 -8 0 -2 -16 D 0 -2 2 0 0 E 12 6 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 -12 B 2 0 8 2 -6 C 0 -8 0 -2 -16 D 0 -2 2 0 0 E 12 6 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 -12 B 2 0 8 2 -6 C 0 -8 0 -2 -16 D 0 -2 2 0 0 E 12 6 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2068: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) D B A C E (7) D B A E C (6) B D C A E (5) B C D E A (5) A E C B D (5) E C A B D (4) E A C D B (4) D A B E C (4) C B E A D (4) D A E B C (3) C B E D A (3) B C D A E (3) B C A D E (3) A E D C B (3) A E D B C (3) E C D A B (2) D E A B C (2) D B C E A (2) C E B A D (2) C E A B D (2) C B D E A (2) B D C E A (2) B C A E D (2) B A C D E (2) A E C D B (2) E D A C B (1) D E C A B (1) D E B A C (1) D C E B A (1) D B E C A (1) C B A E D (1) B D A C E (1) A D E B C (1) A D B E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 10 -4 8 B 6 0 10 -4 10 C -10 -10 0 -4 -4 D 4 4 4 0 6 E -8 -10 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 -4 8 B 6 0 10 -4 10 C -10 -10 0 -4 -4 D 4 4 4 0 6 E -8 -10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=23 E=18 A=17 C=14 so C is eliminated. Round 2 votes counts: B=33 D=28 E=22 A=17 so A is eliminated. Round 3 votes counts: E=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:209 A:204 E:190 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 10 -4 8 B 6 0 10 -4 10 C -10 -10 0 -4 -4 D 4 4 4 0 6 E -8 -10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 -4 8 B 6 0 10 -4 10 C -10 -10 0 -4 -4 D 4 4 4 0 6 E -8 -10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 -4 8 B 6 0 10 -4 10 C -10 -10 0 -4 -4 D 4 4 4 0 6 E -8 -10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2069: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (7) A E C B D (7) B D E A C (6) D C E B A (4) D B C E A (4) C E A D B (4) C A E B D (4) A B C E D (4) E D A C B (3) D C B E A (3) B C D A E (3) A E B D C (3) E A D C B (2) E A C D B (2) D E B C A (2) D C E A B (2) D B E C A (2) D B E A C (2) C E D A B (2) C B D A E (2) C A E D B (2) B D C E A (2) B D A E C (2) B D A C E (2) B C A D E (2) B A E D C (2) B A C D E (2) A C E D B (2) A C B E D (2) A B E C D (2) E D A B C (1) E C A D B (1) D E B A C (1) C D E A B (1) C D B E A (1) C B A E D (1) C B A D E (1) B D E C A (1) B D C A E (1) B A C E D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 6 10 6 8 B -6 0 -8 4 -2 C -10 8 0 8 6 D -6 -4 -8 0 -8 E -8 2 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 6 8 B -6 0 -8 4 -2 C -10 8 0 8 6 D -6 -4 -8 0 -8 E -8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 D=20 C=18 E=9 so E is eliminated. Round 2 votes counts: A=33 D=24 B=24 C=19 so C is eliminated. Round 3 votes counts: A=44 D=28 B=28 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:206 E:198 B:194 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 6 8 B -6 0 -8 4 -2 C -10 8 0 8 6 D -6 -4 -8 0 -8 E -8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 6 8 B -6 0 -8 4 -2 C -10 8 0 8 6 D -6 -4 -8 0 -8 E -8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 6 8 B -6 0 -8 4 -2 C -10 8 0 8 6 D -6 -4 -8 0 -8 E -8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2070: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) C D A E B (7) A B C E D (7) B E A D C (6) B A E D C (6) D C A E B (5) E C D B A (4) D C E A B (4) A C D B E (4) A B D C E (4) B A E C D (3) A B E C D (3) E D C B A (2) E B C D A (2) E B C A D (2) E B A C D (2) D E B C A (2) D C E B A (2) B E D A C (2) B A D E C (2) A D C B E (2) A B C D E (2) E D B C A (1) E C D A B (1) E C B D A (1) E C B A D (1) E B D C A (1) E B D A C (1) D C A B E (1) D B C E A (1) D A B C E (1) C E D A B (1) C A E D B (1) C A E B D (1) C A D E B (1) C A D B E (1) B D E A C (1) A D B C E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 16 -2 2 8 B -16 0 0 -4 -4 C 2 0 0 10 14 D -2 4 -10 0 4 E -8 4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.065067 C: 0.934933 D: 0.000000 E: 0.000000 Sum of squares = 0.87833276762 Cumulative probabilities = A: 0.000000 B: 0.065067 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 2 8 B -16 0 0 -4 -4 C 2 0 0 10 14 D -2 4 -10 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469163207 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=21 B=20 E=18 D=16 so D is eliminated. Round 2 votes counts: C=33 A=26 B=21 E=20 so E is eliminated. Round 3 votes counts: C=42 B=32 A=26 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 A:212 D:198 E:189 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -2 2 8 B -16 0 0 -4 -4 C 2 0 0 10 14 D -2 4 -10 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469163207 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 2 8 B -16 0 0 -4 -4 C 2 0 0 10 14 D -2 4 -10 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469163207 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 2 8 B -16 0 0 -4 -4 C 2 0 0 10 14 D -2 4 -10 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469163207 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2071: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) B A E C D (8) E A B D C (7) D C E A B (6) C D B A E (6) A B E C D (6) C D A B E (5) E D A B C (4) D C E B A (4) B E A C D (4) B A C E D (4) A B C D E (4) E B A C D (3) D E C A B (3) D C A B E (3) C A B D E (3) B C A D E (2) A E B D C (2) E D C B A (1) E D C A B (1) E D A C B (1) E B C D A (1) E B C A D (1) D E C B A (1) D C A E B (1) C D E B A (1) C D B E A (1) C B D E A (1) B C A E D (1) A E D B C (1) A D C B E (1) A C D B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 16 20 0 B -2 0 20 18 4 C -16 -20 0 4 -10 D -20 -18 -4 0 -12 E 0 -4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.692659 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.307341 Sum of squares = 0.574234940494 Cumulative probabilities = A: 0.692659 B: 0.692659 C: 0.692659 D: 0.692659 E: 1.000000 A B C D E A 0 2 16 20 0 B -2 0 20 18 4 C -16 -20 0 4 -10 D -20 -18 -4 0 -12 E 0 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556129258 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=19 D=18 C=17 A=17 so C is eliminated. Round 2 votes counts: D=31 E=29 B=20 A=20 so B is eliminated. Round 3 votes counts: A=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:219 E:209 C:179 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 20 0 B -2 0 20 18 4 C -16 -20 0 4 -10 D -20 -18 -4 0 -12 E 0 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556129258 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 20 0 B -2 0 20 18 4 C -16 -20 0 4 -10 D -20 -18 -4 0 -12 E 0 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556129258 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 20 0 B -2 0 20 18 4 C -16 -20 0 4 -10 D -20 -18 -4 0 -12 E 0 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333332 Sum of squares = 0.555556129258 Cumulative probabilities = A: 0.666668 B: 0.666668 C: 0.666668 D: 0.666668 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2072: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) B D E C A (9) D E A C B (7) B C E A D (7) B E C D A (6) B C A E D (5) A D C E B (4) A C D E B (4) D A B E C (3) C A E B D (3) A D E C B (3) A C E D B (3) E D C A B (2) D B E A C (2) C E B A D (2) C B E A D (2) B D E A C (2) B D A C E (2) A C E B D (2) E C D A B (1) E C B D A (1) E C A D B (1) D E B C A (1) D E B A C (1) D E A B C (1) D B E C A (1) D A E B C (1) C E A B D (1) C A B E D (1) B E D C A (1) B D A E C (1) B C E D A (1) B A D C E (1) B A C E D (1) B A C D E (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 10 -8 2 B -6 0 -4 2 2 C -10 4 0 -10 -10 D 8 -2 10 0 18 E -2 -2 10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999915 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 -8 2 B -6 0 -4 2 2 C -10 4 0 -10 -10 D 8 -2 10 0 18 E -2 -2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999782 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=28 A=21 C=9 E=5 so E is eliminated. Round 2 votes counts: B=37 D=30 A=21 C=12 so C is eliminated. Round 3 votes counts: B=42 D=31 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:217 A:205 B:197 E:194 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -8 2 B -6 0 -4 2 2 C -10 4 0 -10 -10 D 8 -2 10 0 18 E -2 -2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999782 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -8 2 B -6 0 -4 2 2 C -10 4 0 -10 -10 D 8 -2 10 0 18 E -2 -2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999782 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -8 2 B -6 0 -4 2 2 C -10 4 0 -10 -10 D 8 -2 10 0 18 E -2 -2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999782 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2073: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) D C A B E (7) A E B C D (7) D B E C A (6) C D A E B (6) C A D E B (5) A E C B D (5) D C B E A (4) B E D A C (4) B E A D C (4) D B A E C (3) C A E D B (3) B A E D C (3) A C D E B (3) E B C D A (2) C E A B D (2) B D E C A (2) B D E A C (2) A D B E C (2) A D B C E (2) A C E B D (2) E C B A D (1) E B A D C (1) E A C B D (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D E A B (1) C A E B D (1) B E C D A (1) B D A E C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 2 10 6 12 B -2 0 8 -4 -4 C -10 -8 0 2 -10 D -6 4 -2 0 6 E -12 4 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 6 12 B -2 0 8 -4 -4 C -10 -8 0 2 -10 D -6 4 -2 0 6 E -12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=23 C=20 B=17 E=14 so E is eliminated. Round 2 votes counts: B=29 D=26 A=24 C=21 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:201 B:199 E:198 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 6 12 B -2 0 8 -4 -4 C -10 -8 0 2 -10 D -6 4 -2 0 6 E -12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 6 12 B -2 0 8 -4 -4 C -10 -8 0 2 -10 D -6 4 -2 0 6 E -12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 6 12 B -2 0 8 -4 -4 C -10 -8 0 2 -10 D -6 4 -2 0 6 E -12 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2074: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) C D B E A (8) A D C E B (8) C D B A E (7) B C D E A (7) E A B C D (6) D C B A E (6) D C A B E (6) B E C D A (6) A E D C B (6) A E B D C (5) A D C B E (4) E B C D A (2) E A D C B (2) D C E A B (2) D C A E B (2) B E A C D (2) B C E D A (2) A E D B C (2) E B C A D (1) D A C E B (1) D A C B E (1) B E C A D (1) B C D A E (1) B C A D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -4 -2 2 B 8 0 -6 -10 6 C 4 6 0 10 12 D 2 10 -10 0 8 E -2 -6 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -2 2 B 8 0 -6 -10 6 C 4 6 0 10 12 D 2 10 -10 0 8 E -2 -6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=21 B=20 D=18 C=15 so C is eliminated. Round 2 votes counts: D=33 A=26 E=21 B=20 so B is eliminated. Round 3 votes counts: D=41 E=32 A=27 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:205 B:199 A:194 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -4 -2 2 B 8 0 -6 -10 6 C 4 6 0 10 12 D 2 10 -10 0 8 E -2 -6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -2 2 B 8 0 -6 -10 6 C 4 6 0 10 12 D 2 10 -10 0 8 E -2 -6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -2 2 B 8 0 -6 -10 6 C 4 6 0 10 12 D 2 10 -10 0 8 E -2 -6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2075: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) D E A B C (6) C E B A D (5) E D C A B (4) E D A C B (4) D B E A C (4) B C A E D (4) B C A D E (4) E D C B A (3) E C D A B (3) E C B D A (3) D A B E C (3) C B E A D (3) C A B E D (3) A D B C E (3) A C B E D (3) A B D C E (3) E C D B A (2) E C B A D (2) E A D C B (2) E A C D B (2) D E B C A (2) D E B A C (2) D E A C B (2) D A E B C (2) C A E B D (2) B C D A E (2) B A D C E (2) E C A D B (1) E C A B D (1) D B A C E (1) C E A B D (1) C B E D A (1) B D C A E (1) B D A C E (1) B C D E A (1) A E C D B (1) A D E B C (1) A D B E C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -10 6 -8 B 0 0 -14 0 -6 C 10 14 0 6 -2 D -6 0 -6 0 -18 E 8 6 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -10 6 -8 B 0 0 -14 0 -6 C 10 14 0 6 -2 D -6 0 -6 0 -18 E 8 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=22 C=22 B=15 A=14 so A is eliminated. Round 2 votes counts: E=28 D=27 C=27 B=18 so B is eliminated. Round 3 votes counts: C=38 D=34 E=28 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:217 C:214 A:194 B:190 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 6 -8 B 0 0 -14 0 -6 C 10 14 0 6 -2 D -6 0 -6 0 -18 E 8 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 6 -8 B 0 0 -14 0 -6 C 10 14 0 6 -2 D -6 0 -6 0 -18 E 8 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 6 -8 B 0 0 -14 0 -6 C 10 14 0 6 -2 D -6 0 -6 0 -18 E 8 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2076: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (7) E D B C A (6) E D A B C (6) B C A E D (6) D C B E A (5) A C B E D (5) A B C E D (5) E A D B C (4) D E A C B (4) C B D A E (4) C B A D E (4) B C D E A (4) A E D C B (4) A E D B C (4) E B C D A (3) D E C B A (3) A E B C D (3) A C D B E (3) E B D C A (2) D C E B A (2) C D B E A (2) A D C E B (2) E D A C B (1) E B D A C (1) E A D C B (1) D E C A B (1) D E B C A (1) D C B A E (1) D B E C A (1) C B D E A (1) B C E A D (1) B A E C D (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 6 4 2 B -2 0 0 -2 4 C -6 0 0 2 6 D -4 2 -2 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 4 2 B -2 0 0 -2 4 C -6 0 0 2 6 D -4 2 -2 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=24 D=18 B=13 C=11 so C is eliminated. Round 2 votes counts: A=34 E=24 B=22 D=20 so D is eliminated. Round 3 votes counts: E=35 A=34 B=31 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 C:201 B:200 E:199 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 2 B -2 0 0 -2 4 C -6 0 0 2 6 D -4 2 -2 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 2 B -2 0 0 -2 4 C -6 0 0 2 6 D -4 2 -2 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 2 B -2 0 0 -2 4 C -6 0 0 2 6 D -4 2 -2 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2077: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B A E C D (8) B A E D C (7) C D E A B (6) C B A E D (6) E A B C D (4) D E A B C (4) C D B E A (4) A E B D C (4) D E C A B (3) D C B A E (3) C E A D B (3) B C A E D (3) B A C E D (3) E D C A B (2) E A D B C (2) E A C D B (2) E A B D C (2) D E A C B (2) C E A B D (2) C D B A E (2) C B D A E (2) B D A E C (2) B D A C E (2) A E D B C (2) D C B E A (1) D B C A E (1) D A B E C (1) C E D A B (1) C D E B A (1) C B E A D (1) C B A D E (1) B C D A E (1) B A C D E (1) A E B C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -2 8 4 B -2 0 0 4 2 C 2 0 0 6 4 D -8 -4 -6 0 -10 E -4 -2 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.295726 C: 0.704274 D: 0.000000 E: 0.000000 Sum of squares = 0.583455419275 Cumulative probabilities = A: 0.000000 B: 0.295726 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 8 4 B -2 0 0 4 2 C 2 0 0 6 4 D -8 -4 -6 0 -10 E -4 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499758 C: 0.500242 D: 0.000000 E: 0.000000 Sum of squares = 0.500000116645 Cumulative probabilities = A: 0.000000 B: 0.499758 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=27 D=23 E=12 A=9 so A is eliminated. Round 2 votes counts: C=29 B=29 D=23 E=19 so E is eliminated. Round 3 votes counts: B=40 C=31 D=29 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:206 C:206 B:202 E:200 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 8 4 B -2 0 0 4 2 C 2 0 0 6 4 D -8 -4 -6 0 -10 E -4 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499758 C: 0.500242 D: 0.000000 E: 0.000000 Sum of squares = 0.500000116645 Cumulative probabilities = A: 0.000000 B: 0.499758 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 4 B -2 0 0 4 2 C 2 0 0 6 4 D -8 -4 -6 0 -10 E -4 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499758 C: 0.500242 D: 0.000000 E: 0.000000 Sum of squares = 0.500000116645 Cumulative probabilities = A: 0.000000 B: 0.499758 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 4 B -2 0 0 4 2 C 2 0 0 6 4 D -8 -4 -6 0 -10 E -4 -2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499758 C: 0.500242 D: 0.000000 E: 0.000000 Sum of squares = 0.500000116645 Cumulative probabilities = A: 0.000000 B: 0.499758 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2078: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) A B D C E (8) D B A E C (7) B D E C A (7) A C E D B (7) E C D B A (6) A D B C E (6) E C B D A (5) C E A D B (5) B D A E C (5) A C E B D (5) D E C B A (3) E D C B A (2) E C B A D (2) D E C A B (2) D B E A C (2) C E D A B (2) C E A B D (2) B E C D A (2) A B C D E (2) E C D A B (1) E B C D A (1) D E B C A (1) D A B E C (1) B E D C A (1) B D E A C (1) A D C E B (1) A C D E B (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -2 -16 -8 B 8 0 8 -12 8 C 2 -8 0 -12 -16 D 16 12 12 0 14 E 8 -8 16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -16 -8 B 8 0 8 -12 8 C 2 -8 0 -12 -16 D 16 12 12 0 14 E 8 -8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=25 E=17 B=16 C=9 so C is eliminated. Round 2 votes counts: A=33 E=26 D=25 B=16 so B is eliminated. Round 3 votes counts: D=38 A=33 E=29 so E is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 B:206 E:201 A:183 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -16 -8 B 8 0 8 -12 8 C 2 -8 0 -12 -16 D 16 12 12 0 14 E 8 -8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -16 -8 B 8 0 8 -12 8 C 2 -8 0 -12 -16 D 16 12 12 0 14 E 8 -8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -16 -8 B 8 0 8 -12 8 C 2 -8 0 -12 -16 D 16 12 12 0 14 E 8 -8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2079: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) D B C A E (8) A D E B C (8) E C B A D (7) D A B C E (7) C B E D A (6) A D B C E (6) E B C D A (5) A E C B D (5) D B C E A (4) B C D E A (4) D E B C A (3) B C E D A (3) A D B E C (3) E C A B D (2) E A C B D (2) A E D C B (2) A C E B D (2) A C B D E (2) E A D B C (1) D E A B C (1) D C B A E (1) D B E C A (1) D B A C E (1) C E B A D (1) C B D E A (1) B D C E A (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -14 -16 -14 -6 B 14 0 12 4 -2 C 16 -12 0 2 -2 D 14 -4 -2 0 6 E 6 2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.38888888892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 A B C D E A 0 -14 -16 -14 -6 B 14 0 12 4 -2 C 16 -12 0 2 -2 D 14 -4 -2 0 6 E 6 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 D=26 C=8 B=8 so C is eliminated. Round 2 votes counts: A=30 E=29 D=26 B=15 so B is eliminated. Round 3 votes counts: E=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:214 D:207 C:202 E:202 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -14 -6 B 14 0 12 4 -2 C 16 -12 0 2 -2 D 14 -4 -2 0 6 E 6 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -14 -6 B 14 0 12 4 -2 C 16 -12 0 2 -2 D 14 -4 -2 0 6 E 6 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -14 -6 B 14 0 12 4 -2 C 16 -12 0 2 -2 D 14 -4 -2 0 6 E 6 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2080: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (14) B E D A C (12) C A D E B (11) E D A C B (10) B C A D E (8) E C D A B (4) E B D A C (4) C A B D E (4) C A D B E (3) B E C D A (3) E D B A C (2) D A C E B (2) C E D A B (2) C B A D E (2) B C E A D (2) B C A E D (2) B A D C E (2) E D A B C (1) D A E C B (1) B E D C A (1) B E A D C (1) B E A C D (1) B D E A C (1) B A E D C (1) B A C E D (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 16 14 12 B 14 0 16 22 26 C -16 -16 0 18 14 D -14 -22 -18 0 6 E -12 -26 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 16 14 12 B 14 0 16 22 26 C -16 -16 0 18 14 D -14 -22 -18 0 6 E -12 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=49 C=22 E=21 A=5 D=3 so D is eliminated. Round 2 votes counts: B=49 C=22 E=21 A=8 so A is eliminated. Round 3 votes counts: B=51 C=27 E=22 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:239 A:214 C:200 D:176 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 16 14 12 B 14 0 16 22 26 C -16 -16 0 18 14 D -14 -22 -18 0 6 E -12 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 16 14 12 B 14 0 16 22 26 C -16 -16 0 18 14 D -14 -22 -18 0 6 E -12 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 16 14 12 B 14 0 16 22 26 C -16 -16 0 18 14 D -14 -22 -18 0 6 E -12 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2081: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) E B C A D (7) C B A D E (7) E D A B C (6) E D C A B (5) E B A D C (5) C D A B E (5) A D B C E (5) B C A D E (4) E C B D A (3) D C A B E (3) C B D A E (3) E C D B A (2) D C A E B (2) D A E C B (2) D A B E C (2) C E D B A (2) C D E A B (2) C D A E B (2) C B E A D (2) B E C A D (2) B C E A D (2) A D B E C (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D A B (1) E C B A D (1) E B C D A (1) E B A C D (1) D E C A B (1) D E A C B (1) D E A B C (1) D C E A B (1) D A E B C (1) D A C E B (1) C D B A E (1) B E A C D (1) Total count = 100 A B C D E A 0 12 -18 -14 4 B -12 0 -12 -14 2 C 18 12 0 -2 4 D 14 14 2 0 16 E -4 -2 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -18 -14 4 B -12 0 -12 -14 2 C 18 12 0 -2 4 D 14 14 2 0 16 E -4 -2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996517 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=24 D=23 A=11 B=9 so B is eliminated. Round 2 votes counts: E=36 C=30 D=23 A=11 so A is eliminated. Round 3 votes counts: E=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:216 A:192 E:187 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -18 -14 4 B -12 0 -12 -14 2 C 18 12 0 -2 4 D 14 14 2 0 16 E -4 -2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996517 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -18 -14 4 B -12 0 -12 -14 2 C 18 12 0 -2 4 D 14 14 2 0 16 E -4 -2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996517 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -18 -14 4 B -12 0 -12 -14 2 C 18 12 0 -2 4 D 14 14 2 0 16 E -4 -2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996517 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2082: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) C B D A E (9) B C E A D (8) B E C A D (6) B C D A E (6) E B C A D (5) B C E D A (5) A D E C B (5) D A B C E (4) B C D E A (4) A E D C B (4) A D E B C (4) E A D B C (3) E A C B D (3) D B C A E (3) C B D E A (3) E B A C D (2) E A B C D (2) D C B A E (2) D A C B E (2) E C B A D (1) E A B D C (1) D C A E B (1) D C A B E (1) D A C E B (1) C D B A E (1) C B E D A (1) C B E A D (1) B E A C D (1) B D C A E (1) A E D B C (1) Total count = 100 A B C D E A 0 -18 -16 12 -10 B 18 0 12 18 16 C 16 -12 0 16 6 D -12 -18 -16 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 12 -10 B 18 0 12 18 16 C 16 -12 0 16 6 D -12 -18 -16 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=26 C=15 D=14 A=14 so D is eliminated. Round 2 votes counts: B=34 E=26 A=21 C=19 so C is eliminated. Round 3 votes counts: B=51 E=26 A=23 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:213 E:197 A:184 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 12 -10 B 18 0 12 18 16 C 16 -12 0 16 6 D -12 -18 -16 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 12 -10 B 18 0 12 18 16 C 16 -12 0 16 6 D -12 -18 -16 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 12 -10 B 18 0 12 18 16 C 16 -12 0 16 6 D -12 -18 -16 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2083: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) C B D A E (10) B C D A E (8) A E D C B (7) E A D B C (5) E A B C D (5) D C A B E (4) D B C E A (4) C B A D E (4) A E C D B (4) A E C B D (4) D C B E A (3) D C B A E (3) B D C E A (3) E B A C D (2) E A D C B (2) E D A B C (1) E B D C A (1) E B D A C (1) E B A D C (1) D E B C A (1) D E B A C (1) D B E C A (1) D A C E B (1) D A C B E (1) C D A B E (1) C A B D E (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C A D E (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -22 -22 -18 4 B 22 0 4 18 22 C 22 -4 0 16 18 D 18 -18 -16 0 20 E -4 -22 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -22 -18 4 B 22 0 4 18 22 C 22 -4 0 16 18 D 18 -18 -16 0 20 E -4 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=19 E=18 A=18 C=16 so C is eliminated. Round 2 votes counts: B=43 D=20 A=19 E=18 so E is eliminated. Round 3 votes counts: B=48 A=31 D=21 so D is eliminated. Round 4 votes counts: B=61 A=39 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:233 C:226 D:202 A:171 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -22 -18 4 B 22 0 4 18 22 C 22 -4 0 16 18 D 18 -18 -16 0 20 E -4 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -22 -18 4 B 22 0 4 18 22 C 22 -4 0 16 18 D 18 -18 -16 0 20 E -4 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -22 -18 4 B 22 0 4 18 22 C 22 -4 0 16 18 D 18 -18 -16 0 20 E -4 -22 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2084: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (12) D B E C A (6) C A E D B (5) D E B C A (4) D C E B A (4) C D E A B (4) C D A E B (4) B E D A C (4) B D E A C (4) A B E C D (4) D B E A C (3) C D E B A (3) C D A B E (3) C A E B D (3) B A E D C (3) A C B D E (3) E B D A C (2) D B C E A (2) D B A E C (2) C E A D B (2) C A D E B (2) B D A E C (2) A B E D C (2) A B C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E C A B D (1) E B A D C (1) E A B C D (1) D E C B A (1) D E B A C (1) D C B E A (1) C E D A B (1) C A D B E (1) C A B E D (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -2 -10 2 B -6 0 -8 -6 10 C 2 8 0 10 8 D 10 6 -10 0 0 E -2 -10 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -10 2 B -6 0 -8 -6 10 C 2 8 0 10 8 D 10 6 -10 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=25 D=24 B=13 E=9 so E is eliminated. Round 2 votes counts: C=31 D=27 A=26 B=16 so B is eliminated. Round 3 votes counts: D=39 C=31 A=30 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:203 A:198 B:195 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -10 2 B -6 0 -8 -6 10 C 2 8 0 10 8 D 10 6 -10 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -10 2 B -6 0 -8 -6 10 C 2 8 0 10 8 D 10 6 -10 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -10 2 B -6 0 -8 -6 10 C 2 8 0 10 8 D 10 6 -10 0 0 E -2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2085: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) A E C B D (10) D B E C A (6) C E B A D (6) D C B A E (5) D A B E C (5) C B E A D (5) D A E B C (4) A D E C B (4) E C B A D (3) E A C B D (3) B C E D A (3) A C E B D (3) D C A B E (2) D A E C B (2) D A C E B (2) D A C B E (2) C B D E A (2) B D C E A (2) B C E A D (2) A E D B C (2) E C A B D (1) E A B D C (1) D B C A E (1) D A B C E (1) C E A B D (1) C A E B D (1) C A D E B (1) B E C D A (1) B E C A D (1) A E D C B (1) A E B C D (1) A D E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -12 -2 0 B 0 0 -12 -8 2 C 12 12 0 -10 8 D 2 8 10 0 8 E 0 -2 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -2 0 B 0 0 -12 -8 2 C 12 12 0 -10 8 D 2 8 10 0 8 E 0 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 A=24 C=16 B=9 E=8 so E is eliminated. Round 2 votes counts: D=43 A=28 C=20 B=9 so B is eliminated. Round 3 votes counts: D=45 A=28 C=27 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:211 A:193 B:191 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -12 -2 0 B 0 0 -12 -8 2 C 12 12 0 -10 8 D 2 8 10 0 8 E 0 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -2 0 B 0 0 -12 -8 2 C 12 12 0 -10 8 D 2 8 10 0 8 E 0 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -2 0 B 0 0 -12 -8 2 C 12 12 0 -10 8 D 2 8 10 0 8 E 0 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2086: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) A D C E B (7) C B E D A (5) C A E D B (5) C E A B D (4) B C E D A (4) A C D E B (4) E C B A D (3) E B A D C (3) D B A C E (3) D A E B C (3) D A B E C (3) C D B A E (3) C B D E A (3) C A D E B (3) B D E A C (3) B C D E A (3) E A B D C (2) D C A B E (2) D B A E C (2) C D A B E (2) B E D A C (2) B E C A D (2) B D C E A (2) A E C D B (2) A D E C B (2) A D E B C (2) E C A D B (1) E B A C D (1) E A C D B (1) E A B C D (1) D A C E B (1) D A C B E (1) C E B A D (1) C D A E B (1) C B D A E (1) B E D C A (1) B D C A E (1) B D A E C (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 -8 6 2 B 2 0 -6 0 -12 C 8 6 0 14 12 D -6 0 -14 0 6 E -2 12 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 6 2 B 2 0 -6 0 -12 C 8 6 0 14 12 D -6 0 -14 0 6 E -2 12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=19 B=19 A=19 D=15 so D is eliminated. Round 2 votes counts: C=30 A=27 B=24 E=19 so E is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:199 E:196 D:193 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 6 2 B 2 0 -6 0 -12 C 8 6 0 14 12 D -6 0 -14 0 6 E -2 12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 6 2 B 2 0 -6 0 -12 C 8 6 0 14 12 D -6 0 -14 0 6 E -2 12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 6 2 B 2 0 -6 0 -12 C 8 6 0 14 12 D -6 0 -14 0 6 E -2 12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2087: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) C E A B D (8) E D C B A (7) D E B A C (6) D B A E C (6) B A D C E (6) A B C D E (5) E D C A B (4) C A E B D (4) A C B D E (4) A B D C E (4) E C D B A (3) E C D A B (3) D E A B C (3) D B E A C (3) B D A E C (3) B D A C E (3) B A C D E (3) A C B E D (3) E C A D B (2) D A B E C (2) C E B A D (2) C E A D B (2) E C B D A (1) D E B C A (1) D A E B C (1) C A B D E (1) Total count = 100 A B C D E A 0 12 4 8 10 B -12 0 -8 14 6 C -4 8 0 2 10 D -8 -14 -2 0 2 E -10 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 8 10 B -12 0 -8 14 6 C -4 8 0 2 10 D -8 -14 -2 0 2 E -10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 E=20 A=16 B=15 so B is eliminated. Round 2 votes counts: D=28 C=27 A=25 E=20 so E is eliminated. Round 3 votes counts: D=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:217 C:208 B:200 D:189 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 8 10 B -12 0 -8 14 6 C -4 8 0 2 10 D -8 -14 -2 0 2 E -10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 8 10 B -12 0 -8 14 6 C -4 8 0 2 10 D -8 -14 -2 0 2 E -10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 8 10 B -12 0 -8 14 6 C -4 8 0 2 10 D -8 -14 -2 0 2 E -10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2088: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (14) B A C E D (12) D E C B A (9) A B C E D (6) D C E A B (5) C E D A B (5) A C E D B (5) A D B C E (4) E C B D A (3) D E B C A (3) C E B A D (3) A B D C E (3) E D C B A (2) D B A E C (2) D A C E B (2) B E C D A (2) B C E A D (2) B A E C D (2) A D C E B (2) A B C D E (2) E D C A B (1) E C D B A (1) E C B A D (1) E B C D A (1) D B E C A (1) D B E A C (1) D A B E C (1) B E D C A (1) B E C A D (1) B D E C A (1) B A D E C (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -12 -10 -14 B -2 0 -8 -16 -14 C 12 8 0 -6 4 D 10 16 6 0 2 E 14 14 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -10 -14 B -2 0 -8 -16 -14 C 12 8 0 -6 4 D 10 16 6 0 2 E 14 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=23 B=22 E=9 C=8 so C is eliminated. Round 2 votes counts: D=38 A=23 B=22 E=17 so E is eliminated. Round 3 votes counts: D=47 B=30 A=23 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:211 C:209 A:183 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -10 -14 B -2 0 -8 -16 -14 C 12 8 0 -6 4 D 10 16 6 0 2 E 14 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -10 -14 B -2 0 -8 -16 -14 C 12 8 0 -6 4 D 10 16 6 0 2 E 14 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -10 -14 B -2 0 -8 -16 -14 C 12 8 0 -6 4 D 10 16 6 0 2 E 14 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2089: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) D E A B C (7) D E B C A (6) D E B A C (6) D E A C B (6) B C A E D (6) A B C D E (6) C E B A D (5) A C B E D (5) A C B D E (5) E C D B A (3) D B A E C (3) B A C E D (3) B A C D E (3) C E B D A (2) C B A E D (2) C A E D B (2) B A D C E (2) A D B C E (2) A C D E B (2) A B D C E (2) A B C E D (2) E B C D A (1) D E C A B (1) D A E B C (1) C E A D B (1) C E A B D (1) C B E A D (1) C A E B D (1) C A B E D (1) B D E C A (1) B C E D A (1) B C E A D (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 14 10 0 B 6 0 8 2 -8 C -14 -8 0 8 12 D -10 -2 -8 0 10 E 0 8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.568017 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.431983 Sum of squares = 0.509252735004 Cumulative probabilities = A: 0.568017 B: 0.568017 C: 0.568017 D: 0.568017 E: 1.000000 A B C D E A 0 -6 14 10 0 B 6 0 8 2 -8 C -14 -8 0 8 12 D -10 -2 -8 0 10 E 0 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500514 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499486 Sum of squares = 0.500000528077 Cumulative probabilities = A: 0.500514 B: 0.500514 C: 0.500514 D: 0.500514 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=26 B=17 C=16 E=11 so E is eliminated. Round 2 votes counts: D=37 A=26 C=19 B=18 so B is eliminated. Round 3 votes counts: D=38 A=34 C=28 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:204 C:199 D:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -6 14 10 0 B 6 0 8 2 -8 C -14 -8 0 8 12 D -10 -2 -8 0 10 E 0 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500514 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499486 Sum of squares = 0.500000528077 Cumulative probabilities = A: 0.500514 B: 0.500514 C: 0.500514 D: 0.500514 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 10 0 B 6 0 8 2 -8 C -14 -8 0 8 12 D -10 -2 -8 0 10 E 0 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500514 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499486 Sum of squares = 0.500000528077 Cumulative probabilities = A: 0.500514 B: 0.500514 C: 0.500514 D: 0.500514 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 10 0 B 6 0 8 2 -8 C -14 -8 0 8 12 D -10 -2 -8 0 10 E 0 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500514 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499486 Sum of squares = 0.500000528077 Cumulative probabilities = A: 0.500514 B: 0.500514 C: 0.500514 D: 0.500514 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2090: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D B A C E (10) E A B C D (5) D B C A E (5) B A D E C (5) E A C B D (4) A B E D C (4) A B D E C (4) E C A D B (3) E A B D C (3) D C E B A (3) D C B E A (3) D B A E C (3) C E D B A (3) C E A D B (3) C D E B A (3) B D A C E (3) B A D C E (3) E C D A B (2) D C B A E (2) C E D A B (2) C E A B D (2) C D B E A (2) C D B A E (2) C A B E D (2) E D C B A (1) D E C B A (1) D B E A C (1) D B C E A (1) C A E B D (1) B D A E C (1) A E C B D (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -2 4 -4 B 4 0 0 0 4 C 2 0 0 -8 2 D -4 0 8 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.670052 C: 0.000000 D: 0.329948 E: 0.000000 Sum of squares = 0.557835046705 Cumulative probabilities = A: 0.000000 B: 0.670052 C: 0.670052 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 4 -4 B 4 0 0 0 4 C 2 0 0 -8 2 D -4 0 8 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.000000 D: 0.499847 E: 0.000000 Sum of squares = 0.500000046754 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 0.500153 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 C=20 B=12 A=11 so A is eliminated. Round 2 votes counts: E=30 D=29 B=21 C=20 so C is eliminated. Round 3 votes counts: E=41 D=36 B=23 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:204 D:204 C:198 A:197 E:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 4 -4 B 4 0 0 0 4 C 2 0 0 -8 2 D -4 0 8 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.000000 D: 0.499847 E: 0.000000 Sum of squares = 0.500000046754 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 0.500153 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 4 -4 B 4 0 0 0 4 C 2 0 0 -8 2 D -4 0 8 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.000000 D: 0.499847 E: 0.000000 Sum of squares = 0.500000046754 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 0.500153 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 4 -4 B 4 0 0 0 4 C 2 0 0 -8 2 D -4 0 8 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.000000 D: 0.499847 E: 0.000000 Sum of squares = 0.500000046754 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 0.500153 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2091: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) B E C D A (8) D A B C E (6) B C E A D (5) E C A D B (4) E B D A C (4) D A C B E (4) B C E D A (4) A D C E B (4) E C B A D (3) E B C D A (3) B D E A C (3) B C D A E (3) E D B A C (2) E A D C B (2) D A E C B (2) D A B E C (2) C E B A D (2) C B A D E (2) B E D C A (2) B E C A D (2) B D A E C (2) B D A C E (2) B C A D E (2) A E D C B (2) A D C B E (2) E D A C B (1) E D A B C (1) D B A E C (1) D A E B C (1) C E A D B (1) C E A B D (1) C B E A D (1) C B A E D (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -28 -16 -10 -20 B 28 0 30 26 16 C 16 -30 0 6 -10 D 10 -26 -6 0 -14 E 20 -16 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -16 -10 -20 B 28 0 30 26 16 C 16 -30 0 6 -10 D 10 -26 -6 0 -14 E 20 -16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=28 D=16 C=10 A=9 so A is eliminated. Round 2 votes counts: B=37 E=30 D=23 C=10 so C is eliminated. Round 3 votes counts: B=42 E=34 D=24 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:250 E:214 C:191 D:182 A:163 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -16 -10 -20 B 28 0 30 26 16 C 16 -30 0 6 -10 D 10 -26 -6 0 -14 E 20 -16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -16 -10 -20 B 28 0 30 26 16 C 16 -30 0 6 -10 D 10 -26 -6 0 -14 E 20 -16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -16 -10 -20 B 28 0 30 26 16 C 16 -30 0 6 -10 D 10 -26 -6 0 -14 E 20 -16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2092: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) C D E B A (12) C D A E B (9) B E D A C (9) C A D E B (7) A C B E D (7) D E B C A (4) D C E B A (4) B A E D C (4) E D B C A (3) E B D C A (3) B E D C A (3) C D E A B (2) C B E D A (2) B E A D C (2) B C E A D (2) A C D B E (2) A C B D E (2) A B E C D (2) E D B A C (1) E B D A C (1) D C A E B (1) C A B E D (1) C A B D E (1) A E B D C (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -8 -8 4 B 0 0 -2 8 2 C 8 2 0 2 8 D 8 -8 -2 0 -10 E -4 -2 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -8 4 B 0 0 -2 8 2 C 8 2 0 2 8 D 8 -8 -2 0 -10 E -4 -2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=29 B=20 D=9 E=8 so E is eliminated. Round 2 votes counts: C=34 A=29 B=24 D=13 so D is eliminated. Round 3 votes counts: C=39 B=32 A=29 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 B:204 E:198 A:194 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -8 4 B 0 0 -2 8 2 C 8 2 0 2 8 D 8 -8 -2 0 -10 E -4 -2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -8 4 B 0 0 -2 8 2 C 8 2 0 2 8 D 8 -8 -2 0 -10 E -4 -2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -8 4 B 0 0 -2 8 2 C 8 2 0 2 8 D 8 -8 -2 0 -10 E -4 -2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2093: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) C A E B D (10) D E B C A (6) E C A B D (5) D B E C A (4) D B C A E (4) C A E D B (4) B E D A C (4) A C B E D (4) E C A D B (3) C D E A B (3) C A B E D (3) C A B D E (3) B A E C D (3) E B A D C (2) E A C B D (2) D E C B A (2) C D A E B (2) B D A E C (2) B D A C E (2) B A E D C (2) A B C E D (2) A B C D E (2) E D B A C (1) E C D A B (1) E A B C D (1) D E B A C (1) D C E B A (1) D C B A E (1) C E A D B (1) C E A B D (1) C D A B E (1) C B D A E (1) C A D E B (1) B E A D C (1) B D E A C (1) B A D E C (1) B A D C E (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -14 6 0 B 0 0 0 8 4 C 14 0 0 8 -4 D -6 -8 -8 0 -2 E 0 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.732500 C: 0.267500 D: 0.000000 E: 0.000000 Sum of squares = 0.608112268268 Cumulative probabilities = A: 0.000000 B: 0.732500 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 6 0 B 0 0 0 8 4 C 14 0 0 8 -4 D -6 -8 -8 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500290 C: 0.499710 D: 0.000000 E: 0.000000 Sum of squares = 0.500000168262 Cumulative probabilities = A: 0.000000 B: 0.500290 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 B=17 E=15 A=9 so A is eliminated. Round 2 votes counts: C=35 D=29 B=21 E=15 so E is eliminated. Round 3 votes counts: C=46 D=30 B=24 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:206 E:201 A:196 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -14 6 0 B 0 0 0 8 4 C 14 0 0 8 -4 D -6 -8 -8 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500290 C: 0.499710 D: 0.000000 E: 0.000000 Sum of squares = 0.500000168262 Cumulative probabilities = A: 0.000000 B: 0.500290 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 6 0 B 0 0 0 8 4 C 14 0 0 8 -4 D -6 -8 -8 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500290 C: 0.499710 D: 0.000000 E: 0.000000 Sum of squares = 0.500000168262 Cumulative probabilities = A: 0.000000 B: 0.500290 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 6 0 B 0 0 0 8 4 C 14 0 0 8 -4 D -6 -8 -8 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500290 C: 0.499710 D: 0.000000 E: 0.000000 Sum of squares = 0.500000168262 Cumulative probabilities = A: 0.000000 B: 0.500290 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2094: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (15) E D A C B (11) A B C D E (9) E A D C B (5) E A B C D (5) D E C B A (5) D C B A E (5) B C D A E (4) A E B C D (4) A B C E D (4) E D C B A (3) D B C E A (3) E D B C A (2) E B A C D (2) E A D B C (2) D C B E A (2) B A C D E (2) A D E C B (2) A C D B E (2) A B E C D (2) E B D C A (1) E A B D C (1) D C A B E (1) D A C B E (1) C B D A E (1) B C D E A (1) B A E C D (1) A E C D B (1) A D C E B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 14 20 14 B -6 0 16 6 8 C -14 -16 0 10 6 D -20 -6 -10 0 10 E -14 -8 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 20 14 B -6 0 16 6 8 C -14 -16 0 10 6 D -20 -6 -10 0 10 E -14 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=27 B=23 D=17 C=1 so C is eliminated. Round 2 votes counts: E=32 A=27 B=24 D=17 so D is eliminated. Round 3 votes counts: E=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:227 B:212 C:193 D:187 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 20 14 B -6 0 16 6 8 C -14 -16 0 10 6 D -20 -6 -10 0 10 E -14 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 20 14 B -6 0 16 6 8 C -14 -16 0 10 6 D -20 -6 -10 0 10 E -14 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 20 14 B -6 0 16 6 8 C -14 -16 0 10 6 D -20 -6 -10 0 10 E -14 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2095: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) D B C E A (7) C E B D A (6) C B D E A (6) C D B E A (5) E C B D A (4) E C A B D (4) E B C D A (4) A E C D B (4) E C B A D (3) E A C B D (3) D B C A E (3) D B A C E (3) A E B D C (3) A D B E C (3) A C E B D (3) A C D B E (3) D C B E A (2) D A B C E (2) C E A D B (2) C E A B D (2) C D A B E (2) B E D C A (2) B D E A C (2) A D B C E (2) E B D C A (1) E A B C D (1) D B E A C (1) D B A E C (1) C B E D A (1) C A E D B (1) B D E C A (1) B D C E A (1) B C E D A (1) B C D E A (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -18 -12 -20 B 10 0 -20 14 -2 C 18 20 0 30 8 D 12 -14 -30 0 -10 E 20 2 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -12 -20 B 10 0 -20 14 -2 C 18 20 0 30 8 D 12 -14 -30 0 -10 E 20 2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=25 E=20 D=19 B=8 so B is eliminated. Round 2 votes counts: A=28 C=27 D=23 E=22 so E is eliminated. Round 3 votes counts: C=42 A=32 D=26 so D is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:238 E:212 B:201 D:179 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -18 -12 -20 B 10 0 -20 14 -2 C 18 20 0 30 8 D 12 -14 -30 0 -10 E 20 2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -12 -20 B 10 0 -20 14 -2 C 18 20 0 30 8 D 12 -14 -30 0 -10 E 20 2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -12 -20 B 10 0 -20 14 -2 C 18 20 0 30 8 D 12 -14 -30 0 -10 E 20 2 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2096: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (7) C B E A D (5) A B D E C (5) E C D B A (4) D C E A B (4) D A B C E (4) C E B A D (4) B A E C D (4) A D B C E (4) A B C D E (4) E B C A D (3) D E C A B (3) D E A B C (3) D A E B C (3) D A C B E (3) D A B E C (3) C A B D E (3) E D C B A (2) E D B C A (2) E D B A C (2) E C B A D (2) C E D B A (2) B E A C D (2) B C E A D (2) B C A E D (2) E C B D A (1) E B D A C (1) E B C D A (1) E B A D C (1) D E C B A (1) D C E B A (1) D C A B E (1) D A C E B (1) C E B D A (1) C D E B A (1) C B A D E (1) C A D B E (1) B E A D C (1) B A E D C (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 6 12 2 B 8 0 14 8 14 C -6 -14 0 6 8 D -12 -8 -6 0 -4 E -2 -14 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 12 2 B 8 0 14 8 14 C -6 -14 0 6 8 D -12 -8 -6 0 -4 E -2 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=19 B=19 C=18 A=17 so A is eliminated. Round 2 votes counts: D=31 B=30 C=20 E=19 so E is eliminated. Round 3 votes counts: D=37 B=36 C=27 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:206 C:197 E:190 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 12 2 B 8 0 14 8 14 C -6 -14 0 6 8 D -12 -8 -6 0 -4 E -2 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 12 2 B 8 0 14 8 14 C -6 -14 0 6 8 D -12 -8 -6 0 -4 E -2 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 12 2 B 8 0 14 8 14 C -6 -14 0 6 8 D -12 -8 -6 0 -4 E -2 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2097: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (11) B C E D A (9) D E A C B (6) B E C D A (6) A D E C B (6) D E C B A (4) C B A D E (4) B C D E A (4) E D C B A (3) E D B C A (3) E D A C B (3) B C A E D (3) B C A D E (3) A B C E D (3) A B C D E (3) E B C D A (2) D C E B A (2) C B D E A (2) B C E A D (2) A E B D C (2) A B E C D (2) E D A B C (1) E B D C A (1) E B C A D (1) E A D B C (1) D E C A B (1) D C A B E (1) D A E C B (1) C D B E A (1) C D A B E (1) C B D A E (1) B A E C D (1) B A C E D (1) A E D C B (1) A E D B C (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -8 -4 -4 B 6 0 -2 22 20 C 8 2 0 22 8 D 4 -22 -22 0 8 E 4 -20 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -4 -4 B 6 0 -2 22 20 C 8 2 0 22 8 D 4 -22 -22 0 8 E 4 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=29 E=15 D=15 C=9 so C is eliminated. Round 2 votes counts: B=36 A=32 D=17 E=15 so E is eliminated. Round 3 votes counts: B=40 A=33 D=27 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:220 A:189 D:184 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -4 -4 B 6 0 -2 22 20 C 8 2 0 22 8 D 4 -22 -22 0 8 E 4 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -4 -4 B 6 0 -2 22 20 C 8 2 0 22 8 D 4 -22 -22 0 8 E 4 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -4 -4 B 6 0 -2 22 20 C 8 2 0 22 8 D 4 -22 -22 0 8 E 4 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2098: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) E D A B C (5) D C A B E (5) D A E C B (5) C B A D E (5) C A D B E (5) B C E A D (5) E B A C D (4) E A D C B (4) C A B E D (4) D A C E B (3) C A B D E (3) B C E D A (3) B C D E A (3) E B D A C (2) E A B C D (2) D E A C B (2) D E A B C (2) D C B A E (2) D B C E A (2) C B A E D (2) B E D C A (2) B D C A E (2) B C D A E (2) A C D E B (2) A C D B E (2) E B C A D (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E B A C (1) D B E C A (1) C D B A E (1) C D A B E (1) B D C E A (1) B C A D E (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -20 10 12 B -2 0 0 8 22 C 20 0 0 16 26 D -10 -8 -16 0 4 E -12 -22 -26 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.376643 C: 0.623357 D: 0.000000 E: 0.000000 Sum of squares = 0.530433710132 Cumulative probabilities = A: 0.000000 B: 0.376643 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -20 10 12 B -2 0 0 8 22 C 20 0 0 16 26 D -10 -8 -16 0 4 E -12 -22 -26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 E=22 C=21 A=6 so A is eliminated. Round 2 votes counts: B=28 C=25 D=24 E=23 so E is eliminated. Round 3 votes counts: B=38 D=34 C=28 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:231 B:214 A:202 D:185 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -20 10 12 B -2 0 0 8 22 C 20 0 0 16 26 D -10 -8 -16 0 4 E -12 -22 -26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -20 10 12 B -2 0 0 8 22 C 20 0 0 16 26 D -10 -8 -16 0 4 E -12 -22 -26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -20 10 12 B -2 0 0 8 22 C 20 0 0 16 26 D -10 -8 -16 0 4 E -12 -22 -26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2099: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (15) B A C E D (13) E D C A B (8) C A B E D (8) A C B D E (7) D E C A B (6) E D B C A (5) A B C D E (5) C A B D E (4) E D C B A (3) B E D A C (3) E D B A C (2) E C D A B (2) D E B A C (2) D E A C B (2) B A D C E (2) E B D A C (1) E B C A D (1) D E B C A (1) C E D A B (1) C B E A D (1) C A E D B (1) C A D E B (1) B E C A D (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E C D (1) Total count = 100 A B C D E A 0 -10 14 24 16 B 10 0 12 32 28 C -14 -12 0 24 16 D -24 -32 -24 0 -6 E -16 -28 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 24 16 B 10 0 12 32 28 C -14 -12 0 24 16 D -24 -32 -24 0 -6 E -16 -28 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=22 C=16 A=12 D=11 so D is eliminated. Round 2 votes counts: B=39 E=33 C=16 A=12 so A is eliminated. Round 3 votes counts: B=44 E=33 C=23 so C is eliminated. Round 4 votes counts: B=64 E=36 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:241 A:222 C:207 E:173 D:157 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 24 16 B 10 0 12 32 28 C -14 -12 0 24 16 D -24 -32 -24 0 -6 E -16 -28 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 24 16 B 10 0 12 32 28 C -14 -12 0 24 16 D -24 -32 -24 0 -6 E -16 -28 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 24 16 B 10 0 12 32 28 C -14 -12 0 24 16 D -24 -32 -24 0 -6 E -16 -28 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2100: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (13) E B A D C (11) C A D B E (11) E B D A C (9) C D A B E (7) B E C A D (7) B E A C D (5) D A C E B (4) B E C D A (4) A C D B E (4) C A B D E (3) E A D B C (2) D C A E B (2) C D A E B (2) A C D E B (2) E D B A C (1) E D A B C (1) E B D C A (1) D E A C B (1) D E A B C (1) D C E A B (1) C B E D A (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D C A (1) B C E D A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 12 10 24 4 B -12 0 -10 -8 -6 C -10 10 0 2 10 D -24 8 -2 0 10 E -4 6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 24 4 B -12 0 -10 -8 -6 C -10 10 0 2 10 D -24 8 -2 0 10 E -4 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=25 A=20 B=19 D=9 so D is eliminated. Round 2 votes counts: C=30 E=27 A=24 B=19 so B is eliminated. Round 3 votes counts: E=44 C=32 A=24 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:225 C:206 D:196 E:191 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 24 4 B -12 0 -10 -8 -6 C -10 10 0 2 10 D -24 8 -2 0 10 E -4 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 24 4 B -12 0 -10 -8 -6 C -10 10 0 2 10 D -24 8 -2 0 10 E -4 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 24 4 B -12 0 -10 -8 -6 C -10 10 0 2 10 D -24 8 -2 0 10 E -4 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2101: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C B A (8) A B D E C (8) A B C D E (7) C E D B A (5) C E D A B (5) A B D C E (5) E C D B A (3) E C D A B (3) D E C A B (3) D E A C B (3) D B E C A (3) C E A D B (3) B A C E D (3) B A C D E (3) D E B A C (2) D A B E C (2) C E B D A (2) C A E B D (2) B A D E C (2) B A D C E (2) A C B E D (2) E D C B A (1) E D C A B (1) E C A D B (1) D E B C A (1) D B A E C (1) C E A B D (1) C B E A D (1) C B A E D (1) B D E A C (1) B D A E C (1) B C D E A (1) A E D C B (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 18 10 8 4 B -18 0 4 6 6 C -10 -4 0 10 8 D -8 -6 -10 0 6 E -4 -6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 10 8 4 B -18 0 4 6 6 C -10 -4 0 10 8 D -8 -6 -10 0 6 E -4 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=23 C=20 B=13 E=9 so E is eliminated. Round 2 votes counts: A=35 C=27 D=25 B=13 so B is eliminated. Round 3 votes counts: A=45 C=28 D=27 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:202 B:199 D:191 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 8 4 B -18 0 4 6 6 C -10 -4 0 10 8 D -8 -6 -10 0 6 E -4 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 8 4 B -18 0 4 6 6 C -10 -4 0 10 8 D -8 -6 -10 0 6 E -4 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 8 4 B -18 0 4 6 6 C -10 -4 0 10 8 D -8 -6 -10 0 6 E -4 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2102: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) D A B C E (6) B A C E D (6) E D C A B (5) E C B D A (5) E C B A D (5) C B E A D (5) A B D C E (5) D E C A B (4) D C A B E (4) D A B E C (3) C E B D A (3) B C A E D (3) E D C B A (2) E D A C B (2) E D A B C (2) D C E B A (2) C E D B A (2) C B A D E (2) B C A D E (2) B A E C D (2) A D B E C (2) A D B C E (2) A B C E D (2) A B C D E (2) E B A C D (1) E A B D C (1) D E A C B (1) D C E A B (1) D A E B C (1) D A C B E (1) C D B A E (1) C B A E D (1) B A C D E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -14 -10 -2 B 6 0 -12 -2 6 C 14 12 0 8 2 D 10 2 -8 0 -18 E 2 -6 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -10 -2 B 6 0 -12 -2 6 C 14 12 0 8 2 D 10 2 -8 0 -18 E 2 -6 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=23 A=16 C=14 B=14 so C is eliminated. Round 2 votes counts: E=38 D=24 B=22 A=16 so A is eliminated. Round 3 votes counts: E=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:218 E:206 B:199 D:193 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -10 -2 B 6 0 -12 -2 6 C 14 12 0 8 2 D 10 2 -8 0 -18 E 2 -6 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -10 -2 B 6 0 -12 -2 6 C 14 12 0 8 2 D 10 2 -8 0 -18 E 2 -6 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -10 -2 B 6 0 -12 -2 6 C 14 12 0 8 2 D 10 2 -8 0 -18 E 2 -6 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2103: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (15) B A D E C (10) C D E A B (6) B D A C E (5) A E B C D (5) E C D A B (4) E C A D B (4) D C E A B (4) D B C E A (4) B A E D C (4) B A E C D (4) A B E C D (4) D C E B A (3) D B A E C (3) A B E D C (3) E A C D B (2) D E C A B (2) B D A E C (2) B A D C E (2) B A C D E (2) D C B E A (1) D A E B C (1) C E A D B (1) C D E B A (1) C B E A D (1) B D C A E (1) B C E D A (1) B C D E A (1) B A C E D (1) A E C B D (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 8 2 -8 0 B -8 0 10 -4 -2 C -2 -10 0 6 -2 D 8 4 -6 0 -2 E 0 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.144085 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.855915 Sum of squares = 0.753350814047 Cumulative probabilities = A: 0.144085 B: 0.144085 C: 0.144085 D: 0.144085 E: 1.000000 A B C D E A 0 8 2 -8 0 B -8 0 10 -4 -2 C -2 -10 0 6 -2 D 8 4 -6 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000053611 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=24 D=18 A=15 E=10 so E is eliminated. Round 2 votes counts: B=33 C=32 D=18 A=17 so A is eliminated. Round 3 votes counts: B=46 C=35 D=19 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:203 D:202 A:201 B:198 C:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 -8 0 B -8 0 10 -4 -2 C -2 -10 0 6 -2 D 8 4 -6 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000053611 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -8 0 B -8 0 10 -4 -2 C -2 -10 0 6 -2 D 8 4 -6 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000053611 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -8 0 B -8 0 10 -4 -2 C -2 -10 0 6 -2 D 8 4 -6 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000053611 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2104: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) D C A B E (10) B E A C D (9) D A C B E (6) E B C A D (5) B E D A C (5) A D C B E (4) A C D B E (4) E B A C D (3) C D A E B (3) B E A D C (3) A C B E D (3) A B E C D (3) E B D C A (2) E B D A C (2) D E B C A (2) D C E B A (2) D B E A C (2) D A B E C (2) C E B A D (2) C D E B A (2) A C E B D (2) A B E D C (2) E B C D A (1) E B A D C (1) D B E C A (1) D B A E C (1) C E D B A (1) C E B D A (1) C A E B D (1) C A D E B (1) B A E C D (1) A E B C D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 12 -6 12 B -8 0 -4 -4 16 C -12 4 0 -14 4 D 6 4 14 0 4 E -12 -16 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -6 12 B -8 0 -4 -4 16 C -12 4 0 -14 4 D 6 4 14 0 4 E -12 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=21 B=18 E=14 C=11 so C is eliminated. Round 2 votes counts: D=41 A=23 E=18 B=18 so E is eliminated. Round 3 votes counts: D=42 B=35 A=23 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:213 B:200 C:191 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -6 12 B -8 0 -4 -4 16 C -12 4 0 -14 4 D 6 4 14 0 4 E -12 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -6 12 B -8 0 -4 -4 16 C -12 4 0 -14 4 D 6 4 14 0 4 E -12 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -6 12 B -8 0 -4 -4 16 C -12 4 0 -14 4 D 6 4 14 0 4 E -12 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2105: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) E D A B C (7) D C E B A (6) C B A E D (6) C B A D E (5) B A C D E (5) D E C B A (4) D E C A B (4) A B E C D (4) E A C B D (3) D E B C A (3) C A E B D (3) B C A E D (3) E D C A B (2) E C A D B (2) E A C D B (2) D E A B C (2) D C B A E (2) D B E A C (2) C E D A B (2) C A B E D (2) B D A C E (2) A E B C D (2) E D A C B (1) E C D A B (1) E A D C B (1) E A D B C (1) E A B D C (1) E A B C D (1) D C B E A (1) D B E C A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E B A (1) C B D A E (1) B C A D E (1) B A D C E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -2 -8 -18 B 12 0 -4 -14 -22 C 2 4 0 -6 -12 D 8 14 6 0 6 E 18 22 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -8 -18 B 12 0 -4 -14 -22 C 2 4 0 -6 -12 D 8 14 6 0 6 E 18 22 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=22 C=21 B=12 A=8 so A is eliminated. Round 2 votes counts: D=37 E=25 C=22 B=16 so B is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:223 D:217 C:194 B:186 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -8 -18 B 12 0 -4 -14 -22 C 2 4 0 -6 -12 D 8 14 6 0 6 E 18 22 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -8 -18 B 12 0 -4 -14 -22 C 2 4 0 -6 -12 D 8 14 6 0 6 E 18 22 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -8 -18 B 12 0 -4 -14 -22 C 2 4 0 -6 -12 D 8 14 6 0 6 E 18 22 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2106: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) C D E A B (7) C D B E A (7) B D C A E (6) B A E D C (6) E A C B D (5) D C A E B (5) A E B D C (4) A E B C D (4) E A B C D (3) D C B E A (3) D C B A E (3) B E A C D (3) B D C E A (3) A E C D B (3) E A C D B (2) D B C A E (2) D B A C E (2) C B D E A (2) B D A E C (2) B D A C E (2) B C D E A (2) B A D E C (2) A B E D C (2) E C B A D (1) E B A C D (1) D A C E B (1) C E B D A (1) C D E B A (1) C B E D A (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E D A (1) A E D C B (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 -4 -6 B 8 0 -2 10 6 C 8 2 0 6 -2 D 4 -10 -6 0 4 E 6 -6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000096 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -8 -8 -4 -6 B 8 0 -2 10 6 C 8 2 0 6 -2 D 4 -10 -6 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999726 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=19 C=19 D=16 A=16 so D is eliminated. Round 2 votes counts: B=34 C=30 E=19 A=17 so A is eliminated. Round 3 votes counts: B=37 E=32 C=31 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:207 E:199 D:196 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 -6 B 8 0 -2 10 6 C 8 2 0 6 -2 D 4 -10 -6 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999726 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 -6 B 8 0 -2 10 6 C 8 2 0 6 -2 D 4 -10 -6 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999726 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 -6 B 8 0 -2 10 6 C 8 2 0 6 -2 D 4 -10 -6 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999726 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2107: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) A E D B C (7) A D E C B (6) A D E B C (6) E A D C B (5) E C D A B (4) D E C A B (4) B C D A E (4) E D A C B (3) E B A C D (3) B C A D E (3) B A C D E (3) A B E D C (3) E C B D A (2) E C A D B (2) E A B D C (2) D C A E B (2) D A E C B (2) D A C E B (2) C D E A B (2) C D B E A (2) C B E D A (2) C B D E A (2) B C E A D (2) B C D E A (2) B C A E D (2) B A E C D (2) B A D E C (2) E D C A B (1) E C B A D (1) E B A D C (1) E A D B C (1) D C B A E (1) C E D A B (1) C D B A E (1) B E C A D (1) B A D C E (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 18 26 14 B -12 0 10 -6 -16 C -18 -10 0 -18 -36 D -26 6 18 0 -6 E -14 16 36 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 18 26 14 B -12 0 10 -6 -16 C -18 -10 0 -18 -36 D -26 6 18 0 -6 E -14 16 36 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=25 A=25 D=11 C=10 so C is eliminated. Round 2 votes counts: B=33 E=26 A=25 D=16 so D is eliminated. Round 3 votes counts: B=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:235 E:222 D:196 B:188 C:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 18 26 14 B -12 0 10 -6 -16 C -18 -10 0 -18 -36 D -26 6 18 0 -6 E -14 16 36 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 18 26 14 B -12 0 10 -6 -16 C -18 -10 0 -18 -36 D -26 6 18 0 -6 E -14 16 36 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 18 26 14 B -12 0 10 -6 -16 C -18 -10 0 -18 -36 D -26 6 18 0 -6 E -14 16 36 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2108: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) D E A B C (10) C D A B E (6) D C A B E (5) D C A E B (4) E D B A C (3) E B A D C (3) E A B D C (3) C D B A E (3) B E C A D (3) B C E A D (3) B C A E D (3) B A E C D (3) E D A B C (2) E B D A C (2) E B A C D (2) E A D B C (2) D C E A B (2) C B A E D (2) C A D B E (2) B E A C D (2) A B C E D (2) E D B C A (1) E B D C A (1) E B C D A (1) D E C B A (1) D E C A B (1) D C E B A (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) C E B D A (1) C D B E A (1) C B E D A (1) C B D A E (1) C B A D E (1) B A C E D (1) A E B D C (1) A C D B E (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 20 -12 -2 8 B -20 0 2 2 12 C 12 -2 0 6 10 D 2 -2 -6 0 10 E -8 -12 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.058824 B: 0.352941 C: 0.588235 D: 0.000000 E: 0.000000 Sum of squares = 0.474048443044 Cumulative probabilities = A: 0.058824 B: 0.411765 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -12 -2 8 B -20 0 2 2 12 C 12 -2 0 6 10 D 2 -2 -6 0 10 E -8 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.352941 C: 0.588235 D: 0.000000 E: 0.000000 Sum of squares = 0.474048443214 Cumulative probabilities = A: 0.058824 B: 0.411765 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=28 E=20 B=15 A=7 so A is eliminated. Round 2 votes counts: C=32 D=28 E=21 B=19 so B is eliminated. Round 3 votes counts: C=41 E=31 D=28 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:207 D:202 B:198 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -12 -2 8 B -20 0 2 2 12 C 12 -2 0 6 10 D 2 -2 -6 0 10 E -8 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.352941 C: 0.588235 D: 0.000000 E: 0.000000 Sum of squares = 0.474048443214 Cumulative probabilities = A: 0.058824 B: 0.411765 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -12 -2 8 B -20 0 2 2 12 C 12 -2 0 6 10 D 2 -2 -6 0 10 E -8 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.352941 C: 0.588235 D: 0.000000 E: 0.000000 Sum of squares = 0.474048443214 Cumulative probabilities = A: 0.058824 B: 0.411765 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -12 -2 8 B -20 0 2 2 12 C 12 -2 0 6 10 D 2 -2 -6 0 10 E -8 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.352941 C: 0.588235 D: 0.000000 E: 0.000000 Sum of squares = 0.474048443214 Cumulative probabilities = A: 0.058824 B: 0.411765 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2109: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) B E C D A (9) C B D A E (6) B C D E A (6) E A D C B (5) A D C E B (5) A C D B E (5) A D E C B (4) A D C B E (4) E B C D A (3) E B C A D (3) C D A B E (3) C A D B E (3) C A B D E (3) B C E D A (3) E C B A D (2) E B D A C (2) E A C D B (2) E A B D C (2) E A B C D (2) B E D C A (2) B C D A E (2) A E C D B (2) E D B A C (1) E D A B C (1) E B D C A (1) E B A D C (1) E A C B D (1) D C A B E (1) D B E C A (1) D B C A E (1) C B A D E (1) C A E B D (1) B D C A E (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 -4 14 -12 B -10 0 0 2 2 C 4 0 0 16 -8 D -14 -2 -16 0 -6 E 12 -2 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555547 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 A B C D E A 0 10 -4 14 -12 B -10 0 0 2 2 C 4 0 0 16 -8 D -14 -2 -16 0 -6 E 12 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555321 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=23 A=22 C=17 D=3 so D is eliminated. Round 2 votes counts: E=35 B=25 A=22 C=18 so C is eliminated. Round 3 votes counts: E=35 A=33 B=32 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:206 A:204 B:197 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -4 14 -12 B -10 0 0 2 2 C 4 0 0 16 -8 D -14 -2 -16 0 -6 E 12 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555321 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 14 -12 B -10 0 0 2 2 C 4 0 0 16 -8 D -14 -2 -16 0 -6 E 12 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555321 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 14 -12 B -10 0 0 2 2 C 4 0 0 16 -8 D -14 -2 -16 0 -6 E 12 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.416667 Sum of squares = 0.430555555321 Cumulative probabilities = A: 0.083333 B: 0.583333 C: 0.583333 D: 0.583333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2110: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (19) B C E D A (14) D A B E C (13) C E B A D (10) B D A C E (6) D B A C E (5) A E C D B (5) E C A D B (4) E C A B D (4) B D C E A (4) E C B A D (3) D A B C E (3) A D E B C (3) E A C D B (2) B C D E A (2) D A E C B (1) D A E B C (1) B D C A E (1) Total count = 100 A B C D E A 0 10 16 0 14 B -10 0 4 -12 -4 C -16 -4 0 -12 -10 D 0 12 12 0 16 E -14 4 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.486180 B: 0.000000 C: 0.000000 D: 0.513820 E: 0.000000 Sum of squares = 0.500381960476 Cumulative probabilities = A: 0.486180 B: 0.486180 C: 0.486180 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 0 14 B -10 0 4 -12 -4 C -16 -4 0 -12 -10 D 0 12 12 0 16 E -14 4 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 D=23 E=13 C=10 so C is eliminated. Round 2 votes counts: B=27 A=27 E=23 D=23 so E is eliminated. Round 3 votes counts: B=40 A=37 D=23 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:220 E:192 B:189 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 0 14 B -10 0 4 -12 -4 C -16 -4 0 -12 -10 D 0 12 12 0 16 E -14 4 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 0 14 B -10 0 4 -12 -4 C -16 -4 0 -12 -10 D 0 12 12 0 16 E -14 4 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 0 14 B -10 0 4 -12 -4 C -16 -4 0 -12 -10 D 0 12 12 0 16 E -14 4 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2111: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) D A E B C (8) A D B E C (8) D E A C B (7) C B E D A (7) E D A C B (5) D A B E C (4) C B E A D (4) B C A E D (4) E D C A B (3) D A B C E (3) C B D E A (3) B C E A D (3) A D E B C (3) A B D E C (3) E C D A B (2) D E C A B (2) C E B D A (2) C E B A D (2) B C A D E (2) B A E C D (2) A B E D C (2) E C B A D (1) E C A D B (1) E C A B D (1) E A C D B (1) E A B D C (1) D C E A B (1) D B C A E (1) B D A C E (1) B C D A E (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 30 20 -20 8 B -30 0 -4 -18 0 C -20 4 0 -28 -30 D 20 18 28 0 18 E -8 0 30 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 30 20 -20 8 B -30 0 -4 -18 0 C -20 4 0 -28 -30 D 20 18 28 0 18 E -8 0 30 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=18 A=16 E=15 B=15 so E is eliminated. Round 2 votes counts: D=44 C=23 A=18 B=15 so B is eliminated. Round 3 votes counts: D=45 C=33 A=22 so A is eliminated. Round 4 votes counts: D=64 C=36 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:242 A:219 E:202 B:174 C:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 30 20 -20 8 B -30 0 -4 -18 0 C -20 4 0 -28 -30 D 20 18 28 0 18 E -8 0 30 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 20 -20 8 B -30 0 -4 -18 0 C -20 4 0 -28 -30 D 20 18 28 0 18 E -8 0 30 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 20 -20 8 B -30 0 -4 -18 0 C -20 4 0 -28 -30 D 20 18 28 0 18 E -8 0 30 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2112: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) B E C D A (9) B C E D A (8) B E D C A (7) A D C E B (7) C A D B E (6) A C D E B (6) E D B A C (5) D E A B C (5) C B E A D (5) C B A E D (5) E B D A C (4) C A B D E (3) A D E C B (3) E D A B C (2) D A C B E (2) C A D E B (2) B D E A C (2) E B D C A (1) E B C A D (1) D E B A C (1) D A E C B (1) C E A B D (1) C B D A E (1) C B A D E (1) B C E A D (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 -14 -4 B 2 0 10 -2 0 C 2 -10 0 0 -2 D 14 2 0 0 -2 E 4 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.182694 C: 0.000000 D: 0.000000 E: 0.817306 Sum of squares = 0.701365661818 Cumulative probabilities = A: 0.000000 B: 0.182694 C: 0.182694 D: 0.182694 E: 1.000000 A B C D E A 0 -2 -2 -14 -4 B 2 0 10 -2 0 C 2 -10 0 0 -2 D 14 2 0 0 -2 E 4 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499443 C: 0.000000 D: 0.000000 E: 0.500557 Sum of squares = 0.500000620435 Cumulative probabilities = A: 0.000000 B: 0.499443 C: 0.499443 D: 0.499443 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 D=18 A=18 E=13 so E is eliminated. Round 2 votes counts: B=33 D=25 C=24 A=18 so A is eliminated. Round 3 votes counts: D=36 B=33 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:207 B:205 E:204 C:195 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 -14 -4 B 2 0 10 -2 0 C 2 -10 0 0 -2 D 14 2 0 0 -2 E 4 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499443 C: 0.000000 D: 0.000000 E: 0.500557 Sum of squares = 0.500000620435 Cumulative probabilities = A: 0.000000 B: 0.499443 C: 0.499443 D: 0.499443 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -14 -4 B 2 0 10 -2 0 C 2 -10 0 0 -2 D 14 2 0 0 -2 E 4 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499443 C: 0.000000 D: 0.000000 E: 0.500557 Sum of squares = 0.500000620435 Cumulative probabilities = A: 0.000000 B: 0.499443 C: 0.499443 D: 0.499443 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -14 -4 B 2 0 10 -2 0 C 2 -10 0 0 -2 D 14 2 0 0 -2 E 4 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499443 C: 0.000000 D: 0.000000 E: 0.500557 Sum of squares = 0.500000620435 Cumulative probabilities = A: 0.000000 B: 0.499443 C: 0.499443 D: 0.499443 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2113: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) C D E A B (7) C E D A B (5) B A E D C (5) A B E D C (5) A B E C D (5) D E C B A (4) D B C E A (4) C E A D B (4) C D E B A (4) B A D E C (4) A B C E D (4) D B E C A (3) B D E A C (3) B D A E C (3) A E C B D (3) D E B C A (2) D C E B A (2) D C B E A (2) B D C E A (2) A E B D C (2) A E B C D (2) A C E B D (2) E D C A B (1) E C A D B (1) E A D B C (1) E A B D C (1) D B E A C (1) C D B A E (1) C B D A E (1) C B A D E (1) C A D E B (1) C A B E D (1) B E A D C (1) B D C A E (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 -12 8 4 B -10 0 0 -4 -2 C 12 0 0 4 6 D -8 4 -4 0 -6 E -4 2 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.330045 C: 0.669955 D: 0.000000 E: 0.000000 Sum of squares = 0.557769589391 Cumulative probabilities = A: 0.000000 B: 0.330045 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 8 4 B -10 0 0 -4 -2 C 12 0 0 4 6 D -8 4 -4 0 -6 E -4 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499364 C: 0.500636 D: 0.000000 E: 0.000000 Sum of squares = 0.500000807773 Cumulative probabilities = A: 0.000000 B: 0.499364 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=24 B=20 D=18 E=4 so E is eliminated. Round 2 votes counts: C=35 A=26 B=20 D=19 so D is eliminated. Round 3 votes counts: C=44 B=30 A=26 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:211 A:205 E:199 D:193 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 8 4 B -10 0 0 -4 -2 C 12 0 0 4 6 D -8 4 -4 0 -6 E -4 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499364 C: 0.500636 D: 0.000000 E: 0.000000 Sum of squares = 0.500000807773 Cumulative probabilities = A: 0.000000 B: 0.499364 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 8 4 B -10 0 0 -4 -2 C 12 0 0 4 6 D -8 4 -4 0 -6 E -4 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499364 C: 0.500636 D: 0.000000 E: 0.000000 Sum of squares = 0.500000807773 Cumulative probabilities = A: 0.000000 B: 0.499364 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 8 4 B -10 0 0 -4 -2 C 12 0 0 4 6 D -8 4 -4 0 -6 E -4 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499364 C: 0.500636 D: 0.000000 E: 0.000000 Sum of squares = 0.500000807773 Cumulative probabilities = A: 0.000000 B: 0.499364 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2114: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) C B E D A (7) E D B A C (6) B E D A C (6) B C E D A (6) C D E A B (5) C A B D E (5) B E C D A (5) A D C E B (5) D E A B C (4) A B D E C (4) C E D B A (3) C B A E D (3) B A E D C (3) B A C E D (3) E D A B C (2) E B D A C (2) D E A C B (2) D A E C B (2) C E B D A (2) C A D B E (2) B E D C A (2) A D E C B (2) E D C B A (1) E D B C A (1) D E C A B (1) D A E B C (1) C B E A D (1) C A D E B (1) B E C A D (1) B C A E D (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 6 -16 -14 B 6 0 10 4 2 C -6 -10 0 -4 -6 D 16 -4 4 0 -10 E 14 -2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -16 -14 B 6 0 10 4 2 C -6 -10 0 -4 -6 D 16 -4 4 0 -10 E 14 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=27 A=22 E=12 D=10 so D is eliminated. Round 2 votes counts: C=29 B=27 A=25 E=19 so E is eliminated. Round 3 votes counts: B=36 A=33 C=31 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:211 D:203 C:187 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -16 -14 B 6 0 10 4 2 C -6 -10 0 -4 -6 D 16 -4 4 0 -10 E 14 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -16 -14 B 6 0 10 4 2 C -6 -10 0 -4 -6 D 16 -4 4 0 -10 E 14 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -16 -14 B 6 0 10 4 2 C -6 -10 0 -4 -6 D 16 -4 4 0 -10 E 14 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2115: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (11) B D A E C (7) B C D E A (7) A E D C B (7) D E A C B (6) D A E B C (6) C E A B D (6) D B A E C (5) D A E C B (5) C E A D B (4) B C A E D (4) C B E A D (3) B D C E A (3) C E D A B (2) C E B A D (2) B D C A E (2) B C E D A (2) B C D A E (2) E D A C B (1) E C D A B (1) E A C D B (1) D E C A B (1) D E A B C (1) D C E A B (1) D C B E A (1) D B E C A (1) D B C E A (1) C D E A B (1) C B E D A (1) C A E B D (1) B D A C E (1) A E D B C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 -14 -16 -14 B 6 0 10 4 2 C 14 -10 0 -4 10 D 16 -4 4 0 6 E 14 -2 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -16 -14 B 6 0 10 4 2 C 14 -10 0 -4 10 D 16 -4 4 0 6 E 14 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=28 C=20 A=10 E=3 so E is eliminated. Round 2 votes counts: B=39 D=29 C=21 A=11 so A is eliminated. Round 3 votes counts: D=39 B=39 C=22 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:211 C:205 E:198 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 -16 -14 B 6 0 10 4 2 C 14 -10 0 -4 10 D 16 -4 4 0 6 E 14 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -16 -14 B 6 0 10 4 2 C 14 -10 0 -4 10 D 16 -4 4 0 6 E 14 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -16 -14 B 6 0 10 4 2 C 14 -10 0 -4 10 D 16 -4 4 0 6 E 14 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2116: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) B C A D E (10) E D A C B (7) D E A B C (5) C E B D A (4) B A C D E (4) A D E C B (4) A D E B C (4) A D B E C (4) E C D B A (3) C B A E D (3) A B C D E (3) E D C A B (2) C E D A B (2) C B E A D (2) B C E D A (2) B C D E A (2) B C D A E (2) B C A E D (2) B A D E C (2) A E D C B (2) A B D E C (2) A B D C E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D A B (1) D B A E C (1) D A E C B (1) D A E B C (1) C E A D B (1) C A E D B (1) B E D C A (1) B D E A C (1) B D A E C (1) B A D C E (1) Total count = 100 A B C D E A 0 -16 -8 -6 0 B 16 0 4 16 18 C 8 -4 0 12 10 D 6 -16 -12 0 0 E 0 -18 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -6 0 B 16 0 4 16 18 C 8 -4 0 12 10 D 6 -16 -12 0 0 E 0 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=27 A=21 E=16 D=8 so D is eliminated. Round 2 votes counts: B=29 C=27 A=23 E=21 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:227 C:213 D:189 E:186 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -8 -6 0 B 16 0 4 16 18 C 8 -4 0 12 10 D 6 -16 -12 0 0 E 0 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -6 0 B 16 0 4 16 18 C 8 -4 0 12 10 D 6 -16 -12 0 0 E 0 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -6 0 B 16 0 4 16 18 C 8 -4 0 12 10 D 6 -16 -12 0 0 E 0 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2117: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) D A B C E (7) C A B D E (7) E B D C A (6) E C B A D (5) C B A E D (5) E D A C B (4) E D A B C (4) B D A C E (4) E B D A C (3) D A C B E (3) B C A D E (3) E C A D B (2) D E A B C (2) D A B E C (2) C B E A D (2) C B A D E (2) C A B E D (2) B E C D A (2) B C E A D (2) A D C B E (2) A C D B E (2) E D C A B (1) E D B C A (1) E C D A B (1) E C A B D (1) E B C D A (1) E B C A D (1) E A D C B (1) D E A C B (1) D B E A C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C E B (1) C E A B D (1) C A E D B (1) C A E B D (1) B E C A D (1) B D E A C (1) B D C A E (1) B C A E D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 0 -12 -2 B 0 0 4 6 4 C 0 -4 0 -10 0 D 12 -6 10 0 -12 E 2 -4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.232913 B: 0.767087 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.642671227567 Cumulative probabilities = A: 0.232913 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -12 -2 B 0 0 4 6 4 C 0 -4 0 -10 0 D 12 -6 10 0 -12 E 2 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556356 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=21 D=20 B=15 A=6 so A is eliminated. Round 2 votes counts: E=38 C=25 D=22 B=15 so B is eliminated. Round 3 votes counts: E=41 C=31 D=28 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:207 E:205 D:202 A:193 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -12 -2 B 0 0 4 6 4 C 0 -4 0 -10 0 D 12 -6 10 0 -12 E 2 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556356 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -12 -2 B 0 0 4 6 4 C 0 -4 0 -10 0 D 12 -6 10 0 -12 E 2 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556356 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -12 -2 B 0 0 4 6 4 C 0 -4 0 -10 0 D 12 -6 10 0 -12 E 2 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556356 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2118: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) B C A D E (7) C B A D E (6) E A D C B (5) C D B E A (5) B C D A E (4) A E D B C (4) A C B E D (4) D E C B A (3) D E C A B (3) B A C D E (3) A E D C B (3) A E B D C (3) A B C E D (3) C E A D B (2) C D E A B (2) C B D A E (2) B C D E A (2) B A E D C (2) B A D E C (2) B A C E D (2) A E C D B (2) A B E D C (2) A B E C D (2) E D C A B (1) E B A D C (1) E A D B C (1) D E B C A (1) D E B A C (1) D C E B A (1) D B E C A (1) D B C E A (1) C D E B A (1) C B D E A (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D A C (1) B D E C A (1) B D E A C (1) B D C E A (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 4 20 14 B 0 0 -6 8 10 C -4 6 0 8 4 D -20 -8 -8 0 2 E -14 -10 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.744957 B: 0.255043 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.620007814453 Cumulative probabilities = A: 0.744957 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 20 14 B 0 0 -6 8 10 C -4 6 0 8 4 D -20 -8 -8 0 2 E -14 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000027459 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 C=22 E=15 D=11 so D is eliminated. Round 2 votes counts: B=28 A=26 E=23 C=23 so E is eliminated. Round 3 votes counts: A=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:207 B:206 E:185 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 20 14 B 0 0 -6 8 10 C -4 6 0 8 4 D -20 -8 -8 0 2 E -14 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000027459 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 20 14 B 0 0 -6 8 10 C -4 6 0 8 4 D -20 -8 -8 0 2 E -14 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000027459 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 20 14 B 0 0 -6 8 10 C -4 6 0 8 4 D -20 -8 -8 0 2 E -14 -10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000027459 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2119: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (12) B A C E D (12) B A E D C (7) D C E A B (6) B C A E D (5) C D E B A (4) C B A E D (4) B A E C D (4) D C E B A (3) C E D A B (3) C D E A B (3) B D A E C (3) E D C A B (2) D E A C B (2) C B D E A (2) C A E B D (2) C A B E D (2) B C A D E (2) B A D E C (2) B A C D E (2) A E C D B (2) A E B D C (2) A B E D C (2) E D A B C (1) E C D A B (1) E C A D B (1) D E C B A (1) D E A B C (1) D C B E A (1) C E A D B (1) B D E A C (1) B D C A E (1) A E D B C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -12 6 10 B 8 0 -6 10 2 C 12 6 0 4 8 D -6 -10 -4 0 -8 E -10 -2 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 6 10 B 8 0 -6 10 2 C 12 6 0 4 8 D -6 -10 -4 0 -8 E -10 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=26 C=21 A=9 E=5 so E is eliminated. Round 2 votes counts: B=39 D=29 C=23 A=9 so A is eliminated. Round 3 votes counts: B=44 D=30 C=26 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:207 A:198 E:194 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 6 10 B 8 0 -6 10 2 C 12 6 0 4 8 D -6 -10 -4 0 -8 E -10 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 6 10 B 8 0 -6 10 2 C 12 6 0 4 8 D -6 -10 -4 0 -8 E -10 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 6 10 B 8 0 -6 10 2 C 12 6 0 4 8 D -6 -10 -4 0 -8 E -10 -2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2120: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) D B E A C (7) E C B A D (5) E B C A D (5) A C D B E (5) E D B C A (4) E C D B A (4) E B D A C (4) D B A C E (4) E B A C D (3) D A C B E (3) C E A D B (3) C D A E B (3) C A E B D (3) C A D B E (3) B E A C D (3) A C B E D (3) E C A B D (2) E B A D C (2) C D E A B (2) C A B D E (2) B D A E C (2) A C B D E (2) A B D C E (2) A B C E D (2) E B D C A (1) E B C D A (1) E A C B D (1) D E C B A (1) D C E B A (1) D C B E A (1) D C B A E (1) D B A E C (1) D A B C E (1) C A E D B (1) B E A D C (1) B D E A C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 16 -20 B 4 0 -8 8 -8 C 2 8 0 24 4 D -16 -8 -24 0 -12 E 20 8 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 16 -20 B 4 0 -8 8 -8 C 2 8 0 24 4 D -16 -8 -24 0 -12 E 20 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=25 D=20 A=16 B=7 so B is eliminated. Round 2 votes counts: E=36 C=25 D=23 A=16 so A is eliminated. Round 3 votes counts: C=38 E=36 D=26 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:218 B:198 A:195 D:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 16 -20 B 4 0 -8 8 -8 C 2 8 0 24 4 D -16 -8 -24 0 -12 E 20 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 16 -20 B 4 0 -8 8 -8 C 2 8 0 24 4 D -16 -8 -24 0 -12 E 20 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 16 -20 B 4 0 -8 8 -8 C 2 8 0 24 4 D -16 -8 -24 0 -12 E 20 8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2121: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) E C B A D (8) E C A B D (6) D B A C E (6) B E C D A (6) E B C D A (5) D B E C A (5) A C E D B (5) C E A B D (4) B D E C A (4) A E C D B (4) D B A E C (3) A D C E B (3) A C E B D (3) A C D E B (3) D B E A C (2) D A E B C (2) C E B A D (2) C A E B D (2) B D C E A (2) A E C B D (2) A D E C B (2) A D C B E (2) D B C E A (1) D B C A E (1) D A B E C (1) C B E A D (1) C A B D E (1) B C E D A (1) A E D C B (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 6 2 2 6 B -6 0 2 -6 -6 C -2 -2 0 6 -4 D -2 6 -6 0 0 E -6 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 2 6 B -6 0 2 -6 -6 C -2 -2 0 6 -4 D -2 6 -6 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=27 E=19 B=13 C=10 so C is eliminated. Round 2 votes counts: D=31 A=30 E=25 B=14 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:208 E:202 C:199 D:199 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 2 6 B -6 0 2 -6 -6 C -2 -2 0 6 -4 D -2 6 -6 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 6 B -6 0 2 -6 -6 C -2 -2 0 6 -4 D -2 6 -6 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 6 B -6 0 2 -6 -6 C -2 -2 0 6 -4 D -2 6 -6 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2122: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) D A B C E (8) D A C E B (5) C D E A B (5) B E C A D (5) E C B A D (4) D C A E B (4) A D B E C (4) E B C A D (3) E B A C D (3) D C B A E (3) C E D B A (3) C E B D A (3) B E A C D (3) A B E D C (3) E C A B D (2) E A C B D (2) E A B C D (2) D B A E C (2) D B A C E (2) D A B E C (2) C E D A B (2) C E A D B (2) B A E D C (2) B A D E C (2) A D E B C (2) D C B E A (1) D B C E A (1) C E A B D (1) C D E B A (1) C B D E A (1) C A E D B (1) B E D A C (1) B C D E A (1) A E C D B (1) A E B C D (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -6 10 -12 B 2 0 -4 0 -16 C 6 4 0 12 10 D -10 0 -12 0 -8 E 12 16 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 10 -12 B 2 0 -4 0 -16 C 6 4 0 12 10 D -10 0 -12 0 -8 E 12 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=28 E=16 B=14 A=13 so A is eliminated. Round 2 votes counts: D=35 C=29 E=18 B=18 so E is eliminated. Round 3 votes counts: C=38 D=35 B=27 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:213 A:195 B:191 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 10 -12 B 2 0 -4 0 -16 C 6 4 0 12 10 D -10 0 -12 0 -8 E 12 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 10 -12 B 2 0 -4 0 -16 C 6 4 0 12 10 D -10 0 -12 0 -8 E 12 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 10 -12 B 2 0 -4 0 -16 C 6 4 0 12 10 D -10 0 -12 0 -8 E 12 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2123: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) A D B E C (8) D A C B E (7) C E B A D (7) B E A D C (7) C D A E B (5) E C B A D (4) D A B C E (4) C D A B E (4) A D C E B (4) C E B D A (3) C E A D B (3) E C B D A (2) E B C A D (2) E B A D C (2) C E D A B (2) C D B E A (2) B E D A C (2) B E C D A (2) B D A E C (2) E B A C D (1) E A B C D (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C E B (1) C E D B A (1) C D E B A (1) C D E A B (1) C D B A E (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C A D (1) B E A C D (1) B D E A C (1) B C D E A (1) B A D E C (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 6 8 -8 4 B -6 0 -2 -18 12 C -8 2 0 -6 2 D 8 18 6 0 12 E -4 -12 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -8 4 B -6 0 -2 -18 12 C -8 2 0 -6 2 D 8 18 6 0 12 E -4 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=23 B=19 A=14 E=12 so E is eliminated. Round 2 votes counts: C=38 B=24 D=23 A=15 so A is eliminated. Round 3 votes counts: C=38 D=37 B=25 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:205 C:195 B:193 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -8 4 B -6 0 -2 -18 12 C -8 2 0 -6 2 D 8 18 6 0 12 E -4 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 4 B -6 0 -2 -18 12 C -8 2 0 -6 2 D 8 18 6 0 12 E -4 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 4 B -6 0 -2 -18 12 C -8 2 0 -6 2 D 8 18 6 0 12 E -4 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2124: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) E B A C D (8) A B D E C (8) E C B A D (7) C E B D A (6) C D E A B (6) C D A E B (6) D C A B E (5) B E A D C (5) A D B E C (5) D A B C E (4) C D E B A (4) A D B C E (4) D A C B E (3) D A B E C (3) E B C D A (2) C E B A D (2) A B E D C (2) E D B C A (1) D C E B A (1) D C A E B (1) C E D A B (1) C E A D B (1) C E A B D (1) C D A B E (1) B E A C D (1) B A E D C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -8 -6 -10 B -6 0 -12 -12 -14 C 8 12 0 12 14 D 6 12 -12 0 4 E 10 14 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -6 -10 B -6 0 -12 -12 -14 C 8 12 0 12 14 D 6 12 -12 0 4 E 10 14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=21 E=18 D=17 B=7 so B is eliminated. Round 2 votes counts: C=37 E=24 A=22 D=17 so D is eliminated. Round 3 votes counts: C=44 A=32 E=24 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:205 E:203 A:191 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -6 -10 B -6 0 -12 -12 -14 C 8 12 0 12 14 D 6 12 -12 0 4 E 10 14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -6 -10 B -6 0 -12 -12 -14 C 8 12 0 12 14 D 6 12 -12 0 4 E 10 14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -6 -10 B -6 0 -12 -12 -14 C 8 12 0 12 14 D 6 12 -12 0 4 E 10 14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2125: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (20) C E A D B (17) E C A D B (10) B D A C E (5) E C B A D (4) D B A C E (4) C E B D A (4) A E C D B (4) C E B A D (3) C E A B D (3) A D C E B (3) A D B E C (3) D B A E C (2) C E D B A (2) C E D A B (2) B D E C A (2) B D C E A (2) B A D E C (2) A D E C B (2) E C B D A (1) E C A B D (1) D A B C E (1) B E C D A (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -4 8 -4 B 4 0 -14 0 -14 C 4 14 0 6 -8 D -8 0 -6 0 -6 E 4 14 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 8 -4 B 4 0 -14 0 -14 C 4 14 0 6 -8 D -8 0 -6 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=31 E=16 A=14 D=7 so D is eliminated. Round 2 votes counts: B=38 C=31 E=16 A=15 so A is eliminated. Round 3 votes counts: B=43 C=34 E=23 so E is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:216 C:208 A:198 D:190 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 8 -4 B 4 0 -14 0 -14 C 4 14 0 6 -8 D -8 0 -6 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 8 -4 B 4 0 -14 0 -14 C 4 14 0 6 -8 D -8 0 -6 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 8 -4 B 4 0 -14 0 -14 C 4 14 0 6 -8 D -8 0 -6 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2126: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) E A B D C (6) C D B A E (5) C A E B D (5) B E A D C (5) D E A B C (4) C D A E B (4) C B A E D (4) B E A C D (4) D E B A C (3) D C B A E (3) C B D E A (3) E A D B C (2) D C E A B (2) D B E C A (2) D B E A C (2) D A E C B (2) C D A B E (2) B E D A C (2) B D C E A (2) B A E C D (2) A E D B C (2) A E C B D (2) A E B C D (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D C E B A (1) D C A E B (1) D A E B C (1) C B A D E (1) C A D E B (1) C A B E D (1) B D E C A (1) B D E A C (1) B C E D A (1) B A C E D (1) A E C D B (1) A E B D C (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -2 -12 -14 B 14 0 -4 -4 8 C 2 4 0 -18 0 D 12 4 18 0 8 E 14 -8 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -12 -14 B 14 0 -4 -4 8 C 2 4 0 -18 0 D 12 4 18 0 8 E 14 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=26 B=19 E=12 A=10 so A is eliminated. Round 2 votes counts: D=34 C=27 E=20 B=19 so B is eliminated. Round 3 votes counts: D=38 E=33 C=29 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:207 E:199 C:194 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -2 -12 -14 B 14 0 -4 -4 8 C 2 4 0 -18 0 D 12 4 18 0 8 E 14 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -12 -14 B 14 0 -4 -4 8 C 2 4 0 -18 0 D 12 4 18 0 8 E 14 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -12 -14 B 14 0 -4 -4 8 C 2 4 0 -18 0 D 12 4 18 0 8 E 14 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2127: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) E B A C D (9) A B E C D (8) E B A D C (6) E A B D C (6) C D A B E (6) A E B D C (5) D C A E B (4) C D B E A (4) C A B E D (4) B E A C D (4) D E B A C (3) D C A B E (3) C B E A D (3) D E C B A (2) D E B C A (2) D E A B C (2) C D B A E (2) A C B E D (2) E D B A C (1) E B D A C (1) E B C D A (1) D E C A B (1) D E A C B (1) D A C B E (1) D A B E C (1) C D E B A (1) C B E D A (1) C B A E D (1) B C A E D (1) B A E C D (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 6 4 -18 B 8 0 6 8 -14 C -6 -6 0 -4 -12 D -4 -8 4 0 -10 E 18 14 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 6 4 -18 B 8 0 6 8 -14 C -6 -6 0 -4 -12 D -4 -8 4 0 -10 E 18 14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=24 C=22 A=17 B=6 so B is eliminated. Round 2 votes counts: D=31 E=28 C=23 A=18 so A is eliminated. Round 3 votes counts: E=43 D=32 C=25 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:204 A:192 D:191 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 4 -18 B 8 0 6 8 -14 C -6 -6 0 -4 -12 D -4 -8 4 0 -10 E 18 14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 4 -18 B 8 0 6 8 -14 C -6 -6 0 -4 -12 D -4 -8 4 0 -10 E 18 14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 4 -18 B 8 0 6 8 -14 C -6 -6 0 -4 -12 D -4 -8 4 0 -10 E 18 14 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2128: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) E A C D B (8) C B D E A (6) E A D B C (5) D B A E C (5) E C A D B (4) D B A C E (4) C A E B D (4) A D E B C (4) A D B E C (4) E C A B D (3) D A B E C (3) C E B D A (3) C B E D A (3) A E D C B (3) A C B D E (3) E D B A C (2) E C B D A (2) C E A B D (2) C B D A E (2) B D C E A (2) B D C A E (2) E D B C A (1) E D A B C (1) E A D C B (1) D B E C A (1) D B E A C (1) D A E B C (1) D A B C E (1) C E B A D (1) C B E A D (1) C B A D E (1) C A B E D (1) B D E C A (1) B D A C E (1) B C D A E (1) A E C D B (1) A D B C E (1) Total count = 100 A B C D E A 0 20 18 14 4 B -20 0 2 -22 -12 C -18 -2 0 -8 -22 D -14 22 8 0 -12 E -4 12 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 14 4 B -20 0 2 -22 -12 C -18 -2 0 -8 -22 D -14 22 8 0 -12 E -4 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 C=24 D=16 B=7 so B is eliminated. Round 2 votes counts: E=27 A=26 C=25 D=22 so D is eliminated. Round 3 votes counts: A=41 E=30 C=29 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:228 E:221 D:202 C:175 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 14 4 B -20 0 2 -22 -12 C -18 -2 0 -8 -22 D -14 22 8 0 -12 E -4 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 14 4 B -20 0 2 -22 -12 C -18 -2 0 -8 -22 D -14 22 8 0 -12 E -4 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 14 4 B -20 0 2 -22 -12 C -18 -2 0 -8 -22 D -14 22 8 0 -12 E -4 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2129: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) A E D C B (6) E A D C B (5) D E C A B (5) A E D B C (5) A D E B C (5) E D A C B (4) D E A C B (4) D A E B C (4) B C A E D (4) B A C E D (4) C D E B A (3) C B E D A (3) C B E A D (3) B C D E A (3) B C D A E (3) C E D B A (2) C B D E A (2) B D A E C (2) B A D C E (2) B A C D E (2) A D B E C (2) E D C A B (1) E C D A B (1) D E C B A (1) D E A B C (1) D C E B A (1) D B E C A (1) D B A E C (1) C E D A B (1) C E B A D (1) C B A E D (1) C A B E D (1) B D C E A (1) B C E D A (1) B A D E C (1) A E C D B (1) A E C B D (1) A D E C B (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 6 10 12 B 0 0 2 -10 -6 C -6 -2 0 -8 -6 D -10 10 8 0 6 E -12 6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.648823 B: 0.351177 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.54429642575 Cumulative probabilities = A: 0.648823 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 10 12 B 0 0 2 -10 -6 C -6 -2 0 -8 -6 D -10 10 8 0 6 E -12 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500599 B: 0.499401 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000717291 Cumulative probabilities = A: 0.500599 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=23 D=18 C=17 E=11 so E is eliminated. Round 2 votes counts: B=31 A=28 D=23 C=18 so C is eliminated. Round 3 votes counts: B=41 D=30 A=29 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:207 E:197 B:193 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 10 12 B 0 0 2 -10 -6 C -6 -2 0 -8 -6 D -10 10 8 0 6 E -12 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500599 B: 0.499401 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000717291 Cumulative probabilities = A: 0.500599 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 10 12 B 0 0 2 -10 -6 C -6 -2 0 -8 -6 D -10 10 8 0 6 E -12 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500599 B: 0.499401 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000717291 Cumulative probabilities = A: 0.500599 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 10 12 B 0 0 2 -10 -6 C -6 -2 0 -8 -6 D -10 10 8 0 6 E -12 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500599 B: 0.499401 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000717291 Cumulative probabilities = A: 0.500599 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2130: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) C D A E B (8) D A B C E (7) E B C A D (5) D A C B E (5) C E B D A (5) C E A D B (4) C A D E B (4) B E A D C (4) B D A E C (4) E B C D A (3) C E D B A (3) B A E D C (3) B A D E C (3) A D B E C (3) A D B C E (3) D C A B E (2) D B A C E (2) C E D A B (2) C E B A D (2) C D E A B (2) C D A B E (2) A E B D C (2) A D C E B (2) E C A D B (1) E C A B D (1) D A C E B (1) D A B E C (1) C D B E A (1) B E D A C (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -12 -6 12 B -4 0 -8 -8 -8 C 12 8 0 6 18 D 6 8 -6 0 8 E -12 8 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 -6 12 B -4 0 -8 -8 -8 C 12 8 0 6 18 D 6 8 -6 0 8 E -12 8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=19 E=18 D=18 A=12 so A is eliminated. Round 2 votes counts: C=34 D=26 E=20 B=20 so E is eliminated. Round 3 votes counts: C=44 B=30 D=26 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:208 A:199 B:186 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -6 12 B -4 0 -8 -8 -8 C 12 8 0 6 18 D 6 8 -6 0 8 E -12 8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -6 12 B -4 0 -8 -8 -8 C 12 8 0 6 18 D 6 8 -6 0 8 E -12 8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -6 12 B -4 0 -8 -8 -8 C 12 8 0 6 18 D 6 8 -6 0 8 E -12 8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2131: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (10) A C B E D (7) E D B A C (5) D C B E A (5) D B E C A (5) E B A C D (4) D E B C A (4) D B C E A (4) D B C A E (4) A E B C D (4) A C E D B (4) D C B A E (3) B A C E D (3) A C E B D (3) A B C E D (3) E D A B C (2) E A D C B (2) E A C B D (2) D E C B A (2) C D B A E (2) B E A D C (2) B D E C A (2) E D A C B (1) E B A D C (1) E A B D C (1) E A B C D (1) D E A C B (1) D C E A B (1) D C A B E (1) C D A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) B E A C D (1) B D E A C (1) B D C A E (1) B C D A E (1) B C A D E (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 0 8 6 B 4 0 2 2 20 C 0 -2 0 4 14 D -8 -2 -4 0 4 E -6 -20 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998614 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 8 6 B 4 0 2 2 20 C 0 -2 0 4 14 D -8 -2 -4 0 4 E -6 -20 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 E=19 C=16 B=12 so B is eliminated. Round 2 votes counts: D=34 A=26 E=22 C=18 so C is eliminated. Round 3 votes counts: A=40 D=38 E=22 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:214 C:208 A:205 D:195 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 8 6 B 4 0 2 2 20 C 0 -2 0 4 14 D -8 -2 -4 0 4 E -6 -20 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 8 6 B 4 0 2 2 20 C 0 -2 0 4 14 D -8 -2 -4 0 4 E -6 -20 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 8 6 B 4 0 2 2 20 C 0 -2 0 4 14 D -8 -2 -4 0 4 E -6 -20 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2132: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (16) D E A C B (9) E D A C B (6) D E A B C (5) B D C E A (5) B C D E A (5) B C D A E (5) B C A D E (5) D E B A C (4) D B E C A (4) C B A E D (4) A C E D B (4) A E C D B (3) E A D C B (2) C D B A E (2) A E C B D (2) A C B E D (2) E D A B C (1) E A D B C (1) E A B C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) C B A D E (1) C A D E B (1) C A B E D (1) C A B D E (1) B E A C D (1) B D E C A (1) B C E A D (1) A E B C D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -12 -2 4 B 12 0 12 6 12 C 12 -12 0 16 18 D 2 -6 -16 0 6 E -4 -12 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -2 4 B 12 0 12 6 12 C 12 -12 0 16 18 D 2 -6 -16 0 6 E -4 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=26 A=14 E=11 C=10 so C is eliminated. Round 2 votes counts: B=44 D=28 A=17 E=11 so E is eliminated. Round 3 votes counts: B=44 D=35 A=21 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:217 D:193 A:189 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -2 4 B 12 0 12 6 12 C 12 -12 0 16 18 D 2 -6 -16 0 6 E -4 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -2 4 B 12 0 12 6 12 C 12 -12 0 16 18 D 2 -6 -16 0 6 E -4 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -2 4 B 12 0 12 6 12 C 12 -12 0 16 18 D 2 -6 -16 0 6 E -4 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2133: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) D E C A B (8) D A E C B (8) D A B E C (8) C E B A D (8) A D B C E (7) A B D C E (7) E C D B A (6) B A C E D (6) B E C A D (4) D E C B A (3) D E A C B (3) D A E B C (3) B C E A D (3) E D C A B (2) E C B A D (2) C B A E D (2) B C A E D (2) A D C E B (2) D A B C E (1) C B E A D (1) B A D E C (1) B A D C E (1) B A C D E (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -2 2 B -2 0 -10 -4 -8 C 0 10 0 -10 -14 D 2 4 10 0 10 E -2 8 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 2 B -2 0 -10 -4 -8 C 0 10 0 -10 -14 D 2 4 10 0 10 E -2 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=19 B=18 A=18 C=11 so C is eliminated. Round 2 votes counts: D=34 E=27 B=21 A=18 so A is eliminated. Round 3 votes counts: D=44 B=29 E=27 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:205 A:201 C:193 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -2 2 B -2 0 -10 -4 -8 C 0 10 0 -10 -14 D 2 4 10 0 10 E -2 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 2 B -2 0 -10 -4 -8 C 0 10 0 -10 -14 D 2 4 10 0 10 E -2 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 2 B -2 0 -10 -4 -8 C 0 10 0 -10 -14 D 2 4 10 0 10 E -2 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2134: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) E C A B D (6) D B E A C (5) B E D C A (5) A C D E B (5) D A B C E (4) A D C E B (4) E D B A C (3) E B D C A (3) E A C D B (3) D E B A C (3) D B A E C (3) C A B E D (3) B D E A C (3) E D A C B (2) E D A B C (2) E B D A C (2) E B C D A (2) E B C A D (2) D A C B E (2) C B A E D (2) B D E C A (2) A C E D B (2) A C D B E (2) E A D C B (1) D E A C B (1) D E A B C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C E B (1) C A E D B (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D C B (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 12 10 -2 -2 B -12 0 0 -4 -14 C -10 0 0 -8 -10 D 2 4 8 0 -8 E 2 14 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 10 -2 -2 B -12 0 0 -4 -14 C -10 0 0 -8 -10 D 2 4 8 0 -8 E 2 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 B=18 C=17 A=16 so A is eliminated. Round 2 votes counts: D=29 E=27 C=26 B=18 so B is eliminated. Round 3 votes counts: D=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:209 D:203 C:186 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 10 -2 -2 B -12 0 0 -4 -14 C -10 0 0 -8 -10 D 2 4 8 0 -8 E 2 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -2 -2 B -12 0 0 -4 -14 C -10 0 0 -8 -10 D 2 4 8 0 -8 E 2 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -2 -2 B -12 0 0 -4 -14 C -10 0 0 -8 -10 D 2 4 8 0 -8 E 2 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2135: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C A E B D (8) E B A D C (7) C D B A E (7) A E B C D (7) C D A B E (6) A C E B D (6) D C B E A (5) D B E C A (5) A E B D C (5) E A B D C (4) C A E D B (4) E B D A C (3) C D E B A (3) A E C B D (3) D B C E A (2) C D B E A (2) B E D A C (2) A B E D C (2) E A B C D (1) D E B C A (1) C A D E B (1) C A D B E (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E D C (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 10 6 8 B 0 0 4 8 -6 C -10 -4 0 2 -6 D -6 -8 -2 0 -10 E -8 6 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.638809 B: 0.361191 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.538535838649 Cumulative probabilities = A: 0.638809 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 6 8 B 0 0 4 8 -6 C -10 -4 0 2 -6 D -6 -8 -2 0 -10 E -8 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=25 D=22 E=15 B=6 so B is eliminated. Round 2 votes counts: C=32 A=26 D=24 E=18 so E is eliminated. Round 3 votes counts: A=39 C=32 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:207 B:203 C:191 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 6 8 B 0 0 4 8 -6 C -10 -4 0 2 -6 D -6 -8 -2 0 -10 E -8 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 6 8 B 0 0 4 8 -6 C -10 -4 0 2 -6 D -6 -8 -2 0 -10 E -8 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 6 8 B 0 0 4 8 -6 C -10 -4 0 2 -6 D -6 -8 -2 0 -10 E -8 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2136: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) C B D E A (6) C A B D E (6) B E C D A (5) D E B C A (4) B E A C D (4) A C B E D (4) E B D A C (3) D C B E A (3) D A C E B (3) C D A B E (3) C B A D E (3) A E D C B (3) A C D B E (3) A B C E D (3) E D A B C (2) E A D B C (2) D E A C B (2) D E A B C (2) D A E C B (2) C B D A E (2) B E C A D (2) A E D B C (2) A E B D C (2) A E B C D (2) E D B A C (1) D E C B A (1) D E C A B (1) D E B A C (1) D C A E B (1) C D B E A (1) C D B A E (1) C B A E D (1) C A D B E (1) C A B E D (1) B E D C A (1) B D E C A (1) B D C E A (1) B C E D A (1) B C D E A (1) B C A E D (1) A D E C B (1) A D C E B (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -10 -2 B 0 0 -4 16 14 C 8 4 0 8 -2 D 10 -16 -8 0 6 E 2 -14 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000167 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 0 -8 -10 -2 B 0 0 -4 16 14 C 8 4 0 8 -2 D 10 -16 -8 0 6 E 2 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000041 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=24 D=20 B=17 E=14 so E is eliminated. Round 2 votes counts: B=26 A=26 C=25 D=23 so D is eliminated. Round 3 votes counts: A=37 B=32 C=31 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 C:209 D:196 E:192 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -10 -2 B 0 0 -4 16 14 C 8 4 0 8 -2 D 10 -16 -8 0 6 E 2 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000041 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -10 -2 B 0 0 -4 16 14 C 8 4 0 8 -2 D 10 -16 -8 0 6 E 2 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000041 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -10 -2 B 0 0 -4 16 14 C 8 4 0 8 -2 D 10 -16 -8 0 6 E 2 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000041 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2137: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C A D B E (8) E D A B C (6) C A D E B (5) E A D C B (4) C A B D E (4) B E C D A (4) B E C A D (4) B C D A E (4) D A C E B (3) D A C B E (3) B E D A C (3) B D E A C (3) B D C A E (3) B C E A D (3) A D C E B (3) E D B A C (2) E B C A D (2) D E A B C (2) D A E B C (2) D A B C E (2) C B A D E (2) C A E D B (2) A C D E B (2) E C A B D (1) E B A D C (1) E A B C D (1) D B A E C (1) D B A C E (1) D A E C B (1) D A B E C (1) C E A B D (1) C D A B E (1) C B D A E (1) B C E D A (1) B C D E A (1) B C A E D (1) A D E C B (1) Total count = 100 A B C D E A 0 6 4 -10 2 B -6 0 16 0 2 C -4 -16 0 -4 2 D 10 0 4 0 8 E -2 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.374091 C: 0.000000 D: 0.625909 E: 0.000000 Sum of squares = 0.531706049926 Cumulative probabilities = A: 0.000000 B: 0.374091 C: 0.374091 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -10 2 B -6 0 16 0 2 C -4 -16 0 -4 2 D 10 0 4 0 8 E -2 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=27 B=27 C=24 D=16 A=6 so A is eliminated. Round 2 votes counts: E=27 B=27 C=26 D=20 so D is eliminated. Round 3 votes counts: C=35 E=33 B=32 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:211 B:206 A:201 E:193 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -10 2 B -6 0 16 0 2 C -4 -16 0 -4 2 D 10 0 4 0 8 E -2 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -10 2 B -6 0 16 0 2 C -4 -16 0 -4 2 D 10 0 4 0 8 E -2 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -10 2 B -6 0 16 0 2 C -4 -16 0 -4 2 D 10 0 4 0 8 E -2 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2138: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (15) D B A C E (13) D B A E C (8) D C B A E (7) C E A B D (7) E C A D B (6) D B C A E (6) D B E A C (5) B A C D E (4) B D A C E (3) E C D A B (2) E B A D C (2) C D E A B (2) B A D E C (2) A B C E D (2) E D B A C (1) E A C D B (1) E A C B D (1) E A B D C (1) E A B C D (1) D E B C A (1) D C E A B (1) C E A D B (1) C D A B E (1) C A E B D (1) B D C A E (1) B A E D C (1) B A D C E (1) A E C B D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -2 -2 6 B 10 0 8 -10 10 C 2 -8 0 -6 0 D 2 10 6 0 10 E -6 -10 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -2 6 B 10 0 8 -10 10 C 2 -8 0 -6 0 D 2 10 6 0 10 E -6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=30 C=12 B=12 A=5 so A is eliminated. Round 2 votes counts: D=41 E=32 B=15 C=12 so C is eliminated. Round 3 votes counts: D=44 E=41 B=15 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:209 A:196 C:194 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 -2 6 B 10 0 8 -10 10 C 2 -8 0 -6 0 D 2 10 6 0 10 E -6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -2 6 B 10 0 8 -10 10 C 2 -8 0 -6 0 D 2 10 6 0 10 E -6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -2 6 B 10 0 8 -10 10 C 2 -8 0 -6 0 D 2 10 6 0 10 E -6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2139: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) E B A C D (8) D C B E A (7) C D A B E (7) D C A E B (6) B E A C D (6) D A C E B (5) B E D C A (5) E B A D C (4) D B E C A (4) B E C D A (4) B E A D C (3) A D C E B (3) A C E B D (3) A C D E B (3) D E B A C (2) C A D B E (2) B C E A D (2) A E D B C (2) A E B C D (2) E A B D C (1) D C E A B (1) D C B A E (1) D A E B C (1) C D B A E (1) C D A E B (1) C B E A D (1) C A D E B (1) C A B E D (1) B E C A D (1) B D E C A (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -10 -10 0 B 0 0 -6 -14 10 C 10 6 0 -12 10 D 10 14 12 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -10 0 B 0 0 -6 -14 10 C 10 6 0 -12 10 D 10 14 12 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=22 A=15 C=14 E=13 so E is eliminated. Round 2 votes counts: D=36 B=34 A=16 C=14 so C is eliminated. Round 3 votes counts: D=45 B=35 A=20 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:207 B:195 A:190 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -10 0 B 0 0 -6 -14 10 C 10 6 0 -12 10 D 10 14 12 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -10 0 B 0 0 -6 -14 10 C 10 6 0 -12 10 D 10 14 12 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -10 0 B 0 0 -6 -14 10 C 10 6 0 -12 10 D 10 14 12 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2140: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) E D C A B (8) C D E A B (7) B A E D C (5) E D C B A (4) D C E A B (4) C E D A B (4) C D A E B (4) B C A D E (4) B A C D E (4) E D B A C (3) E B A D C (3) E A D B C (3) E A B D C (3) C B A D E (3) C A B D E (3) A B D E C (3) E C D B A (2) D E C A B (2) C E D B A (2) C D A B E (2) B E A D C (2) B A D C E (2) B A C E D (2) E D A C B (1) E C D A B (1) D E A C B (1) D A C B E (1) C D E B A (1) C B E A D (1) C B A E D (1) C A D B E (1) B A D E C (1) A D E B C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 -8 -12 -22 B -20 0 -8 -24 -26 C 8 8 0 -12 -4 D 12 24 12 0 -8 E 22 26 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 -8 -12 -22 B -20 0 -8 -24 -26 C 8 8 0 -12 -4 D 12 24 12 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=29 B=20 D=8 A=6 so A is eliminated. Round 2 votes counts: E=37 C=30 B=24 D=9 so D is eliminated. Round 3 votes counts: E=41 C=35 B=24 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:230 D:220 C:200 A:189 B:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 -8 -12 -22 B -20 0 -8 -24 -26 C 8 8 0 -12 -4 D 12 24 12 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -8 -12 -22 B -20 0 -8 -24 -26 C 8 8 0 -12 -4 D 12 24 12 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -8 -12 -22 B -20 0 -8 -24 -26 C 8 8 0 -12 -4 D 12 24 12 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2141: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) A C E D B (8) B E D C A (6) C B A E D (5) B C A D E (5) D E A B C (4) D B E A C (4) B D A C E (4) E C A D B (3) C B E A D (3) C A E D B (3) B D E A C (3) A D E C B (3) E D C A B (2) E D A C B (2) D E B A C (2) D E A C B (2) D A E C B (2) C E B A D (2) C A B E D (2) B C E D A (2) B A C D E (2) A D B C E (2) A C D E B (2) E D B C A (1) E C D B A (1) E C B D A (1) E C B A D (1) E B C D A (1) E A C D B (1) D B A E C (1) D A B E C (1) D A B C E (1) C E A B D (1) C A E B D (1) B D C E A (1) B D C A E (1) B D A E C (1) B C E A D (1) B C D E A (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -6 -6 -8 B 16 0 6 8 10 C 6 -6 0 -6 -4 D 6 -8 6 0 2 E 8 -10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -6 -8 B 16 0 6 8 10 C 6 -6 0 -6 -4 D 6 -8 6 0 2 E 8 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=17 C=17 A=17 E=13 so E is eliminated. Round 2 votes counts: B=37 C=23 D=22 A=18 so A is eliminated. Round 3 votes counts: B=37 C=35 D=28 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:203 E:200 C:195 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -6 -8 B 16 0 6 8 10 C 6 -6 0 -6 -4 D 6 -8 6 0 2 E 8 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -6 -8 B 16 0 6 8 10 C 6 -6 0 -6 -4 D 6 -8 6 0 2 E 8 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -6 -8 B 16 0 6 8 10 C 6 -6 0 -6 -4 D 6 -8 6 0 2 E 8 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2142: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) B A E C D (6) A D B E C (6) E C B A D (5) D C E B A (5) C E B D A (5) A B E D C (5) D E C A B (4) D A E C B (4) A B D E C (4) E D C A B (3) E C D A B (3) D E A C B (3) D C E A B (3) C E D B A (3) E A C B D (2) E A B C D (2) D A E B C (2) C E B A D (2) B C E A D (2) A E B D C (2) A D E B C (2) E D A C B (1) E C D B A (1) E B C A D (1) E A D B C (1) E A C D B (1) D C B A E (1) D C A E B (1) D C A B E (1) D A C B E (1) D A B C E (1) C D B E A (1) C B D E A (1) C B D A E (1) B E A C D (1) B C A E D (1) B C A D E (1) B A D C E (1) B A C E D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -4 -4 -14 B -8 0 -18 -10 -26 C 4 18 0 -4 -20 D 4 10 4 0 2 E 14 26 20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -4 -14 B -8 0 -18 -10 -26 C 4 18 0 -4 -20 D 4 10 4 0 2 E 14 26 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=21 E=20 C=20 B=13 so B is eliminated. Round 2 votes counts: A=29 D=26 C=24 E=21 so E is eliminated. Round 3 votes counts: A=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:229 D:210 C:199 A:193 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -4 -4 -14 B -8 0 -18 -10 -26 C 4 18 0 -4 -20 D 4 10 4 0 2 E 14 26 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -4 -14 B -8 0 -18 -10 -26 C 4 18 0 -4 -20 D 4 10 4 0 2 E 14 26 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -4 -14 B -8 0 -18 -10 -26 C 4 18 0 -4 -20 D 4 10 4 0 2 E 14 26 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2143: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (18) E B D A C (13) A C D B E (8) C D A B E (6) E B A D C (5) D B A C E (5) B D E A C (5) E C A D B (4) E C A B D (3) A D C B E (3) E C D B A (2) E B D C A (2) E A C B D (2) E A B C D (2) D B C A E (2) D A C B E (2) C E A D B (2) C A E D B (2) C A D E B (2) B E D A C (2) A D B C E (2) E D B C A (1) D B C E A (1) C D B A E (1) B D A E C (1) B D A C E (1) A E B D C (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 18 8 12 12 B -18 0 -14 -24 16 C -8 14 0 8 14 D -12 24 -8 0 16 E -12 -16 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 8 12 12 B -18 0 -14 -24 16 C -8 14 0 8 14 D -12 24 -8 0 16 E -12 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=31 A=16 D=10 B=9 so B is eliminated. Round 2 votes counts: E=36 C=31 D=17 A=16 so A is eliminated. Round 3 votes counts: C=41 E=37 D=22 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:225 C:214 D:210 B:180 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 8 12 12 B -18 0 -14 -24 16 C -8 14 0 8 14 D -12 24 -8 0 16 E -12 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 8 12 12 B -18 0 -14 -24 16 C -8 14 0 8 14 D -12 24 -8 0 16 E -12 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 8 12 12 B -18 0 -14 -24 16 C -8 14 0 8 14 D -12 24 -8 0 16 E -12 -16 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2144: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) D C A E B (10) B E A D C (10) B C D A E (10) E A B D C (5) C D B A E (5) C D A E B (5) B D C A E (5) A E D C B (4) C D E A B (3) B E C A D (3) B E A C D (3) A D E C B (3) E C A D B (2) E B A D C (2) E B A C D (2) D B C A E (2) B E C D A (2) B C D E A (2) B A E D C (2) E A D B C (1) E A C D B (1) D C B A E (1) D A C E B (1) C D A B E (1) C B D A E (1) B C E D A (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 -6 2 4 B 4 0 4 0 -2 C 6 -4 0 -18 -4 D -2 0 18 0 0 E -4 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.35999999993 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -4 -6 2 4 B 4 0 4 0 -2 C 6 -4 0 -18 -4 D -2 0 18 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999896 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=24 C=15 D=14 A=8 so A is eliminated. Round 2 votes counts: B=39 E=29 D=17 C=15 so C is eliminated. Round 3 votes counts: B=40 D=31 E=29 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:208 B:203 E:201 A:198 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 2 4 B 4 0 4 0 -2 C 6 -4 0 -18 -4 D -2 0 18 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999896 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 4 B 4 0 4 0 -2 C 6 -4 0 -18 -4 D -2 0 18 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999896 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 4 B 4 0 4 0 -2 C 6 -4 0 -18 -4 D -2 0 18 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999896 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2145: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) A B E D C (6) C D E B A (5) E D A B C (4) E B C A D (4) E B A D C (4) E B A C D (4) B E A C D (4) B C A E D (4) B A E C D (4) E D C B A (3) D A C B E (3) A C D B E (3) A C B D E (3) E D B A C (2) E D A C B (2) E B D C A (2) E B D A C (2) E B C D A (2) D E C A B (2) D C E B A (2) D C A E B (2) C B A D E (2) B E C A D (2) A D C B E (2) A D B C E (2) A B D E C (2) E D B C A (1) E C B D A (1) D E C B A (1) D E A C B (1) D C E A B (1) D C A B E (1) C E B D A (1) C D B E A (1) C B E D A (1) C A B D E (1) B C E A D (1) B A C D E (1) A E B D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 4 4 -6 B 8 0 8 8 6 C -4 -8 0 8 -10 D -4 -8 -8 0 -12 E 6 -6 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 4 -6 B 8 0 8 8 6 C -4 -8 0 8 -10 D -4 -8 -8 0 -12 E 6 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=21 C=19 B=16 D=13 so D is eliminated. Round 2 votes counts: E=35 C=25 A=24 B=16 so B is eliminated. Round 3 votes counts: E=41 C=30 A=29 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:211 A:197 C:193 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 4 -6 B 8 0 8 8 6 C -4 -8 0 8 -10 D -4 -8 -8 0 -12 E 6 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 4 -6 B 8 0 8 8 6 C -4 -8 0 8 -10 D -4 -8 -8 0 -12 E 6 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 4 -6 B 8 0 8 8 6 C -4 -8 0 8 -10 D -4 -8 -8 0 -12 E 6 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2146: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (16) A D C E B (13) B E C A D (11) D A B E C (9) C E B A D (7) A C E B D (7) E B C A D (5) C E B D A (4) A D B E C (4) D B E C A (3) D A C E B (3) D A C B E (2) C A E B D (2) A E B C D (2) A C D E B (2) E C B A D (1) E B C D A (1) D C A E B (1) C D E B A (1) B E D C A (1) A E C B D (1) A D E C B (1) A C E D B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -6 18 0 B 0 0 8 20 -4 C 6 -8 0 24 -14 D -18 -20 -24 0 -22 E 0 4 14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.432963 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.567037 Sum of squares = 0.508987795032 Cumulative probabilities = A: 0.432963 B: 0.432963 C: 0.432963 D: 0.432963 E: 1.000000 A B C D E A 0 0 -6 18 0 B 0 0 8 20 -4 C 6 -8 0 24 -14 D -18 -20 -24 0 -22 E 0 4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=28 D=18 C=14 E=7 so E is eliminated. Round 2 votes counts: B=34 A=33 D=18 C=15 so C is eliminated. Round 3 votes counts: B=46 A=35 D=19 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:220 B:212 A:206 C:204 D:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 18 0 B 0 0 8 20 -4 C 6 -8 0 24 -14 D -18 -20 -24 0 -22 E 0 4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 18 0 B 0 0 8 20 -4 C 6 -8 0 24 -14 D -18 -20 -24 0 -22 E 0 4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 18 0 B 0 0 8 20 -4 C 6 -8 0 24 -14 D -18 -20 -24 0 -22 E 0 4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2147: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (14) E B D C A (12) C A B D E (11) E D B A C (9) D B E A C (7) E B C D A (4) C A E D B (4) C A B E D (4) A D C B E (4) E B D A C (3) D E B A C (3) D A B C E (3) C E B A D (3) A D B C E (3) D A B E C (2) C B A E D (2) E C B D A (1) E C B A D (1) E C A D B (1) D B A E C (1) D B A C E (1) D A E C B (1) C E A B D (1) C A E B D (1) C A D E B (1) C A D B E (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 4 4 4 8 B -4 0 -2 -12 8 C -4 2 0 0 10 D -4 12 0 0 6 E -8 -8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 4 8 B -4 0 -2 -12 8 C -4 2 0 0 10 D -4 12 0 0 6 E -8 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=28 A=22 D=18 B=1 so B is eliminated. Round 2 votes counts: E=31 C=29 A=22 D=18 so D is eliminated. Round 3 votes counts: E=41 A=30 C=29 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:207 C:204 B:195 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 4 8 B -4 0 -2 -12 8 C -4 2 0 0 10 D -4 12 0 0 6 E -8 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 8 B -4 0 -2 -12 8 C -4 2 0 0 10 D -4 12 0 0 6 E -8 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 8 B -4 0 -2 -12 8 C -4 2 0 0 10 D -4 12 0 0 6 E -8 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2148: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (13) A D B C E (13) D A E C B (8) A B D C E (8) E D C A B (7) E C B D A (6) A D B E C (6) C B E A D (5) D E A C B (4) D A E B C (4) B C A E D (4) E C D B A (3) E C D A B (3) C E B D A (3) B A C D E (3) E D A C B (2) D A B E C (2) B A D C E (2) D E A B C (1) C E B A D (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 10 14 4 B -20 0 14 -8 16 C -10 -14 0 -16 8 D -14 8 16 0 6 E -4 -16 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 14 4 B -20 0 14 -8 16 C -10 -14 0 -16 8 D -14 8 16 0 6 E -4 -16 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=22 E=21 D=19 C=9 so C is eliminated. Round 2 votes counts: A=29 B=27 E=25 D=19 so D is eliminated. Round 3 votes counts: A=43 E=30 B=27 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:208 B:201 C:184 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 14 4 B -20 0 14 -8 16 C -10 -14 0 -16 8 D -14 8 16 0 6 E -4 -16 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 14 4 B -20 0 14 -8 16 C -10 -14 0 -16 8 D -14 8 16 0 6 E -4 -16 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 14 4 B -20 0 14 -8 16 C -10 -14 0 -16 8 D -14 8 16 0 6 E -4 -16 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2149: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) B C E D A (6) D E A B C (5) C B E D A (5) C A B E D (5) E D C B A (4) B D E C A (4) A C E B D (4) A B C D E (4) D E B A C (3) B C A D E (3) A D E C B (3) A C E D B (3) E D C A B (2) E C D B A (2) E B D C A (2) E B C D A (2) D B E C A (2) D B E A C (2) C E B D A (2) B A D C E (2) B A C D E (2) A D C E B (2) A C B D E (2) A B D C E (2) E D B C A (1) E D A C B (1) E C D A B (1) E A C D B (1) D E B C A (1) D E A C B (1) D B A E C (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) B E D C A (1) B D C E A (1) B C A E D (1) A E D C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 2 0 2 -2 B -2 0 -6 16 10 C 0 6 0 12 16 D -2 -16 -12 0 -12 E 2 -10 -16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.505434 B: 0.000000 C: 0.494566 D: 0.000000 E: 0.000000 Sum of squares = 0.500059051746 Cumulative probabilities = A: 0.505434 B: 0.505434 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 -2 B -2 0 -6 16 10 C 0 6 0 12 16 D -2 -16 -12 0 -12 E 2 -10 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=20 C=17 E=16 D=15 so D is eliminated. Round 2 votes counts: A=32 E=26 B=25 C=17 so C is eliminated. Round 3 votes counts: A=39 B=32 E=29 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 B:209 A:201 E:194 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 2 -2 B -2 0 -6 16 10 C 0 6 0 12 16 D -2 -16 -12 0 -12 E 2 -10 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 -2 B -2 0 -6 16 10 C 0 6 0 12 16 D -2 -16 -12 0 -12 E 2 -10 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 -2 B -2 0 -6 16 10 C 0 6 0 12 16 D -2 -16 -12 0 -12 E 2 -10 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2150: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (15) A B D E C (14) E D C B A (10) E C D B A (6) A B C D E (5) C E D A B (4) A B D C E (4) D E C B A (3) A B E D C (3) E D C A B (2) E D A C B (2) C D E B A (2) C D B E A (2) C B D E A (2) C B D A E (2) C A B E D (2) B A D E C (2) B A D C E (2) A E C B D (2) E D B C A (1) E D A B C (1) E A D B C (1) E A C D B (1) D E B C A (1) D E B A C (1) D B E A C (1) D B A E C (1) C E A D B (1) C E A B D (1) C B A D E (1) C A E B D (1) B D C A E (1) B A C D E (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -14 -14 -14 B 8 0 -20 -12 -10 C 14 20 0 -2 -6 D 14 12 2 0 -8 E 14 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -14 -14 -14 B 8 0 -20 -12 -10 C 14 20 0 -2 -6 D 14 12 2 0 -8 E 14 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=30 E=24 D=7 B=6 so B is eliminated. Round 2 votes counts: A=35 C=33 E=24 D=8 so D is eliminated. Round 3 votes counts: A=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:219 C:213 D:210 B:183 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 -14 -14 B 8 0 -20 -12 -10 C 14 20 0 -2 -6 D 14 12 2 0 -8 E 14 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -14 -14 B 8 0 -20 -12 -10 C 14 20 0 -2 -6 D 14 12 2 0 -8 E 14 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -14 -14 B 8 0 -20 -12 -10 C 14 20 0 -2 -6 D 14 12 2 0 -8 E 14 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2151: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) E C D B A (6) E D C B A (4) E D B A C (4) C E B D A (4) A B D C E (4) E C B D A (3) E B D C A (3) D E A C B (3) D C A E B (3) D A E B C (3) C E D B A (3) C D A E B (3) C B E D A (3) B C A E D (3) A B C D E (3) E D C A B (2) E D B C A (2) D E C A B (2) D E A B C (2) D C E A B (2) D A E C B (2) C A D E B (2) B E C D A (2) B A C D E (2) A D C E B (2) A B D E C (2) E D A B C (1) E B D A C (1) E B C D A (1) D E B A C (1) D A C E B (1) C D E A B (1) C B E A D (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A C D (1) B A D E C (1) B A D C E (1) B A C E D (1) A D E B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -6 -24 -4 B 10 0 -4 -8 -24 C 6 4 0 -14 -10 D 24 8 14 0 -4 E 4 24 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 -24 -4 B 10 0 -4 -8 -24 C 6 4 0 -14 -10 D 24 8 14 0 -4 E 4 24 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=20 D=19 B=18 A=16 so A is eliminated. Round 2 votes counts: E=27 B=27 D=25 C=21 so C is eliminated. Round 3 votes counts: E=34 B=34 D=32 so D is eliminated. Round 4 votes counts: E=62 B=38 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:221 C:193 B:187 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 -24 -4 B 10 0 -4 -8 -24 C 6 4 0 -14 -10 D 24 8 14 0 -4 E 4 24 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -24 -4 B 10 0 -4 -8 -24 C 6 4 0 -14 -10 D 24 8 14 0 -4 E 4 24 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -24 -4 B 10 0 -4 -8 -24 C 6 4 0 -14 -10 D 24 8 14 0 -4 E 4 24 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2152: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (7) A D C E B (7) A B E C D (6) D E C B A (5) D A C E B (5) C D E B A (5) E B D C A (4) D C E B A (4) B E A C D (4) E B C D A (3) E B A D C (3) B A E C D (3) A B E D C (3) A B D C E (3) A B C D E (3) E D C B A (2) D C E A B (2) D C A E B (2) C B D E A (2) B E C A D (2) B A C E D (2) A D E C B (2) A D E B C (2) A D C B E (2) A D B C E (2) A C D B E (2) A C B D E (2) E D B C A (1) E C D B A (1) E C B D A (1) D E A C B (1) C E D B A (1) C D A E B (1) C D A B E (1) C B E D A (1) B C E D A (1) B C E A D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 6 0 -2 B 6 0 2 4 -4 C -6 -2 0 -2 -2 D 0 -4 2 0 8 E 2 4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 -6 6 0 -2 B 6 0 2 4 -4 C -6 -2 0 -2 -2 D 0 -4 2 0 8 E 2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000034 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=20 D=19 E=15 C=11 so C is eliminated. Round 2 votes counts: A=35 D=26 B=23 E=16 so E is eliminated. Round 3 votes counts: A=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:204 D:203 E:200 A:199 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 0 -2 B 6 0 2 4 -4 C -6 -2 0 -2 -2 D 0 -4 2 0 8 E 2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000034 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 0 -2 B 6 0 2 4 -4 C -6 -2 0 -2 -2 D 0 -4 2 0 8 E 2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000034 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 0 -2 B 6 0 2 4 -4 C -6 -2 0 -2 -2 D 0 -4 2 0 8 E 2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000034 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2153: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (6) B D E C A (6) C D E A B (5) B E D A C (5) B C D E A (5) B A C E D (5) E D A C B (4) E D A B C (4) D E B C A (4) C A D E B (4) B E D C A (4) D E C B A (3) C B D E A (3) B C A D E (3) B A E D C (3) A E D C B (3) A C E D B (3) A C B D E (3) A B C E D (3) E D B C A (2) E B D A C (2) D E C A B (2) C D E B A (2) C B A D E (2) A E B D C (2) E D C A B (1) E A D B C (1) D C E B A (1) C D B E A (1) C D B A E (1) C A D B E (1) B A E C D (1) B A C D E (1) A E D B C (1) A E C D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -12 -10 -10 B 8 0 6 12 8 C 12 -6 0 4 0 D 10 -12 -4 0 6 E 10 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -10 -10 B 8 0 6 12 8 C 12 -6 0 4 0 D 10 -12 -4 0 6 E 10 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=25 A=18 E=14 D=10 so D is eliminated. Round 2 votes counts: B=33 C=26 E=23 A=18 so A is eliminated. Round 3 votes counts: B=37 C=33 E=30 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:205 D:200 E:198 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -10 -10 B 8 0 6 12 8 C 12 -6 0 4 0 D 10 -12 -4 0 6 E 10 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -10 -10 B 8 0 6 12 8 C 12 -6 0 4 0 D 10 -12 -4 0 6 E 10 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -10 -10 B 8 0 6 12 8 C 12 -6 0 4 0 D 10 -12 -4 0 6 E 10 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2154: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) A E B D C (10) E C A D B (7) E A C D B (7) C D B E A (7) C E D B A (6) B D A C E (6) C E D A B (4) D B C E A (3) B D C E A (3) A E C B D (3) A B E D C (3) A B D E C (3) A B D C E (3) E A C B D (2) E A B D C (2) D B C A E (2) C D E B A (2) E D B C A (1) E C D B A (1) E B D A C (1) E B A D C (1) D C B E A (1) D C B A E (1) C D B A E (1) C D A B E (1) B D A E C (1) B A D E C (1) B A D C E (1) A E C D B (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 -8 4 B 4 0 12 10 2 C 6 -12 0 -12 10 D 8 -10 12 0 0 E -4 -2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -8 4 B 4 0 12 10 2 C 6 -12 0 -12 10 D 8 -10 12 0 0 E -4 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=25 A=25 E=22 C=21 D=7 so D is eliminated. Round 2 votes counts: B=30 A=25 C=23 E=22 so E is eliminated. Round 3 votes counts: A=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:205 C:196 A:193 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -8 4 B 4 0 12 10 2 C 6 -12 0 -12 10 D 8 -10 12 0 0 E -4 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -8 4 B 4 0 12 10 2 C 6 -12 0 -12 10 D 8 -10 12 0 0 E -4 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -8 4 B 4 0 12 10 2 C 6 -12 0 -12 10 D 8 -10 12 0 0 E -4 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2155: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) D C B A E (7) E B C D A (6) B D C E A (6) E B D C A (5) E A B C D (5) A C D B E (5) C D B E A (4) C D B A E (4) E B A C D (3) D C A B E (3) A D C E B (3) A C D E B (3) E B A D C (2) D B C E A (2) D B C A E (2) D A C B E (2) C D A B E (2) C B D E A (2) A E D C B (2) A E C B D (2) A E B D C (2) A D C B E (2) E C B D A (1) E C B A D (1) E A C B D (1) E A B D C (1) D C B E A (1) C E A B D (1) C D A E B (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A E C (1) B C D E A (1) A E D B C (1) A E B C D (1) A D E B C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 -6 10 B 2 0 -14 -12 -4 C 2 14 0 6 6 D 6 12 -6 0 10 E -10 4 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 10 B 2 0 -14 -12 -4 C 2 14 0 6 6 D 6 12 -6 0 10 E -10 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=25 D=17 C=14 B=11 so B is eliminated. Round 2 votes counts: A=33 E=26 D=26 C=15 so C is eliminated. Round 3 votes counts: D=40 A=33 E=27 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:211 A:200 E:189 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 10 B 2 0 -14 -12 -4 C 2 14 0 6 6 D 6 12 -6 0 10 E -10 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 10 B 2 0 -14 -12 -4 C 2 14 0 6 6 D 6 12 -6 0 10 E -10 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 10 B 2 0 -14 -12 -4 C 2 14 0 6 6 D 6 12 -6 0 10 E -10 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2156: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) D B A E C (9) C E A B D (6) D A B C E (5) E C B D A (4) B E C A D (4) E C D A B (3) E C A D B (3) D A E C B (3) C E B A D (3) B A C E D (3) A C E D B (3) E C D B A (2) E C A B D (2) E B C D A (2) D E B C A (2) D B A C E (2) C E A D B (2) B E D C A (2) B D E A C (2) B D A E C (2) B C E A D (2) B A C D E (2) A D C E B (2) A D C B E (2) A B D C E (2) A B C D E (2) E D C B A (1) E D C A B (1) D E C B A (1) D E C A B (1) D B E C A (1) D A C B E (1) D A B E C (1) C A E D B (1) B E C D A (1) B D E C A (1) B D A C E (1) A D B C E (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -8 4 -10 B 12 0 -4 4 -4 C 8 4 0 14 -14 D -4 -4 -14 0 -12 E 10 4 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -8 4 -10 B 12 0 -4 4 -4 C 8 4 0 14 -14 D -4 -4 -14 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 B=20 A=15 C=12 so C is eliminated. Round 2 votes counts: E=38 D=26 B=20 A=16 so A is eliminated. Round 3 votes counts: E=42 D=32 B=26 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:206 B:204 A:187 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 4 -10 B 12 0 -4 4 -4 C 8 4 0 14 -14 D -4 -4 -14 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 4 -10 B 12 0 -4 4 -4 C 8 4 0 14 -14 D -4 -4 -14 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 4 -10 B 12 0 -4 4 -4 C 8 4 0 14 -14 D -4 -4 -14 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2157: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (9) B A E C D (9) D C A E B (8) B A C E D (7) C A D B E (6) B E A C D (6) D C A B E (5) C A B D E (5) A C B D E (5) E D B A C (4) E B D A C (4) E B A C D (3) D E C A B (3) A B C D E (3) E D C B A (2) D E C B A (2) D C E B A (2) D C E A B (2) E D B C A (1) B C A E D (1) A E B C D (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 8 16 24 B 0 0 -2 10 18 C -8 2 0 16 12 D -16 -10 -16 0 2 E -24 -18 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.616581 B: 0.383419 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.527182287251 Cumulative probabilities = A: 0.616581 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 16 24 B 0 0 -2 10 18 C -8 2 0 16 12 D -16 -10 -16 0 2 E -24 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=23 B=23 D=22 C=20 A=12 so A is eliminated. Round 2 votes counts: C=27 B=27 E=24 D=22 so D is eliminated. Round 3 votes counts: C=44 E=29 B=27 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:224 B:213 C:211 D:180 E:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 16 24 B 0 0 -2 10 18 C -8 2 0 16 12 D -16 -10 -16 0 2 E -24 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 16 24 B 0 0 -2 10 18 C -8 2 0 16 12 D -16 -10 -16 0 2 E -24 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 16 24 B 0 0 -2 10 18 C -8 2 0 16 12 D -16 -10 -16 0 2 E -24 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2158: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) B A D E C (11) C D E A B (6) E C D A B (4) D E C A B (4) C D E B A (4) C B E D A (4) B A C E D (4) E A D C B (3) D A E B C (3) C E D B A (3) B C A D E (3) A B D E C (3) E D C A B (2) E D A C B (2) E C A D B (2) D B C E A (2) B C D A E (2) B A D C E (2) A D E B C (2) E A C D B (1) D E A C B (1) D C E B A (1) D C B E A (1) D A B E C (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E A D (1) C B A E D (1) B D C E A (1) B D A C E (1) B C E A D (1) B C D E A (1) B C A E D (1) B A E D C (1) B A E C D (1) A E D B C (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -20 -14 -20 B -6 0 -12 -16 -10 C 20 12 0 12 10 D 14 16 -12 0 0 E 20 10 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -20 -14 -20 B -6 0 -12 -16 -10 C 20 12 0 12 10 D 14 16 -12 0 0 E 20 10 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=29 E=14 D=13 A=9 so A is eliminated. Round 2 votes counts: C=35 B=34 D=16 E=15 so E is eliminated. Round 3 votes counts: C=42 B=34 D=24 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:210 D:209 B:178 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -20 -14 -20 B -6 0 -12 -16 -10 C 20 12 0 12 10 D 14 16 -12 0 0 E 20 10 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -20 -14 -20 B -6 0 -12 -16 -10 C 20 12 0 12 10 D 14 16 -12 0 0 E 20 10 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -20 -14 -20 B -6 0 -12 -16 -10 C 20 12 0 12 10 D 14 16 -12 0 0 E 20 10 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2159: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (18) B C A E D (18) C A B D E (8) B C A D E (8) E D B A C (6) E D A C B (6) E D A B C (5) C B A D E (4) B E D C A (3) B E C D A (3) D A E C B (2) C B A E D (2) A E D C B (2) E B D A C (1) D E B A C (1) D E A B C (1) D B E C A (1) C A D E B (1) C A D B E (1) B E D A C (1) B C E D A (1) B C E A D (1) B A E C D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -2 2 4 B 2 0 2 4 4 C 2 -2 0 2 -4 D -2 -4 -2 0 -4 E -4 -4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 2 4 B 2 0 2 4 4 C 2 -2 0 2 -4 D -2 -4 -2 0 -4 E -4 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=23 E=18 C=16 A=7 so A is eliminated. Round 2 votes counts: B=36 D=25 E=20 C=19 so C is eliminated. Round 3 votes counts: B=50 D=28 E=22 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:206 A:201 E:200 C:199 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 2 4 B 2 0 2 4 4 C 2 -2 0 2 -4 D -2 -4 -2 0 -4 E -4 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 4 B 2 0 2 4 4 C 2 -2 0 2 -4 D -2 -4 -2 0 -4 E -4 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 4 B 2 0 2 4 4 C 2 -2 0 2 -4 D -2 -4 -2 0 -4 E -4 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2160: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) A E B D C (7) D C A E B (5) C D E B A (5) A D E C B (5) D A C E B (4) C D B E A (4) A D B C E (4) A B E D C (4) E A B C D (3) B E A C D (3) E C D B A (2) E B C D A (2) E B C A D (2) E B A C D (2) E A D C B (2) D C E A B (2) D C B E A (2) C E D B A (2) C B D E A (2) B E C D A (2) B E C A D (2) B C E D A (2) B C E A D (2) B C D E A (2) B C D A E (2) A E D C B (2) A D E B C (2) A B D E C (2) E C D A B (1) D E A C B (1) D C E B A (1) D C B A E (1) D B C A E (1) C E B D A (1) C D E A B (1) B C A E D (1) A D C B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 10 4 B 4 0 12 4 -4 C -6 -12 0 6 -8 D -10 -4 -6 0 -4 E -4 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -4 6 10 4 B 4 0 12 4 -4 C -6 -12 0 6 -8 D -10 -4 -6 0 -4 E -4 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=25 D=17 C=15 E=14 so E is eliminated. Round 2 votes counts: A=34 B=31 C=18 D=17 so D is eliminated. Round 3 votes counts: A=39 B=32 C=29 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:208 B:208 E:206 C:190 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 6 10 4 B 4 0 12 4 -4 C -6 -12 0 6 -8 D -10 -4 -6 0 -4 E -4 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 10 4 B 4 0 12 4 -4 C -6 -12 0 6 -8 D -10 -4 -6 0 -4 E -4 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 10 4 B 4 0 12 4 -4 C -6 -12 0 6 -8 D -10 -4 -6 0 -4 E -4 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2161: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (12) B D A E C (7) D B C E A (6) C E A D B (6) E C A B D (5) C E D A B (5) C A E D B (5) B D E A C (4) E A C B D (3) D C B E A (3) C E B D A (3) B A D E C (3) A E C B D (3) A C E D B (3) A B D E C (3) E C B A D (2) E B C D A (2) E B A C D (2) D C B A E (2) D B C A E (2) C E D B A (2) C D B E A (2) B D E C A (2) A B E D C (2) E C B D A (1) E A B C D (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A D C (1) A E B D C (1) A E B C D (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -16 0 -10 -4 B 16 0 6 -4 8 C 0 -6 0 -6 8 D 10 4 6 0 0 E 4 -8 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800604 E: 0.199396 Sum of squares = 0.680725374298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800604 E: 1.000000 A B C D E A 0 -16 0 -10 -4 B 16 0 6 -4 8 C 0 -6 0 -6 8 D 10 4 6 0 0 E 4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 B=19 E=16 A=16 so E is eliminated. Round 2 votes counts: C=32 D=25 B=23 A=20 so A is eliminated. Round 3 votes counts: C=41 B=31 D=28 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:210 C:198 E:194 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 0 -10 -4 B 16 0 6 -4 8 C 0 -6 0 -6 8 D 10 4 6 0 0 E 4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -10 -4 B 16 0 6 -4 8 C 0 -6 0 -6 8 D 10 4 6 0 0 E 4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -10 -4 B 16 0 6 -4 8 C 0 -6 0 -6 8 D 10 4 6 0 0 E 4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2162: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) D B E C A (7) A B E C D (7) D C E B A (5) A C E B D (5) D A C B E (4) C D E A B (4) A C D E B (4) E C B A D (3) C D E B A (3) C D A E B (3) C A E D B (3) A C D B E (3) E B D C A (2) D E B C A (2) D C A B E (2) D B A E C (2) C E D A B (2) C A D E B (2) B D E A C (2) B A E D C (2) A D B C E (2) A B D C E (2) E C D B A (1) E B C D A (1) E B C A D (1) D C B E A (1) D B E A C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E D B A (1) C E A D B (1) C E A B D (1) B E A D C (1) B E A C D (1) B D E C A (1) B A E C D (1) A D C E B (1) A C E D B (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 24 -10 -12 18 B -24 0 -22 -34 -12 C 10 22 0 0 26 D 12 34 0 0 30 E -18 12 -26 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.562542 D: 0.437458 E: 0.000000 Sum of squares = 0.507822900172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.562542 D: 1.000000 E: 1.000000 A B C D E A 0 24 -10 -12 18 B -24 0 -22 -34 -12 C 10 22 0 0 26 D 12 34 0 0 30 E -18 12 -26 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=28 C=20 E=8 B=8 so E is eliminated. Round 2 votes counts: D=36 A=28 C=24 B=12 so B is eliminated. Round 3 votes counts: D=41 A=33 C=26 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:238 C:229 A:210 E:169 B:154 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -10 -12 18 B -24 0 -22 -34 -12 C 10 22 0 0 26 D 12 34 0 0 30 E -18 12 -26 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -10 -12 18 B -24 0 -22 -34 -12 C 10 22 0 0 26 D 12 34 0 0 30 E -18 12 -26 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -10 -12 18 B -24 0 -22 -34 -12 C 10 22 0 0 26 D 12 34 0 0 30 E -18 12 -26 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2163: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) C A B D E (6) B D A E C (6) E C D A B (5) B A D C E (5) E C B D A (4) C E A D B (4) B E C A D (4) E D A B C (3) E C B A D (3) E B D A C (3) E B C D A (3) D A E B C (3) A D C B E (3) A D B C E (3) D B A E C (2) D A C E B (2) C E B A D (2) C B E A D (2) B E D A C (2) B A C D E (2) E D C A B (1) E D B C A (1) E D B A C (1) E C D B A (1) E C A D B (1) E B C A D (1) D E B A C (1) D E A B C (1) D A E C B (1) D A C B E (1) D A B E C (1) C E D A B (1) C E A B D (1) C B A E D (1) C B A D E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E A D C (1) B D E A C (1) B C A D E (1) B A D E C (1) A C D B E (1) Total count = 100 A B C D E A 0 2 8 -8 6 B -2 0 12 0 10 C -8 -12 0 -6 -2 D 8 0 6 0 8 E -6 -10 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.429227 C: 0.000000 D: 0.570773 E: 0.000000 Sum of squares = 0.510017510202 Cumulative probabilities = A: 0.000000 B: 0.429227 C: 0.429227 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -8 6 B -2 0 12 0 10 C -8 -12 0 -6 -2 D 8 0 6 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 D=22 C=21 A=7 so A is eliminated. Round 2 votes counts: D=28 E=27 B=23 C=22 so C is eliminated. Round 3 votes counts: E=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:210 A:204 E:189 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -8 6 B -2 0 12 0 10 C -8 -12 0 -6 -2 D 8 0 6 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -8 6 B -2 0 12 0 10 C -8 -12 0 -6 -2 D 8 0 6 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -8 6 B -2 0 12 0 10 C -8 -12 0 -6 -2 D 8 0 6 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2164: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (7) E A C D B (6) D C E B A (6) D C B E A (5) B D C A E (5) E D A C B (4) D E C A B (4) A B E C D (4) C E A D B (3) B D A E C (3) B D A C E (3) B C A D E (3) B A C E D (3) A E B C D (3) E C D A B (2) E A D C B (2) E A C B D (2) D E A B C (2) D B C E A (2) C D E B A (2) C D E A B (2) C D B E A (2) C B D E A (2) C B A E D (2) B C D A E (2) B A E D C (2) B A D E C (2) A E D B C (2) A E B D C (2) A B E D C (2) E D C A B (1) E C A D B (1) D C E A B (1) D B E A C (1) D B C A E (1) B A D C E (1) A E C D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 8 0 0 B -6 0 -12 0 -6 C -8 12 0 -2 -6 D 0 0 2 0 -2 E 0 6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.475448 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.524552 Sum of squares = 0.501205603897 Cumulative probabilities = A: 0.475448 B: 0.475448 C: 0.475448 D: 0.475448 E: 1.000000 A B C D E A 0 6 8 0 0 B -6 0 -12 0 -6 C -8 12 0 -2 -6 D 0 0 2 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 D=22 E=18 C=13 so C is eliminated. Round 2 votes counts: D=28 B=28 A=23 E=21 so E is eliminated. Round 3 votes counts: A=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:207 D:200 C:198 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 0 0 B -6 0 -12 0 -6 C -8 12 0 -2 -6 D 0 0 2 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 0 0 B -6 0 -12 0 -6 C -8 12 0 -2 -6 D 0 0 2 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 0 0 B -6 0 -12 0 -6 C -8 12 0 -2 -6 D 0 0 2 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2165: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (13) C D A B E (9) E B D A C (7) B E D C A (7) E B D C A (6) A C E B D (6) A C D B E (5) D C B E A (4) A E B C D (4) A C D E B (4) E A B D C (3) D C B A E (3) D B E C A (3) D B C E A (3) E B A D C (2) D E B C A (2) D C A B E (2) C B D E A (2) C A B E D (2) A C E D B (2) E D B C A (1) D E B A C (1) C D B A E (1) C B A D E (1) B E C D A (1) B C E D A (1) B C E A D (1) A E D B C (1) A E C B D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 -24 -6 12 B -8 0 -14 -12 16 C 24 14 0 6 20 D 6 12 -6 0 10 E -12 -16 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -24 -6 12 B -8 0 -14 -12 16 C 24 14 0 6 20 D 6 12 -6 0 10 E -12 -16 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=25 E=19 D=18 B=10 so B is eliminated. Round 2 votes counts: C=30 E=27 A=25 D=18 so D is eliminated. Round 3 votes counts: C=42 E=33 A=25 so A is eliminated. Round 4 votes counts: C=60 E=40 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:232 D:211 A:195 B:191 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -24 -6 12 B -8 0 -14 -12 16 C 24 14 0 6 20 D 6 12 -6 0 10 E -12 -16 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -24 -6 12 B -8 0 -14 -12 16 C 24 14 0 6 20 D 6 12 -6 0 10 E -12 -16 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -24 -6 12 B -8 0 -14 -12 16 C 24 14 0 6 20 D 6 12 -6 0 10 E -12 -16 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2166: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) A B D E C (9) A D B C E (7) E C B D A (6) B A E D C (5) E B C A D (4) E B A C D (4) D C A B E (4) C D E B A (4) B E A D C (4) B A E C D (4) A B E D C (4) A B D C E (4) D C A E B (3) D A C B E (3) C D E A B (3) C D A E B (3) A D B E C (3) E C B A D (2) D C E A B (2) D A B C E (2) C E D A B (2) B E A C D (2) E C D B A (1) C E A B D (1) B A D E C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 8 12 8 B -4 0 8 2 8 C -8 -8 0 -2 0 D -12 -2 2 0 -2 E -8 -8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 12 8 B -4 0 8 2 8 C -8 -8 0 -2 0 D -12 -2 2 0 -2 E -8 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=24 E=17 B=16 D=14 so D is eliminated. Round 2 votes counts: A=34 C=33 E=17 B=16 so B is eliminated. Round 3 votes counts: A=44 C=33 E=23 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:207 D:193 E:193 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 12 8 B -4 0 8 2 8 C -8 -8 0 -2 0 D -12 -2 2 0 -2 E -8 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 12 8 B -4 0 8 2 8 C -8 -8 0 -2 0 D -12 -2 2 0 -2 E -8 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 12 8 B -4 0 8 2 8 C -8 -8 0 -2 0 D -12 -2 2 0 -2 E -8 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2167: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) B A C D E (8) E D A C B (7) A E D B C (7) C B D E A (6) A E D C B (6) C E D B A (5) A B C D E (5) D E C B A (4) A B E D C (4) E D C A B (3) E D A B C (3) D E B A C (3) C B A E D (3) B D C E A (3) B C D E A (3) D E A B C (2) C B A D E (2) B D E A C (2) B C D A E (2) A B C E D (2) E D C B A (1) D E B C A (1) C E D A B (1) C D B E A (1) C B E D A (1) C A B E D (1) B D A E C (1) B A D E C (1) B A D C E (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 10 2 8 B 12 0 14 6 8 C -10 -14 0 0 6 D -2 -6 0 0 8 E -8 -8 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 2 8 B 12 0 14 6 8 C -10 -14 0 0 6 D -2 -6 0 0 8 E -8 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=27 C=20 E=14 D=10 so D is eliminated. Round 2 votes counts: B=29 A=27 E=24 C=20 so C is eliminated. Round 3 votes counts: B=42 E=30 A=28 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:204 D:200 C:191 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 2 8 B 12 0 14 6 8 C -10 -14 0 0 6 D -2 -6 0 0 8 E -8 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 2 8 B 12 0 14 6 8 C -10 -14 0 0 6 D -2 -6 0 0 8 E -8 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 2 8 B 12 0 14 6 8 C -10 -14 0 0 6 D -2 -6 0 0 8 E -8 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2168: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (18) A D C E B (15) B A D C E (11) A D C B E (9) E C D A B (7) A B D C E (7) B E A C D (5) B E C A D (4) B A D E C (4) E B C D A (3) D C A E B (3) A D B C E (3) E C D B A (2) C D E A B (2) E C B D A (1) E B A D C (1) D A C E B (1) C E D B A (1) C E D A B (1) C E B D A (1) B A E D C (1) Total count = 100 A B C D E A 0 -4 14 20 8 B 4 0 14 12 24 C -14 -14 0 -10 8 D -20 -12 10 0 10 E -8 -24 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 20 8 B 4 0 14 12 24 C -14 -14 0 -10 8 D -20 -12 10 0 10 E -8 -24 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 A=34 E=14 C=5 D=4 so D is eliminated. Round 2 votes counts: B=43 A=35 E=14 C=8 so C is eliminated. Round 3 votes counts: B=43 A=38 E=19 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:219 D:194 C:185 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 20 8 B 4 0 14 12 24 C -14 -14 0 -10 8 D -20 -12 10 0 10 E -8 -24 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 20 8 B 4 0 14 12 24 C -14 -14 0 -10 8 D -20 -12 10 0 10 E -8 -24 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 20 8 B 4 0 14 12 24 C -14 -14 0 -10 8 D -20 -12 10 0 10 E -8 -24 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2169: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) A D B E C (7) C E B A D (6) A D B C E (6) D A B E C (5) E B D C A (4) D B E A C (4) A E D C B (4) A D C B E (4) A C B D E (4) D B E C A (3) C E B D A (3) A E C B D (3) A D E B C (3) A C D B E (3) E B C D A (2) E A D B C (2) D E B A C (2) D B A C E (2) C B E D A (2) C B E A D (2) B D C E A (2) B C D E A (2) A D E C B (2) E D B C A (1) D E B C A (1) D B A E C (1) D A E B C (1) D A B C E (1) C E A B D (1) C B D E A (1) C B D A E (1) C A E B D (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E C A (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 10 0 0 B 2 0 4 -4 8 C -10 -4 0 -14 -14 D 0 4 14 0 12 E 0 -8 14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.533544 B: 0.000000 C: 0.000000 D: 0.466456 E: 0.000000 Sum of squares = 0.502250416794 Cumulative probabilities = A: 0.533544 B: 0.533544 C: 0.533544 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 0 0 B 2 0 4 -4 8 C -10 -4 0 -14 -14 D 0 4 14 0 12 E 0 -8 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=20 E=18 C=18 B=7 so B is eliminated. Round 2 votes counts: A=37 D=23 E=20 C=20 so E is eliminated. Round 3 votes counts: A=39 C=32 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 B:205 A:204 E:197 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 0 0 B 2 0 4 -4 8 C -10 -4 0 -14 -14 D 0 4 14 0 12 E 0 -8 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 0 0 B 2 0 4 -4 8 C -10 -4 0 -14 -14 D 0 4 14 0 12 E 0 -8 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 0 0 B 2 0 4 -4 8 C -10 -4 0 -14 -14 D 0 4 14 0 12 E 0 -8 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2170: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) E D B A C (5) D E A B C (5) C A B D E (5) D E A C B (4) B E C D A (4) E D B C A (3) E B D C A (3) D A E C B (3) C E D A B (3) C E B D A (3) C B A E D (3) B C E D A (3) B A C D E (3) A D B E C (3) A C D E B (3) E C B D A (2) D E B A C (2) C D E A B (2) C B E A D (2) C A D B E (2) B E D C A (2) A D C E B (2) A B D E C (2) E D C B A (1) E D C A B (1) E C D B A (1) E C D A B (1) E B C D A (1) D E C A B (1) D A E B C (1) C E D B A (1) C D A E B (1) C B E D A (1) B E C A D (1) B E A C D (1) B C E A D (1) B C A E D (1) B A D E C (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 2 -30 -30 B 12 0 12 6 2 C -2 -12 0 -4 -18 D 30 -6 4 0 -10 E 30 -2 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -30 -30 B 12 0 12 6 2 C -2 -12 0 -4 -18 D 30 -6 4 0 -10 E 30 -2 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 E=18 D=16 A=15 so A is eliminated. Round 2 votes counts: B=32 C=28 D=22 E=18 so E is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:228 B:216 D:209 C:182 A:165 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 -30 -30 B 12 0 12 6 2 C -2 -12 0 -4 -18 D 30 -6 4 0 -10 E 30 -2 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -30 -30 B 12 0 12 6 2 C -2 -12 0 -4 -18 D 30 -6 4 0 -10 E 30 -2 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -30 -30 B 12 0 12 6 2 C -2 -12 0 -4 -18 D 30 -6 4 0 -10 E 30 -2 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2171: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (13) B E A D C (11) C D A E B (9) B A E C D (8) D C E A B (7) C D E A B (7) A B C D E (7) E B D C A (5) A C B D E (5) B A E D C (4) D E C A B (3) A C D B E (3) A B D C E (3) E C D B A (2) D C A E B (2) B E A C D (2) E D B C A (1) E C D A B (1) E B D A C (1) D E C B A (1) D A B C E (1) C E D A B (1) C D A B E (1) C A D B E (1) B A C E D (1) Total count = 100 A B C D E A 0 2 -8 -10 -10 B -2 0 -12 -6 -6 C 8 12 0 -4 -4 D 10 6 4 0 0 E 10 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999967111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 A B C D E A 0 2 -8 -10 -10 B -2 0 -12 -6 -6 C 8 12 0 -4 -4 D 10 6 4 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 C=19 A=18 D=14 so D is eliminated. Round 2 votes counts: C=28 E=27 B=26 A=19 so A is eliminated. Round 3 votes counts: B=37 C=36 E=27 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:210 E:210 C:206 A:187 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -10 -10 B -2 0 -12 -6 -6 C 8 12 0 -4 -4 D 10 6 4 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -10 -10 B -2 0 -12 -6 -6 C 8 12 0 -4 -4 D 10 6 4 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -10 -10 B -2 0 -12 -6 -6 C 8 12 0 -4 -4 D 10 6 4 0 0 E 10 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2172: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) B E C D A (8) B E D A C (6) A D E B C (6) A D C E B (6) C A D E B (5) A D E C B (5) C D E A B (4) B C E D A (4) E D B A C (3) E B D C A (3) B A E D C (3) E D A B C (2) E C D B A (2) E B D A C (2) D A E C B (2) C D A E B (2) C B A E D (2) C B A D E (2) C A B D E (2) B C A E D (2) B A E C D (2) B A C D E (2) A C D E B (2) A B D E C (2) E D C B A (1) E B C D A (1) D E A C B (1) D C E A B (1) C E D B A (1) C E D A B (1) C B E D A (1) C A D B E (1) B E A D C (1) B E A C D (1) B C E A D (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -4 -6 -4 B 12 0 16 6 0 C 4 -16 0 -6 -20 D 6 -6 6 0 -10 E 4 0 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.506781 C: 0.000000 D: 0.000000 E: 0.493219 Sum of squares = 0.500091967276 Cumulative probabilities = A: 0.000000 B: 0.506781 C: 0.506781 D: 0.506781 E: 1.000000 A B C D E A 0 -12 -4 -6 -4 B 12 0 16 6 0 C 4 -16 0 -6 -20 D 6 -6 6 0 -10 E 4 0 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=23 C=21 E=14 D=4 so D is eliminated. Round 2 votes counts: B=38 A=25 C=22 E=15 so E is eliminated. Round 3 votes counts: B=47 A=28 C=25 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:217 D:198 A:187 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 -4 B 12 0 16 6 0 C 4 -16 0 -6 -20 D 6 -6 6 0 -10 E 4 0 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 -4 B 12 0 16 6 0 C 4 -16 0 -6 -20 D 6 -6 6 0 -10 E 4 0 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 -4 B 12 0 16 6 0 C 4 -16 0 -6 -20 D 6 -6 6 0 -10 E 4 0 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2173: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C A D B E (10) E C A B D (7) E B A D C (6) E B A C D (6) C D A B E (6) D C A B E (5) D B A C E (4) B E A D C (3) E C B A D (2) E A B C D (2) D E C B A (2) D C B A E (2) D B E A C (2) D B A E C (2) D A C B E (2) C D A E B (2) C A E D B (2) C A E B D (2) B D E A C (2) A B E C D (2) E D B C A (1) E C B D A (1) D E B C A (1) D C B E A (1) D B E C A (1) D B C A E (1) D A B C E (1) C E D A B (1) C E A B D (1) C A B E D (1) B E D A C (1) B D A E C (1) B A E D C (1) B A D E C (1) A E C B D (1) A C E B D (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 2 0 B 2 0 -2 6 4 C -2 2 0 0 -10 D -2 -6 0 0 -4 E 0 -4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999959 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -2 2 2 0 B 2 0 -2 6 4 C -2 2 0 0 -10 D -2 -6 0 0 -4 E 0 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999321 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=25 D=24 B=9 A=7 so A is eliminated. Round 2 votes counts: E=36 C=28 D=24 B=12 so B is eliminated. Round 3 votes counts: E=43 C=29 D=28 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:205 E:205 A:201 C:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 0 B 2 0 -2 6 4 C -2 2 0 0 -10 D -2 -6 0 0 -4 E 0 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999321 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 0 B 2 0 -2 6 4 C -2 2 0 0 -10 D -2 -6 0 0 -4 E 0 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999321 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 0 B 2 0 -2 6 4 C -2 2 0 0 -10 D -2 -6 0 0 -4 E 0 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999321 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2174: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (18) C E B D A (8) C A E B D (8) E B D C A (7) D B A E C (7) D A B E C (6) A C D E B (6) A C D B E (6) E C B D A (5) D B E A C (5) A D B C E (5) E B C D A (4) C E B A D (4) B E D C A (2) B D E A C (2) A D C B E (2) D B E C A (1) C E A B D (1) B E C A D (1) B D A E C (1) A C B E D (1) Total count = 100 A B C D E A 0 6 18 4 20 B -6 0 18 -12 14 C -18 -18 0 -12 -18 D -4 12 12 0 18 E -20 -14 18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 4 20 B -6 0 18 -12 14 C -18 -18 0 -12 -18 D -4 12 12 0 18 E -20 -14 18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=21 D=19 E=16 B=6 so B is eliminated. Round 2 votes counts: A=38 D=22 C=21 E=19 so E is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:219 B:207 E:183 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 4 20 B -6 0 18 -12 14 C -18 -18 0 -12 -18 D -4 12 12 0 18 E -20 -14 18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 4 20 B -6 0 18 -12 14 C -18 -18 0 -12 -18 D -4 12 12 0 18 E -20 -14 18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 4 20 B -6 0 18 -12 14 C -18 -18 0 -12 -18 D -4 12 12 0 18 E -20 -14 18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2175: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) A C D B E (6) B C E D A (5) E D A B C (4) E B D C A (4) E B C A D (4) D B C E A (4) D A C B E (4) C D B A E (4) B C D E A (4) A D C E B (4) E B A D C (3) E A D B C (3) C B A D E (3) B E C D A (3) E C B A D (2) E B C D A (2) E B A C D (2) E A D C B (2) E A B D C (2) D C A B E (2) C B E A D (2) C A B D E (2) B D C E A (2) A E C D B (2) A E C B D (2) E B D A C (1) E A B C D (1) D E B A C (1) D C B A E (1) D B E C A (1) D B E A C (1) D B A C E (1) D A E B C (1) D A B C E (1) C B D A E (1) C B A E D (1) C A D B E (1) B E D C A (1) B D E C A (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 0 2 -10 B 8 0 4 -4 18 C 0 -4 0 -6 12 D -2 4 6 0 6 E 10 -18 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428582 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 2 -10 B 8 0 4 -4 18 C 0 -4 0 -6 12 D -2 4 6 0 6 E 10 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142851 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=23 D=17 B=16 C=14 so C is eliminated. Round 2 votes counts: E=30 A=26 B=23 D=21 so D is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:207 C:201 A:192 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 2 -10 B 8 0 4 -4 18 C 0 -4 0 -6 12 D -2 4 6 0 6 E 10 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142851 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 2 -10 B 8 0 4 -4 18 C 0 -4 0 -6 12 D -2 4 6 0 6 E 10 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142851 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 2 -10 B 8 0 4 -4 18 C 0 -4 0 -6 12 D -2 4 6 0 6 E 10 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142851 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2176: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) B C D A E (9) A E D C B (7) A E D B C (7) E C B D A (4) D C B A E (4) B C E D A (4) A B E C D (4) E B C D A (3) E A D C B (3) D A C B E (3) C B D E A (3) A D B C E (3) E B C A D (2) D C E B A (2) D C B E A (2) C D B E A (2) B E C A D (2) B C E A D (2) B C A D E (2) A E B D C (2) A D E C B (2) E D A C B (1) E A C B D (1) E A B D C (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) D C A B E (1) D B C A E (1) D A C E B (1) C E B D A (1) C B D A E (1) B E C D A (1) B E A C D (1) B D A C E (1) B C D E A (1) B A C E D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 2 4 B -2 0 16 12 0 C 0 -16 0 6 -6 D -2 -12 -6 0 -12 E -4 0 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.928002 B: 0.000000 C: 0.071998 D: 0.000000 E: 0.000000 Sum of squares = 0.866371028308 Cumulative probabilities = A: 0.928002 B: 0.928002 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 4 B -2 0 16 12 0 C 0 -16 0 6 -6 D -2 -12 -6 0 -12 E -4 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469354712 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=24 B=24 D=18 C=7 so C is eliminated. Round 2 votes counts: B=28 A=27 E=25 D=20 so D is eliminated. Round 3 votes counts: B=37 A=33 E=30 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 E:207 A:204 C:192 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 2 4 B -2 0 16 12 0 C 0 -16 0 6 -6 D -2 -12 -6 0 -12 E -4 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469354712 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 4 B -2 0 16 12 0 C 0 -16 0 6 -6 D -2 -12 -6 0 -12 E -4 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469354712 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 4 B -2 0 16 12 0 C 0 -16 0 6 -6 D -2 -12 -6 0 -12 E -4 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.888889 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.802469354712 Cumulative probabilities = A: 0.888889 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2177: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) E C B A D (9) D A B C E (9) A D B C E (8) C E B D A (6) A E C B D (6) E C A B D (5) D B C E A (5) D B A C E (5) A D B E C (5) E A C B D (4) B D C E A (4) A E C D B (4) B C E D A (3) A D E C B (3) E C A D B (1) E B C A D (1) D C B E A (1) D B C A E (1) C E D B A (1) B E C D A (1) B D A E C (1) B D A C E (1) B C D E A (1) B A E C D (1) A E D B C (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 0 0 -6 B 4 0 -4 10 -4 C 0 4 0 10 -8 D 0 -10 -10 0 -10 E 6 4 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 0 -6 B 4 0 -4 10 -4 C 0 4 0 10 -8 D 0 -10 -10 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=29 D=21 B=12 C=7 so C is eliminated. Round 2 votes counts: E=38 A=29 D=21 B=12 so B is eliminated. Round 3 votes counts: E=42 A=30 D=28 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:203 C:203 A:195 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 0 -6 B 4 0 -4 10 -4 C 0 4 0 10 -8 D 0 -10 -10 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 0 -6 B 4 0 -4 10 -4 C 0 4 0 10 -8 D 0 -10 -10 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 0 -6 B 4 0 -4 10 -4 C 0 4 0 10 -8 D 0 -10 -10 0 -10 E 6 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2178: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (5) B E A C D (5) B C E D A (5) E D C B A (4) D C A E B (4) C D A B E (4) B C A D E (4) E A D B C (3) D E A C B (3) D A E C B (3) B E C D A (3) B A E C D (3) A D E C B (3) E B D C A (2) E B D A C (2) E A B D C (2) D E C A B (2) C D E B A (2) C B D E A (2) C A D B E (2) B E C A D (2) B A E D C (2) B A C D E (2) A E D B C (2) A E B D C (2) A D C E B (2) A C D B E (2) A B E D C (2) A B D E C (2) A B C D E (2) E D B C A (1) E B C D A (1) E B A D C (1) D C E A B (1) D A C E B (1) C D E A B (1) C D B E A (1) C B E D A (1) C B D A E (1) C B A D E (1) B C E A D (1) B C D E A (1) B C A E D (1) B A C E D (1) A E D C B (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 2 6 0 -2 B -2 0 6 4 2 C -6 -6 0 -2 -14 D 0 -4 2 0 -4 E 2 -2 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 2 6 0 -2 B -2 0 6 4 2 C -6 -6 0 -2 -14 D 0 -4 2 0 -4 E 2 -2 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=21 A=20 C=15 D=14 so D is eliminated. Round 2 votes counts: B=30 E=26 A=24 C=20 so C is eliminated. Round 3 votes counts: B=36 A=34 E=30 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:209 B:205 A:203 D:197 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 0 -2 B -2 0 6 4 2 C -6 -6 0 -2 -14 D 0 -4 2 0 -4 E 2 -2 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 -2 B -2 0 6 4 2 C -6 -6 0 -2 -14 D 0 -4 2 0 -4 E 2 -2 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 -2 B -2 0 6 4 2 C -6 -6 0 -2 -14 D 0 -4 2 0 -4 E 2 -2 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2179: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) D A C B E (10) B E C D A (8) C A D B E (5) A D C E B (5) D A B E C (4) B E D C A (4) E C A B D (3) D A B C E (3) C B D A E (3) C A D E B (3) B E D A C (3) A D C B E (3) E C B A D (2) E B D A C (2) E B C D A (2) E B A D C (2) D B A E C (2) C E B A D (2) C E A B D (2) C D A B E (2) B D A C E (2) A C D E B (2) E D A B C (1) E B D C A (1) E A D C B (1) E A D B C (1) E A C D B (1) D A E B C (1) D A C E B (1) C D B A E (1) C B A D E (1) C A E D B (1) B E C A D (1) B C D A E (1) Total count = 100 A B C D E A 0 -2 -12 -2 0 B 2 0 4 6 6 C 12 -4 0 8 -6 D 2 -6 -8 0 -2 E 0 -6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -2 0 B 2 0 4 6 6 C 12 -4 0 8 -6 D 2 -6 -8 0 -2 E 0 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=21 C=20 B=19 A=10 so A is eliminated. Round 2 votes counts: E=30 D=29 C=22 B=19 so B is eliminated. Round 3 votes counts: E=46 D=31 C=23 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:209 C:205 E:201 D:193 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -12 -2 0 B 2 0 4 6 6 C 12 -4 0 8 -6 D 2 -6 -8 0 -2 E 0 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -2 0 B 2 0 4 6 6 C 12 -4 0 8 -6 D 2 -6 -8 0 -2 E 0 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -2 0 B 2 0 4 6 6 C 12 -4 0 8 -6 D 2 -6 -8 0 -2 E 0 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2180: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (12) B D C E A (11) D E A B C (8) C B A E D (8) E A D B C (7) B C D E A (7) A E C D B (7) D B E A C (6) B C D A E (5) A E D C B (5) D E B A C (3) C B D A E (3) C A E D B (3) A C E D B (3) B D E C A (2) B D C A E (2) E D A B C (1) E A D C B (1) D B C E A (1) D B C A E (1) D B A E C (1) C B A D E (1) B E A D C (1) B D E A C (1) Total count = 100 A B C D E A 0 -6 -12 -4 2 B 6 0 14 6 0 C 12 -14 0 -2 14 D 4 -6 2 0 4 E -2 0 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.645592 C: 0.000000 D: 0.000000 E: 0.354408 Sum of squares = 0.542393956789 Cumulative probabilities = A: 0.000000 B: 0.645592 C: 0.645592 D: 0.645592 E: 1.000000 A B C D E A 0 -6 -12 -4 2 B 6 0 14 6 0 C 12 -14 0 -2 14 D 4 -6 2 0 4 E -2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500743 C: 0.000000 D: 0.000000 E: 0.499257 Sum of squares = 0.500001105356 Cumulative probabilities = A: 0.000000 B: 0.500743 C: 0.500743 D: 0.500743 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=27 D=20 A=15 E=9 so E is eliminated. Round 2 votes counts: B=29 C=27 A=23 D=21 so D is eliminated. Round 3 votes counts: B=41 A=32 C=27 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:205 D:202 A:190 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -12 -4 2 B 6 0 14 6 0 C 12 -14 0 -2 14 D 4 -6 2 0 4 E -2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500743 C: 0.000000 D: 0.000000 E: 0.499257 Sum of squares = 0.500001105356 Cumulative probabilities = A: 0.000000 B: 0.500743 C: 0.500743 D: 0.500743 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -4 2 B 6 0 14 6 0 C 12 -14 0 -2 14 D 4 -6 2 0 4 E -2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500743 C: 0.000000 D: 0.000000 E: 0.499257 Sum of squares = 0.500001105356 Cumulative probabilities = A: 0.000000 B: 0.500743 C: 0.500743 D: 0.500743 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -4 2 B 6 0 14 6 0 C 12 -14 0 -2 14 D 4 -6 2 0 4 E -2 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500743 C: 0.000000 D: 0.000000 E: 0.499257 Sum of squares = 0.500001105356 Cumulative probabilities = A: 0.000000 B: 0.500743 C: 0.500743 D: 0.500743 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2181: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (12) E A B D C (10) C D B A E (9) E D A B C (8) C B A D E (7) B A C E D (6) D C E B A (5) A E B D C (4) E A D B C (3) B A E C D (3) A B E C D (3) E C A D B (2) C D E B A (2) C D E A B (2) B C A D E (2) B A E D C (2) A E B C D (2) A B E D C (2) E D C A B (1) E D A C B (1) E A C D B (1) E A B C D (1) D E B C A (1) D E A B C (1) D C E A B (1) D C B E A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D B E A (1) C B D A E (1) B C A E D (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 12 2 4 -8 B -12 0 4 -6 -18 C -2 -4 0 -10 -18 D -4 6 10 0 -4 E 8 18 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 4 -8 B -12 0 4 -6 -18 C -2 -4 0 -10 -18 D -4 6 10 0 -4 E 8 18 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 C=23 B=15 A=12 so A is eliminated. Round 2 votes counts: E=33 C=24 D=23 B=20 so B is eliminated. Round 3 votes counts: E=43 C=33 D=24 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:205 D:204 B:184 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 4 -8 B -12 0 4 -6 -18 C -2 -4 0 -10 -18 D -4 6 10 0 -4 E 8 18 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 4 -8 B -12 0 4 -6 -18 C -2 -4 0 -10 -18 D -4 6 10 0 -4 E 8 18 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 4 -8 B -12 0 4 -6 -18 C -2 -4 0 -10 -18 D -4 6 10 0 -4 E 8 18 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2182: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E B C D A (7) D A E B C (7) B E C D A (7) E B D C A (5) D B E C A (5) C E B A D (5) C B E A D (5) C A B E D (5) A D C E B (5) A D C B E (5) D A B E C (4) A C D B E (4) D E B A C (3) A D E C B (3) A C E B D (3) A C B E D (3) E C B A D (2) D B E A C (2) B D E C A (2) A D E B C (2) E C A B D (1) D E B C A (1) D B C A E (1) C B A E D (1) B E D C A (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 4 -8 8 B -6 0 12 -4 8 C -4 -12 0 -10 -6 D 8 4 10 0 8 E -8 -8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -8 8 B -6 0 12 -4 8 C -4 -12 0 -10 -6 D 8 4 10 0 8 E -8 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=27 C=16 E=15 B=10 so B is eliminated. Round 2 votes counts: D=34 A=27 E=23 C=16 so C is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:205 B:205 E:191 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -8 8 B -6 0 12 -4 8 C -4 -12 0 -10 -6 D 8 4 10 0 8 E -8 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -8 8 B -6 0 12 -4 8 C -4 -12 0 -10 -6 D 8 4 10 0 8 E -8 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -8 8 B -6 0 12 -4 8 C -4 -12 0 -10 -6 D 8 4 10 0 8 E -8 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2183: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) A B C E D (7) D C A E B (6) C D E A B (4) B A E D C (4) B A C E D (4) A C B E D (4) A B E C D (4) E D C B A (3) E B C D A (3) B E D A C (3) B E A D C (3) B A E C D (3) A D B C E (3) A B C D E (3) E D B C A (2) E D B A C (2) E C D B A (2) E B D A C (2) D E C B A (2) D E B C A (2) D C E B A (2) C E B A D (2) C D A E B (2) C A B D E (2) A C D B E (2) A C B D E (2) A B D C E (2) E B D C A (1) D E B A C (1) C E D B A (1) C E B D A (1) C D E B A (1) C A E B D (1) C A B E D (1) B E C A D (1) B E A C D (1) B C E A D (1) A D C B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 6 6 B -6 0 8 14 6 C -6 -8 0 4 18 D -6 -14 -4 0 -14 E -6 -6 -18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 6 B -6 0 8 14 6 C -6 -8 0 4 18 D -6 -14 -4 0 -14 E -6 -6 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=20 B=20 E=15 C=15 so E is eliminated. Round 2 votes counts: A=30 D=27 B=26 C=17 so C is eliminated. Round 3 votes counts: D=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:211 C:204 E:192 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 6 B -6 0 8 14 6 C -6 -8 0 4 18 D -6 -14 -4 0 -14 E -6 -6 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 6 B -6 0 8 14 6 C -6 -8 0 4 18 D -6 -14 -4 0 -14 E -6 -6 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 6 B -6 0 8 14 6 C -6 -8 0 4 18 D -6 -14 -4 0 -14 E -6 -6 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2184: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) D B A C E (6) E A D B C (5) A E C D B (5) A D C B E (5) A C E D B (5) E C B A D (4) C E A B D (4) B E C D A (4) B D E C A (4) D B C A E (3) D A B C E (3) C E B D A (3) A D B C E (3) E C B D A (2) E B D C A (2) E B D A C (2) E B C D A (2) E A C B D (2) D C B A E (2) D B A E C (2) D A B E C (2) C E B A D (2) C A E B D (2) B E D C A (2) B D C E A (2) A D B E C (2) A C D E B (2) A C D B E (2) E B A D C (1) D B E A C (1) D A C B E (1) C D B A E (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D B E (1) B D E A C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 2 6 -2 B -4 0 -4 -6 -2 C -2 4 0 0 0 D -6 6 0 0 -10 E 2 2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.366587 D: 0.000000 E: 0.633413 Sum of squares = 0.535597892938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.366587 D: 0.366587 E: 1.000000 A B C D E A 0 4 2 6 -2 B -4 0 -4 -6 -2 C -2 4 0 0 0 D -6 6 0 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.000000 E: 0.500476 Sum of squares = 0.500000453677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.499524 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=25 D=20 C=16 B=13 so B is eliminated. Round 2 votes counts: E=32 D=27 A=25 C=16 so C is eliminated. Round 3 votes counts: E=42 D=29 A=29 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:205 C:201 D:195 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 6 -2 B -4 0 -4 -6 -2 C -2 4 0 0 0 D -6 6 0 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.000000 E: 0.500476 Sum of squares = 0.500000453677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.499524 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 6 -2 B -4 0 -4 -6 -2 C -2 4 0 0 0 D -6 6 0 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.000000 E: 0.500476 Sum of squares = 0.500000453677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.499524 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 6 -2 B -4 0 -4 -6 -2 C -2 4 0 0 0 D -6 6 0 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.000000 E: 0.500476 Sum of squares = 0.500000453677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499524 D: 0.499524 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2185: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) B E A D C (6) E B D C A (5) C D A E B (5) E B A D C (4) E A C D B (4) D C A B E (4) A C D E B (4) E D C A B (3) E A B C D (3) D C E A B (3) D C B E A (3) D C B A E (3) C D A B E (3) C A D E B (3) B A E C D (3) A C E D B (3) E B A C D (2) D C A E B (2) D B C A E (2) B A D E C (2) A E C D B (2) A E C B D (2) A C D B E (2) A B E C D (2) A B C E D (2) E D C B A (1) E D B C A (1) E B D A C (1) D E C B A (1) D B C E A (1) C A D B E (1) B E D C A (1) B E D A C (1) B D C E A (1) B D C A E (1) B A C E D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 -4 -2 B 2 0 -20 -22 -18 C 8 20 0 -12 12 D 4 22 12 0 4 E 2 18 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -4 -2 B 2 0 -20 -22 -18 C 8 20 0 -12 12 D 4 22 12 0 4 E 2 18 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 A=18 B=17 C=12 so C is eliminated. Round 2 votes counts: D=37 E=24 A=22 B=17 so B is eliminated. Round 3 votes counts: D=39 E=32 A=29 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:214 E:202 A:192 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 -2 B 2 0 -20 -22 -18 C 8 20 0 -12 12 D 4 22 12 0 4 E 2 18 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 -2 B 2 0 -20 -22 -18 C 8 20 0 -12 12 D 4 22 12 0 4 E 2 18 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 -2 B 2 0 -20 -22 -18 C 8 20 0 -12 12 D 4 22 12 0 4 E 2 18 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2186: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) E D C A B (5) D C B A E (5) B C A D E (5) E A B C D (4) D A E B C (4) D A B C E (4) C E B A D (4) E C D B A (3) E C A B D (3) D A B E C (3) C E B D A (3) C D B A E (3) C B D A E (3) A D B C E (3) E C D A B (2) D C E B A (2) D C B E A (2) C E D B A (2) C B D E A (2) C B A D E (2) B C D A E (2) B C A E D (2) A E B C D (2) A D B E C (2) A B E C D (2) A B C D E (2) E D C B A (1) E C B D A (1) E A D C B (1) E A D B C (1) E A B D C (1) D E C B A (1) D E A B C (1) D B C A E (1) C B A E D (1) B D C A E (1) B A D C E (1) A E D B C (1) A E B D C (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -28 -2 6 B 10 0 -8 6 -2 C 28 8 0 14 4 D 2 -6 -14 0 0 E -6 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -28 -2 6 B 10 0 -8 6 -2 C 28 8 0 14 4 D 2 -6 -14 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=23 C=20 A=16 B=11 so B is eliminated. Round 2 votes counts: E=30 C=29 D=24 A=17 so A is eliminated. Round 3 votes counts: E=37 C=32 D=31 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:203 E:196 D:191 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -28 -2 6 B 10 0 -8 6 -2 C 28 8 0 14 4 D 2 -6 -14 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -28 -2 6 B 10 0 -8 6 -2 C 28 8 0 14 4 D 2 -6 -14 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -28 -2 6 B 10 0 -8 6 -2 C 28 8 0 14 4 D 2 -6 -14 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2187: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (19) A B C E D (16) B A C E D (9) B C A E D (6) E C D B A (5) E D C B A (3) D E B C A (3) B C E D A (3) A D E C B (3) D A E C B (2) C E B D A (2) C B E D A (2) C B A E D (2) B D C E A (2) B C E A D (2) B C D E A (2) A D E B C (2) A D B E C (2) A D B C E (2) A B C D E (2) E D C A B (1) E C D A B (1) D E C A B (1) D E A C B (1) D B A E C (1) C B E A D (1) B A C D E (1) A E C D B (1) A C E B D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -26 -10 4 4 B 26 0 8 6 10 C 10 -8 0 14 10 D -4 -6 -14 0 -12 E -4 -10 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -10 4 4 B 26 0 8 6 10 C 10 -8 0 14 10 D -4 -6 -14 0 -12 E -4 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=27 B=25 E=10 C=7 so C is eliminated. Round 2 votes counts: A=31 B=30 D=27 E=12 so E is eliminated. Round 3 votes counts: D=37 B=32 A=31 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:213 E:194 A:186 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -10 4 4 B 26 0 8 6 10 C 10 -8 0 14 10 D -4 -6 -14 0 -12 E -4 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -10 4 4 B 26 0 8 6 10 C 10 -8 0 14 10 D -4 -6 -14 0 -12 E -4 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -10 4 4 B 26 0 8 6 10 C 10 -8 0 14 10 D -4 -6 -14 0 -12 E -4 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2188: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) B C A E D (13) B A E C D (10) C D B A E (9) D C E A B (8) E A D B C (7) D E A C B (6) C B D A E (6) D C B E A (3) C B A D E (3) B A C E D (3) A E B C D (3) D E C A B (2) D C E B A (2) A E B D C (2) A B E C D (2) E D A B C (1) D E C B A (1) D E A B C (1) C D B E A (1) C B D E A (1) C B A E D (1) B E A C D (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 0 18 6 B 8 0 14 16 6 C 0 -14 0 6 0 D -18 -16 -6 0 -14 E -6 -6 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 18 6 B 8 0 14 16 6 C 0 -14 0 6 0 D -18 -16 -6 0 -14 E -6 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 E=21 C=21 A=8 so A is eliminated. Round 2 votes counts: B=29 E=27 D=23 C=21 so C is eliminated. Round 3 votes counts: B=40 D=33 E=27 so E is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:208 E:201 C:196 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 18 6 B 8 0 14 16 6 C 0 -14 0 6 0 D -18 -16 -6 0 -14 E -6 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 18 6 B 8 0 14 16 6 C 0 -14 0 6 0 D -18 -16 -6 0 -14 E -6 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 18 6 B 8 0 14 16 6 C 0 -14 0 6 0 D -18 -16 -6 0 -14 E -6 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2189: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) D B A C E (10) D A B C E (7) B A D C E (6) A D B C E (6) E C D A B (5) E C B D A (4) E C B A D (4) C E A B D (4) B D A C E (4) E D C A B (3) E C D B A (3) D A B E C (3) E C A D B (2) D E B C A (2) D E A C B (2) D B E A C (2) C A B E D (2) B C A E D (2) A C B E D (2) A B C D E (2) D E C A B (1) D B A E C (1) D A E C B (1) C E B A D (1) C B E A D (1) C B A E D (1) C A E B D (1) B D E C A (1) B D C A E (1) B C E D A (1) B A C D E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -2 -2 4 B -10 0 0 4 8 C 2 0 0 -2 8 D 2 -4 2 0 2 E -4 -8 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999997 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -2 4 B -10 0 0 4 8 C 2 0 0 -2 8 D 2 -4 2 0 2 E -4 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999986 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=29 B=16 A=12 C=10 so C is eliminated. Round 2 votes counts: E=38 D=29 B=18 A=15 so A is eliminated. Round 3 votes counts: E=40 D=35 B=25 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:205 C:204 B:201 D:201 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 -2 4 B -10 0 0 4 8 C 2 0 0 -2 8 D 2 -4 2 0 2 E -4 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999986 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -2 4 B -10 0 0 4 8 C 2 0 0 -2 8 D 2 -4 2 0 2 E -4 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999986 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -2 4 B -10 0 0 4 8 C 2 0 0 -2 8 D 2 -4 2 0 2 E -4 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999986 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2190: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (15) E B A C D (10) A B E D C (10) A B D C E (8) E C D B A (7) C D E A B (7) B A E D C (5) B A E C D (5) E C B A D (4) C D E B A (4) B A D C E (4) D A C B E (3) E C D A B (2) C D B A E (2) B E A C D (2) B A C D E (2) A D B C E (2) A B D E C (2) E B C A D (1) D C E A B (1) D C B A E (1) D C A E B (1) C E D B A (1) C E D A B (1) Total count = 100 A B C D E A 0 4 6 10 20 B -4 0 2 6 22 C -6 -2 0 -4 4 D -10 -6 4 0 4 E -20 -22 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 10 20 B -4 0 2 6 22 C -6 -2 0 -4 4 D -10 -6 4 0 4 E -20 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999648 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=22 D=21 B=18 C=15 so C is eliminated. Round 2 votes counts: D=34 E=26 A=22 B=18 so B is eliminated. Round 3 votes counts: A=38 D=34 E=28 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 B:213 C:196 D:196 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 10 20 B -4 0 2 6 22 C -6 -2 0 -4 4 D -10 -6 4 0 4 E -20 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999648 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 10 20 B -4 0 2 6 22 C -6 -2 0 -4 4 D -10 -6 4 0 4 E -20 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999648 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 10 20 B -4 0 2 6 22 C -6 -2 0 -4 4 D -10 -6 4 0 4 E -20 -22 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999648 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2191: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) C D A E B (9) E D C B A (8) A C D B E (8) E B D C A (7) D C A E B (6) A B C D E (6) B E A D C (5) B A E C D (5) E D B C A (4) B E A C D (4) B A C D E (4) C A D B E (3) E B D A C (2) D C E A B (2) B E C D A (2) B C A D E (2) B A E D C (2) B A C E D (2) A C B D E (2) E B C D A (1) C A D E B (1) B E D A C (1) B C E A D (1) A D C E B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -12 -4 6 B 0 0 -2 -6 -2 C 12 2 0 0 -4 D 4 6 0 0 -8 E -6 2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677679 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.454545 D: 0.454545 E: 1.000000 A B C D E A 0 0 -12 -4 6 B 0 0 -2 -6 -2 C 12 2 0 0 -4 D 4 6 0 0 -8 E -6 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677659 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.454545 D: 0.454545 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=28 A=19 C=13 D=8 so D is eliminated. Round 2 votes counts: E=32 B=28 C=21 A=19 so A is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:205 E:204 D:201 A:195 B:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 -4 6 B 0 0 -2 -6 -2 C 12 2 0 0 -4 D 4 6 0 0 -8 E -6 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677659 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.454545 D: 0.454545 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -4 6 B 0 0 -2 -6 -2 C 12 2 0 0 -4 D 4 6 0 0 -8 E -6 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677659 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.454545 D: 0.454545 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -4 6 B 0 0 -2 -6 -2 C 12 2 0 0 -4 D 4 6 0 0 -8 E -6 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677659 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.454545 D: 0.454545 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2192: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C A E D B (6) E B C D A (5) C B E A D (5) D B A E C (4) D A E B C (4) D A B E C (4) D A B C E (4) E B D C A (3) B E D C A (3) B E C D A (3) B D E A C (3) B C E D A (3) A D C E B (3) E C A D B (2) E B D A C (2) D A C B E (2) C E B A D (2) C E A B D (2) C B A E D (2) C A E B D (2) B D A C E (2) B C D A E (2) B C A D E (2) A D E C B (2) E D B A C (1) E D A C B (1) E C A B D (1) E B C A D (1) E A D C B (1) D E B A C (1) D E A B C (1) D B E A C (1) D A C E B (1) C B A D E (1) C A D E B (1) C A B E D (1) B E C A D (1) B C E A D (1) B C A E D (1) A E D C B (1) A E C D B (1) A D C B E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 0 -16 14 B -2 0 4 -2 -6 C 0 -4 0 -6 -8 D 16 2 6 0 -2 E -14 6 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.062500 B: 0.000000 C: 0.000000 D: 0.437500 E: 0.500000 Sum of squares = 0.445312500002 Cumulative probabilities = A: 0.062500 B: 0.062500 C: 0.062500 D: 0.500000 E: 1.000000 A B C D E A 0 2 0 -16 14 B -2 0 4 -2 -6 C 0 -4 0 -6 -8 D 16 2 6 0 -2 E -14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.062500 B: 0.000000 C: 0.000000 D: 0.437500 E: 0.500000 Sum of squares = 0.445312500052 Cumulative probabilities = A: 0.062500 B: 0.062500 C: 0.062500 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=22 B=21 E=17 A=10 so A is eliminated. Round 2 votes counts: D=36 C=24 B=21 E=19 so E is eliminated. Round 3 votes counts: D=40 B=32 C=28 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:201 A:200 B:197 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -16 14 B -2 0 4 -2 -6 C 0 -4 0 -6 -8 D 16 2 6 0 -2 E -14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.062500 B: 0.000000 C: 0.000000 D: 0.437500 E: 0.500000 Sum of squares = 0.445312500052 Cumulative probabilities = A: 0.062500 B: 0.062500 C: 0.062500 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -16 14 B -2 0 4 -2 -6 C 0 -4 0 -6 -8 D 16 2 6 0 -2 E -14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.062500 B: 0.000000 C: 0.000000 D: 0.437500 E: 0.500000 Sum of squares = 0.445312500052 Cumulative probabilities = A: 0.062500 B: 0.062500 C: 0.062500 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -16 14 B -2 0 4 -2 -6 C 0 -4 0 -6 -8 D 16 2 6 0 -2 E -14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.062500 B: 0.000000 C: 0.000000 D: 0.437500 E: 0.500000 Sum of squares = 0.445312500052 Cumulative probabilities = A: 0.062500 B: 0.062500 C: 0.062500 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2193: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (16) A E D B C (10) E A D C B (8) A E D C B (6) E D A C B (5) C B A E D (4) B C D E A (4) A B C E D (4) D E A B C (3) C B A D E (3) B C A E D (3) B C A D E (3) A C B E D (3) E D C B A (2) E D A B C (2) D E B C A (2) D C E B A (2) C B D A E (2) A E B D C (2) E C D B A (1) E A C D B (1) D E C B A (1) D E B A C (1) C D B E A (1) C B E A D (1) C A B E D (1) B D C E A (1) B C D A E (1) A E C D B (1) A E C B D (1) A D B E C (1) A C E B D (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 12 -2 B -4 0 -20 6 2 C -4 20 0 6 4 D -12 -6 -6 0 -14 E 2 -2 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999976 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 4 4 12 -2 B -4 0 -20 6 2 C -4 20 0 6 4 D -12 -6 -6 0 -14 E 2 -2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999925 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 E=19 B=12 D=9 so D is eliminated. Round 2 votes counts: A=32 C=30 E=26 B=12 so B is eliminated. Round 3 votes counts: C=42 A=32 E=26 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:209 E:205 B:192 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 12 -2 B -4 0 -20 6 2 C -4 20 0 6 4 D -12 -6 -6 0 -14 E 2 -2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999925 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 12 -2 B -4 0 -20 6 2 C -4 20 0 6 4 D -12 -6 -6 0 -14 E 2 -2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999925 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 12 -2 B -4 0 -20 6 2 C -4 20 0 6 4 D -12 -6 -6 0 -14 E 2 -2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999925 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2194: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) B E D C A (9) D E B A C (7) C B A E D (5) C A D B E (5) E D B A C (4) E B D A C (4) B E D A C (4) D E B C A (3) C A B D E (3) A C E B D (3) A C D E B (3) D A E B C (2) C D A B E (2) C B D E A (2) B E C D A (2) A D E C B (2) A D E B C (2) A D C E B (2) A C E D B (2) A C D B E (2) E D B C A (1) E D A B C (1) D E A B C (1) D C E A B (1) D C A E B (1) D B E C A (1) D A E C B (1) D A C E B (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D A E (1) C A D E B (1) B E C A D (1) B E A C D (1) B C E A D (1) B A C E D (1) A E D C B (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -6 -2 12 B -2 0 -6 4 10 C 6 6 0 4 2 D 2 -4 -4 0 -10 E -12 -10 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -2 12 B -2 0 -6 4 10 C 6 6 0 4 2 D 2 -4 -4 0 -10 E -12 -10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=20 B=19 D=18 E=10 so E is eliminated. Round 2 votes counts: C=33 D=24 B=23 A=20 so A is eliminated. Round 3 votes counts: C=44 D=31 B=25 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:203 B:203 E:193 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -2 12 B -2 0 -6 4 10 C 6 6 0 4 2 D 2 -4 -4 0 -10 E -12 -10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -2 12 B -2 0 -6 4 10 C 6 6 0 4 2 D 2 -4 -4 0 -10 E -12 -10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -2 12 B -2 0 -6 4 10 C 6 6 0 4 2 D 2 -4 -4 0 -10 E -12 -10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2195: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) E C B A D (9) D A B C E (8) A D E C B (8) B C E A D (7) A D B E C (7) B C E D A (6) C E B D A (5) A D B C E (4) E C D A B (3) D A E C B (3) A E C D B (3) A D E B C (3) E C D B A (2) D C E B A (2) C B E A D (2) B D A C E (2) B C A E D (2) A B E C D (2) E D C A B (1) E D A C B (1) E C A D B (1) E C A B D (1) E A D C B (1) E A C D B (1) D E C B A (1) D E C A B (1) D B C E A (1) C E B A D (1) C B E D A (1) B C D E A (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -12 6 -14 B 2 0 -12 -2 -12 C 12 12 0 14 -14 D -6 2 -14 0 -18 E 14 12 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -12 6 -14 B 2 0 -12 -2 -12 C 12 12 0 14 -14 D -6 2 -14 0 -18 E 14 12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=28 B=18 D=16 C=9 so C is eliminated. Round 2 votes counts: E=35 A=28 B=21 D=16 so D is eliminated. Round 3 votes counts: E=39 A=39 B=22 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:212 A:189 B:188 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -12 6 -14 B 2 0 -12 -2 -12 C 12 12 0 14 -14 D -6 2 -14 0 -18 E 14 12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 6 -14 B 2 0 -12 -2 -12 C 12 12 0 14 -14 D -6 2 -14 0 -18 E 14 12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 6 -14 B 2 0 -12 -2 -12 C 12 12 0 14 -14 D -6 2 -14 0 -18 E 14 12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2196: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) C D B A E (6) B C D E A (6) A E D C B (6) E A B D C (5) E B A C D (4) C D A B E (4) B E C D A (4) A E B C D (4) E D C B A (3) B C A D E (3) B A C D E (3) E D C A B (2) E B D C A (2) E B C D A (2) D E A C B (2) D C E A B (2) D C A E B (2) D C A B E (2) D A C E B (2) C D B E A (2) C B D E A (2) C B D A E (2) B C E D A (2) B C D A E (2) B A E C D (2) A B E C D (2) E B D A C (1) E A D B C (1) D E C B A (1) B E D C A (1) B E C A D (1) B E A C D (1) B C E A D (1) A D E C B (1) A D C E B (1) A C D E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -4 -10 B 2 0 -4 2 -4 C 4 4 0 12 -10 D 4 -2 -12 0 -8 E 10 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -10 B 2 0 -4 2 -4 C 4 4 0 12 -10 D 4 -2 -12 0 -8 E 10 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 A=17 C=16 D=11 so D is eliminated. Round 2 votes counts: E=33 B=26 C=22 A=19 so A is eliminated. Round 3 votes counts: E=44 B=29 C=27 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:205 B:198 D:191 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -10 B 2 0 -4 2 -4 C 4 4 0 12 -10 D 4 -2 -12 0 -8 E 10 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -10 B 2 0 -4 2 -4 C 4 4 0 12 -10 D 4 -2 -12 0 -8 E 10 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -10 B 2 0 -4 2 -4 C 4 4 0 12 -10 D 4 -2 -12 0 -8 E 10 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2197: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (13) E A C B D (7) D C B E A (7) D B C E A (6) E C D A B (5) D E C B A (5) E A C D B (4) D B C A E (4) B D C A E (4) E D C A B (3) E C A D B (3) E A D C B (3) D B A C E (3) B A D C E (3) A B C E D (3) C B A E D (2) B D C E A (2) B D A C E (2) B C D A E (2) A D B E C (2) E D A C B (1) E C D B A (1) E C A B D (1) D E C A B (1) D E B C A (1) D B E A C (1) D A E B C (1) C E B D A (1) C E A B D (1) C B E D A (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B D C (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 0 -2 -8 B -8 0 -22 -6 -10 C 0 22 0 -4 -14 D 2 6 4 0 -10 E 8 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 -2 -8 B -8 0 -22 -6 -10 C 0 22 0 -4 -14 D 2 6 4 0 -10 E 8 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 A=24 B=14 C=5 so C is eliminated. Round 2 votes counts: E=30 D=29 A=24 B=17 so B is eliminated. Round 3 votes counts: D=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:202 D:201 A:199 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 -2 -8 B -8 0 -22 -6 -10 C 0 22 0 -4 -14 D 2 6 4 0 -10 E 8 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -2 -8 B -8 0 -22 -6 -10 C 0 22 0 -4 -14 D 2 6 4 0 -10 E 8 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -2 -8 B -8 0 -22 -6 -10 C 0 22 0 -4 -14 D 2 6 4 0 -10 E 8 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2198: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) C D E B A (7) D C A E B (6) C D E A B (6) B C E D A (6) C D A E B (5) B E C D A (5) B E C A D (4) A B E D C (4) C D B A E (3) C B D E A (3) B C A D E (3) B A E C D (3) A E B D C (3) A D E B C (3) A B D E C (3) E D C A B (2) E A B D C (2) D A E C B (2) B E A D C (2) B A E D C (2) A E D B C (2) A D C E B (2) E D A C B (1) E B D A C (1) E B C D A (1) E B A D C (1) D C E A B (1) D A C E B (1) C E B D A (1) C D B E A (1) C D A B E (1) B A C D E (1) A E D C B (1) A D E C B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -10 -6 -4 B 4 0 12 8 2 C 10 -12 0 18 -2 D 6 -8 -18 0 2 E 4 -2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -4 B 4 0 12 8 2 C 10 -12 0 18 -2 D 6 -8 -18 0 2 E 4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=27 A=21 D=10 E=8 so E is eliminated. Round 2 votes counts: B=37 C=27 A=23 D=13 so D is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:207 E:201 D:191 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -4 B 4 0 12 8 2 C 10 -12 0 18 -2 D 6 -8 -18 0 2 E 4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 12 8 2 C 10 -12 0 18 -2 D 6 -8 -18 0 2 E 4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 12 8 2 C 10 -12 0 18 -2 D 6 -8 -18 0 2 E 4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2199: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (16) C A E B D (10) B D E A C (8) D E A C B (6) C A E D B (6) B E A C D (5) B C A E D (5) E A C D B (4) E B A C D (3) D E B A C (3) E A D C B (2) E A C B D (2) D E C A B (2) D E A B C (2) D C E A B (2) D C A E B (2) C B A E D (2) C A B E D (2) B A E C D (2) A E C D B (2) E D A C B (1) E B A D C (1) E A B C D (1) D B C E A (1) D B C A E (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C E A (1) B D C A E (1) B C D A E (1) B A C E D (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 24 2 -24 B 6 0 8 -4 -4 C -24 -8 0 -2 -24 D -2 4 2 0 -4 E 24 4 24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 24 2 -24 B 6 0 8 -4 -4 C -24 -8 0 -2 -24 D -2 4 2 0 -4 E 24 4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=26 C=22 E=14 A=3 so A is eliminated. Round 2 votes counts: D=35 B=26 C=23 E=16 so E is eliminated. Round 3 votes counts: D=38 C=31 B=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:228 B:203 D:200 A:198 C:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 24 2 -24 B 6 0 8 -4 -4 C -24 -8 0 -2 -24 D -2 4 2 0 -4 E 24 4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 24 2 -24 B 6 0 8 -4 -4 C -24 -8 0 -2 -24 D -2 4 2 0 -4 E 24 4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 24 2 -24 B 6 0 8 -4 -4 C -24 -8 0 -2 -24 D -2 4 2 0 -4 E 24 4 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2200: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) A C E B D (8) E B A C D (6) D B E C A (5) C D A E B (5) B E A C D (5) B D E A C (5) D C E B A (4) D C A B E (4) D C A E B (3) C A E D B (3) B E D A C (3) E A C B D (2) E A B C D (2) D C B E A (2) D C B A E (2) D B A C E (2) C A E B D (2) C A D E B (2) B A E C D (2) A E C B D (2) A E B C D (2) A B E C D (2) D C E A B (1) D B C E A (1) D B C A E (1) D A C B E (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A B E (1) C A D B E (1) B A E D C (1) B A D E C (1) A D B C E (1) A C D B E (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 18 16 0 B 0 0 0 14 8 C -18 0 0 6 4 D -16 -14 -6 0 -10 E 0 -8 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.421492 B: 0.578508 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.512327098203 Cumulative probabilities = A: 0.421492 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 16 0 B 0 0 0 14 8 C -18 0 0 6 4 D -16 -14 -6 0 -10 E 0 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 A=19 C=18 E=10 so E is eliminated. Round 2 votes counts: B=33 D=26 A=23 C=18 so C is eliminated. Round 3 votes counts: D=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:211 E:199 C:196 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 16 0 B 0 0 0 14 8 C -18 0 0 6 4 D -16 -14 -6 0 -10 E 0 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 16 0 B 0 0 0 14 8 C -18 0 0 6 4 D -16 -14 -6 0 -10 E 0 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 16 0 B 0 0 0 14 8 C -18 0 0 6 4 D -16 -14 -6 0 -10 E 0 -8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2201: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) E B A D C (7) C D A B E (7) E A B D C (6) C D B A E (5) D C A E B (4) D A E B C (4) B E C A D (4) B E A C D (4) E A D B C (3) D A C E B (3) C E B D A (3) C B E A D (3) B A E C D (3) E D A B C (2) E B A C D (2) D E A C B (2) D C E A B (2) D C A B E (2) D A B E C (2) B C A D E (2) A D B E C (2) E D A C B (1) E C D A B (1) E C B D A (1) D E C A B (1) D A E C B (1) D A C B E (1) D A B C E (1) C D E B A (1) C D E A B (1) C B A D E (1) B E A D C (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C D E (1) A E D B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 -10 8 B 2 0 0 4 8 C 0 0 0 2 4 D 10 -4 -2 0 12 E -8 -8 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.553124 C: 0.446876 D: 0.000000 E: 0.000000 Sum of squares = 0.505644306254 Cumulative probabilities = A: 0.000000 B: 0.553124 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -10 8 B 2 0 0 4 8 C 0 0 0 2 4 D 10 -4 -2 0 12 E -8 -8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 D=23 B=18 A=5 so A is eliminated. Round 2 votes counts: C=31 D=26 E=24 B=19 so B is eliminated. Round 3 votes counts: E=36 C=36 D=28 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:208 B:207 C:203 A:198 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -10 8 B 2 0 0 4 8 C 0 0 0 2 4 D 10 -4 -2 0 12 E -8 -8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -10 8 B 2 0 0 4 8 C 0 0 0 2 4 D 10 -4 -2 0 12 E -8 -8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -10 8 B 2 0 0 4 8 C 0 0 0 2 4 D 10 -4 -2 0 12 E -8 -8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2202: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (13) D C E B A (9) A B E D C (9) D E C B A (7) C D E B A (7) E B D C A (4) D C A E B (4) E B A D C (3) B A E C D (3) A D C B E (3) A C B D E (3) E D B C A (2) D C E A B (2) C D B E A (2) C D A B E (2) C A D B E (2) A C B E D (2) A B C E D (2) E D C B A (1) E B D A C (1) E B C D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D E B C A (1) D A E B C (1) C E D B A (1) C D B A E (1) C D A E B (1) C B E D A (1) C B A E D (1) C B A D E (1) B E A D C (1) B E A C D (1) B C A E D (1) B A C E D (1) A D E B C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -4 2 6 B 0 0 -4 0 0 C 4 4 0 -8 -6 D -2 0 8 0 0 E -6 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428564 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 6 B 0 0 -4 0 0 C 4 4 0 -8 -6 D -2 0 8 0 0 E -6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=25 C=19 E=14 B=7 so B is eliminated. Round 2 votes counts: A=39 D=25 C=20 E=16 so E is eliminated. Round 3 votes counts: A=46 D=33 C=21 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:203 A:202 E:200 B:198 C:197 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 2 6 B 0 0 -4 0 0 C 4 4 0 -8 -6 D -2 0 8 0 0 E -6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 6 B 0 0 -4 0 0 C 4 4 0 -8 -6 D -2 0 8 0 0 E -6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 6 B 0 0 -4 0 0 C 4 4 0 -8 -6 D -2 0 8 0 0 E -6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2203: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) A E C D B (10) C E D B A (9) C D B E A (6) E C D B A (5) E A C D B (5) B D C A E (5) B D A C E (5) A B D C E (5) A B D E C (4) D B C E A (3) A E C B D (3) E D B C A (2) E C D A B (2) E C A D B (2) B D A E C (2) B A D C E (2) A E B D C (2) A E B C D (2) A B E D C (2) E D C B A (1) E A D C B (1) D C B E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C A E D B (1) B D E C A (1) B D E A C (1) A C E D B (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 -12 -6 B 10 0 -4 -4 0 C 4 4 0 4 10 D 12 4 -4 0 -2 E 6 0 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -12 -6 B 10 0 -4 -4 0 C 4 4 0 4 10 D 12 4 -4 0 -2 E 6 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=27 C=19 E=18 D=4 so D is eliminated. Round 2 votes counts: A=32 B=30 C=20 E=18 so E is eliminated. Round 3 votes counts: A=38 B=32 C=30 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:211 D:205 B:201 E:199 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -12 -6 B 10 0 -4 -4 0 C 4 4 0 4 10 D 12 4 -4 0 -2 E 6 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -12 -6 B 10 0 -4 -4 0 C 4 4 0 4 10 D 12 4 -4 0 -2 E 6 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -12 -6 B 10 0 -4 -4 0 C 4 4 0 4 10 D 12 4 -4 0 -2 E 6 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2204: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) A B E C D (7) A B C D E (6) E D C B A (5) D C E B A (5) C D A E B (5) B E D C A (5) B A E D C (5) A E C D B (5) A E B C D (5) E D C A B (4) D C E A B (4) B D C E A (4) E A D C B (3) D C B E A (3) B C D A E (3) A C D B E (3) E D B C A (2) E D A C B (2) E C D A B (2) E A B D C (2) C D A B E (2) B E A D C (2) B A E C D (2) A B E D C (2) C D E A B (1) C D B A E (1) C A D E B (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 8 4 4 12 B -8 0 8 4 8 C -4 -8 0 2 -6 D -4 -4 -2 0 -6 E -12 -8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 4 12 B -8 0 8 4 8 C -4 -8 0 2 -6 D -4 -4 -2 0 -6 E -12 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=28 E=20 D=12 C=10 so C is eliminated. Round 2 votes counts: B=30 A=29 D=21 E=20 so E is eliminated. Round 3 votes counts: D=36 A=34 B=30 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:206 E:196 C:192 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 4 12 B -8 0 8 4 8 C -4 -8 0 2 -6 D -4 -4 -2 0 -6 E -12 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 4 12 B -8 0 8 4 8 C -4 -8 0 2 -6 D -4 -4 -2 0 -6 E -12 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 4 12 B -8 0 8 4 8 C -4 -8 0 2 -6 D -4 -4 -2 0 -6 E -12 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2205: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) C E B A D (6) C B E A D (6) C B A E D (6) A D C E B (6) D A E B C (5) D A E C B (4) B C E A D (4) A D E C B (4) A D C B E (4) B E C D A (3) A C B D E (3) E D B C A (2) E D A C B (2) E C B A D (2) E B D C A (2) E B C D A (2) E A C D B (2) D E A B C (2) D B A C E (2) B D E C A (2) A E C D B (2) A C B E D (2) E C D B A (1) E C B D A (1) E C A D B (1) E A D C B (1) D E A C B (1) D B E C A (1) D A B E C (1) D A B C E (1) C E A B D (1) C A B E D (1) C A B D E (1) B C D E A (1) B C D A E (1) B C A D E (1) A D B C E (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -8 14 -4 B 4 0 -20 10 2 C 8 20 0 18 18 D -14 -10 -18 0 -14 E 4 -2 -18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 14 -4 B 4 0 -20 10 2 C 8 20 0 18 18 D -14 -10 -18 0 -14 E 4 -2 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=21 B=21 D=17 E=16 so E is eliminated. Round 2 votes counts: A=28 C=26 B=25 D=21 so D is eliminated. Round 3 votes counts: A=44 B=30 C=26 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:232 A:199 E:199 B:198 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 14 -4 B 4 0 -20 10 2 C 8 20 0 18 18 D -14 -10 -18 0 -14 E 4 -2 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 14 -4 B 4 0 -20 10 2 C 8 20 0 18 18 D -14 -10 -18 0 -14 E 4 -2 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 14 -4 B 4 0 -20 10 2 C 8 20 0 18 18 D -14 -10 -18 0 -14 E 4 -2 -18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2206: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) B C A D E (7) D E B A C (6) D B E A C (6) E D A C B (5) E D A B C (5) C B A D E (5) C A E B D (5) B D E C A (5) B C D E A (5) A E C D B (4) E A D C B (3) D E B C A (3) A C E D B (3) A C B D E (3) E D B C A (2) E D B A C (2) E C A D B (2) E A C D B (2) D B E C A (2) C E A B D (2) B D C A E (2) B C D A E (2) A C B E D (2) E B C D A (1) D E A B C (1) D B A C E (1) D A E B C (1) C A B E D (1) B D C E A (1) B C E D A (1) A E D C B (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 8 -2 -8 B 2 0 8 0 -12 C -8 -8 0 6 -2 D 2 0 -6 0 2 E 8 12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 A B C D E A 0 -2 8 -2 -8 B 2 0 8 0 -12 C -8 -8 0 6 -2 D 2 0 -6 0 2 E 8 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=22 A=22 D=20 C=13 so C is eliminated. Round 2 votes counts: B=28 A=28 E=24 D=20 so D is eliminated. Round 3 votes counts: B=37 E=34 A=29 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:199 D:199 A:198 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 -2 -8 B 2 0 8 0 -12 C -8 -8 0 6 -2 D 2 0 -6 0 2 E 8 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -2 -8 B 2 0 8 0 -12 C -8 -8 0 6 -2 D 2 0 -6 0 2 E 8 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -2 -8 B 2 0 8 0 -12 C -8 -8 0 6 -2 D 2 0 -6 0 2 E 8 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2207: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (13) B A E D C (9) A C B E D (9) A B E D C (9) D C E B A (6) D E C B A (5) B E A D C (5) E B D A C (4) A B E C D (4) E D B A C (3) D E B C A (3) C A D E B (3) B E D A C (3) A B C E D (3) E A B D C (2) C D A E B (2) C B D A E (2) C A D B E (2) C A B D E (2) E A D B C (1) D E C A B (1) D E B A C (1) D B E C A (1) C D E A B (1) C D B A E (1) C B A D E (1) C A B E D (1) B C A E D (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 10 8 2 B 16 0 0 12 6 C -10 0 0 -8 -4 D -8 -12 8 0 -12 E -2 -6 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.620333 C: 0.379667 D: 0.000000 E: 0.000000 Sum of squares = 0.528960063351 Cumulative probabilities = A: 0.000000 B: 0.620333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 10 8 2 B 16 0 0 12 6 C -10 0 0 -8 -4 D -8 -12 8 0 -12 E -2 -6 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 B=18 D=17 E=10 so E is eliminated. Round 2 votes counts: A=30 C=28 B=22 D=20 so D is eliminated. Round 3 votes counts: C=40 B=30 A=30 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:217 E:204 A:202 C:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 10 8 2 B 16 0 0 12 6 C -10 0 0 -8 -4 D -8 -12 8 0 -12 E -2 -6 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 8 2 B 16 0 0 12 6 C -10 0 0 -8 -4 D -8 -12 8 0 -12 E -2 -6 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 8 2 B 16 0 0 12 6 C -10 0 0 -8 -4 D -8 -12 8 0 -12 E -2 -6 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2208: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) A E B C D (6) E A B D C (5) D B C E A (4) C D B A E (4) C B D A E (4) A B C E D (4) E D A B C (3) E B A D C (3) C B A D E (3) C A D B E (3) B C D E A (3) B A E C D (3) A E D C B (3) A B E C D (3) E D B C A (2) E D B A C (2) E B D C A (2) E A D B C (2) D E C B A (2) D C E B A (2) D C E A B (2) C D A B E (2) B E C A D (2) B C A E D (2) A E B D C (2) A D E C B (2) A C B E D (2) E A D C B (1) D E B C A (1) D E B A C (1) D E A C B (1) D C B A E (1) C D B E A (1) C B D E A (1) C B A E D (1) C A B E D (1) B E D C A (1) B E A C D (1) B C E A D (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -4 8 0 B 8 0 6 6 10 C 4 -6 0 2 0 D -8 -6 -2 0 -10 E 0 -10 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 8 0 B 8 0 6 6 10 C 4 -6 0 2 0 D -8 -6 -2 0 -10 E 0 -10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=21 E=20 C=20 B=13 so B is eliminated. Round 2 votes counts: A=29 C=26 E=24 D=21 so D is eliminated. Round 3 votes counts: C=42 E=29 A=29 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:200 E:200 A:198 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 8 0 B 8 0 6 6 10 C 4 -6 0 2 0 D -8 -6 -2 0 -10 E 0 -10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 8 0 B 8 0 6 6 10 C 4 -6 0 2 0 D -8 -6 -2 0 -10 E 0 -10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 8 0 B 8 0 6 6 10 C 4 -6 0 2 0 D -8 -6 -2 0 -10 E 0 -10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2209: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) C B A D E (7) E D A B C (6) A B C E D (6) D E B C A (5) D E C B A (4) B C A E D (4) B A C E D (4) E D B A C (3) E D A C B (3) E B D A C (3) E A D B C (3) C A B E D (3) A C B E D (3) A B E C D (3) E B A D C (2) E A B D C (2) E A B C D (2) D E B A C (2) D C B E A (2) C D B A E (2) C B D A E (2) C A D B E (2) C A B D E (2) B C A D E (2) B A E C D (2) E B A C D (1) D C B A E (1) D B C E A (1) C D A E B (1) C D A B E (1) C B A E D (1) B E A D C (1) B C D A E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -4 4 -2 B 0 0 8 6 0 C 4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 0 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.394812 C: 0.000000 D: 0.000000 E: 0.605188 Sum of squares = 0.522129017178 Cumulative probabilities = A: 0.000000 B: 0.394812 C: 0.394812 D: 0.394812 E: 1.000000 A B C D E A 0 0 -4 4 -2 B 0 0 8 6 0 C 4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 C=21 B=14 A=14 so B is eliminated. Round 2 votes counts: C=28 E=26 D=26 A=20 so A is eliminated. Round 3 votes counts: C=42 E=32 D=26 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:207 A:199 C:195 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 4 -2 B 0 0 8 6 0 C 4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 -2 B 0 0 8 6 0 C 4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 -2 B 0 0 8 6 0 C 4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2210: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) D B A C E (11) A B D E C (10) E C A B D (9) C E A D B (8) D B C A E (7) C E D A B (4) B D A E C (4) A B E D C (4) D C B A E (3) B A D E C (3) E C A D B (2) E A C B D (2) D A B C E (2) A E C B D (2) A E B C D (2) E C D B A (1) E C B D A (1) E C B A D (1) E B C A D (1) E A B D C (1) E A B C D (1) D C B E A (1) D B C E A (1) C D E B A (1) B D A C E (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -8 -2 4 B 0 0 -2 -14 -2 C 8 2 0 2 12 D 2 14 -2 0 -6 E -4 2 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -2 4 B 0 0 -2 -14 -2 C 8 2 0 2 12 D 2 14 -2 0 -6 E -4 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 A=21 E=19 B=8 so B is eliminated. Round 2 votes counts: D=30 C=27 A=24 E=19 so E is eliminated. Round 3 votes counts: C=42 D=30 A=28 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:204 A:197 E:196 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -2 4 B 0 0 -2 -14 -2 C 8 2 0 2 12 D 2 14 -2 0 -6 E -4 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 4 B 0 0 -2 -14 -2 C 8 2 0 2 12 D 2 14 -2 0 -6 E -4 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 4 B 0 0 -2 -14 -2 C 8 2 0 2 12 D 2 14 -2 0 -6 E -4 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2211: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) A C E D B (12) A C B D E (12) E D B A C (10) B D E C A (10) E D B C A (7) A E D C B (5) A C E B D (5) E D A B C (4) E A D B C (3) C B D E A (3) D B E C A (2) B C D E A (2) A E D B C (2) A B D E C (2) E D C B A (1) E A D C B (1) D E B C A (1) D B E A C (1) C A E D B (1) B D E A C (1) B D C E A (1) A B C D E (1) Total count = 100 A B C D E A 0 22 18 14 6 B -22 0 -6 0 -4 C -18 6 0 -2 0 D -14 0 2 0 -2 E -6 4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 18 14 6 B -22 0 -6 0 -4 C -18 6 0 -2 0 D -14 0 2 0 -2 E -6 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=26 C=17 B=14 D=4 so D is eliminated. Round 2 votes counts: A=39 E=27 C=17 B=17 so C is eliminated. Round 3 votes counts: A=53 E=27 B=20 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:200 C:193 D:193 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 18 14 6 B -22 0 -6 0 -4 C -18 6 0 -2 0 D -14 0 2 0 -2 E -6 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 14 6 B -22 0 -6 0 -4 C -18 6 0 -2 0 D -14 0 2 0 -2 E -6 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 14 6 B -22 0 -6 0 -4 C -18 6 0 -2 0 D -14 0 2 0 -2 E -6 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2212: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) E A D B C (8) B D C E A (8) A E C D B (8) C B D A E (7) E D B C A (6) D B C E A (6) A C E B D (5) C A B D E (4) E A B D C (3) C B A D E (3) B C D E A (3) A E C B D (3) A C E D B (3) A C B D E (3) E D B A C (2) E B D A C (2) E A D C B (2) D C B E A (2) A E D B C (2) A E B C D (2) E D C B A (1) E D A B C (1) C D B A E (1) B C D A E (1) B A C D E (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 16 20 12 B -14 0 -8 -6 -16 C -16 8 0 -6 0 D -20 6 6 0 -20 E -12 16 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 20 12 B -14 0 -8 -6 -16 C -16 8 0 -6 0 D -20 6 6 0 -20 E -12 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=25 C=15 B=13 D=8 so D is eliminated. Round 2 votes counts: A=39 E=25 B=19 C=17 so C is eliminated. Round 3 votes counts: A=43 B=32 E=25 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:231 E:212 C:193 D:186 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 16 20 12 B -14 0 -8 -6 -16 C -16 8 0 -6 0 D -20 6 6 0 -20 E -12 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 20 12 B -14 0 -8 -6 -16 C -16 8 0 -6 0 D -20 6 6 0 -20 E -12 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 20 12 B -14 0 -8 -6 -16 C -16 8 0 -6 0 D -20 6 6 0 -20 E -12 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2213: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) A D C E B (9) B E C D A (7) B E C A D (6) C E D B A (5) E C B D A (4) E B C D A (4) D A C E B (4) B E A C D (4) A D B C E (4) D C E B A (3) D C E A B (3) B A D E C (3) A C D E B (3) A B D E C (3) E C D B A (2) C E B D A (2) C D E B A (2) B A E C D (2) A D C B E (2) A C E D B (2) A B E C D (2) E C B A D (1) E B C A D (1) D C B E A (1) D C A B E (1) D B C E A (1) D B A C E (1) D A B C E (1) C E D A B (1) C D E A B (1) B E D C A (1) B D E C A (1) B A E D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 -10 0 B 4 0 -12 -12 -14 C 14 12 0 -2 14 D 10 12 2 0 8 E 0 14 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -10 0 B 4 0 -12 -12 -14 C 14 12 0 -2 14 D 10 12 2 0 8 E 0 14 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=25 B=25 E=12 C=11 so C is eliminated. Round 2 votes counts: D=28 A=27 B=25 E=20 so E is eliminated. Round 3 votes counts: B=37 D=36 A=27 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:219 D:216 E:196 A:186 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -14 -10 0 B 4 0 -12 -12 -14 C 14 12 0 -2 14 D 10 12 2 0 8 E 0 14 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -10 0 B 4 0 -12 -12 -14 C 14 12 0 -2 14 D 10 12 2 0 8 E 0 14 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -10 0 B 4 0 -12 -12 -14 C 14 12 0 -2 14 D 10 12 2 0 8 E 0 14 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2214: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) A E C B D (8) E B C A D (6) D B C E A (6) E B D C A (4) D B E C A (4) B D E C A (4) B E D C A (3) B E C D A (3) B E C A D (3) A E B C D (3) E C B A D (2) E B A D C (2) E A C B D (2) E A B C D (2) D E B A C (2) D C B A E (2) D C A B E (2) C B E A D (2) C A B E D (2) C A B D E (2) B C E D A (2) B C E A D (2) A D E C B (2) A D C B E (2) E D B C A (1) E B D A C (1) D C B E A (1) D B C A E (1) D B A E C (1) D A C E B (1) D A C B E (1) D A B E C (1) C D B E A (1) C D A B E (1) C B E D A (1) C B A E D (1) C A E B D (1) A E D B C (1) A E C D B (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -14 14 -8 B 10 0 4 30 -2 C 14 -4 0 14 -12 D -14 -30 -14 0 -26 E 8 2 12 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -14 14 -8 B 10 0 4 30 -2 C 14 -4 0 14 -12 D -14 -30 -14 0 -26 E 8 2 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 E=20 B=17 C=11 so C is eliminated. Round 2 votes counts: A=35 D=24 B=21 E=20 so E is eliminated. Round 3 votes counts: A=39 B=36 D=25 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:224 B:221 C:206 A:191 D:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -14 14 -8 B 10 0 4 30 -2 C 14 -4 0 14 -12 D -14 -30 -14 0 -26 E 8 2 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 14 -8 B 10 0 4 30 -2 C 14 -4 0 14 -12 D -14 -30 -14 0 -26 E 8 2 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 14 -8 B 10 0 4 30 -2 C 14 -4 0 14 -12 D -14 -30 -14 0 -26 E 8 2 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2215: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) A C E D B (8) B D E C A (7) B D A E C (6) E C D B A (5) E C A D B (5) C E A B D (4) C A E B D (4) A D B C E (4) D B A E C (3) D A B E C (3) B D E A C (3) E D B C A (2) E C D A B (2) D E B C A (2) D B E C A (2) C E B D A (2) C A E D B (2) C A B E D (2) B A D C E (2) A D E B C (2) A D B E C (2) A C E B D (2) A B D C E (2) E D C B A (1) E C B D A (1) E B D C A (1) E A D C B (1) D B E A C (1) C E B A D (1) C B A D E (1) B E D C A (1) B E C D A (1) B D C E A (1) B D A C E (1) B C D A E (1) A E D C B (1) A E C D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -12 8 -2 B -10 0 -4 -10 -10 C 12 4 0 4 -6 D -8 10 -4 0 -10 E 2 10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -12 8 -2 B -10 0 -4 -10 -10 C 12 4 0 4 -6 D -8 10 -4 0 -10 E 2 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 B=23 E=18 D=11 so D is eliminated. Round 2 votes counts: B=29 A=27 C=24 E=20 so E is eliminated. Round 3 votes counts: C=38 B=34 A=28 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:214 C:207 A:202 D:194 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -12 8 -2 B -10 0 -4 -10 -10 C 12 4 0 4 -6 D -8 10 -4 0 -10 E 2 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 8 -2 B -10 0 -4 -10 -10 C 12 4 0 4 -6 D -8 10 -4 0 -10 E 2 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 8 -2 B -10 0 -4 -10 -10 C 12 4 0 4 -6 D -8 10 -4 0 -10 E 2 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2216: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) E D A B C (8) C B A D E (8) C A B D E (8) B C D E A (7) A C E B D (7) D E B A C (6) B D E C A (5) D B E C A (4) C B D A E (4) C A B E D (4) B D C E A (4) A E C D B (4) E D B A C (3) E A D B C (3) A C E D B (3) D E A B C (2) C B D E A (2) A E C B D (2) E D B C A (1) E B D C A (1) C D B E A (1) C B A E D (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -20 -16 -14 B 14 0 8 6 -4 C 20 -8 0 4 0 D 16 -6 -4 0 24 E 14 4 0 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.705882 C: 0.000000 D: 0.117647 E: 0.176471 Sum of squares = 0.543252595317 Cumulative probabilities = A: 0.000000 B: 0.705882 C: 0.705882 D: 0.823529 E: 1.000000 A B C D E A 0 -14 -20 -16 -14 B 14 0 8 6 -4 C 20 -8 0 4 0 D 16 -6 -4 0 24 E 14 4 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.705882 C: 0.000000 D: 0.117647 E: 0.176471 Sum of squares = 0.543252595174 Cumulative probabilities = A: 0.000000 B: 0.705882 C: 0.705882 D: 0.823529 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=22 A=18 E=16 B=16 so E is eliminated. Round 2 votes counts: D=34 C=28 A=21 B=17 so B is eliminated. Round 3 votes counts: D=44 C=35 A=21 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:215 B:212 C:208 E:197 A:168 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -20 -16 -14 B 14 0 8 6 -4 C 20 -8 0 4 0 D 16 -6 -4 0 24 E 14 4 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.705882 C: 0.000000 D: 0.117647 E: 0.176471 Sum of squares = 0.543252595174 Cumulative probabilities = A: 0.000000 B: 0.705882 C: 0.705882 D: 0.823529 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -16 -14 B 14 0 8 6 -4 C 20 -8 0 4 0 D 16 -6 -4 0 24 E 14 4 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.705882 C: 0.000000 D: 0.117647 E: 0.176471 Sum of squares = 0.543252595174 Cumulative probabilities = A: 0.000000 B: 0.705882 C: 0.705882 D: 0.823529 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -16 -14 B 14 0 8 6 -4 C 20 -8 0 4 0 D 16 -6 -4 0 24 E 14 4 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.705882 C: 0.000000 D: 0.117647 E: 0.176471 Sum of squares = 0.543252595174 Cumulative probabilities = A: 0.000000 B: 0.705882 C: 0.705882 D: 0.823529 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2217: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) D C E B A (6) B E C A D (5) E D A B C (4) C B D E A (4) B E A C D (4) A E B D C (4) A D E C B (4) E B D C A (3) D C A E B (3) B C E D A (3) A E D B C (3) A D E B C (3) E D B A C (2) E B A D C (2) E A D B C (2) D E C A B (2) D E A B C (2) D A E C B (2) D A C E B (2) C E B D A (2) C D E B A (2) C D B A E (2) C B E A D (2) C B A E D (2) B A C E D (2) A D C E B (2) A C D B E (2) E B D A C (1) E A B D C (1) D E C B A (1) D E A C B (1) D C E A B (1) D A E B C (1) C E D B A (1) C D B E A (1) C D A E B (1) C D A B E (1) C B D A E (1) A D C B E (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 -16 -22 B 10 0 -12 -4 -16 C 4 12 0 -8 4 D 16 4 8 0 -8 E 22 16 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.400000 Sum of squares = 0.36 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.600000 E: 1.000000 A B C D E A 0 -10 -4 -16 -22 B 10 0 -12 -4 -16 C 4 12 0 -8 4 D 16 4 8 0 -8 E 22 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.400000 Sum of squares = 0.360000000037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=22 D=21 E=15 B=14 so B is eliminated. Round 2 votes counts: C=31 E=24 A=24 D=21 so D is eliminated. Round 3 votes counts: C=41 E=30 A=29 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:221 D:210 C:206 B:189 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 -16 -22 B 10 0 -12 -4 -16 C 4 12 0 -8 4 D 16 4 8 0 -8 E 22 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.400000 Sum of squares = 0.360000000037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -16 -22 B 10 0 -12 -4 -16 C 4 12 0 -8 4 D 16 4 8 0 -8 E 22 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.400000 Sum of squares = 0.360000000037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -16 -22 B 10 0 -12 -4 -16 C 4 12 0 -8 4 D 16 4 8 0 -8 E 22 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.400000 Sum of squares = 0.360000000037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2218: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) D B C A E (6) C E A B D (6) B D C E A (6) A D E B C (6) D B A E C (5) D B A C E (5) E C A B D (4) C E B D A (4) A E C D B (4) A C E D B (4) C E B A D (3) C B E D A (3) E A C B D (2) D A B C E (2) C E A D B (2) B E D A C (2) B D E A C (2) B D A E C (2) B C D E A (2) A E D B C (2) A E C B D (2) A D E C B (2) E C B A D (1) E B C A D (1) E B A C D (1) D C B E A (1) D B C E A (1) D A C B E (1) C D E B A (1) C D B E A (1) C D A E B (1) C A E D B (1) B E C D A (1) B D E C A (1) A E B D C (1) A D C E B (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 8 -10 10 B -2 0 10 -12 0 C -8 -10 0 -10 4 D 10 12 10 0 12 E -10 0 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -10 10 B -2 0 10 -12 0 C -8 -10 0 -10 4 D 10 12 10 0 12 E -10 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 C=22 B=16 E=9 so E is eliminated. Round 2 votes counts: D=29 C=27 A=26 B=18 so B is eliminated. Round 3 votes counts: D=42 C=31 A=27 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:205 B:198 C:188 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -10 10 B -2 0 10 -12 0 C -8 -10 0 -10 4 D 10 12 10 0 12 E -10 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -10 10 B -2 0 10 -12 0 C -8 -10 0 -10 4 D 10 12 10 0 12 E -10 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -10 10 B -2 0 10 -12 0 C -8 -10 0 -10 4 D 10 12 10 0 12 E -10 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2219: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) B C D A E (9) C D E A B (6) B D C E A (6) A B E D C (5) C D E B A (4) B C D E A (4) B C A D E (4) A E D C B (4) A E D B C (4) A E B C D (4) D C B E A (3) C B D E A (3) E A D C B (2) E A C D B (2) D E C A B (2) D C E B A (2) D B C E A (2) C E D A B (2) B A D E C (2) B A C D E (2) A E C D B (2) A E B D C (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A D B (1) D E B A C (1) D C E A B (1) D B E C A (1) C B D A E (1) B D C A E (1) B D A E C (1) B D A C E (1) B A E C D (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -24 -22 -6 B 14 0 6 0 16 C 24 -6 0 18 22 D 22 0 -18 0 30 E 6 -16 -22 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.866576 C: 0.000000 D: 0.133424 E: 0.000000 Sum of squares = 0.768756329281 Cumulative probabilities = A: 0.000000 B: 0.866576 C: 0.866576 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -24 -22 -6 B 14 0 6 0 16 C 24 -6 0 18 22 D 22 0 -18 0 30 E 6 -16 -22 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750002 C: 0.000000 D: 0.249998 E: 0.000000 Sum of squares = 0.625001997668 Cumulative probabilities = A: 0.000000 B: 0.750002 C: 0.750002 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=25 A=25 D=12 E=7 so E is eliminated. Round 2 votes counts: B=31 A=29 C=27 D=13 so D is eliminated. Round 3 votes counts: C=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:229 B:218 D:217 E:169 A:167 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -24 -22 -6 B 14 0 6 0 16 C 24 -6 0 18 22 D 22 0 -18 0 30 E 6 -16 -22 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750002 C: 0.000000 D: 0.249998 E: 0.000000 Sum of squares = 0.625001997668 Cumulative probabilities = A: 0.000000 B: 0.750002 C: 0.750002 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -24 -22 -6 B 14 0 6 0 16 C 24 -6 0 18 22 D 22 0 -18 0 30 E 6 -16 -22 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750002 C: 0.000000 D: 0.249998 E: 0.000000 Sum of squares = 0.625001997668 Cumulative probabilities = A: 0.000000 B: 0.750002 C: 0.750002 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -24 -22 -6 B 14 0 6 0 16 C 24 -6 0 18 22 D 22 0 -18 0 30 E 6 -16 -22 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750002 C: 0.000000 D: 0.249998 E: 0.000000 Sum of squares = 0.625001997668 Cumulative probabilities = A: 0.000000 B: 0.750002 C: 0.750002 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2220: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) E A C B D (7) D B C A E (5) D B A C E (5) C E A B D (5) A E B C D (5) E C A B D (4) E A D B C (4) C D B E A (4) C B D A E (4) B D A C E (4) D E A B C (3) D C B E A (3) E D A C B (2) E C D A B (2) E A C D B (2) E A B C D (2) D A E B C (2) C E D A B (2) C B E A D (2) C B A E D (2) B D C A E (2) B A D E C (2) A E D B C (2) E A D C B (1) D E B A C (1) D C E B A (1) D C B A E (1) D B E C A (1) D B E A C (1) D A B E C (1) C E B A D (1) C B D E A (1) B D A E C (1) B C D A E (1) B C A D E (1) B A C E D (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 16 -12 2 B 6 0 12 -6 8 C -16 -12 0 -6 -10 D 12 6 6 0 8 E -2 -8 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 16 -12 2 B 6 0 12 -6 8 C -16 -12 0 -6 -10 D 12 6 6 0 8 E -2 -8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999164 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=24 C=21 B=12 A=10 so A is eliminated. Round 2 votes counts: D=34 E=31 C=21 B=14 so B is eliminated. Round 3 votes counts: D=43 E=33 C=24 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:210 A:200 E:196 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 16 -12 2 B 6 0 12 -6 8 C -16 -12 0 -6 -10 D 12 6 6 0 8 E -2 -8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999164 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 -12 2 B 6 0 12 -6 8 C -16 -12 0 -6 -10 D 12 6 6 0 8 E -2 -8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999164 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 -12 2 B 6 0 12 -6 8 C -16 -12 0 -6 -10 D 12 6 6 0 8 E -2 -8 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999164 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2221: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (18) E C D B A (12) E D C B A (6) E C B D A (4) E C A B D (4) D E C A B (4) D C E B A (4) A B E C D (4) A B C E D (4) E C B A D (3) D E C B A (3) D C B E A (3) A E C D B (3) A D B C E (3) D B A C E (2) C E B D A (2) C B E D A (2) E D C A B (1) E C D A B (1) E A C D B (1) E A C B D (1) D A B C E (1) C D E B A (1) B E C A D (1) B D C E A (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B A D C E (1) B A C E D (1) A E D C B (1) A E D B C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -12 -2 -14 B 2 0 -14 4 -6 C 12 14 0 -4 -4 D 2 -4 4 0 -8 E 14 6 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -12 -2 -14 B 2 0 -14 4 -6 C 12 14 0 -4 -4 D 2 -4 4 0 -8 E 14 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=33 D=17 B=9 C=5 so C is eliminated. Round 2 votes counts: A=36 E=35 D=18 B=11 so B is eliminated. Round 3 votes counts: E=40 A=38 D=22 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:209 D:197 B:193 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -12 -2 -14 B 2 0 -14 4 -6 C 12 14 0 -4 -4 D 2 -4 4 0 -8 E 14 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -2 -14 B 2 0 -14 4 -6 C 12 14 0 -4 -4 D 2 -4 4 0 -8 E 14 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -2 -14 B 2 0 -14 4 -6 C 12 14 0 -4 -4 D 2 -4 4 0 -8 E 14 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2222: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (14) B A D C E (8) B A C D E (7) E D B A C (6) C A B D E (6) D C A B E (5) C D A B E (5) B A C E D (5) E C D A B (4) E B A D C (3) D C E A B (3) C B A D E (3) B A E D C (3) E C B A D (2) D A E B C (2) B A E C D (2) A B C D E (2) E D C B A (1) E D B C A (1) E D A C B (1) E D A B C (1) E C D B A (1) E C B D A (1) E B A C D (1) D E A B C (1) D C A E B (1) D B A E C (1) D A B C E (1) C E D A B (1) C E A B D (1) C D E A B (1) C D A E B (1) C A D B E (1) B E A D C (1) B A D E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 -4 12 B -6 0 -6 -4 6 C 4 6 0 -10 6 D 4 4 10 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -4 12 B -6 0 -6 -4 6 C 4 6 0 -10 6 D 4 4 10 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=27 C=19 D=14 A=4 so A is eliminated. Round 2 votes counts: E=36 B=30 C=20 D=14 so D is eliminated. Round 3 votes counts: E=39 B=32 C=29 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:205 C:203 B:195 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -4 12 B -6 0 -6 -4 6 C 4 6 0 -10 6 D 4 4 10 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 12 B -6 0 -6 -4 6 C 4 6 0 -10 6 D 4 4 10 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 12 B -6 0 -6 -4 6 C 4 6 0 -10 6 D 4 4 10 0 2 E -12 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2223: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (15) A D C E B (11) D E A B C (9) A C D E B (9) B E D C A (8) E D B A C (7) C A B D E (7) B C E A D (6) C B A E D (5) D A E C B (4) C A B E D (4) A C D B E (4) E B D A C (3) D E B A C (3) C A D B E (2) E B D C A (1) B C A E D (1) A D E C B (1) Total count = 100 A B C D E A 0 2 2 0 -4 B -2 0 6 0 4 C -2 -6 0 6 -2 D 0 0 -6 0 0 E 4 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.301432 B: 0.301432 C: 0.000000 D: 0.246419 E: 0.150716 Sum of squares = 0.265160690424 Cumulative probabilities = A: 0.301432 B: 0.602865 C: 0.602865 D: 0.849284 E: 1.000000 A B C D E A 0 2 2 0 -4 B -2 0 6 0 4 C -2 -6 0 6 -2 D 0 0 -6 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.294118 B: 0.294118 C: 0.000000 D: 0.264706 E: 0.147059 Sum of squares = 0.264705882353 Cumulative probabilities = A: 0.294118 B: 0.588235 C: 0.588235 D: 0.852941 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=25 C=18 D=16 E=11 so E is eliminated. Round 2 votes counts: B=34 A=25 D=23 C=18 so C is eliminated. Round 3 votes counts: B=39 A=38 D=23 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:204 E:201 A:200 C:198 D:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 2 0 -4 B -2 0 6 0 4 C -2 -6 0 6 -2 D 0 0 -6 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.294118 B: 0.294118 C: 0.000000 D: 0.264706 E: 0.147059 Sum of squares = 0.264705882353 Cumulative probabilities = A: 0.294118 B: 0.588235 C: 0.588235 D: 0.852941 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 -4 B -2 0 6 0 4 C -2 -6 0 6 -2 D 0 0 -6 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.294118 B: 0.294118 C: 0.000000 D: 0.264706 E: 0.147059 Sum of squares = 0.264705882353 Cumulative probabilities = A: 0.294118 B: 0.588235 C: 0.588235 D: 0.852941 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 -4 B -2 0 6 0 4 C -2 -6 0 6 -2 D 0 0 -6 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.294118 B: 0.294118 C: 0.000000 D: 0.264706 E: 0.147059 Sum of squares = 0.264705882353 Cumulative probabilities = A: 0.294118 B: 0.588235 C: 0.588235 D: 0.852941 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2224: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) D C A E B (8) B A E C D (6) B D A C E (5) A B E C D (5) E A C B D (4) D C E B A (4) D B C E A (4) C D E A B (4) B A E D C (4) C E D A B (3) B E C D A (3) B E A C D (3) E C B D A (2) E C A D B (2) E B A C D (2) D B C A E (2) C D A E B (2) B A D E C (2) A B D C E (2) E C D B A (1) E C D A B (1) E C A B D (1) E B C D A (1) E A C D B (1) D C B E A (1) D C B A E (1) D C A B E (1) D B A C E (1) D A C B E (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E A C (1) B D C E A (1) A E B C D (1) A D C B E (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -4 -16 -2 B -6 0 0 0 6 C 4 0 0 -6 12 D 16 0 6 0 10 E 2 -6 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.501548 C: 0.000000 D: 0.498452 E: 0.000000 Sum of squares = 0.500004790655 Cumulative probabilities = A: 0.000000 B: 0.501548 C: 0.501548 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -16 -2 B -6 0 0 0 6 C 4 0 0 -6 12 D 16 0 6 0 10 E 2 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=16 E=15 C=9 so C is eliminated. Round 2 votes counts: D=38 B=28 E=18 A=16 so A is eliminated. Round 3 votes counts: D=42 B=39 E=19 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:216 C:205 B:200 A:192 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -16 -2 B -6 0 0 0 6 C 4 0 0 -6 12 D 16 0 6 0 10 E 2 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -16 -2 B -6 0 0 0 6 C 4 0 0 -6 12 D 16 0 6 0 10 E 2 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -16 -2 B -6 0 0 0 6 C 4 0 0 -6 12 D 16 0 6 0 10 E 2 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2225: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (20) E B C D A (12) D C A B E (9) E C D B A (8) B A E D C (8) C D A E B (7) B E A D C (7) C D E A B (6) A B D C E (5) E C D A B (3) E B A C D (3) B E A C D (3) B A D C E (3) D C A E B (2) E C B D A (1) E B A D C (1) C E D A B (1) B A D E C (1) Total count = 100 A B C D E A 0 6 2 2 10 B -6 0 -14 -12 12 C -2 14 0 -12 6 D -2 12 12 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 2 10 B -6 0 -14 -12 12 C -2 14 0 -12 6 D -2 12 12 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 B=22 C=14 D=11 so D is eliminated. Round 2 votes counts: E=28 C=25 A=25 B=22 so B is eliminated. Round 3 votes counts: E=38 A=37 C=25 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:210 C:203 B:190 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 2 10 B -6 0 -14 -12 12 C -2 14 0 -12 6 D -2 12 12 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 10 B -6 0 -14 -12 12 C -2 14 0 -12 6 D -2 12 12 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 10 B -6 0 -14 -12 12 C -2 14 0 -12 6 D -2 12 12 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2226: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) E B A C D (6) D C A B E (6) C D E A B (6) C D A E B (6) C D A B E (5) B D C A E (5) E A B D C (4) E C A D B (3) D A C E B (3) C D B E A (3) B A D E C (3) A B D E C (3) E B A D C (2) E A C D B (2) E A B C D (2) D C B A E (2) C E B D A (2) B E C A D (2) B D A C E (2) B C D E A (2) A D C E B (2) A B E D C (2) D B A C E (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) C D B A E (1) C B D E A (1) C A D E B (1) B E D C A (1) B E C D A (1) B E A C D (1) B A E D C (1) A E D C B (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 4 2 0 -4 B -4 0 4 6 12 C -2 -4 0 -8 4 D 0 -6 8 0 10 E 4 -12 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.644305 B: 0.000000 C: 0.000000 D: 0.355695 E: 0.000000 Sum of squares = 0.5416476064 Cumulative probabilities = A: 0.644305 B: 0.644305 C: 0.644305 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 0 -4 B -4 0 4 6 12 C -2 -4 0 -8 4 D 0 -6 8 0 10 E 4 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008009 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=27 E=19 D=14 A=10 so A is eliminated. Round 2 votes counts: B=35 C=27 E=21 D=17 so D is eliminated. Round 3 votes counts: C=41 B=37 E=22 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 D:206 A:201 C:195 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 0 -4 B -4 0 4 6 12 C -2 -4 0 -8 4 D 0 -6 8 0 10 E 4 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008009 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 0 -4 B -4 0 4 6 12 C -2 -4 0 -8 4 D 0 -6 8 0 10 E 4 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008009 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 0 -4 B -4 0 4 6 12 C -2 -4 0 -8 4 D 0 -6 8 0 10 E 4 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008009 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2227: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) E C A D B (9) C E A B D (8) D B E A C (7) D B A E C (7) A C B D E (7) B D A C E (6) E D C B A (5) E C D B A (5) E C D A B (5) D E B C A (5) B A D C E (5) A B D C E (5) B D A E C (4) A B C D E (4) E D B C A (3) C E A D B (2) A C B E D (2) D E B A C (1) Total count = 100 A B C D E A 0 4 -4 4 0 B -4 0 -6 2 -6 C 4 6 0 4 -2 D -4 -2 -4 0 2 E 0 6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.068760 B: 0.000000 C: 0.198430 D: 0.198430 E: 0.534380 Sum of squares = 0.3690388637 Cumulative probabilities = A: 0.068760 B: 0.068760 C: 0.267190 D: 0.465620 E: 1.000000 A B C D E A 0 4 -4 4 0 B -4 0 -6 2 -6 C 4 6 0 4 -2 D -4 -2 -4 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.000000 C: 0.210526 D: 0.210526 E: 0.526316 Sum of squares = 0.36842105263 Cumulative probabilities = A: 0.052632 B: 0.052632 C: 0.263158 D: 0.473684 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=20 C=20 A=18 B=15 so B is eliminated. Round 2 votes counts: D=30 E=27 A=23 C=20 so C is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:206 E:203 A:202 D:196 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 0 B -4 0 -6 2 -6 C 4 6 0 4 -2 D -4 -2 -4 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.000000 C: 0.210526 D: 0.210526 E: 0.526316 Sum of squares = 0.36842105263 Cumulative probabilities = A: 0.052632 B: 0.052632 C: 0.263158 D: 0.473684 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 0 B -4 0 -6 2 -6 C 4 6 0 4 -2 D -4 -2 -4 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.000000 C: 0.210526 D: 0.210526 E: 0.526316 Sum of squares = 0.36842105263 Cumulative probabilities = A: 0.052632 B: 0.052632 C: 0.263158 D: 0.473684 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 0 B -4 0 -6 2 -6 C 4 6 0 4 -2 D -4 -2 -4 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.052632 B: 0.000000 C: 0.210526 D: 0.210526 E: 0.526316 Sum of squares = 0.36842105263 Cumulative probabilities = A: 0.052632 B: 0.052632 C: 0.263158 D: 0.473684 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2228: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) D B C E A (9) D C B A E (8) C D A E B (7) E A B C D (6) A E B C D (6) B E A C D (5) D C B E A (4) C A E D B (4) B D C E A (4) E B A D C (3) E A B D C (3) C A E B D (3) E B A C D (2) D B E A C (2) C A D E B (2) B E A D C (2) B D E A C (2) A E C D B (2) A E C B D (2) E D B A C (1) D E B A C (1) D E A C B (1) D E A B C (1) D C E A B (1) D C A B E (1) D B E C A (1) C D B A E (1) C B A D E (1) B E C A D (1) B C D E A (1) A E D C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -16 -10 0 B -4 0 0 -16 -16 C 16 0 0 -10 16 D 10 16 10 0 14 E 0 16 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -10 0 B -4 0 0 -16 -16 C 16 0 0 -10 16 D 10 16 10 0 14 E 0 16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=18 E=15 B=15 A=13 so A is eliminated. Round 2 votes counts: D=39 E=26 C=20 B=15 so B is eliminated. Round 3 votes counts: D=45 E=34 C=21 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:225 C:211 E:193 A:189 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -16 -10 0 B -4 0 0 -16 -16 C 16 0 0 -10 16 D 10 16 10 0 14 E 0 16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -10 0 B -4 0 0 -16 -16 C 16 0 0 -10 16 D 10 16 10 0 14 E 0 16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -10 0 B -4 0 0 -16 -16 C 16 0 0 -10 16 D 10 16 10 0 14 E 0 16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2229: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) D E B A C (8) C E D B A (8) C E A D B (7) C E D A B (6) E D C B A (5) E C D A B (4) C A E B D (4) B A D E C (4) E D A C B (3) E D A B C (3) D B E A C (3) A B D E C (3) E D C A B (2) E D B A C (2) E A D B C (2) D B E C A (2) C E A B D (2) B D A C E (2) B A D C E (2) B A C D E (2) A D B E C (2) A B C D E (2) E C D B A (1) C D E B A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A E D B (1) C A B E D (1) B D E A C (1) B D C E A (1) B C D A E (1) A E D B C (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 2 -26 -24 B 10 0 4 -22 -22 C -2 -4 0 -14 -12 D 26 22 14 0 -8 E 24 22 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 2 -26 -24 B 10 0 4 -22 -22 C -2 -4 0 -14 -12 D 26 22 14 0 -8 E 24 22 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=22 B=22 D=13 A=10 so A is eliminated. Round 2 votes counts: C=33 B=28 E=24 D=15 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:233 D:227 B:185 C:184 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 -26 -24 B 10 0 4 -22 -22 C -2 -4 0 -14 -12 D 26 22 14 0 -8 E 24 22 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -26 -24 B 10 0 4 -22 -22 C -2 -4 0 -14 -12 D 26 22 14 0 -8 E 24 22 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -26 -24 B 10 0 4 -22 -22 C -2 -4 0 -14 -12 D 26 22 14 0 -8 E 24 22 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2230: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) E C B D A (9) B E D A C (8) C A E D B (7) E B D C A (6) C E A D B (6) C A D E B (6) D B A E C (5) C E B A D (5) A D B C E (4) A C D B E (4) E C B A D (3) C A D B E (3) B D E A C (3) B D A E C (3) A D C B E (3) D A B E C (2) D A B C E (2) C E A B D (2) C A E B D (2) E D B A C (1) E C D A B (1) E B D A C (1) D A E C B (1) C B A E D (1) B E D C A (1) B E C D A (1) Total count = 100 A B C D E A 0 -14 -26 -8 -14 B 14 0 -6 10 -20 C 26 6 0 20 -10 D 8 -10 -20 0 -28 E 14 20 10 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -26 -8 -14 B 14 0 -6 10 -20 C 26 6 0 20 -10 D 8 -10 -20 0 -28 E 14 20 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=31 B=16 A=11 D=10 so D is eliminated. Round 2 votes counts: C=32 E=31 B=21 A=16 so A is eliminated. Round 3 votes counts: C=39 E=32 B=29 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:236 C:221 B:199 D:175 A:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -26 -8 -14 B 14 0 -6 10 -20 C 26 6 0 20 -10 D 8 -10 -20 0 -28 E 14 20 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -26 -8 -14 B 14 0 -6 10 -20 C 26 6 0 20 -10 D 8 -10 -20 0 -28 E 14 20 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -26 -8 -14 B 14 0 -6 10 -20 C 26 6 0 20 -10 D 8 -10 -20 0 -28 E 14 20 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2231: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) B E C A D (7) B D A E C (7) D A C E B (5) A D B E C (5) C E B D A (4) A E D C B (4) E C A D B (3) E C A B D (3) D A B E C (3) C D E A B (3) B D C E A (3) B A E D C (3) A D E C B (3) E C B A D (2) E A C D B (2) D A C B E (2) D A B C E (2) C E D A B (2) C E A D B (2) C E A B D (2) C D B E A (2) B E C D A (2) B E A C D (2) B D E A C (2) B A D E C (2) A E C D B (2) A B D E C (2) E B C A D (1) D C B A E (1) D B A C E (1) C E D B A (1) C D A E B (1) B E D C A (1) B E A D C (1) B D A C E (1) B C D E A (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 2 12 -8 B 4 0 -6 8 0 C -2 6 0 0 -16 D -12 -8 0 0 -4 E 8 0 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.516324 C: 0.000000 D: 0.000000 E: 0.483676 Sum of squares = 0.500532936764 Cumulative probabilities = A: 0.000000 B: 0.516324 C: 0.516324 D: 0.516324 E: 1.000000 A B C D E A 0 -4 2 12 -8 B 4 0 -6 8 0 C -2 6 0 0 -16 D -12 -8 0 0 -4 E 8 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=25 A=18 D=14 E=11 so E is eliminated. Round 2 votes counts: C=33 B=33 A=20 D=14 so D is eliminated. Round 3 votes counts: C=34 B=34 A=32 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:214 B:203 A:201 C:194 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 12 -8 B 4 0 -6 8 0 C -2 6 0 0 -16 D -12 -8 0 0 -4 E 8 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 12 -8 B 4 0 -6 8 0 C -2 6 0 0 -16 D -12 -8 0 0 -4 E 8 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 12 -8 B 4 0 -6 8 0 C -2 6 0 0 -16 D -12 -8 0 0 -4 E 8 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2232: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) C E D A B (6) C D E A B (6) B E A C D (6) B A E D C (6) E A C D B (5) C D B E A (5) A E D B C (5) E A C B D (4) B D C A E (4) E A D C B (3) E A B C D (3) D C A E B (3) C E A D B (3) B C E A D (3) B A D E C (3) A E D C B (3) E C A D B (2) D C E A B (2) D B C A E (2) D A E C B (2) C D B A E (2) B D A C E (2) A B E D C (2) D A C E B (1) C E B A D (1) C B E A D (1) C B D E A (1) B E C A D (1) B C D E A (1) B C D A E (1) B A E C D (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 8 -6 B -4 0 -18 -18 0 C 4 18 0 4 4 D -8 18 -4 0 -10 E 6 0 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 8 -6 B -4 0 -18 -18 0 C 4 18 0 4 4 D -8 18 -4 0 -10 E 6 0 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 D=18 E=17 A=12 so A is eliminated. Round 2 votes counts: B=31 E=25 C=25 D=19 so D is eliminated. Round 3 votes counts: C=39 B=33 E=28 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:206 A:201 D:198 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 8 -6 B -4 0 -18 -18 0 C 4 18 0 4 4 D -8 18 -4 0 -10 E 6 0 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 8 -6 B -4 0 -18 -18 0 C 4 18 0 4 4 D -8 18 -4 0 -10 E 6 0 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 8 -6 B -4 0 -18 -18 0 C 4 18 0 4 4 D -8 18 -4 0 -10 E 6 0 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2233: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (11) B C A E D (10) E D C B A (6) E B D C A (6) A C D B E (6) D E C A B (5) D E A C B (4) B A C D E (4) A D C B E (4) D C A E B (3) D A E C B (3) C B A D E (3) B C E A D (3) B A C E D (3) E D A B C (2) E B D A C (2) E B C D A (2) D A C E B (2) C A B D E (2) B E C D A (2) E D B A C (1) E D A C B (1) B E A C D (1) B C A D E (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -8 0 8 B 10 0 -2 2 2 C 8 2 0 -2 8 D 0 -2 2 0 0 E -8 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 0 8 B 10 0 -2 2 2 C 8 2 0 -2 8 D 0 -2 2 0 0 E -8 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=24 A=23 D=17 C=5 so C is eliminated. Round 2 votes counts: E=31 B=27 A=25 D=17 so D is eliminated. Round 3 votes counts: E=40 A=33 B=27 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:208 B:206 D:200 A:195 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 0 8 B 10 0 -2 2 2 C 8 2 0 -2 8 D 0 -2 2 0 0 E -8 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 0 8 B 10 0 -2 2 2 C 8 2 0 -2 8 D 0 -2 2 0 0 E -8 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 0 8 B 10 0 -2 2 2 C 8 2 0 -2 8 D 0 -2 2 0 0 E -8 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2234: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) C D E B A (7) A B C E D (6) E D B A C (5) E B D A C (5) D E C B A (4) D C E B A (4) B A E D C (4) A C B E D (4) A B E C D (4) E D A B C (3) D E B C A (3) D C E A B (3) C A D B E (3) B E D A C (3) B E A D C (3) B A E C D (3) A E B D C (3) A B E D C (3) E A B D C (2) C D E A B (2) C D A E B (2) C D A B E (2) C A D E B (2) B A C E D (2) A C B D E (2) E D B C A (1) E B A D C (1) D E B A C (1) D E A C B (1) C D B E A (1) B C A E D (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 4 14 6 2 B -4 0 8 8 -2 C -14 -8 0 -2 -2 D -6 -8 2 0 -10 E -2 2 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 6 2 B -4 0 8 8 -2 C -14 -8 0 -2 -2 D -6 -8 2 0 -10 E -2 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 E=17 D=16 B=16 so D is eliminated. Round 2 votes counts: C=34 E=26 A=24 B=16 so B is eliminated. Round 3 votes counts: C=35 A=33 E=32 so E is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:206 B:205 D:189 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 6 2 B -4 0 8 8 -2 C -14 -8 0 -2 -2 D -6 -8 2 0 -10 E -2 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 6 2 B -4 0 8 8 -2 C -14 -8 0 -2 -2 D -6 -8 2 0 -10 E -2 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 6 2 B -4 0 8 8 -2 C -14 -8 0 -2 -2 D -6 -8 2 0 -10 E -2 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2235: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) C D B A E (6) A E D B C (5) A D E C B (5) A D E B C (5) A D B C E (5) E C B D A (4) E B C D A (4) E B C A D (4) E A D C B (4) C B E D A (4) A D C B E (4) E B A C D (3) D C A B E (3) D A C B E (3) C B D E A (3) B E C A D (3) E A B D C (2) D A E C B (2) C E B D A (2) C D E B A (2) C D B E A (2) B C E A D (2) A E B D C (2) A D C E B (2) E D A C B (1) E A D B C (1) D E A C B (1) D C E A B (1) D C B A E (1) D C A E B (1) C E D B A (1) B C E D A (1) B C A E D (1) A D B E C (1) Total count = 100 A B C D E A 0 14 10 -2 10 B -14 0 -22 -30 -22 C -10 22 0 -16 6 D 2 30 16 0 12 E -10 22 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 -2 10 B -14 0 -22 -30 -22 C -10 22 0 -16 6 D 2 30 16 0 12 E -10 22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999935264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=23 D=21 C=20 B=7 so B is eliminated. Round 2 votes counts: A=29 E=26 C=24 D=21 so D is eliminated. Round 3 votes counts: A=43 C=30 E=27 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:230 A:216 C:201 E:197 B:156 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 10 -2 10 B -14 0 -22 -30 -22 C -10 22 0 -16 6 D 2 30 16 0 12 E -10 22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999935264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -2 10 B -14 0 -22 -30 -22 C -10 22 0 -16 6 D 2 30 16 0 12 E -10 22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999935264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -2 10 B -14 0 -22 -30 -22 C -10 22 0 -16 6 D 2 30 16 0 12 E -10 22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999935264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2236: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C A E (7) E A C D B (6) D C A E B (6) E B A C D (5) D C A B E (5) D C B A E (4) D B E C A (4) B D C A E (4) B C D A E (4) B A E C D (4) E D B A C (3) C A D E B (3) B E A C D (3) A C E B D (3) E D A C B (2) E B D A C (2) E A B C D (2) D E C A B (2) D C E A B (2) D B C E A (2) C A B D E (2) B E D A C (2) B D E C A (2) B D E A C (2) E B A D C (1) D E B C A (1) D C E B A (1) C D A E B (1) C B A D E (1) B E A D C (1) B C A E D (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -4 -12 -6 B 8 0 0 0 -4 C 4 0 0 -6 -6 D 12 0 6 0 6 E 6 4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.308696 C: 0.000000 D: 0.691304 E: 0.000000 Sum of squares = 0.573194097014 Cumulative probabilities = A: 0.000000 B: 0.308696 C: 0.308696 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -12 -6 B 8 0 0 0 -4 C 4 0 0 -6 -6 D 12 0 6 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=31 B=23 C=7 A=5 so A is eliminated. Round 2 votes counts: D=34 E=32 B=23 C=11 so C is eliminated. Round 3 votes counts: D=38 E=36 B=26 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:205 B:202 C:196 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 -12 -6 B 8 0 0 0 -4 C 4 0 0 -6 -6 D 12 0 6 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -12 -6 B 8 0 0 0 -4 C 4 0 0 -6 -6 D 12 0 6 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -12 -6 B 8 0 0 0 -4 C 4 0 0 -6 -6 D 12 0 6 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2237: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) A E C D B (7) E A D C B (6) E A C D B (5) D E B A C (4) D B E C A (4) D B C E A (4) C A B E D (4) E D A C B (3) D B E A C (3) B D C A E (3) B C D E A (3) A C E B D (3) A C B E D (3) E D B C A (2) E D A B C (2) E C A D B (2) D E B C A (2) D B A E C (2) C E A B D (2) C B E D A (2) C B A E D (2) C A E B D (2) B C D A E (2) A E D B C (2) A E C B D (2) E D C A B (1) E D B A C (1) E C D B A (1) E C D A B (1) E A D B C (1) E A C B D (1) D E A B C (1) D A E B C (1) D A B E C (1) C E B D A (1) C B E A D (1) C B A D E (1) B D A C E (1) A E D C B (1) A D E B C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 6 -4 -20 B -8 0 -4 -16 -10 C -6 4 0 -8 -14 D 4 16 8 0 -16 E 20 10 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 -4 -20 B -8 0 -4 -16 -10 C -6 4 0 -8 -14 D 4 16 8 0 -16 E 20 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=22 A=21 B=16 C=15 so C is eliminated. Round 2 votes counts: E=29 A=27 D=22 B=22 so D is eliminated. Round 3 votes counts: E=36 B=35 A=29 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 D:206 A:195 C:188 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 -4 -20 B -8 0 -4 -16 -10 C -6 4 0 -8 -14 D 4 16 8 0 -16 E 20 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -4 -20 B -8 0 -4 -16 -10 C -6 4 0 -8 -14 D 4 16 8 0 -16 E 20 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -4 -20 B -8 0 -4 -16 -10 C -6 4 0 -8 -14 D 4 16 8 0 -16 E 20 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2238: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) A B E D C (6) C E B D A (5) D C E A B (4) D A C E B (4) C E D B A (4) B E A C D (4) B A E D C (4) A B D E C (4) A B C D E (4) C E B A D (3) C D E B A (3) C D E A B (3) C D A E B (3) B E A D C (3) E D C B A (2) E C D B A (2) E C B D A (2) E B C D A (2) C B E A D (2) A D B E C (2) A D B C E (2) A B E C D (2) A B C E D (2) D E B C A (1) D E B A C (1) D C E B A (1) D A E B C (1) C E A B D (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B E D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B A E C D (1) B A C E D (1) A D C B E (1) A C D B E (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -6 2 6 B -8 0 -10 12 -2 C 6 10 0 6 18 D -2 -12 -6 0 -8 E -6 2 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 2 6 B -8 0 -10 12 -2 C 6 10 0 6 18 D -2 -12 -6 0 -8 E -6 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 D=20 B=17 E=8 so E is eliminated. Round 2 votes counts: C=32 A=27 D=22 B=19 so B is eliminated. Round 3 votes counts: A=40 C=36 D=24 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:205 B:196 E:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 2 6 B -8 0 -10 12 -2 C 6 10 0 6 18 D -2 -12 -6 0 -8 E -6 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 2 6 B -8 0 -10 12 -2 C 6 10 0 6 18 D -2 -12 -6 0 -8 E -6 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 2 6 B -8 0 -10 12 -2 C 6 10 0 6 18 D -2 -12 -6 0 -8 E -6 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2239: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) D C B A E (8) D C B E A (7) B C D A E (7) E A D C B (6) A E D B C (5) A E D C B (4) A E B C D (4) A D E C B (4) A B D C E (4) E B A C D (3) E A C B D (3) D C E B A (3) D B C A E (3) E D A C B (2) B C E D A (2) B C E A D (2) B C D E A (2) A E B D C (2) A D E B C (2) A B E D C (2) E D C A B (1) E C D A B (1) E C B A D (1) E B C A D (1) E A D B C (1) E A C D B (1) D E A C B (1) D C E A B (1) C B E D A (1) C B D E A (1) C B D A E (1) B D C A E (1) B C A D E (1) B A E C D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 12 16 2 B -8 0 8 0 -10 C -12 -8 0 -14 -8 D -16 0 14 0 -4 E -2 10 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 16 2 B -8 0 8 0 -10 C -12 -8 0 -14 -8 D -16 0 14 0 -4 E -2 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=28 D=23 B=17 C=3 so C is eliminated. Round 2 votes counts: E=29 A=28 D=23 B=20 so B is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:219 E:210 D:197 B:195 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 16 2 B -8 0 8 0 -10 C -12 -8 0 -14 -8 D -16 0 14 0 -4 E -2 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 16 2 B -8 0 8 0 -10 C -12 -8 0 -14 -8 D -16 0 14 0 -4 E -2 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 16 2 B -8 0 8 0 -10 C -12 -8 0 -14 -8 D -16 0 14 0 -4 E -2 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2240: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) C B D E A (9) E A B C D (7) D C B A E (7) A E D B C (5) A E B D C (5) E A B D C (4) D B C A E (4) C D B A E (4) B D C E A (4) A D E B C (4) E A C B D (3) B E A D C (3) A E D C B (3) A D E C B (3) E C A B D (2) E B A C D (2) D C A B E (2) B E C A D (2) B C E D A (2) A E C D B (2) E C B A D (1) E B A D C (1) D A C E B (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A B D (1) C D A E B (1) C B E A D (1) B E D C A (1) B D C A E (1) B C D E A (1) B A D E C (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -6 4 -8 B 8 0 -4 4 6 C 6 4 0 -4 0 D -4 -4 4 0 8 E 8 -6 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 4 -8 B 8 0 -4 4 6 C 6 4 0 -4 0 D -4 -4 4 0 8 E 8 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 E=20 D=16 B=15 so B is eliminated. Round 2 votes counts: C=29 E=26 A=24 D=21 so D is eliminated. Round 3 votes counts: C=47 A=27 E=26 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:207 C:203 D:202 E:197 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 4 -8 B 8 0 -4 4 6 C 6 4 0 -4 0 D -4 -4 4 0 8 E 8 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 4 -8 B 8 0 -4 4 6 C 6 4 0 -4 0 D -4 -4 4 0 8 E 8 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 4 -8 B 8 0 -4 4 6 C 6 4 0 -4 0 D -4 -4 4 0 8 E 8 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2241: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) D A C B E (8) E C A D B (7) C E D A B (6) C E A D B (6) B E C D A (6) B D A C E (6) B D A E C (5) A D C E B (5) D A B C E (4) B E A D C (4) E B C A D (3) A D B C E (3) E B C D A (2) D A C E B (2) C D A E B (2) C A D E B (2) B D C A E (2) B C E D A (2) A D C B E (2) A D B E C (2) E C B D A (1) E C B A D (1) E A D C B (1) D C A B E (1) D B A C E (1) C A E D B (1) B E D C A (1) B E D A C (1) B E C A D (1) B C D E A (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 6 8 -4 14 B -6 0 8 -8 20 C -8 -8 0 -16 10 D 4 8 16 0 12 E -14 -20 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -4 14 B -6 0 8 -8 20 C -8 -8 0 -16 10 D 4 8 16 0 12 E -14 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=17 D=16 E=15 A=13 so A is eliminated. Round 2 votes counts: B=39 D=28 C=17 E=16 so E is eliminated. Round 3 votes counts: B=44 D=30 C=26 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:212 B:207 C:189 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -4 14 B -6 0 8 -8 20 C -8 -8 0 -16 10 D 4 8 16 0 12 E -14 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -4 14 B -6 0 8 -8 20 C -8 -8 0 -16 10 D 4 8 16 0 12 E -14 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -4 14 B -6 0 8 -8 20 C -8 -8 0 -16 10 D 4 8 16 0 12 E -14 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2242: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) B E C D A (7) C E A D B (6) B E D A C (6) B E A D C (6) D C A B E (4) D B A C E (4) A C D E B (4) E C A B D (3) D A C E B (3) D A C B E (3) B D E C A (3) E C B A D (2) E B A C D (2) E A C B D (2) D B C A E (2) D A B C E (2) C A D E B (2) B E D C A (2) B D C A E (2) A E C D B (2) A E B D C (2) A D C E B (2) A B D E C (2) E B C A D (1) E B A D C (1) D C B E A (1) D C B A E (1) D C A E B (1) D A B E C (1) C E D B A (1) C E A B D (1) C D B E A (1) C D A E B (1) C D A B E (1) C A E D B (1) B D C E A (1) B D A E C (1) B D A C E (1) B C E D A (1) A D E C B (1) A D E B C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 10 0 -8 B -8 0 12 6 6 C -10 -12 0 -8 -4 D 0 -6 8 0 -8 E 8 -6 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.338842975207 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 A B C D E A 0 8 10 0 -8 B -8 0 12 6 6 C -10 -12 0 -8 -4 D 0 -6 8 0 -8 E 8 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.338842975208 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=22 E=18 A=16 C=14 so C is eliminated. Round 2 votes counts: B=30 E=26 D=25 A=19 so A is eliminated. Round 3 votes counts: D=36 E=32 B=32 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:207 A:205 D:197 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 0 -8 B -8 0 12 6 6 C -10 -12 0 -8 -4 D 0 -6 8 0 -8 E 8 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.338842975208 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 -8 B -8 0 12 6 6 C -10 -12 0 -8 -4 D 0 -6 8 0 -8 E 8 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.338842975208 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 -8 B -8 0 12 6 6 C -10 -12 0 -8 -4 D 0 -6 8 0 -8 E 8 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.338842975208 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2243: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (31) E C D B A (25) E C A B D (10) D B A C E (7) E C D A B (4) E D C B A (3) E A C B D (3) C A E B D (3) A B C D E (3) C E A B D (2) E D B C A (1) E D B A C (1) E A B C D (1) D B C A E (1) C D B A E (1) B A D C E (1) A E C B D (1) A E B C D (1) A C B D E (1) Total count = 100 A B C D E A 0 20 0 14 0 B -20 0 -6 14 -10 C 0 6 0 10 0 D -14 -14 -10 0 -10 E 0 10 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.365941 B: 0.000000 C: 0.339167 D: 0.000000 E: 0.294892 Sum of squares = 0.335908349374 Cumulative probabilities = A: 0.365941 B: 0.365941 C: 0.705108 D: 0.705108 E: 1.000000 A B C D E A 0 20 0 14 0 B -20 0 -6 14 -10 C 0 6 0 10 0 D -14 -14 -10 0 -10 E 0 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=48 A=37 D=8 C=6 B=1 so B is eliminated. Round 2 votes counts: E=48 A=38 D=8 C=6 so C is eliminated. Round 3 votes counts: E=50 A=41 D=9 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:210 C:208 B:189 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 20 0 14 0 B -20 0 -6 14 -10 C 0 6 0 10 0 D -14 -14 -10 0 -10 E 0 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 14 0 B -20 0 -6 14 -10 C 0 6 0 10 0 D -14 -14 -10 0 -10 E 0 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 14 0 B -20 0 -6 14 -10 C 0 6 0 10 0 D -14 -14 -10 0 -10 E 0 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2244: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (8) E C D B A (6) B D E C A (6) E C A D B (5) A C D B E (5) A B D E C (5) C D E B A (4) E A C D B (3) C E A D B (3) B D A C E (3) B A D E C (3) E D C B A (2) E A C B D (2) D C B A E (2) D B C E A (2) C E D B A (2) C E D A B (2) C D A B E (2) C A D E B (2) B D A E C (2) B A E D C (2) A E C D B (2) A C E D B (2) A B E D C (2) E D B C A (1) E C D A B (1) E B D C A (1) E B C D A (1) E B C A D (1) E A B D C (1) E A B C D (1) D C B E A (1) D C A B E (1) D B C A E (1) D A B C E (1) C D E A B (1) C D B E A (1) C A E D B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E A C (1) B D C A E (1) B A D C E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 6 -4 6 -2 B -6 0 -4 -8 8 C 4 4 0 0 -6 D -6 8 0 0 8 E 2 -8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.375000 Sum of squares = 0.406249999876 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.625000 E: 1.000000 A B C D E A 0 6 -4 6 -2 B -6 0 -4 -8 8 C 4 4 0 0 -6 D -6 8 0 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.375000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 B=22 C=19 D=8 so D is eliminated. Round 2 votes counts: A=27 E=25 B=25 C=23 so C is eliminated. Round 3 votes counts: E=37 A=34 B=29 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:205 A:203 C:201 E:196 B:195 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -4 6 -2 B -6 0 -4 -8 8 C 4 4 0 0 -6 D -6 8 0 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.375000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 6 -2 B -6 0 -4 -8 8 C 4 4 0 0 -6 D -6 8 0 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.375000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.625000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 6 -2 B -6 0 -4 -8 8 C 4 4 0 0 -6 D -6 8 0 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.375000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2245: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) B D C E A (8) A E B C D (7) B A E D C (6) B A E C D (6) A E C D B (6) A E C B D (6) A B E C D (5) E D A C B (3) E A D C B (3) D C B E A (3) B D E A C (3) A E B D C (3) D E C A B (2) D C E B A (2) C D B E A (2) C D A E B (2) C A E D B (2) B D E C A (2) B D C A E (2) B C D A E (2) B A D E C (2) B A C E D (2) E D C A B (1) E D B A C (1) E C A D B (1) E A C D B (1) E A B D C (1) C D B A E (1) C B D A E (1) C A D B E (1) B E D A C (1) B E A D C (1) B A D C E (1) B A C D E (1) A E D C B (1) Total count = 100 A B C D E A 0 6 20 12 14 B -6 0 8 20 0 C -20 -8 0 8 -24 D -12 -20 -8 0 -14 E -14 0 24 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 20 12 14 B -6 0 8 20 0 C -20 -8 0 8 -24 D -12 -20 -8 0 -14 E -14 0 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=28 C=17 E=11 D=7 so D is eliminated. Round 2 votes counts: B=37 A=28 C=22 E=13 so E is eliminated. Round 3 votes counts: B=38 A=36 C=26 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:212 B:211 C:178 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 20 12 14 B -6 0 8 20 0 C -20 -8 0 8 -24 D -12 -20 -8 0 -14 E -14 0 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 20 12 14 B -6 0 8 20 0 C -20 -8 0 8 -24 D -12 -20 -8 0 -14 E -14 0 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 20 12 14 B -6 0 8 20 0 C -20 -8 0 8 -24 D -12 -20 -8 0 -14 E -14 0 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2246: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) E A B D C (6) D E A C B (6) B C A E D (5) B A E C D (5) E A D B C (4) D E B A C (4) D E A B C (4) B A E D C (4) C D B E A (3) C D B A E (3) C B D A E (3) B D E A C (3) B C D A E (3) B C A D E (3) B A C E D (3) A E C B D (3) E D A B C (2) D E C A B (2) D C B E A (2) C D E A B (2) C B A D E (2) C A E D B (2) C A B E D (2) B E A D C (2) B D C A E (2) E D A C B (1) E A D C B (1) E A C D B (1) D B E C A (1) D B C E A (1) C E D A B (1) C D A E B (1) C B A E D (1) B D C E A (1) B D A E C (1) A E B D C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 6 -6 -8 B 4 0 14 4 2 C -6 -14 0 -10 -6 D 6 -4 10 0 8 E 8 -2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -6 -8 B 4 0 14 4 2 C -6 -14 0 -10 -6 D 6 -4 10 0 8 E 8 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=27 C=20 E=15 A=6 so A is eliminated. Round 2 votes counts: B=33 D=27 E=20 C=20 so E is eliminated. Round 3 votes counts: B=41 D=35 C=24 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:210 E:202 A:194 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 -6 -8 B 4 0 14 4 2 C -6 -14 0 -10 -6 D 6 -4 10 0 8 E 8 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -6 -8 B 4 0 14 4 2 C -6 -14 0 -10 -6 D 6 -4 10 0 8 E 8 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -6 -8 B 4 0 14 4 2 C -6 -14 0 -10 -6 D 6 -4 10 0 8 E 8 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2247: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (7) E D B C A (6) E D A B C (6) B C A D E (6) A C B D E (6) A E D B C (5) D C B E A (4) C B D A E (4) C B A D E (4) B C E D A (4) B C A E D (4) E D A C B (3) E B D C A (3) E B C D A (3) A D E C B (3) D E C B A (2) D E A C B (2) D A E C B (2) B E C D A (2) B C E A D (2) B C D E A (2) A E B C D (2) A D C E B (2) A B E C D (2) A B C E D (2) E D C A B (1) E D B A C (1) E B A C D (1) E A D C B (1) E A B D C (1) D E C A B (1) D C E A B (1) D C A B E (1) B E C A D (1) B A E C D (1) B A C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -16 -10 -8 B 16 0 12 18 8 C 16 -12 0 10 2 D 10 -18 -10 0 -4 E 8 -8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -10 -8 B 16 0 12 18 8 C 16 -12 0 10 2 D 10 -18 -10 0 -4 E 8 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=23 A=23 C=15 D=13 so D is eliminated. Round 2 votes counts: E=31 A=25 B=23 C=21 so C is eliminated. Round 3 votes counts: B=42 E=32 A=26 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:208 E:201 D:189 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -16 -10 -8 B 16 0 12 18 8 C 16 -12 0 10 2 D 10 -18 -10 0 -4 E 8 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -10 -8 B 16 0 12 18 8 C 16 -12 0 10 2 D 10 -18 -10 0 -4 E 8 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -10 -8 B 16 0 12 18 8 C 16 -12 0 10 2 D 10 -18 -10 0 -4 E 8 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2248: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) E B D A C (7) C E A B D (6) C A B D E (6) E C D B A (5) E C D A B (5) C A D E B (5) C A D B E (5) E D B C A (4) E B D C A (4) D B A E C (4) C E A D B (4) D B E A C (3) C A E D B (3) A B D C E (3) E C B D A (2) E C A D B (2) E C A B D (2) D A B C E (2) C E D A B (2) B D E A C (2) B D A E C (2) A D C B E (2) A C D B E (2) E C B A D (1) E B C A D (1) D B A C E (1) C A E B D (1) B E D A C (1) B E A D C (1) B A D E C (1) B A D C E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -16 -4 -20 B -4 0 -8 -16 -24 C 16 8 0 6 -10 D 4 16 -6 0 -18 E 20 24 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -16 -4 -20 B -4 0 -8 -16 -24 C 16 8 0 6 -10 D 4 16 -6 0 -18 E 20 24 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=32 D=10 A=9 B=8 so B is eliminated. Round 2 votes counts: E=43 C=32 D=14 A=11 so A is eliminated. Round 3 votes counts: E=43 C=35 D=22 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:236 C:210 D:198 A:182 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -16 -4 -20 B -4 0 -8 -16 -24 C 16 8 0 6 -10 D 4 16 -6 0 -18 E 20 24 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -4 -20 B -4 0 -8 -16 -24 C 16 8 0 6 -10 D 4 16 -6 0 -18 E 20 24 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -4 -20 B -4 0 -8 -16 -24 C 16 8 0 6 -10 D 4 16 -6 0 -18 E 20 24 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2249: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E D C B A (6) E B A C D (6) C D E A B (6) A B C D E (6) E C D B A (5) D C A B E (5) B A E D C (5) D E C B A (4) B A E C D (4) A B E C D (4) D C E B A (3) D C A E B (3) A B C E D (3) D B A E C (2) D A B C E (2) C E D B A (2) B A D E C (2) A D B C E (2) A C B D E (2) A B D E C (2) A B D C E (2) E C B A D (1) E B D A C (1) D B E A C (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E B A (1) C D A E B (1) C D A B E (1) C A D B E (1) C A B E D (1) B E A D C (1) B E A C D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 0 -10 4 B -8 0 -8 -16 2 C 0 8 0 -6 8 D 10 16 6 0 16 E -4 -2 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -10 4 B -8 0 -8 -16 2 C 0 8 0 -6 8 D 10 16 6 0 16 E -4 -2 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=23 E=19 C=14 B=13 so B is eliminated. Round 2 votes counts: A=34 D=31 E=21 C=14 so C is eliminated. Round 3 votes counts: D=40 A=36 E=24 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:205 A:201 B:185 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -10 4 B -8 0 -8 -16 2 C 0 8 0 -6 8 D 10 16 6 0 16 E -4 -2 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -10 4 B -8 0 -8 -16 2 C 0 8 0 -6 8 D 10 16 6 0 16 E -4 -2 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -10 4 B -8 0 -8 -16 2 C 0 8 0 -6 8 D 10 16 6 0 16 E -4 -2 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2250: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (6) C D B E A (5) A E B D C (5) E A B C D (4) D B C A E (4) B D C E A (4) A E C B D (4) E C A D B (3) E A C B D (3) E A B D C (3) C E D B A (3) C D E B A (3) C D B A E (3) B D A C E (3) A E C D B (3) A C E D B (3) A B E D C (3) E C B D A (2) E B D C A (2) E B A D C (2) D C B E A (2) D C B A E (2) D B C E A (2) C A E D B (2) B E D C A (2) A C D B E (2) E C D A B (1) E C B A D (1) E C A B D (1) E B C D A (1) E B A C D (1) E A C D B (1) D B A C E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E A B (1) C D A B E (1) C A D B E (1) B D E C A (1) B D C A E (1) B D A E C (1) B A D E C (1) A E B C D (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 -2 6 -2 B -4 0 -2 6 -6 C 2 2 0 6 -4 D -6 -6 -6 0 -8 E 2 6 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -2 6 -2 B -4 0 -2 6 -6 C 2 2 0 6 -4 D -6 -6 -6 0 -8 E 2 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=25 C=22 B=13 D=11 so D is eliminated. Round 2 votes counts: A=29 C=26 E=25 B=20 so B is eliminated. Round 3 votes counts: C=37 A=35 E=28 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:210 A:203 C:203 B:197 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 6 -2 B -4 0 -2 6 -6 C 2 2 0 6 -4 D -6 -6 -6 0 -8 E 2 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 -2 B -4 0 -2 6 -6 C 2 2 0 6 -4 D -6 -6 -6 0 -8 E 2 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 -2 B -4 0 -2 6 -6 C 2 2 0 6 -4 D -6 -6 -6 0 -8 E 2 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2251: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) E B A D C (8) A C D E B (8) C D A B E (7) A E C D B (7) C D B A E (6) B E D C A (6) B E A D C (5) A E C B D (5) E B A C D (4) E A B C D (4) D C B A E (4) B E D A C (4) D C B E A (3) C D B E A (3) C D A E B (3) B D E C A (3) D C A B E (2) A E B D C (2) E A C B D (1) E A B D C (1) D B C E A (1) C A D E B (1) A E B C D (1) Total count = 100 A B C D E A 0 -16 0 -6 -8 B 16 0 0 10 10 C 0 0 0 0 -2 D 6 -10 0 0 4 E 8 -10 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.441960 C: 0.558040 D: 0.000000 E: 0.000000 Sum of squares = 0.506737279409 Cumulative probabilities = A: 0.000000 B: 0.441960 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -6 -8 B 16 0 0 10 10 C 0 0 0 0 -2 D 6 -10 0 0 4 E 8 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 C=20 E=18 D=10 so D is eliminated. Round 2 votes counts: B=30 C=29 A=23 E=18 so E is eliminated. Round 3 votes counts: B=42 C=29 A=29 so C is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:200 C:199 E:198 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -6 -8 B 16 0 0 10 10 C 0 0 0 0 -2 D 6 -10 0 0 4 E 8 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -6 -8 B 16 0 0 10 10 C 0 0 0 0 -2 D 6 -10 0 0 4 E 8 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -6 -8 B 16 0 0 10 10 C 0 0 0 0 -2 D 6 -10 0 0 4 E 8 -10 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2252: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (17) C A E B D (11) C E B A D (9) C E B D A (8) A D B E C (7) B E D C A (6) D A B E C (5) A C D E B (4) D B E C A (3) D B A E C (3) E B D A C (2) E B C D A (2) C E A B D (2) C B E D A (2) B D E C A (2) A E D B C (2) A D E B C (2) A D C B E (2) A C E B D (2) E A C B D (1) D A C B E (1) C D A B E (1) C A D E B (1) C A D B E (1) B E C D A (1) B D E A C (1) B C E D A (1) A E B C D (1) Total count = 100 A B C D E A 0 -14 0 -10 -14 B 14 0 10 2 6 C 0 -10 0 -6 -10 D 10 -2 6 0 0 E 14 -6 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -10 -14 B 14 0 10 2 6 C 0 -10 0 -6 -10 D 10 -2 6 0 0 E 14 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=29 A=20 B=11 E=5 so E is eliminated. Round 2 votes counts: C=35 D=29 A=21 B=15 so B is eliminated. Round 3 votes counts: D=40 C=39 A=21 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 E:209 D:207 C:187 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -10 -14 B 14 0 10 2 6 C 0 -10 0 -6 -10 D 10 -2 6 0 0 E 14 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -10 -14 B 14 0 10 2 6 C 0 -10 0 -6 -10 D 10 -2 6 0 0 E 14 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -10 -14 B 14 0 10 2 6 C 0 -10 0 -6 -10 D 10 -2 6 0 0 E 14 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2253: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) C B A E D (9) A B D E C (7) E D C A B (6) E C D A B (4) C A B E D (4) B C A D E (4) A B C E D (4) E D A B C (3) D E C B A (3) C E D B A (3) C E D A B (3) C E A B D (3) A C B E D (3) A B C D E (3) E D A C B (2) D B A E C (2) D A E B C (2) C D E B A (2) C B D E A (2) B A D E C (2) B A D C E (2) A D B E C (2) A B E D C (2) E C A D B (1) E A D C B (1) D E B C A (1) D E B A C (1) D E A C B (1) D E A B C (1) D C B E A (1) C B E A D (1) C B A D E (1) B D C A E (1) B D A E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 2 22 22 B -8 0 -2 22 24 C -2 2 0 18 14 D -22 -22 -18 0 -2 E -22 -24 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 22 22 B -8 0 -2 22 24 C -2 2 0 18 14 D -22 -22 -18 0 -2 E -22 -24 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999960057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 B=20 E=17 D=12 so D is eliminated. Round 2 votes counts: C=29 A=25 E=24 B=22 so B is eliminated. Round 3 votes counts: A=42 C=34 E=24 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:227 B:218 C:216 E:171 D:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 22 22 B -8 0 -2 22 24 C -2 2 0 18 14 D -22 -22 -18 0 -2 E -22 -24 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999960057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 22 22 B -8 0 -2 22 24 C -2 2 0 18 14 D -22 -22 -18 0 -2 E -22 -24 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999960057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 22 22 B -8 0 -2 22 24 C -2 2 0 18 14 D -22 -22 -18 0 -2 E -22 -24 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999960057 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2254: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) E D C A B (6) B A C D E (6) E A C B D (4) E A B D C (4) C D B A E (4) B A D C E (4) E D B A C (3) E D A B C (3) D E C B A (3) D E B C A (3) D C E B A (3) B A E D C (3) A E C B D (3) A C E B D (3) A B E C D (3) E C A D B (2) D B E C A (2) D B C E A (2) C A E B D (2) C A B E D (2) C A B D E (2) B D A C E (2) B C A D E (2) A C B E D (2) A B C E D (2) E D C B A (1) E D B C A (1) E C D A B (1) E A C D B (1) E A B C D (1) D E B A C (1) D C B E A (1) D C B A E (1) D B A C E (1) C E A B D (1) C D A E B (1) B D C A E (1) B D A E C (1) B C D A E (1) B A D E C (1) A E B C D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 2 14 B 8 0 12 6 4 C -2 -12 0 -10 2 D -2 -6 10 0 0 E -14 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 2 14 B 8 0 12 6 4 C -2 -12 0 -10 2 D -2 -6 10 0 0 E -14 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 B=21 A=16 C=12 so C is eliminated. Round 2 votes counts: D=29 E=28 A=22 B=21 so B is eliminated. Round 3 votes counts: A=38 D=34 E=28 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:205 D:201 E:190 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 2 14 B 8 0 12 6 4 C -2 -12 0 -10 2 D -2 -6 10 0 0 E -14 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 2 14 B 8 0 12 6 4 C -2 -12 0 -10 2 D -2 -6 10 0 0 E -14 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 2 14 B 8 0 12 6 4 C -2 -12 0 -10 2 D -2 -6 10 0 0 E -14 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2255: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) E B C A D (7) B C E D A (7) A D C E B (6) A D E B C (5) E C B A D (4) B C D E A (4) D C A B E (3) D A C E B (3) D A B C E (3) C E B D A (3) C D B E A (3) C B E D A (3) A D E C B (3) E C B D A (2) E C A B D (2) E A C B D (2) E A B C D (2) D C B A E (2) C D E B A (2) B E C D A (2) B D C A E (2) B A D E C (2) A E D C B (2) A E C D B (2) E C A D B (1) E B A C D (1) D C B E A (1) D B A C E (1) C D E A B (1) C B D E A (1) B E C A D (1) B E A C D (1) B D C E A (1) B D A C E (1) A E D B C (1) A E B D C (1) A D C B E (1) A D B E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -4 -6 -2 B 2 0 -10 0 0 C 4 10 0 2 14 D 6 0 -2 0 10 E 2 0 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -6 -2 B 2 0 -10 0 0 C 4 10 0 2 14 D 6 0 -2 0 10 E 2 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=21 D=21 B=21 C=13 so C is eliminated. Round 2 votes counts: D=27 B=25 E=24 A=24 so E is eliminated. Round 3 votes counts: B=42 A=31 D=27 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:215 D:207 B:196 A:193 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 -2 B 2 0 -10 0 0 C 4 10 0 2 14 D 6 0 -2 0 10 E 2 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 -2 B 2 0 -10 0 0 C 4 10 0 2 14 D 6 0 -2 0 10 E 2 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 -2 B 2 0 -10 0 0 C 4 10 0 2 14 D 6 0 -2 0 10 E 2 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2256: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) C D B A E (7) E B D A C (5) D B E A C (4) C E B D A (4) C A E D B (4) B D E C A (4) B D E A C (4) A E C B D (4) E A C B D (3) E A B C D (3) D B C E A (3) C D A B E (3) B E D C A (3) A D C B E (3) E A B D C (2) D C A B E (2) D B E C A (2) D B C A E (2) D B A E C (2) D B A C E (2) C E A B D (2) C D B E A (2) C A D E B (2) C A D B E (2) B E D A C (2) A C E B D (2) A C D E B (2) E C A B D (1) E B D C A (1) D C B A E (1) D A C B E (1) D A B C E (1) C B D E A (1) C A E B D (1) B E A D C (1) A E C D B (1) A E B D C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 6 -12 6 B 0 0 -12 -10 6 C -6 12 0 6 10 D 12 10 -6 0 2 E -6 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000026 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -12 6 B 0 0 -12 -10 6 C -6 12 0 6 10 D 12 10 -6 0 2 E -6 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 D=20 E=15 B=14 so B is eliminated. Round 2 votes counts: D=28 C=28 A=23 E=21 so E is eliminated. Round 3 votes counts: D=39 A=32 C=29 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:211 D:209 A:200 B:192 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 6 -12 6 B 0 0 -12 -10 6 C -6 12 0 6 10 D 12 10 -6 0 2 E -6 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -12 6 B 0 0 -12 -10 6 C -6 12 0 6 10 D 12 10 -6 0 2 E -6 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -12 6 B 0 0 -12 -10 6 C -6 12 0 6 10 D 12 10 -6 0 2 E -6 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2257: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C E A B D (8) E D C A B (7) D B A C E (7) E C D A B (6) D E B A C (6) B A C D E (6) E C A D B (5) D E B C A (5) C A B E D (5) A C B E D (5) B A D C E (4) E D C B A (3) E D B C A (3) C A E B D (3) B D A C E (3) A B C D E (3) E C A B D (2) D B E A C (2) A C B D E (2) E B C A D (1) D E C B A (1) D E C A B (1) C E A D B (1) C A D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 -6 -2 B 0 0 0 -14 -4 C 4 0 0 -2 -2 D 6 14 2 0 0 E 2 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.215010 E: 0.784990 Sum of squares = 0.662438940903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.215010 E: 1.000000 A B C D E A 0 0 -4 -6 -2 B 0 0 0 -14 -4 C 4 0 0 -2 -2 D 6 14 2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=27 C=18 B=13 A=11 so A is eliminated. Round 2 votes counts: D=31 E=27 C=25 B=17 so B is eliminated. Round 3 votes counts: D=38 C=35 E=27 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:204 C:200 A:194 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -6 -2 B 0 0 0 -14 -4 C 4 0 0 -2 -2 D 6 14 2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -6 -2 B 0 0 0 -14 -4 C 4 0 0 -2 -2 D 6 14 2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -6 -2 B 0 0 0 -14 -4 C 4 0 0 -2 -2 D 6 14 2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2258: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) C B D E A (8) E B C A D (7) A D E B C (7) C B E D A (6) B C E D A (6) A E D B C (6) A D E C B (6) B E C A D (5) B C E A D (5) E B A C D (3) E A B D C (3) D C B A E (3) D C A B E (3) D A C E B (3) C D B A E (3) D A E C B (2) B C D E A (2) E D A C B (1) E C B A D (1) E A D C B (1) E A B C D (1) D B C A E (1) D B A C E (1) D A B C E (1) C E D A B (1) C E B A D (1) C B D A E (1) B C D A E (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 -4 -2 B 8 0 0 2 12 C 8 0 0 2 10 D 4 -2 -2 0 4 E 2 -12 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.361536 C: 0.638464 D: 0.000000 E: 0.000000 Sum of squares = 0.538344411939 Cumulative probabilities = A: 0.000000 B: 0.361536 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -4 -2 B 8 0 0 2 12 C 8 0 0 2 10 D 4 -2 -2 0 4 E 2 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=21 C=20 B=19 E=17 so E is eliminated. Round 2 votes counts: B=29 A=26 D=24 C=21 so C is eliminated. Round 3 votes counts: B=46 D=28 A=26 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:210 D:202 A:189 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 -2 B 8 0 0 2 12 C 8 0 0 2 10 D 4 -2 -2 0 4 E 2 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 -2 B 8 0 0 2 12 C 8 0 0 2 10 D 4 -2 -2 0 4 E 2 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 -2 B 8 0 0 2 12 C 8 0 0 2 10 D 4 -2 -2 0 4 E 2 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2259: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (15) D C A B E (11) B E A C D (9) E B D C A (7) D C A E B (7) C D A B E (7) E B A D C (6) A B C D E (5) E D C B A (3) E B D A C (3) E B C D A (3) A D C B E (3) D A C B E (2) C A D B E (2) A C D B E (2) A B E C D (2) E D C A B (1) E C D B A (1) E B C A D (1) D E C B A (1) D E C A B (1) D C E B A (1) C D E A B (1) C A B D E (1) B A E D C (1) B A E C D (1) A D B C E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 4 2 -6 B 4 0 10 12 -2 C -4 -10 0 2 -12 D -2 -12 -2 0 -8 E 6 2 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 2 -6 B 4 0 10 12 -2 C -4 -10 0 2 -12 D -2 -12 -2 0 -8 E 6 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=23 A=15 C=11 B=11 so C is eliminated. Round 2 votes counts: E=40 D=31 A=18 B=11 so B is eliminated. Round 3 votes counts: E=49 D=31 A=20 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:212 A:198 C:188 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 2 -6 B 4 0 10 12 -2 C -4 -10 0 2 -12 D -2 -12 -2 0 -8 E 6 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 2 -6 B 4 0 10 12 -2 C -4 -10 0 2 -12 D -2 -12 -2 0 -8 E 6 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 2 -6 B 4 0 10 12 -2 C -4 -10 0 2 -12 D -2 -12 -2 0 -8 E 6 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2260: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) B A D C E (6) A B D E C (6) E C D B A (5) D B A E C (5) D A B E C (5) E D C A B (4) D E C B A (4) A B D C E (4) E D C B A (3) E D A C B (3) E C A D B (3) D A E B C (3) C E D B A (3) C E D A B (3) C B A E D (3) C A B E D (3) C E B D A (2) B D A C E (2) B A D E C (2) B A C D E (2) A E D B C (2) A D B E C (2) A B C D E (2) E A D C B (1) D E B A C (1) D E A B C (1) D C E B A (1) D B E C A (1) D B E A C (1) C E B A D (1) C D B E A (1) C B E A D (1) C B D E A (1) C B A D E (1) C A E B D (1) B D A E C (1) B C A D E (1) A E D C B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 -12 8 B -4 0 -2 -18 4 C -4 2 0 -20 -22 D 12 18 20 0 6 E -8 -4 22 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -12 8 B -4 0 -2 -18 4 C -4 2 0 -20 -22 D 12 18 20 0 6 E -8 -4 22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 C=20 A=19 B=14 so B is eliminated. Round 2 votes counts: A=29 E=25 D=25 C=21 so C is eliminated. Round 3 votes counts: A=38 E=35 D=27 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:228 A:202 E:202 B:190 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -12 8 B -4 0 -2 -18 4 C -4 2 0 -20 -22 D 12 18 20 0 6 E -8 -4 22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -12 8 B -4 0 -2 -18 4 C -4 2 0 -20 -22 D 12 18 20 0 6 E -8 -4 22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -12 8 B -4 0 -2 -18 4 C -4 2 0 -20 -22 D 12 18 20 0 6 E -8 -4 22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2261: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (13) D C A B E (10) D C E B A (9) C D A B E (7) A B C E D (6) C D B A E (5) E B A C D (4) C D B E A (4) D E C B A (3) D C E A B (3) B E A C D (3) E B D A C (2) E B C D A (2) E B A D C (2) D C A E B (2) C D E B A (2) B E C A D (2) B A E C D (2) A E B D C (2) E D B A C (1) E B D C A (1) E B C A D (1) E A D B C (1) E A B D C (1) E A B C D (1) D A E C B (1) C E B D A (1) C B E A D (1) C A D B E (1) C A B E D (1) C A B D E (1) A E D B C (1) A E B C D (1) A D C B E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -12 -6 12 B -10 0 -6 -6 18 C 12 6 0 18 12 D 6 6 -18 0 2 E -12 -18 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 -6 12 B -10 0 -6 -6 18 C 12 6 0 18 12 D 6 6 -18 0 2 E -12 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=26 C=23 E=16 B=7 so B is eliminated. Round 2 votes counts: D=28 A=28 C=23 E=21 so E is eliminated. Round 3 votes counts: A=40 D=32 C=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:224 A:202 B:198 D:198 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 -6 12 B -10 0 -6 -6 18 C 12 6 0 18 12 D 6 6 -18 0 2 E -12 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 -6 12 B -10 0 -6 -6 18 C 12 6 0 18 12 D 6 6 -18 0 2 E -12 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 -6 12 B -10 0 -6 -6 18 C 12 6 0 18 12 D 6 6 -18 0 2 E -12 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2262: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) B A C D E (7) D E C A B (5) D E A C B (5) D A B E C (5) C E B D A (4) E D A C B (3) D E C B A (3) C E D B A (3) C E B A D (3) C E A B D (3) B A D C E (3) A D B E C (3) A B D E C (3) A B D C E (3) A B C E D (3) E C A B D (2) D C E B A (2) D C B E A (2) D A E B C (2) C D B E A (2) C B E D A (2) C B A E D (2) B A C E D (2) A D E B C (2) A B E D C (2) A B E C D (2) E D C B A (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D C B (1) C D E B A (1) B D A C E (1) B C A E D (1) A E D C B (1) A E D B C (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 20 2 -6 -10 B -20 0 -18 -10 -12 C -2 18 0 -16 -10 D 6 10 16 0 0 E 10 12 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.381164 E: 0.618836 Sum of squares = 0.528244181142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.381164 E: 1.000000 A B C D E A 0 20 2 -6 -10 B -20 0 -18 -10 -12 C -2 18 0 -16 -10 D 6 10 16 0 0 E 10 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 E=20 C=20 B=14 so B is eliminated. Round 2 votes counts: A=34 D=25 C=21 E=20 so E is eliminated. Round 3 votes counts: D=39 A=35 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:216 A:203 C:195 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 20 2 -6 -10 B -20 0 -18 -10 -12 C -2 18 0 -16 -10 D 6 10 16 0 0 E 10 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 2 -6 -10 B -20 0 -18 -10 -12 C -2 18 0 -16 -10 D 6 10 16 0 0 E 10 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 2 -6 -10 B -20 0 -18 -10 -12 C -2 18 0 -16 -10 D 6 10 16 0 0 E 10 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2263: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) D B C A E (8) B A E C D (8) C A E D B (7) E A C B D (6) B D E A C (5) E A C D B (4) D C A E B (4) D B E C A (4) C D A E B (4) B D C A E (4) D B C E A (3) B E A D C (3) A E C B D (3) A C E B D (3) E C A D B (2) B E A C D (2) B D A C E (2) B A E D C (2) B A C E D (2) E D C A B (1) E A B C D (1) D E C A B (1) D E B C A (1) D C B E A (1) D C A B E (1) D B E A C (1) C E A D B (1) C A E B D (1) C A D E B (1) B E D A C (1) B D E C A (1) B D A E C (1) B C D A E (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 -8 -4 8 B 2 0 4 -4 2 C 8 -4 0 -4 4 D 4 4 4 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -4 8 B 2 0 4 -4 2 C 8 -4 0 -4 4 D 4 4 4 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=32 E=14 C=14 A=7 so A is eliminated. Round 2 votes counts: B=33 D=32 E=18 C=17 so C is eliminated. Round 3 votes counts: D=37 B=33 E=30 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:202 C:202 A:197 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 8 B 2 0 4 -4 2 C 8 -4 0 -4 4 D 4 4 4 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 8 B 2 0 4 -4 2 C 8 -4 0 -4 4 D 4 4 4 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 8 B 2 0 4 -4 2 C 8 -4 0 -4 4 D 4 4 4 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2264: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) C E A B D (6) B E C D A (6) D B A E C (5) E B C D A (4) D B C E A (4) A D B C E (4) D A C B E (3) D A B C E (3) C E D B A (3) C E B D A (3) B E A D C (3) A E B C D (3) A D C E B (3) A C E D B (3) A B D E C (3) E B C A D (2) D C A B E (2) D B E C A (2) D B A C E (2) C E B A D (2) C D B E A (2) B D E A C (2) A E B D C (2) A D C B E (2) A C D E B (2) A B E D C (2) E B A C D (1) D C B E A (1) D B E A C (1) D A B E C (1) C E A D B (1) C D E B A (1) C D A E B (1) C A D E B (1) B E D C A (1) B E C A D (1) B D E C A (1) A C E B D (1) Total count = 100 A B C D E A 0 6 12 4 6 B -6 0 26 -14 22 C -12 -26 0 -14 0 D -4 14 14 0 12 E -6 -22 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 4 6 B -6 0 26 -14 22 C -12 -26 0 -14 0 D -4 14 14 0 12 E -6 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=24 C=20 B=14 E=7 so E is eliminated. Round 2 votes counts: A=35 D=24 B=21 C=20 so C is eliminated. Round 3 votes counts: A=43 D=31 B=26 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:218 A:214 B:214 E:180 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 4 6 B -6 0 26 -14 22 C -12 -26 0 -14 0 D -4 14 14 0 12 E -6 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 4 6 B -6 0 26 -14 22 C -12 -26 0 -14 0 D -4 14 14 0 12 E -6 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 4 6 B -6 0 26 -14 22 C -12 -26 0 -14 0 D -4 14 14 0 12 E -6 -22 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2265: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D E A C B (8) C A E B D (6) B D A C E (6) E C A B D (4) D B A E C (4) A C E D B (4) E C B A D (3) E C A D B (3) D B E C A (3) D B A C E (3) D A E C B (3) B D C A E (3) A C E B D (3) E D A C B (2) E A C D B (2) D E C A B (2) D E B A C (2) D A C E B (2) C B A E D (2) B C A D E (2) B A C E D (2) B A C D E (2) A D C E B (2) A C D E B (2) E D C A B (1) E A D C B (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E A D (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E A D (1) B A D C E (1) A E D C B (1) A E C D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 10 12 24 B -4 0 -12 4 -10 C -10 12 0 4 12 D -12 -4 -4 0 -4 E -24 10 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 12 24 B -4 0 -12 4 -10 C -10 12 0 4 12 D -12 -4 -4 0 -4 E -24 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=29 B=29 E=16 A=15 C=11 so C is eliminated. Round 2 votes counts: B=32 D=29 A=21 E=18 so E is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:209 B:189 E:189 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 12 24 B -4 0 -12 4 -10 C -10 12 0 4 12 D -12 -4 -4 0 -4 E -24 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 12 24 B -4 0 -12 4 -10 C -10 12 0 4 12 D -12 -4 -4 0 -4 E -24 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 12 24 B -4 0 -12 4 -10 C -10 12 0 4 12 D -12 -4 -4 0 -4 E -24 10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2266: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) D E B A C (5) D A B E C (5) C A D E B (5) C E B A D (4) C A E D B (4) C A B E D (4) B E D C A (4) A D B C E (4) E B C D A (3) C D A E B (3) A C D B E (3) E D C B A (2) E D B C A (2) E C B D A (2) E B D C A (2) D E A C B (2) D E A B C (2) D A E C B (2) C E D B A (2) C E D A B (2) C E B D A (2) C E A B D (2) B D A E C (2) A C B D E (2) A B D E C (2) A B D C E (2) D E C A B (1) D E B C A (1) D C A E B (1) D B E A C (1) D B A E C (1) D A E B C (1) C E A D B (1) C B E A D (1) C A E B D (1) B E D A C (1) B E A D C (1) B A E C D (1) B A D E C (1) B A C E D (1) A D C E B (1) A D B E C (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 22 2 2 14 B -22 0 -10 -18 -6 C -2 10 0 -8 10 D -2 18 8 0 12 E -14 6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 2 2 14 B -22 0 -10 -18 -6 C -2 10 0 -8 10 D -2 18 8 0 12 E -14 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=25 D=22 E=11 B=11 so E is eliminated. Round 2 votes counts: C=33 D=26 A=25 B=16 so B is eliminated. Round 3 votes counts: C=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:220 D:218 C:205 E:185 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 2 2 14 B -22 0 -10 -18 -6 C -2 10 0 -8 10 D -2 18 8 0 12 E -14 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 2 2 14 B -22 0 -10 -18 -6 C -2 10 0 -8 10 D -2 18 8 0 12 E -14 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 2 2 14 B -22 0 -10 -18 -6 C -2 10 0 -8 10 D -2 18 8 0 12 E -14 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2267: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B D E A (10) E D A C B (6) A B C D E (6) A B C E D (5) E D C B A (4) E A D C B (4) D E A C B (4) C B E D A (4) C B D A E (4) B C A D E (4) D A B C E (3) A D E B C (3) E C B D A (2) E C B A D (2) D E A B C (2) D C B A E (2) D A E B C (2) C B A D E (2) A E D B C (2) E D A B C (1) E C D B A (1) E B A C D (1) E A C D B (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E B A (1) D A E C B (1) C D B E A (1) C B E A D (1) C B A E D (1) B C A E D (1) B A C E D (1) B A C D E (1) A E B D C (1) A E B C D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 16 0 -16 B -12 0 -6 -2 -4 C -16 6 0 0 -2 D 0 2 0 0 0 E 16 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.579053 E: 0.420947 Sum of squares = 0.512498663911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.579053 E: 1.000000 A B C D E A 0 12 16 0 -16 B -12 0 -6 -2 -4 C -16 6 0 0 -2 D 0 2 0 0 0 E 16 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=23 A=20 D=17 B=7 so B is eliminated. Round 2 votes counts: E=33 C=28 A=22 D=17 so D is eliminated. Round 3 votes counts: E=41 C=31 A=28 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:206 D:201 C:194 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 16 0 -16 B -12 0 -6 -2 -4 C -16 6 0 0 -2 D 0 2 0 0 0 E 16 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 0 -16 B -12 0 -6 -2 -4 C -16 6 0 0 -2 D 0 2 0 0 0 E 16 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 0 -16 B -12 0 -6 -2 -4 C -16 6 0 0 -2 D 0 2 0 0 0 E 16 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2268: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (14) C E B A D (11) E B C D A (10) A C D E B (10) B E C D A (8) A D C E B (6) A D C B E (6) D B E A C (5) C A E B D (5) E C B D A (3) B E D C A (3) B D E A C (3) D B A E C (2) C E B D A (2) C A D E B (2) A D B E C (2) A C B D E (2) E B D C A (1) D A E B C (1) C E A D B (1) C B E A D (1) C B A E D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 4 -4 4 B 0 0 -2 0 -4 C -4 2 0 14 -4 D 4 0 -14 0 8 E -4 4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.181818 E: 0.000000 Sum of squares = 0.47107437991 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -4 4 B 0 0 -2 0 -4 C -4 2 0 14 -4 D 4 0 -14 0 8 E -4 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.181818 E: 0.000000 Sum of squares = 0.471074380132 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 D=22 E=14 B=14 so E is eliminated. Round 2 votes counts: A=27 C=26 B=25 D=22 so D is eliminated. Round 3 votes counts: A=42 B=32 C=26 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:204 A:202 D:199 E:198 B:197 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 -4 4 B 0 0 -2 0 -4 C -4 2 0 14 -4 D 4 0 -14 0 8 E -4 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.181818 E: 0.000000 Sum of squares = 0.471074380132 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -4 4 B 0 0 -2 0 -4 C -4 2 0 14 -4 D 4 0 -14 0 8 E -4 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.181818 E: 0.000000 Sum of squares = 0.471074380132 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -4 4 B 0 0 -2 0 -4 C -4 2 0 14 -4 D 4 0 -14 0 8 E -4 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.181818 E: 0.000000 Sum of squares = 0.471074380132 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2269: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) C E A B D (10) B C A E D (10) D A E C B (6) D E A C B (5) B C E A D (5) A E C B D (5) D B C E A (4) D A E B C (4) B D C A E (4) E C A D B (3) D B A C E (3) D A B E C (3) C E A D B (3) B D C E A (3) B D A C E (3) E A C D B (2) E C A B D (1) D E C A B (1) D C E B A (1) D C E A B (1) D B E A C (1) C E D B A (1) C E D A B (1) C E B A D (1) C B E D A (1) C B A E D (1) B D A E C (1) B C E D A (1) B A C E D (1) A E D C B (1) A E C D B (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -4 -8 10 B 2 0 6 -4 2 C 4 -6 0 -4 10 D 8 4 4 0 2 E -10 -2 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -8 10 B 2 0 6 -4 2 C 4 -6 0 -4 10 D 8 4 4 0 2 E -10 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=28 C=18 A=9 E=6 so E is eliminated. Round 2 votes counts: D=39 B=28 C=22 A=11 so A is eliminated. Round 3 votes counts: D=41 C=31 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:203 C:202 A:198 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -8 10 B 2 0 6 -4 2 C 4 -6 0 -4 10 D 8 4 4 0 2 E -10 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -8 10 B 2 0 6 -4 2 C 4 -6 0 -4 10 D 8 4 4 0 2 E -10 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -8 10 B 2 0 6 -4 2 C 4 -6 0 -4 10 D 8 4 4 0 2 E -10 -2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2270: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) C A D E B (7) B E D A C (6) B D E C A (6) D C A E B (5) A C E D B (5) E B A D C (4) C A D B E (4) B E D C A (4) B E A C D (4) A C D E B (4) E A D C B (3) B D C A E (3) B C D A E (3) A E B C D (3) E D B A C (2) E A C B D (2) D E C A B (2) D C E A B (2) D B C A E (2) C A B D E (2) B A C E D (2) A C E B D (2) E D B C A (1) E D A C B (1) E B D A C (1) E B A C D (1) E A D B C (1) E A C D B (1) D E C B A (1) D E B C A (1) D E A C B (1) D B E C A (1) C D B A E (1) C D A B E (1) B D C E A (1) B C A E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -12 -6 8 B -10 0 -4 -8 0 C 12 4 0 -12 6 D 6 8 12 0 10 E -8 0 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 -6 8 B -10 0 -4 -8 0 C 12 4 0 -12 6 D 6 8 12 0 10 E -8 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 E=17 C=15 A=15 so C is eliminated. Round 2 votes counts: B=30 A=28 D=25 E=17 so E is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:218 C:205 A:200 B:189 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -12 -6 8 B -10 0 -4 -8 0 C 12 4 0 -12 6 D 6 8 12 0 10 E -8 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 -6 8 B -10 0 -4 -8 0 C 12 4 0 -12 6 D 6 8 12 0 10 E -8 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 -6 8 B -10 0 -4 -8 0 C 12 4 0 -12 6 D 6 8 12 0 10 E -8 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2271: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (14) E D A B C (8) C D E B A (7) C A B D E (7) A B C E D (7) D E C A B (6) D C E B A (6) A B E D C (6) E D C B A (5) E D B A C (5) B A C E D (5) A B C D E (5) E A B D C (3) C B A D E (3) C D B A E (2) B A E D C (2) B A C D E (2) E D C A B (1) E D B C A (1) E B A D C (1) D C E A B (1) C D E A B (1) C D A B E (1) B A E C D (1) Total count = 100 A B C D E A 0 -8 -10 -16 -18 B 8 0 -8 -16 -18 C 10 8 0 -18 -6 D 16 16 18 0 10 E 18 18 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -16 -18 B 8 0 -8 -16 -18 C 10 8 0 -18 -6 D 16 16 18 0 10 E 18 18 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 C=21 A=18 B=10 so B is eliminated. Round 2 votes counts: A=28 D=27 E=24 C=21 so C is eliminated. Round 3 votes counts: D=38 A=38 E=24 so E is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:230 E:216 C:197 B:183 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -16 -18 B 8 0 -8 -16 -18 C 10 8 0 -18 -6 D 16 16 18 0 10 E 18 18 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -16 -18 B 8 0 -8 -16 -18 C 10 8 0 -18 -6 D 16 16 18 0 10 E 18 18 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -16 -18 B 8 0 -8 -16 -18 C 10 8 0 -18 -6 D 16 16 18 0 10 E 18 18 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2272: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) D E B C A (6) C B D A E (6) D B E C A (5) A C E B D (5) A C B E D (5) E D C A B (4) E A D C B (4) E A D B C (4) C A B E D (4) B D E C A (4) A E C D B (4) C D B E A (3) C B A D E (3) B D E A C (3) B D C E A (3) A B C E D (3) E D B A C (2) D E B A C (2) C E D A B (2) C B D E A (2) B C D E A (2) B C D A E (2) B A C D E (2) A E D C B (2) E D C B A (1) E A C D B (1) D C B E A (1) C E A D B (1) C A E B D (1) B D C A E (1) B C A D E (1) A E D B C (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -4 -12 -14 B -2 0 -2 -2 4 C 4 2 0 0 0 D 12 2 0 0 -4 E 14 -4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.705973 D: 0.000000 E: 0.294027 Sum of squares = 0.584849457118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.705973 D: 0.705973 E: 1.000000 A B C D E A 0 2 -4 -12 -14 B -2 0 -2 -2 4 C 4 2 0 0 0 D 12 2 0 0 -4 E 14 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555955938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 A=23 C=22 B=18 D=14 so D is eliminated. Round 2 votes counts: E=31 C=23 B=23 A=23 so C is eliminated. Round 3 votes counts: B=38 E=34 A=28 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:207 D:205 C:203 B:199 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -12 -14 B -2 0 -2 -2 4 C 4 2 0 0 0 D 12 2 0 0 -4 E 14 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555955938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -12 -14 B -2 0 -2 -2 4 C 4 2 0 0 0 D 12 2 0 0 -4 E 14 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555955938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -12 -14 B -2 0 -2 -2 4 C 4 2 0 0 0 D 12 2 0 0 -4 E 14 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555955938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2273: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) E B C D A (5) D A C B E (5) A D C B E (5) E D B A C (4) D E A B C (4) C B E A D (4) C B A E D (4) C B A D E (4) B C E D A (4) A C E D B (4) A C D B E (4) E C B A D (3) C A D B E (3) C A B D E (3) B E C D A (3) E B D C A (2) E B C A D (2) D E B A C (2) D C A B E (2) C B D A E (2) B E C A D (2) B C E A D (2) A D C E B (2) E D A B C (1) E C A B D (1) E B D A C (1) D B C A E (1) D B A E C (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A B D (1) C A E D B (1) C A E B D (1) C A B E D (1) B E D C A (1) A E D C B (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -4 0 16 B -6 0 -10 -8 4 C 4 10 0 12 12 D 0 8 -12 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 0 16 B -6 0 -10 -8 4 C 4 10 0 12 12 D 0 8 -12 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 E=19 A=18 B=12 so B is eliminated. Round 2 votes counts: C=30 D=27 E=25 A=18 so A is eliminated. Round 3 votes counts: C=39 D=34 E=27 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:209 D:199 B:190 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 0 16 B -6 0 -10 -8 4 C 4 10 0 12 12 D 0 8 -12 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 16 B -6 0 -10 -8 4 C 4 10 0 12 12 D 0 8 -12 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 16 B -6 0 -10 -8 4 C 4 10 0 12 12 D 0 8 -12 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2274: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) E B A C D (7) E A B C D (7) B D C E A (7) D C B A E (6) B E A C D (6) D C B E A (5) A E C D B (5) B E C A D (4) C D B E A (3) C D A E B (3) B E D C A (3) A E B C D (3) A D C E B (3) A C E D B (3) D C A B E (2) C B D E A (2) B E C D A (2) B C D E A (2) A E C B D (2) A E B D C (2) E B C A D (1) D B C E A (1) D B C A E (1) D A C B E (1) D A B C E (1) C E A D B (1) C D E A B (1) C A D E B (1) B E A D C (1) B D C A E (1) B A E D C (1) A D E C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -10 0 -6 B 6 0 4 6 2 C 10 -4 0 6 6 D 0 -6 -6 0 2 E 6 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 0 -6 B 6 0 4 6 2 C 10 -4 0 6 6 D 0 -6 -6 0 2 E 6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 A=21 E=15 C=11 so C is eliminated. Round 2 votes counts: D=33 B=29 A=22 E=16 so E is eliminated. Round 3 votes counts: B=37 D=33 A=30 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:209 E:198 D:195 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 0 -6 B 6 0 4 6 2 C 10 -4 0 6 6 D 0 -6 -6 0 2 E 6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 0 -6 B 6 0 4 6 2 C 10 -4 0 6 6 D 0 -6 -6 0 2 E 6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 0 -6 B 6 0 4 6 2 C 10 -4 0 6 6 D 0 -6 -6 0 2 E 6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2275: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (14) C A B E D (12) E B A D C (6) D E A B C (6) C B A E D (6) C A B D E (6) C D E B A (4) C D A B E (3) B E A C D (3) B A E C D (3) A B E D C (3) E B D A C (2) D E C B A (2) D C E A B (2) D A E B C (2) D A C E B (2) C B E A D (2) C A D B E (2) A C B E D (2) A B C E D (2) E D B A C (1) E B A C D (1) E A B D C (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E B A (1) D C A E B (1) D C A B E (1) D A C B E (1) C D B A E (1) B C A E D (1) A E D B C (1) A D B E C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 8 4 B -4 0 0 4 2 C -8 0 0 -2 0 D -8 -4 2 0 6 E -4 -2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 4 B -4 0 0 4 2 C -8 0 0 -2 0 D -8 -4 2 0 6 E -4 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=35 E=11 A=11 B=7 so B is eliminated. Round 2 votes counts: C=37 D=35 E=14 A=14 so E is eliminated. Round 3 votes counts: D=38 C=37 A=25 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:212 B:201 D:198 C:195 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 4 B -4 0 0 4 2 C -8 0 0 -2 0 D -8 -4 2 0 6 E -4 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 4 B -4 0 0 4 2 C -8 0 0 -2 0 D -8 -4 2 0 6 E -4 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 4 B -4 0 0 4 2 C -8 0 0 -2 0 D -8 -4 2 0 6 E -4 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2276: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) E B C D A (7) C B D A E (6) A C B D E (6) E D B C A (5) E A D B C (5) A D E B C (5) D E A B C (4) D B C A E (4) C B D E A (4) B C D E A (4) A E D B C (4) A D C B E (4) E D B A C (3) A D E C B (3) E A C B D (2) D E B C A (2) D A B C E (2) C E B D A (2) C B E D A (2) A C E B D (2) A C D B E (2) E D A B C (1) E C B D A (1) E C B A D (1) E B C A D (1) D E B A C (1) D B E A C (1) D B A E C (1) C B E A D (1) C B A D E (1) B D C E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 2 -16 -10 B 10 0 16 0 -20 C -2 -16 0 -10 -14 D 16 0 10 0 4 E 10 20 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.078525 C: 0.000000 D: 0.921475 E: 0.000000 Sum of squares = 0.855282644574 Cumulative probabilities = A: 0.000000 B: 0.078525 C: 0.078525 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -16 -10 B 10 0 16 0 -20 C -2 -16 0 -10 -14 D 16 0 10 0 4 E 10 20 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222238243 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=31 C=16 D=15 B=5 so B is eliminated. Round 2 votes counts: E=33 A=31 C=20 D=16 so D is eliminated. Round 3 votes counts: E=41 A=34 C=25 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:215 B:203 A:183 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 -16 -10 B 10 0 16 0 -20 C -2 -16 0 -10 -14 D 16 0 10 0 4 E 10 20 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222238243 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -16 -10 B 10 0 16 0 -20 C -2 -16 0 -10 -14 D 16 0 10 0 4 E 10 20 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222238243 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -16 -10 B 10 0 16 0 -20 C -2 -16 0 -10 -14 D 16 0 10 0 4 E 10 20 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222238243 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2277: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) B A D E C (8) A B C D E (8) E C D B A (7) D C A B E (5) B A E D C (5) E B C A D (4) D C E A B (4) C D E A B (4) B E A D C (4) A B D C E (4) B A E C D (3) E D C B A (2) E D B C A (2) E B D A C (2) D C E B A (2) D C A E B (2) C D A E B (2) C A E B D (2) B E A C D (2) E D B A C (1) E C B D A (1) E C A B D (1) E B D C A (1) E B A D C (1) E A B C D (1) D B E A C (1) D B A C E (1) D A C B E (1) D A B C E (1) C D A B E (1) C A B E D (1) B E D A C (1) B D A E C (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 0 2 B -2 0 8 6 6 C -2 -8 0 -4 0 D 0 -6 4 0 -2 E -2 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.844051 B: 0.000000 C: 0.000000 D: 0.155949 E: 0.000000 Sum of squares = 0.736742311255 Cumulative probabilities = A: 0.844051 B: 0.844051 C: 0.844051 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 2 B -2 0 8 6 6 C -2 -8 0 -4 0 D 0 -6 4 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000001116 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 C=18 A=18 D=17 so D is eliminated. Round 2 votes counts: C=31 B=26 E=23 A=20 so A is eliminated. Round 3 votes counts: B=42 C=35 E=23 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:203 D:198 E:197 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 2 B -2 0 8 6 6 C -2 -8 0 -4 0 D 0 -6 4 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000001116 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 2 B -2 0 8 6 6 C -2 -8 0 -4 0 D 0 -6 4 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000001116 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 2 B -2 0 8 6 6 C -2 -8 0 -4 0 D 0 -6 4 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000001116 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2278: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) A C D E B (11) B E D A C (6) E D A C B (5) A C B D E (5) E D B A C (4) D E B C A (4) C A D E B (4) B C A D E (4) A C E D B (4) E B D A C (3) D C E A B (3) C A B D E (3) B D E C A (3) B C D A E (3) E D B C A (2) E A D C B (2) E A D B C (2) C A D B E (2) B E A D C (2) B C D E A (2) A E D C B (2) A E C D B (2) E D A B C (1) D E C A B (1) D C E B A (1) D B E C A (1) D B C E A (1) C D A E B (1) C B D E A (1) C B A D E (1) B A E D C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 4 -6 -10 B 0 0 2 -6 -4 C -4 -2 0 -10 -4 D 6 6 10 0 2 E 10 4 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -6 -10 B 0 0 2 -6 -4 C -4 -2 0 -10 -4 D 6 6 10 0 2 E 10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=26 E=19 C=12 D=11 so D is eliminated. Round 2 votes counts: B=34 A=26 E=24 C=16 so C is eliminated. Round 3 votes counts: B=36 A=36 E=28 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:212 E:208 B:196 A:194 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -6 -10 B 0 0 2 -6 -4 C -4 -2 0 -10 -4 D 6 6 10 0 2 E 10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -6 -10 B 0 0 2 -6 -4 C -4 -2 0 -10 -4 D 6 6 10 0 2 E 10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -6 -10 B 0 0 2 -6 -4 C -4 -2 0 -10 -4 D 6 6 10 0 2 E 10 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2279: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) C B E A D (7) C B A E D (7) A E B D C (5) A C B E D (5) A B E C D (5) E B A C D (4) D A E B C (4) A E B C D (4) A D E B C (4) D C B E A (3) D C A B E (3) C D B E A (3) E B D C A (2) D E B C A (2) D E B A C (2) D C B A E (2) D A C E B (2) C B D E A (2) B E C A D (2) B A E C D (2) A E D B C (2) E B D A C (1) E B A D C (1) E A B D C (1) D E C B A (1) D E A B C (1) D C E B A (1) D C A E B (1) D A E C B (1) D A C B E (1) C D B A E (1) C B D A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B C E A D (1) B C A E D (1) B A C E D (1) A D E C B (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 0 18 18 B 10 0 -8 28 20 C 0 8 0 18 10 D -18 -28 -18 0 -20 E -18 -20 -10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.257053 B: 0.000000 C: 0.742947 D: 0.000000 E: 0.000000 Sum of squares = 0.618046743934 Cumulative probabilities = A: 0.257053 B: 0.257053 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 18 18 B 10 0 -8 28 20 C 0 8 0 18 10 D -18 -28 -18 0 -20 E -18 -20 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444440 B: 0.000000 C: 0.555560 D: 0.000000 E: 0.000000 Sum of squares = 0.506173791373 Cumulative probabilities = A: 0.444440 B: 0.444440 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=29 D=24 E=9 B=7 so B is eliminated. Round 2 votes counts: C=33 A=32 D=24 E=11 so E is eliminated. Round 3 votes counts: A=38 C=35 D=27 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:225 C:218 A:213 E:186 D:158 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 18 18 B 10 0 -8 28 20 C 0 8 0 18 10 D -18 -28 -18 0 -20 E -18 -20 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444440 B: 0.000000 C: 0.555560 D: 0.000000 E: 0.000000 Sum of squares = 0.506173791373 Cumulative probabilities = A: 0.444440 B: 0.444440 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 18 18 B 10 0 -8 28 20 C 0 8 0 18 10 D -18 -28 -18 0 -20 E -18 -20 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444440 B: 0.000000 C: 0.555560 D: 0.000000 E: 0.000000 Sum of squares = 0.506173791373 Cumulative probabilities = A: 0.444440 B: 0.444440 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 18 18 B 10 0 -8 28 20 C 0 8 0 18 10 D -18 -28 -18 0 -20 E -18 -20 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444440 B: 0.000000 C: 0.555560 D: 0.000000 E: 0.000000 Sum of squares = 0.506173791373 Cumulative probabilities = A: 0.444440 B: 0.444440 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2280: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (16) A D C B E (8) E B C A D (7) D C A B E (7) E B D C A (5) E B A D C (5) E B A C D (4) D A C E B (4) C B D A E (4) B E C D A (4) C D A B E (3) A D C E B (3) E A B D C (2) D A C B E (2) C D B A E (2) C B E D A (2) B E C A D (2) B C E D A (2) A E D B C (2) A E B D C (2) A E B C D (2) A D E C B (2) A C D B E (2) E D A B C (1) D E A C B (1) D E A B C (1) D A E C B (1) C B A E D (1) C B A D E (1) C A D B E (1) B C A E D (1) Total count = 100 A B C D E A 0 -12 -16 -10 -4 B 12 0 12 20 -16 C 16 -12 0 8 -14 D 10 -20 -8 0 -16 E 4 16 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -16 -10 -4 B 12 0 12 20 -16 C 16 -12 0 8 -14 D 10 -20 -8 0 -16 E 4 16 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=21 D=16 C=14 B=9 so B is eliminated. Round 2 votes counts: E=46 A=21 C=17 D=16 so D is eliminated. Round 3 votes counts: E=48 A=28 C=24 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:214 C:199 D:183 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -16 -10 -4 B 12 0 12 20 -16 C 16 -12 0 8 -14 D 10 -20 -8 0 -16 E 4 16 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -10 -4 B 12 0 12 20 -16 C 16 -12 0 8 -14 D 10 -20 -8 0 -16 E 4 16 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -10 -4 B 12 0 12 20 -16 C 16 -12 0 8 -14 D 10 -20 -8 0 -16 E 4 16 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2281: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (13) C D B A E (11) E A B D C (10) A E B C D (7) D C E B A (6) E D C A B (5) D E C B A (4) D C B A E (4) C B A D E (4) A B C E D (4) B A C E D (3) E D B C A (2) D C B E A (2) D B E C A (2) C B D A E (2) C A B D E (2) B C D A E (2) B C A D E (2) E D C B A (1) E D A C B (1) E D A B C (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E A B (1) D B C E A (1) C D B E A (1) C A E D B (1) B E A D C (1) B A E D C (1) B A E C D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -8 6 18 B 2 0 4 8 14 C 8 -4 0 10 -4 D -6 -8 -10 0 -8 E -18 -14 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 6 18 B 2 0 4 8 14 C 8 -4 0 10 -4 D -6 -8 -10 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997846 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=22 D=21 C=21 B=11 so B is eliminated. Round 2 votes counts: A=31 C=25 E=23 D=21 so D is eliminated. Round 3 votes counts: C=39 A=31 E=30 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:214 A:207 C:205 E:190 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 6 18 B 2 0 4 8 14 C 8 -4 0 10 -4 D -6 -8 -10 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997846 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 6 18 B 2 0 4 8 14 C 8 -4 0 10 -4 D -6 -8 -10 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997846 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 6 18 B 2 0 4 8 14 C 8 -4 0 10 -4 D -6 -8 -10 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997846 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2282: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (6) A B E D C (6) E C A B D (5) D B A E C (5) E B D A C (4) C E D B A (4) C E A D B (4) C E A B D (4) C D B A E (4) B D A E C (4) A B D E C (4) A B D C E (4) E C D B A (3) D B E C A (3) A C E B D (3) E D B C A (2) E B A D C (2) E A C B D (2) E A B D C (2) D B E A C (2) D B C E A (2) D B C A E (2) D B A C E (2) C D E B A (2) C D B E A (2) C A D B E (2) A E B D C (2) E D C B A (1) E D B A C (1) E C D A B (1) E A B C D (1) D C B E A (1) D C B A E (1) C E D A B (1) C A E D B (1) C A B D E (1) B D E A C (1) B A D E C (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -4 4 0 B -2 0 2 8 -4 C 4 -2 0 -4 -6 D -4 -8 4 0 -12 E 0 4 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.377481 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.622519 Sum of squares = 0.530021836458 Cumulative probabilities = A: 0.377481 B: 0.377481 C: 0.377481 D: 0.377481 E: 1.000000 A B C D E A 0 2 -4 4 0 B -2 0 2 8 -4 C 4 -2 0 -4 -6 D -4 -8 4 0 -12 E 0 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=24 A=21 D=18 B=6 so B is eliminated. Round 2 votes counts: C=31 E=24 D=23 A=22 so A is eliminated. Round 3 votes counts: C=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:202 A:201 C:196 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 4 0 B -2 0 2 8 -4 C 4 -2 0 -4 -6 D -4 -8 4 0 -12 E 0 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 0 B -2 0 2 8 -4 C 4 -2 0 -4 -6 D -4 -8 4 0 -12 E 0 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 0 B -2 0 2 8 -4 C 4 -2 0 -4 -6 D -4 -8 4 0 -12 E 0 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2283: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C A E D B (7) B D E C A (6) E D B C A (5) C A B E D (5) E D A B C (4) B A C D E (4) A C B E D (4) A C B D E (4) E D C A B (3) C B A E D (3) C B A D E (3) B C D E A (3) A C E D B (3) E D C B A (2) E D A C B (2) E C D A B (2) C E D B A (2) C B E D A (2) B D C E A (2) B C D A E (2) B A D C E (2) A E D C B (2) A D E B C (2) E C D B A (1) E A D C B (1) D E B C A (1) D E A B C (1) D B E C A (1) D B E A C (1) D A E B C (1) C E D A B (1) C A E B D (1) C A B D E (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B A D E C (1) A E C D B (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -8 -6 4 B 4 0 -2 0 0 C 8 2 0 0 6 D 6 0 0 0 -4 E -4 0 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.594694 D: 0.405306 E: 0.000000 Sum of squares = 0.517933782226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.594694 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -6 4 B 4 0 -2 0 0 C 8 2 0 0 6 D 6 0 0 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=24 E=20 A=19 D=12 so D is eliminated. Round 2 votes counts: E=29 B=26 C=25 A=20 so A is eliminated. Round 3 votes counts: C=37 E=35 B=28 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:201 D:201 E:197 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 4 B 4 0 -2 0 0 C 8 2 0 0 6 D 6 0 0 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 4 B 4 0 -2 0 0 C 8 2 0 0 6 D 6 0 0 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 4 B 4 0 -2 0 0 C 8 2 0 0 6 D 6 0 0 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2284: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E D C A B (5) C A B D E (5) E C D B A (4) E C B A D (4) E B D A C (4) C B A E D (4) B A C E D (4) B A C D E (4) E D A B C (3) E B A D C (3) D C E A B (3) C D E A B (3) B E A C D (3) B C A D E (3) B A E C D (3) B A D C E (3) A B D E C (3) E D B A C (2) E B D C A (2) E B A C D (2) D E C A B (2) D C A B E (2) D A B E C (2) C D A E B (2) C B A D E (2) B C A E D (2) B A D E C (2) E D C B A (1) E D A C B (1) D E A C B (1) D E A B C (1) D A E C B (1) D A C B E (1) C B E A D (1) B A E D C (1) A D B E C (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 6 10 B 8 0 0 14 12 C 4 0 0 10 -2 D -6 -14 -10 0 2 E -10 -12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.554264 C: 0.445736 D: 0.000000 E: 0.000000 Sum of squares = 0.505889148853 Cumulative probabilities = A: 0.000000 B: 0.554264 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 6 10 B 8 0 0 14 12 C 4 0 0 10 -2 D -6 -14 -10 0 2 E -10 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=25 C=24 D=13 A=7 so A is eliminated. Round 2 votes counts: E=31 B=30 C=25 D=14 so D is eliminated. Round 3 votes counts: E=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:206 A:202 E:189 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 6 10 B 8 0 0 14 12 C 4 0 0 10 -2 D -6 -14 -10 0 2 E -10 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 6 10 B 8 0 0 14 12 C 4 0 0 10 -2 D -6 -14 -10 0 2 E -10 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 6 10 B 8 0 0 14 12 C 4 0 0 10 -2 D -6 -14 -10 0 2 E -10 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2285: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (15) B E D C A (10) C D A B E (7) B D C E A (6) E B D A C (5) E B A D C (5) C A D B E (5) C D B A E (4) E A B C D (3) B D E C A (3) A C E D B (3) E B D C A (2) E B A C D (2) E A D C B (2) D C A B E (2) D B C E A (2) B E C D A (2) B E A C D (2) B C D E A (2) A E C D B (2) A C D B E (2) E D A C B (1) E A D B C (1) E A B D C (1) D E B C A (1) D C B E A (1) D C B A E (1) C A D E B (1) B C E D A (1) A E C B D (1) A E B C D (1) A C E B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 0 -4 B -2 0 2 0 6 C 0 -2 0 14 10 D 0 0 -14 0 4 E 4 -6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.572825 B: 0.000000 C: 0.427175 D: 0.000000 E: 0.000000 Sum of squares = 0.510607050305 Cumulative probabilities = A: 0.572825 B: 0.572825 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 -4 B -2 0 2 0 6 C 0 -2 0 14 10 D 0 0 -14 0 4 E 4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500677 B: 0.000000 C: 0.499323 D: 0.000000 E: 0.000000 Sum of squares = 0.500000916108 Cumulative probabilities = A: 0.500677 B: 0.500677 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=26 E=22 C=17 D=7 so D is eliminated. Round 2 votes counts: B=28 A=28 E=23 C=21 so C is eliminated. Round 3 votes counts: A=43 B=34 E=23 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:211 B:203 A:199 D:195 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 0 0 -4 B -2 0 2 0 6 C 0 -2 0 14 10 D 0 0 -14 0 4 E 4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500677 B: 0.000000 C: 0.499323 D: 0.000000 E: 0.000000 Sum of squares = 0.500000916108 Cumulative probabilities = A: 0.500677 B: 0.500677 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 -4 B -2 0 2 0 6 C 0 -2 0 14 10 D 0 0 -14 0 4 E 4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500677 B: 0.000000 C: 0.499323 D: 0.000000 E: 0.000000 Sum of squares = 0.500000916108 Cumulative probabilities = A: 0.500677 B: 0.500677 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 -4 B -2 0 2 0 6 C 0 -2 0 14 10 D 0 0 -14 0 4 E 4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500677 B: 0.000000 C: 0.499323 D: 0.000000 E: 0.000000 Sum of squares = 0.500000916108 Cumulative probabilities = A: 0.500677 B: 0.500677 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2286: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (4) E B A C D (4) D C A B E (4) D A B C E (4) C E D A B (4) C E B D A (4) B D A E C (4) A E C D B (4) A E B D C (4) A D E B C (4) E B A D C (3) D A C B E (3) C D B E A (3) C D B A E (3) A D C B E (3) E C B D A (2) E B C A D (2) E A B C D (2) D C B A E (2) D B A C E (2) C D A E B (2) C B E D A (2) B E A D C (2) B D C E A (2) A D E C B (2) A D C E B (2) A D B E C (2) E C A B D (1) E B C D A (1) E A B D C (1) C E D B A (1) C E A D B (1) C D A B E (1) C B D E A (1) C A D E B (1) B E C D A (1) B D E C A (1) B D E A C (1) B D A C E (1) B C E D A (1) B C D E A (1) B A D E C (1) A E D C B (1) A E D B C (1) A D B C E (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 10 -2 10 B -2 0 -6 -6 -6 C -10 6 0 -4 2 D 2 6 4 0 6 E -10 6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 -2 10 B -2 0 -6 -6 -6 C -10 6 0 -4 2 D 2 6 4 0 6 E -10 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 E=20 D=15 B=15 so D is eliminated. Round 2 votes counts: A=34 C=29 E=20 B=17 so B is eliminated. Round 3 votes counts: A=42 C=33 E=25 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:209 C:197 E:194 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 10 -2 10 B -2 0 -6 -6 -6 C -10 6 0 -4 2 D 2 6 4 0 6 E -10 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -2 10 B -2 0 -6 -6 -6 C -10 6 0 -4 2 D 2 6 4 0 6 E -10 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -2 10 B -2 0 -6 -6 -6 C -10 6 0 -4 2 D 2 6 4 0 6 E -10 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2287: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) D C B E A (8) E B A C D (7) C D B E A (6) D C A B E (5) C D A B E (5) E A B D C (4) A E B D C (4) A C D E B (4) E B A D C (3) E A B C D (3) B E D C A (3) D C B A E (2) D B E C A (2) D B C E A (2) C B D E A (2) B E D A C (2) B E C D A (2) B C E D A (2) B C D E A (2) A E D C B (2) A E D B C (2) A D E C B (2) A C E D B (2) A C E B D (2) D C A E B (1) D B E A C (1) D B A E C (1) D A C E B (1) C D B A E (1) C B E A D (1) B E A D C (1) B E A C D (1) B D E C A (1) B D C E A (1) A E C D B (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 8 0 -8 B 2 0 6 2 2 C -8 -6 0 2 -4 D 0 -2 -2 0 -4 E 8 -2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 0 -8 B 2 0 6 2 2 C -8 -6 0 2 -4 D 0 -2 -2 0 -4 E 8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=23 E=17 C=15 B=15 so C is eliminated. Round 2 votes counts: D=35 A=30 B=18 E=17 so E is eliminated. Round 3 votes counts: A=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:207 B:206 A:199 D:196 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 0 -8 B 2 0 6 2 2 C -8 -6 0 2 -4 D 0 -2 -2 0 -4 E 8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 0 -8 B 2 0 6 2 2 C -8 -6 0 2 -4 D 0 -2 -2 0 -4 E 8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 0 -8 B 2 0 6 2 2 C -8 -6 0 2 -4 D 0 -2 -2 0 -4 E 8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2288: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B A D C E (8) E C D B A (7) A B D C E (7) E B C D A (6) D C E A B (6) C D E A B (5) B A E D C (4) B A E C D (4) B A D E C (4) A D C E B (4) C E D A B (3) B E D C A (3) B E A C D (3) A D C B E (3) B E C D A (2) A C D E B (2) A B C D E (2) E C B A D (1) E C A D B (1) E B C A D (1) E B A C D (1) D C E B A (1) D C A E B (1) D C A B E (1) D A C E B (1) D A B C E (1) B E D A C (1) B D E C A (1) B D E A C (1) A E C D B (1) A D B C E (1) A C E D B (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 2 -4 B -4 0 2 2 -2 C -4 -2 0 4 -2 D -2 -2 -4 0 0 E 4 2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.229199 E: 0.770801 Sum of squares = 0.646665956007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.229199 E: 1.000000 A B C D E A 0 4 4 2 -4 B -4 0 2 2 -2 C -4 -2 0 4 -2 D -2 -2 -4 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555569424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=26 A=24 D=11 C=8 so C is eliminated. Round 2 votes counts: B=31 E=29 A=24 D=16 so D is eliminated. Round 3 votes counts: E=41 B=31 A=28 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:204 A:203 B:199 C:198 D:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 2 -4 B -4 0 2 2 -2 C -4 -2 0 4 -2 D -2 -2 -4 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555569424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 -4 B -4 0 2 2 -2 C -4 -2 0 4 -2 D -2 -2 -4 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555569424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 -4 B -4 0 2 2 -2 C -4 -2 0 4 -2 D -2 -2 -4 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555569424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2289: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) A B D E C (7) D A C E B (5) C B A E D (5) C A B D E (5) E D B A C (4) C E D B A (4) C B E A D (4) C A D B E (4) E B D A C (3) D E C A B (3) D A E B C (3) C D A E B (3) B E A D C (3) B A E D C (3) B A C E D (3) A C D B E (3) E D B C A (2) E C D B A (2) E B D C A (2) C D E A B (2) C D A B E (2) B C A E D (2) B A E C D (2) A C B D E (2) A B D C E (2) A B C D E (2) E D C B A (1) E D A B C (1) E B C D A (1) D A E C B (1) D A B E C (1) C E B A D (1) C B A D E (1) B E A C D (1) B A C D E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 10 12 6 18 B -10 0 4 0 10 C -12 -4 0 0 4 D -6 0 0 0 12 E -18 -10 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 6 18 B -10 0 4 0 10 C -12 -4 0 0 4 D -6 0 0 0 12 E -18 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=20 A=18 E=16 B=15 so B is eliminated. Round 2 votes counts: C=33 A=27 E=20 D=20 so E is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:223 D:203 B:202 C:194 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 6 18 B -10 0 4 0 10 C -12 -4 0 0 4 D -6 0 0 0 12 E -18 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 6 18 B -10 0 4 0 10 C -12 -4 0 0 4 D -6 0 0 0 12 E -18 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 6 18 B -10 0 4 0 10 C -12 -4 0 0 4 D -6 0 0 0 12 E -18 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2290: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) E A C B D (8) D B C A E (5) D C B E A (4) C E B D A (4) C B D E A (4) E C A D B (3) D C B A E (3) D A C E B (3) C E B A D (3) C E A D B (3) B D C E A (3) B A E D C (3) A E C D B (3) A E C B D (3) A E B D C (3) A D B E C (3) E C A B D (2) E B C A D (2) E A B C D (2) D A B E C (2) C E D B A (2) C D B E A (2) C B E D A (2) A D E B C (2) E A C D B (1) D C A B E (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A B D (1) C D E B A (1) B E C D A (1) B E C A D (1) B D C A E (1) B D A C E (1) B C D E A (1) B A D E C (1) A D E C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 0 -6 -2 B 8 0 -12 -4 4 C 0 12 0 0 14 D 6 4 0 0 2 E 2 -4 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.367203 D: 0.632797 E: 0.000000 Sum of squares = 0.535270067497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.367203 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -6 -2 B 8 0 -12 -4 4 C 0 12 0 0 14 D 6 4 0 0 2 E 2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=23 E=18 A=17 B=12 so B is eliminated. Round 2 votes counts: D=35 C=24 A=21 E=20 so E is eliminated. Round 3 votes counts: D=35 C=33 A=32 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:206 B:198 A:192 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 0 -6 -2 B 8 0 -12 -4 4 C 0 12 0 0 14 D 6 4 0 0 2 E 2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -6 -2 B 8 0 -12 -4 4 C 0 12 0 0 14 D 6 4 0 0 2 E 2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -6 -2 B 8 0 -12 -4 4 C 0 12 0 0 14 D 6 4 0 0 2 E 2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2291: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) E A D C B (7) C A B E D (7) B D C A E (6) E A C B D (5) C A E D B (5) A C E D B (5) E D A C B (4) E A C D B (4) C B A D E (4) C A E B D (4) B D C E A (4) D B C A E (3) C A B D E (3) B C D A E (3) E D A B C (2) D E B A C (2) D E A B C (2) C D A B E (2) C B A E D (2) B D E A C (2) A E C D B (2) E B A D C (1) E B A C D (1) D E A C B (1) D C B A E (1) D A E C B (1) C D B A E (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E C A (1) B C A D E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 12 4 8 4 B -12 0 -20 -2 4 C -4 20 0 6 4 D -8 2 -6 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 8 4 B -12 0 -20 -2 4 C -4 20 0 6 4 D -8 2 -6 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=24 B=20 D=19 A=9 so A is eliminated. Round 2 votes counts: C=34 E=27 B=20 D=19 so D is eliminated. Round 3 votes counts: C=35 E=33 B=32 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:213 E:198 D:190 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 8 4 B -12 0 -20 -2 4 C -4 20 0 6 4 D -8 2 -6 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 8 4 B -12 0 -20 -2 4 C -4 20 0 6 4 D -8 2 -6 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 8 4 B -12 0 -20 -2 4 C -4 20 0 6 4 D -8 2 -6 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999823 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2292: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) E C A D B (7) E B D A C (7) D B A C E (7) C A E D B (7) A C D B E (7) E A C D B (5) B D E A C (5) B D A C E (5) E B D C A (4) A D B C E (4) E B A D C (2) D B E A C (2) D B A E C (2) C E A D B (2) C A B D E (2) B D A E C (2) A C E D B (2) E D B A C (1) E C B D A (1) E C B A D (1) E B C D A (1) E B C A D (1) D B C A E (1) C E A B D (1) C D B A E (1) C A E B D (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B D C A E (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 10 14 -6 B -6 0 -2 -6 -10 C -10 2 0 6 -10 D -14 6 -6 0 -12 E 6 10 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 10 14 -6 B -6 0 -2 -6 -10 C -10 2 0 6 -10 D -14 6 -6 0 -12 E 6 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=16 B=16 A=16 D=12 so D is eliminated. Round 2 votes counts: E=40 B=28 C=16 A=16 so C is eliminated. Round 3 votes counts: E=43 B=29 A=28 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:212 C:194 B:188 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 10 14 -6 B -6 0 -2 -6 -10 C -10 2 0 6 -10 D -14 6 -6 0 -12 E 6 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 14 -6 B -6 0 -2 -6 -10 C -10 2 0 6 -10 D -14 6 -6 0 -12 E 6 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 14 -6 B -6 0 -2 -6 -10 C -10 2 0 6 -10 D -14 6 -6 0 -12 E 6 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2293: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) B E D A C (8) C A B D E (7) B C A E D (7) C D E A B (5) B C E D A (5) E B D A C (4) B E D C A (4) B E A D C (4) B C E A D (4) D E A C B (3) C B A D E (3) B A E D C (3) E D B A C (2) E D A B C (2) E A D B C (2) D E C A B (2) D C A E B (2) C D A E B (2) A D E C B (2) A C B D E (2) E D B C A (1) E D A C B (1) E B A D C (1) D C E A B (1) D A E C B (1) C D E B A (1) C D B A E (1) C B D A E (1) C A D B E (1) B E C D A (1) B E C A D (1) B C A D E (1) A C D E B (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -22 6 -4 B 4 0 6 18 12 C 22 -6 0 12 12 D -6 -18 -12 0 -6 E 4 -12 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -22 6 -4 B 4 0 6 18 12 C 22 -6 0 12 12 D -6 -18 -12 0 -6 E 4 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 C=32 E=13 D=9 A=8 so A is eliminated. Round 2 votes counts: B=41 C=35 E=13 D=11 so D is eliminated. Round 3 votes counts: B=41 C=38 E=21 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:220 E:193 A:188 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -22 6 -4 B 4 0 6 18 12 C 22 -6 0 12 12 D -6 -18 -12 0 -6 E 4 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -22 6 -4 B 4 0 6 18 12 C 22 -6 0 12 12 D -6 -18 -12 0 -6 E 4 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -22 6 -4 B 4 0 6 18 12 C 22 -6 0 12 12 D -6 -18 -12 0 -6 E 4 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2294: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) C A E B D (10) C A B D E (10) D B E A C (9) A C E D B (9) C B D A E (7) B D C E A (7) A E C D B (5) E A D B C (4) B D E C A (4) E D A B C (3) D E B A C (3) B D E A C (3) E A C D B (2) D B C E A (2) B D C A E (2) B C D A E (2) A C E B D (2) D E B C A (1) C B A D E (1) C A E D B (1) B A D E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 6 -8 2 B 6 0 4 0 -2 C -6 -4 0 0 8 D 8 0 0 0 4 E -2 2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.444650 C: 0.000000 D: 0.555350 E: 0.000000 Sum of squares = 0.506127135284 Cumulative probabilities = A: 0.000000 B: 0.444650 C: 0.444650 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -8 2 B 6 0 4 0 -2 C -6 -4 0 0 8 D 8 0 0 0 4 E -2 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=20 B=19 A=17 D=15 so D is eliminated. Round 2 votes counts: B=30 C=29 E=24 A=17 so A is eliminated. Round 3 votes counts: C=41 B=30 E=29 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:206 B:204 C:199 A:197 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 6 -8 2 B 6 0 4 0 -2 C -6 -4 0 0 8 D 8 0 0 0 4 E -2 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -8 2 B 6 0 4 0 -2 C -6 -4 0 0 8 D 8 0 0 0 4 E -2 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -8 2 B 6 0 4 0 -2 C -6 -4 0 0 8 D 8 0 0 0 4 E -2 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2295: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) A B E C D (11) C D B E A (9) B E A C D (9) A B E D C (9) C D E B A (8) E B D A C (4) E B A D C (4) C D A B E (4) D C A E B (3) C B E A D (3) C A D B E (2) C A B E D (2) A E B D C (2) A D E B C (2) A C D B E (2) A C B E D (2) E D B A C (1) E B D C A (1) E B C D A (1) D E C B A (1) D E B C A (1) D E B A C (1) D A E B C (1) D A C B E (1) C E B D A (1) C B A E D (1) B E A D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 2 2 -14 B 16 0 -2 4 14 C -2 2 0 12 2 D -2 -4 -12 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.100000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.659999999865 Cumulative probabilities = A: 0.100000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 2 2 -14 B 16 0 -2 4 14 C -2 2 0 12 2 D -2 -4 -12 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.100000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.659999999974 Cumulative probabilities = A: 0.100000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=29 D=20 E=11 B=10 so B is eliminated. Round 2 votes counts: C=30 A=29 E=21 D=20 so D is eliminated. Round 3 votes counts: C=45 A=31 E=24 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 C:207 E:202 D:188 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 2 2 -14 B 16 0 -2 4 14 C -2 2 0 12 2 D -2 -4 -12 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.100000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.659999999974 Cumulative probabilities = A: 0.100000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 2 -14 B 16 0 -2 4 14 C -2 2 0 12 2 D -2 -4 -12 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.100000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.659999999974 Cumulative probabilities = A: 0.100000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 2 -14 B 16 0 -2 4 14 C -2 2 0 12 2 D -2 -4 -12 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.100000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.659999999974 Cumulative probabilities = A: 0.100000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2296: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (20) D A C B E (15) C B E D A (15) E B C A D (11) A D C B E (5) E B A C D (3) D C A B E (3) A E B C D (3) E C B A D (2) E B C D A (2) D A B E C (2) C E B D A (2) C E B A D (2) C B E A D (2) A D B E C (2) D B E C A (1) D B E A C (1) C D B E A (1) C A D B E (1) B E D C A (1) B E C D A (1) A E D B C (1) A E C B D (1) A E B D C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 12 12 12 12 B -12 0 0 -8 2 C -12 0 0 -8 -4 D -12 8 8 0 6 E -12 -2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 12 12 B -12 0 0 -8 2 C -12 0 0 -8 -4 D -12 8 8 0 6 E -12 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=23 D=22 E=18 B=2 so B is eliminated. Round 2 votes counts: A=35 C=23 D=22 E=20 so E is eliminated. Round 3 votes counts: C=39 A=38 D=23 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:205 E:192 B:191 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 12 12 B -12 0 0 -8 2 C -12 0 0 -8 -4 D -12 8 8 0 6 E -12 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 12 12 B -12 0 0 -8 2 C -12 0 0 -8 -4 D -12 8 8 0 6 E -12 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 12 12 B -12 0 0 -8 2 C -12 0 0 -8 -4 D -12 8 8 0 6 E -12 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2297: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) A E C D B (6) A E D C B (5) C E B D A (4) C E B A D (4) C E A B D (4) C B E A D (4) C A E B D (4) A D E B C (4) A D B E C (4) D B E A C (3) D B A E C (3) D A E B C (3) B C D E A (3) E D B C A (2) E C D A B (2) D E B C A (2) D E B A C (2) D B E C A (2) D A B E C (2) C B E D A (2) B D E C A (2) B D A C E (2) B C E D A (2) A D E C B (2) A C E B D (2) E D C B A (1) E C D B A (1) E C A D B (1) E A D C B (1) E A C D B (1) D E A B C (1) C E A D B (1) C B A E D (1) C A B E D (1) B D E A C (1) B C D A E (1) B A D C E (1) B A C D E (1) A D B C E (1) A C E D B (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -2 4 -6 B 2 0 0 -2 -8 C 2 0 0 -2 -2 D -4 2 2 0 -2 E 6 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -2 4 -6 B 2 0 0 -2 -8 C 2 0 0 -2 -2 D -4 2 2 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999035 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=25 B=20 D=18 E=9 so E is eliminated. Round 2 votes counts: A=30 C=29 D=21 B=20 so B is eliminated. Round 3 votes counts: C=35 D=33 A=32 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:209 C:199 D:199 A:197 B:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -6 B 2 0 0 -2 -8 C 2 0 0 -2 -2 D -4 2 2 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999035 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -6 B 2 0 0 -2 -8 C 2 0 0 -2 -2 D -4 2 2 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999035 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -6 B 2 0 0 -2 -8 C 2 0 0 -2 -2 D -4 2 2 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999035 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2298: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) D A B E C (6) E C B A D (5) D C E A B (5) D B A C E (5) A E B C D (5) E C A D B (4) C E D B A (4) C E B D A (4) C D E B A (4) E C A B D (3) B A E C D (3) A E D C B (3) E A B C D (2) D C B A E (2) D B C A E (2) D A E B C (2) D A B C E (2) C D B E A (2) C B E D A (2) C B D E A (2) B C E A D (2) B A D E C (2) A D B E C (2) A B E D C (2) A B E C D (2) E B C A D (1) E A D C B (1) E A C B D (1) D E C A B (1) D E A C B (1) D C E B A (1) D B A E C (1) D A C E B (1) D A C B E (1) C E A B D (1) B E C A D (1) B A C E D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -10 4 -12 B 6 0 -14 -2 -18 C 10 14 0 16 0 D -4 2 -16 0 -12 E 12 18 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.427444 D: 0.000000 E: 0.572556 Sum of squares = 0.510528828411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.427444 D: 0.427444 E: 1.000000 A B C D E A 0 -6 -10 4 -12 B 6 0 -14 -2 -18 C 10 14 0 16 0 D -4 2 -16 0 -12 E 12 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=28 E=17 A=16 B=9 so B is eliminated. Round 2 votes counts: D=30 C=30 A=22 E=18 so E is eliminated. Round 3 votes counts: C=44 D=30 A=26 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:220 A:188 B:186 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 4 -12 B 6 0 -14 -2 -18 C 10 14 0 16 0 D -4 2 -16 0 -12 E 12 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 4 -12 B 6 0 -14 -2 -18 C 10 14 0 16 0 D -4 2 -16 0 -12 E 12 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 4 -12 B 6 0 -14 -2 -18 C 10 14 0 16 0 D -4 2 -16 0 -12 E 12 18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2299: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) C E D A B (8) C B E A D (8) E D A C B (7) D A E B C (7) B A D C E (7) C E B D A (6) C E B A D (6) A D B E C (6) E C D A B (5) B A D E C (5) C E D B A (4) A B D E C (4) D E A B C (3) E D C A B (2) B D A C E (2) E D A B C (1) E B C A D (1) D E A C B (1) D C E A B (1) D A E C B (1) D A C E B (1) C B D A E (1) C B A D E (1) B C A E D (1) B C A D E (1) B A E D C (1) B A C D E (1) Total count = 100 A B C D E A 0 10 10 -16 -6 B -10 0 -4 -10 -8 C -10 4 0 -14 -4 D 16 10 14 0 0 E 6 8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.570727 E: 0.429273 Sum of squares = 0.510004549377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.570727 E: 1.000000 A B C D E A 0 10 10 -16 -6 B -10 0 -4 -10 -8 C -10 4 0 -14 -4 D 16 10 14 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=22 B=18 E=16 A=10 so A is eliminated. Round 2 votes counts: C=34 D=28 B=22 E=16 so E is eliminated. Round 3 votes counts: C=39 D=38 B=23 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:209 A:199 C:188 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -16 -6 B -10 0 -4 -10 -8 C -10 4 0 -14 -4 D 16 10 14 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -16 -6 B -10 0 -4 -10 -8 C -10 4 0 -14 -4 D 16 10 14 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -16 -6 B -10 0 -4 -10 -8 C -10 4 0 -14 -4 D 16 10 14 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2300: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) E D C A B (6) D A B C E (6) E D A C B (5) C B A D E (5) B C A D E (5) E D A B C (4) E C D A B (4) E C B A D (4) B A D C E (4) E B D A C (3) D A C B E (3) C A D B E (3) E D B A C (2) E B C D A (2) D A E B C (2) C E D A B (2) C E B A D (2) C E A D B (2) C A D E B (2) B E C A D (2) B C A E D (2) A D C B E (2) A D B C E (2) E C D B A (1) E C A D B (1) E C A B D (1) D E A C B (1) D C A E B (1) D A E C B (1) D A C E B (1) D A B E C (1) C E A B D (1) C B A E D (1) C A B D E (1) B E A C D (1) B D A C E (1) B C E A D (1) B A C D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -16 8 -8 B -8 0 0 -4 -14 C 16 0 0 8 0 D -8 4 -8 0 -12 E 8 14 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.331787 D: 0.000000 E: 0.668213 Sum of squares = 0.556591250512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.331787 D: 0.331787 E: 1.000000 A B C D E A 0 8 -16 8 -8 B -8 0 0 -4 -14 C 16 0 0 8 0 D -8 4 -8 0 -12 E 8 14 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=19 B=17 D=16 A=6 so A is eliminated. Round 2 votes counts: E=42 D=20 C=19 B=19 so C is eliminated. Round 3 votes counts: E=49 B=26 D=25 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:212 A:196 D:188 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 8 -8 B -8 0 0 -4 -14 C 16 0 0 8 0 D -8 4 -8 0 -12 E 8 14 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 8 -8 B -8 0 0 -4 -14 C 16 0 0 8 0 D -8 4 -8 0 -12 E 8 14 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 8 -8 B -8 0 0 -4 -14 C 16 0 0 8 0 D -8 4 -8 0 -12 E 8 14 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2301: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) C B D E A (7) A B D C E (6) E A C B D (5) E C B A D (4) D B C A E (4) B C D A E (4) E D C A B (3) E C D B A (3) E C B D A (3) C D B E A (3) C B E A D (3) A D E B C (3) A B C D E (3) E D C B A (2) E D A C B (2) E C A D B (2) E C A B D (2) D E C B A (2) D E A B C (2) D C B E A (2) D B C E A (2) D B A C E (2) D A B C E (2) A D B C E (2) A B E C D (2) A B D E C (2) E C D A B (1) E A D B C (1) D E A C B (1) D A E B C (1) C E D B A (1) C E B D A (1) C D E B A (1) B D C A E (1) B D A C E (1) B A D C E (1) B A C E D (1) A E C B D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -2 -2 -6 B -4 0 2 -2 -4 C 2 -2 0 -4 -6 D 2 2 4 0 4 E 6 4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -2 -6 B -4 0 2 -2 -4 C 2 -2 0 -4 -6 D 2 2 4 0 4 E 6 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 D=18 C=16 B=8 so B is eliminated. Round 2 votes counts: A=32 E=28 D=20 C=20 so D is eliminated. Round 3 votes counts: A=38 E=33 C=29 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:206 E:206 A:197 B:196 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -2 -6 B -4 0 2 -2 -4 C 2 -2 0 -4 -6 D 2 2 4 0 4 E 6 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -2 -6 B -4 0 2 -2 -4 C 2 -2 0 -4 -6 D 2 2 4 0 4 E 6 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -2 -6 B -4 0 2 -2 -4 C 2 -2 0 -4 -6 D 2 2 4 0 4 E 6 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2302: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (12) D C A B E (10) E B A C D (7) D C A E B (6) E D C A B (5) B A C D E (5) E C D A B (4) C A D B E (4) D E C B A (3) D C E A B (3) C E A D B (3) C A D E B (3) A C B D E (3) A B C E D (3) E B D C A (2) C D A E B (2) B D A E C (2) B A C E D (2) A C E B D (2) A C D B E (2) E D C B A (1) E D B C A (1) E C A D B (1) E B D A C (1) E A B C D (1) D E C A B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) B E A C D (1) B D E A C (1) B D A C E (1) B A D C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -8 4 22 B -10 0 -12 -10 2 C 8 12 0 14 10 D -4 10 -14 0 8 E -22 -2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 4 22 B -10 0 -12 -10 2 C 8 12 0 14 10 D -4 10 -14 0 8 E -22 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 E=23 C=12 A=12 so C is eliminated. Round 2 votes counts: D=30 E=26 B=25 A=19 so A is eliminated. Round 3 votes counts: D=40 B=32 E=28 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:222 A:214 D:200 B:185 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 4 22 B -10 0 -12 -10 2 C 8 12 0 14 10 D -4 10 -14 0 8 E -22 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 4 22 B -10 0 -12 -10 2 C 8 12 0 14 10 D -4 10 -14 0 8 E -22 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 4 22 B -10 0 -12 -10 2 C 8 12 0 14 10 D -4 10 -14 0 8 E -22 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2303: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (13) C D A E B (11) B E A D C (11) C A D B E (9) A B E C D (9) E B D A C (6) A C B E D (6) D E B C A (4) B E A C D (4) E B A D C (3) D C A E B (3) C D E B A (2) C D A B E (2) B E D A C (2) A B E D C (2) E B D C A (1) E B C D A (1) D E B A C (1) D A C E B (1) C D B E A (1) C B E A D (1) C A B E D (1) B E C A D (1) A E B D C (1) A D E B C (1) A D B C E (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 4 -2 B 2 0 -2 0 4 C 0 2 0 0 6 D -4 0 0 0 0 E 2 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.372266 B: 0.000000 C: 0.627734 D: 0.000000 E: 0.000000 Sum of squares = 0.532631915356 Cumulative probabilities = A: 0.372266 B: 0.372266 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 4 -2 B 2 0 -2 0 4 C 0 2 0 0 6 D -4 0 0 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499925 B: 0.000000 C: 0.500075 D: 0.000000 E: 0.000000 Sum of squares = 0.500000011156 Cumulative probabilities = A: 0.499925 B: 0.499925 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 A=22 B=18 E=11 so E is eliminated. Round 2 votes counts: B=29 C=27 D=22 A=22 so D is eliminated. Round 3 votes counts: C=43 B=34 A=23 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:204 B:202 A:200 D:198 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 4 -2 B 2 0 -2 0 4 C 0 2 0 0 6 D -4 0 0 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499925 B: 0.000000 C: 0.500075 D: 0.000000 E: 0.000000 Sum of squares = 0.500000011156 Cumulative probabilities = A: 0.499925 B: 0.499925 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 -2 B 2 0 -2 0 4 C 0 2 0 0 6 D -4 0 0 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499925 B: 0.000000 C: 0.500075 D: 0.000000 E: 0.000000 Sum of squares = 0.500000011156 Cumulative probabilities = A: 0.499925 B: 0.499925 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 -2 B 2 0 -2 0 4 C 0 2 0 0 6 D -4 0 0 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499925 B: 0.000000 C: 0.500075 D: 0.000000 E: 0.000000 Sum of squares = 0.500000011156 Cumulative probabilities = A: 0.499925 B: 0.499925 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2304: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (12) C B E A D (11) A D C B E (10) E B C D A (6) D A E B C (6) B E C D A (6) A D E C B (6) D E A B C (5) D A B E C (4) C B A D E (3) B C E A D (3) E D A B C (2) E C B A D (2) E B D C A (2) D A E C B (2) D A B C E (2) C E B A D (2) C B A E D (2) B D A C E (2) A D C E B (2) E C B D A (1) E A C D B (1) D E B A C (1) C A B D E (1) B C D E A (1) B C D A E (1) B C A D E (1) A D B C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -8 -6 -10 B 12 0 10 14 24 C 8 -10 0 10 12 D 6 -14 -10 0 0 E 10 -24 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -6 -10 B 12 0 10 14 24 C 8 -10 0 10 12 D 6 -14 -10 0 0 E 10 -24 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=21 D=20 C=19 E=14 so E is eliminated. Round 2 votes counts: B=34 D=22 C=22 A=22 so D is eliminated. Round 3 votes counts: A=43 B=35 C=22 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:230 C:210 D:191 E:187 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -6 -10 B 12 0 10 14 24 C 8 -10 0 10 12 D 6 -14 -10 0 0 E 10 -24 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -6 -10 B 12 0 10 14 24 C 8 -10 0 10 12 D 6 -14 -10 0 0 E 10 -24 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -6 -10 B 12 0 10 14 24 C 8 -10 0 10 12 D 6 -14 -10 0 0 E 10 -24 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2305: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) C D B E A (6) D E B A C (5) C A B E D (5) C B D A E (4) E D A B C (3) E A D B C (3) D E B C A (3) D C B E A (3) C B D E A (3) C A D E B (3) B C A E D (3) A E B C D (3) A C E D B (3) D E A C B (2) D C E A B (2) D B E C A (2) C D A B E (2) C B A D E (2) B E A D C (2) B C D E A (2) B A E D C (2) B A E C D (2) A E D C B (2) A E D B C (2) A E C D B (2) A E C B D (2) A C B E D (2) E D B A C (1) E B A D C (1) D E C B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D B E A C (1) C D B A E (1) C B A E D (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E C A (1) B D C E A (1) B C E A D (1) B A C E D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 0 6 6 B 2 0 -4 -4 -2 C 0 4 0 2 -2 D -6 4 -2 0 -2 E -6 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.446405 B: 0.000000 C: 0.553595 D: 0.000000 E: 0.000000 Sum of squares = 0.505744917429 Cumulative probabilities = A: 0.446405 B: 0.446405 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 6 6 B 2 0 -4 -4 -2 C 0 4 0 2 -2 D -6 4 -2 0 -2 E -6 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=25 D=22 B=16 E=8 so E is eliminated. Round 2 votes counts: C=29 A=28 D=26 B=17 so B is eliminated. Round 3 votes counts: A=36 C=35 D=29 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:205 C:202 E:200 D:197 B:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 6 6 B 2 0 -4 -4 -2 C 0 4 0 2 -2 D -6 4 -2 0 -2 E -6 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 6 6 B 2 0 -4 -4 -2 C 0 4 0 2 -2 D -6 4 -2 0 -2 E -6 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 6 6 B 2 0 -4 -4 -2 C 0 4 0 2 -2 D -6 4 -2 0 -2 E -6 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2306: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) B C A D E (6) B A D C E (6) D A B E C (5) D A E C B (4) D A E B C (4) D A B C E (4) C B E D A (4) E C D A B (3) E C B D A (3) E A D C B (3) E A D B C (3) E A B D C (3) C B E A D (3) C B D A E (3) B C E A D (3) B C A E D (3) A B D E C (3) E C B A D (2) E B C A D (2) C E B D A (2) C D E B A (2) A E B D C (2) A D B E C (2) E D C A B (1) E C D B A (1) E C A D B (1) E C A B D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C B A E (1) D B C A E (1) C E B A D (1) C B D E A (1) B E C A D (1) B D C A E (1) B C D A E (1) B A C D E (1) A E D B C (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 2 2 B -4 0 12 10 4 C -4 -12 0 -10 -10 D -2 -10 10 0 -2 E -2 -4 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 2 2 B -4 0 12 10 4 C -4 -12 0 -10 -10 D -2 -10 10 0 -2 E -2 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=22 D=21 C=16 A=11 so A is eliminated. Round 2 votes counts: E=33 B=27 D=24 C=16 so C is eliminated. Round 3 votes counts: B=38 E=36 D=26 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:206 E:203 D:198 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 2 B -4 0 12 10 4 C -4 -12 0 -10 -10 D -2 -10 10 0 -2 E -2 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 2 B -4 0 12 10 4 C -4 -12 0 -10 -10 D -2 -10 10 0 -2 E -2 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 2 B -4 0 12 10 4 C -4 -12 0 -10 -10 D -2 -10 10 0 -2 E -2 -4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2307: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) A B C D E (7) E D C B A (6) B A C D E (6) C B E D A (5) B D E A C (5) A C E D B (5) A B D E C (5) C A B E D (4) B C D E A (4) E D C A B (3) E C D B A (3) D E B C A (3) D E B A C (3) D E A C B (3) D E A B C (3) C E D B A (3) C A E D B (3) B D E C A (3) C E D A B (2) C E B D A (2) B C E D A (2) B C A D E (2) B A D E C (2) A D E B C (2) E C D A B (1) D B E A C (1) D B A E C (1) B D A E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -8 -4 B 4 0 4 16 16 C -8 -4 0 16 10 D 8 -16 -16 0 4 E 4 -16 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -8 -4 B 4 0 4 16 16 C -8 -4 0 16 10 D 8 -16 -16 0 4 E 4 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999072 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=25 C=19 D=14 E=13 so E is eliminated. Round 2 votes counts: A=29 B=25 D=23 C=23 so D is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:207 A:196 D:190 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 -8 -4 B 4 0 4 16 16 C -8 -4 0 16 10 D 8 -16 -16 0 4 E 4 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999072 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -8 -4 B 4 0 4 16 16 C -8 -4 0 16 10 D 8 -16 -16 0 4 E 4 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999072 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -8 -4 B 4 0 4 16 16 C -8 -4 0 16 10 D 8 -16 -16 0 4 E 4 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999072 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2308: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) C B A E D (10) B C A D E (9) E D A C B (6) B C A E D (6) E D C B A (5) C A B E D (5) E D A B C (4) A C B D E (4) E D C A B (3) D E A C B (3) D A E B C (3) C E B D A (3) B A C D E (3) A B C D E (3) D E B A C (2) B C E A D (2) E D B C A (1) E C D A B (1) E C B D A (1) D B E A C (1) D B A E C (1) D A E C B (1) C E A D B (1) C E A B D (1) C B E D A (1) C B E A D (1) C B A D E (1) C A E B D (1) B D A E C (1) B C E D A (1) B C D E A (1) A D E C B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 0 2 B 0 0 0 8 4 C 8 0 0 12 10 D 0 -8 -12 0 -6 E -2 -4 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.576294 C: 0.423706 D: 0.000000 E: 0.000000 Sum of squares = 0.51164163649 Cumulative probabilities = A: 0.000000 B: 0.576294 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 0 2 B 0 0 0 8 4 C 8 0 0 12 10 D 0 -8 -12 0 -6 E -2 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=23 D=22 E=21 A=10 so A is eliminated. Round 2 votes counts: C=29 B=26 D=24 E=21 so E is eliminated. Round 3 votes counts: D=43 C=31 B=26 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:206 A:197 E:195 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 0 2 B 0 0 0 8 4 C 8 0 0 12 10 D 0 -8 -12 0 -6 E -2 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 0 2 B 0 0 0 8 4 C 8 0 0 12 10 D 0 -8 -12 0 -6 E -2 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 0 2 B 0 0 0 8 4 C 8 0 0 12 10 D 0 -8 -12 0 -6 E -2 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2309: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (18) B E D C A (8) E B D C A (7) B E D A C (7) E D C A B (6) B A E C D (6) A B C D E (6) C D A E B (5) A C D E B (5) A C D B E (5) D C E A B (4) B A C E D (4) E D B C A (3) C A D E B (3) A C B D E (3) B E A D C (2) E D C B A (1) E A D C B (1) D E C A B (1) C D E A B (1) C A D B E (1) B D C E A (1) B A E D C (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 18 12 16 B 16 0 26 28 26 C -18 -26 0 16 14 D -12 -28 -16 0 6 E -16 -26 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 18 12 16 B 16 0 26 28 26 C -18 -26 0 16 14 D -12 -28 -16 0 6 E -16 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=47 A=20 E=18 C=10 D=5 so D is eliminated. Round 2 votes counts: B=47 A=20 E=19 C=14 so C is eliminated. Round 3 votes counts: B=47 A=29 E=24 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:248 A:215 C:193 D:175 E:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 18 12 16 B 16 0 26 28 26 C -18 -26 0 16 14 D -12 -28 -16 0 6 E -16 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 18 12 16 B 16 0 26 28 26 C -18 -26 0 16 14 D -12 -28 -16 0 6 E -16 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 18 12 16 B 16 0 26 28 26 C -18 -26 0 16 14 D -12 -28 -16 0 6 E -16 -26 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2310: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) B C A E D (11) D C A B E (6) E D B A C (5) D A C B E (5) E A C B D (4) D E B C A (4) D C B A E (4) D A E C B (4) B C E A D (3) E D B C A (2) E D A B C (2) E B D C A (2) E B C A D (2) E A B C D (2) D B C A E (2) D A C E B (2) C B A E D (2) C A B E D (2) C A B D E (2) B C A D E (2) A E C B D (2) A C B D E (2) E D A C B (1) E B A C D (1) E A D C B (1) D E A B C (1) D B E C A (1) C B D A E (1) C B A D E (1) B E C A D (1) B D E C A (1) B C D A E (1) A E B C D (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 0 -10 12 B -8 0 -10 -8 2 C 0 10 0 -12 2 D 10 8 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -10 12 B -8 0 -10 -8 2 C 0 10 0 -12 2 D 10 8 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=22 B=19 A=11 C=8 so C is eliminated. Round 2 votes counts: D=40 B=23 E=22 A=15 so A is eliminated. Round 3 votes counts: D=43 B=31 E=26 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:205 C:200 E:189 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -10 12 B -8 0 -10 -8 2 C 0 10 0 -12 2 D 10 8 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -10 12 B -8 0 -10 -8 2 C 0 10 0 -12 2 D 10 8 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -10 12 B -8 0 -10 -8 2 C 0 10 0 -12 2 D 10 8 12 0 6 E -12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2311: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (8) E A D C B (6) A E B D C (6) E C D A B (5) E A D B C (5) B A E D C (5) E A B D C (4) C D E B A (4) C D B E A (4) B A E C D (4) B A D C E (4) A B E D C (4) E A B C D (3) D B A C E (3) C B D A E (3) B A C D E (3) E C A D B (2) E B A C D (2) C E D B A (2) C B D E A (2) B D C A E (2) B C D A E (2) A E D B C (2) A B D C E (2) E D C A B (1) E C D B A (1) E C B D A (1) E A C B D (1) D C B A E (1) D B C A E (1) D A C B E (1) D A B C E (1) B D A C E (1) B C A E D (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 20 14 14 B 8 0 16 2 10 C -20 -16 0 -4 -8 D -14 -2 4 0 -10 E -14 -10 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 20 14 14 B 8 0 16 2 10 C -20 -16 0 -4 -8 D -14 -2 4 0 -10 E -14 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=23 B=22 A=17 D=7 so D is eliminated. Round 2 votes counts: E=31 B=26 C=24 A=19 so A is eliminated. Round 3 votes counts: E=39 B=36 C=25 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:220 B:218 E:197 D:189 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 20 14 14 B 8 0 16 2 10 C -20 -16 0 -4 -8 D -14 -2 4 0 -10 E -14 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 20 14 14 B 8 0 16 2 10 C -20 -16 0 -4 -8 D -14 -2 4 0 -10 E -14 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 20 14 14 B 8 0 16 2 10 C -20 -16 0 -4 -8 D -14 -2 4 0 -10 E -14 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2312: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (13) A D C B E (10) C D E A B (8) A D B C E (7) D C A E B (6) C E D A B (6) B E C A D (6) D A C E B (5) B E A C D (5) B A E D C (5) B A D E C (5) E C D B A (4) E C B D A (3) A B D C E (3) B E A D C (2) A D C E B (2) E C D A B (1) E B D C A (1) E B C A D (1) D C E A B (1) D A C B E (1) C D A E B (1) C A D E B (1) B A C D E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 0 -2 B -8 0 2 -8 -6 C 4 -2 0 2 6 D 0 8 -2 0 6 E 2 6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000000004 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 0 -2 B -8 0 2 -8 -6 C 4 -2 0 2 6 D 0 8 -2 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 E=23 C=16 D=13 so D is eliminated. Round 2 votes counts: A=30 B=24 E=23 C=23 so E is eliminated. Round 3 votes counts: B=39 C=31 A=30 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:206 C:205 A:201 E:198 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 0 -2 B -8 0 2 -8 -6 C 4 -2 0 2 6 D 0 8 -2 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 0 -2 B -8 0 2 -8 -6 C 4 -2 0 2 6 D 0 8 -2 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 0 -2 B -8 0 2 -8 -6 C 4 -2 0 2 6 D 0 8 -2 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2313: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E B C D (8) D C B E A (7) D B C E A (7) A E B D C (5) E A B C D (4) C D A E B (4) C D A B E (4) C A D E B (4) B A E D C (4) A E C B D (4) A B E D C (4) C D E B A (3) A C E B D (3) E B A C D (2) D C B A E (2) D C A B E (2) D B A E C (2) C D E A B (2) C A E D B (2) B E A D C (2) E C A B D (1) E A C B D (1) E A B D C (1) D B E C A (1) D B C A E (1) D A C B E (1) C D B A E (1) B E D C A (1) B E D A C (1) B D E A C (1) B D A E C (1) B A D E C (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -6 -4 10 B -4 0 -6 -12 10 C 6 6 0 10 10 D 4 12 -10 0 12 E -10 -10 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -4 10 B -4 0 -6 -12 10 C 6 6 0 10 10 D 4 12 -10 0 12 E -10 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=26 D=23 B=11 E=9 so E is eliminated. Round 2 votes counts: C=32 A=32 D=23 B=13 so B is eliminated. Round 3 votes counts: A=41 C=32 D=27 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:209 A:202 B:194 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -4 10 B -4 0 -6 -12 10 C 6 6 0 10 10 D 4 12 -10 0 12 E -10 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -4 10 B -4 0 -6 -12 10 C 6 6 0 10 10 D 4 12 -10 0 12 E -10 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -4 10 B -4 0 -6 -12 10 C 6 6 0 10 10 D 4 12 -10 0 12 E -10 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2314: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (6) D A E B C (6) D C E B A (5) D A E C B (5) C B E A D (5) A D E B C (5) E C B D A (4) D A B C E (4) C B E D A (4) C B D E A (4) B C E A D (4) A D B E C (4) D E C B A (3) D E A C B (3) B C A E D (3) A B E C D (3) A B D C E (3) A B C E D (3) A B C D E (3) E D C B A (2) C E B D A (2) B A C E D (2) A E B D C (2) E D C A B (1) E C D B A (1) E B A C D (1) D E C A B (1) D C E A B (1) D C B A E (1) D B C A E (1) D A C E B (1) C D B E A (1) B E C A D (1) B C D A E (1) B C A D E (1) A E D B C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 8 -2 2 B -8 0 12 8 -2 C -8 -12 0 0 -2 D 2 -8 0 0 6 E -2 2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407379 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 -2 2 B -8 0 12 8 -2 C -8 -12 0 0 -2 D 2 -8 0 0 6 E -2 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407423 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=26 C=16 E=15 B=12 so B is eliminated. Round 2 votes counts: D=31 A=28 C=25 E=16 so E is eliminated. Round 3 votes counts: A=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:208 B:205 D:200 E:198 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 -2 2 B -8 0 12 8 -2 C -8 -12 0 0 -2 D 2 -8 0 0 6 E -2 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407423 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -2 2 B -8 0 12 8 -2 C -8 -12 0 0 -2 D 2 -8 0 0 6 E -2 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407423 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -2 2 B -8 0 12 8 -2 C -8 -12 0 0 -2 D 2 -8 0 0 6 E -2 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407423 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2315: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) E A D C B (7) E A C D B (6) C D B E A (6) A E D B C (6) B D A E C (5) C E A D B (4) C E A B D (4) A E C B D (4) E A C B D (3) D B C E A (3) D B A E C (3) C D E A B (3) C B D A E (3) B C D A E (3) A E B C D (3) E C A D B (2) D E A B C (2) D C B E A (2) D A E B C (2) D A B E C (2) C B D E A (2) B C A D E (2) B A D E C (2) E A D B C (1) C E D B A (1) C E D A B (1) C B E A D (1) C B A E D (1) C B A D E (1) B D C E A (1) B C A E D (1) B A E D C (1) B A C D E (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 4 0 4 2 B -4 0 -4 -4 -2 C 0 4 0 6 -2 D -4 4 -6 0 4 E -2 2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.728991 B: 0.000000 C: 0.271009 D: 0.000000 E: 0.000000 Sum of squares = 0.604873681879 Cumulative probabilities = A: 0.728991 B: 0.728991 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 4 2 B -4 0 -4 -4 -2 C 0 4 0 6 -2 D -4 4 -6 0 4 E -2 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045777 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=25 E=19 A=15 D=14 so D is eliminated. Round 2 votes counts: B=31 C=29 E=21 A=19 so A is eliminated. Round 3 votes counts: E=38 B=33 C=29 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:205 C:204 D:199 E:199 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 4 2 B -4 0 -4 -4 -2 C 0 4 0 6 -2 D -4 4 -6 0 4 E -2 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045777 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 4 2 B -4 0 -4 -4 -2 C 0 4 0 6 -2 D -4 4 -6 0 4 E -2 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045777 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 4 2 B -4 0 -4 -4 -2 C 0 4 0 6 -2 D -4 4 -6 0 4 E -2 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045777 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2316: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (9) A D E C B (9) E D A B C (8) A D C E B (8) B C E D A (7) B C D E A (7) E B C D A (6) E A D C B (6) E D B A C (4) A D C B E (4) A C D B E (4) E B D C A (3) D A C B E (3) C B A D E (3) E D A C B (2) E A D B C (2) B C A D E (2) A C B D E (2) E D B C A (1) E B D A C (1) D E B C A (1) D A C E B (1) C D B A E (1) C B D A E (1) C A D B E (1) B E D C A (1) B E C D A (1) B C A E D (1) A E D C B (1) Total count = 100 A B C D E A 0 2 10 -14 0 B -2 0 8 -12 -6 C -10 -8 0 -10 8 D 14 12 10 0 12 E 0 6 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 -14 0 B -2 0 8 -12 -6 C -10 -8 0 -10 8 D 14 12 10 0 12 E 0 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=28 A=28 C=6 D=5 so D is eliminated. Round 2 votes counts: E=34 A=32 B=28 C=6 so C is eliminated. Round 3 votes counts: E=34 B=33 A=33 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:224 A:199 B:194 E:193 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 10 -14 0 B -2 0 8 -12 -6 C -10 -8 0 -10 8 D 14 12 10 0 12 E 0 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -14 0 B -2 0 8 -12 -6 C -10 -8 0 -10 8 D 14 12 10 0 12 E 0 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -14 0 B -2 0 8 -12 -6 C -10 -8 0 -10 8 D 14 12 10 0 12 E 0 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2317: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (11) C B E D A (9) C E B D A (8) C B D A E (7) A D E B C (7) B C E A D (5) B C A D E (5) A D B E C (5) E C B D A (3) E B C A D (3) E A D B C (3) D A E C B (3) E C D A B (2) E B A D C (2) D A C B E (2) C E D A B (2) C D B A E (2) C D A B E (2) B C E D A (2) A B D C E (2) E C D B A (1) E B A C D (1) D A C E B (1) C E D B A (1) C D E A B (1) C B D E A (1) B E C A D (1) B E A D C (1) B C D A E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E D B C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -12 -18 -14 B 12 0 -14 8 0 C 12 14 0 16 10 D 18 -8 -16 0 -12 E 14 0 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -18 -14 B 12 0 -14 8 0 C 12 14 0 16 10 D 18 -8 -16 0 -12 E 14 0 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=26 B=18 A=17 D=6 so D is eliminated. Round 2 votes counts: C=33 E=26 A=23 B=18 so B is eliminated. Round 3 votes counts: C=46 E=28 A=26 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:208 B:203 D:191 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -12 -18 -14 B 12 0 -14 8 0 C 12 14 0 16 10 D 18 -8 -16 0 -12 E 14 0 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -18 -14 B 12 0 -14 8 0 C 12 14 0 16 10 D 18 -8 -16 0 -12 E 14 0 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -18 -14 B 12 0 -14 8 0 C 12 14 0 16 10 D 18 -8 -16 0 -12 E 14 0 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2318: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) B E A D C (7) A B C E D (6) E B D C A (5) E B D A C (5) D E C B A (5) D C E B A (5) D C A E B (5) C D E B A (4) C D A E B (4) A B E C D (4) E B C D A (3) D C E A B (3) C A D B E (3) A C D B E (3) A B C D E (3) E D B C A (2) D A E C B (2) D A C B E (2) B A E C D (2) A C B D E (2) A B E D C (2) E D C B A (1) E D B A C (1) E B A D C (1) E B A C D (1) D E B A C (1) D C A B E (1) D A C E B (1) C D E A B (1) C D A B E (1) B E C A D (1) A D C B E (1) A D B E C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 12 -4 -10 B 6 0 10 6 0 C -12 -10 0 -4 -6 D 4 -6 4 0 -2 E 10 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.381442 C: 0.000000 D: 0.000000 E: 0.618558 Sum of squares = 0.52811222784 Cumulative probabilities = A: 0.000000 B: 0.381442 C: 0.381442 D: 0.381442 E: 1.000000 A B C D E A 0 -6 12 -4 -10 B 6 0 10 6 0 C -12 -10 0 -4 -6 D 4 -6 4 0 -2 E 10 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=24 E=19 B=19 C=13 so C is eliminated. Round 2 votes counts: D=35 A=27 E=19 B=19 so E is eliminated. Round 3 votes counts: D=39 B=34 A=27 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:209 D:200 A:196 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 -4 -10 B 6 0 10 6 0 C -12 -10 0 -4 -6 D 4 -6 4 0 -2 E 10 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -4 -10 B 6 0 10 6 0 C -12 -10 0 -4 -6 D 4 -6 4 0 -2 E 10 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -4 -10 B 6 0 10 6 0 C -12 -10 0 -4 -6 D 4 -6 4 0 -2 E 10 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2319: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) C E B D A (9) D B C A E (6) C E B A D (6) E A D C B (5) D B C E A (5) D A B E C (4) E C A D B (3) D B A C E (3) C B E A D (3) C B D E A (3) B C A D E (3) A E D B C (3) A D E B C (3) A D B E C (3) E C D B A (2) E C B A D (2) E A C D B (2) E A C B D (2) D E A B C (2) C E A B D (2) B D C A E (2) B C D A E (2) B C A E D (2) A E C B D (2) E D C A B (1) E D A C B (1) D E C B A (1) D B E C A (1) D B A E C (1) D A E B C (1) C B E D A (1) B D A C E (1) B C D E A (1) B A C D E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -28 6 -22 B 10 0 -8 6 -14 C 28 8 0 14 0 D -6 -6 -14 0 -14 E 22 14 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.590959 D: 0.000000 E: 0.409041 Sum of squares = 0.516547079647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.590959 D: 0.590959 E: 1.000000 A B C D E A 0 -10 -28 6 -22 B 10 0 -8 6 -14 C 28 8 0 14 0 D -6 -6 -14 0 -14 E 22 14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 C=24 A=13 B=12 so B is eliminated. Round 2 votes counts: C=32 E=27 D=27 A=14 so A is eliminated. Round 3 votes counts: E=34 D=33 C=33 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:225 B:197 D:180 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -28 6 -22 B 10 0 -8 6 -14 C 28 8 0 14 0 D -6 -6 -14 0 -14 E 22 14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -28 6 -22 B 10 0 -8 6 -14 C 28 8 0 14 0 D -6 -6 -14 0 -14 E 22 14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -28 6 -22 B 10 0 -8 6 -14 C 28 8 0 14 0 D -6 -6 -14 0 -14 E 22 14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2320: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (20) B A C E D (10) E C D B A (9) D E C B A (6) D A B E C (6) D A E C B (4) C E B D A (4) B C E A D (4) C E B A D (3) A D B E C (3) A B D C E (3) A B C E D (3) E C D A B (2) D E A C B (2) C E D A B (2) C B E A D (2) B A D E C (2) A D E C B (2) E D C B A (1) E C B D A (1) D E B C A (1) D E A B C (1) D B E A C (1) D B A E C (1) C E A D B (1) B C A E D (1) B A D C E (1) B A C D E (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -14 -22 -20 B -4 0 -20 -26 -18 C 14 20 0 -10 -24 D 22 26 10 0 12 E 20 18 24 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 -22 -20 B -4 0 -20 -26 -18 C 14 20 0 -10 -24 D 22 26 10 0 12 E 20 18 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 B=19 A=14 E=13 C=12 so C is eliminated. Round 2 votes counts: D=42 E=23 B=21 A=14 so A is eliminated. Round 3 votes counts: D=48 B=29 E=23 so E is eliminated. Round 4 votes counts: D=63 B=37 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:235 E:225 C:200 A:174 B:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -14 -22 -20 B -4 0 -20 -26 -18 C 14 20 0 -10 -24 D 22 26 10 0 12 E 20 18 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 -22 -20 B -4 0 -20 -26 -18 C 14 20 0 -10 -24 D 22 26 10 0 12 E 20 18 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 -22 -20 B -4 0 -20 -26 -18 C 14 20 0 -10 -24 D 22 26 10 0 12 E 20 18 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2321: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (8) A D E C B (5) E B D A C (4) B D E C A (4) B D C A E (4) A E C D B (4) A C E D B (4) E B A D C (3) D B C A E (3) C E B A D (3) C E A B D (3) C A E D B (3) C A D B E (3) B E D C A (3) B D C E A (3) B C D E A (3) A E D B C (3) E C A B D (2) E B A C D (2) E A C D B (2) E A B C D (2) D C B A E (2) D B E A C (2) D A B E C (2) D A B C E (2) B E D A C (2) B C E D A (2) E C B A D (1) E B D C A (1) E B C D A (1) E B C A D (1) E A D B C (1) E A C B D (1) E A B D C (1) D C A B E (1) D B A C E (1) D A C B E (1) C E B D A (1) C D B A E (1) C D A B E (1) A E D C B (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 8 -4 -12 B 10 0 20 10 -2 C -8 -20 0 -20 -16 D 4 -10 20 0 -2 E 12 2 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 8 -4 -12 B 10 0 20 10 -2 C -8 -20 0 -20 -16 D 4 -10 20 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=22 A=20 C=15 D=14 so D is eliminated. Round 2 votes counts: B=35 A=25 E=22 C=18 so C is eliminated. Round 3 votes counts: B=38 A=33 E=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:216 D:206 A:191 C:168 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 8 -4 -12 B 10 0 20 10 -2 C -8 -20 0 -20 -16 D 4 -10 20 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -4 -12 B 10 0 20 10 -2 C -8 -20 0 -20 -16 D 4 -10 20 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -4 -12 B 10 0 20 10 -2 C -8 -20 0 -20 -16 D 4 -10 20 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2322: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C B A (9) D E A B C (8) C B A E D (6) A C B E D (5) D A E B C (4) E D C B A (3) D E B C A (3) D E A C B (3) B C E D A (3) B C A E D (3) A D E B C (3) A D C E B (3) A C B D E (3) E D B C A (2) E C B D A (2) E B D C A (2) D E C A B (2) D E B A C (2) D A E C B (2) C E D B A (2) C E B D A (2) C B E A D (2) B C E A D (2) A D E C B (2) A D B E C (2) A B D C E (2) A B C D E (2) E C D B A (1) D C E B A (1) C B E D A (1) C A B E D (1) B E C D A (1) A D C B E (1) Total count = 100 A B C D E A 0 6 4 -6 -2 B -6 0 -2 -6 -12 C -4 2 0 -8 -2 D 6 6 8 0 4 E 2 12 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -6 -2 B -6 0 -2 -6 -12 C -4 2 0 -8 -2 D 6 6 8 0 4 E 2 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=33 C=14 E=10 B=9 so B is eliminated. Round 2 votes counts: D=34 A=33 C=22 E=11 so E is eliminated. Round 3 votes counts: D=41 A=33 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:206 A:201 C:194 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -6 -2 B -6 0 -2 -6 -12 C -4 2 0 -8 -2 D 6 6 8 0 4 E 2 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -6 -2 B -6 0 -2 -6 -12 C -4 2 0 -8 -2 D 6 6 8 0 4 E 2 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -6 -2 B -6 0 -2 -6 -12 C -4 2 0 -8 -2 D 6 6 8 0 4 E 2 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2323: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (12) C A B D E (10) D E C A B (8) E D C B A (5) E D C A B (5) B A C D E (5) A C B D E (5) D E A B C (4) C D E A B (4) B A E D C (4) D C E A B (3) C E D A B (3) C D A E B (3) C A D B E (3) A B C D E (3) E B A D C (2) D E A C B (2) C E B D A (2) B A C E D (2) E D B C A (1) E D A B C (1) E C B D A (1) E B C D A (1) E B A C D (1) D E B A C (1) D C A E B (1) D A B E C (1) C E D B A (1) C D A B E (1) C B A D E (1) B E A C D (1) B A E C D (1) B A D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -6 -20 -16 B -16 0 -16 -18 -22 C 6 16 0 -4 -4 D 20 18 4 0 14 E 16 22 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -6 -20 -16 B -16 0 -16 -18 -22 C 6 16 0 -4 -4 D 20 18 4 0 14 E 16 22 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=28 D=20 B=14 A=9 so A is eliminated. Round 2 votes counts: C=33 E=29 D=20 B=18 so B is eliminated. Round 3 votes counts: C=43 E=35 D=22 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:228 E:214 C:207 A:187 B:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -6 -20 -16 B -16 0 -16 -18 -22 C 6 16 0 -4 -4 D 20 18 4 0 14 E 16 22 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 -20 -16 B -16 0 -16 -18 -22 C 6 16 0 -4 -4 D 20 18 4 0 14 E 16 22 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 -20 -16 B -16 0 -16 -18 -22 C 6 16 0 -4 -4 D 20 18 4 0 14 E 16 22 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2324: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) A B E D C (8) D A C E B (7) B E A C D (7) A D E B C (6) C D B E A (5) B E C D A (5) E D B C A (4) C D E B A (4) D C E B A (3) C B E D A (3) B A E C D (3) A E B D C (3) A D C E B (3) A D B E C (3) A B C E D (3) E B C D A (2) A E D B C (2) A D C B E (2) E D A B C (1) E C D B A (1) E B D C A (1) E B C A D (1) E B A D C (1) E A D B C (1) D E C B A (1) D E C A B (1) D C E A B (1) D A E C B (1) C E D B A (1) C D B A E (1) C D A B E (1) B E C A D (1) B E A D C (1) B C A E D (1) B A C E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 -2 10 B -6 0 12 -14 -6 C -10 -12 0 -16 -10 D 2 14 16 0 -4 E -10 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.468750000023 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 A B C D E A 0 6 10 -2 10 B -6 0 12 -14 -6 C -10 -12 0 -16 -10 D 2 14 16 0 -4 E -10 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.46874999986 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=22 B=19 C=15 E=12 so E is eliminated. Round 2 votes counts: A=33 D=27 B=24 C=16 so C is eliminated. Round 3 votes counts: D=40 A=33 B=27 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:212 E:205 B:193 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 -2 10 B -6 0 12 -14 -6 C -10 -12 0 -16 -10 D 2 14 16 0 -4 E -10 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.46874999986 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -2 10 B -6 0 12 -14 -6 C -10 -12 0 -16 -10 D 2 14 16 0 -4 E -10 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.46874999986 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -2 10 B -6 0 12 -14 -6 C -10 -12 0 -16 -10 D 2 14 16 0 -4 E -10 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.46874999986 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2325: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C E B D (10) D E B C A (8) D B E C A (8) B D E A C (7) A B C D E (7) C A E D B (6) A B D C E (5) C E A D B (4) A C B E D (4) E D C B A (3) E C D B A (3) B D A E C (3) B A D E C (3) A C B D E (3) C A D E B (2) A B D E C (2) E C D A B (1) E C A D B (1) E B D A C (1) D B C E A (1) C E D A B (1) C D E B A (1) C D A B E (1) B A E D C (1) B A D C E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 2 0 B 2 0 20 -2 -4 C 2 -20 0 -10 -6 D -2 2 10 0 4 E 0 4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.491604 B: 0.175063 C: 0.000000 D: 0.175063 E: 0.158271 Sum of squares = 0.328018019438 Cumulative probabilities = A: 0.491604 B: 0.666667 C: 0.666667 D: 0.841729 E: 1.000000 A B C D E A 0 -2 -2 2 0 B 2 0 20 -2 -4 C 2 -20 0 -10 -6 D -2 2 10 0 4 E 0 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.083333 Sum of squares = 0.305555555561 Cumulative probabilities = A: 0.416667 B: 0.666667 C: 0.666667 D: 0.916667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=20 D=17 C=15 B=15 so C is eliminated. Round 2 votes counts: A=41 E=25 D=19 B=15 so B is eliminated. Round 3 votes counts: A=46 D=29 E=25 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:208 D:207 E:203 A:199 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 2 0 B 2 0 20 -2 -4 C 2 -20 0 -10 -6 D -2 2 10 0 4 E 0 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.083333 Sum of squares = 0.305555555561 Cumulative probabilities = A: 0.416667 B: 0.666667 C: 0.666667 D: 0.916667 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 0 B 2 0 20 -2 -4 C 2 -20 0 -10 -6 D -2 2 10 0 4 E 0 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.083333 Sum of squares = 0.305555555561 Cumulative probabilities = A: 0.416667 B: 0.666667 C: 0.666667 D: 0.916667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 0 B 2 0 20 -2 -4 C 2 -20 0 -10 -6 D -2 2 10 0 4 E 0 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.083333 Sum of squares = 0.305555555561 Cumulative probabilities = A: 0.416667 B: 0.666667 C: 0.666667 D: 0.916667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2326: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (12) C D E B A (10) B A E D C (10) B A E C D (9) C D B E A (7) A E B D C (5) C D E A B (4) D E C A B (3) C B D E A (3) C B D A E (3) B C D A E (3) B A C E D (3) E A D C B (2) D E A C B (2) D C B A E (2) C B E D A (2) B C A E D (2) B C A D E (2) E D A C B (1) E C D A B (1) E B A C D (1) E A B D C (1) E A B C D (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D A B (1) C E B D A (1) B E A C D (1) B C E A D (1) B A C D E (1) A E D B C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -22 -14 -16 -8 B 22 0 -12 2 0 C 14 12 0 14 16 D 16 -2 -14 0 10 E 8 0 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -14 -16 -8 B 22 0 -12 2 0 C 14 12 0 14 16 D 16 -2 -14 0 10 E 8 0 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=31 D=22 A=8 E=7 so E is eliminated. Round 2 votes counts: B=33 C=32 D=23 A=12 so A is eliminated. Round 3 votes counts: B=42 C=32 D=26 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:206 D:205 E:191 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -14 -16 -8 B 22 0 -12 2 0 C 14 12 0 14 16 D 16 -2 -14 0 10 E 8 0 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 -16 -8 B 22 0 -12 2 0 C 14 12 0 14 16 D 16 -2 -14 0 10 E 8 0 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 -16 -8 B 22 0 -12 2 0 C 14 12 0 14 16 D 16 -2 -14 0 10 E 8 0 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2327: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) E A C D B (9) C B A D E (6) C A E B D (6) C A B E D (6) A E C B D (5) A C E B D (5) D B E A C (4) B D C A E (4) B C D A E (4) E D C A B (3) E A D C B (3) C A E D B (3) B D C E A (3) B C A D E (3) E D A C B (2) E A D B C (2) C A B D E (2) B A C E D (2) A C B E D (2) E D B A C (1) E D A B C (1) E A B C D (1) D E C B A (1) D E B C A (1) C E A D B (1) C B D A E (1) B D E C A (1) B D E A C (1) B D A C E (1) B C A E D (1) B A D C E (1) B A C D E (1) A E C D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -12 24 12 B -8 0 -12 16 10 C 12 12 0 22 6 D -24 -16 -22 0 -12 E -12 -10 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 24 12 B -8 0 -12 16 10 C 12 12 0 22 6 D -24 -16 -22 0 -12 E -12 -10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=22 B=22 D=16 A=15 so A is eliminated. Round 2 votes counts: C=32 E=28 B=24 D=16 so D is eliminated. Round 3 votes counts: B=38 C=32 E=30 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:216 B:203 E:192 D:163 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 24 12 B -8 0 -12 16 10 C 12 12 0 22 6 D -24 -16 -22 0 -12 E -12 -10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 24 12 B -8 0 -12 16 10 C 12 12 0 22 6 D -24 -16 -22 0 -12 E -12 -10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 24 12 B -8 0 -12 16 10 C 12 12 0 22 6 D -24 -16 -22 0 -12 E -12 -10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2328: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) C B E A D (8) C B A E D (7) E D A B C (5) D A E B C (5) C D A B E (5) C B A D E (5) B A E C D (5) E D C B A (4) E D B A C (3) D A C B E (3) C D E B A (3) A B E D C (3) E B D A C (2) E B C A D (2) E B A D C (2) D E A C B (2) D C A B E (2) C E B A D (2) C A B D E (2) A B C E D (2) A B C D E (2) E D B C A (1) E C D B A (1) E C B D A (1) E B A C D (1) D E B A C (1) D C A E B (1) D A E C B (1) C E D B A (1) C D E A B (1) C D B A E (1) B E A C D (1) B C A E D (1) B A C E D (1) A D E B C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 4 -6 -2 B 6 0 -2 -4 0 C -4 2 0 6 -2 D 6 4 -6 0 -6 E 2 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.339999 C: 0.000000 D: 0.000000 E: 0.660001 Sum of squares = 0.551200347157 Cumulative probabilities = A: 0.000000 B: 0.339999 C: 0.339999 D: 0.339999 E: 1.000000 A B C D E A 0 -6 4 -6 -2 B 6 0 -2 -4 0 C -4 2 0 6 -2 D 6 4 -6 0 -6 E 2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000025227 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=25 E=22 A=10 B=8 so B is eliminated. Round 2 votes counts: C=36 D=25 E=23 A=16 so A is eliminated. Round 3 votes counts: C=42 E=31 D=27 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:205 C:201 B:200 D:199 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 -6 -2 B 6 0 -2 -4 0 C -4 2 0 6 -2 D 6 4 -6 0 -6 E 2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000025227 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -6 -2 B 6 0 -2 -4 0 C -4 2 0 6 -2 D 6 4 -6 0 -6 E 2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000025227 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -6 -2 B 6 0 -2 -4 0 C -4 2 0 6 -2 D 6 4 -6 0 -6 E 2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499888 C: 0.000000 D: 0.000000 E: 0.500112 Sum of squares = 0.500000025227 Cumulative probabilities = A: 0.000000 B: 0.499888 C: 0.499888 D: 0.499888 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2329: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) E A D B C (7) D A E B C (6) D A B E C (6) B A E D C (6) C B E A D (5) C D B A E (4) C B D A E (4) D C A E B (3) D A E C B (3) D A B C E (3) C E D A B (3) B E A D C (3) B D A E C (3) B C E A D (3) B C D A E (3) E A C D B (2) E A B D C (2) C B E D A (2) A B D E C (2) E D C A B (1) E C D A B (1) E C B A D (1) E B C A D (1) E B A D C (1) E A C B D (1) D C E A B (1) D A C E B (1) C E D B A (1) C E B D A (1) C D A B E (1) C B D E A (1) B E C A D (1) B D A C E (1) B C A D E (1) B A D E C (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 2 2 B 8 0 6 10 4 C -2 -6 0 -6 -2 D -2 -10 6 0 -8 E -2 -4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 2 2 B 8 0 6 10 4 C -2 -6 0 -6 -2 D -2 -10 6 0 -8 E -2 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=23 B=22 E=17 A=5 so A is eliminated. Round 2 votes counts: C=33 D=25 B=25 E=17 so E is eliminated. Round 3 votes counts: C=38 D=33 B=29 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:214 E:202 A:199 D:193 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 2 2 B 8 0 6 10 4 C -2 -6 0 -6 -2 D -2 -10 6 0 -8 E -2 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 2 2 B 8 0 6 10 4 C -2 -6 0 -6 -2 D -2 -10 6 0 -8 E -2 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 2 2 B 8 0 6 10 4 C -2 -6 0 -6 -2 D -2 -10 6 0 -8 E -2 -4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2330: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) A D C E B (8) B C E A D (6) D E B A C (4) D E A B C (4) B E C D A (4) A B D E C (4) A B C D E (4) E D C B A (3) C E D B A (3) C B E D A (3) B E D A C (3) B A D E C (3) A D E C B (3) A D B E C (3) A C D E B (3) E D B C A (2) D E A C B (2) C A D E B (2) B E D C A (2) B A C E D (2) A D E B C (2) E D C A B (1) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E B C A (1) D A E C B (1) D A E B C (1) C E D A B (1) C E B D A (1) C B A E D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C A D (1) B C A E D (1) B A E D C (1) A D C B E (1) A D B C E (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 6 4 -6 B -10 0 4 -12 -8 C -6 -4 0 -20 -14 D -4 12 20 0 18 E 6 8 14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.214286 E: 0.142857 Sum of squares = 0.479591836758 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.857143 E: 1.000000 A B C D E A 0 10 6 4 -6 B -10 0 4 -12 -8 C -6 -4 0 -20 -14 D -4 12 20 0 18 E 6 8 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.214286 E: 0.142857 Sum of squares = 0.479591836679 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=23 D=21 C=14 E=10 so E is eliminated. Round 2 votes counts: A=32 D=27 B=25 C=16 so C is eliminated. Round 3 votes counts: A=37 D=32 B=31 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:223 A:207 E:205 B:187 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 4 -6 B -10 0 4 -12 -8 C -6 -4 0 -20 -14 D -4 12 20 0 18 E 6 8 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.214286 E: 0.142857 Sum of squares = 0.479591836679 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.857143 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 -6 B -10 0 4 -12 -8 C -6 -4 0 -20 -14 D -4 12 20 0 18 E 6 8 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.214286 E: 0.142857 Sum of squares = 0.479591836679 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 -6 B -10 0 4 -12 -8 C -6 -4 0 -20 -14 D -4 12 20 0 18 E 6 8 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.000000 C: 0.000000 D: 0.214286 E: 0.142857 Sum of squares = 0.479591836679 Cumulative probabilities = A: 0.642857 B: 0.642857 C: 0.642857 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2331: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) A E D C B (7) B C D E A (6) C A E D B (5) B C A E D (5) E D A C B (4) D E C A B (4) D E A B C (4) B C A D E (4) A C E D B (4) E D A B C (3) D E B C A (3) D E B A C (3) D E A C B (3) C B D A E (3) B D E C A (3) B D C E A (3) B C D A E (3) D C E B A (2) C D A E B (2) C A E B D (2) C A B E D (2) B E D A C (2) B D E A C (2) A E C D B (2) A E C B D (2) E A D C B (1) D C B E A (1) D B E C A (1) D B E A C (1) B A E C D (1) B A C E D (1) A E B C D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -14 -6 8 B 4 0 -8 0 -6 C 14 8 0 6 6 D 6 0 -6 0 -4 E -8 6 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -6 8 B 4 0 -8 0 -6 C 14 8 0 6 6 D 6 0 -6 0 -4 E -8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=22 C=22 A=18 E=8 so E is eliminated. Round 2 votes counts: B=30 D=29 C=22 A=19 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:198 E:198 B:195 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -6 8 B 4 0 -8 0 -6 C 14 8 0 6 6 D 6 0 -6 0 -4 E -8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -6 8 B 4 0 -8 0 -6 C 14 8 0 6 6 D 6 0 -6 0 -4 E -8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -6 8 B 4 0 -8 0 -6 C 14 8 0 6 6 D 6 0 -6 0 -4 E -8 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2332: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (12) C B E D A (8) A D E C B (7) A D E B C (7) D A C E B (6) A D C B E (6) E D B C A (5) E B C A D (5) A C B D E (5) B C E A D (4) D A E C B (3) D A C B E (3) A C B E D (3) A B C E D (3) E A B D C (2) B E C A D (2) E D B A C (1) E D A B C (1) E B D C A (1) E B A C D (1) E A B C D (1) D E C B A (1) D E B C A (1) D E A C B (1) D E A B C (1) D C E B A (1) D C B E A (1) D C A E B (1) D C A B E (1) C B E A D (1) C B D A E (1) C A B D E (1) A D C E B (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 8 2 0 B -10 0 -2 4 -18 C -8 2 0 -2 -6 D -2 -4 2 0 -2 E 0 18 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.779678 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.220322 Sum of squares = 0.656439056974 Cumulative probabilities = A: 0.779678 B: 0.779678 C: 0.779678 D: 0.779678 E: 1.000000 A B C D E A 0 10 8 2 0 B -10 0 -2 4 -18 C -8 2 0 -2 -6 D -2 -4 2 0 -2 E 0 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997699 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=29 D=20 C=11 B=6 so B is eliminated. Round 2 votes counts: A=34 E=31 D=20 C=15 so C is eliminated. Round 3 votes counts: E=44 A=35 D=21 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:213 A:210 D:197 C:193 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 2 0 B -10 0 -2 4 -18 C -8 2 0 -2 -6 D -2 -4 2 0 -2 E 0 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997699 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 2 0 B -10 0 -2 4 -18 C -8 2 0 -2 -6 D -2 -4 2 0 -2 E 0 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997699 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 2 0 B -10 0 -2 4 -18 C -8 2 0 -2 -6 D -2 -4 2 0 -2 E 0 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997699 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2333: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (14) E A B D C (13) C D B A E (13) D C B E A (9) D E A B C (5) D C B A E (5) C D A E B (4) B E A D C (4) C A E B D (3) C A B E D (3) E A B C D (2) D B E C A (2) C D E A B (2) C B D A E (2) C B A D E (2) B A E C D (2) A B C E D (2) E B A D C (1) E A D B C (1) D E C B A (1) D E C A B (1) D E B A C (1) D C E B A (1) D B E A C (1) C E A D B (1) C D A B E (1) C B A E D (1) B E D A C (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -4 0 8 B -6 0 0 4 -2 C 4 0 0 8 2 D 0 -4 -8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.185690 C: 0.814310 D: 0.000000 E: 0.000000 Sum of squares = 0.697581507704 Cumulative probabilities = A: 0.000000 B: 0.185690 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 0 8 B -6 0 0 4 -2 C 4 0 0 8 2 D 0 -4 -8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000008102 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=26 E=17 A=17 B=8 so B is eliminated. Round 2 votes counts: C=33 D=26 E=22 A=19 so A is eliminated. Round 3 votes counts: E=38 C=36 D=26 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:207 A:205 B:198 D:195 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 0 8 B -6 0 0 4 -2 C 4 0 0 8 2 D 0 -4 -8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000008102 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 8 B -6 0 0 4 -2 C 4 0 0 8 2 D 0 -4 -8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000008102 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 8 B -6 0 0 4 -2 C 4 0 0 8 2 D 0 -4 -8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000008102 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2334: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C B A (9) E D B A C (6) B A C E D (6) A B D E C (5) B A E C D (4) A B E D C (4) E B D C A (3) D E A B C (3) D C E B A (3) D A E C B (3) C B A E D (3) A B E C D (3) E D B C A (2) E B D A C (2) D E C A B (2) D C E A B (2) C E D B A (2) C D E B A (2) C B E A D (2) B A E D C (2) A D E B C (2) A C D B E (2) A B C D E (2) E D C B A (1) E C D B A (1) E C B D A (1) D E A C B (1) D C A E B (1) C B E D A (1) C B A D E (1) C A D B E (1) B E D C A (1) B E C D A (1) B E C A D (1) B C E A D (1) B C A E D (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 16 6 6 B 12 0 20 14 8 C -16 -20 0 -6 -14 D -6 -14 6 0 -18 E -6 -8 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 16 6 6 B 12 0 20 14 8 C -16 -20 0 -6 -14 D -6 -14 6 0 -18 E -6 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=24 B=17 E=16 C=12 so C is eliminated. Round 2 votes counts: A=32 D=26 B=24 E=18 so E is eliminated. Round 3 votes counts: D=38 A=32 B=30 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:227 E:209 A:208 D:184 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 16 6 6 B 12 0 20 14 8 C -16 -20 0 -6 -14 D -6 -14 6 0 -18 E -6 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 16 6 6 B 12 0 20 14 8 C -16 -20 0 -6 -14 D -6 -14 6 0 -18 E -6 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 16 6 6 B 12 0 20 14 8 C -16 -20 0 -6 -14 D -6 -14 6 0 -18 E -6 -8 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2335: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) A B E D C (12) B C D E A (6) E D C A B (5) A E D C B (5) C D B E A (4) B A C D E (4) A E B D C (4) E D C B A (3) E C D A B (3) D E C B A (3) B D C E A (3) B C D A E (3) A B C D E (3) E D B C A (2) E D A C B (2) E A D C B (2) C E D A B (2) C D E A B (2) C B D E A (2) B D E C A (2) B C A D E (2) B A E D C (2) A E C D B (2) D C E B A (1) C A D E B (1) C A D B E (1) B A D E C (1) B A D C E (1) A E D B C (1) A C E D B (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -14 -10 -8 B 2 0 -4 -4 -2 C 14 4 0 2 0 D 10 4 -2 0 4 E 8 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.841266 D: 0.000000 E: 0.158734 Sum of squares = 0.732924440065 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.841266 D: 0.841266 E: 1.000000 A B C D E A 0 -2 -14 -10 -8 B 2 0 -4 -4 -2 C 14 4 0 2 0 D 10 4 -2 0 4 E 8 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555934303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=24 B=24 E=17 D=4 so D is eliminated. Round 2 votes counts: A=31 C=25 B=24 E=20 so E is eliminated. Round 3 votes counts: C=39 A=35 B=26 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:208 E:203 B:196 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 -10 -8 B 2 0 -4 -4 -2 C 14 4 0 2 0 D 10 4 -2 0 4 E 8 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555934303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -10 -8 B 2 0 -4 -4 -2 C 14 4 0 2 0 D 10 4 -2 0 4 E 8 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555934303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -10 -8 B 2 0 -4 -4 -2 C 14 4 0 2 0 D 10 4 -2 0 4 E 8 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555934303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2336: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) A C B D E (8) E D C B A (5) E C D A B (4) E C A D B (4) E B D A C (4) D C B A E (4) B D C A E (4) A C E B D (4) E D C A B (3) E D B C A (3) D B C E A (3) C A D E B (3) C A D B E (3) B D E C A (3) B D E A C (3) B A C D E (3) E A C B D (2) D E B C A (2) C D E A B (2) C D A B E (2) C A B D E (2) B E D A C (2) B D A C E (2) A C E D B (2) E D B A C (1) E B D C A (1) E B A D C (1) D C E B A (1) C E D A B (1) C B A D E (1) B C A D E (1) B A E C D (1) B A D E C (1) B A D C E (1) A E C B D (1) A E B C D (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -4 0 -6 B -6 0 -24 -4 -6 C 4 24 0 12 0 D 0 4 -12 0 2 E 6 6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.585907 D: 0.000000 E: 0.414093 Sum of squares = 0.514760047838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.585907 D: 0.585907 E: 1.000000 A B C D E A 0 6 -4 0 -6 B -6 0 -24 -4 -6 C 4 24 0 12 0 D 0 4 -12 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=21 A=19 C=14 D=10 so D is eliminated. Round 2 votes counts: E=38 B=24 C=19 A=19 so C is eliminated. Round 3 votes counts: E=42 B=29 A=29 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:220 E:205 A:198 D:197 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 0 -6 B -6 0 -24 -4 -6 C 4 24 0 12 0 D 0 4 -12 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 -6 B -6 0 -24 -4 -6 C 4 24 0 12 0 D 0 4 -12 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 -6 B -6 0 -24 -4 -6 C 4 24 0 12 0 D 0 4 -12 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2337: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) A B D E C (8) E C A B D (7) C E D B A (6) E A B D C (5) E A D B C (4) D B A E C (4) C D B A E (4) C B D A E (4) B D A C E (4) B C D A E (4) D B C A E (3) D B A C E (3) C D E B A (3) C D B E A (3) E A C B D (2) D C B E A (2) C E D A B (2) C E A B D (2) B D C A E (2) B D A E C (2) B A D C E (2) A D B E C (2) E C D A B (1) E A B C D (1) D E C B A (1) D E A B C (1) D C B A E (1) C E B A D (1) C E A D B (1) C A B E D (1) B A D E C (1) A E D B C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 4 0 16 B 0 0 18 16 2 C -4 -18 0 -14 -4 D 0 -16 14 0 8 E -16 -2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.497529 B: 0.502471 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500012211704 Cumulative probabilities = A: 0.497529 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 0 16 B 0 0 18 16 2 C -4 -18 0 -14 -4 D 0 -16 14 0 8 E -16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=23 E=20 D=15 B=15 so D is eliminated. Round 2 votes counts: C=30 B=25 A=23 E=22 so E is eliminated. Round 3 votes counts: C=39 A=36 B=25 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:218 A:210 D:203 E:189 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 16 B 0 0 18 16 2 C -4 -18 0 -14 -4 D 0 -16 14 0 8 E -16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 16 B 0 0 18 16 2 C -4 -18 0 -14 -4 D 0 -16 14 0 8 E -16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 16 B 0 0 18 16 2 C -4 -18 0 -14 -4 D 0 -16 14 0 8 E -16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2338: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) D E B C A (7) E C D A B (6) D B E A C (6) B A D C E (6) A C B D E (6) E D B C A (5) D E B A C (5) B D E A C (5) A B C D E (5) E D C A B (4) C E A D B (4) C A E D B (4) C A B E D (4) B D A E C (4) C E D A B (3) E D B A C (2) D E C A B (2) D B A E C (2) A B D C E (2) E C A B D (1) E B C A D (1) D A C B E (1) C A E B D (1) C A D E B (1) B E D A C (1) B D A C E (1) B A D E C (1) B A C D E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 0 -22 -18 B 8 0 8 -20 -6 C 0 -8 0 -22 -18 D 22 20 22 0 12 E 18 6 18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -22 -18 B 8 0 8 -20 -6 C 0 -8 0 -22 -18 D 22 20 22 0 12 E 18 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 B=19 C=17 A=15 so A is eliminated. Round 2 votes counts: E=26 B=26 C=25 D=23 so D is eliminated. Round 3 votes counts: E=40 B=34 C=26 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:238 E:215 B:195 A:176 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -22 -18 B 8 0 8 -20 -6 C 0 -8 0 -22 -18 D 22 20 22 0 12 E 18 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -22 -18 B 8 0 8 -20 -6 C 0 -8 0 -22 -18 D 22 20 22 0 12 E 18 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -22 -18 B 8 0 8 -20 -6 C 0 -8 0 -22 -18 D 22 20 22 0 12 E 18 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2339: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) C D A E B (7) D C E B A (6) E B D C A (5) D E C B A (5) C D E A B (5) C A D B E (5) A B E C D (5) E D C B A (4) E D B C A (4) D C A B E (4) A C B D E (4) D C E A B (3) B A E D C (3) A B D C E (3) E C B D A (2) E B A D C (2) D E B C A (2) D A B C E (2) C A E B D (2) B E D A C (2) B A E C D (2) A C D B E (2) E B C D A (1) E B A C D (1) D B E C A (1) D B E A C (1) C D E B A (1) C D A B E (1) C A D E B (1) B E A C D (1) B A D E C (1) A C B E D (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -18 -12 -8 B 4 0 -6 -8 -2 C 18 6 0 -14 -2 D 12 8 14 0 10 E 8 2 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 -12 -8 B 4 0 -6 -8 -2 C 18 6 0 -14 -2 D 12 8 14 0 10 E 8 2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 E=19 A=18 B=17 so B is eliminated. Round 2 votes counts: E=30 D=24 A=24 C=22 so C is eliminated. Round 3 votes counts: D=38 A=32 E=30 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:204 E:201 B:194 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -18 -12 -8 B 4 0 -6 -8 -2 C 18 6 0 -14 -2 D 12 8 14 0 10 E 8 2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -12 -8 B 4 0 -6 -8 -2 C 18 6 0 -14 -2 D 12 8 14 0 10 E 8 2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -12 -8 B 4 0 -6 -8 -2 C 18 6 0 -14 -2 D 12 8 14 0 10 E 8 2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2340: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) B E D A C (10) E B A D C (8) E A C D B (5) D A C B E (5) C D A B E (5) B D C A E (5) E C A B D (3) E B D A C (3) E A C B D (3) B D E A C (3) A C D E B (3) E D A B C (2) E C A D B (2) E A B D C (2) D B A C E (2) C E A D B (2) C A E D B (2) B E D C A (2) B D A E C (2) B D A C E (2) B C D E A (2) A C E D B (2) E B A C D (1) E A D C B (1) D C A B E (1) D B C A E (1) D A B E C (1) C E B A D (1) C B E D A (1) B E C D A (1) B E C A D (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 8 18 2 -6 B -8 0 0 2 -10 C -18 0 0 -6 -8 D -2 -2 6 0 -8 E 6 10 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 18 2 -6 B -8 0 0 2 -10 C -18 0 0 -6 -8 D -2 -2 6 0 -8 E 6 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=29 C=22 D=10 A=9 so A is eliminated. Round 2 votes counts: E=32 B=29 C=27 D=12 so D is eliminated. Round 3 votes counts: E=34 C=33 B=33 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:211 D:197 B:192 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 18 2 -6 B -8 0 0 2 -10 C -18 0 0 -6 -8 D -2 -2 6 0 -8 E 6 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 2 -6 B -8 0 0 2 -10 C -18 0 0 -6 -8 D -2 -2 6 0 -8 E 6 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 2 -6 B -8 0 0 2 -10 C -18 0 0 -6 -8 D -2 -2 6 0 -8 E 6 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2341: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) B E D C A (8) B E A C D (6) D C A E B (5) B E C D A (5) A D C B E (5) D A C E B (4) B A D C E (4) A E C D B (4) A D C E B (4) E B C D A (3) E A C D B (3) B E C A D (3) B A E C D (3) A B D C E (3) E C D A B (2) E C A D B (2) E B A C D (2) D C E B A (2) D C A B E (2) B D C E A (2) A C E D B (2) E C A B D (1) E B C A D (1) E A B C D (1) D C E A B (1) D C B E A (1) D A C B E (1) C E A D B (1) C D A E B (1) C A E D B (1) B E A D C (1) B D C A E (1) B D A C E (1) B A E D C (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 16 22 10 B -12 0 -6 -4 -4 C -16 6 0 8 6 D -22 4 -8 0 -2 E -10 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 22 10 B -12 0 -6 -4 -4 C -16 6 0 8 6 D -22 4 -8 0 -2 E -10 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=31 D=16 E=15 C=3 so C is eliminated. Round 2 votes counts: B=35 A=32 D=17 E=16 so E is eliminated. Round 3 votes counts: B=41 A=40 D=19 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:202 E:195 B:187 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 22 10 B -12 0 -6 -4 -4 C -16 6 0 8 6 D -22 4 -8 0 -2 E -10 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 22 10 B -12 0 -6 -4 -4 C -16 6 0 8 6 D -22 4 -8 0 -2 E -10 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 22 10 B -12 0 -6 -4 -4 C -16 6 0 8 6 D -22 4 -8 0 -2 E -10 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2342: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B A C D E (9) B A E C D (8) E D B C A (5) C D A E B (5) A C B D E (5) A B C D E (5) E D C A B (4) B E A D C (4) B A C E D (4) E B A D C (3) D E C B A (3) A B E C D (3) E D C B A (2) E B D C A (2) D C E B A (2) D C B E A (2) C E D A B (2) C A D B E (2) B E A C D (2) A B C E D (2) E B A C D (1) E A D C B (1) E A C D B (1) E A B D C (1) D E C A B (1) D C B A E (1) D C A B E (1) C D E A B (1) C D A B E (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) B D E A C (1) B A E D C (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 -8 8 10 0 B 8 0 12 12 10 C -8 -12 0 6 6 D -10 -12 -6 0 2 E 0 -10 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 10 0 B 8 0 12 12 10 C -8 -12 0 6 6 D -10 -12 -6 0 2 E 0 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=20 D=19 A=16 C=12 so C is eliminated. Round 2 votes counts: B=33 D=26 E=22 A=19 so A is eliminated. Round 3 votes counts: B=48 D=29 E=23 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:205 C:196 E:191 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 10 0 B 8 0 12 12 10 C -8 -12 0 6 6 D -10 -12 -6 0 2 E 0 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 10 0 B 8 0 12 12 10 C -8 -12 0 6 6 D -10 -12 -6 0 2 E 0 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 10 0 B 8 0 12 12 10 C -8 -12 0 6 6 D -10 -12 -6 0 2 E 0 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2343: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) D A B E C (12) C A D E B (12) E B C D A (9) C E B A D (9) C A E B D (7) B E D A C (7) A D C B E (5) E B C A D (4) C A D B E (4) A D C E B (3) D B E A C (2) C E A B D (2) B E D C A (2) A C D E B (2) E B D C A (1) D E B A C (1) D A C B E (1) D A B C E (1) C B E D A (1) C B E A D (1) C A E D B (1) B D E A C (1) Total count = 100 A B C D E A 0 0 -6 0 -4 B 0 0 4 12 -26 C 6 -4 0 4 -2 D 0 -12 -4 0 -12 E 4 26 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -6 0 -4 B 0 0 4 12 -26 C 6 -4 0 4 -2 D 0 -12 -4 0 -12 E 4 26 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=26 D=17 B=10 A=10 so B is eliminated. Round 2 votes counts: C=37 E=35 D=18 A=10 so A is eliminated. Round 3 votes counts: C=39 E=35 D=26 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:202 A:195 B:195 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 0 -4 B 0 0 4 12 -26 C 6 -4 0 4 -2 D 0 -12 -4 0 -12 E 4 26 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 0 -4 B 0 0 4 12 -26 C 6 -4 0 4 -2 D 0 -12 -4 0 -12 E 4 26 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 0 -4 B 0 0 4 12 -26 C 6 -4 0 4 -2 D 0 -12 -4 0 -12 E 4 26 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2344: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) B C A E D (7) B C D E A (6) C B E D A (5) B A C E D (5) A E D B C (4) A D E B C (4) D C E B A (3) C B D E A (3) B A C D E (3) A B E D C (3) E D A C B (2) E C D A B (2) E A D C B (2) E A C D B (2) D E C B A (2) D E C A B (2) C E D A B (2) C E A D B (2) C E A B D (2) B C A D E (2) B A D E C (2) A E C B D (2) E D C A B (1) D E A C B (1) D E A B C (1) D B E C A (1) D B E A C (1) D B C E A (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E A D (1) C B A E D (1) C A E B D (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D A E (1) B A E D C (1) B A E C D (1) B A D C E (1) A E C D B (1) A E B D C (1) A D B E C (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 20 10 B 4 0 2 8 4 C -2 -2 0 8 4 D -20 -8 -8 0 -18 E -10 -4 -4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 20 10 B 4 0 2 8 4 C -2 -2 0 8 4 D -20 -8 -8 0 -18 E -10 -4 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=27 C=19 D=13 E=9 so E is eliminated. Round 2 votes counts: B=32 A=31 C=21 D=16 so D is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:209 C:204 E:200 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 20 10 B 4 0 2 8 4 C -2 -2 0 8 4 D -20 -8 -8 0 -18 E -10 -4 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 20 10 B 4 0 2 8 4 C -2 -2 0 8 4 D -20 -8 -8 0 -18 E -10 -4 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 20 10 B 4 0 2 8 4 C -2 -2 0 8 4 D -20 -8 -8 0 -18 E -10 -4 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2345: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) C D A B E (7) C A D E B (7) C A D B E (6) A C D E B (6) A E C B D (5) A E B C D (5) A C E B D (5) E B D A C (4) E B A C D (4) D C B A E (4) B E D A C (4) B D E C A (4) E A B D C (3) D B C E A (3) D C A B E (2) C D B A E (2) C A E B D (2) B E C D A (2) E D B A C (1) E B C A D (1) E A B C D (1) D C B E A (1) D B E C A (1) D B E A C (1) D A C E B (1) D A B C E (1) C B E D A (1) C A B E D (1) B E D C A (1) B E C A D (1) B E A C D (1) B D E A C (1) B C D E A (1) A E D B C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 10 6 16 14 B -10 0 -2 8 -10 C -6 2 0 16 0 D -16 -8 -16 0 0 E -14 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 16 14 B -10 0 -2 8 -10 C -6 2 0 16 0 D -16 -8 -16 0 0 E -14 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 E=21 B=15 D=14 so D is eliminated. Round 2 votes counts: C=33 A=26 E=21 B=20 so B is eliminated. Round 3 votes counts: E=37 C=37 A=26 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:223 C:206 E:198 B:193 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 16 14 B -10 0 -2 8 -10 C -6 2 0 16 0 D -16 -8 -16 0 0 E -14 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 16 14 B -10 0 -2 8 -10 C -6 2 0 16 0 D -16 -8 -16 0 0 E -14 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 16 14 B -10 0 -2 8 -10 C -6 2 0 16 0 D -16 -8 -16 0 0 E -14 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2346: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (11) C E A B D (10) A C E B D (9) E C B A D (8) B E C A D (8) D A C E B (7) D B A E C (6) D A B C E (4) A D B C E (4) A B C E D (4) D E C B A (3) C A E B D (3) B E C D A (3) D C E A B (2) B D E C A (2) B D A E C (2) A C E D B (2) E C D B A (1) E C B D A (1) E C A B D (1) E B C A D (1) D E B C A (1) D B A C E (1) C E D A B (1) B A E C D (1) A D C E B (1) A D C B E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -12 10 -6 B -2 0 -2 8 -4 C 12 2 0 8 2 D -10 -8 -8 0 -6 E 6 4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 10 -6 B -2 0 -2 8 -4 C 12 2 0 8 2 D -10 -8 -8 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=23 B=16 C=14 E=12 so E is eliminated. Round 2 votes counts: D=35 C=25 A=23 B=17 so B is eliminated. Round 3 votes counts: D=39 C=37 A=24 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:207 B:200 A:197 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 10 -6 B -2 0 -2 8 -4 C 12 2 0 8 2 D -10 -8 -8 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 10 -6 B -2 0 -2 8 -4 C 12 2 0 8 2 D -10 -8 -8 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 10 -6 B -2 0 -2 8 -4 C 12 2 0 8 2 D -10 -8 -8 0 -6 E 6 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2347: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (11) C D E A B (5) C D A B E (5) B A D E C (5) E B A C D (4) D C A B E (4) D A B C E (4) C E A D B (4) C D B A E (4) C B E D A (4) B D A E C (4) B A E D C (4) E A B D C (3) D A C B E (3) C E B A D (3) B E A D C (3) E C B A D (2) E C A B D (2) E B A D C (2) E A B C D (2) C E D A B (2) C D E B A (2) B C D A E (2) A D B E C (2) A B D E C (2) E B C A D (1) D C A E B (1) D B C A E (1) D B A C E (1) D A E B C (1) C E B D A (1) C D B E A (1) C B D E A (1) B D C A E (1) B D A C E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 6 8 -18 16 B -6 0 12 -4 26 C -8 -12 0 -10 0 D 18 4 10 0 22 E -16 -26 0 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -18 16 B -6 0 12 -4 26 C -8 -12 0 -10 0 D 18 4 10 0 22 E -16 -26 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=26 B=20 E=16 A=6 so A is eliminated. Round 2 votes counts: C=32 D=28 B=22 E=18 so E is eliminated. Round 3 votes counts: C=36 B=35 D=29 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:227 B:214 A:206 C:185 E:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -18 16 B -6 0 12 -4 26 C -8 -12 0 -10 0 D 18 4 10 0 22 E -16 -26 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -18 16 B -6 0 12 -4 26 C -8 -12 0 -10 0 D 18 4 10 0 22 E -16 -26 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -18 16 B -6 0 12 -4 26 C -8 -12 0 -10 0 D 18 4 10 0 22 E -16 -26 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2348: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) D C E A B (8) D C E B A (7) E D A C B (5) B A C E D (5) B A C D E (5) A B E D C (5) E D A B C (4) D E C A B (4) C D E A B (4) B C A D E (4) B A E C D (4) E A B D C (3) D E C B A (3) C D B A E (3) C B D A E (3) E A D B C (2) C D E B A (2) C D B E A (2) C B A D E (2) B A E D C (2) A E B D C (2) A B E C D (2) E D B C A (1) E D B A C (1) E C A D B (1) C D A E B (1) C A E D B (1) C A B D E (1) B E A D C (1) B D E A C (1) B C D A E (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -12 -16 -14 B -6 0 -12 -16 -16 C 12 12 0 -14 0 D 16 16 14 0 2 E 14 16 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -16 -14 B -6 0 -12 -16 -16 C 12 12 0 -14 0 D 16 16 14 0 2 E 14 16 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=23 D=22 C=19 A=11 so A is eliminated. Round 2 votes counts: B=31 E=28 D=22 C=19 so C is eliminated. Round 3 votes counts: B=37 D=34 E=29 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:214 C:205 A:182 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -12 -16 -14 B -6 0 -12 -16 -16 C 12 12 0 -14 0 D 16 16 14 0 2 E 14 16 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -16 -14 B -6 0 -12 -16 -16 C 12 12 0 -14 0 D 16 16 14 0 2 E 14 16 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -16 -14 B -6 0 -12 -16 -16 C 12 12 0 -14 0 D 16 16 14 0 2 E 14 16 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2349: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) B C E D A (10) C B E A D (5) B D E C A (5) A C E D B (5) D A E B C (4) C B E D A (3) B D E A C (3) A D E B C (3) E D C A B (2) E D A C B (2) D E A C B (2) D E A B C (2) D B E A C (2) D B A E C (2) C E A B D (2) C B A E D (2) C A E B D (2) C A B E D (2) B C D E A (2) A C D E B (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D A B (1) E C B D A (1) E C A D B (1) E B C D A (1) D E B A C (1) D A E C B (1) D A B E C (1) C E D B A (1) C E A D B (1) C A E D B (1) B E D C A (1) B D C A E (1) B C A D E (1) B A D C E (1) B A C D E (1) A E D C B (1) A E C D B (1) A D C E B (1) A D B E C (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 6 -2 -2 B -8 0 -10 -10 -8 C -6 10 0 -6 -8 D 2 10 6 0 2 E 2 8 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -2 -2 B -8 0 -10 -10 -8 C -6 10 0 -6 -8 D 2 10 6 0 2 E 2 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=25 C=19 D=15 E=11 so E is eliminated. Round 2 votes counts: A=30 B=26 D=22 C=22 so D is eliminated. Round 3 votes counts: A=42 B=33 C=25 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:210 E:208 A:205 C:195 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -2 -2 B -8 0 -10 -10 -8 C -6 10 0 -6 -8 D 2 10 6 0 2 E 2 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -2 -2 B -8 0 -10 -10 -8 C -6 10 0 -6 -8 D 2 10 6 0 2 E 2 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -2 -2 B -8 0 -10 -10 -8 C -6 10 0 -6 -8 D 2 10 6 0 2 E 2 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2350: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) D E A C B (10) C B D E A (9) A B E D C (6) C B A D E (5) D E C A B (4) A E D B C (4) A D E C B (4) A D E B C (4) A B C E D (4) D E C B A (3) C D E B A (3) C B E D A (3) B C E A D (3) B C A D E (3) A B C D E (3) E D C B A (2) E D A C B (2) E A D B C (2) D C E A B (2) B A C E D (2) A E B D C (2) E D A B C (1) C E B D A (1) C D B E A (1) C D A E B (1) B C E D A (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -8 14 6 B -2 0 -2 14 10 C 8 2 0 8 10 D -14 -14 -8 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 14 6 B -2 0 -2 14 10 C 8 2 0 8 10 D -14 -14 -8 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 B=22 D=19 E=7 so E is eliminated. Round 2 votes counts: A=31 D=24 C=23 B=22 so B is eliminated. Round 3 votes counts: C=43 A=33 D=24 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:210 A:207 D:185 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 14 6 B -2 0 -2 14 10 C 8 2 0 8 10 D -14 -14 -8 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 14 6 B -2 0 -2 14 10 C 8 2 0 8 10 D -14 -14 -8 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 14 6 B -2 0 -2 14 10 C 8 2 0 8 10 D -14 -14 -8 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2351: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C E B A D (7) D A B E C (5) C A D E B (5) D C A B E (4) D B A E C (4) C A E D B (4) E C B A D (3) D A C B E (3) C E A B D (3) C D B E A (3) B E C D A (3) B E A D C (3) B D E A C (3) A D E C B (3) A D C E B (3) E B C A D (2) E B A D C (2) D A C E B (2) C E B D A (2) C D A E B (2) C B E D A (2) C B D E A (2) B D A E C (2) A D E B C (2) A D B E C (2) E B A C D (1) E A C D B (1) D C A E B (1) D B A C E (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B A E (1) B E D C A (1) B E C A D (1) B D C E A (1) B C D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 -12 -2 -12 -8 B 12 0 -8 -2 6 C 2 8 0 -8 2 D 12 2 8 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -12 -8 B 12 0 -8 -2 6 C 2 8 0 -8 2 D 12 2 8 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=25 D=21 A=11 E=9 so E is eliminated. Round 2 votes counts: C=37 B=30 D=21 A=12 so A is eliminated. Round 3 votes counts: C=38 D=32 B=30 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:204 C:202 E:198 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -12 -8 B 12 0 -8 -2 6 C 2 8 0 -8 2 D 12 2 8 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -12 -8 B 12 0 -8 -2 6 C 2 8 0 -8 2 D 12 2 8 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -12 -8 B 12 0 -8 -2 6 C 2 8 0 -8 2 D 12 2 8 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2352: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) E B C D A (6) A D C E B (6) E C B D A (5) D A C E B (5) B E D C A (5) C A D E B (4) B E A C D (4) D C E A B (3) C E D B A (3) C E B D A (3) B E C D A (3) B A E C D (3) A B D E C (3) E C D B A (2) D B E A C (2) D A E C B (2) C E D A B (2) C D E A B (2) B E C A D (2) B E A D C (2) B A E D C (2) A D C B E (2) A C D B E (2) A B E C D (2) A B C E D (2) E D B C A (1) E B D C A (1) D E C A B (1) D C A E B (1) D B A E C (1) D A B E C (1) C E A D B (1) C D A E B (1) C A E B D (1) B E D A C (1) B D E A C (1) B D A E C (1) B A D E C (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 6 -6 0 B -2 0 6 0 0 C -6 -6 0 -2 -6 D 6 0 2 0 -2 E 0 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.178878 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.821122 Sum of squares = 0.706238234482 Cumulative probabilities = A: 0.178878 B: 0.178878 C: 0.178878 D: 0.178878 E: 1.000000 A B C D E A 0 2 6 -6 0 B -2 0 6 0 0 C -6 -6 0 -2 -6 D 6 0 2 0 -2 E 0 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000024439 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 C=17 D=16 E=15 so E is eliminated. Round 2 votes counts: B=32 A=27 C=24 D=17 so D is eliminated. Round 3 votes counts: B=36 A=35 C=29 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:204 D:203 B:202 A:201 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 -6 0 B -2 0 6 0 0 C -6 -6 0 -2 -6 D 6 0 2 0 -2 E 0 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000024439 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -6 0 B -2 0 6 0 0 C -6 -6 0 -2 -6 D 6 0 2 0 -2 E 0 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000024439 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -6 0 B -2 0 6 0 0 C -6 -6 0 -2 -6 D 6 0 2 0 -2 E 0 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000024439 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2353: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) A D E B C (7) D E C A B (6) B A C E D (6) E D C B A (5) D A E C B (5) C B E D A (5) B C A E D (5) A B C E D (5) B E C D A (4) D E C B A (3) D E A C B (3) D C E A B (3) B C E A D (3) A B D E C (3) A B C D E (3) D E A B C (2) D C E B A (2) A D E C B (2) A B E D C (2) A B E C D (2) E D B C A (1) E D B A C (1) E C D B A (1) E B D C A (1) E B C D A (1) D C A E B (1) C E D B A (1) C E B D A (1) C D E A B (1) C B D E A (1) C B A E D (1) B E D C A (1) B C E D A (1) A D C B E (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -10 -14 -8 B 2 0 0 -6 -8 C 10 0 0 -2 -2 D 14 6 2 0 6 E 8 8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -14 -8 B 2 0 0 -6 -8 C 10 0 0 -2 -2 D 14 6 2 0 6 E 8 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 B=20 C=17 E=10 so E is eliminated. Round 2 votes counts: D=32 A=28 B=22 C=18 so C is eliminated. Round 3 votes counts: D=42 B=30 A=28 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:206 C:203 B:194 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -10 -14 -8 B 2 0 0 -6 -8 C 10 0 0 -2 -2 D 14 6 2 0 6 E 8 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -14 -8 B 2 0 0 -6 -8 C 10 0 0 -2 -2 D 14 6 2 0 6 E 8 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -14 -8 B 2 0 0 -6 -8 C 10 0 0 -2 -2 D 14 6 2 0 6 E 8 8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2354: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) B E A C D (8) D C A B E (7) A D C E B (6) E B A C D (5) C D E B A (5) C D B E A (5) E A B D C (4) D C E A B (4) B C D E A (4) A E B D C (4) A D C B E (4) E C D B A (3) E B C D A (3) E B C A D (3) D C A E B (3) C D B A E (3) E A D C B (2) D A C B E (2) C B D E A (2) A E D C B (2) A B D C E (2) E D C A B (1) D A C E B (1) C E D B A (1) C E B D A (1) B E C A D (1) B C D A E (1) B A E C D (1) B A C E D (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 6 6 8 -4 B -6 0 -6 -2 2 C -6 6 0 -6 6 D -8 2 6 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 6 6 8 -4 B -6 0 -6 -2 2 C -6 6 0 -6 6 D -8 2 6 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999768 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=21 D=17 C=17 B=16 so B is eliminated. Round 2 votes counts: A=31 E=30 C=22 D=17 so D is eliminated. Round 3 votes counts: C=36 A=34 E=30 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:201 C:200 E:197 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 8 -4 B -6 0 -6 -2 2 C -6 6 0 -6 6 D -8 2 6 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999768 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 -4 B -6 0 -6 -2 2 C -6 6 0 -6 6 D -8 2 6 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999768 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 -4 B -6 0 -6 -2 2 C -6 6 0 -6 6 D -8 2 6 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999768 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2355: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C E A B D (8) D B E C A (5) D A B E C (5) C A E D B (4) D B E A C (3) D A B C E (3) C E B D A (3) B E D A C (3) B E A D C (3) A E C B D (3) A D C B E (3) E C B A D (2) E B C A D (2) E A B C D (2) D C B A E (2) D B C A E (2) D B A C E (2) D A C B E (2) C E B A D (2) C D A E B (2) B D E A C (2) A D B E C (2) A D B C E (2) A C E B D (2) A B E D C (2) A B D E C (2) E B D C A (1) E B C D A (1) E B A D C (1) E B A C D (1) D C E B A (1) D C B E A (1) D C A B E (1) C E D B A (1) C D E B A (1) C A D E B (1) B E D C A (1) B D A E C (1) B A D E C (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 18 -10 12 B 8 0 18 -12 22 C -18 -18 0 -26 -10 D 10 12 26 0 14 E -12 -22 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 18 -10 12 B 8 0 18 -12 22 C -18 -18 0 -26 -10 D 10 12 26 0 14 E -12 -22 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=22 A=18 B=11 E=10 so E is eliminated. Round 2 votes counts: D=39 C=24 A=20 B=17 so B is eliminated. Round 3 votes counts: D=47 C=27 A=26 so A is eliminated. Round 4 votes counts: D=63 C=37 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:231 B:218 A:206 E:181 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 18 -10 12 B 8 0 18 -12 22 C -18 -18 0 -26 -10 D 10 12 26 0 14 E -12 -22 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 18 -10 12 B 8 0 18 -12 22 C -18 -18 0 -26 -10 D 10 12 26 0 14 E -12 -22 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 18 -10 12 B 8 0 18 -12 22 C -18 -18 0 -26 -10 D 10 12 26 0 14 E -12 -22 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2356: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) B C D A E (6) E C A D B (5) C D A B E (5) C B D A E (5) C D B A E (4) B C E D A (4) A D B E C (4) E C B A D (3) E A D C B (3) E A B C D (3) D A B C E (3) C B E D A (3) E C A B D (2) E B C A D (2) E A D B C (2) E A C D B (2) E A C B D (2) D B A C E (2) D A C B E (2) C E B D A (2) C E A D B (2) C D A E B (2) C B D E A (2) C A E D B (2) B D A E C (2) B D A C E (2) A E D B C (2) A D E C B (2) E A B D C (1) D C A B E (1) D A B E C (1) C E B A D (1) C E A B D (1) C A D E B (1) B E D C A (1) B E C D A (1) B D E C A (1) B D E A C (1) B C D E A (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 0 -28 -16 10 B 0 0 -6 6 16 C 28 6 0 22 16 D 16 -6 -22 0 10 E -10 -16 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -28 -16 10 B 0 0 -6 6 16 C 28 6 0 22 16 D 16 -6 -22 0 10 E -10 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=26 E=25 A=10 D=9 so D is eliminated. Round 2 votes counts: C=31 B=28 E=25 A=16 so A is eliminated. Round 3 votes counts: B=36 C=33 E=31 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:236 B:208 D:199 A:183 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -28 -16 10 B 0 0 -6 6 16 C 28 6 0 22 16 D 16 -6 -22 0 10 E -10 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -28 -16 10 B 0 0 -6 6 16 C 28 6 0 22 16 D 16 -6 -22 0 10 E -10 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -28 -16 10 B 0 0 -6 6 16 C 28 6 0 22 16 D 16 -6 -22 0 10 E -10 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2357: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) D E B A C (9) E C A B D (8) D B E A C (8) C A B E D (5) D C B A E (4) D B A C E (4) C A E B D (4) E B D A C (3) D B C A E (3) C D A B E (3) A B C E D (3) E D C A B (2) E D B A C (2) E C D A B (2) E B A D C (2) E A B C D (2) C E A B D (2) B D A C E (2) B A C D E (2) A C E B D (2) E D A B C (1) E C A D B (1) E A D B C (1) D E B C A (1) D C E B A (1) D B E C A (1) C E A D B (1) C B A D E (1) C A D E B (1) B E D A C (1) B E A D C (1) B A E C D (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -6 6 2 B -8 0 -4 10 6 C 6 4 0 8 4 D -6 -10 -8 0 6 E -2 -6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 6 2 B -8 0 -4 10 6 C 6 4 0 8 4 D -6 -10 -8 0 6 E -2 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=30 E=24 A=8 B=7 so B is eliminated. Round 2 votes counts: D=33 C=30 E=26 A=11 so A is eliminated. Round 3 votes counts: C=38 D=33 E=29 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:205 B:202 D:191 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 6 2 B -8 0 -4 10 6 C 6 4 0 8 4 D -6 -10 -8 0 6 E -2 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 6 2 B -8 0 -4 10 6 C 6 4 0 8 4 D -6 -10 -8 0 6 E -2 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 6 2 B -8 0 -4 10 6 C 6 4 0 8 4 D -6 -10 -8 0 6 E -2 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2358: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (6) D E B A C (5) C A D E B (5) C A B E D (5) B E D A C (5) D C E A B (4) D C A E B (4) C B A D E (4) B C D A E (4) A C B E D (4) D E A C B (3) B C A D E (3) E D A B C (2) E B D A C (2) E A D C B (2) D E B C A (2) D C E B A (2) C D A E B (2) C B A E D (2) C A B D E (2) B D E C A (2) A E D C B (2) A E C D B (2) A E B C D (2) A C E D B (2) A C E B D (2) E D A C B (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C B A (1) D E C A B (1) D E A B C (1) D B E C A (1) D B C E A (1) C D B A E (1) C B D A E (1) C A E D B (1) C A D B E (1) B E A D C (1) B D E A C (1) B D C E A (1) B C D E A (1) B A C E D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -14 4 16 B -4 0 -10 6 -4 C 14 10 0 6 18 D -4 -6 -6 0 6 E -16 4 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 4 16 B -4 0 -10 6 -4 C 14 10 0 6 18 D -4 -6 -6 0 6 E -16 4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 C=24 A=16 E=10 so E is eliminated. Round 2 votes counts: D=28 B=28 C=24 A=20 so A is eliminated. Round 3 votes counts: C=35 D=33 B=32 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:205 D:195 B:194 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 4 16 B -4 0 -10 6 -4 C 14 10 0 6 18 D -4 -6 -6 0 6 E -16 4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 4 16 B -4 0 -10 6 -4 C 14 10 0 6 18 D -4 -6 -6 0 6 E -16 4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 4 16 B -4 0 -10 6 -4 C 14 10 0 6 18 D -4 -6 -6 0 6 E -16 4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2359: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (19) B D C E A (10) D E B C A (9) E A D B C (8) C B A D E (6) A C B E D (6) E D B A C (5) D B C E A (5) C A B D E (5) A E D C B (5) A E C D B (4) C B D A E (3) A C E B D (3) E D A B C (2) D B E C A (2) B D E C A (2) E D B C A (1) E A D C B (1) D E B A C (1) B C D A E (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 -30 -26 -20 -30 B 30 0 34 12 20 C 26 -34 0 -4 18 D 20 -12 4 0 26 E 30 -20 -18 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998675 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -26 -20 -30 B 30 0 34 12 20 C 26 -34 0 -4 18 D 20 -12 4 0 26 E 30 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=19 E=17 D=17 C=14 so C is eliminated. Round 2 votes counts: B=42 A=24 E=17 D=17 so E is eliminated. Round 3 votes counts: B=42 A=33 D=25 so D is eliminated. Round 4 votes counts: B=65 A=35 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:248 D:219 C:203 E:183 A:147 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -26 -20 -30 B 30 0 34 12 20 C 26 -34 0 -4 18 D 20 -12 4 0 26 E 30 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -26 -20 -30 B 30 0 34 12 20 C 26 -34 0 -4 18 D 20 -12 4 0 26 E 30 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -26 -20 -30 B 30 0 34 12 20 C 26 -34 0 -4 18 D 20 -12 4 0 26 E 30 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2360: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C B A (9) C A B D E (8) B E A D C (7) E B D A C (6) D C E B A (6) A B E C D (6) E D B A C (5) C D E A B (5) A C B E D (5) C D A E B (4) C D E B A (3) C A B E D (3) B A E D C (3) A B E D C (3) E D B C A (2) D E B C A (2) D E B A C (2) C A D B E (2) B A C E D (2) A E B D C (2) E A B D C (1) D C E A B (1) C D B E A (1) C D A B E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 6 6 0 B -4 0 2 14 4 C -6 -2 0 2 4 D -6 -14 -2 0 -10 E 0 -4 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.697222 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.302778 Sum of squares = 0.577792860326 Cumulative probabilities = A: 0.697222 B: 0.697222 C: 0.697222 D: 0.697222 E: 1.000000 A B C D E A 0 4 6 6 0 B -4 0 2 14 4 C -6 -2 0 2 4 D -6 -14 -2 0 -10 E 0 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500163 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499837 Sum of squares = 0.500000053111 Cumulative probabilities = A: 0.500163 B: 0.500163 C: 0.500163 D: 0.500163 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=27 A=27 D=20 E=14 B=12 so B is eliminated. Round 2 votes counts: A=32 C=27 E=21 D=20 so D is eliminated. Round 3 votes counts: E=34 C=34 A=32 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:208 B:208 E:201 C:199 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 6 0 B -4 0 2 14 4 C -6 -2 0 2 4 D -6 -14 -2 0 -10 E 0 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500163 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499837 Sum of squares = 0.500000053111 Cumulative probabilities = A: 0.500163 B: 0.500163 C: 0.500163 D: 0.500163 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 6 0 B -4 0 2 14 4 C -6 -2 0 2 4 D -6 -14 -2 0 -10 E 0 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500163 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499837 Sum of squares = 0.500000053111 Cumulative probabilities = A: 0.500163 B: 0.500163 C: 0.500163 D: 0.500163 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 6 0 B -4 0 2 14 4 C -6 -2 0 2 4 D -6 -14 -2 0 -10 E 0 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500163 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499837 Sum of squares = 0.500000053111 Cumulative probabilities = A: 0.500163 B: 0.500163 C: 0.500163 D: 0.500163 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2361: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) A E D C B (8) E A B C D (7) C B D E A (7) E A C B D (6) A E D B C (6) D B A C E (4) A E B D C (4) D A B E C (3) C E B A D (3) C B E D A (3) C B E A D (3) B E A C D (3) B C E D A (3) B C D E A (3) E C A D B (2) D C B A E (2) D A E B C (2) C D B E A (2) B D C E A (2) B D C A E (2) E C A B D (1) E A C D B (1) D C A E B (1) D B A E C (1) D A E C B (1) D A C E B (1) D A C B E (1) C E B D A (1) C E A B D (1) C D E B A (1) B E C A D (1) B D A C E (1) B C E A D (1) B A D E C (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 4 0 -2 B 6 0 6 8 4 C -4 -6 0 0 2 D 0 -8 0 0 -10 E 2 -4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 0 -2 B 6 0 6 8 4 C -4 -6 0 0 2 D 0 -8 0 0 -10 E 2 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=21 A=20 E=17 B=17 so E is eliminated. Round 2 votes counts: A=34 D=25 C=24 B=17 so B is eliminated. Round 3 votes counts: A=38 C=32 D=30 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:203 A:198 C:196 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 0 -2 B 6 0 6 8 4 C -4 -6 0 0 2 D 0 -8 0 0 -10 E 2 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 0 -2 B 6 0 6 8 4 C -4 -6 0 0 2 D 0 -8 0 0 -10 E 2 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 0 -2 B 6 0 6 8 4 C -4 -6 0 0 2 D 0 -8 0 0 -10 E 2 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2362: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) B C D A E (6) B A E C D (5) A E B C D (5) E D A C B (4) E B A D C (4) E A D C B (4) E A B C D (4) E A D B C (3) D C B E A (3) C B D A E (3) B E A C D (3) B A C E D (3) E D B A C (2) E D A B C (2) E A B D C (2) D C E A B (2) D C A E B (2) C D B A E (2) C A B D E (2) B E A D C (2) B D C E A (2) B C A E D (2) A E D C B (2) A E C D B (2) A E C B D (2) A C B D E (2) E D B C A (1) E B A C D (1) D E C B A (1) D E C A B (1) D C E B A (1) D C B A E (1) D A E C B (1) C D A B E (1) C A D E B (1) C A D B E (1) B E D C A (1) B D E C A (1) B C D E A (1) A C D B E (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 14 24 10 B 6 0 20 24 6 C -14 -20 0 16 -8 D -24 -24 -16 0 -14 E -10 -6 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 14 24 10 B 6 0 20 24 6 C -14 -20 0 16 -8 D -24 -24 -16 0 -14 E -10 -6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=27 A=17 D=12 C=10 so C is eliminated. Round 2 votes counts: B=37 E=27 A=21 D=15 so D is eliminated. Round 3 votes counts: B=43 E=32 A=25 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:221 E:203 C:187 D:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 14 24 10 B 6 0 20 24 6 C -14 -20 0 16 -8 D -24 -24 -16 0 -14 E -10 -6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 24 10 B 6 0 20 24 6 C -14 -20 0 16 -8 D -24 -24 -16 0 -14 E -10 -6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 24 10 B 6 0 20 24 6 C -14 -20 0 16 -8 D -24 -24 -16 0 -14 E -10 -6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2363: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) D E A B C (9) B D E A C (8) B C E A D (8) D A E C B (7) E A D C B (5) D B E A C (5) B C D E A (5) C A E D B (4) C A E B D (4) B C D A E (4) A E D C B (4) D E A C B (3) B C A E D (3) A E C D B (3) E A C D B (2) D B A E C (2) C B E A D (2) B D C E A (2) E C A B D (1) E A D B C (1) E A B C D (1) C D A E B (1) C B D A E (1) C B A D E (1) C A B E D (1) B D E C A (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 2 4 -6 B 6 0 -2 6 8 C -2 2 0 6 -4 D -4 -6 -6 0 -2 E 6 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428568 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -6 2 4 -6 B 6 0 -2 6 8 C -2 2 0 6 -4 D -4 -6 -6 0 -2 E 6 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428563 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=26 C=25 E=10 A=8 so A is eliminated. Round 2 votes counts: B=31 D=26 C=26 E=17 so E is eliminated. Round 3 votes counts: D=36 C=32 B=32 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:202 C:201 A:197 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 -6 B 6 0 -2 6 8 C -2 2 0 6 -4 D -4 -6 -6 0 -2 E 6 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428563 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 -6 B 6 0 -2 6 8 C -2 2 0 6 -4 D -4 -6 -6 0 -2 E 6 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428563 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 -6 B 6 0 -2 6 8 C -2 2 0 6 -4 D -4 -6 -6 0 -2 E 6 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428563 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2364: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (19) B E C D A (12) B E D C A (8) D A C E B (5) A D B C E (5) C E D A B (4) E B C D A (3) D B A E C (3) C E D B A (3) B E C A D (3) A B D E C (3) E C B D A (2) D C E A B (2) D A B E C (2) C E A B D (2) B A C E D (2) A C E D B (2) A C D E B (2) E D C B A (1) D E C B A (1) D E C A B (1) D E A C B (1) D E A B C (1) D A E C B (1) C E B D A (1) C E B A D (1) C A E D B (1) C A E B D (1) C A D E B (1) B D E C A (1) B D E A C (1) B C E A D (1) B C A E D (1) B A D E C (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 10 0 -6 2 B -10 0 -4 -12 -12 C 0 4 0 -14 8 D 6 12 14 0 2 E -2 12 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -6 2 B -10 0 -4 -12 -12 C 0 4 0 -14 8 D 6 12 14 0 2 E -2 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=30 D=17 C=14 E=6 so E is eliminated. Round 2 votes counts: B=33 A=33 D=18 C=16 so C is eliminated. Round 3 votes counts: A=38 B=37 D=25 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:203 E:200 C:199 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -6 2 B -10 0 -4 -12 -12 C 0 4 0 -14 8 D 6 12 14 0 2 E -2 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -6 2 B -10 0 -4 -12 -12 C 0 4 0 -14 8 D 6 12 14 0 2 E -2 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -6 2 B -10 0 -4 -12 -12 C 0 4 0 -14 8 D 6 12 14 0 2 E -2 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2365: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) E B D A C (9) C B E D A (9) D A E B C (7) C B A E D (7) B E C D A (7) E B D C A (5) C A B E D (5) A D E C B (5) A D E B C (5) A D C E B (5) A D C B E (4) A C D B E (4) C A B D E (3) B C E D A (3) E D B A C (2) E B C D A (2) D E B A C (2) D E A B C (2) A D B E C (2) C A D B E (1) B E D C A (1) Total count = 100 A B C D E A 0 -14 -6 2 -4 B 14 0 -6 22 12 C 6 6 0 2 2 D -2 -22 -2 0 -20 E 4 -12 -2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 2 -4 B 14 0 -6 22 12 C 6 6 0 2 2 D -2 -22 -2 0 -20 E 4 -12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=25 E=18 D=11 B=11 so D is eliminated. Round 2 votes counts: C=35 A=32 E=22 B=11 so B is eliminated. Round 3 votes counts: C=38 A=32 E=30 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:208 E:205 A:189 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -6 2 -4 B 14 0 -6 22 12 C 6 6 0 2 2 D -2 -22 -2 0 -20 E 4 -12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 2 -4 B 14 0 -6 22 12 C 6 6 0 2 2 D -2 -22 -2 0 -20 E 4 -12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 2 -4 B 14 0 -6 22 12 C 6 6 0 2 2 D -2 -22 -2 0 -20 E 4 -12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2366: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) C B A E D (5) B D C E A (5) A E C B D (5) D E B A C (4) C A E B D (4) C A B E D (4) A E D C B (4) E A C B D (3) D E A B C (3) D C B A E (3) D B E A C (3) D A E C B (3) B D E C A (3) B C D A E (3) A C E D B (3) E A D B C (2) D B E C A (2) D B C E A (2) C B D A E (2) C A E D B (2) C A B D E (2) B C D E A (2) A E C D B (2) A D C E B (2) A C E B D (2) E D B A C (1) E D A C B (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D C B (1) E A C D B (1) E A B D C (1) E A B C D (1) C D B A E (1) C D A B E (1) C A D E B (1) C A D B E (1) B E A C D (1) B D E A C (1) B C E D A (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -4 14 18 B -2 0 -22 10 0 C 4 22 0 12 10 D -14 -10 -12 0 4 E -18 0 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 14 18 B -2 0 -22 10 0 C 4 22 0 12 10 D -14 -10 -12 0 4 E -18 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=20 A=19 B=17 E=14 so E is eliminated. Round 2 votes counts: C=30 A=28 D=23 B=19 so B is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:215 B:193 D:184 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 14 18 B -2 0 -22 10 0 C 4 22 0 12 10 D -14 -10 -12 0 4 E -18 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 14 18 B -2 0 -22 10 0 C 4 22 0 12 10 D -14 -10 -12 0 4 E -18 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 14 18 B -2 0 -22 10 0 C 4 22 0 12 10 D -14 -10 -12 0 4 E -18 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2367: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) E D C A B (6) B A C D E (6) A D E C B (6) B C A E D (5) D E A C B (4) A D E B C (4) E D C B A (3) C E B D A (3) B E C D A (3) B C A D E (3) B A E D C (3) A B D E C (3) E C D B A (2) C E D A B (2) C D E A B (2) B E A C D (2) B C E D A (2) B C E A D (2) B A E C D (2) B A D C E (2) A D C E B (2) A B D C E (2) A B C D E (2) E D B C A (1) E D A B C (1) E C D A B (1) E B D A C (1) E B C D A (1) D E C A B (1) D C E A B (1) D C A E B (1) C E D B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C B A D E (1) C A B D E (1) B A C E D (1) A D C B E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 8 10 0 18 B -8 0 -6 -8 -10 C -10 6 0 -8 -10 D 0 8 8 0 14 E -18 10 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.441442 B: 0.000000 C: 0.000000 D: 0.558558 E: 0.000000 Sum of squares = 0.506857992423 Cumulative probabilities = A: 0.441442 B: 0.441442 C: 0.441442 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 0 18 B -8 0 -6 -8 -10 C -10 6 0 -8 -10 D 0 8 8 0 14 E -18 10 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999696 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=22 D=18 E=16 C=13 so C is eliminated. Round 2 votes counts: B=33 A=23 E=22 D=22 so E is eliminated. Round 3 votes counts: D=39 B=38 A=23 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:215 E:194 C:189 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 0 18 B -8 0 -6 -8 -10 C -10 6 0 -8 -10 D 0 8 8 0 14 E -18 10 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999696 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 18 B -8 0 -6 -8 -10 C -10 6 0 -8 -10 D 0 8 8 0 14 E -18 10 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999696 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 18 B -8 0 -6 -8 -10 C -10 6 0 -8 -10 D 0 8 8 0 14 E -18 10 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999696 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2368: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (12) E C D A B (8) B A D E C (7) C E D A B (6) E C B A D (5) D A B E C (5) B C A D E (5) C E B D A (4) C E B A D (4) B A E D C (4) B A C D E (4) A D B E C (4) E D C A B (3) C B A D E (3) E D A C B (2) D E A C B (2) D C E A B (2) C E D B A (2) C D A B E (2) E D A B C (1) E B A D C (1) E A D B C (1) D C A B E (1) D A E C B (1) D A E B C (1) D A B C E (1) C D E A B (1) C D B A E (1) C D A E B (1) B E C A D (1) B E A D C (1) B E A C D (1) B C A E D (1) B A E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 0 12 10 B 14 0 4 10 10 C 0 -4 0 0 0 D -12 -10 0 0 8 E -10 -10 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 12 10 B 14 0 4 10 10 C 0 -4 0 0 0 D -12 -10 0 0 8 E -10 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=24 E=21 D=13 A=5 so A is eliminated. Round 2 votes counts: B=38 C=24 E=21 D=17 so D is eliminated. Round 3 votes counts: B=48 C=27 E=25 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:204 C:198 D:193 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 12 10 B 14 0 4 10 10 C 0 -4 0 0 0 D -12 -10 0 0 8 E -10 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 12 10 B 14 0 4 10 10 C 0 -4 0 0 0 D -12 -10 0 0 8 E -10 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 12 10 B 14 0 4 10 10 C 0 -4 0 0 0 D -12 -10 0 0 8 E -10 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2369: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (15) E B A D C (10) B E A D C (8) C D A E B (7) B E C D A (6) D C A B E (4) C D B A E (4) E A B D C (3) D A C E B (3) B C E D A (3) A D E C B (3) E B A C D (2) D A C B E (2) C B D E A (2) B C D E A (2) A E D C B (2) A E D B C (2) A D C E B (2) E C A D B (1) E B C A D (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B C D (1) D C A E B (1) C E B D A (1) C D B E A (1) C B E D A (1) C B D A E (1) C A D E B (1) B E D A C (1) B E C A D (1) B E A C D (1) B A E D C (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 8 -4 -8 2 B -8 0 -10 -6 6 C 4 10 0 8 0 D 8 6 -8 0 -2 E -2 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.580587 D: 0.000000 E: 0.419413 Sum of squares = 0.512988512678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.580587 D: 0.580587 E: 1.000000 A B C D E A 0 8 -4 -8 2 B -8 0 -10 -6 6 C 4 10 0 8 0 D 8 6 -8 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=23 E=22 A=12 D=10 so D is eliminated. Round 2 votes counts: C=38 B=23 E=22 A=17 so A is eliminated. Round 3 votes counts: C=45 E=32 B=23 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:202 A:199 E:197 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -8 2 B -8 0 -10 -6 6 C 4 10 0 8 0 D 8 6 -8 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -8 2 B -8 0 -10 -6 6 C 4 10 0 8 0 D 8 6 -8 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -8 2 B -8 0 -10 -6 6 C 4 10 0 8 0 D 8 6 -8 0 -2 E -2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2370: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) A E C B D (8) D A E B C (6) A E D B C (6) A D E B C (6) D A B E C (5) D C B E A (4) C B E D A (4) A E D C B (4) A E C D B (4) A E B C D (4) E A C B D (3) C B D E A (3) B E A C D (3) B D C E A (3) E C B A D (2) E A B C D (2) D A C E B (2) C D B E A (2) C D A E B (2) C B E A D (2) B C E A D (2) B C D E A (2) D C B A E (1) D C A B E (1) D B C A E (1) C E B A D (1) C E A B D (1) C D E B A (1) B E C A D (1) B E A D C (1) B D E C A (1) B D E A C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 12 14 4 4 B -12 0 6 -12 -8 C -14 -6 0 -6 -16 D -4 12 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 4 4 B -12 0 6 -12 -8 C -14 -6 0 -6 -16 D -4 12 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=29 C=16 B=14 E=7 so E is eliminated. Round 2 votes counts: A=39 D=29 C=18 B=14 so B is eliminated. Round 3 votes counts: A=43 D=34 C=23 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:209 E:208 B:187 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 4 4 B -12 0 6 -12 -8 C -14 -6 0 -6 -16 D -4 12 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 4 4 B -12 0 6 -12 -8 C -14 -6 0 -6 -16 D -4 12 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 4 4 B -12 0 6 -12 -8 C -14 -6 0 -6 -16 D -4 12 6 0 4 E -4 8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2371: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) E A C D B (7) B C D A E (7) C B E A D (6) D B C A E (5) B D C A E (5) B C D E A (5) E A D C B (4) E A C B D (4) D A B E C (4) A E D B C (4) C E A B D (3) C B D E A (3) B C E A D (3) A D E B C (3) E C A B D (2) D E A C B (2) D B A C E (2) C E B A D (2) C D E A B (2) C D B E A (2) B A E C D (2) E C D A B (1) E C A D B (1) E A D B C (1) D C E B A (1) D C B E A (1) D A E C B (1) C B E D A (1) B C A D E (1) A E B C D (1) A D E C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -2 -6 -2 B -8 0 12 -6 -2 C 2 -12 0 6 -2 D 6 6 -6 0 14 E 2 2 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -6 -2 B -8 0 12 -6 -2 C 2 -12 0 6 -2 D 6 6 -6 0 14 E 2 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 E=20 C=19 A=11 so A is eliminated. Round 2 votes counts: D=31 E=25 B=25 C=19 so C is eliminated. Round 3 votes counts: D=35 B=35 E=30 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:199 B:198 C:197 E:196 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 -6 -2 B -8 0 12 -6 -2 C 2 -12 0 6 -2 D 6 6 -6 0 14 E 2 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -6 -2 B -8 0 12 -6 -2 C 2 -12 0 6 -2 D 6 6 -6 0 14 E 2 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -6 -2 B -8 0 12 -6 -2 C 2 -12 0 6 -2 D 6 6 -6 0 14 E 2 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2372: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (11) D A E B C (7) A E C B D (7) D B C E A (5) D B C A E (5) B C E A D (5) E A C B D (4) C B E A D (4) C B D E A (4) A E D C B (4) E C A B D (3) E A D C B (3) E A C D B (3) D E A C B (3) D B A E C (3) D A E C B (3) B D C A E (3) B C D A E (3) B C A E D (3) D C B E A (2) C E A B D (2) B D C E A (2) A E B C D (2) A D E B C (2) D E C B A (1) D E C A B (1) D C E A B (1) D B E A C (1) D B A C E (1) D A B E C (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 -10 -14 -10 B 6 0 8 6 6 C 10 -8 0 2 2 D 14 -6 -2 0 20 E 10 -6 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -14 -10 B 6 0 8 6 6 C 10 -8 0 2 2 D 14 -6 -2 0 20 E 10 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=27 A=16 E=13 C=10 so C is eliminated. Round 2 votes counts: B=35 D=34 A=16 E=15 so E is eliminated. Round 3 votes counts: B=35 D=34 A=31 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:213 C:203 E:191 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 -14 -10 B 6 0 8 6 6 C 10 -8 0 2 2 D 14 -6 -2 0 20 E 10 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -14 -10 B 6 0 8 6 6 C 10 -8 0 2 2 D 14 -6 -2 0 20 E 10 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -14 -10 B 6 0 8 6 6 C 10 -8 0 2 2 D 14 -6 -2 0 20 E 10 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2373: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (13) C D E B A (12) A B E C D (9) A B E D C (7) B A E D C (6) D C E B A (5) E B A D C (4) A E B C D (4) E A B C D (3) B A D E C (3) B A D C E (3) A B D E C (3) A B C D E (3) E D C B A (2) D B C A E (2) C E D B A (2) C D A B E (2) B D A C E (2) A C B D E (2) A B D C E (2) E D B C A (1) E A C B D (1) D E C B A (1) D C B E A (1) D C B A E (1) D C A B E (1) C E D A B (1) C E A D B (1) C A D B E (1) B E A D C (1) B D E A C (1) Total count = 100 A B C D E A 0 6 8 6 2 B -6 0 8 8 0 C -8 -8 0 8 8 D -6 -8 -8 0 16 E -2 0 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 6 2 B -6 0 8 8 0 C -8 -8 0 8 8 D -6 -8 -8 0 16 E -2 0 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=30 B=16 E=11 D=11 so E is eliminated. Round 2 votes counts: A=34 C=32 B=20 D=14 so D is eliminated. Round 3 votes counts: C=43 A=34 B=23 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:205 C:200 D:197 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 6 2 B -6 0 8 8 0 C -8 -8 0 8 8 D -6 -8 -8 0 16 E -2 0 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 6 2 B -6 0 8 8 0 C -8 -8 0 8 8 D -6 -8 -8 0 16 E -2 0 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 6 2 B -6 0 8 8 0 C -8 -8 0 8 8 D -6 -8 -8 0 16 E -2 0 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2374: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) B E A C D (7) B D A E C (6) D C A E B (4) B C D E A (4) E C A B D (3) D B A E C (3) D A E C B (3) C E B A D (3) C E A B D (3) C A E D B (3) B E A D C (3) A E D C B (3) A E C D B (3) E A C B D (2) E A B C D (2) D C B A E (2) D B C A E (2) D A B E C (2) D A B C E (2) C D B A E (2) C B E D A (2) C B E A D (2) C A D E B (2) B E C A D (2) B D E A C (2) B C E A D (2) B A E D C (2) A E D B C (2) A E B D C (2) E B A C D (1) E A B D C (1) D A E B C (1) C E B D A (1) C D E B A (1) C D B E A (1) C B D E A (1) B E D A C (1) B E C D A (1) B D C E A (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 2 2 8 B 4 0 2 10 2 C -2 -2 0 10 -10 D -2 -10 -10 0 -2 E -8 -2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 2 8 B 4 0 2 10 2 C -2 -2 0 10 -10 D -2 -10 -10 0 -2 E -8 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=28 D=19 A=13 E=9 so E is eliminated. Round 2 votes counts: B=32 C=31 D=19 A=18 so A is eliminated. Round 3 votes counts: B=38 C=36 D=26 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:204 E:201 C:198 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 2 8 B 4 0 2 10 2 C -2 -2 0 10 -10 D -2 -10 -10 0 -2 E -8 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 2 8 B 4 0 2 10 2 C -2 -2 0 10 -10 D -2 -10 -10 0 -2 E -8 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 2 8 B 4 0 2 10 2 C -2 -2 0 10 -10 D -2 -10 -10 0 -2 E -8 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2375: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E C A B (9) E D C A B (6) B A D C E (6) B A C E D (6) E C D A B (4) D A B E C (4) A B C E D (4) E C A B D (3) D E C B A (3) D E A C B (3) D E A B C (3) D A E B C (3) C B E D A (3) E D C B A (2) E D A C B (2) D C E B A (2) D B A E C (2) D B A C E (2) C E D B A (2) C E B D A (2) C E B A D (2) C E A B D (2) A D B E C (2) A B E C D (2) A B D E C (2) E C A D B (1) E A C D B (1) D B E A C (1) C B A E D (1) B D A C E (1) B C D A E (1) B C A D E (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -8 -10 -2 B -4 0 4 -4 0 C 8 -4 0 -8 -6 D 10 4 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.208018 C: 0.000000 D: 0.000000 E: 0.791982 Sum of squares = 0.670507350762 Cumulative probabilities = A: 0.000000 B: 0.208018 C: 0.208018 D: 0.208018 E: 1.000000 A B C D E A 0 4 -8 -10 -2 B -4 0 4 -4 0 C 8 -4 0 -8 -6 D 10 4 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555558511 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=26 E=19 C=12 A=11 so A is eliminated. Round 2 votes counts: B=35 D=34 E=19 C=12 so C is eliminated. Round 3 votes counts: B=39 D=34 E=27 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:207 B:198 C:195 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -2 B -4 0 4 -4 0 C 8 -4 0 -8 -6 D 10 4 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555558511 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -2 B -4 0 4 -4 0 C 8 -4 0 -8 -6 D 10 4 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555558511 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -2 B -4 0 4 -4 0 C 8 -4 0 -8 -6 D 10 4 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555558511 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2376: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) C B E A D (13) E A C B D (11) D B C A E (9) D A E B C (7) C B D E A (5) A E D C B (4) A E D B C (4) A D E B C (4) E A B C D (3) D A E C B (3) D A B E C (3) B C E D A (3) A D E C B (3) D B A C E (2) D A B C E (2) C B E D A (2) A E C D B (2) E C A B D (1) D C B A E (1) D B A E C (1) D A C E B (1) D A C B E (1) B C E A D (1) Total count = 100 A B C D E A 0 -2 2 -8 -6 B 2 0 6 6 14 C -2 -6 0 10 8 D 8 -6 -10 0 12 E 6 -14 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -8 -6 B 2 0 6 6 14 C -2 -6 0 10 8 D 8 -6 -10 0 12 E 6 -14 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=20 B=18 A=17 E=15 so E is eliminated. Round 2 votes counts: A=31 D=30 C=21 B=18 so B is eliminated. Round 3 votes counts: C=39 A=31 D=30 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:214 C:205 D:202 A:193 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -8 -6 B 2 0 6 6 14 C -2 -6 0 10 8 D 8 -6 -10 0 12 E 6 -14 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -8 -6 B 2 0 6 6 14 C -2 -6 0 10 8 D 8 -6 -10 0 12 E 6 -14 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -8 -6 B 2 0 6 6 14 C -2 -6 0 10 8 D 8 -6 -10 0 12 E 6 -14 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2377: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) E C D A B (10) D E A C B (8) B C E A D (8) A D E B C (8) C E B D A (7) B A D C E (7) A D B E C (6) C B E A D (5) B C E D A (4) B C A E D (4) D A E C B (3) C E D B A (3) E D C A B (1) E D A C B (1) E C A D B (1) D A B E C (1) C B E D A (1) B D A E C (1) B D A C E (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C D E (1) A C E B D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 6 14 0 B -10 0 -8 -10 -12 C -6 8 0 -2 -6 D -14 10 2 0 6 E 0 12 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555316 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.444684 Sum of squares = 0.506119821118 Cumulative probabilities = A: 0.555316 B: 0.555316 C: 0.555316 D: 0.555316 E: 1.000000 A B C D E A 0 10 6 14 0 B -10 0 -8 -10 -12 C -6 8 0 -2 -6 D -14 10 2 0 6 E 0 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=29 C=16 E=13 D=12 so D is eliminated. Round 2 votes counts: A=34 B=29 E=21 C=16 so C is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:206 D:202 C:197 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 14 0 B -10 0 -8 -10 -12 C -6 8 0 -2 -6 D -14 10 2 0 6 E 0 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 14 0 B -10 0 -8 -10 -12 C -6 8 0 -2 -6 D -14 10 2 0 6 E 0 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 14 0 B -10 0 -8 -10 -12 C -6 8 0 -2 -6 D -14 10 2 0 6 E 0 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2378: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) C E A B D (9) A C E D B (7) E B C D A (6) D B A E C (6) D A B C E (6) D A B E C (5) E C B A D (4) C E B A D (4) A C E B D (4) E B C A D (3) D E B C A (3) D A C E B (3) C A E B D (3) A D C B E (3) A D B C E (3) A C D E B (3) E C B D A (2) D A C B E (2) B E D C A (2) B D A E C (2) E D B C A (1) D E C B A (1) D C E A B (1) C A E D B (1) B E C D A (1) B D E C A (1) B D E A C (1) A C D B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 16 -6 2 B -6 0 4 -12 -10 C -16 -4 0 0 4 D 6 12 0 0 4 E -2 10 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.153242 D: 0.846758 E: 0.000000 Sum of squares = 0.740481987703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.153242 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 -6 2 B -6 0 4 -12 -10 C -16 -4 0 0 4 D 6 12 0 0 4 E -2 10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.272727 D: 0.727273 E: 0.000000 Sum of squares = 0.603305863959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.272727 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=23 C=17 E=16 B=7 so B is eliminated. Round 2 votes counts: D=41 A=23 E=19 C=17 so C is eliminated. Round 3 votes counts: D=41 E=32 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:209 E:200 C:192 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 16 -6 2 B -6 0 4 -12 -10 C -16 -4 0 0 4 D 6 12 0 0 4 E -2 10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.272727 D: 0.727273 E: 0.000000 Sum of squares = 0.603305863959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.272727 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 -6 2 B -6 0 4 -12 -10 C -16 -4 0 0 4 D 6 12 0 0 4 E -2 10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.272727 D: 0.727273 E: 0.000000 Sum of squares = 0.603305863959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.272727 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 -6 2 B -6 0 4 -12 -10 C -16 -4 0 0 4 D 6 12 0 0 4 E -2 10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.272727 D: 0.727273 E: 0.000000 Sum of squares = 0.603305863959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.272727 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2379: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (17) D C A E B (9) B A E C D (8) D C E A B (7) A B D C E (7) A B E D C (6) E B C A D (5) C E D B A (5) B E A C D (5) E B A C D (4) D C A B E (4) E C D B A (3) E C B D A (3) D C E B A (3) D A C B E (3) A D B C E (3) A B E C D (2) A B D E C (2) E C B A D (1) C E B D A (1) A E B D C (1) A D C B E (1) Total count = 100 A B C D E A 0 -10 -16 -10 -8 B 10 0 -14 -10 -18 C 16 14 0 8 20 D 10 10 -8 0 12 E 8 18 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -10 -8 B 10 0 -14 -10 -18 C 16 14 0 8 20 D 10 10 -8 0 12 E 8 18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 A=22 E=16 B=13 so B is eliminated. Round 2 votes counts: A=30 D=26 C=23 E=21 so E is eliminated. Round 3 votes counts: A=39 C=35 D=26 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:212 E:197 B:184 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -16 -10 -8 B 10 0 -14 -10 -18 C 16 14 0 8 20 D 10 10 -8 0 12 E 8 18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -10 -8 B 10 0 -14 -10 -18 C 16 14 0 8 20 D 10 10 -8 0 12 E 8 18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -10 -8 B 10 0 -14 -10 -18 C 16 14 0 8 20 D 10 10 -8 0 12 E 8 18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2380: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D E A B C (5) D C B A E (5) B C E A D (5) D C A B E (4) D A C B E (4) C B D E A (4) C B A E D (4) B C A E D (4) E B A C D (3) E A D B C (3) D E C B A (3) D E A C B (3) D C B E A (3) D A E C B (3) C D B A E (3) C B E D A (3) C B A D E (3) B A C E D (3) A B E C D (3) E D B C A (2) E A B D C (2) E A B C D (2) D E C A B (2) D C E B A (2) D A E B C (2) A E D B C (2) A E B C D (2) A D E B C (2) E B C D A (1) D C A E B (1) D A C E B (1) C D B E A (1) C B D A E (1) C A B D E (1) B C E D A (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -6 -18 0 B 2 0 -4 -14 8 C 6 4 0 -10 10 D 18 14 10 0 8 E 0 -8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -18 0 B 2 0 -4 -14 8 C 6 4 0 -10 10 D 18 14 10 0 8 E 0 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=20 E=18 B=13 A=11 so A is eliminated. Round 2 votes counts: D=41 E=22 C=20 B=17 so B is eliminated. Round 3 votes counts: D=41 C=34 E=25 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 C:205 B:196 A:187 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -18 0 B 2 0 -4 -14 8 C 6 4 0 -10 10 D 18 14 10 0 8 E 0 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -18 0 B 2 0 -4 -14 8 C 6 4 0 -10 10 D 18 14 10 0 8 E 0 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -18 0 B 2 0 -4 -14 8 C 6 4 0 -10 10 D 18 14 10 0 8 E 0 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2381: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (11) E D C A B (9) E C D A B (9) C A E B D (9) D E B C A (7) B A C D E (7) D B E A C (6) A B C D E (6) D E B A C (5) B D A E C (5) E D C B A (4) C E A D B (3) C E A B D (3) B D A C E (3) D E C B A (2) D B A E C (2) C A B E D (2) A C B E D (2) E C A D B (1) C B A E D (1) B C D A E (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -2 -6 2 B 10 0 10 4 -4 C 2 -10 0 -10 -2 D 6 -4 10 0 12 E -2 4 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.440000000035 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 -10 -2 -6 2 B 10 0 10 4 -4 C 2 -10 0 -10 -2 D 6 -4 10 0 12 E -2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=23 D=22 C=18 A=9 so A is eliminated. Round 2 votes counts: B=35 E=23 D=22 C=20 so C is eliminated. Round 3 votes counts: B=40 E=38 D=22 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:212 B:210 E:196 A:192 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -6 2 B 10 0 10 4 -4 C 2 -10 0 -10 -2 D 6 -4 10 0 12 E -2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -6 2 B 10 0 10 4 -4 C 2 -10 0 -10 -2 D 6 -4 10 0 12 E -2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -6 2 B 10 0 10 4 -4 C 2 -10 0 -10 -2 D 6 -4 10 0 12 E -2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2382: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) E A D B C (9) D B C E A (8) E A D C B (5) E A C D B (5) B D C E A (5) B C D A E (5) B D C A E (4) C D E A B (3) C A E D B (3) B C A D E (3) A E B C D (3) E A B D C (2) D E A C B (2) D C E A B (2) D C B E A (2) C B D A E (2) B D E A C (2) A B E C D (2) E D C A B (1) E D B A C (1) E D A B C (1) D E C A B (1) D E B A C (1) D C E B A (1) D B E A C (1) C E D A B (1) C D E B A (1) C D B E A (1) C B A D E (1) C A D E B (1) C A B E D (1) B E D A C (1) B D A E C (1) B A E C D (1) B A D E C (1) B A C E D (1) B A C D E (1) A E C B D (1) A E B D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 12 8 6 -12 B -12 0 8 -22 -14 C -8 -8 0 -4 -4 D -6 22 4 0 -2 E 12 14 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 8 6 -12 B -12 0 8 -22 -14 C -8 -8 0 -4 -4 D -6 22 4 0 -2 E 12 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 A=19 D=18 C=14 so C is eliminated. Round 2 votes counts: B=28 E=25 A=24 D=23 so D is eliminated. Round 3 votes counts: B=40 E=36 A=24 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:209 A:207 C:188 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 8 6 -12 B -12 0 8 -22 -14 C -8 -8 0 -4 -4 D -6 22 4 0 -2 E 12 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 6 -12 B -12 0 8 -22 -14 C -8 -8 0 -4 -4 D -6 22 4 0 -2 E 12 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 6 -12 B -12 0 8 -22 -14 C -8 -8 0 -4 -4 D -6 22 4 0 -2 E 12 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2383: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (16) D A C B E (11) B C E A D (9) E D A B C (8) B C A E D (8) D A E C B (5) E B C D A (4) C B A D E (4) A C D B E (4) D A C E B (3) B E C A D (3) A D C B E (3) E D B C A (2) E D B A C (2) D E A C B (2) D E A B C (2) C A B D E (2) A C B D E (2) E B D C A (1) E B A D C (1) E B A C D (1) D C A B E (1) D A E B C (1) C B D A E (1) C B A E D (1) B C E D A (1) B C A D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -8 12 -4 B 10 0 22 10 2 C 8 -22 0 14 2 D -12 -10 -14 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 12 -4 B 10 0 22 10 2 C 8 -22 0 14 2 D -12 -10 -14 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990037 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=25 B=22 A=10 C=8 so C is eliminated. Round 2 votes counts: E=35 B=28 D=25 A=12 so A is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:207 C:201 A:195 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 12 -4 B 10 0 22 10 2 C 8 -22 0 14 2 D -12 -10 -14 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990037 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 12 -4 B 10 0 22 10 2 C 8 -22 0 14 2 D -12 -10 -14 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990037 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 12 -4 B 10 0 22 10 2 C 8 -22 0 14 2 D -12 -10 -14 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990037 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2384: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (12) E A D C B (7) C B E A D (7) A E D B C (6) E C A D B (5) B C D A E (5) E A C D B (4) D E A C B (4) D B A E C (4) C E A D B (4) C B D E A (4) B C A E D (4) E A C B D (3) C E A B D (3) B D A E C (3) D E A B C (2) D A B E C (2) B D C A E (2) B D A C E (2) B A E D C (2) B A D E C (2) A B E C D (2) D C E A B (1) D B C E A (1) D B C A E (1) C E D A B (1) C E B A D (1) C D B E A (1) C B E D A (1) C B A E D (1) C A E B D (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 18 14 8 2 B -18 0 2 -14 -12 C -14 -2 0 -4 -20 D -8 14 4 0 -6 E -2 12 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 14 8 2 B -18 0 2 -14 -12 C -14 -2 0 -4 -20 D -8 14 4 0 -6 E -2 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 B=20 E=19 A=10 so A is eliminated. Round 2 votes counts: D=28 E=26 C=24 B=22 so B is eliminated. Round 3 votes counts: D=37 C=33 E=30 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:221 E:218 D:202 C:180 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 14 8 2 B -18 0 2 -14 -12 C -14 -2 0 -4 -20 D -8 14 4 0 -6 E -2 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 14 8 2 B -18 0 2 -14 -12 C -14 -2 0 -4 -20 D -8 14 4 0 -6 E -2 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 14 8 2 B -18 0 2 -14 -12 C -14 -2 0 -4 -20 D -8 14 4 0 -6 E -2 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2385: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) A B C D E (10) E D C B A (9) E C D A B (9) D E B C A (8) A C B D E (8) C A E B D (7) B A D C E (7) E D B C A (4) C E A D B (4) B D A E C (4) E C D B A (3) E C A D B (3) C E A B D (3) C A B E D (2) A B D C E (2) D E B A C (1) D B A E C (1) C E D A B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -6 -4 -14 B -2 0 0 -10 -4 C 6 0 0 2 -10 D 4 10 -2 0 8 E 14 4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 A B C D E A 0 2 -6 -4 -14 B -2 0 0 -10 -4 C 6 0 0 2 -10 D 4 10 -2 0 8 E 14 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=22 A=22 C=17 B=11 so B is eliminated. Round 2 votes counts: A=29 E=28 D=26 C=17 so C is eliminated. Round 3 votes counts: A=38 E=36 D=26 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:210 C:199 B:192 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -14 B -2 0 0 -10 -4 C 6 0 0 2 -10 D 4 10 -2 0 8 E 14 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -14 B -2 0 0 -10 -4 C 6 0 0 2 -10 D 4 10 -2 0 8 E 14 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -14 B -2 0 0 -10 -4 C 6 0 0 2 -10 D 4 10 -2 0 8 E 14 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2386: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) E C D A B (6) C D E B A (6) B A D C E (6) D C E A B (4) D C B A E (4) D C A B E (4) C D B A E (4) A B E D C (4) E D C A B (3) E A D C B (3) E A B C D (3) D C A E B (3) C E D B A (3) C D B E A (3) B A E D C (3) A E B D C (3) C E D A B (2) B E C A D (2) B A E C D (2) B A C D E (2) A B D C E (2) E C B D A (1) E C B A D (1) E B C D A (1) E B A C D (1) E A B D C (1) D E C A B (1) D A E C B (1) D A C B E (1) C E B D A (1) C B E D A (1) C B D A E (1) B E A C D (1) B C D E A (1) B C D A E (1) B C A D E (1) B A D E C (1) A E D B C (1) A D E B C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -24 -20 -6 B -6 0 -22 -18 -8 C 24 22 0 4 18 D 20 18 -4 0 14 E 6 8 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -24 -20 -6 B -6 0 -22 -18 -8 C 24 22 0 4 18 D 20 18 -4 0 14 E 6 8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=20 B=20 D=18 A=13 so A is eliminated. Round 2 votes counts: C=29 B=27 E=24 D=20 so D is eliminated. Round 3 votes counts: C=45 B=28 E=27 so E is eliminated. Round 4 votes counts: C=61 B=39 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:234 D:224 E:191 A:178 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -24 -20 -6 B -6 0 -22 -18 -8 C 24 22 0 4 18 D 20 18 -4 0 14 E 6 8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -24 -20 -6 B -6 0 -22 -18 -8 C 24 22 0 4 18 D 20 18 -4 0 14 E 6 8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -24 -20 -6 B -6 0 -22 -18 -8 C 24 22 0 4 18 D 20 18 -4 0 14 E 6 8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2387: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) E C A B D (5) D B A C E (5) E C A D B (4) E A C D B (4) D B C A E (4) C D E A B (4) B E A D C (4) A E B D C (4) A D B E C (4) D C B A E (3) C E B D A (3) C D E B A (3) C D B A E (3) C D A E B (3) B D C E A (3) A E D B C (3) A B D E C (3) E A C B D (2) E A B D C (2) D C A B E (2) D A C B E (2) C D B E A (2) B D A E C (2) B D A C E (2) B A D E C (2) A E D C B (2) A B E D C (2) E C B A D (1) D A B C E (1) C D A B E (1) C B D E A (1) B E D C A (1) B E C D A (1) B C D E A (1) A D E C B (1) A D E B C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -4 -8 18 B -4 0 6 -8 14 C 4 -6 0 -22 4 D 8 8 22 0 22 E -18 -14 -4 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -8 18 B -4 0 6 -8 14 C 4 -6 0 -22 4 D 8 8 22 0 22 E -18 -14 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 A=22 C=20 E=18 D=17 so D is eliminated. Round 2 votes counts: B=32 C=25 A=25 E=18 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:230 A:205 B:204 C:190 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -8 18 B -4 0 6 -8 14 C 4 -6 0 -22 4 D 8 8 22 0 22 E -18 -14 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 18 B -4 0 6 -8 14 C 4 -6 0 -22 4 D 8 8 22 0 22 E -18 -14 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 18 B -4 0 6 -8 14 C 4 -6 0 -22 4 D 8 8 22 0 22 E -18 -14 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2388: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (13) D E B A C (12) E A B D C (7) D B E A C (7) C A E B D (7) E D A B C (5) E D A C B (3) E C A B D (3) E A D B C (3) D B C A E (3) D B A E C (3) A E B C D (3) D B A C E (2) C E A B D (2) C D B A E (2) C B A D E (2) B A C D E (2) A B E C D (2) A B C E D (2) E D B A C (1) E C D A B (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E C B A (1) D E B C A (1) D C B A E (1) C B D A E (1) C A B D E (1) B D A E C (1) B C A D E (1) B A E C D (1) B A D C E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 16 20 12 0 B -16 0 16 4 -8 C -20 -16 0 -4 -18 D -12 -4 4 0 -18 E 0 8 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.512756 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.487244 Sum of squares = 0.500325446542 Cumulative probabilities = A: 0.512756 B: 0.512756 C: 0.512756 D: 0.512756 E: 1.000000 A B C D E A 0 16 20 12 0 B -16 0 16 4 -8 C -20 -16 0 -4 -18 D -12 -4 4 0 -18 E 0 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=28 E=27 A=9 B=6 so B is eliminated. Round 2 votes counts: D=31 C=29 E=27 A=13 so A is eliminated. Round 3 votes counts: E=34 C=34 D=32 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:224 E:222 B:198 D:185 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 16 20 12 0 B -16 0 16 4 -8 C -20 -16 0 -4 -18 D -12 -4 4 0 -18 E 0 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 20 12 0 B -16 0 16 4 -8 C -20 -16 0 -4 -18 D -12 -4 4 0 -18 E 0 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 20 12 0 B -16 0 16 4 -8 C -20 -16 0 -4 -18 D -12 -4 4 0 -18 E 0 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2389: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) B E A D C (7) C D A E B (5) C A D E B (5) C A D B E (5) D C E A B (4) D A E C B (4) C B A E D (4) B C E A D (4) A C D E B (4) D E B C A (3) C D E B A (3) C D B A E (3) B A E D C (3) A B E D C (3) D E C B A (2) D E C A B (2) C D B E A (2) C B D E A (2) B E D A C (2) B A C E D (2) A D C E B (2) E D B A C (1) E D A B C (1) E B D A C (1) D E A C B (1) D E A B C (1) D C E B A (1) D C A E B (1) D A E B C (1) D A C E B (1) C D E A B (1) C D A B E (1) C B A D E (1) B C E D A (1) B C A E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -16 -2 8 B 0 0 -14 -14 6 C 16 14 0 -2 14 D 2 14 2 0 16 E -8 -6 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -2 8 B 0 0 -14 -14 6 C 16 14 0 -2 14 D 2 14 2 0 16 E -8 -6 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=27 D=21 A=17 E=3 so E is eliminated. Round 2 votes counts: C=32 B=28 D=23 A=17 so A is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:217 A:195 B:189 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -16 -2 8 B 0 0 -14 -14 6 C 16 14 0 -2 14 D 2 14 2 0 16 E -8 -6 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -2 8 B 0 0 -14 -14 6 C 16 14 0 -2 14 D 2 14 2 0 16 E -8 -6 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -2 8 B 0 0 -14 -14 6 C 16 14 0 -2 14 D 2 14 2 0 16 E -8 -6 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2390: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) C A E B D (7) E C A D B (6) C E B D A (6) D B A E C (5) A D B E C (4) A C E D B (4) A B D C E (4) E D B C A (3) E D A B C (3) C B E D A (3) C A B E D (3) B D A C E (3) B A D C E (3) A E C D B (3) A D B C E (3) E C B D A (2) E A C D B (2) D E B A C (2) C E A B D (2) C B D A E (2) B D E C A (2) B D C A E (2) A E D C B (2) A E D B C (2) A B C D E (2) E D C A B (1) E A D B C (1) D A E B C (1) C B A D E (1) C A E D B (1) C A B D E (1) B D E A C (1) B D C E A (1) B D A E C (1) B C E D A (1) B C D E A (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 8 10 6 12 B -8 0 6 -4 2 C -10 -6 0 -6 2 D -6 4 6 0 -4 E -12 -2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 6 12 B -8 0 6 -4 2 C -10 -6 0 -6 2 D -6 4 6 0 -4 E -12 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 E=18 D=15 B=15 so D is eliminated. Round 2 votes counts: B=27 A=27 C=26 E=20 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:200 B:198 E:194 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 6 12 B -8 0 6 -4 2 C -10 -6 0 -6 2 D -6 4 6 0 -4 E -12 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 6 12 B -8 0 6 -4 2 C -10 -6 0 -6 2 D -6 4 6 0 -4 E -12 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 6 12 B -8 0 6 -4 2 C -10 -6 0 -6 2 D -6 4 6 0 -4 E -12 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2391: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) C D B A E (13) E A B D C (12) A E C B D (10) C A E D B (7) C D B E A (5) B D E C A (5) B D E A C (5) D C B E A (4) E B A D C (3) B D C E A (3) A E C D B (3) A E B D C (3) E A C B D (2) C D A B E (2) C A D E B (2) A E B C D (2) A C E D B (2) E C A B D (1) C E A B D (1) B E D A C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -12 -2 -10 B 4 0 -4 -4 4 C 12 4 0 0 6 D 2 4 0 0 6 E 10 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.409066 D: 0.590934 E: 0.000000 Sum of squares = 0.516538091891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.409066 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 -10 B 4 0 -4 -4 4 C 12 4 0 0 6 D 2 4 0 0 6 E 10 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=21 E=18 D=17 B=14 so B is eliminated. Round 2 votes counts: D=30 C=30 A=21 E=19 so E is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:206 B:200 E:197 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 -10 B 4 0 -4 -4 4 C 12 4 0 0 6 D 2 4 0 0 6 E 10 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 -10 B 4 0 -4 -4 4 C 12 4 0 0 6 D 2 4 0 0 6 E 10 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 -10 B 4 0 -4 -4 4 C 12 4 0 0 6 D 2 4 0 0 6 E 10 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2392: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (20) A B C D E (10) A B D C E (8) A B E D C (6) B A C D E (5) E D C A B (4) E C D B A (4) D C E B A (4) E D A C B (3) D C B E A (3) C E D B A (3) C B D A E (3) B C A D E (3) E C A B D (2) E A D B C (2) D E C B A (2) C D B E A (2) C D B A E (2) A E B C D (2) A B D E C (2) E D A B C (1) E C D A B (1) E A C D B (1) E A B C D (1) D E B A C (1) D E A B C (1) D C B A E (1) C D E B A (1) C B D E A (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -12 -14 -14 B 10 0 -14 -12 -6 C 12 14 0 -16 -8 D 14 12 16 0 -2 E 14 6 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 -14 -14 B 10 0 -14 -12 -6 C 12 14 0 -16 -8 D 14 12 16 0 -2 E 14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=29 D=12 C=12 B=8 so B is eliminated. Round 2 votes counts: E=39 A=34 C=15 D=12 so D is eliminated. Round 3 votes counts: E=43 A=34 C=23 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:220 E:215 C:201 B:189 A:175 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 -14 -14 B 10 0 -14 -12 -6 C 12 14 0 -16 -8 D 14 12 16 0 -2 E 14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -14 -14 B 10 0 -14 -12 -6 C 12 14 0 -16 -8 D 14 12 16 0 -2 E 14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -14 -14 B 10 0 -14 -12 -6 C 12 14 0 -16 -8 D 14 12 16 0 -2 E 14 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2393: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D A C B E (8) E B D C A (6) C A D B E (6) B E D C A (6) A C D E B (6) E B D A C (5) E B A C D (5) A C D B E (5) E B C D A (4) B D E C A (4) A D C B E (4) B D E A C (3) A D C E B (3) E C A B D (2) E A C B D (2) D E B A C (2) D C A B E (2) D B C A E (2) A E C B D (2) A C E D B (2) E D B A C (1) E B A D C (1) E A B C D (1) D B E A C (1) D B C E A (1) D B A C E (1) D A E B C (1) D A B E C (1) C D A B E (1) C A E D B (1) C A D E B (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 0 12 2 -4 B 0 0 6 -2 -10 C -12 -6 0 -8 -14 D -2 2 8 0 6 E 4 10 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888872 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 0 12 2 -4 B 0 0 6 -2 -10 C -12 -6 0 -8 -14 D -2 2 8 0 6 E 4 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888915 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=24 D=19 B=13 C=9 so C is eliminated. Round 2 votes counts: E=35 A=32 D=20 B=13 so B is eliminated. Round 3 votes counts: E=41 A=32 D=27 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:207 A:205 B:197 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 12 2 -4 B 0 0 6 -2 -10 C -12 -6 0 -8 -14 D -2 2 8 0 6 E 4 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888915 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 2 -4 B 0 0 6 -2 -10 C -12 -6 0 -8 -14 D -2 2 8 0 6 E 4 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888915 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 2 -4 B 0 0 6 -2 -10 C -12 -6 0 -8 -14 D -2 2 8 0 6 E 4 10 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888915 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2394: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) B A E C D (6) D C E A B (5) D C B E A (5) D C B A E (5) D C A B E (5) E D C A B (4) D B C E A (4) C D E A B (4) B E A D C (4) B E A C D (4) E B D C A (3) E B A C D (3) B D C A E (3) A B E C D (3) E B A D C (2) E A C D B (2) E A B C D (2) D C E B A (2) B E D A C (2) B D E C A (2) B A E D C (2) A E C B D (2) A E B C D (2) A C D E B (2) A C D B E (2) E D B C A (1) E C A D B (1) D E C A B (1) D C A E B (1) C D B A E (1) C D A B E (1) C B D A E (1) C A D E B (1) C A D B E (1) B E D C A (1) B C A D E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -20 -16 -4 B 4 0 -10 -12 8 C 20 10 0 -4 6 D 16 12 4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -20 -16 -4 B 4 0 -10 -12 8 C 20 10 0 -4 6 D 16 12 4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 E=18 C=16 A=13 so A is eliminated. Round 2 votes counts: D=28 B=28 E=22 C=22 so E is eliminated. Round 3 votes counts: B=40 D=33 C=27 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:216 B:195 E:191 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -20 -16 -4 B 4 0 -10 -12 8 C 20 10 0 -4 6 D 16 12 4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -20 -16 -4 B 4 0 -10 -12 8 C 20 10 0 -4 6 D 16 12 4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -20 -16 -4 B 4 0 -10 -12 8 C 20 10 0 -4 6 D 16 12 4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2395: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) A D B C E (10) E C B D A (9) D A B C E (7) A D C B E (5) A C D B E (5) E C A D B (4) D B E A C (4) C E A B D (4) A C E D B (4) E C D B A (3) E C B A D (3) E B D C A (3) C E A D B (3) C A E D B (3) B D E C A (3) B D A E C (3) A C E B D (3) E C A B D (2) D B A C E (2) A C D E B (2) D E B C A (1) B E D C A (1) B D E A C (1) B D A C E (1) A D C E B (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 22 4 18 B -12 0 -4 -28 10 C -22 4 0 -6 6 D -4 28 6 0 16 E -18 -10 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 22 4 18 B -12 0 -4 -28 10 C -22 4 0 -6 6 D -4 28 6 0 16 E -18 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999036 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=24 D=24 C=10 B=9 so B is eliminated. Round 2 votes counts: A=33 D=32 E=25 C=10 so C is eliminated. Round 3 votes counts: A=36 E=32 D=32 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 D:223 C:191 B:183 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 22 4 18 B -12 0 -4 -28 10 C -22 4 0 -6 6 D -4 28 6 0 16 E -18 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999036 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 22 4 18 B -12 0 -4 -28 10 C -22 4 0 -6 6 D -4 28 6 0 16 E -18 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999036 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 22 4 18 B -12 0 -4 -28 10 C -22 4 0 -6 6 D -4 28 6 0 16 E -18 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999036 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2396: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) A B C D E (8) E C D B A (6) D E C B A (5) A B D C E (5) E D C A B (3) D E A B C (3) D B E C A (3) D B C E A (3) C E B A D (3) C B A E D (3) B C D E A (3) A E C B D (3) A D E B C (3) A D B E C (3) A B C E D (3) E D A C B (2) D E B C A (2) D E B A C (2) D B E A C (2) D B A C E (2) C E B D A (2) C B E D A (2) C A B E D (2) B C A D E (2) A C E B D (2) A C B E D (2) E C B D A (1) E C A B D (1) E A D C B (1) D A B E C (1) C E A B D (1) C D E B A (1) C B D E A (1) B C D A E (1) B C A E D (1) B A C D E (1) A E D C B (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -8 -6 -10 B 8 0 0 -4 -2 C 8 0 0 -2 -2 D 6 4 2 0 4 E 10 2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -6 -10 B 8 0 0 -4 -2 C 8 0 0 -2 -2 D 6 4 2 0 4 E 10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 E=22 C=15 B=8 so B is eliminated. Round 2 votes counts: A=33 D=23 E=22 C=22 so E is eliminated. Round 3 votes counts: D=36 A=34 C=30 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:205 C:202 B:201 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -6 -10 B 8 0 0 -4 -2 C 8 0 0 -2 -2 D 6 4 2 0 4 E 10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -6 -10 B 8 0 0 -4 -2 C 8 0 0 -2 -2 D 6 4 2 0 4 E 10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -6 -10 B 8 0 0 -4 -2 C 8 0 0 -2 -2 D 6 4 2 0 4 E 10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2397: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) E B C D A (7) D A E B C (6) D E A B C (5) A D E B C (5) E B D C A (4) E B C A D (4) C B E A D (4) B E C D A (4) A D C E B (4) A D C B E (4) D A C B E (3) C B E D A (3) A D E C B (3) A C B E D (3) E D B C A (2) E C B A D (2) D B E C A (2) D A B C E (2) C B D E A (2) C B A E D (2) E D B A C (1) E D A B C (1) E B A C D (1) E A D B C (1) E A B C D (1) D E B C A (1) D E B A C (1) D B E A C (1) D B C E A (1) D A B E C (1) C E B A D (1) C B A D E (1) C A B E D (1) B D E C A (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C B D (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -4 -16 -20 B 10 0 26 6 -8 C 4 -26 0 -2 -14 D 16 -6 2 0 -10 E 20 8 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 -16 -20 B 10 0 26 6 -8 C 4 -26 0 -2 -14 D 16 -6 2 0 -10 E 20 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 D=23 B=15 C=14 so C is eliminated. Round 2 votes counts: B=27 E=25 A=25 D=23 so D is eliminated. Round 3 votes counts: A=37 E=32 B=31 so B is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:217 D:201 C:181 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 -16 -20 B 10 0 26 6 -8 C 4 -26 0 -2 -14 D 16 -6 2 0 -10 E 20 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -16 -20 B 10 0 26 6 -8 C 4 -26 0 -2 -14 D 16 -6 2 0 -10 E 20 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -16 -20 B 10 0 26 6 -8 C 4 -26 0 -2 -14 D 16 -6 2 0 -10 E 20 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2398: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C A E D B (7) E C B D A (6) E B D C A (6) C A E B D (6) A C D B E (6) E B C D A (5) D A B C E (5) A D B C E (5) D E B A C (4) C E A B D (4) A D C B E (4) D B E A C (3) B D E A C (3) B D A E C (3) E D B C A (2) E C D B A (2) E C B A D (2) D A B E C (2) C E A D B (2) B E D C A (2) B E D A C (2) E D A C B (1) E C D A B (1) E C A D B (1) E B D A C (1) D A E C B (1) C E B A D (1) C A B E D (1) B E C D A (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 -18 2 B 4 0 8 -10 -4 C -2 -8 0 -8 -14 D 18 10 8 0 -6 E -2 4 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.000000 D: 0.076923 E: 0.692308 Sum of squares = 0.538461538461 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.230769 D: 0.307692 E: 1.000000 A B C D E A 0 -4 2 -18 2 B 4 0 8 -10 -4 C -2 -8 0 -8 -14 D 18 10 8 0 -6 E -2 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.000000 D: 0.076923 E: 0.692308 Sum of squares = 0.538461538451 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.230769 D: 0.307692 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 C=21 A=17 B=11 so B is eliminated. Round 2 votes counts: E=32 D=30 C=21 A=17 so A is eliminated. Round 3 votes counts: D=40 E=32 C=28 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:211 B:199 A:191 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -18 2 B 4 0 8 -10 -4 C -2 -8 0 -8 -14 D 18 10 8 0 -6 E -2 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.000000 D: 0.076923 E: 0.692308 Sum of squares = 0.538461538451 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.230769 D: 0.307692 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -18 2 B 4 0 8 -10 -4 C -2 -8 0 -8 -14 D 18 10 8 0 -6 E -2 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.000000 D: 0.076923 E: 0.692308 Sum of squares = 0.538461538451 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.230769 D: 0.307692 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -18 2 B 4 0 8 -10 -4 C -2 -8 0 -8 -14 D 18 10 8 0 -6 E -2 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.000000 D: 0.076923 E: 0.692308 Sum of squares = 0.538461538451 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.230769 D: 0.307692 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2399: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) A D B E C (9) C E B D A (7) B E C A D (7) A B E C D (7) D C E B A (4) D A B E C (4) B E A C D (4) A D B C E (4) D E B C A (3) D C A E B (3) C E D B A (3) C D E B A (3) B A E D C (3) E C B D A (2) E B C D A (2) C E B A D (2) C D E A B (2) C D A E B (2) B A E C D (2) A D C E B (2) A D C B E (2) A B D E C (2) E C D B A (1) E C B A D (1) D E C B A (1) D A E B C (1) D A B C E (1) C E D A B (1) B E D A C (1) B E A D C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 6 12 -4 10 B -6 0 2 -14 -4 C -12 -2 0 -4 -2 D 4 14 4 0 8 E -10 4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 -4 10 B -6 0 2 -14 -4 C -12 -2 0 -4 -2 D 4 14 4 0 8 E -10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=28 A=28 C=20 B=18 E=6 so E is eliminated. Round 2 votes counts: D=28 A=28 C=24 B=20 so B is eliminated. Round 3 votes counts: A=38 C=33 D=29 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:212 E:194 C:190 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 12 -4 10 B -6 0 2 -14 -4 C -12 -2 0 -4 -2 D 4 14 4 0 8 E -10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 -4 10 B -6 0 2 -14 -4 C -12 -2 0 -4 -2 D 4 14 4 0 8 E -10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 -4 10 B -6 0 2 -14 -4 C -12 -2 0 -4 -2 D 4 14 4 0 8 E -10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2400: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) D A B C E (9) B D A C E (8) A D C E B (8) E C B A D (7) B D A E C (6) B C D A E (6) C A D E B (5) B E C D A (5) E B C A D (4) D A C E B (4) D A C B E (4) E A D C B (3) C E A D B (3) B E C A D (3) D A B E C (2) B E D A C (2) B D E A C (2) A D E C B (2) E B D A C (1) E A C D B (1) D A E C B (1) D A E B C (1) C B D A E (1) C A E D B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 10 12 -4 20 B -10 0 -2 -10 -2 C -12 2 0 -8 2 D 4 10 8 0 22 E -20 2 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 -4 20 B -10 0 -2 -10 -2 C -12 2 0 -8 2 D 4 10 8 0 22 E -20 2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=25 D=21 A=12 C=10 so C is eliminated. Round 2 votes counts: B=33 E=28 D=21 A=18 so A is eliminated. Round 3 votes counts: D=38 B=33 E=29 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:219 C:192 B:188 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 12 -4 20 B -10 0 -2 -10 -2 C -12 2 0 -8 2 D 4 10 8 0 22 E -20 2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -4 20 B -10 0 -2 -10 -2 C -12 2 0 -8 2 D 4 10 8 0 22 E -20 2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -4 20 B -10 0 -2 -10 -2 C -12 2 0 -8 2 D 4 10 8 0 22 E -20 2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2401: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) D C A E B (8) D A B E C (8) C E B A D (7) C E A B D (6) D A B C E (5) D B E C A (4) D A C B E (4) C D E A B (4) A B E C D (4) C E B D A (3) E B C A D (2) D B E A C (2) D A C E B (2) C E D B A (2) C A E B D (2) B E C A D (2) B E A D C (2) B E A C D (2) B D E A C (2) A D B E C (2) A C E B D (2) D C E B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A C E (1) C A D E B (1) B E D C A (1) B E C D A (1) B A E D C (1) B A E C D (1) A E C B D (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 2 -24 14 B -8 0 6 -16 14 C -2 -6 0 -18 6 D 24 16 18 0 18 E -14 -14 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -24 14 B -8 0 6 -16 14 C -2 -6 0 -18 6 D 24 16 18 0 18 E -14 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=49 C=25 B=12 A=12 E=2 so E is eliminated. Round 2 votes counts: D=49 C=25 B=14 A=12 so A is eliminated. Round 3 votes counts: D=51 C=28 B=21 so B is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:238 A:200 B:198 C:190 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -24 14 B -8 0 6 -16 14 C -2 -6 0 -18 6 D 24 16 18 0 18 E -14 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -24 14 B -8 0 6 -16 14 C -2 -6 0 -18 6 D 24 16 18 0 18 E -14 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -24 14 B -8 0 6 -16 14 C -2 -6 0 -18 6 D 24 16 18 0 18 E -14 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2402: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D A C B E (7) E D A B C (6) E B A D C (6) C D A B E (6) E B C D A (5) E D C B A (3) E B A C D (3) D A C E B (3) C D A E B (3) C B E A D (3) C B A D E (3) B E A C D (3) E D C A B (2) E C D B A (2) E C B D A (2) E B D C A (2) D C A E B (2) C E D B A (2) C D E A B (2) C B D A E (2) B E C A D (2) B C E A D (2) B A D E C (2) A D C B E (2) A B D E C (2) E C D A B (1) E C B A D (1) D E A C B (1) D C E A B (1) D A E C B (1) D A B E C (1) C E B D A (1) C E B A D (1) C A D B E (1) B E A D C (1) B A E D C (1) B A C D E (1) A D B E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 -10 -20 B 12 0 -4 4 -16 C 14 4 0 8 -12 D 10 -4 -8 0 -14 E 20 16 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -14 -10 -20 B 12 0 -4 4 -16 C 14 4 0 8 -12 D 10 -4 -8 0 -14 E 20 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=24 D=16 B=12 A=7 so A is eliminated. Round 2 votes counts: E=41 C=24 D=20 B=15 so B is eliminated. Round 3 votes counts: E=48 C=27 D=25 so D is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 C:207 B:198 D:192 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -14 -10 -20 B 12 0 -4 4 -16 C 14 4 0 8 -12 D 10 -4 -8 0 -14 E 20 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -10 -20 B 12 0 -4 4 -16 C 14 4 0 8 -12 D 10 -4 -8 0 -14 E 20 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -10 -20 B 12 0 -4 4 -16 C 14 4 0 8 -12 D 10 -4 -8 0 -14 E 20 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2403: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (15) C A B D E (12) E D B C A (10) A C B D E (9) C A E B D (6) E D A B C (5) E D C B A (4) D E B A C (4) B D A C E (4) A B C D E (3) E D C A B (2) E C D B A (2) D E B C A (2) D B E A C (2) D B A E C (2) D B A C E (2) C A B E D (2) A C B E D (2) E C D A B (1) E C A D B (1) E C A B D (1) E A C B D (1) D B C E A (1) D B C A E (1) C E A B D (1) C D B E A (1) C B D A E (1) B A C D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 4 -18 -6 B 4 0 6 -10 -12 C -4 -6 0 -8 -6 D 18 10 8 0 -10 E 6 12 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 -18 -6 B 4 0 6 -10 -12 C -4 -6 0 -8 -6 D 18 10 8 0 -10 E 6 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=23 A=16 D=14 B=5 so B is eliminated. Round 2 votes counts: E=42 C=23 D=18 A=17 so A is eliminated. Round 3 votes counts: E=43 C=39 D=18 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:213 B:194 A:188 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 -18 -6 B 4 0 6 -10 -12 C -4 -6 0 -8 -6 D 18 10 8 0 -10 E 6 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -18 -6 B 4 0 6 -10 -12 C -4 -6 0 -8 -6 D 18 10 8 0 -10 E 6 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -18 -6 B 4 0 6 -10 -12 C -4 -6 0 -8 -6 D 18 10 8 0 -10 E 6 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2404: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) B C D A E (6) E C D A B (5) E A D C B (5) E A B D C (5) A B E D C (5) E C D B A (4) C D E B A (4) C D E A B (4) B C D E A (4) B A D C E (4) D C E A B (3) C D B E A (3) C D B A E (3) B E C D A (3) B E A C D (3) B A E C D (3) B A C D E (3) E D C A B (2) E A C D B (2) D C A E B (2) A E D C B (2) A E B D C (2) A D C E B (2) E D A C B (1) E B C D A (1) E B C A D (1) C B D E A (1) B E C A D (1) B E A D C (1) B A D E C (1) B A C E D (1) A D E C B (1) A D C B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 6 8 -6 B 12 0 10 12 8 C -6 -10 0 4 -16 D -8 -12 -4 0 -12 E 6 -8 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 8 -6 B 12 0 10 12 8 C -6 -10 0 4 -16 D -8 -12 -4 0 -12 E 6 -8 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=26 C=15 A=15 D=5 so D is eliminated. Round 2 votes counts: B=39 E=26 C=20 A=15 so A is eliminated. Round 3 votes counts: B=46 E=31 C=23 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:213 A:198 C:186 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 8 -6 B 12 0 10 12 8 C -6 -10 0 4 -16 D -8 -12 -4 0 -12 E 6 -8 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 8 -6 B 12 0 10 12 8 C -6 -10 0 4 -16 D -8 -12 -4 0 -12 E 6 -8 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 8 -6 B 12 0 10 12 8 C -6 -10 0 4 -16 D -8 -12 -4 0 -12 E 6 -8 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2405: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) A E C D B (9) B E A D C (8) E A B C D (7) D C B A E (6) B D C E A (6) E A C D B (5) E A C B D (4) D B C E A (4) C D E A B (4) B D C A E (4) A C E D B (4) D C A E B (3) D B C A E (3) A E C B D (3) E B A C D (2) D C B E A (2) B D E C A (2) B D A E C (2) A E B C D (2) E C A D B (1) E B C D A (1) C E A D B (1) C D B E A (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A C D (1) B D A C E (1) Total count = 100 A B C D E A 0 12 -2 -2 0 B -12 0 -12 -12 -18 C 2 12 0 16 4 D 2 12 -16 0 0 E 0 18 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999092 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 -2 0 B -12 0 -12 -12 -18 C 2 12 0 16 4 D 2 12 -16 0 0 E 0 18 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=20 C=19 D=18 A=18 so D is eliminated. Round 2 votes counts: B=32 C=30 E=20 A=18 so A is eliminated. Round 3 votes counts: E=34 C=34 B=32 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:207 A:204 D:199 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 -2 0 B -12 0 -12 -12 -18 C 2 12 0 16 4 D 2 12 -16 0 0 E 0 18 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -2 0 B -12 0 -12 -12 -18 C 2 12 0 16 4 D 2 12 -16 0 0 E 0 18 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -2 0 B -12 0 -12 -12 -18 C 2 12 0 16 4 D 2 12 -16 0 0 E 0 18 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2406: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (14) D E B C A (12) C A D E B (12) E D B C A (8) A C B E D (6) D E C B A (5) D C E A B (5) C D A E B (5) A B C E D (5) D E C A B (3) B D E A C (3) A C B D E (3) E B D A C (2) D C A E B (2) C A E D B (2) C A D B E (2) B E A D C (2) A C D B E (2) E D C A B (1) E C D A B (1) E B D C A (1) C D E A B (1) B A E D C (1) B A E C D (1) B A C D E (1) Total count = 100 A B C D E A 0 0 -20 -26 -16 B 0 0 0 -22 -20 C 20 0 0 -18 -8 D 26 22 18 0 12 E 16 20 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 -26 -16 B 0 0 0 -22 -20 C 20 0 0 -18 -8 D 26 22 18 0 12 E 16 20 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=22 B=22 A=16 E=13 so E is eliminated. Round 2 votes counts: D=36 B=25 C=23 A=16 so A is eliminated. Round 3 votes counts: D=36 C=34 B=30 so B is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:239 E:216 C:197 B:179 A:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -20 -26 -16 B 0 0 0 -22 -20 C 20 0 0 -18 -8 D 26 22 18 0 12 E 16 20 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 -26 -16 B 0 0 0 -22 -20 C 20 0 0 -18 -8 D 26 22 18 0 12 E 16 20 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 -26 -16 B 0 0 0 -22 -20 C 20 0 0 -18 -8 D 26 22 18 0 12 E 16 20 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2407: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) E A D C B (7) B C D A E (7) B A E C D (7) B C D E A (5) A E D C B (5) D A C E B (4) B E A C D (4) A E B D C (4) A B E C D (4) E D C A B (3) D C E A B (3) D C B E A (3) B A C D E (3) A D C E B (3) E C D B A (2) E A B C D (2) D C E B A (2) D C B A E (2) D C A E B (2) D C A B E (2) A E B C D (2) A B D C E (2) A B C D E (2) E C B D A (1) E B C A D (1) E B A C D (1) E A D B C (1) D A C B E (1) C D E B A (1) C B E D A (1) C B D E A (1) B E C A D (1) B C E D A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 6 2 2 B 4 0 -4 0 10 C -6 4 0 10 8 D -2 0 -10 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775566 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 2 B 4 0 -4 0 10 C -6 4 0 10 8 D -2 0 -10 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775516 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 D=19 E=18 C=11 so C is eliminated. Round 2 votes counts: B=31 D=28 A=23 E=18 so E is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:205 A:203 D:196 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 6 2 2 B 4 0 -4 0 10 C -6 4 0 10 8 D -2 0 -10 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775516 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 2 B 4 0 -4 0 10 C -6 4 0 10 8 D -2 0 -10 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775516 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 2 B 4 0 -4 0 10 C -6 4 0 10 8 D -2 0 -10 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775516 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2408: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) C E D A B (8) D A C E B (7) B E C A D (7) A D B E C (7) A D B C E (7) E B C D A (5) D A B E C (5) B C E A D (5) B A D E C (5) D A E B C (4) A D C B E (4) E C D A B (2) D A E C B (2) C E B D A (2) C E B A D (2) C D A E B (2) C A D E B (2) C A D B E (2) B E D A C (2) B E A D C (2) B A D C E (2) E D B A C (1) E D A C B (1) E B D A C (1) C D E A B (1) C B E A D (1) B C A D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 4 -2 4 B -12 0 12 -10 4 C -4 -12 0 -4 -6 D 2 10 4 0 6 E -4 -4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -2 4 B -12 0 12 -10 4 C -4 -12 0 -4 -6 D 2 10 4 0 6 E -4 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=20 A=20 E=18 D=18 so E is eliminated. Round 2 votes counts: C=30 B=30 D=20 A=20 so D is eliminated. Round 3 votes counts: A=39 B=31 C=30 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:209 B:197 E:196 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 4 -2 4 B -12 0 12 -10 4 C -4 -12 0 -4 -6 D 2 10 4 0 6 E -4 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -2 4 B -12 0 12 -10 4 C -4 -12 0 -4 -6 D 2 10 4 0 6 E -4 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -2 4 B -12 0 12 -10 4 C -4 -12 0 -4 -6 D 2 10 4 0 6 E -4 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2409: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) A E D C B (6) A E C D B (6) A E C B D (6) A C E B D (6) E A D B C (5) D E B A C (5) D B E C A (5) B D C E A (5) D B C E A (4) C B A D E (4) B C E A D (4) A C E D B (4) E D B A C (3) C B A E D (3) E B D A C (2) E B A D C (2) C A D B E (2) B D E C A (2) B C D E A (2) E D A B C (1) E B A C D (1) E A D C B (1) E A B C D (1) D E A B C (1) D C B A E (1) D A E C B (1) C D B A E (1) C A E B D (1) C A B E D (1) C A B D E (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 4 10 10 B 12 0 -12 6 -6 C -4 12 0 10 4 D -10 -6 -10 0 -8 E -10 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102042 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 10 10 B 12 0 -12 6 -6 C -4 12 0 10 4 D -10 -6 -10 0 -8 E -10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=24 D=17 E=16 B=14 so B is eliminated. Round 2 votes counts: C=31 A=29 D=24 E=16 so E is eliminated. Round 3 votes counts: A=39 C=31 D=30 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:206 B:200 E:200 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 4 10 10 B 12 0 -12 6 -6 C -4 12 0 10 4 D -10 -6 -10 0 -8 E -10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 10 10 B 12 0 -12 6 -6 C -4 12 0 10 4 D -10 -6 -10 0 -8 E -10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 10 10 B 12 0 -12 6 -6 C -4 12 0 10 4 D -10 -6 -10 0 -8 E -10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2410: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) D E C A B (7) C A D B E (7) E D B A C (6) D C E A B (6) C A B D E (6) A B C E D (6) E B D A C (5) E B A D C (3) C D A E B (3) C B A E D (3) D E B A C (2) D E A B C (2) C D E B A (2) C D A B E (2) C A B E D (2) B E D C A (2) B E A C D (2) B C E A D (2) B C A E D (2) A C D E B (2) A C B E D (2) E D B C A (1) E A B D C (1) D E C B A (1) D E A C B (1) D C E B A (1) D C A E B (1) D A E C B (1) D A C E B (1) C D B E A (1) C D B A E (1) C B A D E (1) B E C A D (1) B A E D C (1) B A C E D (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -4 8 -12 B -4 0 -4 2 6 C 4 4 0 -6 6 D -8 -2 6 0 -2 E 12 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.135135 B: 0.108108 C: 0.270270 D: 0.324324 E: 0.162162 Sum of squares = 0.234477720964 Cumulative probabilities = A: 0.135135 B: 0.243243 C: 0.513514 D: 0.837838 E: 1.000000 A B C D E A 0 4 -4 8 -12 B -4 0 -4 2 6 C 4 4 0 -6 6 D -8 -2 6 0 -2 E 12 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.108108 C: 0.270270 D: 0.324324 E: 0.162162 Sum of squares = 0.234477720964 Cumulative probabilities = A: 0.135135 B: 0.243243 C: 0.513514 D: 0.837838 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 B=21 E=16 A=12 so A is eliminated. Round 2 votes counts: C=32 B=28 D=24 E=16 so E is eliminated. Round 3 votes counts: B=37 C=32 D=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:204 E:201 B:200 A:198 D:197 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -4 8 -12 B -4 0 -4 2 6 C 4 4 0 -6 6 D -8 -2 6 0 -2 E 12 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.108108 C: 0.270270 D: 0.324324 E: 0.162162 Sum of squares = 0.234477720964 Cumulative probabilities = A: 0.135135 B: 0.243243 C: 0.513514 D: 0.837838 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 8 -12 B -4 0 -4 2 6 C 4 4 0 -6 6 D -8 -2 6 0 -2 E 12 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.108108 C: 0.270270 D: 0.324324 E: 0.162162 Sum of squares = 0.234477720964 Cumulative probabilities = A: 0.135135 B: 0.243243 C: 0.513514 D: 0.837838 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 8 -12 B -4 0 -4 2 6 C 4 4 0 -6 6 D -8 -2 6 0 -2 E 12 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.108108 C: 0.270270 D: 0.324324 E: 0.162162 Sum of squares = 0.234477720964 Cumulative probabilities = A: 0.135135 B: 0.243243 C: 0.513514 D: 0.837838 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2411: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) D C E B A (6) B E D A C (6) D C A E B (5) C D E A B (5) B E A D C (5) A C D B E (5) A B E C D (5) C A D E B (4) B A E D C (4) E B D C A (3) A B D C E (3) E C A B D (2) E B C D A (2) E B A C D (2) D E C B A (2) D A C B E (2) C E D A B (2) C D E B A (2) B E D C A (2) B E A C D (2) B A E C D (2) B A D E C (2) A C B D E (2) A B C E D (2) E C D B A (1) E C B D A (1) E B D A C (1) E B C A D (1) D E B C A (1) D C A B E (1) D B E A C (1) C E D B A (1) C D A B E (1) A D C B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -6 -12 4 B -6 0 -8 -2 -4 C 6 8 0 8 8 D 12 2 -8 0 10 E -4 4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -12 4 B -6 0 -8 -2 -4 C 6 8 0 8 8 D 12 2 -8 0 10 E -4 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=23 A=20 D=18 E=13 so E is eliminated. Round 2 votes counts: B=32 C=30 A=20 D=18 so D is eliminated. Round 3 votes counts: C=44 B=34 A=22 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:208 A:196 E:191 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -12 4 B -6 0 -8 -2 -4 C 6 8 0 8 8 D 12 2 -8 0 10 E -4 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -12 4 B -6 0 -8 -2 -4 C 6 8 0 8 8 D 12 2 -8 0 10 E -4 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -12 4 B -6 0 -8 -2 -4 C 6 8 0 8 8 D 12 2 -8 0 10 E -4 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2412: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) D C B E A (7) C A D E B (7) A E C B D (7) B E A D C (5) E B A C D (3) D C A E B (3) C A E D B (3) B E A C D (3) B D E C A (3) B A E C D (3) A E C D B (3) A C E D B (3) E B A D C (2) E A B C D (2) D E C A B (2) D E B C A (2) D C E A B (2) D C B A E (2) D B E C A (2) D B C E A (2) C D A E B (2) B D E A C (2) B A E D C (2) E D C B A (1) E B D A C (1) E A C D B (1) E A C B D (1) D C E B A (1) D B C A E (1) C D A B E (1) C A D B E (1) B D C A E (1) B C A D E (1) B A D C E (1) B A C D E (1) A C E B D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 12 6 -8 B 16 0 2 6 6 C -12 -2 0 -8 -16 D -6 -6 8 0 -10 E 8 -6 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 12 6 -8 B 16 0 2 6 6 C -12 -2 0 -8 -16 D -6 -6 8 0 -10 E 8 -6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=24 A=17 C=14 E=11 so E is eliminated. Round 2 votes counts: B=40 D=25 A=21 C=14 so C is eliminated. Round 3 votes counts: B=40 A=32 D=28 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:214 A:197 D:193 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 12 6 -8 B 16 0 2 6 6 C -12 -2 0 -8 -16 D -6 -6 8 0 -10 E 8 -6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 12 6 -8 B 16 0 2 6 6 C -12 -2 0 -8 -16 D -6 -6 8 0 -10 E 8 -6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 12 6 -8 B 16 0 2 6 6 C -12 -2 0 -8 -16 D -6 -6 8 0 -10 E 8 -6 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2413: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (5) C A B D E (5) E A D C B (4) E A C D B (4) C B D A E (4) A B E D C (4) A B C D E (4) E D C B A (3) E C D A B (3) E C A D B (3) E A D B C (3) D C E B A (3) D C B E A (3) C D B E A (3) B D C A E (3) A E C B D (3) A C E D B (3) A C E B D (3) A B C E D (3) D B E C A (2) C D B A E (2) B D E A C (2) B C D A E (2) B A D E C (2) B A D C E (2) B A C D E (2) A E C D B (2) A E B D C (2) A C B E D (2) A B E C D (2) E D C A B (1) E D B C A (1) E D B A C (1) E A B D C (1) D E C B A (1) C E A D B (1) C D E A B (1) C A D B E (1) B D A E C (1) B D A C E (1) B A E D C (1) A C B D E (1) Total count = 100 A B C D E A 0 12 6 16 10 B -12 0 -12 0 14 C -6 12 0 8 8 D -16 0 -8 0 0 E -10 -14 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 16 10 B -12 0 -12 0 14 C -6 12 0 8 8 D -16 0 -8 0 0 E -10 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 C=17 B=16 D=14 so D is eliminated. Round 2 votes counts: A=29 E=25 C=23 B=23 so C is eliminated. Round 3 votes counts: B=35 A=35 E=30 so E is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:211 B:195 D:188 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 16 10 B -12 0 -12 0 14 C -6 12 0 8 8 D -16 0 -8 0 0 E -10 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 16 10 B -12 0 -12 0 14 C -6 12 0 8 8 D -16 0 -8 0 0 E -10 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 16 10 B -12 0 -12 0 14 C -6 12 0 8 8 D -16 0 -8 0 0 E -10 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2414: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) D C B E A (10) D C A B E (8) A E D B C (7) A E B D C (7) D A C E B (6) B E C A D (6) C D B E A (5) E B A C D (4) E B A D C (3) E A B C D (3) C D A B E (3) C B E A D (3) D E A B C (2) D C E B A (2) D C B A E (2) D C A E B (2) C B E D A (2) C A D B E (2) B E C D A (2) E B D C A (1) D B E C A (1) D A E C B (1) C A B E D (1) B C E A D (1) A D E C B (1) A D E B C (1) A D C E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 16 -2 6 10 B -16 0 0 -8 -4 C 2 0 0 -10 -2 D -6 8 10 0 -4 E -10 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.333333 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765494 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 6 10 B -16 0 0 -8 -4 C 2 0 0 -10 -2 D -6 8 10 0 -4 E -10 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.333333 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=30 C=16 E=11 B=9 so B is eliminated. Round 2 votes counts: D=34 A=30 E=19 C=17 so C is eliminated. Round 3 votes counts: D=42 A=33 E=25 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:204 E:200 C:195 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 -2 6 10 B -16 0 0 -8 -4 C 2 0 0 -10 -2 D -6 8 10 0 -4 E -10 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.333333 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 6 10 B -16 0 0 -8 -4 C 2 0 0 -10 -2 D -6 8 10 0 -4 E -10 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.333333 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 6 10 B -16 0 0 -8 -4 C 2 0 0 -10 -2 D -6 8 10 0 -4 E -10 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.333333 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2415: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) B C D A E (6) D B C A E (5) E D A B C (4) E B D C A (4) E B C A D (4) E A C D B (4) D A C B E (4) E D B A C (3) E A C B D (3) B D C A E (3) B C E A D (3) A D C B E (3) A C D B E (3) E D A C B (2) E A D C B (2) D B E C A (2) D B A C E (2) C A B E D (2) B E C D A (2) B D E C A (2) B D C E A (2) B C A E D (2) B C A D E (2) A E C D B (2) A D E C B (2) A C E B D (2) E C B A D (1) E C A B D (1) D E A C B (1) D E A B C (1) D B E A C (1) D A B C E (1) C E A B D (1) C B E A D (1) C B D A E (1) C B A E D (1) B E C A D (1) B C E D A (1) B C D E A (1) A E C B D (1) A D C E B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 4 -12 6 B 0 0 4 -4 4 C -4 -4 0 -6 -2 D 12 4 6 0 2 E -6 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -12 6 B 0 0 4 -4 4 C -4 -4 0 -6 -2 D 12 4 6 0 2 E -6 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=25 B=25 A=16 C=6 so C is eliminated. Round 2 votes counts: E=29 B=28 D=25 A=18 so A is eliminated. Round 3 votes counts: E=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:202 A:199 E:195 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -12 6 B 0 0 4 -4 4 C -4 -4 0 -6 -2 D 12 4 6 0 2 E -6 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -12 6 B 0 0 4 -4 4 C -4 -4 0 -6 -2 D 12 4 6 0 2 E -6 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -12 6 B 0 0 4 -4 4 C -4 -4 0 -6 -2 D 12 4 6 0 2 E -6 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2416: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) B E A D C (6) C D E A B (5) B A E D C (5) E A B C D (4) D C A B E (4) C D A E B (4) B D C A E (4) E B A C D (3) E A C D B (3) D C E A B (3) D C B A E (3) B E A C D (3) B D E C A (3) B D A C E (3) B A D C E (3) A C D E B (3) E C D B A (2) E B D C A (2) D B C E A (2) D B C A E (2) C A D E B (2) B A E C D (2) B A D E C (2) A E C D B (2) A E B C D (2) A C E D B (2) E D C B A (1) E C D A B (1) E C A D B (1) E C A B D (1) E A C B D (1) D C E B A (1) D A C B E (1) C E D A B (1) C E A D B (1) B D E A C (1) B D C E A (1) B D A E C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 -4 8 B 0 0 0 -2 -4 C 2 0 0 -10 6 D 4 2 10 0 10 E -8 4 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -4 8 B 0 0 0 -2 -4 C 2 0 0 -10 6 D 4 2 10 0 10 E -8 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=23 E=19 C=13 A=11 so A is eliminated. Round 2 votes counts: B=35 E=23 D=23 C=19 so C is eliminated. Round 3 votes counts: D=37 B=36 E=27 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:201 C:199 B:197 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -2 -4 8 B 0 0 0 -2 -4 C 2 0 0 -10 6 D 4 2 10 0 10 E -8 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -4 8 B 0 0 0 -2 -4 C 2 0 0 -10 6 D 4 2 10 0 10 E -8 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -4 8 B 0 0 0 -2 -4 C 2 0 0 -10 6 D 4 2 10 0 10 E -8 4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2417: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) D A C E B (8) C D A B E (7) E B A D C (5) B E D A C (5) B C E D A (5) E A D B C (4) C B E A D (4) C A D E B (4) B E C D A (4) B E C A D (4) E B D A C (3) D C A B E (3) D A E C B (3) C B E D A (3) C B D A E (3) C A D B E (3) D A E B C (2) C B A D E (2) A D E C B (2) A D E B C (2) E D A B C (1) E C B A D (1) D E B A C (1) D E A B C (1) D A C B E (1) C D B A E (1) C A E D B (1) B E D C A (1) B D C A E (1) B C E A D (1) A E D C B (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 12 4 -6 14 B -12 0 -20 -16 -4 C -4 20 0 -10 18 D 6 16 10 0 10 E -14 4 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -6 14 B -12 0 -20 -16 -4 C -4 20 0 -10 18 D 6 16 10 0 10 E -14 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=21 D=19 A=18 E=14 so E is eliminated. Round 2 votes counts: C=29 B=29 A=22 D=20 so D is eliminated. Round 3 votes counts: A=38 C=32 B=30 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:212 C:212 E:181 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 4 -6 14 B -12 0 -20 -16 -4 C -4 20 0 -10 18 D 6 16 10 0 10 E -14 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -6 14 B -12 0 -20 -16 -4 C -4 20 0 -10 18 D 6 16 10 0 10 E -14 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -6 14 B -12 0 -20 -16 -4 C -4 20 0 -10 18 D 6 16 10 0 10 E -14 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2418: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) B D E A C (9) A E D B C (7) C B A D E (6) C A E D B (5) E A D B C (4) C E D A B (4) C B A E D (4) C A E B D (4) C A B E D (4) B A D E C (4) E D A B C (3) D E B A C (3) D E A B C (3) A E D C B (3) A B E D C (3) E A D C B (2) C D E B A (2) C B D A E (2) B D C E A (2) A E C D B (2) E D C A B (1) E D A C B (1) E C A D B (1) D E B C A (1) D B E C A (1) D B E A C (1) C E D B A (1) B D A E C (1) A C E B D (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 6 -2 B -2 0 -12 10 4 C 0 12 0 0 -2 D -6 -10 0 0 -4 E 2 -4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839495 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 2 0 6 -2 B -2 0 -12 10 4 C 0 12 0 0 -2 D -6 -10 0 0 -4 E 2 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839455 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=19 B=16 E=12 D=9 so D is eliminated. Round 2 votes counts: C=44 E=19 A=19 B=18 so B is eliminated. Round 3 votes counts: C=46 E=30 A=24 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:205 A:203 E:202 B:200 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 6 -2 B -2 0 -12 10 4 C 0 12 0 0 -2 D -6 -10 0 0 -4 E 2 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839455 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 -2 B -2 0 -12 10 4 C 0 12 0 0 -2 D -6 -10 0 0 -4 E 2 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839455 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 -2 B -2 0 -12 10 4 C 0 12 0 0 -2 D -6 -10 0 0 -4 E 2 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839455 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2419: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) B E D A C (10) E B C D A (9) D A B C E (9) C E A D B (8) B D A E C (8) E C B A D (7) C A D E B (7) C A D B E (7) D A C B E (4) D A B E C (3) A D C E B (3) E B C A D (2) D B A E C (2) C A E D B (2) E B D C A (1) D B A C E (1) C E B A D (1) C E A B D (1) B E D C A (1) B E C D A (1) B D A C E (1) Total count = 100 A B C D E A 0 12 6 0 18 B -12 0 -4 -16 18 C -6 4 0 -10 12 D 0 16 10 0 14 E -18 -18 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.595890 B: 0.000000 C: 0.000000 D: 0.404110 E: 0.000000 Sum of squares = 0.518389814659 Cumulative probabilities = A: 0.595890 B: 0.595890 C: 0.595890 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 0 18 B -12 0 -4 -16 18 C -6 4 0 -10 12 D 0 16 10 0 14 E -18 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=21 E=19 D=19 A=15 so A is eliminated. Round 2 votes counts: D=34 C=26 B=21 E=19 so E is eliminated. Round 3 votes counts: D=34 C=33 B=33 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:218 C:200 B:193 E:169 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 0 18 B -12 0 -4 -16 18 C -6 4 0 -10 12 D 0 16 10 0 14 E -18 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 0 18 B -12 0 -4 -16 18 C -6 4 0 -10 12 D 0 16 10 0 14 E -18 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 0 18 B -12 0 -4 -16 18 C -6 4 0 -10 12 D 0 16 10 0 14 E -18 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2420: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) E C B D A (11) E B A C D (7) E B A D C (6) D A C B E (6) A D B C E (6) C D B A E (5) E A D B C (4) E C D B A (3) C E D B A (3) C E D A B (3) B E A D C (3) B C D A E (3) B A E D C (3) B A D C E (3) A B D E C (3) E C A D B (2) E B C D A (2) E B C A D (2) E A B D C (2) C D E A B (2) E C D A B (1) D C A B E (1) D A C E B (1) D A B C E (1) C B D A E (1) B E C A D (1) A D E B C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -4 -10 -4 B 6 0 -6 -6 -2 C 4 6 0 16 -2 D 10 6 -16 0 -6 E 4 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -10 -4 B 6 0 -6 -6 -2 C 4 6 0 16 -2 D 10 6 -16 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=26 B=13 A=12 D=9 so D is eliminated. Round 2 votes counts: E=40 C=27 A=20 B=13 so B is eliminated. Round 3 votes counts: E=44 C=30 A=26 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:207 D:197 B:196 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 -4 B 6 0 -6 -6 -2 C 4 6 0 16 -2 D 10 6 -16 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 -4 B 6 0 -6 -6 -2 C 4 6 0 16 -2 D 10 6 -16 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 -4 B 6 0 -6 -6 -2 C 4 6 0 16 -2 D 10 6 -16 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2421: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) E C B D A (6) E C D A B (5) D A B E C (5) D B A E C (4) E C A D B (3) E B D C A (3) C E B A D (3) C E A D B (3) C B A E D (3) C A B D E (3) A D B C E (3) A B C D E (3) E D C A B (2) E D B C A (2) E B D A C (2) E B C D A (2) D E C A B (2) D B A C E (2) D A C E B (2) D A B C E (2) C E A B D (2) C B E A D (2) B E D A C (2) B D A C E (2) B C A E D (2) B A C E D (2) B A C D E (2) A D C B E (2) A B D C E (2) E D C B A (1) E C D B A (1) D E B A C (1) D E A C B (1) D E A B C (1) D B E A C (1) D A C B E (1) C A B E D (1) B E D C A (1) B C E A D (1) B A D E C (1) B A D C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -14 -2 -12 B 12 0 -6 12 -2 C 14 6 0 8 -10 D 2 -12 -8 0 -16 E 12 2 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999018 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -14 -2 -12 B 12 0 -6 12 -2 C 14 6 0 8 -10 D 2 -12 -8 0 -16 E 12 2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=22 C=17 B=14 A=11 so A is eliminated. Round 2 votes counts: E=36 D=27 B=19 C=18 so C is eliminated. Round 3 votes counts: E=44 B=29 D=27 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:209 B:208 D:183 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -14 -2 -12 B 12 0 -6 12 -2 C 14 6 0 8 -10 D 2 -12 -8 0 -16 E 12 2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -2 -12 B 12 0 -6 12 -2 C 14 6 0 8 -10 D 2 -12 -8 0 -16 E 12 2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -2 -12 B 12 0 -6 12 -2 C 14 6 0 8 -10 D 2 -12 -8 0 -16 E 12 2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2422: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) C A D B E (8) E A B D C (7) E A C B D (6) C D B A E (6) D B C E A (5) E B D A C (4) C E B D A (4) C E A B D (4) C B D E A (4) A E C D B (4) D B C A E (3) D B A E C (3) E C A B D (2) E B D C A (2) E B C D A (2) E B A D C (2) D B E A C (2) C B E D A (2) A E C B D (2) A E B D C (2) E C B D A (1) E A B C D (1) D C B A E (1) D B E C A (1) D A B C E (1) C D B E A (1) C D A B E (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) B D E A C (1) B D C E A (1) A E D B C (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -4 4 -10 B -4 0 -18 4 -14 C 4 18 0 20 6 D -4 -4 -20 0 -18 E 10 14 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 4 -10 B -4 0 -18 4 -14 C 4 18 0 20 6 D -4 -4 -20 0 -18 E 10 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=27 A=20 D=16 B=5 so B is eliminated. Round 2 votes counts: C=32 E=29 A=20 D=19 so D is eliminated. Round 3 votes counts: C=42 E=34 A=24 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:218 A:197 B:184 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 -10 B -4 0 -18 4 -14 C 4 18 0 20 6 D -4 -4 -20 0 -18 E 10 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 -10 B -4 0 -18 4 -14 C 4 18 0 20 6 D -4 -4 -20 0 -18 E 10 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 -10 B -4 0 -18 4 -14 C 4 18 0 20 6 D -4 -4 -20 0 -18 E 10 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2423: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (5) A B D C E (5) E C D B A (4) E A B C D (4) D A B C E (4) C E B D A (4) C B E A D (4) C B D A E (4) A D B E C (4) E D A C B (3) E A D B C (3) D C E B A (3) C D B E A (3) C B E D A (3) B C A E D (3) B A D C E (3) E A D C B (2) D B C A E (2) B C D A E (2) B A C D E (2) A D B C E (2) A B E C D (2) A B D E C (2) E D C B A (1) E D C A B (1) E D A B C (1) E C D A B (1) E C B D A (1) E C A B D (1) E B C A D (1) E B A C D (1) E A C D B (1) E A C B D (1) D E C B A (1) D E A C B (1) D C B A E (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C D E B A (1) C D B A E (1) C B D E A (1) C B A E D (1) B C A D E (1) A E D B C (1) A E B D C (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -2 4 -6 B 10 0 -4 6 6 C 2 4 0 8 8 D -4 -6 -8 0 -4 E 6 -6 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 4 -6 B 10 0 -4 6 6 C 2 4 0 8 8 D -4 -6 -8 0 -4 E 6 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=23 A=19 D=16 B=11 so B is eliminated. Round 2 votes counts: E=31 C=29 A=24 D=16 so D is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:209 E:198 A:193 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -2 4 -6 B 10 0 -4 6 6 C 2 4 0 8 8 D -4 -6 -8 0 -4 E 6 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 4 -6 B 10 0 -4 6 6 C 2 4 0 8 8 D -4 -6 -8 0 -4 E 6 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 4 -6 B 10 0 -4 6 6 C 2 4 0 8 8 D -4 -6 -8 0 -4 E 6 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2424: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (14) E B C A D (10) D C A E B (9) C E B A D (9) B E A C D (6) A D B E C (6) C E B D A (5) A B E D C (5) D A C B E (4) D A B E C (4) C D E B A (4) A B E C D (4) E C B A D (3) D A C E B (2) A D B C E (2) A C D B E (2) A B D E C (2) E B A D C (1) D E B C A (1) D E B A C (1) D A E B C (1) D A B C E (1) B E A D C (1) B A E D C (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 -14 -10 8 -10 B 14 0 -6 -4 -20 C 10 6 0 -12 6 D -8 4 12 0 8 E 10 20 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.350649 B: 0.012987 C: 0.142857 D: 0.363636 E: 0.129870 Sum of squares = 0.292629448474 Cumulative probabilities = A: 0.350649 B: 0.363636 C: 0.506494 D: 0.870130 E: 1.000000 A B C D E A 0 -14 -10 8 -10 B 14 0 -6 -4 -20 C 10 6 0 -12 6 D -8 4 12 0 8 E 10 20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.350649 B: 0.012987 C: 0.142857 D: 0.363636 E: 0.129870 Sum of squares = 0.292629448485 Cumulative probabilities = A: 0.350649 B: 0.363636 C: 0.506494 D: 0.870130 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=22 C=18 E=14 B=9 so B is eliminated. Round 2 votes counts: D=37 A=24 E=21 C=18 so C is eliminated. Round 3 votes counts: D=41 E=35 A=24 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:208 C:205 B:192 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 8 -10 B 14 0 -6 -4 -20 C 10 6 0 -12 6 D -8 4 12 0 8 E 10 20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.350649 B: 0.012987 C: 0.142857 D: 0.363636 E: 0.129870 Sum of squares = 0.292629448485 Cumulative probabilities = A: 0.350649 B: 0.363636 C: 0.506494 D: 0.870130 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 8 -10 B 14 0 -6 -4 -20 C 10 6 0 -12 6 D -8 4 12 0 8 E 10 20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.350649 B: 0.012987 C: 0.142857 D: 0.363636 E: 0.129870 Sum of squares = 0.292629448485 Cumulative probabilities = A: 0.350649 B: 0.363636 C: 0.506494 D: 0.870130 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 8 -10 B 14 0 -6 -4 -20 C 10 6 0 -12 6 D -8 4 12 0 8 E 10 20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.350649 B: 0.012987 C: 0.142857 D: 0.363636 E: 0.129870 Sum of squares = 0.292629448485 Cumulative probabilities = A: 0.350649 B: 0.363636 C: 0.506494 D: 0.870130 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2425: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) A C D E B (8) D A C E B (6) E B D A C (5) D A C B E (5) A C D B E (5) E B D C A (4) D C A B E (4) C A D B E (4) C A B E D (4) A D C E B (4) E A B C D (3) D A E C B (3) B E D C A (3) B C E A D (3) E D A B C (2) E B C A D (2) E B A C D (2) D C B A E (2) C B A E D (2) B E C D A (2) A C E D B (2) E D B A C (1) E B C D A (1) E B A D C (1) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) D B E A C (1) D B C A E (1) D A B C E (1) C E A B D (1) C D A B E (1) C B A D E (1) B D E C A (1) B C D A E (1) B C A E D (1) A C E B D (1) Total count = 100 A B C D E A 0 14 8 10 12 B -14 0 -10 -4 0 C -8 10 0 6 14 D -10 4 -6 0 -2 E -12 0 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 10 12 B -14 0 -10 -4 0 C -8 10 0 6 14 D -10 4 -6 0 -2 E -12 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 A=20 B=19 C=13 so C is eliminated. Round 2 votes counts: A=28 E=25 D=25 B=22 so B is eliminated. Round 3 votes counts: E=41 A=32 D=27 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:211 D:193 E:188 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 10 12 B -14 0 -10 -4 0 C -8 10 0 6 14 D -10 4 -6 0 -2 E -12 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 10 12 B -14 0 -10 -4 0 C -8 10 0 6 14 D -10 4 -6 0 -2 E -12 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 10 12 B -14 0 -10 -4 0 C -8 10 0 6 14 D -10 4 -6 0 -2 E -12 0 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2426: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (13) A E B C D (8) B E C D A (7) E B A C D (6) E A B C D (5) D C A B E (5) B C E D A (5) D C B A E (4) C B D E A (4) A E B D C (4) A D C B E (4) A D B E C (4) E B C A D (3) D A C E B (3) C D E B A (3) A D E C B (3) D C A E B (2) B E C A D (2) A E D C B (2) A E D B C (2) A D E B C (2) A D C E B (2) E A D C B (1) D C B E A (1) D B C A E (1) D A C B E (1) C B E D A (1) B E A C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -2 0 -4 B 2 0 2 -6 8 C 2 -2 0 16 -2 D 0 6 -16 0 4 E 4 -8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.250000 D: 0.083333 E: 0.000000 Sum of squares = 0.513888888857 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.916667 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 0 -4 B 2 0 2 -6 8 C 2 -2 0 16 -2 D 0 6 -16 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.250000 D: 0.083333 E: 0.000000 Sum of squares = 0.513888888772 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.916667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=21 D=17 E=15 B=15 so E is eliminated. Round 2 votes counts: A=38 B=24 C=21 D=17 so D is eliminated. Round 3 votes counts: A=42 C=33 B=25 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:203 D:197 E:197 A:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 0 -4 B 2 0 2 -6 8 C 2 -2 0 16 -2 D 0 6 -16 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.250000 D: 0.083333 E: 0.000000 Sum of squares = 0.513888888772 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.916667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 0 -4 B 2 0 2 -6 8 C 2 -2 0 16 -2 D 0 6 -16 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.250000 D: 0.083333 E: 0.000000 Sum of squares = 0.513888888772 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.916667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 0 -4 B 2 0 2 -6 8 C 2 -2 0 16 -2 D 0 6 -16 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.250000 D: 0.083333 E: 0.000000 Sum of squares = 0.513888888772 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.916667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2427: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) E A B C D (8) D A E C B (8) A E D B C (8) D A E B C (7) B C E A D (6) C B E A D (5) E A B D C (4) C D B A E (4) E B A C D (3) B C D E A (3) D C B A E (2) D C A E B (2) D C A B E (2) D A C E B (2) B E C A D (2) B D E A C (2) B C E D A (2) A E D C B (2) A D E B C (2) E B A D C (1) E A C B D (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B E C (1) C E B A D (1) C D B E A (1) C B E D A (1) C A E B D (1) C A B D E (1) B E A C D (1) B D E C A (1) B D C E A (1) A E C B D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 4 8 -2 -8 B -4 0 12 10 -6 C -8 -12 0 0 -10 D 2 -10 0 0 4 E 8 6 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.300000 E: 0.500000 Sum of squares = 0.38000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.500000 E: 1.000000 A B C D E A 0 4 8 -2 -8 B -4 0 12 10 -6 C -8 -12 0 0 -10 D 2 -10 0 0 4 E 8 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.300000 E: 0.500000 Sum of squares = 0.380000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=23 B=18 E=17 A=15 so A is eliminated. Round 2 votes counts: D=30 E=29 C=23 B=18 so B is eliminated. Round 3 votes counts: D=34 C=34 E=32 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:210 B:206 A:201 D:198 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 -2 -8 B -4 0 12 10 -6 C -8 -12 0 0 -10 D 2 -10 0 0 4 E 8 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.300000 E: 0.500000 Sum of squares = 0.380000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 -8 B -4 0 12 10 -6 C -8 -12 0 0 -10 D 2 -10 0 0 4 E 8 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.300000 E: 0.500000 Sum of squares = 0.380000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 -8 B -4 0 12 10 -6 C -8 -12 0 0 -10 D 2 -10 0 0 4 E 8 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.300000 E: 0.500000 Sum of squares = 0.380000000001 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2428: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (14) D E A C B (9) D A B E C (9) B C A E D (8) E C D A B (7) E D C A B (6) C E D B A (5) C E B A D (5) A B D E C (5) B A C D E (4) A D B E C (4) D A E B C (3) C E B D A (3) C B E A D (3) C B A E D (3) E D A C B (2) E C D B A (2) C B E D A (2) E C A D B (1) D E C A B (1) D B A E C (1) B C A D E (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 0 8 B 4 0 2 0 12 C -6 -2 0 -10 0 D 0 0 10 0 4 E -8 -12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.519835 C: 0.000000 D: 0.480165 E: 0.000000 Sum of squares = 0.500786823572 Cumulative probabilities = A: 0.000000 B: 0.519835 C: 0.519835 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 0 8 B 4 0 2 0 12 C -6 -2 0 -10 0 D 0 0 10 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 C=21 E=18 A=10 so A is eliminated. Round 2 votes counts: B=34 D=27 C=21 E=18 so E is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 D:207 A:205 C:191 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 0 8 B 4 0 2 0 12 C -6 -2 0 -10 0 D 0 0 10 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 0 8 B 4 0 2 0 12 C -6 -2 0 -10 0 D 0 0 10 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 0 8 B 4 0 2 0 12 C -6 -2 0 -10 0 D 0 0 10 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2429: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (14) C B A D E (9) D E C B A (7) D C E B A (7) C D B A E (7) B A C D E (6) E A B D C (5) C D E B A (5) E D C A B (4) D B A C E (4) A B E D C (4) A B C E D (4) E C A B D (3) A B E C D (3) E D A C B (2) D E B A C (2) D C B A E (2) C E A B D (2) C A B E D (2) B A D C E (2) E C D A B (1) E A D B C (1) E A B C D (1) D B A E C (1) B C A D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 -12 -8 B 6 0 -2 -14 -8 C 0 2 0 -12 2 D 12 14 12 0 8 E 8 8 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -12 -8 B 6 0 -2 -14 -8 C 0 2 0 -12 2 D 12 14 12 0 8 E 8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=25 D=23 A=12 B=9 so B is eliminated. Round 2 votes counts: E=31 C=26 D=23 A=20 so A is eliminated. Round 3 votes counts: E=38 C=36 D=26 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:223 E:203 C:196 B:191 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -12 -8 B 6 0 -2 -14 -8 C 0 2 0 -12 2 D 12 14 12 0 8 E 8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -12 -8 B 6 0 -2 -14 -8 C 0 2 0 -12 2 D 12 14 12 0 8 E 8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -12 -8 B 6 0 -2 -14 -8 C 0 2 0 -12 2 D 12 14 12 0 8 E 8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2430: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (14) C D A E B (7) B E A D C (7) B E A C D (7) E B A D C (5) C D B E A (5) E A B D C (4) C D A B E (4) A E B D C (4) D C B E A (3) A E B C D (3) E B A C D (2) D C E A B (2) D C B A E (2) C D B A E (2) C A D E B (2) B E D C A (2) B E C D A (2) B E C A D (2) B C D E A (2) A E D C B (2) A C E B D (2) E A D B C (1) E A B C D (1) D C A B E (1) D B E C A (1) D A E C B (1) C B D A E (1) C B A D E (1) B D C E A (1) B C A E D (1) B A E C D (1) A E D B C (1) A E C B D (1) A D C E B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -10 0 6 B -6 0 -6 0 -8 C 10 6 0 -4 6 D 0 0 4 0 2 E -6 8 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.220271 B: 0.000000 C: 0.000000 D: 0.779729 E: 0.000000 Sum of squares = 0.656497106584 Cumulative probabilities = A: 0.220271 B: 0.220271 C: 0.220271 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 0 6 B -6 0 -6 0 -8 C 10 6 0 -4 6 D 0 0 4 0 2 E -6 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836753742 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 C=22 A=16 E=13 so E is eliminated. Round 2 votes counts: B=32 D=24 C=22 A=22 so C is eliminated. Round 3 votes counts: D=42 B=34 A=24 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:209 D:203 A:201 E:197 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -10 0 6 B -6 0 -6 0 -8 C 10 6 0 -4 6 D 0 0 4 0 2 E -6 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836753742 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 0 6 B -6 0 -6 0 -8 C 10 6 0 -4 6 D 0 0 4 0 2 E -6 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836753742 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 0 6 B -6 0 -6 0 -8 C 10 6 0 -4 6 D 0 0 4 0 2 E -6 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836753742 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2431: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) D A C E B (7) B E D A C (6) B E A D C (6) A B E C D (6) A D C B E (5) E B C D A (4) D C A E B (4) C A B E D (4) B E A C D (4) A C B E D (4) D E B A C (3) C E B D A (3) C E B A D (3) C A D E B (3) A B C E D (3) E B D A C (2) D B E A C (2) D A B E C (2) C D E B A (2) A D B E C (2) A C B D E (2) A B D E C (2) E D B C A (1) D A E B C (1) D A C B E (1) C A E B D (1) C A D B E (1) C A B D E (1) B E C A D (1) B A C E D (1) A D C E B (1) A D B C E (1) A C D E B (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 30 10 12 B -10 0 12 24 14 C -30 -12 0 -8 0 D -10 -24 8 0 -14 E -12 -14 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 30 10 12 B -10 0 12 24 14 C -30 -12 0 -8 0 D -10 -24 8 0 -14 E -12 -14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=20 C=18 B=18 E=14 so E is eliminated. Round 2 votes counts: B=31 A=30 D=21 C=18 so C is eliminated. Round 3 votes counts: A=40 B=37 D=23 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:231 B:220 E:194 D:180 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 30 10 12 B -10 0 12 24 14 C -30 -12 0 -8 0 D -10 -24 8 0 -14 E -12 -14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 30 10 12 B -10 0 12 24 14 C -30 -12 0 -8 0 D -10 -24 8 0 -14 E -12 -14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 30 10 12 B -10 0 12 24 14 C -30 -12 0 -8 0 D -10 -24 8 0 -14 E -12 -14 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2432: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) B A C E D (7) B A C D E (6) D E C B A (5) A E C D B (5) A B C E D (5) E D A C B (4) E B D A C (4) A C B E D (4) D C E B A (3) C A B D E (3) B C A D E (3) A C E D B (3) E D B A C (2) E D A B C (2) D E C A B (2) D E B C A (2) D B E C A (2) C D E B A (2) C A D E B (2) B D C E A (2) A E D C B (2) A B E D C (2) A B E C D (2) E D C B A (1) E D B C A (1) E A D C B (1) E A B D C (1) D C B E A (1) C D E A B (1) C D B E A (1) C B D A E (1) C B A D E (1) C A E D B (1) C A D B E (1) B D A C E (1) B C D A E (1) B A D C E (1) A E D B C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 10 6 8 B -6 0 -8 -8 -12 C -10 8 0 2 2 D -6 8 -2 0 -18 E -8 12 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 6 8 B -6 0 -8 -8 -12 C -10 8 0 2 2 D -6 8 -2 0 -18 E -8 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 B=21 D=15 C=13 so C is eliminated. Round 2 votes counts: A=33 E=25 B=23 D=19 so D is eliminated. Round 3 votes counts: E=40 A=33 B=27 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:210 C:201 D:191 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 6 8 B -6 0 -8 -8 -12 C -10 8 0 2 2 D -6 8 -2 0 -18 E -8 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 6 8 B -6 0 -8 -8 -12 C -10 8 0 2 2 D -6 8 -2 0 -18 E -8 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 6 8 B -6 0 -8 -8 -12 C -10 8 0 2 2 D -6 8 -2 0 -18 E -8 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2433: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C E A B D (6) A D E B C (6) C E B A D (5) C E A D B (5) B D E A C (5) B D A E C (5) B C E D A (5) D A E B C (4) D A B E C (4) D A E C B (3) D A C E B (3) C A D E B (3) B E C A D (3) B E A D C (3) A D E C B (3) E C A B D (2) E B C A D (2) E A B D C (2) C B E D A (2) C B E A D (2) B D E C A (2) A D C E B (2) E B A C D (1) E A C D B (1) D C A E B (1) D B A C E (1) D A C B E (1) D A B C E (1) B E D A C (1) B E C D A (1) B E A C D (1) B D C E A (1) B D A C E (1) B C E A D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 18 0 -2 B 2 0 18 2 -2 C -18 -18 0 -16 -16 D 0 -2 16 0 12 E 2 2 16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749999981 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 A B C D E A 0 -2 18 0 -2 B 2 0 18 2 -2 C -18 -18 0 -16 -16 D 0 -2 16 0 12 E 2 2 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749998329 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 C=23 A=13 E=8 so E is eliminated. Round 2 votes counts: B=32 D=27 C=25 A=16 so A is eliminated. Round 3 votes counts: D=38 B=34 C=28 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:210 A:207 E:204 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 18 0 -2 B 2 0 18 2 -2 C -18 -18 0 -16 -16 D 0 -2 16 0 12 E 2 2 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749998329 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 18 0 -2 B 2 0 18 2 -2 C -18 -18 0 -16 -16 D 0 -2 16 0 12 E 2 2 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749998329 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 18 0 -2 B 2 0 18 2 -2 C -18 -18 0 -16 -16 D 0 -2 16 0 12 E 2 2 16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749998329 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2434: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) D E B A C (8) C B A D E (8) D C E A B (6) E D A B C (5) D E A B C (5) C A B E D (5) E D B A C (4) D E A C B (4) B A C E D (4) A C B E D (4) E B A D C (3) D E B C A (3) C D B A E (3) C D A E B (3) C B A E D (3) C A B D E (3) B C A E D (3) B A E C D (3) A B E C D (3) E A B D C (2) C A D B E (2) D E C B A (1) D C E B A (1) C B D A E (1) B E D A C (1) B E A D C (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -2 -8 -6 B -6 0 -8 -10 -10 C 2 8 0 -8 -6 D 8 10 8 0 14 E 6 10 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -8 -6 B -6 0 -8 -10 -10 C 2 8 0 -8 -6 D 8 10 8 0 14 E 6 10 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=28 E=14 B=12 A=9 so A is eliminated. Round 2 votes counts: D=37 C=32 B=16 E=15 so E is eliminated. Round 3 votes counts: D=47 C=32 B=21 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:204 C:198 A:195 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 -8 -6 B -6 0 -8 -10 -10 C 2 8 0 -8 -6 D 8 10 8 0 14 E 6 10 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -8 -6 B -6 0 -8 -10 -10 C 2 8 0 -8 -6 D 8 10 8 0 14 E 6 10 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -8 -6 B -6 0 -8 -10 -10 C 2 8 0 -8 -6 D 8 10 8 0 14 E 6 10 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2435: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) C B E D A (7) C E D A B (6) C A E D B (5) B A D E C (5) A C D E B (5) A B D E C (5) E D C A B (4) B E D C A (4) B D E A C (4) E B C D A (3) D E B A C (3) C A B E D (3) B C E D A (3) A C B D E (3) E D B A C (2) E C D B A (2) E C D A B (2) E B D C A (2) C E B D A (2) C A B D E (2) B C A D E (2) A D E B C (2) A B C D E (2) E D C B A (1) E D B C A (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B A D E (1) C A E B D (1) B D E C A (1) B A C D E (1) A E D C B (1) A D E C B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -24 -12 -14 B 4 0 -12 8 -8 C 24 12 0 20 10 D 12 -8 -20 0 -18 E 14 8 -10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -24 -12 -14 B 4 0 -12 8 -8 C 24 12 0 20 10 D 12 -8 -20 0 -18 E 14 8 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=21 B=20 E=17 D=5 so D is eliminated. Round 2 votes counts: C=37 A=23 E=20 B=20 so E is eliminated. Round 3 votes counts: C=46 B=31 A=23 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:233 E:215 B:196 D:183 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -24 -12 -14 B 4 0 -12 8 -8 C 24 12 0 20 10 D 12 -8 -20 0 -18 E 14 8 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -24 -12 -14 B 4 0 -12 8 -8 C 24 12 0 20 10 D 12 -8 -20 0 -18 E 14 8 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -24 -12 -14 B 4 0 -12 8 -8 C 24 12 0 20 10 D 12 -8 -20 0 -18 E 14 8 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2436: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) E A C D B (6) D B A E C (5) D B A C E (5) A D C B E (5) E B C D A (4) D A E B C (4) D A B C E (4) C A E B D (4) E B D A C (3) E A D C B (3) D B E A C (3) D A B E C (3) C B A D E (3) C A B D E (3) B E D C A (3) B D C A E (3) A D E C B (3) A C E D B (3) E D B A C (2) E C B A D (2) E B D C A (2) C A D B E (2) B E C D A (2) B D E A C (2) B C E D A (2) A D C E B (2) E D A B C (1) E A D B C (1) D E B A C (1) D B C A E (1) C E B A D (1) C A E D B (1) B D E C A (1) B D C E A (1) B C D A E (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 6 16 -6 8 B -6 0 8 -14 -2 C -16 -8 0 -18 -18 D 6 14 18 0 4 E -8 2 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 -6 8 B -6 0 8 -14 -2 C -16 -8 0 -18 -18 D 6 14 18 0 4 E -8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 B=15 A=15 C=14 so C is eliminated. Round 2 votes counts: E=31 D=26 A=25 B=18 so B is eliminated. Round 3 votes counts: E=38 D=34 A=28 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:212 E:204 B:193 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 16 -6 8 B -6 0 8 -14 -2 C -16 -8 0 -18 -18 D 6 14 18 0 4 E -8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 -6 8 B -6 0 8 -14 -2 C -16 -8 0 -18 -18 D 6 14 18 0 4 E -8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 -6 8 B -6 0 8 -14 -2 C -16 -8 0 -18 -18 D 6 14 18 0 4 E -8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2437: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) C D E B A (7) A B E D C (7) B E A D C (6) D A B E C (5) C E B D A (5) A D C B E (5) E B C A D (4) D C A B E (4) D B E A C (4) A E B C D (4) E B A C D (3) C E B A D (3) C D A E B (3) B E D C A (3) E B C D A (2) D C B E A (2) D C B A E (2) D B E C A (2) D A C B E (2) B E D A C (2) A C E B D (2) A B D E C (2) E C B A D (1) D C A E B (1) D B C E A (1) D A B C E (1) C E D A B (1) C E A B D (1) C D E A B (1) C A E B D (1) C A D E B (1) B D E A C (1) B A E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 10 2 2 B -2 0 16 -2 20 C -10 -16 0 -20 -14 D -2 2 20 0 6 E -2 -20 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 2 2 B -2 0 16 -2 20 C -10 -16 0 -20 -14 D -2 2 20 0 6 E -2 -20 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 C=23 B=13 E=10 so E is eliminated. Round 2 votes counts: A=30 D=24 C=24 B=22 so B is eliminated. Round 3 votes counts: A=40 D=30 C=30 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 D:213 A:208 E:193 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 2 2 B -2 0 16 -2 20 C -10 -16 0 -20 -14 D -2 2 20 0 6 E -2 -20 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 2 2 B -2 0 16 -2 20 C -10 -16 0 -20 -14 D -2 2 20 0 6 E -2 -20 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 2 2 B -2 0 16 -2 20 C -10 -16 0 -20 -14 D -2 2 20 0 6 E -2 -20 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2438: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) C B E D A (6) D E C A B (5) D E C B A (4) B C A E D (4) B A C D E (4) A B D E C (4) E D A C B (3) D E A C B (3) C D E B A (3) B D A E C (3) B C E D A (3) B A D C E (3) B A C E D (3) A B E C D (3) E D C A B (2) E C D A B (2) D B C E A (2) D B A E C (2) D A B E C (2) C E D A B (2) C E A D B (2) C B D E A (2) B C D E A (2) A E D C B (2) A E D B C (2) A E C B D (2) A D E B C (2) A B E D C (2) E C A D B (1) D E A B C (1) D B E C A (1) C B E A D (1) C A E B D (1) B D C E A (1) B C D A E (1) B C A D E (1) B A D E C (1) A E C D B (1) A D E C B (1) A D B E C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -6 -14 -6 B 8 0 -2 -2 8 C 6 2 0 6 0 D 14 2 -6 0 0 E 6 -8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.886431 D: 0.000000 E: 0.113569 Sum of squares = 0.798657123187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.886431 D: 0.886431 E: 1.000000 A B C D E A 0 -8 -6 -14 -6 B 8 0 -2 -2 8 C 6 2 0 6 0 D 14 2 -6 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000052563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 A=22 D=20 E=8 so E is eliminated. Round 2 votes counts: C=27 B=26 D=25 A=22 so A is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:207 B:206 D:205 E:199 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -14 -6 B 8 0 -2 -2 8 C 6 2 0 6 0 D 14 2 -6 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000052563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -14 -6 B 8 0 -2 -2 8 C 6 2 0 6 0 D 14 2 -6 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000052563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -14 -6 B 8 0 -2 -2 8 C 6 2 0 6 0 D 14 2 -6 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000052563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2439: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) E D A B C (7) E A D B C (6) D C E B A (5) D C B A E (5) C B A D E (5) D E A C B (4) B C A E D (4) A E D B C (4) A B C E D (4) E D C B A (3) E A B D C (3) D C B E A (3) A B C D E (3) E D A C B (2) E A B C D (2) D C A B E (2) C D B E A (2) C D B A E (2) C B E D A (2) C B A E D (2) B C A D E (2) B A C E D (2) A D E B C (2) A B E C D (2) E D C A B (1) E C D B A (1) E C B D A (1) E B A C D (1) D E C B A (1) D E C A B (1) D C E A B (1) D A E C B (1) D A C E B (1) D A C B E (1) C B D E A (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -8 -14 6 B 4 0 -14 -10 4 C 8 14 0 -6 14 D 14 10 6 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -14 6 B 4 0 -14 -10 4 C 8 14 0 -6 14 D 14 10 6 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 C=24 A=16 B=8 so B is eliminated. Round 2 votes counts: C=30 E=27 D=25 A=18 so A is eliminated. Round 3 votes counts: C=39 E=34 D=27 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:215 B:192 A:190 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -14 6 B 4 0 -14 -10 4 C 8 14 0 -6 14 D 14 10 6 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -14 6 B 4 0 -14 -10 4 C 8 14 0 -6 14 D 14 10 6 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -14 6 B 4 0 -14 -10 4 C 8 14 0 -6 14 D 14 10 6 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2440: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) A C D E B (7) C A D B E (6) B D E C A (6) E B A D C (5) E B A C D (5) E A B C D (4) B E D C A (4) B E C D A (4) A E C D B (4) A E C B D (4) A D C E B (4) A C E D B (4) D A C E B (3) C D A B E (3) C B D A E (3) B C D E A (3) E A B D C (2) D B E C A (2) C D B A E (2) C A D E B (2) B E D A C (2) B D C E A (2) E D A B C (1) E B D A C (1) E A D B C (1) D E A B C (1) D C B A E (1) D B C E A (1) C B A D E (1) B E C A D (1) B C E A D (1) B C D A E (1) A D E C B (1) Total count = 100 A B C D E A 0 10 -2 4 8 B -10 0 -6 -2 2 C 2 6 0 10 4 D -4 2 -10 0 14 E -8 -2 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 4 8 B -10 0 -6 -2 2 C 2 6 0 10 4 D -4 2 -10 0 14 E -8 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 E=19 C=17 D=16 so D is eliminated. Round 2 votes counts: B=27 A=27 C=26 E=20 so E is eliminated. Round 3 votes counts: B=38 A=36 C=26 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:210 D:201 B:192 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 4 8 B -10 0 -6 -2 2 C 2 6 0 10 4 D -4 2 -10 0 14 E -8 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 4 8 B -10 0 -6 -2 2 C 2 6 0 10 4 D -4 2 -10 0 14 E -8 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 4 8 B -10 0 -6 -2 2 C 2 6 0 10 4 D -4 2 -10 0 14 E -8 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2441: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D A B C E (7) E B C A D (5) D A C B E (5) B A C D E (5) E C D A B (4) D E C A B (4) C A B E D (4) E D C A B (3) E D B C A (3) E C A B D (3) D E B A C (3) B D A E C (3) B C A E D (3) B A D C E (3) B A C E D (3) A B C D E (3) E B D A C (2) D B E A C (2) D B A E C (2) D B A C E (2) C A D E B (2) B E A C D (2) E D B A C (1) E B D C A (1) D E A C B (1) D C E A B (1) D C A E B (1) D A E B C (1) D A C E B (1) C E A D B (1) C E A B D (1) C B E A D (1) C A B D E (1) B E D A C (1) B E C A D (1) B D A C E (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 2 4 B 4 0 12 6 8 C -6 -12 0 -2 0 D -2 -6 2 0 6 E -4 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 4 B 4 0 12 6 8 C -6 -12 0 -2 0 D -2 -6 2 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=30 D=30 B=22 C=10 A=8 so A is eliminated. Round 2 votes counts: D=32 E=30 B=26 C=12 so C is eliminated. Round 3 votes counts: D=35 B=33 E=32 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:204 D:200 E:191 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 4 B 4 0 12 6 8 C -6 -12 0 -2 0 D -2 -6 2 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 4 B 4 0 12 6 8 C -6 -12 0 -2 0 D -2 -6 2 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 4 B 4 0 12 6 8 C -6 -12 0 -2 0 D -2 -6 2 0 6 E -4 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2442: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) C B E D A (7) E D B C A (6) D E B A C (5) A C B D E (5) C B A D E (4) A E C D B (4) E B C D A (3) D E B C A (3) D A B E C (3) C B D E A (3) A E D C B (3) E D A B C (2) E B D C A (2) D B E C A (2) D B C E A (2) D A E B C (2) C E B A D (2) C B D A E (2) C A B E D (2) C A B D E (2) A E C B D (2) A D B C E (2) A C E B D (2) A C B E D (2) E A D C B (1) E A D B C (1) D B E A C (1) D B C A E (1) D B A C E (1) C E B D A (1) C B A E D (1) C A E B D (1) B E D C A (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D E A (1) A E D B C (1) A D E C B (1) A D C E B (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 2 -2 10 B 2 0 2 -8 -6 C -2 -2 0 -6 -4 D 2 8 6 0 10 E -10 6 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 10 B 2 0 2 -8 -6 C -2 -2 0 -6 -4 D 2 8 6 0 10 E -10 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=25 D=20 E=15 B=5 so B is eliminated. Round 2 votes counts: A=35 C=27 D=22 E=16 so E is eliminated. Round 3 votes counts: A=37 D=33 C=30 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:204 B:195 E:195 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -2 10 B 2 0 2 -8 -6 C -2 -2 0 -6 -4 D 2 8 6 0 10 E -10 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 10 B 2 0 2 -8 -6 C -2 -2 0 -6 -4 D 2 8 6 0 10 E -10 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 10 B 2 0 2 -8 -6 C -2 -2 0 -6 -4 D 2 8 6 0 10 E -10 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2443: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) B D A C E (9) E D A B C (6) B A D C E (6) E C A D B (5) C E A D B (5) A D B C E (5) E C D A B (3) D E B A C (3) D B A E C (3) D A B E C (3) C E B A D (3) C B A E D (3) B D A E C (3) A B D C E (3) E D C A B (2) C E B D A (2) C A E B D (2) C A B E D (2) C A B D E (2) B A C D E (2) E D B A C (1) E D A C B (1) E C D B A (1) E B C D A (1) E A D C B (1) E A C D B (1) D B E A C (1) D A E B C (1) C E A B D (1) C B E D A (1) C A E D B (1) B D E C A (1) B D E A C (1) B D C A E (1) B C A D E (1) A E D C B (1) A D E C B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 8 12 20 B 4 0 2 8 14 C -8 -2 0 -6 20 D -12 -8 6 0 14 E -20 -14 -20 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 12 20 B 4 0 2 8 14 C -8 -2 0 -6 20 D -12 -8 6 0 14 E -20 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997032 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=24 E=22 A=12 D=11 so D is eliminated. Round 2 votes counts: C=31 B=28 E=25 A=16 so A is eliminated. Round 3 votes counts: B=39 C=33 E=28 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:218 B:214 C:202 D:200 E:166 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 12 20 B 4 0 2 8 14 C -8 -2 0 -6 20 D -12 -8 6 0 14 E -20 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997032 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 12 20 B 4 0 2 8 14 C -8 -2 0 -6 20 D -12 -8 6 0 14 E -20 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997032 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 12 20 B 4 0 2 8 14 C -8 -2 0 -6 20 D -12 -8 6 0 14 E -20 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997032 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2444: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) E C D B A (8) E C A B D (6) B D A C E (6) A B D E C (6) A E B C D (5) D B C A E (4) D B A C E (4) C D E B A (4) B A D C E (4) A B E D C (4) A B C E D (4) A B C D E (4) E C D A B (3) C E D B A (3) C E A B D (3) E D C B A (2) E C A D B (2) D E C B A (2) D C E B A (2) D C B E A (2) D B A E C (2) C D B E A (2) A E C B D (2) A B E C D (2) E A D C B (1) E A C B D (1) D B C E A (1) C E B D A (1) C B D A E (1) Total count = 100 A B C D E A 0 4 8 6 14 B -4 0 10 16 10 C -8 -10 0 2 8 D -6 -16 -2 0 6 E -14 -10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 6 14 B -4 0 10 16 10 C -8 -10 0 2 8 D -6 -16 -2 0 6 E -14 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=23 D=17 C=14 B=10 so B is eliminated. Round 2 votes counts: A=40 E=23 D=23 C=14 so C is eliminated. Round 3 votes counts: A=40 E=30 D=30 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:216 C:196 D:191 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 14 B -4 0 10 16 10 C -8 -10 0 2 8 D -6 -16 -2 0 6 E -14 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 14 B -4 0 10 16 10 C -8 -10 0 2 8 D -6 -16 -2 0 6 E -14 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 14 B -4 0 10 16 10 C -8 -10 0 2 8 D -6 -16 -2 0 6 E -14 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2445: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (14) D E B C A (12) C A B D E (12) E D B A C (10) B A C E D (10) C A B E D (6) A C B E D (5) B A E C D (4) B E D A C (3) B C A D E (3) A B C E D (3) E D A C B (2) E B D A C (2) D E B A C (2) C A D E B (2) B E A D C (2) E A C D B (1) D C B A E (1) D B E C A (1) C D A E B (1) C D A B E (1) C B D A E (1) C A E D B (1) B D E C A (1) Total count = 100 A B C D E A 0 -4 -12 -2 0 B 4 0 6 4 6 C 12 -6 0 0 -8 D 2 -4 0 0 2 E 0 -6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 0 B 4 0 6 4 6 C 12 -6 0 0 -8 D 2 -4 0 0 2 E 0 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=24 B=23 E=15 A=8 so A is eliminated. Round 2 votes counts: D=30 C=29 B=26 E=15 so E is eliminated. Round 3 votes counts: D=42 C=30 B=28 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:210 D:200 E:200 C:199 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 0 B 4 0 6 4 6 C 12 -6 0 0 -8 D 2 -4 0 0 2 E 0 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 0 B 4 0 6 4 6 C 12 -6 0 0 -8 D 2 -4 0 0 2 E 0 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 0 B 4 0 6 4 6 C 12 -6 0 0 -8 D 2 -4 0 0 2 E 0 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2446: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (9) D E A C B (8) B C E A D (8) D A E C B (7) B C A E D (6) D A C B E (5) B C A D E (5) E D A C B (4) E D A B C (4) E B C A D (4) E A D C B (4) D E A B C (4) B E C A D (4) C A B D E (3) E C A B D (2) D A E B C (2) C B A E D (2) C A E B D (2) E C B A D (1) E A C D B (1) D B E A C (1) D B C A E (1) D B A C E (1) D A C E B (1) D A B E C (1) D A B C E (1) C D A B E (1) C B E A D (1) B D C E A (1) B D C A E (1) B C E D A (1) A E D C B (1) A E C D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 8 -4 -6 4 B -8 0 8 2 6 C 4 -8 0 6 2 D 6 -2 -6 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999435 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -6 4 B -8 0 8 2 6 C 4 -8 0 6 2 D 6 -2 -6 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999777 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=32 E=20 C=9 A=4 so A is eliminated. Round 2 votes counts: B=35 D=32 E=22 C=11 so C is eliminated. Round 3 votes counts: B=41 D=34 E=25 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:204 C:202 D:202 A:201 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -4 -6 4 B -8 0 8 2 6 C 4 -8 0 6 2 D 6 -2 -6 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999777 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -6 4 B -8 0 8 2 6 C 4 -8 0 6 2 D 6 -2 -6 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999777 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -6 4 B -8 0 8 2 6 C 4 -8 0 6 2 D 6 -2 -6 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999777 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2447: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (15) C B E D A (9) E D B C A (8) A C B E D (8) C B D E A (7) A E D C B (7) B C D E A (5) A E D B C (5) A C E D B (5) A C E B D (4) C A B E D (3) A C B D E (3) E D B A C (2) E C D B A (2) D E B A C (2) C B A E D (2) C B A D E (2) E D A B C (1) E C B D A (1) E A D B C (1) D E A B C (1) D B E C A (1) C E B D A (1) B D E C A (1) B C D A E (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -16 -14 -14 B 18 0 -12 -6 -16 C 16 12 0 8 0 D 14 6 -8 0 -20 E 14 16 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.698324 D: 0.000000 E: 0.301676 Sum of squares = 0.578664889201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.698324 D: 0.698324 E: 1.000000 A B C D E A 0 -18 -16 -14 -14 B 18 0 -12 -6 -16 C 16 12 0 8 0 D 14 6 -8 0 -20 E 14 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=24 D=19 E=15 B=7 so B is eliminated. Round 2 votes counts: A=35 C=30 D=20 E=15 so E is eliminated. Round 3 votes counts: A=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:225 C:218 D:196 B:192 A:169 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -16 -14 -14 B 18 0 -12 -6 -16 C 16 12 0 8 0 D 14 6 -8 0 -20 E 14 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -14 -14 B 18 0 -12 -6 -16 C 16 12 0 8 0 D 14 6 -8 0 -20 E 14 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -14 -14 B 18 0 -12 -6 -16 C 16 12 0 8 0 D 14 6 -8 0 -20 E 14 16 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2448: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) A C B E D (12) D E B C A (10) D E A C B (7) A C D E B (6) C A B E D (4) B E D C A (4) B D E C A (4) A C E D B (4) E D B C A (3) D B E C A (3) D A C E B (3) A D C E B (3) E D C A B (2) D E A B C (2) D A E C B (2) B E C A D (2) B C E A D (2) B C A D E (2) E B D C A (1) D E B A C (1) D B A E C (1) D A C B E (1) C E B A D (1) C E A D B (1) C B A E D (1) A D B C E (1) A C E B D (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 12 14 B -4 0 0 -2 6 C 6 0 0 4 16 D -12 2 -4 0 -2 E -14 -6 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.248360 C: 0.751640 D: 0.000000 E: 0.000000 Sum of squares = 0.626645220935 Cumulative probabilities = A: 0.000000 B: 0.248360 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 12 14 B -4 0 0 -2 6 C 6 0 0 4 16 D -12 2 -4 0 -2 E -14 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=30 A=30 B=27 C=7 E=6 so E is eliminated. Round 2 votes counts: D=35 A=30 B=28 C=7 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:212 B:200 D:192 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 12 14 B -4 0 0 -2 6 C 6 0 0 4 16 D -12 2 -4 0 -2 E -14 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 12 14 B -4 0 0 -2 6 C 6 0 0 4 16 D -12 2 -4 0 -2 E -14 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 12 14 B -4 0 0 -2 6 C 6 0 0 4 16 D -12 2 -4 0 -2 E -14 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2449: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) B C A D E (9) B A C E D (7) D E C A B (6) D E A B C (6) A B C E D (6) D E C B A (5) D E A C B (5) B C A E D (5) C D E B A (4) C B D E A (3) C B A E D (3) C B A D E (3) E D A B C (2) E A D C B (2) D E B A C (2) C A E D B (2) B D C E A (2) A C B E D (2) A B E D C (2) E C D A B (1) D E B C A (1) D C E B A (1) C E D A B (1) C D B E A (1) C A E B D (1) C A B E D (1) B D E A C (1) B C D A E (1) B A E D C (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 2 0 -2 -4 B -2 0 -4 0 -6 C 0 4 0 6 4 D 2 0 -6 0 0 E 4 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.368608 B: 0.000000 C: 0.631392 D: 0.000000 E: 0.000000 Sum of squares = 0.534527774549 Cumulative probabilities = A: 0.368608 B: 0.368608 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 -4 B -2 0 -4 0 -6 C 0 4 0 6 4 D 2 0 -6 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499863 B: 0.000000 C: 0.500137 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037386 Cumulative probabilities = A: 0.499863 B: 0.499863 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 C=19 A=15 E=14 so E is eliminated. Round 2 votes counts: D=37 B=26 C=20 A=17 so A is eliminated. Round 3 votes counts: D=40 B=36 C=24 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:207 E:203 A:198 D:198 B:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -2 -4 B -2 0 -4 0 -6 C 0 4 0 6 4 D 2 0 -6 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499863 B: 0.000000 C: 0.500137 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037386 Cumulative probabilities = A: 0.499863 B: 0.499863 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 -4 B -2 0 -4 0 -6 C 0 4 0 6 4 D 2 0 -6 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499863 B: 0.000000 C: 0.500137 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037386 Cumulative probabilities = A: 0.499863 B: 0.499863 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 -4 B -2 0 -4 0 -6 C 0 4 0 6 4 D 2 0 -6 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499863 B: 0.000000 C: 0.500137 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037386 Cumulative probabilities = A: 0.499863 B: 0.499863 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2450: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (11) C A D B E (10) E D A C B (9) E B D A C (9) E D A B C (7) C B A D E (6) E A D C B (4) D A C E B (4) B C E A D (4) B C D A E (4) E B C A D (3) D A C B E (3) B E C D A (3) B E C A D (3) E D B A C (2) E B D C A (2) D A E C B (2) B D A C E (2) A D C B E (2) D A E B C (1) C B D A E (1) C A D E B (1) C A B D E (1) B E D A C (1) B D C E A (1) B C A E D (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -2 -2 4 B 6 0 8 4 6 C 2 -8 0 -2 6 D 2 -4 2 0 4 E -4 -6 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -2 4 B 6 0 8 4 6 C 2 -8 0 -2 6 D 2 -4 2 0 4 E -4 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=30 C=19 D=10 A=5 so A is eliminated. Round 2 votes counts: E=36 B=30 C=20 D=14 so D is eliminated. Round 3 votes counts: E=40 C=30 B=30 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:202 C:199 A:197 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -2 4 B 6 0 8 4 6 C 2 -8 0 -2 6 D 2 -4 2 0 4 E -4 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -2 4 B 6 0 8 4 6 C 2 -8 0 -2 6 D 2 -4 2 0 4 E -4 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -2 4 B 6 0 8 4 6 C 2 -8 0 -2 6 D 2 -4 2 0 4 E -4 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2451: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (9) B D E C A (8) D E B C A (6) A C E D B (6) D E C A B (5) B D A C E (5) B A C E D (5) A C B E D (4) E C A D B (3) E B C A D (3) D A C E B (3) E D C A B (2) E C D A B (2) E C A B D (2) D B A E C (2) D B A C E (2) C E A D B (2) B D E A C (2) B A C D E (2) A C D E B (2) A B C E D (2) D E C B A (1) D C E A B (1) D C A E B (1) D B E A C (1) D A C B E (1) C A E D B (1) B E C A D (1) B D A E C (1) B A E C D (1) B A D C E (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 -4 4 B 0 0 4 -2 -4 C -6 -4 0 -4 2 D 4 2 4 0 10 E -4 4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -4 4 B 0 0 4 -2 -4 C -6 -4 0 -4 2 D 4 2 4 0 10 E -4 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=27 B=26 E=12 C=3 so C is eliminated. Round 2 votes counts: D=32 A=28 B=26 E=14 so E is eliminated. Round 3 votes counts: D=36 A=35 B=29 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:203 B:199 C:194 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 -4 4 B 0 0 4 -2 -4 C -6 -4 0 -4 2 D 4 2 4 0 10 E -4 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -4 4 B 0 0 4 -2 -4 C -6 -4 0 -4 2 D 4 2 4 0 10 E -4 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -4 4 B 0 0 4 -2 -4 C -6 -4 0 -4 2 D 4 2 4 0 10 E -4 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2452: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) A E D C B (7) B C D E A (6) B C D A E (6) A E D B C (5) E A B D C (4) D C E B A (4) D A E C B (4) B E C A D (4) E D A C B (3) D C E A B (3) D C B A E (3) C D B E A (3) B C E D A (3) A E B D C (3) A D E C B (3) D E C A B (2) C B D E A (2) C B D A E (2) B E A C D (2) B C A E D (2) B A E C D (2) A B E C D (2) A B C D E (2) E D C B A (1) E C D B A (1) E B C D A (1) E A D B C (1) D E A C B (1) D C B E A (1) D C A E B (1) D A C E B (1) D A C B E (1) B E C D A (1) B C E A D (1) B C A D E (1) B A C E D (1) B A C D E (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 0 -2 B -4 0 0 -6 -6 C -4 0 0 -14 -10 D 0 6 14 0 -2 E 2 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 0 -2 B -4 0 0 -6 -6 C -4 0 0 -14 -10 D 0 6 14 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 D=21 E=18 C=7 so C is eliminated. Round 2 votes counts: B=34 D=24 A=24 E=18 so E is eliminated. Round 3 votes counts: A=36 B=35 D=29 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:210 D:209 A:203 B:192 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 0 -2 B -4 0 0 -6 -6 C -4 0 0 -14 -10 D 0 6 14 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 -2 B -4 0 0 -6 -6 C -4 0 0 -14 -10 D 0 6 14 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 -2 B -4 0 0 -6 -6 C -4 0 0 -14 -10 D 0 6 14 0 -2 E 2 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2453: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) D A C E B (7) A D B C E (6) B E D A C (5) E D C B A (4) E C D B A (4) E C B D A (4) B A D E C (4) E B C D A (3) B C E A D (3) A D C B E (3) E D C A B (2) D C A E B (2) C E B D A (2) C D E A B (2) B E C D A (2) B E A D C (2) B A E D C (2) B A E C D (2) B A C E D (2) B A C D E (2) A C D B E (2) A B D C E (2) E D B C A (1) E C B A D (1) E B D A C (1) E B C A D (1) D E C A B (1) D E A C B (1) D B A E C (1) D A E C B (1) D A E B C (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B A D (1) C E A D B (1) C E A B D (1) C D A E B (1) C B E A D (1) C B A E D (1) C A D E B (1) B E A C D (1) B D E A C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 0 2 -12 B 22 0 8 10 10 C 0 -8 0 0 -12 D -2 -10 0 0 -16 E 12 -10 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 0 2 -12 B 22 0 8 10 10 C 0 -8 0 0 -12 D -2 -10 0 0 -16 E 12 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=21 D=16 A=15 C=12 so C is eliminated. Round 2 votes counts: B=38 E=27 D=19 A=16 so A is eliminated. Round 3 votes counts: B=41 D=32 E=27 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:215 C:190 D:186 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 0 2 -12 B 22 0 8 10 10 C 0 -8 0 0 -12 D -2 -10 0 0 -16 E 12 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 0 2 -12 B 22 0 8 10 10 C 0 -8 0 0 -12 D -2 -10 0 0 -16 E 12 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 0 2 -12 B 22 0 8 10 10 C 0 -8 0 0 -12 D -2 -10 0 0 -16 E 12 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2454: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) D E B A C (5) C E D B A (5) C E D A B (5) D E C B A (4) A C D B E (4) A B C E D (4) E C D B A (3) D C E A B (3) D C A E B (3) C B A E D (3) C A E B D (3) C A B E D (3) B E C A D (3) A C B D E (3) E D C B A (2) D B A E C (2) D A C E B (2) C E B D A (2) C D E A B (2) B D A E C (2) B C A E D (2) B A E D C (2) B A E C D (2) B A D E C (2) A D B E C (2) A B D E C (2) A B D C E (2) E D B C A (1) E C B A D (1) E B D C A (1) D E A C B (1) D B E A C (1) D A E B C (1) C E B A D (1) C A D E B (1) B E D C A (1) B E C D A (1) B C E A D (1) B A C E D (1) A D C B E (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -16 -8 0 B -4 0 -20 -12 -6 C 16 20 0 4 8 D 8 12 -4 0 4 E 0 6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -8 0 B -4 0 -20 -12 -6 C 16 20 0 4 8 D 8 12 -4 0 4 E 0 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 A=21 B=17 E=8 so E is eliminated. Round 2 votes counts: D=32 C=29 A=21 B=18 so B is eliminated. Round 3 votes counts: D=36 C=36 A=28 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 D:210 E:197 A:190 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 -8 0 B -4 0 -20 -12 -6 C 16 20 0 4 8 D 8 12 -4 0 4 E 0 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -8 0 B -4 0 -20 -12 -6 C 16 20 0 4 8 D 8 12 -4 0 4 E 0 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -8 0 B -4 0 -20 -12 -6 C 16 20 0 4 8 D 8 12 -4 0 4 E 0 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2455: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (13) E C B D A (10) E C B A D (7) D A B C E (7) A D B C E (7) A B C E D (6) D E C B A (5) C B E A D (4) B C A D E (4) A B C D E (4) E D A C B (3) D E A C B (3) D E A B C (3) D B C A E (3) D A E B C (3) C B A E D (3) E A C B D (2) D C B E A (2) B C A E D (2) E A D C B (1) D C B A E (1) D B A C E (1) D A B E C (1) C B D E A (1) B D C A E (1) B C D A E (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 -14 -16 -8 B 18 0 -10 -6 0 C 14 10 0 -10 -2 D 16 6 10 0 -2 E 8 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.059607 C: 0.000000 D: 0.000000 E: 0.940393 Sum of squares = 0.887891725849 Cumulative probabilities = A: 0.000000 B: 0.059607 C: 0.059607 D: 0.059607 E: 1.000000 A B C D E A 0 -18 -14 -16 -8 B 18 0 -10 -6 0 C 14 10 0 -10 -2 D 16 6 10 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222324282 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=29 A=18 B=9 C=8 so C is eliminated. Round 2 votes counts: E=36 D=29 A=18 B=17 so B is eliminated. Round 3 votes counts: E=40 D=32 A=28 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:215 C:206 E:206 B:201 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -14 -16 -8 B 18 0 -10 -6 0 C 14 10 0 -10 -2 D 16 6 10 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222324282 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 -16 -8 B 18 0 -10 -6 0 C 14 10 0 -10 -2 D 16 6 10 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222324282 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 -16 -8 B 18 0 -10 -6 0 C 14 10 0 -10 -2 D 16 6 10 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222324282 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2456: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) E C A B D (7) E A C B D (7) A E B C D (7) D B A C E (6) E A C D B (5) C E D A B (5) C D E B A (5) B A D E C (5) A B E D C (5) E C A D B (4) D C B E A (4) C E A D B (4) C D B E A (4) A E B D C (4) C E D B A (3) A B D E C (3) E A B C D (2) B A E C D (2) E D C A B (1) E C D A B (1) D B A E C (1) C E B D A (1) C E A B D (1) C B D E A (1) B D C A E (1) B D A E C (1) B D A C E (1) A E D B C (1) Total count = 100 A B C D E A 0 14 0 14 -10 B -14 0 -6 -4 -16 C 0 6 0 18 -12 D -14 4 -18 0 -20 E 10 16 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 0 14 -10 B -14 0 -6 -4 -16 C 0 6 0 18 -12 D -14 4 -18 0 -20 E 10 16 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=24 A=20 D=19 B=10 so B is eliminated. Round 2 votes counts: E=27 A=27 C=24 D=22 so D is eliminated. Round 3 votes counts: C=37 A=36 E=27 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:229 A:209 C:206 B:180 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 14 -10 B -14 0 -6 -4 -16 C 0 6 0 18 -12 D -14 4 -18 0 -20 E 10 16 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 14 -10 B -14 0 -6 -4 -16 C 0 6 0 18 -12 D -14 4 -18 0 -20 E 10 16 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 14 -10 B -14 0 -6 -4 -16 C 0 6 0 18 -12 D -14 4 -18 0 -20 E 10 16 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2457: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A B E C D (8) E B A C D (7) D C E B A (7) B E A C D (5) A E B C D (5) E D A B C (4) E B A D C (4) E A B D C (3) C D B A E (3) C D A B E (3) B E C D A (3) A E B D C (3) E D B C A (2) E B D A C (2) D E C B A (2) D E C A B (2) D C A E B (2) D C A B E (2) D A E C B (2) B A C E D (2) A C D B E (2) A B C E D (2) E D B A C (1) E B D C A (1) E A D B C (1) E A B C D (1) D C B A E (1) D A C E B (1) C D B E A (1) C B D A E (1) C B A D E (1) C A D B E (1) B C D E A (1) B C A E D (1) B A E C D (1) A E D B C (1) A D E C B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 16 2 -10 B 8 0 16 4 -6 C -16 -16 0 0 -18 D -2 -4 0 0 -14 E 10 6 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 16 2 -10 B 8 0 16 4 -6 C -16 -16 0 0 -18 D -2 -4 0 0 -14 E 10 6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=26 A=24 B=13 C=10 so C is eliminated. Round 2 votes counts: D=34 E=26 A=25 B=15 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:211 A:200 D:190 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 16 2 -10 B 8 0 16 4 -6 C -16 -16 0 0 -18 D -2 -4 0 0 -14 E 10 6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 2 -10 B 8 0 16 4 -6 C -16 -16 0 0 -18 D -2 -4 0 0 -14 E 10 6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 2 -10 B 8 0 16 4 -6 C -16 -16 0 0 -18 D -2 -4 0 0 -14 E 10 6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2458: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) C E A D B (7) E C A D B (6) D B A E C (6) B A D E C (6) D B C E A (5) A E C B D (5) A E C D B (4) A B E C D (4) C E D B A (3) C E A B D (3) B D C E A (3) B D A C E (3) A B E D C (3) E A C B D (2) D E A C B (2) D C E B A (2) D C B E A (2) D B C A E (2) D B A C E (2) C D B E A (2) B D A E C (2) B C D A E (2) A E D C B (2) A E D B C (2) A E B D C (2) A E B C D (2) E A C D B (1) C D E B A (1) C B E D A (1) C B D E A (1) B D C A E (1) B C D E A (1) B A C E D (1) A D E B C (1) Total count = 100 A B C D E A 0 8 0 2 0 B -8 0 -4 -16 -6 C 0 4 0 8 0 D -2 16 -8 0 -12 E 0 6 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.193542 B: 0.000000 C: 0.407349 D: 0.000000 E: 0.399109 Sum of squares = 0.362679870377 Cumulative probabilities = A: 0.193542 B: 0.193542 C: 0.600891 D: 0.600891 E: 1.000000 A B C D E A 0 8 0 2 0 B -8 0 -4 -16 -6 C 0 4 0 8 0 D -2 16 -8 0 -12 E 0 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333334 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=25 D=21 B=19 E=9 so E is eliminated. Round 2 votes counts: C=32 A=28 D=21 B=19 so B is eliminated. Round 3 votes counts: C=35 A=35 D=30 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:209 C:206 A:205 D:197 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 2 0 B -8 0 -4 -16 -6 C 0 4 0 8 0 D -2 16 -8 0 -12 E 0 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333334 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 2 0 B -8 0 -4 -16 -6 C 0 4 0 8 0 D -2 16 -8 0 -12 E 0 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333334 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 2 0 B -8 0 -4 -16 -6 C 0 4 0 8 0 D -2 16 -8 0 -12 E 0 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333334 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2459: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) E C D A B (8) B C E A D (6) B C A E D (6) B A C D E (6) E D C A B (4) E C B D A (4) D A C E B (4) C E B A D (4) B D A E C (4) D E A C B (3) D A B E C (3) D A B C E (3) B A C E D (3) A D B C E (3) E C D B A (2) E C B A D (2) B E C A D (2) A D C B E (2) A B D C E (2) E D C B A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A B C (1) D C E A B (1) D B A E C (1) D A E C B (1) C E A D B (1) C E A B D (1) C B E A D (1) C A E B D (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E A C (1) B A E C D (1) B A D E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 0 8 6 B 16 0 12 18 16 C 0 -12 0 6 12 D -8 -18 -6 0 -4 E -6 -16 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 8 6 B 16 0 12 18 16 C 0 -12 0 6 12 D -8 -18 -6 0 -4 E -6 -16 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=23 D=18 C=9 A=9 so C is eliminated. Round 2 votes counts: B=42 E=29 D=18 A=11 so A is eliminated. Round 3 votes counts: B=45 E=30 D=25 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:231 C:203 A:199 E:185 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 8 6 B 16 0 12 18 16 C 0 -12 0 6 12 D -8 -18 -6 0 -4 E -6 -16 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 8 6 B 16 0 12 18 16 C 0 -12 0 6 12 D -8 -18 -6 0 -4 E -6 -16 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 8 6 B 16 0 12 18 16 C 0 -12 0 6 12 D -8 -18 -6 0 -4 E -6 -16 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2460: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) E A C B D (8) B D A C E (7) E A B C D (6) C D B E A (6) B D C A E (6) A E C D B (6) D B C A E (5) E B A D C (4) C D E B A (3) B D C E A (3) E B D C A (2) D B C E A (2) C D E A B (2) C D A E B (2) C D A B E (2) B E D C A (2) B D E C A (2) B A D E C (2) A E B D C (2) A C E D B (2) A B D C E (2) E C D A B (1) E C A D B (1) E C A B D (1) E A B D C (1) D C B E A (1) D C B A E (1) C E D B A (1) C E A D B (1) C A D E B (1) B E A D C (1) B D E A C (1) B D A E C (1) B A D C E (1) A E C B D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 10 0 -14 B 2 0 2 8 -8 C -10 -2 0 6 -2 D 0 -8 -6 0 2 E 14 8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 A B C D E A 0 -2 10 0 -14 B 2 0 2 8 -8 C -10 -2 0 6 -2 D 0 -8 -6 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=26 C=18 A=15 D=9 so D is eliminated. Round 2 votes counts: B=33 E=32 C=20 A=15 so A is eliminated. Round 3 votes counts: E=41 B=36 C=23 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:202 A:197 C:196 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 10 0 -14 B 2 0 2 8 -8 C -10 -2 0 6 -2 D 0 -8 -6 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 0 -14 B 2 0 2 8 -8 C -10 -2 0 6 -2 D 0 -8 -6 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 0 -14 B 2 0 2 8 -8 C -10 -2 0 6 -2 D 0 -8 -6 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2461: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) E C D A B (11) B A D C E (10) A B D C E (9) E C B D A (8) D C E A B (7) D A C E B (6) B A D E C (4) A D B C E (4) E D C A B (3) D A C B E (3) A B D E C (3) D E A C B (2) D C A E B (2) C E D A B (2) B E C A D (2) B E A C D (2) B A E C D (2) B A C D E (2) E D A C B (1) E B C D A (1) E B C A D (1) E B A C D (1) D E C A B (1) C D A B E (1) C B E D A (1) Total count = 100 A B C D E A 0 10 -2 -20 -8 B -10 0 -18 -8 -14 C 2 18 0 -10 -6 D 20 8 10 0 8 E 8 14 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -20 -8 B -10 0 -18 -8 -14 C 2 18 0 -10 -6 D 20 8 10 0 8 E 8 14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=22 D=21 A=16 C=4 so C is eliminated. Round 2 votes counts: E=39 B=23 D=22 A=16 so A is eliminated. Round 3 votes counts: E=39 B=35 D=26 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:223 E:210 C:202 A:190 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -2 -20 -8 B -10 0 -18 -8 -14 C 2 18 0 -10 -6 D 20 8 10 0 8 E 8 14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -20 -8 B -10 0 -18 -8 -14 C 2 18 0 -10 -6 D 20 8 10 0 8 E 8 14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -20 -8 B -10 0 -18 -8 -14 C 2 18 0 -10 -6 D 20 8 10 0 8 E 8 14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2462: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (25) A E B C D (11) A E B D C (8) D C A E B (5) C D B E A (5) C B E A D (5) D B C E A (4) B E A C D (4) D A E B C (3) D C B A E (2) D B A E C (2) C E A B D (2) C B E D A (2) B E C A D (2) B E A D C (2) A E C B D (2) A D E C B (2) A D E B C (2) A C E B D (2) E A B C D (1) D C A B E (1) D B E C A (1) D B E A C (1) D A C E B (1) D A B E C (1) C E B A D (1) C A E B D (1) B D E A C (1) B C E A D (1) Total count = 100 A B C D E A 0 -16 -14 -8 -14 B 16 0 -12 -10 18 C 14 12 0 -22 14 D 8 10 22 0 12 E 14 -18 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -8 -14 B 16 0 -12 -10 18 C 14 12 0 -22 14 D 8 10 22 0 12 E 14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=46 A=27 C=16 B=10 E=1 so E is eliminated. Round 2 votes counts: D=46 A=28 C=16 B=10 so B is eliminated. Round 3 votes counts: D=47 A=34 C=19 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:209 B:206 E:185 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -14 -8 -14 B 16 0 -12 -10 18 C 14 12 0 -22 14 D 8 10 22 0 12 E 14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -8 -14 B 16 0 -12 -10 18 C 14 12 0 -22 14 D 8 10 22 0 12 E 14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -8 -14 B 16 0 -12 -10 18 C 14 12 0 -22 14 D 8 10 22 0 12 E 14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2463: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (29) B A D C E (15) A D B C E (6) E C D B A (5) B A E D C (4) A B D C E (4) C E D A B (3) C D E A B (3) B E D C A (3) B D A E C (3) B A D E C (3) E B C D A (2) D C E A B (2) D A C E B (2) D A B C E (2) B A E C D (2) A D C B E (2) D E C A B (1) D C A E B (1) C E A D B (1) C A E D B (1) B E C A D (1) B E A C D (1) B D E A C (1) B A C D E (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 18 -4 -14 -4 B -18 0 -4 -20 -4 C 4 4 0 0 -10 D 14 20 0 0 -4 E 4 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 -4 -14 -4 B -18 0 -4 -20 -4 C 4 4 0 0 -10 D 14 20 0 0 -4 E 4 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=34 A=14 D=8 C=8 so D is eliminated. Round 2 votes counts: E=37 B=34 A=18 C=11 so C is eliminated. Round 3 votes counts: E=46 B=34 A=20 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:211 C:199 A:198 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -4 -14 -4 B -18 0 -4 -20 -4 C 4 4 0 0 -10 D 14 20 0 0 -4 E 4 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 -14 -4 B -18 0 -4 -20 -4 C 4 4 0 0 -10 D 14 20 0 0 -4 E 4 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 -14 -4 B -18 0 -4 -20 -4 C 4 4 0 0 -10 D 14 20 0 0 -4 E 4 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2464: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) E D C B A (6) B A D E C (6) E C D B A (5) C A D E B (5) D B A E C (4) C E A D B (4) C A E B D (4) E B D C A (3) D E B C A (3) C E B A D (3) C A E D B (3) A C D E B (3) A B D E C (3) D B E A C (2) B E D C A (2) B D A E C (2) B A E C D (2) B A C E D (2) A D C E B (2) A D C B E (2) A C B E D (2) A B C D E (2) E D C A B (1) E C D A B (1) E C B D A (1) E B C D A (1) D E C B A (1) D B E C A (1) D A C E B (1) D A B E C (1) C D E A B (1) C D A E B (1) B E D A C (1) B E C D A (1) B E A D C (1) B C A E D (1) B A E D C (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -18 0 2 B -2 0 -18 -18 -20 C 18 18 0 12 2 D 0 18 -12 0 -14 E -2 20 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 0 2 B -2 0 -18 -18 -20 C 18 18 0 12 2 D 0 18 -12 0 -14 E -2 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=19 E=18 A=18 D=13 so D is eliminated. Round 2 votes counts: C=32 B=26 E=22 A=20 so A is eliminated. Round 3 votes counts: C=44 B=34 E=22 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:215 D:196 A:193 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -18 0 2 B -2 0 -18 -18 -20 C 18 18 0 12 2 D 0 18 -12 0 -14 E -2 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 0 2 B -2 0 -18 -18 -20 C 18 18 0 12 2 D 0 18 -12 0 -14 E -2 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 0 2 B -2 0 -18 -18 -20 C 18 18 0 12 2 D 0 18 -12 0 -14 E -2 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2465: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (6) D C A E B (6) A D E C B (6) D A E B C (5) C B E D A (5) A E D B C (5) A E B D C (5) D C B E A (4) D C B A E (4) D A C E B (4) C D B E A (4) C B E A D (4) B E A C D (4) D A E C B (3) C B D E A (3) A E C D B (3) A E C B D (3) E B A C D (2) E A C B D (2) D B C E A (2) C D B A E (2) C A E B D (2) B C D E A (2) A E B C D (2) E B C A D (1) E A B D C (1) D C A B E (1) D B E A C (1) C E A B D (1) C D A B E (1) C A E D B (1) C A D E B (1) B D E A C (1) B D C E A (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 20 10 2 12 B -20 0 -22 -8 -22 C -10 22 0 0 -2 D -2 8 0 0 4 E -12 22 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 2 12 B -20 0 -22 -8 -22 C -10 22 0 0 -2 D -2 8 0 0 4 E -12 22 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=26 C=24 E=12 B=8 so B is eliminated. Round 2 votes counts: D=32 C=26 A=26 E=16 so E is eliminated. Round 3 votes counts: A=41 D=32 C=27 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:205 D:205 E:204 B:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 2 12 B -20 0 -22 -8 -22 C -10 22 0 0 -2 D -2 8 0 0 4 E -12 22 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 2 12 B -20 0 -22 -8 -22 C -10 22 0 0 -2 D -2 8 0 0 4 E -12 22 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 2 12 B -20 0 -22 -8 -22 C -10 22 0 0 -2 D -2 8 0 0 4 E -12 22 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2466: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (9) D C A E B (6) B E C A D (6) D A C E B (5) E C D B A (4) A D C B E (4) E C B D A (3) E B C D A (3) E B A C D (3) C D B A E (3) C A D B E (3) B C E A D (3) E D C A B (2) E D A C B (2) E D A B C (2) E B C A D (2) E A B D C (2) D A C B E (2) C D E A B (2) C D A E B (2) C B E D A (2) B E A C D (2) B A D E C (2) A B D C E (2) E B D A C (1) E A D B C (1) D E A C B (1) D A E C B (1) C E D B A (1) C E B D A (1) C B D A E (1) B E A D C (1) B C E D A (1) B C A D E (1) B A D C E (1) A D E B C (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -10 -8 -8 B 0 0 -10 -6 -8 C 10 10 0 6 2 D 8 6 -6 0 -2 E 8 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -8 -8 B 0 0 -10 -6 -8 C 10 10 0 6 2 D 8 6 -6 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=24 B=17 D=15 A=10 so A is eliminated. Round 2 votes counts: E=34 C=25 D=21 B=20 so B is eliminated. Round 3 votes counts: E=43 C=30 D=27 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:208 D:203 B:188 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -8 -8 B 0 0 -10 -6 -8 C 10 10 0 6 2 D 8 6 -6 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -8 -8 B 0 0 -10 -6 -8 C 10 10 0 6 2 D 8 6 -6 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -8 -8 B 0 0 -10 -6 -8 C 10 10 0 6 2 D 8 6 -6 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2467: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) E C D B A (6) E D C A B (5) B C D A E (5) A B D C E (5) E D A C B (3) E A D C B (3) D E A C B (3) D A C B E (3) C D B E A (3) C D B A E (3) A D E C B (3) A B E D C (3) E B C A D (2) D E C A B (2) D C E B A (2) D C B E A (2) C B D E A (2) B A E C D (2) B A C D E (2) A E D B C (2) A E B C D (2) A B D E C (2) A B C E D (2) E C D A B (1) E C B D A (1) E B A C D (1) D C E A B (1) D C B A E (1) D C A B E (1) D B C A E (1) D A E C B (1) D A C E B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D E B A (1) C B E D A (1) C B D A E (1) B E A C D (1) B C E A D (1) B C D E A (1) B C A D E (1) B A C E D (1) A E B D C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 14 8 -8 12 B -14 0 -4 -2 14 C -8 4 0 6 -10 D 8 2 -6 0 0 E -12 -14 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.108108 B: 0.013514 C: 0.148649 D: 0.506757 E: 0.222973 Sum of squares = 0.340485755945 Cumulative probabilities = A: 0.108108 B: 0.121622 C: 0.270270 D: 0.777027 E: 1.000000 A B C D E A 0 14 8 -8 12 B -14 0 -4 -2 14 C -8 4 0 6 -10 D 8 2 -6 0 0 E -12 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.108108 B: 0.013514 C: 0.148649 D: 0.506757 E: 0.222973 Sum of squares = 0.340485755971 Cumulative probabilities = A: 0.108108 B: 0.121622 C: 0.270270 D: 0.777027 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=22 D=19 B=14 C=13 so C is eliminated. Round 2 votes counts: A=32 D=26 E=24 B=18 so B is eliminated. Round 3 votes counts: A=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:213 D:202 B:197 C:196 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 8 -8 12 B -14 0 -4 -2 14 C -8 4 0 6 -10 D 8 2 -6 0 0 E -12 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.108108 B: 0.013514 C: 0.148649 D: 0.506757 E: 0.222973 Sum of squares = 0.340485755971 Cumulative probabilities = A: 0.108108 B: 0.121622 C: 0.270270 D: 0.777027 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -8 12 B -14 0 -4 -2 14 C -8 4 0 6 -10 D 8 2 -6 0 0 E -12 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.108108 B: 0.013514 C: 0.148649 D: 0.506757 E: 0.222973 Sum of squares = 0.340485755971 Cumulative probabilities = A: 0.108108 B: 0.121622 C: 0.270270 D: 0.777027 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -8 12 B -14 0 -4 -2 14 C -8 4 0 6 -10 D 8 2 -6 0 0 E -12 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.108108 B: 0.013514 C: 0.148649 D: 0.506757 E: 0.222973 Sum of squares = 0.340485755971 Cumulative probabilities = A: 0.108108 B: 0.121622 C: 0.270270 D: 0.777027 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2468: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (5) D B C E A (5) C E B D A (5) C A E B D (5) D C B E A (4) D B A E C (4) C D E B A (4) B E D C A (4) B E D A C (4) A D C B E (4) A D B E C (4) A C E B D (4) A C D E B (4) A B E D C (4) E C B A D (3) E B C D A (3) D B E A C (3) C E A B D (3) B D E A C (3) C E B A D (2) C A D E B (2) B D E C A (2) A E C B D (2) A E B C D (2) E C B D A (1) E B C A D (1) E B A C D (1) D C B A E (1) D C A B E (1) D A B C E (1) C E D B A (1) C E D A B (1) C D B E A (1) C D A E B (1) C A E D B (1) B E C D A (1) B E A D C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -18 -14 -10 -16 B 18 0 -4 2 4 C 14 4 0 -2 4 D 10 -2 2 0 0 E 16 -4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999862 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 -10 -16 B 18 0 -4 2 4 C 14 4 0 -2 4 D 10 -2 2 0 0 E 16 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999251 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 D=24 B=15 E=9 so E is eliminated. Round 2 votes counts: C=30 A=26 D=24 B=20 so B is eliminated. Round 3 votes counts: D=37 C=35 A=28 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:210 C:210 D:205 E:204 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -14 -10 -16 B 18 0 -4 2 4 C 14 4 0 -2 4 D 10 -2 2 0 0 E 16 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999251 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 -10 -16 B 18 0 -4 2 4 C 14 4 0 -2 4 D 10 -2 2 0 0 E 16 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999251 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 -10 -16 B 18 0 -4 2 4 C 14 4 0 -2 4 D 10 -2 2 0 0 E 16 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999251 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2469: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) A E D C B (9) B E A C D (8) E B A C D (7) E A B D C (6) C D B E A (6) B C D E A (6) E A D C B (5) A D E C B (5) A D C E B (5) D A C E B (3) C D B A E (3) C B D E A (3) A E B D C (3) E A B C D (2) D C A B E (2) B E C D A (2) B C E D A (2) B A E C D (2) E B C D A (1) D C E B A (1) D C B A E (1) C E B D A (1) C D E B A (1) C B D A E (1) B E C A D (1) B C D A E (1) B A D C E (1) A E D B C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 18 14 -4 B -4 0 -10 -2 -18 C -18 10 0 -4 -8 D -14 2 4 0 -4 E 4 18 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 18 14 -4 B -4 0 -10 -2 -18 C -18 10 0 -4 -8 D -14 2 4 0 -4 E 4 18 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=23 E=21 D=16 C=15 so C is eliminated. Round 2 votes counts: B=27 D=26 A=25 E=22 so E is eliminated. Round 3 votes counts: A=38 B=36 D=26 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 A:216 D:194 C:190 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 18 14 -4 B -4 0 -10 -2 -18 C -18 10 0 -4 -8 D -14 2 4 0 -4 E 4 18 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 14 -4 B -4 0 -10 -2 -18 C -18 10 0 -4 -8 D -14 2 4 0 -4 E 4 18 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 14 -4 B -4 0 -10 -2 -18 C -18 10 0 -4 -8 D -14 2 4 0 -4 E 4 18 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2470: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) D E B A C (9) A C D E B (9) C A B D E (7) B C E D A (7) B E D C A (5) A D E C B (5) E D B A C (4) B C A E D (4) E B D A C (3) D B E C A (3) D A E C B (3) C B A E D (3) A E D C B (3) A C E D B (3) E D B C A (2) E D A B C (2) D E B C A (2) D E A C B (2) B E C D A (2) B D E C A (2) A D C E B (2) D A C E B (1) C B A D E (1) C A D B E (1) B E C A D (1) B C A D E (1) A E C D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -2 6 12 B -2 0 -6 -4 -4 C 2 6 0 4 2 D -6 4 -4 0 -4 E -12 4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 6 12 B -2 0 -6 -4 -4 C 2 6 0 4 2 D -6 4 -4 0 -4 E -12 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=22 B=22 D=20 E=11 so E is eliminated. Round 2 votes counts: D=28 B=25 A=25 C=22 so C is eliminated. Round 3 votes counts: A=43 B=29 D=28 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:207 E:197 D:195 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 6 12 B -2 0 -6 -4 -4 C 2 6 0 4 2 D -6 4 -4 0 -4 E -12 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 6 12 B -2 0 -6 -4 -4 C 2 6 0 4 2 D -6 4 -4 0 -4 E -12 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 6 12 B -2 0 -6 -4 -4 C 2 6 0 4 2 D -6 4 -4 0 -4 E -12 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2471: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) D A C B E (8) C B D A E (7) E A D C B (6) C D B A E (6) B E C D A (6) B C E D A (6) A D E C B (6) A D C E B (6) E B C A D (5) B C D A E (5) A E D C B (5) E A B D C (4) D C A B E (4) B C D E A (4) A D C B E (4) B E C A D (3) E B C D A (2) D C B A E (2) E D A C B (1) E B A C D (1) D A E C B (1) Total count = 100 A B C D E A 0 6 0 -4 8 B -6 0 -12 -14 10 C 0 12 0 -10 4 D 4 14 10 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -4 8 B -6 0 -12 -14 10 C 0 12 0 -10 4 D 4 14 10 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=24 A=21 D=15 C=13 so C is eliminated. Round 2 votes counts: B=31 E=27 D=21 A=21 so D is eliminated. Round 3 votes counts: B=39 A=34 E=27 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:205 C:203 B:189 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -4 8 B -6 0 -12 -14 10 C 0 12 0 -10 4 D 4 14 10 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -4 8 B -6 0 -12 -14 10 C 0 12 0 -10 4 D 4 14 10 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -4 8 B -6 0 -12 -14 10 C 0 12 0 -10 4 D 4 14 10 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2472: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) C A B E D (11) D E C B A (7) C E D B A (7) A C B D E (5) E D C B A (4) A B D E C (4) A B C E D (4) D A B E C (3) C E B A D (3) C E A B D (3) C A E B D (3) C A D B E (3) D E B C A (2) C A B D E (2) B A E D C (2) B A D E C (2) A C B E D (2) A B E C D (2) A B C D E (2) E D B C A (1) E C B A D (1) E B A D C (1) D E C A B (1) D C E B A (1) D C E A B (1) D C A B E (1) D B A E C (1) C E D A B (1) C E B D A (1) C D E A B (1) B E A D C (1) A D C B E (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -8 12 4 B -8 0 -18 2 0 C 8 18 0 2 8 D -12 -2 -2 0 4 E -4 0 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 12 4 B -8 0 -18 2 0 C 8 18 0 2 8 D -12 -2 -2 0 4 E -4 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=29 A=24 E=7 B=5 so B is eliminated. Round 2 votes counts: C=35 D=29 A=28 E=8 so E is eliminated. Round 3 votes counts: C=36 D=34 A=30 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:208 D:194 E:192 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 12 4 B -8 0 -18 2 0 C 8 18 0 2 8 D -12 -2 -2 0 4 E -4 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 12 4 B -8 0 -18 2 0 C 8 18 0 2 8 D -12 -2 -2 0 4 E -4 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 12 4 B -8 0 -18 2 0 C 8 18 0 2 8 D -12 -2 -2 0 4 E -4 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2473: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (13) B D C E A (11) D B A E C (8) B D C A E (8) D A B E C (7) E C A D B (4) C E B A D (4) B D A C E (4) A D E C B (4) E C A B D (3) E A C D B (3) D B A C E (3) C A E D B (3) B C D E A (3) D A B C E (2) C E A B D (2) B E C D A (2) A E C D B (2) A C E D B (2) E C B A D (1) E B D A C (1) E B C A D (1) D A E B C (1) C B E A D (1) C B A D E (1) C A B D E (1) B E C A D (1) B D E A C (1) B C E A D (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -28 -20 -28 -4 B 28 0 34 18 34 C 20 -34 0 -2 20 D 28 -18 2 0 10 E 4 -34 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -20 -28 -4 B 28 0 34 18 34 C 20 -34 0 -2 20 D 28 -18 2 0 10 E 4 -34 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 D=21 E=13 C=12 A=10 so A is eliminated. Round 2 votes counts: B=44 D=25 E=16 C=15 so C is eliminated. Round 3 votes counts: B=47 E=27 D=26 so D is eliminated. Round 4 votes counts: B=67 E=33 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:257 D:211 C:202 E:170 A:160 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -20 -28 -4 B 28 0 34 18 34 C 20 -34 0 -2 20 D 28 -18 2 0 10 E 4 -34 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -20 -28 -4 B 28 0 34 18 34 C 20 -34 0 -2 20 D 28 -18 2 0 10 E 4 -34 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -20 -28 -4 B 28 0 34 18 34 C 20 -34 0 -2 20 D 28 -18 2 0 10 E 4 -34 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2474: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) E D C B A (6) E D C A B (6) A D E B C (6) A B D E C (6) D E A C B (4) D E A B C (4) C E B D A (4) C A B E D (4) A D E C B (4) C E D B A (3) C E D A B (3) B A C E D (3) A C D E B (3) E D B C A (2) D E B C A (2) C E A D B (2) C B E D A (2) C A E B D (2) B D E C A (2) B D E A C (2) B C A E D (2) B A C D E (2) A C E D B (2) A C B E D (2) A B C D E (2) E D A C B (1) D E C A B (1) D E B A C (1) D B E A C (1) D B A E C (1) D A B E C (1) C E B A D (1) C A E D B (1) B D C E A (1) B C A D E (1) A E D C B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -4 -8 -10 B -14 0 -6 -10 -18 C 4 6 0 -6 -2 D 8 10 6 0 -8 E 10 18 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -4 -8 -10 B -14 0 -6 -10 -18 C 4 6 0 -6 -2 D 8 10 6 0 -8 E 10 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 B=20 E=15 D=15 so E is eliminated. Round 2 votes counts: D=30 A=28 C=22 B=20 so B is eliminated. Round 3 votes counts: D=35 A=33 C=32 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:219 D:208 C:201 A:196 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 -8 -10 B -14 0 -6 -10 -18 C 4 6 0 -6 -2 D 8 10 6 0 -8 E 10 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -8 -10 B -14 0 -6 -10 -18 C 4 6 0 -6 -2 D 8 10 6 0 -8 E 10 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -8 -10 B -14 0 -6 -10 -18 C 4 6 0 -6 -2 D 8 10 6 0 -8 E 10 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2475: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) B A E D C (11) C E D A B (10) C D E B A (8) C E A D B (6) B D A E C (5) E A C D B (4) A B E D C (4) E C A D B (3) E A D C B (3) D C E A B (3) D C B E A (3) B A D E C (3) A E B C D (3) E C D A B (2) D B A E C (2) B D C A E (2) A E B D C (2) A B E C D (2) E D C A B (1) E D A C B (1) E A D B C (1) D B C E A (1) D B C A E (1) C E A B D (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D C E (1) A E C D B (1) Total count = 100 A B C D E A 0 18 -12 -4 -18 B -18 0 -16 -24 -22 C 12 16 0 12 2 D 4 24 -12 0 -14 E 18 22 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 -4 -18 B -18 0 -16 -24 -22 C 12 16 0 12 2 D 4 24 -12 0 -14 E 18 22 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=26 E=15 A=12 D=10 so D is eliminated. Round 2 votes counts: C=43 B=30 E=15 A=12 so A is eliminated. Round 3 votes counts: C=43 B=36 E=21 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:226 C:221 D:201 A:192 B:160 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -12 -4 -18 B -18 0 -16 -24 -22 C 12 16 0 12 2 D 4 24 -12 0 -14 E 18 22 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 -4 -18 B -18 0 -16 -24 -22 C 12 16 0 12 2 D 4 24 -12 0 -14 E 18 22 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 -4 -18 B -18 0 -16 -24 -22 C 12 16 0 12 2 D 4 24 -12 0 -14 E 18 22 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2476: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) B C D A E (6) E A C B D (5) C A D B E (5) E B C A D (4) E A D C B (4) D A C B E (4) A C D E B (4) E D A B C (3) E B D A C (3) C A B D E (3) B E D C A (3) B E C D A (3) A E D C B (3) A D E C B (3) A D C E B (3) A D C B E (3) A C D B E (3) E D B A C (2) E A D B C (2) E A C D B (2) E A B D C (2) D C A B E (2) D B C A E (2) D A B C E (2) C D A B E (2) C B E A D (2) B C E D A (2) A C E B D (2) E B A D C (1) E A B C D (1) D B A C E (1) D A B E C (1) C B A E D (1) C B A D E (1) B E C A D (1) B D E C A (1) B C D E A (1) A C E D B (1) Total count = 100 A B C D E A 0 20 10 12 4 B -20 0 -6 -4 -2 C -10 6 0 -2 0 D -12 4 2 0 -6 E -4 2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 12 4 B -20 0 -6 -4 -2 C -10 6 0 -2 0 D -12 4 2 0 -6 E -4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=22 B=17 C=14 D=12 so D is eliminated. Round 2 votes counts: E=35 A=29 B=20 C=16 so C is eliminated. Round 3 votes counts: A=41 E=35 B=24 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:202 C:197 D:194 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 12 4 B -20 0 -6 -4 -2 C -10 6 0 -2 0 D -12 4 2 0 -6 E -4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 12 4 B -20 0 -6 -4 -2 C -10 6 0 -2 0 D -12 4 2 0 -6 E -4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 12 4 B -20 0 -6 -4 -2 C -10 6 0 -2 0 D -12 4 2 0 -6 E -4 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2477: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (20) A C E D B (12) B A D E C (8) A C B E D (8) A B C E D (8) C E D A B (6) B A D C E (6) E D C B A (5) D E C B A (5) E C D A B (3) D E B C A (3) E D C A B (2) C E A D B (2) A C E B D (2) A B E C D (2) A B C D E (2) E D B C A (1) E D A B C (1) D C E B A (1) C D A E B (1) B D E A C (1) B D C E A (1) Total count = 100 A B C D E A 0 -2 0 0 -2 B 2 0 6 16 12 C 0 -6 0 -8 -2 D 0 -16 8 0 -4 E 2 -12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 0 -2 B 2 0 6 16 12 C 0 -6 0 -8 -2 D 0 -16 8 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=34 E=12 D=9 C=9 so D is eliminated. Round 2 votes counts: B=36 A=34 E=20 C=10 so C is eliminated. Round 3 votes counts: B=36 A=35 E=29 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:198 E:198 D:194 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 0 -2 B 2 0 6 16 12 C 0 -6 0 -8 -2 D 0 -16 8 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 -2 B 2 0 6 16 12 C 0 -6 0 -8 -2 D 0 -16 8 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 -2 B 2 0 6 16 12 C 0 -6 0 -8 -2 D 0 -16 8 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2478: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (13) A E B C D (11) D C B A E (8) C B D A E (8) E A B D C (4) B C A E D (4) E A D B C (3) D E A C B (3) D C B E A (3) D B C A E (3) C D B A E (3) C B A E D (3) C B A D E (3) B A E C D (3) B A C E D (3) E D A B C (2) D E C A B (2) D C E B A (2) B C D A E (2) A E B D C (2) A B E C D (2) E D A C B (1) E A D C B (1) E A C D B (1) E A C B D (1) D C E A B (1) C E D A B (1) C E B A D (1) C E A D B (1) C D B E A (1) C B D E A (1) B D C A E (1) B C A D E (1) A E D B C (1) Total count = 100 A B C D E A 0 0 2 16 16 B 0 0 10 26 -2 C -2 -10 0 26 0 D -16 -26 -26 0 -16 E -16 2 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.523882 B: 0.476118 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501140680173 Cumulative probabilities = A: 0.523882 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 16 16 B 0 0 10 26 -2 C -2 -10 0 26 0 D -16 -26 -26 0 -16 E -16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=22 C=22 A=16 B=14 so B is eliminated. Round 2 votes counts: C=29 E=26 D=23 A=22 so A is eliminated. Round 3 votes counts: E=45 C=32 D=23 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:217 B:217 C:207 E:201 D:158 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 16 16 B 0 0 10 26 -2 C -2 -10 0 26 0 D -16 -26 -26 0 -16 E -16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 16 16 B 0 0 10 26 -2 C -2 -10 0 26 0 D -16 -26 -26 0 -16 E -16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 16 16 B 0 0 10 26 -2 C -2 -10 0 26 0 D -16 -26 -26 0 -16 E -16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2479: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) D E A C B (7) C A B E D (6) D E B A C (5) D E A B C (5) C A B D E (5) C B A D E (4) B C D A E (4) A E C D B (4) A C B E D (4) E D B A C (3) D C A E B (3) D A E C B (3) C B D A E (3) C B A E D (3) E D A B C (2) E A B C D (2) D C B E A (2) D B E C A (2) D B C E A (2) B E A C D (2) B C A E D (2) A C E D B (2) A C E B D (2) E D A C B (1) E B D A C (1) E A D C B (1) E A D B C (1) D E C B A (1) D C E B A (1) C D B A E (1) C A D B E (1) B C E A D (1) B A E C D (1) A E D C B (1) A E C B D (1) A C D E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -8 6 B -8 0 -14 -14 -10 C -2 14 0 2 -4 D 8 14 -2 0 16 E -6 10 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.50000000028 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -8 6 B -8 0 -14 -14 -10 C -2 14 0 2 -4 D 8 14 -2 0 16 E -6 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=23 A=17 E=11 B=10 so B is eliminated. Round 2 votes counts: D=39 C=30 A=18 E=13 so E is eliminated. Round 3 votes counts: D=46 C=30 A=24 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:205 A:204 E:196 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -8 6 B -8 0 -14 -14 -10 C -2 14 0 2 -4 D 8 14 -2 0 16 E -6 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -8 6 B -8 0 -14 -14 -10 C -2 14 0 2 -4 D 8 14 -2 0 16 E -6 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -8 6 B -8 0 -14 -14 -10 C -2 14 0 2 -4 D 8 14 -2 0 16 E -6 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2480: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) C D E B A (8) B D C E A (7) A B E D C (7) A E C B D (6) E A C D B (5) C D B E A (5) B D C A E (5) A E B D C (5) B A D E C (4) E C D A B (3) E C A D B (3) B D E C A (3) B A D C E (3) A E B C D (3) E B D A C (2) C E D B A (2) C A D E B (2) A C B D E (2) A B C D E (2) E C D B A (1) E B D C A (1) D C E B A (1) D C B E A (1) D B C E A (1) C E D A B (1) C D E A B (1) C D B A E (1) C A D B E (1) B E D A C (1) B D E A C (1) B D A C E (1) B A E D C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 8 6 B -2 0 -6 8 -8 C -6 6 0 12 -10 D -8 -8 -12 0 -2 E -6 8 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 8 6 B -2 0 -6 8 -8 C -6 6 0 12 -10 D -8 -8 -12 0 -2 E -6 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=26 C=21 E=15 D=3 so D is eliminated. Round 2 votes counts: A=35 B=27 C=23 E=15 so E is eliminated. Round 3 votes counts: A=40 C=30 B=30 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:207 C:201 B:196 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 8 6 B -2 0 -6 8 -8 C -6 6 0 12 -10 D -8 -8 -12 0 -2 E -6 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 8 6 B -2 0 -6 8 -8 C -6 6 0 12 -10 D -8 -8 -12 0 -2 E -6 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 8 6 B -2 0 -6 8 -8 C -6 6 0 12 -10 D -8 -8 -12 0 -2 E -6 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2481: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) B D C A E (6) B C E A D (6) A E C B D (6) E A C D B (5) E C A D B (4) C D B E A (4) C B D E A (4) B C D E A (4) A E D C B (4) A E B C D (4) A B E D C (4) A B E C D (4) C D E B A (3) B D C E A (3) B C D A E (3) A E C D B (3) E D A C B (2) D E A C B (2) C E B A D (2) B C A E D (2) B A E D C (2) B A C E D (2) A E D B C (2) E A D C B (1) E A C B D (1) D C E B A (1) D C B E A (1) D B C A E (1) D B A E C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E A D B (1) C B E A D (1) B A E C D (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -4 12 -2 B 6 0 8 12 12 C 4 -8 0 20 0 D -12 -12 -20 0 -16 E 2 -12 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 12 -2 B 6 0 8 12 12 C 4 -8 0 20 0 D -12 -12 -20 0 -16 E 2 -12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=28 D=15 C=15 E=13 so E is eliminated. Round 2 votes counts: A=35 B=29 C=19 D=17 so D is eliminated. Round 3 votes counts: A=42 B=37 C=21 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:208 E:203 A:200 D:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 12 -2 B 6 0 8 12 12 C 4 -8 0 20 0 D -12 -12 -20 0 -16 E 2 -12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 12 -2 B 6 0 8 12 12 C 4 -8 0 20 0 D -12 -12 -20 0 -16 E 2 -12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 12 -2 B 6 0 8 12 12 C 4 -8 0 20 0 D -12 -12 -20 0 -16 E 2 -12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2482: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) B D C A E (8) E A B D C (6) B D A E C (6) A E D B C (5) A D B E C (5) C B D E A (4) B D A C E (4) E A C D B (3) E A C B D (3) D A B E C (3) C E B D A (3) C E B A D (3) C E A B D (3) C D B A E (3) A E D C B (3) E C A D B (2) E C A B D (2) D B C A E (2) D B A E C (2) D B A C E (2) D A C B E (2) C D B E A (2) C B D A E (2) B C D E A (2) E A D B C (1) E A B C D (1) D C B A E (1) D A B C E (1) C D A E B (1) B D E A C (1) B D C E A (1) B A D E C (1) A D E B C (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 4 0 8 B -6 0 6 2 6 C -4 -6 0 -12 8 D 0 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.578143 B: 0.000000 C: 0.000000 D: 0.421857 E: 0.000000 Sum of squares = 0.512212800573 Cumulative probabilities = A: 0.578143 B: 0.578143 C: 0.578143 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 8 B -6 0 6 2 6 C -4 -6 0 -12 8 D 0 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=23 E=18 A=16 D=13 so D is eliminated. Round 2 votes counts: C=31 B=29 A=22 E=18 so E is eliminated. Round 3 votes counts: A=36 C=35 B=29 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:209 B:204 C:193 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 8 B -6 0 6 2 6 C -4 -6 0 -12 8 D 0 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 8 B -6 0 6 2 6 C -4 -6 0 -12 8 D 0 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 8 B -6 0 6 2 6 C -4 -6 0 -12 8 D 0 -2 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2483: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (13) A D C B E (12) D A C E B (8) D A C B E (7) C B E D A (6) B E C A D (6) E C B D A (4) C D A B E (4) B C E A D (4) E B C A D (3) E B A D C (3) E D C B A (2) E B D C A (2) E A D B C (2) D A E C B (2) B C A E D (2) A D B E C (2) A C D B E (2) A C B D E (2) E D C A B (1) E D A B C (1) E B D A C (1) E B A C D (1) D E A C B (1) D C E A B (1) C E D B A (1) C D E B A (1) C B A E D (1) B E A C D (1) B A E D C (1) B A C E D (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 -10 -8 B 6 0 -10 4 6 C 2 10 0 6 8 D 10 -4 -6 0 -12 E 8 -6 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -10 -8 B 6 0 -10 4 6 C 2 10 0 6 8 D 10 -4 -6 0 -12 E 8 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=20 D=19 B=15 C=13 so C is eliminated. Round 2 votes counts: E=34 D=24 B=22 A=20 so A is eliminated. Round 3 votes counts: D=41 E=34 B=25 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:213 B:203 E:203 D:194 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -10 -8 B 6 0 -10 4 6 C 2 10 0 6 8 D 10 -4 -6 0 -12 E 8 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -10 -8 B 6 0 -10 4 6 C 2 10 0 6 8 D 10 -4 -6 0 -12 E 8 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -10 -8 B 6 0 -10 4 6 C 2 10 0 6 8 D 10 -4 -6 0 -12 E 8 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2484: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) A D C B E (7) A C D E B (7) E B D C A (6) B D E C A (6) D B E C A (5) A C E B D (5) D A B C E (4) A C D B E (4) E C B A D (3) E B C D A (3) E B C A D (3) D E B C A (3) C E A D B (3) C A E D B (3) B D E A C (3) E C A B D (2) D C E B A (2) D C A B E (2) D B E A C (2) C E A B D (2) A C E D B (2) A B E C D (2) E D B C A (1) E C D B A (1) E A C B D (1) D C B E A (1) D A C B E (1) C E D A B (1) C A D E B (1) B E A C D (1) B A E C D (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -16 -2 -18 B 2 0 2 0 0 C 16 -2 0 -6 -8 D 2 0 6 0 -4 E 18 0 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.667783 C: 0.000000 D: 0.000000 E: 0.332217 Sum of squares = 0.556302280539 Cumulative probabilities = A: 0.000000 B: 0.667783 C: 0.667783 D: 0.667783 E: 1.000000 A B C D E A 0 -2 -16 -2 -18 B 2 0 2 0 0 C 16 -2 0 -6 -8 D 2 0 6 0 -4 E 18 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=21 E=20 D=20 C=10 so C is eliminated. Round 2 votes counts: A=33 E=26 B=21 D=20 so D is eliminated. Round 3 votes counts: A=40 E=31 B=29 so B is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:202 D:202 C:200 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -16 -2 -18 B 2 0 2 0 0 C 16 -2 0 -6 -8 D 2 0 6 0 -4 E 18 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -2 -18 B 2 0 2 0 0 C 16 -2 0 -6 -8 D 2 0 6 0 -4 E 18 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -2 -18 B 2 0 2 0 0 C 16 -2 0 -6 -8 D 2 0 6 0 -4 E 18 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2485: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) E C A D B (5) D B C A E (5) B D A C E (5) E C A B D (4) E A C B D (4) D B A E C (4) C E A B D (4) C B D E A (4) B C D A E (4) A E D B C (4) E D A C B (3) D B A C E (3) C E D B A (3) C E D A B (3) A E C B D (3) A E B C D (3) A D B E C (3) E C D A B (2) E A D C B (2) D E A C B (2) C E B D A (2) C D E B A (2) A E B D C (2) A B E C D (2) E A D B C (1) E A C D B (1) D E C A B (1) D C E B A (1) D C B E A (1) D B C E A (1) C E A D B (1) C B E A D (1) B D C A E (1) B C A D E (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 -6 -8 B -4 0 -10 -8 -6 C 4 10 0 14 2 D 6 8 -14 0 -4 E 8 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -6 -8 B -4 0 -10 -8 -6 C 4 10 0 14 2 D 6 8 -14 0 -4 E 8 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=22 A=19 D=18 B=15 so B is eliminated. Round 2 votes counts: C=31 D=24 A=23 E=22 so E is eliminated. Round 3 votes counts: C=42 A=31 D=27 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:208 D:198 A:193 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -6 -8 B -4 0 -10 -8 -6 C 4 10 0 14 2 D 6 8 -14 0 -4 E 8 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 -8 B -4 0 -10 -8 -6 C 4 10 0 14 2 D 6 8 -14 0 -4 E 8 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 -8 B -4 0 -10 -8 -6 C 4 10 0 14 2 D 6 8 -14 0 -4 E 8 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2486: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) E C B A D (5) E B D A C (5) E A D B C (5) C E A D B (5) C B D A E (5) B D E A C (5) B D A C E (5) D A E B C (4) C A E D B (4) E D A B C (3) E C A D B (3) E A C D B (3) D A B E C (3) C A D E B (3) C A D B E (3) B C D A E (3) D B A E C (2) C E B A D (2) B E D A C (2) B E C D A (2) B D C A E (2) B D A E C (2) B C E D A (2) A D E C B (2) E C B D A (1) E B C D A (1) E A D C B (1) D B A C E (1) C E A B D (1) C D A B E (1) C B A D E (1) C A B D E (1) B E D C A (1) A E D B C (1) A D E B C (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -4 -12 -6 B 6 0 2 4 -2 C 4 -2 0 4 -4 D 12 -4 -4 0 -6 E 6 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -12 -6 B 6 0 2 4 -2 C 4 -2 0 4 -4 D 12 -4 -4 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=27 B=24 D=10 A=7 so A is eliminated. Round 2 votes counts: C=32 E=28 B=24 D=16 so D is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:205 C:201 D:199 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -12 -6 B 6 0 2 4 -2 C 4 -2 0 4 -4 D 12 -4 -4 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -12 -6 B 6 0 2 4 -2 C 4 -2 0 4 -4 D 12 -4 -4 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -12 -6 B 6 0 2 4 -2 C 4 -2 0 4 -4 D 12 -4 -4 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2487: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) B A E C D (8) A B C E D (7) B A C D E (6) A B E C D (6) B E D C A (5) D C E B A (4) D C E A B (4) E D C B A (3) E B A D C (3) E A B D C (3) B D C E A (3) A C B D E (3) E D B C A (2) E B D C A (2) E A D C B (2) D E B C A (2) C D E A B (2) B E D A C (2) B A E D C (2) B A D C E (2) A B C D E (2) E D A C B (1) E C D A B (1) E A C D B (1) D E C A B (1) D C B E A (1) C D B E A (1) C D A E B (1) C D A B E (1) C B D A E (1) C A B D E (1) B E A D C (1) B D C A E (1) B C D A E (1) B A C E D (1) A E C D B (1) A E B D C (1) A E B C D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 10 6 -4 B 2 0 22 26 12 C -10 -22 0 -6 -12 D -6 -26 6 0 -26 E 4 -12 12 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 6 -4 B 2 0 22 26 12 C -10 -22 0 -6 -12 D -6 -26 6 0 -26 E 4 -12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=26 A=23 D=12 C=7 so C is eliminated. Round 2 votes counts: B=33 E=26 A=24 D=17 so D is eliminated. Round 3 votes counts: E=39 B=35 A=26 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:215 A:205 C:175 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 6 -4 B 2 0 22 26 12 C -10 -22 0 -6 -12 D -6 -26 6 0 -26 E 4 -12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 6 -4 B 2 0 22 26 12 C -10 -22 0 -6 -12 D -6 -26 6 0 -26 E 4 -12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 6 -4 B 2 0 22 26 12 C -10 -22 0 -6 -12 D -6 -26 6 0 -26 E 4 -12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2488: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) A E C D B (8) E B D C A (7) C D B A E (7) E A D C B (6) B D C A E (6) E B A D C (5) B E D C A (5) E D C B A (4) D C B A E (4) B D E C A (4) E A B D C (3) C A D B E (3) B C D A E (3) A C E D B (3) A C D E B (3) A C D B E (3) E A C D B (2) D C B E A (2) C B D A E (2) E B D A C (1) E A D B C (1) D C A E B (1) D C A B E (1) D B C E A (1) C D A B E (1) B C A D E (1) B A E C D (1) A E C B D (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -22 -16 0 B 24 0 -2 0 8 C 22 2 0 -20 2 D 16 0 20 0 4 E 0 -8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.549951 C: 0.000000 D: 0.450049 E: 0.000000 Sum of squares = 0.504990230095 Cumulative probabilities = A: 0.000000 B: 0.549951 C: 0.549951 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -22 -16 0 B 24 0 -2 0 8 C 22 2 0 -20 2 D 16 0 20 0 4 E 0 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999997633 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=29 B=29 A=20 C=13 D=9 so D is eliminated. Round 2 votes counts: B=30 E=29 C=21 A=20 so A is eliminated. Round 3 votes counts: E=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:215 C:203 E:193 A:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 -22 -16 0 B 24 0 -2 0 8 C 22 2 0 -20 2 D 16 0 20 0 4 E 0 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999997633 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -22 -16 0 B 24 0 -2 0 8 C 22 2 0 -20 2 D 16 0 20 0 4 E 0 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999997633 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -22 -16 0 B 24 0 -2 0 8 C 22 2 0 -20 2 D 16 0 20 0 4 E 0 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999997633 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2489: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) B C A E D (6) B C A D E (6) E C A B D (5) D A B C E (5) E D A C B (4) A C D E B (4) D B E A C (3) D B A C E (3) C A E B D (3) B E C A D (3) B D E C A (3) B C E A D (3) E D B C A (2) E D B A C (2) E C B A D (2) E B D C A (2) E A D C B (2) D E B A C (2) D E A C B (2) D E A B C (2) D A E C B (2) D A C E B (2) C B A E D (2) C A B E D (2) C A B D E (2) B D A C E (2) B C E D A (2) A D C B E (2) E C A D B (1) E A C D B (1) D B A E C (1) D A C B E (1) C E A B D (1) B E D C A (1) B E C D A (1) B D C A E (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 2 12 12 B -4 0 2 16 22 C -2 -2 0 12 18 D -12 -16 -12 0 10 E -12 -22 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 12 12 B -4 0 2 16 22 C -2 -2 0 12 18 D -12 -16 -12 0 10 E -12 -22 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 E=21 A=17 C=10 so C is eliminated. Round 2 votes counts: B=31 A=24 D=23 E=22 so E is eliminated. Round 3 votes counts: B=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:218 A:215 C:213 D:185 E:169 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 12 12 B -4 0 2 16 22 C -2 -2 0 12 18 D -12 -16 -12 0 10 E -12 -22 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 12 12 B -4 0 2 16 22 C -2 -2 0 12 18 D -12 -16 -12 0 10 E -12 -22 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 12 12 B -4 0 2 16 22 C -2 -2 0 12 18 D -12 -16 -12 0 10 E -12 -22 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2490: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A C B E (9) B C A E D (9) E D B C A (8) D E A C B (7) A B C D E (6) E D C B A (4) D A B C E (4) C B A E D (4) B A C E D (4) A C B D E (4) E C B A D (3) C A B E D (3) B C E A D (3) A C B E D (3) D E C B A (2) D E C A B (2) D E A B C (2) D A C E B (2) C A B D E (2) E B D A C (1) E B C D A (1) D E B C A (1) D E B A C (1) D A E C B (1) D A E B C (1) B E C A D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -6 8 8 B 4 0 4 8 8 C 6 -4 0 8 10 D -8 -8 -8 0 -8 E -8 -8 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 8 8 B 4 0 4 8 8 C 6 -4 0 8 10 D -8 -8 -8 0 -8 E -8 -8 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=27 B=17 A=15 C=9 so C is eliminated. Round 2 votes counts: D=32 E=27 B=21 A=20 so A is eliminated. Round 3 votes counts: B=39 D=34 E=27 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:210 A:203 E:191 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 8 8 B 4 0 4 8 8 C 6 -4 0 8 10 D -8 -8 -8 0 -8 E -8 -8 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 8 8 B 4 0 4 8 8 C 6 -4 0 8 10 D -8 -8 -8 0 -8 E -8 -8 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 8 8 B 4 0 4 8 8 C 6 -4 0 8 10 D -8 -8 -8 0 -8 E -8 -8 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2491: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (15) C E A D B (12) C B E A D (7) B C D A E (7) E C A D B (5) E A D C B (4) E A D B C (4) E A C D B (4) C B E D A (4) C B D A E (4) D B A E C (3) D A B E C (3) C B D E A (3) B D C A E (3) C E B A D (2) C E A B D (2) C A E D B (2) B C D E A (2) A E D B C (2) E D A B C (1) E B C D A (1) D A E B C (1) C A D E B (1) B E D A C (1) B D E A C (1) B D C E A (1) B D A C E (1) B C E D A (1) A E D C B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -12 -14 -4 -10 B 12 0 -4 10 12 C 14 4 0 14 4 D 4 -10 -14 0 -6 E 10 -12 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -4 -10 B 12 0 -4 10 12 C 14 4 0 14 4 D 4 -10 -14 0 -6 E 10 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=32 E=19 D=7 A=5 so A is eliminated. Round 2 votes counts: C=37 B=32 E=22 D=9 so D is eliminated. Round 3 votes counts: B=38 C=37 E=25 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:215 E:200 D:187 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -4 -10 B 12 0 -4 10 12 C 14 4 0 14 4 D 4 -10 -14 0 -6 E 10 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -4 -10 B 12 0 -4 10 12 C 14 4 0 14 4 D 4 -10 -14 0 -6 E 10 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -4 -10 B 12 0 -4 10 12 C 14 4 0 14 4 D 4 -10 -14 0 -6 E 10 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2492: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) E A D B C (10) D B A E C (10) C E A B D (8) C B D A E (8) B D A C E (7) E C A D B (5) B C D A E (4) D A B E C (3) B D A E C (3) A D B E C (3) E D A C B (2) E D A B C (2) E A D C B (2) D E A B C (2) D A E B C (2) C B E D A (2) C B E A D (2) B D C A E (2) B A D E C (2) E C A B D (1) E A C D B (1) D B C A E (1) C E D A B (1) C E B D A (1) C E A D B (1) C B D E A (1) C B A E D (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 2 -2 -4 B 10 0 6 8 -2 C -2 -6 0 -6 0 D 2 -8 6 0 -2 E 4 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.125993 D: 0.000000 E: 0.874007 Sum of squares = 0.779762857241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.125993 D: 0.125993 E: 1.000000 A B C D E A 0 -10 2 -2 -4 B 10 0 6 8 -2 C -2 -6 0 -6 0 D 2 -8 6 0 -2 E 4 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500022284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=23 D=18 B=18 A=5 so A is eliminated. Round 2 votes counts: C=36 E=24 D=22 B=18 so B is eliminated. Round 3 votes counts: C=40 D=36 E=24 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:211 E:204 D:199 A:193 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 -2 -4 B 10 0 6 8 -2 C -2 -6 0 -6 0 D 2 -8 6 0 -2 E 4 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500022284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -2 -4 B 10 0 6 8 -2 C -2 -6 0 -6 0 D 2 -8 6 0 -2 E 4 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500022284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -2 -4 B 10 0 6 8 -2 C -2 -6 0 -6 0 D 2 -8 6 0 -2 E 4 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.62500022284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2493: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) C D A E B (8) B A E D C (8) B E A D C (7) E D C B A (6) A B D E C (6) D C E A B (5) C D E A B (4) B A E C D (4) E B D C A (3) C E D B A (3) C A B D E (3) B A C E D (3) A C B D E (3) E C D B A (2) E C B D A (2) E B D A C (2) E B C D A (2) D E C B A (2) D A C E B (2) C B E A D (2) A D C B E (2) A C D B E (2) E D B A C (1) D E C A B (1) D E A C B (1) D C A E B (1) D A E B C (1) C E B D A (1) C B A E D (1) B E C A D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 6 6 10 B 0 0 -2 18 6 C -6 2 0 2 2 D -6 -18 -2 0 2 E -10 -6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.702348 B: 0.297652 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.581889034593 Cumulative probabilities = A: 0.702348 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 6 10 B 0 0 -2 18 6 C -6 2 0 2 2 D -6 -18 -2 0 2 E -10 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 C=22 E=18 D=13 so D is eliminated. Round 2 votes counts: C=28 A=27 B=23 E=22 so E is eliminated. Round 3 votes counts: C=41 B=31 A=28 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:211 B:211 C:200 E:190 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 6 10 B 0 0 -2 18 6 C -6 2 0 2 2 D -6 -18 -2 0 2 E -10 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 10 B 0 0 -2 18 6 C -6 2 0 2 2 D -6 -18 -2 0 2 E -10 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 10 B 0 0 -2 18 6 C -6 2 0 2 2 D -6 -18 -2 0 2 E -10 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2494: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) B D A E C (9) D B E A C (6) B D A C E (5) C B D E A (4) B D E A C (4) A E D B C (4) C E A B D (3) C B D A E (3) C A E B D (3) B D C E A (3) B C D A E (3) A C E D B (3) A C B D E (3) E D A C B (2) E D A B C (2) C B E D A (2) C A B E D (2) B D C A E (2) A E C D B (2) E D B C A (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D B C (1) E A C D B (1) D E B C A (1) D E B A C (1) D B E C A (1) D B C E A (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E B A D (1) C B A E D (1) C B A D E (1) B D E C A (1) B C D E A (1) B C A D E (1) B A D E C (1) A E D C B (1) A E B D C (1) A D E B C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 0 -16 -2 B 14 0 2 14 12 C 0 -2 0 2 12 D 16 -14 -2 0 6 E 2 -12 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -16 -2 B 14 0 2 14 12 C 0 -2 0 2 12 D 16 -14 -2 0 6 E 2 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=30 A=17 D=11 E=10 so E is eliminated. Round 2 votes counts: C=35 B=30 A=19 D=16 so D is eliminated. Round 3 votes counts: B=42 C=35 A=23 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:206 D:203 E:186 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -16 -2 B 14 0 2 14 12 C 0 -2 0 2 12 D 16 -14 -2 0 6 E 2 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -16 -2 B 14 0 2 14 12 C 0 -2 0 2 12 D 16 -14 -2 0 6 E 2 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -16 -2 B 14 0 2 14 12 C 0 -2 0 2 12 D 16 -14 -2 0 6 E 2 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2495: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (13) E A D B C (10) E A C D B (9) C B D A E (7) B D E A C (6) B D C E A (6) A E C D B (6) B C D A E (5) E A C B D (4) B D C A E (4) C D B A E (3) B C D E A (3) D B C A E (2) C A D E B (2) A E D C B (2) E B A D C (1) D C B A E (1) D B E A C (1) D B A E C (1) D B A C E (1) D A C E B (1) C E B A D (1) C E A B D (1) C D A E B (1) C B E D A (1) C B E A D (1) C A E B D (1) B E D A C (1) B D E C A (1) A E D B C (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 8 -6 10 8 B -8 0 -12 -14 -12 C 6 12 0 18 10 D -10 14 -18 0 -6 E -8 12 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 10 8 B -8 0 -12 -14 -12 C 6 12 0 18 10 D -10 14 -18 0 -6 E -8 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 E=24 A=12 D=7 so D is eliminated. Round 2 votes counts: C=32 B=31 E=24 A=13 so A is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:210 E:200 D:190 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 10 8 B -8 0 -12 -14 -12 C 6 12 0 18 10 D -10 14 -18 0 -6 E -8 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 10 8 B -8 0 -12 -14 -12 C 6 12 0 18 10 D -10 14 -18 0 -6 E -8 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 10 8 B -8 0 -12 -14 -12 C 6 12 0 18 10 D -10 14 -18 0 -6 E -8 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2496: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) B D A E C (7) C E D B A (6) C E D A B (6) C B E D A (5) B A D E C (4) B A D C E (4) A D B E C (4) E D C B A (3) E D A B C (3) D B E A C (3) C E A D B (3) C A E B D (3) A D E B C (3) E D C A B (2) E D B A C (2) E C D B A (2) C E B D A (2) C B A D E (2) C A E D B (2) B D E C A (2) B D E A C (2) A E D C B (2) A B D C E (2) A B C D E (2) E D B C A (1) E C D A B (1) E A D B C (1) D E B A C (1) D A E B C (1) C B E A D (1) C B A E D (1) C A B E D (1) C A B D E (1) B D A C E (1) B C D A E (1) B C A D E (1) A E C D B (1) A C E D B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 8 -2 8 B 2 0 6 4 6 C -8 -6 0 -12 -6 D 2 -4 12 0 0 E -8 -6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -2 8 B 2 0 6 4 6 C -8 -6 0 -12 -6 D 2 -4 12 0 0 E -8 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998513 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=25 B=22 E=15 D=5 so D is eliminated. Round 2 votes counts: C=33 A=26 B=25 E=16 so E is eliminated. Round 3 votes counts: C=41 A=30 B=29 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:209 A:206 D:205 E:196 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 -2 8 B 2 0 6 4 6 C -8 -6 0 -12 -6 D 2 -4 12 0 0 E -8 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998513 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -2 8 B 2 0 6 4 6 C -8 -6 0 -12 -6 D 2 -4 12 0 0 E -8 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998513 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -2 8 B 2 0 6 4 6 C -8 -6 0 -12 -6 D 2 -4 12 0 0 E -8 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998513 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2497: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) C E B D A (7) E C A D B (6) B D A C E (6) B A D E C (6) A D B E C (6) E A C D B (4) D A B C E (4) A D B C E (4) D B A C E (3) C E D A B (3) A E D B C (3) A D E B C (3) E C A B D (2) E B C A D (2) E A B D C (2) C D E A B (2) C B E D A (2) B E C A D (2) A B D E C (2) E C B D A (1) E B A D C (1) E B A C D (1) E A C B D (1) E A B C D (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C A E (1) D A C E B (1) D A C B E (1) C E D B A (1) C D A E B (1) B D C A E (1) B D A E C (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) A C D E B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 18 2 B -4 0 8 4 -8 C -8 -8 0 -2 -10 D -18 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999518 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 18 2 B -4 0 8 4 -8 C -8 -8 0 -2 -10 D -18 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=22 B=19 C=16 D=13 so D is eliminated. Round 2 votes counts: E=30 A=28 B=23 C=19 so C is eliminated. Round 3 votes counts: E=43 A=31 B=26 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:209 B:200 D:189 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 18 2 B -4 0 8 4 -8 C -8 -8 0 -2 -10 D -18 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 18 2 B -4 0 8 4 -8 C -8 -8 0 -2 -10 D -18 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 18 2 B -4 0 8 4 -8 C -8 -8 0 -2 -10 D -18 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2498: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) C B A E D (6) B C D A E (5) A C B E D (5) E A D C B (4) E A C D B (4) D E B A C (4) D B E A C (4) D B C E A (4) A E C B D (4) D C B E A (3) D B E C A (3) C E A D B (3) C B D A E (3) C A B E D (3) B D E A C (3) B D A E C (3) E D C A B (2) D E B C A (2) D E A B C (2) D C E B A (2) C E D A B (2) B D C A E (2) B C A D E (2) A E B D C (2) E D A C B (1) E C A D B (1) E A D B C (1) D E A C B (1) C D E A B (1) C D B E A (1) C B D E A (1) C A E D B (1) C A E B D (1) B C D E A (1) B C A E D (1) B A E D C (1) B A D E C (1) B A C E D (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 0 0 0 B 6 0 -16 6 6 C 0 16 0 10 12 D 0 -6 -10 0 -4 E 0 -6 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.509238 B: 0.000000 C: 0.490762 D: 0.000000 E: 0.000000 Sum of squares = 0.50017067086 Cumulative probabilities = A: 0.509238 B: 0.509238 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 0 0 B 6 0 -16 6 6 C 0 16 0 10 12 D 0 -6 -10 0 -4 E 0 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 B=20 A=20 E=13 so E is eliminated. Round 2 votes counts: A=29 D=28 C=23 B=20 so B is eliminated. Round 3 votes counts: D=36 C=32 A=32 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:219 B:201 A:197 E:193 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 0 0 B 6 0 -16 6 6 C 0 16 0 10 12 D 0 -6 -10 0 -4 E 0 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 0 B 6 0 -16 6 6 C 0 16 0 10 12 D 0 -6 -10 0 -4 E 0 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 0 B 6 0 -16 6 6 C 0 16 0 10 12 D 0 -6 -10 0 -4 E 0 -6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2499: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (7) B C D E A (6) D B C E A (5) C E A B D (5) B D C A E (5) A E C B D (5) A D E B C (5) D B E C A (4) B D C E A (4) D B A C E (3) C B D E A (3) C B A D E (3) A C E B D (3) E C B D A (2) E C A B D (2) E A C D B (2) E A C B D (2) D B E A C (2) D B A E C (2) D A E B C (2) C E B A D (2) C B E A D (2) C A E B D (2) A E D B C (2) A D B E C (2) E D B C A (1) E C D B A (1) E C B A D (1) E A D C B (1) D E B A C (1) D E A B C (1) D B C A E (1) C E B D A (1) C B E D A (1) C B D A E (1) C A B E D (1) B D E C A (1) A E D C B (1) A D C E B (1) A D B C E (1) A C B E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 6 0 B 4 0 0 10 0 C 8 0 0 8 6 D -6 -10 -8 0 10 E 0 0 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.702121 C: 0.297879 D: 0.000000 E: 0.000000 Sum of squares = 0.581706100105 Cumulative probabilities = A: 0.000000 B: 0.702121 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 6 0 B 4 0 0 10 0 C 8 0 0 8 6 D -6 -10 -8 0 10 E 0 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=21 C=21 B=16 E=12 so E is eliminated. Round 2 votes counts: A=35 C=27 D=22 B=16 so B is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:207 A:197 D:193 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 6 0 B 4 0 0 10 0 C 8 0 0 8 6 D -6 -10 -8 0 10 E 0 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 6 0 B 4 0 0 10 0 C 8 0 0 8 6 D -6 -10 -8 0 10 E 0 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 6 0 B 4 0 0 10 0 C 8 0 0 8 6 D -6 -10 -8 0 10 E 0 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2500: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) C D B E A (6) A B E D C (6) E A B C D (5) C D E B A (5) C D B A E (5) B A D E C (5) A E B D C (5) D B A E C (4) C E D A B (4) B A E C D (4) E C A B D (3) E A D C B (3) D B A C E (3) C D E A B (3) C B D A E (3) B D A C E (3) E C A D B (2) E A D B C (2) E A C D B (2) E A B D C (2) C E D B A (2) C E B A D (2) C E A D B (2) C B D E A (2) B A E D C (2) A B D E C (2) D E C A B (1) D B C A E (1) D A C B E (1) Total count = 100 A B C D E A 0 -14 -2 -6 8 B 14 0 -12 -12 14 C 2 12 0 0 4 D 6 12 0 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.643701 D: 0.356299 E: 0.000000 Sum of squares = 0.541299799456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.643701 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -6 8 B 14 0 -12 -12 14 C 2 12 0 0 4 D 6 12 0 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=20 E=19 B=14 A=13 so A is eliminated. Round 2 votes counts: C=34 E=24 B=22 D=20 so D is eliminated. Round 3 votes counts: C=45 B=30 E=25 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:209 B:202 A:193 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -2 -6 8 B 14 0 -12 -12 14 C 2 12 0 0 4 D 6 12 0 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -6 8 B 14 0 -12 -12 14 C 2 12 0 0 4 D 6 12 0 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -6 8 B 14 0 -12 -12 14 C 2 12 0 0 4 D 6 12 0 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2501: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) E A B C D (7) B E A D C (7) B D C E A (6) E C A D B (5) C D E B A (5) B E D A C (5) E B A C D (4) D C B E A (4) A C D E B (4) E B C D A (3) E A C D B (3) C D A E B (3) B E D C A (3) B A E D C (3) A D C B E (3) E C D B A (2) D C B A E (2) D C A B E (2) D B C A E (2) C E D A B (2) C D E A B (2) B D A C E (2) A D B C E (2) A C E D B (2) E B C A D (1) D A C B E (1) D A B C E (1) B A D E C (1) B A D C E (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 16 10 -18 B 2 0 0 -10 -8 C -16 0 0 8 -10 D -10 10 -8 0 -16 E 18 8 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 16 10 -18 B 2 0 0 -10 -8 C -16 0 0 8 -10 D -10 10 -8 0 -16 E 18 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 A=23 D=12 C=12 so D is eliminated. Round 2 votes counts: B=30 E=25 A=25 C=20 so C is eliminated. Round 3 votes counts: B=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:203 B:192 C:191 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 16 10 -18 B 2 0 0 -10 -8 C -16 0 0 8 -10 D -10 10 -8 0 -16 E 18 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 10 -18 B 2 0 0 -10 -8 C -16 0 0 8 -10 D -10 10 -8 0 -16 E 18 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 10 -18 B 2 0 0 -10 -8 C -16 0 0 8 -10 D -10 10 -8 0 -16 E 18 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2502: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (10) E B C D A (8) C A B D E (7) E D A B C (6) C B A D E (6) B E C A D (6) E D B A C (5) D E A B C (5) A C D B E (5) D A C E B (4) E D B C A (3) E B D A C (3) D A E C B (3) C B E A D (3) A D C B E (3) D E A C B (2) C E B D A (2) C B A E D (2) B A C E D (2) A D B C E (2) A C B D E (2) E B A D C (1) E A B D C (1) D A E B C (1) C D E B A (1) B E C D A (1) B C E D A (1) B C A E D (1) B C A D E (1) A D E C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -4 10 -16 B 12 0 18 16 6 C 4 -18 0 16 4 D -10 -16 -16 0 -12 E 16 -6 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 10 -16 B 12 0 18 16 6 C 4 -18 0 16 4 D -10 -16 -16 0 -12 E 16 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=22 C=21 D=15 A=15 so D is eliminated. Round 2 votes counts: E=34 A=23 B=22 C=21 so C is eliminated. Round 3 votes counts: E=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:209 C:203 A:189 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 10 -16 B 12 0 18 16 6 C 4 -18 0 16 4 D -10 -16 -16 0 -12 E 16 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 10 -16 B 12 0 18 16 6 C 4 -18 0 16 4 D -10 -16 -16 0 -12 E 16 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 10 -16 B 12 0 18 16 6 C 4 -18 0 16 4 D -10 -16 -16 0 -12 E 16 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2503: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) D E A C B (5) D B C E A (5) B C A E D (5) E A C D B (4) C A D E B (4) E D A C B (3) E B D A C (3) E A D C B (3) D C E A B (3) D C A E B (3) B E A D C (3) B E A C D (3) B D E C A (3) B D C E A (3) B C A D E (3) E D B A C (2) E B A C D (2) E A C B D (2) D E B A C (2) D C B A E (2) D B C A E (2) C D A E B (2) C A B E D (2) B D C A E (2) A C E D B (2) A C E B D (2) E B A D C (1) E A B D C (1) D E C A B (1) D E B C A (1) D C B E A (1) D C A B E (1) D B E C A (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D C A (1) B E D A C (1) B D E A C (1) B C E A D (1) B A C E D (1) A E C D B (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -12 -10 -12 B 14 0 10 2 4 C 12 -10 0 -8 10 D 10 -2 8 0 10 E 12 -4 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -10 -12 B 14 0 10 2 4 C 12 -10 0 -8 10 D 10 -2 8 0 10 E 12 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=27 E=21 C=11 A=7 so A is eliminated. Round 2 votes counts: B=35 D=27 E=23 C=15 so C is eliminated. Round 3 votes counts: B=39 D=34 E=27 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:213 C:202 E:194 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -10 -12 B 14 0 10 2 4 C 12 -10 0 -8 10 D 10 -2 8 0 10 E 12 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -10 -12 B 14 0 10 2 4 C 12 -10 0 -8 10 D 10 -2 8 0 10 E 12 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -10 -12 B 14 0 10 2 4 C 12 -10 0 -8 10 D 10 -2 8 0 10 E 12 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2504: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D E A C B (7) B C A D E (6) A E C B D (5) E A D C B (4) E A C D B (4) D E C B A (4) D C B E A (4) D A E B C (4) B C D A E (4) B A C E D (4) A D E B C (4) A B C E D (4) E D A C B (3) D E A B C (3) D B C E A (3) C B E A D (3) C B D E A (3) A E D B C (3) E C B A D (2) C E B A D (2) C B E D A (2) A E B C D (2) D E B C A (1) D B C A E (1) C E D B A (1) C E B D A (1) C B A E D (1) B D C A E (1) B C D E A (1) A E D C B (1) A E B D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 2 14 4 B 6 0 6 6 -4 C -2 -6 0 8 2 D -14 -6 -8 0 -4 E -4 4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775503 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 -6 2 14 4 B 6 0 6 6 -4 C -2 -6 0 8 2 D -14 -6 -8 0 -4 E -4 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775459 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 A=22 E=13 C=13 so E is eliminated. Round 2 votes counts: D=30 A=30 B=25 C=15 so C is eliminated. Round 3 votes counts: B=39 D=31 A=30 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:207 B:207 C:201 E:201 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 14 4 B 6 0 6 6 -4 C -2 -6 0 8 2 D -14 -6 -8 0 -4 E -4 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775459 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 14 4 B 6 0 6 6 -4 C -2 -6 0 8 2 D -14 -6 -8 0 -4 E -4 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775459 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 14 4 B 6 0 6 6 -4 C -2 -6 0 8 2 D -14 -6 -8 0 -4 E -4 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775459 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2505: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) D E B A C (7) C B E D A (7) D E B C A (6) A C B E D (6) E B D C A (5) B E D C A (5) A D E B C (5) D E A B C (4) C D E A B (4) C B A E D (4) A D E C B (4) A C B D E (4) E D B C A (3) C B D E A (3) C A B E D (3) D A E B C (2) C D E B A (2) C A D E B (2) C A B D E (2) E D B A C (1) D E C B A (1) D E C A B (1) D E A C B (1) C D A E B (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E D C (1) B A C E D (1) A D C E B (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -4 -8 -4 B -2 0 -10 -8 -16 C 4 10 0 4 6 D 8 8 -4 0 20 E 4 16 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -8 -4 B -2 0 -10 -8 -16 C 4 10 0 4 6 D 8 8 -4 0 20 E 4 16 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=28 D=22 B=10 E=9 so E is eliminated. Round 2 votes counts: A=31 C=28 D=26 B=15 so B is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:212 E:197 A:193 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -8 -4 B -2 0 -10 -8 -16 C 4 10 0 4 6 D 8 8 -4 0 20 E 4 16 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -8 -4 B -2 0 -10 -8 -16 C 4 10 0 4 6 D 8 8 -4 0 20 E 4 16 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -8 -4 B -2 0 -10 -8 -16 C 4 10 0 4 6 D 8 8 -4 0 20 E 4 16 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2506: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) E C B A D (6) D B A C E (6) E C A B D (5) E A C B D (5) D B C A E (5) B C E D A (5) E D B C A (4) B D C E A (4) B C D A E (4) A E D C B (4) E B C D A (3) D E A B C (3) C A B E D (3) A D C B E (3) A C E B D (3) E D B A C (2) E A C D B (2) D B E C A (2) B E C D A (2) A E C D B (2) A D C E B (2) A C D E B (2) E D A B C (1) E B D C A (1) E A D C B (1) D E B C A (1) D E B A C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E A D (1) C A E B D (1) B D C A E (1) A E C B D (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 0 -10 -4 B 0 0 12 -6 -6 C 0 -12 0 -4 2 D 10 6 4 0 -8 E 4 6 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000019 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 0 0 -10 -4 B 0 0 12 -6 -6 C 0 -12 0 -4 2 D 10 6 4 0 -8 E 4 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000048 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=28 A=19 B=16 C=7 so C is eliminated. Round 2 votes counts: E=32 D=28 A=23 B=17 so B is eliminated. Round 3 votes counts: E=40 D=37 A=23 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:206 B:200 A:193 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -10 -4 B 0 0 12 -6 -6 C 0 -12 0 -4 2 D 10 6 4 0 -8 E 4 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000048 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -10 -4 B 0 0 12 -6 -6 C 0 -12 0 -4 2 D 10 6 4 0 -8 E 4 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000048 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -10 -4 B 0 0 12 -6 -6 C 0 -12 0 -4 2 D 10 6 4 0 -8 E 4 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000048 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2507: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (11) D A E C B (7) B C A E D (6) A D C B E (6) E B D C A (5) E B C D A (5) B E C D A (5) A D C E B (5) D E A B C (4) C B A D E (4) A D E B C (4) E D A B C (3) E B D A C (3) C A D B E (3) C A B D E (3) E D B C A (2) D A C E B (2) C B E D A (2) C B A E D (2) A C D B E (2) E D C A B (1) E D B A C (1) E C B D A (1) D E A C B (1) D C A E B (1) C E D B A (1) C E B D A (1) C D A E B (1) C B E A D (1) B E C A D (1) B E A D C (1) B C E D A (1) B C A D E (1) A D E C B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -16 6 0 B 8 0 10 8 2 C 16 -10 0 4 10 D -6 -8 -4 0 -6 E 0 -2 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 6 0 B 8 0 10 8 2 C 16 -10 0 4 10 D -6 -8 -4 0 -6 E 0 -2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=21 A=20 C=18 D=15 so D is eliminated. Round 2 votes counts: A=29 E=26 B=26 C=19 so C is eliminated. Round 3 votes counts: A=37 B=35 E=28 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:210 E:197 A:191 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -16 6 0 B 8 0 10 8 2 C 16 -10 0 4 10 D -6 -8 -4 0 -6 E 0 -2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 6 0 B 8 0 10 8 2 C 16 -10 0 4 10 D -6 -8 -4 0 -6 E 0 -2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 6 0 B 8 0 10 8 2 C 16 -10 0 4 10 D -6 -8 -4 0 -6 E 0 -2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2508: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) D E C A B (9) C D E A B (8) E D B A C (7) E D A B C (6) C B A D E (6) C A B D E (6) B A E D C (6) D C E B A (4) D C E A B (4) C D E B A (4) A B C E D (4) E A B D C (3) C D A B E (3) B A C E D (3) A B E D C (3) A B E C D (3) E B A D C (2) C D B A E (2) B A E C D (2) B A C D E (2) E D C A B (1) C E D A B (1) Total count = 100 A B C D E A 0 2 -18 -20 -20 B -2 0 -18 -20 -20 C 18 18 0 -12 -6 D 20 20 12 0 18 E 20 20 6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 -20 -20 B -2 0 -18 -20 -20 C 18 18 0 -12 -6 D 20 20 12 0 18 E 20 20 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=28 E=19 B=13 A=10 so A is eliminated. Round 2 votes counts: C=30 D=28 B=23 E=19 so E is eliminated. Round 3 votes counts: D=42 C=30 B=28 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:235 E:214 C:209 A:172 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -18 -20 -20 B -2 0 -18 -20 -20 C 18 18 0 -12 -6 D 20 20 12 0 18 E 20 20 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -20 -20 B -2 0 -18 -20 -20 C 18 18 0 -12 -6 D 20 20 12 0 18 E 20 20 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -20 -20 B -2 0 -18 -20 -20 C 18 18 0 -12 -6 D 20 20 12 0 18 E 20 20 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2509: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (13) B A C E D (6) A D E B C (6) D E C A B (5) D E A C B (5) C D E B A (5) B C D E A (5) A E D B C (5) E D C A B (4) B C A E D (4) C B D E A (3) B A D C E (3) A B E D C (3) A B C E D (3) D A E B C (2) C E D B A (2) C E D A B (2) C B E D A (2) B D C E A (2) A D E C B (2) A C E B D (2) E D A C B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E A B C (1) D B A E C (1) D A E C B (1) C E B D A (1) C B E A D (1) B D A C E (1) B C A D E (1) B A D E C (1) A E C D B (1) A E C B D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 22 24 12 16 B -22 0 -8 -18 -26 C -24 8 0 -18 -14 D -12 18 18 0 -10 E -16 26 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 24 12 16 B -22 0 -8 -18 -26 C -24 8 0 -18 -14 D -12 18 18 0 -10 E -16 26 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=23 D=16 C=16 E=7 so E is eliminated. Round 2 votes counts: A=40 B=23 D=21 C=16 so C is eliminated. Round 3 votes counts: A=40 D=30 B=30 so D is eliminated. Round 4 votes counts: A=61 B=39 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:237 E:217 D:207 C:176 B:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 24 12 16 B -22 0 -8 -18 -26 C -24 8 0 -18 -14 D -12 18 18 0 -10 E -16 26 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 24 12 16 B -22 0 -8 -18 -26 C -24 8 0 -18 -14 D -12 18 18 0 -10 E -16 26 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 24 12 16 B -22 0 -8 -18 -26 C -24 8 0 -18 -14 D -12 18 18 0 -10 E -16 26 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2510: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) B C A E D (8) D A E C B (7) B C E A D (6) E D C A B (5) B E C D A (5) B A C D E (5) E C D B A (4) B D A E C (4) B A C E D (4) D E C B A (3) D E A C B (3) D A B E C (3) B E D C A (3) B D E A C (3) A D C E B (3) E C A D B (2) D E C A B (2) D E B C A (2) D B E A C (2) C E A B D (2) A C D E B (2) A B D C E (2) E D C B A (1) D E B A C (1) C E A D B (1) C B A E D (1) B D E C A (1) B A D C E (1) A C E D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -14 -20 -14 B 8 0 2 -6 0 C 14 -2 0 8 -24 D 20 6 -8 0 -12 E 14 0 24 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.460098 C: 0.000000 D: 0.000000 E: 0.539902 Sum of squares = 0.503184292332 Cumulative probabilities = A: 0.000000 B: 0.460098 C: 0.460098 D: 0.460098 E: 1.000000 A B C D E A 0 -8 -14 -20 -14 B 8 0 2 -6 0 C 14 -2 0 8 -24 D 20 6 -8 0 -12 E 14 0 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=23 D=23 A=10 C=4 so C is eliminated. Round 2 votes counts: B=41 E=26 D=23 A=10 so A is eliminated. Round 3 votes counts: B=45 D=28 E=27 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:225 D:203 B:202 C:198 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 -20 -14 B 8 0 2 -6 0 C 14 -2 0 8 -24 D 20 6 -8 0 -12 E 14 0 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -20 -14 B 8 0 2 -6 0 C 14 -2 0 8 -24 D 20 6 -8 0 -12 E 14 0 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -20 -14 B 8 0 2 -6 0 C 14 -2 0 8 -24 D 20 6 -8 0 -12 E 14 0 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.500002 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499998 C: 0.499998 D: 0.499998 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2511: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) E B A D C (8) E A B C D (8) D C A B E (7) E A C B D (6) B E D C A (6) A E C D B (6) C D A B E (5) B D C E A (5) B D C A E (5) E B D A C (4) E B A C D (4) E A C D B (3) B E A C D (3) B D E C A (3) A C E D B (3) D B C A E (2) C D B A E (2) E B D C A (1) D E B C A (1) D B C E A (1) C D A E B (1) C A D E B (1) C A D B E (1) C A B D E (1) A E C B D (1) A E B C D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -2 -4 -6 B 8 0 4 14 0 C 2 -4 0 -4 -10 D 4 -14 4 0 -10 E 6 0 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.406223 C: 0.000000 D: 0.000000 E: 0.593777 Sum of squares = 0.517588337628 Cumulative probabilities = A: 0.000000 B: 0.406223 C: 0.406223 D: 0.406223 E: 1.000000 A B C D E A 0 -8 -2 -4 -6 B 8 0 4 14 0 C 2 -4 0 -4 -10 D 4 -14 4 0 -10 E 6 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=22 D=20 A=13 C=11 so C is eliminated. Round 2 votes counts: E=34 D=28 B=22 A=16 so A is eliminated. Round 3 votes counts: E=46 D=31 B=23 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:213 C:192 D:192 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 -6 B 8 0 4 14 0 C 2 -4 0 -4 -10 D 4 -14 4 0 -10 E 6 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 -6 B 8 0 4 14 0 C 2 -4 0 -4 -10 D 4 -14 4 0 -10 E 6 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 -6 B 8 0 4 14 0 C 2 -4 0 -4 -10 D 4 -14 4 0 -10 E 6 0 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2512: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) E C B A D (7) D B C A E (7) A D B E C (6) B D C E A (5) B C D E A (5) A E C B D (5) A D E B C (5) D B A C E (4) C E B D A (4) A D B C E (4) E C A B D (3) E A C B D (3) D C B E A (3) D A B C E (3) C B E D A (3) B D A C E (3) A E D C B (3) E C B D A (2) B D C A E (2) B A D C E (2) B A C D E (2) A E B C D (2) A D E C B (2) A B D C E (2) E C A D B (1) D A E B C (1) C E B A D (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -16 -2 0 10 B 16 0 24 2 18 C 2 -24 0 -22 16 D 0 -2 22 0 28 E -10 -18 -16 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 0 10 B 16 0 24 2 18 C 2 -24 0 -22 16 D 0 -2 22 0 28 E -10 -18 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=26 B=19 E=16 C=8 so C is eliminated. Round 2 votes counts: A=31 D=26 B=22 E=21 so E is eliminated. Round 3 votes counts: A=38 B=36 D=26 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:230 D:224 A:196 C:186 E:164 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 0 10 B 16 0 24 2 18 C 2 -24 0 -22 16 D 0 -2 22 0 28 E -10 -18 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 0 10 B 16 0 24 2 18 C 2 -24 0 -22 16 D 0 -2 22 0 28 E -10 -18 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 0 10 B 16 0 24 2 18 C 2 -24 0 -22 16 D 0 -2 22 0 28 E -10 -18 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2513: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (20) A E D B C (9) E A D B C (7) C B D A E (6) E D B A C (5) A E C D B (5) D B E C A (4) B D C E A (4) E A D C B (3) A D B E C (3) A C E B D (3) E D A B C (2) E C D B A (2) E C A B D (2) E A C D B (2) D E B A C (2) D B E A C (2) C A B D E (2) B D A C E (2) A D E B C (2) E D C B A (1) E D B C A (1) E C B D A (1) E C A D B (1) D E A B C (1) D B A E C (1) C E A B D (1) C A B E D (1) B D C A E (1) B C D A E (1) A E D C B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 4 -12 -22 B 6 0 -6 -8 -2 C -4 6 0 -2 -14 D 12 8 2 0 4 E 22 2 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -12 -22 B 6 0 -6 -8 -2 C -4 6 0 -2 -14 D 12 8 2 0 4 E 22 2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=27 A=25 D=10 B=8 so B is eliminated. Round 2 votes counts: C=31 E=27 A=25 D=17 so D is eliminated. Round 3 votes counts: E=36 C=36 A=28 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:213 B:195 C:193 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 -12 -22 B 6 0 -6 -8 -2 C -4 6 0 -2 -14 D 12 8 2 0 4 E 22 2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -12 -22 B 6 0 -6 -8 -2 C -4 6 0 -2 -14 D 12 8 2 0 4 E 22 2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -12 -22 B 6 0 -6 -8 -2 C -4 6 0 -2 -14 D 12 8 2 0 4 E 22 2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2514: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) D A E B C (10) C B E A D (8) E A C B D (5) D C E A B (4) C E B A D (4) C B D A E (4) E A C D B (3) D B C A E (3) D A B E C (3) C E A B D (3) C D E A B (3) B C A E D (3) D C B A E (2) C E D A B (2) C E B D A (2) C D E B A (2) C B D E A (2) B C D A E (2) B A E C D (2) B A D E C (2) A B E D C (2) E D A C B (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E B A (1) D C B E A (1) D B A C E (1) C E A D B (1) C B E D A (1) C B A E D (1) B D C A E (1) B D A E C (1) B D A C E (1) B A D C E (1) A E D B C (1) A E B D C (1) A E B C D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -2 -16 -12 B -10 0 -22 -2 -16 C 2 22 0 -2 6 D 16 2 2 0 14 E 12 16 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -16 -12 B -10 0 -22 -2 -16 C 2 22 0 -2 6 D 16 2 2 0 14 E 12 16 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=33 B=13 E=11 A=7 so A is eliminated. Round 2 votes counts: D=37 C=33 B=16 E=14 so E is eliminated. Round 3 votes counts: C=41 D=40 B=19 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:214 E:204 A:190 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -2 -16 -12 B -10 0 -22 -2 -16 C 2 22 0 -2 6 D 16 2 2 0 14 E 12 16 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -16 -12 B -10 0 -22 -2 -16 C 2 22 0 -2 6 D 16 2 2 0 14 E 12 16 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -16 -12 B -10 0 -22 -2 -16 C 2 22 0 -2 6 D 16 2 2 0 14 E 12 16 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2515: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (16) A D B E C (12) C E D B A (9) A D E B C (6) A B D E C (6) C B E D A (5) C E D A B (4) C E A D B (4) E C D B A (3) C B D A E (3) B D A E C (3) C A E D B (2) C A D B E (2) C A B D E (2) B E D A C (2) B D E A C (2) A C B D E (2) E D B A C (1) E C B D A (1) C E B A D (1) C B A E D (1) C B A D E (1) B E C D A (1) B D E C A (1) B D C A E (1) B A D E C (1) B A D C E (1) A E D C B (1) A E D B C (1) A D C B E (1) A D B C E (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -14 -4 0 B 6 0 -20 2 -2 C 14 20 0 18 18 D 4 -2 -18 0 -6 E 0 2 -18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -4 0 B 6 0 -20 2 -2 C 14 20 0 18 18 D 4 -2 -18 0 -6 E 0 2 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=50 A=33 B=12 E=5 so D is eliminated. Round 2 votes counts: C=50 A=33 B=12 E=5 so E is eliminated. Round 3 votes counts: C=54 A=33 B=13 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:235 E:195 B:193 D:189 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -4 0 B 6 0 -20 2 -2 C 14 20 0 18 18 D 4 -2 -18 0 -6 E 0 2 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -4 0 B 6 0 -20 2 -2 C 14 20 0 18 18 D 4 -2 -18 0 -6 E 0 2 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -4 0 B 6 0 -20 2 -2 C 14 20 0 18 18 D 4 -2 -18 0 -6 E 0 2 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2516: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) E B C A D (8) E D B C A (6) E D A C B (6) D A C B E (6) B C A D E (6) A C B D E (6) D B A C E (5) B C A E D (5) D A B C E (4) E C B A D (3) D A E C B (3) C B A E D (3) C A B E D (3) B C E A D (3) A C B E D (3) D E B C A (2) D E B A C (2) D E A B C (2) D B C A E (2) C B E A D (2) A D C B E (2) A B C D E (2) E D C A B (1) E D B A C (1) E D A B C (1) D B E C A (1) B D C A E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 8 -4 6 B 0 0 2 -8 8 C -8 -2 0 -8 10 D 4 8 8 0 6 E -6 -8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -4 6 B 0 0 2 -8 8 C -8 -2 0 -8 10 D 4 8 8 0 6 E -6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=26 B=15 A=15 C=8 so C is eliminated. Round 2 votes counts: D=36 E=26 B=20 A=18 so A is eliminated. Round 3 votes counts: D=38 B=34 E=28 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:205 B:201 C:196 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 8 -4 6 B 0 0 2 -8 8 C -8 -2 0 -8 10 D 4 8 8 0 6 E -6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -4 6 B 0 0 2 -8 8 C -8 -2 0 -8 10 D 4 8 8 0 6 E -6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -4 6 B 0 0 2 -8 8 C -8 -2 0 -8 10 D 4 8 8 0 6 E -6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2517: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (13) E C B D A (8) E C A B D (6) E A C D B (5) D B A C E (5) D A B C E (5) B D C A E (5) D E B C A (3) B D C E A (3) A C E B D (3) E C D B A (2) E C B A D (2) E C A D B (2) D B E C A (2) D B E A C (2) D B C A E (2) D A B E C (2) C E A B D (2) C A E B D (2) B D A C E (2) E D C B A (1) E D C A B (1) E D A B C (1) E C D A B (1) E A D C B (1) D E B A C (1) D B C E A (1) D A E B C (1) C E B A D (1) C B E D A (1) C B A E D (1) B D E C A (1) B C E D A (1) B C A D E (1) B A D C E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D B E C (1) A C E D B (1) A C B E D (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 -2 4 B -8 0 10 -12 6 C -2 -10 0 -14 8 D 2 12 14 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -2 4 B -8 0 10 -12 6 C -2 -10 0 -14 8 D 2 12 14 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=25 D=24 B=14 C=7 so C is eliminated. Round 2 votes counts: E=33 A=27 D=24 B=16 so B is eliminated. Round 3 votes counts: E=35 D=35 A=30 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:206 B:198 C:191 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -2 4 B -8 0 10 -12 6 C -2 -10 0 -14 8 D 2 12 14 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 4 B -8 0 10 -12 6 C -2 -10 0 -14 8 D 2 12 14 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 4 B -8 0 10 -12 6 C -2 -10 0 -14 8 D 2 12 14 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2518: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A C E D B (7) B E D C A (6) E D B A C (5) D E B A C (5) A E D C B (5) D E B C A (4) C B A D E (4) B D E C A (4) A D E C B (4) E D B C A (3) E B D C A (3) D B E C A (3) A E C D B (3) A C E B D (3) E D A B C (2) E A B D C (2) D E A B C (2) D C A E B (2) D C A B E (2) C B D E A (2) C B D A E (2) B D C E A (2) B C D E A (2) A C D B E (2) A C B E D (2) E B D A C (1) E B A D C (1) C D B A E (1) C B A E D (1) C A B E D (1) B C E D A (1) B C E A D (1) A E D B C (1) A E B C D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -4 -4 2 B 2 0 -2 -4 -10 C 4 2 0 -14 -10 D 4 4 14 0 2 E -2 10 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 2 B 2 0 -2 -4 -10 C 4 2 0 -14 -10 D 4 4 14 0 2 E -2 10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=19 D=18 E=17 B=16 so B is eliminated. Round 2 votes counts: A=30 D=24 E=23 C=23 so E is eliminated. Round 3 votes counts: D=44 A=33 C=23 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:208 A:196 B:193 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 2 B 2 0 -2 -4 -10 C 4 2 0 -14 -10 D 4 4 14 0 2 E -2 10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 2 B 2 0 -2 -4 -10 C 4 2 0 -14 -10 D 4 4 14 0 2 E -2 10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 2 B 2 0 -2 -4 -10 C 4 2 0 -14 -10 D 4 4 14 0 2 E -2 10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2519: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) A B D C E (9) C D B A E (8) E C B A D (6) E A B D C (4) E A B C D (4) C B D A E (4) A B E D C (4) A B C D E (4) E A D B C (3) D B A C E (3) C D B E A (3) A E B C D (3) E D C B A (2) E C D B A (2) E C B D A (2) E C A B D (2) D C E B A (2) D C B E A (2) D A B C E (2) A D B C E (2) A B D E C (2) A B C E D (2) E D A B C (1) E C D A B (1) E C A D B (1) E A D C B (1) E A C B D (1) D E A C B (1) D E A B C (1) D B C A E (1) C D E B A (1) C B E D A (1) B D C A E (1) B C D A E (1) B C A D E (1) B A D C E (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 0 4 18 B 2 0 2 6 24 C 0 -2 0 -4 16 D -4 -6 4 0 20 E -18 -24 -16 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 4 18 B 2 0 2 6 24 C 0 -2 0 -4 16 D -4 -6 4 0 20 E -18 -24 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=27 D=21 C=17 B=5 so B is eliminated. Round 2 votes counts: E=30 A=29 D=22 C=19 so C is eliminated. Round 3 votes counts: D=39 E=31 A=30 so A is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:217 A:210 D:207 C:205 E:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 4 18 B 2 0 2 6 24 C 0 -2 0 -4 16 D -4 -6 4 0 20 E -18 -24 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 18 B 2 0 2 6 24 C 0 -2 0 -4 16 D -4 -6 4 0 20 E -18 -24 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 18 B 2 0 2 6 24 C 0 -2 0 -4 16 D -4 -6 4 0 20 E -18 -24 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2520: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) A B E C D (10) D C E B A (9) C A E B D (8) D C A E B (4) D B E A C (4) D A B E C (4) C D E B A (4) A E B C D (4) E B A C D (3) D A C B E (3) B E A C D (3) A D B E C (3) A C E B D (3) E B C A D (2) D C B E A (2) D B E C A (2) D B A E C (2) D A B C E (2) C E B A D (2) C D A E B (2) C A D E B (2) B E A D C (2) A B D E C (2) E B C D A (1) C E A B D (1) C D E A B (1) B E D C A (1) B E D A C (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 24 18 16 24 B -24 0 14 10 6 C -18 -14 0 -4 -10 D -16 -10 4 0 -6 E -24 -6 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 18 16 24 B -24 0 14 10 6 C -18 -14 0 -4 -10 D -16 -10 4 0 -6 E -24 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=32 C=20 B=7 E=6 so E is eliminated. Round 2 votes counts: A=35 D=32 C=20 B=13 so B is eliminated. Round 3 votes counts: A=43 D=34 C=23 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:241 B:203 E:193 D:186 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 18 16 24 B -24 0 14 10 6 C -18 -14 0 -4 -10 D -16 -10 4 0 -6 E -24 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 18 16 24 B -24 0 14 10 6 C -18 -14 0 -4 -10 D -16 -10 4 0 -6 E -24 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 18 16 24 B -24 0 14 10 6 C -18 -14 0 -4 -10 D -16 -10 4 0 -6 E -24 -6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2521: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (20) E A D C B (13) E A D B C (9) E A B D C (6) C D B A E (5) E A C D B (4) C B D A E (4) B D C A E (4) B C E D A (4) C D A B E (3) E C B D A (2) C D E A B (2) C D B E A (2) B E C A D (2) B D A C E (2) B A D C E (2) A E D C B (2) E C D A B (1) E C A D B (1) E B A C D (1) E A B C D (1) D C A E B (1) D C A B E (1) D A C E B (1) D A C B E (1) C E D A B (1) C D E B A (1) C D A E B (1) A D E C B (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 -10 -12 0 B -2 0 6 -4 2 C 10 -6 0 10 12 D 12 4 -10 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.200000 D: 0.300000 E: 0.000000 Sum of squares = 0.379999999952 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -12 0 B -2 0 6 -4 2 C 10 -6 0 10 12 D 12 4 -10 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.200000 D: 0.300000 E: 0.000000 Sum of squares = 0.379999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=34 C=19 A=5 D=4 so D is eliminated. Round 2 votes counts: E=38 B=34 C=21 A=7 so A is eliminated. Round 3 votes counts: E=42 B=35 C=23 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:206 B:201 A:190 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -10 -12 0 B -2 0 6 -4 2 C 10 -6 0 10 12 D 12 4 -10 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.200000 D: 0.300000 E: 0.000000 Sum of squares = 0.379999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -12 0 B -2 0 6 -4 2 C 10 -6 0 10 12 D 12 4 -10 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.200000 D: 0.300000 E: 0.000000 Sum of squares = 0.379999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -12 0 B -2 0 6 -4 2 C 10 -6 0 10 12 D 12 4 -10 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.200000 D: 0.300000 E: 0.000000 Sum of squares = 0.379999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2522: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) B C D A E (11) E D A C B (8) D A E B C (7) C B E D A (7) E A D C B (5) D A B C E (5) B C E D A (5) D B A C E (4) C E B A D (4) A D E B C (4) D A B E C (3) B D A C E (3) B C D E A (3) A E D C B (3) C B A E D (2) B C A D E (2) A D E C B (2) E C B D A (1) E C A D B (1) E C A B D (1) D B C E A (1) C E B D A (1) B C E A D (1) B A D C E (1) A E C B D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -4 -18 0 B 16 0 4 12 24 C 4 -4 0 4 26 D 18 -12 -4 0 -4 E 0 -24 -26 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -18 0 B 16 0 4 12 24 C 4 -4 0 4 26 D 18 -12 -4 0 -4 E 0 -24 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992509 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=26 B=26 D=20 E=16 A=12 so A is eliminated. Round 2 votes counts: D=27 B=27 C=26 E=20 so E is eliminated. Round 3 votes counts: D=43 C=30 B=27 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:228 C:215 D:199 A:181 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 -18 0 B 16 0 4 12 24 C 4 -4 0 4 26 D 18 -12 -4 0 -4 E 0 -24 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992509 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -18 0 B 16 0 4 12 24 C 4 -4 0 4 26 D 18 -12 -4 0 -4 E 0 -24 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992509 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -18 0 B 16 0 4 12 24 C 4 -4 0 4 26 D 18 -12 -4 0 -4 E 0 -24 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992509 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2523: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (27) B C A E D (18) E D A C B (6) D E B A C (6) D B E A C (6) E C A B D (5) B D C A E (5) E A C D B (4) E A C B D (4) B C A D E (4) A C E B D (4) D E B C A (2) D B C A E (2) C A E B D (2) D E A B C (1) D B E C A (1) D B C E A (1) C E A B D (1) C A B E D (1) Total count = 100 A B C D E A 0 10 16 -14 -28 B -10 0 -8 -12 -24 C -16 8 0 -14 -24 D 14 12 14 0 10 E 28 24 24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 -14 -28 B -10 0 -8 -12 -24 C -16 8 0 -14 -24 D 14 12 14 0 10 E 28 24 24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=46 B=27 E=19 C=4 A=4 so C is eliminated. Round 2 votes counts: D=46 B=27 E=20 A=7 so A is eliminated. Round 3 votes counts: D=46 B=28 E=26 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:233 D:225 A:192 C:177 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 16 -14 -28 B -10 0 -8 -12 -24 C -16 8 0 -14 -24 D 14 12 14 0 10 E 28 24 24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 -14 -28 B -10 0 -8 -12 -24 C -16 8 0 -14 -24 D 14 12 14 0 10 E 28 24 24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 -14 -28 B -10 0 -8 -12 -24 C -16 8 0 -14 -24 D 14 12 14 0 10 E 28 24 24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2524: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) E A C B D (5) C B D E A (5) B D C E A (5) E C A B D (4) D B C E A (4) A E D B C (4) A D C B E (4) E C B A D (3) E B D C A (3) E B C D A (3) D B A C E (3) D A B C E (3) C A D B E (3) E C B D A (2) E B D A C (2) D B C A E (2) D B A E C (2) C B E D A (2) C A E D B (2) B E D C A (2) B C D E A (2) A E C B D (2) A D B E C (2) A D B C E (2) A C E D B (2) A C D E B (2) A C D B E (2) E B A D C (1) E A B D C (1) E A B C D (1) D C B A E (1) D B E C A (1) D B E A C (1) C E B D A (1) C E A B D (1) C D B A E (1) C A E B D (1) B D E C A (1) A E D C B (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 6 2 8 0 B -6 0 -10 -6 -4 C -2 10 0 6 -2 D -8 6 -6 0 -4 E 0 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.379167 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.620833 Sum of squares = 0.529200983255 Cumulative probabilities = A: 0.379167 B: 0.379167 C: 0.379167 D: 0.379167 E: 1.000000 A B C D E A 0 6 2 8 0 B -6 0 -10 -6 -4 C -2 10 0 6 -2 D -8 6 -6 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=25 D=17 C=16 B=10 so B is eliminated. Round 2 votes counts: A=32 E=27 D=23 C=18 so C is eliminated. Round 3 votes counts: A=38 E=31 D=31 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:206 E:205 D:194 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 8 0 B -6 0 -10 -6 -4 C -2 10 0 6 -2 D -8 6 -6 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 8 0 B -6 0 -10 -6 -4 C -2 10 0 6 -2 D -8 6 -6 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 8 0 B -6 0 -10 -6 -4 C -2 10 0 6 -2 D -8 6 -6 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2525: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) D C B E A (9) A E B C D (8) C D B E A (7) D C A E B (6) E A B C D (4) D C A B E (4) B E A C D (4) A E C B D (4) E B A C D (3) A E D C B (3) A E D B C (3) D B C E A (2) D A E C B (2) D A C E B (2) C D B A E (2) C D A E B (2) C D A B E (2) B E D A C (2) B E C A D (2) A E B D C (2) A D C E B (2) E B A D C (1) E A B D C (1) C B E A D (1) C B D E A (1) C A E D B (1) C A E B D (1) B E D C A (1) B E C D A (1) B E A D C (1) B D E A C (1) B C D E A (1) A D E C B (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -8 -12 16 B 0 0 -24 -20 4 C 8 24 0 -10 10 D 12 20 10 0 12 E -16 -4 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -12 16 B 0 0 -24 -20 4 C 8 24 0 -10 10 D 12 20 10 0 12 E -16 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=25 C=17 B=13 E=9 so E is eliminated. Round 2 votes counts: D=36 A=30 C=17 B=17 so C is eliminated. Round 3 votes counts: D=49 A=32 B=19 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 C:216 A:198 B:180 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -12 16 B 0 0 -24 -20 4 C 8 24 0 -10 10 D 12 20 10 0 12 E -16 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -12 16 B 0 0 -24 -20 4 C 8 24 0 -10 10 D 12 20 10 0 12 E -16 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -12 16 B 0 0 -24 -20 4 C 8 24 0 -10 10 D 12 20 10 0 12 E -16 -4 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2526: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) B C A E D (7) C B A D E (6) B E D C A (6) D E A B C (5) A C E D B (5) D E A C B (4) C A D E B (4) C A D B E (4) D E B A C (3) D B E C A (3) C A B D E (3) B E C D A (3) A E D C B (3) A D C E B (3) E D A B C (2) E B D A C (2) E B A D C (2) C D B A E (2) C B A E D (2) C A B E D (2) A D E C B (2) A C E B D (2) A C D E B (2) E D A C B (1) E A B D C (1) D E C A B (1) D E B C A (1) D C A E B (1) D B C E A (1) D A E C B (1) D A C E B (1) B E D A C (1) B E A C D (1) B D E C A (1) B C D E A (1) B C D A E (1) A E D B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 2 6 B 2 0 2 -16 -10 C -2 -2 0 -8 -4 D -2 16 8 0 0 E -6 10 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.800000 B: 0.100000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.659999999992 Cumulative probabilities = A: 0.800000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 2 6 B 2 0 2 -16 -10 C -2 -2 0 -8 -4 D -2 16 8 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.100000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000008 Cumulative probabilities = A: 0.800000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 D=21 B=21 A=19 E=16 so E is eliminated. Round 2 votes counts: D=32 B=25 C=23 A=20 so A is eliminated. Round 3 votes counts: D=41 C=32 B=27 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:204 E:204 C:192 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 6 B 2 0 2 -16 -10 C -2 -2 0 -8 -4 D -2 16 8 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.100000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000008 Cumulative probabilities = A: 0.800000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 6 B 2 0 2 -16 -10 C -2 -2 0 -8 -4 D -2 16 8 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.100000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000008 Cumulative probabilities = A: 0.800000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 6 B 2 0 2 -16 -10 C -2 -2 0 -8 -4 D -2 16 8 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.100000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000008 Cumulative probabilities = A: 0.800000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2527: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) E C D B A (8) D E C A B (8) C E D B A (7) A B D E C (7) E C B A D (5) D A B C E (5) D C E A B (4) C E B A D (4) C D E A B (4) B E C A D (4) B A D E C (4) A B D C E (4) E D C B A (3) D A C B E (3) B E A C D (3) E C B D A (2) D A B E C (2) B A C E D (2) A D B C E (2) E D C A B (1) E B C A D (1) D E C B A (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E A D (1) C A E D B (1) B A C D E (1) Total count = 100 A B C D E A 0 -10 -10 -4 -12 B 10 0 -10 -6 -6 C 10 10 0 4 -20 D 4 6 -4 0 -2 E 12 6 20 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -10 -4 -12 B 10 0 -10 -6 -6 C 10 10 0 4 -20 D 4 6 -4 0 -2 E 12 6 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 E=20 C=17 A=13 so A is eliminated. Round 2 votes counts: B=34 D=29 E=20 C=17 so C is eliminated. Round 3 votes counts: B=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:220 C:202 D:202 B:194 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -10 -4 -12 B 10 0 -10 -6 -6 C 10 10 0 4 -20 D 4 6 -4 0 -2 E 12 6 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -4 -12 B 10 0 -10 -6 -6 C 10 10 0 4 -20 D 4 6 -4 0 -2 E 12 6 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -4 -12 B 10 0 -10 -6 -6 C 10 10 0 4 -20 D 4 6 -4 0 -2 E 12 6 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2528: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) C E A D B (5) B D E A C (5) D B A C E (4) B E D C A (4) B D E C A (4) B D A E C (4) B A D E C (4) A C D E B (4) A B E D C (4) E B D C A (3) E B C A D (3) E B A C D (3) E A C B D (3) A B D E C (3) E C B D A (2) E B C D A (2) D B E C A (2) C E D A B (2) C A E D B (2) C A D E B (2) A C D B E (2) A B E C D (2) A B D C E (2) E C D B A (1) E C B A D (1) E A B C D (1) D E C B A (1) D C E B A (1) D C A B E (1) D B C E A (1) D B C A E (1) D A C B E (1) C E D B A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D A E B (1) C A E B D (1) B E D A C (1) B A E D C (1) A E C B D (1) A E B C D (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 0 12 12 2 B 0 0 12 12 10 C -12 -12 0 -12 -12 D -12 -12 12 0 8 E -2 -10 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.192000 B: 0.808000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.689727532389 Cumulative probabilities = A: 0.192000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 12 2 B 0 0 12 12 10 C -12 -12 0 -12 -12 D -12 -12 12 0 8 E -2 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 E=19 C=17 D=12 so D is eliminated. Round 2 votes counts: B=31 A=30 E=20 C=19 so C is eliminated. Round 3 votes counts: A=37 E=32 B=31 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:217 A:213 D:198 E:196 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 12 2 B 0 0 12 12 10 C -12 -12 0 -12 -12 D -12 -12 12 0 8 E -2 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 12 2 B 0 0 12 12 10 C -12 -12 0 -12 -12 D -12 -12 12 0 8 E -2 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 12 2 B 0 0 12 12 10 C -12 -12 0 -12 -12 D -12 -12 12 0 8 E -2 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2529: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (17) B C E A D (14) B A C E D (7) E C B D A (6) A D B C E (6) B C A E D (5) E C A D B (3) D A B E C (3) B A D C E (3) E C D B A (2) E C B A D (2) D E C A B (2) D A E B C (2) D A B C E (2) C E B A D (2) C E A B D (2) B A C D E (2) A C E B D (2) A B D C E (2) A B C D E (2) E D C B A (1) E C D A B (1) E B C D A (1) E A C D B (1) D E A B C (1) D B A E C (1) D B A C E (1) C B E A D (1) B C E D A (1) B C D A E (1) A E C D B (1) A D E C B (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 12 16 20 B 0 0 12 8 6 C -12 -12 0 14 8 D -16 -8 -14 0 -6 E -20 -6 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.458119 B: 0.541881 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.503508004432 Cumulative probabilities = A: 0.458119 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 16 20 B 0 0 12 8 6 C -12 -12 0 14 8 D -16 -8 -14 0 -6 E -20 -6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=29 E=17 A=16 C=5 so C is eliminated. Round 2 votes counts: B=34 D=29 E=21 A=16 so A is eliminated. Round 3 votes counts: B=39 D=37 E=24 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:224 B:213 C:199 E:186 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 16 20 B 0 0 12 8 6 C -12 -12 0 14 8 D -16 -8 -14 0 -6 E -20 -6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 16 20 B 0 0 12 8 6 C -12 -12 0 14 8 D -16 -8 -14 0 -6 E -20 -6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 16 20 B 0 0 12 8 6 C -12 -12 0 14 8 D -16 -8 -14 0 -6 E -20 -6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2530: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) A B C E D (8) D E C B A (7) D E B C A (6) C A B E D (6) D E B A C (5) C A E B D (5) E D B A C (4) D B A E C (4) A C B E D (4) D C E A B (3) C A D B E (3) B A E C D (3) E B A D C (2) D B E A C (2) C E D B A (2) C E B A D (2) C D A E B (2) A B D C E (2) A B C D E (2) E D C B A (1) E D B C A (1) E C B A D (1) E B D C A (1) E B D A C (1) D C E B A (1) D A B E C (1) C E D A B (1) C E B D A (1) C E A B D (1) C D E B A (1) C D A B E (1) C A B D E (1) B E D A C (1) B A D E C (1) A D C B E (1) A D B C E (1) A C B D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -10 -8 -4 B -6 0 -6 -10 -12 C 10 6 0 6 14 D 8 10 -6 0 8 E 4 12 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -8 -4 B -6 0 -6 -10 -12 C 10 6 0 6 14 D 8 10 -6 0 8 E 4 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=29 A=21 E=11 B=5 so B is eliminated. Round 2 votes counts: C=34 D=29 A=25 E=12 so E is eliminated. Round 3 votes counts: D=38 C=35 A=27 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:210 E:197 A:192 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 -8 -4 B -6 0 -6 -10 -12 C 10 6 0 6 14 D 8 10 -6 0 8 E 4 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -8 -4 B -6 0 -6 -10 -12 C 10 6 0 6 14 D 8 10 -6 0 8 E 4 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -8 -4 B -6 0 -6 -10 -12 C 10 6 0 6 14 D 8 10 -6 0 8 E 4 12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2531: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) A B E D C (10) D C E A B (8) C D A E B (8) E D C A B (6) B A C D E (6) C D E A B (5) B A E D C (5) E B D C A (4) C D E B A (4) A B C D E (4) D C E B A (3) B E C D A (3) B A E C D (3) E D C B A (2) E A D C B (2) D E C A B (2) C D B A E (2) C D A B E (2) B A C E D (2) A E D C B (2) E B A D C (1) B C D E A (1) A E B D C (1) A D C E B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 0 0 -4 B -6 0 4 4 0 C 0 -4 0 -16 -6 D 0 -4 16 0 -6 E 4 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.287393 C: 0.000000 D: 0.000000 E: 0.712607 Sum of squares = 0.590403601454 Cumulative probabilities = A: 0.000000 B: 0.287393 C: 0.287393 D: 0.287393 E: 1.000000 A B C D E A 0 6 0 0 -4 B -6 0 4 4 0 C 0 -4 0 -16 -6 D 0 -4 16 0 -6 E 4 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000022689 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=21 A=20 E=15 D=13 so D is eliminated. Round 2 votes counts: C=32 B=31 A=20 E=17 so E is eliminated. Round 3 votes counts: C=42 B=36 A=22 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:208 D:203 A:201 B:201 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 0 -4 B -6 0 4 4 0 C 0 -4 0 -16 -6 D 0 -4 16 0 -6 E 4 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000022689 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 -4 B -6 0 4 4 0 C 0 -4 0 -16 -6 D 0 -4 16 0 -6 E 4 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000022689 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 -4 B -6 0 4 4 0 C 0 -4 0 -16 -6 D 0 -4 16 0 -6 E 4 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000022689 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2532: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) A B C D E (7) B D A E C (6) A D B E C (6) C E D B A (5) D B E A C (4) E D B C A (3) C E B D A (3) C E A D B (3) C A B E D (3) B D E C A (3) A C B E D (3) A C B D E (3) A B D E C (3) A B D C E (3) E D C B A (2) E C D A B (2) D B E C A (2) D B A E C (2) C E B A D (2) C E A B D (2) C A E B D (2) B D A C E (2) B C A D E (2) B A C D E (2) A E C D B (2) A D E B C (2) A C E D B (2) E D A C B (1) E A D B C (1) D A B E C (1) C B A D E (1) C A B D E (1) B D C E A (1) B D C A E (1) B A D C E (1) A E D C B (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 8 8 16 B 0 0 6 4 14 C -8 -6 0 8 0 D -8 -4 -8 0 6 E -16 -14 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.336862 B: 0.663138 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.553227919194 Cumulative probabilities = A: 0.336862 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 8 16 B 0 0 6 4 14 C -8 -6 0 8 0 D -8 -4 -8 0 6 E -16 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=22 B=18 E=17 D=9 so D is eliminated. Round 2 votes counts: A=35 B=26 C=22 E=17 so E is eliminated. Round 3 votes counts: A=37 C=34 B=29 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:212 C:197 D:193 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 8 16 B 0 0 6 4 14 C -8 -6 0 8 0 D -8 -4 -8 0 6 E -16 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 8 16 B 0 0 6 4 14 C -8 -6 0 8 0 D -8 -4 -8 0 6 E -16 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 8 16 B 0 0 6 4 14 C -8 -6 0 8 0 D -8 -4 -8 0 6 E -16 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2533: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) A D B E C (9) D A B E C (7) E C B A D (6) C B E D A (6) E C B D A (5) B E C D A (5) A D C E B (5) C E B A D (4) C D A E B (4) A D E B C (4) B C E D A (3) E C A B D (2) E B C A D (2) E B A D C (2) D A C B E (2) B E A D C (2) A D E C B (2) E B C D A (1) E B A C D (1) D C A B E (1) D B A E C (1) D B A C E (1) D A B C E (1) C D B A E (1) C D A B E (1) C B D E A (1) C A D E B (1) B E D C A (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A E C (1) A E D C B (1) A E B D C (1) A D B C E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -14 -10 -6 B 12 0 -6 14 -4 C 14 6 0 8 -10 D 10 -14 -8 0 -4 E 6 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -14 -10 -6 B 12 0 -6 14 -4 C 14 6 0 8 -10 D 10 -14 -8 0 -4 E 6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=25 E=19 B=15 D=13 so D is eliminated. Round 2 votes counts: A=35 C=29 E=19 B=17 so B is eliminated. Round 3 votes counts: A=38 C=34 E=28 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:209 B:208 D:192 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -14 -10 -6 B 12 0 -6 14 -4 C 14 6 0 8 -10 D 10 -14 -8 0 -4 E 6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -10 -6 B 12 0 -6 14 -4 C 14 6 0 8 -10 D 10 -14 -8 0 -4 E 6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -10 -6 B 12 0 -6 14 -4 C 14 6 0 8 -10 D 10 -14 -8 0 -4 E 6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2534: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) C B A E D (8) A E D B C (8) B C A E D (7) B D A E C (5) B A D E C (5) B A E D C (4) E D A C B (3) C D E B A (3) C D B E A (3) B C A D E (3) B A C E D (3) E D C A B (2) E D A B C (2) D E C A B (2) D E A B C (2) D B E A C (2) C E D A B (2) C D E A B (2) C B A D E (2) C A E B D (2) A B E D C (2) E A D B C (1) D E B C A (1) D E B A C (1) D E A C B (1) D C E B A (1) D C E A B (1) C E B D A (1) C E A D B (1) C B E A D (1) C A E D B (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E B C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -6 12 18 B 20 0 6 16 14 C 6 -6 0 8 4 D -12 -16 -8 0 -10 E -18 -14 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 12 18 B 20 0 6 16 14 C 6 -6 0 8 4 D -12 -16 -8 0 -10 E -18 -14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=29 A=18 D=11 E=8 so E is eliminated. Round 2 votes counts: C=34 B=29 A=19 D=18 so D is eliminated. Round 3 votes counts: C=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:228 C:206 A:202 E:187 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -6 12 18 B 20 0 6 16 14 C 6 -6 0 8 4 D -12 -16 -8 0 -10 E -18 -14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 12 18 B 20 0 6 16 14 C 6 -6 0 8 4 D -12 -16 -8 0 -10 E -18 -14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 12 18 B 20 0 6 16 14 C 6 -6 0 8 4 D -12 -16 -8 0 -10 E -18 -14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2535: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (16) A C B E D (14) D C E B A (13) E B D A C (8) A B E D C (7) C D E B A (5) A B E C D (5) E B D C A (4) C A D B E (4) E B C D A (3) C D A E B (3) B E D A C (3) A C D B E (3) D E B C A (2) D C E A B (2) C A B D E (2) E B A D C (1) D E C B A (1) D A C E B (1) C D A B E (1) B E A C D (1) B A E C D (1) Total count = 100 A B C D E A 0 -16 20 8 -18 B 16 0 2 30 14 C -20 -2 0 -16 -4 D -8 -30 16 0 -26 E 18 -14 4 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 20 8 -18 B 16 0 2 30 14 C -20 -2 0 -16 -4 D -8 -30 16 0 -26 E 18 -14 4 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984691 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=21 D=19 E=16 C=15 so C is eliminated. Round 2 votes counts: A=35 D=28 B=21 E=16 so E is eliminated. Round 3 votes counts: B=37 A=35 D=28 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:217 A:197 C:179 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 20 8 -18 B 16 0 2 30 14 C -20 -2 0 -16 -4 D -8 -30 16 0 -26 E 18 -14 4 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984691 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 20 8 -18 B 16 0 2 30 14 C -20 -2 0 -16 -4 D -8 -30 16 0 -26 E 18 -14 4 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984691 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 20 8 -18 B 16 0 2 30 14 C -20 -2 0 -16 -4 D -8 -30 16 0 -26 E 18 -14 4 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984691 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2536: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) E C A D B (8) B C E A D (6) A D C E B (5) C E A D B (4) C E A B D (4) B D E C A (4) B D A C E (4) E D A C B (3) D E A C B (3) D A B E C (3) B D E A C (3) A C D E B (3) E D B A C (2) E C B A D (2) E B C D A (2) E A C D B (2) D B A E C (2) C E B A D (2) C B E A D (2) B E D C A (2) B E C D A (2) B C A D E (2) B A C D E (2) A C E D B (2) E C B D A (1) E C A B D (1) E B D C A (1) E A D C B (1) D E A B C (1) D B E A C (1) D B A C E (1) C B A E D (1) C A E D B (1) C A E B D (1) B D C A E (1) B D A E C (1) B C E D A (1) B C D A E (1) B C A E D (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 6 0 4 -16 B -6 0 -14 -6 -18 C 0 14 0 2 -10 D -4 6 -2 0 -4 E 16 18 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 0 4 -16 B -6 0 -14 -6 -18 C 0 14 0 2 -10 D -4 6 -2 0 -4 E 16 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=23 D=20 C=15 A=12 so A is eliminated. Round 2 votes counts: B=30 D=27 E=23 C=20 so C is eliminated. Round 3 votes counts: E=37 B=33 D=30 so D is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 C:203 D:198 A:197 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 4 -16 B -6 0 -14 -6 -18 C 0 14 0 2 -10 D -4 6 -2 0 -4 E 16 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 4 -16 B -6 0 -14 -6 -18 C 0 14 0 2 -10 D -4 6 -2 0 -4 E 16 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 4 -16 B -6 0 -14 -6 -18 C 0 14 0 2 -10 D -4 6 -2 0 -4 E 16 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2537: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (6) C B D A E (6) B A D C E (6) E A D B C (5) E C A D B (4) C D E B A (4) B D A C E (4) A E B D C (4) E C D A B (3) E A B D C (3) C E D A B (3) C B A E D (3) C B A D E (3) A B E C D (3) A B D E C (3) E D A C B (2) E D A B C (2) E A D C B (2) E A C B D (2) D C B E A (2) D A B E C (2) C E A B D (2) C B E A D (2) B C A D E (2) B A C D E (2) A D B E C (2) E D C A B (1) E C A B D (1) E A C D B (1) D E C A B (1) D B C A E (1) D B A E C (1) D A E B C (1) C E D B A (1) C E B A D (1) C E A D B (1) C D B E A (1) C B D E A (1) B D C A E (1) B D A E C (1) B A D E C (1) A E D B C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 14 12 12 -2 B -14 0 6 6 -4 C -12 -6 0 -6 -8 D -12 -6 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999981 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 A B C D E A 0 14 12 12 -2 B -14 0 6 6 -4 C -12 -6 0 -6 -8 D -12 -6 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999281 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=17 A=15 D=14 so D is eliminated. Round 2 votes counts: E=33 C=30 B=19 A=18 so A is eliminated. Round 3 votes counts: E=40 C=30 B=30 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:218 E:206 B:197 D:195 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 12 12 -2 B -14 0 6 6 -4 C -12 -6 0 -6 -8 D -12 -6 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999281 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 12 -2 B -14 0 6 6 -4 C -12 -6 0 -6 -8 D -12 -6 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999281 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 12 -2 B -14 0 6 6 -4 C -12 -6 0 -6 -8 D -12 -6 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999281 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2538: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) D B E A C (6) B E A C D (6) C D E B A (4) C D A E B (4) B E A D C (4) A C B E D (4) A B E D C (4) A B E C D (4) A B D E C (4) D B E C A (3) D B A E C (3) C E D B A (3) C E A B D (3) C A E B D (3) C A D E B (3) B D A E C (3) A C E B D (3) E B D C A (2) D A C B E (2) B A E D C (2) B A D E C (2) A E B C D (2) E D B C A (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) D E B C A (1) D C E A B (1) D C B E A (1) D B A C E (1) C E D A B (1) C E A D B (1) C D E A B (1) B E D A C (1) B D E C A (1) B D E A C (1) B A E C D (1) A E C B D (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 18 4 -2 B 12 0 16 12 12 C -18 -16 0 -2 -12 D -4 -12 2 0 0 E 2 -12 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 18 4 -2 B 12 0 16 12 12 C -18 -16 0 -2 -12 D -4 -12 2 0 0 E 2 -12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=24 C=23 B=21 E=7 so E is eliminated. Round 2 votes counts: B=27 D=25 A=25 C=23 so C is eliminated. Round 3 votes counts: D=38 A=35 B=27 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:226 A:204 E:201 D:193 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 18 4 -2 B 12 0 16 12 12 C -18 -16 0 -2 -12 D -4 -12 2 0 0 E 2 -12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 18 4 -2 B 12 0 16 12 12 C -18 -16 0 -2 -12 D -4 -12 2 0 0 E 2 -12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 18 4 -2 B 12 0 16 12 12 C -18 -16 0 -2 -12 D -4 -12 2 0 0 E 2 -12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2539: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) D E B A C (7) C A E D B (7) B D E A C (7) D B E A C (6) C A B E D (6) A C B D E (6) E D C A B (5) A C E D B (5) C E A D B (4) C A E B D (4) E D C B A (3) E D B A C (3) E D A C B (3) B D E C A (2) B D A E C (2) B C A D E (2) B A C D E (2) A C E B D (2) E C D A B (1) E C A D B (1) E A D C B (1) D E A B C (1) D B A E C (1) C B A E D (1) C B A D E (1) C A B D E (1) B E C D A (1) B D A C E (1) B C D A E (1) B A D C E (1) A D E C B (1) A D C E B (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 -2 -4 B -4 0 -8 -16 -12 C -6 8 0 -6 -4 D 2 16 6 0 -10 E 4 12 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 -2 -4 B -4 0 -8 -16 -12 C -6 8 0 -6 -4 D 2 16 6 0 -10 E 4 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 B=19 A=18 D=15 so D is eliminated. Round 2 votes counts: E=32 B=26 C=24 A=18 so A is eliminated. Round 3 votes counts: C=39 E=33 B=28 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:207 A:202 C:196 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 -2 -4 B -4 0 -8 -16 -12 C -6 8 0 -6 -4 D 2 16 6 0 -10 E 4 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -2 -4 B -4 0 -8 -16 -12 C -6 8 0 -6 -4 D 2 16 6 0 -10 E 4 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -2 -4 B -4 0 -8 -16 -12 C -6 8 0 -6 -4 D 2 16 6 0 -10 E 4 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2540: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) C A E B D (8) B E A D C (7) C D B E A (5) D C B E A (4) C D A B E (4) C B E A D (4) B E D A C (4) A E C B D (4) A E B D C (4) C A D E B (3) B D E A C (3) A E D B C (3) E C B A D (2) E A B D C (2) D B C E A (2) D A C E B (2) C D B A E (2) C D A E B (2) C B D E A (2) A E D C B (2) A E B C D (2) A D C E B (2) E B C A D (1) E B A D C (1) E A B C D (1) D C B A E (1) D C A B E (1) D B A E C (1) D A B E C (1) C E A B D (1) C B E D A (1) C A E D B (1) B E D C A (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C E A (1) B C E A D (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 0 6 -8 B 8 0 -6 6 12 C 0 6 0 -6 -4 D -6 -6 6 0 -6 E 8 -12 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333332606 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 6 -8 B 8 0 -6 6 12 C 0 6 0 -6 -4 D -6 -6 6 0 -6 E 8 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333297 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=20 B=20 A=20 E=7 so E is eliminated. Round 2 votes counts: C=35 A=23 B=22 D=20 so D is eliminated. Round 3 votes counts: C=41 B=33 A=26 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:210 E:203 C:198 A:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 6 -8 B 8 0 -6 6 12 C 0 6 0 -6 -4 D -6 -6 6 0 -6 E 8 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333297 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 6 -8 B 8 0 -6 6 12 C 0 6 0 -6 -4 D -6 -6 6 0 -6 E 8 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333297 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 6 -8 B 8 0 -6 6 12 C 0 6 0 -6 -4 D -6 -6 6 0 -6 E 8 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333297 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2541: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (5) C D A B E (5) B E C D A (5) B E A D C (5) E B A D C (4) D C A E B (4) C D E B A (4) C D E A B (4) C D A E B (4) B E A C D (4) A B E D C (4) D A C E B (3) A E D C B (3) E D B C A (2) E D A C B (2) E C D B A (2) E C B D A (2) E B C D A (2) E A D C B (2) E A B D C (2) C D B E A (2) C D B A E (2) B C E D A (2) B C D A E (2) B A E D C (2) A E D B C (2) A D C E B (2) A D B C E (2) E D C B A (1) E D C A B (1) E B D C A (1) E B D A C (1) E A D B C (1) C E D B A (1) C B D A E (1) B E C A D (1) B C E A D (1) B C D E A (1) B C A D E (1) B A E C D (1) B A C E D (1) A D E C B (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -12 -18 -16 B 2 0 -4 -12 -12 C 12 4 0 -4 -2 D 18 12 4 0 -10 E 16 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -12 -18 -16 B 2 0 -4 -12 -12 C 12 4 0 -4 -2 D 18 12 4 0 -10 E 16 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 C=23 A=16 D=12 so D is eliminated. Round 2 votes counts: C=32 B=26 E=23 A=19 so A is eliminated. Round 3 votes counts: C=38 B=33 E=29 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:220 D:212 C:205 B:187 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -12 -18 -16 B 2 0 -4 -12 -12 C 12 4 0 -4 -2 D 18 12 4 0 -10 E 16 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -18 -16 B 2 0 -4 -12 -12 C 12 4 0 -4 -2 D 18 12 4 0 -10 E 16 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -18 -16 B 2 0 -4 -12 -12 C 12 4 0 -4 -2 D 18 12 4 0 -10 E 16 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2542: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) E B A C D (9) A C D E B (9) A E C D B (8) D C A B E (6) C D A E B (6) B E D C A (5) B E A D C (5) B E A C D (5) E A B C D (4) E B A D C (3) D C E B A (3) D C B E A (3) A C D B E (3) E A C D B (2) D C B A E (2) C D A B E (2) B A E C D (2) A E C B D (2) A C E D B (2) E D A C B (1) D C E A B (1) D B C E A (1) C A D E B (1) B D E C A (1) B A C D E (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 20 16 16 14 B -20 0 -24 -22 -26 C -16 24 0 16 4 D -16 22 -16 0 0 E -14 26 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 16 16 14 B -20 0 -24 -22 -26 C -16 24 0 16 4 D -16 22 -16 0 0 E -14 26 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 E=19 B=19 C=9 so C is eliminated. Round 2 votes counts: D=35 A=27 E=19 B=19 so E is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:233 C:214 E:204 D:195 B:154 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 16 16 14 B -20 0 -24 -22 -26 C -16 24 0 16 4 D -16 22 -16 0 0 E -14 26 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 16 14 B -20 0 -24 -22 -26 C -16 24 0 16 4 D -16 22 -16 0 0 E -14 26 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 16 14 B -20 0 -24 -22 -26 C -16 24 0 16 4 D -16 22 -16 0 0 E -14 26 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2543: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (13) C B E D A (12) E D C A B (8) B C A D E (7) E D A C B (6) B C A E D (6) D E A C B (5) B C E A D (5) B A D C E (5) E C D A B (4) D A E C B (4) C E D B A (4) B A C D E (4) C E B D A (3) B C E D A (3) A B D E C (3) C B E A D (2) A D E C B (2) A D B E C (2) B A D E C (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -8 2 -4 B 4 0 -2 2 0 C 8 2 0 0 4 D -2 -2 0 0 -6 E 4 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714312 D: 0.285688 E: 0.000000 Sum of squares = 0.591859556862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714312 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 2 -4 B 4 0 -2 2 0 C 8 2 0 0 4 D -2 -2 0 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=21 A=21 E=18 D=9 so D is eliminated. Round 2 votes counts: B=31 A=25 E=23 C=21 so C is eliminated. Round 3 votes counts: B=45 E=30 A=25 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:207 E:203 B:202 D:195 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 2 -4 B 4 0 -2 2 0 C 8 2 0 0 4 D -2 -2 0 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 2 -4 B 4 0 -2 2 0 C 8 2 0 0 4 D -2 -2 0 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 2 -4 B 4 0 -2 2 0 C 8 2 0 0 4 D -2 -2 0 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000000651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2544: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (15) D E C B A (6) B A E C D (6) A B E D C (6) D E A B C (5) D C E A B (5) D E C A B (4) D C E B A (3) D A E B C (3) C B E D A (3) C B A E D (3) A B E C D (3) E B D C A (2) C D B E A (2) C D A B E (2) C B E A D (2) A E D B C (2) A D E B C (2) A B C D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) E B A D C (1) D E A C B (1) D C A E B (1) C E D B A (1) C D E A B (1) C D B A E (1) C B D E A (1) C B A D E (1) C A B D E (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A E D (1) B A E D C (1) B A C E D (1) A D B E C (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -20 -22 -20 B 14 0 -12 -18 -14 C 20 12 0 0 -2 D 22 18 0 0 20 E 20 14 2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.528133 D: 0.471867 E: 0.000000 Sum of squares = 0.501582936248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.528133 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -22 -20 B 14 0 -12 -18 -14 C 20 12 0 0 -2 D 22 18 0 0 20 E 20 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=28 A=19 B=12 E=8 so E is eliminated. Round 2 votes counts: C=34 D=32 A=19 B=15 so B is eliminated. Round 3 votes counts: C=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:230 C:215 E:208 B:185 A:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -20 -22 -20 B 14 0 -12 -18 -14 C 20 12 0 0 -2 D 22 18 0 0 20 E 20 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -22 -20 B 14 0 -12 -18 -14 C 20 12 0 0 -2 D 22 18 0 0 20 E 20 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -22 -20 B 14 0 -12 -18 -14 C 20 12 0 0 -2 D 22 18 0 0 20 E 20 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2545: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) D B A E C (6) B D A C E (6) B C A D E (4) A C E B D (4) E D C A B (3) E C A D B (3) E B C D A (3) C A B E D (3) B D E C A (3) B C A E D (3) B A C D E (3) A D E C B (3) A C E D B (3) E D A C B (2) E C D A B (2) E C B A D (2) E A D C B (2) D E A B C (2) D B E C A (2) D A B C E (2) C E A B D (2) B D C E A (2) B C E D A (2) A C B D E (2) E D B C A (1) E D A B C (1) E C D B A (1) E C A B D (1) E A C D B (1) D E B C A (1) D E B A C (1) D E A C B (1) D B E A C (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B A D (1) C B E A D (1) C B A E D (1) B E D C A (1) B C E A D (1) B C D E A (1) B A D C E (1) A E D C B (1) A E C D B (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 6 12 B -2 0 4 12 0 C 2 -4 0 8 6 D -6 -12 -8 0 -6 E -12 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000075 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 6 12 B -2 0 4 12 0 C 2 -4 0 8 6 D -6 -12 -8 0 -6 E -12 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000011 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=22 D=19 A=17 C=15 so C is eliminated. Round 2 votes counts: B=29 A=27 E=25 D=19 so D is eliminated. Round 3 votes counts: B=39 A=31 E=30 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:207 C:206 E:194 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 6 12 B -2 0 4 12 0 C 2 -4 0 8 6 D -6 -12 -8 0 -6 E -12 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000011 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 6 12 B -2 0 4 12 0 C 2 -4 0 8 6 D -6 -12 -8 0 -6 E -12 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000011 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 6 12 B -2 0 4 12 0 C 2 -4 0 8 6 D -6 -12 -8 0 -6 E -12 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000011 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2546: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) D C A E B (7) D C E A B (6) C D A B E (5) A D C B E (5) A B C D E (5) E B D C A (4) D C A B E (4) B E A C D (4) E D C B A (3) E B A D C (3) E B A C D (3) C D E B A (3) B E C D A (3) B A E C D (3) A D C E B (3) A B D E C (3) E C B D A (2) E B C A D (2) E A B D C (2) B E C A D (2) A D E B C (2) A B E C D (2) E D C A B (1) E D B A C (1) E C D B A (1) E A D B C (1) D C E B A (1) D A E C B (1) C D B E A (1) C D B A E (1) C B D A E (1) B C E A D (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A C D B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -10 -4 -2 B -4 0 8 4 -10 C 10 -8 0 -2 -6 D 4 -4 2 0 2 E 2 10 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.46874999999 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 A B C D E A 0 4 -10 -4 -2 B -4 0 8 4 -10 C 10 -8 0 -2 -6 D 4 -4 2 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468750000221 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=25 D=19 B=15 C=11 so C is eliminated. Round 2 votes counts: E=30 D=29 A=25 B=16 so B is eliminated. Round 3 votes counts: E=40 D=30 A=30 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:202 B:199 C:197 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -10 -4 -2 B -4 0 8 4 -10 C 10 -8 0 -2 -6 D 4 -4 2 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468750000221 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -4 -2 B -4 0 8 4 -10 C 10 -8 0 -2 -6 D 4 -4 2 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468750000221 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -4 -2 B -4 0 8 4 -10 C 10 -8 0 -2 -6 D 4 -4 2 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468750000221 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2547: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) A D E B C (9) C E B A D (7) B D A E C (7) A D B E C (7) C B E D A (6) B E D A C (6) C E B D A (4) C D A B E (4) D A B E C (3) C A D E B (3) B E C D A (3) E C B A D (2) E B A D C (2) E A D B C (2) C E A B D (2) C B D A E (2) B D E A C (2) B C E D A (2) B C D A E (2) A E D B C (2) A D C E B (2) A C D E B (2) E B C A D (1) E B A C D (1) E A D C B (1) D B A C E (1) C A E D B (1) B D E C A (1) B D C A E (1) B D A C E (1) B C D E A (1) Total count = 100 A B C D E A 0 -4 -4 8 -6 B 4 0 8 6 -2 C 4 -8 0 6 2 D -8 -6 -6 0 -4 E 6 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000012 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 -4 -4 8 -6 B 4 0 8 6 -2 C 4 -8 0 6 2 D -8 -6 -6 0 -4 E 6 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=26 A=22 E=9 D=4 so D is eliminated. Round 2 votes counts: C=39 B=27 A=25 E=9 so E is eliminated. Round 3 votes counts: C=41 B=31 A=28 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:205 C:202 A:197 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 8 -6 B 4 0 8 6 -2 C 4 -8 0 6 2 D -8 -6 -6 0 -4 E 6 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 8 -6 B 4 0 8 6 -2 C 4 -8 0 6 2 D -8 -6 -6 0 -4 E 6 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 8 -6 B 4 0 8 6 -2 C 4 -8 0 6 2 D -8 -6 -6 0 -4 E 6 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000061 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2548: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) D C A E B (10) C D A E B (8) B E A D C (7) E C B A D (6) D A C B E (6) E B A C D (5) C E B D A (4) C E B A D (4) C D E A B (4) B D A E C (4) C E A B D (3) E B C D A (2) E B C A D (2) D C B A E (2) D C A B E (2) D B C E A (2) C E D A B (2) A B D E C (2) E C B D A (1) E A B C D (1) D C B E A (1) D B A E C (1) D B A C E (1) C E A D B (1) C A D E B (1) B E D A C (1) B E A C D (1) B D E C A (1) B D E A C (1) B A E D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -12 -28 2 B -6 0 -10 -6 -8 C 12 10 0 -10 24 D 28 6 10 0 16 E -2 8 -24 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -28 2 B -6 0 -10 -6 -8 C 12 10 0 -10 24 D 28 6 10 0 16 E -2 8 -24 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=27 E=17 B=16 A=4 so A is eliminated. Round 2 votes counts: D=37 C=27 B=19 E=17 so E is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 C:218 B:185 A:184 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -12 -28 2 B -6 0 -10 -6 -8 C 12 10 0 -10 24 D 28 6 10 0 16 E -2 8 -24 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -28 2 B -6 0 -10 -6 -8 C 12 10 0 -10 24 D 28 6 10 0 16 E -2 8 -24 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -28 2 B -6 0 -10 -6 -8 C 12 10 0 -10 24 D 28 6 10 0 16 E -2 8 -24 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2549: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) C D E A B (12) A B E D C (12) B A E D C (10) A B D E C (10) E D C B A (6) A B C D E (5) C D E B A (4) A B C E D (4) C A B D E (3) E D B A C (2) D E C B A (2) D E C A B (2) D E B A C (2) C D A E B (2) E C D B A (1) D E A B C (1) D A B E C (1) C E D A B (1) C A B E D (1) B E A D C (1) B D E A C (1) B C E A D (1) B A E C D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 2 -2 0 B -10 0 4 0 2 C -2 -4 0 0 -6 D 2 0 0 0 -10 E 0 -2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.536296 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.463704 Sum of squares = 0.502634780288 Cumulative probabilities = A: 0.536296 B: 0.536296 C: 0.536296 D: 0.536296 E: 1.000000 A B C D E A 0 10 2 -2 0 B -10 0 4 0 2 C -2 -4 0 0 -6 D 2 0 0 0 -10 E 0 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=32 B=14 E=9 D=8 so D is eliminated. Round 2 votes counts: C=37 A=33 E=16 B=14 so B is eliminated. Round 3 votes counts: A=44 C=38 E=18 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:207 A:205 B:198 D:196 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 -2 0 B -10 0 4 0 2 C -2 -4 0 0 -6 D 2 0 0 0 -10 E 0 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -2 0 B -10 0 4 0 2 C -2 -4 0 0 -6 D 2 0 0 0 -10 E 0 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -2 0 B -10 0 4 0 2 C -2 -4 0 0 -6 D 2 0 0 0 -10 E 0 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2550: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) B D E A C (6) E B C D A (5) B D A C E (5) E C A D B (4) E B C A D (4) E B A C D (4) D B A C E (4) B E D A C (4) A D C B E (4) A C D E B (4) E C B D A (3) E C A B D (3) C E A D B (3) C A E D B (3) C A D E B (3) B D E C A (3) B D A E C (3) A C D B E (3) E C D A B (2) B E A D C (2) B A E D C (2) B A D E C (2) A D B C E (2) A B D C E (2) E C D B A (1) D C B E A (1) D C B A E (1) D C A B E (1) D B C E A (1) D B C A E (1) D A C B E (1) D A B C E (1) C D E B A (1) C D A E B (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 2 -6 -10 B 22 0 20 14 16 C -2 -20 0 -10 -16 D 6 -14 10 0 2 E 10 -16 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 2 -6 -10 B 22 0 20 14 16 C -2 -20 0 -10 -16 D 6 -14 10 0 2 E 10 -16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=26 A=17 D=11 C=11 so D is eliminated. Round 2 votes counts: B=41 E=26 A=19 C=14 so C is eliminated. Round 3 votes counts: B=43 E=30 A=27 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:236 E:204 D:202 A:182 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 2 -6 -10 B 22 0 20 14 16 C -2 -20 0 -10 -16 D 6 -14 10 0 2 E 10 -16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 2 -6 -10 B 22 0 20 14 16 C -2 -20 0 -10 -16 D 6 -14 10 0 2 E 10 -16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 2 -6 -10 B 22 0 20 14 16 C -2 -20 0 -10 -16 D 6 -14 10 0 2 E 10 -16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2551: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) D C E B A (7) B A E C D (6) A B E C D (6) E B A C D (5) C D A B E (5) E A D B C (4) D C E A B (4) C D B A E (4) C B D E A (4) B A C E D (4) D E C A B (3) D C A B E (3) C B D A E (3) B E A C D (3) B C A E D (3) E D A B C (2) D E C B A (2) D E A C B (2) D C A E B (2) D A E C B (2) B C E A D (2) E D B C A (1) E D B A C (1) E B C D A (1) E B A D C (1) E A B C D (1) D E A B C (1) D A C E B (1) D A C B E (1) C B E D A (1) C B A E D (1) C B A D E (1) C A B D E (1) B E C A D (1) A E D B C (1) A D E B C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 0 -6 B 2 0 6 6 2 C -2 -6 0 8 -2 D 0 -6 -8 0 -4 E 6 -2 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 0 -6 B 2 0 6 6 2 C -2 -6 0 8 -2 D 0 -6 -8 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 C=20 B=19 A=10 so A is eliminated. Round 2 votes counts: D=29 B=27 E=24 C=20 so C is eliminated. Round 3 votes counts: D=38 B=38 E=24 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:205 C:199 A:197 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 0 -6 B 2 0 6 6 2 C -2 -6 0 8 -2 D 0 -6 -8 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 -6 B 2 0 6 6 2 C -2 -6 0 8 -2 D 0 -6 -8 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 -6 B 2 0 6 6 2 C -2 -6 0 8 -2 D 0 -6 -8 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2552: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) E A D B C (9) B C E D A (8) C B D A E (6) B E C A D (5) A D E C B (5) C D A E B (4) C D A B E (4) C B A D E (4) B E D A C (4) D A E C B (3) C D B A E (3) C B E A D (3) B C D E A (3) A E D C B (3) E D A B C (2) E A B D C (2) D A E B C (2) D A C E B (2) C A D E B (2) B E A D C (2) B C E A D (2) E C B A D (1) E C A D B (1) E B A C D (1) D C A B E (1) D B A E C (1) D B A C E (1) C E A D B (1) C B D E A (1) B E D C A (1) A D E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 0 8 -12 B 12 0 8 6 -2 C 0 -8 0 0 -6 D -8 -6 0 0 -12 E 12 2 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 0 8 -12 B 12 0 8 6 -2 C 0 -8 0 0 -6 D -8 -6 0 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=25 A=11 D=10 so D is eliminated. Round 2 votes counts: C=29 B=27 E=26 A=18 so A is eliminated. Round 3 votes counts: E=40 C=33 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:212 C:193 A:192 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 0 8 -12 B 12 0 8 6 -2 C 0 -8 0 0 -6 D -8 -6 0 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 8 -12 B 12 0 8 6 -2 C 0 -8 0 0 -6 D -8 -6 0 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 8 -12 B 12 0 8 6 -2 C 0 -8 0 0 -6 D -8 -6 0 0 -12 E 12 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2553: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) C E A D B (9) A B D C E (5) E D C A B (4) E C D A B (4) E C A D B (4) D B E A C (4) C E B A D (4) C E A B D (4) A D E C B (4) A C E D B (4) D E B C A (3) C E B D A (3) C A E B D (3) A C E B D (3) E D C B A (2) E C D B A (2) C A E D B (2) B D E C A (2) B D C A E (2) B C A E D (2) B A D C E (2) B A C E D (2) A B C D E (2) E A D C B (1) E A C D B (1) D B E C A (1) D B A E C (1) D A B E C (1) B E C D A (1) B D E A C (1) B D A C E (1) B C E D A (1) B A D E C (1) B A C D E (1) A D B E C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -6 14 -2 B -8 0 -8 2 -14 C 6 8 0 6 4 D -14 -2 -6 0 -14 E 2 14 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 14 -2 B -8 0 -8 2 -14 C 6 8 0 6 4 D -14 -2 -6 0 -14 E 2 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 A=21 E=18 D=10 so D is eliminated. Round 2 votes counts: B=32 C=25 A=22 E=21 so E is eliminated. Round 3 votes counts: C=41 B=35 A=24 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:212 A:207 B:186 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 14 -2 B -8 0 -8 2 -14 C 6 8 0 6 4 D -14 -2 -6 0 -14 E 2 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 14 -2 B -8 0 -8 2 -14 C 6 8 0 6 4 D -14 -2 -6 0 -14 E 2 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 14 -2 B -8 0 -8 2 -14 C 6 8 0 6 4 D -14 -2 -6 0 -14 E 2 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2554: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (23) A C E B D (18) A C E D B (10) C E A B D (8) B D E C A (6) E C B A D (4) D A B C E (4) A B C E D (3) C A E D B (2) B E D C A (2) B E C D A (2) B D A E C (2) A D C B E (2) A C D E B (2) E D B C A (1) E C B D A (1) E C A D B (1) E C A B D (1) D E C B A (1) D B E A C (1) D B A E C (1) D A C E B (1) B A D C E (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -4 10 -2 B -10 0 -4 0 0 C 4 4 0 8 8 D -10 0 -8 0 -8 E 2 0 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 10 -2 B -10 0 -4 0 0 C 4 4 0 8 8 D -10 0 -8 0 -8 E 2 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=31 B=13 C=10 E=8 so E is eliminated. Round 2 votes counts: A=38 D=32 C=17 B=13 so B is eliminated. Round 3 votes counts: D=42 A=39 C=19 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:207 E:201 B:193 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 10 -2 B -10 0 -4 0 0 C 4 4 0 8 8 D -10 0 -8 0 -8 E 2 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 10 -2 B -10 0 -4 0 0 C 4 4 0 8 8 D -10 0 -8 0 -8 E 2 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 10 -2 B -10 0 -4 0 0 C 4 4 0 8 8 D -10 0 -8 0 -8 E 2 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2555: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) B D A E C (8) C A E D B (7) C E B A D (5) B D E A C (5) B E D C A (4) A D E C B (4) A C D E B (4) E C B D A (3) E B D C A (3) C A E B D (3) C A B E D (3) C A B D E (3) A D C B E (3) E D B A C (2) D B A E C (2) B D A C E (2) B C D E A (2) B C D A E (2) A D B C E (2) A C E D B (2) E C A D B (1) E B D A C (1) E A D C B (1) E A C D B (1) D A E B C (1) C E B D A (1) C E A B D (1) C B E D A (1) C B E A D (1) C B A E D (1) C B A D E (1) B E D A C (1) B D E C A (1) B D C A E (1) B C E D A (1) B C A D E (1) B A D C E (1) A D E B C (1) A D C E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -12 18 10 B 0 0 -16 14 -4 C 12 16 0 10 22 D -18 -14 -10 0 -6 E -10 4 -22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 18 10 B 0 0 -16 14 -4 C 12 16 0 10 22 D -18 -14 -10 0 -6 E -10 4 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=29 A=19 E=12 D=3 so D is eliminated. Round 2 votes counts: C=37 B=31 A=20 E=12 so E is eliminated. Round 3 votes counts: C=41 B=37 A=22 so A is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:230 A:208 B:197 E:189 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 18 10 B 0 0 -16 14 -4 C 12 16 0 10 22 D -18 -14 -10 0 -6 E -10 4 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 18 10 B 0 0 -16 14 -4 C 12 16 0 10 22 D -18 -14 -10 0 -6 E -10 4 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 18 10 B 0 0 -16 14 -4 C 12 16 0 10 22 D -18 -14 -10 0 -6 E -10 4 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2556: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) D E B A C (6) C A D E B (6) B E A D C (6) C D A E B (5) E D B A C (4) E B D A C (4) D E B C A (4) C A B E D (4) A C D E B (4) A C B E D (3) E B A D C (2) D C E A B (2) C D E B A (2) C D B E A (2) C B A E D (2) C A D B E (2) C A B D E (2) B E D C A (2) B C E D A (2) A E D B C (2) A E B D C (2) A D E C B (2) A D C E B (2) A B E D C (2) A B C E D (2) D E C B A (1) D E A B C (1) D B E C A (1) D B C E A (1) D A E B C (1) C D A B E (1) C B E D A (1) C B D E A (1) C B A D E (1) B E C D A (1) B E A C D (1) B C E A D (1) B A E D C (1) B A E C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 12 -2 -8 B 12 0 14 2 0 C -12 -14 0 -10 -8 D 2 -2 10 0 -6 E 8 0 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.358635 C: 0.000000 D: 0.000000 E: 0.641365 Sum of squares = 0.539967928779 Cumulative probabilities = A: 0.000000 B: 0.358635 C: 0.358635 D: 0.358635 E: 1.000000 A B C D E A 0 -12 12 -2 -8 B 12 0 14 2 0 C -12 -14 0 -10 -8 D 2 -2 10 0 -6 E 8 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=24 A=20 D=17 E=10 so E is eliminated. Round 2 votes counts: B=30 C=29 D=21 A=20 so A is eliminated. Round 3 votes counts: B=37 C=36 D=27 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:211 D:202 A:195 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 12 -2 -8 B 12 0 14 2 0 C -12 -14 0 -10 -8 D 2 -2 10 0 -6 E 8 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 12 -2 -8 B 12 0 14 2 0 C -12 -14 0 -10 -8 D 2 -2 10 0 -6 E 8 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 12 -2 -8 B 12 0 14 2 0 C -12 -14 0 -10 -8 D 2 -2 10 0 -6 E 8 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2557: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) D E A B C (9) D E B C A (7) C B A E D (6) E D B C A (4) D E B A C (4) B C E D A (4) B C D E A (4) A D E C B (4) D E A C B (3) C A B E D (3) B E C D A (3) B C E A D (3) B C D A E (3) A D C E B (3) A C E B D (3) A C B D E (3) E B D C A (2) D B E C A (2) D A E C B (2) B C A E D (2) A E D C B (2) A E C D B (2) E D B A C (1) E D A C B (1) E A D C B (1) D A C B E (1) C B A D E (1) C A D B E (1) B D E C A (1) B D C E A (1) B C A D E (1) A E C B D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 4 -4 0 B -2 0 2 2 0 C -4 -2 0 2 2 D 4 -2 -2 0 2 E 0 0 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -4 0 B -2 0 2 2 0 C -4 -2 0 2 2 D 4 -2 -2 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999884 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=28 B=22 C=11 E=9 so E is eliminated. Round 2 votes counts: D=34 A=31 B=24 C=11 so C is eliminated. Round 3 votes counts: A=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:201 B:201 D:201 C:199 E:198 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 4 -4 0 B -2 0 2 2 0 C -4 -2 0 2 2 D 4 -2 -2 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999884 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -4 0 B -2 0 2 2 0 C -4 -2 0 2 2 D 4 -2 -2 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999884 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -4 0 B -2 0 2 2 0 C -4 -2 0 2 2 D 4 -2 -2 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999884 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2558: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) A D E C B (5) E C B D A (4) D C A E B (4) B C D A E (4) E B A C D (3) D C A B E (3) D A C E B (3) C D E A B (3) B C D E A (3) B A E D C (3) A E D C B (3) A D B E C (3) A D B C E (3) E C D B A (2) E C B A D (2) E B C A D (2) E A C B D (2) D C E A B (2) C E D B A (2) C E D A B (2) C D B A E (2) C D A B E (2) C B E D A (2) B E C D A (2) B E C A D (2) B D A C E (2) B A D C E (2) A E B D C (2) E D C A B (1) E C A D B (1) E A C D B (1) E A B D C (1) D C B A E (1) D A C B E (1) C D E B A (1) C D A E B (1) C B D E A (1) B E A C D (1) B D C A E (1) B A D E C (1) A D E B C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -22 -20 -2 B 8 0 -4 4 2 C 22 4 0 12 14 D 20 -4 -12 0 0 E 2 -2 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -22 -20 -2 B 8 0 -4 4 2 C 22 4 0 12 14 D 20 -4 -12 0 0 E 2 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=19 A=19 C=16 D=14 so D is eliminated. Round 2 votes counts: B=32 C=26 A=23 E=19 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:205 D:202 E:193 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -22 -20 -2 B 8 0 -4 4 2 C 22 4 0 12 14 D 20 -4 -12 0 0 E 2 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -22 -20 -2 B 8 0 -4 4 2 C 22 4 0 12 14 D 20 -4 -12 0 0 E 2 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -22 -20 -2 B 8 0 -4 4 2 C 22 4 0 12 14 D 20 -4 -12 0 0 E 2 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2559: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) D E A B C (6) D A E B C (6) D C E A B (5) C B A D E (5) B A C E D (5) C D B A E (4) C B D A E (4) B C A E D (4) E D A B C (3) E A D B C (3) D C E B A (3) D C A B E (3) C E B D A (3) C D E B A (3) C B E A D (3) E B A C D (2) D E C A B (2) D E A C B (2) D C A E B (2) C B D E A (2) C B A E D (2) B A E C D (2) A E B D C (2) A B E D C (2) A B E C D (2) E D C B A (1) E D C A B (1) E B C A D (1) E A B C D (1) D C B E A (1) D A E C B (1) D A C B E (1) D A B C E (1) C D B E A (1) B E C A D (1) B C E A D (1) B C A D E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -6 -10 -2 B -2 0 2 0 -6 C 6 -2 0 -6 10 D 10 0 6 0 10 E 2 6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.458159 C: 0.000000 D: 0.541841 E: 0.000000 Sum of squares = 0.503501313865 Cumulative probabilities = A: 0.000000 B: 0.458159 C: 0.458159 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -10 -2 B -2 0 2 0 -6 C 6 -2 0 -6 10 D 10 0 6 0 10 E 2 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=27 E=18 B=14 A=8 so A is eliminated. Round 2 votes counts: D=34 C=27 E=20 B=19 so B is eliminated. Round 3 votes counts: C=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 C:204 B:197 E:194 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -10 -2 B -2 0 2 0 -6 C 6 -2 0 -6 10 D 10 0 6 0 10 E 2 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -10 -2 B -2 0 2 0 -6 C 6 -2 0 -6 10 D 10 0 6 0 10 E 2 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -10 -2 B -2 0 2 0 -6 C 6 -2 0 -6 10 D 10 0 6 0 10 E 2 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2560: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) A D E C B (7) B C E D A (6) D B E A C (4) C B A E D (4) E C B D A (3) D E B A C (3) D A E C B (3) C E B A D (3) C E A B D (3) C B E A D (3) B E C D A (3) A C E D B (3) E D C B A (2) E D A C B (2) E C D A B (2) D A B E C (2) C B E D A (2) B D C E A (2) B D A E C (2) B C D E A (2) B C A D E (2) A E C D B (2) A D B C E (2) A C D E B (2) A C B D E (2) E D B A C (1) E C A D B (1) E C A B D (1) E A C D B (1) D E A B C (1) D B A E C (1) C B A D E (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A C E (1) B C E A D (1) B C A E D (1) B A D C E (1) A E D C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 12 -8 2 B 2 0 -2 -6 -2 C -12 2 0 0 -12 D 8 6 0 0 6 E -2 2 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.214322 D: 0.785678 E: 0.000000 Sum of squares = 0.663223720033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.214322 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 -8 2 B 2 0 -2 -6 -2 C -12 2 0 0 -12 D 8 6 0 0 6 E -2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.55555556424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 D=23 C=16 E=13 so E is eliminated. Round 2 votes counts: D=28 A=25 B=24 C=23 so C is eliminated. Round 3 votes counts: B=40 D=30 A=30 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:210 E:203 A:202 B:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 -8 2 B 2 0 -2 -6 -2 C -12 2 0 0 -12 D 8 6 0 0 6 E -2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.55555556424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -8 2 B 2 0 -2 -6 -2 C -12 2 0 0 -12 D 8 6 0 0 6 E -2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.55555556424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -8 2 B 2 0 -2 -6 -2 C -12 2 0 0 -12 D 8 6 0 0 6 E -2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.55555556424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2561: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) B A E C D (7) E B A C D (6) D C A B E (6) B D A C E (5) D B C A E (4) B E A C D (4) E D C A B (3) E A C B D (3) D C E A B (3) C D A E B (3) C A D E B (3) A C E D B (3) A C E B D (3) A C B D E (3) E C D A B (2) D E C B A (2) D E C A B (2) D C A E B (2) D B A C E (2) B E D C A (2) B A C D E (2) A C D B E (2) A C B E D (2) A B C E D (2) E D C B A (1) E D B C A (1) E C A B D (1) E B D A C (1) D E B C A (1) D C E B A (1) D A B C E (1) B E D A C (1) B E A D C (1) B D C A E (1) B D A E C (1) B A D E C (1) B A C E D (1) A E C B D (1) A E B C D (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 10 16 B -10 0 -8 0 0 C -10 8 0 14 0 D -10 0 -14 0 -8 E -16 0 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 10 16 B -10 0 -8 0 0 C -10 8 0 14 0 D -10 0 -14 0 -8 E -16 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 D=24 A=19 C=6 so C is eliminated. Round 2 votes counts: D=27 B=26 E=25 A=22 so A is eliminated. Round 3 votes counts: B=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:223 C:206 E:196 B:191 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 10 16 B -10 0 -8 0 0 C -10 8 0 14 0 D -10 0 -14 0 -8 E -16 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 10 16 B -10 0 -8 0 0 C -10 8 0 14 0 D -10 0 -14 0 -8 E -16 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 10 16 B -10 0 -8 0 0 C -10 8 0 14 0 D -10 0 -14 0 -8 E -16 0 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2562: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (13) A D E C B (9) E A D C B (7) D A E C B (7) D A E B C (7) C B E A D (5) C B A E D (5) E B C A D (4) B D C A E (4) B C E A D (4) D E A B C (3) C B D A E (3) C B A D E (3) E D A B C (2) D B A E C (2) B C E D A (2) E C A D B (1) E B D A C (1) E A D B C (1) E A C D B (1) D B E A C (1) D A C E B (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E D B (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D C A (1) B E D A C (1) B D C E A (1) B C D E A (1) B C A E D (1) A E D C B (1) Total count = 100 A B C D E A 0 -6 -8 -4 24 B 6 0 0 2 2 C 8 0 0 -2 0 D 4 -2 2 0 18 E -24 -2 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.817269 C: 0.182731 D: 0.000000 E: 0.000000 Sum of squares = 0.701318936684 Cumulative probabilities = A: 0.000000 B: 0.817269 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -4 24 B 6 0 0 2 2 C 8 0 0 -2 0 D 4 -2 2 0 18 E -24 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500799 C: 0.499201 D: 0.000000 E: 0.000000 Sum of squares = 0.500001276397 Cumulative probabilities = A: 0.000000 B: 0.500799 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 C=22 E=17 A=10 so A is eliminated. Round 2 votes counts: D=32 B=28 C=22 E=18 so E is eliminated. Round 3 votes counts: D=43 B=33 C=24 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:205 A:203 C:203 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -4 24 B 6 0 0 2 2 C 8 0 0 -2 0 D 4 -2 2 0 18 E -24 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500799 C: 0.499201 D: 0.000000 E: 0.000000 Sum of squares = 0.500001276397 Cumulative probabilities = A: 0.000000 B: 0.500799 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -4 24 B 6 0 0 2 2 C 8 0 0 -2 0 D 4 -2 2 0 18 E -24 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500799 C: 0.499201 D: 0.000000 E: 0.000000 Sum of squares = 0.500001276397 Cumulative probabilities = A: 0.000000 B: 0.500799 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -4 24 B 6 0 0 2 2 C 8 0 0 -2 0 D 4 -2 2 0 18 E -24 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500799 C: 0.499201 D: 0.000000 E: 0.000000 Sum of squares = 0.500001276397 Cumulative probabilities = A: 0.000000 B: 0.500799 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2563: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) B E D A C (10) D E B C A (9) C D E B A (8) B A E D C (6) A C D E B (6) D E C B A (5) C D E A B (5) C A D E B (5) A C B E D (5) A B C E D (4) D C E B A (3) C D A E B (3) B E A D C (3) A C B D E (3) E B D C A (2) C B E D A (2) C A B D E (2) E D B A C (1) D E B A C (1) D C E A B (1) B E C D A (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 8 -2 -2 B 2 0 2 4 2 C -8 -2 0 -8 -4 D 2 -4 8 0 4 E 2 -2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -2 -2 B 2 0 2 4 2 C -8 -2 0 -8 -4 D 2 -4 8 0 4 E 2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998194 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=25 B=20 D=19 E=3 so E is eliminated. Round 2 votes counts: A=33 C=25 B=22 D=20 so D is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:205 D:205 A:201 E:200 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 -2 -2 B 2 0 2 4 2 C -8 -2 0 -8 -4 D 2 -4 8 0 4 E 2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998194 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -2 -2 B 2 0 2 4 2 C -8 -2 0 -8 -4 D 2 -4 8 0 4 E 2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998194 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -2 -2 B 2 0 2 4 2 C -8 -2 0 -8 -4 D 2 -4 8 0 4 E 2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998194 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2564: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) D B A E C (8) E C A B D (7) D E B A C (7) E C D B A (6) C E A B D (6) B D A C E (6) E C A D B (5) D B A C E (5) A B C D E (4) C A E B D (3) B A D C E (3) A D B E C (3) A C B D E (3) A B D C E (3) E D C B A (2) E D A B C (2) E C D A B (2) D B E C A (2) D A B E C (2) A C E B D (2) E D C A B (1) E D B C A (1) D E B C A (1) D E A B C (1) D B C E A (1) C E B A D (1) C B D A E (1) C A B D E (1) B C D E A (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 18 -16 -10 B 8 0 18 -16 6 C -18 -18 0 -14 -18 D 16 16 14 0 22 E 10 -6 18 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 18 -16 -10 B 8 0 18 -16 6 C -18 -18 0 -14 -18 D 16 16 14 0 22 E 10 -6 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=26 A=16 C=12 B=10 so B is eliminated. Round 2 votes counts: D=42 E=26 A=19 C=13 so C is eliminated. Round 3 votes counts: D=44 E=33 A=23 so A is eliminated. Round 4 votes counts: D=61 E=39 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:234 B:208 E:200 A:192 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 18 -16 -10 B 8 0 18 -16 6 C -18 -18 0 -14 -18 D 16 16 14 0 22 E 10 -6 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 18 -16 -10 B 8 0 18 -16 6 C -18 -18 0 -14 -18 D 16 16 14 0 22 E 10 -6 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 18 -16 -10 B 8 0 18 -16 6 C -18 -18 0 -14 -18 D 16 16 14 0 22 E 10 -6 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2565: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) A B D E C (7) C E D B A (6) B E C D A (6) C E B D A (4) C A D E B (4) A D C E B (4) E D C B A (3) E B D C A (3) C D E A B (3) C A E D B (3) B E D C A (3) A D E B C (3) A C D E B (3) E C D B A (2) D A E B C (2) D A B E C (2) C B E D A (2) C A B E D (2) B E D A C (2) B C A E D (2) B A E D C (2) B A C E D (2) A C D B E (2) E D B C A (1) E B C D A (1) D E C A B (1) D E B A C (1) D E A B C (1) D B E A C (1) D A E C B (1) C E B A D (1) C E A D B (1) C D A E B (1) C B E A D (1) C A E B D (1) B E A D C (1) B D A E C (1) B C E D A (1) B A E C D (1) A D E C B (1) A D C B E (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 4 10 B -6 0 2 -8 -2 C 4 -2 0 2 -6 D -4 8 -2 0 -4 E -10 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.379999999996 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 6 -4 4 10 B -6 0 2 -8 -2 C 4 -2 0 2 -6 D -4 8 -2 0 -4 E -10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.379999999862 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=29 B=21 E=10 D=9 so D is eliminated. Round 2 votes counts: A=36 C=29 B=22 E=13 so E is eliminated. Round 3 votes counts: A=37 C=35 B=28 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:208 E:201 C:199 D:199 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -4 4 10 B -6 0 2 -8 -2 C 4 -2 0 2 -6 D -4 8 -2 0 -4 E -10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.379999999862 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 4 10 B -6 0 2 -8 -2 C 4 -2 0 2 -6 D -4 8 -2 0 -4 E -10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.379999999862 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 4 10 B -6 0 2 -8 -2 C 4 -2 0 2 -6 D -4 8 -2 0 -4 E -10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.379999999862 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2566: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) E D B A C (8) B C A E D (8) C B A D E (7) D A E C B (6) E D A B C (5) C B A E D (5) B C E A D (5) B C A D E (5) A D E C B (5) E B D C A (4) B E D C A (3) B E C D A (3) A C D B E (3) D E A C B (2) A D C E B (2) A C B D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E C A D B (1) E B D A C (1) E B C D A (1) D A E B C (1) C A E D B (1) C A D B E (1) C A B E D (1) C A B D E (1) B C E D A (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 0 0 2 B 6 0 -8 -6 -8 C 0 8 0 -6 -10 D 0 6 6 0 -26 E -2 8 10 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999978 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 0 0 2 B 6 0 -8 -6 -8 C 0 8 0 -6 -10 D 0 6 6 0 -26 E -2 8 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999657 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=25 C=16 A=15 D=9 so D is eliminated. Round 2 votes counts: E=37 B=25 A=22 C=16 so C is eliminated. Round 3 votes counts: E=37 B=37 A=26 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:198 C:196 D:193 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 0 2 B 6 0 -8 -6 -8 C 0 8 0 -6 -10 D 0 6 6 0 -26 E -2 8 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999657 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 2 B 6 0 -8 -6 -8 C 0 8 0 -6 -10 D 0 6 6 0 -26 E -2 8 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999657 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 2 B 6 0 -8 -6 -8 C 0 8 0 -6 -10 D 0 6 6 0 -26 E -2 8 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999657 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2567: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D E A B C (7) D A E C B (7) E B C D A (6) E D A B C (5) E B C A D (5) C B A E D (5) B C E A D (5) A D C E B (5) D A C E B (4) D A C B E (4) C B A D E (4) C B E A D (3) C A B D E (3) B E C D A (3) A C D B E (3) E D B A C (2) E B A D C (2) E B A C D (2) A D E C B (2) A D C B E (2) E B D A C (1) E A D B C (1) D E C B A (1) D C B A E (1) D A E B C (1) C D B E A (1) C A D B E (1) B C E D A (1) B C A E D (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 12 -8 B 4 0 4 6 -2 C -2 -4 0 10 -10 D -12 -6 -10 0 -6 E 8 2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 12 -8 B 4 0 4 6 -2 C -2 -4 0 10 -10 D -12 -6 -10 0 -6 E 8 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=24 B=19 C=17 A=15 so A is eliminated. Round 2 votes counts: D=34 E=24 C=22 B=20 so B is eliminated. Round 3 votes counts: E=37 D=34 C=29 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 B:206 A:201 C:197 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 12 -8 B 4 0 4 6 -2 C -2 -4 0 10 -10 D -12 -6 -10 0 -6 E 8 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 12 -8 B 4 0 4 6 -2 C -2 -4 0 10 -10 D -12 -6 -10 0 -6 E 8 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 12 -8 B 4 0 4 6 -2 C -2 -4 0 10 -10 D -12 -6 -10 0 -6 E 8 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2568: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) E B A D C (6) D A B C E (6) C D B A E (6) E C B A D (5) D A B E C (5) C D A B E (5) A E D B C (5) C E B D A (4) B D A E C (4) A D E B C (4) A D B E C (4) D B A E C (3) D B A C E (3) D A C B E (3) C E B A D (3) C B D E A (3) E B C D A (2) C B E D A (2) B E D A C (2) B E C D A (2) B D A C E (2) B A E D C (2) E C B D A (1) E B C A D (1) D C B A E (1) D C A B E (1) C E D B A (1) C D B E A (1) C D A E B (1) C B D A E (1) B D C A E (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 18 -20 18 B 12 0 20 -2 14 C -18 -20 0 -22 -8 D 20 2 22 0 12 E -18 -14 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 18 -20 18 B 12 0 20 -2 14 C -18 -20 0 -22 -8 D 20 2 22 0 12 E -18 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999978013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=23 D=22 A=15 B=13 so B is eliminated. Round 2 votes counts: D=29 E=27 C=27 A=17 so A is eliminated. Round 3 votes counts: D=38 E=34 C=28 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:222 A:202 E:182 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 18 -20 18 B 12 0 20 -2 14 C -18 -20 0 -22 -8 D 20 2 22 0 12 E -18 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999978013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 18 -20 18 B 12 0 20 -2 14 C -18 -20 0 -22 -8 D 20 2 22 0 12 E -18 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999978013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 18 -20 18 B 12 0 20 -2 14 C -18 -20 0 -22 -8 D 20 2 22 0 12 E -18 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999978013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2569: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) D A B E C (8) E C D A B (7) D E C A B (7) D A E C B (7) B C E A D (7) C E D B A (5) B A C E D (5) E C A D B (4) C E B D A (4) A D B E C (4) B C E D A (3) B A D C E (3) A D E C B (3) D B C E A (2) C E A B D (2) B D C E A (2) A D E B C (2) A B D E C (2) A B D C E (2) E C D B A (1) E C A B D (1) E A C D B (1) D E C B A (1) D C E B A (1) D B A E C (1) D B A C E (1) C E D A B (1) C B E A D (1) C A E B D (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -18 -2 -18 B -8 0 -14 -12 -16 C 18 14 0 8 0 D 2 12 -8 0 -8 E 18 16 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.518457 D: 0.000000 E: 0.481543 Sum of squares = 0.500681327642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.518457 D: 0.518457 E: 1.000000 A B C D E A 0 8 -18 -2 -18 B -8 0 -14 -12 -16 C 18 14 0 8 0 D 2 12 -8 0 -8 E 18 16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 B=20 A=15 E=14 so E is eliminated. Round 2 votes counts: C=36 D=28 B=20 A=16 so A is eliminated. Round 3 votes counts: C=38 D=37 B=25 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:220 D:199 A:185 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 -2 -18 B -8 0 -14 -12 -16 C 18 14 0 8 0 D 2 12 -8 0 -8 E 18 16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -2 -18 B -8 0 -14 -12 -16 C 18 14 0 8 0 D 2 12 -8 0 -8 E 18 16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -2 -18 B -8 0 -14 -12 -16 C 18 14 0 8 0 D 2 12 -8 0 -8 E 18 16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2570: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) B E D C A (8) B E A C D (8) A C D E B (8) D C A E B (7) D C A B E (7) E D A C B (6) B C A D E (6) D A C E B (5) E B A C D (4) B C A E D (4) E B D A C (3) C A B D E (3) D E B C A (2) B E D A C (2) E D B A C (1) E B D C A (1) E B A D C (1) E A C D B (1) E A C B D (1) D E B A C (1) D E A C B (1) D B E C A (1) D B C A E (1) D A E C B (1) C A D E B (1) B E C A D (1) B D C E A (1) B A C E D (1) A C E D B (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -4 4 14 B -8 0 -6 -8 10 C 4 6 0 2 14 D -4 8 -2 0 12 E -14 -10 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 4 14 B -8 0 -6 -8 10 C 4 6 0 2 14 D -4 8 -2 0 12 E -14 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=26 E=18 C=13 A=12 so A is eliminated. Round 2 votes counts: B=32 D=26 C=24 E=18 so E is eliminated. Round 3 votes counts: B=41 D=33 C=26 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:213 A:211 D:207 B:194 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 4 14 B -8 0 -6 -8 10 C 4 6 0 2 14 D -4 8 -2 0 12 E -14 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 4 14 B -8 0 -6 -8 10 C 4 6 0 2 14 D -4 8 -2 0 12 E -14 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 4 14 B -8 0 -6 -8 10 C 4 6 0 2 14 D -4 8 -2 0 12 E -14 -10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2571: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) B C D A E (6) D B A E C (5) A C E D B (5) D A B E C (4) B D A E C (4) B C E D A (4) A D E B C (4) D E A B C (3) D B E A C (3) C E B D A (3) C E B A D (3) C E A D B (3) C A E D B (3) B D A C E (3) A D B E C (3) A C D B E (3) E C D B A (2) E C A D B (2) E B C D A (2) D E B A C (2) D A E B C (2) C E A B D (2) B E C D A (2) B D E C A (2) A D E C B (2) A D C B E (2) A D B C E (2) E D B C A (1) E D A C B (1) E D A B C (1) E B D C A (1) C B E A D (1) C B A D E (1) B E D C A (1) B D C E A (1) B D C A E (1) B A D C E (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 18 -22 6 B 12 0 30 -10 12 C -18 -30 0 -14 -8 D 22 10 14 0 26 E -6 -12 8 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 18 -22 6 B 12 0 30 -10 12 C -18 -30 0 -14 -8 D 22 10 14 0 26 E -6 -12 8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 D=19 C=16 E=10 so E is eliminated. Round 2 votes counts: B=35 A=23 D=22 C=20 so C is eliminated. Round 3 votes counts: B=43 A=33 D=24 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:236 B:222 A:195 E:182 C:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 18 -22 6 B 12 0 30 -10 12 C -18 -30 0 -14 -8 D 22 10 14 0 26 E -6 -12 8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 18 -22 6 B 12 0 30 -10 12 C -18 -30 0 -14 -8 D 22 10 14 0 26 E -6 -12 8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 18 -22 6 B 12 0 30 -10 12 C -18 -30 0 -14 -8 D 22 10 14 0 26 E -6 -12 8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2572: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (13) B C A D E (8) E D A C B (7) E D A B C (7) E B D A C (7) B C D A E (7) C A D E B (6) B E C D A (6) E B C A D (5) D A E C B (5) D A C E B (5) B E C A D (4) B C E A D (4) C D A B E (3) B E D A C (3) C B A D E (2) E A D C B (1) D A B E C (1) C A E D B (1) B E D C A (1) B D A E C (1) B D A C E (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 0 -22 -10 8 B 0 0 12 0 10 C 22 -12 0 20 4 D 10 0 -20 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.727863 C: 0.000000 D: 0.272137 E: 0.000000 Sum of squares = 0.603843077497 Cumulative probabilities = A: 0.000000 B: 0.727863 C: 0.727863 D: 1.000000 E: 1.000000 A B C D E A 0 0 -22 -10 8 B 0 0 12 0 10 C 22 -12 0 20 4 D 10 0 -20 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250177626 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=27 C=25 D=11 A=1 so A is eliminated. Round 2 votes counts: B=36 E=27 C=25 D=12 so D is eliminated. Round 3 votes counts: B=37 E=32 C=31 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:217 B:211 D:198 A:188 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -22 -10 8 B 0 0 12 0 10 C 22 -12 0 20 4 D 10 0 -20 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250177626 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -22 -10 8 B 0 0 12 0 10 C 22 -12 0 20 4 D 10 0 -20 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250177626 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -22 -10 8 B 0 0 12 0 10 C 22 -12 0 20 4 D 10 0 -20 0 6 E -8 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250177626 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2573: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) C D A E B (8) B E A D C (7) C A D E B (6) D A E C B (5) C B D A E (5) B D E A C (5) D E A B C (4) B E C A D (4) E D A B C (3) D C A E B (3) C D B A E (3) C A E D B (3) B E D A C (3) B C E A D (3) B C D E A (3) E B A D C (2) D B E A C (2) C B E A D (2) C A D B E (2) A E D C B (2) A E C D B (2) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) D C B A E (1) D B A C E (1) C D A B E (1) C B A E D (1) C B A D E (1) C A E B D (1) B E C D A (1) B E A C D (1) B D C E A (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 0 -6 B -6 0 0 -16 -6 C -2 0 0 0 -6 D 0 16 0 0 6 E 6 6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.328522 B: 0.000000 C: 0.000000 D: 0.671478 E: 0.000000 Sum of squares = 0.558809606758 Cumulative probabilities = A: 0.328522 B: 0.328522 C: 0.328522 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 0 -6 B -6 0 0 -16 -6 C -2 0 0 0 -6 D 0 16 0 0 6 E 6 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499850 B: 0.000000 C: 0.000000 D: 0.500150 E: 0.000000 Sum of squares = 0.500000044912 Cumulative probabilities = A: 0.499850 B: 0.499850 C: 0.499850 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=28 D=17 E=16 A=6 so A is eliminated. Round 2 votes counts: C=35 B=28 E=20 D=17 so D is eliminated. Round 3 votes counts: C=39 B=31 E=30 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:211 E:206 A:201 C:196 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 0 -6 B -6 0 0 -16 -6 C -2 0 0 0 -6 D 0 16 0 0 6 E 6 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499850 B: 0.000000 C: 0.000000 D: 0.500150 E: 0.000000 Sum of squares = 0.500000044912 Cumulative probabilities = A: 0.499850 B: 0.499850 C: 0.499850 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 -6 B -6 0 0 -16 -6 C -2 0 0 0 -6 D 0 16 0 0 6 E 6 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499850 B: 0.000000 C: 0.000000 D: 0.500150 E: 0.000000 Sum of squares = 0.500000044912 Cumulative probabilities = A: 0.499850 B: 0.499850 C: 0.499850 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 -6 B -6 0 0 -16 -6 C -2 0 0 0 -6 D 0 16 0 0 6 E 6 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499850 B: 0.000000 C: 0.000000 D: 0.500150 E: 0.000000 Sum of squares = 0.500000044912 Cumulative probabilities = A: 0.499850 B: 0.499850 C: 0.499850 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2574: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) A C D B E (7) E B A D C (6) A E C B D (6) E B D C A (5) D C B E A (5) D B E C A (5) C D B E A (5) A C E B D (5) A C D E B (5) C D E B A (4) B D E A C (4) B E D A C (3) A E B D C (3) E B D A C (2) E B C D A (2) E A B D C (2) E A B C D (2) D C B A E (2) B E D C A (2) A C E D B (2) A B D E C (2) D B E A C (1) D B C E A (1) C E D B A (1) C E B D A (1) C E B A D (1) C E A B D (1) C D B A E (1) C A E B D (1) B D E C A (1) B D A E C (1) B A E D C (1) A E B C D (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 10 8 -8 B 8 0 -10 4 0 C -10 10 0 4 0 D -8 -4 -4 0 6 E 8 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.150000 B: 0.070000 C: 0.200000 D: 0.200000 E: 0.380000 Sum of squares = 0.251800000001 Cumulative probabilities = A: 0.150000 B: 0.220000 C: 0.420000 D: 0.620000 E: 1.000000 A B C D E A 0 -8 10 8 -8 B 8 0 -10 4 0 C -10 10 0 4 0 D -8 -4 -4 0 6 E 8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.150000 B: 0.070000 C: 0.200000 D: 0.200000 E: 0.380000 Sum of squares = 0.2518 Cumulative probabilities = A: 0.150000 B: 0.220000 C: 0.420000 D: 0.620000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=22 E=19 D=14 B=12 so B is eliminated. Round 2 votes counts: A=34 E=24 C=22 D=20 so D is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:202 A:201 B:201 E:201 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 10 8 -8 B 8 0 -10 4 0 C -10 10 0 4 0 D -8 -4 -4 0 6 E 8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.150000 B: 0.070000 C: 0.200000 D: 0.200000 E: 0.380000 Sum of squares = 0.2518 Cumulative probabilities = A: 0.150000 B: 0.220000 C: 0.420000 D: 0.620000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 8 -8 B 8 0 -10 4 0 C -10 10 0 4 0 D -8 -4 -4 0 6 E 8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.150000 B: 0.070000 C: 0.200000 D: 0.200000 E: 0.380000 Sum of squares = 0.2518 Cumulative probabilities = A: 0.150000 B: 0.220000 C: 0.420000 D: 0.620000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 8 -8 B 8 0 -10 4 0 C -10 10 0 4 0 D -8 -4 -4 0 6 E 8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.150000 B: 0.070000 C: 0.200000 D: 0.200000 E: 0.380000 Sum of squares = 0.2518 Cumulative probabilities = A: 0.150000 B: 0.220000 C: 0.420000 D: 0.620000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2575: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) E A B C D (7) D E B C A (6) D B C E A (6) C B A D E (6) E A B D C (5) A E C B D (5) E B D A C (4) E D B A C (3) D E C A B (3) B C A D E (3) A C E D B (3) A C E B D (3) E D A C B (2) E B D C A (2) E B A D C (2) E A D C B (2) E A D B C (2) D E C B A (2) D B E C A (2) C D B A E (2) A E C D B (2) A E B C D (2) E D B C A (1) E A C D B (1) E A C B D (1) D C B E A (1) D C A E B (1) D B C A E (1) C B D A E (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C E A (1) B C A E D (1) B A C E D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -4 2 -8 B 12 0 2 -2 -18 C 4 -2 0 -14 -10 D -2 2 14 0 -2 E 8 18 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -4 2 -8 B 12 0 2 -2 -18 C 4 -2 0 -14 -10 D -2 2 14 0 -2 E 8 18 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=32 D=32 A=18 C=11 B=7 so B is eliminated. Round 2 votes counts: D=34 E=32 A=19 C=15 so C is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:206 B:197 A:189 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -4 2 -8 B 12 0 2 -2 -18 C 4 -2 0 -14 -10 D -2 2 14 0 -2 E 8 18 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 2 -8 B 12 0 2 -2 -18 C 4 -2 0 -14 -10 D -2 2 14 0 -2 E 8 18 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 2 -8 B 12 0 2 -2 -18 C 4 -2 0 -14 -10 D -2 2 14 0 -2 E 8 18 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2576: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) B E C D A (9) B E D A C (7) A D C E B (7) C D A E B (5) C A D B E (5) E B D C A (4) E B D A C (4) D A C E B (4) B E D C A (4) D E B A C (3) D E A B C (3) D A E B C (3) C B E A D (3) A B E D C (3) C D E B A (2) C B A E D (2) C A D E B (2) C A B E D (2) B E C A D (2) B E A D C (2) B E A C D (2) A D E B C (2) A C D E B (2) E B A D C (1) D A E C B (1) C D B E A (1) C D A B E (1) C A B D E (1) A E B D C (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -6 -22 -14 B 12 0 0 16 12 C 6 0 0 2 -2 D 22 -16 -2 0 -14 E 14 -12 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.455794 C: 0.544206 D: 0.000000 E: 0.000000 Sum of squares = 0.503908344839 Cumulative probabilities = A: 0.000000 B: 0.455794 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -22 -14 B 12 0 0 16 12 C 6 0 0 2 -2 D 22 -16 -2 0 -14 E 14 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=26 A=17 D=14 E=9 so E is eliminated. Round 2 votes counts: B=35 C=34 A=17 D=14 so D is eliminated. Round 3 votes counts: B=38 C=34 A=28 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:209 C:203 D:195 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -22 -14 B 12 0 0 16 12 C 6 0 0 2 -2 D 22 -16 -2 0 -14 E 14 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -22 -14 B 12 0 0 16 12 C 6 0 0 2 -2 D 22 -16 -2 0 -14 E 14 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -22 -14 B 12 0 0 16 12 C 6 0 0 2 -2 D 22 -16 -2 0 -14 E 14 -12 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2577: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A B C D E (8) B D E C A (7) A C E D B (7) E D C B A (4) C E D A B (4) A E D C B (4) C E D B A (3) C A E D B (3) B C D E A (3) B A D E C (3) A C E B D (3) A C B E D (3) A C B D E (3) A B C E D (3) E D A C B (2) D B E C A (2) B D E A C (2) B C D A E (2) A E C D B (2) A B E D C (2) A B D E C (2) E D C A B (1) E C D B A (1) E C D A B (1) E A D C B (1) D E C B A (1) C E B D A (1) C E A D B (1) C D E B A (1) C B D E A (1) C B A D E (1) C A E B D (1) C A B E D (1) B D C E A (1) B D C A E (1) B C A D E (1) B A D C E (1) A E D B C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -2 6 8 B -10 0 0 2 -4 C 2 0 0 8 8 D -6 -2 -8 0 2 E -8 4 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.074846 C: 0.925154 D: 0.000000 E: 0.000000 Sum of squares = 0.861512175938 Cumulative probabilities = A: 0.000000 B: 0.074846 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 6 8 B -10 0 0 2 -4 C 2 0 0 8 8 D -6 -2 -8 0 2 E -8 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222251577 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=21 C=17 D=12 E=10 so E is eliminated. Round 2 votes counts: A=41 B=21 D=19 C=19 so D is eliminated. Round 3 votes counts: A=43 B=32 C=25 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:209 B:194 D:193 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 6 8 B -10 0 0 2 -4 C 2 0 0 8 8 D -6 -2 -8 0 2 E -8 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222251577 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 6 8 B -10 0 0 2 -4 C 2 0 0 8 8 D -6 -2 -8 0 2 E -8 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222251577 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 6 8 B -10 0 0 2 -4 C 2 0 0 8 8 D -6 -2 -8 0 2 E -8 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222251577 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2578: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) A E D C B (7) B E C A D (6) D C B A E (5) B C D E A (5) A E C D B (5) B D C E A (4) A D E C B (4) E B A C D (3) E A C B D (3) D C A B E (3) C D A E B (3) B E A C D (3) B C E D A (3) E C B A D (2) D C B E A (2) D B C E A (2) D A B E C (2) C E B A D (2) C B D E A (2) C A D E B (2) B C E A D (2) A E D B C (2) A D C E B (2) E B C A D (1) E A B D C (1) E A B C D (1) D C A E B (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C B E (1) D A B C E (1) C E A D B (1) C D B E A (1) C D B A E (1) C B E A D (1) B E D C A (1) B D A E C (1) B A D E C (1) A E B D C (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 -14 4 8 B 14 0 2 -12 12 C 14 -2 0 -4 4 D -4 12 4 0 8 E -8 -12 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074380178 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 4 8 B 14 0 2 -12 12 C 14 -2 0 -4 4 D -4 12 4 0 8 E -8 -12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074380326 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 A=23 C=13 E=11 so E is eliminated. Round 2 votes counts: B=30 A=28 D=27 C=15 so C is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:208 C:206 A:192 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 4 8 B 14 0 2 -12 12 C 14 -2 0 -4 4 D -4 12 4 0 8 E -8 -12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074380326 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 4 8 B 14 0 2 -12 12 C 14 -2 0 -4 4 D -4 12 4 0 8 E -8 -12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074380326 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 4 8 B 14 0 2 -12 12 C 14 -2 0 -4 4 D -4 12 4 0 8 E -8 -12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074380326 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2579: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B A E D C (8) D C A E B (7) B A D E C (7) A B D C E (7) D A B C E (6) E C D B A (4) E B C D A (4) E B C A D (4) B E A C D (4) E C D A B (3) E C B D A (3) C D A E B (3) B E A D C (3) B A E C D (3) B A D C E (3) A D B C E (3) E D C B A (2) E C B A D (2) E B D C A (2) C E D A B (2) C E A D B (2) C E A B D (2) C D E A B (2) A B C D E (2) E D C A B (1) D A C B E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -2 4 4 B -2 0 12 8 -2 C 2 -12 0 -18 0 D -4 -8 18 0 2 E -4 2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593750000267 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 4 4 B -2 0 12 8 -2 C 2 -12 0 -18 0 D -4 -8 18 0 2 E -4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593750000016 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 D=22 A=14 C=11 so C is eliminated. Round 2 votes counts: E=31 B=28 D=27 A=14 so A is eliminated. Round 3 votes counts: B=37 D=32 E=31 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:204 D:204 E:198 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 4 4 B -2 0 12 8 -2 C 2 -12 0 -18 0 D -4 -8 18 0 2 E -4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593750000016 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 4 4 B -2 0 12 8 -2 C 2 -12 0 -18 0 D -4 -8 18 0 2 E -4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593750000016 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 4 4 B -2 0 12 8 -2 C 2 -12 0 -18 0 D -4 -8 18 0 2 E -4 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593750000016 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2580: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) A B E D C (8) C B E D A (6) C D B E A (5) D C E A B (4) B A E C D (4) A D E B C (4) E B C A D (3) D E A C B (3) D A E C B (3) C B D E A (3) B E C A D (3) B C E A D (3) A B E C D (3) E B C D A (2) E B A D C (2) E A B D C (2) D E C B A (2) D E C A B (2) D A C E B (2) C D B A E (2) C B D A E (2) A D B C E (2) A B D E C (2) E D A B C (1) E B A C D (1) E A D B C (1) D C A B E (1) D A C B E (1) C E B D A (1) C D E B A (1) C D A B E (1) C B A D E (1) C A B D E (1) B E C D A (1) B A C E D (1) A E B D C (1) A D E C B (1) A D C B E (1) A D B E C (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 12 14 -10 B 4 0 12 22 28 C -12 -12 0 10 -18 D -14 -22 -10 0 -6 E 10 -28 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 14 -10 B 4 0 12 22 28 C -12 -12 0 10 -18 D -14 -22 -10 0 -6 E 10 -28 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999726 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=23 B=21 D=18 E=12 so E is eliminated. Round 2 votes counts: B=29 A=29 C=23 D=19 so D is eliminated. Round 3 votes counts: A=39 C=32 B=29 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:233 A:206 E:203 C:184 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 14 -10 B 4 0 12 22 28 C -12 -12 0 10 -18 D -14 -22 -10 0 -6 E 10 -28 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999726 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 14 -10 B 4 0 12 22 28 C -12 -12 0 10 -18 D -14 -22 -10 0 -6 E 10 -28 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999726 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 14 -10 B 4 0 12 22 28 C -12 -12 0 10 -18 D -14 -22 -10 0 -6 E 10 -28 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999726 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2581: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) E A C D B (6) D C E A B (5) A E B C D (5) D B E C A (4) C B A D E (4) C A E B D (4) B D C A E (4) B A E D C (4) E D B A C (3) E A B D C (3) D C B E A (3) D B C E A (3) C E A D B (3) C B D A E (3) B D E A C (3) D E B A C (2) D C B A E (2) D B E A C (2) C D B A E (2) B E D A C (2) B D A C E (2) B C A D E (2) B A E C D (2) A E C B D (2) A C E B D (2) E D C A B (1) E D A C B (1) E C D A B (1) E A D C B (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) C A E D B (1) C A B D E (1) B E A D C (1) B D A E C (1) B C D A E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -10 -12 -12 B 6 0 -6 -2 -4 C 10 6 0 0 4 D 12 2 0 0 10 E 12 4 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.684496 D: 0.315504 E: 0.000000 Sum of squares = 0.568077560378 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.684496 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -12 -12 B 6 0 -6 -2 -4 C 10 6 0 0 4 D 12 2 0 0 10 E 12 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=24 B=22 E=17 A=11 so A is eliminated. Round 2 votes counts: C=29 E=24 D=24 B=23 so B is eliminated. Round 3 votes counts: E=34 D=34 C=32 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:210 E:201 B:197 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -12 -12 B 6 0 -6 -2 -4 C 10 6 0 0 4 D 12 2 0 0 10 E 12 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -12 -12 B 6 0 -6 -2 -4 C 10 6 0 0 4 D 12 2 0 0 10 E 12 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -12 -12 B 6 0 -6 -2 -4 C 10 6 0 0 4 D 12 2 0 0 10 E 12 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2582: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) D B A C E (8) D E B C A (7) D E A C B (7) A C E B D (6) E C A B D (5) C E A B D (5) D A E C B (4) C A B E D (4) A C B E D (4) C A E B D (3) B C A E D (3) E D C B A (2) E A C D B (2) D E C A B (2) B D C A E (2) B C E A D (2) B A C D E (2) A E C D B (2) E D C A B (1) E D B C A (1) E D A C B (1) E B D C A (1) E B C D A (1) E B C A D (1) E A D C B (1) D E B A C (1) D E A B C (1) D B E A C (1) D A C E B (1) D A C B E (1) B E C D A (1) B E C A D (1) B D E C A (1) B C A D E (1) A D E C B (1) A C E D B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -8 -8 -10 B -8 0 -8 -10 -14 C 8 8 0 -8 -10 D 8 10 8 0 2 E 10 14 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -8 -10 B -8 0 -8 -10 -14 C 8 8 0 -8 -10 D 8 10 8 0 2 E 10 14 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 E=16 A=16 B=13 C=12 so C is eliminated. Round 2 votes counts: D=43 A=23 E=21 B=13 so B is eliminated. Round 3 votes counts: D=46 A=29 E=25 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:214 C:199 A:191 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -8 -10 B -8 0 -8 -10 -14 C 8 8 0 -8 -10 D 8 10 8 0 2 E 10 14 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -8 -10 B -8 0 -8 -10 -14 C 8 8 0 -8 -10 D 8 10 8 0 2 E 10 14 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -8 -10 B -8 0 -8 -10 -14 C 8 8 0 -8 -10 D 8 10 8 0 2 E 10 14 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2583: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) E B D A C (6) B E C A D (6) D C A E B (5) B E D C A (5) B E A C D (5) B C A E D (5) B E C D A (4) B C A D E (4) E D A C B (3) D E C A B (3) C D A B E (3) C A D B E (3) A C B E D (3) A C B D E (3) E D B C A (2) E B D C A (2) D C E B A (2) D C A B E (2) B C D E A (2) A E B C D (2) A C D E B (2) E D B A C (1) E B A D C (1) E A D B C (1) D E B C A (1) D C E A B (1) D C B E A (1) D B E C A (1) D A E C B (1) C B D A E (1) C A D E B (1) C A B D E (1) B D E C A (1) B C E D A (1) B C D A E (1) B A C E D (1) A E D C B (1) A D C E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -16 -16 -16 B 6 0 6 10 10 C 16 -6 0 0 -10 D 16 -10 0 0 0 E 16 -10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -16 -16 B 6 0 6 10 10 C 16 -6 0 0 -10 D 16 -10 0 0 0 E 16 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=26 E=16 A=14 C=9 so C is eliminated. Round 2 votes counts: B=36 D=29 A=19 E=16 so E is eliminated. Round 3 votes counts: B=45 D=35 A=20 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:208 D:203 C:200 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -16 -16 -16 B 6 0 6 10 10 C 16 -6 0 0 -10 D 16 -10 0 0 0 E 16 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -16 -16 B 6 0 6 10 10 C 16 -6 0 0 -10 D 16 -10 0 0 0 E 16 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -16 -16 B 6 0 6 10 10 C 16 -6 0 0 -10 D 16 -10 0 0 0 E 16 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2584: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) A B D E C (10) C B D A E (6) E C A B D (5) B A C D E (5) D E A B C (4) D A B E C (4) C E D B A (4) E A D B C (3) D E C A B (3) E D A B C (2) E C D A B (2) E C A D B (2) C E D A B (2) C E B A D (2) C D E B A (2) C D B E A (2) C D B A E (2) C B E A D (2) C B A D E (2) B C A D E (2) A E B D C (2) A B E D C (2) A B E C D (2) E D C A B (1) E A B C D (1) D C E B A (1) D C E A B (1) D C B A E (1) D B A C E (1) D A E B C (1) C E B D A (1) C B D E A (1) C B A E D (1) B D A C E (1) B C D A E (1) B A E C D (1) B A D E C (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 8 14 18 B 4 0 14 22 22 C -8 -14 0 -4 6 D -14 -22 4 0 30 E -18 -22 -6 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 14 18 B 4 0 14 22 22 C -8 -14 0 -4 6 D -14 -22 4 0 30 E -18 -22 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 A=17 E=16 D=16 so E is eliminated. Round 2 votes counts: C=36 B=24 A=21 D=19 so D is eliminated. Round 3 votes counts: C=43 A=32 B=25 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:231 A:218 D:199 C:190 E:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 14 18 B 4 0 14 22 22 C -8 -14 0 -4 6 D -14 -22 4 0 30 E -18 -22 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 14 18 B 4 0 14 22 22 C -8 -14 0 -4 6 D -14 -22 4 0 30 E -18 -22 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 14 18 B 4 0 14 22 22 C -8 -14 0 -4 6 D -14 -22 4 0 30 E -18 -22 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2585: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) C E A B D (6) B D A C E (6) E D A B C (5) D A B E C (5) A D E B C (5) C E B A D (4) C E A D B (4) B D C A E (4) A D E C B (4) C B A D E (3) B D A E C (3) E B D C A (2) E B C D A (2) E A D C B (2) E A C D B (2) C A D E B (2) B E D C A (2) B C D E A (2) A D C B E (2) A D B E C (2) A D B C E (2) A C E D B (2) A C D E B (2) E D B A C (1) E C B D A (1) E C B A D (1) E C A B D (1) E A D B C (1) D B A C E (1) D A E B C (1) D A B C E (1) C B E D A (1) C B E A D (1) C B A E D (1) C A B D E (1) B E C D A (1) B D E A C (1) B C E D A (1) B C D A E (1) A E C D B (1) A D C E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 22 4 18 4 B -22 0 0 -8 -14 C -4 0 0 -4 0 D -18 8 4 0 2 E -4 14 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 4 18 4 B -22 0 0 -8 -14 C -4 0 0 -4 0 D -18 8 4 0 2 E -4 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 A=23 B=21 D=8 so D is eliminated. Round 2 votes counts: A=30 E=25 C=23 B=22 so B is eliminated. Round 3 votes counts: A=40 C=31 E=29 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 E:204 D:198 C:196 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 4 18 4 B -22 0 0 -8 -14 C -4 0 0 -4 0 D -18 8 4 0 2 E -4 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 4 18 4 B -22 0 0 -8 -14 C -4 0 0 -4 0 D -18 8 4 0 2 E -4 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 4 18 4 B -22 0 0 -8 -14 C -4 0 0 -4 0 D -18 8 4 0 2 E -4 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2586: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) A B D C E (6) C B E A D (5) E D A C B (4) E D A B C (4) E C B A D (4) E A D B C (4) D E A C B (4) D A E B C (4) C E B D A (4) C E B A D (4) D E A B C (3) C B D A E (3) B C E A D (3) B C A D E (3) B A C E D (3) A D B E C (3) E D C A B (2) E C B D A (2) C E D B A (2) C B D E A (2) C B A E D (2) B A D C E (2) A D B C E (2) A B D E C (2) E C D B A (1) E C D A B (1) E B A C D (1) D E C A B (1) D C E B A (1) D C A B E (1) D A B E C (1) C D E B A (1) C D B E A (1) C D B A E (1) C B E D A (1) C B A D E (1) B E A C D (1) B C A E D (1) B A C D E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 6 0 -12 B 0 0 4 4 4 C -6 -4 0 -4 12 D 0 -4 4 0 0 E 12 -4 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166302 B: 0.833698 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722708172353 Cumulative probabilities = A: 0.166302 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 0 -12 B 0 0 4 4 4 C -6 -4 0 -4 12 D 0 -4 4 0 0 E 12 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000119073 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=23 D=21 A=15 B=14 so B is eliminated. Round 2 votes counts: C=34 E=24 D=21 A=21 so D is eliminated. Round 3 votes counts: C=36 E=32 A=32 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:206 D:200 C:199 E:198 A:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 0 -12 B 0 0 4 4 4 C -6 -4 0 -4 12 D 0 -4 4 0 0 E 12 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000119073 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 -12 B 0 0 4 4 4 C -6 -4 0 -4 12 D 0 -4 4 0 0 E 12 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000119073 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 -12 B 0 0 4 4 4 C -6 -4 0 -4 12 D 0 -4 4 0 0 E 12 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000119073 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2587: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (15) C D B E A (11) E A B C D (8) D C B A E (5) D B C A E (5) C D E B A (4) E A C B D (3) C D E A B (3) A B E D C (3) E B A C D (2) E A C D B (2) E A B D C (2) D A C B E (2) B D E C A (2) B D C E A (2) B D C A E (2) B D A E C (2) B A E D C (2) A E C D B (2) A E B C D (2) E C A D B (1) E C A B D (1) D C B E A (1) D C A E B (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A D B (1) C E A B D (1) C D A E B (1) C B E D A (1) C A E D B (1) B E C D A (1) B E A D C (1) B C D E A (1) A E D B C (1) A E C B D (1) A D E B C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 12 4 4 -2 B -12 0 8 6 -14 C -4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 4 4 -2 B -12 0 8 6 -14 C -4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 E=19 D=15 B=13 so B is eliminated. Round 2 votes counts: A=29 C=27 D=23 E=21 so E is eliminated. Round 3 votes counts: A=47 C=30 D=23 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:209 B:194 C:191 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 4 4 -2 B -12 0 8 6 -14 C -4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 4 -2 B -12 0 8 6 -14 C -4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 4 -2 B -12 0 8 6 -14 C -4 -8 0 2 -8 D -4 -6 -2 0 -8 E 2 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2588: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) C D B E A (8) B C D A E (7) B A E C D (6) A B E D C (6) C B D E A (5) A E D B C (5) A E B D C (5) E D A C B (4) B C A D E (4) E A D B C (3) D C A E B (3) C B D A E (3) B E A C D (3) A D E C B (3) E C D B A (2) E A D C B (2) E A B D C (2) D C A B E (2) D A E C B (2) C D E B A (2) C D B A E (2) B C D E A (2) E D C A B (1) D E A C B (1) D A C E B (1) C D E A B (1) C B E D A (1) B C E D A (1) B C A E D (1) B A C E D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -8 -14 6 B -2 0 -4 -2 8 C 8 4 0 0 8 D 14 2 0 0 14 E -6 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.869643 D: 0.130357 E: 0.000000 Sum of squares = 0.773271363451 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.869643 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -14 6 B -2 0 -4 -2 8 C 8 4 0 0 8 D 14 2 0 0 14 E -6 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=22 A=20 D=18 E=14 so E is eliminated. Round 2 votes counts: A=27 B=26 C=24 D=23 so D is eliminated. Round 3 votes counts: C=39 A=35 B=26 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:215 C:210 B:200 A:193 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -14 6 B -2 0 -4 -2 8 C 8 4 0 0 8 D 14 2 0 0 14 E -6 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -14 6 B -2 0 -4 -2 8 C 8 4 0 0 8 D 14 2 0 0 14 E -6 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -14 6 B -2 0 -4 -2 8 C 8 4 0 0 8 D 14 2 0 0 14 E -6 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2589: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) B A C D E (8) D E A C B (7) E D C A B (6) E C D B A (6) D A E B C (6) E D A B C (5) D E A B C (5) B C A E D (4) A D E B C (4) A B D C E (4) D C E A B (3) D A B C E (3) B C A D E (3) A D B E C (3) E D A C B (2) C E D B A (2) C E B D A (2) C B E D A (2) C B A E D (2) A B D E C (2) E C B A D (1) E B C A D (1) E B A C D (1) E A B D C (1) C E B A D (1) C D B A E (1) C B A D E (1) B A E D C (1) B A C E D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 10 0 -10 B -6 0 8 -8 -6 C -10 -8 0 -8 -4 D 0 8 8 0 2 E 10 6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.119747 B: 0.000000 C: 0.000000 D: 0.880253 E: 0.000000 Sum of squares = 0.789185114953 Cumulative probabilities = A: 0.119747 B: 0.119747 C: 0.119747 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 0 -10 B -6 0 8 -8 -6 C -10 -8 0 -8 -4 D 0 8 8 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222242478 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 C=21 B=17 A=15 so A is eliminated. Round 2 votes counts: D=32 B=24 E=23 C=21 so C is eliminated. Round 3 votes counts: B=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:209 A:203 B:194 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 0 -10 B -6 0 8 -8 -6 C -10 -8 0 -8 -4 D 0 8 8 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222242478 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 0 -10 B -6 0 8 -8 -6 C -10 -8 0 -8 -4 D 0 8 8 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222242478 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 0 -10 B -6 0 8 -8 -6 C -10 -8 0 -8 -4 D 0 8 8 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222242478 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2590: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) E C D B A (7) C A E D B (6) A C E B D (6) A B D C E (6) B A D E C (5) A C B D E (5) D E B C A (4) D B E C A (4) A C B E D (4) A B D E C (4) D C B E A (3) B D E C A (3) A B E D C (3) A B C D E (3) E D C B A (2) E C D A B (2) E C A D B (2) D E C B A (2) C E D A B (2) C E A D B (2) A B E C D (2) A B C E D (2) E A D B C (1) C E D B A (1) C D E B A (1) C B D A E (1) C A D E B (1) C A D B E (1) B D C A E (1) B D A C E (1) B A D C E (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 8 10 12 8 B -8 0 0 16 18 C -10 0 0 0 -6 D -12 -16 0 0 12 E -8 -18 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 12 8 B -8 0 0 16 18 C -10 0 0 0 -6 D -12 -16 0 0 12 E -8 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=21 C=15 E=14 D=13 so D is eliminated. Round 2 votes counts: A=37 B=25 E=20 C=18 so C is eliminated. Round 3 votes counts: A=45 B=29 E=26 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:213 C:192 D:192 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 12 8 B -8 0 0 16 18 C -10 0 0 0 -6 D -12 -16 0 0 12 E -8 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 12 8 B -8 0 0 16 18 C -10 0 0 0 -6 D -12 -16 0 0 12 E -8 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 12 8 B -8 0 0 16 18 C -10 0 0 0 -6 D -12 -16 0 0 12 E -8 -18 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2591: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) D C B A E (7) A E B D C (7) C D E B A (6) C D E A B (6) C D B E A (6) E A C D B (5) E A C B D (4) D B C A E (4) B D C A E (4) D B A E C (3) C E D A B (3) B D A E C (3) B A D E C (3) A E D B C (3) A B E D C (3) E A B D C (2) D C E A B (2) D C B E A (2) C D B A E (2) C B E A D (2) B C A E D (2) B A E D C (2) E D A C B (1) E C A B D (1) E A D C B (1) E A D B C (1) D A E B C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B D E A (1) C B D A E (1) B C D A E (1) B A E C D (1) Total count = 100 A B C D E A 0 -2 -6 -6 -6 B 2 0 -6 -8 -6 C 6 6 0 2 4 D 6 8 -2 0 4 E 6 6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 -6 B 2 0 -6 -8 -6 C 6 6 0 2 4 D 6 8 -2 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=22 D=19 B=16 A=13 so A is eliminated. Round 2 votes counts: E=32 C=30 D=19 B=19 so D is eliminated. Round 3 votes counts: C=41 E=33 B=26 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:208 E:202 B:191 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -6 B 2 0 -6 -8 -6 C 6 6 0 2 4 D 6 8 -2 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -6 B 2 0 -6 -8 -6 C 6 6 0 2 4 D 6 8 -2 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -6 B 2 0 -6 -8 -6 C 6 6 0 2 4 D 6 8 -2 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2592: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D C A B E (7) E B D A C (6) E B A C D (4) E A B C D (4) D B E C A (4) C A D B E (4) A C E B D (4) E A B D C (3) C D B A E (3) B D E C A (3) A E C D B (3) A C D E B (3) E D B A C (2) E B A D C (2) E A C B D (2) D E B C A (2) D C B A E (2) D B C E A (2) D B C A E (2) C D A B E (2) C B D A E (2) C B A D E (2) C A B E D (2) C A B D E (2) B E C A D (2) B D C E A (2) B D C A E (2) A C B E D (2) E D A B C (1) D E B A C (1) D A E C B (1) D A C E B (1) B E D C A (1) B E A C D (1) B C E A D (1) B C A D E (1) A E D C B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 4 8 14 B -6 0 -8 14 0 C -4 8 0 10 -4 D -8 -14 -10 0 -2 E -14 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 8 14 B -6 0 -8 14 0 C -4 8 0 10 -4 D -8 -14 -10 0 -2 E -14 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 D=22 C=17 B=13 so B is eliminated. Round 2 votes counts: D=29 E=28 A=24 C=19 so C is eliminated. Round 3 votes counts: D=36 A=35 E=29 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:205 B:200 E:196 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 8 14 B -6 0 -8 14 0 C -4 8 0 10 -4 D -8 -14 -10 0 -2 E -14 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 8 14 B -6 0 -8 14 0 C -4 8 0 10 -4 D -8 -14 -10 0 -2 E -14 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 8 14 B -6 0 -8 14 0 C -4 8 0 10 -4 D -8 -14 -10 0 -2 E -14 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2593: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (6) D E A B C (5) C B E A D (5) E D C B A (4) E B C D A (4) D E A C B (4) C E B D A (4) A C D B E (4) D C E A B (3) C E D B A (3) C B E D A (3) B E C D A (3) A D E C B (3) A D C B E (3) A D B E C (3) A C B E D (3) E C B D A (2) D E C B A (2) D E B A C (2) C D E B A (2) C D E A B (2) B E D A C (2) B E C A D (2) B A E D C (2) B A E C D (2) A C B D E (2) A B D E C (2) A B D C E (2) A B C D E (2) E C D B A (1) E B D C A (1) E B D A C (1) D E C A B (1) D A E C B (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B D E (1) B C E A D (1) B C A E D (1) B A C E D (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 4 -12 -14 B -2 0 -14 -4 -6 C -4 14 0 4 -8 D 12 4 -4 0 6 E 14 6 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.444444 E: 0.222222 Sum of squares = 0.358024691368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.777778 E: 1.000000 A B C D E A 0 2 4 -12 -14 B -2 0 -14 -4 -6 C -4 14 0 4 -8 D 12 4 -4 0 6 E 14 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.444444 E: 0.222222 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 C=23 B=14 E=13 so E is eliminated. Round 2 votes counts: D=28 C=26 A=26 B=20 so B is eliminated. Round 3 votes counts: C=37 D=32 A=31 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 D:209 C:203 A:190 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -12 -14 B -2 0 -14 -4 -6 C -4 14 0 4 -8 D 12 4 -4 0 6 E 14 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.444444 E: 0.222222 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.777778 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -12 -14 B -2 0 -14 -4 -6 C -4 14 0 4 -8 D 12 4 -4 0 6 E 14 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.444444 E: 0.222222 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.777778 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -12 -14 B -2 0 -14 -4 -6 C -4 14 0 4 -8 D 12 4 -4 0 6 E 14 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.444444 E: 0.222222 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.777778 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2594: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) A E D B C (7) E D A B C (5) C A B E D (5) B C D E A (5) D B E C A (4) C D E B A (4) C A E D B (4) B D C E A (4) D E B A C (3) C E D A B (3) C D B E A (3) B D E C A (3) B C D A E (3) B A D E C (3) B A C D E (3) E A D B C (2) C A E B D (2) B A E D C (2) A E D C B (2) A E B D C (2) A C E B D (2) E D C A B (1) E C D A B (1) E A D C B (1) E A C D B (1) D E B C A (1) D C E B A (1) D B E A C (1) D B C E A (1) C B A D E (1) C A B D E (1) B D E A C (1) B D A E C (1) A E C D B (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -18 -14 -14 B 12 0 8 8 14 C 18 -8 0 8 12 D 14 -8 -8 0 10 E 14 -14 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -14 -14 B 12 0 8 8 14 C 18 -8 0 8 12 D 14 -8 -8 0 10 E 14 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=25 A=18 E=11 D=11 so E is eliminated. Round 2 votes counts: C=36 B=25 A=22 D=17 so D is eliminated. Round 3 votes counts: C=38 B=35 A=27 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:215 D:204 E:189 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -18 -14 -14 B 12 0 8 8 14 C 18 -8 0 8 12 D 14 -8 -8 0 10 E 14 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -14 -14 B 12 0 8 8 14 C 18 -8 0 8 12 D 14 -8 -8 0 10 E 14 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -14 -14 B 12 0 8 8 14 C 18 -8 0 8 12 D 14 -8 -8 0 10 E 14 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2595: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (15) D C A B E (13) B A C E D (11) E D C B A (9) A B C D E (8) D E C A B (7) D A B C E (7) E C B A D (6) B A E C D (4) A B C E D (4) D C E A B (3) E D B A C (2) E B C A D (2) D E C B A (2) D A C B E (1) D A B E C (1) C A B D E (1) B E A C D (1) B C A E D (1) B A D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 12 10 6 B 8 0 16 10 8 C -12 -16 0 6 0 D -10 -10 -6 0 -10 E -6 -8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 10 6 B 8 0 16 10 8 C -12 -16 0 6 0 D -10 -10 -6 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=34 D=34 B=18 A=13 C=1 so C is eliminated. Round 2 votes counts: E=34 D=34 B=18 A=14 so A is eliminated. Round 3 votes counts: E=34 D=34 B=32 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:221 A:210 E:198 C:189 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 10 6 B 8 0 16 10 8 C -12 -16 0 6 0 D -10 -10 -6 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 10 6 B 8 0 16 10 8 C -12 -16 0 6 0 D -10 -10 -6 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 10 6 B 8 0 16 10 8 C -12 -16 0 6 0 D -10 -10 -6 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2596: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) B E C D A (6) A C B D E (6) A D C B E (5) A D B C E (5) D B A E C (4) B D E C A (4) A C E B D (4) A C D E B (4) E C B D A (3) D E B C A (3) D B E C A (3) D B E A C (3) D A E B C (3) C A B E D (3) A C D B E (3) E D B C A (2) E B D C A (2) E B C D A (2) D E B A C (2) D E A C B (2) D A B C E (2) C E B A D (2) C B E A D (2) C A E B D (2) B E D C A (2) B A C D E (2) A D C E B (2) A C E D B (2) E D C A B (1) E C D B A (1) E B C A D (1) D E A B C (1) D B A C E (1) D A E C B (1) C E A B D (1) B C E D A (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 6 2 0 B 4 0 -2 0 6 C -6 2 0 4 -4 D -2 0 -4 0 12 E 0 -6 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888874 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 0 B 4 0 -2 0 6 C -6 2 0 4 -4 D -2 0 -4 0 12 E 0 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=25 E=18 B=15 C=10 so C is eliminated. Round 2 votes counts: A=37 D=25 E=21 B=17 so B is eliminated. Round 3 votes counts: A=39 E=32 D=29 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:204 D:203 A:202 C:198 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 0 B 4 0 -2 0 6 C -6 2 0 4 -4 D -2 0 -4 0 12 E 0 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 0 B 4 0 -2 0 6 C -6 2 0 4 -4 D -2 0 -4 0 12 E 0 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 0 B 4 0 -2 0 6 C -6 2 0 4 -4 D -2 0 -4 0 12 E 0 -6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2597: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) B C D A E (7) C A B D E (5) B C A D E (5) E D B A C (4) C B D A E (4) B D C E A (4) A C E B D (4) A C B E D (4) E D A B C (3) E A D B C (3) D C B E A (3) D B C E A (3) A E C B D (3) E A D C B (2) E A C D B (2) E A B D C (2) D C B A E (2) D C A E B (2) D B C A E (2) B E D A C (2) B D E C A (2) B C A E D (2) E D A C B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A C B D (1) D E C A B (1) D E B C A (1) D E A C B (1) D C A B E (1) D B E C A (1) C D B A E (1) C D A B E (1) C B A D E (1) C A D B E (1) B E D C A (1) B E C A D (1) B E A C D (1) B D C A E (1) B C D E A (1) B A C E D (1) A E C D B (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -6 0 0 B 6 0 16 26 16 C 6 -16 0 10 14 D 0 -26 -10 0 2 E 0 -16 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 0 0 B 6 0 16 26 16 C 6 -16 0 10 14 D 0 -26 -10 0 2 E 0 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996345 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=28 B=28 D=17 A=14 C=13 so C is eliminated. Round 2 votes counts: B=33 E=28 A=20 D=19 so D is eliminated. Round 3 votes counts: B=45 E=31 A=24 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:207 A:194 E:184 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 0 0 B 6 0 16 26 16 C 6 -16 0 10 14 D 0 -26 -10 0 2 E 0 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996345 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 0 0 B 6 0 16 26 16 C 6 -16 0 10 14 D 0 -26 -10 0 2 E 0 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996345 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 0 0 B 6 0 16 26 16 C 6 -16 0 10 14 D 0 -26 -10 0 2 E 0 -16 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996345 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2598: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) E D C B A (7) E C D A B (7) E D C A B (6) D E C B A (4) C E D A B (4) B D A C E (4) A C B D E (4) A B C D E (4) E D B C A (3) E D B A C (3) E C A B D (3) D E B C A (3) D E B A C (3) C A D B E (3) B A E D C (3) A B C E D (3) E B D A C (2) D B A C E (2) C E A D B (2) C A E D B (2) C A E B D (2) C A B E D (2) B A E C D (2) B A D E C (2) A B D C E (2) E B A C D (1) D C E A B (1) D C B A E (1) D B E A C (1) D B A E C (1) C A B D E (1) B D E A C (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -2 -6 -2 B 6 0 0 -6 -6 C 2 0 0 -16 -4 D 6 6 16 0 -6 E 2 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 -6 -2 B 6 0 0 -6 -6 C 2 0 0 -16 -4 D 6 6 16 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=22 D=16 C=16 A=14 so A is eliminated. Round 2 votes counts: E=32 B=31 C=21 D=16 so D is eliminated. Round 3 votes counts: E=42 B=35 C=23 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:209 B:197 A:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -6 -2 B 6 0 0 -6 -6 C 2 0 0 -16 -4 D 6 6 16 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -6 -2 B 6 0 0 -6 -6 C 2 0 0 -16 -4 D 6 6 16 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -6 -2 B 6 0 0 -6 -6 C 2 0 0 -16 -4 D 6 6 16 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2599: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) C A E B D (9) C B D A E (8) A E D B C (6) C B A D E (5) B D C E A (5) B D C A E (5) B D A E C (5) E D A B C (4) E A D B C (4) B D E A C (4) D E B A C (3) C A E D B (3) B D E C A (3) A C E B D (3) E D C B A (2) E A C D B (2) D B A E C (2) C A B E D (2) A E C D B (2) E D B A C (1) E A D C B (1) D E B C A (1) C E B D A (1) C E A D B (1) C D E B A (1) C B D E A (1) C B A E D (1) B C D A E (1) A E D C B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -20 2 -16 10 B 20 0 14 8 6 C -2 -14 0 -20 -8 D 16 -8 20 0 12 E -10 -6 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 2 -16 10 B 20 0 14 8 6 C -2 -14 0 -20 -8 D 16 -8 20 0 12 E -10 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=23 D=17 E=14 A=14 so E is eliminated. Round 2 votes counts: C=32 D=24 B=23 A=21 so A is eliminated. Round 3 votes counts: C=39 D=37 B=24 so B is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:224 D:220 E:190 A:188 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 2 -16 10 B 20 0 14 8 6 C -2 -14 0 -20 -8 D 16 -8 20 0 12 E -10 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 2 -16 10 B 20 0 14 8 6 C -2 -14 0 -20 -8 D 16 -8 20 0 12 E -10 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 2 -16 10 B 20 0 14 8 6 C -2 -14 0 -20 -8 D 16 -8 20 0 12 E -10 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2600: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) E A C B D (5) D E C A B (5) C A E B D (4) B D E A C (4) B D C A E (4) E D A C B (3) E A B C D (3) D B E A C (3) D B C E A (3) D B C A E (3) B D A E C (3) B C A D E (3) B A E D C (3) E D C A B (2) E B A D C (2) D E B C A (2) D E B A C (2) D C E B A (2) D C E A B (2) C E A D B (2) C A E D B (2) B A E C D (2) B A D E C (2) B A C E D (2) A C B E D (2) A B E C D (2) A B C E D (2) E D B A C (1) E C D A B (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D E A C B (1) D C B E A (1) D C B A E (1) C D E A B (1) C D B A E (1) C D A B E (1) C B D A E (1) C A D E B (1) C A B E D (1) B D A C E (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 12 2 2 B -6 0 0 14 -8 C -12 0 0 -6 -16 D -2 -14 6 0 -4 E -2 8 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 2 2 B -6 0 0 14 -8 C -12 0 0 -6 -16 D -2 -14 6 0 -4 E -2 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999508 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 E=21 A=15 C=14 so C is eliminated. Round 2 votes counts: D=28 B=26 E=23 A=23 so E is eliminated. Round 3 votes counts: A=37 D=35 B=28 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:213 A:211 B:200 D:193 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 2 2 B -6 0 0 14 -8 C -12 0 0 -6 -16 D -2 -14 6 0 -4 E -2 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999508 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 2 2 B -6 0 0 14 -8 C -12 0 0 -6 -16 D -2 -14 6 0 -4 E -2 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999508 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 2 2 B -6 0 0 14 -8 C -12 0 0 -6 -16 D -2 -14 6 0 -4 E -2 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999508 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2601: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) D B A C E (11) C E D B A (9) A B D E C (9) E C D A B (7) C D E B A (4) B D A C E (4) E A C B D (3) D C B A E (3) D B C A E (3) C E B A D (3) C E A B D (3) A B E D C (3) E A B C D (2) D C E B A (2) D C B E A (2) D B A E C (2) C E D A B (2) B A D E C (2) B A D C E (2) A D B E C (2) E C A D B (1) E A D C B (1) D E C A B (1) D A B E C (1) C E A D B (1) C D B E A (1) C D B A E (1) C B D A E (1) C B A D E (1) A E D B C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 -12 -8 -6 B 2 0 -14 -10 -4 C 12 14 0 2 6 D 8 10 -2 0 4 E 6 4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -8 -6 B 2 0 -14 -10 -4 C 12 14 0 2 6 D 8 10 -2 0 4 E 6 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 D=25 A=16 B=8 so B is eliminated. Round 2 votes counts: D=29 C=26 E=25 A=20 so A is eliminated. Round 3 votes counts: D=44 E=30 C=26 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:217 D:210 E:200 B:187 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -8 -6 B 2 0 -14 -10 -4 C 12 14 0 2 6 D 8 10 -2 0 4 E 6 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -8 -6 B 2 0 -14 -10 -4 C 12 14 0 2 6 D 8 10 -2 0 4 E 6 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -8 -6 B 2 0 -14 -10 -4 C 12 14 0 2 6 D 8 10 -2 0 4 E 6 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2602: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) B E D A C (7) A C D E B (7) B E D C A (5) E D B C A (4) D E C B A (3) D E C A B (3) D E B C A (3) D A E C B (3) C D E A B (3) C A B E D (3) A D E B C (3) A B D E C (3) A B C E D (3) D E B A C (2) D E A C B (2) D A C E B (2) C E B D A (2) C D E B A (2) C B E A D (2) C B A E D (2) B E C D A (2) B C E D A (2) B A E D C (2) B A E C D (2) A D E C B (2) A C B E D (2) A C B D E (2) A B E C D (2) A B C D E (2) E D C B A (1) E C D B A (1) E B D C A (1) D E A B C (1) D C E B A (1) C E D B A (1) C B E D A (1) B D E A C (1) B C A E D (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 0 -6 0 B -2 0 -4 -2 -8 C 0 4 0 0 -8 D 6 2 0 0 4 E 0 8 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.221432 D: 0.778568 E: 0.000000 Sum of squares = 0.655200156791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.221432 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -6 0 B -2 0 -4 -2 -8 C 0 4 0 0 -8 D 6 2 0 0 4 E 0 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555556578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 B=23 D=20 E=7 so E is eliminated. Round 2 votes counts: A=27 D=25 C=24 B=24 so C is eliminated. Round 3 votes counts: A=37 D=32 B=31 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:206 E:206 A:198 C:198 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -6 0 B -2 0 -4 -2 -8 C 0 4 0 0 -8 D 6 2 0 0 4 E 0 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555556578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -6 0 B -2 0 -4 -2 -8 C 0 4 0 0 -8 D 6 2 0 0 4 E 0 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555556578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -6 0 B -2 0 -4 -2 -8 C 0 4 0 0 -8 D 6 2 0 0 4 E 0 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555556578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2603: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) B A D E C (11) A D B E C (7) C D E A B (6) C D A E B (5) E C D A B (4) B A E D C (4) B A C D E (4) C E D B A (3) E D C B A (2) E C D B A (2) E B C D A (2) D A E B C (2) C E B D A (2) C B E A D (2) C A D E B (2) B E C A D (2) B E A D C (2) B A E C D (2) B A D C E (2) A D B C E (2) A B D E C (2) E D C A B (1) E D B C A (1) E D A B C (1) E C B D A (1) E B D C A (1) D E C A B (1) D A E C B (1) D A C E B (1) C B A E D (1) C B A D E (1) B E C D A (1) B C E A D (1) B C A E D (1) B C A D E (1) B A C E D (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -12 0 4 B 0 0 2 -10 -2 C 12 -2 0 14 0 D 0 10 -14 0 0 E -4 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.383027 D: 0.000000 E: 0.616973 Sum of squares = 0.527365177919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.383027 D: 0.383027 E: 1.000000 A B C D E A 0 0 -12 0 4 B 0 0 2 -10 -2 C 12 -2 0 14 0 D 0 10 -14 0 0 E -4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.000000 E: 0.500294 Sum of squares = 0.50000017304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.499706 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=32 E=15 A=13 D=5 so D is eliminated. Round 2 votes counts: C=35 B=32 A=17 E=16 so E is eliminated. Round 3 votes counts: C=46 B=36 A=18 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 E:199 D:198 A:196 B:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 0 4 B 0 0 2 -10 -2 C 12 -2 0 14 0 D 0 10 -14 0 0 E -4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.000000 E: 0.500294 Sum of squares = 0.50000017304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.499706 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 0 4 B 0 0 2 -10 -2 C 12 -2 0 14 0 D 0 10 -14 0 0 E -4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.000000 E: 0.500294 Sum of squares = 0.50000017304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.499706 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 0 4 B 0 0 2 -10 -2 C 12 -2 0 14 0 D 0 10 -14 0 0 E -4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.000000 E: 0.500294 Sum of squares = 0.50000017304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499706 D: 0.499706 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2604: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) E A D B C (9) E A C B D (6) D B C A E (6) C B A D E (6) C A B D E (5) B D C A E (5) E A C D B (4) D B E A C (4) E D A B C (3) E C A B D (3) C B D A E (3) B C D A E (3) E D C A B (2) E D B C A (2) E A D C B (2) D B C E A (2) D B A E C (2) A E B D C (2) A C B E D (2) A B D E C (2) A B D C E (2) E D B A C (1) E C D B A (1) E C A D B (1) D A B E C (1) C E B A D (1) C D B E A (1) C D B A E (1) C B D E A (1) C A B E D (1) B D A C E (1) A E D B C (1) A E C B D (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 12 2 14 B 0 0 14 -6 20 C -12 -14 0 -14 6 D -2 6 14 0 14 E -14 -20 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.811265 B: 0.188735 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.693772276187 Cumulative probabilities = A: 0.811265 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 2 14 B 0 0 14 -6 20 C -12 -14 0 -14 6 D -2 6 14 0 14 E -14 -20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750001 B: 0.249999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625001226518 Cumulative probabilities = A: 0.750001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=25 C=19 A=13 B=9 so B is eliminated. Round 2 votes counts: E=34 D=31 C=22 A=13 so A is eliminated. Round 3 votes counts: E=38 D=35 C=27 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:214 B:214 C:183 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 2 14 B 0 0 14 -6 20 C -12 -14 0 -14 6 D -2 6 14 0 14 E -14 -20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750001 B: 0.249999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625001226518 Cumulative probabilities = A: 0.750001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 2 14 B 0 0 14 -6 20 C -12 -14 0 -14 6 D -2 6 14 0 14 E -14 -20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750001 B: 0.249999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625001226518 Cumulative probabilities = A: 0.750001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 2 14 B 0 0 14 -6 20 C -12 -14 0 -14 6 D -2 6 14 0 14 E -14 -20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750001 B: 0.249999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625001226518 Cumulative probabilities = A: 0.750001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2605: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) C A E D B (7) B D E A C (6) D C B A E (5) D B C E A (5) C D B E A (5) B E D A C (5) E A C B D (4) A C E D B (4) C D A B E (3) C A D E B (3) B D A E C (3) A E B C D (3) E C A B D (2) E B C D A (2) E B A D C (2) E A B D C (2) D B A E C (2) C E A B D (2) C D E A B (2) B E D C A (2) A E C B D (2) A E B D C (2) A D C B E (2) A D B E C (2) E C B A D (1) E B D A C (1) E A B C D (1) D C B E A (1) D C A B E (1) D B E C A (1) D B A C E (1) D A B C E (1) C D E B A (1) C D B A E (1) C D A E B (1) C A D B E (1) B E A D C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -8 -14 -8 B 6 0 4 0 16 C 8 -4 0 -10 -6 D 14 0 10 0 14 E 8 -16 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.408529 C: 0.000000 D: 0.591471 E: 0.000000 Sum of squares = 0.516733959943 Cumulative probabilities = A: 0.000000 B: 0.408529 C: 0.408529 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -14 -8 B 6 0 4 0 16 C 8 -4 0 -10 -6 D 14 0 10 0 14 E 8 -16 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 D=17 A=17 E=15 so E is eliminated. Round 2 votes counts: B=30 C=29 A=24 D=17 so D is eliminated. Round 3 votes counts: B=39 C=36 A=25 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:219 B:213 C:194 E:192 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -14 -8 B 6 0 4 0 16 C 8 -4 0 -10 -6 D 14 0 10 0 14 E 8 -16 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -14 -8 B 6 0 4 0 16 C 8 -4 0 -10 -6 D 14 0 10 0 14 E 8 -16 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -14 -8 B 6 0 4 0 16 C 8 -4 0 -10 -6 D 14 0 10 0 14 E 8 -16 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2606: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) B D E C A (10) C E D A B (9) A B C E D (8) A B E D C (7) B D E A C (6) B A D E C (6) D E C A B (5) D E C B A (4) D E B C A (4) B C D E A (4) C A E D B (3) B A C D E (3) A E D C B (3) A C B E D (3) C D E B A (2) B A C E D (2) A E C D B (2) A C E B D (2) A B D E C (2) E D C A B (1) D B E C A (1) B C D A E (1) A E D B C (1) Total count = 100 A B C D E A 0 14 12 6 8 B -14 0 10 8 6 C -12 -10 0 0 -4 D -6 -8 0 0 -4 E -8 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 12 6 8 B -14 0 10 8 6 C -12 -10 0 0 -4 D -6 -8 0 0 -4 E -8 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=32 D=14 C=14 E=1 so E is eliminated. Round 2 votes counts: A=39 B=32 D=15 C=14 so C is eliminated. Round 3 votes counts: A=42 B=32 D=26 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 B:205 E:197 D:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 12 6 8 B -14 0 10 8 6 C -12 -10 0 0 -4 D -6 -8 0 0 -4 E -8 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 6 8 B -14 0 10 8 6 C -12 -10 0 0 -4 D -6 -8 0 0 -4 E -8 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 6 8 B -14 0 10 8 6 C -12 -10 0 0 -4 D -6 -8 0 0 -4 E -8 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2607: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (14) A C B D E (10) D E B A C (6) D A B C E (6) E C A B D (5) D B E A C (5) D B A C E (5) C A E B D (5) C A B D E (5) E C B A D (4) E D B A C (3) E A C D B (3) C A B E D (3) A D C B E (3) A C D B E (3) E D A C B (2) D B E C A (2) C E A B D (2) B C A D E (2) E C A D B (1) E B D C A (1) E B C A D (1) D E A B C (1) D B A E C (1) D A C B E (1) C E B A D (1) C B A D E (1) B D E C A (1) B D C A E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 2 2 -4 B -4 0 -2 -14 -2 C -2 2 0 -4 0 D -2 14 4 0 6 E 4 2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888864 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 4 2 2 -4 B -4 0 -2 -14 -2 C -2 2 0 -4 0 D -2 14 4 0 6 E 4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888883 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=27 A=18 C=17 B=4 so B is eliminated. Round 2 votes counts: E=34 D=29 C=19 A=18 so A is eliminated. Round 3 votes counts: E=34 C=34 D=32 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:211 A:202 E:200 C:198 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 2 -4 B -4 0 -2 -14 -2 C -2 2 0 -4 0 D -2 14 4 0 6 E 4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888883 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 -4 B -4 0 -2 -14 -2 C -2 2 0 -4 0 D -2 14 4 0 6 E 4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888883 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 -4 B -4 0 -2 -14 -2 C -2 2 0 -4 0 D -2 14 4 0 6 E 4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888883 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2608: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) D E C B A (10) E D C B A (8) B A C E D (8) A B C E D (8) A C D E B (5) C E D A B (4) A C B E D (4) C A D E B (3) B E D C A (3) B E A D C (3) B D E C A (3) B A E C D (3) B A D E C (3) A C B D E (3) D E B C A (2) D C E A B (2) C D E A B (2) B A E D C (2) E D B C A (1) E C D A B (1) D B E C A (1) D A C E B (1) C E A D B (1) C D A E B (1) B E D A C (1) B D E A C (1) B D A E C (1) A D C B E (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -6 -6 -8 B 0 0 -20 -12 -8 C 6 20 0 -8 -8 D 6 12 8 0 2 E 8 8 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -6 -8 B 0 0 -20 -12 -8 C 6 20 0 -8 -8 D 6 12 8 0 2 E 8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 A=24 C=11 E=10 so E is eliminated. Round 2 votes counts: D=36 B=28 A=24 C=12 so C is eliminated. Round 3 votes counts: D=44 B=28 A=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:211 C:205 A:190 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 -6 -8 B 0 0 -20 -12 -8 C 6 20 0 -8 -8 D 6 12 8 0 2 E 8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -6 -8 B 0 0 -20 -12 -8 C 6 20 0 -8 -8 D 6 12 8 0 2 E 8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -6 -8 B 0 0 -20 -12 -8 C 6 20 0 -8 -8 D 6 12 8 0 2 E 8 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2609: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (7) D A C E B (5) D A B C E (5) B E C D A (5) A D B E C (5) E C B D A (4) E C B A D (4) D A C B E (4) B E D C A (4) B E A C D (4) A D C B E (4) E B C A D (3) D B A E C (3) C E B D A (3) C E A D B (3) C D E A B (3) B E C A D (3) B D A E C (3) A D B C E (3) A C D E B (3) E B C D A (2) D C A E B (2) D B C E A (2) C D B E A (2) C D A E B (2) C A D E B (2) E A C B D (1) D B A C E (1) C E D B A (1) C E D A B (1) C E B A D (1) C A E D B (1) B E A D C (1) B D C E A (1) B A E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 2 -6 4 B -4 0 -6 -18 4 C -2 6 0 -4 12 D 6 18 4 0 16 E -4 -4 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -6 4 B -4 0 -6 -18 4 C -2 6 0 -4 12 D 6 18 4 0 16 E -4 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=22 B=22 C=19 E=14 so E is eliminated. Round 2 votes counts: C=27 B=27 A=24 D=22 so D is eliminated. Round 3 votes counts: A=38 B=33 C=29 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:222 C:206 A:202 B:188 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -6 4 B -4 0 -6 -18 4 C -2 6 0 -4 12 D 6 18 4 0 16 E -4 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -6 4 B -4 0 -6 -18 4 C -2 6 0 -4 12 D 6 18 4 0 16 E -4 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -6 4 B -4 0 -6 -18 4 C -2 6 0 -4 12 D 6 18 4 0 16 E -4 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2610: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D A C B E (6) C E B D A (6) A D B C E (6) E C D A B (4) E B A D C (4) D A B C E (4) C B E D A (4) E A D B C (3) D A C E B (3) C E D B A (3) C D A B E (3) B C D A E (3) B A D C E (3) A D E B C (3) A B D E C (3) E C B A D (2) E C A B D (2) E B C A D (2) E A B D C (2) D C A B E (2) C D B A E (2) C B D A E (2) B E C A D (2) B E A C D (2) B A E D C (2) A D E C B (2) E B A C D (1) E A D C B (1) E A B C D (1) D A E C B (1) C D E A B (1) C B D E A (1) B E C D A (1) B E A D C (1) B D A C E (1) B C A D E (1) B A C D E (1) A E B D C (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 0 6 -6 2 B 0 0 -4 8 4 C -6 4 0 0 6 D 6 -8 0 0 0 E -2 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.490677 B: 0.509323 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500173784786 Cumulative probabilities = A: 0.490677 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -6 2 B 0 0 -4 8 4 C -6 4 0 0 6 D 6 -8 0 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 B=17 A=17 D=16 so D is eliminated. Round 2 votes counts: A=31 E=28 C=24 B=17 so B is eliminated. Round 3 votes counts: A=38 E=34 C=28 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:204 C:202 A:201 D:199 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 -6 2 B 0 0 -4 8 4 C -6 4 0 0 6 D 6 -8 0 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -6 2 B 0 0 -4 8 4 C -6 4 0 0 6 D 6 -8 0 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -6 2 B 0 0 -4 8 4 C -6 4 0 0 6 D 6 -8 0 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999902 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2611: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) E D A C B (5) B C D E A (5) A D E B C (5) E C D B A (4) D E A B C (4) B C A D E (4) A E D B C (4) A D B E C (4) E A D C B (3) C E B A D (3) B D C E A (3) B C D A E (3) A E D C B (3) A E C B D (3) E D C A B (2) E A C D B (2) D B E C A (2) D B E A C (2) D B C E A (2) D A E B C (2) C E B D A (2) C B E A D (2) C B D E A (2) C B A E D (2) B D C A E (2) A B D C E (2) E D C B A (1) E C D A B (1) E C A D B (1) D E C B A (1) D E B C A (1) D E B A C (1) D B A E C (1) C E A B D (1) C B A D E (1) C A E B D (1) C A B E D (1) B A C D E (1) A E C D B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -10 -8 -16 B 6 0 -2 -4 -2 C 10 2 0 -2 -8 D 8 4 2 0 -2 E 16 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -10 -8 -16 B 6 0 -2 -4 -2 C 10 2 0 -2 -8 D 8 4 2 0 -2 E 16 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980215 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 E=19 B=18 D=16 so D is eliminated. Round 2 votes counts: E=26 A=26 B=25 C=23 so C is eliminated. Round 3 votes counts: B=40 E=32 A=28 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:206 C:201 B:199 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 -16 B 6 0 -2 -4 -2 C 10 2 0 -2 -8 D 8 4 2 0 -2 E 16 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980215 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 -16 B 6 0 -2 -4 -2 C 10 2 0 -2 -8 D 8 4 2 0 -2 E 16 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980215 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 -16 B 6 0 -2 -4 -2 C 10 2 0 -2 -8 D 8 4 2 0 -2 E 16 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980215 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2612: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (14) C E A D B (8) B D A E C (7) E C D B A (6) E C A B D (6) E D B C A (5) D B A E C (5) C A D B E (5) E C B D A (4) C E D B A (4) C E A B D (4) C A E B D (4) D A B C E (3) E C B A D (2) E B D A C (2) B A D E C (2) A B D E C (2) A B D C E (2) E D C B A (1) E B D C A (1) E B A D C (1) E A C B D (1) D B C E A (1) D B C A E (1) C D B A E (1) C A E D B (1) C A D E B (1) B E D A C (1) B D A C E (1) A E B C D (1) A C E B D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 -10 -14 6 B 18 0 0 -12 -6 C 10 0 0 0 4 D 14 12 0 0 -8 E -6 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.725444 D: 0.274556 E: 0.000000 Sum of squares = 0.601649836504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.725444 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -14 6 B 18 0 0 -12 -6 C 10 0 0 0 4 D 14 12 0 0 -8 E -6 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555645944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=28 D=24 B=11 A=8 so A is eliminated. Round 2 votes counts: E=30 C=30 D=24 B=16 so B is eliminated. Round 3 votes counts: D=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:209 C:207 E:202 B:200 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -10 -14 6 B 18 0 0 -12 -6 C 10 0 0 0 4 D 14 12 0 0 -8 E -6 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555645944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -14 6 B 18 0 0 -12 -6 C 10 0 0 0 4 D 14 12 0 0 -8 E -6 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555645944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -14 6 B 18 0 0 -12 -6 C 10 0 0 0 4 D 14 12 0 0 -8 E -6 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555645944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2613: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) E A B C D (7) E C D A B (6) D C B A E (6) D B A C E (6) B D A C E (6) D C B E A (5) B A D E C (5) A B E D C (5) E C D B A (4) E C A B D (4) A B D E C (4) A B D C E (4) E C A D B (3) D C E B A (3) D B C A E (2) C D A B E (2) B A D C E (2) A B E C D (2) E D B C A (1) E C B A D (1) E B D A C (1) E B C A D (1) E B A D C (1) E A C B D (1) D E C B A (1) D B E C A (1) C E D A B (1) C E A D B (1) C D E A B (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -8 -14 -8 B 14 0 0 -8 2 C 8 0 0 -6 2 D 14 8 6 0 18 E 8 -2 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -14 -8 B 14 0 0 -8 2 C 8 0 0 -6 2 D 14 8 6 0 18 E 8 -2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=24 A=17 C=16 B=13 so B is eliminated. Round 2 votes counts: E=30 D=30 A=24 C=16 so C is eliminated. Round 3 votes counts: D=44 E=32 A=24 so A is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:204 C:202 E:193 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -8 -14 -8 B 14 0 0 -8 2 C 8 0 0 -6 2 D 14 8 6 0 18 E 8 -2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -14 -8 B 14 0 0 -8 2 C 8 0 0 -6 2 D 14 8 6 0 18 E 8 -2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -14 -8 B 14 0 0 -8 2 C 8 0 0 -6 2 D 14 8 6 0 18 E 8 -2 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2614: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (12) B A C E D (11) E D C A B (5) B A D C E (5) B C A E D (4) B A C D E (4) E C D A B (3) E C A D B (3) D C E B A (3) C E B A D (3) A E B C D (3) A B C E D (3) E D A C B (2) D E C B A (2) D E A C B (2) D B A E C (2) D A E B C (2) C E B D A (2) C D E B A (2) C D B E A (2) B D A C E (2) A E D B C (2) A B E D C (2) A B E C D (2) E A D C B (1) D E A B C (1) D B C E A (1) D A B E C (1) C E D B A (1) C E D A B (1) C E A B D (1) C B E D A (1) C B E A D (1) B D A E C (1) B C D A E (1) B C A D E (1) B A D E C (1) A E D C B (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 2 2 2 B 0 0 4 0 -6 C -2 -4 0 -2 -2 D -2 0 2 0 -6 E -2 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.845828 B: 0.154172 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.739193669375 Cumulative probabilities = A: 0.845828 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 2 2 B 0 0 4 0 -6 C -2 -4 0 -2 -2 D -2 0 2 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000003242 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=26 A=16 E=14 C=14 so E is eliminated. Round 2 votes counts: D=33 B=30 C=20 A=17 so A is eliminated. Round 3 votes counts: B=42 D=38 C=20 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:206 A:203 B:199 D:197 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 2 2 B 0 0 4 0 -6 C -2 -4 0 -2 -2 D -2 0 2 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000003242 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 2 B 0 0 4 0 -6 C -2 -4 0 -2 -2 D -2 0 2 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000003242 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 2 B 0 0 4 0 -6 C -2 -4 0 -2 -2 D -2 0 2 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000003242 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2615: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) E D B C A (7) C A E B D (7) D B A C E (6) D E B A C (5) C A B E D (5) A C B D E (5) E C A D B (4) D B E A C (4) D A C B E (4) E D B A C (3) E B C A D (3) C A B D E (3) B A C D E (3) E D C B A (2) E D C A B (2) E C B A D (2) E B D C A (2) D E B C A (2) D A B C E (2) C A D E B (2) B E A C D (2) B D A C E (2) B A D C E (2) A D B C E (2) E C D A B (1) D C A E B (1) C B A E D (1) C A D B E (1) B E C A D (1) B D A E C (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -12 12 0 B -4 0 -6 0 -6 C 12 6 0 4 -2 D -12 0 -4 0 -6 E 0 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.073400 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.926600 Sum of squares = 0.863974636466 Cumulative probabilities = A: 0.073400 B: 0.073400 C: 0.073400 D: 0.073400 E: 1.000000 A B C D E A 0 4 -12 12 0 B -4 0 -6 0 -6 C 12 6 0 4 -2 D -12 0 -4 0 -6 E 0 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102072641 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=24 C=19 B=11 A=10 so A is eliminated. Round 2 votes counts: E=36 D=27 C=26 B=11 so B is eliminated. Round 3 votes counts: E=39 D=32 C=29 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:207 A:202 B:192 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -12 12 0 B -4 0 -6 0 -6 C 12 6 0 4 -2 D -12 0 -4 0 -6 E 0 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102072641 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 12 0 B -4 0 -6 0 -6 C 12 6 0 4 -2 D -12 0 -4 0 -6 E 0 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102072641 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 12 0 B -4 0 -6 0 -6 C 12 6 0 4 -2 D -12 0 -4 0 -6 E 0 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102072641 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2616: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (15) C B D A E (12) D A E C B (8) E B C A D (7) E B A C D (6) D A C B E (6) C B E D A (6) B C E A D (6) D A C E B (5) B E C A D (5) C D B A E (4) E A B D C (3) D C A B E (3) C B D E A (3) B C E D A (3) A D E C B (3) A E D B C (2) A D E B C (2) E B A D C (1) Total count = 100 A B C D E A 0 -6 2 0 -10 B 6 0 0 4 -4 C -2 0 0 4 -4 D 0 -4 -4 0 -8 E 10 4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 2 0 -10 B 6 0 0 4 -4 C -2 0 0 4 -4 D 0 -4 -4 0 -8 E 10 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=25 D=22 B=14 A=7 so A is eliminated. Round 2 votes counts: E=34 D=27 C=25 B=14 so B is eliminated. Round 3 votes counts: E=39 C=34 D=27 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 B:203 C:199 A:193 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 2 0 -10 B 6 0 0 4 -4 C -2 0 0 4 -4 D 0 -4 -4 0 -8 E 10 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 -10 B 6 0 0 4 -4 C -2 0 0 4 -4 D 0 -4 -4 0 -8 E 10 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 -10 B 6 0 0 4 -4 C -2 0 0 4 -4 D 0 -4 -4 0 -8 E 10 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2617: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) A D E B C (9) C E B D A (7) C E D B A (6) B C E A D (6) E C D A B (5) D A E C B (5) A B D C E (5) E D C A B (4) E C B D A (4) B C A D E (4) D E A C B (3) B A C D E (3) A D B E C (3) A D B C E (3) A B D E C (3) E C D B A (2) B E C A D (2) B A E C D (2) B A D C E (2) E D A C B (1) E B C A D (1) D E C A B (1) D A C B E (1) C E D A B (1) C B D E A (1) B C E D A (1) B C D A E (1) B C A E D (1) A E D B C (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -14 -6 -10 B 6 0 -4 8 -2 C 14 4 0 14 4 D 6 -8 -14 0 -10 E 10 2 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -6 -10 B 6 0 -4 8 -2 C 14 4 0 14 4 D 6 -8 -14 0 -10 E 10 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 B=22 E=17 D=10 so D is eliminated. Round 2 votes counts: A=32 C=25 B=22 E=21 so E is eliminated. Round 3 votes counts: C=41 A=36 B=23 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:209 B:204 D:187 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -6 -10 B 6 0 -4 8 -2 C 14 4 0 14 4 D 6 -8 -14 0 -10 E 10 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -6 -10 B 6 0 -4 8 -2 C 14 4 0 14 4 D 6 -8 -14 0 -10 E 10 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -6 -10 B 6 0 -4 8 -2 C 14 4 0 14 4 D 6 -8 -14 0 -10 E 10 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2618: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (11) E A D B C (10) B C E A D (10) D A E C B (8) C B D A E (8) E B C A D (7) D A C B E (7) A D E B C (7) B C E D A (5) C B D E A (4) A E D B C (4) E B A C D (3) E A B D C (3) E C B D A (2) E A D C B (2) C D B A E (2) E D A C B (1) D C B A E (1) D C A E B (1) D A E B C (1) C B E A D (1) B E C A D (1) A D E C B (1) Total count = 100 A B C D E A 0 -10 -6 -2 -20 B 10 0 2 10 0 C 6 -2 0 8 0 D 2 -10 -8 0 -20 E 20 0 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.828469 C: 0.000000 D: 0.000000 E: 0.171531 Sum of squares = 0.715784214671 Cumulative probabilities = A: 0.000000 B: 0.828469 C: 0.828469 D: 0.828469 E: 1.000000 A B C D E A 0 -10 -6 -2 -20 B 10 0 2 10 0 C 6 -2 0 8 0 D 2 -10 -8 0 -20 E 20 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999991411 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 D=18 B=16 A=12 so A is eliminated. Round 2 votes counts: E=32 D=26 C=26 B=16 so B is eliminated. Round 3 votes counts: C=41 E=33 D=26 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:220 B:211 C:206 D:182 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -2 -20 B 10 0 2 10 0 C 6 -2 0 8 0 D 2 -10 -8 0 -20 E 20 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999991411 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -2 -20 B 10 0 2 10 0 C 6 -2 0 8 0 D 2 -10 -8 0 -20 E 20 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999991411 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -2 -20 B 10 0 2 10 0 C 6 -2 0 8 0 D 2 -10 -8 0 -20 E 20 0 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999991411 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2619: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) B E C A D (6) A C D B E (6) D E A B C (5) C B A E D (5) D C A E B (4) E D C B A (3) E B D A C (3) D E C A B (3) D E A C B (3) D A E C B (3) C B E D A (3) C A B D E (3) B A C E D (3) A D B C E (3) A C B D E (3) E D B A C (2) E B C D A (2) D C E A B (2) C D E A B (2) C D A E B (2) C A D B E (2) B C A E D (2) A D E B C (2) A D C B E (2) A B C E D (2) A B C D E (2) E D B C A (1) E B A D C (1) D A E B C (1) D A C E B (1) C E D B A (1) C E B D A (1) C D E B A (1) C B E A D (1) B E A D C (1) B E A C D (1) B C E A D (1) B A E C D (1) A D E C B (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -2 2 2 B -10 0 -4 -2 0 C 2 4 0 0 6 D -2 2 0 0 8 E -2 0 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.808801 D: 0.191199 E: 0.000000 Sum of squares = 0.690716518033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.808801 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 2 2 B -10 0 -4 -2 0 C 2 4 0 0 6 D -2 2 0 0 8 E -2 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500006 D: 0.499994 E: 0.000000 Sum of squares = 0.500000000083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500006 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=22 C=21 E=18 B=15 so B is eliminated. Round 2 votes counts: A=28 E=26 C=24 D=22 so D is eliminated. Round 3 votes counts: E=37 A=33 C=30 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 C:206 D:204 B:192 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 2 2 B -10 0 -4 -2 0 C 2 4 0 0 6 D -2 2 0 0 8 E -2 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500006 D: 0.499994 E: 0.000000 Sum of squares = 0.500000000083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500006 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 2 2 B -10 0 -4 -2 0 C 2 4 0 0 6 D -2 2 0 0 8 E -2 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500006 D: 0.499994 E: 0.000000 Sum of squares = 0.500000000083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500006 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 2 2 B -10 0 -4 -2 0 C 2 4 0 0 6 D -2 2 0 0 8 E -2 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500006 D: 0.499994 E: 0.000000 Sum of squares = 0.500000000083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500006 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2620: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (16) B D C E A (10) C B D E A (9) A E C D B (9) D B C E A (7) A E C B D (6) E A C D B (5) C B D A E (5) E C D B A (3) E A D B C (3) C A B D E (3) E A D C B (2) D B E A C (2) C D B E A (2) B C D A E (2) A E B D C (2) E D B C A (1) E C A D B (1) D C B E A (1) D B E C A (1) C B A D E (1) C A E B D (1) B D C A E (1) B C D E A (1) B A D C E (1) A E D C B (1) A D B E C (1) A C E B D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 2 10 4 B -6 0 -2 -10 -2 C -2 2 0 0 -8 D -10 10 0 0 -2 E -4 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 10 4 B -6 0 -2 -10 -2 C -2 2 0 0 -8 D -10 10 0 0 -2 E -4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=21 E=15 B=15 D=11 so D is eliminated. Round 2 votes counts: A=38 B=25 C=22 E=15 so E is eliminated. Round 3 votes counts: A=48 C=26 B=26 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:204 D:199 C:196 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 10 4 B -6 0 -2 -10 -2 C -2 2 0 0 -8 D -10 10 0 0 -2 E -4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 10 4 B -6 0 -2 -10 -2 C -2 2 0 0 -8 D -10 10 0 0 -2 E -4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 10 4 B -6 0 -2 -10 -2 C -2 2 0 0 -8 D -10 10 0 0 -2 E -4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2621: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) A C E D B (6) E A D B C (5) C B D A E (5) B D E C A (5) A E C D B (5) E A C B D (4) A E C B D (4) E B C A D (3) D A B C E (3) C E A B D (3) C A B D E (3) B C D E A (3) A D E C B (3) A C D E B (3) E D B A C (2) E B D A C (2) E A D C B (2) D B E A C (2) D B C E A (2) D B A C E (2) C D B A E (2) B E D C A (2) B D C E A (2) B C D A E (2) E C A B D (1) E B D C A (1) E B C D A (1) E A C D B (1) D C B A E (1) D B E C A (1) D B A E C (1) D A E B C (1) C E B A D (1) C D A B E (1) C B A E D (1) C B A D E (1) C A E B D (1) B E C D A (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -4 -2 12 B 4 0 0 -6 -2 C 4 0 0 6 6 D 2 6 -6 0 6 E -12 2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.325849 C: 0.674151 D: 0.000000 E: 0.000000 Sum of squares = 0.560657342706 Cumulative probabilities = A: 0.000000 B: 0.325849 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 12 B 4 0 0 -6 -2 C 4 0 0 6 6 D 2 6 -6 0 6 E -12 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499743 C: 0.500257 D: 0.000000 E: 0.000000 Sum of squares = 0.500000132478 Cumulative probabilities = A: 0.000000 B: 0.499743 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=22 D=22 C=18 B=15 so B is eliminated. Round 2 votes counts: D=29 E=25 C=23 A=23 so C is eliminated. Round 3 votes counts: D=42 E=29 A=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:208 D:204 A:201 B:198 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 12 B 4 0 0 -6 -2 C 4 0 0 6 6 D 2 6 -6 0 6 E -12 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499743 C: 0.500257 D: 0.000000 E: 0.000000 Sum of squares = 0.500000132478 Cumulative probabilities = A: 0.000000 B: 0.499743 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 12 B 4 0 0 -6 -2 C 4 0 0 6 6 D 2 6 -6 0 6 E -12 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499743 C: 0.500257 D: 0.000000 E: 0.000000 Sum of squares = 0.500000132478 Cumulative probabilities = A: 0.000000 B: 0.499743 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 12 B 4 0 0 -6 -2 C 4 0 0 6 6 D 2 6 -6 0 6 E -12 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499743 C: 0.500257 D: 0.000000 E: 0.000000 Sum of squares = 0.500000132478 Cumulative probabilities = A: 0.000000 B: 0.499743 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2622: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) C B D A E (9) A D E B C (7) C B E D A (6) A D E C B (6) E C B A D (4) E B C A D (4) B E C D A (4) B C D E A (4) E B A D C (3) C B D E A (3) C B A D E (3) C A D B E (3) B C E D A (3) E D A B C (2) E B D A C (2) E A D C B (2) E A B D C (2) C D B A E (2) C D A B E (2) C B A E D (2) C A E D B (2) B D C E A (2) E B C D A (1) D B E A C (1) D B A E C (1) D A E B C (1) D A C B E (1) D A B C E (1) C E A B D (1) C B E A D (1) B D E A C (1) B C D A E (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -14 6 -14 B 14 0 4 12 0 C 14 -4 0 12 -6 D -6 -12 -12 0 -2 E 14 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.605830 C: 0.000000 D: 0.000000 E: 0.394170 Sum of squares = 0.522399840468 Cumulative probabilities = A: 0.000000 B: 0.605830 C: 0.605830 D: 0.605830 E: 1.000000 A B C D E A 0 -14 -14 6 -14 B 14 0 4 12 0 C 14 -4 0 12 -6 D -6 -12 -12 0 -2 E 14 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=31 B=15 A=15 D=5 so D is eliminated. Round 2 votes counts: C=34 E=31 A=18 B=17 so B is eliminated. Round 3 votes counts: C=44 E=37 A=19 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:211 C:208 D:184 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -14 6 -14 B 14 0 4 12 0 C 14 -4 0 12 -6 D -6 -12 -12 0 -2 E 14 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 6 -14 B 14 0 4 12 0 C 14 -4 0 12 -6 D -6 -12 -12 0 -2 E 14 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 6 -14 B 14 0 4 12 0 C 14 -4 0 12 -6 D -6 -12 -12 0 -2 E 14 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2623: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (5) D B A E C (5) C A E B D (5) B E D C A (5) D B C E A (4) C E B D A (4) A D B E C (4) A D B C E (4) A C E D B (4) A C E B D (4) E B D A C (3) E B C D A (3) D B E C A (3) D B A C E (3) C E A B D (3) C B D E A (3) B D E C A (3) B C E D A (3) A E C B D (3) A E B D C (3) A C D E B (3) E B D C A (2) E A C B D (2) D B C A E (2) D A B E C (2) D A B C E (2) A D C E B (2) E C B D A (1) D B E A C (1) C E B A D (1) C A D B E (1) B E C D A (1) B D E A C (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 2 0 -4 2 B -2 0 12 12 -4 C 0 -12 0 -4 -2 D 4 -12 4 0 -10 E -2 4 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.250000 Sum of squares = 0.468749999478 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.750000 E: 1.000000 A B C D E A 0 2 0 -4 2 B -2 0 12 12 -4 C 0 -12 0 -4 -2 D 4 -12 4 0 -10 E -2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.250000 Sum of squares = 0.468749999973 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=22 C=17 E=16 B=14 so B is eliminated. Round 2 votes counts: A=31 D=26 E=22 C=21 so C is eliminated. Round 3 votes counts: A=37 E=33 D=30 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:209 E:207 A:200 D:193 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -4 2 B -2 0 12 12 -4 C 0 -12 0 -4 -2 D 4 -12 4 0 -10 E -2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.250000 Sum of squares = 0.468749999973 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 2 B -2 0 12 12 -4 C 0 -12 0 -4 -2 D 4 -12 4 0 -10 E -2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.250000 Sum of squares = 0.468749999973 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 2 B -2 0 12 12 -4 C 0 -12 0 -4 -2 D 4 -12 4 0 -10 E -2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.250000 Sum of squares = 0.468749999973 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2624: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (13) C B A E D (12) B D E C A (10) D E B A C (8) B C D E A (6) C A E D B (5) B D E A C (5) A E D C B (5) E D A C B (4) C A B E D (4) B C A D E (4) E D A B C (3) C B A D E (2) B D C E A (2) B C E D A (2) B C D A E (2) B C A E D (2) A E C D B (2) A C D E B (2) E D B A C (1) D E A C B (1) D A E B C (1) C A D E B (1) C A B D E (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 -6 -16 -10 B 12 0 20 4 4 C 6 -20 0 -8 -8 D 16 -4 8 0 18 E 10 -4 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -16 -10 B 12 0 20 4 4 C 6 -20 0 -8 -8 D 16 -4 8 0 18 E 10 -4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998095 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=25 D=23 A=11 E=8 so E is eliminated. Round 2 votes counts: B=33 D=31 C=25 A=11 so A is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:219 E:198 C:185 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -16 -10 B 12 0 20 4 4 C 6 -20 0 -8 -8 D 16 -4 8 0 18 E 10 -4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998095 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -16 -10 B 12 0 20 4 4 C 6 -20 0 -8 -8 D 16 -4 8 0 18 E 10 -4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998095 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -16 -10 B 12 0 20 4 4 C 6 -20 0 -8 -8 D 16 -4 8 0 18 E 10 -4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998095 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2625: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) A C B D E (9) E B D A C (8) D C A E B (7) D E B C A (5) C D A E B (5) C A D B E (5) E D B C A (4) D E C B A (4) C D A B E (4) B A E C D (4) A C B E D (4) E D B A C (3) A E B D C (3) A B E C D (3) A B C E D (3) D E C A B (2) D C E B A (2) C A B D E (2) B E A C D (2) E B D C A (1) E B A C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D A E C B (1) B E D C A (1) B E A D C (1) B C A E D (1) B C A D E (1) B A C E D (1) A D C E B (1) Total count = 100 A B C D E A 0 0 6 0 10 B 0 0 2 8 -14 C -6 -2 0 -10 -4 D 0 -8 10 0 2 E -10 14 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.686943 B: 0.313057 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.569895657916 Cumulative probabilities = A: 0.686943 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 0 10 B 0 0 2 8 -14 C -6 -2 0 -10 -4 D 0 -8 10 0 2 E -10 14 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.416667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513888901463 Cumulative probabilities = A: 0.583333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 A=23 C=16 B=11 so B is eliminated. Round 2 votes counts: E=30 A=28 D=24 C=18 so C is eliminated. Round 3 votes counts: A=37 D=33 E=30 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:203 D:202 B:198 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 0 10 B 0 0 2 8 -14 C -6 -2 0 -10 -4 D 0 -8 10 0 2 E -10 14 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.416667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513888901463 Cumulative probabilities = A: 0.583333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 10 B 0 0 2 8 -14 C -6 -2 0 -10 -4 D 0 -8 10 0 2 E -10 14 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.416667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513888901463 Cumulative probabilities = A: 0.583333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 10 B 0 0 2 8 -14 C -6 -2 0 -10 -4 D 0 -8 10 0 2 E -10 14 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.416667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513888901463 Cumulative probabilities = A: 0.583333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2626: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) C B D A E (8) E A B D C (7) E A D C B (5) C E B D A (5) C D A B E (5) E B C A D (4) E B A D C (4) C B E D A (4) C B D E A (4) A E D B C (4) A D B E C (4) E C B A D (3) D C A B E (3) B C E D A (3) B A D E C (3) E C A D B (2) D A C E B (2) D A C B E (2) C D E A B (2) B E C A D (2) B E A D C (2) E B A C D (1) D B A C E (1) C E B A D (1) C D B E A (1) C D A E B (1) B D A E C (1) B D A C E (1) B C D A E (1) B A E D C (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -18 -8 0 B 20 0 -18 12 14 C 18 18 0 14 8 D 8 -12 -14 0 4 E 0 -14 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -18 -8 0 B 20 0 -18 12 14 C 18 18 0 14 8 D 8 -12 -14 0 4 E 0 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 E=26 B=14 A=11 D=8 so D is eliminated. Round 2 votes counts: C=44 E=26 B=15 A=15 so B is eliminated. Round 3 votes counts: C=48 E=30 A=22 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:214 D:193 E:187 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -18 -8 0 B 20 0 -18 12 14 C 18 18 0 14 8 D 8 -12 -14 0 4 E 0 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 -8 0 B 20 0 -18 12 14 C 18 18 0 14 8 D 8 -12 -14 0 4 E 0 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 -8 0 B 20 0 -18 12 14 C 18 18 0 14 8 D 8 -12 -14 0 4 E 0 -14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2627: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) B C E A D (8) B C A E D (7) C B E A D (6) D E A B C (5) D B C A E (5) E D A C B (4) D E A C B (4) D A E B C (4) E A D C B (3) D A B C E (3) C B A E D (3) B C E D A (3) B C A D E (3) A E C D B (3) A C B E D (3) D B E C A (2) D B A C E (2) C A E B D (2) B D C A E (2) B C D A E (2) A E D C B (2) A E C B D (2) A C E B D (2) E D C B A (1) E C A B D (1) D E C B A (1) D B C E A (1) D A E C B (1) D A B E C (1) C B E D A (1) B A C D E (1) A D E C B (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 16 2 B -4 0 2 12 10 C -4 -2 0 14 10 D -16 -12 -14 0 -22 E -2 -10 -10 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999044 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 16 2 B -4 0 2 12 10 C -4 -2 0 14 10 D -16 -12 -14 0 -22 E -2 -10 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993843 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 E=18 A=15 C=12 so C is eliminated. Round 2 votes counts: B=36 D=29 E=18 A=17 so A is eliminated. Round 3 votes counts: B=40 D=31 E=29 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:210 C:209 E:200 D:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 16 2 B -4 0 2 12 10 C -4 -2 0 14 10 D -16 -12 -14 0 -22 E -2 -10 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993843 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 16 2 B -4 0 2 12 10 C -4 -2 0 14 10 D -16 -12 -14 0 -22 E -2 -10 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993843 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 16 2 B -4 0 2 12 10 C -4 -2 0 14 10 D -16 -12 -14 0 -22 E -2 -10 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993843 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2628: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) C E B D A (10) A B D C E (10) E C D B A (9) B C E A D (8) D E C A B (7) A D B E C (7) A B C D E (7) D A E C B (6) D E C B A (5) D E A C B (5) E D C B A (3) D A E B C (2) A B C E D (2) E D C A B (1) E C D A B (1) C E B A D (1) C B E D A (1) C B E A D (1) B C A E D (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 4 0 -4 B 2 0 0 6 -2 C -4 0 0 6 6 D 0 -6 -6 0 0 E 4 2 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.675136 C: 0.324864 D: 0.000000 E: 0.000000 Sum of squares = 0.561345155284 Cumulative probabilities = A: 0.000000 B: 0.675136 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 0 -4 B 2 0 0 6 -2 C -4 0 0 6 6 D 0 -6 -6 0 0 E 4 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555613531 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=25 B=21 E=14 C=13 so C is eliminated. Round 2 votes counts: A=27 E=25 D=25 B=23 so B is eliminated. Round 3 votes counts: A=40 E=35 D=25 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:204 B:203 E:200 A:199 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 0 -4 B 2 0 0 6 -2 C -4 0 0 6 6 D 0 -6 -6 0 0 E 4 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555613531 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 0 -4 B 2 0 0 6 -2 C -4 0 0 6 6 D 0 -6 -6 0 0 E 4 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555613531 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 0 -4 B 2 0 0 6 -2 C -4 0 0 6 6 D 0 -6 -6 0 0 E 4 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555613531 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2629: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (13) D E A B C (9) D E B C A (7) E D A B C (6) B C E D A (6) A D E C B (6) C B D E A (5) B C D E A (5) C B A D E (4) B C E A D (4) A E D B C (4) A C B E D (4) E D B C A (3) D A E B C (2) C B D A E (2) A E D C B (2) A E B C D (2) A C B D E (2) E B A C D (1) E A D B C (1) D E C B A (1) D E B A C (1) D E A C B (1) D C B E A (1) D B C E A (1) D A E C B (1) C A B D E (1) B C A E D (1) A D E B C (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -8 -2 -4 B 10 0 8 0 2 C 8 -8 0 2 4 D 2 0 -2 0 6 E 4 -2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500878 C: 0.000000 D: 0.499122 E: 0.000000 Sum of squares = 0.500001538694 Cumulative probabilities = A: 0.000000 B: 0.500878 C: 0.500878 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -2 -4 B 10 0 8 0 2 C 8 -8 0 2 4 D 2 0 -2 0 6 E 4 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 A=24 B=16 E=11 so E is eliminated. Round 2 votes counts: D=33 C=25 A=25 B=17 so B is eliminated. Round 3 votes counts: C=41 D=33 A=26 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:210 C:203 D:203 E:196 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -2 -4 B 10 0 8 0 2 C 8 -8 0 2 4 D 2 0 -2 0 6 E 4 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -2 -4 B 10 0 8 0 2 C 8 -8 0 2 4 D 2 0 -2 0 6 E 4 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -2 -4 B 10 0 8 0 2 C 8 -8 0 2 4 D 2 0 -2 0 6 E 4 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2630: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (14) B A E C D (11) D C E A B (10) E A B D C (6) D E A B C (6) E D A B C (5) D E C A B (4) C D E B A (4) C B D A E (4) C B A D E (4) B C A E D (4) A B E D C (4) E A D B C (2) D E A C B (2) D A B E C (2) C D B E A (2) C D B A E (2) B A E D C (2) B A C E D (2) A E B D C (2) D C E B A (1) D B A E C (1) C E D A B (1) C E A B D (1) C D E A B (1) C B E A D (1) B A D E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -6 10 8 B 6 0 -2 14 10 C 6 2 0 4 2 D -10 -14 -4 0 -12 E -8 -10 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 10 8 B 6 0 -2 14 10 C 6 2 0 4 2 D -10 -14 -4 0 -12 E -8 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=26 B=20 E=13 A=7 so A is eliminated. Round 2 votes counts: C=34 D=26 B=25 E=15 so E is eliminated. Round 3 votes counts: C=34 D=33 B=33 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:207 A:203 E:196 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 10 8 B 6 0 -2 14 10 C 6 2 0 4 2 D -10 -14 -4 0 -12 E -8 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 10 8 B 6 0 -2 14 10 C 6 2 0 4 2 D -10 -14 -4 0 -12 E -8 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 10 8 B 6 0 -2 14 10 C 6 2 0 4 2 D -10 -14 -4 0 -12 E -8 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2631: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) A C D E B (10) A D C E B (7) C D B E A (6) B E C D A (6) A E B C D (6) D C B E A (5) B E D C A (5) C D A B E (4) C A D B E (4) B E A C D (4) E B D A C (3) E B A C D (2) D C A B E (2) C D B A E (2) A C E B D (2) A C B E D (2) A B E C D (2) E D B C A (1) E B D C A (1) E A B D C (1) E A B C D (1) D E B C A (1) D C B A E (1) D B E C A (1) D A C E B (1) C B E D A (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D E B (1) B D C E A (1) B C E D A (1) A E B D C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 8 12 -2 B 8 0 -6 6 4 C -8 6 0 18 10 D -12 -6 -18 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.33884297519 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 12 -2 B 8 0 -6 6 4 C -8 6 0 18 10 D -12 -6 -18 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975209 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=21 E=19 B=17 D=11 so D is eliminated. Round 2 votes counts: A=33 C=29 E=20 B=18 so B is eliminated. Round 3 votes counts: E=36 A=33 C=31 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:213 B:206 A:205 E:193 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 12 -2 B 8 0 -6 6 4 C -8 6 0 18 10 D -12 -6 -18 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975209 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 12 -2 B 8 0 -6 6 4 C -8 6 0 18 10 D -12 -6 -18 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975209 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 12 -2 B 8 0 -6 6 4 C -8 6 0 18 10 D -12 -6 -18 0 2 E 2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.363636 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975209 Cumulative probabilities = A: 0.272727 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2632: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) C A B E D (9) A C D B E (8) B E D C A (7) A C D E B (7) C B E D A (6) D E B A C (5) D A E B C (5) D A B E C (5) A D C E B (5) E B C D A (4) B E C D A (4) C E B A D (3) A D E C B (3) D B E C A (2) D B E A C (2) C B E A D (2) C A E B D (2) A D C B E (2) E D B A C (1) E C B D A (1) E B D A C (1) D E A B C (1) C E A B D (1) C A B D E (1) B C E D A (1) A D E B C (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -4 -8 0 B -4 0 -2 4 2 C 4 2 0 0 -4 D 8 -4 0 0 -4 E 0 -2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.308806 B: 0.147164 C: 0.000000 D: 0.073582 E: 0.470448 Sum of squares = 0.343754016245 Cumulative probabilities = A: 0.308806 B: 0.455970 C: 0.455970 D: 0.529552 E: 1.000000 A B C D E A 0 4 -4 -8 0 B -4 0 -2 4 2 C 4 2 0 0 -4 D 8 -4 0 0 -4 E 0 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.400000 Sum of squares = 0.300000121332 Cumulative probabilities = A: 0.300000 B: 0.500000 C: 0.500000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=24 D=20 E=16 B=12 so B is eliminated. Round 2 votes counts: A=28 E=27 C=25 D=20 so D is eliminated. Round 3 votes counts: A=38 E=37 C=25 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:203 C:201 B:200 D:200 A:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 -8 0 B -4 0 -2 4 2 C 4 2 0 0 -4 D 8 -4 0 0 -4 E 0 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.400000 Sum of squares = 0.300000121332 Cumulative probabilities = A: 0.300000 B: 0.500000 C: 0.500000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 0 B -4 0 -2 4 2 C 4 2 0 0 -4 D 8 -4 0 0 -4 E 0 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.400000 Sum of squares = 0.300000121332 Cumulative probabilities = A: 0.300000 B: 0.500000 C: 0.500000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 0 B -4 0 -2 4 2 C 4 2 0 0 -4 D 8 -4 0 0 -4 E 0 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.400000 Sum of squares = 0.300000121332 Cumulative probabilities = A: 0.300000 B: 0.500000 C: 0.500000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2633: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (20) A E B C D (16) A E B D C (12) C D B E A (9) C B E D A (8) E B A C D (6) D C A B E (6) B E C A D (4) D C A E B (3) D A C E B (3) B E A C D (3) A D E B C (3) E A B C D (2) D C B A E (1) D A E B C (1) B C E A D (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -4 -2 -6 B 4 0 -2 4 4 C 4 2 0 -2 2 D 2 -4 2 0 -6 E 6 -4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999977 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 -6 B 4 0 -2 4 4 C 4 2 0 -2 2 D 2 -4 2 0 -6 E 6 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=33 C=17 E=8 B=8 so E is eliminated. Round 2 votes counts: A=35 D=34 C=17 B=14 so B is eliminated. Round 3 votes counts: A=44 D=34 C=22 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:205 C:203 E:203 D:197 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -6 B 4 0 -2 4 4 C 4 2 0 -2 2 D 2 -4 2 0 -6 E 6 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -6 B 4 0 -2 4 4 C 4 2 0 -2 2 D 2 -4 2 0 -6 E 6 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -6 B 4 0 -2 4 4 C 4 2 0 -2 2 D 2 -4 2 0 -6 E 6 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999983 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2634: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) E A D C B (9) D E A C B (6) D B C E A (6) B A C E D (6) B C D A E (5) B C A D E (5) A B C E D (4) D E C B A (3) D E C A B (3) D E B C A (3) C D B A E (3) B C D E A (3) A E C D B (3) A E B C D (3) E D B A C (2) E B A C D (2) E A B C D (2) B E D C A (2) B E A C D (2) B C A E D (2) A C B E D (2) E B D A C (1) E A B D C (1) D C B E A (1) D C B A E (1) D C A E B (1) C D A B E (1) C B D A E (1) C B A D E (1) B D E C A (1) A E D C B (1) A E C B D (1) A D E C B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 16 -4 -12 B 0 0 0 -10 -6 C -16 0 0 -2 -10 D 4 10 2 0 -6 E 12 6 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 16 -4 -12 B 0 0 0 -10 -6 C -16 0 0 -2 -10 D 4 10 2 0 -6 E 12 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=26 B=26 D=24 A=18 C=6 so C is eliminated. Round 2 votes counts: D=28 B=28 E=26 A=18 so A is eliminated. Round 3 votes counts: E=35 B=34 D=31 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:205 A:200 B:192 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 16 -4 -12 B 0 0 0 -10 -6 C -16 0 0 -2 -10 D 4 10 2 0 -6 E 12 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 -4 -12 B 0 0 0 -10 -6 C -16 0 0 -2 -10 D 4 10 2 0 -6 E 12 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 -4 -12 B 0 0 0 -10 -6 C -16 0 0 -2 -10 D 4 10 2 0 -6 E 12 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2635: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) E D C B A (7) D E A B C (5) D B E C A (5) C B A E D (5) C A B E D (5) D E B C A (4) B C A D E (4) A C E B D (4) A B C D E (4) E C D B A (3) D B A E C (3) A E D C B (3) E D A C B (2) D B E A C (2) D A B E C (2) C E B D A (2) C E B A D (2) C E A B D (2) B D C E A (2) B C D E A (2) B A C D E (2) A E C D B (2) E D B C A (1) E C B D A (1) E C A D B (1) E C A B D (1) E B C D A (1) E A C D B (1) D E B A C (1) D B C E A (1) D B A C E (1) C B E A D (1) C A E B D (1) B D C A E (1) B C A E D (1) B A D C E (1) A D E C B (1) A D E B C (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -6 8 6 B 6 0 -8 6 8 C 6 8 0 10 6 D -8 -6 -10 0 -10 E -6 -8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 8 6 B 6 0 -8 6 8 C 6 8 0 10 6 D -8 -6 -10 0 -10 E -6 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 E=18 C=18 B=13 so B is eliminated. Round 2 votes counts: A=30 D=27 C=25 E=18 so E is eliminated. Round 3 votes counts: D=37 C=32 A=31 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:206 A:201 E:195 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 8 6 B 6 0 -8 6 8 C 6 8 0 10 6 D -8 -6 -10 0 -10 E -6 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 8 6 B 6 0 -8 6 8 C 6 8 0 10 6 D -8 -6 -10 0 -10 E -6 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 8 6 B 6 0 -8 6 8 C 6 8 0 10 6 D -8 -6 -10 0 -10 E -6 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2636: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) C E A B D (7) B A E C D (7) D B A E C (5) D B A C E (5) E C A B D (4) E A B C D (4) D B C A E (4) C E D A B (4) C E A D B (4) C D E A B (4) B D A C E (4) B A D E C (4) D C E B A (3) D B E A C (3) B A E D C (3) A B E C D (3) D C B E A (2) C D A B E (2) A C B E D (2) E D C A B (1) E D A B C (1) E B D A C (1) E B A D C (1) E B A C D (1) E A C D B (1) E A C B D (1) D C B A E (1) D C A B E (1) C A E B D (1) C A B E D (1) B E A C D (1) B C D A E (1) B A C D E (1) A E C B D (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 -2 -4 B -6 0 2 0 2 C -2 -2 0 4 14 D 2 0 -4 0 -2 E 4 -2 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37499999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -2 -4 B -6 0 2 0 2 C -2 -2 0 4 14 D 2 0 -4 0 -2 E 4 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 B=21 E=15 A=8 so A is eliminated. Round 2 votes counts: D=33 C=25 B=25 E=17 so E is eliminated. Round 3 votes counts: D=35 B=33 C=32 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:207 A:201 B:199 D:198 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 2 -2 -4 B -6 0 2 0 2 C -2 -2 0 4 14 D 2 0 -4 0 -2 E 4 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -2 -4 B -6 0 2 0 2 C -2 -2 0 4 14 D 2 0 -4 0 -2 E 4 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -2 -4 B -6 0 2 0 2 C -2 -2 0 4 14 D 2 0 -4 0 -2 E 4 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2637: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (5) C A E D B (5) E C D A B (4) E C A D B (4) D E B C A (4) C D A E B (4) C D A B E (4) A C E B D (4) E A C B D (3) E A B C D (3) D C B E A (3) D B E C A (3) B E A D C (3) B D E A C (3) B D A C E (3) B A E D C (3) B A E C D (3) A B E C D (3) E B D A C (2) E A C D B (2) E A B D C (2) D C B A E (2) D C A B E (2) D B C E A (2) D B C A E (2) B D C A E (2) B D A E C (2) B A D E C (2) A E C B D (2) E D C B A (1) E D B A C (1) E B A D C (1) E B A C D (1) C E A D B (1) C D B A E (1) C A D E B (1) C A D B E (1) B E D A C (1) B A D C E (1) B A C D E (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -2 -2 2 B -6 0 0 -4 0 C 2 0 0 0 -6 D 2 4 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333165 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 6 -2 -2 2 B -6 0 0 -4 0 C 2 0 0 0 -6 D 2 4 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 D=23 C=17 A=12 so A is eliminated. Round 2 votes counts: B=29 E=26 D=23 C=22 so C is eliminated. Round 3 votes counts: E=36 D=34 B=30 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:203 A:202 D:202 C:198 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 -2 2 B -6 0 0 -4 0 C 2 0 0 0 -6 D 2 4 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 2 B -6 0 0 -4 0 C 2 0 0 0 -6 D 2 4 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 2 B -6 0 0 -4 0 C 2 0 0 0 -6 D 2 4 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2638: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) D C A E B (6) D C B A E (5) B E A C D (5) E A B C D (4) D E B A C (4) D E A C B (4) D A E C B (4) C D A B E (4) C B A E D (4) A E D C B (4) E B A D C (3) E B A C D (3) D E A B C (3) C A B E D (3) C A B D E (3) A C E D B (3) E B D A C (2) D B C E A (2) C D B A E (2) C B D A E (2) C A D B E (2) B E C A D (2) B D C E A (2) B C E D A (2) B C E A D (2) B C D E A (2) E D A B C (1) D C E B A (1) D B E C A (1) C A D E B (1) B D E C A (1) B C D A E (1) B C A E D (1) B A E C D (1) A E C D B (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 2 2 0 B -4 0 -2 -10 -6 C -2 2 0 0 -2 D -2 10 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.788141 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.211859 Sum of squares = 0.666050230895 Cumulative probabilities = A: 0.788141 B: 0.788141 C: 0.788141 D: 0.788141 E: 1.000000 A B C D E A 0 4 2 2 0 B -4 0 -2 -10 -6 C -2 2 0 0 -2 D -2 10 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500139 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499861 Sum of squares = 0.500000038542 Cumulative probabilities = A: 0.500139 B: 0.500139 C: 0.500139 D: 0.500139 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=21 E=19 B=19 A=11 so A is eliminated. Round 2 votes counts: D=31 E=25 C=25 B=19 so B is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:205 A:204 E:203 C:199 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 0 B -4 0 -2 -10 -6 C -2 2 0 0 -2 D -2 10 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500139 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499861 Sum of squares = 0.500000038542 Cumulative probabilities = A: 0.500139 B: 0.500139 C: 0.500139 D: 0.500139 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 0 B -4 0 -2 -10 -6 C -2 2 0 0 -2 D -2 10 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500139 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499861 Sum of squares = 0.500000038542 Cumulative probabilities = A: 0.500139 B: 0.500139 C: 0.500139 D: 0.500139 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 0 B -4 0 -2 -10 -6 C -2 2 0 0 -2 D -2 10 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500139 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499861 Sum of squares = 0.500000038542 Cumulative probabilities = A: 0.500139 B: 0.500139 C: 0.500139 D: 0.500139 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2639: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (16) A C D E B (14) B E D A C (6) C D A E B (5) E B D C A (4) D C A E B (4) B E A D C (4) B E A C D (4) D C B E A (3) C A D E B (3) B A E C D (3) A E B C D (3) A C D B E (3) A B E C D (3) E B A D C (2) D E C B A (2) C D A B E (2) B C D E A (2) E D B C A (1) E D B A C (1) E D A C B (1) E A D C B (1) E A B D C (1) D C E B A (1) D C E A B (1) D B C E A (1) C A D B E (1) B D E C A (1) B C A D E (1) B A C E D (1) A D C E B (1) A C E B D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 -2 -4 B 6 0 12 10 8 C -4 -12 0 -2 -6 D 2 -10 2 0 -6 E 4 -8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -2 -4 B 6 0 12 10 8 C -4 -12 0 -2 -6 D 2 -10 2 0 -6 E 4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=28 D=12 E=11 C=11 so E is eliminated. Round 2 votes counts: B=44 A=30 D=15 C=11 so C is eliminated. Round 3 votes counts: B=44 A=34 D=22 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:204 A:196 D:194 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -2 -4 B 6 0 12 10 8 C -4 -12 0 -2 -6 D 2 -10 2 0 -6 E 4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 -4 B 6 0 12 10 8 C -4 -12 0 -2 -6 D 2 -10 2 0 -6 E 4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 -4 B 6 0 12 10 8 C -4 -12 0 -2 -6 D 2 -10 2 0 -6 E 4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2640: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (12) B A C E D (10) E C D A B (9) C E A B D (8) B A C D E (8) E D C A B (6) C B A E D (6) D E C A B (5) C E B A D (4) A B C D E (4) D E A B C (3) D A B E C (3) C A B E D (3) B D A E C (3) B A D E C (2) A B D C E (2) E D C B A (1) E D B C A (1) E C B A D (1) D E B A C (1) D E A C B (1) D B A E C (1) D A C E B (1) D A C B E (1) C E A D B (1) C D A E B (1) C A E B D (1) A D B C E (1) Total count = 100 A B C D E A 0 0 6 26 18 B 0 0 2 28 12 C -6 -2 0 12 26 D -26 -28 -12 0 -2 E -18 -12 -26 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.558045 B: 0.441955 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.506738345811 Cumulative probabilities = A: 0.558045 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 26 18 B 0 0 2 28 12 C -6 -2 0 12 26 D -26 -28 -12 0 -2 E -18 -12 -26 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999136 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=24 E=18 D=16 A=7 so A is eliminated. Round 2 votes counts: B=41 C=24 E=18 D=17 so D is eliminated. Round 3 votes counts: B=46 E=28 C=26 so C is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:225 B:221 C:215 E:173 D:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 26 18 B 0 0 2 28 12 C -6 -2 0 12 26 D -26 -28 -12 0 -2 E -18 -12 -26 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999136 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 26 18 B 0 0 2 28 12 C -6 -2 0 12 26 D -26 -28 -12 0 -2 E -18 -12 -26 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999136 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 26 18 B 0 0 2 28 12 C -6 -2 0 12 26 D -26 -28 -12 0 -2 E -18 -12 -26 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999136 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2641: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (14) A C D E B (9) E B D C A (8) B D E A C (8) B E D C A (6) D B A C E (4) C A E D B (4) C A D E B (4) E C B A D (3) E B C A D (3) D C A E B (3) D A C B E (3) B E C A D (3) B D A C E (3) B A D C E (3) E D B C A (2) C A E B D (2) A C B D E (2) E D C B A (1) E D C A B (1) E C A B D (1) E B C D A (1) D E C A B (1) D E B C A (1) D C E A B (1) D B E A C (1) D A C E B (1) C E A D B (1) C E A B D (1) B E D A C (1) B E A C D (1) B A C E D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 6 8 10 B 2 0 -6 -2 4 C -6 6 0 4 16 D -8 2 -4 0 18 E -10 -4 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102033 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 8 10 B 2 0 -6 -2 4 C -6 6 0 4 16 D -8 2 -4 0 18 E -10 -4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 E=20 D=15 C=12 so C is eliminated. Round 2 votes counts: A=36 B=27 E=22 D=15 so D is eliminated. Round 3 votes counts: A=43 B=32 E=25 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:211 C:210 D:204 B:199 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 8 10 B 2 0 -6 -2 4 C -6 6 0 4 16 D -8 2 -4 0 18 E -10 -4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 8 10 B 2 0 -6 -2 4 C -6 6 0 4 16 D -8 2 -4 0 18 E -10 -4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 8 10 B 2 0 -6 -2 4 C -6 6 0 4 16 D -8 2 -4 0 18 E -10 -4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2642: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (19) B A D E C (18) D E C B A (14) A B C E D (7) C E A D B (4) A C B E D (4) E C D A B (3) D E B C A (3) C D E A B (3) C A E D B (3) B D E A C (3) B A E D C (3) E D C B A (2) C D E B A (2) A C E B D (2) A B E D C (2) A B E C D (2) E B D A C (1) D B E C A (1) B E A D C (1) B D A C E (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 -4 -12 B -2 0 -12 -8 -14 C 8 12 0 2 -8 D 4 8 -2 0 -8 E 12 14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -8 -4 -12 B -2 0 -12 -8 -14 C 8 12 0 2 -8 D 4 8 -2 0 -8 E 12 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 A=19 D=18 E=6 so E is eliminated. Round 2 votes counts: C=34 B=27 D=20 A=19 so A is eliminated. Round 3 votes counts: C=40 B=40 D=20 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:207 D:201 A:189 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 -4 -12 B -2 0 -12 -8 -14 C 8 12 0 2 -8 D 4 8 -2 0 -8 E 12 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -4 -12 B -2 0 -12 -8 -14 C 8 12 0 2 -8 D 4 8 -2 0 -8 E 12 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -4 -12 B -2 0 -12 -8 -14 C 8 12 0 2 -8 D 4 8 -2 0 -8 E 12 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2643: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) C E A B D (9) C E B A D (8) E B A D C (5) D B A E C (4) D A B C E (4) C D A B E (4) C A E D B (4) C A D E B (4) B D E A C (4) E C B A D (3) D A C B E (3) C E B D A (3) B E D A C (3) A D B E C (3) E B C D A (2) E B A C D (2) B A E D C (2) A D B C E (2) A C E D B (2) E C B D A (1) E B D A C (1) E B C A D (1) E A B D C (1) E A B C D (1) D C A B E (1) D B E A C (1) D B C E A (1) C E A D B (1) C D B E A (1) C D A E B (1) C B D E A (1) B E C D A (1) B D A E C (1) B A D E C (1) A D E B C (1) A D C B E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 8 6 0 B -8 0 4 2 -2 C -8 -4 0 0 2 D -6 -2 0 0 0 E 0 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.418233 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.581767 Sum of squares = 0.513371647343 Cumulative probabilities = A: 0.418233 B: 0.418233 C: 0.418233 D: 0.418233 E: 1.000000 A B C D E A 0 8 8 6 0 B -8 0 4 2 -2 C -8 -4 0 0 2 D -6 -2 0 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=24 E=17 B=12 A=11 so A is eliminated. Round 2 votes counts: C=39 D=31 E=17 B=13 so B is eliminated. Round 3 votes counts: C=39 D=38 E=23 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:211 E:200 B:198 D:196 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 6 0 B -8 0 4 2 -2 C -8 -4 0 0 2 D -6 -2 0 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 0 B -8 0 4 2 -2 C -8 -4 0 0 2 D -6 -2 0 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 0 B -8 0 4 2 -2 C -8 -4 0 0 2 D -6 -2 0 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2644: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) C A E D B (8) C B E D A (7) B E C D A (7) B E D C A (6) A D B E C (6) A C D E B (6) E B D C A (5) A D C E B (5) D E B A C (4) A D E B C (4) D B E A C (3) D A B E C (3) C A B E D (3) E D B A C (2) B E D A C (2) B C E D A (2) E B C D A (1) D E A B C (1) D B A E C (1) D A E B C (1) C E B A D (1) C E A B D (1) C B E A D (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B D E (1) B D E C A (1) B D E A C (1) A D B C E (1) A C E D B (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -12 -12 -8 B 6 0 4 2 -2 C 12 -4 0 8 4 D 12 -2 -8 0 -14 E 8 2 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000011 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -6 -12 -12 -8 B 6 0 4 2 -2 C 12 -4 0 8 4 D 12 -2 -8 0 -14 E 8 2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000101 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=26 B=19 D=13 E=8 so E is eliminated. Round 2 votes counts: C=34 A=26 B=25 D=15 so D is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:210 E:210 B:205 D:194 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 -8 B 6 0 4 2 -2 C 12 -4 0 8 4 D 12 -2 -8 0 -14 E 8 2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000101 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 -8 B 6 0 4 2 -2 C 12 -4 0 8 4 D 12 -2 -8 0 -14 E 8 2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000101 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 -8 B 6 0 4 2 -2 C 12 -4 0 8 4 D 12 -2 -8 0 -14 E 8 2 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000101 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2645: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C A E D B (6) A C D B E (5) E D C B A (4) C A D E B (4) B A D E C (4) A C B E D (4) E C B D A (3) E B C D A (3) E B C A D (3) C E D A B (3) C E A D B (3) B E D C A (3) B D E A C (3) B A E D C (3) A C D E B (3) A C B D E (3) E D B C A (2) D B A E C (2) D A B C E (2) C E D B A (2) C E A B D (2) C D E A B (2) C A E B D (2) B E D A C (2) A D C B E (2) A C E D B (2) A B D E C (2) A B D C E (2) E C B A D (1) E C A B D (1) E B D C A (1) D E B C A (1) D C E A B (1) B E A C D (1) B D A E C (1) B A E C D (1) A D B C E (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -8 16 4 B -6 0 -22 -4 -14 C 8 22 0 28 2 D -16 4 -28 0 -24 E -4 14 -2 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 16 4 B -6 0 -22 -4 -14 C 8 22 0 28 2 D -16 4 -28 0 -24 E -4 14 -2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 C=24 B=18 D=6 so D is eliminated. Round 2 votes counts: A=29 E=26 C=25 B=20 so B is eliminated. Round 3 votes counts: A=40 E=35 C=25 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:230 E:216 A:209 B:177 D:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 16 4 B -6 0 -22 -4 -14 C 8 22 0 28 2 D -16 4 -28 0 -24 E -4 14 -2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 16 4 B -6 0 -22 -4 -14 C 8 22 0 28 2 D -16 4 -28 0 -24 E -4 14 -2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 16 4 B -6 0 -22 -4 -14 C 8 22 0 28 2 D -16 4 -28 0 -24 E -4 14 -2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2646: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) B C A E D (7) B A C E D (6) A B D E C (6) D E C A B (5) C D E B A (5) E D C A B (4) E C D A B (4) B A C D E (4) A D B E C (4) E A D C B (3) D A E B C (3) D A B E C (3) B A E C D (3) E C B A D (2) D C E A B (2) D C B A E (2) D B A C E (2) C E B D A (2) C B E D A (2) B C E A D (2) B A D C E (2) A B E D C (2) E D A C B (1) E C B D A (1) E C A D B (1) E B C A D (1) E B A C D (1) E A C D B (1) D C B E A (1) C E D A B (1) C D E A B (1) C D B E A (1) C D B A E (1) C B E A D (1) B E A C D (1) B C A D E (1) A E D B C (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 4 0 B 10 0 4 -10 6 C 8 -4 0 12 4 D -4 10 -12 0 -8 E 0 -6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.384615 D: 0.153846 E: 0.000000 Sum of squares = 0.384615384616 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.846154 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 4 0 B 10 0 4 -10 6 C 8 -4 0 12 4 D -4 10 -12 0 -8 E 0 -6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.384615 D: 0.153846 E: 0.000000 Sum of squares = 0.384615384632 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.846154 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=21 E=19 D=18 A=16 so A is eliminated. Round 2 votes counts: B=35 D=24 C=21 E=20 so E is eliminated. Round 3 votes counts: B=37 D=33 C=30 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:210 B:205 E:199 A:193 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 4 0 B 10 0 4 -10 6 C 8 -4 0 12 4 D -4 10 -12 0 -8 E 0 -6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.384615 D: 0.153846 E: 0.000000 Sum of squares = 0.384615384632 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.846154 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 4 0 B 10 0 4 -10 6 C 8 -4 0 12 4 D -4 10 -12 0 -8 E 0 -6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.384615 D: 0.153846 E: 0.000000 Sum of squares = 0.384615384632 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.846154 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 4 0 B 10 0 4 -10 6 C 8 -4 0 12 4 D -4 10 -12 0 -8 E 0 -6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.384615 D: 0.153846 E: 0.000000 Sum of squares = 0.384615384632 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.846154 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2647: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (16) E C B D A (13) C E A D B (11) E B C D A (10) C E A B D (6) A D C B E (6) A D B C E (6) D A B E C (4) C E B A D (4) B D E A C (4) C E B D A (3) C A E D B (3) A D B E C (3) D B A E C (2) D A B C E (2) C A D E B (2) A C D E B (2) E C B A D (1) B E D A C (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -6 -10 -6 B 8 0 -4 16 -12 C 6 4 0 10 -8 D 10 -16 -10 0 -4 E 6 12 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 -10 -6 B 8 0 -4 16 -12 C 6 4 0 10 -8 D 10 -16 -10 0 -4 E 6 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=24 B=21 A=18 D=8 so D is eliminated. Round 2 votes counts: C=29 E=24 A=24 B=23 so B is eliminated. Round 3 votes counts: A=42 E=29 C=29 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:206 B:204 D:190 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 -6 B 8 0 -4 16 -12 C 6 4 0 10 -8 D 10 -16 -10 0 -4 E 6 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 -6 B 8 0 -4 16 -12 C 6 4 0 10 -8 D 10 -16 -10 0 -4 E 6 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 -6 B 8 0 -4 16 -12 C 6 4 0 10 -8 D 10 -16 -10 0 -4 E 6 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2648: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E D C B (9) D E C A B (7) A E B D C (7) A B E C D (7) A E D B C (6) E D A C B (5) E A D C B (5) D E A C B (5) C B D E A (5) B C D E A (5) D C E B A (4) B C D A E (4) B C A D E (4) B A E C D (4) B A C E D (4) D C E A B (3) B C A E D (2) E D C A B (1) D A C B E (1) C D E B A (1) Total count = 100 A B C D E A 0 12 6 -4 -4 B -12 0 -14 -16 -6 C -6 14 0 -6 -12 D 4 16 6 0 0 E 4 6 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.564050 E: 0.435950 Sum of squares = 0.50820474015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.564050 E: 1.000000 A B C D E A 0 12 6 -4 -4 B -12 0 -14 -16 -6 C -6 14 0 -6 -12 D 4 16 6 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 D=20 C=17 E=11 so E is eliminated. Round 2 votes counts: A=34 D=26 B=23 C=17 so C is eliminated. Round 3 votes counts: D=38 A=34 B=28 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:211 A:205 C:195 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -4 -4 B -12 0 -14 -16 -6 C -6 14 0 -6 -12 D 4 16 6 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 -4 B -12 0 -14 -16 -6 C -6 14 0 -6 -12 D 4 16 6 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 -4 B -12 0 -14 -16 -6 C -6 14 0 -6 -12 D 4 16 6 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2649: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) E B A D C (6) B E D C A (6) A E B D C (6) C D B E A (5) A C D E B (5) E A B D C (4) D C B E A (4) D C B A E (4) B E D A C (4) B D E C A (4) A C D B E (4) E B A C D (3) E A B C D (3) C A D E B (3) A E B C D (3) E B D C A (2) E B C D A (2) D B C E A (2) C D A B E (2) B D C E A (2) A E C D B (2) A D C B E (2) E C B A D (1) E B D A C (1) E B C A D (1) D C A B E (1) D B C A E (1) D A C B E (1) C D B A E (1) C B E D A (1) C B D E A (1) B E C D A (1) B C D E A (1) B A E D C (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 10 8 -8 B 8 0 8 10 -2 C -10 -8 0 -4 -4 D -8 -10 4 0 -14 E 8 2 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 10 8 -8 B 8 0 8 10 -2 C -10 -8 0 -4 -4 D -8 -10 4 0 -14 E 8 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=23 B=19 D=13 C=13 so D is eliminated. Round 2 votes counts: A=33 E=23 C=22 B=22 so C is eliminated. Round 3 votes counts: A=39 B=38 E=23 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:212 A:201 C:187 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 10 8 -8 B 8 0 8 10 -2 C -10 -8 0 -4 -4 D -8 -10 4 0 -14 E 8 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 8 -8 B 8 0 8 10 -2 C -10 -8 0 -4 -4 D -8 -10 4 0 -14 E 8 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 8 -8 B 8 0 8 10 -2 C -10 -8 0 -4 -4 D -8 -10 4 0 -14 E 8 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2650: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) B D E A C (8) C A D B E (7) E B D A C (6) E C B D A (5) E B D C A (5) E C B A D (4) C A D E B (4) A D B C E (4) D A B C E (3) C A E D B (3) C A E B D (3) B E D C A (3) A C D E B (3) E C A B D (2) E B C D A (2) C E A D B (2) C E A B D (2) C B A D E (2) B E D A C (2) B D A E C (2) B D A C E (2) A D C B E (2) A C E D B (2) E D B A C (1) E C A D B (1) E A C D B (1) D E A B C (1) D B A E C (1) D B A C E (1) D A B E C (1) C E B D A (1) C B D A E (1) B D C E A (1) B D C A E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 4 2 6 6 B -4 0 -12 4 0 C -2 12 0 8 8 D -6 -4 -8 0 10 E -6 0 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 6 6 B -4 0 -12 4 0 C -2 12 0 8 8 D -6 -4 -8 0 10 E -6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 A=22 B=19 D=7 so D is eliminated. Round 2 votes counts: E=28 A=26 C=25 B=21 so B is eliminated. Round 3 votes counts: E=41 A=32 C=27 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:209 D:196 B:194 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 6 6 B -4 0 -12 4 0 C -2 12 0 8 8 D -6 -4 -8 0 10 E -6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 6 6 B -4 0 -12 4 0 C -2 12 0 8 8 D -6 -4 -8 0 10 E -6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 6 6 B -4 0 -12 4 0 C -2 12 0 8 8 D -6 -4 -8 0 10 E -6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2651: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) B A E C D (7) D C E B A (6) A B D C E (6) D A B C E (5) B A C E D (5) E C D B A (4) D E A C B (4) E C B A D (3) D B A C E (3) E D C A B (2) E C D A B (2) E C B D A (2) E C A B D (2) E B C A D (2) E A D C B (2) E A C B D (2) E A B C D (2) D E C A B (2) D C B E A (2) D B C A E (2) C E D B A (2) C D E B A (2) B C E A D (2) B A C D E (2) A D B E C (2) A D B C E (2) A B D E C (2) E B A C D (1) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B D A (1) C B E A D (1) B C A E D (1) B A D C E (1) A E B D C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 20 14 6 B -2 0 14 8 10 C -20 -14 0 6 -6 D -14 -8 -6 0 -10 E -6 -10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 20 14 6 B -2 0 14 8 10 C -20 -14 0 6 -6 D -14 -8 -6 0 -10 E -6 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 A=24 B=18 C=6 so C is eliminated. Round 2 votes counts: D=30 E=27 A=24 B=19 so B is eliminated. Round 3 votes counts: A=40 E=30 D=30 so E is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:215 E:200 C:183 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 20 14 6 B -2 0 14 8 10 C -20 -14 0 6 -6 D -14 -8 -6 0 -10 E -6 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 20 14 6 B -2 0 14 8 10 C -20 -14 0 6 -6 D -14 -8 -6 0 -10 E -6 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 20 14 6 B -2 0 14 8 10 C -20 -14 0 6 -6 D -14 -8 -6 0 -10 E -6 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2652: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (10) B D A C E (9) D B C E A (7) C E D B A (7) E C A D B (6) D B C A E (5) A E C B D (5) E A C B D (4) D B A C E (4) E C D A B (3) E C A B D (3) E A C D B (3) D C B E A (3) C D B E A (3) A E B D C (3) A E B C D (3) A B E D C (3) D B A E C (2) C E D A B (2) A B D C E (2) D C E B A (1) D A B E C (1) C E B D A (1) C E A D B (1) C E A B D (1) C D E B A (1) C B E D A (1) C B E A D (1) C B D E A (1) C B A E D (1) B D C E A (1) B D C A E (1) B A D C E (1) Total count = 100 A B C D E A 0 0 0 -6 0 B 0 0 4 2 12 C 0 -4 0 -6 8 D 6 -2 6 0 4 E 0 -12 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.196560 B: 0.803440 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.684151390907 Cumulative probabilities = A: 0.196560 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -6 0 B 0 0 4 2 12 C 0 -4 0 -6 8 D 6 -2 6 0 4 E 0 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000028958 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 C=20 E=19 B=12 so B is eliminated. Round 2 votes counts: D=34 A=27 C=20 E=19 so E is eliminated. Round 3 votes counts: D=34 A=34 C=32 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:209 D:207 C:199 A:197 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -6 0 B 0 0 4 2 12 C 0 -4 0 -6 8 D 6 -2 6 0 4 E 0 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000028958 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -6 0 B 0 0 4 2 12 C 0 -4 0 -6 8 D 6 -2 6 0 4 E 0 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000028958 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -6 0 B 0 0 4 2 12 C 0 -4 0 -6 8 D 6 -2 6 0 4 E 0 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000028958 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2653: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) B E D C A (6) B E C A D (6) A D C E B (5) D B A E C (4) B E D A C (4) B E C D A (4) B A E C D (4) A D C B E (4) A C D E B (4) A C D B E (4) E B C D A (3) D E C B A (3) D E B C A (3) D C A E B (3) D A B E C (3) C A D E B (3) A D B C E (3) E D B C A (2) B E A D C (2) B D E A C (2) A B C E D (2) E C B D A (1) D E A C B (1) D C E A B (1) D B E C A (1) D B E A C (1) D A E C B (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C E B A D (1) C B E A D (1) C A E B D (1) B E A C D (1) A D B E C (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 18 -8 10 B 0 0 12 -14 18 C -18 -12 0 -22 -8 D 8 14 22 0 18 E -10 -18 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 -8 10 B 0 0 12 -14 18 C -18 -12 0 -22 -8 D 8 14 22 0 18 E -10 -18 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=29 A=27 C=8 E=6 so E is eliminated. Round 2 votes counts: D=32 B=32 A=27 C=9 so C is eliminated. Round 3 votes counts: B=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:231 A:210 B:208 E:181 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 18 -8 10 B 0 0 12 -14 18 C -18 -12 0 -22 -8 D 8 14 22 0 18 E -10 -18 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 -8 10 B 0 0 12 -14 18 C -18 -12 0 -22 -8 D 8 14 22 0 18 E -10 -18 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 -8 10 B 0 0 12 -14 18 C -18 -12 0 -22 -8 D 8 14 22 0 18 E -10 -18 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2654: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) A D E C B (5) A D C B E (5) A B E C D (5) E D C B A (4) E B C D A (4) D E C B A (4) D E A C B (4) B C A E D (4) E D A C B (3) D C E B A (3) D C B E A (3) D A C B E (3) C B D E A (3) C B D A E (3) B C E A D (3) A E D B C (3) A E B D C (3) A E B C D (3) E C D B A (2) E A B D C (2) E A B C D (2) D E C A B (2) D C B A E (2) B E C A D (2) B C E D A (2) E C B D A (1) E A D B C (1) D C A B E (1) D A E C B (1) C E D B A (1) C E B D A (1) C B E D A (1) B C D A E (1) B A C E D (1) B A C D E (1) A D E B C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 2 4 B -8 0 -4 2 0 C -6 4 0 -2 -6 D -2 -2 2 0 2 E -4 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 2 4 B -8 0 -4 2 0 C -6 4 0 -2 -6 D -2 -2 2 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=23 E=19 B=14 C=9 so C is eliminated. Round 2 votes counts: A=35 D=23 E=21 B=21 so E is eliminated. Round 3 votes counts: A=40 D=33 B=27 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:200 E:200 B:195 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 2 4 B -8 0 -4 2 0 C -6 4 0 -2 -6 D -2 -2 2 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 2 4 B -8 0 -4 2 0 C -6 4 0 -2 -6 D -2 -2 2 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 2 4 B -8 0 -4 2 0 C -6 4 0 -2 -6 D -2 -2 2 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2655: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (16) D A E B C (9) C E A D B (6) B C D A E (6) A D E B C (5) C B D A E (4) C B A E D (4) B C A D E (4) D A B E C (3) C E B A D (3) C B E D A (3) B D A E C (3) A E D B C (3) E D A C B (2) E C A D B (2) E A D C B (2) E A C D B (2) D E A C B (2) D E A B C (2) D B A E C (2) C E D A B (2) C E B D A (2) B C A E D (2) E D A B C (1) E C A B D (1) E A C B D (1) D C E A B (1) C D E B A (1) C B D E A (1) B D A C E (1) B A E D C (1) B A D E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -16 10 0 B 8 0 -10 10 6 C 16 10 0 20 12 D -10 -10 -20 0 -8 E 0 -6 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 10 0 B 8 0 -10 10 6 C 16 10 0 20 12 D -10 -10 -20 0 -8 E 0 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 D=19 B=18 E=11 A=10 so A is eliminated. Round 2 votes counts: C=42 D=24 B=20 E=14 so E is eliminated. Round 3 votes counts: C=48 D=32 B=20 so B is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:207 E:195 A:193 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -16 10 0 B 8 0 -10 10 6 C 16 10 0 20 12 D -10 -10 -20 0 -8 E 0 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 10 0 B 8 0 -10 10 6 C 16 10 0 20 12 D -10 -10 -20 0 -8 E 0 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 10 0 B 8 0 -10 10 6 C 16 10 0 20 12 D -10 -10 -20 0 -8 E 0 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2656: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) A E B D C (8) B C A D E (7) A B E D C (6) C D B E A (5) D B C E A (4) C B D E A (4) B A D E C (4) D E C A B (3) C D E B A (3) C D E A B (3) C B D A E (3) B A E C D (3) A E D B C (3) E D C A B (2) E D A C B (2) E A D C B (2) D E A C B (2) D E A B C (2) D C E A B (2) C E D A B (2) C B A D E (2) B D C A E (2) B D A E C (2) B C A E D (2) B A E D C (2) B A C E D (2) A B E C D (2) D C E B A (1) C E A D B (1) C A E B D (1) B D A C E (1) B A C D E (1) A E C D B (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 6 14 8 B -4 0 22 8 6 C -6 -22 0 -12 -6 D -14 -8 12 0 2 E -8 -6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 14 8 B -4 0 22 8 6 C -6 -22 0 -12 -6 D -14 -8 12 0 2 E -8 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998405 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 A=22 E=14 D=14 so E is eliminated. Round 2 votes counts: A=32 B=26 C=24 D=18 so D is eliminated. Round 3 votes counts: A=38 C=32 B=30 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:216 D:196 E:195 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 14 8 B -4 0 22 8 6 C -6 -22 0 -12 -6 D -14 -8 12 0 2 E -8 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998405 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 14 8 B -4 0 22 8 6 C -6 -22 0 -12 -6 D -14 -8 12 0 2 E -8 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998405 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 14 8 B -4 0 22 8 6 C -6 -22 0 -12 -6 D -14 -8 12 0 2 E -8 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998405 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2657: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (14) A D E C B (11) E B C A D (8) E D A B C (6) C B D A E (6) B C E D A (5) C B E A D (4) A E D C B (4) E B C D A (3) D C B A E (3) C B D E A (3) C B A D E (3) B C E A D (3) A E D B C (3) A D E B C (3) A D C B E (3) D A E B C (2) D A C B E (2) A D C E B (2) E C B A D (1) E B D A C (1) E B A C D (1) E A B D C (1) D A E C B (1) D A C E B (1) D A B E C (1) C B E D A (1) B E C D A (1) B C D E A (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 14 24 -6 B -10 0 8 -14 -24 C -14 -8 0 -16 -22 D -24 14 16 0 -12 E 6 24 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 24 -6 B -10 0 8 -14 -24 C -14 -8 0 -16 -22 D -24 14 16 0 -12 E 6 24 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=27 C=17 B=11 D=10 so D is eliminated. Round 2 votes counts: E=35 A=34 C=20 B=11 so B is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:221 D:197 B:180 C:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 24 -6 B -10 0 8 -14 -24 C -14 -8 0 -16 -22 D -24 14 16 0 -12 E 6 24 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 24 -6 B -10 0 8 -14 -24 C -14 -8 0 -16 -22 D -24 14 16 0 -12 E 6 24 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 24 -6 B -10 0 8 -14 -24 C -14 -8 0 -16 -22 D -24 14 16 0 -12 E 6 24 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2658: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) E D C B A (9) E C A B D (6) D B A C E (6) E D B A C (5) D B A E C (5) E D B C A (4) D E C B A (4) A B C E D (4) E C D A B (3) D E B A C (3) C E D A B (3) C E A D B (3) C A B E D (3) B A E D C (3) B A D E C (3) D C E B A (2) D B E A C (2) C A E B D (2) B D A E C (2) B A E C D (2) A C B E D (2) A C B D E (2) A B D C E (2) A B C D E (2) E C D B A (1) E B D A C (1) D C E A B (1) C E A B D (1) C D E A B (1) C A E D B (1) C A B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -24 10 -4 2 B 24 0 10 -6 0 C -10 -10 0 -24 -8 D 4 6 24 0 -8 E -2 0 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.322800 C: 0.000000 D: 0.000000 E: 0.677200 Sum of squares = 0.562799929078 Cumulative probabilities = A: 0.000000 B: 0.322800 C: 0.322800 D: 0.322800 E: 1.000000 A B C D E A 0 -24 10 -4 2 B 24 0 10 -6 0 C -10 -10 0 -24 -8 D 4 6 24 0 -8 E -2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=23 B=20 C=15 A=13 so A is eliminated. Round 2 votes counts: E=29 B=29 D=23 C=19 so C is eliminated. Round 3 votes counts: E=39 B=37 D=24 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:213 E:207 A:192 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 10 -4 2 B 24 0 10 -6 0 C -10 -10 0 -24 -8 D 4 6 24 0 -8 E -2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 10 -4 2 B 24 0 10 -6 0 C -10 -10 0 -24 -8 D 4 6 24 0 -8 E -2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 10 -4 2 B 24 0 10 -6 0 C -10 -10 0 -24 -8 D 4 6 24 0 -8 E -2 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2659: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) A C E D B (8) D B E A C (6) D A C E B (6) E C A B D (5) B C E A D (5) E B C A D (4) D A C B E (4) B E C A D (4) E A C B D (3) D B A C E (3) B E D C A (3) B D C E A (3) E D A C B (2) E A C D B (2) C E A B D (2) C A B D E (2) B E C D A (2) B D E C A (2) B D C A E (2) A E C D B (2) A D C E B (2) A C D E B (2) A C D B E (2) E D A B C (1) E B D C A (1) E B D A C (1) E A D C B (1) D E B A C (1) D B C A E (1) D B A E C (1) D A E C B (1) C B A E D (1) C A D B E (1) C A B E D (1) B C D A E (1) B C A E D (1) A E C B D (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 16 2 18 4 B -16 0 -16 4 -8 C -2 16 0 14 12 D -18 -4 -14 0 -16 E -4 8 -12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 18 4 B -16 0 -16 4 -8 C -2 16 0 14 12 D -18 -4 -14 0 -16 E -4 8 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 E=20 A=19 C=15 so C is eliminated. Round 2 votes counts: A=31 B=24 D=23 E=22 so E is eliminated. Round 3 votes counts: A=44 B=30 D=26 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:220 E:204 B:182 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 18 4 B -16 0 -16 4 -8 C -2 16 0 14 12 D -18 -4 -14 0 -16 E -4 8 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 18 4 B -16 0 -16 4 -8 C -2 16 0 14 12 D -18 -4 -14 0 -16 E -4 8 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 18 4 B -16 0 -16 4 -8 C -2 16 0 14 12 D -18 -4 -14 0 -16 E -4 8 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2660: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) A D E B C (6) E C B A D (5) B C E D A (5) D A C B E (4) B D C E A (4) A E C D B (4) A E B C D (4) A D E C B (4) A D B E C (4) E C B D A (3) D B C E A (3) B D C A E (3) B C D E A (3) A E C B D (3) A B D C E (3) E C A B D (2) D C B E A (2) D B C A E (2) D A E C B (2) D A C E B (2) C E B D A (2) B A E C D (2) B A C E D (2) A E D C B (2) A D B C E (2) E D C A B (1) E C D B A (1) E C A D B (1) E B C A D (1) E A C D B (1) D C E B A (1) D C B A E (1) D B A C E (1) B E C A D (1) B C E A D (1) B A C D E (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 16 2 26 B -12 0 18 -8 8 C -16 -18 0 -14 2 D -2 8 14 0 14 E -26 -8 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 2 26 B -12 0 18 -8 8 C -16 -18 0 -14 2 D -2 8 14 0 14 E -26 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=27 B=22 E=15 C=2 so C is eliminated. Round 2 votes counts: A=34 D=27 B=22 E=17 so E is eliminated. Round 3 votes counts: A=38 B=33 D=29 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 D:217 B:203 C:177 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 2 26 B -12 0 18 -8 8 C -16 -18 0 -14 2 D -2 8 14 0 14 E -26 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 2 26 B -12 0 18 -8 8 C -16 -18 0 -14 2 D -2 8 14 0 14 E -26 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 2 26 B -12 0 18 -8 8 C -16 -18 0 -14 2 D -2 8 14 0 14 E -26 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999959261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2661: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (6) D E A C B (5) B E A C D (5) E D A C B (4) C D B A E (4) C B D A E (4) B E C A D (4) B C E D A (4) B A E C D (4) A E D B C (4) D C E A B (3) D A E C B (3) C D B E A (3) B E A D C (3) E D C B A (2) E D A B C (2) E C B D A (2) E B D C A (2) E B A D C (2) E A D C B (2) E A D B C (2) E A B D C (2) D E C A B (2) D C A E B (2) C D E B A (2) C D A B E (2) C B D E A (2) B E C D A (2) A D E B C (2) E D C A B (1) D A C E B (1) C E B D A (1) C B A D E (1) C A D B E (1) C A B D E (1) B C E A D (1) B C A E D (1) A E D C B (1) A E B D C (1) A D E C B (1) A D C B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 -6 -16 B 10 0 -2 0 2 C 6 2 0 2 -18 D 6 0 -2 0 -6 E 16 -2 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.090909 D: 0.000000 E: 0.090909 Sum of squares = 0.685950413164 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 -10 -6 -6 -16 B 10 0 -2 0 2 C 6 2 0 2 -18 D 6 0 -2 0 -6 E 16 -2 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.090909 D: 0.000000 E: 0.090909 Sum of squares = 0.685950412954 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=21 C=21 D=16 A=12 so A is eliminated. Round 2 votes counts: B=32 E=27 C=21 D=20 so D is eliminated. Round 3 votes counts: E=40 B=32 C=28 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:219 B:205 D:199 C:196 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -6 -16 B 10 0 -2 0 2 C 6 2 0 2 -18 D 6 0 -2 0 -6 E 16 -2 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.090909 D: 0.000000 E: 0.090909 Sum of squares = 0.685950412954 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -6 -16 B 10 0 -2 0 2 C 6 2 0 2 -18 D 6 0 -2 0 -6 E 16 -2 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.090909 D: 0.000000 E: 0.090909 Sum of squares = 0.685950412954 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -6 -16 B 10 0 -2 0 2 C 6 2 0 2 -18 D 6 0 -2 0 -6 E 16 -2 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.090909 D: 0.000000 E: 0.090909 Sum of squares = 0.685950412954 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2662: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) A E D C B (8) D B C E A (7) B C A D E (7) A E C D B (6) D E C B A (5) C B E D A (5) B C A E D (5) A B C E D (5) E D C A B (2) D E A C B (2) D E A B C (2) D C B E A (2) D A E C B (2) C E B D A (2) C B E A D (2) B D C A E (2) A D B E C (2) E C D B A (1) E C D A B (1) E C A D B (1) E A C D B (1) E A C B D (1) D E C A B (1) D B E C A (1) D A E B C (1) D A B E C (1) C D E B A (1) B D A C E (1) B C E D A (1) B A C E D (1) B A C D E (1) A E C B D (1) A D E C B (1) A D E B C (1) A C B E D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -18 -6 -2 B 14 0 8 2 20 C 18 -8 0 14 16 D 6 -2 -14 0 10 E 2 -20 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -18 -6 -2 B 14 0 8 2 20 C 18 -8 0 14 16 D 6 -2 -14 0 10 E 2 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=28 D=24 C=10 E=7 so E is eliminated. Round 2 votes counts: B=31 A=30 D=26 C=13 so C is eliminated. Round 3 votes counts: B=40 A=31 D=29 so D is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:220 D:200 A:180 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -18 -6 -2 B 14 0 8 2 20 C 18 -8 0 14 16 D 6 -2 -14 0 10 E 2 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -18 -6 -2 B 14 0 8 2 20 C 18 -8 0 14 16 D 6 -2 -14 0 10 E 2 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -18 -6 -2 B 14 0 8 2 20 C 18 -8 0 14 16 D 6 -2 -14 0 10 E 2 -20 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2663: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (12) B C E A D (10) A D C E B (7) D E B C A (5) D A B E C (5) A D E C B (5) A C E D B (5) B D E C A (4) B C E D A (4) E C B D A (3) D E C A B (3) D B E C A (3) B E C D A (3) A B D C E (3) E D C B A (2) E C D B A (2) D A E B C (2) C E B A D (2) C E A B D (2) C A E B D (2) B E D C A (2) B C A E D (2) A C B E D (2) E C B A D (1) D E C B A (1) D E B A C (1) D E A C B (1) C B E A D (1) B D A E C (1) A D C B E (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -4 -8 0 B -6 0 -4 -12 -12 C 4 4 0 -18 -12 D 8 12 18 0 12 E 0 12 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -8 0 B -6 0 -4 -12 -12 C 4 4 0 -18 -12 D 8 12 18 0 12 E 0 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=26 A=26 E=8 C=7 so C is eliminated. Round 2 votes counts: D=33 A=28 B=27 E=12 so E is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:206 A:197 C:189 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -8 0 B -6 0 -4 -12 -12 C 4 4 0 -18 -12 D 8 12 18 0 12 E 0 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -8 0 B -6 0 -4 -12 -12 C 4 4 0 -18 -12 D 8 12 18 0 12 E 0 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -8 0 B -6 0 -4 -12 -12 C 4 4 0 -18 -12 D 8 12 18 0 12 E 0 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2664: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (12) B D C A E (9) D B E A C (6) D B C A E (6) E A D B C (4) D B C E A (4) C E A D B (4) C A B E D (4) E D A B C (3) E A B D C (3) D C B A E (3) D B E C A (3) C D B A E (3) C D A B E (3) B D A C E (3) B C A D E (3) A C E B D (3) A C B E D (3) E C A D B (2) E A B C D (2) C A E D B (2) E A C D B (1) E A C B D (1) D E C B A (1) D E B A C (1) D C B E A (1) C D E A B (1) C B A D E (1) C A D B E (1) B D E A C (1) B D A E C (1) B A D C E (1) B A C E D (1) A E C B D (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -26 2 24 B -4 0 6 2 16 C 26 -6 0 -2 38 D -2 -2 2 0 4 E -24 -16 -38 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.722222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.561728394935 Cumulative probabilities = A: 0.166667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -26 2 24 B -4 0 6 2 16 C 26 -6 0 -2 38 D -2 -2 2 0 4 E -24 -16 -38 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.722222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.561728394638 Cumulative probabilities = A: 0.166667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 B=19 E=16 A=9 so A is eliminated. Round 2 votes counts: C=37 D=25 B=20 E=18 so E is eliminated. Round 3 votes counts: C=42 D=32 B=26 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:228 B:210 A:202 D:201 E:159 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -26 2 24 B -4 0 6 2 16 C 26 -6 0 -2 38 D -2 -2 2 0 4 E -24 -16 -38 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.722222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.561728394638 Cumulative probabilities = A: 0.166667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -26 2 24 B -4 0 6 2 16 C 26 -6 0 -2 38 D -2 -2 2 0 4 E -24 -16 -38 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.722222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.561728394638 Cumulative probabilities = A: 0.166667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -26 2 24 B -4 0 6 2 16 C 26 -6 0 -2 38 D -2 -2 2 0 4 E -24 -16 -38 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.722222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.561728394638 Cumulative probabilities = A: 0.166667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2665: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) C A D E B (8) C A D B E (7) D E B C A (6) B A C E D (6) A C B E D (6) E B D A C (5) D C A E B (5) D B E C A (4) C A B E D (4) B E A C D (4) E D B A C (3) D E B A C (3) C D A E B (3) C A B D E (3) C D A B E (2) B D C E A (2) B A E C D (2) A E C B D (2) A C E B D (2) A B C E D (2) D E C B A (1) D E C A B (1) D C E A B (1) D B C E A (1) D B C A E (1) B E C D A (1) B E C A D (1) B D E C A (1) B C A D E (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -6 2 14 B 2 0 6 4 16 C 6 -6 0 14 12 D -2 -4 -14 0 2 E -14 -16 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 2 14 B 2 0 6 4 16 C 6 -6 0 14 12 D -2 -4 -14 0 2 E -14 -16 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991331 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=27 B=27 D=23 A=15 E=8 so E is eliminated. Round 2 votes counts: B=32 C=27 D=26 A=15 so A is eliminated. Round 3 votes counts: C=39 B=35 D=26 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:213 A:204 D:191 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 2 14 B 2 0 6 4 16 C 6 -6 0 14 12 D -2 -4 -14 0 2 E -14 -16 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991331 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 14 B 2 0 6 4 16 C 6 -6 0 14 12 D -2 -4 -14 0 2 E -14 -16 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991331 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 14 B 2 0 6 4 16 C 6 -6 0 14 12 D -2 -4 -14 0 2 E -14 -16 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991331 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2666: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (14) E A C D B (13) D E A C B (10) C A E B D (9) B C A E D (7) E D A C B (6) D B E A C (6) E C A D B (4) D B C E A (3) D B C A E (3) A E C D B (3) E A D C B (2) E A C B D (2) D E B C A (2) D E B A C (2) D B E C A (2) C E A B D (2) B C A D E (2) A C E B D (2) E C A B D (1) D E A B C (1) D B A E C (1) B D A C E (1) B C D A E (1) A E C B D (1) Total count = 100 A B C D E A 0 12 0 -4 -12 B -12 0 -10 -16 -20 C 0 10 0 -6 -12 D 4 16 6 0 -4 E 12 20 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 0 -4 -12 B -12 0 -10 -16 -20 C 0 10 0 -6 -12 D 4 16 6 0 -4 E 12 20 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=28 B=25 C=11 A=6 so A is eliminated. Round 2 votes counts: E=32 D=30 B=25 C=13 so C is eliminated. Round 3 votes counts: E=45 D=30 B=25 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:211 A:198 C:196 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 0 -4 -12 B -12 0 -10 -16 -20 C 0 10 0 -6 -12 D 4 16 6 0 -4 E 12 20 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -4 -12 B -12 0 -10 -16 -20 C 0 10 0 -6 -12 D 4 16 6 0 -4 E 12 20 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -4 -12 B -12 0 -10 -16 -20 C 0 10 0 -6 -12 D 4 16 6 0 -4 E 12 20 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2667: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) A B C E D (7) A B E C D (6) B E D A C (5) B A D C E (5) A B C D E (5) D E C B A (4) B A D E C (4) E D C A B (3) E A C D B (3) D C E B A (3) B D E C A (3) B A E C D (3) E D C B A (2) E C D A B (2) E A B D C (2) D E B C A (2) C E D A B (2) C E A D B (2) C D E A B (2) B D E A C (2) B D A E C (2) B D A C E (2) A E C B D (2) E A D B C (1) D E C A B (1) D C E A B (1) D C B E A (1) D B C E A (1) C D B E A (1) C A E D B (1) B E A D C (1) B D C E A (1) B D C A E (1) B C D A E (1) B A E D C (1) B A C E D (1) A E C D B (1) A E B C D (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 32 16 10 B 8 0 30 32 26 C -32 -30 0 4 -2 D -16 -32 -4 0 2 E -10 -26 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 32 16 10 B 8 0 30 32 26 C -32 -30 0 4 -2 D -16 -32 -4 0 2 E -10 -26 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=26 E=13 D=13 C=8 so C is eliminated. Round 2 votes counts: B=40 A=27 E=17 D=16 so D is eliminated. Round 3 votes counts: B=43 E=30 A=27 so A is eliminated. Round 4 votes counts: B=63 E=37 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:248 A:225 E:182 D:175 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 32 16 10 B 8 0 30 32 26 C -32 -30 0 4 -2 D -16 -32 -4 0 2 E -10 -26 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 32 16 10 B 8 0 30 32 26 C -32 -30 0 4 -2 D -16 -32 -4 0 2 E -10 -26 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 32 16 10 B 8 0 30 32 26 C -32 -30 0 4 -2 D -16 -32 -4 0 2 E -10 -26 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2668: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (6) D C A B E (5) A E B C D (5) E B A C D (4) D A C E B (4) C B D E A (4) C A B D E (4) E B D C A (3) D E A B C (3) D C B E A (3) C B A E D (3) A D E B C (3) A C B D E (3) A B E C D (3) E D B C A (2) E D B A C (2) E B C D A (2) E B C A D (2) E B A D C (2) E A B D C (2) E A B C D (2) D E C B A (2) D A C B E (2) B E C A D (2) B C E D A (2) A D E C B (2) A D C B E (2) A C B E D (2) A B C E D (2) E D A B C (1) D E B C A (1) D E A C B (1) D C E A B (1) D C B A E (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C B D A E (1) B E C D A (1) B C E A D (1) B C A E D (1) A E D B C (1) A E B D C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 16 10 -2 10 B -16 0 -4 6 -6 C -10 4 0 -2 -6 D 2 -6 2 0 6 E -10 6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.513888888856 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 -2 10 B -16 0 -4 6 -6 C -10 4 0 -2 -6 D 2 -6 2 0 6 E -10 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.513888888843 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=26 E=22 C=16 B=7 so B is eliminated. Round 2 votes counts: D=29 A=26 E=25 C=20 so C is eliminated. Round 3 votes counts: D=37 A=34 E=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:217 D:202 E:198 C:193 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 -2 10 B -16 0 -4 6 -6 C -10 4 0 -2 -6 D 2 -6 2 0 6 E -10 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.513888888843 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 -2 10 B -16 0 -4 6 -6 C -10 4 0 -2 -6 D 2 -6 2 0 6 E -10 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.513888888843 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 -2 10 B -16 0 -4 6 -6 C -10 4 0 -2 -6 D 2 -6 2 0 6 E -10 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.083333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.513888888843 Cumulative probabilities = A: 0.250000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2669: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (5) D A E B C (5) C B E A D (5) C A E B D (5) B D C E A (5) D A E C B (4) B D E C A (4) A E D C B (4) A D E C B (4) A C E D B (4) D B E A C (3) D B C A E (3) C E B A D (3) B E C A D (3) B C E D A (3) B C E A D (3) B C D E A (3) A D C E B (3) E C A B D (2) D B A E C (2) C E A B D (2) C D A B E (2) C B D A E (2) C A D E B (2) B D E A C (2) A E C D B (2) E C B A D (1) E B D A C (1) E B A C D (1) E A C B D (1) E A B C D (1) D C A B E (1) D B E C A (1) D B A C E (1) D A B E C (1) D A B C E (1) C A E D B (1) C A D B E (1) B E C D A (1) B D C A E (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -8 -2 0 B -4 0 0 0 -4 C 8 0 0 -2 2 D 2 0 2 0 12 E 0 4 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181998 C: 0.000000 D: 0.818002 E: 0.000000 Sum of squares = 0.702250723696 Cumulative probabilities = A: 0.000000 B: 0.181998 C: 0.181998 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -2 0 B -4 0 0 0 -4 C 8 0 0 -2 2 D 2 0 2 0 12 E 0 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555890045 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 C=23 A=18 E=7 so E is eliminated. Round 2 votes counts: D=27 B=27 C=26 A=20 so A is eliminated. Round 3 votes counts: D=38 C=34 B=28 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:208 C:204 A:197 B:196 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -2 0 B -4 0 0 0 -4 C 8 0 0 -2 2 D 2 0 2 0 12 E 0 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555890045 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -2 0 B -4 0 0 0 -4 C 8 0 0 -2 2 D 2 0 2 0 12 E 0 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555890045 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -2 0 B -4 0 0 0 -4 C 8 0 0 -2 2 D 2 0 2 0 12 E 0 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555890045 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2670: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) A E B C D (7) D C B E A (6) C D B E A (6) A B E C D (6) D C B A E (5) A E B D C (5) E B C A D (4) D C A B E (4) C B E A D (4) E B C D A (3) E A B C D (3) D C E B A (3) C D B A E (3) A D E B C (3) D E B A C (2) D A E B C (2) D A C E B (2) C B E D A (2) C B D E A (2) B E C A D (2) B E A C D (2) E D B C A (1) E B A D C (1) E A B D C (1) D E B C A (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C B E (1) C D A B E (1) C A D B E (1) C A B E D (1) B A C E D (1) A E D B C (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 -2 6 -6 B 14 0 8 8 -2 C 2 -8 0 16 -8 D -6 -8 -16 0 -8 E 6 2 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -2 6 -6 B 14 0 8 8 -2 C 2 -8 0 16 -8 D -6 -8 -16 0 -8 E 6 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 E=22 C=20 B=5 so B is eliminated. Round 2 votes counts: D=29 E=26 A=25 C=20 so C is eliminated. Round 3 votes counts: D=41 E=32 A=27 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:212 C:201 A:192 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -2 6 -6 B 14 0 8 8 -2 C 2 -8 0 16 -8 D -6 -8 -16 0 -8 E 6 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 6 -6 B 14 0 8 8 -2 C 2 -8 0 16 -8 D -6 -8 -16 0 -8 E 6 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 6 -6 B 14 0 8 8 -2 C 2 -8 0 16 -8 D -6 -8 -16 0 -8 E 6 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2671: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (13) A B C D E (9) D C B A E (7) C B A D E (7) A B C E D (7) E D A B C (6) A B E C D (6) E D C B A (5) E A B D C (4) E A B C D (4) D E C A B (3) D E A B C (3) D C E B A (3) C D B A E (3) B C A E D (3) A E B D C (3) C B A E D (2) B A C E D (2) B A C D E (2) E D A C B (1) E C D B A (1) D C B E A (1) D A B E C (1) D A B C E (1) C D E B A (1) C B E A D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 2 8 B 2 0 4 2 6 C 0 -4 0 -4 0 D -2 -2 4 0 10 E -8 -6 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 8 B 2 0 4 2 6 C 0 -4 0 -4 0 D -2 -2 4 0 10 E -8 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=26 E=21 C=14 B=7 so B is eliminated. Round 2 votes counts: D=32 A=30 E=21 C=17 so C is eliminated. Round 3 votes counts: A=42 D=36 E=22 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:207 D:205 A:204 C:196 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 2 8 B 2 0 4 2 6 C 0 -4 0 -4 0 D -2 -2 4 0 10 E -8 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 8 B 2 0 4 2 6 C 0 -4 0 -4 0 D -2 -2 4 0 10 E -8 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 8 B 2 0 4 2 6 C 0 -4 0 -4 0 D -2 -2 4 0 10 E -8 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2672: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) A D B E C (8) C E B A D (7) B E D C A (7) C E B D A (6) B D E C A (6) A C E B D (6) D B E A C (5) D B E C A (4) D A B E C (4) E D B C A (3) D E B C A (3) C E A B D (3) B C E D A (3) A D C E B (3) E C B D A (2) C B E A D (2) A D E C B (2) A D E B C (2) A D C B E (2) A D B C E (2) A C E D B (2) E B D C A (1) D B A E C (1) C B E D A (1) C B A E D (1) C A B E D (1) C A B D E (1) B E C D A (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -20 6 -8 B 6 0 2 16 4 C 20 -2 0 -8 2 D -6 -16 8 0 -10 E 8 -4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999621 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 6 -8 B 6 0 2 16 4 C 20 -2 0 -8 2 D -6 -16 8 0 -10 E 8 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=30 A=30 D=17 B=17 E=6 so E is eliminated. Round 2 votes counts: C=32 A=30 D=20 B=18 so B is eliminated. Round 3 votes counts: C=36 D=34 A=30 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:214 C:206 E:206 D:188 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -20 6 -8 B 6 0 2 16 4 C 20 -2 0 -8 2 D -6 -16 8 0 -10 E 8 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 6 -8 B 6 0 2 16 4 C 20 -2 0 -8 2 D -6 -16 8 0 -10 E 8 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 6 -8 B 6 0 2 16 4 C 20 -2 0 -8 2 D -6 -16 8 0 -10 E 8 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2673: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (17) B E A C D (13) C D A E B (6) B D C E A (6) B E A D C (5) A E B C D (5) E A B C D (4) C D B E A (4) A E C D B (4) D C B E A (3) D C B A E (3) C D E A B (3) A D C E B (3) E B A C D (2) E A C B D (2) D C A B E (2) D A C E B (2) B E D C A (2) B C D E A (2) A E B D C (2) A C D E B (2) D B C A E (1) D A C B E (1) C E D B A (1) C D E B A (1) B E D A C (1) B E C D A (1) B D E C A (1) B A E D C (1) Total count = 100 A B C D E A 0 6 -6 -14 -2 B -6 0 -8 -6 -8 C 6 8 0 0 14 D 14 6 0 0 14 E 2 8 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.432016 D: 0.567984 E: 0.000000 Sum of squares = 0.50924374028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.432016 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -14 -2 B -6 0 -8 -6 -8 C 6 8 0 0 14 D 14 6 0 0 14 E 2 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=29 A=16 C=15 E=8 so E is eliminated. Round 2 votes counts: B=34 D=29 A=22 C=15 so C is eliminated. Round 3 votes counts: D=44 B=34 A=22 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:214 A:192 E:191 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -14 -2 B -6 0 -8 -6 -8 C 6 8 0 0 14 D 14 6 0 0 14 E 2 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -14 -2 B -6 0 -8 -6 -8 C 6 8 0 0 14 D 14 6 0 0 14 E 2 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -14 -2 B -6 0 -8 -6 -8 C 6 8 0 0 14 D 14 6 0 0 14 E 2 8 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2674: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) C B A D E (8) E D A B C (7) D E B A C (5) E A D C B (4) D E A B C (4) D B E A C (4) C A B D E (4) B C D A E (4) A C E B D (4) B E C D A (3) B C D E A (3) A E C D B (3) E D A C B (2) E B D C A (2) D B C A E (2) D A C B E (2) C B A E D (2) C A D B E (2) C A B E D (2) B D E C A (2) B D C E A (2) B D C A E (2) A E D C B (2) A C E D B (2) E A D B C (1) E A C D B (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C E A (1) D A E C B (1) D A C E B (1) C B D A E (1) B C E A D (1) B C A D E (1) A E C B D (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 14 -14 -4 B 4 0 6 -12 -6 C -14 -6 0 -10 -6 D 14 12 10 0 6 E 4 6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 -14 -4 B 4 0 6 -12 -6 C -14 -6 0 -10 -6 D 14 12 10 0 6 E 4 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=21 C=19 B=18 A=15 so A is eliminated. Round 2 votes counts: E=33 C=26 D=23 B=18 so B is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:205 A:196 B:196 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 14 -14 -4 B 4 0 6 -12 -6 C -14 -6 0 -10 -6 D 14 12 10 0 6 E 4 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 -14 -4 B 4 0 6 -12 -6 C -14 -6 0 -10 -6 D 14 12 10 0 6 E 4 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 -14 -4 B 4 0 6 -12 -6 C -14 -6 0 -10 -6 D 14 12 10 0 6 E 4 6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2675: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) A D C E B (7) E B A C D (6) D C A E B (5) D C A B E (5) D A B C E (5) C D A E B (5) B D A E C (5) C D E A B (3) B E C A D (3) E C B A D (2) E A C B D (2) D C B A E (2) D A C B E (2) C E D A B (2) C E B D A (2) B E C D A (2) B E A D C (2) B D C E A (2) B A E D C (2) A E D B C (2) A E B D C (2) A C D E B (2) E C B D A (1) E B C D A (1) D C B E A (1) D B C E A (1) D B A C E (1) D A C E B (1) C D E B A (1) C D B E A (1) C B E D A (1) C A D E B (1) B E D C A (1) B E A C D (1) B C E D A (1) B C D E A (1) B A D E C (1) A E D C B (1) A E C D B (1) A E B C D (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -2 -4 12 B -4 0 -2 -8 -16 C 2 2 0 -4 8 D 4 8 4 0 12 E -12 16 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -4 12 B -4 0 -2 -8 -16 C 2 2 0 -4 8 D 4 8 4 0 12 E -12 16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 B=21 A=21 E=19 C=16 so C is eliminated. Round 2 votes counts: D=33 E=23 B=22 A=22 so B is eliminated. Round 3 votes counts: D=41 E=34 A=25 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:205 C:204 E:192 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -4 12 B -4 0 -2 -8 -16 C 2 2 0 -4 8 D 4 8 4 0 12 E -12 16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -4 12 B -4 0 -2 -8 -16 C 2 2 0 -4 8 D 4 8 4 0 12 E -12 16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -4 12 B -4 0 -2 -8 -16 C 2 2 0 -4 8 D 4 8 4 0 12 E -12 16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2676: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) B D C A E (8) B A D C E (7) E C A D B (6) D C B E A (6) A E C B D (6) A E B C D (6) C E D A B (5) C E D B A (4) C D E B A (4) B D A C E (4) E C D A B (3) E A C D B (3) C D B E A (3) A B E D C (3) A B E C D (3) D C E B A (2) D B E C A (2) B A D E C (2) A E C D B (2) A E B D C (2) A B D E C (2) A B C E D (2) E D C A B (1) E A C B D (1) D A B E C (1) B D C E A (1) B C D E A (1) B C A E D (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 -10 -6 0 B 8 0 8 0 10 C 10 -8 0 2 14 D 6 0 -2 0 4 E 0 -10 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.544225 C: 0.000000 D: 0.455775 E: 0.000000 Sum of squares = 0.503911642312 Cumulative probabilities = A: 0.000000 B: 0.544225 C: 0.544225 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 0 B 8 0 8 0 10 C 10 -8 0 2 14 D 6 0 -2 0 4 E 0 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 D=19 C=16 E=14 so E is eliminated. Round 2 votes counts: A=30 C=25 B=25 D=20 so D is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:209 D:204 A:188 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 0 B 8 0 8 0 10 C 10 -8 0 2 14 D 6 0 -2 0 4 E 0 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 0 B 8 0 8 0 10 C 10 -8 0 2 14 D 6 0 -2 0 4 E 0 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 0 B 8 0 8 0 10 C 10 -8 0 2 14 D 6 0 -2 0 4 E 0 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2677: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) B A E C D (8) A C E D B (8) D C E A B (6) B D C A E (6) B C D A E (6) E A D C B (5) E A C D B (5) C D E A B (5) D E C A B (3) D C B E A (3) C A D E B (3) B E A D C (3) A E C D B (3) E D A C B (2) E B A D C (2) D B C E A (2) C D A E B (2) B D E C A (2) B A C E D (2) E D B A C (1) E C D A B (1) E A D B C (1) E A B C D (1) D C B A E (1) D B E C A (1) C D A B E (1) C B D A E (1) C A E D B (1) C A D B E (1) B E D A C (1) B D E A C (1) A E C B D (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -6 -6 -6 B -4 0 -6 -10 -4 C 6 6 0 4 16 D 6 10 -4 0 4 E 6 4 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -6 -6 B -4 0 -6 -10 -4 C 6 6 0 4 16 D 6 10 -4 0 4 E 6 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=18 D=16 A=15 C=14 so C is eliminated. Round 2 votes counts: B=38 D=24 A=20 E=18 so E is eliminated. Round 3 votes counts: B=40 A=32 D=28 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:216 D:208 E:195 A:193 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -6 -6 B -4 0 -6 -10 -4 C 6 6 0 4 16 D 6 10 -4 0 4 E 6 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 -6 B -4 0 -6 -10 -4 C 6 6 0 4 16 D 6 10 -4 0 4 E 6 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 -6 B -4 0 -6 -10 -4 C 6 6 0 4 16 D 6 10 -4 0 4 E 6 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2678: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) A B E C D (7) E D C A B (6) D E C A B (6) A B E D C (6) E A B D C (5) C D E B A (5) C B D A E (5) C B A D E (5) B A C E D (5) E D C B A (4) E D A B C (4) E A D B C (4) D C E A B (3) C D B A E (3) B A C D E (3) E D A C B (2) E C D B A (2) D E C B A (2) D C B A E (2) D A B C E (2) B A E C D (2) A E B D C (2) A B D C E (2) E A B C D (1) D E A C B (1) D E A B C (1) C D B E A (1) B E A C D (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -2 -12 -10 B -6 0 -8 -12 -10 C 2 8 0 -20 -14 D 12 12 20 0 -2 E 10 10 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -2 -12 -10 B -6 0 -8 -12 -10 C 2 8 0 -20 -14 D 12 12 20 0 -2 E 10 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=24 C=19 A=18 B=11 so B is eliminated. Round 2 votes counts: E=29 A=28 D=24 C=19 so C is eliminated. Round 3 votes counts: D=38 A=33 E=29 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:218 A:191 C:188 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 -12 -10 B -6 0 -8 -12 -10 C 2 8 0 -20 -14 D 12 12 20 0 -2 E 10 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -12 -10 B -6 0 -8 -12 -10 C 2 8 0 -20 -14 D 12 12 20 0 -2 E 10 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -12 -10 B -6 0 -8 -12 -10 C 2 8 0 -20 -14 D 12 12 20 0 -2 E 10 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2679: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) D A C B E (9) B C E D A (9) D A E C B (8) C B D A E (8) B C E A D (8) E D A C B (7) B C A D E (7) E B C A D (6) A D E C B (4) E D A B C (3) A E D B C (3) E A D C B (2) E A B C D (2) C B A D E (2) B C D A E (2) E D B C A (1) E B D C A (1) E B C D A (1) D E A C B (1) D C B A E (1) C B E D A (1) C B D E A (1) B C A E D (1) A E D C B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 -6 -4 B -2 0 8 0 2 C -2 -8 0 -4 0 D 6 0 4 0 -10 E 4 -2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000094 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 2 2 -6 -4 B -2 0 8 0 2 C -2 -8 0 -4 0 D 6 0 4 0 -10 E 4 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.37500000015 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=27 D=19 C=12 A=10 so A is eliminated. Round 2 votes counts: E=36 B=28 D=24 C=12 so C is eliminated. Round 3 votes counts: B=40 E=36 D=24 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:206 B:204 D:200 A:197 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 2 -6 -4 B -2 0 8 0 2 C -2 -8 0 -4 0 D 6 0 4 0 -10 E 4 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.37500000015 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -6 -4 B -2 0 8 0 2 C -2 -8 0 -4 0 D 6 0 4 0 -10 E 4 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.37500000015 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -6 -4 B -2 0 8 0 2 C -2 -8 0 -4 0 D 6 0 4 0 -10 E 4 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.37500000015 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2680: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) A E C B D (7) E A C B D (6) D B C A E (6) D A E B C (4) E C B A D (3) D B E C A (3) D A C B E (3) D A B C E (3) C B E A D (3) B C D E A (3) A E D C B (3) A D E C B (3) A C E D B (3) A C E B D (3) A C D E B (3) E B D A C (2) D E B A C (2) D C B A E (2) D C A B E (2) C E B A D (2) C B E D A (2) C B A D E (2) B D E C A (2) E D B C A (1) E D B A C (1) E B C A D (1) E A B C D (1) D B A E C (1) C B D E A (1) C A B E D (1) B E D C A (1) B C E D A (1) A E D B C (1) A E C D B (1) A D E B C (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 4 -4 6 B 2 0 -8 -16 2 C -4 8 0 -10 12 D 4 16 10 0 12 E -6 -2 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -4 6 B 2 0 -8 -16 2 C -4 8 0 -10 12 D 4 16 10 0 12 E -6 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=29 E=15 C=11 B=7 so B is eliminated. Round 2 votes counts: D=40 A=29 E=16 C=15 so C is eliminated. Round 3 votes counts: D=44 A=32 E=24 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:203 A:202 B:190 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -4 6 B 2 0 -8 -16 2 C -4 8 0 -10 12 D 4 16 10 0 12 E -6 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -4 6 B 2 0 -8 -16 2 C -4 8 0 -10 12 D 4 16 10 0 12 E -6 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -4 6 B 2 0 -8 -16 2 C -4 8 0 -10 12 D 4 16 10 0 12 E -6 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2681: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) D C A E B (9) B E A D C (6) E B D A C (5) D A C B E (5) E B D C A (4) E B C D A (4) C E B D A (4) C A B E D (4) B E A C D (4) A B C E D (4) E B C A D (3) B A E C D (3) A B E C D (3) E B A D C (2) D E A B C (2) D A C E B (2) C D E B A (2) C A D B E (2) A D B E C (2) A C D B E (2) A C B E D (2) E D B C A (1) E D B A C (1) E C B D A (1) D E C B A (1) D E B A C (1) D E A C B (1) D C E A B (1) D C A B E (1) D A B E C (1) C E D B A (1) C D A E B (1) C D A B E (1) C B E A D (1) C B A E D (1) C A B D E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -4 -16 -8 B 8 0 -6 8 -10 C 4 6 0 -10 8 D 16 -8 10 0 -8 E 8 10 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.384615 Sum of squares = 0.337278106507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.615385 E: 1.000000 A B C D E A 0 -8 -4 -16 -8 B 8 0 -6 8 -10 C 4 6 0 -10 8 D 16 -8 10 0 -8 E 8 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.384615 Sum of squares = 0.337278106498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.615385 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=21 C=18 A=15 B=13 so B is eliminated. Round 2 votes counts: D=33 E=31 C=18 A=18 so C is eliminated. Round 3 votes counts: E=37 D=37 A=26 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:209 D:205 C:204 B:200 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 -16 -8 B 8 0 -6 8 -10 C 4 6 0 -10 8 D 16 -8 10 0 -8 E 8 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.384615 Sum of squares = 0.337278106498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.615385 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -16 -8 B 8 0 -6 8 -10 C 4 6 0 -10 8 D 16 -8 10 0 -8 E 8 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.384615 Sum of squares = 0.337278106498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.615385 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -16 -8 B 8 0 -6 8 -10 C 4 6 0 -10 8 D 16 -8 10 0 -8 E 8 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.307692 E: 0.384615 Sum of squares = 0.337278106498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.615385 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2682: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) E A D B C (8) E A C D B (6) C E A D B (6) D B C E A (5) B D C A E (5) E A B D C (4) D B C A E (4) C B D A E (4) C B A D E (4) A E B C D (4) D B E C A (3) C D B A E (3) B C D A E (3) A C B E D (3) A B C E D (3) E D A B C (2) E A D C B (2) D E B A C (2) D C E B A (2) C A E B D (2) A C E B D (2) E D B A C (1) E C A D B (1) E A B C D (1) D E C A B (1) D E B C A (1) D C B E A (1) D C B A E (1) D B E A C (1) C A E D B (1) B D E A C (1) B A E D C (1) B A E C D (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 12 6 20 4 B -12 0 4 -2 -10 C -6 -4 0 8 0 D -20 2 -8 0 -16 E -4 10 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 20 4 B -12 0 4 -2 -10 C -6 -4 0 8 0 D -20 2 -8 0 -16 E -4 10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=22 D=21 C=20 B=12 so B is eliminated. Round 2 votes counts: D=27 E=25 A=25 C=23 so C is eliminated. Round 3 votes counts: D=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:211 C:199 B:190 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 20 4 B -12 0 4 -2 -10 C -6 -4 0 8 0 D -20 2 -8 0 -16 E -4 10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 20 4 B -12 0 4 -2 -10 C -6 -4 0 8 0 D -20 2 -8 0 -16 E -4 10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 20 4 B -12 0 4 -2 -10 C -6 -4 0 8 0 D -20 2 -8 0 -16 E -4 10 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2683: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D A C B E (8) C A D B E (7) E B D A C (6) A D C E B (6) D C A B E (5) B E D A C (5) A C D E B (5) C A D E B (4) B E D C A (4) E B C D A (3) E B A D C (3) C A E D B (3) B E C D A (3) E B D C A (2) E A C B D (2) D A B C E (2) C A E B D (2) B D E A C (2) E D A B C (1) E C B A D (1) E C A B D (1) E B A C D (1) E A C D B (1) D E A B C (1) D B E A C (1) D A E B C (1) D A C E B (1) D A B E C (1) C D B A E (1) C B E A D (1) C B A D E (1) C A B E D (1) B D C E A (1) B C E D A (1) A D E C B (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 10 0 2 2 B -10 0 -4 -2 -10 C 0 4 0 -4 0 D -2 2 4 0 0 E -2 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.799269 B: 0.000000 C: 0.200731 D: 0.000000 E: 0.000000 Sum of squares = 0.679123542438 Cumulative probabilities = A: 0.799269 B: 0.799269 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 2 2 B -10 0 -4 -2 -10 C 0 4 0 -4 0 D -2 2 4 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555752698 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=20 C=20 B=16 A=14 so A is eliminated. Round 2 votes counts: E=30 D=28 C=26 B=16 so B is eliminated. Round 3 votes counts: E=42 D=31 C=27 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:207 E:204 D:202 C:200 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 2 2 B -10 0 -4 -2 -10 C 0 4 0 -4 0 D -2 2 4 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555752698 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 2 2 B -10 0 -4 -2 -10 C 0 4 0 -4 0 D -2 2 4 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555752698 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 2 2 B -10 0 -4 -2 -10 C 0 4 0 -4 0 D -2 2 4 0 0 E -2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555752698 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2684: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) D C A E B (8) C E D A B (7) A D B C E (6) A B D C E (6) C D E A B (5) B A E D C (4) E D C A B (3) E C D B A (3) E C D A B (3) E B C D A (3) E B C A D (3) D C E A B (3) D A C B E (3) D A B C E (3) C E D B A (3) B E A D C (3) B E A C D (3) B A C D E (3) E B A D C (2) D C A B E (2) C B A D E (2) B A E C D (2) E D B A C (1) E B A C D (1) D E C A B (1) D A E B C (1) D A C E B (1) C A D B E (1) B E C A D (1) B A D C E (1) A D C B E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -14 -18 -8 B -12 0 -10 -12 -14 C 14 10 0 -2 10 D 18 12 2 0 -2 E 8 14 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 A B C D E A 0 12 -14 -18 -8 B -12 0 -10 -12 -14 C 14 10 0 -2 10 D 18 12 2 0 -2 E 8 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020407859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=22 C=18 B=17 A=15 so A is eliminated. Round 2 votes counts: D=30 E=28 B=24 C=18 so C is eliminated. Round 3 votes counts: E=38 D=36 B=26 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:216 D:215 E:207 A:186 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -14 -18 -8 B -12 0 -10 -12 -14 C 14 10 0 -2 10 D 18 12 2 0 -2 E 8 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020407859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -14 -18 -8 B -12 0 -10 -12 -14 C 14 10 0 -2 10 D 18 12 2 0 -2 E 8 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020407859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -14 -18 -8 B -12 0 -10 -12 -14 C 14 10 0 -2 10 D 18 12 2 0 -2 E 8 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020407859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2685: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) A E D C B (7) E A B D C (6) A E C D B (5) E A D B C (4) C D B A E (4) C D A E B (4) B D C E A (4) B C E A D (4) D C B A E (3) D B E A C (3) C A D E B (3) B E A D C (3) B E A C D (3) A E D B C (3) D C A E B (2) D A E C B (2) D A C E B (2) C D A B E (2) C B D E A (2) C A E B D (2) B D E C A (2) A E C B D (2) E B A C D (1) D E A B C (1) D B E C A (1) D B C A E (1) D B A C E (1) D A E B C (1) C B E A D (1) C B D A E (1) C B A E D (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E A C (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 -6 -6 -6 B 6 0 12 0 8 C 6 -12 0 0 2 D 6 0 0 0 8 E 6 -8 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.505354 C: 0.000000 D: 0.494646 E: 0.000000 Sum of squares = 0.500057329256 Cumulative probabilities = A: 0.000000 B: 0.505354 C: 0.505354 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -6 -6 B 6 0 12 0 8 C 6 -12 0 0 2 D 6 0 0 0 8 E 6 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=20 A=18 D=17 E=11 so E is eliminated. Round 2 votes counts: B=35 A=28 C=20 D=17 so D is eliminated. Round 3 votes counts: B=41 A=34 C=25 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:207 C:198 E:194 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -6 -6 B 6 0 12 0 8 C 6 -12 0 0 2 D 6 0 0 0 8 E 6 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -6 -6 B 6 0 12 0 8 C 6 -12 0 0 2 D 6 0 0 0 8 E 6 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -6 -6 B 6 0 12 0 8 C 6 -12 0 0 2 D 6 0 0 0 8 E 6 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2686: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (21) E D B A C (9) D E B A C (8) A B C D E (8) B A E D C (7) C A B E D (4) B A C E D (4) E D C B A (3) E D B C A (3) C E D A B (3) C D E A B (3) D E C A B (2) D C E A B (2) D A B E C (2) C D A E B (2) C A D B E (2) B C A E D (2) B A D E C (2) E C D B A (1) E B D C A (1) E B D A C (1) D E A B C (1) C E D B A (1) C B E A D (1) C B A E D (1) C A D E B (1) B E D A C (1) B D E A C (1) B A D C E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 12 18 B -6 0 6 14 18 C 6 -6 0 10 14 D -12 -14 -10 0 14 E -18 -18 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 12 18 B -6 0 6 14 18 C 6 -6 0 10 14 D -12 -14 -10 0 14 E -18 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=18 B=18 D=15 A=10 so A is eliminated. Round 2 votes counts: C=39 B=28 E=18 D=15 so D is eliminated. Round 3 votes counts: C=41 B=30 E=29 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:215 C:212 D:189 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -6 12 18 B -6 0 6 14 18 C 6 -6 0 10 14 D -12 -14 -10 0 14 E -18 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 12 18 B -6 0 6 14 18 C 6 -6 0 10 14 D -12 -14 -10 0 14 E -18 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 12 18 B -6 0 6 14 18 C 6 -6 0 10 14 D -12 -14 -10 0 14 E -18 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2687: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (17) C E D A B (13) D E B A C (9) D E C A B (7) E D C B A (6) A B C E D (6) C E D B A (5) B A C E D (5) E D C A B (4) C A B E D (4) B A C D E (3) A C B E D (3) A B D E C (3) D E C B A (2) D E A C B (2) B A D C E (2) E D B C A (1) D E B C A (1) D B E A C (1) C B E A D (1) C B A E D (1) C A E D B (1) B D E A C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 8 -4 -6 B 10 0 2 -6 -2 C -8 -2 0 -14 -10 D 4 6 14 0 0 E 6 2 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.451210 E: 0.548790 Sum of squares = 0.504760929107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.451210 E: 1.000000 A B C D E A 0 -10 8 -4 -6 B 10 0 2 -6 -2 C -8 -2 0 -14 -10 D 4 6 14 0 0 E 6 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 D=22 A=14 E=11 so E is eliminated. Round 2 votes counts: D=33 B=28 C=25 A=14 so A is eliminated. Round 3 votes counts: B=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:209 B:202 A:194 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 -4 -6 B 10 0 2 -6 -2 C -8 -2 0 -14 -10 D 4 6 14 0 0 E 6 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -4 -6 B 10 0 2 -6 -2 C -8 -2 0 -14 -10 D 4 6 14 0 0 E 6 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -4 -6 B 10 0 2 -6 -2 C -8 -2 0 -14 -10 D 4 6 14 0 0 E 6 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2688: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (10) C A E D B (10) B D E A C (7) E B C D A (5) D A B C E (5) B E D C A (5) E C B A D (3) B E D A C (3) E C A B D (2) E B D A C (2) E B C A D (2) D B A C E (2) D A B E C (2) C E B D A (2) C E B A D (2) C A D E B (2) B E C D A (2) A D E B C (2) A D C B E (2) A C E D B (2) A C D E B (2) E B D C A (1) E B A C D (1) E A D B C (1) E A C B D (1) D A C B E (1) C E A D B (1) C B E D A (1) C A D B E (1) B D C E A (1) B D A E C (1) B C E D A (1) B C D E A (1) A E D C B (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -4 -6 -8 B 6 0 12 6 -6 C 4 -12 0 2 -4 D 6 -6 -2 0 -16 E 8 6 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -6 -8 B 6 0 12 6 -6 C 4 -12 0 2 -4 D 6 -6 -2 0 -16 E 8 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 B=21 E=18 A=11 so A is eliminated. Round 2 votes counts: C=33 D=27 B=21 E=19 so E is eliminated. Round 3 votes counts: C=39 B=32 D=29 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:217 B:209 C:195 D:191 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -6 -8 B 6 0 12 6 -6 C 4 -12 0 2 -4 D 6 -6 -2 0 -16 E 8 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -6 -8 B 6 0 12 6 -6 C 4 -12 0 2 -4 D 6 -6 -2 0 -16 E 8 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -6 -8 B 6 0 12 6 -6 C 4 -12 0 2 -4 D 6 -6 -2 0 -16 E 8 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2689: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) B D C E A (8) D C B E A (6) A E B D C (5) A B C D E (5) E D B C A (4) E A D B C (4) A E C D B (4) A C B D E (4) D B E C A (3) C D B E A (3) B C D A E (3) E D C B A (2) D E C B A (2) D C E B A (2) C E D A B (2) C D E B A (2) C D B A E (2) C B D A E (2) B D E C A (2) B D C A E (2) A C E D B (2) A C E B D (2) A B C E D (2) E D C A B (1) E D B A C (1) E D A C B (1) E B A D C (1) E A D C B (1) E A C D B (1) D B C E A (1) C D A E B (1) C D A B E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D C A (1) B D E A C (1) B C A D E (1) B A E D C (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 -6 -14 B 2 0 6 2 6 C 8 -6 0 -14 14 D 6 -2 14 0 12 E 14 -6 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -6 -14 B 2 0 6 2 6 C 8 -6 0 -14 14 D 6 -2 14 0 12 E 14 -6 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999258 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=24 B=19 C=16 D=14 so D is eliminated. Round 2 votes counts: A=27 E=26 C=24 B=23 so B is eliminated. Round 3 votes counts: C=39 E=33 A=28 so A is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:215 B:208 C:201 E:191 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 -6 -14 B 2 0 6 2 6 C 8 -6 0 -14 14 D 6 -2 14 0 12 E 14 -6 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999258 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -6 -14 B 2 0 6 2 6 C 8 -6 0 -14 14 D 6 -2 14 0 12 E 14 -6 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999258 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -6 -14 B 2 0 6 2 6 C 8 -6 0 -14 14 D 6 -2 14 0 12 E 14 -6 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999258 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2690: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) C D A E B (8) E B A D C (7) A E C B D (5) D C B E A (4) D C B A E (4) C D A B E (4) C A D E B (4) A E B C D (4) A C E D B (4) D B C A E (3) C D E B A (3) C A E D B (3) B E D A C (3) B E A D C (3) B D E C A (3) E A B D C (2) D C E B A (2) C D E A B (2) A C E B D (2) A C D B E (2) A B E D C (2) E B D A C (1) E B A C D (1) E A C B D (1) D B E C A (1) D B A C E (1) C E D B A (1) C E A D B (1) C D B A E (1) C A D B E (1) B E D C A (1) B D E A C (1) B D A C E (1) B A E D C (1) A E B D C (1) A D C B E (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 16 8 12 10 B -16 0 -10 -2 -22 C -8 10 0 14 10 D -12 2 -14 0 -4 E -10 22 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 12 10 B -16 0 -10 -2 -22 C -8 10 0 14 10 D -12 2 -14 0 -4 E -10 22 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=24 E=20 D=15 B=13 so B is eliminated. Round 2 votes counts: C=28 E=27 A=25 D=20 so D is eliminated. Round 3 votes counts: C=41 E=32 A=27 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:223 C:213 E:203 D:186 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 12 10 B -16 0 -10 -2 -22 C -8 10 0 14 10 D -12 2 -14 0 -4 E -10 22 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 12 10 B -16 0 -10 -2 -22 C -8 10 0 14 10 D -12 2 -14 0 -4 E -10 22 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 12 10 B -16 0 -10 -2 -22 C -8 10 0 14 10 D -12 2 -14 0 -4 E -10 22 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2691: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) E A B D C (9) D B C E A (8) A E B C D (6) D E B C A (5) D C B E A (5) D B E C A (5) C B D A E (5) E D B C A (4) E A D B C (4) A E C D B (4) A C B D E (4) E A D C B (3) B C D E A (3) B C D A E (3) B C A D E (3) A C B E D (3) E D B A C (2) C D B A E (2) A E D C B (2) E D A C B (1) E D A B C (1) E B D C A (1) E B A D C (1) C D A B E (1) C B A D E (1) C A B D E (1) B D E C A (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 4 6 -6 B -2 0 14 6 -8 C -4 -14 0 -4 -18 D -6 -6 4 0 -4 E 6 8 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 6 -6 B -2 0 14 6 -8 C -4 -14 0 -4 -18 D -6 -6 4 0 -4 E 6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=26 D=23 C=10 B=10 so C is eliminated. Round 2 votes counts: A=32 E=26 D=26 B=16 so B is eliminated. Round 3 votes counts: D=38 A=36 E=26 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:218 B:205 A:203 D:194 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 6 -6 B -2 0 14 6 -8 C -4 -14 0 -4 -18 D -6 -6 4 0 -4 E 6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 6 -6 B -2 0 14 6 -8 C -4 -14 0 -4 -18 D -6 -6 4 0 -4 E 6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 6 -6 B -2 0 14 6 -8 C -4 -14 0 -4 -18 D -6 -6 4 0 -4 E 6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2692: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (17) D B C E A (9) A E C B D (7) A C D B E (7) E A B C D (6) E B D C A (4) D A C B E (4) A E D C B (4) E B C D A (3) A C E B D (3) E B C A D (2) E B A C D (2) E A D B C (2) C B D A E (2) B D C E A (2) A E C D B (2) A D C E B (2) A C E D B (2) A C D E B (2) E B D A C (1) E A B D C (1) D E B C A (1) D E A B C (1) D C A B E (1) D A E C B (1) D A C E B (1) C D B A E (1) C D A B E (1) C B A E D (1) C A B E D (1) C A B D E (1) B E C D A (1) B E C A D (1) B C E D A (1) B C D E A (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 0 -4 24 B -2 0 -24 -18 4 C 0 24 0 -4 20 D 4 18 4 0 12 E -24 -4 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -4 24 B -2 0 -24 -18 4 C 0 24 0 -4 20 D 4 18 4 0 12 E -24 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=31 E=21 C=7 B=6 so B is eliminated. Round 2 votes counts: D=37 A=31 E=23 C=9 so C is eliminated. Round 3 votes counts: D=42 A=34 E=24 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:220 D:219 A:211 B:180 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -4 24 B -2 0 -24 -18 4 C 0 24 0 -4 20 D 4 18 4 0 12 E -24 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 24 B -2 0 -24 -18 4 C 0 24 0 -4 20 D 4 18 4 0 12 E -24 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 24 B -2 0 -24 -18 4 C 0 24 0 -4 20 D 4 18 4 0 12 E -24 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2693: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) B E C D A (6) B D A E C (6) A B D E C (6) E C B D A (5) E C A B D (5) C E D B A (5) E A C B D (4) D B A C E (4) C E B D A (4) A D B C E (4) C E A D B (3) C E A B D (3) B D C E A (3) B D A C E (3) A E C D B (3) E C B A D (2) D B C E A (2) D B A E C (2) C E D A B (2) B D E C A (2) B A D E C (2) E C A D B (1) D C A E B (1) D B C A E (1) D A C B E (1) D A B E C (1) C D A E B (1) C B E D A (1) C A D E B (1) B E D A C (1) B D E A C (1) A E D C B (1) A E C B D (1) A D E B C (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 4 -4 0 B 0 0 6 10 8 C -4 -6 0 -2 -16 D 4 -10 2 0 4 E 0 -8 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.495258 B: 0.504742 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500044967668 Cumulative probabilities = A: 0.495258 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -4 0 B 0 0 6 10 8 C -4 -6 0 -2 -16 D 4 -10 2 0 4 E 0 -8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=24 C=20 E=17 D=12 so D is eliminated. Round 2 votes counts: B=33 A=29 C=21 E=17 so E is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:202 A:200 D:200 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 -4 0 B 0 0 6 10 8 C -4 -6 0 -2 -16 D 4 -10 2 0 4 E 0 -8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -4 0 B 0 0 6 10 8 C -4 -6 0 -2 -16 D 4 -10 2 0 4 E 0 -8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -4 0 B 0 0 6 10 8 C -4 -6 0 -2 -16 D 4 -10 2 0 4 E 0 -8 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2694: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) B C E A D (6) E D C B A (5) C A B D E (5) B E C D A (5) A C D B E (5) E D B C A (4) E B D C A (4) D E A B C (4) C B A E D (4) A D C B E (4) E C B D A (3) D E C A B (3) D A E C B (3) D A E B C (3) C D A E B (3) B A C E D (3) D E A C B (2) C B E A D (2) B E C A D (2) B C A E D (2) B A E D C (2) A C B D E (2) A B C D E (2) E D C A B (1) E C D B A (1) D C E A B (1) C E D B A (1) C E B D A (1) C D E A B (1) C B E D A (1) C A D E B (1) B E D A C (1) B E A D C (1) B A C D E (1) A D E C B (1) A D C E B (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -24 6 2 B -2 0 -14 -2 14 C 24 14 0 14 6 D -6 2 -14 0 2 E -2 -14 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -24 6 2 B -2 0 -14 -2 14 C 24 14 0 14 6 D -6 2 -14 0 2 E -2 -14 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 E=18 A=18 D=16 so D is eliminated. Round 2 votes counts: E=27 C=26 A=24 B=23 so B is eliminated. Round 3 votes counts: E=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:198 A:193 D:192 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -24 6 2 B -2 0 -14 -2 14 C 24 14 0 14 6 D -6 2 -14 0 2 E -2 -14 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -24 6 2 B -2 0 -14 -2 14 C 24 14 0 14 6 D -6 2 -14 0 2 E -2 -14 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -24 6 2 B -2 0 -14 -2 14 C 24 14 0 14 6 D -6 2 -14 0 2 E -2 -14 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2695: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) D C E B A (6) E B D C A (5) B A D E C (5) B E D C A (4) A C E B D (4) A C D E B (4) E C D B A (3) C D E A B (3) B D E A C (3) A E B C D (3) A D C B E (3) A D B C E (3) A C E D B (3) A B D C E (3) E C A B D (2) D C B E A (2) D C B A E (2) D B C E A (2) C E D A B (2) C D A E B (2) C A E D B (2) C A D E B (2) B E A D C (2) B D E C A (2) B D A C E (2) A B E C D (2) E D C B A (1) E B C A D (1) E B A C D (1) E A C B D (1) D E C B A (1) D E B C A (1) D C A B E (1) D B A C E (1) C E D B A (1) B E D A C (1) B E A C D (1) B D A E C (1) A E C B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -8 -10 -8 B 14 0 6 -8 2 C 8 -6 0 -20 0 D 10 8 20 0 20 E 8 -2 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -10 -8 B 14 0 6 -8 2 C 8 -6 0 -20 0 D 10 8 20 0 20 E 8 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 B=21 E=14 C=12 so C is eliminated. Round 2 votes counts: A=32 D=30 B=21 E=17 so E is eliminated. Round 3 votes counts: D=37 A=35 B=28 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 B:207 E:193 C:191 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -8 -10 -8 B 14 0 6 -8 2 C 8 -6 0 -20 0 D 10 8 20 0 20 E 8 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -10 -8 B 14 0 6 -8 2 C 8 -6 0 -20 0 D 10 8 20 0 20 E 8 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -10 -8 B 14 0 6 -8 2 C 8 -6 0 -20 0 D 10 8 20 0 20 E 8 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2696: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (20) E C D A B (15) B A D C E (12) B A D E C (9) E D A C B (7) B A C D E (4) E D C A B (3) C E B A D (3) E C B D A (2) D E A C B (2) D A E B C (2) C B A E D (2) B E A D C (2) B C E A D (2) A D B E C (2) A B D E C (2) D C E A B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A C E B (1) C E B D A (1) C B A D E (1) C A D B E (1) B D E A C (1) B C A D E (1) A D B C E (1) Total count = 100 A B C D E A 0 16 -4 -16 -20 B -16 0 -20 -16 -16 C 4 20 0 4 0 D 16 16 -4 0 -14 E 20 16 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.557112 D: 0.000000 E: 0.442888 Sum of squares = 0.506523472006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.557112 D: 0.557112 E: 1.000000 A B C D E A 0 16 -4 -16 -20 B -16 0 -20 -16 -16 C 4 20 0 4 0 D 16 16 -4 0 -14 E 20 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=28 E=27 D=9 A=5 so A is eliminated. Round 2 votes counts: B=33 C=28 E=27 D=12 so D is eliminated. Round 3 votes counts: B=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:214 D:207 A:188 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 -16 -20 B -16 0 -20 -16 -16 C 4 20 0 4 0 D 16 16 -4 0 -14 E 20 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 -16 -20 B -16 0 -20 -16 -16 C 4 20 0 4 0 D 16 16 -4 0 -14 E 20 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 -16 -20 B -16 0 -20 -16 -16 C 4 20 0 4 0 D 16 16 -4 0 -14 E 20 16 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2697: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) A D E B C (6) A D B E C (6) D E C A B (5) B C E D A (5) B A D E C (5) A B D E C (5) C E D B A (4) C E D A B (4) C B E D A (4) B A C D E (4) E D C A B (3) D E A C B (3) D A E C B (3) B E C D A (3) B C E A D (3) A D E C B (3) C B E A D (2) B D A E C (2) B C A E D (2) A D C E B (2) A C E D B (2) A B D C E (2) E D C B A (1) E C D B A (1) E B C D A (1) D E B C A (1) D E B A C (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E D B (1) C A E B D (1) B D E C A (1) B D E A C (1) B A D C E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -2 -2 -4 B 0 0 0 4 -2 C 2 0 0 -4 -4 D 2 -4 4 0 6 E 4 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888912 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 A B C D E A 0 0 -2 -2 -4 B 0 0 0 4 -2 C 2 0 0 -4 -4 D 2 -4 4 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 C=25 D=14 E=6 so E is eliminated. Round 2 votes counts: B=28 A=28 C=26 D=18 so D is eliminated. Round 3 votes counts: C=35 A=35 B=30 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:204 E:202 B:201 C:197 A:196 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -2 -4 B 0 0 0 4 -2 C 2 0 0 -4 -4 D 2 -4 4 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 -4 B 0 0 0 4 -2 C 2 0 0 -4 -4 D 2 -4 4 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 -4 B 0 0 0 4 -2 C 2 0 0 -4 -4 D 2 -4 4 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) B D A C E (6) D A B C E (5) C D E A B (5) B A D E C (5) A D B E C (5) E C A B D (4) D B A C E (4) C E D B A (4) C D B E A (4) A D E C B (4) A B D E C (4) E C B A D (3) D B C A E (3) D A C B E (3) E B C A D (2) E B A C D (2) E A C D B (2) D C E A B (2) D C B A E (2) D A C E B (2) C E D A B (2) C D E B A (2) B A E D C (2) A B E D C (2) E C D A B (1) E C A D B (1) E A B C D (1) D C B E A (1) D C A E B (1) C B E D A (1) B C E D A (1) B A E C D (1) A E D C B (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 2 -18 4 B 6 0 -10 -10 -2 C -2 10 0 -8 16 D 18 10 8 0 16 E -4 2 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -18 4 B 6 0 -10 -10 -2 C -2 10 0 -8 16 D 18 10 8 0 16 E -4 2 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 A=18 E=16 B=15 so B is eliminated. Round 2 votes counts: D=29 C=29 A=26 E=16 so E is eliminated. Round 3 votes counts: C=40 A=31 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:226 C:208 B:192 A:191 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -18 4 B 6 0 -10 -10 -2 C -2 10 0 -8 16 D 18 10 8 0 16 E -4 2 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -18 4 B 6 0 -10 -10 -2 C -2 10 0 -8 16 D 18 10 8 0 16 E -4 2 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -18 4 B 6 0 -10 -10 -2 C -2 10 0 -8 16 D 18 10 8 0 16 E -4 2 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2699: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) E A C B D (8) D B E C A (8) A C E B D (8) A C B D E (8) E D B C A (7) D B C A E (6) D B C E A (5) B D C A E (5) D E B C A (4) D B A C E (3) C B A D E (3) E C A B D (2) E A D C B (2) A E C B D (2) A C B E D (2) E D C B A (1) E D B A C (1) E D A C B (1) E C A D B (1) D B E A C (1) D B A E C (1) D A B C E (1) C E A B D (1) C A E B D (1) B E D C A (1) B D C E A (1) B C A D E (1) A E D C B (1) A E D B C (1) A E C D B (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 8 -8 B -4 0 -6 -12 -6 C -6 6 0 0 -6 D -8 12 0 0 -4 E 8 6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 8 -8 B -4 0 -6 -12 -6 C -6 6 0 0 -6 D -8 12 0 0 -4 E 8 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=29 A=25 B=8 C=5 so C is eliminated. Round 2 votes counts: E=34 D=29 A=26 B=11 so B is eliminated. Round 3 votes counts: E=35 D=35 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:205 D:200 C:197 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 8 -8 B -4 0 -6 -12 -6 C -6 6 0 0 -6 D -8 12 0 0 -4 E 8 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 8 -8 B -4 0 -6 -12 -6 C -6 6 0 0 -6 D -8 12 0 0 -4 E 8 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 8 -8 B -4 0 -6 -12 -6 C -6 6 0 0 -6 D -8 12 0 0 -4 E 8 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2700: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) E D A C B (8) B E C A D (7) A C D B E (6) D E C A B (5) C A B D E (5) E D C B A (4) D E A C B (4) A D C B E (4) E D B C A (3) E D B A C (3) E D A B C (3) D A C E B (3) C A D B E (3) B A C D E (3) A B C D E (3) E D C A B (2) E B D C A (2) E B A D C (2) E A D B C (2) D C E A B (2) C D A B E (2) B C E A D (2) A D C E B (2) A C B D E (2) E B C A D (1) D C A E B (1) C B D E A (1) B E A C D (1) B C E D A (1) B C A E D (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 16 -4 12 -6 B -16 0 -8 -16 6 C 4 8 0 -2 6 D -12 16 2 0 14 E 6 -6 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839656 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 12 -6 B -16 0 -8 -16 6 C 4 8 0 -2 6 D -12 16 2 0 14 E 6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172841377 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 A=18 D=15 C=11 so C is eliminated. Round 2 votes counts: E=30 B=27 A=26 D=17 so D is eliminated. Round 3 votes counts: E=41 A=32 B=27 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 A:209 C:208 E:190 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 12 -6 B -16 0 -8 -16 6 C 4 8 0 -2 6 D -12 16 2 0 14 E 6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172841377 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 12 -6 B -16 0 -8 -16 6 C 4 8 0 -2 6 D -12 16 2 0 14 E 6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172841377 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 12 -6 B -16 0 -8 -16 6 C 4 8 0 -2 6 D -12 16 2 0 14 E 6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172841377 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2701: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (6) C A D B E (6) C A B D E (6) C D B E A (5) C D B A E (5) E B D A C (4) A E B D C (4) A E B C D (4) A C B E D (4) A B E D C (4) E D B C A (3) D B C E A (3) C E A D B (3) A E C B D (3) A C E B D (3) E D C B A (2) E A D C B (2) E A B D C (2) D B E C A (2) C D E B A (2) C A D E B (2) B E D A C (2) B D E C A (2) B D E A C (2) B D C E A (2) A C E D B (2) A C B D E (2) E D B A C (1) E C A D B (1) E A D B C (1) E A C D B (1) D C E B A (1) D C B E A (1) C D E A B (1) C D A B E (1) C B A D E (1) B C D A E (1) B A D C E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -12 8 2 B -8 0 -8 -2 4 C 12 8 0 10 6 D -8 2 -10 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 8 2 B -8 0 -8 -2 4 C 12 8 0 10 6 D -8 2 -10 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=28 E=17 D=13 B=10 so B is eliminated. Round 2 votes counts: C=33 A=29 E=19 D=19 so E is eliminated. Round 3 votes counts: A=35 C=34 D=31 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:203 D:195 B:193 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 8 2 B -8 0 -8 -2 4 C 12 8 0 10 6 D -8 2 -10 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 8 2 B -8 0 -8 -2 4 C 12 8 0 10 6 D -8 2 -10 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 8 2 B -8 0 -8 -2 4 C 12 8 0 10 6 D -8 2 -10 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2702: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (22) C E A D B (13) D B A C E (5) B D A C E (5) E C A B D (4) D B A E C (4) B E C A D (4) B E A C D (4) E C A D B (3) D A C E B (3) B E A D C (3) E B C A D (2) E A C D B (2) D A B C E (2) C E A B D (2) C A D E B (2) B E D A C (2) B E C D A (2) B D E A C (2) A D C E B (2) E C B A D (1) D C A E B (1) D C A B E (1) D B C A E (1) D A C B E (1) D A B E C (1) C D A E B (1) B E D C A (1) B D C E A (1) B D C A E (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 -20 20 -12 8 B 20 0 24 12 24 C -20 -24 0 -18 -18 D 12 -12 18 0 10 E -8 -24 18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 20 -12 8 B 20 0 24 12 24 C -20 -24 0 -18 -18 D 12 -12 18 0 10 E -8 -24 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=47 D=19 C=18 E=12 A=4 so A is eliminated. Round 2 votes counts: B=47 D=21 C=18 E=14 so E is eliminated. Round 3 votes counts: B=49 C=29 D=22 so D is eliminated. Round 4 votes counts: B=62 C=38 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:240 D:214 A:198 E:188 C:160 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 20 -12 8 B 20 0 24 12 24 C -20 -24 0 -18 -18 D 12 -12 18 0 10 E -8 -24 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 20 -12 8 B 20 0 24 12 24 C -20 -24 0 -18 -18 D 12 -12 18 0 10 E -8 -24 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 20 -12 8 B 20 0 24 12 24 C -20 -24 0 -18 -18 D 12 -12 18 0 10 E -8 -24 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2703: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (7) E C A D B (6) D B E C A (5) C A E D B (5) B D E C A (5) B D A E C (5) B A D C E (5) B D E A C (4) B D A C E (4) A C D B E (4) E B D C A (3) D B C A E (3) A E C B D (3) A C E D B (3) A C B E D (3) E D B C A (2) E C D A B (2) E B A D C (2) D E C B A (2) D E B C A (2) D B C E A (2) C E A D B (2) C D A E B (2) B A D E C (2) A C B D E (2) A B D C E (2) A B C D E (2) E D C B A (1) E C D B A (1) E C B A D (1) E C A B D (1) E A C B D (1) C D E A B (1) C A D E B (1) B E A D C (1) A E B C D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 6 12 12 B 0 0 4 12 2 C -6 -4 0 0 -2 D -12 -12 0 0 6 E -12 -2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.697569 B: 0.302431 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.578066707872 Cumulative probabilities = A: 0.697569 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 12 12 B 0 0 4 12 2 C -6 -4 0 0 -2 D -12 -12 0 0 6 E -12 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=26 E=20 D=14 C=11 so C is eliminated. Round 2 votes counts: A=35 B=26 E=22 D=17 so D is eliminated. Round 3 votes counts: A=37 B=36 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:209 C:194 D:191 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 12 12 B 0 0 4 12 2 C -6 -4 0 0 -2 D -12 -12 0 0 6 E -12 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 12 12 B 0 0 4 12 2 C -6 -4 0 0 -2 D -12 -12 0 0 6 E -12 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 12 12 B 0 0 4 12 2 C -6 -4 0 0 -2 D -12 -12 0 0 6 E -12 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2704: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) B D C A E (6) A C E D B (6) E A C D B (5) C B A D E (5) A E D C B (5) E D A B C (4) C A E B D (4) B C E D A (4) A D E B C (4) C B A E D (3) C A B D E (3) B C D A E (3) A D E C B (3) A D C B E (3) D E B A C (2) D B E C A (2) D A B E C (2) C E B A D (2) C A B E D (2) B D E C A (2) B D C E A (2) A E C D B (2) A C D B E (2) E D B C A (1) E D B A C (1) E C B D A (1) E C B A D (1) E C A B D (1) E B C D A (1) E A C B D (1) D E A B C (1) D B E A C (1) D B C A E (1) D B A E C (1) D A E B C (1) C E A B D (1) C B E A D (1) C B D A E (1) B E C D A (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 14 2 22 16 B -14 0 -18 -8 -8 C -2 18 0 4 2 D -22 8 -4 0 -6 E -16 8 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997274 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 22 16 B -14 0 -18 -8 -8 C -2 18 0 4 2 D -22 8 -4 0 -6 E -16 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967243 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 C=22 B=19 D=11 so D is eliminated. Round 2 votes counts: A=29 E=25 B=24 C=22 so C is eliminated. Round 3 votes counts: A=38 B=34 E=28 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 C:211 E:198 D:188 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 22 16 B -14 0 -18 -8 -8 C -2 18 0 4 2 D -22 8 -4 0 -6 E -16 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967243 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 22 16 B -14 0 -18 -8 -8 C -2 18 0 4 2 D -22 8 -4 0 -6 E -16 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967243 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 22 16 B -14 0 -18 -8 -8 C -2 18 0 4 2 D -22 8 -4 0 -6 E -16 8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967243 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2705: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) B C E D A (6) D A C B E (5) E B C A D (4) D C A B E (4) B E C D A (4) E B D C A (3) E A D B C (3) E A B C D (3) D A E B C (3) D A C E B (3) B D C E A (3) B C E A D (3) B C D E A (3) A E D C B (3) A E C D B (3) A D C E B (3) A C D B E (3) E B A C D (2) E A C B D (2) C D B A E (2) C B E A D (2) C B A E D (2) C B A D E (2) C A D B E (2) A D E B C (2) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) D E A B C (1) D B E C A (1) D B C E A (1) D B C A E (1) D A B E C (1) D A B C E (1) C B D A E (1) C A B E D (1) B E C A D (1) A E D B C (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -6 -6 8 B 2 0 2 -10 14 C 6 -2 0 -4 16 D 6 10 4 0 4 E -8 -14 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 8 B 2 0 2 -10 14 C 6 -2 0 -4 16 D 6 10 4 0 4 E -8 -14 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=21 B=20 A=19 C=12 so C is eliminated. Round 2 votes counts: D=30 B=27 A=22 E=21 so E is eliminated. Round 3 votes counts: B=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:208 B:204 A:197 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 8 B 2 0 2 -10 14 C 6 -2 0 -4 16 D 6 10 4 0 4 E -8 -14 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 8 B 2 0 2 -10 14 C 6 -2 0 -4 16 D 6 10 4 0 4 E -8 -14 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 8 B 2 0 2 -10 14 C 6 -2 0 -4 16 D 6 10 4 0 4 E -8 -14 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2706: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) C E A D B (6) B D A E C (6) A E C B D (6) A C E B D (6) E A C D B (4) D B E C A (4) C D E B A (4) B D A C E (4) E C A D B (3) D E B A C (3) D B C E A (3) C A E B D (3) C A B E D (3) B A D C E (3) B A C E D (3) E A D B C (2) D E C B A (2) D E B C A (2) D B E A C (2) C E D A B (2) B C D A E (2) B C A D E (2) B A C D E (2) A E B C D (2) E D B A C (1) E D A C B (1) E C D A B (1) D E A B C (1) D C E B A (1) D C B E A (1) C D B E A (1) C B A E D (1) C B A D E (1) C A E D B (1) B D C A E (1) B C A E D (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -2 4 0 B 0 0 -8 -2 -14 C 2 8 0 12 6 D -4 2 -12 0 -10 E 0 14 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 4 0 B 0 0 -8 -2 -14 C 2 8 0 12 6 D -4 2 -12 0 -10 E 0 14 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=22 D=19 E=18 A=17 so A is eliminated. Round 2 votes counts: C=29 E=26 B=26 D=19 so D is eliminated. Round 3 votes counts: B=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:209 A:201 B:188 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 4 0 B 0 0 -8 -2 -14 C 2 8 0 12 6 D -4 2 -12 0 -10 E 0 14 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 0 B 0 0 -8 -2 -14 C 2 8 0 12 6 D -4 2 -12 0 -10 E 0 14 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 0 B 0 0 -8 -2 -14 C 2 8 0 12 6 D -4 2 -12 0 -10 E 0 14 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2707: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) D A B C E (6) E D C B A (5) E C B A D (5) E B C A D (5) C A E B D (5) C A D B E (4) C A B E D (4) A D C B E (4) A C B D E (4) D E A C B (3) D A B E C (3) C E D A B (3) C A B D E (3) E C B D A (2) E B D C A (2) D E C A B (2) D E A B C (2) D A C E B (2) D A C B E (2) C E B A D (2) C E A B D (2) C B E A D (2) B C A E D (2) E D B C A (1) E D B A C (1) E C D B A (1) E B C D A (1) D C E A B (1) D C A E B (1) D B A E C (1) C E A D B (1) C D A B E (1) C B A E D (1) C A D E B (1) B D A E C (1) B C E A D (1) B A E D C (1) B A D E C (1) B A C E D (1) B A C D E (1) A D B C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 14 -16 6 4 B -14 0 -26 -6 -8 C 16 26 0 8 14 D -6 6 -8 0 2 E -4 8 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -16 6 4 B -14 0 -26 -6 -8 C 16 26 0 8 14 D -6 6 -8 0 2 E -4 8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=29 C=29 E=23 A=11 B=8 so B is eliminated. Round 2 votes counts: C=32 D=30 E=23 A=15 so A is eliminated. Round 3 votes counts: C=40 D=36 E=24 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:232 A:204 D:197 E:194 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -16 6 4 B -14 0 -26 -6 -8 C 16 26 0 8 14 D -6 6 -8 0 2 E -4 8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -16 6 4 B -14 0 -26 -6 -8 C 16 26 0 8 14 D -6 6 -8 0 2 E -4 8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -16 6 4 B -14 0 -26 -6 -8 C 16 26 0 8 14 D -6 6 -8 0 2 E -4 8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2708: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) D A E B C (8) C E B D A (8) D A C B E (6) C E B A D (6) E B C A D (5) B C E A D (4) E C B A D (3) E B A D C (3) E B A C D (3) D A B E C (3) C B E A D (3) B A C D E (3) A D E B C (3) A B D E C (3) E D C A B (2) E C B D A (2) D A C E B (2) B E C A D (2) B E A C D (2) A D B C E (2) A B E D C (2) E C D B A (1) D A E C B (1) D A B C E (1) C D E B A (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E D A (1) C B D A E (1) C A D B E (1) B C A E D (1) A E D B C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 8 16 4 B 2 0 14 8 -2 C -8 -14 0 6 -6 D -16 -8 -6 0 2 E -4 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 8 16 4 B 2 0 14 8 -2 C -8 -14 0 6 -6 D -16 -8 -6 0 2 E -4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=22 D=21 E=19 B=12 so B is eliminated. Round 2 votes counts: C=31 A=25 E=23 D=21 so D is eliminated. Round 3 votes counts: A=46 C=31 E=23 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:211 E:201 C:189 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 16 4 B 2 0 14 8 -2 C -8 -14 0 6 -6 D -16 -8 -6 0 2 E -4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 16 4 B 2 0 14 8 -2 C -8 -14 0 6 -6 D -16 -8 -6 0 2 E -4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 16 4 B 2 0 14 8 -2 C -8 -14 0 6 -6 D -16 -8 -6 0 2 E -4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2709: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A D C B E (10) D A B C E (7) A C D E B (6) A D C E B (5) E B C A D (4) D B E C A (4) B E C D A (4) A C E B D (4) E C B A D (3) D B A C E (3) D A B E C (3) C E B A D (3) C A E B D (3) B C E D A (3) A D E C B (3) E C A B D (2) D B C A E (2) D A E B C (2) C D B A E (2) A D E B C (2) E A D B C (1) D B E A C (1) D B C E A (1) D B A E C (1) D A C B E (1) C B E A D (1) C B D A E (1) C A B D E (1) B E D C A (1) B C D E A (1) A E C D B (1) A E C B D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 6 4 20 B -8 0 2 -14 -4 C -6 -2 0 6 12 D -4 14 -6 0 14 E -20 4 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 4 20 B -8 0 2 -14 -4 C -6 -2 0 6 12 D -4 14 -6 0 14 E -20 4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=25 E=21 C=11 B=9 so B is eliminated. Round 2 votes counts: A=34 E=26 D=25 C=15 so C is eliminated. Round 3 votes counts: A=38 E=33 D=29 so D is eliminated. Round 4 votes counts: A=60 E=40 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:209 C:205 B:188 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 4 20 B -8 0 2 -14 -4 C -6 -2 0 6 12 D -4 14 -6 0 14 E -20 4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 4 20 B -8 0 2 -14 -4 C -6 -2 0 6 12 D -4 14 -6 0 14 E -20 4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 4 20 B -8 0 2 -14 -4 C -6 -2 0 6 12 D -4 14 -6 0 14 E -20 4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2710: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) C E D B A (8) E C D B A (5) B D A C E (5) E D C B A (3) E D A B C (3) E C A B D (3) E A D C B (3) E A C D B (3) D B C E A (3) D B C A E (3) C E A B D (3) B D C A E (3) B A D C E (3) A B C D E (3) E D B A C (2) E C D A B (2) E A D B C (2) D B E A C (2) C B D E A (2) C A E B D (2) A E D B C (2) A E C B D (2) A C E B D (2) A C B D E (2) A B D E C (2) E C A D B (1) D E C B A (1) D E B A C (1) D B A E C (1) D B A C E (1) D A B E C (1) C D B E A (1) C B D A E (1) C B A D E (1) B C D E A (1) B C A D E (1) A E B C D (1) A D B E C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 12 2 2 B -4 0 8 2 2 C -12 -8 0 -10 16 D -2 -2 10 0 2 E -2 -2 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999168 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 2 2 B -4 0 8 2 2 C -12 -8 0 -10 16 D -2 -2 10 0 2 E -2 -2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999142 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 C=18 D=13 B=13 so D is eliminated. Round 2 votes counts: A=30 E=29 B=23 C=18 so C is eliminated. Round 3 votes counts: E=40 A=32 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:204 D:204 C:193 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 2 2 B -4 0 8 2 2 C -12 -8 0 -10 16 D -2 -2 10 0 2 E -2 -2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999142 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 2 2 B -4 0 8 2 2 C -12 -8 0 -10 16 D -2 -2 10 0 2 E -2 -2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999142 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 2 2 B -4 0 8 2 2 C -12 -8 0 -10 16 D -2 -2 10 0 2 E -2 -2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999142 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2711: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E C B A D (7) E A B C D (6) B A E C D (5) E A B D C (4) D A E B C (4) D E A C B (3) D A B E C (3) C B E D A (3) B C A E D (3) A E B D C (3) E D C A B (2) E D A C B (2) E D A B C (2) E C D B A (2) E C A B D (2) E B A C D (2) C E D B A (2) C E B D A (2) C E B A D (2) C D B E A (2) C D B A E (2) C B E A D (2) C B D A E (2) B C A D E (2) B A D C E (2) B A C D E (2) A E D B C (2) A B D C E (2) D E C A B (1) D C E B A (1) D C B E A (1) D C B A E (1) D B A C E (1) D A E C B (1) D A C B E (1) C D E B A (1) B E A C D (1) B A C E D (1) A D E B C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 20 4 0 B -2 0 16 12 -4 C -20 -16 0 4 -12 D -4 -12 -4 0 -14 E 0 4 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428368 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.571632 Sum of squares = 0.510262352968 Cumulative probabilities = A: 0.428368 B: 0.428368 C: 0.428368 D: 0.428368 E: 1.000000 A B C D E A 0 2 20 4 0 B -2 0 16 12 -4 C -20 -16 0 4 -12 D -4 -12 -4 0 -14 E 0 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 C=18 B=16 A=11 so A is eliminated. Round 2 votes counts: E=34 D=27 B=21 C=18 so C is eliminated. Round 3 votes counts: E=40 D=32 B=28 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:213 B:211 D:183 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 20 4 0 B -2 0 16 12 -4 C -20 -16 0 4 -12 D -4 -12 -4 0 -14 E 0 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 20 4 0 B -2 0 16 12 -4 C -20 -16 0 4 -12 D -4 -12 -4 0 -14 E 0 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 20 4 0 B -2 0 16 12 -4 C -20 -16 0 4 -12 D -4 -12 -4 0 -14 E 0 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2712: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) D C A B E (5) D B A C E (5) C A D B E (5) E C A B D (4) E B A C D (4) D E C B A (4) D E B A C (4) C D A B E (4) A C B D E (4) E C D A B (3) D C B A E (3) D B E A C (3) C E A D B (3) B D E A C (3) E C A D B (2) E A B C D (2) D E B C A (2) D C E B A (2) C E D A B (2) C A E D B (2) B E A D C (2) B D A C E (2) B A D E C (2) A B E C D (2) E D C B A (1) E D B C A (1) E D B A C (1) E C B D A (1) E B D A C (1) E B C D A (1) E B A D C (1) D B A E C (1) D A C B E (1) C E A B D (1) C D E A B (1) C A E B D (1) C A B D E (1) B E D A C (1) B D A E C (1) B A D C E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 -6 0 B -6 0 0 -10 12 C -2 0 0 4 6 D 6 10 -4 0 26 E 0 -12 -6 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -6 0 B -6 0 0 -10 12 C -2 0 0 4 6 D 6 10 -4 0 26 E 0 -12 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=22 C=20 A=16 B=12 so B is eliminated. Round 2 votes counts: D=36 E=25 C=20 A=19 so A is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:204 A:201 B:198 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 2 -6 0 B -6 0 0 -10 12 C -2 0 0 4 6 D 6 10 -4 0 26 E 0 -12 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -6 0 B -6 0 0 -10 12 C -2 0 0 4 6 D 6 10 -4 0 26 E 0 -12 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -6 0 B -6 0 0 -10 12 C -2 0 0 4 6 D 6 10 -4 0 26 E 0 -12 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2713: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) D E B C A (8) E D B A C (7) B D E A C (7) B A C D E (6) A C B E D (6) B A E C D (5) C A D E B (4) D E C B A (3) D E B A C (3) D B E A C (3) C A E D B (3) C A B E D (3) B E D A C (3) B A C E D (3) A C E B D (3) E A C D B (2) D E C A B (2) D B C E A (2) C E A D B (2) B D C A E (2) E D B C A (1) E D A B C (1) E C A D B (1) E B A D C (1) E A D C B (1) E A B D C (1) D C E A B (1) D C B E A (1) D C A E B (1) D B E C A (1) C D E A B (1) C D A B E (1) B A E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 8 4 -4 B 14 0 12 2 8 C -8 -12 0 0 -2 D -4 -2 0 0 10 E 4 -8 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 4 -4 B 14 0 12 2 8 C -8 -12 0 0 -2 D -4 -2 0 0 10 E 4 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 C=23 E=15 A=10 so A is eliminated. Round 2 votes counts: C=32 B=28 D=25 E=15 so E is eliminated. Round 3 votes counts: D=35 C=35 B=30 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:218 D:202 A:197 E:194 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 4 -4 B 14 0 12 2 8 C -8 -12 0 0 -2 D -4 -2 0 0 10 E 4 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 4 -4 B 14 0 12 2 8 C -8 -12 0 0 -2 D -4 -2 0 0 10 E 4 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 4 -4 B 14 0 12 2 8 C -8 -12 0 0 -2 D -4 -2 0 0 10 E 4 -8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2714: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (7) A D E C B (7) B D A E C (6) B A C E D (6) D E C A B (5) E C D A B (4) D B A E C (4) C E D B A (4) B D C E A (4) B D A C E (4) B C E A D (4) A D B E C (4) C E D A B (3) C E B D A (3) C E B A D (3) B A D C E (3) E C D B A (2) E C A D B (2) D B E C A (2) D A E C B (2) D A B E C (2) B C E D A (2) B A D E C (2) A E C D B (2) A C E D B (2) A C E B D (2) E D C B A (1) D E C B A (1) C E A B D (1) C B E A D (1) C A B E D (1) B D E C A (1) A E D C B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 0 0 B 6 0 -8 -10 -4 C 2 8 0 0 4 D 0 10 0 0 -2 E 0 4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.598190 D: 0.401810 E: 0.000000 Sum of squares = 0.519282396732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.598190 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 0 0 B 6 0 -8 -10 -4 C 2 8 0 0 4 D 0 10 0 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=23 A=20 D=16 E=9 so E is eliminated. Round 2 votes counts: B=32 C=31 A=20 D=17 so D is eliminated. Round 3 votes counts: C=38 B=38 A=24 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 D:204 E:201 A:196 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 0 0 B 6 0 -8 -10 -4 C 2 8 0 0 4 D 0 10 0 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 0 0 B 6 0 -8 -10 -4 C 2 8 0 0 4 D 0 10 0 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 0 0 B 6 0 -8 -10 -4 C 2 8 0 0 4 D 0 10 0 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2715: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (6) B E A D C (6) A C D E B (6) D C E B A (5) D C B E A (5) B E D C A (5) B A E D C (5) A B E D C (5) D C B A E (4) D C A B E (4) C D E B A (4) C D A E B (4) A B D C E (4) E B D C A (3) C E D A B (3) A C D B E (3) A B E C D (3) E C D B A (2) E B A C D (2) E A B C D (2) A E B C D (2) A B C D E (2) E D C B A (1) E C D A B (1) E C A D B (1) E B C D A (1) D A C B E (1) C D A B E (1) C A D E B (1) B E D A C (1) B D E C A (1) B A D C E (1) A E C D B (1) A E C B D (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 -4 -4 2 B -8 0 -12 -12 6 C 4 12 0 -6 14 D 4 12 6 0 8 E -2 -6 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -4 2 B -8 0 -12 -12 6 C 4 12 0 -6 14 D 4 12 6 0 8 E -2 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=19 C=19 B=19 E=13 so E is eliminated. Round 2 votes counts: A=32 B=25 C=23 D=20 so D is eliminated. Round 3 votes counts: C=42 A=33 B=25 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:215 C:212 A:201 B:187 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -4 -4 2 B -8 0 -12 -12 6 C 4 12 0 -6 14 D 4 12 6 0 8 E -2 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -4 2 B -8 0 -12 -12 6 C 4 12 0 -6 14 D 4 12 6 0 8 E -2 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -4 2 B -8 0 -12 -12 6 C 4 12 0 -6 14 D 4 12 6 0 8 E -2 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2716: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D E C B A (7) D C E B A (7) D C E A B (6) E C B D A (4) D A C E B (4) C E A B D (4) B E A C D (4) A D B C E (4) A B D E C (4) E C D B A (3) E B C D A (3) C E B A D (3) B A E D C (3) B A E C D (3) A B C E D (3) D B E C A (2) C E D A B (2) C A E B D (2) B E C A D (2) A D C B E (2) A C E B D (2) A B D C E (2) E D C B A (1) E C B A D (1) E B C A D (1) D E B C A (1) D B A E C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E B A (1) B D E A C (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -4 6 -10 B -2 0 -6 6 -8 C 4 6 0 2 -6 D -6 -6 -2 0 -8 E 10 8 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -4 6 -10 B -2 0 -6 6 -8 C 4 6 0 2 -6 D -6 -6 -2 0 -8 E 10 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=29 C=14 E=13 B=13 so E is eliminated. Round 2 votes counts: D=32 A=29 C=22 B=17 so B is eliminated. Round 3 votes counts: A=39 D=33 C=28 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:216 C:203 A:197 B:195 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 6 -10 B -2 0 -6 6 -8 C 4 6 0 2 -6 D -6 -6 -2 0 -8 E 10 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 6 -10 B -2 0 -6 6 -8 C 4 6 0 2 -6 D -6 -6 -2 0 -8 E 10 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 6 -10 B -2 0 -6 6 -8 C 4 6 0 2 -6 D -6 -6 -2 0 -8 E 10 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2717: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) A C D E B (9) B E C D A (8) C A D E B (6) A D C E B (5) A D B E C (5) E B D C A (4) E B C D A (4) B E A D C (4) A D C B E (4) C E D B A (3) C D A E B (3) D C E A B (2) D C A E B (2) D A C E B (2) C E D A B (2) C A B E D (2) B E C A D (2) B A D E C (2) A C B E D (2) A B D E C (2) E D B C A (1) D E C B A (1) D B E A C (1) D B A E C (1) D A E C B (1) D A B E C (1) C E B D A (1) C E B A D (1) C D E B A (1) C D E A B (1) C A E D B (1) C A E B D (1) B A E D C (1) B A C E D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -14 0 6 B -6 0 0 -6 -2 C 14 0 0 -2 2 D 0 6 2 0 0 E -6 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.115270 B: 0.000000 C: 0.000000 D: 0.884730 E: 0.000000 Sum of squares = 0.7960337602 Cumulative probabilities = A: 0.115270 B: 0.115270 C: 0.115270 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 0 6 B -6 0 0 -6 -2 C 14 0 0 -2 2 D 0 6 2 0 0 E -6 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250215413 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=29 A=29 C=22 D=11 E=9 so E is eliminated. Round 2 votes counts: B=37 A=29 C=22 D=12 so D is eliminated. Round 3 votes counts: B=40 A=33 C=27 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:207 D:204 A:199 E:197 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -14 0 6 B -6 0 0 -6 -2 C 14 0 0 -2 2 D 0 6 2 0 0 E -6 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250215413 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 0 6 B -6 0 0 -6 -2 C 14 0 0 -2 2 D 0 6 2 0 0 E -6 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250215413 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 0 6 B -6 0 0 -6 -2 C 14 0 0 -2 2 D 0 6 2 0 0 E -6 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250215413 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2718: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D A E C B (5) C B A D E (5) A C B D E (5) E D B C A (4) D E A C B (4) C A B D E (4) B E C D A (4) B C A E D (4) B A C E D (4) A E D B C (4) E D B A C (3) D E C A B (3) D C E B A (3) B C E A D (3) A C D B E (3) A C B E D (3) D E C B A (2) D E A B C (2) D C A E B (2) C B A E D (2) B E D C A (2) B E A C D (2) B C E D A (2) A E B D C (2) A E B C D (2) A D E C B (2) A B C E D (2) E B D C A (1) E B A D C (1) D E B C A (1) C D B E A (1) C B D A E (1) C A D B E (1) B E C A D (1) B C D E A (1) B A E C D (1) A E D C B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 6 8 10 B -4 0 4 6 4 C -6 -4 0 6 -6 D -8 -6 -6 0 -8 E -10 -4 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 8 10 B -4 0 4 6 4 C -6 -4 0 6 -6 D -8 -6 -6 0 -8 E -10 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=24 D=22 E=14 C=14 so E is eliminated. Round 2 votes counts: D=34 B=26 A=26 C=14 so C is eliminated. Round 3 votes counts: D=35 B=34 A=31 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:205 E:200 C:195 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 8 10 B -4 0 4 6 4 C -6 -4 0 6 -6 D -8 -6 -6 0 -8 E -10 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 8 10 B -4 0 4 6 4 C -6 -4 0 6 -6 D -8 -6 -6 0 -8 E -10 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 8 10 B -4 0 4 6 4 C -6 -4 0 6 -6 D -8 -6 -6 0 -8 E -10 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2719: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) B D C A E (10) C E A D B (8) D B C A E (7) A E B D C (6) E A C B D (5) D C B A E (5) B D A E C (5) B D A C E (4) E C A D B (3) C E D A B (3) B D C E A (3) B A E D C (3) A B E D C (3) E C A B D (2) E A B D C (2) C D E B A (2) C D B E A (2) C D A E B (2) B E A D C (2) B A D E C (2) E B A D C (1) E B A C D (1) D C A B E (1) D B C E A (1) D B A C E (1) C D B A E (1) C B D E A (1) A E C D B (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -2 4 6 B 2 0 6 4 4 C 2 -6 0 -14 2 D -4 -4 14 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 4 6 B 2 0 6 4 4 C 2 -6 0 -14 2 D -4 -4 14 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=25 C=19 D=15 A=12 so A is eliminated. Round 2 votes counts: E=33 B=33 C=19 D=15 so D is eliminated. Round 3 votes counts: B=42 E=33 C=25 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:203 D:201 E:196 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 4 6 B 2 0 6 4 4 C 2 -6 0 -14 2 D -4 -4 14 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 6 B 2 0 6 4 4 C 2 -6 0 -14 2 D -4 -4 14 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 6 B 2 0 6 4 4 C 2 -6 0 -14 2 D -4 -4 14 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2720: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) C E D B A (7) C A E B D (7) E C D B A (6) C A B D E (6) E D C B A (5) E D B A C (5) E A D B C (5) A B D C E (5) A B C D E (5) A C B D E (4) D E B A C (3) C E D A B (3) C E A D B (3) B D A E C (3) B A D E C (3) A B D E C (3) E C D A B (2) D B E A C (2) A B E C D (2) E D A B C (1) E A C D B (1) E A B D C (1) D C B E A (1) C D E B A (1) C A E D B (1) B D E A C (1) B A D C E (1) B A C D E (1) A C E B D (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 0 4 -10 B -6 0 2 -8 -20 C 0 -2 0 4 -4 D -4 8 -4 0 -22 E 10 20 4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 0 4 -10 B -6 0 2 -8 -20 C 0 -2 0 4 -4 D -4 8 -4 0 -22 E 10 20 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=28 A=23 B=9 D=6 so D is eliminated. Round 2 votes counts: E=37 C=29 A=23 B=11 so B is eliminated. Round 3 votes counts: E=40 A=31 C=29 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:200 C:199 D:189 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 4 -10 B -6 0 2 -8 -20 C 0 -2 0 4 -4 D -4 8 -4 0 -22 E 10 20 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 4 -10 B -6 0 2 -8 -20 C 0 -2 0 4 -4 D -4 8 -4 0 -22 E 10 20 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 4 -10 B -6 0 2 -8 -20 C 0 -2 0 4 -4 D -4 8 -4 0 -22 E 10 20 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2721: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) C B D A E (6) E A B C D (4) B C A E D (4) A E B D C (4) A B E D C (4) D E C A B (3) D E A C B (3) C B E A D (3) C B D E A (3) B C E A D (3) B A C E D (3) A B E C D (3) E D A C B (2) E A B D C (2) D E A B C (2) D C E B A (2) D C B A E (2) D B A C E (2) D A E C B (2) D A E B C (2) D A B E C (2) C E B A D (2) C D E B A (2) C B E D A (2) A E D B C (2) A D B E C (2) A B D E C (2) E D C A B (1) E C D B A (1) E C A B D (1) E B A C D (1) E A D C B (1) E A D B C (1) E A C D B (1) D C B E A (1) D C A E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C D B E A (1) C B A E D (1) C B A D E (1) B C A D E (1) B A E C D (1) B A D C E (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 14 2 12 B -10 0 10 8 14 C -14 -10 0 -8 4 D -2 -8 8 0 2 E -12 -14 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 2 12 B -10 0 10 8 14 C -14 -10 0 -8 4 D -2 -8 8 0 2 E -12 -14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=23 A=20 E=15 B=13 so B is eliminated. Round 2 votes counts: C=31 D=29 A=25 E=15 so E is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:211 D:200 C:186 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 2 12 B -10 0 10 8 14 C -14 -10 0 -8 4 D -2 -8 8 0 2 E -12 -14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 2 12 B -10 0 10 8 14 C -14 -10 0 -8 4 D -2 -8 8 0 2 E -12 -14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 2 12 B -10 0 10 8 14 C -14 -10 0 -8 4 D -2 -8 8 0 2 E -12 -14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2722: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (16) D B A E C (11) C E A B D (8) B D A C E (6) C A E B D (5) A B D C E (5) B D E C A (4) B D A E C (3) E D A B C (2) E C B D A (2) E A C D B (2) D B E A C (2) C B E D A (2) B D C E A (2) B D C A E (2) B A D C E (2) A E C D B (2) A D B E C (2) E C D A B (1) E B D C A (1) D E B A C (1) D B E C A (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) C B D E A (1) C B A D E (1) C A E D B (1) B C D E A (1) A E D C B (1) A E D B C (1) A D E B C (1) A D B C E (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 0 8 4 B -10 0 4 2 2 C 0 -4 0 -2 -10 D -8 -2 2 0 2 E -4 -2 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.818900 B: 0.000000 C: 0.181100 D: 0.000000 E: 0.000000 Sum of squares = 0.703394261193 Cumulative probabilities = A: 0.818900 B: 0.818900 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 8 4 B -10 0 4 2 2 C 0 -4 0 -2 -10 D -8 -2 2 0 2 E -4 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836853387 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=20 B=20 D=18 A=18 so D is eliminated. Round 2 votes counts: B=35 E=25 C=20 A=20 so C is eliminated. Round 3 votes counts: B=39 E=35 A=26 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:211 E:201 B:199 D:197 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 8 4 B -10 0 4 2 2 C 0 -4 0 -2 -10 D -8 -2 2 0 2 E -4 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836853387 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 8 4 B -10 0 4 2 2 C 0 -4 0 -2 -10 D -8 -2 2 0 2 E -4 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836853387 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 8 4 B -10 0 4 2 2 C 0 -4 0 -2 -10 D -8 -2 2 0 2 E -4 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836853387 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2723: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) E D A B C (7) D E B A C (7) E D C A B (6) C B A D E (6) C A B E D (6) B D E A C (6) C A E D B (5) A B D E C (5) C E D B A (4) C E D A B (4) E D B A C (3) E D A C B (3) B C D A E (3) A C B E D (3) E D C B A (2) D E B C A (2) C D E B A (2) C B D E A (2) C A E B D (2) E D B C A (1) D E C B A (1) C A B D E (1) B D E C A (1) B D A E C (1) B C A D E (1) B A C D E (1) A E D B C (1) A C E D B (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 2 -10 -2 B 6 0 2 2 -2 C -2 -2 0 -14 -12 D 10 -2 14 0 4 E 2 2 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 -6 2 -10 -2 B 6 0 2 2 -2 C -2 -2 0 -14 -12 D 10 -2 14 0 4 E 2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.374999999141 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=23 E=22 A=13 D=10 so D is eliminated. Round 2 votes counts: E=32 C=32 B=23 A=13 so A is eliminated. Round 3 votes counts: C=37 E=33 B=30 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:206 B:204 A:192 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -10 -2 B 6 0 2 2 -2 C -2 -2 0 -14 -12 D 10 -2 14 0 4 E 2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.374999999141 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -10 -2 B 6 0 2 2 -2 C -2 -2 0 -14 -12 D 10 -2 14 0 4 E 2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.374999999141 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -10 -2 B 6 0 2 2 -2 C -2 -2 0 -14 -12 D 10 -2 14 0 4 E 2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.374999999141 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2724: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) E A B C D (7) E B A C D (5) C D B E A (5) A E B C D (5) E A C B D (4) D C B A E (4) A E D C B (4) A E B D C (4) C B E D A (3) B E C A D (3) A E D B C (3) A E C D B (3) A E C B D (3) E C A B D (2) D C A E B (2) D B C E A (2) D A E C B (2) D A E B C (2) D A C E B (2) D A B E C (2) C B E A D (2) C B D E A (2) B E A C D (2) B D C E A (2) B C E A D (2) B C D E A (2) A D E B C (2) D C A B E (1) D A C B E (1) C E B A D (1) C E A D B (1) C D A E B (1) C A E B D (1) B E D C A (1) B E A D C (1) B D E C A (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 6 6 12 -14 B -6 0 -6 8 -10 C -6 6 0 10 -14 D -12 -8 -10 0 -16 E 14 10 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 12 -14 B -6 0 -6 8 -10 C -6 6 0 10 -14 D -12 -8 -10 0 -16 E 14 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 E=18 C=16 B=15 so B is eliminated. Round 2 votes counts: D=29 E=25 A=25 C=21 so C is eliminated. Round 3 votes counts: D=39 E=35 A=26 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:205 C:198 B:193 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 12 -14 B -6 0 -6 8 -10 C -6 6 0 10 -14 D -12 -8 -10 0 -16 E 14 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 12 -14 B -6 0 -6 8 -10 C -6 6 0 10 -14 D -12 -8 -10 0 -16 E 14 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 12 -14 B -6 0 -6 8 -10 C -6 6 0 10 -14 D -12 -8 -10 0 -16 E 14 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2725: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) A D E B C (8) D A E B C (5) C E B D A (5) C B E D A (5) C B E A D (5) A B D C E (5) E C D B A (4) D A B E C (4) C E B A D (4) C B A D E (4) B C A D E (4) A D B E C (4) E C D A B (3) E D C A B (2) E D A B C (2) C E A D B (2) C E A B D (2) C B A E D (2) B D E C A (2) B D A C E (2) B C D E A (2) A D B C E (2) A C B D E (2) E D C B A (1) E D A C B (1) E C B D A (1) E C A D B (1) E A D C B (1) C A E B D (1) B D C A E (1) B C D A E (1) B A C D E (1) A E D B C (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -4 18 14 B 4 0 4 16 8 C 4 -4 0 0 18 D -18 -16 0 0 12 E -14 -8 -18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 18 14 B 4 0 4 16 8 C 4 -4 0 0 18 D -18 -16 0 0 12 E -14 -8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=24 B=21 E=16 D=9 so D is eliminated. Round 2 votes counts: A=33 C=30 B=21 E=16 so E is eliminated. Round 3 votes counts: C=42 A=37 B=21 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:216 A:212 C:209 D:189 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 18 14 B 4 0 4 16 8 C 4 -4 0 0 18 D -18 -16 0 0 12 E -14 -8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 18 14 B 4 0 4 16 8 C 4 -4 0 0 18 D -18 -16 0 0 12 E -14 -8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 18 14 B 4 0 4 16 8 C 4 -4 0 0 18 D -18 -16 0 0 12 E -14 -8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2726: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (18) D B C A E (12) D E A B C (10) D B C E A (9) B C D A E (6) B C A E D (6) A E C B D (6) D E A C B (5) C B A E D (4) C A B E D (4) E A D C B (3) E A C D B (3) D E B A C (3) D B E C A (3) D B E A C (2) B D C A E (2) B C A D E (2) E B A D C (1) E A D B C (1) Total count = 100 A B C D E A 0 0 4 -4 -16 B 0 0 14 -2 0 C -4 -14 0 -2 -10 D 4 2 2 0 8 E 16 0 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -4 -16 B 0 0 14 -2 0 C -4 -14 0 -2 -10 D 4 2 2 0 8 E 16 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 E=26 B=16 C=8 A=6 so A is eliminated. Round 2 votes counts: D=44 E=32 B=16 C=8 so C is eliminated. Round 3 votes counts: D=44 E=32 B=24 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:209 D:208 B:206 A:192 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -4 -16 B 0 0 14 -2 0 C -4 -14 0 -2 -10 D 4 2 2 0 8 E 16 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -4 -16 B 0 0 14 -2 0 C -4 -14 0 -2 -10 D 4 2 2 0 8 E 16 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -4 -16 B 0 0 14 -2 0 C -4 -14 0 -2 -10 D 4 2 2 0 8 E 16 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2727: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) C B A D E (6) B C D E A (6) B C D A E (5) E B A D C (4) E A B D C (4) D E A C B (4) C D B A E (4) C A D B E (4) B E D C A (4) B E D A C (4) E B D A C (3) C B D A E (3) B E C A D (3) B C A E D (3) E A D B C (2) D C B E A (2) D C A E B (2) D A C E B (2) C D A B E (2) B D E C A (2) A E D C B (2) A E B C D (2) E D B A C (1) E D A B C (1) E B A C D (1) D E B A C (1) D E A B C (1) D C A B E (1) D B E C A (1) C D A E B (1) C A D E B (1) C A B E D (1) C A B D E (1) B D C E A (1) B C E D A (1) B C A D E (1) B A E C D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -28 -10 -4 -6 B 28 0 18 30 28 C 10 -18 0 12 2 D 4 -30 -12 0 6 E 6 -28 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -10 -4 -6 B 28 0 18 30 28 C 10 -18 0 12 2 D 4 -30 -12 0 6 E 6 -28 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 C=23 E=16 D=14 A=9 so A is eliminated. Round 2 votes counts: B=38 C=26 E=20 D=16 so D is eliminated. Round 3 votes counts: B=39 C=34 E=27 so E is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:252 C:203 E:185 D:184 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -10 -4 -6 B 28 0 18 30 28 C 10 -18 0 12 2 D 4 -30 -12 0 6 E 6 -28 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -10 -4 -6 B 28 0 18 30 28 C 10 -18 0 12 2 D 4 -30 -12 0 6 E 6 -28 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -10 -4 -6 B 28 0 18 30 28 C 10 -18 0 12 2 D 4 -30 -12 0 6 E 6 -28 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2728: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (17) B A C D E (11) D E B A C (10) C A B E D (9) E D C A B (6) D B E A C (5) B A D E C (5) C E D A B (4) A B C E D (4) C B A D E (3) B D A E C (3) A B D E C (3) E C D A B (2) D E B C A (2) D E A B C (2) C E D B A (2) C A E B D (2) B A C E D (2) E D C B A (1) E D A C B (1) E A D C B (1) D B C E A (1) C E A B D (1) C B A E D (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 -30 -4 -12 -10 B 30 0 0 -8 -2 C 4 0 0 -16 -16 D 12 8 16 0 28 E 10 2 16 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -4 -12 -10 B 30 0 0 -8 -2 C 4 0 0 -16 -16 D 12 8 16 0 28 E 10 2 16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=23 C=22 E=11 A=7 so A is eliminated. Round 2 votes counts: D=37 B=30 C=22 E=11 so E is eliminated. Round 3 votes counts: D=46 B=30 C=24 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:232 B:210 E:200 C:186 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -30 -4 -12 -10 B 30 0 0 -8 -2 C 4 0 0 -16 -16 D 12 8 16 0 28 E 10 2 16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -4 -12 -10 B 30 0 0 -8 -2 C 4 0 0 -16 -16 D 12 8 16 0 28 E 10 2 16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -4 -12 -10 B 30 0 0 -8 -2 C 4 0 0 -16 -16 D 12 8 16 0 28 E 10 2 16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2729: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) A E B C D (7) D C B E A (6) A B E D C (6) D C A E B (5) C D E B A (5) A B E C D (5) D C A B E (4) D A B C E (4) C E D B A (4) C E B D A (4) B E A C D (4) E C B A D (3) E C A B D (3) E B A C D (3) B E C A D (3) D C E B A (2) C E A D B (2) C E A B D (2) C D E A B (2) C D A E B (2) B A E D C (2) E B C A D (1) E A C B D (1) E A B C D (1) D C B A E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A C B E (1) D A B E C (1) C E D A B (1) C E B A D (1) C B E D A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 16 -4 8 2 B -16 0 -6 -4 -4 C 4 6 0 12 0 D -8 4 -12 0 -10 E -2 4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.715821 D: 0.000000 E: 0.284179 Sum of squares = 0.593157734553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.715821 D: 0.715821 E: 1.000000 A B C D E A 0 16 -4 8 2 B -16 0 -6 -4 -4 C 4 6 0 12 0 D -8 4 -12 0 -10 E -2 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=27 C=24 E=12 B=9 so B is eliminated. Round 2 votes counts: A=30 D=27 C=24 E=19 so E is eliminated. Round 3 votes counts: A=39 C=34 D=27 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:211 E:206 D:187 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 8 2 B -16 0 -6 -4 -4 C 4 6 0 12 0 D -8 4 -12 0 -10 E -2 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 8 2 B -16 0 -6 -4 -4 C 4 6 0 12 0 D -8 4 -12 0 -10 E -2 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 8 2 B -16 0 -6 -4 -4 C 4 6 0 12 0 D -8 4 -12 0 -10 E -2 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2730: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (14) C E A D B (10) B A E C D (10) B A E D C (9) C D E A B (8) D E C A B (4) D B C E A (4) A E C B D (4) D E A B C (3) C A E B D (3) B D A E C (3) D E A C B (2) D B E A C (2) C A E D B (2) B D E A C (2) B C A E D (2) B A D E C (2) A B E C D (2) E D A C B (1) E C A D B (1) E A D C B (1) D C B E A (1) D B E C A (1) C E D A B (1) C D B E A (1) C A B E D (1) B D C A E (1) B A C E D (1) A E C D B (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 22 -8 4 -12 B -22 0 -14 -16 -16 C 8 14 0 0 2 D -4 16 0 0 -4 E 12 16 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.739172 D: 0.260828 E: 0.000000 Sum of squares = 0.614406675063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.739172 D: 1.000000 E: 1.000000 A B C D E A 0 22 -8 4 -12 B -22 0 -14 -16 -16 C 8 14 0 0 2 D -4 16 0 0 -4 E 12 16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555642061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=30 C=26 A=10 E=3 so E is eliminated. Round 2 votes counts: D=32 B=30 C=27 A=11 so A is eliminated. Round 3 votes counts: C=34 D=33 B=33 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:212 D:204 A:203 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -8 4 -12 B -22 0 -14 -16 -16 C 8 14 0 0 2 D -4 16 0 0 -4 E 12 16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555642061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -8 4 -12 B -22 0 -14 -16 -16 C 8 14 0 0 2 D -4 16 0 0 -4 E 12 16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555642061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -8 4 -12 B -22 0 -14 -16 -16 C 8 14 0 0 2 D -4 16 0 0 -4 E 12 16 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555642061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2731: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (14) B D C E A (10) D B C A E (9) B D C A E (7) E A D C B (6) B D E C A (6) B E D C A (4) B C A D E (4) E B D A C (3) C D A B E (3) E A B C D (2) D C B A E (2) D B C E A (2) C D B A E (2) C A D E B (2) C A D B E (2) B E D A C (2) B E A C D (2) B A C E D (2) E D C B A (1) E D A B C (1) E B A C D (1) E A C B D (1) D C A E B (1) D B E C A (1) B E A D C (1) B C D A E (1) B A C D E (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -22 -14 -10 -14 B 22 0 18 0 26 C 14 -18 0 -12 6 D 10 0 12 0 10 E 14 -26 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.496976 C: 0.000000 D: 0.503024 E: 0.000000 Sum of squares = 0.50001828314 Cumulative probabilities = A: 0.000000 B: 0.496976 C: 0.496976 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -14 -10 -14 B 22 0 18 0 26 C 14 -18 0 -12 6 D 10 0 12 0 10 E 14 -26 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=29 D=15 C=9 A=7 so A is eliminated. Round 2 votes counts: B=40 E=31 D=15 C=14 so C is eliminated. Round 3 votes counts: B=41 E=33 D=26 so D is eliminated. Round 4 votes counts: B=63 E=37 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:233 D:216 C:195 E:186 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -14 -10 -14 B 22 0 18 0 26 C 14 -18 0 -12 6 D 10 0 12 0 10 E 14 -26 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 -10 -14 B 22 0 18 0 26 C 14 -18 0 -12 6 D 10 0 12 0 10 E 14 -26 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 -10 -14 B 22 0 18 0 26 C 14 -18 0 -12 6 D 10 0 12 0 10 E 14 -26 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2732: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) D B C A E (6) C D B A E (6) C B D A E (6) D C B E A (5) C D E B A (5) C D B E A (4) B D A C E (4) E D C B A (3) E A D B C (3) C B A D E (3) B C D A E (3) B C A D E (3) A B E D C (3) A B C D E (3) E D C A B (2) E D A B C (2) E A D C B (2) E A C B D (2) D B A E C (2) B D C A E (2) B A C D E (2) A E B C D (2) E C D B A (1) E C D A B (1) E C A D B (1) E A C D B (1) E A B D C (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) D B E C A (1) D B E A C (1) D B A C E (1) C E D B A (1) C E A D B (1) C E A B D (1) B A D C E (1) A C E B D (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -26 -16 -18 16 B 26 0 2 -4 18 C 16 -2 0 0 20 D 18 4 0 0 26 E -16 -18 -20 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.621763 D: 0.378237 E: 0.000000 Sum of squares = 0.529652372117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.621763 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -16 -18 16 B 26 0 2 -4 18 C 16 -2 0 0 20 D 18 4 0 0 26 E -16 -18 -20 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499998 D: 0.500002 E: 0.000000 Sum of squares = 0.499999998021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499998 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=20 D=19 A=19 B=15 so B is eliminated. Round 2 votes counts: C=33 D=25 A=22 E=20 so E is eliminated. Round 3 votes counts: C=36 D=32 A=32 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:224 B:221 C:217 A:178 E:160 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -26 -16 -18 16 B 26 0 2 -4 18 C 16 -2 0 0 20 D 18 4 0 0 26 E -16 -18 -20 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499998 D: 0.500002 E: 0.000000 Sum of squares = 0.499999998021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499998 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -16 -18 16 B 26 0 2 -4 18 C 16 -2 0 0 20 D 18 4 0 0 26 E -16 -18 -20 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499998 D: 0.500002 E: 0.000000 Sum of squares = 0.499999998021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499998 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -16 -18 16 B 26 0 2 -4 18 C 16 -2 0 0 20 D 18 4 0 0 26 E -16 -18 -20 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499998 D: 0.500002 E: 0.000000 Sum of squares = 0.499999998021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499998 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2733: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) E D A C B (11) D E B C A (11) D E A B C (8) A C B E D (7) D B E C A (4) D A B C E (4) A C B D E (4) A B C D E (4) E A C B D (3) C B A E D (3) B C D A E (3) E D C B A (2) E D C A B (2) E D B C A (2) E A D C B (2) D E A C B (2) D B C E A (2) D B C A E (2) D A E B C (2) C B E A D (2) A E C B D (2) A C E B D (2) E C B A D (1) E C A B D (1) E A C D B (1) A D B C E (1) Total count = 100 A B C D E A 0 12 6 -10 -8 B -12 0 10 -12 -4 C -6 -10 0 -10 -8 D 10 12 10 0 18 E 8 4 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -10 -8 B -12 0 10 -12 -4 C -6 -10 0 -10 -8 D 10 12 10 0 18 E 8 4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=25 A=20 B=15 C=5 so C is eliminated. Round 2 votes counts: D=35 E=25 B=20 A=20 so B is eliminated. Round 3 votes counts: D=38 A=35 E=27 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:201 A:200 B:191 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -10 -8 B -12 0 10 -12 -4 C -6 -10 0 -10 -8 D 10 12 10 0 18 E 8 4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -10 -8 B -12 0 10 -12 -4 C -6 -10 0 -10 -8 D 10 12 10 0 18 E 8 4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -10 -8 B -12 0 10 -12 -4 C -6 -10 0 -10 -8 D 10 12 10 0 18 E 8 4 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2734: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (20) B A D C E (18) E C D B A (5) C E D B A (5) A B E D C (5) E C A D B (4) E A C D B (4) C D E B A (4) A B D C E (4) E A B C D (3) D C B A E (3) A E B C D (3) A B D E C (3) D C E B A (2) D B A C E (2) C D B E A (2) B A E C D (2) E C B D A (1) E C A B D (1) E B A C D (1) E A C B D (1) D C E A B (1) D C A B E (1) D B C A E (1) D A C B E (1) B D A C E (1) B A E D C (1) B A D E C (1) Total count = 100 A B C D E A 0 2 0 2 -8 B -2 0 -10 -10 -10 C 0 10 0 12 -10 D -2 10 -12 0 -12 E 8 10 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 2 -8 B -2 0 -10 -10 -10 C 0 10 0 12 -10 D -2 10 -12 0 -12 E 8 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=23 A=15 D=11 C=11 so D is eliminated. Round 2 votes counts: E=40 B=26 C=18 A=16 so A is eliminated. Round 3 votes counts: E=43 B=38 C=19 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:206 A:198 D:192 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 2 -8 B -2 0 -10 -10 -10 C 0 10 0 12 -10 D -2 10 -12 0 -12 E 8 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 -8 B -2 0 -10 -10 -10 C 0 10 0 12 -10 D -2 10 -12 0 -12 E 8 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 -8 B -2 0 -10 -10 -10 C 0 10 0 12 -10 D -2 10 -12 0 -12 E 8 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2735: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (6) C A D B E (6) A C E D B (5) E A C D B (4) D C B A E (4) D B C E A (4) C A D E B (4) A C D E B (4) E D C B A (3) E D B C A (3) E B A D C (3) B E D A C (3) B E A D C (3) A E C D B (3) A C B D E (3) E B D C A (2) E A B C D (2) D C B E A (2) D B E C A (2) B E D C A (2) B A E D C (2) A E C B D (2) A E B C D (2) A C D B E (2) A B E C D (2) A B C D E (2) E D C A B (1) E C D A B (1) E A D C B (1) E A B D C (1) D E C B A (1) D E B C A (1) D C E B A (1) C E D A B (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B C A D E (1) B A D C E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 8 12 -2 B -4 0 -8 -14 -10 C -8 8 0 0 -4 D -12 14 0 0 -10 E 2 10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 12 -2 B -4 0 -8 -14 -10 C -8 8 0 0 -4 D -12 14 0 0 -10 E 2 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 B=16 D=15 C=15 so D is eliminated. Round 2 votes counts: E=29 A=27 C=22 B=22 so C is eliminated. Round 3 votes counts: A=40 E=32 B=28 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:211 C:198 D:196 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 12 -2 B -4 0 -8 -14 -10 C -8 8 0 0 -4 D -12 14 0 0 -10 E 2 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 12 -2 B -4 0 -8 -14 -10 C -8 8 0 0 -4 D -12 14 0 0 -10 E 2 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 12 -2 B -4 0 -8 -14 -10 C -8 8 0 0 -4 D -12 14 0 0 -10 E 2 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2736: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) A E B D C (5) D B C A E (4) C D E B A (4) B E A D C (4) B D E C A (4) B D A E C (4) A E C D B (4) A C D E B (4) D C A B E (3) D B C E A (3) B E D A C (3) A E C B D (3) A E B C D (3) A B E D C (3) E C A B D (2) E B A C D (2) D C B E A (2) D C B A E (2) D A C B E (2) C E A D B (2) C A D E B (2) B D C E A (2) A C E D B (2) E C B A D (1) E C A D B (1) E B C D A (1) E B C A D (1) E A C D B (1) E A C B D (1) E A B C D (1) D B A C E (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B D A (1) C D E A B (1) C D B E A (1) C D A B E (1) C B D E A (1) C A E D B (1) B E D C A (1) B E C D A (1) B C D E A (1) B A E D C (1) B A D E C (1) A D C E B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -2 -4 12 B -6 0 -4 -4 -4 C 2 4 0 2 2 D 4 4 -2 0 6 E -12 4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -4 12 B -6 0 -4 -4 -4 C 2 4 0 2 2 D 4 4 -2 0 6 E -12 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 B=22 D=18 E=11 so E is eliminated. Round 2 votes counts: A=30 C=26 B=26 D=18 so D is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 D:206 C:205 E:192 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -4 12 B -6 0 -4 -4 -4 C 2 4 0 2 2 D 4 4 -2 0 6 E -12 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -4 12 B -6 0 -4 -4 -4 C 2 4 0 2 2 D 4 4 -2 0 6 E -12 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -4 12 B -6 0 -4 -4 -4 C 2 4 0 2 2 D 4 4 -2 0 6 E -12 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2737: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (8) B C D E A (7) E A B D C (5) C D A B E (5) A E B C D (5) E D C B A (3) E D B C A (3) E B D A C (3) E A D B C (3) C D B E A (3) C B D A E (3) B E D C A (3) B D C E A (3) A E D C B (3) A E C D B (3) A C D B E (3) E D A B C (2) E B D C A (2) D E C B A (2) D C E B A (2) D C B E A (2) B C D A E (2) B A E C D (2) A E C B D (2) A E B D C (2) A C B D E (2) A B E C D (2) E D B A C (1) D C A E B (1) C D A E B (1) C B D E A (1) C A D B E (1) C A B D E (1) B E C D A (1) A E D B C (1) A D E C B (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -8 -16 8 B 2 0 -2 0 4 C 8 2 0 16 2 D 16 0 -16 0 0 E -8 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -16 8 B 2 0 -2 0 4 C 8 2 0 16 2 D 16 0 -16 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=23 E=22 B=18 D=7 so D is eliminated. Round 2 votes counts: A=30 C=28 E=24 B=18 so B is eliminated. Round 3 votes counts: C=40 A=32 E=28 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:202 D:200 E:193 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -16 8 B 2 0 -2 0 4 C 8 2 0 16 2 D 16 0 -16 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -16 8 B 2 0 -2 0 4 C 8 2 0 16 2 D 16 0 -16 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -16 8 B 2 0 -2 0 4 C 8 2 0 16 2 D 16 0 -16 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997161 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2738: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) B E A C D (7) A B E D C (7) D C A E B (6) C D E B A (6) A D B E C (6) E B C D A (5) E B C A D (5) A B D E C (5) D A B E C (4) D C E A B (3) C E D B A (3) C E B A D (3) B E C A D (3) E C B D A (2) E B A D C (2) D A C E B (2) C D A B E (2) C B E A D (2) B A E C D (2) E B D C A (1) E B D A C (1) D E C B A (1) D E B A C (1) D E A B C (1) D A E C B (1) D A E B C (1) D A C B E (1) D A B C E (1) C D A E B (1) C A D B E (1) B E A D C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -6 -4 -12 B 8 0 10 12 -8 C 6 -10 0 6 -14 D 4 -12 -6 0 -8 E 12 8 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 -4 -12 B 8 0 10 12 -8 C 6 -10 0 6 -14 D 4 -12 -6 0 -8 E 12 8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 A=22 E=16 B=13 so B is eliminated. Round 2 votes counts: E=27 C=27 A=24 D=22 so D is eliminated. Round 3 votes counts: C=36 A=34 E=30 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:221 B:211 C:194 D:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -12 B 8 0 10 12 -8 C 6 -10 0 6 -14 D 4 -12 -6 0 -8 E 12 8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -12 B 8 0 10 12 -8 C 6 -10 0 6 -14 D 4 -12 -6 0 -8 E 12 8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -12 B 8 0 10 12 -8 C 6 -10 0 6 -14 D 4 -12 -6 0 -8 E 12 8 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2739: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) E C B A D (6) E A D C B (6) D B C A E (6) E D A C B (5) E C D B A (5) D A B C E (5) E C A B D (4) C B E D A (4) A D B E C (4) C B D E A (3) A E B C D (3) A D E B C (3) A D B C E (3) C E B A D (2) C B E A D (2) B C A D E (2) A B E C D (2) A B D C E (2) A B C E D (2) A B C D E (2) E D C B A (1) E D C A B (1) E C D A B (1) E C A D B (1) E A D B C (1) E A C D B (1) E A C B D (1) D E C B A (1) D E A B C (1) D C B E A (1) D A E B C (1) C E D B A (1) C E B D A (1) C B D A E (1) B D C A E (1) B C D A E (1) B A D C E (1) A E D B C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -8 2 -16 B -4 0 -14 2 -14 C 8 14 0 8 -20 D -2 -2 -8 0 -24 E 16 14 20 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -8 2 -16 B -4 0 -14 2 -14 C 8 14 0 8 -20 D -2 -2 -8 0 -24 E 16 14 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 A=24 D=15 C=14 B=5 so B is eliminated. Round 2 votes counts: E=42 A=25 C=17 D=16 so D is eliminated. Round 3 votes counts: E=44 A=31 C=25 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:237 C:205 A:191 B:185 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 2 -16 B -4 0 -14 2 -14 C 8 14 0 8 -20 D -2 -2 -8 0 -24 E 16 14 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 2 -16 B -4 0 -14 2 -14 C 8 14 0 8 -20 D -2 -2 -8 0 -24 E 16 14 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 2 -16 B -4 0 -14 2 -14 C 8 14 0 8 -20 D -2 -2 -8 0 -24 E 16 14 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2740: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) E D C B A (8) A B C D E (8) D C E A B (7) C B A E D (4) C A B D E (4) B A E C D (4) B A C E D (4) A B C E D (4) E C B D A (3) D E C B A (3) C D E A B (3) A B E C D (3) E D A B C (2) D E A C B (2) D C A E B (2) C D E B A (2) C B A D E (2) B A E D C (2) A C B D E (2) E D B A C (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A D C (1) D E A B C (1) D A E B C (1) C E B D A (1) C D A B E (1) C A D B E (1) B E C A D (1) B E A D C (1) B C A E D (1) A E B D C (1) A D C B E (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 -14 -2 -2 B -16 0 -14 4 -6 C 14 14 0 0 -4 D 2 -4 0 0 8 E 2 6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.446633 D: 0.553367 E: 0.000000 Sum of squares = 0.505696137333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.446633 D: 1.000000 E: 1.000000 A B C D E A 0 16 -14 -2 -2 B -16 0 -14 4 -6 C 14 14 0 0 -4 D 2 -4 0 0 8 E 2 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 E=19 C=18 B=13 so B is eliminated. Round 2 votes counts: A=33 D=27 E=21 C=19 so C is eliminated. Round 3 votes counts: A=45 D=33 E=22 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:212 D:203 E:202 A:199 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -14 -2 -2 B -16 0 -14 4 -6 C 14 14 0 0 -4 D 2 -4 0 0 8 E 2 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -14 -2 -2 B -16 0 -14 4 -6 C 14 14 0 0 -4 D 2 -4 0 0 8 E 2 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -14 -2 -2 B -16 0 -14 4 -6 C 14 14 0 0 -4 D 2 -4 0 0 8 E 2 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2741: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) D A B E C (6) A E C B D (5) A D E B C (5) C E A B D (4) C D B E A (4) A E B C D (4) D C B E A (3) D C B A E (3) D B C E A (3) D B A E C (3) C D A E B (3) C A E D B (3) E B C A D (2) E B A C D (2) D B E A C (2) D A C B E (2) C E B D A (2) C B E A D (2) B E D A C (2) B E C D A (2) B E A D C (2) B D E A C (2) B C E D A (2) A E D B C (2) A E B D C (2) A C E D B (2) A C E B D (2) E A C B D (1) E A B C D (1) D C A B E (1) D A C E B (1) D A B C E (1) C E D A B (1) C D E B A (1) C B E D A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E C A D (1) B E A C D (1) B A E D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 0 8 -2 B 2 0 -4 6 -8 C 0 4 0 14 4 D -8 -6 -14 0 -18 E 2 8 -4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.237296 B: 0.000000 C: 0.762704 D: 0.000000 E: 0.000000 Sum of squares = 0.638026565797 Cumulative probabilities = A: 0.237296 B: 0.237296 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 8 -2 B 2 0 -4 6 -8 C 0 4 0 14 4 D -8 -6 -14 0 -18 E 2 8 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=25 A=23 B=14 E=6 so E is eliminated. Round 2 votes counts: C=32 D=25 A=25 B=18 so B is eliminated. Round 3 votes counts: C=39 A=31 D=30 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:212 C:211 A:202 B:198 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 8 -2 B 2 0 -4 6 -8 C 0 4 0 14 4 D -8 -6 -14 0 -18 E 2 8 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 8 -2 B 2 0 -4 6 -8 C 0 4 0 14 4 D -8 -6 -14 0 -18 E 2 8 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 8 -2 B 2 0 -4 6 -8 C 0 4 0 14 4 D -8 -6 -14 0 -18 E 2 8 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2742: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) E C D A B (6) C B E D A (6) B A C D E (6) E C D B A (5) A B D E C (5) D C E A B (4) B C A E D (4) A D E B C (4) A D B C E (4) E D C A B (3) E C B D A (3) D A E C B (3) C D E B A (3) B C E A D (3) A B E D C (3) D C B A E (2) C D B E A (2) A E D B C (2) A B D C E (2) E C B A D (1) E C A D B (1) E B A C D (1) E A D B C (1) E A C B D (1) E A B D C (1) D E C A B (1) D C E B A (1) D B C A E (1) D A C B E (1) C E D A B (1) C E B D A (1) C D B A E (1) C B E A D (1) C B D E A (1) C B D A E (1) B E C A D (1) B D C A E (1) B C D A E (1) B A E C D (1) B A D C E (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -24 -10 -10 B 12 0 -12 -8 0 C 24 12 0 18 12 D 10 8 -18 0 -8 E 10 0 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -24 -10 -10 B 12 0 -12 -8 0 C 24 12 0 18 12 D 10 8 -18 0 -8 E 10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=23 A=21 B=19 D=13 so D is eliminated. Round 2 votes counts: C=31 A=25 E=24 B=20 so B is eliminated. Round 3 votes counts: C=41 A=34 E=25 so E is eliminated. Round 4 votes counts: C=62 A=38 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:233 E:203 B:196 D:196 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -24 -10 -10 B 12 0 -12 -8 0 C 24 12 0 18 12 D 10 8 -18 0 -8 E 10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -24 -10 -10 B 12 0 -12 -8 0 C 24 12 0 18 12 D 10 8 -18 0 -8 E 10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -24 -10 -10 B 12 0 -12 -8 0 C 24 12 0 18 12 D 10 8 -18 0 -8 E 10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2743: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) D E C B A (7) E A B D C (6) E D B C A (5) A E B C D (5) D E C A B (4) E D B A C (3) E B A D C (3) E A D B C (3) D E B C A (3) D C E B A (3) D C B E A (3) C D A B E (3) B E D C A (3) B D C E A (3) A C B D E (3) A B C E D (3) E B D A C (2) E B A C D (2) C A D B E (2) B A C E D (2) A E D C B (2) A E C B D (2) A C E B D (2) A C D B E (2) E D A C B (1) E D A B C (1) E B D C A (1) D C E A B (1) D B C E A (1) D A E C B (1) C D B A E (1) C D A E B (1) C A B D E (1) B E C D A (1) B C D E A (1) B C A E D (1) A E C D B (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 2 -14 B -10 0 0 0 -22 C -10 0 0 -12 -14 D -2 0 12 0 -18 E 14 22 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 10 2 -14 B -10 0 0 0 -22 C -10 0 0 -12 -14 D -2 0 12 0 -18 E 14 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=27 D=23 B=11 C=8 so C is eliminated. Round 2 votes counts: A=34 D=28 E=27 B=11 so B is eliminated. Round 3 votes counts: A=37 D=32 E=31 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:234 A:204 D:196 B:184 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 10 2 -14 B -10 0 0 0 -22 C -10 0 0 -12 -14 D -2 0 12 0 -18 E 14 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 2 -14 B -10 0 0 0 -22 C -10 0 0 -12 -14 D -2 0 12 0 -18 E 14 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 2 -14 B -10 0 0 0 -22 C -10 0 0 -12 -14 D -2 0 12 0 -18 E 14 22 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2744: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) D A E B C (6) C B A E D (6) A D E C B (6) C A E B D (5) C A B E D (5) C A B D E (4) A D C E B (4) A C D E B (4) E D A B C (3) D E B A C (3) D A E C B (3) C B E A D (3) B C E D A (3) A C D B E (3) A C B D E (3) E D B C A (2) E D B A C (2) E D A C B (2) E C B A D (2) E B D C A (2) E B C D A (2) B E C D A (2) B C A D E (2) E C A D B (1) E A D C B (1) E A C D B (1) D B E A C (1) D B A E C (1) D A B E C (1) C E B A D (1) C B A D E (1) C A E D B (1) B E D C A (1) B D E C A (1) B D C A E (1) B C A E D (1) A E D C B (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 26 8 14 20 B -26 0 -18 -10 -22 C -8 18 0 4 -4 D -14 10 -4 0 2 E -20 22 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 8 14 20 B -26 0 -18 -10 -22 C -8 18 0 4 -4 D -14 10 -4 0 2 E -20 22 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 D=22 E=18 B=11 so B is eliminated. Round 2 votes counts: C=32 D=24 A=23 E=21 so E is eliminated. Round 3 votes counts: C=39 D=36 A=25 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:234 C:205 E:202 D:197 B:162 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 8 14 20 B -26 0 -18 -10 -22 C -8 18 0 4 -4 D -14 10 -4 0 2 E -20 22 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 8 14 20 B -26 0 -18 -10 -22 C -8 18 0 4 -4 D -14 10 -4 0 2 E -20 22 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 8 14 20 B -26 0 -18 -10 -22 C -8 18 0 4 -4 D -14 10 -4 0 2 E -20 22 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2745: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) B E C D A (7) B D E C A (6) B C E D A (6) B C E A D (6) D A E C B (4) D A E B C (4) D A B E C (4) A D E C B (4) A D C E B (4) A D B C E (4) C E A B D (3) C B E A D (3) A C E D B (3) E C B D A (2) E C A D B (2) E C A B D (2) D B E A C (2) D B A E C (2) B E D C A (2) A C D E B (2) E D B C A (1) E D A C B (1) E B D C A (1) E B C D A (1) E A D C B (1) D E B A C (1) D E A C B (1) D A B C E (1) C E A D B (1) C B A E D (1) C A E B D (1) C A B E D (1) B D E A C (1) B D C A E (1) B D A C E (1) A E D C B (1) A E C D B (1) A D C B E (1) Total count = 100 A B C D E A 0 -8 -14 2 -20 B 8 0 2 10 -2 C 14 -2 0 4 -2 D -2 -10 -4 0 -14 E 20 2 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -14 2 -20 B 8 0 2 10 -2 C 14 -2 0 4 -2 D -2 -10 -4 0 -14 E 20 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=20 A=20 D=19 E=11 so E is eliminated. Round 2 votes counts: B=32 C=26 D=21 A=21 so D is eliminated. Round 3 votes counts: B=38 A=36 C=26 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:219 B:209 C:207 D:185 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 2 -20 B 8 0 2 10 -2 C 14 -2 0 4 -2 D -2 -10 -4 0 -14 E 20 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 2 -20 B 8 0 2 10 -2 C 14 -2 0 4 -2 D -2 -10 -4 0 -14 E 20 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 2 -20 B 8 0 2 10 -2 C 14 -2 0 4 -2 D -2 -10 -4 0 -14 E 20 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2746: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) B D E C A (7) B D A E C (6) A D C B E (6) C E A D B (5) B E D C A (5) A C E D B (5) A C D E B (5) E B C D A (4) D B C A E (4) A D B C E (4) E C A B D (3) D B A C E (3) D A B C E (3) C A E D B (3) A C D B E (3) A B D C E (3) E C D B A (2) E C B A D (2) E C A D B (2) E B D C A (2) E B C A D (2) B A D E C (2) A C E B D (2) E D B C A (1) E C B D A (1) D C E B A (1) D C A B E (1) C E D B A (1) C E D A B (1) C D E A B (1) B D A C E (1) Total count = 100 A B C D E A 0 -6 4 -6 2 B 6 0 12 -2 14 C -4 -12 0 -16 4 D 6 2 16 0 18 E -2 -14 -4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -6 2 B 6 0 12 -2 14 C -4 -12 0 -16 4 D 6 2 16 0 18 E -2 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=28 E=19 D=12 C=11 so C is eliminated. Round 2 votes counts: A=31 B=30 E=26 D=13 so D is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:221 B:215 A:197 C:186 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 -6 2 B 6 0 12 -2 14 C -4 -12 0 -16 4 D 6 2 16 0 18 E -2 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -6 2 B 6 0 12 -2 14 C -4 -12 0 -16 4 D 6 2 16 0 18 E -2 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -6 2 B 6 0 12 -2 14 C -4 -12 0 -16 4 D 6 2 16 0 18 E -2 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2747: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) E C D B A (6) D A C B E (6) A D B C E (6) E B C D A (5) A D C B E (5) E D C A B (4) E D A B C (4) E C B D A (4) D A E B C (4) B A C D E (4) E D C B A (3) D E A C B (3) B E C A D (3) B E A C D (3) B C E A D (3) E C D A B (2) E C B A D (2) D E C A B (2) D A C E B (2) C E D A B (2) B C A E D (2) B A D C E (2) B A C E D (2) A C D B E (2) E B D A C (1) E B A D C (1) D C E A B (1) D A E C B (1) D A B C E (1) C E D B A (1) C D A B E (1) C B A E D (1) C A B D E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 -6 -14 B 0 0 0 -12 -10 C 0 0 0 4 -12 D 6 12 -4 0 -12 E 14 10 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 -6 -14 B 0 0 0 -12 -10 C 0 0 0 4 -12 D 6 12 -4 0 -12 E 14 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=20 B=19 A=16 C=6 so C is eliminated. Round 2 votes counts: E=42 D=21 B=20 A=17 so A is eliminated. Round 3 votes counts: E=42 D=34 B=24 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:201 C:196 A:190 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -6 -14 B 0 0 0 -12 -10 C 0 0 0 4 -12 D 6 12 -4 0 -12 E 14 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -6 -14 B 0 0 0 -12 -10 C 0 0 0 4 -12 D 6 12 -4 0 -12 E 14 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -6 -14 B 0 0 0 -12 -10 C 0 0 0 4 -12 D 6 12 -4 0 -12 E 14 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2748: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) D E A B C (8) C A E B D (6) D E B A C (5) A E D B C (5) D B C E A (4) B C D E A (4) D C B E A (3) D C A E B (3) D B E C A (3) D A E B C (3) C B D E A (3) B D C E A (3) A E D C B (3) E D A B C (2) E B D A C (2) D A E C B (2) C D B A E (2) C A B E D (2) B E D A C (2) A E C B D (2) A E B D C (2) A E B C D (2) E B A D C (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E B A (1) C D B E A (1) C D A B E (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A D E (1) C A E D B (1) C A B D E (1) B E C A D (1) B E A D C (1) B E A C D (1) B C E D A (1) A E C D B (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -6 -12 -2 B 2 0 4 -2 -8 C 6 -4 0 -14 0 D 12 2 14 0 2 E 2 8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -12 -2 B 2 0 4 -2 -8 C 6 -4 0 -14 0 D 12 2 14 0 2 E 2 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=30 A=17 B=13 E=7 so E is eliminated. Round 2 votes counts: D=35 C=30 A=19 B=16 so B is eliminated. Round 3 votes counts: D=42 C=36 A=22 so A is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:204 B:198 C:194 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -12 -2 B 2 0 4 -2 -8 C 6 -4 0 -14 0 D 12 2 14 0 2 E 2 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -12 -2 B 2 0 4 -2 -8 C 6 -4 0 -14 0 D 12 2 14 0 2 E 2 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -12 -2 B 2 0 4 -2 -8 C 6 -4 0 -14 0 D 12 2 14 0 2 E 2 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2749: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) E B D C A (5) C A D B E (5) B E D C A (5) A C E D B (5) A C D B E (5) E D B A C (4) C A B D E (4) B D E C A (4) A C E B D (4) E B C A D (3) E A C B D (3) A C D E B (3) D E A B C (2) D B E C A (2) D B A C E (2) C B A E D (2) B E C D A (2) B D C E A (2) B C D E A (2) A D C E B (2) A D C B E (2) E D A B C (1) E C B A D (1) E B C D A (1) E B A D C (1) E A D C B (1) E A D B C (1) D E B C A (1) D C A B E (1) D B E A C (1) D B A E C (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) D A B C E (1) C D B A E (1) C B D A E (1) C B A D E (1) C A E B D (1) C A B E D (1) B E C A D (1) B C D A E (1) A E D C B (1) A E C B D (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 6 0 2 B 0 0 0 -6 -2 C -6 0 0 -2 -2 D 0 6 2 0 12 E -2 2 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.551787 B: 0.000000 C: 0.000000 D: 0.448213 E: 0.000000 Sum of squares = 0.505363800841 Cumulative probabilities = A: 0.551787 B: 0.551787 C: 0.551787 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 0 2 B 0 0 0 -6 -2 C -6 0 0 -2 -2 D 0 6 2 0 12 E -2 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 D=21 B=17 C=16 so C is eliminated. Round 2 votes counts: A=36 D=22 E=21 B=21 so E is eliminated. Round 3 votes counts: A=41 B=32 D=27 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:204 B:196 C:195 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 0 2 B 0 0 0 -6 -2 C -6 0 0 -2 -2 D 0 6 2 0 12 E -2 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 2 B 0 0 0 -6 -2 C -6 0 0 -2 -2 D 0 6 2 0 12 E -2 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 2 B 0 0 0 -6 -2 C -6 0 0 -2 -2 D 0 6 2 0 12 E -2 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2750: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A B C E D (7) D E C B A (5) A C D B E (5) E D B A C (4) E C B D A (4) D E C A B (4) D E B C A (4) D E A B C (4) B C A E D (4) B A C E D (4) A C B D E (4) A B C D E (4) E B A C D (3) D E A C B (3) D A C E B (3) C A B D E (3) B A E C D (3) A D C B E (3) E B D C A (2) E A D B C (2) C D A B E (2) C A B E D (2) B E C A D (2) E B D A C (1) E B A D C (1) D C E A B (1) D A E C B (1) D A E B C (1) C E D B A (1) C B A E D (1) C A D B E (1) B E A C D (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 12 4 0 B -4 0 14 -8 -6 C -12 -14 0 2 -10 D -4 8 -2 0 -2 E 0 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.506290 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.493710 Sum of squares = 0.50007913776 Cumulative probabilities = A: 0.506290 B: 0.506290 C: 0.506290 D: 0.506290 E: 1.000000 A B C D E A 0 4 12 4 0 B -4 0 14 -8 -6 C -12 -14 0 2 -10 D -4 8 -2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 A=25 B=14 C=10 so C is eliminated. Round 2 votes counts: A=31 D=28 E=26 B=15 so B is eliminated. Round 3 votes counts: A=43 E=29 D=28 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 E:209 D:200 B:198 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 4 0 B -4 0 14 -8 -6 C -12 -14 0 2 -10 D -4 8 -2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 4 0 B -4 0 14 -8 -6 C -12 -14 0 2 -10 D -4 8 -2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 4 0 B -4 0 14 -8 -6 C -12 -14 0 2 -10 D -4 8 -2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999891 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2751: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) E D A C B (6) E C D A B (6) C B E A D (6) C E D B A (5) C E B D A (5) C E A B D (4) C B D E A (4) D E A B C (3) D A B E C (3) C E A D B (3) C B E D A (3) C B A E D (3) C B A D E (3) B C A D E (3) B A D C E (3) B A C D E (3) E C A D B (2) E A D C B (2) C E B A D (2) C B D A E (2) C A E B D (2) B D A C E (2) A D B E C (2) A B D E C (2) A B C D E (2) E D C B A (1) E D C A B (1) E D A B C (1) D E B C A (1) D E B A C (1) D A E B C (1) B D C A E (1) B D A E C (1) B A D E C (1) A E D B C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -14 -8 -12 B 14 0 -20 6 4 C 14 20 0 16 14 D 8 -6 -16 0 -6 E 12 -4 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 -8 -12 B 14 0 -20 6 4 C 14 20 0 16 14 D 8 -6 -16 0 -6 E 12 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 E=19 D=16 B=14 A=9 so A is eliminated. Round 2 votes counts: C=42 E=20 D=19 B=19 so D is eliminated. Round 3 votes counts: C=42 B=31 E=27 so E is eliminated. Round 4 votes counts: C=60 B=40 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 B:202 E:200 D:190 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 -8 -12 B 14 0 -20 6 4 C 14 20 0 16 14 D 8 -6 -16 0 -6 E 12 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -8 -12 B 14 0 -20 6 4 C 14 20 0 16 14 D 8 -6 -16 0 -6 E 12 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -8 -12 B 14 0 -20 6 4 C 14 20 0 16 14 D 8 -6 -16 0 -6 E 12 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2752: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) D B A E C (6) E C B D A (5) D A B E C (5) C E A D B (5) A C B D E (5) D B E A C (4) C A B E D (4) C E B A D (3) C A E B D (3) B D A E C (3) B A D C E (3) A B D C E (3) A B C D E (3) E D C B A (2) E D B C A (2) E C D B A (2) E C B A D (2) E B D C A (2) E B C D A (2) D E B C A (2) D A C B E (2) D A B C E (2) C E D A B (2) C E A B D (2) C A E D B (2) B E D A C (2) B D E A C (2) A D C B E (2) A C D B E (2) E D B A C (1) E C D A B (1) D E C A B (1) D E B A C (1) D A E B C (1) C B A E D (1) C A D E B (1) B E D C A (1) B A D E C (1) Total count = 100 A B C D E A 0 6 10 -2 12 B -6 0 6 -6 16 C -10 -6 0 -10 4 D 2 6 10 0 12 E -12 -16 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 -2 12 B -6 0 6 -6 16 C -10 -6 0 -10 4 D 2 6 10 0 12 E -12 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 A=22 E=19 B=12 so B is eliminated. Round 2 votes counts: D=29 A=26 C=23 E=22 so E is eliminated. Round 3 votes counts: D=39 C=35 A=26 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:213 B:205 C:189 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -2 12 B -6 0 6 -6 16 C -10 -6 0 -10 4 D 2 6 10 0 12 E -12 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -2 12 B -6 0 6 -6 16 C -10 -6 0 -10 4 D 2 6 10 0 12 E -12 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -2 12 B -6 0 6 -6 16 C -10 -6 0 -10 4 D 2 6 10 0 12 E -12 -16 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2753: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) A D E B C (12) B C A E D (7) B C A D E (6) E D A C B (5) C B A D E (5) E D A B C (4) D E A C B (4) A D B E C (4) E D C B A (3) E C B D A (3) D A E B C (3) C E B D A (3) B A C D E (3) A B D C E (3) C E D B A (2) C B D E A (2) B C E D A (2) B A E D C (2) B A C E D (2) E D C A B (1) E D B C A (1) E C D B A (1) E B C D A (1) D E C A B (1) C D E B A (1) C B E A D (1) C B A E D (1) C A D B E (1) B C E A D (1) A D E C B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -10 2 4 B 18 0 6 12 8 C 10 -6 0 10 6 D -2 -12 -10 0 -4 E -4 -8 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 2 4 B 18 0 6 12 8 C 10 -6 0 10 6 D -2 -12 -10 0 -4 E -4 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=23 A=22 E=19 D=8 so D is eliminated. Round 2 votes counts: C=28 A=25 E=24 B=23 so B is eliminated. Round 3 votes counts: C=44 A=32 E=24 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:222 C:210 E:193 A:189 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 2 4 B 18 0 6 12 8 C 10 -6 0 10 6 D -2 -12 -10 0 -4 E -4 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 2 4 B 18 0 6 12 8 C 10 -6 0 10 6 D -2 -12 -10 0 -4 E -4 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 2 4 B 18 0 6 12 8 C 10 -6 0 10 6 D -2 -12 -10 0 -4 E -4 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2754: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) D B A E C (6) A E D C B (6) A C E D B (5) E A D B C (4) D A E B C (4) C B E A D (4) C A E D B (4) B C D E A (4) D B E A C (3) C E B A D (3) C E A B D (3) C A D E B (3) B D C E A (3) A E D B C (3) E C A B D (2) E B C A D (2) E A C B D (2) E A B D C (2) D E A B C (2) C A B E D (2) A E C D B (2) A C E B D (2) E B A D C (1) E A C D B (1) E A B C D (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C B E (1) D A B E C (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E D A (1) C B D A E (1) C B A D E (1) B E A C D (1) B D E C A (1) B D E A C (1) A E C B D (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 30 4 32 18 B -30 0 -18 -6 -30 C -4 18 0 16 6 D -32 6 -16 0 -22 E -18 30 -6 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 30 4 32 18 B -30 0 -18 -6 -30 C -4 18 0 16 6 D -32 6 -16 0 -22 E -18 30 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986111 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=21 D=20 E=15 B=10 so B is eliminated. Round 2 votes counts: C=38 D=25 A=21 E=16 so E is eliminated. Round 3 votes counts: C=42 A=33 D=25 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:242 C:218 E:214 D:168 B:158 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 30 4 32 18 B -30 0 -18 -6 -30 C -4 18 0 16 6 D -32 6 -16 0 -22 E -18 30 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986111 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 4 32 18 B -30 0 -18 -6 -30 C -4 18 0 16 6 D -32 6 -16 0 -22 E -18 30 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986111 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 4 32 18 B -30 0 -18 -6 -30 C -4 18 0 16 6 D -32 6 -16 0 -22 E -18 30 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986111 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2755: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (13) C D A E B (9) D A C E B (8) B E A C D (7) B E C D A (6) B A E D C (6) B E C A D (5) A D C E B (5) C E B D A (4) C B E D A (4) B C E D A (3) A D B E C (3) E B A C D (2) D C A E B (2) C D E B A (2) C D A B E (2) B A D E C (2) A D E B C (2) A D C B E (2) E C B A D (1) E A B D C (1) D C A B E (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E A B (1) C D B A E (1) C B D E A (1) A E D C B (1) A E B D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 14 6 -2 B 16 0 8 14 20 C -14 -8 0 -2 -2 D -6 -14 2 0 -12 E 2 -20 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 14 6 -2 B 16 0 8 14 20 C -14 -8 0 -2 -2 D -6 -14 2 0 -12 E 2 -20 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 C=25 A=16 D=13 E=4 so E is eliminated. Round 2 votes counts: B=44 C=26 A=17 D=13 so D is eliminated. Round 3 votes counts: B=44 C=29 A=27 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 A:201 E:198 C:187 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 14 6 -2 B 16 0 8 14 20 C -14 -8 0 -2 -2 D -6 -14 2 0 -12 E 2 -20 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 14 6 -2 B 16 0 8 14 20 C -14 -8 0 -2 -2 D -6 -14 2 0 -12 E 2 -20 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 14 6 -2 B 16 0 8 14 20 C -14 -8 0 -2 -2 D -6 -14 2 0 -12 E 2 -20 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2756: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) D A C E B (7) B C D E A (6) A E B D C (6) A E B C D (6) B C E D A (5) B A E C D (5) D C A E B (4) D C A B E (4) B C D A E (4) D C B A E (3) D B C A E (3) C D B E A (3) B E C A D (3) B E A C D (3) B D C A E (3) A E D C B (3) A B E D C (3) E D C A B (2) E A D C B (2) D C B E A (2) C D E B A (2) A D E C B (2) A B E C D (2) E C D A B (1) E C B D A (1) E C B A D (1) E B A C D (1) E A C B D (1) D E A C B (1) D C E A B (1) A E D B C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -10 -18 16 B 4 0 4 0 0 C 10 -4 0 -12 12 D 18 0 12 0 8 E -16 0 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.599808 C: 0.000000 D: 0.400192 E: 0.000000 Sum of squares = 0.519923169928 Cumulative probabilities = A: 0.000000 B: 0.599808 C: 0.599808 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -18 16 B 4 0 4 0 0 C 10 -4 0 -12 12 D 18 0 12 0 8 E -16 0 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 A=25 E=9 C=5 so C is eliminated. Round 2 votes counts: D=37 B=29 A=25 E=9 so E is eliminated. Round 3 votes counts: D=40 B=32 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:219 B:204 C:203 A:192 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -18 16 B 4 0 4 0 0 C 10 -4 0 -12 12 D 18 0 12 0 8 E -16 0 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -18 16 B 4 0 4 0 0 C 10 -4 0 -12 12 D 18 0 12 0 8 E -16 0 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -18 16 B 4 0 4 0 0 C 10 -4 0 -12 12 D 18 0 12 0 8 E -16 0 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2757: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) D A B E C (6) E C B D A (5) C E B D A (5) C B E D A (5) A D B C E (5) E B C D A (4) A D B E C (4) D B A E C (3) C B A D E (3) A D C E B (3) A C D B E (3) E B D C A (2) E A D C B (2) E A D B C (2) D E A B C (2) D B E A C (2) D A E B C (2) C E B A D (2) C E A B D (2) C B E A D (2) C B A E D (2) C A B E D (2) B D E A C (2) E D B A C (1) E D A B C (1) C E A D B (1) C A E D B (1) B E D C A (1) B E C D A (1) B D E C A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 14 4 10 B -4 0 12 -8 0 C -14 -12 0 -8 -10 D -4 8 8 0 10 E -10 0 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 4 10 B -4 0 12 -8 0 C -14 -12 0 -8 -10 D -4 8 8 0 10 E -10 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=25 E=17 D=15 B=12 so B is eliminated. Round 2 votes counts: A=32 C=29 D=20 E=19 so E is eliminated. Round 3 votes counts: C=39 A=36 D=25 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:211 B:200 E:195 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 4 10 B -4 0 12 -8 0 C -14 -12 0 -8 -10 D -4 8 8 0 10 E -10 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 4 10 B -4 0 12 -8 0 C -14 -12 0 -8 -10 D -4 8 8 0 10 E -10 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 4 10 B -4 0 12 -8 0 C -14 -12 0 -8 -10 D -4 8 8 0 10 E -10 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2758: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) B D A C E (7) E D B C A (6) B D E A C (6) A C E B D (6) D E C A B (5) B D A E C (4) E D C A B (3) E C A B D (3) E A C B D (3) D E B C A (3) D B E C A (3) D B C A E (3) B A C D E (3) A C B E D (3) E C D A B (2) E B D C A (2) E B A C D (2) D B C E A (2) D A B C E (2) C A E D B (2) C A D B E (2) B A E C D (2) A C B D E (2) E B D A C (1) E A B C D (1) D E C B A (1) D C E A B (1) D B E A C (1) D A C B E (1) C E A D B (1) C A D E B (1) B E D A C (1) B E A C D (1) B A D C E (1) B A C E D (1) A C E D B (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 -8 -12 B 0 0 8 2 -4 C -4 -8 0 -8 -16 D 8 -2 8 0 2 E 12 4 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999966 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 0 4 -8 -12 B 0 0 8 2 -4 C -4 -8 0 -8 -16 D 8 -2 8 0 2 E 12 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999218 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=26 D=22 A=15 C=6 so C is eliminated. Round 2 votes counts: E=32 B=26 D=22 A=20 so A is eliminated. Round 3 votes counts: E=41 B=33 D=26 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:208 B:203 A:192 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -8 -12 B 0 0 8 2 -4 C -4 -8 0 -8 -16 D 8 -2 8 0 2 E 12 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999218 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -8 -12 B 0 0 8 2 -4 C -4 -8 0 -8 -16 D 8 -2 8 0 2 E 12 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999218 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -8 -12 B 0 0 8 2 -4 C -4 -8 0 -8 -16 D 8 -2 8 0 2 E 12 4 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999218 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2759: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A D C B E (7) D A B C E (6) A C B D E (6) E B C D A (5) D E B C A (5) A D E B C (4) A C B E D (4) E D A B C (3) E C B A D (3) E B D C A (3) E B C A D (3) E A D C B (3) C B E A D (3) A E D C B (3) E A C B D (2) D B E C A (2) D B C E A (2) D B C A E (2) C B A E D (2) C B A D E (2) B C E D A (2) B C D E A (2) A D E C B (2) E D B A C (1) E C A B D (1) D A E B C (1) D A C B E (1) C E B A D (1) C B E D A (1) C B D E A (1) C B D A E (1) B D C E A (1) B C D A E (1) A E D B C (1) A E C D B (1) A E C B D (1) A D B C E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -2 4 -4 B 2 0 6 -8 -4 C 2 -6 0 -12 -4 D -4 8 12 0 -4 E 4 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -2 4 -4 B 2 0 6 -8 -4 C 2 -6 0 -12 -4 D -4 8 12 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=32 A=32 D=19 C=11 B=6 so B is eliminated. Round 2 votes counts: E=32 A=32 D=20 C=16 so C is eliminated. Round 3 votes counts: E=39 A=36 D=25 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:206 A:198 B:198 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -4 B 2 0 6 -8 -4 C 2 -6 0 -12 -4 D -4 8 12 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -4 B 2 0 6 -8 -4 C 2 -6 0 -12 -4 D -4 8 12 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -4 B 2 0 6 -8 -4 C 2 -6 0 -12 -4 D -4 8 12 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2760: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) C D E B A (7) E A B C D (5) C D E A B (5) C D B E A (5) D B C A E (4) B A E D C (4) B A E C D (4) B A D E C (4) D B A C E (3) C E D B A (3) B D C A E (3) B D A C E (3) A E B D C (3) A E B C D (3) A B E D C (3) E C A D B (2) E A C B D (2) D C E A B (2) D A B E C (2) C B E A D (2) B A D C E (2) B A C D E (2) E D C A B (1) E D A B C (1) E C D A B (1) E C A B D (1) E A D C B (1) E A C D B (1) D E C A B (1) D E A C B (1) D C B A E (1) C B E D A (1) C B D A E (1) C B A E D (1) B D A E C (1) B C A D E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -6 -14 -6 B 6 0 0 -4 -2 C 6 0 0 20 16 D 14 4 -20 0 -4 E 6 2 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.586604 C: 0.413396 D: 0.000000 E: 0.000000 Sum of squares = 0.515000357891 Cumulative probabilities = A: 0.000000 B: 0.586604 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -14 -6 B 6 0 0 -4 -2 C 6 0 0 20 16 D 14 4 -20 0 -4 E 6 2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=25 E=15 D=14 A=10 so A is eliminated. Round 2 votes counts: C=36 B=29 E=21 D=14 so D is eliminated. Round 3 votes counts: C=39 B=38 E=23 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:221 B:200 E:198 D:197 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -14 -6 B 6 0 0 -4 -2 C 6 0 0 20 16 D 14 4 -20 0 -4 E 6 2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -14 -6 B 6 0 0 -4 -2 C 6 0 0 20 16 D 14 4 -20 0 -4 E 6 2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -14 -6 B 6 0 0 -4 -2 C 6 0 0 20 16 D 14 4 -20 0 -4 E 6 2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2761: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (7) D B C E A (6) C E A D B (6) E C B A D (5) B D E C A (5) A E C D B (4) A E C B D (4) A D C E B (4) E C A B D (3) E B A C D (3) D A C B E (3) D A B C E (3) C E B D A (3) B D A E C (3) A E B C D (3) A B E D C (3) E B C D A (2) E A B C D (2) C E D B A (2) C D B E A (2) B E D A C (2) B D C E A (2) A D B C E (2) E C B D A (1) E B C A D (1) E A C B D (1) D C B E A (1) D C A B E (1) D B C A E (1) D B A E C (1) C E D A B (1) C D E B A (1) C D E A B (1) C A E D B (1) C A D E B (1) B E A D C (1) B E A C D (1) B D E A C (1) B C D E A (1) B A E D C (1) A D E C B (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -8 0 -22 B 6 0 6 12 -2 C 8 -6 0 14 -14 D 0 -12 -14 0 -14 E 22 2 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 0 -22 B 6 0 6 12 -2 C 8 -6 0 14 -14 D 0 -12 -14 0 -14 E 22 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 E=18 C=18 D=16 so D is eliminated. Round 2 votes counts: B=32 A=30 C=20 E=18 so E is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:226 B:211 C:201 A:182 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 0 -22 B 6 0 6 12 -2 C 8 -6 0 14 -14 D 0 -12 -14 0 -14 E 22 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 -22 B 6 0 6 12 -2 C 8 -6 0 14 -14 D 0 -12 -14 0 -14 E 22 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 -22 B 6 0 6 12 -2 C 8 -6 0 14 -14 D 0 -12 -14 0 -14 E 22 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2762: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) C E D B A (6) E B C D A (5) D A B E C (5) E C B D A (4) C D E A B (4) A C D E B (4) A B D E C (4) D E C B A (3) D C E B A (3) D C A E B (3) C E A B D (3) A C B E D (3) A B E C D (3) E C B A D (2) C E B D A (2) B E C A D (2) B D E A C (2) B A E C D (2) B A D E C (2) A D C E B (2) A D B C E (2) A C E B D (2) A B C E D (2) E D C B A (1) E C D B A (1) D E B C A (1) D B E C A (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B A D (1) C E A D B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E C A (1) B D A E C (1) A D C B E (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -16 -14 -12 B -12 0 -22 -6 -22 C 16 22 0 10 12 D 14 6 -10 0 4 E 12 22 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -16 -14 -12 B -12 0 -22 -6 -22 C 16 22 0 10 12 D 14 6 -10 0 4 E 12 22 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 C=21 B=14 E=13 so E is eliminated. Round 2 votes counts: C=28 D=27 A=26 B=19 so B is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:230 E:209 D:207 A:185 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -16 -14 -12 B -12 0 -22 -6 -22 C 16 22 0 10 12 D 14 6 -10 0 4 E 12 22 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -16 -14 -12 B -12 0 -22 -6 -22 C 16 22 0 10 12 D 14 6 -10 0 4 E 12 22 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -16 -14 -12 B -12 0 -22 -6 -22 C 16 22 0 10 12 D 14 6 -10 0 4 E 12 22 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2763: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (15) E B D C A (13) E B D A C (5) A E C B D (5) E D B A C (4) D B E C A (4) C D B A E (4) A E C D B (4) C B D E A (3) C A B D E (3) A C E D B (3) E B C D A (2) E A B D C (2) D E B C A (2) D B C E A (2) C E B D A (2) C A D B E (2) B E D C A (2) B D E C A (2) B D C E A (2) B C D E A (2) A C E B D (2) A C D E B (2) E B A D C (1) E A D B C (1) E A C B D (1) D A C B E (1) C E A B D (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A E D (1) C B A D E (1) A E B C D (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -2 -4 -6 B 10 0 -10 10 -2 C 2 10 0 16 2 D 4 -10 -16 0 -6 E 6 2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -4 -6 B 10 0 -10 10 -2 C 2 10 0 16 2 D 4 -10 -16 0 -6 E 6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=29 C=20 D=9 B=8 so B is eliminated. Round 2 votes counts: A=34 E=31 C=22 D=13 so D is eliminated. Round 3 votes counts: E=39 A=35 C=26 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:206 B:204 A:189 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -2 -4 -6 B 10 0 -10 10 -2 C 2 10 0 16 2 D 4 -10 -16 0 -6 E 6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -4 -6 B 10 0 -10 10 -2 C 2 10 0 16 2 D 4 -10 -16 0 -6 E 6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -4 -6 B 10 0 -10 10 -2 C 2 10 0 16 2 D 4 -10 -16 0 -6 E 6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2764: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) D A C B E (10) B A E D C (9) C D A E B (8) D C A E B (7) C E D B A (7) C E D A B (6) C D E A B (6) B E A C D (6) A D B C E (6) A B D E C (6) E B C A D (5) E B A C D (3) B E A D C (3) D C A B E (2) C E B D A (2) E C B A D (1) D A B C E (1) B A D E C (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -8 -18 2 B -6 0 -18 -8 -10 C 8 18 0 8 10 D 18 8 -8 0 -4 E -2 10 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -18 2 B -6 0 -18 -8 -10 C 8 18 0 8 10 D 18 8 -8 0 -4 E -2 10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=20 E=19 B=19 A=13 so A is eliminated. Round 2 votes counts: C=29 D=27 B=25 E=19 so E is eliminated. Round 3 votes counts: C=40 B=33 D=27 so D is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:207 E:201 A:191 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -18 2 B -6 0 -18 -8 -10 C 8 18 0 8 10 D 18 8 -8 0 -4 E -2 10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -18 2 B -6 0 -18 -8 -10 C 8 18 0 8 10 D 18 8 -8 0 -4 E -2 10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -18 2 B -6 0 -18 -8 -10 C 8 18 0 8 10 D 18 8 -8 0 -4 E -2 10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2765: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) A D C E B (10) E C B D A (7) A D B C E (7) B E C A D (4) B A E C D (4) A D E C B (4) E C D A B (3) D C E A B (3) D A E C B (3) D A C E B (3) C E B D A (3) B C E D A (3) B A C D E (3) E D C A B (2) E C D B A (2) E B C D A (2) D E C A B (2) D C A E B (2) C E D A B (2) B C E A D (2) B A D E C (2) B A D C E (2) A D C B E (2) A D B E C (2) A B D E C (2) A B D C E (2) E D B A C (1) E D A C B (1) C B E D A (1) B E D C A (1) B E A C D (1) B A E D C (1) A D E B C (1) Total count = 100 A B C D E A 0 2 2 -2 0 B -2 0 0 0 -2 C -2 0 0 -6 -10 D 2 0 6 0 0 E 0 2 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.542312 E: 0.457688 Sum of squares = 0.503580649809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.542312 E: 1.000000 A B C D E A 0 2 2 -2 0 B -2 0 0 0 -2 C -2 0 0 -6 -10 D 2 0 6 0 0 E 0 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=30 E=18 D=13 C=6 so C is eliminated. Round 2 votes counts: B=34 A=30 E=23 D=13 so D is eliminated. Round 3 votes counts: A=38 B=34 E=28 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:204 A:201 B:198 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 2 -2 0 B -2 0 0 0 -2 C -2 0 0 -6 -10 D 2 0 6 0 0 E 0 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -2 0 B -2 0 0 0 -2 C -2 0 0 -6 -10 D 2 0 6 0 0 E 0 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -2 0 B -2 0 0 0 -2 C -2 0 0 -6 -10 D 2 0 6 0 0 E 0 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2766: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (15) E B D C A (14) D A C E B (10) D E B A C (8) E D B A C (7) D A C B E (7) B E C A D (7) E B C A D (5) B C A E D (5) D E A C B (4) C A B E D (4) E D B C A (2) D E A B C (2) C A B D E (2) B C E A D (2) A C B E D (2) A C B D E (2) E B C D A (1) D A E C B (1) Total count = 100 A B C D E A 0 -2 16 -12 -4 B 2 0 6 -12 -8 C -16 -6 0 -10 -2 D 12 12 10 0 2 E 4 8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 -12 -4 B 2 0 6 -12 -8 C -16 -6 0 -10 -2 D 12 12 10 0 2 E 4 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=29 A=19 B=14 C=6 so C is eliminated. Round 2 votes counts: D=32 E=29 A=25 B=14 so B is eliminated. Round 3 votes counts: E=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:206 A:199 B:194 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 16 -12 -4 B 2 0 6 -12 -8 C -16 -6 0 -10 -2 D 12 12 10 0 2 E 4 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 -12 -4 B 2 0 6 -12 -8 C -16 -6 0 -10 -2 D 12 12 10 0 2 E 4 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 -12 -4 B 2 0 6 -12 -8 C -16 -6 0 -10 -2 D 12 12 10 0 2 E 4 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2767: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (12) D B E A C (11) D C E A B (7) D E A B C (6) C A E B D (6) B A E C D (6) D C A E B (5) C D A E B (5) E A C D B (4) B D E A C (4) E A D C B (2) E A D B C (2) E A C B D (2) D E A C B (2) D C B A E (2) D B C E A (2) C B A E D (2) B E A D C (2) B E A C D (2) A E C B D (2) E A B D C (1) E A B C D (1) D B C A E (1) C D B A E (1) C B D A E (1) C A E D B (1) C A B E D (1) B D C A E (1) B C D A E (1) B C A D E (1) B A C E D (1) A E B C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 2 4 B 0 0 12 -2 4 C -2 -12 0 4 2 D -2 2 -4 0 0 E -4 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.742961 B: 0.257039 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.61806025669 Cumulative probabilities = A: 0.742961 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 2 4 B 0 0 12 -2 4 C -2 -12 0 4 2 D -2 2 -4 0 0 E -4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500103 B: 0.499897 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000021254 Cumulative probabilities = A: 0.500103 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=30 C=17 E=12 A=5 so A is eliminated. Round 2 votes counts: D=36 B=31 C=18 E=15 so E is eliminated. Round 3 votes counts: D=40 B=34 C=26 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:207 A:204 D:198 C:196 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 2 4 B 0 0 12 -2 4 C -2 -12 0 4 2 D -2 2 -4 0 0 E -4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500103 B: 0.499897 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000021254 Cumulative probabilities = A: 0.500103 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 4 B 0 0 12 -2 4 C -2 -12 0 4 2 D -2 2 -4 0 0 E -4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500103 B: 0.499897 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000021254 Cumulative probabilities = A: 0.500103 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 4 B 0 0 12 -2 4 C -2 -12 0 4 2 D -2 2 -4 0 0 E -4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500103 B: 0.499897 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000021254 Cumulative probabilities = A: 0.500103 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2768: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) D C B E A (7) D B C E A (7) C D A B E (7) E B A D C (6) B E D C A (6) B D C E A (5) E B D A C (4) E A B D C (4) C D B A E (4) C A D B E (4) A E B C D (4) C A D E B (3) B D E C A (3) A C E D B (3) E A D B C (2) E A B C D (2) D C B A E (2) C B D A E (2) B E D A C (2) A E C D B (2) A C E B D (2) A C D E B (2) A C D B E (2) E D B A C (1) E D A C B (1) E A D C B (1) C A B D E (1) B E A D C (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -2 -2 -4 B 0 0 -6 2 8 C 2 6 0 -6 4 D 2 -2 6 0 0 E 4 -8 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -2 -4 B 0 0 -6 2 8 C 2 6 0 -6 4 D 2 -2 6 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101898 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 C=21 B=17 D=16 so D is eliminated. Round 2 votes counts: C=30 A=25 B=24 E=21 so E is eliminated. Round 3 votes counts: B=35 A=35 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:203 D:203 B:202 A:196 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -2 -2 -4 B 0 0 -6 2 8 C 2 6 0 -6 4 D 2 -2 6 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101898 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 -4 B 0 0 -6 2 8 C 2 6 0 -6 4 D 2 -2 6 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101898 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 -4 B 0 0 -6 2 8 C 2 6 0 -6 4 D 2 -2 6 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101898 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2769: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) E C B A D (7) E C A D B (7) D A B C E (6) A D B E C (6) E C A B D (4) B C E D A (4) B A D E C (4) E A D C B (3) E A D B C (3) E A C D B (3) D B A C E (3) C E A D B (3) C B D E A (3) B D A E C (3) C E D A B (2) C E B D A (2) C D A E B (2) B E C D A (2) B D C A E (2) B C D E A (2) A E D C B (2) A D E B C (2) E B C A D (1) E A B D C (1) E A B C D (1) D C A B E (1) D A C E B (1) C E D B A (1) C E B A D (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) B E A D C (1) B C D A E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 0 2 2 -6 B 0 0 8 2 4 C -2 -8 0 0 -4 D -2 -2 0 0 -2 E 6 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.257690 B: 0.742310 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.617427846665 Cumulative probabilities = A: 0.257690 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 2 -6 B 0 0 8 2 4 C -2 -8 0 0 -4 D -2 -2 0 0 -2 E 6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001417 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=29 C=18 A=12 D=11 so D is eliminated. Round 2 votes counts: B=32 E=30 C=19 A=19 so C is eliminated. Round 3 votes counts: E=39 B=38 A=23 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:207 E:204 A:199 D:197 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 2 -6 B 0 0 8 2 4 C -2 -8 0 0 -4 D -2 -2 0 0 -2 E 6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001417 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 -6 B 0 0 8 2 4 C -2 -8 0 0 -4 D -2 -2 0 0 -2 E 6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001417 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 -6 B 0 0 8 2 4 C -2 -8 0 0 -4 D -2 -2 0 0 -2 E 6 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000001417 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2770: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (10) A B D E C (9) C E D B A (8) E D C B A (5) A B E D C (5) A B C D E (5) B A D C E (4) E C D A B (3) E D B C A (2) E A D C B (2) D E C B A (2) C E A D B (2) C D E B A (2) C B D A E (2) B D C A E (2) B A D E C (2) A B C E D (2) E D C A B (1) E D B A C (1) E A D B C (1) D B E A C (1) D B C E A (1) C E D A B (1) C D B E A (1) C B D E A (1) C B A D E (1) C A E B D (1) C A B D E (1) B D C E A (1) B D A E C (1) B C D E A (1) B A C D E (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 4 8 8 B -2 0 2 8 6 C -4 -2 0 -4 2 D -8 -8 4 0 -2 E -8 -6 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 8 B -2 0 2 8 6 C -4 -2 0 -4 2 D -8 -8 4 0 -2 E -8 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=25 C=20 B=12 D=4 so D is eliminated. Round 2 votes counts: A=39 E=27 C=20 B=14 so B is eliminated. Round 3 votes counts: A=47 E=28 C=25 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:207 C:196 D:193 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 8 B -2 0 2 8 6 C -4 -2 0 -4 2 D -8 -8 4 0 -2 E -8 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 8 B -2 0 2 8 6 C -4 -2 0 -4 2 D -8 -8 4 0 -2 E -8 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 8 B -2 0 2 8 6 C -4 -2 0 -4 2 D -8 -8 4 0 -2 E -8 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2771: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) D B C A E (7) D B C E A (6) E C A D B (5) E D B A C (4) D B E C A (4) B D A C E (4) E A D B C (3) E A C D B (3) C D B E A (3) C A B D E (3) A C E B D (3) A B E D C (3) E D B C A (2) E D A C B (2) E C D A B (2) E A B D C (2) D C B E A (2) D B E A C (2) C E A D B (2) C E A B D (2) C B D A E (2) B D A E C (2) B A D C E (2) A C B E D (2) A B D E C (2) E D C B A (1) E C A B D (1) D E C B A (1) D E B A C (1) D C B A E (1) D B A E C (1) D B A C E (1) C E D B A (1) C D B A E (1) C B A D E (1) C A E D B (1) C A E B D (1) B D C A E (1) A E C B D (1) A E B D C (1) A E B C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 0 -2 -14 B 0 0 0 -12 4 C 0 0 0 -10 -4 D 2 12 10 0 -4 E 14 -4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.44000000004 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 A B C D E A 0 0 0 -2 -14 B 0 0 0 -12 4 C 0 0 0 -10 -4 D 2 12 10 0 -4 E 14 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=26 C=17 A=15 B=9 so B is eliminated. Round 2 votes counts: E=33 D=33 C=17 A=17 so C is eliminated. Round 3 votes counts: D=39 E=38 A=23 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:209 B:196 C:193 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -2 -14 B 0 0 0 -12 4 C 0 0 0 -10 -4 D 2 12 10 0 -4 E 14 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -14 B 0 0 0 -12 4 C 0 0 0 -10 -4 D 2 12 10 0 -4 E 14 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -14 B 0 0 0 -12 4 C 0 0 0 -10 -4 D 2 12 10 0 -4 E 14 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2772: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (9) D B C E A (6) E D B A C (5) E A C D B (5) B D E C A (5) E A B D C (4) A B C D E (4) E D C B A (3) D B E C A (3) C A D B E (3) B D C A E (3) A C E B D (3) E C D B A (2) E C D A B (2) D B C A E (2) C D B A E (2) B E D A C (2) B D E A C (2) B D C E A (2) A E C B D (2) E C A D B (1) E B A D C (1) E A D B C (1) E A C B D (1) C E A D B (1) C D B E A (1) C D A B E (1) C A E D B (1) C A B D E (1) B D A E C (1) B D A C E (1) B A D E C (1) B A D C E (1) A E C D B (1) A E B D C (1) A E B C D (1) A C E D B (1) A C B E D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 0 -8 -16 B 8 0 18 -4 6 C 0 -18 0 -14 -12 D 8 4 14 0 -2 E 16 -6 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888963 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 A B C D E A 0 -8 0 -8 -16 B 8 0 18 -4 6 C 0 -18 0 -14 -12 D 8 4 14 0 -2 E 16 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888515 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=25 B=18 D=11 C=10 so C is eliminated. Round 2 votes counts: E=37 A=30 B=18 D=15 so D is eliminated. Round 3 votes counts: E=37 B=32 A=31 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:212 E:212 A:184 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -8 -16 B 8 0 18 -4 6 C 0 -18 0 -14 -12 D 8 4 14 0 -2 E 16 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888515 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -8 -16 B 8 0 18 -4 6 C 0 -18 0 -14 -12 D 8 4 14 0 -2 E 16 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888515 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -8 -16 B 8 0 18 -4 6 C 0 -18 0 -14 -12 D 8 4 14 0 -2 E 16 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888515 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2773: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (13) C E D A B (10) A B D E C (8) B C E D A (6) A D E C B (6) C E D B A (5) D E A C B (4) D A E C B (4) B A E D C (4) B A C E D (4) A D E B C (4) D E C A B (3) C E B D A (3) B A C D E (3) A D B E C (3) C B E D A (2) C A D E B (2) B E C D A (2) B A D C E (2) A D C E B (2) E C D A B (1) E C B D A (1) E B D C A (1) D C E A B (1) B E D A C (1) B E A D C (1) B C A E D (1) B A E C D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 24 12 18 B 0 0 10 8 6 C -24 -10 0 -14 -14 D -12 -8 14 0 14 E -18 -6 14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.220554 B: 0.779446 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.656179596067 Cumulative probabilities = A: 0.220554 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 24 12 18 B 0 0 10 8 6 C -24 -10 0 -14 -14 D -12 -8 14 0 14 E -18 -6 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=25 C=22 D=12 E=3 so E is eliminated. Round 2 votes counts: B=39 A=25 C=24 D=12 so D is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 B:212 D:204 E:188 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 24 12 18 B 0 0 10 8 6 C -24 -10 0 -14 -14 D -12 -8 14 0 14 E -18 -6 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 24 12 18 B 0 0 10 8 6 C -24 -10 0 -14 -14 D -12 -8 14 0 14 E -18 -6 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 24 12 18 B 0 0 10 8 6 C -24 -10 0 -14 -14 D -12 -8 14 0 14 E -18 -6 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2774: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) D C E A B (5) C B E A D (5) E C B D A (4) B A D E C (4) A B D E C (4) D E A B C (3) D A B E C (3) C D E A B (3) C D A E B (3) B E C A D (3) B E A D C (3) B A E D C (3) B A C E D (3) A D B E C (3) A B D C E (3) D E C A B (2) D A E B C (2) D A C E B (2) D A C B E (2) C E B D A (2) C D A B E (2) C A D B E (2) C A B D E (2) B C A E D (2) A D B C E (2) E D C B A (1) E D C A B (1) E D B C A (1) E D B A C (1) E C D A B (1) E B C D A (1) E B C A D (1) D E A C B (1) D C A E B (1) D B A E C (1) D A E C B (1) C E D B A (1) C E D A B (1) C E B A D (1) C B A E D (1) C B A D E (1) B E A C D (1) B A E C D (1) A D C B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -8 -4 2 B -4 0 -8 -6 8 C 8 8 0 0 -6 D 4 6 0 0 10 E -2 -8 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.389335 D: 0.610665 E: 0.000000 Sum of squares = 0.524493689039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.389335 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -4 2 B -4 0 -8 -6 8 C 8 8 0 0 -6 D 4 6 0 0 10 E -2 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=20 E=18 A=15 so A is eliminated. Round 2 votes counts: D=29 B=28 C=25 E=18 so E is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:205 A:197 B:195 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -4 2 B -4 0 -8 -6 8 C 8 8 0 0 -6 D 4 6 0 0 10 E -2 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -4 2 B -4 0 -8 -6 8 C 8 8 0 0 -6 D 4 6 0 0 10 E -2 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -4 2 B -4 0 -8 -6 8 C 8 8 0 0 -6 D 4 6 0 0 10 E -2 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2775: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) B D C A E (7) A C E B D (7) E A B C D (6) D B C E A (5) C A E D B (5) B D E A C (5) A E C B D (5) E A C D B (4) E A C B D (4) D B E C A (4) C A E B D (4) D C B A E (3) B D A C E (3) A C B E D (3) E A B D C (2) C D B A E (2) C A B D E (2) B E A D C (2) E B A D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E A C (1) C E A D B (1) C D A B E (1) C B D A E (1) C A D E B (1) C A D B E (1) B D E C A (1) B D C E A (1) B A E C D (1) A E C D B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -6 4 16 B 0 0 0 12 8 C 6 0 0 2 18 D -4 -12 -2 0 4 E -16 -8 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500140 C: 0.499860 D: 0.000000 E: 0.000000 Sum of squares = 0.500000038402 Cumulative probabilities = A: 0.000000 B: 0.500140 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 4 16 B 0 0 0 12 8 C 6 0 0 2 18 D -4 -12 -2 0 4 E -16 -8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=20 C=18 A=18 E=17 so E is eliminated. Round 2 votes counts: A=34 D=27 B=21 C=18 so C is eliminated. Round 3 votes counts: A=48 D=30 B=22 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:213 B:210 A:207 D:193 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 4 16 B 0 0 0 12 8 C 6 0 0 2 18 D -4 -12 -2 0 4 E -16 -8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 4 16 B 0 0 0 12 8 C 6 0 0 2 18 D -4 -12 -2 0 4 E -16 -8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 4 16 B 0 0 0 12 8 C 6 0 0 2 18 D -4 -12 -2 0 4 E -16 -8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2776: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D C B E A (7) A E B C D (7) B E A D C (6) D B E A C (5) C A E D B (5) A E B D C (5) E A B C D (4) E C A B D (3) D C B A E (3) D B C E A (3) C D E A B (3) C D A E B (3) B D E A C (3) B A E D C (3) A C E D B (3) E B A C D (2) E A C B D (2) D B A E C (2) C E B A D (2) C D B E A (2) B E D A C (2) A D B E C (2) A C E B D (2) E B C A D (1) D C A B E (1) D A B E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C A D E B (1) B E D C A (1) B D E C A (1) B C E D A (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 10 22 22 0 B -10 0 4 14 -8 C -22 -4 0 4 -22 D -22 -14 -4 0 -22 E 0 8 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.527712 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.472288 Sum of squares = 0.501535887169 Cumulative probabilities = A: 0.527712 B: 0.527712 C: 0.527712 D: 0.527712 E: 1.000000 A B C D E A 0 10 22 22 0 B -10 0 4 14 -8 C -22 -4 0 4 -22 D -22 -14 -4 0 -22 E 0 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=23 C=18 B=18 E=12 so E is eliminated. Round 2 votes counts: A=35 D=23 C=21 B=21 so C is eliminated. Round 3 votes counts: A=46 D=31 B=23 so B is eliminated. Round 4 votes counts: A=61 D=39 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:226 B:200 C:178 D:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 22 22 0 B -10 0 4 14 -8 C -22 -4 0 4 -22 D -22 -14 -4 0 -22 E 0 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 22 22 0 B -10 0 4 14 -8 C -22 -4 0 4 -22 D -22 -14 -4 0 -22 E 0 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 22 22 0 B -10 0 4 14 -8 C -22 -4 0 4 -22 D -22 -14 -4 0 -22 E 0 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2777: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) D C E A B (9) A B C D E (9) B A E C D (8) E B A D C (7) E D C B A (6) A B E C D (5) E D B C A (3) E B D C A (3) D C A E B (3) C D E B A (3) C D A B E (3) B E A D C (3) A C D B E (3) A B E D C (3) A B C E D (3) E B C D A (2) D C A B E (2) B E A C D (2) A C B D E (2) E B A C D (1) E A B D C (1) D E C A B (1) C E B D A (1) C D B A E (1) C D A E B (1) C A D B E (1) B C E A D (1) B A E D C (1) B A C E D (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 6 -4 B 4 0 10 8 -2 C -2 -10 0 -6 6 D -6 -8 6 0 -4 E 4 2 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.43209876543 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 A B C D E A 0 -4 2 6 -4 B 4 0 10 8 -2 C -2 -10 0 -6 6 D -6 -8 6 0 -4 E 4 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765342 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 E=23 B=16 C=10 so C is eliminated. Round 2 votes counts: D=32 A=28 E=24 B=16 so B is eliminated. Round 3 votes counts: A=38 D=32 E=30 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:210 E:202 A:200 C:194 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 6 -4 B 4 0 10 8 -2 C -2 -10 0 -6 6 D -6 -8 6 0 -4 E 4 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765342 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 -4 B 4 0 10 8 -2 C -2 -10 0 -6 6 D -6 -8 6 0 -4 E 4 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765342 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 -4 B 4 0 10 8 -2 C -2 -10 0 -6 6 D -6 -8 6 0 -4 E 4 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765342 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2778: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (11) C B D E A (7) C B A D E (7) C A B D E (7) D E C B A (5) D E B C A (5) A E B D C (4) A C B E D (4) A B C E D (4) E D B C A (3) E D A B C (3) B C A E D (3) E B A D C (2) E A D B C (2) D E A B C (2) D C E B A (2) D C E A B (2) C D B E A (2) C B A E D (2) C A D B E (2) B E D C A (2) B C E D A (2) A E D C B (2) A E C D B (2) A C D E B (2) E D B A C (1) E B D C A (1) C D A B E (1) C A B E D (1) B E C A D (1) B C D E A (1) B A C E D (1) A E B C D (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -12 22 14 B -6 0 0 6 -2 C 12 0 0 6 4 D -22 -6 -6 0 -8 E -14 2 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.270386 C: 0.729614 D: 0.000000 E: 0.000000 Sum of squares = 0.605445561451 Cumulative probabilities = A: 0.000000 B: 0.270386 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 22 14 B -6 0 0 6 -2 C 12 0 0 6 4 D -22 -6 -6 0 -8 E -14 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=29 D=16 E=12 B=10 so B is eliminated. Round 2 votes counts: C=35 A=34 D=16 E=15 so E is eliminated. Round 3 votes counts: A=38 C=36 D=26 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:215 C:211 B:199 E:196 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 22 14 B -6 0 0 6 -2 C 12 0 0 6 4 D -22 -6 -6 0 -8 E -14 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 22 14 B -6 0 0 6 -2 C 12 0 0 6 4 D -22 -6 -6 0 -8 E -14 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 22 14 B -6 0 0 6 -2 C 12 0 0 6 4 D -22 -6 -6 0 -8 E -14 2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2779: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) B E C A D (11) A D E B C (9) C D A B E (8) D A E C B (5) D A C E B (5) D A E B C (4) C B E D A (4) B C E A D (4) D C A B E (3) C D B A E (3) C B D E A (3) E B C D A (2) E B A C D (2) D C A E B (2) C B E A D (2) B E A C D (2) A D B E C (2) E D A C B (1) E D A B C (1) E B D C A (1) E B C A D (1) E A D B C (1) E A B D C (1) D E A C B (1) D E A B C (1) C E B D A (1) C D E A B (1) C D A E B (1) C B A E D (1) C A D B E (1) B E C D A (1) B C E D A (1) A E D B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -2 2 -8 B 2 0 14 -2 -6 C 2 -14 0 -2 -18 D -2 2 2 0 0 E 8 6 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.555741 E: 0.444259 Sum of squares = 0.506214013918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.555741 E: 1.000000 A B C D E A 0 -2 -2 2 -8 B 2 0 14 -2 -6 C 2 -14 0 -2 -18 D -2 2 2 0 0 E 8 6 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=22 D=21 B=19 A=13 so A is eliminated. Round 2 votes counts: D=33 C=25 E=23 B=19 so B is eliminated. Round 3 votes counts: E=37 D=33 C=30 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:216 B:204 D:201 A:195 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 2 -8 B 2 0 14 -2 -6 C 2 -14 0 -2 -18 D -2 2 2 0 0 E 8 6 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 -8 B 2 0 14 -2 -6 C 2 -14 0 -2 -18 D -2 2 2 0 0 E 8 6 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 -8 B 2 0 14 -2 -6 C 2 -14 0 -2 -18 D -2 2 2 0 0 E 8 6 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2780: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) E B C A D (6) D A E B C (6) B E C D A (6) C B E A D (5) B E C A D (5) A D C E B (4) A C D E B (4) E B D A C (3) E B C D A (3) C B A D E (3) C A D E B (3) C A B D E (3) B C E A D (3) A D C B E (3) E B D C A (2) D E B A C (2) D A E C B (2) D A C E B (2) D A B E C (2) C A B E D (2) B E D C A (2) B D E C A (2) A E D C B (2) A C E D B (2) A C D B E (2) E D B A C (1) E D A B C (1) E C B A D (1) D B A C E (1) D A B C E (1) C E B A D (1) C E A B D (1) C B D A E (1) C B A E D (1) C A E B D (1) C A D B E (1) B D C A E (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 -2 -10 6 8 B 2 0 -2 8 6 C 10 2 0 12 8 D -6 -8 -12 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 6 8 B 2 0 -2 8 6 C 10 2 0 12 8 D -6 -8 -12 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 B=21 E=17 A=17 so E is eliminated. Round 2 votes counts: B=35 D=25 C=23 A=17 so A is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:207 A:201 D:188 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 6 8 B 2 0 -2 8 6 C 10 2 0 12 8 D -6 -8 -12 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 6 8 B 2 0 -2 8 6 C 10 2 0 12 8 D -6 -8 -12 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 6 8 B 2 0 -2 8 6 C 10 2 0 12 8 D -6 -8 -12 0 2 E -8 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2781: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (14) D A B C E (5) A D E B C (5) E A C B D (4) A D C B E (4) A C D E B (4) E C B A D (3) D C B A E (3) C B E D A (3) B E C D A (3) A D C E B (3) A C D B E (3) E C B D A (2) E B D C A (2) E B D A C (2) E B C A D (2) D C A B E (2) D B C E A (2) D A C B E (2) C E B D A (2) C E B A D (2) C D B A E (2) C D A B E (2) C B D E A (2) B E D C A (2) B D E A C (2) B D C E A (2) A E D B C (2) A D B E C (2) E B A D C (1) E B A C D (1) E A B D C (1) E A B C D (1) D B E C A (1) D B A E C (1) D B A C E (1) C D B E A (1) C A D B E (1) A E C D B (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -6 -16 -10 B 12 0 6 2 -6 C 6 -6 0 6 -6 D 16 -2 -6 0 4 E 10 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.045463 C: 0.215903 D: 0.392049 E: 0.346586 Sum of squares = 0.322504708248 Cumulative probabilities = A: 0.000000 B: 0.045463 C: 0.261366 D: 0.653414 E: 1.000000 A B C D E A 0 -12 -6 -16 -10 B 12 0 6 2 -6 C 6 -6 0 6 -6 D 16 -2 -6 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.134328 C: 0.149254 D: 0.425373 E: 0.291045 Sum of squares = 0.305970149257 Cumulative probabilities = A: 0.000000 B: 0.134328 C: 0.283582 D: 0.708955 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=26 D=17 C=15 B=9 so B is eliminated. Round 2 votes counts: E=38 A=26 D=21 C=15 so C is eliminated. Round 3 votes counts: E=45 D=28 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:209 B:207 D:206 C:200 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -6 -16 -10 B 12 0 6 2 -6 C 6 -6 0 6 -6 D 16 -2 -6 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.134328 C: 0.149254 D: 0.425373 E: 0.291045 Sum of squares = 0.305970149257 Cumulative probabilities = A: 0.000000 B: 0.134328 C: 0.283582 D: 0.708955 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -16 -10 B 12 0 6 2 -6 C 6 -6 0 6 -6 D 16 -2 -6 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.134328 C: 0.149254 D: 0.425373 E: 0.291045 Sum of squares = 0.305970149257 Cumulative probabilities = A: 0.000000 B: 0.134328 C: 0.283582 D: 0.708955 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -16 -10 B 12 0 6 2 -6 C 6 -6 0 6 -6 D 16 -2 -6 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.134328 C: 0.149254 D: 0.425373 E: 0.291045 Sum of squares = 0.305970149257 Cumulative probabilities = A: 0.000000 B: 0.134328 C: 0.283582 D: 0.708955 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2782: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) E C A B D (8) C D E A B (8) B A E D C (8) D C B E A (7) D B C A E (7) E A C B D (6) D C B A E (6) C D E B A (6) C E D A B (5) B D A E C (4) B A D E C (4) E C A D B (3) E A B C D (3) D C E B A (3) D B A C E (3) C E A D B (3) B D A C E (3) D C A B E (1) B A E C D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -14 -6 -4 B 4 0 -12 -4 8 C 14 12 0 6 4 D 6 4 -6 0 4 E 4 -8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -6 -4 B 4 0 -12 -4 8 C 14 12 0 6 4 D 6 4 -6 0 4 E 4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=22 E=20 B=20 A=11 so A is eliminated. Round 2 votes counts: B=30 D=27 C=22 E=21 so E is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:204 B:198 E:194 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -6 -4 B 4 0 -12 -4 8 C 14 12 0 6 4 D 6 4 -6 0 4 E 4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -6 -4 B 4 0 -12 -4 8 C 14 12 0 6 4 D 6 4 -6 0 4 E 4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -6 -4 B 4 0 -12 -4 8 C 14 12 0 6 4 D 6 4 -6 0 4 E 4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2783: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) A C E D B (7) E D B A C (6) D E A C B (6) B E D C A (6) A C B E D (6) D E B C A (4) B C A E D (4) A C D E B (4) D E C A B (3) B C D E A (3) B C D A E (3) A E D C B (3) A C B D E (3) E D A B C (2) E A D C B (2) D E C B A (2) D C A E B (2) C A B D E (2) B E C D A (2) B D E C A (2) B D C E A (2) B C A D E (2) E D B C A (1) E B D C A (1) E B D A C (1) E A B D C (1) D E B A C (1) D E A B C (1) D C E B A (1) D C E A B (1) D B C E A (1) C B A D E (1) C A D E B (1) C A B E D (1) B E D A C (1) B E A C D (1) B C E A D (1) A E C D B (1) Total count = 100 A B C D E A 0 8 8 -20 -20 B -8 0 -8 -14 -18 C -8 8 0 -16 -10 D 20 14 16 0 -10 E 20 18 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 -20 -20 B -8 0 -8 -14 -18 C -8 8 0 -16 -10 D 20 14 16 0 -10 E 20 18 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 E=22 D=22 C=5 so C is eliminated. Round 2 votes counts: B=28 A=28 E=22 D=22 so E is eliminated. Round 3 votes counts: D=39 A=31 B=30 so B is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:229 D:220 A:188 C:187 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 -20 -20 B -8 0 -8 -14 -18 C -8 8 0 -16 -10 D 20 14 16 0 -10 E 20 18 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -20 -20 B -8 0 -8 -14 -18 C -8 8 0 -16 -10 D 20 14 16 0 -10 E 20 18 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -20 -20 B -8 0 -8 -14 -18 C -8 8 0 -16 -10 D 20 14 16 0 -10 E 20 18 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2784: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) E B D C A (6) C A D B E (6) E B D A C (5) A C D B E (5) C D A B E (4) B E D C A (4) B C D E A (4) A D B C E (4) E B A D C (3) B D C A E (3) B D A C E (3) A E D B C (3) A D C B E (3) A C D E B (3) E C B D A (2) E B C A D (2) E A C D B (2) E A B D C (2) D B A C E (2) D A B C E (2) C D B A E (2) C A D E B (2) E C A D B (1) E C A B D (1) E B A C D (1) E A C B D (1) E A B C D (1) D C A B E (1) D B C A E (1) D B A E C (1) D A C B E (1) C E B A D (1) C B E D A (1) C A E D B (1) B E C D A (1) B D E C A (1) B D C E A (1) B D A E C (1) A D E C B (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -6 -8 4 B 6 0 22 4 4 C 6 -22 0 -2 2 D 8 -4 2 0 8 E -4 -4 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -8 4 B 6 0 22 4 4 C 6 -22 0 -2 2 D 8 -4 2 0 8 E -4 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=22 B=18 C=17 D=8 so D is eliminated. Round 2 votes counts: E=35 A=25 B=22 C=18 so C is eliminated. Round 3 votes counts: A=39 E=36 B=25 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:218 D:207 A:192 C:192 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -8 4 B 6 0 22 4 4 C 6 -22 0 -2 2 D 8 -4 2 0 8 E -4 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -8 4 B 6 0 22 4 4 C 6 -22 0 -2 2 D 8 -4 2 0 8 E -4 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -8 4 B 6 0 22 4 4 C 6 -22 0 -2 2 D 8 -4 2 0 8 E -4 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994521 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2785: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) C E A D B (9) E A D C B (8) C D E A B (6) C D A E B (6) E A D B C (4) D B A E C (4) D A E B C (4) B C E A D (4) E A C B D (3) D C A E B (3) C B E A D (3) A E D C B (3) E C A D B (2) E A C D B (2) D A E C B (2) C D B A E (2) C B D E A (2) B E A D C (2) B E A C D (2) B D C A E (2) B C D E A (2) A E D B C (2) E A B D C (1) E A B C D (1) D C A B E (1) D B C A E (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E D A (1) B C D A E (1) B A E D C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 22 6 2 -8 B -22 0 -10 -22 -20 C -6 10 0 -4 -10 D -2 22 4 0 -2 E 8 20 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 6 2 -8 B -22 0 -10 -22 -20 C -6 10 0 -4 -10 D -2 22 4 0 -2 E 8 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=25 E=21 D=16 A=7 so A is eliminated. Round 2 votes counts: C=31 E=27 B=25 D=17 so D is eliminated. Round 3 votes counts: C=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:211 D:211 C:195 B:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 6 2 -8 B -22 0 -10 -22 -20 C -6 10 0 -4 -10 D -2 22 4 0 -2 E 8 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 6 2 -8 B -22 0 -10 -22 -20 C -6 10 0 -4 -10 D -2 22 4 0 -2 E 8 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 6 2 -8 B -22 0 -10 -22 -20 C -6 10 0 -4 -10 D -2 22 4 0 -2 E 8 20 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2786: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) B E A C D (10) E C B A D (9) D B A C E (9) A B C E D (8) B A E C D (7) E C A B D (6) D C E A B (5) D A C B E (4) D E B C A (3) B A C E D (3) E D B C A (2) E C D B A (2) E B C A D (2) D B E C A (2) D A B C E (2) C D E A B (2) A B D C E (2) E D C B A (1) E B A C D (1) D E C B A (1) D E C A B (1) D C A B E (1) D B E A C (1) D B A E C (1) D A C E B (1) C E A D B (1) B A D E C (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 4 4 2 B 10 0 8 0 6 C -4 -8 0 4 0 D -4 0 -4 0 -6 E -2 -6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.649229 C: 0.000000 D: 0.350771 E: 0.000000 Sum of squares = 0.54453860331 Cumulative probabilities = A: 0.000000 B: 0.649229 C: 0.649229 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 4 2 B 10 0 8 0 6 C -4 -8 0 4 0 D -4 0 -4 0 -6 E -2 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500300 C: 0.000000 D: 0.499700 E: 0.000000 Sum of squares = 0.500000179827 Cumulative probabilities = A: 0.000000 B: 0.500300 C: 0.500300 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=23 B=21 A=12 C=3 so C is eliminated. Round 2 votes counts: D=43 E=24 B=21 A=12 so A is eliminated. Round 3 votes counts: D=44 B=32 E=24 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:200 E:199 C:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 4 2 B 10 0 8 0 6 C -4 -8 0 4 0 D -4 0 -4 0 -6 E -2 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500300 C: 0.000000 D: 0.499700 E: 0.000000 Sum of squares = 0.500000179827 Cumulative probabilities = A: 0.000000 B: 0.500300 C: 0.500300 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 4 2 B 10 0 8 0 6 C -4 -8 0 4 0 D -4 0 -4 0 -6 E -2 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500300 C: 0.000000 D: 0.499700 E: 0.000000 Sum of squares = 0.500000179827 Cumulative probabilities = A: 0.000000 B: 0.500300 C: 0.500300 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 4 2 B 10 0 8 0 6 C -4 -8 0 4 0 D -4 0 -4 0 -6 E -2 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500300 C: 0.000000 D: 0.499700 E: 0.000000 Sum of squares = 0.500000179827 Cumulative probabilities = A: 0.000000 B: 0.500300 C: 0.500300 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2787: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) B C D E A (7) B D C A E (6) A E D C B (6) C B D E A (5) C B D A E (5) E B A D C (4) E A B D C (3) D C B A E (3) D B A C E (3) D A C E B (3) C D B A E (3) B D E A C (3) E A D B C (2) E A B C D (2) D C A B E (2) D B C A E (2) D B A E C (2) D A E B C (2) C D A B E (2) C B E A D (2) C A D E B (2) B E C D A (2) B E C A D (2) B E A D C (2) A D E B C (2) E C A B D (1) E B C A D (1) D A E C B (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E D A (1) C A E B D (1) B D C E A (1) B C E D A (1) A E D B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 -2 -12 0 B 12 0 -2 20 12 C 2 2 0 -2 8 D 12 -20 2 0 14 E 0 -12 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.833333 D: 0.083333 E: 0.000000 Sum of squares = 0.708333333294 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.916667 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -12 0 B 12 0 -2 20 12 C 2 2 0 -2 8 D 12 -20 2 0 14 E 0 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.833333 D: 0.083333 E: 0.000000 Sum of squares = 0.708333333672 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.916667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 E=22 D=20 A=11 so A is eliminated. Round 2 votes counts: E=29 C=24 B=24 D=23 so D is eliminated. Round 3 votes counts: E=34 C=34 B=32 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:205 D:204 A:187 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 -12 0 B 12 0 -2 20 12 C 2 2 0 -2 8 D 12 -20 2 0 14 E 0 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.833333 D: 0.083333 E: 0.000000 Sum of squares = 0.708333333672 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.916667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -12 0 B 12 0 -2 20 12 C 2 2 0 -2 8 D 12 -20 2 0 14 E 0 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.833333 D: 0.083333 E: 0.000000 Sum of squares = 0.708333333672 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.916667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -12 0 B 12 0 -2 20 12 C 2 2 0 -2 8 D 12 -20 2 0 14 E 0 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.833333 D: 0.083333 E: 0.000000 Sum of squares = 0.708333333672 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.916667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2788: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) C E D B A (9) D B C E A (7) C D E B A (7) A C E D B (7) B D E C A (6) C E D A B (5) B D C E A (5) B D A E C (4) A B E C D (4) D C E B A (3) D C B E A (3) B A D C E (3) A B E D C (3) A B D E C (3) A B C E D (3) E C D A B (2) D B E C A (2) C E A D B (2) A E D C B (2) A E C B D (2) A B C D E (2) E C A D B (1) E A C D B (1) D E C B A (1) C B D E A (1) B D A C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -24 -8 -12 -10 B 24 0 6 -4 14 C 8 -6 0 -6 18 D 12 4 6 0 16 E 10 -14 -18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -8 -12 -10 B 24 0 6 -4 14 C 8 -6 0 -6 18 D 12 4 6 0 16 E 10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=27 C=24 D=16 E=4 so E is eliminated. Round 2 votes counts: B=29 A=28 C=27 D=16 so D is eliminated. Round 3 votes counts: B=38 C=34 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:219 C:207 E:181 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 -8 -12 -10 B 24 0 6 -4 14 C 8 -6 0 -6 18 D 12 4 6 0 16 E 10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -8 -12 -10 B 24 0 6 -4 14 C 8 -6 0 -6 18 D 12 4 6 0 16 E 10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -8 -12 -10 B 24 0 6 -4 14 C 8 -6 0 -6 18 D 12 4 6 0 16 E 10 -14 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2789: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) C E D A B (8) B A D E C (7) B A D C E (6) E C D A B (5) D A E B C (5) C E B D A (5) C B E D A (5) B C A D E (5) E D C A B (4) E A D B C (4) D E A C B (4) C B E A D (4) E D A B C (3) C D A B E (3) C B D A E (3) C B A D E (3) A D E B C (2) A D B E C (2) A D B C E (2) A B D E C (2) D E A B C (1) D A E C B (1) D A B C E (1) C D B A E (1) B E C A D (1) B C E A D (1) B A E D C (1) B A E C D (1) A E B D C (1) Total count = 100 A B C D E A 0 14 4 -16 -8 B -14 0 -10 -10 -4 C -4 10 0 -10 -6 D 16 10 10 0 -4 E 8 4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 4 -16 -8 B -14 0 -10 -10 -4 C -4 10 0 -10 -6 D 16 10 10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=25 B=22 D=12 A=9 so A is eliminated. Round 2 votes counts: C=32 E=26 B=24 D=18 so D is eliminated. Round 3 votes counts: E=39 C=32 B=29 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:211 A:197 C:195 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 4 -16 -8 B -14 0 -10 -10 -4 C -4 10 0 -10 -6 D 16 10 10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -16 -8 B -14 0 -10 -10 -4 C -4 10 0 -10 -6 D 16 10 10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -16 -8 B -14 0 -10 -10 -4 C -4 10 0 -10 -6 D 16 10 10 0 -4 E 8 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2790: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (13) C B D A E (10) E A D C B (6) E A D B C (6) D B C A E (6) D E A B C (5) D B E C A (5) A E C B D (5) E A C B D (4) D B C E A (4) C B E A D (4) C B A D E (4) E D A B C (3) B D C E A (3) E A C D B (2) D E B A C (2) C B A E D (2) C A B E D (2) B C D E A (2) A E C D B (2) A D C B E (2) E D B A C (1) E C B A D (1) E C A B D (1) D B A C E (1) D A E B C (1) A E D C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 -14 -12 2 B 16 0 4 6 18 C 14 -4 0 8 10 D 12 -6 -8 0 16 E -2 -18 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -12 2 B 16 0 4 6 18 C 14 -4 0 8 10 D 12 -6 -8 0 16 E -2 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 C=22 B=18 A=12 so A is eliminated. Round 2 votes counts: E=32 D=26 C=24 B=18 so B is eliminated. Round 3 votes counts: C=39 E=32 D=29 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:222 C:214 D:207 A:180 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -14 -12 2 B 16 0 4 6 18 C 14 -4 0 8 10 D 12 -6 -8 0 16 E -2 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -12 2 B 16 0 4 6 18 C 14 -4 0 8 10 D 12 -6 -8 0 16 E -2 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -12 2 B 16 0 4 6 18 C 14 -4 0 8 10 D 12 -6 -8 0 16 E -2 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2791: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) D E B A C (8) C E D B A (7) C A B D E (6) C E D A B (5) C A B E D (5) A C B D E (5) E D C B A (4) E D B C A (4) A B D E C (4) D B E A C (3) D A B E C (3) C E B A D (3) C B A E D (3) C A D B E (3) B A E D C (3) D B A E C (2) C E A B D (2) C A E D B (2) C A E B D (2) B D A E C (2) B A D E C (2) A B C D E (2) E C B D A (1) E B D A C (1) E B C D A (1) D E C A B (1) D E A B C (1) C E B D A (1) C D E A B (1) C B E A D (1) B E C A D (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 -6 -8 -8 B 14 0 -4 -8 -6 C 6 4 0 4 0 D 8 8 -4 0 -10 E 8 6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.769601 D: 0.000000 E: 0.230399 Sum of squares = 0.645369885095 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.769601 D: 0.769601 E: 1.000000 A B C D E A 0 -14 -6 -8 -8 B 14 0 -4 -8 -6 C 6 4 0 4 0 D 8 8 -4 0 -10 E 8 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 E=20 D=18 A=12 B=9 so B is eliminated. Round 2 votes counts: C=41 E=21 D=20 A=18 so A is eliminated. Round 3 votes counts: C=49 D=27 E=24 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:207 D:201 B:198 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -6 -8 -8 B 14 0 -4 -8 -6 C 6 4 0 4 0 D 8 8 -4 0 -10 E 8 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -8 -8 B 14 0 -4 -8 -6 C 6 4 0 4 0 D 8 8 -4 0 -10 E 8 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -8 -8 B 14 0 -4 -8 -6 C 6 4 0 4 0 D 8 8 -4 0 -10 E 8 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2792: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C A D B E (7) B E A D C (7) C D E B A (6) C D B E A (6) B A E C D (6) A B C E D (5) D C E B A (4) A B E C D (4) D E C B A (3) D C E A B (3) C D A B E (3) C A B D E (3) A E D B C (3) A B E D C (3) E D A B C (2) D E A C B (2) C D E A B (2) C D A E B (2) C A B E D (2) B A E D C (2) A C B E D (2) E D B A C (1) E B A D C (1) D E C A B (1) D E B C A (1) D E A B C (1) D C A E B (1) C B D E A (1) C B D A E (1) C B A E D (1) C B A D E (1) B E A C D (1) B C E D A (1) B C A E D (1) B A C E D (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 10 0 18 0 B -10 0 -4 4 18 C 0 4 0 14 8 D -18 -4 -14 0 -2 E 0 -18 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.433537 B: 0.000000 C: 0.566463 D: 0.000000 E: 0.000000 Sum of squares = 0.50883475072 Cumulative probabilities = A: 0.433537 B: 0.433537 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 18 0 B -10 0 -4 4 18 C 0 4 0 14 8 D -18 -4 -14 0 -2 E 0 -18 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=19 A=19 D=16 E=11 so E is eliminated. Round 2 votes counts: C=35 A=26 B=20 D=19 so D is eliminated. Round 3 votes counts: C=47 A=31 B=22 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:213 B:204 E:188 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 18 0 B -10 0 -4 4 18 C 0 4 0 14 8 D -18 -4 -14 0 -2 E 0 -18 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 18 0 B -10 0 -4 4 18 C 0 4 0 14 8 D -18 -4 -14 0 -2 E 0 -18 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 18 0 B -10 0 -4 4 18 C 0 4 0 14 8 D -18 -4 -14 0 -2 E 0 -18 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2793: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (16) E A C B D (15) E A D B C (6) D E A B C (5) D C B E A (4) D C B A E (4) C B E A D (4) C B A E D (4) B D C A E (4) D B A C E (3) B C D A E (3) B C A D E (3) E A C D B (2) D E A C B (2) D B E A C (2) D B C E A (2) D A B E C (2) A E D B C (2) A E B C D (2) E C D A B (1) E C A B D (1) E A D C B (1) E A B C D (1) D C E B A (1) C E D B A (1) C E A B D (1) C D B E A (1) C B D E A (1) C B D A E (1) C B A D E (1) B A C E D (1) B A C D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -6 -6 -2 B 14 0 10 -10 18 C 6 -10 0 -10 14 D 6 10 10 0 14 E 2 -18 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -6 -2 B 14 0 10 -10 18 C 6 -10 0 -10 14 D 6 10 10 0 14 E 2 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=27 C=14 B=12 A=6 so A is eliminated. Round 2 votes counts: D=41 E=31 C=14 B=14 so C is eliminated. Round 3 votes counts: D=42 E=33 B=25 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:216 C:200 A:186 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -6 -2 B 14 0 10 -10 18 C 6 -10 0 -10 14 D 6 10 10 0 14 E 2 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -6 -2 B 14 0 10 -10 18 C 6 -10 0 -10 14 D 6 10 10 0 14 E 2 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -6 -2 B 14 0 10 -10 18 C 6 -10 0 -10 14 D 6 10 10 0 14 E 2 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999671 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2794: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) E B D C A (10) A B C D E (9) B A E C D (8) C D A E B (7) E D C B A (6) D E C A B (6) A C D B E (6) A C B D E (5) E B D A C (4) D C A E B (4) B A C E D (4) E B A D C (3) D C E A B (3) C A D B E (3) B A C D E (3) E D C A B (2) E D B C A (2) E D A C B (1) C B A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 12 10 2 B 6 0 10 18 2 C -12 -10 0 -8 -8 D -10 -18 8 0 -4 E -2 -2 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 10 2 B 6 0 10 18 2 C -12 -10 0 -8 -8 D -10 -18 8 0 -4 E -2 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 A=21 D=13 C=11 so C is eliminated. Round 2 votes counts: E=28 B=28 A=24 D=20 so D is eliminated. Round 3 votes counts: E=37 A=35 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:218 A:209 E:204 D:188 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 10 2 B 6 0 10 18 2 C -12 -10 0 -8 -8 D -10 -18 8 0 -4 E -2 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 10 2 B 6 0 10 18 2 C -12 -10 0 -8 -8 D -10 -18 8 0 -4 E -2 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 10 2 B 6 0 10 18 2 C -12 -10 0 -8 -8 D -10 -18 8 0 -4 E -2 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2795: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (13) B D A C E (12) A D B E C (10) E A D B C (8) C E B D A (6) A D E B C (6) E C A D B (5) D A B E C (4) C E B A D (4) C B D E A (4) E A C D B (3) B D A E C (3) E A D C B (2) D B A E C (2) C E A D B (2) C E A B D (2) C B E D A (2) B A D C E (2) A E D B C (2) A B D E C (2) E D A B C (1) E C D A B (1) C E D A B (1) B D C A E (1) B C D A E (1) B A D E C (1) Total count = 100 A B C D E A 0 -2 16 -2 18 B 2 0 10 6 14 C -16 -10 0 -12 0 D 2 -6 12 0 22 E -18 -14 0 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 -2 18 B 2 0 10 6 14 C -16 -10 0 -12 0 D 2 -6 12 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=20 B=20 A=20 D=6 so D is eliminated. Round 2 votes counts: C=34 A=24 B=22 E=20 so E is eliminated. Round 3 votes counts: C=40 A=38 B=22 so B is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 A:215 D:215 C:181 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 -2 18 B 2 0 10 6 14 C -16 -10 0 -12 0 D 2 -6 12 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 -2 18 B 2 0 10 6 14 C -16 -10 0 -12 0 D 2 -6 12 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 -2 18 B 2 0 10 6 14 C -16 -10 0 -12 0 D 2 -6 12 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2796: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (14) E A D C B (11) E C A D B (7) D A B C E (7) C B E D A (7) E A D B C (6) A D E B C (6) A D B C E (6) E C B A D (5) E C B D A (4) C E B D A (4) C B D A E (4) A D E C B (4) B C D A E (3) B D A C E (2) E C D A B (1) E C A B D (1) E B A D C (1) D C A B E (1) D B A C E (1) D A C E B (1) C D B A E (1) B D C A E (1) B C D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 32 20 24 4 B -32 0 -2 -34 -4 C -20 2 0 -24 -22 D -24 34 24 0 4 E -4 4 22 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 32 20 24 4 B -32 0 -2 -34 -4 C -20 2 0 -24 -22 D -24 34 24 0 4 E -4 4 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=31 C=16 D=10 B=7 so B is eliminated. Round 2 votes counts: E=36 A=31 C=20 D=13 so D is eliminated. Round 3 votes counts: A=42 E=36 C=22 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:240 D:219 E:209 C:168 B:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 32 20 24 4 B -32 0 -2 -34 -4 C -20 2 0 -24 -22 D -24 34 24 0 4 E -4 4 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 32 20 24 4 B -32 0 -2 -34 -4 C -20 2 0 -24 -22 D -24 34 24 0 4 E -4 4 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 32 20 24 4 B -32 0 -2 -34 -4 C -20 2 0 -24 -22 D -24 34 24 0 4 E -4 4 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2797: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) B C E D A (6) C B D E A (5) A B C D E (5) D A E C B (4) B E C D A (4) A D E C B (4) C B E D A (3) C B D A E (3) B C E A D (3) B C A E D (3) A D C E B (3) A D C B E (3) E D B C A (2) E A B D C (2) D E A C B (2) D C B E A (2) D C A B E (2) D A C B E (2) C D A B E (2) C B A D E (2) B E C A D (2) A D E B C (2) A C B D E (2) A B E C D (2) A B C E D (2) E D C B A (1) E C D B A (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E A B (1) D C A E B (1) D A C E B (1) C D B A E (1) C A B D E (1) B E A C D (1) B C D E A (1) A E D C B (1) A E B D C (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -14 -12 -2 B 4 0 0 22 16 C 14 0 0 24 10 D 12 -22 -24 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.448792 C: 0.551208 D: 0.000000 E: 0.000000 Sum of squares = 0.505244602907 Cumulative probabilities = A: 0.000000 B: 0.448792 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -12 -2 B 4 0 0 22 16 C 14 0 0 24 10 D 12 -22 -24 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=20 E=19 D=17 C=17 so D is eliminated. Round 2 votes counts: A=34 E=23 C=23 B=20 so B is eliminated. Round 3 votes counts: C=36 A=34 E=30 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:221 E:186 D:185 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -14 -12 -2 B 4 0 0 22 16 C 14 0 0 24 10 D 12 -22 -24 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -12 -2 B 4 0 0 22 16 C 14 0 0 24 10 D 12 -22 -24 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -12 -2 B 4 0 0 22 16 C 14 0 0 24 10 D 12 -22 -24 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2798: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (6) C D E A B (5) B D C A E (5) B A C E D (5) A E C B D (5) A E B D C (5) E A C D B (4) C E A D B (4) B D A C E (4) A B E C D (4) D C E A B (3) D C B E A (3) D B C E A (3) C D E B A (3) C D B E A (3) B C D A E (3) B A D E C (3) A B E D C (3) E D C A B (2) E C D A B (2) E C A D B (2) C E D B A (2) C E D A B (2) C E A B D (2) B D A E C (2) B A E D C (2) B A C D E (2) A E D B C (2) E C A B D (1) D E C A B (1) D C E B A (1) D A E B C (1) C B A E D (1) B C A E D (1) B A D C E (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 2 10 14 B -12 0 6 12 -8 C -2 -6 0 18 8 D -10 -12 -18 0 -14 E -14 8 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 10 14 B -12 0 6 12 -8 C -2 -6 0 18 8 D -10 -12 -18 0 -14 E -14 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=27 C=22 D=12 E=11 so E is eliminated. Round 2 votes counts: A=31 B=28 C=27 D=14 so D is eliminated. Round 3 votes counts: C=37 A=32 B=31 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:209 E:200 B:199 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 10 14 B -12 0 6 12 -8 C -2 -6 0 18 8 D -10 -12 -18 0 -14 E -14 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 10 14 B -12 0 6 12 -8 C -2 -6 0 18 8 D -10 -12 -18 0 -14 E -14 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 10 14 B -12 0 6 12 -8 C -2 -6 0 18 8 D -10 -12 -18 0 -14 E -14 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998718 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2799: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) C E B D A (11) B D C E A (9) E C B D A (7) E C A B D (6) D B C E A (5) D B A C E (4) A E C D B (4) A C E D B (4) C E A B D (3) B D A E C (3) A D B C E (3) A B D E C (3) E A C B D (2) D B A E C (2) B E C D A (2) B D E A C (2) A D C E B (2) E C B A D (1) E B A D C (1) E A B C D (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D E A (1) C A E D B (1) B E D C A (1) B D E C A (1) A E C B D (1) A E B D C (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -2 -2 -10 B 4 0 2 14 8 C 2 -2 0 -2 -2 D 2 -14 2 0 6 E 10 -8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -2 -10 B 4 0 2 14 8 C 2 -2 0 -2 -2 D 2 -14 2 0 6 E 10 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=19 E=18 B=18 D=11 so D is eliminated. Round 2 votes counts: A=34 B=29 C=19 E=18 so E is eliminated. Round 3 votes counts: A=37 C=33 B=30 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:214 E:199 C:198 D:198 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 -10 B 4 0 2 14 8 C 2 -2 0 -2 -2 D 2 -14 2 0 6 E 10 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 -10 B 4 0 2 14 8 C 2 -2 0 -2 -2 D 2 -14 2 0 6 E 10 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 -10 B 4 0 2 14 8 C 2 -2 0 -2 -2 D 2 -14 2 0 6 E 10 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2800: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) E C B D A (4) E A C B D (4) C B E D A (4) B D C A E (4) B C D E A (4) A E D C B (4) A B D C E (4) E B C A D (3) E A D C B (3) E A B C D (3) D A E C B (3) C E B D A (3) A D E B C (3) A D B C E (3) E D A C B (2) E C D B A (2) E C A B D (2) D C B E A (2) D C B A E (2) D B A C E (2) D A C B E (2) C B D E A (2) B C E A D (2) B A C E D (2) A E D B C (2) A D E C B (2) A B E C D (2) E D C A B (1) E C B A D (1) E C A D B (1) E B C D A (1) E B A C D (1) D C E B A (1) D B C A E (1) D A B C E (1) C D E B A (1) C D B E A (1) B E C A D (1) B E A C D (1) B C D A E (1) B C A E D (1) A E B D C (1) A E B C D (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 -2 -14 B 8 0 6 20 2 C 4 -6 0 12 2 D 2 -20 -12 0 -18 E 14 -2 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -2 -14 B 8 0 6 20 2 C 4 -6 0 12 2 D 2 -20 -12 0 -18 E 14 -2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=24 B=23 D=14 C=11 so C is eliminated. Round 2 votes counts: E=31 B=29 A=24 D=16 so D is eliminated. Round 3 votes counts: B=37 E=33 A=30 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:214 C:206 A:186 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -2 -14 B 8 0 6 20 2 C 4 -6 0 12 2 D 2 -20 -12 0 -18 E 14 -2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -2 -14 B 8 0 6 20 2 C 4 -6 0 12 2 D 2 -20 -12 0 -18 E 14 -2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -2 -14 B 8 0 6 20 2 C 4 -6 0 12 2 D 2 -20 -12 0 -18 E 14 -2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2801: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) A D E C B (8) D A E B C (7) D E A B C (6) C B E A D (6) B C A D E (5) E D A C B (3) D A B C E (3) B C E A D (3) B C D E A (3) B C D A E (3) A E C D B (3) A D B C E (3) E C B D A (2) E C B A D (2) E C A B D (2) E B C D A (2) D E B A C (2) D B C E A (2) D A E C B (2) C E B A D (2) A E D C B (2) A D C B E (2) E D C B A (1) E A D C B (1) E A C D B (1) D E B C A (1) D E A C B (1) D B E C A (1) D B C A E (1) D B A E C (1) D B A C E (1) C B A E D (1) C B A D E (1) B E C D A (1) B D C E A (1) B D A C E (1) B C A E D (1) A E C B D (1) A D C E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 2 -6 -2 B 4 0 12 -6 0 C -2 -12 0 -2 0 D 6 6 2 0 14 E 2 0 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -6 -2 B 4 0 12 -6 0 C -2 -12 0 -2 0 D 6 6 2 0 14 E 2 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=26 A=22 E=14 C=10 so C is eliminated. Round 2 votes counts: B=34 D=28 A=22 E=16 so E is eliminated. Round 3 votes counts: B=42 D=32 A=26 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:205 A:195 E:194 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -6 -2 B 4 0 12 -6 0 C -2 -12 0 -2 0 D 6 6 2 0 14 E 2 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -6 -2 B 4 0 12 -6 0 C -2 -12 0 -2 0 D 6 6 2 0 14 E 2 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -6 -2 B 4 0 12 -6 0 C -2 -12 0 -2 0 D 6 6 2 0 14 E 2 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2802: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (13) B C A E D (6) A C B D E (6) C A B D E (5) B E A C D (5) E D B C A (4) E B D A C (4) C A D B E (4) B A C E D (4) E D A B C (3) D E C A B (3) D C A E B (3) D A C E B (3) B E C A D (3) E D B A C (2) E D A C B (2) E A D C B (2) D C E B A (2) D C B A E (2) D C A B E (2) C D A B E (2) C B A D E (2) B E D C A (2) B C A D E (2) A C B E D (2) E B A D C (1) E B A C D (1) E A B C D (1) D E C B A (1) B E C D A (1) B D C A E (1) B C E A D (1) B C D E A (1) B C D A E (1) B A E C D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 4 -4 -4 B -6 0 -10 4 8 C -4 10 0 0 -2 D 4 -4 0 0 6 E 4 -8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.424022 D: 0.575978 E: 0.000000 Sum of squares = 0.511545429648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.424022 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 -4 B -6 0 -10 4 8 C -4 10 0 0 -2 D 4 -4 0 0 6 E 4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499771 D: 0.500229 E: 0.000000 Sum of squares = 0.500000105188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499771 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=28 E=20 C=13 A=10 so A is eliminated. Round 2 votes counts: D=29 B=29 E=21 C=21 so E is eliminated. Round 3 votes counts: D=42 B=36 C=22 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:203 C:202 A:201 B:198 E:196 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 -4 -4 B -6 0 -10 4 8 C -4 10 0 0 -2 D 4 -4 0 0 6 E 4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499771 D: 0.500229 E: 0.000000 Sum of squares = 0.500000105188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499771 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 -4 B -6 0 -10 4 8 C -4 10 0 0 -2 D 4 -4 0 0 6 E 4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499771 D: 0.500229 E: 0.000000 Sum of squares = 0.500000105188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499771 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 -4 B -6 0 -10 4 8 C -4 10 0 0 -2 D 4 -4 0 0 6 E 4 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499771 D: 0.500229 E: 0.000000 Sum of squares = 0.500000105188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499771 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2803: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) E C A B D (7) D B A C E (6) C E B A D (5) A E C D B (5) D B C E A (4) D A C E B (4) B D A E C (4) A E D C B (4) A E C B D (4) E C B A D (3) D B A E C (3) D A B E C (3) B E C A D (3) B C E D A (3) E C A D B (2) E A C D B (2) D A B C E (2) C E B D A (2) C E A D B (2) B A D E C (2) A D E C B (2) A D B E C (2) A B E C D (2) E A C B D (1) D C E B A (1) D C A E B (1) D B C A E (1) D A E C B (1) C E D B A (1) C D B E A (1) C B E D A (1) B D C A E (1) B D A C E (1) B C D E A (1) A E D B C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 2 -2 2 B 6 0 0 2 2 C -2 0 0 -10 -4 D 2 -2 10 0 2 E -2 -2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.897306 C: 0.102694 D: 0.000000 E: 0.000000 Sum of squares = 0.815703406562 Cumulative probabilities = A: 0.000000 B: 0.897306 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -2 2 B 6 0 0 2 2 C -2 0 0 -10 -4 D 2 -2 10 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222381 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 A=22 E=15 C=12 so C is eliminated. Round 2 votes counts: D=27 B=26 E=25 A=22 so A is eliminated. Round 3 votes counts: E=39 D=32 B=29 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 B:205 E:199 A:198 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -2 2 B 6 0 0 2 2 C -2 0 0 -10 -4 D 2 -2 10 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222381 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 2 B 6 0 0 2 2 C -2 0 0 -10 -4 D 2 -2 10 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222381 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 2 B 6 0 0 2 2 C -2 0 0 -10 -4 D 2 -2 10 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222381 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2804: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) C D A B E (8) C D A E B (7) C A E B D (7) E B A C D (6) D E B C A (6) D C E B A (5) C A D B E (5) A C B E D (5) E B A D C (4) D C B E A (4) D B E C A (4) E B D A C (3) C A D E B (3) C D E B A (2) B E A D C (2) A C E B D (2) E D B C A (1) E C B D A (1) E B C D A (1) E A B C D (1) D E B A C (1) D C B A E (1) D C A B E (1) C E D B A (1) C E B A D (1) C A B E D (1) B E D A C (1) B D E A C (1) B A D E C (1) A E B C D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -18 -18 -12 B 14 0 -8 -22 -6 C 18 8 0 4 8 D 18 22 -4 0 22 E 12 6 -8 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -18 -18 -12 B 14 0 -8 -22 -6 C 18 8 0 4 8 D 18 22 -4 0 22 E 12 6 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=33 E=17 A=10 B=5 so B is eliminated. Round 2 votes counts: C=35 D=34 E=20 A=11 so A is eliminated. Round 3 votes counts: C=42 D=36 E=22 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:229 C:219 E:194 B:189 A:169 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -18 -18 -12 B 14 0 -8 -22 -6 C 18 8 0 4 8 D 18 22 -4 0 22 E 12 6 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -18 -18 -12 B 14 0 -8 -22 -6 C 18 8 0 4 8 D 18 22 -4 0 22 E 12 6 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -18 -18 -12 B 14 0 -8 -22 -6 C 18 8 0 4 8 D 18 22 -4 0 22 E 12 6 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2805: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) A D B C E (8) E B C D A (6) E B C A D (6) E D C B A (5) A D C B E (5) D E C B A (4) D C E B A (4) D E A C B (3) B C E D A (3) A D E B C (3) E D B C A (2) E D A C B (2) E C B D A (2) E A D B C (2) D C B E A (2) D C B A E (2) D C A B E (2) D A C B E (2) B E A C D (2) B C E A D (2) A E D B C (2) A E B D C (2) A B C E D (2) E C D B A (1) E B A C D (1) D E C A B (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C E B (1) C B E D A (1) C B D A E (1) B E C A D (1) B C A E D (1) B C A D E (1) A E B C D (1) A D E C B (1) A D C E B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 2 6 -2 B -6 0 14 -12 -6 C -2 -14 0 -16 2 D -6 12 16 0 10 E 2 6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.333333 Sum of squares = 0.432098765418 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.666667 E: 1.000000 A B C D E A 0 6 2 6 -2 B -6 0 14 -12 -6 C -2 -14 0 -16 2 D -6 12 16 0 10 E 2 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.333333 Sum of squares = 0.432098765236 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=27 D=24 B=10 C=2 so C is eliminated. Round 2 votes counts: A=37 E=27 D=24 B=12 so B is eliminated. Round 3 votes counts: A=39 E=36 D=25 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:216 A:206 E:198 B:195 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 6 -2 B -6 0 14 -12 -6 C -2 -14 0 -16 2 D -6 12 16 0 10 E 2 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.333333 Sum of squares = 0.432098765236 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 6 -2 B -6 0 14 -12 -6 C -2 -14 0 -16 2 D -6 12 16 0 10 E 2 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.333333 Sum of squares = 0.432098765236 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 6 -2 B -6 0 14 -12 -6 C -2 -14 0 -16 2 D -6 12 16 0 10 E 2 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.111111 E: 0.333333 Sum of squares = 0.432098765236 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2806: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (6) A E D B C (6) A E B D C (6) E A B D C (5) E A B C D (5) D C B A E (5) E A D C B (4) D A E C B (4) B C A E D (4) D C A B E (3) D B C A E (3) D A B E C (3) C D E B A (3) C B E D A (3) C B D A E (3) B C D A E (3) A D E B C (3) E C B A D (2) E A C B D (2) D A B C E (2) C E D A B (2) C B E A D (2) B A C E D (2) E C D A B (1) E B A C D (1) D E A C B (1) D C A E B (1) D B A C E (1) D A E B C (1) D A C E B (1) C E D B A (1) C E B D A (1) C E B A D (1) B E A C D (1) B A E D C (1) B A E C D (1) B A D C E (1) A E B C D (1) A D B E C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 12 16 18 B -10 0 20 8 -4 C -12 -20 0 -8 -4 D -16 -8 8 0 -20 E -18 4 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 16 18 B -10 0 20 8 -4 C -12 -20 0 -8 -4 D -16 -8 8 0 -20 E -18 4 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=20 A=20 B=19 C=16 so C is eliminated. Round 2 votes counts: D=28 B=27 E=25 A=20 so A is eliminated. Round 3 votes counts: E=38 D=32 B=30 so B is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:228 B:207 E:205 D:182 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 16 18 B -10 0 20 8 -4 C -12 -20 0 -8 -4 D -16 -8 8 0 -20 E -18 4 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 16 18 B -10 0 20 8 -4 C -12 -20 0 -8 -4 D -16 -8 8 0 -20 E -18 4 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 16 18 B -10 0 20 8 -4 C -12 -20 0 -8 -4 D -16 -8 8 0 -20 E -18 4 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2807: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) D B E C A (8) A C E B D (7) A D B C E (6) C E B D A (5) C B E D A (5) A C E D B (5) B D E C A (4) A E C B D (4) D B A E C (3) C E B A D (3) C E A B D (3) C A E B D (3) B E D C A (3) B D C E A (3) A E C D B (3) A D B E C (3) E B C D A (2) D B A C E (2) D A B E C (2) A E D C B (2) A E D B C (2) A D E B C (2) E D B A C (1) E C B A D (1) E C A B D (1) E A D B C (1) D B C E A (1) D B C A E (1) C B D E A (1) C B A E D (1) B C D E A (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 8 -2 -4 B 8 0 8 -6 10 C -8 -8 0 -8 -2 D 2 6 8 0 -4 E 4 -10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.500000 E: 0.300000 Sum of squares = 0.380000000018 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.700000 E: 1.000000 A B C D E A 0 -8 8 -2 -4 B 8 0 8 -6 10 C -8 -8 0 -8 -2 D 2 6 8 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.500000 E: 0.300000 Sum of squares = 0.37999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=26 C=21 B=11 E=6 so E is eliminated. Round 2 votes counts: A=37 D=27 C=23 B=13 so B is eliminated. Round 3 votes counts: D=37 A=37 C=26 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:206 E:200 A:197 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -2 -4 B 8 0 8 -6 10 C -8 -8 0 -8 -2 D 2 6 8 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.500000 E: 0.300000 Sum of squares = 0.37999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -2 -4 B 8 0 8 -6 10 C -8 -8 0 -8 -2 D 2 6 8 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.500000 E: 0.300000 Sum of squares = 0.37999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.700000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -2 -4 B 8 0 8 -6 10 C -8 -8 0 -8 -2 D 2 6 8 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.500000 E: 0.300000 Sum of squares = 0.37999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2808: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) D C A E B (4) D A C B E (4) C E B A D (4) C D A E B (4) B E A D C (4) E C A B D (3) E B A D C (3) E B A C D (3) D C B A E (3) D A B C E (3) C B D E A (3) B E C A D (3) A E D B C (3) A E B D C (3) E A C B D (2) D C A B E (2) D B A E C (2) D B A C E (2) C E A B D (2) C D B E A (2) C B E D A (2) B E A C D (2) B C E D A (2) B A E D C (2) A E C D B (2) A E B C D (2) A D B E C (2) E C B A D (1) E B C A D (1) E A B D C (1) D B C E A (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D B A (1) C E A D B (1) C D E A B (1) C D A B E (1) C A E D B (1) B C D E A (1) B A D E C (1) A D E B C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 12 12 -2 14 B -12 0 8 -8 6 C -12 -8 0 -12 -6 D 2 8 12 0 4 E -14 -6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 -2 14 B -12 0 8 -8 6 C -12 -8 0 -12 -6 D 2 8 12 0 4 E -14 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=22 B=15 A=15 E=14 so E is eliminated. Round 2 votes counts: D=34 C=26 B=22 A=18 so A is eliminated. Round 3 votes counts: D=41 C=30 B=29 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:213 B:197 E:191 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 12 -2 14 B -12 0 8 -8 6 C -12 -8 0 -12 -6 D 2 8 12 0 4 E -14 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 -2 14 B -12 0 8 -8 6 C -12 -8 0 -12 -6 D 2 8 12 0 4 E -14 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 -2 14 B -12 0 8 -8 6 C -12 -8 0 -12 -6 D 2 8 12 0 4 E -14 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2809: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) D E B C A (7) B C D A E (7) A C B E D (7) E D A B C (6) D B C A E (6) B C D E A (6) A E C B D (6) C B A D E (5) E D B C A (4) D E A B C (4) C B A E D (4) A C E B D (4) E A D C B (3) D B E C A (3) D B C E A (3) D A B C E (3) A C B D E (3) C B E D A (2) C B D A E (2) A D B C E (2) E C B D A (1) E A C D B (1) C B D E A (1) A E D C B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -2 -10 4 B 2 0 4 10 8 C 2 -4 0 12 10 D 10 -10 -12 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -10 4 B 2 0 4 10 8 C 2 -4 0 12 10 D 10 -10 -12 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 E=22 C=14 B=13 so B is eliminated. Round 2 votes counts: C=27 D=26 A=25 E=22 so E is eliminated. Round 3 votes counts: D=36 A=36 C=28 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:212 C:210 D:198 A:195 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 4 B 2 0 4 10 8 C 2 -4 0 12 10 D 10 -10 -12 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 4 B 2 0 4 10 8 C 2 -4 0 12 10 D 10 -10 -12 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 4 B 2 0 4 10 8 C 2 -4 0 12 10 D 10 -10 -12 0 8 E -4 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2810: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) A C D E B (9) B E D C A (7) B D E C A (6) D C A B E (5) E B A C D (4) E A C B D (4) D B C E A (4) D B C A E (4) D A C B E (4) A E C B D (4) A E B C D (4) B E C D A (3) A E C D B (3) A D B C E (3) E C A D B (2) E B C D A (2) E B C A D (2) E A C D B (2) D C A E B (2) C A E D B (2) B D C E A (2) E C D B A (1) E C B A D (1) D C E B A (1) D C B E A (1) D C B A E (1) D B A C E (1) C D E A B (1) C D A E B (1) B E D A C (1) B A E D C (1) B A D E C (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 14 4 6 12 B -14 0 -8 -14 -10 C -4 8 0 10 4 D -6 14 -10 0 -4 E -12 10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 6 12 B -14 0 -8 -14 -10 C -4 8 0 10 4 D -6 14 -10 0 -4 E -12 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=23 B=21 E=18 C=4 so C is eliminated. Round 2 votes counts: A=36 D=25 B=21 E=18 so E is eliminated. Round 3 votes counts: A=44 B=30 D=26 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:209 E:199 D:197 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 6 12 B -14 0 -8 -14 -10 C -4 8 0 10 4 D -6 14 -10 0 -4 E -12 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 6 12 B -14 0 -8 -14 -10 C -4 8 0 10 4 D -6 14 -10 0 -4 E -12 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 6 12 B -14 0 -8 -14 -10 C -4 8 0 10 4 D -6 14 -10 0 -4 E -12 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2811: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E C B A D (6) E B C A D (6) D C A B E (6) C D E A B (6) D A B E C (5) C E D B A (5) E C B D A (4) E B A D C (3) D C E B A (3) D B A E C (3) C E B D A (3) C E B A D (3) C D A E B (3) B E A D C (3) A B D E C (3) E B A C D (2) D E C B A (2) D C A E B (2) C D E B A (2) C D A B E (2) C A D B E (2) B E A C D (2) B A E D C (2) A D B E C (2) A B E C D (2) E B D C A (1) E B C D A (1) D C E A B (1) D A B C E (1) C A B D E (1) B C A E D (1) B A E C D (1) B A D E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -20 -18 -6 B 8 0 -20 -10 -6 C 20 20 0 4 2 D 18 10 -4 0 10 E 6 6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -20 -18 -6 B 8 0 -20 -10 -6 C 20 20 0 4 2 D 18 10 -4 0 10 E 6 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=27 E=23 B=10 A=8 so A is eliminated. Round 2 votes counts: D=35 C=27 E=23 B=15 so B is eliminated. Round 3 votes counts: D=39 E=33 C=28 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:223 D:217 E:200 B:186 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -20 -18 -6 B 8 0 -20 -10 -6 C 20 20 0 4 2 D 18 10 -4 0 10 E 6 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 -18 -6 B 8 0 -20 -10 -6 C 20 20 0 4 2 D 18 10 -4 0 10 E 6 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 -18 -6 B 8 0 -20 -10 -6 C 20 20 0 4 2 D 18 10 -4 0 10 E 6 6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2812: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) A D E B C (10) C B A E D (8) E D A B C (6) C B A D E (6) C D E B A (4) C B E A D (4) A B C D E (4) E D B C A (3) E B D A C (3) D E A C B (3) D A E B C (3) B C E A D (3) A B E D C (3) E D B A C (2) E B D C A (2) D A E C B (2) C B E D A (2) B C A E D (2) B A E C D (2) A B D E C (2) E C B D A (1) E B C D A (1) E B A D C (1) E A B D C (1) D C A E B (1) C E B D A (1) C D E A B (1) C A D B E (1) C A B D E (1) B A C E D (1) A D C E B (1) A D B C E (1) A C B D E (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 18 10 4 B -8 0 26 4 -12 C -18 -26 0 -10 -12 D -10 -4 10 0 4 E -4 12 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 10 4 B -8 0 26 4 -12 C -18 -26 0 -10 -12 D -10 -4 10 0 4 E -4 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=25 E=20 D=19 B=8 so B is eliminated. Round 2 votes counts: C=33 A=28 E=20 D=19 so D is eliminated. Round 3 votes counts: C=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:208 B:205 D:200 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 10 4 B -8 0 26 4 -12 C -18 -26 0 -10 -12 D -10 -4 10 0 4 E -4 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 10 4 B -8 0 26 4 -12 C -18 -26 0 -10 -12 D -10 -4 10 0 4 E -4 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 10 4 B -8 0 26 4 -12 C -18 -26 0 -10 -12 D -10 -4 10 0 4 E -4 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2813: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) A C D E B (6) E B D A C (5) D C B A E (5) B E D C A (5) B E A C D (5) A E B C D (5) D C A B E (4) D B E C A (4) E B A D C (3) E B A C D (3) E A D B C (3) C D A B E (3) B E D A C (3) A C B E D (3) E A B D C (2) E A B C D (2) D E B C A (2) D C A E B (2) C D B A E (2) C D A E B (2) B D E C A (2) A E C B D (2) A D C E B (2) A C E B D (2) E D B A C (1) E D A C B (1) E D A B C (1) E B D C A (1) D E A C B (1) D C E A B (1) D C B E A (1) D B C E A (1) D A E C B (1) C A D E B (1) C A B D E (1) B C A E D (1) A E D C B (1) A E C D B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 12 10 4 6 B -12 0 0 -8 -4 C -10 0 0 -4 -10 D -4 8 4 0 -4 E -6 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 4 6 B -12 0 0 -8 -4 C -10 0 0 -4 -10 D -4 8 4 0 -4 E -6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=22 D=22 C=16 B=16 so C is eliminated. Round 2 votes counts: A=33 D=29 E=22 B=16 so B is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:206 D:202 B:188 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 4 6 B -12 0 0 -8 -4 C -10 0 0 -4 -10 D -4 8 4 0 -4 E -6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 4 6 B -12 0 0 -8 -4 C -10 0 0 -4 -10 D -4 8 4 0 -4 E -6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 4 6 B -12 0 0 -8 -4 C -10 0 0 -4 -10 D -4 8 4 0 -4 E -6 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2814: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) D E C A B (8) C D E B A (7) B A E C D (7) D C E B A (6) B A C D E (6) A E B D C (5) A B E C D (5) D C E A B (4) C D E A B (4) C D B E A (4) B C D E A (4) B C A D E (4) B A E D C (4) B A C E D (4) E D A C B (3) B C D A E (3) E D C A B (2) E A D C B (2) A E D C B (2) D C B E A (1) C E D A B (1) A E D B C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 0 4 6 8 B 0 0 10 10 6 C -4 -10 0 2 -4 D -6 -10 -2 0 2 E -8 -6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.647448 B: 0.352552 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.543481925947 Cumulative probabilities = A: 0.647448 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 6 8 B 0 0 10 10 6 C -4 -10 0 2 -4 D -6 -10 -2 0 2 E -8 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=26 D=19 C=16 E=7 so E is eliminated. Round 2 votes counts: B=32 A=28 D=24 C=16 so C is eliminated. Round 3 votes counts: D=40 B=32 A=28 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:209 E:194 C:192 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 6 8 B 0 0 10 10 6 C -4 -10 0 2 -4 D -6 -10 -2 0 2 E -8 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 8 B 0 0 10 10 6 C -4 -10 0 2 -4 D -6 -10 -2 0 2 E -8 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 8 B 0 0 10 10 6 C -4 -10 0 2 -4 D -6 -10 -2 0 2 E -8 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2815: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) B E D A C (8) B E C A D (8) D A C E B (7) D A E B C (5) C E B A D (4) C A D B E (4) E D B A C (3) E C B D A (3) D A B E C (3) C A D E B (3) B C E A D (3) A D C B E (3) E B D C A (2) E B D A C (2) D C A E B (2) D A B C E (2) C B E A D (2) C A B D E (2) B E A C D (2) B A E D C (2) B A D E C (2) A C D B E (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D A C B (1) D E C A B (1) D E A B C (1) D A E C B (1) C E D A B (1) C D A E B (1) C B A E D (1) C A B E D (1) B E C D A (1) B D E A C (1) B C A E D (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 0 -10 -6 B 8 0 14 14 6 C 0 -14 0 2 -12 D 10 -14 -2 0 -10 E 6 -6 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -10 -6 B 8 0 14 14 6 C 0 -14 0 2 -12 D 10 -14 -2 0 -10 E 6 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=22 E=21 C=19 A=10 so A is eliminated. Round 2 votes counts: B=30 D=26 C=23 E=21 so E is eliminated. Round 3 votes counts: B=42 D=32 C=26 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:211 D:192 A:188 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -10 -6 B 8 0 14 14 6 C 0 -14 0 2 -12 D 10 -14 -2 0 -10 E 6 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -10 -6 B 8 0 14 14 6 C 0 -14 0 2 -12 D 10 -14 -2 0 -10 E 6 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -10 -6 B 8 0 14 14 6 C 0 -14 0 2 -12 D 10 -14 -2 0 -10 E 6 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2816: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D C B A (9) A C B D E (8) E D B C A (6) E C D A B (5) B D E C A (5) A C E B D (5) E D B A C (4) B A D C E (4) A B D E C (4) E C A D B (3) D E B C A (3) B D E A C (3) B D A C E (3) E C D B A (2) D B E C A (2) C E D A B (2) C A E D B (2) B D C E A (2) A E C D B (2) A C E D B (2) A C B E D (2) E D C A B (1) E A D C B (1) E A D B C (1) D E B A C (1) D B E A C (1) C E D B A (1) C E A D B (1) C B D E A (1) B A D E C (1) A E D B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 10 -2 -8 B -4 0 6 0 -4 C -10 -6 0 -8 -12 D 2 0 8 0 -2 E 8 4 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 -2 -8 B -4 0 6 0 -4 C -10 -6 0 -8 -12 D 2 0 8 0 -2 E 8 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=32 B=18 D=7 C=7 so D is eliminated. Round 2 votes counts: E=36 A=36 B=21 C=7 so C is eliminated. Round 3 votes counts: E=40 A=38 B=22 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:204 A:202 B:199 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 -2 -8 B -4 0 6 0 -4 C -10 -6 0 -8 -12 D 2 0 8 0 -2 E 8 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -2 -8 B -4 0 6 0 -4 C -10 -6 0 -8 -12 D 2 0 8 0 -2 E 8 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -2 -8 B -4 0 6 0 -4 C -10 -6 0 -8 -12 D 2 0 8 0 -2 E 8 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2817: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) C A B E D (6) E A D C B (5) C B A E D (5) B D C E A (4) B C E A D (4) B C D A E (4) A D E C B (4) A D C E B (4) E D B A C (3) E D A B C (3) E B D C A (3) D B E C A (3) D B C A E (3) B D E C A (3) B C A D E (3) A E C D B (3) A E C B D (3) A C E B D (3) E A C D B (2) D E B A C (2) D B E A C (2) D A C B E (2) D A B C E (2) C B A D E (2) B D C A E (2) B C A E D (2) E D A C B (1) E B C D A (1) E B C A D (1) E A D B C (1) D E A B C (1) D B A E C (1) D A E C B (1) C A B D E (1) A C E D B (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 6 18 10 B -4 0 -4 8 2 C -6 4 0 0 2 D -18 -8 0 0 -8 E -10 -2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 18 10 B -4 0 -4 8 2 C -6 4 0 0 2 D -18 -8 0 0 -8 E -10 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=22 A=21 D=17 C=14 so C is eliminated. Round 2 votes counts: B=29 A=28 E=26 D=17 so D is eliminated. Round 3 votes counts: B=38 A=33 E=29 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:201 C:200 E:197 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 18 10 B -4 0 -4 8 2 C -6 4 0 0 2 D -18 -8 0 0 -8 E -10 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 18 10 B -4 0 -4 8 2 C -6 4 0 0 2 D -18 -8 0 0 -8 E -10 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 18 10 B -4 0 -4 8 2 C -6 4 0 0 2 D -18 -8 0 0 -8 E -10 -2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2818: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) C B D E A (7) D E C A B (6) D E A C B (6) B C A E D (6) E D A B C (5) C B A D E (5) D E C B A (4) B C D E A (4) C D A E B (3) B C A D E (3) E D B A C (2) E A D B C (2) D E B C A (2) D E A B C (2) D C E B A (2) C B D A E (2) C B A E D (2) C A B E D (2) C A B D E (2) A E D B C (2) A E B D C (2) A B C E D (2) E B D A C (1) D A E C B (1) C D E B A (1) C D B E A (1) C D B A E (1) B E D C A (1) B E A D C (1) B D E C A (1) B D C E A (1) B C E D A (1) B C E A D (1) B A E C D (1) B A C E D (1) A C E D B (1) A C E B D (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -16 -8 -2 B 2 0 -14 0 -4 C 16 14 0 -2 0 D 8 0 2 0 8 E 2 4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.090780 C: 0.000000 D: 0.909220 E: 0.000000 Sum of squares = 0.834922228052 Cumulative probabilities = A: 0.000000 B: 0.090780 C: 0.090780 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -8 -2 B 2 0 -14 0 -4 C 16 14 0 -2 0 D 8 0 2 0 8 E 2 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250021206 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=23 B=21 A=20 E=10 so E is eliminated. Round 2 votes counts: D=30 C=26 B=22 A=22 so B is eliminated. Round 3 votes counts: C=41 D=34 A=25 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:209 E:199 B:192 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -16 -8 -2 B 2 0 -14 0 -4 C 16 14 0 -2 0 D 8 0 2 0 8 E 2 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250021206 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -8 -2 B 2 0 -14 0 -4 C 16 14 0 -2 0 D 8 0 2 0 8 E 2 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250021206 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -8 -2 B 2 0 -14 0 -4 C 16 14 0 -2 0 D 8 0 2 0 8 E 2 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.875000 E: 0.000000 Sum of squares = 0.781250021206 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2819: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) E D C A B (7) A C B D E (7) A B C E D (6) E D C B A (5) E D B C A (5) D E B C A (5) A B C D E (5) C A B D E (4) A C B E D (4) E D A B C (3) C A D B E (3) A B E D C (3) E D B A C (2) E C D A B (2) C D E A B (2) C B D A E (2) B D C E A (2) A E C D B (2) A E B D C (2) A B E C D (2) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) D B E C A (1) C E D A B (1) C D B E A (1) C B A D E (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E C A (1) B D E A C (1) B D C A E (1) B A D C E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 20 10 16 14 B -20 0 8 12 12 C -10 -8 0 14 4 D -16 -12 -14 0 -4 E -14 -12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 16 14 B -20 0 8 12 12 C -10 -8 0 14 4 D -16 -12 -14 0 -4 E -14 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=28 B=17 C=16 D=6 so D is eliminated. Round 2 votes counts: E=33 A=33 B=18 C=16 so C is eliminated. Round 3 votes counts: A=42 E=36 B=22 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 B:206 C:200 E:187 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 16 14 B -20 0 8 12 12 C -10 -8 0 14 4 D -16 -12 -14 0 -4 E -14 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 16 14 B -20 0 8 12 12 C -10 -8 0 14 4 D -16 -12 -14 0 -4 E -14 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 16 14 B -20 0 8 12 12 C -10 -8 0 14 4 D -16 -12 -14 0 -4 E -14 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2820: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E C B A D (8) A D E C B (8) A D C B E (8) E B C D A (5) E C A D B (3) E B D C A (3) E A C D B (3) D A E B C (3) D A B E C (3) C E B A D (3) B E C D A (3) E C A B D (2) D B A C E (2) C E A B D (2) C B A D E (2) C A B E D (2) B D C A E (2) A E D C B (2) A D C E B (2) A C E D B (2) A C D E B (2) A C D B E (2) E D A B C (1) E C B D A (1) E B D A C (1) E A D C B (1) D E B A C (1) D E A B C (1) D B E A C (1) C E B D A (1) C B E D A (1) C B E A D (1) C B A E D (1) B E D C A (1) B E D A C (1) B D E C A (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) A D E B C (1) Total count = 100 A B C D E A 0 14 10 10 6 B -14 0 -14 -10 -12 C -10 14 0 -6 -8 D -10 10 6 0 2 E -6 12 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 10 6 B -14 0 -14 -10 -12 C -10 14 0 -6 -8 D -10 10 6 0 2 E -6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 D=20 C=13 B=12 so B is eliminated. Round 2 votes counts: E=33 A=27 D=24 C=16 so C is eliminated. Round 3 votes counts: E=42 A=32 D=26 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:206 D:204 C:195 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 10 6 B -14 0 -14 -10 -12 C -10 14 0 -6 -8 D -10 10 6 0 2 E -6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 10 6 B -14 0 -14 -10 -12 C -10 14 0 -6 -8 D -10 10 6 0 2 E -6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 10 6 B -14 0 -14 -10 -12 C -10 14 0 -6 -8 D -10 10 6 0 2 E -6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2821: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) C E B A D (5) A E B D C (5) A E B C D (5) A B E D C (5) E A D B C (4) D E A B C (4) E C A B D (3) E A C B D (3) D B A E C (3) D A E B C (3) D A B E C (3) C D E B A (3) C B D A E (3) C B A E D (3) B A D E C (3) B A D C E (3) E C D A B (2) E A B C D (2) D C E B A (2) D C B E A (2) D B C A E (2) D B A C E (2) C E D B A (2) C E D A B (2) C E B D A (2) C E A D B (2) C B D E A (2) C B A D E (2) E D C A B (1) E D A B C (1) E A D C B (1) E A C D B (1) B D C A E (1) B D A E C (1) B C A D E (1) B A E C D (1) A E C B D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 6 18 -2 B -14 0 2 18 -22 C -6 -2 0 4 -8 D -18 -18 -4 0 -16 E 2 22 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 6 18 -2 B -14 0 2 18 -22 C -6 -2 0 4 -8 D -18 -18 -4 0 -16 E 2 22 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=21 E=18 A=18 B=10 so B is eliminated. Round 2 votes counts: C=34 A=25 D=23 E=18 so E is eliminated. Round 3 votes counts: C=39 A=36 D=25 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:224 A:218 C:194 B:192 D:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 6 18 -2 B -14 0 2 18 -22 C -6 -2 0 4 -8 D -18 -18 -4 0 -16 E 2 22 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 18 -2 B -14 0 2 18 -22 C -6 -2 0 4 -8 D -18 -18 -4 0 -16 E 2 22 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 18 -2 B -14 0 2 18 -22 C -6 -2 0 4 -8 D -18 -18 -4 0 -16 E 2 22 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999975855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2822: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) B D A E C (8) D B A E C (7) A D B C E (7) A C E D B (7) B D E C A (6) E B C D A (5) E C B A D (4) C E A B D (4) B E D C A (4) A C D E B (3) A B D C E (3) E C D A B (2) E C B D A (2) E C A B D (2) D E B C A (2) C A E D B (2) C A E B D (2) B D E A C (2) B D A C E (2) A D C E B (2) E D C B A (1) E C D B A (1) D B E C A (1) D A B E C (1) D A B C E (1) C E B A D (1) C B A E D (1) C A B E D (1) B E C D A (1) B E C A D (1) B C E A D (1) B A D C E (1) A D E C B (1) A D C B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -4 8 4 B 2 0 6 6 2 C 4 -6 0 0 -2 D -8 -6 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 8 4 B 2 0 6 6 2 C 4 -6 0 0 -2 D -8 -6 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 C=19 E=17 D=12 so D is eliminated. Round 2 votes counts: B=34 A=28 E=19 C=19 so E is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:203 E:199 C:198 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 8 4 B 2 0 6 6 2 C 4 -6 0 0 -2 D -8 -6 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 4 B 2 0 6 6 2 C 4 -6 0 0 -2 D -8 -6 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 4 B 2 0 6 6 2 C 4 -6 0 0 -2 D -8 -6 0 0 -2 E -4 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2823: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) D B C A E (6) E C A D B (5) B A D E C (5) C E A D B (4) E C D B A (3) E A C B D (3) D C B E A (3) C D E A B (3) B A E D C (3) B A D C E (3) A B D E C (3) E C A B D (2) E A C D B (2) E A B C D (2) D C B A E (2) D C A B E (2) D B A C E (2) D A B C E (2) C D A E B (2) C A E D B (2) C A D E B (2) B D C A E (2) A E C B D (2) A E B D C (2) A D B C E (2) A B E D C (2) A B D C E (2) E C D A B (1) E C B D A (1) E C B A D (1) E B C D A (1) E B A D C (1) E B A C D (1) D B C E A (1) D A C B E (1) C E D B A (1) C D E B A (1) C D B A E (1) B E D A C (1) B D C E A (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 6 2 24 B 4 0 10 2 12 C -6 -10 0 -16 14 D -2 -2 16 0 16 E -24 -12 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 24 B 4 0 10 2 12 C -6 -10 0 -16 14 D -2 -2 16 0 16 E -24 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=23 D=19 C=16 A=15 so A is eliminated. Round 2 votes counts: B=34 E=29 D=21 C=16 so C is eliminated. Round 3 votes counts: E=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:214 D:214 C:191 E:167 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 24 B 4 0 10 2 12 C -6 -10 0 -16 14 D -2 -2 16 0 16 E -24 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 24 B 4 0 10 2 12 C -6 -10 0 -16 14 D -2 -2 16 0 16 E -24 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 24 B 4 0 10 2 12 C -6 -10 0 -16 14 D -2 -2 16 0 16 E -24 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2824: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) C B E A D (12) D E A B C (8) B C D A E (7) A E D B C (7) E D A C B (5) D B C A E (4) A E C B D (4) E C B A D (3) E A C B D (3) D A E B C (3) C B A E D (3) B D C A E (3) B C A D E (3) A D E B C (3) A B C D E (3) B C D E A (2) A E D C B (2) A D B C E (2) E D A B C (1) E C B D A (1) E C A B D (1) E A C D B (1) D B C E A (1) D B A C E (1) D A B C E (1) C B E D A (1) C B D E A (1) B A D C E (1) A E C D B (1) Total count = 100 A B C D E A 0 14 16 22 -4 B -14 0 0 -4 -10 C -16 0 0 -8 -10 D -22 4 8 0 -14 E 4 10 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 16 22 -4 B -14 0 0 -4 -10 C -16 0 0 -8 -10 D -22 4 8 0 -14 E 4 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=22 D=18 C=17 B=16 so B is eliminated. Round 2 votes counts: C=29 E=27 A=23 D=21 so D is eliminated. Round 3 votes counts: C=37 E=35 A=28 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:224 E:219 D:188 B:186 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 16 22 -4 B -14 0 0 -4 -10 C -16 0 0 -8 -10 D -22 4 8 0 -14 E 4 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 22 -4 B -14 0 0 -4 -10 C -16 0 0 -8 -10 D -22 4 8 0 -14 E 4 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 22 -4 B -14 0 0 -4 -10 C -16 0 0 -8 -10 D -22 4 8 0 -14 E 4 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2825: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) D E A B C (7) D C E A B (7) A B E C D (6) D B A E C (5) B A E C D (5) B A D E C (5) B A C E D (5) D C E B A (4) C B A E D (4) E A B D C (3) E A B C D (3) E A C B D (2) D B A C E (2) D A B E C (2) C E D A B (2) C D E B A (2) C D E A B (2) C D B E A (2) B A E D C (2) B A C D E (2) A E B C D (2) A B E D C (2) E D A C B (1) E D A B C (1) E C D A B (1) E C A D B (1) D E B A C (1) D C B A E (1) C E D B A (1) C E A B D (1) C D B A E (1) C B A D E (1) C A E B D (1) B D A E C (1) B A D C E (1) A E B D C (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 10 22 -2 2 B -10 0 14 -4 -2 C -22 -14 0 -10 -20 D 2 4 10 0 10 E -2 2 20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 22 -2 2 B -10 0 14 -4 -2 C -22 -14 0 -10 -20 D 2 4 10 0 10 E -2 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=21 C=17 A=13 E=12 so E is eliminated. Round 2 votes counts: D=39 B=21 A=21 C=19 so C is eliminated. Round 3 votes counts: D=50 B=26 A=24 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:213 E:205 B:199 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 22 -2 2 B -10 0 14 -4 -2 C -22 -14 0 -10 -20 D 2 4 10 0 10 E -2 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 22 -2 2 B -10 0 14 -4 -2 C -22 -14 0 -10 -20 D 2 4 10 0 10 E -2 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 22 -2 2 B -10 0 14 -4 -2 C -22 -14 0 -10 -20 D 2 4 10 0 10 E -2 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2826: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) C B D A E (9) E A D B C (7) E A C B D (6) E C A B D (5) E D A B C (4) E A C D B (4) D B A C E (4) C E B A D (4) B C D A E (4) E C B A D (3) C B E A D (3) C B A D E (3) B D C A E (3) A E D B C (3) E D B C A (2) E C B D A (2) E A D C B (2) D E A B C (2) D A B C E (2) C B A E D (2) A C B D E (2) E D C B A (1) E C D B A (1) D E B A C (1) D B E C A (1) D A B E C (1) C B E D A (1) C A E B D (1) B A C D E (1) A E C B D (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -14 0 2 B 14 0 -2 4 2 C 14 2 0 8 8 D 0 -4 -8 0 -6 E -2 -2 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 0 2 B 14 0 -2 4 2 C 14 2 0 8 8 D 0 -4 -8 0 -6 E -2 -2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=23 C=23 A=9 B=8 so B is eliminated. Round 2 votes counts: E=37 C=27 D=26 A=10 so A is eliminated. Round 3 votes counts: E=41 C=32 D=27 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:209 E:197 D:191 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 0 2 B 14 0 -2 4 2 C 14 2 0 8 8 D 0 -4 -8 0 -6 E -2 -2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 0 2 B 14 0 -2 4 2 C 14 2 0 8 8 D 0 -4 -8 0 -6 E -2 -2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 0 2 B 14 0 -2 4 2 C 14 2 0 8 8 D 0 -4 -8 0 -6 E -2 -2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2827: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) C A B E D (11) E D B A C (9) D E B A C (7) C E D A B (6) C A B D E (6) A B C D E (6) B A D E C (5) C A E B D (4) A C B D E (4) E D B C A (3) E C D A B (3) D B A E C (3) D A B E C (3) A B D C E (3) C E B A D (2) C E A B D (2) B D A E C (2) E D C A B (1) E C D B A (1) D A E B C (1) D A C B E (1) C E A D B (1) C B A E D (1) B E D A C (1) B D E A C (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 4 -4 -6 4 B -4 0 -8 0 -2 C 4 8 0 -4 -2 D 6 0 4 0 -12 E -4 2 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000021 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 4 -4 -6 4 B -4 0 -8 0 -2 C 4 8 0 -4 -2 D 6 0 4 0 -12 E -4 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000049 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=28 D=15 A=13 B=11 so B is eliminated. Round 2 votes counts: C=33 E=29 A=20 D=18 so D is eliminated. Round 3 votes counts: E=37 C=33 A=30 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:206 C:203 A:199 D:199 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -6 4 B -4 0 -8 0 -2 C 4 8 0 -4 -2 D 6 0 4 0 -12 E -4 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000049 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 4 B -4 0 -8 0 -2 C 4 8 0 -4 -2 D 6 0 4 0 -12 E -4 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000049 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 4 B -4 0 -8 0 -2 C 4 8 0 -4 -2 D 6 0 4 0 -12 E -4 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000049 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2828: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) B E A C D (7) B A E D C (7) B E A D C (6) C D E A B (5) C D A E B (5) E B A C D (4) D A C B E (3) B D E C A (3) B D A E C (3) A D C B E (3) A D B C E (3) A C E D B (3) E C A B D (2) E B C D A (2) E B C A D (2) D C A B E (2) D B C A E (2) C D E B A (2) B D E A C (2) B A D E C (2) A E C B D (2) E C D B A (1) E C B D A (1) E C B A D (1) E C A D B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A C E (1) D A B C E (1) C E A D B (1) C A E D B (1) B D A C E (1) B A E C D (1) B A D C E (1) A E B C D (1) A D C E B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 14 8 10 B 4 0 2 2 4 C -14 -2 0 -6 0 D -8 -2 6 0 8 E -10 -4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999277 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 8 10 B 4 0 2 2 4 C -14 -2 0 -6 0 D -8 -2 6 0 8 E -10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=21 E=16 A=16 C=14 so C is eliminated. Round 2 votes counts: D=33 B=33 E=17 A=17 so E is eliminated. Round 3 votes counts: B=43 D=34 A=23 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:206 D:202 C:189 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 8 10 B 4 0 2 2 4 C -14 -2 0 -6 0 D -8 -2 6 0 8 E -10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 8 10 B 4 0 2 2 4 C -14 -2 0 -6 0 D -8 -2 6 0 8 E -10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 8 10 B 4 0 2 2 4 C -14 -2 0 -6 0 D -8 -2 6 0 8 E -10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2829: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) A B C D E (7) E D C A B (6) E A D C B (6) B C A D E (6) B C D A E (4) B A C D E (4) A E D B C (4) A E B D C (4) A B E C D (4) E D C B A (3) E B C A D (3) D C E B A (3) A B D C E (3) A B C E D (3) E D A C B (2) E A B C D (2) D E C B A (2) D E A C B (2) D C B E A (2) B C A E D (2) A E D C B (2) A D B C E (2) E B C D A (1) E A C B D (1) D E C A B (1) D C E A B (1) D C B A E (1) D B C A E (1) D A C B E (1) C D B E A (1) C B D E A (1) B E A C D (1) B C E D A (1) B C E A D (1) B C D E A (1) B A E C D (1) B A C E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 20 18 32 18 B -20 0 30 20 0 C -18 -30 0 4 -8 D -32 -20 -4 0 -12 E -18 0 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 32 18 B -20 0 30 20 0 C -18 -30 0 4 -8 D -32 -20 -4 0 -12 E -18 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=24 B=22 D=14 C=2 so C is eliminated. Round 2 votes counts: A=38 E=24 B=23 D=15 so D is eliminated. Round 3 votes counts: A=39 E=33 B=28 so B is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:244 B:215 E:201 C:174 D:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 32 18 B -20 0 30 20 0 C -18 -30 0 4 -8 D -32 -20 -4 0 -12 E -18 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 32 18 B -20 0 30 20 0 C -18 -30 0 4 -8 D -32 -20 -4 0 -12 E -18 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 32 18 B -20 0 30 20 0 C -18 -30 0 4 -8 D -32 -20 -4 0 -12 E -18 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2830: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) A D B C E (11) D A E B C (6) E D B C A (5) D A E C B (5) A D E C B (5) A C D B E (5) D E A B C (4) E B C D A (3) C B E A D (3) C B A E D (3) B C E D A (3) B C E A D (3) B C A D E (3) A D C B E (3) A D B E C (3) E D C B A (2) C E B D A (2) C B E D A (2) C B A D E (2) B C A E D (2) B A C D E (2) A D C E B (2) A B D C E (2) E D A B C (1) D E A C B (1) C E B A D (1) C E A D B (1) B E D A C (1) B D E A C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 6 12 B -2 0 2 -8 2 C -8 -2 0 -4 4 D -6 8 4 0 14 E -12 -2 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 6 12 B -2 0 2 -8 2 C -8 -2 0 -4 4 D -6 8 4 0 14 E -12 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=22 D=16 B=15 C=14 so C is eliminated. Round 2 votes counts: A=33 E=26 B=25 D=16 so D is eliminated. Round 3 votes counts: A=44 E=31 B=25 so B is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:210 B:197 C:195 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 6 12 B -2 0 2 -8 2 C -8 -2 0 -4 4 D -6 8 4 0 14 E -12 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 6 12 B -2 0 2 -8 2 C -8 -2 0 -4 4 D -6 8 4 0 14 E -12 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 6 12 B -2 0 2 -8 2 C -8 -2 0 -4 4 D -6 8 4 0 14 E -12 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2831: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) B A C D E (10) A B E D C (8) A B E C D (8) E A B C D (6) C D B A E (6) E D C A B (5) D C E B A (4) C B D A E (4) E D A B C (3) E A B D C (3) D E C A B (3) D C B A E (3) C D E B A (3) A E B D C (3) A B D E C (3) C E D B A (2) C D B E A (2) C B A D E (2) B C A D E (2) A B C E D (2) E B A C D (1) E A D B C (1) D E A C B (1) D C E A B (1) D A B C E (1) C E B A D (1) C B E A D (1) B A D C E (1) Total count = 100 A B C D E A 0 -4 22 24 26 B 4 0 24 30 26 C -22 -24 0 20 10 D -24 -30 -20 0 -8 E -26 -26 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 22 24 26 B 4 0 24 30 26 C -22 -24 0 20 10 D -24 -30 -20 0 -8 E -26 -26 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 C=21 E=19 D=13 so D is eliminated. Round 2 votes counts: C=29 A=25 E=23 B=23 so E is eliminated. Round 3 votes counts: A=39 C=37 B=24 so B is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:242 A:234 C:192 E:173 D:159 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 22 24 26 B 4 0 24 30 26 C -22 -24 0 20 10 D -24 -30 -20 0 -8 E -26 -26 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 24 26 B 4 0 24 30 26 C -22 -24 0 20 10 D -24 -30 -20 0 -8 E -26 -26 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 24 26 B 4 0 24 30 26 C -22 -24 0 20 10 D -24 -30 -20 0 -8 E -26 -26 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2832: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) E A B D C (8) B C D E A (7) C D A E B (6) B E C A D (6) D A C E B (5) C D B A E (5) C D A B E (5) B E A D C (5) E B A D C (4) D C A E B (4) B C E D A (4) B C E A D (4) E A D B C (3) D A E C B (3) B E A C D (3) A E D C B (3) D E A C B (2) B C D A E (2) B C A E D (2) B C A D E (2) E A D C B (1) C D B E A (1) B A E D C (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 -20 -12 4 B 16 0 4 20 18 C 20 -4 0 18 18 D 12 -20 -18 0 10 E -4 -18 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -20 -12 4 B 16 0 4 20 18 C 20 -4 0 18 18 D 12 -20 -18 0 10 E -4 -18 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=29 E=16 D=14 A=5 so A is eliminated. Round 2 votes counts: B=36 C=29 E=20 D=15 so D is eliminated. Round 3 votes counts: C=38 B=36 E=26 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 C:226 D:192 A:178 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -20 -12 4 B 16 0 4 20 18 C 20 -4 0 18 18 D 12 -20 -18 0 10 E -4 -18 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -20 -12 4 B 16 0 4 20 18 C 20 -4 0 18 18 D 12 -20 -18 0 10 E -4 -18 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -20 -12 4 B 16 0 4 20 18 C 20 -4 0 18 18 D 12 -20 -18 0 10 E -4 -18 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2833: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) C A E B D (7) A E C D B (7) C E A B D (5) C A E D B (5) B D C E A (5) E A C B D (4) D B A E C (4) E C A B D (3) E B D A C (3) D B A C E (3) B D E C A (3) B D E A C (3) A C E D B (3) E A B D C (2) D B C A E (2) C E B D A (2) C D B A E (2) C B E D A (2) C B D A E (2) C A B D E (2) B C D E A (2) A E D B C (2) E D B A C (1) E C B A D (1) E B D C A (1) E B C D A (1) E A D B C (1) E A C D B (1) D C B A E (1) D B C E A (1) D A E B C (1) D A C B E (1) C B D E A (1) C A D B E (1) B E D C A (1) B D C A E (1) B C E D A (1) A E C B D (1) A D B E C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -4 -4 -4 B 2 0 -4 8 -4 C 4 4 0 8 2 D 4 -8 -8 0 -10 E 4 4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -4 B 2 0 -4 8 -4 C 4 4 0 8 2 D 4 -8 -8 0 -10 E 4 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 E=18 B=16 A=16 so B is eliminated. Round 2 votes counts: D=33 C=32 E=19 A=16 so A is eliminated. Round 3 votes counts: C=36 D=35 E=29 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 E:208 B:201 A:193 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -4 B 2 0 -4 8 -4 C 4 4 0 8 2 D 4 -8 -8 0 -10 E 4 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -4 B 2 0 -4 8 -4 C 4 4 0 8 2 D 4 -8 -8 0 -10 E 4 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -4 B 2 0 -4 8 -4 C 4 4 0 8 2 D 4 -8 -8 0 -10 E 4 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2834: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (9) A B C E D (9) E D B C A (8) A C D B E (8) B E D A C (7) B E A D C (7) B A E D C (7) D E C B A (6) D C E B A (5) C D E A B (5) C D A E B (5) C A D E B (5) E B D C A (4) B E D C A (4) A C B E D (3) A B E C D (3) D E B C A (2) B A E C D (2) A C D E B (1) Total count = 100 A B C D E A 0 -4 12 8 4 B 4 0 6 10 18 C -12 -6 0 0 0 D -8 -10 0 0 -8 E -4 -18 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 8 4 B 4 0 6 10 18 C -12 -6 0 0 0 D -8 -10 0 0 -8 E -4 -18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=27 C=15 D=13 E=12 so E is eliminated. Round 2 votes counts: A=33 B=31 D=21 C=15 so C is eliminated. Round 3 votes counts: A=38 D=31 B=31 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:210 E:193 C:191 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 8 4 B 4 0 6 10 18 C -12 -6 0 0 0 D -8 -10 0 0 -8 E -4 -18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 8 4 B 4 0 6 10 18 C -12 -6 0 0 0 D -8 -10 0 0 -8 E -4 -18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 8 4 B 4 0 6 10 18 C -12 -6 0 0 0 D -8 -10 0 0 -8 E -4 -18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2835: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (7) D A E C B (6) B D C A E (5) D E A C B (4) C E A B D (4) C A E D B (4) B E D A C (4) B C A E D (4) A C E D B (4) E D A C B (3) E C A B D (3) E A C D B (3) D B A C E (3) D A C E B (3) B D E A C (3) B C D A E (3) A D C E B (3) E A D C B (2) D B E A C (2) C B A D E (2) C A E B D (2) C A D E B (2) B C E A D (2) B C A D E (2) E C B A D (1) E B A C D (1) E A C B D (1) D E A B C (1) D C A E B (1) D B A E C (1) D A C B E (1) D A B C E (1) C E B A D (1) C E A D B (1) C D A E B (1) C B A E D (1) C A D B E (1) B E A C D (1) B D E C A (1) B D A E C (1) B D A C E (1) B C D E A (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 8 8 B -6 0 -12 2 -6 C -2 12 0 8 8 D -8 -2 -8 0 0 E -8 6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 8 8 B -6 0 -12 2 -6 C -2 12 0 8 8 D -8 -2 -8 0 0 E -8 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=23 C=19 E=14 A=9 so A is eliminated. Round 2 votes counts: B=35 D=26 C=24 E=15 so E is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:212 E:195 D:191 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 8 8 B -6 0 -12 2 -6 C -2 12 0 8 8 D -8 -2 -8 0 0 E -8 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 8 8 B -6 0 -12 2 -6 C -2 12 0 8 8 D -8 -2 -8 0 0 E -8 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 8 8 B -6 0 -12 2 -6 C -2 12 0 8 8 D -8 -2 -8 0 0 E -8 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2836: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (6) B C D E A (5) A D E C B (5) E A B C D (4) D C B E A (4) D C A B E (4) C D B E A (4) B E C A D (4) B E A C D (4) B D C E A (4) E A C B D (3) D C B A E (3) D C A E B (3) D A C E B (3) A E D C B (3) A E D B C (3) A D E B C (3) E B A C D (2) D B C A E (2) D A B C E (2) C E A D B (2) C D E B A (2) C B E D A (2) B A E D C (2) B A D E C (2) A E B D C (2) A D B E C (2) A B D E C (2) E C A D B (1) E B C A D (1) E A C D B (1) D C E B A (1) D C E A B (1) D A E C B (1) C E B D A (1) C D E A B (1) C B D E A (1) B D C A E (1) B A E C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 6 10 4 B -8 0 8 -2 2 C -6 -8 0 -8 -8 D -10 2 8 0 12 E -4 -2 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 10 4 B -8 0 8 -2 2 C -6 -8 0 -8 -8 D -10 2 8 0 12 E -4 -2 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=24 B=23 C=13 E=12 so E is eliminated. Round 2 votes counts: A=36 B=26 D=24 C=14 so C is eliminated. Round 3 votes counts: A=39 D=31 B=30 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:206 B:200 E:195 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 10 4 B -8 0 8 -2 2 C -6 -8 0 -8 -8 D -10 2 8 0 12 E -4 -2 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 10 4 B -8 0 8 -2 2 C -6 -8 0 -8 -8 D -10 2 8 0 12 E -4 -2 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 10 4 B -8 0 8 -2 2 C -6 -8 0 -8 -8 D -10 2 8 0 12 E -4 -2 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2837: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) C A E B D (7) A B E D C (7) D C E B A (5) C E D B A (5) A C B E D (4) A B D E C (4) D E B C A (3) D B A E C (3) D A C B E (3) C E A B D (3) E D B C A (2) E C D B A (2) E B D C A (2) D C B E A (2) D C A B E (2) D B E A C (2) C A D B E (2) B E A D C (2) B D E A C (2) B A E D C (2) A D B C E (2) A B C E D (2) E D B A C (1) E C B A D (1) E B C D A (1) E B A D C (1) E B A C D (1) D E C B A (1) D E B A C (1) D C B A E (1) D B A C E (1) D A B E C (1) D A B C E (1) C E B D A (1) C E A D B (1) C D E A B (1) C D A B E (1) C A D E B (1) C A B E D (1) A E B C D (1) A D C B E (1) A D B E C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -10 -8 0 B 4 0 -14 -14 -4 C 10 14 0 -6 18 D 8 14 6 0 4 E 0 4 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -8 0 B 4 0 -14 -14 -4 C 10 14 0 -6 18 D 8 14 6 0 4 E 0 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=26 A=24 E=11 B=6 so B is eliminated. Round 2 votes counts: C=33 D=28 A=26 E=13 so E is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:216 E:191 A:189 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -8 0 B 4 0 -14 -14 -4 C 10 14 0 -6 18 D 8 14 6 0 4 E 0 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -8 0 B 4 0 -14 -14 -4 C 10 14 0 -6 18 D 8 14 6 0 4 E 0 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -8 0 B 4 0 -14 -14 -4 C 10 14 0 -6 18 D 8 14 6 0 4 E 0 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2838: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (14) C E B A D (14) B A E D C (7) D A B C E (5) D A B E C (4) C E D B A (4) E C B A D (3) E C A B D (3) E B C A D (3) C E B D A (3) C E A B D (3) B D A C E (3) B A D E C (3) E D C A B (2) E A B D C (2) D C B A E (2) D B A C E (2) C D E A B (2) C D B A E (2) C B A D E (2) E C D A B (1) E C A D B (1) E B A C D (1) E A D B C (1) E A B C D (1) D C A E B (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) B E A D C (1) B D A E C (1) B C D A E (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -34 -6 -28 B 16 0 -26 12 -22 C 34 26 0 28 28 D 6 -12 -28 0 -32 E 28 22 -28 32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -34 -6 -28 B 16 0 -26 12 -22 C 34 26 0 28 28 D 6 -12 -28 0 -32 E 28 22 -28 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=50 E=18 B=16 D=15 A=1 so A is eliminated. Round 2 votes counts: C=50 E=18 B=17 D=15 so D is eliminated. Round 3 votes counts: C=53 B=28 E=19 so E is eliminated. Round 4 votes counts: C=63 B=37 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:258 E:227 B:190 D:167 A:158 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -34 -6 -28 B 16 0 -26 12 -22 C 34 26 0 28 28 D 6 -12 -28 0 -32 E 28 22 -28 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -34 -6 -28 B 16 0 -26 12 -22 C 34 26 0 28 28 D 6 -12 -28 0 -32 E 28 22 -28 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -34 -6 -28 B 16 0 -26 12 -22 C 34 26 0 28 28 D 6 -12 -28 0 -32 E 28 22 -28 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2839: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D B A C E (9) C E A D B (6) B A D E C (6) E C B A D (5) C E D A B (5) B D A E C (5) D A B C E (4) C D E B A (4) A D B E C (4) E C A D B (3) E A B C D (3) D C B A E (3) A B D E C (3) D C B E A (2) D C A E B (2) D B C A E (2) D A B E C (2) B E C D A (2) B D A C E (2) A B E D C (2) E C B D A (1) E B C A D (1) E B A C D (1) E A C B D (1) D C A B E (1) D B A E C (1) D A C B E (1) C E D B A (1) C E B D A (1) C D E A B (1) C D B E A (1) B E D C A (1) B D C E A (1) A E D C B (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 -6 -4 0 B -2 0 2 -8 6 C 6 -2 0 -8 -6 D 4 8 8 0 10 E 0 -6 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -4 0 B -2 0 2 -8 6 C 6 -2 0 -8 -6 D 4 8 8 0 10 E 0 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 C=19 B=17 A=12 so A is eliminated. Round 2 votes counts: D=32 E=27 B=22 C=19 so C is eliminated. Round 3 votes counts: E=40 D=38 B=22 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:199 A:196 C:195 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -4 0 B -2 0 2 -8 6 C 6 -2 0 -8 -6 D 4 8 8 0 10 E 0 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 0 B -2 0 2 -8 6 C 6 -2 0 -8 -6 D 4 8 8 0 10 E 0 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 0 B -2 0 2 -8 6 C 6 -2 0 -8 -6 D 4 8 8 0 10 E 0 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2840: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) A E D B C (7) D E A C B (6) A E D C B (6) C D A E B (5) E A D B C (4) B E A D C (4) B C D E A (4) E D A B C (3) C B D E A (3) C B D A E (3) C A D E B (3) B C E D A (3) B C A E D (3) A C B E D (3) D E C A B (2) D C E A B (2) C A D B E (2) B E D A C (2) B C E A D (2) A E B D C (2) A D E C B (2) A D C E B (2) A C D E B (2) A B E D C (2) E D B A C (1) E A B D C (1) D E C B A (1) D B C E A (1) C D E B A (1) C D E A B (1) C D B E A (1) C A B D E (1) B E C D A (1) B A E D C (1) B A E C D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 18 4 20 14 B -18 0 -12 -6 -2 C -4 12 0 2 6 D -20 6 -2 0 4 E -14 2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 20 14 B -18 0 -12 -6 -2 C -4 12 0 2 6 D -20 6 -2 0 4 E -14 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997055 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=29 A=29 B=21 D=12 E=9 so E is eliminated. Round 2 votes counts: A=34 C=29 B=21 D=16 so D is eliminated. Round 3 votes counts: A=43 C=34 B=23 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:208 D:194 E:189 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 20 14 B -18 0 -12 -6 -2 C -4 12 0 2 6 D -20 6 -2 0 4 E -14 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997055 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 20 14 B -18 0 -12 -6 -2 C -4 12 0 2 6 D -20 6 -2 0 4 E -14 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997055 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 20 14 B -18 0 -12 -6 -2 C -4 12 0 2 6 D -20 6 -2 0 4 E -14 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997055 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2841: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) A D B C E (7) A C E D B (7) E C B D A (5) E B D C A (5) E A C B D (5) C B D E A (5) E A D B C (4) D B C A E (4) C B D A E (4) A C D B E (4) E C A B D (3) D B E A C (3) C E B D A (3) C D B A E (3) C A E B D (3) C A B D E (3) B D C E A (3) A E C D B (3) E A B D C (2) D B A E C (2) A D E B C (2) D B A C E (1) D A B C E (1) C E B A D (1) C B E D A (1) C A D B E (1) B D E C A (1) B C D E A (1) B C D A E (1) A D C E B (1) A D C B E (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 14 6 14 16 B -14 0 -8 -6 -6 C -6 8 0 8 12 D -14 6 -8 0 -2 E -16 6 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 14 16 B -14 0 -8 -6 -6 C -6 8 0 8 12 D -14 6 -8 0 -2 E -16 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=24 C=24 D=11 B=6 so B is eliminated. Round 2 votes counts: A=35 C=26 E=24 D=15 so D is eliminated. Round 3 votes counts: A=39 C=33 E=28 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:211 D:191 E:190 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 14 16 B -14 0 -8 -6 -6 C -6 8 0 8 12 D -14 6 -8 0 -2 E -16 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 14 16 B -14 0 -8 -6 -6 C -6 8 0 8 12 D -14 6 -8 0 -2 E -16 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 14 16 B -14 0 -8 -6 -6 C -6 8 0 8 12 D -14 6 -8 0 -2 E -16 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2842: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) D C E B A (6) D C A E B (5) C D E B A (5) A B E C D (5) D B C E A (4) A C E B D (4) A B D E C (4) E B C D A (3) D B A E C (3) C D E A B (3) B E D C A (3) B E C D A (3) A D B E C (3) A C D E B (3) E B C A D (2) E B A C D (2) E A C B D (2) D A B E C (2) C E D A B (2) B D E A C (2) B A E D C (2) A E C B D (2) A E B C D (2) A D C B E (2) E D C B A (1) E C B D A (1) E C B A D (1) D C B A E (1) D B E C A (1) D B A C E (1) D A C B E (1) D A B C E (1) C E A B D (1) C D A E B (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E C A (1) B A D E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -8 -18 -8 B 8 0 -2 -10 -12 C 8 2 0 6 2 D 18 10 -6 0 2 E 8 12 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -18 -8 B 8 0 -2 -10 -12 C 8 2 0 6 2 D 18 10 -6 0 2 E 8 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=25 C=22 B=14 E=12 so E is eliminated. Round 2 votes counts: A=29 D=26 C=24 B=21 so B is eliminated. Round 3 votes counts: A=36 D=32 C=32 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:209 E:208 B:192 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -18 -8 B 8 0 -2 -10 -12 C 8 2 0 6 2 D 18 10 -6 0 2 E 8 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -18 -8 B 8 0 -2 -10 -12 C 8 2 0 6 2 D 18 10 -6 0 2 E 8 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -18 -8 B 8 0 -2 -10 -12 C 8 2 0 6 2 D 18 10 -6 0 2 E 8 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2843: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) A E C D B (10) A E B C D (8) D B C E A (7) C D B E A (7) A E C B D (6) C A E D B (5) B D E C A (5) E A B D C (4) D C B E A (3) C D B A E (3) C A D E B (3) B E A D C (3) B D E A C (3) A E B D C (3) E C A D B (2) B D A E C (2) A C E D B (2) E D B A C (1) E A D C B (1) C E D A B (1) C E A D B (1) C D E B A (1) C D E A B (1) C D A E B (1) C B A D E (1) C A D B E (1) B E D A C (1) B D C A E (1) B D A C E (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -6 4 -2 B 0 0 2 0 0 C 6 -2 0 6 0 D -4 0 -6 0 2 E 2 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.084845 B: 0.660620 C: 0.000000 D: 0.084845 E: 0.169690 Sum of squares = 0.479610900593 Cumulative probabilities = A: 0.084845 B: 0.745465 C: 0.745465 D: 0.830310 E: 1.000000 A B C D E A 0 0 -6 4 -2 B 0 0 2 0 0 C 6 -2 0 6 0 D -4 0 -6 0 2 E 2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.200000 Sum of squares = 0.420000049562 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 0.700000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=27 C=25 D=10 E=8 so E is eliminated. Round 2 votes counts: A=35 C=27 B=27 D=11 so D is eliminated. Round 3 votes counts: B=35 A=35 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:205 B:201 E:200 A:198 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 4 -2 B 0 0 2 0 0 C 6 -2 0 6 0 D -4 0 -6 0 2 E 2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.200000 Sum of squares = 0.420000049562 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 0.700000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 4 -2 B 0 0 2 0 0 C 6 -2 0 6 0 D -4 0 -6 0 2 E 2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.200000 Sum of squares = 0.420000049562 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 0.700000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 4 -2 B 0 0 2 0 0 C 6 -2 0 6 0 D -4 0 -6 0 2 E 2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.000000 D: 0.100000 E: 0.200000 Sum of squares = 0.420000049562 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 0.700000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2844: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) A E D C B (8) C B E D A (6) E D C B A (5) E D A C B (5) D E B C A (5) A D E B C (5) A C B E D (5) E D B C A (4) D E A B C (4) D B E C A (4) B D C E A (4) D E B A C (3) C E B A D (3) C B A E D (3) B C A D E (3) A E C D B (3) A C E B D (3) E D A B C (2) E A D C B (2) C B E A D (2) B D C A E (2) B C D A E (2) A D B C E (2) E D C A B (1) E C D B A (1) C A B E D (1) B A D C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -10 -14 -20 B 14 0 2 -8 -8 C 10 -2 0 -14 -4 D 14 8 14 0 -8 E 20 8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -10 -14 -20 B 14 0 2 -8 -8 C 10 -2 0 -14 -4 D 14 8 14 0 -8 E 20 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999067 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=21 E=20 D=16 C=15 so C is eliminated. Round 2 votes counts: B=32 A=29 E=23 D=16 so D is eliminated. Round 3 votes counts: B=36 E=35 A=29 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:214 B:200 C:195 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -10 -14 -20 B 14 0 2 -8 -8 C 10 -2 0 -14 -4 D 14 8 14 0 -8 E 20 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999067 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -14 -20 B 14 0 2 -8 -8 C 10 -2 0 -14 -4 D 14 8 14 0 -8 E 20 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999067 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -14 -20 B 14 0 2 -8 -8 C 10 -2 0 -14 -4 D 14 8 14 0 -8 E 20 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999067 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2845: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (12) A B D C E (9) D E B A C (8) E C D B A (7) D A B C E (6) D E C B A (5) C E D A B (5) D C E A B (4) D B A E C (4) C D E A B (4) A B C D E (4) B A E D C (3) B A E C D (3) B A D E C (3) E D B A C (2) E C B A D (2) E B A D C (2) E B A C D (2) C E B A D (2) C E A B D (2) C A B E D (2) E D C B A (1) E B D A C (1) D E C A B (1) D C A B E (1) D A B E C (1) C E D B A (1) C D A B E (1) C A D B E (1) B A C E D (1) Total count = 100 A B C D E A 0 6 22 -4 2 B -6 0 22 -4 2 C -22 -22 0 -2 10 D 4 4 2 0 4 E -2 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 22 -4 2 B -6 0 22 -4 2 C -22 -22 0 -2 10 D 4 4 2 0 4 E -2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=25 C=18 E=17 B=10 so B is eliminated. Round 2 votes counts: A=35 D=30 C=18 E=17 so E is eliminated. Round 3 votes counts: A=39 D=34 C=27 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:213 B:207 D:207 E:191 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 22 -4 2 B -6 0 22 -4 2 C -22 -22 0 -2 10 D 4 4 2 0 4 E -2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 22 -4 2 B -6 0 22 -4 2 C -22 -22 0 -2 10 D 4 4 2 0 4 E -2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 22 -4 2 B -6 0 22 -4 2 C -22 -22 0 -2 10 D 4 4 2 0 4 E -2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2846: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) D A C B E (9) A D C B E (9) E B C D A (8) D A C E B (5) C A D E B (5) B E A D C (5) B A D E C (5) E C B A D (4) C E D A B (4) A D B C E (4) E B C A D (3) C E B D A (3) B E C D A (3) B E C A D (3) A B D E C (3) D A B C E (2) B D A E C (2) B A E D C (2) A D C E B (2) E C B D A (1) C E D B A (1) C E A B D (1) C D E A B (1) B E D A C (1) Total count = 100 A B C D E A 0 18 -2 -8 24 B -18 0 -18 -12 -4 C 2 18 0 2 20 D 8 12 -2 0 22 E -24 4 -20 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -2 -8 24 B -18 0 -18 -12 -4 C 2 18 0 2 20 D 8 12 -2 0 22 E -24 4 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=21 A=18 E=16 D=16 so E is eliminated. Round 2 votes counts: C=34 B=32 A=18 D=16 so D is eliminated. Round 3 votes counts: C=34 A=34 B=32 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:220 A:216 B:174 E:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -2 -8 24 B -18 0 -18 -12 -4 C 2 18 0 2 20 D 8 12 -2 0 22 E -24 4 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -2 -8 24 B -18 0 -18 -12 -4 C 2 18 0 2 20 D 8 12 -2 0 22 E -24 4 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -2 -8 24 B -18 0 -18 -12 -4 C 2 18 0 2 20 D 8 12 -2 0 22 E -24 4 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2847: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (18) D E B C A (13) E B D C A (6) B E D A C (5) D E C B A (4) D A C E B (4) B E C A D (4) A C D B E (4) D E C A B (3) D C E A B (3) C A B E D (3) B A E C D (3) A D C B E (3) E D B C A (2) E B C D A (2) D C A E B (2) C E B A D (2) C B A E D (2) C A D E B (2) B E A C D (2) D E B A C (1) D E A C B (1) D C E B A (1) D B E A C (1) C E B D A (1) C D E A B (1) C B E A D (1) B E D C A (1) B E C D A (1) B E A D C (1) B A C E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -8 -4 -12 B 8 0 -12 10 4 C 8 12 0 -2 0 D 4 -10 2 0 -12 E 12 -4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.524917 D: 0.000000 E: 0.475083 Sum of squares = 0.501241718694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.524917 D: 0.524917 E: 1.000000 A B C D E A 0 -8 -8 -4 -12 B 8 0 -12 10 4 C 8 12 0 -2 0 D 4 -10 2 0 -12 E 12 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=27 B=18 C=12 E=10 so E is eliminated. Round 2 votes counts: D=35 A=27 B=26 C=12 so C is eliminated. Round 3 votes counts: D=36 B=32 A=32 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:210 C:209 B:205 D:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 -12 B 8 0 -12 10 4 C 8 12 0 -2 0 D 4 -10 2 0 -12 E 12 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 -12 B 8 0 -12 10 4 C 8 12 0 -2 0 D 4 -10 2 0 -12 E 12 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 -12 B 8 0 -12 10 4 C 8 12 0 -2 0 D 4 -10 2 0 -12 E 12 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2848: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) E C B D A (8) B D A E C (7) A B D C E (6) E C A B D (5) D B A C E (5) A D B C E (5) E C D B A (4) C E D B A (4) A C D B E (4) E B D A C (3) E B A D C (3) C E A D B (3) C A D B E (3) B D E A C (3) A C B D E (3) E C A D B (2) C D B E A (2) C A E D B (2) E D B C A (1) E C D A B (1) E A C B D (1) E A B C D (1) D C B E A (1) D B C E A (1) D B A E C (1) D A B C E (1) C D E B A (1) C D B A E (1) C D A B E (1) B E D A C (1) B D A C E (1) B A D E C (1) A E C B D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 20 6 10 B -4 0 4 14 18 C -20 -4 0 -4 -10 D -6 -14 4 0 16 E -10 -18 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 20 6 10 B -4 0 4 14 18 C -20 -4 0 -4 -10 D -6 -14 4 0 16 E -10 -18 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=29 C=17 B=13 D=9 so D is eliminated. Round 2 votes counts: A=33 E=29 B=20 C=18 so C is eliminated. Round 3 votes counts: A=39 E=37 B=24 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 B:216 D:200 E:183 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 6 10 B -4 0 4 14 18 C -20 -4 0 -4 -10 D -6 -14 4 0 16 E -10 -18 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 6 10 B -4 0 4 14 18 C -20 -4 0 -4 -10 D -6 -14 4 0 16 E -10 -18 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 6 10 B -4 0 4 14 18 C -20 -4 0 -4 -10 D -6 -14 4 0 16 E -10 -18 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2849: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) E C B D A (8) E B D C A (8) C E A B D (7) A D B C E (7) B D E C A (6) A C E D B (5) A C D B E (4) D B A C E (3) D A B E C (3) B D E A C (3) B D C A E (3) E D B A C (2) E C B A D (2) E C A B D (2) E B C D A (2) E A D B C (2) C E B D A (2) C A B D E (2) B D A C E (2) A D B E C (2) E D B C A (1) E C D A B (1) E C A D B (1) D B E A C (1) C E A D B (1) C B E D A (1) C B D E A (1) C B A D E (1) C A E D B (1) B E D C A (1) B D C E A (1) B C D A E (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -6 -18 -6 B 16 0 16 6 4 C 6 -16 0 -14 -12 D 18 -6 14 0 6 E 6 -4 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -18 -6 B 16 0 16 6 4 C 6 -16 0 -14 -12 D 18 -6 14 0 6 E 6 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 B=17 D=16 C=16 so D is eliminated. Round 2 votes counts: B=30 E=29 A=25 C=16 so C is eliminated. Round 3 votes counts: E=39 B=33 A=28 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:216 E:204 C:182 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -18 -6 B 16 0 16 6 4 C 6 -16 0 -14 -12 D 18 -6 14 0 6 E 6 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -18 -6 B 16 0 16 6 4 C 6 -16 0 -14 -12 D 18 -6 14 0 6 E 6 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -18 -6 B 16 0 16 6 4 C 6 -16 0 -14 -12 D 18 -6 14 0 6 E 6 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2850: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (16) B A C E D (11) E C A D B (7) B D A C E (6) E D C A B (5) D B E C A (5) D B A C E (4) C E A D B (4) B A C D E (4) E C D A B (3) E C A B D (3) D E C B A (3) D B E A C (3) B E C A D (3) B D E A C (3) A C B E D (3) E B C D A (2) D E B C A (2) A B C E D (2) E C D B A (1) E C B A D (1) E B D C A (1) D A C E B (1) D A C B E (1) D A B C E (1) C E A B D (1) C A E D B (1) C A E B D (1) B D A E C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -18 -16 -26 B 0 0 -4 -16 -6 C 18 4 0 -4 -18 D 16 16 4 0 2 E 26 6 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 -16 -26 B 0 0 -4 -16 -6 C 18 4 0 -4 -18 D 16 16 4 0 2 E 26 6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=28 E=23 C=7 A=6 so A is eliminated. Round 2 votes counts: D=36 B=30 E=23 C=11 so C is eliminated. Round 3 votes counts: D=37 B=33 E=30 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:224 D:219 C:200 B:187 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -18 -16 -26 B 0 0 -4 -16 -6 C 18 4 0 -4 -18 D 16 16 4 0 2 E 26 6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 -16 -26 B 0 0 -4 -16 -6 C 18 4 0 -4 -18 D 16 16 4 0 2 E 26 6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 -16 -26 B 0 0 -4 -16 -6 C 18 4 0 -4 -18 D 16 16 4 0 2 E 26 6 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2851: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) B D A E C (7) C E D B A (5) C E B D A (5) E D C A B (4) D E C B A (4) C E A B D (4) E C D A B (3) D E B C A (3) C B E A D (3) B C E A D (3) B A D C E (3) A E D C B (3) A E C D B (3) A D E C B (3) A C E B D (3) A B C E D (3) D B E A C (2) D B A E C (2) C E D A B (2) C B E D A (2) B D E A C (2) B C D E A (2) B C A E D (2) B A C E D (2) A C E D B (2) E D A C B (1) E C D B A (1) E A C D B (1) D E C A B (1) D B E C A (1) C E B A D (1) C E A D B (1) C B A E D (1) B D C A E (1) B D A C E (1) B C E D A (1) B A D E C (1) A E C B D (1) A D E B C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 -14 -14 -18 B 24 0 -10 14 -4 C 14 10 0 4 12 D 14 -14 -4 0 -16 E 18 4 -12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -14 -14 -18 B 24 0 -10 14 -4 C 14 10 0 4 12 D 14 -14 -4 0 -16 E 18 4 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=24 A=21 D=13 E=10 so E is eliminated. Round 2 votes counts: B=32 C=28 A=22 D=18 so D is eliminated. Round 3 votes counts: B=40 C=37 A=23 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:213 B:212 D:190 A:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -24 -14 -14 -18 B 24 0 -10 14 -4 C 14 10 0 4 12 D 14 -14 -4 0 -16 E 18 4 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -14 -14 -18 B 24 0 -10 14 -4 C 14 10 0 4 12 D 14 -14 -4 0 -16 E 18 4 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -14 -14 -18 B 24 0 -10 14 -4 C 14 10 0 4 12 D 14 -14 -4 0 -16 E 18 4 -12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2852: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (11) B C E D A (10) A D E C B (9) E C B D A (7) C B E A D (7) D A E C B (5) E C B A D (4) D A E B C (4) D A B C E (4) E D C B A (3) D E A C B (3) C B E D A (3) B C E A D (3) A D E B C (3) E D C A B (2) D B C A E (2) B A C D E (2) A B D C E (2) A B C D E (2) E D A C B (1) E C A B D (1) E A D C B (1) E A C B D (1) D B A C E (1) D A B E C (1) C E B D A (1) C E B A D (1) B C D E A (1) B C A E D (1) A E D C B (1) A E C D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 8 4 2 B -8 0 -4 -4 4 C -8 4 0 -6 6 D -4 4 6 0 0 E -2 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 4 2 B -8 0 -4 -4 4 C -8 4 0 -6 6 D -4 4 6 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=20 D=20 B=17 C=12 so C is eliminated. Round 2 votes counts: A=31 B=27 E=22 D=20 so D is eliminated. Round 3 votes counts: A=45 B=30 E=25 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:203 C:198 B:194 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 4 2 B -8 0 -4 -4 4 C -8 4 0 -6 6 D -4 4 6 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 4 2 B -8 0 -4 -4 4 C -8 4 0 -6 6 D -4 4 6 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 4 2 B -8 0 -4 -4 4 C -8 4 0 -6 6 D -4 4 6 0 0 E -2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999426 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2853: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (14) B A D C E (10) D E C B A (8) B D A C E (8) E C A D B (6) D B E C A (6) C E A D B (6) B D A E C (6) A B C E D (4) E C D A B (3) B D E C A (3) B A C E D (3) A E C D B (3) D E C A B (2) D B C E A (2) D B A E C (2) C E D A B (2) B D C E A (2) A C E D B (2) D B E A C (1) D A B E C (1) C E D B A (1) C E A B D (1) C A E B D (1) B C D A E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 12 4 14 B 6 0 -2 10 0 C -12 2 0 -2 16 D -4 -10 2 0 4 E -14 0 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000083 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 4 14 B 6 0 -2 10 0 C -12 2 0 -2 16 D -4 -10 2 0 4 E -14 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000106 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=25 D=22 C=11 E=9 so E is eliminated. Round 2 votes counts: B=33 A=25 D=22 C=20 so C is eliminated. Round 3 votes counts: A=39 B=33 D=28 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:212 B:207 C:202 D:196 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 4 14 B 6 0 -2 10 0 C -12 2 0 -2 16 D -4 -10 2 0 4 E -14 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000106 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 4 14 B 6 0 -2 10 0 C -12 2 0 -2 16 D -4 -10 2 0 4 E -14 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000106 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 4 14 B 6 0 -2 10 0 C -12 2 0 -2 16 D -4 -10 2 0 4 E -14 0 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.600000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000106 Cumulative probabilities = A: 0.100000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2854: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) D E B C A (8) C A E B D (7) E D C A B (5) E B D C A (5) B A C D E (5) E C B A D (4) C A B E D (4) A C D B E (4) D E C A B (3) D A C E B (3) D A B C E (3) C A E D B (3) E D B C A (2) E C D A B (2) D E B A C (2) B A C E D (2) A C B D E (2) E C B D A (1) E C A D B (1) E C A B D (1) E B C D A (1) E B C A D (1) D E C B A (1) D E A C B (1) D E A B C (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C B E (1) C E A D B (1) B E C A D (1) B D E A C (1) B A D C E (1) A D C E B (1) A D C B E (1) A D B C E (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 -8 -6 B -4 0 -2 -20 -16 C 2 2 0 -10 -8 D 8 20 10 0 12 E 6 16 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999423 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -8 -6 B -4 0 -2 -20 -16 C 2 2 0 -10 -8 D 8 20 10 0 12 E 6 16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=23 C=15 A=15 B=10 so B is eliminated. Round 2 votes counts: D=38 E=24 A=23 C=15 so C is eliminated. Round 3 votes counts: D=38 A=37 E=25 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:209 A:194 C:193 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -8 -6 B -4 0 -2 -20 -16 C 2 2 0 -10 -8 D 8 20 10 0 12 E 6 16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -8 -6 B -4 0 -2 -20 -16 C 2 2 0 -10 -8 D 8 20 10 0 12 E 6 16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -8 -6 B -4 0 -2 -20 -16 C 2 2 0 -10 -8 D 8 20 10 0 12 E 6 16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2855: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (13) D A E B C (9) C B A E D (7) B A C E D (7) A D E B C (7) E D C B A (6) C E B D A (4) C B E D A (4) C B E A D (4) E D A B C (3) D E C A B (3) D E A C B (3) C E D B A (3) B C A E D (3) A D B E C (3) A B D E C (3) C D E B A (2) C B A D E (2) B A E D C (2) A B E D C (2) E D B A C (1) E B D A C (1) D E C B A (1) C D A E B (1) C A D B E (1) B A E C D (1) A D E C B (1) A D C B E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 18 -8 4 B -4 0 12 -18 -16 C -18 -12 0 -20 -18 D 8 18 20 0 4 E -4 16 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 -8 4 B -4 0 12 -18 -16 C -18 -12 0 -20 -18 D 8 18 20 0 4 E -4 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 A=19 B=13 E=11 so E is eliminated. Round 2 votes counts: D=39 C=28 A=19 B=14 so B is eliminated. Round 3 votes counts: D=40 C=31 A=29 so A is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:213 A:209 B:187 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 18 -8 4 B -4 0 12 -18 -16 C -18 -12 0 -20 -18 D 8 18 20 0 4 E -4 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 -8 4 B -4 0 12 -18 -16 C -18 -12 0 -20 -18 D 8 18 20 0 4 E -4 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 -8 4 B -4 0 12 -18 -16 C -18 -12 0 -20 -18 D 8 18 20 0 4 E -4 16 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2856: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (13) B D E C A (9) B D A C E (7) A C E D B (7) E C D A B (6) D E C A B (5) B A C E D (5) A C E B D (5) D E C B A (4) B A D C E (4) E D C B A (3) E D C A B (3) A C B E D (3) E C A D B (2) D E B C A (2) D B E A C (2) D B A C E (2) C E A D B (2) B C E A D (2) A B C E D (2) E C D B A (1) E C B A D (1) D A E B C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E A B D (1) C A E B D (1) B E D C A (1) B D A E C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 -12 -26 -14 B 14 0 8 -16 8 C 12 -8 0 -24 -10 D 26 16 24 0 10 E 14 -8 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -26 -14 B 14 0 8 -16 8 C 12 -8 0 -24 -10 D 26 16 24 0 10 E 14 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 A=19 E=16 C=4 so C is eliminated. Round 2 votes counts: D=32 B=29 A=20 E=19 so E is eliminated. Round 3 votes counts: D=45 B=30 A=25 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:238 B:207 E:203 C:185 A:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -12 -26 -14 B 14 0 8 -16 8 C 12 -8 0 -24 -10 D 26 16 24 0 10 E 14 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -26 -14 B 14 0 8 -16 8 C 12 -8 0 -24 -10 D 26 16 24 0 10 E 14 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -26 -14 B 14 0 8 -16 8 C 12 -8 0 -24 -10 D 26 16 24 0 10 E 14 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2857: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) E B C A D (8) D A E C B (8) E B C D A (6) C B A D E (6) A D C B E (6) E D A C B (5) D A C E B (5) D A C B E (5) E B D A C (3) D A E B C (3) C A B D E (3) B C A D E (3) C E B D A (2) C D A B E (2) C B E A D (2) B E C A D (2) B C E A D (2) B A C D E (2) E D B C A (1) E C D A B (1) E C B D A (1) E B A D C (1) E A D B C (1) D C A E B (1) C D A E B (1) C B E D A (1) C B D A E (1) C B A E D (1) C A D B E (1) B A D C E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 10 -16 4 B -14 0 -6 -8 -18 C -10 6 0 -8 -4 D 16 8 8 0 2 E -4 18 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 -16 4 B -14 0 -6 -8 -18 C -10 6 0 -8 -4 D 16 8 8 0 2 E -4 18 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=22 C=20 B=10 A=9 so A is eliminated. Round 2 votes counts: E=39 D=29 C=21 B=11 so B is eliminated. Round 3 votes counts: E=41 D=31 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:208 A:206 C:192 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 10 -16 4 B -14 0 -6 -8 -18 C -10 6 0 -8 -4 D 16 8 8 0 2 E -4 18 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -16 4 B -14 0 -6 -8 -18 C -10 6 0 -8 -4 D 16 8 8 0 2 E -4 18 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -16 4 B -14 0 -6 -8 -18 C -10 6 0 -8 -4 D 16 8 8 0 2 E -4 18 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2858: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) C D E A B (10) C E D B A (9) A B D E C (7) E C B A D (6) D C A E B (5) D A B C E (5) A D B E C (4) D A C B E (3) C E B A D (3) B E A D C (3) A D B C E (3) E C B D A (2) E B A D C (2) D E A B C (2) D C E A B (2) D A B E C (2) C E B D A (2) C D A E B (2) B A E C D (2) A B D C E (2) E B C D A (1) E B C A D (1) E B A C D (1) D E C A B (1) D E B A C (1) D C E B A (1) D A E C B (1) C B A E D (1) C A D B E (1) B E A C D (1) B A C E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 6 2 4 B -4 0 2 -4 -4 C -6 -2 0 -10 2 D -2 4 10 0 4 E -4 4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 2 4 B -4 0 2 -4 -4 C -6 -2 0 -10 2 D -2 4 10 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 B=18 A=18 E=13 so E is eliminated. Round 2 votes counts: C=36 D=23 B=23 A=18 so A is eliminated. Round 3 votes counts: C=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:208 D:208 E:197 B:195 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 2 4 B -4 0 2 -4 -4 C -6 -2 0 -10 2 D -2 4 10 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 2 4 B -4 0 2 -4 -4 C -6 -2 0 -10 2 D -2 4 10 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 2 4 B -4 0 2 -4 -4 C -6 -2 0 -10 2 D -2 4 10 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2859: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) A B C E D (6) E C D B A (5) A B D C E (5) C B D E A (4) B C D A E (4) A B C D E (4) E D C A B (3) E D A C B (3) E A D C B (3) D E A C B (3) D B C A E (3) D A B C E (3) B C A D E (3) B A C D E (3) A B E D C (3) E C B A D (2) D E C B A (2) D C B E A (2) D A E C B (2) B C A E D (2) B A C E D (2) A E D C B (2) A E D B C (2) A E B D C (2) A E B C D (2) A D B E C (2) A D B C E (2) E D C B A (1) E C B D A (1) E C A D B (1) E A C D B (1) E A C B D (1) D E C A B (1) D C E B A (1) D B A C E (1) D A C B E (1) C E D B A (1) C D B E A (1) B D C A E (1) B C E D A (1) A B E C D (1) Total count = 100 A B C D E A 0 20 22 12 26 B -20 0 18 -6 8 C -22 -18 0 -10 0 D -12 6 10 0 10 E -26 -8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 22 12 26 B -20 0 18 -6 8 C -22 -18 0 -10 0 D -12 6 10 0 10 E -26 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=21 D=19 B=16 C=6 so C is eliminated. Round 2 votes counts: A=38 E=22 D=20 B=20 so D is eliminated. Round 3 votes counts: A=44 E=29 B=27 so B is eliminated. Round 4 votes counts: A=63 E=37 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:240 D:207 B:200 E:178 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 22 12 26 B -20 0 18 -6 8 C -22 -18 0 -10 0 D -12 6 10 0 10 E -26 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 22 12 26 B -20 0 18 -6 8 C -22 -18 0 -10 0 D -12 6 10 0 10 E -26 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 22 12 26 B -20 0 18 -6 8 C -22 -18 0 -10 0 D -12 6 10 0 10 E -26 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2860: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (13) A C B D E (10) C D E A B (8) B E D A C (8) C A D E B (7) B A E C D (6) A B C D E (6) B E A D C (5) E D B C A (4) D E C A B (4) B A E D C (4) B A C E D (3) A C B E D (3) A B C E D (3) E D C A B (2) E B D A C (2) D C E A B (2) C D A E B (2) B A C D E (2) E C D A B (1) E B D C A (1) D E C B A (1) C A D B E (1) B D E A C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 8 2 -4 B 0 0 -8 10 6 C -8 8 0 4 -4 D -2 -10 -4 0 -10 E 4 -6 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545048 B: 0.454952 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.504058642048 Cumulative probabilities = A: 0.545048 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 -4 B 0 0 -8 10 6 C -8 8 0 4 -4 D -2 -10 -4 0 -10 E 4 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500150 B: 0.499850 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045205 Cumulative probabilities = A: 0.500150 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 A=23 C=18 D=7 so D is eliminated. Round 2 votes counts: B=29 E=28 A=23 C=20 so C is eliminated. Round 3 votes counts: E=38 A=33 B=29 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:206 B:204 A:203 C:200 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 -4 B 0 0 -8 10 6 C -8 8 0 4 -4 D -2 -10 -4 0 -10 E 4 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500150 B: 0.499850 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045205 Cumulative probabilities = A: 0.500150 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 -4 B 0 0 -8 10 6 C -8 8 0 4 -4 D -2 -10 -4 0 -10 E 4 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500150 B: 0.499850 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045205 Cumulative probabilities = A: 0.500150 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 -4 B 0 0 -8 10 6 C -8 8 0 4 -4 D -2 -10 -4 0 -10 E 4 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500150 B: 0.499850 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045205 Cumulative probabilities = A: 0.500150 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2861: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (11) B A E D C (10) C D E A B (8) D B C A E (7) E A C B D (5) D B A E C (4) C E A B D (4) B E A C D (4) E A C D B (3) D C E A B (3) D B C E A (3) C E D A B (3) A E C B D (3) E A B C D (2) D C A E B (2) B D C E A (2) B A D E C (2) E C A B D (1) D C B E A (1) D B A C E (1) D A B E C (1) C D B E A (1) B E A D C (1) B D E A C (1) B D C A E (1) B C E A D (1) B A E C D (1) A E D B C (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 4 2 -8 B 2 0 8 0 4 C -4 -8 0 -4 -4 D -2 0 4 0 -4 E 8 -4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.737206 C: 0.000000 D: 0.262794 E: 0.000000 Sum of squares = 0.612533430824 Cumulative probabilities = A: 0.000000 B: 0.737206 C: 0.737206 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 2 -8 B 2 0 8 0 4 C -4 -8 0 -4 -4 D -2 0 4 0 -4 E 8 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500244 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000119354 Cumulative probabilities = A: 0.000000 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=27 D=22 E=11 A=6 so A is eliminated. Round 2 votes counts: B=34 C=27 D=22 E=17 so E is eliminated. Round 3 votes counts: C=40 B=37 D=23 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:207 E:206 D:199 A:198 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 2 -8 B 2 0 8 0 4 C -4 -8 0 -4 -4 D -2 0 4 0 -4 E 8 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500244 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000119354 Cumulative probabilities = A: 0.000000 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 2 -8 B 2 0 8 0 4 C -4 -8 0 -4 -4 D -2 0 4 0 -4 E 8 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500244 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000119354 Cumulative probabilities = A: 0.000000 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 2 -8 B 2 0 8 0 4 C -4 -8 0 -4 -4 D -2 0 4 0 -4 E 8 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500244 C: 0.000000 D: 0.499756 E: 0.000000 Sum of squares = 0.500000119354 Cumulative probabilities = A: 0.000000 B: 0.500244 C: 0.500244 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2862: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) B A E C D (6) B A C D E (6) E A C D B (5) D C E A B (5) A B C D E (5) D C E B A (4) A C D B E (4) A B E C D (4) E D C B A (3) E B A D C (3) B E D C A (3) A C D E B (3) E A C B D (2) E A B C D (2) D E C A B (2) C D B A E (2) C D A B E (2) B D C E A (2) B D C A E (2) B A D C E (2) A B C E D (2) E D B C A (1) E D A C B (1) E D A B C (1) E C D A B (1) E B A C D (1) E A D C B (1) D E C B A (1) D E B C A (1) D C B E A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C A E (1) C D A E B (1) C B D A E (1) B E D A C (1) B E A D C (1) B E A C D (1) B A D E C (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 8 2 -6 B -8 0 -6 -6 2 C -8 6 0 0 -4 D -2 6 0 0 0 E 6 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.479928 E: 0.520072 Sum of squares = 0.500805804032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.479928 E: 1.000000 A B C D E A 0 8 8 2 -6 B -8 0 -6 -6 2 C -8 6 0 0 -4 D -2 6 0 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=26 A=19 D=18 C=6 so C is eliminated. Round 2 votes counts: E=31 B=27 D=23 A=19 so A is eliminated. Round 3 votes counts: B=39 E=31 D=30 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:206 E:204 D:202 C:197 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 8 2 -6 B -8 0 -6 -6 2 C -8 6 0 0 -4 D -2 6 0 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 2 -6 B -8 0 -6 -6 2 C -8 6 0 0 -4 D -2 6 0 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 2 -6 B -8 0 -6 -6 2 C -8 6 0 0 -4 D -2 6 0 0 0 E 6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2863: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) D C E B A (8) D E B A C (6) C D B E A (5) A B E C D (5) C D A B E (4) C A B E D (4) A D E B C (4) A C B E D (4) E D B A C (3) E B A D C (3) D E C B A (3) C B A E D (3) C A D B E (3) B A E C D (3) A E B D C (3) E A B D C (2) D E A B C (2) D C E A B (2) D C A E B (2) D A E B C (2) B E C D A (2) B E A C D (2) A B C E D (2) E D A B C (1) E B D A C (1) E A D B C (1) D E A C B (1) D C B E A (1) D A E C B (1) D A C E B (1) C B E D A (1) C B D E A (1) C B A D E (1) B C E A D (1) B C A E D (1) A E C B D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 0 -10 -8 B 6 0 4 -20 -12 C 0 -4 0 -12 -8 D 10 20 12 0 14 E 8 12 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -10 -8 B 6 0 4 -20 -12 C 0 -4 0 -12 -8 D 10 20 12 0 14 E 8 12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=22 A=21 E=11 B=9 so B is eliminated. Round 2 votes counts: D=37 C=24 A=24 E=15 so E is eliminated. Round 3 votes counts: D=42 A=32 C=26 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:207 B:189 A:188 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -10 -8 B 6 0 4 -20 -12 C 0 -4 0 -12 -8 D 10 20 12 0 14 E 8 12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -10 -8 B 6 0 4 -20 -12 C 0 -4 0 -12 -8 D 10 20 12 0 14 E 8 12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -10 -8 B 6 0 4 -20 -12 C 0 -4 0 -12 -8 D 10 20 12 0 14 E 8 12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2864: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) E C D A B (6) D A B E C (5) C E A B D (5) E C D B A (4) E C B D A (4) E C A B D (4) D E B C A (4) D E A C B (4) D B A E C (4) A B D C E (4) A B C D E (4) E D C A B (3) D E C B A (3) C A E B D (3) B C A E D (3) B A C D E (3) A C E D B (3) A C B E D (3) D B E A C (2) B D A C E (2) B A D C E (2) B A C E D (2) A C E B D (2) E C A D B (1) E A C D B (1) D E A B C (1) D B E C A (1) D A E B C (1) D A B C E (1) C E B A D (1) C E A D B (1) C B E A D (1) C A B E D (1) B E D C A (1) B D E C A (1) B C E A D (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -8 -6 -10 B -10 0 -14 -6 -16 C 8 14 0 6 -14 D 6 6 -6 0 -12 E 10 16 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -8 -6 -10 B -10 0 -14 -6 -16 C 8 14 0 6 -14 D 6 6 -6 0 -12 E 10 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 A=18 B=15 C=12 so C is eliminated. Round 2 votes counts: E=36 D=26 A=22 B=16 so B is eliminated. Round 3 votes counts: E=39 A=32 D=29 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:207 D:197 A:193 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -8 -6 -10 B -10 0 -14 -6 -16 C 8 14 0 6 -14 D 6 6 -6 0 -12 E 10 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -6 -10 B -10 0 -14 -6 -16 C 8 14 0 6 -14 D 6 6 -6 0 -12 E 10 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -6 -10 B -10 0 -14 -6 -16 C 8 14 0 6 -14 D 6 6 -6 0 -12 E 10 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2865: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) C E D A B (9) C E B A D (8) D B A E C (6) B A D E C (6) C E A B D (5) B A E C D (5) D E C A B (4) D A E C B (4) D C E A B (3) B D A E C (3) B D A C E (3) B C E A D (3) A E C D B (3) D C E B A (2) C E D B A (2) C E B D A (2) B D C A E (2) B C D E A (2) B C A E D (2) E C A D B (1) E A C B D (1) D E A C B (1) D C B E A (1) D B C E A (1) D A E B C (1) C D E B A (1) C D E A B (1) C B E A D (1) B C E D A (1) B A E D C (1) B A D C E (1) B A C E D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -24 2 -20 B 8 0 -22 -4 -22 C 24 22 0 18 24 D -2 4 -18 0 -14 E 20 22 -24 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -24 2 -20 B 8 0 -22 -4 -22 C 24 22 0 18 24 D -2 4 -18 0 -14 E 20 22 -24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=30 D=23 A=5 E=2 so E is eliminated. Round 2 votes counts: C=41 B=30 D=23 A=6 so A is eliminated. Round 3 votes counts: C=45 B=31 D=24 so D is eliminated. Round 4 votes counts: C=61 B=39 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:244 E:216 D:185 B:180 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -24 2 -20 B 8 0 -22 -4 -22 C 24 22 0 18 24 D -2 4 -18 0 -14 E 20 22 -24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -24 2 -20 B 8 0 -22 -4 -22 C 24 22 0 18 24 D -2 4 -18 0 -14 E 20 22 -24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -24 2 -20 B 8 0 -22 -4 -22 C 24 22 0 18 24 D -2 4 -18 0 -14 E 20 22 -24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2866: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) A E B D C (10) C D B E A (7) D C A B E (6) E A B C D (5) D A C B E (5) E B C A D (4) E B A C D (4) D C B E A (4) B E C D A (4) D C B A E (3) D C A E B (3) A E D B C (3) A D C E B (3) A D B E C (3) A B D E C (3) C E D B A (2) C D E B A (2) C B E D A (2) B A E D C (2) E B C D A (1) D B C A E (1) D A C E B (1) C E B D A (1) B E D C A (1) B E D A C (1) B E A C D (1) B D C E A (1) B C E D A (1) A D E C B (1) A D C B E (1) A D B C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 16 14 8 18 B -16 0 16 6 -4 C -14 -16 0 -8 -10 D -8 -6 8 0 -8 E -18 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 8 18 B -16 0 16 6 -4 C -14 -16 0 -8 -10 D -8 -6 8 0 -8 E -18 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=23 E=14 C=14 B=11 so B is eliminated. Round 2 votes counts: A=40 D=24 E=21 C=15 so C is eliminated. Round 3 votes counts: A=40 D=33 E=27 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 E:202 B:201 D:193 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 8 18 B -16 0 16 6 -4 C -14 -16 0 -8 -10 D -8 -6 8 0 -8 E -18 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 8 18 B -16 0 16 6 -4 C -14 -16 0 -8 -10 D -8 -6 8 0 -8 E -18 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 8 18 B -16 0 16 6 -4 C -14 -16 0 -8 -10 D -8 -6 8 0 -8 E -18 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2867: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) B D E A C (8) D B E C A (6) A C B D E (6) C A E D B (5) C A B D E (5) E B D A C (4) E A C B D (4) B D C A E (4) A C E D B (4) E D B A C (3) E A C D B (3) C D A B E (3) B D E C A (3) B D A C E (3) E A D C B (2) D E B C A (2) D B C A E (2) B E D A C (2) B A C D E (2) A E C B D (2) E D B C A (1) E C A D B (1) E A D B C (1) E A B D C (1) E A B C D (1) D C B A E (1) D B C E A (1) C E A D B (1) C D B A E (1) C B A D E (1) C A D E B (1) C A D B E (1) B D C E A (1) B D A E C (1) B A E D C (1) B A D E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 20 8 10 B -4 0 -2 22 10 C -20 2 0 4 6 D -8 -22 -4 0 6 E -10 -10 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 20 8 10 B -4 0 -2 22 10 C -20 2 0 4 6 D -8 -22 -4 0 6 E -10 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=23 E=21 C=18 D=12 so D is eliminated. Round 2 votes counts: B=35 E=23 A=23 C=19 so C is eliminated. Round 3 votes counts: B=38 A=38 E=24 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:213 C:196 D:186 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 8 10 B -4 0 -2 22 10 C -20 2 0 4 6 D -8 -22 -4 0 6 E -10 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 8 10 B -4 0 -2 22 10 C -20 2 0 4 6 D -8 -22 -4 0 6 E -10 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 8 10 B -4 0 -2 22 10 C -20 2 0 4 6 D -8 -22 -4 0 6 E -10 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2868: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (14) C D E A B (9) A B C D E (7) B E C D A (5) E B D C A (4) D E C B A (4) C A D E B (4) B E D C A (4) B A E D C (4) A C D E B (4) D E C A B (3) C D A E B (3) B A E C D (3) A C B D E (3) A B D E C (3) A B D C E (3) E B C D A (2) D C E A B (2) B E D A C (2) B E A C D (2) A D C E B (2) A C D B E (2) A B C E D (2) E D B C A (1) E C D B A (1) D E B A C (1) D C E B A (1) D C A E B (1) C E B D A (1) B E A D C (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -18 -16 -14 B 0 0 -10 -6 -16 C 18 10 0 -4 -12 D 16 6 4 0 6 E 14 16 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 -16 -14 B 0 0 -10 -6 -16 C 18 10 0 -4 -12 D 16 6 4 0 6 E 14 16 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 B=21 C=17 D=12 so D is eliminated. Round 2 votes counts: E=30 A=28 C=21 B=21 so C is eliminated. Round 3 votes counts: E=43 A=36 B=21 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:216 C:206 B:184 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -18 -16 -14 B 0 0 -10 -6 -16 C 18 10 0 -4 -12 D 16 6 4 0 6 E 14 16 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 -16 -14 B 0 0 -10 -6 -16 C 18 10 0 -4 -12 D 16 6 4 0 6 E 14 16 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 -16 -14 B 0 0 -10 -6 -16 C 18 10 0 -4 -12 D 16 6 4 0 6 E 14 16 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2869: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (14) E B C D A (8) A E D C B (6) A B C D E (6) C D B A E (5) B E C D A (5) A D C E B (5) A D C B E (5) E B A C D (4) B C D E A (4) B A C D E (4) E D C B A (3) E A D C B (3) E A B C D (3) D C A B E (3) E D A C B (2) E B D C A (2) D C B E A (2) D C B A E (2) D C A E B (2) E B D A C (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E B A (1) C D B E A (1) C D A B E (1) C B D A E (1) B A E C D (1) A E B D C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -8 -16 16 B 16 0 12 12 12 C 8 -12 0 18 14 D 16 -12 -18 0 16 E -16 -12 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -16 16 B 16 0 12 12 12 C 8 -12 0 18 14 D 16 -12 -18 0 16 E -16 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=28 B=28 A=25 D=11 C=8 so C is eliminated. Round 2 votes counts: B=29 E=28 A=25 D=18 so D is eliminated. Round 3 votes counts: B=39 A=31 E=30 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:214 D:201 A:188 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -8 -16 16 B 16 0 12 12 12 C 8 -12 0 18 14 D 16 -12 -18 0 16 E -16 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -16 16 B 16 0 12 12 12 C 8 -12 0 18 14 D 16 -12 -18 0 16 E -16 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -16 16 B 16 0 12 12 12 C 8 -12 0 18 14 D 16 -12 -18 0 16 E -16 -12 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2870: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) E A B C D (6) A B C E D (6) D C A E B (5) D A E C B (5) D A C E B (5) E B A D C (4) D C A B E (4) D A C B E (4) B C E A D (4) E A B D C (3) D C B E A (3) D C B A E (3) C D B A E (3) C B A D E (3) E D B C A (2) E B C A D (2) D E C B A (2) D E A C B (2) D E A B C (2) D B C E A (2) B E C A D (2) B E A C D (2) A C B E D (2) E B D C A (1) E A D B C (1) D E B C A (1) D E B A C (1) D C E B A (1) D B E C A (1) C B D A E (1) C B A E D (1) B C E D A (1) B C D E A (1) B C A E D (1) A E D C B (1) A D E C B (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 12 0 -4 B 0 0 4 -2 -6 C -12 -4 0 -10 4 D 0 2 10 0 4 E 4 6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250997 B: 0.000000 C: 0.000000 D: 0.749003 E: 0.000000 Sum of squares = 0.624004974611 Cumulative probabilities = A: 0.250997 B: 0.250997 C: 0.250997 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 0 -4 B 0 0 4 -2 -6 C -12 -4 0 -10 4 D 0 2 10 0 4 E 4 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499495 B: 0.000000 C: 0.000000 D: 0.500505 E: 0.000000 Sum of squares = 0.500000510184 Cumulative probabilities = A: 0.499495 B: 0.499495 C: 0.499495 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=27 A=13 B=11 C=8 so C is eliminated. Round 2 votes counts: D=44 E=27 B=16 A=13 so A is eliminated. Round 3 votes counts: D=47 E=28 B=25 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 A:204 E:201 B:198 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 12 0 -4 B 0 0 4 -2 -6 C -12 -4 0 -10 4 D 0 2 10 0 4 E 4 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499495 B: 0.000000 C: 0.000000 D: 0.500505 E: 0.000000 Sum of squares = 0.500000510184 Cumulative probabilities = A: 0.499495 B: 0.499495 C: 0.499495 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 0 -4 B 0 0 4 -2 -6 C -12 -4 0 -10 4 D 0 2 10 0 4 E 4 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499495 B: 0.000000 C: 0.000000 D: 0.500505 E: 0.000000 Sum of squares = 0.500000510184 Cumulative probabilities = A: 0.499495 B: 0.499495 C: 0.499495 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 0 -4 B 0 0 4 -2 -6 C -12 -4 0 -10 4 D 0 2 10 0 4 E 4 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499495 B: 0.000000 C: 0.000000 D: 0.500505 E: 0.000000 Sum of squares = 0.500000510184 Cumulative probabilities = A: 0.499495 B: 0.499495 C: 0.499495 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2871: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (6) D B E C A (5) D A B E C (5) D B C E A (4) C E B A D (4) B D C A E (4) A E D C B (4) A E C B D (4) E A D C B (3) D B A E C (3) D B A C E (3) C B E A D (3) B C D E A (3) A C E B D (3) E C A B D (2) E B C D A (2) E A C D B (2) E A C B D (2) D E B C A (2) C B E D A (2) C B A E D (2) C A B E D (2) B D C E A (2) B C D A E (2) A E C D B (2) A D E C B (2) A D E B C (2) A D B C E (2) A C B E D (2) E D C B A (1) E D B C A (1) E D A C B (1) E C D A B (1) E C B D A (1) E C A D B (1) D E B A C (1) D B C A E (1) D A B C E (1) C E A B D (1) C A E B D (1) B D E C A (1) A D C E B (1) A D B E C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 4 -4 10 B -6 0 4 -10 0 C -4 -4 0 -12 -10 D 4 10 12 0 6 E -10 0 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 10 B -6 0 4 -10 0 C -4 -4 0 -12 -10 D 4 10 12 0 6 E -10 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=25 E=17 C=15 B=12 so B is eliminated. Round 2 votes counts: D=38 A=25 C=20 E=17 so E is eliminated. Round 3 votes counts: D=41 A=32 C=27 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:208 E:197 B:194 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 10 B -6 0 4 -10 0 C -4 -4 0 -12 -10 D 4 10 12 0 6 E -10 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 10 B -6 0 4 -10 0 C -4 -4 0 -12 -10 D 4 10 12 0 6 E -10 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 10 B -6 0 4 -10 0 C -4 -4 0 -12 -10 D 4 10 12 0 6 E -10 0 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2872: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) C D E A B (10) D C E A B (6) C E D B A (6) C B A E D (6) B A E C D (6) A B C D E (6) B A C E D (5) E D C B A (4) A B D E C (4) E D B A C (3) D E C A B (3) E D B C A (2) E B A D C (2) D A E C B (2) C A D B E (2) B E A C D (2) B C A E D (2) A D B E C (2) A B E D C (2) A B D C E (2) E B C D A (1) D E C B A (1) D E A B C (1) D C A E B (1) D A E B C (1) C E B D A (1) C E B A D (1) C D E B A (1) C D A B E (1) C A B D E (1) B E A D C (1) B C E A D (1) Total count = 100 A B C D E A 0 -12 0 12 8 B 12 0 8 8 8 C 0 -8 0 4 4 D -12 -8 -4 0 -12 E -8 -8 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 12 8 B 12 0 8 8 8 C 0 -8 0 4 4 D -12 -8 -4 0 -12 E -8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 A=16 D=15 E=12 so E is eliminated. Round 2 votes counts: B=31 C=29 D=24 A=16 so A is eliminated. Round 3 votes counts: B=45 C=29 D=26 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:204 C:200 E:196 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 12 8 B 12 0 8 8 8 C 0 -8 0 4 4 D -12 -8 -4 0 -12 E -8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 12 8 B 12 0 8 8 8 C 0 -8 0 4 4 D -12 -8 -4 0 -12 E -8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 12 8 B 12 0 8 8 8 C 0 -8 0 4 4 D -12 -8 -4 0 -12 E -8 -8 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2873: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (7) E B D A C (6) E B D C A (5) D E A B C (5) C B E A D (5) B E A C D (5) D C A E B (4) D A C E B (4) D A C B E (4) C A D B E (4) C A B D E (4) D E C B A (3) C E D B A (3) C D E A B (3) C D A B E (3) C B A E D (3) E D C B A (2) E D B C A (2) E D B A C (2) E B C D A (2) D E B A C (2) C A B E D (2) B A E D C (2) A C B D E (2) A B E D C (2) E C D B A (1) E C B A D (1) E B A D C (1) D E A C B (1) D C E B A (1) C E B D A (1) C D A E B (1) B E A D C (1) B C E A D (1) B C A E D (1) B A C E D (1) A D C B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -10 -6 B 2 0 -20 -6 0 C 4 20 0 2 12 D 10 6 -2 0 2 E 6 0 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -10 -6 B 2 0 -20 -6 0 C 4 20 0 2 12 D 10 6 -2 0 2 E 6 0 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 E=22 A=14 B=11 so B is eliminated. Round 2 votes counts: C=31 E=28 D=24 A=17 so A is eliminated. Round 3 votes counts: C=42 E=32 D=26 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 D:208 E:196 A:189 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -10 -6 B 2 0 -20 -6 0 C 4 20 0 2 12 D 10 6 -2 0 2 E 6 0 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -10 -6 B 2 0 -20 -6 0 C 4 20 0 2 12 D 10 6 -2 0 2 E 6 0 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -10 -6 B 2 0 -20 -6 0 C 4 20 0 2 12 D 10 6 -2 0 2 E 6 0 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2874: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (6) E D A B C (5) C E B D A (5) B C D E A (5) E A D B C (4) C B E D A (4) C B A E D (4) A E C D B (4) E D B A C (3) E C A B D (3) D E B A C (3) D E A B C (3) C B D E A (3) C B D A E (3) C B A D E (3) B D C E A (3) B C E D A (3) A E D B C (3) A D E B C (3) E B D C A (2) E A D C B (2) D B E C A (2) D A E B C (2) D A B E C (2) B D E C A (2) A C D B E (2) E C B D A (1) E C B A D (1) E A C D B (1) D B A E C (1) C E A B D (1) C B E A D (1) C A E B D (1) C A B E D (1) C A B D E (1) B E D C A (1) B D C A E (1) B C D A E (1) A D E C B (1) A D C B E (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -4 -10 -16 B 4 0 -2 2 -10 C 4 2 0 0 -10 D 10 -2 0 0 -14 E 16 10 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 -10 -16 B 4 0 -2 2 -10 C 4 2 0 0 -10 D 10 -2 0 0 -14 E 16 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 A=22 B=16 D=13 so D is eliminated. Round 2 votes counts: E=28 C=27 A=26 B=19 so B is eliminated. Round 3 votes counts: C=40 E=33 A=27 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:198 B:197 D:197 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 -10 -16 B 4 0 -2 2 -10 C 4 2 0 0 -10 D 10 -2 0 0 -14 E 16 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -10 -16 B 4 0 -2 2 -10 C 4 2 0 0 -10 D 10 -2 0 0 -14 E 16 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -10 -16 B 4 0 -2 2 -10 C 4 2 0 0 -10 D 10 -2 0 0 -14 E 16 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2875: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) A E C D B (10) C E D A B (7) A E B C D (7) D B C E A (5) E C A D B (3) D C E A B (3) D C A E B (3) D A C E B (3) B D C A E (3) B C E D A (3) B A E C D (3) A E C B D (3) E A C D B (2) E A C B D (2) D B C A E (2) C D E B A (2) C D E A B (2) B E C A D (2) B E A C D (2) B D A C E (2) B A E D C (2) B A D E C (2) A D E C B (2) A B E D C (2) A B E C D (2) E C B A D (1) E C A B D (1) D C E B A (1) D C B E A (1) C E D B A (1) C B D E A (1) B C D E A (1) A E D B C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -6 -2 -2 B -10 0 2 2 -10 C 6 -2 0 10 4 D 2 -2 -10 0 -8 E 2 10 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999996 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 10 -6 -2 -2 B -10 0 2 2 -10 C 6 -2 0 10 4 D 2 -2 -10 0 -8 E 2 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000001 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=29 D=18 C=13 E=9 so E is eliminated. Round 2 votes counts: A=33 B=31 D=18 C=18 so D is eliminated. Round 3 votes counts: B=38 A=36 C=26 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:209 E:208 A:200 B:192 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 -2 -2 B -10 0 2 2 -10 C 6 -2 0 10 4 D 2 -2 -10 0 -8 E 2 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000001 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -2 -2 B -10 0 2 2 -10 C 6 -2 0 10 4 D 2 -2 -10 0 -8 E 2 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000001 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -2 -2 B -10 0 2 2 -10 C 6 -2 0 10 4 D 2 -2 -10 0 -8 E 2 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000001 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2876: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) A B D C E (7) E C D B A (6) E D C A B (5) E C A D B (4) D E A C B (4) D E A B C (4) C B A E D (4) A B C D E (4) E D A C B (3) D E C B A (3) D E B C A (3) D A E B C (3) C E D B A (3) C E B A D (3) B D C E A (3) A D B E C (3) A B D E C (3) E D C B A (2) E A D C B (2) D A B E C (2) B C A E D (2) B C A D E (2) A D E B C (2) E C D A B (1) D E C A B (1) D E B A C (1) D B C E A (1) D B A E C (1) C E B D A (1) C E A B D (1) C B E A D (1) C A B E D (1) B A D C E (1) B A C E D (1) A E C D B (1) A D E C B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 6 -4 B -8 0 4 -12 -8 C -6 -4 0 -10 -10 D -6 12 10 0 14 E 4 8 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.250000 Sum of squares = 0.430555555601 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 0.583333 D: 0.750000 E: 1.000000 A B C D E A 0 8 6 6 -4 B -8 0 4 -12 -8 C -6 -4 0 -10 -10 D -6 12 10 0 14 E 4 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.250000 Sum of squares = 0.430555555551 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 0.583333 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 A=23 B=17 C=14 so C is eliminated. Round 2 votes counts: E=31 A=24 D=23 B=22 so B is eliminated. Round 3 votes counts: A=42 E=32 D=26 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 A:208 E:204 B:188 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 6 -4 B -8 0 4 -12 -8 C -6 -4 0 -10 -10 D -6 12 10 0 14 E 4 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.250000 Sum of squares = 0.430555555551 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 0.583333 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 6 -4 B -8 0 4 -12 -8 C -6 -4 0 -10 -10 D -6 12 10 0 14 E 4 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.250000 Sum of squares = 0.430555555551 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 0.583333 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 6 -4 B -8 0 4 -12 -8 C -6 -4 0 -10 -10 D -6 12 10 0 14 E 4 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.250000 Sum of squares = 0.430555555551 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 0.583333 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2877: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) B C A D E (8) E A D B C (5) A B C E D (5) D C E B A (4) A E D C B (4) A E D B C (4) A E B C D (4) A B E C D (4) E D A B C (3) E A D C B (3) D E C B A (3) B C A E D (3) A B C D E (3) E A B D C (2) D E C A B (2) D E B C A (2) D E A C B (2) D C E A B (2) C D A B E (2) C B D A E (2) C B A D E (2) B C D E A (2) A E B D C (2) A C D E B (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D C A (1) E B D A C (1) D C B E A (1) D C A E B (1) C D B E A (1) B E A D C (1) B D E C A (1) B C D A E (1) B A C D E (1) A D E C B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 26 20 12 2 B -26 0 10 -10 -24 C -20 -10 0 -16 -16 D -12 10 16 0 -12 E -2 24 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 20 12 2 B -26 0 10 -10 -24 C -20 -10 0 -16 -16 D -12 10 16 0 -12 E -2 24 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=28 D=17 B=17 C=7 so C is eliminated. Round 2 votes counts: A=31 E=28 B=21 D=20 so D is eliminated. Round 3 votes counts: E=43 A=34 B=23 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:225 D:201 B:175 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 20 12 2 B -26 0 10 -10 -24 C -20 -10 0 -16 -16 D -12 10 16 0 -12 E -2 24 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 20 12 2 B -26 0 10 -10 -24 C -20 -10 0 -16 -16 D -12 10 16 0 -12 E -2 24 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 20 12 2 B -26 0 10 -10 -24 C -20 -10 0 -16 -16 D -12 10 16 0 -12 E -2 24 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2878: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B A E C (6) A E B D C (6) D B C A E (5) B D A E C (5) D C B A E (4) D B C E A (4) C D A E B (4) B D E A C (4) E C A B D (3) D C B E A (3) C E A D B (3) C E A B D (3) B E A D C (3) A E C B D (3) C D E B A (2) C D E A B (2) C D B E A (2) C A E D B (2) C A D E B (2) B D C E A (2) B A D E C (2) A E D C B (2) A C E D B (2) E C B A D (1) E B A C D (1) D A C E B (1) D A C B E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D B A E (1) B E D C A (1) B E D A C (1) B D E C A (1) B C E D A (1) B C D E A (1) B A E D C (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 0 -6 -2 B 6 0 -10 2 -4 C 0 10 0 -4 -4 D 6 -2 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999934 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 -2 B 6 0 -10 2 -4 C 0 10 0 -4 -4 D 6 -2 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 B=22 E=15 A=15 so E is eliminated. Round 2 votes counts: C=28 A=25 D=24 B=23 so B is eliminated. Round 3 votes counts: D=38 A=32 C=30 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:206 E:203 C:201 B:197 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -6 -2 B 6 0 -10 2 -4 C 0 10 0 -4 -4 D 6 -2 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 -2 B 6 0 -10 2 -4 C 0 10 0 -4 -4 D 6 -2 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 -2 B 6 0 -10 2 -4 C 0 10 0 -4 -4 D 6 -2 4 0 4 E 2 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000054 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2879: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) B A D E C (13) E D C B A (10) D E B C A (9) A C B E D (8) B A C D E (7) A B C D E (7) C A B E D (6) C A E D B (4) D E B A C (3) C A E B D (3) B A D C E (3) E D C A B (2) E C D A B (2) D E C B A (2) C E A D B (2) B D A E C (2) D B E A C (1) B D E A C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -6 10 10 B 2 0 -4 4 -2 C 6 4 0 6 8 D -10 -4 -6 0 -4 E -10 2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 10 10 B 2 0 -4 4 -2 C 6 4 0 6 8 D -10 -4 -6 0 -4 E -10 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=26 A=17 D=15 E=14 so E is eliminated. Round 2 votes counts: C=30 D=27 B=26 A=17 so A is eliminated. Round 3 votes counts: C=38 B=35 D=27 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:206 B:200 E:194 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 10 10 B 2 0 -4 4 -2 C 6 4 0 6 8 D -10 -4 -6 0 -4 E -10 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 10 10 B 2 0 -4 4 -2 C 6 4 0 6 8 D -10 -4 -6 0 -4 E -10 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 10 10 B 2 0 -4 4 -2 C 6 4 0 6 8 D -10 -4 -6 0 -4 E -10 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2880: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) E A D C B (6) B C D A E (6) D E A C B (5) A E C D B (5) E A D B C (4) A E D C B (4) E D A C B (3) D E C A B (3) D E B C A (3) D C B A E (3) C A B D E (3) B E D C A (3) B E A C D (3) B D C E A (3) B C A E D (3) E D A B C (2) D C B E A (2) C B D A E (2) C B A D E (2) B D E C A (2) A E B C D (2) A C E B D (2) A C D E B (2) A C B D E (2) E D B A C (1) E B D A C (1) E B A D C (1) E A B C D (1) D A E C B (1) C D A B E (1) C A D B E (1) B E C D A (1) B E C A D (1) B C E A D (1) B C D E A (1) B C A D E (1) B A E C D (1) B A C E D (1) A E C B D (1) A C E D B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 16 2 0 6 B -16 0 -14 -8 -10 C -2 14 0 -8 -10 D 0 8 8 0 0 E -6 10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.364785 B: 0.000000 C: 0.000000 D: 0.635215 E: 0.000000 Sum of squares = 0.536566124596 Cumulative probabilities = A: 0.364785 B: 0.364785 C: 0.364785 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 0 6 B -16 0 -14 -8 -10 C -2 14 0 -8 -10 D 0 8 8 0 0 E -6 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999622 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=24 A=21 E=19 C=9 so C is eliminated. Round 2 votes counts: B=31 D=25 A=25 E=19 so E is eliminated. Round 3 votes counts: A=36 B=33 D=31 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:208 E:207 C:197 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 0 6 B -16 0 -14 -8 -10 C -2 14 0 -8 -10 D 0 8 8 0 0 E -6 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999622 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 0 6 B -16 0 -14 -8 -10 C -2 14 0 -8 -10 D 0 8 8 0 0 E -6 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999622 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 0 6 B -16 0 -14 -8 -10 C -2 14 0 -8 -10 D 0 8 8 0 0 E -6 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999622 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2881: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) E D C A B (8) A B E D C (8) A E B D C (7) D C E B A (6) E D C B A (4) D C E A B (4) B A C D E (4) A B C E D (4) D E C B A (3) D E C A B (3) C D E A B (3) C D B A E (3) B A E C D (3) A B C D E (3) E D A C B (2) E D A B C (2) E B A D C (2) E A D B C (2) C D E B A (2) B A E D C (2) B A C E D (2) E D B C A (1) E B D C A (1) E B D A C (1) E A D C B (1) D C B E A (1) C D B E A (1) C B D A E (1) C B A D E (1) B E A D C (1) B D E C A (1) B C D E A (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 14 8 2 0 B -14 0 14 8 -4 C -8 -14 0 -20 -24 D -2 -8 20 0 -22 E 0 4 24 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.482861 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.517139 Sum of squares = 0.500587485681 Cumulative probabilities = A: 0.482861 B: 0.482861 C: 0.482861 D: 0.482861 E: 1.000000 A B C D E A 0 14 8 2 0 B -14 0 14 8 -4 C -8 -14 0 -20 -24 D -2 -8 20 0 -22 E 0 4 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=24 D=17 B=16 C=11 so C is eliminated. Round 2 votes counts: A=32 D=26 E=24 B=18 so B is eliminated. Round 3 votes counts: A=45 D=30 E=25 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:225 A:212 B:202 D:194 C:167 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 2 0 B -14 0 14 8 -4 C -8 -14 0 -20 -24 D -2 -8 20 0 -22 E 0 4 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 2 0 B -14 0 14 8 -4 C -8 -14 0 -20 -24 D -2 -8 20 0 -22 E 0 4 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 2 0 B -14 0 14 8 -4 C -8 -14 0 -20 -24 D -2 -8 20 0 -22 E 0 4 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2882: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E A B C D (9) D C B A E (7) E C D A B (6) E A D B C (4) E A B D C (4) D B C A E (4) D A B C E (4) C D E B A (4) B A D C E (4) A E B D C (4) E D A C B (3) D B A C E (3) C E D B A (3) C D B E A (3) E C D B A (2) E C B A D (2) E C A B D (2) D C E A B (2) A D B C E (2) A B E D C (2) A B E C D (2) A B D E C (2) A B D C E (2) E B A C D (1) D E C A B (1) D C A E B (1) C B E A D (1) B D A C E (1) B C D A E (1) B C A D E (1) B A C E D (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 2 0 -10 6 B -2 0 6 -20 2 C 0 -6 0 -2 10 D 10 20 2 0 6 E -6 -2 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -10 6 B -2 0 6 -20 2 C 0 -6 0 -2 10 D 10 20 2 0 6 E -6 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=22 C=21 A=15 B=9 so B is eliminated. Round 2 votes counts: E=33 D=23 C=23 A=21 so A is eliminated. Round 3 votes counts: E=42 D=33 C=25 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:201 A:199 B:193 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -10 6 B -2 0 6 -20 2 C 0 -6 0 -2 10 D 10 20 2 0 6 E -6 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -10 6 B -2 0 6 -20 2 C 0 -6 0 -2 10 D 10 20 2 0 6 E -6 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -10 6 B -2 0 6 -20 2 C 0 -6 0 -2 10 D 10 20 2 0 6 E -6 -2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2883: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (14) B C D E A (11) A E D C B (9) D B C E A (7) D A E B C (6) C B E A D (6) A E C D B (6) E A C B D (5) D E A B C (4) D E B A C (3) C B A E D (3) B D C E A (3) E C A B D (2) E A D C B (2) D B E C A (2) B C E A D (2) B C A E D (2) E A C D B (1) D E B C A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E B D A (1) C E B A D (1) C E A B D (1) C A E B D (1) B D C A E (1) B C D A E (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 8 14 -6 B -6 0 -8 10 -16 C -8 8 0 16 -16 D -14 -10 -16 0 -14 E 6 16 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 14 -6 B -6 0 -8 10 -16 C -8 8 0 16 -16 D -14 -10 -16 0 -14 E 6 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=26 B=20 C=13 E=10 so E is eliminated. Round 2 votes counts: A=39 D=26 B=20 C=15 so C is eliminated. Round 3 votes counts: A=43 B=31 D=26 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:226 A:211 C:200 B:190 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 14 -6 B -6 0 -8 10 -16 C -8 8 0 16 -16 D -14 -10 -16 0 -14 E 6 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 14 -6 B -6 0 -8 10 -16 C -8 8 0 16 -16 D -14 -10 -16 0 -14 E 6 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 14 -6 B -6 0 -8 10 -16 C -8 8 0 16 -16 D -14 -10 -16 0 -14 E 6 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2884: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) A D B E C (11) E C B D A (7) C A E D B (7) E B D C A (5) B D E A C (5) D E B A C (4) D B A E C (4) C A D E B (4) A C D B E (4) A D E B C (3) A D B C E (3) E B D A C (2) D B E A C (2) C E B A D (2) C E A D B (2) C E A B D (2) B D A E C (2) A B D C E (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A C B (1) E C D B A (1) E C D A B (1) D A B E C (1) C A E B D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D C A (1) B E D A C (1) B D A C E (1) B A D E C (1) A D C E B (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 2 -4 0 B 4 0 2 -8 -16 C -2 -2 0 -8 -12 D 4 8 8 0 2 E 0 16 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -4 0 B 4 0 2 -8 -16 C -2 -2 0 -8 -12 D 4 8 8 0 2 E 0 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=26 E=20 D=11 B=11 so D is eliminated. Round 2 votes counts: C=32 A=27 E=24 B=17 so B is eliminated. Round 3 votes counts: A=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:213 D:211 A:197 B:191 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -4 0 B 4 0 2 -8 -16 C -2 -2 0 -8 -12 D 4 8 8 0 2 E 0 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -4 0 B 4 0 2 -8 -16 C -2 -2 0 -8 -12 D 4 8 8 0 2 E 0 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -4 0 B 4 0 2 -8 -16 C -2 -2 0 -8 -12 D 4 8 8 0 2 E 0 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2885: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) C D A E B (8) B E A D C (7) C A D E B (6) D C B E A (5) D C A E B (5) D B E C A (4) A E B C D (4) C D B E A (3) C A E D B (3) E B D A C (2) D E B A C (2) D E A B C (2) D C E B A (2) D B E A C (2) D A E B C (2) C D A B E (2) C B E A D (2) C A E B D (2) B C E A D (2) B A E C D (2) A E D B C (2) A C D E B (2) A B E C D (2) E D B A C (1) E B A D C (1) E A D B C (1) D A C E B (1) C B D E A (1) C B A E D (1) C A D B E (1) C A B E D (1) B E D C A (1) B D E C A (1) B D E A C (1) B C D E A (1) B C A E D (1) A E C D B (1) A E C B D (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -4 -10 0 B 2 0 0 -14 -2 C 4 0 0 -4 2 D 10 14 4 0 4 E 0 2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -10 0 B 2 0 0 -14 -2 C 4 0 0 -4 2 D 10 14 4 0 4 E 0 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 B=25 A=15 E=5 so E is eliminated. Round 2 votes counts: C=30 B=28 D=26 A=16 so A is eliminated. Round 3 votes counts: C=36 B=34 D=30 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:216 C:201 E:198 B:193 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -10 0 B 2 0 0 -14 -2 C 4 0 0 -4 2 D 10 14 4 0 4 E 0 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -10 0 B 2 0 0 -14 -2 C 4 0 0 -4 2 D 10 14 4 0 4 E 0 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -10 0 B 2 0 0 -14 -2 C 4 0 0 -4 2 D 10 14 4 0 4 E 0 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2886: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (15) B A C D E (11) D E C A B (7) B E A C D (5) E C A D B (4) D C A E B (4) B D E C A (4) B D A C E (4) B A D C E (4) A C E D B (4) E D B C A (3) B A E C D (3) B A C E D (3) A C D E B (3) A C D B E (3) E B A C D (2) E A C B D (2) D C A B E (2) C A D E B (2) A C B D E (2) E D C B A (1) D E B C A (1) D C B A E (1) D B E C A (1) C E A D B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E A C (1) B A D E C (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 8 0 B -4 0 -6 -8 0 C -4 6 0 -4 -8 D -8 8 4 0 6 E 0 0 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.640733 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.359267 Sum of squares = 0.539611603974 Cumulative probabilities = A: 0.640733 B: 0.640733 C: 0.640733 D: 0.640733 E: 1.000000 A B C D E A 0 4 4 8 0 B -4 0 -6 -8 0 C -4 6 0 -4 -8 D -8 8 4 0 6 E 0 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=27 D=16 A=14 C=4 so C is eliminated. Round 2 votes counts: B=39 E=28 A=17 D=16 so D is eliminated. Round 3 votes counts: B=41 E=36 A=23 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:208 D:205 E:201 C:195 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 8 0 B -4 0 -6 -8 0 C -4 6 0 -4 -8 D -8 8 4 0 6 E 0 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 8 0 B -4 0 -6 -8 0 C -4 6 0 -4 -8 D -8 8 4 0 6 E 0 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 8 0 B -4 0 -6 -8 0 C -4 6 0 -4 -8 D -8 8 4 0 6 E 0 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2887: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) A E C B D (8) D C B E A (6) D B C E A (6) C B A D E (6) B C D A E (6) B C A D E (6) E D A C B (5) E D A B C (5) A E B C D (5) A C E B D (4) A B C E D (4) C B D A E (3) D E C B A (2) D E B C A (2) D B E C A (2) C D B A E (2) B D C E A (2) A E C D B (2) E A D B C (1) E A C D B (1) E A B D C (1) D E C A B (1) D E B A C (1) D C E B A (1) D C B A E (1) C D A E B (1) C A B D E (1) B D C A E (1) B C A E D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 0 6 6 B -4 0 -14 -2 -4 C 0 14 0 4 6 D -6 2 -4 0 0 E -6 4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.592658 B: 0.000000 C: 0.407342 D: 0.000000 E: 0.000000 Sum of squares = 0.517170826154 Cumulative probabilities = A: 0.592658 B: 0.592658 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 6 6 B -4 0 -14 -2 -4 C 0 14 0 4 6 D -6 2 -4 0 0 E -6 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 D=22 B=16 C=13 so C is eliminated. Round 2 votes counts: A=26 D=25 B=25 E=24 so E is eliminated. Round 3 votes counts: A=40 D=35 B=25 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:208 D:196 E:196 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 6 6 B -4 0 -14 -2 -4 C 0 14 0 4 6 D -6 2 -4 0 0 E -6 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 6 B -4 0 -14 -2 -4 C 0 14 0 4 6 D -6 2 -4 0 0 E -6 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 6 B -4 0 -14 -2 -4 C 0 14 0 4 6 D -6 2 -4 0 0 E -6 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2888: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D E A C B (5) C B E D A (5) B A E D C (5) E B A D C (4) D A E C B (4) C B A D E (4) B A C E D (4) B A C D E (4) E D C A B (3) E C D B A (3) C E D B A (3) C E B D A (3) B A E C D (3) E D A C B (2) E C B D A (2) E B D C A (2) E B C D A (2) E A D B C (2) D E C A B (2) D C E A B (2) D A E B C (2) D A C B E (2) C B D E A (2) C B A E D (2) B C E A D (2) B C A D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E B C A D (1) D C A B E (1) D A C E B (1) C D A E B (1) C B E A D (1) B E C A D (1) B C A E D (1) B A D E C (1) A E B D C (1) A D E B C (1) A D C B E (1) A C D B E (1) A B E D C (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 6 -10 -10 B 20 0 -2 12 -8 C -6 2 0 -4 -12 D 10 -12 4 0 -22 E 10 8 12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 6 -10 -10 B 20 0 -2 12 -8 C -6 2 0 -4 -12 D 10 -12 4 0 -22 E 10 8 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=23 C=21 D=19 A=8 so A is eliminated. Round 2 votes counts: E=30 B=27 C=22 D=21 so D is eliminated. Round 3 votes counts: E=44 C=29 B=27 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:211 C:190 D:190 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 6 -10 -10 B 20 0 -2 12 -8 C -6 2 0 -4 -12 D 10 -12 4 0 -22 E 10 8 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -10 -10 B 20 0 -2 12 -8 C -6 2 0 -4 -12 D 10 -12 4 0 -22 E 10 8 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -10 -10 B 20 0 -2 12 -8 C -6 2 0 -4 -12 D 10 -12 4 0 -22 E 10 8 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2889: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) A E B D C (6) D B C A E (5) C D B E A (5) A E C B D (5) D B C E A (4) C A E D B (4) E C A D B (3) E A C B D (3) C E D B A (3) C E A D B (3) C D B A E (3) C A D E B (3) B D C E A (3) B D C A E (3) B D A C E (3) A E B C D (3) E B A D C (2) D C B E A (2) C E D A B (2) C A D B E (2) B D A E C (2) B A D E C (2) A E C D B (2) A C E D B (2) A C E B D (2) A B E D C (2) E C D B A (1) E B D A C (1) E A B C D (1) D C B A E (1) D B E C A (1) C D E B A (1) B E D A C (1) B E A D C (1) B D E C A (1) B D E A C (1) B A E D C (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 0 14 8 B -6 0 4 6 -8 C 0 -4 0 -2 6 D -14 -6 2 0 -10 E -8 8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.604247 B: 0.000000 C: 0.395753 D: 0.000000 E: 0.000000 Sum of squares = 0.521734784854 Cumulative probabilities = A: 0.604247 B: 0.604247 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 14 8 B -6 0 4 6 -8 C 0 -4 0 -2 6 D -14 -6 2 0 -10 E -8 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=25 E=18 B=18 D=13 so D is eliminated. Round 2 votes counts: C=29 B=28 A=25 E=18 so E is eliminated. Round 3 votes counts: A=36 C=33 B=31 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:202 C:200 B:198 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 14 8 B -6 0 4 6 -8 C 0 -4 0 -2 6 D -14 -6 2 0 -10 E -8 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 14 8 B -6 0 4 6 -8 C 0 -4 0 -2 6 D -14 -6 2 0 -10 E -8 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 14 8 B -6 0 4 6 -8 C 0 -4 0 -2 6 D -14 -6 2 0 -10 E -8 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2890: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) D E C B A (7) C D E B A (6) C A B D E (6) D C E A B (5) C D A B E (4) A C B D E (4) E D B C A (3) E D B A C (3) B A E C D (3) A B E C D (3) A B C E D (3) E B D A C (2) E B C A D (2) D E A B C (2) C D E A B (2) C D A E B (2) C A D B E (2) C A B E D (2) B C A E D (2) B A C E D (2) A E B D C (2) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E B C A (1) D C E B A (1) D C A E B (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C B E (1) C E D B A (1) C E B D A (1) C B E D A (1) C B E A D (1) C B A E D (1) C B A D E (1) B E A D C (1) B C E A D (1) B A E D C (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -18 -2 -10 B 10 0 -10 6 -16 C 18 10 0 6 4 D 2 -6 -6 0 0 E 10 16 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -2 -10 B 10 0 -10 6 -16 C 18 10 0 6 4 D 2 -6 -6 0 0 E 10 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=24 D=22 A=14 B=10 so B is eliminated. Round 2 votes counts: C=33 E=25 D=22 A=20 so A is eliminated. Round 3 votes counts: C=42 E=35 D=23 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:211 B:195 D:195 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -18 -2 -10 B 10 0 -10 6 -16 C 18 10 0 6 4 D 2 -6 -6 0 0 E 10 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -2 -10 B 10 0 -10 6 -16 C 18 10 0 6 4 D 2 -6 -6 0 0 E 10 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -2 -10 B 10 0 -10 6 -16 C 18 10 0 6 4 D 2 -6 -6 0 0 E 10 16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2891: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) D B E A C (5) C E A D B (5) B D A E C (5) D E B A C (4) D B E C A (4) B D C E A (4) B C A D E (4) A B C E D (4) E D C A B (3) E C D A B (3) C E D B A (3) C E D A B (3) C E A B D (3) B D A C E (3) E D A C B (2) E A D C B (2) D E B C A (2) D E A B C (2) D B C E A (2) C E B A D (2) C B A E D (2) B C D E A (2) B C D A E (2) B A D E C (2) A C E B D (2) E C A D B (1) E A C D B (1) D B A E C (1) D A E B C (1) D A B E C (1) C E B D A (1) C D E B A (1) C B E A D (1) C A E B D (1) B D E C A (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D B C (1) A E C D B (1) A D B E C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -22 -14 -24 -28 B 22 0 10 -12 -2 C 14 -10 0 -8 -2 D 24 12 8 0 12 E 28 2 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -14 -24 -28 B 22 0 10 -12 -2 C 14 -10 0 -8 -2 D 24 12 8 0 12 E 28 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 C=22 E=12 A=11 so A is eliminated. Round 2 votes counts: B=31 D=30 C=25 E=14 so E is eliminated. Round 3 votes counts: D=38 C=31 B=31 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:210 B:209 C:197 A:156 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -14 -24 -28 B 22 0 10 -12 -2 C 14 -10 0 -8 -2 D 24 12 8 0 12 E 28 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 -24 -28 B 22 0 10 -12 -2 C 14 -10 0 -8 -2 D 24 12 8 0 12 E 28 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 -24 -28 B 22 0 10 -12 -2 C 14 -10 0 -8 -2 D 24 12 8 0 12 E 28 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2892: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (14) B E C A D (11) D A C B E (9) E B D A C (5) E B C A D (5) C A D E B (5) B E D A C (5) B D A C E (5) D A E C B (4) E A C D B (3) C A E D B (3) A D C E B (3) E D A B C (2) E C A D B (2) E B A D C (2) D C A B E (2) C B A D E (2) A C D E B (2) E D A C B (1) E C A B D (1) E B A C D (1) E A D C B (1) D E A B C (1) D B A E C (1) C E A D B (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D C A (1) B E C D A (1) B D A E C (1) B C E A D (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 14 20 -8 12 B -14 0 -12 -12 -12 C -20 12 0 -14 4 D 8 12 14 0 8 E -12 12 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 -8 12 B -14 0 -12 -12 -12 C -20 12 0 -14 4 D 8 12 14 0 8 E -12 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=27 E=23 C=14 A=5 so A is eliminated. Round 2 votes counts: D=34 B=27 E=23 C=16 so C is eliminated. Round 3 votes counts: D=43 B=30 E=27 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:219 E:194 C:191 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 20 -8 12 B -14 0 -12 -12 -12 C -20 12 0 -14 4 D 8 12 14 0 8 E -12 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 -8 12 B -14 0 -12 -12 -12 C -20 12 0 -14 4 D 8 12 14 0 8 E -12 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 -8 12 B -14 0 -12 -12 -12 C -20 12 0 -14 4 D 8 12 14 0 8 E -12 12 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2893: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) D E A B C (7) B A D C E (7) E C D B A (6) E C D A B (6) C B E A D (5) C B A E D (5) D A E B C (4) B C A D E (4) B A D E C (4) B A C D E (4) A D B E C (4) A B D E C (4) E D C A B (3) E D A C B (3) D A B E C (2) C B A D E (2) B D E A C (2) A B D C E (2) E D C B A (1) E D A B C (1) E C B D A (1) D E B A C (1) D E A C B (1) C E D B A (1) C E D A B (1) C E A D B (1) C E A B D (1) C A E D B (1) C A B D E (1) B D A E C (1) B C E D A (1) A D E C B (1) A D E B C (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 2 -2 -2 B 8 0 -2 8 -2 C -2 2 0 0 -6 D 2 -8 0 0 8 E 2 2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407411 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 A B C D E A 0 -8 2 -2 -2 B 8 0 -2 8 -2 C -2 2 0 0 -6 D 2 -8 0 0 8 E 2 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=23 E=21 D=15 A=14 so A is eliminated. Round 2 votes counts: B=29 C=28 D=22 E=21 so E is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:206 D:201 E:201 C:197 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -2 -2 B 8 0 -2 8 -2 C -2 2 0 0 -6 D 2 -8 0 0 8 E 2 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 -2 B 8 0 -2 8 -2 C -2 2 0 0 -6 D 2 -8 0 0 8 E 2 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 -2 B 8 0 -2 8 -2 C -2 2 0 0 -6 D 2 -8 0 0 8 E 2 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2894: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (13) A B C E D (11) E D B A C (10) C A B D E (10) B A E C D (8) B A C E D (7) D E B C A (5) C D A B E (5) D C A E B (4) B E A D C (4) E D B C A (3) D E C A B (3) D C E A B (3) C D A E B (3) C A D B E (3) E B D A C (2) A C B E D (2) A C B D E (2) E B A D C (1) C A D E B (1) Total count = 100 A B C D E A 0 -6 -6 -2 12 B 6 0 2 -6 4 C 6 -2 0 4 2 D 2 6 -4 0 4 E -12 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -2 12 B 6 0 2 -6 4 C 6 -2 0 4 2 D 2 6 -4 0 4 E -12 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=22 B=19 E=16 A=15 so A is eliminated. Round 2 votes counts: B=30 D=28 C=26 E=16 so E is eliminated. Round 3 votes counts: D=41 B=33 C=26 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:205 D:204 B:203 A:199 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -2 12 B 6 0 2 -6 4 C 6 -2 0 4 2 D 2 6 -4 0 4 E -12 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -2 12 B 6 0 2 -6 4 C 6 -2 0 4 2 D 2 6 -4 0 4 E -12 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -2 12 B 6 0 2 -6 4 C 6 -2 0 4 2 D 2 6 -4 0 4 E -12 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2895: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) C A E B D (10) D B E A C (9) E A C D B (8) C B D E A (6) A E C D B (6) B D E C A (5) B D C A E (5) E A D B C (3) D B A E C (3) C B D A E (3) A C E D B (3) E D B A C (2) E D A B C (2) C E B A D (2) C B A D E (2) C A E D B (2) C A B E D (2) C A B D E (2) B D A E C (2) A D B E C (2) E C A D B (1) E A D C B (1) D E B A C (1) C E A B D (1) C B E D A (1) B D A C E (1) B C D E A (1) B C D A E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -12 -8 -10 B 12 0 -2 14 16 C 12 2 0 4 10 D 8 -14 -4 0 12 E 10 -16 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -8 -10 B 12 0 -2 14 16 C 12 2 0 4 10 D 8 -14 -4 0 12 E 10 -16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 E=17 D=13 A=12 so A is eliminated. Round 2 votes counts: C=35 B=27 E=23 D=15 so D is eliminated. Round 3 votes counts: B=41 C=35 E=24 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:220 C:214 D:201 E:186 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -12 -8 -10 B 12 0 -2 14 16 C 12 2 0 4 10 D 8 -14 -4 0 12 E 10 -16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -8 -10 B 12 0 -2 14 16 C 12 2 0 4 10 D 8 -14 -4 0 12 E 10 -16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -8 -10 B 12 0 -2 14 16 C 12 2 0 4 10 D 8 -14 -4 0 12 E 10 -16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2896: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (20) C E B A D (20) D A E B C (8) C E D A B (6) C B E A D (5) B A D E C (5) C D E A B (4) B E A C D (4) A D B E C (4) B E C A D (3) A B D E C (3) C E B D A (2) B E A D C (2) E C B A D (1) E C A B D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E A B (1) D C A B E (1) D A C E B (1) D A C B E (1) C E D B A (1) C B A E D (1) B C E A D (1) B A E D C (1) Total count = 100 A B C D E A 0 6 6 8 -10 B -6 0 8 2 2 C -6 -8 0 -2 -12 D -8 -2 2 0 -2 E 10 -2 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765436 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 6 6 8 -10 B -6 0 8 2 2 C -6 -8 0 -2 -12 D -8 -2 2 0 -2 E 10 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765372 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=33 B=16 A=7 E=5 so E is eliminated. Round 2 votes counts: C=41 D=33 B=17 A=9 so A is eliminated. Round 3 votes counts: C=41 D=38 B=21 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:211 A:205 B:203 D:195 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 8 -10 B -6 0 8 2 2 C -6 -8 0 -2 -12 D -8 -2 2 0 -2 E 10 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765372 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 -10 B -6 0 8 2 2 C -6 -8 0 -2 -12 D -8 -2 2 0 -2 E 10 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765372 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 -10 B -6 0 8 2 2 C -6 -8 0 -2 -12 D -8 -2 2 0 -2 E 10 -2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765372 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2897: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) B D A E C (6) B A D E C (6) C D E B A (5) E C D B A (4) C E D B A (4) A B E D C (4) E C D A B (3) E C A D B (3) E A B D C (3) D C B E A (3) D B C A E (3) C D B A E (3) B D A C E (3) B A D C E (3) A E C B D (3) A C E B D (3) E D C B A (2) E D B A C (2) D B E C A (2) D B E A C (2) C E A B D (2) C A E B D (2) B D C A E (2) A B E C D (2) A B D C E (2) E D C A B (1) E C A B D (1) E B D A C (1) E A C B D (1) D E B C A (1) D C E B A (1) D B C E A (1) D B A C E (1) C E D A B (1) C E A D B (1) C D B E A (1) A E B D C (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 8 -4 10 B 12 0 12 12 8 C -8 -12 0 -18 -16 D 4 -12 18 0 10 E -10 -8 16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 8 -4 10 B 12 0 12 12 8 C -8 -12 0 -18 -16 D 4 -12 18 0 10 E -10 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=21 B=20 C=19 D=14 so D is eliminated. Round 2 votes counts: B=29 A=26 C=23 E=22 so E is eliminated. Round 3 votes counts: C=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:210 A:201 E:194 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 8 -4 10 B 12 0 12 12 8 C -8 -12 0 -18 -16 D 4 -12 18 0 10 E -10 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 8 -4 10 B 12 0 12 12 8 C -8 -12 0 -18 -16 D 4 -12 18 0 10 E -10 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 8 -4 10 B 12 0 12 12 8 C -8 -12 0 -18 -16 D 4 -12 18 0 10 E -10 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2898: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) B E C D A (6) B E A D C (6) B E A C D (6) E B D C A (5) C D B E A (4) C D A B E (4) B E D C A (4) A E B D C (4) A D C E B (4) E B A D C (3) D C E B A (3) D C A E B (3) C B D E A (3) B C E A D (3) A C D B E (3) A C B E D (3) A C B D E (3) E B D A C (2) D E C B A (2) C A D E B (2) C A D B E (2) B C E D A (2) A D E C B (2) A D E B C (2) A C D E B (2) A B E C D (2) E D B A C (1) D E B C A (1) D E A C B (1) C D E B A (1) C A B E D (1) B E D A C (1) B E C A D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 2 -10 B 8 0 2 14 12 C 6 -2 0 10 0 D -2 -14 -10 0 -4 E 10 -12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 2 -10 B 8 0 2 14 12 C 6 -2 0 10 0 D -2 -14 -10 0 -4 E 10 -12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=27 C=23 E=11 D=10 so D is eliminated. Round 2 votes counts: C=29 B=29 A=27 E=15 so E is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:207 E:201 A:189 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 2 -10 B 8 0 2 14 12 C 6 -2 0 10 0 D -2 -14 -10 0 -4 E 10 -12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 2 -10 B 8 0 2 14 12 C 6 -2 0 10 0 D -2 -14 -10 0 -4 E 10 -12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 2 -10 B 8 0 2 14 12 C 6 -2 0 10 0 D -2 -14 -10 0 -4 E 10 -12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2899: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (8) B D C E A (7) A E C B D (7) D B C E A (6) B D C A E (5) A C B D E (5) E C D B A (4) A E C D B (4) D E B C A (3) D E B A C (3) B C D E A (3) A E D B C (3) E D B C A (2) E C A B D (2) E A D C B (2) D B C A E (2) B D A C E (2) B C A D E (2) A C B E D (2) E D C B A (1) E D B A C (1) E D A B C (1) E C B D A (1) E C B A D (1) E A C D B (1) D B E A C (1) D B A E C (1) C E A B D (1) C B A D E (1) C A E B D (1) C A B E D (1) B A D C E (1) A E D C B (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -2 -2 4 B 10 0 8 8 -2 C 2 -8 0 -6 2 D 2 -8 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 A B C D E A 0 -10 -2 -2 4 B 10 0 8 8 -2 C 2 -8 0 -6 2 D 2 -8 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999955 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=24 B=20 E=16 C=4 so C is eliminated. Round 2 votes counts: A=38 D=24 B=21 E=17 so E is eliminated. Round 3 votes counts: A=44 D=33 B=23 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:203 A:195 C:195 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -2 4 B 10 0 8 8 -2 C 2 -8 0 -6 2 D 2 -8 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999955 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -2 4 B 10 0 8 8 -2 C 2 -8 0 -6 2 D 2 -8 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999955 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -2 4 B 10 0 8 8 -2 C 2 -8 0 -6 2 D 2 -8 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999955 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2900: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (5) D E A C B (5) B C E A D (5) D E C B A (4) C E B D A (4) B C A E D (4) B A C E D (4) A D E B C (4) A C B D E (4) A B D E C (4) A B C D E (4) E D C B A (3) E C D B A (3) E B C D A (3) D A E C B (3) C B E D A (3) C B E A D (3) C B A E D (3) C A B E D (3) A D E C B (3) A D C E B (3) D E C A B (2) D E A B C (2) B E C D A (2) A D C B E (2) A D B C E (2) A C D B E (2) E C B D A (1) E B D C A (1) D E B C A (1) D A E B C (1) D A B E C (1) C E D B A (1) C A D B E (1) B A D E C (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 10 4 B 2 0 -8 4 2 C 4 8 0 4 2 D -10 -4 -4 0 0 E -4 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 10 4 B 2 0 -8 4 2 C 4 8 0 4 2 D -10 -4 -4 0 0 E -4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=19 C=18 E=16 B=16 so E is eliminated. Round 2 votes counts: A=31 D=27 C=22 B=20 so B is eliminated. Round 3 votes counts: C=36 A=36 D=28 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:204 B:200 E:196 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 10 4 B 2 0 -8 4 2 C 4 8 0 4 2 D -10 -4 -4 0 0 E -4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 4 B 2 0 -8 4 2 C 4 8 0 4 2 D -10 -4 -4 0 0 E -4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 4 B 2 0 -8 4 2 C 4 8 0 4 2 D -10 -4 -4 0 0 E -4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2901: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (12) B C E A D (12) B C D A E (8) A E D B C (7) C B D E A (6) A D E B C (5) C B E D A (4) A D E C B (4) D A C E B (3) D A B E C (3) C D B E A (3) E A D C B (2) B C D E A (2) B A E D C (2) A E D C B (2) E D A C B (1) E C B A D (1) E C A D B (1) E B C A D (1) E B A C D (1) E A D B C (1) E A C D B (1) E A B D C (1) E A B C D (1) D C A E B (1) D C A B E (1) D A E B C (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A E B (1) C B D A E (1) B E C A D (1) B C E D A (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 2 -10 12 B -8 0 0 -10 -4 C -2 0 0 0 0 D 10 10 0 0 12 E -12 4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.553910 D: 0.446090 E: 0.000000 Sum of squares = 0.505812580753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.553910 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -10 12 B -8 0 0 -10 -4 C -2 0 0 0 0 D 10 10 0 0 12 E -12 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=23 C=20 A=20 E=11 so E is eliminated. Round 2 votes counts: B=28 A=26 D=24 C=22 so C is eliminated. Round 3 votes counts: B=41 D=32 A=27 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:206 C:199 E:190 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -10 12 B -8 0 0 -10 -4 C -2 0 0 0 0 D 10 10 0 0 12 E -12 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -10 12 B -8 0 0 -10 -4 C -2 0 0 0 0 D 10 10 0 0 12 E -12 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -10 12 B -8 0 0 -10 -4 C -2 0 0 0 0 D 10 10 0 0 12 E -12 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2902: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (15) A E B C D (8) E D C A B (7) B A C D E (7) A B E C D (7) E A D C B (6) D C E B A (5) E A C D B (4) E A C B D (4) B C D A E (4) A E B D C (4) E A B D C (3) D B C A E (3) C D B E A (3) E C D A B (2) E A D B C (2) E A B C D (2) B D C A E (2) E D A C B (1) E C D B A (1) E C A D B (1) D E C B A (1) D E C A B (1) D C B A E (1) C D E B A (1) C B D A E (1) B C A D E (1) B A C E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 4 -18 B -8 0 -8 -8 -6 C -2 8 0 -4 -10 D -4 8 4 0 -10 E 18 6 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 4 -18 B -8 0 -8 -8 -6 C -2 8 0 -4 -10 D -4 8 4 0 -10 E 18 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=26 A=21 B=15 C=5 so C is eliminated. Round 2 votes counts: E=33 D=30 A=21 B=16 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:199 A:198 C:196 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 4 -18 B -8 0 -8 -8 -6 C -2 8 0 -4 -10 D -4 8 4 0 -10 E 18 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 4 -18 B -8 0 -8 -8 -6 C -2 8 0 -4 -10 D -4 8 4 0 -10 E 18 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 4 -18 B -8 0 -8 -8 -6 C -2 8 0 -4 -10 D -4 8 4 0 -10 E 18 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2903: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) E D B A C (8) D A B E C (7) E D A C B (6) D A E B C (6) E C B D A (5) E B C D A (4) C B E A D (4) A D C B E (4) C B A D E (3) C A D E B (3) C A B D E (3) B C E D A (3) E D A B C (2) E B D C A (2) E B D A C (2) D E A B C (2) D A E C B (2) C B A E D (2) B E C D A (2) B C A D E (2) B A D C E (2) A D E C B (2) E D C A B (1) E C D B A (1) E C A D B (1) E A D C B (1) D B A E C (1) D A B C E (1) C E A B D (1) C A E D B (1) B D A E C (1) B C A E D (1) A D C E B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 0 -12 -12 B 8 0 -6 -4 -24 C 0 6 0 -4 -12 D 12 4 4 0 -16 E 12 24 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 -12 -12 B 8 0 -6 -4 -24 C 0 6 0 -4 -12 D 12 4 4 0 -16 E 12 24 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=28 D=19 B=11 A=9 so A is eliminated. Round 2 votes counts: E=33 C=29 D=27 B=11 so B is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:232 D:202 C:195 B:187 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -12 -12 B 8 0 -6 -4 -24 C 0 6 0 -4 -12 D 12 4 4 0 -16 E 12 24 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -12 -12 B 8 0 -6 -4 -24 C 0 6 0 -4 -12 D 12 4 4 0 -16 E 12 24 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -12 -12 B 8 0 -6 -4 -24 C 0 6 0 -4 -12 D 12 4 4 0 -16 E 12 24 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2904: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) C B E A D (8) D A E B C (7) B C A E D (6) A E D B C (6) C D B E A (5) B C E A D (5) E A B C D (4) C B D E A (4) B A C E D (4) A E B D C (4) D A E C B (3) C E B D A (3) E D A C B (2) E B A C D (2) E A D B C (2) D C B E A (2) C B E D A (2) A D E B C (2) E D C A B (1) E C D B A (1) E C B A D (1) E C A B D (1) E A C D B (1) D E A B C (1) D C E A B (1) D C A E B (1) D B C A E (1) C E D B A (1) C E B A D (1) C D B A E (1) C B D A E (1) B E C A D (1) B C D A E (1) B C A D E (1) B A D C E (1) A E B C D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 6 -16 B 4 0 2 4 -10 C -2 -2 0 12 -2 D -6 -4 -12 0 -16 E 16 10 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 6 -16 B 4 0 2 4 -10 C -2 -2 0 12 -2 D -6 -4 -12 0 -16 E 16 10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 B=19 E=15 A=15 so E is eliminated. Round 2 votes counts: C=29 D=28 A=22 B=21 so B is eliminated. Round 3 votes counts: C=43 A=29 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:222 C:203 B:200 A:194 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 6 -16 B 4 0 2 4 -10 C -2 -2 0 12 -2 D -6 -4 -12 0 -16 E 16 10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 -16 B 4 0 2 4 -10 C -2 -2 0 12 -2 D -6 -4 -12 0 -16 E 16 10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 -16 B 4 0 2 4 -10 C -2 -2 0 12 -2 D -6 -4 -12 0 -16 E 16 10 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2905: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (13) A D E C B (9) B C D A E (5) B C A D E (5) C D A E B (4) B E A D C (4) B C E D A (4) B C D E A (4) B A C D E (4) E A D C B (3) E A D B C (3) B E C D A (3) A E D C B (3) E D C B A (2) E D B A C (2) C D E A B (2) C B D A E (2) C B A D E (2) C A D E B (2) B C E A D (2) A D C E B (2) A C D E B (2) E D C A B (1) E D B C A (1) E D A B C (1) E B D A C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C A E B (1) D A E C B (1) C D B A E (1) C A D B E (1) B E D C A (1) B C A E D (1) B A E C D (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 -2 0 B -10 0 -10 -14 -12 C -10 10 0 -4 -6 D 2 14 4 0 4 E 0 12 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -2 0 B -10 0 -10 -14 -12 C -10 10 0 -4 -6 D 2 14 4 0 4 E 0 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=28 A=20 C=14 D=4 so D is eliminated. Round 2 votes counts: B=34 E=30 A=21 C=15 so C is eliminated. Round 3 votes counts: B=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:212 A:209 E:207 C:195 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -2 0 B -10 0 -10 -14 -12 C -10 10 0 -4 -6 D 2 14 4 0 4 E 0 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -2 0 B -10 0 -10 -14 -12 C -10 10 0 -4 -6 D 2 14 4 0 4 E 0 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -2 0 B -10 0 -10 -14 -12 C -10 10 0 -4 -6 D 2 14 4 0 4 E 0 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2906: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) A D C B E (9) C E B A D (8) C A E D B (7) B E D A C (7) A C D E B (7) A C D B E (7) D B E A C (6) D B A E C (5) E B C D A (4) D A B E C (4) D A B C E (4) C E A B D (4) B D E A C (4) C A E B D (3) E B D C A (2) B E D C A (2) A D B C E (2) E C B A D (1) E B C A D (1) D E B C A (1) C D A E B (1) B A E D C (1) Total count = 100 A B C D E A 0 16 12 20 20 B -16 0 -14 -26 2 C -12 14 0 6 24 D -20 26 -6 0 20 E -20 -2 -24 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 20 20 B -16 0 -14 -26 2 C -12 14 0 6 24 D -20 26 -6 0 20 E -20 -2 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=25 D=20 B=14 E=8 so E is eliminated. Round 2 votes counts: C=34 A=25 B=21 D=20 so D is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:234 C:216 D:210 B:173 E:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 20 20 B -16 0 -14 -26 2 C -12 14 0 6 24 D -20 26 -6 0 20 E -20 -2 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 20 20 B -16 0 -14 -26 2 C -12 14 0 6 24 D -20 26 -6 0 20 E -20 -2 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 20 20 B -16 0 -14 -26 2 C -12 14 0 6 24 D -20 26 -6 0 20 E -20 -2 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2907: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (11) A D B E C (10) C E B D A (8) E C B D A (7) C D B E A (7) A E C B D (6) E C A B D (4) D A B C E (4) B D E C A (4) A E C D B (4) A D C B E (4) C E A B D (3) A D B C E (3) E A C B D (2) D B C A E (2) D B A C E (2) C E D B A (2) A E B D C (2) A B D E C (2) E C B A D (1) E B C D A (1) D C B A E (1) D A C B E (1) C E B A D (1) B E D C A (1) B E D A C (1) B E A D C (1) B D C E A (1) B A E D C (1) A E B C D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -8 -6 -10 B 4 0 -4 -6 12 C 8 4 0 -4 2 D 6 6 4 0 6 E 10 -12 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -6 -10 B 4 0 -4 -6 12 C 8 4 0 -4 2 D 6 6 4 0 6 E 10 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=21 C=21 E=15 B=9 so B is eliminated. Round 2 votes counts: A=35 D=26 C=21 E=18 so E is eliminated. Round 3 votes counts: A=38 C=34 D=28 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:211 C:205 B:203 E:195 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 -10 B 4 0 -4 -6 12 C 8 4 0 -4 2 D 6 6 4 0 6 E 10 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 -10 B 4 0 -4 -6 12 C 8 4 0 -4 2 D 6 6 4 0 6 E 10 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 -10 B 4 0 -4 -6 12 C 8 4 0 -4 2 D 6 6 4 0 6 E 10 -12 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2908: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) B A E C D (5) E A B C D (4) D B C E A (4) B E D A C (4) B E A D C (4) D C E B A (3) D C B A E (3) D C A E B (3) D C A B E (3) C D E A B (3) C A D E B (3) B D E A C (3) B A D E C (3) A E C B D (3) A C E B D (3) A B E C D (3) E B D C A (2) E A C B D (2) D C E A B (2) D B C A E (2) C D A E B (2) C A E D B (2) B E A C D (2) B A E D C (2) A B C E D (2) E D C B A (1) E C B D A (1) E C A D B (1) E C A B D (1) E B D A C (1) E B A D C (1) D E C B A (1) D C B E A (1) D B E C A (1) D B A E C (1) D B A C E (1) C E D A B (1) C E A D B (1) C E A B D (1) B D E C A (1) B D A E C (1) B A D C E (1) A E B C D (1) A D C B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 14 10 -6 B 12 0 14 20 0 C -14 -14 0 0 -12 D -10 -20 0 0 -10 E 6 0 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.319553 C: 0.000000 D: 0.000000 E: 0.680447 Sum of squares = 0.565122437089 Cumulative probabilities = A: 0.000000 B: 0.319553 C: 0.319553 D: 0.319553 E: 1.000000 A B C D E A 0 -12 14 10 -6 B 12 0 14 20 0 C -14 -14 0 0 -12 D -10 -20 0 0 -10 E 6 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 E=21 A=15 C=13 so C is eliminated. Round 2 votes counts: D=30 B=26 E=24 A=20 so A is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:214 A:203 C:180 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 14 10 -6 B 12 0 14 20 0 C -14 -14 0 0 -12 D -10 -20 0 0 -10 E 6 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 14 10 -6 B 12 0 14 20 0 C -14 -14 0 0 -12 D -10 -20 0 0 -10 E 6 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 14 10 -6 B 12 0 14 20 0 C -14 -14 0 0 -12 D -10 -20 0 0 -10 E 6 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2909: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (15) C A D B E (13) A C D B E (11) D B E A C (9) E B D C A (5) C D B A E (3) C A E B D (3) B E D C A (3) D A B E C (2) C E B A D (2) C B E D A (2) C B D E A (2) B E D A C (2) B D E C A (2) B D E A C (2) A D C B E (2) A C D E B (2) E D B A C (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A D C (1) E A B D C (1) E A B C D (1) D B E C A (1) D B C E A (1) D B C A E (1) D B A E C (1) D B A C E (1) C E B D A (1) C E A B D (1) C A E D B (1) C A B D E (1) A E C B D (1) A D C E B (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 10 -10 -10 B 14 0 4 -4 20 C -10 -4 0 -4 0 D 10 4 4 0 12 E 10 -20 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 -10 -10 B 14 0 4 -4 20 C -10 -4 0 -4 0 D 10 4 4 0 12 E 10 -20 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=27 A=19 D=16 B=9 so B is eliminated. Round 2 votes counts: E=32 C=29 D=20 A=19 so A is eliminated. Round 3 votes counts: C=43 E=33 D=24 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:217 D:215 C:191 E:189 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 10 -10 -10 B 14 0 4 -4 20 C -10 -4 0 -4 0 D 10 4 4 0 12 E 10 -20 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 -10 -10 B 14 0 4 -4 20 C -10 -4 0 -4 0 D 10 4 4 0 12 E 10 -20 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 -10 -10 B 14 0 4 -4 20 C -10 -4 0 -4 0 D 10 4 4 0 12 E 10 -20 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2910: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) D B C A E (7) E A C D B (6) E B D C A (5) E A B C D (5) C D A B E (5) E C A D B (4) D A C B E (4) B D E C A (4) A C E D B (4) A C D B E (4) E B C D A (3) D C A B E (3) B E D A C (3) A C D E B (3) E C B D A (2) E B A D C (2) E A C B D (2) D C B A E (2) B E D C A (2) B D C E A (2) B D A E C (2) A E C D B (2) E B D A C (1) E B C A D (1) D B A C E (1) C D B A E (1) C D A E B (1) C A E D B (1) B D E A C (1) B D A C E (1) B A D C E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -12 -26 14 B 8 0 12 2 16 C 12 -12 0 -12 10 D 26 -2 12 0 14 E -14 -16 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -26 14 B 8 0 12 2 16 C 12 -12 0 -12 10 D 26 -2 12 0 14 E -14 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=29 D=17 A=15 C=8 so C is eliminated. Round 2 votes counts: E=31 B=29 D=24 A=16 so A is eliminated. Round 3 votes counts: E=38 D=32 B=30 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:225 B:219 C:199 A:184 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -26 14 B 8 0 12 2 16 C 12 -12 0 -12 10 D 26 -2 12 0 14 E -14 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -26 14 B 8 0 12 2 16 C 12 -12 0 -12 10 D 26 -2 12 0 14 E -14 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -26 14 B 8 0 12 2 16 C 12 -12 0 -12 10 D 26 -2 12 0 14 E -14 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2911: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) C A D B E (7) E B A D C (6) C D A B E (6) E A B C D (5) D E C A B (5) D E B C A (5) E D B A C (4) E B D A C (4) D C A E B (4) A C B E D (4) E B A C D (3) D C E B A (3) C A D E B (3) B E D A C (3) B D E C A (3) A E C B D (3) D C B A E (2) B E A D C (2) B E A C D (2) B C A D E (2) B A E C D (2) E D A C B (1) E A D C B (1) E A C B D (1) D E C B A (1) D C E A B (1) D C A B E (1) C D A E B (1) C B A D E (1) B D C A E (1) B C D A E (1) B A C E D (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -10 8 0 B -8 0 -8 10 -6 C 10 8 0 6 -4 D -8 -10 -6 0 10 E 0 6 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.380000000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 A B C D E A 0 8 -10 8 0 B -8 0 -8 10 -6 C 10 8 0 6 -4 D -8 -10 -6 0 10 E 0 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 D=22 B=17 A=10 so A is eliminated. Round 2 votes counts: C=31 E=29 D=22 B=18 so B is eliminated. Round 3 votes counts: E=38 C=36 D=26 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:210 A:203 E:200 B:194 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 8 0 B -8 0 -8 10 -6 C 10 8 0 6 -4 D -8 -10 -6 0 10 E 0 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 8 0 B -8 0 -8 10 -6 C 10 8 0 6 -4 D -8 -10 -6 0 10 E 0 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 8 0 B -8 0 -8 10 -6 C 10 8 0 6 -4 D -8 -10 -6 0 10 E 0 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2912: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) D A C B E (9) A D C B E (6) C B E D A (5) A E B C D (5) E A B C D (4) D C B E A (4) C D A B E (4) A D E C B (4) A D E B C (4) A C B D E (4) E B A C D (3) A D C E B (3) E B C A D (2) D E C B A (2) D C E B A (2) D C A B E (2) D A C E B (2) C D B E A (2) C D B A E (2) C B D E A (2) B E C D A (2) B E A C D (2) B C E D A (2) E D C B A (1) E D B C A (1) E B D C A (1) E A B D C (1) D E A C B (1) D C B A E (1) D A E C B (1) C B D A E (1) C B A D E (1) C A D B E (1) B C E A D (1) A E D B C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 4 -12 6 B -8 0 -22 -8 6 C -4 22 0 8 12 D 12 8 -8 0 18 E -6 -6 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888964 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -12 6 B -8 0 -22 -8 6 C -4 22 0 8 12 D 12 8 -8 0 18 E -6 -6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=24 E=22 C=18 B=7 so B is eliminated. Round 2 votes counts: A=29 E=26 D=24 C=21 so C is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:219 D:215 A:203 B:184 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 4 -12 6 B -8 0 -22 -8 6 C -4 22 0 8 12 D 12 8 -8 0 18 E -6 -6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -12 6 B -8 0 -22 -8 6 C -4 22 0 8 12 D 12 8 -8 0 18 E -6 -6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -12 6 B -8 0 -22 -8 6 C -4 22 0 8 12 D 12 8 -8 0 18 E -6 -6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2913: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (5) B D E A C (5) A C D B E (5) E D B C A (4) D E B A C (4) B E D C A (4) D A E C B (3) D A C E B (3) C E B A D (3) C E A B D (3) C A E D B (3) C A D E B (3) B E D A C (3) B E C D A (3) E D C B A (2) E C D B A (2) E B C D A (2) D E A C B (2) D E A B C (2) D B A E C (2) D A B E C (2) C E D A B (2) C E B D A (2) C E A D B (2) C B E A D (2) C A B E D (2) B D A E C (2) B A D C E (2) B A C E D (2) A D C E B (2) A D C B E (2) A D B C E (2) A C B D E (2) A B D C E (2) E C B D A (1) D A E B C (1) C A E B D (1) B E C A D (1) B E A C D (1) B C E A D (1) B A C D E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 4 -12 -12 B 8 0 4 0 -6 C -4 -4 0 -10 -4 D 12 0 10 0 -2 E 12 6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 4 -12 -12 B 8 0 4 0 -6 C -4 -4 0 -10 -4 D 12 0 10 0 -2 E 12 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 D=19 A=17 E=16 so E is eliminated. Round 2 votes counts: B=32 C=26 D=25 A=17 so A is eliminated. Round 3 votes counts: C=34 B=34 D=32 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:212 D:210 B:203 C:189 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 -12 -12 B 8 0 4 0 -6 C -4 -4 0 -10 -4 D 12 0 10 0 -2 E 12 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -12 -12 B 8 0 4 0 -6 C -4 -4 0 -10 -4 D 12 0 10 0 -2 E 12 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -12 -12 B 8 0 4 0 -6 C -4 -4 0 -10 -4 D 12 0 10 0 -2 E 12 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2914: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) E D C A B (6) C E D A B (6) B C D A E (6) D C E B A (4) C D E B A (4) A E C D B (4) A B E D C (4) D E C B A (3) D C B E A (3) C D B E A (3) B D A C E (3) B C D E A (3) B A D E C (3) B A C D E (3) A B E C D (3) E D A C B (2) D E C A B (2) D C E A B (2) C D E A B (2) B D C A E (2) E C A D B (1) E A D C B (1) E A D B C (1) D E B A C (1) D B C E A (1) C E A D B (1) C B D E A (1) C B D A E (1) C B A D E (1) B D E C A (1) B D E A C (1) B D A E C (1) B C A D E (1) B A E D C (1) B A E C D (1) B A D C E (1) A E D C B (1) A E D B C (1) A E C B D (1) A E B D C (1) A E B C D (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -24 -32 -14 B 14 0 -2 2 8 C 24 2 0 -8 18 D 32 -2 8 0 22 E 14 -8 -18 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -24 -32 -14 B 14 0 -2 2 8 C 24 2 0 -8 18 D 32 -2 8 0 22 E 14 -8 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998458 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=19 A=19 D=16 E=11 so E is eliminated. Round 2 votes counts: B=35 D=24 A=21 C=20 so C is eliminated. Round 3 votes counts: D=39 B=38 A=23 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:230 C:218 B:211 E:183 A:158 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -24 -32 -14 B 14 0 -2 2 8 C 24 2 0 -8 18 D 32 -2 8 0 22 E 14 -8 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998458 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -24 -32 -14 B 14 0 -2 2 8 C 24 2 0 -8 18 D 32 -2 8 0 22 E 14 -8 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998458 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -24 -32 -14 B 14 0 -2 2 8 C 24 2 0 -8 18 D 32 -2 8 0 22 E 14 -8 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999998458 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2915: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) A C E D B (8) D B E A C (7) A D C B E (7) A C E B D (7) A C D B E (7) D B E C A (6) E B C D A (5) D B A E C (5) C E B A D (5) E B D C A (4) D A B C E (4) C E A B D (4) B D E C A (3) A C D E B (3) A C B D E (3) E D B C A (2) D A B E C (2) E C B A D (1) D A E C B (1) D A C B E (1) C B E A D (1) C A B E D (1) B E D C A (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 20 14 18 22 B -20 0 -20 -10 -2 C -14 20 0 10 26 D -18 10 -10 0 2 E -22 2 -26 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 14 18 22 B -20 0 -20 -10 -2 C -14 20 0 10 26 D -18 10 -10 0 2 E -22 2 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=26 C=21 E=12 B=4 so B is eliminated. Round 2 votes counts: A=37 D=29 C=21 E=13 so E is eliminated. Round 3 votes counts: A=37 D=36 C=27 so C is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:237 C:221 D:192 E:176 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 14 18 22 B -20 0 -20 -10 -2 C -14 20 0 10 26 D -18 10 -10 0 2 E -22 2 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 14 18 22 B -20 0 -20 -10 -2 C -14 20 0 10 26 D -18 10 -10 0 2 E -22 2 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 14 18 22 B -20 0 -20 -10 -2 C -14 20 0 10 26 D -18 10 -10 0 2 E -22 2 -26 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2916: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) E C A D B (6) D A C B E (6) A D C E B (6) E C B A D (5) E C A B D (5) E B C A D (5) B E C D A (5) D A B C E (4) B E D C A (4) B E C A D (4) E A C D B (3) C E A D B (3) B E D A C (3) A C E D B (3) A C D E B (3) E B D A C (2) E B C D A (2) D A E B C (2) D A C E B (2) C A D E B (2) B D A C E (2) B C E D A (2) E D A B C (1) E A D C B (1) D B E A C (1) C E A B D (1) C D A B E (1) C A E D B (1) C A D B E (1) B D E A C (1) B D C E A (1) B D C A E (1) B C E A D (1) B C D E A (1) B C D A E (1) A D E C B (1) Total count = 100 A B C D E A 0 4 -4 2 -14 B -4 0 0 -8 -8 C 4 0 0 10 -2 D -2 8 -10 0 -14 E 14 8 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -4 2 -14 B -4 0 0 -8 -8 C 4 0 0 10 -2 D -2 8 -10 0 -14 E 14 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 D=22 A=13 C=9 so C is eliminated. Round 2 votes counts: E=34 B=26 D=23 A=17 so A is eliminated. Round 3 votes counts: E=38 D=36 B=26 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:206 A:194 D:191 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 2 -14 B -4 0 0 -8 -8 C 4 0 0 10 -2 D -2 8 -10 0 -14 E 14 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 2 -14 B -4 0 0 -8 -8 C 4 0 0 10 -2 D -2 8 -10 0 -14 E 14 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 2 -14 B -4 0 0 -8 -8 C 4 0 0 10 -2 D -2 8 -10 0 -14 E 14 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993358 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2917: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (6) E C B D A (5) A B D E C (5) D A E C B (4) B A E C D (4) E B C A D (3) D C E A B (3) D C A E B (3) D C A B E (3) C B E D A (3) B E C A D (3) B A C E D (3) A D C B E (3) A B E C D (3) A B D C E (3) A B C D E (3) E D A B C (2) E C D B A (2) E B C D A (2) D E C A B (2) D C E B A (2) D A C B E (2) C E D B A (2) C E B D A (2) C B A D E (2) C A D B E (2) B C E A D (2) A D E B C (2) A B C E D (2) E D C B A (1) E B A C D (1) D E A C B (1) D A E B C (1) D A C E B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D E A (1) C B D A E (1) B C E D A (1) A E D B C (1) A E B D C (1) A D B E C (1) A C D B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -8 -6 10 B -4 0 -12 4 6 C 8 12 0 16 10 D 6 -4 -16 0 12 E -10 -6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -6 10 B -4 0 -12 4 6 C 8 12 0 16 10 D 6 -4 -16 0 12 E -10 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=22 C=22 E=16 B=13 so B is eliminated. Round 2 votes counts: A=34 C=25 D=22 E=19 so E is eliminated. Round 3 votes counts: C=40 A=35 D=25 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:200 D:199 B:197 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -6 10 B -4 0 -12 4 6 C 8 12 0 16 10 D 6 -4 -16 0 12 E -10 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -6 10 B -4 0 -12 4 6 C 8 12 0 16 10 D 6 -4 -16 0 12 E -10 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -6 10 B -4 0 -12 4 6 C 8 12 0 16 10 D 6 -4 -16 0 12 E -10 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2918: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) A B D E C (13) E C D B A (10) C E A D B (8) A E C D B (7) A B D C E (5) A E D B C (4) C E B D A (3) C B D E A (3) A D B E C (3) E C D A B (2) E C A D B (2) D B E C A (2) C D B E A (2) B D E C A (2) B D C E A (2) B D C A E (2) B D A E C (2) B A D E C (2) A C E B D (2) E D C B A (1) E A C D B (1) D B C E A (1) C B E D A (1) C A E B D (1) B D A C E (1) B C D E A (1) A E C B D (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -14 2 -10 B -2 0 -18 -16 -14 C 14 18 0 18 -6 D -2 16 -18 0 -14 E 10 14 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -14 2 -10 B -2 0 -18 -16 -14 C 14 18 0 18 -6 D -2 16 -18 0 -14 E 10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=32 E=16 B=12 D=3 so D is eliminated. Round 2 votes counts: A=37 C=32 E=16 B=15 so B is eliminated. Round 3 votes counts: A=42 C=38 E=20 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:222 D:191 A:190 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -14 2 -10 B -2 0 -18 -16 -14 C 14 18 0 18 -6 D -2 16 -18 0 -14 E 10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 2 -10 B -2 0 -18 -16 -14 C 14 18 0 18 -6 D -2 16 -18 0 -14 E 10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 2 -10 B -2 0 -18 -16 -14 C 14 18 0 18 -6 D -2 16 -18 0 -14 E 10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2919: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (7) D E C B A (6) B E A D C (6) B E D C A (5) B A C E D (5) D C E A B (4) B A E D C (4) A E D B C (4) E D B C A (3) E D A C B (3) E D A B C (3) E B D A C (3) C D E B A (3) C D A E B (3) C A D B E (3) E D C B A (2) E D B A C (2) D C E B A (2) C D E A B (2) C B D E A (2) C B A D E (2) B C A E D (2) B A E C D (2) A E D C B (2) A E B D C (2) A C B D E (2) E D C A B (1) E B D C A (1) E A D B C (1) D E C A B (1) D E A C B (1) D C A E B (1) D A C E B (1) C B D A E (1) C A B D E (1) B E D A C (1) B E A C D (1) B C A D E (1) A D C E B (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 8 -2 -6 B 8 0 8 -2 -6 C -8 -8 0 -22 -10 D 2 2 22 0 -22 E 6 6 10 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 8 -2 -6 B 8 0 8 -2 -6 C -8 -8 0 -22 -10 D 2 2 22 0 -22 E 6 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=21 E=19 C=17 D=16 so D is eliminated. Round 2 votes counts: E=27 B=27 C=24 A=22 so A is eliminated. Round 3 votes counts: E=35 B=35 C=30 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:204 D:202 A:196 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 8 -2 -6 B 8 0 8 -2 -6 C -8 -8 0 -22 -10 D 2 2 22 0 -22 E 6 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -2 -6 B 8 0 8 -2 -6 C -8 -8 0 -22 -10 D 2 2 22 0 -22 E 6 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -2 -6 B 8 0 8 -2 -6 C -8 -8 0 -22 -10 D 2 2 22 0 -22 E 6 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2920: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) A B C D E (8) B C A D E (7) A E D C B (6) E D B C A (5) E D C B A (4) A D C E B (4) E B D C A (3) E B C D A (3) E A B C D (3) B C E D A (3) A E B C D (3) A C D B E (3) E D A B C (2) E B D A C (2) E A D C B (2) D C B E A (2) C D B E A (2) C B D A E (2) C B A D E (2) B C D E A (2) B C D A E (2) A D E C B (2) E D C A B (1) E B A C D (1) E A D B C (1) D E C B A (1) D C E A B (1) D C A E B (1) D C A B E (1) D A C E B (1) C D A B E (1) C A B D E (1) B E D C A (1) B E C D A (1) B C E A D (1) A D C B E (1) A C E B D (1) A C D E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 18 18 B -12 0 0 16 4 C -6 0 0 18 16 D -18 -16 -18 0 10 E -18 -4 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 18 18 B -12 0 0 16 4 C -6 0 0 18 16 D -18 -16 -18 0 10 E -18 -4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 E=27 B=17 C=8 D=7 so D is eliminated. Round 2 votes counts: A=42 E=28 B=17 C=13 so C is eliminated. Round 3 votes counts: A=46 E=29 B=25 so B is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 C:214 B:204 D:179 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 18 18 B -12 0 0 16 4 C -6 0 0 18 16 D -18 -16 -18 0 10 E -18 -4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 18 18 B -12 0 0 16 4 C -6 0 0 18 16 D -18 -16 -18 0 10 E -18 -4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 18 18 B -12 0 0 16 4 C -6 0 0 18 16 D -18 -16 -18 0 10 E -18 -4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2921: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (7) E A C B D (6) B E D A C (6) B E D C A (5) B D E A C (5) A C E D B (5) E B D A C (4) D B C A E (4) C E A D B (4) A E C B D (4) E B D C A (3) C A E D B (3) C A D E B (3) C A D B E (3) B D A C E (3) A C D B E (3) E C A B D (2) E B C A D (2) E B A D C (2) E B A C D (2) D C A B E (2) A C D E B (2) E C D B A (1) E C A D B (1) E A B C D (1) D C B A E (1) D B C E A (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D B A (1) C E D A B (1) C D E B A (1) C D E A B (1) C D A B E (1) B E A D C (1) B D C A E (1) B A E D C (1) A E B C D (1) A D B C E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 4 -2 -14 B 4 0 6 16 -2 C -4 -6 0 -2 -8 D 2 -16 2 0 -14 E 14 2 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 -2 -14 B 4 0 6 16 -2 C -4 -6 0 -2 -8 D 2 -16 2 0 -14 E 14 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=24 C=18 A=18 D=11 so D is eliminated. Round 2 votes counts: B=35 E=24 C=21 A=20 so A is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:219 B:212 A:192 C:190 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 -2 -14 B 4 0 6 16 -2 C -4 -6 0 -2 -8 D 2 -16 2 0 -14 E 14 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -2 -14 B 4 0 6 16 -2 C -4 -6 0 -2 -8 D 2 -16 2 0 -14 E 14 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -2 -14 B 4 0 6 16 -2 C -4 -6 0 -2 -8 D 2 -16 2 0 -14 E 14 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2922: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) A C B D E (10) C A B D E (9) D E A C B (7) B A C E D (6) E B D C A (5) A D C E B (5) B E D A C (4) B C A E D (4) A C D B E (4) E D C A B (3) E D B A C (3) D A C E B (3) B A C D E (3) D E C A B (2) D A E C B (2) C B A E D (2) B E D C A (2) B E C D A (2) B E C A D (2) A C D E B (2) E B D A C (1) D E B A C (1) D E A B C (1) D C E A B (1) D C A E B (1) C A D E B (1) C A B E D (1) B E A C D (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 4 10 4 10 B -4 0 -6 8 4 C -10 6 0 -4 4 D -4 -8 4 0 6 E -10 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 4 10 B -4 0 -6 8 4 C -10 6 0 -4 4 D -4 -8 4 0 6 E -10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 A=21 D=18 C=13 so C is eliminated. Round 2 votes counts: A=32 B=28 E=22 D=18 so D is eliminated. Round 3 votes counts: A=38 E=34 B=28 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:201 D:199 C:198 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 4 10 B -4 0 -6 8 4 C -10 6 0 -4 4 D -4 -8 4 0 6 E -10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 4 10 B -4 0 -6 8 4 C -10 6 0 -4 4 D -4 -8 4 0 6 E -10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 4 10 B -4 0 -6 8 4 C -10 6 0 -4 4 D -4 -8 4 0 6 E -10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2923: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) D A C E B (6) B D E C A (6) B E C A D (5) D C B E A (4) D B A C E (4) B D C E A (4) E C A B D (3) D C A E B (3) D B C E A (3) C D E A B (3) B A E C D (3) A E C D B (3) A E C B D (3) A D C E B (3) E B C A D (2) D B C A E (2) D A C B E (2) C E D A B (2) C A E D B (2) B E D C A (2) B E A C D (2) B D A E C (2) B A E D C (2) A E B C D (2) A B E D C (2) E C B A D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C B A E (1) D A B E C (1) D A B C E (1) C E D B A (1) C D E B A (1) B E C D A (1) B E A D C (1) B D E A C (1) B C E D A (1) B C D E A (1) B A D E C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -16 -8 -10 B 4 0 2 -6 6 C 16 -2 0 -6 10 D 8 6 6 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -8 -10 B 4 0 2 -6 6 C 16 -2 0 -6 10 D 8 6 6 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=29 C=17 A=15 E=7 so E is eliminated. Round 2 votes counts: B=34 D=29 C=21 A=16 so A is eliminated. Round 3 votes counts: B=40 D=32 C=28 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:209 B:203 E:196 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -16 -8 -10 B 4 0 2 -6 6 C 16 -2 0 -6 10 D 8 6 6 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -8 -10 B 4 0 2 -6 6 C 16 -2 0 -6 10 D 8 6 6 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -8 -10 B 4 0 2 -6 6 C 16 -2 0 -6 10 D 8 6 6 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2924: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) D C E A B (8) C E A B D (6) D C E B A (5) D C B A E (5) C E D A B (5) D B C A E (4) C B A E D (4) B A E C D (4) B A D E C (4) E A B C D (3) D B A C E (3) C E B A D (3) C D E A B (3) E C A D B (2) E A C B D (2) D B A E C (2) C E D B A (2) B D A C E (2) E D C A B (1) E A D C B (1) E A D B C (1) E A B D C (1) D E A C B (1) D C B E A (1) D A B E C (1) C E A D B (1) C D E B A (1) C D B E A (1) C B D A E (1) B D A E C (1) B A E D C (1) B A D C E (1) A E D B C (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -30 6 -20 B -10 0 -30 0 -22 C 30 30 0 4 12 D -6 0 -4 0 -8 E 20 22 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -30 6 -20 B -10 0 -30 0 -22 C 30 30 0 4 12 D -6 0 -4 0 -8 E 20 22 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=27 E=23 B=13 A=7 so A is eliminated. Round 2 votes counts: D=31 C=27 E=26 B=16 so B is eliminated. Round 3 votes counts: D=40 E=33 C=27 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:238 E:219 D:191 A:183 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -30 6 -20 B -10 0 -30 0 -22 C 30 30 0 4 12 D -6 0 -4 0 -8 E 20 22 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -30 6 -20 B -10 0 -30 0 -22 C 30 30 0 4 12 D -6 0 -4 0 -8 E 20 22 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -30 6 -20 B -10 0 -30 0 -22 C 30 30 0 4 12 D -6 0 -4 0 -8 E 20 22 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2925: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) E A B D C (9) E A B C D (9) C D B A E (7) E B A D C (6) A B D C E (6) E D C B A (5) E C A D B (5) B A D C E (5) C D E B A (4) D B C A E (3) C A B D E (3) B D A C E (3) A E B C D (3) A B C D E (3) E C D A B (2) E A C B D (2) C D A B E (2) A B E D C (2) A B D E C (2) E C D B A (1) D C E B A (1) D C B E A (1) C E D A B (1) C D B E A (1) C B D A E (1) C A D B E (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 2 4 16 6 B -2 0 4 12 2 C -4 -4 0 -8 6 D -16 -12 8 0 8 E -6 -2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 16 6 B -2 0 4 12 2 C -4 -4 0 -8 6 D -16 -12 8 0 8 E -6 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=20 A=17 D=15 B=9 so B is eliminated. Round 2 votes counts: E=39 A=23 C=20 D=18 so D is eliminated. Round 3 votes counts: E=39 C=35 A=26 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 B:208 C:195 D:194 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 16 6 B -2 0 4 12 2 C -4 -4 0 -8 6 D -16 -12 8 0 8 E -6 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 16 6 B -2 0 4 12 2 C -4 -4 0 -8 6 D -16 -12 8 0 8 E -6 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 16 6 B -2 0 4 12 2 C -4 -4 0 -8 6 D -16 -12 8 0 8 E -6 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2926: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) E B A C D (6) C D A E B (6) B E C D A (5) B E A D C (5) B E D A C (4) A C E D B (4) E C A B D (3) D B A E C (3) D A C B E (3) C E A D B (3) C A E D B (3) B E C A D (3) E C B D A (2) E B C D A (2) D A C E B (2) D A B E C (2) D A B C E (2) C E D A B (2) C E B D A (2) C D E A B (2) C A D E B (2) B E D C A (2) B E A C D (2) B D E A C (2) A D C E B (2) A D C B E (2) A D B E C (2) E C B A D (1) E A B C D (1) D C B E A (1) D C A E B (1) D C A B E (1) C E D B A (1) C E A B D (1) C D B E A (1) B D E C A (1) B D A E C (1) A E C B D (1) A E B D C (1) A E B C D (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -4 4 -18 B 2 0 6 8 -14 C 4 -6 0 24 -16 D -4 -8 -24 0 -24 E 18 14 16 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 4 -18 B 2 0 6 8 -14 C 4 -6 0 24 -16 D -4 -8 -24 0 -24 E 18 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 E=22 D=15 A=15 so D is eliminated. Round 2 votes counts: B=28 C=26 A=24 E=22 so E is eliminated. Round 3 votes counts: B=43 C=32 A=25 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:236 C:203 B:201 A:190 D:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 4 -18 B 2 0 6 8 -14 C 4 -6 0 24 -16 D -4 -8 -24 0 -24 E 18 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 -18 B 2 0 6 8 -14 C 4 -6 0 24 -16 D -4 -8 -24 0 -24 E 18 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 -18 B 2 0 6 8 -14 C 4 -6 0 24 -16 D -4 -8 -24 0 -24 E 18 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2927: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (16) B E A D C (12) E B D A C (8) E B D C A (5) C A D E B (5) A C D E B (5) C D A B E (4) B E D A C (4) B E A C D (4) E D B C A (3) D E C B A (3) B A C E D (3) A B C E D (3) E B A D C (2) D C A E B (2) D A C E B (2) C A D B E (2) B E D C A (2) B A E C D (2) A D E C B (2) A C D B E (2) D C E B A (1) D C E A B (1) C B D E A (1) C A B D E (1) B C A D E (1) B A E D C (1) A E D C B (1) A E D B C (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 6 -4 8 B 4 0 2 0 -14 C -6 -2 0 0 0 D 4 0 0 0 -2 E -8 14 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428566 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 A B C D E A 0 -4 6 -4 8 B 4 0 2 0 -14 C -6 -2 0 0 0 D 4 0 0 0 -2 E -8 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=29 B=29 E=18 A=15 D=9 so D is eliminated. Round 2 votes counts: C=33 B=29 E=21 A=17 so A is eliminated. Round 3 votes counts: C=43 B=32 E=25 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:204 A:203 D:201 B:196 C:196 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -4 8 B 4 0 2 0 -14 C -6 -2 0 0 0 D 4 0 0 0 -2 E -8 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 8 B 4 0 2 0 -14 C -6 -2 0 0 0 D 4 0 0 0 -2 E -8 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 8 B 4 0 2 0 -14 C -6 -2 0 0 0 D 4 0 0 0 -2 E -8 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2928: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) C E D A B (9) B A D C E (8) B A D E C (7) E D C B A (6) A B C D E (6) D E B C A (5) B A E D C (5) A B C E D (5) E D C A B (4) C A E D B (4) B A C D E (4) E D B C A (3) C D E A B (3) C A B D E (3) B D E A C (3) D E C B A (2) C E A D B (2) E C D A B (1) D E B A C (1) D B E A C (1) C D E B A (1) C B A D E (1) C A E B D (1) C A D E B (1) C A B E D (1) A E B D C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 6 22 18 B -6 0 0 14 12 C -6 0 0 6 20 D -22 -14 -6 0 -6 E -18 -12 -20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 22 18 B -6 0 0 14 12 C -6 0 0 6 20 D -22 -14 -6 0 -6 E -18 -12 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 A=24 E=14 D=9 so D is eliminated. Round 2 votes counts: B=28 C=26 A=24 E=22 so E is eliminated. Round 3 votes counts: C=39 B=37 A=24 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:226 B:210 C:210 E:178 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 22 18 B -6 0 0 14 12 C -6 0 0 6 20 D -22 -14 -6 0 -6 E -18 -12 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 22 18 B -6 0 0 14 12 C -6 0 0 6 20 D -22 -14 -6 0 -6 E -18 -12 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 22 18 B -6 0 0 14 12 C -6 0 0 6 20 D -22 -14 -6 0 -6 E -18 -12 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2929: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) D B A C E (8) C E D B A (8) D B C E A (5) D B C A E (5) C E B D A (5) E C A B D (4) E A C B D (4) D A B C E (4) B C E A D (4) A E B C D (4) A B E C D (4) C D E B A (3) B D A C E (3) A E C D B (3) A D B E C (3) A B E D C (3) A B D E C (3) C B E D A (2) B D C E A (2) B D C A E (2) E C B A D (1) D C E B A (1) D C B A E (1) D A E C B (1) C E B A D (1) C B D E A (1) B C E D A (1) B A D C E (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 8 -4 16 B 8 0 4 12 4 C -8 -4 0 12 14 D 4 -12 -12 0 -12 E -16 -4 -14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -4 16 B 8 0 4 12 4 C -8 -4 0 12 14 D 4 -12 -12 0 -12 E -16 -4 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=25 C=20 B=13 E=9 so E is eliminated. Round 2 votes counts: A=37 D=25 C=25 B=13 so B is eliminated. Round 3 votes counts: A=38 D=32 C=30 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:214 C:207 A:206 E:189 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -4 16 B 8 0 4 12 4 C -8 -4 0 12 14 D 4 -12 -12 0 -12 E -16 -4 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -4 16 B 8 0 4 12 4 C -8 -4 0 12 14 D 4 -12 -12 0 -12 E -16 -4 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -4 16 B 8 0 4 12 4 C -8 -4 0 12 14 D 4 -12 -12 0 -12 E -16 -4 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2930: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) A D E B C (7) D A E B C (6) E B C D A (5) D E B A C (5) C B E A D (4) C B A E D (4) A C B E D (4) E B D C A (3) E B C A D (3) E B A C D (3) D E B C A (3) D C A B E (3) D A E C B (3) C B E D A (3) B E C A D (3) A E B C D (3) A D C E B (3) E B D A C (2) D E A B C (2) D A C E B (2) C D A B E (2) B E C D A (2) A C B D E (2) A B E C D (2) A B C E D (2) E D B C A (1) E D B A C (1) D C E B A (1) D C B E A (1) D C B A E (1) D B C E A (1) C D B E A (1) C A D B E (1) C A B D E (1) B E A C D (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 0 -4 4 8 B 0 0 14 12 -6 C 4 -14 0 10 -10 D -4 -12 -10 0 -10 E -8 6 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545544 B: 0.454456 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.504148547177 Cumulative probabilities = A: 0.545544 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 4 8 B 0 0 14 12 -6 C 4 -14 0 10 -10 D -4 -12 -10 0 -10 E -8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 A=23 E=18 B=8 so B is eliminated. Round 2 votes counts: D=28 C=25 E=24 A=23 so A is eliminated. Round 3 votes counts: D=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:210 E:209 A:204 C:195 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -4 4 8 B 0 0 14 12 -6 C 4 -14 0 10 -10 D -4 -12 -10 0 -10 E -8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 8 B 0 0 14 12 -6 C 4 -14 0 10 -10 D -4 -12 -10 0 -10 E -8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 8 B 0 0 14 12 -6 C 4 -14 0 10 -10 D -4 -12 -10 0 -10 E -8 6 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2931: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) E B C D A (6) E A B C D (6) E B C A D (5) B E C D A (5) A D C B E (5) E A D C B (4) E A C B D (4) D B C A E (4) A E D C B (4) A D E C B (4) A D C E B (4) E B D C A (3) D C A B E (3) C B D A E (3) B D C E A (3) B C D A E (3) A E C D B (3) E B A D C (2) E A D B C (2) E A B D C (2) D A C B E (2) A E C B D (2) A C D B E (2) A C B D E (2) E B A C D (1) D E B C A (1) D C B A E (1) D B C E A (1) C B A D E (1) B E C A D (1) B C E D A (1) A C B E D (1) Total count = 100 A B C D E A 0 0 0 10 -12 B 0 0 10 20 -6 C 0 -10 0 10 -10 D -10 -20 -10 0 -4 E 12 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 10 -12 B 0 0 10 20 -6 C 0 -10 0 10 -10 D -10 -20 -10 0 -4 E 12 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=27 B=22 D=12 C=4 so C is eliminated. Round 2 votes counts: E=35 A=27 B=26 D=12 so D is eliminated. Round 3 votes counts: E=36 B=32 A=32 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:212 A:199 C:195 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 10 -12 B 0 0 10 20 -6 C 0 -10 0 10 -10 D -10 -20 -10 0 -4 E 12 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 10 -12 B 0 0 10 20 -6 C 0 -10 0 10 -10 D -10 -20 -10 0 -4 E 12 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 10 -12 B 0 0 10 20 -6 C 0 -10 0 10 -10 D -10 -20 -10 0 -4 E 12 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2932: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) E C B A D (6) C B E D A (6) E A D C B (5) A D B C E (5) C E B D A (4) B C D A E (4) A E D C B (4) E A C D B (3) E A C B D (3) D A E B C (3) B C A E D (3) A E B C D (3) A D E B C (3) E D A C B (2) D E C A B (2) D B A C E (2) D A E C B (2) D A B C E (2) C B E A D (2) B C E D A (2) B C D E A (2) B C A D E (2) A D E C B (2) A D B E C (2) E C D B A (1) E C B D A (1) E C A B D (1) E B C A D (1) E A B C D (1) D E A C B (1) D C B E A (1) D C A E B (1) D B C A E (1) D A B E C (1) C B D E A (1) B E C A D (1) B D A C E (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 20 -8 B 2 0 4 12 -2 C 0 -4 0 16 -2 D -20 -12 -16 0 -22 E 8 2 2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 20 -8 B 2 0 4 12 -2 C 0 -4 0 16 -2 D -20 -12 -16 0 -22 E 8 2 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 A=22 D=16 C=13 so C is eliminated. Round 2 votes counts: B=34 E=28 A=22 D=16 so D is eliminated. Round 3 votes counts: B=38 E=31 A=31 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:217 B:208 A:205 C:205 D:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 20 -8 B 2 0 4 12 -2 C 0 -4 0 16 -2 D -20 -12 -16 0 -22 E 8 2 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 20 -8 B 2 0 4 12 -2 C 0 -4 0 16 -2 D -20 -12 -16 0 -22 E 8 2 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 20 -8 B 2 0 4 12 -2 C 0 -4 0 16 -2 D -20 -12 -16 0 -22 E 8 2 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2933: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) C E B A D (8) C E A B D (7) A B D E C (7) D A B E C (6) C E D B A (4) E C B A D (3) E B D A C (3) D E B C A (3) D C A B E (3) D B A E C (3) B A D E C (3) A D C B E (3) E B A C D (2) D C E B A (2) C D A E B (2) C D A B E (2) B E A D C (2) A D B E C (2) E D C B A (1) E D B C A (1) E C B D A (1) E B D C A (1) E B C A D (1) E B A D C (1) D E B A C (1) D C B E A (1) D C A E B (1) D B E A C (1) D A C B E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E B A (1) C D E A B (1) C A E B D (1) C A D B E (1) B E D A C (1) B D A E C (1) B A E D C (1) A D B C E (1) A C B D E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -4 2 B -6 0 6 -4 4 C -4 -6 0 -22 6 D 4 4 22 0 14 E -2 -4 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 2 B -6 0 6 -4 4 C -4 -6 0 -22 6 D 4 4 22 0 14 E -2 -4 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=30 A=17 E=14 B=8 so B is eliminated. Round 2 votes counts: D=32 C=30 A=21 E=17 so E is eliminated. Round 3 votes counts: D=39 C=35 A=26 so A is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:204 B:200 C:187 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 2 B -6 0 6 -4 4 C -4 -6 0 -22 6 D 4 4 22 0 14 E -2 -4 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 2 B -6 0 6 -4 4 C -4 -6 0 -22 6 D 4 4 22 0 14 E -2 -4 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 2 B -6 0 6 -4 4 C -4 -6 0 -22 6 D 4 4 22 0 14 E -2 -4 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2934: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (12) D A C B E (10) B E D A C (10) C A D E B (7) A D C B E (6) E B C D A (5) C D A E B (5) B E C D A (5) A D B C E (5) B E A D C (4) B D A E C (4) A C D E B (4) E B C A D (3) C E A D B (3) E C B D A (2) E C A B D (2) D A C E B (2) B E D C A (2) E C B A D (1) D C A E B (1) D A B C E (1) B E C A D (1) B E A C D (1) B D E A C (1) B D C E A (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 18 24 2 18 B -18 0 -10 -14 6 C -24 10 0 -22 14 D -2 14 22 0 20 E -18 -6 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 24 2 18 B -18 0 -10 -14 6 C -24 10 0 -22 14 D -2 14 22 0 20 E -18 -6 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974308 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=28 C=15 D=14 E=13 so E is eliminated. Round 2 votes counts: B=38 A=28 C=20 D=14 so D is eliminated. Round 3 votes counts: A=41 B=38 C=21 so C is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:231 D:227 C:189 B:182 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 24 2 18 B -18 0 -10 -14 6 C -24 10 0 -22 14 D -2 14 22 0 20 E -18 -6 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974308 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 24 2 18 B -18 0 -10 -14 6 C -24 10 0 -22 14 D -2 14 22 0 20 E -18 -6 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974308 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 24 2 18 B -18 0 -10 -14 6 C -24 10 0 -22 14 D -2 14 22 0 20 E -18 -6 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974308 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2935: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (15) B A C D E (14) E C D B A (6) E C D A B (6) B A E D C (4) B A D C E (4) A E D C B (4) A B E D C (4) C D E B A (3) C D B E A (3) B C D A E (3) A E D B C (3) A B D E C (3) A B D C E (3) E A D C B (2) C D E A B (2) C D B A E (2) A E B D C (2) E D A C B (1) E B D C A (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E A B (1) D A C E B (1) C E D B A (1) C E D A B (1) B C E D A (1) B C D E A (1) B C A D E (1) B A D E C (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 6 4 0 2 B -6 0 0 -10 -12 C -4 0 0 -12 -16 D 0 10 12 0 -10 E -2 12 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.888800 B: 0.000000 C: 0.000000 D: 0.111200 E: 0.000000 Sum of squares = 0.802330760149 Cumulative probabilities = A: 0.888800 B: 0.888800 C: 0.888800 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 2 B -6 0 0 -10 -12 C -4 0 0 -12 -16 D 0 10 12 0 -10 E -2 12 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222271338 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=29 A=21 C=12 D=3 so D is eliminated. Round 2 votes counts: E=36 B=29 A=22 C=13 so C is eliminated. Round 3 votes counts: E=44 B=34 A=22 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:206 D:206 B:186 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 2 B -6 0 0 -10 -12 C -4 0 0 -12 -16 D 0 10 12 0 -10 E -2 12 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222271338 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 2 B -6 0 0 -10 -12 C -4 0 0 -12 -16 D 0 10 12 0 -10 E -2 12 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222271338 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 2 B -6 0 0 -10 -12 C -4 0 0 -12 -16 D 0 10 12 0 -10 E -2 12 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222271338 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2936: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) B E C A D (9) E C B D A (8) A D B C E (7) C E D A B (6) E B C D A (5) D A C E B (5) B E A D C (5) A D C B E (5) E B A D C (4) C E B D A (4) B A D E C (4) B A D C E (4) E D A C B (2) E C D A B (2) E B C A D (2) E A D B C (2) D A C B E (2) A D B E C (2) C D E A B (1) C D A B E (1) C B D A E (1) C A D B E (1) B E C D A (1) B C E A D (1) B C A D E (1) B A E D C (1) A D E C B (1) A D C E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -6 4 -4 B 0 0 0 4 -6 C 6 0 0 6 2 D -4 -4 -6 0 -4 E 4 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.132380 C: 0.867620 D: 0.000000 E: 0.000000 Sum of squares = 0.770288511197 Cumulative probabilities = A: 0.000000 B: 0.132380 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 4 -4 B 0 0 0 4 -6 C 6 0 0 6 2 D -4 -4 -6 0 -4 E 4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000618 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 C=24 A=18 D=7 so D is eliminated. Round 2 votes counts: B=26 E=25 A=25 C=24 so C is eliminated. Round 3 votes counts: A=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:207 E:206 B:199 A:197 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 4 -4 B 0 0 0 4 -6 C 6 0 0 6 2 D -4 -4 -6 0 -4 E 4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000618 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 4 -4 B 0 0 0 4 -6 C 6 0 0 6 2 D -4 -4 -6 0 -4 E 4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000618 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 4 -4 B 0 0 0 4 -6 C 6 0 0 6 2 D -4 -4 -6 0 -4 E 4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000618 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2937: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) E D A B C (7) E B C D A (6) D A E C B (6) C B A D E (6) A C B D E (6) A B C E D (6) D E C B A (5) B C E D A (5) A D C B E (5) E B C A D (4) D A C B E (3) B C E A D (3) B C A E D (3) E A B C D (2) C B E D A (2) C B A E D (2) C A B D E (2) A D E C B (2) E D C B A (1) E C B D A (1) E B D C A (1) E B A C D (1) E A D B C (1) D E A C B (1) D E A B C (1) D C B E A (1) D C B A E (1) D C A B E (1) D A E B C (1) B A C E D (1) A D E B C (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 -4 -2 B 4 0 8 4 0 C 6 -8 0 4 0 D 4 -4 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.544862 C: 0.000000 D: 0.000000 E: 0.455138 Sum of squares = 0.504025166396 Cumulative probabilities = A: 0.000000 B: 0.544862 C: 0.544862 D: 0.544862 E: 1.000000 A B C D E A 0 -4 -6 -4 -2 B 4 0 8 4 0 C 6 -8 0 4 0 D 4 -4 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999786 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=23 D=20 C=12 B=12 so C is eliminated. Round 2 votes counts: E=33 A=25 B=22 D=20 so D is eliminated. Round 3 votes counts: E=40 A=36 B=24 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:208 E:206 C:201 D:193 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 -2 B 4 0 8 4 0 C 6 -8 0 4 0 D 4 -4 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999786 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 -2 B 4 0 8 4 0 C 6 -8 0 4 0 D 4 -4 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999786 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 -2 B 4 0 8 4 0 C 6 -8 0 4 0 D 4 -4 -4 0 -10 E 2 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999786 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2938: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (6) B E D C A (6) B C E D A (6) C B E D A (5) A B D E C (5) D A E B C (4) C B E A D (4) A D E C B (4) A D B E C (4) D E C B A (3) C E B D A (3) A D E B C (3) A B C D E (3) E D C B A (2) E C B D A (2) D E C A B (2) D E B A C (2) D C E A B (2) C A D E B (2) C A B E D (2) B E C D A (2) B C E A D (2) A D C E B (2) A C D E B (2) E D B C A (1) E C D B A (1) E B D C A (1) D E B C A (1) D E A B C (1) D C A E B (1) D A C E B (1) D A B E C (1) C E D A B (1) C E A B D (1) C D E A B (1) C D A E B (1) B E D A C (1) B E C A D (1) B D E A C (1) B C A E D (1) B A E D C (1) B A C E D (1) A D B C E (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -20 -16 -16 B 6 0 -2 0 0 C 20 2 0 0 2 D 16 0 0 0 -2 E 16 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.744269 D: 0.255731 E: 0.000000 Sum of squares = 0.619334411923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.744269 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -16 -16 B 6 0 -2 0 0 C 20 2 0 0 2 D 16 0 0 0 -2 E 16 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.970880 D: 0.029120 E: 0.000000 Sum of squares = 0.943455875614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.970880 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 B=22 D=18 E=7 so E is eliminated. Round 2 votes counts: C=29 A=27 B=23 D=21 so D is eliminated. Round 3 votes counts: C=39 A=34 B=27 so B is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:208 D:207 B:202 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -20 -16 -16 B 6 0 -2 0 0 C 20 2 0 0 2 D 16 0 0 0 -2 E 16 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.970880 D: 0.029120 E: 0.000000 Sum of squares = 0.943455875614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.970880 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -16 -16 B 6 0 -2 0 0 C 20 2 0 0 2 D 16 0 0 0 -2 E 16 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.970880 D: 0.029120 E: 0.000000 Sum of squares = 0.943455875614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.970880 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -16 -16 B 6 0 -2 0 0 C 20 2 0 0 2 D 16 0 0 0 -2 E 16 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.970880 D: 0.029120 E: 0.000000 Sum of squares = 0.943455875614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.970880 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2939: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (18) C E A D B (13) B D A C E (8) D A B E C (7) E A D C B (5) C E B A D (5) E C A D B (4) C B E D A (4) B D C A E (4) A D E B C (4) B C D A E (3) C E B D A (2) B D E C A (2) B C D E A (2) A D B E C (2) E D A B C (1) E C D A B (1) E C B D A (1) E A C D B (1) D E A B C (1) D B E A C (1) D B A E C (1) D A E B C (1) C E A B D (1) C B E A D (1) C B D A E (1) C A E B D (1) C A D E B (1) B E D A C (1) B D E A C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 8 -20 6 B 10 0 14 10 12 C -8 -14 0 -18 -6 D 20 -10 18 0 18 E -6 -12 6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -20 6 B 10 0 14 10 12 C -8 -14 0 -18 -6 D 20 -10 18 0 18 E -6 -12 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=29 E=13 D=11 A=8 so A is eliminated. Round 2 votes counts: B=39 C=29 D=19 E=13 so E is eliminated. Round 3 votes counts: B=39 C=36 D=25 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:223 A:192 E:185 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 -20 6 B 10 0 14 10 12 C -8 -14 0 -18 -6 D 20 -10 18 0 18 E -6 -12 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -20 6 B 10 0 14 10 12 C -8 -14 0 -18 -6 D 20 -10 18 0 18 E -6 -12 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -20 6 B 10 0 14 10 12 C -8 -14 0 -18 -6 D 20 -10 18 0 18 E -6 -12 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2940: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) B A D E C (9) C D E B A (7) A B E C D (7) B D C E A (6) B D A C E (6) E A C D B (5) D B C E A (5) C E D A B (5) B D C A E (5) A E B C D (5) B A E D C (4) E C D A B (3) A E C B D (3) E C A D B (2) B D A E C (2) A B E D C (2) D E C B A (1) D E C A B (1) D C E B A (1) D B E C A (1) C E A D B (1) C D E A B (1) C A B E D (1) B D E C A (1) B A D C E (1) B A C E D (1) A E D B C (1) A E B D C (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 0 18 10 20 B 0 0 16 10 4 C -18 -16 0 6 -16 D -10 -10 -6 0 -6 E -20 -4 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.287832 B: 0.712168 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.59003035762 Cumulative probabilities = A: 0.287832 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 10 20 B 0 0 16 10 4 C -18 -16 0 6 -16 D -10 -10 -6 0 -6 E -20 -4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=31 C=15 E=10 D=9 so D is eliminated. Round 2 votes counts: B=41 A=31 C=16 E=12 so E is eliminated. Round 3 votes counts: B=41 A=36 C=23 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:215 E:199 D:184 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 10 20 B 0 0 16 10 4 C -18 -16 0 6 -16 D -10 -10 -6 0 -6 E -20 -4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 10 20 B 0 0 16 10 4 C -18 -16 0 6 -16 D -10 -10 -6 0 -6 E -20 -4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 10 20 B 0 0 16 10 4 C -18 -16 0 6 -16 D -10 -10 -6 0 -6 E -20 -4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2941: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (10) E B A C D (8) D C A B E (8) E A B C D (6) D C A E B (6) E B D A C (5) E B A D C (5) A C B D E (5) E D B A C (3) E A C D B (3) D E C A B (3) C D A B E (3) B A C D E (3) E D A C B (2) D E B C A (2) D C E A B (2) B E D C A (2) B D C A E (2) B C D A E (2) B A E C D (2) E D C B A (1) E D C A B (1) E D B C A (1) E B D C A (1) E A D C B (1) D C B E A (1) D C B A E (1) D B C A E (1) B E D A C (1) B E A D C (1) B C D E A (1) B A C E D (1) A E C D B (1) A D E C B (1) A C D B E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 24 2 -20 B 8 0 20 16 -4 C -24 -20 0 -2 -22 D -2 -16 2 0 -14 E 20 4 22 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 24 2 -20 B 8 0 20 16 -4 C -24 -20 0 -2 -22 D -2 -16 2 0 -14 E 20 4 22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=25 D=24 A=11 C=3 so C is eliminated. Round 2 votes counts: E=37 D=27 B=25 A=11 so A is eliminated. Round 3 votes counts: E=38 B=33 D=29 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 B:220 A:199 D:185 C:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 24 2 -20 B 8 0 20 16 -4 C -24 -20 0 -2 -22 D -2 -16 2 0 -14 E 20 4 22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 24 2 -20 B 8 0 20 16 -4 C -24 -20 0 -2 -22 D -2 -16 2 0 -14 E 20 4 22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 24 2 -20 B 8 0 20 16 -4 C -24 -20 0 -2 -22 D -2 -16 2 0 -14 E 20 4 22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2942: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) C A D E B (6) C A B D E (6) E D B A C (5) A B E C D (5) D E B C A (4) B A C D E (4) A C E D B (4) E A B D C (3) D C B E A (3) B A E D C (3) A C B D E (3) E D C A B (2) E B D A C (2) D C E A B (2) D B C E A (2) C D B A E (2) C B D A E (2) C A D B E (2) B E A D C (2) B D C E A (2) B C D A E (2) B A E C D (2) B A C E D (2) A E C D B (2) A C B E D (2) A B C E D (2) A B C D E (2) E D B C A (1) E B A D C (1) E A D B C (1) E A C D B (1) D C E B A (1) D B E C A (1) C D E A B (1) C D A E B (1) C B A D E (1) B E D A C (1) B D E C A (1) B C A D E (1) A E C B D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 0 16 14 B 4 0 0 2 8 C 0 0 0 10 8 D -16 -2 -10 0 14 E -14 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.445292 C: 0.554708 D: 0.000000 E: 0.000000 Sum of squares = 0.505985869739 Cumulative probabilities = A: 0.000000 B: 0.445292 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 16 14 B 4 0 0 2 8 C 0 0 0 10 8 D -16 -2 -10 0 14 E -14 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 C=21 D=20 B=20 E=16 so E is eliminated. Round 2 votes counts: D=28 A=28 B=23 C=21 so C is eliminated. Round 3 votes counts: A=42 D=32 B=26 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:209 B:207 D:193 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 16 14 B 4 0 0 2 8 C 0 0 0 10 8 D -16 -2 -10 0 14 E -14 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 16 14 B 4 0 0 2 8 C 0 0 0 10 8 D -16 -2 -10 0 14 E -14 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 16 14 B 4 0 0 2 8 C 0 0 0 10 8 D -16 -2 -10 0 14 E -14 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2943: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) C E A D B (8) E C D B A (7) A B D E C (7) C E D B A (6) C E D A B (4) C A B E D (4) B D A E C (4) D B E A C (3) D B A E C (3) C A E B D (3) B A D E C (3) B A D C E (3) A D B E C (3) A B C D E (3) E D C B A (2) E D B A C (2) E B D C A (2) D E B A C (2) D A B E C (2) C E B D A (2) C E B A D (2) C A B D E (2) B D E A C (2) A D B C E (2) A C B D E (2) E D B C A (1) E C B D A (1) C E A B D (1) C B E A D (1) C A E D B (1) A E C D B (1) Total count = 100 A B C D E A 0 8 6 14 8 B -8 0 6 6 10 C -6 -6 0 -4 10 D -14 -6 4 0 4 E -8 -10 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 14 8 B -8 0 6 6 10 C -6 -6 0 -4 10 D -14 -6 4 0 4 E -8 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=29 E=15 B=12 D=10 so D is eliminated. Round 2 votes counts: C=34 A=31 B=18 E=17 so E is eliminated. Round 3 votes counts: C=44 A=31 B=25 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:207 C:197 D:194 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 14 8 B -8 0 6 6 10 C -6 -6 0 -4 10 D -14 -6 4 0 4 E -8 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 14 8 B -8 0 6 6 10 C -6 -6 0 -4 10 D -14 -6 4 0 4 E -8 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 14 8 B -8 0 6 6 10 C -6 -6 0 -4 10 D -14 -6 4 0 4 E -8 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2944: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) C A B D E (9) E D C A B (6) D E B C A (6) B A D E C (6) A B E D C (6) A B C D E (6) E D B A C (5) C D E B A (5) B D E A C (4) A B C E D (4) C B D A E (3) A C B D E (3) A B D E C (3) E D A B C (2) D E C B A (2) D E B A C (2) C E D B A (2) C E D A B (2) C E A D B (2) C A B E D (2) A C B E D (2) E D B C A (1) D C E B A (1) C D E A B (1) C A E D B (1) B D A E C (1) B A E D C (1) B A D C E (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 0 -4 -4 0 B 0 0 0 6 4 C 4 0 0 -14 -10 D 4 -6 14 0 8 E 0 -4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.811689 C: 0.188311 D: 0.000000 E: 0.000000 Sum of squares = 0.694299439157 Cumulative probabilities = A: 0.000000 B: 0.811689 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -4 0 B 0 0 0 6 4 C 4 0 0 -14 -10 D 4 -6 14 0 8 E 0 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836744215 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=25 E=23 B=14 D=11 so D is eliminated. Round 2 votes counts: E=33 C=28 A=25 B=14 so B is eliminated. Round 3 votes counts: E=37 A=35 C=28 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:210 B:205 E:199 A:196 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -4 0 B 0 0 0 6 4 C 4 0 0 -14 -10 D 4 -6 14 0 8 E 0 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836744215 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 0 B 0 0 0 6 4 C 4 0 0 -14 -10 D 4 -6 14 0 8 E 0 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836744215 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 0 B 0 0 0 6 4 C 4 0 0 -14 -10 D 4 -6 14 0 8 E 0 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836744215 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2945: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) C B D A E (6) A E B D C (6) E D C B A (5) E D C A B (5) D C E B A (5) B C A D E (5) C D B E A (4) B A C E D (4) E B A D C (3) E A D C B (3) E A B D C (3) D C E A B (3) C D B A E (3) C B D E A (3) B E A C D (3) B C D A E (3) B A C D E (3) E D A C B (2) D C A E B (2) B C E D A (2) B C D E A (2) A E D C B (2) A E D B C (2) A D E C B (2) A D C E B (2) A B E D C (2) A B E C D (2) A B C D E (2) D E C A B (1) C D A B E (1) B E C D A (1) A C D B E (1) Total count = 100 A B C D E A 0 -18 -2 4 10 B 18 0 0 14 8 C 2 0 0 4 2 D -4 -14 -4 0 -4 E -10 -8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.261046 C: 0.738954 D: 0.000000 E: 0.000000 Sum of squares = 0.614197863445 Cumulative probabilities = A: 0.000000 B: 0.261046 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -2 4 10 B 18 0 0 14 8 C 2 0 0 4 2 D -4 -14 -4 0 -4 E -10 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=21 A=21 C=17 D=11 so D is eliminated. Round 2 votes counts: B=30 C=27 E=22 A=21 so A is eliminated. Round 3 votes counts: B=36 E=34 C=30 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:204 A:197 E:192 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -2 4 10 B 18 0 0 14 8 C 2 0 0 4 2 D -4 -14 -4 0 -4 E -10 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 4 10 B 18 0 0 14 8 C 2 0 0 4 2 D -4 -14 -4 0 -4 E -10 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 4 10 B 18 0 0 14 8 C 2 0 0 4 2 D -4 -14 -4 0 -4 E -10 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2946: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) E B A C D (8) E C A D B (6) D C A B E (6) E C A B D (4) E B C A D (4) B D A E C (4) B D A C E (4) E C B A D (3) D A C B E (3) D A B C E (3) C D E A B (3) A D B C E (3) E B C D A (2) E A C B D (2) D C B A E (2) D C A E B (2) D B C A E (2) C E D B A (2) C E D A B (2) C D A E B (2) C A D E B (2) B A D E C (2) A E C D B (2) A C E D B (2) A B D E C (2) E C D B A (1) E A B C D (1) D C B E A (1) D B A C E (1) C E A D B (1) B E D A C (1) B E A C D (1) B D E C A (1) B D E A C (1) B D C E A (1) A D C E B (1) A D C B E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 6 12 -8 B 0 0 2 2 -2 C -6 -2 0 -2 -10 D -12 -2 2 0 -4 E 8 2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 6 12 -8 B 0 0 2 2 -2 C -6 -2 0 -2 -10 D -12 -2 2 0 -4 E 8 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998406 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=24 D=20 A=13 C=12 so C is eliminated. Round 2 votes counts: E=36 D=25 B=24 A=15 so A is eliminated. Round 3 votes counts: E=40 D=33 B=27 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:205 B:201 D:192 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 12 -8 B 0 0 2 2 -2 C -6 -2 0 -2 -10 D -12 -2 2 0 -4 E 8 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998406 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 12 -8 B 0 0 2 2 -2 C -6 -2 0 -2 -10 D -12 -2 2 0 -4 E 8 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998406 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 12 -8 B 0 0 2 2 -2 C -6 -2 0 -2 -10 D -12 -2 2 0 -4 E 8 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998406 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2947: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) C A B D E (8) A C B E D (8) B E D C A (7) A C D E B (6) C A D B E (5) A C D B E (5) E B D C A (4) C A D E B (3) B A E C D (3) A D E C B (3) E D B C A (2) E D B A C (2) E B D A C (2) D E C B A (2) D E A C B (2) D C E A B (2) C B A D E (2) B E D A C (2) B E A C D (2) B C A E D (2) B A C E D (2) A E B D C (2) A C B D E (2) A B C E D (2) E D A B C (1) E B A D C (1) D C A E B (1) D A E C B (1) C D B E A (1) C D A E B (1) C B A E D (1) C A B E D (1) B E C A D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -4 22 20 B -10 0 -10 6 10 C 4 10 0 14 6 D -22 -6 -14 0 6 E -20 -10 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 22 20 B -10 0 -10 6 10 C 4 10 0 14 6 D -22 -6 -14 0 6 E -20 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=22 B=19 D=17 E=12 so E is eliminated. Round 2 votes counts: A=30 B=26 D=22 C=22 so D is eliminated. Round 3 votes counts: B=39 A=34 C=27 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:217 B:198 D:182 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 22 20 B -10 0 -10 6 10 C 4 10 0 14 6 D -22 -6 -14 0 6 E -20 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 22 20 B -10 0 -10 6 10 C 4 10 0 14 6 D -22 -6 -14 0 6 E -20 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 22 20 B -10 0 -10 6 10 C 4 10 0 14 6 D -22 -6 -14 0 6 E -20 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997045 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2948: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A D E B C (11) B E C A D (10) E B A C D (7) C B E A D (7) A E B D C (7) D C A E B (6) B E A C D (5) B C E A D (5) C D B A E (4) C B E D A (4) D C A B E (3) D A E C B (3) D A C E B (3) E A B C D (2) D C B A E (2) D A E B C (2) A E B C D (2) E B A D C (1) D E A B C (1) D C B E A (1) D B E C A (1) C B D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 -18 -10 16 -12 B 18 0 10 2 8 C 10 -10 0 16 -6 D -16 -2 -16 0 -2 E 12 -8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 16 -12 B 18 0 10 2 8 C 10 -10 0 16 -6 D -16 -2 -16 0 -2 E 12 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 A=21 B=20 E=10 so E is eliminated. Round 2 votes counts: B=28 C=27 A=23 D=22 so D is eliminated. Round 3 votes counts: C=39 A=32 B=29 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:219 E:206 C:205 A:188 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 16 -12 B 18 0 10 2 8 C 10 -10 0 16 -6 D -16 -2 -16 0 -2 E 12 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 16 -12 B 18 0 10 2 8 C 10 -10 0 16 -6 D -16 -2 -16 0 -2 E 12 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 16 -12 B 18 0 10 2 8 C 10 -10 0 16 -6 D -16 -2 -16 0 -2 E 12 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2949: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) B C A E D (9) D B E A C (8) D B E C A (7) C B A E D (7) C A E B D (6) B D C A E (5) E D A C B (4) E A C D B (4) D E A B C (4) C A B E D (4) B C D A E (4) B C A D E (4) A C E B D (4) E A C B D (3) D E B A C (3) A E C D B (3) B D C E A (2) B C E A D (2) E D B A C (1) E A D C B (1) D B C A E (1) D A E B C (1) D A C E B (1) B E C A D (1) B D E C A (1) A E C B D (1) Total count = 100 A B C D E A 0 -10 -6 -2 0 B 10 0 6 6 10 C 6 -6 0 4 -2 D 2 -6 -4 0 0 E 0 -10 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -2 0 B 10 0 6 6 10 C 6 -6 0 4 -2 D 2 -6 -4 0 0 E 0 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=28 C=17 E=13 A=8 so A is eliminated. Round 2 votes counts: D=34 B=28 C=21 E=17 so E is eliminated. Round 3 votes counts: D=40 C=32 B=28 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:216 C:201 D:196 E:196 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -2 0 B 10 0 6 6 10 C 6 -6 0 4 -2 D 2 -6 -4 0 0 E 0 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -2 0 B 10 0 6 6 10 C 6 -6 0 4 -2 D 2 -6 -4 0 0 E 0 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -2 0 B 10 0 6 6 10 C 6 -6 0 4 -2 D 2 -6 -4 0 0 E 0 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2950: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) C A B E D (9) D B E A C (6) C E A B D (6) E D B C A (5) D E B C A (5) D E B A C (5) D B A E C (5) A C B D E (5) E D C B A (3) C A B D E (3) A C B E D (3) A B C E D (3) E D B A C (2) E C D B A (2) E C B D A (2) E C A B D (2) E B C A D (2) C A D B E (2) B A D C E (2) E D C A B (1) E C D A B (1) E C A D B (1) E B C D A (1) E B A C D (1) D E C B A (1) D C A B E (1) D B A C E (1) D A B C E (1) C E A D B (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A C D (1) B D A E C (1) B D A C E (1) B A E C D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -20 10 4 B -6 0 -12 6 -6 C 20 12 0 20 4 D -10 -6 -20 0 -14 E -4 6 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999726 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -20 10 4 B -6 0 -12 6 -6 C 20 12 0 20 4 D -10 -6 -20 0 -14 E -4 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=25 E=23 A=13 B=6 so B is eliminated. Round 2 votes counts: C=33 D=27 E=24 A=16 so A is eliminated. Round 3 votes counts: C=46 D=29 E=25 so E is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:206 A:200 B:191 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -20 10 4 B -6 0 -12 6 -6 C 20 12 0 20 4 D -10 -6 -20 0 -14 E -4 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -20 10 4 B -6 0 -12 6 -6 C 20 12 0 20 4 D -10 -6 -20 0 -14 E -4 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -20 10 4 B -6 0 -12 6 -6 C 20 12 0 20 4 D -10 -6 -20 0 -14 E -4 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2951: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (12) D A B C E (11) E D A C B (6) A B D C E (5) A B C D E (5) E D A B C (4) E C D B A (4) D A B E C (4) C B A E D (4) B C A D E (4) A D B C E (4) E D C B A (3) E C B D A (3) D E A B C (3) D C B A E (3) D A E B C (3) C B E A D (3) B A C D E (3) E D C A B (2) C E B A D (2) A B C E D (2) E C D A B (1) E C A B D (1) E A D C B (1) E A B C D (1) D E C B A (1) D E C A B (1) D B C A E (1) C E B D A (1) C B A D E (1) A E D B C (1) Total count = 100 A B C D E A 0 10 6 -2 2 B -10 0 2 -6 0 C -6 -2 0 -6 -2 D 2 6 6 0 -2 E -2 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 10 6 -2 2 B -10 0 2 -6 0 C -6 -2 0 -6 -2 D 2 6 6 0 -2 E -2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=27 A=17 C=11 B=7 so B is eliminated. Round 2 votes counts: E=38 D=27 A=20 C=15 so C is eliminated. Round 3 votes counts: E=44 A=29 D=27 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:206 E:201 B:193 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 -2 2 B -10 0 2 -6 0 C -6 -2 0 -6 -2 D 2 6 6 0 -2 E -2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -2 2 B -10 0 2 -6 0 C -6 -2 0 -6 -2 D 2 6 6 0 -2 E -2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -2 2 B -10 0 2 -6 0 C -6 -2 0 -6 -2 D 2 6 6 0 -2 E -2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2952: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) C D A E B (8) A B D C E (8) E B D C A (7) C E D A B (6) B E A D C (5) E B C D A (4) C D E A B (4) B A E D C (4) B E D A C (3) A D C B E (3) A D B C E (3) A B C E D (3) D E B C A (2) D B E A C (2) D B A C E (2) C D E B A (2) C A D E B (2) B D E A C (2) A C D B E (2) A C B D E (2) A B E C D (2) A B C D E (2) E D C B A (1) E D B C A (1) E C B D A (1) E C A B D (1) E B C A D (1) D C A B E (1) D B A E C (1) D A C B E (1) C D A B E (1) C A E D B (1) B E A C D (1) B A D E C (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -2 -14 -2 B -2 0 12 -2 2 C 2 -12 0 2 4 D 14 2 -2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000019 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -14 -2 B -2 0 12 -2 2 C 2 -12 0 2 4 D 14 2 -2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749998942 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 C=24 B=17 D=9 so D is eliminated. Round 2 votes counts: A=27 E=26 C=25 B=22 so B is eliminated. Round 3 votes counts: E=39 A=36 C=25 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:208 B:205 C:198 E:197 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 -14 -2 B -2 0 12 -2 2 C 2 -12 0 2 4 D 14 2 -2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749998942 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -14 -2 B -2 0 12 -2 2 C 2 -12 0 2 4 D 14 2 -2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749998942 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -14 -2 B -2 0 12 -2 2 C 2 -12 0 2 4 D 14 2 -2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749998942 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2953: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) A B D C E (8) A D B C E (6) A C D E B (6) A D C B E (5) E C D B A (4) E C D A B (4) C E D B A (4) C E A D B (4) A C E D B (4) E C B D A (3) E C A B D (3) E A C B D (3) B E D C A (3) B E C D A (3) B A D C E (3) E C A D B (2) E B C D A (2) D C E B A (2) D B A C E (2) C A D E B (2) B D A C E (2) B A D E C (2) A E C D B (2) E B A C D (1) D E B C A (1) D B E C A (1) D B C E A (1) D A C E B (1) C E D A B (1) B E A C D (1) B A E D C (1) B A E C D (1) A E C B D (1) A E B C D (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 4 16 -2 B -10 0 -4 -6 -4 C -4 4 0 4 4 D -16 6 -4 0 4 E 2 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999997 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 10 4 16 -2 B -10 0 -4 -6 -4 C -4 4 0 4 4 D -16 6 -4 0 4 E 2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999934 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=24 E=22 C=11 D=8 so D is eliminated. Round 2 votes counts: A=36 B=28 E=23 C=13 so C is eliminated. Round 3 votes counts: A=38 E=34 B=28 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:214 C:204 E:199 D:195 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 16 -2 B -10 0 -4 -6 -4 C -4 4 0 4 4 D -16 6 -4 0 4 E 2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999934 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 16 -2 B -10 0 -4 -6 -4 C -4 4 0 4 4 D -16 6 -4 0 4 E 2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999934 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 16 -2 B -10 0 -4 -6 -4 C -4 4 0 4 4 D -16 6 -4 0 4 E 2 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999934 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2954: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) E C D B A (6) B D A E C (5) E C D A B (4) C E A D B (4) B E C D A (4) B E C A D (4) B A D E C (4) E C B D A (3) E B C D A (3) D C A E B (3) D A C B E (3) B A E D C (3) B A D C E (3) A D C B E (3) A C D E B (3) A B D C E (3) E C B A D (2) E C A D B (2) D E C A B (2) D C E A B (2) C E D A B (2) C E A B D (2) B D E A C (2) B A E C D (2) A B C E D (2) E D C B A (1) E D B C A (1) E C A B D (1) E B C A D (1) D E C B A (1) D A B C E (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A C D (1) B D A C E (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 -6 2 B -4 0 -14 -4 -10 C 4 14 0 4 -6 D 6 4 -4 0 -2 E -2 10 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.440000000043 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 A B C D E A 0 4 -4 -6 2 B -4 0 -14 -4 -10 C 4 14 0 4 -6 D 6 4 -4 0 -2 E -2 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999985 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=24 D=19 A=16 C=11 so C is eliminated. Round 2 votes counts: E=32 B=30 D=20 A=18 so A is eliminated. Round 3 votes counts: B=36 E=34 D=30 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:208 E:208 D:202 A:198 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 -6 2 B -4 0 -14 -4 -10 C 4 14 0 4 -6 D 6 4 -4 0 -2 E -2 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999985 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 2 B -4 0 -14 -4 -10 C 4 14 0 4 -6 D 6 4 -4 0 -2 E -2 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999985 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 2 B -4 0 -14 -4 -10 C 4 14 0 4 -6 D 6 4 -4 0 -2 E -2 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999985 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2955: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (12) B D C E A (7) A C E D B (7) E A D C B (6) C A E D B (4) B A C D E (4) A B E D C (4) E A C D B (3) D B C E A (3) B D E C A (3) B C D E A (3) B C A D E (3) E A D B C (2) D E B C A (2) D B E C A (2) C D E B A (2) B C D A E (2) A E D B C (2) A E B C D (2) A C B D E (2) A B C E D (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D A B (1) E B D A C (1) D E C B A (1) D C E B A (1) C E D A B (1) C E A D B (1) C D B E A (1) C B D A E (1) C A D E B (1) C A D B E (1) C A B D E (1) B D E A C (1) B A D E C (1) B A D C E (1) A E D C B (1) A E C B D (1) A E B D C (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 18 12 30 12 B -18 0 -2 -14 -10 C -12 2 0 16 0 D -30 14 -16 0 -12 E -12 10 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 12 30 12 B -18 0 -2 -14 -10 C -12 2 0 16 0 D -30 14 -16 0 -12 E -12 10 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=25 E=16 C=13 D=9 so D is eliminated. Round 2 votes counts: A=37 B=30 E=19 C=14 so C is eliminated. Round 3 votes counts: A=44 B=32 E=24 so E is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:236 E:205 C:203 B:178 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 12 30 12 B -18 0 -2 -14 -10 C -12 2 0 16 0 D -30 14 -16 0 -12 E -12 10 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 30 12 B -18 0 -2 -14 -10 C -12 2 0 16 0 D -30 14 -16 0 -12 E -12 10 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 30 12 B -18 0 -2 -14 -10 C -12 2 0 16 0 D -30 14 -16 0 -12 E -12 10 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2956: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) E B C D A (5) D A C E B (5) C A B E D (5) D E B A C (4) C E B A D (4) B D E A C (4) B C E A D (4) A D B C E (4) A C D B E (4) D A B E C (3) C B E A D (3) C B A E D (3) B E C A D (3) B A D E C (3) A D C E B (3) D E A C B (2) D E A B C (2) C E A B D (2) B C A E D (2) B A C E D (2) B A C D E (2) A C D E B (2) A C B D E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C B D A (1) E B D A C (1) D C E A B (1) D B E A C (1) D A E C B (1) D A E B C (1) C E D A B (1) C E B D A (1) C D E A B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B E A C D (1) B A D C E (1) A D C B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 6 8 -8 B 14 0 8 22 8 C -6 -8 0 2 10 D -8 -22 -2 0 -4 E 8 -8 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 8 -8 B 14 0 8 22 8 C -6 -8 0 2 10 D -8 -22 -2 0 -4 E 8 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=20 C=20 A=18 E=16 so E is eliminated. Round 2 votes counts: B=38 D=22 C=22 A=18 so A is eliminated. Round 3 votes counts: B=39 C=31 D=30 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:199 E:197 A:196 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 6 8 -8 B 14 0 8 22 8 C -6 -8 0 2 10 D -8 -22 -2 0 -4 E 8 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 8 -8 B 14 0 8 22 8 C -6 -8 0 2 10 D -8 -22 -2 0 -4 E 8 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 8 -8 B 14 0 8 22 8 C -6 -8 0 2 10 D -8 -22 -2 0 -4 E 8 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2957: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (13) E D B C A (7) C B A E D (7) C A B E D (6) B C A E D (6) E B D C A (5) E B C D A (5) D E A B C (5) A C B D E (5) D E B A C (4) D A E C B (4) B E C D A (4) A C D B E (4) A C B E D (4) D E A C B (3) A D C E B (3) D E B C A (2) D A C E B (2) B E C A D (2) A D C B E (2) E D C B A (1) E C B D A (1) E B C A D (1) D A E B C (1) C B E A D (1) B E D C A (1) B D E C A (1) Total count = 100 A B C D E A 0 -22 -26 8 -12 B 22 0 14 24 12 C 26 -14 0 18 6 D -8 -24 -18 0 -28 E 12 -12 -6 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -26 8 -12 B 22 0 14 24 12 C 26 -14 0 18 6 D -8 -24 -18 0 -28 E 12 -12 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=21 E=20 A=18 C=14 so C is eliminated. Round 2 votes counts: B=35 A=24 D=21 E=20 so E is eliminated. Round 3 votes counts: B=47 D=29 A=24 so A is eliminated. Round 4 votes counts: B=62 D=38 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:236 C:218 E:211 A:174 D:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -26 8 -12 B 22 0 14 24 12 C 26 -14 0 18 6 D -8 -24 -18 0 -28 E 12 -12 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -26 8 -12 B 22 0 14 24 12 C 26 -14 0 18 6 D -8 -24 -18 0 -28 E 12 -12 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -26 8 -12 B 22 0 14 24 12 C 26 -14 0 18 6 D -8 -24 -18 0 -28 E 12 -12 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2958: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) C E D A B (5) A C B D E (5) D E C B A (4) C D E A B (4) C A D E B (4) B A D E C (4) A B C E D (4) E D C B A (3) E C D A B (3) D E B C A (3) C D A E B (3) B E D A C (3) B A E D C (3) B A D C E (3) A C D B E (3) A B D C E (3) A B C D E (3) E D B C A (2) E C B D A (2) E C A B D (2) D B E C A (2) D B A C E (2) C E A D B (2) C A E D B (2) C A D B E (2) B A E C D (2) A C E B D (2) A C B E D (2) A B E C D (2) E C D B A (1) E B D A C (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A E B (1) C D A B E (1) C A E B D (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 14 -12 2 10 B -14 0 -14 -2 -8 C 12 14 0 12 12 D -2 2 -12 0 2 E -10 8 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -12 2 10 B -14 0 -14 -2 -8 C 12 14 0 12 12 D -2 2 -12 0 2 E -10 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 E=20 D=15 B=15 so D is eliminated. Round 2 votes counts: C=28 E=27 A=26 B=19 so B is eliminated. Round 3 votes counts: A=40 E=32 C=28 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:225 A:207 D:195 E:192 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -12 2 10 B -14 0 -14 -2 -8 C 12 14 0 12 12 D -2 2 -12 0 2 E -10 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -12 2 10 B -14 0 -14 -2 -8 C 12 14 0 12 12 D -2 2 -12 0 2 E -10 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -12 2 10 B -14 0 -14 -2 -8 C 12 14 0 12 12 D -2 2 -12 0 2 E -10 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2959: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (13) C A B D E (12) E C A D B (11) E D B C A (5) E C D B A (5) A C B D E (5) E A D B C (4) D B E A C (4) C A E B D (4) B D A E C (4) A B D C E (4) E A C B D (3) D B E C A (3) D B C A E (3) E C A B D (2) C E D B A (2) C E A D B (2) C B D A E (2) B D A C E (2) A C E B D (2) A B D E C (2) E A C D B (1) D B A E C (1) C D B E A (1) B D C A E (1) B A D E C (1) A E C B D (1) Total count = 100 A B C D E A 0 6 -6 8 -12 B -6 0 -6 -10 -10 C 6 6 0 6 -20 D -8 10 -6 0 -10 E 12 10 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -6 8 -12 B -6 0 -6 -10 -10 C 6 6 0 6 -20 D -8 10 -6 0 -10 E 12 10 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 C=23 A=14 D=11 B=8 so B is eliminated. Round 2 votes counts: E=44 C=23 D=18 A=15 so A is eliminated. Round 3 votes counts: E=45 C=30 D=25 so D is eliminated. Round 4 votes counts: E=60 C=40 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:199 A:198 D:193 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 8 -12 B -6 0 -6 -10 -10 C 6 6 0 6 -20 D -8 10 -6 0 -10 E 12 10 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 8 -12 B -6 0 -6 -10 -10 C 6 6 0 6 -20 D -8 10 -6 0 -10 E 12 10 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 8 -12 B -6 0 -6 -10 -10 C 6 6 0 6 -20 D -8 10 -6 0 -10 E 12 10 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2960: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) B C E D A (7) E B A C D (6) D A C E B (6) A D C E B (6) D C B A E (5) A E B D C (5) E B C A D (4) E A B D C (4) C B D E A (4) B E C D A (4) E B C D A (3) D C A B E (3) E A D B C (2) E A B C D (2) D A E C B (2) D A C B E (2) C B E D A (2) C B D A E (2) B E C A D (2) B E A C D (2) B C D E A (2) A E D C B (2) A E D B C (2) A D C B E (2) E C B D A (1) D C B E A (1) D C A E B (1) C D B E A (1) C D B A E (1) B C E A D (1) B C D A E (1) A E B C D (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 10 4 4 B -2 0 2 6 -12 C -10 -2 0 -8 -4 D -4 -6 8 0 0 E -4 12 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 4 B -2 0 2 6 -12 C -10 -2 0 -8 -4 D -4 -6 8 0 0 E -4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995885 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=22 D=20 B=19 C=10 so C is eliminated. Round 2 votes counts: A=29 B=27 E=22 D=22 so E is eliminated. Round 3 votes counts: B=41 A=37 D=22 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 E:206 D:199 B:197 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 4 B -2 0 2 6 -12 C -10 -2 0 -8 -4 D -4 -6 8 0 0 E -4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995885 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 4 B -2 0 2 6 -12 C -10 -2 0 -8 -4 D -4 -6 8 0 0 E -4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995885 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 4 B -2 0 2 6 -12 C -10 -2 0 -8 -4 D -4 -6 8 0 0 E -4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995885 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2961: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (13) D C B A E (13) A E C D B (12) B D C E A (9) B C D E A (8) B E A C D (6) A E D C B (6) D C A E B (5) D B C E A (4) E A C B D (3) E B A C D (2) B D E C A (2) A E B D C (2) A E B C D (2) D C B E A (1) D B C A E (1) D B A E C (1) C D E B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C B D E A (1) B E A D C (1) B C E D A (1) A E D B C (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 2 0 -8 B 6 0 6 4 0 C -2 -6 0 6 -6 D 0 -4 -6 0 -2 E 8 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.563427 C: 0.000000 D: 0.000000 E: 0.436573 Sum of squares = 0.50804588945 Cumulative probabilities = A: 0.000000 B: 0.563427 C: 0.563427 D: 0.563427 E: 1.000000 A B C D E A 0 -6 2 0 -8 B 6 0 6 4 0 C -2 -6 0 6 -6 D 0 -4 -6 0 -2 E 8 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 A=25 E=18 C=5 so C is eliminated. Round 2 votes counts: B=29 D=28 A=25 E=18 so E is eliminated. Round 3 votes counts: A=41 B=31 D=28 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:208 C:196 A:194 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 0 -8 B 6 0 6 4 0 C -2 -6 0 6 -6 D 0 -4 -6 0 -2 E 8 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 -8 B 6 0 6 4 0 C -2 -6 0 6 -6 D 0 -4 -6 0 -2 E 8 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 -8 B 6 0 6 4 0 C -2 -6 0 6 -6 D 0 -4 -6 0 -2 E 8 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999938 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2962: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) C D A E B (8) B E D A C (7) B A E D C (7) B A C E D (5) A B D E C (4) A B C D E (4) D E C A B (3) C E D B A (3) C A D E B (3) B A C D E (3) A C D B E (3) A C B D E (3) E D C B A (2) E D B A C (2) E B D A C (2) D C E A B (2) C B E D A (2) C A B D E (2) B E D C A (2) B E A D C (2) E D C A B (1) E D B C A (1) E B A D C (1) D E A C B (1) D A C E B (1) C D E B A (1) C B E A D (1) C B D E A (1) C B A D E (1) C A D B E (1) B E C D A (1) B C E D A (1) B C E A D (1) B C A E D (1) B C A D E (1) B A E C D (1) A D E C B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -2 -6 2 B -2 0 -6 6 8 C 2 6 0 20 22 D 6 -6 -20 0 14 E -2 -8 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -6 2 B -2 0 -6 6 8 C 2 6 0 20 22 D 6 -6 -20 0 14 E -2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=32 A=17 E=9 D=7 so D is eliminated. Round 2 votes counts: C=37 B=32 A=18 E=13 so E is eliminated. Round 3 votes counts: C=43 B=38 A=19 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:203 A:198 D:197 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 -6 2 B -2 0 -6 6 8 C 2 6 0 20 22 D 6 -6 -20 0 14 E -2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -6 2 B -2 0 -6 6 8 C 2 6 0 20 22 D 6 -6 -20 0 14 E -2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -6 2 B -2 0 -6 6 8 C 2 6 0 20 22 D 6 -6 -20 0 14 E -2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2963: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) E B C A D (6) C E D B A (6) D B E C A (5) D B A E C (5) A B E D C (5) A B D E C (5) E C B D A (3) D C E B A (3) D A B E C (3) C E A D B (3) C D E A B (3) A E B C D (3) A D B C E (3) A C E B D (3) E C B A D (2) C E D A B (2) C E B D A (2) C E A B D (2) C A E B D (2) B A E D C (2) B A E C D (2) A E C B D (2) A D C B E (2) A B D C E (2) E B D C A (1) E A C B D (1) E A B C D (1) D E B C A (1) D C B E A (1) D C A B E (1) D B E A C (1) D B A C E (1) C D E B A (1) C D A E B (1) C A D E B (1) B E D C A (1) B D A E C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 12 6 16 10 B -12 0 16 10 0 C -6 -16 0 12 -20 D -16 -10 -12 0 -16 E -10 0 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 16 10 B -12 0 16 10 0 C -6 -16 0 12 -20 D -16 -10 -12 0 -16 E -10 0 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=23 D=21 E=14 B=6 so B is eliminated. Round 2 votes counts: A=40 C=23 D=22 E=15 so E is eliminated. Round 3 votes counts: A=42 C=34 D=24 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:213 B:207 C:185 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 16 10 B -12 0 16 10 0 C -6 -16 0 12 -20 D -16 -10 -12 0 -16 E -10 0 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 16 10 B -12 0 16 10 0 C -6 -16 0 12 -20 D -16 -10 -12 0 -16 E -10 0 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 16 10 B -12 0 16 10 0 C -6 -16 0 12 -20 D -16 -10 -12 0 -16 E -10 0 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2964: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) E B D C A (7) A C D B E (7) B E C A D (6) B A C E D (6) B A C D E (6) E B D A C (5) B E A C D (5) E B C D A (4) C A B D E (4) A B C D E (4) E D C A B (3) D C A E B (3) B E A D C (3) E D B C A (2) D E C A B (2) D E A C B (2) D A C E B (2) C D A E B (2) E D B A C (1) E D A B C (1) E C D B A (1) E C B D A (1) D E B A C (1) D C E A B (1) D B A E C (1) D A E C B (1) D A E B C (1) C B A E D (1) C B A D E (1) B C E A D (1) B C A E D (1) B A E D C (1) B A E C D (1) A D C B E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 2 18 8 B 10 0 16 16 20 C -2 -16 0 22 2 D -18 -16 -22 0 0 E -8 -20 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 18 8 B 10 0 16 16 20 C -2 -16 0 22 2 D -18 -16 -22 0 0 E -8 -20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=25 C=17 D=14 A=14 so D is eliminated. Round 2 votes counts: B=31 E=30 C=21 A=18 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:209 C:203 E:185 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 18 8 B 10 0 16 16 20 C -2 -16 0 22 2 D -18 -16 -22 0 0 E -8 -20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 18 8 B 10 0 16 16 20 C -2 -16 0 22 2 D -18 -16 -22 0 0 E -8 -20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 18 8 B 10 0 16 16 20 C -2 -16 0 22 2 D -18 -16 -22 0 0 E -8 -20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2965: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) D B C E A (6) E D A C B (4) E A D C B (4) B D C A E (4) B D A E C (4) E D C A B (3) D C B E A (3) D B E A C (3) C B A E D (3) C A E B D (3) B D A C E (3) A E D B C (3) A B E D C (3) E C A D B (2) D E B A C (2) D B E C A (2) D A B E C (2) C E A D B (2) C E A B D (2) C D B E A (2) C B D E A (2) B D C E A (2) B C D A E (2) B C A D E (2) B A C D E (2) A E D C B (2) A E C D B (2) A C E B D (2) E C D A B (1) E A D B C (1) E A C D B (1) D E C B A (1) D E C A B (1) C E D B A (1) C E B D A (1) C D E B A (1) C A B E D (1) B C D E A (1) B C A E D (1) A E B D C (1) A D E B C (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 -2 4 B -4 0 -4 0 2 C -2 4 0 -12 -6 D 2 0 12 0 -6 E -4 -2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 4 2 -2 4 B -4 0 -4 0 2 C -2 4 0 -12 -6 D 2 0 12 0 -6 E -4 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888631 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=21 D=20 C=18 E=16 so E is eliminated. Round 2 votes counts: A=31 D=27 C=21 B=21 so C is eliminated. Round 3 votes counts: A=41 D=32 B=27 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:204 D:204 E:203 B:197 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 -2 4 B -4 0 -4 0 2 C -2 4 0 -12 -6 D 2 0 12 0 -6 E -4 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888631 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -2 4 B -4 0 -4 0 2 C -2 4 0 -12 -6 D 2 0 12 0 -6 E -4 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888631 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -2 4 B -4 0 -4 0 2 C -2 4 0 -12 -6 D 2 0 12 0 -6 E -4 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888631 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2966: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) B E A C D (8) A E B D C (8) C D B E A (7) D C A B E (5) E A B C D (4) C D E B A (4) B E A D C (4) D C B A E (3) C D A E B (3) B A E D C (3) A D E C B (3) A B E D C (3) E B A C D (2) E A C B D (2) D C B E A (2) C E D A B (2) C D E A B (2) C B E D A (2) B E C D A (2) B C D E A (2) A E C D B (2) A D B E C (2) E C A B D (1) E B A D C (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C E B (1) D A C B E (1) C E A D B (1) C B D E A (1) B E C A D (1) B D E A C (1) B D C E A (1) B C E D A (1) B A D E C (1) A E D C B (1) Total count = 100 A B C D E A 0 4 -2 -6 -2 B -4 0 -8 -4 4 C 2 8 0 -6 0 D 6 4 6 0 4 E 2 -4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -6 -2 B -4 0 -8 -4 4 C 2 8 0 -6 0 D 6 4 6 0 4 E 2 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=24 C=22 A=19 E=10 so E is eliminated. Round 2 votes counts: B=27 D=25 A=25 C=23 so C is eliminated. Round 3 votes counts: D=43 B=30 A=27 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:202 A:197 E:197 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -6 -2 B -4 0 -8 -4 4 C 2 8 0 -6 0 D 6 4 6 0 4 E 2 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -6 -2 B -4 0 -8 -4 4 C 2 8 0 -6 0 D 6 4 6 0 4 E 2 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -6 -2 B -4 0 -8 -4 4 C 2 8 0 -6 0 D 6 4 6 0 4 E 2 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2967: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (11) C B A D E (10) C B E D A (9) E D B A C (5) D E A B C (5) C B A E D (5) B E D A C (5) A D E C B (5) E D A B C (4) C E B D A (3) C A D E B (3) B A D E C (3) E D B C A (2) E D A C B (2) C E D B A (2) C E D A B (2) C B E A D (2) B E D C A (2) B C E D A (2) A C D E B (2) E C D B A (1) D E B A C (1) D E A C B (1) D A E C B (1) C E A D B (1) C A B D E (1) B D E A C (1) B C A D E (1) B A C D E (1) A D B E C (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 8 4 0 B 10 0 -4 -2 -2 C -8 4 0 -4 -2 D -4 2 4 0 6 E 0 2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999975 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 4 0 B 10 0 -4 -2 -2 C -8 4 0 -4 -2 D -4 2 4 0 6 E 0 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000219 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=25 B=15 E=14 D=8 so D is eliminated. Round 2 votes counts: C=38 A=26 E=21 B=15 so B is eliminated. Round 3 votes counts: C=41 A=30 E=29 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:204 A:201 B:201 E:199 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 4 0 B 10 0 -4 -2 -2 C -8 4 0 -4 -2 D -4 2 4 0 6 E 0 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000219 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 4 0 B 10 0 -4 -2 -2 C -8 4 0 -4 -2 D -4 2 4 0 6 E 0 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000219 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 4 0 B 10 0 -4 -2 -2 C -8 4 0 -4 -2 D -4 2 4 0 6 E 0 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000219 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2968: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (18) E D A C B (9) C B A D E (8) C B E A D (5) E C B A D (4) C E B D A (4) B C A D E (4) A D B E C (4) E D C A B (3) E C B D A (3) E A D B C (3) D A E B C (3) C B D A E (3) B A C D E (3) E A B D C (2) C E D B A (2) C E B A D (2) C B A E D (2) A D B C E (2) A B D C E (2) E D C B A (1) E B A D C (1) E B A C D (1) D E A C B (1) D C A B E (1) D A C B E (1) D A B C E (1) C D E A B (1) C D B A E (1) C D A B E (1) C B D E A (1) B E A C D (1) B C E A D (1) A E D B C (1) Total count = 100 A B C D E A 0 6 6 -8 -26 B -6 0 -6 -6 -18 C -6 6 0 -6 -10 D 8 6 6 0 -26 E 26 18 10 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 -8 -26 B -6 0 -6 -6 -18 C -6 6 0 -6 -10 D 8 6 6 0 -26 E 26 18 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=45 C=30 B=9 A=9 D=7 so D is eliminated. Round 2 votes counts: E=46 C=31 A=14 B=9 so B is eliminated. Round 3 votes counts: E=47 C=36 A=17 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:240 D:197 C:192 A:189 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -8 -26 B -6 0 -6 -6 -18 C -6 6 0 -6 -10 D 8 6 6 0 -26 E 26 18 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -8 -26 B -6 0 -6 -6 -18 C -6 6 0 -6 -10 D 8 6 6 0 -26 E 26 18 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -8 -26 B -6 0 -6 -6 -18 C -6 6 0 -6 -10 D 8 6 6 0 -26 E 26 18 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2969: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) C D E B A (9) B E A D C (7) E B D C A (6) D C E B A (5) A C B E D (5) A B E D C (5) C D B E A (4) B E C D A (4) A B E C D (4) E B D A C (3) B E A C D (3) A E B D C (3) E B A D C (2) D E B C A (2) D C A E B (2) C D A B E (2) C A D E B (2) C A D B E (2) B E D C A (2) B E D A C (2) B E C A D (2) A C D E B (2) A B C E D (2) D E C B A (1) D C E A B (1) D C B E A (1) C B D E A (1) B A E D C (1) B A E C D (1) A D E B C (1) A D C E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -12 -10 -10 B 12 0 0 8 0 C 12 0 0 10 2 D 10 -8 -10 0 -4 E 10 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.327970 C: 0.672030 D: 0.000000 E: 0.000000 Sum of squares = 0.559188346845 Cumulative probabilities = A: 0.000000 B: 0.327970 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -10 -10 B 12 0 0 8 0 C 12 0 0 10 2 D 10 -8 -10 0 -4 E 10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=25 B=22 D=12 E=11 so E is eliminated. Round 2 votes counts: B=33 C=30 A=25 D=12 so D is eliminated. Round 3 votes counts: C=40 B=35 A=25 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 B:210 E:206 D:194 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -10 -10 B 12 0 0 8 0 C 12 0 0 10 2 D 10 -8 -10 0 -4 E 10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -10 -10 B 12 0 0 8 0 C 12 0 0 10 2 D 10 -8 -10 0 -4 E 10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -10 -10 B 12 0 0 8 0 C 12 0 0 10 2 D 10 -8 -10 0 -4 E 10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2970: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) C A D E B (8) C A D B E (8) E A C D B (7) E C A D B (5) E C A B D (4) E B D C A (4) C A E D B (4) A C D B E (4) E B C D A (3) C D B A E (3) B D E A C (3) B D A C E (3) A D C B E (3) A D B C E (3) E A B D C (2) E A B C D (2) D C B A E (2) C E A D B (2) B E D A C (2) B D A E C (2) A C D E B (2) E B A D C (1) E B A C D (1) E A C B D (1) D B C A E (1) D B A C E (1) D A C B E (1) D A B C E (1) C D A B E (1) C B D A E (1) B D E C A (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 20 6 18 0 B -20 0 -14 -16 -18 C -6 14 0 14 -2 D -18 16 -14 0 -2 E 0 18 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.404628 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.595372 Sum of squares = 0.518191717449 Cumulative probabilities = A: 0.404628 B: 0.404628 C: 0.404628 D: 0.404628 E: 1.000000 A B C D E A 0 20 6 18 0 B -20 0 -14 -16 -18 C -6 14 0 14 -2 D -18 16 -14 0 -2 E 0 18 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=27 A=14 B=11 D=6 so D is eliminated. Round 2 votes counts: E=42 C=29 A=16 B=13 so B is eliminated. Round 3 votes counts: E=48 C=30 A=22 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:222 E:211 C:210 D:191 B:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 18 0 B -20 0 -14 -16 -18 C -6 14 0 14 -2 D -18 16 -14 0 -2 E 0 18 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 18 0 B -20 0 -14 -16 -18 C -6 14 0 14 -2 D -18 16 -14 0 -2 E 0 18 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 18 0 B -20 0 -14 -16 -18 C -6 14 0 14 -2 D -18 16 -14 0 -2 E 0 18 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2971: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (12) E C B A D (5) C E B D A (5) C B E D A (5) C A E D B (5) D B C E A (4) D B A E C (4) B E D A C (4) D B A C E (3) D A B C E (3) C E B A D (3) B D C E A (3) A E D B C (3) A D E B C (3) A D C B E (3) A D B C E (3) E C B D A (2) E B A D C (2) D B C A E (2) C E A B D (2) B E D C A (2) B D E A C (2) E C A B D (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) E A B C D (1) D C A B E (1) D A C B E (1) D A B E C (1) C D B E A (1) C D B A E (1) C B D E A (1) B E C D A (1) B C E D A (1) A E C B D (1) A E B D C (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 6 2 2 B 10 0 18 -8 18 C -6 -18 0 -18 2 D -2 8 18 0 2 E -2 -18 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.100000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999999 Cumulative probabilities = A: 0.400000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 2 2 B 10 0 18 -8 18 C -6 -18 0 -18 2 D -2 8 18 0 2 E -2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.100000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999922 Cumulative probabilities = A: 0.400000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=23 D=19 E=15 B=13 so B is eliminated. Round 2 votes counts: A=30 D=24 C=24 E=22 so E is eliminated. Round 3 votes counts: C=35 A=34 D=31 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:219 D:213 A:200 E:188 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 6 2 2 B 10 0 18 -8 18 C -6 -18 0 -18 2 D -2 8 18 0 2 E -2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.100000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999922 Cumulative probabilities = A: 0.400000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 2 2 B 10 0 18 -8 18 C -6 -18 0 -18 2 D -2 8 18 0 2 E -2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.100000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999922 Cumulative probabilities = A: 0.400000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 2 2 B 10 0 18 -8 18 C -6 -18 0 -18 2 D -2 8 18 0 2 E -2 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.100000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999922 Cumulative probabilities = A: 0.400000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2972: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (13) E C D A B (11) B A D C E (7) C D E A B (6) A D E C B (6) A D C E B (6) B A D E C (4) B A C D E (4) E C B D A (3) B E A D C (3) B C D E A (3) E D C A B (2) E D A C B (2) E B A D C (2) E A D C B (2) E A B D C (2) C D A E B (2) B E A C D (2) B C D A E (2) B A E D C (2) A B D C E (2) E D C B A (1) E C D B A (1) E B C D A (1) E A D B C (1) D C A E B (1) D A C E B (1) C E D A B (1) B C E A D (1) B C A D E (1) B A E C D (1) B A C E D (1) A E D B C (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 2 0 -14 B 4 0 8 8 -6 C -2 -8 0 6 -22 D 0 -8 -6 0 -6 E 14 6 22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 0 -14 B 4 0 8 8 -6 C -2 -8 0 6 -22 D 0 -8 -6 0 -6 E 14 6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 E=28 A=17 C=9 D=2 so D is eliminated. Round 2 votes counts: B=44 E=28 A=18 C=10 so C is eliminated. Round 3 votes counts: B=44 E=35 A=21 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:207 A:192 D:190 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 0 -14 B 4 0 8 8 -6 C -2 -8 0 6 -22 D 0 -8 -6 0 -6 E 14 6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 0 -14 B 4 0 8 8 -6 C -2 -8 0 6 -22 D 0 -8 -6 0 -6 E 14 6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 0 -14 B 4 0 8 8 -6 C -2 -8 0 6 -22 D 0 -8 -6 0 -6 E 14 6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2973: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (15) C A E B D (12) E A C D B (10) D B C E A (7) D B E A C (6) C B A E D (6) B C D A E (5) D E B A C (4) A E C B D (4) A C E B D (4) E A D C B (3) D E A B C (3) C A B E D (3) B D A C E (3) E A D B C (2) D B C A E (2) C B A D E (2) B D C E A (2) E D A C B (1) E A C B D (1) D B E C A (1) C D B E A (1) C B D A E (1) B D A E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -14 -4 18 B 12 0 4 20 12 C 14 -4 0 0 26 D 4 -20 0 0 6 E -18 -12 -26 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -4 18 B 12 0 4 20 12 C 14 -4 0 0 26 D 4 -20 0 0 6 E -18 -12 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 D=23 E=17 A=9 so A is eliminated. Round 2 votes counts: C=29 B=27 D=23 E=21 so E is eliminated. Round 3 votes counts: C=44 D=29 B=27 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:218 D:195 A:194 E:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 -4 18 B 12 0 4 20 12 C 14 -4 0 0 26 D 4 -20 0 0 6 E -18 -12 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -4 18 B 12 0 4 20 12 C 14 -4 0 0 26 D 4 -20 0 0 6 E -18 -12 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -4 18 B 12 0 4 20 12 C 14 -4 0 0 26 D 4 -20 0 0 6 E -18 -12 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2974: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) C E B D A (7) D B C A E (5) C E D B A (5) C B D E A (5) A E D B C (5) D A B C E (4) B D A C E (4) B C D A E (4) B A C D E (4) E D A C B (3) E C D A B (3) E C A B D (3) E A C D B (3) E A C B D (3) D C B E A (3) D B A C E (3) E C D B A (2) E C A D B (2) E A D C B (2) E A B C D (2) D A B E C (2) B D C A E (2) A B E D C (2) E C B D A (1) E C B A D (1) D E C A B (1) D C B A E (1) D A E B C (1) C D B E A (1) C B E D A (1) C B E A D (1) B C A D E (1) A E B D C (1) A D B C E (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 2 -16 2 B 2 0 4 -10 10 C -2 -4 0 0 6 D 16 10 0 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.520409 D: 0.479591 E: 0.000000 Sum of squares = 0.500833064585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.520409 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -16 2 B 2 0 4 -10 10 C -2 -4 0 0 6 D 16 10 0 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=20 C=20 A=20 B=15 so B is eliminated. Round 2 votes counts: D=26 E=25 C=25 A=24 so A is eliminated. Round 3 votes counts: D=37 E=34 C=29 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:203 C:200 A:193 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -16 2 B 2 0 4 -10 10 C -2 -4 0 0 6 D 16 10 0 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -16 2 B 2 0 4 -10 10 C -2 -4 0 0 6 D 16 10 0 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -16 2 B 2 0 4 -10 10 C -2 -4 0 0 6 D 16 10 0 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2975: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (14) A E C B D (10) C B E D A (9) E A C B D (7) E C B A D (6) E C B D A (5) C E B A D (5) B C D E A (5) D B C A E (4) E C A B D (3) A E D C B (3) A D E B C (3) D A E B C (2) C B E A D (2) C B D E A (2) A E D B C (2) A E C D B (2) A D B C E (2) A C E B D (2) E B C D A (1) E A C D B (1) D E A B C (1) D B E A C (1) D B A C E (1) D A B E C (1) D A B C E (1) B D C A E (1) B C E D A (1) B C D A E (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 -18 0 -26 B 16 0 -18 22 -6 C 18 18 0 26 4 D 0 -22 -26 0 -20 E 26 6 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -18 0 -26 B 16 0 -18 22 -6 C 18 18 0 26 4 D 0 -22 -26 0 -20 E 26 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 E=23 C=18 B=8 so B is eliminated. Round 2 votes counts: D=26 A=26 C=25 E=23 so E is eliminated. Round 3 votes counts: C=40 A=34 D=26 so D is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:233 E:224 B:207 A:170 D:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -18 0 -26 B 16 0 -18 22 -6 C 18 18 0 26 4 D 0 -22 -26 0 -20 E 26 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 0 -26 B 16 0 -18 22 -6 C 18 18 0 26 4 D 0 -22 -26 0 -20 E 26 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 0 -26 B 16 0 -18 22 -6 C 18 18 0 26 4 D 0 -22 -26 0 -20 E 26 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2976: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) B D E C A (9) B D E A C (9) E C A B D (6) D B E A C (6) B D C A E (5) E A C D B (4) D B A C E (4) D A C B E (4) C A E B D (4) C A B E D (3) A C D B E (3) E C A D B (2) E B C A D (2) D E B A C (2) C E A B D (2) C A D B E (2) C A B D E (2) B E D C A (2) B C A D E (2) A C E D B (2) A C D E B (2) E D B A C (1) E D A C B (1) E B D C A (1) D E A B C (1) D B A E C (1) D A E C B (1) D A C E B (1) C B A E D (1) C B A D E (1) C A D E B (1) B E D A C (1) B D C E A (1) B C E A D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 -12 0 -2 B -2 0 -4 4 16 C 12 4 0 -2 2 D 0 -4 2 0 16 E 2 -16 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 0 -2 B -2 0 -4 4 16 C 12 4 0 -2 2 D 0 -4 2 0 16 E 2 -16 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000059 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=25 D=20 E=17 A=8 so A is eliminated. Round 2 votes counts: C=32 B=30 D=21 E=17 so E is eliminated. Round 3 votes counts: C=44 B=33 D=23 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:207 D:207 A:194 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 0 -2 B -2 0 -4 4 16 C 12 4 0 -2 2 D 0 -4 2 0 16 E 2 -16 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000059 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 0 -2 B -2 0 -4 4 16 C 12 4 0 -2 2 D 0 -4 2 0 16 E 2 -16 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000059 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 0 -2 B -2 0 -4 4 16 C 12 4 0 -2 2 D 0 -4 2 0 16 E 2 -16 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000059 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2977: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) D A C E B (9) E B C A D (8) D A C B E (7) E D B A C (4) E B D A C (4) D C A E B (4) B E C A D (4) A D C B E (4) E B D C A (3) D C A B E (3) C A D B E (3) B C E A D (3) A B E D C (3) E D C B A (2) D E A B C (2) D A E B C (2) C B E D A (2) B E C D A (2) A C B D E (2) A B C E D (2) E B A D C (1) E B A C D (1) E A D B C (1) D E A C B (1) D C E B A (1) D C E A B (1) D A E C B (1) C D E B A (1) C D A B E (1) C B A E D (1) C B A D E (1) C A B D E (1) B E A D C (1) B C A E D (1) A D E B C (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 -18 -2 B -2 0 8 0 -12 C 2 -8 0 -12 0 D 18 0 12 0 -4 E 2 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.189398 D: 0.000000 E: 0.810602 Sum of squares = 0.692947700547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.189398 D: 0.189398 E: 1.000000 A B C D E A 0 2 -2 -18 -2 B -2 0 8 0 -12 C 2 -8 0 -12 0 D 18 0 12 0 -4 E 2 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000013884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=31 A=15 B=11 C=10 so C is eliminated. Round 2 votes counts: E=33 D=33 A=19 B=15 so B is eliminated. Round 3 votes counts: E=45 D=33 A=22 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:209 B:197 C:191 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 -18 -2 B -2 0 8 0 -12 C 2 -8 0 -12 0 D 18 0 12 0 -4 E 2 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000013884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -18 -2 B -2 0 8 0 -12 C 2 -8 0 -12 0 D 18 0 12 0 -4 E 2 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000013884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -18 -2 B -2 0 8 0 -12 C 2 -8 0 -12 0 D 18 0 12 0 -4 E 2 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000013884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2978: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B E D A (10) D A B C E (9) E C B A D (8) A D E B C (7) B C D A E (6) C B D A E (5) E A D C B (4) B C E A D (4) E C B D A (3) C E B D A (3) B A D C E (3) D A E C B (2) D A C B E (2) C B E A D (2) B A D E C (2) A E D B C (2) A D B E C (2) E D C A B (1) E C D A B (1) E C A B D (1) D C A E B (1) D A E B C (1) D A C E B (1) D A B E C (1) C D B A E (1) C B D E A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C A D E (1) B A C D E (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 2 -2 B 6 0 6 8 8 C 0 -6 0 0 2 D -2 -8 0 0 -4 E 2 -8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 2 -2 B 6 0 6 8 8 C 0 -6 0 0 2 D -2 -8 0 0 -4 E 2 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 B=20 D=17 A=13 so A is eliminated. Round 2 votes counts: E=30 D=27 C=22 B=21 so B is eliminated. Round 3 votes counts: C=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:198 E:198 A:197 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 2 -2 B 6 0 6 8 8 C 0 -6 0 0 2 D -2 -8 0 0 -4 E 2 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 2 -2 B 6 0 6 8 8 C 0 -6 0 0 2 D -2 -8 0 0 -4 E 2 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 2 -2 B 6 0 6 8 8 C 0 -6 0 0 2 D -2 -8 0 0 -4 E 2 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2979: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) B A C E D (8) D E C A B (6) C A B D E (5) B A E C D (5) A B E D C (5) E D C A B (4) E D B A C (4) D E C B A (4) D C E A B (4) C B A D E (4) A B C E D (4) E D A B C (3) C A D B E (3) B A E D C (3) B A C D E (3) A C D E B (3) E D C B A (2) E D B C A (2) C D A E B (2) B E A D C (2) B C A D E (2) E D A C B (1) E B D C A (1) E B D A C (1) E A D B C (1) C D E B A (1) C B D A E (1) C A D E B (1) B E D C A (1) B E D A C (1) B C D E A (1) A E D B C (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 6 6 B -8 0 -2 -2 2 C 4 2 0 8 6 D -6 2 -8 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 6 6 B -8 0 -2 -2 2 C 4 2 0 8 6 D -6 2 -8 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 E=19 A=16 D=14 so D is eliminated. Round 2 votes counts: E=29 C=29 B=26 A=16 so A is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:208 B:195 D:195 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 6 6 B -8 0 -2 -2 2 C 4 2 0 8 6 D -6 2 -8 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 6 6 B -8 0 -2 -2 2 C 4 2 0 8 6 D -6 2 -8 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 6 6 B -8 0 -2 -2 2 C 4 2 0 8 6 D -6 2 -8 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2980: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (14) B E C A D (11) D A C E B (10) C E B D A (8) D A B E C (6) C E B A D (5) C D A E B (5) E B C D A (4) C A D E B (4) A D C E B (4) E C B D A (3) B E D A C (3) B E C D A (3) B E A C D (3) B E A D C (2) E D B C A (1) E B D C A (1) E B C A D (1) D E B C A (1) D C E B A (1) D C A E B (1) D A E B C (1) D A C B E (1) C E D B A (1) C D E B A (1) C A E B D (1) B A E D C (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -4 -2 2 B 0 0 8 -6 -6 C 4 -8 0 0 -12 D 2 6 0 0 2 E -2 6 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.097077 D: 0.902923 E: 0.000000 Sum of squares = 0.82469320979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.097077 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -2 2 B 0 0 8 -6 -6 C 4 -8 0 0 -12 D 2 6 0 0 2 E -2 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102049973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 D=21 A=21 E=10 so E is eliminated. Round 2 votes counts: B=29 C=28 D=22 A=21 so A is eliminated. Round 3 votes counts: D=42 B=30 C=28 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:207 D:205 A:198 B:198 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -2 2 B 0 0 8 -6 -6 C 4 -8 0 0 -12 D 2 6 0 0 2 E -2 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102049973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 2 B 0 0 8 -6 -6 C 4 -8 0 0 -12 D 2 6 0 0 2 E -2 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102049973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 2 B 0 0 8 -6 -6 C 4 -8 0 0 -12 D 2 6 0 0 2 E -2 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102049973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2981: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) C E B A D (11) D B A E C (10) B A D C E (10) B A C D E (6) C E A B D (5) E D A B C (4) D A B E C (4) C E D A B (4) B A D E C (4) D E A B C (3) C E A D B (3) B D A E C (3) B C A D E (3) E D C A B (2) E D A C B (2) E C A D B (2) D E B A C (2) C B E A D (2) C B A E D (2) C B A D E (2) D A E B C (1) C E D B A (1) C B D E A (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 2 4 -6 B 14 0 4 0 -2 C -2 -4 0 6 0 D -4 0 -6 0 2 E 6 2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.296842 D: 0.000000 E: 0.703158 Sum of squares = 0.582546066371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.296842 D: 0.296842 E: 1.000000 A B C D E A 0 -14 2 4 -6 B 14 0 4 0 -2 C -2 -4 0 6 0 D -4 0 -6 0 2 E 6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555573844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 E=21 D=20 A=2 so A is eliminated. Round 2 votes counts: C=31 B=27 E=21 D=21 so E is eliminated. Round 3 votes counts: C=44 D=29 B=27 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:208 E:203 C:200 D:196 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 4 -6 B 14 0 4 0 -2 C -2 -4 0 6 0 D -4 0 -6 0 2 E 6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555573844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 4 -6 B 14 0 4 0 -2 C -2 -4 0 6 0 D -4 0 -6 0 2 E 6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555573844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 4 -6 B 14 0 4 0 -2 C -2 -4 0 6 0 D -4 0 -6 0 2 E 6 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555573844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2982: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) E A B D C (6) C D E A B (6) D C B A E (5) C D B E A (5) A E B D C (5) E A B C D (4) C D B A E (4) A E D B C (4) E C B A D (3) E A D C B (3) D C A B E (3) C D E B A (3) B A E D C (3) B A E C D (3) B A D E C (3) B A D C E (3) A B D E C (3) E A C B D (2) D C A E B (2) D A E C B (2) C E D A B (2) C E B D A (2) B E C A D (2) E D A C B (1) E C A B D (1) E B A D C (1) E A D B C (1) E A C D B (1) D E C A B (1) D A E B C (1) D A B C E (1) C E D B A (1) C B E D A (1) C B D E A (1) B E A C D (1) B A C E D (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 16 2 4 -10 B -16 0 -12 -8 -20 C -2 12 0 -14 -6 D -4 8 14 0 2 E 10 20 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999961 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 A B C D E A 0 16 2 4 -10 B -16 0 -12 -8 -20 C -2 12 0 -14 -6 D -4 8 14 0 2 E 10 20 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999968 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=23 D=22 B=16 A=14 so A is eliminated. Round 2 votes counts: E=32 C=25 D=23 B=20 so B is eliminated. Round 3 votes counts: E=42 D=32 C=26 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:217 D:210 A:206 C:195 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 2 4 -10 B -16 0 -12 -8 -20 C -2 12 0 -14 -6 D -4 8 14 0 2 E 10 20 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999968 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 4 -10 B -16 0 -12 -8 -20 C -2 12 0 -14 -6 D -4 8 14 0 2 E 10 20 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999968 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 4 -10 B -16 0 -12 -8 -20 C -2 12 0 -14 -6 D -4 8 14 0 2 E 10 20 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.250000 Sum of squares = 0.468749999968 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2983: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (14) D C A E B (12) E B D C A (10) A C B E D (9) D E B C A (8) D E B A C (6) C A D B E (5) E B A C D (4) E B D A C (3) E B C A D (3) C A B E D (3) A C D B E (3) D E C B A (2) D E C A B (2) D C A B E (2) D B E A C (2) A B C E D (2) E D B C A (1) E D B A C (1) D A C B E (1) C A E D B (1) C A D E B (1) C A B D E (1) B E D A C (1) B E C A D (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 -16 -4 -2 -16 B 16 0 16 6 -8 C 4 -16 0 -2 -18 D 2 -6 2 0 -10 E 16 8 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -4 -2 -16 B 16 0 16 6 -8 C 4 -16 0 -2 -18 D 2 -6 2 0 -10 E 16 8 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=22 B=18 A=14 C=11 so C is eliminated. Round 2 votes counts: D=35 A=25 E=22 B=18 so B is eliminated. Round 3 votes counts: E=38 D=35 A=27 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:215 D:194 C:184 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -4 -2 -16 B 16 0 16 6 -8 C 4 -16 0 -2 -18 D 2 -6 2 0 -10 E 16 8 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -2 -16 B 16 0 16 6 -8 C 4 -16 0 -2 -18 D 2 -6 2 0 -10 E 16 8 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -2 -16 B 16 0 16 6 -8 C 4 -16 0 -2 -18 D 2 -6 2 0 -10 E 16 8 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2984: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) E A D C B (8) D E C A B (7) E D A C B (6) D C B E A (6) B C A D E (6) A E B C D (6) C B D E A (5) B C D E A (5) B C D A E (5) B A C E D (4) D E A C B (3) C D B E A (3) B D C E A (3) A E D B C (3) D B C E A (2) B C A E D (2) B A E C D (2) A E B D C (2) A B E C D (2) E A D B C (1) D C E B A (1) C D E A B (1) C B D A E (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E C A (1) B A E D C (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 4 0 2 -6 B -4 0 -8 -4 -2 C 0 8 0 -8 -6 D -2 4 8 0 0 E 6 2 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.425545 E: 0.574455 Sum of squares = 0.511087081165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.425545 E: 1.000000 A B C D E A 0 4 0 2 -6 B -4 0 -8 -4 -2 C 0 8 0 -8 -6 D -2 4 8 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 D=19 E=15 C=12 so C is eliminated. Round 2 votes counts: B=36 A=26 D=23 E=15 so E is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:207 D:205 A:200 C:197 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 2 -6 B -4 0 -8 -4 -2 C 0 8 0 -8 -6 D -2 4 8 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 -6 B -4 0 -8 -4 -2 C 0 8 0 -8 -6 D -2 4 8 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 -6 B -4 0 -8 -4 -2 C 0 8 0 -8 -6 D -2 4 8 0 0 E 6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2985: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (5) B D C A E (5) E A C D B (4) D E B C A (4) D C B E A (4) C B D A E (4) A E C B D (4) E D C B A (3) E D B A C (3) E A D C B (3) E A D B C (3) C D E B A (3) B C D A E (3) A E B C D (3) E B D A C (2) D B C A E (2) C D B E A (2) C D B A E (2) C B A D E (2) C A E D B (2) B D E A C (2) B D C E A (2) B A D E C (2) B A D C E (2) A E C D B (2) A E B D C (2) A C E D B (2) A C E B D (2) A C B E D (2) A C B D E (2) E D C A B (1) E D B C A (1) E D A B C (1) E B A D C (1) E A B D C (1) D E C B A (1) D C E B A (1) D B E C A (1) C E D A B (1) C D E A B (1) C D A B E (1) C A D B E (1) B E A D C (1) B D A E C (1) B A E D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -20 -4 -12 -2 B 20 0 0 -8 -2 C 4 0 0 -12 2 D 12 8 12 0 6 E 2 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -12 -2 B 20 0 0 -8 -2 C 4 0 0 -12 2 D 12 8 12 0 6 E 2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=21 C=19 B=19 D=18 so D is eliminated. Round 2 votes counts: E=28 B=27 C=24 A=21 so A is eliminated. Round 3 votes counts: E=39 C=32 B=29 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:219 B:205 E:198 C:197 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -4 -12 -2 B 20 0 0 -8 -2 C 4 0 0 -12 2 D 12 8 12 0 6 E 2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -12 -2 B 20 0 0 -8 -2 C 4 0 0 -12 2 D 12 8 12 0 6 E 2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -12 -2 B 20 0 0 -8 -2 C 4 0 0 -12 2 D 12 8 12 0 6 E 2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 2986: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) B A D C E (9) E C D A B (6) C E B D A (6) A D E B C (6) A D B E C (6) E D A C B (5) E C A D B (5) B D A C E (5) B C A D E (5) E A D C B (4) D A B C E (4) C E B A D (4) C B E D A (4) B A D E C (4) D A E B C (3) C B E A D (3) E C B A D (2) D B A C E (2) E A D B C (1) C E D A B (1) C E A D B (1) C D B E A (1) C D B A E (1) B C D A E (1) B A E D C (1) B A C D E (1) Total count = 100 A B C D E A 0 2 20 4 14 B -2 0 14 -10 12 C -20 -14 0 -18 -4 D -4 10 18 0 14 E -14 -12 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 20 4 14 B -2 0 14 -10 12 C -20 -14 0 -18 -4 D -4 10 18 0 14 E -14 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994407 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 C=21 D=18 A=12 so A is eliminated. Round 2 votes counts: D=30 B=26 E=23 C=21 so C is eliminated. Round 3 votes counts: E=35 B=33 D=32 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:220 D:219 B:207 E:182 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 20 4 14 B -2 0 14 -10 12 C -20 -14 0 -18 -4 D -4 10 18 0 14 E -14 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994407 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 20 4 14 B -2 0 14 -10 12 C -20 -14 0 -18 -4 D -4 10 18 0 14 E -14 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994407 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 20 4 14 B -2 0 14 -10 12 C -20 -14 0 -18 -4 D -4 10 18 0 14 E -14 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994407 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2987: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) D E A B C (4) C B E A D (4) B D C E A (4) B C E D A (4) E B D C A (3) D B E C A (3) D B C A E (3) D B A C E (3) D A E C B (3) C A B E D (3) B D E C A (3) B C D E A (3) A D C E B (3) A C D E B (3) E D B C A (2) E B C D A (2) E A C B D (2) D E B A C (2) D A E B C (2) C B D A E (2) C B A E D (2) C B A D E (2) C A E B D (2) C A B D E (2) B E D C A (2) B C E A D (2) B C D A E (2) A D E C B (2) A C E D B (2) E D B A C (1) E D A B C (1) E C B A D (1) E B D A C (1) E A B C D (1) D E B C A (1) D C B A E (1) D B C E A (1) D A C B E (1) C E B A D (1) C A D B E (1) B E C D A (1) B D C A E (1) A E D C B (1) A E C B D (1) A C E B D (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -26 -28 -12 -10 B 26 0 16 18 4 C 28 -16 0 4 10 D 12 -18 -4 0 6 E 10 -4 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -28 -12 -10 B 26 0 16 18 4 C 28 -16 0 4 10 D 12 -18 -4 0 6 E 10 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 E=20 C=19 A=15 so A is eliminated. Round 2 votes counts: D=29 C=27 E=22 B=22 so E is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:213 D:198 E:195 A:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -28 -12 -10 B 26 0 16 18 4 C 28 -16 0 4 10 D 12 -18 -4 0 6 E 10 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -28 -12 -10 B 26 0 16 18 4 C 28 -16 0 4 10 D 12 -18 -4 0 6 E 10 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -28 -12 -10 B 26 0 16 18 4 C 28 -16 0 4 10 D 12 -18 -4 0 6 E 10 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2988: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) B E D A C (7) D A C E B (6) E B A D C (5) E B C A D (4) C D A B E (4) B D A E C (4) B C D A E (4) E C A D B (3) E A D C B (3) B C E A D (3) E B A C D (2) E A C D B (2) E A B D C (2) D C A E B (2) D B A E C (2) D A E C B (2) D A C B E (2) D A B C E (2) C E A B D (2) C D A E B (2) C B A E D (2) C A D E B (2) B E A D C (2) B D A C E (2) A D C E B (2) E D A C B (1) E D A B C (1) E C A B D (1) E A D B C (1) E A C B D (1) D E A B C (1) D C A B E (1) D B C A E (1) D B A C E (1) D A B E C (1) C E A D B (1) C A E D B (1) C A D B E (1) B D E A C (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 8 -2 -8 B 4 0 16 8 4 C -8 -16 0 -10 -10 D 2 -8 10 0 -6 E 8 -4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -2 -8 B 4 0 16 8 4 C -8 -16 0 -10 -10 D 2 -8 10 0 -6 E 8 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=26 D=21 C=15 A=3 so A is eliminated. Round 2 votes counts: B=35 E=26 D=24 C=15 so C is eliminated. Round 3 votes counts: B=37 D=33 E=30 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:210 D:199 A:197 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 -2 -8 B 4 0 16 8 4 C -8 -16 0 -10 -10 D 2 -8 10 0 -6 E 8 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -2 -8 B 4 0 16 8 4 C -8 -16 0 -10 -10 D 2 -8 10 0 -6 E 8 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -2 -8 B 4 0 16 8 4 C -8 -16 0 -10 -10 D 2 -8 10 0 -6 E 8 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 2989: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) C E B D A (8) E A C B D (6) D B A C E (6) B D C E A (6) A D B E C (6) A E C D B (5) A B D E C (4) C E D B A (3) B E C D A (3) A E C B D (3) A B E D C (3) E C B D A (2) E B C A D (2) E A B C D (2) D B C A E (2) C B E D A (2) B D E A C (2) B C D E A (2) A E D B C (2) A E B D C (2) A D C E B (2) E C B A D (1) E C A D B (1) E B C D A (1) E B A C D (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C E A (1) C E D A B (1) C D E A B (1) C B D E A (1) B E C A D (1) B D A C E (1) B C E D A (1) B A E D C (1) A E B C D (1) A D E C B (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 0 8 -12 B -4 0 2 28 -10 C 0 -2 0 14 -16 D -8 -28 -14 0 -22 E 12 10 16 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 8 -12 B -4 0 2 28 -10 C 0 -2 0 14 -16 D -8 -28 -14 0 -22 E 12 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 B=17 C=16 D=12 so D is eliminated. Round 2 votes counts: A=31 B=26 E=24 C=19 so C is eliminated. Round 3 votes counts: E=37 A=33 B=30 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:230 B:208 A:200 C:198 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 8 -12 B -4 0 2 28 -10 C 0 -2 0 14 -16 D -8 -28 -14 0 -22 E 12 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 8 -12 B -4 0 2 28 -10 C 0 -2 0 14 -16 D -8 -28 -14 0 -22 E 12 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 8 -12 B -4 0 2 28 -10 C 0 -2 0 14 -16 D -8 -28 -14 0 -22 E 12 10 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2990: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (13) D E A B C (6) E D B A C (5) E B D A C (5) E D A B C (4) D A E C B (4) C D E A B (4) B E A D C (4) B C E A D (4) B C A E D (4) B A E C D (4) E B D C A (3) D E C A B (3) D E A C B (2) D A E B C (2) C D E B A (2) C D A E B (2) C B A E D (2) C A D E B (2) B A E D C (2) B A C E D (2) A D C E B (2) A B E D C (2) E D C B A (1) E D B C A (1) E C D B A (1) E A D B C (1) D C A E B (1) C E D B A (1) C E B D A (1) C B E A D (1) B E D C A (1) B E C D A (1) B A C D E (1) A E B D C (1) A D B E C (1) A C D E B (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 4 0 -2 B 18 0 12 8 -10 C -4 -12 0 -4 -10 D 0 -8 4 0 -2 E 2 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 4 0 -2 B 18 0 12 8 -10 C -4 -12 0 -4 -10 D 0 -8 4 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=23 E=21 D=18 A=10 so A is eliminated. Round 2 votes counts: C=30 B=27 E=22 D=21 so D is eliminated. Round 3 votes counts: E=39 C=33 B=28 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:212 D:197 A:192 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 4 0 -2 B 18 0 12 8 -10 C -4 -12 0 -4 -10 D 0 -8 4 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 0 -2 B 18 0 12 8 -10 C -4 -12 0 -4 -10 D 0 -8 4 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 0 -2 B 18 0 12 8 -10 C -4 -12 0 -4 -10 D 0 -8 4 0 -2 E 2 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2991: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (13) B D C E A (9) C D E A B (8) A E C D B (8) D C E A B (5) B A E C D (5) A E C B D (5) A E B C D (5) D C B E A (4) C D B E A (4) E A C D B (3) C A E D B (3) B D C A E (3) E A D C B (2) D B C E A (2) C E D A B (2) C D E B A (2) B D E A C (2) B C A D E (2) A E B D C (2) A B E D C (2) E C A D B (1) D E C A B (1) D C E B A (1) D B E C A (1) C E A D B (1) B C D E A (1) B A D E C (1) B A D C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 0 10 2 B 2 0 -2 2 0 C 0 2 0 2 -2 D -10 -2 -2 0 -6 E -2 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.232993 B: 0.267007 C: 0.232993 D: 0.000000 E: 0.267007 Sum of squares = 0.251156970063 Cumulative probabilities = A: 0.232993 B: 0.500000 C: 0.732993 D: 0.732993 E: 1.000000 A B C D E A 0 -2 0 10 2 B 2 0 -2 2 0 C 0 2 0 2 -2 D -10 -2 -2 0 -6 E -2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=23 C=20 D=14 E=6 so E is eliminated. Round 2 votes counts: B=37 A=28 C=21 D=14 so D is eliminated. Round 3 votes counts: B=40 C=32 A=28 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:205 E:203 B:201 C:201 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 10 2 B 2 0 -2 2 0 C 0 2 0 2 -2 D -10 -2 -2 0 -6 E -2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 10 2 B 2 0 -2 2 0 C 0 2 0 2 -2 D -10 -2 -2 0 -6 E -2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 10 2 B 2 0 -2 2 0 C 0 2 0 2 -2 D -10 -2 -2 0 -6 E -2 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2992: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D C B A E (8) A E C D B (7) E A B C D (6) D C A B E (6) B E A D C (6) B D C A E (6) C D A E B (5) B D C E A (5) E C A D B (4) B E D C A (4) A C D E B (4) A C D B E (4) E B A C D (2) E A C B D (2) C A D E B (2) B E D A C (2) B A E D C (2) A B E D C (2) E B C D A (1) E B A D C (1) D C B E A (1) D B C E A (1) D B C A E (1) D B A C E (1) C D E A B (1) C A E D B (1) C A D B E (1) B D A E C (1) B D A C E (1) A E C B D (1) A D C B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 14 6 12 12 B -14 0 -12 -12 10 C -6 12 0 0 0 D -12 12 0 0 0 E -12 -10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 12 12 B -14 0 -12 -12 10 C -6 12 0 0 0 D -12 12 0 0 0 E -12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=24 A=21 D=18 C=10 so C is eliminated. Round 2 votes counts: B=27 A=25 E=24 D=24 so E is eliminated. Round 3 votes counts: A=45 B=31 D=24 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:203 D:200 E:189 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 12 12 B -14 0 -12 -12 10 C -6 12 0 0 0 D -12 12 0 0 0 E -12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 12 12 B -14 0 -12 -12 10 C -6 12 0 0 0 D -12 12 0 0 0 E -12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 12 12 B -14 0 -12 -12 10 C -6 12 0 0 0 D -12 12 0 0 0 E -12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2993: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) D A E B C (9) C B A D E (9) D A E C B (5) E B A D C (4) C B E A D (4) E D B A C (3) E B D A C (3) D E A B C (3) C D A E B (3) C A D B E (3) B E C A D (3) B C E A D (3) A D E B C (3) A D B E C (3) E C B D A (2) D E A C B (2) C D E A B (2) C D A B E (2) C B A E D (2) C A B D E (2) B A E D C (2) B A D C E (2) B A C D E (2) A D C B E (2) E D C A B (1) E D B C A (1) E B C D A (1) E A D B C (1) D C A E B (1) C B E D A (1) C B D A E (1) B E A D C (1) B C A E D (1) B C A D E (1) B A D E C (1) A D B C E (1) Total count = 100 A B C D E A 0 6 14 0 10 B -6 0 16 -10 -8 C -14 -16 0 -16 -16 D 0 10 16 0 14 E -10 8 16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.573499 B: 0.000000 C: 0.000000 D: 0.426501 E: 0.000000 Sum of squares = 0.510804291633 Cumulative probabilities = A: 0.573499 B: 0.573499 C: 0.573499 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 0 10 B -6 0 16 -10 -8 C -14 -16 0 -16 -16 D 0 10 16 0 14 E -10 8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=26 D=20 B=16 A=9 so A is eliminated. Round 2 votes counts: D=29 C=29 E=26 B=16 so B is eliminated. Round 3 votes counts: C=36 E=32 D=32 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:215 E:200 B:196 C:169 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 0 10 B -6 0 16 -10 -8 C -14 -16 0 -16 -16 D 0 10 16 0 14 E -10 8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 0 10 B -6 0 16 -10 -8 C -14 -16 0 -16 -16 D 0 10 16 0 14 E -10 8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 0 10 B -6 0 16 -10 -8 C -14 -16 0 -16 -16 D 0 10 16 0 14 E -10 8 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2994: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) A D C B E (7) D C E A B (6) B E C D A (6) A B E D C (6) D C A E B (5) C D E B A (5) A C D E B (5) E B C D A (4) D C E B A (4) B A E C D (4) A D C E B (4) E D C B A (3) E C D B A (3) B E A C D (3) A B D C E (3) E C B D A (2) E B D C A (2) C D E A B (2) B E A D C (2) B A E D C (2) A B C D E (2) E D B C A (1) D E C B A (1) D E B C A (1) D A C E B (1) C E D B A (1) C E B D A (1) C A D E B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 4 4 6 B -10 0 -4 -4 -4 C -4 4 0 2 -2 D -4 4 -2 0 -2 E -6 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 4 6 B -10 0 -4 -4 -4 C -4 4 0 2 -2 D -4 4 -2 0 -2 E -6 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 D=18 B=17 E=15 C=10 so C is eliminated. Round 2 votes counts: A=41 D=25 E=17 B=17 so E is eliminated. Round 3 votes counts: A=41 D=33 B=26 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:201 C:200 D:198 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 4 6 B -10 0 -4 -4 -4 C -4 4 0 2 -2 D -4 4 -2 0 -2 E -6 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 4 6 B -10 0 -4 -4 -4 C -4 4 0 2 -2 D -4 4 -2 0 -2 E -6 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 4 6 B -10 0 -4 -4 -4 C -4 4 0 2 -2 D -4 4 -2 0 -2 E -6 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2995: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) C E B D A (9) C D B A E (7) E A D B C (5) A E D B C (5) A E B D C (5) A D B E C (5) D B A C E (4) E C B A D (3) E C A D B (3) E C A B D (3) E A B D C (3) D A B C E (3) C E D B A (3) C B D E A (3) E A D C B (2) E A C D B (2) D A B E C (2) C B E A D (2) B D C A E (2) B C D A E (2) A D E B C (2) E C D A B (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C A E (1) C E B A D (1) C B E D A (1) B D A C E (1) B C E A D (1) B A E C D (1) B A D E C (1) B A D C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -8 -2 12 B 10 0 -6 8 2 C 8 6 0 10 6 D 2 -8 -10 0 -6 E -12 -2 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -2 12 B 10 0 -6 8 2 C 8 6 0 10 6 D 2 -8 -10 0 -6 E -12 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=24 A=19 D=11 B=9 so B is eliminated. Round 2 votes counts: C=40 E=24 A=22 D=14 so D is eliminated. Round 3 votes counts: C=44 A=32 E=24 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:207 A:196 E:193 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -2 12 B 10 0 -6 8 2 C 8 6 0 10 6 D 2 -8 -10 0 -6 E -12 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -2 12 B 10 0 -6 8 2 C 8 6 0 10 6 D 2 -8 -10 0 -6 E -12 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -2 12 B 10 0 -6 8 2 C 8 6 0 10 6 D 2 -8 -10 0 -6 E -12 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 2996: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) B A E D C (8) C D E A B (7) A B E C D (7) A B E D C (6) E D B A C (5) A E B D C (5) E A B D C (4) D E B A C (4) C A B D E (4) E D A B C (3) E B D A C (3) D E C B A (3) D C E B A (3) B D A E C (3) C E D A B (2) C D B E A (2) C D B A E (2) C A E D B (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) D C B E A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E A D B (1) C B D A E (1) C B A D E (1) C A E B D (1) C A D E B (1) C A B E D (1) B E A D C (1) B C D A E (1) B A E C D (1) Total count = 100 A B C D E A 0 -2 6 -10 -8 B 2 0 8 -4 -16 C -6 -8 0 -8 -20 D 10 4 8 0 -10 E 8 16 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 -10 -8 B 2 0 8 -4 -16 C -6 -8 0 -8 -20 D 10 4 8 0 -10 E 8 16 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=20 A=18 D=14 B=14 so D is eliminated. Round 2 votes counts: C=38 E=27 A=18 B=17 so B is eliminated. Round 3 votes counts: C=40 A=31 E=29 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:227 D:206 B:195 A:193 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 -10 -8 B 2 0 8 -4 -16 C -6 -8 0 -8 -20 D 10 4 8 0 -10 E 8 16 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -10 -8 B 2 0 8 -4 -16 C -6 -8 0 -8 -20 D 10 4 8 0 -10 E 8 16 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -10 -8 B 2 0 8 -4 -16 C -6 -8 0 -8 -20 D 10 4 8 0 -10 E 8 16 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2997: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) E B D A C (5) E B A D C (5) D B A E C (5) C D A B E (5) E B D C A (4) D C B E A (4) C D B E A (4) C A E B D (4) E B C D A (3) E B A C D (3) D C A B E (3) D B E A C (3) C A E D B (3) A E B C D (3) A C E B D (3) D B E C A (2) D B C E A (2) C E B D A (2) C A D E B (2) C A D B E (2) B E D A C (2) A E C B D (2) A D C B E (2) A D B C E (2) A C E D B (2) A C D E B (2) A B E D C (2) E C B D A (1) E A B D C (1) E A B C D (1) D C B A E (1) D A B E C (1) C E A B D (1) C D E B A (1) B D E A C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 12 2 10 B -4 0 -6 -12 2 C -12 6 0 6 8 D -2 12 -6 0 4 E -10 -2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 2 10 B -4 0 -6 -12 2 C -12 6 0 6 8 D -2 12 -6 0 4 E -10 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=24 E=23 D=21 B=3 so B is eliminated. Round 2 votes counts: A=29 E=25 C=24 D=22 so D is eliminated. Round 3 votes counts: A=35 C=34 E=31 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:204 D:204 B:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 2 10 B -4 0 -6 -12 2 C -12 6 0 6 8 D -2 12 -6 0 4 E -10 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 2 10 B -4 0 -6 -12 2 C -12 6 0 6 8 D -2 12 -6 0 4 E -10 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 2 10 B -4 0 -6 -12 2 C -12 6 0 6 8 D -2 12 -6 0 4 E -10 -2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 2998: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) E A D C B (6) A E D C B (6) A E B D C (6) E D C B A (5) D C E A B (5) B E C A D (4) B A C D E (4) A B D C E (4) D E C A B (3) C D E B A (3) C D B E A (3) A B E C D (3) E C D B A (2) E C B D A (2) D A C E B (2) C B D E A (2) B E C D A (2) B C E D A (2) B C E A D (2) B C D A E (2) B C A D E (2) A E D B C (2) A B E D C (2) E D C A B (1) E B D C A (1) E B C D A (1) E B A C D (1) E A D B C (1) E A B D C (1) D E C B A (1) D C E B A (1) D C B A E (1) D C A E B (1) C D B A E (1) B E A C D (1) B C D E A (1) B A C E D (1) A D E C B (1) A D C E B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 4 16 -2 B 6 0 4 6 -6 C -4 -4 0 -6 -22 D -16 -6 6 0 -20 E 2 6 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 16 -2 B 6 0 4 6 -6 C -4 -4 0 -6 -22 D -16 -6 6 0 -20 E 2 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980239 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=27 E=21 D=14 C=9 so C is eliminated. Round 2 votes counts: B=31 A=27 E=21 D=21 so E is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:225 A:206 B:205 C:182 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 16 -2 B 6 0 4 6 -6 C -4 -4 0 -6 -22 D -16 -6 6 0 -20 E 2 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980239 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 16 -2 B 6 0 4 6 -6 C -4 -4 0 -6 -22 D -16 -6 6 0 -20 E 2 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980239 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 16 -2 B 6 0 4 6 -6 C -4 -4 0 -6 -22 D -16 -6 6 0 -20 E 2 6 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980239 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 2999: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) A E C D B (8) D B A E C (6) D B C A E (5) C D B A E (5) B D E C A (5) C A E D B (4) C A D E B (4) B E D C A (4) B D E A C (4) A C E D B (4) D B A C E (3) C B E D A (3) B E D A C (3) E C A B D (2) E B A C D (2) D A B E C (2) C E A B D (2) C B E A D (2) B E C D A (2) B D C E A (2) A E D C B (2) E C B A D (1) E A D B C (1) E A B D C (1) E A B C D (1) D C B A E (1) D B E A C (1) D A C B E (1) D A B C E (1) C E B A D (1) C A E B D (1) C A D B E (1) B D C A E (1) B C E D A (1) B C D E A (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 4 -2 2 B 6 0 -6 -2 8 C -4 6 0 12 -12 D 2 2 -12 0 -12 E -2 -8 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.307692 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745559 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.769231 D: 0.769231 E: 1.000000 A B C D E A 0 -6 4 -2 2 B 6 0 -6 -2 8 C -4 6 0 12 -12 D 2 2 -12 0 -12 E -2 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.307692 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745469 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.769231 D: 0.769231 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 D=20 E=18 A=16 so A is eliminated. Round 2 votes counts: E=29 C=28 B=23 D=20 so D is eliminated. Round 3 votes counts: B=41 C=30 E=29 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:207 B:203 C:201 A:199 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -2 2 B 6 0 -6 -2 8 C -4 6 0 12 -12 D 2 2 -12 0 -12 E -2 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.307692 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745469 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.769231 D: 0.769231 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 2 B 6 0 -6 -2 8 C -4 6 0 12 -12 D 2 2 -12 0 -12 E -2 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.307692 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745469 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.769231 D: 0.769231 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 2 B 6 0 -6 -2 8 C -4 6 0 12 -12 D 2 2 -12 0 -12 E -2 -8 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.307692 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745469 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.769231 D: 0.769231 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3000: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (7) B C E D A (6) A D B C E (6) E C B A D (5) D E A B C (5) A B C D E (5) E D C B A (4) D E A C B (4) D A E B C (4) C B E D A (4) C B E A D (4) B C A E D (4) A D E C B (4) A C B E D (4) E C B D A (3) D E B C A (3) B C D A E (3) A E C B D (3) D E C B A (2) D B C E A (2) D B C A E (2) D A B C E (2) B C E A D (2) A D E B C (2) A B C E D (2) E D C A B (1) E D B C A (1) E D A C B (1) E A D C B (1) D A E C B (1) D A B E C (1) B C D E A (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -8 0 2 B 6 0 16 8 10 C 8 -16 0 8 10 D 0 -8 -8 0 8 E -2 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 0 2 B 6 0 16 8 10 C 8 -16 0 8 10 D 0 -8 -8 0 8 E -2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 B=23 E=16 C=8 so C is eliminated. Round 2 votes counts: B=31 A=27 D=26 E=16 so E is eliminated. Round 3 votes counts: B=39 D=33 A=28 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:205 D:196 A:194 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 0 2 B 6 0 16 8 10 C 8 -16 0 8 10 D 0 -8 -8 0 8 E -2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 2 B 6 0 16 8 10 C 8 -16 0 8 10 D 0 -8 -8 0 8 E -2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 2 B 6 0 16 8 10 C 8 -16 0 8 10 D 0 -8 -8 0 8 E -2 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3001: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (15) C B D E A (8) D E A C B (7) B C A D E (6) B C A E D (5) A D E B C (5) E D A C B (4) C B E D A (4) C B D A E (4) A E B D C (4) A D E C B (4) A B E D C (4) E A D B C (3) D C E A B (3) E D A B C (2) E A D C B (2) C D B E A (2) B C E D A (2) B C E A D (2) B A E C D (2) A B C D E (2) E B A D C (1) D E C A B (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D A B E (1) B A C E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 20 20 16 14 B -20 0 12 -4 -10 C -20 -12 0 -14 -12 D -16 4 14 0 -6 E -14 10 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 20 16 14 B -20 0 12 -4 -10 C -20 -12 0 -14 -12 D -16 4 14 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=21 B=19 D=13 E=12 so E is eliminated. Round 2 votes counts: A=40 C=21 B=20 D=19 so D is eliminated. Round 3 votes counts: A=55 C=25 B=20 so B is eliminated. Round 4 votes counts: A=60 C=40 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:235 E:207 D:198 B:189 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 20 16 14 B -20 0 12 -4 -10 C -20 -12 0 -14 -12 D -16 4 14 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 20 16 14 B -20 0 12 -4 -10 C -20 -12 0 -14 -12 D -16 4 14 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 20 16 14 B -20 0 12 -4 -10 C -20 -12 0 -14 -12 D -16 4 14 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3002: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (6) B A D E C (6) A B D E C (6) A B D C E (6) D C E A B (5) D A B E C (5) B C A E D (5) E C B D A (4) C E D A B (4) A D B E C (4) E C D B A (3) D E C A B (3) D E A C B (3) B A C D E (3) A B C D E (3) E D C B A (2) E C D A B (2) D A E C B (2) C E D B A (2) C E B A D (2) C B E A D (2) B E C A D (2) B A E C D (2) B A C E D (2) A D B C E (2) E D C A B (1) E B C D A (1) D E C B A (1) D E B A C (1) D E A B C (1) D A B C E (1) C B A E D (1) C A D B E (1) B E C D A (1) B E A C D (1) B C E A D (1) B A D C E (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 2 2 4 4 B -2 0 8 10 12 C -2 -8 0 -4 -2 D -4 -10 4 0 12 E -4 -12 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 4 4 B -2 0 8 10 12 C -2 -8 0 -4 -2 D -4 -10 4 0 12 E -4 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 D=22 C=18 E=13 so E is eliminated. Round 2 votes counts: C=27 D=25 B=25 A=23 so A is eliminated. Round 3 votes counts: B=40 D=33 C=27 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:206 D:201 C:192 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 4 B -2 0 8 10 12 C -2 -8 0 -4 -2 D -4 -10 4 0 12 E -4 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 4 B -2 0 8 10 12 C -2 -8 0 -4 -2 D -4 -10 4 0 12 E -4 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 4 B -2 0 8 10 12 C -2 -8 0 -4 -2 D -4 -10 4 0 12 E -4 -12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999611 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3003: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (6) B C E D A (6) E B D A C (5) E D A B C (4) E A C B D (4) D A E B C (4) D A C B E (4) B E D C A (4) A C D E B (4) E B C A D (3) E A D B C (3) C A B D E (3) B E C D A (3) B C D A E (3) A D C E B (3) A C E D B (3) D E B A C (2) D E A B C (2) D A B E C (2) C D A B E (2) C B A E D (2) C A D B E (2) C A B E D (2) B D C E A (2) B C D E A (2) A E C D B (2) A D E C B (2) E C B A D (1) E B D C A (1) E A D C B (1) E A B D C (1) D B E C A (1) D A B C E (1) C E B A D (1) C D B A E (1) C B E A D (1) C A E D B (1) B E C A D (1) B D E C A (1) B C E A D (1) A E D C B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 6 0 -12 0 B -6 0 4 6 4 C 0 -4 0 10 4 D 12 -6 -10 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.410287 B: 0.000000 C: 0.589713 D: 0.000000 E: 0.000000 Sum of squares = 0.516096976826 Cumulative probabilities = A: 0.410287 B: 0.410287 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -12 0 B -6 0 4 6 4 C 0 -4 0 10 4 D 12 -6 -10 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.000000 Sum of squares = 0.504132244642 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=23 B=23 C=21 A=17 D=16 so D is eliminated. Round 2 votes counts: A=28 E=27 B=24 C=21 so C is eliminated. Round 3 votes counts: A=38 B=34 E=28 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:205 B:204 A:197 D:197 E:197 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 -12 0 B -6 0 4 6 4 C 0 -4 0 10 4 D 12 -6 -10 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.000000 Sum of squares = 0.504132244642 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -12 0 B -6 0 4 6 4 C 0 -4 0 10 4 D 12 -6 -10 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.000000 Sum of squares = 0.504132244642 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -12 0 B -6 0 4 6 4 C 0 -4 0 10 4 D 12 -6 -10 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.000000 Sum of squares = 0.504132244642 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3004: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B C D A E (7) E D A C B (5) E A D B C (5) B C E D A (5) A D E C B (5) E B C D A (4) E B A C D (4) D A C E B (4) C D B A E (4) C B D A E (4) B E C A D (4) A E D C B (4) A D C E B (4) B C E A D (3) B C A D E (3) D C A E B (2) C B E D A (2) B E A C D (2) A E D B C (2) A D C B E (2) E D C B A (1) E B D C A (1) E B C A D (1) E A B D C (1) D C A B E (1) D A C B E (1) C D B E A (1) C D A B E (1) C B D E A (1) C B A D E (1) B E C D A (1) B A C D E (1) A E B D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 6 10 -2 B 0 0 -6 -4 -8 C -6 6 0 0 -2 D -10 4 0 0 -14 E 2 8 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 6 10 -2 B 0 0 -6 -4 -8 C -6 6 0 0 -2 D -10 4 0 0 -14 E 2 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=26 A=20 C=14 D=8 so D is eliminated. Round 2 votes counts: E=32 B=26 A=25 C=17 so C is eliminated. Round 3 votes counts: B=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:207 C:199 B:191 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 10 -2 B 0 0 -6 -4 -8 C -6 6 0 0 -2 D -10 4 0 0 -14 E 2 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 10 -2 B 0 0 -6 -4 -8 C -6 6 0 0 -2 D -10 4 0 0 -14 E 2 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 10 -2 B 0 0 -6 -4 -8 C -6 6 0 0 -2 D -10 4 0 0 -14 E 2 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3005: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (7) C E A D B (7) C A D B E (7) E C B D A (6) E B D A C (6) B D A C E (6) A D B C E (6) E C A D B (5) E B D C A (5) D B A C E (4) E A C D B (3) C E B D A (3) C B D A E (3) C A E D B (3) B D C A E (3) B D C E A (2) B D A E C (2) A E D B C (2) A D C B E (2) A D B E C (2) A C D B E (2) E C B A D (1) E B C D A (1) E A D B C (1) D B A E C (1) D A B E C (1) D A B C E (1) C E A B D (1) C D A B E (1) C B D E A (1) C A D E B (1) C A B D E (1) B D E A C (1) B C D E A (1) A E C D B (1) Total count = 100 A B C D E A 0 8 -18 4 -2 B -8 0 -10 0 -6 C 18 10 0 10 10 D -4 0 -10 0 -4 E 2 6 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 4 -2 B -8 0 -10 0 -6 C 18 10 0 10 10 D -4 0 -10 0 -4 E 2 6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=28 B=15 A=15 D=7 so D is eliminated. Round 2 votes counts: E=35 C=28 B=20 A=17 so A is eliminated. Round 3 votes counts: E=38 C=32 B=30 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:201 A:196 D:191 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 4 -2 B -8 0 -10 0 -6 C 18 10 0 10 10 D -4 0 -10 0 -4 E 2 6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 4 -2 B -8 0 -10 0 -6 C 18 10 0 10 10 D -4 0 -10 0 -4 E 2 6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 4 -2 B -8 0 -10 0 -6 C 18 10 0 10 10 D -4 0 -10 0 -4 E 2 6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3006: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (13) A D E C B (10) C E B A D (6) B C E A D (6) B D C E A (5) A E C D B (5) D B E C A (4) D A E C B (4) E C A D B (3) E C A B D (3) D B C E A (3) D A B E C (3) B C A E D (3) B A C E D (3) A D B E C (3) E C D A B (2) D E C B A (2) D E C A B (2) D B A C E (2) C E A B D (2) C B E A D (2) B D A C E (2) A E C B D (2) E C B D A (1) D E A C B (1) D B A E C (1) D A B C E (1) C B E D A (1) B C D E A (1) B A D C E (1) A D E B C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -18 4 -14 B 12 0 6 6 10 C 18 -6 0 10 6 D -4 -6 -10 0 -8 E 14 -10 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 4 -14 B 12 0 6 6 10 C 18 -6 0 10 6 D -4 -6 -10 0 -8 E 14 -10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=23 A=23 C=11 E=9 so E is eliminated. Round 2 votes counts: B=34 D=23 A=23 C=20 so C is eliminated. Round 3 votes counts: B=44 A=31 D=25 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:214 E:203 D:186 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -18 4 -14 B 12 0 6 6 10 C 18 -6 0 10 6 D -4 -6 -10 0 -8 E 14 -10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 4 -14 B 12 0 6 6 10 C 18 -6 0 10 6 D -4 -6 -10 0 -8 E 14 -10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 4 -14 B 12 0 6 6 10 C 18 -6 0 10 6 D -4 -6 -10 0 -8 E 14 -10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3007: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) C B A E D (8) C E B D A (7) C E B A D (7) E D A C B (6) E C D A B (5) A D B E C (5) E D C A B (4) E C D B A (4) D E A B C (4) D A B E C (4) B A C D E (4) A B D C E (4) A B C D E (4) E C A D B (3) D A E B C (3) E D A B C (2) E A C D B (2) D B A C E (2) C B E D A (2) C B A D E (2) B D A C E (2) E D C B A (1) E C B D A (1) E C A B D (1) E A D C B (1) B C D A E (1) B A D C E (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -10 4 0 B 2 0 -8 6 -4 C 10 8 0 20 6 D -4 -6 -20 0 -8 E 0 4 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 4 0 B 2 0 -8 6 -4 C 10 8 0 20 6 D -4 -6 -20 0 -8 E 0 4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 B=17 A=14 D=13 so D is eliminated. Round 2 votes counts: E=34 C=26 A=21 B=19 so B is eliminated. Round 3 votes counts: C=36 E=34 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:203 B:198 A:196 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 4 0 B 2 0 -8 6 -4 C 10 8 0 20 6 D -4 -6 -20 0 -8 E 0 4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 4 0 B 2 0 -8 6 -4 C 10 8 0 20 6 D -4 -6 -20 0 -8 E 0 4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 4 0 B 2 0 -8 6 -4 C 10 8 0 20 6 D -4 -6 -20 0 -8 E 0 4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3008: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (12) D C E A B (10) D A E B C (10) C B E A D (5) D E C A B (4) D E A C B (4) A E B D C (4) D E A B C (3) C D B E A (3) B A E D C (3) B A C E D (3) A B E D C (3) E A D B C (2) D C B E A (2) D A E C B (2) D A B E C (2) C E D B A (2) C E B A D (2) C D E B A (2) B C A E D (2) B A D E C (2) E D C A B (1) E A C D B (1) E A B C D (1) D B C A E (1) D B A C E (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E A B (1) C B E D A (1) C B D A E (1) C B A E D (1) C B A D E (1) B C A D E (1) A E D B C (1) A E B C D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 14 -2 6 B -10 0 8 -8 -8 C -14 -8 0 -12 -16 D 2 8 12 0 2 E -6 8 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 -2 6 B -10 0 8 -8 -8 C -14 -8 0 -12 -16 D 2 8 12 0 2 E -6 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=23 C=22 A=11 E=5 so E is eliminated. Round 2 votes counts: D=40 B=23 C=22 A=15 so A is eliminated. Round 3 votes counts: D=43 B=34 C=23 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:212 E:208 B:191 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 14 -2 6 B -10 0 8 -8 -8 C -14 -8 0 -12 -16 D 2 8 12 0 2 E -6 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -2 6 B -10 0 8 -8 -8 C -14 -8 0 -12 -16 D 2 8 12 0 2 E -6 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -2 6 B -10 0 8 -8 -8 C -14 -8 0 -12 -16 D 2 8 12 0 2 E -6 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3009: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) E C D B A (7) A C B E D (6) D B A C E (5) D A B E C (5) D A B C E (5) C B A E D (5) A B D C E (5) D E A B C (4) A D B C E (4) E D C B A (3) C E B A D (3) E D C A B (2) E C B D A (2) E A C B D (2) D B C E A (2) D A E B C (2) C B E A D (2) C B A D E (2) B C D A E (2) B A C D E (2) A D B E C (2) A B C D E (2) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) E A D C B (1) E A C D B (1) D E B A C (1) D B E A C (1) D B C A E (1) C E B D A (1) C A B E D (1) B D C A E (1) B C A D E (1) B A D C E (1) A E D B C (1) Total count = 100 A B C D E A 0 -6 2 8 6 B 6 0 -6 0 10 C -2 6 0 6 2 D -8 0 -6 0 -4 E -6 -10 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102153 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 8 6 B 6 0 -6 0 10 C -2 6 0 6 2 D -8 0 -6 0 -4 E -6 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=26 A=20 C=14 B=7 so B is eliminated. Round 2 votes counts: E=33 D=27 A=23 C=17 so C is eliminated. Round 3 votes counts: E=39 A=32 D=29 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:206 A:205 B:205 E:193 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 8 6 B 6 0 -6 0 10 C -2 6 0 6 2 D -8 0 -6 0 -4 E -6 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 8 6 B 6 0 -6 0 10 C -2 6 0 6 2 D -8 0 -6 0 -4 E -6 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 8 6 B 6 0 -6 0 10 C -2 6 0 6 2 D -8 0 -6 0 -4 E -6 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3010: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) D A B E C (11) A D B E C (8) C A B E D (7) D E B A C (6) C E B D A (6) C E B A D (5) C A D B E (5) B E D A C (5) E B C D A (4) A D C B E (4) E B D C A (3) E B C A D (2) D B A E C (2) C D E A B (2) C B E A D (2) C A D E B (2) E D B C A (1) E C B D A (1) D E B C A (1) D B E A C (1) D A E B C (1) C E D B A (1) C E A B D (1) C B A E D (1) C A B D E (1) B E A D C (1) B E A C D (1) B D E A C (1) B A E D C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 10 -14 -10 B 12 0 22 6 6 C -10 -22 0 -16 -22 D 14 -6 16 0 -6 E 10 -6 22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -14 -10 B 12 0 22 6 6 C -10 -22 0 -16 -22 D 14 -6 16 0 -6 E 10 -6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=22 D=22 A=14 B=9 so B is eliminated. Round 2 votes counts: C=33 E=29 D=23 A=15 so A is eliminated. Round 3 votes counts: D=36 C=34 E=30 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:223 E:216 D:209 A:187 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 -14 -10 B 12 0 22 6 6 C -10 -22 0 -16 -22 D 14 -6 16 0 -6 E 10 -6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -14 -10 B 12 0 22 6 6 C -10 -22 0 -16 -22 D 14 -6 16 0 -6 E 10 -6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -14 -10 B 12 0 22 6 6 C -10 -22 0 -16 -22 D 14 -6 16 0 -6 E 10 -6 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3011: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (12) B E A C D (10) D B E C A (9) D B E A C (9) C A E B D (9) B E D A C (5) D A C E B (4) E B A C D (3) B E C A D (3) B E A D C (3) A C E B D (3) E A B C D (2) D C A B E (2) D B C E A (2) D B C A E (2) D B A E C (2) D A E B C (2) D A C B E (2) C E A B D (2) C A E D B (2) B D E A C (2) E C B A D (1) D E B A C (1) D C B E A (1) D A B E C (1) C D A E B (1) C A D E B (1) B D E C A (1) A E B C D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 4 -16 -8 B 8 0 16 -10 8 C -4 -16 0 -22 -10 D 16 10 22 0 10 E 8 -8 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -16 -8 B 8 0 16 -10 8 C -4 -16 0 -22 -10 D 16 10 22 0 10 E 8 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=49 B=24 C=15 E=6 A=6 so E is eliminated. Round 2 votes counts: D=49 B=27 C=16 A=8 so A is eliminated. Round 3 votes counts: D=50 B=30 C=20 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:229 B:211 E:200 A:186 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 4 -16 -8 B 8 0 16 -10 8 C -4 -16 0 -22 -10 D 16 10 22 0 10 E 8 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -16 -8 B 8 0 16 -10 8 C -4 -16 0 -22 -10 D 16 10 22 0 10 E 8 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -16 -8 B 8 0 16 -10 8 C -4 -16 0 -22 -10 D 16 10 22 0 10 E 8 -8 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3012: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) A B D E C (8) C D A E B (7) C E B D A (5) B A E D C (5) C D E A B (4) E D C B A (3) E C D B A (3) E B D A C (3) D C E A B (3) C E D B A (3) C B E D A (3) B E C D A (3) B C A E D (3) B A C E D (3) A D C E B (3) A D B E C (3) D E C A B (2) D A E C B (2) C D E B A (2) B E D A C (2) B C E A D (2) A D E B C (2) A C D B E (2) A C B D E (2) A B E D C (2) E D A B C (1) E B D C A (1) E B A D C (1) D C A E B (1) C D A B E (1) C A D B E (1) B E C A D (1) B E A C D (1) B A E C D (1) A D B C E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 4 2 -2 B 6 0 6 12 8 C -4 -6 0 -4 -4 D -2 -12 4 0 -10 E 2 -8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 2 -2 B 6 0 6 12 8 C -4 -6 0 -4 -4 D -2 -12 4 0 -10 E 2 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=26 A=25 E=12 D=8 so D is eliminated. Round 2 votes counts: C=30 B=29 A=27 E=14 so E is eliminated. Round 3 votes counts: C=38 B=34 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:204 A:199 C:191 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 2 -2 B 6 0 6 12 8 C -4 -6 0 -4 -4 D -2 -12 4 0 -10 E 2 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 2 -2 B 6 0 6 12 8 C -4 -6 0 -4 -4 D -2 -12 4 0 -10 E 2 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 2 -2 B 6 0 6 12 8 C -4 -6 0 -4 -4 D -2 -12 4 0 -10 E 2 -8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3013: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (7) E C A D B (6) A C E B D (6) A C B E D (6) E C D A B (5) B A D C E (5) D E C B A (4) D E C A B (4) B D E A C (4) B D A E C (4) B A C D E (4) A B C E D (4) E D C B A (3) D E B C A (3) C E A D B (3) B D A C E (3) B A C E D (3) A B C D E (3) E D B C A (2) E A C B D (2) D B E C A (2) B A E C D (2) B A D E C (2) E D C A B (1) E B A C D (1) E A B C D (1) D C E A B (1) D B A C E (1) C E D A B (1) C D E A B (1) C A D E B (1) B D E C A (1) B A E D C (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 20 10 B -10 0 -6 8 -6 C -10 6 0 18 2 D -20 -8 -18 0 -12 E -10 6 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 20 10 B -10 0 -6 8 -6 C -10 6 0 18 2 D -20 -8 -18 0 -12 E -10 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=22 E=21 D=15 C=13 so C is eliminated. Round 2 votes counts: A=30 B=29 E=25 D=16 so D is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:225 C:208 E:203 B:193 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 20 10 B -10 0 -6 8 -6 C -10 6 0 18 2 D -20 -8 -18 0 -12 E -10 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 20 10 B -10 0 -6 8 -6 C -10 6 0 18 2 D -20 -8 -18 0 -12 E -10 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 20 10 B -10 0 -6 8 -6 C -10 6 0 18 2 D -20 -8 -18 0 -12 E -10 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3014: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) E C D A B (9) A B C D E (7) D E B A C (6) C E D A B (6) C E A B D (6) A B D E C (6) E D C B A (5) C A E B D (5) A C B E D (5) C E A D B (4) B A D E C (4) B A D C E (4) D B E A C (3) C E D B A (3) C A B E D (3) E C D B A (2) B D A E C (2) E D C A B (1) D E C B A (1) D B A C E (1) C B D A E (1) C A B D E (1) B D A C E (1) B C A D E (1) B A C D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 4 -2 8 B -10 0 -4 -2 4 C -4 4 0 10 2 D 2 2 -10 0 0 E -8 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000026 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -2 8 B -10 0 -4 -2 4 C -4 4 0 10 2 D 2 2 -10 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.46874999994 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 A=20 E=17 B=13 so B is eliminated. Round 2 votes counts: C=30 A=29 D=24 E=17 so E is eliminated. Round 3 votes counts: C=41 D=30 A=29 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:210 C:206 D:197 B:194 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 -2 8 B -10 0 -4 -2 4 C -4 4 0 10 2 D 2 2 -10 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.46874999994 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -2 8 B -10 0 -4 -2 4 C -4 4 0 10 2 D 2 2 -10 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.46874999994 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -2 8 B -10 0 -4 -2 4 C -4 4 0 10 2 D 2 2 -10 0 0 E -8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.46874999994 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3015: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (6) A E D C B (6) B C A E D (5) E D B A C (4) E D A B C (4) C B A D E (4) B E D C A (4) B E C D A (4) B E C A D (4) A E D B C (4) A D E C B (4) A D C E B (4) A B C E D (4) E D B C A (3) C A B D E (3) E B D C A (2) E A D B C (2) D E C B A (2) D C E B A (2) D C E A B (2) C D A E B (2) C A D B E (2) B C E D A (2) B C D E A (2) B C D A E (2) A C B D E (2) E D A C B (1) D E A C B (1) D C A E B (1) C D E A B (1) C D B E A (1) C D A B E (1) C B D E A (1) C A D E B (1) B E D A C (1) B E A D C (1) B E A C D (1) B A E D C (1) B A C E D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -2 2 -2 B 6 0 16 2 -6 C 2 -16 0 -12 -12 D -2 -2 12 0 -22 E 2 6 12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 2 -2 B 6 0 16 2 -6 C 2 -16 0 -12 -12 D -2 -2 12 0 -22 E 2 6 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=26 E=22 C=16 D=8 so D is eliminated. Round 2 votes counts: B=28 A=26 E=25 C=21 so C is eliminated. Round 3 votes counts: A=36 B=34 E=30 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:221 B:209 A:196 D:193 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 2 -2 B 6 0 16 2 -6 C 2 -16 0 -12 -12 D -2 -2 12 0 -22 E 2 6 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 -2 B 6 0 16 2 -6 C 2 -16 0 -12 -12 D -2 -2 12 0 -22 E 2 6 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 -2 B 6 0 16 2 -6 C 2 -16 0 -12 -12 D -2 -2 12 0 -22 E 2 6 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3016: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (15) B D C E A (12) C D B E A (8) A E C B D (8) D B C E A (6) D C B E A (5) E C D B A (4) C D E B A (4) A B E D C (4) A B D C E (4) A E B D C (3) A E B C D (3) E B C D A (2) B E D C A (2) B D C A E (2) B D A C E (2) A C E D B (2) E C B D A (1) E C A D B (1) E B D C A (1) E A C D B (1) C E D B A (1) C E D A B (1) B D E C A (1) B D A E C (1) B A D C E (1) A E D C B (1) A D B C E (1) A C D E B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -2 -6 0 B 6 0 -8 -4 2 C 2 8 0 6 2 D 6 4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -6 0 B 6 0 -8 -4 2 C 2 8 0 6 2 D 6 4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 B=21 C=14 D=11 E=10 so E is eliminated. Round 2 votes counts: A=45 B=24 C=20 D=11 so D is eliminated. Round 3 votes counts: A=45 B=30 C=25 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:209 D:202 B:198 E:198 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -6 0 B 6 0 -8 -4 2 C 2 8 0 6 2 D 6 4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -6 0 B 6 0 -8 -4 2 C 2 8 0 6 2 D 6 4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -6 0 B 6 0 -8 -4 2 C 2 8 0 6 2 D 6 4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3017: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) E C D A B (6) D B A E C (6) C E A B D (6) B A D E C (6) D E C A B (5) B A C E D (5) A B C E D (5) D E C B A (4) C D E B A (4) B A D C E (4) D C E B A (3) B D A E C (3) A B E C D (3) E D C A B (2) E C A D B (2) D E B C A (2) D E A B C (2) D C B E A (2) C B A E D (2) C A E B D (2) C A B E D (2) B A C D E (2) A B D E C (2) E D A C B (1) E A C B D (1) D E B A C (1) D B A C E (1) D A B E C (1) C E D B A (1) C E A D B (1) C B E D A (1) B C A E D (1) B C A D E (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -10 -6 -4 B -2 0 -8 -4 -4 C 10 8 0 8 4 D 6 4 -8 0 -2 E 4 4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -6 -4 B -2 0 -8 -4 -4 C 10 8 0 8 4 D 6 4 -8 0 -2 E 4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 B=22 E=12 A=12 so E is eliminated. Round 2 votes counts: C=35 D=30 B=22 A=13 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:203 D:200 A:191 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -6 -4 B -2 0 -8 -4 -4 C 10 8 0 8 4 D 6 4 -8 0 -2 E 4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -6 -4 B -2 0 -8 -4 -4 C 10 8 0 8 4 D 6 4 -8 0 -2 E 4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -6 -4 B -2 0 -8 -4 -4 C 10 8 0 8 4 D 6 4 -8 0 -2 E 4 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3018: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (16) E C A D B (11) A D B C E (11) B D C A E (10) B D A C E (10) E A C D B (7) C E B D A (4) A E C D B (4) B D C E A (3) B C D E A (3) A D B E C (3) E A D B C (2) C B E D A (2) E B C D A (1) D B A C E (1) D A B C E (1) C E D B A (1) C A E D B (1) C A D E B (1) B E D C A (1) B D E C A (1) B D A E C (1) B C E D A (1) B C D A E (1) A E D B C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 -14 -6 B 12 0 6 10 0 C 14 -6 0 6 2 D 14 -10 -6 0 -4 E 6 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.642657 C: 0.000000 D: 0.000000 E: 0.357343 Sum of squares = 0.5407019604 Cumulative probabilities = A: 0.000000 B: 0.642657 C: 0.642657 D: 0.642657 E: 1.000000 A B C D E A 0 -12 -14 -14 -6 B 12 0 6 10 0 C 14 -6 0 6 2 D 14 -10 -6 0 -4 E 6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=31 A=21 C=9 D=2 so D is eliminated. Round 2 votes counts: E=37 B=32 A=22 C=9 so C is eliminated. Round 3 votes counts: E=42 B=34 A=24 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:208 E:204 D:197 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 -14 -6 B 12 0 6 10 0 C 14 -6 0 6 2 D 14 -10 -6 0 -4 E 6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -14 -6 B 12 0 6 10 0 C 14 -6 0 6 2 D 14 -10 -6 0 -4 E 6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -14 -6 B 12 0 6 10 0 C 14 -6 0 6 2 D 14 -10 -6 0 -4 E 6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3019: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) E A C D B (10) B D C A E (7) E A D B C (5) A E D B C (5) A E C D B (5) D B A E C (4) D B A C E (4) C E A B D (4) B D E A C (4) E C A B D (3) E A C B D (3) D A B E C (3) C B D A E (3) B D C E A (3) A D E B C (3) A D B E C (3) E A D C B (2) D B C A E (2) C E B D A (2) C E B A D (2) C B E D A (2) B D E C A (2) E B D A C (1) D B E A C (1) D A B C E (1) C A D B E (1) B E D C A (1) B C D E A (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 10 -6 -16 B 2 0 0 0 10 C -10 0 0 -2 -10 D 6 0 2 0 10 E 16 -10 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.465453 C: 0.000000 D: 0.534547 E: 0.000000 Sum of squares = 0.502387001447 Cumulative probabilities = A: 0.000000 B: 0.465453 C: 0.465453 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -6 -16 B 2 0 0 0 10 C -10 0 0 -2 -10 D 6 0 2 0 10 E 16 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999742 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 B=18 A=17 D=15 so D is eliminated. Round 2 votes counts: B=29 C=26 E=24 A=21 so A is eliminated. Round 3 votes counts: E=37 B=36 C=27 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:209 B:206 E:203 A:193 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 -6 -16 B 2 0 0 0 10 C -10 0 0 -2 -10 D 6 0 2 0 10 E 16 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999742 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -6 -16 B 2 0 0 0 10 C -10 0 0 -2 -10 D 6 0 2 0 10 E 16 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999742 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -6 -16 B 2 0 0 0 10 C -10 0 0 -2 -10 D 6 0 2 0 10 E 16 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999742 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3020: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) B D C E A (6) E B C D A (5) E A C B D (5) E B D C A (4) D C B A E (4) B C D E A (4) A C E D B (4) A C D B E (4) E B C A D (3) E A D B C (3) E A B C D (3) C D B A E (3) C A B D E (3) A E D C B (3) A E C D B (3) A D E C B (3) E D B A C (2) E B D A C (2) E B A C D (2) E A B D C (2) D E B A C (2) C A B E D (2) B D C A E (2) A D C B E (2) A C B D E (2) E D B C A (1) E C B A D (1) D B C E A (1) D B A C E (1) D A C B E (1) D A B C E (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) C A D B E (1) B E D C A (1) A E D B C (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -4 0 0 B 10 0 6 4 -2 C 4 -6 0 2 6 D 0 -4 -2 0 0 E 0 2 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755102038 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 -10 -4 0 0 B 10 0 6 4 -2 C 4 -6 0 2 6 D 0 -4 -2 0 0 E 0 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101884 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=24 D=17 C=13 B=13 so C is eliminated. Round 2 votes counts: E=33 A=30 D=20 B=17 so B is eliminated. Round 3 votes counts: E=36 D=34 A=30 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:209 C:203 E:198 D:197 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 0 0 B 10 0 6 4 -2 C 4 -6 0 2 6 D 0 -4 -2 0 0 E 0 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101884 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 0 0 B 10 0 6 4 -2 C 4 -6 0 2 6 D 0 -4 -2 0 0 E 0 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101884 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 0 0 B 10 0 6 4 -2 C 4 -6 0 2 6 D 0 -4 -2 0 0 E 0 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101884 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3021: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) D C A B E (6) D B C A E (6) B E D C A (6) E B A C D (5) D C A E B (5) A C D E B (5) E A B C D (4) C A D E B (4) B E D A C (4) B D E C A (4) A C E D B (4) E A C D B (3) E A C B D (3) B A C D E (3) A E C D B (3) E D C A B (2) E B D C A (2) D E B C A (2) B E A D C (2) B D C A E (2) B A E C D (2) B A D C E (2) A E C B D (2) E D C B A (1) E C D A B (1) E C A D B (1) D E C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B A E (1) D B C E A (1) C D A E B (1) B D C E A (1) Total count = 100 A B C D E A 0 -8 0 2 -8 B 8 0 8 0 -4 C 0 -8 0 -2 -14 D -2 0 2 0 -6 E 8 4 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 2 -8 B 8 0 8 0 -4 C 0 -8 0 -2 -14 D -2 0 2 0 -6 E 8 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=25 E=22 A=14 C=5 so C is eliminated. Round 2 votes counts: B=34 D=26 E=22 A=18 so A is eliminated. Round 3 votes counts: D=35 B=34 E=31 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:206 D:197 A:193 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 2 -8 B 8 0 8 0 -4 C 0 -8 0 -2 -14 D -2 0 2 0 -6 E 8 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 2 -8 B 8 0 8 0 -4 C 0 -8 0 -2 -14 D -2 0 2 0 -6 E 8 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 2 -8 B 8 0 8 0 -4 C 0 -8 0 -2 -14 D -2 0 2 0 -6 E 8 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3022: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) D E C B A (8) B A E D C (7) A B C D E (6) C E A D B (5) C A E B D (5) B A D C E (5) C E D A B (4) C E A B D (4) B D A E C (4) A C B E D (4) E C D B A (3) E C D A B (3) D E B A C (3) D E B C A (2) D C E A B (2) C A D E B (2) C A B E D (2) B D E A C (2) B A D E C (2) A B C E D (2) E D C A B (1) E D B C A (1) E D B A C (1) E C B D A (1) E B D A C (1) D C A E B (1) D B E A C (1) D B A E C (1) C D E A B (1) C D A E B (1) C A E D B (1) C A D B E (1) B E D A C (1) B E A D C (1) B A E C D (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -12 0 -6 B 8 0 -14 0 -16 C 12 14 0 -6 -4 D 0 0 6 0 -12 E 6 16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 0 -6 B 8 0 -14 0 -16 C 12 14 0 -6 -4 D 0 0 6 0 -12 E 6 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 E=19 D=18 A=13 so A is eliminated. Round 2 votes counts: B=33 C=30 E=19 D=18 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:219 C:208 D:197 B:189 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 0 -6 B 8 0 -14 0 -16 C 12 14 0 -6 -4 D 0 0 6 0 -12 E 6 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 0 -6 B 8 0 -14 0 -16 C 12 14 0 -6 -4 D 0 0 6 0 -12 E 6 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 0 -6 B 8 0 -14 0 -16 C 12 14 0 -6 -4 D 0 0 6 0 -12 E 6 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3023: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) A C B E D (6) D A C E B (5) C E A B D (5) C A E D B (5) B C E A D (5) E D C A B (4) E D B C A (4) B A C E D (4) A C D E B (4) E C D B A (3) D E A C B (3) D A E C B (3) B E C D A (3) B D E A C (3) E C D A B (2) E B D C A (2) D E B C A (2) D E B A C (2) D B E A C (2) C E B A D (2) C E A D B (2) C A B E D (2) B E C A D (2) B A C D E (2) A C D B E (2) E D C B A (1) E B C D A (1) D A B E C (1) C E D A B (1) C B E A D (1) C B A E D (1) C A E B D (1) B D A E C (1) B C A E D (1) A D C E B (1) A D B C E (1) A C E D B (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -12 2 -12 B -4 0 -12 2 -8 C 12 12 0 18 10 D -2 -2 -18 0 -32 E 12 8 -10 32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 2 -12 B -4 0 -12 2 -8 C 12 12 0 18 10 D -2 -2 -18 0 -32 E 12 8 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=20 D=18 A=18 E=17 so E is eliminated. Round 2 votes counts: B=30 D=27 C=25 A=18 so A is eliminated. Round 3 votes counts: C=39 B=32 D=29 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:221 A:191 B:189 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 2 -12 B -4 0 -12 2 -8 C 12 12 0 18 10 D -2 -2 -18 0 -32 E 12 8 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 2 -12 B -4 0 -12 2 -8 C 12 12 0 18 10 D -2 -2 -18 0 -32 E 12 8 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 2 -12 B -4 0 -12 2 -8 C 12 12 0 18 10 D -2 -2 -18 0 -32 E 12 8 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3024: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E B C D A (8) C B E D A (8) D C A B E (7) A D C B E (7) E B C A D (6) A E D B C (6) A D C E B (6) C B D E A (5) C D B A E (4) B C E D A (3) A E D C B (3) A D E B C (3) E B A C D (2) E A C B D (2) D A B C E (2) C D A B E (2) C A D B E (2) B E C D A (2) A D E C B (2) E C B A D (1) E C A B D (1) E B D A C (1) E B A D C (1) E A B C D (1) D C B A E (1) C E A B D (1) C B D A E (1) B E D C A (1) B C D E A (1) A E C B D (1) Total count = 100 A B C D E A 0 10 -8 -10 12 B -10 0 -26 -8 10 C 8 26 0 2 18 D 10 8 -2 0 4 E -12 -10 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 -10 12 B -10 0 -26 -8 10 C 8 26 0 2 18 D 10 8 -2 0 4 E -12 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 C=23 D=19 B=7 so B is eliminated. Round 2 votes counts: A=28 C=27 E=26 D=19 so D is eliminated. Round 3 votes counts: A=39 C=35 E=26 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:210 A:202 B:183 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 -10 12 B -10 0 -26 -8 10 C 8 26 0 2 18 D 10 8 -2 0 4 E -12 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -10 12 B -10 0 -26 -8 10 C 8 26 0 2 18 D 10 8 -2 0 4 E -12 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -10 12 B -10 0 -26 -8 10 C 8 26 0 2 18 D 10 8 -2 0 4 E -12 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3025: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (7) A B E D C (6) A B E C D (6) C B A E D (5) C B A D E (5) E A D B C (4) E A B D C (4) D C E A B (4) C D E B A (4) B A E C D (4) B A C E D (4) A E B D C (4) E D A B C (3) D E C A B (3) D A B E C (3) C E D B A (3) C B D A E (3) B A C D E (3) A B D E C (3) E C D A B (2) E A B C D (2) D C B A E (2) C E B A D (2) C D B E A (2) B A D C E (2) E D A C B (1) D E A C B (1) D E A B C (1) D C E B A (1) D C A B E (1) D B A C E (1) C B E A D (1) B C A E D (1) B C A D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 6 16 24 B 2 0 6 12 20 C -6 -6 0 10 4 D -16 -12 -10 0 -4 E -24 -20 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 16 24 B 2 0 6 12 20 C -6 -6 0 10 4 D -16 -12 -10 0 -4 E -24 -20 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=20 D=17 E=16 B=15 so B is eliminated. Round 2 votes counts: C=34 A=33 D=17 E=16 so E is eliminated. Round 3 votes counts: A=43 C=36 D=21 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:220 C:201 D:179 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 16 24 B 2 0 6 12 20 C -6 -6 0 10 4 D -16 -12 -10 0 -4 E -24 -20 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 16 24 B 2 0 6 12 20 C -6 -6 0 10 4 D -16 -12 -10 0 -4 E -24 -20 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 16 24 B 2 0 6 12 20 C -6 -6 0 10 4 D -16 -12 -10 0 -4 E -24 -20 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3026: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) C E A D B (8) A B E D C (7) A B D E C (6) C E D A B (5) B A D E C (4) A E C B D (4) A B D C E (4) D C E B A (3) D C B E A (3) D B E C A (3) B D A E C (3) B A E D C (3) A E B C D (3) E C A B D (2) E B C D A (2) D C A B E (2) C D E B A (2) C D E A B (2) C D A E B (2) B D E C A (2) B D A C E (2) A B E C D (2) E C D B A (1) E C B A D (1) E C A D B (1) E B D A C (1) E B C A D (1) E A C B D (1) D B C E A (1) D B C A E (1) D B A E C (1) C E D B A (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E A C (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 10 6 14 B -8 0 14 4 12 C -10 -14 0 -14 -2 D -6 -4 14 0 6 E -14 -12 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 6 14 B -8 0 14 4 12 C -10 -14 0 -14 -2 D -6 -4 14 0 6 E -14 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 C=22 B=17 E=10 so E is eliminated. Round 2 votes counts: A=30 C=27 D=22 B=21 so B is eliminated. Round 3 votes counts: A=38 D=32 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:211 D:205 E:185 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 6 14 B -8 0 14 4 12 C -10 -14 0 -14 -2 D -6 -4 14 0 6 E -14 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 6 14 B -8 0 14 4 12 C -10 -14 0 -14 -2 D -6 -4 14 0 6 E -14 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 6 14 B -8 0 14 4 12 C -10 -14 0 -14 -2 D -6 -4 14 0 6 E -14 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3027: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) E B C D A (5) D E A B C (5) A B E D C (5) E B D C A (4) D C E B A (4) C E B D A (4) C D E B A (4) B A E C D (4) A D C B E (4) A B C E D (4) D C E A B (3) D A C E B (3) D A C B E (3) C D A E B (3) A C D B E (3) E D B A C (2) D A E C B (2) C E D B A (2) C A D B E (2) B E A D C (2) B E A C D (2) A D B C E (2) A B E C D (2) E D C B A (1) E D B C A (1) E B D A C (1) E B A D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D E B A C (1) D E A C B (1) D A E B C (1) C D B E A (1) C D A B E (1) C B E D A (1) C B E A D (1) C B A E D (1) B E C A D (1) B C A E D (1) B A E D C (1) A C B E D (1) Total count = 100 A B C D E A 0 6 0 -26 0 B -6 0 -8 -18 -16 C 0 8 0 -14 10 D 26 18 14 0 6 E 0 16 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -26 0 B -6 0 -8 -18 -16 C 0 8 0 -14 10 D 26 18 14 0 6 E 0 16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=21 C=20 E=15 B=11 so B is eliminated. Round 2 votes counts: D=33 A=26 C=21 E=20 so E is eliminated. Round 3 votes counts: D=42 A=31 C=27 so C is eliminated. Round 4 votes counts: D=63 A=37 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:232 C:202 E:200 A:190 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -26 0 B -6 0 -8 -18 -16 C 0 8 0 -14 10 D 26 18 14 0 6 E 0 16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -26 0 B -6 0 -8 -18 -16 C 0 8 0 -14 10 D 26 18 14 0 6 E 0 16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -26 0 B -6 0 -8 -18 -16 C 0 8 0 -14 10 D 26 18 14 0 6 E 0 16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3028: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) C E A D B (8) D B C A E (6) C D A E B (6) B D A E C (6) B E A D C (5) B D C A E (5) E A B C D (4) A E C D B (4) C B E A D (3) B D C E A (3) A E D C B (3) D C B A E (2) D C A E B (2) C E A B D (2) C D B A E (2) C A E D B (2) B E C A D (2) B E A C D (2) B D E C A (2) B C E A D (2) B C D E A (2) E C A B D (1) E A D C B (1) E A D B C (1) D C A B E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B C E (1) C D E A B (1) C D B E A (1) C B D E A (1) C A D E B (1) A E B D C (1) Total count = 100 A B C D E A 0 8 -10 8 -6 B -8 0 -10 -18 -4 C 10 10 0 12 6 D -8 18 -12 0 -6 E 6 4 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 8 -6 B -8 0 -10 -18 -4 C 10 10 0 12 6 D -8 18 -12 0 -6 E 6 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=27 E=19 D=17 A=8 so A is eliminated. Round 2 votes counts: B=29 E=27 C=27 D=17 so D is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:205 A:200 D:196 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 8 -6 B -8 0 -10 -18 -4 C 10 10 0 12 6 D -8 18 -12 0 -6 E 6 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 8 -6 B -8 0 -10 -18 -4 C 10 10 0 12 6 D -8 18 -12 0 -6 E 6 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 8 -6 B -8 0 -10 -18 -4 C 10 10 0 12 6 D -8 18 -12 0 -6 E 6 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3029: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) E A D C B (6) B D C E A (6) A E D C B (5) E D C A B (4) E C D A B (4) D B C E A (4) B C D A E (4) E A C D B (3) D C E B A (3) D C B E A (3) C E B D A (3) B D C A E (3) B C E D A (3) A E C D B (3) A D E C B (3) E D A C B (2) E C D B A (2) B D A C E (2) B C E A D (2) B A D C E (2) B A C E D (2) A E C B D (2) A B E D C (2) A B E C D (2) E C A D B (1) E C A B D (1) D E C B A (1) D E C A B (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E B A (1) C B E A D (1) C B D E A (1) B C A E D (1) B A C D E (1) A E D B C (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -20 -16 -26 B 12 0 -4 -2 2 C 20 4 0 -4 8 D 16 2 4 0 -4 E 26 -2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 -12 -20 -16 -26 B 12 0 -4 -2 2 C 20 4 0 -4 8 D 16 2 4 0 -4 E 26 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=23 A=21 D=13 C=8 so C is eliminated. Round 2 votes counts: B=37 E=28 A=21 D=14 so D is eliminated. Round 3 votes counts: B=44 E=34 A=22 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:214 E:210 D:209 B:204 A:163 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -20 -16 -26 B 12 0 -4 -2 2 C 20 4 0 -4 8 D 16 2 4 0 -4 E 26 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -16 -26 B 12 0 -4 -2 2 C 20 4 0 -4 8 D 16 2 4 0 -4 E 26 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -16 -26 B 12 0 -4 -2 2 C 20 4 0 -4 8 D 16 2 4 0 -4 E 26 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3030: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (15) E A D B C (14) C B A D E (6) C D B E A (5) C D B A E (5) E D C B A (4) E C D B A (4) E A B D C (4) A E B C D (4) E D C A B (3) E D A B C (3) E A C B D (3) A E B D C (3) D E C B A (2) D B C A E (2) D B A E C (2) D B A C E (2) C E D B A (2) C D E B A (2) C B D E A (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A C B (1) D E A B C (1) D C E B A (1) D C B E A (1) D B C E A (1) C B A E D (1) B D C A E (1) B D A C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -16 -24 -10 B 22 0 -14 -14 -6 C 16 14 0 2 0 D 24 14 -2 0 4 E 10 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.839633 D: 0.000000 E: 0.160367 Sum of squares = 0.730700711035 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.839633 D: 0.839633 E: 1.000000 A B C D E A 0 -22 -16 -24 -10 B 22 0 -14 -14 -6 C 16 14 0 2 0 D 24 14 -2 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555598547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=38 C=38 D=12 A=10 B=2 so B is eliminated. Round 2 votes counts: E=38 C=38 D=14 A=10 so A is eliminated. Round 3 votes counts: E=45 C=40 D=15 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:216 E:206 B:194 A:164 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -16 -24 -10 B 22 0 -14 -14 -6 C 16 14 0 2 0 D 24 14 -2 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555598547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -16 -24 -10 B 22 0 -14 -14 -6 C 16 14 0 2 0 D 24 14 -2 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555598547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -16 -24 -10 B 22 0 -14 -14 -6 C 16 14 0 2 0 D 24 14 -2 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555598547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3031: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) B A D E C (9) E C B A D (6) D C E A B (6) E C D B A (5) D A B C E (5) B E C A D (5) B A E C D (5) A D B E C (5) A B D E C (5) A B D C E (5) E C B D A (3) D A B E C (3) C E D A B (3) C E B A D (3) E B C A D (2) D E C A B (2) D C A E B (2) D A C E B (2) D A C B E (2) C D E A B (2) B E A C D (2) B A E D C (2) D E A C B (1) D A E B C (1) C E B D A (1) C B A E D (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -12 -2 4 -2 B 12 0 2 2 2 C 2 -2 0 -2 -12 D -4 -2 2 0 2 E 2 -2 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 4 -2 B 12 0 2 2 2 C 2 -2 0 -2 -12 D -4 -2 2 0 2 E 2 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 C=20 E=16 A=15 so A is eliminated. Round 2 votes counts: B=35 D=29 C=20 E=16 so E is eliminated. Round 3 votes counts: B=37 C=34 D=29 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:205 D:199 A:194 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 4 -2 B 12 0 2 2 2 C 2 -2 0 -2 -12 D -4 -2 2 0 2 E 2 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 4 -2 B 12 0 2 2 2 C 2 -2 0 -2 -12 D -4 -2 2 0 2 E 2 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 4 -2 B 12 0 2 2 2 C 2 -2 0 -2 -12 D -4 -2 2 0 2 E 2 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3032: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) D E A B C (5) C D E B A (5) B E C D A (4) B C E A D (4) A B C E D (4) E C D B A (3) E B D C A (3) D E C B A (3) D E A C B (3) D A E B C (3) C D A E B (3) C B A E D (3) A D E B C (3) A D B C E (3) A C B D E (3) A B C D E (3) E D C B A (2) D E C A B (2) D C E B A (2) D A E C B (2) B E A D C (2) B C A E D (2) B A C E D (2) A D B E C (2) A B E D C (2) A B E C D (2) A B D E C (2) E D B C A (1) E D B A C (1) E C B D A (1) D C A E B (1) D A C E B (1) C D E A B (1) C B E A D (1) C B A D E (1) C A B D E (1) B E A C D (1) B C E D A (1) A E D B C (1) A D E C B (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -2 -8 -4 B -2 0 2 -2 4 C 2 -2 0 6 2 D 8 2 -6 0 6 E 4 -4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999988 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -8 -4 B -2 0 2 -2 4 C 2 -2 0 6 2 D 8 2 -6 0 6 E 4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 C=22 B=16 E=11 so E is eliminated. Round 2 votes counts: A=29 D=26 C=26 B=19 so B is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:205 C:204 B:201 E:196 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 -8 -4 B -2 0 2 -2 4 C 2 -2 0 6 2 D 8 2 -6 0 6 E 4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -8 -4 B -2 0 2 -2 4 C 2 -2 0 6 2 D 8 2 -6 0 6 E 4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -8 -4 B -2 0 2 -2 4 C 2 -2 0 6 2 D 8 2 -6 0 6 E 4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999952 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3033: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (11) C B D E A (10) A E D B C (9) A E B D C (9) E A D B C (6) D E A C B (6) C D B E A (4) B A E C D (4) D E A B C (3) C B D A E (3) B C E A D (3) E A B D C (2) D C E A B (2) D B C E A (2) D A E C B (2) C B A E D (2) B E A C D (2) B C E D A (2) B C A E D (2) A E D C B (2) A D E C B (2) E D A B C (1) D C E B A (1) D C A E B (1) D B E A C (1) C D B A E (1) C D A E B (1) C D A B E (1) C A E D B (1) B D C E A (1) B A E D C (1) B A C E D (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 4 -6 -14 B 2 0 22 8 2 C -4 -22 0 -2 -2 D 6 -8 2 0 4 E 14 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -6 -14 B 2 0 22 8 2 C -4 -22 0 -2 -2 D 6 -8 2 0 4 E 14 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 A=23 D=18 E=9 so E is eliminated. Round 2 votes counts: A=31 B=27 C=23 D=19 so D is eliminated. Round 3 votes counts: A=43 B=30 C=27 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:205 D:202 A:191 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 -6 -14 B 2 0 22 8 2 C -4 -22 0 -2 -2 D 6 -8 2 0 4 E 14 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -6 -14 B 2 0 22 8 2 C -4 -22 0 -2 -2 D 6 -8 2 0 4 E 14 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -6 -14 B 2 0 22 8 2 C -4 -22 0 -2 -2 D 6 -8 2 0 4 E 14 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3034: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (14) E B C D A (12) B E C D A (8) D A C B E (7) E B C A D (6) E B A C D (6) D C A B E (6) C D A B E (6) A D C E B (6) E B A D C (4) E A B D C (4) A E D B C (4) A D E B C (4) C D B A E (3) C B E D A (2) C B D E A (2) C B D A E (2) B C E D A (2) A D E C B (2) Total count = 100 A B C D E A 0 6 2 0 8 B -6 0 0 -4 4 C -2 0 0 -2 0 D 0 4 2 0 4 E -8 -4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250573 B: 0.000000 C: 0.000000 D: 0.749427 E: 0.000000 Sum of squares = 0.62442800731 Cumulative probabilities = A: 0.250573 B: 0.250573 C: 0.250573 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 0 8 B -6 0 0 -4 4 C -2 0 0 -2 0 D 0 4 2 0 4 E -8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=30 C=15 D=13 B=10 so B is eliminated. Round 2 votes counts: E=40 A=30 C=17 D=13 so D is eliminated. Round 3 votes counts: E=40 A=37 C=23 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:205 C:198 B:197 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 0 8 B -6 0 0 -4 4 C -2 0 0 -2 0 D 0 4 2 0 4 E -8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 8 B -6 0 0 -4 4 C -2 0 0 -2 0 D 0 4 2 0 4 E -8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 8 B -6 0 0 -4 4 C -2 0 0 -2 0 D 0 4 2 0 4 E -8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3035: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) E C B A D (5) B D A C E (5) B C E D A (5) E A D C B (4) D A B C E (4) B D C A E (4) A D E B C (4) D B A C E (3) C E D A B (3) C E B D A (3) B E C A D (3) B E A C D (3) B A E D C (3) A E D C B (3) A B E D C (3) E B A C D (2) E A C D B (2) D A C E B (2) D A C B E (2) C D B E A (2) C B E D A (2) C B D A E (2) B C D E A (2) B C D A E (2) B A D E C (2) A E D B C (2) A E B D C (2) E C D A B (1) E C A D B (1) E C A B D (1) E B C A D (1) E A B D C (1) E A B C D (1) D C B A E (1) D B C A E (1) C E D B A (1) C D E A B (1) C D B A E (1) B A D C E (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 16 6 12 B 8 0 24 6 18 C -16 -24 0 -12 -4 D -6 -6 12 0 -4 E -12 -18 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 16 6 12 B 8 0 24 6 18 C -16 -24 0 -12 -4 D -6 -6 12 0 -4 E -12 -18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=23 E=19 C=15 D=13 so D is eliminated. Round 2 votes counts: B=34 A=31 E=19 C=16 so C is eliminated. Round 3 votes counts: B=42 A=31 E=27 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:213 D:198 E:189 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 16 6 12 B 8 0 24 6 18 C -16 -24 0 -12 -4 D -6 -6 12 0 -4 E -12 -18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 6 12 B 8 0 24 6 18 C -16 -24 0 -12 -4 D -6 -6 12 0 -4 E -12 -18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 6 12 B 8 0 24 6 18 C -16 -24 0 -12 -4 D -6 -6 12 0 -4 E -12 -18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996048 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3036: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) E C D B A (7) A E D C B (6) A D B C E (6) B D C A E (5) A E B D C (5) A B D C E (5) B A D C E (4) E B C D A (3) E A C D B (3) E A C B D (3) E A B C D (3) D C B A E (3) C B E D A (3) B C D E A (3) A D C B E (3) E C B A D (2) E B A C D (2) D B C A E (2) C D B E A (2) B D C E A (2) A D C E B (2) A B D E C (2) E C A D B (1) D C A B E (1) D B A C E (1) D A B C E (1) C E D A B (1) C D E B A (1) C D B A E (1) C D A E B (1) B E C D A (1) B D A C E (1) B C E D A (1) A D E B C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 0 0 4 B 8 0 -2 12 -4 C 0 2 0 -2 0 D 0 -12 2 0 -6 E -4 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.135135 B: 0.054054 C: 0.648649 D: 0.054054 E: 0.108108 Sum of squares = 0.456537618835 Cumulative probabilities = A: 0.135135 B: 0.189189 C: 0.837838 D: 0.891892 E: 1.000000 A B C D E A 0 -8 0 0 4 B 8 0 -2 12 -4 C 0 2 0 -2 0 D 0 -12 2 0 -6 E -4 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.054054 C: 0.648649 D: 0.054054 E: 0.108108 Sum of squares = 0.456537618699 Cumulative probabilities = A: 0.135135 B: 0.189189 C: 0.837838 D: 0.891892 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=32 B=17 C=9 D=8 so D is eliminated. Round 2 votes counts: E=34 A=33 B=20 C=13 so C is eliminated. Round 3 votes counts: E=36 A=35 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:207 E:203 C:200 A:198 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 0 4 B 8 0 -2 12 -4 C 0 2 0 -2 0 D 0 -12 2 0 -6 E -4 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.054054 C: 0.648649 D: 0.054054 E: 0.108108 Sum of squares = 0.456537618699 Cumulative probabilities = A: 0.135135 B: 0.189189 C: 0.837838 D: 0.891892 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 0 4 B 8 0 -2 12 -4 C 0 2 0 -2 0 D 0 -12 2 0 -6 E -4 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.054054 C: 0.648649 D: 0.054054 E: 0.108108 Sum of squares = 0.456537618699 Cumulative probabilities = A: 0.135135 B: 0.189189 C: 0.837838 D: 0.891892 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 0 4 B 8 0 -2 12 -4 C 0 2 0 -2 0 D 0 -12 2 0 -6 E -4 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.135135 B: 0.054054 C: 0.648649 D: 0.054054 E: 0.108108 Sum of squares = 0.456537618699 Cumulative probabilities = A: 0.135135 B: 0.189189 C: 0.837838 D: 0.891892 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3037: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) C E D A B (6) C A E D B (6) A D B C E (5) D B A E C (4) C E A D B (4) C E A B D (4) B D A E C (4) E C B A D (3) E B C D A (3) D C A B E (3) D B E A C (3) C D E A B (3) B E A D C (3) A D C B E (3) A C D E B (3) E C B D A (2) E B D C A (2) D B E C A (2) D A C B E (2) D A B E C (2) C E D B A (2) C D E B A (2) A E B C D (2) A B E D C (2) E C A B D (1) E B C A D (1) D C B E A (1) D C A E B (1) D B C E A (1) D B A C E (1) C E B D A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E A C (1) B A E D C (1) B A D E C (1) A E C B D (1) A C E D B (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 20 -2 -8 6 B -20 0 -4 -24 -2 C 2 4 0 -2 16 D 8 24 2 0 0 E -6 2 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.937416 E: 0.062584 Sum of squares = 0.88266618602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.937416 E: 1.000000 A B C D E A 0 20 -2 -8 6 B -20 0 -4 -24 -2 C 2 4 0 -2 16 D 8 24 2 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 0.111111 Sum of squares = 0.802469375929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=26 A=20 E=12 B=12 so E is eliminated. Round 2 votes counts: C=36 D=26 A=20 B=18 so B is eliminated. Round 3 votes counts: C=40 D=35 A=25 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:210 A:208 E:190 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 -2 -8 6 B -20 0 -4 -24 -2 C 2 4 0 -2 16 D 8 24 2 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 0.111111 Sum of squares = 0.802469375929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -2 -8 6 B -20 0 -4 -24 -2 C 2 4 0 -2 16 D 8 24 2 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 0.111111 Sum of squares = 0.802469375929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -2 -8 6 B -20 0 -4 -24 -2 C 2 4 0 -2 16 D 8 24 2 0 0 E -6 2 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 0.111111 Sum of squares = 0.802469375929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.888889 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3038: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (8) E D C A B (7) B A D E C (7) B A C D E (7) E A D C B (5) C D E B A (5) B A E D C (5) A B E C D (5) C D E A B (4) E C D A B (3) C D B E A (3) B C D A E (3) B A D C E (3) A E C D B (3) A E B D C (3) E A C D B (2) D E C A B (2) D C E B A (2) D C E A B (2) D C B E A (2) D B C E A (2) C E D A B (2) B C A D E (2) A B C E D (2) E D A C B (1) E C A D B (1) D E C B A (1) D E A C B (1) C E D B A (1) C B D E A (1) B D C A E (1) B D A C E (1) B A E C D (1) B A C E D (1) A E B C D (1) Total count = 100 A B C D E A 0 4 12 12 6 B -4 0 4 2 8 C -12 -4 0 -6 -12 D -12 -2 6 0 -2 E -6 -8 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 12 6 B -4 0 4 2 8 C -12 -4 0 -6 -12 D -12 -2 6 0 -2 E -6 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=22 E=19 C=16 D=12 so D is eliminated. Round 2 votes counts: B=33 E=23 C=22 A=22 so C is eliminated. Round 3 votes counts: E=39 B=39 A=22 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:205 E:200 D:195 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 12 6 B -4 0 4 2 8 C -12 -4 0 -6 -12 D -12 -2 6 0 -2 E -6 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 12 6 B -4 0 4 2 8 C -12 -4 0 -6 -12 D -12 -2 6 0 -2 E -6 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 12 6 B -4 0 4 2 8 C -12 -4 0 -6 -12 D -12 -2 6 0 -2 E -6 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3039: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (16) A D C B E (10) C D B A E (8) E C B D A (7) E B D C A (7) A E D B C (7) E A B C D (5) C B D E A (5) C D B E A (4) A E B D C (4) A D B C E (4) E A B D C (3) D C B A E (2) D A C B E (2) C E B D A (2) A D E C B (2) A C D B E (2) E B D A C (1) E A C B D (1) D C A B E (1) D B C E A (1) C D A B E (1) C B E D A (1) C A D B E (1) B D C E A (1) B C D E A (1) A E C D B (1) Total count = 100 A B C D E A 0 -12 -16 -20 -10 B 12 0 0 8 -12 C 16 0 0 10 -8 D 20 -8 -10 0 -10 E 10 12 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -16 -20 -10 B 12 0 0 8 -12 C 16 0 0 10 -8 D 20 -8 -10 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=30 C=22 D=6 B=2 so B is eliminated. Round 2 votes counts: E=40 A=30 C=23 D=7 so D is eliminated. Round 3 votes counts: E=40 A=32 C=28 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:209 B:204 D:196 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -16 -20 -10 B 12 0 0 8 -12 C 16 0 0 10 -8 D 20 -8 -10 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -20 -10 B 12 0 0 8 -12 C 16 0 0 10 -8 D 20 -8 -10 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -20 -10 B 12 0 0 8 -12 C 16 0 0 10 -8 D 20 -8 -10 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3040: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) B E A C D (9) A D B E C (9) D A B E C (7) D C A E B (6) C E B D A (6) D E B C A (5) E B C A D (4) D A C B E (4) B E C A D (4) A C B E D (4) D A C E B (3) C D E B A (3) A B E D C (3) A B D E C (3) E B D C A (2) D C E B A (2) D B E A C (2) A D C B E (2) A B E C D (2) E D C B A (1) E C B D A (1) E B C D A (1) D C E A B (1) D A E B C (1) D A B C E (1) C E D A B (1) C D E A B (1) C B E A D (1) C A E B D (1) A D B C E (1) Total count = 100 A B C D E A 0 0 2 4 -6 B 0 0 8 0 4 C -2 -8 0 -6 -8 D -4 0 6 0 2 E 6 -4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.273943 B: 0.726057 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.602203666535 Cumulative probabilities = A: 0.273943 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 -6 B 0 0 8 0 4 C -2 -8 0 -6 -8 D -4 0 6 0 2 E 6 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000004977 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=24 C=22 B=13 E=9 so E is eliminated. Round 2 votes counts: D=33 A=24 C=23 B=20 so B is eliminated. Round 3 votes counts: D=35 A=33 C=32 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:206 E:204 D:202 A:200 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 4 -6 B 0 0 8 0 4 C -2 -8 0 -6 -8 D -4 0 6 0 2 E 6 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000004977 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 -6 B 0 0 8 0 4 C -2 -8 0 -6 -8 D -4 0 6 0 2 E 6 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000004977 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 -6 B 0 0 8 0 4 C -2 -8 0 -6 -8 D -4 0 6 0 2 E 6 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000004977 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3041: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) A D C E B (9) D A C E B (8) A D B E C (8) D A B E C (7) C E B D A (6) D A C B E (5) C E A B D (5) E C B A D (4) C E B A D (4) B E C A D (4) E B C A D (2) D B E C A (2) D B C E A (2) D A B C E (2) C A E D B (2) B E D C A (2) A C E B D (2) A B E C D (2) D C E B A (1) D C A E B (1) D B A E C (1) C E D A B (1) C D E B A (1) C A D E B (1) B D E C A (1) B D E A C (1) B A D E C (1) A E C B D (1) A D E C B (1) A D C B E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 2 -2 8 B -16 0 -6 -8 2 C -2 6 0 -10 4 D 2 8 10 0 10 E -8 -2 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 -2 8 B -16 0 -6 -8 2 C -2 6 0 -10 4 D 2 8 10 0 10 E -8 -2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=26 C=20 B=19 E=6 so E is eliminated. Round 2 votes counts: D=29 A=26 C=24 B=21 so B is eliminated. Round 3 votes counts: C=40 D=33 A=27 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:212 C:199 E:188 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 2 -2 8 B -16 0 -6 -8 2 C -2 6 0 -10 4 D 2 8 10 0 10 E -8 -2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 -2 8 B -16 0 -6 -8 2 C -2 6 0 -10 4 D 2 8 10 0 10 E -8 -2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 -2 8 B -16 0 -6 -8 2 C -2 6 0 -10 4 D 2 8 10 0 10 E -8 -2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3042: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) B D A C E (6) D A B C E (5) B D C A E (5) E A C D B (4) E A B D C (4) C D B A E (4) A D B E C (4) A D B C E (4) A B D E C (4) E C A D B (3) E A C B D (3) D B A C E (3) C E D B A (3) C E B D A (3) C B D E A (3) C B D A E (3) A E D B C (3) E C A B D (2) E A D C B (2) C E D A B (2) C D B E A (2) C B E D A (2) B C D E A (2) E C B A D (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B C D (1) D C B A E (1) D C A B E (1) D B C A E (1) C D A B E (1) B E A D C (1) B C E D A (1) B A D E C (1) A E D C B (1) A E B D C (1) A D E B C (1) A D C E B (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 6 -8 4 B 0 0 2 4 12 C -6 -2 0 -6 10 D 8 -4 6 0 8 E -4 -12 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.204928 B: 0.795072 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.674135503449 Cumulative probabilities = A: 0.204928 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -8 4 B 0 0 2 4 12 C -6 -2 0 -6 10 D 8 -4 6 0 8 E -4 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555718382 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=23 A=21 B=16 D=11 so D is eliminated. Round 2 votes counts: E=29 A=26 C=25 B=20 so B is eliminated. Round 3 votes counts: A=36 C=34 E=30 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:209 D:209 A:201 C:198 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 -8 4 B 0 0 2 4 12 C -6 -2 0 -6 10 D 8 -4 6 0 8 E -4 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555718382 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -8 4 B 0 0 2 4 12 C -6 -2 0 -6 10 D 8 -4 6 0 8 E -4 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555718382 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -8 4 B 0 0 2 4 12 C -6 -2 0 -6 10 D 8 -4 6 0 8 E -4 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555718382 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3043: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) A D C B E (8) E B D C A (6) E B C D A (6) C E B D A (6) C A D E B (6) A B D E C (5) A D B E C (4) A C D E B (4) A C D B E (4) D E B C A (3) C E D B A (3) C E B A D (3) C D A E B (3) A B E D C (3) D B E A C (2) D A C E B (2) D A B E C (2) C A E D B (2) C A E B D (2) B E D C A (2) B E C A D (2) B E A D C (2) B E A C D (2) A C B E D (2) E C B D A (1) D C E B A (1) D C A E B (1) D A C B E (1) C D E A B (1) B D E A C (1) B D A E C (1) A C E B D (1) Total count = 100 A B C D E A 0 2 4 0 2 B -2 0 -2 6 -2 C -4 2 0 -4 0 D 0 -6 4 0 -2 E -2 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.836923 B: 0.000000 C: 0.000000 D: 0.163077 E: 0.000000 Sum of squares = 0.727033695265 Cumulative probabilities = A: 0.836923 B: 0.836923 C: 0.836923 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 0 2 B -2 0 -2 6 -2 C -4 2 0 -4 0 D 0 -6 4 0 -2 E -2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.62500000357 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=26 B=18 E=13 D=12 so D is eliminated. Round 2 votes counts: A=36 C=28 B=20 E=16 so E is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:204 E:201 B:200 D:198 C:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 2 B -2 0 -2 6 -2 C -4 2 0 -4 0 D 0 -6 4 0 -2 E -2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.62500000357 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 2 B -2 0 -2 6 -2 C -4 2 0 -4 0 D 0 -6 4 0 -2 E -2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.62500000357 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 2 B -2 0 -2 6 -2 C -4 2 0 -4 0 D 0 -6 4 0 -2 E -2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.62500000357 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3044: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) A B D C E (8) B A D C E (7) B A E D C (6) E B A C D (5) D C A B E (5) C D E A B (5) C D A E B (5) C D A B E (5) E C D B A (4) B A D E C (4) A D C B E (4) E B D C A (3) E B A D C (3) D C E A B (3) E B C D A (2) E B C A D (2) D C A E B (2) C E D A B (2) B E A D C (2) B E A C D (2) A B C D E (2) E D C B A (1) E D C A B (1) E C B D A (1) E C B A D (1) D B A C E (1) D A C B E (1) D A B C E (1) B A E C D (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -4 -4 6 B -10 0 0 -2 0 C 4 0 0 -6 4 D 4 2 6 0 8 E -6 0 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -4 6 B -10 0 0 -2 0 C 4 0 0 -6 4 D 4 2 6 0 8 E -6 0 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=22 C=17 A=15 D=13 so D is eliminated. Round 2 votes counts: E=33 C=27 B=23 A=17 so A is eliminated. Round 3 votes counts: B=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:204 C:201 B:194 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 -4 6 B -10 0 0 -2 0 C 4 0 0 -6 4 D 4 2 6 0 8 E -6 0 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -4 6 B -10 0 0 -2 0 C 4 0 0 -6 4 D 4 2 6 0 8 E -6 0 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -4 6 B -10 0 0 -2 0 C 4 0 0 -6 4 D 4 2 6 0 8 E -6 0 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3045: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) E B A D C (6) E A B C D (5) C D A B E (5) A C E D B (5) D C B E A (4) D C B A E (4) D B C E A (4) B E D A C (4) B E A D C (4) B D C E A (4) D B C A E (3) C D A E B (3) B D E C A (3) A E B D C (3) A C D E B (3) A C D B E (3) A B E D C (3) E C A D B (2) E B D C A (2) D A B C E (2) C D E B A (2) B E D C A (2) B A D E C (2) A E C D B (2) A E C B D (2) E B C A D (1) E B A C D (1) E A C D B (1) E A B D C (1) D C A B E (1) D A C B E (1) C E A D B (1) C D B E A (1) B D A C E (1) B A E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 16 8 4 B -2 0 20 6 6 C -16 -20 0 -12 -4 D -8 -6 12 0 -6 E -4 -6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 8 4 B -2 0 20 6 6 C -16 -20 0 -12 -4 D -8 -6 12 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=21 E=19 D=19 C=12 so C is eliminated. Round 2 votes counts: D=30 A=29 B=21 E=20 so E is eliminated. Round 3 votes counts: A=39 B=31 D=30 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:215 E:200 D:196 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 8 4 B -2 0 20 6 6 C -16 -20 0 -12 -4 D -8 -6 12 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 8 4 B -2 0 20 6 6 C -16 -20 0 -12 -4 D -8 -6 12 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 8 4 B -2 0 20 6 6 C -16 -20 0 -12 -4 D -8 -6 12 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3046: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (15) C B E A D (13) A D E B C (12) D E A B C (5) A E D B C (5) A C B E D (4) E A D B C (3) C B E D A (3) C B D E A (3) C B A E D (3) C A B E D (3) A D E C B (3) E A B C D (2) D C B E A (2) A C E B D (2) E B C A D (1) E B A D C (1) E A B D C (1) D E B C A (1) D E B A C (1) D C B A E (1) D C A B E (1) D B C E A (1) D A E C B (1) D A C B E (1) C D A B E (1) C B D A E (1) B E D C A (1) B E C A D (1) B D E C A (1) B C E D A (1) B C E A D (1) A E D C B (1) A E C B D (1) A E B D C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 28 22 20 16 B -28 0 8 -12 -14 C -22 -8 0 -18 -14 D -20 12 18 0 4 E -16 14 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 22 20 16 B -28 0 8 -12 -14 C -22 -8 0 -18 -14 D -20 12 18 0 4 E -16 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=29 C=27 E=8 B=5 so B is eliminated. Round 2 votes counts: A=31 D=30 C=29 E=10 so E is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:243 D:207 E:204 B:177 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 22 20 16 B -28 0 8 -12 -14 C -22 -8 0 -18 -14 D -20 12 18 0 4 E -16 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 22 20 16 B -28 0 8 -12 -14 C -22 -8 0 -18 -14 D -20 12 18 0 4 E -16 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 22 20 16 B -28 0 8 -12 -14 C -22 -8 0 -18 -14 D -20 12 18 0 4 E -16 14 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3047: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D A C E B (8) A D B E C (6) D C B E A (5) B E C D A (5) B E C A D (5) E C B A D (4) D B C E A (4) C E B D A (4) A E B C D (4) A D C E B (4) A B E D C (4) D C E B A (3) C D E B A (3) B E A D C (3) D B E C A (2) D A B C E (2) C E D B A (2) C E B A D (2) A E C B D (2) E C B D A (1) E C A B D (1) E B A C D (1) D C E A B (1) D C A E B (1) D B A E C (1) D A C B E (1) D A B E C (1) C E A B D (1) C D B E A (1) B E D C A (1) B E D A C (1) B E A C D (1) B D E C A (1) B C E D A (1) A D E C B (1) A D C B E (1) A C E D B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -12 2 -22 B 18 0 6 4 -4 C 12 -6 0 -4 -10 D -2 -4 4 0 -6 E 22 4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -12 2 -22 B 18 0 6 4 -4 C 12 -6 0 -4 -10 D -2 -4 4 0 -6 E 22 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=25 B=18 E=15 C=13 so C is eliminated. Round 2 votes counts: D=33 A=25 E=24 B=18 so B is eliminated. Round 3 votes counts: E=41 D=34 A=25 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:212 C:196 D:196 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -12 2 -22 B 18 0 6 4 -4 C 12 -6 0 -4 -10 D -2 -4 4 0 -6 E 22 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 2 -22 B 18 0 6 4 -4 C 12 -6 0 -4 -10 D -2 -4 4 0 -6 E 22 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 2 -22 B 18 0 6 4 -4 C 12 -6 0 -4 -10 D -2 -4 4 0 -6 E 22 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3048: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C B A E D (9) C B A D E (8) B C A E D (8) E B D A C (7) D E A C B (7) D E A B C (7) E D B A C (4) B E A C D (3) B C E A D (3) A C B D E (3) D A E C B (2) C D A E B (2) C B E A D (2) A D E B C (2) E D B C A (1) E B D C A (1) E B A D C (1) E A B D C (1) D C E B A (1) D C E A B (1) D C A E B (1) D A E B C (1) D A C E B (1) C D B A E (1) C B E D A (1) C B D A E (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E D A (1) B A E C D (1) B A C E D (1) A E D B C (1) A D C E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 10 0 -6 B 14 0 12 12 -2 C -10 -12 0 -2 -4 D 0 -12 2 0 -16 E 6 2 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 10 0 -6 B 14 0 12 12 -2 C -10 -12 0 -2 -4 D 0 -12 2 0 -16 E 6 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 D=21 B=20 A=9 so A is eliminated. Round 2 votes counts: C=30 E=25 D=24 B=21 so B is eliminated. Round 3 votes counts: C=43 E=33 D=24 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:218 E:214 A:195 D:187 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 10 0 -6 B 14 0 12 12 -2 C -10 -12 0 -2 -4 D 0 -12 2 0 -16 E 6 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 0 -6 B 14 0 12 12 -2 C -10 -12 0 -2 -4 D 0 -12 2 0 -16 E 6 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 0 -6 B 14 0 12 12 -2 C -10 -12 0 -2 -4 D 0 -12 2 0 -16 E 6 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3049: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) D B A E C (10) D B E C A (7) E C B A D (6) C E A B D (6) C A E B D (6) D A B C E (5) A D C B E (5) D E B C A (4) D A C B E (4) E B C A D (3) B D E A C (3) B D A E C (3) A C E B D (3) A C B E D (3) E C B D A (2) E C A B D (2) B E D C A (2) A D B C E (2) A C D B E (2) E D B C A (1) E B D C A (1) D C E A B (1) B E C D A (1) B D E C A (1) B A D C E (1) A D C E B (1) A C D E B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 14 -14 -4 B 14 0 16 -10 26 C -14 -16 0 -26 -16 D 14 10 26 0 26 E 4 -26 16 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 14 -14 -4 B 14 0 16 -10 26 C -14 -16 0 -26 -16 D 14 10 26 0 26 E 4 -26 16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 A=19 E=15 C=12 B=11 so B is eliminated. Round 2 votes counts: D=50 A=20 E=18 C=12 so C is eliminated. Round 3 votes counts: D=50 A=26 E=24 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:238 B:223 A:191 E:184 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 14 -14 -4 B 14 0 16 -10 26 C -14 -16 0 -26 -16 D 14 10 26 0 26 E 4 -26 16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 14 -14 -4 B 14 0 16 -10 26 C -14 -16 0 -26 -16 D 14 10 26 0 26 E 4 -26 16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 14 -14 -4 B 14 0 16 -10 26 C -14 -16 0 -26 -16 D 14 10 26 0 26 E 4 -26 16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3050: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) E C B A D (5) D C A E B (5) E B A D C (4) E B A C D (4) D C A B E (4) B E A C D (4) A B D E C (4) D A B C E (3) C D E B A (3) C D A B E (3) C B A E D (3) C A B D E (3) B A E C D (3) E D C B A (2) E D B A C (2) E C D B A (2) E C B D A (2) E B C A D (2) D E A C B (2) D C E A B (2) D A E B C (2) D A C B E (2) D A B E C (2) C E D B A (2) C D E A B (2) C B E A D (2) A D B C E (2) E D B C A (1) E D A B C (1) E B D C A (1) E B D A C (1) E A B D C (1) D E C A B (1) D E A B C (1) C D B A E (1) C D A E B (1) C B E D A (1) C B A D E (1) C A D B E (1) B E A D C (1) B A E D C (1) B A C E D (1) A E B D C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -12 2 -10 B 10 0 -12 6 -12 C 12 12 0 6 -2 D -2 -6 -6 0 -8 E 10 12 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 2 -10 B 10 0 -12 6 -12 C 12 12 0 6 -2 D -2 -6 -6 0 -8 E 10 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=28 D=24 B=10 A=9 so A is eliminated. Round 2 votes counts: E=29 C=29 D=26 B=16 so B is eliminated. Round 3 votes counts: E=39 C=31 D=30 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:214 B:196 D:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 2 -10 B 10 0 -12 6 -12 C 12 12 0 6 -2 D -2 -6 -6 0 -8 E 10 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 2 -10 B 10 0 -12 6 -12 C 12 12 0 6 -2 D -2 -6 -6 0 -8 E 10 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 2 -10 B 10 0 -12 6 -12 C 12 12 0 6 -2 D -2 -6 -6 0 -8 E 10 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3051: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (15) B A D C E (12) C E D A B (10) E C D A B (9) E C B D A (9) D A C E B (6) A D B C E (6) D A C B E (5) D C A E B (4) B E C A D (4) B E A C D (4) E B C A D (3) D A B C E (3) E C B A D (2) C D A E B (2) A B D C E (2) B E D C A (1) B E A D C (1) B A E D C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 12 2 14 B 4 0 4 8 10 C -12 -4 0 -14 2 D -2 -8 14 0 12 E -14 -10 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 2 14 B 4 0 4 8 10 C -12 -4 0 -14 2 D -2 -8 14 0 12 E -14 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=23 D=18 C=12 A=9 so A is eliminated. Round 2 votes counts: B=40 D=25 E=23 C=12 so C is eliminated. Round 3 votes counts: B=40 E=33 D=27 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:212 D:208 C:186 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 2 14 B 4 0 4 8 10 C -12 -4 0 -14 2 D -2 -8 14 0 12 E -14 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 2 14 B 4 0 4 8 10 C -12 -4 0 -14 2 D -2 -8 14 0 12 E -14 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 2 14 B 4 0 4 8 10 C -12 -4 0 -14 2 D -2 -8 14 0 12 E -14 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3052: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) D C B E A (6) D C E B A (5) C D B A E (5) A E B C D (4) E A B C D (3) D E B C A (3) D B C E A (3) C D A E B (3) C A D B E (3) A E C D B (3) E D B C A (2) E D B A C (2) E B D A C (2) E B A D C (2) C D B E A (2) C A D E B (2) C A B D E (2) B E D C A (2) B E A D C (2) B D C A E (2) B A E D C (2) B A C D E (2) A E B D C (2) A C E D B (2) A C B E D (2) A C B D E (2) A B E C D (2) E D C A B (1) E D A B C (1) E B D C A (1) E A D C B (1) E A C D B (1) D E C B A (1) D B E C A (1) C D E B A (1) C D E A B (1) C D A B E (1) C B D A E (1) B E D A C (1) B D E C A (1) B D C E A (1) B A E C D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 2 -14 B 2 0 8 -2 -10 C 0 -8 0 -10 -4 D -2 2 10 0 -2 E 14 10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 2 -14 B 2 0 8 -2 -10 C 0 -8 0 -10 -4 D -2 2 10 0 -2 E 14 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=21 D=19 A=19 B=14 so B is eliminated. Round 2 votes counts: E=32 A=24 D=23 C=21 so C is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:204 B:199 A:193 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 2 -14 B 2 0 8 -2 -10 C 0 -8 0 -10 -4 D -2 2 10 0 -2 E 14 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 -14 B 2 0 8 -2 -10 C 0 -8 0 -10 -4 D -2 2 10 0 -2 E 14 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 -14 B 2 0 8 -2 -10 C 0 -8 0 -10 -4 D -2 2 10 0 -2 E 14 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3053: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (13) E D C B A (8) B A D C E (6) E D B C A (5) D B A E C (5) C A E B D (5) C A B E D (5) E D C A B (4) E C D B A (4) D E B A C (4) D B E A C (4) A C B D E (4) D E B C A (3) C E A D B (3) C E A B D (3) B A C D E (3) E C B A D (2) D B A C E (2) B D A E C (2) A B D C E (2) E C D A B (1) E C A B D (1) E B D C A (1) D A C E B (1) D A B E C (1) B E D A C (1) B E C A D (1) B E A C D (1) B D E A C (1) B D A C E (1) B C A E D (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 6 4 6 B 12 0 16 10 10 C -6 -16 0 -2 2 D -4 -10 2 0 4 E -6 -10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 4 6 B 12 0 16 10 10 C -6 -16 0 -2 2 D -4 -10 2 0 4 E -6 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=20 A=20 B=18 C=16 so C is eliminated. Round 2 votes counts: E=32 A=30 D=20 B=18 so B is eliminated. Round 3 votes counts: A=41 E=35 D=24 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:224 A:202 D:196 C:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 4 6 B 12 0 16 10 10 C -6 -16 0 -2 2 D -4 -10 2 0 4 E -6 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 4 6 B 12 0 16 10 10 C -6 -16 0 -2 2 D -4 -10 2 0 4 E -6 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 4 6 B 12 0 16 10 10 C -6 -16 0 -2 2 D -4 -10 2 0 4 E -6 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3054: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) D C A E B (5) D A C E B (5) C D A E B (5) B E A C D (5) A C D B E (5) B E A D C (4) B A E C D (4) A D C B E (4) E D B C A (3) E B D C A (3) C D E A B (3) B E C A D (3) A C B E D (3) E D C B A (2) E C B D A (2) E B D A C (2) E B C A D (2) D C E A B (2) C E B A D (2) C A D B E (2) B E D A C (2) A B E D C (2) A B D C E (2) A B C E D (2) E C B A D (1) D E C B A (1) D E A C B (1) D E A B C (1) D B E A C (1) D A E B C (1) D A C B E (1) C E B D A (1) C A E B D (1) C A D E B (1) B A E D C (1) B A C E D (1) A D B E C (1) A D B C E (1) A C B D E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 6 2 2 B -4 0 4 10 -4 C -6 -4 0 8 -4 D -2 -10 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 2 2 B -4 0 4 10 -4 C -6 -4 0 8 -4 D -2 -10 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995255 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 B=20 D=18 C=15 so C is eliminated. Round 2 votes counts: A=28 E=26 D=26 B=20 so B is eliminated. Round 3 votes counts: E=40 A=34 D=26 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:207 B:203 C:197 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 2 2 B -4 0 4 10 -4 C -6 -4 0 8 -4 D -2 -10 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995255 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 2 2 B -4 0 4 10 -4 C -6 -4 0 8 -4 D -2 -10 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995255 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 2 2 B -4 0 4 10 -4 C -6 -4 0 8 -4 D -2 -10 -8 0 -10 E -2 4 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995255 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3055: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (16) E B A C D (14) D C A E B (14) B E A C D (11) B E D A C (4) E B A D C (3) D E B C A (3) D B E C A (3) D B C A E (3) C D A B E (3) C A B D E (3) E B D A C (2) E A B C D (2) D E C A B (2) C A D B E (2) A C E D B (2) A C E B D (2) E D B A C (1) E D A C B (1) E A B D C (1) D C E A B (1) D C B A E (1) D B C E A (1) C A D E B (1) B A E C D (1) B A C E D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -6 -10 2 B -4 0 2 -8 2 C 6 -2 0 -12 2 D 10 8 12 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -10 2 B -4 0 2 -8 2 C 6 -2 0 -12 2 D 10 8 12 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 E=24 B=17 C=9 A=6 so A is eliminated. Round 2 votes counts: D=44 E=24 B=18 C=14 so C is eliminated. Round 3 votes counts: D=50 E=28 B=22 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:197 B:196 A:195 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 -10 2 B -4 0 2 -8 2 C 6 -2 0 -12 2 D 10 8 12 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -10 2 B -4 0 2 -8 2 C 6 -2 0 -12 2 D 10 8 12 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -10 2 B -4 0 2 -8 2 C 6 -2 0 -12 2 D 10 8 12 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3056: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (13) E B D A C (11) B E A D C (6) E D B A C (5) E C D A B (5) E B C A D (5) B A D C E (5) D A C B E (4) C E A D B (3) C D A E B (3) B E D A C (3) E D B C A (2) E C A D B (2) E B A D C (2) D C A E B (2) C E D A B (2) C D A B E (2) C A E D B (2) C A E B D (2) C A D E B (2) B D A E C (2) B D A C E (2) B A C D E (2) E D C B A (1) E D A C B (1) E C D B A (1) E C B A D (1) E B D C A (1) D E B A C (1) D E A B C (1) D A B C E (1) C A B D E (1) B E A C D (1) B C A E D (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -2 0 -8 B 4 0 4 -8 -10 C 2 -4 0 -4 -2 D 0 8 4 0 -14 E 8 10 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 0 -8 B 4 0 4 -8 -10 C 2 -4 0 -4 -2 D 0 8 4 0 -14 E 8 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=30 B=22 D=9 A=2 so A is eliminated. Round 2 votes counts: E=37 C=30 B=23 D=10 so D is eliminated. Round 3 votes counts: E=39 C=37 B=24 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:199 C:196 B:195 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 0 -8 B 4 0 4 -8 -10 C 2 -4 0 -4 -2 D 0 8 4 0 -14 E 8 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 0 -8 B 4 0 4 -8 -10 C 2 -4 0 -4 -2 D 0 8 4 0 -14 E 8 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 0 -8 B 4 0 4 -8 -10 C 2 -4 0 -4 -2 D 0 8 4 0 -14 E 8 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3057: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) D E C B A (8) B D E C A (7) D E B C A (6) A C E D B (6) A B C E D (6) B D E A C (5) E D A C B (4) A C E B D (4) A C B E D (4) C E D A B (3) C A E D B (3) B D A E C (3) B C A D E (3) B A D C E (3) A E D C B (3) E D C B A (2) D E B A C (2) D B E C A (2) C B A E D (2) B A D E C (2) A E D B C (2) E D C A B (1) E C D A B (1) D B E A C (1) C D E B A (1) C A B E D (1) B C D E A (1) B C D A E (1) B C A E D (1) A E C D B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 14 4 12 B 18 0 12 6 6 C -14 -12 0 -6 -4 D -4 -6 6 0 10 E -12 -6 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 14 4 12 B 18 0 12 6 6 C -14 -12 0 -6 -4 D -4 -6 6 0 10 E -12 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=28 D=19 C=10 E=8 so E is eliminated. Round 2 votes counts: B=35 A=28 D=26 C=11 so C is eliminated. Round 3 votes counts: B=37 A=32 D=31 so D is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:206 D:203 E:188 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 14 4 12 B 18 0 12 6 6 C -14 -12 0 -6 -4 D -4 -6 6 0 10 E -12 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 14 4 12 B 18 0 12 6 6 C -14 -12 0 -6 -4 D -4 -6 6 0 10 E -12 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 14 4 12 B 18 0 12 6 6 C -14 -12 0 -6 -4 D -4 -6 6 0 10 E -12 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3058: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) E A B C D (5) D C B E A (5) D C B A E (5) B C E A D (5) E B A C D (4) D E A B C (4) D C A B E (4) C B A E D (4) D E B C A (3) D A E B C (3) D A C B E (3) C B E A D (3) B E C A D (3) A E D B C (3) A C B E D (3) E D A B C (2) E B D C A (2) E B C A D (2) E A B D C (2) D E B A C (2) D C E B A (2) D A E C B (2) D A C E B (2) C D B E A (2) C D B A E (2) A D E B C (2) E B A D C (1) C B E D A (1) C B D A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B C A E D (1) B A C E D (1) A E C B D (1) A D E C B (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 4 10 4 B 0 0 10 4 -4 C -4 -10 0 2 -4 D -10 -4 -2 0 -8 E -4 4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.743266 B: 0.256734 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.618356978545 Cumulative probabilities = A: 0.743266 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 10 4 B 0 0 10 4 -4 C -4 -10 0 2 -4 D -10 -4 -2 0 -8 E -4 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.499863 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037665 Cumulative probabilities = A: 0.500137 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=21 E=18 C=16 B=10 so B is eliminated. Round 2 votes counts: D=35 C=22 A=22 E=21 so E is eliminated. Round 3 votes counts: D=39 A=34 C=27 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:206 B:205 C:192 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 10 4 B 0 0 10 4 -4 C -4 -10 0 2 -4 D -10 -4 -2 0 -8 E -4 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.499863 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037665 Cumulative probabilities = A: 0.500137 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 4 B 0 0 10 4 -4 C -4 -10 0 2 -4 D -10 -4 -2 0 -8 E -4 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.499863 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037665 Cumulative probabilities = A: 0.500137 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 4 B 0 0 10 4 -4 C -4 -10 0 2 -4 D -10 -4 -2 0 -8 E -4 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500137 B: 0.499863 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000037665 Cumulative probabilities = A: 0.500137 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3059: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) E A B C D (7) B A E D C (7) A E B C D (7) C D E A B (5) A B C D E (5) D C E B A (4) B E A D C (4) E C D A B (3) E A B D C (3) D C B A E (3) C D B A E (3) B D A E C (3) B A D C E (3) E D B C A (2) E A C D B (2) E A C B D (2) D E C B A (2) D B C E A (2) C D A E B (2) C D A B E (2) C A D B E (2) B D A C E (2) B A C D E (2) A E C B D (2) A B E C D (2) E D C B A (1) E B D A C (1) D B C A E (1) C E D A B (1) C B A D E (1) C A D E B (1) B E D A C (1) B A E C D (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 30 24 2 B 4 0 26 26 -8 C -30 -26 0 4 -18 D -24 -26 -4 0 -12 E -2 8 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428675 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 -4 30 24 2 B 4 0 26 26 -8 C -30 -26 0 4 -18 D -24 -26 -4 0 -12 E -2 8 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571432994 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=23 A=18 C=17 D=12 so D is eliminated. Round 2 votes counts: E=32 B=26 C=24 A=18 so A is eliminated. Round 3 votes counts: E=41 B=34 C=25 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:226 B:224 E:218 D:167 C:165 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 30 24 2 B 4 0 26 26 -8 C -30 -26 0 4 -18 D -24 -26 -4 0 -12 E -2 8 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571432994 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 30 24 2 B 4 0 26 26 -8 C -30 -26 0 4 -18 D -24 -26 -4 0 -12 E -2 8 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571432994 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 30 24 2 B 4 0 26 26 -8 C -30 -26 0 4 -18 D -24 -26 -4 0 -12 E -2 8 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.428571432994 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3060: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) A E C B D (8) C A E D B (7) E A C D B (6) B D E A C (6) C D E A B (5) B D A E C (5) D C E A B (4) B A E D C (4) A E C D B (4) E A B C D (3) D B E C A (3) B D E C A (3) B A E C D (3) A E B C D (3) A C E D B (3) E C A D B (2) D C B A E (2) D B C A E (2) C E D A B (2) B E A D C (2) B D A C E (2) B A C E D (2) A C E B D (2) E B A D C (1) E A C B D (1) D C E B A (1) C D B A E (1) C D A E B (1) C A E B D (1) B D C A E (1) B C A D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 12 8 6 B -6 0 0 -2 -8 C -12 0 0 12 -8 D -8 2 -12 0 -10 E -6 8 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 8 6 B -6 0 0 -2 -8 C -12 0 0 12 -8 D -8 2 -12 0 -10 E -6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=21 D=20 C=17 E=13 so E is eliminated. Round 2 votes counts: A=31 B=30 D=20 C=19 so C is eliminated. Round 3 votes counts: A=41 B=30 D=29 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:210 C:196 B:192 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 8 6 B -6 0 0 -2 -8 C -12 0 0 12 -8 D -8 2 -12 0 -10 E -6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 8 6 B -6 0 0 -2 -8 C -12 0 0 12 -8 D -8 2 -12 0 -10 E -6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 8 6 B -6 0 0 -2 -8 C -12 0 0 12 -8 D -8 2 -12 0 -10 E -6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3061: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) D C B A E (6) A E D C B (6) A E C B D (6) E A B C D (5) D C A B E (5) B E C D A (5) B C E D A (5) E B A C D (4) D A C B E (4) D C B E A (3) D B C E A (3) C D B A E (3) C B D E A (3) A E D B C (3) E B C A D (2) C B A E D (2) B D E C A (2) B D C E A (2) A E B D C (2) A E B C D (2) A C E B D (2) E D A B C (1) E B D A C (1) E A B D C (1) D B E C A (1) D B E A C (1) D B A E C (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E D A (1) C B D A E (1) B E D C A (1) B E C A D (1) A E C D B (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 -8 -20 0 B 12 0 6 12 16 C 8 -6 0 2 0 D 20 -12 -2 0 -2 E 0 -16 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -20 0 B 12 0 6 12 16 C 8 -6 0 2 0 D 20 -12 -2 0 -2 E 0 -16 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 B=24 E=14 C=10 so C is eliminated. Round 2 votes counts: B=31 D=30 A=25 E=14 so E is eliminated. Round 3 votes counts: B=38 D=31 A=31 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:202 D:202 E:193 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -20 0 B 12 0 6 12 16 C 8 -6 0 2 0 D 20 -12 -2 0 -2 E 0 -16 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -20 0 B 12 0 6 12 16 C 8 -6 0 2 0 D 20 -12 -2 0 -2 E 0 -16 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -20 0 B 12 0 6 12 16 C 8 -6 0 2 0 D 20 -12 -2 0 -2 E 0 -16 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3062: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) E B A D C (6) B E D A C (6) A C D B E (6) A B D C E (6) E B D C A (5) C D A B E (4) B D A E C (4) E C B D A (3) E A B C D (3) D B C A E (3) C E D A B (3) C A D B E (3) B A D E C (3) A D C B E (3) A D B C E (3) E B C D A (2) E A C B D (2) C E A D B (2) C D A E B (2) B E A D C (2) A E C D B (2) E C D B A (1) E C D A B (1) E A C D B (1) E A B D C (1) D C B A E (1) D B A C E (1) D A C B E (1) C E D B A (1) C D B E A (1) C D B A E (1) C A E D B (1) C A D E B (1) B E D C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A C E (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 24 -4 0 B 4 0 16 12 8 C -24 -16 0 -16 -4 D 4 -12 16 0 -2 E 0 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 24 -4 0 B 4 0 16 12 8 C -24 -16 0 -16 -4 D 4 -12 16 0 -2 E 0 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=23 B=20 C=19 D=6 so D is eliminated. Round 2 votes counts: E=32 B=24 A=24 C=20 so C is eliminated. Round 3 votes counts: E=38 A=35 B=27 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:208 D:203 E:199 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 24 -4 0 B 4 0 16 12 8 C -24 -16 0 -16 -4 D 4 -12 16 0 -2 E 0 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 24 -4 0 B 4 0 16 12 8 C -24 -16 0 -16 -4 D 4 -12 16 0 -2 E 0 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 24 -4 0 B 4 0 16 12 8 C -24 -16 0 -16 -4 D 4 -12 16 0 -2 E 0 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3063: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) E A D C B (8) C B D A E (7) D A E B C (6) C B E A D (6) E D A C B (5) E A D B C (5) B C A D E (5) C B E D A (4) B A C D E (4) E D A B C (3) E C A D B (3) D A B E C (3) C E B D A (3) C B A D E (3) A D E B C (3) D E A B C (2) C E B A D (2) C E A B D (2) B D C A E (2) B C D A E (2) B A D E C (2) B A D C E (2) A D B E C (2) E C A B D (1) D E A C B (1) C E D A B (1) C D E B A (1) C D E A B (1) C B D E A (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -4 0 -14 B -12 0 -14 -6 -12 C 4 14 0 8 -8 D 0 6 -8 0 -4 E 14 12 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -4 0 -14 B -12 0 -14 -6 -12 C 4 14 0 8 -8 D 0 6 -8 0 -4 E 14 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=31 B=17 D=12 A=7 so A is eliminated. Round 2 votes counts: E=34 C=31 B=18 D=17 so D is eliminated. Round 3 votes counts: E=46 C=31 B=23 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:209 A:197 D:197 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -4 0 -14 B -12 0 -14 -6 -12 C 4 14 0 8 -8 D 0 6 -8 0 -4 E 14 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 0 -14 B -12 0 -14 -6 -12 C 4 14 0 8 -8 D 0 6 -8 0 -4 E 14 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 0 -14 B -12 0 -14 -6 -12 C 4 14 0 8 -8 D 0 6 -8 0 -4 E 14 12 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3064: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (13) E D B C A (11) C A E D B (11) B D E A C (10) D E B A C (8) D B E A C (5) C A B D E (5) A B C D E (5) E D C B A (4) B D A E C (4) C E D A B (3) C A B E D (3) B A D E C (3) B A D C E (3) A B D C E (3) E D B A C (2) E C D A B (2) C E A D B (2) C A E B D (2) A C B E D (1) Total count = 100 A B C D E A 0 0 14 2 6 B 0 0 8 4 10 C -14 -8 0 -6 2 D -2 -4 6 0 18 E -6 -10 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.197676 B: 0.802324 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.682799900872 Cumulative probabilities = A: 0.197676 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 2 6 B 0 0 8 4 10 C -14 -8 0 -6 2 D -2 -4 6 0 18 E -6 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=22 B=20 E=19 D=13 so D is eliminated. Round 2 votes counts: E=27 C=26 B=25 A=22 so A is eliminated. Round 3 votes counts: C=40 B=33 E=27 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 B:211 D:209 C:187 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 2 6 B 0 0 8 4 10 C -14 -8 0 -6 2 D -2 -4 6 0 18 E -6 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 2 6 B 0 0 8 4 10 C -14 -8 0 -6 2 D -2 -4 6 0 18 E -6 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 2 6 B 0 0 8 4 10 C -14 -8 0 -6 2 D -2 -4 6 0 18 E -6 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3065: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) D E A C B (6) C E B D A (6) A D E B C (6) A B C D E (5) D E C A B (4) D C E A B (4) D A E C B (4) C D E B A (4) E B D C A (3) D C A E B (3) C D E A B (3) B C E D A (3) B C A E D (3) C D A E B (2) C A D B E (2) B E C A D (2) B E A D C (2) B C E A D (2) B A E D C (2) B A C E D (2) E D C B A (1) E D C A B (1) E D A C B (1) E D A B C (1) E B D A C (1) E B C D A (1) D A C E B (1) C E D B A (1) C D A B E (1) C B E D A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B A E C D (1) B A C D E (1) A D C E B (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -8 -12 -4 B -14 0 -16 -8 -24 C 8 16 0 -6 2 D 12 8 6 0 20 E 4 24 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 -12 -4 B -14 0 -16 -8 -24 C 8 16 0 -6 2 D 12 8 6 0 20 E 4 24 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=22 B=22 A=22 E=9 so E is eliminated. Round 2 votes counts: B=27 D=26 C=25 A=22 so A is eliminated. Round 3 votes counts: D=40 B=35 C=25 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:210 E:203 A:195 B:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -8 -12 -4 B -14 0 -16 -8 -24 C 8 16 0 -6 2 D 12 8 6 0 20 E 4 24 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 -12 -4 B -14 0 -16 -8 -24 C 8 16 0 -6 2 D 12 8 6 0 20 E 4 24 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 -12 -4 B -14 0 -16 -8 -24 C 8 16 0 -6 2 D 12 8 6 0 20 E 4 24 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3066: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C A B D E (10) E D B A C (6) A E D C B (6) D B A E C (4) C B A D E (4) B D C E A (4) A C D B E (4) C A B E D (3) B D E C A (3) B D C A E (3) B C D E A (3) B C D A E (3) A E C D B (3) E B D A C (2) E A D C B (2) D E A B C (2) C B D A E (2) C A E D B (2) C A E B D (2) C A D B E (2) B E D C A (2) B D E A C (2) A C D E B (2) E C B A D (1) E B D C A (1) E B C D A (1) E A D B C (1) E A C D B (1) D B E A C (1) D B A C E (1) D A B E C (1) D A B C E (1) C B E D A (1) C B E A D (1) A D C E B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 10 4 -6 12 B -10 0 2 -4 12 C -4 -2 0 -8 2 D 6 4 8 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -6 12 B -10 0 2 -4 12 C -4 -2 0 -8 2 D 6 4 8 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=25 B=20 A=18 D=10 so D is eliminated. Round 2 votes counts: E=27 C=27 B=26 A=20 so A is eliminated. Round 3 votes counts: E=36 C=36 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:213 A:210 B:200 C:194 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -6 12 B -10 0 2 -4 12 C -4 -2 0 -8 2 D 6 4 8 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -6 12 B -10 0 2 -4 12 C -4 -2 0 -8 2 D 6 4 8 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -6 12 B -10 0 2 -4 12 C -4 -2 0 -8 2 D 6 4 8 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3067: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E C D B A (9) C E D A B (8) A B D C E (8) E C A B D (6) D E C B A (6) B D A E C (6) C E A D B (5) C E A B D (5) B A D E C (5) A B C D E (4) D B E C A (3) D B A C E (3) A B C E D (3) B A D C E (2) A C E B D (2) E D C B A (1) E C B A D (1) E C A D B (1) E B D C A (1) E B C A D (1) D C E B A (1) D B E A C (1) D B A E C (1) C E D B A (1) C D E B A (1) C A E D B (1) B E A D C (1) A C D B E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -22 -4 -24 B -12 0 -18 -6 -20 C 22 18 0 20 -10 D 4 6 -20 0 -16 E 24 20 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -22 -4 -24 B -12 0 -18 -6 -20 C 22 18 0 20 -10 D 4 6 -20 0 -16 E 24 20 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=21 A=20 D=15 B=14 so B is eliminated. Round 2 votes counts: E=31 A=27 D=21 C=21 so D is eliminated. Round 3 votes counts: E=41 A=37 C=22 so C is eliminated. Round 4 votes counts: E=62 A=38 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:235 C:225 D:187 A:181 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -22 -4 -24 B -12 0 -18 -6 -20 C 22 18 0 20 -10 D 4 6 -20 0 -16 E 24 20 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -22 -4 -24 B -12 0 -18 -6 -20 C 22 18 0 20 -10 D 4 6 -20 0 -16 E 24 20 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -22 -4 -24 B -12 0 -18 -6 -20 C 22 18 0 20 -10 D 4 6 -20 0 -16 E 24 20 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3068: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) B C E D A (11) E D A C B (9) A E D B C (7) D E A C B (4) C B D E A (4) C B D A E (4) A D E C B (4) E A D B C (3) C D A E B (3) B E C D A (3) A D E B C (3) E D C A B (2) E D A B C (2) E A D C B (2) D C A E B (2) D A E C B (2) C E D B A (2) C B A D E (2) E D C B A (1) E D B A C (1) E C D B A (1) E B D A C (1) D E C A B (1) C E B D A (1) C D A B E (1) C B E D A (1) C A B D E (1) B E A D C (1) B E A C D (1) B C D E A (1) B C A E D (1) B A E D C (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 -14 -4 B -2 0 0 -8 -6 C 8 0 0 2 0 D 14 8 -2 0 -2 E 4 6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.559728 D: 0.000000 E: 0.440272 Sum of squares = 0.507134895732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.559728 D: 0.559728 E: 1.000000 A B C D E A 0 2 -8 -14 -4 B -2 0 0 -8 -6 C 8 0 0 2 0 D 14 8 -2 0 -2 E 4 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=22 C=19 A=19 D=9 so D is eliminated. Round 2 votes counts: B=31 E=27 C=21 A=21 so C is eliminated. Round 3 votes counts: B=42 E=30 A=28 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:209 E:206 C:205 B:192 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -14 -4 B -2 0 0 -8 -6 C 8 0 0 2 0 D 14 8 -2 0 -2 E 4 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -14 -4 B -2 0 0 -8 -6 C 8 0 0 2 0 D 14 8 -2 0 -2 E 4 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -14 -4 B -2 0 0 -8 -6 C 8 0 0 2 0 D 14 8 -2 0 -2 E 4 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3069: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) D B E A C (7) B A E D C (6) C D E B A (5) D E C B A (4) D C E B A (4) C E D B A (4) A B E C D (4) A B D C E (4) E D B A C (3) C D E A B (3) C D A B E (3) C A E B D (3) A B D E C (3) A B C E D (3) A B C D E (3) E D C B A (2) E B D A C (2) E A B C D (2) D E B A C (2) C A D E B (2) B E A D C (2) B D E A C (2) A E B C D (2) A B E D C (2) E C D B A (1) E B A D C (1) E B A C D (1) D C B A E (1) D B E C A (1) D B A E C (1) C E D A B (1) C E A D B (1) C E A B D (1) C D A E B (1) C A E D B (1) C A B E D (1) C A B D E (1) B E D A C (1) B A D C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 6 -10 -14 B 16 0 20 -8 -8 C -6 -20 0 -12 -12 D 10 8 12 0 10 E 14 8 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 -10 -14 B 16 0 20 -8 -8 C -6 -20 0 -12 -12 D 10 8 12 0 10 E 14 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 A=22 E=12 B=12 so E is eliminated. Round 2 votes counts: D=32 C=28 A=24 B=16 so B is eliminated. Round 3 votes counts: D=37 A=35 C=28 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:212 B:210 A:183 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 6 -10 -14 B 16 0 20 -8 -8 C -6 -20 0 -12 -12 D 10 8 12 0 10 E 14 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -10 -14 B 16 0 20 -8 -8 C -6 -20 0 -12 -12 D 10 8 12 0 10 E 14 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -10 -14 B 16 0 20 -8 -8 C -6 -20 0 -12 -12 D 10 8 12 0 10 E 14 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3070: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) E D A B C (8) E C A B D (6) D E B A C (6) D B A C E (6) D B A E C (5) C D B A E (5) B A C D E (5) E D C A B (4) E A B C D (4) C B A D E (4) E D B A C (3) D E C B A (3) D E B C A (3) D C B A E (3) B A D C E (3) A B C E D (3) E A D B C (2) D B C A E (2) C A B E D (2) A C B E D (2) E D B C A (1) E C D A B (1) D B E A C (1) C D E B A (1) C B D A E (1) C B A E D (1) C A B D E (1) B D A C E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -28 4 -28 -6 B 28 0 12 -28 -4 C -4 -12 0 -24 -18 D 28 28 24 0 4 E 6 4 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 4 -28 -6 B 28 0 12 -28 -4 C -4 -12 0 -24 -18 D 28 28 24 0 4 E 6 4 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=29 C=15 B=9 A=8 so A is eliminated. Round 2 votes counts: E=39 D=29 C=17 B=15 so B is eliminated. Round 3 votes counts: E=40 D=34 C=26 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:242 E:212 B:204 A:171 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -28 4 -28 -6 B 28 0 12 -28 -4 C -4 -12 0 -24 -18 D 28 28 24 0 4 E 6 4 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 4 -28 -6 B 28 0 12 -28 -4 C -4 -12 0 -24 -18 D 28 28 24 0 4 E 6 4 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 4 -28 -6 B 28 0 12 -28 -4 C -4 -12 0 -24 -18 D 28 28 24 0 4 E 6 4 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999980743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3071: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (8) D C A B E (6) D B E C A (6) A C B E D (6) C A E B D (5) E D B C A (4) E B D A C (4) D B C A E (4) C A D E B (4) B A C D E (4) A C E B D (4) E C A D B (3) C A E D B (3) C A B D E (3) B E D A C (3) B D A C E (3) B A C E D (3) A C B D E (3) E D B A C (2) E A C B D (2) D E B C A (2) D B E A C (2) C D A B E (2) A B C D E (2) E D C A B (1) E B A C D (1) E A D C B (1) D E C B A (1) D E C A B (1) D C E A B (1) C E A D B (1) C A D B E (1) B E A C D (1) B A D C E (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 4 0 12 B -2 0 2 10 18 C -4 -2 0 0 14 D 0 -10 0 0 8 E -12 -18 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.904624 B: 0.000000 C: 0.000000 D: 0.095376 E: 0.000000 Sum of squares = 0.827441446017 Cumulative probabilities = A: 0.904624 B: 0.904624 C: 0.904624 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 0 12 B -2 0 2 10 18 C -4 -2 0 0 14 D 0 -10 0 0 8 E -12 -18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222308329 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 C=19 E=18 A=17 so A is eliminated. Round 2 votes counts: C=32 B=26 D=23 E=19 so E is eliminated. Round 3 votes counts: C=38 D=31 B=31 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:209 C:204 D:199 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 12 B -2 0 2 10 18 C -4 -2 0 0 14 D 0 -10 0 0 8 E -12 -18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222308329 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 12 B -2 0 2 10 18 C -4 -2 0 0 14 D 0 -10 0 0 8 E -12 -18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222308329 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 12 B -2 0 2 10 18 C -4 -2 0 0 14 D 0 -10 0 0 8 E -12 -18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222308329 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3072: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) D C A B E (7) C D A E B (6) C D A B E (6) D B C A E (5) C A D E B (5) D E B C A (4) C A D B E (4) E B D A C (3) D B E A C (3) C A E B D (3) A C B E D (3) E D C B A (2) E B A D C (2) E A B C D (2) D E C B A (2) D B E C A (2) D B C E A (2) B E D A C (2) B E A D C (2) B D E A C (2) B A E D C (2) A C E B D (2) A B C E D (2) E D B C A (1) E D B A C (1) E C D A B (1) E C A B D (1) E B C A D (1) E A C B D (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) D B A E C (1) C E A B D (1) C A E D B (1) B D A E C (1) B A D E C (1) A C D B E (1) A C B D E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -16 -10 12 B -4 0 -2 -16 4 C 16 2 0 0 10 D 10 16 0 0 18 E -12 -4 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.460990 D: 0.539010 E: 0.000000 Sum of squares = 0.503043620523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.460990 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -10 12 B -4 0 -2 -16 4 C 16 2 0 0 10 D 10 16 0 0 18 E -12 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=26 E=22 A=12 B=10 so B is eliminated. Round 2 votes counts: D=33 E=26 C=26 A=15 so A is eliminated. Round 3 votes counts: C=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:222 C:214 A:195 B:191 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 -10 12 B -4 0 -2 -16 4 C 16 2 0 0 10 D 10 16 0 0 18 E -12 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -10 12 B -4 0 -2 -16 4 C 16 2 0 0 10 D 10 16 0 0 18 E -12 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -10 12 B -4 0 -2 -16 4 C 16 2 0 0 10 D 10 16 0 0 18 E -12 -4 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3073: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) E B A D C (10) C D A B E (9) B D C E A (8) A E C D B (7) D C B A E (6) A C D E B (6) E A B D C (4) E A B C D (4) D B C A E (4) C D B A E (4) C A D E B (3) B D E C A (3) A D C E B (3) E A D C B (2) D C A B E (2) B D C A E (2) A C E D B (2) E B D A C (1) E B A C D (1) E A C D B (1) E A C B D (1) C B D A E (1) C A D B E (1) B E C D A (1) B E A D C (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 -12 -6 4 B 8 0 2 -2 8 C 12 -2 0 -16 4 D 6 2 16 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -6 4 B 8 0 2 -2 8 C 12 -2 0 -16 4 D 6 2 16 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=24 A=19 C=18 D=12 so D is eliminated. Round 2 votes counts: B=31 C=26 E=24 A=19 so A is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:215 B:208 C:199 A:189 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 -6 4 B 8 0 2 -2 8 C 12 -2 0 -16 4 D 6 2 16 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -6 4 B 8 0 2 -2 8 C 12 -2 0 -16 4 D 6 2 16 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -6 4 B 8 0 2 -2 8 C 12 -2 0 -16 4 D 6 2 16 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3074: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) E A D B C (6) D B A E C (6) C B E D A (6) C B D E A (6) B C D E A (6) A E D B C (6) A E D C B (5) C A E B D (4) A C E D B (4) A D E B C (3) E B D A C (2) D B E A C (2) D A E B C (2) C E B A D (2) C E A B D (2) C B E A D (2) C A E D B (2) B E D A C (2) B D C E A (2) A E C D B (2) A D E C B (2) E D B A C (1) E B A D C (1) E A D C B (1) E A C B D (1) E A B C D (1) D E A B C (1) D B C A E (1) D A B E C (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E A C (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 2 -4 6 B 6 0 -8 4 0 C -2 8 0 6 4 D 4 -4 -6 0 -6 E -6 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999904 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -4 6 B 6 0 -8 4 0 C -2 8 0 6 4 D 4 -4 -6 0 -6 E -6 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999733 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=23 E=13 D=13 B=13 so E is eliminated. Round 2 votes counts: C=38 A=32 B=16 D=14 so D is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:208 B:201 A:199 E:198 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 -4 6 B 6 0 -8 4 0 C -2 8 0 6 4 D 4 -4 -6 0 -6 E -6 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999733 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -4 6 B 6 0 -8 4 0 C -2 8 0 6 4 D 4 -4 -6 0 -6 E -6 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999733 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -4 6 B 6 0 -8 4 0 C -2 8 0 6 4 D 4 -4 -6 0 -6 E -6 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999733 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3075: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) A B C D E (5) E D C B A (4) E C D B A (4) E C A D B (4) D E B C A (4) D B A E C (4) C B A D E (4) E D B A C (3) D B C E A (3) C B D A E (3) C A B D E (3) B C D A E (3) A D B E C (3) A B D E C (3) E D B C A (2) E D A B C (2) E C D A B (2) C E D B A (2) C E B D A (2) C E A B D (2) C A E B D (2) B D C A E (2) B A C D E (2) A E D B C (2) A E B D C (2) A C B E D (2) A B D C E (2) E D C A B (1) E A D C B (1) E A C D B (1) D E B A C (1) D E A B C (1) D C B E A (1) D B E A C (1) C E D A B (1) C D B E A (1) C B D E A (1) C A B E D (1) B D C E A (1) B D A E C (1) B A D C E (1) A C E B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -6 0 -4 B 0 0 12 -10 -2 C 6 -12 0 -4 -6 D 0 10 4 0 2 E 4 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.249285 B: 0.000000 C: 0.000000 D: 0.750715 E: 0.000000 Sum of squares = 0.625716098956 Cumulative probabilities = A: 0.249285 B: 0.249285 C: 0.249285 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 0 -4 B 0 0 12 -10 -2 C 6 -12 0 -4 -6 D 0 10 4 0 2 E 4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555560373 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 A=22 D=15 B=10 so B is eliminated. Round 2 votes counts: E=31 C=25 A=25 D=19 so D is eliminated. Round 3 votes counts: E=38 C=32 A=30 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:208 E:205 B:200 A:195 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 0 -4 B 0 0 12 -10 -2 C 6 -12 0 -4 -6 D 0 10 4 0 2 E 4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555560373 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 0 -4 B 0 0 12 -10 -2 C 6 -12 0 -4 -6 D 0 10 4 0 2 E 4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555560373 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 0 -4 B 0 0 12 -10 -2 C 6 -12 0 -4 -6 D 0 10 4 0 2 E 4 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555560373 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3076: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) D A E C B (8) B E C D A (8) A D C E B (7) E C B D A (6) E D C A B (5) B C A E D (5) E B C D A (4) B A C D E (4) E B D C A (3) D E B A C (3) D A E B C (3) C B E A D (3) C B A E D (3) E D B C A (2) E C D B A (2) D E A C B (2) A D B E C (2) E D C B A (1) E D B A C (1) E D A B C (1) E C D A B (1) D E A B C (1) D A C E B (1) C E B D A (1) C E B A D (1) C A D E B (1) C A B E D (1) B E D C A (1) B D A E C (1) B C E D A (1) B A D C E (1) A D E C B (1) A D C B E (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -16 -12 -12 B 20 0 4 8 -12 C 16 -4 0 6 -12 D 12 -8 -6 0 -18 E 12 12 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -16 -12 -12 B 20 0 4 8 -12 C 16 -4 0 6 -12 D 12 -8 -6 0 -18 E 12 12 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 D=18 A=16 C=10 so C is eliminated. Round 2 votes counts: B=36 E=28 D=18 A=18 so D is eliminated. Round 3 votes counts: B=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:210 C:203 D:190 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -16 -12 -12 B 20 0 4 8 -12 C 16 -4 0 6 -12 D 12 -8 -6 0 -18 E 12 12 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 -12 -12 B 20 0 4 8 -12 C 16 -4 0 6 -12 D 12 -8 -6 0 -18 E 12 12 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 -12 -12 B 20 0 4 8 -12 C 16 -4 0 6 -12 D 12 -8 -6 0 -18 E 12 12 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3077: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) E D C A B (6) C B D A E (6) B C A D E (6) B A C D E (5) E C D B A (4) C D B A E (4) B C E D A (4) D C E A B (3) C B A D E (3) A B D C E (3) A B C D E (3) E D C B A (2) E D A B C (2) E B C D A (2) E A D B C (2) D E C A B (2) D E A C B (2) D C A E B (2) C D E B A (2) B C E A D (2) B A E C D (2) A E D B C (2) A D C B E (2) A D B E C (2) A B E D C (2) A B D E C (2) E D B C A (1) E C B D A (1) E B A D C (1) D C A B E (1) D A E C B (1) D A C E B (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D E A (1) B E A D C (1) B E A C D (1) B A E D C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -10 -16 2 B 2 0 -8 -6 10 C 10 8 0 -2 4 D 16 6 2 0 10 E -2 -10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -16 2 B 2 0 -8 -6 10 C 10 8 0 -2 4 D 16 6 2 0 10 E -2 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=22 C=19 A=18 D=12 so D is eliminated. Round 2 votes counts: E=33 C=25 B=22 A=20 so A is eliminated. Round 3 votes counts: E=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:217 C:210 B:199 A:187 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -10 -16 2 B 2 0 -8 -6 10 C 10 8 0 -2 4 D 16 6 2 0 10 E -2 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -16 2 B 2 0 -8 -6 10 C 10 8 0 -2 4 D 16 6 2 0 10 E -2 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -16 2 B 2 0 -8 -6 10 C 10 8 0 -2 4 D 16 6 2 0 10 E -2 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3078: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (13) E B C A D (7) B E C D A (7) E B C D A (6) C E B D A (6) A D C E B (6) E B A D C (5) C D A B E (5) A D B C E (4) E C B D A (3) E C B A D (3) B C E D A (3) E A B D C (2) D C A B E (2) C E A D B (2) C B E D A (2) C B D E A (2) C A D E B (2) A D E C B (2) A D E B C (2) A C D E B (2) E C A B D (1) E B A C D (1) D C B A E (1) D A C B E (1) C E D B A (1) C D E B A (1) C D B E A (1) C D A E B (1) B E D A C (1) B D A E C (1) B D A C E (1) B C D E A (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -6 -14 8 -10 B 6 0 -16 4 -10 C 14 16 0 14 14 D -8 -4 -14 0 -4 E 10 10 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 8 -10 B 6 0 -16 4 -10 C 14 16 0 14 14 D -8 -4 -14 0 -4 E 10 10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=28 C=23 B=14 D=4 so D is eliminated. Round 2 votes counts: A=32 E=28 C=26 B=14 so B is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:229 E:205 B:192 A:189 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 8 -10 B 6 0 -16 4 -10 C 14 16 0 14 14 D -8 -4 -14 0 -4 E 10 10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 8 -10 B 6 0 -16 4 -10 C 14 16 0 14 14 D -8 -4 -14 0 -4 E 10 10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 8 -10 B 6 0 -16 4 -10 C 14 16 0 14 14 D -8 -4 -14 0 -4 E 10 10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3079: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) A D C E B (6) C D E B A (5) C B E D A (5) B C E D A (5) E D B C A (4) A B E D C (4) D C A E B (3) C E B D A (3) C D E A B (3) C D A E B (3) B E D A C (3) B E A C D (3) E B D C A (2) E B D A C (2) D E C A B (2) D A E C B (2) C E D B A (2) C B A E D (2) C A B D E (2) B E A D C (2) B C A E D (2) B A C E D (2) A D B C E (2) A C D B E (2) A B D E C (2) E D C B A (1) E D B A C (1) E C B D A (1) D E C B A (1) D E A B C (1) D C E B A (1) D C E A B (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B A E D C (1) B A E C D (1) A D B E C (1) A C D E B (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -2 B 4 0 4 -4 -10 C 4 -4 0 -6 8 D 6 4 6 0 2 E 2 10 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -2 B 4 0 4 -4 -10 C 4 -4 0 -6 8 D 6 4 6 0 2 E 2 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 B=22 E=11 D=11 so E is eliminated. Round 2 votes counts: A=30 C=27 B=26 D=17 so D is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:209 C:201 E:201 B:197 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -2 B 4 0 4 -4 -10 C 4 -4 0 -6 8 D 6 4 6 0 2 E 2 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 4 -4 -10 C 4 -4 0 -6 8 D 6 4 6 0 2 E 2 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 4 -4 -10 C 4 -4 0 -6 8 D 6 4 6 0 2 E 2 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3080: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) E D A C B (7) D E A B C (7) C A B E D (7) E A D C B (5) C B A E D (5) B C D A E (5) E D A B C (4) D E B C A (4) E A C D B (3) D B C E A (3) D B C A E (3) A E D B C (3) A E C B D (3) A C B E D (3) E A D B C (2) D E B A C (2) D B E C A (2) C E B A D (2) C E A B D (2) C B D A E (2) B D C A E (2) E D C B A (1) E C A D B (1) D B A E C (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B D A (1) C B E A D (1) C A E B D (1) B D A C E (1) A E D C B (1) A C E B D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -2 4 6 B -10 0 -10 -4 -2 C 2 10 0 -6 2 D -4 4 6 0 -6 E -6 2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 4 6 B -10 0 -10 -4 -2 C 2 10 0 -6 2 D -4 4 6 0 -6 E -6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 E=23 A=14 B=8 so B is eliminated. Round 2 votes counts: C=35 D=28 E=23 A=14 so A is eliminated. Round 3 votes counts: C=40 E=30 D=30 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:209 C:204 D:200 E:200 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 4 6 B -10 0 -10 -4 -2 C 2 10 0 -6 2 D -4 4 6 0 -6 E -6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 4 6 B -10 0 -10 -4 -2 C 2 10 0 -6 2 D -4 4 6 0 -6 E -6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 4 6 B -10 0 -10 -4 -2 C 2 10 0 -6 2 D -4 4 6 0 -6 E -6 2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888812 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3081: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) D B C A E (7) E A C B D (6) A E D B C (5) A E C D B (5) A D C B E (5) E B C D A (4) E A B C D (4) D C B A E (4) C E B D A (4) A D E B C (4) E C B D A (3) E C B A D (3) E C A B D (3) D B C E A (3) D B A C E (3) A E C B D (3) E A B D C (2) C D B E A (2) C B E D A (2) B D E C A (2) B D C E A (2) A C E D B (2) E D B A C (1) E A D B C (1) D A B E C (1) D A B C E (1) C D B A E (1) C D A B E (1) C B D A E (1) B E D C A (1) A E B D C (1) A E B C D (1) A D C E B (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -2 -2 -2 B 2 0 -10 0 -6 C 2 10 0 8 -2 D 2 0 -8 0 -2 E 2 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -2 -2 -2 B 2 0 -10 0 -6 C 2 10 0 8 -2 D 2 0 -8 0 -2 E 2 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=27 D=19 C=19 B=5 so B is eliminated. Round 2 votes counts: A=30 E=28 D=23 C=19 so C is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:206 A:196 D:196 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 -2 -2 B 2 0 -10 0 -6 C 2 10 0 8 -2 D 2 0 -8 0 -2 E 2 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -2 -2 B 2 0 -10 0 -6 C 2 10 0 8 -2 D 2 0 -8 0 -2 E 2 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -2 -2 B 2 0 -10 0 -6 C 2 10 0 8 -2 D 2 0 -8 0 -2 E 2 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3082: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (7) B E C D A (6) A D C E B (6) A D C B E (6) C D A B E (5) C B D A E (4) A E D C B (4) A D E C B (4) E B D A C (3) E B C A D (3) E B A C D (3) C D B A E (3) C A D B E (3) B C A D E (3) A C D E B (3) E D A B C (2) E B C D A (2) E B A D C (2) E A D C B (2) D C A B E (2) D A C E B (2) D A C B E (2) C B A D E (2) B E C A D (2) E D B C A (1) E D B A C (1) E B D C A (1) E A D B C (1) E A C B D (1) E A B C D (1) D E A C B (1) D C B A E (1) D C A E B (1) D A E C B (1) C A B E D (1) B D E C A (1) B D C E A (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B C A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 12 6 14 26 B -12 0 -24 -18 8 C -6 24 0 10 16 D -14 18 -10 0 20 E -26 -8 -16 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 14 26 B -12 0 -24 -18 8 C -6 24 0 10 16 D -14 18 -10 0 20 E -26 -8 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=23 C=18 B=18 D=10 so D is eliminated. Round 2 votes counts: A=36 E=24 C=22 B=18 so B is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: A=63 E=37 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:222 D:207 B:177 E:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 14 26 B -12 0 -24 -18 8 C -6 24 0 10 16 D -14 18 -10 0 20 E -26 -8 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 14 26 B -12 0 -24 -18 8 C -6 24 0 10 16 D -14 18 -10 0 20 E -26 -8 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 14 26 B -12 0 -24 -18 8 C -6 24 0 10 16 D -14 18 -10 0 20 E -26 -8 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3083: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) D A E B C (5) A D C E B (5) E B C A D (4) D C A B E (4) D A C E B (4) C B E A D (4) B E C A D (4) A E C B D (4) D B E C A (3) C B E D A (3) C A E B D (3) E B A C D (2) D E B A C (2) D B E A C (2) B D E A C (2) B D C E A (2) B C E D A (2) A E D B C (2) A E C D B (2) A D E B C (2) A C E D B (2) A C E B D (2) E D B A C (1) E C A B D (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A B C (1) D C B E A (1) D C B A E (1) D B C E A (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C B E (1) D A B E C (1) D A B C E (1) C D B E A (1) C D B A E (1) C D A B E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B D E C A (1) B C E A D (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 2 -4 B 4 0 -2 -4 -12 C 2 2 0 -2 6 D -2 4 2 0 -2 E 4 12 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.186131 B: 0.000000 C: 0.274452 D: 0.451096 E: 0.088322 Sum of squares = 0.321256527708 Cumulative probabilities = A: 0.186131 B: 0.186131 C: 0.460583 D: 0.911678 E: 1.000000 A B C D E A 0 -4 -2 2 -4 B 4 0 -2 -4 -12 C 2 2 0 -2 6 D -2 4 2 0 -2 E 4 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.240740 B: 0.000000 C: 0.296296 D: 0.407408 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.240740 B: 0.240740 C: 0.537037 D: 0.944444 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=22 A=20 B=16 E=12 so E is eliminated. Round 2 votes counts: D=31 B=24 C=23 A=22 so A is eliminated. Round 3 votes counts: D=41 C=34 B=25 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:206 C:204 D:201 A:196 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 2 -4 B 4 0 -2 -4 -12 C 2 2 0 -2 6 D -2 4 2 0 -2 E 4 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.240740 B: 0.000000 C: 0.296296 D: 0.407408 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.240740 B: 0.240740 C: 0.537037 D: 0.944444 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 2 -4 B 4 0 -2 -4 -12 C 2 2 0 -2 6 D -2 4 2 0 -2 E 4 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.240740 B: 0.000000 C: 0.296296 D: 0.407408 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.240740 B: 0.240740 C: 0.537037 D: 0.944444 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 2 -4 B 4 0 -2 -4 -12 C 2 2 0 -2 6 D -2 4 2 0 -2 E 4 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.240740 B: 0.000000 C: 0.296296 D: 0.407408 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.240740 B: 0.240740 C: 0.537037 D: 0.944444 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3084: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) D A E C B (7) C B A E D (6) E D A C B (5) E B C D A (5) B C A E D (5) A C B D E (5) E C B A D (4) B C A D E (4) E D C B A (3) E D A B C (3) D E A B C (3) D A B C E (3) C A B E D (3) B E C D A (3) B D A C E (3) E B C A D (2) D E B A C (2) D E A C B (2) B E C A D (2) B C E A D (2) A D C B E (2) E D C A B (1) E D B A C (1) E B D C A (1) D B A E C (1) D B A C E (1) D A E B C (1) D A C B E (1) C E A B D (1) C B A D E (1) C A B D E (1) B E D C A (1) B D E C A (1) B D C E A (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E C B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -8 -12 0 B 16 0 8 6 -2 C 8 -8 0 -6 -14 D 12 -6 6 0 -12 E 0 2 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.055840 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.944160 Sum of squares = 0.894556524151 Cumulative probabilities = A: 0.055840 B: 0.055840 C: 0.055840 D: 0.055840 E: 1.000000 A B C D E A 0 -16 -8 -12 0 B 16 0 8 6 -2 C 8 -8 0 -6 -14 D 12 -6 6 0 -12 E 0 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469151736 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=24 D=21 C=12 A=11 so A is eliminated. Round 2 votes counts: E=33 D=24 B=24 C=19 so C is eliminated. Round 3 votes counts: B=40 E=34 D=26 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:214 D:200 C:190 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -8 -12 0 B 16 0 8 6 -2 C 8 -8 0 -6 -14 D 12 -6 6 0 -12 E 0 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469151736 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -12 0 B 16 0 8 6 -2 C 8 -8 0 -6 -14 D 12 -6 6 0 -12 E 0 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469151736 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -12 0 B 16 0 8 6 -2 C 8 -8 0 -6 -14 D 12 -6 6 0 -12 E 0 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469151736 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3085: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (7) E A D C B (6) E A D B C (6) C D A E B (6) B E A D C (6) C B D A E (5) B E A C D (5) B C D A E (5) E B A D C (4) B C E A D (4) E A B D C (3) D A E C B (3) D A C E B (3) A D C E B (3) D C A E B (2) C D B A E (2) C A D E B (2) B C E D A (2) B C A E D (2) E D A C B (1) E D A B C (1) E B D A C (1) E A B C D (1) D E A C B (1) D C E A B (1) D B E C A (1) C A D B E (1) B E D C A (1) B E C D A (1) B D C E A (1) B C A D E (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 6 6 -12 B 2 0 12 4 -6 C -6 -12 0 -6 0 D -6 -4 6 0 -4 E 12 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.202064 D: 0.000000 E: 0.797936 Sum of squares = 0.677531179485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.202064 D: 0.202064 E: 1.000000 A B C D E A 0 -2 6 6 -12 B 2 0 12 4 -6 C -6 -12 0 -6 0 D -6 -4 6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=23 C=16 A=12 D=11 so D is eliminated. Round 2 votes counts: B=39 E=24 C=19 A=18 so A is eliminated. Round 3 votes counts: B=39 E=36 C=25 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:206 A:199 D:196 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 6 -12 B 2 0 12 4 -6 C -6 -12 0 -6 0 D -6 -4 6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 -12 B 2 0 12 4 -6 C -6 -12 0 -6 0 D -6 -4 6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 -12 B 2 0 12 4 -6 C -6 -12 0 -6 0 D -6 -4 6 0 -4 E 12 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3086: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (16) A E B D C (10) A C D B E (9) C D B E A (8) B E D C A (7) A C D E B (7) A E D B C (4) A C B D E (4) E D B C A (3) C A D B E (3) E B D A C (2) E B A D C (2) D C E B A (2) C D A B E (2) C B D E A (2) B E C D A (2) A D C E B (2) A B E C D (2) A B C E D (2) E A B D C (1) D E C B A (1) D E B C A (1) D C B E A (1) D A E C B (1) C A B D E (1) B E D A C (1) B D E C A (1) B D C E A (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -2 -2 -2 B 0 0 12 10 -8 C 2 -12 0 -14 -12 D 2 -10 14 0 -6 E 2 8 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 -2 -2 B 0 0 12 10 -8 C 2 -12 0 -14 -12 D 2 -10 14 0 -6 E 2 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=24 C=16 B=12 D=6 so D is eliminated. Round 2 votes counts: A=43 E=26 C=19 B=12 so B is eliminated. Round 3 votes counts: A=43 E=37 C=20 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:207 D:200 A:197 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 -2 -2 B 0 0 12 10 -8 C 2 -12 0 -14 -12 D 2 -10 14 0 -6 E 2 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 -2 B 0 0 12 10 -8 C 2 -12 0 -14 -12 D 2 -10 14 0 -6 E 2 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 -2 B 0 0 12 10 -8 C 2 -12 0 -14 -12 D 2 -10 14 0 -6 E 2 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3087: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) C E A B D (7) B D A C E (7) E D C A B (4) D A B C E (4) A B D C E (4) E D A C B (3) E C D A B (3) E C A D B (3) D E B A C (3) D A B E C (3) C E B A D (3) C A E B D (3) B D C A E (3) A C B E D (3) E C D B A (2) E C A B D (2) D B E A C (2) D B A E C (2) B A D C E (2) B A C D E (2) A E C D B (2) A D E B C (2) A D B C E (2) A B C D E (2) E D C B A (1) E C B D A (1) E C B A D (1) E A D C B (1) D E B C A (1) D E A C B (1) D B E C A (1) D A E B C (1) C E B D A (1) C B E A D (1) C B A E D (1) B E C D A (1) B D C E A (1) B C E D A (1) B C D A E (1) B C A E D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 14 -12 12 B -4 0 10 -2 6 C -14 -10 0 -16 18 D 12 2 16 0 8 E -12 -6 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 -12 12 B -4 0 10 -2 6 C -14 -10 0 -16 18 D 12 2 16 0 8 E -12 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=21 B=19 A=17 C=16 so C is eliminated. Round 2 votes counts: E=32 D=27 B=21 A=20 so A is eliminated. Round 3 votes counts: E=38 D=32 B=30 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:209 B:205 C:189 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 14 -12 12 B -4 0 10 -2 6 C -14 -10 0 -16 18 D 12 2 16 0 8 E -12 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -12 12 B -4 0 10 -2 6 C -14 -10 0 -16 18 D 12 2 16 0 8 E -12 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -12 12 B -4 0 10 -2 6 C -14 -10 0 -16 18 D 12 2 16 0 8 E -12 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3088: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) A E C B D (7) C E A D B (6) C A E B D (6) A E B D C (6) D B C E A (5) D B A E C (5) E A D B C (4) C E D B A (4) C D B E A (4) B D C A E (4) E C A D B (3) E A C D B (3) D B E A C (3) C E A B D (3) C B D A E (3) A B D E C (3) C E D A B (2) C A B E D (2) B D A C E (2) B A D E C (2) E D B C A (1) E C D A B (1) E A D C B (1) E A C B D (1) E A B D C (1) D E C B A (1) D B E C A (1) D B C A E (1) D B A C E (1) C D E B A (1) C D B A E (1) C B A D E (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 0 2 10 B -2 0 0 2 -4 C 0 0 0 -4 -8 D -2 -2 4 0 -6 E -10 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.772329 B: 0.000000 C: 0.227671 D: 0.000000 E: 0.000000 Sum of squares = 0.648326178378 Cumulative probabilities = A: 0.772329 B: 0.772329 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 10 B -2 0 0 2 -4 C 0 0 0 -4 -8 D -2 -2 4 0 -6 E -10 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556351 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=18 D=17 B=17 E=15 so E is eliminated. Round 2 votes counts: C=37 A=28 D=18 B=17 so B is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:207 E:204 B:198 D:197 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 2 10 B -2 0 0 2 -4 C 0 0 0 -4 -8 D -2 -2 4 0 -6 E -10 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556351 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 10 B -2 0 0 2 -4 C 0 0 0 -4 -8 D -2 -2 4 0 -6 E -10 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556351 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 10 B -2 0 0 2 -4 C 0 0 0 -4 -8 D -2 -2 4 0 -6 E -10 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556351 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3089: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (9) C D E A B (7) C D B E A (6) B A E D C (5) A B E D C (5) E D A B C (4) D E C B A (4) C D E B A (4) C B D A E (4) E D C A B (3) C D B A E (3) C A D E B (3) A E B D C (3) E D B A C (2) E B A D C (2) C D A E B (2) B D C E A (2) B A E C D (2) A E B C D (2) E D C B A (1) E D B C A (1) E D A C B (1) E A D B C (1) E A B D C (1) D E C A B (1) D E B C A (1) D C B E A (1) D B E C A (1) D B C E A (1) C B D E A (1) C A D B E (1) C A B D E (1) B E A D C (1) B C D A E (1) B A C E D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -16 -20 -10 B 8 0 -6 -14 -4 C 16 6 0 0 -2 D 20 14 0 0 8 E 10 4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.313844 D: 0.686156 E: 0.000000 Sum of squares = 0.569308145678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.313844 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -20 -10 B 8 0 -6 -14 -4 C 16 6 0 0 -2 D 20 14 0 0 8 E 10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=21 D=18 E=16 B=13 so B is eliminated. Round 2 votes counts: C=33 A=30 D=20 E=17 so E is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:221 C:210 E:204 B:192 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -16 -20 -10 B 8 0 -6 -14 -4 C 16 6 0 0 -2 D 20 14 0 0 8 E 10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -20 -10 B 8 0 -6 -14 -4 C 16 6 0 0 -2 D 20 14 0 0 8 E 10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -20 -10 B 8 0 -6 -14 -4 C 16 6 0 0 -2 D 20 14 0 0 8 E 10 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3090: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (14) B D E A C (10) E D B C A (8) E C D B A (8) C A E D B (6) C E A D B (5) C A E B D (5) A C E D B (5) A C B D E (5) B D E C A (4) B D A E C (4) A C E B D (3) E B D C A (2) D B E C A (2) D B A E C (2) A C D B E (2) A B D C E (2) A B C D E (2) E C D A B (1) E C B D A (1) E B C D A (1) E A D C B (1) D E B C A (1) C A B E D (1) C A B D E (1) B A D C E (1) A D B C E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 8 -16 -16 B 16 0 10 -12 6 C -8 -10 0 -6 -20 D 16 12 6 0 4 E 16 -6 20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 8 -16 -16 B 16 0 10 -12 6 C -8 -10 0 -6 -20 D 16 12 6 0 4 E 16 -6 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=22 A=22 D=19 B=19 C=18 so C is eliminated. Round 2 votes counts: A=35 E=27 D=19 B=19 so D is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:219 E:213 B:210 A:180 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 8 -16 -16 B 16 0 10 -12 6 C -8 -10 0 -6 -20 D 16 12 6 0 4 E 16 -6 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 8 -16 -16 B 16 0 10 -12 6 C -8 -10 0 -6 -20 D 16 12 6 0 4 E 16 -6 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 8 -16 -16 B 16 0 10 -12 6 C -8 -10 0 -6 -20 D 16 12 6 0 4 E 16 -6 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3091: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C A D B E (9) B D E A C (7) C A E D B (6) A C D E B (6) D B E A C (5) C A E B D (4) B E D C A (4) A C E D B (4) E D B A C (3) B E C D A (3) B D E C A (3) B D C A E (3) D E B A C (2) D B A E C (2) D B A C E (2) D A C B E (2) C A B D E (2) B C D A E (2) B C A D E (2) A E C D B (2) E C A B D (1) E B C A D (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D E A B C (1) D C A B E (1) D A B C E (1) C E B A D (1) C B A D E (1) B E D A C (1) B D C E A (1) B D A C E (1) B C E A D (1) A E D C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -10 10 -6 4 B 10 0 12 -2 8 C -10 -12 0 -6 0 D 6 2 6 0 8 E -4 -8 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -6 4 B 10 0 12 -2 8 C -10 -12 0 -6 0 D 6 2 6 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 E=19 D=16 A=14 so A is eliminated. Round 2 votes counts: C=33 B=28 E=22 D=17 so D is eliminated. Round 3 votes counts: B=38 C=37 E=25 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:211 A:199 E:190 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 10 -6 4 B 10 0 12 -2 8 C -10 -12 0 -6 0 D 6 2 6 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -6 4 B 10 0 12 -2 8 C -10 -12 0 -6 0 D 6 2 6 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -6 4 B 10 0 12 -2 8 C -10 -12 0 -6 0 D 6 2 6 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3092: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (14) C D E B A (10) D C E B A (9) C D A B E (8) D C A B E (7) E B A D C (6) D C A E B (5) A B E D C (5) E B A C D (4) A D C B E (3) A C B E D (3) D E C B A (2) D C E A B (2) D A C B E (2) C A D B E (2) E D B A C (1) E B D C A (1) E B D A C (1) E B C D A (1) D E B A C (1) D A E B C (1) D A C E B (1) C E B D A (1) C D B E A (1) C D B A E (1) C B A E D (1) B E A C D (1) B A E C D (1) A D E B C (1) A D B E C (1) A D B C E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 16 -2 -10 18 B -16 0 -16 -18 6 C 2 16 0 -2 16 D 10 18 2 0 18 E -18 -6 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -10 18 B -16 0 -16 -18 6 C 2 16 0 -2 16 D 10 18 2 0 18 E -18 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=30 A=30 C=24 E=14 B=2 so B is eliminated. Round 2 votes counts: A=31 D=30 C=24 E=15 so E is eliminated. Round 3 votes counts: A=42 D=33 C=25 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:216 A:211 B:178 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -2 -10 18 B -16 0 -16 -18 6 C 2 16 0 -2 16 D 10 18 2 0 18 E -18 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -10 18 B -16 0 -16 -18 6 C 2 16 0 -2 16 D 10 18 2 0 18 E -18 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -10 18 B -16 0 -16 -18 6 C 2 16 0 -2 16 D 10 18 2 0 18 E -18 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3093: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) D E C B A (7) C E D A B (7) D B E C A (6) B A D E C (6) E D C B A (5) A C B E D (5) A B C D E (5) D E B C A (4) C A E D B (4) B D E A C (4) A C E B D (4) A B D C E (4) A B C E D (4) E C A D B (3) C E A D B (3) B D A E C (3) E C D B A (2) E C D A B (2) D B E A C (2) D B C E A (2) B D A C E (2) A B D E C (2) D C E B A (1) C E D B A (1) C A E B D (1) B A E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 4 4 2 B 10 0 10 2 12 C -4 -10 0 -16 4 D -4 -2 16 0 14 E -2 -12 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 4 2 B 10 0 10 2 12 C -4 -10 0 -16 4 D -4 -2 16 0 14 E -2 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=25 A=25 D=22 C=16 E=12 so E is eliminated. Round 2 votes counts: D=27 B=25 A=25 C=23 so C is eliminated. Round 3 votes counts: D=39 A=36 B=25 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:217 D:212 A:200 C:187 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 4 2 B 10 0 10 2 12 C -4 -10 0 -16 4 D -4 -2 16 0 14 E -2 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 4 2 B 10 0 10 2 12 C -4 -10 0 -16 4 D -4 -2 16 0 14 E -2 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 4 2 B 10 0 10 2 12 C -4 -10 0 -16 4 D -4 -2 16 0 14 E -2 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3094: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (14) D B A E C (12) E C A D B (9) E C D B A (8) C E A B D (6) B A D C E (5) D B E A C (4) A C E B D (4) C E D B A (3) C E B A D (3) E D B A C (2) E A D C B (2) D B A C E (2) C E B D A (2) C A E B D (2) C A B D E (2) B D A C E (2) E D B C A (1) E D A B C (1) E C D A B (1) E A C D B (1) D E B C A (1) D E B A C (1) D B E C A (1) D A B E C (1) C B D E A (1) C B A D E (1) C A B E D (1) B D C E A (1) B D C A E (1) A E D B C (1) A E C D B (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 12 10 4 B 2 0 2 -4 2 C -12 -2 0 -4 6 D -10 4 4 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.625000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.468749999987 Cumulative probabilities = A: 0.250000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 10 4 B 2 0 2 -4 2 C -12 -2 0 -4 6 D -10 4 4 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.625000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999971 Cumulative probabilities = A: 0.250000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 D=22 C=21 B=9 so B is eliminated. Round 2 votes counts: A=28 D=26 E=25 C=21 so C is eliminated. Round 3 votes counts: E=39 A=34 D=27 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:201 D:200 C:194 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 12 10 4 B 2 0 2 -4 2 C -12 -2 0 -4 6 D -10 4 4 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.625000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999971 Cumulative probabilities = A: 0.250000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 10 4 B 2 0 2 -4 2 C -12 -2 0 -4 6 D -10 4 4 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.625000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999971 Cumulative probabilities = A: 0.250000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 10 4 B 2 0 2 -4 2 C -12 -2 0 -4 6 D -10 4 4 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.625000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46874999971 Cumulative probabilities = A: 0.250000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3095: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) D A C B E (7) D C E A B (5) B E A D C (5) D C E B A (4) C D A E B (4) B E D A C (4) B E A C D (4) B A E D C (4) B A E C D (4) A D C B E (4) A C D B E (4) E C D B A (3) E C B D A (3) D C A B E (3) C D E A B (3) B A D E C (3) A B E C D (3) A B D C E (3) E D C B A (2) D E C B A (2) D C A E B (2) A C B E D (2) A B C D E (2) E C B A D (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) D B E A C (1) D B A E C (1) C E D A B (1) C E A D B (1) B D E A C (1) A D B C E (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 14 -10 -2 B 6 0 -4 4 14 C -14 4 0 -22 -6 D 10 -4 22 0 2 E 2 -14 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.733333 C: 0.133333 D: 0.133333 E: 0.000000 Sum of squares = 0.573333333409 Cumulative probabilities = A: 0.000000 B: 0.733333 C: 0.866667 D: 1.000000 E: 1.000000 A B C D E A 0 -6 14 -10 -2 B 6 0 -4 4 14 C -14 4 0 -22 -6 D 10 -4 22 0 2 E 2 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.733333 C: 0.133333 D: 0.133333 E: 0.000000 Sum of squares = 0.573333333576 Cumulative probabilities = A: 0.000000 B: 0.733333 C: 0.866667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 A=21 E=20 C=9 so C is eliminated. Round 2 votes counts: D=32 B=25 E=22 A=21 so A is eliminated. Round 3 votes counts: D=41 B=36 E=23 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:215 B:210 A:198 E:196 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 14 -10 -2 B 6 0 -4 4 14 C -14 4 0 -22 -6 D 10 -4 22 0 2 E 2 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.733333 C: 0.133333 D: 0.133333 E: 0.000000 Sum of squares = 0.573333333576 Cumulative probabilities = A: 0.000000 B: 0.733333 C: 0.866667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 -10 -2 B 6 0 -4 4 14 C -14 4 0 -22 -6 D 10 -4 22 0 2 E 2 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.733333 C: 0.133333 D: 0.133333 E: 0.000000 Sum of squares = 0.573333333576 Cumulative probabilities = A: 0.000000 B: 0.733333 C: 0.866667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 -10 -2 B 6 0 -4 4 14 C -14 4 0 -22 -6 D 10 -4 22 0 2 E 2 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.733333 C: 0.133333 D: 0.133333 E: 0.000000 Sum of squares = 0.573333333576 Cumulative probabilities = A: 0.000000 B: 0.733333 C: 0.866667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3096: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (13) C A B D E (10) B C A E D (10) E D B A C (7) E B D C A (7) E D A B C (6) A C B D E (6) D A C E B (5) D A C B E (5) E D B C A (3) E B C A D (3) D E C A B (3) D A E C B (3) D C A E B (2) B E C A D (2) B E A C D (2) B C E A D (2) B C A D E (2) A B C D E (2) E D C A B (1) E B A C D (1) D C E A B (1) C A D B E (1) B A C E D (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 18 6 -12 -2 B -18 0 0 0 -10 C -6 0 0 -14 -2 D 12 0 14 0 10 E 2 10 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.266544 C: 0.000000 D: 0.733456 E: 0.000000 Sum of squares = 0.609003045987 Cumulative probabilities = A: 0.000000 B: 0.266544 C: 0.266544 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 -12 -2 B -18 0 0 0 -10 C -6 0 0 -14 -2 D 12 0 14 0 10 E 2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000016641 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=28 B=20 C=11 A=9 so A is eliminated. Round 2 votes counts: D=32 E=28 B=23 C=17 so C is eliminated. Round 3 votes counts: B=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:218 A:205 E:202 C:189 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 6 -12 -2 B -18 0 0 0 -10 C -6 0 0 -14 -2 D 12 0 14 0 10 E 2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000016641 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 -12 -2 B -18 0 0 0 -10 C -6 0 0 -14 -2 D 12 0 14 0 10 E 2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000016641 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 -12 -2 B -18 0 0 0 -10 C -6 0 0 -14 -2 D 12 0 14 0 10 E 2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000016641 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3097: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) D E A B C (6) C D A B E (5) E D A B C (4) C D B A E (4) C B A E D (4) C B A D E (4) B A C E D (4) E D C A B (3) E C D B A (3) E C B A D (3) E B A D C (3) E B A C D (3) D A B E C (3) D A B C E (3) C E D B A (3) C E B A D (3) A B D C E (3) E B C A D (2) E A D B C (2) D E A C B (2) D C E A B (2) B A D C E (2) B A C D E (2) A B D E C (2) E D C B A (1) E D A C B (1) E C D A B (1) E A B D C (1) D E C A B (1) D C A E B (1) D A C E B (1) D A C B E (1) C E D A B (1) C D E B A (1) C B D A E (1) C A D B E (1) B E A D C (1) B C A E D (1) B A E D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -6 -12 6 B -8 0 -10 -18 4 C 6 10 0 -8 10 D 12 18 8 0 8 E -6 -4 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 -12 6 B -8 0 -10 -18 4 C 6 10 0 -8 10 D 12 18 8 0 8 E -6 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 C=27 B=11 A=7 so A is eliminated. Round 2 votes counts: D=29 E=27 C=27 B=17 so B is eliminated. Round 3 votes counts: D=36 C=34 E=30 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:209 A:198 E:186 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -6 -12 6 B -8 0 -10 -18 4 C 6 10 0 -8 10 D 12 18 8 0 8 E -6 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -12 6 B -8 0 -10 -18 4 C 6 10 0 -8 10 D 12 18 8 0 8 E -6 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -12 6 B -8 0 -10 -18 4 C 6 10 0 -8 10 D 12 18 8 0 8 E -6 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3098: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) B D E A C (10) C A E D B (9) A C D E B (8) A C E D B (7) B D E C A (6) D A C E B (5) B E C A D (5) A C D B E (4) B D A C E (3) A C E B D (3) E B C D A (2) D E C A B (2) D B E A C (2) D B A C E (2) B E C D A (2) B A C E D (2) A C B E D (2) E C A D B (1) E C A B D (1) E B D C A (1) D E B C A (1) D E A C B (1) D B E C A (1) C E A D B (1) C E A B D (1) C A E B D (1) B E A C D (1) B D A E C (1) B A C D E (1) A D C E B (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 12 2 4 B 0 0 2 8 10 C -12 -2 0 4 6 D -2 -8 -4 0 2 E -4 -10 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.425434 B: 0.574566 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.511120039004 Cumulative probabilities = A: 0.425434 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 2 4 B 0 0 2 8 10 C -12 -2 0 4 6 D -2 -8 -4 0 2 E -4 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 A=28 D=14 C=12 E=5 so E is eliminated. Round 2 votes counts: B=44 A=28 D=14 C=14 so D is eliminated. Round 3 votes counts: B=50 A=34 C=16 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:209 C:198 D:194 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 2 4 B 0 0 2 8 10 C -12 -2 0 4 6 D -2 -8 -4 0 2 E -4 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 2 4 B 0 0 2 8 10 C -12 -2 0 4 6 D -2 -8 -4 0 2 E -4 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 2 4 B 0 0 2 8 10 C -12 -2 0 4 6 D -2 -8 -4 0 2 E -4 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3099: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) A E D C B (8) C B D E A (7) B C D E A (7) A E C D B (7) A E D B C (6) B A D E C (5) E A C D B (4) E C A D B (3) C E A B D (3) A E C B D (3) A E B C D (3) C E D A B (2) B C D A E (2) B A C D E (2) A B D E C (2) E D A C B (1) E C D A B (1) E A D C B (1) D E C B A (1) D E C A B (1) D E B C A (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C E A (1) C E D B A (1) C E B D A (1) C E A D B (1) C D E B A (1) C D B E A (1) C B E D A (1) C B A E D (1) B D C A E (1) B D A E C (1) B D A C E (1) B C A D E (1) B A D C E (1) A E B D C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -2 8 -6 B 2 0 -2 12 -2 C 2 2 0 6 -4 D -8 -12 -6 0 0 E 6 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.083966 E: 0.916034 Sum of squares = 0.846168935301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.083966 E: 1.000000 A B C D E A 0 -2 -2 8 -6 B 2 0 -2 12 -2 C 2 2 0 6 -4 D -8 -12 -6 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102071908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=32 A=32 C=19 E=10 D=7 so D is eliminated. Round 2 votes counts: B=35 A=32 C=20 E=13 so E is eliminated. Round 3 votes counts: A=38 B=36 C=26 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:206 B:205 C:203 A:199 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 8 -6 B 2 0 -2 12 -2 C 2 2 0 6 -4 D -8 -12 -6 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102071908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 8 -6 B 2 0 -2 12 -2 C 2 2 0 6 -4 D -8 -12 -6 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102071908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 8 -6 B 2 0 -2 12 -2 C 2 2 0 6 -4 D -8 -12 -6 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102071908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3100: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C D A E B (10) B E C D A (9) B E A C D (9) A D C E B (9) D C A E B (6) A E B D C (5) C D E B A (4) E B A D C (3) B C D E A (3) A B E D C (3) E A B D C (2) D A C E B (2) C B D E A (2) B A E D C (2) B A C D E (2) A E D B C (2) A C D E B (2) A C D B E (2) E D A C B (1) E C D B A (1) C D E A B (1) C D A B E (1) C B A D E (1) B E D C A (1) B E C A D (1) B C E D A (1) B C A D E (1) A D E C B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 16 16 2 B 2 0 12 14 2 C -16 -12 0 0 -4 D -16 -14 0 0 -2 E -2 -2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 16 2 B 2 0 12 14 2 C -16 -12 0 0 -4 D -16 -14 0 0 -2 E -2 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=26 C=19 D=8 E=7 so E is eliminated. Round 2 votes counts: B=43 A=28 C=20 D=9 so D is eliminated. Round 3 votes counts: B=43 A=31 C=26 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:215 E:201 C:184 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 16 2 B 2 0 12 14 2 C -16 -12 0 0 -4 D -16 -14 0 0 -2 E -2 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 16 2 B 2 0 12 14 2 C -16 -12 0 0 -4 D -16 -14 0 0 -2 E -2 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 16 2 B 2 0 12 14 2 C -16 -12 0 0 -4 D -16 -14 0 0 -2 E -2 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3101: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) C B E D A (6) A D C E B (6) E B C D A (5) D A B E C (5) B E C D A (5) A C D B E (5) E B D C A (4) E B C A D (4) D B C E A (4) D A E B C (4) D A C B E (3) C E B A D (3) C A E B D (3) A D B E C (3) E C B A D (2) E B D A C (2) E B A C D (2) D C B A E (2) C D A B E (2) B E D C A (2) A E C B D (2) A D E B C (2) E D B A C (1) E A D B C (1) E A C B D (1) D C B E A (1) D C A B E (1) D B E C A (1) D B E A C (1) D A B C E (1) C D B E A (1) C B E A D (1) C B D A E (1) C A D B E (1) A E D C B (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 2 -4 6 B -4 0 -4 -10 8 C -2 4 0 -8 0 D 4 10 8 0 6 E -6 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -4 6 B -4 0 -4 -10 8 C -2 4 0 -8 0 D 4 10 8 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=23 E=22 C=18 B=7 so B is eliminated. Round 2 votes counts: A=30 E=29 D=23 C=18 so C is eliminated. Round 3 votes counts: E=39 A=34 D=27 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:204 C:197 B:195 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -4 6 B -4 0 -4 -10 8 C -2 4 0 -8 0 D 4 10 8 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -4 6 B -4 0 -4 -10 8 C -2 4 0 -8 0 D 4 10 8 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -4 6 B -4 0 -4 -10 8 C -2 4 0 -8 0 D 4 10 8 0 6 E -6 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3102: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (6) C A B D E (6) B E D C A (6) B C A D E (6) A C E B D (5) E D B A C (4) D E B A C (4) D E A C B (4) B D E C A (4) B C A E D (4) A C B E D (4) D E A B C (3) C B A E D (3) B E C D A (3) A D C E B (3) A C E D B (3) A C D E B (3) E D B C A (2) E D A B C (2) E A D C B (2) E A C B D (2) D B E C A (2) D A E C B (2) D A C E B (2) B C E A D (2) A C D B E (2) E B D C A (1) E B D A C (1) E A B C D (1) C A D B E (1) B E C A D (1) B D C A E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 8 0 14 10 B -8 0 -2 16 6 C 0 2 0 10 6 D -14 -16 -10 0 -8 E -10 -6 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.531432 B: 0.000000 C: 0.468568 D: 0.000000 E: 0.000000 Sum of squares = 0.501975991132 Cumulative probabilities = A: 0.531432 B: 0.531432 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 14 10 B -8 0 -2 16 6 C 0 2 0 10 6 D -14 -16 -10 0 -8 E -10 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 D=17 C=16 E=15 so E is eliminated. Round 2 votes counts: B=31 A=28 D=25 C=16 so C is eliminated. Round 3 votes counts: A=41 B=34 D=25 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:209 B:206 E:193 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 14 10 B -8 0 -2 16 6 C 0 2 0 10 6 D -14 -16 -10 0 -8 E -10 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 14 10 B -8 0 -2 16 6 C 0 2 0 10 6 D -14 -16 -10 0 -8 E -10 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 14 10 B -8 0 -2 16 6 C 0 2 0 10 6 D -14 -16 -10 0 -8 E -10 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3103: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) D A E B C (7) D A C B E (5) E C D B A (4) E C B D A (4) D E A C B (4) D A E C B (4) A D B C E (4) E D A B C (3) E C D A B (3) E B C A D (3) E B A D C (3) D C A E B (3) C B A D E (3) B E A D C (3) B A D E C (3) A D C B E (3) E D A C B (2) E B C D A (2) D A B E C (2) C B E A D (2) B E C A D (2) B C E A D (2) B C A D E (2) B A D C E (2) A D E B C (2) A B D C E (2) E D C A B (1) E D B A C (1) E C B A D (1) E B D C A (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D A B E (1) C B E D A (1) C A D B E (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 0 6 -18 -6 B 0 0 -6 -8 -18 C -6 6 0 -16 -14 D 18 8 16 0 0 E 6 18 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.437677 E: 0.562323 Sum of squares = 0.507768208291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.437677 E: 1.000000 A B C D E A 0 0 6 -18 -6 B 0 0 -6 -8 -18 C -6 6 0 -16 -14 D 18 8 16 0 0 E 6 18 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 C=19 B=15 A=12 so A is eliminated. Round 2 votes counts: D=36 E=28 C=19 B=17 so B is eliminated. Round 3 votes counts: D=43 E=34 C=23 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:219 A:191 C:185 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 -18 -6 B 0 0 -6 -8 -18 C -6 6 0 -16 -14 D 18 8 16 0 0 E 6 18 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -18 -6 B 0 0 -6 -8 -18 C -6 6 0 -16 -14 D 18 8 16 0 0 E 6 18 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -18 -6 B 0 0 -6 -8 -18 C -6 6 0 -16 -14 D 18 8 16 0 0 E 6 18 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3104: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) D B A C E (5) C D E B A (5) A E B D C (5) E C B A D (4) E C A D B (4) E A D C B (4) E A B C D (4) D C B A E (4) B D A C E (4) E C D A B (3) B C D A E (3) B A E D C (3) A D B E C (3) A B D E C (3) E D A C B (2) E C B D A (2) E A C D B (2) D C E A B (2) D B C A E (2) C E D B A (2) C E B D A (2) C B E D A (2) B D C A E (2) B C E A D (2) B C D E A (2) B A D C E (2) A E D B C (2) E B A C D (1) E A D B C (1) E A C B D (1) D C B E A (1) C B D E A (1) B C A D E (1) B A E C D (1) B A C E D (1) A D E B C (1) A D B C E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 -2 -2 -8 B 18 0 4 -2 6 C 2 -4 0 2 4 D 2 2 -2 0 0 E 8 -6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999996 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -2 -2 -8 B 18 0 4 -2 6 C 2 -4 0 2 4 D 2 2 -2 0 0 E 8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=21 C=19 A=18 D=14 so D is eliminated. Round 2 votes counts: E=28 B=28 C=26 A=18 so A is eliminated. Round 3 votes counts: B=38 E=36 C=26 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:202 D:201 E:199 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -2 -2 -8 B 18 0 4 -2 6 C 2 -4 0 2 4 D 2 2 -2 0 0 E 8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 -2 -8 B 18 0 4 -2 6 C 2 -4 0 2 4 D 2 2 -2 0 0 E 8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 -2 -8 B 18 0 4 -2 6 C 2 -4 0 2 4 D 2 2 -2 0 0 E 8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3105: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (6) A D E B C (6) D C E A B (5) C B D E A (5) B A C E D (5) A D B C E (5) A B D E C (5) E C D A B (4) C B E D A (4) B C E A D (4) B A E C D (4) D E C A B (3) D A E C B (3) C E B D A (3) C B D A E (3) A E D B C (3) E C B D A (2) D C A E B (2) D A C E B (2) C D E A B (2) B E C A D (2) B C A E D (2) B C A D E (2) B A E D C (2) E D C A B (1) E D A C B (1) E D A B C (1) E B C A D (1) E B A D C (1) E B A C D (1) E A B D C (1) D E A C B (1) C D E B A (1) C B E A D (1) C B A D E (1) B A D E C (1) B A D C E (1) B A C D E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -8 2 0 B 6 0 0 6 0 C 8 0 0 8 10 D -2 -6 -8 0 0 E 0 0 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.484498 C: 0.515502 D: 0.000000 E: 0.000000 Sum of squares = 0.500480607921 Cumulative probabilities = A: 0.000000 B: 0.484498 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 2 0 B 6 0 0 6 0 C 8 0 0 8 10 D -2 -6 -8 0 0 E 0 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 A=21 D=16 E=13 so E is eliminated. Round 2 votes counts: C=32 B=27 A=22 D=19 so D is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:206 E:195 A:194 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 2 0 B 6 0 0 6 0 C 8 0 0 8 10 D -2 -6 -8 0 0 E 0 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 2 0 B 6 0 0 6 0 C 8 0 0 8 10 D -2 -6 -8 0 0 E 0 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 2 0 B 6 0 0 6 0 C 8 0 0 8 10 D -2 -6 -8 0 0 E 0 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3106: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) A D B C E (10) A D E B C (9) C B E D A (7) D A C B E (6) D A E C B (5) E C B D A (4) E C B A D (4) C E B D A (4) E D C B A (3) D C B A E (3) D A B C E (3) B C E A D (3) E A B C D (2) D E A C B (2) D C A B E (2) C D B E A (2) A D B E C (2) A B D C E (2) A B C E D (2) A B C D E (2) E D A C B (1) D E C B A (1) D C E B A (1) D C B E A (1) D A E B C (1) D A C E B (1) D A B E C (1) C B D A E (1) B E C A D (1) B E A C D (1) B C A E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 2 6 B -4 0 4 -8 4 C -4 -4 0 -8 4 D -2 8 8 0 12 E -6 -4 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 2 6 B -4 0 4 -8 4 C -4 -4 0 -8 4 D -2 8 8 0 12 E -6 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=27 E=24 C=14 B=7 so B is eliminated. Round 2 votes counts: A=29 D=27 E=26 C=18 so C is eliminated. Round 3 votes counts: E=40 D=30 A=30 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:208 B:198 C:194 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 6 B -4 0 4 -8 4 C -4 -4 0 -8 4 D -2 8 8 0 12 E -6 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 6 B -4 0 4 -8 4 C -4 -4 0 -8 4 D -2 8 8 0 12 E -6 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 6 B -4 0 4 -8 4 C -4 -4 0 -8 4 D -2 8 8 0 12 E -6 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3107: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) E D B C A (6) A B C E D (6) D E C B A (5) C A D E B (5) B D E C A (5) A B E C D (5) C E D A B (4) B A D E C (4) A B E D C (4) B E D C A (3) B E D A C (3) B E A D C (3) B A E D C (3) A C E D B (3) A B D E C (3) A B C D E (3) D C E B A (2) D B E C A (2) C D E B A (2) B D A E C (2) A C D E B (2) A C B D E (2) E D C B A (1) E A B C D (1) D A C B E (1) C E D B A (1) C E A D B (1) C D A E B (1) C A E D B (1) B D E A C (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -2 -2 -4 B -14 0 10 0 6 C 2 -10 0 2 -2 D 2 0 -2 0 8 E 4 -6 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.298701 B: 0.090909 C: 0.402597 D: 0.181818 E: 0.025974 Sum of squares = 0.293304098493 Cumulative probabilities = A: 0.298701 B: 0.389610 C: 0.792208 D: 0.974026 E: 1.000000 A B C D E A 0 14 -2 -2 -4 B -14 0 10 0 6 C 2 -10 0 2 -2 D 2 0 -2 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.298701 B: 0.090909 C: 0.402597 D: 0.181818 E: 0.025974 Sum of squares = 0.293304098395 Cumulative probabilities = A: 0.298701 B: 0.389610 C: 0.792208 D: 0.974026 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=27 B=24 D=10 E=8 so E is eliminated. Round 2 votes counts: A=32 C=27 B=24 D=17 so D is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:204 A:203 B:201 C:196 E:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -2 -4 B -14 0 10 0 6 C 2 -10 0 2 -2 D 2 0 -2 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.298701 B: 0.090909 C: 0.402597 D: 0.181818 E: 0.025974 Sum of squares = 0.293304098395 Cumulative probabilities = A: 0.298701 B: 0.389610 C: 0.792208 D: 0.974026 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -2 -4 B -14 0 10 0 6 C 2 -10 0 2 -2 D 2 0 -2 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.298701 B: 0.090909 C: 0.402597 D: 0.181818 E: 0.025974 Sum of squares = 0.293304098395 Cumulative probabilities = A: 0.298701 B: 0.389610 C: 0.792208 D: 0.974026 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -2 -4 B -14 0 10 0 6 C 2 -10 0 2 -2 D 2 0 -2 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.298701 B: 0.090909 C: 0.402597 D: 0.181818 E: 0.025974 Sum of squares = 0.293304098395 Cumulative probabilities = A: 0.298701 B: 0.389610 C: 0.792208 D: 0.974026 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3108: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (15) E B A D C (9) D E C B A (9) D C E B A (9) B A E C D (9) C D A B E (6) C A D B E (4) A B E D C (4) E D B A C (3) E A B D C (3) D C E A B (3) A B C E D (3) E D B C A (2) D E C A B (2) C D B A E (2) C D A E B (2) C A B D E (2) A E B D C (2) A B C D E (2) E B D C A (1) E B D A C (1) C D E B A (1) C D B E A (1) C B A D E (1) B E C A D (1) B A E D C (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 8 16 10 B 2 0 14 12 6 C -8 -14 0 2 -24 D -16 -12 -2 0 -10 E -10 -6 24 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 16 10 B 2 0 14 12 6 C -8 -14 0 2 -24 D -16 -12 -2 0 -10 E -10 -6 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 E=19 C=19 B=12 so B is eliminated. Round 2 votes counts: A=38 D=23 E=20 C=19 so C is eliminated. Round 3 votes counts: A=45 D=35 E=20 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:217 A:216 E:209 D:180 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 16 10 B 2 0 14 12 6 C -8 -14 0 2 -24 D -16 -12 -2 0 -10 E -10 -6 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 16 10 B 2 0 14 12 6 C -8 -14 0 2 -24 D -16 -12 -2 0 -10 E -10 -6 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 16 10 B 2 0 14 12 6 C -8 -14 0 2 -24 D -16 -12 -2 0 -10 E -10 -6 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3109: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (18) E C D A B (8) C E D A B (6) A B C D E (6) B A D E C (5) E B D A C (4) E B A C D (4) C D E A B (4) C D A B E (4) E C D B A (3) C A D B E (3) E D C B A (2) E C A D B (2) D B A C E (2) D A B C E (2) C D A E B (2) C A B D E (2) B E A D C (2) B A E C D (2) B A C D E (2) A C B D E (2) E D C A B (1) E D B A C (1) E B D C A (1) E B C D A (1) E B C A D (1) E B A D C (1) D C A E B (1) D C A B E (1) D B A E C (1) C E A D B (1) C A D E B (1) C A B E D (1) B E D A C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 10 10 14 B 2 0 10 8 12 C -10 -10 0 10 20 D -10 -8 -10 0 16 E -14 -12 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 10 14 B 2 0 10 8 12 C -10 -10 0 10 20 D -10 -8 -10 0 16 E -14 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=29 C=24 A=10 D=7 so D is eliminated. Round 2 votes counts: B=33 E=29 C=26 A=12 so A is eliminated. Round 3 votes counts: B=42 E=29 C=29 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:216 C:205 D:194 E:169 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 10 14 B 2 0 10 8 12 C -10 -10 0 10 20 D -10 -8 -10 0 16 E -14 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 10 14 B 2 0 10 8 12 C -10 -10 0 10 20 D -10 -8 -10 0 16 E -14 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 10 14 B 2 0 10 8 12 C -10 -10 0 10 20 D -10 -8 -10 0 16 E -14 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3110: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (15) B D E C A (10) A D C E B (10) D B A E C (6) C E B A D (5) C E A D B (5) D A E C B (4) B D A E C (4) B C E A D (4) B A D C E (4) E C A D B (3) A D B C E (3) A C E D B (3) E C D A B (2) E C B A D (2) D B E C A (2) D A C E B (2) C E A B D (2) B D A C E (2) B A C E D (2) A B D C E (2) E C D B A (1) D E C A B (1) D A E B C (1) D A B E C (1) B E C A D (1) B A C D E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -18 -6 -2 -6 B 18 0 16 10 14 C 6 -16 0 -4 -6 D 2 -10 4 0 8 E 6 -14 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -2 -6 B 18 0 16 10 14 C 6 -16 0 -4 -6 D 2 -10 4 0 8 E 6 -14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 A=20 D=17 C=12 E=8 so E is eliminated. Round 2 votes counts: B=43 C=20 A=20 D=17 so D is eliminated. Round 3 votes counts: B=51 A=28 C=21 so C is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:202 E:195 C:190 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -6 -2 -6 B 18 0 16 10 14 C 6 -16 0 -4 -6 D 2 -10 4 0 8 E 6 -14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -2 -6 B 18 0 16 10 14 C 6 -16 0 -4 -6 D 2 -10 4 0 8 E 6 -14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -2 -6 B 18 0 16 10 14 C 6 -16 0 -4 -6 D 2 -10 4 0 8 E 6 -14 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3111: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) D E A C B (6) B A C D E (6) E D A C B (5) D A E B C (5) E C B A D (4) E A D B C (4) D C B A E (4) A D B E C (4) E D C A B (3) D A B E C (3) D A B C E (3) A B C D E (3) E C D B A (2) E C D A B (2) E C B D A (2) E A D C B (2) E A B C D (2) D E A B C (2) D B C A E (2) D A E C B (2) C E D B A (2) C E B D A (2) C B E D A (2) C B D A E (2) B A D C E (2) A D E B C (2) A D B C E (2) E D C B A (1) E D A B C (1) E A C D B (1) E A C B D (1) D E C B A (1) D C B E A (1) C B D E A (1) C B A E D (1) C B A D E (1) B C A E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 16 -8 -10 B -10 0 -12 -20 -4 C -16 12 0 -12 -12 D 8 20 12 0 6 E 10 4 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 -8 -10 B -10 0 -12 -20 -4 C -16 12 0 -12 -12 D 8 20 12 0 6 E 10 4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=29 C=19 A=13 B=9 so B is eliminated. Round 2 votes counts: E=30 D=29 A=21 C=20 so C is eliminated. Round 3 votes counts: E=44 D=32 A=24 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:210 A:204 C:186 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 16 -8 -10 B -10 0 -12 -20 -4 C -16 12 0 -12 -12 D 8 20 12 0 6 E 10 4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 -8 -10 B -10 0 -12 -20 -4 C -16 12 0 -12 -12 D 8 20 12 0 6 E 10 4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 -8 -10 B -10 0 -12 -20 -4 C -16 12 0 -12 -12 D 8 20 12 0 6 E 10 4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3112: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (13) C A D E B (8) E B C A D (7) E B A D C (6) B E C D A (6) C A D B E (5) E C B A D (4) C E B A D (4) B E D C A (4) A D C B E (4) E B A C D (3) D A C B E (3) D A B E C (3) D A B C E (3) C D A B E (3) C A E D B (3) C A E B D (2) B D E A C (2) A D C E B (2) E C A B D (1) E B D A C (1) E B C D A (1) E A B C D (1) D C B A E (1) D B E A C (1) D B A E C (1) C E A B D (1) C B E D A (1) C B E A D (1) C B D A E (1) B D E C A (1) B C E D A (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -20 -12 6 -20 B 20 0 10 24 10 C 12 -10 0 8 -12 D -6 -24 -8 0 -20 E 20 -10 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -12 6 -20 B 20 0 10 24 10 C 12 -10 0 8 -12 D -6 -24 -8 0 -20 E 20 -10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 E=24 D=12 A=7 so A is eliminated. Round 2 votes counts: C=29 B=28 E=24 D=19 so D is eliminated. Round 3 votes counts: C=39 B=36 E=25 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 E:221 C:199 A:177 D:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -12 6 -20 B 20 0 10 24 10 C 12 -10 0 8 -12 D -6 -24 -8 0 -20 E 20 -10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -12 6 -20 B 20 0 10 24 10 C 12 -10 0 8 -12 D -6 -24 -8 0 -20 E 20 -10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -12 6 -20 B 20 0 10 24 10 C 12 -10 0 8 -12 D -6 -24 -8 0 -20 E 20 -10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3113: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) A D B C E (7) C B E A D (6) B C A D E (6) D E A C B (5) D A E C B (5) E D A C B (4) E C B A D (4) B C A E D (4) E D C B A (3) E D B C A (3) E C B D A (3) C B A E D (3) B C E A D (3) E D C A B (2) E C A B D (2) E B C D A (2) E A D C B (2) E A C D B (2) D E B A C (2) D A E B C (2) D A B E C (2) C A B E D (2) B E C D A (2) A E C D B (2) A B C D E (2) E D A B C (1) E B D C A (1) D E B C A (1) D E A B C (1) D B E A C (1) D B C A E (1) D B A E C (1) C E B A D (1) B D C A E (1) B D A C E (1) B A C D E (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 0 -2 -2 B 0 0 4 -12 2 C 0 -4 0 -8 -6 D 2 12 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.50617283951 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 A B C D E A 0 0 0 -2 -2 B 0 0 4 -12 2 C 0 -4 0 -8 -6 D 2 12 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=28 B=18 A=13 C=12 so C is eliminated. Round 2 votes counts: E=30 D=28 B=27 A=15 so A is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:209 E:205 A:198 B:197 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -2 -2 B 0 0 4 -12 2 C 0 -4 0 -8 -6 D 2 12 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -2 B 0 0 4 -12 2 C 0 -4 0 -8 -6 D 2 12 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -2 B 0 0 4 -12 2 C 0 -4 0 -8 -6 D 2 12 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3114: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (9) E A B C D (7) C D E A B (6) D E C A B (5) B A E C D (5) E C D A B (4) D E A B C (4) D C A B E (4) C D E B A (4) C B A D E (4) B A E D C (4) B A D C E (4) B A C D E (4) A B E D C (4) D C E A B (3) E D C A B (2) E D A B C (2) E B A C D (2) D A B C E (2) C E B A D (2) E C A B D (1) E A D B C (1) D C B A E (1) D A E B C (1) C E D B A (1) C D B E A (1) C B D A E (1) B C A D E (1) B A D E C (1) B A C E D (1) Total count = 100 A B C D E A 0 10 2 0 -8 B -10 0 4 0 -8 C -2 -4 0 6 -4 D 0 0 -6 0 10 E 8 8 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 A B C D E A 0 10 2 0 -8 B -10 0 4 0 -8 C -2 -4 0 6 -4 D 0 0 -6 0 10 E 8 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 D=20 B=20 A=4 so A is eliminated. Round 2 votes counts: E=28 C=28 B=24 D=20 so D is eliminated. Round 3 votes counts: E=38 C=36 B=26 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:205 A:202 D:202 C:198 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 2 0 -8 B -10 0 4 0 -8 C -2 -4 0 6 -4 D 0 0 -6 0 10 E 8 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 0 -8 B -10 0 4 0 -8 C -2 -4 0 6 -4 D 0 0 -6 0 10 E 8 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 0 -8 B -10 0 4 0 -8 C -2 -4 0 6 -4 D 0 0 -6 0 10 E 8 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.300000 Sum of squares = 0.379999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.700000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3115: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) C B E A D (6) A D C B E (6) E B C D A (5) D E B C A (5) D A C B E (5) D A B C E (5) B E C D A (5) A C B E D (5) E B C A D (4) D B E C A (4) A D E C B (4) A D C E B (4) E B D C A (3) D E B A C (3) D B C E A (3) A C D B E (3) B C E D A (2) A E C B D (2) A C E B D (2) A C D E B (2) E D A B C (1) E C B A D (1) D B C A E (1) D B A C E (1) D A E C B (1) D A B E C (1) C E B A D (1) C B A E D (1) C A B E D (1) B E D C A (1) B D E C A (1) B C E A D (1) A E C D B (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 4 10 -8 8 B -4 0 8 -16 6 C -10 -8 0 -14 0 D 8 16 14 0 16 E -8 -6 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 -8 8 B -4 0 8 -16 6 C -10 -8 0 -14 0 D 8 16 14 0 16 E -8 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=31 E=14 B=10 C=9 so C is eliminated. Round 2 votes counts: D=36 A=32 B=17 E=15 so E is eliminated. Round 3 votes counts: D=37 A=32 B=31 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:207 B:197 E:185 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 10 -8 8 B -4 0 8 -16 6 C -10 -8 0 -14 0 D 8 16 14 0 16 E -8 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -8 8 B -4 0 8 -16 6 C -10 -8 0 -14 0 D 8 16 14 0 16 E -8 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -8 8 B -4 0 8 -16 6 C -10 -8 0 -14 0 D 8 16 14 0 16 E -8 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3116: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) B E C D A (8) A D C B E (8) C E B A D (6) A D C E B (6) A D B E C (5) E B C D A (4) A D B C E (4) D A C E B (3) D A B E C (3) C E A B D (3) C A E D B (3) B E D C A (3) B E D A C (3) B E C A D (3) E C B D A (2) E C B A D (2) E B C A D (2) D B C E A (2) D A C B E (2) D A B C E (2) C E D B A (2) B E A D C (2) A C D E B (2) D C A E B (1) D B E C A (1) D B E A C (1) D B A E C (1) C E D A B (1) C D A E B (1) C A D E B (1) B A D E C (1) A E B C D (1) A C E D B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -6 4 -6 B 2 0 -6 0 0 C 6 6 0 2 14 D -4 0 -2 0 -12 E 6 0 -14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 4 -6 B 2 0 -6 0 0 C 6 6 0 2 14 D -4 0 -2 0 -12 E 6 0 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 B=20 D=16 E=10 so E is eliminated. Round 2 votes counts: C=29 A=29 B=26 D=16 so D is eliminated. Round 3 votes counts: A=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:214 E:202 B:198 A:195 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 4 -6 B 2 0 -6 0 0 C 6 6 0 2 14 D -4 0 -2 0 -12 E 6 0 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 4 -6 B 2 0 -6 0 0 C 6 6 0 2 14 D -4 0 -2 0 -12 E 6 0 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 4 -6 B 2 0 -6 0 0 C 6 6 0 2 14 D -4 0 -2 0 -12 E 6 0 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3117: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) E D C A B (7) D E B A C (7) E D C B A (6) D B E A C (6) B D A E C (6) B A C D E (6) E C D A B (5) E C A D B (5) A C B D E (5) A B C D E (5) E D B C A (4) C A B E D (4) B A D C E (4) D E B C A (3) C E A D B (3) C A E D B (2) B D E C A (2) B D E A C (2) B D A C E (2) A C B E D (2) D E A B C (1) D B A E C (1) C E B D A (1) B C E D A (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -4 -8 -6 B 2 0 0 -2 -8 C 4 0 0 -2 -10 D 8 2 2 0 0 E 6 8 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.540115 E: 0.459885 Sum of squares = 0.503218408824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.540115 E: 1.000000 A B C D E A 0 -2 -4 -8 -6 B 2 0 0 -2 -8 C 4 0 0 -2 -10 D 8 2 2 0 0 E 6 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 C=19 D=18 A=13 so A is eliminated. Round 2 votes counts: B=28 E=27 C=27 D=18 so D is eliminated. Round 3 votes counts: E=38 B=35 C=27 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:206 B:196 C:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -8 -6 B 2 0 0 -2 -8 C 4 0 0 -2 -10 D 8 2 2 0 0 E 6 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -8 -6 B 2 0 0 -2 -8 C 4 0 0 -2 -10 D 8 2 2 0 0 E 6 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -8 -6 B 2 0 0 -2 -8 C 4 0 0 -2 -10 D 8 2 2 0 0 E 6 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3118: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (12) A D E B C (7) A D C B E (6) B D C A E (5) B C E D A (5) A E D C B (5) A D B C E (5) E C B A D (4) E B C A D (4) D B A C E (4) A D E C B (4) E A D C B (3) D A C B E (3) D A B C E (3) C E B D A (3) E C B D A (2) E C A B D (2) E B A C D (2) E A C B D (2) C B D E A (2) E C A D B (1) E B D C A (1) E B D A C (1) E A D B C (1) E A C D B (1) E A B C D (1) C E A D B (1) C B E D A (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E A C (1) B C D E A (1) B C D A E (1) A E D B C (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 6 6 -6 B 2 0 16 4 -20 C -6 -16 0 -6 -16 D -6 -4 6 0 -12 E 6 20 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 6 -6 B 2 0 16 4 -20 C -6 -16 0 -6 -16 D -6 -4 6 0 -12 E 6 20 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=30 B=15 D=10 C=8 so C is eliminated. Round 2 votes counts: E=41 A=31 B=18 D=10 so D is eliminated. Round 3 votes counts: E=41 A=37 B=22 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:202 B:201 D:192 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 6 -6 B 2 0 16 4 -20 C -6 -16 0 -6 -16 D -6 -4 6 0 -12 E 6 20 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 -6 B 2 0 16 4 -20 C -6 -16 0 -6 -16 D -6 -4 6 0 -12 E 6 20 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 -6 B 2 0 16 4 -20 C -6 -16 0 -6 -16 D -6 -4 6 0 -12 E 6 20 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3119: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) C D E A B (9) E A B C D (6) E A C B D (4) D C B A E (4) C D B E A (4) B A E D C (4) B A D E C (4) D B C A E (3) D A B E C (3) C E B A D (3) C E A D B (3) B E A C D (3) B D A C E (3) B A E C D (3) E C A B D (2) E B A C D (2) D C A E B (2) C E D A B (2) C D E B A (2) C B E D A (2) C B D E A (2) B E C A D (2) A E B D C (2) A D B E C (2) A B E D C (2) E C B A D (1) E A C D B (1) D C A B E (1) D A E C B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D A E B (1) B D C E A (1) B D C A E (1) B D A E C (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 6 -8 -2 B 12 0 6 0 12 C -6 -6 0 8 10 D 8 0 -8 0 10 E 2 -12 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.739556 C: 0.000000 D: 0.260444 E: 0.000000 Sum of squares = 0.614773772044 Cumulative probabilities = A: 0.000000 B: 0.739556 C: 0.739556 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -8 -2 B 12 0 6 0 12 C -6 -6 0 8 10 D 8 0 -8 0 10 E 2 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204091635 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=24 B=22 E=16 A=8 so A is eliminated. Round 2 votes counts: C=30 D=26 B=25 E=19 so E is eliminated. Round 3 votes counts: C=38 B=35 D=27 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:205 C:203 A:192 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 -8 -2 B 12 0 6 0 12 C -6 -6 0 8 10 D 8 0 -8 0 10 E 2 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204091635 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -8 -2 B 12 0 6 0 12 C -6 -6 0 8 10 D 8 0 -8 0 10 E 2 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204091635 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -8 -2 B 12 0 6 0 12 C -6 -6 0 8 10 D 8 0 -8 0 10 E 2 -12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204091635 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3120: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) D B C A E (5) A D C B E (5) A C E B D (5) E C B A D (4) E B C D A (4) C B A E D (4) C A B E D (4) A C B D E (4) D E B A C (3) D B A C E (3) D A B E C (3) B D C E A (3) A E D C B (3) A D E C B (3) E D B C A (2) E A D C B (2) D E A B C (2) D B E C A (2) D A E C B (2) D A E B C (2) C E B A D (2) C B A D E (2) B E C D A (2) B D C A E (2) B C E D A (2) B C A D E (2) A C D E B (2) A C D B E (2) E D B A C (1) E D A C B (1) E C A B D (1) E B D C A (1) E A C D B (1) E A C B D (1) D B E A C (1) D B C E A (1) C B E A D (1) B D E C A (1) B C D E A (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 2 8 0 22 B -2 0 -2 -8 12 C -8 2 0 -8 14 D 0 8 8 0 14 E -22 -12 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.356879 B: 0.000000 C: 0.000000 D: 0.643121 E: 0.000000 Sum of squares = 0.540967519083 Cumulative probabilities = A: 0.356879 B: 0.356879 C: 0.356879 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 0 22 B -2 0 -2 -8 12 C -8 2 0 -8 14 D 0 8 8 0 14 E -22 -12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=26 E=18 C=13 B=13 so C is eliminated. Round 2 votes counts: D=30 A=30 E=20 B=20 so E is eliminated. Round 3 votes counts: A=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:215 B:200 C:200 E:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 0 22 B -2 0 -2 -8 12 C -8 2 0 -8 14 D 0 8 8 0 14 E -22 -12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 0 22 B -2 0 -2 -8 12 C -8 2 0 -8 14 D 0 8 8 0 14 E -22 -12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 0 22 B -2 0 -2 -8 12 C -8 2 0 -8 14 D 0 8 8 0 14 E -22 -12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3121: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) C A D E B (7) C B D E A (6) C A B D E (6) A E D B C (6) A C E B D (6) E B D A C (5) C B D A E (5) B E D A C (5) B D E C A (5) A C B E D (5) E D B A C (3) E A D B C (3) D B E C A (3) C D B E A (3) A E B D C (3) D E B C A (2) C D B A E (2) C D A E B (2) C A E D B (2) C A D B E (2) B D E A C (2) A E C D B (2) E D A C B (1) E A B D C (1) D B C E A (1) C B A D E (1) B D C E A (1) Total count = 100 A B C D E A 0 12 4 8 18 B -12 0 -20 2 -6 C -4 20 0 18 18 D -8 -2 -18 0 -4 E -18 6 -18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 8 18 B -12 0 -20 2 -6 C -4 20 0 18 18 D -8 -2 -18 0 -4 E -18 6 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=32 E=13 B=13 D=6 so D is eliminated. Round 2 votes counts: C=36 A=32 B=17 E=15 so E is eliminated. Round 3 votes counts: A=37 C=36 B=27 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:226 A:221 E:187 D:184 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 8 18 B -12 0 -20 2 -6 C -4 20 0 18 18 D -8 -2 -18 0 -4 E -18 6 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 8 18 B -12 0 -20 2 -6 C -4 20 0 18 18 D -8 -2 -18 0 -4 E -18 6 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 8 18 B -12 0 -20 2 -6 C -4 20 0 18 18 D -8 -2 -18 0 -4 E -18 6 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3122: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) D E C B A (9) A E D B C (7) A B C E D (7) D C B E A (6) E B C D A (4) D E A B C (4) D C B A E (4) C B E D A (4) C B D E A (4) C B A E D (4) E B C A D (3) C B D A E (3) A D E B C (3) A C B D E (3) E A B C D (2) D E A C B (2) D A C B E (2) A E B D C (2) E D A B C (1) E C B D A (1) E B A C D (1) D E C A B (1) D E B C A (1) D C E B A (1) D C A B E (1) D A E B C (1) D A C E B (1) C B E A D (1) C B A D E (1) B C E D A (1) B C E A D (1) B C A E D (1) A D C B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 -2 6 B 0 0 0 10 -8 C 2 0 0 6 -6 D 2 -10 -6 0 -4 E -6 8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 0 -2 -2 6 B 0 0 0 10 -8 C 2 0 0 6 -6 D 2 -10 -6 0 -4 E -6 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102065 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=33 C=17 E=12 B=3 so B is eliminated. Round 2 votes counts: A=35 D=33 C=20 E=12 so E is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:206 A:201 B:201 C:201 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 -2 6 B 0 0 0 10 -8 C 2 0 0 6 -6 D 2 -10 -6 0 -4 E -6 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102065 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 6 B 0 0 0 10 -8 C 2 0 0 6 -6 D 2 -10 -6 0 -4 E -6 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102065 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 6 B 0 0 0 10 -8 C 2 0 0 6 -6 D 2 -10 -6 0 -4 E -6 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102065 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3123: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) C E B A D (7) E C D B A (5) E C A D B (5) E C A B D (5) B A D C E (5) A B D C E (5) D B A C E (4) A D B C E (4) E D A C B (3) E C B D A (3) E C B A D (3) C E B D A (3) C B E A D (3) B D C A E (3) E C D A B (2) D E A C B (2) D A B E C (2) D A B C E (2) B C E A D (2) B C A D E (2) A E C B D (2) A D E B C (2) A C B E D (2) E D C A B (1) E A C D B (1) D E C B A (1) D E C A B (1) D B C E A (1) D B C A E (1) D B A E C (1) C B E D A (1) C A B E D (1) B D A C E (1) B C E D A (1) B C D E A (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 2 -4 22 -2 B -2 0 -4 2 6 C 4 4 0 2 0 D -22 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.732613 D: 0.000000 E: 0.267387 Sum of squares = 0.608217178468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.732613 D: 0.732613 E: 1.000000 A B C D E A 0 2 -4 22 -2 B -2 0 -4 2 6 C 4 4 0 2 0 D -22 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.000000 E: 0.399995 Sum of squares = 0.520002174664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.600005 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 B=16 D=15 C=15 so D is eliminated. Round 2 votes counts: E=32 A=30 B=23 C=15 so C is eliminated. Round 3 votes counts: E=42 A=31 B=27 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:209 C:205 B:201 E:199 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 22 -2 B -2 0 -4 2 6 C 4 4 0 2 0 D -22 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.000000 E: 0.399995 Sum of squares = 0.520002174664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.600005 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 22 -2 B -2 0 -4 2 6 C 4 4 0 2 0 D -22 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.000000 E: 0.399995 Sum of squares = 0.520002174664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.600005 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 22 -2 B -2 0 -4 2 6 C 4 4 0 2 0 D -22 -2 -2 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.000000 E: 0.399995 Sum of squares = 0.520002174664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600005 D: 0.600005 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3124: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (6) B C A E D (5) A B D E C (5) A B D C E (5) E D C A B (4) E C D B A (4) E C B D A (4) D E C A B (4) C D E B A (4) E C D A B (3) B C E A D (3) A B E C D (3) A B C E D (3) A B C D E (3) E D C B A (2) E D A C B (2) E A B C D (2) D E C B A (2) D E A C B (2) D A E C B (2) D A E B C (2) C B E D A (2) C B D E A (2) B C E D A (2) B C D A E (2) B C A D E (2) A E D B C (2) A D E B C (2) A D B E C (2) A D B C E (2) A B E D C (2) E B A C D (1) D C E B A (1) D A B C E (1) C E D B A (1) C E B A D (1) C B E A D (1) B D C A E (1) B A D C E (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 6 0 6 6 B -6 0 16 14 6 C 0 -16 0 10 -4 D -6 -14 -10 0 -10 E -6 -6 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.841080 B: 0.000000 C: 0.158920 D: 0.000000 E: 0.000000 Sum of squares = 0.732671338224 Cumulative probabilities = A: 0.841080 B: 0.841080 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 6 6 B -6 0 16 14 6 C 0 -16 0 10 -4 D -6 -14 -10 0 -10 E -6 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845949 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=22 B=22 D=14 C=11 so C is eliminated. Round 2 votes counts: A=31 B=27 E=24 D=18 so D is eliminated. Round 3 votes counts: E=37 A=36 B=27 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:209 E:201 C:195 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 6 6 B -6 0 16 14 6 C 0 -16 0 10 -4 D -6 -14 -10 0 -10 E -6 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845949 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 6 6 B -6 0 16 14 6 C 0 -16 0 10 -4 D -6 -14 -10 0 -10 E -6 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845949 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 6 6 B -6 0 16 14 6 C 0 -16 0 10 -4 D -6 -14 -10 0 -10 E -6 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845949 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3125: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) A C B D E (8) D B E A C (7) D B A C E (7) C A E B D (6) E D B C A (5) D E B A C (5) C A B E D (5) B C A D E (4) E C D A B (3) E C B A D (3) E C A D B (3) C E A B D (3) B D A E C (3) B D A C E (3) B A C D E (3) E D C A B (2) D B A E C (2) B A D C E (2) A B D C E (2) E D B A C (1) E D A C B (1) E C B D A (1) E B D C A (1) E B C D A (1) D E A B C (1) C E A D B (1) C A B D E (1) B E C D A (1) B D E C A (1) B D E A C (1) B C A E D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -4 8 -2 B 4 0 2 20 2 C 4 -2 0 10 -4 D -8 -20 -10 0 2 E 2 -2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 8 -2 B 4 0 2 20 2 C 4 -2 0 10 -4 D -8 -20 -10 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=22 B=19 C=16 A=12 so A is eliminated. Round 2 votes counts: E=31 C=25 D=23 B=21 so B is eliminated. Round 3 votes counts: D=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:204 E:201 A:199 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 8 -2 B 4 0 2 20 2 C 4 -2 0 10 -4 D -8 -20 -10 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 8 -2 B 4 0 2 20 2 C 4 -2 0 10 -4 D -8 -20 -10 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 8 -2 B 4 0 2 20 2 C 4 -2 0 10 -4 D -8 -20 -10 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3126: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) D C A E B (7) D A C E B (7) D A C B E (6) B A E C D (6) E C B D A (5) C E D B A (5) B E C A D (4) A D B C E (4) D C E A B (3) C E B D A (3) C D E A B (3) B E A D C (3) E C B A D (2) E B C A D (2) D A B E C (2) D A B C E (2) C E D A B (2) C D A E B (2) B A D E C (2) A D B E C (2) A C D B E (2) E D B C A (1) E C D B A (1) E B D C A (1) E B D A C (1) E B C D A (1) D E C B A (1) D E C A B (1) D E B A C (1) D B E A C (1) D B A E C (1) D A E C B (1) C A D E B (1) B D A E C (1) B C E A D (1) B C A E D (1) A D C B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 6 -18 0 B 4 0 -8 -14 -2 C -6 8 0 0 2 D 18 14 0 0 6 E 0 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.412935 D: 0.587065 E: 0.000000 Sum of squares = 0.515160706483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.412935 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -18 0 B 4 0 -8 -14 -2 C -6 8 0 0 2 D 18 14 0 0 6 E 0 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=26 C=16 E=14 A=11 so A is eliminated. Round 2 votes counts: D=40 B=27 C=19 E=14 so E is eliminated. Round 3 votes counts: D=41 B=32 C=27 so C is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:202 E:197 A:192 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -18 0 B 4 0 -8 -14 -2 C -6 8 0 0 2 D 18 14 0 0 6 E 0 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -18 0 B 4 0 -8 -14 -2 C -6 8 0 0 2 D 18 14 0 0 6 E 0 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -18 0 B 4 0 -8 -14 -2 C -6 8 0 0 2 D 18 14 0 0 6 E 0 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3127: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) E B C D A (6) D E A B C (6) D A E C B (6) A D C B E (6) A C B D E (6) C B A E D (5) E D B C A (4) E B D C A (4) D E B C A (4) D E B A C (4) A C B E D (4) E D A B C (3) E B C A D (3) C A B D E (3) B C E A D (3) A C D B E (3) E D B A C (2) D A C B E (2) C B E A D (2) C B A D E (2) C A B E D (2) B C E D A (2) A D C E B (2) A C E B D (2) D B E C A (1) D B E A C (1) B E D C A (1) B E C D A (1) B D C A E (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 6 10 -12 4 B -6 0 8 -4 -8 C -10 -8 0 -10 -8 D 12 4 10 0 12 E -4 8 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 -12 4 B -6 0 8 -4 -8 C -10 -8 0 -10 -8 D 12 4 10 0 12 E -4 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=24 E=22 C=14 B=9 so B is eliminated. Round 2 votes counts: D=32 E=24 A=24 C=20 so C is eliminated. Round 3 votes counts: A=36 D=33 E=31 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:204 E:200 B:195 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -12 4 B -6 0 8 -4 -8 C -10 -8 0 -10 -8 D 12 4 10 0 12 E -4 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -12 4 B -6 0 8 -4 -8 C -10 -8 0 -10 -8 D 12 4 10 0 12 E -4 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -12 4 B -6 0 8 -4 -8 C -10 -8 0 -10 -8 D 12 4 10 0 12 E -4 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3128: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A E C (6) D A C E B (6) B E C A D (5) D B C E A (4) C E A D B (4) C E A B D (4) B D A E C (4) D C A E B (3) D A B C E (3) C D E A B (3) B E C D A (3) B E A C D (3) B A E C D (3) B A D E C (3) A C E D B (3) A B E C D (3) E A C B D (2) D C B E A (2) D C A B E (2) D B C A E (2) C E D B A (2) C E B A D (2) B D E A C (2) A E C B D (2) A C D E B (2) E B A C D (1) D C E B A (1) D B A C E (1) C E D A B (1) C D A E B (1) B E D C A (1) B D E C A (1) B C E D A (1) B A E D C (1) A E B C D (1) A D C E B (1) A D B E C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 0 2 0 B -4 0 0 4 4 C 0 0 0 10 -2 D -2 -4 -10 0 -2 E 0 -4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625827 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.374173 Sum of squares = 0.531665074155 Cumulative probabilities = A: 0.625827 B: 0.625827 C: 0.625827 D: 0.625827 E: 1.000000 A B C D E A 0 4 0 2 0 B -4 0 0 4 4 C 0 0 0 10 -2 D -2 -4 -10 0 -2 E 0 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500444 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499556 Sum of squares = 0.500000394631 Cumulative probabilities = A: 0.500444 B: 0.500444 C: 0.500444 D: 0.500444 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=27 C=17 A=15 E=11 so E is eliminated. Round 2 votes counts: D=30 B=28 C=25 A=17 so A is eliminated. Round 3 votes counts: C=35 B=33 D=32 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:204 A:203 B:202 E:200 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 0 B -4 0 0 4 4 C 0 0 0 10 -2 D -2 -4 -10 0 -2 E 0 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500444 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499556 Sum of squares = 0.500000394631 Cumulative probabilities = A: 0.500444 B: 0.500444 C: 0.500444 D: 0.500444 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 0 B -4 0 0 4 4 C 0 0 0 10 -2 D -2 -4 -10 0 -2 E 0 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500444 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499556 Sum of squares = 0.500000394631 Cumulative probabilities = A: 0.500444 B: 0.500444 C: 0.500444 D: 0.500444 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 0 B -4 0 0 4 4 C 0 0 0 10 -2 D -2 -4 -10 0 -2 E 0 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500444 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499556 Sum of squares = 0.500000394631 Cumulative probabilities = A: 0.500444 B: 0.500444 C: 0.500444 D: 0.500444 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3129: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (13) B A D C E (12) A B E D C (11) C D E B A (10) A E B C D (10) E A C D B (9) A B E C D (8) E C D B A (4) D C E B A (4) B D C A E (4) A E C D B (4) D C B E A (3) A B D C E (2) E A C B D (1) C E D B A (1) C D B E A (1) B C D E A (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 20 18 18 6 B -20 0 -2 0 -14 C -18 2 0 24 -24 D -18 0 -24 0 -24 E -6 14 24 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 18 6 B -20 0 -2 0 -14 C -18 2 0 24 -24 D -18 0 -24 0 -24 E -6 14 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=27 B=17 C=12 D=7 so D is eliminated. Round 2 votes counts: A=37 E=27 C=19 B=17 so B is eliminated. Round 3 votes counts: A=49 E=27 C=24 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 E:228 C:192 B:182 D:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 18 6 B -20 0 -2 0 -14 C -18 2 0 24 -24 D -18 0 -24 0 -24 E -6 14 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 18 6 B -20 0 -2 0 -14 C -18 2 0 24 -24 D -18 0 -24 0 -24 E -6 14 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 18 6 B -20 0 -2 0 -14 C -18 2 0 24 -24 D -18 0 -24 0 -24 E -6 14 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3130: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) C D E B A (10) E A B C D (9) E C A D B (6) C D E A B (5) B A D E C (5) E A C B D (4) D C B E A (4) D B C A E (4) E C D A B (3) C E D A B (3) B A D C E (3) E A C D B (2) E A B D C (2) D C E B A (2) C D B E A (2) C D B A E (2) B D A E C (2) B D A C E (2) B A E D C (2) E D B C A (1) E C D B A (1) E B D A C (1) D C B A E (1) D B C E A (1) C E D B A (1) C D A B E (1) B E A D C (1) B D E A C (1) B D C A E (1) A E B C D (1) A C E D B (1) A C D E B (1) A C D B E (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 4 -18 B -6 0 0 -4 -6 C -4 0 0 8 -8 D -4 4 -8 0 0 E 18 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.348310 E: 0.651690 Sum of squares = 0.546019525681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.348310 E: 1.000000 A B C D E A 0 6 4 4 -18 B -6 0 0 -4 -6 C -4 0 0 8 -8 D -4 4 -8 0 0 E 18 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 0.500465 Sum of squares = 0.500000432901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=24 A=18 B=17 D=12 so D is eliminated. Round 2 votes counts: C=31 E=29 B=22 A=18 so A is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:216 A:198 C:198 D:196 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 4 -18 B -6 0 0 -4 -6 C -4 0 0 8 -8 D -4 4 -8 0 0 E 18 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 0.500465 Sum of squares = 0.500000432901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 4 -18 B -6 0 0 -4 -6 C -4 0 0 8 -8 D -4 4 -8 0 0 E 18 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 0.500465 Sum of squares = 0.500000432901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 4 -18 B -6 0 0 -4 -6 C -4 0 0 8 -8 D -4 4 -8 0 0 E 18 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 0.500465 Sum of squares = 0.500000432901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499535 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3131: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (13) E B C D A (11) B E D A C (11) D A B E C (9) C A D E B (7) A D C B E (7) E B D C A (5) C A E B D (5) B D E A C (5) A C D B E (4) D B E A C (3) D B A E C (3) C A E D B (2) B E D C A (2) A D B E C (2) A D B C E (2) A C D E B (2) E C B D A (1) D E B A C (1) D E A C B (1) C E B D A (1) C E A B D (1) C A D B E (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 2 -6 -10 B 12 0 8 10 -2 C -2 -8 0 -4 -8 D 6 -10 4 0 -4 E 10 2 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 2 -6 -10 B 12 0 8 10 -2 C -2 -8 0 -4 -8 D 6 -10 4 0 -4 E 10 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=18 A=18 E=17 D=17 so E is eliminated. Round 2 votes counts: B=34 C=31 A=18 D=17 so D is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:212 D:198 C:189 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 2 -6 -10 B 12 0 8 10 -2 C -2 -8 0 -4 -8 D 6 -10 4 0 -4 E 10 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -6 -10 B 12 0 8 10 -2 C -2 -8 0 -4 -8 D 6 -10 4 0 -4 E 10 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -6 -10 B 12 0 8 10 -2 C -2 -8 0 -4 -8 D 6 -10 4 0 -4 E 10 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3132: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) A E B C D (7) D B C A E (6) C E D A B (5) C E A B D (5) D C E B A (4) D A B E C (4) C D E B A (4) B A D E C (4) A B E D C (4) A B D E C (4) E A C B D (3) D C E A B (3) D B A C E (3) D A E B C (3) B D A E C (3) B A E D C (3) E C A B D (2) E A C D B (2) D B A E C (2) C E A D B (2) C D E A B (2) C B D E A (2) B D C A E (2) B A E C D (2) A B E C D (2) E C B A D (1) E C A D B (1) D E C A B (1) D B C E A (1) C E D B A (1) C D B E A (1) B D A C E (1) B C D A E (1) B A C E D (1) A D E B C (1) Total count = 100 A B C D E A 0 2 -2 -12 6 B -2 0 8 -6 6 C 2 -8 0 -12 2 D 12 6 12 0 18 E -6 -6 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -12 6 B -2 0 8 -6 6 C 2 -8 0 -12 2 D 12 6 12 0 18 E -6 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=22 A=18 B=17 E=9 so E is eliminated. Round 2 votes counts: D=34 C=26 A=23 B=17 so B is eliminated. Round 3 votes counts: D=40 A=33 C=27 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:203 A:197 C:192 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -12 6 B -2 0 8 -6 6 C 2 -8 0 -12 2 D 12 6 12 0 18 E -6 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -12 6 B -2 0 8 -6 6 C 2 -8 0 -12 2 D 12 6 12 0 18 E -6 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -12 6 B -2 0 8 -6 6 C 2 -8 0 -12 2 D 12 6 12 0 18 E -6 -6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3133: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (17) E B A C D (15) D C A E B (10) E A C D B (8) D C A B E (8) B E D C A (4) B C A D E (4) E D A C B (3) E B D C A (3) B D C A E (3) B A C D E (3) A C D E B (3) E D C A B (2) E A B C D (2) D E C A B (2) C D A B E (2) C A D B E (2) B A E C D (2) E B D A C (1) D C B A E (1) C D A E B (1) B E D A C (1) B D E C A (1) B A C E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 14 16 -18 B 12 0 14 14 -2 C -14 -14 0 22 -22 D -16 -14 -22 0 -20 E 18 2 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 14 16 -18 B 12 0 14 14 -2 C -14 -14 0 22 -22 D -16 -14 -22 0 -20 E 18 2 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=34 D=21 C=5 A=4 so A is eliminated. Round 2 votes counts: B=36 E=34 D=21 C=9 so C is eliminated. Round 3 votes counts: B=36 E=35 D=29 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:219 A:200 C:186 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 14 16 -18 B 12 0 14 14 -2 C -14 -14 0 22 -22 D -16 -14 -22 0 -20 E 18 2 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 14 16 -18 B 12 0 14 14 -2 C -14 -14 0 22 -22 D -16 -14 -22 0 -20 E 18 2 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 14 16 -18 B 12 0 14 14 -2 C -14 -14 0 22 -22 D -16 -14 -22 0 -20 E 18 2 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999994437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3134: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) D B A E C (5) E A D C B (4) C E A D B (4) C A B D E (4) B D C E A (4) B C D A E (4) A E D C B (4) A E C D B (4) A D B E C (4) E A D B C (3) E A C D B (3) D B E A C (3) C E B D A (3) C E B A D (3) C E A B D (3) C B E D A (3) C A E B D (3) B D E A C (3) E B D C A (2) E B D A C (2) D A B E C (2) C B D E A (2) C B A E D (2) B D E C A (2) B C D E A (2) A C E D B (2) E D A B C (1) E C B D A (1) D B A C E (1) C B D A E (1) C B A D E (1) C A E D B (1) C A B E D (1) B D A E C (1) B D A C E (1) A E D B C (1) A D E C B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 8 14 4 B -8 0 -4 -4 -6 C -8 4 0 -2 -6 D -14 4 2 0 0 E -4 6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 14 4 B -8 0 -4 -4 -6 C -8 4 0 -2 -6 D -14 4 2 0 0 E -4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=25 B=17 E=16 D=11 so D is eliminated. Round 2 votes counts: C=31 A=27 B=26 E=16 so E is eliminated. Round 3 votes counts: A=38 C=32 B=30 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:204 D:196 C:194 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 14 4 B -8 0 -4 -4 -6 C -8 4 0 -2 -6 D -14 4 2 0 0 E -4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 14 4 B -8 0 -4 -4 -6 C -8 4 0 -2 -6 D -14 4 2 0 0 E -4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 14 4 B -8 0 -4 -4 -6 C -8 4 0 -2 -6 D -14 4 2 0 0 E -4 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3135: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) A B D E C (8) C D E B A (6) D C E B A (5) D C A E B (5) D A C B E (5) D A B C E (5) C E B D A (5) E C B A D (4) B E A C D (4) A B E C D (4) D C E A B (3) B A E C D (3) E B C A D (2) D A B E C (2) C E D B A (2) C E A B D (2) B E C A D (2) B A E D C (2) A D B E C (2) A C B E D (2) A B E D C (2) A B C E D (2) E C D B A (1) E C B D A (1) E B C D A (1) D B E A C (1) C D E A B (1) C D A E B (1) C A E B D (1) B E A D C (1) A D C B E (1) A D B C E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -6 12 -4 B 2 0 -16 16 -4 C 6 16 0 14 20 D -12 -16 -14 0 -6 E 4 4 -20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 12 -4 B 2 0 -16 16 -4 C 6 16 0 14 20 D -12 -16 -14 0 -6 E 4 4 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 A=24 B=12 E=9 so E is eliminated. Round 2 votes counts: C=35 D=26 A=24 B=15 so B is eliminated. Round 3 votes counts: C=40 A=34 D=26 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:200 B:199 E:197 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 12 -4 B 2 0 -16 16 -4 C 6 16 0 14 20 D -12 -16 -14 0 -6 E 4 4 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 12 -4 B 2 0 -16 16 -4 C 6 16 0 14 20 D -12 -16 -14 0 -6 E 4 4 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 12 -4 B 2 0 -16 16 -4 C 6 16 0 14 20 D -12 -16 -14 0 -6 E 4 4 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3136: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) B E D A C (6) E B A D C (5) B E D C A (5) E A B D C (4) C B E D A (4) C A D E B (4) A D C E B (4) E B C D A (3) E B C A D (3) E A D B C (3) D A B C E (3) C D B A E (3) B D C A E (3) B D A E C (3) A D E C B (3) A D C B E (3) E B D A C (2) D C A B E (2) C A D B E (2) B E C D A (2) B D E A C (2) A E D C B (2) A C D E B (2) E C A D B (1) E A D C B (1) E A C D B (1) D C B A E (1) D B A C E (1) D A C B E (1) C E B D A (1) C B D E A (1) C B D A E (1) C A E D B (1) B D C E A (1) B C E D A (1) B C D A E (1) A E D B C (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 0 -14 8 B -2 0 4 -4 16 C 0 -4 0 -18 2 D 14 4 18 0 8 E -8 -16 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -14 8 B -2 0 4 -4 16 C 0 -4 0 -18 2 D 14 4 18 0 8 E -8 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 E=23 A=18 D=8 so D is eliminated. Round 2 votes counts: C=30 B=25 E=23 A=22 so A is eliminated. Round 3 votes counts: C=40 E=30 B=30 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:207 A:198 C:190 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -14 8 B -2 0 4 -4 16 C 0 -4 0 -18 2 D 14 4 18 0 8 E -8 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -14 8 B -2 0 4 -4 16 C 0 -4 0 -18 2 D 14 4 18 0 8 E -8 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -14 8 B -2 0 4 -4 16 C 0 -4 0 -18 2 D 14 4 18 0 8 E -8 -16 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3137: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) E C A D B (9) D B E A C (9) B D A C E (9) C A B E D (7) A C B E D (7) E D C A B (6) C A E B D (4) D E B C A (3) D E B A C (3) B D A E C (3) B C A E D (3) D E C B A (2) D E A C B (2) B A C E D (2) E D C B A (1) E D A C B (1) E C D A B (1) E C A B D (1) E A D C B (1) D E C A B (1) D B E C A (1) D B A E C (1) D B A C E (1) D A C E B (1) C E A B D (1) C B A E D (1) B E D C A (1) B D E C A (1) B A D C E (1) A D B C E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 14 6 12 B 10 0 4 12 24 C -14 -4 0 2 6 D -6 -12 -2 0 6 E -12 -24 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999067 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 6 12 B 10 0 4 12 24 C -14 -4 0 2 6 D -6 -12 -2 0 6 E -12 -24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=24 E=20 C=13 A=10 so A is eliminated. Round 2 votes counts: B=33 D=25 C=22 E=20 so E is eliminated. Round 3 votes counts: D=34 C=33 B=33 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:211 C:195 D:193 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 6 12 B 10 0 4 12 24 C -14 -4 0 2 6 D -6 -12 -2 0 6 E -12 -24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 6 12 B 10 0 4 12 24 C -14 -4 0 2 6 D -6 -12 -2 0 6 E -12 -24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 6 12 B 10 0 4 12 24 C -14 -4 0 2 6 D -6 -12 -2 0 6 E -12 -24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3138: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) C E D B A (9) A B E D C (8) C A B E D (6) D E B A C (5) D E A B C (5) C E B D A (5) E D C B A (4) D E C B A (4) C D E A B (4) A B C E D (4) C B E A D (3) B A E D C (3) E D B A C (2) E B D A C (2) D E A C B (2) C E B A D (2) C D A E B (2) C B A E D (2) C A B D E (2) B E A D C (2) E D B C A (1) E C B A D (1) E B C D A (1) E B A D C (1) D E B C A (1) D C A E B (1) D A E B C (1) C D E B A (1) C A D B E (1) B E C A D (1) A D B E C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 0 -12 B 0 0 0 12 -8 C 2 0 0 -8 -14 D 0 -12 8 0 -16 E 12 8 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 0 -12 B 0 0 0 12 -8 C 2 0 0 -8 -14 D 0 -12 8 0 -16 E 12 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=26 D=19 E=12 B=6 so B is eliminated. Round 2 votes counts: C=37 A=29 D=19 E=15 so E is eliminated. Round 3 votes counts: C=40 A=32 D=28 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:225 B:202 A:193 C:190 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 0 -12 B 0 0 0 12 -8 C 2 0 0 -8 -14 D 0 -12 8 0 -16 E 12 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 0 -12 B 0 0 0 12 -8 C 2 0 0 -8 -14 D 0 -12 8 0 -16 E 12 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 0 -12 B 0 0 0 12 -8 C 2 0 0 -8 -14 D 0 -12 8 0 -16 E 12 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3139: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) B D A E C (10) E A D C B (7) C E A D B (7) B C D A E (7) D E A C B (6) C B A E D (6) C B E A D (5) B C A E D (5) A E D C B (5) E D A C B (3) D B E A C (3) B C D E A (3) A E C D B (3) D A E B C (2) C E A B D (2) C A E D B (2) C A B E D (2) B D E A C (2) D B A E C (1) D A E C B (1) C A E B D (1) B D C E A (1) B D C A E (1) B D A C E (1) B C A D E (1) B A D E C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 6 14 -2 2 B -6 0 -2 -4 -2 C -14 2 0 -12 -12 D 2 4 12 0 4 E -2 2 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 -2 2 B -6 0 -2 -4 -2 C -14 2 0 -12 -12 D 2 4 12 0 4 E -2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=25 D=23 E=10 A=10 so E is eliminated. Round 2 votes counts: B=32 D=26 C=25 A=17 so A is eliminated. Round 3 votes counts: D=40 B=32 C=28 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:210 E:204 B:193 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 14 -2 2 B -6 0 -2 -4 -2 C -14 2 0 -12 -12 D 2 4 12 0 4 E -2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -2 2 B -6 0 -2 -4 -2 C -14 2 0 -12 -12 D 2 4 12 0 4 E -2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -2 2 B -6 0 -2 -4 -2 C -14 2 0 -12 -12 D 2 4 12 0 4 E -2 2 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3140: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (13) D A C B E (11) E B C A D (10) E B D C A (9) A C D E B (6) A C E B D (5) B E C A D (3) A D C B E (3) A C B E D (3) A C B D E (3) D E B A C (2) D E A B C (2) D B E C A (2) D B A C E (2) B C A E D (2) B C A D E (2) A E C D B (2) A C E D B (2) E D B A C (1) E D A B C (1) E C B A D (1) E C A B D (1) E B D A C (1) E B C D A (1) D E B C A (1) D B C A E (1) D A C E B (1) D A B C E (1) C D A B E (1) C B A D E (1) C A E B D (1) C A D B E (1) C A B E D (1) B C E D A (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 18 20 24 26 B -18 0 -14 -8 4 C -20 14 0 22 26 D -24 8 -22 0 8 E -26 -4 -26 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 20 24 26 B -18 0 -14 -8 4 C -20 14 0 22 26 D -24 8 -22 0 8 E -26 -4 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=25 D=23 B=9 C=5 so C is eliminated. Round 2 votes counts: A=41 E=25 D=24 B=10 so B is eliminated. Round 3 votes counts: A=46 E=30 D=24 so D is eliminated. Round 4 votes counts: A=63 E=37 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:244 C:221 D:185 B:182 E:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 20 24 26 B -18 0 -14 -8 4 C -20 14 0 22 26 D -24 8 -22 0 8 E -26 -4 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 24 26 B -18 0 -14 -8 4 C -20 14 0 22 26 D -24 8 -22 0 8 E -26 -4 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 24 26 B -18 0 -14 -8 4 C -20 14 0 22 26 D -24 8 -22 0 8 E -26 -4 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3141: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (13) B D C E A (10) B D C A E (8) E A B C D (6) E A B D C (5) C D B E A (5) E A C D B (4) D C B A E (4) A E B D C (4) B E D C A (3) A B D E C (3) E C B D A (2) E A C B D (2) D B C A E (2) C D A B E (2) B D E C A (2) B D A C E (2) A E C B D (2) A C E D B (2) A C D E B (2) A C D B E (2) E C A D B (1) E C A B D (1) E B D C A (1) E B C D A (1) E B A D C (1) D C B E A (1) D B A C E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D B A E (1) C A D E B (1) B D E A C (1) B C D E A (1) A D C B E (1) Total count = 100 A B C D E A 0 4 4 0 2 B -4 0 4 10 0 C -4 -4 0 0 -4 D 0 -10 0 0 2 E -2 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.809402 B: 0.000000 C: 0.000000 D: 0.190598 E: 0.000000 Sum of squares = 0.691458952768 Cumulative probabilities = A: 0.809402 B: 0.809402 C: 0.809402 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 2 B -4 0 4 10 0 C -4 -4 0 0 -4 D 0 -10 0 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735423 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 E=24 C=11 D=9 so D is eliminated. Round 2 votes counts: B=30 A=30 E=24 C=16 so C is eliminated. Round 3 votes counts: B=41 A=33 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:205 B:205 E:200 D:196 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 2 B -4 0 4 10 0 C -4 -4 0 0 -4 D 0 -10 0 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735423 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 2 B -4 0 4 10 0 C -4 -4 0 0 -4 D 0 -10 0 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735423 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 2 B -4 0 4 10 0 C -4 -4 0 0 -4 D 0 -10 0 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735423 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3142: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) E A C D B (8) A E B D C (8) E C D A B (6) E A D C B (6) C B D E A (5) D C B E A (4) C E D B A (4) B A C D E (4) A E C B D (4) E A D B C (3) C D E B A (3) B D C A E (3) B C D E A (3) B A D E C (3) B A D C E (3) A B E D C (3) E D C A B (2) D E C B A (2) C E D A B (2) C B D A E (2) E D A C B (1) E C A D B (1) D E B C A (1) D E B A C (1) D B E A C (1) D B C E A (1) C E A D B (1) C E A B D (1) B C D A E (1) A E D B C (1) A E C D B (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -2 -2 -30 B 0 0 -24 -16 -14 C 2 24 0 12 -8 D 2 16 -12 0 -6 E 30 14 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 -2 -30 B 0 0 -24 -16 -14 C 2 24 0 12 -8 D 2 16 -12 0 -6 E 30 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=27 C=27 A=19 B=17 D=10 so D is eliminated. Round 2 votes counts: E=31 C=31 B=19 A=19 so B is eliminated. Round 3 votes counts: C=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:215 D:200 A:183 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 -2 -30 B 0 0 -24 -16 -14 C 2 24 0 12 -8 D 2 16 -12 0 -6 E 30 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 -30 B 0 0 -24 -16 -14 C 2 24 0 12 -8 D 2 16 -12 0 -6 E 30 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 -30 B 0 0 -24 -16 -14 C 2 24 0 12 -8 D 2 16 -12 0 -6 E 30 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3143: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) B A D E C (7) A D B E C (7) B A D C E (5) A B D E C (5) E C D B A (4) E C B D A (4) E C B A D (4) D E C A B (4) C E B D A (4) B A E D C (4) A B D C E (4) D E A C B (3) D C E A B (3) D A C B E (3) B A C E D (3) D E A B C (2) D A E B C (2) D A C E B (2) D A B E C (2) D A B C E (2) C D E A B (2) B A E C D (2) B A C D E (2) A D B C E (2) E D C A B (1) E C D A B (1) E A B D C (1) D A E C B (1) C E D A B (1) C E B A D (1) C D E B A (1) B E C A D (1) B E A C D (1) B C A E D (1) Total count = 100 A B C D E A 0 -4 20 0 8 B 4 0 6 -2 2 C -20 -6 0 -20 -12 D 0 2 20 0 18 E -8 -2 12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.233560 B: 0.000000 C: 0.000000 D: 0.766440 E: 0.000000 Sum of squares = 0.641980093034 Cumulative probabilities = A: 0.233560 B: 0.233560 C: 0.233560 D: 1.000000 E: 1.000000 A B C D E A 0 -4 20 0 8 B 4 0 6 -2 2 C -20 -6 0 -20 -12 D 0 2 20 0 18 E -8 -2 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555556109229 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=24 A=18 C=17 E=15 so E is eliminated. Round 2 votes counts: C=30 B=26 D=25 A=19 so A is eliminated. Round 3 votes counts: B=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:212 B:205 E:192 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 20 0 8 B 4 0 6 -2 2 C -20 -6 0 -20 -12 D 0 2 20 0 18 E -8 -2 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555556109229 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 20 0 8 B 4 0 6 -2 2 C -20 -6 0 -20 -12 D 0 2 20 0 18 E -8 -2 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555556109229 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 20 0 8 B 4 0 6 -2 2 C -20 -6 0 -20 -12 D 0 2 20 0 18 E -8 -2 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555556109229 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3144: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) C D A B E (6) B E A C D (6) E D B C A (5) D E C A B (5) D C A E B (5) D C A B E (5) E B A D C (4) C A D B E (4) E B D C A (3) E B D A C (3) C A B D E (3) A C B D E (3) E D B A C (2) E D A C B (2) E A D C B (2) D E B C A (2) D C B E A (2) C B A D E (2) B E C A D (2) A B E C D (2) A B C E D (2) E A C B D (1) D E C B A (1) D E A C B (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C A E (1) D A C E B (1) C D B A E (1) C D A E B (1) C B D A E (1) B E D C A (1) B D E C A (1) B C A E D (1) B A C E D (1) A E C B D (1) A E B C D (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -10 -4 -6 B 0 0 -8 -4 -4 C 10 8 0 2 -8 D 4 4 -2 0 0 E 6 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.400040 E: 0.599960 Sum of squares = 0.51998388844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.400040 E: 1.000000 A B C D E A 0 0 -10 -4 -6 B 0 0 -8 -4 -4 C 10 8 0 2 -8 D 4 4 -2 0 0 E 6 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 C=18 A=14 B=12 so B is eliminated. Round 2 votes counts: E=39 D=27 C=19 A=15 so A is eliminated. Round 3 votes counts: E=43 C=30 D=27 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:206 D:203 B:192 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 -4 -6 B 0 0 -8 -4 -4 C 10 8 0 2 -8 D 4 4 -2 0 0 E 6 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 -6 B 0 0 -8 -4 -4 C 10 8 0 2 -8 D 4 4 -2 0 0 E 6 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 -6 B 0 0 -8 -4 -4 C 10 8 0 2 -8 D 4 4 -2 0 0 E 6 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3145: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) B E D A C (6) C A E D B (5) B E D C A (4) E B D A C (3) E B C A D (3) E A C D B (3) D A C E B (3) D A C B E (3) D A B E C (3) C E A D B (3) C A D E B (3) C A B D E (3) B D C A E (3) B C E D A (3) E C A D B (2) D B E A C (2) D B A E C (2) D B A C E (2) D A E B C (2) C E B A D (2) C B A E D (2) B E C D A (2) B D E A C (2) B D A C E (2) B C D A E (2) A E C D B (2) A D E C B (2) A D C E B (2) E D B A C (1) E D A B C (1) E C A B D (1) E A D B C (1) D B C A E (1) C E A B D (1) C D B A E (1) C B E A D (1) C B A D E (1) B E C A D (1) B D E C A (1) B D A E C (1) B C E A D (1) A E D C B (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -6 0 12 B -2 0 -2 -10 16 C 6 2 0 2 8 D 0 10 -2 0 2 E -12 -16 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 0 12 B -2 0 -2 -10 16 C 6 2 0 2 8 D 0 10 -2 0 2 E -12 -16 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 D=18 E=15 A=10 so A is eliminated. Round 2 votes counts: C=31 B=28 D=23 E=18 so E is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:205 A:204 B:201 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 0 12 B -2 0 -2 -10 16 C 6 2 0 2 8 D 0 10 -2 0 2 E -12 -16 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 12 B -2 0 -2 -10 16 C 6 2 0 2 8 D 0 10 -2 0 2 E -12 -16 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 12 B -2 0 -2 -10 16 C 6 2 0 2 8 D 0 10 -2 0 2 E -12 -16 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3146: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (14) D E C A B (11) E D B A C (9) B C A E D (8) B A C E D (8) C A B E D (7) A B C E D (5) D E B A C (4) D E A C B (4) E B D A C (3) A C B E D (3) D E C B A (2) D E A B C (2) C A D B E (2) B E D A C (2) B A E C D (2) A C B D E (2) E A D B C (1) D E B C A (1) D C E A B (1) C D A E B (1) C D A B E (1) C B A E D (1) C A D E B (1) B E A D C (1) B E A C D (1) B C E A D (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 14 -2 18 14 B -14 0 0 18 18 C 2 0 0 16 10 D -18 -18 -16 0 -8 E -14 -18 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.086196 C: 0.913804 D: 0.000000 E: 0.000000 Sum of squares = 0.842467276139 Cumulative probabilities = A: 0.000000 B: 0.086196 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 18 14 B -14 0 0 18 18 C 2 0 0 16 10 D -18 -18 -16 0 -8 E -14 -18 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250132638 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 B=23 E=13 A=12 so A is eliminated. Round 2 votes counts: C=32 B=29 D=25 E=14 so E is eliminated. Round 3 votes counts: D=36 C=32 B=32 so C is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:222 C:214 B:211 E:183 D:170 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 18 14 B -14 0 0 18 18 C 2 0 0 16 10 D -18 -18 -16 0 -8 E -14 -18 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250132638 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 18 14 B -14 0 0 18 18 C 2 0 0 16 10 D -18 -18 -16 0 -8 E -14 -18 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250132638 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 18 14 B -14 0 0 18 18 C 2 0 0 16 10 D -18 -18 -16 0 -8 E -14 -18 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250132638 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3147: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (17) D A C E B (12) C A D E B (9) B D E A C (6) B E D A C (4) B C E A D (4) E C A B D (3) D A C B E (3) C E A B D (3) C A E D B (3) B D C A E (3) E C B A D (2) E B C A D (2) E A C D B (2) D B E A C (2) D A E C B (2) B E D C A (2) B E C D A (2) B D E C A (2) A D C E B (2) E D B A C (1) E D A C B (1) E B A D C (1) E A D C B (1) D E A C B (1) D C A B E (1) D B C A E (1) D B A C E (1) C D A E B (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D B E (1) B E A D C (1) B C A E D (1) Total count = 100 A B C D E A 0 -8 -20 10 -16 B 8 0 0 12 6 C 20 0 0 6 -4 D -10 -12 -6 0 -6 E 16 -6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.648488 C: 0.351512 D: 0.000000 E: 0.000000 Sum of squares = 0.544097612551 Cumulative probabilities = A: 0.000000 B: 0.648488 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -20 10 -16 B 8 0 0 12 6 C 20 0 0 6 -4 D -10 -12 -6 0 -6 E 16 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 D=23 C=20 E=13 A=2 so A is eliminated. Round 2 votes counts: B=42 D=25 C=20 E=13 so E is eliminated. Round 3 votes counts: B=45 D=28 C=27 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:211 E:210 A:183 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -20 10 -16 B 8 0 0 12 6 C 20 0 0 6 -4 D -10 -12 -6 0 -6 E 16 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 10 -16 B 8 0 0 12 6 C 20 0 0 6 -4 D -10 -12 -6 0 -6 E 16 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 10 -16 B 8 0 0 12 6 C 20 0 0 6 -4 D -10 -12 -6 0 -6 E 16 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3148: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) E D B A C (7) D E A C B (7) D A C E B (6) C A B D E (6) B E C A D (6) E D A C B (5) C B A D E (5) B E C D A (5) B C A D E (5) E B D C A (4) B C A E D (4) A C D B E (4) C A D B E (3) B C E A D (3) C E D B A (2) B E A D C (2) B A C E D (2) B A C D E (2) A C B D E (2) E D C A B (1) E D B C A (1) E C B D A (1) E B D A C (1) E B C D A (1) D E A B C (1) C D A E B (1) B A E D C (1) A D C B E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 4 -4 6 B 4 0 -8 2 4 C -4 8 0 6 -6 D 4 -2 -6 0 8 E -6 -4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.468237 B: 0.138829 C: 0.265882 D: 0.127053 E: 0.000000 Sum of squares = 0.32535458443 Cumulative probabilities = A: 0.468237 B: 0.607066 C: 0.872947 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -4 6 B 4 0 -8 2 4 C -4 8 0 6 -6 D 4 -2 -6 0 8 E -6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.457627 B: 0.101695 C: 0.271186 D: 0.169491 E: 0.000000 Sum of squares = 0.322033898328 Cumulative probabilities = A: 0.457627 B: 0.559323 C: 0.830509 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 E=21 C=17 A=9 so A is eliminated. Round 2 votes counts: B=31 D=24 C=24 E=21 so E is eliminated. Round 3 votes counts: D=38 B=37 C=25 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:202 D:202 A:201 B:201 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 4 -4 6 B 4 0 -8 2 4 C -4 8 0 6 -6 D 4 -2 -6 0 8 E -6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.457627 B: 0.101695 C: 0.271186 D: 0.169491 E: 0.000000 Sum of squares = 0.322033898328 Cumulative probabilities = A: 0.457627 B: 0.559323 C: 0.830509 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -4 6 B 4 0 -8 2 4 C -4 8 0 6 -6 D 4 -2 -6 0 8 E -6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.457627 B: 0.101695 C: 0.271186 D: 0.169491 E: 0.000000 Sum of squares = 0.322033898328 Cumulative probabilities = A: 0.457627 B: 0.559323 C: 0.830509 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -4 6 B 4 0 -8 2 4 C -4 8 0 6 -6 D 4 -2 -6 0 8 E -6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.457627 B: 0.101695 C: 0.271186 D: 0.169491 E: 0.000000 Sum of squares = 0.322033898328 Cumulative probabilities = A: 0.457627 B: 0.559323 C: 0.830509 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3149: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (18) A C E B D (17) A C B E D (9) E C A D B (8) C A E D B (7) B D A C E (7) B A C D E (7) D E B C A (6) E D C A B (4) B A D C E (4) E C D A B (3) B D E A C (2) A C E D B (2) E D B C A (1) E A C B D (1) C E A D B (1) B D E C A (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 6 2 14 10 B -6 0 -6 0 0 C -2 6 0 12 10 D -14 0 -12 0 -6 E -10 0 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 14 10 B -6 0 -6 0 0 C -2 6 0 12 10 D -14 0 -12 0 -6 E -10 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=24 B=22 E=17 C=8 so C is eliminated. Round 2 votes counts: A=36 D=24 B=22 E=18 so E is eliminated. Round 3 votes counts: A=46 D=32 B=22 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:213 B:194 E:193 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 14 10 B -6 0 -6 0 0 C -2 6 0 12 10 D -14 0 -12 0 -6 E -10 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 14 10 B -6 0 -6 0 0 C -2 6 0 12 10 D -14 0 -12 0 -6 E -10 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 14 10 B -6 0 -6 0 0 C -2 6 0 12 10 D -14 0 -12 0 -6 E -10 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3150: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) D E C A B (9) B A D C E (9) A B D E C (9) D A E B C (5) B C A E D (5) B A C E D (5) A B D C E (5) E C D A B (4) D A B E C (4) C E B D A (4) A D B E C (4) E D C A B (3) D E C B A (3) C E D B A (3) A B C E D (3) D E A C B (2) C E B A D (2) C B E A D (2) B A C D E (2) E C A D B (1) D E A B C (1) D B E A C (1) D B A E C (1) D A E C B (1) C E A B D (1) B D C E A (1) B D A E C (1) Total count = 100 A B C D E A 0 4 6 -4 8 B -4 0 12 -2 4 C -6 -12 0 -18 -16 D 4 2 18 0 16 E -8 -4 16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -4 8 B -4 0 12 -2 4 C -6 -12 0 -18 -16 D 4 2 18 0 16 E -8 -4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 A=21 E=17 C=12 so C is eliminated. Round 2 votes counts: E=27 D=27 B=25 A=21 so A is eliminated. Round 3 votes counts: B=42 D=31 E=27 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:207 B:205 E:194 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -4 8 B -4 0 12 -2 4 C -6 -12 0 -18 -16 D 4 2 18 0 16 E -8 -4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -4 8 B -4 0 12 -2 4 C -6 -12 0 -18 -16 D 4 2 18 0 16 E -8 -4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -4 8 B -4 0 12 -2 4 C -6 -12 0 -18 -16 D 4 2 18 0 16 E -8 -4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3151: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (12) B D A C E (12) C A E D B (9) E C A B D (7) C A D B E (7) A C D B E (6) C A D E B (5) E B D C A (4) B E D A C (4) E C B A D (3) E B C D A (3) B D E A C (3) E B D A C (2) E B C A D (2) E A C B D (2) D B A C E (2) C D A E B (2) A B D C E (2) E C B D A (1) D B E C A (1) D B C A E (1) D A B C E (1) C E D A B (1) B E A D C (1) B D A E C (1) B A D E C (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 18 -16 24 8 B -18 0 -14 2 -10 C 16 14 0 24 0 D -24 -2 -24 0 -8 E -8 10 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545962 D: 0.000000 E: 0.454038 Sum of squares = 0.504224991629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545962 D: 0.545962 E: 1.000000 A B C D E A 0 18 -16 24 8 B -18 0 -14 2 -10 C 16 14 0 24 0 D -24 -2 -24 0 -8 E -8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=24 B=22 A=13 D=5 so D is eliminated. Round 2 votes counts: E=36 B=26 C=24 A=14 so A is eliminated. Round 3 votes counts: E=38 C=31 B=31 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:227 A:217 E:205 B:180 D:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -16 24 8 B -18 0 -14 2 -10 C 16 14 0 24 0 D -24 -2 -24 0 -8 E -8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -16 24 8 B -18 0 -14 2 -10 C 16 14 0 24 0 D -24 -2 -24 0 -8 E -8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -16 24 8 B -18 0 -14 2 -10 C 16 14 0 24 0 D -24 -2 -24 0 -8 E -8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3152: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) A D E B C (11) B C A E D (9) C B E D A (8) D A E C B (7) C E B D A (5) B C E A D (5) E D C A B (4) A D B C E (4) D A E B C (3) B A C E D (3) B A C D E (3) A D B E C (3) A B D E C (3) E A D C B (2) C E D B A (2) C B E A D (2) C B D E A (2) A B D C E (2) E C D B A (1) E C D A B (1) E C B D A (1) E A B D C (1) D E C A B (1) D C E B A (1) D A B C E (1) C B D A E (1) B C A D E (1) B A E C D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 12 2 6 B -10 0 2 -4 -2 C -12 -2 0 -8 -2 D -2 4 8 0 8 E -6 2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 2 6 B -10 0 2 -4 -2 C -12 -2 0 -8 -2 D -2 4 8 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 B=22 C=20 E=10 so E is eliminated. Round 2 votes counts: D=28 A=27 C=23 B=22 so B is eliminated. Round 3 votes counts: C=38 A=34 D=28 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:209 E:195 B:193 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 2 6 B -10 0 2 -4 -2 C -12 -2 0 -8 -2 D -2 4 8 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 2 6 B -10 0 2 -4 -2 C -12 -2 0 -8 -2 D -2 4 8 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 2 6 B -10 0 2 -4 -2 C -12 -2 0 -8 -2 D -2 4 8 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3153: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (19) C E B A D (15) E B A C D (10) E B A D C (6) D C A B E (6) E C B A D (5) E B C A D (4) C E B D A (4) A D B E C (4) C D A B E (3) D C A E B (2) C D E A B (2) C D A E B (2) B A E D C (2) A B E D C (2) A B D E C (2) E B D A C (1) D E B A C (1) D C E B A (1) D B E A C (1) D B A E C (1) D A C B E (1) C E D B A (1) C E A B D (1) C D E B A (1) B E A D C (1) B E A C D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 6 8 -10 B 10 0 10 8 -12 C -6 -10 0 0 -20 D -8 -8 0 0 -6 E 10 12 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 6 8 -10 B 10 0 10 8 -12 C -6 -10 0 0 -20 D -8 -8 0 0 -6 E 10 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=29 E=26 A=9 B=4 so B is eliminated. Round 2 votes counts: D=32 C=29 E=28 A=11 so A is eliminated. Round 3 votes counts: D=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:208 A:197 D:189 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 6 8 -10 B 10 0 10 8 -12 C -6 -10 0 0 -20 D -8 -8 0 0 -6 E 10 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 8 -10 B 10 0 10 8 -12 C -6 -10 0 0 -20 D -8 -8 0 0 -6 E 10 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 8 -10 B 10 0 10 8 -12 C -6 -10 0 0 -20 D -8 -8 0 0 -6 E 10 12 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3154: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) C B E A D (8) A C E B D (8) D E B A C (6) C B E D A (6) D B E C A (5) B E C D A (5) A D E C B (5) A C D B E (4) D E A B C (3) D A C B E (3) A C D E B (3) E B C A D (2) E A B D C (2) D A E C B (2) C D B E A (2) C A B E D (2) B C E D A (2) B C D E A (2) A D E B C (2) E B D C A (1) E B C D A (1) E B A D C (1) E B A C D (1) E A B C D (1) D E B C A (1) D C B E A (1) D C B A E (1) D C A B E (1) D B C E A (1) D B A E C (1) C D A B E (1) C B A E D (1) C A B D E (1) A E D B C (1) A E C B D (1) A E B D C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 12 -8 -2 B -4 0 -4 -8 -4 C -12 4 0 2 -2 D 8 8 -2 0 12 E 2 4 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.363636 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528925 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.454545 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 -8 -2 B -4 0 -4 -8 -4 C -12 4 0 2 -2 D 8 8 -2 0 12 E 2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.363636 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528926 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.454545 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=27 C=21 E=9 B=9 so E is eliminated. Round 2 votes counts: D=34 A=30 C=21 B=15 so B is eliminated. Round 3 votes counts: D=35 C=33 A=32 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:213 A:203 E:198 C:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 12 -8 -2 B -4 0 -4 -8 -4 C -12 4 0 2 -2 D 8 8 -2 0 12 E 2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.363636 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528926 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.454545 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 -8 -2 B -4 0 -4 -8 -4 C -12 4 0 2 -2 D 8 8 -2 0 12 E 2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.363636 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528926 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.454545 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 -8 -2 B -4 0 -4 -8 -4 C -12 4 0 2 -2 D 8 8 -2 0 12 E 2 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.363636 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528926 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.454545 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3155: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (14) B C E A D (13) C E B D A (12) A D B E C (9) B A D C E (7) E C D A B (6) E C B D A (6) B C A E D (6) B A C D E (5) C B E A D (3) B C A D E (3) A D E B C (3) E D C A B (2) E D A C B (2) D E A C B (2) D A E B C (2) A B D C E (2) E C D B A (1) D A B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -4 4 6 B 12 0 4 14 0 C 4 -4 0 10 4 D -4 -14 -10 0 -2 E -6 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.698770 C: 0.000000 D: 0.000000 E: 0.301230 Sum of squares = 0.579019188867 Cumulative probabilities = A: 0.000000 B: 0.698770 C: 0.698770 D: 0.698770 E: 1.000000 A B C D E A 0 -12 -4 4 6 B 12 0 4 14 0 C 4 -4 0 10 4 D -4 -14 -10 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500311 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000192808 Cumulative probabilities = A: 0.000000 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=19 E=17 C=15 A=15 so C is eliminated. Round 2 votes counts: B=37 E=29 D=19 A=15 so A is eliminated. Round 3 votes counts: B=39 D=32 E=29 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:207 A:197 E:196 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 4 6 B 12 0 4 14 0 C 4 -4 0 10 4 D -4 -14 -10 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500311 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000192808 Cumulative probabilities = A: 0.000000 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 4 6 B 12 0 4 14 0 C 4 -4 0 10 4 D -4 -14 -10 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500311 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000192808 Cumulative probabilities = A: 0.000000 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 4 6 B 12 0 4 14 0 C 4 -4 0 10 4 D -4 -14 -10 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500311 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000192808 Cumulative probabilities = A: 0.000000 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3156: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (19) C B A E D (19) D E A C B (5) B C A E D (5) B A C E D (5) B A C D E (5) E D C A B (4) D E C A B (4) C E D B A (3) A D E B C (3) D A E B C (2) C D E B A (2) C B D A E (2) C B A D E (2) B C A D E (2) A B E C D (2) A B D E C (2) A B C E D (2) E D A C B (1) E D A B C (1) E A C B D (1) E A B D C (1) D E C B A (1) D C E B A (1) C E B D A (1) C E B A D (1) C D B A E (1) C B E A D (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 2 6 8 B 2 0 2 2 -2 C -2 -2 0 8 4 D -6 -2 -8 0 4 E -8 2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 -2 2 6 8 B 2 0 2 2 -2 C -2 -2 0 8 4 D -6 -2 -8 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=32 C=32 B=17 A=11 E=8 so E is eliminated. Round 2 votes counts: D=38 C=32 B=17 A=13 so A is eliminated. Round 3 votes counts: D=43 C=33 B=24 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:207 C:204 B:202 D:194 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 6 8 B 2 0 2 2 -2 C -2 -2 0 8 4 D -6 -2 -8 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 6 8 B 2 0 2 2 -2 C -2 -2 0 8 4 D -6 -2 -8 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 6 8 B 2 0 2 2 -2 C -2 -2 0 8 4 D -6 -2 -8 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3157: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) B C E A D (9) A D E B C (9) E D A B C (7) C B E D A (6) D A E B C (5) E B C A D (4) B C A D E (4) E C B D A (3) E B D C A (3) D A C E B (3) C B E A D (3) B E C A D (3) E D C B A (2) E D A C B (2) E A D B C (2) D A C B E (2) C D B A E (2) C B A D E (2) A D C B E (2) E D C A B (1) E D B C A (1) E B D A C (1) E B C D A (1) E B A D C (1) D E A B C (1) C E B D A (1) C D E B A (1) C B D E A (1) C B A E D (1) B C A E D (1) B A C D E (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 -6 -6 B 2 0 16 -4 -16 C -2 -16 0 -12 -16 D 6 4 12 0 -6 E 6 16 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 -6 -6 B 2 0 16 -4 -16 C -2 -16 0 -12 -16 D 6 4 12 0 -6 E 6 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=21 B=18 C=17 A=16 so A is eliminated. Round 2 votes counts: D=34 E=28 B=21 C=17 so C is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:222 D:208 B:199 A:194 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -6 -6 B 2 0 16 -4 -16 C -2 -16 0 -12 -16 D 6 4 12 0 -6 E 6 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -6 -6 B 2 0 16 -4 -16 C -2 -16 0 -12 -16 D 6 4 12 0 -6 E 6 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -6 -6 B 2 0 16 -4 -16 C -2 -16 0 -12 -16 D 6 4 12 0 -6 E 6 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3158: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) D B A C E (7) A B D C E (6) B D A C E (5) E C A B D (4) D C A E B (4) D B C A E (4) E D C B A (3) E B C D A (3) D C B A E (3) D C A B E (3) C E A D B (3) B A D E C (3) A B E C D (3) E C D B A (2) E C A D B (2) D C E A B (2) D B E C A (2) C E D A B (2) C D E A B (2) C D A E B (2) C A D E B (2) B D E A C (2) B A E D C (2) A E C B D (2) A D B C E (2) E D B C A (1) E C B D A (1) E C B A D (1) E B D C A (1) E B A C D (1) E A C B D (1) E A B C D (1) D E B C A (1) D A C B E (1) D A B C E (1) B E A D C (1) B E A C D (1) B D A E C (1) A D C B E (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -10 -20 12 B -10 0 -4 -16 0 C 10 4 0 -12 8 D 20 16 12 0 10 E -12 0 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -10 -20 12 B -10 0 -4 -16 0 C 10 4 0 -12 8 D 20 16 12 0 10 E -12 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 A=18 B=15 C=11 so C is eliminated. Round 2 votes counts: E=33 D=32 A=20 B=15 so B is eliminated. Round 3 votes counts: D=40 E=35 A=25 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 C:205 A:196 B:185 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -10 -20 12 B -10 0 -4 -16 0 C 10 4 0 -12 8 D 20 16 12 0 10 E -12 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 -20 12 B -10 0 -4 -16 0 C 10 4 0 -12 8 D 20 16 12 0 10 E -12 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 -20 12 B -10 0 -4 -16 0 C 10 4 0 -12 8 D 20 16 12 0 10 E -12 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3159: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (13) C D A B E (13) D C E B A (6) A B E C D (6) E A B D C (5) E B A C D (4) C D B A E (4) B E A C D (4) A B C E D (4) D C A E B (3) D C A B E (3) A C D B E (3) E D B A C (2) E B D C A (2) E A B C D (2) D C E A B (2) D A C E B (2) C D B E A (2) B E C A D (2) B A E C D (2) A C B D E (2) A B E D C (2) A B C D E (2) E D C B A (1) E D B C A (1) E B D A C (1) C D E B A (1) C B E D A (1) C A B D E (1) B E C D A (1) B A C E D (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 14 10 0 B -4 0 12 12 6 C -14 -12 0 10 0 D -10 -12 -10 0 -10 E 0 -6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.783242 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.216758 Sum of squares = 0.660451652059 Cumulative probabilities = A: 0.783242 B: 0.783242 C: 0.783242 D: 0.783242 E: 1.000000 A B C D E A 0 4 14 10 0 B -4 0 12 12 6 C -14 -12 0 10 0 D -10 -12 -10 0 -10 E 0 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399998 Sum of squares = 0.520000602277 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 0.600002 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 A=21 D=16 B=10 so B is eliminated. Round 2 votes counts: E=38 A=24 C=22 D=16 so D is eliminated. Round 3 votes counts: E=38 C=36 A=26 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 B:213 E:202 C:192 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 0 B -4 0 12 12 6 C -14 -12 0 10 0 D -10 -12 -10 0 -10 E 0 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399998 Sum of squares = 0.520000602277 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 0.600002 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 0 B -4 0 12 12 6 C -14 -12 0 10 0 D -10 -12 -10 0 -10 E 0 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399998 Sum of squares = 0.520000602277 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 0.600002 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 0 B -4 0 12 12 6 C -14 -12 0 10 0 D -10 -12 -10 0 -10 E 0 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399998 Sum of squares = 0.520000602277 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 0.600002 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3160: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (13) B E D A C (13) C A D B E (8) C B E A D (7) D E A B C (5) B E D C A (5) C B A E D (4) B C E D A (4) A D E C B (4) D E B A C (3) D A E B C (3) C A B E D (3) C A B D E (3) A D E B C (3) A C D E B (3) E B D A C (2) C B E D A (2) B E C A D (2) B C E A D (2) A D C E B (2) E D B A C (1) E A D B C (1) D A C E B (1) C D B E A (1) C B A D E (1) B E C D A (1) B D E C A (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 2 -14 16 0 B -2 0 -6 0 14 C 14 6 0 8 8 D -16 0 -8 0 2 E 0 -14 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 16 0 B -2 0 -6 0 14 C 14 6 0 8 8 D -16 0 -8 0 2 E 0 -14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 B=28 A=14 D=12 E=4 so E is eliminated. Round 2 votes counts: C=42 B=30 A=15 D=13 so D is eliminated. Round 3 votes counts: C=42 B=34 A=24 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:203 A:202 D:189 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 16 0 B -2 0 -6 0 14 C 14 6 0 8 8 D -16 0 -8 0 2 E 0 -14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 16 0 B -2 0 -6 0 14 C 14 6 0 8 8 D -16 0 -8 0 2 E 0 -14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 16 0 B -2 0 -6 0 14 C 14 6 0 8 8 D -16 0 -8 0 2 E 0 -14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3161: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) A C B E D (10) B C A D E (9) C A B D E (6) C B A D E (5) A E D C B (5) D B E C A (4) B C D A E (4) E D B C A (3) E D B A C (3) E D A B C (3) D E B C A (3) C B D A E (3) A B C D E (3) E A D B C (2) D E C B A (2) D E B A C (2) B D E C A (2) B D C E A (2) B C D E A (2) A E C D B (2) A B E C D (2) E D C B A (1) D B C E A (1) C E D A B (1) C B D E A (1) C A E B D (1) C A D B E (1) C A B E D (1) B D A E C (1) A E D B C (1) A C E B D (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -4 4 16 B -4 0 -2 12 20 C 4 2 0 8 6 D -4 -12 -8 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 4 16 B -4 0 -2 12 20 C 4 2 0 8 6 D -4 -12 -8 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 B=20 C=19 D=12 so D is eliminated. Round 2 votes counts: E=29 A=27 B=25 C=19 so C is eliminated. Round 3 votes counts: A=36 B=34 E=30 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:210 C:210 D:190 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 16 B -4 0 -2 12 20 C 4 2 0 8 6 D -4 -12 -8 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 16 B -4 0 -2 12 20 C 4 2 0 8 6 D -4 -12 -8 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 16 B -4 0 -2 12 20 C 4 2 0 8 6 D -4 -12 -8 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3162: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) E A B D C (9) C D E A B (8) B C D A E (8) B A E D C (7) C D A E B (6) E A D C B (5) A E D C B (5) D C A E B (4) C D B E A (4) D C E A B (3) C D B A E (3) B E A C D (3) B C D E A (3) E B A D C (2) B C E A D (2) B C A D E (2) B A E C D (2) A E D B C (2) E D A C B (1) E A D B C (1) D E C A B (1) D A E C B (1) D A C E B (1) C E D B A (1) C B D E A (1) B C E D A (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 6 8 -14 B 2 0 12 8 -2 C -6 -12 0 -12 -6 D -8 -8 12 0 -8 E 14 2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 8 -14 B 2 0 12 8 -2 C -6 -12 0 -12 -6 D -8 -8 12 0 -8 E 14 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 C=23 E=18 D=10 A=9 so A is eliminated. Round 2 votes counts: B=41 E=26 C=23 D=10 so D is eliminated. Round 3 votes counts: B=41 C=31 E=28 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:210 A:199 D:194 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 8 -14 B 2 0 12 8 -2 C -6 -12 0 -12 -6 D -8 -8 12 0 -8 E 14 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 8 -14 B 2 0 12 8 -2 C -6 -12 0 -12 -6 D -8 -8 12 0 -8 E 14 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 8 -14 B 2 0 12 8 -2 C -6 -12 0 -12 -6 D -8 -8 12 0 -8 E 14 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3163: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) A D E C B (10) C A B D E (9) E D B C A (8) B C E D A (7) A C B D E (6) C B A E D (5) B E C D A (5) E D B A C (4) A D C E B (4) D A E C B (3) C B E D A (3) A D E B C (3) A C D B E (3) E B D C A (2) D E B A C (2) D A E B C (2) C B E A D (2) B C A E D (2) A C D E B (2) E D A B C (1) E C B D A (1) C E A D B (1) C B A D E (1) B E D C A (1) B C E A D (1) Total count = 100 A B C D E A 0 12 4 -2 0 B -12 0 0 -10 -10 C -4 0 0 -4 -8 D 2 10 4 0 14 E 0 10 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -2 0 B -12 0 0 -10 -10 C -4 0 0 -4 -8 D 2 10 4 0 14 E 0 10 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998516 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=21 D=19 E=16 B=16 so E is eliminated. Round 2 votes counts: D=32 A=28 C=22 B=18 so B is eliminated. Round 3 votes counts: C=37 D=35 A=28 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:207 E:202 C:192 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 4 -2 0 B -12 0 0 -10 -10 C -4 0 0 -4 -8 D 2 10 4 0 14 E 0 10 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998516 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -2 0 B -12 0 0 -10 -10 C -4 0 0 -4 -8 D 2 10 4 0 14 E 0 10 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998516 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -2 0 B -12 0 0 -10 -10 C -4 0 0 -4 -8 D 2 10 4 0 14 E 0 10 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998516 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3164: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (16) A B D C E (11) B D A C E (10) A B D E C (10) E C D B A (8) E C A D B (7) A E C B D (7) D B C E A (5) A E C D B (5) E C A B D (4) B D C E A (4) E C D A B (2) C E B D A (2) B A D C E (2) A D B E C (2) E A C D B (1) D B C A E (1) A E B C D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 2 4 2 B -4 0 -6 4 -8 C -2 6 0 8 2 D -4 -4 -8 0 -8 E -2 8 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 4 2 B -4 0 -6 4 -8 C -2 6 0 8 2 D -4 -4 -8 0 -8 E -2 8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=22 C=18 B=16 D=6 so D is eliminated. Round 2 votes counts: A=38 E=22 B=22 C=18 so C is eliminated. Round 3 votes counts: E=40 A=38 B=22 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:206 E:206 B:193 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 4 2 B -4 0 -6 4 -8 C -2 6 0 8 2 D -4 -4 -8 0 -8 E -2 8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 2 B -4 0 -6 4 -8 C -2 6 0 8 2 D -4 -4 -8 0 -8 E -2 8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 2 B -4 0 -6 4 -8 C -2 6 0 8 2 D -4 -4 -8 0 -8 E -2 8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3165: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (7) E C A D B (6) D B C A E (6) E C A B D (4) E B D C A (4) B D C E A (4) A D B C E (4) A C E D B (4) E B C D A (3) E A C B D (3) D B A C E (3) B D E C A (3) B D E A C (3) A D C B E (3) A C D E B (3) A B D E C (3) E C D B A (2) E C B A D (2) E B A D C (2) E B A C D (2) D C B A E (2) D B C E A (2) C D A E B (2) B E D C A (2) B E D A C (2) B E A D C (2) A E C D B (2) A E C B D (2) A B D C E (2) E C D A B (1) E A C D B (1) D C B E A (1) D A C B E (1) D A B C E (1) C E A D B (1) C D E B A (1) C A E D B (1) B D A C E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 6 6 12 -2 B -6 0 2 -8 4 C -6 -2 0 -4 -2 D -12 8 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 6 6 12 -2 B -6 0 2 -8 4 C -6 -2 0 -4 -2 D -12 8 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=30 B=17 D=16 C=5 so C is eliminated. Round 2 votes counts: A=33 E=31 D=19 B=17 so B is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:211 D:202 E:198 B:196 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 12 -2 B -6 0 2 -8 4 C -6 -2 0 -4 -2 D -12 8 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 12 -2 B -6 0 2 -8 4 C -6 -2 0 -4 -2 D -12 8 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 12 -2 B -6 0 2 -8 4 C -6 -2 0 -4 -2 D -12 8 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3166: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) C D A B E (6) E D A B C (4) D A C E B (4) C B E D A (4) C B D E A (4) C B A E D (4) B E C A D (4) B E A C D (4) A E D B C (4) A E B C D (4) E B C D A (3) E B A D C (3) D C E B A (3) D C A B E (3) C B A D E (3) B C E A D (3) D E B A C (2) C B D A E (2) C A B D E (2) A D E B C (2) A D C E B (2) A B E C D (2) E D B A C (1) E B A C D (1) D E C B A (1) D E B C A (1) D E A B C (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C B E (1) C D B E A (1) C D B A E (1) C A D B E (1) B E C D A (1) B C E D A (1) B A E C D (1) B A C E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 0 12 B 2 0 4 12 6 C 4 -4 0 10 4 D 0 -12 -10 0 -6 E -12 -6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 0 12 B 2 0 4 12 6 C 4 -4 0 10 4 D 0 -12 -10 0 -6 E -12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 D=22 B=15 E=12 so E is eliminated. Round 2 votes counts: C=28 D=27 A=23 B=22 so B is eliminated. Round 3 votes counts: C=40 A=33 D=27 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:212 C:207 A:203 E:192 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 0 12 B 2 0 4 12 6 C 4 -4 0 10 4 D 0 -12 -10 0 -6 E -12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 0 12 B 2 0 4 12 6 C 4 -4 0 10 4 D 0 -12 -10 0 -6 E -12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 0 12 B 2 0 4 12 6 C 4 -4 0 10 4 D 0 -12 -10 0 -6 E -12 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3167: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) D B E A C (9) D B C A E (6) D C B A E (5) D C A B E (5) D B E C A (5) B E D A C (5) B E A C D (5) D E B A C (4) C A D E B (4) B D E A C (4) A C E B D (4) C A E D B (3) E A C B D (2) D C A E B (2) B E A D C (2) B C A E D (2) A E C B D (2) E B D A C (1) E B A D C (1) E B A C D (1) E A C D B (1) D E C A B (1) D E A C B (1) D B C E A (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D B E (1) C A B D E (1) B E D C A (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D E A (1) A E C D B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -10 -10 4 B 16 0 2 -4 18 C 10 -2 0 -8 4 D 10 4 8 0 8 E -4 -18 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -10 4 B 16 0 2 -4 18 C 10 -2 0 -8 4 D 10 4 8 0 8 E -4 -18 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=23 B=23 A=9 E=6 so E is eliminated. Round 2 votes counts: D=39 B=26 C=23 A=12 so A is eliminated. Round 3 votes counts: D=39 C=35 B=26 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:215 C:202 A:184 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -10 -10 4 B 16 0 2 -4 18 C 10 -2 0 -8 4 D 10 4 8 0 8 E -4 -18 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -10 4 B 16 0 2 -4 18 C 10 -2 0 -8 4 D 10 4 8 0 8 E -4 -18 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -10 4 B 16 0 2 -4 18 C 10 -2 0 -8 4 D 10 4 8 0 8 E -4 -18 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3168: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (11) B A D E C (7) D C E A B (6) C E D A B (6) E C A D B (4) D B A C E (4) A B D E C (4) E C D A B (3) E C B A D (3) E A C D B (3) D A C E B (3) C E D B A (3) C D E A B (3) B D A C E (3) B C D E A (3) B A E D C (3) B A D C E (3) E C A B D (2) D C E B A (2) D B C A E (2) D A B C E (2) A E C D B (2) A E B D C (2) A D B E C (2) A B E C D (2) E C B D A (1) E B C A D (1) E B A C D (1) E A C B D (1) C D E B A (1) C D B E A (1) B E C A D (1) B D C A E (1) B C E A D (1) A E D C B (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 12 12 8 B 4 0 10 2 2 C -12 -10 0 6 -12 D -12 -2 -6 0 -6 E -8 -2 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 12 8 B 4 0 10 2 2 C -12 -10 0 6 -12 D -12 -2 -6 0 -6 E -8 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=19 D=19 A=15 C=14 so C is eliminated. Round 2 votes counts: B=33 E=28 D=24 A=15 so A is eliminated. Round 3 votes counts: B=40 E=34 D=26 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:209 E:204 D:187 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 12 8 B 4 0 10 2 2 C -12 -10 0 6 -12 D -12 -2 -6 0 -6 E -8 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 12 8 B 4 0 10 2 2 C -12 -10 0 6 -12 D -12 -2 -6 0 -6 E -8 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 12 8 B 4 0 10 2 2 C -12 -10 0 6 -12 D -12 -2 -6 0 -6 E -8 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996717 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3169: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A E B D C (11) A D C B E (11) C D B E A (8) E C B D A (4) E A B C D (4) D C B E A (4) A D B E C (4) E B A C D (3) A E B C D (3) A B D E C (3) E B D C A (2) C E D B A (2) C E B D A (2) C D B A E (2) B D C E A (2) A E C D B (2) A E C B D (2) A D B C E (2) E C B A D (1) E B D A C (1) E B C A D (1) E A C B D (1) D C B A E (1) D B C E A (1) C E A D B (1) C D E B A (1) C D A E B (1) B E D C A (1) B E D A C (1) B D E C A (1) B D A E C (1) B A D E C (1) A D C E B (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 8 8 -4 B 2 0 8 14 -10 C -8 -8 0 2 -18 D -8 -14 -2 0 -8 E 4 10 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 8 -4 B 2 0 8 14 -10 C -8 -8 0 2 -18 D -8 -14 -2 0 -8 E 4 10 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999168 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=28 C=17 B=7 D=6 so D is eliminated. Round 2 votes counts: A=42 E=28 C=22 B=8 so B is eliminated. Round 3 votes counts: A=44 E=31 C=25 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:207 A:205 C:184 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 8 -4 B 2 0 8 14 -10 C -8 -8 0 2 -18 D -8 -14 -2 0 -8 E 4 10 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999168 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 8 -4 B 2 0 8 14 -10 C -8 -8 0 2 -18 D -8 -14 -2 0 -8 E 4 10 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999168 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 8 -4 B 2 0 8 14 -10 C -8 -8 0 2 -18 D -8 -14 -2 0 -8 E 4 10 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999168 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3170: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) E D B A C (5) D E A B C (5) D C A E B (5) C D A B E (5) B A E C D (5) A B C E D (5) D E B A C (4) C B E A D (4) B E A C D (4) D C E B A (3) D C E A B (3) C D E B A (3) C B A E D (3) C A D B E (3) A B E D C (3) E B A D C (2) D A E B C (2) D A C E B (2) C B E D A (2) C A B E D (2) C A B D E (2) A D E B C (2) A D B E C (2) A C B D E (2) A B E C D (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B A D (1) E B D C A (1) E B D A C (1) E B A C D (1) D E B C A (1) D C A B E (1) B C A E D (1) B A C E D (1) A D C B E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 2 -4 2 B 2 0 -2 -12 0 C -2 2 0 -4 -2 D 4 12 4 0 6 E -2 0 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 2 B 2 0 -2 -12 0 C -2 2 0 -4 -2 D 4 12 4 0 6 E -2 0 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=24 A=19 E=14 B=11 so B is eliminated. Round 2 votes counts: D=32 C=25 A=25 E=18 so E is eliminated. Round 3 votes counts: D=41 A=32 C=27 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:199 C:197 E:197 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -4 2 B 2 0 -2 -12 0 C -2 2 0 -4 -2 D 4 12 4 0 6 E -2 0 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 2 B 2 0 -2 -12 0 C -2 2 0 -4 -2 D 4 12 4 0 6 E -2 0 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 2 B 2 0 -2 -12 0 C -2 2 0 -4 -2 D 4 12 4 0 6 E -2 0 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3171: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) C E A B D (9) D A B C E (8) B A D C E (7) E B A C D (5) D C A B E (5) C A B E D (5) B A C E D (5) E D B A C (4) D E C B A (4) D E B A C (4) E C D A B (3) D B A E C (3) A B C D E (3) E D C B A (2) E B C A D (2) E B A D C (2) D E C A B (2) D C E A B (2) B E A C D (2) B A C D E (2) E D C A B (1) E C A B D (1) D E B C A (1) D C A E B (1) C E D A B (1) C D A B E (1) B E A D C (1) B A E C D (1) B A D E C (1) A D B C E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 2 16 -10 B 10 0 8 14 -6 C -2 -8 0 0 4 D -16 -14 0 0 -8 E 10 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691358 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 A B C D E A 0 -10 2 16 -10 B 10 0 8 14 -6 C -2 -8 0 0 4 D -16 -14 0 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=29 B=19 C=16 A=6 so A is eliminated. Round 2 votes counts: D=31 E=29 B=24 C=16 so C is eliminated. Round 3 votes counts: E=39 D=32 B=29 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:210 A:199 C:197 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 16 -10 B 10 0 8 14 -6 C -2 -8 0 0 4 D -16 -14 0 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 16 -10 B 10 0 8 14 -6 C -2 -8 0 0 4 D -16 -14 0 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 16 -10 B 10 0 8 14 -6 C -2 -8 0 0 4 D -16 -14 0 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3172: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) A C D B E (8) B D E C A (7) C D B E A (6) D B C A E (5) E B D A C (4) A D B C E (4) A C E D B (4) E A B D C (3) C E A D B (3) B D C E A (3) A B E D C (3) E C A B D (2) E B D C A (2) E B A D C (2) E A B C D (2) D B C E A (2) D B A C E (2) C E D B A (2) C D B A E (2) C D A B E (2) C A E D B (2) B E D C A (2) B E D A C (2) B D A E C (2) A E C D B (2) A E C B D (2) E C B D A (1) E B C D A (1) E A C B D (1) D C B E A (1) D C B A E (1) D A B C E (1) C D E B A (1) C A D E B (1) C A D B E (1) B D E A C (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 6 0 4 B -2 0 14 -2 12 C -6 -14 0 -12 6 D 0 2 12 0 4 E -4 -12 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.631829 B: 0.000000 C: 0.000000 D: 0.368171 E: 0.000000 Sum of squares = 0.534757912708 Cumulative probabilities = A: 0.631829 B: 0.631829 C: 0.631829 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 0 4 B -2 0 14 -2 12 C -6 -14 0 -12 6 D 0 2 12 0 4 E -4 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=20 E=18 B=17 D=12 so D is eliminated. Round 2 votes counts: A=34 B=26 C=22 E=18 so E is eliminated. Round 3 votes counts: A=40 B=35 C=25 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:211 D:209 A:206 C:187 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 0 4 B -2 0 14 -2 12 C -6 -14 0 -12 6 D 0 2 12 0 4 E -4 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 4 B -2 0 14 -2 12 C -6 -14 0 -12 6 D 0 2 12 0 4 E -4 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 4 B -2 0 14 -2 12 C -6 -14 0 -12 6 D 0 2 12 0 4 E -4 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3173: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) D A C B E (7) A B C D E (7) B C A E D (6) A B C E D (6) E B C A D (5) D E C A B (5) D E A C B (4) A C B D E (4) E D A B C (3) D E C B A (3) D E A B C (3) B C E A D (3) E D B C A (2) E C D B A (2) E B A C D (2) D C B A E (2) D C A B E (2) C D B A E (2) C B D A E (2) C B A D E (2) B C A D E (2) A D B C E (2) A B E C D (2) E D B A C (1) E D A C B (1) E B C D A (1) E A D B C (1) E A B C D (1) D C E B A (1) D C E A B (1) D A E B C (1) C E B D A (1) C D A B E (1) C B D E A (1) B A C E D (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 6 -4 -8 2 B -6 0 0 -8 6 C 4 0 0 2 8 D 8 8 -2 0 8 E -2 -6 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.096923 C: 0.903077 D: 0.000000 E: 0.000000 Sum of squares = 0.824941748431 Cumulative probabilities = A: 0.000000 B: 0.096923 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -8 2 B -6 0 0 -8 6 C 4 0 0 2 8 D 8 8 -2 0 8 E -2 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000685 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=27 A=23 B=12 C=9 so C is eliminated. Round 2 votes counts: D=32 E=28 A=23 B=17 so B is eliminated. Round 3 votes counts: D=35 A=34 E=31 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:207 A:198 B:196 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -8 2 B -6 0 0 -8 6 C 4 0 0 2 8 D 8 8 -2 0 8 E -2 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000685 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -8 2 B -6 0 0 -8 6 C 4 0 0 2 8 D 8 8 -2 0 8 E -2 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000685 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -8 2 B -6 0 0 -8 6 C 4 0 0 2 8 D 8 8 -2 0 8 E -2 -6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000685 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3174: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (11) C B A D E (7) B C E A D (6) B C A E D (6) B C A D E (6) E D C A B (5) E D A B C (5) A E D B C (5) C D A E B (4) B A C E D (4) E A D B C (3) D E A C B (3) D A E C B (3) C D E A B (3) C B E D A (3) A D E B C (3) C B D A E (2) C A D B E (2) A D C E B (2) E D C B A (1) E C D B A (1) E B C D A (1) E B A D C (1) E A B D C (1) D E C A B (1) D C E A B (1) D A E B C (1) C D B E A (1) C B E A D (1) B E C D A (1) B E A D C (1) B A E D C (1) B A E C D (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -4 6 0 B -12 0 -2 -10 -10 C 4 2 0 -2 -2 D -6 10 2 0 -18 E 0 10 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.315241 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.684759 Sum of squares = 0.56827150013 Cumulative probabilities = A: 0.315241 B: 0.315241 C: 0.315241 D: 0.315241 E: 1.000000 A B C D E A 0 12 -4 6 0 B -12 0 -2 -10 -10 C 4 2 0 -2 -2 D -6 10 2 0 -18 E 0 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555630684 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=26 C=23 A=13 D=9 so D is eliminated. Round 2 votes counts: E=33 B=26 C=24 A=17 so A is eliminated. Round 3 votes counts: E=45 B=29 C=26 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:207 C:201 D:194 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -4 6 0 B -12 0 -2 -10 -10 C 4 2 0 -2 -2 D -6 10 2 0 -18 E 0 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555630684 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 6 0 B -12 0 -2 -10 -10 C 4 2 0 -2 -2 D -6 10 2 0 -18 E 0 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555630684 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 6 0 B -12 0 -2 -10 -10 C 4 2 0 -2 -2 D -6 10 2 0 -18 E 0 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555630684 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3175: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) B E A D C (6) E B A C D (5) D C E B A (5) C E B D A (5) A B E D C (5) D B E C A (4) D A B E C (4) C E D B A (4) C E A B D (4) A B D E C (4) E C B D A (3) E C B A D (3) D C A B E (3) C E B A D (3) C D E A B (3) A D B E C (3) A C E B D (3) D C A E B (2) D B E A C (2) D B A E C (2) C D E B A (2) C D A E B (2) C A E D B (2) B A D E C (2) A E B C D (2) E A B C D (1) D E C B A (1) D C B A E (1) D B A C E (1) D A C B E (1) C E A D B (1) C A E B D (1) B D A E C (1) B A E D C (1) Total count = 100 A B C D E A 0 -2 2 8 -4 B 2 0 2 14 -4 C -2 -2 0 4 -14 D -8 -14 -4 0 -14 E 4 4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 8 -4 B 2 0 2 14 -4 C -2 -2 0 4 -14 D -8 -14 -4 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 A=25 E=12 B=10 so B is eliminated. Round 2 votes counts: A=28 D=27 C=27 E=18 so E is eliminated. Round 3 votes counts: A=40 C=33 D=27 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:218 B:207 A:202 C:193 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 8 -4 B 2 0 2 14 -4 C -2 -2 0 4 -14 D -8 -14 -4 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 8 -4 B 2 0 2 14 -4 C -2 -2 0 4 -14 D -8 -14 -4 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 8 -4 B 2 0 2 14 -4 C -2 -2 0 4 -14 D -8 -14 -4 0 -14 E 4 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3176: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C D A E B (6) D E B A C (5) D C A E B (5) B A E C D (5) A B C E D (5) A B C D E (5) D E C B A (4) D E B C A (4) C A B E D (4) A C B E D (4) D C E B A (3) D B E A C (3) C A D B E (3) A C B D E (3) E D B C A (2) E D B A C (2) E C D B A (2) E B A C D (2) C A B D E (2) B E A C D (2) B A E D C (2) A B E C D (2) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A D C (1) D C E A B (1) D B A E C (1) D A C B E (1) C E A D B (1) C D E A B (1) C A E D B (1) B E D A C (1) B E A D C (1) B D E A C (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 14 -4 2 B 8 0 14 10 -6 C -14 -14 0 4 -6 D 4 -10 -4 0 0 E -2 6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.353853 E: 0.646147 Sum of squares = 0.542718139677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.353853 E: 1.000000 A B C D E A 0 -8 14 -4 2 B 8 0 14 10 -6 C -14 -14 0 4 -6 D 4 -10 -4 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.625000 Sum of squares = 0.531250038573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=22 A=20 C=18 B=13 so B is eliminated. Round 2 votes counts: D=28 A=28 E=26 C=18 so C is eliminated. Round 3 votes counts: A=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:213 E:205 A:202 D:195 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 14 -4 2 B 8 0 14 10 -6 C -14 -14 0 4 -6 D 4 -10 -4 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.625000 Sum of squares = 0.531250038573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 -4 2 B 8 0 14 10 -6 C -14 -14 0 4 -6 D 4 -10 -4 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.625000 Sum of squares = 0.531250038573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 -4 2 B 8 0 14 10 -6 C -14 -14 0 4 -6 D 4 -10 -4 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.625000 Sum of squares = 0.531250038573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3177: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (23) B E D C A (17) C A D E B (11) B E D A C (7) D C E A B (6) D E B C A (5) A C B D E (4) E D B C A (3) C D A E B (3) B A E C D (3) A C B E D (3) E B D C A (2) C D E A B (2) A C D B E (2) D E C B A (1) D E C A B (1) D B E C A (1) C A E D B (1) C A E B D (1) B E A C D (1) B A C E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 18 -8 4 8 B -18 0 -16 -18 -18 C 8 16 0 12 18 D -4 18 -12 0 20 E -8 18 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -8 4 8 B -18 0 -16 -18 -18 C 8 16 0 12 18 D -4 18 -12 0 20 E -8 18 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=29 C=18 D=14 E=5 so E is eliminated. Round 2 votes counts: A=34 B=31 C=18 D=17 so D is eliminated. Round 3 votes counts: B=40 A=34 C=26 so C is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:227 A:211 D:211 E:186 B:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -8 4 8 B -18 0 -16 -18 -18 C 8 16 0 12 18 D -4 18 -12 0 20 E -8 18 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 4 8 B -18 0 -16 -18 -18 C 8 16 0 12 18 D -4 18 -12 0 20 E -8 18 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 4 8 B -18 0 -16 -18 -18 C 8 16 0 12 18 D -4 18 -12 0 20 E -8 18 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3178: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) E D C B A (9) B C A D E (8) D E C B A (7) A B C E D (6) A B C D E (4) E D A C B (3) E C B D A (3) D E A C B (3) D C E B A (3) B C E D A (3) B C D A E (3) B A C E D (3) A D B C E (3) E C D B A (2) B C E A D (2) B A C D E (2) A E D B C (2) A E B D C (2) A D E C B (2) A D E B C (2) E D C A B (1) E B C A D (1) D C B E A (1) D A E C B (1) C E D B A (1) C D E B A (1) C D B E A (1) C B E D A (1) C B D E A (1) C B A D E (1) B C D E A (1) B A E C D (1) A E D C B (1) A D C B E (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -32 -22 10 12 B 32 0 14 10 12 C 22 -14 0 14 14 D -10 -10 -14 0 -8 E -12 -12 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 -22 10 12 B 32 0 14 10 12 C 22 -14 0 14 14 D -10 -10 -14 0 -8 E -12 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998367 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=26 E=19 D=15 C=6 so C is eliminated. Round 2 votes counts: B=37 A=26 E=20 D=17 so D is eliminated. Round 3 votes counts: B=39 E=34 A=27 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:234 C:218 E:185 A:184 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -32 -22 10 12 B 32 0 14 10 12 C 22 -14 0 14 14 D -10 -10 -14 0 -8 E -12 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998367 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 -22 10 12 B 32 0 14 10 12 C 22 -14 0 14 14 D -10 -10 -14 0 -8 E -12 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998367 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 -22 10 12 B 32 0 14 10 12 C 22 -14 0 14 14 D -10 -10 -14 0 -8 E -12 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998367 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3179: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) B D C E A (9) B D E C A (8) A E B D C (8) C D B E A (7) A B E D C (5) C A E D B (4) A E D B C (4) A E C D B (4) E D A B C (3) E A D B C (3) C D E B A (3) B E D A C (3) B D E A C (3) A E D C B (3) D C B E A (2) B A E D C (2) A E B C D (2) E D B C A (1) E D B A C (1) E D A C B (1) E C D A B (1) E A B D C (1) D E B C A (1) D E B A C (1) D B C E A (1) C D E A B (1) C B D A E (1) C A D E B (1) C A B D E (1) B E A D C (1) B C D A E (1) B C A D E (1) A C E D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 -14 -20 B 10 0 22 14 12 C 4 -22 0 -22 -12 D 14 -14 22 0 2 E 20 -12 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -14 -20 B 10 0 22 14 12 C 4 -22 0 -22 -12 D 14 -14 22 0 2 E 20 -12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 C=27 E=11 D=5 so D is eliminated. Round 2 votes counts: C=29 B=29 A=29 E=13 so E is eliminated. Round 3 votes counts: A=37 B=33 C=30 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:212 E:209 A:176 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -14 -20 B 10 0 22 14 12 C 4 -22 0 -22 -12 D 14 -14 22 0 2 E 20 -12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -14 -20 B 10 0 22 14 12 C 4 -22 0 -22 -12 D 14 -14 22 0 2 E 20 -12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -14 -20 B 10 0 22 14 12 C 4 -22 0 -22 -12 D 14 -14 22 0 2 E 20 -12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3180: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) C A B D E (7) E D B A C (6) E B C D A (6) E D A B C (5) C A D B E (5) B C D A E (5) B C A D E (5) A C D B E (5) E B D C A (4) D A E B C (4) D A B C E (4) E B D A C (3) C B A E D (3) A D C B E (3) E D A C B (2) D E A B C (2) D B A C E (2) C E B A D (2) B E C D A (2) B D C A E (2) E C B D A (1) E C A B D (1) E B C A D (1) E A D C B (1) D B E A C (1) D A E C B (1) D A C B E (1) D A B E C (1) C B E A D (1) B D C E A (1) B C E A D (1) B C D E A (1) A E C D B (1) A D E C B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -10 -8 18 B 8 0 12 6 14 C 10 -12 0 10 14 D 8 -6 -10 0 18 E -18 -14 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -8 18 B 8 0 12 6 14 C 10 -12 0 10 14 D 8 -6 -10 0 18 E -18 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=25 B=17 D=16 A=12 so A is eliminated. Round 2 votes counts: E=31 C=31 D=21 B=17 so B is eliminated. Round 3 votes counts: C=43 E=33 D=24 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:220 C:211 D:205 A:196 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -8 18 B 8 0 12 6 14 C 10 -12 0 10 14 D 8 -6 -10 0 18 E -18 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -8 18 B 8 0 12 6 14 C 10 -12 0 10 14 D 8 -6 -10 0 18 E -18 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -8 18 B 8 0 12 6 14 C 10 -12 0 10 14 D 8 -6 -10 0 18 E -18 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3181: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) E A B D C (7) D B C A E (7) A E C D B (7) C D B A E (6) E B A D C (4) B D C E A (4) A C E D B (4) E C A B D (3) E A C D B (3) D C B A E (3) C B D A E (3) B E D C A (3) B D E A C (3) B D C A E (3) A E D C B (3) E A D C B (2) E A D B C (2) C B D E A (2) B D E C A (2) A E D B C (2) E C B A D (1) E B A C D (1) E A B C D (1) D B A E C (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A B D (1) C D A B E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E C D A (1) B C E D A (1) B C D E A (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 6 8 14 -6 B -6 0 -12 4 -10 C -8 12 0 0 -16 D -14 -4 0 0 -16 E 6 10 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 14 -6 B -6 0 -12 4 -10 C -8 12 0 0 -16 D -14 -4 0 0 -16 E 6 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=18 A=18 C=17 D=13 so D is eliminated. Round 2 votes counts: E=34 B=26 C=20 A=20 so C is eliminated. Round 3 votes counts: B=40 E=36 A=24 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:211 C:194 B:188 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 14 -6 B -6 0 -12 4 -10 C -8 12 0 0 -16 D -14 -4 0 0 -16 E 6 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 14 -6 B -6 0 -12 4 -10 C -8 12 0 0 -16 D -14 -4 0 0 -16 E 6 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 14 -6 B -6 0 -12 4 -10 C -8 12 0 0 -16 D -14 -4 0 0 -16 E 6 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3182: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (5) E A D C B (5) D B E A C (5) C A E B D (5) B D E C A (5) A C E D B (5) E A C D B (4) D A E C B (4) C B A D E (4) B C D A E (4) E D B A C (3) E D A B C (3) E B D A C (3) D E B A C (3) C B A E D (3) B E D C A (3) B D C E A (3) B D C A E (3) A E C D B (3) E B C A D (2) E B A C D (2) D E A B C (2) C A B E D (2) C A B D E (2) B C E A D (2) B C A D E (2) A C D E B (2) E C A B D (1) D E A C B (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A C B E (1) B E C D A (1) B C E D A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 12 -8 -8 B 2 0 -2 -4 -12 C -12 2 0 -10 -16 D 8 4 10 0 -8 E 8 12 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 12 -8 -8 B 2 0 -2 -4 -12 C -12 2 0 -10 -16 D 8 4 10 0 -8 E 8 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=24 D=20 C=16 A=12 so A is eliminated. Round 2 votes counts: E=32 B=24 C=23 D=21 so D is eliminated. Round 3 votes counts: E=43 B=31 C=26 so C is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:207 A:197 B:192 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 12 -8 -8 B 2 0 -2 -4 -12 C -12 2 0 -10 -16 D 8 4 10 0 -8 E 8 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -8 -8 B 2 0 -2 -4 -12 C -12 2 0 -10 -16 D 8 4 10 0 -8 E 8 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -8 -8 B 2 0 -2 -4 -12 C -12 2 0 -10 -16 D 8 4 10 0 -8 E 8 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3183: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) D E C A B (7) D C E B A (7) B C A D E (6) B C D A E (5) A E D B C (5) A B E D C (5) E D A C B (4) C B D E A (4) C B A E D (4) E D C A B (3) C E D B A (3) C D E B A (3) C D B E A (3) B C A E D (3) A E B C D (3) E C A B D (2) E A D C B (2) E A D B C (2) D E A C B (2) B A D E C (2) B A D C E (2) A E B D C (2) A B D E C (2) E C D A B (1) E C A D B (1) E A C D B (1) D C B E A (1) D B C E A (1) C E A B D (1) C B E A D (1) C B D A E (1) B D A C E (1) B A E C D (1) B A C D E (1) Total count = 100 A B C D E A 0 -14 -14 8 2 B 14 0 -2 8 2 C 14 2 0 4 10 D -8 -8 -4 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 8 2 B 14 0 -2 8 2 C 14 2 0 4 10 D -8 -8 -4 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=20 D=18 A=17 E=16 so E is eliminated. Round 2 votes counts: B=29 D=25 C=24 A=22 so A is eliminated. Round 3 votes counts: B=41 D=34 C=25 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:211 E:195 A:191 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 8 2 B 14 0 -2 8 2 C 14 2 0 4 10 D -8 -8 -4 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 8 2 B 14 0 -2 8 2 C 14 2 0 4 10 D -8 -8 -4 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 8 2 B 14 0 -2 8 2 C 14 2 0 4 10 D -8 -8 -4 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3184: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (12) E A B D C (8) D C B E A (8) C D B A E (8) D C E A B (7) D C B A E (7) D C E B A (6) D E C A B (5) C B A E D (5) B C A E D (5) B A C E D (5) E D A B C (3) C B D A E (3) C B A D E (3) B A E C D (3) A B E C D (3) E A D B C (2) E B A C D (1) D E A C B (1) D E A B C (1) C D B E A (1) B C A D E (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -18 0 -10 B 12 0 -8 2 6 C 18 8 0 4 20 D 0 -2 -4 0 4 E 10 -6 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 0 -10 B 12 0 -8 2 6 C 18 8 0 4 20 D 0 -2 -4 0 4 E 10 -6 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=26 C=20 B=14 A=5 so A is eliminated. Round 2 votes counts: D=35 E=27 C=20 B=18 so B is eliminated. Round 3 votes counts: D=35 E=33 C=32 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:225 B:206 D:199 E:190 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -18 0 -10 B 12 0 -8 2 6 C 18 8 0 4 20 D 0 -2 -4 0 4 E 10 -6 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 0 -10 B 12 0 -8 2 6 C 18 8 0 4 20 D 0 -2 -4 0 4 E 10 -6 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 0 -10 B 12 0 -8 2 6 C 18 8 0 4 20 D 0 -2 -4 0 4 E 10 -6 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3185: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) D C E A B (7) A B D E C (7) D E C A B (6) B A E C D (6) A D B C E (6) E B A C D (5) C E D B A (5) C D E A B (5) A B D C E (4) D C A B E (3) D A B C E (3) C D A B E (3) C A B D E (3) A B C D E (3) E D B A C (2) E C B A D (2) E B A D C (2) D A C B E (2) D A B E C (2) B E A D C (2) B C A E D (2) E D C B A (1) E B D A C (1) D E A B C (1) D C A E B (1) C B E A D (1) C A D B E (1) B E C A D (1) B E A C D (1) B A E D C (1) B A C E D (1) A D B E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 20 4 2 2 B -20 0 2 -14 10 C -4 -2 0 -4 4 D -2 14 4 0 20 E -2 -10 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 4 2 2 B -20 0 2 -14 10 C -4 -2 0 -4 4 D -2 14 4 0 20 E -2 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=23 E=20 C=18 B=14 so B is eliminated. Round 2 votes counts: A=31 D=25 E=24 C=20 so C is eliminated. Round 3 votes counts: A=37 D=33 E=30 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:218 A:214 C:197 B:189 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 4 2 2 B -20 0 2 -14 10 C -4 -2 0 -4 4 D -2 14 4 0 20 E -2 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 4 2 2 B -20 0 2 -14 10 C -4 -2 0 -4 4 D -2 14 4 0 20 E -2 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 4 2 2 B -20 0 2 -14 10 C -4 -2 0 -4 4 D -2 14 4 0 20 E -2 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3186: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) A E D C B (8) B C D E A (7) B E A D C (5) D C B A E (4) D B C A E (4) D A C E B (4) B E A C D (4) B C E D A (4) A D E C B (4) E A C D B (3) D C A E B (3) C B E D A (3) E C A B D (2) D C A B E (2) D A B C E (2) C E B D A (2) C D A E B (2) B C E A D (2) E C B A D (1) E B C A D (1) E B A C D (1) E A D C B (1) E A B D C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A D B (1) C D E A B (1) C D B A E (1) C B E A D (1) C B D E A (1) B E D A C (1) B D E A C (1) B D C A E (1) B D A E C (1) B A E D C (1) B A D E C (1) A E B D C (1) A D E B C (1) A D C E B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -4 2 -4 B 14 0 4 4 18 C 4 -4 0 -8 -2 D -2 -4 8 0 -6 E 4 -18 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 2 -4 B 14 0 4 4 18 C 4 -4 0 -8 -2 D -2 -4 8 0 -6 E 4 -18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=24 A=17 C=13 E=10 so E is eliminated. Round 2 votes counts: B=38 D=24 A=22 C=16 so C is eliminated. Round 3 votes counts: B=47 D=28 A=25 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:198 E:197 C:195 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 2 -4 B 14 0 4 4 18 C 4 -4 0 -8 -2 D -2 -4 8 0 -6 E 4 -18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 2 -4 B 14 0 4 4 18 C 4 -4 0 -8 -2 D -2 -4 8 0 -6 E 4 -18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 2 -4 B 14 0 4 4 18 C 4 -4 0 -8 -2 D -2 -4 8 0 -6 E 4 -18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997162 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3187: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (13) A E C B D (13) C B A D E (10) E A D C B (9) D B C E A (9) B C D A E (7) A C B E D (7) E D A B C (5) D B E C A (4) C B D A E (4) E D B A C (3) E A D B C (3) C A B D E (3) A C B D E (3) E D B C A (1) E D A C B (1) E A C D B (1) B C A D E (1) A E D B C (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -4 6 2 B 4 0 -6 -2 -4 C 4 6 0 2 -10 D -6 2 -2 0 8 E -2 4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.413043 B: 0.173913 C: 0.065217 D: 0.065217 E: 0.282609 Sum of squares = 0.289224952822 Cumulative probabilities = A: 0.413043 B: 0.586957 C: 0.652174 D: 0.717391 E: 1.000000 A B C D E A 0 -4 -4 6 2 B 4 0 -6 -2 -4 C 4 6 0 2 -10 D -6 2 -2 0 8 E -2 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.413043 B: 0.173913 C: 0.065217 D: 0.065217 E: 0.282609 Sum of squares = 0.289224952747 Cumulative probabilities = A: 0.413043 B: 0.586957 C: 0.652174 D: 0.717391 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 E=23 C=17 B=8 so B is eliminated. Round 2 votes counts: D=26 A=26 C=25 E=23 so E is eliminated. Round 3 votes counts: A=39 D=36 C=25 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:202 C:201 D:201 A:200 B:196 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 -4 6 2 B 4 0 -6 -2 -4 C 4 6 0 2 -10 D -6 2 -2 0 8 E -2 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.413043 B: 0.173913 C: 0.065217 D: 0.065217 E: 0.282609 Sum of squares = 0.289224952747 Cumulative probabilities = A: 0.413043 B: 0.586957 C: 0.652174 D: 0.717391 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 6 2 B 4 0 -6 -2 -4 C 4 6 0 2 -10 D -6 2 -2 0 8 E -2 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.413043 B: 0.173913 C: 0.065217 D: 0.065217 E: 0.282609 Sum of squares = 0.289224952747 Cumulative probabilities = A: 0.413043 B: 0.586957 C: 0.652174 D: 0.717391 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 6 2 B 4 0 -6 -2 -4 C 4 6 0 2 -10 D -6 2 -2 0 8 E -2 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.413043 B: 0.173913 C: 0.065217 D: 0.065217 E: 0.282609 Sum of squares = 0.289224952747 Cumulative probabilities = A: 0.413043 B: 0.586957 C: 0.652174 D: 0.717391 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3188: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) E C D B A (8) C E B D A (8) A D B E C (7) A B D C E (7) B C A E D (4) B C A D E (4) E D C A B (3) D E A C B (3) D A E B C (3) C B E A D (3) B C D A E (3) B A D C E (3) A B C E D (3) E C D A B (2) D E A B C (2) D A B E C (2) C E B A D (2) C B A E D (2) B D C A E (2) B D A C E (2) B A C D E (2) E D A C B (1) E C A D B (1) D E C B A (1) D C E B A (1) D B E A C (1) C B E D A (1) C B D E A (1) C A B E D (1) B C D E A (1) B A C E D (1) A E D C B (1) A E B C D (1) A D B C E (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 10 22 B 0 0 20 8 8 C -4 -20 0 0 8 D -10 -8 0 0 14 E -22 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.532571 B: 0.467429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502121696935 Cumulative probabilities = A: 0.532571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 10 22 B 0 0 20 8 8 C -4 -20 0 0 8 D -10 -8 0 0 14 E -22 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=22 C=18 E=15 D=13 so D is eliminated. Round 2 votes counts: A=37 B=23 E=21 C=19 so C is eliminated. Round 3 votes counts: A=38 E=32 B=30 so B is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:218 D:198 C:192 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 10 22 B 0 0 20 8 8 C -4 -20 0 0 8 D -10 -8 0 0 14 E -22 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 22 B 0 0 20 8 8 C -4 -20 0 0 8 D -10 -8 0 0 14 E -22 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 22 B 0 0 20 8 8 C -4 -20 0 0 8 D -10 -8 0 0 14 E -22 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3189: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) C B A D E (9) E D B C A (8) E D C B A (6) A D B C E (6) A B C D E (6) E D A B C (5) E D B A C (3) E D A C B (3) C E B D A (3) C B D A E (3) A E D B C (3) A E C B D (3) E A D C B (2) D B C A E (2) D B A C E (2) D A E B C (2) C B D E A (2) A B D C E (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A B D (1) E A D B C (1) E A C D B (1) D E B C A (1) D E B A C (1) D B C E A (1) D A B E C (1) D A B C E (1) C A E B D (1) C A B D E (1) B D C A E (1) B C A D E (1) A E C D B (1) A E B C D (1) A D E B C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 10 14 6 18 B -10 0 -2 -6 -2 C -14 2 0 -4 6 D -6 6 4 0 6 E -18 2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 6 18 B -10 0 -2 -6 -2 C -14 2 0 -4 6 D -6 6 4 0 6 E -18 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=33 C=19 D=11 B=2 so B is eliminated. Round 2 votes counts: A=35 E=33 C=20 D=12 so D is eliminated. Round 3 votes counts: A=41 E=35 C=24 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:205 C:195 B:190 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 6 18 B -10 0 -2 -6 -2 C -14 2 0 -4 6 D -6 6 4 0 6 E -18 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 6 18 B -10 0 -2 -6 -2 C -14 2 0 -4 6 D -6 6 4 0 6 E -18 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 6 18 B -10 0 -2 -6 -2 C -14 2 0 -4 6 D -6 6 4 0 6 E -18 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3190: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) E A B C D (10) E D A C B (7) D E C B A (7) B A C E D (7) C B D A E (5) B C A D E (5) E A B D C (4) D E C A B (4) D C B E A (4) C D B A E (4) A B E C D (4) E A D C B (3) E A C D B (3) D C E B A (3) C B A D E (3) A E B C D (3) E A D B C (2) A B C E D (2) E D A B C (1) E A C B D (1) D E B C A (1) D B E C A (1) D B C E A (1) D B C A E (1) C A B D E (1) B C D A E (1) B C A E D (1) Total count = 100 A B C D E A 0 -10 -6 -2 -4 B 10 0 -12 -6 2 C 6 12 0 0 -2 D 2 6 0 0 4 E 4 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.349194 D: 0.650806 E: 0.000000 Sum of squares = 0.54548483655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.349194 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -2 -4 B 10 0 -12 -6 2 C 6 12 0 0 -2 D 2 6 0 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=31 B=14 C=13 A=9 so A is eliminated. Round 2 votes counts: E=34 D=33 B=20 C=13 so C is eliminated. Round 3 votes counts: D=37 E=34 B=29 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:208 D:206 E:200 B:197 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -6 -2 -4 B 10 0 -12 -6 2 C 6 12 0 0 -2 D 2 6 0 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -2 -4 B 10 0 -12 -6 2 C 6 12 0 0 -2 D 2 6 0 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -2 -4 B 10 0 -12 -6 2 C 6 12 0 0 -2 D 2 6 0 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3191: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) E D C A B (7) A B C E D (6) B A D E C (5) D E B C A (4) C E D A B (4) B C A D E (4) B A E D C (4) A C E D B (4) E D A C B (3) D C E B A (3) C D E A B (3) C A E D B (3) B D E A C (3) A E C D B (3) D E C B A (2) D E C A B (2) D C E A B (2) B D E C A (2) B C D E A (2) A E B C D (2) A C E B D (2) E D A B C (1) E A D C B (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C E A (1) C D E B A (1) C D B E A (1) C B D A E (1) C A B E D (1) C A B D E (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) B A D C E (1) B A C E D (1) A E D C B (1) A E D B C (1) A E B D C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 4 2 8 B 0 0 6 0 0 C -4 -6 0 -4 8 D -2 0 4 0 6 E -8 0 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571272 B: 0.428728 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510159468506 Cumulative probabilities = A: 0.571272 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 2 8 B 0 0 6 0 0 C -4 -6 0 -4 8 D -2 0 4 0 6 E -8 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999863 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=22 D=17 C=15 E=12 so E is eliminated. Round 2 votes counts: B=34 D=28 A=23 C=15 so C is eliminated. Round 3 votes counts: D=37 B=35 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:207 D:204 B:203 C:197 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 2 8 B 0 0 6 0 0 C -4 -6 0 -4 8 D -2 0 4 0 6 E -8 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999863 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 2 8 B 0 0 6 0 0 C -4 -6 0 -4 8 D -2 0 4 0 6 E -8 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999863 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 2 8 B 0 0 6 0 0 C -4 -6 0 -4 8 D -2 0 4 0 6 E -8 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999863 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3192: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (13) E A C D B (10) E A C B D (9) B D C A E (9) C B A E D (6) B C D A E (6) A E C B D (6) E D A C B (5) C A E B D (4) D E B A C (3) D E A C B (3) D E A B C (3) D B E A C (3) B C A E D (3) B C A D E (3) E A D C B (2) D B E C A (2) D B C E A (2) D B A C E (2) C A B E D (2) D A E B C (1) D A B E C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 0 -6 16 B 4 0 2 0 6 C 0 -2 0 2 4 D 6 0 -2 0 2 E -16 -6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666838 C: 0.000000 D: 0.333162 E: 0.000000 Sum of squares = 0.555669817798 Cumulative probabilities = A: 0.000000 B: 0.666838 C: 0.666838 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -6 16 B 4 0 2 0 6 C 0 -2 0 2 4 D 6 0 -2 0 2 E -16 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500258 C: 0.000000 D: 0.499742 E: 0.000000 Sum of squares = 0.500000133422 Cumulative probabilities = A: 0.000000 B: 0.500258 C: 0.500258 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=26 B=21 C=12 A=8 so A is eliminated. Round 2 votes counts: D=33 E=32 B=21 C=14 so C is eliminated. Round 3 votes counts: E=37 D=33 B=30 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:206 A:203 D:203 C:202 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 -6 16 B 4 0 2 0 6 C 0 -2 0 2 4 D 6 0 -2 0 2 E -16 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500258 C: 0.000000 D: 0.499742 E: 0.000000 Sum of squares = 0.500000133422 Cumulative probabilities = A: 0.000000 B: 0.500258 C: 0.500258 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -6 16 B 4 0 2 0 6 C 0 -2 0 2 4 D 6 0 -2 0 2 E -16 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500258 C: 0.000000 D: 0.499742 E: 0.000000 Sum of squares = 0.500000133422 Cumulative probabilities = A: 0.000000 B: 0.500258 C: 0.500258 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -6 16 B 4 0 2 0 6 C 0 -2 0 2 4 D 6 0 -2 0 2 E -16 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500258 C: 0.000000 D: 0.499742 E: 0.000000 Sum of squares = 0.500000133422 Cumulative probabilities = A: 0.000000 B: 0.500258 C: 0.500258 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3193: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) A D B E C (8) E A D B C (6) D A E B C (6) B A D C E (6) C E B D A (5) A B D E C (5) E D A C B (4) C E D B A (4) C B E A D (4) B C A D E (4) E C D B A (3) D E A C B (3) D A B E C (3) B A D E C (3) B A C D E (3) E C B A D (2) D A C B E (2) C E D A B (2) C D B A E (2) C B E D A (2) C B D A E (2) B A E D C (2) E B C A D (1) D E C A B (1) C D B E A (1) C B A D E (1) B E C A D (1) B C E A D (1) B C A E D (1) B A C E D (1) A E D B C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 10 4 4 B -2 0 8 -12 6 C -10 -8 0 -4 -16 D -4 12 4 0 4 E -4 -6 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 4 B -2 0 8 -12 6 C -10 -8 0 -4 -16 D -4 12 4 0 4 E -4 -6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=23 B=22 A=16 D=15 so D is eliminated. Round 2 votes counts: E=28 A=27 C=23 B=22 so B is eliminated. Round 3 votes counts: A=42 E=29 C=29 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:208 E:201 B:200 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 4 B -2 0 8 -12 6 C -10 -8 0 -4 -16 D -4 12 4 0 4 E -4 -6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 4 B -2 0 8 -12 6 C -10 -8 0 -4 -16 D -4 12 4 0 4 E -4 -6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 4 B -2 0 8 -12 6 C -10 -8 0 -4 -16 D -4 12 4 0 4 E -4 -6 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3194: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (7) A D B E C (6) A D B C E (6) B A D E C (5) C E B D A (4) C E B A D (4) C E A D B (4) A D C E B (4) E C D B A (3) E C D A B (3) E C B D A (3) D B A E C (3) B D A E C (3) D E A C B (2) D A B E C (2) C E A B D (2) C B E D A (2) C A E B D (2) B E C D A (2) B D E C A (2) B C E D A (2) B C E A D (2) B A C D E (2) A D E C B (2) A D E B C (2) A C D B E (2) A B C D E (2) E D C A B (1) E D A C B (1) D E B C A (1) D E A B C (1) D B E A C (1) D A E B C (1) C E D B A (1) C E D A B (1) C B E A D (1) C A E D B (1) C A B E D (1) B D E A C (1) B C A E D (1) B A D C E (1) B A C E D (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 10 14 20 12 B -10 0 8 2 12 C -14 -8 0 -4 10 D -20 -2 4 0 14 E -12 -12 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 20 12 B -10 0 8 2 12 C -14 -8 0 -4 10 D -20 -2 4 0 14 E -12 -12 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=23 B=22 E=11 D=11 so E is eliminated. Round 2 votes counts: A=33 C=32 B=22 D=13 so D is eliminated. Round 3 votes counts: A=40 C=33 B=27 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:206 D:198 C:192 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 20 12 B -10 0 8 2 12 C -14 -8 0 -4 10 D -20 -2 4 0 14 E -12 -12 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 20 12 B -10 0 8 2 12 C -14 -8 0 -4 10 D -20 -2 4 0 14 E -12 -12 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 20 12 B -10 0 8 2 12 C -14 -8 0 -4 10 D -20 -2 4 0 14 E -12 -12 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3195: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (17) C B E A D (15) C B E D A (5) D B E A C (4) D B A E C (4) C B D E A (4) A D E B C (4) D A E C B (3) C B A E D (3) C A E B D (3) B C E D A (3) A E C D B (3) A D E C B (3) D B C E A (2) C A D E B (2) B C E A D (2) B C D E A (2) A E D B C (2) E B C A D (1) E B A D C (1) E A D B C (1) E A B C D (1) D E B A C (1) D E A B C (1) D B C A E (1) D A C E B (1) C E B A D (1) C B A D E (1) C A E D B (1) C A B E D (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E C A (1) A E C B D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 2 -2 4 B 8 0 2 -2 2 C -2 -2 0 4 -2 D 2 2 -4 0 4 E -4 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -2 4 B 8 0 2 -2 2 C -2 -2 0 4 -2 D 2 2 -4 0 4 E -4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000003 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=34 A=15 B=11 E=4 so E is eliminated. Round 2 votes counts: C=36 D=34 A=17 B=13 so B is eliminated. Round 3 votes counts: C=45 D=36 A=19 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:205 D:202 C:199 A:198 E:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -2 4 B 8 0 2 -2 2 C -2 -2 0 4 -2 D 2 2 -4 0 4 E -4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000003 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 4 B 8 0 2 -2 2 C -2 -2 0 4 -2 D 2 2 -4 0 4 E -4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000003 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 4 B 8 0 2 -2 2 C -2 -2 0 4 -2 D 2 2 -4 0 4 E -4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000003 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3196: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (13) D A E B C (7) C B E A D (7) B C E D A (7) B D C A E (6) E A D C B (5) D B A E C (5) D A B E C (5) E C A D B (4) C E A B D (4) B D A C E (4) B C D E A (4) E A C D B (3) D A E C B (3) C E D A B (3) C B E D A (3) B C D A E (3) C E B A D (2) B C E A D (2) B A C D E (2) A D E C B (2) D E A C B (1) D C B E A (1) D B C E A (1) B D C E A (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 4 -8 4 B -4 0 -2 -10 4 C -4 2 0 -12 0 D 8 10 12 0 -6 E -4 -4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691353 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 A B C D E A 0 4 4 -8 4 B -4 0 -2 -10 4 C -4 2 0 -12 0 D 8 10 12 0 -6 E -4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691336 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 C=19 A=17 E=12 so E is eliminated. Round 2 votes counts: B=29 A=25 D=23 C=23 so D is eliminated. Round 3 votes counts: A=41 B=35 C=24 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:202 E:199 B:194 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 -8 4 B -4 0 -2 -10 4 C -4 2 0 -12 0 D 8 10 12 0 -6 E -4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691336 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -8 4 B -4 0 -2 -10 4 C -4 2 0 -12 0 D 8 10 12 0 -6 E -4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691336 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -8 4 B -4 0 -2 -10 4 C -4 2 0 -12 0 D 8 10 12 0 -6 E -4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.444444 Sum of squares = 0.358024691336 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.555556 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3197: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D A B E C (6) C E B A D (6) C E A D B (6) A D B C E (5) E B D A C (4) E B C D A (4) D A B C E (4) C A D E B (4) A B D C E (4) E C B A D (3) C A D B E (3) B D A E C (3) B A D C E (3) A D C B E (3) E B C A D (2) D E B A C (2) D B E A C (2) D B A E C (2) D A E C B (2) D A E B C (2) D A C B E (2) C E A B D (2) C B A D E (2) B E D A C (2) E D B A C (1) E C D B A (1) E C D A B (1) D E A C B (1) C E D A B (1) C D A E B (1) C B E A D (1) C A E B D (1) C A B D E (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A D E (1) B A D E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 4 6 4 B -2 0 2 0 0 C -4 -2 0 0 6 D -6 0 0 0 12 E -4 0 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 6 4 B -2 0 2 0 0 C -4 -2 0 0 6 D -6 0 0 0 12 E -4 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 E=22 A=14 B=13 so B is eliminated. Round 2 votes counts: C=30 E=26 D=26 A=18 so A is eliminated. Round 3 votes counts: D=42 C=32 E=26 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:208 D:203 B:200 C:200 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 6 4 B -2 0 2 0 0 C -4 -2 0 0 6 D -6 0 0 0 12 E -4 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 6 4 B -2 0 2 0 0 C -4 -2 0 0 6 D -6 0 0 0 12 E -4 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 6 4 B -2 0 2 0 0 C -4 -2 0 0 6 D -6 0 0 0 12 E -4 0 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3198: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) A C E B D (8) D B E C A (7) D B C A E (6) C A B D E (6) B D E C A (6) B D C E A (5) D B A C E (4) D E B A C (3) D B E A C (3) D B C E A (3) C B A D E (3) C A E B D (3) B D C A E (3) A C E D B (3) E D B A C (2) E C A B D (2) E B D C A (2) E A D C B (2) E A C D B (2) D A B C E (2) B C D A E (2) A E C D B (2) E D A B C (1) E C B A D (1) E B C D A (1) D E B C A (1) C E A B D (1) C B D A E (1) C A B E D (1) B E D C A (1) B C D E A (1) A D C E B (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -12 -8 -4 B 10 0 6 12 12 C 12 -6 0 -4 12 D 8 -12 4 0 20 E 4 -12 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 -4 B 10 0 6 12 12 C 12 -6 0 -4 12 D 8 -12 4 0 20 E 4 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 B=18 A=17 C=15 so C is eliminated. Round 2 votes counts: D=29 A=27 E=22 B=22 so E is eliminated. Round 3 votes counts: A=42 D=32 B=26 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:220 D:210 C:207 A:183 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 -4 B 10 0 6 12 12 C 12 -6 0 -4 12 D 8 -12 4 0 20 E 4 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 -4 B 10 0 6 12 12 C 12 -6 0 -4 12 D 8 -12 4 0 20 E 4 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 -4 B 10 0 6 12 12 C 12 -6 0 -4 12 D 8 -12 4 0 20 E 4 -12 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3199: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) C B D E A (7) D E C B A (6) A E D B C (6) A B C E D (6) E D A C B (5) B C D E A (5) A E D C B (5) E D A B C (3) E A D C B (3) D E B C A (3) D E A C B (3) D E A B C (3) C B D A E (3) A E B D C (3) D E C A B (2) D B E C A (2) A E C B D (2) A E B C D (2) A C E B D (2) A B E D C (2) A B E C D (2) E A D B C (1) D C E B A (1) D C B E A (1) C D E B A (1) C D B E A (1) C B A E D (1) C B A D E (1) C A E D B (1) B D E A C (1) B D C E A (1) B C D A E (1) B C A E D (1) B A C E D (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 4 2 2 B -6 0 6 4 -6 C -4 -6 0 -2 -10 D -2 -4 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 2 B -6 0 6 4 -6 C -4 -6 0 -2 -10 D -2 -4 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=21 B=20 C=15 E=12 so E is eliminated. Round 2 votes counts: A=36 D=29 B=20 C=15 so C is eliminated. Round 3 votes counts: A=37 B=32 D=31 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:205 D:200 B:199 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 2 B -6 0 6 4 -6 C -4 -6 0 -2 -10 D -2 -4 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 2 B -6 0 6 4 -6 C -4 -6 0 -2 -10 D -2 -4 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 2 B -6 0 6 4 -6 C -4 -6 0 -2 -10 D -2 -4 2 0 4 E -2 6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3200: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) C D A B E (6) C A D E B (6) E B A D C (5) C D B A E (5) C B D A E (5) E A D B C (4) B E D C A (4) B E C A D (4) B C E D A (4) A E D C B (4) C A E D B (3) B E A C D (3) B C E A D (3) E C B A D (2) E A D C B (2) E A B D C (2) D C A E B (2) D A C E B (2) C E B A D (2) C D A E B (2) B D E A C (2) B D C E A (2) E C A B D (1) E A C D B (1) D B A C E (1) D A E B C (1) D A C B E (1) C E A D B (1) C E A B D (1) C B E D A (1) C B D E A (1) C A D B E (1) B E A D C (1) B D A E C (1) B D A C E (1) B C D E A (1) A E C D B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -14 2 -8 B 10 0 -8 6 8 C 14 8 0 12 8 D -2 -6 -12 0 -14 E 8 -8 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 2 -8 B 10 0 -8 6 8 C 14 8 0 12 8 D -2 -6 -12 0 -14 E 8 -8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=33 E=17 A=9 D=7 so D is eliminated. Round 2 votes counts: C=36 B=34 E=17 A=13 so A is eliminated. Round 3 votes counts: C=42 B=34 E=24 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:208 E:203 A:185 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 2 -8 B 10 0 -8 6 8 C 14 8 0 12 8 D -2 -6 -12 0 -14 E 8 -8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 2 -8 B 10 0 -8 6 8 C 14 8 0 12 8 D -2 -6 -12 0 -14 E 8 -8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 2 -8 B 10 0 -8 6 8 C 14 8 0 12 8 D -2 -6 -12 0 -14 E 8 -8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3201: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) A B E D C (7) E A B C D (6) A D C E B (6) A E B C D (5) D C B E A (4) D C B A E (4) D C A B E (4) C D E A B (4) C D B E A (4) E B C D A (3) B D C A E (3) B C D E A (3) A E D B C (3) A E C D B (3) A D C B E (3) D B C E A (2) D B C A E (2) B E A D C (2) A D E C B (2) A D B C E (2) A B D C E (2) E C D B A (1) E B C A D (1) E B A C D (1) E A C D B (1) D C A E B (1) D A C B E (1) C D E B A (1) C D A E B (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E D A (1) B A E D C (1) B A D C E (1) A E D C B (1) A E C B D (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 26 16 16 26 B -26 0 14 -4 2 C -16 -14 0 -24 2 D -16 4 24 0 8 E -26 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 16 16 26 B -26 0 14 -4 2 C -16 -14 0 -24 2 D -16 4 24 0 8 E -26 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 D=18 B=14 E=13 C=10 so C is eliminated. Round 2 votes counts: A=45 D=28 B=14 E=13 so E is eliminated. Round 3 votes counts: A=52 D=29 B=19 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:242 D:210 B:193 E:181 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 16 16 26 B -26 0 14 -4 2 C -16 -14 0 -24 2 D -16 4 24 0 8 E -26 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 16 16 26 B -26 0 14 -4 2 C -16 -14 0 -24 2 D -16 4 24 0 8 E -26 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 16 16 26 B -26 0 14 -4 2 C -16 -14 0 -24 2 D -16 4 24 0 8 E -26 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3202: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) C E D B A (8) B E A C D (7) E B C A D (6) A B D E C (5) E C B D A (4) D A C B E (4) D A B C E (4) A D C E B (4) A D C B E (4) C D E B A (3) C D E A B (3) C D A E B (3) A D B C E (3) A B E D C (3) D C B E A (2) D B A E C (2) D A B E C (2) C E D A B (2) C E B D A (2) B E C A D (2) B A E C D (2) A D B E C (2) A C E B D (2) A C D E B (2) A B E C D (2) D C A E B (1) D C A B E (1) C E A D B (1) B E D C A (1) B E C D A (1) B E A D C (1) B A E D C (1) B A D E C (1) Total count = 100 A B C D E A 0 -4 2 -4 -4 B 4 0 8 -2 0 C -2 -8 0 18 -2 D 4 2 -18 0 -8 E 4 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.371608 C: 0.000000 D: 0.000000 E: 0.628392 Sum of squares = 0.532969230859 Cumulative probabilities = A: 0.000000 B: 0.371608 C: 0.371608 D: 0.371608 E: 1.000000 A B C D E A 0 -4 2 -4 -4 B 4 0 8 -2 0 C -2 -8 0 18 -2 D 4 2 -18 0 -8 E 4 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 E=19 D=16 B=16 so D is eliminated. Round 2 votes counts: A=37 C=26 E=19 B=18 so B is eliminated. Round 3 votes counts: A=43 E=31 C=26 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:207 B:205 C:203 A:195 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -4 -4 B 4 0 8 -2 0 C -2 -8 0 18 -2 D 4 2 -18 0 -8 E 4 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -4 -4 B 4 0 8 -2 0 C -2 -8 0 18 -2 D 4 2 -18 0 -8 E 4 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -4 -4 B 4 0 8 -2 0 C -2 -8 0 18 -2 D 4 2 -18 0 -8 E 4 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3203: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (13) B A D C E (10) B D E A C (7) B D A E C (7) E C D A B (5) C A E D B (5) E D C A B (4) C A B D E (4) E C A D B (3) D A B E C (3) C E A B D (3) C A D E B (3) B D A C E (3) A D C E B (3) E D A C B (2) E C B D A (2) D B E A C (2) C A E B D (2) B C A D E (2) A D B C E (2) E D C B A (1) E D B A C (1) E B D C A (1) E B C D A (1) E A C D B (1) D E B A C (1) D E A B C (1) D B A E C (1) C B A E D (1) B E C D A (1) B C E A D (1) A D C B E (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -4 14 0 B -16 0 -10 -6 -4 C 4 10 0 -2 12 D -14 6 2 0 6 E 0 4 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.000000 Sum of squares = 0.540000000065 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 14 0 B -16 0 -10 -6 -4 C 4 10 0 -2 12 D -14 6 2 0 6 E 0 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.000000 Sum of squares = 0.53999999996 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=31 B=31 E=21 A=9 D=8 so D is eliminated. Round 2 votes counts: B=34 C=31 E=23 A=12 so A is eliminated. Round 3 votes counts: B=40 C=37 E=23 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:212 D:200 E:193 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 14 0 B -16 0 -10 -6 -4 C 4 10 0 -2 12 D -14 6 2 0 6 E 0 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.000000 Sum of squares = 0.53999999996 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 14 0 B -16 0 -10 -6 -4 C 4 10 0 -2 12 D -14 6 2 0 6 E 0 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.000000 Sum of squares = 0.53999999996 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 14 0 B -16 0 -10 -6 -4 C 4 10 0 -2 12 D -14 6 2 0 6 E 0 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.000000 Sum of squares = 0.53999999996 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3204: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (10) A B C D E (8) E D C A B (7) C D E A B (6) B A E C D (6) B E D A C (5) B E A D C (5) A C D E B (5) A C B D E (4) E D B C A (3) C A D E B (3) B A C D E (3) E D C B A (2) E C D A B (2) D C E A B (2) D B E C A (2) C D A E B (2) C D A B E (2) B A E D C (2) E C A D B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A C D B (1) E A C B D (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E B A (1) D C B A E (1) D C A B E (1) C A E D B (1) C A D B E (1) B E A C D (1) B D E C A (1) B D E A C (1) B A D C E (1) B A C E D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 -4 -14 B 0 0 8 6 -8 C 2 -8 0 2 -14 D 4 -6 -2 0 -6 E 14 8 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 -4 -14 B 0 0 8 6 -8 C 2 -8 0 2 -14 D 4 -6 -2 0 -6 E 14 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 A=19 C=15 D=10 so D is eliminated. Round 2 votes counts: E=33 B=28 C=20 A=19 so A is eliminated. Round 3 votes counts: B=37 E=33 C=30 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:203 D:195 C:191 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 -4 -14 B 0 0 8 6 -8 C 2 -8 0 2 -14 D 4 -6 -2 0 -6 E 14 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -4 -14 B 0 0 8 6 -8 C 2 -8 0 2 -14 D 4 -6 -2 0 -6 E 14 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -4 -14 B 0 0 8 6 -8 C 2 -8 0 2 -14 D 4 -6 -2 0 -6 E 14 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3205: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (15) A E C D B (11) E A C D B (8) B D C A E (7) B E C D A (5) B E A C D (4) B D A C E (4) A D C E B (4) B E D C A (3) A E B C D (3) A D C B E (3) A C D E B (3) E A C B D (2) D C A B E (2) B D E C A (2) E C D A B (1) E C B D A (1) E C A D B (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D C B A E (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E D B (1) B E A D C (1) B A D E C (1) A E C B D (1) A E B D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 6 2 -4 B -4 0 6 12 6 C -6 -6 0 0 -2 D -2 -12 0 0 -2 E 4 -6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 4 6 2 -4 B -4 0 6 12 6 C -6 -6 0 0 -2 D -2 -12 0 0 -2 E 4 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=28 E=18 D=7 C=5 so C is eliminated. Round 2 votes counts: B=42 A=29 E=20 D=9 so D is eliminated. Round 3 votes counts: B=44 A=35 E=21 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:204 E:201 C:193 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 2 -4 B -4 0 6 12 6 C -6 -6 0 0 -2 D -2 -12 0 0 -2 E 4 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 2 -4 B -4 0 6 12 6 C -6 -6 0 0 -2 D -2 -12 0 0 -2 E 4 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 2 -4 B -4 0 6 12 6 C -6 -6 0 0 -2 D -2 -12 0 0 -2 E 4 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3206: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) D E B A C (7) D E A B C (5) D A E B C (5) A C D B E (5) B E C A D (4) A C B D E (4) A B C E D (4) E D B C A (3) E C B D A (3) E B D C A (3) E B C D A (3) D E C B A (3) D C A E B (3) A D C B E (3) A C B E D (3) D E B C A (2) D C E A B (2) D A C E B (2) C E D B A (2) C E B D A (2) C B A E D (2) B A C E D (2) A B E C D (2) E D C B A (1) E D B A C (1) E C B A D (1) E B C A D (1) D E C A B (1) D E A C B (1) D A B E C (1) C E B A D (1) C B E A D (1) C A E D B (1) B E D A C (1) B E A D C (1) B A E D C (1) A D B C E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 2 -2 2 B -10 0 -2 2 -6 C -2 2 0 4 -2 D 2 -2 -4 0 -6 E -2 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999862 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 10 2 -2 2 B -10 0 -2 2 -6 C -2 2 0 4 -2 D 2 -2 -4 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999989 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=24 C=19 E=16 B=9 so B is eliminated. Round 2 votes counts: D=32 A=27 E=22 C=19 so C is eliminated. Round 3 votes counts: A=40 D=32 E=28 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:206 E:206 C:201 D:195 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 -2 2 B -10 0 -2 2 -6 C -2 2 0 4 -2 D 2 -2 -4 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999989 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -2 2 B -10 0 -2 2 -6 C -2 2 0 4 -2 D 2 -2 -4 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999989 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -2 2 B -10 0 -2 2 -6 C -2 2 0 4 -2 D 2 -2 -4 0 -6 E -2 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999989 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3207: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (13) B E A C D (10) E B D C A (9) D E B C A (8) C A D E B (8) D B E A C (6) C A E B D (6) D C A E B (5) B E D A C (5) D C A B E (4) D A C B E (4) A C D B E (4) A C B E D (4) E B D A C (3) E B A C D (2) C D A E B (2) C A D B E (2) E C B A D (1) D B A E C (1) D B A C E (1) D A B C E (1) B E A D C (1) Total count = 100 A B C D E A 0 -20 -16 2 -16 B 20 0 20 8 -14 C 16 -20 0 4 -18 D -2 -8 -4 0 -8 E 16 14 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -16 2 -16 B 20 0 20 8 -14 C 16 -20 0 4 -18 D -2 -8 -4 0 -8 E 16 14 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=28 C=18 B=16 A=8 so A is eliminated. Round 2 votes counts: D=30 E=28 C=26 B=16 so B is eliminated. Round 3 votes counts: E=44 D=30 C=26 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:228 B:217 C:191 D:189 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -16 2 -16 B 20 0 20 8 -14 C 16 -20 0 4 -18 D -2 -8 -4 0 -8 E 16 14 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 2 -16 B 20 0 20 8 -14 C 16 -20 0 4 -18 D -2 -8 -4 0 -8 E 16 14 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 2 -16 B 20 0 20 8 -14 C 16 -20 0 4 -18 D -2 -8 -4 0 -8 E 16 14 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999422 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3208: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E D A B C (4) E B A C D (4) D A C B E (4) C B D E A (4) A D C B E (4) A C B D E (4) A B C E D (4) E D B C A (3) D E A C B (3) D A E C B (3) D A C E B (3) C D B A E (3) C B E D A (3) B E C A D (3) A E D B C (3) A C D B E (3) A B E C D (3) E D C B A (2) E D B A C (2) E B D C A (2) D E C B A (2) C B E A D (2) C B A D E (2) B C A E D (2) A E B C D (2) A B C D E (2) E A D B C (1) D E A B C (1) D C E B A (1) D C B E A (1) D A E B C (1) C E B D A (1) C D B E A (1) C D A B E (1) C B D A E (1) C B A E D (1) B E A C D (1) B C E A D (1) B A C E D (1) A E B D C (1) A D E C B (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 0 14 -6 2 B 0 0 -2 2 4 C -14 2 0 12 2 D 6 -2 -12 0 -6 E -2 -4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200851 B: 0.799149 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.678979808102 Cumulative probabilities = A: 0.200851 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 -6 2 B 0 0 -2 2 4 C -14 2 0 12 2 D 6 -2 -12 0 -6 E -2 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042854 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=25 D=19 C=19 B=8 so B is eliminated. Round 2 votes counts: A=30 E=29 C=22 D=19 so D is eliminated. Round 3 votes counts: A=41 E=35 C=24 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:205 B:202 C:201 E:199 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 14 -6 2 B 0 0 -2 2 4 C -14 2 0 12 2 D 6 -2 -12 0 -6 E -2 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042854 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 -6 2 B 0 0 -2 2 4 C -14 2 0 12 2 D 6 -2 -12 0 -6 E -2 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042854 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 -6 2 B 0 0 -2 2 4 C -14 2 0 12 2 D 6 -2 -12 0 -6 E -2 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042854 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3209: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) E A C D B (8) E C D B A (6) E B A D C (6) C D A B E (6) B D E C A (5) D C B A E (4) B A D E C (4) E B D C A (3) E A B D C (3) C D A E B (3) B E D A C (3) B E A D C (3) B A D C E (3) A B D C E (3) E A C B D (2) D C B E A (2) C A E D B (2) B D C E A (2) B D A C E (2) A E C D B (2) A E B D C (2) A B E D C (2) E D B C A (1) E C D A B (1) E C A D B (1) E B C D A (1) D B C A E (1) C E D A B (1) C D E B A (1) C D B A E (1) B E D C A (1) B D E A C (1) B D C A E (1) B D A E C (1) B A E D C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 14 6 -20 B 6 0 16 16 -6 C -14 -16 0 -8 -32 D -6 -16 8 0 -16 E 20 6 32 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 14 6 -20 B 6 0 16 16 -6 C -14 -16 0 -8 -32 D -6 -16 8 0 -16 E 20 6 32 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=27 C=14 A=11 D=7 so D is eliminated. Round 2 votes counts: E=41 B=28 C=20 A=11 so A is eliminated. Round 3 votes counts: E=45 B=33 C=22 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:237 B:216 A:197 D:185 C:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 14 6 -20 B 6 0 16 16 -6 C -14 -16 0 -8 -32 D -6 -16 8 0 -16 E 20 6 32 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 6 -20 B 6 0 16 16 -6 C -14 -16 0 -8 -32 D -6 -16 8 0 -16 E 20 6 32 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 6 -20 B 6 0 16 16 -6 C -14 -16 0 -8 -32 D -6 -16 8 0 -16 E 20 6 32 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3210: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) D B C A E (5) C D E B A (5) C B D A E (5) A B E D C (5) E C A D B (4) E C A B D (4) E A B C D (4) D B A C E (4) C E D A B (4) A E B C D (4) E D C A B (3) E A C B D (3) A E B D C (3) A B D E C (3) E D A C B (2) E A C D B (2) D C E B A (2) C E B D A (2) C D E A B (2) B D C A E (2) B C A E D (2) B C A D E (2) B A D E C (2) A B E C D (2) E C D A B (1) E A D B C (1) E A B D C (1) D E A B C (1) D C B E A (1) D C B A E (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A B D (1) C D B E A (1) C B E A D (1) C B D E A (1) C B A E D (1) B D A C E (1) B A D C E (1) B A C E D (1) A D E B C (1) Total count = 100 A B C D E A 0 16 -14 -2 -6 B -16 0 -6 4 -14 C 14 6 0 6 -8 D 2 -4 -6 0 -4 E 6 14 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -14 -2 -6 B -16 0 -6 4 -14 C 14 6 0 6 -8 D 2 -4 -6 0 -4 E 6 14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 D=22 A=18 B=11 so B is eliminated. Round 2 votes counts: C=28 E=25 D=25 A=22 so A is eliminated. Round 3 votes counts: E=39 D=32 C=29 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:209 A:197 D:194 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -14 -2 -6 B -16 0 -6 4 -14 C 14 6 0 6 -8 D 2 -4 -6 0 -4 E 6 14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -14 -2 -6 B -16 0 -6 4 -14 C 14 6 0 6 -8 D 2 -4 -6 0 -4 E 6 14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -14 -2 -6 B -16 0 -6 4 -14 C 14 6 0 6 -8 D 2 -4 -6 0 -4 E 6 14 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3211: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) C D A B E (11) C D B E A (8) E B A D C (6) A D C E B (5) B E A D C (4) A D B E C (4) E B A C D (3) D C A B E (3) D A C B E (3) C E B A D (3) C E A B D (3) C D E B A (3) C A D E B (3) B E D A C (3) B E C D A (3) E B C A D (2) D A B E C (2) B C E D A (2) A E B D C (2) A D C B E (2) E C B A D (1) E A B D C (1) E A B C D (1) D C B E A (1) D C A E B (1) C E A D B (1) C A E D B (1) B D E A C (1) B D C E A (1) B A D E C (1) A E C B D (1) A D B C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 16 -16 -6 6 B -16 0 -24 -22 2 C 16 24 0 16 28 D 6 22 -16 0 24 E -6 -2 -28 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -16 -6 6 B -16 0 -24 -22 2 C 16 24 0 16 28 D 6 22 -16 0 24 E -6 -2 -28 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=17 B=15 E=14 D=10 so D is eliminated. Round 2 votes counts: C=49 A=22 B=15 E=14 so E is eliminated. Round 3 votes counts: C=50 B=26 A=24 so A is eliminated. Round 4 votes counts: C=62 B=38 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:242 D:218 A:200 B:170 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -16 -6 6 B -16 0 -24 -22 2 C 16 24 0 16 28 D 6 22 -16 0 24 E -6 -2 -28 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -16 -6 6 B -16 0 -24 -22 2 C 16 24 0 16 28 D 6 22 -16 0 24 E -6 -2 -28 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -16 -6 6 B -16 0 -24 -22 2 C 16 24 0 16 28 D 6 22 -16 0 24 E -6 -2 -28 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3212: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) E A C B D (7) D C B E A (7) E A C D B (6) A E B D C (6) E C D A B (5) B D A C E (5) B A D E C (5) A B E C D (5) C D B E A (4) E C A D B (3) E A D C B (3) D C E B A (3) D C E A B (3) A E B C D (3) A B E D C (3) D B C A E (2) C B D E A (2) B A E C D (2) B A C E D (2) A E D C B (2) A E C B D (2) E D A C B (1) D E C A B (1) D A C E B (1) C E D B A (1) C B E D A (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C D E (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 14 8 2 B -8 0 -4 12 4 C -14 4 0 -8 -14 D -8 -12 8 0 -6 E -2 -4 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 8 2 B -8 0 -4 12 4 C -14 4 0 -8 -14 D -8 -12 8 0 -6 E -2 -4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996505 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 A=24 D=17 C=8 so C is eliminated. Round 2 votes counts: B=29 E=26 A=24 D=21 so D is eliminated. Round 3 votes counts: B=42 E=33 A=25 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 E:207 B:202 D:191 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 8 2 B -8 0 -4 12 4 C -14 4 0 -8 -14 D -8 -12 8 0 -6 E -2 -4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996505 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 8 2 B -8 0 -4 12 4 C -14 4 0 -8 -14 D -8 -12 8 0 -6 E -2 -4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996505 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 8 2 B -8 0 -4 12 4 C -14 4 0 -8 -14 D -8 -12 8 0 -6 E -2 -4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996505 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3213: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (11) C E D A B (9) E C A B D (8) E B A D C (8) D B A E C (8) C D B A E (6) E A B D C (4) A B D E C (4) E C B A D (3) D B E A C (3) C D A B E (3) C A B D E (3) B A E D C (3) B A D E C (3) C E D B A (2) B E D A C (2) A B E D C (2) A B D C E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E B D A C (1) E B A C D (1) E A B C D (1) D E B C A (1) D E B A C (1) D B A C E (1) C D E B A (1) C D B E A (1) C A D B E (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -6 10 -24 B 2 0 -4 18 -12 C 6 4 0 6 -18 D -10 -18 -6 0 -20 E 24 12 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 10 -24 B 2 0 -4 18 -12 C 6 4 0 6 -18 D -10 -18 -6 0 -20 E 24 12 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=31 D=14 B=9 A=9 so B is eliminated. Round 2 votes counts: C=37 E=33 D=15 A=15 so D is eliminated. Round 3 votes counts: E=38 C=37 A=25 so A is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:237 B:202 C:199 A:189 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 10 -24 B 2 0 -4 18 -12 C 6 4 0 6 -18 D -10 -18 -6 0 -20 E 24 12 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 10 -24 B 2 0 -4 18 -12 C 6 4 0 6 -18 D -10 -18 -6 0 -20 E 24 12 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 10 -24 B 2 0 -4 18 -12 C 6 4 0 6 -18 D -10 -18 -6 0 -20 E 24 12 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3214: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) B D A C E (8) A B E C D (6) D B A C E (5) A B E D C (5) A B D E C (5) E C A D B (4) E A C B D (4) D B C A E (4) E C B A D (3) E C A B D (3) E A C D B (3) D C B E A (3) D A B C E (3) B A E C D (3) D C E A B (2) D A C E B (2) C E D B A (2) C D E B A (2) B E A C D (2) B D C A E (2) B A D E C (2) B A D C E (2) A E C D B (2) E C D B A (1) E C D A B (1) E C B D A (1) E B C A D (1) D C A E B (1) D B C E A (1) C E D A B (1) C D E A B (1) C B E D A (1) B E C A D (1) B D C E A (1) B A C E D (1) A E C B D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 12 0 8 B 8 0 8 6 14 C -12 -8 0 -12 4 D 0 -6 12 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 0 8 B 8 0 8 6 14 C -12 -8 0 -12 4 D 0 -6 12 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=22 E=21 A=21 C=7 so C is eliminated. Round 2 votes counts: D=32 E=24 B=23 A=21 so A is eliminated. Round 3 votes counts: B=40 D=33 E=27 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:207 A:206 C:186 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 0 8 B 8 0 8 6 14 C -12 -8 0 -12 4 D 0 -6 12 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 0 8 B 8 0 8 6 14 C -12 -8 0 -12 4 D 0 -6 12 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 0 8 B 8 0 8 6 14 C -12 -8 0 -12 4 D 0 -6 12 0 8 E -8 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3215: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) E D C B A (7) E D C A B (7) E B D C A (7) D C A E B (7) A C D B E (6) A B C D E (5) E B D A C (4) B E C D A (4) B C A D E (4) C D A E B (3) C D A B E (3) B E C A D (3) B C E A D (3) E A D C B (2) D A C E B (2) C A D B E (2) B E D C A (2) B E A D C (2) B C D A E (2) B A C D E (2) A C B D E (2) E D B C A (1) E C B D A (1) E B C D A (1) D E C A B (1) C E D B A (1) C D E B A (1) C D B A E (1) B E A C D (1) B C E D A (1) B C D E A (1) B A C E D (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -26 -14 0 B 0 0 -12 -8 -10 C 26 12 0 -4 12 D 14 8 4 0 4 E 0 10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -26 -14 0 B 0 0 -12 -8 -10 C 26 12 0 -4 12 D 14 8 4 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 A=23 C=11 D=10 so D is eliminated. Round 2 votes counts: E=31 B=26 A=25 C=18 so C is eliminated. Round 3 votes counts: A=40 E=33 B=27 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:223 D:215 E:197 B:185 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -26 -14 0 B 0 0 -12 -8 -10 C 26 12 0 -4 12 D 14 8 4 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -26 -14 0 B 0 0 -12 -8 -10 C 26 12 0 -4 12 D 14 8 4 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -26 -14 0 B 0 0 -12 -8 -10 C 26 12 0 -4 12 D 14 8 4 0 4 E 0 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3216: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) B A E C D (7) D A B E C (6) C E D B A (6) C E B A D (6) A B E D C (5) E B C A D (4) D C A B E (4) D C B A E (3) D C A E B (3) D B A E C (3) D A C B E (3) D A B C E (3) B A E D C (3) A E B C D (3) A B E C D (3) E C B A D (2) E C A B D (2) D C B E A (2) D B C A E (2) C E B D A (2) C E A D B (2) B E A C D (2) A B D E C (2) E A B C D (1) D C E B A (1) D C E A B (1) D B C E A (1) D A C E B (1) C E D A B (1) C E A B D (1) C D A E B (1) C B E D A (1) B D C E A (1) B D A E C (1) B A D E C (1) A E C B D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -6 -6 14 B 10 0 0 -4 8 C 6 0 0 4 4 D 6 4 -4 0 -4 E -14 -8 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.393773 C: 0.606227 D: 0.000000 E: 0.000000 Sum of squares = 0.522568380135 Cumulative probabilities = A: 0.000000 B: 0.393773 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -6 14 B 10 0 0 -4 8 C 6 0 0 4 4 D 6 4 -4 0 -4 E -14 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499737 C: 0.500263 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138861 Cumulative probabilities = A: 0.000000 B: 0.499737 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=27 A=16 B=15 E=9 so E is eliminated. Round 2 votes counts: D=33 C=31 B=19 A=17 so A is eliminated. Round 3 votes counts: D=35 B=33 C=32 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:207 C:207 D:201 A:196 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 -6 14 B 10 0 0 -4 8 C 6 0 0 4 4 D 6 4 -4 0 -4 E -14 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499737 C: 0.500263 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138861 Cumulative probabilities = A: 0.000000 B: 0.499737 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -6 14 B 10 0 0 -4 8 C 6 0 0 4 4 D 6 4 -4 0 -4 E -14 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499737 C: 0.500263 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138861 Cumulative probabilities = A: 0.000000 B: 0.499737 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -6 14 B 10 0 0 -4 8 C 6 0 0 4 4 D 6 4 -4 0 -4 E -14 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499737 C: 0.500263 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138861 Cumulative probabilities = A: 0.000000 B: 0.499737 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3217: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (5) E B A D C (5) D E C A B (5) B A C E D (5) E A B C D (4) C B D A E (4) B C D A E (4) B C A D E (4) D E C B A (3) D C E A B (3) D C A E B (3) C D A B E (3) B E A C D (3) A B C E D (3) E D B A C (2) E D A B C (2) E A D C B (2) E A D B C (2) E A C D B (2) D E B C A (2) D C B A E (2) D C A B E (2) C D A E B (2) C A D B E (2) B E D A C (2) A E C B D (2) A E B C D (2) A C B E D (2) E B D A C (1) E A B D C (1) D C B E A (1) D B E C A (1) D B C A E (1) C D B A E (1) C B A D E (1) C A D E B (1) C A B D E (1) B E A D C (1) B D E C A (1) B D C E A (1) B C A E D (1) B A E C D (1) A E C D B (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 -2 2 B -6 0 -2 0 -6 C -2 2 0 4 0 D 2 0 -4 0 -2 E -2 6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -2 2 B -6 0 -2 0 -6 C -2 2 0 4 0 D 2 0 -4 0 -2 E -2 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 B=23 C=15 A=13 so A is eliminated. Round 2 votes counts: E=31 B=26 D=23 C=20 so C is eliminated. Round 3 votes counts: B=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:204 E:203 C:202 D:198 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 -2 2 B -6 0 -2 0 -6 C -2 2 0 4 0 D 2 0 -4 0 -2 E -2 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -2 2 B -6 0 -2 0 -6 C -2 2 0 4 0 D 2 0 -4 0 -2 E -2 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -2 2 B -6 0 -2 0 -6 C -2 2 0 4 0 D 2 0 -4 0 -2 E -2 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3218: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) D E B C A (7) D C E B A (6) C D A E B (6) C A D B E (6) E B A D C (5) B E A D C (5) A B C E D (5) E D B A C (4) C D A B E (4) C A E B D (4) A B E C D (4) D E C B A (3) C A B D E (3) E B A C D (2) D B E A C (2) C E D A B (2) C D E B A (2) C A D E B (2) C A B E D (2) B A E D C (2) A C B D E (2) E D C B A (1) E C D B A (1) E C A B D (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A E C (1) C D E A B (1) B E D A C (1) B D E A C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -8 -10 -8 B 10 0 2 -2 -14 C 8 -2 0 -2 0 D 10 2 2 0 0 E 8 14 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.660224 E: 0.339776 Sum of squares = 0.551343752424 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.660224 E: 1.000000 A B C D E A 0 -10 -8 -10 -8 B 10 0 2 -2 -14 C 8 -2 0 -2 0 D 10 2 2 0 0 E 8 14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=24 D=22 A=13 B=9 so B is eliminated. Round 2 votes counts: C=32 E=30 D=23 A=15 so A is eliminated. Round 3 votes counts: C=41 E=36 D=23 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:211 D:207 C:202 B:198 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -8 -10 -8 B 10 0 2 -2 -14 C 8 -2 0 -2 0 D 10 2 2 0 0 E 8 14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -10 -8 B 10 0 2 -2 -14 C 8 -2 0 -2 0 D 10 2 2 0 0 E 8 14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -10 -8 B 10 0 2 -2 -14 C 8 -2 0 -2 0 D 10 2 2 0 0 E 8 14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3219: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) E A D B C (5) C B D E A (5) B C A D E (5) A E D C B (5) A E C D B (5) E D A C B (4) B D C E A (4) B C D E A (4) A B E C D (4) A B C E D (4) D E C B A (3) B A E C D (3) A E C B D (3) A C E D B (3) E D A B C (2) E A D C B (2) D E B C A (2) D E B A C (2) D C B E A (2) D B E C A (2) C D E B A (2) C A B D E (2) B A E D C (2) B A C E D (2) A E B D C (2) E D C A B (1) E D B A C (1) E B D A C (1) D E C A B (1) D B C E A (1) C D E A B (1) C B D A E (1) C B A D E (1) C A E D B (1) C A D B E (1) B E A D C (1) B D E C A (1) B D E A C (1) B C D A E (1) A E D B C (1) Total count = 100 A B C D E A 0 -6 6 4 -8 B 6 0 2 -6 6 C -6 -2 0 8 -8 D -4 6 -8 0 -4 E 8 -6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999995 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 6 4 -8 B 6 0 2 -6 6 C -6 -2 0 8 -8 D -4 6 -8 0 -4 E 8 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999864 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=24 C=20 E=16 D=13 so D is eliminated. Round 2 votes counts: B=27 A=27 E=24 C=22 so C is eliminated. Round 3 votes counts: B=42 A=31 E=27 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:207 B:204 A:198 C:196 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 4 -8 B 6 0 2 -6 6 C -6 -2 0 8 -8 D -4 6 -8 0 -4 E 8 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999864 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 4 -8 B 6 0 2 -6 6 C -6 -2 0 8 -8 D -4 6 -8 0 -4 E 8 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999864 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 4 -8 B 6 0 2 -6 6 C -6 -2 0 8 -8 D -4 6 -8 0 -4 E 8 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999864 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3220: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) B A E D C (8) A B E D C (8) C D E B A (7) D E B A C (6) C D B E A (5) A E B D C (5) B A D E C (4) E D A B C (3) E A D B C (3) D C B E A (3) C A E B D (3) A E B C D (3) A B E C D (3) E A C B D (2) E A B D C (2) D B E A C (2) C D B A E (2) C A E D B (2) C A B E D (2) A B C E D (2) E B A D C (1) D E C B A (1) D E C A B (1) D C E B A (1) D B E C A (1) D B C E A (1) C E D A B (1) C B A D E (1) C A B D E (1) B E A D C (1) B D E A C (1) B D A E C (1) B A D C E (1) Total count = 100 A B C D E A 0 6 12 4 -8 B -6 0 12 -2 -6 C -12 -12 0 -8 -12 D -4 2 8 0 2 E 8 6 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428551 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 A B C D E A 0 6 12 4 -8 B -6 0 12 -2 -6 C -12 -12 0 -8 -12 D -4 2 8 0 2 E 8 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428535 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=21 D=16 B=16 E=11 so E is eliminated. Round 2 votes counts: C=36 A=28 D=19 B=17 so B is eliminated. Round 3 votes counts: A=43 C=36 D=21 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:212 A:207 D:204 B:199 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 4 -8 B -6 0 12 -2 -6 C -12 -12 0 -8 -12 D -4 2 8 0 2 E 8 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428535 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 4 -8 B -6 0 12 -2 -6 C -12 -12 0 -8 -12 D -4 2 8 0 2 E 8 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428535 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 4 -8 B -6 0 12 -2 -6 C -12 -12 0 -8 -12 D -4 2 8 0 2 E 8 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428535 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3221: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) D E C A B (6) D A E C B (6) A B C E D (6) E C B D A (5) B C E D A (5) B C E A D (5) D E A C B (4) D B E C A (4) C E B A D (4) B C A E D (4) A B D C E (4) C B E A D (3) B A C D E (3) A D B E C (3) A B C D E (3) D E C B A (2) D A B E C (2) A E D C B (2) A E C B D (2) A D B C E (2) A C E B D (2) E D C A B (1) E C A B D (1) D E B C A (1) D E B A C (1) D B C E A (1) D B A C E (1) D A E B C (1) D A B C E (1) C E B D A (1) C E A B D (1) C B A E D (1) B D C E A (1) B C D E A (1) B C A D E (1) B A C E D (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 0 -6 B 2 0 2 10 6 C 8 -2 0 12 4 D 0 -10 -12 0 -2 E 6 -6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 0 -6 B 2 0 2 10 6 C 8 -2 0 12 4 D 0 -10 -12 0 -2 E 6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994366 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=26 B=21 E=13 C=10 so C is eliminated. Round 2 votes counts: D=30 A=26 B=25 E=19 so E is eliminated. Round 3 votes counts: D=37 B=35 A=28 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:210 E:199 A:192 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 0 -6 B 2 0 2 10 6 C 8 -2 0 12 4 D 0 -10 -12 0 -2 E 6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994366 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 0 -6 B 2 0 2 10 6 C 8 -2 0 12 4 D 0 -10 -12 0 -2 E 6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994366 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 0 -6 B 2 0 2 10 6 C 8 -2 0 12 4 D 0 -10 -12 0 -2 E 6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994366 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3222: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) B A C E D (6) B A C D E (5) D E C A B (4) D C E B A (4) C B E A D (4) E C D B A (3) E A C B D (3) C D E B A (3) C D B E A (3) B C D A E (3) B C A E D (3) A E B D C (3) A D E B C (3) E D C A B (2) E D A C B (2) D E A C B (2) D C B E A (2) D C B A E (2) D A B C E (2) C E B D A (2) C B E D A (2) B C A D E (2) B A D C E (2) A E D B C (2) A E B C D (2) A D B C E (2) A B E D C (2) E C B A D (1) E B C A D (1) E B A C D (1) E A D C B (1) E A B C D (1) D C E A B (1) D B A C E (1) D A E B C (1) D A B E C (1) C B D A E (1) B E C A D (1) B A E C D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 12 18 14 B 6 0 16 16 16 C -12 -16 0 -2 22 D -18 -16 2 0 12 E -14 -16 -22 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 18 14 B 6 0 16 16 16 C -12 -16 0 -2 22 D -18 -16 2 0 12 E -14 -16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=23 D=20 E=15 C=15 so E is eliminated. Round 2 votes counts: A=32 B=25 D=24 C=19 so C is eliminated. Round 3 votes counts: B=35 D=33 A=32 so A is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:219 C:196 D:190 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 18 14 B 6 0 16 16 16 C -12 -16 0 -2 22 D -18 -16 2 0 12 E -14 -16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 18 14 B 6 0 16 16 16 C -12 -16 0 -2 22 D -18 -16 2 0 12 E -14 -16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 18 14 B 6 0 16 16 16 C -12 -16 0 -2 22 D -18 -16 2 0 12 E -14 -16 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3223: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E C B D A (6) A D B C E (6) E B C A D (4) E A D B C (4) D A C E B (4) D A C B E (4) B E A C D (4) A E D B C (4) E B A C D (3) D C A B E (3) C D B A E (3) C B E D A (3) A D B E C (3) E D A C B (2) E D A B C (2) E B A D C (2) D A E C B (2) D A E B C (2) C D A B E (2) C B D E A (2) B E C A D (2) B C A E D (2) B C A D E (2) B A C E D (2) A D E B C (2) A B D C E (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B A D (1) D E A C B (1) C E B D A (1) C D E A B (1) C D B E A (1) B C E A D (1) B A E D C (1) B A E C D (1) B A C D E (1) A E B D C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 10 0 14 B 2 0 6 2 10 C -10 -6 0 4 -4 D 0 -2 -4 0 0 E -14 -10 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 0 14 B 2 0 6 2 10 C -10 -6 0 4 -4 D 0 -2 -4 0 0 E -14 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993656 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=21 A=20 D=16 B=16 so D is eliminated. Round 2 votes counts: A=32 E=28 C=24 B=16 so B is eliminated. Round 3 votes counts: A=37 E=34 C=29 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:210 D:197 C:192 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 0 14 B 2 0 6 2 10 C -10 -6 0 4 -4 D 0 -2 -4 0 0 E -14 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993656 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 0 14 B 2 0 6 2 10 C -10 -6 0 4 -4 D 0 -2 -4 0 0 E -14 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993656 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 0 14 B 2 0 6 2 10 C -10 -6 0 4 -4 D 0 -2 -4 0 0 E -14 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993656 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3224: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (7) A E D B C (7) C D B E A (6) C B D A E (6) B E A C D (6) B C D E A (5) E A D B C (4) D A E C B (4) B E A D C (4) D E A C B (3) D C A E B (3) C B A E D (3) B A E C D (3) E B A D C (2) D E A B C (2) D C E A B (2) C D A E B (2) B C E D A (2) B C E A D (2) B C A E D (2) A E D C B (2) A D E C B (2) A C D E B (2) A B E C D (2) E D A B C (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D A C E B (1) C D B A E (1) C D A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B D E C A (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -4 -2 -6 B 10 0 -6 2 14 C 4 6 0 14 2 D 2 -2 -14 0 12 E 6 -14 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -2 -6 B 10 0 -6 2 14 C 4 6 0 14 2 D 2 -2 -14 0 12 E 6 -14 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=27 D=18 A=17 E=8 so E is eliminated. Round 2 votes counts: C=30 B=29 A=22 D=19 so D is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:210 D:199 A:189 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -2 -6 B 10 0 -6 2 14 C 4 6 0 14 2 D 2 -2 -14 0 12 E 6 -14 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -2 -6 B 10 0 -6 2 14 C 4 6 0 14 2 D 2 -2 -14 0 12 E 6 -14 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -2 -6 B 10 0 -6 2 14 C 4 6 0 14 2 D 2 -2 -14 0 12 E 6 -14 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3225: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E D C A B (6) E A B D C (6) D E C A B (6) D C E B A (6) C D E A B (6) B A C D E (6) B A E C D (5) C D B E A (4) B A E D C (4) A B E C D (4) C D B A E (3) C D A B E (3) A E B D C (3) E B A D C (2) E A D C B (2) C D E B A (2) C D A E B (2) B E A D C (2) B D C A E (2) B C D A E (2) B C A D E (2) A E B C D (2) A C D E B (2) E D C B A (1) E D A B C (1) E A D B C (1) E A C D B (1) B A C E D (1) A E C D B (1) A C E D B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 16 -8 -6 -10 B -16 0 -12 -14 -20 C 8 12 0 -4 4 D 6 14 4 0 12 E 10 20 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -8 -6 -10 B -16 0 -12 -14 -20 C 8 12 0 -4 4 D 6 14 4 0 12 E 10 20 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=21 E=20 C=20 A=15 so A is eliminated. Round 2 votes counts: B=29 E=26 C=24 D=21 so D is eliminated. Round 3 votes counts: C=39 E=32 B=29 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:210 E:207 A:196 B:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -8 -6 -10 B -16 0 -12 -14 -20 C 8 12 0 -4 4 D 6 14 4 0 12 E 10 20 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 -6 -10 B -16 0 -12 -14 -20 C 8 12 0 -4 4 D 6 14 4 0 12 E 10 20 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 -6 -10 B -16 0 -12 -14 -20 C 8 12 0 -4 4 D 6 14 4 0 12 E 10 20 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3226: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (15) B E D A C (13) E B C A D (10) D A C B E (10) E B A D C (6) C E A D B (5) E C B A D (4) E C A B D (4) A C D E B (4) D A C E B (3) D A B C E (3) A D C E B (3) E C A D B (2) E B A C D (2) B D E A C (2) B D C A E (2) B D A E C (2) E A C D B (1) D B A C E (1) C D A E B (1) C D A B E (1) B E D C A (1) B E C D A (1) B E C A D (1) B D C E A (1) B D A C E (1) B C D A E (1) Total count = 100 A B C D E A 0 4 2 14 -6 B -4 0 -6 2 -20 C -2 6 0 4 2 D -14 -2 -4 0 0 E 6 20 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999991 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 4 2 14 -6 B -4 0 -6 2 -20 C -2 6 0 4 2 D -14 -2 -4 0 0 E 6 20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000243 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=25 C=22 D=17 A=7 so A is eliminated. Round 2 votes counts: E=29 C=26 B=25 D=20 so D is eliminated. Round 3 votes counts: C=42 E=29 B=29 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 A:207 C:205 D:190 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 2 14 -6 B -4 0 -6 2 -20 C -2 6 0 4 2 D -14 -2 -4 0 0 E 6 20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000243 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 14 -6 B -4 0 -6 2 -20 C -2 6 0 4 2 D -14 -2 -4 0 0 E 6 20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000243 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 14 -6 B -4 0 -6 2 -20 C -2 6 0 4 2 D -14 -2 -4 0 0 E 6 20 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000243 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3227: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) E C B D A (5) E C A D B (5) D C E B A (5) B A D E C (5) A E C D B (5) A D C E B (5) B D A C E (4) A E C B D (4) A D B C E (4) A C E D B (4) A B E C D (4) E C D A B (3) D B C E A (3) B D E C A (3) A D C B E (3) A B D C E (3) C E D B A (2) C E D A B (2) B E D C A (2) B E C D A (2) B A D C E (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C D A (1) D C E A B (1) D A C E B (1) D A C B E (1) D A B C E (1) C D E B A (1) C A E D B (1) B A E D C (1) B A E C D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 6 6 B -4 0 -22 -16 -16 C -4 22 0 8 -10 D -6 16 -8 0 -8 E -6 16 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 6 6 B -4 0 -22 -16 -16 C -4 22 0 8 -10 D -6 16 -8 0 -8 E -6 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=26 B=20 D=12 C=6 so C is eliminated. Round 2 votes counts: A=37 E=30 B=20 D=13 so D is eliminated. Round 3 votes counts: A=40 E=37 B=23 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:214 A:210 C:208 D:197 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 6 6 B -4 0 -22 -16 -16 C -4 22 0 8 -10 D -6 16 -8 0 -8 E -6 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 6 6 B -4 0 -22 -16 -16 C -4 22 0 8 -10 D -6 16 -8 0 -8 E -6 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 6 6 B -4 0 -22 -16 -16 C -4 22 0 8 -10 D -6 16 -8 0 -8 E -6 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3228: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (11) A E C B D (10) D B C E A (9) D B C A E (9) E A C D B (7) E A B D C (6) A C E B D (6) D B E C A (5) E D B A C (4) E D B C A (3) D E B C A (3) C B D A E (3) C A D B E (3) C A B D E (3) E A D C B (2) E A D B C (2) E A B C D (2) B D E C A (2) B D C E A (2) E B D C A (1) E B D A C (1) D C B A E (1) C D B A E (1) B D C A E (1) A E C D B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 8 10 -20 B -10 0 0 -2 -18 C -8 0 0 -2 -20 D -10 2 2 0 -12 E 20 18 20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 8 10 -20 B -10 0 0 -2 -18 C -8 0 0 -2 -20 D -10 2 2 0 -12 E 20 18 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=27 A=19 C=10 B=5 so B is eliminated. Round 2 votes counts: E=39 D=32 A=19 C=10 so C is eliminated. Round 3 votes counts: E=39 D=36 A=25 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:235 A:204 D:191 B:185 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 8 10 -20 B -10 0 0 -2 -18 C -8 0 0 -2 -20 D -10 2 2 0 -12 E 20 18 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 10 -20 B -10 0 0 -2 -18 C -8 0 0 -2 -20 D -10 2 2 0 -12 E 20 18 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 10 -20 B -10 0 0 -2 -18 C -8 0 0 -2 -20 D -10 2 2 0 -12 E 20 18 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3229: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (7) E D B C A (6) E D A C B (6) A C B E D (6) E D B A C (5) B C A E D (5) B A C E D (5) A C B D E (5) D E C A B (4) B E A D C (4) B C A D E (4) D E B C A (3) C B A D E (3) E B D A C (2) E B A D C (2) C A D B E (2) A E C D B (2) A C E B D (2) A C D E B (2) A B C E D (2) E D A B C (1) E A D C B (1) E A B D C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E C A (1) C D B A E (1) C D A E B (1) C B D E A (1) C B D A E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E C A (1) B C E D A (1) B C D E A (1) A E C B D (1) A E B D C (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 16 4 B 6 0 2 22 12 C -4 -2 0 12 6 D -16 -22 -12 0 -16 E -4 -12 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998086 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 16 4 B 6 0 2 22 12 C -4 -2 0 12 6 D -16 -22 -12 0 -16 E -4 -12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995251 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 A=23 C=16 D=12 so D is eliminated. Round 2 votes counts: E=32 B=26 A=23 C=19 so C is eliminated. Round 3 votes counts: E=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:221 A:209 C:206 E:197 D:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 16 4 B 6 0 2 22 12 C -4 -2 0 12 6 D -16 -22 -12 0 -16 E -4 -12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995251 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 16 4 B 6 0 2 22 12 C -4 -2 0 12 6 D -16 -22 -12 0 -16 E -4 -12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995251 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 16 4 B 6 0 2 22 12 C -4 -2 0 12 6 D -16 -22 -12 0 -16 E -4 -12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995251 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3230: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) E A D C B (6) A C D B E (5) E D A B C (4) D B C A E (4) C A B D E (4) B D C A E (4) A E C D B (4) A C D E B (4) E B D C A (3) E A C D B (3) D C A B E (3) B E D C A (3) B D C E A (3) B C D A E (3) E D B A C (2) E A D B C (2) D B C E A (2) C D B A E (2) C B A D E (2) C A D B E (2) B E C D A (2) B D E C A (2) A C E D B (2) A C E B D (2) A C B E D (2) E D B C A (1) E D A C B (1) E B A C D (1) E A C B D (1) E A B C D (1) D E A C B (1) D C A E B (1) D A C E B (1) C D A B E (1) B E C A D (1) B E A C D (1) B C D E A (1) B C A D E (1) A E D C B (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 6 -8 -6 18 B -6 0 -18 -24 14 C 8 18 0 -8 16 D 6 24 8 0 12 E -18 -14 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -6 18 B -6 0 -18 -24 14 C 8 18 0 -8 16 D 6 24 8 0 12 E -18 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=22 D=21 B=21 C=11 so C is eliminated. Round 2 votes counts: A=28 E=25 D=24 B=23 so B is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:225 C:217 A:205 B:183 E:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -6 18 B -6 0 -18 -24 14 C 8 18 0 -8 16 D 6 24 8 0 12 E -18 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -6 18 B -6 0 -18 -24 14 C 8 18 0 -8 16 D 6 24 8 0 12 E -18 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -6 18 B -6 0 -18 -24 14 C 8 18 0 -8 16 D 6 24 8 0 12 E -18 -14 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3231: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) A B E D C (8) C D E B A (5) E C B D A (4) E B C D A (4) E B A C D (4) A E B C D (4) A D B C E (4) E C D B A (3) D B C A E (3) C D E A B (3) B E D C A (3) B E C D A (3) A D C E B (3) A C D E B (3) A B D E C (3) A B D C E (3) E A B C D (2) D C B E A (2) B D C E A (2) B A E D C (2) A E C D B (2) E C A D B (1) E B C A D (1) E A C D B (1) D C A E B (1) D B C E A (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C D A E B (1) B D E C A (1) B A D E C (1) B A D C E (1) A E D B C (1) A E C B D (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 16 20 18 16 B -16 0 8 -2 4 C -20 -8 0 -10 0 D -18 2 10 0 4 E -16 -4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 20 18 16 B -16 0 8 -2 4 C -20 -8 0 -10 0 D -18 2 10 0 4 E -16 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 E=20 B=13 C=11 D=10 so D is eliminated. Round 2 votes counts: A=48 E=20 B=18 C=14 so C is eliminated. Round 3 votes counts: A=50 E=30 B=20 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:235 D:199 B:197 E:188 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 20 18 16 B -16 0 8 -2 4 C -20 -8 0 -10 0 D -18 2 10 0 4 E -16 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 20 18 16 B -16 0 8 -2 4 C -20 -8 0 -10 0 D -18 2 10 0 4 E -16 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 20 18 16 B -16 0 8 -2 4 C -20 -8 0 -10 0 D -18 2 10 0 4 E -16 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3232: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (24) A D C B E (24) E B C A D (6) C D A B E (5) B E D A C (5) B E C D A (4) B E A D C (4) A D B C E (4) E C B D A (3) D A C B E (3) A D C E B (3) E B A D C (2) C D A E B (2) E C B A D (1) E A D B C (1) D A B C E (1) C E B D A (1) C E A D B (1) C D B A E (1) C A D E B (1) B E D C A (1) B D A E C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 0 -2 -6 B 6 0 8 4 8 C 0 -8 0 0 -6 D 2 -4 0 0 -6 E 6 -8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 -6 B 6 0 8 4 8 C 0 -8 0 0 -6 D 2 -4 0 0 -6 E 6 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=33 B=15 C=11 D=4 so D is eliminated. Round 2 votes counts: E=37 A=37 B=15 C=11 so C is eliminated. Round 3 votes counts: A=45 E=39 B=16 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:205 D:196 A:193 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -2 -6 B 6 0 8 4 8 C 0 -8 0 0 -6 D 2 -4 0 0 -6 E 6 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 -6 B 6 0 8 4 8 C 0 -8 0 0 -6 D 2 -4 0 0 -6 E 6 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 -6 B 6 0 8 4 8 C 0 -8 0 0 -6 D 2 -4 0 0 -6 E 6 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3233: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (16) A B C E D (13) A B C D E (7) D E A C B (6) D A B C E (6) C B E A D (6) E C B D A (5) D A E B C (5) A D B C E (5) A B D C E (5) E C B A D (3) D E C A B (3) B A C E D (3) E D C B A (2) E C D B A (2) D A B E C (2) B C E A D (2) A D B E C (2) D E A B C (1) D A E C B (1) C E D B A (1) C E B A D (1) C B D A E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 16 16 -2 4 B -16 0 6 -4 8 C -16 -6 0 -10 0 D 2 4 10 0 22 E -4 -8 0 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 -2 4 B -16 0 6 -4 8 C -16 -6 0 -10 0 D 2 4 10 0 22 E -4 -8 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=34 E=12 C=9 B=5 so B is eliminated. Round 2 votes counts: D=40 A=37 E=12 C=11 so C is eliminated. Round 3 votes counts: D=41 A=37 E=22 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:217 B:197 C:184 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 16 -2 4 B -16 0 6 -4 8 C -16 -6 0 -10 0 D 2 4 10 0 22 E -4 -8 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 -2 4 B -16 0 6 -4 8 C -16 -6 0 -10 0 D 2 4 10 0 22 E -4 -8 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 -2 4 B -16 0 6 -4 8 C -16 -6 0 -10 0 D 2 4 10 0 22 E -4 -8 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3234: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (21) C B A D E (15) B C D E A (11) E D B A C (8) E D A B C (8) C A B D E (8) B D E C A (7) A C E D B (6) A C B E D (4) A E D C B (3) A E D B C (3) E A D B C (2) D E B A C (2) C B D E A (1) C B D A E (1) Total count = 100 A B C D E A 0 -32 -28 -18 -20 B 32 0 24 -6 -6 C 28 -24 0 -8 -8 D 18 6 8 0 32 E 20 6 8 -32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 -28 -18 -20 B 32 0 24 -6 -6 C 28 -24 0 -8 -8 D 18 6 8 0 32 E 20 6 8 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=23 E=18 B=18 A=16 so A is eliminated. Round 2 votes counts: C=35 E=24 D=23 B=18 so B is eliminated. Round 3 votes counts: C=46 D=30 E=24 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:232 B:222 E:201 C:194 A:151 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -32 -28 -18 -20 B 32 0 24 -6 -6 C 28 -24 0 -8 -8 D 18 6 8 0 32 E 20 6 8 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 -28 -18 -20 B 32 0 24 -6 -6 C 28 -24 0 -8 -8 D 18 6 8 0 32 E 20 6 8 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 -28 -18 -20 B 32 0 24 -6 -6 C 28 -24 0 -8 -8 D 18 6 8 0 32 E 20 6 8 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3235: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) E A D B C (6) E A C B D (6) E C A B D (5) A B C E D (4) E D A B C (3) E C A D B (3) D C B E A (3) C B D A E (3) C A B E D (3) A E B D C (3) A E B C D (3) A B E D C (3) A B C D E (3) E C D A B (2) E A B D C (2) E A B C D (2) D E C B A (2) D E B A C (2) D B C A E (2) C E D B A (2) C E B D A (2) C B A D E (2) B D A C E (2) B C D A E (2) B A C D E (2) A E C B D (2) A B D E C (2) A B D C E (2) E D C B A (1) E D A C B (1) E A D C B (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C E A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E B A (1) C D B E A (1) C A E B D (1) B A D C E (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 20 8 18 -12 B -20 0 0 14 -20 C -8 0 0 14 -18 D -18 -14 -14 0 -26 E 12 20 18 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 8 18 -12 B -20 0 0 14 -20 C -8 0 0 14 -18 D -18 -14 -14 0 -26 E 12 20 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=24 C=16 D=15 B=7 so B is eliminated. Round 2 votes counts: E=38 A=27 C=18 D=17 so D is eliminated. Round 3 votes counts: E=44 A=31 C=25 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:238 A:217 C:194 B:187 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 8 18 -12 B -20 0 0 14 -20 C -8 0 0 14 -18 D -18 -14 -14 0 -26 E 12 20 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 8 18 -12 B -20 0 0 14 -20 C -8 0 0 14 -18 D -18 -14 -14 0 -26 E 12 20 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 8 18 -12 B -20 0 0 14 -20 C -8 0 0 14 -18 D -18 -14 -14 0 -26 E 12 20 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3236: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) C D A E B (9) E B D C A (7) E B D A C (6) B E A D C (6) B A E C D (6) B A C E D (6) D E C A B (5) D C E A B (5) C A D B E (5) A C B D E (5) E D B C A (4) D C A E B (4) D A C E B (3) E D C B A (2) E B C D A (2) D E C B A (2) B E A C D (2) A B C D E (2) E D B A C (1) E C D B A (1) D E A B C (1) C A D E B (1) B E C A D (1) B A D E C (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 6 4 -4 10 B -6 0 -10 -10 -6 C -4 10 0 4 6 D 4 10 -4 0 10 E -10 6 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 10 B -6 0 -10 -10 -6 C -4 10 0 4 6 D 4 10 -4 0 10 E -10 6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 B=22 D=20 A=20 C=15 so C is eliminated. Round 2 votes counts: D=29 A=26 E=23 B=22 so B is eliminated. Round 3 votes counts: A=39 E=32 D=29 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:208 C:208 E:190 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 -4 10 B -6 0 -10 -10 -6 C -4 10 0 4 6 D 4 10 -4 0 10 E -10 6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 10 B -6 0 -10 -10 -6 C -4 10 0 4 6 D 4 10 -4 0 10 E -10 6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 10 B -6 0 -10 -10 -6 C -4 10 0 4 6 D 4 10 -4 0 10 E -10 6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3237: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) D A C B E (10) C A D B E (10) C B E A D (9) B E C A D (7) E B C D A (6) A D C B E (6) E B D A C (5) E B C A D (5) D A E B C (5) B C E A D (4) E B D C A (3) D A E C B (3) C E B A D (2) E D C A B (1) E C D A B (1) E C B D A (1) E B A D C (1) D E A B C (1) D C A E B (1) D A B E C (1) D A B C E (1) C D E A B (1) C D A E B (1) C B A D E (1) C A B D E (1) B A C D E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -8 -2 6 B -10 0 -18 -8 6 C 8 18 0 2 20 D 2 8 -2 0 10 E -6 -6 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 -2 6 B -10 0 -18 -8 6 C 8 18 0 2 20 D 2 8 -2 0 10 E -6 -6 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=25 E=23 B=12 A=8 so A is eliminated. Round 2 votes counts: D=39 C=26 E=23 B=12 so B is eliminated. Round 3 votes counts: D=39 C=31 E=30 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 D:209 A:203 B:185 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 -2 6 B -10 0 -18 -8 6 C 8 18 0 2 20 D 2 8 -2 0 10 E -6 -6 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -2 6 B -10 0 -18 -8 6 C 8 18 0 2 20 D 2 8 -2 0 10 E -6 -6 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -2 6 B -10 0 -18 -8 6 C 8 18 0 2 20 D 2 8 -2 0 10 E -6 -6 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3238: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) B D C A E (9) D B C E A (8) D A B E C (5) C E A B D (5) D A E B C (4) C E B A D (4) A E C B D (4) E A C B D (3) C B E D A (3) B D C E A (3) B C D A E (3) E C A D B (2) E C A B D (2) D E A B C (2) C B E A D (2) C B D E A (2) C A E B D (2) B D A E C (2) A E D B C (2) A E B C D (2) A D E B C (2) E C D A B (1) E A D C B (1) E A C D B (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A C E (1) C E D B A (1) C E D A B (1) C D B E A (1) C A B E D (1) B D A C E (1) B A D E C (1) A E D C B (1) A E C D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -8 -24 6 B 12 0 20 2 14 C 8 -20 0 -18 0 D 24 -2 18 0 20 E -6 -14 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998472 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -24 6 B 12 0 20 2 14 C 8 -20 0 -18 0 D 24 -2 18 0 20 E -6 -14 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=22 B=19 A=14 E=10 so E is eliminated. Round 2 votes counts: D=35 C=27 B=19 A=19 so B is eliminated. Round 3 votes counts: D=50 C=30 A=20 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 B:224 C:185 A:181 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -24 6 B 12 0 20 2 14 C 8 -20 0 -18 0 D 24 -2 18 0 20 E -6 -14 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -24 6 B 12 0 20 2 14 C 8 -20 0 -18 0 D 24 -2 18 0 20 E -6 -14 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -24 6 B 12 0 20 2 14 C 8 -20 0 -18 0 D 24 -2 18 0 20 E -6 -14 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3239: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (13) E A B D C (12) C D B A E (11) B D C A E (8) C D A B E (5) A B D C E (5) E B D C A (4) E B A D C (3) D C B A E (3) A E C D B (3) A D C B E (3) E C D B A (2) E B D A C (2) E A C B D (2) D B C A E (2) C D B E A (2) B E D C A (2) B D C E A (2) B A D C E (2) A E B D C (2) A C D B E (2) E C D A B (1) E C A D B (1) E A B C D (1) C A D E B (1) B E C D A (1) B C D E A (1) A E D C B (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 8 8 2 B -10 0 -4 -6 0 C -8 4 0 -4 0 D -8 6 4 0 -2 E -2 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 8 2 B -10 0 -4 -6 0 C -8 4 0 -4 0 D -8 6 4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999248 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=19 A=19 B=16 D=5 so D is eliminated. Round 2 votes counts: E=41 C=22 A=19 B=18 so B is eliminated. Round 3 votes counts: E=44 C=35 A=21 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 D:200 E:200 C:196 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 8 2 B -10 0 -4 -6 0 C -8 4 0 -4 0 D -8 6 4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999248 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 8 2 B -10 0 -4 -6 0 C -8 4 0 -4 0 D -8 6 4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999248 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 8 2 B -10 0 -4 -6 0 C -8 4 0 -4 0 D -8 6 4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999248 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3240: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) C B E A D (6) A B C D E (6) D E C B A (5) A C B E D (5) D E C A B (4) D E B C A (4) B A C E D (4) A D C E B (4) B C E D A (3) B C E A D (3) A D E C B (3) A D C B E (3) A D B E C (3) E D C B A (2) E D C A B (2) E D B C A (2) E C D A B (2) D E B A C (2) D A E C B (2) D A E B C (2) A D B C E (2) A C E D B (2) A C B D E (2) A B C E D (2) E C D B A (1) E C B D A (1) E B C D A (1) D E A C B (1) D E A B C (1) D B E C A (1) D A B E C (1) C E B D A (1) C E A D B (1) B E C D A (1) B D E A C (1) B A D C E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 20 10 B 0 0 4 0 12 C 2 -4 0 6 16 D -20 0 -6 0 -2 E -10 -12 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.376348 B: 0.623652 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.530579600972 Cumulative probabilities = A: 0.376348 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 20 10 B 0 0 4 0 12 C 2 -4 0 6 16 D -20 0 -6 0 -2 E -10 -12 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999366 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=24 D=23 E=11 C=8 so C is eliminated. Round 2 votes counts: A=34 B=30 D=23 E=13 so E is eliminated. Round 3 votes counts: A=35 B=33 D=32 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:210 B:208 D:186 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 20 10 B 0 0 4 0 12 C 2 -4 0 6 16 D -20 0 -6 0 -2 E -10 -12 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999366 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 20 10 B 0 0 4 0 12 C 2 -4 0 6 16 D -20 0 -6 0 -2 E -10 -12 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999366 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 20 10 B 0 0 4 0 12 C 2 -4 0 6 16 D -20 0 -6 0 -2 E -10 -12 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999366 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3241: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (7) E D B A C (5) E B D A C (5) D A C E B (5) B E D A C (5) B E C D A (4) A D C E B (4) A C D E B (4) E D A B C (3) D A E C B (3) C D A B E (3) C B A E D (3) C A E D B (3) C A E B D (3) B E D C A (3) E A C B D (2) D E B A C (2) D B A E C (2) C E B A D (2) C B E A D (2) C B A D E (2) C A D E B (2) C A D B E (2) B D E C A (2) B C E D A (2) B C E A D (2) B C D A E (2) A E C D B (2) E C A B D (1) E B A D C (1) E B A C D (1) E A D C B (1) E A D B C (1) E A C D B (1) D B A C E (1) D A C B E (1) D A B E C (1) C E A B D (1) C B D A E (1) B D E A C (1) B D C A E (1) B C D E A (1) Total count = 100 A B C D E A 0 -14 2 -6 -10 B 14 0 4 8 -4 C -2 -4 0 6 -6 D 6 -8 -6 0 -20 E 10 4 6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 2 -6 -10 B 14 0 4 8 -4 C -2 -4 0 6 -6 D 6 -8 -6 0 -20 E 10 4 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=24 E=21 D=15 A=10 so A is eliminated. Round 2 votes counts: B=30 C=28 E=23 D=19 so D is eliminated. Round 3 votes counts: C=38 B=34 E=28 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:220 B:211 C:197 A:186 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 2 -6 -10 B 14 0 4 8 -4 C -2 -4 0 6 -6 D 6 -8 -6 0 -20 E 10 4 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -6 -10 B 14 0 4 8 -4 C -2 -4 0 6 -6 D 6 -8 -6 0 -20 E 10 4 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -6 -10 B 14 0 4 8 -4 C -2 -4 0 6 -6 D 6 -8 -6 0 -20 E 10 4 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3242: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (20) E B C A D (13) E D B C A (5) D E B C A (5) E C B A D (4) D A B C E (4) B E C A D (4) A B C D E (4) B E A C D (3) E C B D A (2) D E C B A (2) D E B A C (2) D A E B C (2) D A C E B (2) C E B A D (2) C E A B D (2) C B E A D (2) C A B E D (2) A D C B E (2) A C B E D (2) E D C B A (1) E B D A C (1) E B C D A (1) D E C A B (1) D E A B C (1) D C A E B (1) D C A B E (1) D A B E C (1) C B A E D (1) C A D B E (1) C A B D E (1) B C E A D (1) B A C E D (1) A D B C E (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 -4 -4 B 0 0 0 -6 6 C 4 0 0 -4 4 D 4 6 4 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -4 -4 B 0 0 0 -6 6 C 4 0 0 -4 4 D 4 6 4 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 E=27 C=11 A=11 B=9 so B is eliminated. Round 2 votes counts: D=42 E=34 C=12 A=12 so C is eliminated. Round 3 votes counts: D=42 E=41 A=17 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:209 C:202 B:200 E:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -4 -4 B 0 0 0 -6 6 C 4 0 0 -4 4 D 4 6 4 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 -4 B 0 0 0 -6 6 C 4 0 0 -4 4 D 4 6 4 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 -4 B 0 0 0 -6 6 C 4 0 0 -4 4 D 4 6 4 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3243: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (6) E D C B A (5) E B C A D (5) D E C A B (5) D A E C B (5) D A C E B (5) A B D C E (5) E C D B A (4) B E C A D (4) E C B D A (3) D C A E B (3) D A C B E (3) B A E C D (3) B A C E D (3) E B A D C (2) D E A B C (2) D C E A B (2) D A E B C (2) C E D B A (2) C B E A D (2) B E A C D (2) B C E A D (2) B C A E D (2) B A E D C (2) A D B C E (2) A B C E D (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A B C (1) D C E B A (1) C E B D A (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A B E (1) C B A E D (1) C B A D E (1) A D E B C (1) A D B E C (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 4 0 6 B -2 0 -8 -14 -6 C -4 8 0 -12 0 D 0 14 12 0 4 E -6 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.584071 B: 0.000000 C: 0.000000 D: 0.415929 E: 0.000000 Sum of squares = 0.514135801986 Cumulative probabilities = A: 0.584071 B: 0.584071 C: 0.584071 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 0 6 B -2 0 -8 -14 -6 C -4 8 0 -12 0 D 0 14 12 0 4 E -6 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=22 A=21 B=18 C=11 so C is eliminated. Round 2 votes counts: D=32 E=25 B=22 A=21 so A is eliminated. Round 3 votes counts: D=43 B=32 E=25 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:206 E:198 C:196 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 6 B -2 0 -8 -14 -6 C -4 8 0 -12 0 D 0 14 12 0 4 E -6 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 6 B -2 0 -8 -14 -6 C -4 8 0 -12 0 D 0 14 12 0 4 E -6 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 6 B -2 0 -8 -14 -6 C -4 8 0 -12 0 D 0 14 12 0 4 E -6 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3244: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) A E B C D (6) E A C D B (5) D C B E A (5) B E C A D (5) A E D C B (5) A E B D C (5) C D B E A (4) B C E A D (4) B C D E A (4) E C A B D (3) D C B A E (3) D C A E B (3) D B C A E (3) B D C A E (3) B C E D A (3) B A E D C (3) A E D B C (3) D C A B E (2) D B A C E (2) C D E A B (2) C B D E A (2) B A E C D (2) A B E D C (2) E C A D B (1) E B A C D (1) E A C B D (1) E A B C D (1) D A C E B (1) D A B E C (1) C E D B A (1) C B E D A (1) B E A C D (1) B D A E C (1) B D A C E (1) A D E B C (1) Total count = 100 A B C D E A 0 2 2 -2 12 B -2 0 4 -2 4 C -2 -4 0 -6 -12 D 2 2 6 0 -6 E -12 -4 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.100000 Sum of squares = 0.459999999991 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.900000 E: 1.000000 A B C D E A 0 2 2 -2 12 B -2 0 4 -2 4 C -2 -4 0 -6 -12 D 2 2 6 0 -6 E -12 -4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.100000 Sum of squares = 0.459999999883 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=27 A=22 E=12 C=10 so C is eliminated. Round 2 votes counts: D=35 B=30 A=22 E=13 so E is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:207 B:202 D:202 E:201 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 -2 12 B -2 0 4 -2 4 C -2 -4 0 -6 -12 D 2 2 6 0 -6 E -12 -4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.100000 Sum of squares = 0.459999999883 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -2 12 B -2 0 4 -2 4 C -2 -4 0 -6 -12 D 2 2 6 0 -6 E -12 -4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.100000 Sum of squares = 0.459999999883 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.900000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -2 12 B -2 0 4 -2 4 C -2 -4 0 -6 -12 D 2 2 6 0 -6 E -12 -4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.100000 Sum of squares = 0.459999999883 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.900000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3245: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) E B D A C (7) A C D E B (6) C A D E B (5) A C B E D (5) E B A D C (4) D C B E A (4) B E D C A (4) B E A C D (4) A C E B D (4) D C A E B (3) C D A E B (3) C A D B E (3) B D E C A (3) A C E D B (3) D E B C A (2) D B E C A (2) D B C E A (2) C D A B E (2) B E A D C (2) A E C D B (2) A E C B D (2) D E C B A (1) D E C A B (1) D E B A C (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A B E (1) C D B A E (1) B E D A C (1) B E C D A (1) B D C E A (1) B C D A E (1) B A C E D (1) A E D C B (1) A E B D C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 14 12 16 B -12 0 -2 6 -18 C -14 2 0 12 -2 D -12 -6 -12 0 -8 E -16 18 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 12 16 B -12 0 -2 6 -18 C -14 2 0 12 -2 D -12 -6 -12 0 -8 E -16 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=20 B=18 C=14 E=11 so E is eliminated. Round 2 votes counts: A=37 B=29 D=20 C=14 so C is eliminated. Round 3 votes counts: A=45 B=29 D=26 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:206 C:199 B:187 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 12 16 B -12 0 -2 6 -18 C -14 2 0 12 -2 D -12 -6 -12 0 -8 E -16 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 12 16 B -12 0 -2 6 -18 C -14 2 0 12 -2 D -12 -6 -12 0 -8 E -16 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 12 16 B -12 0 -2 6 -18 C -14 2 0 12 -2 D -12 -6 -12 0 -8 E -16 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3246: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) A D E B C (10) C A D E B (7) B E A D C (6) A D C E B (6) E B D A C (5) C D A E B (5) B E D A C (5) B E C D A (5) C B A E D (4) C A D B E (4) B E A C D (4) A B E D C (4) B C E A D (3) A D E C B (3) D A C E B (2) C B E A D (2) B E D C A (2) B C E D A (2) A C D E B (2) E D B A C (1) D E B A C (1) D E A B C (1) D A E B C (1) C E B D A (1) C D E B A (1) B A E D C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 8 16 2 B 6 0 6 12 6 C -8 -6 0 2 0 D -16 -12 -2 0 -14 E -2 -6 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 16 2 B 6 0 6 12 6 C -8 -6 0 2 0 D -16 -12 -2 0 -14 E -2 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=28 A=27 E=6 D=5 so D is eliminated. Round 2 votes counts: C=34 A=30 B=28 E=8 so E is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:210 E:203 C:194 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 16 2 B 6 0 6 12 6 C -8 -6 0 2 0 D -16 -12 -2 0 -14 E -2 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 16 2 B 6 0 6 12 6 C -8 -6 0 2 0 D -16 -12 -2 0 -14 E -2 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 16 2 B 6 0 6 12 6 C -8 -6 0 2 0 D -16 -12 -2 0 -14 E -2 -6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3247: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (25) C B E D A (19) E D A C B (11) A E D C B (10) B C A D E (7) B C E D A (4) B C A E D (3) A D B E C (3) D E A C B (2) D A E B C (2) B A C D E (2) A D E C B (2) E D C A B (1) E C D B A (1) E C D A B (1) E A D C B (1) C E D B A (1) C B E A D (1) B D A E C (1) B C E A D (1) B C D A E (1) A B C D E (1) Total count = 100 A B C D E A 0 18 20 12 14 B -18 0 0 -20 -14 C -20 0 0 -16 -20 D -12 20 16 0 -8 E -14 14 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 20 12 14 B -18 0 0 -20 -14 C -20 0 0 -16 -20 D -12 20 16 0 -8 E -14 14 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=21 B=19 E=15 D=4 so D is eliminated. Round 2 votes counts: A=43 C=21 B=19 E=17 so E is eliminated. Round 3 votes counts: A=57 C=24 B=19 so B is eliminated. Round 4 votes counts: A=60 C=40 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:232 E:214 D:208 B:174 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 20 12 14 B -18 0 0 -20 -14 C -20 0 0 -16 -20 D -12 20 16 0 -8 E -14 14 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 12 14 B -18 0 0 -20 -14 C -20 0 0 -16 -20 D -12 20 16 0 -8 E -14 14 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 12 14 B -18 0 0 -20 -14 C -20 0 0 -16 -20 D -12 20 16 0 -8 E -14 14 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3248: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (16) A B C D E (12) B A C D E (11) E D C B A (10) E A B D C (6) E A D C B (4) D C E B A (4) C D B A E (4) A B E C D (4) D E C A B (3) D C A E B (3) C D A B E (3) E B D C A (2) E B A D C (2) E A D B C (2) D C E A B (2) C B D E A (2) E D B C A (1) C D B E A (1) C B D A E (1) C A D B E (1) B E A D C (1) B C D E A (1) B A C E D (1) A E D C B (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 18 -8 -6 -14 B -18 0 -12 -12 -14 C 8 12 0 -16 -6 D 6 12 16 0 -2 E 14 14 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 -8 -6 -14 B -18 0 -12 -12 -14 C 8 12 0 -16 -6 D 6 12 16 0 -2 E 14 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 A=19 B=14 D=12 C=12 so D is eliminated. Round 2 votes counts: E=46 C=21 A=19 B=14 so B is eliminated. Round 3 votes counts: E=47 A=31 C=22 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:216 C:199 A:195 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -8 -6 -14 B -18 0 -12 -12 -14 C 8 12 0 -16 -6 D 6 12 16 0 -2 E 14 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 -6 -14 B -18 0 -12 -12 -14 C 8 12 0 -16 -6 D 6 12 16 0 -2 E 14 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 -6 -14 B -18 0 -12 -12 -14 C 8 12 0 -16 -6 D 6 12 16 0 -2 E 14 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3249: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) B C E D A (8) E D C A B (7) B D E A C (7) C E D A B (6) C B A E D (5) B C A E D (5) B E D C A (4) A C D E B (4) D E A C B (3) C E D B A (3) C E A D B (3) A D E C B (3) E C D A B (2) C A D E B (2) B D A E C (2) B C E A D (2) B A D E C (2) B A C D E (2) A D E B C (2) A D C E B (2) A B D E C (2) E D B C A (1) E D B A C (1) E C D B A (1) E B D C A (1) E B C D A (1) D E A B C (1) D A E C B (1) D A E B C (1) C E B D A (1) C A E B D (1) C A B D E (1) B E C D A (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -28 -4 -6 B -6 0 -12 -6 -12 C 28 12 0 16 14 D 4 6 -16 0 -24 E 6 12 -14 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -28 -4 -6 B -6 0 -12 -6 -12 C 28 12 0 16 14 D 4 6 -16 0 -24 E 6 12 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=31 A=16 E=14 D=6 so D is eliminated. Round 2 votes counts: B=33 C=31 E=18 A=18 so E is eliminated. Round 3 votes counts: C=41 B=37 A=22 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:235 E:214 D:185 A:184 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -28 -4 -6 B -6 0 -12 -6 -12 C 28 12 0 16 14 D 4 6 -16 0 -24 E 6 12 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -28 -4 -6 B -6 0 -12 -6 -12 C 28 12 0 16 14 D 4 6 -16 0 -24 E 6 12 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -28 -4 -6 B -6 0 -12 -6 -12 C 28 12 0 16 14 D 4 6 -16 0 -24 E 6 12 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3250: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) E B A D C (7) C D A B E (7) E A B D C (6) D B E C A (6) B E D A C (5) A C E B D (5) D C B E A (4) A C E D B (4) E B D A C (3) D C A B E (3) D B E A C (3) D B C E A (3) C D B E A (3) C A E D B (3) A E C B D (3) A E B D C (3) D C B A E (2) D A C B E (2) C D B A E (2) C A E B D (2) C A D B E (2) B E D C A (2) A E B C D (2) E B C D A (1) E B A C D (1) E A B C D (1) C E B A D (1) C E A B D (1) B D E C A (1) B D E A C (1) B C E D A (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -6 0 -2 B -6 0 -6 -6 -4 C 6 6 0 -2 10 D 0 6 2 0 -2 E 2 4 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 A B C D E A 0 6 -6 0 -2 B -6 0 -6 -6 -4 C 6 6 0 -2 10 D 0 6 2 0 -2 E 2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=23 E=19 A=18 B=11 so B is eliminated. Round 2 votes counts: C=31 E=26 D=25 A=18 so A is eliminated. Round 3 votes counts: C=41 E=34 D=25 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:203 A:199 E:199 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 0 -2 B -6 0 -6 -6 -4 C 6 6 0 -2 10 D 0 6 2 0 -2 E 2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 0 -2 B -6 0 -6 -6 -4 C 6 6 0 -2 10 D 0 6 2 0 -2 E 2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 0 -2 B -6 0 -6 -6 -4 C 6 6 0 -2 10 D 0 6 2 0 -2 E 2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3251: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) E A C B D (8) B D C A E (8) E B D C A (6) D B C A E (6) E D B A C (5) C A B D E (5) E D A C B (4) D B E A C (4) D A C B E (4) B D E C A (4) E B C A D (3) D B E C A (3) A C E B D (3) E B A C D (2) E A D C B (2) D E A C B (2) C A E B D (2) B E C A D (2) E D A B C (1) E C A B D (1) E B D A C (1) E B C D A (1) D E B A C (1) D B C E A (1) D B A C E (1) D A E C B (1) C A B E D (1) B C D A E (1) B C A E D (1) B C A D E (1) A C E D B (1) A C D E B (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 8 -8 -22 B 2 0 4 4 -10 C -8 -4 0 -8 -22 D 8 -4 8 0 -10 E 22 10 22 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 -8 -22 B 2 0 4 4 -10 C -8 -4 0 -8 -22 D 8 -4 8 0 -10 E 22 10 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 D=23 B=17 C=8 A=8 so C is eliminated. Round 2 votes counts: E=44 D=23 B=17 A=16 so A is eliminated. Round 3 votes counts: E=50 D=25 B=25 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:232 D:201 B:200 A:188 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 -8 -22 B 2 0 4 4 -10 C -8 -4 0 -8 -22 D 8 -4 8 0 -10 E 22 10 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -8 -22 B 2 0 4 4 -10 C -8 -4 0 -8 -22 D 8 -4 8 0 -10 E 22 10 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -8 -22 B 2 0 4 4 -10 C -8 -4 0 -8 -22 D 8 -4 8 0 -10 E 22 10 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3252: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) C E A D B (9) B C D A E (7) E C A D B (6) D A B E C (6) B E D A C (5) C E B A D (4) C D A B E (4) A D C E B (4) E A D C B (3) C B D A E (3) C A D E B (3) B D E A C (3) E C B A D (2) E B C A D (2) E A D B C (2) D A C B E (2) C B E A D (2) B E C D A (2) B D A C E (2) B C E D A (2) E B A D C (1) E A C D B (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A B D (1) C D B A E (1) C D A E B (1) C B E D A (1) C A E D B (1) B D C A E (1) B C E A D (1) B C D E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -8 -12 4 B 4 0 -4 0 12 C 8 4 0 8 6 D 12 0 -8 0 8 E -4 -12 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -12 4 B 4 0 -4 0 12 C 8 4 0 8 6 D 12 0 -8 0 8 E -4 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=30 E=17 D=13 A=6 so A is eliminated. Round 2 votes counts: B=34 C=30 E=18 D=18 so E is eliminated. Round 3 votes counts: C=39 B=37 D=24 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:206 D:206 A:190 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -12 4 B 4 0 -4 0 12 C 8 4 0 8 6 D 12 0 -8 0 8 E -4 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -12 4 B 4 0 -4 0 12 C 8 4 0 8 6 D 12 0 -8 0 8 E -4 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -12 4 B 4 0 -4 0 12 C 8 4 0 8 6 D 12 0 -8 0 8 E -4 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3253: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (13) C B D E A (11) B C D E A (8) A E D B C (5) E D A C B (4) D E A B C (4) B C A D E (4) B A D E C (4) B A C D E (4) A C E D B (4) A B C E D (4) C E D B A (3) A D E B C (3) A B D E C (3) D E C B A (2) C E D A B (2) C B A E D (2) C A B E D (2) B D C E A (2) E D C B A (1) E D C A B (1) E C D A B (1) E A D C B (1) D E B A C (1) D C E B A (1) C D E B A (1) C B E A D (1) B D E C A (1) B D E A C (1) B D C A E (1) B C A E D (1) B A D C E (1) A E C D B (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 10 10 8 B 0 0 -4 4 4 C -10 4 0 0 6 D -10 -4 0 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.588452 B: 0.411548 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.515647419353 Cumulative probabilities = A: 0.588452 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 10 8 B 0 0 -4 4 4 C -10 4 0 0 6 D -10 -4 0 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=27 C=22 E=8 D=8 so E is eliminated. Round 2 votes counts: A=36 B=27 C=23 D=14 so D is eliminated. Round 3 votes counts: A=44 C=28 B=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:202 C:200 D:195 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 10 8 B 0 0 -4 4 4 C -10 4 0 0 6 D -10 -4 0 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 10 8 B 0 0 -4 4 4 C -10 4 0 0 6 D -10 -4 0 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 10 8 B 0 0 -4 4 4 C -10 4 0 0 6 D -10 -4 0 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3254: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) C E B D A (7) C E B A D (7) D A B E C (6) D A B C E (6) A D B E C (6) D A C E B (5) D B A E C (4) D B A C E (4) C E A B D (4) A E C B D (4) D C E B A (3) D B C E A (3) C E D B A (3) C B E D A (3) B E C D A (3) C E A D B (2) A C E D B (2) A B E C D (2) E C A B D (1) E B C A D (1) E B A C D (1) D C E A B (1) D C B E A (1) C E D A B (1) C D E B A (1) C A E D B (1) B E C A D (1) B E A C D (1) B D E C A (1) B D C E A (1) B C D E A (1) A E D C B (1) A E B C D (1) A D E C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 -8 -10 B 8 0 -12 -2 -10 C 8 12 0 10 12 D 8 2 -10 0 -10 E 10 10 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -8 -10 B 8 0 -12 -2 -10 C 8 12 0 10 12 D 8 2 -10 0 -10 E 10 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=29 A=19 E=11 B=8 so B is eliminated. Round 2 votes counts: D=35 C=30 A=19 E=16 so E is eliminated. Round 3 votes counts: C=44 D=35 A=21 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:221 E:209 D:195 B:192 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -8 -10 B 8 0 -12 -2 -10 C 8 12 0 10 12 D 8 2 -10 0 -10 E 10 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -8 -10 B 8 0 -12 -2 -10 C 8 12 0 10 12 D 8 2 -10 0 -10 E 10 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -8 -10 B 8 0 -12 -2 -10 C 8 12 0 10 12 D 8 2 -10 0 -10 E 10 10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3255: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) B D A E C (6) B D E C A (5) B D C E A (5) E D A C B (4) B C D E A (4) A D E B C (4) E C D A B (3) D E B C A (3) D B E C A (3) C E A D B (3) B C A E D (3) B A C E D (3) A C E B D (3) A B C E D (3) E D C A B (2) E C A D B (2) E A C D B (2) D E A C B (2) D A E B C (2) C E B D A (2) C A E B D (2) B C A D E (2) A C B E D (2) A B D E C (2) D E C B A (1) D E B A C (1) D B E A C (1) D B A E C (1) D A E C B (1) C E D B A (1) C B D E A (1) C B A E D (1) B D E A C (1) B D C A E (1) B D A C E (1) B C D A E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E C B (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 10 -4 8 B 0 0 12 2 0 C -10 -12 0 -2 -18 D 4 -2 2 0 6 E -8 0 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.325404 B: 0.674596 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.560967598819 Cumulative probabilities = A: 0.325404 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 -4 8 B 0 0 12 2 0 C -10 -12 0 -2 -18 D 4 -2 2 0 6 E -8 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556117211 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=27 D=15 E=13 C=10 so C is eliminated. Round 2 votes counts: B=37 A=29 E=19 D=15 so D is eliminated. Round 3 votes counts: B=42 A=32 E=26 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:207 B:207 D:205 E:202 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 10 -4 8 B 0 0 12 2 0 C -10 -12 0 -2 -18 D 4 -2 2 0 6 E -8 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556117211 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -4 8 B 0 0 12 2 0 C -10 -12 0 -2 -18 D 4 -2 2 0 6 E -8 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556117211 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -4 8 B 0 0 12 2 0 C -10 -12 0 -2 -18 D 4 -2 2 0 6 E -8 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333332 B: 0.666668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555556117211 Cumulative probabilities = A: 0.333332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3256: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) D C A B E (6) E C A B D (5) C E A D B (5) A C E B D (5) B E D A C (4) B E A D C (4) B A D C E (4) E B C D A (3) D A C B E (3) C E D A B (3) C D A E B (3) B D E A C (3) E B D C A (2) E B C A D (2) E A C B D (2) E A B C D (2) D C A E B (2) D B A C E (2) D A B C E (2) C A E D B (2) B A D E C (2) A C D E B (2) A B D C E (2) E D C B A (1) E C D B A (1) E C D A B (1) E B D A C (1) D E C B A (1) D C B A E (1) D B E C A (1) D B C A E (1) B D E C A (1) B A E D C (1) B A E C D (1) A E C B D (1) A E B C D (1) A D C B E (1) A C E D B (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 18 16 -4 B -8 0 2 20 -12 C -18 -2 0 10 -2 D -16 -20 -10 0 -22 E 4 12 2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 18 16 -4 B -8 0 2 20 -12 C -18 -2 0 10 -2 D -16 -20 -10 0 -22 E 4 12 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=20 D=19 A=18 C=13 so C is eliminated. Round 2 votes counts: E=38 D=22 B=20 A=20 so B is eliminated. Round 3 votes counts: E=46 A=28 D=26 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:219 B:201 C:194 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 18 16 -4 B -8 0 2 20 -12 C -18 -2 0 10 -2 D -16 -20 -10 0 -22 E 4 12 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 16 -4 B -8 0 2 20 -12 C -18 -2 0 10 -2 D -16 -20 -10 0 -22 E 4 12 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 16 -4 B -8 0 2 20 -12 C -18 -2 0 10 -2 D -16 -20 -10 0 -22 E 4 12 2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3257: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B C E A (8) D E B A C (6) D B E C A (6) E A D B C (4) C B D E A (4) C B D A E (4) E B A C D (3) C D B A E (3) C B A D E (3) C A B D E (3) B D C E A (3) A E D C B (3) A E D B C (3) A E C D B (3) E D B A C (2) E B D A C (2) C B E A D (2) C B A E D (2) C A B E D (2) B E D C A (2) A E C B D (2) A C E D B (2) E B D C A (1) E B C A D (1) E A B C D (1) D C B E A (1) D A E B C (1) C D A B E (1) C B E D A (1) C A E B D (1) C A D B E (1) B C E D A (1) B C E A D (1) B C D E A (1) A E B C D (1) A D E C B (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -14 -4 6 0 B 14 0 -6 4 0 C 4 6 0 12 16 D -6 -4 -12 0 -2 E 0 0 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 6 0 B 14 0 -6 4 0 C 4 6 0 12 16 D -6 -4 -12 0 -2 E 0 0 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 D=22 E=14 B=8 so B is eliminated. Round 2 votes counts: C=30 A=29 D=25 E=16 so E is eliminated. Round 3 votes counts: A=37 D=32 C=31 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:219 B:206 A:194 E:193 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -4 6 0 B 14 0 -6 4 0 C 4 6 0 12 16 D -6 -4 -12 0 -2 E 0 0 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 6 0 B 14 0 -6 4 0 C 4 6 0 12 16 D -6 -4 -12 0 -2 E 0 0 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 6 0 B 14 0 -6 4 0 C 4 6 0 12 16 D -6 -4 -12 0 -2 E 0 0 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3258: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) D E B C A (8) B E C D A (8) E D B C A (6) A C B D E (6) C B E A D (5) C B A E D (5) E B D C A (4) E B C D A (4) D E C B A (4) B C E D A (4) A D E B C (4) A C D B E (4) D E A B C (3) D A E B C (3) C B E D A (3) D E B A C (2) D E A C B (2) D A E C B (2) B C E A D (2) A D C E B (2) A B C E D (2) E D C B A (1) C E D B A (1) B A C E D (1) A D E C B (1) A D C B E (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -10 -10 -14 B 16 0 8 10 6 C 10 -8 0 10 -10 D 10 -10 -10 0 -12 E 14 -6 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -10 -14 B 16 0 8 10 6 C 10 -8 0 10 -10 D 10 -10 -10 0 -12 E 14 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=24 E=15 B=15 C=14 so C is eliminated. Round 2 votes counts: A=32 B=28 D=24 E=16 so E is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:215 C:201 D:189 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -10 -14 B 16 0 8 10 6 C 10 -8 0 10 -10 D 10 -10 -10 0 -12 E 14 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -10 -14 B 16 0 8 10 6 C 10 -8 0 10 -10 D 10 -10 -10 0 -12 E 14 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -10 -14 B 16 0 8 10 6 C 10 -8 0 10 -10 D 10 -10 -10 0 -12 E 14 -6 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3259: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (8) D B A E C (6) E C B A D (5) D C A E B (5) C D E A B (5) A B D E C (5) D C E B A (4) D A C B E (4) C E D B A (4) B E A D C (4) A D B C E (4) E C B D A (3) E C A B D (3) C E A D B (3) B E A C D (3) D C A B E (2) C E D A B (2) C E A B D (2) C D A E B (2) B E D C A (2) B D E A C (2) B A E D C (2) B A D E C (2) A D C B E (2) A C D E B (2) A B E C D (2) E C D B A (1) E B C D A (1) E B C A D (1) D C E A B (1) D B A C E (1) C A E D B (1) B E D A C (1) B D A E C (1) B A E C D (1) A E C B D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 12 6 -10 6 B -12 0 -4 -16 8 C -6 4 0 -16 4 D 10 16 16 0 14 E -6 -8 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -10 6 B -12 0 -4 -16 8 C -6 4 0 -16 4 D 10 16 16 0 14 E -6 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=19 B=18 A=18 E=14 so E is eliminated. Round 2 votes counts: D=31 C=31 B=20 A=18 so A is eliminated. Round 3 votes counts: D=38 C=34 B=28 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:228 A:207 C:193 B:188 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -10 6 B -12 0 -4 -16 8 C -6 4 0 -16 4 D 10 16 16 0 14 E -6 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -10 6 B -12 0 -4 -16 8 C -6 4 0 -16 4 D 10 16 16 0 14 E -6 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -10 6 B -12 0 -4 -16 8 C -6 4 0 -16 4 D 10 16 16 0 14 E -6 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3260: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (14) E B C D A (10) D A C B E (7) E B C A D (5) E B D C A (4) E A C B D (4) B C D E A (4) B C D A E (4) A C D B E (4) A D E C B (3) E A D C B (2) E A D B C (2) E A B D C (2) D C B A E (2) D B C A E (2) C B D A E (2) A E D C B (2) A E C D B (2) A D C E B (2) A C D E B (2) A C B D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B A D (1) E B A D C (1) E A C D B (1) D C A B E (1) D B E C A (1) D A E B C (1) C D A B E (1) C B E A D (1) C B A E D (1) C B A D E (1) C A B E D (1) C A B D E (1) B D C E A (1) B D C A E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 10 12 16 B -14 0 -18 -6 2 C -10 18 0 -2 10 D -12 6 2 0 12 E -16 -2 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 12 16 B -14 0 -18 -6 2 C -10 18 0 -2 10 D -12 6 2 0 12 E -16 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=33 D=14 B=10 C=8 so C is eliminated. Round 2 votes counts: E=35 A=35 D=15 B=15 so D is eliminated. Round 3 votes counts: A=45 E=35 B=20 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 C:208 D:204 B:182 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 12 16 B -14 0 -18 -6 2 C -10 18 0 -2 10 D -12 6 2 0 12 E -16 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 12 16 B -14 0 -18 -6 2 C -10 18 0 -2 10 D -12 6 2 0 12 E -16 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 12 16 B -14 0 -18 -6 2 C -10 18 0 -2 10 D -12 6 2 0 12 E -16 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3261: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (6) B E C D A (6) B E A D C (6) D A B C E (5) C D A E B (5) E C A B D (4) D B C A E (4) E C B D A (3) E B C A D (3) E B A C D (3) E A B C D (3) C E A D B (3) C D E A B (3) B D A C E (3) B A E D C (3) A D C B E (3) E C B A D (2) E C A D B (2) E A C B D (2) D C A B E (2) D B A C E (2) C D A B E (2) B D E C A (2) B D E A C (2) B D A E C (2) E B C D A (1) D C B A E (1) C E D A B (1) C E B D A (1) C D B E A (1) B E D C A (1) B E D A C (1) B E C A D (1) B D C E A (1) B D C A E (1) B C D E A (1) B A D E C (1) A E D B C (1) A E B D C (1) A D E C B (1) A D B C E (1) A C E D B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -2 -14 -6 B 4 0 12 10 18 C 2 -12 0 -2 -4 D 14 -10 2 0 0 E 6 -18 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -14 -6 B 4 0 12 10 18 C 2 -12 0 -2 -4 D 14 -10 2 0 0 E 6 -18 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=23 D=20 C=16 A=10 so A is eliminated. Round 2 votes counts: B=32 E=25 D=25 C=18 so C is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:203 E:196 C:192 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -14 -6 B 4 0 12 10 18 C 2 -12 0 -2 -4 D 14 -10 2 0 0 E 6 -18 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -14 -6 B 4 0 12 10 18 C 2 -12 0 -2 -4 D 14 -10 2 0 0 E 6 -18 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -14 -6 B 4 0 12 10 18 C 2 -12 0 -2 -4 D 14 -10 2 0 0 E 6 -18 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3262: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) C D A B E (9) B A C D E (7) B E A D C (6) B A D C E (6) E B A D C (5) C D A E B (5) E D C A B (3) E D A C B (3) C D E A B (3) C D B A E (3) C B D A E (3) B C D A E (3) B A E D C (3) A B D C E (3) E A D C B (2) E A D B C (2) E A B D C (2) D A E C B (2) B E C A D (2) B E A C D (2) B C E D A (2) B A E C D (2) E C D B A (1) E C B D A (1) E B C D A (1) D C A E B (1) C E D A B (1) C D E B A (1) B C D E A (1) B C A E D (1) B C A D E (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -6 -8 2 B 4 0 0 4 12 C 6 0 0 20 2 D 8 -4 -20 0 0 E -2 -12 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.462305 C: 0.537695 D: 0.000000 E: 0.000000 Sum of squares = 0.502841777995 Cumulative probabilities = A: 0.000000 B: 0.462305 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -8 2 B 4 0 0 4 12 C 6 0 0 20 2 D 8 -4 -20 0 0 E -2 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999608 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=31 C=25 A=4 D=3 so D is eliminated. Round 2 votes counts: B=37 E=31 C=26 A=6 so A is eliminated. Round 3 votes counts: B=40 E=33 C=27 so C is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:214 B:210 A:192 D:192 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -8 2 B 4 0 0 4 12 C 6 0 0 20 2 D 8 -4 -20 0 0 E -2 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999608 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -8 2 B 4 0 0 4 12 C 6 0 0 20 2 D 8 -4 -20 0 0 E -2 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999608 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -8 2 B 4 0 0 4 12 C 6 0 0 20 2 D 8 -4 -20 0 0 E -2 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999608 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3263: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) E A D B C (7) A D E B C (7) E A B D C (6) B E C A D (6) D A C E B (5) C D A B E (5) C B E D A (5) B E C D A (5) B C E D A (5) E B A D C (4) C B D A E (4) E B C A D (3) B C E A D (3) A E D B C (3) A D E C B (3) D C A B E (2) C B D E A (2) A D C E B (2) A D C B E (2) E B C D A (1) E B A C D (1) D C A E B (1) D A E C B (1) C D B E A (1) C D B A E (1) B E A D C (1) B A D E C (1) B A D C E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 12 4 0 B -12 0 12 -4 12 C -12 -12 0 -16 0 D -4 4 16 0 0 E 0 -12 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.668400 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.331600 Sum of squares = 0.556716795209 Cumulative probabilities = A: 0.668400 B: 0.668400 C: 0.668400 D: 0.668400 E: 1.000000 A B C D E A 0 12 12 4 0 B -12 0 12 -4 12 C -12 -12 0 -16 0 D -4 4 16 0 0 E 0 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500311 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000193306 Cumulative probabilities = A: 0.500311 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 D=19 A=19 C=18 so C is eliminated. Round 2 votes counts: B=33 D=26 E=22 A=19 so A is eliminated. Round 3 votes counts: D=42 B=33 E=25 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:208 B:204 E:194 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 4 0 B -12 0 12 -4 12 C -12 -12 0 -16 0 D -4 4 16 0 0 E 0 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500311 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000193306 Cumulative probabilities = A: 0.500311 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 4 0 B -12 0 12 -4 12 C -12 -12 0 -16 0 D -4 4 16 0 0 E 0 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500311 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000193306 Cumulative probabilities = A: 0.500311 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 4 0 B -12 0 12 -4 12 C -12 -12 0 -16 0 D -4 4 16 0 0 E 0 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500311 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499689 Sum of squares = 0.500000193306 Cumulative probabilities = A: 0.500311 B: 0.500311 C: 0.500311 D: 0.500311 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3264: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) B E A C D (6) B A E D C (6) D C A B E (5) B E A D C (5) B A E C D (5) E B A C D (4) D C E A B (4) D C A E B (4) C D E A B (4) A C D B E (4) E D C B A (3) E B D C A (3) E B C D A (3) D C B A E (3) B A D C E (3) A D C B E (3) E A B C D (2) D B C A E (2) C D A B E (2) B D A C E (2) E C D A B (1) E C A D B (1) E B D A C (1) E B C A D (1) E A C D B (1) D E C B A (1) D E C A B (1) D C B E A (1) D B A C E (1) D A C B E (1) C E D A B (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) A E C B D (1) A E B C D (1) A D B C E (1) A C D E B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 2 -4 8 B 6 0 2 -4 10 C -2 -2 0 -6 2 D 4 4 6 0 4 E -8 -10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -4 8 B 6 0 2 -4 10 C -2 -2 0 -6 2 D 4 4 6 0 4 E -8 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 E=20 C=14 A=13 so A is eliminated. Round 2 votes counts: B=32 D=27 E=22 C=19 so C is eliminated. Round 3 votes counts: D=45 B=32 E=23 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:207 A:200 C:196 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -4 8 B 6 0 2 -4 10 C -2 -2 0 -6 2 D 4 4 6 0 4 E -8 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -4 8 B 6 0 2 -4 10 C -2 -2 0 -6 2 D 4 4 6 0 4 E -8 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -4 8 B 6 0 2 -4 10 C -2 -2 0 -6 2 D 4 4 6 0 4 E -8 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3265: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) E A B D C (6) C D B A E (6) A B E D C (6) D B A E C (5) E A B C D (4) D C B E A (4) D C B A E (4) D B C A E (4) B A D E C (4) D B A C E (3) C E D A B (3) C D B E A (3) C A B E D (3) A E B C D (3) E D C A B (2) E C A B D (2) D E B A C (2) D C E B A (2) C D E B A (2) C D E A B (2) B D A E C (2) B A E D C (2) A B E C D (2) E D B A C (1) E D A B C (1) E C D A B (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D E C A B (1) C E A D B (1) C D A B E (1) C B D A E (1) C B A D E (1) C A E B D (1) B D A C E (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 2 -6 -2 2 B -2 0 -2 -2 8 C 6 2 0 -6 4 D 2 2 6 0 0 E -2 -8 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.870299 E: 0.129701 Sum of squares = 0.774243347344 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.870299 E: 1.000000 A B C D E A 0 2 -6 -2 2 B -2 0 -2 -2 8 C 6 2 0 -6 4 D 2 2 6 0 0 E -2 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=25 E=21 B=11 A=11 so B is eliminated. Round 2 votes counts: C=33 D=28 E=21 A=18 so A is eliminated. Round 3 votes counts: E=34 D=33 C=33 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:205 C:203 B:201 A:198 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -2 2 B -2 0 -2 -2 8 C 6 2 0 -6 4 D 2 2 6 0 0 E -2 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -2 2 B -2 0 -2 -2 8 C 6 2 0 -6 4 D 2 2 6 0 0 E -2 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -2 2 B -2 0 -2 -2 8 C 6 2 0 -6 4 D 2 2 6 0 0 E -2 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000000859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3266: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (5) B C E A D (5) B C A E D (5) D E A C B (4) D A E B C (4) A D B C E (4) A B E D C (4) D E C A B (3) D C E A B (3) D C A B E (3) C E D B A (3) C D E B A (3) B A E C D (3) B A C E D (3) A D E B C (3) E D C B A (2) E D A B C (2) E C D B A (2) E B C A D (2) E A D B C (2) D C A E B (2) D A C B E (2) C D B A E (2) C B D E A (2) C B D A E (2) B E C A D (2) B A C D E (2) A B D E C (2) A B D C E (2) E D A C B (1) E C B D A (1) E B D C A (1) E B C D A (1) E B A C D (1) E A B D C (1) D E C B A (1) D C E B A (1) D A E C B (1) D A C E B (1) C E B D A (1) C B E A D (1) C B A D E (1) B E A C D (1) A E D B C (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 -6 -2 B 6 0 4 -2 6 C 8 -4 0 -2 10 D 6 2 2 0 0 E 2 -6 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.916349 E: 0.083651 Sum of squares = 0.846693804257 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.916349 E: 1.000000 A B C D E A 0 -6 -8 -6 -2 B 6 0 4 -2 6 C 8 -4 0 -2 10 D 6 2 2 0 0 E 2 -6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222236367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=21 C=20 A=18 E=16 so E is eliminated. Round 2 votes counts: D=30 B=26 C=23 A=21 so A is eliminated. Round 3 votes counts: D=41 B=36 C=23 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:207 C:206 D:205 E:193 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -8 -6 -2 B 6 0 4 -2 6 C 8 -4 0 -2 10 D 6 2 2 0 0 E 2 -6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222236367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -6 -2 B 6 0 4 -2 6 C 8 -4 0 -2 10 D 6 2 2 0 0 E 2 -6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222236367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -6 -2 B 6 0 4 -2 6 C 8 -4 0 -2 10 D 6 2 2 0 0 E 2 -6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222236367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3267: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (13) B D C A E (11) E A C D B (9) B A E C D (9) A E C B D (8) B A C E D (7) D B C E A (6) A E B C D (6) A B E C D (5) E C A D B (3) E D A C B (2) D C B E A (2) C E A D B (2) C D E A B (2) B A C D E (2) E A D C B (1) E A D B C (1) D E C A B (1) D E A B C (1) D B E C A (1) C D B E A (1) C D B A E (1) B D C E A (1) B D A E C (1) B D A C E (1) B C A E D (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 8 10 12 8 B -8 0 10 8 2 C -10 -10 0 12 2 D -12 -8 -12 0 -10 E -8 -2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 12 8 B -8 0 10 8 2 C -10 -10 0 12 2 D -12 -8 -12 0 -10 E -8 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=24 A=19 E=16 C=6 so C is eliminated. Round 2 votes counts: B=35 D=28 A=19 E=18 so E is eliminated. Round 3 votes counts: B=35 A=35 D=30 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:206 E:199 C:197 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 12 8 B -8 0 10 8 2 C -10 -10 0 12 2 D -12 -8 -12 0 -10 E -8 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 12 8 B -8 0 10 8 2 C -10 -10 0 12 2 D -12 -8 -12 0 -10 E -8 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 12 8 B -8 0 10 8 2 C -10 -10 0 12 2 D -12 -8 -12 0 -10 E -8 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3268: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (16) E C B D A (13) C B E D A (9) A D B C E (8) E D A C B (6) D A B C E (5) C B E A D (4) B C D E A (4) B C D A E (4) B C A D E (4) A D B E C (4) E A D C B (3) C E B D A (3) A E D C B (3) D E A B C (2) B D C A E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C B A D (1) E C A B D (1) E A C D B (1) D B C A E (1) D A E B C (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 -6 0 B -4 0 6 -6 -6 C -2 -6 0 -6 -8 D 6 6 6 0 6 E 0 6 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -6 0 B -4 0 6 -6 -6 C -2 -6 0 -6 -8 D 6 6 6 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=28 C=16 B=15 D=9 so D is eliminated. Round 2 votes counts: A=38 E=30 C=16 B=16 so C is eliminated. Round 3 votes counts: A=38 E=33 B=29 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:212 E:204 A:200 B:195 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -6 0 B -4 0 6 -6 -6 C -2 -6 0 -6 -8 D 6 6 6 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -6 0 B -4 0 6 -6 -6 C -2 -6 0 -6 -8 D 6 6 6 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -6 0 B -4 0 6 -6 -6 C -2 -6 0 -6 -8 D 6 6 6 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3269: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E A C B (8) C A D B E (6) C A B D E (6) D C A B E (5) D A C E B (5) E D A C B (4) E D A B C (4) E B D A C (4) B C E A D (4) D E B C A (3) C B A D E (3) A C D E B (3) E D B A C (2) D C A E B (2) D A E C B (2) C D A B E (2) C B A E D (2) C A B E D (2) B E D C A (2) B E C D A (2) B E C A D (2) A C D B E (2) A C B E D (2) E B A D C (1) E B A C D (1) E A D C B (1) E A B C D (1) D E B A C (1) D E A B C (1) D C E A B (1) D C B A E (1) C B D A E (1) B E A C D (1) B A C E D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 18 -8 0 14 B -18 0 -18 -8 10 C 8 18 0 6 20 D 0 8 -6 0 4 E -14 -10 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -8 0 14 B -18 0 -18 -8 10 C 8 18 0 6 20 D 0 8 -6 0 4 E -14 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=22 B=22 E=18 A=9 so A is eliminated. Round 2 votes counts: C=30 D=29 B=23 E=18 so E is eliminated. Round 3 votes counts: D=40 C=30 B=30 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:226 A:212 D:203 B:183 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -8 0 14 B -18 0 -18 -8 10 C 8 18 0 6 20 D 0 8 -6 0 4 E -14 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 0 14 B -18 0 -18 -8 10 C 8 18 0 6 20 D 0 8 -6 0 4 E -14 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 0 14 B -18 0 -18 -8 10 C 8 18 0 6 20 D 0 8 -6 0 4 E -14 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3270: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) B C E D A (7) D E A B C (5) C E B D A (5) B D E A C (5) A D E B C (5) C B A D E (4) A D B E C (4) E D A C B (3) E D A B C (3) C B A E D (3) B D E C A (3) B C D E A (3) A C E D B (3) E D B A C (2) E B C D A (2) D E B A C (2) D A E B C (2) C E A D B (2) C B E A D (2) B D C E A (2) B D A E C (2) A D C E B (2) A C D E B (2) A B D C E (2) E C D A B (1) E C B D A (1) E A D C B (1) E A C D B (1) D B E A C (1) C E D B A (1) C E B A D (1) C A E D B (1) C A E B D (1) B E D C A (1) B C D A E (1) B C A D E (1) B A D C E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 2 -18 -22 B 14 0 12 10 2 C -2 -12 0 0 4 D 18 -10 0 0 0 E 22 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -18 -22 B 14 0 12 10 2 C -2 -12 0 0 4 D 18 -10 0 0 0 E 22 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=26 A=23 E=14 D=10 so D is eliminated. Round 2 votes counts: C=27 B=27 A=25 E=21 so E is eliminated. Round 3 votes counts: A=38 B=33 C=29 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:208 D:204 C:195 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 -18 -22 B 14 0 12 10 2 C -2 -12 0 0 4 D 18 -10 0 0 0 E 22 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -18 -22 B 14 0 12 10 2 C -2 -12 0 0 4 D 18 -10 0 0 0 E 22 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -18 -22 B 14 0 12 10 2 C -2 -12 0 0 4 D 18 -10 0 0 0 E 22 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987588 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3271: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) C E A B D (8) B D A E C (8) E C D B A (7) E C B D A (7) A D B C E (7) D A B E C (6) D B A E C (5) C E A D B (5) C E D A B (4) D B E A C (3) C A E B D (3) A B C D E (3) E D C B A (2) E C B A D (2) C A E D B (2) B D E A C (2) A C D B E (2) A C B E D (2) E B C D A (1) D E B A C (1) D A B C E (1) C E B A D (1) B E C D A (1) B A D E C (1) B A C E D (1) A D B E C (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 16 14 4 12 B -16 0 6 6 12 C -14 -6 0 2 6 D -4 -6 -2 0 8 E -12 -12 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 4 12 B -16 0 6 6 12 C -14 -6 0 2 6 D -4 -6 -2 0 8 E -12 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 E=19 D=16 B=13 so B is eliminated. Round 2 votes counts: A=31 D=26 C=23 E=20 so E is eliminated. Round 3 votes counts: C=41 A=31 D=28 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 B:204 D:198 C:194 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 4 12 B -16 0 6 6 12 C -14 -6 0 2 6 D -4 -6 -2 0 8 E -12 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 4 12 B -16 0 6 6 12 C -14 -6 0 2 6 D -4 -6 -2 0 8 E -12 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 4 12 B -16 0 6 6 12 C -14 -6 0 2 6 D -4 -6 -2 0 8 E -12 -12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3272: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D C B A (6) E C D B A (6) C B A D E (6) E A D B C (5) E D A B C (4) E C A B D (4) A B D C E (4) E A C B D (3) D B A C E (3) C B E D A (3) C B D A E (3) B C D A E (3) B C A D E (3) A E D B C (3) A D E B C (3) E D C A B (2) E D A C B (2) E C B D A (2) D E B A C (2) D A B E C (2) C E B A D (2) C B D E A (2) A D B E C (2) A C B E D (2) E C B A D (1) E C A D B (1) E A C D B (1) D E A B C (1) D B C A E (1) D A E B C (1) C E B D A (1) C D B E A (1) C B A E D (1) C A B E D (1) B A C D E (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 2 10 2 B -6 0 0 6 0 C -2 0 0 16 -4 D -10 -6 -16 0 -2 E -2 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 10 2 B -6 0 0 6 0 C -2 0 0 16 -4 D -10 -6 -16 0 -2 E -2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998599 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=26 C=20 D=10 B=7 so B is eliminated. Round 2 votes counts: E=37 A=27 C=26 D=10 so D is eliminated. Round 3 votes counts: E=40 A=33 C=27 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:205 E:202 B:200 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 10 2 B -6 0 0 6 0 C -2 0 0 16 -4 D -10 -6 -16 0 -2 E -2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998599 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 10 2 B -6 0 0 6 0 C -2 0 0 16 -4 D -10 -6 -16 0 -2 E -2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998599 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 10 2 B -6 0 0 6 0 C -2 0 0 16 -4 D -10 -6 -16 0 -2 E -2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998599 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3273: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) D B E A C (8) E C D A B (7) A C B D E (6) D E B A C (5) B D E C A (5) E D B C A (4) C A E D B (4) B D A C E (4) C A E B D (3) C A B E D (3) B D E A C (3) B D A E C (3) A D E C B (3) E D A C B (2) D E B C A (2) D B E C A (2) D B A E C (2) C E A D B (2) B C E D A (2) E D C A B (1) E D B A C (1) E A D C B (1) E A C D B (1) D A B E C (1) D A B C E (1) C E A B D (1) C B A E D (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D C B (1) A E C D B (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 20 -10 4 B -8 0 0 -26 2 C -20 0 0 -6 -8 D 10 26 6 0 10 E -4 -2 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 20 -10 4 B -8 0 0 -26 2 C -20 0 0 -6 -8 D 10 26 6 0 10 E -4 -2 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=21 B=21 E=17 C=14 so C is eliminated. Round 2 votes counts: A=37 B=22 D=21 E=20 so E is eliminated. Round 3 votes counts: A=42 D=36 B=22 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 A:211 E:196 B:184 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 20 -10 4 B -8 0 0 -26 2 C -20 0 0 -6 -8 D 10 26 6 0 10 E -4 -2 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 20 -10 4 B -8 0 0 -26 2 C -20 0 0 -6 -8 D 10 26 6 0 10 E -4 -2 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 20 -10 4 B -8 0 0 -26 2 C -20 0 0 -6 -8 D 10 26 6 0 10 E -4 -2 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3274: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) C A E D B (7) B D E A C (5) E C B A D (4) D B E A C (4) B E D C A (4) A C E B D (4) E C B D A (3) E B A C D (3) E A C B D (3) D C A B E (3) D B C A E (3) D B A C E (3) C E D A B (3) C A D E B (3) B E D A C (3) B E A D C (3) A B D C E (3) D E B C A (2) D B E C A (2) D A C B E (2) C E A D B (2) B E A C D (2) B D A E C (2) B A D E C (2) A C E D B (2) A C D B E (2) E D C B A (1) E C D B A (1) E B C A D (1) D C E A B (1) C D E A B (1) C D A E B (1) C A E B D (1) B D E C A (1) B A E D C (1) B A E C D (1) A E C B D (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -4 10 -14 B 2 0 -6 12 -4 C 4 6 0 8 -16 D -10 -12 -8 0 -18 E 14 4 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 10 -14 B 2 0 -6 12 -4 C 4 6 0 8 -16 D -10 -12 -8 0 -18 E 14 4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 D=20 C=18 A=14 so A is eliminated. Round 2 votes counts: B=28 C=26 E=25 D=21 so D is eliminated. Round 3 votes counts: B=41 C=32 E=27 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:226 B:202 C:201 A:195 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 10 -14 B 2 0 -6 12 -4 C 4 6 0 8 -16 D -10 -12 -8 0 -18 E 14 4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 -14 B 2 0 -6 12 -4 C 4 6 0 8 -16 D -10 -12 -8 0 -18 E 14 4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 -14 B 2 0 -6 12 -4 C 4 6 0 8 -16 D -10 -12 -8 0 -18 E 14 4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3275: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) E A B C D (4) C E D A B (4) C D E A B (4) B D A E C (4) B A D E C (4) A D B E C (4) E A C B D (3) C E B D A (3) B E C D A (3) B A E D C (3) A B E D C (3) A B D E C (3) E C B A D (2) E B C A D (2) D C B A E (2) D B C E A (2) D B C A E (2) D A B C E (2) C E D B A (2) B E A D C (2) B E A C D (2) B D C A E (2) B C E D A (2) A E C D B (2) A E B D C (2) A D E C B (2) A C E D B (2) E C A D B (1) D C E A B (1) D C A E B (1) D C A B E (1) D B A C E (1) D A C B E (1) C E A D B (1) C D E B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C A E D B (1) B E D C A (1) B E C A D (1) B D E C A (1) B D E A C (1) B D C E A (1) A E D C B (1) A E D B C (1) A E B C D (1) A D C B E (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 12 10 -2 B 2 0 20 12 4 C -12 -20 0 0 -16 D -10 -12 0 0 -10 E 2 -4 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 10 -2 B 2 0 20 12 4 C -12 -20 0 0 -16 D -10 -12 0 0 -10 E 2 -4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 C=19 E=17 D=13 so D is eliminated. Round 2 votes counts: B=32 A=27 C=24 E=17 so E is eliminated. Round 3 votes counts: B=39 A=34 C=27 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:212 A:209 D:184 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 12 10 -2 B 2 0 20 12 4 C -12 -20 0 0 -16 D -10 -12 0 0 -10 E 2 -4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 10 -2 B 2 0 20 12 4 C -12 -20 0 0 -16 D -10 -12 0 0 -10 E 2 -4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 10 -2 B 2 0 20 12 4 C -12 -20 0 0 -16 D -10 -12 0 0 -10 E 2 -4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3276: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) C E A D B (7) C E B D A (6) C A E B D (6) E D B C A (5) C A E D B (5) D B E A C (4) A C B D E (4) E C D B A (3) D B A E C (3) B D E C A (3) D E B C A (2) D E B A C (2) C E D A B (2) C E A B D (2) C B A E D (2) B A C D E (2) A D B E C (2) A D B C E (2) A C E D B (2) A C D E B (2) A B D C E (2) A B C D E (2) E D C B A (1) E D B A C (1) E D A C B (1) E C D A B (1) D E A B C (1) C E D B A (1) C E B A D (1) C A B E D (1) B E D C A (1) B D E A C (1) B D A C E (1) B C E D A (1) B A D E C (1) B A D C E (1) A E D C B (1) A D E C B (1) A D E B C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 -2 8 B 6 0 0 0 -10 C 0 0 0 2 6 D 2 0 -2 0 -2 E -8 10 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.278912 C: 0.721088 D: 0.000000 E: 0.000000 Sum of squares = 0.597759840379 Cumulative probabilities = A: 0.000000 B: 0.278912 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 8 B 6 0 0 0 -10 C 0 0 0 2 6 D 2 0 -2 0 -2 E -8 10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.374999 C: 0.625001 D: 0.000000 E: 0.000000 Sum of squares = 0.531250376422 Cumulative probabilities = A: 0.000000 B: 0.374999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=22 A=21 E=12 D=12 so E is eliminated. Round 2 votes counts: C=37 B=22 A=21 D=20 so D is eliminated. Round 3 votes counts: B=39 C=38 A=23 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:204 A:200 D:199 E:199 B:198 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -2 8 B 6 0 0 0 -10 C 0 0 0 2 6 D 2 0 -2 0 -2 E -8 10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.374999 C: 0.625001 D: 0.000000 E: 0.000000 Sum of squares = 0.531250376422 Cumulative probabilities = A: 0.000000 B: 0.374999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 8 B 6 0 0 0 -10 C 0 0 0 2 6 D 2 0 -2 0 -2 E -8 10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.374999 C: 0.625001 D: 0.000000 E: 0.000000 Sum of squares = 0.531250376422 Cumulative probabilities = A: 0.000000 B: 0.374999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 8 B 6 0 0 0 -10 C 0 0 0 2 6 D 2 0 -2 0 -2 E -8 10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.374999 C: 0.625001 D: 0.000000 E: 0.000000 Sum of squares = 0.531250376422 Cumulative probabilities = A: 0.000000 B: 0.374999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3277: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (15) E D B A C (14) A C E D B (12) B D E C A (10) D E B C A (7) C B D E A (5) A C E B D (4) A C B E D (4) D B E C A (3) C A D E B (3) C A D B E (3) A E D B C (3) E D A B C (2) E A D B C (2) B E D A C (2) A E C D B (2) E B D A C (1) D E C B A (1) D C E A B (1) C D B E A (1) C B D A E (1) C A B E D (1) B E D C A (1) B D C E A (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 0 -2 B -6 0 -6 -8 -4 C 6 6 0 4 4 D 0 8 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 0 -2 B -6 0 -6 -8 -4 C 6 6 0 4 4 D 0 8 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 E=19 B=14 D=12 so D is eliminated. Round 2 votes counts: C=30 E=27 A=26 B=17 so B is eliminated. Round 3 votes counts: E=43 C=31 A=26 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:203 E:200 A:199 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 0 -2 B -6 0 -6 -8 -4 C 6 6 0 4 4 D 0 8 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 0 -2 B -6 0 -6 -8 -4 C 6 6 0 4 4 D 0 8 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 0 -2 B -6 0 -6 -8 -4 C 6 6 0 4 4 D 0 8 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3278: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) B E A C D (9) A E B C D (8) C D A E B (7) D C B E A (5) A E C B D (5) C A D E B (4) B E A D C (4) A E C D B (4) E A D B C (3) C D B A E (3) E B A C D (2) E A B D C (2) D C E B A (2) D C E A B (2) C A E D B (2) B D E C A (2) B D E A C (2) B D C E A (2) B C D E A (2) B A E C D (2) A E D C B (2) A C E D B (2) A B E C D (2) E D A B C (1) E B D A C (1) E B A D C (1) D E A B C (1) D C A B E (1) D B E A C (1) D B C E A (1) C A E B D (1) B E D C A (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 16 4 10 10 B -16 0 -2 -4 -22 C -4 2 0 10 -6 D -10 4 -10 0 -8 E -10 22 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 10 10 B -16 0 -2 -4 -22 C -4 2 0 10 -6 D -10 4 -10 0 -8 E -10 22 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998386 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=24 A=23 C=17 E=10 so E is eliminated. Round 2 votes counts: B=30 A=28 D=25 C=17 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:213 C:201 D:188 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 4 10 10 B -16 0 -2 -4 -22 C -4 2 0 10 -6 D -10 4 -10 0 -8 E -10 22 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998386 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 10 10 B -16 0 -2 -4 -22 C -4 2 0 10 -6 D -10 4 -10 0 -8 E -10 22 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998386 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 10 10 B -16 0 -2 -4 -22 C -4 2 0 10 -6 D -10 4 -10 0 -8 E -10 22 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998386 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3279: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) A D E B C (14) B C A D E (12) C E B D A (8) E D A C B (7) D E A B C (6) E D C A B (4) D A E B C (4) B A C D E (4) E D C B A (3) E D A B C (3) C B A E D (3) C B E A D (2) C B A D E (2) A C B D E (2) A B D E C (2) A B D C E (2) A B C D E (2) E C D B A (1) D E A C B (1) C A E D B (1) B C E D A (1) B C D E A (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -4 -6 -2 B 2 0 4 10 -4 C 4 -4 0 6 8 D 6 -10 -6 0 6 E 2 4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 -4 -6 -2 B 2 0 4 10 -4 C 4 -4 0 6 8 D 6 -10 -6 0 6 E 2 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=23 E=18 B=18 D=11 so D is eliminated. Round 2 votes counts: C=30 A=27 E=25 B=18 so B is eliminated. Round 3 votes counts: C=44 A=31 E=25 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:206 D:198 E:196 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 -2 B 2 0 4 10 -4 C 4 -4 0 6 8 D 6 -10 -6 0 6 E 2 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 -2 B 2 0 4 10 -4 C 4 -4 0 6 8 D 6 -10 -6 0 6 E 2 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 -2 B 2 0 4 10 -4 C 4 -4 0 6 8 D 6 -10 -6 0 6 E 2 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3280: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (6) A D E C B (5) D A B C E (4) C D B A E (4) B C E D A (4) B C D E A (4) E C A B D (3) E B A C D (3) E A C B D (3) D A C E B (3) C D A E B (3) C B E D A (3) C A E D B (3) B E C A D (3) A E D C B (3) A D E B C (3) A D C E B (3) E C B A D (2) E B C A D (2) E B A D C (2) D C A B E (2) D B A E C (2) D A C B E (2) D A B E C (2) B E C D A (2) B D C E A (2) A C D E B (2) E A C D B (1) E A B D C (1) E A B C D (1) D B C A E (1) D B A C E (1) D A E B C (1) C E B D A (1) C E B A D (1) C E A D B (1) C E A B D (1) C D A B E (1) C B D E A (1) B E D A C (1) B E A D C (1) B D E A C (1) B D C A E (1) B D A C E (1) B C D A E (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 12 8 4 12 B -12 0 -10 -10 -12 C -8 10 0 14 2 D -4 10 -14 0 0 E -12 12 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 4 12 B -12 0 -10 -10 -12 C -8 10 0 14 2 D -4 10 -14 0 0 E -12 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=21 C=19 E=18 D=18 so E is eliminated. Round 2 votes counts: A=30 B=28 C=24 D=18 so D is eliminated. Round 3 votes counts: A=42 B=32 C=26 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:209 E:199 D:196 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 4 12 B -12 0 -10 -10 -12 C -8 10 0 14 2 D -4 10 -14 0 0 E -12 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 4 12 B -12 0 -10 -10 -12 C -8 10 0 14 2 D -4 10 -14 0 0 E -12 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 4 12 B -12 0 -10 -10 -12 C -8 10 0 14 2 D -4 10 -14 0 0 E -12 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3281: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (11) C B D E A (9) A D E B C (9) D A E B C (7) A E D C B (6) B C D E A (5) A E D B C (4) E A D B C (3) D E A B C (3) C B E D A (3) C B A E D (3) E C D B A (2) E C D A B (2) C E B D A (2) C E B A D (2) C B E A D (2) C B D A E (2) C B A D E (2) B D C A E (2) E D C B A (1) E D A C B (1) E C B A D (1) E C A D B (1) E A C D B (1) D B A C E (1) D A B E C (1) A D B E C (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 0 -4 2 B -4 0 -4 -10 -12 C 0 4 0 -2 -8 D 4 10 2 0 8 E -2 12 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -4 2 B -4 0 -4 -10 -12 C 0 4 0 -2 -8 D 4 10 2 0 8 E -2 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=23 A=22 B=18 D=12 so D is eliminated. Round 2 votes counts: A=30 E=26 C=25 B=19 so B is eliminated. Round 3 votes counts: C=43 A=31 E=26 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:212 E:205 A:201 C:197 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -4 2 B -4 0 -4 -10 -12 C 0 4 0 -2 -8 D 4 10 2 0 8 E -2 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -4 2 B -4 0 -4 -10 -12 C 0 4 0 -2 -8 D 4 10 2 0 8 E -2 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -4 2 B -4 0 -4 -10 -12 C 0 4 0 -2 -8 D 4 10 2 0 8 E -2 12 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3282: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (12) B C D E A (11) D B C E A (9) E C A B D (5) E A C D B (5) E D C B A (3) D B A C E (3) C E D B A (3) C B D E A (3) B D C A E (3) B A D C E (3) A E D C B (3) A D E B C (3) E D C A B (2) E C D A B (2) E A C B D (2) A E D B C (2) A E C D B (2) A D B E C (2) A D B C E (2) A B D E C (2) E D A C B (1) E C D B A (1) E A D C B (1) D E C B A (1) D C E B A (1) D C B E A (1) D B E C A (1) D A B E C (1) C B E D A (1) B D A C E (1) B C E A D (1) B C D A E (1) B C A D E (1) B A C E D (1) B A C D E (1) A E B C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 2 -8 B 0 0 2 2 0 C 0 -2 0 8 -6 D -2 -2 -8 0 2 E 8 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.725874 C: 0.000000 D: 0.000000 E: 0.274126 Sum of squares = 0.602037724134 Cumulative probabilities = A: 0.000000 B: 0.725874 C: 0.725874 D: 0.725874 E: 1.000000 A B C D E A 0 0 0 2 -8 B 0 0 2 2 0 C 0 -2 0 8 -6 D -2 -2 -8 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=23 E=22 D=17 C=7 so C is eliminated. Round 2 votes counts: A=31 B=27 E=25 D=17 so D is eliminated. Round 3 votes counts: B=41 A=32 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 B:202 C:200 A:197 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 2 -8 B 0 0 2 2 0 C 0 -2 0 8 -6 D -2 -2 -8 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -8 B 0 0 2 2 0 C 0 -2 0 8 -6 D -2 -2 -8 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -8 B 0 0 2 2 0 C 0 -2 0 8 -6 D -2 -2 -8 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.499998 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 0.500002 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3283: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) C B E D A (7) A D E B C (7) E B C A D (6) A B C E D (6) D E B C A (5) D A C B E (5) C B E A D (5) A D C B E (5) C E B D A (4) A E D B C (4) E B C D A (3) A D B C E (3) A C B E D (3) E B A C D (2) D E C B A (2) D C E B A (2) D C B E A (2) D A E C B (2) D A C E B (2) C B A E D (2) E D B C A (1) E D B A C (1) E C B D A (1) E B D C A (1) E B D A C (1) E A B C D (1) D E C A B (1) D E A C B (1) D E A B C (1) C D E B A (1) C B D E A (1) C B D A E (1) C A B E D (1) A E B C D (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 4 8 -4 2 B -4 0 2 -6 -14 C -8 -2 0 -8 2 D 4 6 8 0 0 E -2 14 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.767827 E: 0.232173 Sum of squares = 0.643462238074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.767827 E: 1.000000 A B C D E A 0 4 8 -4 2 B -4 0 2 -6 -14 C -8 -2 0 -8 2 D 4 6 8 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=30 C=22 E=17 so B is eliminated. Round 2 votes counts: A=31 D=30 C=22 E=17 so E is eliminated. Round 3 votes counts: D=34 A=34 C=32 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:209 A:205 E:205 C:192 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -4 2 B -4 0 2 -6 -14 C -8 -2 0 -8 2 D 4 6 8 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -4 2 B -4 0 2 -6 -14 C -8 -2 0 -8 2 D 4 6 8 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -4 2 B -4 0 2 -6 -14 C -8 -2 0 -8 2 D 4 6 8 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3284: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (5) B C E D A (5) B C A D E (5) A D E C B (5) E D A C B (4) E A D B C (4) D C E A B (4) B E A D C (4) A D E B C (4) E D C A B (3) E D A B C (3) E B A D C (3) D A E C B (3) D A C E B (3) C D B A E (3) C D A B E (3) B E A C D (3) A D C B E (3) E C D B A (2) D E A C B (2) C E B D A (2) C D E B A (2) C D E A B (2) C D B E A (2) C B D A E (2) B E C D A (2) B C E A D (2) A B D E C (2) E B D C A (1) E A B D C (1) D C A E B (1) C E D B A (1) C D A E B (1) C B E D A (1) C B A D E (1) B C D A E (1) B A E D C (1) B A C E D (1) A D B E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -2 -6 -16 B -2 0 -4 -14 -2 C 2 4 0 -4 -6 D 6 14 4 0 4 E 16 2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -6 -16 B -2 0 -4 -14 -2 C 2 4 0 -4 -6 D 6 14 4 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=21 C=20 A=17 D=13 so D is eliminated. Round 2 votes counts: B=29 C=25 E=23 A=23 so E is eliminated. Round 3 votes counts: A=37 B=33 C=30 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:214 E:210 C:198 A:189 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -6 -16 B -2 0 -4 -14 -2 C 2 4 0 -4 -6 D 6 14 4 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -6 -16 B -2 0 -4 -14 -2 C 2 4 0 -4 -6 D 6 14 4 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -6 -16 B -2 0 -4 -14 -2 C 2 4 0 -4 -6 D 6 14 4 0 4 E 16 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3285: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (12) B D C E A (8) A E C B D (8) D B C E A (7) E C A D B (6) D B A E C (6) D B E C A (4) B D A C E (4) C E D B A (3) C E A B D (3) A E D C B (3) E C D A B (2) E C A B D (2) E A C D B (2) D A E B C (2) D A B E C (2) C E B D A (2) C E B A D (2) A E D B C (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D B A (1) E A C B D (1) D E A B C (1) D C E B A (1) D B C A E (1) D B A C E (1) C E A D B (1) C B E D A (1) C B D E A (1) B D A E C (1) B C D E A (1) B C D A E (1) A D E B C (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 10 6 -2 0 B -10 0 -6 -20 -14 C -6 6 0 0 -20 D 2 20 0 0 -6 E 0 14 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.529431 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.470569 Sum of squares = 0.501732325886 Cumulative probabilities = A: 0.529431 B: 0.529431 C: 0.529431 D: 0.529431 E: 1.000000 A B C D E A 0 10 6 -2 0 B -10 0 -6 -20 -14 C -6 6 0 0 -20 D 2 20 0 0 -6 E 0 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999907 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=25 E=15 B=15 C=13 so C is eliminated. Round 2 votes counts: A=32 E=26 D=25 B=17 so B is eliminated. Round 3 votes counts: D=41 A=32 E=27 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:208 A:207 C:190 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 -2 0 B -10 0 -6 -20 -14 C -6 6 0 0 -20 D 2 20 0 0 -6 E 0 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999907 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -2 0 B -10 0 -6 -20 -14 C -6 6 0 0 -20 D 2 20 0 0 -6 E 0 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999907 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -2 0 B -10 0 -6 -20 -14 C -6 6 0 0 -20 D 2 20 0 0 -6 E 0 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999907 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3286: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) E C A D B (6) B E D A C (6) B D A E C (6) C E A D B (5) E C D A B (4) B D E A C (4) B D A C E (4) B A D C E (4) A C D B E (4) E B D C A (3) E B C A D (3) D A C E B (3) C A D B E (3) B E C A D (3) A D C B E (3) E B D A C (2) D A C B E (2) D A B C E (2) C A E D B (2) B C A D E (2) B A C D E (2) A D B C E (2) E D C A B (1) E C B D A (1) E B C D A (1) D E C A B (1) D E B A C (1) D E A C B (1) D B E A C (1) D B A C E (1) D A E C B (1) D A B E C (1) C E B A D (1) C A E B D (1) B C E A D (1) B A C E D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 8 10 B -6 0 2 -8 6 C -6 -2 0 2 8 D -8 8 -2 0 18 E -10 -6 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 8 10 B -6 0 2 -8 6 C -6 -2 0 2 8 D -8 8 -2 0 18 E -10 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=21 C=21 D=14 A=11 so A is eliminated. Round 2 votes counts: B=34 C=26 E=21 D=19 so D is eliminated. Round 3 votes counts: B=41 C=34 E=25 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:215 D:208 C:201 B:197 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 8 10 B -6 0 2 -8 6 C -6 -2 0 2 8 D -8 8 -2 0 18 E -10 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 10 B -6 0 2 -8 6 C -6 -2 0 2 8 D -8 8 -2 0 18 E -10 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 10 B -6 0 2 -8 6 C -6 -2 0 2 8 D -8 8 -2 0 18 E -10 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) D C E A B (8) D E A C B (7) C B A E D (7) B A C E D (7) C B A D E (6) B C A E D (6) C D B E A (5) C B D A E (5) B A E C D (5) A E B D C (5) E D A B C (4) E A B D C (3) D E A B C (3) C D E B A (3) C D E A B (3) B A E D C (3) A E D B C (3) C B E D A (2) A B E D C (2) D E C A B (1) C E D B A (1) C D B A E (1) C B D E A (1) B C A D E (1) Total count = 100 A B C D E A 0 -6 0 12 2 B 6 0 0 6 2 C 0 0 0 6 12 D -12 -6 -6 0 -12 E -2 -2 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.596886 C: 0.403114 D: 0.000000 E: 0.000000 Sum of squares = 0.518773781321 Cumulative probabilities = A: 0.000000 B: 0.596886 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 12 2 B 6 0 0 6 2 C 0 0 0 6 12 D -12 -6 -6 0 -12 E -2 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=22 D=19 E=15 A=10 so A is eliminated. Round 2 votes counts: C=34 B=24 E=23 D=19 so D is eliminated. Round 3 votes counts: C=42 E=34 B=24 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:207 A:204 E:198 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 12 2 B 6 0 0 6 2 C 0 0 0 6 12 D -12 -6 -6 0 -12 E -2 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 12 2 B 6 0 0 6 2 C 0 0 0 6 12 D -12 -6 -6 0 -12 E -2 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 12 2 B 6 0 0 6 2 C 0 0 0 6 12 D -12 -6 -6 0 -12 E -2 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3288: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) A B D C E (6) D C E A B (5) B E C A D (5) E D C A B (4) E C D B A (4) E C D A B (4) D A C B E (4) C D E B A (4) B C E A D (4) B C A D E (4) B A C D E (4) E B C A D (3) D A C E B (3) B A E C D (3) B A D C E (3) A D E B C (3) E B A C D (2) E A B D C (2) D E C A B (2) D C A B E (2) D A E C B (2) C E B D A (2) C B E D A (2) B A C E D (2) A D B E C (2) E C B A D (1) E A D B C (1) D E A C B (1) D C B A E (1) C E D B A (1) B C A E D (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -20 -4 -16 B 14 0 -6 12 -12 C 20 6 0 14 -4 D 4 -12 -14 0 -6 E 16 12 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -20 -4 -16 B 14 0 -6 12 -12 C 20 6 0 14 -4 D 4 -12 -14 0 -6 E 16 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=26 D=20 A=13 C=9 so C is eliminated. Round 2 votes counts: E=35 B=28 D=24 A=13 so A is eliminated. Round 3 votes counts: E=36 B=35 D=29 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:218 B:204 D:186 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -20 -4 -16 B 14 0 -6 12 -12 C 20 6 0 14 -4 D 4 -12 -14 0 -6 E 16 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -4 -16 B 14 0 -6 12 -12 C 20 6 0 14 -4 D 4 -12 -14 0 -6 E 16 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -4 -16 B 14 0 -6 12 -12 C 20 6 0 14 -4 D 4 -12 -14 0 -6 E 16 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3289: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) C A E D B (8) E B D A C (7) C A E B D (7) E B D C A (6) B D E C A (6) B D E A C (5) D B A E C (4) C E A B D (4) C B D E A (4) A C E D B (4) E C A B D (3) D B E A C (3) D B A C E (3) A C D B E (3) E C B D A (2) E A D B C (2) E A C B D (2) C A D E B (2) B E D C A (2) A D C B E (2) E C B A D (1) E B A D C (1) D B C A E (1) D A B C E (1) C E B A D (1) C B D A E (1) B D C E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 -16 8 0 B -4 0 -12 8 -6 C 16 12 0 6 4 D -8 -8 -6 0 -6 E 0 6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 8 0 B -4 0 -12 8 -6 C 16 12 0 6 4 D -8 -8 -6 0 -6 E 0 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=24 B=14 A=14 D=12 so D is eliminated. Round 2 votes counts: C=36 B=25 E=24 A=15 so A is eliminated. Round 3 votes counts: C=45 B=28 E=27 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:204 A:198 B:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 8 0 B -4 0 -12 8 -6 C 16 12 0 6 4 D -8 -8 -6 0 -6 E 0 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 8 0 B -4 0 -12 8 -6 C 16 12 0 6 4 D -8 -8 -6 0 -6 E 0 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 8 0 B -4 0 -12 8 -6 C 16 12 0 6 4 D -8 -8 -6 0 -6 E 0 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3290: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) A C B E D (8) D E B C A (6) A B C D E (6) D B E A C (5) B D A E C (4) B A D E C (4) D B E C A (3) C A B D E (3) A C E B D (3) A B E C D (3) A B C E D (3) E D C B A (2) E D B C A (2) E C D A B (2) E C A D B (2) E C A B D (2) E B D A C (2) E B A D C (2) D E B A C (2) D C E B A (2) D B C E A (2) C E A D B (2) C D E A B (2) C A D B E (2) B A E D C (2) A E B C D (2) A B D C E (2) E D B A C (1) E A C B D (1) D C E A B (1) D B C A E (1) D B A C E (1) C E D A B (1) C D A E B (1) C A B E D (1) B E A D C (1) B D E A C (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 14 10 18 12 B -14 0 10 20 6 C -10 -10 0 8 0 D -18 -20 -8 0 -2 E -12 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 18 12 B -14 0 10 20 6 C -10 -10 0 8 0 D -18 -20 -8 0 -2 E -12 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=23 C=20 E=16 B=12 so B is eliminated. Round 2 votes counts: A=35 D=28 C=20 E=17 so E is eliminated. Round 3 votes counts: A=39 D=35 C=26 so C is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:227 B:211 C:194 E:192 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 18 12 B -14 0 10 20 6 C -10 -10 0 8 0 D -18 -20 -8 0 -2 E -12 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 18 12 B -14 0 10 20 6 C -10 -10 0 8 0 D -18 -20 -8 0 -2 E -12 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 18 12 B -14 0 10 20 6 C -10 -10 0 8 0 D -18 -20 -8 0 -2 E -12 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3291: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) C E D B A (8) E C A B D (7) A B E C D (7) D C B E A (6) B A D E C (5) D B C E A (4) A E B C D (4) A B E D C (4) E A C B D (3) D C E B A (3) C D E B A (3) E C B D A (2) E C A D B (2) E B C A D (2) D B C A E (2) D B A C E (2) C E B D A (2) B D C E A (2) B D A C E (2) A E C B D (2) E C B A D (1) E A B C D (1) D C E A B (1) D C A E B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C B D E A (1) B A E C D (1) B A D C E (1) A E D B C (1) A E C D B (1) A D E C B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 6 4 14 2 B -6 0 6 18 6 C -4 -6 0 -4 -14 D -14 -18 4 0 2 E -2 -6 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 14 2 B -6 0 6 18 6 C -4 -6 0 -4 -14 D -14 -18 4 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=22 E=18 C=15 B=11 so B is eliminated. Round 2 votes counts: A=41 D=26 E=18 C=15 so C is eliminated. Round 3 votes counts: A=41 D=30 E=29 so E is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:212 E:202 D:187 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 14 2 B -6 0 6 18 6 C -4 -6 0 -4 -14 D -14 -18 4 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 14 2 B -6 0 6 18 6 C -4 -6 0 -4 -14 D -14 -18 4 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 14 2 B -6 0 6 18 6 C -4 -6 0 -4 -14 D -14 -18 4 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3292: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (16) E A C B D (12) B D E A C (9) E A B C D (7) C D A E B (6) D C B A E (5) D B C E A (5) C A E D B (5) E A C D B (4) B D C A E (4) A C E B D (4) B E A D C (3) C E A D B (2) B A E D C (2) A E C B D (2) E D C A B (1) E C A D B (1) E B D A C (1) E B A C D (1) E A D B C (1) D C E B A (1) D C E A B (1) D C A B E (1) C A D E B (1) B E A C D (1) B D A E C (1) B D A C E (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 2 -4 0 B 2 0 6 0 0 C -2 -6 0 -4 6 D 4 0 4 0 2 E 0 0 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.532634 C: 0.000000 D: 0.467366 E: 0.000000 Sum of squares = 0.50212995814 Cumulative probabilities = A: 0.000000 B: 0.532634 C: 0.532634 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 0 B 2 0 6 0 0 C -2 -6 0 -4 6 D 4 0 4 0 2 E 0 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 B=22 C=14 A=7 so A is eliminated. Round 2 votes counts: E=30 D=29 B=22 C=19 so C is eliminated. Round 3 votes counts: E=41 D=36 B=23 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:205 B:204 A:198 C:197 E:196 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -4 0 B 2 0 6 0 0 C -2 -6 0 -4 6 D 4 0 4 0 2 E 0 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 0 B 2 0 6 0 0 C -2 -6 0 -4 6 D 4 0 4 0 2 E 0 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 0 B 2 0 6 0 0 C -2 -6 0 -4 6 D 4 0 4 0 2 E 0 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3293: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (18) B D C A E (16) E A B D C (7) A E C D B (6) A E C B D (5) C D B A E (4) E B A D C (3) C D E B A (3) B D C E A (3) A E B C D (3) E A C B D (2) D B C E A (2) C D A B E (2) B E D A C (2) A E B D C (2) A B D C E (2) E D B C A (1) E C D B A (1) E C D A B (1) E C A D B (1) E A B C D (1) D C B E A (1) C E D B A (1) C D B E A (1) C D A E B (1) C A D B E (1) B E D C A (1) B E A D C (1) B D A C E (1) B A D E C (1) B A D C E (1) A C E D B (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 14 20 18 0 B -14 0 -2 8 -14 C -20 2 0 10 -14 D -18 -8 -10 0 -16 E 0 14 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.338861 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.661139 Sum of squares = 0.551931873821 Cumulative probabilities = A: 0.338861 B: 0.338861 C: 0.338861 D: 0.338861 E: 1.000000 A B C D E A 0 14 20 18 0 B -14 0 -2 8 -14 C -20 2 0 10 -14 D -18 -8 -10 0 -16 E 0 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=26 A=23 C=13 D=3 so D is eliminated. Round 2 votes counts: E=35 B=28 A=23 C=14 so C is eliminated. Round 3 votes counts: E=39 B=34 A=27 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:226 E:222 B:189 C:189 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 18 0 B -14 0 -2 8 -14 C -20 2 0 10 -14 D -18 -8 -10 0 -16 E 0 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 18 0 B -14 0 -2 8 -14 C -20 2 0 10 -14 D -18 -8 -10 0 -16 E 0 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 18 0 B -14 0 -2 8 -14 C -20 2 0 10 -14 D -18 -8 -10 0 -16 E 0 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3294: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) E A B D C (7) B C E D A (7) E A D B C (6) C B D E A (6) D A C E B (5) A E D B C (5) E B A C D (4) D A E C B (4) B C E A D (4) D A C B E (3) C D A B E (3) B E C D A (3) B E C A D (3) E B D C A (2) E B C A D (2) D C B E A (2) D C B A E (2) A E D C B (2) A E B C D (2) A D E C B (2) A D C E B (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D C A B E (1) D B C E A (1) C D B A E (1) C A B D E (1) B C A E D (1) A D E B C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -2 -8 -2 B 4 0 4 14 4 C 2 -4 0 2 6 D 8 -14 -2 0 -4 E 2 -4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 -2 B 4 0 4 14 4 C 2 -4 0 2 6 D 8 -14 -2 0 -4 E 2 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 D=18 B=18 A=16 so A is eliminated. Round 2 votes counts: E=34 D=24 C=24 B=18 so B is eliminated. Round 3 votes counts: E=40 C=36 D=24 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:213 C:203 E:198 D:194 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 -2 B 4 0 4 14 4 C 2 -4 0 2 6 D 8 -14 -2 0 -4 E 2 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 -2 B 4 0 4 14 4 C 2 -4 0 2 6 D 8 -14 -2 0 -4 E 2 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 -2 B 4 0 4 14 4 C 2 -4 0 2 6 D 8 -14 -2 0 -4 E 2 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3295: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (16) A E B D C (14) C D B E A (9) D B C E A (7) B D E C A (7) E B D A C (6) C D B A E (5) B D C E A (5) A E C B D (4) A C E D B (4) D C B E A (3) C A E D B (3) C A D B E (3) B D E A C (3) C D A B E (2) A E C D B (2) E B C D A (1) E B A D C (1) D B E A C (1) D B C A E (1) B D A E C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 8 -2 -18 B 0 0 28 20 -4 C -8 -28 0 -32 -14 D 2 -20 32 0 -4 E 18 4 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 -2 -18 B 0 0 28 20 -4 C -8 -28 0 -32 -14 D 2 -20 32 0 -4 E 18 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 C=22 B=16 D=12 so D is eliminated. Round 2 votes counts: A=26 C=25 B=25 E=24 so E is eliminated. Round 3 votes counts: A=42 B=33 C=25 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:222 E:220 D:205 A:194 C:159 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 -2 -18 B 0 0 28 20 -4 C -8 -28 0 -32 -14 D 2 -20 32 0 -4 E 18 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -2 -18 B 0 0 28 20 -4 C -8 -28 0 -32 -14 D 2 -20 32 0 -4 E 18 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -2 -18 B 0 0 28 20 -4 C -8 -28 0 -32 -14 D 2 -20 32 0 -4 E 18 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3296: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) D C B A E (9) B D C E A (8) D B C A E (6) C D B A E (6) E A B D C (5) E A B C D (4) C D B E A (4) E B A D C (3) E A C B D (3) C D A B E (3) C A E D B (3) B E A D C (3) B D E C A (3) E A C D B (2) C D A E B (2) C A D E B (2) B E D A C (2) A B E D C (2) E B A C D (1) D C B E A (1) D C A B E (1) D B A C E (1) C E D B A (1) C E B D A (1) C B D E A (1) B E D C A (1) B E C D A (1) B D A E C (1) B C E D A (1) A E D C B (1) A E D B C (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -8 -6 10 B 8 0 -8 -14 12 C 8 8 0 -4 2 D 6 14 4 0 2 E -10 -12 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -6 10 B 8 0 -8 -14 12 C 8 8 0 -4 2 D 6 14 4 0 2 E -10 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 A=21 B=20 E=18 D=18 so E is eliminated. Round 2 votes counts: A=35 B=24 C=23 D=18 so D is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:207 B:199 A:194 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -6 10 B 8 0 -8 -14 12 C 8 8 0 -4 2 D 6 14 4 0 2 E -10 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -6 10 B 8 0 -8 -14 12 C 8 8 0 -4 2 D 6 14 4 0 2 E -10 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -6 10 B 8 0 -8 -14 12 C 8 8 0 -4 2 D 6 14 4 0 2 E -10 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3297: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (7) A E D B C (7) C D E A B (6) D E C A B (5) C B A D E (5) B C D E A (5) C D B E A (4) C B D E A (4) C B A E D (4) B A E D C (4) E A D B C (3) D E B A C (3) D E A C B (3) D E A B C (3) D C E A B (3) C D E B A (3) C B D A E (3) B A E C D (3) B E D A C (2) B C D A E (2) B A C E D (2) A E B D C (2) A B E D C (2) A B C E D (2) E D A C B (1) E D A B C (1) D E C B A (1) C A E B D (1) C A D E B (1) C A B E D (1) B E A D C (1) B D E C A (1) B C E D A (1) B C A D E (1) A E D C B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -16 -2 0 B 12 0 4 8 10 C 16 -4 0 14 14 D 2 -8 -14 0 6 E 0 -10 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -2 0 B 12 0 4 8 10 C 16 -4 0 14 14 D 2 -8 -14 0 6 E 0 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=29 D=18 A=16 E=5 so E is eliminated. Round 2 votes counts: C=32 B=29 D=20 A=19 so A is eliminated. Round 3 votes counts: B=35 C=34 D=31 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:217 D:193 A:185 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -16 -2 0 B 12 0 4 8 10 C 16 -4 0 14 14 D 2 -8 -14 0 6 E 0 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -2 0 B 12 0 4 8 10 C 16 -4 0 14 14 D 2 -8 -14 0 6 E 0 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -2 0 B 12 0 4 8 10 C 16 -4 0 14 14 D 2 -8 -14 0 6 E 0 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3298: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) D A E B C (7) E B D A C (6) B E C A D (6) A D E B C (6) C B E D A (5) C B A E D (5) C A D B E (5) B C E A D (5) E B A D C (4) D E A B C (4) E D A B C (3) D A C E B (3) C B A D E (3) C A B D E (3) A D C E B (3) E D B A C (2) E B D C A (2) D E A C B (2) B E A C D (2) B C E D A (2) A B E D C (2) E B C D A (1) C E B D A (1) C D E B A (1) C D B E A (1) C D A B E (1) C B E A D (1) C B D E A (1) B E C D A (1) A E D B C (1) A D E C B (1) A D C B E (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 2 12 0 0 B -2 0 10 2 -8 C -12 -10 0 -10 -16 D 0 -2 10 0 2 E 0 8 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.808564 B: 0.000000 C: 0.000000 D: 0.191436 E: 0.000000 Sum of squares = 0.690423737234 Cumulative probabilities = A: 0.808564 B: 0.808564 C: 0.808564 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 0 0 B -2 0 10 2 -8 C -12 -10 0 -10 -16 D 0 -2 10 0 2 E 0 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500637 B: 0.000000 C: 0.000000 D: 0.499363 E: 0.000000 Sum of squares = 0.500000811439 Cumulative probabilities = A: 0.500637 B: 0.500637 C: 0.500637 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=23 E=18 B=16 A=16 so B is eliminated. Round 2 votes counts: C=34 E=27 D=23 A=16 so A is eliminated. Round 3 votes counts: D=35 C=35 E=30 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:211 A:207 D:205 B:201 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 0 0 B -2 0 10 2 -8 C -12 -10 0 -10 -16 D 0 -2 10 0 2 E 0 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500637 B: 0.000000 C: 0.000000 D: 0.499363 E: 0.000000 Sum of squares = 0.500000811439 Cumulative probabilities = A: 0.500637 B: 0.500637 C: 0.500637 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 0 0 B -2 0 10 2 -8 C -12 -10 0 -10 -16 D 0 -2 10 0 2 E 0 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500637 B: 0.000000 C: 0.000000 D: 0.499363 E: 0.000000 Sum of squares = 0.500000811439 Cumulative probabilities = A: 0.500637 B: 0.500637 C: 0.500637 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 0 0 B -2 0 10 2 -8 C -12 -10 0 -10 -16 D 0 -2 10 0 2 E 0 8 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500637 B: 0.000000 C: 0.000000 D: 0.499363 E: 0.000000 Sum of squares = 0.500000811439 Cumulative probabilities = A: 0.500637 B: 0.500637 C: 0.500637 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3299: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (15) B D C A E (12) E A D C B (8) D B E A C (8) E A C D B (6) B C D A E (6) C B D A E (5) B D C E A (5) A E C D B (5) E D A B C (4) A C E D B (4) C E A B D (3) C B A E D (3) C A B E D (3) E A D B C (2) D E B A C (2) C B A D E (2) E D B A C (1) E C A D B (1) D B A E C (1) C E A D B (1) B D E C A (1) B D E A C (1) B C D E A (1) Total count = 100 A B C D E A 0 4 -16 6 12 B -4 0 -12 14 -4 C 16 12 0 10 20 D -6 -14 -10 0 -12 E -12 4 -20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 6 12 B -4 0 -12 14 -4 C 16 12 0 10 20 D -6 -14 -10 0 -12 E -12 4 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=26 E=22 D=11 A=9 so A is eliminated. Round 2 votes counts: C=36 E=27 B=26 D=11 so D is eliminated. Round 3 votes counts: C=36 B=35 E=29 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 A:203 B:197 E:192 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 6 12 B -4 0 -12 14 -4 C 16 12 0 10 20 D -6 -14 -10 0 -12 E -12 4 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 6 12 B -4 0 -12 14 -4 C 16 12 0 10 20 D -6 -14 -10 0 -12 E -12 4 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 6 12 B -4 0 -12 14 -4 C 16 12 0 10 20 D -6 -14 -10 0 -12 E -12 4 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3300: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) A D C E B (8) E B C D A (7) E B A C D (7) B E C D A (7) E A D C B (6) D A C B E (6) C D B A E (6) B C E D A (6) A E D C B (6) E B C A D (5) D C A B E (5) C B D A E (5) A D E C B (5) E A B D C (4) A D C B E (4) E A D B C (2) B C D A E (2) E A B C D (1) Total count = 100 A B C D E A 0 -6 -2 -4 -6 B 6 0 -2 4 -2 C 2 2 0 8 0 D 4 -4 -8 0 -2 E 6 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.460716 D: 0.000000 E: 0.539284 Sum of squares = 0.503086531252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.460716 D: 0.460716 E: 1.000000 A B C D E A 0 -6 -2 -4 -6 B 6 0 -2 4 -2 C 2 2 0 8 0 D 4 -4 -8 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=23 A=23 D=11 C=11 so D is eliminated. Round 2 votes counts: E=32 A=29 B=23 C=16 so C is eliminated. Round 3 votes counts: B=34 A=34 E=32 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:206 E:205 B:203 D:195 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 -6 B 6 0 -2 4 -2 C 2 2 0 8 0 D 4 -4 -8 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 -6 B 6 0 -2 4 -2 C 2 2 0 8 0 D 4 -4 -8 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 -6 B 6 0 -2 4 -2 C 2 2 0 8 0 D 4 -4 -8 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3301: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) B D A E C (9) E A B C D (7) B D E A C (7) D B A E C (6) C E A D B (6) C A E D B (6) B D C E A (6) A E C D B (6) E A C B D (5) D C B A E (4) C E A B D (3) A E C B D (3) C D E A B (2) B A E D C (2) A E D B C (2) A E B D C (2) E C A B D (1) E A C D B (1) E A B D C (1) D C B E A (1) D C A E B (1) D C A B E (1) D B A C E (1) C D B E A (1) C D B A E (1) C D A E B (1) C D A B E (1) B E A D C (1) B D E C A (1) B D C A E (1) B C E A D (1) Total count = 100 A B C D E A 0 -2 6 -6 12 B 2 0 12 0 6 C -6 -12 0 -10 -8 D 6 0 10 0 6 E -12 -6 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.468910 C: 0.000000 D: 0.531090 E: 0.000000 Sum of squares = 0.501933175774 Cumulative probabilities = A: 0.000000 B: 0.468910 C: 0.468910 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -6 12 B 2 0 12 0 6 C -6 -12 0 -10 -8 D 6 0 10 0 6 E -12 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 C=21 E=15 A=13 so A is eliminated. Round 2 votes counts: E=28 B=28 D=23 C=21 so C is eliminated. Round 3 votes counts: E=43 D=29 B=28 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:210 A:205 E:192 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 -6 12 B 2 0 12 0 6 C -6 -12 0 -10 -8 D 6 0 10 0 6 E -12 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -6 12 B 2 0 12 0 6 C -6 -12 0 -10 -8 D 6 0 10 0 6 E -12 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -6 12 B 2 0 12 0 6 C -6 -12 0 -10 -8 D 6 0 10 0 6 E -12 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3302: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) E A C B D (9) D B C E A (9) B C D E A (8) A E C B D (7) A D E C B (6) B C E D A (5) A E D C B (4) A E C D B (4) C B D E A (3) E B C A D (2) E A B D C (2) D C B A E (2) D A C B E (2) D A B C E (2) C B E D A (2) C B E A D (2) A E D B C (2) E C B A D (1) E C A B D (1) E B D C A (1) E B A C D (1) E A D B C (1) D E B A C (1) D B A C E (1) D A B E C (1) C E B A D (1) C B D A E (1) C B A E D (1) C B A D E (1) C A E B D (1) B D C E A (1) B C E A D (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 -10 -2 -2 B 12 0 2 4 8 C 10 -2 0 4 12 D 2 -4 -4 0 2 E 2 -8 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -2 -2 B 12 0 2 4 8 C 10 -2 0 4 12 D 2 -4 -4 0 2 E 2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=25 E=18 B=15 C=12 so C is eliminated. Round 2 votes counts: D=30 A=26 B=25 E=19 so E is eliminated. Round 3 votes counts: A=39 B=31 D=30 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:212 D:198 E:190 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 -2 -2 B 12 0 2 4 8 C 10 -2 0 4 12 D 2 -4 -4 0 2 E 2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -2 -2 B 12 0 2 4 8 C 10 -2 0 4 12 D 2 -4 -4 0 2 E 2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -2 -2 B 12 0 2 4 8 C 10 -2 0 4 12 D 2 -4 -4 0 2 E 2 -8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3303: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C D A E B (8) C D E B A (7) E B D A C (4) E B A D C (4) C D E A B (4) C D A B E (4) B A E D C (4) E B D C A (3) C A D B E (3) B A C E D (3) A B C E D (3) E B C D A (2) D E C A B (2) D C E B A (2) D C A E B (2) D A C E B (2) C A B D E (2) A D E B C (2) A D C B E (2) A C B D E (2) A B E D C (2) A B D E C (2) E D B C A (1) E D B A C (1) E A B D C (1) D E C B A (1) D E A B C (1) D C E A B (1) D A E C B (1) D A E B C (1) C E D B A (1) C E B D A (1) C D B A E (1) C B A D E (1) B E A C D (1) B C E A D (1) A D C E B (1) A D B E C (1) A D B C E (1) A C D B E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 0 2 B -2 0 2 -2 -6 C -6 -2 0 -6 8 D 0 2 6 0 12 E -2 6 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.726062 B: 0.000000 C: 0.000000 D: 0.273938 E: 0.000000 Sum of squares = 0.602207974078 Cumulative probabilities = A: 0.726062 B: 0.726062 C: 0.726062 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 0 2 B -2 0 2 -2 -6 C -6 -2 0 -6 8 D 0 2 6 0 12 E -2 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=20 A=19 E=16 D=13 so D is eliminated. Round 2 votes counts: C=37 A=23 E=20 B=20 so E is eliminated. Round 3 votes counts: C=40 B=35 A=25 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:205 C:197 B:196 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 0 2 B -2 0 2 -2 -6 C -6 -2 0 -6 8 D 0 2 6 0 12 E -2 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 2 B -2 0 2 -2 -6 C -6 -2 0 -6 8 D 0 2 6 0 12 E -2 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 2 B -2 0 2 -2 -6 C -6 -2 0 -6 8 D 0 2 6 0 12 E -2 6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3304: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (16) A C E B D (9) D E C A B (7) D E C B A (5) D B E C A (5) C E A B D (5) E C D A B (4) D B C E A (4) B C A E D (4) A D E C B (4) D E A C B (3) D B A E C (3) D A E C B (3) B D A C E (3) E C A D B (2) D B E A C (2) B A D C E (2) B A C D E (2) A E C D B (2) A B C E D (2) E D C A B (1) E A C D B (1) D C E B A (1) D A B E C (1) C E D A B (1) B D C E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D E A (1) B A D E C (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 12 6 10 B 6 0 2 2 2 C -12 -2 0 4 8 D -6 -2 -4 0 0 E -10 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 6 10 B 6 0 2 2 2 C -12 -2 0 4 8 D -6 -2 -4 0 0 E -10 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=33 A=19 E=8 C=6 so C is eliminated. Round 2 votes counts: D=34 B=33 A=19 E=14 so E is eliminated. Round 3 votes counts: D=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:211 B:206 C:199 D:194 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 6 10 B 6 0 2 2 2 C -12 -2 0 4 8 D -6 -2 -4 0 0 E -10 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 6 10 B 6 0 2 2 2 C -12 -2 0 4 8 D -6 -2 -4 0 0 E -10 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 6 10 B 6 0 2 2 2 C -12 -2 0 4 8 D -6 -2 -4 0 0 E -10 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3305: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (14) D C E A B (7) E A B C D (6) D C A E B (5) E B A D C (4) B E A C D (4) B D C A E (4) B A E C D (4) A E C D B (4) E A B D C (3) D C B E A (3) C D A E B (3) C D A B E (3) D C E B A (2) D B C E A (2) C D B A E (2) C A D B E (2) B E D A C (2) B D C E A (2) A E B C D (2) A C E D B (2) A C D E B (2) A B E C D (2) E D C B A (1) E D C A B (1) E B D A C (1) E B A C D (1) E A D C B (1) E A C D B (1) E A C B D (1) D E C A B (1) B E A D C (1) B D E C A (1) B C D A E (1) B A C E D (1) B A C D E (1) A E C B D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -8 -10 10 B 2 0 -14 -14 2 C 8 14 0 -10 16 D 10 14 10 0 14 E -10 -2 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -10 10 B 2 0 -14 -14 2 C 8 14 0 -10 16 D 10 14 10 0 14 E -10 -2 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=21 E=20 A=15 C=10 so C is eliminated. Round 2 votes counts: D=42 B=21 E=20 A=17 so A is eliminated. Round 3 votes counts: D=47 E=29 B=24 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:214 A:195 B:188 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -10 10 B 2 0 -14 -14 2 C 8 14 0 -10 16 D 10 14 10 0 14 E -10 -2 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -10 10 B 2 0 -14 -14 2 C 8 14 0 -10 16 D 10 14 10 0 14 E -10 -2 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -10 10 B 2 0 -14 -14 2 C 8 14 0 -10 16 D 10 14 10 0 14 E -10 -2 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3306: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) C E B A D (5) C B A D E (5) C A E D B (4) B E D C A (4) B E C D A (4) B C E A D (4) A D C E B (4) A C D B E (4) E D A C B (3) D A B E C (3) C A E B D (3) C A D E B (3) B D E A C (3) B C A D E (3) A D B C E (3) E D A B C (2) D E B A C (2) D E A B C (2) D A E C B (2) C E A D B (2) C E A B D (2) C B A E D (2) B D A E C (2) B C E D A (2) B C A E D (2) A D E C B (2) A D C B E (2) A C D E B (2) E D B A C (1) E C B D A (1) E B D A C (1) E B C D A (1) D E A C B (1) D B E A C (1) D B A E C (1) D A E B C (1) C B E A D (1) B D E C A (1) B C D A E (1) B A D C E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -12 22 14 B -4 0 -10 -2 12 C 12 10 0 16 24 D -22 2 -16 0 12 E -14 -12 -24 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 22 14 B -4 0 -10 -2 12 C 12 10 0 16 24 D -22 2 -16 0 12 E -14 -12 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=27 A=18 D=13 E=9 so E is eliminated. Round 2 votes counts: C=34 B=29 D=19 A=18 so A is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:231 A:214 B:198 D:188 E:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 22 14 B -4 0 -10 -2 12 C 12 10 0 16 24 D -22 2 -16 0 12 E -14 -12 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 22 14 B -4 0 -10 -2 12 C 12 10 0 16 24 D -22 2 -16 0 12 E -14 -12 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 22 14 B -4 0 -10 -2 12 C 12 10 0 16 24 D -22 2 -16 0 12 E -14 -12 -24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3307: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) A E D B C (9) A D E B C (7) E A D B C (6) C B E A D (6) B C D E A (6) E A D C B (5) D A E B C (5) D E A B C (4) A E D C B (4) A D E C B (4) E C B A D (3) D A B C E (3) C B D A E (3) E C A B D (2) E B C D A (2) E A C D B (2) D E B C A (2) C E B A D (2) C B D E A (2) C B A D E (2) B C D A E (2) E D B C A (1) E D A B C (1) E C B D A (1) E A C B D (1) D A B E C (1) C A B E D (1) B D C E A (1) B C E D A (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 14 8 12 -14 B -14 0 4 -12 -22 C -8 -4 0 -10 -20 D -12 12 10 0 -12 E 14 22 20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 8 12 -14 B -14 0 4 -12 -22 C -8 -4 0 -10 -20 D -12 12 10 0 -12 E 14 22 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 E=24 D=15 B=10 so B is eliminated. Round 2 votes counts: C=34 A=26 E=24 D=16 so D is eliminated. Round 3 votes counts: C=35 A=35 E=30 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:234 A:210 D:199 C:179 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 8 12 -14 B -14 0 4 -12 -22 C -8 -4 0 -10 -20 D -12 12 10 0 -12 E 14 22 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 12 -14 B -14 0 4 -12 -22 C -8 -4 0 -10 -20 D -12 12 10 0 -12 E 14 22 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 12 -14 B -14 0 4 -12 -22 C -8 -4 0 -10 -20 D -12 12 10 0 -12 E 14 22 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3308: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) B D A C E (8) B D C A E (7) A C E B D (7) D B C E A (6) D B A E C (6) D B E C A (5) C A E B D (5) E A C D B (4) B D C E A (4) E D C B A (3) D E B C A (3) D B E A C (3) A E C D B (3) E C D B A (2) E C D A B (2) A E C B D (2) A B D E C (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C A B D (1) E A D C B (1) E A C B D (1) D B A C E (1) C E B A D (1) C E A B D (1) C B D E A (1) C B A D E (1) C A B E D (1) B C D E A (1) B A D C E (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -8 -10 -2 B 10 0 6 -2 4 C 8 -6 0 -8 -2 D 10 2 8 0 4 E 2 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -10 -2 B 10 0 6 -2 4 C 8 -6 0 -8 -2 D 10 2 8 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 B=21 A=19 C=10 so C is eliminated. Round 2 votes counts: E=28 A=25 D=24 B=23 so B is eliminated. Round 3 votes counts: D=45 E=28 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:209 E:198 C:196 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -8 -10 -2 B 10 0 6 -2 4 C 8 -6 0 -8 -2 D 10 2 8 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -10 -2 B 10 0 6 -2 4 C 8 -6 0 -8 -2 D 10 2 8 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -10 -2 B 10 0 6 -2 4 C 8 -6 0 -8 -2 D 10 2 8 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3309: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) A E B C D (10) A E B D C (9) A E D C B (6) B C D E A (5) E D C A B (4) E A D C B (4) E A B D C (4) A C D E B (4) A C D B E (4) D C B E A (3) A B E C D (3) E B D C A (2) E B A D C (2) D C E A B (2) C D B A E (2) C D A B E (2) B C D A E (2) B C A D E (2) B A E C D (2) B A C D E (2) A E C D B (2) A B C D E (2) E D C B A (1) E A D B C (1) D C E B A (1) D C A E B (1) D B E C A (1) C D A E B (1) C B D A E (1) C A D B E (1) B E C D A (1) B E A D C (1) B A C E D (1) A C E D B (1) Total count = 100 A B C D E A 0 22 16 22 16 B -22 0 0 -2 -10 C -16 0 0 16 -6 D -22 2 -16 0 -8 E -16 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 16 22 16 B -22 0 0 -2 -10 C -16 0 0 16 -6 D -22 2 -16 0 -8 E -16 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 E=18 C=17 B=16 D=8 so D is eliminated. Round 2 votes counts: A=41 C=24 E=18 B=17 so B is eliminated. Round 3 votes counts: A=46 C=33 E=21 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:238 E:204 C:197 B:183 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 16 22 16 B -22 0 0 -2 -10 C -16 0 0 16 -6 D -22 2 -16 0 -8 E -16 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 22 16 B -22 0 0 -2 -10 C -16 0 0 16 -6 D -22 2 -16 0 -8 E -16 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 22 16 B -22 0 0 -2 -10 C -16 0 0 16 -6 D -22 2 -16 0 -8 E -16 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3310: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) A B D C E (9) B A D E C (8) E C B D A (7) A D B C E (7) E C D B A (6) D C A B E (5) D A C B E (4) C D E A B (4) E C B A D (3) E B C A D (3) C E D A B (3) B E A C D (3) E B A C D (2) D C A E B (2) C E D B A (2) C D A E B (2) B E A D C (2) B A E C D (2) B A D C E (2) A B E D C (2) A B D E C (2) E D A B C (1) E C D A B (1) E B A D C (1) D E C A B (1) D C E A B (1) D A B C E (1) C E B D A (1) C D A B E (1) C B D A E (1) Total count = 100 A B C D E A 0 -8 14 14 18 B 8 0 12 18 20 C -14 -12 0 -18 -10 D -14 -18 18 0 0 E -18 -20 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 14 14 18 B 8 0 12 18 20 C -14 -12 0 -18 -10 D -14 -18 18 0 0 E -18 -20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 A=20 D=14 C=14 so D is eliminated. Round 2 votes counts: B=28 E=25 A=25 C=22 so C is eliminated. Round 3 votes counts: E=36 A=35 B=29 so B is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:229 A:219 D:193 E:186 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 14 14 18 B 8 0 12 18 20 C -14 -12 0 -18 -10 D -14 -18 18 0 0 E -18 -20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 14 18 B 8 0 12 18 20 C -14 -12 0 -18 -10 D -14 -18 18 0 0 E -18 -20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 14 18 B 8 0 12 18 20 C -14 -12 0 -18 -10 D -14 -18 18 0 0 E -18 -20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3311: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) D A C B E (6) C D A B E (5) C B E D A (5) A D E B C (5) E C A B D (4) C D A E B (4) E A D C B (3) E A B D C (3) D A B C E (3) C D B A E (3) C B D E A (3) B D C A E (3) B C E D A (3) B C D A E (3) E C A D B (2) E B C A D (2) E B A C D (2) E A D B C (2) D A C E B (2) D A B E C (2) C E B D A (2) C E A D B (2) B E C D A (2) B E A D C (2) B D A E C (2) A E D B C (2) E C B A D (1) E B A D C (1) E A C D B (1) D B A E C (1) D B A C E (1) C E B A D (1) C A D E B (1) B E C A D (1) B E A C D (1) B D E A C (1) B D A C E (1) B C D E A (1) B A E D C (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -12 -22 10 B 2 0 -8 6 18 C 12 8 0 14 14 D 22 -6 -14 0 14 E -10 -18 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -22 10 B 2 0 -8 6 18 C 12 8 0 14 14 D 22 -6 -14 0 14 E -10 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=21 B=21 D=15 A=9 so A is eliminated. Round 2 votes counts: C=34 E=23 D=22 B=21 so B is eliminated. Round 3 votes counts: C=41 E=30 D=29 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:209 D:208 A:187 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -22 10 B 2 0 -8 6 18 C 12 8 0 14 14 D 22 -6 -14 0 14 E -10 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -22 10 B 2 0 -8 6 18 C 12 8 0 14 14 D 22 -6 -14 0 14 E -10 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -22 10 B 2 0 -8 6 18 C 12 8 0 14 14 D 22 -6 -14 0 14 E -10 -18 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3312: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) B E C A D (13) E C B A D (7) D A C E B (7) D B A C E (6) B D A E C (6) E C A B D (5) C E A D B (5) B E C D A (4) E C B D A (3) C A E D B (3) B D A C E (3) E C D A B (2) E C A D B (2) D C E A B (2) D B A E C (2) C E A B D (2) B E D C A (2) A C D E B (2) E B C A D (1) D E C A B (1) D A C B E (1) C E D A B (1) C A E B D (1) B E A C D (1) B D E A C (1) B A D E C (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -8 -8 -4 B 0 0 8 0 10 C 8 -8 0 4 -2 D 8 0 -4 0 -4 E 4 -10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.570131 C: 0.000000 D: 0.429869 E: 0.000000 Sum of squares = 0.509836583663 Cumulative probabilities = A: 0.000000 B: 0.570131 C: 0.570131 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -8 -4 B 0 0 8 0 10 C 8 -8 0 4 -2 D 8 0 -4 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=31 E=20 C=12 A=5 so A is eliminated. Round 2 votes counts: D=35 B=31 E=20 C=14 so C is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:209 C:201 D:200 E:200 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -8 -4 B 0 0 8 0 10 C 8 -8 0 4 -2 D 8 0 -4 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -8 -4 B 0 0 8 0 10 C 8 -8 0 4 -2 D 8 0 -4 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -8 -4 B 0 0 8 0 10 C 8 -8 0 4 -2 D 8 0 -4 0 -4 E 4 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3313: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (12) B A D C E (10) B D A C E (8) D B E C A (6) E D C A B (5) B A C D E (5) A C E B D (5) E A C B D (4) D B C A E (4) E D C B A (3) E A B C D (3) D E B C A (3) C E A D B (3) E D B C A (2) E C D A B (2) E C A B D (2) E B A C D (2) D E C A B (2) C A E D B (2) C A D E B (2) A C B D E (2) E B D A C (1) D C E B A (1) D C E A B (1) C D A B E (1) C A E B D (1) C A D B E (1) B E D A C (1) B E A C D (1) B D A E C (1) B A C E D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -6 18 -8 B -2 0 0 0 -12 C 6 0 0 4 0 D -18 0 -4 0 -4 E 8 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636823 D: 0.000000 E: 0.363177 Sum of squares = 0.537441223277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636823 D: 0.636823 E: 1.000000 A B C D E A 0 2 -6 18 -8 B -2 0 0 0 -12 C 6 0 0 4 0 D -18 0 -4 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=27 D=17 C=10 A=10 so C is eliminated. Round 2 votes counts: E=39 B=27 D=18 A=16 so A is eliminated. Round 3 votes counts: E=47 B=32 D=21 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:205 A:203 B:193 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 18 -8 B -2 0 0 0 -12 C 6 0 0 4 0 D -18 0 -4 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 18 -8 B -2 0 0 0 -12 C 6 0 0 4 0 D -18 0 -4 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 18 -8 B -2 0 0 0 -12 C 6 0 0 4 0 D -18 0 -4 0 -4 E 8 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3314: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) C D B A E (6) C A E D B (5) C A D E B (5) B E D A C (5) C E B D A (4) C D B E A (4) E C A B D (3) E B A C D (3) E A B D C (3) C B D E A (3) B D E A C (3) A E B D C (3) A D B E C (3) E C B D A (2) E B C A D (2) E B A D C (2) E A C B D (2) E A B C D (2) D C B A E (2) D C A B E (2) D A B E C (2) C E B A D (2) C E A B D (2) C D A B E (2) C B E D A (2) B D E C A (2) A E D B C (2) A D E C B (2) A D E B C (2) E B C D A (1) D B C E A (1) D A C B E (1) C D E B A (1) C D A E B (1) C A D B E (1) B E D C A (1) B C E D A (1) B C D E A (1) A D B C E (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -12 -6 -4 B 8 0 -6 2 0 C 12 6 0 12 -4 D 6 -2 -12 0 4 E 4 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 A B C D E A 0 -8 -12 -6 -4 B 8 0 -6 2 0 C 12 6 0 12 -4 D 6 -2 -12 0 4 E 4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=20 A=15 D=14 B=13 so B is eliminated. Round 2 votes counts: C=40 E=26 D=19 A=15 so A is eliminated. Round 3 votes counts: C=41 E=31 D=28 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:213 B:202 E:202 D:198 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -6 -4 B 8 0 -6 2 0 C 12 6 0 12 -4 D 6 -2 -12 0 4 E 4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -6 -4 B 8 0 -6 2 0 C 12 6 0 12 -4 D 6 -2 -12 0 4 E 4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -6 -4 B 8 0 -6 2 0 C 12 6 0 12 -4 D 6 -2 -12 0 4 E 4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.600000 Sum of squares = 0.439999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3315: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (22) C B E A D (13) D A B C E (9) E D A B C (5) C E B A D (5) C D B A E (5) D A C B E (4) C B A D E (4) E C B A D (3) E A D B C (3) D C A B E (3) D A B E C (3) E B C A D (2) D E C A B (2) D E A B C (2) D A E C B (2) E D C A B (1) E C D B A (1) E C B D A (1) E B A C D (1) D E A C B (1) C E B D A (1) B E C A D (1) B C A D E (1) B A D C E (1) A E D B C (1) A E B C D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 22 14 -24 16 B -22 0 8 -32 -10 C -14 -8 0 -22 -8 D 24 32 22 0 22 E -16 10 8 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 14 -24 16 B -22 0 8 -32 -10 C -14 -8 0 -22 -8 D 24 32 22 0 22 E -16 10 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=48 C=28 E=17 A=4 B=3 so B is eliminated. Round 2 votes counts: D=48 C=29 E=18 A=5 so A is eliminated. Round 3 votes counts: D=51 C=29 E=20 so E is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:250 A:214 E:190 C:174 B:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 14 -24 16 B -22 0 8 -32 -10 C -14 -8 0 -22 -8 D 24 32 22 0 22 E -16 10 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 14 -24 16 B -22 0 8 -32 -10 C -14 -8 0 -22 -8 D 24 32 22 0 22 E -16 10 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 14 -24 16 B -22 0 8 -32 -10 C -14 -8 0 -22 -8 D 24 32 22 0 22 E -16 10 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3316: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) E C D B A (8) B E D C A (8) A B D C E (7) C E D A B (6) C D E B A (4) B D E C A (4) B A D C E (4) A E C D B (4) A C E D B (4) E C D A B (3) E B D C A (3) B D C A E (3) A C D E B (3) A B E D C (3) A B E C D (3) A B C D E (3) E B C D A (2) C E D B A (2) C D E A B (2) B A D E C (2) A C D B E (2) E D C B A (1) E D B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) C D A E B (1) B E D A C (1) B D A C E (1) A E C B D (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -20 -24 -14 B 12 0 14 12 6 C 20 -14 0 -2 6 D 24 -12 2 0 -2 E 14 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -20 -24 -14 B 12 0 14 12 6 C 20 -14 0 -2 6 D 24 -12 2 0 -2 E 14 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=32 A=32 E=18 C=15 D=3 so D is eliminated. Round 2 votes counts: B=33 A=32 E=18 C=17 so C is eliminated. Round 3 votes counts: B=34 E=33 A=33 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:206 C:205 E:202 A:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -20 -24 -14 B 12 0 14 12 6 C 20 -14 0 -2 6 D 24 -12 2 0 -2 E 14 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -24 -14 B 12 0 14 12 6 C 20 -14 0 -2 6 D 24 -12 2 0 -2 E 14 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -24 -14 B 12 0 14 12 6 C 20 -14 0 -2 6 D 24 -12 2 0 -2 E 14 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3317: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (13) B E D A C (8) B E C A D (7) D C A E B (6) B E A C D (5) C A E D B (4) B D E A C (4) A C E D B (4) E A C D B (3) D C A B E (3) D B C A E (3) D A C E B (3) E B D A C (2) E B A D C (2) E B A C D (2) E A C B D (2) C D A E B (2) B E A D C (2) B D E C A (2) B D C A E (2) A E C D B (2) A C D E B (2) E C A B D (1) E A D C B (1) E A B C D (1) D B E A C (1) D A C B E (1) D A B C E (1) C D A B E (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D C A (1) B D C E A (1) B C E A D (1) B C D A E (1) A D E C B (1) Total count = 100 A B C D E A 0 8 -6 16 8 B -8 0 -8 -4 -2 C 6 8 0 12 6 D -16 4 -12 0 0 E -8 2 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 16 8 B -8 0 -8 -4 -2 C 6 8 0 12 6 D -16 4 -12 0 0 E -8 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=25 D=18 E=14 A=9 so A is eliminated. Round 2 votes counts: B=34 C=31 D=19 E=16 so E is eliminated. Round 3 votes counts: B=41 C=39 D=20 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:213 E:194 B:189 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 16 8 B -8 0 -8 -4 -2 C 6 8 0 12 6 D -16 4 -12 0 0 E -8 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 16 8 B -8 0 -8 -4 -2 C 6 8 0 12 6 D -16 4 -12 0 0 E -8 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 16 8 B -8 0 -8 -4 -2 C 6 8 0 12 6 D -16 4 -12 0 0 E -8 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3318: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) E A D B C (7) A E D C B (6) B C E D A (5) E B A D C (4) D C A E B (4) C D B A E (4) C D A E B (4) C D A B E (4) B E D C A (4) B E C A D (4) B E A D C (4) A D E C B (4) E A D C B (3) D B C E A (3) C B D A E (3) C A D E B (3) E D A B C (2) E B A C D (2) B E C D A (2) B D C E A (2) B C E A D (2) B C D A E (2) E D A C B (1) E B D A C (1) E A B D C (1) D C B E A (1) D C B A E (1) D A E C B (1) D A C E B (1) B E D A C (1) B E A C D (1) B C A D E (1) B A C E D (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 -14 -16 -10 -18 B 14 0 18 0 8 C 16 -18 0 -4 2 D 10 0 4 0 -4 E 18 -8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.662107 C: 0.000000 D: 0.337893 E: 0.000000 Sum of squares = 0.552557212692 Cumulative probabilities = A: 0.000000 B: 0.662107 C: 0.662107 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -10 -18 B 14 0 18 0 8 C 16 -18 0 -4 2 D 10 0 4 0 -4 E 18 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=21 C=18 A=12 D=11 so D is eliminated. Round 2 votes counts: B=41 C=24 E=21 A=14 so A is eliminated. Round 3 votes counts: B=41 E=33 C=26 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:206 D:205 C:198 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -10 -18 B 14 0 18 0 8 C 16 -18 0 -4 2 D 10 0 4 0 -4 E 18 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -10 -18 B 14 0 18 0 8 C 16 -18 0 -4 2 D 10 0 4 0 -4 E 18 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -10 -18 B 14 0 18 0 8 C 16 -18 0 -4 2 D 10 0 4 0 -4 E 18 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3319: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) E D A B C (5) D E B C A (5) A C E B D (5) D E B A C (4) D E A B C (4) D B E C A (4) C B A D E (4) B D C E A (4) B C E D A (4) B C D E A (4) E D B A C (3) B C D A E (3) B C A E D (3) A E B C D (3) E B D A C (2) E B A D C (2) D B C E A (2) C A B D E (2) B E A C D (2) A E D C B (2) A E C D B (2) A E C B D (2) A D E C B (2) A B C E D (2) E A D B C (1) E A B D C (1) E A B C D (1) D E C A B (1) D E A C B (1) D C A E B (1) D A C E B (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D B E (1) B E D A C (1) B E C A D (1) B D E C A (1) B C E A D (1) B C A D E (1) B A C E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -4 -4 -8 B 8 0 30 18 2 C 4 -30 0 6 0 D 4 -18 -6 0 -6 E 8 -2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -4 -8 B 8 0 30 18 2 C 4 -30 0 6 0 D 4 -18 -6 0 -6 E 8 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=23 A=19 C=17 E=15 so E is eliminated. Round 2 votes counts: D=31 B=30 A=22 C=17 so C is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 E:206 C:190 A:188 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -4 -8 B 8 0 30 18 2 C 4 -30 0 6 0 D 4 -18 -6 0 -6 E 8 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -4 -8 B 8 0 30 18 2 C 4 -30 0 6 0 D 4 -18 -6 0 -6 E 8 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -4 -8 B 8 0 30 18 2 C 4 -30 0 6 0 D 4 -18 -6 0 -6 E 8 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3320: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (5) C A B E D (5) B D A E C (5) C E A B D (4) B C A D E (4) B A D C E (4) E D B A C (3) E D A B C (3) E C D B A (3) E C A D B (3) C E D B A (3) C E A D B (3) C B E A D (3) C B A D E (3) B A D E C (3) E C D A B (2) D E B A C (2) D E A B C (2) D B A E C (2) C E D A B (2) C B D A E (2) C A B D E (2) B D E A C (2) B C D E A (2) B C D A E (2) B A C D E (2) A D E B C (2) A C E D B (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E D A C B (1) E B D C A (1) E A D C B (1) E A C D B (1) D A E B C (1) C B E D A (1) C A E B D (1) B D C A E (1) B C E D A (1) A E D C B (1) A D B E C (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -12 2 -2 B 12 0 -6 16 0 C 12 6 0 22 16 D -2 -16 -22 0 -4 E 2 0 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 2 -2 B 12 0 -6 16 0 C 12 6 0 22 16 D -2 -16 -22 0 -4 E 2 0 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=26 E=21 A=12 D=7 so D is eliminated. Round 2 votes counts: C=34 B=28 E=25 A=13 so A is eliminated. Round 3 votes counts: C=39 B=32 E=29 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:211 E:195 A:188 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -12 2 -2 B 12 0 -6 16 0 C 12 6 0 22 16 D -2 -16 -22 0 -4 E 2 0 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 2 -2 B 12 0 -6 16 0 C 12 6 0 22 16 D -2 -16 -22 0 -4 E 2 0 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 2 -2 B 12 0 -6 16 0 C 12 6 0 22 16 D -2 -16 -22 0 -4 E 2 0 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3321: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) A E C B D (7) D B E A C (6) C B D A E (5) C A E B D (5) E D A C B (4) C B A E D (4) B C D A E (4) E A C D B (3) D E A B C (3) D B C E A (3) C D B E A (3) C B A D E (3) E C A D B (2) E A D C B (2) E A D B C (2) D C B E A (2) D B A E C (2) C E A D B (2) C D E A B (2) C A B E D (2) B D A C E (2) B A D E C (2) A E B D C (2) A E B C D (2) A C E B D (2) A C B E D (2) A B E C D (2) E C D A B (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D E A C B (1) D C E A B (1) D B E C A (1) C E D A B (1) C D E B A (1) C D B A E (1) B D A E C (1) B C A D E (1) Total count = 100 A B C D E A 0 2 -6 -6 12 B -2 0 -16 10 6 C 6 16 0 12 6 D 6 -10 -12 0 6 E -12 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -6 12 B -2 0 -16 10 6 C 6 16 0 12 6 D 6 -10 -12 0 6 E -12 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 B=17 A=17 E=16 so E is eliminated. Round 2 votes counts: C=32 A=26 D=25 B=17 so B is eliminated. Round 3 votes counts: C=37 D=35 A=28 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:201 B:199 D:195 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -6 12 B -2 0 -16 10 6 C 6 16 0 12 6 D 6 -10 -12 0 6 E -12 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -6 12 B -2 0 -16 10 6 C 6 16 0 12 6 D 6 -10 -12 0 6 E -12 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -6 12 B -2 0 -16 10 6 C 6 16 0 12 6 D 6 -10 -12 0 6 E -12 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3322: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (10) C A D B E (9) E B D A C (7) A C D E B (6) A C B E D (6) E B A D C (5) D E B C A (5) D C A E B (5) A E B C D (5) B E C A D (4) E B A C D (3) D C E B A (3) D B E C A (3) C D A B E (3) A B E C D (3) E D B A C (2) D E B A C (2) C D B A E (2) B E D C A (2) B E A D C (2) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) C D A E B (1) C B E A D (1) C B D A E (1) C B A E D (1) C A D E B (1) C A B E D (1) B D E C A (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 10 18 -2 B 8 0 10 6 4 C -10 -10 0 18 -10 D -18 -6 -18 0 -6 E 2 -4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 18 -2 B 8 0 10 6 4 C -10 -10 0 18 -10 D -18 -6 -18 0 -6 E 2 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=22 A=22 C=20 B=19 E=17 so E is eliminated. Round 2 votes counts: B=34 D=24 A=22 C=20 so C is eliminated. Round 3 votes counts: B=37 A=33 D=30 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:209 E:207 C:194 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 18 -2 B 8 0 10 6 4 C -10 -10 0 18 -10 D -18 -6 -18 0 -6 E 2 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 18 -2 B 8 0 10 6 4 C -10 -10 0 18 -10 D -18 -6 -18 0 -6 E 2 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 18 -2 B 8 0 10 6 4 C -10 -10 0 18 -10 D -18 -6 -18 0 -6 E 2 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3323: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (7) E D A B C (6) D B E C A (6) D E B A C (5) D B E A C (5) A E C D B (5) E D B A C (4) E A D C B (4) D A C B E (4) A C E B D (4) E A C D B (3) D B C A E (3) C A B E D (3) A C E D B (3) A C D E B (3) E B D C A (2) E B C A D (2) D B C E A (2) C D B A E (2) C B A E D (2) C A E B D (2) B D C A E (2) B C D A E (2) A E D C B (2) A D E C B (2) A C D B E (2) E D A C B (1) E C B A D (1) E C A B D (1) E B D A C (1) E A C B D (1) D B A C E (1) D A B C E (1) C E B A D (1) C A D B E (1) B E C A D (1) B D E C A (1) B C D E A (1) A E C B D (1) Total count = 100 A B C D E A 0 12 16 2 4 B -12 0 -10 -30 -8 C -16 10 0 -4 -8 D -2 30 4 0 0 E -4 8 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 2 4 B -12 0 -10 -30 -8 C -16 10 0 -4 -8 D -2 30 4 0 0 E -4 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=26 A=22 C=18 B=7 so B is eliminated. Round 2 votes counts: D=30 E=27 A=22 C=21 so C is eliminated. Round 3 votes counts: A=37 D=35 E=28 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:216 E:206 C:191 B:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 2 4 B -12 0 -10 -30 -8 C -16 10 0 -4 -8 D -2 30 4 0 0 E -4 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 2 4 B -12 0 -10 -30 -8 C -16 10 0 -4 -8 D -2 30 4 0 0 E -4 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 2 4 B -12 0 -10 -30 -8 C -16 10 0 -4 -8 D -2 30 4 0 0 E -4 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3324: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) A E B C D (8) D C B E A (7) C B E A D (7) C D B E A (5) C B E D A (5) D A E C B (4) C B A E D (4) B C E A D (4) A E B D C (4) A B C E D (4) E A D B C (3) D E A C B (3) D E A B C (3) D C E B A (3) D C B A E (3) C B D A E (3) A E D B C (3) E B A C D (2) D E C B A (2) D A C B E (2) A C B D E (2) E D B C A (1) E B C D A (1) D A E B C (1) D A C E B (1) C B D E A (1) C B A D E (1) B E C A D (1) B C A E D (1) B A C E D (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 10 -12 B 4 0 -6 16 4 C -2 6 0 18 8 D -10 -16 -18 0 -16 E 12 -4 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.090909 Sum of squares = 0.43801652885 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 -4 2 10 -12 B 4 0 -6 16 4 C -2 6 0 18 8 D -10 -16 -18 0 -16 E 12 -4 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.090909 Sum of squares = 0.438016527818 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 A=23 E=15 B=7 so B is eliminated. Round 2 votes counts: C=31 D=29 A=24 E=16 so E is eliminated. Round 3 votes counts: A=37 C=33 D=30 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:215 B:209 E:208 A:198 D:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 10 -12 B 4 0 -6 16 4 C -2 6 0 18 8 D -10 -16 -18 0 -16 E 12 -4 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.090909 Sum of squares = 0.438016527818 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 10 -12 B 4 0 -6 16 4 C -2 6 0 18 8 D -10 -16 -18 0 -16 E 12 -4 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.090909 Sum of squares = 0.438016527818 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 10 -12 B 4 0 -6 16 4 C -2 6 0 18 8 D -10 -16 -18 0 -16 E 12 -4 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.545455 D: 0.000000 E: 0.090909 Sum of squares = 0.438016527818 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3325: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (17) C B D A E (15) C B A D E (5) A B C D E (5) A B D E C (4) E C D B A (3) E A C B D (3) D C B E A (3) D B C E A (3) A D B E C (3) A C B D E (3) E C A D B (2) C E A B D (2) C D B E A (2) C D B A E (2) C A B D E (2) A E D B C (2) A E B D C (2) A E B C D (2) E D B C A (1) E D B A C (1) E D A B C (1) E C A B D (1) E A D C B (1) E A C D B (1) D E B C A (1) D E B A C (1) D B E A C (1) D B C A E (1) D B A E C (1) D B A C E (1) C E D B A (1) C B D E A (1) C A B E D (1) B C D A E (1) B C A D E (1) B A C D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 4 20 8 B -8 0 2 2 14 C -4 -2 0 12 2 D -20 -2 -12 0 14 E -8 -14 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 20 8 B -8 0 2 2 14 C -4 -2 0 12 2 D -20 -2 -12 0 14 E -8 -14 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=31 C=31 A=23 D=12 B=3 so B is eliminated. Round 2 votes counts: C=33 E=31 A=24 D=12 so D is eliminated. Round 3 votes counts: C=40 E=34 A=26 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:220 B:205 C:204 D:190 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 20 8 B -8 0 2 2 14 C -4 -2 0 12 2 D -20 -2 -12 0 14 E -8 -14 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 20 8 B -8 0 2 2 14 C -4 -2 0 12 2 D -20 -2 -12 0 14 E -8 -14 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 20 8 B -8 0 2 2 14 C -4 -2 0 12 2 D -20 -2 -12 0 14 E -8 -14 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3326: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) E D A B C (8) C E A B D (5) B D A E C (5) D B A E C (4) D A B E C (4) C E A D B (4) C A E D B (4) E D A C B (3) D B E A C (3) C B A E D (3) C A B D E (3) B C A D E (3) A D E B C (3) E C A D B (2) D E A B C (2) D A E B C (2) C E B D A (2) C B E D A (2) B D E C A (2) B D A C E (2) B C E D A (2) B A D C E (2) A D B E C (2) E D C B A (1) E C D B A (1) E C D A B (1) E A D C B (1) E A D B C (1) C E B A D (1) C B E A D (1) C B A D E (1) C A B E D (1) B E D C A (1) B E C D A (1) B D C A E (1) B C D A E (1) B A C D E (1) A E D C B (1) A D C B E (1) A D B C E (1) A C E D B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 14 -12 -4 B -4 0 20 -16 -2 C -14 -20 0 -18 -12 D 12 16 18 0 -10 E 4 2 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 14 -12 -4 B -4 0 20 -16 -2 C -14 -20 0 -18 -12 D 12 16 18 0 -10 E 4 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 B=21 D=15 A=11 so A is eliminated. Round 2 votes counts: C=29 E=27 D=22 B=22 so D is eliminated. Round 3 votes counts: B=36 E=34 C=30 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:214 A:201 B:199 C:168 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 14 -12 -4 B -4 0 20 -16 -2 C -14 -20 0 -18 -12 D 12 16 18 0 -10 E 4 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -12 -4 B -4 0 20 -16 -2 C -14 -20 0 -18 -12 D 12 16 18 0 -10 E 4 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -12 -4 B -4 0 20 -16 -2 C -14 -20 0 -18 -12 D 12 16 18 0 -10 E 4 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3327: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) A B E D C (6) A B C D E (6) C D E B A (5) B A C E D (5) B A C D E (5) B C E D A (4) A D E B C (4) A B D E C (4) E D A C B (3) C B D E A (3) B E A D C (3) B C A D E (3) A E D B C (3) A D E C B (3) E D C B A (2) E D C A B (2) E B D C A (2) D C E A B (2) C E D B A (2) C D E A B (2) B A E D C (2) A C B D E (2) A B D C E (2) A B C E D (2) E D A B C (1) E C D B A (1) D E A C B (1) D C E B A (1) D A E C B (1) C D A E B (1) C D A B E (1) C B E D A (1) B E C D A (1) B E C A D (1) B C D E A (1) B C D A E (1) B C A E D (1) B A E C D (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 10 12 10 10 B -10 0 18 14 10 C -12 -18 0 -2 0 D -10 -14 2 0 10 E -10 -10 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 10 10 B -10 0 18 14 10 C -12 -18 0 -2 0 D -10 -14 2 0 10 E -10 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=28 C=15 D=12 E=11 so E is eliminated. Round 2 votes counts: A=34 B=30 D=20 C=16 so C is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:216 D:194 E:185 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 10 10 B -10 0 18 14 10 C -12 -18 0 -2 0 D -10 -14 2 0 10 E -10 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 10 10 B -10 0 18 14 10 C -12 -18 0 -2 0 D -10 -14 2 0 10 E -10 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 10 10 B -10 0 18 14 10 C -12 -18 0 -2 0 D -10 -14 2 0 10 E -10 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3328: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) D C A E B (6) D C E B A (5) D A C E B (5) B E A C D (4) A E B C D (4) A C E B D (4) E C B A D (3) C D E B A (3) B E C A D (3) B A E D C (3) A D B E C (3) E C B D A (2) D C B E A (2) D B E C A (2) D B C E A (2) D B A E C (2) D A C B E (2) C E B D A (2) C E B A D (2) C D E A B (2) C D A E B (2) C A D E B (2) B E D C A (2) B E C D A (2) A D C E B (2) A C D E B (2) A B E D C (2) E B C A D (1) E B A C D (1) D C A B E (1) D B E A C (1) D B C A E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E A B D (1) C A E D B (1) B D E C A (1) B A E C D (1) A E C B D (1) A D C B E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 0 4 14 B -8 0 -8 -2 -8 C 0 8 0 10 2 D -4 2 -10 0 -2 E -14 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.511146 B: 0.000000 C: 0.488854 D: 0.000000 E: 0.000000 Sum of squares = 0.500248474498 Cumulative probabilities = A: 0.511146 B: 0.511146 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 4 14 B -8 0 -8 -2 -8 C 0 8 0 10 2 D -4 2 -10 0 -2 E -14 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=30 A=30 C=17 B=16 E=7 so E is eliminated. Round 2 votes counts: D=30 A=30 C=22 B=18 so B is eliminated. Round 3 votes counts: A=39 D=33 C=28 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:210 E:197 D:193 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 4 14 B -8 0 -8 -2 -8 C 0 8 0 10 2 D -4 2 -10 0 -2 E -14 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 4 14 B -8 0 -8 -2 -8 C 0 8 0 10 2 D -4 2 -10 0 -2 E -14 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 4 14 B -8 0 -8 -2 -8 C 0 8 0 10 2 D -4 2 -10 0 -2 E -14 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999905 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3329: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) D C A E B (7) C A B D E (7) A C B E D (7) A C D E B (6) E D B C A (5) B E D C A (5) B A C E D (5) A C B D E (5) B E A C D (4) A C D B E (4) E D B A C (3) E B D C A (3) E B D A C (3) C A B E D (3) A C E B D (3) E B A C D (2) D E C A B (2) D E A C B (2) D A C E B (2) C A D E B (2) C A D B E (2) B E D A C (2) B C A E D (2) D E B A C (1) D C E A B (1) C D A E B (1) B E C A D (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 10 0 10 16 B -10 0 -10 6 -4 C 0 10 0 10 18 D -10 -6 -10 0 4 E -16 4 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.562564 B: 0.000000 C: 0.437436 D: 0.000000 E: 0.000000 Sum of squares = 0.507828614353 Cumulative probabilities = A: 0.562564 B: 0.562564 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 10 16 B -10 0 -10 6 -4 C 0 10 0 10 18 D -10 -6 -10 0 4 E -16 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 B=20 E=16 C=15 so C is eliminated. Round 2 votes counts: A=40 D=24 B=20 E=16 so E is eliminated. Round 3 votes counts: A=40 D=32 B=28 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:219 A:218 B:191 D:189 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 10 16 B -10 0 -10 6 -4 C 0 10 0 10 18 D -10 -6 -10 0 4 E -16 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 10 16 B -10 0 -10 6 -4 C 0 10 0 10 18 D -10 -6 -10 0 4 E -16 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 10 16 B -10 0 -10 6 -4 C 0 10 0 10 18 D -10 -6 -10 0 4 E -16 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3330: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) E B D C A (6) D E B A C (6) E B C D A (5) D A C B E (5) A D C B E (5) A C B E D (5) E D B C A (4) D E B C A (4) D B E C A (4) B E D C A (4) A D C E B (4) A C B D E (4) C A B E D (3) B E C D A (3) E B C A D (2) D B E A C (2) D A E B C (2) D A B C E (2) C B A D E (2) C A E B D (2) A C E B D (2) D B C A E (1) D B A C E (1) D A C E B (1) D A B E C (1) C E A B D (1) C B A E D (1) C A B D E (1) B E C A D (1) B D E C A (1) B C E D A (1) B C E A D (1) B C D E A (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 4 -8 8 B 0 0 4 -8 18 C -4 -4 0 -6 10 D 8 8 6 0 16 E -8 -18 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -8 8 B 0 0 4 -8 18 C -4 -4 0 -6 10 D 8 8 6 0 16 E -8 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=29 E=17 B=12 C=10 so C is eliminated. Round 2 votes counts: A=38 D=29 E=18 B=15 so B is eliminated. Round 3 votes counts: A=41 D=31 E=28 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:207 A:202 C:198 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -8 8 B 0 0 4 -8 18 C -4 -4 0 -6 10 D 8 8 6 0 16 E -8 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -8 8 B 0 0 4 -8 18 C -4 -4 0 -6 10 D 8 8 6 0 16 E -8 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -8 8 B 0 0 4 -8 18 C -4 -4 0 -6 10 D 8 8 6 0 16 E -8 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3331: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) E D B A C (7) C B A D E (6) C A B D E (6) B E D C A (6) E D A B C (5) C A B E D (5) D E B A C (4) A C B D E (4) D A E B C (3) B D E C A (3) A C D E B (3) E B C D A (2) D B E A C (2) C E A B D (2) C B E D A (2) C B E A D (2) B C D E A (2) B C A D E (2) A D C E B (2) A D C B E (2) A C D B E (2) E C D B A (1) E B D C A (1) E A D C B (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E B D (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C E D A (1) B C D A E (1) A E D C B (1) A D E C B (1) A D B E C (1) A D B C E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -12 -8 -6 B 12 0 12 4 10 C 12 -12 0 -6 2 D 8 -4 6 0 2 E 6 -10 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -8 -6 B 12 0 12 4 10 C 12 -12 0 -6 2 D 8 -4 6 0 2 E 6 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 B=19 A=19 D=11 so D is eliminated. Round 2 votes counts: E=29 C=26 A=23 B=22 so B is eliminated. Round 3 votes counts: E=41 C=34 A=25 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:219 D:206 C:198 E:196 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -8 -6 B 12 0 12 4 10 C 12 -12 0 -6 2 D 8 -4 6 0 2 E 6 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -8 -6 B 12 0 12 4 10 C 12 -12 0 -6 2 D 8 -4 6 0 2 E 6 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -8 -6 B 12 0 12 4 10 C 12 -12 0 -6 2 D 8 -4 6 0 2 E 6 -10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3332: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) A E B C D (7) E A D B C (5) C D B E A (5) D B C A E (4) B D C A E (4) A E B D C (4) E A C B D (3) D C E B A (3) D C B E A (3) D B A E C (3) C B A D E (3) B A D E C (3) B A D C E (3) E A D C B (2) D E C B A (2) D E B A C (2) C E D B A (2) C E D A B (2) C D E B A (2) C B D A E (2) C A E B D (2) B D A C E (2) A E C B D (2) A B C E D (2) E D C A B (1) E D A C B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C A D B (1) E A C D B (1) E A B D C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E A B D (1) C D B A E (1) C A B E D (1) B C A D E (1) A E D B C (1) A D B E C (1) A C B E D (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 -4 4 B 8 0 14 -10 2 C -2 -14 0 -14 6 D 4 10 14 0 12 E -4 -2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -4 4 B 8 0 14 -10 2 C -2 -14 0 -14 6 D 4 10 14 0 12 E -4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=21 A=21 E=18 B=13 so B is eliminated. Round 2 votes counts: D=33 A=27 C=22 E=18 so E is eliminated. Round 3 votes counts: A=39 D=36 C=25 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:207 A:197 C:188 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -4 4 B 8 0 14 -10 2 C -2 -14 0 -14 6 D 4 10 14 0 12 E -4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -4 4 B 8 0 14 -10 2 C -2 -14 0 -14 6 D 4 10 14 0 12 E -4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -4 4 B 8 0 14 -10 2 C -2 -14 0 -14 6 D 4 10 14 0 12 E -4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3333: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (5) B C E D A (5) A D E C B (5) C B E A D (4) C A B E D (4) B E D A C (4) E A D B C (3) D B C A E (3) D A E C B (3) D A C E B (3) C B D A E (3) C B A D E (3) C A E D B (3) B D E C A (3) B D C E A (3) B C D E A (3) A E D C B (3) A E D B C (3) E B A C D (2) D E A B C (2) D C A B E (2) D B E A C (2) C E A B D (2) C D B A E (2) C A D E B (2) B E C A D (2) A E C D B (2) A D E B C (2) E D A B C (1) E C A B D (1) E B A D C (1) E A B D C (1) D C A E B (1) D B E C A (1) D A B E C (1) C D A E B (1) C A E B D (1) C A D B E (1) C A B D E (1) B E A C D (1) B D C A E (1) B C E A D (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 12 -6 2 16 B -12 0 0 -8 0 C 6 0 0 -8 4 D -2 8 8 0 10 E -16 0 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000061 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 2 16 B -12 0 0 -8 0 C 6 0 0 -8 4 D -2 8 8 0 10 E -16 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000025 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=23 B=23 A=18 E=9 so E is eliminated. Round 2 votes counts: C=28 B=26 D=24 A=22 so A is eliminated. Round 3 votes counts: D=41 C=32 B=27 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:212 D:212 C:201 B:190 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -6 2 16 B -12 0 0 -8 0 C 6 0 0 -8 4 D -2 8 8 0 10 E -16 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000025 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 2 16 B -12 0 0 -8 0 C 6 0 0 -8 4 D -2 8 8 0 10 E -16 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000025 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 2 16 B -12 0 0 -8 0 C 6 0 0 -8 4 D -2 8 8 0 10 E -16 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000025 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3334: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) D B C A E (6) D B A C E (6) C A B E D (5) B D C A E (5) D B E C A (4) C B A D E (4) E D A C B (3) E C A D B (3) D E B A C (3) C E A B D (3) B C D A E (3) E D C B A (2) E D A B C (2) E A D C B (2) E A D B C (2) D E B C A (2) D B E A C (2) D B C E A (2) D A B E C (2) C E B D A (2) C E B A D (2) B C A D E (2) A E C B D (2) A C B E D (2) A C B D E (2) A B C D E (2) E D C A B (1) E D B A C (1) E C D B A (1) E C A B D (1) E A C D B (1) E A C B D (1) D E A B C (1) C B D E A (1) C B D A E (1) C B A E D (1) C A E B D (1) B D A C E (1) B A D C E (1) B A C D E (1) A E D B C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -4 -18 16 B 22 0 16 -10 24 C 4 -16 0 -16 10 D 18 10 16 0 20 E -16 -24 -10 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -4 -18 16 B 22 0 16 -10 24 C 4 -16 0 -16 10 D 18 10 16 0 20 E -16 -24 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=20 C=20 B=13 A=11 so A is eliminated. Round 2 votes counts: D=36 C=25 E=23 B=16 so B is eliminated. Round 3 votes counts: D=44 C=33 E=23 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:232 B:226 C:191 A:186 E:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -4 -18 16 B 22 0 16 -10 24 C 4 -16 0 -16 10 D 18 10 16 0 20 E -16 -24 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -4 -18 16 B 22 0 16 -10 24 C 4 -16 0 -16 10 D 18 10 16 0 20 E -16 -24 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -4 -18 16 B 22 0 16 -10 24 C 4 -16 0 -16 10 D 18 10 16 0 20 E -16 -24 -10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3335: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (12) D A C B E (8) E C B A D (7) E B A D C (6) D A B E C (6) D A B C E (6) C E B A D (4) C B E A D (4) A D B C E (4) E D A B C (3) E C B D A (3) C B A D E (3) E B A C D (2) D C A B E (2) D A E B C (2) C E B D A (2) C D A B E (2) B C E A D (2) A D B E C (2) A B D C E (2) E D C A B (1) E D A C B (1) E C D A B (1) E B D C A (1) E A D B C (1) D E A C B (1) D E A B C (1) D C A E B (1) D A E C B (1) D A C E B (1) C B E D A (1) B E C A D (1) B E A D C (1) B E A C D (1) B A C D E (1) A E D B C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 6 12 -12 B 2 0 14 10 -6 C -6 -14 0 -8 -14 D -12 -10 8 0 -12 E 12 6 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 12 -12 B 2 0 14 10 -6 C -6 -14 0 -8 -14 D -12 -10 8 0 -12 E 12 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=29 C=16 A=11 B=6 so B is eliminated. Round 2 votes counts: E=41 D=29 C=18 A=12 so A is eliminated. Round 3 votes counts: E=43 D=38 C=19 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:210 A:202 D:187 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 12 -12 B 2 0 14 10 -6 C -6 -14 0 -8 -14 D -12 -10 8 0 -12 E 12 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 12 -12 B 2 0 14 10 -6 C -6 -14 0 -8 -14 D -12 -10 8 0 -12 E 12 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 12 -12 B 2 0 14 10 -6 C -6 -14 0 -8 -14 D -12 -10 8 0 -12 E 12 6 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3336: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) D E B C A (6) C A E D B (5) A C B E D (5) D B E A C (4) B E C A D (4) A C D E B (4) A C D B E (4) A C B D E (4) E D B C A (3) E C A D B (3) D E A C B (3) D A C E B (3) D A B C E (3) C A E B D (3) C A D E B (3) B E C D A (3) B D E A C (3) E D C A B (2) E C D A B (2) E C A B D (2) D E C A B (2) C E A B D (2) C B A E D (2) B E D A C (2) B D A E C (2) B A D C E (2) E B D C A (1) E B C A D (1) D E B A C (1) D C A E B (1) D A C B E (1) C A B E D (1) B C A E D (1) B A C E D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 10 -12 -2 -6 B -10 0 -8 -2 2 C 12 8 0 2 -6 D 2 2 -2 0 -6 E 6 -2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999991 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 10 -12 -2 -6 B -10 0 -8 -2 2 C 12 8 0 2 -6 D 2 2 -2 0 -6 E 6 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=24 A=19 C=16 E=14 so E is eliminated. Round 2 votes counts: D=29 B=29 C=23 A=19 so A is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:208 E:208 D:198 A:195 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -12 -2 -6 B -10 0 -8 -2 2 C 12 8 0 2 -6 D 2 2 -2 0 -6 E 6 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 -2 -6 B -10 0 -8 -2 2 C 12 8 0 2 -6 D 2 2 -2 0 -6 E 6 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 -2 -6 B -10 0 -8 -2 2 C 12 8 0 2 -6 D 2 2 -2 0 -6 E 6 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3337: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E A C B (7) E D B A C (6) C A B D E (6) B E D C A (6) B C A D E (6) D C A E B (5) D A C E B (4) C A D B E (4) E D A B C (3) B E D A C (3) B E C D A (3) B E C A D (3) A C D E B (3) E D A C B (2) E B D A C (2) E B A D C (2) E A D C B (2) D E B C A (2) C D A E B (2) C B A D E (2) B C D E A (2) B A C E D (2) A E C B D (2) E D B C A (1) D E C A B (1) D C E A B (1) D B E C A (1) D B C E A (1) D A E C B (1) C A D E B (1) B C E D A (1) A D E C B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -16 -8 2 B 6 0 8 2 2 C 16 -8 0 -2 4 D 8 -2 2 0 0 E -2 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -8 2 B 6 0 8 2 2 C 16 -8 0 -2 4 D 8 -2 2 0 0 E -2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=23 E=18 C=15 A=8 so A is eliminated. Round 2 votes counts: B=36 D=24 E=20 C=20 so E is eliminated. Round 3 votes counts: B=40 D=38 C=22 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:205 D:204 E:196 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -16 -8 2 B 6 0 8 2 2 C 16 -8 0 -2 4 D 8 -2 2 0 0 E -2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -8 2 B 6 0 8 2 2 C 16 -8 0 -2 4 D 8 -2 2 0 0 E -2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -8 2 B 6 0 8 2 2 C 16 -8 0 -2 4 D 8 -2 2 0 0 E -2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3338: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) E D C A B (8) C B A D E (7) D E C B A (6) A E B D C (6) E D C B A (5) E D A C B (5) E D A B C (5) B A C D E (5) A B C D E (5) E A D B C (4) C D E B A (4) C D B E A (4) D C E B A (3) C B D A E (3) A B E C D (3) E D B C A (2) E A B D C (2) B C A D E (2) A B E D C (2) E D B A C (1) E A D C B (1) C D E A B (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D E A (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 10 2 -2 -4 B -10 0 -4 -6 -8 C -2 4 0 0 0 D 2 6 0 0 -10 E 4 8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.439942 D: 0.000000 E: 0.560058 Sum of squares = 0.507213920052 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.439942 D: 0.439942 E: 1.000000 A B C D E A 0 10 2 -2 -4 B -10 0 -4 -6 -8 C -2 4 0 0 0 D 2 6 0 0 -10 E 4 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=27 C=23 D=9 B=8 so B is eliminated. Round 2 votes counts: E=33 A=33 C=25 D=9 so D is eliminated. Round 3 votes counts: E=39 A=33 C=28 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:203 C:201 D:199 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 -2 -4 B -10 0 -4 -6 -8 C -2 4 0 0 0 D 2 6 0 0 -10 E 4 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -2 -4 B -10 0 -4 -6 -8 C -2 4 0 0 0 D 2 6 0 0 -10 E 4 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -2 -4 B -10 0 -4 -6 -8 C -2 4 0 0 0 D 2 6 0 0 -10 E 4 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3339: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) C E B A D (6) A E D B C (6) D B A C E (5) C E A D B (5) C B E D A (5) B C E D A (5) A E D C B (4) E C A B D (3) E A B C D (3) D A B E C (3) B D C E A (3) A E C D B (3) E A C D B (2) D B C A E (2) D B A E C (2) D A E C B (2) D A C B E (2) C E A B D (2) C B E A D (2) C B D E A (2) B D C A E (2) B D A E C (2) A D E B C (2) A D C E B (2) E C A D B (1) E B A C D (1) D C B A E (1) D A E B C (1) D A C E B (1) C E D B A (1) C D B A E (1) C D A B E (1) B E D A C (1) B E C A D (1) B E A D C (1) B D E A C (1) B D A C E (1) B C E A D (1) B C D E A (1) A E B D C (1) A D E C B (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 10 10 -6 B -6 0 -8 2 -8 C -10 8 0 6 4 D -10 -2 -6 0 -24 E 6 8 -4 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.500000 Sum of squares = 0.379999999998 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 6 10 10 -6 B -6 0 -8 2 -8 C -10 8 0 6 4 D -10 -2 -6 0 -24 E 6 8 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000094 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=21 D=19 B=19 E=16 so E is eliminated. Round 2 votes counts: A=32 C=29 B=20 D=19 so D is eliminated. Round 3 votes counts: A=41 C=30 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:217 A:210 C:204 B:190 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 10 10 -6 B -6 0 -8 2 -8 C -10 8 0 6 4 D -10 -2 -6 0 -24 E 6 8 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000094 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 10 -6 B -6 0 -8 2 -8 C -10 8 0 6 4 D -10 -2 -6 0 -24 E 6 8 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000094 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 10 -6 B -6 0 -8 2 -8 C -10 8 0 6 4 D -10 -2 -6 0 -24 E 6 8 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.500000 Sum of squares = 0.380000000094 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3340: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) E A C D B (7) D C A B E (7) B E A D C (6) B D C A E (6) A D C E B (6) B A E D C (5) C D E A B (4) B E D C A (4) E B A C D (3) D C B A E (3) A E D C B (3) E C D A B (2) E C A D B (2) E A B C D (2) C A D E B (2) B E C D A (2) B E A C D (2) B A D C E (2) A D C B E (2) A C D E B (2) A B E D C (2) A B D C E (2) E C D B A (1) E B C D A (1) E B A D C (1) E A B D C (1) D C A E B (1) C E D A B (1) C D B A E (1) B D C E A (1) B C D E A (1) A E C D B (1) Total count = 100 A B C D E A 0 22 -6 2 18 B -22 0 -18 -18 -8 C 6 18 0 -4 10 D -2 18 4 0 8 E -18 8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888986 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -6 2 18 B -22 0 -18 -18 -8 C 6 18 0 -4 10 D -2 18 4 0 8 E -18 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888887856 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 E=20 A=18 D=11 so D is eliminated. Round 2 votes counts: C=33 B=29 E=20 A=18 so A is eliminated. Round 3 votes counts: C=43 B=33 E=24 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:218 C:215 D:214 E:186 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 -6 2 18 B -22 0 -18 -18 -8 C 6 18 0 -4 10 D -2 18 4 0 8 E -18 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888887856 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -6 2 18 B -22 0 -18 -18 -8 C 6 18 0 -4 10 D -2 18 4 0 8 E -18 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888887856 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -6 2 18 B -22 0 -18 -18 -8 C 6 18 0 -4 10 D -2 18 4 0 8 E -18 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888887856 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3341: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) B D A E C (5) B A D C E (5) A C D B E (5) E D C A B (4) E D B C A (4) C A E D B (4) A C B D E (4) E D C B A (3) E C D A B (3) D B E A C (3) D A C E B (3) C A D E B (3) B E D C A (3) B E C A D (3) B E A C D (3) B A C E D (3) E B D C A (2) D E C A B (2) D E B A C (2) D E A C B (2) D C A E B (2) B E D A C (2) B D E A C (2) A C B E D (2) A B D C E (2) E D B A C (1) E B C D A (1) D E A B C (1) D C E A B (1) D B A E C (1) D B A C E (1) D A E B C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E D A B (1) C E A D B (1) B E A D C (1) B C A E D (1) A D C B E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 24 -6 10 B 6 0 14 -6 16 C -24 -14 0 -14 0 D 6 6 14 0 16 E -10 -16 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 24 -6 10 B 6 0 14 -6 16 C -24 -14 0 -14 0 D 6 6 14 0 16 E -10 -16 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=22 E=18 A=16 C=9 so C is eliminated. Round 2 votes counts: B=35 A=23 D=22 E=20 so E is eliminated. Round 3 votes counts: D=38 B=38 A=24 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:215 A:211 E:179 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 24 -6 10 B 6 0 14 -6 16 C -24 -14 0 -14 0 D 6 6 14 0 16 E -10 -16 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 24 -6 10 B 6 0 14 -6 16 C -24 -14 0 -14 0 D 6 6 14 0 16 E -10 -16 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 24 -6 10 B 6 0 14 -6 16 C -24 -14 0 -14 0 D 6 6 14 0 16 E -10 -16 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3342: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) C D B A E (7) C B D A E (7) E D A C B (6) B A E C D (6) B A C E D (6) E A B D C (5) D C E B A (5) E A D B C (4) D E C A B (4) B C A D E (4) A B E C D (4) E A B C D (3) C B A D E (3) B E A C D (3) A E B C D (3) E D A B C (2) D E A C B (2) D C B E A (2) D C B A E (2) B C D A E (2) B C A E D (2) E D B C A (1) E D B A C (1) E B D A C (1) D C A B E (1) C D B E A (1) C D A B E (1) B C E D A (1) B A C D E (1) A E D C B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 0 -6 4 B 10 0 0 4 8 C 0 0 0 12 8 D 6 -4 -12 0 0 E -4 -8 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.592236 C: 0.407764 D: 0.000000 E: 0.000000 Sum of squares = 0.517015092407 Cumulative probabilities = A: 0.000000 B: 0.592236 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -6 4 B 10 0 0 4 8 C 0 0 0 12 8 D 6 -4 -12 0 0 E -4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 D=23 C=19 A=10 so A is eliminated. Round 2 votes counts: B=30 E=27 D=23 C=20 so C is eliminated. Round 3 votes counts: B=40 D=33 E=27 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:210 D:195 A:194 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -6 4 B 10 0 0 4 8 C 0 0 0 12 8 D 6 -4 -12 0 0 E -4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -6 4 B 10 0 0 4 8 C 0 0 0 12 8 D 6 -4 -12 0 0 E -4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -6 4 B 10 0 0 4 8 C 0 0 0 12 8 D 6 -4 -12 0 0 E -4 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3343: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) C E D A B (8) D B E C A (7) B A D E C (7) D E C A B (5) C A E D B (5) B A E D C (5) A C E D B (5) A B E C D (5) B D E C A (4) C E A D B (3) A B C E D (3) E D C A B (2) D E C B A (2) D C E A B (2) C A D E B (2) B E D A C (2) B E A D C (2) B D E A C (2) B A E C D (2) A E C B D (2) E C D A B (1) E A C D B (1) D E B C A (1) D C E B A (1) D B C E A (1) C D E B A (1) C D A E B (1) C A D B E (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) B A C E D (1) A E B C D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 16 -16 -6 -10 B -16 0 -6 -16 -6 C 16 6 0 6 -4 D 6 16 -6 0 2 E 10 6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 1.000000 A B C D E A 0 16 -16 -6 -10 B -16 0 -6 -16 -6 C 16 6 0 6 -4 D 6 16 -6 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=29 D=19 A=18 E=4 so E is eliminated. Round 2 votes counts: C=31 B=29 D=21 A=19 so A is eliminated. Round 3 votes counts: C=41 B=38 D=21 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:209 E:209 A:192 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -16 -6 -10 B -16 0 -6 -16 -6 C 16 6 0 6 -4 D 6 16 -6 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -16 -6 -10 B -16 0 -6 -16 -6 C 16 6 0 6 -4 D 6 16 -6 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -16 -6 -10 B -16 0 -6 -16 -6 C 16 6 0 6 -4 D 6 16 -6 0 2 E 10 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3344: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) C D E B A (7) D E C B A (6) D A C E B (5) B A E C D (5) A B E D C (5) C E B D A (4) C D E A B (4) C D A E B (4) A D C E B (4) A B D E C (4) E D B C A (3) D E B C A (3) D C A E B (3) C E D B A (3) C A B D E (3) E C D B A (2) D E B A C (2) D C E A B (2) B E D A C (2) B E A C D (2) B A E D C (2) A D C B E (2) A D B E C (2) A C B E D (2) A B E C D (2) E D C B A (1) E D B A C (1) C A D E B (1) C A B E D (1) B E C D A (1) B E A D C (1) A D E B C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -12 -22 -4 B 6 0 -10 -16 -28 C 12 10 0 -16 -6 D 22 16 16 0 10 E 4 28 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -22 -4 B 6 0 -10 -16 -28 C 12 10 0 -16 -6 D 22 16 16 0 10 E 4 28 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 D=21 E=15 B=13 so B is eliminated. Round 2 votes counts: A=31 C=27 E=21 D=21 so E is eliminated. Round 3 votes counts: D=36 A=34 C=30 so C is eliminated. Round 4 votes counts: D=61 A=39 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:232 E:214 C:200 A:178 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -22 -4 B 6 0 -10 -16 -28 C 12 10 0 -16 -6 D 22 16 16 0 10 E 4 28 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -22 -4 B 6 0 -10 -16 -28 C 12 10 0 -16 -6 D 22 16 16 0 10 E 4 28 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -22 -4 B 6 0 -10 -16 -28 C 12 10 0 -16 -6 D 22 16 16 0 10 E 4 28 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3345: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) D C A E B (8) C D B A E (8) D C B A E (6) D C A B E (6) A E B C D (6) E D A C B (4) E B A C D (4) E A D C B (4) E A B C D (4) B E A C D (4) A B E C D (4) E D C A B (3) E B A D C (3) D C B E A (3) E A D B C (2) D E A C B (2) B C D A E (2) B A E C D (2) B A C D E (2) A B C D E (2) D E C A B (1) D C E B A (1) D C E A B (1) D B C E A (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) B C D E A (1) B A C E D (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 20 10 2 8 B -20 0 -8 -8 -4 C -10 8 0 -6 -2 D -2 8 6 0 2 E -8 4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 2 8 B -20 0 -8 -8 -4 C -10 8 0 -6 -2 D -2 8 6 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=29 A=15 C=12 B=12 so C is eliminated. Round 2 votes counts: D=38 E=32 A=16 B=14 so B is eliminated. Round 3 votes counts: D=42 E=36 A=22 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:220 D:207 E:198 C:195 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 2 8 B -20 0 -8 -8 -4 C -10 8 0 -6 -2 D -2 8 6 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 2 8 B -20 0 -8 -8 -4 C -10 8 0 -6 -2 D -2 8 6 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 2 8 B -20 0 -8 -8 -4 C -10 8 0 -6 -2 D -2 8 6 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3346: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (14) E A D B C (13) C D B A E (5) E A D C B (4) E A C D B (4) D A B C E (4) C B D E A (4) A D C B E (4) E C B A D (3) C E B A D (3) A D B E C (3) E C A D B (2) E A B D C (2) D B C A E (2) D B A C E (2) D A B E C (2) C E B D A (2) C D A B E (2) C B E D A (2) B E D A C (2) B C E D A (2) A E D C B (2) A D E C B (2) E C A B D (1) E B A D C (1) E A C B D (1) D C B A E (1) D B A E C (1) D A C B E (1) C E A D B (1) C E A B D (1) C A D E B (1) B E D C A (1) B D C A E (1) B D A E C (1) B C D A E (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 2 0 2 B -4 0 -20 -16 10 C -2 20 0 -2 6 D 0 16 2 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.603212 B: 0.000000 C: 0.000000 D: 0.396788 E: 0.000000 Sum of squares = 0.521305471815 Cumulative probabilities = A: 0.603212 B: 0.603212 C: 0.603212 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 0 2 B -4 0 -20 -16 10 C -2 20 0 -2 6 D 0 16 2 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=31 D=13 A=13 B=8 so B is eliminated. Round 2 votes counts: C=38 E=34 D=15 A=13 so A is eliminated. Round 3 votes counts: C=38 E=37 D=25 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:211 A:204 E:189 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 0 2 B -4 0 -20 -16 10 C -2 20 0 -2 6 D 0 16 2 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 0 2 B -4 0 -20 -16 10 C -2 20 0 -2 6 D 0 16 2 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 0 2 B -4 0 -20 -16 10 C -2 20 0 -2 6 D 0 16 2 0 4 E -2 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3347: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) B C D E A (8) B C E D A (6) A E D C B (6) A D E C B (6) C B D A E (5) C B E A D (4) B C D A E (4) E B A C D (3) E A B D C (3) D C A B E (3) D A C B E (3) B E C A D (3) E C A B D (2) E B D A C (2) E B C A D (2) E A C B D (2) E A B C D (2) D A B C E (2) C D B A E (2) C B D E A (2) B E C D A (2) A D C E B (2) E D A B C (1) E C B A D (1) E B C D A (1) E B A D C (1) E A D C B (1) E A C D B (1) D E A B C (1) D C B A E (1) D A E B C (1) D A C E B (1) C D A B E (1) C B E D A (1) C A E D B (1) C A D E B (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 0 6 -18 B -4 0 8 8 -6 C 0 -8 0 8 -6 D -6 -8 -8 0 -12 E 18 6 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 6 -18 B -4 0 8 8 -6 C 0 -8 0 8 -6 D -6 -8 -8 0 -12 E 18 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=23 C=17 A=16 D=12 so D is eliminated. Round 2 votes counts: E=33 B=23 A=23 C=21 so C is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:203 C:197 A:196 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 6 -18 B -4 0 8 8 -6 C 0 -8 0 8 -6 D -6 -8 -8 0 -12 E 18 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 -18 B -4 0 8 8 -6 C 0 -8 0 8 -6 D -6 -8 -8 0 -12 E 18 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 -18 B -4 0 8 8 -6 C 0 -8 0 8 -6 D -6 -8 -8 0 -12 E 18 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3348: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) A C D B E (7) C B A E D (6) E B D C A (5) C A D B E (5) C B E A D (4) C A D E B (4) B E D A C (4) A D C B E (4) E D B C A (3) E C B D A (3) E B C D A (3) C A B D E (3) B E D C A (3) B D E A C (3) A D C E B (3) E D B A C (2) E B D A C (2) D E A B C (2) D A E C B (2) C A E D B (2) C A B E D (2) B E C D A (2) A D B E C (2) E D A B C (1) E C D B A (1) D E B A C (1) D E A C B (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) C E A B D (1) C B A D E (1) C A E B D (1) B E C A D (1) B C A E D (1) B C A D E (1) B A D E C (1) B A C D E (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -22 18 -2 B 14 0 -6 14 18 C 22 6 0 18 12 D -18 -14 -18 0 -12 E 2 -18 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -22 18 -2 B 14 0 -6 14 18 C 22 6 0 18 12 D -18 -14 -18 0 -12 E 2 -18 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=24 E=20 A=17 D=8 so D is eliminated. Round 2 votes counts: C=31 E=24 B=24 A=21 so A is eliminated. Round 3 votes counts: C=46 E=27 B=27 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:220 E:192 A:190 D:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -22 18 -2 B 14 0 -6 14 18 C 22 6 0 18 12 D -18 -14 -18 0 -12 E 2 -18 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -22 18 -2 B 14 0 -6 14 18 C 22 6 0 18 12 D -18 -14 -18 0 -12 E 2 -18 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -22 18 -2 B 14 0 -6 14 18 C 22 6 0 18 12 D -18 -14 -18 0 -12 E 2 -18 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3349: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) D B E C A (9) C A E B D (9) A C E B D (8) D B E A C (7) C A B E D (7) A E C B D (5) C D B A E (4) E A B D C (3) D C B E A (3) D B C E A (3) C D B E A (3) C A D B E (3) B D E C A (3) A E B C D (3) C A B D E (2) A E D B C (2) A E B D C (2) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) D A E B C (1) C D A E B (1) C D A B E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A E D B (1) C A D E B (1) B E C D A (1) B D E A C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -8 0 6 B -2 0 -4 -2 0 C 8 4 0 6 0 D 0 2 -6 0 6 E -6 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.645141 D: 0.000000 E: 0.354859 Sum of squares = 0.542131628004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.645141 D: 0.645141 E: 1.000000 A B C D E A 0 2 -8 0 6 B -2 0 -4 -2 0 C 8 4 0 6 0 D 0 2 -6 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.000000 E: 0.499046 Sum of squares = 0.50000182093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.500954 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=32 A=21 E=7 B=5 so B is eliminated. Round 2 votes counts: D=36 C=35 A=21 E=8 so E is eliminated. Round 3 votes counts: D=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:201 A:200 B:196 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 0 6 B -2 0 -4 -2 0 C 8 4 0 6 0 D 0 2 -6 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.000000 E: 0.499046 Sum of squares = 0.50000182093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.500954 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 0 6 B -2 0 -4 -2 0 C 8 4 0 6 0 D 0 2 -6 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.000000 E: 0.499046 Sum of squares = 0.50000182093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.500954 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 0 6 B -2 0 -4 -2 0 C 8 4 0 6 0 D 0 2 -6 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.000000 E: 0.499046 Sum of squares = 0.50000182093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500954 D: 0.500954 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3350: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) D A C E B (6) C A B D E (6) B E A C D (6) D C A E B (5) C A D B E (5) D C E A B (4) C D B A E (4) B E C A D (4) E D B A C (3) E D A B C (3) E B A C D (3) D E A B C (3) D C A B E (3) C D A B E (3) B C A E D (3) B A C E D (3) D E C A B (2) D E B C A (2) D A E C B (2) C B D A E (2) C B A E D (2) B E C D A (2) B C E A D (2) B A E C D (2) E B D C A (1) E A D B C (1) E A B C D (1) D E A C B (1) D C E B A (1) D C B E A (1) C B E A D (1) C B D E A (1) C B A D E (1) A E B D C (1) A D C E B (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -10 2 4 B 0 0 -6 -2 8 C 10 6 0 8 14 D -2 2 -8 0 10 E -4 -8 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 2 4 B 0 0 -6 -2 8 C 10 6 0 8 14 D -2 2 -8 0 10 E -4 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=25 B=22 E=18 A=5 so A is eliminated. Round 2 votes counts: D=31 C=27 B=23 E=19 so E is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:219 D:201 B:200 A:198 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 2 4 B 0 0 -6 -2 8 C 10 6 0 8 14 D -2 2 -8 0 10 E -4 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 2 4 B 0 0 -6 -2 8 C 10 6 0 8 14 D -2 2 -8 0 10 E -4 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 2 4 B 0 0 -6 -2 8 C 10 6 0 8 14 D -2 2 -8 0 10 E -4 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3351: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (6) D C B E A (5) D C E B A (4) B C A D E (4) D E B A C (3) D C B A E (3) D B E C A (3) C D E A B (3) C A E D B (3) C A B E D (3) B E D A C (3) B D E A C (3) B A E C D (3) A C E B D (3) A B E C D (3) E A C B D (2) E A B D C (2) D E C B A (2) D E C A B (2) D C E A B (2) D B E A C (2) C D B A E (2) C B D A E (2) C B A D E (2) C A B D E (2) A E C B D (2) A B C E D (2) E D C A B (1) E D B A C (1) E D A C B (1) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D E B C A (1) D B C E A (1) D B C A E (1) C E D A B (1) C E A D B (1) C D A E B (1) C D A B E (1) C A E B D (1) C A D B E (1) B E A D C (1) B D C A E (1) B A E D C (1) B A D E C (1) B A C E D (1) B A C D E (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -6 -2 4 B 6 0 -4 6 6 C 6 4 0 6 4 D 2 -6 -6 0 6 E -4 -6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -2 4 B 6 0 -4 6 6 C 6 4 0 6 4 D 2 -6 -6 0 6 E -4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=23 B=19 A=18 E=11 so E is eliminated. Round 2 votes counts: D=33 C=23 A=23 B=21 so B is eliminated. Round 3 votes counts: D=41 A=32 C=27 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:210 B:207 D:198 A:195 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -2 4 B 6 0 -4 6 6 C 6 4 0 6 4 D 2 -6 -6 0 6 E -4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -2 4 B 6 0 -4 6 6 C 6 4 0 6 4 D 2 -6 -6 0 6 E -4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -2 4 B 6 0 -4 6 6 C 6 4 0 6 4 D 2 -6 -6 0 6 E -4 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3352: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) D E C B A (8) A B C E D (8) C A B E D (7) A B D E C (7) D E B A C (6) C D E B A (6) C D E A B (4) B E A D C (4) C E B D A (3) B A E C D (3) A D E B C (3) A C B E D (3) A B C D E (3) D E A B C (2) D A E B C (2) D A C E B (2) C B E D A (2) B A E D C (2) A C B D E (2) A B E C D (2) E D C B A (1) E B D C A (1) D E A C B (1) D C E B A (1) D A E C B (1) C E D B A (1) C D A E B (1) C A D B E (1) C A B D E (1) B E C D A (1) A D B E C (1) Total count = 100 A B C D E A 0 22 24 14 18 B -22 0 10 18 14 C -24 -10 0 -4 -10 D -14 -18 4 0 4 E -18 -14 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 24 14 18 B -22 0 10 18 14 C -24 -10 0 -4 -10 D -14 -18 4 0 4 E -18 -14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=26 D=23 B=10 E=2 so E is eliminated. Round 2 votes counts: A=39 C=26 D=24 B=11 so B is eliminated. Round 3 votes counts: A=48 C=27 D=25 so D is eliminated. Round 4 votes counts: A=62 C=38 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:239 B:210 D:188 E:187 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 24 14 18 B -22 0 10 18 14 C -24 -10 0 -4 -10 D -14 -18 4 0 4 E -18 -14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 24 14 18 B -22 0 10 18 14 C -24 -10 0 -4 -10 D -14 -18 4 0 4 E -18 -14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 24 14 18 B -22 0 10 18 14 C -24 -10 0 -4 -10 D -14 -18 4 0 4 E -18 -14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3353: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) C B A D E (6) E D A B C (5) E B A D C (4) D E C A B (4) C E D B A (4) C E B D A (4) C B E D A (4) B A E D C (4) B A C E D (4) A D E B C (4) A B E D C (4) D A E C B (3) C D E A B (3) B E A C D (3) E D A C B (2) E A D B C (2) D C E A B (2) D A C E B (2) C D E B A (2) C B D A E (2) B C A E D (2) B C A D E (2) A E D B C (2) A D C B E (2) A B D E C (2) E D C A B (1) E D B C A (1) E B A C D (1) D E A C B (1) D C A E B (1) D A E B C (1) C D A B E (1) C B D E A (1) C A D B E (1) B E A D C (1) B C E D A (1) B C E A D (1) B A E C D (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 14 10 6 B 10 0 8 10 0 C -14 -8 0 0 4 D -10 -10 0 0 -4 E -6 0 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.698831 C: 0.000000 D: 0.000000 E: 0.301169 Sum of squares = 0.579067550346 Cumulative probabilities = A: 0.000000 B: 0.698831 C: 0.698831 D: 0.698831 E: 1.000000 A B C D E A 0 -10 14 10 6 B 10 0 8 10 0 C -14 -8 0 0 4 D -10 -10 0 0 -4 E -6 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=26 E=16 A=16 D=14 so D is eliminated. Round 2 votes counts: C=31 B=26 A=22 E=21 so E is eliminated. Round 3 votes counts: C=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:214 A:210 E:197 C:191 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 10 6 B 10 0 8 10 0 C -14 -8 0 0 4 D -10 -10 0 0 -4 E -6 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 10 6 B 10 0 8 10 0 C -14 -8 0 0 4 D -10 -10 0 0 -4 E -6 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 10 6 B 10 0 8 10 0 C -14 -8 0 0 4 D -10 -10 0 0 -4 E -6 0 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3354: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) E D A B C (7) C B A E D (6) E D C A B (5) A B D E C (5) D E C A B (4) D A E B C (4) C D E B A (4) B A C E D (4) B A C D E (4) A B E D C (4) E C D B A (3) C E B D A (3) C B E A D (3) B A E C D (3) E A D B C (2) E A B D C (2) D C E A B (2) D A C E B (2) D A C B E (2) C E D B A (2) C D B E A (2) C B A D E (2) B C A E D (2) A E B D C (2) E D A C B (1) E B A C D (1) D E A C B (1) D C A B E (1) D A E C B (1) D A B C E (1) C D B A E (1) C B E D A (1) C B D E A (1) C B D A E (1) B C A D E (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 10 12 -14 -4 B -10 0 4 -8 -8 C -12 -4 0 -12 -10 D 14 8 12 0 -4 E 4 8 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 12 -14 -4 B -10 0 4 -8 -8 C -12 -4 0 -12 -10 D 14 8 12 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 E=21 B=15 A=12 so A is eliminated. Round 2 votes counts: D=27 C=26 B=24 E=23 so E is eliminated. Round 3 votes counts: D=42 C=29 B=29 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:213 A:202 B:189 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 12 -14 -4 B -10 0 4 -8 -8 C -12 -4 0 -12 -10 D 14 8 12 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -14 -4 B -10 0 4 -8 -8 C -12 -4 0 -12 -10 D 14 8 12 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -14 -4 B -10 0 4 -8 -8 C -12 -4 0 -12 -10 D 14 8 12 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3355: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) D B C A E (6) B E C A D (5) B C D A E (5) E A C D B (4) E A B D C (4) B C E A D (4) E B A D C (3) E A C B D (3) E A B C D (3) D C B A E (3) D C A E B (3) D A E C B (3) C D A B E (3) C B D A E (3) B E D A C (3) B D C E A (3) B D C A E (3) A E C D B (3) D C A B E (2) D A C E B (2) B D E C A (2) B C D E A (2) A C E D B (2) A C D E B (2) E D A B C (1) E B D A C (1) E B A C D (1) D E B A C (1) D E A B C (1) D B E A C (1) C E B A D (1) C D A E B (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B E D (1) B E A C D (1) B D E A C (1) B C E D A (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 -4 -2 -4 B 2 0 4 4 0 C 4 -4 0 -2 4 D 2 -4 2 0 2 E 4 0 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.677660 C: 0.000000 D: 0.000000 E: 0.322340 Sum of squares = 0.563126198471 Cumulative probabilities = A: 0.000000 B: 0.677660 C: 0.677660 D: 0.677660 E: 1.000000 A B C D E A 0 -2 -4 -2 -4 B 2 0 4 4 0 C 4 -4 0 -2 4 D 2 -4 2 0 2 E 4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500108 C: 0.000000 D: 0.000000 E: 0.499892 Sum of squares = 0.500000023434 Cumulative probabilities = A: 0.000000 B: 0.500108 C: 0.500108 D: 0.500108 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 D=22 C=13 A=9 so A is eliminated. Round 2 votes counts: B=30 E=29 D=24 C=17 so C is eliminated. Round 3 votes counts: B=35 E=34 D=31 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:205 C:201 D:201 E:199 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 -4 B 2 0 4 4 0 C 4 -4 0 -2 4 D 2 -4 2 0 2 E 4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500108 C: 0.000000 D: 0.000000 E: 0.499892 Sum of squares = 0.500000023434 Cumulative probabilities = A: 0.000000 B: 0.500108 C: 0.500108 D: 0.500108 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 -4 B 2 0 4 4 0 C 4 -4 0 -2 4 D 2 -4 2 0 2 E 4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500108 C: 0.000000 D: 0.000000 E: 0.499892 Sum of squares = 0.500000023434 Cumulative probabilities = A: 0.000000 B: 0.500108 C: 0.500108 D: 0.500108 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 -4 B 2 0 4 4 0 C 4 -4 0 -2 4 D 2 -4 2 0 2 E 4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500108 C: 0.000000 D: 0.000000 E: 0.499892 Sum of squares = 0.500000023434 Cumulative probabilities = A: 0.000000 B: 0.500108 C: 0.500108 D: 0.500108 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3356: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) D C A B E (5) E D C A B (4) E D B A C (4) D E B A C (4) D C E A B (4) D B E A C (4) C E A B D (4) C A B E D (4) E D C B A (3) E C D A B (3) E C A B D (3) B D E A C (3) B D A E C (3) B A E C D (3) B A C E D (3) A B C E D (3) E D B C A (2) E B D A C (2) E B A D C (2) E B A C D (2) D E C B A (2) D E B C A (2) D B A C E (2) C E A D B (2) C D E A B (2) C D A E B (2) C A E B D (2) B A D C E (2) B A C D E (2) A C B D E (2) E C B D A (1) E B C A D (1) D C A E B (1) D B A E C (1) C A D B E (1) C A B D E (1) B D A C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -12 -24 -22 B -2 0 -6 -10 -18 C 12 6 0 -16 -12 D 24 10 16 0 4 E 22 18 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -24 -22 B -2 0 -6 -10 -18 C 12 6 0 -16 -12 D 24 10 16 0 4 E 22 18 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=27 C=18 B=17 A=6 so A is eliminated. Round 2 votes counts: D=32 E=27 B=21 C=20 so C is eliminated. Round 3 votes counts: D=37 E=35 B=28 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:224 C:195 B:182 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -24 -22 B -2 0 -6 -10 -18 C 12 6 0 -16 -12 D 24 10 16 0 4 E 22 18 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -24 -22 B -2 0 -6 -10 -18 C 12 6 0 -16 -12 D 24 10 16 0 4 E 22 18 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -24 -22 B -2 0 -6 -10 -18 C 12 6 0 -16 -12 D 24 10 16 0 4 E 22 18 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3357: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) E B D C A (6) C A E B D (6) D B E A C (5) C B E A D (4) B E D C A (4) A D C E B (4) E D B C A (3) E B C D A (3) D E A B C (3) D A E C B (3) D A B E C (3) C A B E D (3) B C E A D (3) A D C B E (3) A C E D B (3) A C B D E (3) E D A C B (2) E C B D A (2) D E B C A (2) D E B A C (2) D B A E C (2) D A E B C (2) D A C B E (2) D A B C E (2) C E B A D (2) C E A B D (2) C B A E D (2) B E C D A (2) B D E C A (2) A C D B E (2) A B D C E (2) E D C A B (1) B D C E A (1) B D A E C (1) B D A C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 -8 -10 B 6 0 0 12 -4 C 8 0 0 -12 -8 D 8 -12 12 0 -8 E 10 4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -8 -10 B 6 0 0 12 -4 C 8 0 0 -12 -8 D 8 -12 12 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 C=19 A=18 B=14 so B is eliminated. Round 2 votes counts: D=31 E=29 C=22 A=18 so A is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:215 B:207 D:200 C:194 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -8 -10 B 6 0 0 12 -4 C 8 0 0 -12 -8 D 8 -12 12 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -8 -10 B 6 0 0 12 -4 C 8 0 0 -12 -8 D 8 -12 12 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -8 -10 B 6 0 0 12 -4 C 8 0 0 -12 -8 D 8 -12 12 0 -8 E 10 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3358: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A D E C B (7) A B C D E (7) A D E B C (6) B C A E D (5) D E A B C (4) B C E D A (4) A C B D E (4) E B D C A (3) D E C B A (3) D E A C B (3) D A E C B (3) C B A E D (3) C B A D E (3) B C E A D (3) A D B E C (3) E D C B A (2) D E C A B (2) D A E B C (2) C D A E B (2) C B E D A (2) C A B D E (2) B A C E D (2) A D B C E (2) A C D B E (2) E D B A C (1) E D A B C (1) E B C D A (1) D C E B A (1) D C E A B (1) D C A E B (1) D A C E B (1) C D E B A (1) C A D B E (1) B E C D A (1) A E D B C (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 14 2 6 18 B -14 0 10 -18 -10 C -2 -10 0 -14 -2 D -6 18 14 0 26 E -18 10 2 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 6 18 B -14 0 10 -18 -10 C -2 -10 0 -14 -2 D -6 18 14 0 26 E -18 10 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=21 E=16 B=15 C=14 so C is eliminated. Round 2 votes counts: A=37 D=24 B=23 E=16 so E is eliminated. Round 3 votes counts: A=37 D=36 B=27 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:226 A:220 C:186 B:184 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 6 18 B -14 0 10 -18 -10 C -2 -10 0 -14 -2 D -6 18 14 0 26 E -18 10 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 6 18 B -14 0 10 -18 -10 C -2 -10 0 -14 -2 D -6 18 14 0 26 E -18 10 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 6 18 B -14 0 10 -18 -10 C -2 -10 0 -14 -2 D -6 18 14 0 26 E -18 10 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3359: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) E C D B A (7) C B A D E (7) D A B E C (5) A B D C E (5) A B C D E (5) E D A B C (4) D B A E C (4) C E A B D (4) E D B C A (3) E C A B D (3) E A C B D (3) D B C A E (3) A E B D C (3) E A D B C (2) E A C D B (2) E A B C D (2) D C B A E (2) C E B A D (2) C A E B D (2) B D A C E (2) A D B E C (2) A C B E D (2) E D A C B (1) E C D A B (1) E A B D C (1) D E B C A (1) D E B A C (1) D A B C E (1) C E D B A (1) C E B D A (1) C D B A E (1) C A B E D (1) C A B D E (1) B C A D E (1) B A D C E (1) B A C D E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 18 4 22 B -4 0 14 -2 10 C -18 -14 0 -2 10 D -4 2 2 0 8 E -22 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 4 22 B -4 0 14 -2 10 C -18 -14 0 -2 10 D -4 2 2 0 8 E -22 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992185 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=27 C=20 A=19 B=5 so B is eliminated. Round 2 votes counts: E=29 D=29 C=21 A=21 so C is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:209 D:204 C:188 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 18 4 22 B -4 0 14 -2 10 C -18 -14 0 -2 10 D -4 2 2 0 8 E -22 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992185 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 4 22 B -4 0 14 -2 10 C -18 -14 0 -2 10 D -4 2 2 0 8 E -22 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992185 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 4 22 B -4 0 14 -2 10 C -18 -14 0 -2 10 D -4 2 2 0 8 E -22 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992185 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3360: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) C D B E A (9) E B C D A (6) A E C D B (6) A B E D C (6) D C B E A (5) D C B A E (5) B D C E A (5) E A B C D (4) C D A B E (4) B E D C A (4) E B A D C (3) E B A C D (3) C D A E B (3) B D C A E (3) B A E D C (3) E A C D B (2) C D E B A (2) B D A C E (2) A E B C D (2) A D C B E (2) E C D B A (1) E C D A B (1) E B D C A (1) D B C A E (1) C D E A B (1) B C D E A (1) B A D C E (1) A D B C E (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 -8 4 B 10 0 14 10 6 C 4 -14 0 -6 -4 D 8 -10 6 0 -6 E -4 -6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -8 4 B 10 0 14 10 6 C 4 -14 0 -6 -4 D 8 -10 6 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=21 C=19 B=19 D=11 so D is eliminated. Round 2 votes counts: A=30 C=29 E=21 B=20 so B is eliminated. Round 3 votes counts: C=39 A=36 E=25 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:220 E:200 D:199 A:191 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -8 4 B 10 0 14 10 6 C 4 -14 0 -6 -4 D 8 -10 6 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -8 4 B 10 0 14 10 6 C 4 -14 0 -6 -4 D 8 -10 6 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -8 4 B 10 0 14 10 6 C 4 -14 0 -6 -4 D 8 -10 6 0 -6 E -4 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3361: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) E C A D B (8) B D A C E (8) E A C D B (6) D B A C E (5) C E A D B (5) B D C A E (5) E C A B D (4) D B C A E (4) C D A B E (4) B E D A C (3) B D E A C (3) E B A D C (2) E A C B D (2) D C B A E (2) C D B A E (2) C A E D B (2) C A D E B (2) B D C E A (2) B C D E A (2) A E D B C (2) A E C D B (2) A C E D B (2) E B C A D (1) E B A C D (1) D A C B E (1) D A B C E (1) C B E D A (1) B E D C A (1) B D E C A (1) B A D E C (1) A E D C B (1) A D E C B (1) A D C B E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 8 -10 16 B 8 0 8 -4 18 C -8 -8 0 -12 -2 D 10 4 12 0 14 E -16 -18 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -10 16 B 8 0 8 -4 18 C -8 -8 0 -12 -2 D 10 4 12 0 14 E -16 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=24 C=16 D=13 A=11 so A is eliminated. Round 2 votes counts: B=37 E=29 C=18 D=16 so D is eliminated. Round 3 votes counts: B=48 E=30 C=22 so C is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:215 A:203 C:185 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -10 16 B 8 0 8 -4 18 C -8 -8 0 -12 -2 D 10 4 12 0 14 E -16 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -10 16 B 8 0 8 -4 18 C -8 -8 0 -12 -2 D 10 4 12 0 14 E -16 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -10 16 B 8 0 8 -4 18 C -8 -8 0 -12 -2 D 10 4 12 0 14 E -16 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3362: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (9) D B A C E (9) E C B A D (6) D C E A B (6) D A B C E (5) A B D E C (5) C E B D A (4) C E A D B (4) B A E C D (4) B A D E C (4) E C A B D (3) D B C E A (3) C E D B A (3) C E D A B (3) A E C B D (3) C E B A D (2) C E A B D (2) C D E B A (2) B A E D C (2) A D B E C (2) E A C B D (1) D B A E C (1) D A C B E (1) C A E D B (1) B D C E A (1) B C E D A (1) A D C E B (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 0 4 0 B 2 0 -4 -2 -2 C 0 4 0 0 16 D -4 2 0 0 -2 E 0 2 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.496179 B: 0.000000 C: 0.503821 D: 0.000000 E: 0.000000 Sum of squares = 0.500029197407 Cumulative probabilities = A: 0.496179 B: 0.496179 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 4 0 B 2 0 -4 -2 -2 C 0 4 0 0 16 D -4 2 0 0 -2 E 0 2 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=23 C=21 B=12 E=10 so E is eliminated. Round 2 votes counts: D=34 C=30 A=24 B=12 so B is eliminated. Round 3 votes counts: D=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:210 A:201 D:198 B:197 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 4 0 B 2 0 -4 -2 -2 C 0 4 0 0 16 D -4 2 0 0 -2 E 0 2 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 0 B 2 0 -4 -2 -2 C 0 4 0 0 16 D -4 2 0 0 -2 E 0 2 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 0 B 2 0 -4 -2 -2 C 0 4 0 0 16 D -4 2 0 0 -2 E 0 2 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3363: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) E C B D A (6) C E A D B (6) D A B C E (5) E B D A C (4) C E B A D (4) E C B A D (3) D B A E C (3) D A B E C (3) B D A E C (3) B D A C E (3) A D C E B (3) A D C B E (3) A D B C E (3) E D A C B (2) E D A B C (2) E C A D B (2) E B D C A (2) D E B A C (2) D A E C B (2) D A E B C (2) C B E A D (2) C B A E D (2) C B A D E (2) C A D B E (2) C A B D E (2) B E C D A (2) B C E A D (2) B C A D E (2) B A D C E (2) A C D B E (2) E D B A C (1) E B C A D (1) E A C D B (1) D E A C B (1) C E A B D (1) C A D E B (1) B C D A E (1) Total count = 100 A B C D E A 0 -14 -6 -8 -8 B 14 0 6 8 -12 C 6 -6 0 8 -4 D 8 -8 -8 0 -6 E 8 12 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -6 -8 -8 B 14 0 6 8 -12 C 6 -6 0 8 -4 D 8 -8 -8 0 -6 E 8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=22 D=18 B=15 A=11 so A is eliminated. Round 2 votes counts: E=34 D=27 C=24 B=15 so B is eliminated. Round 3 votes counts: E=36 D=35 C=29 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:208 C:202 D:193 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -6 -8 -8 B 14 0 6 8 -12 C 6 -6 0 8 -4 D 8 -8 -8 0 -6 E 8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -8 -8 B 14 0 6 8 -12 C 6 -6 0 8 -4 D 8 -8 -8 0 -6 E 8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -8 -8 B 14 0 6 8 -12 C 6 -6 0 8 -4 D 8 -8 -8 0 -6 E 8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3364: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) D C E B A (9) D A C E B (6) A D B E C (6) E C B A D (5) D A B E C (5) B E C A D (5) C E D B A (4) B E A C D (4) A D C E B (4) D C E A B (3) D A B C E (3) C E B A D (3) B D A E C (3) A E C B D (3) E B C A D (2) D C B E A (2) D B A E C (2) C E A D B (2) B A E C D (2) A B D E C (2) D C A E B (1) D B E C A (1) D B E A C (1) D B C A E (1) C E A B D (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C D A (1) B A E D C (1) B A D E C (1) A E B C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -8 -8 -10 B 18 0 -12 -4 -14 C 8 12 0 -6 4 D 8 4 6 0 2 E 10 14 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 -8 -10 B 18 0 -12 -4 -14 C 8 12 0 -6 4 D 8 4 6 0 2 E 10 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=23 B=18 A=18 E=7 so E is eliminated. Round 2 votes counts: D=34 C=28 B=20 A=18 so A is eliminated. Round 3 votes counts: D=44 C=31 B=25 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:209 E:209 B:194 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -8 -8 -10 B 18 0 -12 -4 -14 C 8 12 0 -6 4 D 8 4 6 0 2 E 10 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -8 -10 B 18 0 -12 -4 -14 C 8 12 0 -6 4 D 8 4 6 0 2 E 10 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -8 -10 B 18 0 -12 -4 -14 C 8 12 0 -6 4 D 8 4 6 0 2 E 10 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3365: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) E A D C B (7) C B D E A (7) A E D B C (6) C B E D A (5) A E D C B (4) A D B E C (4) E D C B A (3) E D C A B (3) E C A D B (3) E A D B C (3) B C A D E (3) A B C D E (3) E D A C B (2) E C D B A (2) E C A B D (2) E A C D B (2) D E C B A (2) D B C E A (2) D B A C E (2) C B E A D (2) A B D C E (2) E D A B C (1) E C D A B (1) E A C B D (1) D E B C A (1) D E A B C (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) D B A E C (1) D A B E C (1) C E D B A (1) C E B D A (1) C B A E D (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) B A C D E (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 14 6 10 -14 B -14 0 -8 -28 -14 C -6 8 0 -20 -20 D -10 28 20 0 -6 E 14 14 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 6 10 -14 B -14 0 -8 -28 -14 C -6 8 0 -20 -20 D -10 28 20 0 -6 E 14 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=30 A=30 C=17 D=14 B=9 so B is eliminated. Round 2 votes counts: A=31 E=30 C=22 D=17 so D is eliminated. Round 3 votes counts: A=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:216 A:208 C:181 B:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 6 10 -14 B -14 0 -8 -28 -14 C -6 8 0 -20 -20 D -10 28 20 0 -6 E 14 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 10 -14 B -14 0 -8 -28 -14 C -6 8 0 -20 -20 D -10 28 20 0 -6 E 14 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 10 -14 B -14 0 -8 -28 -14 C -6 8 0 -20 -20 D -10 28 20 0 -6 E 14 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3366: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (20) B A D C E (14) E A D C B (9) B C D A E (6) B E C D A (4) B A E D C (4) A D E C B (4) E D C A B (3) C D B A E (3) B C D E A (3) B A D E C (3) E A B D C (2) D C A E B (2) C E D B A (2) C D A E B (2) B E A D C (2) B A C D E (2) E C D B A (1) E A D B C (1) D C A B E (1) C E D A B (1) C D E B A (1) C D A B E (1) C B E D A (1) C B D A E (1) B E C A D (1) B C A D E (1) A D E B C (1) A D C E B (1) A D C B E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -8 -4 -2 B -2 0 -8 -10 0 C 8 8 0 0 -14 D 4 10 0 0 -2 E 2 0 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.109252 C: 0.000000 D: 0.000000 E: 0.890748 Sum of squares = 0.805368686108 Cumulative probabilities = A: 0.000000 B: 0.109252 C: 0.109252 D: 0.109252 E: 1.000000 A B C D E A 0 2 -8 -4 -2 B -2 0 -8 -10 0 C 8 8 0 0 -14 D 4 10 0 0 -2 E 2 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222828 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=36 C=12 A=9 D=3 so D is eliminated. Round 2 votes counts: B=40 E=36 C=15 A=9 so A is eliminated. Round 3 votes counts: B=42 E=41 C=17 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:209 D:206 C:201 A:194 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 -4 -2 B -2 0 -8 -10 0 C 8 8 0 0 -14 D 4 10 0 0 -2 E 2 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222828 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -4 -2 B -2 0 -8 -10 0 C 8 8 0 0 -14 D 4 10 0 0 -2 E 2 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222828 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -4 -2 B -2 0 -8 -10 0 C 8 8 0 0 -14 D 4 10 0 0 -2 E 2 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222828 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3367: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) D E B A C (9) E D C A B (7) C A B D E (7) B A C D E (7) E C A B D (4) D B A E C (4) D B A C E (4) C E A B D (4) A C B D E (4) E D C B A (3) E D B A C (3) C A E B D (3) E D B C A (2) E C D A B (2) E C A D B (2) E B D A C (2) D E C A B (2) D A B C E (2) B C A E D (2) B A C E D (2) E C D B A (1) E B C A D (1) D E A C B (1) D E A B C (1) D C A E B (1) D B E A C (1) C B A E D (1) C A D E B (1) C A D B E (1) B E C A D (1) B D E A C (1) B D A C E (1) B A D C E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -10 6 6 B -8 0 -8 4 2 C 10 8 0 8 6 D -6 -4 -8 0 0 E -6 -2 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 6 6 B -8 0 -8 4 2 C 10 8 0 8 6 D -6 -4 -8 0 0 E -6 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=27 C=27 D=25 B=15 A=6 so A is eliminated. Round 2 votes counts: C=31 E=27 D=26 B=16 so B is eliminated. Round 3 votes counts: C=43 D=29 E=28 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:205 B:195 E:193 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 6 6 B -8 0 -8 4 2 C 10 8 0 8 6 D -6 -4 -8 0 0 E -6 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 6 6 B -8 0 -8 4 2 C 10 8 0 8 6 D -6 -4 -8 0 0 E -6 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 6 6 B -8 0 -8 4 2 C 10 8 0 8 6 D -6 -4 -8 0 0 E -6 -2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3368: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) B A C D E (6) E D C A B (5) B A C E D (5) A B C E D (5) D C E B A (4) B C D A E (4) E D A C B (3) E A D C B (3) D E C A B (3) D E B C A (3) D E A B C (3) C E D B A (3) C D B E A (3) C B D E A (3) C B A E D (3) B C A D E (3) A C B E D (3) D A E B C (2) C E D A B (2) B C A E D (2) A E C B D (2) A B E C D (2) E C D B A (1) E C D A B (1) E C A D B (1) E A C D B (1) D B E A C (1) D B C E A (1) D B A E C (1) C E B A D (1) C E A B D (1) C B E A D (1) C A B E D (1) B D C E A (1) B D C A E (1) B C D E A (1) B A D E C (1) B A D C E (1) A E D C B (1) A E D B C (1) A E B D C (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -10 -4 -4 B 12 0 -4 2 4 C 10 4 0 12 12 D 4 -2 -12 0 0 E 4 -4 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -4 -4 B 12 0 -4 2 4 C 10 4 0 12 12 D 4 -2 -12 0 0 E 4 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 C=18 A=18 E=15 so E is eliminated. Round 2 votes counts: D=32 B=25 A=22 C=21 so C is eliminated. Round 3 votes counts: D=42 B=33 A=25 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:219 B:207 D:195 E:194 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 -4 -4 B 12 0 -4 2 4 C 10 4 0 12 12 D 4 -2 -12 0 0 E 4 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -4 -4 B 12 0 -4 2 4 C 10 4 0 12 12 D 4 -2 -12 0 0 E 4 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -4 -4 B 12 0 -4 2 4 C 10 4 0 12 12 D 4 -2 -12 0 0 E 4 -4 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3369: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) A D B E C (6) A B D E C (6) C E D B A (5) C B E A D (5) C D E A B (4) C A D E B (4) D E B C A (3) D E B A C (3) D A B E C (3) B D E A C (3) B A D E C (3) A C D E B (3) A C B D E (3) D E C B A (2) D B E A C (2) D A E B C (2) C E A D B (2) C B A E D (2) C A E D B (2) C A B E D (2) B E A C D (2) B A E C D (2) A D C E B (2) A C B E D (2) E D B C A (1) E C B D A (1) E B D C A (1) D E A C B (1) C E D A B (1) C E B A D (1) C D E B A (1) C D A E B (1) C B E D A (1) B E D C A (1) B E D A C (1) B E C D A (1) B C E D A (1) B A C E D (1) A D E B C (1) A D C B E (1) A D B C E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 6 -2 B 2 0 -8 -2 2 C 0 8 0 12 10 D -6 2 -12 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.480291 B: 0.000000 C: 0.519709 D: 0.000000 E: 0.000000 Sum of squares = 0.500776906111 Cumulative probabilities = A: 0.480291 B: 0.480291 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 6 -2 B 2 0 -8 -2 2 C 0 8 0 12 10 D -6 2 -12 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=27 D=16 B=15 E=3 so E is eliminated. Round 2 votes counts: C=40 A=27 D=17 B=16 so B is eliminated. Round 3 votes counts: C=42 A=35 D=23 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:201 D:198 B:197 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 6 -2 B 2 0 -8 -2 2 C 0 8 0 12 10 D -6 2 -12 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 6 -2 B 2 0 -8 -2 2 C 0 8 0 12 10 D -6 2 -12 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 6 -2 B 2 0 -8 -2 2 C 0 8 0 12 10 D -6 2 -12 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3370: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) C E D A B (6) E B D C A (5) D B A C E (5) C D A B E (5) B A D E C (5) A D C B E (5) A D B C E (5) C A D B E (4) E C A B D (3) E B C D A (3) D A C B E (3) C D A E B (3) B E A D C (3) B A E D C (3) E C D B A (2) E C B A D (2) E B C A D (2) E B A D C (2) C D E A B (2) C A D E B (2) B E D C A (2) B D A C E (2) A C D B E (2) A B D C E (2) E C D A B (1) E C A D B (1) E B D A C (1) E B A C D (1) D C B A E (1) D B C E A (1) D B C A E (1) D A B C E (1) C E D B A (1) C E A D B (1) B E D A C (1) B D E C A (1) B D E A C (1) B D A E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -12 -12 2 B 6 0 -2 -4 10 C 12 2 0 -4 4 D 12 4 4 0 4 E -2 -10 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -12 2 B 6 0 -2 -4 10 C 12 2 0 -4 4 D 12 4 4 0 4 E -2 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=24 B=19 A=15 D=12 so D is eliminated. Round 2 votes counts: E=30 B=26 C=25 A=19 so A is eliminated. Round 3 votes counts: C=35 B=35 E=30 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:207 B:205 E:190 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 2 B 6 0 -2 -4 10 C 12 2 0 -4 4 D 12 4 4 0 4 E -2 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 2 B 6 0 -2 -4 10 C 12 2 0 -4 4 D 12 4 4 0 4 E -2 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 2 B 6 0 -2 -4 10 C 12 2 0 -4 4 D 12 4 4 0 4 E -2 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3371: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) D C A B E (6) D B E C A (6) A C E B D (6) A B E D C (5) D C B E A (4) C E A B D (4) A E B C D (4) E C A B D (3) E A B C D (3) D B E A C (3) D B A E C (3) C E D A B (3) C D E B A (3) C D E A B (3) C A E D B (3) B E A D C (3) B D E A C (3) E B A D C (2) E B A C D (2) D C E B A (2) D B C A E (2) C E D B A (2) C A E B D (2) C A D E B (2) B A E D C (2) E C B A D (1) E B D A C (1) D B C E A (1) D A C B E (1) D A B C E (1) C E B D A (1) C D A E B (1) C D A B E (1) A E C B D (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 0 -8 -14 B -4 0 -2 4 2 C 0 2 0 -4 0 D 8 -4 4 0 -14 E 14 -2 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.621484 D: 0.000000 E: 0.378516 Sum of squares = 0.529516528471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.621484 D: 0.621484 E: 1.000000 A B C D E A 0 4 0 -8 -14 B -4 0 -2 4 2 C 0 2 0 -4 0 D 8 -4 4 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.000000 E: 0.499794 Sum of squares = 0.500000085028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.500206 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 A=19 B=15 E=12 so E is eliminated. Round 2 votes counts: D=29 C=29 A=22 B=20 so B is eliminated. Round 3 votes counts: D=40 A=31 C=29 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:213 B:200 C:199 D:197 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -8 -14 B -4 0 -2 4 2 C 0 2 0 -4 0 D 8 -4 4 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.000000 E: 0.499794 Sum of squares = 0.500000085028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.500206 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -8 -14 B -4 0 -2 4 2 C 0 2 0 -4 0 D 8 -4 4 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.000000 E: 0.499794 Sum of squares = 0.500000085028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.500206 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -8 -14 B -4 0 -2 4 2 C 0 2 0 -4 0 D 8 -4 4 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.000000 E: 0.499794 Sum of squares = 0.500000085028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500206 D: 0.500206 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3372: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) B A D C E (8) E C D A B (5) C D E A B (5) D C A E B (4) B E C D A (4) B E A C D (4) A B D C E (4) E C D B A (3) D C A B E (3) D A C B E (3) A D C E B (3) A D B C E (3) A B D E C (3) E C B D A (2) E A B D C (2) C D E B A (2) B E C A D (2) B E A D C (2) B A E C D (2) A E B D C (2) A D C B E (2) A D B E C (2) E D C A B (1) E D A C B (1) E B C D A (1) E B C A D (1) E B A C D (1) E A B C D (1) D C B A E (1) D A C E B (1) C E D B A (1) C E D A B (1) C E B D A (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D E A (1) B C E D A (1) B C D E A (1) B C D A E (1) B A D E C (1) B A C D E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 12 8 10 B 2 0 12 10 20 C -12 -12 0 -12 0 D -8 -10 12 0 2 E -10 -20 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 8 10 B 2 0 12 10 20 C -12 -12 0 -12 0 D -8 -10 12 0 2 E -10 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=21 E=18 C=14 D=12 so D is eliminated. Round 2 votes counts: B=35 A=25 C=22 E=18 so E is eliminated. Round 3 votes counts: B=38 C=33 A=29 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:214 D:198 E:184 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 12 8 10 B 2 0 12 10 20 C -12 -12 0 -12 0 D -8 -10 12 0 2 E -10 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 8 10 B 2 0 12 10 20 C -12 -12 0 -12 0 D -8 -10 12 0 2 E -10 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 8 10 B 2 0 12 10 20 C -12 -12 0 -12 0 D -8 -10 12 0 2 E -10 -20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3373: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) C E D B A (5) C B A E D (5) E D A B C (4) D B A C E (4) C E A B D (4) B A D C E (4) B A C E D (4) E C D A B (3) E C A D B (3) E C A B D (3) D B C A E (3) D A B E C (3) C E B A D (3) A E B D C (3) E D C A B (2) E A D B C (2) D C E B A (2) D C B E A (2) D C B A E (2) D A E B C (2) C B D A E (2) B C A E D (2) A E B C D (2) A B E C D (2) A B D E C (2) A B D C E (2) E A C B D (1) E A B C D (1) D E C B A (1) D E C A B (1) D E A C B (1) C E B D A (1) C D B E A (1) C B E A D (1) C B A D E (1) B D A C E (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D B C (1) A D E B C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 2 2 B -6 0 12 -4 -8 C -2 -12 0 -4 6 D -2 4 4 0 -6 E -2 8 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 2 2 B -6 0 12 -4 -8 C -2 -12 0 -4 6 D -2 4 4 0 -6 E -2 8 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=23 E=19 A=15 B=14 so B is eliminated. Round 2 votes counts: D=30 C=27 A=24 E=19 so E is eliminated. Round 3 votes counts: D=36 C=36 A=28 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:206 E:203 D:200 B:197 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 2 2 B -6 0 12 -4 -8 C -2 -12 0 -4 6 D -2 4 4 0 -6 E -2 8 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 2 B -6 0 12 -4 -8 C -2 -12 0 -4 6 D -2 4 4 0 -6 E -2 8 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 2 B -6 0 12 -4 -8 C -2 -12 0 -4 6 D -2 4 4 0 -6 E -2 8 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3374: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) D C B E A (8) D B C E A (7) A C D E B (7) A C E B D (6) B D E C A (5) B E A D C (4) B E D C A (3) B E A C D (3) A C E D B (3) E B A C D (2) E A B C D (2) D C E B A (2) D C A B E (2) D B A C E (2) C D E B A (2) C D A E B (2) B E D A C (2) B E C D A (2) B D E A C (2) A E C B D (2) A E B C D (2) A D B E C (2) A D B C E (2) A B E D C (2) E C A B D (1) E B C D A (1) E A C B D (1) D C A E B (1) D B C A E (1) D A C B E (1) C E D B A (1) C E B A D (1) C E A B D (1) C A E D B (1) C A D E B (1) A D C E B (1) A D C B E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -2 -6 -18 B 14 0 10 -12 20 C 2 -10 0 -16 6 D 6 12 16 0 18 E 18 -20 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -6 -18 B 14 0 10 -12 20 C 2 -10 0 -16 6 D 6 12 16 0 18 E 18 -20 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=30 B=21 C=9 E=7 so E is eliminated. Round 2 votes counts: D=33 A=33 B=24 C=10 so C is eliminated. Round 3 votes counts: D=38 A=37 B=25 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:216 C:191 E:187 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -2 -6 -18 B 14 0 10 -12 20 C 2 -10 0 -16 6 D 6 12 16 0 18 E 18 -20 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -6 -18 B 14 0 10 -12 20 C 2 -10 0 -16 6 D 6 12 16 0 18 E 18 -20 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -6 -18 B 14 0 10 -12 20 C 2 -10 0 -16 6 D 6 12 16 0 18 E 18 -20 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3375: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (19) B D E A C (16) E C A D B (8) D B E C A (8) A C E B D (6) C E A D B (5) E D B C A (3) E A C D B (3) C A E B D (3) B D A C E (3) E D C B A (2) D E B C A (2) D B E A C (2) B D C A E (2) A C B E D (2) E D C A B (1) E D B A C (1) E D A C B (1) E A C B D (1) D E C B A (1) D E B A C (1) D B C E A (1) D B C A E (1) C B D A E (1) C A D E B (1) C A D B E (1) C A B E D (1) C A B D E (1) B A D E C (1) B A D C E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 -22 8 -12 B -8 0 -16 -24 -18 C 22 16 0 6 -2 D -8 24 -6 0 -14 E 12 18 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -22 8 -12 B -8 0 -16 -24 -18 C 22 16 0 6 -2 D -8 24 -6 0 -14 E 12 18 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=23 E=20 D=16 A=9 so A is eliminated. Round 2 votes counts: C=41 B=23 E=20 D=16 so D is eliminated. Round 3 votes counts: C=41 B=35 E=24 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:221 D:198 A:191 B:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -22 8 -12 B -8 0 -16 -24 -18 C 22 16 0 6 -2 D -8 24 -6 0 -14 E 12 18 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -22 8 -12 B -8 0 -16 -24 -18 C 22 16 0 6 -2 D -8 24 -6 0 -14 E 12 18 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -22 8 -12 B -8 0 -16 -24 -18 C 22 16 0 6 -2 D -8 24 -6 0 -14 E 12 18 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3376: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) D B C A E (8) C B D A E (7) A E C D B (7) D B E C A (6) E A C B D (5) C A D B E (5) E D B A C (4) C A B D E (4) B D C E A (4) C D B A E (3) B D E C A (3) A E C B D (3) A C E D B (3) A C E B D (3) E B D C A (2) E B D A C (2) E A B D C (2) D E A B C (2) D B C E A (2) E A D C B (1) E A C D B (1) E A B C D (1) D E B A C (1) C D A B E (1) C B D E A (1) C B A E D (1) C B A D E (1) B E D C A (1) B D C A E (1) B C D E A (1) B C D A E (1) A E D C B (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 0 4 B -2 0 0 -14 4 C 4 0 0 0 -2 D 0 14 0 0 8 E -4 -4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.543181 D: 0.456819 E: 0.000000 Sum of squares = 0.503729153773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.543181 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 0 4 B -2 0 0 -14 4 C 4 0 0 0 -2 D 0 14 0 0 8 E -4 -4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=23 A=20 D=19 B=11 so B is eliminated. Round 2 votes counts: E=28 D=27 C=25 A=20 so A is eliminated. Round 3 votes counts: E=39 C=33 D=28 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:211 A:201 C:201 B:194 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 0 4 B -2 0 0 -14 4 C 4 0 0 0 -2 D 0 14 0 0 8 E -4 -4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 4 B -2 0 0 -14 4 C 4 0 0 0 -2 D 0 14 0 0 8 E -4 -4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 4 B -2 0 0 -14 4 C 4 0 0 0 -2 D 0 14 0 0 8 E -4 -4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3377: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) B D E A C (6) A B C D E (6) E D C B A (5) E D B C A (5) C A E D B (5) E A C B D (4) D E B C A (4) C A B D E (4) E D B A C (3) D B C A E (3) B E D A C (3) B D A C E (3) B A D C E (3) E B D A C (2) E A B C D (2) D E C B A (2) D C E A B (2) C E A D B (2) C D E A B (2) B A D E C (2) A C B E D (2) A B C E D (2) E D C A B (1) E C D A B (1) E B A D C (1) D C E B A (1) D B E C A (1) D B C E A (1) C E D A B (1) C D A B E (1) C A E B D (1) C A D E B (1) C A D B E (1) B E A D C (1) B D A E C (1) B A E C D (1) B A C D E (1) A E B C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 12 4 0 B -2 0 6 16 6 C -12 -6 0 0 6 D -4 -16 0 0 10 E 0 -6 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.840924 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.159076 Sum of squares = 0.732457866968 Cumulative probabilities = A: 0.840924 B: 0.840924 C: 0.840924 D: 0.840924 E: 1.000000 A B C D E A 0 2 12 4 0 B -2 0 6 16 6 C -12 -6 0 0 6 D -4 -16 0 0 10 E 0 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000020706 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=23 B=21 C=18 D=14 so D is eliminated. Round 2 votes counts: E=30 B=26 A=23 C=21 so C is eliminated. Round 3 votes counts: E=38 A=36 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:209 D:195 C:194 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 4 0 B -2 0 6 16 6 C -12 -6 0 0 6 D -4 -16 0 0 10 E 0 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000020706 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 4 0 B -2 0 6 16 6 C -12 -6 0 0 6 D -4 -16 0 0 10 E 0 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000020706 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 4 0 B -2 0 6 16 6 C -12 -6 0 0 6 D -4 -16 0 0 10 E 0 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000020706 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3378: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) A E C D B (7) E A D C B (6) E D A B C (5) D B C E A (5) B D E C A (5) B D C E A (5) E D B A C (4) E A D B C (4) E A B D C (4) C A B D E (4) A E C B D (4) A E D C B (3) A C E D B (3) A C E B D (3) A C B E D (3) E D A C B (2) D E A C B (2) D B E C A (2) C D B A E (2) C B D A E (2) C B A D E (2) C A B E D (2) B C D A E (2) B C A E D (2) A E B C D (2) E B A D C (1) D E C B A (1) D E C A B (1) D E B A C (1) C D A E B (1) B E A C D (1) B C D E A (1) A B E C D (1) Total count = 100 A B C D E A 0 14 12 4 -14 B -14 0 4 -12 -22 C -12 -4 0 -16 -26 D -4 12 16 0 -14 E 14 22 26 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 12 4 -14 B -14 0 4 -12 -22 C -12 -4 0 -16 -26 D -4 12 16 0 -14 E 14 22 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 A=26 D=19 B=16 C=13 so C is eliminated. Round 2 votes counts: A=32 E=26 D=22 B=20 so B is eliminated. Round 3 votes counts: D=37 A=36 E=27 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:238 A:208 D:205 B:178 C:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 12 4 -14 B -14 0 4 -12 -22 C -12 -4 0 -16 -26 D -4 12 16 0 -14 E 14 22 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 4 -14 B -14 0 4 -12 -22 C -12 -4 0 -16 -26 D -4 12 16 0 -14 E 14 22 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 4 -14 B -14 0 4 -12 -22 C -12 -4 0 -16 -26 D -4 12 16 0 -14 E 14 22 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3379: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) C E D B A (6) E C A B D (5) D B A E C (5) E A B C D (4) D C B A E (4) D C A B E (4) D B C A E (4) C E D A B (4) C D E B A (4) C D E A B (4) B A E D C (4) D C E B A (3) D B A C E (3) C E A B D (3) A D B C E (3) E B D C A (2) E B C D A (2) E B A C D (2) E A C B D (2) D C B E A (2) D B E C A (2) D A B C E (2) C E B D A (2) B D A E C (2) A B D E C (2) E C B D A (1) E C B A D (1) E B D A C (1) E B C A D (1) C E A D B (1) C D A E B (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E A C (1) A E B C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -16 -20 -10 B 8 0 2 -8 -2 C 16 -2 0 -6 4 D 20 8 6 0 -6 E 10 2 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.625000 E: 1.000000 A B C D E A 0 -8 -16 -20 -10 B 8 0 2 -8 -2 C 16 -2 0 -6 4 D 20 8 6 0 -6 E 10 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 E=21 A=15 B=9 so B is eliminated. Round 2 votes counts: D=32 C=26 E=23 A=19 so A is eliminated. Round 3 votes counts: D=37 E=36 C=27 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:214 E:207 C:206 B:200 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -16 -20 -10 B 8 0 2 -8 -2 C 16 -2 0 -6 4 D 20 8 6 0 -6 E 10 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.625000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -20 -10 B 8 0 2 -8 -2 C 16 -2 0 -6 4 D 20 8 6 0 -6 E 10 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.625000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -20 -10 B 8 0 2 -8 -2 C 16 -2 0 -6 4 D 20 8 6 0 -6 E 10 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3380: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (7) D E C B A (6) B A C D E (6) A B C D E (6) E D C A B (5) E D A C B (5) C B E A D (5) E D C B A (4) D E A C B (4) D E A B C (4) C B A E D (4) E C D B A (3) C B E D A (3) C B D E A (3) A B E C D (3) D E C A B (2) D C B E A (2) D A E B C (2) D A B C E (2) C E D B A (2) C D B E A (2) B C A E D (2) A D B E C (2) A B D E C (2) A B D C E (2) E C A B D (1) E A D B C (1) D C E B A (1) D B C A E (1) C D E B A (1) C B D A E (1) B C A D E (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 2 -6 -8 B -4 0 -8 -4 12 C -2 8 0 2 6 D 6 4 -2 0 4 E 8 -12 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000013 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -6 -8 B -4 0 -8 -4 12 C -2 8 0 2 6 D 6 4 -2 0 4 E 8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000008 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 C=21 E=19 B=10 so B is eliminated. Round 2 votes counts: A=33 D=24 C=24 E=19 so E is eliminated. Round 3 votes counts: D=38 A=34 C=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:206 B:198 A:196 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 2 -6 -8 B -4 0 -8 -4 12 C -2 8 0 2 6 D 6 4 -2 0 4 E 8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000008 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -6 -8 B -4 0 -8 -4 12 C -2 8 0 2 6 D 6 4 -2 0 4 E 8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000008 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -6 -8 B -4 0 -8 -4 12 C -2 8 0 2 6 D 6 4 -2 0 4 E 8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000008 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3381: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) D C B E A (6) C D E B A (6) E B A C D (5) E A B C D (5) C D B E A (5) B E A C D (5) A E B D C (5) A B E D C (5) D C B A E (4) D C A B E (4) D A C E B (3) D A C B E (3) D A B C E (3) B D C E A (3) B C E D A (3) E B C A D (2) B E C D A (2) B A E D C (2) A E D B C (2) A E C D B (2) A D C E B (2) D C E A B (1) D C A E B (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) B E D A C (1) B E A D C (1) A E D C B (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 18 4 4 B -10 0 12 0 -6 C -18 -12 0 4 0 D -4 0 -4 0 -10 E -4 6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 18 4 4 B -10 0 12 0 -6 C -18 -12 0 4 0 D -4 0 -4 0 -10 E -4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=25 B=17 C=15 E=12 so E is eliminated. Round 2 votes counts: A=36 D=25 B=24 C=15 so C is eliminated. Round 3 votes counts: D=39 A=37 B=24 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:206 B:198 D:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 18 4 4 B -10 0 12 0 -6 C -18 -12 0 4 0 D -4 0 -4 0 -10 E -4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 4 4 B -10 0 12 0 -6 C -18 -12 0 4 0 D -4 0 -4 0 -10 E -4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 4 4 B -10 0 12 0 -6 C -18 -12 0 4 0 D -4 0 -4 0 -10 E -4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3382: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) B D C A E (9) E D A C B (7) B C A D E (7) D B E C A (6) B C D A E (6) B C A E D (6) E A C D B (5) D E B A C (5) A C E B D (5) E A D C B (4) C B A E D (4) D E C A B (3) D E B C A (3) D E A B C (3) C A E B D (3) D B C E A (2) C A E D B (2) B A C E D (2) D E C B A (1) C A B E D (1) B D C E A (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 -16 -4 B 4 0 2 -6 -8 C 8 -2 0 -12 0 D 16 6 12 0 16 E 4 8 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -16 -4 B 4 0 2 -6 -8 C 8 -2 0 -12 0 D 16 6 12 0 16 E 4 8 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=31 E=16 C=10 A=8 so A is eliminated. Round 2 votes counts: D=35 B=32 E=17 C=16 so C is eliminated. Round 3 votes counts: B=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:198 C:197 B:196 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -16 -4 B 4 0 2 -6 -8 C 8 -2 0 -12 0 D 16 6 12 0 16 E 4 8 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -16 -4 B 4 0 2 -6 -8 C 8 -2 0 -12 0 D 16 6 12 0 16 E 4 8 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -16 -4 B 4 0 2 -6 -8 C 8 -2 0 -12 0 D 16 6 12 0 16 E 4 8 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3383: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) D B E C A (8) A C E B D (8) A E C B D (7) D B C E A (6) C A D E B (6) A E B C D (6) C D A E B (5) B D E A C (5) A E B D C (5) D C B E A (4) D C B A E (3) D C A B E (3) C A E B D (3) B D E C A (3) E B A C D (2) E A B C D (2) C D A B E (2) B E D A C (2) B E A D C (2) A B D E C (2) D B E A C (1) D B A E C (1) C D E A B (1) C D B A E (1) B E D C A (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 22 -10 8 26 B -22 0 -8 -4 -12 C 10 8 0 8 6 D -8 4 -8 0 4 E -26 12 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -10 8 26 B -22 0 -8 -4 -12 C 10 8 0 8 6 D -8 4 -8 0 4 E -26 12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=27 D=26 B=13 E=4 so E is eliminated. Round 2 votes counts: A=32 C=27 D=26 B=15 so B is eliminated. Round 3 votes counts: D=37 A=36 C=27 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:216 D:196 E:188 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -10 8 26 B -22 0 -8 -4 -12 C 10 8 0 8 6 D -8 4 -8 0 4 E -26 12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -10 8 26 B -22 0 -8 -4 -12 C 10 8 0 8 6 D -8 4 -8 0 4 E -26 12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -10 8 26 B -22 0 -8 -4 -12 C 10 8 0 8 6 D -8 4 -8 0 4 E -26 12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3384: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) B C D A E (9) B A E C D (9) E A D C B (7) C D E A B (7) C D E B A (5) C D B A E (5) A E B D C (5) E A D B C (4) D C E A B (4) C D B E A (4) C B D A E (3) B D C A E (3) B A E D C (3) A B E D C (3) E A C D B (2) E A B C D (2) D C B A E (2) B C A D E (2) E D C A B (1) E D A C B (1) E C D A B (1) E A C B D (1) D E C A B (1) D C E B A (1) C E D A B (1) B A C E D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 2 4 -6 B -4 0 8 8 -8 C -2 -8 0 6 -4 D -4 -8 -6 0 -4 E 6 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 4 -6 B -4 0 8 8 -8 C -2 -8 0 6 -4 D -4 -8 -6 0 -4 E 6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=28 C=25 A=9 D=8 so D is eliminated. Round 2 votes counts: C=32 E=31 B=28 A=9 so A is eliminated. Round 3 votes counts: E=36 C=32 B=32 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:202 B:202 C:196 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 4 -6 B -4 0 8 8 -8 C -2 -8 0 6 -4 D -4 -8 -6 0 -4 E 6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 -6 B -4 0 8 8 -8 C -2 -8 0 6 -4 D -4 -8 -6 0 -4 E 6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 -6 B -4 0 8 8 -8 C -2 -8 0 6 -4 D -4 -8 -6 0 -4 E 6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3385: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) E B A C D (8) C D B A E (7) E A B D C (6) E B C D A (5) E A D C B (5) D A C B E (5) C B D A E (5) A B D C E (5) E D C A B (4) C D A B E (3) B E A C D (3) B C A D E (3) A E B D C (3) A D B C E (3) D C A B E (2) C D E A B (2) C B D E A (2) B E C A D (2) B A D C E (2) A D C B E (2) A B E D C (2) E D A C B (1) E C D A B (1) E C B D A (1) E B C A D (1) E A D B C (1) D C A E B (1) C E D B A (1) C D E B A (1) C D B E A (1) B C A E D (1) B A C D E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -2 0 -6 B 4 0 -8 0 -2 C 2 8 0 12 -4 D 0 0 -12 0 -6 E 6 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 0 -6 B 4 0 -8 0 -2 C 2 8 0 12 -4 D 0 0 -12 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=22 A=17 B=12 D=8 so D is eliminated. Round 2 votes counts: E=41 C=25 A=22 B=12 so B is eliminated. Round 3 votes counts: E=46 C=29 A=25 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:209 B:197 A:194 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 0 -6 B 4 0 -8 0 -2 C 2 8 0 12 -4 D 0 0 -12 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 0 -6 B 4 0 -8 0 -2 C 2 8 0 12 -4 D 0 0 -12 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 0 -6 B 4 0 -8 0 -2 C 2 8 0 12 -4 D 0 0 -12 0 -6 E 6 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3386: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) B D C E A (6) B A D C E (6) B D C A E (5) C D A E B (4) A C D E B (4) E D B C A (3) E C A D B (3) D C E A B (3) C D E A B (3) C A D E B (3) B E D C A (3) B E D A C (3) A E C D B (3) A C E D B (3) E D C B A (2) E B D A C (2) E A C B D (2) D B E C A (2) D B C E A (2) D B C A E (2) B E A D C (2) B D E A C (2) B A E D C (2) B A C D E (2) A E C B D (2) A C D B E (2) E D C A B (1) E C D A B (1) D E C B A (1) D E C A B (1) D E B C A (1) D C B A E (1) C E A D B (1) C D A B E (1) B E A C D (1) B D E C A (1) B D A C E (1) B A E C D (1) B A D E C (1) B A C E D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 0 -2 -4 B 6 0 2 -6 2 C 0 -2 0 -6 6 D 2 6 6 0 12 E 4 -2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 -4 B 6 0 2 -6 2 C 0 -2 0 -6 6 D 2 6 6 0 12 E 4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=20 A=18 D=13 C=12 so C is eliminated. Round 2 votes counts: B=37 E=21 D=21 A=21 so E is eliminated. Round 3 votes counts: B=39 A=33 D=28 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:202 C:199 A:194 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -2 -4 B 6 0 2 -6 2 C 0 -2 0 -6 6 D 2 6 6 0 12 E 4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 -4 B 6 0 2 -6 2 C 0 -2 0 -6 6 D 2 6 6 0 12 E 4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 -4 B 6 0 2 -6 2 C 0 -2 0 -6 6 D 2 6 6 0 12 E 4 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3387: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (15) D B E A C (9) E A C B D (6) C A E D B (6) C B A E D (4) B C A E D (4) B A E C D (4) B A C E D (4) E A B C D (3) D E A C B (3) D E A B C (3) D B C A E (3) B E A C D (3) B D E A C (3) D E C A B (2) D E B A C (2) D C E A B (2) D C B A E (2) D C A E B (2) D C A B E (2) D B C E A (2) C D A E B (2) C A D E B (2) C A B E D (2) A E C B D (2) E D A C B (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B D C (1) C B D A E (1) C A D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -4 22 14 B -14 0 -10 10 -10 C 4 10 0 22 10 D -22 -10 -22 0 -18 E -14 10 -10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 22 14 B -14 0 -10 10 -10 C 4 10 0 22 10 D -22 -10 -22 0 -18 E -14 10 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=32 B=18 E=14 A=3 so A is eliminated. Round 2 votes counts: C=33 D=32 B=19 E=16 so E is eliminated. Round 3 votes counts: C=41 D=34 B=25 so B is eliminated. Round 4 votes counts: C=61 D=39 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:223 C:223 E:202 B:188 D:164 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 22 14 B -14 0 -10 10 -10 C 4 10 0 22 10 D -22 -10 -22 0 -18 E -14 10 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 22 14 B -14 0 -10 10 -10 C 4 10 0 22 10 D -22 -10 -22 0 -18 E -14 10 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 22 14 B -14 0 -10 10 -10 C 4 10 0 22 10 D -22 -10 -22 0 -18 E -14 10 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3388: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) A E C B D (11) B D A C E (10) D B A E C (8) E C A B D (6) C E A B D (6) E C A D B (5) C E B D A (4) B D C E A (4) C B E D A (3) B D A E C (3) A E C D B (3) A B D E C (3) C E D B A (2) B D C A E (2) A D B E C (2) A C E B D (2) E C D A B (1) E A C D B (1) E A C B D (1) D B A C E (1) D A E B C (1) D A B E C (1) C E D A B (1) C E B A D (1) C D B E A (1) C B D E A (1) B C D E A (1) B A D E C (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -12 0 B 8 0 0 20 8 C 0 0 0 2 4 D 12 -20 -2 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.366739 C: 0.633261 D: 0.000000 E: 0.000000 Sum of squares = 0.535516745438 Cumulative probabilities = A: 0.000000 B: 0.366739 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -12 0 B 8 0 0 20 8 C 0 0 0 2 4 D 12 -20 -2 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 A=23 B=21 C=19 E=14 so E is eliminated. Round 2 votes counts: C=31 A=25 D=23 B=21 so B is eliminated. Round 3 votes counts: D=42 C=32 A=26 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:203 D:196 E:193 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -12 0 B 8 0 0 20 8 C 0 0 0 2 4 D 12 -20 -2 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -12 0 B 8 0 0 20 8 C 0 0 0 2 4 D 12 -20 -2 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -12 0 B 8 0 0 20 8 C 0 0 0 2 4 D 12 -20 -2 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3389: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) E A B C D (7) E A C B D (5) D C B A E (5) D B C A E (4) D B A C E (4) D A B E C (4) C E A D B (4) E C A D B (3) D B A E C (3) D A E B C (3) C D E B A (3) C D E A B (3) B D C A E (3) A E B D C (3) E C A B D (2) E A C D B (2) D C B E A (2) C E D B A (2) C E D A B (2) C E A B D (2) B D A E C (2) B D A C E (2) B A D E C (2) A B E D C (2) E D A C B (1) E B A C D (1) E A B D C (1) D E C A B (1) D C E A B (1) D A E C B (1) D A B C E (1) C E B D A (1) C B E A D (1) C B D E A (1) B C D A E (1) B A E D C (1) B A C E D (1) B A C D E (1) A E B C D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 -18 -8 B -2 0 -2 -16 2 C 0 2 0 6 6 D 18 16 -6 0 14 E 8 -2 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.167413 B: 0.000000 C: 0.832587 D: 0.000000 E: 0.000000 Sum of squares = 0.721228249586 Cumulative probabilities = A: 0.167413 B: 0.167413 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -18 -8 B -2 0 -2 -16 2 C 0 2 0 6 6 D 18 16 -6 0 14 E 8 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000021677 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 E=22 B=13 A=8 so A is eliminated. Round 2 votes counts: D=29 C=28 E=26 B=17 so B is eliminated. Round 3 votes counts: D=39 C=31 E=30 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:221 C:207 E:193 B:191 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -18 -8 B -2 0 -2 -16 2 C 0 2 0 6 6 D 18 16 -6 0 14 E 8 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000021677 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -18 -8 B -2 0 -2 -16 2 C 0 2 0 6 6 D 18 16 -6 0 14 E 8 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000021677 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -18 -8 B -2 0 -2 -16 2 C 0 2 0 6 6 D 18 16 -6 0 14 E 8 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000021677 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3390: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (18) E B C A D (11) D A C B E (10) A D E B C (10) C B E D A (5) B C E D A (5) A D E C B (5) D C A B E (4) B E C D A (4) A E D B C (4) A D C B E (4) E A B C D (3) D C B A E (3) A D C E B (3) E C B D A (1) E C B A D (1) E B A C D (1) E A C B D (1) E A B D C (1) C D B E A (1) C D B A E (1) B E A D C (1) B D C E A (1) B C D E A (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -12 -8 -10 B 8 0 22 8 -18 C 12 -22 0 6 -22 D 8 -8 -6 0 -12 E 10 18 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 -8 -10 B 8 0 22 8 -18 C 12 -22 0 6 -22 D 8 -8 -6 0 -12 E 10 18 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=27 D=17 B=12 C=7 so C is eliminated. Round 2 votes counts: E=37 A=27 D=19 B=17 so B is eliminated. Round 3 votes counts: E=52 A=27 D=21 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:210 D:191 C:187 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 -8 -10 B 8 0 22 8 -18 C 12 -22 0 6 -22 D 8 -8 -6 0 -12 E 10 18 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -8 -10 B 8 0 22 8 -18 C 12 -22 0 6 -22 D 8 -8 -6 0 -12 E 10 18 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -8 -10 B 8 0 22 8 -18 C 12 -22 0 6 -22 D 8 -8 -6 0 -12 E 10 18 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999484 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3391: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) A E B C D (10) A E B D C (7) E A B C D (6) D C B E A (6) C D A E B (5) B E A D C (5) D C B A E (4) A E C B D (4) E B A D C (3) E B A C D (3) B D C E A (3) D C A E B (2) D B C E A (2) D A E B C (2) C D B A E (2) C D A B E (2) B D E C A (2) B D E A C (2) A E C D B (2) A C E D B (2) A C D E B (2) E A B D C (1) D A C E B (1) D A C B E (1) C E A D B (1) C E A B D (1) C B D E A (1) B E D C A (1) B E D A C (1) B E C A D (1) B C E D A (1) A E D C B (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 6 12 4 0 B -6 0 2 4 -12 C -12 -2 0 6 -6 D -4 -4 -6 0 0 E 0 12 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.469228 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.530772 Sum of squares = 0.501893803733 Cumulative probabilities = A: 0.469228 B: 0.469228 C: 0.469228 D: 0.469228 E: 1.000000 A B C D E A 0 6 12 4 0 B -6 0 2 4 -12 C -12 -2 0 6 -6 D -4 -4 -6 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=22 D=18 B=16 E=13 so E is eliminated. Round 2 votes counts: A=38 C=22 B=22 D=18 so D is eliminated. Round 3 votes counts: A=42 C=34 B=24 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:209 B:194 C:193 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 4 0 B -6 0 2 4 -12 C -12 -2 0 6 -6 D -4 -4 -6 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 4 0 B -6 0 2 4 -12 C -12 -2 0 6 -6 D -4 -4 -6 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 4 0 B -6 0 2 4 -12 C -12 -2 0 6 -6 D -4 -4 -6 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3392: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) D E B A C (9) C B A E D (9) E D B C A (7) D E A B C (5) A D E C B (5) A C D E B (5) A C B D E (5) B D E C A (4) B C D E A (4) C E B D A (3) C B E D A (3) C B A D E (3) C A B E D (3) E D C B A (2) E D B A C (2) C A B D E (2) B D E A C (2) A C E D B (2) A C B E D (2) E D C A B (1) E D A C B (1) E D A B C (1) E C D A B (1) D E B C A (1) B E D C A (1) B E C D A (1) B C A D E (1) B A C D E (1) A E D C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -28 -14 -18 -18 B 28 0 2 10 4 C 14 -2 0 12 10 D 18 -10 -12 0 -2 E 18 -4 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -14 -18 -18 B 28 0 2 10 4 C 14 -2 0 12 10 D 18 -10 -12 0 -2 E 18 -4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 A=22 E=15 D=15 so E is eliminated. Round 2 votes counts: D=29 B=25 C=24 A=22 so A is eliminated. Round 3 votes counts: C=38 D=37 B=25 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:222 C:217 E:203 D:197 A:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -14 -18 -18 B 28 0 2 10 4 C 14 -2 0 12 10 D 18 -10 -12 0 -2 E 18 -4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -14 -18 -18 B 28 0 2 10 4 C 14 -2 0 12 10 D 18 -10 -12 0 -2 E 18 -4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -14 -18 -18 B 28 0 2 10 4 C 14 -2 0 12 10 D 18 -10 -12 0 -2 E 18 -4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3393: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) E C A B D (5) D E A C B (5) B A C E D (5) E C D B A (4) D A E B C (4) C B A E D (4) B C A D E (4) B A C D E (4) E C B A D (3) E C A D B (3) D E A B C (3) D C B E A (3) D A B E C (3) C B D E A (3) A B C E D (3) E D C A B (2) E A C B D (2) D E C B A (2) D B C A E (2) D A B C E (2) C B E D A (2) B D C A E (2) B C D A E (2) B C A E D (2) A E D B C (2) A B E D C (2) E D C B A (1) E D A C B (1) E C B D A (1) D E C A B (1) D E B C A (1) D B C E A (1) D B A C E (1) C E B A D (1) C D E B A (1) B A D C E (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 -20 6 -10 B 20 0 -10 16 14 C 20 10 0 22 8 D -6 -16 -22 0 -8 E 10 -14 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -20 6 -10 B 20 0 -10 16 14 C 20 10 0 22 8 D -6 -16 -22 0 -8 E 10 -14 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=22 C=21 B=20 A=9 so A is eliminated. Round 2 votes counts: D=28 B=26 E=25 C=21 so C is eliminated. Round 3 votes counts: B=45 D=29 E=26 so E is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:230 B:220 E:198 A:178 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -20 6 -10 B 20 0 -10 16 14 C 20 10 0 22 8 D -6 -16 -22 0 -8 E 10 -14 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -20 6 -10 B 20 0 -10 16 14 C 20 10 0 22 8 D -6 -16 -22 0 -8 E 10 -14 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -20 6 -10 B 20 0 -10 16 14 C 20 10 0 22 8 D -6 -16 -22 0 -8 E 10 -14 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3394: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) C B A D E (7) E D A B C (6) E B D A C (6) A D E B C (6) E D B A C (5) C A D B E (4) C A B D E (4) E D A C B (3) E B D C A (3) D E A C B (3) D A E C B (3) C D A E B (3) B C E D A (3) B C A E D (3) E B C D A (2) D E A B C (2) C E D B A (2) B E C D A (2) B A E D C (2) B A D E C (2) A D C E B (2) A D C B E (2) A B C D E (2) E D C B A (1) E D C A B (1) E C D A B (1) E C B D A (1) D A E B C (1) C E B D A (1) C B E D A (1) C B D E A (1) B C A D E (1) B A E C D (1) B A D C E (1) B A C D E (1) A D B E C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 2 -2 -4 B 8 0 18 6 -4 C -2 -18 0 -2 -4 D 2 -6 2 0 -4 E 4 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 2 -2 -4 B 8 0 18 6 -4 C -2 -18 0 -2 -4 D 2 -6 2 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 C=23 A=15 D=9 so D is eliminated. Round 2 votes counts: E=34 B=24 C=23 A=19 so A is eliminated. Round 3 votes counts: E=44 C=28 B=28 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:208 D:197 A:194 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 2 -2 -4 B 8 0 18 6 -4 C -2 -18 0 -2 -4 D 2 -6 2 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 -4 B 8 0 18 6 -4 C -2 -18 0 -2 -4 D 2 -6 2 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 -4 B 8 0 18 6 -4 C -2 -18 0 -2 -4 D 2 -6 2 0 -4 E 4 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3395: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (16) B C D A E (16) E A D B C (10) C B D E A (6) A E D B C (6) E B C A D (5) C B D A E (5) E C B A D (3) E A B D C (3) B C E D A (3) E A C B D (2) D C B A E (2) D B C A E (2) D B A C E (2) D A C B E (2) C E B D A (2) C B E D A (2) A D E B C (2) E C D A B (1) E C B D A (1) E B C D A (1) D C A B E (1) D A C E B (1) D A B C E (1) C D B A E (1) B C D E A (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -4 0 -12 B 4 0 8 0 -6 C 4 -8 0 -2 -2 D 0 0 2 0 -10 E 12 6 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 0 -12 B 4 0 8 0 -6 C 4 -8 0 -2 -2 D 0 0 2 0 -10 E 12 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 B=20 C=16 D=11 A=11 so D is eliminated. Round 2 votes counts: E=42 B=24 C=19 A=15 so A is eliminated. Round 3 votes counts: E=50 B=27 C=23 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:203 C:196 D:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 0 -12 B 4 0 8 0 -6 C 4 -8 0 -2 -2 D 0 0 2 0 -10 E 12 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 -12 B 4 0 8 0 -6 C 4 -8 0 -2 -2 D 0 0 2 0 -10 E 12 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 -12 B 4 0 8 0 -6 C 4 -8 0 -2 -2 D 0 0 2 0 -10 E 12 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3396: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) A D E B C (9) E D A C B (7) C B D E A (6) A B D E C (5) C E D B A (4) B C A D E (4) A E D C B (4) A E D B C (4) E A D C B (3) A D B E C (3) E D C A B (2) E D A B C (2) E C A D B (2) D E C B A (2) D E B C A (2) D E A B C (2) D A E B C (2) C B E A D (2) C B A E D (2) C A E B D (2) B A C D E (2) E D C B A (1) E C D A B (1) E A C D B (1) D B E C A (1) D A B E C (1) C E B D A (1) C D B E A (1) C B D A E (1) C B A D E (1) B D A C E (1) B C D E A (1) B C D A E (1) B C A E D (1) B A D E C (1) A E C D B (1) A E B D C (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 4 0 B -12 0 -6 -10 -6 C -6 6 0 -10 -18 D -4 10 10 0 -6 E 0 6 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.546533 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.453467 Sum of squares = 0.504330568406 Cumulative probabilities = A: 0.546533 B: 0.546533 C: 0.546533 D: 0.546533 E: 1.000000 A B C D E A 0 12 6 4 0 B -12 0 -6 -10 -6 C -6 6 0 -10 -18 D -4 10 10 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=29 E=19 B=11 D=10 so D is eliminated. Round 2 votes counts: A=34 C=29 E=25 B=12 so B is eliminated. Round 3 votes counts: A=38 C=36 E=26 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:211 D:205 C:186 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 4 0 B -12 0 -6 -10 -6 C -6 6 0 -10 -18 D -4 10 10 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 4 0 B -12 0 -6 -10 -6 C -6 6 0 -10 -18 D -4 10 10 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 4 0 B -12 0 -6 -10 -6 C -6 6 0 -10 -18 D -4 10 10 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3397: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) E B C D A (8) B E C D A (8) A D E C B (8) B C E D A (7) C B E A D (6) E D B C A (5) D A E C B (5) D E A B C (4) D A E B C (4) E D B A C (3) E D A B C (3) C A B E D (3) E B D C A (2) D E B A C (2) D B A C E (2) D A B E C (2) C B A E D (2) C B A D E (2) C A B D E (2) A C B D E (2) E C B A D (1) D B E C A (1) D B E A C (1) C E B A D (1) B E D C A (1) B C E A D (1) B C D E A (1) A E C D B (1) A E C B D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -2 -18 -10 B 8 0 10 -4 2 C 2 -10 0 -6 -20 D 18 4 6 0 -6 E 10 -2 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.388888888759 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 A B C D E A 0 -8 -2 -18 -10 B 8 0 10 -4 2 C 2 -10 0 -6 -20 D 18 4 6 0 -6 E 10 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.38888888892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=22 D=21 B=18 C=16 so C is eliminated. Round 2 votes counts: B=28 A=28 E=23 D=21 so D is eliminated. Round 3 votes counts: A=39 B=32 E=29 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:217 D:211 B:208 C:183 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 -18 -10 B 8 0 10 -4 2 C 2 -10 0 -6 -20 D 18 4 6 0 -6 E 10 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.38888888892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -18 -10 B 8 0 10 -4 2 C 2 -10 0 -6 -20 D 18 4 6 0 -6 E 10 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.38888888892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -18 -10 B 8 0 10 -4 2 C 2 -10 0 -6 -20 D 18 4 6 0 -6 E 10 -2 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.333333 Sum of squares = 0.38888888892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3398: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (12) D C A B E (9) B E C D A (7) C D B E A (6) A E B D C (6) D C B A E (5) D C A E B (5) B E A D C (5) A E D C B (4) C D A E B (3) C B D E A (3) A D C E B (3) E A B D C (2) E A B C D (2) B E A C D (2) B D C E A (2) B A E D C (2) A E D B C (2) A E C D B (2) A B E D C (2) E B C D A (1) E B C A D (1) E B A D C (1) D C B E A (1) D A C B E (1) C E D B A (1) C D A B E (1) C B E D A (1) B D C A E (1) B C E D A (1) A E B C D (1) A D E C B (1) A D E B C (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 4 4 4 B 4 0 4 4 0 C -4 -4 0 -8 -8 D -4 -4 8 0 -12 E -4 0 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.681080 C: 0.000000 D: 0.000000 E: 0.318920 Sum of squares = 0.565579676263 Cumulative probabilities = A: 0.000000 B: 0.681080 C: 0.681080 D: 0.681080 E: 1.000000 A B C D E A 0 -4 4 4 4 B 4 0 4 4 0 C -4 -4 0 -8 -8 D -4 -4 8 0 -12 E -4 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500203 C: 0.000000 D: 0.000000 E: 0.499797 Sum of squares = 0.50000008249 Cumulative probabilities = A: 0.000000 B: 0.500203 C: 0.500203 D: 0.500203 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=21 B=20 E=19 C=15 so C is eliminated. Round 2 votes counts: D=31 A=25 B=24 E=20 so E is eliminated. Round 3 votes counts: B=39 D=32 A=29 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:208 B:206 A:204 D:194 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 4 4 B 4 0 4 4 0 C -4 -4 0 -8 -8 D -4 -4 8 0 -12 E -4 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500203 C: 0.000000 D: 0.000000 E: 0.499797 Sum of squares = 0.50000008249 Cumulative probabilities = A: 0.000000 B: 0.500203 C: 0.500203 D: 0.500203 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 4 B 4 0 4 4 0 C -4 -4 0 -8 -8 D -4 -4 8 0 -12 E -4 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500203 C: 0.000000 D: 0.000000 E: 0.499797 Sum of squares = 0.50000008249 Cumulative probabilities = A: 0.000000 B: 0.500203 C: 0.500203 D: 0.500203 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 4 B 4 0 4 4 0 C -4 -4 0 -8 -8 D -4 -4 8 0 -12 E -4 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500203 C: 0.000000 D: 0.000000 E: 0.499797 Sum of squares = 0.50000008249 Cumulative probabilities = A: 0.000000 B: 0.500203 C: 0.500203 D: 0.500203 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3399: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (15) B A C E D (12) E C D A B (10) D E C B A (6) D E C A B (6) B A D C E (6) A B C E D (5) E C A D B (4) C E A D B (4) B D A E C (4) A C E B D (4) D E B C A (3) C A E B D (3) B D A C E (3) A C B E D (3) D B E A C (2) C E D A B (2) A E C B D (2) E D C A B (1) D B C E A (1) B D C A E (1) B A E C D (1) B A D E C (1) A E C D B (1) Total count = 100 A B C D E A 0 -10 -12 -8 -8 B 10 0 8 -10 8 C 12 -8 0 2 -12 D 8 10 -2 0 -4 E 8 -8 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826455 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 A B C D E A 0 -10 -12 -8 -8 B 10 0 8 -10 8 C 12 -8 0 2 -12 D 8 10 -2 0 -4 E 8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826448 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=28 E=15 A=15 C=9 so C is eliminated. Round 2 votes counts: D=33 B=28 E=21 A=18 so A is eliminated. Round 3 votes counts: B=36 D=33 E=31 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:208 E:208 D:206 C:197 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 -8 B 10 0 8 -10 8 C 12 -8 0 2 -12 D 8 10 -2 0 -4 E 8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826448 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 -8 B 10 0 8 -10 8 C 12 -8 0 2 -12 D 8 10 -2 0 -4 E 8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826448 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 -8 B 10 0 8 -10 8 C 12 -8 0 2 -12 D 8 10 -2 0 -4 E 8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.454545 Sum of squares = 0.371900826448 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.545455 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3400: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) C D E B A (8) A D B E C (7) C D A B E (6) A B E D C (6) E B A C D (5) D C A B E (5) C E B A D (5) B E A D C (5) A D C B E (4) E B C D A (3) D C E B A (3) D A C B E (3) E B D C A (2) E B D A C (2) E B C A D (2) C E B D A (2) C D A E B (2) B A E D C (2) A C D B E (2) A C B E D (2) D B E C A (1) D B E A C (1) D B A E C (1) D A B E C (1) C E D B A (1) C A E B D (1) C A D E B (1) C A D B E (1) A E B D C (1) A E B C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 14 18 -4 B 10 0 8 6 -2 C -14 -8 0 -14 -6 D -18 -6 14 0 -6 E 4 2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 14 18 -4 B 10 0 8 6 -2 C -14 -8 0 -14 -6 D -18 -6 14 0 -6 E 4 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 A=25 D=15 B=7 so B is eliminated. Round 2 votes counts: E=31 C=27 A=27 D=15 so D is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:211 A:209 E:209 D:192 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 14 18 -4 B 10 0 8 6 -2 C -14 -8 0 -14 -6 D -18 -6 14 0 -6 E 4 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 18 -4 B 10 0 8 6 -2 C -14 -8 0 -14 -6 D -18 -6 14 0 -6 E 4 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 18 -4 B 10 0 8 6 -2 C -14 -8 0 -14 -6 D -18 -6 14 0 -6 E 4 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3401: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) B E D C A (6) E B A D C (5) D E C B A (5) A B E C D (5) E D C A B (4) E B D A C (4) E A B D C (4) A C B D E (4) D E B C A (3) D C B E A (3) C A B D E (3) B D C E A (3) E D B A C (2) E D A C B (2) E A D C B (2) C D B A E (2) C D A E B (2) C D A B E (2) C A D E B (2) B D E C A (2) B C A D E (2) B A E C D (2) B A C D E (2) A C D E B (2) A C B E D (2) A B C E D (2) E D B C A (1) E D A B C (1) E B D C A (1) E A D B C (1) D E C A B (1) D C E B A (1) D B E C A (1) C B A D E (1) B E D A C (1) B E A D C (1) B E A C D (1) B C D A E (1) A E C D B (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -6 -10 -24 B 0 0 4 8 -8 C 6 -4 0 -22 -16 D 10 -8 22 0 -2 E 24 8 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -6 -10 -24 B 0 0 4 8 -8 C 6 -4 0 -22 -16 D 10 -8 22 0 -2 E 24 8 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=21 B=21 A=19 C=12 so C is eliminated. Round 2 votes counts: E=27 D=27 A=24 B=22 so B is eliminated. Round 3 votes counts: E=36 D=33 A=31 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:211 B:202 C:182 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 -10 -24 B 0 0 4 8 -8 C 6 -4 0 -22 -16 D 10 -8 22 0 -2 E 24 8 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -10 -24 B 0 0 4 8 -8 C 6 -4 0 -22 -16 D 10 -8 22 0 -2 E 24 8 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -10 -24 B 0 0 4 8 -8 C 6 -4 0 -22 -16 D 10 -8 22 0 -2 E 24 8 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3402: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (10) B E D C A (9) B C E D A (7) A D E C B (7) C B A E D (6) C A B D E (6) D E A B C (5) C A B E D (5) B E D A C (5) B E C D A (4) B C A E D (4) E D B A C (3) D E B A C (3) C A E D B (3) C A D B E (2) B D E A C (2) A D E B C (2) E D B C A (1) E D A C B (1) E D A B C (1) D E A C B (1) D A E C B (1) C E D A B (1) C E B D A (1) C B E A D (1) C A E B D (1) C A D E B (1) B C E A D (1) B C A D E (1) B A D E C (1) A D C E B (1) A C D B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -8 10 8 B -2 0 0 12 14 C 8 0 0 12 6 D -10 -12 -12 0 -8 E -8 -14 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.448436 C: 0.551564 D: 0.000000 E: 0.000000 Sum of squares = 0.505317627328 Cumulative probabilities = A: 0.000000 B: 0.448436 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 10 8 B -2 0 0 12 14 C 8 0 0 12 6 D -10 -12 -12 0 -8 E -8 -14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=27 A=23 D=10 E=6 so E is eliminated. Round 2 votes counts: B=34 C=27 A=23 D=16 so D is eliminated. Round 3 votes counts: B=41 A=32 C=27 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 B:212 A:206 E:190 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 10 8 B -2 0 0 12 14 C 8 0 0 12 6 D -10 -12 -12 0 -8 E -8 -14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 10 8 B -2 0 0 12 14 C 8 0 0 12 6 D -10 -12 -12 0 -8 E -8 -14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 10 8 B -2 0 0 12 14 C 8 0 0 12 6 D -10 -12 -12 0 -8 E -8 -14 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3403: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E B C A D (7) E A B C D (7) D C B A E (7) A D E C B (6) A E D B C (5) C B E A D (4) C B D E A (4) B C E D A (4) D A E B C (3) A E C D B (3) A E C B D (3) E B C D A (2) D B C E A (2) D B C A E (2) C B D A E (2) B C D E A (2) A E D C B (2) A D C E B (2) A D C B E (2) E D B A C (1) E D A B C (1) E B A C D (1) E A D B C (1) E A C B D (1) E A B D C (1) D E A B C (1) D B E C A (1) D B A C E (1) D A B C E (1) C E B A D (1) C D B A E (1) C B E D A (1) C B A E D (1) C A B E D (1) B E D C A (1) B E C D A (1) B E C A D (1) A D E B C (1) A C E D B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 6 10 6 10 B -6 0 -6 -8 0 C -10 6 0 0 0 D -6 8 0 0 -4 E -10 0 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 6 10 B -6 0 -6 -8 0 C -10 6 0 0 0 D -6 8 0 0 -4 E -10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 E=22 C=15 B=9 so B is eliminated. Round 2 votes counts: D=27 A=27 E=25 C=21 so C is eliminated. Round 3 votes counts: D=36 E=35 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:216 D:199 C:198 E:197 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 6 10 B -6 0 -6 -8 0 C -10 6 0 0 0 D -6 8 0 0 -4 E -10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 6 10 B -6 0 -6 -8 0 C -10 6 0 0 0 D -6 8 0 0 -4 E -10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 6 10 B -6 0 -6 -8 0 C -10 6 0 0 0 D -6 8 0 0 -4 E -10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3404: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) C E B D A (6) A D B C E (6) D C E B A (5) E C D B A (4) E C B A D (4) E B C A D (4) D A B C E (4) A D B E C (4) D A E C B (3) B E C A D (3) B C E A D (3) A B E D C (3) A B E C D (3) E D C A B (2) E C D A B (2) E C B D A (2) E A C D B (2) D E A C B (2) D A C E B (2) D A C B E (2) C B E D A (2) B A C E D (2) A E B C D (2) A B D C E (2) E A C B D (1) D E C B A (1) D E C A B (1) D C E A B (1) D C B A E (1) D C A B E (1) C D E B A (1) C B D E A (1) B E A C D (1) B D C A E (1) B D A C E (1) B C D E A (1) B A E C D (1) A E D C B (1) A E D B C (1) A E C B D (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -6 -6 -12 B 2 0 2 4 0 C 6 -2 0 6 -2 D 6 -4 -6 0 -14 E 12 0 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.688097 C: 0.000000 D: 0.000000 E: 0.311903 Sum of squares = 0.570760799442 Cumulative probabilities = A: 0.000000 B: 0.688097 C: 0.688097 D: 0.688097 E: 1.000000 A B C D E A 0 -2 -6 -6 -12 B 2 0 2 4 0 C 6 -2 0 6 -2 D 6 -4 -6 0 -14 E 12 0 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 E=21 B=20 C=10 so C is eliminated. Round 2 votes counts: E=27 A=26 D=24 B=23 so B is eliminated. Round 3 votes counts: E=43 A=29 D=28 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:204 C:204 D:191 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -12 B 2 0 2 4 0 C 6 -2 0 6 -2 D 6 -4 -6 0 -14 E 12 0 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -12 B 2 0 2 4 0 C 6 -2 0 6 -2 D 6 -4 -6 0 -14 E 12 0 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -12 B 2 0 2 4 0 C 6 -2 0 6 -2 D 6 -4 -6 0 -14 E 12 0 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3405: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) B D E C A (7) C E D A B (6) B A D E C (6) A B D E C (6) C A E D B (5) B D A E C (5) A C E D B (5) A B C D E (4) E C D A B (3) D E C B A (3) D B E C A (3) A C E B D (3) A B D C E (3) A B C E D (3) E D C B A (2) D E B C A (2) C E D B A (2) C E A B D (2) C B E D A (2) B D E A C (2) B C E D A (2) B A D C E (2) B A C D E (2) E D C A B (1) E C D B A (1) D B E A C (1) D A B E C (1) C E B D A (1) C B E A D (1) B D C A E (1) B D A C E (1) B C A E D (1) A E C D B (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -6 8 2 B -6 0 4 10 8 C 6 -4 0 6 10 D -8 -10 -6 0 0 E -2 -8 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 8 2 B -6 0 4 10 8 C 6 -4 0 6 10 D -8 -10 -6 0 0 E -2 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=27 A=27 D=10 E=7 so E is eliminated. Round 2 votes counts: C=31 B=29 A=27 D=13 so D is eliminated. Round 3 votes counts: C=37 B=35 A=28 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:208 A:205 E:190 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 8 2 B -6 0 4 10 8 C 6 -4 0 6 10 D -8 -10 -6 0 0 E -2 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 8 2 B -6 0 4 10 8 C 6 -4 0 6 10 D -8 -10 -6 0 0 E -2 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 8 2 B -6 0 4 10 8 C 6 -4 0 6 10 D -8 -10 -6 0 0 E -2 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343750000002 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3406: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) D A E C B (9) B A D C E (9) B C E A D (8) A D B E C (8) D A B E C (7) C E B D A (7) B C A E D (6) A B D E C (6) D A E B C (5) D E A C B (4) C B E A D (4) B A C E D (4) D E C A B (3) C E D B A (3) E D C A B (2) B A C D E (2) C E B A D (1) B D A C E (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 6 10 -4 14 B -6 0 16 0 14 C -10 -16 0 -8 -6 D 4 0 8 0 10 E -14 -14 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.259176 C: 0.000000 D: 0.740824 E: 0.000000 Sum of squares = 0.615992646214 Cumulative probabilities = A: 0.000000 B: 0.259176 C: 0.259176 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 -4 14 B -6 0 16 0 14 C -10 -16 0 -8 -6 D 4 0 8 0 10 E -14 -14 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000059334 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=28 C=15 A=14 E=11 so E is eliminated. Round 2 votes counts: B=32 D=30 C=24 A=14 so A is eliminated. Round 3 votes counts: D=38 B=38 C=24 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:212 D:211 E:184 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -4 14 B -6 0 16 0 14 C -10 -16 0 -8 -6 D 4 0 8 0 10 E -14 -14 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000059334 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -4 14 B -6 0 16 0 14 C -10 -16 0 -8 -6 D 4 0 8 0 10 E -14 -14 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000059334 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -4 14 B -6 0 16 0 14 C -10 -16 0 -8 -6 D 4 0 8 0 10 E -14 -14 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000059334 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3407: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (17) E B A C D (14) E D C A B (7) B A C E D (7) E B A D C (6) E D C B A (5) D E C A B (5) C A B D E (5) D C E A B (4) C D A B E (4) C A D B E (4) B E A C D (4) B A E C D (3) E D B A C (2) D E C B A (2) D C A E B (2) B C A D E (2) B A C D E (2) E D B C A (1) E B D A C (1) D C E B A (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -18 -2 -4 B 0 0 -12 -8 0 C 18 12 0 -6 0 D 2 8 6 0 -2 E 4 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.249613 D: 0.000000 E: 0.750387 Sum of squares = 0.625387251362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.249613 D: 0.249613 E: 1.000000 A B C D E A 0 0 -18 -2 -4 B 0 0 -12 -8 0 C 18 12 0 -6 0 D 2 8 6 0 -2 E 4 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000247846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=31 B=18 C=13 A=2 so A is eliminated. Round 2 votes counts: E=36 D=31 B=20 C=13 so C is eliminated. Round 3 votes counts: D=39 E=36 B=25 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:212 D:207 E:203 B:190 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -18 -2 -4 B 0 0 -12 -8 0 C 18 12 0 -6 0 D 2 8 6 0 -2 E 4 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000247846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 -2 -4 B 0 0 -12 -8 0 C 18 12 0 -6 0 D 2 8 6 0 -2 E 4 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000247846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 -2 -4 B 0 0 -12 -8 0 C 18 12 0 -6 0 D 2 8 6 0 -2 E 4 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000247846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3408: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) B E C D A (7) E B D A C (6) C E B A D (6) E B C A D (5) D A E B C (5) A D C E B (4) E B A D C (3) C B D E A (3) B E D C A (3) B D E A C (3) A D E C B (3) A D E B C (3) E A B D C (2) E A B C D (2) D B A C E (2) D A B E C (2) C E A B D (2) C D B A E (2) C D A B E (2) C A D E B (2) C A D B E (2) B E D A C (2) B C E D A (2) A E D B C (2) A C D E B (2) E C B A D (1) E B D C A (1) E B C D A (1) E A D B C (1) E A C B D (1) D C A B E (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B C E (1) C B E A D (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B D E (1) A E D C B (1) Total count = 100 A B C D E A 0 -4 8 -8 -6 B 4 0 12 8 -10 C -8 -12 0 -10 -12 D 8 -8 10 0 -2 E 6 10 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 8 -8 -6 B 4 0 12 8 -10 C -8 -12 0 -10 -12 D 8 -8 10 0 -2 E 6 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=23 D=21 B=17 A=15 so A is eliminated. Round 2 votes counts: D=31 E=26 C=26 B=17 so B is eliminated. Round 3 votes counts: E=38 D=34 C=28 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:207 D:204 A:195 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 8 -8 -6 B 4 0 12 8 -10 C -8 -12 0 -10 -12 D 8 -8 10 0 -2 E 6 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -8 -6 B 4 0 12 8 -10 C -8 -12 0 -10 -12 D 8 -8 10 0 -2 E 6 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -8 -6 B 4 0 12 8 -10 C -8 -12 0 -10 -12 D 8 -8 10 0 -2 E 6 10 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3409: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) E B A D C (6) C B D E A (6) A D B E C (6) C D B A E (5) C B D A E (5) E A D B C (4) C D A B E (4) E C B A D (3) E B C A D (3) D A C E B (3) D A B C E (3) B E C D A (3) B E A D C (3) B C E D A (3) E C A D B (2) E C A B D (2) E B A C D (2) D B A C E (2) C E B D A (2) C D A E B (2) C B E D A (2) B D A C E (2) B C D A E (2) A D E C B (2) A D E B C (2) E B C D A (1) E A D C B (1) E A C D B (1) E A C B D (1) D C B A E (1) D C A B E (1) C E D A B (1) B E C A D (1) B D C A E (1) B D A E C (1) B A E D C (1) B A D E C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 0 0 -8 B 12 0 8 16 8 C 0 -8 0 2 -6 D 0 -16 -2 0 2 E 8 -8 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 0 -8 B 12 0 8 16 8 C 0 -8 0 2 -6 D 0 -16 -2 0 2 E 8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=27 B=18 A=12 D=10 so D is eliminated. Round 2 votes counts: E=33 C=29 B=20 A=18 so A is eliminated. Round 3 votes counts: E=37 C=34 B=29 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:202 C:194 D:192 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 0 -8 B 12 0 8 16 8 C 0 -8 0 2 -6 D 0 -16 -2 0 2 E 8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 0 -8 B 12 0 8 16 8 C 0 -8 0 2 -6 D 0 -16 -2 0 2 E 8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 0 -8 B 12 0 8 16 8 C 0 -8 0 2 -6 D 0 -16 -2 0 2 E 8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3410: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (16) E A D B C (10) C B A D E (9) E D A C B (8) B C A D E (7) A D E B C (6) E D A B C (5) D E A B C (4) C B E A D (3) C B D E A (3) B C A E D (3) A E D B C (3) E D C A B (2) D E A C B (2) C E D B A (2) C D B A E (2) A D B E C (2) E C B A D (1) E A D C B (1) E A B C D (1) D E C A B (1) D C E A B (1) D A B C E (1) C E B D A (1) C E B A D (1) C B A E D (1) B D C A E (1) B A C D E (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -8 2 8 B 2 0 -8 0 0 C 8 8 0 4 4 D -2 0 -4 0 12 E -8 0 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 2 8 B 2 0 -8 0 0 C 8 8 0 4 4 D -2 0 -4 0 12 E -8 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=28 A=13 B=12 D=9 so D is eliminated. Round 2 votes counts: C=39 E=35 A=14 B=12 so B is eliminated. Round 3 votes counts: C=50 E=35 A=15 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:203 A:200 B:197 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 2 8 B 2 0 -8 0 0 C 8 8 0 4 4 D -2 0 -4 0 12 E -8 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 2 8 B 2 0 -8 0 0 C 8 8 0 4 4 D -2 0 -4 0 12 E -8 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 2 8 B 2 0 -8 0 0 C 8 8 0 4 4 D -2 0 -4 0 12 E -8 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3411: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) C E A D B (8) E A B D C (5) C A D B E (5) A C E B D (5) E C A B D (4) D B E A C (4) D B A C E (4) B D E A C (4) E D B C A (3) E B D A C (3) E A C B D (3) D B C A E (3) C A E B D (3) A E C B D (3) E D B A C (2) E A B C D (2) D B C E A (2) C E D B A (2) C D B E A (2) C D B A E (2) C A E D B (2) B D A E C (2) A C B D E (2) A B D C E (2) E C D A B (1) E C A D B (1) E B A D C (1) D E B C A (1) D B A E C (1) C E D A B (1) C D A B E (1) C A B D E (1) B A D E C (1) B A D C E (1) A E B D C (1) A E B C D (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 8 -14 B -8 0 -8 -6 -20 C 0 8 0 16 -6 D -8 6 -16 0 -20 E 14 20 6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 8 -14 B -8 0 -8 -6 -20 C 0 8 0 16 -6 D -8 6 -16 0 -20 E 14 20 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=27 A=17 D=15 B=8 so B is eliminated. Round 2 votes counts: E=33 C=27 D=21 A=19 so A is eliminated. Round 3 votes counts: E=39 C=35 D=26 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:230 C:209 A:201 D:181 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 8 -14 B -8 0 -8 -6 -20 C 0 8 0 16 -6 D -8 6 -16 0 -20 E 14 20 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 8 -14 B -8 0 -8 -6 -20 C 0 8 0 16 -6 D -8 6 -16 0 -20 E 14 20 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 8 -14 B -8 0 -8 -6 -20 C 0 8 0 16 -6 D -8 6 -16 0 -20 E 14 20 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3412: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (12) A C E B D (12) D E A C B (11) D B E C A (7) C A B E D (7) E A C B D (5) D E B A C (5) B C A D E (5) D E A B C (4) E D B A C (3) E A C D B (3) C B A E D (3) C A B D E (3) D A C E B (2) B E C D A (2) B C E A D (2) A E C D B (2) A C D E B (2) E A D C B (1) D A E C B (1) D A C B E (1) C A D B E (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D A E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 8 20 10 B -14 0 -12 12 -6 C -8 12 0 24 8 D -20 -12 -24 0 -8 E -10 6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 20 10 B -14 0 -12 12 -6 C -8 12 0 24 8 D -20 -12 -24 0 -8 E -10 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=25 A=18 C=14 E=12 so E is eliminated. Round 2 votes counts: D=34 A=27 B=25 C=14 so C is eliminated. Round 3 votes counts: A=38 D=34 B=28 so B is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 C:218 E:198 B:190 D:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 20 10 B -14 0 -12 12 -6 C -8 12 0 24 8 D -20 -12 -24 0 -8 E -10 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 20 10 B -14 0 -12 12 -6 C -8 12 0 24 8 D -20 -12 -24 0 -8 E -10 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 20 10 B -14 0 -12 12 -6 C -8 12 0 24 8 D -20 -12 -24 0 -8 E -10 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3413: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) C E D B A (9) D E A B C (7) E D A C B (6) B A C E D (6) D E A C B (5) C B E D A (5) B A C D E (5) D E C B A (4) B A D C E (4) A D E B C (4) A B D E C (4) C D E B A (3) C B D E A (3) B C A E D (3) B A D E C (3) E C D A B (2) D E C A B (2) A B E D C (2) E D C B A (1) D E B C A (1) D A B E C (1) C E B D A (1) C B E A D (1) C A B E D (1) A E D B C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 2 -26 -26 B -2 0 -12 -18 -18 C -2 12 0 -18 -12 D 26 18 18 0 -4 E 26 18 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 -26 -26 B -2 0 -12 -18 -18 C -2 12 0 -18 -12 D 26 18 18 0 -4 E 26 18 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 E=22 B=21 D=20 A=14 so A is eliminated. Round 2 votes counts: B=30 D=24 E=23 C=23 so E is eliminated. Round 3 votes counts: D=45 B=30 C=25 so C is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:230 D:229 C:190 A:176 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 -26 -26 B -2 0 -12 -18 -18 C -2 12 0 -18 -12 D 26 18 18 0 -4 E 26 18 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -26 -26 B -2 0 -12 -18 -18 C -2 12 0 -18 -12 D 26 18 18 0 -4 E 26 18 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -26 -26 B -2 0 -12 -18 -18 C -2 12 0 -18 -12 D 26 18 18 0 -4 E 26 18 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3414: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) D C B A E (8) C B A E D (8) A B C E D (8) E A B C D (7) D C E B A (6) E A B D C (4) E B A C D (3) D C B E A (3) D A E B C (3) C B D A E (3) C B A D E (3) B A C E D (3) E D A B C (2) E B C A D (2) E A D B C (2) D E C B A (2) C B E A D (2) A E B C D (2) A B E C D (2) E C B A D (1) D E C A B (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A C B E (1) C D B A E (1) C D A B E (1) C B E D A (1) C B D E A (1) C A B E D (1) B E C A D (1) B C E A D (1) B C A E D (1) A D E B C (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 0 0 8 4 B 0 0 2 8 4 C 0 -2 0 4 14 D -8 -8 -4 0 -2 E -4 -4 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.392268 B: 0.607732 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.523212148133 Cumulative probabilities = A: 0.392268 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 8 4 B 0 0 2 8 4 C 0 -2 0 4 14 D -8 -8 -4 0 -2 E -4 -4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=21 C=21 A=15 B=6 so B is eliminated. Round 2 votes counts: D=37 C=23 E=22 A=18 so A is eliminated. Round 3 votes counts: D=40 C=34 E=26 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:207 A:206 E:190 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 8 4 B 0 0 2 8 4 C 0 -2 0 4 14 D -8 -8 -4 0 -2 E -4 -4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 8 4 B 0 0 2 8 4 C 0 -2 0 4 14 D -8 -8 -4 0 -2 E -4 -4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 8 4 B 0 0 2 8 4 C 0 -2 0 4 14 D -8 -8 -4 0 -2 E -4 -4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3415: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (12) B A E D C (11) E A C B D (9) E A B C D (8) C E A D B (7) C D E A B (6) B A D E C (5) A E B D C (5) D C B E A (4) D B C A E (4) C E D A B (4) B D A E C (4) A E B C D (4) A B E D C (4) E A B D C (2) D B A C E (2) C D E B A (2) C D B E A (2) E C A D B (1) E C A B D (1) D B A E C (1) C D B A E (1) B D A C E (1) Total count = 100 A B C D E A 0 2 12 14 8 B -2 0 2 8 2 C -12 -2 0 -10 -10 D -14 -8 10 0 -12 E -8 -2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 14 8 B -2 0 2 8 2 C -12 -2 0 -10 -10 D -14 -8 10 0 -12 E -8 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 E=21 B=21 A=13 so A is eliminated. Round 2 votes counts: E=30 B=25 D=23 C=22 so C is eliminated. Round 3 votes counts: E=41 D=34 B=25 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:218 E:206 B:205 D:188 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 14 8 B -2 0 2 8 2 C -12 -2 0 -10 -10 D -14 -8 10 0 -12 E -8 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 14 8 B -2 0 2 8 2 C -12 -2 0 -10 -10 D -14 -8 10 0 -12 E -8 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 14 8 B -2 0 2 8 2 C -12 -2 0 -10 -10 D -14 -8 10 0 -12 E -8 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3416: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) B C D A E (6) A E C D B (6) E B D C A (5) E A D B C (5) B D C E A (5) E A B D C (4) C D A B E (4) B D E C A (4) A C D E B (4) E D A C B (3) E A D C B (3) C D B A E (3) A E C B D (3) E B D A C (2) E A C B D (2) E A B C D (2) D C E A B (2) D C B A E (2) D B E C A (2) D B C E A (2) D B C A E (2) C B D A E (2) C A D B E (2) B C A D E (2) A C D B E (2) E D B C A (1) D E A C B (1) D C B E A (1) C B A D E (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B A C E D (1) A E D C B (1) A C E D B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 2 0 -10 B -10 0 -6 -10 -8 C -2 6 0 6 -10 D 0 10 -6 0 -2 E 10 8 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 2 0 -10 B -10 0 -6 -10 -8 C -2 6 0 6 -10 D 0 10 -6 0 -2 E 10 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=22 A=19 C=13 D=12 so D is eliminated. Round 2 votes counts: E=35 B=28 A=19 C=18 so C is eliminated. Round 3 votes counts: E=37 B=37 A=26 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:201 D:201 C:200 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 0 -10 B -10 0 -6 -10 -8 C -2 6 0 6 -10 D 0 10 -6 0 -2 E 10 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 0 -10 B -10 0 -6 -10 -8 C -2 6 0 6 -10 D 0 10 -6 0 -2 E 10 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 0 -10 B -10 0 -6 -10 -8 C -2 6 0 6 -10 D 0 10 -6 0 -2 E 10 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3417: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) C E A B D (6) A E B C D (6) A B E D C (6) C E D A B (5) E C B A D (4) E C A B D (4) E B D A C (4) E B A C D (4) D B A C E (4) D B A E C (3) D A B C E (3) B A D E C (3) A D B C E (3) A B D E C (3) E D B C A (2) E A B C D (2) D C E B A (2) D C B A E (2) D C A B E (2) D B E A C (2) C E D B A (2) C D E B A (2) C D A B E (2) C A D B E (2) B D A E C (2) A B E C D (2) E D B A C (1) E B D C A (1) E B C A D (1) E B A D C (1) D E C B A (1) D C B E A (1) C E A D B (1) C D E A B (1) C A D E B (1) B D E A C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 16 14 4 B 0 0 22 16 -4 C -16 -22 0 -8 -20 D -14 -16 8 0 -20 E -4 4 20 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.557045 B: 0.442955 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.506508372159 Cumulative probabilities = A: 0.557045 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 14 4 B 0 0 22 16 -4 C -16 -22 0 -8 -20 D -14 -16 8 0 -20 E -4 4 20 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500617 B: 0.499383 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000761522 Cumulative probabilities = A: 0.500617 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=22 A=21 D=20 B=13 so B is eliminated. Round 2 votes counts: A=31 E=24 D=23 C=22 so C is eliminated. Round 3 votes counts: E=38 A=34 D=28 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:220 A:217 B:217 D:179 C:167 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 16 14 4 B 0 0 22 16 -4 C -16 -22 0 -8 -20 D -14 -16 8 0 -20 E -4 4 20 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500617 B: 0.499383 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000761522 Cumulative probabilities = A: 0.500617 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 14 4 B 0 0 22 16 -4 C -16 -22 0 -8 -20 D -14 -16 8 0 -20 E -4 4 20 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500617 B: 0.499383 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000761522 Cumulative probabilities = A: 0.500617 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 14 4 B 0 0 22 16 -4 C -16 -22 0 -8 -20 D -14 -16 8 0 -20 E -4 4 20 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500617 B: 0.499383 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000761522 Cumulative probabilities = A: 0.500617 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3418: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) C D A E B (8) A E B C D (7) B E A D C (6) D C B E A (5) D C A B E (5) C D A B E (5) B E D A C (5) A C D E B (5) E A B C D (4) E B A C D (3) B E D C A (3) B D E C A (3) B D C E A (3) A E B D C (3) A B E D C (3) D B C E A (2) C D E B A (2) C D B E A (2) B A E D C (2) E C B D A (1) E C B A D (1) E B C D A (1) E B A D C (1) E A B D C (1) D B A C E (1) C E D A B (1) C E A D B (1) C D B A E (1) C A E D B (1) B D A C E (1) A E C D B (1) A E C B D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -8 -16 10 B 4 0 -2 -2 14 C 8 2 0 -8 8 D 16 2 8 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -16 10 B 4 0 -2 -2 14 C 8 2 0 -8 8 D 16 2 8 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 D=22 A=22 C=21 E=12 so E is eliminated. Round 2 votes counts: B=28 A=27 C=23 D=22 so D is eliminated. Round 3 votes counts: C=42 B=31 A=27 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:216 B:207 C:205 A:191 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -16 10 B 4 0 -2 -2 14 C 8 2 0 -8 8 D 16 2 8 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -16 10 B 4 0 -2 -2 14 C 8 2 0 -8 8 D 16 2 8 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -16 10 B 4 0 -2 -2 14 C 8 2 0 -8 8 D 16 2 8 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3419: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) E B A C D (7) B A E C D (6) B A D C E (6) D C A B E (5) E C D B A (4) D C E A B (4) C E D A B (4) A B C E D (4) E C D A B (3) C D E A B (3) B D A C E (3) B A E D C (3) E C A D B (2) E C A B D (2) E B D C A (2) E B C A D (2) D C A E B (2) D B A C E (2) B E A D C (2) B E A C D (2) B A D E C (2) A C D B E (2) A C B E D (2) E D B C A (1) E C B A D (1) E B D A C (1) E B A D C (1) D E C B A (1) D B E C A (1) D A C B E (1) D A B C E (1) C E A B D (1) C D A B E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D A C (1) A E B C D (1) A D C B E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 2 6 -4 B 10 0 2 4 -2 C -2 -2 0 4 -6 D -6 -4 -4 0 -22 E 4 2 6 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 2 6 -4 B 10 0 2 4 -2 C -2 -2 0 4 -6 D -6 -4 -4 0 -22 E 4 2 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=25 D=17 A=13 C=12 so C is eliminated. Round 2 votes counts: E=38 B=25 D=21 A=16 so A is eliminated. Round 3 votes counts: E=40 B=34 D=26 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:207 A:197 C:197 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 6 -4 B 10 0 2 4 -2 C -2 -2 0 4 -6 D -6 -4 -4 0 -22 E 4 2 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 6 -4 B 10 0 2 4 -2 C -2 -2 0 4 -6 D -6 -4 -4 0 -22 E 4 2 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 6 -4 B 10 0 2 4 -2 C -2 -2 0 4 -6 D -6 -4 -4 0 -22 E 4 2 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3420: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (11) E D C A B (9) E D A B C (7) E B A D C (7) C B A D E (5) D A C B E (4) C E D B A (4) A D B E C (4) A B D C E (4) E C D B A (3) E C D A B (3) E D A C B (2) E C B D A (2) E A D B C (2) D E C A B (2) D E A B C (2) D C E A B (2) C D E A B (2) C D B A E (2) C D A B E (2) C B E A D (2) C B D A E (2) B C A D E (2) B A E D C (2) B A E C D (2) E A B D C (1) D E A C B (1) D C A E B (1) D A B E C (1) C E B A D (1) B C E A D (1) B C A E D (1) B A D C E (1) A D E B C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 8 -2 -6 B -4 0 2 -10 -4 C -8 -2 0 -10 -4 D 2 10 10 0 2 E 6 4 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -2 -6 B -4 0 2 -10 -4 C -8 -2 0 -10 -4 D 2 10 10 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=20 B=20 D=13 A=11 so A is eliminated. Round 2 votes counts: E=36 B=25 C=20 D=19 so D is eliminated. Round 3 votes counts: E=42 B=31 C=27 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:206 A:202 B:192 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -2 -6 B -4 0 2 -10 -4 C -8 -2 0 -10 -4 D 2 10 10 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 -6 B -4 0 2 -10 -4 C -8 -2 0 -10 -4 D 2 10 10 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 -6 B -4 0 2 -10 -4 C -8 -2 0 -10 -4 D 2 10 10 0 2 E 6 4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3421: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (11) D C B A E (10) E A D B C (7) C B D E A (6) D A E C B (5) B C D E A (5) E A B D C (4) B C E A D (4) A E D B C (4) E B A C D (3) E A D C B (3) D C A B E (3) D A C E B (3) B E A C D (3) B C D A E (3) A E D C B (3) A D E C B (3) D C A E B (2) D A C B E (2) C D B E A (2) C D B A E (2) C B D A E (2) B C E D A (2) A E B D C (2) E C A D B (1) E B C A D (1) C B E D A (1) B E C A D (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 8 10 4 -8 B -8 0 4 0 -6 C -10 -4 0 -4 -4 D -4 0 4 0 -2 E 8 6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 10 4 -8 B -8 0 4 0 -6 C -10 -4 0 -4 -4 D -4 0 4 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=25 B=19 C=13 A=13 so C is eliminated. Round 2 votes counts: E=30 D=29 B=28 A=13 so A is eliminated. Round 3 votes counts: E=40 D=32 B=28 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:207 D:199 B:195 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 4 -8 B -8 0 4 0 -6 C -10 -4 0 -4 -4 D -4 0 4 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 4 -8 B -8 0 4 0 -6 C -10 -4 0 -4 -4 D -4 0 4 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 4 -8 B -8 0 4 0 -6 C -10 -4 0 -4 -4 D -4 0 4 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3422: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (10) D C A B E (9) B A E D C (9) A D C B E (8) E B C D A (6) A C D B E (6) C D A E B (5) B E A D C (5) A C D E B (5) E B A C D (4) C D E A B (4) E C D B A (3) B A D C E (3) A B D C E (3) E C D A B (2) D A C B E (2) B E D C A (2) B D C A E (2) B A D E C (2) A E C D B (2) E C B D A (1) E A B C D (1) D C B E A (1) D C A E B (1) C A D E B (1) B A E C D (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 4 22 B -2 0 -2 -2 8 C -6 2 0 -16 4 D -4 2 16 0 6 E -22 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999129 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 4 22 B -2 0 -2 -2 8 C -6 2 0 -16 4 D -4 2 16 0 6 E -22 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 B=24 D=13 C=10 so C is eliminated. Round 2 votes counts: E=27 A=27 B=24 D=22 so D is eliminated. Round 3 votes counts: A=44 E=31 B=25 so B is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:210 B:201 C:192 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 22 B -2 0 -2 -2 8 C -6 2 0 -16 4 D -4 2 16 0 6 E -22 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 22 B -2 0 -2 -2 8 C -6 2 0 -16 4 D -4 2 16 0 6 E -22 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 22 B -2 0 -2 -2 8 C -6 2 0 -16 4 D -4 2 16 0 6 E -22 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3423: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) B A E C D (6) A C B E D (5) E D B C A (4) E C D B A (4) E B D A C (4) C A D E B (4) B E D A C (4) E B C A D (3) D E C B A (3) D E B C A (3) D C A E B (3) C E A B D (3) C A E B D (3) A B C E D (3) E C B D A (2) E C B A D (2) E C A B D (2) E B D C A (2) D E B A C (2) D C E A B (2) D B E A C (2) D B A C E (2) D A B C E (2) B E A C D (2) B D E A C (2) A C B D E (2) A B C D E (2) E D C B A (1) E B C D A (1) D B A E C (1) D A C B E (1) C E A D B (1) C D A E B (1) B D A C E (1) B A D E C (1) B A C E D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -10 6 2 B 6 0 0 26 0 C 10 0 0 16 -2 D -6 -26 -16 0 -28 E -2 0 2 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.681911 C: 0.000000 D: 0.000000 E: 0.318089 Sum of squares = 0.56618342494 Cumulative probabilities = A: 0.000000 B: 0.681911 C: 0.681911 D: 0.681911 E: 1.000000 A B C D E A 0 -6 -10 6 2 B 6 0 0 26 0 C 10 0 0 16 -2 D -6 -26 -16 0 -28 E -2 0 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500003 C: 0.000000 D: 0.000000 E: 0.499997 Sum of squares = 0.499999991086 Cumulative probabilities = A: 0.000000 B: 0.500003 C: 0.500003 D: 0.500003 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 D=21 B=17 A=14 so A is eliminated. Round 2 votes counts: C=30 E=25 B=23 D=22 so D is eliminated. Round 3 votes counts: C=36 E=33 B=31 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:216 E:214 C:212 A:196 D:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 6 2 B 6 0 0 26 0 C 10 0 0 16 -2 D -6 -26 -16 0 -28 E -2 0 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500003 C: 0.000000 D: 0.000000 E: 0.499997 Sum of squares = 0.499999991086 Cumulative probabilities = A: 0.000000 B: 0.500003 C: 0.500003 D: 0.500003 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 6 2 B 6 0 0 26 0 C 10 0 0 16 -2 D -6 -26 -16 0 -28 E -2 0 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500003 C: 0.000000 D: 0.000000 E: 0.499997 Sum of squares = 0.499999991086 Cumulative probabilities = A: 0.000000 B: 0.500003 C: 0.500003 D: 0.500003 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 6 2 B 6 0 0 26 0 C 10 0 0 16 -2 D -6 -26 -16 0 -28 E -2 0 2 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500003 C: 0.000000 D: 0.000000 E: 0.499997 Sum of squares = 0.499999991086 Cumulative probabilities = A: 0.000000 B: 0.500003 C: 0.500003 D: 0.500003 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3424: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) E D A B C (6) E B A D C (6) B E A C D (5) E D C A B (4) E C B A D (4) E B A C D (4) C D A B E (4) A B E D C (4) D E A C B (3) D A E B C (3) D A B C E (3) C B E A D (3) C B D A E (3) C B A D E (3) A B D C E (3) E C D B A (2) E B C A D (2) D E A B C (2) D C A E B (2) D C A B E (2) D A C B E (2) C E D B A (2) B E A D C (2) B A C D E (2) A D B E C (2) A B D E C (2) E D A C B (1) E C B D A (1) D C E A B (1) C E B A D (1) C D E B A (1) C D E A B (1) C B A E D (1) B C E A D (1) B C A E D (1) B A E D C (1) B A D C E (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 6 22 0 -4 B -6 0 18 2 8 C -22 -18 0 -16 -22 D 0 -2 16 0 -4 E 4 -8 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.358024691352 Cumulative probabilities = A: 0.444444 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 6 22 0 -4 B -6 0 18 2 8 C -22 -18 0 -16 -22 D 0 -2 16 0 -4 E 4 -8 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.358024690808 Cumulative probabilities = A: 0.444444 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=25 C=19 B=14 A=12 so A is eliminated. Round 2 votes counts: E=30 D=28 B=23 C=19 so C is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:212 B:211 E:211 D:205 C:161 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 22 0 -4 B -6 0 18 2 8 C -22 -18 0 -16 -22 D 0 -2 16 0 -4 E 4 -8 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.358024690808 Cumulative probabilities = A: 0.444444 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 22 0 -4 B -6 0 18 2 8 C -22 -18 0 -16 -22 D 0 -2 16 0 -4 E 4 -8 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.358024690808 Cumulative probabilities = A: 0.444444 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 22 0 -4 B -6 0 18 2 8 C -22 -18 0 -16 -22 D 0 -2 16 0 -4 E 4 -8 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.358024690808 Cumulative probabilities = A: 0.444444 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3425: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D E A (10) B C D A E (10) E C B D A (6) E B C D A (6) C D B A E (6) E A B D C (5) A E D C B (5) E A C D B (4) B D C A E (4) A E D B C (4) A D B C E (4) A B D C E (4) E C D B A (3) D C B A E (3) C B D A E (3) E A D B C (2) A D C B E (2) E C D A B (1) E C B A D (1) E A B C D (1) D C A B E (1) C B D E A (1) A D E C B (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -10 -8 -2 B 6 0 2 4 -2 C 10 -2 0 4 -2 D 8 -4 -4 0 0 E 2 2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.149328 E: 0.850672 Sum of squares = 0.745941885701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.149328 E: 1.000000 A B C D E A 0 -6 -10 -8 -2 B 6 0 2 4 -2 C 10 -2 0 4 -2 D 8 -4 -4 0 0 E 2 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555578081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=24 A=22 C=10 D=4 so D is eliminated. Round 2 votes counts: E=40 B=24 A=22 C=14 so C is eliminated. Round 3 votes counts: E=40 B=37 A=23 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:205 C:205 E:203 D:200 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 -2 B 6 0 2 4 -2 C 10 -2 0 4 -2 D 8 -4 -4 0 0 E 2 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555578081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 -2 B 6 0 2 4 -2 C 10 -2 0 4 -2 D 8 -4 -4 0 0 E 2 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555578081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 -2 B 6 0 2 4 -2 C 10 -2 0 4 -2 D 8 -4 -4 0 0 E 2 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555578081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3426: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) E D C B A (5) E C B D A (5) B C A E D (5) B C A D E (5) A D B C E (5) A B C D E (5) E D C A B (4) E D A C B (4) E B C A D (4) D E A C B (4) D A E C B (4) E A D B C (3) C B E A D (3) B A C D E (3) E C D B A (2) D E C B A (2) D E C A B (2) D A E B C (2) D A C B E (2) D A B C E (2) C B A D E (2) A B E C D (2) A B C E D (2) E D B C A (1) E B C D A (1) E B A D C (1) E A B C D (1) D E A B C (1) D C E A B (1) D C B E A (1) D C B A E (1) C B D E A (1) C B D A E (1) C A D B E (1) B A C E D (1) A E D B C (1) A E B D C (1) A E B C D (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 6 -4 -4 B -12 0 10 -10 -14 C -6 -10 0 -10 -16 D 4 10 10 0 -6 E 4 14 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 6 -4 -4 B -12 0 10 -10 -14 C -6 -10 0 -10 -16 D 4 10 10 0 -6 E 4 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=22 A=19 B=14 C=8 so C is eliminated. Round 2 votes counts: E=37 D=22 B=21 A=20 so A is eliminated. Round 3 votes counts: E=40 B=31 D=29 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:209 A:205 B:187 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 6 -4 -4 B -12 0 10 -10 -14 C -6 -10 0 -10 -16 D 4 10 10 0 -6 E 4 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 -4 B -12 0 10 -10 -14 C -6 -10 0 -10 -16 D 4 10 10 0 -6 E 4 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 -4 B -12 0 10 -10 -14 C -6 -10 0 -10 -16 D 4 10 10 0 -6 E 4 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3427: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (7) E B A C D (5) A E D B C (5) A D C B E (5) E D A B C (4) E B A D C (4) C B D A E (4) B E A C D (4) E D A C B (3) E B D C A (3) E B D A C (3) E A D B C (3) D E A C B (3) D A C E B (3) C A B D E (3) B E C D A (3) A D C E B (3) A C D B E (3) E D B A C (2) E B C D A (2) D A E C B (2) D A C B E (2) C B D E A (2) C B A D E (2) C A D B E (2) B C E A D (2) A E B D C (2) A D E C B (2) A D E B C (2) A C B D E (2) E D B C A (1) E A B D C (1) D E C B A (1) D C A B E (1) C D B E A (1) B C E D A (1) B C A E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 28 18 -10 B -4 0 12 4 -8 C -28 -12 0 -10 -24 D -18 -4 10 0 -12 E 10 8 24 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 28 18 -10 B -4 0 12 4 -8 C -28 -12 0 -10 -24 D -18 -4 10 0 -12 E 10 8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=25 B=18 C=14 D=12 so D is eliminated. Round 2 votes counts: E=35 A=32 B=18 C=15 so C is eliminated. Round 3 votes counts: A=38 E=35 B=27 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:220 B:202 D:188 C:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 28 18 -10 B -4 0 12 4 -8 C -28 -12 0 -10 -24 D -18 -4 10 0 -12 E 10 8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 28 18 -10 B -4 0 12 4 -8 C -28 -12 0 -10 -24 D -18 -4 10 0 -12 E 10 8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 28 18 -10 B -4 0 12 4 -8 C -28 -12 0 -10 -24 D -18 -4 10 0 -12 E 10 8 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3428: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (13) E D C B A (9) E B A D C (9) B A E C D (7) A B C D E (6) E B A C D (5) D C E B A (5) D C E A B (5) E D B C A (4) E B D A C (4) A B E C D (4) D C A E B (3) C D A E B (3) A B C E D (3) D C A B E (2) D B C A E (2) C A D B E (2) B E A C D (2) B A C E D (2) A C B D E (2) E D C A B (1) E A B C D (1) D C B E A (1) D C B A E (1) C D E A B (1) B E A D C (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -4 -8 4 B 6 0 2 -6 0 C 4 -2 0 6 6 D 8 6 -6 0 -4 E -4 0 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -8 4 B 6 0 2 -6 0 C 4 -2 0 6 6 D 8 6 -6 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102025 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=19 C=19 A=16 B=13 so B is eliminated. Round 2 votes counts: E=36 A=26 D=19 C=19 so D is eliminated. Round 3 votes counts: C=38 E=36 A=26 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:207 D:202 B:201 E:197 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 4 B 6 0 2 -6 0 C 4 -2 0 6 6 D 8 6 -6 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102025 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 4 B 6 0 2 -6 0 C 4 -2 0 6 6 D 8 6 -6 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102025 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 4 B 6 0 2 -6 0 C 4 -2 0 6 6 D 8 6 -6 0 -4 E -4 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102025 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3429: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) D C B A E (8) B C D A E (8) A E B C D (8) B C D E A (5) E A B C D (4) C B D E A (4) E A D C B (3) D E C A B (3) D A C B E (3) C E B D A (3) E D A C B (2) D E A C B (2) D C E B A (2) D A E C B (2) C D B E A (2) A E D C B (2) A E D B C (2) A E B D C (2) A D E C B (2) A B E C D (2) E C D B A (1) E C D A B (1) E C B D A (1) E C A B D (1) E A C D B (1) D C A E B (1) D C A B E (1) D B A C E (1) D A B C E (1) B E C A D (1) B E A C D (1) B D C A E (1) B C E D A (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -16 -30 0 B 10 0 -14 -4 14 C 16 14 0 -4 14 D 30 4 4 0 22 E 0 -14 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -30 0 B 10 0 -14 -4 14 C 16 14 0 -4 14 D 30 4 4 0 22 E 0 -14 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999975474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=21 A=20 E=14 C=9 so C is eliminated. Round 2 votes counts: D=38 B=25 A=20 E=17 so E is eliminated. Round 3 votes counts: D=42 B=29 A=29 so B is eliminated. Round 4 votes counts: D=65 A=35 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:230 C:220 B:203 E:175 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -16 -30 0 B 10 0 -14 -4 14 C 16 14 0 -4 14 D 30 4 4 0 22 E 0 -14 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999975474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -30 0 B 10 0 -14 -4 14 C 16 14 0 -4 14 D 30 4 4 0 22 E 0 -14 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999975474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -30 0 B 10 0 -14 -4 14 C 16 14 0 -4 14 D 30 4 4 0 22 E 0 -14 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999975474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3430: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) C E A B D (7) A B D C E (6) E D B C A (5) E D B A C (5) D E B A C (5) D B E A C (5) C A B D E (5) B A D E C (5) E C D B A (4) D B A E C (4) C E A D B (4) C A B E D (4) E C D A B (3) A B C D E (3) C E D B A (2) C E D A B (2) C A E B D (2) B D A E C (2) B D A C E (2) B A D C E (2) A C B E D (2) A C B D E (2) E D A B C (1) E A D B C (1) E A C B D (1) D E C B A (1) D E B C A (1) C D B E A (1) C D B A E (1) B C D A E (1) B A C D E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 0 -4 -10 B 8 0 4 -4 -2 C 0 -4 0 -8 -6 D 4 4 8 0 -4 E 10 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 -4 -10 B 8 0 4 -4 -2 C 0 -4 0 -8 -6 D 4 4 8 0 -4 E 10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 D=16 A=16 B=13 so B is eliminated. Round 2 votes counts: C=29 E=27 A=24 D=20 so D is eliminated. Round 3 votes counts: E=39 A=32 C=29 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:206 B:203 C:191 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -4 -10 B 8 0 4 -4 -2 C 0 -4 0 -8 -6 D 4 4 8 0 -4 E 10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -4 -10 B 8 0 4 -4 -2 C 0 -4 0 -8 -6 D 4 4 8 0 -4 E 10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -4 -10 B 8 0 4 -4 -2 C 0 -4 0 -8 -6 D 4 4 8 0 -4 E 10 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3431: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) D E A C B (11) E D A B C (7) A E D B C (7) E A D B C (5) A D E B C (5) E D A C B (4) D A E C B (4) B C E A D (4) B C A E D (4) B C A D E (4) E D C B A (3) C B D A E (3) B A C E D (3) E A B D C (2) C D E B A (2) C B D E A (2) C B A D E (2) A B E D C (2) A B E C D (2) A B C D E (2) E D C A B (1) E C D B A (1) E B C D A (1) D A C E B (1) D A C B E (1) B A C D E (1) A E B D C (1) A D E C B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 22 -6 -10 B -16 0 2 -8 -12 C -22 -2 0 -14 -16 D 6 8 14 0 -18 E 10 12 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 22 -6 -10 B -16 0 2 -8 -12 C -22 -2 0 -14 -16 D 6 8 14 0 -18 E 10 12 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=22 C=21 D=17 B=16 so B is eliminated. Round 2 votes counts: C=33 A=26 E=24 D=17 so D is eliminated. Round 3 votes counts: E=35 C=33 A=32 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:211 D:205 B:183 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 22 -6 -10 B -16 0 2 -8 -12 C -22 -2 0 -14 -16 D 6 8 14 0 -18 E 10 12 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 22 -6 -10 B -16 0 2 -8 -12 C -22 -2 0 -14 -16 D 6 8 14 0 -18 E 10 12 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 22 -6 -10 B -16 0 2 -8 -12 C -22 -2 0 -14 -16 D 6 8 14 0 -18 E 10 12 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3432: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) D E B A C (6) D E A C B (6) D E A B C (6) D B E C A (6) D A E C B (6) D A C B E (6) C A B E D (5) D E B C A (4) A D C E B (4) E B C A D (3) D B C E A (3) B E C A D (3) B C E A D (3) B C A E D (3) A D E C B (3) A C E B D (3) A C B E D (3) B C D E A (2) B C A D E (2) A E C B D (2) A C D B E (2) E D B C A (1) E D A B C (1) E B C D A (1) E B A C D (1) E A C B D (1) D B C A E (1) D A C E B (1) B E D C A (1) B D E C A (1) B D C E A (1) A C B D E (1) Total count = 100 A B C D E A 0 0 4 -6 0 B 0 0 -2 -12 2 C -4 2 0 -14 -4 D 6 12 14 0 22 E 0 -2 4 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -6 0 B 0 0 -2 -12 2 C -4 2 0 -14 -4 D 6 12 14 0 22 E 0 -2 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 A=18 B=16 C=13 E=8 so E is eliminated. Round 2 votes counts: D=47 B=21 A=19 C=13 so C is eliminated. Round 3 votes counts: D=47 B=29 A=24 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:199 B:194 C:190 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -6 0 B 0 0 -2 -12 2 C -4 2 0 -14 -4 D 6 12 14 0 22 E 0 -2 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -6 0 B 0 0 -2 -12 2 C -4 2 0 -14 -4 D 6 12 14 0 22 E 0 -2 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -6 0 B 0 0 -2 -12 2 C -4 2 0 -14 -4 D 6 12 14 0 22 E 0 -2 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3433: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) D A C E B (7) C A D B E (7) E B C A D (6) E D A B C (5) D A E C B (5) E D A C B (4) E C B A D (4) E B C D A (4) C A B D E (4) B C E A D (4) B C A E D (4) E D B A C (3) E B D A C (3) A C B D E (3) D E A C B (2) D C A E B (2) D A B C E (2) C B A D E (2) B C A D E (2) A C D B E (2) E D B C A (1) E C B D A (1) E C A B D (1) E B A D C (1) D E B A C (1) D E A B C (1) D C A B E (1) D A B E C (1) C E B A D (1) C D A E B (1) C B A E D (1) B E C A D (1) B E A C D (1) B D A C E (1) A B C D E (1) Total count = 100 A B C D E A 0 18 6 -10 12 B -18 0 -16 -10 -6 C -6 16 0 0 10 D 10 10 0 0 10 E -12 6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.438966 D: 0.561034 E: 0.000000 Sum of squares = 0.507450255709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.438966 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 -10 12 B -18 0 -16 -10 -6 C -6 16 0 0 10 D 10 10 0 0 10 E -12 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=32 C=16 B=13 A=6 so A is eliminated. Round 2 votes counts: E=33 D=32 C=21 B=14 so B is eliminated. Round 3 votes counts: E=35 D=33 C=32 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:213 C:210 E:187 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 6 -10 12 B -18 0 -16 -10 -6 C -6 16 0 0 10 D 10 10 0 0 10 E -12 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 -10 12 B -18 0 -16 -10 -6 C -6 16 0 0 10 D 10 10 0 0 10 E -12 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 -10 12 B -18 0 -16 -10 -6 C -6 16 0 0 10 D 10 10 0 0 10 E -12 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3434: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (14) E B C D A (10) E A B C D (6) D C B A E (6) B D C E A (6) C D B E A (5) A E B D C (5) A E B C D (5) A D B C E (5) B E C D A (4) A E C D B (4) D B C E A (3) A E D C B (3) E C B D A (2) E A C B D (2) D C B E A (2) C B D E A (2) B D C A E (2) A E C B D (2) A C E D B (2) E B D C A (1) E B C A D (1) E B A D C (1) D B C A E (1) C D A B E (1) B E D C A (1) A D C E B (1) A C D E B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 8 8 B -6 0 4 2 8 C -6 -4 0 -4 4 D -8 -2 4 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 8 8 B -6 0 4 2 8 C -6 -4 0 -4 4 D -8 -2 4 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 E=23 B=13 D=12 C=8 so C is eliminated. Round 2 votes counts: A=44 E=23 D=18 B=15 so B is eliminated. Round 3 votes counts: A=44 E=28 D=28 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:204 D:198 C:195 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 8 8 B -6 0 4 2 8 C -6 -4 0 -4 4 D -8 -2 4 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 8 B -6 0 4 2 8 C -6 -4 0 -4 4 D -8 -2 4 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 8 B -6 0 4 2 8 C -6 -4 0 -4 4 D -8 -2 4 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3435: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) D A B C E (7) A D E B C (7) D A E B C (6) C E B D A (6) B C E A D (5) A D B E C (5) E C B A D (4) D C A B E (4) C B E D A (4) C B E A D (4) A B E D C (3) E C D B A (2) E C B D A (2) E B A C D (2) E A D B C (2) D C B A E (2) D A C E B (2) C E B A D (2) C D E B A (2) C D B A E (2) C B D A E (2) B E C A D (2) E D C A B (1) E A B D C (1) E A B C D (1) D E C A B (1) D C E A B (1) D C A E B (1) D A E C B (1) D A B E C (1) C B D E A (1) C B A D E (1) B E A C D (1) B C A D E (1) B A D C E (1) A E B D C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 0 -10 12 B -8 0 -4 -10 10 C 0 4 0 -12 14 D 10 10 12 0 14 E -12 -10 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -10 12 B -8 0 -4 -10 10 C 0 4 0 -12 14 D 10 10 12 0 14 E -12 -10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=24 A=18 E=15 B=10 so B is eliminated. Round 2 votes counts: D=33 C=30 A=19 E=18 so E is eliminated. Round 3 votes counts: C=40 D=34 A=26 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:205 C:203 B:194 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -10 12 B -8 0 -4 -10 10 C 0 4 0 -12 14 D 10 10 12 0 14 E -12 -10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -10 12 B -8 0 -4 -10 10 C 0 4 0 -12 14 D 10 10 12 0 14 E -12 -10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -10 12 B -8 0 -4 -10 10 C 0 4 0 -12 14 D 10 10 12 0 14 E -12 -10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3436: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) D E B A C (10) E D B A C (7) D A E B C (7) C B A E D (7) C B E A D (5) C A B D E (5) A C B E D (5) E B C A D (4) D E B C A (4) D E A B C (4) E B D C A (3) D A B E C (2) C E B D A (2) A D C B E (2) A C B D E (2) A B C E D (2) E D B C A (1) E C B A D (1) E B C D A (1) E B A D C (1) D E A C B (1) D C E B A (1) D C A B E (1) D A C E B (1) D A C B E (1) D A B C E (1) C E B A D (1) C D E B A (1) C A D B E (1) B E A C D (1) B A E C D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 4 4 B 2 0 0 8 0 C 0 0 0 6 4 D -4 -8 -6 0 -8 E -4 0 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.526672 C: 0.473328 D: 0.000000 E: 0.000000 Sum of squares = 0.501422781598 Cumulative probabilities = A: 0.000000 B: 0.526672 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 4 4 B 2 0 0 8 0 C 0 0 0 6 4 D -4 -8 -6 0 -8 E -4 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=33 E=18 A=13 B=2 so B is eliminated. Round 2 votes counts: C=34 D=33 E=19 A=14 so A is eliminated. Round 3 votes counts: C=45 D=35 E=20 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:205 C:205 A:203 E:200 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 4 4 B 2 0 0 8 0 C 0 0 0 6 4 D -4 -8 -6 0 -8 E -4 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 4 B 2 0 0 8 0 C 0 0 0 6 4 D -4 -8 -6 0 -8 E -4 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 4 B 2 0 0 8 0 C 0 0 0 6 4 D -4 -8 -6 0 -8 E -4 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3437: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (7) E A D B C (6) D E A B C (6) C B A E D (6) A E D B C (6) E A D C B (5) D E A C B (5) D C E A B (4) C B D E A (4) B D C E A (4) C D B E A (3) B C D A E (3) A E C D B (3) D E B A C (2) D B E C A (2) C A E B D (2) B A E C D (2) A E B D C (2) A C E B D (2) E D A B C (1) D C E B A (1) D C B E A (1) D B E A C (1) D B C E A (1) C D E A B (1) C D B A E (1) C A E D B (1) C A B E D (1) B D E C A (1) B D E A C (1) B D C A E (1) B C A E D (1) A E C B D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 4 -6 -18 B -10 0 4 -14 -12 C -4 -4 0 -14 -6 D 6 14 14 0 4 E 18 12 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -6 -18 B -10 0 4 -14 -12 C -4 -4 0 -14 -6 D 6 14 14 0 4 E 18 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 A=23 C=19 E=12 so E is eliminated. Round 2 votes counts: A=34 D=24 B=23 C=19 so C is eliminated. Round 3 votes counts: A=38 B=33 D=29 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:219 E:216 A:195 C:186 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -6 -18 B -10 0 4 -14 -12 C -4 -4 0 -14 -6 D 6 14 14 0 4 E 18 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -6 -18 B -10 0 4 -14 -12 C -4 -4 0 -14 -6 D 6 14 14 0 4 E 18 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -6 -18 B -10 0 4 -14 -12 C -4 -4 0 -14 -6 D 6 14 14 0 4 E 18 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3438: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) C B A D E (5) E B C A D (4) E A D B C (4) C B D A E (4) B E C D A (4) A E C D B (4) A D C E B (4) D E A C B (3) D A E C B (3) B D C E A (3) A D E C B (3) E B D A C (2) E B A C D (2) E A D C B (2) E A B D C (2) E A B C D (2) D E A B C (2) D C B A E (2) D B C A E (2) D A C E B (2) C A D B E (2) C A B D E (2) B E C A D (2) B D E C A (2) B C E D A (2) B C D E A (2) A E D C B (2) A C E D B (2) A C D E B (2) E D B A C (1) E B A D C (1) E A C D B (1) D C A B E (1) D B E A C (1) D B C E A (1) D A C B E (1) C E B A D (1) C D B A E (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D E B (1) B C E A D (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 6 6 6 -6 B -6 0 -2 -8 -16 C -6 2 0 -2 -8 D -6 8 2 0 2 E 6 16 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102029 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 A B C D E A 0 6 6 6 -6 B -6 0 -2 -8 -16 C -6 2 0 -2 -8 D -6 8 2 0 2 E 6 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102031 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=19 D=18 B=18 A=17 so A is eliminated. Round 2 votes counts: E=34 D=25 C=23 B=18 so B is eliminated. Round 3 votes counts: E=40 D=30 C=30 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:206 D:203 C:193 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 6 -6 B -6 0 -2 -8 -16 C -6 2 0 -2 -8 D -6 8 2 0 2 E 6 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102031 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 -6 B -6 0 -2 -8 -16 C -6 2 0 -2 -8 D -6 8 2 0 2 E 6 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102031 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 -6 B -6 0 -2 -8 -16 C -6 2 0 -2 -8 D -6 8 2 0 2 E 6 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102031 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3439: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) C D E A B (7) E D A B C (6) E A D B C (6) E A B D C (6) C E A B D (5) C B A E D (5) D A B E C (4) C E D A B (3) C D B A E (3) C B E A D (3) B C A D E (3) B A E D C (3) B A C D E (3) E D A C B (2) D E A B C (2) D B A E C (2) C E B A D (2) C B A D E (2) B D A C E (2) B C D A E (2) B C A E D (2) B A D C E (2) E C D A B (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E A C B (1) D A E B C (1) C D B E A (1) C B E D A (1) B E A C D (1) B D A E C (1) B A D E C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -4 0 -2 B 0 0 0 16 6 C 4 0 0 16 14 D 0 -16 -16 0 -4 E 2 -6 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.591508 C: 0.408492 D: 0.000000 E: 0.000000 Sum of squares = 0.516747415018 Cumulative probabilities = A: 0.000000 B: 0.591508 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 0 -2 B 0 0 0 16 6 C 4 0 0 16 14 D 0 -16 -16 0 -4 E 2 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=43 E=25 B=20 D=10 A=2 so A is eliminated. Round 2 votes counts: C=43 E=26 B=21 D=10 so D is eliminated. Round 3 votes counts: C=43 E=30 B=27 so B is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:211 A:197 E:193 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 0 -2 B 0 0 0 16 6 C 4 0 0 16 14 D 0 -16 -16 0 -4 E 2 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 0 -2 B 0 0 0 16 6 C 4 0 0 16 14 D 0 -16 -16 0 -4 E 2 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 0 -2 B 0 0 0 16 6 C 4 0 0 16 14 D 0 -16 -16 0 -4 E 2 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3440: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) A B E C D (9) D E C A B (8) B A C E D (7) C E D B A (6) E A C D B (5) C D E B A (5) A B D E C (5) E C D A B (4) B A C D E (4) A B E D C (4) D C E B A (3) B C A E D (3) B C A D E (3) A D E B C (3) E D C A B (2) D C B E A (2) D A E B C (2) C D B E A (2) A B C E D (2) E D A C B (1) E C D B A (1) E A C B D (1) D E C B A (1) D B C E A (1) C B E D A (1) C B E A D (1) C B A E D (1) B A D E C (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 12 22 12 B 4 0 10 6 12 C -12 -10 0 10 2 D -22 -6 -10 0 0 E -12 -12 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 22 12 B 4 0 10 6 12 C -12 -10 0 10 2 D -22 -6 -10 0 0 E -12 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997467 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=25 D=17 C=16 E=14 so E is eliminated. Round 2 votes counts: A=31 B=28 C=21 D=20 so D is eliminated. Round 3 votes counts: C=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:216 C:195 E:187 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 22 12 B 4 0 10 6 12 C -12 -10 0 10 2 D -22 -6 -10 0 0 E -12 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997467 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 22 12 B 4 0 10 6 12 C -12 -10 0 10 2 D -22 -6 -10 0 0 E -12 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997467 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 22 12 B 4 0 10 6 12 C -12 -10 0 10 2 D -22 -6 -10 0 0 E -12 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997467 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3441: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (6) B C E A D (6) A C B D E (6) E B D C A (5) B E D C A (5) B C A E D (5) E B C D A (4) D E B A C (4) A C E D B (4) D E A C B (3) D B E A C (3) D A E C B (3) C E A D B (3) C A B E D (3) B E C D A (3) A C D E B (3) E D C B A (2) E D B C A (2) E C D A B (2) D E B C A (2) D B E C A (2) D A B E C (2) D A B C E (2) B E C A D (2) B D E C A (2) B D E A C (2) A D C B E (2) E D C A B (1) D B A E C (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) B D A C E (1) B A C D E (1) A D C E B (1) A D B C E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -18 -2 -12 B 10 0 10 12 4 C 18 -10 0 8 0 D 2 -12 -8 0 -16 E 12 -4 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -2 -12 B 10 0 10 12 4 C 18 -10 0 8 0 D 2 -12 -8 0 -16 E 12 -4 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 A=19 E=16 C=16 so E is eliminated. Round 2 votes counts: B=36 D=27 A=19 C=18 so C is eliminated. Round 3 votes counts: B=39 A=32 D=29 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:212 C:208 D:183 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -18 -2 -12 B 10 0 10 12 4 C 18 -10 0 8 0 D 2 -12 -8 0 -16 E 12 -4 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -2 -12 B 10 0 10 12 4 C 18 -10 0 8 0 D 2 -12 -8 0 -16 E 12 -4 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -2 -12 B 10 0 10 12 4 C 18 -10 0 8 0 D 2 -12 -8 0 -16 E 12 -4 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3442: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) A B E C D (10) A B E D C (8) D C A B E (7) C D E A B (5) B A E D C (5) D C E A B (4) C D E B A (4) A B D C E (4) E B C A D (3) E B A C D (3) B E A D C (3) B A E C D (3) A E B C D (3) E C D B A (2) E C B A D (2) D E C B A (2) D C A E B (2) D A B C E (2) A D B C E (2) E D B C A (1) E C A B D (1) E B D C A (1) E B C D A (1) D B E C A (1) C E D A B (1) C D A E B (1) C D A B E (1) C A D E B (1) B E A C D (1) B D A E C (1) A E C B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -4 4 6 B -10 0 8 4 0 C 4 -8 0 -12 -6 D -4 -4 12 0 2 E -6 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000009 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 4 6 B -10 0 8 4 0 C 4 -8 0 -12 -6 D -4 -4 12 0 2 E -6 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000002 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=30 A=30 E=14 C=13 B=13 so C is eliminated. Round 2 votes counts: D=41 A=31 E=15 B=13 so B is eliminated. Round 3 votes counts: D=42 A=39 E=19 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:203 B:201 E:199 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -4 4 6 B -10 0 8 4 0 C 4 -8 0 -12 -6 D -4 -4 12 0 2 E -6 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000002 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 4 6 B -10 0 8 4 0 C 4 -8 0 -12 -6 D -4 -4 12 0 2 E -6 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000002 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 4 6 B -10 0 8 4 0 C 4 -8 0 -12 -6 D -4 -4 12 0 2 E -6 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000002 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3443: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) A D B E C (5) A B E C D (5) E B A C D (4) D C E B A (4) D A E B C (4) C D E B A (4) B E C A D (4) D E C B A (3) D E A C B (3) D A C E B (3) D A C B E (3) A E B C D (3) A D B C E (3) A B E D C (3) A B D E C (3) E D C B A (2) E A B C D (2) D C B A E (2) D A B C E (2) C B E D A (2) B E A C D (2) B C E A D (2) B A C E D (2) A E B D C (2) E D A B C (1) E A B D C (1) D E C A B (1) D C E A B (1) D C B E A (1) D C A B E (1) D A E C B (1) D A B E C (1) C E D B A (1) C E B D A (1) C E B A D (1) C D B A E (1) C B D A E (1) C B A D E (1) B A E C D (1) A D E B C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 14 14 0 B -2 0 6 4 14 C -14 -6 0 -4 -4 D -14 -4 4 0 0 E 0 -14 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.939154 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.060846 Sum of squares = 0.885712528918 Cumulative probabilities = A: 0.939154 B: 0.939154 C: 0.939154 D: 0.939154 E: 1.000000 A B C D E A 0 2 14 14 0 B -2 0 6 4 14 C -14 -6 0 -4 -4 D -14 -4 4 0 0 E 0 -14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250016867 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=27 C=22 B=11 E=10 so E is eliminated. Round 2 votes counts: D=33 A=30 C=22 B=15 so B is eliminated. Round 3 votes counts: A=39 D=33 C=28 so C is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:211 E:195 D:193 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 14 0 B -2 0 6 4 14 C -14 -6 0 -4 -4 D -14 -4 4 0 0 E 0 -14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250016867 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 14 0 B -2 0 6 4 14 C -14 -6 0 -4 -4 D -14 -4 4 0 0 E 0 -14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250016867 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 14 0 B -2 0 6 4 14 C -14 -6 0 -4 -4 D -14 -4 4 0 0 E 0 -14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250016867 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3444: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) D B C A E (8) A E D C B (8) D A B C E (5) E A D C B (4) D A B E C (4) C B E D A (4) C B E A D (4) B D C A E (4) A E D B C (4) D A E B C (3) C E B A D (3) B C D E A (3) A E C D B (3) A D E B C (3) E D A B C (2) E C B A D (2) E A D B C (2) E A B D C (2) D B E A C (2) D B A E C (2) C D B A E (2) C D A B E (2) B C E D A (2) B C D A E (2) E C A B D (1) E B D C A (1) E B C D A (1) E A B C D (1) D B A C E (1) D A C B E (1) C B D E A (1) C B D A E (1) B E D C A (1) B D C E A (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 14 -6 10 B -10 0 8 -16 0 C -14 -8 0 -18 -10 D 6 16 18 0 -6 E -10 0 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.454545 E: 0.272727 Sum of squares = 0.355371900873 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.727273 E: 1.000000 A B C D E A 0 10 14 -6 10 B -10 0 8 -16 0 C -14 -8 0 -18 -10 D 6 16 18 0 -6 E -10 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.454545 E: 0.272727 Sum of squares = 0.355371900807 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=24 A=20 C=17 B=13 so B is eliminated. Round 2 votes counts: D=31 E=25 C=24 A=20 so A is eliminated. Round 3 votes counts: E=40 D=35 C=25 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:217 A:214 E:203 B:191 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 -6 10 B -10 0 8 -16 0 C -14 -8 0 -18 -10 D 6 16 18 0 -6 E -10 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.454545 E: 0.272727 Sum of squares = 0.355371900807 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.727273 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -6 10 B -10 0 8 -16 0 C -14 -8 0 -18 -10 D 6 16 18 0 -6 E -10 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.454545 E: 0.272727 Sum of squares = 0.355371900807 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.727273 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -6 10 B -10 0 8 -16 0 C -14 -8 0 -18 -10 D 6 16 18 0 -6 E -10 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.454545 E: 0.272727 Sum of squares = 0.355371900807 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.727273 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3445: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) D C A B E (7) C B E D A (7) A D E B C (7) D A C E B (6) B E C A D (6) B C E D A (5) E B C A D (4) D A E C B (4) D A C B E (4) C D B A E (4) E C B A D (3) E B A D C (3) C B D A E (3) B C E A D (3) A E D B C (3) A D E C B (3) E C D A B (2) E A D C B (2) D A B C E (2) C D A B E (2) B C D A E (2) E B A C D (1) E A B D C (1) D C A E B (1) C D E A B (1) C B D E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B A E D C (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 8 -6 -8 6 B -8 0 0 -12 2 C 6 0 0 -10 4 D 8 12 10 0 2 E -6 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 -8 6 B -8 0 0 -12 2 C 6 0 0 -10 4 D 8 12 10 0 2 E -6 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 B=20 C=18 A=15 so A is eliminated. Round 2 votes counts: D=35 E=27 B=20 C=18 so C is eliminated. Round 3 votes counts: D=42 B=31 E=27 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:200 C:200 E:193 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -6 -8 6 B -8 0 0 -12 2 C 6 0 0 -10 4 D 8 12 10 0 2 E -6 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -8 6 B -8 0 0 -12 2 C 6 0 0 -10 4 D 8 12 10 0 2 E -6 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -8 6 B -8 0 0 -12 2 C 6 0 0 -10 4 D 8 12 10 0 2 E -6 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3446: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) A E D B C (12) B C D E A (9) B C A E D (8) A E D C B (7) D E C B A (5) D E A C B (4) B C A D E (4) A E B C D (4) A D E C B (4) E D B C A (3) E A D B C (3) B C E D A (3) E D A C B (2) D E C A B (2) D C E B A (2) C B A D E (2) A B C E D (2) E A B D C (1) D C B E A (1) D A E C B (1) D A C E B (1) C D B E A (1) B E C D A (1) B C E A D (1) B A C E D (1) B A C D E (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -8 6 0 B 8 0 6 0 -6 C 8 -6 0 2 2 D -6 0 -2 0 4 E 0 6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.083333 B: 0.305556 C: 0.027778 D: 0.444444 E: 0.138889 Sum of squares = 0.317901234858 Cumulative probabilities = A: 0.083333 B: 0.388889 C: 0.416667 D: 0.861111 E: 1.000000 A B C D E A 0 -8 -8 6 0 B 8 0 6 0 -6 C 8 -6 0 2 2 D -6 0 -2 0 4 E 0 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.305556 C: 0.027778 D: 0.444444 E: 0.138889 Sum of squares = 0.317901234685 Cumulative probabilities = A: 0.083333 B: 0.388889 C: 0.416667 D: 0.861111 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=28 D=16 C=15 E=9 so E is eliminated. Round 2 votes counts: A=36 B=28 D=21 C=15 so C is eliminated. Round 3 votes counts: B=42 A=36 D=22 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:204 C:203 E:200 D:198 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 6 0 B 8 0 6 0 -6 C 8 -6 0 2 2 D -6 0 -2 0 4 E 0 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.305556 C: 0.027778 D: 0.444444 E: 0.138889 Sum of squares = 0.317901234685 Cumulative probabilities = A: 0.083333 B: 0.388889 C: 0.416667 D: 0.861111 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 6 0 B 8 0 6 0 -6 C 8 -6 0 2 2 D -6 0 -2 0 4 E 0 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.305556 C: 0.027778 D: 0.444444 E: 0.138889 Sum of squares = 0.317901234685 Cumulative probabilities = A: 0.083333 B: 0.388889 C: 0.416667 D: 0.861111 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 6 0 B 8 0 6 0 -6 C 8 -6 0 2 2 D -6 0 -2 0 4 E 0 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.305556 C: 0.027778 D: 0.444444 E: 0.138889 Sum of squares = 0.317901234685 Cumulative probabilities = A: 0.083333 B: 0.388889 C: 0.416667 D: 0.861111 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3447: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) E C A B D (7) C E A B D (7) C A E B D (7) B D C A E (7) B C D E A (6) E A C D B (5) C B E D A (4) B D C E A (4) A E C D B (4) D B E A C (3) D B A C E (3) C E B A D (3) B D E C A (3) A E D C B (3) E A D B C (2) D E B A C (2) D A B E C (2) B D A C E (2) A D E B C (2) A D B E C (2) E D A B C (1) E C A D B (1) E A D C B (1) D E A B C (1) D A E B C (1) C E B D A (1) C B D A E (1) C B A E D (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -2 B 4 0 10 6 2 C 4 -10 0 -6 -6 D 6 -6 6 0 4 E 2 -2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -2 B 4 0 10 6 2 C 4 -10 0 -6 -6 D 6 -6 6 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 B=22 E=17 A=13 so A is eliminated. Round 2 votes counts: D=29 E=25 C=24 B=22 so B is eliminated. Round 3 votes counts: D=45 C=30 E=25 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:211 D:205 E:201 A:192 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -2 B 4 0 10 6 2 C 4 -10 0 -6 -6 D 6 -6 6 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 10 6 2 C 4 -10 0 -6 -6 D 6 -6 6 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 10 6 2 C 4 -10 0 -6 -6 D 6 -6 6 0 4 E 2 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3448: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) E C B D A (5) E C B A D (5) E B C A D (5) D A B C E (5) A D B C E (5) A B D E C (5) C E A D B (4) C D E B A (4) B E A C D (4) B A E D C (4) A D B E C (4) E B C D A (3) D A C B E (3) C E D B A (3) D C E B A (2) D C A E B (2) C E D A B (2) B E C D A (2) B E C A D (2) B E A D C (2) B D E C A (2) B D E A C (2) B A D E C (2) A E C D B (2) E C A B D (1) E A C B D (1) D C A B E (1) C E B D A (1) C D E A B (1) C D A E B (1) B E D C A (1) A E C B D (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 6 4 -4 B 14 0 14 14 8 C -6 -14 0 -2 -24 D -4 -14 2 0 0 E 4 -8 24 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 4 -4 B 14 0 14 14 8 C -6 -14 0 -2 -24 D -4 -14 2 0 0 E 4 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=22 E=20 C=16 D=13 so D is eliminated. Round 2 votes counts: A=30 B=29 C=21 E=20 so E is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:210 A:196 D:192 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 6 4 -4 B 14 0 14 14 8 C -6 -14 0 -2 -24 D -4 -14 2 0 0 E 4 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 4 -4 B 14 0 14 14 8 C -6 -14 0 -2 -24 D -4 -14 2 0 0 E 4 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 4 -4 B 14 0 14 14 8 C -6 -14 0 -2 -24 D -4 -14 2 0 0 E 4 -8 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3449: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (14) C A B E D (13) E B A D C (10) D C E B A (10) A B E C D (9) C D A B E (5) B E A D C (4) A C B E D (4) C D E B A (3) C D E A B (3) C A B D E (3) B A E D C (3) A B C E D (3) E B D A C (2) D E C B A (2) A E B C D (2) E D B A C (1) E A B D C (1) E A B C D (1) D E B C A (1) D C B A E (1) D B E A C (1) C D B E A (1) C D B A E (1) C D A E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 12 8 -8 B 8 0 6 12 -2 C -12 -6 0 -2 -4 D -8 -12 2 0 -8 E 8 2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 12 8 -8 B 8 0 6 12 -2 C -12 -6 0 -2 -4 D -8 -12 2 0 -8 E 8 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 A=19 E=15 B=7 so B is eliminated. Round 2 votes counts: C=30 D=29 A=22 E=19 so E is eliminated. Round 3 votes counts: A=38 D=32 C=30 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:211 A:202 C:188 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 12 8 -8 B 8 0 6 12 -2 C -12 -6 0 -2 -4 D -8 -12 2 0 -8 E 8 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 8 -8 B 8 0 6 12 -2 C -12 -6 0 -2 -4 D -8 -12 2 0 -8 E 8 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 8 -8 B 8 0 6 12 -2 C -12 -6 0 -2 -4 D -8 -12 2 0 -8 E 8 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3450: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) B E D A C (8) D B A C E (6) B D A E C (6) D A B C E (5) E C B A D (4) E B C D A (4) E B A C D (4) C D A E B (4) C A E D B (4) C A D E B (4) B E A D C (4) E B D C A (3) C E A D B (3) B D E A C (3) E B C A D (2) D B A E C (2) D A C B E (2) C E A B D (2) B A D E C (2) A C D B E (2) A B E D C (2) E C D B A (1) E C B D A (1) E B A D C (1) E A B C D (1) C E D B A (1) C D E A B (1) C A E B D (1) C A D B E (1) B A E D C (1) A E C B D (1) A D B C E (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 8 6 -6 B 6 0 12 22 -8 C -8 -12 0 6 -20 D -6 -22 -6 0 -18 E 6 8 20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 8 6 -6 B 6 0 12 22 -8 C -8 -12 0 6 -20 D -6 -22 -6 0 -18 E 6 8 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=24 C=21 D=15 A=9 so A is eliminated. Round 2 votes counts: E=32 B=27 C=25 D=16 so D is eliminated. Round 3 votes counts: B=41 E=32 C=27 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:216 A:201 C:183 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 8 6 -6 B 6 0 12 22 -8 C -8 -12 0 6 -20 D -6 -22 -6 0 -18 E 6 8 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 6 -6 B 6 0 12 22 -8 C -8 -12 0 6 -20 D -6 -22 -6 0 -18 E 6 8 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 6 -6 B 6 0 12 22 -8 C -8 -12 0 6 -20 D -6 -22 -6 0 -18 E 6 8 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3451: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) C D B E A (9) E A B D C (8) D C B E A (6) A E C B D (6) A E B C D (6) A E B D C (5) D B C E A (4) A E C D B (4) A C E D B (4) C A D E B (3) B D E C A (3) B D E A C (3) A C E B D (3) E A D B C (2) C D E A B (2) C D A E B (2) C A E D B (2) C A E B D (2) B E A D C (2) B D C E A (2) E B A D C (1) D C E A B (1) D B E C A (1) C B D E A (1) C B D A E (1) C B A E D (1) C A D B E (1) B C D E A (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -8 4 8 B -6 0 -20 -8 -4 C 8 20 0 22 14 D -4 8 -22 0 4 E -8 4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 4 8 B -6 0 -20 -8 -4 C 8 20 0 22 14 D -4 8 -22 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=30 D=12 E=11 B=11 so E is eliminated. Round 2 votes counts: A=40 C=36 D=12 B=12 so D is eliminated. Round 3 votes counts: C=43 A=40 B=17 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:232 A:205 D:193 E:189 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 4 8 B -6 0 -20 -8 -4 C 8 20 0 22 14 D -4 8 -22 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 4 8 B -6 0 -20 -8 -4 C 8 20 0 22 14 D -4 8 -22 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 4 8 B -6 0 -20 -8 -4 C 8 20 0 22 14 D -4 8 -22 0 4 E -8 4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3452: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (17) A B D C E (11) A C E B D (9) B D A E C (7) D B E A C (6) C E D A B (6) D B E C A (5) C E A D B (5) A B C D E (5) D E B C A (4) C E A B D (4) B A D E C (4) A C B E D (4) E D C B A (3) D B A E C (2) C A E B D (2) A B C E D (2) E C B A D (1) C A E D B (1) B E D C A (1) A D B C E (1) Total count = 100 A B C D E A 0 0 2 -2 -4 B 0 0 -4 0 -4 C -2 4 0 12 0 D 2 0 -12 0 -10 E 4 4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.327119 D: 0.000000 E: 0.672881 Sum of squares = 0.559775374734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.327119 D: 0.327119 E: 1.000000 A B C D E A 0 0 2 -2 -4 B 0 0 -4 0 -4 C -2 4 0 12 0 D 2 0 -12 0 -10 E 4 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=21 C=18 D=17 B=12 so B is eliminated. Round 2 votes counts: A=36 D=24 E=22 C=18 so C is eliminated. Round 3 votes counts: A=39 E=37 D=24 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:207 A:198 B:196 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -2 -4 B 0 0 -4 0 -4 C -2 4 0 12 0 D 2 0 -12 0 -10 E 4 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -4 B 0 0 -4 0 -4 C -2 4 0 12 0 D 2 0 -12 0 -10 E 4 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -4 B 0 0 -4 0 -4 C -2 4 0 12 0 D 2 0 -12 0 -10 E 4 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3453: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B D A E (10) D A B C E (8) C B E D A (8) E C B A D (7) E C A B D (7) E A D C B (7) A D E B C (6) E C B D A (5) B C D A E (5) E A C D B (4) B D A C E (4) A D B C E (3) D B A C E (2) D A B E C (2) C B A D E (2) A D B E C (2) D A E B C (1) C E B D A (1) C E B A D (1) C B E A D (1) C B D E A (1) C B A E D (1) C A B D E (1) A D C B E (1) Total count = 100 A B C D E A 0 4 0 6 -4 B -4 0 -14 8 2 C 0 14 0 8 -2 D -6 -8 -8 0 -4 E 4 -2 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629681 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 A B C D E A 0 4 0 6 -4 B -4 0 -14 8 2 C 0 14 0 8 -2 D -6 -8 -8 0 -4 E 4 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629597 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=26 D=13 A=12 B=9 so B is eliminated. Round 2 votes counts: E=40 C=31 D=17 A=12 so A is eliminated. Round 3 votes counts: E=40 C=31 D=29 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:204 A:203 B:196 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 6 -4 B -4 0 -14 8 2 C 0 14 0 8 -2 D -6 -8 -8 0 -4 E 4 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629597 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 -4 B -4 0 -14 8 2 C 0 14 0 8 -2 D -6 -8 -8 0 -4 E 4 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629597 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 -4 B -4 0 -14 8 2 C 0 14 0 8 -2 D -6 -8 -8 0 -4 E 4 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629597 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3454: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (15) D A C B E (9) B E C A D (7) B C A E D (6) E B D C A (5) D A C E B (5) C A B D E (5) E D B A C (3) E C D A B (3) E C B A D (3) D E B A C (3) D E A C B (3) D E A B C (3) C A D B E (3) A C D B E (3) E B D A C (2) E B C D A (2) D B A E C (2) B D E A C (2) A D C B E (2) A C B D E (2) E D C B A (1) E C A B D (1) D B E A C (1) D B A C E (1) D A E C B (1) C B A E D (1) C A B E D (1) B E A D C (1) B A D C E (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -6 8 -10 B 16 0 14 14 0 C 6 -14 0 10 -16 D -8 -14 -10 0 -4 E 10 0 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.643826 C: 0.000000 D: 0.000000 E: 0.356174 Sum of squares = 0.541372104059 Cumulative probabilities = A: 0.000000 B: 0.643826 C: 0.643826 D: 0.643826 E: 1.000000 A B C D E A 0 -16 -6 8 -10 B 16 0 14 14 0 C 6 -14 0 10 -16 D -8 -14 -10 0 -4 E 10 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=28 B=19 C=10 A=8 so A is eliminated. Round 2 votes counts: E=35 D=30 B=20 C=15 so C is eliminated. Round 3 votes counts: D=36 E=35 B=29 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:215 C:193 A:188 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 8 -10 B 16 0 14 14 0 C 6 -14 0 10 -16 D -8 -14 -10 0 -4 E 10 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 8 -10 B 16 0 14 14 0 C 6 -14 0 10 -16 D -8 -14 -10 0 -4 E 10 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 8 -10 B 16 0 14 14 0 C 6 -14 0 10 -16 D -8 -14 -10 0 -4 E 10 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3455: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) D E A B C (8) E A C B D (7) D E B A C (7) C A E B D (6) C A B E D (6) B A C E D (5) D C B A E (4) B C A D E (4) D C A E B (3) D B C A E (3) B E A C D (3) B D C A E (3) A E C B D (3) E A C D B (2) E A B C D (2) D E C A B (2) D B E C A (2) D B E A C (2) B E D A C (2) B D E A C (2) B C D A E (2) B C A E D (2) E D A C B (1) E D A B C (1) E B D A C (1) E B A C D (1) D B C E A (1) C D A E B (1) C A E D B (1) C A D E B (1) B A E C D (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 18 -6 -4 B -10 0 6 6 -12 C -18 -6 0 0 -12 D 6 -6 0 0 6 E 4 12 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999963 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 10 18 -6 -4 B -10 0 6 6 -12 C -18 -6 0 0 -12 D 6 -6 0 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999922 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=24 E=15 C=15 A=6 so A is eliminated. Round 2 votes counts: D=40 B=25 E=18 C=17 so C is eliminated. Round 3 votes counts: D=42 B=32 E=26 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:211 A:209 D:203 B:195 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 18 -6 -4 B -10 0 6 6 -12 C -18 -6 0 0 -12 D 6 -6 0 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999922 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 -6 -4 B -10 0 6 6 -12 C -18 -6 0 0 -12 D 6 -6 0 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999922 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 -6 -4 B -10 0 6 6 -12 C -18 -6 0 0 -12 D 6 -6 0 0 6 E 4 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999922 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3456: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) A D C B E (8) E B C D A (5) B E C A D (5) B A D E C (5) A D B C E (5) E C B D A (4) E D C B A (3) E C D B A (3) D B A E C (3) C D E A B (3) B E D A C (3) B D A E C (3) D C A E B (2) D A C E B (2) D A B E C (2) C A D E B (2) B E A D C (2) B A D C E (2) B A C E D (2) E C D A B (1) E C B A D (1) E B D C A (1) E B C A D (1) D B E A C (1) D A E B C (1) D A C B E (1) D A B C E (1) C E B A D (1) C D A E B (1) C B A E D (1) C A E D B (1) B E D C A (1) B D E A C (1) B A E D C (1) B A C D E (1) A D C E B (1) A C D B E (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 -10 2 B 0 0 0 -10 8 C -2 0 0 -2 4 D 10 10 2 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 0 2 -10 2 B 0 0 0 -10 8 C -2 0 0 -2 4 D 10 10 2 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=22 A=20 E=19 D=13 so D is eliminated. Round 2 votes counts: B=30 A=27 C=24 E=19 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:210 C:200 B:199 A:197 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 2 -10 2 B 0 0 0 -10 8 C -2 0 0 -2 4 D 10 10 2 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -10 2 B 0 0 0 -10 8 C -2 0 0 -2 4 D 10 10 2 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -10 2 B 0 0 0 -10 8 C -2 0 0 -2 4 D 10 10 2 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3457: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (13) E A C B D (12) A E C D B (8) B D C E A (7) D B E A C (5) C A E B D (5) E C A B D (3) D C B A E (3) C D B A E (3) C A D E B (3) B D C A E (3) E A B D C (2) D B A E C (2) C A E D B (2) C A D B E (2) B D E C A (2) A E D C B (2) A E C B D (2) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B C D (1) D C A B E (1) D B E C A (1) D B C E A (1) D B A C E (1) D A E B C (1) C E A B D (1) C D A B E (1) C B A D E (1) C A B D E (1) B E D C A (1) B D E A C (1) B C E A D (1) B C D A E (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -12 6 16 B -6 0 -10 -8 2 C 12 10 0 0 6 D -6 8 0 0 8 E -16 -2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.607599 D: 0.392401 E: 0.000000 Sum of squares = 0.523155024232 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.607599 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 6 16 B -6 0 -10 -8 2 C 12 10 0 0 6 D -6 8 0 0 8 E -16 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=22 C=19 B=16 A=15 so A is eliminated. Round 2 votes counts: E=34 D=29 C=21 B=16 so B is eliminated. Round 3 votes counts: D=42 E=35 C=23 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:214 A:208 D:205 B:189 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 6 16 B -6 0 -10 -8 2 C 12 10 0 0 6 D -6 8 0 0 8 E -16 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 6 16 B -6 0 -10 -8 2 C 12 10 0 0 6 D -6 8 0 0 8 E -16 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 6 16 B -6 0 -10 -8 2 C 12 10 0 0 6 D -6 8 0 0 8 E -16 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3458: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (18) E C A D B (14) D B A E C (10) D B E A C (9) C A E B D (8) C E A B D (7) E A C D B (5) B D C A E (4) E D B C A (3) E C D B A (3) D B E C A (3) C A B E D (2) C A B D E (2) A B C D E (2) E D B A C (1) E C D A B (1) E A D C B (1) C E A D B (1) C B D E A (1) C B D A E (1) B D E C A (1) B A D C E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -2 -10 0 B 10 0 6 -4 10 C 2 -6 0 -4 -2 D 10 4 4 0 6 E 0 -10 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -10 0 B 10 0 6 -4 10 C 2 -6 0 -4 -2 D 10 4 4 0 6 E 0 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=24 D=22 C=22 A=4 so A is eliminated. Round 2 votes counts: E=28 B=27 C=23 D=22 so D is eliminated. Round 3 votes counts: B=49 E=28 C=23 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:212 B:211 C:195 E:193 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 -10 0 B 10 0 6 -4 10 C 2 -6 0 -4 -2 D 10 4 4 0 6 E 0 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -10 0 B 10 0 6 -4 10 C 2 -6 0 -4 -2 D 10 4 4 0 6 E 0 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -10 0 B 10 0 6 -4 10 C 2 -6 0 -4 -2 D 10 4 4 0 6 E 0 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3459: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) A E D B C (10) C B A E D (9) D E A B C (6) A E B C D (6) D C B E A (5) E D A B C (4) C D B A E (4) E A D B C (3) E A B D C (3) D A E C B (3) D A E B C (3) C B E A D (3) C B D A E (3) B C E A D (3) B E C A D (2) B C D E A (2) A C B E D (2) E B D A C (1) E B A D C (1) D E A C B (1) D C E B A (1) D C A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D C A (1) B E A C D (1) B C E D A (1) B C A E D (1) A E C B D (1) A E B D C (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 4 0 B 2 0 0 12 8 C 4 0 0 12 6 D -4 -12 -12 0 -10 E 0 -8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.594457 C: 0.405543 D: 0.000000 E: 0.000000 Sum of squares = 0.517844079485 Cumulative probabilities = A: 0.000000 B: 0.594457 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 4 0 B 2 0 0 12 8 C 4 0 0 12 6 D -4 -12 -12 0 -10 E 0 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=22 D=20 E=12 B=11 so B is eliminated. Round 2 votes counts: C=42 A=22 D=20 E=16 so E is eliminated. Round 3 votes counts: C=44 A=30 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:211 A:199 E:198 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 4 0 B 2 0 0 12 8 C 4 0 0 12 6 D -4 -12 -12 0 -10 E 0 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 0 B 2 0 0 12 8 C 4 0 0 12 6 D -4 -12 -12 0 -10 E 0 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 0 B 2 0 0 12 8 C 4 0 0 12 6 D -4 -12 -12 0 -10 E 0 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3460: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) E B A D C (6) E B C A D (5) E D B C A (4) D C E A B (4) C D A B E (4) C B A D E (4) B A E C D (4) E D C B A (3) E D B A C (3) D E A B C (3) C E D B A (3) C D E B A (3) B A C E D (3) D C A E B (2) C B E A D (2) C B A E D (2) C A B D E (2) B C E A D (2) B C A E D (2) B C A D E (2) A B E C D (2) A B D E C (2) E D C A B (1) E D A B C (1) E C B A D (1) E B D A C (1) E B C D A (1) E B A C D (1) E A B D C (1) D E C A B (1) D C E B A (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B C E (1) C E B D A (1) C D B A E (1) B E C A D (1) B E A C D (1) B A E D C (1) A E D B C (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -6 16 0 B 16 0 18 14 0 C 6 -18 0 14 6 D -16 -14 -14 0 -8 E 0 0 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.519204 C: 0.000000 D: 0.000000 E: 0.480796 Sum of squares = 0.500737602442 Cumulative probabilities = A: 0.000000 B: 0.519204 C: 0.519204 D: 0.519204 E: 1.000000 A B C D E A 0 -16 -6 16 0 B 16 0 18 14 0 C 6 -18 0 14 6 D -16 -14 -14 0 -8 E 0 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 A=18 D=16 B=16 so D is eliminated. Round 2 votes counts: E=32 C=30 A=22 B=16 so B is eliminated. Round 3 votes counts: C=36 E=34 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:204 E:201 A:197 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 16 0 B 16 0 18 14 0 C 6 -18 0 14 6 D -16 -14 -14 0 -8 E 0 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 16 0 B 16 0 18 14 0 C 6 -18 0 14 6 D -16 -14 -14 0 -8 E 0 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 16 0 B 16 0 18 14 0 C 6 -18 0 14 6 D -16 -14 -14 0 -8 E 0 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3461: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (12) A B C D E (11) C D E B A (9) B A E D C (7) A B C E D (7) C D E A B (6) B A C D E (6) E D C A B (5) C D B E A (4) B A D C E (4) B A D E C (3) A B E D C (3) C E D A B (2) B C A D E (2) E D B C A (1) E D A C B (1) E C D A B (1) E B A D C (1) E A D C B (1) D E C B A (1) D C E B A (1) C D A E B (1) C B A D E (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) B C D A E (1) A E D C B (1) A E D B C (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 0 4 4 B 12 0 0 2 8 C 0 0 0 10 18 D -4 -2 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.617289 C: 0.382711 D: 0.000000 E: 0.000000 Sum of squares = 0.527513597285 Cumulative probabilities = A: 0.000000 B: 0.617289 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 4 4 B 12 0 0 2 8 C 0 0 0 10 18 D -4 -2 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 C=24 E=22 D=2 so D is eliminated. Round 2 votes counts: B=26 A=26 C=25 E=23 so E is eliminated. Round 3 votes counts: C=44 B=28 A=28 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:214 B:211 A:198 D:195 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 4 4 B 12 0 0 2 8 C 0 0 0 10 18 D -4 -2 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 4 4 B 12 0 0 2 8 C 0 0 0 10 18 D -4 -2 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 4 4 B 12 0 0 2 8 C 0 0 0 10 18 D -4 -2 -10 0 6 E -4 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3462: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D C B A (8) C E D A B (8) D C E B A (7) B A D E C (5) E C D A B (4) A B E C D (4) A B C E D (4) E D B A C (3) E A C B D (3) C E A D B (3) C D E B A (3) A B E D C (3) E D C A B (2) E A B D C (2) D E C B A (2) D E B A C (2) C D A B E (2) C A B D E (2) B D A C E (2) B A E D C (2) E C A D B (1) E C A B D (1) E B D A C (1) E B A D C (1) E A B C D (1) D E B C A (1) D B E A C (1) D B A E C (1) D B A C E (1) C D E A B (1) C D B A E (1) C D A E B (1) C A E B D (1) C A D B E (1) C A B E D (1) B A D C E (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 0 -2 -10 B -16 0 -8 -6 -12 C 0 8 0 8 2 D 2 6 -8 0 -8 E 10 12 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.091223 B: 0.000000 C: 0.908777 D: 0.000000 E: 0.000000 Sum of squares = 0.834197301619 Cumulative probabilities = A: 0.091223 B: 0.091223 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 -2 -10 B -16 0 -8 -6 -12 C 0 8 0 8 2 D 2 6 -8 0 -8 E 10 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222306873 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=24 A=24 D=15 B=10 so B is eliminated. Round 2 votes counts: A=32 E=27 C=24 D=17 so D is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:209 A:202 D:196 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 0 -2 -10 B -16 0 -8 -6 -12 C 0 8 0 8 2 D 2 6 -8 0 -8 E 10 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222306873 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 -2 -10 B -16 0 -8 -6 -12 C 0 8 0 8 2 D 2 6 -8 0 -8 E 10 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222306873 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 -2 -10 B -16 0 -8 -6 -12 C 0 8 0 8 2 D 2 6 -8 0 -8 E 10 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222306873 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3463: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) B D C A E (7) E A C B D (6) E A C D B (5) E A B C D (5) E A D B C (4) C D B A E (4) A E B C D (4) E A B D C (3) D C B A E (3) C E D A B (3) C D E B A (3) C D B E A (3) B A D E C (3) A B E D C (3) E D A B C (2) E C A D B (2) D B C A E (2) C E A B D (2) B D A C E (2) A E D B C (2) E D A C B (1) E C D A B (1) E C A B D (1) E A D C B (1) D E C A B (1) D E A B C (1) D C B E A (1) C E D B A (1) C E B A D (1) C B D E A (1) C B D A E (1) C B A D E (1) B C D A E (1) B C A D E (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C E D (1) A D E B C (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 22 20 24 4 B -22 0 16 16 -20 C -20 -16 0 -2 -22 D -24 -16 2 0 -22 E -4 20 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 20 24 4 B -22 0 16 16 -20 C -20 -16 0 -2 -22 D -24 -16 2 0 -22 E -4 20 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=23 C=20 B=18 D=8 so D is eliminated. Round 2 votes counts: E=33 C=24 A=23 B=20 so B is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=61 C=39 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:235 E:230 B:195 C:170 D:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 20 24 4 B -22 0 16 16 -20 C -20 -16 0 -2 -22 D -24 -16 2 0 -22 E -4 20 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 20 24 4 B -22 0 16 16 -20 C -20 -16 0 -2 -22 D -24 -16 2 0 -22 E -4 20 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 20 24 4 B -22 0 16 16 -20 C -20 -16 0 -2 -22 D -24 -16 2 0 -22 E -4 20 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3464: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) A E B D C (8) E A C B D (6) C D B E A (5) A E C B D (5) E C D B A (4) C E D B A (4) C D B A E (4) B D A C E (4) A B E D C (4) A B D E C (4) E C A D B (3) C D E B A (3) E C D A B (2) E C B D A (2) E A B D C (2) E A B C D (2) D B C A E (2) C E D A B (2) C D A B E (2) B D E C A (2) B D E A C (2) B D C E A (2) B A D E C (2) A B D C E (2) E D B C A (1) E B D C A (1) E B A D C (1) E A C D B (1) D C B E A (1) D C B A E (1) D B E C A (1) D A B C E (1) C E A D B (1) C A D E B (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -6 -10 -14 B 2 0 2 -2 0 C 6 -2 0 -4 -6 D 10 2 4 0 0 E 14 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.670402 E: 0.329598 Sum of squares = 0.558073779486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.670402 E: 1.000000 A B C D E A 0 -2 -6 -10 -14 B 2 0 2 -2 0 C 6 -2 0 -4 -6 D 10 2 4 0 0 E 14 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 C=22 D=15 B=12 so B is eliminated. Round 2 votes counts: A=28 E=25 D=25 C=22 so C is eliminated. Round 3 votes counts: D=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:210 D:208 B:201 C:197 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -10 -14 B 2 0 2 -2 0 C 6 -2 0 -4 -6 D 10 2 4 0 0 E 14 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -10 -14 B 2 0 2 -2 0 C 6 -2 0 -4 -6 D 10 2 4 0 0 E 14 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -10 -14 B 2 0 2 -2 0 C 6 -2 0 -4 -6 D 10 2 4 0 0 E 14 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3465: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) E D A B C (9) D E A C B (7) E D B A C (6) E D A C B (6) C A B D E (5) E B D A C (4) D A E C B (4) D A C E B (3) B E A D C (3) B C E A D (3) B C A E D (3) A D C E B (3) A C B D E (3) C D A E B (2) C A D B E (2) B E C D A (2) B E A C D (2) B C E D A (2) B C A D E (2) A D E C B (2) E D C A B (1) E B C D A (1) E A D B C (1) D E C A B (1) D E A B C (1) D C E A B (1) D C A E B (1) C D A B E (1) C B E D A (1) C B A E D (1) B E D A C (1) B E C A D (1) A E D B C (1) A D E B C (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 14 18 -8 -6 B -14 0 -12 -10 -10 C -18 12 0 -14 -10 D 8 10 14 0 4 E 6 10 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 -8 -6 B -14 0 -12 -10 -10 C -18 12 0 -14 -10 D 8 10 14 0 4 E 6 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 B=19 D=18 A=12 so A is eliminated. Round 2 votes counts: E=29 C=27 D=25 B=19 so B is eliminated. Round 3 votes counts: E=38 C=37 D=25 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:211 A:209 C:185 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 18 -8 -6 B -14 0 -12 -10 -10 C -18 12 0 -14 -10 D 8 10 14 0 4 E 6 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 -8 -6 B -14 0 -12 -10 -10 C -18 12 0 -14 -10 D 8 10 14 0 4 E 6 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 -8 -6 B -14 0 -12 -10 -10 C -18 12 0 -14 -10 D 8 10 14 0 4 E 6 10 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3466: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (14) A D C B E (8) A D C E B (6) E C B D A (4) E B A C D (4) D C B A E (4) D A C B E (4) C B D E A (4) A D E B C (4) E B C A D (3) E A B C D (3) A E D B C (3) A E B D C (3) E B A D C (2) D B C E A (2) D A B C E (2) C D A B E (2) C A D B E (2) A E C D B (2) A E B C D (2) A C D E B (2) E B D A C (1) E A C B D (1) E A B D C (1) D E A B C (1) D C B E A (1) D C A B E (1) D B E C A (1) C E B D A (1) C D B E A (1) C D B A E (1) C B E D A (1) C A E B D (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D E A (1) A E C B D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 2 4 0 0 B -2 0 2 2 -22 C -4 -2 0 4 -8 D 0 -2 -4 0 0 E 0 22 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.399747 B: 0.000000 C: 0.000000 D: 0.334304 E: 0.265949 Sum of squares = 0.342285813812 Cumulative probabilities = A: 0.399747 B: 0.399747 C: 0.399747 D: 0.734051 E: 1.000000 A B C D E A 0 2 4 0 0 B -2 0 2 2 -22 C -4 -2 0 4 -8 D 0 -2 -4 0 0 E 0 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=33 A=33 D=16 C=13 B=5 so B is eliminated. Round 2 votes counts: E=35 A=33 D=18 C=14 so C is eliminated. Round 3 votes counts: E=37 A=36 D=27 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:203 D:197 C:195 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 0 B -2 0 2 2 -22 C -4 -2 0 4 -8 D 0 -2 -4 0 0 E 0 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 0 B -2 0 2 2 -22 C -4 -2 0 4 -8 D 0 -2 -4 0 0 E 0 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 0 B -2 0 2 2 -22 C -4 -2 0 4 -8 D 0 -2 -4 0 0 E 0 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3467: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (8) E D B C A (7) D E B C A (6) D B E A C (6) A C E B D (6) E C A D B (5) C B A D E (5) A C B E D (5) E D C B A (4) D B E C A (4) B A C D E (4) C E A D B (3) C A E B D (3) B D C A E (3) B D A E C (3) B C A D E (3) A B C D E (3) E D A B C (2) E C D A B (2) D B C E A (2) B D C E A (2) B D A C E (2) A E C D B (2) E D B A C (1) E D A C B (1) E C D B A (1) C E D B A (1) C A E D B (1) C A B E D (1) C A B D E (1) B C D A E (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -10 4 6 B 12 0 2 4 10 C 10 -2 0 10 10 D -4 -4 -10 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 4 6 B 12 0 2 4 10 C 10 -2 0 10 10 D -4 -4 -10 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 B=19 D=18 C=15 so C is eliminated. Round 2 votes counts: A=31 E=27 B=24 D=18 so D is eliminated. Round 3 votes counts: B=36 E=33 A=31 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:214 D:195 A:194 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 4 6 B 12 0 2 4 10 C 10 -2 0 10 10 D -4 -4 -10 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 4 6 B 12 0 2 4 10 C 10 -2 0 10 10 D -4 -4 -10 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 4 6 B 12 0 2 4 10 C 10 -2 0 10 10 D -4 -4 -10 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3468: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (15) C D A B E (15) E B C A D (4) C D A E B (4) E C A D B (3) D C A B E (3) C D B A E (3) B D A C E (3) B C E D A (3) B C D E A (3) A D C E B (3) A C D E B (3) E B C D A (2) E B A C D (2) E A C D B (2) D A C B E (2) D A B C E (2) C B D E A (2) C B D A E (2) C A D E B (2) B E C D A (2) B E A D C (2) B C D A E (2) A E D B C (2) A D E C B (2) E C B D A (1) E C A B D (1) E A D C B (1) E A D B C (1) E A B D C (1) C E A D B (1) C A E D B (1) B E D C A (1) B D C A E (1) B D A E C (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -12 -4 6 B -2 0 -4 -4 -2 C 12 4 0 16 12 D 4 4 -16 0 10 E -6 2 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -4 6 B -2 0 -4 -4 -2 C 12 4 0 16 12 D 4 4 -16 0 10 E -6 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=30 B=18 A=12 D=7 so D is eliminated. Round 2 votes counts: E=33 C=33 B=18 A=16 so A is eliminated. Round 3 votes counts: C=42 E=37 B=21 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:201 A:196 B:194 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -4 6 B -2 0 -4 -4 -2 C 12 4 0 16 12 D 4 4 -16 0 10 E -6 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -4 6 B -2 0 -4 -4 -2 C 12 4 0 16 12 D 4 4 -16 0 10 E -6 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -4 6 B -2 0 -4 -4 -2 C 12 4 0 16 12 D 4 4 -16 0 10 E -6 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3469: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) A C D E B (6) D A C E B (5) C D A B E (5) C B D E A (5) C B A D E (5) B C E D A (5) E D A B C (4) E B D A C (4) E B A D C (4) C D B A E (4) C B D A E (4) D A E C B (3) C D A E B (3) B C E A D (3) A D E C B (3) D C A E B (2) C A B D E (2) B E A D C (2) B C A E D (2) A E D C B (2) A E D B C (2) A E B D C (2) A D C E B (2) E D B A C (1) E A B D C (1) D E C A B (1) D E B C A (1) D E B A C (1) D E A C B (1) C A D E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B C D E A (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -6 -16 12 B 4 0 -14 4 0 C 6 14 0 4 16 D 16 -4 -4 0 12 E -12 0 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -16 12 B 4 0 -14 4 0 C 6 14 0 4 16 D 16 -4 -4 0 12 E -12 0 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=23 A=19 E=14 D=14 so E is eliminated. Round 2 votes counts: B=31 C=30 A=20 D=19 so D is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:220 D:210 B:197 A:193 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -16 12 B 4 0 -14 4 0 C 6 14 0 4 16 D 16 -4 -4 0 12 E -12 0 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -16 12 B 4 0 -14 4 0 C 6 14 0 4 16 D 16 -4 -4 0 12 E -12 0 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -16 12 B 4 0 -14 4 0 C 6 14 0 4 16 D 16 -4 -4 0 12 E -12 0 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3470: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) D E C A B (6) D C E A B (6) D E C B A (5) C B A E D (4) A D C B E (4) A B C E D (4) A B C D E (4) C E D B A (3) A D C E B (3) A D B C E (3) A C D E B (3) A B D E C (3) E D B C A (2) E B D C A (2) D E A B C (2) D A E C B (2) D A E B C (2) D A C E B (2) C E B D A (2) C D E A B (2) C B E A D (2) B E A C D (2) B C E A D (2) B A E D C (2) A D B E C (2) A C D B E (2) A C B D E (2) E C D B A (1) D E B C A (1) D E B A C (1) D E A C B (1) D C A E B (1) C A D E B (1) C A B E D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B C A E D (1) B A E C D (1) B A D E C (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -2 0 0 B -14 0 -18 -22 -8 C 2 18 0 -20 8 D 0 22 20 0 20 E 0 8 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.487314 B: 0.000000 C: 0.000000 D: 0.512686 E: 0.000000 Sum of squares = 0.500321867779 Cumulative probabilities = A: 0.487314 B: 0.487314 C: 0.487314 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 0 0 B -14 0 -18 -22 -8 C 2 18 0 -20 8 D 0 22 20 0 20 E 0 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=29 C=15 B=14 E=11 so E is eliminated. Round 2 votes counts: D=37 A=31 C=16 B=16 so C is eliminated. Round 3 votes counts: D=43 A=33 B=24 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:231 A:206 C:204 E:190 B:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 0 0 B -14 0 -18 -22 -8 C 2 18 0 -20 8 D 0 22 20 0 20 E 0 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 0 0 B -14 0 -18 -22 -8 C 2 18 0 -20 8 D 0 22 20 0 20 E 0 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 0 0 B -14 0 -18 -22 -8 C 2 18 0 -20 8 D 0 22 20 0 20 E 0 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3471: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) D E C B A (8) C B A D E (8) C B D E A (7) A B E D C (6) E D A B C (5) C B D A E (5) B C A D E (5) C D E B A (4) E D A C B (3) D E A B C (3) B A D E C (3) B A C D E (3) E A D C B (2) D E C A B (2) C E D B A (2) C D B E A (2) C B A E D (2) B A C E D (2) A B E C D (2) A B D E C (2) A B C E D (2) E C D A B (1) E A D B C (1) D E B C A (1) D E B A C (1) C E A B D (1) B A D C E (1) A E D B C (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -22 -14 -12 B 8 0 -22 2 0 C 22 22 0 -6 -10 D 14 -2 6 0 10 E 12 0 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.066667 D: 0.733333 E: 0.000000 Sum of squares = 0.582222222267 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.266667 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -22 -14 -12 B 8 0 -22 2 0 C 22 22 0 -6 -10 D 14 -2 6 0 10 E 12 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.066667 D: 0.733333 E: 0.000000 Sum of squares = 0.582222222221 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.266667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=25 D=15 A=15 B=14 so B is eliminated. Round 2 votes counts: C=36 E=25 A=24 D=15 so D is eliminated. Round 3 votes counts: E=40 C=36 A=24 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:214 D:214 E:206 B:194 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -22 -14 -12 B 8 0 -22 2 0 C 22 22 0 -6 -10 D 14 -2 6 0 10 E 12 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.066667 D: 0.733333 E: 0.000000 Sum of squares = 0.582222222221 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.266667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -22 -14 -12 B 8 0 -22 2 0 C 22 22 0 -6 -10 D 14 -2 6 0 10 E 12 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.066667 D: 0.733333 E: 0.000000 Sum of squares = 0.582222222221 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.266667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -22 -14 -12 B 8 0 -22 2 0 C 22 22 0 -6 -10 D 14 -2 6 0 10 E 12 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.066667 D: 0.733333 E: 0.000000 Sum of squares = 0.582222222221 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.266667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3472: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (12) E D A B C (6) E A D B C (6) B C D E A (6) A C E D B (6) C B A D E (5) D E A C B (4) C B D A E (4) C A B E D (4) D E A B C (3) D B C E A (3) C B A E D (3) B D E C A (3) B D C E A (3) A E C D B (3) E A D C B (2) D E B C A (2) D E B A C (2) D B E C A (2) C D B E A (2) C D A E B (2) C A D E B (2) E D B A C (1) E D A C B (1) E A B D C (1) D C E B A (1) D B E A C (1) D A E C B (1) C B D E A (1) C A E D B (1) C A D B E (1) B E D A C (1) B E C D A (1) B A C E D (1) A E D B C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 16 8 0 -4 B -16 0 -12 -32 -18 C -8 12 0 -12 -8 D 0 32 12 0 -2 E 4 18 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 8 0 -4 B -16 0 -12 -32 -18 C -8 12 0 -12 -8 D 0 32 12 0 -2 E 4 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=24 D=19 E=17 B=15 so B is eliminated. Round 2 votes counts: C=31 D=25 A=25 E=19 so E is eliminated. Round 3 votes counts: D=34 A=34 C=32 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:221 E:216 A:210 C:192 B:161 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 8 0 -4 B -16 0 -12 -32 -18 C -8 12 0 -12 -8 D 0 32 12 0 -2 E 4 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 0 -4 B -16 0 -12 -32 -18 C -8 12 0 -12 -8 D 0 32 12 0 -2 E 4 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 0 -4 B -16 0 -12 -32 -18 C -8 12 0 -12 -8 D 0 32 12 0 -2 E 4 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3473: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) E B C A D (9) B E A D C (7) A D B C E (7) A B D E C (7) D A C B E (5) C D E A B (5) E C B D A (4) E B A C D (4) D C A E B (4) D A B C E (4) C E D B A (4) B E A C D (4) B A E D C (4) A D C B E (4) D C A B E (3) C E D A B (3) B A D E C (3) E C B A D (2) A D C E B (2) E C D B A (1) E B C D A (1) D A C E B (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 14 8 10 12 B -14 0 4 -6 -2 C -8 -4 0 -4 6 D -10 6 4 0 12 E -12 2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 10 12 B -14 0 4 -6 -2 C -8 -4 0 -4 6 D -10 6 4 0 12 E -12 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=22 A=22 E=21 B=18 D=17 so D is eliminated. Round 2 votes counts: A=32 C=29 E=21 B=18 so B is eliminated. Round 3 votes counts: A=39 E=32 C=29 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:206 C:195 B:191 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 10 12 B -14 0 4 -6 -2 C -8 -4 0 -4 6 D -10 6 4 0 12 E -12 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 10 12 B -14 0 4 -6 -2 C -8 -4 0 -4 6 D -10 6 4 0 12 E -12 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 10 12 B -14 0 4 -6 -2 C -8 -4 0 -4 6 D -10 6 4 0 12 E -12 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3474: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (14) E A C B D (11) E A B D C (9) E A B C D (9) C E A D B (8) C E A B D (6) B D A E C (5) E C A B D (4) D C B A E (4) D B A E C (4) A E B D C (4) D B C A E (3) C E D A B (3) C D E A B (3) A B E D C (3) D B A C E (2) C D E B A (2) C D B E A (2) E C A D B (1) D C B E A (1) B D E A C (1) B A E D C (1) Total count = 100 A B C D E A 0 22 -2 12 -20 B -22 0 -18 6 -20 C 2 18 0 26 -4 D -12 -6 -26 0 -18 E 20 20 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 -2 12 -20 B -22 0 -18 6 -20 C 2 18 0 26 -4 D -12 -6 -26 0 -18 E 20 20 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=34 D=14 B=7 A=7 so B is eliminated. Round 2 votes counts: C=38 E=34 D=20 A=8 so A is eliminated. Round 3 votes counts: E=42 C=38 D=20 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 C:221 A:206 B:173 D:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 -2 12 -20 B -22 0 -18 6 -20 C 2 18 0 26 -4 D -12 -6 -26 0 -18 E 20 20 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -2 12 -20 B -22 0 -18 6 -20 C 2 18 0 26 -4 D -12 -6 -26 0 -18 E 20 20 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -2 12 -20 B -22 0 -18 6 -20 C 2 18 0 26 -4 D -12 -6 -26 0 -18 E 20 20 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3475: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (6) A D E C B (6) B D A E C (5) E C B A D (4) D B C A E (4) C E B D A (4) E A C B D (3) D A C E B (3) D A C B E (3) C E B A D (3) B E C D A (3) B C E D A (3) A D C E B (3) A B D E C (3) E C A B D (2) D B A C E (2) D A B E C (2) C E D A B (2) C E A B D (2) C B E D A (2) C B D E A (2) B D C E A (2) B D A C E (2) B C D E A (2) B A D E C (2) A E C D B (2) A E B D C (2) A D E B C (2) A D B E C (2) E B C A D (1) E A C D B (1) E A B C D (1) D C A E B (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E B A (1) C D B E A (1) C D A E B (1) B E D A C (1) B D E C A (1) B D C A E (1) A E D B C (1) A E B C D (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 6 0 0 2 B -6 0 -10 4 -10 C 0 10 0 0 -2 D 0 -4 0 0 6 E -2 10 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.328054 B: 0.000000 C: 0.269275 D: 0.402672 E: 0.000000 Sum of squares = 0.342272496057 Cumulative probabilities = A: 0.328054 B: 0.328054 C: 0.597328 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 2 B -6 0 -10 4 -10 C 0 10 0 0 -2 D 0 -4 0 0 6 E -2 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333334 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333334 B: 0.333334 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=22 C=20 E=18 D=16 so D is eliminated. Round 2 votes counts: A=33 B=28 C=21 E=18 so E is eliminated. Round 3 votes counts: A=38 C=33 B=29 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:204 C:204 E:202 D:201 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 0 2 B -6 0 -10 4 -10 C 0 10 0 0 -2 D 0 -4 0 0 6 E -2 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333334 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333334 B: 0.333334 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 2 B -6 0 -10 4 -10 C 0 10 0 0 -2 D 0 -4 0 0 6 E -2 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333334 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333334 B: 0.333334 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 2 B -6 0 -10 4 -10 C 0 10 0 0 -2 D 0 -4 0 0 6 E -2 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333334 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333334 B: 0.333334 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3476: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) B A C D E (8) B C A E D (6) E D C A B (5) D E C A B (5) B A D C E (5) A D C E B (5) A B C D E (5) E C D B A (4) D E A C B (4) B E C A D (4) E C B D A (3) D A E C B (3) C E D A B (3) C E A D B (2) C A D E B (2) B E D A C (2) B E C D A (2) B C E A D (2) B A D E C (2) E B D C A (1) E B D A C (1) E B C D A (1) D E A B C (1) D C A E B (1) D A E B C (1) D A B E C (1) C E B A D (1) C D E A B (1) C B E A D (1) C B A E D (1) C A E D B (1) B E D C A (1) B E A D C (1) B E A C D (1) B D E A C (1) A D B C E (1) A C D E B (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 6 18 6 B 12 0 12 18 8 C -6 -12 0 16 10 D -18 -18 -16 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 18 6 B 12 0 12 18 8 C -6 -12 0 16 10 D -18 -18 -16 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 D=16 E=15 A=14 C=12 so C is eliminated. Round 2 votes counts: B=45 E=21 D=17 A=17 so D is eliminated. Round 3 votes counts: B=45 E=32 A=23 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:209 C:204 E:189 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 18 6 B 12 0 12 18 8 C -6 -12 0 16 10 D -18 -18 -16 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 18 6 B 12 0 12 18 8 C -6 -12 0 16 10 D -18 -18 -16 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 18 6 B 12 0 12 18 8 C -6 -12 0 16 10 D -18 -18 -16 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3477: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) C B E D A (12) A D E C B (11) B C E D A (9) D E A B C (7) B E D A C (6) C B A D E (4) C A B D E (4) A D C E B (4) A C D E B (4) E D B A C (3) C B A E D (3) E D A B C (2) C A D E B (2) B E D C A (2) A C B D E (2) A B D E C (2) E D C B A (1) E D A C B (1) C E D B A (1) C E D A B (1) C A D B E (1) B E C D A (1) B C A E D (1) B C A D E (1) A D B E C (1) Total count = 100 A B C D E A 0 12 14 8 8 B -12 0 -2 -6 -2 C -14 2 0 -8 -2 D -8 6 8 0 14 E -8 2 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 8 8 B -12 0 -2 -6 -2 C -14 2 0 -8 -2 D -8 6 8 0 14 E -8 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=28 B=20 E=7 D=7 so E is eliminated. Round 2 votes counts: A=38 C=28 B=20 D=14 so D is eliminated. Round 3 votes counts: A=48 C=29 B=23 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:210 E:191 B:189 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 8 8 B -12 0 -2 -6 -2 C -14 2 0 -8 -2 D -8 6 8 0 14 E -8 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 8 8 B -12 0 -2 -6 -2 C -14 2 0 -8 -2 D -8 6 8 0 14 E -8 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 8 8 B -12 0 -2 -6 -2 C -14 2 0 -8 -2 D -8 6 8 0 14 E -8 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3478: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (13) B E A C D (6) A C E B D (6) B D E A C (5) E C A B D (4) D E B C A (4) D B A C E (4) C A E B D (4) B D E C A (4) B A E C D (4) E B C A D (3) D B E A C (3) A C E D B (3) E D B C A (2) D C A E B (2) D A C B E (2) C D A E B (2) C A E D B (2) C A D E B (2) B E D A C (2) A C D E B (2) A C B E D (2) A B C E D (2) E D C B A (1) E C D B A (1) E B D C A (1) E B C D A (1) D E C B A (1) D E C A B (1) D B C E A (1) D B A E C (1) D A C E B (1) C E A D B (1) B E C A D (1) B D A E C (1) B A D C E (1) B A C D E (1) A D C B E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -2 -6 -10 B 22 0 22 -2 12 C 2 -22 0 -4 -18 D 6 2 4 0 8 E 10 -12 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -2 -6 -10 B 22 0 22 -2 12 C 2 -22 0 -4 -18 D 6 2 4 0 8 E 10 -12 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=25 A=18 E=13 C=11 so C is eliminated. Round 2 votes counts: D=35 A=26 B=25 E=14 so E is eliminated. Round 3 votes counts: D=39 A=31 B=30 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:227 D:210 E:204 A:180 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -2 -6 -10 B 22 0 22 -2 12 C 2 -22 0 -4 -18 D 6 2 4 0 8 E 10 -12 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -2 -6 -10 B 22 0 22 -2 12 C 2 -22 0 -4 -18 D 6 2 4 0 8 E 10 -12 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -2 -6 -10 B 22 0 22 -2 12 C 2 -22 0 -4 -18 D 6 2 4 0 8 E 10 -12 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3479: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) B E C D A (7) B E D A C (6) B A E D C (6) A D C E B (6) A D B E C (4) A B D E C (4) E C D B A (3) E B D C A (3) C E D B A (3) B A D E C (3) A D E B C (3) E C B D A (2) E B C D A (2) D A E C B (2) C B E D A (2) C B E A D (2) C A D E B (2) B E D C A (2) B E C A D (2) B C E D A (2) A D C B E (2) A C D E B (2) A C B D E (2) A B C E D (2) D E B A C (1) D C E A B (1) D C A E B (1) D B A E C (1) D A E B C (1) D A C E B (1) D A B E C (1) C E D A B (1) C E B A D (1) C D E A B (1) C A B E D (1) B E A D C (1) B D A E C (1) B A E C D (1) B A C E D (1) A D E C B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 8 -4 0 B 20 0 10 22 10 C -8 -10 0 -4 -14 D 4 -22 4 0 -16 E 0 -10 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999435 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 8 -4 0 B 20 0 10 22 10 C -8 -10 0 -4 -14 D 4 -22 4 0 -16 E 0 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=28 C=21 E=10 D=9 so D is eliminated. Round 2 votes counts: B=33 A=33 C=23 E=11 so E is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:210 A:192 D:185 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 8 -4 0 B 20 0 10 22 10 C -8 -10 0 -4 -14 D 4 -22 4 0 -16 E 0 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 8 -4 0 B 20 0 10 22 10 C -8 -10 0 -4 -14 D 4 -22 4 0 -16 E 0 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 8 -4 0 B 20 0 10 22 10 C -8 -10 0 -4 -14 D 4 -22 4 0 -16 E 0 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3480: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (6) E D A B C (5) D E C A B (5) C B A D E (5) B A C E D (5) E D C A B (4) E D A C B (4) D B E A C (4) C D E A B (4) A E B C D (4) E A C D B (3) D C E B A (3) D B E C A (3) C D B A E (3) C A B E D (3) B D A E C (3) B C A D E (3) E B A D C (2) E A D C B (2) D B C E A (2) D B C A E (2) C E A B D (2) C B D A E (2) B D C A E (2) B A D E C (2) B A C D E (2) E C D A B (1) E A C B D (1) E A B D C (1) E A B C D (1) D E B C A (1) D E B A C (1) D C B A E (1) C E D A B (1) C E A D B (1) C A D B E (1) B C A E D (1) B A E D C (1) B A D C E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 0 -14 -2 B 10 0 6 -2 6 C 0 -6 0 -10 4 D 14 2 10 0 12 E 2 -6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -14 -2 B 10 0 6 -2 6 C 0 -6 0 -10 4 D 14 2 10 0 12 E 2 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=24 D=22 C=22 A=6 so A is eliminated. Round 2 votes counts: E=28 B=27 C=23 D=22 so D is eliminated. Round 3 votes counts: B=38 E=35 C=27 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:219 B:210 C:194 E:190 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 0 -14 -2 B 10 0 6 -2 6 C 0 -6 0 -10 4 D 14 2 10 0 12 E 2 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -14 -2 B 10 0 6 -2 6 C 0 -6 0 -10 4 D 14 2 10 0 12 E 2 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -14 -2 B 10 0 6 -2 6 C 0 -6 0 -10 4 D 14 2 10 0 12 E 2 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3481: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) A B E C D (7) C E D A B (6) B A D C E (6) E C D A B (4) D E C B A (4) D C E B A (4) D B C E A (4) B A D E C (4) A E C D B (4) A E B C D (4) C E D B A (3) B D C E A (3) B D A C E (3) B A C D E (3) E D A C B (2) D C B E A (2) D B E C A (2) C B E D A (2) A E D C B (2) A E D B C (2) A D B E C (2) A B E D C (2) A B D E C (2) E C A D B (1) E A C D B (1) D E C A B (1) D E A C B (1) C E B D A (1) C E A D B (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C E D (1) A E C B D (1) A C E D B (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -2 0 B -8 0 0 -8 -2 C -2 0 0 -6 -6 D 2 8 6 0 -12 E 0 2 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.515016 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.484984 Sum of squares = 0.500450971102 Cumulative probabilities = A: 0.515016 B: 0.515016 C: 0.515016 D: 0.515016 E: 1.000000 A B C D E A 0 8 2 -2 0 B -8 0 0 -8 -2 C -2 0 0 -6 -6 D 2 8 6 0 -12 E 0 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 D=18 E=15 C=13 so C is eliminated. Round 2 votes counts: A=30 E=26 B=26 D=18 so D is eliminated. Round 3 votes counts: E=36 B=34 A=30 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:204 D:202 C:193 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 -2 0 B -8 0 0 -8 -2 C -2 0 0 -6 -6 D 2 8 6 0 -12 E 0 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 0 B -8 0 0 -8 -2 C -2 0 0 -6 -6 D 2 8 6 0 -12 E 0 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 0 B -8 0 0 -8 -2 C -2 0 0 -6 -6 D 2 8 6 0 -12 E 0 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3482: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (12) A B E D C (11) C D E B A (9) A D B E C (7) C D A E B (6) E B A C D (4) D C E B A (4) D A C B E (4) C E B A D (4) E B C A D (3) B E A D C (3) B A E D C (3) D B E A C (2) C E D B A (2) C E A B D (2) C D E A B (2) A C D B E (2) A B E C D (2) A B D E C (2) E D B C A (1) E B C D A (1) E A C B D (1) D E C B A (1) D E B C A (1) D B A E C (1) D A B E C (1) C D A B E (1) C A E B D (1) C A D E B (1) B E D A C (1) B E A C D (1) A E B C D (1) A D C B E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -2 2 -8 B 6 0 -8 6 -12 C 2 8 0 12 6 D -2 -6 -12 0 -8 E 8 12 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 2 -8 B 6 0 -8 6 -12 C 2 8 0 12 6 D -2 -6 -12 0 -8 E 8 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 A=28 D=14 E=10 B=8 so B is eliminated. Round 2 votes counts: C=40 A=31 E=15 D=14 so D is eliminated. Round 3 votes counts: C=44 A=37 E=19 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:211 B:196 A:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 2 -8 B 6 0 -8 6 -12 C 2 8 0 12 6 D -2 -6 -12 0 -8 E 8 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 -8 B 6 0 -8 6 -12 C 2 8 0 12 6 D -2 -6 -12 0 -8 E 8 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 -8 B 6 0 -8 6 -12 C 2 8 0 12 6 D -2 -6 -12 0 -8 E 8 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3483: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) A B C E D (11) D E B C A (9) C A E D B (9) B E D A C (8) C A D E B (6) A C B E D (6) D E C B A (5) C D E A B (5) B D E A C (4) B A C E D (4) E D C B A (3) A C B D E (3) C E D A B (2) B E D C A (2) B A D E C (2) E D C A B (1) E B D C A (1) D E C A B (1) D C E A B (1) C E A D B (1) B D E C A (1) A C E D B (1) A C E B D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 14 12 B 0 0 8 10 6 C -6 -8 0 2 4 D -14 -10 -2 0 -22 E -12 -6 -4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.476400 B: 0.523600 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501113895702 Cumulative probabilities = A: 0.476400 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 14 12 B 0 0 8 10 6 C -6 -8 0 2 4 D -14 -10 -2 0 -22 E -12 -6 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=24 C=23 D=16 E=5 so E is eliminated. Round 2 votes counts: B=33 A=24 C=23 D=20 so D is eliminated. Round 3 votes counts: B=42 C=34 A=24 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:212 E:200 C:196 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 14 12 B 0 0 8 10 6 C -6 -8 0 2 4 D -14 -10 -2 0 -22 E -12 -6 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 14 12 B 0 0 8 10 6 C -6 -8 0 2 4 D -14 -10 -2 0 -22 E -12 -6 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 14 12 B 0 0 8 10 6 C -6 -8 0 2 4 D -14 -10 -2 0 -22 E -12 -6 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3484: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) E B D C A (8) C A B D E (8) C B D A E (7) A D B E C (7) A C D B E (6) E C B D A (5) A D E B C (5) E B C D A (3) D A B E C (3) B D A E C (3) A D B C E (3) E B D A C (2) D B A E C (2) C B E D A (2) C B A D E (2) B C D A E (2) A C D E B (2) E D C B A (1) E D B C A (1) E D A B C (1) E C A D B (1) E A D C B (1) E A D B C (1) D E B A C (1) D B E A C (1) C E B D A (1) C E A D B (1) C E A B D (1) B E D C A (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) A E D C B (1) A E C D B (1) A D C B E (1) Total count = 100 A B C D E A 0 -14 6 -18 8 B 14 0 18 -2 4 C -6 -18 0 -16 -24 D 18 2 16 0 14 E -8 -4 24 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 -18 8 B 14 0 18 -2 4 C -6 -18 0 -16 -24 D 18 2 16 0 14 E -8 -4 24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=26 C=22 B=10 D=7 so D is eliminated. Round 2 votes counts: E=36 A=29 C=22 B=13 so B is eliminated. Round 3 votes counts: E=40 A=34 C=26 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:225 B:217 E:199 A:191 C:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 6 -18 8 B 14 0 18 -2 4 C -6 -18 0 -16 -24 D 18 2 16 0 14 E -8 -4 24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 -18 8 B 14 0 18 -2 4 C -6 -18 0 -16 -24 D 18 2 16 0 14 E -8 -4 24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 -18 8 B 14 0 18 -2 4 C -6 -18 0 -16 -24 D 18 2 16 0 14 E -8 -4 24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3485: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) C B E D A (5) B C A D E (5) B A D C E (5) A D E B C (5) E D A C B (4) C B A E D (4) B C D E A (4) E D C A B (3) D A B E C (3) B D A C E (3) E D B C A (2) E C A D B (2) D B E A C (2) D B A E C (2) C E B D A (2) C E B A D (2) C B D E A (2) C B A D E (2) C A B E D (2) B D E C A (2) B D C E A (2) B D A E C (2) A E D C B (2) A E D B C (2) A E C D B (2) A D B E C (2) A B C D E (2) E D A B C (1) E C D B A (1) E C D A B (1) E A C D B (1) D E A B C (1) D B E C A (1) D A E B C (1) C E A D B (1) C E A B D (1) C B E A D (1) C A E D B (1) C A E B D (1) B D C A E (1) B C D A E (1) B A D E C (1) A C E B D (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 8 6 B 4 0 4 8 14 C -2 -4 0 -10 0 D -8 -8 10 0 4 E -6 -14 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 8 6 B 4 0 4 8 14 C -2 -4 0 -10 0 D -8 -8 10 0 4 E -6 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 E=21 A=19 D=10 so D is eliminated. Round 2 votes counts: B=31 C=24 A=23 E=22 so E is eliminated. Round 3 votes counts: A=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:206 D:199 C:192 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 8 6 B 4 0 4 8 14 C -2 -4 0 -10 0 D -8 -8 10 0 4 E -6 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 6 B 4 0 4 8 14 C -2 -4 0 -10 0 D -8 -8 10 0 4 E -6 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 6 B 4 0 4 8 14 C -2 -4 0 -10 0 D -8 -8 10 0 4 E -6 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3486: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) D B C A E (7) E A C D B (6) D A E B C (6) C E B A D (6) C B E A D (5) E C A B D (4) D B A E C (4) D B A C E (4) D A B E C (4) C B E D A (4) B C A E D (4) B D A C E (3) B C D A E (3) A E D C B (3) E C A D B (2) E A C B D (2) D E A C B (2) C E A D B (2) C B D E A (2) B D C A E (2) A E B C D (2) A B E D C (2) E D A C B (1) E C D A B (1) E A D C B (1) D C E B A (1) D C E A B (1) D C B E A (1) C D E A B (1) C D B E A (1) B D A E C (1) B C E A D (1) B C D E A (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 0 -14 0 -6 B 0 0 -8 2 -2 C 14 8 0 10 14 D 0 -2 -10 0 -10 E 6 2 -14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 0 -6 B 0 0 -8 2 -2 C 14 8 0 10 14 D 0 -2 -10 0 -10 E 6 2 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=29 E=17 B=15 A=9 so A is eliminated. Round 2 votes counts: D=31 C=29 E=23 B=17 so B is eliminated. Round 3 votes counts: C=38 D=37 E=25 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 E:202 B:196 A:190 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 0 -6 B 0 0 -8 2 -2 C 14 8 0 10 14 D 0 -2 -10 0 -10 E 6 2 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 0 -6 B 0 0 -8 2 -2 C 14 8 0 10 14 D 0 -2 -10 0 -10 E 6 2 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 0 -6 B 0 0 -8 2 -2 C 14 8 0 10 14 D 0 -2 -10 0 -10 E 6 2 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3487: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (12) E D A C B (9) A B C E D (8) B C D E A (7) E D C B A (6) E A D C B (5) D C E B A (5) B C A D E (5) A E D B C (5) A B C D E (4) E D C A B (3) D E C B A (3) A E B D C (3) A B E C D (3) B C D A E (2) B A C D E (2) A E B C D (2) A C B D E (2) A B E D C (2) E D A B C (1) E A D B C (1) D E B C A (1) D B C E A (1) C D E A B (1) C D B E A (1) C B D E A (1) C B A D E (1) C A B D E (1) B D C E A (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 28 22 16 8 B -28 0 -4 -8 -16 C -22 4 0 -16 -14 D -16 8 16 0 -24 E -8 16 14 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 22 16 8 B -28 0 -4 -8 -16 C -22 4 0 -16 -14 D -16 8 16 0 -24 E -8 16 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 E=25 B=17 D=10 C=5 so C is eliminated. Round 2 votes counts: A=44 E=25 B=19 D=12 so D is eliminated. Round 3 votes counts: A=44 E=35 B=21 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:237 E:223 D:192 C:176 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 22 16 8 B -28 0 -4 -8 -16 C -22 4 0 -16 -14 D -16 8 16 0 -24 E -8 16 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 22 16 8 B -28 0 -4 -8 -16 C -22 4 0 -16 -14 D -16 8 16 0 -24 E -8 16 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 22 16 8 B -28 0 -4 -8 -16 C -22 4 0 -16 -14 D -16 8 16 0 -24 E -8 16 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3488: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) E C D B A (8) C D B A E (7) E C A B D (6) E B D A C (5) C D A B E (5) B D A E C (5) A B D C E (5) D B A C E (4) C A D B E (4) E D B A C (3) E C A D B (3) C E A D B (3) B E A D C (3) B D A C E (3) E D B C A (2) C E D A B (2) B A D E C (2) A D B C E (2) A C B D E (2) A B D E C (2) E C B D A (1) E A C B D (1) E A B D C (1) D C B E A (1) D B C A E (1) C E A B D (1) C D E B A (1) C D B E A (1) C A B D E (1) B A E D C (1) A E B D C (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 6 2 -4 B 16 0 4 4 4 C -6 -4 0 -4 -10 D -2 -4 4 0 -4 E 4 -4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 2 -4 B 16 0 4 4 4 C -6 -4 0 -4 -10 D -2 -4 4 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=25 A=15 B=14 D=6 so D is eliminated. Round 2 votes counts: E=40 C=26 B=19 A=15 so A is eliminated. Round 3 votes counts: E=41 B=30 C=29 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:207 D:197 A:194 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 2 -4 B 16 0 4 4 4 C -6 -4 0 -4 -10 D -2 -4 4 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 2 -4 B 16 0 4 4 4 C -6 -4 0 -4 -10 D -2 -4 4 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 2 -4 B 16 0 4 4 4 C -6 -4 0 -4 -10 D -2 -4 4 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3489: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) B E D C A (10) C A D B E (9) A C D E B (8) E B A C D (6) B D E C A (4) E B D A C (3) E B A D C (3) E A B D C (3) D B E C A (3) D B C E A (3) C A D E B (3) A D C B E (3) A C E D B (3) E C B A D (2) E B C D A (2) E B C A D (2) E A C B D (2) D C A B E (2) D B E A C (2) D A C B E (2) B E C D A (2) A E B C D (2) E B D C A (1) D C B A E (1) D B A E C (1) D A B E C (1) C E B A D (1) C D B A E (1) C B D E A (1) C A E B D (1) B E D A C (1) B D E A C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -16 -2 -4 B 0 0 0 -4 14 C 16 0 0 12 -2 D 2 4 -12 0 10 E 4 -14 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.503937 C: 0.496063 D: 0.000000 E: 0.000000 Sum of squares = 0.500030989369 Cumulative probabilities = A: 0.000000 B: 0.503937 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -2 -4 B 0 0 0 -4 14 C 16 0 0 12 -2 D 2 4 -12 0 10 E 4 -14 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 B=18 A=17 D=15 so D is eliminated. Round 2 votes counts: C=29 B=27 E=24 A=20 so A is eliminated. Round 3 votes counts: C=46 B=28 E=26 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:205 D:202 E:191 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -2 -4 B 0 0 0 -4 14 C 16 0 0 12 -2 D 2 4 -12 0 10 E 4 -14 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -2 -4 B 0 0 0 -4 14 C 16 0 0 12 -2 D 2 4 -12 0 10 E 4 -14 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -2 -4 B 0 0 0 -4 14 C 16 0 0 12 -2 D 2 4 -12 0 10 E 4 -14 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3490: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (14) B A C E D (10) D E C A B (8) E A B D C (7) D C E B A (7) C B A E D (6) E A B C D (5) D C B A E (5) D C B E A (3) C D B A E (3) C B A D E (3) B C A E D (3) A E B C D (3) E D A B C (2) E A D B C (2) D E A C B (2) D E A B C (2) C D E B A (2) C B D A E (2) A E B D C (2) A B E D C (2) E D C A B (1) E D A C B (1) E B A C D (1) D A E B C (1) D A B C E (1) C B D E A (1) A B D E C (1) Total count = 100 A B C D E A 0 8 12 18 12 B -8 0 12 20 8 C -12 -12 0 6 -8 D -18 -20 -6 0 -18 E -12 -8 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 18 12 B -8 0 12 20 8 C -12 -12 0 6 -8 D -18 -20 -6 0 -18 E -12 -8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=22 E=19 C=17 B=13 so B is eliminated. Round 2 votes counts: A=32 D=29 C=20 E=19 so E is eliminated. Round 3 votes counts: A=47 D=33 C=20 so C is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:225 B:216 E:203 C:187 D:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 18 12 B -8 0 12 20 8 C -12 -12 0 6 -8 D -18 -20 -6 0 -18 E -12 -8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 18 12 B -8 0 12 20 8 C -12 -12 0 6 -8 D -18 -20 -6 0 -18 E -12 -8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 18 12 B -8 0 12 20 8 C -12 -12 0 6 -8 D -18 -20 -6 0 -18 E -12 -8 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3491: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) E A B D C (6) B E D C A (6) E A C D B (5) D C A B E (5) C D A E B (5) C A D E B (5) B E A D C (5) D C B A E (4) C D A B E (4) A E C D B (4) E B A D C (3) E A C B D (3) B D C E A (3) A E D C B (3) A D C E B (3) E B A C D (2) C D B E A (2) C D B A E (2) B E C D A (2) B D C A E (2) B A D E C (2) A E B D C (2) A C D E B (2) E C B D A (1) D C A E B (1) D B C A E (1) D A C B E (1) C E D B A (1) C E A D B (1) C D E B A (1) B E D A C (1) B D E C A (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D B C (1) Total count = 100 A B C D E A 0 16 2 10 -4 B -16 0 -6 -2 -12 C -2 6 0 -2 -10 D -10 2 2 0 -10 E 4 12 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 2 10 -4 B -16 0 -6 -2 -12 C -2 6 0 -2 -10 D -10 2 2 0 -10 E 4 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 C=21 A=15 D=12 so D is eliminated. Round 2 votes counts: C=31 E=27 B=26 A=16 so A is eliminated. Round 3 votes counts: E=37 C=37 B=26 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:212 C:196 D:192 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 2 10 -4 B -16 0 -6 -2 -12 C -2 6 0 -2 -10 D -10 2 2 0 -10 E 4 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 10 -4 B -16 0 -6 -2 -12 C -2 6 0 -2 -10 D -10 2 2 0 -10 E 4 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 10 -4 B -16 0 -6 -2 -12 C -2 6 0 -2 -10 D -10 2 2 0 -10 E 4 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3492: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) D C E B A (6) E A D C B (5) E D C A B (4) E D B C A (4) E D B A C (4) A C B E D (4) D C B E A (3) C D B E A (3) C B D A E (3) C A D B E (3) C A B D E (3) B D C E A (3) B C A D E (3) B A C D E (3) A E C D B (3) A C E D B (3) E D A C B (2) E D A B C (2) D E C A B (2) D E B C A (2) D B C E A (2) B D E C A (2) B C D A E (2) A E B D C (2) A B E C D (2) A B C D E (2) E D C B A (1) E A D B C (1) C D E B A (1) C D E A B (1) C A D E B (1) B E A D C (1) B D E A C (1) B A E D C (1) B A E C D (1) B A D C E (1) A E C B D (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -16 -14 -18 B 12 0 -18 -24 -8 C 16 18 0 -18 -2 D 14 24 18 0 14 E 18 8 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -14 -18 B 12 0 -18 -24 -8 C 16 18 0 -18 -2 D 14 24 18 0 14 E 18 8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 A=20 B=18 C=15 so C is eliminated. Round 2 votes counts: D=29 A=27 E=23 B=21 so B is eliminated. Round 3 votes counts: D=40 A=36 E=24 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:235 C:207 E:207 B:181 A:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -16 -14 -18 B 12 0 -18 -24 -8 C 16 18 0 -18 -2 D 14 24 18 0 14 E 18 8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -14 -18 B 12 0 -18 -24 -8 C 16 18 0 -18 -2 D 14 24 18 0 14 E 18 8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -14 -18 B 12 0 -18 -24 -8 C 16 18 0 -18 -2 D 14 24 18 0 14 E 18 8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3493: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (13) D E B A C (11) C A B E D (9) D A E B C (8) C B E D A (8) A C D E B (7) A C B E D (7) A D C E B (5) C B E A D (4) B E C D A (4) D E A B C (3) E D B A C (2) D E B C A (2) C A D B E (2) A D E C B (2) E B D A C (1) D C E B A (1) D B E C A (1) D A C E B (1) C D B E A (1) C D B A E (1) C B A E D (1) C A B D E (1) B E A D C (1) B C E D A (1) A E B D C (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 2 -16 -6 B 4 0 -2 2 10 C -2 2 0 -6 0 D 16 -2 6 0 -4 E 6 -10 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -16 -6 B 4 0 -2 2 10 C -2 2 0 -6 0 D 16 -2 6 0 -4 E 6 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999998 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 A=24 B=19 E=3 so E is eliminated. Round 2 votes counts: D=29 C=27 A=24 B=20 so B is eliminated. Round 3 votes counts: D=43 C=32 A=25 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:207 E:200 C:197 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -16 -6 B 4 0 -2 2 10 C -2 2 0 -6 0 D 16 -2 6 0 -4 E 6 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999998 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -16 -6 B 4 0 -2 2 10 C -2 2 0 -6 0 D 16 -2 6 0 -4 E 6 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999998 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -16 -6 B 4 0 -2 2 10 C -2 2 0 -6 0 D 16 -2 6 0 -4 E 6 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999998 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3494: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) D C B A E (10) C D A E B (7) E A B C D (6) D B C A E (6) B E A D C (6) B E A C D (6) E A C B D (4) D C A B E (4) A D C E B (4) E B A C D (3) A E D C B (3) A E C D B (3) E C A D B (2) D C B E A (2) C A E D B (2) B D C E A (2) B D C A E (2) A C E D B (2) E B C A D (1) E A C D B (1) D B A C E (1) D A B C E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A B E (1) C A D E B (1) B E C A D (1) B D E C A (1) B D E A C (1) B D A C E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 14 -12 -4 22 B -14 0 -20 -28 -10 C 12 20 0 -12 20 D 4 28 12 0 12 E -22 10 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -12 -4 22 B -14 0 -20 -28 -10 C 12 20 0 -12 20 D 4 28 12 0 12 E -22 10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=20 E=17 C=14 A=14 so C is eliminated. Round 2 votes counts: D=44 B=20 E=19 A=17 so A is eliminated. Round 3 votes counts: D=49 E=31 B=20 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 C:220 A:210 E:178 B:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -12 -4 22 B -14 0 -20 -28 -10 C 12 20 0 -12 20 D 4 28 12 0 12 E -22 10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -12 -4 22 B -14 0 -20 -28 -10 C 12 20 0 -12 20 D 4 28 12 0 12 E -22 10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -12 -4 22 B -14 0 -20 -28 -10 C 12 20 0 -12 20 D 4 28 12 0 12 E -22 10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3495: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (20) D A E C B (12) B D A E C (8) D A C E B (5) C E A D B (5) C E A B D (5) B D E A C (5) C B E A D (4) D A B E C (3) B D A C E (3) A D E C B (3) E A D C B (2) E A C D B (2) A C E D B (2) A C D E B (2) E C B A D (1) D B C A E (1) D B A C E (1) D A E B C (1) C E B A D (1) C B A D E (1) C A E D B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C A E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 6 6 -4 B 8 0 2 14 10 C -6 -2 0 0 12 D -6 -14 0 0 0 E 4 -10 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 6 -4 B 8 0 2 14 10 C -6 -2 0 0 12 D -6 -14 0 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998325 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=45 D=23 C=17 A=10 E=5 so E is eliminated. Round 2 votes counts: B=45 D=23 C=18 A=14 so A is eliminated. Round 3 votes counts: B=45 D=30 C=25 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:202 A:200 E:191 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 6 -4 B 8 0 2 14 10 C -6 -2 0 0 12 D -6 -14 0 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998325 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 6 -4 B 8 0 2 14 10 C -6 -2 0 0 12 D -6 -14 0 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998325 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 6 -4 B 8 0 2 14 10 C -6 -2 0 0 12 D -6 -14 0 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998325 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3496: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) E D B A C (11) E C D B A (7) D B A E C (7) E D C B A (5) D E B A C (5) C A B E D (5) A B C D E (5) E C D A B (3) E B A D C (3) E A B C D (3) C E A B D (3) B A D C E (3) A B D C E (3) E C A B D (2) D E C B A (2) D C E B A (2) D C B A E (2) D B E A C (2) C E D A B (2) C D E A B (2) E B D A C (1) E A B D C (1) D B C A E (1) D B A C E (1) C D E B A (1) B D A E C (1) B A D E C (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 0 -10 -10 B 10 0 2 -6 -6 C 0 -2 0 -4 -12 D 10 6 4 0 4 E 10 6 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -10 -10 B 10 0 2 -6 -6 C 0 -2 0 -4 -12 D 10 6 4 0 4 E 10 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=26 D=22 A=11 B=5 so B is eliminated. Round 2 votes counts: E=36 C=26 D=23 A=15 so A is eliminated. Round 3 votes counts: E=37 C=32 D=31 so D is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:212 B:200 C:191 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 0 -10 -10 B 10 0 2 -6 -6 C 0 -2 0 -4 -12 D 10 6 4 0 4 E 10 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -10 -10 B 10 0 2 -6 -6 C 0 -2 0 -4 -12 D 10 6 4 0 4 E 10 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -10 -10 B 10 0 2 -6 -6 C 0 -2 0 -4 -12 D 10 6 4 0 4 E 10 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3497: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) E B C D A (10) E B A C D (10) A E B D C (8) C D B A E (6) A D C B E (6) C D B E A (5) E A B D C (4) A B D C E (4) E B C A D (3) D A C B E (3) A B E D C (3) E C D B A (2) C D E B A (2) B E C D A (2) B E A C D (2) A D C E B (2) A B D E C (2) E C B D A (1) E B A D C (1) E A B C D (1) D B C A E (1) C E D B A (1) C D E A B (1) C D A B E (1) B E A D C (1) B D C A E (1) B A E D C (1) A E D C B (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 2 2 2 8 B -2 0 10 8 4 C -2 -10 0 -6 -6 D -2 -8 6 0 -2 E -8 -4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 2 8 B -2 0 10 8 4 C -2 -10 0 -6 -6 D -2 -8 6 0 -2 E -8 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=28 D=17 C=16 B=7 so B is eliminated. Round 2 votes counts: E=37 A=29 D=18 C=16 so C is eliminated. Round 3 votes counts: E=38 D=33 A=29 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:210 A:207 E:198 D:197 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 2 8 B -2 0 10 8 4 C -2 -10 0 -6 -6 D -2 -8 6 0 -2 E -8 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 8 B -2 0 10 8 4 C -2 -10 0 -6 -6 D -2 -8 6 0 -2 E -8 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 8 B -2 0 10 8 4 C -2 -10 0 -6 -6 D -2 -8 6 0 -2 E -8 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3498: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (21) C D B A E (18) B A E D C (7) E A C D B (5) D C B A E (5) C D E A B (5) C D B E A (5) E A B C D (4) B C D A E (3) B A D C E (3) A E B D C (3) E C A D B (2) E A D B C (2) E A C B D (2) D C B E A (2) E A D C B (1) D B C A E (1) C E A D B (1) C B D A E (1) C B A E D (1) C A B E D (1) B D A E C (1) B C A E D (1) B C A D E (1) B A C D E (1) A E B C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 6 18 0 B 0 0 2 6 6 C -6 -2 0 6 -2 D -18 -6 -6 0 -8 E 0 -6 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.393488 B: 0.606512 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.522689777451 Cumulative probabilities = A: 0.393488 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 18 0 B 0 0 2 6 6 C -6 -2 0 6 -2 D -18 -6 -6 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=32 B=17 D=8 A=6 so A is eliminated. Round 2 votes counts: E=41 C=32 B=19 D=8 so D is eliminated. Round 3 votes counts: E=41 C=39 B=20 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:212 B:207 E:202 C:198 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 18 0 B 0 0 2 6 6 C -6 -2 0 6 -2 D -18 -6 -6 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 18 0 B 0 0 2 6 6 C -6 -2 0 6 -2 D -18 -6 -6 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 18 0 B 0 0 2 6 6 C -6 -2 0 6 -2 D -18 -6 -6 0 -8 E 0 -6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3499: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B A E C D (8) D C E B A (7) B A E D C (7) B A D E C (7) C E D A B (6) E A C B D (5) E C A D B (4) D B C A E (4) D B A C E (3) C E A D B (3) B D A E C (3) A E C B D (3) A E B C D (3) A B E C D (3) E C A B D (2) E A B C D (2) D C B E A (2) D C B A E (2) D B C E A (2) C E D B A (2) C D E A B (2) B E A C D (2) E C B D A (1) E B A C D (1) D C A E B (1) B E C D A (1) B D E C A (1) B D A C E (1) B C E D A (1) A E C D B (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 2 6 -4 B 10 0 0 4 -4 C -2 0 0 0 -10 D -6 -4 0 0 -10 E 4 4 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 2 6 -4 B 10 0 0 4 -4 C -2 0 0 0 -10 D -6 -4 0 0 -10 E 4 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 E=15 C=13 A=12 so A is eliminated. Round 2 votes counts: B=35 D=30 E=22 C=13 so C is eliminated. Round 3 votes counts: B=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:205 A:197 C:194 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 6 -4 B 10 0 0 4 -4 C -2 0 0 0 -10 D -6 -4 0 0 -10 E 4 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 6 -4 B 10 0 0 4 -4 C -2 0 0 0 -10 D -6 -4 0 0 -10 E 4 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 6 -4 B 10 0 0 4 -4 C -2 0 0 0 -10 D -6 -4 0 0 -10 E 4 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3500: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) E C D B A (9) B A D C E (8) B A E D C (7) B A E C D (7) A D C B E (7) C D E A B (6) E C D A B (5) D C A B E (4) D C A E B (3) C E D A B (3) A C D E B (3) E C B D A (2) E B C A D (2) D C E B A (2) D C E A B (2) C D A E B (2) A E C D B (2) A E B C D (2) E C A D B (1) E B C D A (1) E A B C D (1) D C B A E (1) D B C A E (1) D A C B E (1) C E D B A (1) C D E B A (1) B E A D C (1) B E A C D (1) B D E C A (1) B D C A E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 8 22 B -8 0 -10 -8 4 C -4 10 0 2 14 D -8 8 -2 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 8 22 B -8 0 -10 -8 4 C -4 10 0 2 14 D -8 8 -2 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 E=21 D=14 C=13 so C is eliminated. Round 2 votes counts: B=26 A=26 E=25 D=23 so D is eliminated. Round 3 votes counts: E=36 A=36 B=28 so B is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:211 D:203 B:189 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 8 22 B -8 0 -10 -8 4 C -4 10 0 2 14 D -8 8 -2 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 8 22 B -8 0 -10 -8 4 C -4 10 0 2 14 D -8 8 -2 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 8 22 B -8 0 -10 -8 4 C -4 10 0 2 14 D -8 8 -2 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3501: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) A B E D C (8) D C E A B (6) D C A B E (6) E B C A D (5) B A E C D (5) E C D B A (4) D C A E B (4) C D E A B (4) A D C B E (4) A B D C E (4) E B A D C (3) C D E B A (3) C D A B E (3) B E A C D (3) A B C D E (3) E B D C A (2) D A C B E (2) C E D B A (2) C A D B E (2) C A B D E (2) B C A E D (2) E D C B A (1) E D B C A (1) E B C D A (1) E A B D C (1) D E C B A (1) D A E B C (1) D A C E B (1) C B A E D (1) C B A D E (1) B E A D C (1) B A E D C (1) B A C E D (1) A D B C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -2 16 8 B -8 0 2 6 4 C 2 -2 0 4 8 D -16 -6 -4 0 0 E -8 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000155 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 16 8 B -8 0 2 6 4 C 2 -2 0 4 8 D -16 -6 -4 0 0 E -8 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999639 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 D=21 C=18 B=13 so B is eliminated. Round 2 votes counts: E=30 A=29 D=21 C=20 so C is eliminated. Round 3 votes counts: A=37 E=32 D=31 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:206 B:202 E:190 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 16 8 B -8 0 2 6 4 C 2 -2 0 4 8 D -16 -6 -4 0 0 E -8 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999639 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 16 8 B -8 0 2 6 4 C 2 -2 0 4 8 D -16 -6 -4 0 0 E -8 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999639 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 16 8 B -8 0 2 6 4 C 2 -2 0 4 8 D -16 -6 -4 0 0 E -8 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999639 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3502: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) B A C E D (7) D E C A B (6) D A E C B (6) B E C D A (6) B C E D A (6) A D C E B (6) A D E C B (5) A D B C E (5) E C B D A (4) A B D C E (4) E C D B A (3) D E A C B (3) C E B D A (3) E D C B A (2) D E C B A (2) C E D B A (2) C E A D B (2) C B E A D (2) B D E C A (2) B A D E C (2) A D B E C (2) A B C D E (2) E D C A B (1) E B D C A (1) D B E A C (1) D A C E B (1) D A B E C (1) C E D A B (1) B D E A C (1) B D A E C (1) B A C D E (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 0 -6 -10 B 6 0 2 0 2 C 0 -2 0 -6 0 D 6 0 6 0 4 E 10 -2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.672458 C: 0.000000 D: 0.327542 E: 0.000000 Sum of squares = 0.559483663076 Cumulative probabilities = A: 0.000000 B: 0.672458 C: 0.672458 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 -10 B 6 0 2 0 2 C 0 -2 0 -6 0 D 6 0 6 0 4 E 10 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=26 D=20 E=11 C=10 so C is eliminated. Round 2 votes counts: B=35 A=26 D=20 E=19 so E is eliminated. Round 3 votes counts: B=43 D=29 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:208 B:205 E:202 C:196 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -6 -10 B 6 0 2 0 2 C 0 -2 0 -6 0 D 6 0 6 0 4 E 10 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 -10 B 6 0 2 0 2 C 0 -2 0 -6 0 D 6 0 6 0 4 E 10 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 -10 B 6 0 2 0 2 C 0 -2 0 -6 0 D 6 0 6 0 4 E 10 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3503: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (14) E B D A C (10) A C D B E (7) E C A B D (6) C A D B E (5) D B A E C (4) D B A C E (4) C A E D B (4) C A E B D (4) E D B A C (3) B D E A C (3) A C B D E (3) E D B C A (2) E C B D A (2) E A C B D (2) D E B A C (2) C E A B D (2) C A D E B (2) A B D C E (2) A B C D E (2) E C D B A (1) E B D C A (1) E B C D A (1) E B C A D (1) D C B A E (1) D C A B E (1) D B C A E (1) D A C B E (1) C D A B E (1) C A B D E (1) B E D A C (1) B E A D C (1) B D A E C (1) A D C B E (1) A D B C E (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 28 -8 -4 B 6 0 10 -10 12 C -28 -10 0 -10 -12 D 8 10 10 0 14 E 4 -12 12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 28 -8 -4 B 6 0 10 -10 12 C -28 -10 0 -10 -12 D 8 10 10 0 14 E 4 -12 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=28 C=19 A=18 B=6 so B is eliminated. Round 2 votes counts: D=32 E=31 C=19 A=18 so A is eliminated. Round 3 votes counts: D=36 E=32 C=32 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:209 A:205 E:195 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 28 -8 -4 B 6 0 10 -10 12 C -28 -10 0 -10 -12 D 8 10 10 0 14 E 4 -12 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 28 -8 -4 B 6 0 10 -10 12 C -28 -10 0 -10 -12 D 8 10 10 0 14 E 4 -12 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 28 -8 -4 B 6 0 10 -10 12 C -28 -10 0 -10 -12 D 8 10 10 0 14 E 4 -12 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3504: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) C E D B A (9) D B C E A (8) A B D E C (8) E C A D B (7) E C A B D (7) D B A C E (7) A E C B D (5) E C D A B (3) C D E B A (3) E A C D B (2) D C B E A (2) D B C A E (2) D B A E C (2) D A B E C (2) C D B E A (2) A D B E C (2) A B E C D (2) E D C A B (1) E C D B A (1) E A D C B (1) E A C B D (1) D C E B A (1) C E B D A (1) C E A B D (1) C B D E A (1) C B A D E (1) B D C E A (1) B D C A E (1) B A D E C (1) A E B C D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 -18 -4 B 10 0 2 -10 12 C 4 -2 0 -6 6 D 18 10 6 0 14 E 4 -12 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -18 -4 B 10 0 2 -10 12 C 4 -2 0 -6 6 D 18 10 6 0 14 E 4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 A=20 C=18 B=15 so B is eliminated. Round 2 votes counts: D=38 E=23 A=21 C=18 so C is eliminated. Round 3 votes counts: D=44 E=34 A=22 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:207 C:201 E:186 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 -18 -4 B 10 0 2 -10 12 C 4 -2 0 -6 6 D 18 10 6 0 14 E 4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -18 -4 B 10 0 2 -10 12 C 4 -2 0 -6 6 D 18 10 6 0 14 E 4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -18 -4 B 10 0 2 -10 12 C 4 -2 0 -6 6 D 18 10 6 0 14 E 4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3505: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) E D B C A (6) D A C E B (6) A D B C E (6) E B C D A (5) D A E C B (4) C B A E D (4) A C B D E (4) E B D C A (3) E B C A D (3) B E C A D (3) A D C B E (3) A B C D E (3) E C D B A (2) E C B D A (2) D E C B A (2) D C E B A (2) D C E A B (2) C D A B E (2) C B E A D (2) C A B D E (2) B E A C D (2) B C E A D (2) A D B E C (2) A B C E D (2) E D C B A (1) E D B A C (1) E B A C D (1) D E C A B (1) D E B A C (1) D E A C B (1) D A E B C (1) C E D B A (1) C B E D A (1) C B A D E (1) C A D B E (1) B E A D C (1) B A E C D (1) A E B D C (1) A D E B C (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -4 -6 B -6 0 10 -10 -10 C -4 -10 0 -8 -8 D 4 10 8 0 10 E 6 10 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 -6 B -6 0 10 -10 -10 C -4 -10 0 -8 -8 D 4 10 8 0 10 E 6 10 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=25 E=24 C=14 B=9 so B is eliminated. Round 2 votes counts: E=30 D=28 A=26 C=16 so C is eliminated. Round 3 votes counts: E=36 A=34 D=30 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:207 A:200 B:192 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 -6 B -6 0 10 -10 -10 C -4 -10 0 -8 -8 D 4 10 8 0 10 E 6 10 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 -6 B -6 0 10 -10 -10 C -4 -10 0 -8 -8 D 4 10 8 0 10 E 6 10 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 -6 B -6 0 10 -10 -10 C -4 -10 0 -8 -8 D 4 10 8 0 10 E 6 10 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3506: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (13) C D E A B (12) C D E B A (6) D E C A B (5) C B A E D (5) C E D A B (4) E A D B C (3) E A B D C (3) D C E A B (3) C D B A E (3) B A E C D (3) E D C A B (2) D A B E C (2) C E B A D (2) C B D A E (2) B E A C D (2) B C A E D (2) B A D E C (2) A B E D C (2) E D A B C (1) E C D A B (1) E C B A D (1) E C A B D (1) E A B C D (1) D E A C B (1) D E A B C (1) D C B A E (1) D C A E B (1) D C A B E (1) D B A C E (1) D A E B C (1) C E D B A (1) C E B D A (1) C E A B D (1) C D B E A (1) C B E D A (1) C B E A D (1) C B A D E (1) B D A E C (1) B C A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 -20 -6 -10 B 8 0 -16 -2 -4 C 20 16 0 10 8 D 6 2 -10 0 -4 E 10 4 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -20 -6 -10 B 8 0 -16 -2 -4 C 20 16 0 10 8 D 6 2 -10 0 -4 E 10 4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 B=27 D=17 E=13 A=2 so A is eliminated. Round 2 votes counts: C=41 B=29 D=17 E=13 so E is eliminated. Round 3 votes counts: C=44 B=33 D=23 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:205 D:197 B:193 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -20 -6 -10 B 8 0 -16 -2 -4 C 20 16 0 10 8 D 6 2 -10 0 -4 E 10 4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 -6 -10 B 8 0 -16 -2 -4 C 20 16 0 10 8 D 6 2 -10 0 -4 E 10 4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 -6 -10 B 8 0 -16 -2 -4 C 20 16 0 10 8 D 6 2 -10 0 -4 E 10 4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3507: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) C D A B E (6) B D A E C (6) B D A C E (6) E C A D B (5) E B C D A (5) E A D C B (4) C B D A E (4) E C B D A (3) E A D B C (3) D A B C E (3) C E A D B (3) C A D B E (3) B E D A C (3) B C E D A (3) A E D C B (3) A D E C B (3) A D B C E (3) E C B A D (2) E B D A C (2) E B A D C (2) C A E D B (2) A D E B C (2) A D B E C (2) E C A B D (1) E B A C D (1) E A C D B (1) E A B D C (1) D B A E C (1) D B A C E (1) C E B A D (1) C B E D A (1) B E C D A (1) B D E A C (1) B D C A E (1) A E D B C (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 2 -10 -2 B 4 0 -2 6 -6 C -2 2 0 0 -8 D 10 -6 0 0 -14 E 2 6 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 -10 -2 B 4 0 -2 6 -6 C -2 2 0 0 -8 D 10 -6 0 0 -14 E 2 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=28 B=21 A=16 D=5 so D is eliminated. Round 2 votes counts: E=30 C=28 B=23 A=19 so A is eliminated. Round 3 votes counts: E=39 B=32 C=29 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:201 C:196 D:195 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -10 -2 B 4 0 -2 6 -6 C -2 2 0 0 -8 D 10 -6 0 0 -14 E 2 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -10 -2 B 4 0 -2 6 -6 C -2 2 0 0 -8 D 10 -6 0 0 -14 E 2 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -10 -2 B 4 0 -2 6 -6 C -2 2 0 0 -8 D 10 -6 0 0 -14 E 2 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3508: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D A B C E (10) D B A C E (9) A C E D B (9) C E A B D (6) A D C B E (6) B E D C A (5) B D E C A (5) E C B A D (4) D B E C A (4) D B E A C (4) D A C B E (3) C E A D B (3) C A E D B (3) C A E B D (3) A C D E B (3) E C D B A (2) E B C D A (2) D B A E C (2) E B C A D (1) D E B C A (1) B E C D A (1) B D A E C (1) A D B C E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 18 0 2 4 B -18 0 -8 -22 4 C 0 8 0 -2 16 D -2 22 2 0 0 E -4 -4 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600443 B: 0.000000 C: 0.399557 D: 0.000000 E: 0.000000 Sum of squares = 0.520177562748 Cumulative probabilities = A: 0.600443 B: 0.600443 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 2 4 B -18 0 -8 -22 4 C 0 8 0 -2 16 D -2 22 2 0 0 E -4 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045444 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=21 E=19 C=15 B=12 so B is eliminated. Round 2 votes counts: D=39 E=25 A=21 C=15 so C is eliminated. Round 3 votes counts: D=39 E=34 A=27 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:212 C:211 D:211 E:188 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 2 4 B -18 0 -8 -22 4 C 0 8 0 -2 16 D -2 22 2 0 0 E -4 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045444 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 2 4 B -18 0 -8 -22 4 C 0 8 0 -2 16 D -2 22 2 0 0 E -4 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045444 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 2 4 B -18 0 -8 -22 4 C 0 8 0 -2 16 D -2 22 2 0 0 E -4 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500151 B: 0.000000 C: 0.499849 D: 0.000000 E: 0.000000 Sum of squares = 0.500000045444 Cumulative probabilities = A: 0.500151 B: 0.500151 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3509: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) B C A E D (10) E B D C A (7) B A C D E (6) A C D B E (6) E D B C A (5) D E A C B (5) E D C A B (4) B E C A D (4) B C E A D (4) E B C D A (3) A C B D E (3) E C B D A (2) C E A B D (2) C B A E D (2) C B A D E (2) C A D E B (2) C A B E D (2) C A B D E (2) B E D A C (2) A D C B E (2) E D C B A (1) E B D A C (1) E B C A D (1) D E A B C (1) D C A E B (1) D B A E C (1) D A C E B (1) D A B E C (1) C B E A D (1) C A D B E (1) B E A C D (1) B D E A C (1) B C A D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -32 -14 6 -12 B 32 0 22 16 6 C 14 -22 0 12 -2 D -6 -16 -12 0 -26 E 12 -6 2 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 -14 6 -12 B 32 0 22 16 6 C 14 -22 0 12 -2 D -6 -16 -12 0 -26 E 12 -6 2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=29 C=14 A=12 D=10 so D is eliminated. Round 2 votes counts: E=41 B=30 C=15 A=14 so A is eliminated. Round 3 votes counts: E=41 B=32 C=27 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:238 E:217 C:201 A:174 D:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -32 -14 6 -12 B 32 0 22 16 6 C 14 -22 0 12 -2 D -6 -16 -12 0 -26 E 12 -6 2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 -14 6 -12 B 32 0 22 16 6 C 14 -22 0 12 -2 D -6 -16 -12 0 -26 E 12 -6 2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 -14 6 -12 B 32 0 22 16 6 C 14 -22 0 12 -2 D -6 -16 -12 0 -26 E 12 -6 2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3510: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) D A B E C (7) C E B A D (6) C B E A D (6) C A D E B (6) D A E B C (5) A D C E B (5) A C D E B (5) E B D A C (4) D A B C E (4) B E D A C (4) B D E A C (4) C A E B D (3) D B E A C (2) D A C B E (2) C E A B D (2) C B E D A (2) C B D E A (2) B E D C A (2) A E D B C (2) A D E C B (2) E C B A D (1) E B C A D (1) E B A D C (1) E A C B D (1) E A B C D (1) D C A B E (1) D B A E C (1) C E B D A (1) C D B A E (1) C A E D B (1) C A D B E (1) B E C D A (1) B C E D A (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 20 24 12 16 B -20 0 0 -14 -16 C -24 0 0 -14 2 D -12 14 14 0 18 E -16 16 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 24 12 16 B -20 0 0 -14 -16 C -24 0 0 -14 2 D -12 14 14 0 18 E -16 16 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=26 D=22 B=12 E=9 so E is eliminated. Round 2 votes counts: C=32 A=28 D=22 B=18 so B is eliminated. Round 3 votes counts: D=36 C=35 A=29 so A is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:236 D:217 E:190 C:182 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 24 12 16 B -20 0 0 -14 -16 C -24 0 0 -14 2 D -12 14 14 0 18 E -16 16 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 24 12 16 B -20 0 0 -14 -16 C -24 0 0 -14 2 D -12 14 14 0 18 E -16 16 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 24 12 16 B -20 0 0 -14 -16 C -24 0 0 -14 2 D -12 14 14 0 18 E -16 16 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3511: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) E D C B A (7) E D B A C (7) E B D A C (5) D E B A C (5) B A D C E (5) E C D A B (4) E C A B D (4) A B C D E (4) E D C A B (3) E C A D B (3) E B A C D (3) D C A B E (3) D B A C E (3) C A B D E (3) B D A E C (3) B A E D C (3) E B A D C (2) C E A B D (2) B E A D C (2) B A C D E (2) E D B C A (1) E C B A D (1) E B C A D (1) D C E A B (1) D C B A E (1) D C A E B (1) D B E A C (1) D A C B E (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E D B (1) C A E B D (1) C A B E D (1) B D E A C (1) B D A C E (1) B A D E C (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 2 0 -10 B 10 0 0 -8 -10 C -2 0 0 -16 -14 D 0 8 16 0 -4 E 10 10 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 2 0 -10 B 10 0 0 -8 -10 C -2 0 0 -16 -14 D 0 8 16 0 -4 E 10 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=19 B=18 D=16 A=6 so A is eliminated. Round 2 votes counts: E=41 B=22 C=20 D=17 so D is eliminated. Round 3 votes counts: E=46 C=28 B=26 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:210 B:196 A:191 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 2 0 -10 B 10 0 0 -8 -10 C -2 0 0 -16 -14 D 0 8 16 0 -4 E 10 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 0 -10 B 10 0 0 -8 -10 C -2 0 0 -16 -14 D 0 8 16 0 -4 E 10 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 0 -10 B 10 0 0 -8 -10 C -2 0 0 -16 -14 D 0 8 16 0 -4 E 10 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3512: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) C D A E B (7) C D A B E (7) C D B A E (6) A E D B C (6) E A B D C (5) D C A E B (5) C B D E A (5) C D E A B (4) B E A D C (4) B C E A D (4) E B A D C (3) E B A C D (3) D A C E B (3) B E C A D (3) A D E C B (3) E A D B C (2) D A E C B (2) C B E D A (2) B C E D A (2) B A E D C (2) A E B D C (2) A D E B C (2) E B C A D (1) D E A C B (1) D C E A B (1) C D B E A (1) B C D A E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 6 -6 B -2 0 6 0 0 C -2 -6 0 16 -4 D -6 0 -16 0 -2 E 6 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.351305 C: 0.000000 D: 0.000000 E: 0.648695 Sum of squares = 0.544220547671 Cumulative probabilities = A: 0.000000 B: 0.351305 C: 0.351305 D: 0.351305 E: 1.000000 A B C D E A 0 2 2 6 -6 B -2 0 6 0 0 C -2 -6 0 16 -4 D -6 0 -16 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=28 E=14 A=14 D=12 so D is eliminated. Round 2 votes counts: C=38 B=28 A=19 E=15 so E is eliminated. Round 3 votes counts: C=38 B=35 A=27 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:206 A:202 B:202 C:202 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 6 -6 B -2 0 6 0 0 C -2 -6 0 16 -4 D -6 0 -16 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 6 -6 B -2 0 6 0 0 C -2 -6 0 16 -4 D -6 0 -16 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 6 -6 B -2 0 6 0 0 C -2 -6 0 16 -4 D -6 0 -16 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3513: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (15) D B A C E (14) C A E B D (10) E C A D B (8) D E B C A (8) D B E A C (7) B D A C E (7) B A C E D (6) E D C A B (4) B A C D E (4) D B E C A (3) C E A B D (3) A B C E D (3) D E C A B (2) B A D C E (2) A C E B D (2) D C A B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 8 0 B 2 0 8 6 -4 C 8 -8 0 4 6 D -8 -6 -4 0 -4 E 0 4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.222222 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691322 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.555556 E: 1.000000 A B C D E A 0 -2 -8 8 0 B 2 0 8 6 -4 C 8 -8 0 4 6 D -8 -6 -4 0 -4 E 0 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.222222 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=27 B=19 C=13 A=6 so A is eliminated. Round 2 votes counts: D=35 E=27 B=22 C=16 so C is eliminated. Round 3 votes counts: E=42 D=35 B=23 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:206 C:205 E:201 A:199 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 8 0 B 2 0 8 6 -4 C 8 -8 0 4 6 D -8 -6 -4 0 -4 E 0 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.222222 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 8 0 B 2 0 8 6 -4 C 8 -8 0 4 6 D -8 -6 -4 0 -4 E 0 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.222222 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 8 0 B 2 0 8 6 -4 C 8 -8 0 4 6 D -8 -6 -4 0 -4 E 0 4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.222222 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3514: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) B D C A E (8) D B A E C (6) B D C E A (6) E C A B D (5) A E C D B (5) E D A B C (4) C B E A D (4) D B E A C (3) C A E B D (3) B C D E A (3) A D E B C (3) A C E D B (3) E C B D A (2) E C A D B (2) E A D C B (2) D E A B C (2) D A E B C (2) D A B E C (2) C E B A D (2) C E A B D (2) C B A D E (2) C A B D E (2) B D E C A (2) B C E D A (2) B C D A E (2) A E D B C (2) A D E C B (2) E D B A C (1) E B C D A (1) C E B D A (1) B E D C A (1) A E D C B (1) A D C B E (1) A D B E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 0 4 -6 B -8 0 2 -4 -8 C 0 -2 0 2 -14 D -4 4 -2 0 -2 E 6 8 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 4 -6 B -8 0 2 -4 -8 C 0 -2 0 2 -14 D -4 4 -2 0 -2 E 6 8 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=24 A=20 C=16 D=15 so D is eliminated. Round 2 votes counts: B=33 E=27 A=24 C=16 so C is eliminated. Round 3 votes counts: B=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:203 D:198 C:193 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 4 -6 B -8 0 2 -4 -8 C 0 -2 0 2 -14 D -4 4 -2 0 -2 E 6 8 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 4 -6 B -8 0 2 -4 -8 C 0 -2 0 2 -14 D -4 4 -2 0 -2 E 6 8 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 4 -6 B -8 0 2 -4 -8 C 0 -2 0 2 -14 D -4 4 -2 0 -2 E 6 8 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3515: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) D C E B A (10) E C D A B (6) C E D B A (6) B A C E D (6) D E C A B (5) D B C E A (5) B A D C E (5) A B E C D (4) E D C A B (3) D E C B A (3) A D E B C (3) D C B E A (2) D B E C A (2) D A E C B (2) C E B A D (2) B C D E A (2) B C A E D (2) A E B C D (2) E C A B D (1) E A C D B (1) D E A C B (1) D B A E C (1) D A E B C (1) C E B D A (1) C D E B A (1) C D B E A (1) C B E A D (1) B D C A E (1) B D A C E (1) B C E D A (1) B A C D E (1) A E D C B (1) A E C D B (1) A E C B D (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -10 -10 -8 B 8 0 2 -10 -4 C 10 -2 0 -18 -2 D 10 10 18 0 18 E 8 4 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -10 -8 B 8 0 2 -10 -4 C 10 -2 0 -18 -2 D 10 10 18 0 18 E 8 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=26 B=19 C=12 E=11 so E is eliminated. Round 2 votes counts: D=35 A=27 C=19 B=19 so C is eliminated. Round 3 votes counts: D=49 A=28 B=23 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:198 E:198 C:194 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -10 -8 B 8 0 2 -10 -4 C 10 -2 0 -18 -2 D 10 10 18 0 18 E 8 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -10 -8 B 8 0 2 -10 -4 C 10 -2 0 -18 -2 D 10 10 18 0 18 E 8 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -10 -8 B 8 0 2 -10 -4 C 10 -2 0 -18 -2 D 10 10 18 0 18 E 8 4 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3516: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (13) C B D E A (12) B C A E D (12) C B A D E (6) A E D C B (6) E D A B C (5) D E A B C (5) D E A C B (4) C B D A E (4) C B A E D (4) B C D E A (4) D E C A B (3) A D E C B (3) D E B C A (2) B E D A C (2) A B E D C (2) E D A C B (1) D E B A C (1) C D E B A (1) C D B E A (1) C D A E B (1) C A D E B (1) C A B E D (1) B D C E A (1) B A E C D (1) B A C E D (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -6 6 16 B 4 0 0 6 6 C 6 0 0 4 2 D -6 -6 -4 0 -2 E -16 -6 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.592698 C: 0.407302 D: 0.000000 E: 0.000000 Sum of squares = 0.517185706261 Cumulative probabilities = A: 0.000000 B: 0.592698 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 6 16 B 4 0 0 6 6 C 6 0 0 4 2 D -6 -6 -4 0 -2 E -16 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=27 B=21 D=15 E=6 so E is eliminated. Round 2 votes counts: C=31 A=27 D=21 B=21 so D is eliminated. Round 3 votes counts: A=42 C=34 B=24 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:208 A:206 C:206 D:191 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 6 16 B 4 0 0 6 6 C 6 0 0 4 2 D -6 -6 -4 0 -2 E -16 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 16 B 4 0 0 6 6 C 6 0 0 4 2 D -6 -6 -4 0 -2 E -16 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 16 B 4 0 0 6 6 C 6 0 0 4 2 D -6 -6 -4 0 -2 E -16 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3517: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) B D A E C (10) D B A E C (9) B A E D C (9) C E A D B (8) E A B C D (6) C E A B D (6) D C B A E (5) C D E A B (5) B A E C D (4) A E B C D (4) D C E A B (3) A E C B D (3) E A C B D (2) D C E B A (2) D C A E B (2) D B C A E (2) C E D A B (2) A B E C D (2) D B E C A (1) D B C E A (1) D B A C E (1) C D A E B (1) B E C A D (1) B D E A C (1) Total count = 100 A B C D E A 0 -12 2 -10 4 B 12 0 2 -4 12 C -2 -2 0 -12 -4 D 10 4 12 0 6 E -4 -12 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -10 4 B 12 0 2 -4 12 C -2 -2 0 -12 -4 D 10 4 12 0 6 E -4 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=25 C=22 A=9 E=8 so E is eliminated. Round 2 votes counts: D=36 B=25 C=22 A=17 so A is eliminated. Round 3 votes counts: B=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:211 A:192 E:191 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -10 4 B 12 0 2 -4 12 C -2 -2 0 -12 -4 D 10 4 12 0 6 E -4 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -10 4 B 12 0 2 -4 12 C -2 -2 0 -12 -4 D 10 4 12 0 6 E -4 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -10 4 B 12 0 2 -4 12 C -2 -2 0 -12 -4 D 10 4 12 0 6 E -4 -12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3518: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D B C A E (7) A E D C B (7) E A C B D (6) D A C B E (6) C B D A E (5) B D C E A (5) B C E D A (5) B C D E A (4) D C B A E (3) D C A B E (3) D B C E A (3) C A B E D (3) B E D C A (3) A E C B D (3) E B C D A (2) C B E A D (2) B C E A D (2) B C D A E (2) A C D B E (2) E A D B C (1) E A B C D (1) D B E C A (1) D B E A C (1) D A E C B (1) D A E B C (1) D A B C E (1) C D B A E (1) C B A E D (1) C B A D E (1) B E C D A (1) B D E C A (1) A E C D B (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -16 -26 -12 8 B 16 0 -2 12 30 C 26 2 0 6 22 D 12 -12 -6 0 6 E -8 -30 -22 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -26 -12 8 B 16 0 -2 12 30 C 26 2 0 6 22 D 12 -12 -6 0 6 E -8 -30 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 A=19 E=18 C=13 so C is eliminated. Round 2 votes counts: B=32 D=28 A=22 E=18 so E is eliminated. Round 3 votes counts: B=42 A=30 D=28 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:228 C:228 D:200 A:177 E:167 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -26 -12 8 B 16 0 -2 12 30 C 26 2 0 6 22 D 12 -12 -6 0 6 E -8 -30 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -26 -12 8 B 16 0 -2 12 30 C 26 2 0 6 22 D 12 -12 -6 0 6 E -8 -30 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -26 -12 8 B 16 0 -2 12 30 C 26 2 0 6 22 D 12 -12 -6 0 6 E -8 -30 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3519: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (14) A E C B D (9) E C A D B (6) D B C A E (6) D B A E C (5) B D C A E (5) B D A E C (5) E A C D B (3) D C E B A (3) D B E A C (3) C E A D B (3) A E D B C (3) A B E C D (3) E C A B D (2) D E C B A (2) C E D A B (2) C D E B A (2) C D B E A (2) B D A C E (2) B C A E D (2) B A D E C (2) A E D C B (2) E C D A B (1) E A D C B (1) E A C B D (1) D E A C B (1) D C B E A (1) D B E C A (1) D A B E C (1) C E B A D (1) C B D E A (1) B C D E A (1) B A E D C (1) A E B C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 -10 -16 -2 B 18 0 12 -24 12 C 10 -12 0 -18 -8 D 16 24 18 0 14 E 2 -12 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -16 -2 B 18 0 12 -24 12 C 10 -12 0 -18 -8 D 16 24 18 0 14 E 2 -12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=20 B=18 E=14 C=11 so C is eliminated. Round 2 votes counts: D=41 E=20 A=20 B=19 so B is eliminated. Round 3 votes counts: D=55 A=25 E=20 so E is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:236 B:209 E:192 C:186 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -10 -16 -2 B 18 0 12 -24 12 C 10 -12 0 -18 -8 D 16 24 18 0 14 E 2 -12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -16 -2 B 18 0 12 -24 12 C 10 -12 0 -18 -8 D 16 24 18 0 14 E 2 -12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -16 -2 B 18 0 12 -24 12 C 10 -12 0 -18 -8 D 16 24 18 0 14 E 2 -12 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3520: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) E D A C B (7) E D A B C (5) E C A D B (5) D A B E C (5) E D B A C (4) D E A C B (4) C B A D E (4) B D A C E (4) E C D A B (3) B D A E C (3) B A C D E (3) E D C B A (2) E D C A B (2) D E A B C (2) D A E C B (2) C E A D B (2) C B A E D (2) C A E D B (2) C A D E B (2) C A D B E (2) B E D A C (2) B E C D A (2) B A D C E (2) A D B C E (2) E C B A D (1) E B D C A (1) E B D A C (1) E A D C B (1) D B E A C (1) D B A C E (1) D A B C E (1) C E A B D (1) C B E A D (1) B E C A D (1) B D E A C (1) B C E A D (1) B C D A E (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 4 -8 0 B 2 0 12 -12 4 C -4 -12 0 -6 -10 D 8 12 6 0 6 E 0 -4 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -8 0 B 2 0 12 -12 4 C -4 -12 0 -6 -10 D 8 12 6 0 6 E 0 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=32 D=16 C=16 A=3 so A is eliminated. Round 2 votes counts: B=33 E=32 D=18 C=17 so C is eliminated. Round 3 votes counts: B=40 E=37 D=23 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:203 E:200 A:197 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -8 0 B 2 0 12 -12 4 C -4 -12 0 -6 -10 D 8 12 6 0 6 E 0 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -8 0 B 2 0 12 -12 4 C -4 -12 0 -6 -10 D 8 12 6 0 6 E 0 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -8 0 B 2 0 12 -12 4 C -4 -12 0 -6 -10 D 8 12 6 0 6 E 0 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3521: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) E B A D C (7) B E C A D (7) E A D B C (6) D A E C B (6) E B A C D (5) D A C E B (5) C D A B E (5) D E A B C (4) C B D A E (4) B E C D A (4) C B E A D (3) C B D E A (3) A D E B C (3) A D C E B (3) C D B A E (2) C B E D A (2) C B A E D (2) C A D B E (2) C A B E D (2) B C E D A (2) B C E A D (2) A E D B C (2) D E B A C (1) D C A E B (1) D C A B E (1) C B A D E (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 14 0 8 B -10 0 6 -10 -16 C -14 -6 0 -2 -18 D 0 10 2 0 6 E -8 16 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.481262 B: 0.000000 C: 0.000000 D: 0.518738 E: 0.000000 Sum of squares = 0.500702255562 Cumulative probabilities = A: 0.481262 B: 0.481262 C: 0.481262 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 0 8 B -10 0 6 -10 -16 C -14 -6 0 -2 -18 D 0 10 2 0 6 E -8 16 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 E=18 B=15 A=13 so A is eliminated. Round 2 votes counts: D=35 C=27 E=23 B=15 so B is eliminated. Round 3 votes counts: D=35 E=34 C=31 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:216 E:210 D:209 B:185 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 0 8 B -10 0 6 -10 -16 C -14 -6 0 -2 -18 D 0 10 2 0 6 E -8 16 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 0 8 B -10 0 6 -10 -16 C -14 -6 0 -2 -18 D 0 10 2 0 6 E -8 16 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 0 8 B -10 0 6 -10 -16 C -14 -6 0 -2 -18 D 0 10 2 0 6 E -8 16 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3522: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) E B D C A (7) D B C E A (7) A E B D C (7) C D B E A (6) C A D B E (5) B D C E A (5) A C D B E (5) E D B C A (4) A C E D B (4) C D E B A (3) C D B A E (3) A B E D C (3) E D B A C (2) E B A D C (2) E A D B C (2) C D A B E (2) A E C D B (2) A E B C D (2) A C D E B (2) E C A D B (1) E B D A C (1) E A C D B (1) D C B E A (1) C E D B A (1) C D E A B (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E C A (1) B D C A E (1) B D A C E (1) B A D E C (1) B A D C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 4 -12 B 2 0 18 -6 -2 C 4 -18 0 -14 6 D -4 6 14 0 -2 E 12 2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 A B C D E A 0 -2 -4 4 -12 B 2 0 18 -6 -2 C 4 -18 0 -14 6 D -4 6 14 0 -2 E 12 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 C=24 B=13 D=8 so D is eliminated. Round 2 votes counts: E=28 A=27 C=25 B=20 so B is eliminated. Round 3 votes counts: C=38 E=32 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:207 B:206 E:205 A:193 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 4 -12 B 2 0 18 -6 -2 C 4 -18 0 -14 6 D -4 6 14 0 -2 E 12 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 -12 B 2 0 18 -6 -2 C 4 -18 0 -14 6 D -4 6 14 0 -2 E 12 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 -12 B 2 0 18 -6 -2 C 4 -18 0 -14 6 D -4 6 14 0 -2 E 12 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3523: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) A C E B D (7) D B E C A (6) C E B D A (6) C A E B D (6) D A B E C (5) C D B E A (5) C E B A D (4) C E A B D (4) C D A B E (4) A E C B D (4) E B C A D (3) D C B E A (3) D B E A C (3) B E D A C (3) A E B D C (3) D C B A E (2) D A C B E (2) B E D C A (2) B E A D C (2) E C A B D (1) E B C D A (1) E B A D C (1) E A C B D (1) E A B D C (1) E A B C D (1) D C A B E (1) D B C E A (1) D B A E C (1) C D A E B (1) C A E D B (1) C A D E B (1) B E C D A (1) A E B C D (1) A D E B C (1) A D C B E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 12 -6 6 2 B -12 0 -10 6 2 C 6 10 0 6 0 D -6 -6 -6 0 -8 E -2 -2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.614911 D: 0.000000 E: 0.385089 Sum of squares = 0.526409193966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.614911 D: 0.614911 E: 1.000000 A B C D E A 0 12 -6 6 2 B -12 0 -10 6 2 C 6 10 0 6 0 D -6 -6 -6 0 -8 E -2 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=27 D=24 E=9 B=8 so B is eliminated. Round 2 votes counts: C=32 A=27 D=24 E=17 so E is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:207 E:202 B:193 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 6 2 B -12 0 -10 6 2 C 6 10 0 6 0 D -6 -6 -6 0 -8 E -2 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 6 2 B -12 0 -10 6 2 C 6 10 0 6 0 D -6 -6 -6 0 -8 E -2 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 6 2 B -12 0 -10 6 2 C 6 10 0 6 0 D -6 -6 -6 0 -8 E -2 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3524: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) E B C D A (7) D A B C E (7) A D B C E (7) E C B A D (6) C E A D B (5) B C A D E (5) A D C B E (5) D A B E C (4) C E B A D (4) C B A D E (4) E D A C B (3) E C D A B (3) E C B D A (3) E B D A C (3) D A E C B (3) B D A C E (2) B C A E D (2) B A D C E (2) E D B A C (1) E D A B C (1) E C D B A (1) D A E B C (1) D A C E B (1) C B A E D (1) C A E D B (1) C A D E B (1) C A D B E (1) C A B D E (1) B E C D A (1) B D E A C (1) B C E A D (1) A D C E B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -12 16 2 B 6 0 -8 8 10 C 12 8 0 14 24 D -16 -8 -14 0 -4 E -2 -10 -24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 16 2 B 6 0 -8 8 10 C 12 8 0 14 24 D -16 -8 -14 0 -4 E -2 -10 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 D=16 A=15 B=14 so B is eliminated. Round 2 votes counts: C=35 E=29 D=19 A=17 so A is eliminated. Round 3 votes counts: C=36 D=35 E=29 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:208 A:200 E:184 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 16 2 B 6 0 -8 8 10 C 12 8 0 14 24 D -16 -8 -14 0 -4 E -2 -10 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 16 2 B 6 0 -8 8 10 C 12 8 0 14 24 D -16 -8 -14 0 -4 E -2 -10 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 16 2 B 6 0 -8 8 10 C 12 8 0 14 24 D -16 -8 -14 0 -4 E -2 -10 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3525: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (7) B A D C E (7) C B E D A (6) B C E D A (6) A D E B C (6) E D A C B (4) D E B A C (4) B A D E C (4) A B D E C (4) D A E B C (3) C E B D A (3) B D E A C (3) B C A D E (3) B A C D E (3) A D E C B (3) E C D B A (2) C E A B D (2) C B E A D (2) C A E B D (2) B D E C A (2) B C D E A (2) A E D C B (2) A C E D B (2) E D C B A (1) E D C A B (1) E D B A C (1) E C B D A (1) E A D C B (1) D E A B C (1) D A B E C (1) C E D A B (1) C B A E D (1) C A B E D (1) B D A C E (1) B C D A E (1) A E C D B (1) A D C E B (1) A D B E C (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 26 -2 14 B 20 0 24 26 16 C -26 -24 0 -18 -6 D 2 -26 18 0 20 E -14 -16 6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 26 -2 14 B 20 0 24 26 16 C -26 -24 0 -18 -6 D 2 -26 18 0 20 E -14 -16 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=23 C=18 E=11 D=9 so D is eliminated. Round 2 votes counts: B=39 A=27 C=18 E=16 so E is eliminated. Round 3 votes counts: B=44 A=33 C=23 so C is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:243 A:209 D:207 E:178 C:163 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 26 -2 14 B 20 0 24 26 16 C -26 -24 0 -18 -6 D 2 -26 18 0 20 E -14 -16 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 26 -2 14 B 20 0 24 26 16 C -26 -24 0 -18 -6 D 2 -26 18 0 20 E -14 -16 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 26 -2 14 B 20 0 24 26 16 C -26 -24 0 -18 -6 D 2 -26 18 0 20 E -14 -16 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3526: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) B E C D A (7) C D B A E (6) C D A B E (6) E B D A C (5) E A B D C (5) D C A E B (5) B E D C A (5) B C E D A (5) A C D E B (5) B E A C D (4) B C D A E (4) D A C E B (3) C A D B E (3) B C D E A (3) E D B A C (2) E B D C A (2) D C B E A (2) C D A E B (2) C B A D E (2) A E C D B (2) A D E C B (2) A D C E B (2) E D B C A (1) E A D B C (1) D E A C B (1) D C A B E (1) C B D A E (1) C A B D E (1) B E D A C (1) A E D C B (1) A E B C D (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 -12 -24 -2 B 14 0 6 8 4 C 12 -6 0 8 6 D 24 -8 -8 0 0 E 2 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -24 -2 B 14 0 6 8 4 C 12 -6 0 8 6 D 24 -8 -8 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 C=21 A=15 D=12 so D is eliminated. Round 2 votes counts: C=29 B=29 E=24 A=18 so A is eliminated. Round 3 votes counts: C=41 E=30 B=29 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:216 C:210 D:204 E:196 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -24 -2 B 14 0 6 8 4 C 12 -6 0 8 6 D 24 -8 -8 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -24 -2 B 14 0 6 8 4 C 12 -6 0 8 6 D 24 -8 -8 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -24 -2 B 14 0 6 8 4 C 12 -6 0 8 6 D 24 -8 -8 0 0 E 2 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3527: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) E A C D B (6) A E B D C (6) A B E D C (5) C D B E A (4) C B D E A (4) A E D B C (4) E C A D B (3) D B C A E (3) D B A C E (3) C E B A D (3) C D E B A (3) B C D E A (3) A B D E C (3) E C A B D (2) E A C B D (2) D C B E A (2) D A E C B (2) D A E B C (2) C E B D A (2) B D A C E (2) B A E C D (2) A E D C B (2) A E B C D (2) E C D A B (1) E A D C B (1) E A B C D (1) D C E A B (1) D C B A E (1) D C A E B (1) D A B E C (1) C E D B A (1) C E D A B (1) C E A D B (1) C D E A B (1) C B E D A (1) C B E A D (1) B E C A D (1) B E A C D (1) B D C E A (1) B D C A E (1) B C E A D (1) B C D A E (1) B C A E D (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 10 16 4 B 2 0 8 10 4 C -10 -8 0 0 -12 D -16 -10 0 0 -2 E -4 -4 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 16 4 B 2 0 8 10 4 C -10 -8 0 0 -12 D -16 -10 0 0 -2 E -4 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=23 A=23 C=22 E=16 D=16 so E is eliminated. Round 2 votes counts: A=33 C=28 B=23 D=16 so D is eliminated. Round 3 votes counts: A=38 C=33 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:212 E:203 D:186 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 16 4 B 2 0 8 10 4 C -10 -8 0 0 -12 D -16 -10 0 0 -2 E -4 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 16 4 B 2 0 8 10 4 C -10 -8 0 0 -12 D -16 -10 0 0 -2 E -4 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 16 4 B 2 0 8 10 4 C -10 -8 0 0 -12 D -16 -10 0 0 -2 E -4 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3528: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (12) B C A D E (10) A D B C E (9) E D A C B (8) D A B C E (8) A B C D E (5) E D A B C (4) E C B A D (4) C B A D E (4) E D C B A (3) D E A C B (3) D A E B C (3) C B E A D (3) A D E B C (3) D A E C B (2) D A C B E (2) C B A E D (2) B C E A D (2) A B D C E (2) E C D B A (1) E B C A D (1) E A B C D (1) D E C B A (1) D C E B A (1) D C B A E (1) C E B D A (1) C B D E A (1) B E C A D (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 2 4 -2 6 B -2 0 2 2 2 C -4 -2 0 0 4 D 2 -2 0 0 12 E -6 -2 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333329 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -2 6 B -2 0 2 2 2 C -4 -2 0 0 4 D 2 -2 0 0 12 E -6 -2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=21 A=20 B=14 C=11 so C is eliminated. Round 2 votes counts: E=35 B=24 D=21 A=20 so A is eliminated. Round 3 votes counts: E=36 D=33 B=31 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 A:205 B:202 C:199 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 -2 6 B -2 0 2 2 2 C -4 -2 0 0 4 D 2 -2 0 0 12 E -6 -2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 6 B -2 0 2 2 2 C -4 -2 0 0 4 D 2 -2 0 0 12 E -6 -2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 6 B -2 0 2 2 2 C -4 -2 0 0 4 D 2 -2 0 0 12 E -6 -2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3529: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C D B A E (9) D C A B E (5) C B D A E (5) A D E B C (5) D A B C E (4) B C D A E (4) E C B A D (3) E A B D C (3) D A E B C (3) C E D B A (3) C E D A B (3) C B E A D (3) A E D B C (3) D C B A E (2) D C A E B (2) D A E C B (2) D A B E C (2) C D E B A (2) C D E A B (2) C B D E A (2) B D C A E (2) B C E A D (2) B A E D C (2) B A D E C (2) A D B E C (2) E C D A B (1) E C A D B (1) E C A B D (1) E B C A D (1) E B A C D (1) E A C B D (1) D E C A B (1) D B A C E (1) D A C B E (1) C E B D A (1) C E B A D (1) C D A B E (1) C B E D A (1) B E A C D (1) Total count = 100 A B C D E A 0 4 -16 -18 14 B -4 0 -6 -28 2 C 16 6 0 -2 12 D 18 28 2 0 18 E -14 -2 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -18 14 B -4 0 -6 -28 2 C 16 6 0 -2 12 D 18 28 2 0 18 E -14 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999913661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=23 E=21 B=13 A=10 so A is eliminated. Round 2 votes counts: C=33 D=30 E=24 B=13 so B is eliminated. Round 3 votes counts: C=39 D=34 E=27 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:233 C:216 A:192 B:182 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -16 -18 14 B -4 0 -6 -28 2 C 16 6 0 -2 12 D 18 28 2 0 18 E -14 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999913661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -18 14 B -4 0 -6 -28 2 C 16 6 0 -2 12 D 18 28 2 0 18 E -14 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999913661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -18 14 B -4 0 -6 -28 2 C 16 6 0 -2 12 D 18 28 2 0 18 E -14 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999913661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3530: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (14) D B C E A (11) A E C B D (11) D E A B C (10) E A D B C (7) C B D E A (6) A E D B C (6) D C B A E (5) D A E C B (5) E A B C D (4) B C E A D (3) A C B E D (3) D C B E A (2) C B D A E (2) C A B E D (2) B C D E A (2) A E B C D (2) E A B D C (1) D A E B C (1) C B E A D (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 4 12 6 B -8 0 -6 4 2 C -4 6 0 2 4 D -12 -4 -2 0 -12 E -6 -2 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 12 6 B -8 0 -6 4 2 C -4 6 0 2 4 D -12 -4 -2 0 -12 E -6 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=25 A=24 E=12 B=5 so B is eliminated. Round 2 votes counts: D=34 C=30 A=24 E=12 so E is eliminated. Round 3 votes counts: A=36 D=34 C=30 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:204 E:200 B:196 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 12 6 B -8 0 -6 4 2 C -4 6 0 2 4 D -12 -4 -2 0 -12 E -6 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 12 6 B -8 0 -6 4 2 C -4 6 0 2 4 D -12 -4 -2 0 -12 E -6 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 12 6 B -8 0 -6 4 2 C -4 6 0 2 4 D -12 -4 -2 0 -12 E -6 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3531: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) A E C D B (7) B D C E A (6) C A E D B (5) A E D B C (5) C D E A B (4) B D E C A (4) B C D E A (4) E A D B C (3) D B E A C (3) C B D E A (3) C B A D E (3) B A E D C (3) E D A B C (2) E A D C B (2) D B C E A (2) C A B E D (2) B C A E D (2) A E C B D (2) A E B D C (2) A E B C D (2) A C E D B (2) A B C E D (2) E D A C B (1) E B D A C (1) D E C B A (1) D E C A B (1) D E A C B (1) D B E C A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D B E A (1) C D A E B (1) C B D A E (1) C B A E D (1) C A E B D (1) B E D A C (1) B D A E C (1) B C D A E (1) B C A D E (1) B A D E C (1) B A C E D (1) B A C D E (1) A E D C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 4 2 -2 B 2 0 14 8 6 C -4 -14 0 0 -6 D -2 -8 0 0 -2 E 2 -6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 2 -2 B 2 0 14 8 6 C -4 -14 0 0 -6 D -2 -8 0 0 -2 E 2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=25 A=24 E=9 D=9 so E is eliminated. Round 2 votes counts: B=34 A=29 C=25 D=12 so D is eliminated. Round 3 votes counts: B=40 A=33 C=27 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:202 A:201 D:194 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 2 -2 B 2 0 14 8 6 C -4 -14 0 0 -6 D -2 -8 0 0 -2 E 2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 2 -2 B 2 0 14 8 6 C -4 -14 0 0 -6 D -2 -8 0 0 -2 E 2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 2 -2 B 2 0 14 8 6 C -4 -14 0 0 -6 D -2 -8 0 0 -2 E 2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3532: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C D B A E (7) C B D E A (6) C B D A E (6) B C E A D (5) B C A E D (5) A D E C B (5) A E D B C (4) A E B D C (4) D E A C B (3) D C A E B (3) D A C E B (3) B C D E A (3) E D C B A (2) E B A C D (2) E A D C B (2) E A B D C (2) D C B E A (2) D A E C B (2) C D B E A (2) B E C A D (2) B C E D A (2) A E D C B (2) A E B C D (2) D E C A B (1) D C E B A (1) D C B A E (1) D C A B E (1) D A C B E (1) B E C D A (1) B E A C D (1) B C D A E (1) B C A D E (1) B A E C D (1) A D C E B (1) A D C B E (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 4 8 B 2 0 -6 -6 4 C 4 6 0 0 8 D -4 6 0 0 6 E -8 -4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.765416 D: 0.234584 E: 0.000000 Sum of squares = 0.640891321327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.765416 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 4 8 B 2 0 -6 -6 4 C 4 6 0 0 8 D -4 6 0 0 6 E -8 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500103 D: 0.499897 E: 0.000000 Sum of squares = 0.500000021288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500103 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=22 A=22 C=21 D=18 E=17 so E is eliminated. Round 2 votes counts: A=35 B=24 C=21 D=20 so D is eliminated. Round 3 votes counts: A=44 C=32 B=24 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:204 A:203 B:197 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 4 8 B 2 0 -6 -6 4 C 4 6 0 0 8 D -4 6 0 0 6 E -8 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500103 D: 0.499897 E: 0.000000 Sum of squares = 0.500000021288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500103 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 8 B 2 0 -6 -6 4 C 4 6 0 0 8 D -4 6 0 0 6 E -8 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500103 D: 0.499897 E: 0.000000 Sum of squares = 0.500000021288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500103 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 8 B 2 0 -6 -6 4 C 4 6 0 0 8 D -4 6 0 0 6 E -8 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500103 D: 0.499897 E: 0.000000 Sum of squares = 0.500000021288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500103 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3533: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (14) B A D C E (11) B A C E D (11) E C D A B (6) E C B A D (5) E D C A B (4) D A C E B (4) E B C A D (3) D A C B E (3) B E C A D (3) A D B C E (3) A B C D E (3) E C B D A (2) D C E A B (2) D C A E B (2) D B A C E (2) B A E C D (2) B A C D E (2) A B D C E (2) E C D B A (1) E C A B D (1) E B D C A (1) D E C B A (1) D E B C A (1) D B E A C (1) D A B C E (1) C E B A D (1) C E A B D (1) C D E A B (1) C A E B D (1) B D A E C (1) B D A C E (1) B A E D C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 0 4 4 B 0 0 0 8 -2 C 0 0 0 -10 6 D -4 -8 10 0 10 E -4 2 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.280621 B: 0.438199 C: 0.281180 D: 0.000000 E: 0.000000 Sum of squares = 0.349828783815 Cumulative probabilities = A: 0.280621 B: 0.718820 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 4 B 0 0 0 8 -2 C 0 0 0 -10 6 D -4 -8 10 0 10 E -4 2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=31 E=23 A=10 C=4 so C is eliminated. Round 2 votes counts: D=32 B=32 E=25 A=11 so A is eliminated. Round 3 votes counts: B=39 D=35 E=26 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:204 D:204 B:203 C:198 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 4 B 0 0 0 8 -2 C 0 0 0 -10 6 D -4 -8 10 0 10 E -4 2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 4 B 0 0 0 8 -2 C 0 0 0 -10 6 D -4 -8 10 0 10 E -4 2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 4 B 0 0 0 8 -2 C 0 0 0 -10 6 D -4 -8 10 0 10 E -4 2 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3534: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) B E D C A (6) B E D A C (5) B E C D A (5) E B A D C (4) E A B D C (4) D C A B E (4) C D A B E (4) E C B A D (3) E B D A C (3) D A C B E (3) C E A B D (3) B D E C A (3) B D E A C (3) A E C D B (3) A D E B C (3) E B C A D (2) E A C B D (2) E A B C D (2) D B C A E (2) C E B A D (2) C A D E B (2) C A D B E (2) B C D E A (2) A E D B C (2) A E C B D (2) A C D E B (2) E B A C D (1) E A D B C (1) D B E A C (1) D B A E C (1) D A E B C (1) D A B E C (1) C B D A E (1) C A E B D (1) B D C E A (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -4 -10 -6 B 8 0 6 10 6 C 4 -6 0 -2 -20 D 10 -10 2 0 -6 E 6 -6 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -10 -6 B 8 0 6 10 6 C 4 -6 0 -2 -20 D 10 -10 2 0 -6 E 6 -6 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=24 E=22 A=16 D=13 so D is eliminated. Round 2 votes counts: B=29 C=28 E=22 A=21 so A is eliminated. Round 3 votes counts: E=35 C=35 B=30 so B is eliminated. Round 4 votes counts: E=60 C=40 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:213 D:198 C:188 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -10 -6 B 8 0 6 10 6 C 4 -6 0 -2 -20 D 10 -10 2 0 -6 E 6 -6 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -10 -6 B 8 0 6 10 6 C 4 -6 0 -2 -20 D 10 -10 2 0 -6 E 6 -6 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -10 -6 B 8 0 6 10 6 C 4 -6 0 -2 -20 D 10 -10 2 0 -6 E 6 -6 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3535: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) A C E B D (7) D C B E A (6) C D B A E (6) C A D E B (6) E B A D C (5) E A B D C (5) D B C E A (5) A C E D B (5) D E B C A (4) C D A E B (4) C D A B E (4) B E D A C (4) B E A D C (4) A E C D B (3) E A D C B (2) C D B E A (2) C A E D B (2) C A D B E (2) B D E A C (2) A C B E D (2) A B E C D (2) E D A C B (1) D E C B A (1) D B E C A (1) D B C A E (1) C D E A B (1) C B A D E (1) B D E C A (1) B D C A E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 12 4 12 12 B -12 0 -26 -12 -12 C -4 26 0 12 10 D -12 12 -12 0 -4 E -12 12 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 12 12 B -12 0 -26 -12 -12 C -4 26 0 12 10 D -12 12 -12 0 -4 E -12 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=28 D=18 E=13 B=12 so B is eliminated. Round 2 votes counts: A=29 C=28 D=22 E=21 so E is eliminated. Round 3 votes counts: A=45 C=28 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:222 A:220 E:197 D:192 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 12 12 B -12 0 -26 -12 -12 C -4 26 0 12 10 D -12 12 -12 0 -4 E -12 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 12 12 B -12 0 -26 -12 -12 C -4 26 0 12 10 D -12 12 -12 0 -4 E -12 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 12 12 B -12 0 -26 -12 -12 C -4 26 0 12 10 D -12 12 -12 0 -4 E -12 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3536: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) A E B D C (8) C D E B A (7) B A D C E (7) C D B E A (6) A B E D C (6) E A C D B (5) A E B C D (5) C E D A B (4) A E C D B (4) E C D A B (3) D E C B A (3) B D C A E (3) B D A C E (3) B C D A E (3) B A D E C (3) E D C A B (2) E D A C B (2) D C E B A (2) C B D A E (2) E D C B A (1) E D A B C (1) E C A D B (1) E A D C B (1) D C B E A (1) C E D B A (1) C B D E A (1) B D E A C (1) A E D C B (1) A E C B D (1) A C B E D (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 4 -8 0 B 4 0 2 10 -4 C -4 -2 0 -8 0 D 8 -10 8 0 4 E 0 4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.407407407406 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 A B C D E A 0 -4 4 -8 0 B 4 0 2 10 -4 C -4 -2 0 -8 0 D 8 -10 8 0 4 E 0 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.407407407412 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 C=21 E=16 D=6 so D is eliminated. Round 2 votes counts: A=29 B=28 C=24 E=19 so E is eliminated. Round 3 votes counts: A=38 C=34 B=28 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:206 D:205 E:200 A:196 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 -8 0 B 4 0 2 10 -4 C -4 -2 0 -8 0 D 8 -10 8 0 4 E 0 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.407407407412 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -8 0 B 4 0 2 10 -4 C -4 -2 0 -8 0 D 8 -10 8 0 4 E 0 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.407407407412 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -8 0 B 4 0 2 10 -4 C -4 -2 0 -8 0 D 8 -10 8 0 4 E 0 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.222222 E: 0.555556 Sum of squares = 0.407407407412 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.444444 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3537: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) C D E A B (6) A B E D C (5) E C B D A (4) E B C A D (4) D C A B E (4) C D E B A (4) E A B C D (3) D C E A B (3) D C A E B (3) D A C B E (3) B E A C D (3) B D C A E (3) B C D A E (3) A E B D C (3) A B D C E (3) E C D B A (2) E B A C D (2) E A B D C (2) D E C A B (2) D C B A E (2) D A C E B (2) C E D B A (2) C E B D A (2) C D B E A (2) C D B A E (2) B A E C D (2) A D B E C (2) E C D A B (1) E C A D B (1) E A D B C (1) E A C D B (1) D A B C E (1) C B E D A (1) B E C A D (1) B D A C E (1) B A E D C (1) B A D C E (1) B A C D E (1) A E D B C (1) A D C E B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -4 -6 6 B -14 0 2 -8 0 C 4 -2 0 -6 16 D 6 8 6 0 16 E -6 0 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 -6 6 B -14 0 2 -8 0 C 4 -2 0 -6 16 D 6 8 6 0 16 E -6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=21 D=20 C=19 B=16 so B is eliminated. Round 2 votes counts: A=29 E=25 D=24 C=22 so C is eliminated. Round 3 votes counts: D=41 E=30 A=29 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:206 A:205 B:190 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -4 -6 6 B -14 0 2 -8 0 C 4 -2 0 -6 16 D 6 8 6 0 16 E -6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -6 6 B -14 0 2 -8 0 C 4 -2 0 -6 16 D 6 8 6 0 16 E -6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -6 6 B -14 0 2 -8 0 C 4 -2 0 -6 16 D 6 8 6 0 16 E -6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3538: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) C A E D B (5) C E B A D (4) B D E A C (4) B A D C E (4) A C D E B (4) E D C B A (3) E D C A B (3) E C D A B (3) E C B D A (3) E B C D A (3) D B E A C (3) D B A E C (3) C E A D B (3) C E A B D (3) B E D A C (3) B E C A D (3) A D C E B (3) A D C B E (3) A C D B E (3) A B D C E (3) D C A E B (2) D A E C B (2) D A B E C (2) B E C D A (2) B D A E C (2) B A D E C (2) A C B D E (2) E D B C A (1) E B D C A (1) D E B A C (1) D A E B C (1) D A B C E (1) C E D A B (1) C D E A B (1) C B A E D (1) C A B E D (1) C A B D E (1) B E D C A (1) B C A E D (1) Total count = 100 A B C D E A 0 10 8 8 8 B -10 0 -2 -12 6 C -8 2 0 -12 8 D -8 12 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 8 8 B -10 0 -2 -12 6 C -8 2 0 -12 8 D -8 12 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=22 C=20 E=17 D=15 so D is eliminated. Round 2 votes counts: A=32 B=28 C=22 E=18 so E is eliminated. Round 3 votes counts: C=34 B=34 A=32 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:217 D:213 C:195 B:191 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 8 8 B -10 0 -2 -12 6 C -8 2 0 -12 8 D -8 12 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 8 8 B -10 0 -2 -12 6 C -8 2 0 -12 8 D -8 12 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 8 8 B -10 0 -2 -12 6 C -8 2 0 -12 8 D -8 12 12 0 10 E -8 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3539: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) E C D B A (7) C E D B A (5) C D B E A (5) B D C A E (5) A E C B D (5) E C A D B (4) A E B C D (4) A B D E C (4) E C D A B (3) E A C D B (3) D B C E A (3) B D C E A (3) A E B D C (3) A B E D C (3) E A B D C (2) D C B E A (2) D C B A E (2) C E A D B (2) C D E B A (2) B D A C E (2) B A D C E (2) A E C D B (2) A B E C D (2) E D C B A (1) E D B C A (1) E B A D C (1) E A C B D (1) E A B C D (1) D B E C A (1) C E D A B (1) C D B A E (1) C D A E B (1) C A D B E (1) B D E C A (1) B D E A C (1) B A E D C (1) B A D E C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -2 6 0 B -6 0 2 6 2 C 2 -2 0 4 -4 D -6 -6 -4 0 -6 E 0 -2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.447012 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.552988 Sum of squares = 0.505615368864 Cumulative probabilities = A: 0.447012 B: 0.447012 C: 0.447012 D: 0.447012 E: 1.000000 A B C D E A 0 6 -2 6 0 B -6 0 2 6 2 C 2 -2 0 4 -4 D -6 -6 -4 0 -6 E 0 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=24 C=18 B=16 D=8 so D is eliminated. Round 2 votes counts: A=34 E=24 C=22 B=20 so B is eliminated. Round 3 votes counts: A=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:205 E:204 B:202 C:200 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 6 0 B -6 0 2 6 2 C 2 -2 0 4 -4 D -6 -6 -4 0 -6 E 0 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 6 0 B -6 0 2 6 2 C 2 -2 0 4 -4 D -6 -6 -4 0 -6 E 0 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 6 0 B -6 0 2 6 2 C 2 -2 0 4 -4 D -6 -6 -4 0 -6 E 0 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3540: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) C E A D B (5) C A E D B (5) B A D C E (5) A C E D B (5) D E A C B (4) C E B A D (4) B D E A C (4) E D C A B (3) D E A B C (3) C A B E D (3) B D E C A (3) B C E D A (3) B C E A D (3) B A D E C (3) B A C D E (3) E D C B A (2) E C D A B (2) D B E A C (2) D A E C B (2) C B E D A (2) C B E A D (2) B E D C A (2) B C A E D (2) A D C E B (2) A D B E C (2) A C B D E (2) A B C D E (2) E C B D A (1) E B C D A (1) D E B A C (1) D B A E C (1) D A B E C (1) C E D B A (1) C E D A B (1) C E A B D (1) B C A D E (1) A D E B C (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 6 6 0 B 8 0 4 12 10 C -6 -4 0 0 8 D -6 -12 0 0 4 E 0 -10 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 6 0 B 8 0 4 12 10 C -6 -4 0 0 8 D -6 -12 0 0 4 E 0 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=24 A=16 D=14 E=9 so E is eliminated. Round 2 votes counts: B=38 C=27 D=19 A=16 so A is eliminated. Round 3 votes counts: B=41 C=35 D=24 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:202 C:199 D:193 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 6 0 B 8 0 4 12 10 C -6 -4 0 0 8 D -6 -12 0 0 4 E 0 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 6 0 B 8 0 4 12 10 C -6 -4 0 0 8 D -6 -12 0 0 4 E 0 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 6 0 B 8 0 4 12 10 C -6 -4 0 0 8 D -6 -12 0 0 4 E 0 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3541: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (13) A D E C B (9) D A B C E (6) C E B D A (5) B E C A D (5) A E D C B (5) D A E C B (4) A D E B C (4) E C B A D (3) D A B E C (3) C B E D A (3) B D C A E (3) B C D E A (3) E C A D B (2) E C A B D (2) E A C D B (2) D C B A E (2) D A E B C (2) D A C E B (2) C D E A B (2) C B D E A (2) B E A C D (2) B C E A D (2) B A E D C (2) E B C A D (1) D C A B E (1) D B A C E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D A E B (1) C B E A D (1) B D A E C (1) B D A C E (1) B C D A E (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -10 -16 -2 B 6 0 2 2 6 C 10 -2 0 6 4 D 16 -2 -6 0 -2 E 2 -6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -16 -2 B 6 0 2 2 6 C 10 -2 0 6 4 D 16 -2 -6 0 -2 E 2 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999585 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=21 A=19 C=17 E=10 so E is eliminated. Round 2 votes counts: B=34 C=24 D=21 A=21 so D is eliminated. Round 3 votes counts: A=38 B=35 C=27 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:208 D:203 E:197 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 -16 -2 B 6 0 2 2 6 C 10 -2 0 6 4 D 16 -2 -6 0 -2 E 2 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999585 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -16 -2 B 6 0 2 2 6 C 10 -2 0 6 4 D 16 -2 -6 0 -2 E 2 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999585 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -16 -2 B 6 0 2 2 6 C 10 -2 0 6 4 D 16 -2 -6 0 -2 E 2 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999585 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3542: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) E A D B C (5) D A C E B (5) E B A D C (4) E A B D C (4) C D B A E (4) C B D E A (4) C B D A E (4) B E C A D (4) B C E D A (4) B C D E A (4) E B A C D (3) D C A B E (3) D A E B C (3) D A B E C (3) C B E D A (3) A E D B C (3) E B C A D (2) E A D C B (2) D A E C B (2) C D A B E (2) C B E A D (2) B E C D A (2) B D C A E (2) A E D C B (2) A D C E B (2) E C B A D (1) E C A D B (1) E A C B D (1) D C B A E (1) D C A E B (1) D B C A E (1) D A C B E (1) D A B C E (1) C E A B D (1) C A D E B (1) B E A D C (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 4 2 -2 2 B -4 0 -4 -4 -4 C -2 4 0 -10 -4 D 2 4 10 0 8 E -2 4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -2 2 B -4 0 -4 -4 -4 C -2 4 0 -10 -4 D 2 4 10 0 8 E -2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 D=21 C=21 B=19 A=16 so A is eliminated. Round 2 votes counts: D=32 E=28 C=21 B=19 so B is eliminated. Round 3 votes counts: E=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:203 E:199 C:194 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -2 2 B -4 0 -4 -4 -4 C -2 4 0 -10 -4 D 2 4 10 0 8 E -2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -2 2 B -4 0 -4 -4 -4 C -2 4 0 -10 -4 D 2 4 10 0 8 E -2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -2 2 B -4 0 -4 -4 -4 C -2 4 0 -10 -4 D 2 4 10 0 8 E -2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3543: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) B A C E D (9) C B A D E (7) B C A D E (6) D E C B A (5) D C B A E (4) B A E C D (4) A B E C D (4) E D B A C (3) E B D A C (3) E A D C B (3) E A B D C (3) E A B C D (3) D E C A B (3) D C A E B (3) E B A D C (2) D C E A B (2) D B C A E (2) C A D B E (2) B E D A C (2) B D E C A (2) B D C A E (2) B C D A E (2) A C B E D (2) E D A B C (1) E B A C D (1) E A D B C (1) E A C B D (1) D E B C A (1) D E A C B (1) D C A B E (1) D B C E A (1) C D A B E (1) B E D C A (1) B E A D C (1) B E A C D (1) B C A E D (1) Total count = 100 A B C D E A 0 -20 8 2 0 B 20 0 12 14 10 C -8 -12 0 -12 -10 D -2 -14 12 0 -10 E 0 -10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 8 2 0 B 20 0 12 14 10 C -8 -12 0 -12 -10 D -2 -14 12 0 -10 E 0 -10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=30 D=23 C=10 A=6 so A is eliminated. Round 2 votes counts: B=35 E=30 D=23 C=12 so C is eliminated. Round 3 votes counts: B=44 E=30 D=26 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:228 E:205 A:195 D:193 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 8 2 0 B 20 0 12 14 10 C -8 -12 0 -12 -10 D -2 -14 12 0 -10 E 0 -10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 8 2 0 B 20 0 12 14 10 C -8 -12 0 -12 -10 D -2 -14 12 0 -10 E 0 -10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 8 2 0 B 20 0 12 14 10 C -8 -12 0 -12 -10 D -2 -14 12 0 -10 E 0 -10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3544: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (15) D C E A B (7) B A E C D (7) E C B A D (4) A C E B D (4) A C B E D (4) A B C E D (4) A B C D E (4) D E C B A (3) D C A B E (3) C A D E B (3) B E A C D (3) E D C B A (2) E C A B D (2) E B D C A (2) D C E B A (2) C A E D B (2) B E D C A (2) B A E D C (2) B A D E C (2) A C D B E (2) A C B D E (2) E D C A B (1) E D B C A (1) E C A D B (1) E B C A D (1) E B A C D (1) D E C A B (1) D E B C A (1) D B E C A (1) D B E A C (1) D B A C E (1) C D A E B (1) B E D A C (1) B E A D C (1) B D A C E (1) B A D C E (1) B A C E D (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 18 -10 8 24 B -18 0 -22 0 -12 C 10 22 0 -4 20 D -8 0 4 0 6 E -24 12 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.363636 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826445 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.545455 D: 1.000000 E: 1.000000 A B C D E A 0 18 -10 8 24 B -18 0 -22 0 -12 C 10 22 0 -4 20 D -8 0 4 0 6 E -24 12 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.363636 D: 0.454545 E: 0.000000 Sum of squares = 0.371900825667 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.545455 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=23 B=21 E=15 C=6 so C is eliminated. Round 2 votes counts: D=36 A=28 B=21 E=15 so E is eliminated. Round 3 votes counts: D=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:224 A:220 D:201 E:181 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -10 8 24 B -18 0 -22 0 -12 C 10 22 0 -4 20 D -8 0 4 0 6 E -24 12 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.363636 D: 0.454545 E: 0.000000 Sum of squares = 0.371900825667 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.545455 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -10 8 24 B -18 0 -22 0 -12 C 10 22 0 -4 20 D -8 0 4 0 6 E -24 12 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.363636 D: 0.454545 E: 0.000000 Sum of squares = 0.371900825667 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.545455 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -10 8 24 B -18 0 -22 0 -12 C 10 22 0 -4 20 D -8 0 4 0 6 E -24 12 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.363636 D: 0.454545 E: 0.000000 Sum of squares = 0.371900825667 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.545455 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3545: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) C A B D E (7) D A C E B (6) A D C B E (6) E D A B C (4) E C B D A (4) C B A E D (4) B E A D C (4) E D B C A (3) E B D A C (3) D E A C B (3) A C D B E (3) A C B D E (3) E D C B A (2) D E C A B (2) D E B A C (2) C B E A D (2) C B A D E (2) C A D E B (2) C A B E D (2) B E C A D (2) B D A E C (2) B C A E D (2) B C A D E (2) B A C D E (2) E D C A B (1) E C D A B (1) E B D C A (1) D E A B C (1) D C A E B (1) D B E A C (1) D A E C B (1) D A E B C (1) D A C B E (1) D A B C E (1) C E B D A (1) C D A E B (1) C A D B E (1) B E D A C (1) B D E A C (1) B C E A D (1) B A C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 12 -6 4 B 2 0 -12 -4 4 C -12 12 0 -12 4 D 6 4 12 0 6 E -4 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 -6 4 B 2 0 -12 -4 4 C -12 12 0 -12 4 D 6 4 12 0 6 E -4 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=22 D=20 B=18 A=13 so A is eliminated. Round 2 votes counts: C=28 E=27 D=26 B=19 so B is eliminated. Round 3 votes counts: C=37 E=34 D=29 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:214 A:204 C:196 B:195 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 -6 4 B 2 0 -12 -4 4 C -12 12 0 -12 4 D 6 4 12 0 6 E -4 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -6 4 B 2 0 -12 -4 4 C -12 12 0 -12 4 D 6 4 12 0 6 E -4 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -6 4 B 2 0 -12 -4 4 C -12 12 0 -12 4 D 6 4 12 0 6 E -4 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3546: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (6) D E C A B (5) B A C E D (5) E C D A B (4) E B D C A (4) D B E A C (4) C A B E D (4) A C D B E (4) A B C D E (4) E D C B A (3) E D C A B (3) E C B D A (3) D E A B C (3) D A C E B (3) C E D A B (3) A B D C E (3) D E B C A (2) D A E B C (2) C E A D B (2) C E A B D (2) C B A E D (2) B E D C A (2) B D E A C (2) B C A E D (2) A D C E B (2) A C D E B (2) E D B C A (1) E C D B A (1) E B C D A (1) D E C B A (1) D E B A C (1) D E A C B (1) D A E C B (1) D A B E C (1) C E B D A (1) C B E A D (1) C A E D B (1) B E C D A (1) B E A D C (1) B C E A D (1) B A E C D (1) B A D E C (1) B A D C E (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 14 0 -6 -6 B -14 0 -10 -2 -4 C 0 10 0 4 2 D 6 2 -4 0 2 E 6 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125472 B: 0.000000 C: 0.874528 D: 0.000000 E: 0.000000 Sum of squares = 0.780542119593 Cumulative probabilities = A: 0.125472 B: 0.125472 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 -6 -6 B -14 0 -10 -2 -4 C 0 10 0 4 2 D 6 2 -4 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000009852 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 E=20 B=18 C=16 so C is eliminated. Round 2 votes counts: E=28 A=27 D=24 B=21 so B is eliminated. Round 3 votes counts: A=40 E=34 D=26 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:208 D:203 E:203 A:201 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 0 -6 -6 B -14 0 -10 -2 -4 C 0 10 0 4 2 D 6 2 -4 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000009852 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 -6 -6 B -14 0 -10 -2 -4 C 0 10 0 4 2 D 6 2 -4 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000009852 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 -6 -6 B -14 0 -10 -2 -4 C 0 10 0 4 2 D 6 2 -4 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000009852 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3547: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (6) B D E A C (6) E B A D C (5) B E D A C (5) A C D E B (5) D C B A E (4) D B A E C (4) D A C B E (4) C E A B D (4) C A D E B (4) E C A B D (3) E B C A D (3) E B A C D (3) E A C B D (3) D B C A E (3) C D A B E (3) B E D C A (3) A C E D B (3) E A B C D (2) D B E C A (2) D B E A C (2) D B C E A (2) D B A C E (2) C D B E A (2) B D E C A (2) A D C B E (2) E C B A D (1) E B C D A (1) D C A B E (1) C E B D A (1) C E A D B (1) C A E B D (1) B E A D C (1) B D C E A (1) B D A E C (1) A E D C B (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 4 2 -6 B 8 0 2 -6 4 C -4 -2 0 -8 -2 D -2 6 8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999912 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 2 -6 B 8 0 2 -6 4 C -4 -2 0 -8 -2 D -2 6 8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999938 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 E=21 B=19 A=14 so A is eliminated. Round 2 votes counts: C=30 D=28 E=22 B=20 so B is eliminated. Round 3 votes counts: D=38 E=32 C=30 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:204 E:200 A:196 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 4 2 -6 B 8 0 2 -6 4 C -4 -2 0 -8 -2 D -2 6 8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999938 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 2 -6 B 8 0 2 -6 4 C -4 -2 0 -8 -2 D -2 6 8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999938 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 2 -6 B 8 0 2 -6 4 C -4 -2 0 -8 -2 D -2 6 8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999938 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3548: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) C B D E A (7) B C E A D (7) B C D E A (7) D B C A E (6) D A E C B (5) C B E A D (5) A D E B C (4) E A C D B (3) E A B C D (3) D B A E C (3) D B A C E (3) C E B A D (3) C D E A B (3) E C B A D (2) E A C B D (2) D C A E B (2) D A C E B (2) B A E C D (2) A E B D C (2) A D E C B (2) E C A D B (1) E C A B D (1) E B C A D (1) E A D C B (1) D E C A B (1) D B C E A (1) D A E B C (1) C E B D A (1) C D B E A (1) B E C A D (1) B E A C D (1) B D C A E (1) B C A D E (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -4 12 -4 B 8 0 -2 4 -2 C 4 2 0 8 0 D -12 -4 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.527873 D: 0.000000 E: 0.472127 Sum of squares = 0.50155376592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.527873 D: 0.527873 E: 1.000000 A B C D E A 0 -8 -4 12 -4 B 8 0 -2 4 -2 C 4 2 0 8 0 D -12 -4 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=21 A=21 C=20 E=14 so E is eliminated. Round 2 votes counts: A=30 D=24 C=24 B=22 so B is eliminated. Round 3 votes counts: C=41 A=34 D=25 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:204 E:201 A:198 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -4 12 -4 B 8 0 -2 4 -2 C 4 2 0 8 0 D -12 -4 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 12 -4 B 8 0 -2 4 -2 C 4 2 0 8 0 D -12 -4 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 12 -4 B 8 0 -2 4 -2 C 4 2 0 8 0 D -12 -4 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3549: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) E D C A B (9) C E D B A (8) D E A B C (6) C B A E D (6) A B D E C (6) B A C E D (5) B A C D E (5) A B D C E (5) E D A C B (4) D A E B C (4) C B A D E (4) B C A D E (4) D E C A B (3) D E A C B (2) C D B A E (2) C B E A D (2) A D E B C (2) E C D B A (1) E C B A D (1) E A D B C (1) E A B D C (1) D C A B E (1) D B A C E (1) C E B D A (1) C B E D A (1) C B D A E (1) B C A E D (1) B A D C E (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 12 10 -8 0 B -12 0 8 -10 -8 C -10 -8 0 -16 -4 D 8 10 16 0 -4 E 0 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.165811 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.834189 Sum of squares = 0.723364804448 Cumulative probabilities = A: 0.165811 B: 0.165811 C: 0.165811 D: 0.165811 E: 1.000000 A B C D E A 0 12 10 -8 0 B -12 0 8 -10 -8 C -10 -8 0 -16 -4 D 8 10 16 0 -4 E 0 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555599002 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 D=17 B=16 A=15 so A is eliminated. Round 2 votes counts: E=28 B=27 C=25 D=20 so D is eliminated. Round 3 votes counts: E=45 B=29 C=26 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:208 A:207 B:189 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 10 -8 0 B -12 0 8 -10 -8 C -10 -8 0 -16 -4 D 8 10 16 0 -4 E 0 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555599002 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -8 0 B -12 0 8 -10 -8 C -10 -8 0 -16 -4 D 8 10 16 0 -4 E 0 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555599002 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -8 0 B -12 0 8 -10 -8 C -10 -8 0 -16 -4 D 8 10 16 0 -4 E 0 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555599002 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3550: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) C D E B A (6) A B D E C (6) E C D A B (5) D C E A B (5) B A E C D (5) E D C A B (4) D E C A B (4) D E A C B (4) C E D B A (4) B A C D E (4) A D E B C (4) E D A C B (2) E D A B C (2) E A B C D (2) D A E C B (2) D A B C E (2) C D B E A (2) C B E D A (2) C B E A D (2) B C E A D (2) B C A D E (2) B A C E D (2) A B D C E (2) E C D B A (1) E A D C B (1) D E A B C (1) D C E B A (1) D C A E B (1) D C A B E (1) D A E B C (1) C D E A B (1) C D B A E (1) B A E D C (1) B A D C E (1) A E D B C (1) A E B D C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 28 12 -4 -2 B -28 0 2 -12 -8 C -12 -2 0 -14 -18 D 4 12 14 0 2 E 2 8 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 12 -4 -2 B -28 0 2 -12 -8 C -12 -2 0 -14 -18 D 4 12 14 0 2 E 2 8 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=22 C=18 E=17 B=17 so E is eliminated. Round 2 votes counts: D=30 A=29 C=24 B=17 so B is eliminated. Round 3 votes counts: A=42 D=30 C=28 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:217 D:216 E:213 B:177 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 28 12 -4 -2 B -28 0 2 -12 -8 C -12 -2 0 -14 -18 D 4 12 14 0 2 E 2 8 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 12 -4 -2 B -28 0 2 -12 -8 C -12 -2 0 -14 -18 D 4 12 14 0 2 E 2 8 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 12 -4 -2 B -28 0 2 -12 -8 C -12 -2 0 -14 -18 D 4 12 14 0 2 E 2 8 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992028 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3551: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) A E D B C (6) A E C B D (6) D B C E A (5) C E A B D (5) E A B D C (4) E A B C D (4) D C B A E (4) C B D E A (4) D C B E A (3) D C A E B (3) D B A E C (3) D A E C B (3) C D B E A (3) C A E D B (3) B E A D C (3) B D C E A (3) D B C A E (2) D A C E B (2) D A B E C (2) C E B A D (2) C D A E B (2) B D E A C (2) A E C D B (2) A D E C B (2) E B A C D (1) E A C B D (1) D A E B C (1) D A C B E (1) C D B A E (1) C B E D A (1) C B E A D (1) C A E B D (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B A E D C (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 14 10 6 8 B -14 0 -2 2 -14 C -10 2 0 -18 -8 D -6 -2 18 0 -8 E -8 14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 6 8 B -14 0 -2 2 -14 C -10 2 0 -18 -8 D -6 -2 18 0 -8 E -8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=25 C=23 B=13 E=10 so E is eliminated. Round 2 votes counts: A=34 D=29 C=23 B=14 so B is eliminated. Round 3 votes counts: A=40 D=35 C=25 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:211 D:201 B:186 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 6 8 B -14 0 -2 2 -14 C -10 2 0 -18 -8 D -6 -2 18 0 -8 E -8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 6 8 B -14 0 -2 2 -14 C -10 2 0 -18 -8 D -6 -2 18 0 -8 E -8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 6 8 B -14 0 -2 2 -14 C -10 2 0 -18 -8 D -6 -2 18 0 -8 E -8 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3552: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) C A E D B (9) B A E D C (8) C A B E D (6) B D E A C (6) D B E A C (4) D E B C A (3) C D A E B (3) C A E B D (3) B E A D C (3) A C E B D (3) A B C E D (3) E A B D C (2) D E B A C (2) D C E B A (2) D C E A B (2) D C B E A (2) C D E B A (2) B A E C D (2) A E C B D (2) A E B D C (2) E D B A C (1) E D A C B (1) E D A B C (1) E C D A B (1) E B D A C (1) E A D C B (1) D E A B C (1) D B C E A (1) C D B A E (1) C B D A E (1) C B A E D (1) C A B D E (1) B E D A C (1) B D C E A (1) B D C A E (1) B C A D E (1) B A D E C (1) B A C E D (1) A E B C D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -2 4 4 B -8 0 -4 6 -6 C 2 4 0 4 10 D -4 -6 -4 0 -10 E -4 6 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 4 4 B -8 0 -4 6 -6 C 2 4 0 4 10 D -4 -6 -4 0 -10 E -4 6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=25 D=17 A=13 E=8 so E is eliminated. Round 2 votes counts: C=38 B=26 D=20 A=16 so A is eliminated. Round 3 votes counts: C=44 B=35 D=21 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:207 E:201 B:194 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 4 4 B -8 0 -4 6 -6 C 2 4 0 4 10 D -4 -6 -4 0 -10 E -4 6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 4 4 B -8 0 -4 6 -6 C 2 4 0 4 10 D -4 -6 -4 0 -10 E -4 6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 4 4 B -8 0 -4 6 -6 C 2 4 0 4 10 D -4 -6 -4 0 -10 E -4 6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3553: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) C D E B A (7) E D B C A (6) E B D A C (6) C E D B A (6) C A D B E (6) E D C B A (4) B D E A C (4) A B D E C (4) E B A D C (3) D B E C A (3) C E D A B (3) A C B E D (3) A B E D C (3) A B D C E (3) D E B C A (2) D B E A C (2) C D E A B (2) C D B E A (2) B E D A C (2) B D A C E (2) B A D E C (2) A B C D E (2) E C A D B (1) E A B D C (1) D C E B A (1) D C B E A (1) D B C E A (1) C A E D B (1) C A D E B (1) B A E D C (1) B A D C E (1) A E C B D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 6 -8 -14 B 12 0 0 0 10 C -6 0 0 -4 8 D 8 0 4 0 14 E 14 -10 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.858667 C: 0.000000 D: 0.141333 E: 0.000000 Sum of squares = 0.757283796726 Cumulative probabilities = A: 0.000000 B: 0.858667 C: 0.858667 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -8 -14 B 12 0 0 0 10 C -6 0 0 -4 8 D 8 0 4 0 14 E 14 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999215 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=28 E=21 B=12 D=10 so D is eliminated. Round 2 votes counts: C=30 A=29 E=23 B=18 so B is eliminated. Round 3 votes counts: A=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:213 B:211 C:199 E:191 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 -8 -14 B 12 0 0 0 10 C -6 0 0 -4 8 D 8 0 4 0 14 E 14 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999215 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -8 -14 B 12 0 0 0 10 C -6 0 0 -4 8 D 8 0 4 0 14 E 14 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999215 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -8 -14 B 12 0 0 0 10 C -6 0 0 -4 8 D 8 0 4 0 14 E 14 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999215 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3554: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) D A B C E (8) C E D B A (8) B A E C D (7) D C B A E (5) D A B E C (5) C D E B A (5) A B E C D (5) A B D E C (5) E C B A D (4) E C A B D (4) A B E D C (4) B A C E D (3) A D B E C (3) E B A C D (2) E A B C D (2) D C E A B (2) D B A C E (2) D A E B C (2) C E B A D (2) E D C A B (1) E A D B C (1) E A C B D (1) D E A B C (1) D C B E A (1) D A C B E (1) C E B D A (1) C D B E A (1) C B A E D (1) B E A C D (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 -10 10 -6 6 B 10 0 6 -12 8 C -10 -6 0 -4 4 D 6 12 4 0 6 E -6 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -6 6 B 10 0 6 -12 8 C -10 -6 0 -4 4 D 6 12 4 0 6 E -6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=18 A=17 E=15 B=13 so B is eliminated. Round 2 votes counts: D=37 A=29 C=18 E=16 so E is eliminated. Round 3 votes counts: D=38 A=36 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:206 A:200 C:192 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 10 -6 6 B 10 0 6 -12 8 C -10 -6 0 -4 4 D 6 12 4 0 6 E -6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -6 6 B 10 0 6 -12 8 C -10 -6 0 -4 4 D 6 12 4 0 6 E -6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -6 6 B 10 0 6 -12 8 C -10 -6 0 -4 4 D 6 12 4 0 6 E -6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3555: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (18) E D A B C (11) C B A D E (10) A B C D E (10) D E C B A (9) C B A E D (8) A B C E D (5) E D C B A (3) C B D A E (3) B C A E D (3) E D A C B (2) E A D B C (2) D A E B C (2) A D B C E (2) E D C A B (1) E C B D A (1) E A C B D (1) D E A C B (1) D C B E A (1) D B C A E (1) D A B C E (1) C D E B A (1) C B E D A (1) A E B D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 18 16 -12 -4 B -18 0 16 -12 -6 C -16 -16 0 -14 -6 D 12 12 14 0 22 E 4 6 6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 -12 -4 B -18 0 16 -12 -6 C -16 -16 0 -14 -6 D 12 12 14 0 22 E 4 6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 E=21 A=20 B=3 so B is eliminated. Round 2 votes counts: D=33 C=26 E=21 A=20 so A is eliminated. Round 3 votes counts: C=41 D=37 E=22 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:209 E:197 B:190 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 16 -12 -4 B -18 0 16 -12 -6 C -16 -16 0 -14 -6 D 12 12 14 0 22 E 4 6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 -12 -4 B -18 0 16 -12 -6 C -16 -16 0 -14 -6 D 12 12 14 0 22 E 4 6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 -12 -4 B -18 0 16 -12 -6 C -16 -16 0 -14 -6 D 12 12 14 0 22 E 4 6 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3556: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) C D E A B (7) D A B E C (6) E B C A D (4) E B A C D (4) D B A E C (4) D A C B E (4) C E A B D (4) B A D E C (4) E C A B D (3) D C A B E (3) D A B C E (3) C E B D A (3) B D E A C (3) A E B D C (3) A B E D C (3) E A B C D (2) D C A E B (2) D B A C E (2) C E D B A (2) C E D A B (2) C E A D B (2) C D E B A (2) C D B E A (2) B E A D C (2) B D A E C (2) B A E D C (2) A D B E C (2) E C B A D (1) E B A D C (1) E A C B D (1) E A B D C (1) D C B A E (1) C D A B E (1) C B D E A (1) C A E D B (1) B E C A D (1) A D E B C (1) Total count = 100 A B C D E A 0 2 0 0 -12 B -2 0 0 6 -8 C 0 0 0 2 0 D 0 -6 -2 0 0 E 12 8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.711167 D: 0.000000 E: 0.288833 Sum of squares = 0.589183328686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.711167 D: 0.711167 E: 1.000000 A B C D E A 0 2 0 0 -12 B -2 0 0 6 -8 C 0 0 0 2 0 D 0 -6 -2 0 0 E 12 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=25 E=17 B=14 A=9 so A is eliminated. Round 2 votes counts: C=35 D=28 E=20 B=17 so B is eliminated. Round 3 votes counts: D=37 C=35 E=28 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:210 C:201 B:198 D:196 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 0 -12 B -2 0 0 6 -8 C 0 0 0 2 0 D 0 -6 -2 0 0 E 12 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 -12 B -2 0 0 6 -8 C 0 0 0 2 0 D 0 -6 -2 0 0 E 12 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 -12 B -2 0 0 6 -8 C 0 0 0 2 0 D 0 -6 -2 0 0 E 12 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3557: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) D E A C B (6) D C A E B (6) E A C B D (5) C A D E B (5) C A B D E (5) A C D E B (5) E B D A C (4) E B A C D (4) D E B C A (4) B E A C D (4) B D E C A (4) A C E D B (4) E D B A C (3) D E B A C (3) B D C A E (3) B C A E D (3) E D A C B (2) D E C A B (2) D B E C A (2) C D A B E (2) B E D C A (2) B C A D E (2) A C E B D (2) E A C D B (1) D C B A E (1) D B E A C (1) D B C A E (1) C A E D B (1) C A D B E (1) C A B E D (1) B D C E A (1) B C E A D (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 6 -10 -14 B 2 0 0 -2 -16 C -6 0 0 -6 -12 D 10 2 6 0 8 E 14 16 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -10 -14 B 2 0 0 -2 -16 C -6 0 0 -6 -12 D 10 2 6 0 8 E 14 16 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=26 E=19 C=15 A=12 so A is eliminated. Round 2 votes counts: B=28 D=26 C=26 E=20 so E is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:217 D:213 B:192 A:190 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -10 -14 B 2 0 0 -2 -16 C -6 0 0 -6 -12 D 10 2 6 0 8 E 14 16 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -10 -14 B 2 0 0 -2 -16 C -6 0 0 -6 -12 D 10 2 6 0 8 E 14 16 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -10 -14 B 2 0 0 -2 -16 C -6 0 0 -6 -12 D 10 2 6 0 8 E 14 16 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3558: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) D B E C A (6) A E C D B (5) A C E B D (5) D E C B A (4) B D C A E (4) B A D E C (4) B A C E D (4) A E C B D (4) A C E D B (4) E D C A B (3) D C E B A (3) C E D A B (3) C D E A B (3) B D E C A (3) B A E D C (3) B A D C E (3) E C D A B (2) C E A D B (2) C D E B A (2) B D A E C (2) B D A C E (2) B A E C D (2) B A C D E (2) A B E D C (2) A B E C D (2) A B C E D (2) E A C D B (1) D E C A B (1) D E B C A (1) D C E A B (1) D B C E A (1) C D B E A (1) C A E D B (1) B D E A C (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 0 -4 6 B 14 0 8 12 8 C 0 -8 0 -6 6 D 4 -12 6 0 6 E -6 -8 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -4 6 B 14 0 8 12 8 C 0 -8 0 -6 6 D 4 -12 6 0 6 E -6 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=26 D=17 C=12 E=6 so E is eliminated. Round 2 votes counts: B=39 A=27 D=20 C=14 so C is eliminated. Round 3 votes counts: B=39 D=31 A=30 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:202 C:196 A:194 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -4 6 B 14 0 8 12 8 C 0 -8 0 -6 6 D 4 -12 6 0 6 E -6 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -4 6 B 14 0 8 12 8 C 0 -8 0 -6 6 D 4 -12 6 0 6 E -6 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -4 6 B 14 0 8 12 8 C 0 -8 0 -6 6 D 4 -12 6 0 6 E -6 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3559: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) E D A C B (7) C B A E D (7) D E A C B (6) B C A E D (6) D E A B C (5) B C D E A (5) A E D C B (5) C E D A B (4) C A B E D (4) B C A D E (4) C A E D B (3) B D E A C (3) B D A E C (3) A C E D B (3) D E C B A (2) D E B C A (2) D E B A C (2) D B E A C (2) C E D B A (2) C B E D A (2) B A C D E (2) A C B E D (2) A B D E C (2) E D C A B (1) E C D A B (1) E A D C B (1) C E A D B (1) C D E B A (1) B D E C A (1) B C D A E (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 10 6 0 4 B -10 0 -6 -14 -12 C -6 6 0 -2 -4 D 0 14 2 0 -2 E -4 12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.676134 B: 0.000000 C: 0.000000 D: 0.323866 E: 0.000000 Sum of squares = 0.562046425542 Cumulative probabilities = A: 0.676134 B: 0.676134 C: 0.676134 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 0 4 B -10 0 -6 -14 -12 C -6 6 0 -2 -4 D 0 14 2 0 -2 E -4 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=24 A=22 D=19 E=10 so E is eliminated. Round 2 votes counts: D=27 C=25 B=25 A=23 so A is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:207 E:207 C:197 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 0 4 B -10 0 -6 -14 -12 C -6 6 0 -2 -4 D 0 14 2 0 -2 E -4 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 0 4 B -10 0 -6 -14 -12 C -6 6 0 -2 -4 D 0 14 2 0 -2 E -4 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 0 4 B -10 0 -6 -14 -12 C -6 6 0 -2 -4 D 0 14 2 0 -2 E -4 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3560: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) B E C D A (7) B E A D C (5) A E D C B (5) A D C E B (5) E A B D C (4) C D A E B (4) B E A C D (4) A E B D C (4) E B A C D (3) D C B A E (3) D C A B E (3) D B C A E (3) D A C E B (3) D A C B E (3) C D E A B (3) E C A D B (2) C D B E A (2) C D A B E (2) B E C A D (2) B D C A E (2) B D A C E (2) B C E D A (2) B A E D C (2) A D E C B (2) E C D A B (1) E C B D A (1) E C A B D (1) E A D C B (1) E A C B D (1) E A B C D (1) C D B A E (1) C B E D A (1) B D A E C (1) B A D E C (1) B A D C E (1) A E D B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 14 6 -2 18 B -14 0 -6 -10 0 C -6 6 0 -22 0 D 2 10 22 0 2 E -18 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -2 18 B -14 0 -6 -10 0 C -6 6 0 -22 0 D 2 10 22 0 2 E -18 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=22 A=21 E=15 C=13 so C is eliminated. Round 2 votes counts: D=34 B=30 A=21 E=15 so E is eliminated. Round 3 votes counts: D=35 B=34 A=31 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:218 E:190 C:189 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -2 18 B -14 0 -6 -10 0 C -6 6 0 -22 0 D 2 10 22 0 2 E -18 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -2 18 B -14 0 -6 -10 0 C -6 6 0 -22 0 D 2 10 22 0 2 E -18 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -2 18 B -14 0 -6 -10 0 C -6 6 0 -22 0 D 2 10 22 0 2 E -18 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3561: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) B C E A D (7) D A C E B (5) B D C A E (5) D B E A C (4) C A B D E (4) B D E C A (4) E D A C B (3) E B C A D (3) E A D C B (3) D E B A C (3) D E A B C (3) D B A C E (3) D A E B C (3) C A E D B (3) B D E A C (3) B C A D E (3) E D B A C (2) E C A D B (2) E C A B D (2) E A C D B (2) D A E C B (2) C A B E D (2) B E D C A (2) B E C D A (2) B E C A D (2) B C D A E (2) B C A E D (2) A C D E B (2) E D A B C (1) D E A C B (1) D B A E C (1) D A B C E (1) C B A E D (1) C A E B D (1) C A D B E (1) B D C E A (1) B D A E C (1) Total count = 100 A B C D E A 0 -2 2 -4 0 B 2 0 16 -6 2 C -2 -16 0 -2 2 D 4 6 2 0 4 E 0 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 0 B 2 0 16 -6 2 C -2 -16 0 -2 2 D 4 6 2 0 4 E 0 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=26 E=18 C=12 A=10 so A is eliminated. Round 2 votes counts: B=34 D=26 C=22 E=18 so E is eliminated. Round 3 votes counts: B=37 D=35 C=28 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:207 A:198 E:196 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -4 0 B 2 0 16 -6 2 C -2 -16 0 -2 2 D 4 6 2 0 4 E 0 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 0 B 2 0 16 -6 2 C -2 -16 0 -2 2 D 4 6 2 0 4 E 0 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 0 B 2 0 16 -6 2 C -2 -16 0 -2 2 D 4 6 2 0 4 E 0 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3562: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) E D C A B (8) C A B E D (8) C A E B D (7) B A C D E (7) A C B D E (7) E D A C B (6) D E B A C (6) E D B A C (4) E D C B A (3) E C A D B (3) D E B C A (3) C A B D E (3) B A D C E (3) E D A B C (2) E A C D B (2) D B E A C (2) A C E B D (2) A C B E D (2) E C D A B (1) E A D C B (1) C E A D B (1) C B E D A (1) C B A E D (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D E A (1) B C A D E (1) A E C D B (1) Total count = 100 A B C D E A 0 8 -8 -2 -16 B -8 0 -14 -8 -22 C 8 14 0 -4 -10 D 2 8 4 0 -28 E 16 22 10 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -8 -2 -16 B -8 0 -14 -8 -22 C 8 14 0 -4 -10 D 2 8 4 0 -28 E 16 22 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=21 B=15 A=12 D=11 so D is eliminated. Round 2 votes counts: E=50 C=21 B=17 A=12 so A is eliminated. Round 3 votes counts: E=51 C=32 B=17 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:238 C:204 D:193 A:191 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -8 -2 -16 B -8 0 -14 -8 -22 C 8 14 0 -4 -10 D 2 8 4 0 -28 E 16 22 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -2 -16 B -8 0 -14 -8 -22 C 8 14 0 -4 -10 D 2 8 4 0 -28 E 16 22 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -2 -16 B -8 0 -14 -8 -22 C 8 14 0 -4 -10 D 2 8 4 0 -28 E 16 22 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3563: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) E A D B C (8) C B D A E (8) C A E B D (8) E A B D C (5) D C B E A (5) D B E A C (4) D B C E A (4) E A C B D (3) D C E A B (3) D C B A E (3) D B C A E (3) C D B A E (3) C A B E D (3) A E B C D (3) D E B A C (2) C E A B D (2) C D E A B (2) C B A E D (2) B D C A E (2) A E B D C (2) A C E B D (2) E D B A C (1) E D A C B (1) E D A B C (1) E A D C B (1) E A C D B (1) E A B C D (1) D C E B A (1) D B E C A (1) C B A D E (1) B D E A C (1) B A E D C (1) B A D E C (1) A E C B D (1) Total count = 100 A B C D E A 0 14 -2 -10 -14 B -14 0 0 -8 -16 C 2 0 0 -20 4 D 10 8 20 0 8 E 14 16 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -10 -14 B -14 0 0 -8 -16 C 2 0 0 -20 4 D 10 8 20 0 8 E 14 16 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=29 E=22 A=8 B=5 so B is eliminated. Round 2 votes counts: D=39 C=29 E=22 A=10 so A is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:209 A:194 C:193 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -10 -14 B -14 0 0 -8 -16 C 2 0 0 -20 4 D 10 8 20 0 8 E 14 16 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -10 -14 B -14 0 0 -8 -16 C 2 0 0 -20 4 D 10 8 20 0 8 E 14 16 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -10 -14 B -14 0 0 -8 -16 C 2 0 0 -20 4 D 10 8 20 0 8 E 14 16 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3564: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) C D A B E (6) E B A C D (5) D E B A C (5) C A D B E (5) D C A E B (4) D A C B E (4) B E A C D (4) A B C E D (4) E D B A C (3) E B D A C (3) E B C D A (3) D C E B A (3) D C E A B (3) D A B E C (3) C D E B A (3) C B E A D (3) B A E C D (3) A C B E D (3) A B E D C (3) D C A B E (2) D A C E B (2) C E B A D (2) C D A E B (2) C A B E D (2) C A B D E (2) A D B E C (2) A D B C E (2) A B E C D (2) E D B C A (1) E B C A D (1) D E C B A (1) C E B D A (1) C B A E D (1) Total count = 100 A B C D E A 0 2 10 2 4 B -2 0 2 -2 2 C -10 -2 0 4 8 D -2 2 -4 0 -2 E -4 -2 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 2 4 B -2 0 2 -2 2 C -10 -2 0 4 8 D -2 2 -4 0 -2 E -4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999212 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 E=23 A=16 B=7 so B is eliminated. Round 2 votes counts: E=27 D=27 C=27 A=19 so A is eliminated. Round 3 votes counts: E=35 C=34 D=31 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:209 B:200 C:200 D:197 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 2 4 B -2 0 2 -2 2 C -10 -2 0 4 8 D -2 2 -4 0 -2 E -4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999212 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 2 4 B -2 0 2 -2 2 C -10 -2 0 4 8 D -2 2 -4 0 -2 E -4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999212 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 2 4 B -2 0 2 -2 2 C -10 -2 0 4 8 D -2 2 -4 0 -2 E -4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999212 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3565: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E A D B C (9) C E B D A (8) A D E B C (8) D A B C E (6) C B E D A (6) C B D A E (6) E C A D B (5) E C B A D (4) E A D C B (4) B C D A E (4) A D B E C (4) C E B A D (3) C E A D B (3) B D C A E (2) A D E C B (2) A D B C E (2) E C A B D (1) E B C A D (1) E B A D C (1) E A C D B (1) D C A B E (1) D B A C E (1) C D A B E (1) B E C D A (1) B D E A C (1) B D A E C (1) B C E D A (1) B C D E A (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 4 -2 0 B 4 0 8 4 0 C -4 -8 0 -8 14 D 2 -4 8 0 4 E 0 0 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.754191 C: 0.000000 D: 0.000000 E: 0.245809 Sum of squares = 0.629225835336 Cumulative probabilities = A: 0.000000 B: 0.754191 C: 0.754191 D: 0.754191 E: 1.000000 A B C D E A 0 -4 4 -2 0 B 4 0 8 4 0 C -4 -8 0 -8 14 D 2 -4 8 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190087013 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 B=22 A=17 D=8 so D is eliminated. Round 2 votes counts: C=28 E=26 B=23 A=23 so B is eliminated. Round 3 votes counts: C=36 A=36 E=28 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:208 D:205 A:199 C:197 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 -2 0 B 4 0 8 4 0 C -4 -8 0 -8 14 D 2 -4 8 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190087013 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -2 0 B 4 0 8 4 0 C -4 -8 0 -8 14 D 2 -4 8 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190087013 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -2 0 B 4 0 8 4 0 C -4 -8 0 -8 14 D 2 -4 8 0 4 E 0 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190087013 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3566: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) B A D E C (9) B C E D A (7) B A D C E (7) A D C E B (6) D A C E B (5) C E D B A (5) C D E A B (5) A D E C B (5) A D E B C (4) B E C D A (3) B A C D E (3) A D B E C (3) A B D E C (3) E D A B C (2) E C D B A (2) E C B D A (2) D E A C B (2) D A E C B (2) C E B D A (2) B C A E D (2) B A E D C (2) E D C A B (1) E C D A B (1) C A D E B (1) B E D A C (1) B E C A D (1) B C E A D (1) B A C E D (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 14 0 10 B -4 0 0 -12 -10 C -14 0 0 -8 14 D 0 12 8 0 14 E -10 10 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.423860 B: 0.000000 C: 0.000000 D: 0.576140 E: 0.000000 Sum of squares = 0.511594636853 Cumulative probabilities = A: 0.423860 B: 0.423860 C: 0.423860 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 0 10 B -4 0 0 -12 -10 C -14 0 0 -8 14 D 0 12 8 0 14 E -10 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=23 A=23 D=9 E=8 so E is eliminated. Round 2 votes counts: B=37 C=28 A=23 D=12 so D is eliminated. Round 3 votes counts: B=37 A=34 C=29 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:214 C:196 B:187 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 0 10 B -4 0 0 -12 -10 C -14 0 0 -8 14 D 0 12 8 0 14 E -10 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 0 10 B -4 0 0 -12 -10 C -14 0 0 -8 14 D 0 12 8 0 14 E -10 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 0 10 B -4 0 0 -12 -10 C -14 0 0 -8 14 D 0 12 8 0 14 E -10 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3567: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (15) A C B D E (8) E D C B A (6) E D C A B (5) E C D A B (4) A C B E D (4) D E C B A (3) C D E B A (3) C B D A E (3) B A D E C (3) A B C E D (3) C D E A B (2) C D B A E (2) C B A D E (2) C A E D B (2) C A B D E (2) B E D A C (2) B A D C E (2) A B E D C (2) A B D E C (2) E D B C A (1) E D B A C (1) E D A C B (1) E C D B A (1) E C A D B (1) E A C D B (1) E A B D C (1) D E B C A (1) D C E B A (1) D C B E A (1) C E D B A (1) C A D B E (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D A E (1) B C A D E (1) B A C D E (1) A E C B D (1) A E B C D (1) A C E B D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 18 6 12 22 B -18 0 -12 24 24 C -6 12 0 24 18 D -12 -24 -24 0 20 E -22 -24 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 12 22 B -18 0 -12 24 24 C -6 12 0 24 18 D -12 -24 -24 0 20 E -22 -24 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=22 C=18 B=15 D=6 so D is eliminated. Round 2 votes counts: A=39 E=26 C=20 B=15 so B is eliminated. Round 3 votes counts: A=46 E=30 C=24 so C is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:224 B:209 D:180 E:158 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 12 22 B -18 0 -12 24 24 C -6 12 0 24 18 D -12 -24 -24 0 20 E -22 -24 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 12 22 B -18 0 -12 24 24 C -6 12 0 24 18 D -12 -24 -24 0 20 E -22 -24 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 12 22 B -18 0 -12 24 24 C -6 12 0 24 18 D -12 -24 -24 0 20 E -22 -24 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) B C D A E (7) A E C B D (7) E A D C B (6) D E B C A (5) C B D A E (5) E D A B C (4) A E D B C (4) A E C D B (4) E A D B C (3) D B E C A (3) B D C E A (3) A C E B D (3) A C B E D (3) A B C E D (3) D E B A C (2) D E A B C (2) B D C A E (2) B C A D E (2) A B E D C (2) E D C A B (1) E D A C B (1) E C A D B (1) E C A B D (1) E A C D B (1) D E C B A (1) D C E B A (1) D C B E A (1) D B C A E (1) D B A E C (1) C D B E A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B E D (1) B C D E A (1) B C A E D (1) B A C D E (1) A E D C B (1) A E B D C (1) A E B C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 0 8 B -4 0 16 -2 0 C -4 -16 0 -8 -6 D 0 2 8 0 0 E -8 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.435987 B: 0.000000 C: 0.000000 D: 0.564013 E: 0.000000 Sum of squares = 0.508195316439 Cumulative probabilities = A: 0.435987 B: 0.435987 C: 0.435987 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 8 B -4 0 16 -2 0 C -4 -16 0 -8 -6 D 0 2 8 0 0 E -8 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=24 E=18 B=17 C=10 so C is eliminated. Round 2 votes counts: A=32 D=25 B=25 E=18 so E is eliminated. Round 3 votes counts: A=44 D=31 B=25 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:205 D:205 E:199 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 8 B -4 0 16 -2 0 C -4 -16 0 -8 -6 D 0 2 8 0 0 E -8 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 8 B -4 0 16 -2 0 C -4 -16 0 -8 -6 D 0 2 8 0 0 E -8 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 8 B -4 0 16 -2 0 C -4 -16 0 -8 -6 D 0 2 8 0 0 E -8 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3569: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) D B A C E (7) A B D E C (7) E C A B D (6) C D B A E (6) A B E D C (6) E A B C D (5) B A D E C (4) B A D C E (4) A B D C E (4) E C D B A (3) C E D B A (3) C D A B E (3) A E B D C (3) E A C B D (2) E A B D C (2) D C B E A (2) D C B A E (2) C E D A B (2) C E A D B (2) C A B D E (2) B D A E C (2) A B E C D (2) E D C B A (1) E C D A B (1) E C A D B (1) E B D A C (1) E B A D C (1) D E C B A (1) D C E B A (1) D B C A E (1) C E A B D (1) B D A C E (1) A E C B D (1) Total count = 100 A B C D E A 0 0 4 6 10 B 0 0 0 8 6 C -4 0 0 0 2 D -6 -8 0 0 14 E -10 -6 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.443298 B: 0.556702 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.506430131826 Cumulative probabilities = A: 0.443298 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 6 10 B 0 0 0 8 6 C -4 0 0 0 2 D -6 -8 0 0 14 E -10 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=23 A=23 D=14 B=11 so B is eliminated. Round 2 votes counts: A=31 C=29 E=23 D=17 so D is eliminated. Round 3 votes counts: A=41 C=35 E=24 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:207 D:200 C:199 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 6 10 B 0 0 0 8 6 C -4 0 0 0 2 D -6 -8 0 0 14 E -10 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 10 B 0 0 0 8 6 C -4 0 0 0 2 D -6 -8 0 0 14 E -10 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 10 B 0 0 0 8 6 C -4 0 0 0 2 D -6 -8 0 0 14 E -10 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3570: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) A D C B E (8) D A B E C (7) E B D C A (6) C E B A D (6) D A C E B (5) E B C A D (4) D E B C A (4) C A E B D (4) B E D C A (4) A D C E B (4) A C D E B (4) A C D B E (4) E B C D A (3) D E B A C (3) D B E C A (3) D A C B E (3) B E C D A (3) A C B E D (3) B E C A D (2) B A E C D (2) A C B D E (2) E C B A D (1) D A E B C (1) C B A E D (1) C A B E D (1) B D E A C (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 16 -6 0 B 6 0 6 -12 8 C -16 -6 0 -18 -8 D 6 12 18 0 18 E 0 -8 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 16 -6 0 B 6 0 6 -12 8 C -16 -6 0 -18 -8 D 6 12 18 0 18 E 0 -8 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=26 E=14 C=12 B=12 so C is eliminated. Round 2 votes counts: D=36 A=31 E=20 B=13 so B is eliminated. Round 3 votes counts: D=37 A=34 E=29 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 B:204 A:202 E:191 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 16 -6 0 B 6 0 6 -12 8 C -16 -6 0 -18 -8 D 6 12 18 0 18 E 0 -8 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 -6 0 B 6 0 6 -12 8 C -16 -6 0 -18 -8 D 6 12 18 0 18 E 0 -8 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 -6 0 B 6 0 6 -12 8 C -16 -6 0 -18 -8 D 6 12 18 0 18 E 0 -8 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3571: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) E B D A C (8) C D B E A (6) C A E D B (6) A C E B D (6) E B A D C (4) D B C E A (4) C A D B E (4) A E C B D (4) A E B D C (4) A E B C D (4) A D B E C (4) D B E C A (3) D B E A C (3) C D B A E (3) B E D A C (3) E B D C A (2) E B C D A (2) C D A B E (2) C A E B D (2) C A D E B (2) B E D C A (2) A C E D B (2) A C D B E (2) E C B A D (1) E B C A D (1) E A B C D (1) D C B E A (1) C E B A D (1) C B D E A (1) B D E C A (1) B D E A C (1) A D C B E (1) Total count = 100 A B C D E A 0 6 12 16 -8 B -6 0 12 14 -18 C -12 -12 0 0 -14 D -16 -14 0 0 -24 E 8 18 14 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 12 16 -8 B -6 0 12 14 -18 C -12 -12 0 0 -14 D -16 -14 0 0 -24 E 8 18 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 A=27 D=11 B=7 so B is eliminated. Round 2 votes counts: E=33 C=27 A=27 D=13 so D is eliminated. Round 3 votes counts: E=41 C=32 A=27 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:213 B:201 C:181 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 16 -8 B -6 0 12 14 -18 C -12 -12 0 0 -14 D -16 -14 0 0 -24 E 8 18 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 16 -8 B -6 0 12 14 -18 C -12 -12 0 0 -14 D -16 -14 0 0 -24 E 8 18 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 16 -8 B -6 0 12 14 -18 C -12 -12 0 0 -14 D -16 -14 0 0 -24 E 8 18 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3572: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) E A B C D (9) D A E B C (9) D C B A E (6) E A B D C (4) D A B E C (4) C E B A D (4) C B D E A (4) D E A C B (3) D A B C E (3) C D E B A (3) C B E D A (3) A E B D C (3) D C B E A (2) D C A B E (2) D B C A E (2) D A E C B (2) C D B A E (2) C B A E D (2) B A E C D (2) A E D B C (2) A D B E C (2) E D A C B (1) E C B A D (1) E B C A D (1) E A D B C (1) E A C B D (1) D C E A B (1) D C A E B (1) D A C E B (1) C E D B A (1) C E B D A (1) C D B E A (1) B D C A E (1) B C A D E (1) A E B C D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 2 -6 -2 B -6 0 -4 2 0 C -2 4 0 -4 2 D 6 -2 4 0 2 E 2 0 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101761 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -6 -2 B -6 0 -4 2 0 C -2 4 0 -4 2 D 6 -2 4 0 2 E 2 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102019 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=31 E=18 A=11 B=4 so B is eliminated. Round 2 votes counts: D=37 C=32 E=18 A=13 so A is eliminated. Round 3 votes counts: D=40 C=32 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:205 A:200 C:200 E:199 B:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -6 -2 B -6 0 -4 2 0 C -2 4 0 -4 2 D 6 -2 4 0 2 E 2 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102019 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -6 -2 B -6 0 -4 2 0 C -2 4 0 -4 2 D 6 -2 4 0 2 E 2 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102019 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -6 -2 B -6 0 -4 2 0 C -2 4 0 -4 2 D 6 -2 4 0 2 E 2 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102019 Cumulative probabilities = A: 0.142857 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3573: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) B C D A E (12) A E D C B (11) D C B E A (8) B A E C D (6) A E B C D (6) E D A C B (5) C D B E A (4) B C D E A (4) B C A D E (4) D E C A B (3) E D C A B (2) D C E B A (2) C B D A E (2) B E A C D (2) B A C D E (2) A C B D E (2) A B E C D (2) E D B C A (1) E B D A C (1) E A D B C (1) D E A C B (1) D C E A B (1) D B C E A (1) B E C D A (1) B A C E D (1) A E C D B (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 10 4 2 B 2 0 -8 -6 4 C -10 8 0 0 -12 D -4 6 0 0 -8 E -2 -4 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888887 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 4 2 B 2 0 -8 -6 4 C -10 8 0 0 -12 D -4 6 0 0 -8 E -2 -4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888865 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=24 E=22 D=16 C=6 so C is eliminated. Round 2 votes counts: B=34 A=24 E=22 D=20 so D is eliminated. Round 3 votes counts: B=47 E=29 A=24 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:207 E:207 D:197 B:196 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 10 4 2 B 2 0 -8 -6 4 C -10 8 0 0 -12 D -4 6 0 0 -8 E -2 -4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888865 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 4 2 B 2 0 -8 -6 4 C -10 8 0 0 -12 D -4 6 0 0 -8 E -2 -4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888865 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 4 2 B 2 0 -8 -6 4 C -10 8 0 0 -12 D -4 6 0 0 -8 E -2 -4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888865 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3574: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (13) D A E C B (11) E A D C B (8) B C E D A (6) A D E C B (6) D A E B C (5) C B E A D (4) B C D A E (4) E B C A D (3) E A C D B (3) D B A E C (3) E C A D B (2) E C A B D (2) C E A B D (2) C D A B E (2) C B D A E (2) B E C D A (2) B E A D C (2) B D A E C (2) B C D E A (2) A E D C B (2) E D A B C (1) E C B A D (1) E B A D C (1) E A D B C (1) D E A B C (1) D A C E B (1) D A C B E (1) D A B E C (1) C A D E B (1) B E D A C (1) B E C A D (1) B D E A C (1) B D C A E (1) B D A C E (1) Total count = 100 A B C D E A 0 0 4 4 -14 B 0 0 4 2 -2 C -4 -4 0 0 -20 D -4 -2 0 0 -10 E 14 2 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 4 -14 B 0 0 4 2 -2 C -4 -4 0 0 -20 D -4 -2 0 0 -10 E 14 2 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=23 E=22 C=11 A=8 so A is eliminated. Round 2 votes counts: B=36 D=29 E=24 C=11 so C is eliminated. Round 3 votes counts: B=42 D=32 E=26 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:223 B:202 A:197 D:192 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 4 -14 B 0 0 4 2 -2 C -4 -4 0 0 -20 D -4 -2 0 0 -10 E 14 2 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 -14 B 0 0 4 2 -2 C -4 -4 0 0 -20 D -4 -2 0 0 -10 E 14 2 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 -14 B 0 0 4 2 -2 C -4 -4 0 0 -20 D -4 -2 0 0 -10 E 14 2 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3575: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) C D A E B (9) A E B D C (6) C E A B D (5) E A B D C (4) C B E D A (4) C B D E A (4) B E A D C (4) A E C B D (4) A C E D B (4) C D A B E (3) C A E D B (3) C A D E B (3) B D E A C (3) E B A C D (2) D C B E A (2) D B C E A (2) D B A E C (2) D A C E B (2) C D B A E (2) B E C D A (2) B C D E A (2) A E D B C (2) A E C D B (2) A E B C D (2) D C B A E (1) D C A B E (1) D A C B E (1) D A B C E (1) C A E B D (1) B E D C A (1) B D E C A (1) A E D C B (1) A D E C B (1) A C E B D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 16 -12 -6 8 B -16 0 -30 -6 -6 C 12 30 0 28 24 D 6 6 -28 0 2 E -8 6 -24 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -12 -6 8 B -16 0 -30 -6 -6 C 12 30 0 28 24 D 6 6 -28 0 2 E -8 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=25 B=13 D=12 E=6 so E is eliminated. Round 2 votes counts: C=44 A=29 B=15 D=12 so D is eliminated. Round 3 votes counts: C=48 A=33 B=19 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:247 A:203 D:193 E:186 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -12 -6 8 B -16 0 -30 -6 -6 C 12 30 0 28 24 D 6 6 -28 0 2 E -8 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -12 -6 8 B -16 0 -30 -6 -6 C 12 30 0 28 24 D 6 6 -28 0 2 E -8 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -12 -6 8 B -16 0 -30 -6 -6 C 12 30 0 28 24 D 6 6 -28 0 2 E -8 6 -24 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3576: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) E A C B D (9) D B C A E (9) D B A E C (9) A E C D B (9) C E A B D (8) C B D E A (7) B D C A E (6) E C A B D (4) B D C E A (3) A E D C B (3) E A C D B (2) E A B D C (2) E A B C D (2) D C B A E (2) D B A C E (2) C D B A E (2) E C A D B (1) E A D B C (1) D C A B E (1) C D A B E (1) C B E D A (1) C B E A D (1) C B D A E (1) B D E A C (1) B D A E C (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 8 4 6 14 B -8 0 -4 -6 -4 C -4 4 0 -2 -10 D -6 6 2 0 -6 E -14 4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 6 14 B -8 0 -4 -6 -4 C -4 4 0 -2 -10 D -6 6 2 0 -6 E -14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 A=23 E=21 C=21 B=12 so B is eliminated. Round 2 votes counts: D=34 A=23 C=22 E=21 so E is eliminated. Round 3 votes counts: A=39 D=34 C=27 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:203 D:198 C:194 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 6 14 B -8 0 -4 -6 -4 C -4 4 0 -2 -10 D -6 6 2 0 -6 E -14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 6 14 B -8 0 -4 -6 -4 C -4 4 0 -2 -10 D -6 6 2 0 -6 E -14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 6 14 B -8 0 -4 -6 -4 C -4 4 0 -2 -10 D -6 6 2 0 -6 E -14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3577: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) D A E C B (5) D A B C E (5) B C E A D (5) B A C E D (5) E C D B A (4) D C E B A (4) A D E C B (4) A B E C D (4) A B D C E (4) A E B C D (3) A D B C E (3) A B C D E (3) E B A C D (2) E A C B D (2) E A B C D (2) D E C B A (2) D C B E A (2) D A C E B (2) C E D B A (2) C E B D A (2) B E C A D (2) B D C A E (2) B A E C D (2) B A C D E (2) A D B E C (2) E D C B A (1) E D C A B (1) E D A C B (1) E C B A D (1) E B C A D (1) D C A E B (1) C D E B A (1) C D B E A (1) B E A C D (1) B C E D A (1) B C A E D (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 14 8 6 B 6 0 8 12 -8 C -14 -8 0 12 -8 D -8 -12 -12 0 -12 E -6 8 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000014 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 -6 14 8 6 B 6 0 8 12 -8 C -14 -8 0 12 -8 D -8 -12 -12 0 -12 E -6 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000003 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 D=21 B=21 C=6 so C is eliminated. Round 2 votes counts: E=28 A=28 D=23 B=21 so B is eliminated. Round 3 votes counts: A=38 E=37 D=25 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:211 B:209 C:191 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -6 14 8 6 B 6 0 8 12 -8 C -14 -8 0 12 -8 D -8 -12 -12 0 -12 E -6 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000003 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 8 6 B 6 0 8 12 -8 C -14 -8 0 12 -8 D -8 -12 -12 0 -12 E -6 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000003 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 8 6 B 6 0 8 12 -8 C -14 -8 0 12 -8 D -8 -12 -12 0 -12 E -6 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.000000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000003 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3578: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) B D E C A (7) E C A B D (6) D B A E C (6) C E A B D (6) D A B C E (5) E B C D A (4) D B A C E (4) A C E D B (4) E B A C D (3) D B C E A (3) D A C B E (3) C E A D B (3) B E D C A (3) B D A E C (3) A C D E B (3) E A C B D (2) C E B A D (2) C A D E B (2) B E D A C (2) B D C E A (2) E B C A D (1) E B A D C (1) E A B C D (1) D C B E A (1) D C B A E (1) D C A B E (1) D B E C A (1) D B E A C (1) D B C A E (1) C E B D A (1) C D B E A (1) C D A E B (1) C A E D B (1) B E A D C (1) A E B C D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 4 -22 -24 B 18 0 24 14 16 C -4 -24 0 -16 -10 D 22 -14 16 0 14 E 24 -16 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 4 -22 -24 B 18 0 24 14 16 C -4 -24 0 -16 -10 D 22 -14 16 0 14 E 24 -16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 E=18 C=17 A=10 so A is eliminated. Round 2 votes counts: B=29 D=28 C=24 E=19 so E is eliminated. Round 3 votes counts: B=40 C=32 D=28 so D is eliminated. Round 4 votes counts: B=62 C=38 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:236 D:219 E:202 C:173 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 4 -22 -24 B 18 0 24 14 16 C -4 -24 0 -16 -10 D 22 -14 16 0 14 E 24 -16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 -22 -24 B 18 0 24 14 16 C -4 -24 0 -16 -10 D 22 -14 16 0 14 E 24 -16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 -22 -24 B 18 0 24 14 16 C -4 -24 0 -16 -10 D 22 -14 16 0 14 E 24 -16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3579: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) C B A E D (8) E D A B C (5) D C E B A (5) D C B E A (5) C B D A E (5) B C A D E (5) A E B C D (5) C D B E A (4) D E C A B (3) D E A C B (3) D E A B C (3) A B C E D (3) D E C B A (2) D B E C A (2) C B D E A (2) B E A D C (2) B D E C A (2) B C D E A (2) B C D A E (2) B A E C D (2) B A C E D (2) A E C D B (2) A E C B D (2) A E B D C (2) A B E C D (2) E D B A C (1) E D A C B (1) E A D C B (1) E A B D C (1) D E B A C (1) D C E A B (1) D B E A C (1) C D B A E (1) C B A D E (1) B D C E A (1) B C A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -4 -4 -12 B 14 0 6 0 6 C 4 -6 0 0 -2 D 4 0 0 0 2 E 12 -6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.398317 C: 0.000000 D: 0.601683 E: 0.000000 Sum of squares = 0.520678746323 Cumulative probabilities = A: 0.000000 B: 0.398317 C: 0.398317 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -4 -12 B 14 0 6 0 6 C 4 -6 0 0 -2 D 4 0 0 0 2 E 12 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999947 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=21 B=19 E=17 A=17 so E is eliminated. Round 2 votes counts: D=33 A=27 C=21 B=19 so B is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:213 D:203 E:203 C:198 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 -4 -12 B 14 0 6 0 6 C 4 -6 0 0 -2 D 4 0 0 0 2 E 12 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999947 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -4 -12 B 14 0 6 0 6 C 4 -6 0 0 -2 D 4 0 0 0 2 E 12 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999947 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -4 -12 B 14 0 6 0 6 C 4 -6 0 0 -2 D 4 0 0 0 2 E 12 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999947 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3580: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (12) B A E C D (10) E A B C D (7) D C E A B (6) D C B A E (6) D E A B C (5) D A E B C (5) C D B E A (4) C B E A D (4) D E A C B (3) C D B A E (3) C B A E D (3) A B E D C (3) E D A B C (2) D C A E B (2) C D E B A (2) C B D A E (2) B C A E D (2) B A C E D (2) A E B D C (2) A E B C D (2) E D C A B (1) E C A B D (1) D E C A B (1) D C E B A (1) D C B E A (1) D C A B E (1) D A E C B (1) C E D B A (1) C E B A D (1) C B A D E (1) B D A C E (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 10 14 4 -4 B -10 0 10 10 -10 C -14 -10 0 -6 -12 D -4 -10 6 0 -8 E 4 10 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 4 -4 B -10 0 10 10 -10 C -14 -10 0 -6 -12 D -4 -10 6 0 -8 E 4 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=23 C=21 B=16 A=8 so A is eliminated. Round 2 votes counts: D=32 E=27 C=21 B=20 so B is eliminated. Round 3 votes counts: E=41 D=33 C=26 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:212 B:200 D:192 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 4 -4 B -10 0 10 10 -10 C -14 -10 0 -6 -12 D -4 -10 6 0 -8 E 4 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 4 -4 B -10 0 10 10 -10 C -14 -10 0 -6 -12 D -4 -10 6 0 -8 E 4 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 4 -4 B -10 0 10 10 -10 C -14 -10 0 -6 -12 D -4 -10 6 0 -8 E 4 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3581: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) E D C A B (7) E C D B A (6) C B E A D (6) A B D C E (6) C B A D E (5) B A D C E (5) A D B C E (5) E D A C B (4) D A E B C (4) C E B D A (4) E D A B C (3) B C A E D (3) A B D E C (3) E C B D A (2) E C B A D (2) D E A C B (2) D E A B C (2) D C A E B (2) C E B A D (2) C D E A B (2) C B A E D (2) B C A D E (2) A D B E C (2) E B C A D (1) D C A B E (1) D A B E C (1) C E D B A (1) C D E B A (1) B E C A D (1) B E A C D (1) B C E A D (1) B A E C D (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -2 16 4 B 10 0 2 14 10 C 2 -2 0 4 14 D -16 -14 -4 0 2 E -4 -10 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 16 4 B 10 0 2 14 10 C 2 -2 0 4 14 D -16 -14 -4 0 2 E -4 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 B=23 A=17 D=12 so D is eliminated. Round 2 votes counts: E=29 C=26 B=23 A=22 so A is eliminated. Round 3 votes counts: B=41 E=33 C=26 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:209 A:204 E:185 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 16 4 B 10 0 2 14 10 C 2 -2 0 4 14 D -16 -14 -4 0 2 E -4 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 16 4 B 10 0 2 14 10 C 2 -2 0 4 14 D -16 -14 -4 0 2 E -4 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 16 4 B 10 0 2 14 10 C 2 -2 0 4 14 D -16 -14 -4 0 2 E -4 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3582: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) B D A C E (10) D C B A E (8) D B C A E (6) C E A D B (6) E A C B D (5) C E D A B (4) C D E B A (4) A E C B D (4) A E B C D (4) D B A C E (3) C A D E B (3) B D C A E (3) B A D C E (3) E C A B D (2) D C B E A (2) D C A B E (2) B D A E C (2) B A D E C (2) A B E D C (2) A B D C E (2) E C D B A (1) E C D A B (1) E A B C D (1) D B C E A (1) D A C B E (1) C E D B A (1) C D E A B (1) C D A E B (1) C A D B E (1) B D E A C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -16 -4 18 B -6 0 -16 -16 2 C 16 16 0 2 28 D 4 16 -2 0 14 E -18 -2 -28 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -16 -4 18 B -6 0 -16 -16 2 C 16 16 0 2 28 D 4 16 -2 0 14 E -18 -2 -28 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=21 C=21 B=21 A=14 so A is eliminated. Round 2 votes counts: E=29 B=26 D=24 C=21 so C is eliminated. Round 3 votes counts: E=40 D=34 B=26 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:231 D:216 A:202 B:182 E:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -16 -4 18 B -6 0 -16 -16 2 C 16 16 0 2 28 D 4 16 -2 0 14 E -18 -2 -28 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -16 -4 18 B -6 0 -16 -16 2 C 16 16 0 2 28 D 4 16 -2 0 14 E -18 -2 -28 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -16 -4 18 B -6 0 -16 -16 2 C 16 16 0 2 28 D 4 16 -2 0 14 E -18 -2 -28 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999925995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3583: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) A C D B E (8) E D C B A (7) B E D A C (7) A C B D E (7) E D B C A (5) D E C A B (5) B A E D C (5) B A C E D (5) B E D C A (4) D E C B A (3) D E B C A (3) C D E A B (3) B A E C D (3) A B C D E (3) E B D C A (2) D E B A C (2) C E D A B (2) B E A D C (2) B E A C D (2) A C D E B (2) E D C A B (1) D C E A B (1) D A B E C (1) C D A E B (1) C B E A D (1) B E C D A (1) B E C A D (1) B D E A C (1) B A C D E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 2 2 -6 B 10 0 -2 -6 8 C -2 2 0 2 -10 D -2 6 -2 0 0 E 6 -8 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.631134 E: 0.368866 Sum of squares = 0.534392001112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.631134 E: 1.000000 A B C D E A 0 -10 2 2 -6 B 10 0 -2 -6 8 C -2 2 0 2 -10 D -2 6 -2 0 0 E 6 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.510204085019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=22 C=16 E=15 D=15 so E is eliminated. Round 2 votes counts: B=34 D=28 A=22 C=16 so C is eliminated. Round 3 votes counts: B=35 D=34 A=31 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:205 E:204 D:201 C:196 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 2 -6 B 10 0 -2 -6 8 C -2 2 0 2 -10 D -2 6 -2 0 0 E 6 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.510204085019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 2 -6 B 10 0 -2 -6 8 C -2 2 0 2 -10 D -2 6 -2 0 0 E 6 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.510204085019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 2 -6 B 10 0 -2 -6 8 C -2 2 0 2 -10 D -2 6 -2 0 0 E 6 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.510204085019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3584: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (14) C B A E D (9) C B E D A (8) D E A B C (7) B C A D E (7) A D E B C (7) E D C A B (6) C E D B A (5) E D A B C (4) D A E B C (3) C B E A D (3) C B A D E (3) A D B E C (3) A B D E C (3) E C D A B (2) B A C D E (2) A B D C E (2) E D C B A (1) D E B A C (1) D B E A C (1) C E B D A (1) C E B A D (1) C E A D B (1) C E A B D (1) C B D E A (1) C A B E D (1) B A D C E (1) A E D B C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 0 -8 -14 B -12 0 -14 -12 -10 C 0 14 0 -8 -6 D 8 12 8 0 -16 E 14 10 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 0 -8 -14 B -12 0 -14 -12 -10 C 0 14 0 -8 -6 D 8 12 8 0 -16 E 14 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=27 A=17 D=12 B=10 so B is eliminated. Round 2 votes counts: C=41 E=27 A=20 D=12 so D is eliminated. Round 3 votes counts: C=41 E=36 A=23 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:206 C:200 A:195 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 0 -8 -14 B -12 0 -14 -12 -10 C 0 14 0 -8 -6 D 8 12 8 0 -16 E 14 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -8 -14 B -12 0 -14 -12 -10 C 0 14 0 -8 -6 D 8 12 8 0 -16 E 14 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -8 -14 B -12 0 -14 -12 -10 C 0 14 0 -8 -6 D 8 12 8 0 -16 E 14 10 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3585: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (12) D B E C A (10) C B A D E (9) C A B E D (9) D E B A C (8) A C E B D (6) D B C E A (5) E A D C B (4) B D C E A (4) A E C D B (4) E D A B C (3) E A C D B (3) D E A B C (3) C A E B D (3) B C A D E (3) E D A C B (2) D B E A C (2) C B A E D (2) B D C A E (2) A E C B D (2) A C B E D (2) E C A B D (1) D E B C A (1) Total count = 100 A B C D E A 0 -16 -22 -4 8 B 16 0 6 10 20 C 22 -6 0 12 14 D 4 -10 -12 0 18 E -8 -20 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -22 -4 8 B 16 0 6 10 20 C 22 -6 0 12 14 D 4 -10 -12 0 18 E -8 -20 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=23 B=21 A=14 E=13 so E is eliminated. Round 2 votes counts: D=34 C=24 B=21 A=21 so B is eliminated. Round 3 votes counts: D=40 C=39 A=21 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:226 C:221 D:200 A:183 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -22 -4 8 B 16 0 6 10 20 C 22 -6 0 12 14 D 4 -10 -12 0 18 E -8 -20 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -22 -4 8 B 16 0 6 10 20 C 22 -6 0 12 14 D 4 -10 -12 0 18 E -8 -20 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -22 -4 8 B 16 0 6 10 20 C 22 -6 0 12 14 D 4 -10 -12 0 18 E -8 -20 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3586: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) D E C A B (8) B A C E D (7) E D C B A (6) E C D B A (5) E B D A C (4) C A D B E (4) A C B D E (4) E D B C A (3) E C B D A (3) D E C B A (3) D E B A C (3) D C E A B (3) D C A E B (3) C D E A B (3) C A B D E (3) B A E C D (3) E C D A B (2) C D A E B (2) B E D A C (2) B A D E C (2) A D B C E (2) A C B E D (2) E D C A B (1) E B D C A (1) E B A D C (1) D E B C A (1) D E A B C (1) D A C B E (1) C A D E B (1) C A B E D (1) B E A D C (1) B E A C D (1) B A C D E (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 -10 -4 B -6 0 -12 -6 -8 C 6 12 0 6 -2 D 10 6 -6 0 12 E 4 8 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.300000 Sum of squares = 0.46000000016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.700000 E: 1.000000 A B C D E A 0 6 -6 -10 -4 B -6 0 -12 -6 -8 C 6 12 0 6 -2 D 10 6 -6 0 12 E 4 8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.300000 Sum of squares = 0.4600000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 A=20 B=17 C=14 so C is eliminated. Round 2 votes counts: A=29 D=28 E=26 B=17 so B is eliminated. Round 3 votes counts: A=42 E=30 D=28 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:211 D:211 E:201 A:193 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -10 -4 B -6 0 -12 -6 -8 C 6 12 0 6 -2 D 10 6 -6 0 12 E 4 8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.300000 Sum of squares = 0.4600000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.700000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -10 -4 B -6 0 -12 -6 -8 C 6 12 0 6 -2 D 10 6 -6 0 12 E 4 8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.300000 Sum of squares = 0.4600000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -10 -4 B -6 0 -12 -6 -8 C 6 12 0 6 -2 D 10 6 -6 0 12 E 4 8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.100000 E: 0.300000 Sum of squares = 0.4600000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.700000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3587: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) C E A B D (10) C A E D B (6) B E A D C (4) B D E A C (4) A E C B D (4) A D B E C (4) E A B C D (3) D C B E A (3) B D E C A (3) A E C D B (3) A E B D C (3) E C B A D (2) E B A C D (2) E A C B D (2) D C B A E (2) D B C E A (2) D B A C E (2) D A B E C (2) C E B A D (2) C D B E A (2) C B D E A (2) B D C E A (2) A E B C D (2) A D E B C (2) E C A B D (1) E A B D C (1) D C A B E (1) D A C B E (1) D A B C E (1) C E D A B (1) C D A E B (1) C B E D A (1) C A E B D (1) B E C A D (1) B E A C D (1) B D A E C (1) B C D E A (1) B A E D C (1) B A D E C (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 12 16 0 B -2 0 8 12 6 C -12 -8 0 -2 -16 D -16 -12 2 0 -4 E 0 -6 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.871819 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.128181 Sum of squares = 0.77649897996 Cumulative probabilities = A: 0.871819 B: 0.871819 C: 0.871819 D: 0.871819 E: 1.000000 A B C D E A 0 2 12 16 0 B -2 0 8 12 6 C -12 -8 0 -2 -16 D -16 -12 2 0 -4 E 0 -6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000025211 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=24 A=20 B=19 E=11 so E is eliminated. Round 2 votes counts: C=29 A=26 D=24 B=21 so B is eliminated. Round 3 votes counts: A=35 D=34 C=31 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:212 E:207 D:185 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 16 0 B -2 0 8 12 6 C -12 -8 0 -2 -16 D -16 -12 2 0 -4 E 0 -6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000025211 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 16 0 B -2 0 8 12 6 C -12 -8 0 -2 -16 D -16 -12 2 0 -4 E 0 -6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000025211 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 16 0 B -2 0 8 12 6 C -12 -8 0 -2 -16 D -16 -12 2 0 -4 E 0 -6 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000025211 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3588: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) B C A D E (6) E D B C A (5) C B A E D (5) C A B E D (5) E D A C B (4) A C B D E (4) E C A D B (3) E B D C A (3) D E B A C (3) D E A C B (3) D E A B C (3) D B E A C (3) D B A E C (3) C E A B D (3) B D E A C (3) D A B E C (2) C A E B D (2) C A B D E (2) B D E C A (2) B D A C E (2) B C A E D (2) B A D C E (2) A C E D B (2) A C D B E (2) A B C D E (2) E D C B A (1) E D B A C (1) E D A B C (1) E C D A B (1) E C B D A (1) E B C D A (1) C E A D B (1) C B E A D (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E D A (1) B C E A D (1) A E C D B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -14 -8 -12 B -2 0 -2 2 2 C 14 2 0 -4 -10 D 8 -2 4 0 -10 E 12 -2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408131 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 2 -14 -8 -12 B -2 0 -2 2 2 C 14 2 0 -4 -10 D 8 -2 4 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408116 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=22 C=19 D=17 A=13 so A is eliminated. Round 2 votes counts: E=30 C=28 B=24 D=18 so D is eliminated. Round 3 votes counts: E=40 B=32 C=28 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:215 C:201 B:200 D:200 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -14 -8 -12 B -2 0 -2 2 2 C 14 2 0 -4 -10 D 8 -2 4 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408116 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -8 -12 B -2 0 -2 2 2 C 14 2 0 -4 -10 D 8 -2 4 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408116 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -8 -12 B -2 0 -2 2 2 C 14 2 0 -4 -10 D 8 -2 4 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408116 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3589: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) B E A C D (9) C A E D B (8) E A C B D (7) D B C E A (7) B D E A C (7) A E C B D (7) D B E A C (5) D C B A E (4) D C A B E (4) D B E C A (4) C A E B D (4) A C E B D (3) E C A B D (2) D B C A E (2) C E A B D (2) C A D E B (2) E A B C D (1) D C E B A (1) D C E A B (1) D B A E C (1) D B A C E (1) C D A E B (1) B E D C A (1) B E C A D (1) B E A D C (1) B D A E C (1) B A E D C (1) B A E C D (1) A E B C D (1) Total count = 100 A B C D E A 0 6 -8 0 2 B -6 0 -12 -2 0 C 8 12 0 -2 0 D 0 2 2 0 2 E -2 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.128604 B: 0.000000 C: 0.000000 D: 0.871396 E: 0.000000 Sum of squares = 0.775870193775 Cumulative probabilities = A: 0.128604 B: 0.128604 C: 0.128604 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 0 2 B -6 0 -12 -2 0 C 8 12 0 -2 0 D 0 2 2 0 2 E -2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000015981 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=22 C=17 A=11 E=10 so E is eliminated. Round 2 votes counts: D=40 B=22 C=19 A=19 so C is eliminated. Round 3 votes counts: D=41 A=37 B=22 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:209 D:203 A:200 E:198 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 0 2 B -6 0 -12 -2 0 C 8 12 0 -2 0 D 0 2 2 0 2 E -2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000015981 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 0 2 B -6 0 -12 -2 0 C 8 12 0 -2 0 D 0 2 2 0 2 E -2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000015981 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 0 2 B -6 0 -12 -2 0 C 8 12 0 -2 0 D 0 2 2 0 2 E -2 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000015981 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3590: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C A E B D (6) E D C B A (5) C A B E D (5) B A D C E (5) D E B C A (4) D B A E C (4) B A C D E (4) A C B E D (4) E D A C B (3) E C A D B (3) D E A B C (3) B C A D E (3) A C E B D (3) E D C A B (2) E C D B A (2) D A E B C (2) D A B E C (2) C E A B D (2) B D E C A (2) B D A C E (2) B C E A D (2) A D E C B (2) A B D C E (2) A B C D E (2) E D B C A (1) E C D A B (1) E A D C B (1) D E A C B (1) D B E C A (1) D B E A C (1) C E A D B (1) C B E A D (1) B D C E A (1) B D A E C (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E B C (1) A D B E C (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 10 4 8 B -2 0 8 -2 -6 C -10 -8 0 -10 -4 D -4 2 10 0 8 E -8 6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 8 B -2 0 8 -2 -6 C -10 -8 0 -10 -4 D -4 2 10 0 8 E -8 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=23 A=19 E=18 C=15 so C is eliminated. Round 2 votes counts: A=30 D=25 B=24 E=21 so E is eliminated. Round 3 votes counts: D=39 A=37 B=24 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:208 B:199 E:197 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 8 B -2 0 8 -2 -6 C -10 -8 0 -10 -4 D -4 2 10 0 8 E -8 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 8 B -2 0 8 -2 -6 C -10 -8 0 -10 -4 D -4 2 10 0 8 E -8 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 8 B -2 0 8 -2 -6 C -10 -8 0 -10 -4 D -4 2 10 0 8 E -8 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3591: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (15) E B C D A (12) A D C B E (12) C B E A D (8) B E C A D (8) A D B E C (6) D A E B C (4) E B D C A (3) E B D A C (3) D A E C B (3) C E B D A (3) C A D B E (3) C D A E B (2) A C D B E (2) E D B C A (1) E C B D A (1) D C A E B (1) D A C B E (1) C D E B A (1) C D E A B (1) C B E D A (1) C A B E D (1) B E C D A (1) B E A D C (1) B E A C D (1) A D C E B (1) A D B C E (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 -6 10 B -12 0 -14 -8 -2 C -6 14 0 -4 10 D 6 8 4 0 6 E -10 2 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -6 10 B -12 0 -14 -8 -2 C -6 14 0 -4 10 D 6 8 4 0 6 E -10 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=24 E=20 C=20 B=11 so B is eliminated. Round 2 votes counts: E=31 A=25 D=24 C=20 so C is eliminated. Round 3 votes counts: E=43 A=29 D=28 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:211 C:207 E:188 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -6 10 B -12 0 -14 -8 -2 C -6 14 0 -4 10 D 6 8 4 0 6 E -10 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -6 10 B -12 0 -14 -8 -2 C -6 14 0 -4 10 D 6 8 4 0 6 E -10 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -6 10 B -12 0 -14 -8 -2 C -6 14 0 -4 10 D 6 8 4 0 6 E -10 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3592: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) C D A B E (11) B E C D A (9) B E A D C (8) B C D E A (8) C D B A E (7) A D C E B (6) B C E D A (5) C D A E B (4) B C D A E (4) A E D C B (4) E A D C B (3) E A B D C (3) D A C E B (3) E A D B C (2) D C A E B (2) B E C A D (2) A D E C B (2) E C A D B (1) E B C A D (1) D C A B E (1) D A C B E (1) C B D A E (1) B E A C D (1) Total count = 100 A B C D E A 0 -14 -12 -12 -8 B 14 0 8 6 16 C 12 -8 0 8 6 D 12 -6 -8 0 0 E 8 -16 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -12 -8 B 14 0 8 6 16 C 12 -8 0 8 6 D 12 -6 -8 0 0 E 8 -16 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=23 E=21 A=12 D=7 so D is eliminated. Round 2 votes counts: B=37 C=26 E=21 A=16 so A is eliminated. Round 3 votes counts: B=37 C=36 E=27 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:209 D:199 E:193 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -12 -8 B 14 0 8 6 16 C 12 -8 0 8 6 D 12 -6 -8 0 0 E 8 -16 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -12 -8 B 14 0 8 6 16 C 12 -8 0 8 6 D 12 -6 -8 0 0 E 8 -16 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -12 -8 B 14 0 8 6 16 C 12 -8 0 8 6 D 12 -6 -8 0 0 E 8 -16 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3593: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (14) E B C D A (7) D A C B E (6) E B C A D (5) E B A D C (5) C D A B E (5) E C B D A (4) C B E D A (4) B E D A C (4) A D E B C (4) A D B E C (4) C D B E A (3) C D B A E (3) B D A E C (3) A E D B C (3) A D C E B (3) A D B C E (3) E B A C D (2) D A B C E (2) E C A B D (1) E B D A C (1) E A C B D (1) E A B D C (1) D C A B E (1) C E D B A (1) C E D A B (1) C E B D A (1) C E B A D (1) C D A E B (1) C B D E A (1) C A E D B (1) C A D E B (1) C A D B E (1) B C E D A (1) A C D E B (1) Total count = 100 A B C D E A 0 8 14 2 12 B -8 0 -10 -16 10 C -14 10 0 -8 10 D -2 16 8 0 12 E -12 -10 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 2 12 B -8 0 -10 -16 10 C -14 10 0 -8 10 D -2 16 8 0 12 E -12 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=27 C=24 D=9 B=8 so B is eliminated. Round 2 votes counts: A=32 E=31 C=25 D=12 so D is eliminated. Round 3 votes counts: A=43 E=31 C=26 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:217 C:199 B:188 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 2 12 B -8 0 -10 -16 10 C -14 10 0 -8 10 D -2 16 8 0 12 E -12 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 2 12 B -8 0 -10 -16 10 C -14 10 0 -8 10 D -2 16 8 0 12 E -12 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 2 12 B -8 0 -10 -16 10 C -14 10 0 -8 10 D -2 16 8 0 12 E -12 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3594: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) C B E D A (7) E C A D B (6) A B D C E (6) C E B A D (5) C B E A D (5) B A D C E (5) A D E C B (5) D A B E C (4) C E B D A (4) B C A E D (4) A E D C B (4) D A E C B (3) B C D E A (3) A D E B C (3) E D A C B (2) E C B D A (2) E A C D B (2) B D A C E (2) B C E A D (2) B C A D E (2) B A C D E (2) A B C E D (2) E D C A B (1) E C D A B (1) E C B A D (1) E C A B D (1) E A D C B (1) D E C B A (1) D E A C B (1) B D C E A (1) B D C A E (1) B C E D A (1) B A C E D (1) A E C B D (1) Total count = 100 A B C D E A 0 2 4 32 6 B -2 0 -6 16 12 C -4 6 0 4 6 D -32 -16 -4 0 -6 E -6 -12 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 32 6 B -2 0 -6 16 12 C -4 6 0 4 6 D -32 -16 -4 0 -6 E -6 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 C=21 E=17 D=9 so D is eliminated. Round 2 votes counts: A=36 B=24 C=21 E=19 so E is eliminated. Round 3 votes counts: A=42 C=34 B=24 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:210 C:206 E:191 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 32 6 B -2 0 -6 16 12 C -4 6 0 4 6 D -32 -16 -4 0 -6 E -6 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 32 6 B -2 0 -6 16 12 C -4 6 0 4 6 D -32 -16 -4 0 -6 E -6 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 32 6 B -2 0 -6 16 12 C -4 6 0 4 6 D -32 -16 -4 0 -6 E -6 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3595: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) C A E D B (11) B D E A C (10) E B C A D (8) D B E A C (7) E C A B D (6) C E A B D (6) B E D C A (6) A C D B E (6) A C D E B (4) D B A E C (3) D A B C E (3) C A E B D (3) D B A C E (2) C A D E B (2) C A B E D (2) B E D A C (2) E C B A D (1) E C A D B (1) E B D A C (1) E B C D A (1) D E B A C (1) D A C B E (1) C E B A D (1) A D C B E (1) Total count = 100 A B C D E A 0 -8 -18 4 -24 B 8 0 10 16 -14 C 18 -10 0 4 -16 D -4 -16 -4 0 -20 E 24 14 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -18 4 -24 B 8 0 10 16 -14 C 18 -10 0 4 -16 D -4 -16 -4 0 -20 E 24 14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=25 B=18 D=17 A=11 so A is eliminated. Round 2 votes counts: C=35 E=29 D=18 B=18 so D is eliminated. Round 3 votes counts: C=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:237 B:210 C:198 D:178 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -18 4 -24 B 8 0 10 16 -14 C 18 -10 0 4 -16 D -4 -16 -4 0 -20 E 24 14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 4 -24 B 8 0 10 16 -14 C 18 -10 0 4 -16 D -4 -16 -4 0 -20 E 24 14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 4 -24 B 8 0 10 16 -14 C 18 -10 0 4 -16 D -4 -16 -4 0 -20 E 24 14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3596: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) B E D C A (6) E C B D A (5) E B A C D (5) D C B E A (5) C A D E B (4) B D E C A (4) A D C B E (4) A B D E C (4) E C B A D (3) E B C D A (3) C D E A B (3) A B E D C (3) E C D B A (2) E A B C D (2) D C A B E (2) D B C E A (2) C E D B A (2) C E D A B (2) B E A D C (2) B D E A C (2) B A E D C (2) B A D E C (2) A E C B D (2) A E B C D (2) A D B C E (2) A C E D B (2) A C D E B (2) E C A B D (1) D C B A E (1) D C A E B (1) D B C A E (1) D A C B E (1) C D E B A (1) C D A E B (1) C A E D B (1) B E D A C (1) B E A C D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -14 -4 -18 B 14 0 -8 6 -2 C 14 8 0 -6 -6 D 4 -6 6 0 4 E 18 2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888879 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 A B C D E A 0 -14 -14 -4 -18 B 14 0 -8 6 -2 C 14 8 0 -6 -6 D 4 -6 6 0 4 E 18 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888793 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 D=20 B=20 C=14 so C is eliminated. Round 2 votes counts: A=30 E=25 D=25 B=20 so B is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:205 C:205 D:204 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -14 -4 -18 B 14 0 -8 6 -2 C 14 8 0 -6 -6 D 4 -6 6 0 4 E 18 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888793 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -4 -18 B 14 0 -8 6 -2 C 14 8 0 -6 -6 D 4 -6 6 0 4 E 18 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888793 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -4 -18 B 14 0 -8 6 -2 C 14 8 0 -6 -6 D 4 -6 6 0 4 E 18 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888793 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3597: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) B C D A E (6) A B E D C (5) E C A D B (4) E A D C B (4) D B C A E (4) B C A E D (4) A E B D C (4) E D A C B (3) E A B C D (3) D C B A E (3) D A E B C (3) C E D A B (3) C D B E A (3) C B D E A (3) B A D C E (3) E C A B D (2) D E C A B (2) D C E A B (2) D B A C E (2) D A B E C (2) C D E B A (2) C B E D A (2) B D C A E (2) B D A C E (2) B A D E C (2) B A C E D (2) E A D B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E A C B (1) D C E B A (1) D A E C B (1) C E B A D (1) C D E A B (1) C D B A E (1) C B D A E (1) B C A D E (1) B A E D C (1) B A E C D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 4 0 18 B -6 0 16 -4 4 C -4 -16 0 -16 -2 D 0 4 16 0 0 E -18 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.461305 B: 0.000000 C: 0.000000 D: 0.538695 E: 0.000000 Sum of squares = 0.502994560983 Cumulative probabilities = A: 0.461305 B: 0.461305 C: 0.461305 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 18 B -6 0 16 -4 4 C -4 -16 0 -16 -2 D 0 4 16 0 0 E -18 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=21 E=20 A=18 C=17 so C is eliminated. Round 2 votes counts: B=30 D=28 E=24 A=18 so A is eliminated. Round 3 votes counts: B=36 E=35 D=29 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:214 D:210 B:205 E:190 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 18 B -6 0 16 -4 4 C -4 -16 0 -16 -2 D 0 4 16 0 0 E -18 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 18 B -6 0 16 -4 4 C -4 -16 0 -16 -2 D 0 4 16 0 0 E -18 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 18 B -6 0 16 -4 4 C -4 -16 0 -16 -2 D 0 4 16 0 0 E -18 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3598: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) D A B E C (9) B A D E C (9) C E B A D (8) A D B E C (8) E C A B D (7) C E A B D (6) D B A E C (4) C E D A B (4) C E B D A (4) E C B A D (3) C E D B A (3) B D A E C (3) D B C E A (2) D A C E B (2) C E A D B (2) B A E D C (2) A E B C D (2) D C E B A (1) D C E A B (1) D C B E A (1) D A C B E (1) D A B C E (1) C D E B A (1) B D A C E (1) B A E C D (1) A E D C B (1) A E C D B (1) A E C B D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 14 6 14 B 4 0 6 -2 6 C -14 -6 0 -14 -6 D -6 2 14 0 8 E -14 -6 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 6 14 B 4 0 6 -2 6 C -14 -6 0 -14 -6 D -6 2 14 0 8 E -14 -6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=28 B=16 A=15 E=10 so E is eliminated. Round 2 votes counts: C=38 D=31 B=16 A=15 so A is eliminated. Round 3 votes counts: D=40 C=40 B=20 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:215 D:209 B:207 E:189 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 6 14 B 4 0 6 -2 6 C -14 -6 0 -14 -6 D -6 2 14 0 8 E -14 -6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 6 14 B 4 0 6 -2 6 C -14 -6 0 -14 -6 D -6 2 14 0 8 E -14 -6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 6 14 B 4 0 6 -2 6 C -14 -6 0 -14 -6 D -6 2 14 0 8 E -14 -6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3599: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) C B E A D (7) C B A E D (7) D B A E C (6) C E B A D (6) B D A E C (6) B C D A E (6) D A E B C (5) E C A D B (4) E A D B C (4) C E A D B (4) E A D C B (3) E A C D B (3) D E A B C (3) C B D A E (3) B D A C E (3) E D A B C (2) D A B E C (2) E A C B D (1) D E C A B (1) D C E A B (1) D B C A E (1) C E B D A (1) C B E D A (1) C B D E A (1) C B A D E (1) B D C A E (1) B C A D E (1) B A E D C (1) B A D E C (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -6 14 2 B 10 0 -2 18 4 C 6 2 0 10 6 D -14 -18 -10 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 14 2 B 10 0 -2 18 4 C 6 2 0 10 6 D -14 -18 -10 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=21 D=19 E=17 A=5 so A is eliminated. Round 2 votes counts: C=38 B=23 D=20 E=19 so E is eliminated. Round 3 votes counts: C=46 D=30 B=24 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:212 A:200 E:199 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 14 2 B 10 0 -2 18 4 C 6 2 0 10 6 D -14 -18 -10 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 14 2 B 10 0 -2 18 4 C 6 2 0 10 6 D -14 -18 -10 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 14 2 B 10 0 -2 18 4 C 6 2 0 10 6 D -14 -18 -10 0 -10 E -2 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3600: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (13) E B A D C (10) C D A B E (9) A D E B C (9) B E C D A (6) B E C A D (5) A D E C B (5) B E D A C (4) E B C A D (3) D A C E B (3) D A C B E (3) C D B A E (3) B C E D A (3) E A D B C (2) D A E B C (2) D A B E C (2) C E B A D (2) C B E A D (2) C B D E A (2) C B D A E (2) E C B A D (1) E B A C D (1) E A B D C (1) D C A E B (1) D C A B E (1) D A B C E (1) B E D C A (1) B C E A D (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -18 -10 -12 -14 B 18 0 4 14 16 C 10 -4 0 6 -6 D 12 -14 -6 0 -12 E 14 -16 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -12 -14 B 18 0 4 14 16 C 10 -4 0 6 -6 D 12 -14 -6 0 -12 E 14 -16 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=20 E=18 A=16 D=13 so D is eliminated. Round 2 votes counts: C=35 A=27 B=20 E=18 so E is eliminated. Round 3 votes counts: C=36 B=34 A=30 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:208 C:203 D:190 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 -12 -14 B 18 0 4 14 16 C 10 -4 0 6 -6 D 12 -14 -6 0 -12 E 14 -16 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -12 -14 B 18 0 4 14 16 C 10 -4 0 6 -6 D 12 -14 -6 0 -12 E 14 -16 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -12 -14 B 18 0 4 14 16 C 10 -4 0 6 -6 D 12 -14 -6 0 -12 E 14 -16 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3601: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) D C E A B (6) B A E D C (6) D B A E C (5) B C E A D (5) B A E C D (5) E A C D B (4) E A C B D (4) D C B E A (4) D B C E A (4) D A E C B (4) C E A B D (4) E C A D B (3) B D C E A (3) A E B D C (3) D B C A E (2) D A B E C (2) C E A D B (2) C D E A B (2) B D C A E (2) B C D E A (2) A E D C B (2) A E C D B (2) A D E C B (2) A B D E C (2) E A D C B (1) D C B A E (1) D C A B E (1) D A E B C (1) C E D B A (1) C E D A B (1) C D B E A (1) C B E D A (1) C B E A D (1) B E A C D (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 8 -2 0 B 4 0 6 -4 12 C -8 -6 0 -20 -14 D 2 4 20 0 6 E 0 -12 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -2 0 B 4 0 6 -4 12 C -8 -6 0 -20 -14 D 2 4 20 0 6 E 0 -12 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=30 C=13 A=13 E=12 so E is eliminated. Round 2 votes counts: B=32 D=30 A=22 C=16 so C is eliminated. Round 3 votes counts: D=35 B=34 A=31 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:209 A:201 E:198 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -2 0 B 4 0 6 -4 12 C -8 -6 0 -20 -14 D 2 4 20 0 6 E 0 -12 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -2 0 B 4 0 6 -4 12 C -8 -6 0 -20 -14 D 2 4 20 0 6 E 0 -12 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -2 0 B 4 0 6 -4 12 C -8 -6 0 -20 -14 D 2 4 20 0 6 E 0 -12 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3602: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) A C B D E (8) E A C B D (6) A E C B D (6) E A D C B (5) C B D A E (5) E C B A D (4) E A D B C (4) D B C A E (4) B C D E A (4) E D B A C (3) E D A B C (3) D E B C A (3) B C D A E (3) A E D C B (3) E D B C A (2) E C B D A (2) D B A C E (2) D A E B C (2) C E B A D (2) C B E D A (2) A E D B C (2) A E C D B (2) E A C D B (1) D E A B C (1) D B C E A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B E D (1) C A B D E (1) B D C E A (1) B D C A E (1) B C E D A (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 14 20 18 8 B -14 0 -10 2 -6 C -20 10 0 6 0 D -18 -2 -6 0 -2 E -8 6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 18 8 B -14 0 -10 2 -6 C -20 10 0 6 0 D -18 -2 -6 0 -2 E -8 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=30 C=14 D=13 B=10 so B is eliminated. Round 2 votes counts: A=33 E=30 C=22 D=15 so D is eliminated. Round 3 votes counts: A=37 E=34 C=29 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:200 C:198 B:186 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 18 8 B -14 0 -10 2 -6 C -20 10 0 6 0 D -18 -2 -6 0 -2 E -8 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 18 8 B -14 0 -10 2 -6 C -20 10 0 6 0 D -18 -2 -6 0 -2 E -8 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 18 8 B -14 0 -10 2 -6 C -20 10 0 6 0 D -18 -2 -6 0 -2 E -8 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3603: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (15) D C B A E (10) D B C A E (8) E A D B C (7) D A E B C (4) C B E A D (4) E C A B D (3) E A D C B (3) E A C B D (3) D E A B C (3) D A B C E (3) C B E D A (3) C B D E A (3) C B D A E (3) C B A E D (3) A E D B C (3) E C B A D (2) D B A C E (2) D A E C B (2) C E B A D (2) A E B C D (2) E D A C B (1) D E C B A (1) D E A C B (1) D A B E C (1) C D B E A (1) B C D A E (1) B A C E D (1) B A C D E (1) A E B D C (1) A D E B C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 12 6 -4 B -10 0 10 -4 -8 C -12 -10 0 -4 -6 D -6 4 4 0 -8 E 4 8 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 12 6 -4 B -10 0 10 -4 -8 C -12 -10 0 -4 -6 D -6 4 4 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=34 C=19 A=9 B=3 so B is eliminated. Round 2 votes counts: D=35 E=34 C=20 A=11 so A is eliminated. Round 3 votes counts: E=40 D=37 C=23 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:212 D:197 B:194 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 12 6 -4 B -10 0 10 -4 -8 C -12 -10 0 -4 -6 D -6 4 4 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 6 -4 B -10 0 10 -4 -8 C -12 -10 0 -4 -6 D -6 4 4 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 6 -4 B -10 0 10 -4 -8 C -12 -10 0 -4 -6 D -6 4 4 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3604: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) A C B D E (10) E B D C A (5) E B D A C (5) D E B C A (5) C D A E B (4) C A D B E (4) B E D C A (4) B E D A C (4) B A C E D (4) A C D E B (4) A C B E D (4) D E C A B (3) C A B D E (3) B E A C D (3) A C D B E (3) E B A D C (2) D E C B A (2) D C A E B (2) B C A D E (2) A C E D B (2) A C E B D (2) A B E C D (2) E D C A B (1) E D B A C (1) D C E A B (1) D B E C A (1) C B A D E (1) C A D E B (1) B E A D C (1) B D E C A (1) A E C D B (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 2 2 B 2 0 2 10 -4 C 0 -2 0 4 -4 D -2 -10 -4 0 -4 E -2 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999702 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 0 2 2 B 2 0 2 10 -4 C 0 -2 0 4 -4 D -2 -10 -4 0 -4 E -2 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999968 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=24 B=19 D=14 C=13 so C is eliminated. Round 2 votes counts: A=38 E=24 B=20 D=18 so D is eliminated. Round 3 votes counts: A=44 E=35 B=21 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:205 E:205 A:201 C:199 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 2 2 B 2 0 2 10 -4 C 0 -2 0 4 -4 D -2 -10 -4 0 -4 E -2 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999968 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 2 B 2 0 2 10 -4 C 0 -2 0 4 -4 D -2 -10 -4 0 -4 E -2 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999968 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 2 B 2 0 2 10 -4 C 0 -2 0 4 -4 D -2 -10 -4 0 -4 E -2 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999968 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3605: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (14) E A C D B (7) D B C E A (7) A E C B D (7) E C A D B (6) B D C A E (6) B D C E A (5) D B C A E (4) B D A E C (4) A C E D B (4) E C D B A (3) B D E C A (3) A E B C D (3) E C D A B (2) E A C B D (2) D C B E A (2) C E D A B (2) C D E B A (2) B A D C E (2) A B E D C (2) A B D C E (2) A B C D E (2) E B D C A (1) D C A B E (1) D B E C A (1) C E A D B (1) B A D E C (1) A E C D B (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 8 -14 12 B 10 0 16 14 16 C -8 -16 0 -10 12 D 14 -14 10 0 14 E -12 -16 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -14 12 B 10 0 16 14 16 C -8 -16 0 -10 12 D 14 -14 10 0 14 E -12 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=24 E=21 D=15 C=5 so C is eliminated. Round 2 votes counts: B=35 E=24 A=24 D=17 so D is eliminated. Round 3 votes counts: B=49 E=26 A=25 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:228 D:212 A:198 C:189 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 -14 12 B 10 0 16 14 16 C -8 -16 0 -10 12 D 14 -14 10 0 14 E -12 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -14 12 B 10 0 16 14 16 C -8 -16 0 -10 12 D 14 -14 10 0 14 E -12 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -14 12 B 10 0 16 14 16 C -8 -16 0 -10 12 D 14 -14 10 0 14 E -12 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3606: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) D E A B C (11) E D C B A (9) C B A E D (9) A B C D E (9) E D A B C (6) B A C D E (6) C B A D E (5) E C D B A (4) A B D C E (4) D E C A B (3) D A E B C (2) D A B E C (2) C E D B A (2) C E B A D (2) C B E A D (2) A B D E C (2) E D A C B (1) E C B A D (1) E A B D C (1) D E A C B (1) D A B C E (1) C E B D A (1) C B D A E (1) B C A E D (1) B C A D E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 -4 -10 -10 B -12 0 -4 -10 -10 C 4 4 0 -12 -10 D 10 10 12 0 0 E 10 10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.413721 E: 0.586279 Sum of squares = 0.514888199503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.413721 E: 1.000000 A B C D E A 0 12 -4 -10 -10 B -12 0 -4 -10 -10 C 4 4 0 -12 -10 D 10 10 12 0 0 E 10 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=22 D=20 A=17 B=8 so B is eliminated. Round 2 votes counts: E=33 C=24 A=23 D=20 so D is eliminated. Round 3 votes counts: E=48 A=28 C=24 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:215 A:194 C:193 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -4 -10 -10 B -12 0 -4 -10 -10 C 4 4 0 -12 -10 D 10 10 12 0 0 E 10 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 -10 -10 B -12 0 -4 -10 -10 C 4 4 0 -12 -10 D 10 10 12 0 0 E 10 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 -10 -10 B -12 0 -4 -10 -10 C 4 4 0 -12 -10 D 10 10 12 0 0 E 10 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3607: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) D A B C E (8) C B E A D (7) E C D B A (6) D A B E C (6) D E C A B (5) B C A E D (5) D E C B A (4) A B C E D (4) A B C D E (4) E D C B A (3) E C B D A (3) D E A C B (3) D B E C A (3) D A E C B (3) C E B A D (3) B A C E D (3) E C B A D (2) B A D C E (2) A E C B D (2) A D B E C (2) E D A C B (1) D E B C A (1) D B C E A (1) D A E B C (1) C E A B D (1) C B E D A (1) B D E C A (1) B D C A E (1) B C E D A (1) A D E C B (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -4 8 B -6 0 10 -18 20 C -4 -10 0 -16 4 D 4 18 16 0 16 E -8 -20 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 8 B -6 0 10 -18 20 C -4 -10 0 -16 4 D 4 18 16 0 16 E -8 -20 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=25 E=15 B=13 C=12 so C is eliminated. Round 2 votes counts: D=35 A=25 B=21 E=19 so E is eliminated. Round 3 votes counts: D=45 B=29 A=26 so A is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:207 B:203 C:187 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 8 B -6 0 10 -18 20 C -4 -10 0 -16 4 D 4 18 16 0 16 E -8 -20 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 8 B -6 0 10 -18 20 C -4 -10 0 -16 4 D 4 18 16 0 16 E -8 -20 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 8 B -6 0 10 -18 20 C -4 -10 0 -16 4 D 4 18 16 0 16 E -8 -20 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3608: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) D B E A C (9) D B A E C (6) D A B E C (6) C E B A D (6) D C A B E (5) A D B E C (5) C D A B E (4) C A D B E (4) E B A D C (3) D E B C A (3) D A B C E (3) A B E D C (3) D E B A C (2) C E B D A (2) C E A B D (2) C D E B A (2) C A D E B (2) C A B E D (2) A B D E C (2) E B D C A (1) E B D A C (1) E B C A D (1) E B A C D (1) E A B C D (1) D C E B A (1) D B C A E (1) D A C B E (1) C E D B A (1) C E A D B (1) C D E A B (1) C D A E B (1) B E A D C (1) B D E A C (1) A D B C E (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 16 -2 -2 20 B -16 0 4 -20 14 C 2 -4 0 -10 6 D 2 20 10 0 22 E -20 -14 -6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -2 20 B -16 0 4 -20 14 C 2 -4 0 -10 6 D 2 20 10 0 22 E -20 -14 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=37 A=14 E=8 B=2 so B is eliminated. Round 2 votes counts: C=39 D=38 A=14 E=9 so E is eliminated. Round 3 votes counts: D=40 C=40 A=20 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:216 C:197 B:191 E:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -2 -2 20 B -16 0 4 -20 14 C 2 -4 0 -10 6 D 2 20 10 0 22 E -20 -14 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -2 20 B -16 0 4 -20 14 C 2 -4 0 -10 6 D 2 20 10 0 22 E -20 -14 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -2 20 B -16 0 4 -20 14 C 2 -4 0 -10 6 D 2 20 10 0 22 E -20 -14 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3609: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) D B C A E (6) E C A D B (5) E B A C D (5) E A B C D (5) B E A D C (5) B D E C A (5) B D A E C (5) B D C A E (4) B D A C E (4) E A C B D (3) D C A E B (3) C E D A B (3) C D A E B (3) C A E D B (3) A E C D B (3) E C D A B (2) D C E B A (2) D B C E A (2) B E D C A (2) B A D E C (2) A E C B D (2) A D B C E (2) A C E D B (2) E C D B A (1) E C B A D (1) E B C A D (1) D C E A B (1) D C B A E (1) D C A B E (1) C E A D B (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E A C (1) B D C E A (1) B A D C E (1) A E B C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 2 -8 B 2 0 12 4 -8 C 0 -12 0 0 -16 D -2 -4 0 0 -8 E 8 8 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 2 -8 B 2 0 12 4 -8 C 0 -12 0 0 -16 D -2 -4 0 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=29 D=16 A=12 C=11 so C is eliminated. Round 2 votes counts: E=33 B=32 D=19 A=16 so A is eliminated. Round 3 votes counts: E=44 B=34 D=22 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:205 A:196 D:193 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 2 -8 B 2 0 12 4 -8 C 0 -12 0 0 -16 D -2 -4 0 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 -8 B 2 0 12 4 -8 C 0 -12 0 0 -16 D -2 -4 0 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 -8 B 2 0 12 4 -8 C 0 -12 0 0 -16 D -2 -4 0 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3610: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (17) D C E A B (12) B D C A E (6) E A C D B (5) D C B E A (4) B A E D C (4) B A C E D (4) E A D C B (3) D C E B A (3) C D E A B (3) B D A E C (3) A E B C D (3) E A D B C (2) E A B D C (2) D E C A B (2) B E A D C (2) B C D A E (2) B C A E D (2) B C A D E (2) A E C B D (2) A B E C D (2) E D C A B (1) E C A D B (1) E A C B D (1) E A B C D (1) D E A B C (1) D C B A E (1) D B E A C (1) D B C E A (1) D B C A E (1) C D B A E (1) C A E D B (1) B D E A C (1) B D A C E (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 -14 14 12 8 B 14 0 18 12 12 C -14 -18 0 -2 -10 D -12 -12 2 0 -8 E -8 -12 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 14 12 8 B 14 0 18 12 12 C -14 -18 0 -2 -10 D -12 -12 2 0 -8 E -8 -12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=45 D=26 E=16 A=8 C=5 so C is eliminated. Round 2 votes counts: B=45 D=30 E=16 A=9 so A is eliminated. Round 3 votes counts: B=47 D=30 E=23 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:210 E:199 D:185 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 14 12 8 B 14 0 18 12 12 C -14 -18 0 -2 -10 D -12 -12 2 0 -8 E -8 -12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 14 12 8 B 14 0 18 12 12 C -14 -18 0 -2 -10 D -12 -12 2 0 -8 E -8 -12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 14 12 8 B 14 0 18 12 12 C -14 -18 0 -2 -10 D -12 -12 2 0 -8 E -8 -12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3611: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) A D B C E (6) E C A B D (5) E B C D A (5) D A B C E (5) C E D B A (5) B D A E C (5) A B D E C (5) E C B D A (4) E A B C D (4) E B A D C (3) D C B E A (3) D B C A E (3) C D E B A (3) C D B E A (3) A E C D B (3) A E B D C (3) D C B A E (2) D B C E A (2) D B A C E (2) B D A C E (2) A E C B D (2) A E B C D (2) A D C B E (2) E C B A D (1) E B C A D (1) D C A B E (1) D A C B E (1) C E A D B (1) C D E A B (1) C D A E B (1) B E D A C (1) B E A D C (1) B D E A C (1) B A E D C (1) B A D E C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 18 0 12 B -2 0 22 8 10 C -18 -22 0 -18 -12 D 0 -8 18 0 2 E -12 -10 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.835453 B: 0.000000 C: 0.000000 D: 0.164547 E: 0.000000 Sum of squares = 0.725057752515 Cumulative probabilities = A: 0.835453 B: 0.835453 C: 0.835453 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 0 12 B -2 0 22 8 10 C -18 -22 0 -18 -12 D 0 -8 18 0 2 E -12 -10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000013206 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=23 D=19 C=14 B=12 so B is eliminated. Round 2 votes counts: A=34 D=27 E=25 C=14 so C is eliminated. Round 3 votes counts: D=35 A=34 E=31 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:219 A:216 D:206 E:194 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 18 0 12 B -2 0 22 8 10 C -18 -22 0 -18 -12 D 0 -8 18 0 2 E -12 -10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000013206 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 0 12 B -2 0 22 8 10 C -18 -22 0 -18 -12 D 0 -8 18 0 2 E -12 -10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000013206 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 0 12 B -2 0 22 8 10 C -18 -22 0 -18 -12 D 0 -8 18 0 2 E -12 -10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000013206 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3612: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) B C D E A (6) E B D A C (5) C B D A E (5) E B C A D (4) E A C D B (4) C A D E B (4) C A D B E (4) B E D C A (4) B D C E A (4) E A C B D (3) C D B A E (3) C B A D E (3) B E D A C (3) B D C A E (3) B C D A E (3) A D E C B (3) A C D E B (3) E C A B D (2) E B A D C (2) E B A C D (2) E A D C B (2) C D A B E (2) A C E D B (2) E A B D C (1) E A B C D (1) D B E A C (1) D B C A E (1) D A C E B (1) D A C B E (1) D A B C E (1) C B E D A (1) C B E A D (1) C A E D B (1) C A B D E (1) B E C A D (1) B D E A C (1) B D A C E (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 -8 6 -14 B 10 0 6 16 2 C 8 -6 0 14 6 D -6 -16 -14 0 4 E 14 -2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 6 -14 B 10 0 6 16 2 C 8 -6 0 14 6 D -6 -16 -14 0 4 E 14 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994426 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=27 C=25 A=9 D=5 so D is eliminated. Round 2 votes counts: E=34 B=29 C=25 A=12 so A is eliminated. Round 3 votes counts: E=37 C=33 B=30 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:217 C:211 E:201 A:187 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 6 -14 B 10 0 6 16 2 C 8 -6 0 14 6 D -6 -16 -14 0 4 E 14 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994426 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 6 -14 B 10 0 6 16 2 C 8 -6 0 14 6 D -6 -16 -14 0 4 E 14 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994426 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 6 -14 B 10 0 6 16 2 C 8 -6 0 14 6 D -6 -16 -14 0 4 E 14 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994426 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3613: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) D E C A B (6) D B A C E (6) E C A B D (5) C A E B D (5) A C B E D (5) E B C A D (4) D C A E B (4) D B E C A (4) D A C B E (4) E D C A B (3) E C D A B (3) E C A D B (3) D C E A B (3) D B E A C (3) D B A E C (3) E C B A D (2) D E B C A (2) D C A B E (2) C E A B D (2) C A E D B (2) B D A C E (2) B A E C D (2) B A C D E (2) A C E B D (2) A B C E D (2) E D B C A (1) E B A C D (1) D E C B A (1) D B C E A (1) D A B C E (1) C E D A B (1) C D E A B (1) B E A D C (1) B E A C D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -10 -2 4 B -12 0 -10 -10 -2 C 10 10 0 4 10 D 2 10 -4 0 -6 E -4 2 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999297 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 -2 4 B -12 0 -10 -10 -2 C 10 10 0 4 10 D 2 10 -4 0 -6 E -4 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=22 B=16 C=11 A=11 so C is eliminated. Round 2 votes counts: D=41 E=25 A=18 B=16 so B is eliminated. Round 3 votes counts: D=43 A=30 E=27 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:217 A:202 D:201 E:197 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 -2 4 B -12 0 -10 -10 -2 C 10 10 0 4 10 D 2 10 -4 0 -6 E -4 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 -2 4 B -12 0 -10 -10 -2 C 10 10 0 4 10 D 2 10 -4 0 -6 E -4 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 -2 4 B -12 0 -10 -10 -2 C 10 10 0 4 10 D 2 10 -4 0 -6 E -4 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3614: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) D A C B E (7) C A B E D (7) A C D B E (6) D E C A B (5) B E C A D (5) D E B A C (4) B C A E D (4) B A C D E (4) A B C D E (4) D E A C B (3) E D C A B (2) E D B C A (2) E C D A B (2) E B D C A (2) E B C D A (2) E B C A D (2) D B E A C (2) C E B A D (2) C A E B D (2) B E D C A (2) B E D A C (2) B A D C E (2) B A C E D (2) A D C B E (2) E D C B A (1) E D B A C (1) D E A B C (1) D C A E B (1) D B A E C (1) D A E C B (1) C E A D B (1) C B E A D (1) C A D E B (1) B E A C D (1) B D E A C (1) B C E A D (1) B A D E C (1) A D C E B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 10 10 2 10 B -10 0 -6 -4 12 C -10 6 0 -4 14 D -2 4 4 0 12 E -10 -12 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 2 10 B -10 0 -6 -4 12 C -10 6 0 -4 14 D -2 4 4 0 12 E -10 -12 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=25 A=15 E=14 C=14 so E is eliminated. Round 2 votes counts: D=38 B=31 C=16 A=15 so A is eliminated. Round 3 votes counts: D=41 B=36 C=23 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:209 C:203 B:196 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 2 10 B -10 0 -6 -4 12 C -10 6 0 -4 14 D -2 4 4 0 12 E -10 -12 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 2 10 B -10 0 -6 -4 12 C -10 6 0 -4 14 D -2 4 4 0 12 E -10 -12 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 2 10 B -10 0 -6 -4 12 C -10 6 0 -4 14 D -2 4 4 0 12 E -10 -12 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3615: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) E D A C B (6) C B D A E (6) A B E D C (6) D E C A B (5) C D E B A (5) C B A E D (5) B A C D E (5) C D B A E (4) C B D E A (4) C B A D E (4) B A C E D (4) D E A C B (3) B C A E D (3) B C A D E (3) E B A C D (2) E A D B C (2) D E A B C (2) D C E B A (2) C B E D A (2) A E D B C (2) A B D E C (2) A B C D E (2) E D C B A (1) E D C A B (1) E C B A D (1) E B D C A (1) E B A D C (1) D C E A B (1) D A E C B (1) D A E B C (1) B E A C D (1) A E B D C (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 4 -6 4 B 8 0 -2 8 6 C -4 2 0 2 2 D 6 -8 -2 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.42857142857 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -6 4 B 8 0 -2 8 6 C -4 2 0 2 2 D 6 -8 -2 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428254 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=23 B=16 A=16 D=15 so D is eliminated. Round 2 votes counts: E=33 C=33 A=18 B=16 so B is eliminated. Round 3 votes counts: C=39 E=34 A=27 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:210 C:201 D:201 A:197 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -6 4 B 8 0 -2 8 6 C -4 2 0 2 2 D 6 -8 -2 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428254 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -6 4 B 8 0 -2 8 6 C -4 2 0 2 2 D 6 -8 -2 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428254 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -6 4 B 8 0 -2 8 6 C -4 2 0 2 2 D 6 -8 -2 0 6 E -4 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428254 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3616: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) A B E D C (8) D B E C A (6) A E C B D (6) C D E B A (5) E C D B A (4) B A D E C (4) A B D C E (4) E C D A B (3) E A C B D (3) D B C E A (3) A E B C D (3) A C E B D (3) A C B D E (3) E C A D B (2) D C B A E (2) D B C A E (2) C E D B A (2) C E D A B (2) C E A D B (2) B D A C E (2) B A E D C (2) A B C D E (2) E D C B A (1) E B D A C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E B C A (1) D B A C E (1) C D E A B (1) C D B E A (1) C D B A E (1) C A E D B (1) C A E B D (1) B E A D C (1) A E B D C (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 18 2 18 B 0 0 10 18 10 C -18 -10 0 -4 -20 D -2 -18 4 0 0 E -18 -10 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.576089 B: 0.423911 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.511579161572 Cumulative probabilities = A: 0.576089 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 2 18 B 0 0 10 18 10 C -18 -10 0 -4 -20 D -2 -18 4 0 0 E -18 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=19 E=16 D=16 C=16 so E is eliminated. Round 2 votes counts: A=38 C=25 B=20 D=17 so D is eliminated. Round 3 votes counts: A=38 B=33 C=29 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:219 E:196 D:192 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 2 18 B 0 0 10 18 10 C -18 -10 0 -4 -20 D -2 -18 4 0 0 E -18 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 2 18 B 0 0 10 18 10 C -18 -10 0 -4 -20 D -2 -18 4 0 0 E -18 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 2 18 B 0 0 10 18 10 C -18 -10 0 -4 -20 D -2 -18 4 0 0 E -18 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3617: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (6) C B E A D (6) D A B E C (5) C E D A B (5) C E B A D (5) B A D E C (5) E C D A B (4) B C A E D (4) B A C E D (4) A B E D C (4) E C A D B (3) D C E A B (3) B C E A D (3) B A E D C (3) E D A C B (2) E A D C B (2) C E D B A (2) C B E D A (2) C B D E A (2) B D A C E (2) B A E C D (2) B A C D E (2) A E D B C (2) A E B D C (2) A E B C D (2) A B D E C (2) E A B C D (1) D E C A B (1) D E A C B (1) D C E B A (1) D C A B E (1) D B C A E (1) D B A C E (1) D A E C B (1) C E B D A (1) C E A B D (1) C D B E A (1) C D B A E (1) B C D A E (1) B C A D E (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 0 2 12 8 B 0 0 10 10 8 C -2 -10 0 6 0 D -12 -10 -6 0 -20 E -8 -8 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.627402 B: 0.372598 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.53246254422 Cumulative probabilities = A: 0.627402 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 12 8 B 0 0 10 10 8 C -2 -10 0 6 0 D -12 -10 -6 0 -20 E -8 -8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 D=21 A=14 E=12 so E is eliminated. Round 2 votes counts: C=33 B=27 D=23 A=17 so A is eliminated. Round 3 votes counts: B=38 C=33 D=29 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:211 E:202 C:197 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 12 8 B 0 0 10 10 8 C -2 -10 0 6 0 D -12 -10 -6 0 -20 E -8 -8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 12 8 B 0 0 10 10 8 C -2 -10 0 6 0 D -12 -10 -6 0 -20 E -8 -8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 12 8 B 0 0 10 10 8 C -2 -10 0 6 0 D -12 -10 -6 0 -20 E -8 -8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3618: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B A D E (9) C A B E D (9) E A B D C (6) E D A B C (5) D E B A C (5) C A E B D (5) C B D A E (4) D C E B A (3) D B E A C (3) C D B A E (3) A E B D C (3) A E B C D (3) D C B A E (2) D B E C A (2) C E D A B (2) C D E A B (2) C D B E A (2) B D C A E (2) A C E B D (2) A B E D C (2) E C A B D (1) E A C D B (1) D E B C A (1) D E A B C (1) D C B E A (1) D B C E A (1) D B A E C (1) C E A D B (1) C E A B D (1) B D A C E (1) B C D A E (1) B A E D C (1) B A D C E (1) B A C D E (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -4 16 6 B -12 0 2 10 -2 C 4 -2 0 0 10 D -16 -10 0 0 -8 E -6 2 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839629 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 16 6 B -12 0 2 10 -2 C 4 -2 0 0 10 D -16 -10 0 0 -8 E -6 2 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839513 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=22 D=20 A=13 B=7 so B is eliminated. Round 2 votes counts: C=39 D=23 E=22 A=16 so A is eliminated. Round 3 votes counts: C=44 E=32 D=24 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:215 C:206 B:199 E:197 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 16 6 B -12 0 2 10 -2 C 4 -2 0 0 10 D -16 -10 0 0 -8 E -6 2 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839513 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 16 6 B -12 0 2 10 -2 C 4 -2 0 0 10 D -16 -10 0 0 -8 E -6 2 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839513 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 16 6 B -12 0 2 10 -2 C 4 -2 0 0 10 D -16 -10 0 0 -8 E -6 2 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839513 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3619: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) A D C B E (6) A D E B C (5) A B D E C (5) D A E B C (4) A D C E B (4) A B E D C (4) A B D C E (4) E B D C A (3) E B C D A (3) E B A D C (3) D A C E B (3) B C E A D (3) E D C B A (2) E B D A C (2) D E C A B (2) D C E A B (2) D C A E B (2) D A E C B (2) C E D B A (2) C D E B A (2) C B E D A (2) B C A E D (2) A D B E C (2) A B C D E (2) E D B C A (1) E C D B A (1) E C B D A (1) E B A C D (1) E A D B C (1) E A B D C (1) D E A B C (1) D C A B E (1) D A C B E (1) C D B E A (1) C D A B E (1) C B E A D (1) C B A E D (1) C B A D E (1) C A D B E (1) B E A D C (1) B E A C D (1) B A C D E (1) A E D B C (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 14 14 22 10 B -14 0 22 2 0 C -14 -22 0 -30 -12 D -22 -2 30 0 8 E -10 0 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 22 10 B -14 0 22 2 0 C -14 -22 0 -30 -12 D -22 -2 30 0 8 E -10 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999204 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=19 D=18 B=16 C=12 so C is eliminated. Round 2 votes counts: A=36 D=22 E=21 B=21 so E is eliminated. Round 3 votes counts: A=38 B=34 D=28 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 D:207 B:205 E:197 C:161 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 22 10 B -14 0 22 2 0 C -14 -22 0 -30 -12 D -22 -2 30 0 8 E -10 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999204 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 22 10 B -14 0 22 2 0 C -14 -22 0 -30 -12 D -22 -2 30 0 8 E -10 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999204 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 22 10 B -14 0 22 2 0 C -14 -22 0 -30 -12 D -22 -2 30 0 8 E -10 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999204 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3620: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) A E D C B (8) E D B C A (6) B D E C A (6) A C E D B (6) E D A B C (5) D E B A C (5) C B A D E (5) A E D B C (5) E D C B A (3) C A E D B (3) A D E B C (3) D E B C A (2) D B E C A (2) C E A D B (2) C B A E D (2) C A B E D (2) C A B D E (2) B D A E C (2) B C D E A (2) B C A D E (2) A C B E D (2) A C B D E (2) E D B A C (1) E D A C B (1) E A D C B (1) E A D B C (1) D E A B C (1) D B E A C (1) D A E B C (1) C E D B A (1) C B E D A (1) B D C E A (1) B D A C E (1) A C E B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 -2 -2 B 4 0 -2 -16 -12 C 2 2 0 -14 -10 D 2 16 14 0 -2 E 2 12 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 -2 -2 B 4 0 -2 -16 -12 C 2 2 0 -14 -10 D 2 16 14 0 -2 E 2 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 E=18 B=14 D=12 so D is eliminated. Round 2 votes counts: A=30 C=27 E=26 B=17 so B is eliminated. Round 3 votes counts: E=35 A=33 C=32 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:213 A:195 C:190 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 -2 B 4 0 -2 -16 -12 C 2 2 0 -14 -10 D 2 16 14 0 -2 E 2 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 -2 B 4 0 -2 -16 -12 C 2 2 0 -14 -10 D 2 16 14 0 -2 E 2 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 -2 B 4 0 -2 -16 -12 C 2 2 0 -14 -10 D 2 16 14 0 -2 E 2 12 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3621: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) B A D C E (6) E D C A B (5) D B E A C (5) A C B E D (5) C A E B D (4) C A B E D (4) A C B D E (4) E D C B A (3) E D A C B (3) E C D B A (3) E C D A B (3) D E B A C (3) D E A C B (3) C B A E D (3) B E D C A (3) A B C D E (3) E D B C A (2) D E B C A (2) D E A B C (2) D B A E C (2) C E B A D (2) C E A D B (2) A B C E D (2) E C A D B (1) E A D C B (1) D B E C A (1) D A B E C (1) C E B D A (1) C E A B D (1) C B E A D (1) C A E D B (1) B D C A E (1) B D A E C (1) B C D E A (1) B C A E D (1) B C A D E (1) A D E C B (1) A D C B E (1) A C E D B (1) A C E B D (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 8 10 4 B -2 0 -10 6 8 C -8 10 0 6 10 D -10 -6 -6 0 -6 E -4 -8 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 10 4 B -2 0 -10 6 8 C -8 10 0 6 10 D -10 -6 -6 0 -6 E -4 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=21 B=21 A=20 D=19 C=19 so D is eliminated. Round 2 votes counts: E=31 B=29 A=21 C=19 so C is eliminated. Round 3 votes counts: E=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:212 C:209 B:201 E:192 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 10 4 B -2 0 -10 6 8 C -8 10 0 6 10 D -10 -6 -6 0 -6 E -4 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 10 4 B -2 0 -10 6 8 C -8 10 0 6 10 D -10 -6 -6 0 -6 E -4 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 10 4 B -2 0 -10 6 8 C -8 10 0 6 10 D -10 -6 -6 0 -6 E -4 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3622: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) C A B E D (7) E B D C A (6) D E B A C (6) D E A B C (6) D E C B A (5) A C B D E (5) D E A C B (4) C B A E D (4) B E D A C (4) B C A E D (4) C B E A D (3) C A B D E (3) A D E B C (3) A C B E D (3) E D B A C (2) D E B C A (2) D A E B C (2) C D E A B (2) B C E A D (2) B A E D C (2) B A C E D (2) E C B D A (1) D A C E B (1) C E D B A (1) C D E B A (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B A E C D (1) A C D E B (1) A C D B E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -6 -4 -12 B 14 0 12 8 -4 C 6 -12 0 -6 -10 D 4 -8 6 0 -10 E 12 4 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -6 -4 -12 B 14 0 12 8 -4 C 6 -12 0 -6 -10 D 4 -8 6 0 -10 E 12 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 B=19 E=16 A=16 so E is eliminated. Round 2 votes counts: D=35 B=25 C=24 A=16 so A is eliminated. Round 3 votes counts: D=38 C=34 B=28 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:218 B:215 D:196 C:189 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 -12 B 14 0 12 8 -4 C 6 -12 0 -6 -10 D 4 -8 6 0 -10 E 12 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 -12 B 14 0 12 8 -4 C 6 -12 0 -6 -10 D 4 -8 6 0 -10 E 12 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 -12 B 14 0 12 8 -4 C 6 -12 0 -6 -10 D 4 -8 6 0 -10 E 12 4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3623: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) A D B C E (11) E B C D A (9) A D B E C (9) A D C B E (7) E C B D A (4) C E A D B (4) B E C D A (4) A B D E C (4) D A B C E (3) A D C E B (3) D C A B E (2) C E A B D (2) C A D E B (2) B D E A C (2) B D A E C (2) B C D E A (2) A B E D C (2) E C B A D (1) E C A B D (1) E B A C D (1) D B A E C (1) D B A C E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B A D (1) C D B E A (1) C A E D B (1) B E D A C (1) B E A D C (1) B D E C A (1) B A D E C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 10 4 6 4 B -10 0 14 2 12 C -4 -14 0 -4 8 D -6 -2 4 0 8 E -4 -12 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 6 4 B -10 0 14 2 12 C -4 -14 0 -4 8 D -6 -2 4 0 8 E -4 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=24 E=16 B=14 D=8 so D is eliminated. Round 2 votes counts: A=42 C=26 E=16 B=16 so E is eliminated. Round 3 votes counts: A=42 C=32 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:209 D:202 C:193 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 6 4 B -10 0 14 2 12 C -4 -14 0 -4 8 D -6 -2 4 0 8 E -4 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 6 4 B -10 0 14 2 12 C -4 -14 0 -4 8 D -6 -2 4 0 8 E -4 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 6 4 B -10 0 14 2 12 C -4 -14 0 -4 8 D -6 -2 4 0 8 E -4 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3624: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) A C E B D (9) D C B E A (8) C B E A D (7) D B E A C (5) E B A C D (4) C D B E A (4) B E A C D (4) E B A D C (3) D B E C A (3) D A B E C (3) C B E D A (3) C A E B D (3) C A D E B (3) B E A D C (3) A D E B C (3) D E B A C (2) D A E B C (2) B E D A C (2) B E C A D (2) A E C B D (2) A D C E B (2) E A B C D (1) D B C E A (1) D B A E C (1) D A E C B (1) D A C E B (1) C E B A D (1) C B D E A (1) C A B E D (1) B E C D A (1) B D C E A (1) A E D B C (1) A E B D C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 22 22 -12 B 12 0 4 16 0 C -22 -4 0 14 -6 D -22 -16 -14 0 -16 E 12 0 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.451032 C: 0.000000 D: 0.000000 E: 0.548968 Sum of squares = 0.504795638425 Cumulative probabilities = A: 0.000000 B: 0.451032 C: 0.451032 D: 0.451032 E: 1.000000 A B C D E A 0 -12 22 22 -12 B 12 0 4 16 0 C -22 -4 0 14 -6 D -22 -16 -14 0 -16 E 12 0 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999804 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=27 C=23 B=13 E=8 so E is eliminated. Round 2 votes counts: A=30 D=27 C=23 B=20 so B is eliminated. Round 3 votes counts: A=44 D=30 C=26 so C is eliminated. Round 4 votes counts: A=61 D=39 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:217 B:216 A:210 C:191 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 22 22 -12 B 12 0 4 16 0 C -22 -4 0 14 -6 D -22 -16 -14 0 -16 E 12 0 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999804 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 22 22 -12 B 12 0 4 16 0 C -22 -4 0 14 -6 D -22 -16 -14 0 -16 E 12 0 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999804 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 22 22 -12 B 12 0 4 16 0 C -22 -4 0 14 -6 D -22 -16 -14 0 -16 E 12 0 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999804 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3625: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) E D A B C (8) C B A D E (8) A B C E D (7) D E C A B (5) D C E A B (5) C A B D E (5) B A C E D (5) E B A D C (4) D C E B A (4) B C A D E (4) E D B A C (3) E C A D B (3) E D A C B (2) E A B C D (2) D E B C A (2) D E B A C (2) D C B A E (2) C D A B E (2) B A E C D (2) A B E C D (2) E B D A C (1) E A D C B (1) E A B D C (1) D C B E A (1) D B E A C (1) C D E A B (1) C A E B D (1) C A B E D (1) B D C A E (1) B C D A E (1) B C A E D (1) B A E D C (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 -12 0 -10 B 8 0 0 -2 -8 C 12 0 0 -8 2 D 0 2 8 0 10 E 10 8 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.148978 B: 0.000000 C: 0.000000 D: 0.851022 E: 0.000000 Sum of squares = 0.746432773612 Cumulative probabilities = A: 0.148978 B: 0.148978 C: 0.148978 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 0 -10 B 8 0 0 -2 -8 C 12 0 0 -8 2 D 0 2 8 0 10 E 10 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000005409 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=25 C=18 B=17 A=9 so A is eliminated. Round 2 votes counts: D=31 B=26 E=25 C=18 so C is eliminated. Round 3 votes counts: B=40 D=34 E=26 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:203 E:203 B:199 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 0 -10 B 8 0 0 -2 -8 C 12 0 0 -8 2 D 0 2 8 0 10 E 10 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000005409 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 0 -10 B 8 0 0 -2 -8 C 12 0 0 -8 2 D 0 2 8 0 10 E 10 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000005409 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 0 -10 B 8 0 0 -2 -8 C 12 0 0 -8 2 D 0 2 8 0 10 E 10 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000005409 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3626: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (21) E C D A B (17) C D A B E (7) B A D E C (7) B A E D C (5) E B C A D (4) C D A E B (4) E D C A B (3) E C D B A (3) D A C B E (3) B E A D C (3) A D B C E (3) E D A C B (2) D C A E B (2) C D E A B (2) A B D C E (2) E D A B C (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) D A C E B (1) D A B E C (1) D A B C E (1) C A D B E (1) B E A C D (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 2 8 -2 20 B -2 0 6 -4 14 C -8 -6 0 -16 -2 D 2 4 16 0 14 E -20 -14 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -2 20 B -2 0 6 -4 14 C -8 -6 0 -16 -2 D 2 4 16 0 14 E -20 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=34 C=14 D=8 A=6 so A is eliminated. Round 2 votes counts: B=40 E=34 C=14 D=12 so D is eliminated. Round 3 votes counts: B=45 E=34 C=21 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:218 A:214 B:207 C:184 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -2 20 B -2 0 6 -4 14 C -8 -6 0 -16 -2 D 2 4 16 0 14 E -20 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -2 20 B -2 0 6 -4 14 C -8 -6 0 -16 -2 D 2 4 16 0 14 E -20 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -2 20 B -2 0 6 -4 14 C -8 -6 0 -16 -2 D 2 4 16 0 14 E -20 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3627: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (12) D B E C A (7) D B E A C (7) A C E D B (7) A C D B E (7) E B D C A (6) E B C D A (6) E D B C A (5) E C A B D (5) D B A E C (5) D B A C E (5) E C B A D (3) B E D C A (3) A D C B E (3) A C E B D (3) D E B C A (2) D A B C E (2) C E A B D (2) B D E C A (2) A C B D E (2) E D A C B (1) C E B A D (1) C A B E D (1) B E C D A (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -12 -4 -2 B 6 0 2 -6 -10 C 12 -2 0 4 -8 D 4 6 -4 0 -14 E 2 10 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -4 -2 B 6 0 2 -6 -10 C 12 -2 0 4 -8 D 4 6 -4 0 -14 E 2 10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 A=24 C=16 B=6 so B is eliminated. Round 2 votes counts: E=30 D=30 A=24 C=16 so C is eliminated. Round 3 votes counts: A=37 E=33 D=30 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:203 B:196 D:196 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -4 -2 B 6 0 2 -6 -10 C 12 -2 0 4 -8 D 4 6 -4 0 -14 E 2 10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -4 -2 B 6 0 2 -6 -10 C 12 -2 0 4 -8 D 4 6 -4 0 -14 E 2 10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -4 -2 B 6 0 2 -6 -10 C 12 -2 0 4 -8 D 4 6 -4 0 -14 E 2 10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3628: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) A C D E B (12) A E C D B (11) D C B A E (8) E B A D C (7) E B A C D (6) B E D C A (5) E A B D C (4) E A B C D (4) A E D C B (4) C D A B E (3) C A D E B (2) B E D A C (2) B E A C D (2) A D E C B (2) E A D C B (1) E A C D B (1) E A C B D (1) D C A E B (1) D C A B E (1) D B C E A (1) D B C A E (1) C D B A E (1) C B D A E (1) B E C D A (1) B E A D C (1) B D E C A (1) B D C A E (1) B C E D A (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 20 20 0 B 2 0 -2 -2 -16 C -20 2 0 -6 -6 D -20 2 6 0 -4 E 0 16 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.401833 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.598167 Sum of squares = 0.519273424912 Cumulative probabilities = A: 0.401833 B: 0.401833 C: 0.401833 D: 0.401833 E: 1.000000 A B C D E A 0 -2 20 20 0 B 2 0 -2 -2 -16 C -20 2 0 -6 -6 D -20 2 6 0 -4 E 0 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=26 E=24 D=12 C=7 so C is eliminated. Round 2 votes counts: A=33 B=27 E=24 D=16 so D is eliminated. Round 3 votes counts: B=38 A=38 E=24 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:219 E:213 D:192 B:191 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 20 20 0 B 2 0 -2 -2 -16 C -20 2 0 -6 -6 D -20 2 6 0 -4 E 0 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 20 20 0 B 2 0 -2 -2 -16 C -20 2 0 -6 -6 D -20 2 6 0 -4 E 0 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 20 20 0 B 2 0 -2 -2 -16 C -20 2 0 -6 -6 D -20 2 6 0 -4 E 0 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3629: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (19) D B A E C (13) D B A C E (7) E A B D C (5) C A E B D (5) E D B A C (4) D B E A C (4) C E D A B (4) E A B C D (3) D C B E A (3) C D E B A (3) E C A B D (2) E B D A C (2) E B A D C (2) E A C B D (2) D C B A E (2) C A B E D (2) C A B D E (2) B E A D C (2) B D A E C (2) A B D E C (2) E C D B A (1) E C B A D (1) D E B A C (1) D C A B E (1) C D B A E (1) C D A B E (1) C A D E B (1) C A D B E (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 2 2 2 -16 B -2 0 -2 6 -12 C -2 2 0 -4 4 D -2 -6 4 0 -12 E 16 12 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 2 2 2 -16 B -2 0 -2 6 -12 C -2 2 0 -4 4 D -2 -6 4 0 -12 E 16 12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=31 E=22 B=5 A=3 so A is eliminated. Round 2 votes counts: C=39 D=31 E=23 B=7 so B is eliminated. Round 3 votes counts: C=39 D=35 E=26 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:218 C:200 A:195 B:195 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 2 2 -16 B -2 0 -2 6 -12 C -2 2 0 -4 4 D -2 -6 4 0 -12 E 16 12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 -16 B -2 0 -2 6 -12 C -2 2 0 -4 4 D -2 -6 4 0 -12 E 16 12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 -16 B -2 0 -2 6 -12 C -2 2 0 -4 4 D -2 -6 4 0 -12 E 16 12 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3630: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) C D B E A (6) C A D B E (6) A E D B C (6) C B A E D (5) B E A D C (5) D C E A B (4) C D A E B (4) C B D E A (4) D E B C A (3) D E A C B (3) C D E B A (3) A B C E D (3) E D B A C (2) E D A B C (2) D E C B A (2) D E A B C (2) D C E B A (2) C B A D E (2) B E D A C (2) B E A C D (2) B A E C D (2) A C B E D (2) A B E D C (2) E D B C A (1) E B D A C (1) E B A D C (1) E A D B C (1) D E C A B (1) D C A E B (1) D B E C A (1) D A E C B (1) C D A B E (1) C A B E D (1) C A B D E (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E D C (1) A E D C B (1) A D C E B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -2 6 0 B -4 0 -4 -10 0 C 2 4 0 -6 0 D -6 10 6 0 -4 E 0 0 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.282733 D: 0.000000 E: 0.717267 Sum of squares = 0.594410276952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.282733 D: 0.282733 E: 1.000000 A B C D E A 0 4 -2 6 0 B -4 0 -4 -10 0 C 2 4 0 -6 0 D -6 10 6 0 -4 E 0 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000090619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=24 D=20 B=15 E=8 so E is eliminated. Round 2 votes counts: C=33 D=25 A=25 B=17 so B is eliminated. Round 3 votes counts: C=36 A=36 D=28 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:204 D:203 E:202 C:200 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 6 0 B -4 0 -4 -10 0 C 2 4 0 -6 0 D -6 10 6 0 -4 E 0 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000090619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 0 B -4 0 -4 -10 0 C 2 4 0 -6 0 D -6 10 6 0 -4 E 0 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000090619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 0 B -4 0 -4 -10 0 C 2 4 0 -6 0 D -6 10 6 0 -4 E 0 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000090619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3631: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) D A E C B (10) B C E A D (9) B D A C E (6) E C A B D (5) D B A E C (5) C E A B D (5) D E C A B (4) C E B A D (4) D B E C A (3) B D C E A (3) B C E D A (3) B C A E D (3) B A C E D (3) A E C D B (3) D B A C E (2) C A E B D (2) B D C A E (2) A D B C E (2) E C D B A (1) E C D A B (1) E C B D A (1) D E C B A (1) D A E B C (1) D A B C E (1) C B E A D (1) B D E C A (1) B C D A E (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E C B (1) A C E D B (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -20 8 -4 B 2 0 -4 6 -4 C 20 4 0 10 4 D -8 -6 -10 0 -8 E 4 4 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -20 8 -4 B 2 0 -4 6 -4 C 20 4 0 10 4 D -8 -6 -10 0 -8 E 4 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=27 E=18 C=12 A=10 so A is eliminated. Round 2 votes counts: B=34 D=30 E=22 C=14 so C is eliminated. Round 3 votes counts: E=35 B=35 D=30 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:206 B:200 A:191 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -20 8 -4 B 2 0 -4 6 -4 C 20 4 0 10 4 D -8 -6 -10 0 -8 E 4 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -20 8 -4 B 2 0 -4 6 -4 C 20 4 0 10 4 D -8 -6 -10 0 -8 E 4 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -20 8 -4 B 2 0 -4 6 -4 C 20 4 0 10 4 D -8 -6 -10 0 -8 E 4 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3632: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (5) A B E D C (5) E D B C A (4) C D E B A (4) C D E A B (4) B D E C A (4) B D C E A (4) B A E D C (4) A C B D E (4) C A D E B (3) B C D E A (3) B A D E C (3) B A C D E (3) A C E D B (3) A B C E D (3) A B C D E (3) E B D A C (2) D E C B A (2) C D B E A (2) C B D A E (2) B E D C A (2) B E D A C (2) B C D A E (2) A E C D B (2) A C D E B (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A C B (1) E B A D C (1) E A D C B (1) E A D B C (1) D E C A B (1) D E B C A (1) D C E B A (1) D B C E A (1) C E D A B (1) C D A E B (1) C B D E A (1) C B A D E (1) C A E D B (1) C A D B E (1) B D A E C (1) B C A D E (1) B A D C E (1) B A C E D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -10 2 2 8 B 10 0 10 8 8 C -2 -10 0 -4 6 D -2 -8 4 0 12 E -8 -8 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 2 8 B 10 0 10 8 8 C -2 -10 0 -4 6 D -2 -8 4 0 12 E -8 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=29 C=21 E=13 D=6 so D is eliminated. Round 2 votes counts: B=32 A=29 C=22 E=17 so E is eliminated. Round 3 votes counts: B=41 A=32 C=27 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:203 A:201 C:195 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 2 8 B 10 0 10 8 8 C -2 -10 0 -4 6 D -2 -8 4 0 12 E -8 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 2 8 B 10 0 10 8 8 C -2 -10 0 -4 6 D -2 -8 4 0 12 E -8 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 2 8 B 10 0 10 8 8 C -2 -10 0 -4 6 D -2 -8 4 0 12 E -8 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3633: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) A D C B E (10) D C A B E (9) A B E D C (7) C D E B A (6) B E A D C (6) E B C A D (4) C E D B A (4) A C D B E (4) E B C D A (3) D A C B E (3) A D B E C (3) A B D E C (3) E C B D A (2) E C B A D (2) D C E B A (2) C D E A B (2) C D A E B (2) C A D E B (2) A C D E B (2) E C D B A (1) E A C B D (1) D C A E B (1) D B E C A (1) D B A E C (1) C D A B E (1) C A E D B (1) B E A C D (1) B D A E C (1) B A E D C (1) A E C B D (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 8 14 22 8 B -8 0 -14 -12 4 C -14 14 0 2 2 D -22 12 -2 0 8 E -8 -4 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 22 8 B -8 0 -14 -12 4 C -14 14 0 2 2 D -22 12 -2 0 8 E -8 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=24 C=18 D=17 B=9 so B is eliminated. Round 2 votes counts: A=33 E=31 D=18 C=18 so D is eliminated. Round 3 votes counts: A=38 E=32 C=30 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 C:202 D:198 E:189 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 22 8 B -8 0 -14 -12 4 C -14 14 0 2 2 D -22 12 -2 0 8 E -8 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 22 8 B -8 0 -14 -12 4 C -14 14 0 2 2 D -22 12 -2 0 8 E -8 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 22 8 B -8 0 -14 -12 4 C -14 14 0 2 2 D -22 12 -2 0 8 E -8 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3634: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) C E B D A (6) C B D E A (6) A E C B D (6) E B D A C (5) E B C D A (5) B D E C A (5) E A C B D (4) D B C A E (4) D B A C E (4) C B E D A (4) E C B D A (3) E B D C A (3) D B E A C (3) D B C E A (3) A E D B C (3) A E C D B (3) A C E D B (3) A C D B E (3) E C A B D (2) D B A E C (2) C A E B D (2) B E D C A (2) A D B E C (2) D C B A E (1) D A B E C (1) D A B C E (1) C E B A D (1) C D B A E (1) C D A B E (1) B D C E A (1) A E B D C (1) A D C B E (1) Total count = 100 A B C D E A 0 -18 0 -22 -6 B 18 0 6 12 6 C 0 -6 0 0 0 D 22 -12 0 0 -6 E 6 -6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 0 -22 -6 B 18 0 6 12 6 C 0 -6 0 0 0 D 22 -12 0 0 -6 E 6 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995335 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=22 C=21 D=19 B=8 so B is eliminated. Round 2 votes counts: A=30 D=25 E=24 C=21 so C is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:221 E:203 D:202 C:197 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 0 -22 -6 B 18 0 6 12 6 C 0 -6 0 0 0 D 22 -12 0 0 -6 E 6 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995335 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -22 -6 B 18 0 6 12 6 C 0 -6 0 0 0 D 22 -12 0 0 -6 E 6 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995335 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -22 -6 B 18 0 6 12 6 C 0 -6 0 0 0 D 22 -12 0 0 -6 E 6 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995335 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3635: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) A E D C B (10) B C D E A (8) B C E D A (7) E C A B D (6) D B C A E (6) D B A C E (6) B C E A D (6) D A E B C (5) A D E C B (5) D A E C B (4) B D C A E (4) C E B A D (3) E C B A D (2) E A C B D (2) D A B E C (2) D A B C E (2) C B D A E (2) A E C D B (2) E A C D B (1) D E A B C (1) D B E A C (1) C B D E A (1) C B A E D (1) B D C E A (1) B C D A E (1) Total count = 100 A B C D E A 0 -20 -18 -2 0 B 20 0 0 10 18 C 18 0 0 6 18 D 2 -10 -6 0 -2 E 0 -18 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.471432 C: 0.528568 D: 0.000000 E: 0.000000 Sum of squares = 0.501632234696 Cumulative probabilities = A: 0.000000 B: 0.471432 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -18 -2 0 B 20 0 0 10 18 C 18 0 0 6 18 D 2 -10 -6 0 -2 E 0 -18 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 C=18 A=17 E=11 so E is eliminated. Round 2 votes counts: D=27 B=27 C=26 A=20 so A is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:221 D:192 E:183 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -18 -2 0 B 20 0 0 10 18 C 18 0 0 6 18 D 2 -10 -6 0 -2 E 0 -18 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 -2 0 B 20 0 0 10 18 C 18 0 0 6 18 D 2 -10 -6 0 -2 E 0 -18 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 -2 0 B 20 0 0 10 18 C 18 0 0 6 18 D 2 -10 -6 0 -2 E 0 -18 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3636: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) E A D C B (7) C E B A D (7) E C B D A (4) C B E D A (4) B D C A E (4) B C D A E (4) A D B E C (4) E D A B C (3) E A D B C (3) E A C D B (3) D B A E C (3) D B A C E (3) D A B E C (3) C E A B D (3) B D C E A (3) B D A E C (3) B D A C E (3) A D E B C (3) A D B C E (3) E C B A D (2) D B E A C (2) A E D C B (2) A D C B E (2) E C A B D (1) D E B A C (1) D A B C E (1) C E B D A (1) C B D E A (1) C B D A E (1) C B A D E (1) C A E B D (1) B D E C A (1) B D E A C (1) B C E D A (1) A E D B C (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 6 6 -12 B 0 0 0 -8 -4 C -6 0 0 -14 -12 D -6 8 14 0 -2 E 12 4 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 6 6 -12 B 0 0 0 -8 -4 C -6 0 0 -14 -12 D -6 8 14 0 -2 E 12 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=20 C=19 A=17 D=13 so D is eliminated. Round 2 votes counts: E=32 B=28 A=21 C=19 so C is eliminated. Round 3 votes counts: E=43 B=35 A=22 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:207 A:200 B:194 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 6 -12 B 0 0 0 -8 -4 C -6 0 0 -14 -12 D -6 8 14 0 -2 E 12 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 -12 B 0 0 0 -8 -4 C -6 0 0 -14 -12 D -6 8 14 0 -2 E 12 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 -12 B 0 0 0 -8 -4 C -6 0 0 -14 -12 D -6 8 14 0 -2 E 12 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3637: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (12) A C D E B (12) B E D C A (9) B E D A C (9) A C E B D (7) D C A B E (5) E B C A D (3) E B A C D (3) E A B C D (3) D B E A C (3) D B C E A (3) C A E B D (3) C A D E B (3) B D E C A (3) E C B A D (2) D C B A E (2) D B C A E (2) A E C B D (2) A E B C D (2) A C E D B (2) E B A D C (1) E A C B D (1) D C B E A (1) D A C B E (1) D A B E C (1) D A B C E (1) C E A B D (1) C D A B E (1) B E C D A (1) A E B D C (1) Total count = 100 A B C D E A 0 -8 -2 -8 -10 B 8 0 14 2 8 C 2 -14 0 -8 -12 D 8 -2 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -8 -10 B 8 0 14 2 8 C 2 -14 0 -8 -12 D 8 -2 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=26 B=22 E=13 C=8 so C is eliminated. Round 2 votes counts: D=32 A=32 B=22 E=14 so E is eliminated. Round 3 votes counts: A=37 D=32 B=31 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:207 E:207 A:186 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -8 -10 B 8 0 14 2 8 C 2 -14 0 -8 -12 D 8 -2 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -8 -10 B 8 0 14 2 8 C 2 -14 0 -8 -12 D 8 -2 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -8 -10 B 8 0 14 2 8 C 2 -14 0 -8 -12 D 8 -2 8 0 0 E 10 -8 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3638: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) C E D A B (6) D A B C E (5) C E B A D (5) E A B D C (4) D C A B E (4) C D A E B (4) B E C A D (4) B D A E C (4) E C A B D (3) B A E D C (3) A E D B C (3) E B A D C (2) E B A C D (2) D C A E B (2) D A E B C (2) C E A D B (2) C B E D A (2) C B E A D (2) B D C A E (2) B D A C E (2) B A D E C (2) A E D C B (2) E C B A D (1) E C A D B (1) E B C A D (1) E A D B C (1) E A C B D (1) D B C A E (1) D B A E C (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B E C (1) C E A B D (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D E A (1) C B D A E (1) B E A C D (1) B C E A D (1) A E B D C (1) A D E C B (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 4 10 -2 B -4 0 8 10 2 C -4 -8 0 -14 -6 D -10 -10 14 0 -14 E 2 -2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 4 10 -2 B -4 0 8 10 2 C -4 -8 0 -14 -6 D -10 -10 14 0 -14 E 2 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=27 D=19 E=16 A=10 so A is eliminated. Round 2 votes counts: C=28 B=28 E=22 D=22 so E is eliminated. Round 3 votes counts: B=38 C=34 D=28 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:210 A:208 B:208 D:190 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 10 -2 B -4 0 8 10 2 C -4 -8 0 -14 -6 D -10 -10 14 0 -14 E 2 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 10 -2 B -4 0 8 10 2 C -4 -8 0 -14 -6 D -10 -10 14 0 -14 E 2 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 10 -2 B -4 0 8 10 2 C -4 -8 0 -14 -6 D -10 -10 14 0 -14 E 2 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3639: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) D E B A C (7) C A E B D (7) C A B E D (7) A C B D E (7) B C A D E (6) E D C A B (5) E D B A C (4) D B E A C (4) B A C D E (4) E C A D B (3) C A B D E (3) B D E C A (3) B C A E D (3) A B C D E (3) E D A C B (2) D E A C B (2) D B E C A (2) C E B A D (2) B E D C A (2) B D A C E (2) B C E A D (2) A C D E B (2) E C B A D (1) E C A B D (1) D E A B C (1) D B A C E (1) D A B C E (1) C B A E D (1) C B A D E (1) B A D C E (1) A C E D B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -12 14 4 B 6 0 6 14 10 C 12 -6 0 12 12 D -14 -14 -12 0 2 E -4 -10 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 14 4 B 6 0 6 14 10 C 12 -6 0 12 12 D -14 -14 -12 0 2 E -4 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=23 B=23 C=21 D=18 A=15 so A is eliminated. Round 2 votes counts: C=33 B=26 E=23 D=18 so D is eliminated. Round 3 votes counts: B=34 E=33 C=33 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:215 A:200 E:186 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -12 14 4 B 6 0 6 14 10 C 12 -6 0 12 12 D -14 -14 -12 0 2 E -4 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 14 4 B 6 0 6 14 10 C 12 -6 0 12 12 D -14 -14 -12 0 2 E -4 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 14 4 B 6 0 6 14 10 C 12 -6 0 12 12 D -14 -14 -12 0 2 E -4 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3640: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) E D A B C (8) C B A E D (6) C A D E B (6) B E A C D (6) D E A B C (5) B C E D A (5) A D E C B (5) D E A C B (4) C B A D E (4) E B D A C (3) C D A E B (3) B E D A C (3) B E C D A (3) B C E A D (3) B C A E D (3) E A D B C (2) B E D C A (2) B E C A D (2) B E A D C (2) E D B A C (1) E B A D C (1) D C A E B (1) D A C E B (1) C D A B E (1) C B E A D (1) C A D B E (1) C A B D E (1) B C D E A (1) A E D C B (1) A D C E B (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 8 14 -4 -4 B -8 0 2 -6 -10 C -14 -2 0 -2 -20 D 4 6 2 0 -8 E 4 10 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 14 -4 -4 B -8 0 2 -6 -10 C -14 -2 0 -2 -20 D 4 6 2 0 -8 E 4 10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=23 D=22 E=15 A=10 so A is eliminated. Round 2 votes counts: B=31 D=28 C=25 E=16 so E is eliminated. Round 3 votes counts: D=40 B=35 C=25 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:221 A:207 D:202 B:189 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 14 -4 -4 B -8 0 2 -6 -10 C -14 -2 0 -2 -20 D 4 6 2 0 -8 E 4 10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 -4 -4 B -8 0 2 -6 -10 C -14 -2 0 -2 -20 D 4 6 2 0 -8 E 4 10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 -4 -4 B -8 0 2 -6 -10 C -14 -2 0 -2 -20 D 4 6 2 0 -8 E 4 10 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3641: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) E A D B C (6) A B D E C (5) E A C D B (4) D B C A E (4) C B D A E (4) A E D B C (4) A E B D C (4) E D A B C (3) E C D B A (3) D C E B A (3) D B A C E (3) D A E B C (3) C E D B A (3) C D E B A (3) C B D E A (3) E D C A B (2) E C A B D (2) E A D C B (2) D C B E A (2) D C B A E (2) C D B E A (2) B D C A E (2) B D A C E (2) B C D A E (2) B C A D E (2) B A C D E (2) A B E D C (2) A B E C D (2) E A B D C (1) D E C B A (1) D B C E A (1) C E B A D (1) C B E A D (1) C B A D E (1) B C A E D (1) B A D C E (1) B A C E D (1) A E C B D (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 0 10 4 0 B 0 0 6 -4 -10 C -10 -6 0 -8 -8 D -4 4 8 0 -2 E 0 10 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.503875 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.496125 Sum of squares = 0.500030032719 Cumulative probabilities = A: 0.503875 B: 0.503875 C: 0.503875 D: 0.503875 E: 1.000000 A B C D E A 0 0 10 4 0 B 0 0 6 -4 -10 C -10 -6 0 -8 -8 D -4 4 8 0 -2 E 0 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=20 D=19 C=18 B=13 so B is eliminated. Round 2 votes counts: E=30 A=24 D=23 C=23 so D is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:207 D:203 B:196 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 4 0 B 0 0 6 -4 -10 C -10 -6 0 -8 -8 D -4 4 8 0 -2 E 0 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 4 0 B 0 0 6 -4 -10 C -10 -6 0 -8 -8 D -4 4 8 0 -2 E 0 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 4 0 B 0 0 6 -4 -10 C -10 -6 0 -8 -8 D -4 4 8 0 -2 E 0 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3642: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) D A E C B (6) C E B D A (6) C B E D A (5) E C D A B (4) D E C A B (4) D E A C B (4) D A B E C (4) B C E D A (4) A D E C B (4) E D A C B (3) E C D B A (3) C E D B A (3) B C A E D (3) B A D C E (3) A D B E C (3) A B D E C (3) E D C A B (2) E C B A D (2) E C A D B (2) D A E B C (2) B D C A E (2) B C A D E (2) B A C E D (2) B A C D E (2) A D E B C (2) A B D C E (2) E C B D A (1) E A D C B (1) D C E B A (1) C E B A D (1) B D A C E (1) B C D E A (1) B C D A E (1) A E D C B (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -12 -14 -12 B 4 0 -6 0 -4 C 12 6 0 2 -2 D 14 0 -2 0 -4 E 12 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -12 -14 -12 B 4 0 -6 0 -4 C 12 6 0 2 -2 D 14 0 -2 0 -4 E 12 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=21 E=18 A=16 C=15 so C is eliminated. Round 2 votes counts: B=35 E=28 D=21 A=16 so A is eliminated. Round 3 votes counts: B=40 D=31 E=29 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:211 C:209 D:204 B:197 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -12 -14 -12 B 4 0 -6 0 -4 C 12 6 0 2 -2 D 14 0 -2 0 -4 E 12 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -14 -12 B 4 0 -6 0 -4 C 12 6 0 2 -2 D 14 0 -2 0 -4 E 12 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -14 -12 B 4 0 -6 0 -4 C 12 6 0 2 -2 D 14 0 -2 0 -4 E 12 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3643: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) C B D A E (6) D A C B E (5) E D C B A (4) E B C A D (4) E A D B C (4) D A E C B (4) B C E A D (4) A E D B C (4) A D B C E (4) A B C D E (4) E C D B A (3) E A B C D (3) B C A D E (3) A D E C B (3) A D E B C (3) E D A C B (2) E B A C D (2) D C B A E (2) C E B D A (2) C D B A E (2) C B E D A (2) C B D E A (2) A E B D C (2) A E B C D (2) E D C A B (1) E B C D A (1) E A D C B (1) D E C A B (1) D E A C B (1) D C A E B (1) D C A B E (1) B C D A E (1) B C A E D (1) B A C D E (1) A D C B E (1) A D B E C (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 0 8 B -2 0 -6 4 -14 C 0 6 0 8 -12 D 0 -4 -8 0 -6 E -8 14 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.707026 B: 0.000000 C: 0.292974 D: 0.000000 E: 0.000000 Sum of squares = 0.585719238598 Cumulative probabilities = A: 0.707026 B: 0.707026 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 8 B -2 0 -6 4 -14 C 0 6 0 8 -12 D 0 -4 -8 0 -6 E -8 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000429867 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=27 D=15 C=14 B=10 so B is eliminated. Round 2 votes counts: E=34 A=28 C=23 D=15 so D is eliminated. Round 3 votes counts: A=37 E=36 C=27 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:212 A:205 C:201 B:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 0 8 B -2 0 -6 4 -14 C 0 6 0 8 -12 D 0 -4 -8 0 -6 E -8 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000429867 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 8 B -2 0 -6 4 -14 C 0 6 0 8 -12 D 0 -4 -8 0 -6 E -8 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000429867 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 8 B -2 0 -6 4 -14 C 0 6 0 8 -12 D 0 -4 -8 0 -6 E -8 14 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.399999 D: 0.000000 E: 0.000000 Sum of squares = 0.520000429867 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3644: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) D A C E B (5) D A C B E (5) C E B D A (4) B A E D C (4) B A D C E (4) A D B E C (4) E C B D A (3) E C B A D (3) E B A C D (3) E A D B C (3) D A E C B (3) B E C A D (3) A D E B C (3) A B D E C (3) E C D B A (2) E B C A D (2) D C A E B (2) C D E B A (2) C D E A B (2) C D A E B (2) C B E D A (2) C B D E A (2) B E A C D (2) B C D A E (2) B C A E D (2) E D C A B (1) E C D A B (1) E A B D C (1) D C E A B (1) D C A B E (1) D A B C E (1) C E D A B (1) C D B A E (1) C D A B E (1) C B D A E (1) B D A C E (1) B C E D A (1) B C E A D (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C D E (1) A E D B C (1) A E B D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 12 6 20 B -2 0 10 2 8 C -12 -10 0 -8 8 D -6 -2 8 0 14 E -20 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999328 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 6 20 B -2 0 10 2 8 C -12 -10 0 -8 8 D -6 -2 8 0 14 E -20 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969233 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=21 E=19 D=18 C=18 so D is eliminated. Round 2 votes counts: A=35 B=24 C=22 E=19 so E is eliminated. Round 3 votes counts: A=39 C=32 B=29 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 B:209 D:207 C:189 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 6 20 B -2 0 10 2 8 C -12 -10 0 -8 8 D -6 -2 8 0 14 E -20 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969233 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 6 20 B -2 0 10 2 8 C -12 -10 0 -8 8 D -6 -2 8 0 14 E -20 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969233 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 6 20 B -2 0 10 2 8 C -12 -10 0 -8 8 D -6 -2 8 0 14 E -20 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999969233 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3645: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C B E D A (8) A D E C B (7) A D E B C (7) E D A C B (6) E A D B C (6) E D C A B (4) C D E A B (4) B C E D A (4) B C A D E (4) A D B E C (4) C E B D A (3) B C E A D (3) B A C D E (3) E C B D A (2) C E D A B (2) C B D E A (2) C B D A E (2) C B A D E (2) B A E D C (2) B A D E C (2) A B D E C (2) E D A B C (1) E B D A C (1) E B A D C (1) C E D B A (1) C D A E B (1) C D A B E (1) C A D B E (1) B E C D A (1) B E C A D (1) B E A D C (1) B A D C E (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 12 8 -2 -2 B -12 0 -10 -10 -12 C -8 10 0 -10 -14 D 2 10 10 0 4 E 2 12 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -2 -2 B -12 0 -10 -10 -12 C -8 10 0 -10 -14 D 2 10 10 0 4 E 2 12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=22 A=22 E=21 D=8 so D is eliminated. Round 2 votes counts: A=30 C=27 B=22 E=21 so E is eliminated. Round 3 votes counts: A=43 C=33 B=24 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:213 E:212 A:208 C:189 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -2 -2 B -12 0 -10 -10 -12 C -8 10 0 -10 -14 D 2 10 10 0 4 E 2 12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -2 -2 B -12 0 -10 -10 -12 C -8 10 0 -10 -14 D 2 10 10 0 4 E 2 12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -2 -2 B -12 0 -10 -10 -12 C -8 10 0 -10 -14 D 2 10 10 0 4 E 2 12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3646: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) B D A C E (8) E C D A B (7) B A C D E (6) E C A D B (5) B A D C E (5) D E C A B (4) C A E B D (4) A C D B E (4) B E D A C (3) B D E A C (3) B A C E D (3) A C B D E (3) A B C D E (3) E D B C A (2) E C B A D (2) D E C B A (2) D E B C A (2) D C A E B (2) C A D B E (2) B D A E C (2) A C B E D (2) E D C B A (1) E B D C A (1) E B D A C (1) D E B A C (1) D C E A B (1) D B E A C (1) D B A C E (1) C E A D B (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A C D (1) A D C B E (1) Total count = 100 A B C D E A 0 10 -4 -12 -2 B -10 0 -14 -6 -4 C 4 14 0 -8 -2 D 12 6 8 0 6 E 2 4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -12 -2 B -10 0 -14 -6 -4 C 4 14 0 -8 -2 D 12 6 8 0 6 E 2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=31 D=14 A=13 C=10 so C is eliminated. Round 2 votes counts: E=33 B=31 A=21 D=15 so D is eliminated. Round 3 votes counts: E=43 B=33 A=24 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:216 C:204 E:201 A:196 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 -12 -2 B -10 0 -14 -6 -4 C 4 14 0 -8 -2 D 12 6 8 0 6 E 2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -12 -2 B -10 0 -14 -6 -4 C 4 14 0 -8 -2 D 12 6 8 0 6 E 2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -12 -2 B -10 0 -14 -6 -4 C 4 14 0 -8 -2 D 12 6 8 0 6 E 2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3647: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D C A E (8) D B E C A (6) D B E A C (6) C B A D E (6) B C D A E (6) A E C D B (6) D B C E A (5) C A E B D (5) A E C B D (5) E A D B C (4) A C E B D (4) E A D C B (3) D E B A C (3) C B A E D (3) C A B E D (3) E D A B C (2) C B D A E (2) B D E C A (2) B D E A C (2) B D C E A (2) D E C A B (1) D E A C B (1) D E A B C (1) D C B E A (1) C D B A E (1) B D A C E (1) B A E D C (1) B A C D E (1) Total count = 100 A B C D E A 0 -12 -2 0 4 B 12 0 0 2 12 C 2 0 0 2 -4 D 0 -2 -2 0 10 E -4 -12 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.455243 C: 0.544757 D: 0.000000 E: 0.000000 Sum of squares = 0.504006446297 Cumulative probabilities = A: 0.000000 B: 0.455243 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 0 4 B 12 0 0 2 12 C 2 0 0 2 -4 D 0 -2 -2 0 10 E -4 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=23 C=20 E=18 A=15 so A is eliminated. Round 2 votes counts: E=29 D=24 C=24 B=23 so B is eliminated. Round 3 votes counts: D=39 C=31 E=30 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:213 D:203 C:200 A:195 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 0 4 B 12 0 0 2 12 C 2 0 0 2 -4 D 0 -2 -2 0 10 E -4 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 0 4 B 12 0 0 2 12 C 2 0 0 2 -4 D 0 -2 -2 0 10 E -4 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 0 4 B 12 0 0 2 12 C 2 0 0 2 -4 D 0 -2 -2 0 10 E -4 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3648: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (12) C B E A D (8) D A E B C (7) C E B A D (7) E C B D A (6) D A B C E (6) E D C B A (5) A B C D E (5) E C B A D (4) C B A E D (4) A C B D E (4) A B D C E (4) E D C A B (3) D E A C B (3) D E A B C (3) D A B E C (3) B C E A D (3) B C A E D (3) E D B C A (2) D A E C B (2) B E C A D (2) E C D B A (1) E C D A B (1) D A C E B (1) B A C D E (1) Total count = 100 A B C D E A 0 8 2 14 4 B -8 0 2 2 10 C -2 -2 0 -2 16 D -14 -2 2 0 2 E -4 -10 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 14 4 B -8 0 2 2 10 C -2 -2 0 -2 16 D -14 -2 2 0 2 E -4 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 E=22 C=19 B=9 so B is eliminated. Round 2 votes counts: A=26 D=25 C=25 E=24 so E is eliminated. Round 3 votes counts: C=39 D=35 A=26 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:214 C:205 B:203 D:194 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 14 4 B -8 0 2 2 10 C -2 -2 0 -2 16 D -14 -2 2 0 2 E -4 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 14 4 B -8 0 2 2 10 C -2 -2 0 -2 16 D -14 -2 2 0 2 E -4 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 14 4 B -8 0 2 2 10 C -2 -2 0 -2 16 D -14 -2 2 0 2 E -4 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3649: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (11) E C D B A (8) E D C A B (7) A B D C E (7) C D E B A (4) A E D C B (4) A D C E B (4) A D C B E (4) A B E D C (4) E D A C B (3) D C E A B (3) B C E D A (3) B A E C D (3) A E B D C (3) E C B D A (2) E B C D A (2) E A D C B (2) D C A E B (2) B C D E A (2) B C A E D (2) E D C B A (1) E B D C A (1) E A D B C (1) E A B D C (1) D E C A B (1) D C A B E (1) D A C E B (1) C D E A B (1) C B D A E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C D A E (1) B C A D E (1) B A E D C (1) B A D C E (1) A E D B C (1) A D E C B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 10 10 10 B -6 0 0 2 -6 C -10 0 0 -10 0 D -10 -2 10 0 -4 E -10 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 10 10 B -6 0 0 2 -6 C -10 0 0 -10 0 D -10 -2 10 0 -4 E -10 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 B=28 D=8 C=6 so C is eliminated. Round 2 votes counts: A=30 B=29 E=28 D=13 so D is eliminated. Round 3 votes counts: E=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:200 D:197 B:195 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 10 10 B -6 0 0 2 -6 C -10 0 0 -10 0 D -10 -2 10 0 -4 E -10 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 10 10 B -6 0 0 2 -6 C -10 0 0 -10 0 D -10 -2 10 0 -4 E -10 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 10 10 B -6 0 0 2 -6 C -10 0 0 -10 0 D -10 -2 10 0 -4 E -10 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3650: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) C B D E A (9) B E A C D (9) E A B D C (8) D A E C B (7) D A E B C (6) C D B E A (6) C B E A D (6) A E B D C (6) D A C E B (5) A E D B C (4) D C A E B (3) D C A B E (3) D C B A E (2) C D B A E (2) C B E D A (2) B E C A D (2) A D E B C (2) E B C A D (1) E A B C D (1) D C E A B (1) D A C B E (1) C B A D E (1) B C E A D (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 22 6 -14 B 4 0 4 14 -10 C -22 -4 0 2 -16 D -6 -14 -2 0 -2 E 14 10 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 22 6 -14 B 4 0 4 14 -10 C -22 -4 0 2 -16 D -6 -14 -2 0 -2 E 14 10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 E=21 A=13 B=12 so B is eliminated. Round 2 votes counts: E=32 D=28 C=27 A=13 so A is eliminated. Round 3 votes counts: E=42 D=31 C=27 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:206 A:205 D:188 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 22 6 -14 B 4 0 4 14 -10 C -22 -4 0 2 -16 D -6 -14 -2 0 -2 E 14 10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 6 -14 B 4 0 4 14 -10 C -22 -4 0 2 -16 D -6 -14 -2 0 -2 E 14 10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 6 -14 B 4 0 4 14 -10 C -22 -4 0 2 -16 D -6 -14 -2 0 -2 E 14 10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999974007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3651: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) D E A C B (7) A E B C D (7) B C A E D (5) E D A C B (4) D C B E A (4) C B D A E (4) B D C E A (4) B C D E A (4) B C D A E (4) B A E C D (4) D E B C A (3) C D B E A (3) A E D C B (3) E B D A C (2) E A D C B (2) D E C A B (2) D C E B A (2) C A B E D (2) B A C E D (2) A E C B D (2) A D E C B (2) A B E C D (2) D E C B A (1) D C E A B (1) D B E C A (1) D A E C B (1) C D A E B (1) C B D E A (1) B E D C A (1) B E C D A (1) B E A D C (1) B E A C D (1) B C E D A (1) B C A D E (1) A E D B C (1) A E C D B (1) A E B D C (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 8 -4 -8 B 0 0 10 4 -4 C -8 -10 0 -2 -16 D 4 -4 2 0 -6 E 8 4 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 -4 -8 B 0 0 10 4 -4 C -8 -10 0 -2 -16 D 4 -4 2 0 -6 E 8 4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=22 A=22 E=16 C=11 so C is eliminated. Round 2 votes counts: B=34 D=26 A=24 E=16 so E is eliminated. Round 3 votes counts: B=36 A=34 D=30 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 B:205 A:198 D:198 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 -4 -8 B 0 0 10 4 -4 C -8 -10 0 -2 -16 D 4 -4 2 0 -6 E 8 4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -4 -8 B 0 0 10 4 -4 C -8 -10 0 -2 -16 D 4 -4 2 0 -6 E 8 4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -4 -8 B 0 0 10 4 -4 C -8 -10 0 -2 -16 D 4 -4 2 0 -6 E 8 4 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3652: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (8) A B D E C (7) E C D B A (6) B A D E C (6) B A D C E (6) A E C B D (6) A B E D C (6) C E D B A (4) E C D A B (3) D B C E A (3) D B C A E (3) C D E B A (3) B D A C E (3) A E B C D (3) E D C B A (2) E D B A C (2) E C A D B (2) D C E B A (2) C E D A B (2) B D A E C (2) A C E B D (2) A B E C D (2) A B C E D (2) A B C D E (2) E C A B D (1) E A D B C (1) E A C B D (1) E A B D C (1) D C B E A (1) D B E C A (1) D B E A C (1) D B A E C (1) D B A C E (1) C E A D B (1) C D B E A (1) C B D A E (1) C A E B D (1) Total count = 100 A B C D E A 0 2 26 16 24 B -2 0 22 20 14 C -26 -22 0 -14 -8 D -16 -20 14 0 4 E -24 -14 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 26 16 24 B -2 0 22 20 14 C -26 -22 0 -14 -8 D -16 -20 14 0 4 E -24 -14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=19 B=17 D=13 C=13 so D is eliminated. Round 2 votes counts: A=38 B=27 E=19 C=16 so C is eliminated. Round 3 votes counts: A=39 E=31 B=30 so B is eliminated. Round 4 votes counts: A=62 E=38 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:234 B:227 D:191 E:183 C:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 26 16 24 B -2 0 22 20 14 C -26 -22 0 -14 -8 D -16 -20 14 0 4 E -24 -14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 26 16 24 B -2 0 22 20 14 C -26 -22 0 -14 -8 D -16 -20 14 0 4 E -24 -14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 26 16 24 B -2 0 22 20 14 C -26 -22 0 -14 -8 D -16 -20 14 0 4 E -24 -14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3653: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) B D E C A (6) C D E B A (5) E D C B A (4) E D B A C (4) C B D E A (4) B A D E C (4) A B E D C (4) E D C A B (3) D E B C A (3) C A D E B (3) B D E A C (3) B D C E A (3) B A E D C (3) A C B E D (3) E D B C A (2) E D A B C (2) E C D A B (2) E B D A C (2) C E D A B (2) C D B E A (2) C A E D B (2) C A B D E (2) A E B D C (2) A B E C D (2) A B C E D (2) A B C D E (2) E D A C B (1) E C D B A (1) E A D C B (1) E A D B C (1) D C B E A (1) D B E C A (1) C E D B A (1) C E A D B (1) C D A B E (1) C B A D E (1) B C D E A (1) B C D A E (1) B A D C E (1) B A C D E (1) A E D C B (1) A E D B C (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -16 -22 -24 B 20 0 4 -4 -2 C 16 -4 0 -20 -22 D 22 4 20 0 4 E 24 2 22 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -16 -22 -24 B 20 0 4 -4 -2 C 16 -4 0 -20 -22 D 22 4 20 0 4 E 24 2 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=23 B=23 A=19 D=11 so D is eliminated. Round 2 votes counts: E=32 C=25 B=24 A=19 so A is eliminated. Round 3 votes counts: E=37 B=35 C=28 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:225 E:222 B:209 C:185 A:159 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -16 -22 -24 B 20 0 4 -4 -2 C 16 -4 0 -20 -22 D 22 4 20 0 4 E 24 2 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 -22 -24 B 20 0 4 -4 -2 C 16 -4 0 -20 -22 D 22 4 20 0 4 E 24 2 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 -22 -24 B 20 0 4 -4 -2 C 16 -4 0 -20 -22 D 22 4 20 0 4 E 24 2 22 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995172 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3654: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) C D A E B (8) C A D E B (8) B E D A C (8) A C E D B (7) D C E B A (5) B E A D C (5) A C D E B (4) A B E C D (4) D C E A B (3) D C B E A (3) D B C E A (3) C D E A B (3) E D A C B (2) D E B C A (2) B E D C A (2) B C D A E (2) B A E D C (2) A C E B D (2) E D B C A (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D C B (1) E A C D B (1) E A B D C (1) D E C B A (1) D E C A B (1) C D A B E (1) B D C E A (1) B A E C D (1) B A D C E (1) A E C D B (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 -16 -12 B -2 0 -4 -12 -8 C 8 4 0 -10 8 D 16 12 10 0 16 E 12 8 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -16 -12 B -2 0 -4 -12 -8 C 8 4 0 -10 8 D 16 12 10 0 16 E 12 8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=21 C=20 D=18 E=9 so E is eliminated. Round 2 votes counts: B=34 A=24 D=22 C=20 so C is eliminated. Round 3 votes counts: D=34 B=34 A=32 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 C:205 E:198 B:187 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -16 -12 B -2 0 -4 -12 -8 C 8 4 0 -10 8 D 16 12 10 0 16 E 12 8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -16 -12 B -2 0 -4 -12 -8 C 8 4 0 -10 8 D 16 12 10 0 16 E 12 8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -16 -12 B -2 0 -4 -12 -8 C 8 4 0 -10 8 D 16 12 10 0 16 E 12 8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3655: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) E C B A D (7) D A B E C (7) A E D C B (7) A E C D B (7) C E B A D (6) B D C E A (6) A D E B C (6) E A C B D (4) D B C A E (4) C B E D A (4) D B E A C (3) D A B C E (3) C E A B D (3) E C A B D (2) E A D B C (2) D B A C E (2) C A E B D (2) A D B E C (2) A C E D B (2) E B C D A (1) E A C D B (1) D B A E C (1) D A E B C (1) C E B D A (1) C B E A D (1) C B D E A (1) C A D B E (1) B C E D A (1) A E D B C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 6 2 12 -4 B -6 0 0 -4 -8 C -2 0 0 6 -4 D -12 4 -6 0 -4 E 4 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 2 12 -4 B -6 0 0 -4 -8 C -2 0 0 6 -4 D -12 4 -6 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=21 C=19 E=17 B=16 so B is eliminated. Round 2 votes counts: C=29 D=27 A=27 E=17 so E is eliminated. Round 3 votes counts: C=39 A=34 D=27 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:208 C:200 B:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 12 -4 B -6 0 0 -4 -8 C -2 0 0 6 -4 D -12 4 -6 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 12 -4 B -6 0 0 -4 -8 C -2 0 0 6 -4 D -12 4 -6 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 12 -4 B -6 0 0 -4 -8 C -2 0 0 6 -4 D -12 4 -6 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3656: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) C E A D B (6) C A E D B (6) E B D C A (4) D B E C A (4) C E A B D (4) C A D E B (4) B E D A C (4) B D E A C (4) A C D B E (4) E C D B A (3) E C B D A (3) E C B A D (3) E B D A C (3) D B A C E (3) D A B C E (3) A B D C E (3) E B A C D (2) D B E A C (2) D B C E A (2) D B C A E (2) B D E C A (2) A C E D B (2) A C D E B (2) A C B E D (2) E D C B A (1) E B C D A (1) E B A D C (1) E A C B D (1) D E C B A (1) D A C B E (1) C E D B A (1) C D E B A (1) C D A B E (1) B E D C A (1) B D A E C (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B C E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -12 4 -20 B -2 0 -12 -2 -16 C 12 12 0 12 2 D -4 2 -12 0 -14 E 20 16 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 4 -20 B -2 0 -12 -2 -16 C 12 12 0 12 2 D -4 2 -12 0 -14 E 20 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 A=19 D=18 B=12 so B is eliminated. Round 2 votes counts: E=33 D=25 C=23 A=19 so A is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:224 C:219 A:187 D:186 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 4 -20 B -2 0 -12 -2 -16 C 12 12 0 12 2 D -4 2 -12 0 -14 E 20 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 4 -20 B -2 0 -12 -2 -16 C 12 12 0 12 2 D -4 2 -12 0 -14 E 20 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 4 -20 B -2 0 -12 -2 -16 C 12 12 0 12 2 D -4 2 -12 0 -14 E 20 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3657: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) D C E B A (10) C D E B A (9) A B E C D (7) C D A E B (6) D C A B E (5) B E A D C (5) B A E D C (5) C D E A B (4) C D A B E (3) A C D B E (3) E D B C A (2) E C D B A (2) E B D C A (2) E B A D C (2) D C A E B (2) B A E C D (2) A D C B E (2) A B D E C (2) A B C E D (2) E D C B A (1) E C B D A (1) E B C D A (1) E B C A D (1) E B A C D (1) D E B C A (1) D C B E A (1) C E D B A (1) C E B A D (1) C A D E B (1) B E D A C (1) B E A C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 -4 6 B 0 0 -4 -6 4 C 8 4 0 -6 2 D 4 6 6 0 0 E -6 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.721187 E: 0.278813 Sum of squares = 0.59784747583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.721187 E: 1.000000 A B C D E A 0 0 -8 -4 6 B 0 0 -4 -6 4 C 8 4 0 -6 2 D 4 6 6 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000002677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 D=19 B=14 E=13 so E is eliminated. Round 2 votes counts: A=29 C=28 D=22 B=21 so B is eliminated. Round 3 votes counts: A=45 C=30 D=25 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:204 A:197 B:197 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -4 6 B 0 0 -4 -6 4 C 8 4 0 -6 2 D 4 6 6 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000002677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 6 B 0 0 -4 -6 4 C 8 4 0 -6 2 D 4 6 6 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000002677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 6 B 0 0 -4 -6 4 C 8 4 0 -6 2 D 4 6 6 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000002677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3658: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) E C D B A (7) C E D A B (6) C E D B A (5) C E A D B (5) A D C B E (5) A D B C E (5) A B D E C (5) D B A E C (4) C A E B D (3) B E D C A (3) B A D E C (3) A B D C E (3) E D C B A (2) E D B C A (2) E B C D A (2) D E B C A (2) D B E A C (2) D A B E C (2) C E B D A (2) C E B A D (2) B D E A C (2) B D A E C (2) A D B E C (2) A B C D E (2) E B C A D (1) C D E A B (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C A D (1) B E A C D (1) B A E D C (1) B A E C D (1) A C E B D (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -10 -8 -10 B 8 0 -4 -8 -6 C 10 4 0 8 -8 D 8 8 -8 0 -14 E 10 6 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -10 -8 -10 B 8 0 -4 -8 -6 C 10 4 0 8 -8 D 8 8 -8 0 -14 E 10 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 E=22 B=15 D=10 so D is eliminated. Round 2 votes counts: A=28 C=27 E=24 B=21 so B is eliminated. Round 3 votes counts: A=39 E=34 C=27 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:207 D:197 B:195 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -8 -10 B 8 0 -4 -8 -6 C 10 4 0 8 -8 D 8 8 -8 0 -14 E 10 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -8 -10 B 8 0 -4 -8 -6 C 10 4 0 8 -8 D 8 8 -8 0 -14 E 10 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -8 -10 B 8 0 -4 -8 -6 C 10 4 0 8 -8 D 8 8 -8 0 -14 E 10 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3659: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) B D E A C (7) B A E D C (7) E A B D C (6) D C E B A (4) D B E C A (4) D B C E A (4) B E D A C (4) A E C B D (4) D E B C A (3) C D E A B (3) C D B A E (3) C D A E B (3) B E A D C (3) A E B D C (3) E A D B C (2) E A C D B (2) D C B E A (2) C A D B E (2) B D A C E (2) A C E D B (2) A B C E D (2) E D B A C (1) E C D A B (1) E A D C B (1) D E C B A (1) D B C A E (1) C E D A B (1) C D B E A (1) C D A B E (1) C A D E B (1) C A B D E (1) B D C A E (1) B D A E C (1) B C A D E (1) B A C D E (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 24 4 0 B 2 0 30 14 6 C -24 -30 0 -16 -22 D -4 -14 16 0 -2 E 0 -6 22 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 24 4 0 B 2 0 30 14 6 C -24 -30 0 -16 -22 D -4 -14 16 0 -2 E 0 -6 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=25 D=19 C=16 E=13 so E is eliminated. Round 2 votes counts: A=36 B=27 D=20 C=17 so C is eliminated. Round 3 votes counts: A=40 D=33 B=27 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:226 A:213 E:209 D:198 C:154 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 24 4 0 B 2 0 30 14 6 C -24 -30 0 -16 -22 D -4 -14 16 0 -2 E 0 -6 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 24 4 0 B 2 0 30 14 6 C -24 -30 0 -16 -22 D -4 -14 16 0 -2 E 0 -6 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 24 4 0 B 2 0 30 14 6 C -24 -30 0 -16 -22 D -4 -14 16 0 -2 E 0 -6 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3660: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) B E A D C (9) D C B E A (6) C A E D B (6) B D E A C (6) A E C B D (6) D C B A E (5) E A C B D (4) D B E C A (4) E A B C D (3) D C E A B (3) D B C A E (3) C D E A B (3) C D A B E (3) E B A D C (2) C D B A E (2) C A D E B (2) B E A C D (2) B D A E C (2) B D A C E (2) B A E D C (2) B A E C D (2) A E B C D (2) E D A C B (1) E B D A C (1) E A D B C (1) E A C D B (1) D B E A C (1) D B C E A (1) C A D B E (1) C A B E D (1) C A B D E (1) B A C D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 -4 4 B 2 0 -10 -4 10 C 0 10 0 2 0 D 4 4 -2 0 10 E -4 -10 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250457 B: 0.000000 C: 0.749543 D: 0.000000 E: 0.000000 Sum of squares = 0.624543786335 Cumulative probabilities = A: 0.250457 B: 0.250457 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -4 4 B 2 0 -10 -4 10 C 0 10 0 2 0 D 4 4 -2 0 10 E -4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555714444 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=26 D=23 E=13 A=10 so A is eliminated. Round 2 votes counts: C=29 B=27 D=23 E=21 so E is eliminated. Round 3 votes counts: C=40 B=35 D=25 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:206 A:199 B:199 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 -4 4 B 2 0 -10 -4 10 C 0 10 0 2 0 D 4 4 -2 0 10 E -4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555714444 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 4 B 2 0 -10 -4 10 C 0 10 0 2 0 D 4 4 -2 0 10 E -4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555714444 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 4 B 2 0 -10 -4 10 C 0 10 0 2 0 D 4 4 -2 0 10 E -4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555714444 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3661: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) E A B D C (7) C D B A E (6) D C A B E (5) C D E B A (5) A E B D C (5) E C D B A (4) C B D A E (4) E B A C D (3) D C E A B (3) B A E C D (3) A D B C E (3) A B E C D (3) A B D E C (3) E D C A B (2) E B C A D (2) E A D B C (2) E A B C D (2) D A C B E (2) B E C A D (2) B E A C D (2) B C A E D (2) B A C E D (2) A D E B C (2) E C B A D (1) E C A D B (1) E A D C B (1) D E C A B (1) D C E B A (1) D C B A E (1) D C A E B (1) C E D B A (1) C D B E A (1) C B E D A (1) C B D E A (1) B C E D A (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -12 2 -4 B 8 0 0 12 -6 C 12 0 0 16 -14 D -2 -12 -16 0 -14 E 4 6 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 2 -4 B 8 0 0 12 -6 C 12 0 0 16 -14 D -2 -12 -16 0 -14 E 4 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=19 A=19 B=15 D=14 so D is eliminated. Round 2 votes counts: E=34 C=30 A=21 B=15 so B is eliminated. Round 3 votes counts: E=38 C=35 A=27 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:207 C:207 A:189 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 2 -4 B 8 0 0 12 -6 C 12 0 0 16 -14 D -2 -12 -16 0 -14 E 4 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 2 -4 B 8 0 0 12 -6 C 12 0 0 16 -14 D -2 -12 -16 0 -14 E 4 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 2 -4 B 8 0 0 12 -6 C 12 0 0 16 -14 D -2 -12 -16 0 -14 E 4 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3662: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) E D A C B (6) C E A D B (6) D E A C B (5) C E D A B (5) B A D E C (5) B A C D E (5) C A E D B (4) A D E C B (4) A D E B C (4) C E A B D (3) C B E D A (3) C B E A D (3) B C A E D (3) B A D C E (3) D E A B C (2) C E D B A (2) C E B D A (2) C E B A D (2) B E D C A (2) B E C D A (2) B D E A C (2) B D A E C (2) B C E D A (2) B C E A D (2) A D C E B (2) A D B E C (2) E C D A B (1) D E B A C (1) D A E C B (1) D A E B C (1) C B A E D (1) B D E C A (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -6 2 -22 B -14 0 -20 -12 -22 C 6 20 0 -4 0 D -2 12 4 0 -16 E 22 22 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.398927 D: 0.000000 E: 0.601073 Sum of squares = 0.520431509934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.398927 D: 0.398927 E: 1.000000 A B C D E A 0 14 -6 2 -22 B -14 0 -20 -12 -22 C 6 20 0 -4 0 D -2 12 4 0 -16 E 22 22 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=29 E=16 A=14 D=10 so D is eliminated. Round 2 votes counts: C=31 B=29 E=24 A=16 so A is eliminated. Round 3 votes counts: E=34 C=34 B=32 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:230 C:211 D:199 A:194 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -6 2 -22 B -14 0 -20 -12 -22 C 6 20 0 -4 0 D -2 12 4 0 -16 E 22 22 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 2 -22 B -14 0 -20 -12 -22 C 6 20 0 -4 0 D -2 12 4 0 -16 E 22 22 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 2 -22 B -14 0 -20 -12 -22 C 6 20 0 -4 0 D -2 12 4 0 -16 E 22 22 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3663: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) D B A E C (7) D B E A C (6) E C D A B (5) E C A D B (5) E D C A B (4) E C B D A (4) C E A B D (4) E C D B A (3) C E B A D (3) C A E B D (3) B D A E C (3) A C B E D (3) A B D C E (3) A B C D E (3) E D C B A (2) E D B C A (2) E C B A D (2) E C A B D (2) D E C A B (2) D E B C A (2) D A E C B (2) D A B E C (2) C E A D B (2) B C E A D (2) A D B C E (2) A C E B D (2) E B D C A (1) E B C D A (1) D E B A C (1) D E A C B (1) D A E B C (1) C A E D B (1) B E C D A (1) B D E C A (1) B D E A C (1) B D A C E (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -4 -6 -14 B 4 0 -2 -2 -12 C 4 2 0 -8 -22 D 6 2 8 0 -4 E 14 12 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -14 B 4 0 -2 -2 -12 C 4 2 0 -8 -22 D 6 2 8 0 -4 E 14 12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=24 B=18 A=14 C=13 so C is eliminated. Round 2 votes counts: E=40 D=24 B=18 A=18 so B is eliminated. Round 3 votes counts: E=43 D=30 A=27 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:206 B:194 C:188 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -14 B 4 0 -2 -2 -12 C 4 2 0 -8 -22 D 6 2 8 0 -4 E 14 12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -14 B 4 0 -2 -2 -12 C 4 2 0 -8 -22 D 6 2 8 0 -4 E 14 12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -14 B 4 0 -2 -2 -12 C 4 2 0 -8 -22 D 6 2 8 0 -4 E 14 12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3664: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (12) A B C E D (10) E D C B A (6) E D A B C (6) A B C D E (5) E D B C A (4) E D A C B (4) D E A C B (4) C B A D E (4) B C A D E (4) A E B C D (4) E D B A C (3) E A D B C (3) D C E B A (3) D C B E A (3) C B E D A (3) A E B D C (3) D E C A B (2) D C A B E (2) A C B D E (2) A B E C D (2) E D C A B (1) E C D B A (1) E A B D C (1) D C B A E (1) D A C E B (1) C D B E A (1) C B D E A (1) C B D A E (1) B C D E A (1) A E D B C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 0 -20 -18 B -4 0 -6 -18 -18 C 0 6 0 -20 -14 D 20 18 20 0 -4 E 18 18 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 -20 -18 B -4 0 -6 -18 -18 C 0 6 0 -20 -14 D 20 18 20 0 -4 E 18 18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=28 A=28 C=10 B=5 so B is eliminated. Round 2 votes counts: E=29 D=28 A=28 C=15 so C is eliminated. Round 3 votes counts: A=36 E=32 D=32 so E is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:227 C:186 A:183 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -20 -18 B -4 0 -6 -18 -18 C 0 6 0 -20 -14 D 20 18 20 0 -4 E 18 18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -20 -18 B -4 0 -6 -18 -18 C 0 6 0 -20 -14 D 20 18 20 0 -4 E 18 18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -20 -18 B -4 0 -6 -18 -18 C 0 6 0 -20 -14 D 20 18 20 0 -4 E 18 18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3665: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) C A B E D (9) B A C D E (9) E D C A B (8) E D C B A (7) B A C E D (6) E C B A D (5) A B C D E (5) E C A B D (4) D E C A B (3) E D B C A (2) E D B A C (2) D E A C B (2) D E A B C (2) C D A B E (2) C B A E D (2) B C A E D (2) B A D C E (2) A C B D E (2) A B C E D (2) D E B C A (1) D C A B E (1) D B E A C (1) D B A E C (1) D B A C E (1) D A C B E (1) D A B E C (1) C A D B E (1) C A B D E (1) B E C A D (1) B D A C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 2 4 0 B 10 0 2 4 4 C -2 -2 0 4 -4 D -4 -4 -4 0 -2 E 0 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 4 0 B 10 0 2 4 4 C -2 -2 0 4 -4 D -4 -4 -4 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 B=21 C=15 A=10 so A is eliminated. Round 2 votes counts: E=28 B=28 D=26 C=18 so C is eliminated. Round 3 votes counts: B=43 D=29 E=28 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 E:201 A:198 C:198 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 4 0 B 10 0 2 4 4 C -2 -2 0 4 -4 D -4 -4 -4 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 4 0 B 10 0 2 4 4 C -2 -2 0 4 -4 D -4 -4 -4 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 4 0 B 10 0 2 4 4 C -2 -2 0 4 -4 D -4 -4 -4 0 -2 E 0 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3666: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (6) A D C B E (6) E B C D A (5) D A E C B (5) D A B E C (5) C E B A D (5) D B A C E (4) B C D A E (4) A D C E B (4) E A C D B (3) D A E B C (3) C E A B D (3) C B E A D (3) B C E D A (3) A D E C B (3) A D B C E (3) E C B A D (2) E C A B D (2) D B E A C (2) D B A E C (2) B E D C A (2) B E C D A (2) B D A C E (2) B C D E A (2) A E D C B (2) A E C D B (2) E D C A B (1) E D A C B (1) E C B D A (1) E C A D B (1) E A D C B (1) D E A B C (1) D A C B E (1) D A B C E (1) C B A E D (1) C A E B D (1) B D E C A (1) B D C E A (1) B C A E D (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 -2 4 2 4 B 2 0 4 -2 8 C -4 -4 0 -2 6 D -2 2 2 0 4 E -4 -8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 2 4 B 2 0 4 -2 8 C -4 -4 0 -2 6 D -2 2 2 0 4 E -4 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=24 A=20 E=17 C=13 so C is eliminated. Round 2 votes counts: B=30 E=25 D=24 A=21 so A is eliminated. Round 3 votes counts: D=40 E=30 B=30 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:206 A:204 D:203 C:198 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 2 4 B 2 0 4 -2 8 C -4 -4 0 -2 6 D -2 2 2 0 4 E -4 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 2 4 B 2 0 4 -2 8 C -4 -4 0 -2 6 D -2 2 2 0 4 E -4 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 2 4 B 2 0 4 -2 8 C -4 -4 0 -2 6 D -2 2 2 0 4 E -4 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3667: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D B E A C (11) D B C A E (8) C A E D B (6) B D E A C (6) B D E C A (5) E B A D C (4) C A E B D (4) A C E D B (4) E A B C D (3) D C B A E (2) D C A B E (2) D A E C B (2) C D A B E (2) C B A E D (2) C A D E B (2) A E C B D (2) E B D A C (1) E B C A D (1) E A D C B (1) E A C D B (1) E A B D C (1) D E B A C (1) D C A E B (1) D B E C A (1) D B A C E (1) D A C E B (1) D A B C E (1) C E A B D (1) C D B A E (1) C B D A E (1) B E D A C (1) B E C A D (1) B D C E A (1) B D C A E (1) B C E A D (1) B C A E D (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 12 0 -8 B 2 0 0 0 -2 C -12 0 0 -6 -14 D 0 0 6 0 0 E 8 2 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.612905 E: 0.387095 Sum of squares = 0.525495151756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.612905 E: 1.000000 A B C D E A 0 -2 12 0 -8 B 2 0 0 0 -2 C -12 0 0 -6 -14 D 0 0 6 0 0 E 8 2 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=25 C=19 B=17 A=8 so A is eliminated. Round 2 votes counts: D=31 E=29 C=23 B=17 so B is eliminated. Round 3 votes counts: D=44 E=31 C=25 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:212 D:203 A:201 B:200 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 0 -8 B 2 0 0 0 -2 C -12 0 0 -6 -14 D 0 0 6 0 0 E 8 2 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 0 -8 B 2 0 0 0 -2 C -12 0 0 -6 -14 D 0 0 6 0 0 E 8 2 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 0 -8 B 2 0 0 0 -2 C -12 0 0 -6 -14 D 0 0 6 0 0 E 8 2 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3668: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) E D B A C (5) E B A D C (5) D C A E B (5) C A D B E (5) C D A E B (4) C A D E B (4) B E D A C (4) B D E C A (4) E B D A C (3) E A C B D (3) E A B C D (3) D E C A B (3) D E B C A (3) C A B D E (3) B A C E D (3) A C E D B (3) E D A C B (2) D C E A B (2) D C B A E (2) C B D A E (2) B C D A E (2) B A E C D (2) A E C B D (2) A B C E D (2) E A D C B (1) E A C D B (1) E A B D C (1) D E C B A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C E A (1) C D A B E (1) B E A C D (1) B D E A C (1) B D C E A (1) B D C A E (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 12 6 0 4 B -12 0 -10 -8 -22 C -6 10 0 4 -12 D 0 8 -4 0 -4 E -4 22 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.832645 B: 0.000000 C: 0.000000 D: 0.167355 E: 0.000000 Sum of squares = 0.721305090061 Cumulative probabilities = A: 0.832645 B: 0.832645 C: 0.832645 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 0 4 B -12 0 -10 -8 -22 C -6 10 0 4 -12 D 0 8 -4 0 -4 E -4 22 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500315 B: 0.000000 C: 0.000000 D: 0.499685 E: 0.000000 Sum of squares = 0.500000197983 Cumulative probabilities = A: 0.500315 B: 0.500315 C: 0.500315 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=20 C=19 B=19 A=18 so A is eliminated. Round 2 votes counts: E=35 C=23 B=22 D=20 so D is eliminated. Round 3 votes counts: E=42 C=34 B=24 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:211 D:200 C:198 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 0 4 B -12 0 -10 -8 -22 C -6 10 0 4 -12 D 0 8 -4 0 -4 E -4 22 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500315 B: 0.000000 C: 0.000000 D: 0.499685 E: 0.000000 Sum of squares = 0.500000197983 Cumulative probabilities = A: 0.500315 B: 0.500315 C: 0.500315 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 0 4 B -12 0 -10 -8 -22 C -6 10 0 4 -12 D 0 8 -4 0 -4 E -4 22 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500315 B: 0.000000 C: 0.000000 D: 0.499685 E: 0.000000 Sum of squares = 0.500000197983 Cumulative probabilities = A: 0.500315 B: 0.500315 C: 0.500315 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 0 4 B -12 0 -10 -8 -22 C -6 10 0 4 -12 D 0 8 -4 0 -4 E -4 22 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500315 B: 0.000000 C: 0.000000 D: 0.499685 E: 0.000000 Sum of squares = 0.500000197983 Cumulative probabilities = A: 0.500315 B: 0.500315 C: 0.500315 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3669: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) B A E D C (6) B A E C D (6) E B A D C (5) C D B A E (5) B E A C D (5) A B E D C (5) E D C A B (4) D C A B E (4) C D A B E (4) E C D B A (3) E B A C D (3) D E C A B (3) D C E A B (3) B A C E D (3) A B D E C (3) E D A C B (2) E C D A B (2) E A D B C (2) C D E B A (2) C D A E B (2) C B A D E (2) B A D C E (2) E D A B C (1) E C B A D (1) E B C A D (1) D C A E B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C B E D A (1) C B E A D (1) B C E A D (1) B C A E D (1) B A D E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 0 0 B 0 0 -4 -2 8 C 2 4 0 6 -6 D 0 2 -6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691029 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 0 -2 0 0 B 0 0 -4 -2 8 C 2 4 0 6 -6 D 0 2 -6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.35802469128 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=25 E=24 D=14 A=10 so A is eliminated. Round 2 votes counts: B=34 C=27 E=24 D=15 so D is eliminated. Round 3 votes counts: C=37 B=36 E=27 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:204 C:203 B:201 A:199 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 0 0 B 0 0 -4 -2 8 C 2 4 0 6 -6 D 0 2 -6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.35802469128 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 0 0 B 0 0 -4 -2 8 C 2 4 0 6 -6 D 0 2 -6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.35802469128 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 0 0 B 0 0 -4 -2 8 C 2 4 0 6 -6 D 0 2 -6 0 -10 E 0 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.000000 E: 0.222222 Sum of squares = 0.35802469128 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3670: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) A E D C B (9) E B C D A (8) C D B A E (8) D C B A E (7) D C A B E (6) E B A C D (5) E A D C B (4) C B D A E (4) B C D E A (4) E A C D B (3) E A B D C (3) B D C A E (3) A D C E B (3) D A C B E (2) C D B E A (2) B C D A E (2) A D B C E (2) E C D B A (1) E C D A B (1) E C B D A (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B C D (1) C D A B E (1) B E C D A (1) B A D C E (1) A E D B C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 2 0 -2 26 B -2 0 -30 -28 10 C 0 30 0 -14 14 D 2 28 14 0 16 E -26 -10 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 26 B -2 0 -30 -28 10 C 0 30 0 -14 14 D 2 28 14 0 16 E -26 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999948861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=29 D=15 C=15 B=11 so B is eliminated. Round 2 votes counts: E=31 A=30 C=21 D=18 so D is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:230 C:215 A:213 B:175 E:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -2 26 B -2 0 -30 -28 10 C 0 30 0 -14 14 D 2 28 14 0 16 E -26 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999948861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 26 B -2 0 -30 -28 10 C 0 30 0 -14 14 D 2 28 14 0 16 E -26 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999948861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 26 B -2 0 -30 -28 10 C 0 30 0 -14 14 D 2 28 14 0 16 E -26 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999948861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3671: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) D B A C E (7) C A E B D (7) B D E C A (7) E B C A D (6) D A C B E (6) B D E A C (6) E B D A C (5) C A E D B (5) A C E D B (5) C A D E B (4) C A D B E (4) B E D C A (4) E C A B D (3) E B D C A (3) D B E A C (3) E B C D A (2) E A C B D (2) C E A B D (2) A D C E B (2) A C D B E (2) D E A B C (1) D B C A E (1) D B A E C (1) B E D A C (1) B E C A D (1) B D C E A (1) B C E A D (1) Total count = 100 A B C D E A 0 2 -2 4 4 B -2 0 0 2 -10 C 2 0 0 4 10 D -4 -2 -4 0 6 E -4 10 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.334990 C: 0.665010 D: 0.000000 E: 0.000000 Sum of squares = 0.554456459936 Cumulative probabilities = A: 0.000000 B: 0.334990 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 4 4 B -2 0 0 2 -10 C 2 0 0 4 10 D -4 -2 -4 0 6 E -4 10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499387 C: 0.500613 D: 0.000000 E: 0.000000 Sum of squares = 0.500000750332 Cumulative probabilities = A: 0.000000 B: 0.499387 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 E=21 B=21 D=19 A=17 so A is eliminated. Round 2 votes counts: C=37 E=21 D=21 B=21 so E is eliminated. Round 3 votes counts: C=42 B=37 D=21 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:208 A:204 D:198 B:195 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 4 4 B -2 0 0 2 -10 C 2 0 0 4 10 D -4 -2 -4 0 6 E -4 10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499387 C: 0.500613 D: 0.000000 E: 0.000000 Sum of squares = 0.500000750332 Cumulative probabilities = A: 0.000000 B: 0.499387 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 4 4 B -2 0 0 2 -10 C 2 0 0 4 10 D -4 -2 -4 0 6 E -4 10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499387 C: 0.500613 D: 0.000000 E: 0.000000 Sum of squares = 0.500000750332 Cumulative probabilities = A: 0.000000 B: 0.499387 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 4 4 B -2 0 0 2 -10 C 2 0 0 4 10 D -4 -2 -4 0 6 E -4 10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499387 C: 0.500613 D: 0.000000 E: 0.000000 Sum of squares = 0.500000750332 Cumulative probabilities = A: 0.000000 B: 0.499387 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3672: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (13) A B D E C (10) D E C A B (8) D E A B C (8) C B A E D (8) C B E A D (6) D E A C B (5) B C A E D (5) B A C E D (5) C E B D A (4) A D B E C (4) A B C D E (4) E D C B A (3) E D C A B (3) E C D B A (3) E C D A B (2) B A C D E (2) A B D C E (2) D A E B C (1) C E D A B (1) C E B A D (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 -4 -14 -2 -14 B 4 0 -14 -2 -4 C 14 14 0 10 2 D 2 2 -10 0 -10 E 14 4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -2 -14 B 4 0 -14 -2 -4 C 14 14 0 10 2 D 2 2 -10 0 -10 E 14 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 A=20 B=14 E=11 so E is eliminated. Round 2 votes counts: C=38 D=28 A=20 B=14 so B is eliminated. Round 3 votes counts: C=43 A=29 D=28 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:213 B:192 D:192 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -2 -14 B 4 0 -14 -2 -4 C 14 14 0 10 2 D 2 2 -10 0 -10 E 14 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -2 -14 B 4 0 -14 -2 -4 C 14 14 0 10 2 D 2 2 -10 0 -10 E 14 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -2 -14 B 4 0 -14 -2 -4 C 14 14 0 10 2 D 2 2 -10 0 -10 E 14 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3673: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) D C E B A (9) A B C E D (9) E D B A C (6) D E B C A (6) D C A B E (5) C D B A E (5) C D A B E (5) C A B D E (5) E B A D C (4) E A B D C (3) D E C B A (3) D C B E A (3) D C B A E (3) A E B C D (3) A C B E D (3) E B A C D (2) E A B C D (2) C B E D A (2) C B A D E (2) A E B D C (2) A C B D E (2) D E B A C (1) D E A B C (1) D C E A B (1) C B D A E (1) C A B E D (1) Total count = 100 A B C D E A 0 6 -2 -2 14 B -6 0 0 4 14 C 2 0 0 6 12 D 2 -4 -6 0 4 E -14 -14 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.174125 C: 0.825875 D: 0.000000 E: 0.000000 Sum of squares = 0.712389147877 Cumulative probabilities = A: 0.000000 B: 0.174125 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -2 14 B -6 0 0 4 14 C 2 0 0 6 12 D 2 -4 -6 0 4 E -14 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000190213 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=30 C=21 E=17 so B is eliminated. Round 2 votes counts: D=32 A=30 C=21 E=17 so E is eliminated. Round 3 votes counts: A=41 D=38 C=21 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:210 A:208 B:206 D:198 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -2 14 B -6 0 0 4 14 C 2 0 0 6 12 D 2 -4 -6 0 4 E -14 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000190213 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 14 B -6 0 0 4 14 C 2 0 0 6 12 D 2 -4 -6 0 4 E -14 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000190213 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 14 B -6 0 0 4 14 C 2 0 0 6 12 D 2 -4 -6 0 4 E -14 -14 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000190213 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3674: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) A C D E B (11) E D C A B (5) B E A C D (5) B D A C E (5) D C A B E (4) B D C A E (4) E D C B A (3) E C D A B (3) E B D C A (3) D C A E B (3) B A E C D (3) B A C D E (3) E C A D B (2) E B C D A (2) E A C D B (2) E A C B D (2) C A E D B (2) B E D A C (2) B A D C E (2) A D C B E (2) A C D B E (2) A C B D E (2) E C D B A (1) E B C A D (1) E B A C D (1) D E C A B (1) D B C A E (1) C D A E B (1) C A D E B (1) B E A D C (1) B D E C A (1) B D C E A (1) A E C D B (1) A E C B D (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -6 -8 0 B 6 0 0 8 6 C 6 0 0 -4 -8 D 8 -8 4 0 -10 E 0 -6 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.682614 C: 0.317386 D: 0.000000 E: 0.000000 Sum of squares = 0.566696030713 Cumulative probabilities = A: 0.000000 B: 0.682614 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -8 0 B 6 0 0 8 6 C 6 0 0 -4 -8 D 8 -8 4 0 -10 E 0 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.51020421409 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=25 A=21 D=9 C=4 so C is eliminated. Round 2 votes counts: B=41 E=25 A=24 D=10 so D is eliminated. Round 3 votes counts: B=42 A=32 E=26 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:210 E:206 C:197 D:197 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -8 0 B 6 0 0 8 6 C 6 0 0 -4 -8 D 8 -8 4 0 -10 E 0 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.51020421409 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -8 0 B 6 0 0 8 6 C 6 0 0 -4 -8 D 8 -8 4 0 -10 E 0 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.51020421409 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -8 0 B 6 0 0 8 6 C 6 0 0 -4 -8 D 8 -8 4 0 -10 E 0 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.51020421409 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3675: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (15) E D A C B (14) B C A D E (11) E D A B C (10) A D E C B (10) A D E B C (6) B C E D A (5) E D B A C (4) E B D A C (2) D E A C B (2) C B A E D (2) C A B D E (2) A B D E C (2) E D C B A (1) E D C A B (1) D E A B C (1) D A E C B (1) C E D B A (1) C B E A D (1) C A D E B (1) B E D C A (1) B C E A D (1) B C A E D (1) B A D E C (1) B A C D E (1) A D C E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 14 14 12 B -6 0 -6 -8 -10 C -14 6 0 -14 -12 D -14 8 14 0 12 E -12 10 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 14 12 B -6 0 -6 -8 -10 C -14 6 0 -14 -12 D -14 8 14 0 12 E -12 10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=22 B=21 A=21 D=4 so D is eliminated. Round 2 votes counts: E=35 C=22 A=22 B=21 so B is eliminated. Round 3 votes counts: C=40 E=36 A=24 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:223 D:210 E:199 B:185 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 14 12 B -6 0 -6 -8 -10 C -14 6 0 -14 -12 D -14 8 14 0 12 E -12 10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 14 12 B -6 0 -6 -8 -10 C -14 6 0 -14 -12 D -14 8 14 0 12 E -12 10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 14 12 B -6 0 -6 -8 -10 C -14 6 0 -14 -12 D -14 8 14 0 12 E -12 10 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3676: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) B A C E D (8) D E C B A (6) D B E C A (5) D A B E C (5) B A D E C (5) E C B A D (4) D C E A B (4) C E D A B (4) A C E B D (4) A B C E D (4) C E B A D (3) C E A D B (3) C E A B D (3) B D E C A (3) A D C E B (3) E C D B A (2) E C B D A (2) D E B C A (2) D B E A C (2) D A E C B (2) A D B C E (2) E C D A B (1) E B D C A (1) D E C A B (1) D C A E B (1) D B A E C (1) D A C E B (1) D A C B E (1) C D A E B (1) B E D C A (1) B D E A C (1) B D A E C (1) B A E C D (1) A C D E B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 6 4 4 B -4 0 4 4 2 C -6 -4 0 -16 6 D -4 -4 16 0 16 E -4 -2 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 4 B -4 0 4 4 2 C -6 -4 0 -16 6 D -4 -4 16 0 16 E -4 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=25 B=20 C=14 E=10 so E is eliminated. Round 2 votes counts: D=31 A=25 C=23 B=21 so B is eliminated. Round 3 votes counts: A=39 D=38 C=23 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:209 B:203 C:190 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 4 B -4 0 4 4 2 C -6 -4 0 -16 6 D -4 -4 16 0 16 E -4 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 4 B -4 0 4 4 2 C -6 -4 0 -16 6 D -4 -4 16 0 16 E -4 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 4 B -4 0 4 4 2 C -6 -4 0 -16 6 D -4 -4 16 0 16 E -4 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3677: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) A C B E D (7) D E C B A (5) A B C E D (5) E D A B C (4) E B A D C (4) C B A E D (4) A B E C D (4) E B C A D (3) E B A C D (3) D E B C A (3) D E B A C (3) D E A B C (3) C B E D A (3) C A D B E (3) C A B D E (3) B A E C D (3) D C A B E (2) C D A B E (2) C B D E A (2) C A B E D (2) B E C A D (2) A D C B E (2) E D B C A (1) E B D C A (1) E B D A C (1) E B C D A (1) E A D B C (1) D E A C B (1) D C B E A (1) D C B A E (1) D C A E B (1) D A E C B (1) D A C B E (1) C D B A E (1) C B E A D (1) B C E D A (1) A E D B C (1) A E B C D (1) A D E B C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -4 4 -6 B 6 0 -8 2 2 C 4 8 0 4 6 D -4 -2 -4 0 -6 E 6 -2 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 4 -6 B 6 0 -8 2 2 C 4 8 0 4 6 D -4 -2 -4 0 -6 E 6 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=23 C=21 E=19 B=6 so B is eliminated. Round 2 votes counts: D=31 A=26 C=22 E=21 so E is eliminated. Round 3 votes counts: D=38 A=34 C=28 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:211 E:202 B:201 A:194 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 4 -6 B 6 0 -8 2 2 C 4 8 0 4 6 D -4 -2 -4 0 -6 E 6 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 4 -6 B 6 0 -8 2 2 C 4 8 0 4 6 D -4 -2 -4 0 -6 E 6 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 4 -6 B 6 0 -8 2 2 C 4 8 0 4 6 D -4 -2 -4 0 -6 E 6 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3678: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (13) C E A B D (10) E C A B D (8) D B A E C (8) D B A C E (7) B A D E C (7) E C D B A (5) E C B A D (4) C E D A B (4) A B E C D (4) A B D E C (4) E C B D A (3) D C E B A (3) D A B C E (3) C E A D B (3) A B D C E (3) D C E A B (2) C D E B A (2) E B C A D (1) E A B C D (1) D E C B A (1) C A D E B (1) B D A E C (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -20 -4 -20 B 12 0 -18 -6 -22 C 20 18 0 18 4 D 4 6 -18 0 -14 E 20 22 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -20 -4 -20 B 12 0 -18 -6 -22 C 20 18 0 18 4 D 4 6 -18 0 -14 E 20 22 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=24 E=22 A=12 B=9 so B is eliminated. Round 2 votes counts: C=33 D=25 E=22 A=20 so A is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:230 E:226 D:189 B:183 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -20 -4 -20 B 12 0 -18 -6 -22 C 20 18 0 18 4 D 4 6 -18 0 -14 E 20 22 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -4 -20 B 12 0 -18 -6 -22 C 20 18 0 18 4 D 4 6 -18 0 -14 E 20 22 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -4 -20 B 12 0 -18 -6 -22 C 20 18 0 18 4 D 4 6 -18 0 -14 E 20 22 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3679: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) B D C A E (9) D A C E B (8) B E C A D (7) B D A C E (5) E C A B D (4) E B C A D (4) D B A C E (4) D A C B E (4) B E D C A (4) C A E D B (3) B E D A C (3) B C E A D (3) A D C E B (3) E C A D B (2) D B A E C (2) D A E C B (2) C A D E B (2) B E C D A (2) B D E C A (2) A C D E B (2) E D A C B (1) E C B A D (1) E B A C D (1) E A D C B (1) D E A C B (1) D E A B C (1) D A B C E (1) C E A D B (1) C D A B E (1) C B D A E (1) C A D B E (1) B D C E A (1) B D A E C (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 2 -8 2 B 2 0 2 -2 4 C -2 -2 0 -6 2 D 8 2 6 0 4 E -2 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -8 2 B 2 0 2 -2 4 C -2 -2 0 -6 2 D 8 2 6 0 4 E -2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=24 D=23 C=9 A=6 so A is eliminated. Round 2 votes counts: B=38 D=26 E=24 C=12 so C is eliminated. Round 3 votes counts: B=39 D=32 E=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:203 A:197 C:196 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -8 2 B 2 0 2 -2 4 C -2 -2 0 -6 2 D 8 2 6 0 4 E -2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -8 2 B 2 0 2 -2 4 C -2 -2 0 -6 2 D 8 2 6 0 4 E -2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -8 2 B 2 0 2 -2 4 C -2 -2 0 -6 2 D 8 2 6 0 4 E -2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3680: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) B D C A E (8) D B E A C (6) D B C E A (6) E A B D C (5) A C E B D (5) E D C A B (4) E D A B C (4) E A C D B (4) A E C B D (4) D B E C A (3) C B A E D (3) C A B E D (3) B D A C E (3) E D A C B (2) D E C B A (2) D E B A C (2) C E A D B (2) C B A D E (2) B D A E C (2) B C A D E (2) E A D C B (1) E A D B C (1) E A C B D (1) E A B C D (1) D E A B C (1) D C E A B (1) D C B E A (1) D B C A E (1) C D B E A (1) C B D A E (1) B E A D C (1) B D E A C (1) B C D A E (1) B A E D C (1) B A C E D (1) B A C D E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -2 0 0 B -2 0 6 16 -2 C 2 -6 0 -12 4 D 0 -16 12 0 -10 E 0 2 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999916 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 0 0 B -2 0 6 16 -2 C 2 -6 0 -12 4 D 0 -16 12 0 -10 E 0 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999931 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 C=22 B=21 A=11 so A is eliminated. Round 2 votes counts: E=28 C=27 D=23 B=22 so B is eliminated. Round 3 votes counts: D=37 C=32 E=31 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:209 E:204 A:200 C:194 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 0 0 B -2 0 6 16 -2 C 2 -6 0 -12 4 D 0 -16 12 0 -10 E 0 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999931 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 0 B -2 0 6 16 -2 C 2 -6 0 -12 4 D 0 -16 12 0 -10 E 0 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999931 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 0 B -2 0 6 16 -2 C 2 -6 0 -12 4 D 0 -16 12 0 -10 E 0 2 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999931 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3681: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (12) E A B D C (10) E A B C D (7) E D A C B (6) D E C B A (5) C D B A E (5) C B A D E (5) A E B C D (5) D C E B A (4) B C D A E (4) E A C B D (3) D E B C A (3) D C B E A (3) B A C D E (3) A B E C D (3) A B C E D (3) E D A B C (2) D C E A B (2) D B C A E (2) C D E A B (2) B A C E D (2) A C B E D (2) E D C A B (1) E A D B C (1) E A C D B (1) D E B A C (1) C E D A B (1) B C A D E (1) B A E D C (1) Total count = 100 A B C D E A 0 -2 0 -6 -4 B 2 0 -4 -2 -8 C 0 4 0 -6 2 D 6 2 6 0 4 E 4 8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -6 -4 B 2 0 -4 -2 -8 C 0 4 0 -6 2 D 6 2 6 0 4 E 4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=31 C=13 A=13 B=11 so B is eliminated. Round 2 votes counts: D=32 E=31 A=19 C=18 so C is eliminated. Round 3 votes counts: D=43 E=32 A=25 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:203 C:200 A:194 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -6 -4 B 2 0 -4 -2 -8 C 0 4 0 -6 2 D 6 2 6 0 4 E 4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 -4 B 2 0 -4 -2 -8 C 0 4 0 -6 2 D 6 2 6 0 4 E 4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 -4 B 2 0 -4 -2 -8 C 0 4 0 -6 2 D 6 2 6 0 4 E 4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3682: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) D B E A C (8) E C A B D (7) B D A C E (7) B A C D E (7) A C B E D (6) B D E A C (5) E D B C A (4) D E B C A (4) C A E B D (4) B A C E D (4) D E C A B (3) D C A E B (3) C A B E D (3) A C B D E (3) E D C A B (2) E B A D C (2) D E B A C (2) D B E C A (2) C A E D B (2) B E D A C (2) E D C B A (1) E B D A C (1) E B A C D (1) D E C B A (1) D C A B E (1) D B A C E (1) C E A D B (1) B A D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 2 6 -14 B 6 0 4 8 2 C -2 -4 0 0 -12 D -6 -8 0 0 -4 E 14 -2 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 6 -14 B 6 0 4 8 2 C -2 -4 0 0 -12 D -6 -8 0 0 -4 E 14 -2 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=26 D=25 C=10 A=10 so C is eliminated. Round 2 votes counts: E=30 B=26 D=25 A=19 so A is eliminated. Round 3 votes counts: B=39 E=36 D=25 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:210 A:194 C:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 6 -14 B 6 0 4 8 2 C -2 -4 0 0 -12 D -6 -8 0 0 -4 E 14 -2 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 6 -14 B 6 0 4 8 2 C -2 -4 0 0 -12 D -6 -8 0 0 -4 E 14 -2 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 6 -14 B 6 0 4 8 2 C -2 -4 0 0 -12 D -6 -8 0 0 -4 E 14 -2 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3683: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (9) E D B C A (8) D B E C A (7) D B A C E (7) E C A B D (6) C E B A D (5) A C E B D (5) D E B A C (4) E C B D A (3) B D C E A (3) A D E B C (3) A D B C E (3) E C B A D (2) D E B C A (2) D A B E C (2) D A B C E (2) C A B E D (2) B E D C A (2) B C A D E (2) A E C D B (2) A E C B D (2) E D B A C (1) E D A C B (1) E C D A B (1) E B D C A (1) E B C D A (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A E C (1) D A E B C (1) C B E D A (1) C B E A D (1) C B A D E (1) B C D E A (1) A E D C B (1) A D C E B (1) A C E D B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -2 -4 -4 B 12 0 10 -2 -2 C 2 -10 0 -6 -4 D 4 2 6 0 6 E 4 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -4 -4 B 12 0 10 -2 -2 C 2 -10 0 -6 -4 D 4 2 6 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 E=24 C=10 B=8 so B is eliminated. Round 2 votes counts: D=32 A=29 E=26 C=13 so C is eliminated. Round 3 votes counts: A=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:209 D:209 E:202 C:191 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -4 -4 B 12 0 10 -2 -2 C 2 -10 0 -6 -4 D 4 2 6 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -4 -4 B 12 0 10 -2 -2 C 2 -10 0 -6 -4 D 4 2 6 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -4 -4 B 12 0 10 -2 -2 C 2 -10 0 -6 -4 D 4 2 6 0 6 E 4 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3684: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (12) B E A D C (8) D C B A E (5) D C A E B (5) D C A B E (5) E B A C D (4) C D E A B (4) E A B C D (3) D B C E A (3) B D C A E (3) B D A C E (3) B A E D C (3) E C A D B (2) D B C A E (2) C E D A B (2) C A D E B (2) B E D C A (2) B E D A C (2) B D E C A (2) B D C E A (2) B A D E C (2) A E C D B (2) A D B C E (2) A C E D B (2) A B E D C (2) E C B D A (1) E C B A D (1) E B C D A (1) E B C A D (1) E A C D B (1) D C B E A (1) D A C B E (1) C E A D B (1) C D E B A (1) B E C D A (1) B E A C D (1) B D E A C (1) A E B C D (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -18 -18 10 B 0 0 -2 -12 4 C 18 2 0 -12 18 D 18 12 12 0 18 E -10 -4 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 -18 10 B 0 0 -2 -12 4 C 18 2 0 -12 18 D 18 12 12 0 18 E -10 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=22 C=22 E=14 A=12 so A is eliminated. Round 2 votes counts: B=32 D=26 C=25 E=17 so E is eliminated. Round 3 votes counts: B=42 C=32 D=26 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:230 C:213 B:195 A:187 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -18 -18 10 B 0 0 -2 -12 4 C 18 2 0 -12 18 D 18 12 12 0 18 E -10 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 -18 10 B 0 0 -2 -12 4 C 18 2 0 -12 18 D 18 12 12 0 18 E -10 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 -18 10 B 0 0 -2 -12 4 C 18 2 0 -12 18 D 18 12 12 0 18 E -10 -4 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3685: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (19) E D B C A (12) A C E B D (6) D B E C A (5) B D C A E (5) A E B D C (5) D B C E A (4) E D C B A (3) A E C B D (3) E C D A B (2) E C A D B (2) E A D B C (2) D E B C A (2) D C B E A (2) C E D B A (2) C E A D B (2) C A B D E (2) B D E A C (2) B A C D E (2) E D C A B (1) E D B A C (1) E D A B C (1) E C D B A (1) E B D A C (1) E A C D B (1) E A B D C (1) C E D A B (1) C B D A E (1) C A E D B (1) B D E C A (1) B D A E C (1) B D A C E (1) B A D E C (1) B A D C E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 2 2 B -4 0 2 10 0 C -2 -2 0 -8 2 D -2 -10 8 0 4 E -2 0 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999451 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 2 B -4 0 2 10 0 C -2 -2 0 -8 2 D -2 -10 8 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=28 B=14 D=13 C=9 so C is eliminated. Round 2 votes counts: A=39 E=33 B=15 D=13 so D is eliminated. Round 3 votes counts: A=39 E=35 B=26 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:205 B:204 D:200 E:196 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 2 B -4 0 2 10 0 C -2 -2 0 -8 2 D -2 -10 8 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 2 B -4 0 2 10 0 C -2 -2 0 -8 2 D -2 -10 8 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 2 B -4 0 2 10 0 C -2 -2 0 -8 2 D -2 -10 8 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3686: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (19) C D B A E (12) B A E D C (10) D C B A E (8) B A E C D (6) E A B C D (5) D E A B C (5) C E A B D (5) C B A E D (4) E A D B C (3) D C E A B (3) D B A C E (3) C D B E A (3) D B A E C (2) C D E A B (2) C B D A E (2) B C A E D (2) E C A B D (1) E A C B D (1) D E A C B (1) D B C A E (1) C E A D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 12 16 2 B 6 0 14 12 6 C -12 -14 0 -10 -8 D -16 -12 10 0 -16 E -2 -6 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 16 2 B 6 0 14 12 6 C -12 -14 0 -10 -8 D -16 -12 10 0 -16 E -2 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=29 C=29 D=23 B=18 A=1 so A is eliminated. Round 2 votes counts: E=30 C=29 D=23 B=18 so B is eliminated. Round 3 votes counts: E=46 C=31 D=23 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:219 A:212 E:208 D:183 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 16 2 B 6 0 14 12 6 C -12 -14 0 -10 -8 D -16 -12 10 0 -16 E -2 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 16 2 B 6 0 14 12 6 C -12 -14 0 -10 -8 D -16 -12 10 0 -16 E -2 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 16 2 B 6 0 14 12 6 C -12 -14 0 -10 -8 D -16 -12 10 0 -16 E -2 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3687: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) C E B D A (7) E C B D A (6) D E A C B (6) A B D C E (6) E D C A B (5) C B E A D (5) A D B E C (5) D A E B C (4) B A C D E (4) E D A C B (3) C E D B A (3) B C E A D (3) B C A E D (3) A B D E C (3) D E C A B (2) D C B A E (2) D A E C B (2) D A B E C (2) A D E B C (2) A B C D E (2) E C B A D (1) E B A C D (1) E A D B C (1) E A B D C (1) D E C B A (1) D C B E A (1) C E B A D (1) C D B E A (1) C B E D A (1) C B D E A (1) C B D A E (1) B A E C D (1) B A D C E (1) A E B C D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -4 -12 -16 B 4 0 -14 4 -10 C 4 14 0 4 -14 D 12 -4 -4 0 -8 E 16 10 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 -12 -16 B 4 0 -14 4 -10 C 4 14 0 4 -14 D 12 -4 -4 0 -8 E 16 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 D=20 C=20 B=12 so B is eliminated. Round 2 votes counts: A=28 E=26 C=26 D=20 so D is eliminated. Round 3 votes counts: A=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 C:204 D:198 B:192 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 -12 -16 B 4 0 -14 4 -10 C 4 14 0 4 -14 D 12 -4 -4 0 -8 E 16 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -12 -16 B 4 0 -14 4 -10 C 4 14 0 4 -14 D 12 -4 -4 0 -8 E 16 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -12 -16 B 4 0 -14 4 -10 C 4 14 0 4 -14 D 12 -4 -4 0 -8 E 16 10 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3688: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (7) E D A B C (6) C A D E B (6) D A E C B (5) C B E A D (5) B E D A C (5) B E C A D (5) D E A B C (4) D A E B C (4) C A E D B (4) A D E C B (4) E A B D C (3) B C E D A (3) A E C D B (3) E A D C B (2) C D B A E (2) C D A B E (2) C B D A E (2) C B A E D (2) C B A D E (2) B E A D C (2) B D E A C (2) B C E A D (2) B C D A E (2) A E D C B (2) E C A B D (1) E B A D C (1) E A D B C (1) E A C B D (1) E A B C D (1) D B E A C (1) D B A E C (1) C D A E B (1) C A B D E (1) B E D C A (1) B D C E A (1) B D C A E (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 6 0 0 -8 B -6 0 6 0 -2 C 0 -6 0 6 -10 D 0 0 -6 0 -2 E 8 2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 0 0 -8 B -6 0 6 0 -2 C 0 -6 0 6 -10 D 0 0 -6 0 -2 E 8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999152 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=27 E=16 D=15 A=11 so A is eliminated. Round 2 votes counts: B=31 C=28 E=21 D=20 so D is eliminated. Round 3 votes counts: E=38 B=33 C=29 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:199 B:199 D:196 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 0 -8 B -6 0 6 0 -2 C 0 -6 0 6 -10 D 0 0 -6 0 -2 E 8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999152 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 -8 B -6 0 6 0 -2 C 0 -6 0 6 -10 D 0 0 -6 0 -2 E 8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999152 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 -8 B -6 0 6 0 -2 C 0 -6 0 6 -10 D 0 0 -6 0 -2 E 8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999152 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3689: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) E D B C A (6) D E A B C (6) C B E D A (6) C A B E D (6) B C E D A (6) A D E C B (6) A D E B C (6) A C B D E (6) D E B A C (5) B E D C A (5) B D E A C (5) D E B C A (3) A C D B E (3) E D C B A (2) C B A E D (2) B E C D A (2) A B D E C (2) A B C D E (2) E C D B A (1) E B D C A (1) E B C D A (1) C E D B A (1) C A E D B (1) B D A E C (1) B C E A D (1) B C A E D (1) B A D E C (1) B A C E D (1) A D C E B (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 10 -2 -2 B 2 0 14 4 6 C -10 -14 0 -2 -6 D 2 -4 2 0 10 E 2 -6 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -2 -2 B 2 0 14 4 6 C -10 -14 0 -2 -6 D 2 -4 2 0 10 E 2 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=23 C=16 D=14 E=11 so E is eliminated. Round 2 votes counts: A=36 B=25 D=22 C=17 so C is eliminated. Round 3 votes counts: A=43 B=33 D=24 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:205 A:202 E:196 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 -2 -2 B 2 0 14 4 6 C -10 -14 0 -2 -6 D 2 -4 2 0 10 E 2 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -2 -2 B 2 0 14 4 6 C -10 -14 0 -2 -6 D 2 -4 2 0 10 E 2 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -2 -2 B 2 0 14 4 6 C -10 -14 0 -2 -6 D 2 -4 2 0 10 E 2 -6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3690: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) E D B A C (6) D E B C A (6) A C B E D (6) C E D A B (5) A B E D C (5) D E C B A (4) D C B E A (4) B D E A C (4) E D C B A (3) D C E B A (3) D B E C A (3) D B C E A (3) C A B E D (3) C A B D E (3) A B C D E (3) D B E A C (2) C E A D B (2) C A E D B (2) B D C A E (2) B A D E C (2) A E B D C (2) A C E B D (2) E D C A B (1) E D B C A (1) E C D B A (1) E C D A B (1) E C A D B (1) E B A D C (1) E A D C B (1) D E B A C (1) C D E A B (1) C D B E A (1) C B A D E (1) C A E B D (1) B D A C E (1) B A D C E (1) B A C D E (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -4 -22 -10 B 18 0 6 -4 8 C 4 -6 0 -28 -8 D 22 4 28 0 8 E 10 -8 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -4 -22 -10 B 18 0 6 -4 8 C 4 -6 0 -28 -8 D 22 4 28 0 8 E 10 -8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=20 C=19 B=19 E=16 so E is eliminated. Round 2 votes counts: D=37 C=22 A=21 B=20 so B is eliminated. Round 3 votes counts: D=52 A=26 C=22 so C is eliminated. Round 4 votes counts: D=61 A=39 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:231 B:214 E:201 C:181 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -4 -22 -10 B 18 0 6 -4 8 C 4 -6 0 -28 -8 D 22 4 28 0 8 E 10 -8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -22 -10 B 18 0 6 -4 8 C 4 -6 0 -28 -8 D 22 4 28 0 8 E 10 -8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -22 -10 B 18 0 6 -4 8 C 4 -6 0 -28 -8 D 22 4 28 0 8 E 10 -8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3691: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (12) C E B A D (11) A D B E C (9) D A C E B (5) D A B C E (5) B E C D A (5) B D A E C (5) A D E C B (5) C B E D A (4) E C B A D (3) C E B D A (3) C E A D B (3) C D E A B (3) B E C A D (3) E B C A D (2) D A C B E (2) B C E D A (2) B C E A D (2) E B A C D (1) E A C D B (1) D C A B E (1) D B A E C (1) D A E B C (1) C E A B D (1) C D A E B (1) C B E A D (1) C A D E B (1) B E A D C (1) B E A C D (1) B C D E A (1) A E B D C (1) A D E B C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 8 6 -2 4 B -8 0 8 -4 12 C -6 -8 0 -2 -6 D 2 4 2 0 8 E -4 -12 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -2 4 B -8 0 8 -4 12 C -6 -8 0 -2 -6 D 2 4 2 0 8 E -4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=27 B=20 A=18 E=7 so E is eliminated. Round 2 votes counts: C=31 D=27 B=23 A=19 so A is eliminated. Round 3 votes counts: D=43 C=32 B=25 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:208 D:208 B:204 E:191 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -2 4 B -8 0 8 -4 12 C -6 -8 0 -2 -6 D 2 4 2 0 8 E -4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -2 4 B -8 0 8 -4 12 C -6 -8 0 -2 -6 D 2 4 2 0 8 E -4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -2 4 B -8 0 8 -4 12 C -6 -8 0 -2 -6 D 2 4 2 0 8 E -4 -12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3692: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) C D B E A (7) A E B D C (6) E A C D B (4) B D C A E (4) B D A C E (4) B C D E A (4) B A D E C (4) E C A D B (3) E A C B D (3) E A B C D (3) D C A B E (3) C E D B A (3) C B D E A (3) A D B E C (3) E C B D A (2) D B A C E (2) C E B D A (2) C E A D B (2) C B E D A (2) A E D C B (2) A D E B C (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C D A (1) E B A D C (1) E B A C D (1) E A B D C (1) D B C A E (1) D A C E B (1) D A B C E (1) C E D A B (1) C D E A B (1) C D B A E (1) C D A E B (1) B E C A D (1) B D C E A (1) B D A E C (1) B A E D C (1) B A D C E (1) A E D B C (1) A E C D B (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -6 -8 4 B 12 0 -6 2 10 C 6 6 0 -4 6 D 8 -2 4 0 12 E -4 -10 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -8 4 B 12 0 -6 2 10 C 6 6 0 -4 6 D 8 -2 4 0 12 E -4 -10 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 E=21 B=21 A=19 D=16 so D is eliminated. Round 2 votes counts: C=34 B=24 E=21 A=21 so E is eliminated. Round 3 votes counts: C=41 A=32 B=27 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:211 B:209 C:207 A:189 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -8 4 B 12 0 -6 2 10 C 6 6 0 -4 6 D 8 -2 4 0 12 E -4 -10 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -8 4 B 12 0 -6 2 10 C 6 6 0 -4 6 D 8 -2 4 0 12 E -4 -10 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -8 4 B 12 0 -6 2 10 C 6 6 0 -4 6 D 8 -2 4 0 12 E -4 -10 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3693: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A B E D C (9) E D A C B (7) B A C E D (7) B A C D E (7) E D C A B (6) D E C A B (6) A E D B C (6) B C D E A (5) B C A E D (4) D E C B A (3) C B E D A (3) C B D E A (3) A E D C B (3) A B D E C (3) D E A C B (2) D C E B A (2) C E D B A (2) E D C B A (1) E A D C B (1) E A C D B (1) D B E C A (1) B D E A C (1) B C E D A (1) B C A D E (1) B A D E C (1) B A D C E (1) A D E B C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 4 -6 -10 B 6 0 0 -4 -2 C -4 0 0 -10 -4 D 6 4 10 0 -4 E 10 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 -6 -10 B 6 0 0 -4 -2 C -4 0 0 -10 -4 D 6 4 10 0 -4 E 10 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=24 C=18 E=16 D=14 so D is eliminated. Round 2 votes counts: B=29 E=27 A=24 C=20 so C is eliminated. Round 3 votes counts: E=41 B=35 A=24 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 D:208 B:200 A:191 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 -6 -10 B 6 0 0 -4 -2 C -4 0 0 -10 -4 D 6 4 10 0 -4 E 10 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -6 -10 B 6 0 0 -4 -2 C -4 0 0 -10 -4 D 6 4 10 0 -4 E 10 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -6 -10 B 6 0 0 -4 -2 C -4 0 0 -10 -4 D 6 4 10 0 -4 E 10 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3694: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) A B D C E (9) E C D B A (7) B D A E C (7) A D B C E (7) C E A D B (5) E C B D A (4) D B E A C (4) B D E C A (4) A C E D B (4) D B E C A (3) D B A E C (3) C A E D B (3) B D E A C (3) B A D E C (3) A C B D E (3) E B C D A (2) D E B C A (2) D A B C E (2) C E D B A (2) A C E B D (2) A C B E D (2) A B D E C (2) E D B C A (1) E C B A D (1) E C A B D (1) C E D A B (1) C D A E B (1) C A E B D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 8 6 8 2 B -8 0 4 6 8 C -6 -4 0 -2 6 D -8 -6 2 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 8 2 B -8 0 4 6 8 C -6 -4 0 -2 6 D -8 -6 2 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=22 B=17 E=16 D=14 so D is eliminated. Round 2 votes counts: A=33 B=27 C=22 E=18 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:205 D:199 C:197 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 8 2 B -8 0 4 6 8 C -6 -4 0 -2 6 D -8 -6 2 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 8 2 B -8 0 4 6 8 C -6 -4 0 -2 6 D -8 -6 2 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 8 2 B -8 0 4 6 8 C -6 -4 0 -2 6 D -8 -6 2 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3695: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) B C D E A (7) B E C D A (6) C D A B E (5) B D E C A (5) B D C E A (5) E A D B C (4) C B D A E (4) B E D C A (4) B E D A C (4) A E C D B (4) E A D C B (3) B D C A E (3) B C E D A (3) B C D A E (3) A E D B C (3) A D E C B (3) E C B A D (2) E A C D B (2) C D B A E (2) A E D C B (2) A D C E B (2) E C A D B (1) E C A B D (1) E A C B D (1) E A B D C (1) D C B A E (1) D C A B E (1) D A C B E (1) D A B E C (1) C E A B D (1) C A E D B (1) C A D E B (1) C A D B E (1) B D E A C (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -14 -12 -18 B 16 0 18 18 14 C 14 -18 0 -6 -14 D 12 -18 6 0 -4 E 18 -14 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -12 -18 B 16 0 18 18 14 C 14 -18 0 -6 -14 D 12 -18 6 0 -4 E 18 -14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=23 A=17 C=15 D=4 so D is eliminated. Round 2 votes counts: B=41 E=23 A=19 C=17 so C is eliminated. Round 3 votes counts: B=48 A=28 E=24 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:211 D:198 C:188 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -14 -12 -18 B 16 0 18 18 14 C 14 -18 0 -6 -14 D 12 -18 6 0 -4 E 18 -14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -12 -18 B 16 0 18 18 14 C 14 -18 0 -6 -14 D 12 -18 6 0 -4 E 18 -14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -12 -18 B 16 0 18 18 14 C 14 -18 0 -6 -14 D 12 -18 6 0 -4 E 18 -14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3696: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) D B C E A (6) A B C E D (6) D E C A B (5) B A C E D (5) A E B C D (5) A B E C D (5) B A C D E (4) E C A B D (3) E A C B D (3) D E C B A (3) D C B E A (3) D B C A E (3) A B D C E (3) E C D B A (2) E C D A B (2) E C B A D (2) E C A D B (2) D A E C B (2) D A B C E (2) C E B D A (2) B D A C E (2) A E D C B (2) A E C D B (2) A E C B D (2) A D B E C (2) A D B C E (2) A B C D E (2) E D C B A (1) E A D C B (1) E A C D B (1) D E A C B (1) D C A E B (1) D B A C E (1) C E B A D (1) C B E D A (1) B D C E A (1) B C E D A (1) B A D C E (1) Total count = 100 A B C D E A 0 8 8 8 4 B -8 0 2 -2 0 C -8 -2 0 2 8 D -8 2 -2 0 2 E -4 0 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 4 B -8 0 2 -2 0 C -8 -2 0 2 8 D -8 2 -2 0 2 E -4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=31 E=17 B=14 C=4 so C is eliminated. Round 2 votes counts: D=34 A=31 E=20 B=15 so B is eliminated. Round 3 votes counts: A=41 D=37 E=22 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:200 D:197 B:196 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 4 B -8 0 2 -2 0 C -8 -2 0 2 8 D -8 2 -2 0 2 E -4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 4 B -8 0 2 -2 0 C -8 -2 0 2 8 D -8 2 -2 0 2 E -4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 4 B -8 0 2 -2 0 C -8 -2 0 2 8 D -8 2 -2 0 2 E -4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3697: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) E B A D C (5) D E C A B (5) B A C E D (5) A D B C E (5) C E B D A (4) B A E C D (4) A B D C E (4) A B C D E (4) E D C B A (3) D C E A B (3) D A E B C (3) D A C B E (3) C B E A D (3) A D B E C (3) A B C E D (3) E D A B C (2) E C D B A (2) E C B A D (2) D E A C B (2) D C A E B (2) C E D B A (2) C E B A D (2) C D E B A (2) C D B A E (2) C B A E D (2) C B A D E (2) B C E A D (2) E D B C A (1) E D B A C (1) E C B D A (1) E B A C D (1) D E C B A (1) D C A B E (1) D A B E C (1) D A B C E (1) C A D B E (1) B E A C D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 14 0 -6 B -4 0 10 -8 -4 C -14 -10 0 -8 6 D 0 8 8 0 6 E 6 4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.388624 B: 0.000000 C: 0.000000 D: 0.611376 E: 0.000000 Sum of squares = 0.524809336831 Cumulative probabilities = A: 0.388624 B: 0.388624 C: 0.388624 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 0 -6 B -4 0 10 -8 -4 C -14 -10 0 -8 6 D 0 8 8 0 6 E 6 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499778 B: 0.000000 C: 0.000000 D: 0.500222 E: 0.000000 Sum of squares = 0.500000098382 Cumulative probabilities = A: 0.499778 B: 0.499778 C: 0.499778 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=20 A=20 E=18 B=12 so B is eliminated. Round 2 votes counts: D=30 A=29 C=22 E=19 so E is eliminated. Round 3 votes counts: D=37 A=36 C=27 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:206 E:199 B:197 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 14 0 -6 B -4 0 10 -8 -4 C -14 -10 0 -8 6 D 0 8 8 0 6 E 6 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499778 B: 0.000000 C: 0.000000 D: 0.500222 E: 0.000000 Sum of squares = 0.500000098382 Cumulative probabilities = A: 0.499778 B: 0.499778 C: 0.499778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 0 -6 B -4 0 10 -8 -4 C -14 -10 0 -8 6 D 0 8 8 0 6 E 6 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499778 B: 0.000000 C: 0.000000 D: 0.500222 E: 0.000000 Sum of squares = 0.500000098382 Cumulative probabilities = A: 0.499778 B: 0.499778 C: 0.499778 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 0 -6 B -4 0 10 -8 -4 C -14 -10 0 -8 6 D 0 8 8 0 6 E 6 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499778 B: 0.000000 C: 0.000000 D: 0.500222 E: 0.000000 Sum of squares = 0.500000098382 Cumulative probabilities = A: 0.499778 B: 0.499778 C: 0.499778 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) A C D E B (6) E B D A C (5) C D A E B (5) B E D A C (5) B E A D C (5) A E D C B (5) E D B A C (4) B C E D A (4) B A E D C (4) A D E C B (4) C A B D E (3) B E D C A (3) B E C D A (3) B C A E D (3) B A E C D (3) A B E D C (3) E D A C B (2) C D E B A (2) C D E A B (2) C B D E A (2) C B A D E (2) C A D B E (2) B C A D E (2) A D C E B (2) E D C B A (1) E D B C A (1) E D A B C (1) D E C B A (1) D E C A B (1) D E A C B (1) D C E A B (1) D A C E B (1) C D B E A (1) C B D A E (1) B C D E A (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 4 4 8 B 6 0 -6 0 -4 C -4 6 0 0 -4 D -4 0 0 0 -4 E -8 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999984 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 4 8 B 6 0 -6 0 -4 C -4 6 0 0 -4 D -4 0 0 0 -4 E -8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=27 A=21 E=14 D=5 so D is eliminated. Round 2 votes counts: B=33 C=28 A=22 E=17 so E is eliminated. Round 3 votes counts: B=43 C=31 A=26 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:205 E:202 C:199 B:198 D:196 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 4 4 8 B 6 0 -6 0 -4 C -4 6 0 0 -4 D -4 0 0 0 -4 E -8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 4 8 B 6 0 -6 0 -4 C -4 6 0 0 -4 D -4 0 0 0 -4 E -8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 4 8 B 6 0 -6 0 -4 C -4 6 0 0 -4 D -4 0 0 0 -4 E -8 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3699: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) A C D B E (7) C B D E A (5) A E C D B (5) D B A C E (4) D A B E C (4) C E B A D (4) C A B D E (4) A E D C B (4) A C E D B (4) E D B A C (3) E A D B C (3) D A B C E (3) C E B D A (3) C E A B D (3) C B D A E (3) C B A D E (3) B C D E A (3) E B D C A (2) C B E D A (2) C A E B D (2) C A B E D (2) B D E C A (2) B D C E A (2) A E D B C (2) A D B C E (2) E D B C A (1) E C B A D (1) E C A B D (1) E B C D A (1) E A C D B (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A E C (1) B D C A E (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -8 0 8 B 2 0 -24 4 6 C 8 24 0 24 18 D 0 -4 -24 0 -2 E -8 -6 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 0 8 B 2 0 -24 4 6 C 8 24 0 24 18 D 0 -4 -24 0 -2 E -8 -6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=26 E=20 D=15 B=8 so B is eliminated. Round 2 votes counts: C=34 A=26 E=20 D=20 so E is eliminated. Round 3 votes counts: C=44 A=30 D=26 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:237 A:199 B:194 D:185 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 0 8 B 2 0 -24 4 6 C 8 24 0 24 18 D 0 -4 -24 0 -2 E -8 -6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 0 8 B 2 0 -24 4 6 C 8 24 0 24 18 D 0 -4 -24 0 -2 E -8 -6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 0 8 B 2 0 -24 4 6 C 8 24 0 24 18 D 0 -4 -24 0 -2 E -8 -6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3700: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) B A E C D (8) C B A E D (7) C A E B D (6) B E A D C (5) B E A C D (5) D E A B C (4) D C E A B (4) D C A E B (4) D B C E A (4) C D A E B (4) B C A E D (4) D E A C B (3) C A E D B (3) E A D C B (2) E A B C D (2) D C B A E (2) C D B A E (2) B D E A C (2) B D A E C (2) B C D A E (2) A E C B D (2) A B E C D (2) E A C B D (1) E A B D C (1) D E C A B (1) D C E B A (1) D B C A E (1) C E A D B (1) C E A B D (1) C D E A B (1) B D C A E (1) B A E D C (1) Total count = 100 A B C D E A 0 -16 2 2 2 B 16 0 10 4 18 C -2 -10 0 2 -4 D -2 -4 -2 0 -2 E -2 -18 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 2 2 2 B 16 0 10 4 18 C -2 -10 0 2 -4 D -2 -4 -2 0 -2 E -2 -18 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=30 C=25 E=6 A=4 so A is eliminated. Round 2 votes counts: D=35 B=32 C=25 E=8 so E is eliminated. Round 3 votes counts: D=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:195 D:195 C:193 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 2 2 2 B 16 0 10 4 18 C -2 -10 0 2 -4 D -2 -4 -2 0 -2 E -2 -18 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 2 2 B 16 0 10 4 18 C -2 -10 0 2 -4 D -2 -4 -2 0 -2 E -2 -18 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 2 2 B 16 0 10 4 18 C -2 -10 0 2 -4 D -2 -4 -2 0 -2 E -2 -18 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3701: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) B D C A E (9) C D A B E (8) A E C D B (8) E A C D B (7) E A B D C (6) E A B C D (6) A C D E B (5) A C D B E (5) D C B A E (4) B E A D C (4) E B D C A (3) C D B A E (3) E C D B A (2) B E D C A (2) A E B D C (2) E C D A B (1) E B A D C (1) E A C B D (1) D C B E A (1) C D E B A (1) C D A E B (1) C A D E B (1) C A D B E (1) B E D A C (1) B D E C A (1) B A D E C (1) A D B C E (1) A C E D B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 2 4 4 B -12 0 0 0 8 C -2 0 0 2 6 D -4 0 -2 0 8 E -4 -8 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 4 4 B -12 0 0 0 8 C -2 0 0 2 6 D -4 0 -2 0 8 E -4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=27 A=24 C=15 D=5 so D is eliminated. Round 2 votes counts: B=29 E=27 A=24 C=20 so C is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:203 D:201 B:198 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 4 4 B -12 0 0 0 8 C -2 0 0 2 6 D -4 0 -2 0 8 E -4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 4 4 B -12 0 0 0 8 C -2 0 0 2 6 D -4 0 -2 0 8 E -4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 4 4 B -12 0 0 0 8 C -2 0 0 2 6 D -4 0 -2 0 8 E -4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998343 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3702: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (11) D B C A E (8) A E C B D (8) E A C B D (7) C D B E A (6) A E B C D (6) C D E B A (5) D C B E A (4) D B C E A (4) C E D A B (4) E C A B D (3) B A E D C (3) A B E D C (3) C E A B D (2) C D E A B (2) A E B D C (2) E B C A D (1) D C A B E (1) D B A E C (1) D B A C E (1) C E D B A (1) C D A E B (1) C A E D B (1) C A D E B (1) B D C E A (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -12 0 -2 B -6 0 -14 -2 -10 C 12 14 0 18 6 D 0 2 -18 0 -4 E 2 10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 0 -2 B -6 0 -14 -2 -10 C 12 14 0 18 6 D 0 2 -18 0 -4 E 2 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=20 D=19 B=16 E=11 so E is eliminated. Round 2 votes counts: C=37 A=27 D=19 B=17 so B is eliminated. Round 3 votes counts: C=38 D=31 A=31 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:205 A:196 D:190 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 0 -2 B -6 0 -14 -2 -10 C 12 14 0 18 6 D 0 2 -18 0 -4 E 2 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 0 -2 B -6 0 -14 -2 -10 C 12 14 0 18 6 D 0 2 -18 0 -4 E 2 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 0 -2 B -6 0 -14 -2 -10 C 12 14 0 18 6 D 0 2 -18 0 -4 E 2 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3703: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (11) D A E C B (10) B E D A C (10) D A C E B (9) B E C A D (9) E B D A C (7) E D A B C (4) C A D B E (4) E B C A D (3) C D A B E (3) C B A E D (3) C A D E B (3) B E D C A (3) B E C D A (3) C B D A E (2) B D E A C (2) B C A E D (2) E D A C B (1) E C A D B (1) E A D B C (1) E A C D B (1) E A B D C (1) D E A B C (1) D A E B C (1) C D B A E (1) C B A D E (1) B D C A E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -16 0 -16 -16 B 16 0 18 16 10 C 0 -18 0 -6 -18 D 16 -16 6 0 -20 E 16 -10 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -16 -16 B 16 0 18 16 10 C 0 -18 0 -6 -18 D 16 -16 6 0 -20 E 16 -10 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=21 E=19 C=17 A=2 so A is eliminated. Round 2 votes counts: B=41 D=23 E=19 C=17 so C is eliminated. Round 3 votes counts: B=47 D=34 E=19 so E is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:230 E:222 D:193 C:179 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -16 -16 B 16 0 18 16 10 C 0 -18 0 -6 -18 D 16 -16 6 0 -20 E 16 -10 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -16 -16 B 16 0 18 16 10 C 0 -18 0 -6 -18 D 16 -16 6 0 -20 E 16 -10 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -16 -16 B 16 0 18 16 10 C 0 -18 0 -6 -18 D 16 -16 6 0 -20 E 16 -10 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3704: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E B A C D (7) E A B D C (7) E A B C D (7) C B E A D (7) A E B D C (7) E A D B C (6) D A C E B (5) B C E A D (5) D C A B E (4) C B D A E (4) D A E B C (3) E D A B C (2) E B C A D (2) D E A C B (2) D C A E B (2) D A E C B (2) D A C B E (2) C D B E A (2) C B D E A (2) B E A C D (2) E D C A B (1) E D A C B (1) E C B A D (1) D E C A B (1) D C E A B (1) C B A E D (1) C B A D E (1) B C A D E (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 10 10 12 -12 B -10 0 2 8 -16 C -10 -2 0 4 -4 D -12 -8 -4 0 -12 E 12 16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 10 12 -12 B -10 0 2 8 -16 C -10 -2 0 4 -4 D -12 -8 -4 0 -12 E 12 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=27 D=22 A=9 B=8 so B is eliminated. Round 2 votes counts: E=36 C=33 D=22 A=9 so A is eliminated. Round 3 votes counts: E=43 C=33 D=24 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:210 C:194 B:192 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 10 12 -12 B -10 0 2 8 -16 C -10 -2 0 4 -4 D -12 -8 -4 0 -12 E 12 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 12 -12 B -10 0 2 8 -16 C -10 -2 0 4 -4 D -12 -8 -4 0 -12 E 12 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 12 -12 B -10 0 2 8 -16 C -10 -2 0 4 -4 D -12 -8 -4 0 -12 E 12 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3705: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) D C E B A (10) E B C D A (6) A B E C D (6) B E A C D (5) A B E D C (5) D A C E B (4) A D C B E (4) E D B A C (3) D C E A B (3) D C A E B (3) D C A B E (3) A C B D E (3) A B C E D (3) E C B D A (2) D A E C B (2) C D A B E (2) C B E D A (2) C B E A D (2) A B C D E (2) E D B C A (1) E B D C A (1) E B D A C (1) E B C A D (1) E B A D C (1) D E C B A (1) D E B C A (1) D A E B C (1) D A C B E (1) C E B D A (1) B E C D A (1) B E C A D (1) B C A E D (1) B A E C D (1) A D E B C (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 -22 -10 B 8 0 -10 -8 -10 C 8 10 0 2 14 D 22 8 -2 0 12 E 10 10 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -22 -10 B 8 0 -10 -8 -10 C 8 10 0 2 14 D 22 8 -2 0 12 E 10 10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 C=19 E=16 B=9 so B is eliminated. Round 2 votes counts: D=29 A=28 E=23 C=20 so C is eliminated. Round 3 votes counts: D=43 A=29 E=28 so E is eliminated. Round 4 votes counts: D=61 A=39 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:217 E:197 B:190 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -22 -10 B 8 0 -10 -8 -10 C 8 10 0 2 14 D 22 8 -2 0 12 E 10 10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -22 -10 B 8 0 -10 -8 -10 C 8 10 0 2 14 D 22 8 -2 0 12 E 10 10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -22 -10 B 8 0 -10 -8 -10 C 8 10 0 2 14 D 22 8 -2 0 12 E 10 10 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3706: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) D E B A C (9) A B E D C (8) C A B E D (7) D E B C A (5) C D E B A (5) E B D A C (4) C D E A B (4) A C B D E (4) A B E C D (4) D B A E C (3) C A D B E (3) C A B D E (3) B E A D C (3) A C B E D (3) A B C D E (3) D E C B A (2) D B E A C (2) C E D B A (2) C A D E B (2) B A E D C (2) E D C B A (1) E D B C A (1) E C D B A (1) D C B A E (1) C E A B D (1) C D A E B (1) C D A B E (1) C A E D B (1) B E D A C (1) B D E A C (1) A D B C E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 18 -6 -2 B 4 0 16 -8 4 C -18 -16 0 -8 -14 D 6 8 8 0 2 E 2 -4 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 18 -6 -2 B 4 0 16 -8 4 C -18 -16 0 -8 -14 D 6 8 8 0 2 E 2 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=25 D=22 E=16 B=7 so B is eliminated. Round 2 votes counts: C=30 A=27 D=23 E=20 so E is eliminated. Round 3 votes counts: D=39 C=31 A=30 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:208 E:205 A:203 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 18 -6 -2 B 4 0 16 -8 4 C -18 -16 0 -8 -14 D 6 8 8 0 2 E 2 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 18 -6 -2 B 4 0 16 -8 4 C -18 -16 0 -8 -14 D 6 8 8 0 2 E 2 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 18 -6 -2 B 4 0 16 -8 4 C -18 -16 0 -8 -14 D 6 8 8 0 2 E 2 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3707: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) A B C D E (11) D C A B E (10) E D C A B (7) E B A C D (5) E A B C D (5) C D A B E (5) E D C B A (4) E D B C A (4) D C E A B (4) D C B A E (4) B A C D E (3) E B D A C (2) D C E B A (2) D C B E A (2) C A D B E (2) B E A C D (2) B A E C D (2) B A D C E (2) A C B D E (2) E B D C A (1) E A C D B (1) D E C A B (1) D C A E B (1) C D B A E (1) B E A D C (1) B A E D C (1) A E C B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 4 -6 B -4 0 6 4 0 C -4 -6 0 -16 0 D -4 -4 16 0 0 E 6 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.486776 C: 0.000000 D: 0.000000 E: 0.513224 Sum of squares = 0.50034975307 Cumulative probabilities = A: 0.000000 B: 0.486776 C: 0.486776 D: 0.486776 E: 1.000000 A B C D E A 0 4 4 4 -6 B -4 0 6 4 0 C -4 -6 0 -16 0 D -4 -4 16 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 D=24 A=16 B=11 C=8 so C is eliminated. Round 2 votes counts: E=41 D=30 A=18 B=11 so B is eliminated. Round 3 votes counts: E=44 D=30 A=26 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:204 A:203 B:203 E:203 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 4 -6 B -4 0 6 4 0 C -4 -6 0 -16 0 D -4 -4 16 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 -6 B -4 0 6 4 0 C -4 -6 0 -16 0 D -4 -4 16 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 -6 B -4 0 6 4 0 C -4 -6 0 -16 0 D -4 -4 16 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3708: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (18) B D A C E (14) B D E A C (9) E C A B D (6) C A E D B (5) E D C A B (4) B D A E C (4) E B C A D (3) D A B C E (3) B E D C A (3) E C D A B (2) E C B A D (2) E B C D A (2) D B A C E (2) C E A D B (2) C A E B D (2) B E C D A (2) B C A E D (2) B A C D E (2) A D C E B (2) A C E D B (2) A C D E B (2) E B D C A (1) D E A C B (1) D A E C B (1) D A C E B (1) B E C A D (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -10 2 -12 B -6 0 -4 8 -12 C 10 4 0 8 -18 D -2 -8 -8 0 -14 E 12 12 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -10 2 -12 B -6 0 -4 8 -12 C 10 4 0 8 -18 D -2 -8 -8 0 -14 E 12 12 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=37 C=9 D=8 A=8 so D is eliminated. Round 2 votes counts: E=39 B=39 A=13 C=9 so C is eliminated. Round 3 votes counts: E=41 B=39 A=20 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:202 A:193 B:193 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -10 2 -12 B -6 0 -4 8 -12 C 10 4 0 8 -18 D -2 -8 -8 0 -14 E 12 12 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 2 -12 B -6 0 -4 8 -12 C 10 4 0 8 -18 D -2 -8 -8 0 -14 E 12 12 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 2 -12 B -6 0 -4 8 -12 C 10 4 0 8 -18 D -2 -8 -8 0 -14 E 12 12 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3709: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (12) B D A C E (11) D A B E C (8) B C D A E (7) E A D C B (5) D A E B C (5) C E B A D (5) B C A D E (5) A D E C B (5) E A C D B (4) C B E A D (4) B D A E C (4) D B A E C (3) B D C A E (3) B C E D A (3) A E D C B (3) C B E D A (2) E D A C B (1) D A E C B (1) C B A D E (1) C A D B E (1) C A B E D (1) B E C D A (1) B C D E A (1) B C A E D (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 6 0 24 B 2 0 10 -2 16 C -6 -10 0 -4 -8 D 0 2 4 0 16 E -24 -16 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.228440 B: 0.000000 C: 0.000000 D: 0.771560 E: 0.000000 Sum of squares = 0.647489442387 Cumulative probabilities = A: 0.228440 B: 0.228440 C: 0.228440 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 0 24 B 2 0 10 -2 16 C -6 -10 0 -4 -8 D 0 2 4 0 16 E -24 -16 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499644 B: 0.000000 C: 0.000000 D: 0.500356 E: 0.000000 Sum of squares = 0.500000253398 Cumulative probabilities = A: 0.499644 B: 0.499644 C: 0.499644 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=22 D=17 C=14 A=11 so A is eliminated. Round 2 votes counts: B=36 E=25 D=25 C=14 so C is eliminated. Round 3 votes counts: B=44 E=30 D=26 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:213 D:211 C:186 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 0 24 B 2 0 10 -2 16 C -6 -10 0 -4 -8 D 0 2 4 0 16 E -24 -16 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499644 B: 0.000000 C: 0.000000 D: 0.500356 E: 0.000000 Sum of squares = 0.500000253398 Cumulative probabilities = A: 0.499644 B: 0.499644 C: 0.499644 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 24 B 2 0 10 -2 16 C -6 -10 0 -4 -8 D 0 2 4 0 16 E -24 -16 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499644 B: 0.000000 C: 0.000000 D: 0.500356 E: 0.000000 Sum of squares = 0.500000253398 Cumulative probabilities = A: 0.499644 B: 0.499644 C: 0.499644 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 24 B 2 0 10 -2 16 C -6 -10 0 -4 -8 D 0 2 4 0 16 E -24 -16 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499644 B: 0.000000 C: 0.000000 D: 0.500356 E: 0.000000 Sum of squares = 0.500000253398 Cumulative probabilities = A: 0.499644 B: 0.499644 C: 0.499644 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3710: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (13) A B D E C (13) D B C E A (5) D B A C E (5) B D A E C (5) E C A B D (4) D B C A E (4) C D B E A (4) A B D C E (4) D C B A E (3) E A C B D (2) E A B D C (2) D B A E C (2) D A B C E (2) C E D A B (2) C E A D B (2) C E A B D (2) C D E B A (2) C A D B E (2) B D E A C (2) A E B D C (2) A C D B E (2) A B E D C (2) E C B D A (1) E C B A D (1) E B D A C (1) E A B C D (1) D C B E A (1) D B E A C (1) D A C B E (1) C D B A E (1) C A E D B (1) C A D E B (1) B A D E C (1) A D B C E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 2 -10 8 B 4 0 6 -10 24 C -2 -6 0 -14 20 D 10 10 14 0 24 E -8 -24 -20 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -10 8 B 4 0 6 -10 24 C -2 -6 0 -14 20 D 10 10 14 0 24 E -8 -24 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=26 D=24 E=12 B=8 so B is eliminated. Round 2 votes counts: D=31 C=30 A=27 E=12 so E is eliminated. Round 3 votes counts: C=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:229 B:212 C:199 A:198 E:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -10 8 B 4 0 6 -10 24 C -2 -6 0 -14 20 D 10 10 14 0 24 E -8 -24 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -10 8 B 4 0 6 -10 24 C -2 -6 0 -14 20 D 10 10 14 0 24 E -8 -24 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -10 8 B 4 0 6 -10 24 C -2 -6 0 -14 20 D 10 10 14 0 24 E -8 -24 -20 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3711: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C A D E B (9) A C B E D (8) E B D C A (6) B E D C A (6) D E B C A (5) A C D E B (5) A C D B E (5) D C A E B (4) B E A D C (4) B A E C D (4) A B E C D (4) E D B C A (3) D C E B A (3) D C E A B (3) B E A C D (3) E B A C D (2) A C B D E (2) E D C B A (1) E B D A C (1) E B C D A (1) E A B C D (1) D E C B A (1) D B E C A (1) D B A E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C D A E B (1) B D E C A (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 6 2 -8 B 6 0 10 12 2 C -6 -10 0 -4 -10 D -2 -12 4 0 -14 E 8 -2 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 2 -8 B 6 0 10 12 2 C -6 -10 0 -4 -10 D -2 -12 4 0 -14 E 8 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=26 D=19 E=15 C=12 so C is eliminated. Round 2 votes counts: A=35 B=28 D=20 E=17 so E is eliminated. Round 3 votes counts: B=38 A=38 D=24 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:215 A:197 D:188 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 2 -8 B 6 0 10 12 2 C -6 -10 0 -4 -10 D -2 -12 4 0 -14 E 8 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 2 -8 B 6 0 10 12 2 C -6 -10 0 -4 -10 D -2 -12 4 0 -14 E 8 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 2 -8 B 6 0 10 12 2 C -6 -10 0 -4 -10 D -2 -12 4 0 -14 E 8 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3712: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) A D B C E (11) A B C E D (10) D E C B A (7) E C B A D (5) D A B C E (5) E C D B A (4) B E C A D (4) B C A E D (4) E C B D A (3) D E C A B (3) B A C E D (3) A D C B E (3) E B C A D (2) D C E A B (2) D A C B E (2) D A B E C (2) C E B A D (2) C B A E D (2) B C E A D (2) B A E C D (2) E D B C A (1) E B C D A (1) D E B C A (1) D B E C A (1) D B A E C (1) D A E C B (1) C E A B D (1) C B E A D (1) B E A C D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 10 18 B -6 0 4 -10 12 C -8 -4 0 -4 22 D -10 10 4 0 2 E -18 -12 -22 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 10 18 B -6 0 4 -10 12 C -8 -4 0 -4 22 D -10 10 4 0 2 E -18 -12 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=26 E=16 B=16 C=6 so C is eliminated. Round 2 votes counts: D=36 A=26 E=19 B=19 so E is eliminated. Round 3 votes counts: D=41 B=32 A=27 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:221 C:203 D:203 B:200 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 10 18 B -6 0 4 -10 12 C -8 -4 0 -4 22 D -10 10 4 0 2 E -18 -12 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 10 18 B -6 0 4 -10 12 C -8 -4 0 -4 22 D -10 10 4 0 2 E -18 -12 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 10 18 B -6 0 4 -10 12 C -8 -4 0 -4 22 D -10 10 4 0 2 E -18 -12 -22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3713: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) D B A E C (6) C B A E D (5) E D C A B (4) D E C A B (4) B A C D E (4) E D A C B (3) E C D A B (3) E C A D B (3) C A B E D (3) B D A C E (3) B C A D E (3) B A D C E (3) B A C E D (3) E C A B D (2) D E C B A (2) D E A C B (2) D C E B A (2) D B E A C (2) D A B E C (2) C E A D B (2) C D E B A (2) C B D A E (2) B D C A E (2) B D A E C (2) A E C B D (2) A E B D C (2) A E B C D (2) A B C E D (2) E A D C B (1) E A D B C (1) D E B C A (1) D E B A C (1) D C B E A (1) D B C E A (1) C E D A B (1) C E B D A (1) C E A B D (1) C B E A D (1) C A E B D (1) B C D A E (1) B C A E D (1) B A E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 2 -10 4 B 0 0 4 -2 0 C -2 -4 0 -8 -10 D 10 2 8 0 8 E -4 0 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -10 4 B 0 0 4 -2 0 C -2 -4 0 -8 -10 D 10 2 8 0 8 E -4 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=23 C=19 E=17 A=10 so A is eliminated. Round 2 votes counts: D=31 B=27 E=23 C=19 so C is eliminated. Round 3 votes counts: B=38 D=33 E=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:201 E:199 A:198 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -10 4 B 0 0 4 -2 0 C -2 -4 0 -8 -10 D 10 2 8 0 8 E -4 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -10 4 B 0 0 4 -2 0 C -2 -4 0 -8 -10 D 10 2 8 0 8 E -4 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -10 4 B 0 0 4 -2 0 C -2 -4 0 -8 -10 D 10 2 8 0 8 E -4 0 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3714: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (6) B D A C E (5) D A E B C (4) C E B A D (4) A D B E C (4) A C E D B (4) E C B D A (3) D B A E C (3) C E A D B (3) C B E D A (3) C A B E D (3) B D A E C (3) A D B C E (3) A B C D E (3) E D A C B (2) E C D A B (2) E C A D B (2) D E B A C (2) C B E A D (2) B E C D A (2) B D E C A (2) B D E A C (2) B D C A E (2) B C D A E (2) B C A D E (2) B A D C E (2) B A C D E (2) A D E C B (2) A D E B C (2) A B D C E (2) E D C B A (1) E D C A B (1) E D B C A (1) E B C D A (1) E A C D B (1) D E A C B (1) D E A B C (1) D B E A C (1) D A B E C (1) C B A E D (1) C A E B D (1) B D C E A (1) B C E D A (1) A E D C B (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 8 -6 10 B 8 0 8 14 6 C -8 -8 0 2 10 D 6 -14 -2 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -6 10 B 8 0 8 14 6 C -8 -8 0 2 10 D 6 -14 -2 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 C=23 E=14 D=13 so D is eliminated. Round 2 votes counts: B=30 A=29 C=23 E=18 so E is eliminated. Round 3 votes counts: B=34 A=34 C=32 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:202 C:198 D:198 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -6 10 B 8 0 8 14 6 C -8 -8 0 2 10 D 6 -14 -2 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -6 10 B 8 0 8 14 6 C -8 -8 0 2 10 D 6 -14 -2 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -6 10 B 8 0 8 14 6 C -8 -8 0 2 10 D 6 -14 -2 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3715: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) D C E B A (6) C B D A E (6) B A C D E (6) D C B E A (5) D C B A E (4) C D E B A (4) B C A D E (4) B A C E D (4) E D A B C (3) D E C A B (3) D B C A E (3) D B A C E (3) C D B A E (3) B C A E D (3) A E B D C (3) A B E D C (3) E D A C B (2) E C A D B (2) E C A B D (2) E A D B C (2) E A C D B (2) D A E B C (2) B A D C E (2) A E B C D (2) A B E C D (2) A B D C E (2) E C D A B (1) E A D C B (1) E A C B D (1) E A B C D (1) D E C B A (1) C B E A D (1) B D C A E (1) B D A C E (1) B C D A E (1) B A E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -12 -10 14 B 18 0 0 -6 12 C 12 0 0 -8 18 D 10 6 8 0 16 E -14 -12 -18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -12 -10 14 B 18 0 0 -6 12 C 12 0 0 -8 18 D 10 6 8 0 16 E -14 -12 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=23 B=23 C=14 A=13 so A is eliminated. Round 2 votes counts: B=31 E=28 D=27 C=14 so C is eliminated. Round 3 votes counts: B=38 D=34 E=28 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:212 C:211 A:187 E:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -12 -10 14 B 18 0 0 -6 12 C 12 0 0 -8 18 D 10 6 8 0 16 E -14 -12 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -10 14 B 18 0 0 -6 12 C 12 0 0 -8 18 D 10 6 8 0 16 E -14 -12 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -10 14 B 18 0 0 -6 12 C 12 0 0 -8 18 D 10 6 8 0 16 E -14 -12 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3716: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (7) A B D E C (7) E C B A D (5) C E D B A (5) E C D B A (4) D A C B E (4) B A E D C (4) B A E C D (4) B A C E D (4) E D C A B (3) D A E B C (3) D A C E B (3) C D A B E (3) A D B C E (3) A B D C E (3) E D C B A (2) E C B D A (2) D E A C B (2) C E D A B (2) B E C A D (2) B E A C D (2) B C E A D (2) B A C D E (2) A D B E C (2) E D B A C (1) E C D A B (1) E B C A D (1) E B A D C (1) D E C A B (1) D E A B C (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) D A B C E (1) C E B A D (1) C D A E B (1) C B E A D (1) C B A D E (1) B E A D C (1) B A D E C (1) B A D C E (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 0 4 B -6 0 -4 -8 0 C -6 4 0 2 -4 D 0 8 -2 0 4 E -4 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.616951 B: 0.000000 C: 0.000000 D: 0.383049 E: 0.000000 Sum of squares = 0.527355227282 Cumulative probabilities = A: 0.616951 B: 0.616951 C: 0.616951 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 0 4 B -6 0 -4 -8 0 C -6 4 0 2 -4 D 0 8 -2 0 4 E -4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 C=21 E=20 D=19 A=17 so A is eliminated. Round 2 votes counts: B=34 D=25 C=21 E=20 so E is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:208 D:205 C:198 E:198 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 0 4 B -6 0 -4 -8 0 C -6 4 0 2 -4 D 0 8 -2 0 4 E -4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 0 4 B -6 0 -4 -8 0 C -6 4 0 2 -4 D 0 8 -2 0 4 E -4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 0 4 B -6 0 -4 -8 0 C -6 4 0 2 -4 D 0 8 -2 0 4 E -4 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3717: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) B C D E A (6) D C B A E (5) D A B E C (5) A E D C B (5) A E C D B (5) E C A B D (4) C B D E A (4) B D C E A (4) B C E D A (4) D B A C E (3) D A E C B (3) A D E C B (3) A D E B C (3) E A C D B (2) D C B E A (2) D C A E B (2) D C A B E (2) D B C A E (2) D A C E B (2) C E B A D (2) C D B E A (2) B D A E C (2) B C E A D (2) A E D B C (2) A E B D C (2) E B C A D (1) E A B C D (1) D B A E C (1) D A B C E (1) C D E B A (1) C D E A B (1) C B E D A (1) C B E A D (1) C A E D B (1) B E A D C (1) B D C A E (1) B D A C E (1) B A E D C (1) A E C B D (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 6 4 -10 10 B -6 0 -10 -8 4 C -4 10 0 -8 0 D 10 8 8 0 14 E -10 -4 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -10 10 B -6 0 -10 -8 4 C -4 10 0 -8 0 D 10 8 8 0 14 E -10 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=23 B=22 E=14 C=13 so C is eliminated. Round 2 votes counts: D=32 B=28 A=24 E=16 so E is eliminated. Round 3 votes counts: A=37 D=32 B=31 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:205 C:199 B:190 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -10 10 B -6 0 -10 -8 4 C -4 10 0 -8 0 D 10 8 8 0 14 E -10 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -10 10 B -6 0 -10 -8 4 C -4 10 0 -8 0 D 10 8 8 0 14 E -10 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -10 10 B -6 0 -10 -8 4 C -4 10 0 -8 0 D 10 8 8 0 14 E -10 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3718: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) D C A B E (8) D C B A E (6) C D A B E (6) B D C A E (6) E D C A B (5) E B A D C (5) B A C D E (5) A C D B E (5) B A D C E (4) E C D A B (3) E B D C A (3) E A C D B (3) E A B C D (3) D C E A B (3) B A E C D (3) B E A D C (2) B D C E A (2) A E B C D (2) A C D E B (2) A C B D E (2) E D C B A (1) E D B C A (1) E B D A C (1) E A D C B (1) D C E B A (1) D C B E A (1) C D B A E (1) C D A E B (1) B E D C A (1) A E C D B (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 2 0 10 B 4 0 0 2 10 C -2 0 0 -2 10 D 0 -2 2 0 8 E -10 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.809381 C: 0.190619 D: 0.000000 E: 0.000000 Sum of squares = 0.691433030942 Cumulative probabilities = A: 0.000000 B: 0.809381 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 0 10 B 4 0 0 2 10 C -2 0 0 -2 10 D 0 -2 2 0 8 E -10 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.499847 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047124 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=23 D=19 A=15 C=8 so C is eliminated. Round 2 votes counts: E=35 D=27 B=23 A=15 so A is eliminated. Round 3 votes counts: E=38 D=34 B=28 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:208 A:204 D:204 C:203 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 0 10 B 4 0 0 2 10 C -2 0 0 -2 10 D 0 -2 2 0 8 E -10 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.499847 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047124 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 0 10 B 4 0 0 2 10 C -2 0 0 -2 10 D 0 -2 2 0 8 E -10 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.499847 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047124 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 0 10 B 4 0 0 2 10 C -2 0 0 -2 10 D 0 -2 2 0 8 E -10 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500153 C: 0.499847 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047124 Cumulative probabilities = A: 0.000000 B: 0.500153 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3719: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (19) D A E C B (11) C B D A E (10) E A D B C (8) E A B C D (7) D C B A E (6) C B D E A (6) A E D B C (6) C B E A D (4) D A E B C (3) A E B D C (3) C B A E D (2) B E C A D (2) A D E B C (2) E B A C D (1) E A B D C (1) D E A C B (1) D E A B C (1) D C B E A (1) D C A B E (1) D A C E B (1) C D B A E (1) C B E D A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 -6 14 -4 B 6 0 6 12 6 C 6 -6 0 6 4 D -14 -12 -6 0 -10 E 4 -6 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 14 -4 B 6 0 6 12 6 C 6 -6 0 6 4 D -14 -12 -6 0 -10 E 4 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 B=21 E=17 A=13 so A is eliminated. Round 2 votes counts: D=28 E=27 C=24 B=21 so B is eliminated. Round 3 votes counts: C=43 E=29 D=28 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:205 E:202 A:199 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 14 -4 B 6 0 6 12 6 C 6 -6 0 6 4 D -14 -12 -6 0 -10 E 4 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 14 -4 B 6 0 6 12 6 C 6 -6 0 6 4 D -14 -12 -6 0 -10 E 4 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 14 -4 B 6 0 6 12 6 C 6 -6 0 6 4 D -14 -12 -6 0 -10 E 4 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3720: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (27) C B D E A (18) C E D A B (6) C B D A E (6) E D A B C (5) C E D B A (3) C B A D E (3) B C D E A (3) B A D E C (3) E A D B C (2) D E A B C (2) C E A D B (2) C D E B A (2) C A E D B (2) C A B E D (2) B D E A C (2) B A C D E (2) E D A C B (1) E A D C B (1) C B E D A (1) B D E C A (1) B D C A E (1) B D A E C (1) A E D C B (1) A D E B C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 0 -4 2 B -8 0 2 -12 -12 C 0 -2 0 2 4 D 4 12 -2 0 -8 E -2 12 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272747 B: 0.000000 C: 0.727253 D: 0.000000 E: 0.000000 Sum of squares = 0.603287397324 Cumulative probabilities = A: 0.272747 B: 0.272747 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -4 2 B -8 0 2 -12 -12 C 0 -2 0 2 4 D 4 12 -2 0 -8 E -2 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555725 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=45 A=31 B=13 E=9 D=2 so D is eliminated. Round 2 votes counts: C=45 A=31 B=13 E=11 so E is eliminated. Round 3 votes counts: C=45 A=42 B=13 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:207 A:203 D:203 C:202 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 0 -4 2 B -8 0 2 -12 -12 C 0 -2 0 2 4 D 4 12 -2 0 -8 E -2 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555725 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -4 2 B -8 0 2 -12 -12 C 0 -2 0 2 4 D 4 12 -2 0 -8 E -2 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555725 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -4 2 B -8 0 2 -12 -12 C 0 -2 0 2 4 D 4 12 -2 0 -8 E -2 12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555725 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3721: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) C A B E D (5) C A B D E (5) B E D C A (5) E D B A C (4) C B A D E (4) A C D E B (4) E D A B C (3) E B D C A (3) D E A B C (3) B D E C A (3) B C E D A (3) B C A D E (3) A E D C B (3) A D E C B (3) A C E D B (3) A C B D E (3) E D C A B (2) E D B C A (2) E D A C B (2) D A E B C (2) D A B E C (2) C B E D A (2) C B E A D (2) C B A E D (2) C A E D B (2) B E C D A (2) B C D A E (2) A D C E B (2) A D B C E (2) E B C D A (1) D B E A C (1) C E A B D (1) B D E A C (1) B D A E C (1) B C D E A (1) B A C D E (1) A E C D B (1) A D E B C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -2 6 B -2 0 6 2 4 C 0 -6 0 -2 -2 D 2 -2 2 0 4 E -6 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 6 B -2 0 6 2 4 C 0 -6 0 -2 -2 D 2 -2 2 0 4 E -6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 B=22 E=17 D=14 so D is eliminated. Round 2 votes counts: A=28 E=26 C=23 B=23 so C is eliminated. Round 3 votes counts: A=40 B=33 E=27 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:205 A:203 D:203 C:195 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 -2 6 B -2 0 6 2 4 C 0 -6 0 -2 -2 D 2 -2 2 0 4 E -6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 6 B -2 0 6 2 4 C 0 -6 0 -2 -2 D 2 -2 2 0 4 E -6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 6 B -2 0 6 2 4 C 0 -6 0 -2 -2 D 2 -2 2 0 4 E -6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3722: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (11) C D B E A (10) A E B C D (6) E A B D C (5) C D E B A (5) E A D B C (4) D C B E A (4) A E C D B (4) D C E B A (3) B C D A E (3) B A E D C (3) A B E D C (3) E A C D B (2) C D B A E (2) C B D A E (2) C A D E B (2) C A D B E (2) B D C E A (2) B D C A E (2) B D A E C (2) B C A D E (2) B A C D E (2) A B C D E (2) E D A C B (1) E C D A B (1) E B A D C (1) E A D C B (1) D E B C A (1) D B C E A (1) C E D A B (1) C D A E B (1) C D A B E (1) C B A D E (1) C A E D B (1) B E D A C (1) B D A C E (1) B C D E A (1) A E D C B (1) A E D B C (1) A C B E D (1) Total count = 100 A B C D E A 0 2 4 10 12 B -2 0 8 2 -4 C -4 -8 0 4 4 D -10 -2 -4 0 4 E -12 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 10 12 B -2 0 8 2 -4 C -4 -8 0 4 4 D -10 -2 -4 0 4 E -12 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=28 B=19 E=15 D=9 so D is eliminated. Round 2 votes counts: C=35 A=29 B=20 E=16 so E is eliminated. Round 3 votes counts: A=42 C=36 B=22 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:202 C:198 D:194 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 10 12 B -2 0 8 2 -4 C -4 -8 0 4 4 D -10 -2 -4 0 4 E -12 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 10 12 B -2 0 8 2 -4 C -4 -8 0 4 4 D -10 -2 -4 0 4 E -12 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 10 12 B -2 0 8 2 -4 C -4 -8 0 4 4 D -10 -2 -4 0 4 E -12 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991475 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3723: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) C E B A D (7) C A E D B (6) D A E B C (5) C A D E B (5) B D A E C (5) A D E C B (5) E A D C B (4) D A B E C (4) C B E A D (4) B C D A E (4) A D C E B (4) E C A D B (3) E B D A C (3) B D E A C (3) B C D E A (3) E D A B C (2) C E B D A (2) C B E D A (2) C B A D E (2) B E D A C (2) B E C D A (2) A E D C B (2) A D E B C (2) E C B D A (1) E B C D A (1) E A D B C (1) E A C D B (1) D E A B C (1) D B E A C (1) D B A E C (1) C E A D B (1) C A E B D (1) C A B D E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -6 0 -4 B 2 0 -4 2 -16 C 6 4 0 10 2 D 0 -2 -10 0 -8 E 4 16 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 0 -4 B 2 0 -4 2 -16 C 6 4 0 10 2 D 0 -2 -10 0 -8 E 4 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 E=16 A=14 D=12 so D is eliminated. Round 2 votes counts: C=31 B=29 A=23 E=17 so E is eliminated. Round 3 votes counts: C=35 B=33 A=32 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:211 A:194 B:192 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 0 -4 B 2 0 -4 2 -16 C 6 4 0 10 2 D 0 -2 -10 0 -8 E 4 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 0 -4 B 2 0 -4 2 -16 C 6 4 0 10 2 D 0 -2 -10 0 -8 E 4 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 0 -4 B 2 0 -4 2 -16 C 6 4 0 10 2 D 0 -2 -10 0 -8 E 4 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3724: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (6) E D A B C (5) B C A E D (5) A C D E B (5) E B D A C (4) D C A E B (4) C A B E D (4) E A D B C (3) D E B C A (3) D E A B C (3) C B D A E (3) C B A E D (3) C A B D E (3) B D C E A (3) A D E C B (3) E D B A C (2) E B A D C (2) D E B A C (2) D E A C B (2) D C B A E (2) D B C E A (2) C D A B E (2) B E A D C (2) B E A C D (2) B D E C A (2) B C A D E (2) A E C D B (2) E B A C D (1) E A B C D (1) D C B E A (1) D C A B E (1) D B E C A (1) D A E C B (1) C D B A E (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B A C E D (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -12 0 4 B 16 0 8 6 10 C 12 -8 0 -2 8 D 0 -6 2 0 10 E -4 -10 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 0 4 B 16 0 8 6 10 C 12 -8 0 -2 8 D 0 -6 2 0 10 E -4 -10 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 D=22 E=18 A=12 so A is eliminated. Round 2 votes counts: C=29 D=25 B=25 E=21 so E is eliminated. Round 3 votes counts: D=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:205 D:203 A:188 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 0 4 B 16 0 8 6 10 C 12 -8 0 -2 8 D 0 -6 2 0 10 E -4 -10 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 0 4 B 16 0 8 6 10 C 12 -8 0 -2 8 D 0 -6 2 0 10 E -4 -10 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 0 4 B 16 0 8 6 10 C 12 -8 0 -2 8 D 0 -6 2 0 10 E -4 -10 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3725: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) B D E A C (10) A B D E C (10) B A D E C (8) A C D E B (8) C E D A B (7) C A E D B (7) C E D B A (6) C E B D A (3) A D B E C (3) A C E D B (3) A C B D E (3) A B C D E (3) E C D B A (2) B E C D A (2) E D C B A (1) E D B C A (1) D E B A C (1) D E A C B (1) C E A D B (1) C A E B D (1) B D E C A (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 8 2 0 B -2 0 8 10 10 C -8 -8 0 -6 -12 D -2 -10 6 0 4 E 0 -10 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.940233 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.059767 Sum of squares = 0.8876103756 Cumulative probabilities = A: 0.940233 B: 0.940233 C: 0.940233 D: 0.940233 E: 1.000000 A B C D E A 0 2 8 2 0 B -2 0 8 10 10 C -8 -8 0 -6 -12 D -2 -10 6 0 4 E 0 -10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222223937 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=34 C=25 E=4 D=2 so D is eliminated. Round 2 votes counts: B=35 A=34 C=25 E=6 so E is eliminated. Round 3 votes counts: B=37 A=35 C=28 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:206 D:199 E:199 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 2 0 B -2 0 8 10 10 C -8 -8 0 -6 -12 D -2 -10 6 0 4 E 0 -10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222223937 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 2 0 B -2 0 8 10 10 C -8 -8 0 -6 -12 D -2 -10 6 0 4 E 0 -10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222223937 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 2 0 B -2 0 8 10 10 C -8 -8 0 -6 -12 D -2 -10 6 0 4 E 0 -10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222223937 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3726: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) E C B D A (6) A B C E D (6) D E C B A (5) E C D B A (4) D B A C E (4) D A E C B (4) D A B E C (4) D A B C E (4) B C E D A (4) A D B C E (4) D E B C A (3) D A E B C (3) C E B A D (3) B E D C A (3) A B D C E (3) E D C B A (2) E C B A D (2) D E C A B (2) D E A B C (2) C B E A D (2) B C E A D (2) B A D C E (2) B A C E D (2) B A C D E (2) A D E C B (2) A D C B E (2) E D C A B (1) E D A C B (1) E C D A B (1) D E A C B (1) C E A B D (1) B E C D A (1) B D E C A (1) B C D A E (1) B C A E D (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 10 -14 6 B 0 0 6 2 12 C -10 -6 0 -6 4 D 14 -2 6 0 0 E -6 -12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.046601 B: 0.953399 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.911140999256 Cumulative probabilities = A: 0.046601 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 -14 6 B 0 0 6 2 12 C -10 -6 0 -6 4 D 14 -2 6 0 0 E -6 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250009976 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=26 B=19 E=17 C=6 so C is eliminated. Round 2 votes counts: D=32 A=26 E=21 B=21 so E is eliminated. Round 3 votes counts: D=41 B=32 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:209 A:201 C:191 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 10 -14 6 B 0 0 6 2 12 C -10 -6 0 -6 4 D 14 -2 6 0 0 E -6 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250009976 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -14 6 B 0 0 6 2 12 C -10 -6 0 -6 4 D 14 -2 6 0 0 E -6 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250009976 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -14 6 B 0 0 6 2 12 C -10 -6 0 -6 4 D 14 -2 6 0 0 E -6 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250009976 Cumulative probabilities = A: 0.125000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3727: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) D A B E C (6) C B E D A (6) D A C E B (5) B C E A D (5) D A E B C (4) B E A C D (4) B A D E C (4) E C B A D (3) D C A E B (3) C E D B A (3) C E D A B (3) C E B A D (3) C D B E A (3) B E C A D (3) B A E D C (3) A D B E C (3) E B C A D (2) E B A C D (2) E A C B D (2) E A B D C (2) D C A B E (2) D A E C B (2) C D E A B (2) C D B A E (2) A B D E C (2) E C D A B (1) E A D C B (1) E A D B C (1) D E A C B (1) C D A E B (1) C D A B E (1) C B E A D (1) C B D E A (1) C B D A E (1) B E A D C (1) B D A C E (1) A E D B C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -6 -10 -14 B 10 0 -8 8 0 C 6 8 0 12 0 D 10 -8 -12 0 -10 E 14 0 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.560428 D: 0.000000 E: 0.439572 Sum of squares = 0.507303047197 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.560428 D: 0.560428 E: 1.000000 A B C D E A 0 -10 -6 -10 -14 B 10 0 -8 8 0 C 6 8 0 12 0 D 10 -8 -12 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=23 B=21 E=14 A=8 so A is eliminated. Round 2 votes counts: C=34 D=27 B=24 E=15 so E is eliminated. Round 3 votes counts: C=40 D=30 B=30 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:212 B:205 D:190 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 -10 -14 B 10 0 -8 8 0 C 6 8 0 12 0 D 10 -8 -12 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -10 -14 B 10 0 -8 8 0 C 6 8 0 12 0 D 10 -8 -12 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -10 -14 B 10 0 -8 8 0 C 6 8 0 12 0 D 10 -8 -12 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3728: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C D E A B (8) C D A B E (8) C D A E B (7) C A D B E (7) B E A D C (7) E C D B A (5) C E D A B (4) B A E D C (4) C E D B A (3) C A B D E (3) B A D E C (3) E C B D A (2) E B C D A (2) E B A D C (2) D C E A B (2) D A C B E (2) D A B E C (2) C E B A D (2) A C B D E (2) A B D E C (2) A B D C E (2) E D C B A (1) E D B C A (1) D E B A C (1) D C A E B (1) D C A B E (1) D B A E C (1) C A B E D (1) B A C E D (1) B A C D E (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 8 -16 -22 0 B -8 0 -20 -12 -2 C 16 20 0 12 12 D 22 12 -12 0 10 E 0 2 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -16 -22 0 B -8 0 -20 -12 -2 C 16 20 0 12 12 D 22 12 -12 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=43 E=23 B=16 D=10 A=8 so A is eliminated. Round 2 votes counts: C=45 E=23 B=20 D=12 so D is eliminated. Round 3 votes counts: C=52 E=24 B=24 so E is eliminated. Round 4 votes counts: C=60 B=40 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:230 D:216 E:190 A:185 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 -22 0 B -8 0 -20 -12 -2 C 16 20 0 12 12 D 22 12 -12 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 -22 0 B -8 0 -20 -12 -2 C 16 20 0 12 12 D 22 12 -12 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 -22 0 B -8 0 -20 -12 -2 C 16 20 0 12 12 D 22 12 -12 0 10 E 0 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3729: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) E C A D B (6) B D A C E (5) B A D C E (5) E C D A B (4) C E D B A (4) B C E D A (4) B C D E A (4) A E D C B (4) A D E C B (4) A D E B C (4) E A C D B (3) D E A C B (3) D C E B A (3) C E B D A (3) C B E D A (3) B C E A D (3) E C B A D (2) D B A C E (2) D A E C B (2) D A B C E (2) B E C A D (2) B A C E D (2) A E D B C (2) E D C A B (1) E C B D A (1) E A D C B (1) D E C A B (1) D B C E A (1) D B C A E (1) D A E B C (1) D A B E C (1) C E B A D (1) C D E B A (1) C D E A B (1) B D C A E (1) B A C D E (1) A E B D C (1) A E B C D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 6 2 -4 B -2 0 4 -20 -8 C -6 -4 0 -8 -2 D -2 20 8 0 4 E 4 8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 A B C D E A 0 2 6 2 -4 B -2 0 4 -20 -8 C -6 -4 0 -8 -2 D -2 20 8 0 4 E 4 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=25 E=18 D=17 C=13 so C is eliminated. Round 2 votes counts: B=30 E=26 A=25 D=19 so D is eliminated. Round 3 votes counts: E=35 B=34 A=31 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:205 A:203 C:190 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 2 -4 B -2 0 4 -20 -8 C -6 -4 0 -8 -2 D -2 20 8 0 4 E 4 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 2 -4 B -2 0 4 -20 -8 C -6 -4 0 -8 -2 D -2 20 8 0 4 E 4 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 2 -4 B -2 0 4 -20 -8 C -6 -4 0 -8 -2 D -2 20 8 0 4 E 4 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3730: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) E A B D C (6) E A D B C (5) C B D E A (5) A E B D C (5) D E B A C (4) D E A B C (4) B D E C A (4) B C D E A (4) A E D C B (4) A E D B C (4) C B A E D (3) C A D E B (3) E D B A C (2) E B A D C (2) C D B A E (2) C B D A E (2) C A E D B (2) B E A D C (2) B D E A C (2) B A E C D (2) A E C D B (2) A C E D B (2) A C D E B (2) E D A B C (1) E B D A C (1) D E C A B (1) D E B C A (1) D C B E A (1) D C A E B (1) D B E A C (1) D A E B C (1) C D B E A (1) C D A E B (1) C D A B E (1) C B A D E (1) C A B E D (1) B E D C A (1) B E A C D (1) A E C B D (1) A D E C B (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 30 2 -14 B 0 0 24 4 -16 C -30 -24 0 -26 -34 D -2 -4 26 0 -12 E 14 16 34 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 30 2 -14 B 0 0 24 4 -16 C -30 -24 0 -26 -34 D -2 -4 26 0 -12 E 14 16 34 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 C=22 E=17 D=14 so D is eliminated. Round 2 votes counts: E=27 B=25 C=24 A=24 so C is eliminated. Round 3 votes counts: B=40 A=33 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:238 A:209 B:206 D:204 C:143 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 30 2 -14 B 0 0 24 4 -16 C -30 -24 0 -26 -34 D -2 -4 26 0 -12 E 14 16 34 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 30 2 -14 B 0 0 24 4 -16 C -30 -24 0 -26 -34 D -2 -4 26 0 -12 E 14 16 34 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 30 2 -14 B 0 0 24 4 -16 C -30 -24 0 -26 -34 D -2 -4 26 0 -12 E 14 16 34 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3731: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) B E C D A (6) E B D A C (5) C B A D E (5) D E A B C (4) D C A E B (4) D A C E B (4) B C E A D (4) E B D C A (3) D E C B A (3) C B D A E (3) C A D B E (3) C A B E D (3) B E A C D (3) B A C E D (3) A E D B C (3) E D B A C (2) E D A B C (2) E A B D C (2) D E B A C (2) D C E A B (2) D A E C B (2) D A E B C (2) C D A B E (2) C B A E D (2) C A B D E (2) B E C A D (2) A C D B E (2) E A D B C (1) D E C A B (1) D E B C A (1) C B E D A (1) C B E A D (1) B E D C A (1) B E A D C (1) B C E D A (1) B C A E D (1) A E C D B (1) A D E B C (1) A D C E B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -2 -2 -8 B 12 0 12 12 -6 C 2 -12 0 -6 -8 D 2 -12 6 0 -10 E 8 6 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -2 -2 -8 B 12 0 12 12 -6 C 2 -12 0 -6 -8 D 2 -12 6 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 B=22 E=21 A=10 so A is eliminated. Round 2 votes counts: D=27 C=26 E=25 B=22 so B is eliminated. Round 3 votes counts: E=38 C=35 D=27 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:215 D:193 A:188 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -2 -2 -8 B 12 0 12 12 -6 C 2 -12 0 -6 -8 D 2 -12 6 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -2 -8 B 12 0 12 12 -6 C 2 -12 0 -6 -8 D 2 -12 6 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -2 -8 B 12 0 12 12 -6 C 2 -12 0 -6 -8 D 2 -12 6 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3732: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) D C E A B (7) B E A D C (6) B E D A C (5) E C D B A (4) B A E C D (4) A B D C E (4) D C A B E (3) C E D A B (3) C A D E B (3) B E A C D (3) B D E A C (3) B A E D C (3) A D B C E (3) A C D B E (3) E C B A D (2) E B A C D (2) D E C B A (2) D C A E B (2) D B A C E (2) C E A D B (2) C D E A B (2) C A E D B (2) A C B D E (2) E D C B A (1) E D B C A (1) E B D C A (1) E B D A C (1) E B C D A (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C A E (1) D B A E C (1) D A C B E (1) D A B C E (1) C D A E B (1) C A E B D (1) B A D E C (1) B A D C E (1) A E B C D (1) A D C B E (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 2 8 -12 B 10 0 12 2 2 C -2 -12 0 -8 -4 D -8 -2 8 0 -6 E 12 -2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 8 -12 B 10 0 12 2 2 C -2 -12 0 -8 -4 D -8 -2 8 0 -6 E 12 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=23 E=20 A=17 C=14 so C is eliminated. Round 2 votes counts: D=26 B=26 E=25 A=23 so A is eliminated. Round 3 votes counts: D=36 B=34 E=30 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:210 D:196 A:194 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 8 -12 B 10 0 12 2 2 C -2 -12 0 -8 -4 D -8 -2 8 0 -6 E 12 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 8 -12 B 10 0 12 2 2 C -2 -12 0 -8 -4 D -8 -2 8 0 -6 E 12 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 8 -12 B 10 0 12 2 2 C -2 -12 0 -8 -4 D -8 -2 8 0 -6 E 12 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3733: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (11) D C E B A (10) E A D B C (9) C D B E A (8) B C A D E (8) A B C E D (8) B A C D E (6) E D C A B (5) E D A C B (5) D E C B A (5) C B D A E (5) A B E C D (5) D E C A B (3) D C B E A (3) A E B C D (3) E D A B C (2) B A C E D (2) E A B D C (1) A E D B C (1) Total count = 100 A B C D E A 0 6 6 8 -2 B -6 0 12 -2 -10 C -6 -12 0 -10 0 D -8 2 10 0 -4 E 2 10 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.165553 D: 0.000000 E: 0.834447 Sum of squares = 0.723709818822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.165553 D: 0.165553 E: 1.000000 A B C D E A 0 6 6 8 -2 B -6 0 12 -2 -10 C -6 -12 0 -10 0 D -8 2 10 0 -4 E 2 10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 D=21 B=16 C=13 so C is eliminated. Round 2 votes counts: D=29 A=28 E=22 B=21 so B is eliminated. Round 3 votes counts: A=44 D=34 E=22 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:208 D:200 B:197 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 8 -2 B -6 0 12 -2 -10 C -6 -12 0 -10 0 D -8 2 10 0 -4 E 2 10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 -2 B -6 0 12 -2 -10 C -6 -12 0 -10 0 D -8 2 10 0 -4 E 2 10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 -2 B -6 0 12 -2 -10 C -6 -12 0 -10 0 D -8 2 10 0 -4 E 2 10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3734: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) A E B D C (11) B D C E A (10) A E B C D (9) D C B E A (7) E A B D C (6) C D B A E (6) D C B A E (5) E B A D C (3) E A B C D (3) C D B E A (3) C D A B E (3) C A D E B (3) B E D C A (3) E B C D A (2) C D A E B (2) B D E C A (2) A E C B D (2) A C E D B (2) A C D E B (2) D C A B E (1) C D E B A (1) B E C D A (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 14 2 8 18 B -14 0 2 6 -16 C -2 -2 0 0 -8 D -8 -6 0 0 -8 E -18 16 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 8 18 B -14 0 2 6 -16 C -2 -2 0 0 -8 D -8 -6 0 0 -8 E -18 16 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=18 B=16 E=14 D=13 so D is eliminated. Round 2 votes counts: A=39 C=31 B=16 E=14 so E is eliminated. Round 3 votes counts: A=48 C=31 B=21 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:207 C:194 B:189 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 8 18 B -14 0 2 6 -16 C -2 -2 0 0 -8 D -8 -6 0 0 -8 E -18 16 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 8 18 B -14 0 2 6 -16 C -2 -2 0 0 -8 D -8 -6 0 0 -8 E -18 16 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 8 18 B -14 0 2 6 -16 C -2 -2 0 0 -8 D -8 -6 0 0 -8 E -18 16 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3735: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) C D B A E (7) D C B E A (6) E A B D C (5) B D C A E (5) E D C A B (4) E A C D B (4) D C E B A (4) C D B E A (4) B A C D E (4) E A B C D (3) C B D A E (3) A E B D C (3) A C E B D (3) E D B C A (2) E D A C B (2) E A D B C (2) E A C B D (2) D C B A E (2) D B C E A (2) C A B D E (2) B C D A E (2) B A D C E (2) A E C B D (2) A C B E D (2) E C D A B (1) E A D C B (1) D B C A E (1) C D A B E (1) C A E D B (1) C A E B D (1) B D E A C (1) B D C E A (1) B D A C E (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 2 12 B -6 0 -6 12 0 C -2 6 0 10 12 D -2 -12 -10 0 -2 E -12 0 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 2 12 B -6 0 -6 12 0 C -2 6 0 10 12 D -2 -12 -10 0 -2 E -12 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=24 C=19 B=16 D=15 so D is eliminated. Round 2 votes counts: C=31 E=26 A=24 B=19 so B is eliminated. Round 3 votes counts: C=42 A=31 E=27 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:211 B:200 E:189 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 2 12 B -6 0 -6 12 0 C -2 6 0 10 12 D -2 -12 -10 0 -2 E -12 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 12 B -6 0 -6 12 0 C -2 6 0 10 12 D -2 -12 -10 0 -2 E -12 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 12 B -6 0 -6 12 0 C -2 6 0 10 12 D -2 -12 -10 0 -2 E -12 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993781 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3736: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) E C B D A (8) E B D C A (5) D A B E C (5) A C E D B (5) C A E D B (4) A C D B E (4) A C B E D (4) A B D C E (4) E B C D A (3) D B E A C (3) C E B A D (3) B E D C A (3) B D E A C (3) A D C B E (3) E C D B A (2) E C B A D (2) C E A B D (2) C A E B D (2) B E C D A (2) B D E C A (2) B D A E C (2) A D B C E (2) A C D E B (2) E D B C A (1) E C D A B (1) D E C B A (1) D E B C A (1) D E B A C (1) D B A E C (1) D A E C B (1) C A B E D (1) B D A C E (1) B C E A D (1) B C A E D (1) B A D C E (1) A D C E B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -10 8 -8 B -6 0 -16 4 -12 C 10 16 0 18 6 D -8 -4 -18 0 -22 E 8 12 -6 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 8 -8 B -6 0 -16 4 -12 C 10 16 0 18 6 D -8 -4 -18 0 -22 E 8 12 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 C=22 B=16 D=13 so D is eliminated. Round 2 votes counts: A=33 E=25 C=22 B=20 so B is eliminated. Round 3 votes counts: E=38 A=38 C=24 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:225 E:218 A:198 B:185 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 8 -8 B -6 0 -16 4 -12 C 10 16 0 18 6 D -8 -4 -18 0 -22 E 8 12 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 8 -8 B -6 0 -16 4 -12 C 10 16 0 18 6 D -8 -4 -18 0 -22 E 8 12 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 8 -8 B -6 0 -16 4 -12 C 10 16 0 18 6 D -8 -4 -18 0 -22 E 8 12 -6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3737: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) C D B A E (9) E B D A C (8) A E D B C (8) E A B D C (7) C B D E A (5) E B D C A (4) E B C D A (4) D B C E A (4) B D E C A (4) C B D A E (3) A C D B E (3) E A D B C (2) E A C B D (2) B E C D A (2) A E D C B (2) A E B D C (2) A D E B C (2) A C D E B (2) E C B A D (1) E A B C D (1) D B E C A (1) D B C A E (1) D A B C E (1) C D B E A (1) C D A B E (1) C A E B D (1) C A B D E (1) B E D C A (1) B D C E A (1) B C D E A (1) A E C B D (1) A D C B E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -12 -2 2 B 0 0 8 0 2 C 12 -8 0 2 -4 D 2 0 -2 0 4 E -2 -2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.413825 C: 0.000000 D: 0.586175 E: 0.000000 Sum of squares = 0.514852404548 Cumulative probabilities = A: 0.000000 B: 0.413825 C: 0.413825 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -2 2 B 0 0 8 0 2 C 12 -8 0 2 -4 D 2 0 -2 0 4 E -2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=29 A=23 B=9 D=7 so D is eliminated. Round 2 votes counts: C=32 E=29 A=24 B=15 so B is eliminated. Round 3 votes counts: C=39 E=37 A=24 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:205 D:202 C:201 E:198 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -12 -2 2 B 0 0 8 0 2 C 12 -8 0 2 -4 D 2 0 -2 0 4 E -2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -2 2 B 0 0 8 0 2 C 12 -8 0 2 -4 D 2 0 -2 0 4 E -2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -2 2 B 0 0 8 0 2 C 12 -8 0 2 -4 D 2 0 -2 0 4 E -2 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3738: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) B E C D A (9) B C E D A (7) D A E B C (6) E B D A C (5) C A D B E (5) B E D C A (5) A C D E B (5) E D B A C (4) D E A B C (4) C B A E D (4) A D C E B (4) B C E A D (3) E D A B C (2) E B D C A (2) D B E A C (2) D A E C B (2) D A C E B (2) C B D A E (2) C B A D E (2) B E C A D (2) E B C D A (1) E A D B C (1) E A C D B (1) E A C B D (1) D E B A C (1) D C A B E (1) D B C E A (1) D B A E C (1) D A C B E (1) C A B E D (1) C A B D E (1) B E D A C (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -22 -10 -18 -22 B 22 0 14 10 16 C 10 -14 0 6 -2 D 18 -10 -6 0 -16 E 22 -16 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -10 -18 -22 B 22 0 14 10 16 C 10 -14 0 6 -2 D 18 -10 -6 0 -16 E 22 -16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 D=21 E=17 A=11 so A is eliminated. Round 2 votes counts: C=29 D=27 B=27 E=17 so E is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:212 C:200 D:193 A:164 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -10 -18 -22 B 22 0 14 10 16 C 10 -14 0 6 -2 D 18 -10 -6 0 -16 E 22 -16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -10 -18 -22 B 22 0 14 10 16 C 10 -14 0 6 -2 D 18 -10 -6 0 -16 E 22 -16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -10 -18 -22 B 22 0 14 10 16 C 10 -14 0 6 -2 D 18 -10 -6 0 -16 E 22 -16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3739: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) D B E C A (8) B D C E A (7) E A D C B (5) E A D B C (5) D B C E A (5) C A B D E (5) B C D A E (5) E D B A C (4) E A C B D (4) C B D A E (4) A E C D B (4) A C E B D (4) A C B E D (4) A E C B D (3) E D B C A (2) D E B A C (2) D E A B C (2) A C E D B (2) E D A C B (1) E D A B C (1) E B A C D (1) E A C D B (1) D E B C A (1) D C B A E (1) D B C A E (1) C A B E D (1) B D E C A (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 -10 8 -2 B 12 0 -4 6 12 C 10 4 0 4 8 D -8 -6 -4 0 12 E 2 -12 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 8 -2 B 12 0 -4 6 12 C 10 4 0 4 8 D -8 -6 -4 0 12 E 2 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=21 D=20 A=19 B=16 so B is eliminated. Round 2 votes counts: C=29 D=28 E=24 A=19 so A is eliminated. Round 3 votes counts: C=39 E=32 D=29 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:213 C:213 D:197 A:192 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 8 -2 B 12 0 -4 6 12 C 10 4 0 4 8 D -8 -6 -4 0 12 E 2 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 8 -2 B 12 0 -4 6 12 C 10 4 0 4 8 D -8 -6 -4 0 12 E 2 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 8 -2 B 12 0 -4 6 12 C 10 4 0 4 8 D -8 -6 -4 0 12 E 2 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3740: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) D E B C A (11) D E A C B (8) D B E C A (6) A C E B D (6) C A B E D (5) A C B E D (5) C B A E D (4) B D E C A (4) E D A C B (3) D A E C B (3) B C D E A (3) B A C D E (3) A C E D B (3) E D B C A (2) D E C A B (2) B C D A E (2) B A C E D (2) E D C A B (1) E C A D B (1) E A D C B (1) E A C D B (1) D E A B C (1) D B E A C (1) D A B E C (1) C E A B D (1) C B E A D (1) C A E B D (1) B D C A E (1) B C E D A (1) B C E A D (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -10 -20 0 2 B 10 0 4 6 6 C 20 -4 0 8 4 D 0 -6 -8 0 -6 E -2 -6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 0 2 B 10 0 4 6 6 C 20 -4 0 8 4 D 0 -6 -8 0 -6 E -2 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=30 A=16 C=12 E=9 so E is eliminated. Round 2 votes counts: D=39 B=30 A=18 C=13 so C is eliminated. Round 3 votes counts: D=39 B=35 A=26 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:214 B:213 E:197 D:190 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -20 0 2 B 10 0 4 6 6 C 20 -4 0 8 4 D 0 -6 -8 0 -6 E -2 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 0 2 B 10 0 4 6 6 C 20 -4 0 8 4 D 0 -6 -8 0 -6 E -2 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 0 2 B 10 0 4 6 6 C 20 -4 0 8 4 D 0 -6 -8 0 -6 E -2 -6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3741: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) D B A C E (6) D A B C E (5) C E A D B (5) B D E C A (5) A D B C E (5) A C E D B (5) E B C D A (4) C E B D A (4) B D A E C (4) A D C E B (4) A D B E C (4) E C B A D (3) E C A B D (3) C A E D B (3) B E C D A (3) A E C D B (3) E C A D B (2) E A C B D (2) D B A E C (2) D A B E C (2) C E B A D (2) C D A B E (2) B D C E A (2) A D E B C (2) A C D E B (2) E B C A D (1) C E D B A (1) C E A B D (1) C A D E B (1) B E D C A (1) B D E A C (1) B D A C E (1) B C E D A (1) A D C B E (1) Total count = 100 A B C D E A 0 4 -2 -2 4 B -4 0 -2 -10 -10 C 2 2 0 10 2 D 2 10 -10 0 -2 E -4 10 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -2 4 B -4 0 -2 -10 -10 C 2 2 0 10 2 D 2 10 -10 0 -2 E -4 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 C=19 B=18 D=15 so D is eliminated. Round 2 votes counts: A=33 B=26 E=22 C=19 so C is eliminated. Round 3 votes counts: A=39 E=35 B=26 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:208 E:203 A:202 D:200 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -2 4 B -4 0 -2 -10 -10 C 2 2 0 10 2 D 2 10 -10 0 -2 E -4 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -2 4 B -4 0 -2 -10 -10 C 2 2 0 10 2 D 2 10 -10 0 -2 E -4 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -2 4 B -4 0 -2 -10 -10 C 2 2 0 10 2 D 2 10 -10 0 -2 E -4 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3742: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D B C A E (9) B D C A E (9) D B A C E (7) E A C D B (6) E A D C B (5) E A C B D (5) E C B A D (4) C E A B D (4) A E C B D (4) D A B E C (3) B C D A E (3) A E C D B (3) E C B D A (2) D B C E A (2) D B A E C (2) D A E B C (2) C A B E D (2) B D C E A (2) B C A D E (2) E D C B A (1) E D B C A (1) E C D A B (1) E C A D B (1) D E B C A (1) D E A B C (1) D B E A C (1) D A B C E (1) B E D C A (1) B D E C A (1) B D A C E (1) B C D E A (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -14 -4 0 B 0 0 0 4 -4 C 14 0 0 0 -10 D 4 -4 0 0 -2 E 0 4 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.225048 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.774952 Sum of squares = 0.651197117766 Cumulative probabilities = A: 0.225048 B: 0.225048 C: 0.225048 D: 0.225048 E: 1.000000 A B C D E A 0 0 -14 -4 0 B 0 0 0 4 -4 C 14 0 0 0 -10 D 4 -4 0 0 -2 E 0 4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555561733 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=29 B=20 A=9 C=6 so C is eliminated. Round 2 votes counts: E=40 D=29 B=20 A=11 so A is eliminated. Round 3 votes counts: E=48 D=30 B=22 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:208 C:202 B:200 D:199 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -14 -4 0 B 0 0 0 4 -4 C 14 0 0 0 -10 D 4 -4 0 0 -2 E 0 4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555561733 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -4 0 B 0 0 0 4 -4 C 14 0 0 0 -10 D 4 -4 0 0 -2 E 0 4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555561733 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -4 0 B 0 0 0 4 -4 C 14 0 0 0 -10 D 4 -4 0 0 -2 E 0 4 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555561733 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3743: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (7) E A D B C (6) E A B C D (6) D C B A E (5) E B A C D (4) C E B A D (4) C D A B E (4) A D E B C (4) E C B A D (3) E B C A D (3) C E B D A (3) C B D E A (3) C B D A E (3) E A D C B (2) E A C D B (2) D B A C E (2) D A C B E (2) B E A D C (2) B D C A E (2) B D A C E (2) B C D A E (2) B A D E C (2) A E B D C (2) A D B E C (2) E C A D B (1) E A C B D (1) D B C A E (1) D A E B C (1) D A C E B (1) D A B C E (1) C E D B A (1) C E D A B (1) C D E A B (1) C D B E A (1) B E C A D (1) B C E D A (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 6 12 -10 B 4 0 10 8 -12 C -6 -10 0 4 -6 D -12 -8 -4 0 -6 E 10 12 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 6 12 -10 B 4 0 10 8 -12 C -6 -10 0 4 -6 D -12 -8 -4 0 -6 E 10 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=28 D=13 B=12 A=10 so A is eliminated. Round 2 votes counts: E=40 C=28 D=19 B=13 so B is eliminated. Round 3 votes counts: E=43 C=31 D=26 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:205 A:202 C:191 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 12 -10 B 4 0 10 8 -12 C -6 -10 0 4 -6 D -12 -8 -4 0 -6 E 10 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 12 -10 B 4 0 10 8 -12 C -6 -10 0 4 -6 D -12 -8 -4 0 -6 E 10 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 12 -10 B 4 0 10 8 -12 C -6 -10 0 4 -6 D -12 -8 -4 0 -6 E 10 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3744: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (15) C B A D E (10) D E A C B (7) E A D B C (6) E D A C B (4) E D A B C (4) C B D A E (4) A E B C D (4) A C E B D (4) C D B A E (3) B C D E A (3) A E D C B (3) E A B D C (2) D E A B C (2) D C A E B (2) C B A E D (2) C A D B E (2) B C D A E (2) E A D C B (1) E A B C D (1) D E B C A (1) D E B A C (1) D C B A E (1) D B E C A (1) D A E C B (1) C D B E A (1) C A B E D (1) B E A D C (1) B E A C D (1) B D C E A (1) B C E A D (1) B C A D E (1) B A E C D (1) A E C D B (1) A E C B D (1) A D E C B (1) A C D E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 24 24 B 0 0 -2 14 6 C 2 2 0 22 10 D -24 -14 -22 0 -10 E -24 -6 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 24 24 B 0 0 -2 14 6 C 2 2 0 22 10 D -24 -14 -22 0 -10 E -24 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=23 E=18 A=17 D=16 so D is eliminated. Round 2 votes counts: E=29 B=27 C=26 A=18 so A is eliminated. Round 3 votes counts: E=40 C=32 B=28 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:223 C:218 B:209 E:185 D:165 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 24 24 B 0 0 -2 14 6 C 2 2 0 22 10 D -24 -14 -22 0 -10 E -24 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 24 24 B 0 0 -2 14 6 C 2 2 0 22 10 D -24 -14 -22 0 -10 E -24 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 24 24 B 0 0 -2 14 6 C 2 2 0 22 10 D -24 -14 -22 0 -10 E -24 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3745: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (14) A D E B C (6) E D B C A (5) E B D C A (5) D E A B C (5) C B E A D (5) C B A E D (4) C A B E D (4) B E D A C (4) A C B D E (4) E C B D A (3) D A E B C (3) A D C B E (3) E D B A C (2) D C A E B (2) C E B D A (2) C A D B E (2) C A B D E (2) B E C A D (2) B E A D C (2) A D B C E (2) A C D B E (2) A C B E D (2) E B D A C (1) E B C D A (1) D E C B A (1) D E B C A (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C A D E B (1) B C E A D (1) B A E C D (1) A D C E B (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 16 -6 -10 B 10 0 16 -12 -14 C -16 -16 0 -24 -20 D 6 12 24 0 10 E 10 14 20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 16 -6 -10 B 10 0 16 -12 -14 C -16 -16 0 -24 -20 D 6 12 24 0 10 E 10 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=23 C=21 E=17 B=10 so B is eliminated. Round 2 votes counts: D=29 E=25 A=24 C=22 so C is eliminated. Round 3 votes counts: A=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:226 E:217 B:200 A:195 C:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 16 -6 -10 B 10 0 16 -12 -14 C -16 -16 0 -24 -20 D 6 12 24 0 10 E 10 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 -6 -10 B 10 0 16 -12 -14 C -16 -16 0 -24 -20 D 6 12 24 0 10 E 10 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 -6 -10 B 10 0 16 -12 -14 C -16 -16 0 -24 -20 D 6 12 24 0 10 E 10 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3746: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E C D B A (10) B A E D C (6) E B A C D (5) D C B E A (5) B A D C E (5) C D E B A (4) C D E A B (4) A B E D C (4) E A C B D (3) E A B C D (3) B D E C A (3) B D A C E (3) B A D E C (3) A B E C D (3) E C D A B (2) E C A D B (2) D C E B A (2) D C B A E (2) D B C A E (2) D B A C E (2) B E A D C (2) E C A B D (1) E B D C A (1) E A C D B (1) D E B C A (1) D C A B E (1) C E D B A (1) C E D A B (1) C D A E B (1) C A D E B (1) B E D C A (1) B D C E A (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 8 6 -6 B 18 0 18 16 12 C -8 -18 0 -14 -6 D -6 -16 14 0 6 E 6 -12 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 8 6 -6 B 18 0 18 16 12 C -8 -18 0 -14 -6 D -6 -16 14 0 6 E 6 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=24 A=21 D=15 C=12 so C is eliminated. Round 2 votes counts: E=30 D=24 B=24 A=22 so A is eliminated. Round 3 votes counts: B=44 E=31 D=25 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:232 D:199 E:197 A:195 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 8 6 -6 B 18 0 18 16 12 C -8 -18 0 -14 -6 D -6 -16 14 0 6 E 6 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 8 6 -6 B 18 0 18 16 12 C -8 -18 0 -14 -6 D -6 -16 14 0 6 E 6 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 8 6 -6 B 18 0 18 16 12 C -8 -18 0 -14 -6 D -6 -16 14 0 6 E 6 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3747: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) B E A D C (9) E B A C D (8) D C A E B (7) B A E D C (6) E C D B A (5) E C B D A (5) D C A B E (5) C D E A B (5) E B C D A (4) D C E A B (4) C D A E B (4) C E D B A (3) B E A C D (3) A D B C E (3) A B D E C (3) A B D C E (3) B A E C D (2) E D B C A (1) E B D C A (1) E B C A D (1) D B A E C (1) C E B A D (1) C D E B A (1) C D A B E (1) C A D E B (1) C A D B E (1) C A B E D (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -2 -12 -2 B 2 0 -8 -4 -2 C 2 8 0 -8 0 D 12 4 8 0 -2 E 2 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.128125 D: 0.000000 E: 0.871875 Sum of squares = 0.776581641572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.128125 D: 0.128125 E: 1.000000 A B C D E A 0 -2 -2 -12 -2 B 2 0 -8 -4 -2 C 2 8 0 -8 0 D 12 4 8 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.68000000337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 B=20 C=18 A=11 so A is eliminated. Round 2 votes counts: D=30 B=27 E=25 C=18 so C is eliminated. Round 3 votes counts: D=43 E=29 B=28 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:203 C:201 B:194 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 -12 -2 B 2 0 -8 -4 -2 C 2 8 0 -8 0 D 12 4 8 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.68000000337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -12 -2 B 2 0 -8 -4 -2 C 2 8 0 -8 0 D 12 4 8 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.68000000337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -12 -2 B 2 0 -8 -4 -2 C 2 8 0 -8 0 D 12 4 8 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.800000 Sum of squares = 0.68000000337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3748: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (11) C D B A E (7) B E A C D (7) D A C E B (6) E A B D C (5) C D A E B (5) C D A B E (5) B E C A D (5) D C B A E (4) D C A E B (4) C B D E A (4) B E A D C (4) A D E B C (4) C E B A D (3) C D E A B (3) C D B E A (3) A E D B C (3) E B C A D (2) E B A C D (2) D C A B E (2) E C A B D (1) E B A D C (1) D A E C B (1) D A E B C (1) D A B C E (1) C D E B A (1) C B E D A (1) B D A E C (1) B C E A D (1) B C D E A (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -4 0 12 B -6 0 0 0 -6 C 4 0 0 2 2 D 0 0 -2 0 6 E -12 6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.139436 C: 0.860564 D: 0.000000 E: 0.000000 Sum of squares = 0.760013032676 Cumulative probabilities = A: 0.000000 B: 0.139436 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 0 12 B -6 0 0 0 -6 C 4 0 0 2 2 D 0 0 -2 0 6 E -12 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000399 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=19 B=19 A=19 E=11 so E is eliminated. Round 2 votes counts: C=33 B=24 A=24 D=19 so D is eliminated. Round 3 votes counts: C=43 A=33 B=24 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:207 C:204 D:202 B:194 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 0 12 B -6 0 0 0 -6 C 4 0 0 2 2 D 0 0 -2 0 6 E -12 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000399 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 12 B -6 0 0 0 -6 C 4 0 0 2 2 D 0 0 -2 0 6 E -12 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000399 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 12 B -6 0 0 0 -6 C 4 0 0 2 2 D 0 0 -2 0 6 E -12 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000399 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3749: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) E C A D B (9) C A B E D (7) B D A C E (7) E D C A B (6) D E B A C (6) D E B C A (5) D B E A C (5) C E A B D (5) D B E C A (4) D B A E C (4) C A E B D (4) B C A D E (4) B A D C E (3) B A C D E (3) A B C D E (3) E D B C A (2) E C D A B (2) D E A B C (2) A C E B D (2) E D A B C (1) E A C D B (1) C B A E D (1) C B A D E (1) B D E C A (1) B D C A E (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -4 -10 -16 B -6 0 2 -14 -10 C 4 -2 0 -12 -14 D 10 14 12 0 0 E 16 10 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.549606 E: 0.450394 Sum of squares = 0.504921588248 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.549606 E: 1.000000 A B C D E A 0 6 -4 -10 -16 B -6 0 2 -14 -10 C 4 -2 0 -12 -14 D 10 14 12 0 0 E 16 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 B=19 C=18 A=7 so A is eliminated. Round 2 votes counts: E=30 D=26 C=22 B=22 so C is eliminated. Round 3 votes counts: E=41 B=32 D=27 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:218 A:188 C:188 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -10 -16 B -6 0 2 -14 -10 C 4 -2 0 -12 -14 D 10 14 12 0 0 E 16 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -10 -16 B -6 0 2 -14 -10 C 4 -2 0 -12 -14 D 10 14 12 0 0 E 16 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -10 -16 B -6 0 2 -14 -10 C 4 -2 0 -12 -14 D 10 14 12 0 0 E 16 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3750: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) B A C D E (8) D E A C B (7) C E B D A (7) B A C E D (7) E C D B A (6) E D C A B (5) D E C A B (5) B C A E D (5) C B E D A (4) C D E B A (3) C B E A D (3) A D E B C (3) A D B E C (3) E C B A D (2) D A E B C (2) C E D B A (2) C E B A D (2) A D B C E (2) A B D E C (2) A B C E D (2) E D C B A (1) E C D A B (1) E C B D A (1) D C E B A (1) D A E C B (1) D A B E C (1) B E C A D (1) B C E A D (1) B A E C D (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 0 6 -4 B 10 0 -2 12 0 C 0 2 0 14 12 D -6 -12 -14 0 -6 E 4 0 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.110236 B: 0.000000 C: 0.889764 D: 0.000000 E: 0.000000 Sum of squares = 0.803831610257 Cumulative probabilities = A: 0.110236 B: 0.110236 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 6 -4 B 10 0 -2 12 0 C 0 2 0 14 12 D -6 -12 -14 0 -6 E 4 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222597 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=23 A=23 C=21 D=17 E=16 so E is eliminated. Round 2 votes counts: C=31 D=23 B=23 A=23 so D is eliminated. Round 3 votes counts: C=43 A=34 B=23 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:214 B:210 E:199 A:196 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 6 -4 B 10 0 -2 12 0 C 0 2 0 14 12 D -6 -12 -14 0 -6 E 4 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222597 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 -4 B 10 0 -2 12 0 C 0 2 0 14 12 D -6 -12 -14 0 -6 E 4 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222597 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 -4 B 10 0 -2 12 0 C 0 2 0 14 12 D -6 -12 -14 0 -6 E 4 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222222597 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3751: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (13) E A C D B (10) D B C A E (8) B E A C D (6) B D C E A (5) A E C D B (5) E A D C B (4) E A C B D (4) D A E C B (4) E A B C D (3) C A E D B (3) C A E B D (3) B E D A C (3) B D C A E (3) E B A C D (2) D B E A C (2) C D A E B (2) B D E C A (2) B C D A E (2) E A B D C (1) D E A B C (1) D C B A E (1) D B C E A (1) C D B A E (1) C B D A E (1) C B A E D (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E A C (1) B C E A D (1) B C D E A (1) B C A E D (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 -2 -4 0 B -10 0 -8 -14 -14 C 2 8 0 -2 -4 D 4 14 2 0 -4 E 0 14 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.218243 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.781757 Sum of squares = 0.658774371051 Cumulative probabilities = A: 0.218243 B: 0.218243 C: 0.218243 D: 0.218243 E: 1.000000 A B C D E A 0 10 -2 -4 0 B -10 0 -8 -14 -14 C 2 8 0 -2 -4 D 4 14 2 0 -4 E 0 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499804 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500196 Sum of squares = 0.500000076882 Cumulative probabilities = A: 0.499804 B: 0.499804 C: 0.499804 D: 0.499804 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=28 E=24 C=11 A=7 so A is eliminated. Round 2 votes counts: E=30 D=30 B=28 C=12 so C is eliminated. Round 3 votes counts: E=37 D=33 B=30 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:208 A:202 C:202 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -2 -4 0 B -10 0 -8 -14 -14 C 2 8 0 -2 -4 D 4 14 2 0 -4 E 0 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499804 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500196 Sum of squares = 0.500000076882 Cumulative probabilities = A: 0.499804 B: 0.499804 C: 0.499804 D: 0.499804 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -4 0 B -10 0 -8 -14 -14 C 2 8 0 -2 -4 D 4 14 2 0 -4 E 0 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499804 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500196 Sum of squares = 0.500000076882 Cumulative probabilities = A: 0.499804 B: 0.499804 C: 0.499804 D: 0.499804 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -4 0 B -10 0 -8 -14 -14 C 2 8 0 -2 -4 D 4 14 2 0 -4 E 0 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499804 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500196 Sum of squares = 0.500000076882 Cumulative probabilities = A: 0.499804 B: 0.499804 C: 0.499804 D: 0.499804 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3752: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) A E C B D (9) D E B C A (7) C B A E D (7) A C B D E (7) E A D C B (6) E A D B C (5) B C D A E (5) D B C A E (4) C B A D E (4) A C B E D (4) E A C B D (3) C B D A E (3) E D B C A (2) E D A B C (2) D B E C A (2) C B E D A (2) B D C E A (2) A D E B C (2) E D C B A (1) D E A B C (1) C B E A D (1) C B D E A (1) C A B E D (1) B D C A E (1) B C D E A (1) A E D C B (1) A E D B C (1) A D B C E (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -12 8 4 B 10 0 -2 6 18 C 12 2 0 0 16 D -8 -6 0 0 8 E -4 -18 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.798831 D: 0.201169 E: 0.000000 Sum of squares = 0.678599571162 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.798831 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 8 4 B 10 0 -2 6 18 C 12 2 0 0 16 D -8 -6 0 0 8 E -4 -18 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000246037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 E=19 C=19 B=9 so B is eliminated. Round 2 votes counts: D=29 A=27 C=25 E=19 so E is eliminated. Round 3 votes counts: A=41 D=34 C=25 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:216 C:215 D:197 A:195 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 8 4 B 10 0 -2 6 18 C 12 2 0 0 16 D -8 -6 0 0 8 E -4 -18 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000246037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 8 4 B 10 0 -2 6 18 C 12 2 0 0 16 D -8 -6 0 0 8 E -4 -18 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000246037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 8 4 B 10 0 -2 6 18 C 12 2 0 0 16 D -8 -6 0 0 8 E -4 -18 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000246037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3753: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) D B E C A (7) D A B E C (7) D B A E C (6) B E A C D (6) A C E B D (6) D C A E B (5) E B C A D (3) D C E A B (3) C E B A D (3) C E A B D (3) B E C A D (3) E C B A D (2) D C E B A (2) D C B E A (2) D B E A C (2) D B C E A (2) D A C E B (2) D A C B E (2) D A B C E (2) C A E B D (2) B D A E C (2) A E C B D (2) A D B E C (2) E B A C D (1) D C A B E (1) D B C A E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E A B (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D E A C (1) B A E D C (1) B A D E C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 8 -4 6 B -2 0 20 0 20 C -8 -20 0 -10 -18 D 4 0 10 0 4 E -6 -20 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.297025 C: 0.000000 D: 0.702975 E: 0.000000 Sum of squares = 0.582397370756 Cumulative probabilities = A: 0.000000 B: 0.297025 C: 0.297025 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -4 6 B -2 0 20 0 20 C -8 -20 0 -10 -18 D 4 0 10 0 4 E -6 -20 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 A=20 B=17 C=13 E=6 so E is eliminated. Round 2 votes counts: D=44 B=21 A=20 C=15 so C is eliminated. Round 3 votes counts: D=46 B=27 A=27 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 D:209 A:206 E:194 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -4 6 B -2 0 20 0 20 C -8 -20 0 -10 -18 D 4 0 10 0 4 E -6 -20 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -4 6 B -2 0 20 0 20 C -8 -20 0 -10 -18 D 4 0 10 0 4 E -6 -20 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -4 6 B -2 0 20 0 20 C -8 -20 0 -10 -18 D 4 0 10 0 4 E -6 -20 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3754: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (16) B A D C E (14) D C E B A (10) E C A D B (6) B A D E C (6) A B E C D (6) C E D A B (5) A B C E D (5) B D A C E (4) B A E C D (4) D E C B A (3) B D A E C (3) E D C B A (2) E A C B D (2) D B E C A (2) D B A C E (2) A E C B D (2) A B D C E (2) D C E A B (1) D C B E A (1) D B C E A (1) C D E A B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 4 -2 0 B 4 0 0 0 0 C -4 0 0 -2 -6 D 2 0 2 0 0 E 0 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.429639 C: 0.000000 D: 0.314044 E: 0.256317 Sum of squares = 0.348911617251 Cumulative probabilities = A: 0.000000 B: 0.429639 C: 0.429639 D: 0.743683 E: 1.000000 A B C D E A 0 -4 4 -2 0 B 4 0 0 0 0 C -4 0 0 -2 -6 D 2 0 2 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=26 D=20 A=17 C=6 so C is eliminated. Round 2 votes counts: E=31 B=31 D=21 A=17 so A is eliminated. Round 3 votes counts: B=44 E=35 D=21 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:203 B:202 D:202 A:199 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 -2 0 B 4 0 0 0 0 C -4 0 0 -2 -6 D 2 0 2 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -2 0 B 4 0 0 0 0 C -4 0 0 -2 -6 D 2 0 2 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -2 0 B 4 0 0 0 0 C -4 0 0 -2 -6 D 2 0 2 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3755: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) C D A B E (6) C A D E B (6) C D A E B (5) B E A D C (5) A E B D C (5) E A D B C (4) A E D C B (4) E D A B C (3) D E A B C (3) D A E C B (3) B E D C A (3) B E A C D (3) E D B A C (2) E B A D C (2) D E B A C (2) D C B E A (2) D B E A C (2) C D B E A (2) C D B A E (2) C B A E D (2) C A B E D (2) B D E C A (2) B C E D A (2) B C D E A (2) A D E C B (2) A C E B D (2) A C D E B (2) E A B D C (1) D B C E A (1) C B D A E (1) C A E B D (1) C A B D E (1) B D E A C (1) B C E A D (1) B C A E D (1) B A C E D (1) A E D B C (1) A E C B D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 8 16 -2 0 B -8 0 8 -6 0 C -16 -8 0 -10 -12 D 2 6 10 0 -10 E 0 0 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.440044 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.559956 Sum of squares = 0.507189545566 Cumulative probabilities = A: 0.440044 B: 0.440044 C: 0.440044 D: 0.440044 E: 1.000000 A B C D E A 0 8 16 -2 0 B -8 0 8 -6 0 C -16 -8 0 -10 -12 D 2 6 10 0 -10 E 0 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=28 B=28 A=19 D=13 E=12 so E is eliminated. Round 2 votes counts: B=30 C=28 A=24 D=18 so D is eliminated. Round 3 votes counts: B=37 A=33 C=30 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:211 D:204 B:197 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 -2 0 B -8 0 8 -6 0 C -16 -8 0 -10 -12 D 2 6 10 0 -10 E 0 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -2 0 B -8 0 8 -6 0 C -16 -8 0 -10 -12 D 2 6 10 0 -10 E 0 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -2 0 B -8 0 8 -6 0 C -16 -8 0 -10 -12 D 2 6 10 0 -10 E 0 0 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3756: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) C B A E D (6) D E A B C (5) D C B E A (4) D C A E B (4) C A E D B (4) A E B C D (4) A C E B D (4) E A B D C (3) D A C E B (3) C A D E B (3) B E A D C (3) B C E A D (3) E D B A C (2) E A D B C (2) E A B C D (2) D E B A C (2) D B E C A (2) D B E A C (2) D B C E A (2) C D A B E (2) C B D A E (2) C A E B D (2) C A B E D (2) A E D C B (2) A C D E B (2) E B D A C (1) E B A D C (1) E B A C D (1) D E B C A (1) D A E C B (1) C D B E A (1) C B D E A (1) C B A D E (1) C A B D E (1) B E C D A (1) B E C A D (1) B D E C A (1) B C E D A (1) B A E C D (1) A E D B C (1) A E C B D (1) A D E C B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 10 24 -2 B -2 0 4 6 -6 C -10 -4 0 14 -2 D -24 -6 -14 0 -18 E 2 6 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 10 24 -2 B -2 0 4 6 -6 C -10 -4 0 14 -2 D -24 -6 -14 0 -18 E 2 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=25 B=20 A=17 E=12 so E is eliminated. Round 2 votes counts: D=28 C=25 A=24 B=23 so B is eliminated. Round 3 votes counts: A=39 C=31 D=30 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:214 B:201 C:199 D:169 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 10 24 -2 B -2 0 4 6 -6 C -10 -4 0 14 -2 D -24 -6 -14 0 -18 E 2 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 24 -2 B -2 0 4 6 -6 C -10 -4 0 14 -2 D -24 -6 -14 0 -18 E 2 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 24 -2 B -2 0 4 6 -6 C -10 -4 0 14 -2 D -24 -6 -14 0 -18 E 2 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3757: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (14) E B C A D (9) D A C E B (5) C B E D A (5) C B E A D (5) E A D B C (4) C D A B E (4) E B A D C (3) D A C B E (3) B E A C D (3) E D A B C (2) E B A C D (2) D C A B E (2) D A E C B (2) C E D B A (2) C D E B A (2) C D B A E (2) B E C A D (2) B C A E D (2) B A E C D (2) A D B E C (2) E D C A B (1) E D A C B (1) E C D B A (1) E C B D A (1) E C B A D (1) E B D C A (1) E A B D C (1) D C E A B (1) D C A E B (1) D A B E C (1) D A B C E (1) C E B D A (1) C E B A D (1) C D E A B (1) C D B E A (1) C D A E B (1) C B D A E (1) B E A D C (1) B A E D C (1) B A C D E (1) A E D B C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 4 -14 -4 B 0 0 10 -12 -20 C -4 -10 0 0 -16 D 14 12 0 0 -8 E 4 20 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 -14 -4 B 0 0 10 -12 -20 C -4 -10 0 0 -16 D 14 12 0 0 -8 E 4 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 C=26 B=12 A=5 so A is eliminated. Round 2 votes counts: D=33 E=28 C=26 B=13 so B is eliminated. Round 3 votes counts: E=38 D=33 C=29 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:209 A:193 B:189 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 -14 -4 B 0 0 10 -12 -20 C -4 -10 0 0 -16 D 14 12 0 0 -8 E 4 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -14 -4 B 0 0 10 -12 -20 C -4 -10 0 0 -16 D 14 12 0 0 -8 E 4 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -14 -4 B 0 0 10 -12 -20 C -4 -10 0 0 -16 D 14 12 0 0 -8 E 4 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3758: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) E D C A B (10) D E A C B (9) B C A D E (8) D A B C E (6) E C B A D (5) D A E C B (5) B A D C E (4) B A C D E (4) D E B A C (3) A D B C E (3) E D C B A (2) E B D C A (2) E B C D A (2) D E A B C (2) D A C E B (2) C B A D E (2) C A B E D (2) B C A E D (2) A D C B E (2) E D B A C (1) E D A B C (1) E C D B A (1) E C A B D (1) D A E B C (1) C A D E B (1) C A B D E (1) B E D A C (1) B D A C E (1) B C E A D (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 22 20 -22 -6 B -22 0 -12 -24 -22 C -20 12 0 -36 -16 D 22 24 36 0 14 E 6 22 16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 20 -22 -6 B -22 0 -12 -24 -22 C -20 12 0 -36 -16 D 22 24 36 0 14 E 6 22 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=28 B=21 A=8 C=6 so C is eliminated. Round 2 votes counts: E=37 D=28 B=23 A=12 so A is eliminated. Round 3 votes counts: E=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:248 E:215 A:207 C:170 B:160 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 20 -22 -6 B -22 0 -12 -24 -22 C -20 12 0 -36 -16 D 22 24 36 0 14 E 6 22 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 20 -22 -6 B -22 0 -12 -24 -22 C -20 12 0 -36 -16 D 22 24 36 0 14 E 6 22 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 20 -22 -6 B -22 0 -12 -24 -22 C -20 12 0 -36 -16 D 22 24 36 0 14 E 6 22 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3759: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B E C A D (7) A D C B E (7) E D C B A (6) D A C E B (6) E B C D A (5) B A E C D (5) A B D C E (5) E C D B A (3) D C A E B (3) C D E A B (3) B E A C D (3) B A C E D (3) A D C E B (3) A B D E C (3) A B C E D (3) E B D C A (2) D C E B A (2) C E D B A (2) C D E B A (2) B E C D A (2) B A E D C (2) A D B E C (2) A B E C D (2) A B C D E (2) E D B C A (1) E C B D A (1) D E C B A (1) D E A C B (1) C A D E B (1) B C E A D (1) B C A E D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -2 4 0 B -2 0 0 -4 0 C 2 0 0 -6 6 D -4 4 6 0 0 E 0 0 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 4 0 B -2 0 0 -4 0 C 2 0 0 -6 6 D -4 4 6 0 0 E 0 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 D=21 E=18 C=8 so C is eliminated. Round 2 votes counts: A=30 D=26 B=24 E=20 so E is eliminated. Round 3 votes counts: D=38 B=32 A=30 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:203 A:202 C:201 B:197 E:197 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 4 0 B -2 0 0 -4 0 C 2 0 0 -6 6 D -4 4 6 0 0 E 0 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 4 0 B -2 0 0 -4 0 C 2 0 0 -6 6 D -4 4 6 0 0 E 0 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 4 0 B -2 0 0 -4 0 C 2 0 0 -6 6 D -4 4 6 0 0 E 0 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3760: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C A B E D (7) E D B C A (6) D B E A C (6) D A B E C (6) A C B D E (6) C A B D E (4) B D E C A (4) A C D B E (4) E C B D A (3) D B A E C (3) C E A B D (3) A D B C E (3) A C D E B (3) E C D B A (2) E B D C A (2) E B C D A (2) C E B D A (2) C B E D A (2) C B A E D (2) C A E B D (2) B E C D A (2) A D E C B (2) E D C A B (1) E D B A C (1) E D A C B (1) E D A B C (1) D E A B C (1) D B E C A (1) D A E B C (1) C E B A D (1) B E D C A (1) B D E A C (1) B D A E C (1) B D A C E (1) A D E B C (1) A D C E B (1) A D C B E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 -16 0 B 0 0 6 -6 14 C -6 -6 0 -8 -14 D 16 6 8 0 18 E 0 -14 14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -16 0 B 0 0 6 -6 14 C -6 -6 0 -8 -14 D 16 6 8 0 18 E 0 -14 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 A=23 E=19 B=10 so B is eliminated. Round 2 votes counts: D=32 C=23 A=23 E=22 so E is eliminated. Round 3 votes counts: D=45 C=32 A=23 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:207 A:195 E:191 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 -16 0 B 0 0 6 -6 14 C -6 -6 0 -8 -14 D 16 6 8 0 18 E 0 -14 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -16 0 B 0 0 6 -6 14 C -6 -6 0 -8 -14 D 16 6 8 0 18 E 0 -14 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -16 0 B 0 0 6 -6 14 C -6 -6 0 -8 -14 D 16 6 8 0 18 E 0 -14 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3761: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) C D B E A (6) A E B D C (6) E A C D B (5) B D C A E (5) B D C E A (4) B C D A E (4) A C B D E (4) A B C D E (4) E D C A B (3) E D B C A (3) E D B A C (3) E A D B C (3) D E B C A (3) A E C B D (3) A C E B D (3) A B D E C (3) C D E B A (2) C B D A E (2) C B A D E (2) B C A D E (2) A C B E D (2) A B C E D (2) E D A C B (1) E D A B C (1) E C D A B (1) E C A D B (1) E B D A C (1) E B A D C (1) E A D C B (1) E A B D C (1) D B E C A (1) D B C E A (1) C E D A B (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B D E C A (1) B A D C E (1) A E C D B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -6 0 -2 B -2 0 2 8 -8 C 6 -2 0 -2 -2 D 0 -8 2 0 -6 E 2 8 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -6 0 -2 B -2 0 2 8 -8 C 6 -2 0 -2 -2 D 0 -8 2 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=29 C=17 B=17 D=5 so D is eliminated. Round 2 votes counts: E=35 A=29 B=19 C=17 so C is eliminated. Round 3 votes counts: E=38 A=33 B=29 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:200 C:200 A:197 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 0 -2 B -2 0 2 8 -8 C 6 -2 0 -2 -2 D 0 -8 2 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 -2 B -2 0 2 8 -8 C 6 -2 0 -2 -2 D 0 -8 2 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 -2 B -2 0 2 8 -8 C 6 -2 0 -2 -2 D 0 -8 2 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3762: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) D B C E A (6) E A C D B (5) A E C B D (5) E C A D B (4) C E D B A (4) C E D A B (4) B D C A E (4) B D A C E (4) B A D C E (4) D B A E C (3) C E A B D (3) A B D E C (3) E D C B A (2) E D C A B (2) D E C B A (2) D E B C A (2) C E A D B (2) B D A E C (2) B A C D E (2) A E B C D (2) A C E B D (2) A B E C D (2) A B D C E (2) A B C D E (2) E D A C B (1) E D A B C (1) D C B E A (1) D B E C A (1) D B E A C (1) C D E B A (1) C D B E A (1) C B D E A (1) C B D A E (1) C B A E D (1) C A E B D (1) C A B E D (1) B D C E A (1) B A D E C (1) A E D B C (1) A E C D B (1) A E B D C (1) A D B E C (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -2 -2 -2 B -10 0 -4 -4 -4 C 2 4 0 6 0 D 2 4 -6 0 -8 E 2 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.597960 D: 0.000000 E: 0.402040 Sum of squares = 0.519192153989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.597960 D: 0.597960 E: 1.000000 A B C D E A 0 10 -2 -2 -2 B -10 0 -4 -4 -4 C 2 4 0 6 0 D 2 4 -6 0 -8 E 2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 C=20 B=18 D=16 so D is eliminated. Round 2 votes counts: B=29 E=25 A=25 C=21 so C is eliminated. Round 3 votes counts: E=39 B=34 A=27 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:207 C:206 A:202 D:196 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 -2 -2 B -10 0 -4 -4 -4 C 2 4 0 6 0 D 2 4 -6 0 -8 E 2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -2 -2 B -10 0 -4 -4 -4 C 2 4 0 6 0 D 2 4 -6 0 -8 E 2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -2 -2 B -10 0 -4 -4 -4 C 2 4 0 6 0 D 2 4 -6 0 -8 E 2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3763: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) B E A C D (9) D C E A B (8) E B D C A (7) B E D A C (7) D E C A B (6) D C A E B (5) C A D E B (5) E D B C A (4) B E D C A (4) B A E C D (4) A B C E D (4) A B C D E (4) E D C B A (3) B E A D C (3) D E C B A (2) D A C B E (2) C D A E B (2) B A C E D (2) E D C A B (1) C E A D B (1) C D E A B (1) B D A E C (1) A D C B E (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 2 -6 -12 B -8 0 -2 -6 8 C -2 2 0 -8 -2 D 6 6 8 0 0 E 12 -8 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.722258 E: 0.277742 Sum of squares = 0.598797146878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.722258 E: 1.000000 A B C D E A 0 8 2 -6 -12 B -8 0 -2 -6 8 C -2 2 0 -8 -2 D 6 6 8 0 0 E 12 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.51020410194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 A=23 E=15 C=9 so C is eliminated. Round 2 votes counts: B=30 A=28 D=26 E=16 so E is eliminated. Round 3 votes counts: B=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:203 A:196 B:196 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -6 -12 B -8 0 -2 -6 8 C -2 2 0 -8 -2 D 6 6 8 0 0 E 12 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.51020410194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -6 -12 B -8 0 -2 -6 8 C -2 2 0 -8 -2 D 6 6 8 0 0 E 12 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.51020410194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -6 -12 B -8 0 -2 -6 8 C -2 2 0 -8 -2 D 6 6 8 0 0 E 12 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.428571 Sum of squares = 0.51020410194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571429 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3764: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) E A B C D (6) D C B A E (6) D B A C E (6) C E D B A (6) E A B D C (5) A B D E C (5) E C A B D (4) E A C B D (3) D B C A E (3) C E B A D (3) C E A B D (3) C D B A E (3) B A D C E (3) A E B D C (3) A B E D C (3) E D C A B (2) E C A D B (2) E A D B C (2) D B A E C (2) D A B E C (2) C D E B A (2) C D B E A (2) C B D A E (2) C B A D E (2) B D A C E (2) E D A C B (1) E D A B C (1) E A D C B (1) D E C B A (1) D A E B C (1) C B A E D (1) B D C A E (1) B C A D E (1) B A C D E (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 0 2 0 B -6 0 -2 2 -4 C 0 2 0 0 -2 D -2 -2 0 0 -8 E 0 4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.395992 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.604008 Sum of squares = 0.521635199063 Cumulative probabilities = A: 0.395992 B: 0.395992 C: 0.395992 D: 0.395992 E: 1.000000 A B C D E A 0 6 0 2 0 B -6 0 -2 2 -4 C 0 2 0 0 -2 D -2 -2 0 0 -8 E 0 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=24 D=21 A=14 B=8 so B is eliminated. Round 2 votes counts: E=33 C=25 D=24 A=18 so A is eliminated. Round 3 votes counts: E=40 D=32 C=28 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:204 C:200 B:195 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 2 0 B -6 0 -2 2 -4 C 0 2 0 0 -2 D -2 -2 0 0 -8 E 0 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 2 0 B -6 0 -2 2 -4 C 0 2 0 0 -2 D -2 -2 0 0 -8 E 0 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 2 0 B -6 0 -2 2 -4 C 0 2 0 0 -2 D -2 -2 0 0 -8 E 0 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3765: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) C D E A B (11) A B E C D (10) B A E D C (9) D C E A B (7) D C B E A (6) B A E C D (6) E C D A B (5) A E C D B (5) B A C D E (4) A B E D C (4) E A D C B (3) C D E B A (3) B D C A E (3) B A D C E (3) E A C D B (2) C E D A B (2) E D C A B (1) B D C E A (1) B D A C E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -2 -4 -6 B -4 0 -16 -14 -6 C 2 16 0 0 6 D 4 14 0 0 2 E 6 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.602824 D: 0.397176 E: 0.000000 Sum of squares = 0.521145437778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.602824 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -4 -6 B -4 0 -16 -14 -6 C 2 16 0 0 6 D 4 14 0 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 A=21 C=16 E=11 so E is eliminated. Round 2 votes counts: B=27 D=26 A=26 C=21 so C is eliminated. Round 3 votes counts: D=47 B=27 A=26 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:212 D:210 E:202 A:196 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -4 -6 B -4 0 -16 -14 -6 C 2 16 0 0 6 D 4 14 0 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -4 -6 B -4 0 -16 -14 -6 C 2 16 0 0 6 D 4 14 0 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -4 -6 B -4 0 -16 -14 -6 C 2 16 0 0 6 D 4 14 0 0 2 E 6 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3766: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (17) B D A E C (17) E C D B A (6) A B D C E (6) E C D A B (4) E C B D A (4) C E B D A (4) C A E D B (3) C A D B E (3) B E D C A (3) B D E A C (3) B A D C E (3) E B D C A (2) E B D A C (2) C A E B D (2) B E D A C (2) A D C B E (2) A C D B E (2) E D C A B (1) E C A D B (1) D B A E C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B A D (1) C B A D E (1) C A D E B (1) C A B E D (1) B D A C E (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -12 -8 -4 B 2 0 -10 6 -2 C 12 10 0 6 2 D 8 -6 -6 0 -10 E 4 2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -8 -4 B 2 0 -10 6 -2 C 12 10 0 6 2 D 8 -6 -6 0 -10 E 4 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=29 E=20 A=13 D=3 so D is eliminated. Round 2 votes counts: C=35 B=30 E=20 A=15 so A is eliminated. Round 3 votes counts: B=40 C=39 E=21 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:207 B:198 D:193 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -8 -4 B 2 0 -10 6 -2 C 12 10 0 6 2 D 8 -6 -6 0 -10 E 4 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -8 -4 B 2 0 -10 6 -2 C 12 10 0 6 2 D 8 -6 -6 0 -10 E 4 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -8 -4 B 2 0 -10 6 -2 C 12 10 0 6 2 D 8 -6 -6 0 -10 E 4 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3767: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) B D C A E (7) C A B D E (6) E D B A C (5) E D A B C (5) E A D C B (4) D B E A C (4) C B A D E (4) A C E B D (4) C E A D B (3) C B D E A (3) C A E B D (3) B D A E C (3) A C B E D (3) E D C B A (2) D E B A C (2) D B C E A (2) C E D A B (2) C D E B A (2) C B D A E (2) C A B E D (2) B D A C E (2) B C D A E (2) A B E D C (2) E D C A B (1) E C D B A (1) E C D A B (1) E C A D B (1) E A C D B (1) D B E C A (1) C E D B A (1) C D B E A (1) B A D C E (1) A E D B C (1) A E C B D (1) A E B D C (1) A D E B C (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 6 2 -4 B -10 0 2 -2 -4 C -6 -2 0 -8 4 D -2 2 8 0 -12 E 4 4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 10 6 2 -4 B -10 0 2 -2 -4 C -6 -2 0 -8 4 D -2 2 8 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775456 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=29 A=16 B=15 D=9 so D is eliminated. Round 2 votes counts: E=33 C=29 B=22 A=16 so A is eliminated. Round 3 votes counts: E=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:208 A:207 D:198 C:194 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 2 -4 B -10 0 2 -2 -4 C -6 -2 0 -8 4 D -2 2 8 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775456 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 -4 B -10 0 2 -2 -4 C -6 -2 0 -8 4 D -2 2 8 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775456 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 -4 B -10 0 2 -2 -4 C -6 -2 0 -8 4 D -2 2 8 0 -12 E 4 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.346938775456 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3768: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (12) C B D A E (12) B C D E A (7) B D C E A (6) D B C E A (5) C B A E D (5) E A B D C (4) A E C B D (4) D E A B C (3) C B D E A (3) A E D B C (3) E B A C D (2) D C B A E (2) D A E C B (2) C D B A E (2) C B A D E (2) B E C D A (2) A E C D B (2) A E B C D (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D E B A C (1) D E A C B (1) D C B E A (1) D C A B E (1) D A C E B (1) C D B E A (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D C A (1) B E A C D (1) B C E D A (1) B C E A D (1) A E D C B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -10 -8 -12 B 14 0 10 14 10 C 10 -10 0 4 8 D 8 -14 -4 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -8 -12 B 14 0 10 14 10 C 10 -10 0 4 8 D 8 -14 -4 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=22 B=19 D=17 A=14 so A is eliminated. Round 2 votes counts: E=34 C=29 B=19 D=18 so D is eliminated. Round 3 votes counts: E=42 C=34 B=24 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:206 D:199 E:193 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -10 -8 -12 B 14 0 10 14 10 C 10 -10 0 4 8 D 8 -14 -4 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -8 -12 B 14 0 10 14 10 C 10 -10 0 4 8 D 8 -14 -4 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -8 -12 B 14 0 10 14 10 C 10 -10 0 4 8 D 8 -14 -4 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3769: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) A C B E D (12) B C A E D (9) D E A C B (6) B E C A D (6) A C E B D (6) B C E A D (5) E D A C B (4) B D C A E (4) E A C D B (3) D B C A E (3) D A C E B (3) A C B D E (3) E A C B D (2) D E A B C (2) D B E C A (2) D B A C E (2) D A E C B (2) B C A D E (2) E D B C A (1) E C B A D (1) E B D C A (1) E B C A D (1) D E B A C (1) D B E A C (1) D A C B E (1) C A B E D (1) B E C D A (1) B D C E A (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 0 6 0 B 6 0 8 10 8 C 0 -8 0 8 8 D -6 -10 -8 0 -6 E 0 -8 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 6 0 B 6 0 8 10 8 C 0 -8 0 8 8 D -6 -10 -8 0 -6 E 0 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=28 A=23 E=13 C=1 so C is eliminated. Round 2 votes counts: D=35 B=28 A=24 E=13 so E is eliminated. Round 3 votes counts: D=40 B=31 A=29 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:204 A:200 E:195 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 6 0 B 6 0 8 10 8 C 0 -8 0 8 8 D -6 -10 -8 0 -6 E 0 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 6 0 B 6 0 8 10 8 C 0 -8 0 8 8 D -6 -10 -8 0 -6 E 0 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 6 0 B 6 0 8 10 8 C 0 -8 0 8 8 D -6 -10 -8 0 -6 E 0 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3770: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) A B E C D (8) A E B C D (5) E A C D B (4) D E C A B (4) D C B E A (4) B C A D E (4) B A C D E (4) A E B D C (4) A B D E C (4) E C D B A (3) E A D C B (3) C D E B A (3) B A C E D (3) E D C A B (2) E C B D A (2) D C E B A (2) D C B A E (2) C D B E A (2) B E C A D (2) B C D E A (2) B C D A E (2) A D B C E (2) A B E D C (2) A B C D E (2) E D C B A (1) E D A C B (1) E C D A B (1) E B A C D (1) E A C B D (1) D E C B A (1) D C A E B (1) D B C A E (1) B D C E A (1) B A E C D (1) B A D C E (1) A E D C B (1) A E C B D (1) A D E C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 16 20 30 20 B -16 0 20 18 14 C -20 -20 0 2 -10 D -30 -18 -2 0 8 E -20 -14 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 20 30 20 B -16 0 20 18 14 C -20 -20 0 2 -10 D -30 -18 -2 0 8 E -20 -14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 B=20 E=19 D=15 C=5 so C is eliminated. Round 2 votes counts: A=41 D=20 B=20 E=19 so E is eliminated. Round 3 votes counts: A=49 D=28 B=23 so B is eliminated. Round 4 votes counts: A=65 D=35 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:243 B:218 E:184 D:179 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 20 30 20 B -16 0 20 18 14 C -20 -20 0 2 -10 D -30 -18 -2 0 8 E -20 -14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 20 30 20 B -16 0 20 18 14 C -20 -20 0 2 -10 D -30 -18 -2 0 8 E -20 -14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 20 30 20 B -16 0 20 18 14 C -20 -20 0 2 -10 D -30 -18 -2 0 8 E -20 -14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3771: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) E B D C A (6) E B D A C (6) E B C A D (6) E A C D B (6) B E D C A (6) A C D B E (6) E B C D A (4) E A D C B (3) D B A C E (3) B D E C A (3) B D C A E (3) A D C B E (3) A C E D B (3) A C D E B (3) E C A D B (2) E B A D C (2) D B C A E (2) D A C B E (2) B C D E A (2) A D C E B (2) E D A B C (1) E C B A D (1) E B A C D (1) E A B D C (1) D B A E C (1) D A B C E (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) C A E D B (1) B E D A C (1) B D C E A (1) B D A E C (1) B D A C E (1) B C E D A (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -2 4 -6 B 8 0 6 -4 0 C 2 -6 0 0 -6 D -4 4 0 0 -4 E 6 0 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.289811 C: 0.000000 D: 0.000000 E: 0.710189 Sum of squares = 0.58835924721 Cumulative probabilities = A: 0.000000 B: 0.289811 C: 0.289811 D: 0.289811 E: 1.000000 A B C D E A 0 -8 -2 4 -6 B 8 0 6 -4 0 C 2 -6 0 0 -6 D -4 4 0 0 -4 E 6 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.500102 Sum of squares = 0.500000020732 Cumulative probabilities = A: 0.000000 B: 0.499898 C: 0.499898 D: 0.499898 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=19 A=19 C=14 D=9 so D is eliminated. Round 2 votes counts: E=39 B=25 A=22 C=14 so C is eliminated. Round 3 votes counts: E=39 A=33 B=28 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 B:205 D:198 C:195 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 4 -6 B 8 0 6 -4 0 C 2 -6 0 0 -6 D -4 4 0 0 -4 E 6 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.500102 Sum of squares = 0.500000020732 Cumulative probabilities = A: 0.000000 B: 0.499898 C: 0.499898 D: 0.499898 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 4 -6 B 8 0 6 -4 0 C 2 -6 0 0 -6 D -4 4 0 0 -4 E 6 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.500102 Sum of squares = 0.500000020732 Cumulative probabilities = A: 0.000000 B: 0.499898 C: 0.499898 D: 0.499898 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 4 -6 B 8 0 6 -4 0 C 2 -6 0 0 -6 D -4 4 0 0 -4 E 6 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.500102 Sum of squares = 0.500000020732 Cumulative probabilities = A: 0.000000 B: 0.499898 C: 0.499898 D: 0.499898 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3772: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) A D B E C (8) C E A B D (7) E C B D A (6) A D B C E (6) E D A C B (5) C A B D E (5) A B D C E (5) E D A B C (3) E C D B A (3) C E B A D (3) B D A C E (3) A C B D E (3) E D B C A (2) E D B A C (2) E C A D B (2) E A C D B (2) D E A B C (2) D A B C E (2) B D E C A (2) B A D C E (2) A E D C B (2) E D C B A (1) E C D A B (1) E C B A D (1) D B E A C (1) C E B D A (1) C B E D A (1) C B E A D (1) C B A D E (1) C A E B D (1) B D C E A (1) B D C A E (1) B C E D A (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 34 18 8 4 B -34 0 0 -8 6 C -18 0 0 -18 -4 D -8 8 18 0 8 E -4 -6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999118 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 34 18 8 4 B -34 0 0 -8 6 C -18 0 0 -18 -4 D -8 8 18 0 8 E -4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=28 C=20 D=13 B=10 so B is eliminated. Round 2 votes counts: A=31 E=28 C=21 D=20 so D is eliminated. Round 3 votes counts: A=44 E=33 C=23 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:232 D:213 E:193 B:182 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 34 18 8 4 B -34 0 0 -8 6 C -18 0 0 -18 -4 D -8 8 18 0 8 E -4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 34 18 8 4 B -34 0 0 -8 6 C -18 0 0 -18 -4 D -8 8 18 0 8 E -4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 34 18 8 4 B -34 0 0 -8 6 C -18 0 0 -18 -4 D -8 8 18 0 8 E -4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3773: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (12) D A B C E (10) E C B A D (9) D A E C B (6) B C E D A (6) C B E D A (5) C B E A D (4) A D E C B (4) A D E B C (4) E C D B A (3) E A D C B (3) D A E B C (3) C E B A D (3) A D B E C (3) E D A C B (2) E C B D A (2) D E A C B (2) C E B D A (2) C B D E A (2) B C D A E (2) E D C A B (1) E C D A B (1) E C A B D (1) E B C A D (1) D B C A E (1) D B A C E (1) D A C E B (1) B D C A E (1) B C D E A (1) B C A D E (1) B A C E D (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 -16 -4 -20 B 14 0 -2 8 2 C 16 2 0 14 8 D 4 -8 -14 0 -14 E 20 -2 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -4 -20 B 14 0 -2 8 2 C 16 2 0 14 8 D 4 -8 -14 0 -14 E 20 -2 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999966492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=24 B=24 E=23 C=16 A=13 so A is eliminated. Round 2 votes counts: D=36 E=24 B=24 C=16 so C is eliminated. Round 3 votes counts: D=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:220 E:212 B:211 D:184 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -16 -4 -20 B 14 0 -2 8 2 C 16 2 0 14 8 D 4 -8 -14 0 -14 E 20 -2 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999966492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -4 -20 B 14 0 -2 8 2 C 16 2 0 14 8 D 4 -8 -14 0 -14 E 20 -2 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999966492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -4 -20 B 14 0 -2 8 2 C 16 2 0 14 8 D 4 -8 -14 0 -14 E 20 -2 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999966492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3774: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) D E C A B (8) B A C E D (8) A B C E D (8) B E C D A (7) B A D E C (5) A C D E B (5) E C D B A (4) D A E C B (4) B D E C A (4) B C E D A (4) A C B E D (4) A B D E C (4) B E D C A (3) A D C E B (3) C E D A B (2) C E B D A (2) C B E D A (2) A D E C B (2) A C E D B (2) A B C D E (2) C E A B D (1) B D E A C (1) B D A E C (1) B C A E D (1) B A E D C (1) B A E C D (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 6 -2 6 B 6 0 0 20 14 C -6 0 0 8 -8 D 2 -20 -8 0 -2 E -6 -14 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.647606 C: 0.352394 D: 0.000000 E: 0.000000 Sum of squares = 0.543575154926 Cumulative probabilities = A: 0.000000 B: 0.647606 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -2 6 B 6 0 0 20 14 C -6 0 0 8 -8 D 2 -20 -8 0 -2 E -6 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500562 C: 0.499438 D: 0.000000 E: 0.000000 Sum of squares = 0.500000631252 Cumulative probabilities = A: 0.000000 B: 0.500562 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=32 D=21 C=7 E=4 so E is eliminated. Round 2 votes counts: B=36 A=32 D=21 C=11 so C is eliminated. Round 3 votes counts: B=40 A=33 D=27 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:202 C:197 E:195 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -2 6 B 6 0 0 20 14 C -6 0 0 8 -8 D 2 -20 -8 0 -2 E -6 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500562 C: 0.499438 D: 0.000000 E: 0.000000 Sum of squares = 0.500000631252 Cumulative probabilities = A: 0.000000 B: 0.500562 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -2 6 B 6 0 0 20 14 C -6 0 0 8 -8 D 2 -20 -8 0 -2 E -6 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500562 C: 0.499438 D: 0.000000 E: 0.000000 Sum of squares = 0.500000631252 Cumulative probabilities = A: 0.000000 B: 0.500562 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -2 6 B 6 0 0 20 14 C -6 0 0 8 -8 D 2 -20 -8 0 -2 E -6 -14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500562 C: 0.499438 D: 0.000000 E: 0.000000 Sum of squares = 0.500000631252 Cumulative probabilities = A: 0.000000 B: 0.500562 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3775: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (13) D C E B A (9) A D B E C (9) D A B E C (8) C E B D A (8) C E B A D (6) E C B A D (4) D C E A B (4) D A B C E (4) C E A B D (4) D B A E C (3) C E D B A (3) D B C E A (2) D A C E B (2) D A C B E (2) B D E C A (2) B D E A C (2) A E C B D (2) A D C E B (2) A B D E C (2) D C B E A (1) C A E D B (1) B E D C A (1) B E C D A (1) B E C A D (1) B A D E C (1) A E B C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 6 -4 4 B -12 0 2 0 6 C -6 -2 0 -10 -2 D 4 0 10 0 6 E -4 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.128571 C: 0.000000 D: 0.871429 E: 0.000000 Sum of squares = 0.775919033542 Cumulative probabilities = A: 0.000000 B: 0.128571 C: 0.128571 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -4 4 B -12 0 2 0 6 C -6 -2 0 -10 -2 D 4 0 10 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000018646 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=31 C=22 B=8 E=4 so E is eliminated. Round 2 votes counts: D=35 A=31 C=26 B=8 so B is eliminated. Round 3 votes counts: D=40 A=32 C=28 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:209 B:198 E:193 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -4 4 B -12 0 2 0 6 C -6 -2 0 -10 -2 D 4 0 10 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000018646 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 4 B -12 0 2 0 6 C -6 -2 0 -10 -2 D 4 0 10 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000018646 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 4 B -12 0 2 0 6 C -6 -2 0 -10 -2 D 4 0 10 0 6 E -4 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000018646 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3776: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) E A B D C (5) D A B E C (5) B D C A E (5) A D E B C (5) E C B A D (4) E A D B C (4) C B E D A (4) A E D B C (4) E A D C B (3) D B A E C (3) D A C B E (3) D A B C E (3) C B D E A (3) B D A E C (3) B C E D A (3) B C D E A (3) A D B E C (3) E B C A D (2) E B A D C (2) E A C D B (2) C E D A B (2) C E B D A (2) C E B A D (2) C D B A E (2) A E B D C (2) A D E C B (2) A D C E B (2) E C A B D (1) D C B A E (1) D C A B E (1) D B C A E (1) C E A D B (1) C E A B D (1) C D A B E (1) B E C D A (1) B E C A D (1) B D E A C (1) Total count = 100 A B C D E A 0 0 18 -8 6 B 0 0 26 -10 8 C -18 -26 0 -30 -6 D 8 10 30 0 8 E -6 -8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 -8 6 B 0 0 26 -10 8 C -18 -26 0 -30 -6 D 8 10 30 0 8 E -6 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 C=18 A=18 B=17 so B is eliminated. Round 2 votes counts: D=33 E=25 C=24 A=18 so A is eliminated. Round 3 votes counts: D=45 E=31 C=24 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:212 A:208 E:192 C:160 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 18 -8 6 B 0 0 26 -10 8 C -18 -26 0 -30 -6 D 8 10 30 0 8 E -6 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 -8 6 B 0 0 26 -10 8 C -18 -26 0 -30 -6 D 8 10 30 0 8 E -6 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 -8 6 B 0 0 26 -10 8 C -18 -26 0 -30 -6 D 8 10 30 0 8 E -6 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3777: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) E B C A D (6) D C E A B (6) A D B E C (6) D A C B E (5) D A B E C (5) E B A C D (4) D C E B A (4) C E B D A (4) C D E B A (4) A B E D C (4) D E C B A (3) D C A E B (3) C E D B A (3) C E B A D (3) B C A E D (3) A B D E C (3) D A E B C (2) C A B E D (2) B E A C D (2) B A E C D (2) A B E C D (2) E C D B A (1) E C B D A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C A B E (1) D A E C B (1) D A B C E (1) C D B E A (1) C B E A D (1) C B A E D (1) B C E A D (1) B A C E D (1) A D B C E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -12 4 -8 B 4 0 -6 0 -10 C 12 6 0 2 -4 D -4 0 -2 0 2 E 8 10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 -12 4 -8 B 4 0 -6 0 -10 C 12 6 0 2 -4 D -4 0 -2 0 2 E 8 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.37499999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=19 C=19 A=19 B=9 so B is eliminated. Round 2 votes counts: D=34 C=23 A=22 E=21 so E is eliminated. Round 3 votes counts: C=38 D=34 A=28 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:210 C:208 D:198 B:194 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -12 4 -8 B 4 0 -6 0 -10 C 12 6 0 2 -4 D -4 0 -2 0 2 E 8 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.37499999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 4 -8 B 4 0 -6 0 -10 C 12 6 0 2 -4 D -4 0 -2 0 2 E 8 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.37499999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 4 -8 B 4 0 -6 0 -10 C 12 6 0 2 -4 D -4 0 -2 0 2 E 8 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.37499999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3778: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (21) C A B D E (12) E D B C A (8) E D A B C (7) A C B D E (6) C B A D E (4) A C B E D (4) E D C B A (3) D E B A C (3) A C E B D (3) E D A C B (2) E A D C B (2) E A D B C (2) E A C D B (2) D B E A C (2) C B D A E (2) C A B E D (2) B D C E A (2) B C A D E (2) B A C D E (2) D E B C A (1) C E A D B (1) C A E B D (1) B D E C A (1) B D C A E (1) B A D E C (1) B A D C E (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 20 -6 -14 B 8 0 10 -10 -14 C -20 -10 0 -16 -12 D 6 10 16 0 -20 E 14 14 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 20 -6 -14 B 8 0 10 -10 -14 C -20 -10 0 -16 -12 D 6 10 16 0 -20 E 14 14 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=47 C=22 A=15 B=10 D=6 so D is eliminated. Round 2 votes counts: E=51 C=22 A=15 B=12 so B is eliminated. Round 3 votes counts: E=54 C=27 A=19 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:230 D:206 B:197 A:196 C:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 20 -6 -14 B 8 0 10 -10 -14 C -20 -10 0 -16 -12 D 6 10 16 0 -20 E 14 14 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 20 -6 -14 B 8 0 10 -10 -14 C -20 -10 0 -16 -12 D 6 10 16 0 -20 E 14 14 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 20 -6 -14 B 8 0 10 -10 -14 C -20 -10 0 -16 -12 D 6 10 16 0 -20 E 14 14 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3779: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (13) B C E A D (11) D E C A B (7) E C D A B (5) E C B D A (5) B E C D A (5) B C A E D (5) A D B C E (5) E D C B A (4) D E A C B (4) B A C E D (4) A D C E B (4) E C D B A (3) B A D C E (3) A D C B E (3) E D C A B (2) D E C B A (2) C B E A D (2) E B C D A (1) D E A B C (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E D B (1) C A E B D (1) B D E C A (1) B A C D E (1) A D E C B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -14 -10 -10 B -4 0 -18 -16 -14 C 14 18 0 -4 -12 D 10 16 4 0 -4 E 10 14 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -14 -10 -10 B -4 0 -18 -16 -14 C 14 18 0 -4 -12 D 10 16 4 0 -4 E 10 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=29 E=20 A=15 C=6 so C is eliminated. Round 2 votes counts: B=32 D=29 E=22 A=17 so A is eliminated. Round 3 votes counts: D=42 B=34 E=24 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:213 C:208 A:185 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -14 -10 -10 B -4 0 -18 -16 -14 C 14 18 0 -4 -12 D 10 16 4 0 -4 E 10 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 -10 -10 B -4 0 -18 -16 -14 C 14 18 0 -4 -12 D 10 16 4 0 -4 E 10 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 -10 -10 B -4 0 -18 -16 -14 C 14 18 0 -4 -12 D 10 16 4 0 -4 E 10 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3780: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) A D E C B (9) A D B C E (9) E C B D A (6) D A E B C (6) D A B E C (6) D A E C B (5) D A B C E (4) C E B A D (4) B C E A D (4) A D B E C (4) B A C D E (3) A B D C E (3) E D C A B (2) E C D B A (2) E C A D B (2) C E B D A (2) C E A B D (2) C B E A D (2) B D C E A (2) B D A C E (2) B C A D E (2) A D C E B (2) E C D A B (1) E C A B D (1) D E B A C (1) D E A C B (1) C B A E D (1) B D C A E (1) B C D E A (1) B A D C E (1) Total count = 100 A B C D E A 0 14 12 -2 16 B -14 0 16 -8 8 C -12 -16 0 -16 8 D 2 8 16 0 24 E -16 -8 -8 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 12 -2 16 B -14 0 16 -8 8 C -12 -16 0 -16 8 D 2 8 16 0 24 E -16 -8 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 D=23 E=14 C=11 so C is eliminated. Round 2 votes counts: B=28 A=27 D=23 E=22 so E is eliminated. Round 3 votes counts: B=40 A=32 D=28 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:225 A:220 B:201 C:182 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 12 -2 16 B -14 0 16 -8 8 C -12 -16 0 -16 8 D 2 8 16 0 24 E -16 -8 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 -2 16 B -14 0 16 -8 8 C -12 -16 0 -16 8 D 2 8 16 0 24 E -16 -8 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 -2 16 B -14 0 16 -8 8 C -12 -16 0 -16 8 D 2 8 16 0 24 E -16 -8 -8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3781: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) D B C A E (6) E A B D C (5) A E C D B (5) A C E D B (5) E A D B C (4) C D B A E (4) B D E C A (4) A D E B C (4) E B D A C (3) D B A C E (3) C B D E A (3) C B D A E (3) B E D C A (3) B C D E A (3) C B E A D (2) C A E B D (2) C A D E B (2) C A D B E (2) B E C D A (2) A E D B C (2) A C D E B (2) A C D B E (2) E C B A D (1) E C A B D (1) E B D C A (1) E B A D C (1) E B A C D (1) E A B C D (1) D E B A C (1) D C A B E (1) D B E A C (1) D B C E A (1) D A E B C (1) D A B E C (1) D A B C E (1) C D A B E (1) C A E D B (1) C A B E D (1) C A B D E (1) B E D A C (1) B D C E A (1) A E D C B (1) A D E C B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 10 8 10 8 B -10 0 2 -8 -4 C -8 -2 0 2 -2 D -10 8 -2 0 2 E -8 4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 10 8 B -10 0 2 -8 -4 C -8 -2 0 2 -2 D -10 8 -2 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 C=22 D=16 B=14 so B is eliminated. Round 2 votes counts: E=30 C=25 A=24 D=21 so D is eliminated. Round 3 votes counts: E=36 C=34 A=30 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:218 D:199 E:198 C:195 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 10 8 B -10 0 2 -8 -4 C -8 -2 0 2 -2 D -10 8 -2 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 10 8 B -10 0 2 -8 -4 C -8 -2 0 2 -2 D -10 8 -2 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 10 8 B -10 0 2 -8 -4 C -8 -2 0 2 -2 D -10 8 -2 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3782: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) C D E B A (9) A B D C E (7) E C D A B (6) E A B C D (6) C D B A E (6) A B D E C (6) D B C A E (5) C E D B A (5) E A B D C (4) D B A C E (4) B A D C E (4) A B E D C (4) E C D B A (3) E C A B D (2) E A C B D (2) C E D A B (2) C E A B D (2) C A B D E (2) B D A C E (2) E D B A C (1) E C A D B (1) D E C B A (1) D C E B A (1) D B A E C (1) C E A D B (1) B A D E C (1) A E B D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -12 -12 8 B 6 0 -6 -12 6 C 12 6 0 -4 22 D 12 12 4 0 20 E -8 -6 -22 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -12 8 B 6 0 -6 -12 6 C 12 6 0 -4 22 D 12 12 4 0 20 E -8 -6 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=25 D=22 A=19 B=7 so B is eliminated. Round 2 votes counts: C=27 E=25 D=24 A=24 so D is eliminated. Round 3 votes counts: C=43 A=31 E=26 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:224 C:218 B:197 A:189 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 8 B 6 0 -6 -12 6 C 12 6 0 -4 22 D 12 12 4 0 20 E -8 -6 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 8 B 6 0 -6 -12 6 C 12 6 0 -4 22 D 12 12 4 0 20 E -8 -6 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 8 B 6 0 -6 -12 6 C 12 6 0 -4 22 D 12 12 4 0 20 E -8 -6 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3783: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (22) E C B A D (21) C E B A D (8) E B C A D (4) E C D B A (3) D E C A B (3) C B A E D (3) B A C E D (3) E D C B A (2) E D B C A (2) D E A C B (2) D E A B C (2) D A E C B (2) D A B E C (2) C A B E D (2) B C E A D (2) A D C B E (2) A D B C E (2) A B C D E (2) D E B A C (1) D A E B C (1) D A C E B (1) C D A E B (1) C B E A D (1) C A E B D (1) C A B D E (1) B E A C D (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -8 12 -4 B 2 0 -10 2 -8 C 8 10 0 10 8 D -12 -2 -10 0 -6 E 4 8 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 12 -4 B 2 0 -10 2 -8 C 8 10 0 10 8 D -12 -2 -10 0 -6 E 4 8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=32 C=17 A=9 B=6 so B is eliminated. Round 2 votes counts: D=36 E=33 C=19 A=12 so A is eliminated. Round 3 votes counts: D=41 E=33 C=26 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:218 E:205 A:199 B:193 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 12 -4 B 2 0 -10 2 -8 C 8 10 0 10 8 D -12 -2 -10 0 -6 E 4 8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 12 -4 B 2 0 -10 2 -8 C 8 10 0 10 8 D -12 -2 -10 0 -6 E 4 8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 12 -4 B 2 0 -10 2 -8 C 8 10 0 10 8 D -12 -2 -10 0 -6 E 4 8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3784: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (8) D E A C B (5) B A C D E (5) A D C B E (5) E D C A B (4) E D A B C (4) B C A D E (4) E D A C B (3) E B C D A (3) E A D B C (3) D C E A B (3) C D B A E (3) B A E C D (3) B A C E D (3) A E D B C (3) A B D C E (3) A B C D E (3) E D C B A (2) E C D B A (2) E C B D A (2) D E C A B (2) D A E C B (2) C E B D A (2) C B A D E (2) C A D B E (2) B E C A D (2) B E A C D (2) B C E A D (2) A B D E C (2) E B A D C (1) E A B D C (1) D A C E B (1) C D E A B (1) C D A B E (1) C B D A E (1) B E C D A (1) A E B D C (1) A D E B C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 6 16 10 B -4 0 12 6 8 C -6 -12 0 6 0 D -16 -6 -6 0 -6 E -10 -8 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 16 10 B -4 0 12 6 8 C -6 -12 0 6 0 D -16 -6 -6 0 -6 E -10 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=25 A=20 D=13 C=12 so C is eliminated. Round 2 votes counts: B=33 E=27 A=22 D=18 so D is eliminated. Round 3 votes counts: E=38 B=36 A=26 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:218 B:211 C:194 E:194 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 16 10 B -4 0 12 6 8 C -6 -12 0 6 0 D -16 -6 -6 0 -6 E -10 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 16 10 B -4 0 12 6 8 C -6 -12 0 6 0 D -16 -6 -6 0 -6 E -10 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 16 10 B -4 0 12 6 8 C -6 -12 0 6 0 D -16 -6 -6 0 -6 E -10 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3785: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) E B A C D (8) C D E A B (8) B A E D C (8) E B A D C (7) D A B C E (7) C D A B E (7) E C D B A (6) E C B A D (6) D C A B E (3) C E D A B (3) C E A B D (3) B A D E C (3) A B D C E (3) E D C B A (2) D C E A B (2) D B A E C (2) D A B E C (2) B A E C D (2) E C B D A (1) E B D A C (1) E B C A D (1) D E B A C (1) D A C B E (1) C E D B A (1) C E A D B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 12 6 -2 B -2 0 12 8 -2 C -12 -12 0 -4 -22 D -6 -8 4 0 -4 E 2 2 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 12 6 -2 B -2 0 12 8 -2 C -12 -12 0 -4 -22 D -6 -8 4 0 -4 E 2 2 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=23 D=18 A=14 B=13 so B is eliminated. Round 2 votes counts: E=32 A=27 C=23 D=18 so D is eliminated. Round 3 votes counts: A=39 E=33 C=28 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:209 B:208 D:193 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 12 6 -2 B -2 0 12 8 -2 C -12 -12 0 -4 -22 D -6 -8 4 0 -4 E 2 2 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 6 -2 B -2 0 12 8 -2 C -12 -12 0 -4 -22 D -6 -8 4 0 -4 E 2 2 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 6 -2 B -2 0 12 8 -2 C -12 -12 0 -4 -22 D -6 -8 4 0 -4 E 2 2 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3786: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) A C D E B (9) B D C A E (8) C A D B E (7) E B D C A (6) B E D C A (6) A C E D B (6) E B D A C (5) E A C B D (5) B D C E A (5) A C D B E (5) E B A C D (4) E A C D B (4) D B C A E (4) D B C E A (3) A E C D B (3) D C B A E (2) D C A B E (2) B E A D C (2) E D B C A (1) E B A D C (1) E A B C D (1) C E A D B (1) B E D A C (1) Total count = 100 A B C D E A 0 -14 -8 -4 -8 B 14 0 12 6 8 C 8 -12 0 -10 4 D 4 -6 10 0 8 E 8 -8 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -4 -8 B 14 0 12 6 8 C 8 -12 0 -10 4 D 4 -6 10 0 8 E 8 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=27 A=23 D=11 C=8 so C is eliminated. Round 2 votes counts: B=31 A=30 E=28 D=11 so D is eliminated. Round 3 votes counts: B=40 A=32 E=28 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:208 C:195 E:194 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 -4 -8 B 14 0 12 6 8 C 8 -12 0 -10 4 D 4 -6 10 0 8 E 8 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -4 -8 B 14 0 12 6 8 C 8 -12 0 -10 4 D 4 -6 10 0 8 E 8 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -4 -8 B 14 0 12 6 8 C 8 -12 0 -10 4 D 4 -6 10 0 8 E 8 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3787: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) C E B A D (8) D C E A B (7) D C B A E (7) C D B E A (7) B A E C D (6) D A B E C (5) A E B D C (4) D C A E B (3) D C A B E (3) D A E B C (3) E B A C D (2) C E D B A (2) C D E A B (2) C B E A D (2) B E A C D (2) B C A E D (2) B A E D C (2) A E D B C (2) A B D E C (2) E A D C B (1) E A D B C (1) E A C B D (1) E A B D C (1) E A B C D (1) D E C A B (1) D C B E A (1) D B C A E (1) D B A C E (1) D A E C B (1) C E D A B (1) C E A D B (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B C E A D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -26 -16 -8 B 16 0 -22 -22 -4 C 26 22 0 6 24 D 16 22 -6 0 14 E 8 4 -24 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -26 -16 -8 B 16 0 -22 -22 -4 C 26 22 0 6 24 D 16 22 -6 0 14 E 8 4 -24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=33 B=14 A=10 E=7 so E is eliminated. Round 2 votes counts: C=36 D=33 B=16 A=15 so A is eliminated. Round 3 votes counts: D=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:239 D:223 E:187 B:184 A:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -26 -16 -8 B 16 0 -22 -22 -4 C 26 22 0 6 24 D 16 22 -6 0 14 E 8 4 -24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -26 -16 -8 B 16 0 -22 -22 -4 C 26 22 0 6 24 D 16 22 -6 0 14 E 8 4 -24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -26 -16 -8 B 16 0 -22 -22 -4 C 26 22 0 6 24 D 16 22 -6 0 14 E 8 4 -24 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3788: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) E A C D B (9) B D C A E (7) B D A C E (6) D C B E A (5) C D B E A (5) C D E B A (4) C D E A B (4) B A D E C (4) A B E D C (4) E A C B D (3) C E D A B (3) B C D E A (3) A E D C B (3) A E B D C (3) D C A E B (2) B A E C D (2) A E B C D (2) A D E C B (2) A B E C D (2) E C D A B (1) E C A D B (1) E A B C D (1) D C E A B (1) D C B A E (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B C E (1) C B D E A (1) B D C E A (1) B A D C E (1) A E D B C (1) A E C B D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 20 2 16 B 4 0 4 4 14 C -20 -4 0 -16 -4 D -2 -4 16 0 8 E -16 -14 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 20 2 16 B 4 0 4 4 14 C -20 -4 0 -16 -4 D -2 -4 16 0 8 E -16 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991578 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=20 C=17 E=15 D=14 so D is eliminated. Round 2 votes counts: B=36 C=26 A=23 E=15 so E is eliminated. Round 3 votes counts: B=36 A=36 C=28 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:213 D:209 E:183 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 20 2 16 B 4 0 4 4 14 C -20 -4 0 -16 -4 D -2 -4 16 0 8 E -16 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991578 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 20 2 16 B 4 0 4 4 14 C -20 -4 0 -16 -4 D -2 -4 16 0 8 E -16 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991578 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 20 2 16 B 4 0 4 4 14 C -20 -4 0 -16 -4 D -2 -4 16 0 8 E -16 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991578 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3789: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) C B E D A (12) A D E C B (7) C B A E D (6) C E D B A (5) B C E D A (5) D E A B C (4) C E B D A (4) A B D E C (4) E D C B A (3) B D E C A (3) B C D E A (3) A C D E B (3) A C B D E (3) E D B C A (2) E D A C B (2) D A E B C (2) C A E D B (2) C A B E D (2) C A B D E (2) B E D C A (2) B A D E C (2) E D B A C (1) E C D B A (1) D B E A C (1) C E D A B (1) C E A D B (1) B D E A C (1) B D C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 -4 -10 -6 -4 B 4 0 -10 0 -6 C 10 10 0 0 0 D 6 0 0 0 0 E 4 6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.094952 D: 0.419999 E: 0.485049 Sum of squares = 0.420687251326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.094952 D: 0.514951 E: 1.000000 A B C D E A 0 -4 -10 -6 -4 B 4 0 -10 0 -6 C 10 10 0 0 0 D 6 0 0 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=32 B=17 E=9 D=7 so D is eliminated. Round 2 votes counts: C=35 A=34 B=18 E=13 so E is eliminated. Round 3 votes counts: A=40 C=39 B=21 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 E:205 D:203 B:194 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -4 B 4 0 -10 0 -6 C 10 10 0 0 0 D 6 0 0 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 -10 0 -6 C 10 10 0 0 0 D 6 0 0 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 -10 0 -6 C 10 10 0 0 0 D 6 0 0 0 0 E 4 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333334 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3790: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (15) D B E C A (10) A C D B E (9) A C E B D (7) C A D B E (6) C D B A E (5) E B D A C (4) D E B C A (4) A E B D C (4) E D B A C (3) A C D E B (3) C B D E A (2) C A B E D (2) C A B D E (2) B E D C A (2) B E C D A (2) A D E B C (2) A D C E B (2) A D C B E (2) E D B C A (1) E B C D A (1) E A B D C (1) D C B E A (1) C E B A D (1) C D A B E (1) C B E D A (1) C B D A E (1) B D E C A (1) A E D C B (1) A E C B D (1) A E B C D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -16 -8 2 B 8 0 2 -4 -6 C 16 -2 0 -8 -8 D 8 4 8 0 4 E -2 6 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -8 2 B 8 0 2 -4 -6 C 16 -2 0 -8 -8 D 8 4 8 0 4 E -2 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=25 C=21 D=15 B=5 so B is eliminated. Round 2 votes counts: A=34 E=29 C=21 D=16 so D is eliminated. Round 3 votes counts: E=44 A=34 C=22 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:212 E:204 B:200 C:199 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 2 B 8 0 2 -4 -6 C 16 -2 0 -8 -8 D 8 4 8 0 4 E -2 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 2 B 8 0 2 -4 -6 C 16 -2 0 -8 -8 D 8 4 8 0 4 E -2 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 2 B 8 0 2 -4 -6 C 16 -2 0 -8 -8 D 8 4 8 0 4 E -2 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3791: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) E A B C D (7) D C A E B (6) D A E C B (6) A E C D B (5) D C B A E (4) C B E A D (4) C A E B D (4) B C D E A (4) E A C B D (3) D A C E B (3) C D A E B (3) C A E D B (3) B E C D A (3) B D E A C (3) E B A C D (2) D B E A C (2) D B C E A (2) D B A E C (2) D A E B C (2) C D B A E (2) B E C A D (2) B E A C D (2) B D C E A (2) B C E A D (2) A E D B C (2) E A B D C (1) D B A C E (1) C E A B D (1) C D A B E (1) C B D A E (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A D C (1) A E D C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -8 -12 16 B -4 0 0 -14 -4 C 8 0 0 4 8 D 12 14 -4 0 10 E -16 4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.151680 C: 0.848320 D: 0.000000 E: 0.000000 Sum of squares = 0.742654309668 Cumulative probabilities = A: 0.000000 B: 0.151680 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -12 16 B -4 0 0 -14 -4 C 8 0 0 4 8 D 12 14 -4 0 10 E -16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.654321039739 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=21 B=20 E=13 A=10 so A is eliminated. Round 2 votes counts: D=37 C=22 E=21 B=20 so B is eliminated. Round 3 votes counts: D=42 E=30 C=28 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:210 A:200 B:189 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -12 16 B -4 0 0 -14 -4 C 8 0 0 4 8 D 12 14 -4 0 10 E -16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.654321039739 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -12 16 B -4 0 0 -14 -4 C 8 0 0 4 8 D 12 14 -4 0 10 E -16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.654321039739 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -12 16 B -4 0 0 -14 -4 C 8 0 0 4 8 D 12 14 -4 0 10 E -16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.654321039739 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3792: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) D E A C B (6) D C E A B (6) B E A D C (6) A E B C D (5) E A D B C (4) D E C A B (4) C B D A E (4) C B A E D (4) C A E D B (4) B C A E D (4) B A E D C (4) E B A D C (2) E A D C B (2) E A B D C (2) D E A B C (2) D C E B A (2) D B E A C (2) C D A E B (2) B D E A C (2) B C D A E (2) B C A D E (2) B A E C D (2) A E D B C (2) E D B A C (1) E D A B C (1) D C B E A (1) D B E C A (1) D B C E A (1) C D E A B (1) C D B A E (1) C B A D E (1) C A E B D (1) C A D E B (1) C A B D E (1) B E D A C (1) A E D C B (1) A E C B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 14 20 6 B 2 0 12 10 0 C -14 -12 0 -6 -4 D -20 -10 6 0 -16 E -6 0 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.837423 C: 0.000000 D: 0.000000 E: 0.162577 Sum of squares = 0.727707990699 Cumulative probabilities = A: 0.000000 B: 0.837423 C: 0.837423 D: 0.837423 E: 1.000000 A B C D E A 0 -2 14 20 6 B 2 0 12 10 0 C -14 -12 0 -6 -4 D -20 -10 6 0 -16 E -6 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000202976 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=25 C=20 E=12 A=12 so E is eliminated. Round 2 votes counts: B=33 D=27 C=20 A=20 so C is eliminated. Round 3 votes counts: B=42 D=31 A=27 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:212 E:207 C:182 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 14 20 6 B 2 0 12 10 0 C -14 -12 0 -6 -4 D -20 -10 6 0 -16 E -6 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000202976 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 20 6 B 2 0 12 10 0 C -14 -12 0 -6 -4 D -20 -10 6 0 -16 E -6 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000202976 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 20 6 B 2 0 12 10 0 C -14 -12 0 -6 -4 D -20 -10 6 0 -16 E -6 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000202976 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3793: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) E C A B D (7) E B D A C (5) D B E A C (5) D B A E C (5) C E A B D (5) A C B E D (4) A C B D E (4) E B A D C (3) D E C B A (3) B D A E C (3) B A E D C (3) E D B C A (2) E C D B A (2) E B D C A (2) D E B A C (2) D B A C E (2) D A B C E (2) C E D A B (2) C E A D B (2) C A E D B (2) B E A D C (2) A D B C E (2) A B D C E (2) E D C B A (1) E D B A C (1) E C B D A (1) E B A C D (1) D E B C A (1) D C E B A (1) D C B A E (1) D C A E B (1) D B E C A (1) D A C B E (1) C D E A B (1) C D A B E (1) C A D E B (1) C A B D E (1) B E D A C (1) B D E A C (1) B A D E C (1) B A D C E (1) A C E B D (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 8 4 -4 B 2 0 -2 14 -10 C -8 2 0 -10 -6 D -4 -14 10 0 -12 E 4 10 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 4 -4 B 2 0 -2 14 -10 C -8 2 0 -10 -6 D -4 -14 10 0 -12 E 4 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 C=23 A=15 B=12 so B is eliminated. Round 2 votes counts: D=29 E=28 C=23 A=20 so A is eliminated. Round 3 votes counts: D=35 C=34 E=31 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:216 A:203 B:202 D:190 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 4 -4 B 2 0 -2 14 -10 C -8 2 0 -10 -6 D -4 -14 10 0 -12 E 4 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 4 -4 B 2 0 -2 14 -10 C -8 2 0 -10 -6 D -4 -14 10 0 -12 E 4 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 4 -4 B 2 0 -2 14 -10 C -8 2 0 -10 -6 D -4 -14 10 0 -12 E 4 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3794: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) B C A E D (7) C B A E D (6) E A D B C (5) E A B D C (5) D E A C B (5) D C B A E (5) C B D A E (5) B A C E D (5) A B E C D (5) C D B A E (4) E D A B C (3) E A B C D (3) D E A B C (3) C D B E A (3) A B D E C (3) E B A C D (2) D C E B A (2) D C E A B (2) D A E B C (2) C B E A D (2) B E A C D (2) E C D B A (1) D E C A B (1) D C A E B (1) D A E C B (1) D A C B E (1) C E B D A (1) C D E B A (1) C B D E A (1) B E C A D (1) B A E C D (1) A E D B C (1) A E B C D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 16 14 B 12 0 2 18 20 C 0 -2 0 18 8 D -16 -18 -18 0 -6 E -14 -20 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998463 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 16 14 B 12 0 2 18 20 C 0 -2 0 18 8 D -16 -18 -18 0 -6 E -14 -20 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=23 E=19 B=16 A=12 so A is eliminated. Round 2 votes counts: C=30 B=26 D=23 E=21 so E is eliminated. Round 3 votes counts: B=37 D=32 C=31 so C is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:212 A:209 E:182 D:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 16 14 B 12 0 2 18 20 C 0 -2 0 18 8 D -16 -18 -18 0 -6 E -14 -20 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 16 14 B 12 0 2 18 20 C 0 -2 0 18 8 D -16 -18 -18 0 -6 E -14 -20 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 16 14 B 12 0 2 18 20 C 0 -2 0 18 8 D -16 -18 -18 0 -6 E -14 -20 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3795: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D E A C B (6) E C D A B (4) E A C D B (4) D A B E C (4) B C E A D (4) A B D C E (4) D A E C B (3) C E B A D (3) C B E D A (3) C B E A D (3) B C E D A (3) A E C B D (3) A D E C B (3) E C B D A (2) E C A D B (2) E A D C B (2) D E B A C (2) D B A E C (2) D B A C E (2) C E A B D (2) B D C E A (2) B D C A E (2) B D A C E (2) B A C E D (2) A E D C B (2) A B C D E (2) E D C B A (1) E D A C B (1) E C D B A (1) E C A B D (1) D E C B A (1) D E C A B (1) D E B C A (1) D B E A C (1) D B C E A (1) D A E B C (1) C B A E D (1) C A E B D (1) B C D E A (1) B C D A E (1) B C A D E (1) A D E B C (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -2 4 -4 B 2 0 -2 6 4 C 2 2 0 8 0 D -4 -6 -8 0 -10 E 4 -4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.851987 D: 0.000000 E: 0.148013 Sum of squares = 0.747789432207 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.851987 D: 0.851987 E: 1.000000 A B C D E A 0 -2 -2 4 -4 B 2 0 -2 6 4 C 2 2 0 8 0 D -4 -6 -8 0 -10 E 4 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555659489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 E=18 A=17 C=13 so C is eliminated. Round 2 votes counts: B=34 D=25 E=23 A=18 so A is eliminated. Round 3 votes counts: B=41 D=30 E=29 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:206 B:205 E:205 A:198 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -4 B 2 0 -2 6 4 C 2 2 0 8 0 D -4 -6 -8 0 -10 E 4 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555659489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -4 B 2 0 -2 6 4 C 2 2 0 8 0 D -4 -6 -8 0 -10 E 4 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555659489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -4 B 2 0 -2 6 4 C 2 2 0 8 0 D -4 -6 -8 0 -10 E 4 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.333333 Sum of squares = 0.555555659489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3796: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (16) A D B C E (9) D A B C E (7) E C B A D (5) D A E B C (5) D A B E C (4) A D B E C (4) E D A C B (3) E C D B A (3) E A C D B (3) C E B A D (3) C B E A D (3) C B A D E (3) B C A D E (3) E C A D B (2) E B C D A (2) D E A B C (2) C E A B D (2) C B E D A (2) C A B E D (2) B D A C E (2) A D E B C (2) E D C B A (1) E D A B C (1) E C D A B (1) E B D C A (1) E A D C B (1) D E B A C (1) C A E B D (1) B C D E A (1) B C A E D (1) A D E C B (1) A D C E B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 -4 -6 B -6 0 -8 -2 -14 C 4 8 0 8 -16 D 4 2 -8 0 -6 E 6 14 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 -4 -6 B -6 0 -8 -2 -14 C 4 8 0 8 -16 D 4 2 -8 0 -6 E 6 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=19 A=19 C=16 B=7 so B is eliminated. Round 2 votes counts: E=39 D=21 C=21 A=19 so A is eliminated. Round 3 votes counts: E=39 D=39 C=22 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:202 A:196 D:196 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 -4 -6 B -6 0 -8 -2 -14 C 4 8 0 8 -16 D 4 2 -8 0 -6 E 6 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 -6 B -6 0 -8 -2 -14 C 4 8 0 8 -16 D 4 2 -8 0 -6 E 6 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 -6 B -6 0 -8 -2 -14 C 4 8 0 8 -16 D 4 2 -8 0 -6 E 6 14 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3797: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) A D B C E (6) E C A B D (5) D B C A E (5) A E D B C (5) A D C B E (5) E B C D A (4) E A C B D (4) D B A C E (4) D A B C E (4) A E D C B (4) E C B A D (3) E B C A D (3) D A C B E (3) C E B D A (3) C D A B E (3) B C E D A (3) A E C D B (3) A D B E C (3) E B A D C (2) C B E D A (2) C B D E A (2) C A E D B (2) B D C E A (2) A E B D C (2) A D E B C (2) E B A C D (1) E A B D C (1) E A B C D (1) D C B A E (1) D B A E C (1) C D B A E (1) C A D B E (1) B E D A C (1) B D C A E (1) B C D E A (1) Total count = 100 A B C D E A 0 8 4 6 12 B -8 0 4 -6 -2 C -4 -4 0 -4 -2 D -6 6 4 0 -10 E -12 2 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 6 12 B -8 0 4 -6 -2 C -4 -4 0 -4 -2 D -6 6 4 0 -10 E -12 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=30 A=30 D=18 C=14 B=8 so B is eliminated. Round 2 votes counts: E=31 A=30 D=21 C=18 so C is eliminated. Round 3 votes counts: E=39 A=33 D=28 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:201 D:197 B:194 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 6 12 B -8 0 4 -6 -2 C -4 -4 0 -4 -2 D -6 6 4 0 -10 E -12 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 6 12 B -8 0 4 -6 -2 C -4 -4 0 -4 -2 D -6 6 4 0 -10 E -12 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 6 12 B -8 0 4 -6 -2 C -4 -4 0 -4 -2 D -6 6 4 0 -10 E -12 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3798: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) A C B E D (7) D B E C A (6) A B E D C (6) D B E A C (4) C D E B A (4) C D B E A (4) C A D E B (4) C A D B E (4) A C E B D (4) E D B C A (3) E B A D C (3) D C B E A (3) C D A E B (3) C A E D B (3) C A B D E (3) B A E D C (3) A B D E C (3) E D B A C (2) E A B D C (2) C E D B A (2) C E A D B (2) B E D A C (2) B D E A C (2) A E B D C (2) A E B C D (2) A B C E D (2) A B C D E (2) D B C E A (1) C D B A E (1) C D A B E (1) C A E B D (1) C A B E D (1) B D A E C (1) Total count = 100 A B C D E A 0 4 8 8 6 B -4 0 6 6 12 C -8 -6 0 0 4 D -8 -6 0 0 -8 E -6 -12 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 6 B -4 0 6 6 12 C -8 -6 0 0 4 D -8 -6 0 0 -8 E -6 -12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=28 E=17 D=14 B=8 so B is eliminated. Round 2 votes counts: C=33 A=31 E=19 D=17 so D is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:210 C:195 E:193 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 6 B -4 0 6 6 12 C -8 -6 0 0 4 D -8 -6 0 0 -8 E -6 -12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 6 B -4 0 6 6 12 C -8 -6 0 0 4 D -8 -6 0 0 -8 E -6 -12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 6 B -4 0 6 6 12 C -8 -6 0 0 4 D -8 -6 0 0 -8 E -6 -12 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3799: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) B C D A E (6) E A D C B (5) E A C D B (5) D A C B E (5) C D A B E (5) A D C E B (5) E B A C D (4) E A D B C (4) D C A B E (4) B E C D A (4) B D C A E (4) B E D C A (3) B E C A D (3) B C E D A (3) E C A B D (2) E B A D C (2) E A B D C (2) D B C A E (2) D A E B C (2) C B D A E (2) C B A D E (2) A E D C B (2) A C E D B (2) E D A C B (1) E B D A C (1) D C B A E (1) D B E A C (1) D A E C B (1) D A B C E (1) C E B A D (1) C A E D B (1) C A D B E (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 14 6 -14 10 B -14 0 -8 -14 2 C -6 8 0 -16 10 D 14 14 16 0 8 E -10 -2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -14 10 B -14 0 -8 -14 2 C -6 8 0 -16 10 D 14 14 16 0 8 E -10 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=26 D=25 C=12 A=10 so A is eliminated. Round 2 votes counts: D=31 E=28 B=27 C=14 so C is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 A:208 C:198 E:185 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -14 10 B -14 0 -8 -14 2 C -6 8 0 -16 10 D 14 14 16 0 8 E -10 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -14 10 B -14 0 -8 -14 2 C -6 8 0 -16 10 D 14 14 16 0 8 E -10 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -14 10 B -14 0 -8 -14 2 C -6 8 0 -16 10 D 14 14 16 0 8 E -10 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3800: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) D B C E A (8) E A D B C (6) A E C B D (6) E A C B D (5) E D A B C (4) B C D E A (4) A C E B D (4) E C B A D (3) E A D C B (3) C A B D E (3) B C D A E (3) A E C D B (3) E A C D B (2) D E B C A (2) D B E C A (2) C B E A D (2) C B D A E (2) C B A E D (2) B C E D A (2) A D E C B (2) A D E B C (2) A D C B E (2) A C B E D (2) E D B C A (1) E B C A D (1) D E A B C (1) D A C B E (1) D A B C E (1) C E B A D (1) C E A B D (1) C D B A E (1) C B E D A (1) C B A D E (1) B E C D A (1) B D C E A (1) B D C A E (1) A E D C B (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -6 8 -2 B -2 0 0 -8 2 C 6 0 0 2 10 D -8 8 -2 0 -2 E 2 -2 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.135113 C: 0.864887 D: 0.000000 E: 0.000000 Sum of squares = 0.766285621192 Cumulative probabilities = A: 0.000000 B: 0.135113 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 8 -2 B -2 0 0 -8 2 C 6 0 0 2 10 D -8 8 -2 0 -2 E 2 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017503 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 A=24 C=14 B=12 so B is eliminated. Round 2 votes counts: D=27 E=26 A=24 C=23 so C is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 A:201 D:198 B:196 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 8 -2 B -2 0 0 -8 2 C 6 0 0 2 10 D -8 8 -2 0 -2 E 2 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017503 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 8 -2 B -2 0 0 -8 2 C 6 0 0 2 10 D -8 8 -2 0 -2 E 2 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017503 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 8 -2 B -2 0 0 -8 2 C 6 0 0 2 10 D -8 8 -2 0 -2 E 2 -2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000017503 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3801: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) B E C D A (7) A D E C B (7) E A D B C (6) A D C E B (6) C B A D E (5) E D A B C (4) E B C D A (4) C A D B E (4) B C E D A (4) E A B D C (3) D E A B C (3) D A E C B (3) D A C E B (3) C B E D A (3) A E D B C (3) E B D C A (2) E B A D C (2) D A E B C (2) C D B A E (2) C D A B E (2) C B E A D (2) C B D A E (2) B E C A D (2) A D C B E (2) A C D B E (2) E B C A D (1) E A C B D (1) D A C B E (1) C B D E A (1) B E D C A (1) B C E A D (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 8 14 -2 -8 B -8 0 6 -4 -18 C -14 -6 0 -12 -18 D 2 4 12 0 -6 E 8 18 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 14 -2 -8 B -8 0 6 -4 -18 C -14 -6 0 -12 -18 D 2 4 12 0 -6 E 8 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=22 C=21 B=15 D=12 so D is eliminated. Round 2 votes counts: E=33 A=31 C=21 B=15 so B is eliminated. Round 3 votes counts: E=43 A=31 C=26 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:206 D:206 B:188 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 14 -2 -8 B -8 0 6 -4 -18 C -14 -6 0 -12 -18 D 2 4 12 0 -6 E 8 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 -2 -8 B -8 0 6 -4 -18 C -14 -6 0 -12 -18 D 2 4 12 0 -6 E 8 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 -2 -8 B -8 0 6 -4 -18 C -14 -6 0 -12 -18 D 2 4 12 0 -6 E 8 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3802: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) E C D B A (5) D B A E C (5) B D E A C (5) D B E A C (4) C E D A B (4) C A E D B (4) C A E B D (4) E C B D A (3) E B C D A (3) C E A D B (3) C A D E B (3) B A D E C (3) A C B D E (3) A B D C E (3) E C D A B (2) E C B A D (2) E B D C A (2) D E B C A (2) C E A B D (2) C D A E B (2) B E D A C (2) B E C A D (2) B A D C E (2) E D C B A (1) E D C A B (1) E D B C A (1) E B D A C (1) D E C A B (1) D E B A C (1) D E A C B (1) D C A E B (1) D A C B E (1) D A B E C (1) C D E A B (1) C A D B E (1) B E A C D (1) B A E D C (1) B A C D E (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 0 -22 0 B 16 0 4 4 0 C 0 -4 0 -4 -22 D 22 -4 4 0 10 E 0 0 22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.816264 C: 0.000000 D: 0.000000 E: 0.183736 Sum of squares = 0.700046383672 Cumulative probabilities = A: 0.000000 B: 0.816264 C: 0.816264 D: 0.816264 E: 1.000000 A B C D E A 0 -16 0 -22 0 B 16 0 4 4 0 C 0 -4 0 -4 -22 D 22 -4 4 0 10 E 0 0 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.591836790248 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=24 E=21 D=17 A=10 so A is eliminated. Round 2 votes counts: B=31 C=29 E=21 D=19 so D is eliminated. Round 3 votes counts: B=42 C=32 E=26 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:212 E:206 C:185 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -22 0 B 16 0 4 4 0 C 0 -4 0 -4 -22 D 22 -4 4 0 10 E 0 0 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.591836790248 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -22 0 B 16 0 4 4 0 C 0 -4 0 -4 -22 D 22 -4 4 0 10 E 0 0 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.591836790248 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -22 0 B 16 0 4 4 0 C 0 -4 0 -4 -22 D 22 -4 4 0 10 E 0 0 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.285714 Sum of squares = 0.591836790248 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3803: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) E D C A B (9) E A C B D (9) B A C D E (9) B A C E D (8) E B A C D (7) E A C D B (7) B D C A E (6) D C A B E (5) D E C A B (4) E C A D B (3) D C E A B (3) D C B A E (3) D C A E B (3) B E A C D (2) B D E C A (2) A C B D E (2) E D A C B (1) D C B E A (1) B D A C E (1) B C D A E (1) B C A D E (1) B A E C D (1) B A D C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -2 2 4 B 6 0 -2 2 6 C 2 2 0 2 10 D -2 -2 -2 0 4 E -4 -6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 2 4 B 6 0 -2 2 6 C 2 2 0 2 10 D -2 -2 -2 0 4 E -4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=32 D=29 A=3 so C is eliminated. Round 2 votes counts: E=36 B=32 D=29 A=3 so A is eliminated. Round 3 votes counts: E=37 B=34 D=29 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:206 A:199 D:199 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 2 4 B 6 0 -2 2 6 C 2 2 0 2 10 D -2 -2 -2 0 4 E -4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 4 B 6 0 -2 2 6 C 2 2 0 2 10 D -2 -2 -2 0 4 E -4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 4 B 6 0 -2 2 6 C 2 2 0 2 10 D -2 -2 -2 0 4 E -4 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3804: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) D B E C A (8) A D B E C (7) D B A E C (6) A C D B E (6) E C B D A (5) D B E A C (5) A C E B D (5) A C D E B (5) E B D C A (4) C E B D A (4) C A E B D (4) B D E C A (4) C E D B A (3) A D B C E (3) E C D B A (2) E C A B D (2) C E A D B (2) C A E D B (2) B D E A C (2) A B D E C (2) D E B C A (1) D C A B E (1) C E B A D (1) C D E B A (1) C A D E B (1) B E D C A (1) B D A E C (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -10 4 -8 B -4 0 -12 -8 -4 C 10 12 0 10 0 D -4 8 -10 0 8 E 8 4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.685970 D: 0.000000 E: 0.314030 Sum of squares = 0.569169803626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.685970 D: 0.685970 E: 1.000000 A B C D E A 0 4 -10 4 -8 B -4 0 -12 -8 -4 C 10 12 0 10 0 D -4 8 -10 0 8 E 8 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=27 D=21 E=13 B=8 so B is eliminated. Round 2 votes counts: A=31 D=28 C=27 E=14 so E is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:202 D:201 A:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 4 -8 B -4 0 -12 -8 -4 C 10 12 0 10 0 D -4 8 -10 0 8 E 8 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 4 -8 B -4 0 -12 -8 -4 C 10 12 0 10 0 D -4 8 -10 0 8 E 8 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 4 -8 B -4 0 -12 -8 -4 C 10 12 0 10 0 D -4 8 -10 0 8 E 8 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3805: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (14) B D C E A (12) D C B E A (9) E A C D B (7) A E B C D (7) B D C A E (6) D C E B A (4) B A E D C (4) E C D A B (3) C D E A B (3) B E D C A (3) B E D A C (3) B A D C E (3) A B E D C (3) B E A D C (2) A D B C E (2) A C D E B (2) E C D B A (1) E C A D B (1) E B D C A (1) E B C D A (1) E A B C D (1) D C A B E (1) D B C A E (1) D A C B E (1) A E C B D (1) A C E D B (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 8 2 -2 B 0 0 2 -2 6 C -8 -2 0 -14 -6 D -2 2 14 0 -6 E 2 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.688726 B: 0.311274 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.571234654376 Cumulative probabilities = A: 0.688726 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 -2 B 0 0 2 -2 6 C -8 -2 0 -14 -6 D -2 2 14 0 -6 E 2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500263 B: 0.499737 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138738 Cumulative probabilities = A: 0.500263 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=33 A=33 D=16 E=15 C=3 so C is eliminated. Round 2 votes counts: B=33 A=33 D=19 E=15 so E is eliminated. Round 3 votes counts: A=42 B=35 D=23 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:204 D:204 E:204 B:203 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 -2 B 0 0 2 -2 6 C -8 -2 0 -14 -6 D -2 2 14 0 -6 E 2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500263 B: 0.499737 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138738 Cumulative probabilities = A: 0.500263 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 -2 B 0 0 2 -2 6 C -8 -2 0 -14 -6 D -2 2 14 0 -6 E 2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500263 B: 0.499737 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138738 Cumulative probabilities = A: 0.500263 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 -2 B 0 0 2 -2 6 C -8 -2 0 -14 -6 D -2 2 14 0 -6 E 2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500263 B: 0.499737 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000138738 Cumulative probabilities = A: 0.500263 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3806: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) C A E B D (10) E C A D B (9) D B A C E (8) E C D A B (6) E D B C A (4) D B E A C (4) C A B E D (4) B D A C E (4) B A C D E (4) E C B A D (3) D E B A C (3) D B A E C (3) B A D C E (3) A B C D E (3) E D C A B (2) D E B C A (2) D E A C B (2) D A B C E (2) B D E C A (2) A C B E D (2) E C B D A (1) E B C A D (1) C E B A D (1) C A E D B (1) B E D C A (1) B E C D A (1) B D A E C (1) B C A E D (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -18 8 -4 B -6 0 -6 6 -12 C 18 6 0 18 -10 D -8 -6 -18 0 -16 E 4 12 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -18 8 -4 B -6 0 -6 6 -12 C 18 6 0 18 -10 D -8 -6 -18 0 -16 E 4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=24 B=17 C=16 A=7 so A is eliminated. Round 2 votes counts: E=36 D=24 C=20 B=20 so C is eliminated. Round 3 votes counts: E=49 B=27 D=24 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:216 A:196 B:191 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -18 8 -4 B -6 0 -6 6 -12 C 18 6 0 18 -10 D -8 -6 -18 0 -16 E 4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 8 -4 B -6 0 -6 6 -12 C 18 6 0 18 -10 D -8 -6 -18 0 -16 E 4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 8 -4 B -6 0 -6 6 -12 C 18 6 0 18 -10 D -8 -6 -18 0 -16 E 4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3807: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) D E C B A (10) A B C E D (8) A B C D E (8) A D B C E (6) D E A C B (5) D A E B C (5) B C A E D (5) E A C B D (4) C B E A D (4) A B D C E (4) E C B D A (3) D A E C B (3) B C E A D (3) A E B C D (3) E A D C B (2) D C B E A (2) C B E D A (2) B C A D E (2) B A C E D (2) E D C B A (1) E D A C B (1) E C A B D (1) D E C A B (1) D C E B A (1) D A B C E (1) C E B D A (1) A D E B C (1) Total count = 100 A B C D E A 0 6 6 28 -4 B -6 0 -4 22 -6 C -6 4 0 14 -2 D -28 -22 -14 0 -2 E 4 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 28 -4 B -6 0 -4 22 -6 C -6 4 0 14 -2 D -28 -22 -14 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=28 E=23 B=12 C=7 so C is eliminated. Round 2 votes counts: A=30 D=28 E=24 B=18 so B is eliminated. Round 3 votes counts: A=39 E=33 D=28 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:218 E:207 C:205 B:203 D:167 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 28 -4 B -6 0 -4 22 -6 C -6 4 0 14 -2 D -28 -22 -14 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 28 -4 B -6 0 -4 22 -6 C -6 4 0 14 -2 D -28 -22 -14 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 28 -4 B -6 0 -4 22 -6 C -6 4 0 14 -2 D -28 -22 -14 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3808: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) E A C B D (6) D B E A C (6) D B A C E (6) D B C A E (5) B D A C E (5) E C A B D (4) E A B C D (4) D B A E C (4) C E D A B (3) C E A D B (3) B D E A C (3) B D A E C (3) B A E C D (3) B A D C E (3) A B C E D (3) E C D A B (2) E C A D B (2) D E C B A (2) D E B C A (2) D C E A B (2) D C B A E (2) C D A B E (2) A E C B D (2) E D C A B (1) E B D A C (1) E B A D C (1) D E B A C (1) D C B E A (1) D B E C A (1) D B C E A (1) C E A B D (1) C D E A B (1) C A E D B (1) C A B D E (1) B E A D C (1) B A D E C (1) B A C D E (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 12 -8 2 B 6 0 12 4 6 C -12 -12 0 -4 -2 D 8 -4 4 0 6 E -2 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -8 2 B 6 0 12 4 6 C -12 -12 0 -4 -2 D 8 -4 4 0 6 E -2 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=21 B=20 C=19 A=7 so A is eliminated. Round 2 votes counts: D=33 E=24 B=23 C=20 so C is eliminated. Round 3 votes counts: E=39 D=36 B=25 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:214 D:207 A:200 E:194 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 -8 2 B 6 0 12 4 6 C -12 -12 0 -4 -2 D 8 -4 4 0 6 E -2 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -8 2 B 6 0 12 4 6 C -12 -12 0 -4 -2 D 8 -4 4 0 6 E -2 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -8 2 B 6 0 12 4 6 C -12 -12 0 -4 -2 D 8 -4 4 0 6 E -2 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3809: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (11) C B D E A (10) D B C A E (8) A E D B C (8) D B C E A (6) C B D A E (6) A E C B D (5) D A B E C (4) B C D E A (4) C A B E D (3) A E C D B (3) A D E B C (3) E D A B C (2) E C B A D (2) E C A B D (2) E A D B C (2) C E A B D (2) C B E D A (2) C B E A D (2) A C E B D (2) E D B C A (1) E B D A C (1) E A B D C (1) E A B C D (1) D E B A C (1) D B E C A (1) D A C B E (1) C A E B D (1) C A D B E (1) B D C E A (1) A D C E B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -4 4 -4 B -10 0 -8 12 2 C 4 8 0 14 2 D -4 -12 -14 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 4 -4 B -10 0 -8 12 2 C 4 8 0 14 2 D -4 -12 -14 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 E=23 D=21 B=5 so B is eliminated. Round 2 votes counts: C=31 A=24 E=23 D=22 so D is eliminated. Round 3 votes counts: C=46 A=29 E=25 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:203 E:201 B:198 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 4 -4 B -10 0 -8 12 2 C 4 8 0 14 2 D -4 -12 -14 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 4 -4 B -10 0 -8 12 2 C 4 8 0 14 2 D -4 -12 -14 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 4 -4 B -10 0 -8 12 2 C 4 8 0 14 2 D -4 -12 -14 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3810: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) D E B A C (7) C A B E D (7) A B C D E (7) D E A B C (5) C E D B A (5) D E C A B (4) C B E A D (4) A C B D E (4) A B D E C (4) A B D C E (4) E D C B A (3) D B E A C (3) C E B A D (3) B A D E C (3) E D B C A (2) E C D B A (2) C E B D A (2) C B A E D (2) C A B D E (2) B A E C D (2) A B C E D (2) E B D C A (1) D B A E C (1) D A E C B (1) C E D A B (1) C A E D B (1) C A E B D (1) B E D C A (1) B E D A C (1) B E A D C (1) B C E D A (1) B C E A D (1) B C A E D (1) B A C D E (1) A D C E B (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -2 6 -8 B 8 0 12 12 8 C 2 -12 0 0 2 D -6 -12 0 0 10 E 8 -8 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 6 -8 B 8 0 12 12 8 C 2 -12 0 0 2 D -6 -12 0 0 10 E 8 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 A=24 B=12 E=8 so E is eliminated. Round 2 votes counts: D=33 C=30 A=24 B=13 so B is eliminated. Round 3 votes counts: D=36 C=33 A=31 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:220 C:196 D:196 A:194 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 6 -8 B 8 0 12 12 8 C 2 -12 0 0 2 D -6 -12 0 0 10 E 8 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 6 -8 B 8 0 12 12 8 C 2 -12 0 0 2 D -6 -12 0 0 10 E 8 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 6 -8 B 8 0 12 12 8 C 2 -12 0 0 2 D -6 -12 0 0 10 E 8 -8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3811: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E B D C (10) D A E C B (8) A E D B C (8) B E C A D (6) B C E A D (6) D C A E B (5) C B D E A (5) B E A C D (5) A D E B C (4) E B A C D (3) D A C E B (3) A E B C D (3) A D E C B (3) E A B C D (2) D C E A B (2) D C B E A (2) D C B A E (2) B A C E D (2) E A B D C (1) D E C B A (1) D C A B E (1) C B E D A (1) C B D A E (1) C B A D E (1) B C E D A (1) A D C B E (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 14 8 B -6 0 6 -4 -6 C -10 -6 0 -4 -10 D -14 4 4 0 2 E -8 6 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 14 8 B -6 0 6 -4 -6 C -10 -6 0 -4 -10 D -14 4 4 0 2 E -8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=24 B=20 C=19 E=6 so E is eliminated. Round 2 votes counts: A=34 D=24 B=23 C=19 so C is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:203 D:198 B:195 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 14 8 B -6 0 6 -4 -6 C -10 -6 0 -4 -10 D -14 4 4 0 2 E -8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 14 8 B -6 0 6 -4 -6 C -10 -6 0 -4 -10 D -14 4 4 0 2 E -8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 14 8 B -6 0 6 -4 -6 C -10 -6 0 -4 -10 D -14 4 4 0 2 E -8 6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3812: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) B A D C E (10) B E C A D (7) B A C E D (6) B A C D E (5) E D C A B (4) D E C A B (4) D C E A B (4) D A C E B (4) A D B C E (4) D A B C E (3) C E D A B (3) A B C D E (3) E C D B A (2) E C B D A (2) E C B A D (2) E B C A D (2) D B A E C (2) D A E C B (2) D A B E C (2) C E B A D (2) B E D C A (2) B E C D A (2) B C E A D (2) B A D E C (2) A B D C E (2) D A C B E (1) C D E A B (1) C A E B D (1) B D E A C (1) B A E C D (1) Total count = 100 A B C D E A 0 0 -4 -2 -4 B 0 0 12 4 10 C 4 -12 0 6 2 D 2 -4 -6 0 0 E 4 -10 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.455612 B: 0.544388 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.503940509771 Cumulative probabilities = A: 0.455612 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -2 -4 B 0 0 12 4 10 C 4 -12 0 6 2 D 2 -4 -6 0 0 E 4 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=24 D=22 A=9 C=7 so C is eliminated. Round 2 votes counts: B=38 E=29 D=23 A=10 so A is eliminated. Round 3 votes counts: B=43 E=30 D=27 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:200 D:196 E:196 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -2 -4 B 0 0 12 4 10 C 4 -12 0 6 2 D 2 -4 -6 0 0 E 4 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 -4 B 0 0 12 4 10 C 4 -12 0 6 2 D 2 -4 -6 0 0 E 4 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 -4 B 0 0 12 4 10 C 4 -12 0 6 2 D 2 -4 -6 0 0 E 4 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3813: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) B C A E D (8) C A B E D (7) E D B C A (6) D E B A C (6) D E A B C (6) D A E C B (6) C B A E D (6) B C E A D (6) B E C D A (5) D A E B C (4) C A B D E (4) E B D C A (3) D E A C B (3) C B E A D (3) A D C E B (3) E B C D A (2) D A C E B (2) C E B D A (2) E D C B A (1) E D B A C (1) E B D A C (1) B E C A D (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -8 4 8 B 4 0 0 22 8 C 8 0 0 16 10 D -4 -22 -16 0 -6 E -8 -8 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.304486 C: 0.695514 D: 0.000000 E: 0.000000 Sum of squares = 0.576451592412 Cumulative probabilities = A: 0.000000 B: 0.304486 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 4 8 B 4 0 0 22 8 C 8 0 0 16 10 D -4 -22 -16 0 -6 E -8 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=22 B=21 A=16 E=14 so E is eliminated. Round 2 votes counts: D=35 B=27 C=22 A=16 so A is eliminated. Round 3 votes counts: D=38 C=35 B=27 so B is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:217 C:217 A:200 E:190 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 4 8 B 4 0 0 22 8 C 8 0 0 16 10 D -4 -22 -16 0 -6 E -8 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 4 8 B 4 0 0 22 8 C 8 0 0 16 10 D -4 -22 -16 0 -6 E -8 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 4 8 B 4 0 0 22 8 C 8 0 0 16 10 D -4 -22 -16 0 -6 E -8 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3814: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (20) E B A D C (12) D E B A C (12) C D A B E (11) D C E A B (9) A B E C D (7) D C E B A (6) A B C E D (4) E D B A C (3) C D A E B (3) C A E B D (2) B A E D C (2) B A E C D (2) A C B E D (2) E A B D C (1) E A B C D (1) D C B E A (1) C D E A B (1) C D B A E (1) Total count = 100 A B C D E A 0 22 -8 6 8 B -22 0 -12 6 0 C 8 12 0 8 20 D -6 -6 -8 0 -12 E -8 0 -20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -8 6 8 B -22 0 -12 6 0 C 8 12 0 8 20 D -6 -6 -8 0 -12 E -8 0 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=28 E=17 A=13 B=4 so B is eliminated. Round 2 votes counts: C=38 D=28 E=17 A=17 so E is eliminated. Round 3 votes counts: C=38 D=31 A=31 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:214 E:192 B:186 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -8 6 8 B -22 0 -12 6 0 C 8 12 0 8 20 D -6 -6 -8 0 -12 E -8 0 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -8 6 8 B -22 0 -12 6 0 C 8 12 0 8 20 D -6 -6 -8 0 -12 E -8 0 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -8 6 8 B -22 0 -12 6 0 C 8 12 0 8 20 D -6 -6 -8 0 -12 E -8 0 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3815: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) D E B A C (7) C B E D A (6) E B D A C (5) D E A B C (5) A D E B C (5) C B E A D (4) C A B D E (4) E B D C A (3) D A E B C (3) C D E B A (3) C B A E D (3) B E C D A (3) A D C E B (3) A D B E C (3) A B E D C (3) E D B A C (2) D E C B A (2) C D E A B (2) B E D C A (2) A D E C B (2) A D C B E (2) E D B C A (1) D E B C A (1) D E A C B (1) D C E B A (1) D C E A B (1) D C A E B (1) D A E C B (1) C E D B A (1) C E B D A (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A C D (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 0 -6 -6 B -6 0 -4 0 -4 C 0 4 0 -12 -6 D 6 0 12 0 6 E 6 4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.327803 C: 0.000000 D: 0.672197 E: 0.000000 Sum of squares = 0.559303543718 Cumulative probabilities = A: 0.000000 B: 0.327803 C: 0.327803 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -6 -6 B -6 0 -4 0 -4 C 0 4 0 -12 -6 D 6 0 12 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499695 C: 0.000000 D: 0.500305 E: 0.000000 Sum of squares = 0.500000186386 Cumulative probabilities = A: 0.000000 B: 0.499695 C: 0.499695 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=24 D=23 E=11 B=7 so B is eliminated. Round 2 votes counts: C=35 A=24 D=23 E=18 so E is eliminated. Round 3 votes counts: C=38 D=37 A=25 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:205 A:197 B:193 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -6 -6 B -6 0 -4 0 -4 C 0 4 0 -12 -6 D 6 0 12 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499695 C: 0.000000 D: 0.500305 E: 0.000000 Sum of squares = 0.500000186386 Cumulative probabilities = A: 0.000000 B: 0.499695 C: 0.499695 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -6 -6 B -6 0 -4 0 -4 C 0 4 0 -12 -6 D 6 0 12 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499695 C: 0.000000 D: 0.500305 E: 0.000000 Sum of squares = 0.500000186386 Cumulative probabilities = A: 0.000000 B: 0.499695 C: 0.499695 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -6 -6 B -6 0 -4 0 -4 C 0 4 0 -12 -6 D 6 0 12 0 6 E 6 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499695 C: 0.000000 D: 0.500305 E: 0.000000 Sum of squares = 0.500000186386 Cumulative probabilities = A: 0.000000 B: 0.499695 C: 0.499695 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3816: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (12) C E B D A (8) B A D E C (7) E B C A D (6) D A B E C (6) C D E A B (5) B E C A D (5) D C A E B (4) D A C B E (4) E C B A D (3) C E D B A (3) C B E D A (3) B E A D C (3) B A E D C (3) E C A D B (2) E B A C D (2) E A C B D (2) D A B C E (2) C E D A B (2) C E A D B (2) C D A E B (2) B E A C D (2) B D C A E (2) C E B A D (1) C D E B A (1) C D B A E (1) B D A C E (1) B C E D A (1) A E B D C (1) A D E C B (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -2 -14 -2 B 4 0 -12 4 -14 C 2 12 0 2 8 D 14 -4 -2 0 0 E 2 14 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -14 -2 B 4 0 -12 4 -14 C 2 12 0 2 8 D 14 -4 -2 0 0 E 2 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 B=24 E=15 A=5 so A is eliminated. Round 2 votes counts: D=30 C=28 B=26 E=16 so E is eliminated. Round 3 votes counts: C=35 B=35 D=30 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:204 E:204 B:191 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -14 -2 B 4 0 -12 4 -14 C 2 12 0 2 8 D 14 -4 -2 0 0 E 2 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -14 -2 B 4 0 -12 4 -14 C 2 12 0 2 8 D 14 -4 -2 0 0 E 2 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -14 -2 B 4 0 -12 4 -14 C 2 12 0 2 8 D 14 -4 -2 0 0 E 2 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3817: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) B A D C E (7) A B D C E (7) B E C D A (6) E C D B A (5) E C D A B (4) E B C D A (4) C E D A B (4) B E A C D (4) B A E D C (4) E C B D A (3) D A C E B (3) C D E A B (3) B E C A D (3) A D E C B (3) A D C B E (3) A B D E C (3) E D C A B (2) E D A C B (2) D C A E B (2) B E A D C (2) B A C D E (2) E D B C A (1) E B D A C (1) E B A D C (1) D C E A B (1) C E D B A (1) C D A E B (1) B C E A D (1) B C D A E (1) B C A D E (1) B A E C D (1) B A D E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 14 12 4 B -2 0 4 6 -4 C -14 -4 0 -12 -2 D -12 -6 12 0 0 E -4 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 12 4 B -2 0 4 6 -4 C -14 -4 0 -12 -2 D -12 -6 12 0 0 E -4 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=29 E=23 C=9 D=6 so D is eliminated. Round 2 votes counts: B=33 A=32 E=23 C=12 so C is eliminated. Round 3 votes counts: A=35 B=33 E=32 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:202 E:201 D:197 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 12 4 B -2 0 4 6 -4 C -14 -4 0 -12 -2 D -12 -6 12 0 0 E -4 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 12 4 B -2 0 4 6 -4 C -14 -4 0 -12 -2 D -12 -6 12 0 0 E -4 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 12 4 B -2 0 4 6 -4 C -14 -4 0 -12 -2 D -12 -6 12 0 0 E -4 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3818: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (6) B D E A C (6) C D A E B (5) B A C E D (5) E D B C A (4) D B E A C (4) A C B E D (4) A B C D E (4) E B D A C (3) D E C B A (3) D E B A C (3) D C E A B (3) C D E A B (3) C A D E B (3) C A D B E (3) B A E C D (3) B A D E C (3) A C B D E (3) E D C B A (2) E D C A B (2) E B D C A (2) D E C A B (2) D E B C A (2) C E D A B (2) C E A B D (2) C A E B D (2) C A B E D (2) B E D A C (2) E C D B A (1) E C B A D (1) E B C D A (1) D C A E B (1) D B A E C (1) C A E D B (1) B D A E C (1) B A E D C (1) B A D C E (1) B A C D E (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 6 -6 -8 B 12 0 10 8 4 C -6 -10 0 -6 -6 D 6 -8 6 0 6 E 8 -4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -6 -8 B 12 0 10 8 4 C -6 -10 0 -6 -6 D 6 -8 6 0 6 E 8 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=23 D=19 E=16 A=13 so A is eliminated. Round 2 votes counts: B=34 C=30 D=20 E=16 so E is eliminated. Round 3 votes counts: B=40 C=32 D=28 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:205 E:202 A:190 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 -6 -8 B 12 0 10 8 4 C -6 -10 0 -6 -6 D 6 -8 6 0 6 E 8 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -6 -8 B 12 0 10 8 4 C -6 -10 0 -6 -6 D 6 -8 6 0 6 E 8 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -6 -8 B 12 0 10 8 4 C -6 -10 0 -6 -6 D 6 -8 6 0 6 E 8 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3819: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) E C B A D (6) E C A B D (6) D A B C E (6) A D B C E (5) A B D C E (5) A B C E D (5) E C D B A (4) E C B D A (4) D E C B A (4) C B E A D (4) A E C B D (4) D B A C E (3) D E C A B (2) D B C E A (2) D B C A E (2) D A E B C (2) C E B D A (2) C B E D A (2) B D A C E (2) B C E A D (2) B A C E D (2) E D C B A (1) E D C A B (1) E D A C B (1) E C A D B (1) E A C D B (1) E A C B D (1) D E A C B (1) D A B E C (1) C E A B D (1) B D C E A (1) B C E D A (1) B C A E D (1) B A D C E (1) B A C D E (1) A D E B C (1) A D B E C (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -10 16 -12 B 6 0 -10 22 -2 C 10 10 0 14 14 D -16 -22 -14 0 -18 E 12 2 -14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 16 -12 B 6 0 -10 22 -2 C 10 10 0 14 14 D -16 -22 -14 0 -18 E 12 2 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 A=23 C=17 B=11 so B is eliminated. Round 2 votes counts: A=27 E=26 D=26 C=21 so C is eliminated. Round 3 votes counts: E=46 A=28 D=26 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:224 E:209 B:208 A:194 D:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 16 -12 B 6 0 -10 22 -2 C 10 10 0 14 14 D -16 -22 -14 0 -18 E 12 2 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 16 -12 B 6 0 -10 22 -2 C 10 10 0 14 14 D -16 -22 -14 0 -18 E 12 2 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 16 -12 B 6 0 -10 22 -2 C 10 10 0 14 14 D -16 -22 -14 0 -18 E 12 2 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3820: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) B A E D C (7) B A C E D (7) C E D A B (6) B C D E A (6) B C A E D (5) B A E C D (5) E A D C B (4) A B E D C (4) D E A C B (3) C E D B A (3) C D E A B (3) B A D E C (3) B A C D E (3) D C E A B (2) D B C E A (2) C E A D B (2) C D E B A (2) C A E D B (2) B D A E C (2) B C D A E (2) B A D C E (2) A E D C B (2) A E C D B (2) A E B D C (2) E D C A B (1) E A C D B (1) D C E B A (1) D A E C B (1) C B D E A (1) A E D B C (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 2 8 4 B 2 0 2 0 -2 C -2 -2 0 2 0 D -8 0 -2 0 -10 E -4 2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999756 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 2 8 4 B 2 0 2 0 -2 C -2 -2 0 2 0 D -8 0 -2 0 -10 E -4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999926 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 D=20 C=19 A=13 E=6 so E is eliminated. Round 2 votes counts: B=42 D=21 C=19 A=18 so A is eliminated. Round 3 votes counts: B=48 D=29 C=23 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:206 E:204 B:201 C:199 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 8 4 B 2 0 2 0 -2 C -2 -2 0 2 0 D -8 0 -2 0 -10 E -4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999926 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 8 4 B 2 0 2 0 -2 C -2 -2 0 2 0 D -8 0 -2 0 -10 E -4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999926 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 8 4 B 2 0 2 0 -2 C -2 -2 0 2 0 D -8 0 -2 0 -10 E -4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999926 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3821: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) E C D A B (8) A B C E D (7) E C D B A (6) D B A E C (6) B A C E D (6) D E C B A (5) D E A B C (5) C E A B D (5) C E B A D (4) D A B E C (3) C E D A B (3) B A D E C (3) B A D C E (3) A B C D E (3) E D C A B (2) D E B C A (2) D B E A C (2) C B E A D (2) B D A E C (2) A C B E D (2) E C B D A (1) D E B A C (1) C E D B A (1) C E B D A (1) C A E B D (1) B D E C A (1) B C A E D (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -8 -20 -20 B -6 0 -4 -12 -12 C 8 4 0 2 -18 D 20 12 -2 0 0 E 20 12 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.517480 E: 0.482520 Sum of squares = 0.500611105349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.517480 E: 1.000000 A B C D E A 0 6 -8 -20 -20 B -6 0 -4 -12 -12 C 8 4 0 2 -18 D 20 12 -2 0 0 E 20 12 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=17 C=17 B=16 A=15 so A is eliminated. Round 2 votes counts: D=36 B=28 C=19 E=17 so E is eliminated. Round 3 votes counts: D=38 C=34 B=28 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:225 D:215 C:198 B:183 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 -20 -20 B -6 0 -4 -12 -12 C 8 4 0 2 -18 D 20 12 -2 0 0 E 20 12 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -20 -20 B -6 0 -4 -12 -12 C 8 4 0 2 -18 D 20 12 -2 0 0 E 20 12 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -20 -20 B -6 0 -4 -12 -12 C 8 4 0 2 -18 D 20 12 -2 0 0 E 20 12 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3822: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) D E A C B (7) A B C D E (7) A D E C B (6) D E C A B (5) B A C E D (5) E D C B A (4) E D B A C (4) C B A D E (4) B C E A D (4) B C A E D (4) B E C D A (3) A C D B E (3) D A E B C (2) C D E B A (2) C D B E A (2) C B E D A (2) B E C A D (2) A D B E C (2) A C B D E (2) A B E D C (2) A B C E D (2) E D A C B (1) E D A B C (1) E C B D A (1) E B D A C (1) E B C D A (1) D E C B A (1) D C A E B (1) C E B D A (1) C D E A B (1) C D A B E (1) C B A E D (1) C A D B E (1) B E A D C (1) B A E C D (1) A E D B C (1) A D E B C (1) A D C E B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 4 4 -2 B 2 0 4 -10 0 C -4 -4 0 2 -10 D -4 10 -2 0 2 E 2 0 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -2 4 4 -2 B 2 0 4 -10 0 C -4 -4 0 2 -10 D -4 10 -2 0 2 E 2 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999953 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=20 B=20 D=16 C=15 so C is eliminated. Round 2 votes counts: A=30 B=27 D=22 E=21 so E is eliminated. Round 3 votes counts: D=39 B=31 A=30 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:205 D:203 A:202 B:198 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 4 -2 B 2 0 4 -10 0 C -4 -4 0 2 -10 D -4 10 -2 0 2 E 2 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999953 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 4 -2 B 2 0 4 -10 0 C -4 -4 0 2 -10 D -4 10 -2 0 2 E 2 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999953 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 4 -2 B 2 0 4 -10 0 C -4 -4 0 2 -10 D -4 10 -2 0 2 E 2 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999953 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3823: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (8) E A D C B (7) B C D A E (6) A C D B E (5) E D A C B (4) B E C D A (4) B C A D E (4) A D E C B (4) A C D E B (4) E D A B C (3) E B D C A (3) C B D A E (3) C B A D E (3) E D B C A (2) E A D B C (2) E A B D C (2) D E A C B (2) D B C E A (2) B E D C A (2) A E C D B (2) A C B D E (2) E D B A C (1) E B D A C (1) E A B C D (1) D C A E B (1) D A C E B (1) C D B A E (1) C A B D E (1) B E C A D (1) B D C E A (1) B C E A D (1) B C A E D (1) A E C B D (1) A D C E B (1) A C E D B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 8 6 2 B -8 0 -4 -2 -2 C -8 4 0 6 -2 D -6 2 -6 0 2 E -2 2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 6 2 B -8 0 -4 -2 -2 C -8 4 0 6 -2 D -6 2 -6 0 2 E -2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=30 A=30 E=26 C=8 D=6 so D is eliminated. Round 2 votes counts: B=32 A=31 E=28 C=9 so C is eliminated. Round 3 votes counts: B=39 A=33 E=28 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:200 E:200 D:196 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 6 2 B -8 0 -4 -2 -2 C -8 4 0 6 -2 D -6 2 -6 0 2 E -2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 2 B -8 0 -4 -2 -2 C -8 4 0 6 -2 D -6 2 -6 0 2 E -2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 2 B -8 0 -4 -2 -2 C -8 4 0 6 -2 D -6 2 -6 0 2 E -2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998112 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3824: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) A C B E D (6) A B C E D (6) E D B C A (5) E B D C A (5) D C E B A (5) C A D E B (5) A C B D E (5) E B D A C (4) D E B C A (4) C D E B A (4) B E A D C (4) B A E D C (4) B A E C D (4) D E C B A (3) C D E A B (3) A C D B E (3) D E B A C (2) D C E A B (2) D A E C B (2) C B A E D (2) C A B E D (2) A B E D C (2) E D B A C (1) E B C D A (1) E B C A D (1) D C A E B (1) C E D B A (1) C E B D A (1) C B E A D (1) C A D B E (1) B E C A D (1) B E A C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -8 0 2 B 8 0 -6 4 -12 C 8 6 0 10 10 D 0 -4 -10 0 -6 E -2 12 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 0 2 B 8 0 -6 4 -12 C 8 6 0 10 10 D 0 -4 -10 0 -6 E -2 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 D=19 E=17 B=14 so B is eliminated. Round 2 votes counts: A=32 C=26 E=23 D=19 so D is eliminated. Round 3 votes counts: C=34 A=34 E=32 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:203 B:197 A:193 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 0 2 B 8 0 -6 4 -12 C 8 6 0 10 10 D 0 -4 -10 0 -6 E -2 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 0 2 B 8 0 -6 4 -12 C 8 6 0 10 10 D 0 -4 -10 0 -6 E -2 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 0 2 B 8 0 -6 4 -12 C 8 6 0 10 10 D 0 -4 -10 0 -6 E -2 12 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3825: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) E D B A C (5) E D A B C (5) C B A E D (5) C B A D E (5) C A B D E (5) B C A E D (5) E D B C A (4) D A E C B (4) A C B D E (4) D E C B A (3) D E C A B (3) D E A C B (3) C D A B E (3) C A D B E (3) B E C A D (3) E B D C A (2) E B A D C (2) D E A B C (2) C B D E A (2) B E C D A (2) B C E A D (2) A E D B C (2) A E B D C (2) A C D B E (2) E A B D C (1) D E B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) C D B A E (1) B E D C A (1) B E A C D (1) B A E C D (1) B A C E D (1) A E D C B (1) A E B C D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -8 10 8 B -2 0 -12 4 10 C 8 12 0 8 2 D -10 -4 -8 0 -6 E -8 -10 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 10 8 B -2 0 -12 4 10 C 8 12 0 8 2 D -10 -4 -8 0 -6 E -8 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998076 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=21 D=20 E=19 B=16 so B is eliminated. Round 2 votes counts: C=31 E=26 A=23 D=20 so D is eliminated. Round 3 votes counts: E=38 C=35 A=27 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:206 B:200 E:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 10 8 B -2 0 -12 4 10 C 8 12 0 8 2 D -10 -4 -8 0 -6 E -8 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998076 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 10 8 B -2 0 -12 4 10 C 8 12 0 8 2 D -10 -4 -8 0 -6 E -8 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998076 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 10 8 B -2 0 -12 4 10 C 8 12 0 8 2 D -10 -4 -8 0 -6 E -8 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998076 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3826: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (13) E A B D C (12) B D A E C (9) D B A E C (7) C D B A E (6) D B A C E (4) C E A B D (4) C B D E A (4) C A E D B (4) B E A D C (4) A E D B C (4) D B C A E (3) B D E A C (3) B D C A E (3) D C B A E (2) B D C E A (2) E C A B D (1) E B C A D (1) E B A D C (1) E B A C D (1) E A C B D (1) E A B C D (1) D C A B E (1) D A E B C (1) D A B C E (1) C D A E B (1) C D A B E (1) C B E A D (1) C B D A E (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 4 2 2 B 4 0 18 0 4 C -4 -18 0 -18 2 D -2 0 18 0 0 E -2 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.678454 C: 0.000000 D: 0.321546 E: 0.000000 Sum of squares = 0.563691957366 Cumulative probabilities = A: 0.000000 B: 0.678454 C: 0.678454 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 2 2 B 4 0 18 0 4 C -4 -18 0 -18 2 D -2 0 18 0 0 E -2 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=21 D=19 E=18 A=7 so A is eliminated. Round 2 votes counts: C=35 E=24 B=21 D=20 so D is eliminated. Round 3 votes counts: C=38 B=36 E=26 so E is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:208 A:202 E:196 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 2 2 B 4 0 18 0 4 C -4 -18 0 -18 2 D -2 0 18 0 0 E -2 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 2 2 B 4 0 18 0 4 C -4 -18 0 -18 2 D -2 0 18 0 0 E -2 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 2 2 B 4 0 18 0 4 C -4 -18 0 -18 2 D -2 0 18 0 0 E -2 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3827: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (14) E C A D B (11) A C D B E (9) B D E A C (8) E B D C A (6) C A E D B (6) A D B C E (6) D B A C E (5) C E A D B (5) E C A B D (4) E C B D A (3) E D B C A (2) E B D A C (2) C A D E B (2) C A D B E (2) A B D C E (2) E C D B A (1) E A B C D (1) D C B A E (1) D B E C A (1) D A B C E (1) C A E B D (1) B D E C A (1) B D A E C (1) B A D E C (1) B A D C E (1) A E B C D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 8 8 10 B -6 0 6 -4 8 C -8 -6 0 -4 14 D -8 4 4 0 12 E -10 -8 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 8 10 B -6 0 6 -4 8 C -8 -6 0 -4 14 D -8 4 4 0 12 E -10 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 A=20 C=16 D=8 so D is eliminated. Round 2 votes counts: B=32 E=30 A=21 C=17 so C is eliminated. Round 3 votes counts: E=35 B=33 A=32 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 D:206 B:202 C:198 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 8 10 B -6 0 6 -4 8 C -8 -6 0 -4 14 D -8 4 4 0 12 E -10 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 8 10 B -6 0 6 -4 8 C -8 -6 0 -4 14 D -8 4 4 0 12 E -10 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 8 10 B -6 0 6 -4 8 C -8 -6 0 -4 14 D -8 4 4 0 12 E -10 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3828: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) D E B A C (6) C D A B E (5) B E D C A (5) D E B C A (4) D C E B A (4) A C B E D (4) A B C E D (4) D E A B C (3) D C E A B (3) D C A E B (3) C D B A E (3) C B A E D (3) C A D B E (3) C A B E D (3) C A B D E (3) B E A D C (3) D E C B A (2) D B C E A (2) C D A E B (2) C B D E A (2) B C D E A (2) B A E C D (2) A E B D C (2) A B E C D (2) E D B A C (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C B E A (1) D A E C B (1) C D B E A (1) B E C D A (1) B E A C D (1) B C E A D (1) B C A E D (1) B A C E D (1) A E D C B (1) A E C D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -10 -20 -8 B 8 0 2 0 6 C 10 -2 0 -6 6 D 20 0 6 0 6 E 8 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.679139 C: 0.000000 D: 0.320861 E: 0.000000 Sum of squares = 0.564181241464 Cumulative probabilities = A: 0.000000 B: 0.679139 C: 0.679139 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -20 -8 B 8 0 2 0 6 C 10 -2 0 -6 6 D 20 0 6 0 6 E 8 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=25 B=17 A=16 E=11 so E is eliminated. Round 2 votes counts: D=32 C=25 B=25 A=18 so A is eliminated. Round 3 votes counts: D=34 B=34 C=32 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:208 C:204 E:195 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -20 -8 B 8 0 2 0 6 C 10 -2 0 -6 6 D 20 0 6 0 6 E 8 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -20 -8 B 8 0 2 0 6 C 10 -2 0 -6 6 D 20 0 6 0 6 E 8 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -20 -8 B 8 0 2 0 6 C 10 -2 0 -6 6 D 20 0 6 0 6 E 8 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3829: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) A D B E C (9) C A D E B (8) E B C D A (6) A D B C E (6) E C B D A (5) E C B A D (5) D A B C E (5) C A D B E (4) B E D A C (4) E C A B D (3) E B D A C (3) D B A C E (3) D A B E C (3) C E A D B (3) B D A E C (3) E B D C A (2) C E B D A (2) C D B A E (2) C A E D B (2) E B C A D (1) E B A D C (1) E A B D C (1) C D E B A (1) C D A B E (1) C B D A E (1) B E D C A (1) B D C A E (1) B C D E A (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 16 2 12 22 B -16 0 0 -20 8 C -2 0 0 -4 6 D -12 20 4 0 20 E -22 -8 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998324 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 12 22 B -16 0 0 -20 8 C -2 0 0 -4 6 D -12 20 4 0 20 E -22 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 C=24 D=11 B=10 so B is eliminated. Round 2 votes counts: E=32 A=28 C=25 D=15 so D is eliminated. Round 3 votes counts: A=42 E=32 C=26 so C is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:216 C:200 B:186 E:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 12 22 B -16 0 0 -20 8 C -2 0 0 -4 6 D -12 20 4 0 20 E -22 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 12 22 B -16 0 0 -20 8 C -2 0 0 -4 6 D -12 20 4 0 20 E -22 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 12 22 B -16 0 0 -20 8 C -2 0 0 -4 6 D -12 20 4 0 20 E -22 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3830: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) C A B E D (7) E D B C A (6) D E A B C (6) A C B D E (6) A B C D E (6) D E B A C (5) D A B E C (5) C E B A D (5) E D C B A (4) E C D B A (4) C B E A D (4) D E A C B (3) A D B C E (3) A C D B E (3) E C B D A (2) D E C B A (2) D A E B C (2) B E D A C (2) B C A E D (2) B A C E D (2) E D C A B (1) E D B A C (1) E B D C A (1) E B C D A (1) D C A E B (1) C E B D A (1) C A E B D (1) B E C A D (1) B A C D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 6 2 B 8 0 -8 8 8 C 6 8 0 14 8 D -6 -8 -14 0 -12 E -2 -8 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 6 2 B 8 0 -8 8 8 C 6 8 0 14 8 D -6 -8 -14 0 -12 E -2 -8 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 E=20 A=20 B=8 so B is eliminated. Round 2 votes counts: C=30 D=24 E=23 A=23 so E is eliminated. Round 3 votes counts: D=39 C=38 A=23 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:208 A:197 E:197 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 6 2 B 8 0 -8 8 8 C 6 8 0 14 8 D -6 -8 -14 0 -12 E -2 -8 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 6 2 B 8 0 -8 8 8 C 6 8 0 14 8 D -6 -8 -14 0 -12 E -2 -8 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 6 2 B 8 0 -8 8 8 C 6 8 0 14 8 D -6 -8 -14 0 -12 E -2 -8 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3831: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (16) C A E B D (13) E B A C D (6) C D A B E (6) B E A D C (6) E B A D C (5) C D A E B (5) C A D E B (5) B E D A C (3) A E C B D (3) A C E B D (3) E B D A C (2) D E B C A (2) D C B E A (2) D C B A E (2) D C A B E (2) D B E C A (2) C E A B D (2) C D E B A (2) A E B C D (2) E C A B D (1) E A B C D (1) D E B A C (1) D C E B A (1) D B C E A (1) D B C A E (1) D B A E C (1) D A C B E (1) D A B E C (1) C A D B E (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 4 -2 -6 B 6 0 0 -6 -8 C -4 0 0 0 -4 D 2 6 0 0 6 E 6 8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.230445 D: 0.769555 E: 0.000000 Sum of squares = 0.645319795827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.230445 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -2 -6 B 6 0 0 -6 -8 C -4 0 0 0 -4 D 2 6 0 0 6 E 6 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555643533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=33 E=15 B=9 A=9 so B is eliminated. Round 2 votes counts: C=34 D=33 E=24 A=9 so A is eliminated. Round 3 votes counts: C=37 D=34 E=29 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:207 E:206 B:196 C:196 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 -2 -6 B 6 0 0 -6 -8 C -4 0 0 0 -4 D 2 6 0 0 6 E 6 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555643533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 -6 B 6 0 0 -6 -8 C -4 0 0 0 -4 D 2 6 0 0 6 E 6 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555643533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 -6 B 6 0 0 -6 -8 C -4 0 0 0 -4 D 2 6 0 0 6 E 6 8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555643533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3832: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) A D B C E (9) E C B A D (6) D E A C B (6) D A E C B (5) D A E B C (5) A B D C E (5) A B C E D (5) D A B E C (4) B C A E D (4) B A C E D (4) D A B C E (3) C E B A D (3) B C E A D (3) A B C D E (3) E D C B A (2) E C D B A (2) D E C A B (2) A D E C B (2) A D E B C (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A B D (1) D E C B A (1) D B C E A (1) D B C A E (1) C E B D A (1) C B E A D (1) B D C A E (1) B C E D A (1) B A D C E (1) A E D C B (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 16 16 4 14 B -16 0 6 0 -6 C -16 -6 0 -8 -8 D -4 0 8 0 4 E -14 6 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 4 14 B -16 0 6 0 -6 C -16 -6 0 -8 -8 D -4 0 8 0 4 E -14 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=28 E=24 B=14 C=5 so C is eliminated. Round 2 votes counts: A=29 E=28 D=28 B=15 so B is eliminated. Round 3 votes counts: A=38 E=33 D=29 so D is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:204 E:198 B:192 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 4 14 B -16 0 6 0 -6 C -16 -6 0 -8 -8 D -4 0 8 0 4 E -14 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 4 14 B -16 0 6 0 -6 C -16 -6 0 -8 -8 D -4 0 8 0 4 E -14 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 4 14 B -16 0 6 0 -6 C -16 -6 0 -8 -8 D -4 0 8 0 4 E -14 6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3833: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) A B E D C (6) A D B E C (5) B E D A C (4) A C D E B (4) A B D E C (4) E B D C A (3) D E C B A (3) D E B C A (3) C D E B A (3) C D E A B (3) C D A E B (3) C A E B D (3) C A D E B (3) B E D C A (3) B D E A C (3) B A E D C (3) A B D C E (3) E D C B A (2) E C B D A (2) D B E A C (2) D A C E B (2) C E D A B (2) C E B A D (2) C E A D B (2) B E A D C (2) A D B C E (2) A C D B E (2) E D B C A (1) E C D B A (1) D C E B A (1) D C E A B (1) D B A E C (1) C A E D B (1) B E C D A (1) B E A C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 -10 -12 B 4 0 -4 -16 -12 C 8 4 0 -8 0 D 10 16 8 0 -4 E 12 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.202815 D: 0.000000 E: 0.797185 Sum of squares = 0.67663825684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.202815 D: 0.202815 E: 1.000000 A B C D E A 0 -4 -8 -10 -12 B 4 0 -4 -16 -12 C 8 4 0 -8 0 D 10 16 8 0 -4 E 12 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555638633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=28 B=17 D=13 E=9 so E is eliminated. Round 2 votes counts: C=36 A=28 B=20 D=16 so D is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:215 E:214 C:202 B:186 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 -10 -12 B 4 0 -4 -16 -12 C 8 4 0 -8 0 D 10 16 8 0 -4 E 12 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555638633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -10 -12 B 4 0 -4 -16 -12 C 8 4 0 -8 0 D 10 16 8 0 -4 E 12 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555638633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -10 -12 B 4 0 -4 -16 -12 C 8 4 0 -8 0 D 10 16 8 0 -4 E 12 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555638633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3834: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) E B C D A (7) E B C A D (5) D A C B E (5) C D E A B (5) B E A C D (5) D C A E B (4) A D B C E (4) A B D E C (4) E C B D A (3) E C B A D (3) D B A E C (3) B E A D C (3) E B D C A (2) D E C B A (2) D C E B A (2) D B E A C (2) D A B C E (2) C E D B A (2) C E A D B (2) C E A B D (2) C A D E B (2) B A E D C (2) D C A B E (1) D B E C A (1) D A C E B (1) D A B E C (1) C E B A D (1) C A E D B (1) C A E B D (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E A C (1) B D A E C (1) B A E C D (1) B A D E C (1) A D B E C (1) A C E D B (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 4 6 -2 B 0 0 0 -6 6 C -4 0 0 -10 -2 D -6 6 10 0 12 E 2 -6 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.592743 B: 0.407257 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.517202500479 Cumulative probabilities = A: 0.592743 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 6 -2 B 0 0 0 -6 6 C -4 0 0 -10 -2 D -6 6 10 0 12 E 2 -6 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500332 B: 0.499668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000220824 Cumulative probabilities = A: 0.500332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=23 E=20 B=17 C=16 so C is eliminated. Round 2 votes counts: D=29 E=27 A=27 B=17 so B is eliminated. Round 3 votes counts: E=38 D=31 A=31 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:211 A:204 B:200 E:193 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 6 -2 B 0 0 0 -6 6 C -4 0 0 -10 -2 D -6 6 10 0 12 E 2 -6 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500332 B: 0.499668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000220824 Cumulative probabilities = A: 0.500332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 -2 B 0 0 0 -6 6 C -4 0 0 -10 -2 D -6 6 10 0 12 E 2 -6 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500332 B: 0.499668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000220824 Cumulative probabilities = A: 0.500332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 -2 B 0 0 0 -6 6 C -4 0 0 -10 -2 D -6 6 10 0 12 E 2 -6 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500332 B: 0.499668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000220824 Cumulative probabilities = A: 0.500332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3835: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) E C B D A (8) C B D A E (8) C E B D A (7) C B D E A (7) A D B C E (7) C B E D A (6) A D E B C (6) E A D B C (5) B D C A E (4) E C D B A (3) E A D C B (3) D A B C E (3) B C D A E (3) A B D C E (3) E D A B C (2) E C B A D (2) A E C D B (2) E D A C B (1) E C A B D (1) D E A B C (1) D B C E A (1) D B A C E (1) C E B A D (1) C B A D E (1) B D C E A (1) B D A C E (1) B C D E A (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -8 -16 0 B 10 0 -2 10 -4 C 8 2 0 2 10 D 16 -10 -2 0 -2 E 0 4 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -16 0 B 10 0 -2 10 -4 C 8 2 0 2 10 D 16 -10 -2 0 -2 E 0 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=29 E=25 B=10 D=6 so D is eliminated. Round 2 votes counts: A=32 C=30 E=26 B=12 so B is eliminated. Round 3 votes counts: C=40 A=34 E=26 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:207 D:201 E:198 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -16 0 B 10 0 -2 10 -4 C 8 2 0 2 10 D 16 -10 -2 0 -2 E 0 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -16 0 B 10 0 -2 10 -4 C 8 2 0 2 10 D 16 -10 -2 0 -2 E 0 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -16 0 B 10 0 -2 10 -4 C 8 2 0 2 10 D 16 -10 -2 0 -2 E 0 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3836: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) C D E A B (6) C D B A E (4) C B A D E (4) C A D E B (4) B C D A E (4) A E B C D (4) E B D A C (3) C A E D B (3) B D E A C (3) A E C D B (3) A C B E D (3) A B E C D (3) A B C E D (3) E D B A C (2) E A D B C (2) E A C D B (2) E A B D C (2) D E C B A (2) D C E A B (2) D B E C A (2) C B D A E (2) B D E C A (2) B A C E D (2) A C E B D (2) E D C A B (1) E D A B C (1) E B A D C (1) E A D C B (1) D E C A B (1) D E B C A (1) D E B A C (1) D C E B A (1) D B E A C (1) D B C E A (1) C D B E A (1) C D A E B (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D A C (1) B E A D C (1) B D C A E (1) B D A E C (1) B A E C D (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 8 8 18 B 2 0 6 10 6 C -8 -6 0 12 -6 D -8 -10 -12 0 -4 E -18 -6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 8 18 B 2 0 6 10 6 C -8 -6 0 12 -6 D -8 -10 -12 0 -4 E -18 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 A=20 E=15 D=12 so D is eliminated. Round 2 votes counts: C=31 B=29 E=20 A=20 so E is eliminated. Round 3 votes counts: B=37 C=35 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:212 C:196 E:193 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 8 18 B 2 0 6 10 6 C -8 -6 0 12 -6 D -8 -10 -12 0 -4 E -18 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 8 18 B 2 0 6 10 6 C -8 -6 0 12 -6 D -8 -10 -12 0 -4 E -18 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 8 18 B 2 0 6 10 6 C -8 -6 0 12 -6 D -8 -10 -12 0 -4 E -18 -6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3837: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) B A D E C (7) C E A D B (6) C E D A B (5) B D A E C (5) C B D E A (4) B D C A E (4) E C A D B (3) D B A E C (3) A E B D C (3) A C E B D (3) A B C E D (3) E A C D B (2) D E C A B (2) D C B E A (2) C E D B A (2) C E A B D (2) C D E B A (2) B D C E A (2) B C D E A (2) B C A E D (2) B A C E D (2) A E D B C (2) A E C D B (2) A E B C D (2) A B E D C (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A B D (1) E A D C B (1) D E A B C (1) D C E B A (1) D B C E A (1) D B A C E (1) C E B D A (1) C E B A D (1) C B E D A (1) C B A E D (1) C A E B D (1) B C D A E (1) B A D C E (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 4 2 12 B 6 0 8 22 6 C -4 -8 0 6 16 D -2 -22 -6 0 -6 E -12 -6 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 2 12 B 6 0 8 22 6 C -4 -8 0 6 16 D -2 -22 -6 0 -6 E -12 -6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=26 A=21 D=11 E=9 so E is eliminated. Round 2 votes counts: B=33 C=31 A=24 D=12 so D is eliminated. Round 3 votes counts: B=38 C=36 A=26 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:206 C:205 E:186 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 2 12 B 6 0 8 22 6 C -4 -8 0 6 16 D -2 -22 -6 0 -6 E -12 -6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 2 12 B 6 0 8 22 6 C -4 -8 0 6 16 D -2 -22 -6 0 -6 E -12 -6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 2 12 B 6 0 8 22 6 C -4 -8 0 6 16 D -2 -22 -6 0 -6 E -12 -6 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3838: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) D A C E B (7) B E A C D (7) A C D B E (7) E B D C A (6) C D A B E (5) E D B C A (4) D E A C B (4) C A D B E (4) B E C D A (4) B A E C D (4) E B A D C (3) C A B D E (3) B E C A D (3) E D B A C (2) E D A B C (2) E B C D A (2) D E A B C (2) D C E B A (2) D A E C B (2) C D B A E (2) B C A E D (2) A D C E B (2) A C B D E (2) E B D A C (1) E B C A D (1) D E C A B (1) D C E A B (1) D C A B E (1) C D E B A (1) B A C E D (1) A D C B E (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 -14 8 B -10 0 -8 -16 -2 C 0 8 0 2 2 D 14 16 -2 0 10 E -8 2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.072855 B: 0.000000 C: 0.927145 D: 0.000000 E: 0.000000 Sum of squares = 0.86490615631 Cumulative probabilities = A: 0.072855 B: 0.072855 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -14 8 B -10 0 -8 -16 -2 C 0 8 0 2 2 D 14 16 -2 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250181052 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=21 B=21 C=15 A=15 so C is eliminated. Round 2 votes counts: D=36 A=22 E=21 B=21 so E is eliminated. Round 3 votes counts: D=44 B=34 A=22 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:206 A:202 E:191 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 0 -14 8 B -10 0 -8 -16 -2 C 0 8 0 2 2 D 14 16 -2 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250181052 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -14 8 B -10 0 -8 -16 -2 C 0 8 0 2 2 D 14 16 -2 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250181052 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -14 8 B -10 0 -8 -16 -2 C 0 8 0 2 2 D 14 16 -2 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.875000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250181052 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3839: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (17) D E B C A (10) E D B A C (5) D E B A C (5) C A B E D (4) C A B D E (4) E D B C A (3) E B D C A (3) D C E B A (3) C A D B E (3) B C E A D (3) B C A E D (3) A C D B E (3) A B E C D (3) C B D E A (2) B E C D A (2) B E A C D (2) A E B C D (2) A C B D E (2) A B C E D (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B D C (1) D E C B A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C B E A (1) D C A E B (1) C D B A E (1) C B A E D (1) C B A D E (1) B E D C A (1) B E C A D (1) B C E D A (1) B A E C D (1) B A C E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 0 12 0 B 6 0 6 18 18 C 0 -6 0 20 8 D -12 -18 -20 0 -20 E 0 -18 -8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 12 0 B 6 0 6 18 18 C 0 -6 0 20 8 D -12 -18 -20 0 -20 E 0 -18 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 C=16 E=15 B=15 so E is eliminated. Round 2 votes counts: D=33 A=31 B=20 C=16 so C is eliminated. Round 3 votes counts: A=42 D=34 B=24 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:224 C:211 A:203 E:197 D:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 12 0 B 6 0 6 18 18 C 0 -6 0 20 8 D -12 -18 -20 0 -20 E 0 -18 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 12 0 B 6 0 6 18 18 C 0 -6 0 20 8 D -12 -18 -20 0 -20 E 0 -18 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 12 0 B 6 0 6 18 18 C 0 -6 0 20 8 D -12 -18 -20 0 -20 E 0 -18 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3840: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A B E D C (9) C D E A B (8) B D E C A (5) A E D B C (5) E D C A B (4) C E D A B (4) C A E D B (4) B A E D C (4) A C E D B (4) A C B E D (4) D E C B A (3) D E B C A (3) C B D E A (3) C B A D E (3) B A D E C (3) B A C D E (3) A B C E D (3) E D B A C (2) B D E A C (2) B C D E A (2) B C A D E (2) A E D C B (2) E D A B C (1) D E B A C (1) C D B E A (1) C B D A E (1) C A B E D (1) B D A E C (1) B C D A E (1) B A C E D (1) Total count = 100 A B C D E A 0 -2 -10 -4 2 B 2 0 -4 -4 -2 C 10 4 0 10 10 D 4 4 -10 0 4 E -2 2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -4 2 B 2 0 -4 -4 -2 C 10 4 0 10 10 D 4 4 -10 0 4 E -2 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=27 B=24 E=7 D=7 so E is eliminated. Round 2 votes counts: C=35 A=27 B=24 D=14 so D is eliminated. Round 3 votes counts: C=42 B=30 A=28 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:201 B:196 A:193 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 -4 2 B 2 0 -4 -4 -2 C 10 4 0 10 10 D 4 4 -10 0 4 E -2 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -4 2 B 2 0 -4 -4 -2 C 10 4 0 10 10 D 4 4 -10 0 4 E -2 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -4 2 B 2 0 -4 -4 -2 C 10 4 0 10 10 D 4 4 -10 0 4 E -2 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3841: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (14) E B D A C (12) C E D A B (11) C A D B E (9) C D A B E (6) B E A D C (6) C D A E B (4) A D B E C (4) A D B C E (4) E D A B C (3) C E B A D (3) E B C D A (2) E B A D C (2) D A E B C (2) D A B E C (2) C E B D A (2) C D E A B (2) E D C A B (1) E D A C B (1) E C D A B (1) E C B D A (1) D A C B E (1) C E A D B (1) C B E A D (1) C B A D E (1) C A B D E (1) B E A C D (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 6 6 -2 0 B -6 0 8 -4 4 C -6 -8 0 -6 -4 D 2 4 6 0 4 E 0 -4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -2 0 B -6 0 8 -4 4 C -6 -8 0 -6 -4 D 2 4 6 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 E=23 B=23 A=8 D=5 so D is eliminated. Round 2 votes counts: C=41 E=23 B=23 A=13 so A is eliminated. Round 3 votes counts: C=42 B=33 E=25 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:208 A:205 B:201 E:198 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -2 0 B -6 0 8 -4 4 C -6 -8 0 -6 -4 D 2 4 6 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -2 0 B -6 0 8 -4 4 C -6 -8 0 -6 -4 D 2 4 6 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -2 0 B -6 0 8 -4 4 C -6 -8 0 -6 -4 D 2 4 6 0 4 E 0 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3842: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (13) C B E D A (13) C B A D E (11) A D E C B (9) B C E D A (8) B E D A C (6) A D E B C (5) E B D A C (4) C A D B E (4) A D C E B (4) A C D E B (4) C B A E D (3) B E C D A (3) E A D B C (2) D A E B C (2) C B D A E (2) E B C A D (1) D E A B C (1) D B E A C (1) C E B A D (1) C B E A D (1) C A B D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 4 -6 -8 B 8 0 -6 8 6 C -4 6 0 4 4 D 6 -8 -4 0 -12 E 8 -6 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.358024691353 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -6 -8 B 8 0 -6 8 6 C -4 6 0 4 4 D 6 -8 -4 0 -12 E 8 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.358024691295 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=23 E=20 B=17 D=4 so D is eliminated. Round 2 votes counts: C=36 A=25 E=21 B=18 so B is eliminated. Round 3 votes counts: C=44 E=31 A=25 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:208 C:205 E:205 A:191 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 4 -6 -8 B 8 0 -6 8 6 C -4 6 0 4 4 D 6 -8 -4 0 -12 E 8 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.358024691295 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -6 -8 B 8 0 -6 8 6 C -4 6 0 4 4 D 6 -8 -4 0 -12 E 8 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.358024691295 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -6 -8 B 8 0 -6 8 6 C -4 6 0 4 4 D 6 -8 -4 0 -12 E 8 -6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.358024691295 Cumulative probabilities = A: 0.333333 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3843: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) B D C A E (11) D C B A E (9) B D C E A (9) B E A D C (8) A E C D B (8) C D A E B (5) E A B C D (4) C D B A E (4) E A B D C (3) D C B E A (3) A B E C D (3) E A D C B (2) D B C A E (2) C D A B E (2) B E D A C (2) B C D A E (2) A E B C D (2) A C E D B (2) A C D E B (2) E B A D C (1) D E C A B (1) C A D E B (1) B E D C A (1) B D E C A (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 -2 -4 8 B 8 0 0 -4 16 C 2 0 0 -6 6 D 4 4 6 0 6 E -8 -16 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -4 8 B 8 0 0 -4 16 C 2 0 0 -6 6 D 4 4 6 0 6 E -8 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=21 A=17 D=15 C=12 so C is eliminated. Round 2 votes counts: B=35 D=26 E=21 A=18 so A is eliminated. Round 3 votes counts: B=38 E=33 D=29 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:210 C:201 A:197 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 8 B 8 0 0 -4 16 C 2 0 0 -6 6 D 4 4 6 0 6 E -8 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 8 B 8 0 0 -4 16 C 2 0 0 -6 6 D 4 4 6 0 6 E -8 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 8 B 8 0 0 -4 16 C 2 0 0 -6 6 D 4 4 6 0 6 E -8 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3844: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (11) C B A D E (8) C D A E B (7) B E D A C (7) B C E D A (7) B A D E C (6) E D A C B (5) B E A D C (5) A D E C B (5) E D A B C (4) E B D A C (4) C B E D A (4) C E D A B (3) B A D C E (3) A D E B C (3) E C D A B (2) D A E C B (2) C E D B A (2) B C E A D (2) B A C D E (2) D E A C B (1) C E B D A (1) C D E A B (1) C B D E A (1) C A D B E (1) B E C D A (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -30 -2 -4 0 B 30 0 14 26 20 C 2 -14 0 6 8 D 4 -26 -6 0 4 E 0 -20 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -2 -4 0 B 30 0 14 26 20 C 2 -14 0 6 8 D 4 -26 -6 0 4 E 0 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=45 C=28 E=15 A=9 D=3 so D is eliminated. Round 2 votes counts: B=45 C=28 E=16 A=11 so A is eliminated. Round 3 votes counts: B=46 C=28 E=26 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:245 C:201 D:188 E:184 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -2 -4 0 B 30 0 14 26 20 C 2 -14 0 6 8 D 4 -26 -6 0 4 E 0 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -2 -4 0 B 30 0 14 26 20 C 2 -14 0 6 8 D 4 -26 -6 0 4 E 0 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -2 -4 0 B 30 0 14 26 20 C 2 -14 0 6 8 D 4 -26 -6 0 4 E 0 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3845: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (13) D C B A E (9) D B C E A (7) D E A B C (6) D C B E A (6) B C D E A (5) A E D B C (5) C B A E D (4) B D C E A (4) D A E C B (3) C D B A E (3) C B D E A (3) A E C D B (3) A E B C D (3) E A D B C (2) E A B C D (2) C D A B E (2) C B D A E (2) C A E B D (2) C A B E D (2) A E D C B (2) E D A B C (1) E A B D C (1) D E B A C (1) D C A E B (1) D B E A C (1) D A E B C (1) C B E A D (1) C B A D E (1) B E A C D (1) B C E A D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -6 -10 16 B -2 0 -16 -8 4 C 6 16 0 -4 6 D 10 8 4 0 12 E -16 -4 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -10 16 B -2 0 -16 -8 4 C 6 16 0 -4 6 D 10 8 4 0 12 E -16 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=28 C=20 B=11 E=6 so E is eliminated. Round 2 votes counts: D=36 A=33 C=20 B=11 so B is eliminated. Round 3 votes counts: D=40 A=34 C=26 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:212 A:201 B:189 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -10 16 B -2 0 -16 -8 4 C 6 16 0 -4 6 D 10 8 4 0 12 E -16 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -10 16 B -2 0 -16 -8 4 C 6 16 0 -4 6 D 10 8 4 0 12 E -16 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -10 16 B -2 0 -16 -8 4 C 6 16 0 -4 6 D 10 8 4 0 12 E -16 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3846: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) E D A C B (9) A B E C D (9) D C E A B (7) C D B A E (7) B A C D E (6) D C E B A (5) B A E C D (5) A E B C D (4) D C B E A (3) C D B E A (3) E B A C D (2) E A D B C (2) D E C B A (2) D E C A B (2) C D A B E (2) B E A C D (2) B C A D E (2) B A C E D (2) E D C B A (1) E D C A B (1) E D B C A (1) E B D C A (1) E B C D A (1) E A B C D (1) B E C D A (1) B C D E A (1) B C D A E (1) B C A E D (1) A E B D C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 16 4 -16 B -6 0 16 10 -6 C -16 -16 0 2 -18 D -4 -10 -2 0 -18 E 16 6 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 16 4 -16 B -6 0 16 10 -6 C -16 -16 0 2 -18 D -4 -10 -2 0 -18 E 16 6 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=21 D=19 A=16 C=12 so C is eliminated. Round 2 votes counts: E=32 D=31 B=21 A=16 so A is eliminated. Round 3 votes counts: E=37 B=32 D=31 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:229 B:207 A:205 D:183 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 16 4 -16 B -6 0 16 10 -6 C -16 -16 0 2 -18 D -4 -10 -2 0 -18 E 16 6 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 4 -16 B -6 0 16 10 -6 C -16 -16 0 2 -18 D -4 -10 -2 0 -18 E 16 6 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 4 -16 B -6 0 16 10 -6 C -16 -16 0 2 -18 D -4 -10 -2 0 -18 E 16 6 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3847: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) D B E A C (8) C E A B D (6) C A E B D (6) A E B C D (6) B E A D C (5) B D E A C (5) E A B C D (4) C E A D B (4) B D A E C (4) D B A E C (3) C D E A B (3) C D A B E (3) A B E D C (3) E C A B D (2) E B A D C (2) E A C B D (2) D C B A E (2) D B C E A (2) C D B A E (2) C A E D B (2) C A D E B (2) B A D E C (2) A C B E D (2) E B D A C (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C A E (1) A E B D C (1) A C E B D (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 6 -14 B 0 0 4 10 12 C -4 -4 0 -4 -2 D -6 -10 4 0 2 E 14 -12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.328331 B: 0.671669 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.558940382438 Cumulative probabilities = A: 0.328331 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 6 -14 B 0 0 4 10 12 C -4 -4 0 -4 -2 D -6 -10 4 0 2 E 14 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461536 B: 0.538464 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502958966216 Cumulative probabilities = A: 0.461536 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 B=16 A=16 E=11 so E is eliminated. Round 2 votes counts: C=30 D=29 A=22 B=19 so B is eliminated. Round 3 votes counts: D=39 A=31 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:213 E:201 A:198 D:195 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 6 -14 B 0 0 4 10 12 C -4 -4 0 -4 -2 D -6 -10 4 0 2 E 14 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461536 B: 0.538464 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502958966216 Cumulative probabilities = A: 0.461536 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 -14 B 0 0 4 10 12 C -4 -4 0 -4 -2 D -6 -10 4 0 2 E 14 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461536 B: 0.538464 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502958966216 Cumulative probabilities = A: 0.461536 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 -14 B 0 0 4 10 12 C -4 -4 0 -4 -2 D -6 -10 4 0 2 E 14 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461536 B: 0.538464 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502958966216 Cumulative probabilities = A: 0.461536 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3848: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A B E D (6) D A C B E (5) D A B C E (5) E D B C A (4) E D B A C (4) E C B A D (4) E B D A C (4) D E C A B (4) B A C E D (4) E B D C A (3) E B C A D (3) D C A E B (3) D B A E C (3) B E D A C (3) A C B D E (3) E D C A B (2) E C A B D (2) D B E A C (2) A B C D E (2) E D C B A (1) E C D A B (1) E B C D A (1) D E A B C (1) D C E A B (1) D C A B E (1) D B A C E (1) D A E C B (1) C E A B D (1) C D E A B (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E C A D (1) B D E A C (1) B D A E C (1) B C A E D (1) B A E C D (1) B A D E C (1) B A D C E (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 8 -26 -6 B 8 0 14 -6 -8 C -8 -14 0 -22 -14 D 26 6 22 0 2 E 6 8 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -26 -6 B 8 0 14 -6 -8 C -8 -14 0 -22 -14 D 26 6 22 0 2 E 6 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=29 B=15 C=14 A=6 so A is eliminated. Round 2 votes counts: D=36 E=29 C=18 B=17 so B is eliminated. Round 3 votes counts: D=40 E=34 C=26 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:213 B:204 A:184 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -26 -6 B 8 0 14 -6 -8 C -8 -14 0 -22 -14 D 26 6 22 0 2 E 6 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -26 -6 B 8 0 14 -6 -8 C -8 -14 0 -22 -14 D 26 6 22 0 2 E 6 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -26 -6 B 8 0 14 -6 -8 C -8 -14 0 -22 -14 D 26 6 22 0 2 E 6 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3849: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) C B A E D (10) E A D B C (6) B C E A D (6) A E D C B (6) A E C D B (6) D B E A C (4) C D A E B (4) C B D E A (4) C A E D B (4) C A E B D (4) B D E A C (4) B C D E A (4) C D B A E (3) C B D A E (3) B D C E A (3) C B A D E (2) A E D B C (2) E D A B C (1) E B A D C (1) E A B D C (1) D E A C B (1) D C B E A (1) D C A E B (1) D B E C A (1) D A E C B (1) C A D E B (1) B E D A C (1) B D E C A (1) B C E D A (1) A E C B D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -6 4 0 B -2 0 -4 -4 -2 C 6 4 0 8 2 D -4 4 -8 0 -4 E 0 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 4 0 B -2 0 -4 -4 -2 C 6 4 0 8 2 D -4 4 -8 0 -4 E 0 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=20 D=19 A=17 E=9 so E is eliminated. Round 2 votes counts: C=35 A=24 B=21 D=20 so D is eliminated. Round 3 votes counts: C=37 A=37 B=26 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 E:202 A:200 B:194 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 4 0 B -2 0 -4 -4 -2 C 6 4 0 8 2 D -4 4 -8 0 -4 E 0 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 4 0 B -2 0 -4 -4 -2 C 6 4 0 8 2 D -4 4 -8 0 -4 E 0 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 4 0 B -2 0 -4 -4 -2 C 6 4 0 8 2 D -4 4 -8 0 -4 E 0 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3850: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) D B A E C (8) D A B C E (6) B D E C A (6) D B E A C (5) C A E B D (5) B E D C A (5) A D C B E (5) A C D E B (5) D A B E C (4) C E A B D (4) C A E D B (4) E C B A D (3) E C A D B (2) E C A B D (2) D A C B E (2) B D E A C (2) A D C E B (2) A C E D B (2) A C D B E (2) A C B D E (2) E D A C B (1) E C B D A (1) E B D C A (1) E B C A D (1) D E B A C (1) D B A C E (1) D A E B C (1) C E B A D (1) C E A D B (1) C B A E D (1) B E C D A (1) B E C A D (1) B D A C E (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 -10 6 B -4 0 8 -6 6 C -4 -8 0 -4 -10 D 10 6 4 0 8 E -6 -6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -10 6 B -4 0 8 -6 6 C -4 -8 0 -4 -10 D 10 6 4 0 8 E -6 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=20 A=20 C=16 B=16 so C is eliminated. Round 2 votes counts: A=29 D=28 E=26 B=17 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:202 B:202 E:195 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -10 6 B -4 0 8 -6 6 C -4 -8 0 -4 -10 D 10 6 4 0 8 E -6 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -10 6 B -4 0 8 -6 6 C -4 -8 0 -4 -10 D 10 6 4 0 8 E -6 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -10 6 B -4 0 8 -6 6 C -4 -8 0 -4 -10 D 10 6 4 0 8 E -6 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3851: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) B E D A C (7) C B E A D (6) E D B A C (5) B E C D A (5) B C E A D (5) D E A B C (4) D A E B C (4) D A C E B (4) C B A E D (4) C A D E B (4) C A D B E (4) C A B D E (4) E B D C A (3) C B E D A (3) B E A D C (3) A D C E B (3) A C D B E (3) D A E C B (2) C E B D A (2) B A E D C (2) A D B E C (2) E D A B C (1) D E C A B (1) C E D B A (1) C D E A B (1) B E D C A (1) B C E D A (1) B A E C D (1) A D E B C (1) A D C B E (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 10 -8 -16 B 16 0 10 16 8 C -10 -10 0 -6 -2 D 8 -16 6 0 -18 E 16 -8 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 10 -8 -16 B 16 0 10 16 8 C -10 -10 0 -6 -2 D 8 -16 6 0 -18 E 16 -8 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=25 E=18 D=15 A=13 so A is eliminated. Round 2 votes counts: C=34 B=26 D=22 E=18 so E is eliminated. Round 3 votes counts: B=38 C=34 D=28 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:214 D:190 C:186 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 10 -8 -16 B 16 0 10 16 8 C -10 -10 0 -6 -2 D 8 -16 6 0 -18 E 16 -8 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 -8 -16 B 16 0 10 16 8 C -10 -10 0 -6 -2 D 8 -16 6 0 -18 E 16 -8 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 -8 -16 B 16 0 10 16 8 C -10 -10 0 -6 -2 D 8 -16 6 0 -18 E 16 -8 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3852: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) A C B D E (9) B D C E A (7) E D B C A (5) E A D C B (5) A C E D B (5) E D C B A (4) D E B C A (4) C A B D E (4) B D E C A (4) A C E B D (4) E D C A B (3) E D B A C (3) B C D A E (3) A E C D B (3) A B C D E (3) E D A B C (2) E A D B C (2) C B D A E (2) C B A D E (2) B C A D E (2) A E D B C (2) D B E C A (1) D B C E A (1) C D E B A (1) C A E D B (1) C A E B D (1) B D C A E (1) B D A C E (1) B A D E C (1) B A C D E (1) A E D C B (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 14 8 16 16 B -14 0 -12 14 6 C -8 12 0 4 16 D -16 -14 -4 0 -4 E -16 -6 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 16 16 B -14 0 -12 14 6 C -8 12 0 4 16 D -16 -14 -4 0 -4 E -16 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=24 B=20 C=11 D=6 so D is eliminated. Round 2 votes counts: A=39 E=28 B=22 C=11 so C is eliminated. Round 3 votes counts: A=45 E=29 B=26 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 C:212 B:197 E:183 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 16 16 B -14 0 -12 14 6 C -8 12 0 4 16 D -16 -14 -4 0 -4 E -16 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 16 16 B -14 0 -12 14 6 C -8 12 0 4 16 D -16 -14 -4 0 -4 E -16 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 16 16 B -14 0 -12 14 6 C -8 12 0 4 16 D -16 -14 -4 0 -4 E -16 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3853: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) E B A C D (4) D C A B E (4) D B A C E (4) C A D E B (4) B D E C A (4) A C E B D (4) E B D C A (3) D A C B E (3) B E D A C (3) B D E A C (3) B A E D C (3) A C D B E (3) A B E C D (3) E B C D A (2) E B C A D (2) D C B A E (2) D C A E B (2) D B C E A (2) D B C A E (2) C A E D B (2) B E A C D (2) B A E C D (2) B A D E C (2) A D C B E (2) A C E D B (2) A C D E B (2) A C B E D (2) E C D B A (1) E C B D A (1) E C B A D (1) E C A B D (1) E A C B D (1) D C E B A (1) D C E A B (1) D B E C A (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) B E A D C (1) B D A E C (1) B A D C E (1) A E C B D (1) A E B C D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 6 0 12 B 10 0 10 16 20 C -6 -10 0 -4 0 D 0 -16 4 0 -4 E -12 -20 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 0 12 B 10 0 10 16 20 C -6 -10 0 -4 0 D 0 -16 4 0 -4 E -12 -20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 D=22 E=16 C=10 so C is eliminated. Round 2 votes counts: B=29 A=29 D=24 E=18 so E is eliminated. Round 3 votes counts: B=42 A=32 D=26 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:204 D:192 C:190 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 0 12 B 10 0 10 16 20 C -6 -10 0 -4 0 D 0 -16 4 0 -4 E -12 -20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 0 12 B 10 0 10 16 20 C -6 -10 0 -4 0 D 0 -16 4 0 -4 E -12 -20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 0 12 B 10 0 10 16 20 C -6 -10 0 -4 0 D 0 -16 4 0 -4 E -12 -20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3854: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A C E B D (8) D B E C A (6) C E B D A (6) A B D E C (6) C E B A D (5) A B E D C (5) D B E A C (4) D B A E C (4) A D B E C (4) E B C D A (3) C A D E B (3) E C B D A (2) E B D A C (2) C E D B A (2) C E A B D (2) C D E B A (2) C D A B E (2) C A E B D (2) B E A D C (2) A D C B E (2) A C D B E (2) A C B E D (2) A B E C D (2) E C B A D (1) E B A D C (1) D E B C A (1) D B C E A (1) D A B E C (1) C E A D B (1) C A D B E (1) B E D A C (1) B D E A C (1) A E B D C (1) A E B C D (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 4 8 -2 B 4 0 12 26 -2 C -4 -12 0 -4 -12 D -8 -26 4 0 -14 E 2 2 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 8 -2 B 4 0 12 26 -2 C -4 -12 0 -4 -12 D -8 -26 4 0 -14 E 2 2 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=26 E=17 D=17 B=4 so B is eliminated. Round 2 votes counts: A=36 C=26 E=20 D=18 so D is eliminated. Round 3 votes counts: A=41 E=32 C=27 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:215 A:203 C:184 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 8 -2 B 4 0 12 26 -2 C -4 -12 0 -4 -12 D -8 -26 4 0 -14 E 2 2 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 8 -2 B 4 0 12 26 -2 C -4 -12 0 -4 -12 D -8 -26 4 0 -14 E 2 2 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 8 -2 B 4 0 12 26 -2 C -4 -12 0 -4 -12 D -8 -26 4 0 -14 E 2 2 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3855: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) A C B D E (12) C B A E D (11) B C A E D (10) E D B C A (5) E D B A C (5) E D A B C (5) B C E D A (5) A C D E B (5) D E A C B (4) C A B D E (4) B E D C A (4) E D A C B (3) D E A B C (3) C B A D E (3) B C A D E (3) D E B A C (2) C A B E D (1) B E C D A (1) A D C E B (1) Total count = 100 A B C D E A 0 2 6 26 26 B -2 0 -14 8 8 C -6 14 0 10 10 D -26 -8 -10 0 0 E -26 -8 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 26 26 B -2 0 -14 8 8 C -6 14 0 10 10 D -26 -8 -10 0 0 E -26 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=23 C=19 E=18 D=9 so D is eliminated. Round 2 votes counts: A=31 E=27 B=23 C=19 so C is eliminated. Round 3 votes counts: B=37 A=36 E=27 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:214 B:200 D:178 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 26 26 B -2 0 -14 8 8 C -6 14 0 10 10 D -26 -8 -10 0 0 E -26 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 26 26 B -2 0 -14 8 8 C -6 14 0 10 10 D -26 -8 -10 0 0 E -26 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 26 26 B -2 0 -14 8 8 C -6 14 0 10 10 D -26 -8 -10 0 0 E -26 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3856: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) C A D E B (7) A C D E B (7) E C B A D (6) E B C D A (6) D A C B E (5) D A B C E (5) B E D A C (5) B D E A C (5) D C A B E (4) C A E D B (4) B E D C A (4) A D C E B (4) C E A B D (3) C D A E B (3) B E A D C (3) A D C B E (3) D A B E C (2) C E B A D (2) B E C D A (2) A B D E C (2) D B A E C (1) C E B D A (1) C D A B E (1) C A E B D (1) B D E C A (1) B D A E C (1) B A D E C (1) A E B C D (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 8 -6 8 8 B -8 0 -4 4 -8 C 6 4 0 4 2 D -8 -4 -4 0 8 E -8 8 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 8 8 B -8 0 -4 4 -8 C 6 4 0 4 2 D -8 -4 -4 0 8 E -8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=22 B=22 E=20 A=19 D=17 so D is eliminated. Round 2 votes counts: A=31 C=26 B=23 E=20 so E is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:209 C:208 D:196 E:195 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 8 8 B -8 0 -4 4 -8 C 6 4 0 4 2 D -8 -4 -4 0 8 E -8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 8 8 B -8 0 -4 4 -8 C 6 4 0 4 2 D -8 -4 -4 0 8 E -8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 8 8 B -8 0 -4 4 -8 C 6 4 0 4 2 D -8 -4 -4 0 8 E -8 8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3857: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (13) D C E A B (12) A E B D C (7) E A B C D (6) C D E A B (6) E A D B C (5) D C B A E (5) B A E D C (5) C D B A E (4) C D B E A (3) C B D A E (3) B C D A E (3) D E C A B (2) B E A C D (2) B D A C E (2) A E B C D (2) A B E D C (2) E C D A B (1) E C A D B (1) E A D C B (1) D C E B A (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A E B C (1) C E D A B (1) B D C A E (1) B C A E D (1) B A D E C (1) B A C E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 6 -2 18 B -4 0 14 2 4 C -6 -14 0 -4 -6 D 2 -2 4 0 2 E -18 -4 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -2 18 B -4 0 14 2 4 C -6 -14 0 -4 -6 D 2 -2 4 0 2 E -18 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999996 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 C=17 E=14 A=13 so A is eliminated. Round 2 votes counts: B=33 D=27 E=23 C=17 so C is eliminated. Round 3 votes counts: D=40 B=36 E=24 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:208 D:203 E:191 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 -2 18 B -4 0 14 2 4 C -6 -14 0 -4 -6 D 2 -2 4 0 2 E -18 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999996 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -2 18 B -4 0 14 2 4 C -6 -14 0 -4 -6 D 2 -2 4 0 2 E -18 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999996 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -2 18 B -4 0 14 2 4 C -6 -14 0 -4 -6 D 2 -2 4 0 2 E -18 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999996 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3858: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) D A E C B (7) C D E A B (7) C B D A E (7) D A E B C (5) C D B A E (4) C B E A D (4) C B D E A (4) B C A D E (4) E B A D C (3) E A D C B (3) C D A E B (3) B D A C E (3) B C D A E (3) E B C A D (2) E A B D C (2) D A C E B (2) C D A B E (2) C B E D A (2) B E A D C (2) B C A E D (2) B A E D C (2) A D E B C (2) E D C A B (1) E D A C B (1) E C A B D (1) E B A C D (1) E A C D B (1) D C A E B (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B A D (1) B E C A D (1) B D C A E (1) B C E A D (1) B A D E C (1) B A D C E (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -4 -12 8 B -2 0 -4 -2 -6 C 4 4 0 2 8 D 12 2 -2 0 20 E -8 6 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -12 8 B -2 0 -4 -2 -6 C 4 4 0 2 8 D 12 2 -2 0 20 E -8 6 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=23 B=21 D=17 A=4 so A is eliminated. Round 2 votes counts: C=35 E=24 B=22 D=19 so D is eliminated. Round 3 votes counts: E=38 C=38 B=24 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:216 C:209 A:197 B:193 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -12 8 B -2 0 -4 -2 -6 C 4 4 0 2 8 D 12 2 -2 0 20 E -8 6 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -12 8 B -2 0 -4 -2 -6 C 4 4 0 2 8 D 12 2 -2 0 20 E -8 6 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -12 8 B -2 0 -4 -2 -6 C 4 4 0 2 8 D 12 2 -2 0 20 E -8 6 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3859: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (15) E C B A D (9) B D A C E (8) E C A B D (7) D B A C E (6) A D E C B (6) E C A D B (4) E A C D B (4) D A B E C (4) C E B A D (4) C E B D A (3) C B E D A (3) B C E D A (3) E C D A B (2) E C B D A (2) D B C E A (2) D A E C B (2) B C E A D (2) B A D C E (2) A E D C B (2) A D B E C (2) D E C A B (1) D B C A E (1) D A E B C (1) C E D B A (1) B C D E A (1) B C A E D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 -10 4 B -4 0 0 -8 4 C -8 0 0 -8 8 D 10 8 8 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -10 4 B -4 0 0 -8 4 C -8 0 0 -8 8 D 10 8 8 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=28 B=17 A=12 C=11 so C is eliminated. Round 2 votes counts: E=36 D=32 B=20 A=12 so A is eliminated. Round 3 votes counts: D=41 E=38 B=21 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:203 B:196 C:196 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -10 4 B -4 0 0 -8 4 C -8 0 0 -8 8 D 10 8 8 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -10 4 B -4 0 0 -8 4 C -8 0 0 -8 8 D 10 8 8 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -10 4 B -4 0 0 -8 4 C -8 0 0 -8 8 D 10 8 8 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3860: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) C D B E A (7) A B E D C (7) E D C A B (5) D E C B A (5) C D E B A (5) E D A B C (4) E A D B C (4) D C E B A (4) C D E A B (4) A B E C D (4) B A D E C (3) B A C E D (3) A E B C D (3) E D A C B (2) E C A D B (2) D C E A B (2) D C B E A (2) C E D A B (2) C E A D B (2) C B A D E (2) C A B E D (2) B A E D C (2) A E B D C (2) A B C E D (2) E C D A B (1) E A C B D (1) D E C A B (1) D E B C A (1) D E B A C (1) D B E C A (1) D B E A C (1) D B C E A (1) C B D A E (1) B D C A E (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 0 -4 -2 -16 B 0 0 0 -14 -2 C 4 0 0 0 0 D 2 14 0 0 4 E 16 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.669737 D: 0.330263 E: 0.000000 Sum of squares = 0.557621187805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.669737 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -2 -16 B 0 0 0 -14 -2 C 4 0 0 0 0 D 2 14 0 0 4 E 16 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=19 D=19 B=19 A=18 so A is eliminated. Round 2 votes counts: B=32 C=25 E=24 D=19 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:210 E:207 C:202 B:192 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -2 -16 B 0 0 0 -14 -2 C 4 0 0 0 0 D 2 14 0 0 4 E 16 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 -16 B 0 0 0 -14 -2 C 4 0 0 0 0 D 2 14 0 0 4 E 16 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 -16 B 0 0 0 -14 -2 C 4 0 0 0 0 D 2 14 0 0 4 E 16 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3861: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) A D E C B (8) E D A B C (5) D A E B C (5) C B A E D (4) A D C E B (4) E A D C B (3) D A B C E (3) C B E A D (3) C A B D E (3) B C E D A (3) B C E A D (3) E D B C A (2) E D B A C (2) E C A B D (2) E B C A D (2) D E B A C (2) D B E C A (2) D A E C B (2) C B A D E (2) C A E B D (2) B D C E A (2) B C D E A (2) B C A D E (2) A D C B E (2) A C B D E (2) E D A C B (1) E C D A B (1) E C B A D (1) E B C D A (1) D E A C B (1) D B A C E (1) D A C E B (1) D A C B E (1) D A B E C (1) C E B A D (1) C A B E D (1) B E C D A (1) B D E C A (1) B D A C E (1) B C D A E (1) A E D C B (1) A E C D B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 22 16 0 2 B -22 0 0 -20 -16 C -16 0 0 -18 -6 D 0 20 18 0 18 E -2 16 6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.562072 B: 0.000000 C: 0.000000 D: 0.437928 E: 0.000000 Sum of squares = 0.507705845996 Cumulative probabilities = A: 0.562072 B: 0.562072 C: 0.562072 D: 1.000000 E: 1.000000 A B C D E A 0 22 16 0 2 B -22 0 0 -20 -16 C -16 0 0 -18 -6 D 0 20 18 0 18 E -2 16 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=21 E=20 C=16 B=16 so C is eliminated. Round 2 votes counts: D=27 A=27 B=25 E=21 so E is eliminated. Round 3 votes counts: D=38 A=32 B=30 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:228 A:220 E:201 C:180 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 22 16 0 2 B -22 0 0 -20 -16 C -16 0 0 -18 -6 D 0 20 18 0 18 E -2 16 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 0 2 B -22 0 0 -20 -16 C -16 0 0 -18 -6 D 0 20 18 0 18 E -2 16 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 0 2 B -22 0 0 -20 -16 C -16 0 0 -18 -6 D 0 20 18 0 18 E -2 16 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3862: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) E C A D B (6) C D B A E (6) C D A B E (6) A D B C E (6) A E B D C (5) E A B C D (4) B A D C E (4) E B A C D (3) E A C B D (3) D A C B E (3) B D C A E (3) B A D E C (3) A C D E B (3) A B D C E (3) E C D A B (2) E B C D A (2) C E D A B (2) C E B D A (2) B E D A C (2) B D C E A (2) A C E D B (2) E C B D A (1) E C B A D (1) E A C D B (1) E A B D C (1) D B C E A (1) D B C A E (1) C E D B A (1) C D E B A (1) C D E A B (1) C D B E A (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A D C (1) B D A C E (1) B A E D C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 6 14 20 4 B -6 0 6 4 -6 C -14 -6 0 4 8 D -20 -4 -4 0 -2 E -4 6 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 20 4 B -6 0 6 4 -6 C -14 -6 0 4 8 D -20 -4 -4 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997491 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=23 A=21 B=18 D=5 so D is eliminated. Round 2 votes counts: E=33 A=24 C=23 B=20 so B is eliminated. Round 3 votes counts: E=37 A=33 C=30 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:199 E:198 C:196 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 20 4 B -6 0 6 4 -6 C -14 -6 0 4 8 D -20 -4 -4 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997491 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 20 4 B -6 0 6 4 -6 C -14 -6 0 4 8 D -20 -4 -4 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997491 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 20 4 B -6 0 6 4 -6 C -14 -6 0 4 8 D -20 -4 -4 0 -2 E -4 6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997491 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3863: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) D C B A E (6) E C B A D (5) B E A C D (5) A D B E C (5) A D E B C (4) E A D C B (3) E A B C D (3) D B C A E (3) D A C B E (3) C B E D A (3) A E D B C (3) A E B D C (3) E B A C D (2) E A D B C (2) E A B D C (2) D A E C B (2) C E D B A (2) C E D A B (2) C D B E A (2) C D B A E (2) C B D A E (2) B E C A D (2) A D B C E (2) E D C A B (1) E C D A B (1) E C A D B (1) E B C A D (1) E A C D B (1) E A C B D (1) D C E A B (1) D C A E B (1) D C A B E (1) D B A C E (1) D A E B C (1) D A C E B (1) C E B D A (1) C D E B A (1) C D E A B (1) C B E A D (1) B C D E A (1) B C D A E (1) B C A D E (1) B A E D C (1) B A E C D (1) B A D E C (1) B A C D E (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 8 14 6 10 B -8 0 8 -22 6 C -14 -8 0 -12 -6 D -6 22 12 0 4 E -10 -6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 6 10 B -8 0 8 -22 6 C -14 -8 0 -12 -6 D -6 22 12 0 4 E -10 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=23 A=19 C=17 B=14 so B is eliminated. Round 2 votes counts: E=30 D=27 A=23 C=20 so C is eliminated. Round 3 votes counts: E=39 D=37 A=24 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:219 D:216 E:193 B:192 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 6 10 B -8 0 8 -22 6 C -14 -8 0 -12 -6 D -6 22 12 0 4 E -10 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 6 10 B -8 0 8 -22 6 C -14 -8 0 -12 -6 D -6 22 12 0 4 E -10 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 6 10 B -8 0 8 -22 6 C -14 -8 0 -12 -6 D -6 22 12 0 4 E -10 -6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3864: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (11) A D E C B (8) C B A E D (7) C B A D E (6) D E A B C (5) D E B C A (4) C B E D A (4) C B D E A (4) B C E D A (4) E D B A C (3) E D A B C (3) D E B A C (3) B C E A D (3) A C E D B (3) A C E B D (3) A C D B E (3) A C B E D (3) E A D B C (2) B E C D A (2) B C D E A (2) A E C B D (2) A D E B C (2) E B D C A (1) E B C A D (1) E A B C D (1) D C A B E (1) D B E C A (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E A D (1) C A B E D (1) B E A C D (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 16 20 10 B -6 0 2 -6 -14 C -16 -2 0 6 -6 D -20 6 -6 0 -14 E -10 14 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 20 10 B -6 0 2 -6 -14 C -16 -2 0 6 -6 D -20 6 -6 0 -14 E -10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=23 D=17 B=12 E=11 so E is eliminated. Round 2 votes counts: A=40 D=23 C=23 B=14 so B is eliminated. Round 3 votes counts: A=41 C=35 D=24 so D is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:212 C:191 B:188 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 20 10 B -6 0 2 -6 -14 C -16 -2 0 6 -6 D -20 6 -6 0 -14 E -10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 20 10 B -6 0 2 -6 -14 C -16 -2 0 6 -6 D -20 6 -6 0 -14 E -10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 20 10 B -6 0 2 -6 -14 C -16 -2 0 6 -6 D -20 6 -6 0 -14 E -10 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3865: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (5) D A E C B (4) D A C E B (4) C D B E A (4) A D E B C (4) E C D A B (3) E B C A D (3) E B A C D (3) D C E A B (3) C E D B A (3) C E B D A (3) A E D C B (3) A E D B C (3) A D B C E (3) E C B A D (2) E A D C B (2) E A B C D (2) D E C A B (2) D C B A E (2) D C A E B (2) D A C B E (2) D A B C E (2) C B D E A (2) B E C A D (2) B C E D A (2) B A E C D (2) B A D C E (2) B A C D E (2) A E B D C (2) A B D E C (2) E D C A B (1) E C B D A (1) E C A B D (1) E A D B C (1) D C A B E (1) D B C A E (1) D B A C E (1) C D B A E (1) C B E D A (1) C B D A E (1) B E A C D (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E D C (1) B A C E D (1) A D E C B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 4 4 B 0 0 2 -8 -6 C 0 -2 0 -2 4 D -4 8 2 0 0 E -4 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.756079 B: 0.243921 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.631152695911 Cumulative probabilities = A: 0.756079 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 4 B 0 0 2 -8 -6 C 0 -2 0 -2 4 D -4 8 2 0 0 E -4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555593 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 A=20 E=19 C=15 so C is eliminated. Round 2 votes counts: D=29 B=26 E=25 A=20 so A is eliminated. Round 3 votes counts: D=38 E=33 B=29 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:204 D:203 C:200 E:199 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 4 B 0 0 2 -8 -6 C 0 -2 0 -2 4 D -4 8 2 0 0 E -4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555593 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 4 B 0 0 2 -8 -6 C 0 -2 0 -2 4 D -4 8 2 0 0 E -4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555593 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 4 B 0 0 2 -8 -6 C 0 -2 0 -2 4 D -4 8 2 0 0 E -4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555555593 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3866: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) B D A C E (6) C E D A B (5) A E C B D (5) E A C D B (4) D B A E C (4) B D C A E (4) A B E C D (4) E C A D B (3) D C E B A (3) C D B E A (3) B D A E C (3) B A C E D (3) A C E B D (3) A B E D C (3) D E C A B (2) D E B C A (2) D E A B C (2) D C E A B (2) D C B E A (2) D B E C A (2) C E A D B (2) C A E B D (2) B C A D E (2) B A E D C (2) A E C D B (2) A E B D C (2) E D C A B (1) E A D C B (1) D E B A C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E A B D (1) C D E B A (1) C B E A D (1) B A E C D (1) B A D E C (1) A E D C B (1) A E D B C (1) A E B C D (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 12 -4 10 B -4 0 10 -6 4 C -12 -10 0 -10 -4 D 4 6 10 0 0 E -10 -4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.844874 E: 0.155126 Sum of squares = 0.737876521584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.844874 E: 1.000000 A B C D E A 0 4 12 -4 10 B -4 0 10 -6 4 C -12 -10 0 -10 -4 D 4 6 10 0 0 E -10 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836735371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=25 B=22 C=15 E=9 so E is eliminated. Round 2 votes counts: D=30 A=30 B=22 C=18 so C is eliminated. Round 3 votes counts: D=39 A=38 B=23 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:211 D:210 B:202 E:195 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 12 -4 10 B -4 0 10 -6 4 C -12 -10 0 -10 -4 D 4 6 10 0 0 E -10 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836735371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 -4 10 B -4 0 10 -6 4 C -12 -10 0 -10 -4 D 4 6 10 0 0 E -10 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836735371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 -4 10 B -4 0 10 -6 4 C -12 -10 0 -10 -4 D 4 6 10 0 0 E -10 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.285714 Sum of squares = 0.591836735371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3867: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (7) D C E B A (5) C B D E A (4) B C E D A (4) B A C E D (4) B A C D E (4) E A D B C (3) E A B D C (3) D E C A B (3) C B D A E (3) B E A C D (3) B C A E D (3) A E D B C (3) A E B D C (3) A D E C B (3) A B E C D (3) E D C B A (2) E D B C A (2) E D A C B (2) E D A B C (2) E B C D A (2) D E C B A (2) D E A C B (2) C D E B A (2) C D B E A (2) C D B A E (2) B A E C D (2) A E D C B (2) A D C B E (2) A B C E D (2) E D C A B (1) E C B D A (1) E B D C A (1) E B D A C (1) E B A D C (1) D C A B E (1) D A E C B (1) D A C E B (1) C B E D A (1) C B A D E (1) B E C A D (1) A D E B C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 0 6 0 B 20 0 14 12 2 C 0 -14 0 4 -2 D -6 -12 -4 0 -6 E 0 -2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 0 6 0 B 20 0 14 12 2 C 0 -14 0 4 -2 D -6 -12 -4 0 -6 E 0 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=21 A=21 D=15 C=15 so D is eliminated. Round 2 votes counts: E=28 B=28 A=23 C=21 so C is eliminated. Round 3 votes counts: B=41 E=35 A=24 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:203 C:194 A:193 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 0 6 0 B 20 0 14 12 2 C 0 -14 0 4 -2 D -6 -12 -4 0 -6 E 0 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 0 6 0 B 20 0 14 12 2 C 0 -14 0 4 -2 D -6 -12 -4 0 -6 E 0 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 0 6 0 B 20 0 14 12 2 C 0 -14 0 4 -2 D -6 -12 -4 0 -6 E 0 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999955957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3868: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) B A E D C (9) E A B D C (8) E A B C D (7) D B A E C (7) A E B D C (7) D C B A E (6) C B D A E (5) C E A B D (4) B D A E C (4) D C E A B (3) C D B A E (3) E A D B C (2) D A E B C (2) C D B E A (2) C B E A D (2) B A E C D (2) B A D E C (2) E D A C B (1) E D A B C (1) E C A B D (1) E A C B D (1) D E A B C (1) D C A B E (1) D B C A E (1) D B A C E (1) C E A D B (1) B E A C D (1) B D A C E (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 6 16 -2 6 B -6 0 16 12 -2 C -16 -16 0 -16 -14 D 2 -12 16 0 2 E -6 2 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.100000 C: 0.000000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000043 Cumulative probabilities = A: 0.600000 B: 0.700000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 -2 6 B -6 0 16 12 -2 C -16 -16 0 -16 -14 D 2 -12 16 0 2 E -6 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.100000 C: 0.000000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000337 Cumulative probabilities = A: 0.600000 B: 0.700000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=22 E=21 B=20 A=8 so A is eliminated. Round 2 votes counts: C=29 E=28 D=22 B=21 so B is eliminated. Round 3 votes counts: E=41 C=30 D=29 so D is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:213 B:210 D:204 E:204 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 -2 6 B -6 0 16 12 -2 C -16 -16 0 -16 -14 D 2 -12 16 0 2 E -6 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.100000 C: 0.000000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000337 Cumulative probabilities = A: 0.600000 B: 0.700000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 -2 6 B -6 0 16 12 -2 C -16 -16 0 -16 -14 D 2 -12 16 0 2 E -6 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.100000 C: 0.000000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000337 Cumulative probabilities = A: 0.600000 B: 0.700000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 -2 6 B -6 0 16 12 -2 C -16 -16 0 -16 -14 D 2 -12 16 0 2 E -6 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.100000 C: 0.000000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000337 Cumulative probabilities = A: 0.600000 B: 0.700000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3869: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D A B E C (7) E B D A C (5) E B C A D (5) C A E B D (5) C A B E D (5) A C D B E (5) C E A B D (4) E D B C A (3) D C A E B (3) D B E A C (3) C A D E B (3) A C B E D (3) E C B A D (2) E B D C A (2) E B C D A (2) D E B A C (2) C E D B A (2) C D A E B (2) C A D B E (2) B E A C D (2) B A E D C (2) A D C B E (2) A D B E C (2) A B E D C (2) A B D E C (2) A B C E D (2) E D B A C (1) E C D B A (1) E C B D A (1) D E C B A (1) D C E A B (1) D B A E C (1) C E B D A (1) B E D A C (1) B E C A D (1) B D E A C (1) B D A E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -8 18 2 B -4 0 -4 18 -8 C 8 4 0 14 0 D -18 -18 -14 0 -20 E -2 8 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.564639 D: 0.000000 E: 0.435361 Sum of squares = 0.508356331582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.564639 D: 0.564639 E: 1.000000 A B C D E A 0 4 -8 18 2 B -4 0 -4 18 -8 C 8 4 0 14 0 D -18 -18 -14 0 -20 E -2 8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=22 A=20 D=18 B=8 so B is eliminated. Round 2 votes counts: C=32 E=26 A=22 D=20 so D is eliminated. Round 3 votes counts: C=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:213 A:208 B:201 D:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 18 2 B -4 0 -4 18 -8 C 8 4 0 14 0 D -18 -18 -14 0 -20 E -2 8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 18 2 B -4 0 -4 18 -8 C 8 4 0 14 0 D -18 -18 -14 0 -20 E -2 8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 18 2 B -4 0 -4 18 -8 C 8 4 0 14 0 D -18 -18 -14 0 -20 E -2 8 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3870: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (15) D B C A E (14) E A C B D (11) B D C A E (11) A C E B D (10) D B E C A (9) D B C E A (4) A E C B D (4) E A D C B (3) C A B D E (2) A C B E D (2) E D B A C (1) E D A C B (1) E D A B C (1) E A D B C (1) D E B A C (1) D B E A C (1) C D B A E (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B E D (1) B D E C A (1) B C D A E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 4 8 2 B -10 0 -10 -8 0 C -4 10 0 4 2 D -8 8 -4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998469 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 8 2 B -10 0 -10 -8 0 C -4 10 0 4 2 D -8 8 -4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991169 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=29 A=17 B=13 C=8 so C is eliminated. Round 2 votes counts: E=33 D=30 A=23 B=14 so B is eliminated. Round 3 votes counts: D=43 E=33 A=24 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:212 C:206 E:200 D:196 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 8 2 B -10 0 -10 -8 0 C -4 10 0 4 2 D -8 8 -4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991169 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 8 2 B -10 0 -10 -8 0 C -4 10 0 4 2 D -8 8 -4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991169 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 8 2 B -10 0 -10 -8 0 C -4 10 0 4 2 D -8 8 -4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991169 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3871: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) E A B D C (10) D E C A B (10) C D B A E (10) A B E C D (10) B A C D E (6) E D C A B (5) E A B C D (5) C B A D E (5) B A C E D (4) D C B A E (3) C D E B A (3) B A E C D (3) A B E D C (2) A B D E C (2) E D A B C (1) E C D B A (1) D E A C B (1) D C E A B (1) D C A B E (1) C E B A D (1) C B D A E (1) B C A D E (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -8 2 0 B 0 0 -8 4 -2 C 8 8 0 2 -2 D -2 -4 -2 0 14 E 0 2 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.111111 E: 0.111111 Sum of squares = 0.629629629758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.888889 E: 1.000000 A B C D E A 0 0 -8 2 0 B 0 0 -8 4 -2 C 8 8 0 2 -2 D -2 -4 -2 0 14 E 0 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.111111 E: 0.111111 Sum of squares = 0.629629629704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=22 C=20 A=16 B=14 so B is eliminated. Round 2 votes counts: A=29 D=28 E=22 C=21 so C is eliminated. Round 3 votes counts: D=42 A=35 E=23 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:208 D:203 A:197 B:197 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 2 0 B 0 0 -8 4 -2 C 8 8 0 2 -2 D -2 -4 -2 0 14 E 0 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.111111 E: 0.111111 Sum of squares = 0.629629629704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.888889 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 0 B 0 0 -8 4 -2 C 8 8 0 2 -2 D -2 -4 -2 0 14 E 0 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.111111 E: 0.111111 Sum of squares = 0.629629629704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.888889 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 0 B 0 0 -8 4 -2 C 8 8 0 2 -2 D -2 -4 -2 0 14 E 0 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.111111 E: 0.111111 Sum of squares = 0.629629629704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.888889 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3872: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) C E B A D (7) C D A E B (6) D A C B E (5) A D B E C (5) E B A D C (4) D A B E C (4) C E D B A (4) C A E D B (4) C A D E B (4) E C B A D (3) E B C D A (3) E B A C D (3) D A B C E (3) C E B D A (3) C E A B D (3) C D A B E (3) E C B D A (2) E B D A C (2) B E D A C (2) B E A D C (2) A B E D C (2) A B D E C (2) E B D C A (1) D E B C A (1) D B E A C (1) D B A E C (1) C E D A B (1) C E A D B (1) C D E A B (1) C A D B E (1) B D E A C (1) B D A E C (1) B A E D C (1) B A E C D (1) B A D E C (1) A D C B E (1) A D B C E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -10 10 -4 B 2 0 0 6 -20 C 10 0 0 18 0 D -10 -6 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.519602 D: 0.000000 E: 0.480398 Sum of squares = 0.500768463299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.519602 D: 0.519602 E: 1.000000 A B C D E A 0 -2 -10 10 -4 B 2 0 0 6 -20 C 10 0 0 18 0 D -10 -6 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=25 D=15 A=13 B=9 so B is eliminated. Round 2 votes counts: C=38 E=29 D=17 A=16 so A is eliminated. Round 3 votes counts: C=40 E=33 D=27 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:220 C:214 A:197 B:194 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 10 -4 B 2 0 0 6 -20 C 10 0 0 18 0 D -10 -6 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 10 -4 B 2 0 0 6 -20 C 10 0 0 18 0 D -10 -6 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 10 -4 B 2 0 0 6 -20 C 10 0 0 18 0 D -10 -6 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3873: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) E C B A D (7) D C E A B (5) D C A E B (5) C E B D A (5) E B C A D (4) D A C E B (4) B E C A D (4) B A E C D (4) A B D E C (4) D C E B A (3) C E D B A (3) A D B C E (3) E C A D B (2) D C B E A (2) D A C B E (2) D A B C E (2) C B E D A (2) B E A C D (2) B C E D A (2) A E D C B (2) A E B C D (2) A B E D C (2) E C D B A (1) E C D A B (1) E C B D A (1) E B A C D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C A B E (1) D B C E A (1) D B A C E (1) C D E A B (1) B C E A D (1) B C D E A (1) B C D A E (1) B A D C E (1) A E D B C (1) A D E C B (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -8 8 -4 B -6 0 0 12 -6 C 8 0 0 12 -6 D -8 -12 -12 0 -16 E 4 6 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 8 -4 B -6 0 0 12 -6 C 8 0 0 12 -6 D -8 -12 -12 0 -16 E 4 6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=27 E=18 B=16 C=11 so C is eliminated. Round 2 votes counts: D=29 A=27 E=26 B=18 so B is eliminated. Round 3 votes counts: E=37 A=32 D=31 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:207 A:201 B:200 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 8 -4 B -6 0 0 12 -6 C 8 0 0 12 -6 D -8 -12 -12 0 -16 E 4 6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 8 -4 B -6 0 0 12 -6 C 8 0 0 12 -6 D -8 -12 -12 0 -16 E 4 6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 8 -4 B -6 0 0 12 -6 C 8 0 0 12 -6 D -8 -12 -12 0 -16 E 4 6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3874: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) E B A C D (7) D A C B E (7) D C A B E (6) C B E A D (6) A E B D C (6) B E C A D (5) E B C A D (4) D C A E B (4) C D E B A (4) A D E B C (4) E A B D C (3) D A E B C (3) C D B A E (3) C B D E A (3) C B D A E (3) B E A C D (3) A D B E C (3) D C E A B (2) D A E C B (2) E D C B A (1) E C B A D (1) E B A D C (1) E A D B C (1) D E A C B (1) C D E A B (1) C B E D A (1) C B A D E (1) B C E A D (1) B C A E D (1) B A E C D (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -10 2 -6 B 8 0 -8 0 10 C 10 8 0 8 4 D -2 0 -8 0 14 E 6 -10 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 2 -6 B 8 0 -8 0 10 C 10 8 0 8 4 D -2 0 -8 0 14 E 6 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 E=18 A=16 B=11 so B is eliminated. Round 2 votes counts: C=32 E=26 D=25 A=17 so A is eliminated. Round 3 votes counts: E=34 D=33 C=33 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:205 D:202 A:189 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 2 -6 B 8 0 -8 0 10 C 10 8 0 8 4 D -2 0 -8 0 14 E 6 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 2 -6 B 8 0 -8 0 10 C 10 8 0 8 4 D -2 0 -8 0 14 E 6 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 2 -6 B 8 0 -8 0 10 C 10 8 0 8 4 D -2 0 -8 0 14 E 6 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3875: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) E C D B A (9) D C E A B (9) A D C E B (9) D A C E B (6) B E C A D (6) B A E C D (6) A D B C E (6) E C B D A (5) A B D E C (5) E B C D A (4) A B D C E (4) C D E A B (3) A B E C D (3) D C A E B (2) B A E D C (2) A B C E D (2) D E C B A (1) D C E B A (1) C E D B A (1) C E A B D (1) B E D C A (1) B D A E C (1) B A D E C (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -6 -6 -2 B -4 0 2 2 -4 C 6 -2 0 2 -8 D 6 -2 -2 0 -2 E 2 4 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -6 -6 -2 B -4 0 2 2 -4 C 6 -2 0 2 -8 D 6 -2 -2 0 -2 E 2 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=27 D=19 E=18 C=5 so C is eliminated. Round 2 votes counts: A=31 B=27 D=22 E=20 so E is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:208 D:200 C:199 B:198 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -6 -6 -2 B -4 0 2 2 -4 C 6 -2 0 2 -8 D 6 -2 -2 0 -2 E 2 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 -2 B -4 0 2 2 -4 C 6 -2 0 2 -8 D 6 -2 -2 0 -2 E 2 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 -2 B -4 0 2 2 -4 C 6 -2 0 2 -8 D 6 -2 -2 0 -2 E 2 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3876: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) D B E C A (7) B D C A E (7) B D A C E (6) D C B E A (5) A C E B D (5) D B C E A (4) B D E A C (4) E C A D B (3) E A C D B (3) C E A D B (3) B A D E C (3) E A C B D (2) D E B C A (2) D B C A E (2) C E D A B (2) C D E A B (2) C A B D E (2) B A C D E (2) A B C E D (2) E D C A B (1) E D B A C (1) E D A C B (1) E D A B C (1) E C D A B (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B A E (1) C D B A E (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D A C (1) B D C E A (1) B D A E C (1) B C A D E (1) B A D C E (1) A E B D C (1) A E B C D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -4 -10 2 B 4 0 -2 6 4 C 4 2 0 -8 8 D 10 -6 8 0 14 E -2 -4 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000008 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -10 2 B 4 0 -2 6 4 C 4 2 0 -8 8 D 10 -6 8 0 14 E -2 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000006 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 A=21 E=14 C=13 so C is eliminated. Round 2 votes counts: D=28 B=27 A=26 E=19 so E is eliminated. Round 3 votes counts: A=38 D=35 B=27 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:206 C:203 A:192 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -10 2 B 4 0 -2 6 4 C 4 2 0 -8 8 D 10 -6 8 0 14 E -2 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000006 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -10 2 B 4 0 -2 6 4 C 4 2 0 -8 8 D 10 -6 8 0 14 E -2 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000006 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -10 2 B 4 0 -2 6 4 C 4 2 0 -8 8 D 10 -6 8 0 14 E -2 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.375000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000006 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3877: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D B A E C (7) B E C A D (7) B D A E C (6) E C A B D (4) D B C A E (4) D B A C E (4) D A E B C (4) D A C E B (4) C E A D B (4) C D B E A (3) B A D E C (3) A E D C B (3) E B C A D (2) D A E C B (2) C E A B D (2) C D E A B (2) C D A E B (2) B D C A E (2) B D A C E (2) B C E D A (2) A D E C B (2) E A C D B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A E B (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E B A (1) C B E D A (1) C A E D B (1) C A D E B (1) B E A C D (1) B C E A D (1) B C D E A (1) A E D B C (1) A E C D B (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -4 -6 10 B 12 0 6 -6 -4 C 4 -6 0 -2 0 D 6 6 2 0 12 E -10 4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -6 10 B 12 0 6 -6 -4 C 4 -6 0 -2 0 D 6 6 2 0 12 E -10 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 B=25 A=10 E=9 so E is eliminated. Round 2 votes counts: C=31 D=29 B=27 A=13 so A is eliminated. Round 3 votes counts: D=36 C=34 B=30 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:204 C:198 A:194 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 10 B 12 0 6 -6 -4 C 4 -6 0 -2 0 D 6 6 2 0 12 E -10 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 10 B 12 0 6 -6 -4 C 4 -6 0 -2 0 D 6 6 2 0 12 E -10 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 10 B 12 0 6 -6 -4 C 4 -6 0 -2 0 D 6 6 2 0 12 E -10 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3878: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (13) A B C E D (13) E D B C A (10) C A B D E (9) E D B A C (7) B A C E D (6) E B A D C (4) A C B D E (4) D E C A B (3) D C E B A (3) C B A D E (3) B A E C D (3) E D A B C (2) D E A C B (2) D C A E B (2) A C B E D (2) E D A C B (1) E B D C A (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) E A B C D (1) D E B C A (1) D C E A B (1) D C B A E (1) C D A B E (1) C A D B E (1) B C A D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 0 2 -6 B 10 0 8 2 -10 C 0 -8 0 -8 -6 D -2 -2 8 0 -10 E 6 10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 0 2 -6 B 10 0 8 2 -10 C 0 -8 0 -8 -6 D -2 -2 8 0 -10 E 6 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 A=20 C=14 B=10 so B is eliminated. Round 2 votes counts: E=30 A=29 D=26 C=15 so C is eliminated. Round 3 votes counts: A=43 E=30 D=27 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:205 D:197 A:193 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 0 2 -6 B 10 0 8 2 -10 C 0 -8 0 -8 -6 D -2 -2 8 0 -10 E 6 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 2 -6 B 10 0 8 2 -10 C 0 -8 0 -8 -6 D -2 -2 8 0 -10 E 6 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 2 -6 B 10 0 8 2 -10 C 0 -8 0 -8 -6 D -2 -2 8 0 -10 E 6 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3879: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) B D A C E (6) E C A D B (5) D B A E C (5) D A C B E (5) D B A C E (4) D A B C E (4) C E A B D (4) B E D A C (4) E C A B D (3) E B D A C (3) D A C E B (3) C A B E D (3) B D A E C (3) E D C A B (2) E D B A C (2) E C B A D (2) D E B A C (2) D E A C B (2) D B E A C (2) D A B E C (2) C A D B E (2) B E C D A (2) B A D C E (2) A D B C E (2) A C D B E (2) A B D C E (2) E D C B A (1) E D A C B (1) E D A B C (1) E C D A B (1) E C B D A (1) E B C D A (1) E B C A D (1) D A E C B (1) D A E B C (1) C E A D B (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B D E A C (1) A C B D E (1) Total count = 100 A B C D E A 0 4 22 -32 4 B -4 0 12 -6 6 C -22 -12 0 -34 -10 D 32 6 34 0 8 E -4 -6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 22 -32 4 B -4 0 12 -6 6 C -22 -12 0 -34 -10 D 32 6 34 0 8 E -4 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=30 B=18 C=14 A=7 so A is eliminated. Round 2 votes counts: D=33 E=30 B=20 C=17 so C is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:240 B:204 A:199 E:196 C:161 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 22 -32 4 B -4 0 12 -6 6 C -22 -12 0 -34 -10 D 32 6 34 0 8 E -4 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 22 -32 4 B -4 0 12 -6 6 C -22 -12 0 -34 -10 D 32 6 34 0 8 E -4 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 22 -32 4 B -4 0 12 -6 6 C -22 -12 0 -34 -10 D 32 6 34 0 8 E -4 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3880: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (18) C B A D E (17) B C E D A (14) E D A B C (7) D E A B C (5) B C A D E (5) D E A C B (4) D A E C B (3) D A E B C (3) C B E D A (3) E D B C A (2) C B A E D (2) C A B D E (2) B E C D A (2) B C E A D (2) B C A E D (2) A D C B E (2) A C B D E (2) E D C B A (1) E D B A C (1) E D A C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -4 8 16 B 2 0 -10 2 8 C 4 10 0 2 2 D -8 -2 -2 0 26 E -16 -8 -2 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 8 16 B 2 0 -10 2 8 C 4 10 0 2 2 D -8 -2 -2 0 26 E -16 -8 -2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=24 A=24 D=15 E=12 so E is eliminated. Round 2 votes counts: D=27 B=25 C=24 A=24 so C is eliminated. Round 3 votes counts: B=47 D=27 A=26 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:209 C:209 D:207 B:201 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 8 16 B 2 0 -10 2 8 C 4 10 0 2 2 D -8 -2 -2 0 26 E -16 -8 -2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 16 B 2 0 -10 2 8 C 4 10 0 2 2 D -8 -2 -2 0 26 E -16 -8 -2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 16 B 2 0 -10 2 8 C 4 10 0 2 2 D -8 -2 -2 0 26 E -16 -8 -2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3881: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) E D C A B (7) E D B C A (6) E A D C B (6) A B E C D (6) A C E D B (4) E D A C B (3) D C E B A (3) B D C E A (3) A E C D B (3) A B C D E (3) E A D B C (2) E A B D C (2) D E C B A (2) D B C E A (2) C D E A B (2) C D B E A (2) B E D C A (2) B E D A C (2) B D E C A (2) B C A D E (2) B A E D C (2) B A E C D (2) A C B E D (2) A C B D E (2) A B C E D (2) E D C B A (1) E D A B C (1) E C D A B (1) E B D A C (1) E A C D B (1) D C B E A (1) D B E C A (1) C D A B E (1) C A D E B (1) C A D B E (1) B E A D C (1) A E C B D (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 20 14 -8 B -6 0 10 -2 2 C -20 -10 0 0 -12 D -14 2 0 0 -20 E 8 -2 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999995 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 6 20 14 -8 B -6 0 10 -2 2 C -20 -10 0 0 -12 D -14 2 0 0 -20 E 8 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999998 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=28 A=25 D=9 C=7 so C is eliminated. Round 2 votes counts: E=31 B=28 A=27 D=14 so D is eliminated. Round 3 votes counts: E=38 B=34 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:219 A:216 B:202 D:184 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 20 14 -8 B -6 0 10 -2 2 C -20 -10 0 0 -12 D -14 2 0 0 -20 E 8 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999998 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 20 14 -8 B -6 0 10 -2 2 C -20 -10 0 0 -12 D -14 2 0 0 -20 E 8 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999998 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 20 14 -8 B -6 0 10 -2 2 C -20 -10 0 0 -12 D -14 2 0 0 -20 E 8 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999998 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3882: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) C B D A E (5) B C A E D (5) E D A B C (4) E A B D C (4) D C B E A (4) D C A B E (4) A B C E D (4) E B A C D (3) E A D B C (3) E A B C D (3) D E C B A (3) C B A D E (3) B C E A D (3) B C D E A (3) B A C E D (3) E D B C A (2) D E C A B (2) D E A C B (2) D C E B A (2) C D B A E (2) B E C D A (2) B E C A D (2) B C A D E (2) B A E C D (2) A E D C B (2) A D E C B (2) E D B A C (1) E D A C B (1) E B C D A (1) E B A D C (1) E A D C B (1) D C E A B (1) D C B A E (1) D A C E B (1) D A C B E (1) C D B E A (1) C D A B E (1) C A B D E (1) B E A C D (1) B C D A E (1) A D C E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -2 10 0 B 6 0 14 16 6 C 2 -14 0 14 2 D -10 -16 -14 0 -12 E 0 -6 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 10 0 B 6 0 14 16 6 C 2 -14 0 14 2 D -10 -16 -14 0 -12 E 0 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 D=21 A=18 C=13 so C is eliminated. Round 2 votes counts: B=32 D=25 E=24 A=19 so A is eliminated. Round 3 votes counts: B=39 E=33 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:202 E:202 A:201 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 10 0 B 6 0 14 16 6 C 2 -14 0 14 2 D -10 -16 -14 0 -12 E 0 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 10 0 B 6 0 14 16 6 C 2 -14 0 14 2 D -10 -16 -14 0 -12 E 0 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 10 0 B 6 0 14 16 6 C 2 -14 0 14 2 D -10 -16 -14 0 -12 E 0 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3883: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (6) B A C D E (6) A B E D C (6) B C E D A (5) B A C E D (5) A B D C E (5) A B C D E (5) E D A C B (4) E D C B A (3) E B D C A (3) C E D B A (3) C D E B A (3) C D A E B (3) C B D E A (3) B E A D C (3) A D C E B (3) E D B A C (2) E B A D C (2) D A C E B (2) C D E A B (2) B E D A C (2) B E C D A (2) A D E C B (2) A B D E C (2) E D C A B (1) E D B C A (1) E D A B C (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) C B D A E (1) C B A D E (1) C A B D E (1) B E D C A (1) B E A C D (1) B C D E A (1) B C A E D (1) B A E C D (1) A D B E C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 24 6 8 B 12 0 26 30 22 C -24 -26 0 -8 8 D -6 -30 8 0 -8 E -8 -22 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 24 6 8 B 12 0 26 30 22 C -24 -26 0 -8 8 D -6 -30 8 0 -8 E -8 -22 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=26 E=17 C=17 D=6 so D is eliminated. Round 2 votes counts: B=34 A=28 E=19 C=19 so E is eliminated. Round 3 votes counts: B=42 A=34 C=24 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:245 A:213 E:185 D:182 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 24 6 8 B 12 0 26 30 22 C -24 -26 0 -8 8 D -6 -30 8 0 -8 E -8 -22 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 24 6 8 B 12 0 26 30 22 C -24 -26 0 -8 8 D -6 -30 8 0 -8 E -8 -22 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 24 6 8 B 12 0 26 30 22 C -24 -26 0 -8 8 D -6 -30 8 0 -8 E -8 -22 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3884: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) E C B A D (8) E D C B A (5) E A D B C (5) C B D A E (5) E C D B A (4) E C B D A (4) D A B C E (4) C B E D A (4) E D A C B (3) E C D A B (3) E A C B D (3) A E B C D (3) A B D C E (3) E C A B D (2) E A D C B (2) E A C D B (2) D B A C E (2) C E D B A (2) C E B A D (2) B C A D E (2) A D E B C (2) A D B C E (2) A B C D E (2) E D A B C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E A B C (1) D C E B A (1) D C B A E (1) D B C A E (1) C E B D A (1) C D B E A (1) C B A E D (1) C B A D E (1) B C D A E (1) B A D C E (1) B A C D E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 10 -14 B -2 0 -12 -6 -32 C 0 12 0 6 -24 D -10 6 -6 0 -32 E 14 32 24 32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 10 -14 B -2 0 -12 -6 -32 C 0 12 0 6 -24 D -10 6 -6 0 -32 E 14 32 24 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 A=23 C=17 D=11 B=5 so B is eliminated. Round 2 votes counts: E=44 A=25 C=20 D=11 so D is eliminated. Round 3 votes counts: E=46 A=31 C=23 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:251 A:199 C:197 D:179 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 10 -14 B -2 0 -12 -6 -32 C 0 12 0 6 -24 D -10 6 -6 0 -32 E 14 32 24 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 10 -14 B -2 0 -12 -6 -32 C 0 12 0 6 -24 D -10 6 -6 0 -32 E 14 32 24 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 10 -14 B -2 0 -12 -6 -32 C 0 12 0 6 -24 D -10 6 -6 0 -32 E 14 32 24 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3885: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) E B D C A (9) D E A B C (9) C B A E D (9) E D B A C (8) B E C A D (8) A C D B E (8) C A B D E (7) B C E A D (7) B C A E D (5) E B C D A (3) D E B A C (3) A D C E B (3) A D C B E (3) E D B C A (2) D A C E B (2) A C B D E (2) D E A C B (1) B D E A C (1) Total count = 100 A B C D E A 0 -10 0 4 -2 B 10 0 10 2 0 C 0 -10 0 -2 -8 D -4 -2 2 0 -2 E 2 0 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.527240 C: 0.000000 D: 0.000000 E: 0.472760 Sum of squares = 0.501484013839 Cumulative probabilities = A: 0.000000 B: 0.527240 C: 0.527240 D: 0.527240 E: 1.000000 A B C D E A 0 -10 0 4 -2 B 10 0 10 2 0 C 0 -10 0 -2 -8 D -4 -2 2 0 -2 E 2 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=22 B=21 C=16 A=16 so C is eliminated. Round 2 votes counts: B=30 D=25 A=23 E=22 so E is eliminated. Round 3 votes counts: B=42 D=35 A=23 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:206 D:197 A:196 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 4 -2 B 10 0 10 2 0 C 0 -10 0 -2 -8 D -4 -2 2 0 -2 E 2 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 4 -2 B 10 0 10 2 0 C 0 -10 0 -2 -8 D -4 -2 2 0 -2 E 2 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 4 -2 B 10 0 10 2 0 C 0 -10 0 -2 -8 D -4 -2 2 0 -2 E 2 0 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3886: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) B A E D C (7) D C B A E (6) E A B C D (5) D C B E A (5) B A D E C (5) A E B C D (5) E B A C D (4) C D E B A (4) B D A C E (4) E A C B D (3) E B C D A (2) E B C A D (2) D C E A B (2) D C A E B (2) D B C A E (2) D B A C E (2) C E D B A (2) C D A E B (2) B E A D C (2) A E C D B (2) A D B C E (2) A B E C D (2) A B D C E (2) E C D A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E A C D B (1) D C A B E (1) D A C B E (1) C E D A B (1) C A D E B (1) B E D C A (1) B E A C D (1) B D C A E (1) B D A E C (1) A D C B E (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 6 0 4 B 6 0 4 0 -4 C -6 -4 0 2 2 D 0 0 -2 0 12 E -4 4 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.581369 C: 0.000000 D: 0.418631 E: 0.000000 Sum of squares = 0.513241861868 Cumulative probabilities = A: 0.000000 B: 0.581369 C: 0.581369 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 0 4 B 6 0 4 0 -4 C -6 -4 0 2 2 D 0 0 -2 0 12 E -4 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 E=21 D=21 C=19 A=17 so A is eliminated. Round 2 votes counts: E=28 B=27 D=25 C=20 so C is eliminated. Round 3 votes counts: D=42 E=31 B=27 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:205 B:203 A:202 C:197 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 6 0 4 B 6 0 4 0 -4 C -6 -4 0 2 2 D 0 0 -2 0 12 E -4 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 0 4 B 6 0 4 0 -4 C -6 -4 0 2 2 D 0 0 -2 0 12 E -4 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 0 4 B 6 0 4 0 -4 C -6 -4 0 2 2 D 0 0 -2 0 12 E -4 4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3887: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) D C A B E (6) C D A B E (6) B E A D C (6) A B D E C (6) E B C D A (5) E B A C D (5) A B E D C (5) D C E B A (4) C D A E B (4) A B E C D (4) E C B D A (3) E B D C A (3) E B A D C (3) C E D B A (3) C E B D A (3) C D E A B (3) E C B A D (2) E B D A C (2) E B C A D (2) A D C B E (2) A D B E C (2) D E C B A (1) D E B C A (1) D A B E C (1) C E B A D (1) C A D E B (1) C A D B E (1) B A E D C (1) B A E C D (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -22 -14 -18 B 16 0 -6 4 -16 C 22 6 0 14 -6 D 14 -4 -14 0 2 E 18 16 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 A B C D E A 0 -16 -22 -14 -18 B 16 0 -6 4 -16 C 22 6 0 14 -6 D 14 -4 -14 0 2 E 18 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=25 A=20 D=13 B=8 so B is eliminated. Round 2 votes counts: C=34 E=31 A=22 D=13 so D is eliminated. Round 3 votes counts: C=44 E=33 A=23 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:218 B:199 D:199 A:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -22 -14 -18 B 16 0 -6 4 -16 C 22 6 0 14 -6 D 14 -4 -14 0 2 E 18 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -22 -14 -18 B 16 0 -6 4 -16 C 22 6 0 14 -6 D 14 -4 -14 0 2 E 18 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -22 -14 -18 B 16 0 -6 4 -16 C 22 6 0 14 -6 D 14 -4 -14 0 2 E 18 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.272727 E: 0.636364 Sum of squares = 0.487603305778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.090909 D: 0.363636 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3888: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D C A E (8) B C E D A (8) E C A D B (7) D B A C E (6) A E C D B (6) E C A B D (5) D A B C E (5) B D A C E (5) A D E C B (5) B E C A D (4) B D C E A (4) C E A D B (3) B D A E C (3) E C B A D (2) D A E C B (2) D A C E B (2) C E B A D (2) B C D E A (2) E A C B D (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A E C (1) D A C B E (1) D A B E C (1) C D E A B (1) B E C D A (1) B E A C D (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 4 0 -6 0 B -4 0 2 -8 4 C 0 -2 0 4 0 D 6 8 -4 0 -2 E 0 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.372716 B: 0.000000 C: 0.574052 D: 0.000000 E: 0.053232 Sum of squares = 0.471286124129 Cumulative probabilities = A: 0.372716 B: 0.372716 C: 0.946768 D: 0.946768 E: 1.000000 A B C D E A 0 4 0 -6 0 B -4 0 2 -8 4 C 0 -2 0 4 0 D 6 8 -4 0 -2 E 0 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250009016 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=24 D=21 A=13 C=6 so C is eliminated. Round 2 votes counts: B=36 E=29 D=22 A=13 so A is eliminated. Round 3 votes counts: E=37 B=36 D=27 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:204 C:201 A:199 E:199 B:197 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -6 0 B -4 0 2 -8 4 C 0 -2 0 4 0 D 6 8 -4 0 -2 E 0 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250009016 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -6 0 B -4 0 2 -8 4 C 0 -2 0 4 0 D 6 8 -4 0 -2 E 0 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250009016 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -6 0 B -4 0 2 -8 4 C 0 -2 0 4 0 D 6 8 -4 0 -2 E 0 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250009016 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3889: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (6) B D E A C (6) A D C B E (6) E B C D A (5) D A B E C (5) C E B A D (5) C E A D B (5) B E C D A (5) A D C E B (5) A D B C E (5) E B D C A (4) B E D C A (4) B E D A C (4) B D A E C (4) D B A E C (3) C E A B D (3) A C D E B (3) A C D B E (3) E D B A C (2) C A D E B (2) C A D B E (2) B C E A D (2) E D A C B (1) E D A B C (1) E C B A D (1) E C A D B (1) D E B A C (1) D E A B C (1) D A E B C (1) D A B C E (1) B A D C E (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 10 4 -2 B -6 0 10 -10 4 C -10 -10 0 -12 0 D -4 10 12 0 2 E 2 -4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 6 10 4 -2 B -6 0 10 -10 4 C -10 -10 0 -12 0 D -4 10 12 0 2 E 2 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 C=23 E=15 D=12 so D is eliminated. Round 2 votes counts: A=31 B=29 C=23 E=17 so E is eliminated. Round 3 votes counts: B=41 A=34 C=25 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:209 B:199 E:198 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 4 -2 B -6 0 10 -10 4 C -10 -10 0 -12 0 D -4 10 12 0 2 E 2 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 4 -2 B -6 0 10 -10 4 C -10 -10 0 -12 0 D -4 10 12 0 2 E 2 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 4 -2 B -6 0 10 -10 4 C -10 -10 0 -12 0 D -4 10 12 0 2 E 2 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3890: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) E A C B D (6) D C B A E (6) D B C E A (6) D C B E A (5) A E B C D (5) A B E D C (5) D B C A E (4) D B A C E (4) C D E A B (4) E B A C D (3) C E D A B (3) C E A D B (3) C D E B A (3) B D A E C (3) B A D E C (3) E A B C D (2) C E B D A (2) C D B E A (2) C D A E B (2) B D E C A (2) A E C D B (2) E C B D A (1) E C A D B (1) E B C A D (1) E B A D C (1) E A B D C (1) D C A E B (1) D A C B E (1) C A E D B (1) B E A D C (1) B D C E A (1) A E C B D (1) A D E B C (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 4 4 0 4 B -4 0 8 -6 -8 C -4 -8 0 -12 2 D 0 6 12 0 4 E -4 8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.849953 B: 0.000000 C: 0.000000 D: 0.150047 E: 0.000000 Sum of squares = 0.744934750775 Cumulative probabilities = A: 0.849953 B: 0.849953 C: 0.849953 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 4 B -4 0 8 -6 -8 C -4 -8 0 -12 2 D 0 6 12 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 C=20 E=16 B=10 so B is eliminated. Round 2 votes counts: D=33 A=30 C=20 E=17 so E is eliminated. Round 3 votes counts: A=44 D=33 C=23 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:206 E:199 B:195 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 4 B -4 0 8 -6 -8 C -4 -8 0 -12 2 D 0 6 12 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 4 B -4 0 8 -6 -8 C -4 -8 0 -12 2 D 0 6 12 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 4 B -4 0 8 -6 -8 C -4 -8 0 -12 2 D 0 6 12 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3891: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (12) C E D A B (9) A B C E D (8) B A D E C (7) B D A E C (6) B A D C E (6) B A C E D (6) D B A E C (5) E D C B A (4) C E A D B (4) B A E C D (4) A B C D E (4) E C D A B (3) C E A B D (3) B D E A C (3) A B D C E (3) D E C A B (2) A C B E D (2) E C D B A (1) E C B D A (1) E C B A D (1) D B E C A (1) D B E A C (1) D A C E B (1) C A E B D (1) B E A C D (1) B A E D C (1) Total count = 100 A B C D E A 0 -20 16 2 8 B 20 0 12 14 16 C -16 -12 0 -4 -6 D -2 -14 4 0 2 E -8 -16 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 16 2 8 B 20 0 12 14 16 C -16 -12 0 -4 -6 D -2 -14 4 0 2 E -8 -16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=22 C=17 A=17 E=10 so E is eliminated. Round 2 votes counts: B=34 D=26 C=23 A=17 so A is eliminated. Round 3 votes counts: B=49 D=26 C=25 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:203 D:195 E:190 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 16 2 8 B 20 0 12 14 16 C -16 -12 0 -4 -6 D -2 -14 4 0 2 E -8 -16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 16 2 8 B 20 0 12 14 16 C -16 -12 0 -4 -6 D -2 -14 4 0 2 E -8 -16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 16 2 8 B 20 0 12 14 16 C -16 -12 0 -4 -6 D -2 -14 4 0 2 E -8 -16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3892: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (16) E C B D A (7) B C E D A (7) D A C E B (6) B A D C E (6) E C B A D (5) C E B D A (5) A D B C E (5) C E D B A (4) B E C A D (4) A D B E C (4) E B C A D (3) D A B C E (3) B C D A E (3) B E C D A (2) B D A C E (2) A E D C B (2) A D E B C (2) E C D B A (1) E C D A B (1) E A D C B (1) D C B A E (1) D B A C E (1) D A E C B (1) D A C B E (1) C E D A B (1) C D E B A (1) B E A C D (1) B D C A E (1) B A E D C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 8 4 14 B 10 0 -8 -4 -14 C -8 8 0 -10 -4 D -4 4 10 0 10 E -14 14 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.40740740742 Cumulative probabilities = A: 0.222222 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 4 14 B 10 0 -8 -4 -14 C -8 8 0 -10 -4 D -4 4 10 0 10 E -14 14 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.4074074075 Cumulative probabilities = A: 0.222222 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=27 E=18 D=13 C=11 so C is eliminated. Round 2 votes counts: A=31 E=28 B=27 D=14 so D is eliminated. Round 3 votes counts: A=42 E=29 B=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:208 E:197 C:193 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 4 14 B 10 0 -8 -4 -14 C -8 8 0 -10 -4 D -4 4 10 0 10 E -14 14 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.4074074075 Cumulative probabilities = A: 0.222222 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 4 14 B 10 0 -8 -4 -14 C -8 8 0 -10 -4 D -4 4 10 0 10 E -14 14 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.4074074075 Cumulative probabilities = A: 0.222222 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 4 14 B 10 0 -8 -4 -14 C -8 8 0 -10 -4 D -4 4 10 0 10 E -14 14 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.4074074075 Cumulative probabilities = A: 0.222222 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3893: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) B E A C D (9) C A B E D (8) D C A E B (6) D C A B E (6) C A E B D (5) D E A C B (4) B E D A C (4) E B A C D (3) D B E A C (3) C A E D B (3) E A B C D (2) D E C A B (2) D C E A B (2) D B E C A (2) C A D E B (2) C A D B E (2) C A B D E (2) B E A D C (2) B D E A C (2) A E C B D (2) E D A C B (1) E B D A C (1) D E C B A (1) D E B C A (1) D C B E A (1) D C B A E (1) D B C A E (1) C D A E B (1) C D A B E (1) C B A D E (1) B C D A E (1) B C A E D (1) B A E C D (1) B A C E D (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 0 -6 -4 B -4 0 -6 -6 -2 C 0 6 0 -4 -6 D 6 6 4 0 8 E 4 2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -6 -4 B -4 0 -6 -6 -2 C 0 6 0 -4 -6 D 6 6 4 0 8 E 4 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 C=25 B=21 E=7 A=5 so A is eliminated. Round 2 votes counts: D=42 C=27 B=21 E=10 so E is eliminated. Round 3 votes counts: D=43 C=29 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:202 C:198 A:197 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -6 -4 B -4 0 -6 -6 -2 C 0 6 0 -4 -6 D 6 6 4 0 8 E 4 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -6 -4 B -4 0 -6 -6 -2 C 0 6 0 -4 -6 D 6 6 4 0 8 E 4 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -6 -4 B -4 0 -6 -6 -2 C 0 6 0 -4 -6 D 6 6 4 0 8 E 4 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3894: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) B D E C A (9) E C A B D (8) D A B C E (8) A C E D B (8) E C B A D (7) D B E C A (6) A D C E B (6) D B A E C (5) B E C D A (5) E B C D A (4) A C E B D (4) D B E A C (3) D B A C E (3) E B C A D (2) C A E B D (2) B E D C A (2) A C D E B (2) E C B D A (1) D A C E B (1) D A C B E (1) C B E A D (1) A E D C B (1) A E C D B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 -12 4 -14 B -4 0 -6 8 -12 C 12 6 0 8 -8 D -4 -8 -8 0 -10 E 14 12 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -12 4 -14 B -4 0 -6 8 -12 C 12 6 0 8 -8 D -4 -8 -8 0 -10 E 14 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 E=22 B=16 C=12 so C is eliminated. Round 2 votes counts: E=31 D=27 A=25 B=17 so B is eliminated. Round 3 votes counts: E=39 D=36 A=25 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:209 B:193 A:191 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -12 4 -14 B -4 0 -6 8 -12 C 12 6 0 8 -8 D -4 -8 -8 0 -10 E 14 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 4 -14 B -4 0 -6 8 -12 C 12 6 0 8 -8 D -4 -8 -8 0 -10 E 14 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 4 -14 B -4 0 -6 8 -12 C 12 6 0 8 -8 D -4 -8 -8 0 -10 E 14 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3895: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) D E B A C (7) E B D C A (6) B C A E D (6) C B A E D (5) A D C E B (5) E B C D A (4) D A E B C (4) C E B D A (4) C A B E D (4) B E C D A (4) A D B C E (4) A C B D E (4) A B C D E (4) E D B C A (3) E C D B A (3) D E A C B (3) D A E C B (3) E C B D A (2) B C E A D (2) A B D C E (2) E D C B A (1) E D B A C (1) D E A B C (1) C E D A B (1) C E B A D (1) C A D E B (1) B A D E C (1) B A C D E (1) A D C B E (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -20 -12 6 -4 B 20 0 0 20 -2 C 12 0 0 16 14 D -6 -20 -16 0 -14 E 4 2 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.361696 C: 0.638304 D: 0.000000 E: 0.000000 Sum of squares = 0.538256029383 Cumulative probabilities = A: 0.000000 B: 0.361696 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -12 6 -4 B 20 0 0 20 -2 C 12 0 0 16 14 D -6 -20 -16 0 -14 E 4 2 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 E=20 D=18 B=14 so B is eliminated. Round 2 votes counts: C=33 A=25 E=24 D=18 so D is eliminated. Round 3 votes counts: E=35 C=33 A=32 so A is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:219 E:203 A:185 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -12 6 -4 B 20 0 0 20 -2 C 12 0 0 16 14 D -6 -20 -16 0 -14 E 4 2 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -12 6 -4 B 20 0 0 20 -2 C 12 0 0 16 14 D -6 -20 -16 0 -14 E 4 2 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -12 6 -4 B 20 0 0 20 -2 C 12 0 0 16 14 D -6 -20 -16 0 -14 E 4 2 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3896: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) D C A B E (5) D A C E B (5) C D E A B (5) C D B E A (5) C B E D A (5) B E C A D (5) B E A C D (5) A D E C B (5) B C D E A (4) A E B D C (4) E B A C D (3) C B D E A (3) B A E D C (3) A E D C B (3) A E D B C (3) A D E B C (3) E A C B D (2) D C B A E (2) D A B E C (2) C D B A E (2) B E A D C (2) B C E D A (2) B A D E C (2) E C A B D (1) E A C D B (1) D B C A E (1) D A E C B (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E B A (1) B D C A E (1) B C E A D (1) B C D A E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -4 -10 8 B -4 0 -10 -10 10 C 4 10 0 -6 6 D 10 10 6 0 16 E -8 -10 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -10 8 B -4 0 -10 -10 10 C 4 10 0 -6 6 D 10 10 6 0 16 E -8 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 C=22 A=20 E=7 so E is eliminated. Round 2 votes counts: B=29 D=25 C=23 A=23 so C is eliminated. Round 3 votes counts: D=39 B=37 A=24 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:207 A:199 B:193 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -10 8 B -4 0 -10 -10 10 C 4 10 0 -6 6 D 10 10 6 0 16 E -8 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -10 8 B -4 0 -10 -10 10 C 4 10 0 -6 6 D 10 10 6 0 16 E -8 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -10 8 B -4 0 -10 -10 10 C 4 10 0 -6 6 D 10 10 6 0 16 E -8 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3897: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (11) B C A D E (8) E D A C B (7) C D E B A (7) B C D E A (7) A E D C B (6) B A C E D (4) A B E D C (4) D E C A B (3) D E A C B (3) B C A E D (3) B A C D E (3) A B C E D (3) E A D C B (2) D E C B A (2) D E B C A (2) C E D B A (2) C D B E A (2) C B A E D (2) A E C D B (2) A C B E D (2) E D C A B (1) D E B A C (1) D C E B A (1) C E D A B (1) C E A D B (1) C B E D A (1) C A E B D (1) B D E C A (1) B D E A C (1) B A D E C (1) A E D B C (1) A E B D C (1) A D E B C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -12 -6 -12 B 18 0 -16 8 8 C 12 16 0 22 20 D 6 -8 -22 0 10 E 12 -8 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -12 -6 -12 B 18 0 -16 8 8 C 12 16 0 22 20 D 6 -8 -22 0 10 E 12 -8 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=28 B=28 A=22 D=12 E=10 so E is eliminated. Round 2 votes counts: C=28 B=28 A=24 D=20 so D is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:235 B:209 D:193 E:187 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -12 -6 -12 B 18 0 -16 8 8 C 12 16 0 22 20 D 6 -8 -22 0 10 E 12 -8 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -6 -12 B 18 0 -16 8 8 C 12 16 0 22 20 D 6 -8 -22 0 10 E 12 -8 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -6 -12 B 18 0 -16 8 8 C 12 16 0 22 20 D 6 -8 -22 0 10 E 12 -8 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3898: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) C D E A B (11) B A E D C (9) A B E D C (7) E A D B C (5) C D E B A (5) D C E A B (4) C E D A B (4) C B A E D (4) E A B C D (3) D E A B C (3) B A D E C (3) E C D A B (2) D E C A B (2) D C B A E (2) C E A B D (2) C D B E A (2) C B D A E (2) B A D C E (2) A E B D C (2) E A C B D (1) E A B D C (1) D B A C E (1) C E B A D (1) C B E A D (1) C B A D E (1) B D C A E (1) B D A E C (1) B D A C E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C E D (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -16 -6 6 B 4 0 -12 -6 6 C 16 12 0 12 18 D 6 6 -12 0 8 E -6 -6 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -6 6 B 4 0 -12 -6 6 C 16 12 0 12 18 D 6 6 -12 0 8 E -6 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=45 B=21 E=12 D=12 A=10 so A is eliminated. Round 2 votes counts: C=45 B=28 E=15 D=12 so D is eliminated. Round 3 votes counts: C=51 B=29 E=20 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:204 B:196 A:190 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 -6 6 B 4 0 -12 -6 6 C 16 12 0 12 18 D 6 6 -12 0 8 E -6 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -6 6 B 4 0 -12 -6 6 C 16 12 0 12 18 D 6 6 -12 0 8 E -6 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -6 6 B 4 0 -12 -6 6 C 16 12 0 12 18 D 6 6 -12 0 8 E -6 -6 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3899: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) E A B C D (8) D C B A E (7) D C A E B (5) D C A B E (5) D B E C A (5) D B C E A (5) C A E B D (5) B E A C D (5) A E C B D (5) E B A D C (4) D B E A C (4) E A C B D (3) D E B A C (3) C A D E B (3) C A B E D (3) A C E B D (3) D E A B C (2) B E D A C (2) E D B A C (1) E D A B C (1) E A D C B (1) D E C A B (1) D E A C B (1) D B C A E (1) C D A E B (1) C B D A E (1) C B A E D (1) C A D B E (1) B D C E A (1) B C E A D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 8 8 -16 B 2 0 6 6 -14 C -8 -6 0 2 -12 D -8 -6 -2 0 -8 E 16 14 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 8 -16 B 2 0 6 6 -14 C -8 -6 0 2 -12 D -8 -6 -2 0 -8 E 16 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=28 C=15 B=9 A=9 so B is eliminated. Round 2 votes counts: D=40 E=35 C=16 A=9 so A is eliminated. Round 3 votes counts: E=40 D=40 C=20 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:200 A:199 C:188 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 8 -16 B 2 0 6 6 -14 C -8 -6 0 2 -12 D -8 -6 -2 0 -8 E 16 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 8 -16 B 2 0 6 6 -14 C -8 -6 0 2 -12 D -8 -6 -2 0 -8 E 16 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 8 -16 B 2 0 6 6 -14 C -8 -6 0 2 -12 D -8 -6 -2 0 -8 E 16 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3900: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) A B D C E (9) E D C A B (7) C B E D A (7) B C A E D (7) E D C B A (6) B A C E D (5) B A C D E (5) E C D B A (4) D A E C B (4) D E C A B (3) D E A C B (3) D A E B C (3) C E D B A (3) A D B E C (3) E C B D A (2) C E B D A (2) B C E A D (2) B C A D E (2) A B D E C (2) A B C D E (2) E D A C B (1) E A D B C (1) D E C B A (1) D C E B A (1) D A C E B (1) B A E C D (1) A E D B C (1) A E B D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 6 4 14 B -4 0 10 -2 -6 C -6 -10 0 -12 -6 D -4 2 12 0 -4 E -14 6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 14 B -4 0 10 -2 -6 C -6 -10 0 -12 -6 D -4 2 12 0 -4 E -14 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997046 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=22 E=21 D=16 C=12 so C is eliminated. Round 2 votes counts: B=29 A=29 E=26 D=16 so D is eliminated. Round 3 votes counts: A=37 E=34 B=29 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:203 E:201 B:199 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 14 B -4 0 10 -2 -6 C -6 -10 0 -12 -6 D -4 2 12 0 -4 E -14 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997046 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 14 B -4 0 10 -2 -6 C -6 -10 0 -12 -6 D -4 2 12 0 -4 E -14 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997046 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 14 B -4 0 10 -2 -6 C -6 -10 0 -12 -6 D -4 2 12 0 -4 E -14 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997046 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3901: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) C D E B A (9) B A C D E (9) B A E C D (7) A B E D C (7) D C E A B (5) B C D A E (5) E D C B A (4) C D B E A (4) E D A C B (3) E A B D C (3) C D E A B (3) B C A D E (3) A E B D C (3) E A D C B (2) B A C E D (2) A B D C E (2) A B C D E (2) E B C D A (1) E A D B C (1) D E C A B (1) D E A C B (1) D C A E B (1) C D B A E (1) C D A B E (1) C B D A E (1) B E C D A (1) B E C A D (1) B C D E A (1) B A E D C (1) A E D C B (1) A D C E B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -6 -6 -2 B 0 0 0 0 0 C 6 0 0 2 2 D 6 0 -2 0 4 E 2 0 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.524904 C: 0.475096 D: 0.000000 E: 0.000000 Sum of squares = 0.501240376243 Cumulative probabilities = A: 0.000000 B: 0.524904 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -6 -2 B 0 0 0 0 0 C 6 0 0 2 2 D 6 0 -2 0 4 E 2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=25 C=19 A=18 D=8 so D is eliminated. Round 2 votes counts: B=30 E=27 C=25 A=18 so A is eliminated. Round 3 votes counts: B=42 E=31 C=27 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:205 D:204 B:200 E:198 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 -6 -2 B 0 0 0 0 0 C 6 0 0 2 2 D 6 0 -2 0 4 E 2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -6 -2 B 0 0 0 0 0 C 6 0 0 2 2 D 6 0 -2 0 4 E 2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -6 -2 B 0 0 0 0 0 C 6 0 0 2 2 D 6 0 -2 0 4 E 2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3902: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) E C B D A (8) A D E C B (7) A D B C E (6) C E B A D (5) E D C B A (4) D E A C B (4) D A E C B (4) D A B E C (4) B C E D A (4) D A E B C (3) C B E A D (3) B C A E D (3) A B C D E (3) C B E D A (2) B D C E A (2) B D A C E (2) A E C D B (2) E D C A B (1) E C B A D (1) E C A D B (1) E B C D A (1) E A C B D (1) D E C A B (1) D E B C A (1) D B E C A (1) D B A E C (1) C E B D A (1) C E A B D (1) C B A E D (1) C A E B D (1) B E C D A (1) B C D E A (1) B A D C E (1) B A C E D (1) B A C D E (1) A D E B C (1) A D C E B (1) A D C B E (1) A C E B D (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 8 -6 B 8 0 -6 12 0 C 6 6 0 10 6 D -8 -12 -10 0 -8 E 6 0 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 8 -6 B 8 0 -6 12 0 C 6 6 0 10 6 D -8 -12 -10 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=25 A=25 D=19 E=17 C=14 so C is eliminated. Round 2 votes counts: B=31 A=26 E=24 D=19 so D is eliminated. Round 3 votes counts: A=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:214 B:207 E:204 A:194 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 8 -6 B 8 0 -6 12 0 C 6 6 0 10 6 D -8 -12 -10 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 8 -6 B 8 0 -6 12 0 C 6 6 0 10 6 D -8 -12 -10 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 8 -6 B 8 0 -6 12 0 C 6 6 0 10 6 D -8 -12 -10 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) D C E B A (8) B A E D C (8) B A E C D (7) C D E B A (6) B E A D C (6) A B E C D (6) D C E A B (5) C D A B E (5) D E C B A (4) C D E A B (4) A B E D C (4) C B A E D (3) A E B D C (3) E D B A C (2) E B D A C (2) E A B D C (2) D E A B C (2) C D A E B (2) E D A B C (1) D E C A B (1) D E B C A (1) D E B A C (1) D C A E B (1) C B E D A (1) C B D E A (1) C B A D E (1) C A B D E (1) B A C E D (1) A D E B C (1) Total count = 100 A B C D E A 0 -24 12 6 -14 B 24 0 14 12 -12 C -12 -14 0 -24 -22 D -6 -12 24 0 -12 E 14 12 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 12 6 -14 B 24 0 14 12 -12 C -12 -14 0 -24 -22 D -6 -12 24 0 -12 E 14 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=22 E=17 A=14 so A is eliminated. Round 2 votes counts: B=32 D=24 C=24 E=20 so E is eliminated. Round 3 votes counts: B=49 D=27 C=24 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:230 B:219 D:197 A:190 C:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 12 6 -14 B 24 0 14 12 -12 C -12 -14 0 -24 -22 D -6 -12 24 0 -12 E 14 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 12 6 -14 B 24 0 14 12 -12 C -12 -14 0 -24 -22 D -6 -12 24 0 -12 E 14 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 12 6 -14 B 24 0 14 12 -12 C -12 -14 0 -24 -22 D -6 -12 24 0 -12 E 14 12 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3904: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (14) C D E A B (11) B A E D C (8) D C E B A (7) C D A E B (5) C A E B D (5) A E B C D (4) A C B E D (4) A B C E D (4) D E B C A (3) D B E A C (3) B D A E C (3) D E C B A (2) D C B E A (2) C E D A B (2) C D A B E (2) C A B D E (2) B E D A C (2) E D C B A (1) E D B A C (1) E B D A C (1) E B A D C (1) E A B C D (1) D C B A E (1) D B E C A (1) D B A E C (1) D B A C E (1) C E A D B (1) C D E B A (1) C A B E D (1) B D A C E (1) B A D E C (1) A E C B D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 18 6 -2 20 B -18 0 0 10 4 C -6 0 0 20 4 D 2 -10 -20 0 -4 E -20 -4 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.071429 D: 0.214286 E: 0.000000 Sum of squares = 0.561224489788 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.785714 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 -2 20 B -18 0 0 10 4 C -6 0 0 20 4 D 2 -10 -20 0 -4 E -20 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.071429 D: 0.214286 E: 0.000000 Sum of squares = 0.561224489794 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.785714 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=29 D=21 B=15 E=5 so E is eliminated. Round 2 votes counts: C=30 A=30 D=23 B=17 so B is eliminated. Round 3 votes counts: A=40 D=30 C=30 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:209 B:198 E:188 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 -2 20 B -18 0 0 10 4 C -6 0 0 20 4 D 2 -10 -20 0 -4 E -20 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.071429 D: 0.214286 E: 0.000000 Sum of squares = 0.561224489794 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.785714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 -2 20 B -18 0 0 10 4 C -6 0 0 20 4 D 2 -10 -20 0 -4 E -20 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.071429 D: 0.214286 E: 0.000000 Sum of squares = 0.561224489794 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.785714 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 -2 20 B -18 0 0 10 4 C -6 0 0 20 4 D 2 -10 -20 0 -4 E -20 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.071429 D: 0.214286 E: 0.000000 Sum of squares = 0.561224489794 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.785714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3905: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (15) A C E B D (12) D C A B E (9) C A E B D (9) E B A C D (5) D B E C A (5) D A C E B (5) C A D E B (5) E B C A D (3) C B E A D (3) B D E A C (3) E A C B D (2) E A B C D (2) D C A E B (2) D A E C B (2) B E A C D (2) B C E D A (2) A D C E B (2) E B D A C (1) D C B A E (1) C D B A E (1) C B D E A (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E C A (1) A E C B D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 10 0 4 B -8 0 -16 0 -6 C -10 16 0 4 10 D 0 0 -4 0 8 E -4 6 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.596787 B: 0.000000 C: 0.000000 D: 0.403213 E: 0.000000 Sum of squares = 0.518735555674 Cumulative probabilities = A: 0.596787 B: 0.596787 C: 0.596787 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 0 4 B -8 0 -16 0 -6 C -10 16 0 4 10 D 0 0 -4 0 8 E -4 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=20 A=17 E=13 B=11 so B is eliminated. Round 2 votes counts: D=43 C=22 E=18 A=17 so A is eliminated. Round 3 votes counts: D=45 C=36 E=19 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:210 D:202 E:192 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 0 4 B -8 0 -16 0 -6 C -10 16 0 4 10 D 0 0 -4 0 8 E -4 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 4 B -8 0 -16 0 -6 C -10 16 0 4 10 D 0 0 -4 0 8 E -4 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 4 B -8 0 -16 0 -6 C -10 16 0 4 10 D 0 0 -4 0 8 E -4 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3906: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (15) A E C D B (15) B D A C E (7) B E D C A (5) E C A D B (4) E B C D A (4) C E D B A (4) A E B C D (4) E C D B A (3) E A C D B (3) E A C B D (3) B D C A E (3) A D B C E (3) A C D E B (3) A B D C E (3) D C B E A (2) B A D C E (2) A B E D C (2) E C B A D (1) E B A C D (1) E A B C D (1) D C E B A (1) D C A B E (1) D B C E A (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E B A (1) C D B E A (1) B D A E C (1) A E C B D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 6 -4 -2 B 6 0 6 6 -2 C -6 -6 0 2 4 D 4 -6 -2 0 -6 E 2 2 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888883 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -6 6 -4 -2 B 6 0 6 6 -2 C -6 -6 0 2 4 D 4 -6 -2 0 -6 E 2 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=33 A=33 E=20 D=7 C=7 so D is eliminated. Round 2 votes counts: B=35 A=34 E=20 C=11 so C is eliminated. Round 3 votes counts: B=38 A=35 E=27 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:203 A:197 C:197 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -4 -2 B 6 0 6 6 -2 C -6 -6 0 2 4 D 4 -6 -2 0 -6 E 2 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -4 -2 B 6 0 6 6 -2 C -6 -6 0 2 4 D 4 -6 -2 0 -6 E 2 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -4 -2 B 6 0 6 6 -2 C -6 -6 0 2 4 D 4 -6 -2 0 -6 E 2 2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888873 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3907: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) C E B A D (6) D A E B C (5) B D A E C (5) E A B C D (4) D C A B E (4) D A B C E (4) C E A D B (4) C E A B D (4) A E D B C (4) D B C A E (3) B E D A C (3) B D C E A (3) A E D C B (3) E C A B D (2) E A B D C (2) D C B A E (2) D A B E C (2) C D B A E (2) C B E D A (2) C A E D B (2) B E C A D (2) B E A D C (2) B C E D A (2) B C D E A (2) A D E B C (2) E B C A D (1) E A C D B (1) E A C B D (1) D C A E B (1) D A C E B (1) C D E B A (1) C D A E B (1) C B E A D (1) C B D E A (1) B D E C A (1) B D E A C (1) B D C A E (1) B D A C E (1) A E C D B (1) A E B D C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 4 -10 8 B -2 0 16 -4 2 C -4 -16 0 -20 -2 D 10 4 20 0 4 E -8 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -10 8 B -2 0 16 -4 2 C -4 -16 0 -20 -2 D 10 4 20 0 4 E -8 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=24 B=23 A=13 E=11 so E is eliminated. Round 2 votes counts: D=29 C=26 B=24 A=21 so A is eliminated. Round 3 votes counts: D=40 B=31 C=29 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:206 A:202 E:194 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -10 8 B -2 0 16 -4 2 C -4 -16 0 -20 -2 D 10 4 20 0 4 E -8 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -10 8 B -2 0 16 -4 2 C -4 -16 0 -20 -2 D 10 4 20 0 4 E -8 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -10 8 B -2 0 16 -4 2 C -4 -16 0 -20 -2 D 10 4 20 0 4 E -8 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3908: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (15) A E D C B (10) E D A B C (5) A C D E B (5) E D B A C (4) B E D C A (4) A E D B C (4) C B A D E (3) C A B D E (3) B C D A E (3) B A C E D (3) E D B C A (2) E D A C B (2) E A D C B (2) D E C B A (2) D E B C A (2) C D A E B (2) C B D E A (2) C B D A E (2) B D E C A (2) B D C E A (2) B C A D E (2) B A E C D (2) A C E D B (2) A C B D E (2) E B A D C (1) D C B E A (1) C D E A B (1) C D B E A (1) B E D A C (1) B E A D C (1) B C A E D (1) B A C D E (1) A E C D B (1) A E B D C (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 0 -6 0 B 14 0 16 8 8 C 0 -16 0 6 4 D 6 -8 -6 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -6 0 B 14 0 16 8 8 C 0 -16 0 6 4 D 6 -8 -6 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=28 E=16 C=14 D=5 so D is eliminated. Round 2 votes counts: B=37 A=28 E=20 C=15 so C is eliminated. Round 3 votes counts: B=46 A=33 E=21 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:197 D:197 E:193 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -6 0 B 14 0 16 8 8 C 0 -16 0 6 4 D 6 -8 -6 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -6 0 B 14 0 16 8 8 C 0 -16 0 6 4 D 6 -8 -6 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -6 0 B 14 0 16 8 8 C 0 -16 0 6 4 D 6 -8 -6 0 2 E 0 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3909: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) B C D A E (10) E A D B C (8) E A D C B (7) C B D E A (6) E A B C D (5) E B C A D (4) D C B A E (4) B C E D A (4) A E D B C (4) A D E C B (4) C B E D A (3) E C B A D (2) E A C B D (2) E A B D C (2) D E A C B (2) D A E C B (2) D A C B E (2) D A B C E (2) C D B A E (2) B C E A D (2) B C D E A (2) A E D C B (2) E D C B A (1) E D A C B (1) E C B D A (1) E B A C D (1) D C A B E (1) D A C E B (1) B E C A D (1) A E B C D (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -6 -8 -8 B 6 0 -8 12 0 C 6 8 0 12 0 D 8 -12 -12 0 -2 E 8 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.467931 D: 0.000000 E: 0.532069 Sum of squares = 0.502056836681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.467931 D: 0.467931 E: 1.000000 A B C D E A 0 -6 -6 -8 -8 B 6 0 -8 12 0 C 6 8 0 12 0 D 8 -12 -12 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=21 B=19 D=14 A=12 so A is eliminated. Round 2 votes counts: E=41 C=21 D=19 B=19 so D is eliminated. Round 3 votes counts: E=49 C=30 B=21 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:205 E:205 D:191 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -8 -8 B 6 0 -8 12 0 C 6 8 0 12 0 D 8 -12 -12 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -8 -8 B 6 0 -8 12 0 C 6 8 0 12 0 D 8 -12 -12 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -8 -8 B 6 0 -8 12 0 C 6 8 0 12 0 D 8 -12 -12 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3910: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) E A B C D (11) B C D E A (10) D C B A E (8) D C A B E (8) A D E C B (8) C B D E A (6) B E C D A (6) A E B D C (5) E A B D C (4) A E D C B (4) A D C E B (4) D A C E B (3) C D B A E (3) B E C A D (3) C D B E A (2) B C E D A (2) B E A C D (1) A E D B C (1) Total count = 100 A B C D E A 0 -4 4 4 -12 B 4 0 8 18 -2 C -4 -8 0 10 -8 D -4 -18 -10 0 4 E 12 2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.083333 E: 0.750000 Sum of squares = 0.597222222189 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.250000 E: 1.000000 A B C D E A 0 -4 4 4 -12 B 4 0 8 18 -2 C -4 -8 0 10 -8 D -4 -18 -10 0 4 E 12 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.083333 E: 0.750000 Sum of squares = 0.597222222202 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.250000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=22 A=22 D=19 C=11 so C is eliminated. Round 2 votes counts: B=28 E=26 D=24 A=22 so A is eliminated. Round 3 votes counts: E=36 D=36 B=28 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:214 E:209 A:196 C:195 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 4 -12 B 4 0 8 18 -2 C -4 -8 0 10 -8 D -4 -18 -10 0 4 E 12 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.083333 E: 0.750000 Sum of squares = 0.597222222202 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.250000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 -12 B 4 0 8 18 -2 C -4 -8 0 10 -8 D -4 -18 -10 0 4 E 12 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.083333 E: 0.750000 Sum of squares = 0.597222222202 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 -12 B 4 0 8 18 -2 C -4 -8 0 10 -8 D -4 -18 -10 0 4 E 12 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.083333 E: 0.750000 Sum of squares = 0.597222222202 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3911: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (14) E C D A B (12) C E B D A (9) C E D A B (7) E A D C B (5) C B D A E (5) B C A D E (5) A D B E C (5) E D A C B (4) D A B C E (4) B C D A E (4) E C B A D (3) E C A D B (3) C E B A D (2) C B E D A (2) C B E A D (2) B D A C E (2) E A D B C (1) E A C B D (1) D A E B C (1) C E D B A (1) C D E A B (1) C D B A E (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D E C (1) B A C D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -14 -6 -10 B 10 0 -16 8 -2 C 14 16 0 20 20 D 6 -8 -20 0 -10 E 10 2 -20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -6 -10 B 10 0 -16 8 -2 C 14 16 0 20 20 D 6 -8 -20 0 -10 E 10 2 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=30 B=30 E=29 A=6 D=5 so D is eliminated. Round 2 votes counts: C=30 B=30 E=29 A=11 so A is eliminated. Round 3 votes counts: B=39 E=31 C=30 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:235 E:201 B:200 D:184 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -6 -10 B 10 0 -16 8 -2 C 14 16 0 20 20 D 6 -8 -20 0 -10 E 10 2 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -6 -10 B 10 0 -16 8 -2 C 14 16 0 20 20 D 6 -8 -20 0 -10 E 10 2 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -6 -10 B 10 0 -16 8 -2 C 14 16 0 20 20 D 6 -8 -20 0 -10 E 10 2 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3912: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) A B C E D (7) D E C B A (6) C A B D E (6) A C B E D (6) C D E A B (5) B A E C D (5) E D C A B (4) E D B A C (4) C D A E B (4) C A B E D (4) B A E D C (4) B A C E D (4) D E B C A (3) D C E A B (3) B A C D E (3) E B D A C (2) E B A D C (2) E A B D C (2) B E A D C (2) A B E C D (2) E D C B A (1) D C E B A (1) D C B E A (1) D B C E A (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D A C (1) B D E A C (1) B C A D E (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 14 -6 6 2 B -14 0 -10 8 2 C 6 10 0 2 0 D -6 -8 -2 0 -2 E -2 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.532139 D: 0.000000 E: 0.467861 Sum of squares = 0.50206578224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.532139 D: 0.532139 E: 1.000000 A B C D E A 0 14 -6 6 2 B -14 0 -10 8 2 C 6 10 0 2 0 D -6 -8 -2 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 B=22 A=16 E=15 so E is eliminated. Round 2 votes counts: D=34 B=26 C=22 A=18 so A is eliminated. Round 3 votes counts: B=37 D=34 C=29 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:209 A:208 E:199 B:193 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 6 2 B -14 0 -10 8 2 C 6 10 0 2 0 D -6 -8 -2 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 6 2 B -14 0 -10 8 2 C 6 10 0 2 0 D -6 -8 -2 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 6 2 B -14 0 -10 8 2 C 6 10 0 2 0 D -6 -8 -2 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3913: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) B D E C A (6) C D B A E (5) C D A B E (5) B D C E A (5) A E D B C (5) A C E D B (5) D B C E A (4) C A D E B (4) C A D B E (4) D B E C A (3) C A E D B (3) A E C B D (3) E B D A C (2) E B A D C (2) E A D B C (2) D A C E B (2) D A C B E (2) C A E B D (2) B E D A C (2) B D E A C (2) B C D E A (2) A E D C B (2) A E B C D (2) A C E B D (2) E D B A C (1) E A B C D (1) D E B A C (1) D C B E A (1) D B E A C (1) D B A E C (1) D A E B C (1) C B E A D (1) C B D E A (1) C B A D E (1) B E D C A (1) B E C D A (1) B C E D A (1) A E C D B (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 12 0 0 4 B -12 0 10 -8 -2 C 0 -10 0 -10 2 D 0 8 10 0 4 E -4 2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.290205 B: 0.000000 C: 0.000000 D: 0.709795 E: 0.000000 Sum of squares = 0.588028195478 Cumulative probabilities = A: 0.290205 B: 0.290205 C: 0.290205 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 0 4 B -12 0 10 -8 -2 C 0 -10 0 -10 2 D 0 8 10 0 4 E -4 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=22 B=20 E=16 D=16 so E is eliminated. Round 2 votes counts: A=33 C=26 B=24 D=17 so D is eliminated. Round 3 votes counts: A=38 B=35 C=27 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:208 E:196 B:194 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 0 4 B -12 0 10 -8 -2 C 0 -10 0 -10 2 D 0 8 10 0 4 E -4 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 0 4 B -12 0 10 -8 -2 C 0 -10 0 -10 2 D 0 8 10 0 4 E -4 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 0 4 B -12 0 10 -8 -2 C 0 -10 0 -10 2 D 0 8 10 0 4 E -4 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3914: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) C B A E D (10) E D A B C (9) D E A B C (9) C A B E D (8) B C A E D (7) B C A D E (7) B C D E A (4) D E B C A (3) D B E C A (3) A E C D B (3) A C B E D (3) D E A C B (2) C B D E A (2) C A E D B (2) B D C E A (2) B C D A E (2) A E D C B (2) A C E B D (2) E D A C B (1) E A D C B (1) E A D B C (1) D E C A B (1) C B D A E (1) C B A D E (1) C A E B D (1) B D E C A (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -10 -2 0 B 8 0 20 4 4 C 10 -20 0 6 4 D 2 -4 -6 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -2 0 B 8 0 20 4 4 C 10 -20 0 6 4 D 2 -4 -6 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 B=23 E=12 A=11 so A is eliminated. Round 2 votes counts: C=30 D=29 B=24 E=17 so E is eliminated. Round 3 votes counts: D=43 C=33 B=24 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:200 E:197 D:195 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -2 0 B 8 0 20 4 4 C 10 -20 0 6 4 D 2 -4 -6 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -2 0 B 8 0 20 4 4 C 10 -20 0 6 4 D 2 -4 -6 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -2 0 B 8 0 20 4 4 C 10 -20 0 6 4 D 2 -4 -6 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3915: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) A E D B C (5) D E A C B (4) C B D E A (4) B A C E D (4) A E B D C (4) E D A C B (3) E A B C D (3) D E C B A (3) D C E A B (3) D C B E A (3) D A C B E (3) C D B E A (3) B A E C D (3) A E B C D (3) A B E C D (3) E D C B A (2) E C B A D (2) E B C A D (2) D C B A E (2) D A E C B (2) D A E B C (2) C D B A E (2) C B E D A (2) C B E A D (2) C B D A E (2) B C A D E (2) B A C D E (2) A D E B C (2) A D B E C (2) E C B D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D A C E B (1) D A B C E (1) C E D B A (1) C E B D A (1) B C E A D (1) B C A E D (1) A D B C E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 6 -6 0 B 4 0 -8 -8 -8 C -6 8 0 -10 0 D 6 8 10 0 8 E 0 8 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -6 0 B 4 0 -8 -8 -8 C -6 8 0 -10 0 D 6 8 10 0 8 E 0 8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=23 C=17 E=15 B=13 so B is eliminated. Round 2 votes counts: D=32 A=32 C=21 E=15 so E is eliminated. Round 3 votes counts: D=37 A=37 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:200 A:198 C:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -6 0 B 4 0 -8 -8 -8 C -6 8 0 -10 0 D 6 8 10 0 8 E 0 8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -6 0 B 4 0 -8 -8 -8 C -6 8 0 -10 0 D 6 8 10 0 8 E 0 8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -6 0 B 4 0 -8 -8 -8 C -6 8 0 -10 0 D 6 8 10 0 8 E 0 8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3916: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (6) B E C D A (6) B C D E A (6) E C D B A (5) A E D C B (5) A E B C D (5) E C B D A (4) D C A B E (4) C D B E A (4) B A D C E (4) A E B D C (4) A B D C E (4) B E A C D (3) B D C A E (3) A D E C B (3) E C A D B (2) E B C A D (2) E A C B D (2) E A B C D (2) D C B E A (2) D B C A E (2) C D E A B (2) C B D E A (2) A D C E B (2) A D B C E (2) A B E D C (2) E C D A B (1) E A C D B (1) D C E B A (1) D C A E B (1) D A C E B (1) D A B C E (1) C E D B A (1) C D E B A (1) B D C E A (1) B A E D C (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -12 -8 4 B 10 0 0 6 10 C 12 0 0 -2 0 D 8 -6 2 0 8 E -4 -10 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.670016 C: 0.329984 D: 0.000000 E: 0.000000 Sum of squares = 0.557811092248 Cumulative probabilities = A: 0.000000 B: 0.670016 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 4 B 10 0 0 6 10 C 12 0 0 -2 0 D 8 -6 2 0 8 E -4 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 E=19 D=18 C=10 so C is eliminated. Round 2 votes counts: A=28 B=27 D=25 E=20 so E is eliminated. Round 3 votes counts: A=35 B=33 D=32 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:206 C:205 E:189 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 4 B 10 0 0 6 10 C 12 0 0 -2 0 D 8 -6 2 0 8 E -4 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 4 B 10 0 0 6 10 C 12 0 0 -2 0 D 8 -6 2 0 8 E -4 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 4 B 10 0 0 6 10 C 12 0 0 -2 0 D 8 -6 2 0 8 E -4 -10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999929 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3917: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (13) D C A E B (10) B E A D C (10) C D A E B (7) A E B C D (7) D C B E A (6) D C B A E (6) C A D E B (6) A E C B D (6) E B A C D (4) D C A B E (4) D B C E A (4) B E D A C (4) A C E B D (4) E A B C D (2) D B E A C (2) D B E C A (1) C A E D B (1) B E C D A (1) B D E A C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 8 8 4 B 4 0 -2 4 4 C -8 2 0 4 -2 D -8 -4 -4 0 -4 E -4 -4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428571 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 8 4 B 4 0 -2 4 4 C -8 2 0 4 -2 D -8 -4 -4 0 -4 E -4 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428555 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=29 A=18 C=14 E=6 so E is eliminated. Round 2 votes counts: D=33 B=33 A=20 C=14 so C is eliminated. Round 3 votes counts: D=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:208 B:205 E:199 C:198 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 8 4 B 4 0 -2 4 4 C -8 2 0 4 -2 D -8 -4 -4 0 -4 E -4 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428555 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 8 4 B 4 0 -2 4 4 C -8 2 0 4 -2 D -8 -4 -4 0 -4 E -4 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428555 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 8 4 B 4 0 -2 4 4 C -8 2 0 4 -2 D -8 -4 -4 0 -4 E -4 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428555 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3918: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) D A B C E (11) D C A B E (9) E B A C D (8) C D A B E (8) E C B A D (5) B A E D C (5) B A D E C (5) D B A E C (4) C E A B D (3) A B D E C (3) E D B A C (2) E C D B A (2) E B D A C (2) E B C A D (2) D A C B E (2) D A B E C (2) C E D A B (2) C D A E B (2) A D B C E (2) A B D C E (2) A B C E D (2) E D B C A (1) C E D B A (1) C E B A D (1) C D E A B (1) C A B E D (1) B E A D C (1) Total count = 100 A B C D E A 0 0 24 2 16 B 0 0 26 2 14 C -24 -26 0 -24 -6 D -2 -2 24 0 2 E -16 -14 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.524756 B: 0.475244 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501225637307 Cumulative probabilities = A: 0.524756 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 24 2 16 B 0 0 26 2 14 C -24 -26 0 -24 -6 D -2 -2 24 0 2 E -16 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999990398 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=28 C=19 B=11 A=9 so A is eliminated. Round 2 votes counts: E=33 D=30 C=19 B=18 so B is eliminated. Round 3 votes counts: D=40 E=39 C=21 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:221 B:221 D:211 E:187 C:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 24 2 16 B 0 0 26 2 14 C -24 -26 0 -24 -6 D -2 -2 24 0 2 E -16 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999990398 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 24 2 16 B 0 0 26 2 14 C -24 -26 0 -24 -6 D -2 -2 24 0 2 E -16 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999990398 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 24 2 16 B 0 0 26 2 14 C -24 -26 0 -24 -6 D -2 -2 24 0 2 E -16 -14 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999990398 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3919: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) B A C D E (7) B D E A C (6) A C B D E (6) E D B A C (5) E B D A C (4) D E B C A (4) D B E A C (4) C A E B D (4) C A B D E (4) E D C B A (3) E D C A B (3) E C D A B (3) E C A D B (3) E C A B D (3) D B A C E (3) C A D E B (3) C A B E D (3) B D A C E (3) B D A E C (2) A C B E D (2) E B C D A (1) E B A D C (1) D E B A C (1) D C A B E (1) C E A D B (1) C D E A B (1) C A D B E (1) B E D A C (1) B E A D C (1) B E A C D (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -24 -2 -16 -18 B 24 0 18 2 -6 C 2 -18 0 -12 -18 D 16 -2 12 0 -6 E 18 6 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 -2 -16 -18 B 24 0 18 2 -6 C 2 -18 0 -12 -18 D 16 -2 12 0 -6 E 18 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=23 C=17 D=13 A=8 so A is eliminated. Round 2 votes counts: E=39 C=25 B=23 D=13 so D is eliminated. Round 3 votes counts: E=44 B=30 C=26 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:219 D:210 C:177 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 -2 -16 -18 B 24 0 18 2 -6 C 2 -18 0 -12 -18 D 16 -2 12 0 -6 E 18 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -2 -16 -18 B 24 0 18 2 -6 C 2 -18 0 -12 -18 D 16 -2 12 0 -6 E 18 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -2 -16 -18 B 24 0 18 2 -6 C 2 -18 0 -12 -18 D 16 -2 12 0 -6 E 18 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3920: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D A C (9) E B D C A (8) E B A C D (8) D C A B E (8) C A D E B (7) C D A E B (6) C A D B E (5) E B A D C (3) B A E C D (3) E B C A D (2) D C E A B (2) D B C A E (2) C D A B E (2) B E D A C (2) B E A C D (2) E C B A D (1) E C A D B (1) E C A B D (1) E B C D A (1) D E C A B (1) D C A E B (1) D B E C A (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E A C (1) B D A C E (1) B A E D C (1) B A D C E (1) B A C D E (1) A C E B D (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 8 10 B -2 0 0 2 -6 C 2 0 0 14 4 D -8 -2 -14 0 6 E -10 6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.219277 C: 0.780723 D: 0.000000 E: 0.000000 Sum of squares = 0.657610812314 Cumulative probabilities = A: 0.000000 B: 0.219277 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 8 10 B -2 0 0 2 -6 C 2 0 0 14 4 D -8 -2 -14 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000000581 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=21 A=16 D=15 B=14 so B is eliminated. Round 2 votes counts: E=40 A=22 C=21 D=17 so D is eliminated. Round 3 votes counts: E=43 C=34 A=23 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:209 B:197 E:193 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 8 10 B -2 0 0 2 -6 C 2 0 0 14 4 D -8 -2 -14 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000000581 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 10 B -2 0 0 2 -6 C 2 0 0 14 4 D -8 -2 -14 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000000581 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 10 B -2 0 0 2 -6 C 2 0 0 14 4 D -8 -2 -14 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000000581 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3921: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (11) D C A E B (8) C D B E A (8) C B D E A (8) A E D B C (8) A D E C B (7) B E C A D (6) B E A C D (6) E B A C D (5) D C B A E (5) D C A B E (5) D A C E B (5) E A B C D (4) C D B A E (2) C D A E B (2) B C D E A (2) A E D C B (2) A D E B C (2) E B A D C (1) D A C B E (1) C B E D A (1) B C E D A (1) Total count = 100 A B C D E A 0 10 4 4 16 B -10 0 -8 -10 -10 C -4 8 0 -10 -4 D -4 10 10 0 10 E -16 10 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 4 16 B -10 0 -8 -10 -10 C -4 8 0 -10 -4 D -4 10 10 0 10 E -16 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 C=21 B=15 E=10 so E is eliminated. Round 2 votes counts: A=34 D=24 C=21 B=21 so C is eliminated. Round 3 votes counts: D=36 A=34 B=30 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:213 C:195 E:194 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 4 16 B -10 0 -8 -10 -10 C -4 8 0 -10 -4 D -4 10 10 0 10 E -16 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 4 16 B -10 0 -8 -10 -10 C -4 8 0 -10 -4 D -4 10 10 0 10 E -16 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 4 16 B -10 0 -8 -10 -10 C -4 8 0 -10 -4 D -4 10 10 0 10 E -16 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3922: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) E B A D C (11) D C A B E (10) B A E C D (9) E B A C D (8) D C A E B (8) D C E A B (6) A B C D E (5) D C E B A (4) B E A C D (4) E B D A C (3) A C B D E (3) A B E C D (3) E D C B A (2) C A D B E (2) B A C D E (2) A C D B E (2) E D B C A (1) E B D C A (1) D E C B A (1) B E A D C (1) B A C E D (1) A C E B D (1) Total count = 100 A B C D E A 0 4 6 4 16 B -4 0 -2 4 8 C -6 2 0 4 12 D -4 -4 -4 0 10 E -16 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 16 B -4 0 -2 4 8 C -6 2 0 4 12 D -4 -4 -4 0 10 E -16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=26 B=17 C=14 A=14 so C is eliminated. Round 2 votes counts: D=41 E=26 B=17 A=16 so A is eliminated. Round 3 votes counts: D=45 B=28 E=27 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:215 C:206 B:203 D:199 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 16 B -4 0 -2 4 8 C -6 2 0 4 12 D -4 -4 -4 0 10 E -16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 16 B -4 0 -2 4 8 C -6 2 0 4 12 D -4 -4 -4 0 10 E -16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 16 B -4 0 -2 4 8 C -6 2 0 4 12 D -4 -4 -4 0 10 E -16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3923: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (15) D B E A C (11) B A C D E (10) D E B A C (9) C A E B D (8) B D A C E (8) C A B E D (7) D E B C A (4) B D A E C (4) A B C E D (4) E C A D B (3) D E C B A (3) A C B E D (3) E C D A B (2) C E A D B (2) A B C D E (2) E D B C A (1) E B D A C (1) C E A B D (1) C A E D B (1) B A D C E (1) Total count = 100 A B C D E A 0 -4 6 -16 -4 B 4 0 10 -2 0 C -6 -10 0 -14 -6 D 16 2 14 0 4 E 4 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -16 -4 B 4 0 10 -2 0 C -6 -10 0 -14 -6 D 16 2 14 0 4 E 4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 E=22 C=19 A=9 so A is eliminated. Round 2 votes counts: B=29 D=27 E=22 C=22 so E is eliminated. Round 3 votes counts: D=43 B=30 C=27 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:206 E:203 A:191 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -16 -4 B 4 0 10 -2 0 C -6 -10 0 -14 -6 D 16 2 14 0 4 E 4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -16 -4 B 4 0 10 -2 0 C -6 -10 0 -14 -6 D 16 2 14 0 4 E 4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -16 -4 B 4 0 10 -2 0 C -6 -10 0 -14 -6 D 16 2 14 0 4 E 4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3924: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (6) D C E A B (5) C D E B A (5) C D A E B (4) B E D C A (4) A B C E D (4) A B C D E (4) E D C B A (3) C D E A B (3) C B D A E (3) C A B D E (3) B E A D C (3) B A E C D (3) A B E C D (3) E D C A B (2) E B D C A (2) E B A D C (2) E A D C B (2) E A D B C (2) D E C B A (2) D C B E A (2) C A D B E (2) B D C E A (2) B C D E A (2) A E D C B (2) A C D E B (2) A C D B E (2) A C B D E (2) E D A C B (1) E B D A C (1) D E C A B (1) D E A C B (1) D C E B A (1) D B E C A (1) D A C E B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D E A (1) C B A D E (1) B E D A C (1) B E C D A (1) B C D A E (1) B C A D E (1) B A E D C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 10 -10 -6 -2 B -10 0 -10 4 2 C 10 10 0 0 10 D 6 -4 0 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.640000 D: 0.360000 E: 0.000000 Sum of squares = 0.539199984722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.640000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -10 -6 -2 B -10 0 -10 4 2 C 10 10 0 0 10 D 6 -4 0 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=25 B=19 E=15 D=14 so D is eliminated. Round 2 votes counts: C=33 A=28 B=20 E=19 so E is eliminated. Round 3 votes counts: C=41 A=34 B=25 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:207 A:196 B:193 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 -6 -2 B -10 0 -10 4 2 C 10 10 0 0 10 D 6 -4 0 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 -6 -2 B -10 0 -10 4 2 C 10 10 0 0 10 D 6 -4 0 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 -6 -2 B -10 0 -10 4 2 C 10 10 0 0 10 D 6 -4 0 0 12 E 2 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3925: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) A C D B E (7) C B A E D (5) C A B E D (5) B C E A D (5) A D C E B (5) A C B D E (5) C B E A D (4) B E C D A (4) E D B C A (3) E D B A C (3) E B D C A (3) E B C D A (3) D B E C A (3) D B E A C (3) B C E D A (3) A C E B D (3) E B C A D (2) D E B C A (2) D A E C B (2) D A C E B (2) C B A D E (2) C A E B D (2) E D A C B (1) E C B A D (1) E A D C B (1) E A C D B (1) E A B C D (1) D E A C B (1) D E A B C (1) D B A E C (1) D A E B C (1) D A C B E (1) C B D A E (1) B E D C A (1) B C D A E (1) A E D C B (1) A E C B D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 0 6 -6 B 14 0 -6 4 4 C 0 6 0 14 6 D -6 -4 -14 0 -8 E 6 -4 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.167934 B: 0.000000 C: 0.832066 D: 0.000000 E: 0.000000 Sum of squares = 0.720535332465 Cumulative probabilities = A: 0.167934 B: 0.167934 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 6 -6 B 14 0 -6 4 4 C 0 6 0 14 6 D -6 -4 -14 0 -8 E 6 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.700000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000000774 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=19 C=19 B=14 so B is eliminated. Round 2 votes counts: C=28 E=24 D=24 A=24 so E is eliminated. Round 3 votes counts: C=38 D=35 A=27 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:208 E:202 A:193 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 0 6 -6 B 14 0 -6 4 4 C 0 6 0 14 6 D -6 -4 -14 0 -8 E 6 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.700000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000000774 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 6 -6 B 14 0 -6 4 4 C 0 6 0 14 6 D -6 -4 -14 0 -8 E 6 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.700000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000000774 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 6 -6 B 14 0 -6 4 4 C 0 6 0 14 6 D -6 -4 -14 0 -8 E 6 -4 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.700000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000000774 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3926: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) A E D C B (8) C B D E A (7) B C E D A (7) A D C E B (6) D E C A B (5) A E B D C (5) E D C A B (4) C D B E A (4) A D E C B (4) E D C B A (3) E D A C B (3) B E D C A (3) B E A D C (3) B A C E D (3) C D E B A (2) B A E D C (2) A C D E B (2) A B E D C (2) E D B A C (1) E A D B C (1) E A B D C (1) D C E A B (1) D C A E B (1) D A C E B (1) C B D A E (1) C B A D E (1) C A D E B (1) B E A C D (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E D B C (1) A D C B E (1) A C D B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -2 -8 B 0 0 -12 0 0 C 0 12 0 -12 4 D 2 0 12 0 -2 E 8 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.222222 E: 0.666667 Sum of squares = 0.506172839482 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.333333 E: 1.000000 A B C D E A 0 0 0 -2 -8 B 0 0 -12 0 0 C 0 12 0 -12 4 D 2 0 12 0 -2 E 8 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.222222 E: 0.666667 Sum of squares = 0.506172839527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=31 C=16 E=13 D=8 so D is eliminated. Round 2 votes counts: A=33 B=31 E=18 C=18 so E is eliminated. Round 3 votes counts: A=38 B=32 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:206 E:203 C:202 A:195 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -2 -8 B 0 0 -12 0 0 C 0 12 0 -12 4 D 2 0 12 0 -2 E 8 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.222222 E: 0.666667 Sum of squares = 0.506172839527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.333333 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -8 B 0 0 -12 0 0 C 0 12 0 -12 4 D 2 0 12 0 -2 E 8 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.222222 E: 0.666667 Sum of squares = 0.506172839527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -8 B 0 0 -12 0 0 C 0 12 0 -12 4 D 2 0 12 0 -2 E 8 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.222222 E: 0.666667 Sum of squares = 0.506172839527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.111111 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3927: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (6) B E A C D (6) B C E D A (6) B E A D C (5) B C D A E (5) B A D C E (5) A D C B E (5) E A D C B (4) C D A B E (4) B E C D A (4) E B A D C (3) D C A E B (3) D A C E B (3) C D E A B (3) B E C A D (3) E D C A B (2) E C B D A (2) C E D A B (2) C D A E B (2) C B D A E (2) B C A D E (2) B A E D C (2) A D C E B (2) A D B C E (2) A B D C E (2) E D A C B (1) E C D B A (1) E C D A B (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) D E C A B (1) C D B A E (1) C B D E A (1) C A D B E (1) B A D E C (1) B A C D E (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -4 -2 -10 B 16 0 16 20 18 C 4 -16 0 8 4 D 2 -20 -8 0 -4 E 10 -18 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -2 -10 B 16 0 16 20 18 C 4 -16 0 8 4 D 2 -20 -8 0 -4 E 10 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=24 C=16 A=13 D=7 so D is eliminated. Round 2 votes counts: B=40 E=25 C=19 A=16 so A is eliminated. Round 3 votes counts: B=45 C=29 E=26 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:235 C:200 E:196 D:185 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 -2 -10 B 16 0 16 20 18 C 4 -16 0 8 4 D 2 -20 -8 0 -4 E 10 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -2 -10 B 16 0 16 20 18 C 4 -16 0 8 4 D 2 -20 -8 0 -4 E 10 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -2 -10 B 16 0 16 20 18 C 4 -16 0 8 4 D 2 -20 -8 0 -4 E 10 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3928: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A E B C D (8) A B E D C (7) E A B C D (5) B D C A E (5) C D E A B (4) B A E D C (4) E C D A B (3) E A C D B (3) D C E B A (3) D C B A E (3) C D E B A (3) B E A C D (3) A E C D B (3) E B C D A (2) E B A C D (2) D C E A B (2) D C A E B (2) D B C A E (2) C E D A B (2) C D B E A (2) B E C D A (2) B C D E A (2) A E B D C (2) A D B C E (2) E C D B A (1) E B C A D (1) E A C B D (1) D C A B E (1) D A C E B (1) D A C B E (1) C B D E A (1) B E D C A (1) B E A D C (1) B D C E A (1) B D A C E (1) B A E C D (1) B A D C E (1) A D C E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 -6 -6 B 0 0 10 6 2 C 2 -10 0 -2 -2 D 6 -6 2 0 -4 E 6 -2 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.177108 B: 0.822892 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.708518179454 Cumulative probabilities = A: 0.177108 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -6 -6 B 0 0 10 6 2 C 2 -10 0 -2 -2 D 6 -6 2 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000194 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 B=22 E=18 C=12 so C is eliminated. Round 2 votes counts: D=32 A=25 B=23 E=20 so E is eliminated. Round 3 votes counts: D=38 A=34 B=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:209 E:205 D:199 C:194 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -6 -6 B 0 0 10 6 2 C 2 -10 0 -2 -2 D 6 -6 2 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000194 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -6 -6 B 0 0 10 6 2 C 2 -10 0 -2 -2 D 6 -6 2 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000194 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -6 -6 B 0 0 10 6 2 C 2 -10 0 -2 -2 D 6 -6 2 0 -4 E 6 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000194 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3929: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) C E A D B (6) E D A C B (5) D E A C B (5) D A E B C (5) C E A B D (5) B C A E D (5) D E A B C (4) C B E A D (4) C B D E A (4) B D A E C (4) B C A D E (4) E A C D B (3) A E D C B (3) A E D B C (3) D E C A B (2) D B A E C (2) D A B E C (2) C E D A B (2) C B A E D (2) B D C E A (2) B D C A E (2) A E C B D (2) A D E B C (2) E C D A B (1) E C A D B (1) D E C B A (1) D B E C A (1) D B C E A (1) C E D B A (1) C E B D A (1) C D B E A (1) C A B E D (1) B D A C E (1) B C D E A (1) B C D A E (1) B A C E D (1) A E B C D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 22 2 2 -14 B -22 0 -12 -14 -18 C -2 12 0 -4 -10 D -2 14 4 0 -8 E 14 18 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 2 2 -14 B -22 0 -12 -14 -18 C -2 12 0 -4 -10 D -2 14 4 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=23 B=21 E=16 A=13 so A is eliminated. Round 2 votes counts: C=27 E=25 D=25 B=23 so B is eliminated. Round 3 votes counts: C=39 D=35 E=26 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:225 A:206 D:204 C:198 B:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 2 2 -14 B -22 0 -12 -14 -18 C -2 12 0 -4 -10 D -2 14 4 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 2 2 -14 B -22 0 -12 -14 -18 C -2 12 0 -4 -10 D -2 14 4 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 2 2 -14 B -22 0 -12 -14 -18 C -2 12 0 -4 -10 D -2 14 4 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3930: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) C E D A B (6) D B A E C (5) A D B C E (5) D A B C E (4) C A D E B (4) B E D C A (4) B E D A C (4) A C D E B (4) D B E C A (3) D A C B E (3) C E A B D (3) B D E C A (3) E C B D A (2) E C B A D (2) E B C D A (2) D C E B A (2) D C A E B (2) C E A D B (2) C D E A B (2) B E C D A (2) B A E D C (2) A C E D B (2) A B D E C (2) E A C B D (1) D E C B A (1) D E B C A (1) D C E A B (1) D C B E A (1) D C A B E (1) D B A C E (1) D A C E B (1) D A B E C (1) C E B D A (1) C D E B A (1) C A E D B (1) C A E B D (1) B E C A D (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E C D (1) B A D E C (1) A D C E B (1) A D C B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -10 -12 -8 B 0 0 8 -12 2 C 10 -8 0 -8 2 D 12 12 8 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -12 -8 B 0 0 8 -12 2 C 10 -8 0 -8 2 D 12 12 8 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=22 C=21 A=17 E=13 so E is eliminated. Round 2 votes counts: B=30 D=27 C=25 A=18 so A is eliminated. Round 3 votes counts: D=34 B=34 C=32 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:199 C:198 E:198 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -12 -8 B 0 0 8 -12 2 C 10 -8 0 -8 2 D 12 12 8 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -12 -8 B 0 0 8 -12 2 C 10 -8 0 -8 2 D 12 12 8 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -12 -8 B 0 0 8 -12 2 C 10 -8 0 -8 2 D 12 12 8 0 8 E 8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3931: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) D A C B E (8) A E D B C (8) A D E B C (7) E B C A D (6) E A B D C (6) C B E D A (6) A E B D C (5) A D C E B (5) C D B A E (4) C D A B E (4) B E C D A (4) E B A D C (3) D C A B E (3) C D B E A (3) D C B A E (2) C B D E A (2) B C E D A (2) A D E C B (2) E A B C D (1) D A C E B (1) D A B C E (1) C E B D A (1) C D A E B (1) B E D C A (1) B D C A E (1) B C D E A (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 8 18 10 8 B -8 0 14 -2 -14 C -18 -14 0 -10 -10 D -10 2 10 0 -8 E -8 14 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 10 8 B -8 0 14 -2 -14 C -18 -14 0 -10 -10 D -10 2 10 0 -8 E -8 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=26 C=21 D=15 B=9 so B is eliminated. Round 2 votes counts: E=31 A=29 C=24 D=16 so D is eliminated. Round 3 votes counts: A=39 E=31 C=30 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:212 D:197 B:195 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 10 8 B -8 0 14 -2 -14 C -18 -14 0 -10 -10 D -10 2 10 0 -8 E -8 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 10 8 B -8 0 14 -2 -14 C -18 -14 0 -10 -10 D -10 2 10 0 -8 E -8 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 10 8 B -8 0 14 -2 -14 C -18 -14 0 -10 -10 D -10 2 10 0 -8 E -8 14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3932: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (15) A E B C D (15) E A C D B (13) B D C A E (8) C D E A B (7) B A D E C (6) E C A D B (3) E A C B D (3) D C E A B (3) D B C E A (3) B C D A E (3) A E B D C (3) C E A D B (2) C D E B A (2) B D A C E (2) B A E D C (2) B A E C D (2) D E C A B (1) D C E B A (1) C E D A B (1) C D B E A (1) B D A E C (1) B A D C E (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 6 0 4 -10 B -6 0 -6 -6 -12 C 0 6 0 6 -2 D -4 6 -6 0 8 E 10 12 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.125000 E: 0.375000 Sum of squares = 0.406250000017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.625000 E: 1.000000 A B C D E A 0 6 0 4 -10 B -6 0 -6 -6 -12 C 0 6 0 6 -2 D -4 6 -6 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.125000 E: 0.375000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 A=20 E=19 C=13 so C is eliminated. Round 2 votes counts: D=33 B=25 E=22 A=20 so A is eliminated. Round 3 votes counts: E=42 D=33 B=25 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:208 C:205 D:202 A:200 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 4 -10 B -6 0 -6 -6 -12 C 0 6 0 6 -2 D -4 6 -6 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.125000 E: 0.375000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.625000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 4 -10 B -6 0 -6 -6 -12 C 0 6 0 6 -2 D -4 6 -6 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.125000 E: 0.375000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 4 -10 B -6 0 -6 -6 -12 C 0 6 0 6 -2 D -4 6 -6 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.125000 E: 0.375000 Sum of squares = 0.406250000074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3933: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) B D C E A (8) D B E C A (7) A E C D B (7) C E B D A (6) A D E B C (6) D B A E C (5) C E A B D (5) A D E C B (5) D B E A C (3) D B A C E (3) C B E A D (3) E D B C A (2) E C A B D (2) C E B A D (2) C B E D A (2) C B A E D (2) A E D C B (2) A D B E C (2) A B D C E (2) E C B D A (1) E B C D A (1) E A D C B (1) E A C D B (1) D E B A C (1) D E A B C (1) D B C E A (1) D A E B C (1) D A B E C (1) C A B E D (1) B D E C A (1) B C E D A (1) B C D E A (1) B C A E D (1) A D B C E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 6 8 0 B 2 0 -2 -2 -8 C -6 2 0 -6 0 D -8 2 6 0 -2 E 0 8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.522042 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.477958 Sum of squares = 0.500971668486 Cumulative probabilities = A: 0.522042 B: 0.522042 C: 0.522042 D: 0.522042 E: 1.000000 A B C D E A 0 -2 6 8 0 B 2 0 -2 -2 -8 C -6 2 0 -6 0 D -8 2 6 0 -2 E 0 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=23 C=21 B=12 E=8 so E is eliminated. Round 2 votes counts: A=38 D=25 C=24 B=13 so B is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:205 D:199 B:195 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 8 0 B 2 0 -2 -2 -8 C -6 2 0 -6 0 D -8 2 6 0 -2 E 0 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 8 0 B 2 0 -2 -2 -8 C -6 2 0 -6 0 D -8 2 6 0 -2 E 0 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 8 0 B 2 0 -2 -2 -8 C -6 2 0 -6 0 D -8 2 6 0 -2 E 0 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3934: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) D B E A C (9) C E A D B (8) E A C D B (7) C A E D B (7) A E C D B (7) B D C E A (6) B D A E C (5) E D A B C (3) D E B A C (3) B D E A C (3) A E D B C (3) E C A D B (2) D B E C A (2) D B A E C (2) C B D E A (2) B D C A E (2) B D A C E (2) A C E D B (2) E D C B A (1) E A D C B (1) D C B E A (1) C E D B A (1) C B E D A (1) C B A D E (1) C A B E D (1) C A B D E (1) B D E C A (1) B C D E A (1) B C D A E (1) B A C D E (1) A D B E C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 2 8 -4 B -10 0 -8 -20 -12 C -2 8 0 8 0 D -8 20 -8 0 -10 E 4 12 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.440261 D: 0.000000 E: 0.559739 Sum of squares = 0.50713752343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.440261 D: 0.440261 E: 1.000000 A B C D E A 0 10 2 8 -4 B -10 0 -8 -20 -12 C -2 8 0 8 0 D -8 20 -8 0 -10 E 4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=22 D=17 A=15 E=14 so E is eliminated. Round 2 votes counts: C=34 A=23 B=22 D=21 so D is eliminated. Round 3 votes counts: B=38 C=36 A=26 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 A:208 C:207 D:197 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 8 -4 B -10 0 -8 -20 -12 C -2 8 0 8 0 D -8 20 -8 0 -10 E 4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 8 -4 B -10 0 -8 -20 -12 C -2 8 0 8 0 D -8 20 -8 0 -10 E 4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 8 -4 B -10 0 -8 -20 -12 C -2 8 0 8 0 D -8 20 -8 0 -10 E 4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3935: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) A B E C D (8) D C E A B (7) D C B E A (6) C D B A E (6) A E B C D (6) D C E B A (5) C D A B E (5) B C D A E (4) D C A E B (3) D B C E A (3) C D A E B (3) C B D A E (3) C A D E B (3) B E A D C (3) E D C A B (2) E D A C B (2) E B A D C (2) D E C A B (2) B D C E A (2) A C D E B (2) E D A B C (1) E B D A C (1) E A D C B (1) E A B C D (1) D E C B A (1) D C A B E (1) D B E C A (1) C A B D E (1) B E D A C (1) B E A C D (1) B A C E D (1) A E C D B (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -16 -18 16 B -4 0 -12 -10 14 C 16 12 0 12 16 D 18 10 -12 0 18 E -16 -14 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 -18 16 B -4 0 -12 -10 14 C 16 12 0 12 16 D 18 10 -12 0 18 E -16 -14 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=21 B=20 A=20 E=10 so E is eliminated. Round 2 votes counts: D=34 B=23 A=22 C=21 so C is eliminated. Round 3 votes counts: D=48 B=26 A=26 so B is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:228 D:217 B:194 A:193 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 -18 16 B -4 0 -12 -10 14 C 16 12 0 12 16 D 18 10 -12 0 18 E -16 -14 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -18 16 B -4 0 -12 -10 14 C 16 12 0 12 16 D 18 10 -12 0 18 E -16 -14 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -18 16 B -4 0 -12 -10 14 C 16 12 0 12 16 D 18 10 -12 0 18 E -16 -14 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3936: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) D B E A C (7) B D E A C (7) C A E B D (6) D E A B C (5) D B C E A (4) B E A D C (4) B D C E A (4) A E D B C (4) C B A E D (3) C A D E B (3) B C E A D (3) E B A D C (2) D C B A E (2) C D B E A (2) C D A E B (2) C D A B E (2) C A E D B (2) C A B E D (2) B E D A C (2) B E A C D (2) B D E C A (2) A E C D B (2) A E B D C (2) A D C E B (2) A C E D B (2) E D A B C (1) D E B A C (1) D C B E A (1) D A E B C (1) C D B A E (1) C B E A D (1) C B D E A (1) C B D A E (1) B E C D A (1) B E C A D (1) A E D C B (1) A E C B D (1) A E B C D (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 12 6 -16 B 4 0 24 8 6 C -12 -24 0 -20 -10 D -6 -8 20 0 -2 E 16 -6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 6 -16 B 4 0 24 8 6 C -12 -24 0 -20 -10 D -6 -8 20 0 -2 E 16 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=26 B=26 D=21 A=17 E=10 so E is eliminated. Round 2 votes counts: B=28 C=26 A=24 D=22 so D is eliminated. Round 3 votes counts: B=40 A=31 C=29 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:211 D:202 A:199 C:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 6 -16 B 4 0 24 8 6 C -12 -24 0 -20 -10 D -6 -8 20 0 -2 E 16 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 6 -16 B 4 0 24 8 6 C -12 -24 0 -20 -10 D -6 -8 20 0 -2 E 16 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 6 -16 B 4 0 24 8 6 C -12 -24 0 -20 -10 D -6 -8 20 0 -2 E 16 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3937: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (5) A D C B E (5) E B C A D (4) D C A E B (4) D A C B E (4) C E B A D (4) A D B E C (4) A C E B D (4) E B C D A (3) D C B E A (3) D B E C A (3) D B E A C (3) D A B E C (3) C A E D B (3) C A E B D (3) C A D E B (3) B E D A C (3) B D E A C (3) A B E C D (3) E C B A D (2) D A C E B (2) D A B C E (2) C E D B A (2) C E B D A (2) C D E A B (2) C D A E B (2) B E C A D (2) B E A C D (2) A D B C E (2) A C D E B (2) A B D E C (2) D B C E A (1) D B A E C (1) C E A B D (1) B E C D A (1) B E A D C (1) B D A E C (1) B A E D C (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 6 0 0 6 B -6 0 2 -4 10 C 0 -2 0 -8 4 D 0 4 8 0 6 E -6 -10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.534349 B: 0.000000 C: 0.000000 D: 0.465651 E: 0.000000 Sum of squares = 0.502359642528 Cumulative probabilities = A: 0.534349 B: 0.534349 C: 0.534349 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 6 B -6 0 2 -4 10 C 0 -2 0 -8 4 D 0 4 8 0 6 E -6 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=24 C=22 B=19 E=9 so E is eliminated. Round 2 votes counts: D=26 B=26 C=24 A=24 so C is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:209 A:206 B:201 C:197 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 0 6 B -6 0 2 -4 10 C 0 -2 0 -8 4 D 0 4 8 0 6 E -6 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 6 B -6 0 2 -4 10 C 0 -2 0 -8 4 D 0 4 8 0 6 E -6 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 6 B -6 0 2 -4 10 C 0 -2 0 -8 4 D 0 4 8 0 6 E -6 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3938: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) E D C B A (7) C E D B A (6) D E B C A (5) C A E D B (5) C E D A B (4) C E A D B (4) B D E A C (4) A C E B D (4) A B D C E (4) A B C D E (4) D E B A C (3) B D E C A (3) B D A E C (3) B A D E C (3) A C B E D (3) E D B A C (2) E C D A B (2) D B E C A (2) C D B E A (2) C A E B D (2) B D C E A (2) A C B D E (2) A B C E D (2) E D C A B (1) E D A B C (1) E C D B A (1) E A D B C (1) D E C B A (1) D B C E A (1) C D E B A (1) B A D C E (1) A E C B D (1) A E B D C (1) A C E D B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 2 -2 -6 B -6 0 6 0 -6 C -2 -6 0 -10 -4 D 2 0 10 0 0 E 6 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.332387 E: 0.667613 Sum of squares = 0.55618834167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.332387 E: 1.000000 A B C D E A 0 6 2 -2 -6 B -6 0 6 0 -6 C -2 -6 0 -10 -4 D 2 0 10 0 0 E 6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 B=16 E=15 D=12 so D is eliminated. Round 2 votes counts: A=33 E=24 C=24 B=19 so B is eliminated. Round 3 votes counts: A=40 E=33 C=27 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:206 A:200 B:197 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -2 -6 B -6 0 6 0 -6 C -2 -6 0 -10 -4 D 2 0 10 0 0 E 6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -2 -6 B -6 0 6 0 -6 C -2 -6 0 -10 -4 D 2 0 10 0 0 E 6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -2 -6 B -6 0 6 0 -6 C -2 -6 0 -10 -4 D 2 0 10 0 0 E 6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3939: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) D B C A E (8) C A E D B (7) B D E A C (7) A E C B D (7) E A B D C (6) D C B A E (6) A E C D B (6) E A B C D (5) B D C E A (5) E A C B D (4) D B E A C (4) C D B A E (4) E B A D C (2) E A C D B (2) D E A B C (2) D B C E A (2) B E D A C (2) B E A D C (2) E A D B C (1) D E B A C (1) C D A E B (1) C B A E D (1) C B A D E (1) C A D E B (1) B E A C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 8 16 8 B -8 0 -4 8 -14 C -8 4 0 4 -4 D -16 -8 -4 0 -16 E -8 14 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 16 8 B -8 0 -4 8 -14 C -8 4 0 4 -4 D -16 -8 -4 0 -16 E -8 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=23 E=20 B=17 A=15 so A is eliminated. Round 2 votes counts: E=33 C=27 D=23 B=17 so B is eliminated. Round 3 votes counts: E=38 D=35 C=27 so C is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:220 E:213 C:198 B:191 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 16 8 B -8 0 -4 8 -14 C -8 4 0 4 -4 D -16 -8 -4 0 -16 E -8 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 16 8 B -8 0 -4 8 -14 C -8 4 0 4 -4 D -16 -8 -4 0 -16 E -8 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 16 8 B -8 0 -4 8 -14 C -8 4 0 4 -4 D -16 -8 -4 0 -16 E -8 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3940: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (13) D C B A E (13) A E B C D (7) D C B E A (5) E C B A D (4) E A C B D (4) C B D A E (4) A B C E D (4) E A D B C (3) D E C B A (3) D E A B C (3) D A B C E (3) C B A E D (3) A B C D E (3) E D C B A (2) E D A B C (2) D A E B C (2) C B D E A (2) A E D B C (2) E D A C B (1) E C D A B (1) E C A B D (1) D E C A B (1) D E A C B (1) D C A B E (1) D A B E C (1) C E B A D (1) C D E B A (1) C D B A E (1) C B E D A (1) C B A D E (1) B D C A E (1) B C A E D (1) B A C D E (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 6 2 2 B -12 0 -2 2 -6 C -6 2 0 6 -6 D -2 -2 -6 0 0 E -2 6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 2 2 B -12 0 -2 2 -6 C -6 2 0 6 -6 D -2 -2 -6 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=31 A=19 C=14 B=3 so B is eliminated. Round 2 votes counts: D=34 E=31 A=20 C=15 so C is eliminated. Round 3 votes counts: D=42 E=33 A=25 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:211 E:205 C:198 D:195 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 2 2 B -12 0 -2 2 -6 C -6 2 0 6 -6 D -2 -2 -6 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 2 2 B -12 0 -2 2 -6 C -6 2 0 6 -6 D -2 -2 -6 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 2 2 B -12 0 -2 2 -6 C -6 2 0 6 -6 D -2 -2 -6 0 0 E -2 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3941: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) D C A B E (7) B D E C A (7) D E B A C (4) D B C E A (4) B D C E A (4) E B A C D (3) E A C B D (3) E A B C D (3) D B C A E (3) C D A B E (3) C A B E D (3) C A B D E (3) B E D A C (3) A C E D B (3) A C E B D (3) E D A B C (2) D E B C A (2) D E A C B (2) D B E C A (2) C B A E D (2) C A D E B (2) C A D B E (2) B E A C D (2) B C E A D (2) B C A E D (2) A E C D B (2) E D B A C (1) E D A C B (1) E B D A C (1) E B A D C (1) E A D C B (1) E A C D B (1) D E C A B (1) D A C E B (1) B E D C A (1) B E C A D (1) B D C A E (1) B C D A E (1) B C A D E (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -22 -16 -4 B 10 0 2 -4 24 C 22 -2 0 -12 12 D 16 4 12 0 16 E 4 -24 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999155 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 -16 -4 B 10 0 2 -4 24 C 22 -2 0 -12 12 D 16 4 12 0 16 E 4 -24 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=25 E=17 C=15 A=10 so A is eliminated. Round 2 votes counts: D=33 B=25 C=23 E=19 so E is eliminated. Round 3 votes counts: D=38 B=33 C=29 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:216 C:210 E:176 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -22 -16 -4 B 10 0 2 -4 24 C 22 -2 0 -12 12 D 16 4 12 0 16 E 4 -24 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 -16 -4 B 10 0 2 -4 24 C 22 -2 0 -12 12 D 16 4 12 0 16 E 4 -24 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 -16 -4 B 10 0 2 -4 24 C 22 -2 0 -12 12 D 16 4 12 0 16 E 4 -24 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3942: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) D C A B E (7) E A B D C (6) A D C E B (6) D C A E B (5) D A C E B (5) C B E D A (4) B E C D A (4) B E C A D (4) B E A C D (4) E B D A C (3) C D B A E (3) C D A B E (3) C A D B E (3) A E B D C (3) A B E C D (3) E B D C A (2) D A E C B (2) C B A D E (2) B C E D A (2) B C E A D (2) A C D B E (2) A C B E D (2) A C B D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E B C D A (1) E A D B C (1) D E C B A (1) D E A B C (1) D C E A B (1) D A E B C (1) C B D E A (1) C B A E D (1) C A B D E (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 14 4 2 6 B -14 0 -4 8 2 C -4 4 0 -12 6 D -2 -8 12 0 -4 E -6 -2 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 2 6 B -14 0 -4 8 2 C -4 4 0 -12 6 D -2 -8 12 0 -4 E -6 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998094 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 A=20 C=18 B=16 so B is eliminated. Round 2 votes counts: E=35 D=23 C=22 A=20 so A is eliminated. Round 3 votes counts: E=41 D=31 C=28 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:213 D:199 C:197 B:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 2 6 B -14 0 -4 8 2 C -4 4 0 -12 6 D -2 -8 12 0 -4 E -6 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998094 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 2 6 B -14 0 -4 8 2 C -4 4 0 -12 6 D -2 -8 12 0 -4 E -6 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998094 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 2 6 B -14 0 -4 8 2 C -4 4 0 -12 6 D -2 -8 12 0 -4 E -6 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998094 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3943: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (7) C D B A E (6) A E B C D (6) E A B D C (4) C D B E A (4) C B D E A (4) C B D A E (4) A E C D B (4) A E B D C (4) E D B A C (3) C D A E B (3) B E D A C (3) B D E C A (3) B D C E A (3) B C D E A (3) A E D B C (3) A E C B D (3) E B D A C (2) E B A D C (2) E A D B C (2) D E A C B (2) D C B E A (2) C D A B E (2) C B A D E (2) C A B E D (2) B E D C A (2) E D A B C (1) E A D C B (1) D E C A B (1) D E B A C (1) D E A B C (1) D B E C A (1) C A D B E (1) B E A D C (1) B E A C D (1) B D E A C (1) B C D A E (1) B C A E D (1) B A E C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 10 -6 4 B 2 0 2 10 -2 C -10 -2 0 0 -22 D 6 -10 0 0 -10 E -4 2 22 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000025 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 10 -6 4 B 2 0 2 10 -2 C -10 -2 0 0 -22 D 6 -10 0 0 -10 E -4 2 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=28 B=20 E=15 D=8 so D is eliminated. Round 2 votes counts: C=30 A=29 B=21 E=20 so E is eliminated. Round 3 votes counts: A=40 C=31 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:215 B:206 A:203 D:193 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 -6 4 B 2 0 2 10 -2 C -10 -2 0 0 -22 D 6 -10 0 0 -10 E -4 2 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -6 4 B 2 0 2 10 -2 C -10 -2 0 0 -22 D 6 -10 0 0 -10 E -4 2 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -6 4 B 2 0 2 10 -2 C -10 -2 0 0 -22 D 6 -10 0 0 -10 E -4 2 22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3944: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) E B C D A (8) A B D C E (7) B A D E C (5) E C A B D (4) E B A C D (4) E A C B D (4) D C A E B (4) B A E D C (4) A B E C D (4) E C D B A (3) D A C B E (3) C D A E B (3) E C B A D (2) E A B C D (2) D C E B A (2) C E D A B (2) B E A D C (2) A E B C D (2) A D B C E (2) A C D B E (2) A B C E D (2) E C B D A (1) E C A D B (1) E B D C A (1) E B D A C (1) D E C B A (1) D E B C A (1) D C E A B (1) D B C A E (1) D B A C E (1) D A B C E (1) C D E B A (1) C D E A B (1) C A E D B (1) C A D E B (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A E C (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 16 0 2 10 B -16 0 6 18 -4 C 0 -6 0 -2 -12 D -2 -18 2 0 -2 E -10 4 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.699076 B: 0.000000 C: 0.300924 D: 0.000000 E: 0.000000 Sum of squares = 0.579262359663 Cumulative probabilities = A: 0.699076 B: 0.699076 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 2 10 B -16 0 6 18 -4 C 0 -6 0 -2 -12 D -2 -18 2 0 -2 E -10 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.504132291102 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=24 A=21 B=15 C=9 so C is eliminated. Round 2 votes counts: E=33 D=29 A=23 B=15 so B is eliminated. Round 3 votes counts: E=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:204 B:202 C:190 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 2 10 B -16 0 6 18 -4 C 0 -6 0 -2 -12 D -2 -18 2 0 -2 E -10 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.504132291102 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 2 10 B -16 0 6 18 -4 C 0 -6 0 -2 -12 D -2 -18 2 0 -2 E -10 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.504132291102 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 2 10 B -16 0 6 18 -4 C 0 -6 0 -2 -12 D -2 -18 2 0 -2 E -10 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.454545 D: 0.000000 E: 0.000000 Sum of squares = 0.504132291102 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3945: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) E D C B A (7) B A C D E (7) E D A C B (6) C B A E D (6) C E B A D (5) E D B C A (4) E C B D A (4) D A B E C (4) A B D C E (4) E C D B A (3) D B A C E (3) D A E B C (3) A D B C E (3) A B C D E (3) E D C A B (2) E C D A B (2) E C A D B (2) D E B A C (2) D E A C B (2) D A B C E (2) C E A B D (2) C B E A D (2) C B A D E (2) B C E D A (2) B C A E D (2) D B A E C (1) C E B D A (1) C A B E D (1) B D A E C (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 -8 4 -18 -12 B 8 0 6 -12 -10 C -4 -6 0 -10 -6 D 18 12 10 0 -2 E 12 10 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 4 -18 -12 B 8 0 6 -12 -10 C -4 -6 0 -10 -6 D 18 12 10 0 -2 E 12 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 C=19 B=14 A=10 so A is eliminated. Round 2 votes counts: E=30 D=30 B=21 C=19 so C is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:219 E:215 B:196 C:187 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 -18 -12 B 8 0 6 -12 -10 C -4 -6 0 -10 -6 D 18 12 10 0 -2 E 12 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -18 -12 B 8 0 6 -12 -10 C -4 -6 0 -10 -6 D 18 12 10 0 -2 E 12 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -18 -12 B 8 0 6 -12 -10 C -4 -6 0 -10 -6 D 18 12 10 0 -2 E 12 10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3946: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (13) E A B D C (9) C D E A B (6) E A B C D (5) C D B E A (5) D C B A E (4) C D E B A (4) B C D A E (4) B A D E C (4) C E D A B (3) C E A B D (3) C B D A E (3) A E B D C (3) E A D C B (2) E A D B C (2) D A E B C (2) B D C A E (2) B D A C E (2) B C E A D (2) B A E C D (2) A E D B C (2) A B E D C (2) A B D E C (2) E C A D B (1) E C A B D (1) E A C D B (1) D E A C B (1) D B C A E (1) D B A E C (1) D B A C E (1) D A B C E (1) C E D B A (1) C E B A D (1) C E A D B (1) C B D E A (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 -6 -12 -10 2 B 6 0 -2 -4 4 C 12 2 0 14 18 D 10 4 -14 0 16 E -2 -4 -18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -10 2 B 6 0 -2 -4 4 C 12 2 0 14 18 D 10 4 -14 0 16 E -2 -4 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 E=21 B=18 D=11 A=9 so A is eliminated. Round 2 votes counts: C=41 E=26 B=22 D=11 so D is eliminated. Round 3 votes counts: C=45 E=29 B=26 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:208 B:202 A:187 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 -10 2 B 6 0 -2 -4 4 C 12 2 0 14 18 D 10 4 -14 0 16 E -2 -4 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -10 2 B 6 0 -2 -4 4 C 12 2 0 14 18 D 10 4 -14 0 16 E -2 -4 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -10 2 B 6 0 -2 -4 4 C 12 2 0 14 18 D 10 4 -14 0 16 E -2 -4 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3947: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (6) C D A E B (5) B E A D C (5) A D E B C (5) D C A E B (4) D A C E B (4) B E C D A (4) B C E A D (4) A D B E C (4) E B D C A (3) E B A D C (3) C B E A D (3) C B A E D (3) C A D B E (3) B C A E D (3) B A C E D (3) A D C B E (3) A C D B E (3) E D C B A (2) E D B A C (2) E B D A C (2) D A E C B (2) C B A D E (2) C A B D E (2) B E C A D (2) A B D E C (2) E D C A B (1) E C D B A (1) E B C D A (1) D E C A B (1) D E A C B (1) D C E A B (1) C E D B A (1) C E B D A (1) C D E A B (1) C B E D A (1) C A D E B (1) B E A C D (1) B C E D A (1) B A E C D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -2 22 16 B -2 0 -4 -4 4 C 2 4 0 -6 12 D -22 4 6 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.733333 D: 0.066667 E: 0.000000 Sum of squares = 0.5822222222 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.933333 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 22 16 B -2 0 -4 -4 4 C 2 4 0 -6 12 D -22 4 6 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.733333 D: 0.066667 E: 0.000000 Sum of squares = 0.582222222195 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.933333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=24 C=23 E=15 D=13 so D is eliminated. Round 2 votes counts: A=31 C=28 B=24 E=17 so E is eliminated. Round 3 votes counts: B=35 C=33 A=32 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:206 B:197 D:195 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 22 16 B -2 0 -4 -4 4 C 2 4 0 -6 12 D -22 4 6 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.733333 D: 0.066667 E: 0.000000 Sum of squares = 0.582222222195 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.933333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 22 16 B -2 0 -4 -4 4 C 2 4 0 -6 12 D -22 4 6 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.733333 D: 0.066667 E: 0.000000 Sum of squares = 0.582222222195 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.933333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 22 16 B -2 0 -4 -4 4 C 2 4 0 -6 12 D -22 4 6 0 2 E -16 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.733333 D: 0.066667 E: 0.000000 Sum of squares = 0.582222222195 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.933333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3948: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) B D E C A (6) A E B D C (6) A B C E D (5) E D A B C (4) C A E D B (4) B D E A C (4) B D C E A (4) A C E D B (4) E A D B C (3) C A D E B (3) C A B D E (3) A B C D E (3) E D C B A (2) E D B C A (2) E D B A C (2) E B D A C (2) E A D C B (2) D E C B A (2) D B C E A (2) C D E B A (2) C B D E A (2) C B D A E (2) B C D A E (2) B A D E C (2) A E C D B (2) A C E B D (2) A C B E D (2) E D A C B (1) D E B C A (1) D C E B A (1) D B E C A (1) C E D A B (1) C B A D E (1) B E A D C (1) B C A D E (1) B A E D C (1) A E D B C (1) A E C B D (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 0 -4 B 0 0 10 6 4 C -2 -10 0 -2 4 D 0 -6 2 0 0 E 4 -4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.334332 B: 0.665668 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.554891768774 Cumulative probabilities = A: 0.334332 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 0 -4 B 0 0 10 6 4 C -2 -10 0 -2 4 D 0 -6 2 0 0 E 4 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499782 B: 0.500218 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000094866 Cumulative probabilities = A: 0.499782 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 B=21 E=18 D=7 so D is eliminated. Round 2 votes counts: A=29 C=26 B=24 E=21 so E is eliminated. Round 3 votes counts: A=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:199 D:198 E:198 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 0 -4 B 0 0 10 6 4 C -2 -10 0 -2 4 D 0 -6 2 0 0 E 4 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499782 B: 0.500218 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000094866 Cumulative probabilities = A: 0.499782 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 0 -4 B 0 0 10 6 4 C -2 -10 0 -2 4 D 0 -6 2 0 0 E 4 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499782 B: 0.500218 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000094866 Cumulative probabilities = A: 0.499782 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 0 -4 B 0 0 10 6 4 C -2 -10 0 -2 4 D 0 -6 2 0 0 E 4 -4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499782 B: 0.500218 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000094866 Cumulative probabilities = A: 0.499782 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3949: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) C D A B E (8) C D B E A (7) E B C D A (5) E B A D C (5) E A B C D (5) C D B A E (5) B E D C A (5) A E B D C (5) A C D E B (5) A E C D B (4) A C D B E (4) A E C B D (3) A E B C D (3) E C B D A (2) E A B D C (2) D C B E A (2) C A D E B (2) A E D C B (2) A E D B C (2) A D C B E (2) E B D A C (1) D C B A E (1) D C A B E (1) D B C E A (1) C A E D B (1) C A D B E (1) B E D A C (1) B E C D A (1) B D E C A (1) B D E A C (1) B D C E A (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -6 -4 2 B -4 0 -2 0 -14 C 6 2 0 14 -14 D 4 0 -14 0 -14 E -2 14 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305773 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.727273 D: 0.727273 E: 1.000000 A B C D E A 0 4 -6 -4 2 B -4 0 -2 0 -14 C 6 2 0 14 -14 D 4 0 -14 0 -14 E -2 14 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.727273 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=29 C=24 B=10 D=5 so D is eliminated. Round 2 votes counts: A=32 E=29 C=28 B=11 so B is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:220 C:204 A:198 B:190 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -6 -4 2 B -4 0 -2 0 -14 C 6 2 0 14 -14 D 4 0 -14 0 -14 E -2 14 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.727273 D: 0.727273 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -4 2 B -4 0 -2 0 -14 C 6 2 0 14 -14 D 4 0 -14 0 -14 E -2 14 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.727273 D: 0.727273 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -4 2 B -4 0 -2 0 -14 C 6 2 0 14 -14 D 4 0 -14 0 -14 E -2 14 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305762 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.727273 D: 0.727273 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3950: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) C A B E D (6) A C B E D (6) A B C E D (6) C B A D E (5) D E A B C (4) C B D E A (4) C B D A E (4) C B A E D (4) B C E D A (4) A E B D C (4) E D A B C (3) D E B C A (3) C A D E B (3) A E D B C (3) A B E C D (3) D E B A C (2) D B E C A (2) D B C E A (2) C D A E B (2) C A B D E (2) B C A E D (2) E D B A C (1) E A D B C (1) E A B D C (1) D E C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) D A E C B (1) C D B A E (1) C B E D A (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C A D (1) B A C E D (1) A E D C B (1) A E B C D (1) A D E C B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -16 8 18 B -6 0 -6 16 28 C 16 6 0 28 26 D -8 -16 -28 0 -4 E -18 -28 -26 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -16 8 18 B -6 0 -6 16 28 C 16 6 0 28 26 D -8 -16 -28 0 -4 E -18 -28 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=27 D=18 B=10 E=6 so E is eliminated. Round 2 votes counts: C=39 A=29 D=22 B=10 so B is eliminated. Round 3 votes counts: C=46 A=30 D=24 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:238 B:216 A:208 D:172 E:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -16 8 18 B -6 0 -6 16 28 C 16 6 0 28 26 D -8 -16 -28 0 -4 E -18 -28 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -16 8 18 B -6 0 -6 16 28 C 16 6 0 28 26 D -8 -16 -28 0 -4 E -18 -28 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -16 8 18 B -6 0 -6 16 28 C 16 6 0 28 26 D -8 -16 -28 0 -4 E -18 -28 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3951: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) D C A B E (6) B E C A D (6) E B A D C (4) E A D B C (4) D E A C B (4) D C B A E (4) D C A E B (4) D A C E B (4) D C B E A (3) D A E C B (3) C D B A E (3) C D A B E (3) C B D A E (3) B C E D A (3) B C E A D (3) A D E C B (3) E D A B C (2) D E C B A (2) D C E B A (2) C B A E D (2) B E A C D (2) A E D B C (2) A E B C D (2) A C B E D (2) E D B A C (1) E B D C A (1) E B D A C (1) E B A C D (1) E A B C D (1) D E C A B (1) D E B C A (1) D B C E A (1) C D B E A (1) C B D E A (1) C B A D E (1) C A B E D (1) B E C D A (1) B C A E D (1) B A E C D (1) A E B D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -8 0 C 8 6 0 -20 -2 D 10 8 20 0 0 E 4 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.410582 E: 0.589418 Sum of squares = 0.51599131531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.410582 E: 1.000000 A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -8 0 C 8 6 0 -20 -2 D 10 8 20 0 0 E 4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=21 B=17 C=15 A=12 so A is eliminated. Round 2 votes counts: D=38 E=26 B=19 C=17 so C is eliminated. Round 3 votes counts: D=45 B=29 E=26 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:203 C:196 B:192 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -8 0 C 8 6 0 -20 -2 D 10 8 20 0 0 E 4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -8 0 C 8 6 0 -20 -2 D 10 8 20 0 0 E 4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -8 0 C 8 6 0 -20 -2 D 10 8 20 0 0 E 4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3952: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (12) C B E D A (11) C B E A D (9) E B C A D (6) D A C B E (6) D A B E C (4) C D B A E (4) E B A D C (3) E B A C D (3) D C A B E (3) D A E C B (3) D A E B C (3) C D A B E (3) A D B E C (3) E C B A D (2) E A D B C (2) C D B E A (2) C B D E A (2) B E C A D (2) B C E A D (2) A E B D C (2) E C D A B (1) E A B D C (1) D C A E B (1) D A C E B (1) D A B C E (1) C E B D A (1) C E B A D (1) C D E B A (1) C B D A E (1) B E A C D (1) B A E D C (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -4 4 0 B 6 0 -4 0 14 C 4 4 0 4 0 D -4 0 -4 0 4 E 0 -14 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.849957 D: 0.000000 E: 0.150043 Sum of squares = 0.744939568082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.849957 D: 0.849957 E: 1.000000 A B C D E A 0 -6 -4 4 0 B 6 0 -4 0 14 C 4 4 0 4 0 D -4 0 -4 0 4 E 0 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.000000 E: 0.222222 Sum of squares = 0.654321032575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=22 E=18 A=18 B=7 so B is eliminated. Round 2 votes counts: C=37 D=22 E=21 A=20 so A is eliminated. Round 3 votes counts: D=39 C=37 E=24 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:208 C:206 D:198 A:197 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 4 0 B 6 0 -4 0 14 C 4 4 0 4 0 D -4 0 -4 0 4 E 0 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.000000 E: 0.222222 Sum of squares = 0.654321032575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 4 0 B 6 0 -4 0 14 C 4 4 0 4 0 D -4 0 -4 0 4 E 0 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.000000 E: 0.222222 Sum of squares = 0.654321032575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 4 0 B 6 0 -4 0 14 C 4 4 0 4 0 D -4 0 -4 0 4 E 0 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.000000 E: 0.222222 Sum of squares = 0.654321032575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3953: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (15) A B E C D (14) A E B C D (7) D C E A B (6) B A E C D (6) D C B E A (5) C D E A B (5) B A E D C (5) C D E B A (4) B D C E A (4) E C D A B (3) B A D E C (3) A B E D C (3) E C A D B (2) E A C D B (2) E A C B D (2) A E C D B (2) E C D B A (1) D C A E B (1) D B C E A (1) C E D A B (1) C D A E B (1) B D E C A (1) B D C A E (1) B A D C E (1) A E C B D (1) A D C E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -2 2 -4 B -6 0 -4 0 -8 C 2 4 0 2 -6 D -2 0 -2 0 2 E 4 8 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.088272 B: 0.000000 C: 0.129382 D: 0.564691 E: 0.217654 Sum of squares = 0.390781266749 Cumulative probabilities = A: 0.088272 B: 0.088272 C: 0.217654 D: 0.782346 E: 1.000000 A B C D E A 0 6 -2 2 -4 B -6 0 -4 0 -8 C 2 4 0 2 -6 D -2 0 -2 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.000000 C: 0.043478 D: 0.521739 E: 0.239130 Sum of squares = 0.369565217388 Cumulative probabilities = A: 0.195652 B: 0.195652 C: 0.239130 D: 0.760870 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=28 B=21 C=11 E=10 so E is eliminated. Round 2 votes counts: A=34 D=28 B=21 C=17 so C is eliminated. Round 3 votes counts: D=43 A=36 B=21 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:201 C:201 D:199 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 2 -4 B -6 0 -4 0 -8 C 2 4 0 2 -6 D -2 0 -2 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.000000 C: 0.043478 D: 0.521739 E: 0.239130 Sum of squares = 0.369565217388 Cumulative probabilities = A: 0.195652 B: 0.195652 C: 0.239130 D: 0.760870 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 2 -4 B -6 0 -4 0 -8 C 2 4 0 2 -6 D -2 0 -2 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.000000 C: 0.043478 D: 0.521739 E: 0.239130 Sum of squares = 0.369565217388 Cumulative probabilities = A: 0.195652 B: 0.195652 C: 0.239130 D: 0.760870 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 2 -4 B -6 0 -4 0 -8 C 2 4 0 2 -6 D -2 0 -2 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.195652 B: 0.000000 C: 0.043478 D: 0.521739 E: 0.239130 Sum of squares = 0.369565217388 Cumulative probabilities = A: 0.195652 B: 0.195652 C: 0.239130 D: 0.760870 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3954: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) E A D C B (7) B C D A E (7) E B A D C (5) C D A B E (5) E B D A C (4) D A E C B (4) D A C E B (4) B E A C D (4) B C E A D (4) E D A B C (3) E A B D C (3) C A D B E (3) B C A D E (3) E D A C B (2) E B A C D (2) D E A C B (2) D C A E B (2) D C A B E (2) B E C D A (2) B C A E D (2) A E D C B (2) A E C D B (2) A D C E B (2) A C D E B (2) E D B A C (1) E A D B C (1) E A C D B (1) D C B A E (1) D B C A E (1) C D B A E (1) C D A E B (1) C A B D E (1) B E D C A (1) B E D A C (1) B D C E A (1) B C E D A (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 0 6 6 -8 B 0 0 10 0 -2 C -6 -10 0 0 -12 D -6 0 0 0 -12 E 8 2 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 6 6 -8 B 0 0 10 0 -2 C -6 -10 0 0 -12 D -6 0 0 0 -12 E 8 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=29 D=16 C=11 A=9 so A is eliminated. Round 2 votes counts: B=35 E=33 D=19 C=13 so C is eliminated. Round 3 votes counts: B=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:204 A:202 D:191 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 6 -8 B 0 0 10 0 -2 C -6 -10 0 0 -12 D -6 0 0 0 -12 E 8 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 6 -8 B 0 0 10 0 -2 C -6 -10 0 0 -12 D -6 0 0 0 -12 E 8 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 6 -8 B 0 0 10 0 -2 C -6 -10 0 0 -12 D -6 0 0 0 -12 E 8 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3955: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) E D C B A (8) E B A D C (8) B A E C D (6) E D B A C (5) D C E A B (5) C D A B E (5) C A B D E (5) C D E A B (4) C A D B E (4) E D B C A (3) D E C A B (3) D C A B E (3) C D A E B (3) A B C E D (3) E D C A B (2) E C D A B (2) E B D A C (2) E B A C D (2) B E A D C (2) B A E D C (2) B A D E C (2) B A D C E (2) A C B D E (2) E C D B A (1) E C B A D (1) D E C B A (1) C E D A B (1) C A E B D (1) C A B E D (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -6 4 0 B -8 0 -4 0 -4 C 6 4 0 2 0 D -4 0 -2 0 -2 E 0 4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.507822 D: 0.000000 E: 0.492178 Sum of squares = 0.500122355776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.507822 D: 0.507822 E: 1.000000 A B C D E A 0 8 -6 4 0 B -8 0 -4 0 -4 C 6 4 0 2 0 D -4 0 -2 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=24 B=15 A=15 D=12 so D is eliminated. Round 2 votes counts: E=38 C=32 B=15 A=15 so B is eliminated. Round 3 votes counts: E=40 C=32 A=28 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 A:203 E:203 D:196 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 4 0 B -8 0 -4 0 -4 C 6 4 0 2 0 D -4 0 -2 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 4 0 B -8 0 -4 0 -4 C 6 4 0 2 0 D -4 0 -2 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 4 0 B -8 0 -4 0 -4 C 6 4 0 2 0 D -4 0 -2 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3956: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) B C D A E (7) C B D E A (6) C B D A E (6) B C E A D (6) A E B D C (6) E A D C B (5) D A E B C (4) C D B E A (4) E A D B C (3) E A B D C (3) D E A C B (3) D C B A E (3) B C A E D (3) A E D B C (3) E A B C D (2) D C E A B (2) D C A B E (2) C E B D A (2) C B E D A (2) C B E A D (2) B C A D E (2) B A E D C (2) E D A C B (1) E C D A B (1) E A C B D (1) D C E B A (1) D C B E A (1) D B C A E (1) D A C E B (1) D A C B E (1) B E A C D (1) B D A C E (1) B A C E D (1) B A C D E (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -2 -14 8 B 4 0 -6 8 4 C 2 6 0 -6 10 D 14 -8 6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.300000 E: 0.000000 Sum of squares = 0.339999999962 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -14 8 B 4 0 -6 8 4 C 2 6 0 -6 10 D 14 -8 6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=24 C=22 E=16 A=11 so A is eliminated. Round 2 votes counts: D=28 E=26 B=24 C=22 so C is eliminated. Round 3 votes counts: B=40 D=32 E=28 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:211 C:206 B:205 A:194 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -14 8 B 4 0 -6 8 4 C 2 6 0 -6 10 D 14 -8 6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -14 8 B 4 0 -6 8 4 C 2 6 0 -6 10 D 14 -8 6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -14 8 B 4 0 -6 8 4 C 2 6 0 -6 10 D 14 -8 6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3957: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) C E D B A (7) C E B D A (7) B A C E D (6) D E C A B (5) A B D E C (5) C D E B A (4) D E A C B (3) D A B E C (3) C E B A D (3) B A E C D (3) A D B E C (3) A D B C E (3) A B D C E (3) E C D B A (2) E C B A D (2) D C E A B (2) D C A B E (2) D A E B C (2) D A B C E (2) C B E A D (2) C B A E D (2) B E A C D (2) A B C D E (2) E D C A B (1) E B C A D (1) E B A C D (1) E A D B C (1) D E C B A (1) D E A B C (1) D C E B A (1) D A E C B (1) D A C E B (1) D A C B E (1) C B E D A (1) B E C A D (1) B C E A D (1) B C A E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 8 6 2 B -2 0 4 8 8 C -8 -4 0 0 4 D -6 -8 0 0 -8 E -2 -8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 6 2 B -2 0 4 8 8 C -8 -4 0 0 4 D -6 -8 0 0 -8 E -2 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 D=25 B=15 E=8 so E is eliminated. Round 2 votes counts: C=30 A=27 D=26 B=17 so B is eliminated. Round 3 votes counts: A=40 C=34 D=26 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:209 E:197 C:196 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 6 2 B -2 0 4 8 8 C -8 -4 0 0 4 D -6 -8 0 0 -8 E -2 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 6 2 B -2 0 4 8 8 C -8 -4 0 0 4 D -6 -8 0 0 -8 E -2 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 6 2 B -2 0 4 8 8 C -8 -4 0 0 4 D -6 -8 0 0 -8 E -2 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3958: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) E C A B D (7) B A D C E (7) E D C B A (5) E D B C A (5) E C D A B (4) D C B A E (4) C A B D E (4) B D A C E (4) A B D C E (4) A B C D E (4) E D B A C (3) E B A D C (3) C D A B E (3) E B D A C (2) E A B C D (2) D E B C A (2) D B E A C (2) D B A E C (2) C E D A B (2) C E A D B (2) C A E B D (2) C A D B E (2) A B C E D (2) E C A D B (1) E A C B D (1) D E C B A (1) D B E C A (1) C E A B D (1) C D E A B (1) C A B E D (1) B E A D C (1) B A E D C (1) A D B C E (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 4 -2 8 B 6 0 14 -2 12 C -4 -14 0 -16 12 D 2 2 16 0 6 E -8 -12 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -2 8 B 6 0 14 -2 12 C -4 -14 0 -16 12 D 2 2 16 0 6 E -8 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=22 C=18 A=14 B=13 so B is eliminated. Round 2 votes counts: E=34 D=26 A=22 C=18 so C is eliminated. Round 3 votes counts: E=39 A=31 D=30 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 D:213 A:202 C:189 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 -2 8 B 6 0 14 -2 12 C -4 -14 0 -16 12 D 2 2 16 0 6 E -8 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 8 B 6 0 14 -2 12 C -4 -14 0 -16 12 D 2 2 16 0 6 E -8 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 8 B 6 0 14 -2 12 C -4 -14 0 -16 12 D 2 2 16 0 6 E -8 -12 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3959: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (15) E A C D B (13) D B E A C (8) B D C A E (8) C E A B D (7) C B D A E (6) D B A E C (5) E D A B C (4) E A D B C (4) C B A D E (4) B C D A E (4) E A D C B (3) D E B A C (2) D E A B C (2) C E A D B (2) C B A E D (2) A E C D B (2) E D A C B (1) E C A D B (1) D B C E A (1) C E D A B (1) C A B E D (1) B D A E C (1) B C A D E (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 16 -6 14 2 B -16 0 -18 0 -18 C 6 18 0 20 6 D -14 0 -20 0 -16 E -2 18 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -6 14 2 B -16 0 -18 0 -18 C 6 18 0 20 6 D -14 0 -20 0 -16 E -2 18 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=26 D=18 B=14 A=4 so A is eliminated. Round 2 votes counts: C=39 E=29 D=18 B=14 so B is eliminated. Round 3 votes counts: C=44 E=29 D=27 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:225 A:213 E:213 D:175 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -6 14 2 B -16 0 -18 0 -18 C 6 18 0 20 6 D -14 0 -20 0 -16 E -2 18 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 14 2 B -16 0 -18 0 -18 C 6 18 0 20 6 D -14 0 -20 0 -16 E -2 18 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 14 2 B -16 0 -18 0 -18 C 6 18 0 20 6 D -14 0 -20 0 -16 E -2 18 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3960: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E B C A D (8) B C D A E (8) B D C A E (7) A D C E B (6) E A C D B (4) C A D B E (4) B E C D A (4) E A D C B (3) E A B C D (3) B E D C A (3) B C E A D (3) E C A B D (2) E B A D C (2) D B E A C (2) D A E C B (2) D A C E B (2) D A B C E (2) C D A B E (2) C A B D E (2) A E C D B (2) A D E C B (2) E C B A D (1) E B D C A (1) E B D A C (1) E B C D A (1) E B A C D (1) D C B A E (1) D C A B E (1) D B A C E (1) C E B A D (1) C B E A D (1) C B A D E (1) C A D E B (1) B E C A D (1) B D E A C (1) B D A C E (1) B C D E A (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 0 12 B 4 0 4 10 12 C 10 -4 0 6 12 D 0 -10 -6 0 14 E -12 -12 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 0 12 B 4 0 4 10 12 C 10 -4 0 6 12 D 0 -10 -6 0 14 E -12 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 D=20 C=12 A=11 so A is eliminated. Round 2 votes counts: B=30 E=29 D=28 C=13 so C is eliminated. Round 3 votes counts: D=36 B=34 E=30 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:212 A:199 D:199 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 0 12 B 4 0 4 10 12 C 10 -4 0 6 12 D 0 -10 -6 0 14 E -12 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 0 12 B 4 0 4 10 12 C 10 -4 0 6 12 D 0 -10 -6 0 14 E -12 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 0 12 B 4 0 4 10 12 C 10 -4 0 6 12 D 0 -10 -6 0 14 E -12 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3961: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D B E A (7) C D A B E (7) C A E D B (6) A D C B E (6) E C B D A (4) D C B A E (4) E B C A D (3) C A D E B (3) A C D E B (3) E B D C A (2) E B C D A (2) E A B D C (2) D C A B E (2) D B A E C (2) D A C B E (2) C E A D B (2) C A D B E (2) A E C B D (2) A E B D C (2) A C D B E (2) A B E D C (2) A B D E C (2) E B A C D (1) E A C B D (1) D B C E A (1) D B C A E (1) D B A C E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D B A E (1) B E A D C (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A D C E B (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 10 -12 10 12 B -10 0 -26 -24 -2 C 12 26 0 10 22 D -10 24 -10 0 8 E -12 2 -22 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 10 12 B -10 0 -26 -24 -2 C 12 26 0 10 22 D -10 24 -10 0 8 E -12 2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=24 A=24 D=14 B=4 so B is eliminated. Round 2 votes counts: C=35 E=25 A=24 D=16 so D is eliminated. Round 3 votes counts: C=44 A=30 E=26 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:235 A:210 D:206 E:180 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 10 12 B -10 0 -26 -24 -2 C 12 26 0 10 22 D -10 24 -10 0 8 E -12 2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 10 12 B -10 0 -26 -24 -2 C 12 26 0 10 22 D -10 24 -10 0 8 E -12 2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 10 12 B -10 0 -26 -24 -2 C 12 26 0 10 22 D -10 24 -10 0 8 E -12 2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3962: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) C D E B A (7) B D C A E (7) E C D B A (5) E A C D B (5) B A D C E (5) A E B D C (5) E D C B A (4) E C D A B (4) D C B A E (4) C D B A E (4) E A B D C (3) D C B E A (3) B D A C E (3) A C D B E (3) E A B C D (2) C E D A B (2) C D B E A (2) B D E C A (2) E D B C A (1) E C A D B (1) E B D A C (1) E B A D C (1) E A C B D (1) D B C E A (1) D B C A E (1) C E D B A (1) C D E A B (1) C D A E B (1) C D A B E (1) C A D B E (1) B A D E C (1) A E C B D (1) A E B C D (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -6 -10 6 B 6 0 -4 -4 6 C 6 4 0 -10 22 D 10 4 10 0 22 E -6 -6 -22 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -10 6 B 6 0 -4 -4 6 C 6 4 0 -10 22 D 10 4 10 0 22 E -6 -6 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 C=20 B=18 D=9 so D is eliminated. Round 2 votes counts: E=28 C=27 A=25 B=20 so B is eliminated. Round 3 votes counts: C=36 A=34 E=30 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:223 C:211 B:202 A:192 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -10 6 B 6 0 -4 -4 6 C 6 4 0 -10 22 D 10 4 10 0 22 E -6 -6 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -10 6 B 6 0 -4 -4 6 C 6 4 0 -10 22 D 10 4 10 0 22 E -6 -6 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -10 6 B 6 0 -4 -4 6 C 6 4 0 -10 22 D 10 4 10 0 22 E -6 -6 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3963: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) C A D B E (7) B D E A C (7) E D B A C (6) D B A C E (6) C A E B D (5) E C A B D (4) E B D C A (4) E B D A C (4) C A E D B (4) A C D B E (4) D B E A C (3) D A C E B (3) B E D C A (3) E C A D B (2) E B C A D (2) E A C D B (2) D A B C E (2) C A B D E (2) B E D A C (2) B E C A D (2) A C D E B (2) E D A C B (1) E D A B C (1) E B C D A (1) E A D C B (1) D E B A C (1) D E A C B (1) D A E C B (1) D A E B C (1) D A C B E (1) D A B E C (1) C B A E D (1) C B A D E (1) B D C A E (1) B D A C E (1) B C A D E (1) A D C B E (1) Total count = 100 A B C D E A 0 8 4 -2 6 B -8 0 -2 -18 -8 C -4 2 0 -4 0 D 2 18 4 0 10 E -6 8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -2 6 B -8 0 -2 -18 -8 C -4 2 0 -4 0 D 2 18 4 0 10 E -6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 D=20 B=17 A=7 so A is eliminated. Round 2 votes counts: C=34 E=28 D=21 B=17 so B is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:217 A:208 C:197 E:196 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -2 6 B -8 0 -2 -18 -8 C -4 2 0 -4 0 D 2 18 4 0 10 E -6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -2 6 B -8 0 -2 -18 -8 C -4 2 0 -4 0 D 2 18 4 0 10 E -6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -2 6 B -8 0 -2 -18 -8 C -4 2 0 -4 0 D 2 18 4 0 10 E -6 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3964: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) D C B A E (6) E C D B A (5) E C B A D (5) C B A E D (5) D E C A B (4) D A B E C (4) A B E D C (4) A B C D E (4) E C A B D (3) E A B D C (3) C B A D E (3) B A C E D (3) B A C D E (3) A B E C D (3) A B D E C (3) A B D C E (3) E A B C D (2) D E A B C (2) D B A C E (2) D A E B C (2) D A B C E (2) C E D B A (2) C E B A D (2) C D B A E (2) E D C B A (1) E D A C B (1) E A D B C (1) E A C B D (1) D E C B A (1) D E A C B (1) D B C A E (1) D A C B E (1) C E B D A (1) C D E B A (1) C D B E A (1) C B D A E (1) B C A E D (1) B C A D E (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 4 12 B -6 0 -8 4 10 C 6 8 0 0 -8 D -4 -4 0 0 -2 E -12 -10 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.461538 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745261 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.769231 D: 0.769231 E: 1.000000 A B C D E A 0 6 -6 4 12 B -6 0 -8 4 10 C 6 8 0 0 -8 D -4 -4 0 0 -2 E -12 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.461538 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745528 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.769231 D: 0.769231 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 A=19 C=18 B=8 so B is eliminated. Round 2 votes counts: E=29 D=26 A=25 C=20 so C is eliminated. Round 3 votes counts: A=35 E=34 D=31 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:203 B:200 D:195 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -6 4 12 B -6 0 -8 4 10 C 6 8 0 0 -8 D -4 -4 0 0 -2 E -12 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.461538 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745528 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.769231 D: 0.769231 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 4 12 B -6 0 -8 4 10 C 6 8 0 0 -8 D -4 -4 0 0 -2 E -12 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.461538 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745528 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.769231 D: 0.769231 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 4 12 B -6 0 -8 4 10 C 6 8 0 0 -8 D -4 -4 0 0 -2 E -12 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.000000 C: 0.461538 D: 0.000000 E: 0.230769 Sum of squares = 0.360946745528 Cumulative probabilities = A: 0.307692 B: 0.307692 C: 0.769231 D: 0.769231 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3965: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) B E C D A (10) E C D A B (7) D A C E B (7) B E C A D (6) A D C B E (5) E B C D A (3) C E D A B (3) B E D C A (3) B A D E C (3) B A D C E (3) A D B C E (3) E C B A D (2) D B A C E (2) C E A D B (2) C D E A B (2) C A E D B (2) B E A D C (2) B D E A C (2) B A E C D (2) A C E D B (2) E D C B A (1) E C D B A (1) E C B D A (1) E C A D B (1) D E C A B (1) D C E A B (1) D C A E B (1) D A C B E (1) D A B C E (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A C D (1) B D E C A (1) B A E D C (1) A C D E B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 0 0 -2 B -10 0 -10 -14 -2 C 0 10 0 -2 2 D 0 14 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 10 0 0 -2 B -10 0 -10 -14 -2 C 0 10 0 -2 2 D 0 14 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=24 E=16 D=14 C=11 so C is eliminated. Round 2 votes counts: B=35 A=27 E=21 D=17 so D is eliminated. Round 3 votes counts: A=38 B=37 E=25 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:207 C:205 A:204 E:202 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 0 0 -2 B -10 0 -10 -14 -2 C 0 10 0 -2 2 D 0 14 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 0 -2 B -10 0 -10 -14 -2 C 0 10 0 -2 2 D 0 14 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 0 -2 B -10 0 -10 -14 -2 C 0 10 0 -2 2 D 0 14 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3966: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (13) C A B D E (9) C A D E B (8) C A B E D (8) A C D E B (7) E D B A C (6) D E B A C (5) B E D A C (5) A C E D B (5) B D E C A (4) E A B D C (3) D E A B C (3) C B A E D (3) B C A E D (3) D B E C A (2) C A D B E (2) B C E D A (2) B C E A D (2) E D A B C (1) E B D A C (1) D E A C B (1) D A E C B (1) D A C E B (1) C A E B D (1) B E C D A (1) B C D E A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -18 6 0 B -4 0 4 14 10 C 18 -4 0 6 6 D -6 -14 -6 0 -10 E 0 -10 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627219072 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -18 6 0 B -4 0 4 14 10 C 18 -4 0 6 6 D -6 -14 -6 0 -10 E 0 -10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627219607 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=31 B=31 A=14 D=13 E=11 so E is eliminated. Round 2 votes counts: B=32 C=31 D=20 A=17 so A is eliminated. Round 3 votes counts: C=44 B=35 D=21 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:212 E:197 A:196 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -18 6 0 B -4 0 4 14 10 C 18 -4 0 6 6 D -6 -14 -6 0 -10 E 0 -10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627219607 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -18 6 0 B -4 0 4 14 10 C 18 -4 0 6 6 D -6 -14 -6 0 -10 E 0 -10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627219607 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -18 6 0 B -4 0 4 14 10 C 18 -4 0 6 6 D -6 -14 -6 0 -10 E 0 -10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627219607 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3967: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) C D B A E (10) A E B D C (7) E D B A C (6) E D C B A (5) C A B D E (5) A B E D C (5) C D E B A (4) C D B E A (4) E C A D B (3) D C E B A (3) E D B C A (2) E C D A B (2) D E C B A (2) C E D B A (2) C B D A E (2) B A D E C (2) A B E C D (2) A B D E C (2) A B D C E (2) E D C A B (1) E C D B A (1) E B A D C (1) E A D B C (1) D E B C A (1) D E B A C (1) D B E A C (1) D B C A E (1) C E D A B (1) C D A B E (1) C B A D E (1) B D A C E (1) B A E D C (1) B A D C E (1) B A C D E (1) A E B C D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 -2 -4 B 6 0 4 -4 -10 C 2 -4 0 -14 -16 D 2 4 14 0 -8 E 4 10 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 -2 -4 B 6 0 4 -4 -10 C 2 -4 0 -14 -16 D 2 4 14 0 -8 E 4 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=30 A=22 D=9 B=6 so B is eliminated. Round 2 votes counts: E=33 C=30 A=27 D=10 so D is eliminated. Round 3 votes counts: E=38 C=34 A=28 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:206 B:198 A:193 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -2 -4 B 6 0 4 -4 -10 C 2 -4 0 -14 -16 D 2 4 14 0 -8 E 4 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -2 -4 B 6 0 4 -4 -10 C 2 -4 0 -14 -16 D 2 4 14 0 -8 E 4 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -2 -4 B 6 0 4 -4 -10 C 2 -4 0 -14 -16 D 2 4 14 0 -8 E 4 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3968: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) A E B C D (7) D C E B A (5) C B D E A (5) E A D C B (4) E A C B D (4) D E C A B (4) A E D B C (4) A E B D C (4) E D C A B (3) A E D C B (3) A D E B C (3) E C B A D (2) E A C D B (2) D C E A B (2) D B C A E (2) D A B C E (2) C B E A D (2) B C E A D (2) B C D A E (2) B C A E D (2) B A D C E (2) B A C E D (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) E A B C D (1) D E C B A (1) D E A C B (1) D B C E A (1) D B A C E (1) D A E C B (1) D A C E B (1) D A C B E (1) C E D B A (1) C B E D A (1) B D C A E (1) B D A C E (1) B C A D E (1) B A E C D (1) B A C D E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 2 6 -8 B -10 0 -12 -10 -14 C -2 12 0 -18 -4 D -6 10 18 0 -2 E 8 14 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 2 6 -8 B -10 0 -12 -10 -14 C -2 12 0 -18 -4 D -6 10 18 0 -2 E 8 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=25 E=20 B=15 C=9 so C is eliminated. Round 2 votes counts: D=31 A=25 B=23 E=21 so E is eliminated. Round 3 votes counts: A=38 D=37 B=25 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:214 D:210 A:205 C:194 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 6 -8 B -10 0 -12 -10 -14 C -2 12 0 -18 -4 D -6 10 18 0 -2 E 8 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 6 -8 B -10 0 -12 -10 -14 C -2 12 0 -18 -4 D -6 10 18 0 -2 E 8 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 6 -8 B -10 0 -12 -10 -14 C -2 12 0 -18 -4 D -6 10 18 0 -2 E 8 14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3969: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) D C A B E (7) E B A C D (6) D C A E B (6) B E A C D (6) D C B A E (5) D A B E C (5) C D B E A (5) E A B D C (4) C D E B A (4) C D E A B (4) C E B A D (3) B A E D C (3) E C D A B (2) D C E A B (2) C E D B A (2) C B A E D (2) A E B D C (2) A B E D C (2) E B C A D (1) E A D B C (1) D E A C B (1) D E A B C (1) D B A C E (1) D A E B C (1) D A C B E (1) D A B C E (1) C E B D A (1) C D A B E (1) C B E A D (1) B C E A D (1) B C A E D (1) B A E C D (1) B A D E C (1) A E D B C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -20 -24 10 B 12 0 -18 -28 14 C 20 18 0 6 20 D 24 28 -6 0 20 E -10 -14 -20 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -20 -24 10 B 12 0 -18 -28 14 C 20 18 0 6 20 D 24 28 -6 0 20 E -10 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=31 E=14 B=13 A=7 so A is eliminated. Round 2 votes counts: C=35 D=32 E=17 B=16 so B is eliminated. Round 3 votes counts: C=37 D=34 E=29 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:233 C:232 B:190 A:177 E:168 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -20 -24 10 B 12 0 -18 -28 14 C 20 18 0 6 20 D 24 28 -6 0 20 E -10 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -24 10 B 12 0 -18 -28 14 C 20 18 0 6 20 D 24 28 -6 0 20 E -10 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -24 10 B 12 0 -18 -28 14 C 20 18 0 6 20 D 24 28 -6 0 20 E -10 -14 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3970: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (19) E A D C B (18) C B E A D (10) C B D A E (10) E A D B C (9) D A E B C (6) B C D E A (3) A E D B C (3) D C A B E (2) D B C A E (2) D A E C B (2) B D C A E (2) A E D C B (2) E A C D B (1) D B A E C (1) D A C B E (1) D A B E C (1) D A B C E (1) C B E D A (1) C B D E A (1) C B A E D (1) B C E D A (1) B C E A D (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -6 -6 10 B 4 0 0 -2 14 C 6 0 0 -4 10 D 6 2 4 0 6 E -10 -14 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -6 10 B 4 0 0 -2 14 C 6 0 0 -4 10 D 6 2 4 0 6 E -10 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=26 C=23 D=16 A=7 so A is eliminated. Round 2 votes counts: E=33 B=26 C=23 D=18 so D is eliminated. Round 3 votes counts: E=43 B=31 C=26 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:209 B:208 C:206 A:197 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -6 10 B 4 0 0 -2 14 C 6 0 0 -4 10 D 6 2 4 0 6 E -10 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -6 10 B 4 0 0 -2 14 C 6 0 0 -4 10 D 6 2 4 0 6 E -10 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -6 10 B 4 0 0 -2 14 C 6 0 0 -4 10 D 6 2 4 0 6 E -10 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3971: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (17) D B A E C (13) C B A E D (6) C B A D E (5) E C A D B (4) D A E B C (4) B D C A E (4) E A D B C (3) E A C D B (3) C E B A D (3) C B D E A (3) B D A C E (3) E C D A B (2) D E A B C (2) D B C A E (2) D A B E C (2) C B E D A (2) C B E A D (2) C B D A E (2) A E D B C (2) E D C A B (1) E D A C B (1) E C A B D (1) E A D C B (1) D E B C A (1) D E B A C (1) D C B E A (1) D B C E A (1) D B A C E (1) C A B E D (1) B C D A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -18 6 2 B 6 0 -14 6 4 C 18 14 0 10 12 D -6 -6 -10 0 -2 E -2 -4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 6 2 B 6 0 -14 6 4 C 18 14 0 10 12 D -6 -6 -10 0 -2 E -2 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 D=28 E=16 B=10 A=5 so A is eliminated. Round 2 votes counts: C=41 D=29 E=20 B=10 so B is eliminated. Round 3 votes counts: C=43 D=37 E=20 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:201 A:192 E:192 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -18 6 2 B 6 0 -14 6 4 C 18 14 0 10 12 D -6 -6 -10 0 -2 E -2 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 6 2 B 6 0 -14 6 4 C 18 14 0 10 12 D -6 -6 -10 0 -2 E -2 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 6 2 B 6 0 -14 6 4 C 18 14 0 10 12 D -6 -6 -10 0 -2 E -2 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3972: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) A B E D C (9) C D E A B (8) B A E D C (8) C D E B A (6) A C B D E (6) E D B A C (5) A C B E D (5) A B E C D (5) D E B C A (4) A B C E D (4) D E B A C (3) D C E B A (3) C D A E B (3) C A B D E (3) B E D A C (3) A B C D E (3) E D B C A (2) C A D E B (2) E D C B A (1) E B D A C (1) E B C D A (1) C E D B A (1) C A D B E (1) C A B E D (1) B D E A C (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 10 -2 4 B -2 0 2 4 2 C -10 -2 0 -2 -8 D 2 -4 2 0 8 E -4 -2 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 -2 4 B -2 0 2 4 2 C -10 -2 0 -2 -8 D 2 -4 2 0 8 E -4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=25 D=19 B=13 E=10 so E is eliminated. Round 2 votes counts: A=33 D=27 C=25 B=15 so B is eliminated. Round 3 votes counts: A=42 D=32 C=26 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:204 B:203 E:197 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 -2 4 B -2 0 2 4 2 C -10 -2 0 -2 -8 D 2 -4 2 0 8 E -4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -2 4 B -2 0 2 4 2 C -10 -2 0 -2 -8 D 2 -4 2 0 8 E -4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -2 4 B -2 0 2 4 2 C -10 -2 0 -2 -8 D 2 -4 2 0 8 E -4 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3973: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) E B D A C (12) E D C A B (9) B A C E D (7) D E C A B (6) D C E A B (6) B A C D E (6) A B C D E (6) C A D B E (5) A C B D E (5) E D C B A (4) C A D E B (4) B E A D C (4) C D A E B (3) E B D C A (2) B A E C D (2) E D B A C (1) B E D A C (1) B E A C D (1) B A E D C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -4 -14 -18 B 8 0 14 -4 -20 C 4 -14 0 -18 -12 D 14 4 18 0 -16 E 18 20 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -4 -14 -18 B 8 0 14 -4 -20 C 4 -14 0 -18 -12 D 14 4 18 0 -16 E 18 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=22 A=13 D=12 C=12 so D is eliminated. Round 2 votes counts: E=47 B=22 C=18 A=13 so A is eliminated. Round 3 votes counts: E=47 B=29 C=24 so C is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:233 D:210 B:199 C:180 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -14 -18 B 8 0 14 -4 -20 C 4 -14 0 -18 -12 D 14 4 18 0 -16 E 18 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -14 -18 B 8 0 14 -4 -20 C 4 -14 0 -18 -12 D 14 4 18 0 -16 E 18 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -14 -18 B 8 0 14 -4 -20 C 4 -14 0 -18 -12 D 14 4 18 0 -16 E 18 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3974: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) E D B A C (7) B D A E C (6) E D B C A (5) E A D B C (5) D B E C A (5) D B E A C (5) C E A D B (5) C D B E A (5) A E B D C (5) E B D A C (4) A C B D E (4) E A B D C (3) D B C E A (3) C A E D B (3) C A B D E (3) A C E B D (3) A B E D C (3) C A E B D (2) B D E A C (2) A B D E C (2) E C D B A (1) E C A D B (1) E A C D B (1) C D B A E (1) C B D A E (1) B D C A E (1) B A D E C (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 6 -14 -26 B 16 0 18 -16 -14 C -6 -18 0 -16 -14 D 14 16 16 0 -20 E 26 14 14 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 6 -14 -26 B 16 0 18 -16 -14 C -6 -18 0 -16 -14 D 14 16 16 0 -20 E 26 14 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=27 A=19 D=13 B=10 so B is eliminated. Round 2 votes counts: C=31 E=27 D=22 A=20 so A is eliminated. Round 3 votes counts: C=38 E=36 D=26 so D is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:237 D:213 B:202 A:175 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 6 -14 -26 B 16 0 18 -16 -14 C -6 -18 0 -16 -14 D 14 16 16 0 -20 E 26 14 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -14 -26 B 16 0 18 -16 -14 C -6 -18 0 -16 -14 D 14 16 16 0 -20 E 26 14 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -14 -26 B 16 0 18 -16 -14 C -6 -18 0 -16 -14 D 14 16 16 0 -20 E 26 14 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3975: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) C B E D A (11) C B A E D (11) E D B A C (10) C A D E B (9) B E D A C (7) A D E B C (7) C A B D E (4) E D B C A (3) D A E B C (3) B E D C A (3) A D E C B (3) A D C E B (3) A C D E B (3) B C E D A (2) E D A B C (1) D E A C B (1) D C A E B (1) D A E C B (1) C B E A D (1) C A D B E (1) B E C D A (1) B E C A D (1) B C A E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 2 -10 -4 B 2 0 0 -14 -12 C -2 0 0 -8 -4 D 10 14 8 0 -4 E 4 12 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 -10 -4 B 2 0 0 -14 -12 C -2 0 0 -8 -4 D 10 14 8 0 -4 E 4 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=17 A=17 B=15 E=14 so E is eliminated. Round 2 votes counts: C=37 D=31 A=17 B=15 so B is eliminated. Round 3 votes counts: C=42 D=41 A=17 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:212 A:193 C:193 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -10 -4 B 2 0 0 -14 -12 C -2 0 0 -8 -4 D 10 14 8 0 -4 E 4 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -10 -4 B 2 0 0 -14 -12 C -2 0 0 -8 -4 D 10 14 8 0 -4 E 4 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -10 -4 B 2 0 0 -14 -12 C -2 0 0 -8 -4 D 10 14 8 0 -4 E 4 12 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3976: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) D C E B A (6) A E B D C (6) A B E D C (6) D C E A B (5) C B D E A (5) B C E D A (5) A B C E D (5) D E C B A (3) D C A E B (3) C D E B A (3) C D B E A (3) B E C D A (3) B E A C D (3) E D B C A (2) E D A B C (2) E A D B C (2) D E C A B (2) D E A C B (2) B C A E D (2) A D E C B (2) A C B D E (2) E D B A C (1) E D A C B (1) E B D C A (1) E B D A C (1) E A B D C (1) D A E C B (1) C D A E B (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) B E D A C (1) B C A D E (1) B A E C D (1) B A C E D (1) A D E B C (1) A D C E B (1) A D B E C (1) A C D E B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 2 -6 -4 B -12 0 8 -6 -10 C -2 -8 0 -16 -2 D 6 6 16 0 -4 E 4 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 -6 -4 B -12 0 8 -6 -10 C -2 -8 0 -16 -2 D 6 6 16 0 -4 E 4 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=22 B=17 C=16 E=11 so E is eliminated. Round 2 votes counts: A=37 D=28 B=19 C=16 so C is eliminated. Round 3 votes counts: A=38 D=36 B=26 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:210 A:202 B:190 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 -6 -4 B -12 0 8 -6 -10 C -2 -8 0 -16 -2 D 6 6 16 0 -4 E 4 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -6 -4 B -12 0 8 -6 -10 C -2 -8 0 -16 -2 D 6 6 16 0 -4 E 4 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -6 -4 B -12 0 8 -6 -10 C -2 -8 0 -16 -2 D 6 6 16 0 -4 E 4 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3977: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) D E B C A (7) C A B E D (6) A C B E D (6) E D B C A (5) D E B A C (5) C A D E B (5) E B D C A (4) A C D E B (4) A C B D E (4) D E C B A (3) C B A E D (3) B E D C A (3) B E C D A (3) A D C E B (3) A C D B E (3) D E C A B (2) D E A C B (2) D B E A C (2) C E B D A (2) B C E A D (2) E C D B A (1) D C E A B (1) D A E C B (1) D A C E B (1) C E D B A (1) C E A D B (1) C D E A B (1) C A E B D (1) B E C A D (1) B D E A C (1) B A E D C (1) B A E C D (1) B A D E C (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -4 -6 -10 B 8 0 -6 -4 -4 C 4 6 0 -8 -8 D 6 4 8 0 0 E 10 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.746684 E: 0.253316 Sum of squares = 0.621705505864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.746684 E: 1.000000 A B C D E A 0 -8 -4 -6 -10 B 8 0 -6 -4 -4 C 4 6 0 -8 -8 D 6 4 8 0 0 E 10 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=24 B=21 C=20 E=10 so E is eliminated. Round 2 votes counts: D=29 B=25 A=25 C=21 so C is eliminated. Round 3 votes counts: A=38 D=32 B=30 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:209 B:197 C:197 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 -6 -10 B 8 0 -6 -4 -4 C 4 6 0 -8 -8 D 6 4 8 0 0 E 10 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -6 -10 B 8 0 -6 -4 -4 C 4 6 0 -8 -8 D 6 4 8 0 0 E 10 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -6 -10 B 8 0 -6 -4 -4 C 4 6 0 -8 -8 D 6 4 8 0 0 E 10 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3978: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) B E C A D (6) E D B C A (5) E B A C D (5) D C A B E (5) D A C E B (5) C B E D A (5) A D C B E (5) E B C A D (4) C D B E A (4) B E A C D (4) A E B D C (4) E A B D C (3) D C E B A (3) D A C B E (3) C A B E D (3) A B E C D (3) E B A D C (2) D A E C B (2) C D A B E (2) A D E B C (2) A B C E D (2) E C B D A (1) E B D C A (1) D E C B A (1) D E B A C (1) D E A B C (1) D C E A B (1) D C B E A (1) D C A E B (1) D A E B C (1) C B D A E (1) C B A D E (1) A D E C B (1) A D C E B (1) A D B E C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -4 -2 -10 B 4 0 6 8 -4 C 4 -6 0 -2 -12 D 2 -8 2 0 -14 E 10 4 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 -10 B 4 0 6 8 -4 C 4 -6 0 -2 -12 D 2 -8 2 0 -14 E 10 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=25 A=21 C=16 B=10 so B is eliminated. Round 2 votes counts: E=38 D=25 A=21 C=16 so C is eliminated. Round 3 votes counts: E=43 D=32 A=25 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:207 C:192 D:191 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -10 B 4 0 6 8 -4 C 4 -6 0 -2 -12 D 2 -8 2 0 -14 E 10 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -10 B 4 0 6 8 -4 C 4 -6 0 -2 -12 D 2 -8 2 0 -14 E 10 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -10 B 4 0 6 8 -4 C 4 -6 0 -2 -12 D 2 -8 2 0 -14 E 10 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3979: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) D B E C A (7) C B A E D (6) C A B E D (6) D A E C B (5) E C A B D (4) D E B A C (4) E D A B C (3) E C B A D (3) D E B C A (3) D E A B C (3) D A C B E (3) C A E B D (3) A D C E B (3) A C B E D (3) E D B C A (2) E B D C A (2) E A C B D (2) D B C E A (2) B D E C A (2) B D C A E (2) A E C D B (2) A C B D E (2) E D B A C (1) E B C A D (1) E A D C B (1) E A C D B (1) D B E A C (1) D B C A E (1) D A C E B (1) C E A B D (1) C B E A D (1) C B A D E (1) C A B D E (1) B E D C A (1) B E C A D (1) B D C E A (1) B C E A D (1) B C D E A (1) A D E C B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 12 -6 10 2 B -12 0 -22 8 -12 C 6 22 0 0 0 D -10 -8 0 0 -8 E -2 12 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.558187 D: 0.000000 E: 0.441813 Sum of squares = 0.506771552733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.558187 D: 0.558187 E: 1.000000 A B C D E A 0 12 -6 10 2 B -12 0 -22 8 -12 C 6 22 0 0 0 D -10 -8 0 0 -8 E -2 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=22 E=20 C=19 B=9 so B is eliminated. Round 2 votes counts: D=35 E=22 A=22 C=21 so C is eliminated. Round 3 votes counts: A=39 D=36 E=25 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:214 A:209 E:209 D:187 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 10 2 B -12 0 -22 8 -12 C 6 22 0 0 0 D -10 -8 0 0 -8 E -2 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 10 2 B -12 0 -22 8 -12 C 6 22 0 0 0 D -10 -8 0 0 -8 E -2 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 10 2 B -12 0 -22 8 -12 C 6 22 0 0 0 D -10 -8 0 0 -8 E -2 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3980: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) E A C D B (9) E A C B D (9) D B E C A (9) D B C A E (9) D B C E A (6) E D A C B (4) E C A B D (4) E A D C B (4) D E B A C (4) D B A C E (4) D E A B C (3) A E C B D (3) E D A B C (2) C A B E D (2) B D C E A (2) B D C A E (2) B C D A E (2) D B E A C (1) D A C B E (1) C B A E D (1) C A E B D (1) B E D C A (1) B C E D A (1) B C A E D (1) B C A D E (1) A E C D B (1) A D C E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 12 16 -2 -18 B -12 0 -4 -16 -10 C -16 4 0 -6 -8 D 2 16 6 0 -8 E 18 10 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 16 -2 -18 B -12 0 -4 -16 -10 C -16 4 0 -6 -8 D 2 16 6 0 -8 E 18 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=32 A=17 B=10 C=4 so C is eliminated. Round 2 votes counts: D=37 E=32 A=20 B=11 so B is eliminated. Round 3 votes counts: D=43 E=34 A=23 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:208 A:204 C:187 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 16 -2 -18 B -12 0 -4 -16 -10 C -16 4 0 -6 -8 D 2 16 6 0 -8 E 18 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 -2 -18 B -12 0 -4 -16 -10 C -16 4 0 -6 -8 D 2 16 6 0 -8 E 18 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 -2 -18 B -12 0 -4 -16 -10 C -16 4 0 -6 -8 D 2 16 6 0 -8 E 18 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3981: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (7) A E B C D (6) D A E C B (5) D A C B E (5) A D E C B (5) E B C A D (4) E A B C D (4) D E A C B (4) D C B A E (4) B C A E D (4) A E D B C (4) E C B D A (3) E A D B C (3) C D B E A (3) C B D E A (3) B C E A D (3) A B C D E (3) E D A C B (2) E A D C B (2) D E C B A (2) D C E B A (2) B C D E A (2) B A C D E (2) A D B C E (2) E D C B A (1) E B C D A (1) D E C A B (1) C D B A E (1) C B E D A (1) C B D A E (1) B E C D A (1) B E C A D (1) B C E D A (1) B C D A E (1) B C A D E (1) A E B D C (1) A D C B E (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 16 18 12 12 B -16 0 -6 -10 -14 C -18 6 0 -6 -16 D -12 10 6 0 0 E -12 14 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 12 12 B -16 0 -6 -10 -14 C -18 6 0 -6 -16 D -12 10 6 0 0 E -12 14 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 E=20 B=16 C=9 so C is eliminated. Round 2 votes counts: A=32 D=27 B=21 E=20 so E is eliminated. Round 3 votes counts: A=41 D=30 B=29 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:209 D:202 C:183 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 12 12 B -16 0 -6 -10 -14 C -18 6 0 -6 -16 D -12 10 6 0 0 E -12 14 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 12 12 B -16 0 -6 -10 -14 C -18 6 0 -6 -16 D -12 10 6 0 0 E -12 14 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 12 12 B -16 0 -6 -10 -14 C -18 6 0 -6 -16 D -12 10 6 0 0 E -12 14 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3982: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (13) D A C E B (10) C D A E B (8) B E A D C (7) B E C A D (6) B A D E C (6) E B C D A (5) C E D A B (5) C E B D A (5) A D B E C (5) E B C A D (4) C E B A D (3) B E A C D (3) E C B D A (2) E C B A D (2) D C A E B (2) D A C B E (2) B A E D C (2) D A B C E (1) C E D B A (1) C E A B D (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C D A (1) A D C E B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 6 14 8 B -4 0 -14 -2 -2 C -6 14 0 -4 10 D -14 2 4 0 2 E -8 2 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 14 8 B -4 0 -14 -2 -2 C -6 14 0 -4 10 D -14 2 4 0 2 E -8 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 A=21 D=15 E=13 so E is eliminated. Round 2 votes counts: B=35 C=29 A=21 D=15 so D is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:207 D:197 E:191 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 14 8 B -4 0 -14 -2 -2 C -6 14 0 -4 10 D -14 2 4 0 2 E -8 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 14 8 B -4 0 -14 -2 -2 C -6 14 0 -4 10 D -14 2 4 0 2 E -8 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 14 8 B -4 0 -14 -2 -2 C -6 14 0 -4 10 D -14 2 4 0 2 E -8 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3983: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) D C B A E (9) D B C E A (7) A E D C B (5) E B A C D (4) B E A C D (4) B C D E A (4) A E C B D (4) E B A D C (3) D C B E A (3) C D B E A (3) C A D E B (3) B E A D C (3) A E D B C (3) A E C D B (3) A C E D B (3) D C A E B (2) D C A B E (2) D B E A C (2) B E C D A (2) B D C E A (2) A E B D C (2) E C B A D (1) E A C B D (1) E A B D C (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B E C (1) C D B A E (1) C D A E B (1) C D A B E (1) C A E D B (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E A C (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 8 4 -8 B 8 0 8 -10 0 C -8 -8 0 -6 -10 D -4 10 6 0 -6 E 8 0 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.177263 C: 0.000000 D: 0.000000 E: 0.822737 Sum of squares = 0.708317835132 Cumulative probabilities = A: 0.000000 B: 0.177263 C: 0.177263 D: 0.177263 E: 1.000000 A B C D E A 0 -8 8 4 -8 B 8 0 8 -10 0 C -8 -8 0 -6 -10 D -4 10 6 0 -6 E 8 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250032621 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=21 B=20 E=19 C=10 so C is eliminated. Round 2 votes counts: D=36 A=25 B=20 E=19 so E is eliminated. Round 3 votes counts: D=36 A=36 B=28 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:212 B:203 D:203 A:198 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 8 4 -8 B 8 0 8 -10 0 C -8 -8 0 -6 -10 D -4 10 6 0 -6 E 8 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250032621 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 4 -8 B 8 0 8 -10 0 C -8 -8 0 -6 -10 D -4 10 6 0 -6 E 8 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250032621 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 4 -8 B 8 0 8 -10 0 C -8 -8 0 -6 -10 D -4 10 6 0 -6 E 8 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250032621 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3984: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) A D B E C (6) E C B D A (5) E B A D C (5) B E A D C (5) A D C B E (5) C D A B E (4) A D B C E (4) A B D E C (4) E B C D A (3) E B C A D (3) E B A C D (3) D C A B E (3) C E D A B (3) C D E B A (3) B D E A C (3) B D A E C (3) E C B A D (2) E A B C D (2) D A C B E (2) D A B C E (2) C E B A D (2) C E A D B (2) C E A B D (2) C D A E B (2) C A D E B (2) E B D C A (1) E A C B D (1) C E D B A (1) C D B A E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D C E A (1) A E B D C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -2 4 -16 B 4 0 -4 14 -6 C 2 4 0 6 2 D -4 -14 -6 0 -4 E 16 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999052 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 4 -16 B 4 0 -4 14 -6 C 2 4 0 6 2 D -4 -14 -6 0 -4 E 16 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=25 A=22 B=14 D=7 so D is eliminated. Round 2 votes counts: C=35 A=26 E=25 B=14 so B is eliminated. Round 3 votes counts: C=36 E=35 A=29 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:207 B:204 A:191 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 4 -16 B 4 0 -4 14 -6 C 2 4 0 6 2 D -4 -14 -6 0 -4 E 16 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 4 -16 B 4 0 -4 14 -6 C 2 4 0 6 2 D -4 -14 -6 0 -4 E 16 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 4 -16 B 4 0 -4 14 -6 C 2 4 0 6 2 D -4 -14 -6 0 -4 E 16 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3985: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D A E B C (6) D A B E C (6) E A B C D (5) C D E A B (5) B C E A D (5) A E D B C (5) E C B A D (4) B E C A D (4) A E B D C (4) A D E B C (4) E B A C D (3) C D B E A (3) C B E D A (3) B D A E C (3) E B C A D (2) D C E A B (2) D C A E B (2) C E D A B (2) C E B D A (2) C B D E A (2) B C D A E (2) E A B D C (1) D C B A E (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D B A E (1) C B E A D (1) C B D A E (1) B C D E A (1) B A D E C (1) B A D C E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -8 0 -10 B 4 0 16 8 -18 C 8 -16 0 12 -2 D 0 -8 -12 0 0 E 10 18 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.089727 E: 0.910273 Sum of squares = 0.836648638196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.089727 E: 1.000000 A B C D E A 0 -4 -8 0 -10 B 4 0 16 8 -18 C 8 -16 0 12 -2 D 0 -8 -12 0 0 E 10 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102050668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=23 B=17 E=15 A=15 so E is eliminated. Round 2 votes counts: C=34 D=23 B=22 A=21 so A is eliminated. Round 3 votes counts: C=34 D=33 B=33 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:205 C:201 D:190 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 0 -10 B 4 0 16 8 -18 C 8 -16 0 12 -2 D 0 -8 -12 0 0 E 10 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102050668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 0 -10 B 4 0 16 8 -18 C 8 -16 0 12 -2 D 0 -8 -12 0 0 E 10 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102050668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 0 -10 B 4 0 16 8 -18 C 8 -16 0 12 -2 D 0 -8 -12 0 0 E 10 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.755102050668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3986: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) E A C B D (10) D B C E A (10) A E C B D (5) D E A B C (4) C B D A E (4) E D A B C (3) E C A B D (3) E A D C B (3) E A C D B (3) D A E B C (3) C B A E D (3) B D C E A (3) B C D E A (3) B C D A E (3) A E C D B (3) A D C B E (3) E C B A D (2) E B D C A (2) C B A D E (2) E A D B C (1) D B E C A (1) D B E A C (1) D B A C E (1) D A B C E (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D E A (1) C A B E D (1) B E C D A (1) B C E D A (1) A E D C B (1) A E D B C (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -8 -8 -10 B 4 0 0 0 8 C 8 0 0 2 6 D 8 0 -2 0 4 E 10 -8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.349316 C: 0.650684 D: 0.000000 E: 0.000000 Sum of squares = 0.545411201667 Cumulative probabilities = A: 0.000000 B: 0.349316 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -8 -10 B 4 0 0 0 8 C 8 0 0 2 6 D 8 0 -2 0 4 E 10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=27 A=16 C=14 B=11 so B is eliminated. Round 2 votes counts: D=35 E=28 C=21 A=16 so A is eliminated. Round 3 votes counts: E=38 D=38 C=24 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:208 B:206 D:205 E:196 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -8 -10 B 4 0 0 0 8 C 8 0 0 2 6 D 8 0 -2 0 4 E 10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -8 -10 B 4 0 0 0 8 C 8 0 0 2 6 D 8 0 -2 0 4 E 10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -8 -10 B 4 0 0 0 8 C 8 0 0 2 6 D 8 0 -2 0 4 E 10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999836 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3987: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) B D C E A (7) E D A B C (6) D B E C A (6) C B A D E (6) C A B D E (6) B D C A E (5) E A D C B (4) C B D A E (4) A E D B C (4) A C B D E (4) E A D B C (3) D E B C A (3) D B E A C (3) C B D E A (3) E A C D B (2) C E A B D (2) B C D E A (2) B C D A E (2) A E C B D (2) A C B E D (2) E D B C A (1) E D A C B (1) E C D B A (1) E C A D B (1) D E B A C (1) D B C E A (1) D B A E C (1) D B A C E (1) C A E B D (1) C A B E D (1) A E C D B (1) A D E B C (1) A D B C E (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -4 -14 -12 B 12 0 16 -2 12 C 4 -16 0 -16 0 D 14 2 16 0 16 E 12 -12 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -14 -12 B 12 0 16 -2 12 C 4 -16 0 -16 0 D 14 2 16 0 16 E 12 -12 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 A=17 D=16 B=16 so D is eliminated. Round 2 votes counts: E=32 B=28 C=23 A=17 so A is eliminated. Round 3 votes counts: E=40 C=30 B=30 so C is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:224 B:219 E:192 C:186 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -4 -14 -12 B 12 0 16 -2 12 C 4 -16 0 -16 0 D 14 2 16 0 16 E 12 -12 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -14 -12 B 12 0 16 -2 12 C 4 -16 0 -16 0 D 14 2 16 0 16 E 12 -12 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -14 -12 B 12 0 16 -2 12 C 4 -16 0 -16 0 D 14 2 16 0 16 E 12 -12 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 3988: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) C E D A B (10) C A D B E (9) B A D E C (9) E B D C A (7) C E A D B (7) A B D C E (7) C A D E B (5) A D B C E (4) B D E A C (3) E D C B A (2) E C B D A (2) E B D A C (2) D B A E C (2) C D A E B (2) B D A E C (2) A C D B E (2) A C B D E (2) A B C D E (2) E C B A D (1) E B C A D (1) E B A D C (1) D E C B A (1) D B E A C (1) C E A B D (1) C A E B D (1) C A B D E (1) B E A D C (1) A D C B E (1) Total count = 100 A B C D E A 0 8 -22 10 -2 B -8 0 -16 -14 -8 C 22 16 0 14 8 D -10 14 -14 0 6 E 2 8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -22 10 -2 B -8 0 -16 -14 -8 C 22 16 0 14 8 D -10 14 -14 0 6 E 2 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=27 A=18 B=15 D=4 so D is eliminated. Round 2 votes counts: C=36 E=28 B=18 A=18 so B is eliminated. Round 3 votes counts: C=36 E=33 A=31 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:230 D:198 E:198 A:197 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -22 10 -2 B -8 0 -16 -14 -8 C 22 16 0 14 8 D -10 14 -14 0 6 E 2 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -22 10 -2 B -8 0 -16 -14 -8 C 22 16 0 14 8 D -10 14 -14 0 6 E 2 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -22 10 -2 B -8 0 -16 -14 -8 C 22 16 0 14 8 D -10 14 -14 0 6 E 2 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3989: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) E A B D C (7) A B E C D (6) E A B C D (5) D E C B A (5) A E B C D (5) D C E B A (4) C E D A B (4) C D B A E (4) B A E C D (4) E D B A C (3) E B A D C (3) D C E A B (3) D C B E A (3) C D E A B (3) B A E D C (3) B A C D E (3) E D A B C (2) E A C B D (2) C B A D E (2) C A E B D (2) E C A B D (1) E B D A C (1) D E C A B (1) D E B C A (1) D E B A C (1) D B E A C (1) C E A D B (1) C D A B E (1) C B D A E (1) C A B E D (1) C A B D E (1) B E A D C (1) B D A E C (1) B A D E C (1) B A C E D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 4 2 -4 B 6 0 0 6 -8 C -4 0 0 -2 -8 D -2 -6 2 0 -8 E 4 8 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 2 -4 B 6 0 0 6 -8 C -4 0 0 -2 -8 D -2 -6 2 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 C=20 B=14 A=13 so A is eliminated. Round 2 votes counts: E=29 D=29 C=21 B=21 so C is eliminated. Round 3 votes counts: D=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:202 A:198 C:193 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 2 -4 B 6 0 0 6 -8 C -4 0 0 -2 -8 D -2 -6 2 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 2 -4 B 6 0 0 6 -8 C -4 0 0 -2 -8 D -2 -6 2 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 2 -4 B 6 0 0 6 -8 C -4 0 0 -2 -8 D -2 -6 2 0 -8 E 4 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3990: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) B C A D E (7) A B C E D (7) D C B E A (6) E C A B D (5) B C D A E (4) A E D B C (4) A B E C D (4) E D C A B (3) E D A C B (3) E A C B D (3) D E C B A (3) D E C A B (3) D E A C B (3) D C E B A (3) D B C A E (3) D A B E C (3) D E B C A (2) D B A C E (2) C D B E A (2) C B A E D (2) E D A B C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E A B C (1) D C B A E (1) D B C E A (1) D A E B C (1) C E B A D (1) C D E B A (1) C B E A D (1) C B D E A (1) B D C A E (1) B C A E D (1) B A D C E (1) B A C E D (1) A D E B C (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -2 4 8 B -12 0 12 0 2 C 2 -12 0 0 -6 D -4 0 0 0 2 E -8 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.461538 B: 0.076923 C: 0.461538 D: 0.000000 E: 0.000000 Sum of squares = 0.43195266265 Cumulative probabilities = A: 0.461538 B: 0.538462 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 4 8 B -12 0 12 0 2 C 2 -12 0 0 -6 D -4 0 0 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461538 B: 0.076923 C: 0.461538 D: 0.000000 E: 0.000000 Sum of squares = 0.431952662697 Cumulative probabilities = A: 0.461538 B: 0.538462 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=27 E=18 B=15 C=8 so C is eliminated. Round 2 votes counts: D=35 A=27 E=19 B=19 so E is eliminated. Round 3 votes counts: D=42 A=38 B=20 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:201 D:199 E:197 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 -2 4 8 B -12 0 12 0 2 C 2 -12 0 0 -6 D -4 0 0 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461538 B: 0.076923 C: 0.461538 D: 0.000000 E: 0.000000 Sum of squares = 0.431952662697 Cumulative probabilities = A: 0.461538 B: 0.538462 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 4 8 B -12 0 12 0 2 C 2 -12 0 0 -6 D -4 0 0 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461538 B: 0.076923 C: 0.461538 D: 0.000000 E: 0.000000 Sum of squares = 0.431952662697 Cumulative probabilities = A: 0.461538 B: 0.538462 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 4 8 B -12 0 12 0 2 C 2 -12 0 0 -6 D -4 0 0 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.461538 B: 0.076923 C: 0.461538 D: 0.000000 E: 0.000000 Sum of squares = 0.431952662697 Cumulative probabilities = A: 0.461538 B: 0.538462 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3991: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (5) B A E D C (5) A C E D B (5) A B E C D (5) E D C B A (4) D E C B A (4) B D E C A (4) B A D E C (4) E B D C A (3) E A C D B (3) D C E B A (3) C D A E B (3) B E D C A (3) B E D A C (3) B A D C E (3) A C D B E (3) A B C D E (3) E B A C D (2) E A B C D (2) D C E A B (2) C E D A B (2) C D E A B (2) C A D E B (2) B E A D C (2) A E C D B (2) A E B C D (2) A C B E D (2) A C B D E (2) E D B C A (1) E C D B A (1) E C D A B (1) E C A B D (1) E B C D A (1) E B A D C (1) D E B C A (1) D C B E A (1) B D E A C (1) B D C E A (1) B D A C E (1) A E C B D (1) A C E B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 10 16 -8 B -2 0 0 10 -10 C -10 0 0 4 -26 D -16 -10 -4 0 -20 E 8 10 26 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 10 16 -8 B -2 0 0 10 -10 C -10 0 0 4 -26 D -16 -10 -4 0 -20 E 8 10 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 E=25 D=11 C=9 so C is eliminated. Round 2 votes counts: A=30 E=27 B=27 D=16 so D is eliminated. Round 3 votes counts: E=39 A=33 B=28 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:210 B:199 C:184 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 10 16 -8 B -2 0 0 10 -10 C -10 0 0 4 -26 D -16 -10 -4 0 -20 E 8 10 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 16 -8 B -2 0 0 10 -10 C -10 0 0 4 -26 D -16 -10 -4 0 -20 E 8 10 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 16 -8 B -2 0 0 10 -10 C -10 0 0 4 -26 D -16 -10 -4 0 -20 E 8 10 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3992: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B C D A (6) C E B A D (6) B E D A C (6) E B D C A (5) D B E A C (5) C A E B D (5) C A D E B (5) D B A E C (4) D A E B C (4) C E A B D (4) B E D C A (4) A D C B E (4) A C D E B (4) E D B C A (3) C A B E D (3) D E B A C (2) D A B E C (2) C B E A D (2) A D C E B (2) E D B A C (1) E C B A D (1) E B D A C (1) E B C A D (1) D E A B C (1) D A E C B (1) D A C B E (1) D A B C E (1) C E A D B (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B D E A C (1) B C A E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -2 2 -4 B 2 0 0 0 -6 C 2 0 0 4 0 D -2 0 -4 0 -4 E 4 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.530637 D: 0.000000 E: 0.469363 Sum of squares = 0.501877218758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.530637 D: 0.530637 E: 1.000000 A B C D E A 0 -2 -2 2 -4 B 2 0 0 0 -6 C 2 0 0 4 0 D -2 0 -4 0 -4 E 4 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=21 A=19 E=18 B=14 so B is eliminated. Round 2 votes counts: E=30 C=29 D=22 A=19 so A is eliminated. Round 3 votes counts: C=42 E=30 D=28 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:207 C:203 B:198 A:197 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 2 -4 B 2 0 0 0 -6 C 2 0 0 4 0 D -2 0 -4 0 -4 E 4 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 -4 B 2 0 0 0 -6 C 2 0 0 4 0 D -2 0 -4 0 -4 E 4 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 -4 B 2 0 0 0 -6 C 2 0 0 4 0 D -2 0 -4 0 -4 E 4 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3993: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) E B A D C (9) E B A C D (8) D C B E A (8) A C E D B (8) C D A B E (7) C D A E B (6) B E D A C (6) D C B A E (5) C A D E B (5) A E B C D (5) D B C E A (4) B E D C A (4) D C A B E (3) B D E C A (2) A E C B D (2) A C E B D (2) A C D E B (2) E D B C A (1) E B D C A (1) E B D A C (1) C A D B E (1) Total count = 100 A B C D E A 0 2 6 4 -8 B -2 0 2 0 -20 C -6 -2 0 12 2 D -4 0 -12 0 -14 E 8 20 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999989 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 2 6 4 -8 B -2 0 2 0 -20 C -6 -2 0 12 2 D -4 0 -12 0 -14 E 8 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999964 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=20 C=19 A=19 B=12 so B is eliminated. Round 2 votes counts: E=40 D=22 C=19 A=19 so C is eliminated. Round 3 votes counts: E=40 D=35 A=25 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:203 A:202 B:190 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 4 -8 B -2 0 2 0 -20 C -6 -2 0 12 2 D -4 0 -12 0 -14 E 8 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999964 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 -8 B -2 0 2 0 -20 C -6 -2 0 12 2 D -4 0 -12 0 -14 E 8 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999964 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 -8 B -2 0 2 0 -20 C -6 -2 0 12 2 D -4 0 -12 0 -14 E 8 20 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999964 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3994: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) C E D A B (8) E C D B A (5) D A B E C (4) C E A D B (4) B E C A D (4) B A D C E (4) A B D C E (4) E C B D A (3) D A C B E (3) B A C E D (3) A D C B E (3) A D B C E (3) E D B A C (2) E C D A B (2) E B D C A (2) D B A E C (2) D A C E B (2) C E B A D (2) C E A B D (2) C D E A B (2) C D A E B (2) B E D A C (2) B E A D C (2) B E A C D (2) B A E D C (2) B A E C D (2) A C B D E (2) E D C B A (1) E C B A D (1) E B C A D (1) D E B A C (1) D E A B C (1) D C A E B (1) D B E A C (1) D A E B C (1) D A B C E (1) C E D B A (1) C A D E B (1) B D A E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 16 4 2 B 6 0 8 -4 10 C -16 -8 0 -4 -2 D -4 4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 -6 16 4 2 B 6 0 8 -4 10 C -16 -8 0 -4 -2 D -4 4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=22 E=17 D=17 A=13 so A is eliminated. Round 2 votes counts: B=35 C=25 D=23 E=17 so E is eliminated. Round 3 votes counts: B=38 C=36 D=26 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:208 D:201 E:196 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 16 4 2 B 6 0 8 -4 10 C -16 -8 0 -4 -2 D -4 4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 4 2 B 6 0 8 -4 10 C -16 -8 0 -4 -2 D -4 4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 4 2 B 6 0 8 -4 10 C -16 -8 0 -4 -2 D -4 4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.285714 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3995: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (14) E C A D B (10) B D A E C (10) E C B D A (6) C A E D B (6) B D A C E (6) B D E A C (4) D A B C E (3) B A D C E (3) A C E D B (3) E D C B A (2) E C D B A (2) E C A B D (2) D B A E C (2) D B A C E (2) C B A E D (2) B E D C A (2) B D E C A (2) A D C E B (2) A D B C E (2) E C D A B (1) E C B A D (1) E B D C A (1) E B C D A (1) E A C D B (1) D E B A C (1) D B E A C (1) D A E C B (1) D A E B C (1) D A B E C (1) C A E B D (1) B E C D A (1) A D E C B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 0 -4 B -2 0 -12 -14 -16 C 8 12 0 6 -8 D 0 14 -6 0 -12 E 4 16 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -8 0 -4 B -2 0 -12 -14 -16 C 8 12 0 6 -8 D 0 14 -6 0 -12 E 4 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=27 C=23 D=12 A=10 so A is eliminated. Round 2 votes counts: B=29 E=27 C=27 D=17 so D is eliminated. Round 3 votes counts: B=40 E=31 C=29 so C is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:209 D:198 A:195 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 0 -4 B -2 0 -12 -14 -16 C 8 12 0 6 -8 D 0 14 -6 0 -12 E 4 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 0 -4 B -2 0 -12 -14 -16 C 8 12 0 6 -8 D 0 14 -6 0 -12 E 4 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 0 -4 B -2 0 -12 -14 -16 C 8 12 0 6 -8 D 0 14 -6 0 -12 E 4 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 3996: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) B C A E D (7) E D C B A (6) D E C A B (6) A B C E D (5) B C E A D (4) A D E B C (4) D E C B A (3) C E D B A (3) C B E D A (3) C B D A E (3) B C A D E (3) A E B D C (3) A B E C D (3) A B C D E (3) E D C A B (2) E D A B C (2) E C D B A (2) E B A D C (2) C D E B A (2) C B D E A (2) B A C D E (2) A B E D C (2) E D A C B (1) E C B D A (1) E B C A D (1) E A D B C (1) E A B C D (1) D E A C B (1) D C E A B (1) C E B D A (1) C D B E A (1) C D B A E (1) C B E A D (1) C B A E D (1) B E C A D (1) B E A C D (1) B A E C D (1) A E D B C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 -10 18 2 B 24 0 16 24 12 C 10 -16 0 28 10 D -18 -24 -28 0 -32 E -2 -12 -10 32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -10 18 2 B 24 0 16 24 12 C 10 -16 0 28 10 D -18 -24 -28 0 -32 E -2 -12 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 E=19 C=18 D=11 so D is eliminated. Round 2 votes counts: E=29 B=29 A=23 C=19 so C is eliminated. Round 3 votes counts: B=41 E=36 A=23 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:238 C:216 E:204 A:193 D:149 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -10 18 2 B 24 0 16 24 12 C 10 -16 0 28 10 D -18 -24 -28 0 -32 E -2 -12 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -10 18 2 B 24 0 16 24 12 C 10 -16 0 28 10 D -18 -24 -28 0 -32 E -2 -12 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -10 18 2 B 24 0 16 24 12 C 10 -16 0 28 10 D -18 -24 -28 0 -32 E -2 -12 -10 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 3997: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (18) C E B D A (15) A C D B E (6) C A B E D (4) E D B A C (3) E B D C A (3) D E B A C (3) D A B E C (3) C A B D E (3) A D E B C (3) E D B C A (2) E C D B A (2) C E B A D (2) C E A D B (2) C B E D A (2) C A E B D (2) C A D E B (2) B E D C A (2) B D E A C (2) B A D E C (2) B A C D E (2) A D E C B (2) A D B C E (2) A C B D E (2) D E A B C (1) D A E B C (1) C E D B A (1) C E D A B (1) C E A B D (1) C B E A D (1) C B A E D (1) B E C D A (1) B D A E C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 6 14 12 B -10 0 -2 -8 4 C -6 2 0 2 2 D -14 8 -2 0 10 E -12 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 14 12 B -10 0 -2 -8 4 C -6 2 0 2 2 D -14 8 -2 0 10 E -12 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=35 E=10 B=10 D=8 so D is eliminated. Round 2 votes counts: A=39 C=37 E=14 B=10 so B is eliminated. Round 3 votes counts: A=44 C=37 E=19 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:201 C:200 B:192 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 14 12 B -10 0 -2 -8 4 C -6 2 0 2 2 D -14 8 -2 0 10 E -12 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 14 12 B -10 0 -2 -8 4 C -6 2 0 2 2 D -14 8 -2 0 10 E -12 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 14 12 B -10 0 -2 -8 4 C -6 2 0 2 2 D -14 8 -2 0 10 E -12 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 3998: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (13) C A B E D (13) A C E B D (7) D B E C A (5) B C D A E (5) A C E D B (5) E A D C B (4) C A B D E (4) B D E C A (4) E D B A C (3) C B A D E (3) B D C E A (3) E D A B C (2) D E B C A (2) C D B A E (2) C A E D B (2) B E A C D (2) B C A E D (2) B C A D E (2) A E C D B (2) A C B E D (2) E D A C B (1) E A B C D (1) D E A C B (1) D E A B C (1) D C B A E (1) D B C E A (1) C D A B E (1) C B A E D (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C A E (1) A E C B D (1) Total count = 100 A B C D E A 0 -4 -6 6 10 B 4 0 -2 6 8 C 6 2 0 12 12 D -6 -6 -12 0 0 E -10 -8 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 6 10 B 4 0 -2 6 8 C 6 2 0 12 12 D -6 -6 -12 0 0 E -10 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=24 B=21 A=17 E=11 so E is eliminated. Round 2 votes counts: D=30 C=27 A=22 B=21 so B is eliminated. Round 3 votes counts: D=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:208 A:203 D:188 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 6 10 B 4 0 -2 6 8 C 6 2 0 12 12 D -6 -6 -12 0 0 E -10 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 10 B 4 0 -2 6 8 C 6 2 0 12 12 D -6 -6 -12 0 0 E -10 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 10 B 4 0 -2 6 8 C 6 2 0 12 12 D -6 -6 -12 0 0 E -10 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 3999: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) A C E B D (9) D B C E A (7) D B C A E (7) C A E D B (7) C A D E B (7) D B E C A (6) B D E A C (6) E A B C D (4) C A E B D (4) B E A D C (4) E A C B D (3) D B E A C (3) B E D A C (3) B D A E C (3) D C B E A (2) D C A B E (2) D B A C E (2) C D A B E (2) A E C B D (2) A C D E B (2) E C A B D (1) E B A D C (1) C E A D B (1) C D A E B (1) B D E C A (1) A E B D C (1) Total count = 100 A B C D E A 0 -8 -14 -8 16 B 8 0 -4 -16 14 C 14 4 0 -14 24 D 8 16 14 0 20 E -16 -14 -24 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -8 16 B 8 0 -4 -16 14 C 14 4 0 -14 24 D 8 16 14 0 20 E -16 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=22 B=17 A=14 E=9 so E is eliminated. Round 2 votes counts: D=38 C=23 A=21 B=18 so B is eliminated. Round 3 votes counts: D=51 A=26 C=23 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 C:214 B:201 A:193 E:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -14 -8 16 B 8 0 -4 -16 14 C 14 4 0 -14 24 D 8 16 14 0 20 E -16 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -8 16 B 8 0 -4 -16 14 C 14 4 0 -14 24 D 8 16 14 0 20 E -16 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -8 16 B 8 0 -4 -16 14 C 14 4 0 -14 24 D 8 16 14 0 20 E -16 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4000: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) D A E B C (8) C B E D A (7) A D E C B (7) D E A B C (5) C B E A D (5) A D E B C (5) E C B D A (4) C B A E D (4) B C A D E (3) A D B C E (3) A B D C E (3) E D C B A (2) E D A C B (2) E C A D B (2) E B D C A (2) D E B A C (2) D A E C B (2) C A B E D (2) B D E C A (2) B C D A E (2) A D C B E (2) A C B D E (2) E C D B A (1) E C D A B (1) E A D C B (1) D E B C A (1) D E A C B (1) D B E C A (1) D A B E C (1) C E A B D (1) C A E B D (1) B D C E A (1) B C E A D (1) B C A E D (1) A E D C B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -6 -8 -2 B -4 0 0 2 2 C 6 0 0 -4 -2 D 8 -2 4 0 4 E 2 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428391 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -8 -2 B -4 0 0 2 2 C 6 0 0 -4 -2 D 8 -2 4 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428413 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=21 C=20 B=19 E=15 so E is eliminated. Round 2 votes counts: C=28 A=26 D=25 B=21 so B is eliminated. Round 3 votes counts: C=44 D=30 A=26 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:207 B:200 C:200 E:199 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 -8 -2 B -4 0 0 2 2 C 6 0 0 -4 -2 D 8 -2 4 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428413 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -8 -2 B -4 0 0 2 2 C 6 0 0 -4 -2 D 8 -2 4 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428413 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -8 -2 B -4 0 0 2 2 C 6 0 0 -4 -2 D 8 -2 4 0 4 E 2 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428413 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4001: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) E B D A C (6) C B D A E (5) B D E C A (5) E D B A C (4) D E A B C (4) B E D C A (4) B E D A C (4) A D E C B (4) D B E A C (3) C B D E A (3) C A E B D (3) C A B E D (3) B C D E A (3) A E D C B (3) A E D B C (3) A D E B C (3) A C E B D (3) E B A C D (2) C B E D A (2) C B A E D (2) C A D E B (2) C A B D E (2) A D C E B (2) A C E D B (2) E B A D C (1) E A D B C (1) D C B E A (1) D B C E A (1) D A E B C (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E A D (1) C B A D E (1) C A E D B (1) C A D B E (1) B E C D A (1) B E C A D (1) B D E A C (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 10 -2 2 B 6 0 -4 6 -6 C -10 4 0 -2 -4 D 2 -6 2 0 4 E -2 6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 10 -2 2 B 6 0 -4 6 -6 C -10 4 0 -2 -4 D 2 -6 2 0 4 E -2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999935 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=28 B=19 E=14 D=10 so D is eliminated. Round 2 votes counts: C=30 A=29 B=23 E=18 so E is eliminated. Round 3 votes counts: B=36 A=34 C=30 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:202 E:202 B:201 D:201 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 10 -2 2 B 6 0 -4 6 -6 C -10 4 0 -2 -4 D 2 -6 2 0 4 E -2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999935 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 -2 2 B 6 0 -4 6 -6 C -10 4 0 -2 -4 D 2 -6 2 0 4 E -2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999935 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 -2 2 B 6 0 -4 6 -6 C -10 4 0 -2 -4 D 2 -6 2 0 4 E -2 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999935 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4002: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) B E A D C (6) D C A E B (5) E B D A C (4) D E B A C (4) D A C E B (4) C D A E B (4) C A D B E (4) A D C B E (4) A B D E C (4) E D B C A (3) C B E A D (3) C A D E B (3) E D C B A (2) E D B A C (2) E C B D A (2) E B D C A (2) D E A B C (2) D C E A B (2) D A B E C (2) C E D B A (2) C A B E D (2) B E D A C (2) B E A C D (2) A D B E C (2) A D B C E (2) A C D B E (2) A B C D E (2) E C D B A (1) D E C A B (1) D A E C B (1) D A C B E (1) C E B D A (1) C E A B D (1) C D E B A (1) B C A E D (1) B A E C D (1) B A C E D (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 20 2 -14 -4 B -20 0 -12 -26 -12 C -2 12 0 -12 14 D 14 26 12 0 22 E 4 12 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 2 -14 -4 B -20 0 -12 -26 -12 C -2 12 0 -12 14 D 14 26 12 0 22 E 4 12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 A=19 E=16 B=13 so B is eliminated. Round 2 votes counts: C=31 E=26 D=22 A=21 so A is eliminated. Round 3 votes counts: C=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:237 C:206 A:202 E:190 B:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 2 -14 -4 B -20 0 -12 -26 -12 C -2 12 0 -12 14 D 14 26 12 0 22 E 4 12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 2 -14 -4 B -20 0 -12 -26 -12 C -2 12 0 -12 14 D 14 26 12 0 22 E 4 12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 2 -14 -4 B -20 0 -12 -26 -12 C -2 12 0 -12 14 D 14 26 12 0 22 E 4 12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4003: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) C E B A D (8) D A B E C (6) A D C B E (6) E C B A D (5) D A B C E (5) C D A B E (5) B E C D A (5) C A D E B (4) A D C E B (4) A D B E C (4) B E D A C (3) B D A E C (3) E C A D B (2) D B A C E (2) D A C B E (2) C A E D B (2) B C E D A (2) A E D B C (2) A D E C B (2) A C D E B (2) E B D C A (1) E B D A C (1) E B C A D (1) D C A B E (1) D B A E C (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E D A (1) C B D E A (1) C A D B E (1) B D E A C (1) B D A C E (1) B C D E A (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 4 -6 -6 10 B -4 0 0 -8 2 C 6 0 0 6 2 D 6 8 -6 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.304735 C: 0.695265 D: 0.000000 E: 0.000000 Sum of squares = 0.57625660093 Cumulative probabilities = A: 0.000000 B: 0.304735 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -6 10 B -4 0 0 -8 2 C 6 0 0 6 2 D 6 8 -6 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204133458 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=22 E=20 D=17 B=16 so B is eliminated. Round 2 votes counts: E=28 C=28 D=22 A=22 so D is eliminated. Round 3 votes counts: A=42 E=29 C=29 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 D:207 A:201 B:195 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -6 10 B -4 0 0 -8 2 C 6 0 0 6 2 D 6 8 -6 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204133458 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 10 B -4 0 0 -8 2 C 6 0 0 6 2 D 6 8 -6 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204133458 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 10 B -4 0 0 -8 2 C 6 0 0 6 2 D 6 8 -6 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204133458 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4004: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) D A E C B (6) D A C B E (6) C B D A E (6) E A C B D (5) B C E D A (5) E A D C B (4) B C D E A (4) A E D C B (4) A D C B E (4) E B C A D (3) D A E B C (3) C B E A D (3) C B A D E (3) B C D A E (3) E A B C D (2) D E B A C (2) D B A C E (2) C A D B E (2) B D C E A (2) A D E B C (2) A D C E B (2) A C D B E (2) E D B A C (1) E D A B C (1) E C B A D (1) E C A B D (1) E B C D A (1) E B A C D (1) E A D B C (1) D E A B C (1) D C A B E (1) D A C E B (1) D A B C E (1) C E B A D (1) C D B A E (1) C B A E D (1) C A B E D (1) A E D B C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 20 22 6 22 B -20 0 -28 -12 -6 C -22 28 0 -4 6 D -6 12 4 0 24 E -22 6 -6 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 22 6 22 B -20 0 -28 -12 -6 C -22 28 0 -4 6 D -6 12 4 0 24 E -22 6 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 E=21 C=18 B=14 so B is eliminated. Round 2 votes counts: C=30 D=25 A=24 E=21 so E is eliminated. Round 3 votes counts: A=37 C=36 D=27 so D is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:235 D:217 C:204 E:177 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 22 6 22 B -20 0 -28 -12 -6 C -22 28 0 -4 6 D -6 12 4 0 24 E -22 6 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 22 6 22 B -20 0 -28 -12 -6 C -22 28 0 -4 6 D -6 12 4 0 24 E -22 6 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 22 6 22 B -20 0 -28 -12 -6 C -22 28 0 -4 6 D -6 12 4 0 24 E -22 6 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4005: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) E D C B A (6) C A D E B (6) A C B D E (6) E D B C A (5) B E A D C (5) B A E D C (5) C D E B A (4) C D E A B (4) C A D B E (4) B A E C D (4) A B E C D (4) D E C B A (3) C D B E A (3) B E D C A (3) A C D E B (3) E D A B C (2) E B D A C (2) D C E B A (2) B A C E D (2) A E B D C (2) A C D B E (2) A B E D C (2) E D B A C (1) E D A C B (1) D C E A B (1) C D A E B (1) B E C D A (1) B E A C D (1) B C A D E (1) B A C D E (1) A E D C B (1) A E D B C (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 12 6 -4 B 14 0 4 0 10 C -12 -4 0 0 -14 D -6 0 0 0 -16 E 4 -10 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.756312 C: 0.000000 D: 0.243688 E: 0.000000 Sum of squares = 0.631392119018 Cumulative probabilities = A: 0.000000 B: 0.756312 C: 0.756312 D: 1.000000 E: 1.000000 A B C D E A 0 -14 12 6 -4 B 14 0 4 0 10 C -12 -4 0 0 -14 D -6 0 0 0 -16 E 4 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.615385 C: 0.000000 D: 0.384615 E: 0.000000 Sum of squares = 0.526627219644 Cumulative probabilities = A: 0.000000 B: 0.615385 C: 0.615385 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=24 C=22 E=17 D=6 so D is eliminated. Round 2 votes counts: B=31 C=25 A=24 E=20 so E is eliminated. Round 3 votes counts: B=39 C=34 A=27 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:212 A:200 D:189 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 12 6 -4 B 14 0 4 0 10 C -12 -4 0 0 -14 D -6 0 0 0 -16 E 4 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.615385 C: 0.000000 D: 0.384615 E: 0.000000 Sum of squares = 0.526627219644 Cumulative probabilities = A: 0.000000 B: 0.615385 C: 0.615385 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 12 6 -4 B 14 0 4 0 10 C -12 -4 0 0 -14 D -6 0 0 0 -16 E 4 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.615385 C: 0.000000 D: 0.384615 E: 0.000000 Sum of squares = 0.526627219644 Cumulative probabilities = A: 0.000000 B: 0.615385 C: 0.615385 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 12 6 -4 B 14 0 4 0 10 C -12 -4 0 0 -14 D -6 0 0 0 -16 E 4 -10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.615385 C: 0.000000 D: 0.384615 E: 0.000000 Sum of squares = 0.526627219644 Cumulative probabilities = A: 0.000000 B: 0.615385 C: 0.615385 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4006: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (14) C D E B A (14) E A B C D (11) A B E D C (8) B A E D C (6) B A D E C (5) E C A B D (4) C D E A B (4) C D B E A (4) B A D C E (4) E C D A B (3) E A B D C (3) D C A B E (3) E A C B D (2) D B C A E (2) B D A C E (2) E C D B A (1) E C B A D (1) E C A D B (1) E B C A D (1) D B A C E (1) D A B C E (1) C E D A B (1) C D B A E (1) C D A B E (1) C A E D B (1) A E B D C (1) Total count = 100 A B C D E A 0 -12 -12 -4 0 B 12 0 -10 -4 4 C 12 10 0 0 6 D 4 4 0 0 12 E 0 -4 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.618119 D: 0.381881 E: 0.000000 Sum of squares = 0.527904231065 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.618119 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -4 0 B 12 0 -10 -4 4 C 12 10 0 0 6 D 4 4 0 0 12 E 0 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=26 D=21 B=17 A=9 so A is eliminated. Round 2 votes counts: E=28 C=26 B=25 D=21 so D is eliminated. Round 3 votes counts: C=43 B=29 E=28 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:210 B:201 E:189 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -12 -4 0 B 12 0 -10 -4 4 C 12 10 0 0 6 D 4 4 0 0 12 E 0 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -4 0 B 12 0 -10 -4 4 C 12 10 0 0 6 D 4 4 0 0 12 E 0 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -4 0 B 12 0 -10 -4 4 C 12 10 0 0 6 D 4 4 0 0 12 E 0 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4007: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C A B (8) B D E C A (6) B A E C D (6) E D B C A (5) D E C B A (5) E D C B A (4) A B C E D (4) E B D C A (3) D E C A B (3) D C E A B (3) C D A B E (3) B E D A C (3) B A E D C (3) B A C E D (3) A B C D E (3) E C D A B (2) C D E A B (2) C A D E B (2) B E D C A (2) B E A D C (2) A C E D B (2) A C D E B (2) E C A D B (1) E A C D B (1) D C E B A (1) D C B A E (1) C D A E B (1) C A E D B (1) B D C E A (1) B D C A E (1) B A D C E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -10 -8 -4 B 12 0 6 2 2 C 10 -6 0 -4 -14 D 8 -2 4 0 -8 E 4 -2 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999041 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -8 -4 B 12 0 6 2 2 C 10 -6 0 -4 -14 D 8 -2 4 0 -8 E 4 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=24 A=17 D=13 C=9 so C is eliminated. Round 2 votes counts: B=37 E=24 A=20 D=19 so D is eliminated. Round 3 votes counts: E=38 B=38 A=24 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:212 B:211 D:201 C:193 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 -8 -4 B 12 0 6 2 2 C 10 -6 0 -4 -14 D 8 -2 4 0 -8 E 4 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -8 -4 B 12 0 6 2 2 C 10 -6 0 -4 -14 D 8 -2 4 0 -8 E 4 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -8 -4 B 12 0 6 2 2 C 10 -6 0 -4 -14 D 8 -2 4 0 -8 E 4 -2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4008: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) C E D B A (10) C E B D A (10) A B D E C (6) C D E A B (5) B E D C A (4) B E D A C (4) A D C E B (4) D E B A C (3) C A D E B (3) B E C D A (3) B E A D C (3) A D E B C (3) A D B C E (3) A C D E B (3) E B C D A (2) C E D A B (2) C D E B A (2) B A E D C (2) A D C B E (2) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) D C A E B (1) D A C E B (1) C D A E B (1) B E C A D (1) B C A E D (1) B A E C D (1) B A C E D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 4 -4 -6 B 2 0 6 -16 -10 C -4 -6 0 -4 -2 D 4 16 4 0 2 E 6 10 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -4 -6 B 2 0 6 -16 -10 C -4 -6 0 -4 -2 D 4 16 4 0 2 E 6 10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=33 B=20 E=6 D=5 so D is eliminated. Round 2 votes counts: A=37 C=34 B=20 E=9 so E is eliminated. Round 3 votes counts: A=37 C=36 B=27 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:213 E:208 A:196 C:192 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -4 -6 B 2 0 6 -16 -10 C -4 -6 0 -4 -2 D 4 16 4 0 2 E 6 10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -4 -6 B 2 0 6 -16 -10 C -4 -6 0 -4 -2 D 4 16 4 0 2 E 6 10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -4 -6 B 2 0 6 -16 -10 C -4 -6 0 -4 -2 D 4 16 4 0 2 E 6 10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4009: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) D C A B E (6) D A C B E (6) E C D B A (5) D A B C E (5) E D C B A (4) E B C A D (4) A B D C E (4) E D C A B (3) E C B D A (3) D C E A B (3) B A C D E (3) A D B C E (3) E D A B C (2) E B A D C (2) E B A C D (2) D E A C B (2) D C A E B (2) D A B E C (2) C B A D E (2) B E A C D (2) B C E A D (2) B C A E D (2) A B D E C (2) E D A C B (1) E C B A D (1) E A D B C (1) D E C A B (1) D E A B C (1) D A E C B (1) C E D A B (1) C E B D A (1) C E B A D (1) C D E B A (1) C D A E B (1) C B E A D (1) C A D B E (1) C A B D E (1) B E C A D (1) B C A D E (1) B A E D C (1) B A D E C (1) A D C B E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 20 -12 -18 10 B -20 0 -14 -24 14 C 12 14 0 -10 14 D 18 24 10 0 20 E -10 -14 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -12 -18 10 B -20 0 -14 -24 14 C 12 14 0 -10 14 D 18 24 10 0 20 E -10 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 C=18 B=13 A=12 so A is eliminated. Round 2 votes counts: D=34 E=28 B=20 C=18 so C is eliminated. Round 3 votes counts: D=45 E=31 B=24 so B is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:236 C:215 A:200 B:178 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 -12 -18 10 B -20 0 -14 -24 14 C 12 14 0 -10 14 D 18 24 10 0 20 E -10 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -12 -18 10 B -20 0 -14 -24 14 C 12 14 0 -10 14 D 18 24 10 0 20 E -10 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -12 -18 10 B -20 0 -14 -24 14 C 12 14 0 -10 14 D 18 24 10 0 20 E -10 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4010: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) D E C A B (9) A B C E D (9) B A C E D (8) E D C B A (5) A C B E D (5) C E D A B (4) C E B A D (4) C A B E D (4) A B C D E (4) E C D B A (3) D B E A C (3) B A D E C (3) A B D C E (3) E D B C A (2) D E B C A (2) D B A E C (2) A D B C E (2) D E A B C (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) D A B E C (1) C E A D B (1) C E A B D (1) C A E D B (1) B E D A C (1) B E C A D (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E C D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 4 0 0 B -2 0 -2 -2 6 C -4 2 0 -6 0 D 0 2 6 0 -2 E 0 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.803375 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.196625 Sum of squares = 0.684072457501 Cumulative probabilities = A: 0.803375 B: 0.803375 C: 0.803375 D: 0.803375 E: 1.000000 A B C D E A 0 2 4 0 0 B -2 0 -2 -2 6 C -4 2 0 -6 0 D 0 2 6 0 -2 E 0 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000044201 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=24 B=18 C=15 E=10 so E is eliminated. Round 2 votes counts: D=40 A=24 C=18 B=18 so C is eliminated. Round 3 votes counts: D=47 A=31 B=22 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:203 D:203 B:200 E:198 C:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 0 B -2 0 -2 -2 6 C -4 2 0 -6 0 D 0 2 6 0 -2 E 0 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000044201 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 0 B -2 0 -2 -2 6 C -4 2 0 -6 0 D 0 2 6 0 -2 E 0 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000044201 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 0 B -2 0 -2 -2 6 C -4 2 0 -6 0 D 0 2 6 0 -2 E 0 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000044201 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4011: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (15) B E C D A (11) B E A C D (10) A C D B E (8) D C A E B (7) D A C E B (6) E B A D C (5) C D A B E (4) A D C E B (4) E B D A C (3) C D A E B (3) B E C A D (3) E D C B A (2) C A D B E (2) B C E D A (2) B A C D E (2) A D E C B (2) E B A C D (1) E A B D C (1) D E C B A (1) D C E A B (1) C D E B A (1) C D B A E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -10 -18 -14 B 18 0 14 12 -8 C 10 -14 0 2 -14 D 18 -12 -2 0 -12 E 14 8 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -10 -18 -14 B 18 0 14 12 -8 C 10 -14 0 2 -14 D 18 -12 -2 0 -12 E 14 8 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 A=17 D=15 C=11 so C is eliminated. Round 2 votes counts: B=30 E=27 D=24 A=19 so A is eliminated. Round 3 votes counts: D=40 B=31 E=29 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:224 B:218 D:196 C:192 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -10 -18 -14 B 18 0 14 12 -8 C 10 -14 0 2 -14 D 18 -12 -2 0 -12 E 14 8 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -18 -14 B 18 0 14 12 -8 C 10 -14 0 2 -14 D 18 -12 -2 0 -12 E 14 8 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -18 -14 B 18 0 14 12 -8 C 10 -14 0 2 -14 D 18 -12 -2 0 -12 E 14 8 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4012: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (16) B C D E A (8) B D C E A (7) A E C D B (7) D B C A E (6) E A D B C (5) A E D C B (5) D A E B C (4) E A D C B (3) D C B A E (3) D B E A C (3) C B D A E (3) B E C A D (3) E B C A D (2) D B C E A (2) C E A B D (2) C A E B D (2) A E C B D (2) E A B D C (1) E A B C D (1) D E B A C (1) D C A B E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C B E (1) D A B C E (1) C D B A E (1) C D A B E (1) B E D A C (1) B E C D A (1) A E D B C (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 14 16 6 -12 B -14 0 -2 0 -12 C -16 2 0 4 -16 D -6 0 -4 0 -8 E 12 12 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 16 6 -12 B -14 0 -2 0 -12 C -16 2 0 4 -16 D -6 0 -4 0 -8 E 12 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=25 B=20 A=18 C=9 so C is eliminated. Round 2 votes counts: E=30 D=27 B=23 A=20 so A is eliminated. Round 3 votes counts: E=49 D=28 B=23 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:212 D:191 C:187 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 16 6 -12 B -14 0 -2 0 -12 C -16 2 0 4 -16 D -6 0 -4 0 -8 E 12 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 6 -12 B -14 0 -2 0 -12 C -16 2 0 4 -16 D -6 0 -4 0 -8 E 12 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 6 -12 B -14 0 -2 0 -12 C -16 2 0 4 -16 D -6 0 -4 0 -8 E 12 12 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4013: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) B E A D C (6) E A D C B (5) E A B D C (5) B A E D C (5) D C A E B (4) C E D B A (4) C D E A B (4) C B D E A (4) B E A C D (4) A E D B C (4) A D E B C (4) E C A D B (3) D C A B E (3) C D A E B (3) B E C A D (3) B D A C E (3) E A C D B (2) D A E C B (2) C E D A B (2) C E B A D (2) C B E D A (2) C B D A E (2) B C E D A (2) B C D A E (2) E C D A B (1) E B A D C (1) D A C B E (1) D A B E C (1) C E A D B (1) C D E B A (1) B D C A E (1) B D A E C (1) B C E A D (1) A E B D C (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 10 14 -6 B 4 0 6 6 0 C -10 -6 0 -14 -16 D -14 -6 14 0 -8 E 6 0 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.797990 C: 0.000000 D: 0.000000 E: 0.202010 Sum of squares = 0.677595573719 Cumulative probabilities = A: 0.000000 B: 0.797990 C: 0.797990 D: 0.797990 E: 1.000000 A B C D E A 0 -4 10 14 -6 B 4 0 6 6 0 C -10 -6 0 -14 -16 D -14 -6 14 0 -8 E 6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=25 E=17 D=11 A=11 so D is eliminated. Round 2 votes counts: B=36 C=32 E=17 A=15 so A is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:208 A:207 D:193 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 14 -6 B 4 0 6 6 0 C -10 -6 0 -14 -16 D -14 -6 14 0 -8 E 6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 14 -6 B 4 0 6 6 0 C -10 -6 0 -14 -16 D -14 -6 14 0 -8 E 6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 14 -6 B 4 0 6 6 0 C -10 -6 0 -14 -16 D -14 -6 14 0 -8 E 6 0 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4014: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) B E A C D (8) E A B D C (7) B A E D C (7) D C E A B (6) C D B E A (5) C D B A E (5) C B E A D (5) B C E A D (5) B A E C D (5) D E A C B (4) C B D E A (4) B C A E D (4) A E D B C (4) C D E B A (3) D E C A B (2) D A E C B (2) C B D A E (2) A E B D C (2) E B A C D (1) E A D C B (1) E A D B C (1) D E A B C (1) D B C A E (1) D A E B C (1) C E D A B (1) C D A E B (1) C B A E D (1) B E A D C (1) B C A D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -8 8 -26 B 16 0 0 8 10 C 8 0 0 18 4 D -8 -8 -18 0 -8 E 26 -10 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.755116 C: 0.244884 D: 0.000000 E: 0.000000 Sum of squares = 0.630168601383 Cumulative probabilities = A: 0.000000 B: 0.755116 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 8 -26 B 16 0 0 8 10 C 8 0 0 18 4 D -8 -8 -18 0 -8 E 26 -10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=31 D=17 E=10 A=7 so A is eliminated. Round 2 votes counts: C=35 B=32 D=17 E=16 so E is eliminated. Round 3 votes counts: B=42 C=35 D=23 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:215 E:210 A:179 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -8 8 -26 B 16 0 0 8 10 C 8 0 0 18 4 D -8 -8 -18 0 -8 E 26 -10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 8 -26 B 16 0 0 8 10 C 8 0 0 18 4 D -8 -8 -18 0 -8 E 26 -10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 8 -26 B 16 0 0 8 10 C 8 0 0 18 4 D -8 -8 -18 0 -8 E 26 -10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4015: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (12) B E C D A (11) D A C E B (10) A D C B E (10) A D C E B (9) B E C A D (5) A D E C B (4) E A D B C (3) B C E A D (3) B C A D E (3) A D E B C (3) E B A D C (2) D A E C B (2) C B E D A (2) B C E D A (2) E D C B A (1) E D A C B (1) E D A B C (1) E B D C A (1) E A B D C (1) D E C A B (1) D C A E B (1) D C A B E (1) C E B D A (1) C D E B A (1) C D B A E (1) C B D A E (1) C A D B E (1) C A B D E (1) B E A C D (1) B C A E D (1) B A D C E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 0 2 B -2 0 4 -2 -8 C 0 -4 0 -8 0 D 0 2 8 0 4 E -2 8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.686963 B: 0.000000 C: 0.000000 D: 0.313037 E: 0.000000 Sum of squares = 0.569910604009 Cumulative probabilities = A: 0.686963 B: 0.686963 C: 0.686963 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 2 B -2 0 4 -2 -8 C 0 -4 0 -8 0 D 0 2 8 0 4 E -2 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 E=22 D=15 C=8 so C is eliminated. Round 2 votes counts: B=30 A=30 E=23 D=17 so D is eliminated. Round 3 votes counts: A=44 B=31 E=25 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:207 A:202 E:201 B:196 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 0 2 B -2 0 4 -2 -8 C 0 -4 0 -8 0 D 0 2 8 0 4 E -2 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 2 B -2 0 4 -2 -8 C 0 -4 0 -8 0 D 0 2 8 0 4 E -2 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 2 B -2 0 4 -2 -8 C 0 -4 0 -8 0 D 0 2 8 0 4 E -2 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4016: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (12) C B D A E (9) B D C A E (6) E A D C B (5) C A E D B (5) E A C D B (4) D A E B C (4) C E A D B (4) E A B D C (3) D A E C B (3) B C D A E (3) A E D C B (3) E B A D C (2) D B A E C (2) C E B A D (2) C E A B D (2) B E C A D (2) B D A E C (2) B C E D A (2) A E D B C (2) A D E C B (2) E C A B D (1) E B C A D (1) E A C B D (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C A E (1) D B A C E (1) D A C E B (1) D A B E C (1) C D A B E (1) C B E D A (1) C A D E B (1) C A D B E (1) C A B D E (1) B D C E A (1) B D A C E (1) B C D E A (1) A E C D B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 22 8 16 8 B -22 0 0 -16 -26 C -8 0 0 -12 -8 D -16 16 12 0 -8 E -8 26 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 8 16 8 B -22 0 0 -16 -26 C -8 0 0 -12 -8 D -16 16 12 0 -8 E -8 26 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=27 B=18 D=15 A=10 so A is eliminated. Round 2 votes counts: E=36 C=27 D=19 B=18 so B is eliminated. Round 3 votes counts: E=38 C=33 D=29 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:227 E:217 D:202 C:186 B:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 8 16 8 B -22 0 0 -16 -26 C -8 0 0 -12 -8 D -16 16 12 0 -8 E -8 26 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 8 16 8 B -22 0 0 -16 -26 C -8 0 0 -12 -8 D -16 16 12 0 -8 E -8 26 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 8 16 8 B -22 0 0 -16 -26 C -8 0 0 -12 -8 D -16 16 12 0 -8 E -8 26 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4017: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (9) D E B A C (8) C A B E D (7) E C B A D (5) A B E C D (5) E B A D C (4) D B E A C (4) B A E D C (4) D C A B E (3) D A B E C (3) C D A B E (3) A B E D C (3) E C A B D (2) E B A C D (2) D E B C A (2) D C B A E (2) D A B C E (2) C E B A D (2) C A E B D (2) A B D C E (2) A B C E D (2) E D B A C (1) E C D B A (1) E B D A C (1) D E C B A (1) D C E B A (1) D C E A B (1) D B C A E (1) D B A C E (1) C D E B A (1) C D A E B (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 12 4 10 B 4 0 14 6 12 C -12 -14 0 -12 -16 D -4 -6 12 0 -2 E -10 -12 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 4 10 B 4 0 14 6 12 C -12 -14 0 -12 -16 D -4 -6 12 0 -2 E -10 -12 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 C=25 E=16 A=15 B=4 so B is eliminated. Round 2 votes counts: D=40 C=25 A=19 E=16 so E is eliminated. Round 3 votes counts: D=42 C=33 A=25 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:218 A:211 D:200 E:198 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 4 10 B 4 0 14 6 12 C -12 -14 0 -12 -16 D -4 -6 12 0 -2 E -10 -12 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 4 10 B 4 0 14 6 12 C -12 -14 0 -12 -16 D -4 -6 12 0 -2 E -10 -12 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 4 10 B 4 0 14 6 12 C -12 -14 0 -12 -16 D -4 -6 12 0 -2 E -10 -12 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4018: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) D A E C B (7) B C E A D (7) D E C A B (6) D C E A B (6) C E D B A (6) B A E C D (5) B A C E D (5) A B E D C (5) A B E C D (5) A D B C E (4) E D C B A (3) D A C E B (3) C B E D A (3) A D E C B (3) E C D B A (2) C D E B A (2) B C E D A (2) A D E B C (2) A B D E C (2) A B D C E (2) A B C D E (2) E D C A B (1) E C B D A (1) E B A C D (1) D E C B A (1) D E A C B (1) D C E B A (1) C E B D A (1) B E C A D (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 16 14 8 10 B -16 0 6 -12 6 C -14 -6 0 -10 -8 D -8 12 10 0 2 E -10 -6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 8 10 B -16 0 6 -12 6 C -14 -6 0 -10 -8 D -8 12 10 0 2 E -10 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=25 B=21 C=12 E=8 so E is eliminated. Round 2 votes counts: A=34 D=29 B=22 C=15 so C is eliminated. Round 3 votes counts: D=39 A=34 B=27 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:208 E:195 B:192 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 8 10 B -16 0 6 -12 6 C -14 -6 0 -10 -8 D -8 12 10 0 2 E -10 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 8 10 B -16 0 6 -12 6 C -14 -6 0 -10 -8 D -8 12 10 0 2 E -10 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 8 10 B -16 0 6 -12 6 C -14 -6 0 -10 -8 D -8 12 10 0 2 E -10 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4019: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E C D B A (9) C B A E D (8) D E A B C (7) B A C D E (6) A D E B C (5) E D C B A (4) E D A B C (4) B A D C E (4) E D B C A (3) D A E B C (3) B C A D E (3) A D B E C (3) E D C A B (2) D E B C A (2) D A B E C (2) C E B D A (2) C E B A D (2) C B E D A (2) C B E A D (2) B D C E A (2) A B D C E (2) A B C D E (2) E A D C B (1) D B A E C (1) C E A B D (1) C B A D E (1) C A E B D (1) B D C A E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 6 -12 -8 B 8 0 6 -18 -18 C -6 -6 0 -16 -18 D 12 18 16 0 -8 E 8 18 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 6 -12 -8 B 8 0 6 -18 -18 C -6 -6 0 -16 -18 D 12 18 16 0 -8 E 8 18 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=19 B=18 D=15 A=15 so D is eliminated. Round 2 votes counts: E=42 A=20 C=19 B=19 so C is eliminated. Round 3 votes counts: E=47 B=32 A=21 so A is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:219 A:189 B:189 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 -12 -8 B 8 0 6 -18 -18 C -6 -6 0 -16 -18 D 12 18 16 0 -8 E 8 18 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -12 -8 B 8 0 6 -18 -18 C -6 -6 0 -16 -18 D 12 18 16 0 -8 E 8 18 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -12 -8 B 8 0 6 -18 -18 C -6 -6 0 -16 -18 D 12 18 16 0 -8 E 8 18 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4020: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) E A C B D (6) C E D A B (6) E C A D B (5) B A E D C (5) E C A B D (4) D B C A E (4) C D E A B (4) C D A E B (4) B E A C D (4) B D A E C (4) E B A C D (3) C E A D B (3) C D E B A (3) B D A C E (3) A B D E C (3) E C D A B (2) E A B C D (2) D C B A E (2) C E D B A (2) C A D E B (2) B E D C A (2) A D C B E (2) A D B C E (2) E C B D A (1) E B C A D (1) D B C E A (1) D B A C E (1) D A C B E (1) D A B C E (1) B E D A C (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) A E C D B (1) A E C B D (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 2 8 -6 B -2 0 0 6 -2 C -2 0 0 10 -12 D -8 -6 -10 0 -2 E 6 2 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 8 -6 B -2 0 0 6 -2 C -2 0 0 10 -12 D -8 -6 -10 0 -2 E 6 2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=24 C=24 A=11 D=10 so D is eliminated. Round 2 votes counts: B=37 C=26 E=24 A=13 so A is eliminated. Round 3 votes counts: B=44 C=30 E=26 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:211 A:203 B:201 C:198 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 8 -6 B -2 0 0 6 -2 C -2 0 0 10 -12 D -8 -6 -10 0 -2 E 6 2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 8 -6 B -2 0 0 6 -2 C -2 0 0 10 -12 D -8 -6 -10 0 -2 E 6 2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 8 -6 B -2 0 0 6 -2 C -2 0 0 10 -12 D -8 -6 -10 0 -2 E 6 2 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4021: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) E B D A C (8) E A C B D (7) C A E B D (7) D B E A C (5) A C E D B (5) A C D E B (5) C A E D B (4) C A D E B (4) E B C A D (3) C A D B E (3) B D E A C (3) A C D B E (3) E C A B D (2) E A B D C (2) D A C B E (2) C D A B E (2) B E D C A (2) B E D A C (2) B D C E A (2) B C D E A (2) A D C B E (2) E B D C A (1) E B C D A (1) E B A C D (1) E A D B C (1) E A B C D (1) D C A B E (1) D B C A E (1) D B A E C (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C A B D E (1) A E C D B (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 16 8 10 -6 B -16 0 -4 14 -14 C -8 4 0 10 -6 D -10 -14 -10 0 -2 E 6 14 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 8 10 -6 B -16 0 -4 14 -14 C -8 4 0 10 -6 D -10 -14 -10 0 -2 E 6 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=22 B=20 A=18 D=13 so D is eliminated. Round 2 votes counts: E=27 B=27 C=23 A=23 so C is eliminated. Round 3 votes counts: A=45 E=28 B=27 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:214 E:214 C:200 B:190 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 8 10 -6 B -16 0 -4 14 -14 C -8 4 0 10 -6 D -10 -14 -10 0 -2 E 6 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 10 -6 B -16 0 -4 14 -14 C -8 4 0 10 -6 D -10 -14 -10 0 -2 E 6 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 10 -6 B -16 0 -4 14 -14 C -8 4 0 10 -6 D -10 -14 -10 0 -2 E 6 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4022: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) A D E B C (10) E D B C A (9) C B E D A (9) B C E D A (8) C B A E D (6) A D E C B (6) B E C D A (4) A C B D E (4) E D B A C (3) D A E B C (3) C A B E D (3) C A B D E (3) A D C E B (3) E B D C A (2) D E B C A (2) D E B A C (2) C B E A D (2) C B A D E (2) A C B E D (2) A B C E D (2) E D A B C (1) C B D E A (1) B E D C A (1) B C E A D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -6 -10 -10 B 4 0 16 0 -2 C 6 -16 0 -4 -6 D 10 0 4 0 -6 E 10 2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -6 -10 -10 B 4 0 16 0 -2 C 6 -16 0 -4 -6 D 10 0 4 0 -6 E 10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=26 D=17 E=15 B=14 so B is eliminated. Round 2 votes counts: C=35 A=28 E=20 D=17 so D is eliminated. Round 3 votes counts: C=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:209 D:204 C:190 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -6 -10 -10 B 4 0 16 0 -2 C 6 -16 0 -4 -6 D 10 0 4 0 -6 E 10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -10 -10 B 4 0 16 0 -2 C 6 -16 0 -4 -6 D 10 0 4 0 -6 E 10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -10 -10 B 4 0 16 0 -2 C 6 -16 0 -4 -6 D 10 0 4 0 -6 E 10 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4023: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) C D A B E (7) A C D E B (6) A C B D E (6) E D C A B (5) B C A D E (5) E A D C B (4) B E A D C (4) B A E C D (4) B A C E D (4) B A C D E (4) E D A C B (3) E B D A C (3) D C E B A (3) D C E A B (3) C D A E B (3) E D B C A (2) D C A E B (2) C A D E B (2) C A D B E (2) B C D A E (2) A B E C D (2) E D B A C (1) E D A B C (1) E B D C A (1) E B A D C (1) E A B D C (1) D E C B A (1) D C A B E (1) C B A D E (1) C A B D E (1) B D E C A (1) B D C E A (1) A E C D B (1) A D C E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 24 -4 2 12 B -24 0 -24 -16 -6 C 4 24 0 4 12 D -2 16 -4 0 24 E -12 6 -12 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 -4 2 12 B -24 0 -24 -16 -6 C 4 24 0 4 12 D -2 16 -4 0 24 E -12 6 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=22 D=19 A=18 C=16 so C is eliminated. Round 2 votes counts: D=29 B=26 A=23 E=22 so E is eliminated. Round 3 votes counts: D=41 B=31 A=28 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:222 A:217 D:217 E:179 B:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -4 2 12 B -24 0 -24 -16 -6 C 4 24 0 4 12 D -2 16 -4 0 24 E -12 6 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -4 2 12 B -24 0 -24 -16 -6 C 4 24 0 4 12 D -2 16 -4 0 24 E -12 6 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -4 2 12 B -24 0 -24 -16 -6 C 4 24 0 4 12 D -2 16 -4 0 24 E -12 6 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997571 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4024: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) E B C A D (7) B C E A D (7) C B E D A (6) B E C A D (6) A D B E C (6) A D B C E (6) D A E C B (4) D A C E B (4) C E B D A (4) C D B E A (4) C D B A E (4) E D C B A (3) D C A B E (3) E B C D A (2) A D E B C (2) E D A B C (1) E C B D A (1) E B A C D (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C E A B (1) D A E B C (1) C E D B A (1) C B D E A (1) C B D A E (1) B C A D E (1) B A E C D (1) B A C E D (1) A E B D C (1) A E B C D (1) A D C B E (1) A B E D C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 -6 2 B 2 0 0 -8 24 C 6 0 0 2 16 D 6 8 -2 0 6 E -2 -24 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.072417 C: 0.927583 D: 0.000000 E: 0.000000 Sum of squares = 0.86565412739 Cumulative probabilities = A: 0.000000 B: 0.072417 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 2 B 2 0 0 -8 24 C 6 0 0 2 16 D 6 8 -2 0 6 E -2 -24 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000166958 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=21 A=21 E=17 B=16 so B is eliminated. Round 2 votes counts: C=29 D=25 E=23 A=23 so E is eliminated. Round 3 votes counts: C=45 D=29 A=26 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:209 D:209 A:194 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 2 B 2 0 0 -8 24 C 6 0 0 2 16 D 6 8 -2 0 6 E -2 -24 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000166958 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 2 B 2 0 0 -8 24 C 6 0 0 2 16 D 6 8 -2 0 6 E -2 -24 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000166958 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 2 B 2 0 0 -8 24 C 6 0 0 2 16 D 6 8 -2 0 6 E -2 -24 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000166958 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4025: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (18) B E D A C (15) B C A D E (11) E D A C B (6) C A D B E (6) E B D A C (5) B D C A E (4) B D A C E (4) E D A B C (3) E D B A C (2) C B A D E (2) C A B D E (2) B D E A C (2) B D C E A (2) A D C E B (2) A C D E B (2) E C A D B (1) C E A D B (1) C A E D B (1) B E D C A (1) B E C D A (1) B E C A D (1) B C E A D (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -8 6 16 B 6 0 8 6 10 C 8 -8 0 2 20 D -6 -6 -2 0 18 E -16 -10 -20 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 6 16 B 6 0 8 6 10 C 8 -8 0 2 20 D -6 -6 -2 0 18 E -16 -10 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 C=30 E=17 A=9 so D is eliminated. Round 2 votes counts: B=44 C=30 E=17 A=9 so A is eliminated. Round 3 votes counts: B=44 C=36 E=20 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:211 A:204 D:202 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 6 16 B 6 0 8 6 10 C 8 -8 0 2 20 D -6 -6 -2 0 18 E -16 -10 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 6 16 B 6 0 8 6 10 C 8 -8 0 2 20 D -6 -6 -2 0 18 E -16 -10 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 6 16 B 6 0 8 6 10 C 8 -8 0 2 20 D -6 -6 -2 0 18 E -16 -10 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4026: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) E B D A C (6) D B E A C (5) D A B E C (5) C A D B E (5) B D E C A (5) A C D E B (5) C B E D A (4) C B D E A (4) A C D B E (4) E B D C A (3) D A E B C (3) C E B A D (3) C A B E D (3) A D E C B (3) E A C B D (2) C D B A E (2) C A E B D (2) C A B D E (2) B E C D A (2) A D E B C (2) A D C E B (2) A C E D B (2) E D B A C (1) E C B D A (1) E C A B D (1) E B C D A (1) D E B A C (1) D C B E A (1) D C A B E (1) D B E C A (1) D B C A E (1) D A B C E (1) C E A B D (1) C B A E D (1) C B A D E (1) B C E D A (1) B C D E A (1) A E D B C (1) A E B D C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -8 -14 -2 B 4 0 -4 6 16 C 8 4 0 -2 -2 D 14 -6 2 0 12 E 2 -16 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888931 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -14 -2 B 4 0 -4 6 16 C 8 4 0 -2 -2 D 14 -6 2 0 12 E 2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888941 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=22 D=19 B=16 E=15 so E is eliminated. Round 2 votes counts: C=30 B=26 A=24 D=20 so D is eliminated. Round 3 votes counts: B=35 A=33 C=32 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:211 C:204 E:188 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -14 -2 B 4 0 -4 6 16 C 8 4 0 -2 -2 D 14 -6 2 0 12 E 2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888941 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -14 -2 B 4 0 -4 6 16 C 8 4 0 -2 -2 D 14 -6 2 0 12 E 2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888941 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -14 -2 B 4 0 -4 6 16 C 8 4 0 -2 -2 D 14 -6 2 0 12 E 2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888941 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4027: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) E D C A B (6) B A D C E (6) E D A C B (5) C B A E D (5) B A C D E (5) D E A B C (4) D B A E C (4) C E A D B (4) C E B A D (3) C E A B D (3) C B E A D (3) B C A D E (3) B A D E C (3) A B D E C (3) D E B A C (2) D A E B C (2) C E B D A (2) C A E B D (2) C A B E D (2) B D A E C (2) A B D C E (2) A B C D E (2) E D C B A (1) E D A B C (1) E C D B A (1) E C B D A (1) D A B E C (1) C E D B A (1) C E D A B (1) C A E D B (1) C A B D E (1) B D C E A (1) B C D E A (1) B C D A E (1) A D E B C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -10 2 -4 B -10 0 -10 4 -6 C 10 10 0 10 0 D -2 -4 -10 0 -8 E 4 6 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.277222 D: 0.000000 E: 0.722778 Sum of squares = 0.599259679061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.277222 D: 0.277222 E: 1.000000 A B C D E A 0 10 -10 2 -4 B -10 0 -10 4 -6 C 10 10 0 10 0 D -2 -4 -10 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 B=22 D=13 A=10 so A is eliminated. Round 2 votes counts: C=29 B=29 E=27 D=15 so D is eliminated. Round 3 votes counts: E=36 B=35 C=29 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:209 A:199 B:189 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 2 -4 B -10 0 -10 4 -6 C 10 10 0 10 0 D -2 -4 -10 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 2 -4 B -10 0 -10 4 -6 C 10 10 0 10 0 D -2 -4 -10 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 2 -4 B -10 0 -10 4 -6 C 10 10 0 10 0 D -2 -4 -10 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4028: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (12) D C A B E (9) C D A B E (9) B E C A D (8) A D C E B (8) D A C E B (7) E B A D C (5) C D B A E (4) C A D B E (4) B E C D A (4) B E D C A (3) A E D B C (3) A E B D C (3) A D E C B (3) E A B D C (2) D C A E B (2) C B E D A (2) C B D E A (2) E D B A C (1) E B D A C (1) E A B C D (1) D A E B C (1) C B E A D (1) B E A C D (1) B C E D A (1) B C E A D (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 0 8 10 B -8 0 -6 -4 -2 C 0 6 0 4 2 D -8 4 -4 0 0 E -10 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.390185 B: 0.000000 C: 0.609815 D: 0.000000 E: 0.000000 Sum of squares = 0.524118779639 Cumulative probabilities = A: 0.390185 B: 0.390185 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 8 10 B -8 0 -6 -4 -2 C 0 6 0 4 2 D -8 4 -4 0 0 E -10 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=22 C=22 D=19 A=19 B=18 so B is eliminated. Round 2 votes counts: E=38 C=24 D=19 A=19 so D is eliminated. Round 3 votes counts: E=38 C=35 A=27 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:206 D:196 E:195 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 8 10 B -8 0 -6 -4 -2 C 0 6 0 4 2 D -8 4 -4 0 0 E -10 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 8 10 B -8 0 -6 -4 -2 C 0 6 0 4 2 D -8 4 -4 0 0 E -10 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 8 10 B -8 0 -6 -4 -2 C 0 6 0 4 2 D -8 4 -4 0 0 E -10 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4029: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (9) E B A D C (7) D A C B E (6) A D E C B (6) E C B D A (5) B C D A E (5) E A D B C (4) B E C A D (4) B A D C E (4) A D B E C (4) E C B A D (3) E B C A D (3) C D A E B (3) A D E B C (3) E C D A B (2) E C A D B (2) E A D C B (2) D A C E B (2) C E D A B (2) C E B D A (2) B E C D A (2) B C E A D (2) B C A D E (2) B A D E C (2) A E D B C (2) E D A C B (1) E B C D A (1) E A C D B (1) E A B D C (1) D A E C B (1) D A B C E (1) C D A B E (1) C B E D A (1) C B D A E (1) B E A D C (1) B D A C E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 18 26 8 B -8 0 18 -4 -6 C -18 -18 0 -16 -14 D -26 4 16 0 4 E -8 6 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 26 8 B -8 0 18 -4 -6 C -18 -18 0 -16 -14 D -26 4 16 0 4 E -8 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=25 B=23 D=10 C=10 so D is eliminated. Round 2 votes counts: A=35 E=32 B=23 C=10 so C is eliminated. Round 3 votes counts: A=39 E=36 B=25 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:204 B:200 D:199 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 26 8 B -8 0 18 -4 -6 C -18 -18 0 -16 -14 D -26 4 16 0 4 E -8 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 26 8 B -8 0 18 -4 -6 C -18 -18 0 -16 -14 D -26 4 16 0 4 E -8 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 26 8 B -8 0 18 -4 -6 C -18 -18 0 -16 -14 D -26 4 16 0 4 E -8 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4030: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) A C D B E (12) E D B A C (9) E B D C A (8) C A E B D (7) E B D A C (5) D B A E C (5) E C B D A (4) D B E A C (4) E C A B D (3) D B A C E (3) C A B E D (3) E D B C A (2) D A B C E (2) C A E D B (2) B D A E C (2) B C D A E (2) B A D C E (2) E D A C B (1) E D A B C (1) E C A D B (1) E A C D B (1) C E A B D (1) C B E D A (1) C A D B E (1) B E D C A (1) B D E C A (1) B D A C E (1) A D C B E (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 2 -4 14 B 0 0 -4 8 8 C -2 4 0 4 4 D 4 -8 -4 0 -2 E -14 -8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.457666 B: 0.072083 C: 0.313501 D: 0.156750 E: 0.000000 Sum of squares = 0.337507624051 Cumulative probabilities = A: 0.457666 B: 0.529749 C: 0.843250 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -4 14 B 0 0 -4 8 8 C -2 4 0 4 4 D 4 -8 -4 0 -2 E -14 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.465116 B: 0.081395 C: 0.302326 D: 0.151163 E: 0.000000 Sum of squares = 0.337209302326 Cumulative probabilities = A: 0.465116 B: 0.546512 C: 0.848837 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=27 A=15 D=14 B=9 so B is eliminated. Round 2 votes counts: E=36 C=29 D=18 A=17 so A is eliminated. Round 3 votes counts: C=43 E=36 D=21 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:206 B:206 C:205 D:195 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 2 -4 14 B 0 0 -4 8 8 C -2 4 0 4 4 D 4 -8 -4 0 -2 E -14 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.465116 B: 0.081395 C: 0.302326 D: 0.151163 E: 0.000000 Sum of squares = 0.337209302326 Cumulative probabilities = A: 0.465116 B: 0.546512 C: 0.848837 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -4 14 B 0 0 -4 8 8 C -2 4 0 4 4 D 4 -8 -4 0 -2 E -14 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.465116 B: 0.081395 C: 0.302326 D: 0.151163 E: 0.000000 Sum of squares = 0.337209302326 Cumulative probabilities = A: 0.465116 B: 0.546512 C: 0.848837 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -4 14 B 0 0 -4 8 8 C -2 4 0 4 4 D 4 -8 -4 0 -2 E -14 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.465116 B: 0.081395 C: 0.302326 D: 0.151163 E: 0.000000 Sum of squares = 0.337209302326 Cumulative probabilities = A: 0.465116 B: 0.546512 C: 0.848837 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4031: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (14) B E D C A (9) C A D E B (6) A C B E D (6) E D B C A (5) D E C A B (5) D E B C A (4) B D E A C (4) B A E C D (4) B E D A C (3) B A C E D (3) A C B D E (3) E D C B A (2) E B D C A (2) E B C A D (2) D E C B A (2) D C E A B (2) C A E B D (2) B E C A D (2) B D A E C (2) A B C D E (2) E D C A B (1) E C D A B (1) D E B A C (1) D C A E B (1) D B E C A (1) D B A E C (1) D A E C B (1) D A B C E (1) C B A E D (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E C A (1) B A C D E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 0 2 2 B 4 0 0 -2 -6 C 0 0 0 2 -10 D -2 2 -2 0 6 E -2 6 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.266667 B: 0.200000 C: 0.133333 D: 0.333333 E: 0.066667 Sum of squares = 0.24444444446 Cumulative probabilities = A: 0.266667 B: 0.466667 C: 0.600000 D: 0.933333 E: 1.000000 A B C D E A 0 -4 0 2 2 B 4 0 0 -2 -6 C 0 0 0 2 -10 D -2 2 -2 0 6 E -2 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.200000 C: 0.133333 D: 0.333333 E: 0.066667 Sum of squares = 0.244444444446 Cumulative probabilities = A: 0.266667 B: 0.466667 C: 0.600000 D: 0.933333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=27 D=19 E=13 C=10 so C is eliminated. Round 2 votes counts: A=36 B=32 D=19 E=13 so E is eliminated. Round 3 votes counts: B=36 A=36 D=28 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:204 D:202 A:200 B:198 C:196 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 2 2 B 4 0 0 -2 -6 C 0 0 0 2 -10 D -2 2 -2 0 6 E -2 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.200000 C: 0.133333 D: 0.333333 E: 0.066667 Sum of squares = 0.244444444446 Cumulative probabilities = A: 0.266667 B: 0.466667 C: 0.600000 D: 0.933333 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 2 B 4 0 0 -2 -6 C 0 0 0 2 -10 D -2 2 -2 0 6 E -2 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.200000 C: 0.133333 D: 0.333333 E: 0.066667 Sum of squares = 0.244444444446 Cumulative probabilities = A: 0.266667 B: 0.466667 C: 0.600000 D: 0.933333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 2 B 4 0 0 -2 -6 C 0 0 0 2 -10 D -2 2 -2 0 6 E -2 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.200000 C: 0.133333 D: 0.333333 E: 0.066667 Sum of squares = 0.244444444446 Cumulative probabilities = A: 0.266667 B: 0.466667 C: 0.600000 D: 0.933333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4032: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) E A C B D (8) C D B E A (8) D B C A E (7) A D E B C (6) C B E D A (5) C B D E A (5) E C A B D (4) D C B A E (4) D A E C B (4) C E B A D (4) E A C D B (3) D B A C E (3) D A B E C (3) B C E A D (3) E A D C B (2) D A B C E (2) C E B D A (2) B C D A E (2) A E B D C (2) A D B E C (2) E C B A D (1) D C B E A (1) D A C B E (1) C E D A B (1) C B E A D (1) B C E D A (1) B C A D E (1) B A E D C (1) A E D C B (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 2 -18 B -2 0 -10 2 2 C 0 10 0 18 2 D -2 -2 -18 0 0 E 18 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.054330 B: 0.000000 C: 0.945670 D: 0.000000 E: 0.000000 Sum of squares = 0.897243541425 Cumulative probabilities = A: 0.054330 B: 0.054330 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 -18 B -2 0 -10 2 2 C 0 10 0 18 2 D -2 -2 -18 0 0 E 18 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000009856 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 D=25 A=13 B=8 so B is eliminated. Round 2 votes counts: C=33 E=28 D=25 A=14 so A is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:207 B:196 A:193 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 2 -18 B -2 0 -10 2 2 C 0 10 0 18 2 D -2 -2 -18 0 0 E 18 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000009856 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 -18 B -2 0 -10 2 2 C 0 10 0 18 2 D -2 -2 -18 0 0 E 18 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000009856 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 -18 B -2 0 -10 2 2 C 0 10 0 18 2 D -2 -2 -18 0 0 E 18 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000009856 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4033: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) B C D E A (8) E A C B D (7) E A C D B (6) D B C A E (6) E C B A D (4) C B D E A (4) B D C A E (4) A E D B C (4) A D E B C (4) E A D C B (3) D B A C E (3) C E B A D (3) C E A B D (3) E C A B D (2) E A D B C (2) D B A E C (2) D A E B C (2) D A C B E (2) D A B E C (2) C B D A E (2) B D C E A (2) B C E D A (2) E B C A D (1) E A B D C (1) E A B C D (1) D C B A E (1) D C A B E (1) D B E A C (1) D A B C E (1) C D B A E (1) C B E D A (1) B E C A D (1) B C E A D (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 4 8 -6 B -4 0 -2 -4 -8 C -4 2 0 -2 -6 D -8 4 2 0 -6 E 6 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 8 -6 B -4 0 -2 -4 -8 C -4 2 0 -2 -6 D -8 4 2 0 -6 E 6 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=21 B=19 A=19 C=14 so C is eliminated. Round 2 votes counts: E=33 B=26 D=22 A=19 so A is eliminated. Round 3 votes counts: E=48 D=26 B=26 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:205 D:196 C:195 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 8 -6 B -4 0 -2 -4 -8 C -4 2 0 -2 -6 D -8 4 2 0 -6 E 6 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 8 -6 B -4 0 -2 -4 -8 C -4 2 0 -2 -6 D -8 4 2 0 -6 E 6 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 8 -6 B -4 0 -2 -4 -8 C -4 2 0 -2 -6 D -8 4 2 0 -6 E 6 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4034: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) B A D C E (8) E C D A B (6) E D C A B (4) E A D B C (4) A B E D C (4) E C A D B (3) E A C D B (3) E A B D C (3) D A B C E (3) C E D B A (3) C B D E A (3) B C A D E (3) A D B C E (3) E C B A D (2) E B C A D (2) E A C B D (2) C B E D A (2) C B D A E (2) B D C A E (2) B C E A D (2) B A E C D (2) B A C D E (2) A D B E C (2) A B D C E (2) E D A C B (1) E C D B A (1) E C B D A (1) E A D C B (1) E A B C D (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C B E (1) C E D A B (1) C E B D A (1) C E B A D (1) C D E A B (1) C D B E A (1) C D B A E (1) B D A C E (1) B A D E C (1) A E D B C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -2 8 2 B 2 0 16 14 12 C 2 -16 0 8 6 D -8 -14 -8 0 -4 E -2 -12 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 8 2 B 2 0 16 14 12 C 2 -16 0 8 6 D -8 -14 -8 0 -4 E -2 -12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988789 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=29 C=16 A=14 D=7 so D is eliminated. Round 2 votes counts: E=34 B=31 A=19 C=16 so C is eliminated. Round 3 votes counts: E=41 B=40 A=19 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:203 C:200 E:192 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 8 2 B 2 0 16 14 12 C 2 -16 0 8 6 D -8 -14 -8 0 -4 E -2 -12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988789 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 8 2 B 2 0 16 14 12 C 2 -16 0 8 6 D -8 -14 -8 0 -4 E -2 -12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988789 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 8 2 B 2 0 16 14 12 C 2 -16 0 8 6 D -8 -14 -8 0 -4 E -2 -12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988789 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4035: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) E B D C A (8) C B E A D (8) A D C E B (7) A C D B E (7) E B C D A (6) D E B A C (5) D A C E B (5) C A B E D (5) B E C D A (5) B E C A D (4) D A E C B (3) C E B A D (3) B E D C A (3) A D C B E (3) D A C B E (2) B C E A D (2) A C D E B (2) E C B D A (1) E C B A D (1) E B C A D (1) D E A B C (1) D B E A C (1) D A B E C (1) C E A B D (1) C B A E D (1) C A E D B (1) C A E B D (1) C A B D E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -4 -2 0 B -2 0 -6 4 -12 C 4 6 0 2 0 D 2 -4 -2 0 -2 E 0 12 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.661760 D: 0.000000 E: 0.338240 Sum of squares = 0.552332405255 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.661760 D: 0.661760 E: 1.000000 A B C D E A 0 2 -4 -2 0 B -2 0 -6 4 -12 C 4 6 0 2 0 D 2 -4 -2 0 -2 E 0 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=21 A=20 E=17 B=14 so B is eliminated. Round 2 votes counts: E=29 D=28 C=23 A=20 so A is eliminated. Round 3 votes counts: D=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:207 C:206 A:198 D:197 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -2 0 B -2 0 -6 4 -12 C 4 6 0 2 0 D 2 -4 -2 0 -2 E 0 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 0 B -2 0 -6 4 -12 C 4 6 0 2 0 D 2 -4 -2 0 -2 E 0 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 0 B -2 0 -6 4 -12 C 4 6 0 2 0 D 2 -4 -2 0 -2 E 0 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4036: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) A B E C D (7) A B C E D (6) E A C B D (5) D C E B A (5) B A D C E (5) E D C A B (4) B A C D E (4) C E A B D (3) C B D A E (3) B C A D E (3) A B D E C (3) E D A B C (2) E C D A B (2) E C A B D (2) E A D B C (2) E A B C D (2) D C B E A (2) D C B A E (2) D B C A E (2) D B A C E (2) C D B E A (2) A E B C D (2) D E C A B (1) D B E A C (1) D B C E A (1) D B A E C (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E B A (1) C B E A D (1) C A E B D (1) C A B E D (1) B D C A E (1) B D A C E (1) B A D E C (1) B A C E D (1) A E C B D (1) A D B E C (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 10 4 B 2 0 4 14 10 C -2 -4 0 0 2 D -10 -14 0 0 10 E -4 -10 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 10 4 B 2 0 4 14 10 C -2 -4 0 0 2 D -10 -14 0 0 10 E -4 -10 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989746 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=23 E=19 B=16 C=14 so C is eliminated. Round 2 votes counts: D=31 A=25 E=24 B=20 so B is eliminated. Round 3 votes counts: A=39 D=36 E=25 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:207 C:198 D:193 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 10 4 B 2 0 4 14 10 C -2 -4 0 0 2 D -10 -14 0 0 10 E -4 -10 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989746 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 10 4 B 2 0 4 14 10 C -2 -4 0 0 2 D -10 -14 0 0 10 E -4 -10 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989746 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 10 4 B 2 0 4 14 10 C -2 -4 0 0 2 D -10 -14 0 0 10 E -4 -10 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989746 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4037: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) B A D C E (8) E C D A B (7) B A C D E (6) E C B A D (5) C E D B A (5) E D A C B (4) D A E B C (4) C B D E A (3) B D A C E (3) B C E A D (3) B C A D E (3) A D B E C (3) A D B C E (3) A B D E C (3) E D C A B (2) E C D B A (2) E A D C B (2) D A B C E (2) B C D A E (2) B A C E D (2) A D E B C (2) A B D C E (2) E C B D A (1) E B C A D (1) E A D B C (1) E A C D B (1) E A B C D (1) D C E B A (1) D C B E A (1) D C A E B (1) D C A B E (1) D B A C E (1) C E B A D (1) C D E B A (1) C B E D A (1) C B E A D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 0 0 -4 B 18 0 4 12 0 C 0 -4 0 10 18 D 0 -12 -10 0 0 E 4 0 -18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.902365 C: 0.000000 D: 0.000000 E: 0.097635 Sum of squares = 0.823794978127 Cumulative probabilities = A: 0.000000 B: 0.902365 C: 0.902365 D: 0.902365 E: 1.000000 A B C D E A 0 -18 0 0 -4 B 18 0 4 12 0 C 0 -4 0 10 18 D 0 -12 -10 0 0 E 4 0 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.702479408679 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=27 B=27 C=20 A=15 D=11 so D is eliminated. Round 2 votes counts: B=28 E=27 C=24 A=21 so A is eliminated. Round 3 votes counts: B=43 E=33 C=24 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:212 E:193 A:189 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 0 0 -4 B 18 0 4 12 0 C 0 -4 0 10 18 D 0 -12 -10 0 0 E 4 0 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.702479408679 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 0 -4 B 18 0 4 12 0 C 0 -4 0 10 18 D 0 -12 -10 0 0 E 4 0 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.702479408679 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 0 -4 B 18 0 4 12 0 C 0 -4 0 10 18 D 0 -12 -10 0 0 E 4 0 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.818182 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.702479408679 Cumulative probabilities = A: 0.000000 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4038: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) A C B D E (7) E D B A C (6) E D C A B (5) D E B A C (5) D E A C B (5) C A B D E (5) A C D B E (5) B C A E D (4) E C D A B (3) E B D C A (3) D E A B C (3) D A E C B (3) A D C E B (3) E C B D A (2) D A C E B (2) C E B A D (2) C B E A D (2) C B A E D (2) B E D A C (2) B E C D A (2) B D E A C (2) B A C D E (2) E B D A C (1) D E C A B (1) D B E A C (1) D B A E C (1) D A E B C (1) C B A D E (1) C A D E B (1) C A B E D (1) B E D C A (1) B C E A D (1) B C A D E (1) B A D E C (1) B A D C E (1) A D C B E (1) A D B C E (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 12 -12 -8 B 0 0 -4 -10 -8 C -12 4 0 -14 -10 D 12 10 14 0 12 E 8 8 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 -12 -8 B 0 0 -4 -10 -8 C -12 4 0 -14 -10 D 12 10 14 0 12 E 8 8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=22 A=20 B=17 C=14 so C is eliminated. Round 2 votes counts: E=29 A=27 D=22 B=22 so D is eliminated. Round 3 votes counts: E=43 A=33 B=24 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:224 E:207 A:196 B:189 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 12 -12 -8 B 0 0 -4 -10 -8 C -12 4 0 -14 -10 D 12 10 14 0 12 E 8 8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -12 -8 B 0 0 -4 -10 -8 C -12 4 0 -14 -10 D 12 10 14 0 12 E 8 8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -12 -8 B 0 0 -4 -10 -8 C -12 4 0 -14 -10 D 12 10 14 0 12 E 8 8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4039: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (5) E B D C A (4) D B E C A (4) B E D A C (4) A E C B D (4) A C E D B (4) A C E B D (4) E C D B A (3) E A C B D (3) C E D B A (3) C D E B A (3) C A D E B (3) B D E A C (3) A C D B E (3) A B E D C (3) E D B C A (2) E B C D A (2) E B A D C (2) D C A B E (2) D B C E A (2) D B C A E (2) C E D A B (2) C E A D B (2) C A E D B (2) C A D B E (2) B E D C A (2) B D A E C (2) B A E D C (2) B A D E C (2) A E B C D (2) E C B D A (1) E C A B D (1) E B D A C (1) E B C A D (1) E B A C D (1) D E C B A (1) D E B C A (1) D C B A E (1) D A C B E (1) C D A E B (1) C D A B E (1) B E A D C (1) A D B C E (1) A C D E B (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -6 -6 -8 B 10 0 -10 -4 -8 C 6 10 0 2 -6 D 6 4 -2 0 -14 E 8 8 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 -6 -8 B 10 0 -10 -4 -8 C 6 10 0 2 -6 D 6 4 -2 0 -14 E 8 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 D=19 C=19 B=16 so B is eliminated. Round 2 votes counts: A=29 E=28 D=24 C=19 so C is eliminated. Round 3 votes counts: A=36 E=35 D=29 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:206 D:197 B:194 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 -6 -8 B 10 0 -10 -4 -8 C 6 10 0 2 -6 D 6 4 -2 0 -14 E 8 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -6 -8 B 10 0 -10 -4 -8 C 6 10 0 2 -6 D 6 4 -2 0 -14 E 8 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -6 -8 B 10 0 -10 -4 -8 C 6 10 0 2 -6 D 6 4 -2 0 -14 E 8 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4040: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A B D C E (8) B D A C E (7) E C D B A (6) B A D E C (5) E C B D A (4) A D C E B (4) A C D E B (4) E C A D B (3) E B C D A (3) E B C A D (3) D C A B E (3) C D E A B (3) C D A E B (3) B E C D A (3) B A E C D (3) E C D A B (2) E B D C A (2) B D C E A (2) A E C D B (2) A D C B E (2) A D B C E (2) A C E D B (2) E B A C D (1) E A C D B (1) D E C B A (1) D C E B A (1) D C E A B (1) D B A C E (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) B E A C D (1) B D E A C (1) B D A E C (1) B A D C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 0 -12 4 B 8 0 8 8 2 C 0 -8 0 -6 -2 D 12 -8 6 0 6 E -4 -2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -12 4 B 8 0 8 8 2 C 0 -8 0 -6 -2 D 12 -8 6 0 6 E -4 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995165 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=25 A=25 D=10 C=8 so C is eliminated. Round 2 votes counts: B=32 E=27 A=25 D=16 so D is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:208 E:195 A:192 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -12 4 B 8 0 8 8 2 C 0 -8 0 -6 -2 D 12 -8 6 0 6 E -4 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995165 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -12 4 B 8 0 8 8 2 C 0 -8 0 -6 -2 D 12 -8 6 0 6 E -4 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995165 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -12 4 B 8 0 8 8 2 C 0 -8 0 -6 -2 D 12 -8 6 0 6 E -4 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995165 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4041: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) A B D E C (9) C E D B A (7) B C A E D (7) E C D A B (6) D E A C B (6) C E D A B (6) B A C D E (6) A D E B C (6) B A D C E (5) C B E D A (4) A D B E C (4) C E B D A (3) B A D E C (3) E C D B A (2) E B A D C (2) D E C A B (2) D A E B C (2) C B E A D (2) A B D C E (2) E A D B C (1) D E A B C (1) D C A B E (1) D A E C B (1) B C E A D (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 14 0 0 -4 B -14 0 2 -8 -8 C 0 -2 0 -10 -10 D 0 8 10 0 -4 E 4 8 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 0 0 -4 B -14 0 2 -8 -8 C 0 -2 0 -10 -10 D 0 8 10 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 C=22 A=22 E=20 D=13 so D is eliminated. Round 2 votes counts: E=29 A=25 C=23 B=23 so C is eliminated. Round 3 votes counts: E=45 B=29 A=26 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:207 A:205 C:189 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 0 -4 B -14 0 2 -8 -8 C 0 -2 0 -10 -10 D 0 8 10 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 0 -4 B -14 0 2 -8 -8 C 0 -2 0 -10 -10 D 0 8 10 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 0 -4 B -14 0 2 -8 -8 C 0 -2 0 -10 -10 D 0 8 10 0 -4 E 4 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4042: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (20) C D B E A (10) C D B A E (8) E A B D C (6) D C A E B (6) C D A E B (5) A E D B C (5) E B A C D (4) D C A B E (3) C D A B E (3) C B D E A (3) B E A C D (3) B D E A C (3) A E B D C (3) E A B C D (2) C B E D A (2) A E D C B (2) E B A D C (1) D C B A E (1) D B A E C (1) D A E C B (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E A D (1) C A D E B (1) B E C A D (1) B D C E A (1) B C D E A (1) Total count = 100 A B C D E A 0 -22 2 4 -22 B 22 0 2 6 22 C -2 -2 0 -6 -4 D -4 -6 6 0 -6 E 22 -22 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 2 4 -22 B 22 0 2 6 22 C -2 -2 0 -6 -4 D -4 -6 6 0 -6 E 22 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=29 E=13 D=12 A=10 so A is eliminated. Round 2 votes counts: C=36 B=29 E=23 D=12 so D is eliminated. Round 3 votes counts: C=46 B=30 E=24 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:205 D:195 C:193 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 2 4 -22 B 22 0 2 6 22 C -2 -2 0 -6 -4 D -4 -6 6 0 -6 E 22 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 2 4 -22 B 22 0 2 6 22 C -2 -2 0 -6 -4 D -4 -6 6 0 -6 E 22 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 2 4 -22 B 22 0 2 6 22 C -2 -2 0 -6 -4 D -4 -6 6 0 -6 E 22 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971622 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4043: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (11) A D B E C (8) D A E C B (7) B C E D A (7) B C A E D (7) B A C D E (7) C E D B A (5) C E B D A (5) B A C E D (5) A D E C B (5) E C B D A (4) E C D B A (3) D E A C B (3) A D E B C (3) D E C A B (2) C B E A D (2) B A D C E (2) E D C A B (1) E C D A B (1) E C B A D (1) D E C B A (1) D C E B A (1) D A B C E (1) C B E D A (1) B D A C E (1) B C D A E (1) B C A D E (1) A E C B D (1) A E B C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -30 -8 12 4 B 30 0 14 18 12 C 8 -14 0 26 16 D -12 -18 -26 0 -10 E -4 -12 -16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -8 12 4 B 30 0 14 18 12 C 8 -14 0 26 16 D -12 -18 -26 0 -10 E -4 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=20 D=15 C=13 E=10 so E is eliminated. Round 2 votes counts: B=42 C=22 A=20 D=16 so D is eliminated. Round 3 votes counts: B=42 A=31 C=27 so C is eliminated. Round 4 votes counts: B=65 A=35 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:237 C:218 A:189 E:189 D:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -8 12 4 B 30 0 14 18 12 C 8 -14 0 26 16 D -12 -18 -26 0 -10 E -4 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -8 12 4 B 30 0 14 18 12 C 8 -14 0 26 16 D -12 -18 -26 0 -10 E -4 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -8 12 4 B 30 0 14 18 12 C 8 -14 0 26 16 D -12 -18 -26 0 -10 E -4 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4044: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) C E D B A (10) E D C A B (6) A B D E C (6) B C D E A (5) C E D A B (4) C D E B A (4) C B D E A (4) B A C D E (4) A B C E D (3) A B C D E (3) E D C B A (2) E D A C B (2) D E C B A (2) D C E B A (2) C E A D B (2) C A E D B (2) B D C E A (2) B A D E C (2) A E D C B (2) A C B E D (2) A B E D C (2) E C D A B (1) E C A D B (1) D E B C A (1) D E B A C (1) D E A B C (1) D B E C A (1) C D B E A (1) C B E D A (1) C B E A D (1) C A B E D (1) B D E C A (1) B D A E C (1) B C D A E (1) B C A D E (1) B A D C E (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -12 -6 -10 B -4 0 -2 -18 -12 C 12 2 0 4 10 D 6 18 -4 0 -10 E 10 12 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 -6 -10 B -4 0 -2 -18 -12 C 12 2 0 4 10 D 6 18 -4 0 -10 E 10 12 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=30 B=18 E=12 D=8 so D is eliminated. Round 2 votes counts: C=32 A=32 B=19 E=17 so E is eliminated. Round 3 votes counts: C=44 A=35 B=21 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:211 D:205 A:188 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -6 -10 B -4 0 -2 -18 -12 C 12 2 0 4 10 D 6 18 -4 0 -10 E 10 12 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -6 -10 B -4 0 -2 -18 -12 C 12 2 0 4 10 D 6 18 -4 0 -10 E 10 12 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -6 -10 B -4 0 -2 -18 -12 C 12 2 0 4 10 D 6 18 -4 0 -10 E 10 12 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4045: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) B C A D E (9) B C D E A (7) E D A C B (6) A E D C B (6) A E D B C (6) A D E C B (5) E A D C B (4) C D B E A (4) C B D E A (4) E D C B A (3) E D C A B (3) B C E D A (3) A B E D C (3) E A B D C (2) C D E B A (2) C D E A B (2) B E D C A (2) B A E D C (2) B A C E D (2) B A C D E (2) E D B C A (1) E D A B C (1) E C B D A (1) E B D C A (1) D E A C B (1) D A E C B (1) C B D A E (1) B E C D A (1) B C A E D (1) B A E C D (1) A D C E B (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -8 -6 -14 B -6 0 -10 -12 -12 C 8 10 0 -14 -18 D 6 12 14 0 2 E 14 12 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -6 -14 B -6 0 -10 -12 -12 C 8 10 0 -14 -18 D 6 12 14 0 2 E 14 12 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 E=22 C=13 D=11 so D is eliminated. Round 2 votes counts: E=32 B=30 A=25 C=13 so C is eliminated. Round 3 votes counts: B=39 E=36 A=25 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 D:217 C:193 A:189 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -6 -14 B -6 0 -10 -12 -12 C 8 10 0 -14 -18 D 6 12 14 0 2 E 14 12 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -6 -14 B -6 0 -10 -12 -12 C 8 10 0 -14 -18 D 6 12 14 0 2 E 14 12 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -6 -14 B -6 0 -10 -12 -12 C 8 10 0 -14 -18 D 6 12 14 0 2 E 14 12 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4046: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (16) D E B C A (13) B E C D A (7) A D C E B (5) A C D E B (5) E B D C A (4) D E C B A (4) B E C A D (4) B E A C D (4) A C D B E (4) A B C E D (4) D A E B C (3) D A C E B (3) C D E B A (3) E B C D A (2) D A E C B (2) A B E C D (2) E D B C A (1) D E C A B (1) D E B A C (1) D E A B C (1) D C E B A (1) C E B D A (1) C E B A D (1) C B E A D (1) C A B E D (1) B E D C A (1) B C E A D (1) B A E D C (1) A D B E C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 8 4 -2 B 0 0 2 4 -2 C -8 -2 0 14 -6 D -4 -4 -14 0 -2 E 2 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 4 -2 B 0 0 2 4 -2 C -8 -2 0 14 -6 D -4 -4 -14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 D=29 B=18 E=7 C=7 so E is eliminated. Round 2 votes counts: A=39 D=30 B=24 C=7 so C is eliminated. Round 3 votes counts: A=40 D=33 B=27 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:205 B:202 C:199 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 4 -2 B 0 0 2 4 -2 C -8 -2 0 14 -6 D -4 -4 -14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 4 -2 B 0 0 2 4 -2 C -8 -2 0 14 -6 D -4 -4 -14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 4 -2 B 0 0 2 4 -2 C -8 -2 0 14 -6 D -4 -4 -14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4047: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) A D B C E (6) E C B D A (5) D E B C A (5) D A E B C (5) A E C B D (5) D B E C A (4) D A B C E (4) C B E D A (4) C B E A D (4) A E D C B (4) E D C B A (3) E D A C B (3) B D C E A (3) E C D B A (2) E C A B D (2) E A C B D (2) D E A B C (2) D B C A E (2) D B A C E (2) D A E C B (2) B C E D A (2) B C D E A (2) B C D A E (2) B C A D E (2) A D E C B (2) A D E B C (2) E D B C A (1) E B C D A (1) D E C B A (1) D E A C B (1) D B C E A (1) D A B E C (1) C E B A D (1) C B A E D (1) C A E B D (1) B D A C E (1) A E C D B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -10 -18 -10 B 10 0 -2 -8 -14 C 10 2 0 -12 -20 D 18 8 12 0 2 E 10 14 20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -18 -10 B 10 0 -2 -8 -14 C 10 2 0 -12 -20 D 18 8 12 0 2 E 10 14 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=25 A=22 B=12 C=11 so C is eliminated. Round 2 votes counts: D=30 E=26 A=23 B=21 so B is eliminated. Round 3 votes counts: D=38 E=36 A=26 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:221 D:220 B:193 C:190 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 -18 -10 B 10 0 -2 -8 -14 C 10 2 0 -12 -20 D 18 8 12 0 2 E 10 14 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -18 -10 B 10 0 -2 -8 -14 C 10 2 0 -12 -20 D 18 8 12 0 2 E 10 14 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -18 -10 B 10 0 -2 -8 -14 C 10 2 0 -12 -20 D 18 8 12 0 2 E 10 14 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4048: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) E B A D C (7) D E B A C (7) D A C E B (7) C A B E D (7) C D A B E (6) B E C A D (6) E B C D A (5) D A E B C (5) C A D B E (5) D E A B C (3) D C A E B (3) C D E B A (3) C B A E D (3) A D C B E (3) D A E C B (2) C A B D E (2) B C E A D (2) E D B A C (1) E D A B C (1) E B D C A (1) E B C A D (1) D E C B A (1) D E B C A (1) D C E B A (1) C D B E A (1) C B E D A (1) C B E A D (1) A D E C B (1) A D E B C (1) A D B E C (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 0 -12 0 B -2 0 0 -8 -16 C 0 0 0 -6 -4 D 12 8 6 0 10 E 0 16 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -12 0 B -2 0 0 -8 -16 C 0 0 0 -6 -4 D 12 8 6 0 10 E 0 16 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=29 E=23 A=10 B=8 so B is eliminated. Round 2 votes counts: C=31 D=30 E=29 A=10 so A is eliminated. Round 3 votes counts: D=36 C=34 E=30 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:205 A:195 C:195 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -12 0 B -2 0 0 -8 -16 C 0 0 0 -6 -4 D 12 8 6 0 10 E 0 16 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -12 0 B -2 0 0 -8 -16 C 0 0 0 -6 -4 D 12 8 6 0 10 E 0 16 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -12 0 B -2 0 0 -8 -16 C 0 0 0 -6 -4 D 12 8 6 0 10 E 0 16 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4049: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B D C A (5) D A E B C (5) B E D C A (5) E B D A C (4) A D E C B (4) E C A B D (3) C B D A E (3) C A E B D (3) C A B D E (3) B C E D A (3) A D C B E (3) A C D E B (3) E D B A C (2) E B C D A (2) E B C A D (2) E A D B C (2) E A C B D (2) D E B A C (2) D E A B C (2) D C B A E (2) D A C B E (2) C B E A D (2) C B A D E (2) C A D B E (2) B E C D A (2) B D C A E (2) A E D C B (2) A D C E B (2) E D A B C (1) E C B A D (1) E A D C B (1) E A C D B (1) E A B D C (1) D B A E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C D A B E (1) C A B E D (1) B D E C A (1) B D C E A (1) B C D A E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 12 2 4 6 B -12 0 -10 4 -8 C -2 10 0 -2 -4 D -4 -4 2 0 2 E -6 8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 4 6 B -12 0 -10 4 -8 C -2 10 0 -2 -4 D -4 -4 2 0 2 E -6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=24 C=19 D=15 B=15 so D is eliminated. Round 2 votes counts: A=32 E=31 C=21 B=16 so B is eliminated. Round 3 votes counts: E=39 A=33 C=28 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:202 C:201 D:198 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 4 6 B -12 0 -10 4 -8 C -2 10 0 -2 -4 D -4 -4 2 0 2 E -6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 4 6 B -12 0 -10 4 -8 C -2 10 0 -2 -4 D -4 -4 2 0 2 E -6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 4 6 B -12 0 -10 4 -8 C -2 10 0 -2 -4 D -4 -4 2 0 2 E -6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4050: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) E A C D B (9) E D A C B (8) C A B E D (7) B D E A C (7) B D C A E (7) E A D C B (4) D E A B C (4) D B E A C (4) C A E B D (4) B C D A E (4) D E B A C (3) C B A E D (3) B D E C A (3) B C A D E (3) E D B A C (2) D E A C B (2) C B A D E (2) C A B D E (2) B C A E D (2) E C A D B (1) E C A B D (1) E B D A C (1) D B C A E (1) D A E C B (1) C E A D B (1) C A D B E (1) B E C A D (1) B D A C E (1) A C E D B (1) Total count = 100 A B C D E A 0 12 -6 4 -2 B -12 0 -14 -4 -4 C 6 14 0 4 -2 D -4 4 -4 0 -10 E 2 4 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -6 4 -2 B -12 0 -14 -4 -4 C 6 14 0 4 -2 D -4 4 -4 0 -10 E 2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=28 E=26 D=15 A=1 so A is eliminated. Round 2 votes counts: C=31 B=28 E=26 D=15 so D is eliminated. Round 3 votes counts: E=36 B=33 C=31 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:211 E:209 A:204 D:193 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -6 4 -2 B -12 0 -14 -4 -4 C 6 14 0 4 -2 D -4 4 -4 0 -10 E 2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 4 -2 B -12 0 -14 -4 -4 C 6 14 0 4 -2 D -4 4 -4 0 -10 E 2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 4 -2 B -12 0 -14 -4 -4 C 6 14 0 4 -2 D -4 4 -4 0 -10 E 2 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4051: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) E A C B D (5) B C A D E (5) E D A C B (4) C A B E D (4) B D C A E (4) B A C D E (4) A E B C D (4) E D C A B (3) D E B C A (3) D B E A C (3) C D E A B (3) B D A C E (3) B C D A E (3) A E C B D (3) A C B E D (3) E A D B C (2) E A C D B (2) E A B C D (2) D E C B A (2) D E B A C (2) D C E B A (2) D B E C A (2) D B C E A (2) B A D C E (2) A C E B D (2) A B C E D (2) E C D A B (1) E C A D B (1) E B A D C (1) E A D C B (1) E A B D C (1) D E C A B (1) D C E A B (1) D C B E A (1) D B C A E (1) C E A D B (1) C D B A E (1) C A E D B (1) C A E B D (1) C A B D E (1) B E D A C (1) B D A E C (1) B A D E C (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 12 2 -4 B -10 0 12 10 -8 C -12 -12 0 2 -4 D -2 -10 -2 0 -4 E 4 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 12 2 -4 B -10 0 12 10 -8 C -12 -12 0 2 -4 D -2 -10 -2 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=25 D=20 A=15 C=12 so C is eliminated. Round 2 votes counts: E=29 B=25 D=24 A=22 so A is eliminated. Round 3 votes counts: E=40 B=36 D=24 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:210 E:210 B:202 D:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 12 2 -4 B -10 0 12 10 -8 C -12 -12 0 2 -4 D -2 -10 -2 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 2 -4 B -10 0 12 10 -8 C -12 -12 0 2 -4 D -2 -10 -2 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 2 -4 B -10 0 12 10 -8 C -12 -12 0 2 -4 D -2 -10 -2 0 -4 E 4 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4052: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) A B C D E (9) E D C A B (7) A B E D C (6) A B C E D (6) E D C B A (5) C D E B A (5) E D A B C (3) E C D A B (3) E A D B C (3) D E C B A (3) E D A C B (2) D B E A C (2) C E D B A (2) C E D A B (2) C D B E A (2) C B D E A (2) C B D A E (2) B C A D E (2) B A D E C (2) A E D B C (2) A E B D C (2) A C B E D (2) A B E C D (2) E A D C B (1) D E B C A (1) D E B A C (1) D C E B A (1) D B C E A (1) C E A D B (1) B D C A E (1) B C D A E (1) B A D C E (1) A E D C B (1) A B D C E (1) Total count = 100 A B C D E A 0 6 18 8 6 B -6 0 18 4 10 C -18 -18 0 8 8 D -8 -4 -8 0 0 E -6 -10 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 8 6 B -6 0 18 4 10 C -18 -18 0 8 8 D -8 -4 -8 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 B=20 C=16 D=9 so D is eliminated. Round 2 votes counts: A=31 E=29 B=23 C=17 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:213 C:190 D:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 8 6 B -6 0 18 4 10 C -18 -18 0 8 8 D -8 -4 -8 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 8 6 B -6 0 18 4 10 C -18 -18 0 8 8 D -8 -4 -8 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 8 6 B -6 0 18 4 10 C -18 -18 0 8 8 D -8 -4 -8 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4053: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) E C B A D (7) E B C D A (6) E A C B D (6) D B E C A (5) D A B C E (5) A C B D E (5) D B A C E (4) C B D E A (4) A E C B D (4) A D E B C (4) E C B D A (3) E A D B C (3) A D B C E (3) A C E B D (3) E D B C A (2) E B D C A (2) D E A B C (2) D B C E A (2) D A E B C (2) C B E D A (2) A D E C B (2) A C D B E (2) E D A B C (1) E C A B D (1) D E B C A (1) D E B A C (1) D B C A E (1) D A B E C (1) C E B D A (1) C B D A E (1) C B A E D (1) C B A D E (1) B E D C A (1) B C E D A (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 6 14 2 -4 B -6 0 -4 0 -4 C -14 4 0 -2 -10 D -2 0 2 0 10 E 4 4 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468749999999 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 A B C D E A 0 6 14 2 -4 B -6 0 -4 0 -4 C -14 4 0 -2 -10 D -2 0 2 0 10 E 4 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000014 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=31 D=24 C=10 B=3 so B is eliminated. Round 2 votes counts: E=32 A=32 D=24 C=12 so C is eliminated. Round 3 votes counts: E=36 A=34 D=30 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:209 D:205 E:204 B:193 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 14 2 -4 B -6 0 -4 0 -4 C -14 4 0 -2 -10 D -2 0 2 0 10 E 4 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000014 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 2 -4 B -6 0 -4 0 -4 C -14 4 0 -2 -10 D -2 0 2 0 10 E 4 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000014 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 2 -4 B -6 0 -4 0 -4 C -14 4 0 -2 -10 D -2 0 2 0 10 E 4 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000014 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4054: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (13) C E D B A (8) E B A C D (6) C D E A B (6) E C B D A (5) C D E B A (5) A D B C E (5) E C B A D (4) D C E B A (4) D C A B E (4) A B E D C (4) C D A E B (3) B A E D C (3) A D C B E (3) E C D B A (2) E B C D A (2) E B C A D (2) D A C B E (2) B E A C D (2) A B E C D (2) E C A B D (1) E A B C D (1) D E C B A (1) D C B A E (1) D C A E B (1) D B A C E (1) D A B C E (1) C E D A B (1) C A D E B (1) C A D B E (1) B E D A C (1) B A D E C (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -4 4 -2 B -4 0 -10 -4 -8 C 4 10 0 8 0 D -4 4 -8 0 10 E 2 8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.721544 D: 0.000000 E: 0.278456 Sum of squares = 0.598163620922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.721544 D: 0.721544 E: 1.000000 A B C D E A 0 4 -4 4 -2 B -4 0 -10 -4 -8 C 4 10 0 8 0 D -4 4 -8 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.444444 Sum of squares = 0.506172884365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=25 E=23 D=15 B=7 so B is eliminated. Round 2 votes counts: A=34 E=26 C=25 D=15 so D is eliminated. Round 3 votes counts: A=38 C=35 E=27 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:201 D:201 E:200 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 -2 B -4 0 -10 -4 -8 C 4 10 0 8 0 D -4 4 -8 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.444444 Sum of squares = 0.506172884365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 -2 B -4 0 -10 -4 -8 C 4 10 0 8 0 D -4 4 -8 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.444444 Sum of squares = 0.506172884365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 -2 B -4 0 -10 -4 -8 C 4 10 0 8 0 D -4 4 -8 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.444444 Sum of squares = 0.506172884365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4055: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (13) E D C B A (9) B A E C D (9) E B D C A (8) A C D B E (8) A B C D E (7) C D A E B (6) B E D C A (6) B E A D C (5) E D C A B (3) D C E B A (3) A B E C D (3) A B C E D (3) E D B C A (2) B E A C D (2) E A D C B (1) D E C B A (1) D E C A B (1) D C E A B (1) D C A E B (1) C D A B E (1) C A D B E (1) B E D A C (1) B C D A E (1) B A C D E (1) A E D C B (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 12 12 14 B -4 0 -4 -6 -2 C -12 4 0 12 -4 D -12 6 -12 0 -6 E -14 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 12 14 B -4 0 -4 -6 -2 C -12 4 0 12 -4 D -12 6 -12 0 -6 E -14 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=25 E=23 C=8 D=7 so D is eliminated. Round 2 votes counts: A=37 E=25 B=25 C=13 so C is eliminated. Round 3 votes counts: A=46 E=29 B=25 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:200 E:199 B:192 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 12 14 B -4 0 -4 -6 -2 C -12 4 0 12 -4 D -12 6 -12 0 -6 E -14 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 12 14 B -4 0 -4 -6 -2 C -12 4 0 12 -4 D -12 6 -12 0 -6 E -14 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 12 14 B -4 0 -4 -6 -2 C -12 4 0 12 -4 D -12 6 -12 0 -6 E -14 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4056: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) D A E B C (6) C B A D E (6) E B C A D (5) D E A B C (5) C A D B E (5) C A B D E (5) E D A B C (4) E B D A C (4) B C A E D (4) E B A D C (3) D E A C B (3) D A C B E (3) C D A B E (3) B E C A D (3) B E A C D (3) E B C D A (2) D E C A B (2) D C E A B (2) D A E C B (2) D A C E B (2) C B A E D (2) A C B D E (2) E D C A B (1) E B D C A (1) E B A C D (1) E A B D C (1) D C A E B (1) D A B C E (1) C E B A D (1) C D E A B (1) C D A E B (1) C B E A D (1) B E A D C (1) B C E A D (1) B A E C D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 6 -8 -10 B -2 0 12 -6 -16 C -6 -12 0 -4 -16 D 8 6 4 0 2 E 10 16 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -8 -10 B -2 0 12 -6 -16 C -6 -12 0 -4 -16 D 8 6 4 0 2 E 10 16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=27 C=25 B=13 A=3 so A is eliminated. Round 2 votes counts: E=32 C=28 D=27 B=13 so B is eliminated. Round 3 votes counts: E=40 C=33 D=27 so D is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:210 A:195 B:194 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -8 -10 B -2 0 12 -6 -16 C -6 -12 0 -4 -16 D 8 6 4 0 2 E 10 16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -8 -10 B -2 0 12 -6 -16 C -6 -12 0 -4 -16 D 8 6 4 0 2 E 10 16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -8 -10 B -2 0 12 -6 -16 C -6 -12 0 -4 -16 D 8 6 4 0 2 E 10 16 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4057: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) E B D C A (7) C A B E D (7) D E B C A (6) D E B A C (6) A C B E D (6) E D B C A (5) A C B D E (5) D E C A B (4) D C A E B (4) B E C A D (4) D E C B A (3) C A D B E (3) B E D A C (3) A B C E D (3) D E A C B (2) D C E A B (2) D A C E B (2) C D A E B (2) C A D E B (2) B A E C D (2) B A C E D (2) E D B A C (1) E B D A C (1) E B C D A (1) C D E B A (1) C D E A B (1) B E D C A (1) B E C D A (1) B E A C D (1) B C E A D (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -12 -6 -2 B -6 0 -8 -8 -2 C 12 8 0 6 4 D 6 8 -6 0 6 E 2 2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -6 -2 B -6 0 -8 -8 -2 C 12 8 0 6 4 D 6 8 -6 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 C=16 B=16 E=15 so E is eliminated. Round 2 votes counts: D=35 B=25 A=24 C=16 so C is eliminated. Round 3 votes counts: D=39 A=36 B=25 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:207 E:197 A:193 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -6 -2 B -6 0 -8 -8 -2 C 12 8 0 6 4 D 6 8 -6 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -6 -2 B -6 0 -8 -8 -2 C 12 8 0 6 4 D 6 8 -6 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -6 -2 B -6 0 -8 -8 -2 C 12 8 0 6 4 D 6 8 -6 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4058: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (14) B A E C D (10) A B C E D (9) D C E A B (7) A C B E D (5) D C A E B (4) B E D C A (4) B E C A D (4) E C D B A (3) E C B D A (3) C E D B A (3) B A D E C (3) A B C D E (3) D C E B A (2) D B E C A (2) D A C E B (2) C E D A B (2) B E C D A (2) B D E A C (2) B A E D C (2) A D B E C (2) A C E D B (2) E B C D A (1) D E B C A (1) C E B A D (1) C A E B D (1) B E A C D (1) A D C E B (1) A C E B D (1) A C D E B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -8 -4 -4 B 16 0 -6 6 2 C 8 6 0 8 -10 D 4 -6 -8 0 -10 E 4 -2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.111111 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765471 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -16 -8 -4 -4 B 16 0 -6 6 2 C 8 6 0 8 -10 D 4 -6 -8 0 -10 E 4 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.111111 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765365 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=26 E=7 C=7 so E is eliminated. Round 2 votes counts: D=32 B=29 A=26 C=13 so C is eliminated. Round 3 votes counts: D=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:211 B:209 C:206 D:190 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -8 -4 -4 B 16 0 -6 6 2 C 8 6 0 8 -10 D 4 -6 -8 0 -10 E 4 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.111111 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765365 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -4 -4 B 16 0 -6 6 2 C 8 6 0 8 -10 D 4 -6 -8 0 -10 E 4 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.111111 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765365 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -4 -4 B 16 0 -6 6 2 C 8 6 0 8 -10 D 4 -6 -8 0 -10 E 4 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.111111 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765365 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4059: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (8) A B C E D (8) C E B A D (6) A C E B D (6) D E C B A (5) E D C B A (4) D B A E C (3) D A B E C (3) B C E D A (3) A D B E C (3) A C B E D (3) E C D A B (2) E C B D A (2) D E B C A (2) D B E A C (2) D A E C B (2) C E A D B (2) C E A B D (2) C A E B D (2) B D A C E (2) B A C D E (2) A D B C E (2) A B C D E (2) E C A D B (1) E B C D A (1) D E C A B (1) D E A C B (1) C E B D A (1) C B E A D (1) B E D C A (1) B D C E A (1) B C E A D (1) B A D C E (1) B A C E D (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 4 8 2 B -2 0 -6 8 -6 C -4 6 0 16 10 D -8 -8 -16 0 -18 E -2 6 -10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 2 B -2 0 -6 8 -6 C -4 6 0 16 10 D -8 -8 -16 0 -18 E -2 6 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=20 D=19 C=14 B=12 so B is eliminated. Round 2 votes counts: A=39 D=22 E=21 C=18 so C is eliminated. Round 3 votes counts: A=41 E=37 D=22 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:214 A:208 E:206 B:197 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 2 B -2 0 -6 8 -6 C -4 6 0 16 10 D -8 -8 -16 0 -18 E -2 6 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 2 B -2 0 -6 8 -6 C -4 6 0 16 10 D -8 -8 -16 0 -18 E -2 6 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 2 B -2 0 -6 8 -6 C -4 6 0 16 10 D -8 -8 -16 0 -18 E -2 6 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4060: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) D C B E A (9) E C D A B (8) B D C A E (8) E A C D B (7) A E B C D (7) A B E D C (7) D B C E A (6) B D C E A (5) A E C B D (5) B A D C E (4) A E C D B (4) E C D B A (3) E C A D B (3) A B D E C (3) B D A C E (2) E A B C D (1) D C E B A (1) D C E A B (1) D B C A E (1) C E D B A (1) C E D A B (1) C D E A B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -16 -14 -14 B 0 0 -8 -14 -6 C 16 8 0 2 2 D 14 14 -2 0 6 E 14 6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -14 -14 B 0 0 -8 -14 -6 C 16 8 0 2 2 D 14 14 -2 0 6 E 14 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 B=19 D=18 C=13 so C is eliminated. Round 2 votes counts: D=29 A=28 E=24 B=19 so B is eliminated. Round 3 votes counts: D=44 A=32 E=24 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:214 E:206 B:186 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -14 -14 B 0 0 -8 -14 -6 C 16 8 0 2 2 D 14 14 -2 0 6 E 14 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -14 -14 B 0 0 -8 -14 -6 C 16 8 0 2 2 D 14 14 -2 0 6 E 14 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -14 -14 B 0 0 -8 -14 -6 C 16 8 0 2 2 D 14 14 -2 0 6 E 14 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4061: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C B D A E (7) C B A D E (7) B C D A E (7) B D C E A (6) B C D E A (6) E A D C B (5) E A C D B (5) D E B A C (4) A E C D B (4) E A D B C (3) B D E C A (3) B D C A E (3) A C D B E (3) E B D C A (2) D E A B C (2) D B C E A (2) C A B E D (2) B E D C A (2) A D E C B (2) A C E B D (2) E D B A C (1) E A C B D (1) E A B D C (1) D A E B C (1) D A B E C (1) C A D B E (1) C A B D E (1) B E C D A (1) B C E A D (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -2 -20 -14 B 4 0 16 6 8 C 2 -16 0 -2 -2 D 20 -6 2 0 14 E 14 -8 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999361 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -20 -14 B 4 0 16 6 8 C 2 -16 0 -2 -2 D 20 -6 2 0 14 E 14 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=29 C=18 A=13 D=10 so D is eliminated. Round 2 votes counts: E=36 B=31 C=18 A=15 so A is eliminated. Round 3 votes counts: E=44 B=32 C=24 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:215 E:197 C:191 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -20 -14 B 4 0 16 6 8 C 2 -16 0 -2 -2 D 20 -6 2 0 14 E 14 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -20 -14 B 4 0 16 6 8 C 2 -16 0 -2 -2 D 20 -6 2 0 14 E 14 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -20 -14 B 4 0 16 6 8 C 2 -16 0 -2 -2 D 20 -6 2 0 14 E 14 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4062: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) B A E C D (6) E B C A D (5) D C E A B (5) D A C B E (5) C D E B A (5) C D E A B (5) B E A C D (5) A B E D C (5) A B D E C (5) E C B A D (4) D C A E B (4) C E D B A (4) B E C A D (4) E C B D A (3) E B A C D (3) C E D A B (3) B A E D C (3) D C B E A (2) D B A C E (2) A D B C E (2) A B E C D (2) E B C D A (1) D B E C A (1) D B C E A (1) D A C E B (1) D A B C E (1) C E A D B (1) B E C D A (1) B D E C A (1) B A D E C (1) A E B C D (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 -12 -2 -20 B 16 0 2 0 -4 C 12 -2 0 8 -4 D 2 0 -8 0 -4 E 20 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -12 -2 -20 B 16 0 2 0 -4 C 12 -2 0 8 -4 D 2 0 -8 0 -4 E 20 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=21 C=18 A=17 E=16 so E is eliminated. Round 2 votes counts: B=30 D=28 C=25 A=17 so A is eliminated. Round 3 votes counts: B=43 D=31 C=26 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:207 C:207 D:195 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -12 -2 -20 B 16 0 2 0 -4 C 12 -2 0 8 -4 D 2 0 -8 0 -4 E 20 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -2 -20 B 16 0 2 0 -4 C 12 -2 0 8 -4 D 2 0 -8 0 -4 E 20 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -2 -20 B 16 0 2 0 -4 C 12 -2 0 8 -4 D 2 0 -8 0 -4 E 20 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4063: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) D A C E B (7) A D B C E (7) D A E B C (6) C D E B A (5) E C D B A (4) E C B D A (4) E B C A D (4) D C A E B (4) D A C B E (4) C E D B A (4) C E B D A (4) E B C D A (3) D C E B A (3) A D B E C (3) E D C B A (2) D C E A B (2) C E B A D (2) B E C A D (2) B C E A D (2) A B C E D (2) A B C D E (2) E C B A D (1) D E C B A (1) D E C A B (1) D E A C B (1) D C A B E (1) D A E C B (1) D A B E C (1) D A B C E (1) C D E A B (1) C B E A D (1) B A C E D (1) A D C B E (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -2 -20 6 B 4 0 -10 -20 -20 C 2 10 0 4 12 D 20 20 -4 0 8 E -6 20 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -20 6 B 4 0 -10 -20 -20 C 2 10 0 4 12 D 20 20 -4 0 8 E -6 20 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=18 A=18 C=17 B=14 so B is eliminated. Round 2 votes counts: D=33 A=28 E=20 C=19 so C is eliminated. Round 3 votes counts: D=39 E=33 A=28 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:214 E:197 A:190 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -20 6 B 4 0 -10 -20 -20 C 2 10 0 4 12 D 20 20 -4 0 8 E -6 20 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -20 6 B 4 0 -10 -20 -20 C 2 10 0 4 12 D 20 20 -4 0 8 E -6 20 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -20 6 B 4 0 -10 -20 -20 C 2 10 0 4 12 D 20 20 -4 0 8 E -6 20 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4064: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) D E A B C (8) D A E B C (8) C B A E D (8) E B D C A (5) E B C D A (5) D A C E B (5) C A B E D (4) A C D B E (4) E D B C A (3) E B D A C (3) D A E C B (3) C B E A D (3) C A B D E (3) B C E A D (3) A D C B E (3) A C B E D (3) E B C A D (2) E B A D C (2) C D A B E (2) B E C A D (2) E D B A C (1) E B A C D (1) D E B C A (1) D C A E B (1) C E B D A (1) C D E B A (1) C D B E A (1) C B E D A (1) B E A C D (1) A D B E C (1) A D B C E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -14 -4 B 4 0 12 -2 -16 C -8 -12 0 -6 -8 D 14 2 6 0 2 E 4 16 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -14 -4 B 4 0 12 -2 -16 C -8 -12 0 -6 -8 D 14 2 6 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=24 E=22 A=14 B=6 so B is eliminated. Round 2 votes counts: D=34 C=27 E=25 A=14 so A is eliminated. Round 3 votes counts: D=39 C=36 E=25 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:212 B:199 A:193 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -14 -4 B 4 0 12 -2 -16 C -8 -12 0 -6 -8 D 14 2 6 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -14 -4 B 4 0 12 -2 -16 C -8 -12 0 -6 -8 D 14 2 6 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -14 -4 B 4 0 12 -2 -16 C -8 -12 0 -6 -8 D 14 2 6 0 2 E 4 16 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4065: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) D A C E B (6) B E D C A (6) E D B C A (5) D A E C B (4) C A E B D (4) C A B E D (4) B D E A C (4) A D C E B (4) A C B D E (4) D E C A B (3) D E B A C (3) C A E D B (3) B E C A D (3) B C E A D (3) E D C A B (2) E B D C A (2) D E B C A (2) D B E A C (2) D A B C E (2) C E B A D (2) C B A E D (2) B E C D A (2) B D A E C (2) B C A E D (2) B A C D E (2) E D C B A (1) E B C D A (1) D E A C B (1) D E A B C (1) D B A E C (1) D A B E C (1) C E A B D (1) B A C E D (1) A D C B E (1) A C E D B (1) A C D B E (1) A C B E D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 -2 12 B -8 0 -4 -2 -6 C -4 4 0 -8 8 D 2 2 8 0 6 E -12 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -2 12 B -8 0 -4 -2 -6 C -4 4 0 -8 8 D 2 2 8 0 6 E -12 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 A=22 C=16 E=11 so E is eliminated. Round 2 votes counts: D=34 B=28 A=22 C=16 so C is eliminated. Round 3 votes counts: D=34 A=34 B=32 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:211 D:209 C:200 B:190 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -2 12 B -8 0 -4 -2 -6 C -4 4 0 -8 8 D 2 2 8 0 6 E -12 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -2 12 B -8 0 -4 -2 -6 C -4 4 0 -8 8 D 2 2 8 0 6 E -12 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -2 12 B -8 0 -4 -2 -6 C -4 4 0 -8 8 D 2 2 8 0 6 E -12 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4066: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (13) B C A E D (12) C A E D B (11) D E A C B (7) B D C E A (6) C B A E D (5) B C D A E (5) A E C D B (4) D E C A B (3) D B E A C (3) C A B E D (3) E A D C B (2) C B D A E (2) C A E B D (2) B E A D C (2) B D C A E (2) E A B D C (1) D E B C A (1) D E B A C (1) D E A B C (1) D B E C A (1) D B C E A (1) C E A D B (1) C D E A B (1) C D A E B (1) B E D A C (1) B C A D E (1) B A E D C (1) B A E C D (1) B A C E D (1) A E C B D (1) A E B D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 -16 2 10 B 18 0 10 22 20 C 16 -10 0 6 12 D -2 -22 -6 0 -2 E -10 -20 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 2 10 B 18 0 10 22 20 C 16 -10 0 6 12 D -2 -22 -6 0 -2 E -10 -20 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=45 C=26 D=18 A=8 E=3 so E is eliminated. Round 2 votes counts: B=45 C=26 D=18 A=11 so A is eliminated. Round 3 votes counts: B=47 C=33 D=20 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:235 C:212 A:189 D:184 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 2 10 B 18 0 10 22 20 C 16 -10 0 6 12 D -2 -22 -6 0 -2 E -10 -20 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 2 10 B 18 0 10 22 20 C 16 -10 0 6 12 D -2 -22 -6 0 -2 E -10 -20 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 2 10 B 18 0 10 22 20 C 16 -10 0 6 12 D -2 -22 -6 0 -2 E -10 -20 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4067: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (15) C A B D E (13) C A E D B (6) A C B E D (6) E D C A B (5) D B E C A (5) B D E A C (5) B A C D E (5) A C E B D (5) A C B D E (5) E D B A C (3) D E B C A (3) B C A D E (3) C A E B D (2) B D E C A (2) B A E D C (2) A E C D B (2) A C E D B (2) E D A C B (1) D C E B A (1) D C B A E (1) D B E A C (1) C E A D B (1) C D E A B (1) C A D E B (1) B D C A E (1) B D A C E (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 0 -20 10 14 B 0 0 -4 4 4 C 20 4 0 6 12 D -10 -4 -6 0 -2 E -14 -4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 10 14 B 0 0 -4 4 4 C 20 4 0 6 12 D -10 -4 -6 0 -2 E -14 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 B=21 A=20 D=11 so D is eliminated. Round 2 votes counts: E=27 B=27 C=26 A=20 so A is eliminated. Round 3 votes counts: C=44 E=29 B=27 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:221 A:202 B:202 D:189 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -20 10 14 B 0 0 -4 4 4 C 20 4 0 6 12 D -10 -4 -6 0 -2 E -14 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 10 14 B 0 0 -4 4 4 C 20 4 0 6 12 D -10 -4 -6 0 -2 E -14 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 10 14 B 0 0 -4 4 4 C 20 4 0 6 12 D -10 -4 -6 0 -2 E -14 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4068: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) E D B C A (6) D E B A C (5) C A B E D (5) B A C D E (5) A D B E C (5) C B E A D (4) C B A E D (4) B E D C A (4) E C D B A (3) E B D C A (3) D E A B C (3) A D E C B (3) A B D C E (3) E D C A B (2) E C B D A (2) D E A C B (2) D B A E C (2) D A E B C (2) C E D A B (2) C E B A D (2) B E C D A (2) B D E A C (2) B C E D A (2) A D E B C (2) A D C E B (2) A C B D E (2) A B C D E (2) E D A C B (1) D A B E C (1) C E B D A (1) C B E D A (1) B D A E C (1) B C E A D (1) B C D E A (1) B C A E D (1) B C A D E (1) B A D C E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -6 -8 -10 B 20 0 10 0 2 C 6 -10 0 -12 -14 D 8 0 12 0 -6 E 10 -2 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.844120 C: 0.000000 D: 0.155880 E: 0.000000 Sum of squares = 0.73683698329 Cumulative probabilities = A: 0.000000 B: 0.844120 C: 0.844120 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -8 -10 B 20 0 10 0 2 C 6 -10 0 -12 -14 D 8 0 12 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000081035 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=22 B=21 C=19 D=15 so D is eliminated. Round 2 votes counts: E=33 A=25 B=23 C=19 so C is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:214 D:207 C:185 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -6 -8 -10 B 20 0 10 0 2 C 6 -10 0 -12 -14 D 8 0 12 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000081035 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -8 -10 B 20 0 10 0 2 C 6 -10 0 -12 -14 D 8 0 12 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000081035 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -8 -10 B 20 0 10 0 2 C 6 -10 0 -12 -14 D 8 0 12 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000081035 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4069: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (32) E A B D C (30) E D B A C (4) C A B D E (4) D B C A E (3) E A C B D (2) E A B C D (2) D E B A C (2) C D B E A (2) C A E B D (2) A E B D C (2) E C D B A (1) E C A B D (1) E B D A C (1) E A D B C (1) D E B C A (1) D C B E A (1) D C B A E (1) D B E A C (1) D B A C E (1) C E A B D (1) C A B E D (1) B A D C E (1) A E C B D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 0 0 0 B 2 0 0 0 -4 C 0 0 0 0 0 D 0 0 0 0 -2 E 0 4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307193 B: 0.000000 C: 0.317621 D: 0.000000 E: 0.375186 Sum of squares = 0.336015182586 Cumulative probabilities = A: 0.307193 B: 0.307193 C: 0.624814 D: 0.624814 E: 1.000000 A B C D E A 0 -2 0 0 0 B 2 0 0 0 -4 C 0 0 0 0 0 D 0 0 0 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=42 C=42 D=10 A=5 B=1 so B is eliminated. Round 2 votes counts: E=42 C=42 D=10 A=6 so A is eliminated. Round 3 votes counts: E=46 C=43 D=11 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:203 C:200 A:199 B:199 D:199 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 0 0 B 2 0 0 0 -4 C 0 0 0 0 0 D 0 0 0 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 0 B 2 0 0 0 -4 C 0 0 0 0 0 D 0 0 0 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 0 B 2 0 0 0 -4 C 0 0 0 0 0 D 0 0 0 0 -2 E 0 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333334 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 0.666666 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4070: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) A B C E D (12) D E C B A (9) B A D E C (8) A B D E C (8) C E D A B (5) A C B E D (5) A B C D E (5) B D E C A (4) E D C A B (3) D B E C A (3) B A C D E (3) E C D B A (2) D E B C A (2) C A E D B (2) A D B E C (2) A C E D B (2) E D C B A (1) D E C A B (1) D B E A C (1) C E A D B (1) C B E D A (1) C A B E D (1) B D A E C (1) B C E D A (1) A D E B C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 4 4 B 0 0 4 2 12 C 0 -4 0 10 8 D -4 -2 -10 0 -2 E -4 -12 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.641179 B: 0.358821 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.539862822367 Cumulative probabilities = A: 0.641179 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 4 B 0 0 4 2 12 C 0 -4 0 10 8 D -4 -2 -10 0 -2 E -4 -12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=24 B=17 D=16 E=6 so E is eliminated. Round 2 votes counts: A=37 C=26 D=20 B=17 so B is eliminated. Round 3 votes counts: A=48 C=27 D=25 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:209 C:207 A:204 D:191 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 4 B 0 0 4 2 12 C 0 -4 0 10 8 D -4 -2 -10 0 -2 E -4 -12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 4 B 0 0 4 2 12 C 0 -4 0 10 8 D -4 -2 -10 0 -2 E -4 -12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 4 B 0 0 4 2 12 C 0 -4 0 10 8 D -4 -2 -10 0 -2 E -4 -12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4071: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) D A E C B (9) C B E D A (7) A D E B C (7) B C E A D (6) C E B D A (5) B C D A E (5) A D B C E (5) E C D B A (4) D A B C E (4) A D B E C (4) E C B D A (3) E C A B D (3) D B C A E (3) B C E D A (3) E D A C B (2) E C D A B (2) C B D E A (2) B A C D E (2) E A D C B (1) D C A E B (1) D B A C E (1) D A C E B (1) D A C B E (1) C B D A E (1) B D C A E (1) B D A C E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D C E (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 -16 -12 4 B 16 0 -6 6 0 C 16 6 0 12 4 D 12 -6 -12 0 2 E -4 0 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -12 4 B 16 0 -6 6 0 C 16 6 0 12 4 D 12 -6 -12 0 2 E -4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=22 D=20 A=18 C=15 so C is eliminated. Round 2 votes counts: B=32 E=30 D=20 A=18 so A is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:219 B:208 D:198 E:195 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -16 -12 4 B 16 0 -6 6 0 C 16 6 0 12 4 D 12 -6 -12 0 2 E -4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -12 4 B 16 0 -6 6 0 C 16 6 0 12 4 D 12 -6 -12 0 2 E -4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -12 4 B 16 0 -6 6 0 C 16 6 0 12 4 D 12 -6 -12 0 2 E -4 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4072: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) A E D B C (8) E A C B D (7) E C A B D (6) E A C D B (6) A E D C B (6) A E C B D (6) D B C E A (5) D B C A E (5) C B E D A (5) C E B A D (4) E C B D A (3) D A B E C (3) B D C E A (3) A D E B C (3) D B A C E (2) B C D E A (2) A E C D B (2) E C B A D (1) D C B E A (1) D B A E C (1) D A E B C (1) C E B D A (1) C E A B D (1) C B E A D (1) C B A E D (1) B D C A E (1) B C D A E (1) A E B C D (1) A D B E C (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 14 -10 B -8 0 -22 10 -12 C 0 22 0 18 -10 D -14 -10 -18 0 -20 E 10 12 10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 14 -10 B -8 0 -22 10 -12 C 0 22 0 18 -10 D -14 -10 -18 0 -20 E 10 12 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 C=22 D=18 B=7 so B is eliminated. Round 2 votes counts: A=30 C=25 E=23 D=22 so D is eliminated. Round 3 votes counts: C=40 A=37 E=23 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:226 C:215 A:206 B:184 D:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 14 -10 B -8 0 -22 10 -12 C 0 22 0 18 -10 D -14 -10 -18 0 -20 E 10 12 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 14 -10 B -8 0 -22 10 -12 C 0 22 0 18 -10 D -14 -10 -18 0 -20 E 10 12 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 14 -10 B -8 0 -22 10 -12 C 0 22 0 18 -10 D -14 -10 -18 0 -20 E 10 12 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4073: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) D B E A C (7) B D C A E (7) C B A E D (6) B D A E C (5) B C A E D (5) D C E A B (4) D C B E A (4) D B C E A (4) C E A D B (4) C A E B D (4) A E C B D (4) E A D C B (3) D E A B C (3) D B E C A (3) B A C E D (3) E C A D B (2) D E A C B (2) D B C A E (2) D B A E C (2) C D E A B (2) C A B E D (2) B C D A E (2) B A E D C (2) C E A B D (1) C D E B A (1) C D B E A (1) B C A D E (1) B A E C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 -10 2 -6 B 12 0 -6 -12 14 C 10 6 0 4 8 D -2 12 -4 0 0 E 6 -14 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 2 -6 B 12 0 -6 -12 14 C 10 6 0 4 8 D -2 12 -4 0 0 E 6 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 C=21 E=17 A=5 so A is eliminated. Round 2 votes counts: D=31 B=26 C=22 E=21 so E is eliminated. Round 3 votes counts: C=40 D=34 B=26 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:204 D:203 E:192 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 2 -6 B 12 0 -6 -12 14 C 10 6 0 4 8 D -2 12 -4 0 0 E 6 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 2 -6 B 12 0 -6 -12 14 C 10 6 0 4 8 D -2 12 -4 0 0 E 6 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 2 -6 B 12 0 -6 -12 14 C 10 6 0 4 8 D -2 12 -4 0 0 E 6 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4074: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) E C B A D (8) D A E C B (6) E A D C B (5) C B E D A (5) B C D A E (5) A D E B C (5) A D B E C (5) D A B C E (4) C E B A D (4) B C E D A (4) B C E A D (4) E C A D B (3) B E C A D (3) A D E C B (3) E A B C D (2) D A C E B (2) D A B E C (2) B D A C E (2) B C A D E (2) A E D C B (2) E B A C D (1) E A D B C (1) E A C D B (1) E A C B D (1) D C E A B (1) D C A E B (1) D B C A E (1) D A C B E (1) C B E A D (1) B A E C D (1) B A D C E (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -6 10 -10 B 6 0 -10 12 -16 C 6 10 0 12 -2 D -10 -12 -12 0 -16 E 10 16 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -6 10 -10 B 6 0 -10 12 -16 C 6 10 0 12 -2 D -10 -12 -12 0 -16 E 10 16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 C=21 D=18 A=17 so A is eliminated. Round 2 votes counts: D=31 E=25 B=23 C=21 so C is eliminated. Round 3 votes counts: E=40 D=31 B=29 so B is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:213 B:196 A:194 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -6 10 -10 B 6 0 -10 12 -16 C 6 10 0 12 -2 D -10 -12 -12 0 -16 E 10 16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 10 -10 B 6 0 -10 12 -16 C 6 10 0 12 -2 D -10 -12 -12 0 -16 E 10 16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 10 -10 B 6 0 -10 12 -16 C 6 10 0 12 -2 D -10 -12 -12 0 -16 E 10 16 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4075: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (7) D A E B C (5) E D B C A (4) D B E A C (4) A D C B E (4) A D B C E (4) E C D B A (3) E C B D A (3) E B D C A (3) D B E C A (3) D A B E C (3) D A B C E (3) C E B A D (3) C E A B D (3) C B A E D (3) C A E B D (3) B D A C E (3) A C E D B (3) E D C B A (2) E C A D B (2) D E B C A (2) D E A B C (2) D B A E C (2) D B A C E (2) C B E A D (2) B D C E A (2) A C D B E (2) E D C A B (1) E C D A B (1) E C B A D (1) E C A B D (1) E A C D B (1) D E B A C (1) C E B D A (1) C B A D E (1) C A B E D (1) B E D C A (1) B E C D A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D A E (1) B C A D E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -4 -10 2 B 4 0 2 -12 8 C 4 -2 0 -6 6 D 10 12 6 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -10 2 B 4 0 2 -12 8 C 4 -2 0 -6 6 D 10 12 6 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=22 A=22 C=17 B=12 so B is eliminated. Round 2 votes counts: D=33 E=24 A=22 C=21 so C is eliminated. Round 3 votes counts: E=35 D=34 A=31 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:201 C:201 A:192 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -10 2 B 4 0 2 -12 8 C 4 -2 0 -6 6 D 10 12 6 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -10 2 B 4 0 2 -12 8 C 4 -2 0 -6 6 D 10 12 6 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -10 2 B 4 0 2 -12 8 C 4 -2 0 -6 6 D 10 12 6 0 10 E -2 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4076: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E B C A D (6) D E A C B (5) D A C E B (5) C B A D E (5) E D B C A (4) E D A B C (4) D E A B C (4) B C A E D (4) A D C B E (4) A B C D E (4) E D C B A (3) E B C D A (3) D E C B A (3) B C E A D (3) A C D B E (3) E B A C D (2) D A E C B (2) D A E B C (2) C B D A E (2) C B A E D (2) A E D B C (2) A E B D C (2) A C B D E (2) A B C E D (2) E C B D A (1) E A B D C (1) E A B C D (1) D C E B A (1) D C B A E (1) C D B E A (1) C D B A E (1) C B E A D (1) C A B D E (1) B E C A D (1) B A C E D (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 10 14 -2 12 B -10 0 -4 -12 -2 C -14 4 0 -8 6 D 2 12 8 0 14 E -12 2 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 -2 12 B -10 0 -4 -12 -2 C -14 4 0 -8 6 D 2 12 8 0 14 E -12 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=25 A=21 C=13 B=9 so B is eliminated. Round 2 votes counts: D=32 E=26 A=22 C=20 so C is eliminated. Round 3 votes counts: D=36 A=34 E=30 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:217 C:194 B:186 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 14 -2 12 B -10 0 -4 -12 -2 C -14 4 0 -8 6 D 2 12 8 0 14 E -12 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -2 12 B -10 0 -4 -12 -2 C -14 4 0 -8 6 D 2 12 8 0 14 E -12 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -2 12 B -10 0 -4 -12 -2 C -14 4 0 -8 6 D 2 12 8 0 14 E -12 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4077: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) B A C E D (6) E A C D B (5) D E B C A (5) B D C A E (5) E A D C B (4) D E C B A (4) D E A B C (4) D B E C A (4) B D C E A (4) B D A C E (4) A E C B D (4) E D A C B (3) D E C A B (3) D C E B A (3) D B C E A (3) E C D A B (2) E C A D B (2) C B D E A (2) C B A E D (2) C A B E D (2) B D A E C (2) B A D E C (2) A E B C D (2) A C E B D (2) A B C E D (2) D E A C B (1) D C B E A (1) D B E A C (1) C E D A B (1) C E A D B (1) C B D A E (1) C A E D B (1) B C A E D (1) B C A D E (1) A E C D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -12 -22 -6 B 18 0 10 2 4 C 12 -10 0 -6 0 D 22 -2 6 0 16 E 6 -4 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -12 -22 -6 B 18 0 10 2 4 C 12 -10 0 -6 0 D 22 -2 6 0 16 E 6 -4 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=29 E=16 A=12 C=10 so C is eliminated. Round 2 votes counts: B=38 D=29 E=18 A=15 so A is eliminated. Round 3 votes counts: B=43 D=29 E=28 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:221 B:217 C:198 E:193 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -12 -22 -6 B 18 0 10 2 4 C 12 -10 0 -6 0 D 22 -2 6 0 16 E 6 -4 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -22 -6 B 18 0 10 2 4 C 12 -10 0 -6 0 D 22 -2 6 0 16 E 6 -4 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -22 -6 B 18 0 10 2 4 C 12 -10 0 -6 0 D 22 -2 6 0 16 E 6 -4 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4078: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) E D B A C (6) A E B D C (6) D B E A C (5) A B E D C (5) C E A D B (4) C D E B A (4) C D B A E (4) C A E B D (4) E D A B C (3) D C B E A (3) D B C A E (3) C B A D E (3) C A B E D (3) C A B D E (3) A E C B D (3) A C B E D (3) E D C A B (2) E C A D B (2) E A C D B (2) D E B A C (2) C E A B D (2) C D B E A (2) B C D A E (2) B A D E C (2) A B D E C (2) E D C B A (1) E C A B D (1) E A D B C (1) D E B C A (1) D C E B A (1) D B A E C (1) D B A C E (1) C B D A E (1) C A E D B (1) B D A C E (1) B A D C E (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 6 14 2 B -12 0 2 2 -8 C -6 -2 0 -8 -6 D -14 -2 8 0 -14 E -2 8 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 14 2 B -12 0 2 2 -8 C -6 -2 0 -8 -6 D -14 -2 8 0 -14 E -2 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=25 A=21 D=17 B=6 so B is eliminated. Round 2 votes counts: C=33 E=25 A=24 D=18 so D is eliminated. Round 3 votes counts: C=40 E=33 A=27 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:217 E:213 B:192 C:189 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 14 2 B -12 0 2 2 -8 C -6 -2 0 -8 -6 D -14 -2 8 0 -14 E -2 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 14 2 B -12 0 2 2 -8 C -6 -2 0 -8 -6 D -14 -2 8 0 -14 E -2 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 14 2 B -12 0 2 2 -8 C -6 -2 0 -8 -6 D -14 -2 8 0 -14 E -2 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4079: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) E D C A B (6) E D B C A (6) C D E A B (6) C A D E B (5) A C B D E (5) D C E A B (3) C E D A B (3) B E D C A (3) B E A D C (3) B A E D C (3) B A D E C (3) B A C E D (3) B A C D E (3) A B C D E (3) E D B A C (2) E B D C A (2) E B D A C (2) E B C D A (2) D E C A B (2) D C A E B (2) D A C E B (2) C A D B E (2) C A B D E (2) B E D A C (2) B A E C D (2) B A D C E (2) A D C B E (2) E C D B A (1) E C D A B (1) D E A B C (1) D A E C B (1) C E B D A (1) B E A C D (1) B C A E D (1) A D C E B (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -12 -12 -10 B 0 0 -10 -12 -16 C 12 10 0 -14 -2 D 12 12 14 0 -4 E 10 16 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -12 -12 -10 B 0 0 -10 -12 -16 C 12 10 0 -14 -2 D 12 12 14 0 -4 E 10 16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=26 C=19 A=14 D=11 so D is eliminated. Round 2 votes counts: E=33 B=26 C=24 A=17 so A is eliminated. Round 3 votes counts: C=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:217 E:216 C:203 A:183 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -12 -12 -10 B 0 0 -10 -12 -16 C 12 10 0 -14 -2 D 12 12 14 0 -4 E 10 16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -12 -10 B 0 0 -10 -12 -16 C 12 10 0 -14 -2 D 12 12 14 0 -4 E 10 16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -12 -10 B 0 0 -10 -12 -16 C 12 10 0 -14 -2 D 12 12 14 0 -4 E 10 16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4080: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (12) A B D C E (9) C D B A E (6) A B C D E (6) B A C D E (5) E D C A B (4) E D A B C (4) E A B D C (4) E D C B A (3) E C D B A (3) E C A B D (3) C E D B A (3) C D E B A (3) C B D A E (3) A E B D C (3) A B E D C (3) E D A C B (2) E C D A B (2) D E C B A (2) C D B E A (2) A B D E C (2) E A D C B (1) E A B C D (1) D E A B C (1) D C E B A (1) D C B A E (1) D B C A E (1) C E B D A (1) C E B A D (1) C B A D E (1) B D C A E (1) B C D A E (1) B A D C E (1) A D E B C (1) A D B E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 22 16 12 -6 B -22 0 14 -6 -10 C -16 -14 0 -14 -4 D -12 6 14 0 -4 E 6 10 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 16 12 -6 B -22 0 14 -6 -10 C -16 -14 0 -14 -4 D -12 6 14 0 -4 E 6 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=27 C=20 B=8 D=6 so D is eliminated. Round 2 votes counts: E=42 A=27 C=22 B=9 so B is eliminated. Round 3 votes counts: E=42 A=33 C=25 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:222 E:212 D:202 B:188 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 16 12 -6 B -22 0 14 -6 -10 C -16 -14 0 -14 -4 D -12 6 14 0 -4 E 6 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 12 -6 B -22 0 14 -6 -10 C -16 -14 0 -14 -4 D -12 6 14 0 -4 E 6 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 12 -6 B -22 0 14 -6 -10 C -16 -14 0 -14 -4 D -12 6 14 0 -4 E 6 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4081: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) B C D E A (9) B C D A E (7) D A B C E (6) C B E A D (6) B C E D A (5) B C E A D (5) A D E C B (5) E D A B C (4) D B A C E (4) C E B A D (4) B D C A E (3) A E D C B (3) E A C D B (2) D E A B C (2) D B E A C (2) C E A B D (2) C B A D E (2) B E C A D (2) A D C E B (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A D C (1) E A D C B (1) E A B C D (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B E C (1) C D A B E (1) C B D A E (1) C B A E D (1) B E C D A (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -4 -16 0 B 10 0 26 6 14 C 4 -26 0 8 22 D 16 -6 -8 0 14 E 0 -14 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -16 0 B 10 0 26 6 14 C 4 -26 0 8 22 D 16 -6 -8 0 14 E 0 -14 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=27 C=17 E=12 A=12 so E is eliminated. Round 2 votes counts: B=34 D=31 C=19 A=16 so A is eliminated. Round 3 votes counts: D=42 B=35 C=23 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:228 D:208 C:204 A:185 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -16 0 B 10 0 26 6 14 C 4 -26 0 8 22 D 16 -6 -8 0 14 E 0 -14 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -16 0 B 10 0 26 6 14 C 4 -26 0 8 22 D 16 -6 -8 0 14 E 0 -14 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -16 0 B 10 0 26 6 14 C 4 -26 0 8 22 D 16 -6 -8 0 14 E 0 -14 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4082: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (11) A E C B D (8) E B D C A (5) E A B C D (5) A C E B D (5) D C B A E (4) B C D E A (4) A E D B C (4) A E B C D (4) E B C D A (3) D B C A E (3) D A C B E (3) C B E D A (3) A D E B C (3) A D C B E (3) E B C A D (2) D E B C A (2) D B E C A (2) C B E A D (2) C B D E A (2) B E C D A (2) A C D B E (2) E C A B D (1) E B A D C (1) E B A C D (1) E A D B C (1) E A C B D (1) D E A B C (1) D C B E A (1) D B A C E (1) D A E B C (1) D A B E C (1) C A B E D (1) C A B D E (1) B C E D A (1) A E D C B (1) A E B D C (1) A D E C B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 0 0 -2 B 0 0 18 8 -4 C 0 -18 0 0 -2 D 0 -8 0 0 -6 E 2 4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 0 -2 B 0 0 18 8 -4 C 0 -18 0 0 -2 D 0 -8 0 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=30 E=20 C=9 B=7 so B is eliminated. Round 2 votes counts: A=34 D=30 E=22 C=14 so C is eliminated. Round 3 votes counts: D=36 A=36 E=28 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:211 E:207 A:199 D:193 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 0 -2 B 0 0 18 8 -4 C 0 -18 0 0 -2 D 0 -8 0 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 -2 B 0 0 18 8 -4 C 0 -18 0 0 -2 D 0 -8 0 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 -2 B 0 0 18 8 -4 C 0 -18 0 0 -2 D 0 -8 0 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4083: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) C B D E A (7) A E B D C (6) A D B C E (6) E C A B D (5) A D B E C (5) E C B D A (4) E C B A D (4) D B C A E (4) E C A D B (3) D B A C E (3) C E B D A (3) B D C A E (3) A E D C B (3) A D C B E (3) E B A D C (2) E A B D C (2) C E D B A (2) C D B A E (2) C D A B E (2) B D C E A (2) B D A C E (2) A E D B C (2) A E C D B (2) A B D E C (2) E B D A C (1) E B C D A (1) E A B C D (1) D A B C E (1) C E D A B (1) C E A D B (1) C D B E A (1) C B E D A (1) C A E D B (1) C A D E B (1) B D A E C (1) B C E D A (1) B C D E A (1) B A D E C (1) Total count = 100 A B C D E A 0 8 0 14 0 B -8 0 -6 14 -4 C 0 6 0 2 -4 D -14 -14 -2 0 -6 E 0 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.368632 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.631368 Sum of squares = 0.534515222549 Cumulative probabilities = A: 0.368632 B: 0.368632 C: 0.368632 D: 0.368632 E: 1.000000 A B C D E A 0 8 0 14 0 B -8 0 -6 14 -4 C 0 6 0 2 -4 D -14 -14 -2 0 -6 E 0 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=29 C=22 B=11 D=8 so D is eliminated. Round 2 votes counts: E=30 A=30 C=22 B=18 so B is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:207 C:202 B:198 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 14 0 B -8 0 -6 14 -4 C 0 6 0 2 -4 D -14 -14 -2 0 -6 E 0 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 14 0 B -8 0 -6 14 -4 C 0 6 0 2 -4 D -14 -14 -2 0 -6 E 0 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 14 0 B -8 0 -6 14 -4 C 0 6 0 2 -4 D -14 -14 -2 0 -6 E 0 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4084: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) A E C D B (6) A C E B D (6) A C B E D (6) C B A E D (5) C B E A D (4) B C A D E (4) D E B A C (3) D B A C E (3) C A E B D (3) B D C E A (3) E D C B A (2) E D C A B (2) E C D B A (2) E A D C B (2) D E B C A (2) D E A B C (2) D A E B C (2) C E B A D (2) B D A C E (2) B C E D A (2) B C D E A (2) B A C D E (2) A D E C B (2) A B D C E (2) A B C D E (2) E D B C A (1) E D A C B (1) E C B D A (1) E C A D B (1) E C A B D (1) E B C D A (1) E A C D B (1) D E A C B (1) D B E A C (1) D B C E A (1) D B A E C (1) D A B E C (1) C E A B D (1) C A B E D (1) B D C A E (1) B C D A E (1) B C A E D (1) A E D C B (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -2 10 8 B 8 0 -4 8 6 C 2 4 0 12 8 D -10 -8 -12 0 -8 E -8 -6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 10 8 B 8 0 -4 8 6 C 2 4 0 12 8 D -10 -8 -12 0 -8 E -8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 B=18 C=16 E=15 so E is eliminated. Round 2 votes counts: D=30 A=30 C=21 B=19 so B is eliminated. Round 3 votes counts: D=36 C=32 A=32 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:213 B:209 A:204 E:193 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 10 8 B 8 0 -4 8 6 C 2 4 0 12 8 D -10 -8 -12 0 -8 E -8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 10 8 B 8 0 -4 8 6 C 2 4 0 12 8 D -10 -8 -12 0 -8 E -8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 10 8 B 8 0 -4 8 6 C 2 4 0 12 8 D -10 -8 -12 0 -8 E -8 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4085: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) E A B C D (8) B E D C A (8) E B D C A (7) A C D B E (7) E B A C D (6) B E C D A (6) D C A B E (4) C D A B E (4) B D C E A (4) B C D E A (4) A C D E B (4) E B D A C (3) D C B E A (3) D C B A E (3) A D C B E (3) A E D C B (2) A E C D B (2) E A D B C (1) E A B D C (1) D E A C B (1) D C A E B (1) D A C E B (1) C D B A E (1) C A D B E (1) A D C E B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 8 0 -28 B 14 0 22 20 -2 C -8 -22 0 -10 -16 D 0 -20 10 0 -16 E 28 2 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 8 0 -28 B 14 0 22 20 -2 C -8 -22 0 -10 -16 D 0 -20 10 0 -16 E 28 2 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=22 A=21 D=13 C=6 so C is eliminated. Round 2 votes counts: E=38 B=22 A=22 D=18 so D is eliminated. Round 3 votes counts: E=39 A=32 B=29 so B is eliminated. Round 4 votes counts: E=64 A=36 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:227 D:187 A:183 C:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 8 0 -28 B 14 0 22 20 -2 C -8 -22 0 -10 -16 D 0 -20 10 0 -16 E 28 2 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 0 -28 B 14 0 22 20 -2 C -8 -22 0 -10 -16 D 0 -20 10 0 -16 E 28 2 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 0 -28 B 14 0 22 20 -2 C -8 -22 0 -10 -16 D 0 -20 10 0 -16 E 28 2 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4086: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (5) C E B D A (4) C E A B D (4) C B A D E (4) B C D A E (4) B C A D E (4) E D A B C (3) E C A B D (3) E A D B C (3) D B C A E (3) C B A E D (3) C A B E D (3) B D C A E (3) B D A C E (3) B A D C E (3) E C D A B (2) E C A D B (2) E A D C B (2) D E B C A (2) D E B A C (2) D E A B C (2) D B A E C (2) D A E B C (2) C E D B A (2) C E B A D (2) C B D E A (2) C B D A E (2) B A C D E (2) A C E B D (2) A B D C E (2) E D C B A (1) E D B C A (1) E A C D B (1) E A C B D (1) D E C B A (1) D C E B A (1) D B E C A (1) D B E A C (1) D B C E A (1) C D E B A (1) C B E D A (1) C B E A D (1) A E D B C (1) A E C B D (1) A E B D C (1) A D B E C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -16 -4 6 B 24 0 6 14 6 C 16 -6 0 6 28 D 4 -14 -6 0 10 E -6 -6 -28 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -16 -4 6 B 24 0 6 14 6 C 16 -6 0 6 28 D 4 -14 -6 0 10 E -6 -6 -28 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995318 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=23 E=19 B=19 A=10 so A is eliminated. Round 2 votes counts: C=32 D=24 E=22 B=22 so E is eliminated. Round 3 votes counts: C=42 D=35 B=23 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:225 C:222 D:197 A:181 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -16 -4 6 B 24 0 6 14 6 C 16 -6 0 6 28 D 4 -14 -6 0 10 E -6 -6 -28 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995318 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -16 -4 6 B 24 0 6 14 6 C 16 -6 0 6 28 D 4 -14 -6 0 10 E -6 -6 -28 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995318 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -16 -4 6 B 24 0 6 14 6 C 16 -6 0 6 28 D 4 -14 -6 0 10 E -6 -6 -28 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995318 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4087: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B D E A (9) B C D A E (9) D A E B C (7) C B D A E (6) E A B C D (5) A E D B C (5) E A C B D (4) D B C A E (4) C D B A E (4) E A D C B (3) D C B A E (3) E A C D B (2) E A B D C (2) C D B E A (2) C B E A D (2) A E B D C (2) E D C A B (1) E C A D B (1) E B A C D (1) D E A C B (1) D C A E B (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A B C E (1) C E D A B (1) C E A B D (1) C D E B A (1) C D E A B (1) B E C A D (1) B E A C D (1) B C E A D (1) B C A E D (1) B A E D C (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 0 0 -10 0 B 0 0 10 -4 0 C 0 -10 0 8 0 D 10 4 -8 0 8 E 0 0 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826447 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -10 0 B 0 0 10 -4 0 C 0 -10 0 8 0 D 10 4 -8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826461 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=27 D=21 B=16 A=7 so A is eliminated. Round 2 votes counts: E=36 C=27 D=21 B=16 so B is eliminated. Round 3 votes counts: E=40 C=38 D=22 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:207 B:203 C:199 E:196 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -10 0 B 0 0 10 -4 0 C 0 -10 0 8 0 D 10 4 -8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826461 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -10 0 B 0 0 10 -4 0 C 0 -10 0 8 0 D 10 4 -8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826461 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -10 0 B 0 0 10 -4 0 C 0 -10 0 8 0 D 10 4 -8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826461 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4088: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) C E B D A (7) E C D B A (6) C E A B D (5) A C B E D (5) D A B E C (4) C E A D B (4) A D B E C (4) E C D A B (3) D B E C A (3) C A E B D (3) B D A C E (3) B A C D E (3) A B C D E (3) E C A D B (2) D E B A C (2) C B E D A (2) C B E A D (2) B D E C A (2) B D C E A (2) B D C A E (2) B C E D A (2) A C E D B (2) D E C B A (1) D E B C A (1) D E A C B (1) D E A B C (1) D B A E C (1) D A E B C (1) C E D B A (1) C E D A B (1) C B A E D (1) C A B E D (1) B E D C A (1) B C A E D (1) B C A D E (1) B A D C E (1) A E C D B (1) A D E B C (1) A D B C E (1) A C E B D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -8 6 2 B -10 0 2 18 12 C 8 -2 0 16 28 D -6 -18 -16 0 -4 E -2 -12 -28 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.400000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999995 Cumulative probabilities = A: 0.100000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 6 2 B -10 0 2 18 12 C 8 -2 0 16 28 D -6 -18 -16 0 -4 E -2 -12 -28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.400000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000024 Cumulative probabilities = A: 0.100000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 B=18 D=15 E=11 so E is eliminated. Round 2 votes counts: C=38 A=29 B=18 D=15 so D is eliminated. Round 3 votes counts: C=39 A=36 B=25 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:211 A:205 E:181 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 6 2 B -10 0 2 18 12 C 8 -2 0 16 28 D -6 -18 -16 0 -4 E -2 -12 -28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.400000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000024 Cumulative probabilities = A: 0.100000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 6 2 B -10 0 2 18 12 C 8 -2 0 16 28 D -6 -18 -16 0 -4 E -2 -12 -28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.400000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000024 Cumulative probabilities = A: 0.100000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 6 2 B -10 0 2 18 12 C 8 -2 0 16 28 D -6 -18 -16 0 -4 E -2 -12 -28 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.400000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000024 Cumulative probabilities = A: 0.100000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4089: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) E A C B D (7) D B C A E (7) C E A D B (7) B D A C E (7) B D C E A (6) E C A D B (5) E C A B D (4) E A C D B (3) D B C E A (3) D B A C E (3) B D E A C (3) B D A E C (3) A E C B D (3) A E B C D (3) E C B A D (2) D C B E A (2) D A C B E (2) C D E B A (2) C A E D B (2) B E C D A (2) B E A C D (2) E A B C D (1) D C B A E (1) D C A E B (1) D C A B E (1) C E D A B (1) C E B D A (1) B D E C A (1) B D C A E (1) A E D C B (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 2 6 -4 B -8 0 -16 -8 -12 C -2 16 0 14 -2 D -6 8 -14 0 -10 E 4 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 6 -4 B -8 0 -16 -8 -12 C -2 16 0 14 -2 D -6 8 -14 0 -10 E 4 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=22 D=20 A=20 C=13 so C is eliminated. Round 2 votes counts: E=31 B=25 D=22 A=22 so D is eliminated. Round 3 votes counts: B=41 E=33 A=26 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:213 A:206 D:189 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 6 -4 B -8 0 -16 -8 -12 C -2 16 0 14 -2 D -6 8 -14 0 -10 E 4 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 6 -4 B -8 0 -16 -8 -12 C -2 16 0 14 -2 D -6 8 -14 0 -10 E 4 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 6 -4 B -8 0 -16 -8 -12 C -2 16 0 14 -2 D -6 8 -14 0 -10 E 4 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4090: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) D E A C B (6) B C A E D (6) C B E A D (5) B C A D E (5) B A C E D (5) A D B E C (5) E D C A B (4) C D E A B (4) D A E B C (3) C E D B A (3) B A E C D (3) B A D E C (3) B A C D E (3) A D E B C (3) E D A C B (2) E C D A B (2) D E C A B (2) D E A B C (2) D C E A B (2) C B D E A (2) C B A E D (2) C B A D E (2) A B D E C (2) E D A B C (1) E C B A D (1) E A D B C (1) E A B C D (1) D C A B E (1) D A E C B (1) D A B C E (1) C E D A B (1) C D E B A (1) C B E D A (1) C B D A E (1) B E A C D (1) B D A C E (1) B C E A D (1) A E D B C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -6 4 0 B 6 0 0 8 2 C 6 0 0 14 8 D -4 -8 -14 0 0 E 0 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.533311 C: 0.466689 D: 0.000000 E: 0.000000 Sum of squares = 0.502219220757 Cumulative probabilities = A: 0.000000 B: 0.533311 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 4 0 B 6 0 0 8 2 C 6 0 0 14 8 D -4 -8 -14 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 D=18 A=13 E=12 so E is eliminated. Round 2 votes counts: C=32 B=28 D=25 A=15 so A is eliminated. Round 3 votes counts: D=35 B=33 C=32 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:214 B:208 A:196 E:195 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 4 0 B 6 0 0 8 2 C 6 0 0 14 8 D -4 -8 -14 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 4 0 B 6 0 0 8 2 C 6 0 0 14 8 D -4 -8 -14 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 4 0 B 6 0 0 8 2 C 6 0 0 14 8 D -4 -8 -14 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4091: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (6) E C D B A (4) D C E A B (4) D A B C E (4) C D A E B (4) A D C B E (4) A C D B E (4) A B C D E (4) D E C B A (3) C E B A D (3) C A B E D (3) B E A D C (3) B E A C D (3) A D B C E (3) E D B A C (2) E C B A D (2) E B D C A (2) E B C A D (2) D E B A C (2) D C A E B (2) D A E B C (2) C E D A B (2) C A D B E (2) B E D A C (2) B E C A D (2) B A E C D (2) B A C E D (2) A C B E D (2) A B D E C (2) A B D C E (2) A B C E D (2) E D C B A (1) E C B D A (1) D E C A B (1) D E B C A (1) D E A B C (1) D C E B A (1) D A C E B (1) D A C B E (1) D A B E C (1) C E A B D (1) C A B D E (1) B D A E C (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 8 16 2 2 B -8 0 4 0 4 C -16 -4 0 -6 6 D -2 0 6 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 2 2 B -8 0 4 0 4 C -16 -4 0 -6 6 D -2 0 6 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=20 C=16 B=16 so C is eliminated. Round 2 votes counts: A=30 D=28 E=26 B=16 so B is eliminated. Round 3 votes counts: E=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:205 B:200 E:191 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 2 2 B -8 0 4 0 4 C -16 -4 0 -6 6 D -2 0 6 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 2 2 B -8 0 4 0 4 C -16 -4 0 -6 6 D -2 0 6 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 2 2 B -8 0 4 0 4 C -16 -4 0 -6 6 D -2 0 6 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4092: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) A D E B C (11) A E D B C (10) B E C A D (8) D C B A E (7) D A C E B (6) B C E A D (6) E B A C D (5) D A E C B (5) A E B D C (5) E A B C D (4) D C B E A (3) D C A B E (3) D A C B E (3) C D B E A (3) C B D E A (3) E B C A D (2) A E B C D (2) Total count = 100 A B C D E A 0 -2 2 6 4 B 2 0 6 -2 0 C -2 -6 0 -6 -4 D -6 2 6 0 -12 E -4 0 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.713601 C: 0.000000 D: 0.000000 E: 0.286399 Sum of squares = 0.59125056272 Cumulative probabilities = A: 0.000000 B: 0.713601 C: 0.713601 D: 0.713601 E: 1.000000 A B C D E A 0 -2 2 6 4 B 2 0 6 -2 0 C -2 -6 0 -6 -4 D -6 2 6 0 -12 E -4 0 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555718622 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=27 C=20 B=14 E=11 so E is eliminated. Round 2 votes counts: A=32 D=27 B=21 C=20 so C is eliminated. Round 3 votes counts: B=38 A=32 D=30 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:206 A:205 B:203 D:195 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 6 4 B 2 0 6 -2 0 C -2 -6 0 -6 -4 D -6 2 6 0 -12 E -4 0 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555718622 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 6 4 B 2 0 6 -2 0 C -2 -6 0 -6 -4 D -6 2 6 0 -12 E -4 0 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555718622 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 6 4 B 2 0 6 -2 0 C -2 -6 0 -6 -4 D -6 2 6 0 -12 E -4 0 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555718622 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4093: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) A B D E C (7) C E D B A (6) C D E A B (5) B C A E D (5) C B A E D (4) A D E B C (4) E C D B A (3) D E C A B (3) D E A C B (3) A B C D E (3) E D A B C (2) E B D A C (2) C E B D A (2) C D A E B (2) C B E D A (2) C A D B E (2) C A B D E (2) B E C A D (2) B C A D E (2) A D B E C (2) A D B C E (2) A C B D E (2) A B E D C (2) A B D C E (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A C B (1) E C B D A (1) E B D C A (1) D E A B C (1) D C E A B (1) D A E B C (1) D A C E B (1) C B A D E (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B C E D A (1) B A E C D (1) B A C D E (1) A E D B C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 6 0 12 14 B -6 0 10 8 8 C 0 -10 0 -2 -4 D -12 -8 2 0 0 E -14 -8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.796776 B: 0.000000 C: 0.203224 D: 0.000000 E: 0.000000 Sum of squares = 0.676152060761 Cumulative probabilities = A: 0.796776 B: 0.796776 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 12 14 B -6 0 10 8 8 C 0 -10 0 -2 -4 D -12 -8 2 0 0 E -14 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250085456 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=27 A=27 B=23 E=13 D=10 so D is eliminated. Round 2 votes counts: A=29 C=28 B=23 E=20 so E is eliminated. Round 3 votes counts: C=36 A=36 B=28 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:210 C:192 D:191 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 12 14 B -6 0 10 8 8 C 0 -10 0 -2 -4 D -12 -8 2 0 0 E -14 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250085456 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 12 14 B -6 0 10 8 8 C 0 -10 0 -2 -4 D -12 -8 2 0 0 E -14 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250085456 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 12 14 B -6 0 10 8 8 C 0 -10 0 -2 -4 D -12 -8 2 0 0 E -14 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250085456 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4094: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) E C A D B (8) D B A C E (8) B D A C E (8) D E B C A (7) C A E B D (7) B A C D E (7) E C A B D (5) D B E A C (5) A C B E D (5) E D C A B (4) D B A E C (4) B A D C E (4) B A C E D (4) E C D A B (3) D E C A B (2) C A B E D (2) A B C E D (2) E A C B D (1) D E B A C (1) D B E C A (1) C E A B D (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 2 -4 6 B 18 0 4 -4 2 C -2 -4 0 -8 -2 D 4 4 8 0 14 E -6 -2 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 2 -4 6 B 18 0 4 -4 2 C -2 -4 0 -8 -2 D 4 4 8 0 14 E -6 -2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=24 E=21 C=10 A=8 so A is eliminated. Round 2 votes counts: D=37 B=26 E=21 C=16 so C is eliminated. Round 3 votes counts: D=37 B=33 E=30 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:210 A:193 C:192 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 2 -4 6 B 18 0 4 -4 2 C -2 -4 0 -8 -2 D 4 4 8 0 14 E -6 -2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 2 -4 6 B 18 0 4 -4 2 C -2 -4 0 -8 -2 D 4 4 8 0 14 E -6 -2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 2 -4 6 B 18 0 4 -4 2 C -2 -4 0 -8 -2 D 4 4 8 0 14 E -6 -2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4095: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (5) A D E B C (5) E C D B A (4) E C B A D (4) E A B C D (4) C B D A E (4) A B E C D (4) A B D C E (4) E D C A B (3) E D A C B (3) D E C A B (3) D E A C B (3) D C E B A (3) D C B E A (3) C D B E A (3) C B E D A (3) B C A D E (3) B A C D E (3) E C B D A (2) E A D C B (2) E A D B C (2) D C B A E (2) C D E B A (2) B A C E D (2) A E B D C (2) A B C E D (2) A B C D E (2) E A C D B (1) E A C B D (1) D E C B A (1) D A E C B (1) D A C B E (1) D A B C E (1) C E B D A (1) C E B A D (1) C D B A E (1) C B D E A (1) B C E A D (1) B C D A E (1) B C A E D (1) A E D B C (1) A E B C D (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -4 -2 -12 B 2 0 -16 -2 -10 C 4 16 0 4 -10 D 2 2 -4 0 -2 E 12 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -2 -12 B 2 0 -16 -2 -10 C 4 16 0 4 -10 D 2 2 -4 0 -2 E 12 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=24 D=18 C=16 B=11 so B is eliminated. Round 2 votes counts: E=31 A=29 C=22 D=18 so D is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:207 D:199 A:190 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 -12 B 2 0 -16 -2 -10 C 4 16 0 4 -10 D 2 2 -4 0 -2 E 12 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 -12 B 2 0 -16 -2 -10 C 4 16 0 4 -10 D 2 2 -4 0 -2 E 12 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 -12 B 2 0 -16 -2 -10 C 4 16 0 4 -10 D 2 2 -4 0 -2 E 12 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4096: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) E D C A B (11) D E C A B (8) B A C E D (7) E D B C A (6) E D B A C (4) C A B D E (4) E D C B A (3) E C D A B (3) E B A C D (3) D E C B A (3) D C A E B (3) C D A B E (3) A C B D E (3) A B C E D (3) A B C D E (3) E C A D B (2) E B D A C (2) D C E A B (2) C A D B E (2) B A E C D (2) E B A D C (1) D E B C A (1) D C A B E (1) C E D A B (1) C D A E B (1) C A B E D (1) B E D A C (1) B E A D C (1) B E A C D (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -10 -6 -6 B -4 0 -4 -8 -8 C 10 4 0 4 -4 D 6 8 -4 0 -6 E 6 8 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -10 -6 -6 B -4 0 -4 -8 -8 C 10 4 0 4 -4 D 6 8 -4 0 -6 E 6 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=25 D=18 C=12 A=10 so A is eliminated. Round 2 votes counts: E=35 B=31 D=18 C=16 so C is eliminated. Round 3 votes counts: B=40 E=36 D=24 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:207 D:202 A:191 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -10 -6 -6 B -4 0 -4 -8 -8 C 10 4 0 4 -4 D 6 8 -4 0 -6 E 6 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -6 -6 B -4 0 -4 -8 -8 C 10 4 0 4 -4 D 6 8 -4 0 -6 E 6 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -6 -6 B -4 0 -4 -8 -8 C 10 4 0 4 -4 D 6 8 -4 0 -6 E 6 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4097: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (13) C D B E A (9) A D B E C (8) E C B A D (4) D A B C E (4) E C A B D (3) E B C A D (3) E B A C D (3) C E B A D (3) C E A D B (3) C B E D A (3) A D C B E (3) E C B D A (2) E B C D A (2) E B A D C (2) D C B A E (2) D C A B E (2) D B C E A (2) C D E B A (2) B E C D A (2) A D C E B (2) A C D E B (2) A B E D C (2) E A C B D (1) E A B C D (1) D C B E A (1) D B C A E (1) D A C B E (1) D A B E C (1) C E D A B (1) C E A B D (1) C D E A B (1) C D B A E (1) C D A B E (1) C B D E A (1) B E D C A (1) B D A E C (1) B A E D C (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -28 -8 -28 B 18 0 -28 2 -6 C 28 28 0 28 18 D 8 -2 -28 0 -8 E 28 6 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -28 -8 -28 B 18 0 -28 2 -6 C 28 28 0 28 18 D 8 -2 -28 0 -8 E 28 6 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=21 A=21 D=14 B=5 so B is eliminated. Round 2 votes counts: C=39 E=24 A=22 D=15 so D is eliminated. Round 3 votes counts: C=47 A=29 E=24 so E is eliminated. Round 4 votes counts: C=64 A=36 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:251 E:212 B:193 D:185 A:159 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -28 -8 -28 B 18 0 -28 2 -6 C 28 28 0 28 18 D 8 -2 -28 0 -8 E 28 6 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -28 -8 -28 B 18 0 -28 2 -6 C 28 28 0 28 18 D 8 -2 -28 0 -8 E 28 6 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -28 -8 -28 B 18 0 -28 2 -6 C 28 28 0 28 18 D 8 -2 -28 0 -8 E 28 6 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4098: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) E A D C B (7) E A C D B (7) B C D A E (5) E C A D B (4) E B C D A (4) C A E D B (4) B C D E A (4) A C D E B (4) E D B A C (3) D B A C E (3) D A E B C (3) C B A D E (3) B D E A C (3) A E D C B (3) A D E C B (3) E D A B C (2) E B D C A (2) D A B C E (2) C A E B D (2) C A D E B (2) C A D B E (2) C A B E D (2) B D C E A (2) E B C A D (1) E A C B D (1) D B E A C (1) D B A E C (1) D A E C B (1) D A B E C (1) C B E A D (1) C B D A E (1) B E D C A (1) B E D A C (1) B D C A E (1) B D A C E (1) B C E D A (1) A D E B C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 6 16 -2 -6 B -6 0 2 -12 -24 C -16 -2 0 -2 -16 D 2 12 2 0 -8 E 6 24 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 16 -2 -6 B -6 0 2 -12 -24 C -16 -2 0 -2 -16 D 2 12 2 0 -8 E 6 24 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=19 C=17 A=13 D=12 so D is eliminated. Round 2 votes counts: E=39 B=24 A=20 C=17 so C is eliminated. Round 3 votes counts: E=39 A=32 B=29 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:207 D:204 C:182 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 16 -2 -6 B -6 0 2 -12 -24 C -16 -2 0 -2 -16 D 2 12 2 0 -8 E 6 24 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 -2 -6 B -6 0 2 -12 -24 C -16 -2 0 -2 -16 D 2 12 2 0 -8 E 6 24 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 -2 -6 B -6 0 2 -12 -24 C -16 -2 0 -2 -16 D 2 12 2 0 -8 E 6 24 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4099: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) C A E D B (8) C A D B E (8) D B C E A (7) B E D A C (6) B D E A C (6) A C E D B (6) D B E C A (5) A E C B D (5) E B D A C (4) E A B C D (4) D C B A E (4) B D E C A (4) A C E B D (4) C A D E B (3) E B A D C (2) E A C D B (2) E A C B D (2) D E B A C (2) E A B D C (1) D B C A E (1) C A E B D (1) C A B E D (1) C A B D E (1) B E A C D (1) B D C A E (1) A E C D B (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 -4 4 8 B -14 0 -8 -10 8 C 4 8 0 14 6 D -4 10 -14 0 0 E -8 -8 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 4 8 B -14 0 -8 -10 8 C 4 8 0 14 6 D -4 10 -14 0 0 E -8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=19 B=18 A=18 E=15 so E is eliminated. Round 2 votes counts: C=30 A=27 B=24 D=19 so D is eliminated. Round 3 votes counts: B=39 C=34 A=27 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:211 D:196 E:189 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 4 8 B -14 0 -8 -10 8 C 4 8 0 14 6 D -4 10 -14 0 0 E -8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 4 8 B -14 0 -8 -10 8 C 4 8 0 14 6 D -4 10 -14 0 0 E -8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 4 8 B -14 0 -8 -10 8 C 4 8 0 14 6 D -4 10 -14 0 0 E -8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4100: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) D C A E B (7) D C A B E (6) C D B A E (5) B C D A E (5) E A B D C (4) B E A C D (4) E C D A B (3) D A C B E (3) C D E A B (3) C D A E B (3) B C A D E (3) A B E D C (3) E D A C B (2) E A D C B (2) E A D B C (2) D E C A B (2) D C E A B (2) D A E C B (2) C D E B A (2) C D B E A (2) B C A E D (2) B A E C D (2) A E D B C (2) E C B D A (1) E B A D C (1) E A B C D (1) D B A C E (1) D A C E B (1) C E D B A (1) C D A B E (1) C B E D A (1) C B D E A (1) B C E D A (1) B C E A D (1) B A D E C (1) B A D C E (1) B A C D E (1) A E B D C (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -4 -10 28 B -6 0 -2 -8 12 C 4 2 0 -14 6 D 10 8 14 0 10 E -28 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -10 28 B -6 0 -2 -8 12 C 4 2 0 -14 6 D 10 8 14 0 10 E -28 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=24 C=19 E=16 A=9 so A is eliminated. Round 2 votes counts: B=36 D=26 E=19 C=19 so E is eliminated. Round 3 votes counts: B=43 D=34 C=23 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:210 C:199 B:198 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -10 28 B -6 0 -2 -8 12 C 4 2 0 -14 6 D 10 8 14 0 10 E -28 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -10 28 B -6 0 -2 -8 12 C 4 2 0 -14 6 D 10 8 14 0 10 E -28 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -10 28 B -6 0 -2 -8 12 C 4 2 0 -14 6 D 10 8 14 0 10 E -28 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4101: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (13) A B D C E (10) E C D B A (9) C E D B A (9) A C E B D (7) D B E C A (5) C E D A B (5) C E A D B (5) D E B C A (4) B D A E C (4) A C B E D (4) E D B C A (3) D E C B A (3) A E B D C (3) B D E A C (2) B D A C E (2) B A D C E (2) A C B D E (2) E D C B A (1) E D A B C (1) D B C E A (1) D B C A E (1) C A E D B (1) C A E B D (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 4 0 4 B -6 0 6 4 -4 C -4 -6 0 -12 2 D 0 -4 12 0 0 E -4 4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.621600 B: 0.000000 C: 0.000000 D: 0.378400 E: 0.000000 Sum of squares = 0.529573233504 Cumulative probabilities = A: 0.621600 B: 0.621600 C: 0.621600 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 4 B -6 0 6 4 -4 C -4 -6 0 -12 2 D 0 -4 12 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=21 E=14 D=14 B=11 so B is eliminated. Round 2 votes counts: A=43 D=22 C=21 E=14 so E is eliminated. Round 3 votes counts: A=43 C=30 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:207 D:204 B:200 E:199 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 4 B -6 0 6 4 -4 C -4 -6 0 -12 2 D 0 -4 12 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 4 B -6 0 6 4 -4 C -4 -6 0 -12 2 D 0 -4 12 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 4 B -6 0 6 4 -4 C -4 -6 0 -12 2 D 0 -4 12 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4102: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) B D C A E (7) A E D C B (6) E A B D C (5) E A B C D (5) D C B A E (5) A C D E B (5) E B C D A (4) E A C B D (4) D B C A E (4) E B A D C (3) E B A C D (3) C D B A E (3) C B D E A (3) B E C D A (3) A D C E B (3) D A C B E (2) B E D C A (2) B E D A C (2) B E A D C (2) B D C E A (2) A D B C E (2) E C B D A (1) E C B A D (1) E C A D B (1) D B A C E (1) C E D B A (1) C E B D A (1) C E A D B (1) C D B E A (1) C D A B E (1) B C D E A (1) A E D B C (1) A E C D B (1) A E B D C (1) A D C B E (1) A D B E C (1) A C E D B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 16 12 -6 B 0 0 2 4 -10 C -16 -2 0 -4 -8 D -12 -4 4 0 -12 E 6 10 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 16 12 -6 B 0 0 2 4 -10 C -16 -2 0 -4 -8 D -12 -4 4 0 -12 E 6 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=24 B=19 D=12 C=11 so C is eliminated. Round 2 votes counts: E=37 A=24 B=22 D=17 so D is eliminated. Round 3 votes counts: E=37 B=36 A=27 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:211 B:198 D:188 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 16 12 -6 B 0 0 2 4 -10 C -16 -2 0 -4 -8 D -12 -4 4 0 -12 E 6 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 12 -6 B 0 0 2 4 -10 C -16 -2 0 -4 -8 D -12 -4 4 0 -12 E 6 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 12 -6 B 0 0 2 4 -10 C -16 -2 0 -4 -8 D -12 -4 4 0 -12 E 6 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4103: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (16) B A D C E (16) B E A D C (8) E B C D A (7) C D A E B (7) E C B D A (6) E B C A D (4) A D C B E (4) D A C B E (3) C D A B E (3) B E A C D (3) B A D E C (3) E C D B A (2) D C A E B (2) C E D A B (2) B E C A D (2) B A E D C (2) A D B C E (2) A B D C E (2) E D C A B (1) E B D A C (1) E A D C B (1) B E C D A (1) B A C D E (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -6 -2 -8 B 12 0 4 12 0 C 6 -4 0 8 -16 D 2 -12 -8 0 -12 E 8 0 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.654788 C: 0.000000 D: 0.000000 E: 0.345212 Sum of squares = 0.547918933238 Cumulative probabilities = A: 0.000000 B: 0.654788 C: 0.654788 D: 0.654788 E: 1.000000 A B C D E A 0 -12 -6 -2 -8 B 12 0 4 12 0 C 6 -4 0 8 -16 D 2 -12 -8 0 -12 E 8 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=36 C=12 A=9 D=5 so D is eliminated. Round 2 votes counts: E=38 B=36 C=14 A=12 so A is eliminated. Round 3 votes counts: B=40 E=39 C=21 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:214 C:197 A:186 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -2 -8 B 12 0 4 12 0 C 6 -4 0 8 -16 D 2 -12 -8 0 -12 E 8 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -2 -8 B 12 0 4 12 0 C 6 -4 0 8 -16 D 2 -12 -8 0 -12 E 8 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -2 -8 B 12 0 4 12 0 C 6 -4 0 8 -16 D 2 -12 -8 0 -12 E 8 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4104: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) E C D B A (9) D B E C A (9) A B D C E (9) C E A D B (8) A B D E C (7) D B A C E (5) B D E A C (4) C D E B A (3) C A E D B (3) B D A E C (3) B D A C E (3) B A D E C (3) A E C B D (3) E C D A B (2) E C A D B (2) B D E C A (2) A B E C D (2) E D C B A (1) E C A B D (1) E B D C A (1) E A C B D (1) D C B A E (1) D B E A C (1) D B C E A (1) D B C A E (1) C E D A B (1) A C E D B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 10 6 8 B -6 0 4 2 6 C -10 -4 0 -4 -4 D -6 -2 4 0 6 E -8 -6 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 6 8 B -6 0 4 2 6 C -10 -4 0 -4 -4 D -6 -2 4 0 6 E -8 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=18 E=17 C=15 B=15 so C is eliminated. Round 2 votes counts: A=38 E=26 D=21 B=15 so B is eliminated. Round 3 votes counts: A=41 D=33 E=26 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:203 D:201 E:192 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 6 8 B -6 0 4 2 6 C -10 -4 0 -4 -4 D -6 -2 4 0 6 E -8 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 6 8 B -6 0 4 2 6 C -10 -4 0 -4 -4 D -6 -2 4 0 6 E -8 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 6 8 B -6 0 4 2 6 C -10 -4 0 -4 -4 D -6 -2 4 0 6 E -8 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4105: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) D B A E C (8) C E A B D (7) B D C A E (6) B C D A E (6) E C A D B (5) D B A C E (5) E D A B C (4) D E A B C (4) C E B A D (4) B D A C E (4) E C A B D (3) C A B E D (3) B C A D E (3) A D E B C (3) D E B A C (2) D A E B C (2) D A B E C (2) C B E A D (2) C A E B D (2) A E C D B (2) A B D C E (2) E D B A C (1) E C D A B (1) E C B D A (1) E A D C B (1) D B E A C (1) D A B C E (1) C B A E D (1) C B A D E (1) B D C E A (1) B A D C E (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 8 2 8 B -6 0 12 -4 -6 C -8 -12 0 2 0 D -2 4 -2 0 4 E -8 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 2 8 B -6 0 12 -4 -6 C -8 -12 0 2 0 D -2 4 -2 0 4 E -8 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 B=21 C=20 A=9 so A is eliminated. Round 2 votes counts: E=28 D=28 B=23 C=21 so C is eliminated. Round 3 votes counts: E=42 B=30 D=28 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 D:202 B:198 E:197 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 8 B -6 0 12 -4 -6 C -8 -12 0 2 0 D -2 4 -2 0 4 E -8 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 8 B -6 0 12 -4 -6 C -8 -12 0 2 0 D -2 4 -2 0 4 E -8 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 8 B -6 0 12 -4 -6 C -8 -12 0 2 0 D -2 4 -2 0 4 E -8 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4106: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) D E C B A (7) C B D E A (7) A B E C D (6) D C E B A (5) C B D A E (5) B C A D E (5) A B C E D (5) E D A C B (4) D C B E A (4) B A C E D (4) A E B D C (4) A E B C D (4) E A B C D (3) D A E C B (3) A C B D E (3) E D C B A (2) E A B D C (2) C D B E A (2) C D B A E (2) B C A E D (2) A B C D E (2) E D A B C (1) E A D C B (1) D E A C B (1) D C B A E (1) D C A B E (1) C D A B E (1) B E C A D (1) B C D E A (1) B C D A E (1) B A C D E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 0 6 4 4 B 0 0 0 12 8 C -6 0 0 10 4 D -4 -12 -10 0 4 E -4 -8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.535508 B: 0.464492 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502521702075 Cumulative probabilities = A: 0.535508 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 4 B 0 0 0 12 8 C -6 0 0 10 4 D -4 -12 -10 0 4 E -4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=22 E=20 C=17 B=15 so B is eliminated. Round 2 votes counts: A=31 C=26 D=22 E=21 so E is eliminated. Round 3 votes counts: A=44 D=29 C=27 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:207 C:204 E:190 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 4 B 0 0 0 12 8 C -6 0 0 10 4 D -4 -12 -10 0 4 E -4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 4 B 0 0 0 12 8 C -6 0 0 10 4 D -4 -12 -10 0 4 E -4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 4 B 0 0 0 12 8 C -6 0 0 10 4 D -4 -12 -10 0 4 E -4 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4107: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) E A D C B (7) B D C A E (7) B C D A E (7) E B D A C (5) C A D B E (5) E D B A C (4) E D A B C (4) D B E A C (4) C A D E B (4) C A B D E (4) B D E A C (4) B D C E A (4) C B A D E (3) B E D C A (3) B D E C A (3) E D A C B (2) C D A B E (2) C A E D B (2) A E C D B (2) A C E D B (2) A C D E B (2) E A D B C (1) D B C A E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A B C E (1) C A B E D (1) B E D A C (1) B E C A D (1) B C A D E (1) Total count = 100 A B C D E A 0 0 4 -10 -6 B 0 0 6 -12 8 C -4 -6 0 -8 -4 D 10 12 8 0 10 E 6 -8 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -10 -6 B 0 0 6 -12 8 C -4 -6 0 -8 -4 D 10 12 8 0 10 E 6 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=31 C=21 D=9 A=6 so A is eliminated. Round 2 votes counts: E=35 B=31 C=25 D=9 so D is eliminated. Round 3 votes counts: B=39 E=35 C=26 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:201 E:196 A:194 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -10 -6 B 0 0 6 -12 8 C -4 -6 0 -8 -4 D 10 12 8 0 10 E 6 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -10 -6 B 0 0 6 -12 8 C -4 -6 0 -8 -4 D 10 12 8 0 10 E 6 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -10 -6 B 0 0 6 -12 8 C -4 -6 0 -8 -4 D 10 12 8 0 10 E 6 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4108: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) C D E B A (9) A D C B E (8) E B A C D (7) C E D B A (6) A B D E C (5) E B D C A (4) D C B E A (4) D A C B E (4) E C B D A (3) D C A B E (3) A E C B D (3) A E B C D (3) A D B C E (3) E B C D A (2) E A B C D (2) C D B E A (2) C D A B E (2) B E D C A (2) A C E D B (2) E C B A D (1) E C A B D (1) D C B A E (1) D B E C A (1) D B C E A (1) D B A C E (1) C D E A B (1) C A D E B (1) B E A D C (1) A D B E C (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 12 8 6 B -10 0 -6 -4 8 C -12 6 0 -6 2 D -8 4 6 0 0 E -6 -8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 8 6 B -10 0 -6 -4 8 C -12 6 0 -6 2 D -8 4 6 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=21 E=20 D=15 B=3 so B is eliminated. Round 2 votes counts: A=41 E=23 C=21 D=15 so D is eliminated. Round 3 votes counts: A=46 C=30 E=24 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:201 C:195 B:194 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 8 6 B -10 0 -6 -4 8 C -12 6 0 -6 2 D -8 4 6 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 8 6 B -10 0 -6 -4 8 C -12 6 0 -6 2 D -8 4 6 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 8 6 B -10 0 -6 -4 8 C -12 6 0 -6 2 D -8 4 6 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4109: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) E B D C A (7) D C B E A (7) B D C E A (6) B C D E A (6) A E D C B (6) D C B A E (5) A C D E B (5) E A B D C (4) A C D B E (4) E B A D C (3) C D B A E (3) C D A B E (3) C B D A E (3) B E C D A (3) A E C D B (3) A D C E B (3) E B A C D (2) E A D B C (2) C A D B E (2) B C D A E (2) A B E C D (2) E B D A C (1) E A B C D (1) D C A E B (1) D C A B E (1) D A C E B (1) B E D C A (1) B E C A D (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -2 0 12 B 0 0 4 6 -2 C 2 -4 0 2 6 D 0 -6 -2 0 4 E -12 2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.370492 B: 0.629508 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.533544466529 Cumulative probabilities = A: 0.370492 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 0 12 B 0 0 4 6 -2 C 2 -4 0 2 6 D 0 -6 -2 0 4 E -12 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=20 B=19 D=15 C=11 so C is eliminated. Round 2 votes counts: A=37 B=22 D=21 E=20 so E is eliminated. Round 3 votes counts: A=44 B=35 D=21 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:205 B:204 C:203 D:198 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 0 12 B 0 0 4 6 -2 C 2 -4 0 2 6 D 0 -6 -2 0 4 E -12 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 0 12 B 0 0 4 6 -2 C 2 -4 0 2 6 D 0 -6 -2 0 4 E -12 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 0 12 B 0 0 4 6 -2 C 2 -4 0 2 6 D 0 -6 -2 0 4 E -12 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4110: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) E B A D C (6) B A E D C (6) A B E D C (6) E A B C D (5) C D E B A (5) B D A E C (5) C D A B E (4) E D B C A (3) D B A C E (3) C D E A B (3) B A D E C (3) A E B C D (3) E C D B A (2) E C B A D (2) E B D A C (2) E B C A D (2) D C E B A (2) D B A E C (2) C E D B A (2) C E D A B (2) C E A B D (2) C A E B D (2) B E A D C (2) A B D E C (2) E C A B D (1) E B A C D (1) D C B E A (1) D C A B E (1) D B C A E (1) C D B A E (1) C D A E B (1) C A D B E (1) C A B E D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 0 -2 14 B 24 0 8 10 8 C 0 -8 0 -14 -8 D 2 -10 14 0 -4 E -14 -8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 0 -2 14 B 24 0 8 10 8 C 0 -8 0 -14 -8 D 2 -10 14 0 -4 E -14 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 D=21 B=16 A=15 so A is eliminated. Round 2 votes counts: E=27 C=26 B=26 D=21 so D is eliminated. Round 3 votes counts: C=41 B=32 E=27 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:201 E:195 A:194 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 0 -2 14 B 24 0 8 10 8 C 0 -8 0 -14 -8 D 2 -10 14 0 -4 E -14 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 0 -2 14 B 24 0 8 10 8 C 0 -8 0 -14 -8 D 2 -10 14 0 -4 E -14 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 0 -2 14 B 24 0 8 10 8 C 0 -8 0 -14 -8 D 2 -10 14 0 -4 E -14 -8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4111: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) D E C B A (8) B C E D A (7) A E D B C (6) A D E C B (5) E D C B A (4) E B C A D (4) A E B C D (4) A D C B E (4) A B E C D (4) D C E B A (3) D C B E A (3) D A E C B (3) C B D E A (3) A B C D E (3) D E A C B (2) D C E A B (2) B C E A D (2) B C D A E (2) B C A D E (2) A D C E B (2) A D B C E (2) A B C E D (2) E D B A C (1) E D A B C (1) E B A C D (1) E A D B C (1) E A B D C (1) D C B A E (1) D A C E B (1) C B D A E (1) B E C D A (1) B E C A D (1) A E B D C (1) A D E B C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -6 -4 -8 B 6 0 12 -2 -20 C 6 -12 0 -4 -16 D 4 2 4 0 0 E 8 20 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.723387 E: 0.276613 Sum of squares = 0.599803310109 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.723387 E: 1.000000 A B C D E A 0 -6 -6 -4 -8 B 6 0 12 -2 -20 C 6 -12 0 -4 -16 D 4 2 4 0 0 E 8 20 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=23 E=22 B=15 C=4 so C is eliminated. Round 2 votes counts: A=36 D=23 E=22 B=19 so B is eliminated. Round 3 votes counts: A=38 E=33 D=29 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:205 B:198 A:188 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 -8 B 6 0 12 -2 -20 C 6 -12 0 -4 -16 D 4 2 4 0 0 E 8 20 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 -8 B 6 0 12 -2 -20 C 6 -12 0 -4 -16 D 4 2 4 0 0 E 8 20 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 -8 B 6 0 12 -2 -20 C 6 -12 0 -4 -16 D 4 2 4 0 0 E 8 20 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4112: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (14) A B E D C (11) D C A B E (10) D A B E C (7) C D E B A (7) E B A D C (5) D A C B E (5) E B C A D (3) D C E B A (3) C E B D A (3) C D A B E (3) C A B E D (3) B E A D C (3) A D B E C (3) C E D B A (2) C E B A D (2) C D E A B (2) C D A E B (2) B E A C D (2) E B D A C (1) E B C D A (1) D A B C E (1) C A E B D (1) C A D B E (1) B A E D C (1) A D B C E (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 14 6 4 B -6 0 10 6 8 C -14 -10 0 -4 -6 D -6 -6 4 0 -6 E -4 -8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 6 4 B -6 0 10 6 8 C -14 -10 0 -4 -6 D -6 -6 4 0 -6 E -4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 E=24 A=18 B=6 so B is eliminated. Round 2 votes counts: E=29 D=26 C=26 A=19 so A is eliminated. Round 3 votes counts: E=42 D=31 C=27 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:215 B:209 E:200 D:193 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 6 4 B -6 0 10 6 8 C -14 -10 0 -4 -6 D -6 -6 4 0 -6 E -4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 6 4 B -6 0 10 6 8 C -14 -10 0 -4 -6 D -6 -6 4 0 -6 E -4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 6 4 B -6 0 10 6 8 C -14 -10 0 -4 -6 D -6 -6 4 0 -6 E -4 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4113: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) A E D C B (12) D A E B C (8) A D E C B (8) E A D C B (7) C B E A D (7) E A C B D (5) B C D A E (5) B C A E D (3) A E C D B (3) A D E B C (3) D B C A E (2) D A E C B (2) C B E D A (2) B D C A E (2) B C E A D (2) E D C B A (1) E D A C B (1) E C B A D (1) D E C A B (1) D E A C B (1) D E A B C (1) D C E B A (1) D B C E A (1) D B A C E (1) D A B E C (1) C E B A D (1) C D B E A (1) C B D E A (1) B C E D A (1) A E D B C (1) Total count = 100 A B C D E A 0 8 8 6 2 B -8 0 -10 -12 -14 C -8 10 0 -8 -12 D -6 12 8 0 6 E -2 14 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 6 2 B -8 0 -10 -12 -14 C -8 10 0 -8 -12 D -6 12 8 0 6 E -2 14 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 D=19 E=15 C=12 so C is eliminated. Round 2 votes counts: B=37 A=27 D=20 E=16 so E is eliminated. Round 3 votes counts: B=39 A=39 D=22 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:210 E:209 C:191 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 6 2 B -8 0 -10 -12 -14 C -8 10 0 -8 -12 D -6 12 8 0 6 E -2 14 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 2 B -8 0 -10 -12 -14 C -8 10 0 -8 -12 D -6 12 8 0 6 E -2 14 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 2 B -8 0 -10 -12 -14 C -8 10 0 -8 -12 D -6 12 8 0 6 E -2 14 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4114: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (6) A E B D C (6) A B E D C (6) E D A C B (5) C D E B A (5) C D E A B (5) B A E C D (5) D C E A B (4) C D B E A (4) B A C E D (4) A B E C D (4) C D A E B (3) B E A D C (3) A E D C B (3) A B C E D (3) E D C A B (2) D E C B A (2) D C E B A (2) C D A B E (2) C B A D E (2) C A D E B (2) C A D B E (2) B C D E A (2) E D B C A (1) E D B A C (1) E B A D C (1) E A D B C (1) D E C A B (1) D C A E B (1) D A E C B (1) C D B A E (1) C B D E A (1) C B D A E (1) C A B D E (1) B E D C A (1) B E D A C (1) B C A D E (1) A E D B C (1) A E C D B (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 12 8 8 16 B -12 0 -6 -2 2 C -8 6 0 0 -4 D -8 2 0 0 -10 E -16 -2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 8 16 B -12 0 -6 -2 2 C -8 6 0 0 -4 D -8 2 0 0 -10 E -16 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 B=23 E=11 D=11 so E is eliminated. Round 2 votes counts: C=29 A=27 B=24 D=20 so D is eliminated. Round 3 votes counts: C=41 A=33 B=26 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:198 C:197 D:192 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 8 16 B -12 0 -6 -2 2 C -8 6 0 0 -4 D -8 2 0 0 -10 E -16 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 8 16 B -12 0 -6 -2 2 C -8 6 0 0 -4 D -8 2 0 0 -10 E -16 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 8 16 B -12 0 -6 -2 2 C -8 6 0 0 -4 D -8 2 0 0 -10 E -16 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4115: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) C B A E D (6) C B D E A (5) B C A D E (5) B A D E C (5) B A C D E (5) D E C A B (4) D E B C A (4) C A B E D (4) D E C B A (3) D E A B C (3) D B C E A (3) A B E D C (3) A B C E D (3) E A D C B (2) D E B A C (2) D C E B A (2) C E D B A (2) C E D A B (2) C D E B A (2) C B E A D (2) B C D A E (2) B A C E D (2) A E B C D (2) A B E C D (2) E D A B C (1) E A D B C (1) D B A E C (1) C E A B D (1) C D B E A (1) C B D A E (1) C A E B D (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C A E D (1) A E D C B (1) A E C D B (1) A E C B D (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -6 0 2 B 16 0 0 14 10 C 6 0 0 4 8 D 0 -14 -4 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.388004 C: 0.611996 D: 0.000000 E: 0.000000 Sum of squares = 0.525086375452 Cumulative probabilities = A: 0.000000 B: 0.388004 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 0 2 B 16 0 0 14 10 C 6 0 0 4 8 D 0 -14 -4 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 D=22 A=15 E=12 so E is eliminated. Round 2 votes counts: D=31 C=27 B=24 A=18 so A is eliminated. Round 3 votes counts: D=35 B=35 C=30 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:209 D:193 A:190 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 0 2 B 16 0 0 14 10 C 6 0 0 4 8 D 0 -14 -4 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 0 2 B 16 0 0 14 10 C 6 0 0 4 8 D 0 -14 -4 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 0 2 B 16 0 0 14 10 C 6 0 0 4 8 D 0 -14 -4 0 4 E -2 -10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4116: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) B A E D C (8) B E D C A (7) A E D B C (7) C D E A B (6) A C D E B (6) C D E B A (5) A B E D C (5) C A D E B (4) A B C E D (4) D E C A B (3) C D B E A (3) B C E D A (3) B C D E A (3) B C A D E (3) C B D A E (2) C A D B E (2) B E D A C (2) B A C E D (2) A E D C B (2) A C B D E (2) E D A C B (1) D E A C B (1) D B E C A (1) C D A E B (1) C B A D E (1) B E C D A (1) B E C A D (1) B E A D C (1) B A C D E (1) A E B D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -12 2 4 B 8 0 2 16 26 C 12 -2 0 22 18 D -2 -16 -22 0 10 E -4 -26 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 2 4 B 8 0 2 16 26 C 12 -2 0 22 18 D -2 -16 -22 0 10 E -4 -26 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=32 A=28 D=5 E=1 so E is eliminated. Round 2 votes counts: C=34 B=32 A=28 D=6 so D is eliminated. Round 3 votes counts: C=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:225 A:193 D:185 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 2 4 B 8 0 2 16 26 C 12 -2 0 22 18 D -2 -16 -22 0 10 E -4 -26 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 2 4 B 8 0 2 16 26 C 12 -2 0 22 18 D -2 -16 -22 0 10 E -4 -26 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 2 4 B 8 0 2 16 26 C 12 -2 0 22 18 D -2 -16 -22 0 10 E -4 -26 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4117: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) E A C D B (6) A E C D B (6) E C B D A (5) B C D E A (5) A E D C B (5) E C A B D (4) D B C A E (4) B E C D A (4) A D B E C (4) E B C D A (3) E B C A D (3) E A C B D (3) D B A C E (3) C E A D B (3) C B E D A (3) A D E C B (3) A D B C E (3) E C B A D (2) C D A E B (2) B E D C A (2) B D C A E (2) B D A C E (2) A D C E B (2) E C A D B (1) E A B C D (1) C E B D A (1) C D E B A (1) C D B A E (1) C B D E A (1) B E C A D (1) B D E A C (1) B D C E A (1) B D A E C (1) B C E D A (1) A E B D C (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 6 0 0 -4 B -6 0 0 -6 -6 C 0 0 0 14 -14 D 0 6 -14 0 -10 E 4 6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 0 0 -4 B -6 0 0 -6 -6 C 0 0 0 14 -14 D 0 6 -14 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 B=20 D=14 C=12 so C is eliminated. Round 2 votes counts: E=32 A=26 B=24 D=18 so D is eliminated. Round 3 votes counts: A=35 E=33 B=32 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:201 C:200 B:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 0 -4 B -6 0 0 -6 -6 C 0 0 0 14 -14 D 0 6 -14 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 -4 B -6 0 0 -6 -6 C 0 0 0 14 -14 D 0 6 -14 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 -4 B -6 0 0 -6 -6 C 0 0 0 14 -14 D 0 6 -14 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4118: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) A D C E B (8) B E C D A (7) B D C E A (6) A E C D B (6) E C D B A (4) C D E B A (4) A B D C E (4) E C B D A (3) E B C A D (3) D C B E A (3) D C A E B (3) A B E D C (3) E C A D B (2) E A C D B (2) D C B A E (2) C D E A B (2) B D E C A (2) B C D E A (2) B A D C E (2) A D C B E (2) A D B C E (2) E C D A B (1) E A B C D (1) D C E B A (1) D B C E A (1) D A C B E (1) C E D A B (1) C D B E A (1) C D A E B (1) B E D C A (1) B E C A D (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D E C (1) A E D C B (1) A E C B D (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -20 -10 -18 B 12 0 -4 -2 -12 C 20 4 0 8 -4 D 10 2 -8 0 0 E 18 12 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.215853 E: 0.784147 Sum of squares = 0.6614790958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.215853 E: 1.000000 A B C D E A 0 -12 -20 -10 -18 B 12 0 -4 -2 -12 C 20 4 0 8 -4 D 10 2 -8 0 0 E 18 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555646472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 B=25 D=11 C=9 so C is eliminated. Round 2 votes counts: A=30 E=26 B=25 D=19 so D is eliminated. Round 3 votes counts: A=35 E=33 B=32 so B is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:214 D:202 B:197 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -20 -10 -18 B 12 0 -4 -2 -12 C 20 4 0 8 -4 D 10 2 -8 0 0 E 18 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555646472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -10 -18 B 12 0 -4 -2 -12 C 20 4 0 8 -4 D 10 2 -8 0 0 E 18 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555646472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -10 -18 B 12 0 -4 -2 -12 C 20 4 0 8 -4 D 10 2 -8 0 0 E 18 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555646472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4119: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) D A C B E (8) E B C A D (6) E B A D C (6) D C A B E (6) E B A C D (5) A B D C E (5) B A E C D (4) A B D E C (4) A B C D E (4) D C A E B (3) C A B D E (3) A B E D C (3) E D B C A (2) E C B A D (2) E B C D A (2) C E D B A (2) C E B D A (2) C D E A B (2) C D A E B (2) C A D B E (2) B E A C D (2) E C D B A (1) E C B D A (1) E B D A C (1) D E C B A (1) D E C A B (1) D A C E B (1) D A B E C (1) D A B C E (1) C E B A D (1) C D A B E (1) B C E A D (1) B C A E D (1) B A E D C (1) B A C E D (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 16 0 8 8 B -16 0 4 10 2 C 0 -4 0 -8 12 D -8 -10 8 0 12 E -8 -2 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.696854 B: 0.000000 C: 0.303146 D: 0.000000 E: 0.000000 Sum of squares = 0.577503208406 Cumulative probabilities = A: 0.696854 B: 0.696854 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 8 8 B -16 0 4 10 2 C 0 -4 0 -8 12 D -8 -10 8 0 12 E -8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500735 B: 0.000000 C: 0.499265 D: 0.000000 E: 0.000000 Sum of squares = 0.500001079816 Cumulative probabilities = A: 0.500735 B: 0.500735 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=26 A=19 C=15 B=10 so B is eliminated. Round 2 votes counts: D=30 E=28 A=25 C=17 so C is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:201 B:200 C:200 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 8 8 B -16 0 4 10 2 C 0 -4 0 -8 12 D -8 -10 8 0 12 E -8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500735 B: 0.000000 C: 0.499265 D: 0.000000 E: 0.000000 Sum of squares = 0.500001079816 Cumulative probabilities = A: 0.500735 B: 0.500735 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 8 8 B -16 0 4 10 2 C 0 -4 0 -8 12 D -8 -10 8 0 12 E -8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500735 B: 0.000000 C: 0.499265 D: 0.000000 E: 0.000000 Sum of squares = 0.500001079816 Cumulative probabilities = A: 0.500735 B: 0.500735 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 8 8 B -16 0 4 10 2 C 0 -4 0 -8 12 D -8 -10 8 0 12 E -8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500735 B: 0.000000 C: 0.499265 D: 0.000000 E: 0.000000 Sum of squares = 0.500001079816 Cumulative probabilities = A: 0.500735 B: 0.500735 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4120: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (15) E B A C D (7) E A B C D (7) D C B A E (7) B E A C D (5) D E C A B (4) E A C B D (3) D B C E A (3) D B C A E (3) C D A B E (3) B E C A D (3) A C B D E (3) E D A C B (2) E B D C A (2) E B A D C (2) D E A C B (2) D A C E B (2) B C E A D (2) B C A E D (2) A E C B D (2) A C E B D (2) A C D B E (2) E B C A D (1) E A D C B (1) D E C B A (1) D E B C A (1) D C E A B (1) D C A E B (1) D A E C B (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D C A (1) B D C E A (1) B C D A E (1) B C A D E (1) A E B C D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -12 -2 2 B -12 0 -12 -2 12 C 12 12 0 0 6 D 2 2 0 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.374205 D: 0.625795 E: 0.000000 Sum of squares = 0.531648726879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.374205 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 -2 2 B -12 0 -12 -2 12 C 12 12 0 0 6 D 2 2 0 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=25 B=16 A=12 C=6 so C is eliminated. Round 2 votes counts: D=44 E=25 B=17 A=14 so A is eliminated. Round 3 votes counts: D=47 E=31 B=22 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:207 A:200 B:193 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -12 -2 2 B -12 0 -12 -2 12 C 12 12 0 0 6 D 2 2 0 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 -2 2 B -12 0 -12 -2 12 C 12 12 0 0 6 D 2 2 0 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 -2 2 B -12 0 -12 -2 12 C 12 12 0 0 6 D 2 2 0 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4121: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) B D C E A (8) B D E A C (7) B C D A E (7) C E A B D (6) C A E D B (6) A E C D B (6) A C E D B (6) C B A E D (5) A E D C B (5) D E A B C (4) B C D E A (4) C B E A D (3) B D E C A (3) B D C A E (3) E D A B C (2) E A D C B (2) D A E B C (2) C B D E A (2) C B A D E (2) C A B E D (2) E B D A C (1) D C A B E (1) D A B E C (1) C B D A E (1) B E C D A (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 4 -26 4 12 B -4 0 -12 28 2 C 26 12 0 20 30 D -4 -28 -20 0 -8 E -12 -2 -30 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -26 4 12 B -4 0 -12 28 2 C 26 12 0 20 30 D -4 -28 -20 0 -8 E -12 -2 -30 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=34 A=18 D=8 E=5 so E is eliminated. Round 2 votes counts: C=35 B=35 A=20 D=10 so D is eliminated. Round 3 votes counts: C=36 B=35 A=29 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:244 B:207 A:197 E:182 D:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -26 4 12 B -4 0 -12 28 2 C 26 12 0 20 30 D -4 -28 -20 0 -8 E -12 -2 -30 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -26 4 12 B -4 0 -12 28 2 C 26 12 0 20 30 D -4 -28 -20 0 -8 E -12 -2 -30 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -26 4 12 B -4 0 -12 28 2 C 26 12 0 20 30 D -4 -28 -20 0 -8 E -12 -2 -30 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4122: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (15) E D A B C (12) C B A E D (8) A B C D E (7) B C A D E (6) E D C B A (5) E D B C A (4) D E A B C (4) C A B E D (4) A C B E D (4) E C B A D (3) D E B C A (3) D E B A C (3) D B C A E (3) E D B A C (2) E C B D A (2) E A C B D (2) D B A C E (2) D A E B C (2) C B E A D (2) E D A C B (1) E A C D B (1) E A B C D (1) D B A E C (1) D A B E C (1) B D C A E (1) A D B C E (1) Total count = 100 A B C D E A 0 -20 -12 8 10 B 20 0 6 10 10 C 12 -6 0 10 6 D -8 -10 -10 0 -2 E -10 -10 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -12 8 10 B 20 0 6 10 10 C 12 -6 0 10 6 D -8 -10 -10 0 -2 E -10 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=29 D=19 A=12 B=7 so B is eliminated. Round 2 votes counts: C=35 E=33 D=20 A=12 so A is eliminated. Round 3 votes counts: C=46 E=33 D=21 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:223 C:211 A:193 E:188 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -12 8 10 B 20 0 6 10 10 C 12 -6 0 10 6 D -8 -10 -10 0 -2 E -10 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -12 8 10 B 20 0 6 10 10 C 12 -6 0 10 6 D -8 -10 -10 0 -2 E -10 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -12 8 10 B 20 0 6 10 10 C 12 -6 0 10 6 D -8 -10 -10 0 -2 E -10 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4123: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) D C E A B (10) B A E C D (10) B A D E C (10) E C A B D (6) D C E B A (6) D A B C E (6) C E A D B (6) D B A C E (4) A B D E C (4) C E D B A (3) B A E D C (3) A B E C D (3) A B D C E (3) C D E A B (2) A E B C D (2) A D B C E (2) E C D B A (1) E C B A D (1) E B C A D (1) E A C B D (1) D C B E A (1) B E A C D (1) A B E D C (1) Total count = 100 A B C D E A 0 18 0 8 -4 B -18 0 0 -8 -4 C 0 0 0 0 12 D -8 8 0 0 -4 E 4 4 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.475815 B: 0.000000 C: 0.524185 D: 0.000000 E: 0.000000 Sum of squares = 0.501169782575 Cumulative probabilities = A: 0.475815 B: 0.475815 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 8 -4 B -18 0 0 -8 -4 C 0 0 0 0 12 D -8 8 0 0 -4 E 4 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 B=24 A=15 E=10 so E is eliminated. Round 2 votes counts: C=32 D=27 B=25 A=16 so A is eliminated. Round 3 votes counts: B=38 C=33 D=29 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 C:206 E:200 D:198 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 0 8 -4 B -18 0 0 -8 -4 C 0 0 0 0 12 D -8 8 0 0 -4 E 4 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 8 -4 B -18 0 0 -8 -4 C 0 0 0 0 12 D -8 8 0 0 -4 E 4 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 8 -4 B -18 0 0 -8 -4 C 0 0 0 0 12 D -8 8 0 0 -4 E 4 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999958 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4124: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) B A D E C (6) A B C D E (6) E D C B A (5) E B D C A (5) A C B D E (5) C E D A B (4) C D A E B (4) C A D E B (4) B E A D C (4) A B D E C (4) C E D B A (3) B E D C A (3) E D C A B (2) E D B A C (2) E C D B A (2) E B C D A (2) C B E A D (2) C A D B E (2) B E D A C (2) B A E D C (2) B A C E D (2) A D B C E (2) A C D E B (2) A C D B E (2) A B D C E (2) E D B C A (1) E B D A C (1) D E C A B (1) D C E A B (1) D A E C B (1) C E B D A (1) C E A D B (1) C A E B D (1) C A B D E (1) B D E A C (1) B C E A D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 10 -8 2 -4 B -10 0 -8 2 -4 C 8 8 0 6 12 D -2 -2 -6 0 8 E 4 4 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 2 -4 B -10 0 -8 2 -4 C 8 8 0 6 12 D -2 -2 -6 0 8 E 4 4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=25 B=21 E=20 D=3 so D is eliminated. Round 2 votes counts: C=32 A=26 E=21 B=21 so E is eliminated. Round 3 votes counts: C=42 B=32 A=26 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:200 D:199 E:194 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 2 -4 B -10 0 -8 2 -4 C 8 8 0 6 12 D -2 -2 -6 0 8 E 4 4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 2 -4 B -10 0 -8 2 -4 C 8 8 0 6 12 D -2 -2 -6 0 8 E 4 4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 2 -4 B -10 0 -8 2 -4 C 8 8 0 6 12 D -2 -2 -6 0 8 E 4 4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4125: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (13) E A D C B (12) C D B E A (6) B A C D E (6) A E B D C (6) B A C E D (5) E D C A B (4) B A E C D (4) D C E A B (3) B C A D E (3) A E D C B (3) E D C B A (2) E D A C B (2) E A B D C (2) D E C A B (2) D E A C B (2) C D B A E (2) C D A E B (2) C D A B E (2) B C D E A (2) A E C D B (2) A B E C D (2) E D B C A (1) D C A E B (1) C B D E A (1) C B D A E (1) C A B D E (1) B E D C A (1) B E C D A (1) B E A D C (1) B D C E A (1) B C E A D (1) B C A E D (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 2 12 B 4 0 2 6 8 C 2 -2 0 14 4 D -2 -6 -14 0 -2 E -12 -8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 2 12 B 4 0 2 6 8 C 2 -2 0 14 4 D -2 -6 -14 0 -2 E -12 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=23 C=15 A=15 D=8 so D is eliminated. Round 2 votes counts: B=39 E=27 C=19 A=15 so A is eliminated. Round 3 votes counts: B=41 E=39 C=20 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:209 A:204 E:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 2 12 B 4 0 2 6 8 C 2 -2 0 14 4 D -2 -6 -14 0 -2 E -12 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 2 12 B 4 0 2 6 8 C 2 -2 0 14 4 D -2 -6 -14 0 -2 E -12 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 2 12 B 4 0 2 6 8 C 2 -2 0 14 4 D -2 -6 -14 0 -2 E -12 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4126: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) A D C E B (9) E C B D A (6) D A B C E (5) A D C B E (5) A D B E C (5) A D B C E (5) D A C B E (4) C E B D A (4) A D E C B (4) E C B A D (2) E C A B D (2) E B C D A (2) E B C A D (2) E A B C D (2) D C B E A (2) C D A E B (2) C B E D A (2) B E D C A (2) B D A E C (2) B C E D A (2) B C D E A (2) A E C D B (2) A E B D C (2) A D E B C (2) D B C A E (1) D B A C E (1) C E D B A (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E B A (1) C B D E A (1) B D C E A (1) B A E D C (1) A E D B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 10 4 -4 6 B -10 0 0 -4 6 C -4 0 0 -8 2 D 4 4 8 0 6 E -6 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -4 6 B -10 0 0 -4 6 C -4 0 0 -8 2 D 4 4 8 0 6 E -6 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=20 E=16 C=14 D=13 so D is eliminated. Round 2 votes counts: A=46 B=22 E=16 C=16 so E is eliminated. Round 3 votes counts: A=48 C=26 B=26 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:208 B:196 C:195 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -4 6 B -10 0 0 -4 6 C -4 0 0 -8 2 D 4 4 8 0 6 E -6 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -4 6 B -10 0 0 -4 6 C -4 0 0 -8 2 D 4 4 8 0 6 E -6 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -4 6 B -10 0 0 -4 6 C -4 0 0 -8 2 D 4 4 8 0 6 E -6 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4127: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (14) D E A C B (11) C B E A D (11) B C A D E (5) D B A E C (4) D A E B C (4) C E B A D (4) E C A D B (3) E A D C B (3) C B A E D (3) B D A C E (3) B C D A E (3) E A C B D (2) D E C A B (2) D E A B C (2) D C B E A (2) D B C E A (2) D B C A E (2) C E D A B (2) C E A D B (2) B D C A E (2) A E D B C (2) E D C A B (1) E D A C B (1) E A C D B (1) D B E A C (1) D A B E C (1) C E A B D (1) C D E B A (1) C A E B D (1) B C E D A (1) B A C E D (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -24 10 -6 B 18 0 -4 4 10 C 24 4 0 12 20 D -10 -4 -12 0 -8 E 6 -10 -20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -24 10 -6 B 18 0 -4 4 10 C 24 4 0 12 20 D -10 -4 -12 0 -8 E 6 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=29 C=25 E=11 A=4 so A is eliminated. Round 2 votes counts: D=32 B=29 C=25 E=14 so E is eliminated. Round 3 votes counts: D=39 C=32 B=29 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:214 E:192 D:183 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -24 10 -6 B 18 0 -4 4 10 C 24 4 0 12 20 D -10 -4 -12 0 -8 E 6 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -24 10 -6 B 18 0 -4 4 10 C 24 4 0 12 20 D -10 -4 -12 0 -8 E 6 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -24 10 -6 B 18 0 -4 4 10 C 24 4 0 12 20 D -10 -4 -12 0 -8 E 6 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4128: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A B C E (9) A D B E C (7) E B A C D (6) C D E B A (6) C E B D A (5) D C A E B (4) D C A B E (4) D A B E C (4) C D A E B (4) B A E D C (4) A B D E C (4) E C B A D (3) C D E A B (3) B E A C D (3) A D C E B (3) E A B C D (2) D A C B E (2) C E A D B (2) B D A E C (2) A B E D C (2) D C B E A (1) D C B A E (1) D B A E C (1) C E D B A (1) C E D A B (1) C A E D B (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E A C (1) B A D E C (1) Total count = 100 A B C D E A 0 4 4 0 6 B -4 0 18 -8 -2 C -4 -18 0 -2 -6 D 0 8 2 0 14 E -6 2 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.889874 B: 0.000000 C: 0.000000 D: 0.110126 E: 0.000000 Sum of squares = 0.804003177362 Cumulative probabilities = A: 0.889874 B: 0.889874 C: 0.889874 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 6 B -4 0 18 -8 -2 C -4 -18 0 -2 -6 D 0 8 2 0 14 E -6 2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 E=21 A=16 B=14 so B is eliminated. Round 2 votes counts: D=29 E=27 C=23 A=21 so A is eliminated. Round 3 votes counts: D=44 E=33 C=23 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 B:202 E:194 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 6 B -4 0 18 -8 -2 C -4 -18 0 -2 -6 D 0 8 2 0 14 E -6 2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 6 B -4 0 18 -8 -2 C -4 -18 0 -2 -6 D 0 8 2 0 14 E -6 2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 6 B -4 0 18 -8 -2 C -4 -18 0 -2 -6 D 0 8 2 0 14 E -6 2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4129: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) A B D E C (9) D A C B E (8) D A B E C (6) C E D B A (6) B E A C D (5) E C B A D (4) B A E D C (4) A B E D C (4) E C B D A (3) D C E A B (3) D C A E B (3) D A B C E (3) C E B D A (3) C D E B A (3) C D E A B (3) B E C A D (3) B A E C D (3) E C D B A (2) E B C A D (2) E D C A B (1) E D A B C (1) E B A C D (1) D E C A B (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B A D (1) C B E A D (1) B A D E C (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 20 2 14 B -14 0 10 -8 22 C -20 -10 0 -20 -24 D -2 8 20 0 12 E -14 -22 24 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 2 14 B -14 0 10 -8 22 C -20 -10 0 -20 -24 D -2 8 20 0 12 E -14 -22 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 C=17 B=17 E=14 so E is eliminated. Round 2 votes counts: D=29 C=26 A=25 B=20 so B is eliminated. Round 3 votes counts: A=40 C=31 D=29 so D is eliminated. Round 4 votes counts: A=60 C=40 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:219 B:205 E:188 C:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 2 14 B -14 0 10 -8 22 C -20 -10 0 -20 -24 D -2 8 20 0 12 E -14 -22 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 2 14 B -14 0 10 -8 22 C -20 -10 0 -20 -24 D -2 8 20 0 12 E -14 -22 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 2 14 B -14 0 10 -8 22 C -20 -10 0 -20 -24 D -2 8 20 0 12 E -14 -22 24 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4130: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) D B E A C (5) C E A B D (5) C B E A D (5) A D C E B (5) C B A E D (4) B C E D A (4) A E C D B (4) E A B C D (3) C B D A E (3) B E C D A (3) B C D E A (3) E B C A D (2) D C B A E (2) D B C E A (2) D A E B C (2) D A C E B (2) D A B E C (2) C D A B E (2) C B E D A (2) C B D E A (2) C A B E D (2) B E D C A (2) B E C A D (2) A E D B C (2) A E C B D (2) A C E B D (2) A C D E B (2) E C B A D (1) E C A B D (1) E B D A C (1) E B A D C (1) E A B D C (1) D E B A C (1) D E A B C (1) D C A B E (1) D B E C A (1) D A C B E (1) D A B C E (1) C D B A E (1) C A D E B (1) B E D A C (1) B D E C A (1) B D C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 0 -20 6 -2 B 0 0 -16 22 6 C 20 16 0 26 20 D -6 -22 -26 0 -16 E 2 -6 -20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 6 -2 B 0 0 -16 22 6 C 20 16 0 26 20 D -6 -22 -26 0 -16 E 2 -6 -20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=21 A=18 B=17 E=10 so E is eliminated. Round 2 votes counts: C=36 A=22 D=21 B=21 so D is eliminated. Round 3 votes counts: C=39 A=31 B=30 so B is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:241 B:206 E:196 A:192 D:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -20 6 -2 B 0 0 -16 22 6 C 20 16 0 26 20 D -6 -22 -26 0 -16 E 2 -6 -20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 6 -2 B 0 0 -16 22 6 C 20 16 0 26 20 D -6 -22 -26 0 -16 E 2 -6 -20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 6 -2 B 0 0 -16 22 6 C 20 16 0 26 20 D -6 -22 -26 0 -16 E 2 -6 -20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4131: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) C B A E D (10) D E B C A (9) D E A B C (8) A C B E D (7) D E B A C (6) A C B D E (6) D E C B A (3) D E A C B (3) A D E C B (3) A D E B C (3) A D C E B (3) A D C B E (3) E B C D A (2) D A E B C (2) C B E D A (2) B E C D A (2) B C E A D (2) A C D B E (2) A B C E D (2) E D A B C (1) E B D C A (1) D C E B A (1) D A C E B (1) C A B E D (1) C A B D E (1) B E C A D (1) B C E D A (1) A E D B C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 6 -6 -6 B 2 0 8 -22 -16 C -6 -8 0 -20 -14 D 6 22 20 0 10 E 6 16 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -6 -6 B 2 0 8 -22 -16 C -6 -8 0 -20 -14 D 6 22 20 0 10 E 6 16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=32 E=15 C=14 B=6 so B is eliminated. Round 2 votes counts: D=33 A=32 E=18 C=17 so C is eliminated. Round 3 votes counts: A=44 D=33 E=23 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:213 A:196 B:186 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -6 -6 B 2 0 8 -22 -16 C -6 -8 0 -20 -14 D 6 22 20 0 10 E 6 16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -6 -6 B 2 0 8 -22 -16 C -6 -8 0 -20 -14 D 6 22 20 0 10 E 6 16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -6 -6 B 2 0 8 -22 -16 C -6 -8 0 -20 -14 D 6 22 20 0 10 E 6 16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4132: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) E C B A D (7) D C B E A (7) A E C B D (6) D A C E B (4) D A B C E (4) B C E D A (4) A D B C E (4) E C B D A (3) E A C B D (3) E A B C D (3) D B C E A (3) D B C A E (3) A E C D B (3) A D E C B (3) E C A B D (2) E A C D B (2) D C E B A (2) D B A C E (2) C E B D A (2) B D C A E (2) B C D E A (2) A E D C B (2) A D E B C (2) A D B E C (2) A B D C E (2) E B C A D (1) D A E C B (1) C E D B A (1) C D E B A (1) C D B E A (1) B E C D A (1) B E A C D (1) B D C E A (1) B D A C E (1) B A D C E (1) A E D B C (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 14 10 6 B -8 0 0 4 -16 C -14 0 0 4 -6 D -10 -4 -4 0 -4 E -6 16 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 10 6 B -8 0 0 4 -16 C -14 0 0 4 -6 D -10 -4 -4 0 -4 E -6 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=26 E=21 B=13 C=5 so C is eliminated. Round 2 votes counts: A=35 D=28 E=24 B=13 so B is eliminated. Round 3 votes counts: A=36 D=34 E=30 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:210 C:192 B:190 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 10 6 B -8 0 0 4 -16 C -14 0 0 4 -6 D -10 -4 -4 0 -4 E -6 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 10 6 B -8 0 0 4 -16 C -14 0 0 4 -6 D -10 -4 -4 0 -4 E -6 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 10 6 B -8 0 0 4 -16 C -14 0 0 4 -6 D -10 -4 -4 0 -4 E -6 16 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4133: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B D C A (6) C A D E B (6) B E A D C (6) E C D B A (5) D C E A B (4) C D E A B (4) C A D B E (4) B A E D C (4) A C B D E (4) D E B C A (3) C D A E B (3) B A E C D (3) E D B C A (2) E B C D A (2) E B C A D (2) E B A C D (2) D E C B A (2) D C A E B (2) B E D A C (2) B A D E C (2) A C B E D (2) A B D C E (2) A B C E D (2) A B C D E (2) E D C B A (1) E B A D C (1) D C E B A (1) D C A B E (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C E A B D (1) C D A B E (1) B E A C D (1) A D B C E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -6 10 4 B -2 0 -2 -8 4 C 6 2 0 8 4 D -10 8 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 10 4 B -2 0 -2 -8 4 C 6 2 0 8 4 D -10 8 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=21 C=20 D=18 B=18 so D is eliminated. Round 2 votes counts: C=28 E=26 A=24 B=22 so B is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:210 A:205 D:200 B:196 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 10 4 B -2 0 -2 -8 4 C 6 2 0 8 4 D -10 8 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 10 4 B -2 0 -2 -8 4 C 6 2 0 8 4 D -10 8 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 10 4 B -2 0 -2 -8 4 C 6 2 0 8 4 D -10 8 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4134: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) B E D A C (7) D C A B E (6) B E D C A (6) E B C A D (4) D C A E B (4) C A D E B (4) B E A C D (4) E B D C A (3) C D A E B (3) C A E D B (3) A C D E B (3) E D C B A (2) E D B C A (2) E C D B A (2) E C A B D (2) E B A C D (2) E A B C D (2) D C E A B (2) D B C A E (2) D A C B E (2) B E A D C (2) A C E D B (2) A C D B E (2) A B C E D (2) E C A D B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B A C E (1) D A C E B (1) C E A D B (1) B E C A D (1) B D E C A (1) B D E A C (1) B D A E C (1) B A E C D (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B C E (1) A C E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -20 -14 -12 B 6 0 8 -12 4 C 20 -8 0 -12 -10 D 14 12 12 0 -6 E 12 -4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.181818 E: 0.545455 Sum of squares = 0.404958677745 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.454545 E: 1.000000 A B C D E A 0 -6 -20 -14 -12 B 6 0 8 -12 4 C 20 -8 0 -12 -10 D 14 12 12 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.181818 E: 0.545455 Sum of squares = 0.404958677634 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.454545 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=24 E=20 A=16 C=11 so C is eliminated. Round 2 votes counts: D=32 B=24 A=23 E=21 so E is eliminated. Round 3 votes counts: D=38 B=33 A=29 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:212 B:203 C:195 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -20 -14 -12 B 6 0 8 -12 4 C 20 -8 0 -12 -10 D 14 12 12 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.181818 E: 0.545455 Sum of squares = 0.404958677634 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.454545 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -14 -12 B 6 0 8 -12 4 C 20 -8 0 -12 -10 D 14 12 12 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.181818 E: 0.545455 Sum of squares = 0.404958677634 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.454545 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -14 -12 B 6 0 8 -12 4 C 20 -8 0 -12 -10 D 14 12 12 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.181818 E: 0.545455 Sum of squares = 0.404958677634 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.454545 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4135: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) E C A D B (7) A B D C E (7) E C B D A (6) A D B E C (5) A D B C E (5) E C D B A (4) B A D C E (4) A D E B C (4) E C D A B (3) E C A B D (3) D B A C E (3) E C B A D (2) E A C B D (2) D E C B A (2) D A E B C (2) D A B E C (2) D A B C E (2) C E D B A (2) B C E D A (2) B C D E A (2) B A C E D (2) A E C B D (2) D E A C B (1) D C E B A (1) D C B E A (1) D B C E A (1) D B C A E (1) C E B A D (1) C D B E A (1) C B E D A (1) C B E A D (1) B D C E A (1) B C E A D (1) B A C D E (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -6 4 -10 B 0 0 -4 0 -12 C 6 4 0 12 2 D -4 0 -12 0 -6 E 10 12 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 4 -10 B 0 0 -4 0 -12 C 6 4 0 12 2 D -4 0 -12 0 -6 E 10 12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 D=16 C=16 B=13 so B is eliminated. Round 2 votes counts: A=35 E=27 C=21 D=17 so D is eliminated. Round 3 votes counts: A=44 E=30 C=26 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 C:212 A:194 B:192 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 4 -10 B 0 0 -4 0 -12 C 6 4 0 12 2 D -4 0 -12 0 -6 E 10 12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 4 -10 B 0 0 -4 0 -12 C 6 4 0 12 2 D -4 0 -12 0 -6 E 10 12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 4 -10 B 0 0 -4 0 -12 C 6 4 0 12 2 D -4 0 -12 0 -6 E 10 12 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4136: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (7) D E B C A (6) B C E A D (6) A C E B D (6) D E A C B (4) D A E C B (4) C B A E D (4) C A B E D (4) B E C D A (4) A C B D E (4) E A D C B (3) D E A B C (3) D B E C A (3) B E C A D (3) A D C E B (3) A C D E B (3) E B C D A (2) E B C A D (2) E A C D B (2) D E B A C (2) D B A C E (2) B D E C A (2) B D C E A (2) A C E D B (2) A C D B E (2) A C B E D (2) E D C B A (1) E C B A D (1) E B D C A (1) E A C B D (1) D B E A C (1) D A E B C (1) D A C E B (1) C E A B D (1) B C E D A (1) A E C D B (1) A D C B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 20 -2 B 0 0 0 10 0 C 0 0 0 18 6 D -20 -10 -18 0 -8 E 2 0 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.093634 B: 0.590434 C: 0.315931 D: 0.000000 E: 0.000000 Sum of squares = 0.457192748198 Cumulative probabilities = A: 0.093634 B: 0.684069 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 20 -2 B 0 0 0 10 0 C 0 0 0 18 6 D -20 -10 -18 0 -8 E 2 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 B=25 E=13 C=9 so C is eliminated. Round 2 votes counts: A=30 B=29 D=27 E=14 so E is eliminated. Round 3 votes counts: A=37 B=35 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:209 B:205 E:202 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 20 -2 B 0 0 0 10 0 C 0 0 0 18 6 D -20 -10 -18 0 -8 E 2 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 20 -2 B 0 0 0 10 0 C 0 0 0 18 6 D -20 -10 -18 0 -8 E 2 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 20 -2 B 0 0 0 10 0 C 0 0 0 18 6 D -20 -10 -18 0 -8 E 2 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333313 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4137: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (11) E A B D C (7) C D B A E (7) C B D A E (6) B D C E A (6) E A D B C (5) E A B C D (4) D B E C A (4) C A E D B (4) B C D E A (4) D E A B C (3) D C B E A (3) D B E A C (3) C A E B D (3) A E C B D (3) A E B C D (3) D C B A E (2) C D A E B (2) B E A C D (2) B D E C A (2) A E D C B (2) A E C D B (2) E D B A C (1) E B A D C (1) E A D C B (1) D E B A C (1) D C A E B (1) C A D B E (1) C A B E D (1) B C E D A (1) B C A E D (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -18 -14 -18 B 10 0 18 -8 10 C 18 -18 0 -6 12 D 14 8 6 0 12 E 18 -10 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -14 -18 B 10 0 18 -8 10 C 18 -18 0 -6 12 D 14 8 6 0 12 E 18 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 E=19 B=16 A=13 so A is eliminated. Round 2 votes counts: E=29 D=28 C=27 B=16 so B is eliminated. Round 3 votes counts: D=36 C=33 E=31 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:215 C:203 E:192 A:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -18 -14 -18 B 10 0 18 -8 10 C 18 -18 0 -6 12 D 14 8 6 0 12 E 18 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -14 -18 B 10 0 18 -8 10 C 18 -18 0 -6 12 D 14 8 6 0 12 E 18 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -14 -18 B 10 0 18 -8 10 C 18 -18 0 -6 12 D 14 8 6 0 12 E 18 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4138: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) C A B E D (7) A B C E D (7) E B A D C (6) D C E B A (6) D C A B E (6) C A B D E (6) D E C B A (5) D C E A B (5) C D A B E (5) B E A C D (5) A C B E D (5) B A E C D (4) A B E C D (4) E B D A C (2) D C A E B (2) B E A D C (2) E D B C A (1) E D B A C (1) E C B A D (1) E B A C D (1) D E B A C (1) D A B E C (1) C E B A D (1) C D E B A (1) B E C A D (1) B D E A C (1) B A E D C (1) A D C B E (1) A C D B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -10 10 4 B -4 0 -4 12 18 C 10 4 0 -2 6 D -10 -12 2 0 0 E -4 -18 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.50617283959 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 10 4 B -4 0 -4 12 18 C 10 4 0 -2 6 D -10 -12 2 0 0 E -4 -18 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839508 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=20 A=20 B=14 E=12 so E is eliminated. Round 2 votes counts: D=36 B=23 C=21 A=20 so A is eliminated. Round 3 votes counts: D=37 B=36 C=27 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:209 A:204 D:190 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 10 4 B -4 0 -4 12 18 C 10 4 0 -2 6 D -10 -12 2 0 0 E -4 -18 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839508 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 10 4 B -4 0 -4 12 18 C 10 4 0 -2 6 D -10 -12 2 0 0 E -4 -18 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839508 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 10 4 B -4 0 -4 12 18 C 10 4 0 -2 6 D -10 -12 2 0 0 E -4 -18 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839508 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4139: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B D E A C (7) A C D B E (7) D B A C E (6) E C B A D (5) A C E D B (5) E C B D A (4) E B D C A (4) B D E C A (4) E A C B D (3) D B E C A (3) C E A D B (3) A C E B D (3) E C A D B (2) E C A B D (2) E B C D A (2) E B C A D (2) C A D E B (2) B E D C A (2) B D A E C (2) A D B C E (2) E C D B A (1) E B A D C (1) D E B C A (1) D C E B A (1) D B E A C (1) D B C E A (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E A B (1) C D A E B (1) C A E B D (1) B E D A C (1) B D A C E (1) B A E D C (1) B A D E C (1) B A D C E (1) A E C B D (1) A D C B E (1) A C D E B (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 8 -4 B 4 0 -8 0 -10 C 2 8 0 14 -2 D -8 0 -14 0 -6 E 4 10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 8 -4 B 4 0 -8 0 -10 C 2 8 0 14 -2 D -8 0 -14 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 B=20 C=17 D=15 so D is eliminated. Round 2 votes counts: B=31 E=27 A=24 C=18 so C is eliminated. Round 3 votes counts: A=36 E=33 B=31 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:211 E:211 A:199 B:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 8 -4 B 4 0 -8 0 -10 C 2 8 0 14 -2 D -8 0 -14 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 8 -4 B 4 0 -8 0 -10 C 2 8 0 14 -2 D -8 0 -14 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 8 -4 B 4 0 -8 0 -10 C 2 8 0 14 -2 D -8 0 -14 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4140: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C A E B D (7) C A D B E (6) B E D C A (6) A C D B E (6) A C D E B (5) E B D C A (4) E B C A D (4) C A D E B (4) B D E A C (4) D B E A C (3) B D E C A (3) B C E D A (3) A C E D B (3) D B A E C (2) D B A C E (2) D A C B E (2) D A B C E (2) C B D E A (2) C A E D B (2) A D C E B (2) A D C B E (2) E D A B C (1) E C B A D (1) E B C D A (1) E A D B C (1) E A B C D (1) D C A B E (1) D B E C A (1) D B C A E (1) D A B E C (1) C E B A D (1) C D B A E (1) C B E D A (1) C B E A D (1) C A B E D (1) C A B D E (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -4 0 6 B 0 0 0 2 4 C 4 0 0 4 14 D 0 -2 -4 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.632506 C: 0.367493 D: 0.000000 E: 0.000000 Sum of squares = 0.535115920175 Cumulative probabilities = A: 0.000000 B: 0.632507 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 0 6 B 0 0 0 2 4 C 4 0 0 4 14 D 0 -2 -4 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 A=20 B=16 D=15 so D is eliminated. Round 2 votes counts: C=28 B=25 A=25 E=22 so E is eliminated. Round 3 votes counts: B=43 C=29 A=28 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:203 A:201 D:199 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 0 6 B 0 0 0 2 4 C 4 0 0 4 14 D 0 -2 -4 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 0 6 B 0 0 0 2 4 C 4 0 0 4 14 D 0 -2 -4 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 0 6 B 0 0 0 2 4 C 4 0 0 4 14 D 0 -2 -4 0 4 E -6 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4141: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) D A C B E (7) B A C E D (7) E C A D B (6) D C A E B (6) D E C A B (5) E C D A B (4) E B C A D (4) B E D C A (4) B E A C D (4) C A E D B (3) B A C D E (3) A C E B D (3) E C A B D (2) E B D C A (2) D E B C A (2) D A C E B (2) B E C A D (2) B E A D C (2) B D A C E (2) B A E C D (2) A C D B E (2) A C B D E (2) A B C D E (2) E D B C A (1) E C B A D (1) D C E A B (1) D B E C A (1) D B A C E (1) D A B C E (1) C E A D B (1) C D A E B (1) B D E C A (1) B D E A C (1) B D A E C (1) B A D C E (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 16 -8 0 -2 B -16 0 -8 -4 -4 C 8 8 0 2 -4 D 0 4 -2 0 -12 E 2 4 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -8 0 -2 B -16 0 -8 -4 -4 C 8 8 0 2 -4 D 0 4 -2 0 -12 E 2 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 D=26 A=12 C=5 so C is eliminated. Round 2 votes counts: B=30 E=28 D=27 A=15 so A is eliminated. Round 3 votes counts: B=36 E=34 D=30 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:207 A:203 D:195 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -8 0 -2 B -16 0 -8 -4 -4 C 8 8 0 2 -4 D 0 4 -2 0 -12 E 2 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 0 -2 B -16 0 -8 -4 -4 C 8 8 0 2 -4 D 0 4 -2 0 -12 E 2 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 0 -2 B -16 0 -8 -4 -4 C 8 8 0 2 -4 D 0 4 -2 0 -12 E 2 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4142: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) C B A E D (9) E D A B C (6) E A C B D (6) B C A D E (6) A B C E D (6) E D A C B (5) C A B E D (4) B A C D E (4) D E B C A (3) D E B A C (3) D B A C E (3) C B A D E (3) E C D A B (2) E C A B D (2) E A C D B (2) D E A C B (2) D B C A E (2) C A E B D (2) B C D A E (2) A C B E D (2) E D C A B (1) E C A D B (1) D B E C A (1) D B E A C (1) D B A E C (1) D A B E C (1) C E D B A (1) C E A B D (1) C B D A E (1) A E C B D (1) A E B D C (1) A D B E C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 20 18 6 2 B -20 0 10 4 -2 C -18 -10 0 14 -6 D -6 -4 -14 0 -6 E -2 2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 6 2 B -20 0 10 4 -2 C -18 -10 0 14 -6 D -6 -4 -14 0 -6 E -2 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 C=21 A=13 B=12 so B is eliminated. Round 2 votes counts: D=29 C=29 E=25 A=17 so A is eliminated. Round 3 votes counts: C=42 D=30 E=28 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:223 E:206 B:196 C:190 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 6 2 B -20 0 10 4 -2 C -18 -10 0 14 -6 D -6 -4 -14 0 -6 E -2 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 6 2 B -20 0 10 4 -2 C -18 -10 0 14 -6 D -6 -4 -14 0 -6 E -2 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 6 2 B -20 0 10 4 -2 C -18 -10 0 14 -6 D -6 -4 -14 0 -6 E -2 2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4143: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (11) C B E D A (10) C E D B A (7) D E A C B (6) B A C E D (6) B A E D C (5) C E B D A (4) A B D E C (4) D A E C B (3) C D E B A (3) C B A D E (3) B C E D A (3) B C E A D (3) B C A E D (3) A D B E C (3) E C B D A (2) C D E A B (2) B E A C D (2) B C A D E (2) B A E C D (2) A D E C B (2) E D C A B (1) E D A B C (1) E C D B A (1) E B C D A (1) D E A B C (1) D A C E B (1) C B D E A (1) C B A E D (1) B E C A D (1) B A C D E (1) A D C E B (1) A D B C E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -22 4 6 2 B 22 0 2 12 6 C -4 -2 0 18 6 D -6 -12 -18 0 -8 E -2 -6 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 4 6 2 B 22 0 2 12 6 C -4 -2 0 18 6 D -6 -12 -18 0 -8 E -2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=28 A=24 D=11 E=6 so E is eliminated. Round 2 votes counts: C=34 B=29 A=24 D=13 so D is eliminated. Round 3 votes counts: A=36 C=35 B=29 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:221 C:209 E:197 A:195 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 4 6 2 B 22 0 2 12 6 C -4 -2 0 18 6 D -6 -12 -18 0 -8 E -2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 4 6 2 B 22 0 2 12 6 C -4 -2 0 18 6 D -6 -12 -18 0 -8 E -2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 4 6 2 B 22 0 2 12 6 C -4 -2 0 18 6 D -6 -12 -18 0 -8 E -2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4144: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (13) D B E A C (11) D B A E C (7) C A E B D (7) A B D C E (5) E B D A C (4) D E B C A (4) D C E B A (4) E D B C A (3) E C D B A (3) C E A D B (3) B D A E C (3) A D B C E (3) D A B C E (2) C E D B A (2) C E D A B (2) C A B E D (2) B A E D C (2) B A D E C (2) A D C B E (2) A C B D E (2) A B D E C (2) E D C B A (1) E D B A C (1) E C B D A (1) D E C B A (1) D E B A C (1) D C E A B (1) C A E D B (1) C A D B E (1) C A B D E (1) B D E A C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 -4 -12 B 2 0 4 -6 -4 C 0 -4 0 -20 6 D 4 6 20 0 6 E 12 4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -4 -12 B 2 0 4 -6 -4 C 0 -4 0 -20 6 D 4 6 20 0 6 E 12 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=31 A=16 E=13 B=8 so B is eliminated. Round 2 votes counts: D=35 C=32 A=20 E=13 so E is eliminated. Round 3 votes counts: D=44 C=36 A=20 so A is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:202 B:198 A:191 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -4 -12 B 2 0 4 -6 -4 C 0 -4 0 -20 6 D 4 6 20 0 6 E 12 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 -12 B 2 0 4 -6 -4 C 0 -4 0 -20 6 D 4 6 20 0 6 E 12 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 -12 B 2 0 4 -6 -4 C 0 -4 0 -20 6 D 4 6 20 0 6 E 12 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4145: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (14) A E D B C (14) C E B D A (7) C B D A E (7) A D B E C (7) C B E D A (6) E D B A C (5) E B D C A (5) A C D B E (5) A D E B C (4) A D B C E (4) E A D B C (3) C A B D E (3) E D A B C (2) E A C D B (2) D B E A C (2) B D E C A (2) C E A B D (1) C A E D B (1) C A E B D (1) C A D B E (1) B D E A C (1) B C D A E (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 2 -4 0 B 0 0 0 -4 6 C -2 0 0 0 4 D 4 4 0 0 2 E 0 -6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.437681 D: 0.562319 E: 0.000000 Sum of squares = 0.507767365292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.437681 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -4 0 B 0 0 0 -4 6 C -2 0 0 0 4 D 4 4 0 0 2 E 0 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=36 E=17 B=4 D=2 so D is eliminated. Round 2 votes counts: C=41 A=36 E=17 B=6 so B is eliminated. Round 3 votes counts: C=42 A=36 E=22 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:205 B:201 C:201 A:199 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -4 0 B 0 0 0 -4 6 C -2 0 0 0 4 D 4 4 0 0 2 E 0 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -4 0 B 0 0 0 -4 6 C -2 0 0 0 4 D 4 4 0 0 2 E 0 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -4 0 B 0 0 0 -4 6 C -2 0 0 0 4 D 4 4 0 0 2 E 0 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4146: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) B E C D A (8) E C B A D (7) E B C A D (6) D A C B E (5) C A D E B (5) B D A E C (5) B E D C A (4) A D C E B (4) A D C B E (4) A C D E B (4) D C A B E (3) B E C A D (3) B D E C A (3) C E A D B (2) C D A E B (2) C A E D B (2) B E D A C (2) B E A D C (2) A C E D B (2) E C B D A (1) E C A B D (1) E B C D A (1) E A B C D (1) D C A E B (1) D B A E C (1) D A C E B (1) C E B D A (1) C E A B D (1) B E A C D (1) B D E A C (1) B A D E C (1) A E C D B (1) A E B C D (1) A D E C B (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -2 2 10 B -6 0 4 2 10 C 2 -4 0 0 -6 D -2 -2 0 0 6 E -10 -10 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888884 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 2 10 B -6 0 4 2 10 C 2 -4 0 0 -6 D -2 -2 0 0 6 E -10 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888841 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=21 A=19 E=17 C=13 so C is eliminated. Round 2 votes counts: B=30 A=26 D=23 E=21 so E is eliminated. Round 3 votes counts: B=46 A=31 D=23 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:205 D:201 C:196 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 2 10 B -6 0 4 2 10 C 2 -4 0 0 -6 D -2 -2 0 0 6 E -10 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888841 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 2 10 B -6 0 4 2 10 C 2 -4 0 0 -6 D -2 -2 0 0 6 E -10 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888841 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 2 10 B -6 0 4 2 10 C 2 -4 0 0 -6 D -2 -2 0 0 6 E -10 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888841 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4147: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) C A B E D (8) B D C A E (8) C B A D E (7) E D A C B (5) C A E B D (5) B C A D E (5) E A C D B (4) D B E A C (4) A C B E D (4) E A D C B (3) D E B C A (3) D E A B C (3) D B A E C (3) A E C B D (3) E D A B C (2) E C A D B (2) C E A B D (2) B C D A E (2) A C E B D (2) E D C B A (1) E D C A B (1) E C D A B (1) E A C B D (1) D E C B A (1) D B E C A (1) D B C E A (1) C E A D B (1) C B D A E (1) B D A E C (1) B D A C E (1) A E B C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 0 6 B 0 0 -4 6 -4 C 0 4 0 2 -2 D 0 -6 -2 0 6 E -6 4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.478694 B: 0.000000 C: 0.521306 D: 0.000000 E: 0.000000 Sum of squares = 0.500907857253 Cumulative probabilities = A: 0.478694 B: 0.478694 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 0 6 B 0 0 -4 6 -4 C 0 4 0 2 -2 D 0 -6 -2 0 6 E -6 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 E=20 B=17 A=12 so A is eliminated. Round 2 votes counts: C=30 D=27 E=24 B=19 so B is eliminated. Round 3 votes counts: C=39 D=37 E=24 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:203 C:202 B:199 D:199 E:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 0 6 B 0 0 -4 6 -4 C 0 4 0 2 -2 D 0 -6 -2 0 6 E -6 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 6 B 0 0 -4 6 -4 C 0 4 0 2 -2 D 0 -6 -2 0 6 E -6 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 6 B 0 0 -4 6 -4 C 0 4 0 2 -2 D 0 -6 -2 0 6 E -6 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4148: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) A E B D C (7) E A B D C (6) C E D B A (6) C D B E A (6) A B D C E (6) E A C B D (5) D C B A E (5) B D C A E (4) B D A C E (4) E C D A B (3) E C A D B (3) E C A B D (3) E A B C D (3) D B A C E (3) C D B A E (3) C B D A E (3) B A D C E (3) A B E D C (3) A B D E C (3) E A C D B (2) D B C A E (2) A D B C E (2) E C B D A (1) D E C B A (1) D C E B A (1) D A B C E (1) C D E B A (1) C B A D E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 -2 4 B 2 0 -2 6 2 C 0 2 0 -4 4 D 2 -6 4 0 2 E -4 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888749 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -2 4 B 2 0 -2 6 2 C 0 2 0 -4 4 D 2 -6 4 0 2 E -4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=23 C=20 D=13 B=11 so B is eliminated. Round 2 votes counts: E=33 A=26 D=21 C=20 so C is eliminated. Round 3 votes counts: E=39 D=34 A=27 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:204 C:201 D:201 A:200 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -2 4 B 2 0 -2 6 2 C 0 2 0 -4 4 D 2 -6 4 0 2 E -4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -2 4 B 2 0 -2 6 2 C 0 2 0 -4 4 D 2 -6 4 0 2 E -4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -2 4 B 2 0 -2 6 2 C 0 2 0 -4 4 D 2 -6 4 0 2 E -4 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4149: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (8) D E C B A (6) B C A D E (6) E D A C B (5) D E A B C (5) C B D A E (5) C B A E D (5) A B C E D (5) D E B A C (4) D C E B A (4) B C A E D (4) A B E C D (4) D E B C A (3) C A B E D (3) B A C E D (3) A E D B C (3) A E B C D (3) E A D B C (2) D E A C B (2) C D B E A (2) B C D E A (2) D C B E A (1) C B D E A (1) B D E A C (1) B C D A E (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -2 -4 4 B 12 0 14 6 4 C 2 -14 0 6 2 D 4 -6 -6 0 2 E -4 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -4 4 B 12 0 14 6 4 C 2 -14 0 6 2 D 4 -6 -6 0 2 E -4 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 E=17 B=17 A=17 so E is eliminated. Round 2 votes counts: D=40 C=24 A=19 B=17 so B is eliminated. Round 3 votes counts: D=41 C=37 A=22 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:198 D:197 E:194 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 -4 4 B 12 0 14 6 4 C 2 -14 0 6 2 D 4 -6 -6 0 2 E -4 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -4 4 B 12 0 14 6 4 C 2 -14 0 6 2 D 4 -6 -6 0 2 E -4 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -4 4 B 12 0 14 6 4 C 2 -14 0 6 2 D 4 -6 -6 0 2 E -4 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4150: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) B C E A D (11) A E B D C (4) A D E B C (4) D A E B C (3) D A B E C (3) C E B A D (3) C E A D B (3) C D E A B (3) B C D E A (3) B C D A E (3) B C A E D (3) B A D E C (3) E D A C B (2) E C A D B (2) E A D C B (2) D E C A B (2) D C B A E (2) D C A E B (2) C B E D A (2) C B D E A (2) B E C A D (2) B E A C D (2) B D A E C (2) B D A C E (2) B C E D A (2) B A E D C (2) A E D C B (2) A E D B C (2) A D E C B (2) E A B C D (1) D C E A B (1) D B A E C (1) D B A C E (1) D A C E B (1) C D E B A (1) C B E A D (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 4 2 10 B -2 0 12 0 -2 C -4 -12 0 -10 -8 D -2 0 10 0 4 E -10 2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 2 10 B -2 0 12 0 -2 C -4 -12 0 -10 -8 D -2 0 10 0 4 E -10 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=27 C=15 A=15 E=7 so E is eliminated. Round 2 votes counts: B=36 D=29 A=18 C=17 so C is eliminated. Round 3 votes counts: B=44 D=33 A=23 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:209 D:206 B:204 E:198 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 2 10 B -2 0 12 0 -2 C -4 -12 0 -10 -8 D -2 0 10 0 4 E -10 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 10 B -2 0 12 0 -2 C -4 -12 0 -10 -8 D -2 0 10 0 4 E -10 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 10 B -2 0 12 0 -2 C -4 -12 0 -10 -8 D -2 0 10 0 4 E -10 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4151: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) D C E B A (9) C D A E B (7) E B C D A (6) A C D B E (5) D B E C A (4) C A D E B (4) B E A C D (4) A D C B E (4) A C D E B (4) E B D C A (3) D C A E B (3) C D E B A (3) E C B A D (2) D C B E A (2) D C A B E (2) D A C B E (2) C D E A B (2) B A E D C (2) A C B E D (2) A B E D C (2) A B E C D (2) E B C A D (1) E B A C D (1) E A B C D (1) D C B A E (1) D A C E B (1) D A B E C (1) C E D B A (1) C A E B D (1) B E D A C (1) B D E C A (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -32 -26 -10 B 10 0 -10 -10 0 C 32 10 0 -4 8 D 26 10 4 0 10 E 10 0 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -32 -26 -10 B 10 0 -10 -10 0 C 32 10 0 -4 8 D 26 10 4 0 10 E 10 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=22 A=21 C=18 E=14 so E is eliminated. Round 2 votes counts: B=33 D=25 A=22 C=20 so C is eliminated. Round 3 votes counts: D=38 B=35 A=27 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 C:223 E:196 B:195 A:161 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -32 -26 -10 B 10 0 -10 -10 0 C 32 10 0 -4 8 D 26 10 4 0 10 E 10 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -32 -26 -10 B 10 0 -10 -10 0 C 32 10 0 -4 8 D 26 10 4 0 10 E 10 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -32 -26 -10 B 10 0 -10 -10 0 C 32 10 0 -4 8 D 26 10 4 0 10 E 10 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4152: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) C D B A E (8) C A D B E (5) A E C D B (5) E B D C A (4) E A C B D (4) A E D C B (4) A D C B E (4) A C D E B (4) E B C D A (3) D C B A E (3) D B C E A (3) C D A B E (3) B D C E A (3) A E D B C (3) E B C A D (2) E B A D C (2) E A D B C (2) E A B D C (2) E A B C D (2) D B E A C (2) D B C A E (2) C B D E A (2) C B D A E (2) A E C B D (2) A D C E B (2) E C B A D (1) D C A B E (1) D B E C A (1) D A C B E (1) C E B D A (1) C B E D A (1) B E D C A (1) B E C D A (1) B D E C A (1) B C D E A (1) A D E C B (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 12 2 12 22 B -12 0 -28 -30 10 C -2 28 0 14 14 D -12 30 -14 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 12 22 B -12 0 -28 -30 10 C -2 28 0 14 14 D -12 30 -14 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=22 C=22 D=13 B=7 so B is eliminated. Round 2 votes counts: A=36 E=24 C=23 D=17 so D is eliminated. Round 3 votes counts: A=37 C=35 E=28 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:227 A:224 D:211 B:170 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 12 22 B -12 0 -28 -30 10 C -2 28 0 14 14 D -12 30 -14 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 12 22 B -12 0 -28 -30 10 C -2 28 0 14 14 D -12 30 -14 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 12 22 B -12 0 -28 -30 10 C -2 28 0 14 14 D -12 30 -14 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4153: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C A D (9) B E A C D (8) A D E B C (7) A D B E C (7) C E B D A (5) D C A E B (3) D A B C E (3) C E D B A (3) C D E A B (3) B C E D A (3) A D E C B (3) E B C A D (2) D C A B E (2) D A C B E (2) C D E B A (2) C D B E A (2) C B E D A (2) B D A E C (2) B A E D C (2) B A E C D (2) A E B D C (2) A B D E C (2) E C B A D (1) E B A C D (1) E A B C D (1) D C B A E (1) D B C A E (1) D B A C E (1) D A B E C (1) C E D A B (1) C E B A D (1) C B D E A (1) B D C E A (1) B D C A E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 12 -2 8 B 2 0 14 -6 8 C -12 -14 0 -6 -2 D 2 6 6 0 12 E -8 -8 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 -2 8 B 2 0 14 -6 8 C -12 -14 0 -6 -2 D 2 6 6 0 12 E -8 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=25 A=22 C=20 E=5 so E is eliminated. Round 2 votes counts: B=31 D=25 A=23 C=21 so C is eliminated. Round 3 votes counts: B=41 D=36 A=23 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:209 A:208 E:187 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 -2 8 B 2 0 14 -6 8 C -12 -14 0 -6 -2 D 2 6 6 0 12 E -8 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -2 8 B 2 0 14 -6 8 C -12 -14 0 -6 -2 D 2 6 6 0 12 E -8 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -2 8 B 2 0 14 -6 8 C -12 -14 0 -6 -2 D 2 6 6 0 12 E -8 -8 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4154: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (11) C A B D E (10) A C D E B (8) C B D E A (7) A C B D E (7) B E D C A (6) E D B A C (5) A C B E D (5) E D B C A (4) D E B C A (4) D B E C A (3) A D E C B (3) E D A B C (2) E B D A C (2) D E B A C (2) C B A E D (2) C B A D E (2) B E C D A (2) A C E B D (2) A C D B E (2) E B D C A (1) C D A E B (1) C B E D A (1) C A B E D (1) B D E C A (1) B C E D A (1) B C A E D (1) A E B D C (1) A E B C D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 12 6 16 18 B -12 0 -4 6 2 C -6 4 0 8 2 D -16 -6 -8 0 2 E -18 -2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 16 18 B -12 0 -4 6 2 C -6 4 0 8 2 D -16 -6 -8 0 2 E -18 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 C=24 E=14 B=11 D=9 so D is eliminated. Round 2 votes counts: A=42 C=24 E=20 B=14 so B is eliminated. Round 3 votes counts: A=42 E=32 C=26 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 C:204 B:196 E:188 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 16 18 B -12 0 -4 6 2 C -6 4 0 8 2 D -16 -6 -8 0 2 E -18 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 16 18 B -12 0 -4 6 2 C -6 4 0 8 2 D -16 -6 -8 0 2 E -18 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 16 18 B -12 0 -4 6 2 C -6 4 0 8 2 D -16 -6 -8 0 2 E -18 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4155: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) C E A B D (8) E C A D B (6) B C D E A (6) E C D A B (5) D B A E C (5) B D A E C (5) A B D E C (5) D B E A C (4) C E B A D (4) C E A D B (4) D E A C B (3) B D C A E (3) B D A C E (3) A C E B D (3) E A C D B (2) C B E D A (2) C B E A D (2) B D C E A (2) B A D C E (2) A D E C B (2) E D C A B (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A B C (1) D A B E C (1) C E B D A (1) B D E C A (1) B C E A D (1) B A D E C (1) B A C E D (1) A E D C B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 4 8 -12 B -8 0 -12 4 -8 C -4 12 0 14 -12 D -8 -4 -14 0 -6 E 12 8 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 4 8 -12 B -8 0 -12 4 -8 C -4 12 0 14 -12 D -8 -4 -14 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=23 C=21 D=17 E=14 so E is eliminated. Round 2 votes counts: C=32 B=25 A=25 D=18 so D is eliminated. Round 3 votes counts: B=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:219 C:205 A:204 B:188 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 4 8 -12 B -8 0 -12 4 -8 C -4 12 0 14 -12 D -8 -4 -14 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 8 -12 B -8 0 -12 4 -8 C -4 12 0 14 -12 D -8 -4 -14 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 8 -12 B -8 0 -12 4 -8 C -4 12 0 14 -12 D -8 -4 -14 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4156: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) B C E D A (7) E B A C D (5) B E C D A (5) A D E C B (5) A D E B C (5) A D C E B (5) E A B C D (4) C D B E A (4) B E D A C (4) D C B E A (3) D C B A E (3) D C A B E (3) C B E D A (3) A E D B C (3) A E C B D (3) A E B D C (3) A E B C D (3) E C B A D (2) E B C A D (2) D B C E A (2) C E B A D (2) C B D E A (2) A C E B D (2) A C D E B (2) E B D A C (1) E B C D A (1) E A B D C (1) D B A E C (1) D A B E C (1) D A B C E (1) C D B A E (1) C D A B E (1) C A E B D (1) B E D C A (1) A E D C B (1) Total count = 100 A B C D E A 0 2 14 -2 2 B -2 0 0 4 -2 C -14 0 0 0 -2 D 2 -4 0 0 -8 E -2 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999603 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 2 14 -2 2 B -2 0 0 4 -2 C -14 0 0 0 -2 D 2 -4 0 0 -8 E -2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=21 B=17 E=16 C=14 so C is eliminated. Round 2 votes counts: A=33 D=27 B=22 E=18 so E is eliminated. Round 3 votes counts: A=38 B=35 D=27 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:205 B:200 D:195 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 -2 2 B -2 0 0 4 -2 C -14 0 0 0 -2 D 2 -4 0 0 -8 E -2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -2 2 B -2 0 0 4 -2 C -14 0 0 0 -2 D 2 -4 0 0 -8 E -2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -2 2 B -2 0 0 4 -2 C -14 0 0 0 -2 D 2 -4 0 0 -8 E -2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4157: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) E C B A D (6) E B C D A (6) D A B C E (6) C E A D B (5) C A D E B (5) E B D A C (4) C A E D B (4) C A D B E (4) B E D A C (4) A D E C B (4) A D C E B (4) D A E B C (3) D A C B E (3) D A B E C (3) C E B A D (3) B C E D A (3) A C D B E (3) C E A B D (2) B E C D A (2) A E D C B (2) A C D E B (2) E C A D B (1) E C A B D (1) E B D C A (1) E B C A D (1) E B A D C (1) E A D B C (1) E A C D B (1) D B A E C (1) D B A C E (1) C B E A D (1) C A B D E (1) B E D C A (1) B D E A C (1) B D A E C (1) Total count = 100 A B C D E A 0 26 6 20 10 B -26 0 -20 -22 -14 C -6 20 0 2 10 D -20 22 -2 0 0 E -10 14 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 6 20 10 B -26 0 -20 -22 -14 C -6 20 0 2 10 D -20 22 -2 0 0 E -10 14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997074 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=23 A=23 D=17 B=12 so B is eliminated. Round 2 votes counts: E=30 C=28 A=23 D=19 so D is eliminated. Round 3 votes counts: A=41 E=31 C=28 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 C:213 D:200 E:197 B:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 6 20 10 B -26 0 -20 -22 -14 C -6 20 0 2 10 D -20 22 -2 0 0 E -10 14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997074 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 6 20 10 B -26 0 -20 -22 -14 C -6 20 0 2 10 D -20 22 -2 0 0 E -10 14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997074 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 6 20 10 B -26 0 -20 -22 -14 C -6 20 0 2 10 D -20 22 -2 0 0 E -10 14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997074 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4158: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) E D B A C (6) D E B C A (6) B A C D E (6) E A C B D (5) C A B D E (5) E B A C D (4) D E C B A (4) D E C A B (4) A B C E D (4) E C A D B (3) D C B A E (3) C A E B D (3) A C E B D (3) A C B E D (3) A B C D E (3) E C A B D (2) D C E A B (2) D C A B E (2) D B E A C (2) D B C A E (2) D B A C E (2) B A C E D (2) A E B C D (2) A C B D E (2) E D B C A (1) E D A C B (1) E B D A C (1) E B A D C (1) E A B C D (1) D B A E C (1) C D A B E (1) C B A D E (1) C A D B E (1) C A B E D (1) B E A C D (1) B D A C E (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 8 6 10 0 B -8 0 -4 6 -10 C -6 4 0 6 -4 D -10 -6 -6 0 0 E 0 10 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.614037 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.385963 Sum of squares = 0.526008908478 Cumulative probabilities = A: 0.614037 B: 0.614037 C: 0.614037 D: 0.614037 E: 1.000000 A B C D E A 0 8 6 10 0 B -8 0 -4 6 -10 C -6 4 0 6 -4 D -10 -6 -6 0 0 E 0 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=28 A=17 C=12 B=12 so C is eliminated. Round 2 votes counts: E=31 D=29 A=27 B=13 so B is eliminated. Round 3 votes counts: A=38 E=32 D=30 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:207 C:200 B:192 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 10 0 B -8 0 -4 6 -10 C -6 4 0 6 -4 D -10 -6 -6 0 0 E 0 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 10 0 B -8 0 -4 6 -10 C -6 4 0 6 -4 D -10 -6 -6 0 0 E 0 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 10 0 B -8 0 -4 6 -10 C -6 4 0 6 -4 D -10 -6 -6 0 0 E 0 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4159: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) E B C D A (6) D A C E B (6) C A E D B (5) A D C B E (5) D A E C B (4) D A C B E (4) B E C A D (4) E B C A D (3) C E A D B (3) C E A B D (3) C A E B D (3) E C B A D (2) E B D C A (2) D B E A C (2) D B A C E (2) C A D E B (2) B E D C A (2) B D E A C (2) B C A E D (2) B A C D E (2) A D C E B (2) A C D E B (2) E D C A B (1) E C D B A (1) E C A D B (1) E C A B D (1) D E C A B (1) D E B A C (1) D E A C B (1) D E A B C (1) D B A E C (1) D A B E C (1) C E B A D (1) C B E A D (1) C A B E D (1) B E D A C (1) B E C D A (1) B E A C D (1) B D A E C (1) B D A C E (1) B C E A D (1) B A D C E (1) A C D B E (1) Total count = 100 A B C D E A 0 18 6 -6 14 B -18 0 -2 -16 -4 C -6 2 0 -6 18 D 6 16 6 0 8 E -14 4 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 -6 14 B -18 0 -2 -16 -4 C -6 2 0 -6 18 D 6 16 6 0 8 E -14 4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=19 B=19 E=17 A=10 so A is eliminated. Round 2 votes counts: D=42 C=22 B=19 E=17 so E is eliminated. Round 3 votes counts: D=43 B=30 C=27 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:216 C:204 E:182 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 6 -6 14 B -18 0 -2 -16 -4 C -6 2 0 -6 18 D 6 16 6 0 8 E -14 4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 -6 14 B -18 0 -2 -16 -4 C -6 2 0 -6 18 D 6 16 6 0 8 E -14 4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 -6 14 B -18 0 -2 -16 -4 C -6 2 0 -6 18 D 6 16 6 0 8 E -14 4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4160: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) E B D C A (8) C E D B A (7) C D E B A (7) C A D B E (7) A B E D C (7) A E B D C (5) D B E C A (4) C A D E B (4) B E D A C (4) E B D A C (3) D B C E A (3) E D B C A (2) E B A D C (2) B E D C A (2) B D E A C (2) A E C B D (2) A E B C D (2) A C E D B (2) A C E B D (2) A B D E C (2) E B A C D (1) D C B E A (1) D B E A C (1) D B A E C (1) D B A C E (1) D A C B E (1) C E A B D (1) C D B E A (1) C A E D B (1) B D E C A (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 2 2 0 B 2 0 2 -10 2 C -2 -2 0 -2 2 D -2 10 2 0 -2 E 0 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.408649 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.305637 Sum of squares = 0.301224145149 Cumulative probabilities = A: 0.408649 B: 0.551506 C: 0.551506 D: 0.694363 E: 1.000000 A B C D E A 0 -2 2 2 0 B 2 0 2 -10 2 C -2 -2 0 -2 2 D -2 10 2 0 -2 E 0 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357144 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.357142 Sum of squares = 0.295918367347 Cumulative probabilities = A: 0.357144 B: 0.500001 C: 0.500001 D: 0.642858 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=28 E=16 D=12 B=9 so B is eliminated. Round 2 votes counts: A=35 C=28 E=22 D=15 so D is eliminated. Round 3 votes counts: A=38 C=32 E=30 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:204 A:201 E:199 B:198 C:198 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 0 B 2 0 2 -10 2 C -2 -2 0 -2 2 D -2 10 2 0 -2 E 0 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357144 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.357142 Sum of squares = 0.295918367347 Cumulative probabilities = A: 0.357144 B: 0.500001 C: 0.500001 D: 0.642858 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 0 B 2 0 2 -10 2 C -2 -2 0 -2 2 D -2 10 2 0 -2 E 0 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357144 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.357142 Sum of squares = 0.295918367347 Cumulative probabilities = A: 0.357144 B: 0.500001 C: 0.500001 D: 0.642858 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 0 B 2 0 2 -10 2 C -2 -2 0 -2 2 D -2 10 2 0 -2 E 0 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357144 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.357142 Sum of squares = 0.295918367347 Cumulative probabilities = A: 0.357144 B: 0.500001 C: 0.500001 D: 0.642858 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4161: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) E C B A D (6) C B E D A (5) E B C A D (4) C E D B A (4) C E A D B (4) B C E D A (4) A E D B C (4) A D E B C (4) D C A E B (3) D B A C E (3) D A C E B (3) B D C A E (3) B A D E C (3) A D E C B (3) A D B E C (3) E A C D B (2) E A C B D (2) E A B C D (2) D C A B E (2) D A C B E (2) D A B C E (2) C E B D A (2) C E B A D (2) C D E A B (2) B C D A E (2) A E D C B (2) E C A D B (1) E C A B D (1) E B A C D (1) E A D B C (1) E A B D C (1) D A B E C (1) C E D A B (1) C D E B A (1) C D B E A (1) B D A C E (1) B C E A D (1) B A E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -16 16 -14 B 6 0 2 2 -12 C 16 -2 0 14 -4 D -16 -2 -14 0 -20 E 14 12 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -16 16 -14 B 6 0 2 2 -12 C 16 -2 0 14 -4 D -16 -2 -14 0 -20 E 14 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=22 E=21 A=17 D=16 so D is eliminated. Round 2 votes counts: C=27 B=27 A=25 E=21 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:225 C:212 B:199 A:190 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -16 16 -14 B 6 0 2 2 -12 C 16 -2 0 14 -4 D -16 -2 -14 0 -20 E 14 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 16 -14 B 6 0 2 2 -12 C 16 -2 0 14 -4 D -16 -2 -14 0 -20 E 14 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 16 -14 B 6 0 2 2 -12 C 16 -2 0 14 -4 D -16 -2 -14 0 -20 E 14 12 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4162: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) A C D E B (8) D B E C A (7) A D C B E (6) E C B D A (5) E B C D A (5) D B E A C (5) B E D C A (5) A D B E C (4) A C B E D (4) C E D B A (3) C A E D B (3) D E B C A (2) D B A E C (2) C E B D A (2) C E B A D (2) C E A B D (2) C A E B D (2) B E C D A (2) B D A E C (2) A C D B E (2) A B E C D (2) A B C E D (2) E D C B A (1) E D B C A (1) D C E A B (1) C E D A B (1) C D E B A (1) C D A E B (1) C A D E B (1) B E C A D (1) B E A C D (1) B D E A C (1) B A E D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 4 -14 4 B -2 0 6 -20 14 C -4 -6 0 4 -14 D 14 20 -4 0 6 E -4 -14 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.583333 E: 0.166667 Sum of squares = 0.430555555604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.833333 E: 1.000000 A B C D E A 0 2 4 -14 4 B -2 0 6 -20 14 C -4 -6 0 4 -14 D 14 20 -4 0 6 E -4 -14 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.583333 E: 0.166667 Sum of squares = 0.430555555552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=27 C=18 B=13 E=12 so E is eliminated. Round 2 votes counts: A=30 D=29 C=23 B=18 so B is eliminated. Round 3 votes counts: D=37 A=32 C=31 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:199 A:198 E:195 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -14 4 B -2 0 6 -20 14 C -4 -6 0 4 -14 D 14 20 -4 0 6 E -4 -14 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.583333 E: 0.166667 Sum of squares = 0.430555555552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.833333 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -14 4 B -2 0 6 -20 14 C -4 -6 0 4 -14 D 14 20 -4 0 6 E -4 -14 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.583333 E: 0.166667 Sum of squares = 0.430555555552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -14 4 B -2 0 6 -20 14 C -4 -6 0 4 -14 D 14 20 -4 0 6 E -4 -14 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.583333 E: 0.166667 Sum of squares = 0.430555555552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4163: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) A D E C B (6) B C E A D (5) A E D C B (5) A E D B C (5) E C D A B (4) E A D C B (4) B C D E A (4) E C D B A (3) E C B D A (3) E A C D B (3) C E D B A (3) C D B E A (3) B A E C D (3) B A D C E (3) D A E C B (2) C B D E A (2) B E C A D (2) A E B C D (2) A D B E C (2) A B D E C (2) E D C A B (1) E D A C B (1) E C B A D (1) E B C D A (1) E B C A D (1) E A C B D (1) E A B C D (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) D B C A E (1) D B A C E (1) C D E B A (1) B E A C D (1) B C D A E (1) B A D E C (1) B A C E D (1) A D E B C (1) A D C E B (1) A D C B E (1) A D B C E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 2 12 -12 B 2 0 -2 -10 -8 C -2 2 0 8 -18 D -12 10 -8 0 -20 E 12 8 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 12 -12 B 2 0 -2 -10 -8 C -2 2 0 8 -18 D -12 10 -8 0 -20 E 12 8 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 E=24 D=10 C=9 so C is eliminated. Round 2 votes counts: B=30 A=29 E=27 D=14 so D is eliminated. Round 3 votes counts: B=37 A=32 E=31 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:229 A:200 C:195 B:191 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 12 -12 B 2 0 -2 -10 -8 C -2 2 0 8 -18 D -12 10 -8 0 -20 E 12 8 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 12 -12 B 2 0 -2 -10 -8 C -2 2 0 8 -18 D -12 10 -8 0 -20 E 12 8 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 12 -12 B 2 0 -2 -10 -8 C -2 2 0 8 -18 D -12 10 -8 0 -20 E 12 8 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4164: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) B E C D A (8) C D B E A (6) B C E D A (6) A D C E B (6) E B C D A (5) E B A C D (5) B E A C D (5) D C B E A (4) D C A E B (4) D C A B E (4) D A C B E (4) A E B C D (4) E A B C D (3) A D E C B (3) A D C B E (3) C B E D A (2) C B D E A (2) B E C A D (2) B A E C D (2) A E D B C (2) E B C A D (1) E A B D C (1) D C E B A (1) D C B A E (1) D A C E B (1) C E D B A (1) C D E B A (1) B A C D E (1) A E D C B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 4 0 -6 B 6 0 12 14 4 C -4 -12 0 8 -6 D 0 -14 -8 0 -16 E 6 -4 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 0 -6 B 6 0 12 14 4 C -4 -12 0 8 -6 D 0 -14 -8 0 -16 E 6 -4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 D=19 E=15 C=12 so C is eliminated. Round 2 votes counts: A=30 B=28 D=26 E=16 so E is eliminated. Round 3 votes counts: B=39 A=34 D=27 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:212 A:196 C:193 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 0 -6 B 6 0 12 14 4 C -4 -12 0 8 -6 D 0 -14 -8 0 -16 E 6 -4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 0 -6 B 6 0 12 14 4 C -4 -12 0 8 -6 D 0 -14 -8 0 -16 E 6 -4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 0 -6 B 6 0 12 14 4 C -4 -12 0 8 -6 D 0 -14 -8 0 -16 E 6 -4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999171 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4165: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) E A D B C (8) D B C A E (7) C B D A E (7) C B A D E (7) E C D B A (4) E A C D B (4) B D C A E (4) E C A B D (3) E A D C B (3) E A C B D (3) D B A C E (3) D A B E C (3) C E B A D (3) C B E D A (3) C A B D E (3) E D C B A (2) D B E A C (2) D A E B C (2) C B D E A (2) B C D A E (2) A E C B D (2) A C B E D (2) A B C D E (2) E D B A C (1) E C D A B (1) E C B D A (1) E C B A D (1) D E B C A (1) C E B D A (1) C E A B D (1) C B A E D (1) C A B E D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 -8 -4 B 4 0 -10 0 2 C 10 10 0 8 2 D 8 0 -8 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -8 -4 B 4 0 -10 0 2 C 10 10 0 8 2 D 8 0 -8 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=29 D=18 A=8 B=6 so B is eliminated. Round 2 votes counts: E=39 C=31 D=22 A=8 so A is eliminated. Round 3 votes counts: E=41 C=35 D=24 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:203 B:198 D:197 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -8 -4 B 4 0 -10 0 2 C 10 10 0 8 2 D 8 0 -8 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -8 -4 B 4 0 -10 0 2 C 10 10 0 8 2 D 8 0 -8 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -8 -4 B 4 0 -10 0 2 C 10 10 0 8 2 D 8 0 -8 0 -6 E 4 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4166: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) A B E C D (8) D C E B A (6) D B E C A (6) B E A C D (6) D B E A C (5) C A E B D (5) B E D A C (5) A C B E D (5) D E B C A (4) D C E A B (4) E B D C A (3) B A E C D (3) A C E B D (3) E B C A D (2) D A C B E (2) C E B A D (2) C E A B D (2) C D E A B (2) C A E D B (2) C A D E B (2) A D C B E (2) E C B A D (1) D E C B A (1) D B A E C (1) D A B E C (1) C E D B A (1) C D E B A (1) C D A E B (1) B D A E C (1) B A E D C (1) A C D E B (1) A C D B E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 -4 -2 B -2 0 -4 -2 -2 C 6 4 0 -2 2 D 4 2 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 2 -6 -4 -2 B -2 0 -4 -2 -2 C 6 4 0 -2 2 D 4 2 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=22 C=18 B=16 E=6 so E is eliminated. Round 2 votes counts: D=38 A=22 B=21 C=19 so C is eliminated. Round 3 votes counts: D=43 A=33 B=24 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:205 D:203 E:202 A:195 B:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -2 B -2 0 -4 -2 -2 C 6 4 0 -2 2 D 4 2 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -2 B -2 0 -4 -2 -2 C 6 4 0 -2 2 D 4 2 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -2 B -2 0 -4 -2 -2 C 6 4 0 -2 2 D 4 2 2 0 -2 E 2 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4167: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) A E C B D (10) A E B C D (9) D C B E A (6) E A C D B (5) B D A C E (5) D E C A B (4) D C E A B (4) B D C A E (4) B A E D C (4) C E D A B (3) C D E B A (3) B A D E C (3) E C D A B (2) E A C B D (2) D C E B A (2) B D C E A (2) B D A E C (2) B A D C E (2) B A C E D (2) A E C D B (2) A B E C D (2) E A D C B (1) D E C B A (1) C D E A B (1) C B D E A (1) C B A E D (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E C D (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 4 -4 4 B 6 0 2 8 -2 C -4 -2 0 -4 0 D 4 -8 4 0 8 E -4 2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.40740740739 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 A B C D E A 0 -6 4 -4 4 B 6 0 2 8 -2 C -4 -2 0 -4 0 D 4 -8 4 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407389 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 A=25 E=10 C=9 so C is eliminated. Round 2 votes counts: D=31 B=31 A=25 E=13 so E is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:207 D:204 A:199 C:195 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -4 4 B 6 0 2 8 -2 C -4 -2 0 -4 0 D 4 -8 4 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407389 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -4 4 B 6 0 2 8 -2 C -4 -2 0 -4 0 D 4 -8 4 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407389 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -4 4 B 6 0 2 8 -2 C -4 -2 0 -4 0 D 4 -8 4 0 8 E -4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407389 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4168: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) A B C D E (7) E D C B A (6) D C E B A (5) D A C E B (5) A D C B E (5) D A C B E (4) C D E B A (4) B E C A D (4) A E D B C (4) E C B D A (3) C D B E A (3) B A C E D (3) A D C E B (3) A C D B E (3) E C D B A (2) D E C B A (2) D E C A B (2) C E D B A (2) B E A C D (2) B C A D E (2) A E B D C (2) A D B C E (2) A B E C D (2) A B D C E (2) E D A C B (1) E B A C D (1) D C A E B (1) D C A B E (1) D A E C B (1) C B E D A (1) C B D A E (1) B C E D A (1) B C E A D (1) B C A E D (1) A D E C B (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 -4 6 B 2 0 -14 -14 -4 C -2 14 0 4 18 D 4 14 -4 0 10 E -6 4 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 6 B 2 0 -14 -14 -4 C -2 14 0 4 18 D 4 14 -4 0 10 E -6 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999988 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=21 E=20 B=14 C=11 so C is eliminated. Round 2 votes counts: A=34 D=28 E=22 B=16 so B is eliminated. Round 3 votes counts: A=40 E=31 D=29 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:217 D:212 A:201 B:185 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 2 -4 6 B 2 0 -14 -14 -4 C -2 14 0 4 18 D 4 14 -4 0 10 E -6 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999988 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 6 B 2 0 -14 -14 -4 C -2 14 0 4 18 D 4 14 -4 0 10 E -6 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999988 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 6 B 2 0 -14 -14 -4 C -2 14 0 4 18 D 4 14 -4 0 10 E -6 4 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999988 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4169: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) E A B C D (7) D C B A E (6) D C A B E (6) D A C B E (6) E B C A D (5) C D B A E (5) E D C B A (4) E D A C B (3) C B D E A (3) C B A D E (3) B C A D E (3) E B C D A (2) E B A C D (2) D E C B A (2) D C E B A (2) D C E A B (2) D A E C B (2) C B E D A (2) C B D A E (2) B E C A D (2) A E B D C (2) A D E B C (2) A B C D E (2) E D B C A (1) E B D C A (1) E A D B C (1) D E C A B (1) D E A C B (1) D A C E B (1) C E B D A (1) C D B E A (1) B E A C D (1) B C E A D (1) B A E C D (1) B A C E D (1) B A C D E (1) A D B C E (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 -8 -6 B 4 0 -6 6 0 C 10 6 0 -6 2 D 8 -6 6 0 6 E 6 0 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -8 -6 B 4 0 -6 6 0 C 10 6 0 -6 2 D 8 -6 6 0 6 E 6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=29 C=17 B=10 A=10 so B is eliminated. Round 2 votes counts: E=37 D=29 C=21 A=13 so A is eliminated. Round 3 votes counts: E=41 D=33 C=26 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:207 C:206 B:202 E:199 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -8 -6 B 4 0 -6 6 0 C 10 6 0 -6 2 D 8 -6 6 0 6 E 6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -8 -6 B 4 0 -6 6 0 C 10 6 0 -6 2 D 8 -6 6 0 6 E 6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -8 -6 B 4 0 -6 6 0 C 10 6 0 -6 2 D 8 -6 6 0 6 E 6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4170: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) A C B E D (6) D E A B C (5) B C E D A (5) A D E C B (4) A D B C E (4) E D B C A (3) E C D B A (3) E C B D A (3) E B C D A (3) D E B C A (3) D E B A C (3) C B A E D (3) B C A E D (3) A D C B E (3) A C D B E (3) A B D C E (3) A B C D E (3) D A E C B (2) C B E A D (2) C A E D B (2) C A B E D (2) B E C D A (2) B D C A E (2) A D C E B (2) A C B D E (2) E D C B A (1) E D A C B (1) D E C B A (1) D E A C B (1) D B E C A (1) D B E A C (1) D A B E C (1) C E B D A (1) C B E D A (1) B E D C A (1) B D A E C (1) B C E A D (1) B A C E D (1) A E D C B (1) A E C D B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 10 14 -4 16 B -10 0 8 -10 2 C -14 -8 0 -4 0 D 4 10 4 0 8 E -16 -2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 -4 16 B -10 0 8 -10 2 C -14 -8 0 -4 0 D 4 10 4 0 8 E -16 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=25 B=16 E=14 C=11 so C is eliminated. Round 2 votes counts: A=38 D=25 B=22 E=15 so E is eliminated. Round 3 votes counts: A=38 D=33 B=29 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:213 B:195 C:187 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 14 -4 16 B -10 0 8 -10 2 C -14 -8 0 -4 0 D 4 10 4 0 8 E -16 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -4 16 B -10 0 8 -10 2 C -14 -8 0 -4 0 D 4 10 4 0 8 E -16 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -4 16 B -10 0 8 -10 2 C -14 -8 0 -4 0 D 4 10 4 0 8 E -16 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4171: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (15) B D C E A (14) A E B D C (10) A E D C B (7) E A C D B (4) E A B D C (4) C D B E A (4) B C D E A (4) E A D C B (3) C D E A B (3) C A E D B (3) A B E C D (3) B E D A C (2) B D E A C (2) A E D B C (2) A E C B D (2) E C A D B (1) D E C B A (1) D C B E A (1) D B C E A (1) C D E B A (1) C D B A E (1) C D A B E (1) C B D E A (1) C A D E B (1) B D E C A (1) B D A E C (1) B D A C E (1) B C D A E (1) B A E D C (1) B A C D E (1) A E B C D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 24 22 20 6 B -24 0 2 2 -16 C -22 -2 0 -4 -22 D -20 -2 4 0 -20 E -6 16 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 22 20 6 B -24 0 2 2 -16 C -22 -2 0 -4 -22 D -20 -2 4 0 -20 E -6 16 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 B=28 C=15 E=12 D=3 so D is eliminated. Round 2 votes counts: A=42 B=29 C=16 E=13 so E is eliminated. Round 3 votes counts: A=53 B=29 C=18 so C is eliminated. Round 4 votes counts: A=62 B=38 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:236 E:226 B:182 D:181 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 22 20 6 B -24 0 2 2 -16 C -22 -2 0 -4 -22 D -20 -2 4 0 -20 E -6 16 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 22 20 6 B -24 0 2 2 -16 C -22 -2 0 -4 -22 D -20 -2 4 0 -20 E -6 16 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 22 20 6 B -24 0 2 2 -16 C -22 -2 0 -4 -22 D -20 -2 4 0 -20 E -6 16 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4172: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) A C B D E (8) E D B C A (7) D E B C A (7) A C D B E (7) E D C B A (5) B D E A C (5) A B D E C (5) B E D A C (4) E B D C A (3) D E C B A (3) C A D E B (3) B E D C A (3) A C B E D (3) A B E D C (3) D E B A C (2) C E D B A (2) C E A D B (2) C A E B D (2) A C D E B (2) E B C D A (1) D C E A B (1) D B E A C (1) D A E B C (1) C E D A B (1) C E B D A (1) C E A B D (1) C D E B A (1) C D E A B (1) C B E A D (1) C A B E D (1) B A D E C (1) A D C E B (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -8 2 -4 B -6 0 -8 -12 -10 C 8 8 0 -8 -4 D -2 12 8 0 2 E 4 10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000045 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 6 -8 2 -4 B -6 0 -8 -12 -10 C 8 8 0 -8 -4 D -2 12 8 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000012 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=24 E=16 D=15 B=13 so B is eliminated. Round 2 votes counts: A=33 C=24 E=23 D=20 so D is eliminated. Round 3 votes counts: E=41 A=34 C=25 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:208 C:202 A:198 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 2 -4 B -6 0 -8 -12 -10 C 8 8 0 -8 -4 D -2 12 8 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000012 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 2 -4 B -6 0 -8 -12 -10 C 8 8 0 -8 -4 D -2 12 8 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000012 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 2 -4 B -6 0 -8 -12 -10 C 8 8 0 -8 -4 D -2 12 8 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000012 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4173: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (6) B D E A C (6) D B E A C (5) B E D C A (5) B C D A E (5) A C E D B (5) E A D C B (4) B E C D A (4) E D A C B (3) E D A B C (3) D E A B C (3) C A B E D (3) B D E C A (3) E B C A D (2) E A C D B (2) D E B A C (2) D B A E C (2) D A E C B (2) D A E B C (2) C B A D E (2) C A E D B (2) C A E B D (2) B E D A C (2) B D C A E (2) B C E A D (2) A E D C B (2) A E C D B (2) A D E C B (2) E B D A C (1) E A C B D (1) D A C E B (1) D A C B E (1) D A B E C (1) C E A B D (1) C B E A D (1) C B D A E (1) C A D E B (1) C A B D E (1) B E C A D (1) B D C E A (1) B C E D A (1) B C A D E (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 6 -12 -6 B 10 0 8 8 12 C -6 -8 0 -6 -20 D 12 -8 6 0 -10 E 6 -12 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -12 -6 B 10 0 8 8 12 C -6 -8 0 -6 -20 D 12 -8 6 0 -10 E 6 -12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=20 D=19 E=16 A=12 so A is eliminated. Round 2 votes counts: B=33 C=26 D=21 E=20 so E is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:212 D:200 A:189 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 -12 -6 B 10 0 8 8 12 C -6 -8 0 -6 -20 D 12 -8 6 0 -10 E 6 -12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -12 -6 B 10 0 8 8 12 C -6 -8 0 -6 -20 D 12 -8 6 0 -10 E 6 -12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -12 -6 B 10 0 8 8 12 C -6 -8 0 -6 -20 D 12 -8 6 0 -10 E 6 -12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4174: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) C B D A E (7) B C A D E (7) C D B A E (6) B C D A E (5) B A C D E (5) E A D B C (4) B A E C D (4) E D C A B (3) E D A C B (3) E A B D C (3) D A C B E (3) C B D E A (3) B C D E A (3) A D C B E (3) A B D C E (3) E C D B A (2) D C E A B (2) B C E D A (2) B C E A D (2) A E D C B (2) A E D B C (2) A E B D C (2) A B E D C (2) E D C B A (1) E C B D A (1) E B C A D (1) E B A C D (1) D E A C B (1) D C E B A (1) D C A E B (1) D C A B E (1) C D E B A (1) C D B E A (1) B E C D A (1) B E A C D (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -2 4 10 B 12 0 0 8 22 C 2 0 0 8 16 D -4 -8 -8 0 8 E -10 -22 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.502352 C: 0.497648 D: 0.000000 E: 0.000000 Sum of squares = 0.500011058127 Cumulative probabilities = A: 0.000000 B: 0.502352 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 4 10 B 12 0 0 8 22 C 2 0 0 8 16 D -4 -8 -8 0 8 E -10 -22 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=27 C=18 A=15 D=9 so D is eliminated. Round 2 votes counts: B=31 E=28 C=23 A=18 so A is eliminated. Round 3 votes counts: B=37 E=34 C=29 so C is eliminated. Round 4 votes counts: B=61 E=39 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:213 A:200 D:194 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 4 10 B 12 0 0 8 22 C 2 0 0 8 16 D -4 -8 -8 0 8 E -10 -22 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 4 10 B 12 0 0 8 22 C 2 0 0 8 16 D -4 -8 -8 0 8 E -10 -22 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 4 10 B 12 0 0 8 22 C 2 0 0 8 16 D -4 -8 -8 0 8 E -10 -22 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999972 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4175: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) E C D B A (9) B A E C D (9) D A C B E (8) D C A E B (7) B A D E C (6) E C D A B (4) E C B D A (4) E C B A D (4) E B C A D (4) C E D A B (4) B A E D C (4) A B D E C (4) D C E A B (3) A D B C E (3) E B A C D (2) D A C E B (2) C E D B A (2) B E A C D (2) B A D C E (2) E B C D A (1) C D E B A (1) C D E A B (1) C D B E A (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 10 6 16 B 2 0 -2 8 4 C -10 2 0 -4 -8 D -6 -8 4 0 0 E -16 -4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408227 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 6 16 B 2 0 -2 8 4 C -10 2 0 -4 -8 D -6 -8 4 0 0 E -16 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408141 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=23 D=20 A=20 C=9 so C is eliminated. Round 2 votes counts: E=34 D=23 B=23 A=20 so A is eliminated. Round 3 votes counts: B=39 E=34 D=27 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:206 D:195 E:194 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 10 6 16 B 2 0 -2 8 4 C -10 2 0 -4 -8 D -6 -8 4 0 0 E -16 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408141 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 6 16 B 2 0 -2 8 4 C -10 2 0 -4 -8 D -6 -8 4 0 0 E -16 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408141 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 6 16 B 2 0 -2 8 4 C -10 2 0 -4 -8 D -6 -8 4 0 0 E -16 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408141 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4176: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) A D C E B (8) C D A E B (6) C A D B E (5) A D C B E (5) E D C A B (4) E D A B C (4) B E D A C (4) B C E A D (4) D A E C B (3) B E A D C (3) B C A D E (3) A D B C E (3) E D A C B (2) E B D A C (2) D A C E B (2) C E D A B (2) C B A D E (2) C A D E B (2) C A B D E (2) B E C D A (2) B E C A D (2) B A E D C (2) B A D E C (2) A D E B C (2) A D B E C (2) E D B A C (1) E C D A B (1) E B D C A (1) D E A C B (1) D A E B C (1) C E B D A (1) C D A B E (1) C B E D A (1) B A D C E (1) B A C D E (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 18 4 4 12 B -18 0 0 -14 -6 C -4 0 0 -8 4 D -4 14 8 0 10 E -12 6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 4 12 B -18 0 0 -14 -6 C -4 0 0 -8 4 D -4 14 8 0 10 E -12 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 A=23 C=22 D=7 so D is eliminated. Round 2 votes counts: A=29 E=25 B=24 C=22 so C is eliminated. Round 3 votes counts: A=45 E=28 B=27 so B is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:214 C:196 E:190 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 4 12 B -18 0 0 -14 -6 C -4 0 0 -8 4 D -4 14 8 0 10 E -12 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 4 12 B -18 0 0 -14 -6 C -4 0 0 -8 4 D -4 14 8 0 10 E -12 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 4 12 B -18 0 0 -14 -6 C -4 0 0 -8 4 D -4 14 8 0 10 E -12 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997424 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4177: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) B C E D A (9) C B E D A (8) D E A C B (6) A B C D E (6) E B C D A (5) B C A E D (5) E D C B A (4) C B D A E (4) B C E A D (4) A D C B E (3) E A D B C (2) C D E B A (2) C B D E A (2) A E D B C (2) A D E B C (2) A D B E C (2) A D B C E (2) E D B C A (1) E D B A C (1) E D A C B (1) E B D C A (1) E B D A C (1) E B A D C (1) D E C B A (1) D E A B C (1) D C A E B (1) D A E C B (1) D A C E B (1) C E D B A (1) C D B E A (1) C D A B E (1) C A B D E (1) B A E D C (1) B A C E D (1) B A C D E (1) A D C E B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -2 -6 -4 B 8 0 0 4 6 C 2 0 0 2 10 D 6 -4 -2 0 2 E 4 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.636633 C: 0.363367 D: 0.000000 E: 0.000000 Sum of squares = 0.537337378713 Cumulative probabilities = A: 0.000000 B: 0.636633 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -6 -4 B 8 0 0 4 6 C 2 0 0 2 10 D 6 -4 -2 0 2 E 4 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=21 C=20 E=17 D=11 so D is eliminated. Round 2 votes counts: A=33 E=25 C=21 B=21 so C is eliminated. Round 3 votes counts: B=36 A=36 E=28 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:207 D:201 E:193 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -6 -4 B 8 0 0 4 6 C 2 0 0 2 10 D 6 -4 -2 0 2 E 4 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -6 -4 B 8 0 0 4 6 C 2 0 0 2 10 D 6 -4 -2 0 2 E 4 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -6 -4 B 8 0 0 4 6 C 2 0 0 2 10 D 6 -4 -2 0 2 E 4 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4178: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) B E C D A (9) C B A E D (8) B C E A D (8) D A E C B (6) D A C E B (6) E B D C A (5) E D B A C (4) D E A B C (4) C B E A D (4) C A D B E (4) C A B D E (3) A D C E B (3) E B D A C (2) D E B C A (2) D E B A C (2) C A B E D (2) B E C A D (2) B C E D A (2) A D E C B (2) A D C B E (2) A C B D E (2) E D B C A (1) E D A B C (1) E B A C D (1) D A E B C (1) B E D C A (1) B E A C D (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 8 2 B 4 0 -6 0 16 C 2 6 0 14 10 D -8 0 -14 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 8 2 B 4 0 -6 0 16 C 2 6 0 14 10 D -8 0 -14 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 D=21 C=21 A=21 E=14 so E is eliminated. Round 2 votes counts: B=31 D=27 C=21 A=21 so C is eliminated. Round 3 votes counts: B=43 A=30 D=27 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:207 A:202 D:188 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 8 2 B 4 0 -6 0 16 C 2 6 0 14 10 D -8 0 -14 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 8 2 B 4 0 -6 0 16 C 2 6 0 14 10 D -8 0 -14 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 8 2 B 4 0 -6 0 16 C 2 6 0 14 10 D -8 0 -14 0 -2 E -2 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4179: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) C D B A E (6) B D A C E (5) B A D E C (5) A E B D C (5) E C D B A (4) D C B E A (4) C D E A B (4) E A B C D (3) D B C A E (3) C E D A B (3) B D E A C (3) B A E D C (3) A B E D C (3) A B E C D (3) A B C D E (3) E A C B D (2) D E C B A (2) D C B A E (2) C D A B E (2) B D A E C (2) B A D C E (2) A E B C D (2) A C B E D (2) A B D E C (2) A B D C E (2) E D B C A (1) E D B A C (1) E C A D B (1) E C A B D (1) E A B D C (1) D B A C E (1) C E A D B (1) C D B E A (1) C A E B D (1) C A D B E (1) B E A D C (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 6 -8 14 B 12 0 2 6 12 C -6 -2 0 4 8 D 8 -6 -4 0 20 E -14 -12 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -8 14 B 12 0 2 6 12 C -6 -2 0 4 8 D 8 -6 -4 0 20 E -14 -12 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=24 B=21 E=14 D=12 so D is eliminated. Round 2 votes counts: C=35 B=25 A=24 E=16 so E is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 D:209 C:202 A:200 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 -8 14 B 12 0 2 6 12 C -6 -2 0 4 8 D 8 -6 -4 0 20 E -14 -12 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -8 14 B 12 0 2 6 12 C -6 -2 0 4 8 D 8 -6 -4 0 20 E -14 -12 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -8 14 B 12 0 2 6 12 C -6 -2 0 4 8 D 8 -6 -4 0 20 E -14 -12 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4180: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) B D C E A (7) B A C D E (7) B C D E A (5) D E B C A (4) A E D B C (4) A C E D B (4) A B C E D (4) E A D C B (3) D B E C A (3) C B A E D (3) B C A D E (3) A E C D B (3) A C B E D (3) E D C B A (2) D E C B A (2) D E A B C (2) D C E B A (2) C E D B A (2) C B D E A (2) B D E C A (2) B D A E C (2) B C D A E (2) A C E B D (2) A B E D C (2) E D A C B (1) E D A B C (1) E C A D B (1) D E B A C (1) D B E A C (1) C E D A B (1) C D E B A (1) C A E D B (1) C A B E D (1) B D E A C (1) B D C A E (1) B A D E C (1) B A D C E (1) A E C B D (1) A D E B C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -6 -6 -4 B 10 0 14 4 6 C 6 -14 0 -6 4 D 6 -4 6 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -6 -4 B 10 0 14 4 6 C 6 -14 0 -6 4 D 6 -4 6 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=26 E=16 D=15 C=11 so C is eliminated. Round 2 votes counts: B=37 A=28 E=19 D=16 so D is eliminated. Round 3 votes counts: B=41 E=31 A=28 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:206 C:195 E:195 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -6 -4 B 10 0 14 4 6 C 6 -14 0 -6 4 D 6 -4 6 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -6 -4 B 10 0 14 4 6 C 6 -14 0 -6 4 D 6 -4 6 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -6 -4 B 10 0 14 4 6 C 6 -14 0 -6 4 D 6 -4 6 0 4 E 4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4181: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) B A D C E (8) B C D E A (7) A B E C D (7) A E D C B (6) E A C D B (5) B E C D A (5) D C E B A (4) D C A E B (4) B E A C D (4) B D C E A (4) B D C A E (4) B A E C D (4) E C D A B (3) D C B E A (3) C D B E A (3) E B C D A (2) D C B A E (2) A E B D C (2) A D C E B (2) A B E D C (2) E C D B A (1) E C B D A (1) E B A C D (1) E A D C B (1) E A B C D (1) D C E A B (1) D C A B E (1) B A C D E (1) A E C D B (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 -6 -6 -8 B 24 0 6 8 14 C 6 -6 0 8 8 D 6 -8 -8 0 8 E 8 -14 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -6 -6 -8 B 24 0 6 8 14 C 6 -6 0 8 8 D 6 -8 -8 0 8 E 8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=22 E=15 D=15 C=11 so C is eliminated. Round 2 votes counts: B=37 D=26 A=22 E=15 so E is eliminated. Round 3 votes counts: B=41 D=30 A=29 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:208 D:199 E:189 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -6 -6 -8 B 24 0 6 8 14 C 6 -6 0 8 8 D 6 -8 -8 0 8 E 8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -6 -6 -8 B 24 0 6 8 14 C 6 -6 0 8 8 D 6 -8 -8 0 8 E 8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -6 -6 -8 B 24 0 6 8 14 C 6 -6 0 8 8 D 6 -8 -8 0 8 E 8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4182: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (12) A D C E B (7) D A B E C (6) C B E D A (6) C A D E B (5) B E C D A (5) B E C A D (5) A D E B C (5) E B C A D (3) D C A B E (3) D A C B E (3) A D E C B (3) D B E A C (2) D A E B C (2) D A C E B (2) C D B E A (2) C D A E B (2) C A E B D (2) B E A C D (2) B D E C A (2) A E B D C (2) A C E D B (2) A C E B D (2) E B A D C (1) E B A C D (1) E A B D C (1) D C B E A (1) D B C A E (1) D B A E C (1) D A E C B (1) C E B D A (1) C B E A D (1) B E D C A (1) B E D A C (1) B D E A C (1) B C E D A (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -6 12 2 B 0 0 -12 0 -12 C 6 12 0 6 8 D -12 0 -6 0 0 E -2 12 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 12 2 B 0 0 -12 0 -12 C 6 12 0 6 8 D -12 0 -6 0 0 E -2 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=23 D=22 B=18 E=6 so E is eliminated. Round 2 votes counts: C=31 A=24 B=23 D=22 so D is eliminated. Round 3 votes counts: A=38 C=35 B=27 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:204 E:201 D:191 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 12 2 B 0 0 -12 0 -12 C 6 12 0 6 8 D -12 0 -6 0 0 E -2 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 12 2 B 0 0 -12 0 -12 C 6 12 0 6 8 D -12 0 -6 0 0 E -2 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 12 2 B 0 0 -12 0 -12 C 6 12 0 6 8 D -12 0 -6 0 0 E -2 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4183: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D A C E B (7) C A D E B (7) B E D A C (7) D A E C B (5) C A D B E (4) E B D A C (3) E B C D A (3) E B C A D (3) D A C B E (3) A D C E B (3) A D C B E (3) A C D E B (3) E D A C B (2) E C B A D (2) E B D C A (2) D E A B C (2) C E A D B (2) C B A E D (2) C A B E D (2) C A B D E (2) B E D C A (2) B E C D A (2) B D A C E (2) B C A E D (2) A C D B E (2) E D C B A (1) E D B A C (1) E C B D A (1) E C A D B (1) D E A C B (1) D B A E C (1) D A B C E (1) C E B A D (1) C B A D E (1) C A E D B (1) C A E B D (1) B D E A C (1) B C E A D (1) A D B C E (1) Total count = 100 A B C D E A 0 6 -4 6 6 B -6 0 -14 -2 -4 C 4 14 0 4 2 D -6 2 -4 0 -2 E -6 4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 6 6 B -6 0 -14 -2 -4 C 4 14 0 4 2 D -6 2 -4 0 -2 E -6 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=23 D=20 E=19 A=12 so A is eliminated. Round 2 votes counts: C=28 D=27 B=26 E=19 so E is eliminated. Round 3 votes counts: B=37 C=32 D=31 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:207 E:199 D:195 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 6 6 B -6 0 -14 -2 -4 C 4 14 0 4 2 D -6 2 -4 0 -2 E -6 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 6 6 B -6 0 -14 -2 -4 C 4 14 0 4 2 D -6 2 -4 0 -2 E -6 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 6 6 B -6 0 -14 -2 -4 C 4 14 0 4 2 D -6 2 -4 0 -2 E -6 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4184: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) E C B A D (6) C B A E D (6) C B E A D (5) C B A D E (5) A D B C E (5) A B C D E (5) D E A C B (4) E D C B A (3) E D A B C (3) E B C A D (3) D C A B E (3) D A C B E (3) D A B C E (3) C A B D E (3) A B D C E (3) E D B C A (2) E C B D A (2) E B A C D (2) D E C A B (2) D A B E C (2) C D A B E (2) B C A E D (2) B A C E D (2) B A C D E (2) E D C A B (1) E D B A C (1) E C D B A (1) E B C D A (1) D C A E B (1) D A E B C (1) D A C E B (1) C D E A B (1) C D B A E (1) B E C A D (1) B C E A D (1) B A E C D (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -4 8 6 B -6 0 -2 0 14 C 4 2 0 6 10 D -8 0 -6 0 14 E -6 -14 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 8 6 B -6 0 -2 0 14 C 4 2 0 6 10 D -8 0 -6 0 14 E -6 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=25 C=23 A=15 B=9 so B is eliminated. Round 2 votes counts: D=28 E=26 C=26 A=20 so A is eliminated. Round 3 votes counts: D=37 C=36 E=27 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:208 B:203 D:200 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 8 6 B -6 0 -2 0 14 C 4 2 0 6 10 D -8 0 -6 0 14 E -6 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 8 6 B -6 0 -2 0 14 C 4 2 0 6 10 D -8 0 -6 0 14 E -6 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 8 6 B -6 0 -2 0 14 C 4 2 0 6 10 D -8 0 -6 0 14 E -6 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4185: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (7) A B E C D (6) D C E B A (5) D B E C A (5) C E D B A (4) B D A E C (4) C E D A B (3) C E A B D (3) B A D E C (3) A E B C D (3) A B D E C (3) A B C E D (3) E D C B A (2) E B D C A (2) D C A E B (2) D B A E C (2) D B A C E (2) C D A E B (2) C A D E B (2) B D E A C (2) B A E D C (2) A C E B D (2) E D B C A (1) E C B A D (1) E C A B D (1) D E C B A (1) D E B C A (1) D C B A E (1) D C A B E (1) D B C E A (1) D A C B E (1) D A B C E (1) C E A D B (1) C D E B A (1) C D E A B (1) B E D C A (1) B E D A C (1) B E A D C (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 -14 4 B 6 0 6 -6 4 C 4 -6 0 -10 -6 D 14 6 10 0 4 E -4 -4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -14 4 B 6 0 6 -6 4 C 4 -6 0 -10 -6 D 14 6 10 0 4 E -4 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=23 E=17 C=17 B=14 so B is eliminated. Round 2 votes counts: A=34 D=29 E=20 C=17 so C is eliminated. Round 3 votes counts: A=36 D=33 E=31 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 B:205 E:197 C:191 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -14 4 B 6 0 6 -6 4 C 4 -6 0 -10 -6 D 14 6 10 0 4 E -4 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -14 4 B 6 0 6 -6 4 C 4 -6 0 -10 -6 D 14 6 10 0 4 E -4 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -14 4 B 6 0 6 -6 4 C 4 -6 0 -10 -6 D 14 6 10 0 4 E -4 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4186: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) E B D A C (8) A C D E B (6) E B C A D (5) E B D C A (4) C A D E B (4) B E D C A (4) E B C D A (3) C B E D A (3) B D C A E (3) E D A B C (2) E C B A D (2) E B A D C (2) E A D B C (2) D B C A E (2) D B A C E (2) D A C B E (2) D A B C E (2) C E A B D (2) C D B A E (2) C B D E A (2) C A D B E (2) B E C D A (2) B D E C A (2) B C D E A (2) A E D B C (2) A E C D B (2) A D E B C (2) A C D B E (2) E A D C B (1) E A B D C (1) D B A E C (1) D A E B C (1) C D A B E (1) C B D A E (1) C A B E D (1) C A B D E (1) B E D A C (1) B D C E A (1) A E D C B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 2 -2 2 B 4 0 8 0 -4 C -2 -8 0 -12 2 D 2 0 12 0 2 E -2 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.281847 C: 0.000000 D: 0.718153 E: 0.000000 Sum of squares = 0.59518171218 Cumulative probabilities = A: 0.000000 B: 0.281847 C: 0.281847 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -2 2 B 4 0 8 0 -4 C -2 -8 0 -12 2 D 2 0 12 0 2 E -2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555680216 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=26 C=19 B=15 D=10 so D is eliminated. Round 2 votes counts: A=31 E=30 B=20 C=19 so C is eliminated. Round 3 votes counts: A=40 E=32 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:208 B:204 A:199 E:199 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -2 2 B 4 0 8 0 -4 C -2 -8 0 -12 2 D 2 0 12 0 2 E -2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555680216 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 2 B 4 0 8 0 -4 C -2 -8 0 -12 2 D 2 0 12 0 2 E -2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555680216 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 2 B 4 0 8 0 -4 C -2 -8 0 -12 2 D 2 0 12 0 2 E -2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555680216 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4187: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) C E B A D (9) D E A B C (8) E B C D A (7) E D B A C (6) E B D A C (6) A C D B E (6) C A D B E (5) C A B D E (5) D A E B C (4) C A D E B (4) D A B E C (3) B E C D A (3) E B D C A (2) C E B D A (2) C E A B D (2) C B A E D (2) C B A D E (2) C A E D B (2) B E D A C (2) A D C B E (2) D B A E C (1) D A E C B (1) C A E B D (1) B D A E C (1) B A E D C (1) A D C E B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -12 8 -14 B 8 0 -8 12 -10 C 12 8 0 22 10 D -8 -12 -22 0 -10 E 14 10 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 8 -14 B 8 0 -8 12 -10 C 12 8 0 22 10 D -8 -12 -22 0 -10 E 14 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 E=21 D=17 A=11 B=7 so B is eliminated. Round 2 votes counts: C=44 E=26 D=18 A=12 so A is eliminated. Round 3 votes counts: C=51 E=27 D=22 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:212 B:201 A:187 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 8 -14 B 8 0 -8 12 -10 C 12 8 0 22 10 D -8 -12 -22 0 -10 E 14 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 8 -14 B 8 0 -8 12 -10 C 12 8 0 22 10 D -8 -12 -22 0 -10 E 14 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 8 -14 B 8 0 -8 12 -10 C 12 8 0 22 10 D -8 -12 -22 0 -10 E 14 10 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4188: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C A D (8) D C A B E (7) B E A C D (6) E B A C D (4) D C B A E (4) D C A E B (4) D B C E A (4) C D B E A (4) D A B E C (3) C B E A D (3) C A E B D (3) B E A D C (3) A E B D C (3) E A B C D (2) D C B E A (2) D A C B E (2) C D A E B (2) C A D E B (2) B A E D C (2) A E D B C (2) A E C B D (2) A C D E B (2) D B E C A (1) C E B A D (1) C E A B D (1) C B E D A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C E D A (1) B C E A D (1) B C D E A (1) A E B C D (1) A D E C B (1) A D E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -6 0 8 B -2 0 -8 -8 12 C 6 8 0 -6 18 D 0 8 6 0 8 E -8 -12 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.243928 B: 0.000000 C: 0.000000 D: 0.756072 E: 0.000000 Sum of squares = 0.631145447716 Cumulative probabilities = A: 0.243928 B: 0.243928 C: 0.243928 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 0 8 B -2 0 -8 -8 12 C 6 8 0 -6 18 D 0 8 6 0 8 E -8 -12 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.497352 B: 0.000000 C: 0.000000 D: 0.502648 E: 0.000000 Sum of squares = 0.500014027424 Cumulative probabilities = A: 0.497352 B: 0.497352 C: 0.497352 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=25 C=17 A=14 E=6 so E is eliminated. Round 2 votes counts: D=38 B=29 C=17 A=16 so A is eliminated. Round 3 votes counts: D=43 B=35 C=22 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:211 A:202 B:197 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 0 8 B -2 0 -8 -8 12 C 6 8 0 -6 18 D 0 8 6 0 8 E -8 -12 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.497352 B: 0.000000 C: 0.000000 D: 0.502648 E: 0.000000 Sum of squares = 0.500014027424 Cumulative probabilities = A: 0.497352 B: 0.497352 C: 0.497352 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 8 B -2 0 -8 -8 12 C 6 8 0 -6 18 D 0 8 6 0 8 E -8 -12 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.497352 B: 0.000000 C: 0.000000 D: 0.502648 E: 0.000000 Sum of squares = 0.500014027424 Cumulative probabilities = A: 0.497352 B: 0.497352 C: 0.497352 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 8 B -2 0 -8 -8 12 C 6 8 0 -6 18 D 0 8 6 0 8 E -8 -12 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.497352 B: 0.000000 C: 0.000000 D: 0.502648 E: 0.000000 Sum of squares = 0.500014027424 Cumulative probabilities = A: 0.497352 B: 0.497352 C: 0.497352 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4189: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (15) C E A B D (12) D E C A B (9) B A C E D (9) E C D A B (8) B A D C E (8) E C A B D (6) C E B A D (5) D B A C E (4) A B C E D (4) E D C A B (3) D E A B C (3) C B A E D (3) C E D B A (2) C A B E D (2) D E C B A (1) D E B A C (1) D C E B A (1) D A B E C (1) C E D A B (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -6 2 -4 B 0 0 -8 2 -4 C 6 8 0 8 6 D -2 -2 -8 0 -12 E 4 4 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 2 -4 B 0 0 -8 2 -4 C 6 8 0 8 6 D -2 -2 -8 0 -12 E 4 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=25 B=18 E=17 A=5 so A is eliminated. Round 2 votes counts: D=35 C=26 B=22 E=17 so E is eliminated. Round 3 votes counts: C=40 D=38 B=22 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:207 A:196 B:195 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 2 -4 B 0 0 -8 2 -4 C 6 8 0 8 6 D -2 -2 -8 0 -12 E 4 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 -4 B 0 0 -8 2 -4 C 6 8 0 8 6 D -2 -2 -8 0 -12 E 4 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 -4 B 0 0 -8 2 -4 C 6 8 0 8 6 D -2 -2 -8 0 -12 E 4 4 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4190: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (7) D C E B A (5) A B E C D (5) E B A C D (4) C E A B D (4) B E A C D (4) A B D C E (4) E C D B A (3) E B D C A (3) D C E A B (3) D B E A C (3) D A B C E (3) C D E A B (3) C A E D B (3) B E D A C (3) B A E D C (3) B A D E C (3) A C D B E (3) A B C D E (3) E B C A D (2) D E C B A (2) D A C B E (2) C A E B D (2) C A D E B (2) A C B E D (2) A B C E D (2) E D B C A (1) E C B D A (1) D E B C A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B E C A (1) D B C E A (1) C E D B A (1) C E A D B (1) C D A E B (1) C A D B E (1) B E A D C (1) B D A E C (1) B A E C D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -2 4 -8 B -10 0 2 -2 0 C 2 -2 0 10 14 D -4 2 -10 0 -6 E 8 0 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.551020408152 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 4 -8 B -10 0 2 -2 0 C 2 -2 0 10 14 D -4 2 -10 0 -6 E 8 0 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040827 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 A=21 B=16 E=14 so E is eliminated. Round 2 votes counts: C=29 D=25 B=25 A=21 so A is eliminated. Round 3 votes counts: B=40 C=34 D=26 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 A:202 E:200 B:195 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 4 -8 B -10 0 2 -2 0 C 2 -2 0 10 14 D -4 2 -10 0 -6 E 8 0 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040827 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 4 -8 B -10 0 2 -2 0 C 2 -2 0 10 14 D -4 2 -10 0 -6 E 8 0 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040827 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 4 -8 B -10 0 2 -2 0 C 2 -2 0 10 14 D -4 2 -10 0 -6 E 8 0 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.55102040827 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4191: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) C B E A D (8) B D A C E (8) A D E C B (8) D A E B C (7) E C A D B (6) B C E D A (6) C E A B D (4) C B E D A (3) E B D C A (2) D B A C E (2) D A B E C (2) C E B D A (2) C B A E D (2) C A E D B (2) B C E A D (2) E D C B A (1) E D A C B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C B D A (1) E A D C B (1) E A C D B (1) D E A C B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A B C E (1) C B A D E (1) C A B E D (1) B D E C A (1) B D C E A (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) A D C B E (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -14 6 -8 B 10 0 -16 12 0 C 14 16 0 12 22 D -6 -12 -12 0 -12 E 8 0 -22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999542 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 6 -8 B 10 0 -16 12 0 C 14 16 0 12 22 D -6 -12 -12 0 -12 E 8 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=22 E=16 D=16 A=15 so A is eliminated. Round 2 votes counts: C=34 D=27 B=23 E=16 so E is eliminated. Round 3 votes counts: C=44 D=31 B=25 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:232 B:203 E:199 A:187 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 6 -8 B 10 0 -16 12 0 C 14 16 0 12 22 D -6 -12 -12 0 -12 E 8 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 6 -8 B 10 0 -16 12 0 C 14 16 0 12 22 D -6 -12 -12 0 -12 E 8 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 6 -8 B 10 0 -16 12 0 C 14 16 0 12 22 D -6 -12 -12 0 -12 E 8 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4192: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) D E B A C (10) A B E C D (9) D B E A C (7) C A B E D (7) C D E A B (4) C A E B D (4) B D E A C (4) A E B C D (4) E B A D C (3) D E B C A (3) C D A E B (3) C D A B E (3) B E A D C (3) B A E D C (3) E B D A C (2) D C E A B (2) D C B E A (2) C A E D B (2) C A D B E (2) C A B D E (2) A C B E D (2) E D A B C (1) E A B C D (1) D C B A E (1) D B E C A (1) C D B A E (1) B E D A C (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 4 -12 -10 B 4 0 6 -6 -4 C -4 -6 0 -10 -6 D 12 6 10 0 12 E 10 4 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -12 -10 B 4 0 6 -6 -4 C -4 -6 0 -10 -6 D 12 6 10 0 12 E 10 4 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=28 A=17 B=11 E=7 so E is eliminated. Round 2 votes counts: D=38 C=28 A=18 B=16 so B is eliminated. Round 3 votes counts: D=45 C=28 A=27 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:204 B:200 A:189 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -12 -10 B 4 0 6 -6 -4 C -4 -6 0 -10 -6 D 12 6 10 0 12 E 10 4 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -12 -10 B 4 0 6 -6 -4 C -4 -6 0 -10 -6 D 12 6 10 0 12 E 10 4 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -12 -10 B 4 0 6 -6 -4 C -4 -6 0 -10 -6 D 12 6 10 0 12 E 10 4 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4193: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) B A E C D (10) B A C D E (9) A B C D E (7) E D C A B (6) C D A B E (6) D C E A B (5) E B A D C (4) D E C A B (4) C D A E B (4) B A C E D (4) A C B D E (4) E D B C A (3) B E A D C (3) E C D B A (2) E B D A C (2) E B A C D (2) D A B C E (2) B A D C E (2) E D B A C (1) E C B A D (1) D E B C A (1) C D E A B (1) C A D B E (1) C A B D E (1) B E A C D (1) B A E D C (1) B A D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 10 6 8 B 14 0 10 8 8 C -10 -10 0 8 -4 D -6 -8 -8 0 -2 E -8 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 6 8 B 14 0 10 8 8 C -10 -10 0 8 -4 D -6 -8 -8 0 -2 E -8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=31 B=31 C=13 A=13 D=12 so D is eliminated. Round 2 votes counts: E=36 B=31 C=18 A=15 so A is eliminated. Round 3 votes counts: B=42 E=36 C=22 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:205 E:195 C:192 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 6 8 B 14 0 10 8 8 C -10 -10 0 8 -4 D -6 -8 -8 0 -2 E -8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 6 8 B 14 0 10 8 8 C -10 -10 0 8 -4 D -6 -8 -8 0 -2 E -8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 6 8 B 14 0 10 8 8 C -10 -10 0 8 -4 D -6 -8 -8 0 -2 E -8 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4194: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (14) A B E D C (13) C D A B E (11) E B A D C (6) C E D B A (5) C A D B E (5) C D E A B (4) D C E B A (3) D B E A C (3) B E A D C (3) A E B C D (3) E B D A C (2) E B A C D (2) D C A B E (2) C D A E B (2) A C D B E (2) A B E C D (2) A B D E C (2) E D B C A (1) E C D B A (1) E C B D A (1) E B D C A (1) E A C B D (1) D C B E A (1) D C B A E (1) D B C E A (1) D A C B E (1) C E B D A (1) C A E D B (1) C A E B D (1) B A E D C (1) A E B D C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -12 -10 0 B -6 0 -14 -18 0 C 12 14 0 14 14 D 10 18 -14 0 6 E 0 0 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -10 0 B -6 0 -14 -18 0 C 12 14 0 14 14 D 10 18 -14 0 6 E 0 0 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=25 E=15 D=12 B=4 so B is eliminated. Round 2 votes counts: C=44 A=26 E=18 D=12 so D is eliminated. Round 3 votes counts: C=52 A=27 E=21 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:210 A:192 E:190 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -10 0 B -6 0 -14 -18 0 C 12 14 0 14 14 D 10 18 -14 0 6 E 0 0 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -10 0 B -6 0 -14 -18 0 C 12 14 0 14 14 D 10 18 -14 0 6 E 0 0 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -10 0 B -6 0 -14 -18 0 C 12 14 0 14 14 D 10 18 -14 0 6 E 0 0 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4195: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) C E D B A (7) B A C D E (7) D E C B A (6) B C A D E (5) C E D A B (4) A E D C B (4) A B D E C (4) A B C E D (4) C E A D B (3) C D E B A (3) C B A E D (3) B C D E A (3) A B E D C (3) E A D C B (2) D E A B C (2) D B E C A (2) C B D E A (2) B D C E A (2) B A D C E (2) A C B E D (2) A B E C D (2) E D A C B (1) E D A B C (1) E C D A B (1) D E B A C (1) D E A C B (1) C D B E A (1) C B E A D (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D A E (1) B A D E C (1) B A C E D (1) A E C D B (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 10 16 12 B 4 0 12 0 0 C -10 -12 0 8 8 D -16 0 -8 0 -4 E -12 0 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.814710 C: 0.000000 D: 0.000000 E: 0.185290 Sum of squares = 0.698084314066 Cumulative probabilities = A: 0.000000 B: 0.814710 C: 0.814710 D: 0.814710 E: 1.000000 A B C D E A 0 -4 10 16 12 B 4 0 12 0 0 C -10 -12 0 8 8 D -16 0 -8 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500000497 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=26 C=24 D=12 E=5 so E is eliminated. Round 2 votes counts: A=35 B=26 C=25 D=14 so D is eliminated. Round 3 votes counts: A=40 C=31 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:208 C:197 E:192 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 16 12 B 4 0 12 0 0 C -10 -12 0 8 8 D -16 0 -8 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500000497 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 16 12 B 4 0 12 0 0 C -10 -12 0 8 8 D -16 0 -8 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500000497 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 16 12 B 4 0 12 0 0 C -10 -12 0 8 8 D -16 0 -8 0 -4 E -12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.62500000497 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4196: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (6) B E D C A (6) B D E A C (6) A D C B E (6) A C E D B (6) D B A C E (5) A C D B E (5) E B A C D (4) D B C A E (4) E B D C A (3) E B C D A (3) E A C B D (3) D C A B E (3) C D A B E (3) B E D A C (3) A D B C E (3) E C A B D (2) E B D A C (2) E B C A D (2) E B A D C (2) E A B C D (2) D A C B E (2) C E A D B (2) C A E D B (2) C A D B E (2) B D E C A (2) B D A E C (2) A C D E B (2) E C B D A (1) E C B A D (1) D C B E A (1) D C B A E (1) D B C E A (1) B E A D C (1) A E C B D (1) Total count = 100 A B C D E A 0 0 10 4 6 B 0 0 2 -8 12 C -10 -2 0 -6 8 D -4 8 6 0 8 E -6 -12 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.811853 B: 0.188147 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.694504075431 Cumulative probabilities = A: 0.811853 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 4 6 B 0 0 2 -8 12 C -10 -2 0 -6 8 D -4 8 6 0 8 E -6 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555565029 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 B=20 D=17 C=15 so C is eliminated. Round 2 votes counts: A=33 E=27 D=20 B=20 so D is eliminated. Round 3 votes counts: A=41 B=32 E=27 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:209 B:203 C:195 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 4 6 B 0 0 2 -8 12 C -10 -2 0 -6 8 D -4 8 6 0 8 E -6 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555565029 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 4 6 B 0 0 2 -8 12 C -10 -2 0 -6 8 D -4 8 6 0 8 E -6 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555565029 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 4 6 B 0 0 2 -8 12 C -10 -2 0 -6 8 D -4 8 6 0 8 E -6 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555565029 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4197: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (15) B D A E C (11) C E A D B (7) C A E D B (7) B D E A C (7) A C E D B (7) A C E B D (7) E C D A B (4) B D E C A (4) A B D C E (4) E D B C A (3) A C B D E (3) E D C B A (2) E C D B A (2) E C A D B (2) D E B C A (2) A C B E D (2) A B D E C (2) D B A E C (1) C D B E A (1) C B D E A (1) C A E B D (1) B D C E A (1) B D A C E (1) B A D C E (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -10 -2 B 4 0 6 -6 10 C 4 -6 0 -8 -12 D 10 6 8 0 10 E 2 -10 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -10 -2 B 4 0 6 -6 10 C 4 -6 0 -8 -12 D 10 6 8 0 10 E 2 -10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 D=18 C=17 E=13 so E is eliminated. Round 2 votes counts: A=27 C=25 B=25 D=23 so D is eliminated. Round 3 votes counts: B=46 C=27 A=27 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:217 B:207 E:197 A:190 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -10 -2 B 4 0 6 -6 10 C 4 -6 0 -8 -12 D 10 6 8 0 10 E 2 -10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -10 -2 B 4 0 6 -6 10 C 4 -6 0 -8 -12 D 10 6 8 0 10 E 2 -10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -10 -2 B 4 0 6 -6 10 C 4 -6 0 -8 -12 D 10 6 8 0 10 E 2 -10 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4198: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) B A C D E (7) A E D B C (7) B C A D E (6) A B C E D (6) A B E D C (5) A B D E C (5) E D C A B (4) E D A C B (3) C E B D A (3) C B D E A (3) B C D A E (3) A D E B C (3) A D B E C (3) E A C D B (2) D E C B A (2) D E A B C (2) C E D A B (2) C B A D E (2) B C A E D (2) E C D A B (1) E A D C B (1) D E B C A (1) D C E B A (1) D C B E A (1) C E A D B (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E D A (1) C B E A D (1) C B A E D (1) C A E B D (1) B D A E C (1) B A D C E (1) B A C E D (1) A E D C B (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 18 14 B 0 0 8 2 2 C 0 -8 0 18 16 D -18 -2 -18 0 -12 E -14 -2 -16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.408733 B: 0.591267 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.516659316671 Cumulative probabilities = A: 0.408733 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 18 14 B 0 0 8 2 2 C 0 -8 0 18 16 D -18 -2 -18 0 -12 E -14 -2 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=29 B=21 E=11 D=7 so D is eliminated. Round 2 votes counts: A=32 C=31 B=21 E=16 so E is eliminated. Round 3 votes counts: A=40 C=38 B=22 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:213 B:206 E:190 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 18 14 B 0 0 8 2 2 C 0 -8 0 18 16 D -18 -2 -18 0 -12 E -14 -2 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 18 14 B 0 0 8 2 2 C 0 -8 0 18 16 D -18 -2 -18 0 -12 E -14 -2 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 18 14 B 0 0 8 2 2 C 0 -8 0 18 16 D -18 -2 -18 0 -12 E -14 -2 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4199: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (14) E C B A D (12) E C A B D (9) C B E D A (8) A D E B C (7) D B A C E (6) A E D C B (5) E A C D B (4) E A C B D (4) C E B D A (4) A D B E C (4) D A B E C (3) C B D E A (3) B D C A E (3) D A C E B (2) B C D E A (2) A E D B C (2) E C A D B (1) E B A C D (1) D B C A E (1) B E D A C (1) B D C E A (1) B C E D A (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 14 10 2 -2 B -14 0 -6 0 -6 C -10 6 0 0 -10 D -2 0 0 0 -8 E 2 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 10 2 -2 B -14 0 -6 0 -6 C -10 6 0 0 -10 D -2 0 0 0 -8 E 2 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=26 A=20 C=15 B=8 so B is eliminated. Round 2 votes counts: E=32 D=30 A=20 C=18 so C is eliminated. Round 3 votes counts: E=45 D=35 A=20 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:212 D:195 C:193 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 10 2 -2 B -14 0 -6 0 -6 C -10 6 0 0 -10 D -2 0 0 0 -8 E 2 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 2 -2 B -14 0 -6 0 -6 C -10 6 0 0 -10 D -2 0 0 0 -8 E 2 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 2 -2 B -14 0 -6 0 -6 C -10 6 0 0 -10 D -2 0 0 0 -8 E 2 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4200: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) D C A B E (9) A E B D C (9) A D C E B (8) D A C E B (6) B E A D C (5) E B A C D (4) D C A E B (4) E A B C D (3) C D A E B (3) A D E C B (3) E B C A D (2) D C B E A (2) D C B A E (2) C D B E A (2) C D B A E (2) B E D C A (2) B E C A D (2) B E A C D (2) B C D E A (2) A E B C D (2) A D E B C (2) E B A D C (1) D A C B E (1) D A B C E (1) C E B A D (1) C E A B D (1) C B E D A (1) C A E B D (1) C A D E B (1) A E D C B (1) A E D B C (1) A E C B D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 18 4 4 18 B -18 0 2 2 -8 C -4 -2 0 -18 -6 D -4 -2 18 0 -2 E -18 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998565 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 4 18 B -18 0 2 2 -8 C -4 -2 0 -18 -6 D -4 -2 18 0 -2 E -18 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=25 B=24 C=12 E=10 so E is eliminated. Round 2 votes counts: A=32 B=31 D=25 C=12 so C is eliminated. Round 3 votes counts: A=35 B=33 D=32 so D is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:205 E:199 B:189 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 4 18 B -18 0 2 2 -8 C -4 -2 0 -18 -6 D -4 -2 18 0 -2 E -18 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 4 18 B -18 0 2 2 -8 C -4 -2 0 -18 -6 D -4 -2 18 0 -2 E -18 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 4 18 B -18 0 2 2 -8 C -4 -2 0 -18 -6 D -4 -2 18 0 -2 E -18 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4201: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C D B E A (9) A E B D C (7) E D B A C (5) E A C D B (5) C B D A E (5) A E C D B (5) E D A B C (4) D B E C A (4) C B A D E (4) A E C B D (4) D E B C A (3) C E D B A (3) C A B D E (3) A C B D E (3) E D C B A (2) D C B E A (2) C E A D B (2) C B D E A (2) B D C A E (2) A E D B C (2) A C B E D (2) E D B C A (1) E C D A B (1) E C A D B (1) E A D C B (1) D B C E A (1) C A E B D (1) B D E A C (1) B D A E C (1) B D A C E (1) B A D C E (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 6 -12 B -6 0 -12 -20 -14 C -8 12 0 2 -14 D -6 20 -2 0 -12 E 12 14 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 6 -12 B -6 0 -12 -20 -14 C -8 12 0 2 -14 D -6 20 -2 0 -12 E 12 14 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=29 C=29 A=26 D=10 B=6 so B is eliminated. Round 2 votes counts: E=29 C=29 A=27 D=15 so D is eliminated. Round 3 votes counts: E=37 C=34 A=29 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:204 D:200 C:196 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 6 -12 B -6 0 -12 -20 -14 C -8 12 0 2 -14 D -6 20 -2 0 -12 E 12 14 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 6 -12 B -6 0 -12 -20 -14 C -8 12 0 2 -14 D -6 20 -2 0 -12 E 12 14 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 6 -12 B -6 0 -12 -20 -14 C -8 12 0 2 -14 D -6 20 -2 0 -12 E 12 14 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4202: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) A D B C E (10) A E C D B (7) C E B D A (6) B D C E A (6) E A C B D (5) E C A B D (4) D B A C E (4) A D E B C (4) A D B E C (4) D B C A E (3) C E A B D (3) C B D E A (3) E B D C A (2) D B C E A (2) D A B C E (2) B D E C A (2) A E D C B (2) A E D B C (2) A E C B D (2) A D E C B (2) A D C B E (2) A C E D B (2) E B D A C (1) E B C D A (1) D B A E C (1) C B D A E (1) C A D B E (1) B E D C A (1) B C E D A (1) B C D E A (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 4 4 2 B -8 0 -6 0 -10 C -4 6 0 0 -2 D -4 0 0 0 0 E -2 10 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 4 2 B -8 0 -6 0 -10 C -4 6 0 0 -2 D -4 0 0 0 0 E -2 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=24 C=14 D=12 B=11 so B is eliminated. Round 2 votes counts: A=39 E=25 D=20 C=16 so C is eliminated. Round 3 votes counts: A=40 E=35 D=25 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:205 C:200 D:198 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 4 2 B -8 0 -6 0 -10 C -4 6 0 0 -2 D -4 0 0 0 0 E -2 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 4 2 B -8 0 -6 0 -10 C -4 6 0 0 -2 D -4 0 0 0 0 E -2 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 4 2 B -8 0 -6 0 -10 C -4 6 0 0 -2 D -4 0 0 0 0 E -2 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4203: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) C D A B E (6) C B E D A (6) B E C D A (5) D C A E B (4) C B D E A (4) C B A E D (4) B E C A D (4) B E A D C (4) B C E D A (4) A E D B C (4) A E B D C (4) A D E C B (4) A D E B C (4) E B D A C (3) E B A D C (3) C D B E A (3) C D B A E (3) C A D E B (3) B E A C D (3) D A C E B (2) C A D B E (2) B E D A C (2) A C D E B (2) E A B D C (1) D E C B A (1) D E B A C (1) D C E A B (1) D A E C B (1) D A E B C (1) C B E A D (1) C A B E D (1) B E D C A (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -18 -8 6 B 4 0 -10 2 8 C 18 10 0 16 6 D 8 -2 -16 0 -2 E -6 -8 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 -8 6 B 4 0 -10 2 8 C 18 10 0 16 6 D 8 -2 -16 0 -2 E -6 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=23 A=20 D=11 E=7 so E is eliminated. Round 2 votes counts: C=39 B=29 A=21 D=11 so D is eliminated. Round 3 votes counts: C=45 B=30 A=25 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:202 D:194 E:191 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -18 -8 6 B 4 0 -10 2 8 C 18 10 0 16 6 D 8 -2 -16 0 -2 E -6 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -8 6 B 4 0 -10 2 8 C 18 10 0 16 6 D 8 -2 -16 0 -2 E -6 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -8 6 B 4 0 -10 2 8 C 18 10 0 16 6 D 8 -2 -16 0 -2 E -6 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4204: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) B D E A C (9) E D A B C (6) E D B A C (4) E A D C B (4) D E B A C (4) D E A B C (4) C A E B D (4) C A B E D (4) B E D C A (4) B D E C A (4) B D C E A (4) C B A D E (3) A D C B E (3) E C A D B (2) D A E C B (2) C E A B D (2) C B A E D (2) C A E D B (2) B C D E A (2) B C D A E (2) A E D C B (2) A C D E B (2) E D B C A (1) E A C D B (1) D E A C B (1) D B E A C (1) C E B A D (1) C E A D B (1) C B E A D (1) C A B D E (1) B E C D A (1) B D C A E (1) B D A E C (1) B D A C E (1) A E C D B (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 16 -4 -14 B -8 0 -2 -4 -10 C -16 2 0 -14 -6 D 4 4 14 0 -6 E 14 10 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 16 -4 -14 B -8 0 -2 -4 -10 C -16 2 0 -14 -6 D 4 4 14 0 -6 E 14 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=21 A=20 E=18 D=12 so D is eliminated. Round 2 votes counts: B=30 E=27 A=22 C=21 so C is eliminated. Round 3 votes counts: B=36 A=33 E=31 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:218 D:208 A:203 B:188 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 -4 -14 B -8 0 -2 -4 -10 C -16 2 0 -14 -6 D 4 4 14 0 -6 E 14 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -4 -14 B -8 0 -2 -4 -10 C -16 2 0 -14 -6 D 4 4 14 0 -6 E 14 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -4 -14 B -8 0 -2 -4 -10 C -16 2 0 -14 -6 D 4 4 14 0 -6 E 14 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4205: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (15) D A B E C (10) D B E C A (8) A E B C D (5) A D B E C (4) A C E B D (4) A B D E C (4) D C B E A (3) D C A E B (3) D B A E C (3) A C D E B (3) A B E C D (3) E B C A D (2) E B A C D (2) D B E A C (2) D A B C E (2) C E B D A (2) B E D C A (2) A B E D C (2) E C B D A (1) D C E B A (1) D B C E A (1) D A C E B (1) D A C B E (1) C E D A B (1) C E A B D (1) C D E B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C A E D B (1) C A E B D (1) B E C D A (1) B E C A D (1) B D E C A (1) B D E A C (1) B A E D C (1) B A E C D (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 2 4 4 B 2 0 12 4 6 C -2 -12 0 -2 -10 D -4 -4 2 0 4 E -4 -6 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 4 4 B 2 0 12 4 6 C -2 -12 0 -2 -10 D -4 -4 2 0 4 E -4 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=27 C=25 B=8 E=5 so E is eliminated. Round 2 votes counts: D=35 A=27 C=26 B=12 so B is eliminated. Round 3 votes counts: D=39 A=31 C=30 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:212 A:204 D:199 E:198 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 4 4 B 2 0 12 4 6 C -2 -12 0 -2 -10 D -4 -4 2 0 4 E -4 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 4 B 2 0 12 4 6 C -2 -12 0 -2 -10 D -4 -4 2 0 4 E -4 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 4 B 2 0 12 4 6 C -2 -12 0 -2 -10 D -4 -4 2 0 4 E -4 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4206: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (5) B D E C A (5) A E B D C (5) A C D B E (5) E A C B D (4) D C B A E (4) D B C E A (4) A E C D B (4) E B D C A (3) D B A C E (3) C E B D A (3) C D B A E (3) B D C E A (3) A E C B D (3) A D B C E (3) A C E D B (3) A C D E B (3) A B D E C (3) E B A D C (2) E A B C D (2) D C B E A (2) D B C A E (2) D A B C E (2) C D B E A (2) C A D B E (2) A E D B C (2) A E B C D (2) A D B E C (2) E C A B D (1) E B D A C (1) E B C D A (1) C E D B A (1) C D A B E (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E A C (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 8 12 6 8 B -8 0 10 0 6 C -12 -10 0 -8 4 D -6 0 8 0 6 E -8 -6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 6 8 B -8 0 10 0 6 C -12 -10 0 -8 4 D -6 0 8 0 6 E -8 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=19 D=17 C=16 B=13 so B is eliminated. Round 2 votes counts: A=35 D=26 E=21 C=18 so C is eliminated. Round 3 votes counts: A=39 D=34 E=27 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:204 D:204 E:188 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 6 8 B -8 0 10 0 6 C -12 -10 0 -8 4 D -6 0 8 0 6 E -8 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 6 8 B -8 0 10 0 6 C -12 -10 0 -8 4 D -6 0 8 0 6 E -8 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 6 8 B -8 0 10 0 6 C -12 -10 0 -8 4 D -6 0 8 0 6 E -8 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4207: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (13) B D E A C (8) C B A E D (6) D B E A C (5) C A E B D (4) C A D E B (4) B D C E A (4) B C A E D (4) A E B D C (4) C D E B A (3) C A B E D (3) B E D A C (3) B E A D C (3) A E B C D (3) D E B A C (2) C D E A B (2) C B D E A (2) C B D A E (2) B D E C A (2) B C D E A (2) A E D B C (2) A E C D B (2) A E C B D (2) A C E D B (2) E D B A C (1) E A D B C (1) D C E B A (1) D C B E A (1) D B C E A (1) C D B E A (1) C D A E B (1) B A E C D (1) B A C E D (1) A C E B D (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -12 18 16 B 6 0 -2 16 4 C 12 2 0 22 18 D -18 -16 -22 0 -18 E -16 -4 -18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 18 16 B 6 0 -2 16 4 C 12 2 0 22 18 D -18 -16 -22 0 -18 E -16 -4 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 B=28 A=19 D=10 E=2 so E is eliminated. Round 2 votes counts: C=41 B=28 A=20 D=11 so D is eliminated. Round 3 votes counts: C=43 B=37 A=20 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:212 A:208 E:190 D:163 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 18 16 B 6 0 -2 16 4 C 12 2 0 22 18 D -18 -16 -22 0 -18 E -16 -4 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 18 16 B 6 0 -2 16 4 C 12 2 0 22 18 D -18 -16 -22 0 -18 E -16 -4 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 18 16 B 6 0 -2 16 4 C 12 2 0 22 18 D -18 -16 -22 0 -18 E -16 -4 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999956078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4208: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) E A C D B (7) C D B E A (7) B C D A E (7) A E B D C (7) E A D C B (6) A B E C D (6) D C B E A (5) B D C A E (5) A B E D C (5) C D E B A (4) B A D C E (4) B A C D E (4) D E C B A (3) A E C B D (3) E D C A B (2) E C D A B (2) E A B D C (2) D C E B A (2) D B C E A (2) C D B A E (2) C B D A E (2) E D A C B (1) E D A B C (1) D B E C A (1) B C A D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 10 8 10 B -2 0 8 10 4 C -10 -8 0 8 -8 D -8 -10 -8 0 0 E -10 -4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 8 10 B -2 0 8 10 4 C -10 -8 0 8 -8 D -8 -10 -8 0 0 E -10 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=21 B=21 C=15 D=13 so D is eliminated. Round 2 votes counts: A=30 E=24 B=24 C=22 so C is eliminated. Round 3 votes counts: B=40 E=30 A=30 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:210 E:197 C:191 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 10 B -2 0 8 10 4 C -10 -8 0 8 -8 D -8 -10 -8 0 0 E -10 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 10 B -2 0 8 10 4 C -10 -8 0 8 -8 D -8 -10 -8 0 0 E -10 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 10 B -2 0 8 10 4 C -10 -8 0 8 -8 D -8 -10 -8 0 0 E -10 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4209: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) B E D C A (6) E B D A C (5) E A D C B (4) D E B C A (4) C A B D E (4) A C B D E (4) E D A C B (3) E B D C A (3) E B A C D (3) E A D B C (3) D E A C B (3) C A D B E (3) B E C A D (3) B D E C A (3) A C E B D (3) E D B C A (2) E D B A C (2) E A C D B (2) E A B D C (2) D C A B E (2) D A C E B (2) C D B A E (2) C D A B E (2) B E C D A (2) B D C E A (2) B C A E D (2) B C A D E (2) A D C E B (2) E D A B C (1) E B A D C (1) E A C B D (1) E A B C D (1) D B C E A (1) C B A D E (1) B D C A E (1) B C E D A (1) A E D C B (1) A E C B D (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -6 -8 -16 B 6 0 14 18 0 C 6 -14 0 -6 -12 D 8 -18 6 0 -6 E 16 0 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.316073 C: 0.000000 D: 0.000000 E: 0.683927 Sum of squares = 0.567658362237 Cumulative probabilities = A: 0.000000 B: 0.316073 C: 0.316073 D: 0.316073 E: 1.000000 A B C D E A 0 -6 -6 -8 -16 B 6 0 14 18 0 C 6 -14 0 -6 -12 D 8 -18 6 0 -6 E 16 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=29 A=14 D=12 C=12 so D is eliminated. Round 2 votes counts: E=40 B=30 A=16 C=14 so C is eliminated. Round 3 votes counts: E=40 B=33 A=27 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:217 D:195 C:187 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -8 -16 B 6 0 14 18 0 C 6 -14 0 -6 -12 D 8 -18 6 0 -6 E 16 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -8 -16 B 6 0 14 18 0 C 6 -14 0 -6 -12 D 8 -18 6 0 -6 E 16 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -8 -16 B 6 0 14 18 0 C 6 -14 0 -6 -12 D 8 -18 6 0 -6 E 16 0 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4210: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) E C D B A (8) C E B D A (7) D A E B C (6) D E A C B (5) A B D C E (4) D A B E C (3) C E D B A (3) C D A B E (3) C B A E D (3) B A C D E (3) A D B E C (3) A D B C E (3) A B D E C (3) E D C A B (2) E C D A B (2) D E A B C (2) D A E C B (2) D A C E B (2) C E B A D (2) C D E A B (2) B C E A D (2) B C A E D (2) B A D C E (2) A E B D C (2) E D A C B (1) E C B D A (1) E A D B C (1) D E C A B (1) D C E A B (1) D C A E B (1) D A B C E (1) C E D A B (1) C B D E A (1) C A B D E (1) B E C A D (1) B E A D C (1) B E A C D (1) B C A D E (1) B A C E D (1) Total count = 100 A B C D E A 0 4 -8 -10 -8 B -4 0 -16 -6 -4 C 8 16 0 8 10 D 10 6 -8 0 0 E 8 4 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -10 -8 B -4 0 -16 -6 -4 C 8 16 0 8 10 D 10 6 -8 0 0 E 8 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 E=15 A=15 B=14 so B is eliminated. Round 2 votes counts: C=37 D=24 A=21 E=18 so E is eliminated. Round 3 votes counts: C=49 D=27 A=24 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:204 E:201 A:189 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -8 B -4 0 -16 -6 -4 C 8 16 0 8 10 D 10 6 -8 0 0 E 8 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -8 B -4 0 -16 -6 -4 C 8 16 0 8 10 D 10 6 -8 0 0 E 8 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -8 B -4 0 -16 -6 -4 C 8 16 0 8 10 D 10 6 -8 0 0 E 8 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4211: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (13) B C D A E (9) E A D B C (7) B C D E A (6) C D B A E (5) A E D C B (5) E A B D C (4) D A E C B (4) C B D A E (4) E A C D B (3) D C B A E (3) C D A E B (3) B E A D C (3) B D A E C (3) E B A C D (2) E A B C D (2) D C A B E (2) D A C E B (2) B E C D A (2) B E A C D (2) E C B A D (1) E C A B D (1) D B C A E (1) D A C B E (1) D A B E C (1) C E D A B (1) C D A B E (1) C B E D A (1) C B E A D (1) C B D E A (1) B E C A D (1) B D C A E (1) B C E D A (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 6 10 -4 -4 B -6 0 -8 -10 -2 C -10 8 0 -6 -12 D 4 10 6 0 0 E 4 2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.475843 E: 0.524157 Sum of squares = 0.50116708069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.475843 E: 1.000000 A B C D E A 0 6 10 -4 -4 B -6 0 -8 -10 -2 C -10 8 0 -6 -12 D 4 10 6 0 0 E 4 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=28 C=17 D=14 A=8 so A is eliminated. Round 2 votes counts: E=38 B=28 D=17 C=17 so D is eliminated. Round 3 votes counts: E=44 B=30 C=26 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:209 A:204 C:190 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -4 -4 B -6 0 -8 -10 -2 C -10 8 0 -6 -12 D 4 10 6 0 0 E 4 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -4 -4 B -6 0 -8 -10 -2 C -10 8 0 -6 -12 D 4 10 6 0 0 E 4 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -4 -4 B -6 0 -8 -10 -2 C -10 8 0 -6 -12 D 4 10 6 0 0 E 4 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4212: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (15) A B E C D (11) D C E B A (10) E C D B A (9) A B E D C (9) A B D C E (9) C D E B A (8) B A E C D (5) E C B A D (4) D C A B E (3) C D E A B (3) E B C A D (2) D C A E B (2) D A B C E (2) C E D B A (2) E B A C D (1) D A C B E (1) B E A C D (1) B A E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -16 -10 -10 B -14 0 -14 -10 -12 C 16 14 0 -6 12 D 10 10 6 0 10 E 10 12 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -16 -10 -10 B -14 0 -14 -10 -12 C 16 14 0 -6 12 D 10 10 6 0 10 E 10 12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=31 E=16 C=13 B=7 so B is eliminated. Round 2 votes counts: A=37 D=33 E=17 C=13 so C is eliminated. Round 3 votes counts: D=44 A=37 E=19 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:218 E:200 A:189 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -16 -10 -10 B -14 0 -14 -10 -12 C 16 14 0 -6 12 D 10 10 6 0 10 E 10 12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -16 -10 -10 B -14 0 -14 -10 -12 C 16 14 0 -6 12 D 10 10 6 0 10 E 10 12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -16 -10 -10 B -14 0 -14 -10 -12 C 16 14 0 -6 12 D 10 10 6 0 10 E 10 12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4213: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) C E B A D (7) C A B E D (7) D A B E C (5) C E D B A (5) C D A E B (5) E B D A C (4) D E C B A (4) A C B D E (4) A B D E C (4) D E A B C (3) A C B E D (3) A B E D C (3) E D B C A (2) E D B A C (2) E C D B A (2) E C B D A (2) C A E D B (2) C A B D E (2) B E D A C (2) A B D C E (2) A B C E D (2) E B C D A (1) E B C A D (1) E B A D C (1) D E C A B (1) D C A E B (1) D A E C B (1) D A E B C (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E A B (1) C B E A D (1) C B A E D (1) B E C A D (1) B E A C D (1) B A E D C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 2 -4 -4 B -6 0 -8 4 -14 C -2 8 0 6 0 D 4 -4 -6 0 -8 E 4 14 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.537185 D: 0.000000 E: 0.462815 Sum of squares = 0.502765417872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.537185 D: 0.537185 E: 1.000000 A B C D E A 0 6 2 -4 -4 B -6 0 -8 4 -14 C -2 8 0 6 0 D 4 -4 -6 0 -8 E 4 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=24 A=22 E=15 B=5 so B is eliminated. Round 2 votes counts: C=34 D=24 A=23 E=19 so E is eliminated. Round 3 votes counts: C=41 D=34 A=25 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:206 A:200 D:193 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 -4 -4 B -6 0 -8 4 -14 C -2 8 0 6 0 D 4 -4 -6 0 -8 E 4 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -4 -4 B -6 0 -8 4 -14 C -2 8 0 6 0 D 4 -4 -6 0 -8 E 4 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -4 -4 B -6 0 -8 4 -14 C -2 8 0 6 0 D 4 -4 -6 0 -8 E 4 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4214: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (12) A D C E B (12) B E C D A (11) D A E C B (10) B E D A C (7) C A B D E (5) B E C A D (4) E B D A C (3) D A E B C (3) C B E A D (3) C A D E B (3) C A D B E (3) B E D C A (3) D E A B C (2) D A B E C (2) B C A D E (2) E D A B C (1) E A D C B (1) D E B A C (1) D E A C B (1) D B A E C (1) D A C E B (1) C B A E D (1) C B A D E (1) C A E D B (1) C A B E D (1) B D E A C (1) B D A C E (1) B C A E D (1) B A D C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -2 4 0 B 6 0 12 14 20 C 2 -12 0 -2 -2 D -4 -14 2 0 2 E 0 -20 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 4 0 B 6 0 12 14 20 C 2 -12 0 -2 -2 D -4 -14 2 0 2 E 0 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 D=21 C=18 A=13 E=5 so E is eliminated. Round 2 votes counts: B=46 D=22 C=18 A=14 so A is eliminated. Round 3 votes counts: B=46 D=35 C=19 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:198 C:193 D:193 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 4 0 B 6 0 12 14 20 C 2 -12 0 -2 -2 D -4 -14 2 0 2 E 0 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 4 0 B 6 0 12 14 20 C 2 -12 0 -2 -2 D -4 -14 2 0 2 E 0 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 4 0 B 6 0 12 14 20 C 2 -12 0 -2 -2 D -4 -14 2 0 2 E 0 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4215: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) D E B A C (8) B C D E A (7) C E D B A (6) B D E C A (6) B D E A C (6) C A E D B (5) B A D E C (5) A B C D E (5) D E B C A (4) A E D C B (4) A C B E D (4) E D C A B (3) C A B E D (3) B C A D E (3) A B D E C (3) E D C B A (2) E D A B C (2) D E C B A (2) C E D A B (2) B A C D E (2) A E D B C (2) E C D A B (1) D E A B C (1) D B E A C (1) C D E B A (1) C B D E A (1) C B A E D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 6 -22 -22 B 10 0 12 -8 -6 C -6 -12 0 -16 -18 D 22 8 16 0 12 E 22 6 18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999501 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -22 -22 B 10 0 12 -8 -6 C -6 -12 0 -16 -18 D 22 8 16 0 12 E 22 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=20 C=19 E=16 D=16 so E is eliminated. Round 2 votes counts: D=31 B=29 C=20 A=20 so C is eliminated. Round 3 votes counts: D=41 B=31 A=28 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:217 B:204 A:176 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 6 -22 -22 B 10 0 12 -8 -6 C -6 -12 0 -16 -18 D 22 8 16 0 12 E 22 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -22 -22 B 10 0 12 -8 -6 C -6 -12 0 -16 -18 D 22 8 16 0 12 E 22 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -22 -22 B 10 0 12 -8 -6 C -6 -12 0 -16 -18 D 22 8 16 0 12 E 22 6 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4216: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) D A B E C (6) E A D C B (4) D B A E C (4) C E B D A (4) C E B A D (4) C B E D A (4) B D A E C (4) B D A C E (4) B C D A E (4) E D C A B (3) E C D A B (3) D A E B C (3) C E A D B (3) C B E A D (3) C B D E A (3) B A D E C (3) B A D C E (3) A B D E C (3) E D A C B (2) D B C A E (2) C E D B A (2) C E D A B (2) C E A B D (2) C A E B D (2) B A C E D (2) A D E B C (2) A D B E C (2) E C A D B (1) D E B A C (1) D E A C B (1) C D B E A (1) C B A E D (1) B D C A E (1) B C A E D (1) B C A D E (1) A E D B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 6 -22 0 B 4 0 10 0 6 C -6 -10 0 -12 0 D 22 0 12 0 10 E 0 -6 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.692183 C: 0.000000 D: 0.307817 E: 0.000000 Sum of squares = 0.573868476224 Cumulative probabilities = A: 0.000000 B: 0.692183 C: 0.692183 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -22 0 B 4 0 10 0 6 C -6 -10 0 -12 0 D 22 0 12 0 10 E 0 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=24 B=23 E=13 A=9 so A is eliminated. Round 2 votes counts: C=31 D=28 B=27 E=14 so E is eliminated. Round 3 votes counts: D=38 C=35 B=27 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:210 E:192 A:190 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 -22 0 B 4 0 10 0 6 C -6 -10 0 -12 0 D 22 0 12 0 10 E 0 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -22 0 B 4 0 10 0 6 C -6 -10 0 -12 0 D 22 0 12 0 10 E 0 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -22 0 B 4 0 10 0 6 C -6 -10 0 -12 0 D 22 0 12 0 10 E 0 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4217: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) E A B D C (7) B D E C A (6) A E B C D (6) B D C E A (5) A C E B D (5) E D B A C (3) D C B E A (3) C D B A E (3) C D A E B (3) C D A B E (3) C A B D E (3) B E D A C (3) A E C D B (3) A C E D B (3) A C B E D (3) E A D C B (2) E A D B C (2) D E C A B (2) D E B C A (2) D B C E A (2) C B A D E (2) C A D B E (2) B E A C D (2) B C D A E (2) B C A E D (2) A E C B D (2) E B D A C (1) E A B C D (1) D E C B A (1) D E B A C (1) D C E B A (1) D C E A B (1) C A B E D (1) B C A D E (1) B A E C D (1) A E D C B (1) A B E C D (1) Total count = 100 A B C D E A 0 18 -6 16 10 B -18 0 -4 8 -10 C 6 4 0 14 6 D -16 -8 -14 0 2 E -10 10 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -6 16 10 B -18 0 -4 8 -10 C 6 4 0 14 6 D -16 -8 -14 0 2 E -10 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=24 B=22 E=16 D=13 so D is eliminated. Round 2 votes counts: C=30 B=24 A=24 E=22 so E is eliminated. Round 3 votes counts: A=36 C=33 B=31 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:215 E:196 B:188 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -6 16 10 B -18 0 -4 8 -10 C 6 4 0 14 6 D -16 -8 -14 0 2 E -10 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -6 16 10 B -18 0 -4 8 -10 C 6 4 0 14 6 D -16 -8 -14 0 2 E -10 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -6 16 10 B -18 0 -4 8 -10 C 6 4 0 14 6 D -16 -8 -14 0 2 E -10 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4218: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) B C A E D (10) A E B D C (8) D E A C B (6) D B C E A (6) D C B E A (5) A E C D B (5) E A D B C (4) D E A B C (4) C B A E D (4) E A B D C (3) C A E B D (3) B A E C D (3) D C E A B (2) C D B E A (2) C D A E B (2) C A E D B (2) B E A D C (2) B D C E A (2) B C D A E (2) A E D B C (2) A E C B D (2) A E B C D (2) E A D C B (1) D E C A B (1) D C E B A (1) D B E C A (1) D B E A C (1) C D A B E (1) C B D E A (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -10 6 16 B 0 0 0 6 0 C 10 0 0 0 8 D -6 -6 0 0 -6 E -16 0 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.469161 C: 0.530839 D: 0.000000 E: 0.000000 Sum of squares = 0.50190204688 Cumulative probabilities = A: 0.000000 B: 0.469161 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 6 16 B 0 0 0 6 0 C 10 0 0 0 8 D -6 -6 0 0 -6 E -16 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 A=21 B=19 E=8 so E is eliminated. Round 2 votes counts: A=29 D=27 C=25 B=19 so B is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:206 B:203 D:191 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -10 6 16 B 0 0 0 6 0 C 10 0 0 0 8 D -6 -6 0 0 -6 E -16 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 6 16 B 0 0 0 6 0 C 10 0 0 0 8 D -6 -6 0 0 -6 E -16 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 6 16 B 0 0 0 6 0 C 10 0 0 0 8 D -6 -6 0 0 -6 E -16 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4219: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (18) D E B C A (14) D E C B A (6) D E B A C (5) C A B E D (5) A C B D E (4) A B C E D (4) A C D B E (3) A B E C D (3) E D B C A (2) D E A B C (2) D C E B A (2) D C A E B (2) D A C E B (2) C B E D A (2) C B E A D (2) C A D B E (2) B E C A D (2) B E A C D (2) B A E C D (2) B A C E D (2) A D C E B (2) E D B A C (1) E B D C A (1) E B D A C (1) D E A C B (1) D C E A B (1) C D B A E (1) C B A E D (1) C A B D E (1) B E C D A (1) B E A D C (1) B C E A D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 8 12 6 B -2 0 -12 6 16 C -8 12 0 12 12 D -12 -6 -12 0 -2 E -6 -16 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999558 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 12 6 B -2 0 -12 6 16 C -8 12 0 12 12 D -12 -6 -12 0 -2 E -6 -16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=35 A=35 C=14 B=11 E=5 so E is eliminated. Round 2 votes counts: D=38 A=35 C=14 B=13 so B is eliminated. Round 3 votes counts: A=42 D=40 C=18 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:214 B:204 D:184 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 12 6 B -2 0 -12 6 16 C -8 12 0 12 12 D -12 -6 -12 0 -2 E -6 -16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 12 6 B -2 0 -12 6 16 C -8 12 0 12 12 D -12 -6 -12 0 -2 E -6 -16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 12 6 B -2 0 -12 6 16 C -8 12 0 12 12 D -12 -6 -12 0 -2 E -6 -16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4220: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (10) D A E B C (6) C B E D A (6) C B E A D (6) B E C D A (6) E B D A C (5) E B C A D (4) D A C B E (4) D A B E C (4) C E B A D (4) B C E D A (4) E C B A D (3) C E A B D (3) C A E B D (3) C A D B E (3) A D E C B (3) A D E B C (3) A D C B E (3) E B D C A (2) C D A B E (2) C B D E A (2) E D A B C (1) E B A D C (1) E B A C D (1) E A B D C (1) D E B A C (1) D B E A C (1) D B A C E (1) C B D A E (1) C B A D E (1) C A B E D (1) B E D C A (1) B E D A C (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 -4 4 -6 B 2 0 -12 12 -6 C 4 12 0 0 8 D -4 -12 0 0 -10 E 6 6 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.707037 D: 0.292963 E: 0.000000 Sum of squares = 0.585728905131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.707037 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 4 -6 B 2 0 -12 12 -6 C 4 12 0 0 8 D -4 -12 0 0 -10 E 6 6 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.444444 E: 0.000000 Sum of squares = 0.50617285291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=21 E=18 D=17 B=12 so B is eliminated. Round 2 votes counts: C=36 E=26 A=21 D=17 so D is eliminated. Round 3 votes counts: C=36 A=36 E=28 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:207 B:198 A:196 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 4 -6 B 2 0 -12 12 -6 C 4 12 0 0 8 D -4 -12 0 0 -10 E 6 6 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.444444 E: 0.000000 Sum of squares = 0.50617285291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 -6 B 2 0 -12 12 -6 C 4 12 0 0 8 D -4 -12 0 0 -10 E 6 6 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.444444 E: 0.000000 Sum of squares = 0.50617285291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 -6 B 2 0 -12 12 -6 C 4 12 0 0 8 D -4 -12 0 0 -10 E 6 6 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.444444 E: 0.000000 Sum of squares = 0.50617285291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4221: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) E C A D B (8) D A B C E (8) C E A B D (6) E C A B D (5) D A C E B (5) B E D C A (5) D B A C E (4) B E C A D (4) B A C D E (4) D A C B E (3) B D E A C (3) E D B C A (2) E C B A D (2) D B E A C (2) D B A E C (2) C A E D B (2) B C E A D (2) A D C E B (2) A C D E B (2) E D C B A (1) E D C A B (1) E C D A B (1) E C B D A (1) E B C A D (1) D E B C A (1) D E A C B (1) D E A B C (1) D A B E C (1) C B E A D (1) C A E B D (1) C A B E D (1) B E C D A (1) B D E C A (1) B C A E D (1) B A C E D (1) A C E B D (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 -6 0 B -4 0 10 6 12 C -4 -10 0 -6 12 D 6 -6 6 0 4 E 0 -12 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.343750000007 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -6 0 B -4 0 10 6 12 C -4 -10 0 -6 12 D 6 -6 6 0 4 E 0 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999961 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=28 E=22 C=11 A=8 so A is eliminated. Round 2 votes counts: B=33 D=30 E=22 C=15 so C is eliminated. Round 3 votes counts: B=36 E=32 D=32 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:205 A:201 C:196 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 4 -6 0 B -4 0 10 6 12 C -4 -10 0 -6 12 D 6 -6 6 0 4 E 0 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999961 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -6 0 B -4 0 10 6 12 C -4 -10 0 -6 12 D 6 -6 6 0 4 E 0 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999961 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -6 0 B -4 0 10 6 12 C -4 -10 0 -6 12 D 6 -6 6 0 4 E 0 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999961 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4222: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) E A B C D (8) E B A C D (7) D C A B E (7) D A C E B (6) A D C E B (6) A C D E B (6) D C B A E (5) B E C A D (5) B D C E A (5) A E D C B (4) D A C B E (3) C D A B E (3) B E C D A (3) E B C A D (2) E A B D C (2) C D B A E (2) C D A E B (2) C B D E A (2) C A D E B (2) A E C B D (2) E B D A C (1) D C B E A (1) D B C E A (1) B E D C A (1) B E D A C (1) B C E D A (1) B C D E A (1) A E C D B (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 8 14 10 2 B -8 0 -4 2 -18 C -14 4 0 -4 6 D -10 -2 4 0 4 E -2 18 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999273 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 10 2 B -8 0 -4 2 -18 C -14 4 0 -4 6 D -10 -2 4 0 4 E -2 18 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 A=21 B=17 C=11 so C is eliminated. Round 2 votes counts: D=30 E=28 A=23 B=19 so B is eliminated. Round 3 votes counts: E=39 D=38 A=23 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:217 E:203 D:198 C:196 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 10 2 B -8 0 -4 2 -18 C -14 4 0 -4 6 D -10 -2 4 0 4 E -2 18 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 10 2 B -8 0 -4 2 -18 C -14 4 0 -4 6 D -10 -2 4 0 4 E -2 18 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 10 2 B -8 0 -4 2 -18 C -14 4 0 -4 6 D -10 -2 4 0 4 E -2 18 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4223: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (13) D B E C A (12) C D E B A (6) B D E A C (6) A E B C D (6) A B E D C (6) D C B E A (5) C A D E B (5) D B C E A (3) C E A B D (3) A E C B D (3) A C E D B (3) E A B C D (2) D B E A C (2) D A B E C (2) C E D B A (2) C A E D B (2) C A E B D (2) E C B D A (1) E C B A D (1) E B D C A (1) E B C D A (1) E B C A D (1) D C B A E (1) D B A E C (1) D B A C E (1) C E D A B (1) C E B D A (1) B E D C A (1) B E A D C (1) B D A E C (1) B A D E C (1) A D C B E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 4 0 B -2 0 -2 4 -10 C -2 2 0 8 0 D -4 -4 -8 0 -2 E 0 10 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.716836 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.283164 Sum of squares = 0.594035346263 Cumulative probabilities = A: 0.716836 B: 0.716836 C: 0.716836 D: 0.716836 E: 1.000000 A B C D E A 0 2 2 4 0 B -2 0 -2 4 -10 C -2 2 0 8 0 D -4 -4 -8 0 -2 E 0 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=27 C=22 B=10 E=7 so E is eliminated. Round 2 votes counts: A=36 D=27 C=24 B=13 so B is eliminated. Round 3 votes counts: A=38 D=36 C=26 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:204 C:204 B:195 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 0 B -2 0 -2 4 -10 C -2 2 0 8 0 D -4 -4 -8 0 -2 E 0 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 0 B -2 0 -2 4 -10 C -2 2 0 8 0 D -4 -4 -8 0 -2 E 0 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 0 B -2 0 -2 4 -10 C -2 2 0 8 0 D -4 -4 -8 0 -2 E 0 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4224: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) C A E D B (6) A C D B E (6) A C B D E (6) B D E A C (5) D C A E B (4) D C A B E (4) B E D A C (4) A B C E D (4) E D B C A (3) D E B C A (3) A B C D E (3) E D C A B (2) E C A B D (2) D E C B A (2) D C E A B (2) D B E C A (2) D A C B E (2) C A D E B (2) B E A C D (2) B D A E C (2) B A E C D (2) B A C D E (2) A C E B D (2) A C D E B (2) E D C B A (1) E C B A D (1) E C A D B (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C B D (1) D E C A B (1) D B A C E (1) C E A D B (1) C A D B E (1) B E A D C (1) B D E C A (1) B A E D C (1) B A C E D (1) A E C B D (1) A D B C E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 12 6 6 10 B -12 0 -2 6 6 C -6 2 0 0 4 D -6 -6 0 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 6 10 B -12 0 -2 6 6 C -6 2 0 0 4 D -6 -6 0 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=21 D=21 B=21 C=10 so C is eliminated. Round 2 votes counts: A=36 E=22 D=21 B=21 so D is eliminated. Round 3 votes counts: A=46 E=30 B=24 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:200 B:199 D:197 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 6 10 B -12 0 -2 6 6 C -6 2 0 0 4 D -6 -6 0 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 6 10 B -12 0 -2 6 6 C -6 2 0 0 4 D -6 -6 0 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 6 10 B -12 0 -2 6 6 C -6 2 0 0 4 D -6 -6 0 0 6 E -10 -6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4225: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) A C D E B (9) E B D A C (8) D C A B E (7) C D A B E (5) C A D B E (5) E B A D C (4) E B A C D (4) D C B A E (4) A D C E B (4) D A C B E (3) B D E C A (3) A C E D B (3) E B D C A (2) E A B D C (2) E A B C D (2) C D B A E (2) C B D E A (2) B E C D A (2) B D C E A (2) A E D B C (2) A E C D B (2) A D C B E (2) E B C D A (1) E B C A D (1) D B E C A (1) D A E B C (1) D A C E B (1) C B E A D (1) C A D E B (1) B E D A C (1) B C D E A (1) A E D C B (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 2 -10 8 B -4 0 -8 -8 2 C -2 8 0 -14 6 D 10 8 14 0 8 E -8 -2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -10 8 B -4 0 -8 -8 2 C -2 8 0 -14 6 D 10 8 14 0 8 E -8 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 B=18 D=17 C=16 so C is eliminated. Round 2 votes counts: A=31 E=24 D=24 B=21 so B is eliminated. Round 3 votes counts: E=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:202 C:199 B:191 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -10 8 B -4 0 -8 -8 2 C -2 8 0 -14 6 D 10 8 14 0 8 E -8 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -10 8 B -4 0 -8 -8 2 C -2 8 0 -14 6 D 10 8 14 0 8 E -8 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -10 8 B -4 0 -8 -8 2 C -2 8 0 -14 6 D 10 8 14 0 8 E -8 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4226: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) D A B C E (7) E C B A D (5) B C E D A (5) A C D B E (5) A D E C B (4) A D E B C (4) A D C B E (4) A D B E C (4) A D B C E (4) D A B E C (3) C B E D A (3) C B D A E (3) B C D E A (3) A C E D B (3) E C B D A (2) E B D C A (2) D B E A C (2) D B A C E (2) C E B A D (2) C A E B D (2) C A D B E (2) C A B E D (2) C A B D E (2) B E D C A (2) B D C E A (2) A C D E B (2) E D A B C (1) E A D B C (1) D E B A C (1) D A E B C (1) C E B D A (1) C E A B D (1) C B A D E (1) B E C D A (1) B D E C A (1) B D C A E (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 8 0 0 16 B -8 0 10 -4 18 C 0 -10 0 6 14 D 0 4 -6 0 16 E -16 -18 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.692988 B: 0.000000 C: 0.307012 D: 0.000000 E: 0.000000 Sum of squares = 0.574488644537 Cumulative probabilities = A: 0.692988 B: 0.692988 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 0 16 B -8 0 10 -4 18 C 0 -10 0 6 14 D 0 4 -6 0 16 E -16 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555566 B: 0.000000 C: 0.444434 D: 0.000000 E: 0.000000 Sum of squares = 0.506175083331 Cumulative probabilities = A: 0.555566 B: 0.555566 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=19 E=18 D=16 B=15 so B is eliminated. Round 2 votes counts: A=32 C=27 E=21 D=20 so D is eliminated. Round 3 votes counts: A=45 C=30 E=25 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:208 D:207 C:205 E:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 0 16 B -8 0 10 -4 18 C 0 -10 0 6 14 D 0 4 -6 0 16 E -16 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555566 B: 0.000000 C: 0.444434 D: 0.000000 E: 0.000000 Sum of squares = 0.506175083331 Cumulative probabilities = A: 0.555566 B: 0.555566 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 0 16 B -8 0 10 -4 18 C 0 -10 0 6 14 D 0 4 -6 0 16 E -16 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555566 B: 0.000000 C: 0.444434 D: 0.000000 E: 0.000000 Sum of squares = 0.506175083331 Cumulative probabilities = A: 0.555566 B: 0.555566 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 0 16 B -8 0 10 -4 18 C 0 -10 0 6 14 D 0 4 -6 0 16 E -16 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555566 B: 0.000000 C: 0.444434 D: 0.000000 E: 0.000000 Sum of squares = 0.506175083331 Cumulative probabilities = A: 0.555566 B: 0.555566 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4227: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (9) A B D C E (8) E C B D A (5) C D E A B (5) B A E D C (5) A D C B E (5) E B C D A (4) E B A C D (4) D C A B E (4) B E A D C (4) A D B C E (4) E C D B A (3) E B C A D (3) D C A E B (3) C D A E B (3) E C D A B (2) C E D B A (2) C E D A B (2) C D E B A (2) B A C D E (2) A E D B C (2) A B E D C (2) A B D E C (2) E D C A B (1) E D A C B (1) C D B A E (1) C B E D A (1) B C E D A (1) B A D C E (1) Total count = 100 A B C D E A 0 6 -2 2 2 B -6 0 2 2 -2 C 2 -2 0 -2 6 D -2 -2 2 0 -2 E -2 2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333311 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 2 2 B -6 0 2 2 -2 C 2 -2 0 -2 6 D -2 -2 2 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=25 A=23 B=13 D=7 so D is eliminated. Round 2 votes counts: E=32 C=32 A=23 B=13 so B is eliminated. Round 3 votes counts: E=36 C=33 A=31 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:204 C:202 B:198 D:198 E:198 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 2 2 B -6 0 2 2 -2 C 2 -2 0 -2 6 D -2 -2 2 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 2 2 B -6 0 2 2 -2 C 2 -2 0 -2 6 D -2 -2 2 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 2 2 B -6 0 2 2 -2 C 2 -2 0 -2 6 D -2 -2 2 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.33333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4228: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (11) C E D A B (9) B A C E D (8) A C B E D (8) D E B C A (6) D E C A B (5) A C B D E (5) B A C D E (4) A C E D B (4) A B C E D (4) E D C B A (3) E D C A B (3) C A E D B (3) B E D C A (3) B A D E C (3) D E C B A (2) D E B A C (2) C E D B A (2) C D E A B (2) B D E C A (2) A C D E B (2) A B D E C (2) E D B C A (1) D E A C B (1) D B E C A (1) C A B E D (1) B E D A C (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 14 -8 -8 B -2 0 -2 6 8 C -14 2 0 6 6 D 8 -6 -6 0 0 E 8 -8 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.406249999981 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 -8 -8 B -2 0 -2 6 8 C -14 2 0 6 6 D 8 -6 -6 0 0 E 8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000064 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=27 D=17 C=17 E=7 so E is eliminated. Round 2 votes counts: B=32 A=27 D=24 C=17 so C is eliminated. Round 3 votes counts: D=37 B=32 A=31 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:205 A:200 C:200 D:198 E:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 14 -8 -8 B -2 0 -2 6 8 C -14 2 0 6 6 D 8 -6 -6 0 0 E 8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000064 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -8 -8 B -2 0 -2 6 8 C -14 2 0 6 6 D 8 -6 -6 0 0 E 8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000064 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -8 -8 B -2 0 -2 6 8 C -14 2 0 6 6 D 8 -6 -6 0 0 E 8 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000064 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4229: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (12) B C E D A (9) D E A C B (7) C E B D A (7) D A E C B (6) B C A E D (5) A B D C E (5) E D C B A (4) C B E D A (4) A D B E C (4) A B C E D (4) A B C D E (4) D E C A B (3) A D E B C (3) E C D B A (2) E C B D A (2) C B A E D (2) B E C D A (2) A D B C E (2) A C D E B (2) E D B C A (1) E B C D A (1) D E C B A (1) D B A E C (1) D A E B C (1) C B E A D (1) C A B E D (1) B C E A D (1) B A C E D (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 12 8 -2 10 B -12 0 -12 0 -6 C -8 12 0 -2 0 D 2 0 2 0 6 E -10 6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.039403 C: 0.000000 D: 0.960597 E: 0.000000 Sum of squares = 0.924298982993 Cumulative probabilities = A: 0.000000 B: 0.039403 C: 0.039403 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -2 10 B -12 0 -12 0 -6 C -8 12 0 -2 0 D 2 0 2 0 6 E -10 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102206556 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=19 B=18 C=15 E=10 so E is eliminated. Round 2 votes counts: A=38 D=24 C=19 B=19 so C is eliminated. Round 3 votes counts: A=39 B=35 D=26 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:205 C:201 E:195 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -2 10 B -12 0 -12 0 -6 C -8 12 0 -2 0 D 2 0 2 0 6 E -10 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102206556 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -2 10 B -12 0 -12 0 -6 C -8 12 0 -2 0 D 2 0 2 0 6 E -10 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102206556 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -2 10 B -12 0 -12 0 -6 C -8 12 0 -2 0 D 2 0 2 0 6 E -10 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102206556 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4230: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (12) D A C E B (7) B E C A D (7) D C A E B (6) C D E B A (6) B E A C D (6) C E B D A (5) C D A E B (5) A B E D C (5) C B E D A (4) A D B E C (4) C D E A B (3) B E A D C (3) E B C D A (2) E B A D C (2) D A E C B (2) D A C B E (2) C D B E A (2) B C E A D (2) B A E D C (2) B A E C D (2) E B A C D (1) E A B D C (1) D C E A B (1) D C A B E (1) D A E B C (1) C D A B E (1) B C E D A (1) B C A E D (1) A E D B C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 6 2 8 B -8 0 10 -8 -12 C -6 -10 0 -4 -6 D -2 8 4 0 8 E -8 12 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 2 8 B -8 0 10 -8 -12 C -6 -10 0 -4 -6 D -2 8 4 0 8 E -8 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 A=24 D=20 E=6 so E is eliminated. Round 2 votes counts: B=29 C=26 A=25 D=20 so D is eliminated. Round 3 votes counts: A=37 C=34 B=29 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:209 E:201 B:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 2 8 B -8 0 10 -8 -12 C -6 -10 0 -4 -6 D -2 8 4 0 8 E -8 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 2 8 B -8 0 10 -8 -12 C -6 -10 0 -4 -6 D -2 8 4 0 8 E -8 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 2 8 B -8 0 10 -8 -12 C -6 -10 0 -4 -6 D -2 8 4 0 8 E -8 12 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4231: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) E D A B C (5) D E A B C (5) C B A D E (5) A E D C B (5) E A D C B (4) D E B A C (4) D A E B C (4) C E B D A (4) B C A D E (4) E D A C B (3) E C D B A (3) E C A D B (3) E A C D B (3) C E A B D (3) B C D E A (3) B C D A E (3) C E B A D (2) C B E A D (2) C A E B D (2) C A B E D (2) B D A C E (2) A E D B C (2) A D E B C (2) A D B E C (2) A C E B D (2) E C D A B (1) E A D B C (1) D B A E C (1) D A B E C (1) C E A D B (1) C B D E A (1) C B D A E (1) C B A E D (1) B D E C A (1) B A D C E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 0 -14 B -8 0 -14 0 -18 C 0 14 0 12 0 D 0 0 -12 0 -14 E 14 18 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.567789 D: 0.000000 E: 0.432211 Sum of squares = 0.509190710468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.567789 D: 0.567789 E: 1.000000 A B C D E A 0 8 0 0 -14 B -8 0 -14 0 -18 C 0 14 0 12 0 D 0 0 -12 0 -14 E 14 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=23 A=16 D=15 B=14 so B is eliminated. Round 2 votes counts: C=42 E=23 D=18 A=17 so A is eliminated. Round 3 votes counts: C=46 E=30 D=24 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:213 A:197 D:187 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 0 0 -14 B -8 0 -14 0 -18 C 0 14 0 12 0 D 0 0 -12 0 -14 E 14 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 0 -14 B -8 0 -14 0 -18 C 0 14 0 12 0 D 0 0 -12 0 -14 E 14 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 0 -14 B -8 0 -14 0 -18 C 0 14 0 12 0 D 0 0 -12 0 -14 E 14 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4232: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) D B E A C (8) C A B D E (8) C E A B D (7) E C A B D (6) E C D A B (5) C A B E D (5) B A D C E (5) A B C D E (5) D E B A C (4) E D C B A (3) E D B C A (3) B D A C E (3) A C B D E (3) E D B A C (2) E C A D B (2) C E D A B (2) B D A E C (2) A C B E D (2) E D C A B (1) E D A B C (1) E A D B C (1) E A C B D (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) C E A D B (1) C D B A E (1) B E A D C (1) B A E D C (1) B A D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 22 -18 20 -4 B -22 0 -18 22 -2 C 18 18 0 16 12 D -20 -22 -16 0 -8 E 4 2 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -18 20 -4 B -22 0 -18 22 -2 C 18 18 0 16 12 D -20 -22 -16 0 -8 E 4 2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=25 D=17 B=13 A=11 so A is eliminated. Round 2 votes counts: C=39 E=25 B=19 D=17 so D is eliminated. Round 3 votes counts: C=41 E=30 B=29 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:232 A:210 E:201 B:190 D:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -18 20 -4 B -22 0 -18 22 -2 C 18 18 0 16 12 D -20 -22 -16 0 -8 E 4 2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -18 20 -4 B -22 0 -18 22 -2 C 18 18 0 16 12 D -20 -22 -16 0 -8 E 4 2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -18 20 -4 B -22 0 -18 22 -2 C 18 18 0 16 12 D -20 -22 -16 0 -8 E 4 2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4233: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) B E C A D (8) D A C E B (5) B C E A D (5) E B C D A (4) C D E B A (4) C B E D A (4) A D C B E (4) A B E D C (4) A B D E C (4) E B C A D (3) E A B D C (3) D E C B A (3) D C A B E (3) E C B D A (2) D A E C B (2) C E B D A (2) C D B E A (2) B E A C D (2) A E D B C (2) A E B D C (2) A D B E C (2) A D B C E (2) A B E C D (2) E D B C A (1) E D A B C (1) E C D B A (1) E B D C A (1) E B A C D (1) E A B C D (1) D E A C B (1) D C E A B (1) D C A E B (1) D A C B E (1) C B E A D (1) B C A E D (1) B A E C D (1) A D E B C (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -12 4 -20 B 10 0 6 6 -2 C 12 -6 0 -8 -4 D -4 -6 8 0 -6 E 20 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 4 -20 B 10 0 6 6 -2 C 12 -6 0 -8 -4 D -4 -6 8 0 -6 E 20 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 E=18 B=17 C=13 so C is eliminated. Round 2 votes counts: D=32 A=26 B=22 E=20 so E is eliminated. Round 3 votes counts: D=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:210 C:197 D:196 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 4 -20 B 10 0 6 6 -2 C 12 -6 0 -8 -4 D -4 -6 8 0 -6 E 20 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 4 -20 B 10 0 6 6 -2 C 12 -6 0 -8 -4 D -4 -6 8 0 -6 E 20 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 4 -20 B 10 0 6 6 -2 C 12 -6 0 -8 -4 D -4 -6 8 0 -6 E 20 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4234: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (6) E B C A D (5) E B A D C (4) D C A B E (4) D A E B C (4) D A C B E (4) B E C D A (4) B C E D A (4) A E D C B (4) A D E C B (4) A D E B C (4) C B D E A (3) B E D A C (3) B C D E A (3) E B D A C (2) E B A C D (2) E A D B C (2) E A C B D (2) D C B A E (2) D B E A C (2) D B C E A (2) D B C A E (2) C E B A D (2) C D B A E (2) C B E A D (2) C B D A E (2) C B A E D (2) A E D B C (2) A E C D B (2) A D C E B (2) A C E D B (2) E C B A D (1) E C A B D (1) E A B D C (1) E A B C D (1) C B E D A (1) C B A D E (1) C A E B D (1) B E C A D (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -2 0 -8 B 16 0 8 10 0 C 2 -8 0 -8 -4 D 0 -10 8 0 -4 E 8 0 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.064985 C: 0.000000 D: 0.000000 E: 0.935015 Sum of squares = 0.8784757146 Cumulative probabilities = A: 0.000000 B: 0.064985 C: 0.064985 D: 0.064985 E: 1.000000 A B C D E A 0 -16 -2 0 -8 B 16 0 8 10 0 C 2 -8 0 -8 -4 D 0 -10 8 0 -4 E 8 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=22 E=21 B=21 D=20 C=16 so C is eliminated. Round 2 votes counts: B=32 E=23 A=23 D=22 so D is eliminated. Round 3 votes counts: B=42 A=35 E=23 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:208 D:197 C:191 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 0 -8 B 16 0 8 10 0 C 2 -8 0 -8 -4 D 0 -10 8 0 -4 E 8 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 0 -8 B 16 0 8 10 0 C 2 -8 0 -8 -4 D 0 -10 8 0 -4 E 8 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 0 -8 B 16 0 8 10 0 C 2 -8 0 -8 -4 D 0 -10 8 0 -4 E 8 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4235: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (5) C B D A E (5) C B A E D (5) B E A C D (5) B E D A C (4) E B A D C (3) E B A C D (3) D C A E B (3) D A E C B (3) C A B E D (3) B C E D A (3) B C D E A (3) B C A E D (3) A E D C B (3) E B D A C (2) E A B D C (2) E A B C D (2) D C B E A (2) D B C E A (2) D A C E B (2) C D B A E (2) C D A E B (2) C B D E A (2) C B A D E (2) C A D B E (2) B C E A D (2) A E C B D (2) A D E C B (2) A C D E B (2) E D B A C (1) E D A B C (1) E A D C B (1) E A D B C (1) D E C A B (1) D E B A C (1) D E A C B (1) D E A B C (1) D C B A E (1) D B E C A (1) B E D C A (1) B E C D A (1) B E C A D (1) A E D B C (1) A E B C D (1) A D C E B (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -4 0 6 B 10 0 -6 16 16 C 4 6 0 18 8 D 0 -16 -18 0 -8 E -6 -16 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 0 6 B 10 0 -6 16 16 C 4 6 0 18 8 D 0 -16 -18 0 -8 E -6 -16 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=23 D=18 E=16 A=15 so A is eliminated. Round 2 votes counts: C=31 B=25 E=23 D=21 so D is eliminated. Round 3 votes counts: C=40 E=32 B=28 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:218 A:196 E:189 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 0 6 B 10 0 -6 16 16 C 4 6 0 18 8 D 0 -16 -18 0 -8 E -6 -16 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 0 6 B 10 0 -6 16 16 C 4 6 0 18 8 D 0 -16 -18 0 -8 E -6 -16 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 0 6 B 10 0 -6 16 16 C 4 6 0 18 8 D 0 -16 -18 0 -8 E -6 -16 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4236: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (5) D B C E A (5) B A E C D (5) A E C B D (5) D E C B A (4) D C E A B (4) D C B E A (4) C D E A B (4) B D C A E (4) E A C B D (3) D B C A E (3) C E A D B (3) B D E A C (3) A C E B D (3) A B E C D (3) E D C A B (2) E B A D C (2) D B E C A (2) C D A E B (2) C A E D B (2) C A E B D (2) C A B D E (2) B A E D C (2) B A D E C (2) B A D C E (2) A C B E D (2) A B C E D (2) E D B A C (1) E D A B C (1) E C A D B (1) E B D A C (1) E A D B C (1) E A B C D (1) D E B C A (1) D E B A C (1) D C E B A (1) D C B A E (1) C D B A E (1) C D A B E (1) C A B E D (1) B E D A C (1) B E A D C (1) B C D A E (1) B A C E D (1) A E B C D (1) Total count = 100 A B C D E A 0 2 -12 -6 -4 B -2 0 -6 0 -2 C 12 6 0 -8 2 D 6 0 8 0 6 E 4 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.369303 C: 0.000000 D: 0.630697 E: 0.000000 Sum of squares = 0.534163663319 Cumulative probabilities = A: 0.000000 B: 0.369303 C: 0.369303 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -6 -4 B -2 0 -6 0 -2 C 12 6 0 -8 2 D 6 0 8 0 6 E 4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=22 C=18 A=16 E=13 so E is eliminated. Round 2 votes counts: D=35 B=25 A=21 C=19 so C is eliminated. Round 3 votes counts: D=43 A=32 B=25 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:206 E:199 B:195 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -6 -4 B -2 0 -6 0 -2 C 12 6 0 -8 2 D 6 0 8 0 6 E 4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -6 -4 B -2 0 -6 0 -2 C 12 6 0 -8 2 D 6 0 8 0 6 E 4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -6 -4 B -2 0 -6 0 -2 C 12 6 0 -8 2 D 6 0 8 0 6 E 4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4237: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) B E C D A (10) A D C E B (10) D A B C E (7) B C E D A (7) D A C B E (6) E C B A D (5) A D E B C (5) E C B D A (3) E B C A D (3) D A C E B (3) B C D E A (3) A D B C E (3) E B C D A (2) D C E B A (2) D C B E A (2) C E B A D (2) C B E D A (2) A E C B D (2) A E B C D (2) D C A E B (1) D B A C E (1) D A B E C (1) C D E B A (1) B E C A D (1) B D C E A (1) A E C D B (1) A D C B E (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 -4 12 B -10 0 -4 -12 -6 C -10 4 0 -10 4 D 4 12 10 0 18 E -12 6 -4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -4 12 B -10 0 -4 -12 -6 C -10 4 0 -10 4 D 4 12 10 0 18 E -12 6 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=23 B=22 E=13 C=5 so C is eliminated. Round 2 votes counts: A=37 D=24 B=24 E=15 so E is eliminated. Round 3 votes counts: B=39 A=37 D=24 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:222 A:214 C:194 E:186 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -4 12 B -10 0 -4 -12 -6 C -10 4 0 -10 4 D 4 12 10 0 18 E -12 6 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -4 12 B -10 0 -4 -12 -6 C -10 4 0 -10 4 D 4 12 10 0 18 E -12 6 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -4 12 B -10 0 -4 -12 -6 C -10 4 0 -10 4 D 4 12 10 0 18 E -12 6 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999135 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4238: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) C A B D E (6) B A E D C (6) C D E A B (5) A C B D E (5) E D B C A (4) A C B E D (4) A B C E D (4) A B C D E (4) E D C B A (3) E B D A C (3) D C E A B (3) C D A E B (3) C A D B E (3) B E A D C (3) E D B A C (2) E C D A B (2) E B A D C (2) E B A C D (2) D E B A C (2) D C E B A (2) C E D A B (2) C E A D B (2) C D A B E (2) C A E B D (2) B E D A C (2) E D C A B (1) E A B C D (1) D E B C A (1) D C B A E (1) D C A B E (1) D B A C E (1) C A E D B (1) C A D E B (1) C A B E D (1) B D E A C (1) B A E C D (1) B A D E C (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -8 2 -2 B -8 0 -16 0 -4 C 8 16 0 6 8 D -2 0 -6 0 0 E 2 4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 2 -2 B -8 0 -16 0 -4 C 8 16 0 6 8 D -2 0 -6 0 0 E 2 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=20 D=19 A=18 B=15 so B is eliminated. Round 2 votes counts: C=28 A=27 E=25 D=20 so D is eliminated. Round 3 votes counts: E=37 C=35 A=28 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:200 E:199 D:196 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 2 -2 B -8 0 -16 0 -4 C 8 16 0 6 8 D -2 0 -6 0 0 E 2 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 2 -2 B -8 0 -16 0 -4 C 8 16 0 6 8 D -2 0 -6 0 0 E 2 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 2 -2 B -8 0 -16 0 -4 C 8 16 0 6 8 D -2 0 -6 0 0 E 2 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4239: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (16) D A B C E (15) A D C B E (10) E B C D A (6) E C A B D (5) E D A B C (4) C B A D E (4) E D B C A (3) D A E B C (3) C A B D E (3) E A D C B (2) D B A C E (2) D A E C B (2) C B E A D (2) B C E D A (2) A C B D E (2) E D B A C (1) E D A C B (1) E C B D A (1) E B D C A (1) D B E C A (1) D B C A E (1) D A B E C (1) C E B A D (1) C E A B D (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D A E (1) B C A D E (1) A D E C B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -8 2 -2 B -6 0 -6 4 4 C 8 6 0 -2 -2 D -2 -4 2 0 2 E 2 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888844 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 2 -2 B -6 0 -6 4 4 C 8 6 0 -2 -2 D -2 -4 2 0 2 E 2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=25 A=15 C=12 B=8 so B is eliminated. Round 2 votes counts: E=42 D=27 C=16 A=15 so A is eliminated. Round 3 votes counts: E=42 D=38 C=20 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:205 A:199 D:199 E:199 B:198 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 2 -2 B -6 0 -6 4 4 C 8 6 0 -2 -2 D -2 -4 2 0 2 E 2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 2 -2 B -6 0 -6 4 4 C 8 6 0 -2 -2 D -2 -4 2 0 2 E 2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 2 -2 B -6 0 -6 4 4 C 8 6 0 -2 -2 D -2 -4 2 0 2 E 2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4240: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (14) D E B C A (14) C B E D A (9) A C B D E (9) E D C B A (7) A C B E D (6) E D A B C (5) E D B A C (4) C B A E D (4) C B A D E (4) B C A D E (3) A E D C B (3) A D E B C (3) A E D B C (2) E D A C B (1) E C D A B (1) E C B D A (1) E A D C B (1) D E A B C (1) C A B D E (1) B D C E A (1) B D A C E (1) A E C D B (1) A D B C E (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -18 -18 -18 B 24 0 2 -16 -18 C 18 -2 0 -18 -16 D 18 16 18 0 -20 E 18 18 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 -18 -18 -18 B 24 0 2 -16 -18 C 18 -2 0 -18 -16 D 18 16 18 0 -20 E 18 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=28 C=18 D=15 B=5 so B is eliminated. Round 2 votes counts: E=34 A=28 C=21 D=17 so D is eliminated. Round 3 votes counts: E=49 A=29 C=22 so C is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:236 D:216 B:196 C:191 A:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 -18 -18 -18 B 24 0 2 -16 -18 C 18 -2 0 -18 -16 D 18 16 18 0 -20 E 18 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -18 -18 -18 B 24 0 2 -16 -18 C 18 -2 0 -18 -16 D 18 16 18 0 -20 E 18 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -18 -18 -18 B 24 0 2 -16 -18 C 18 -2 0 -18 -16 D 18 16 18 0 -20 E 18 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4241: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) E C D B A (9) C E B A D (8) C E D B A (7) B A C D E (7) D E A B C (6) C B A D E (6) A B D E C (6) A B D C E (6) C B A E D (5) D A B E C (4) E D A B C (3) D A E B C (3) B A E D C (3) A B C D E (3) E C D A B (2) D E C A B (2) C B E A D (2) E D A C B (1) E C B A D (1) C D A B E (1) B C A E D (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 -2 -12 0 -6 B 2 0 -12 0 -8 C 12 12 0 4 -6 D 0 0 -4 0 -8 E 6 8 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -12 0 -6 B 2 0 -12 0 -8 C 12 12 0 4 -6 D 0 0 -4 0 -8 E 6 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=28 D=15 A=15 B=13 so B is eliminated. Round 2 votes counts: C=30 E=28 A=27 D=15 so D is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:211 D:194 B:191 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -12 0 -6 B 2 0 -12 0 -8 C 12 12 0 4 -6 D 0 0 -4 0 -8 E 6 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 0 -6 B 2 0 -12 0 -8 C 12 12 0 4 -6 D 0 0 -4 0 -8 E 6 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 0 -6 B 2 0 -12 0 -8 C 12 12 0 4 -6 D 0 0 -4 0 -8 E 6 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4242: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) B D C A E (10) D B A C E (8) A D B E C (8) C E B D A (7) E C A D B (6) E A C D B (6) B D A C E (6) A E D B C (6) C E D B A (5) E C A B D (4) E A D C B (4) A D E B C (4) E A D B C (2) C B E A D (2) B A D C E (2) E D C B A (1) E D C A B (1) E A C B D (1) D E A B C (1) D B C A E (1) C E A B D (1) C B D A E (1) A E B D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 2 -2 -2 B 6 0 2 -6 0 C -2 -2 0 -14 8 D 2 6 14 0 6 E 2 0 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -2 -2 B 6 0 2 -6 0 C -2 -2 0 -14 8 D 2 6 14 0 6 E 2 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 A=21 B=18 D=10 so D is eliminated. Round 2 votes counts: B=27 E=26 C=26 A=21 so A is eliminated. Round 3 votes counts: E=37 B=37 C=26 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:214 B:201 A:196 C:195 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -2 -2 B 6 0 2 -6 0 C -2 -2 0 -14 8 D 2 6 14 0 6 E 2 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 -2 B 6 0 2 -6 0 C -2 -2 0 -14 8 D 2 6 14 0 6 E 2 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 -2 B 6 0 2 -6 0 C -2 -2 0 -14 8 D 2 6 14 0 6 E 2 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4243: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) D E B C A (6) A B C D E (6) E D C B A (5) E C A D B (5) E A C D B (5) D E C B A (4) D B C E A (4) C B A D E (4) C A B D E (4) A C E B D (4) A B C E D (4) B D C A E (3) B D A C E (3) A E C B D (3) E D A B C (2) E C D A B (2) E A C B D (2) D C B E A (2) D B C A E (2) B C A D E (2) B A C D E (2) A C B E D (2) E D B C A (1) E D B A C (1) E D A C B (1) E A D C B (1) E A D B C (1) D B E A C (1) C D B E A (1) C B D A E (1) C A E B D (1) C A B E D (1) B D A E C (1) B A D E C (1) B A D C E (1) A E B D C (1) A E B C D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 6 -2 B 4 0 2 -2 8 C 10 -2 0 2 -2 D -6 2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 6 -2 B 4 0 2 -2 8 C 10 -2 0 2 -2 D -6 2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=26 D=26 A=23 B=13 C=12 so C is eliminated. Round 2 votes counts: A=29 D=27 E=26 B=18 so B is eliminated. Round 3 votes counts: A=39 D=35 E=26 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:206 C:204 D:204 A:195 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 6 -2 B 4 0 2 -2 8 C 10 -2 0 2 -2 D -6 2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 6 -2 B 4 0 2 -2 8 C 10 -2 0 2 -2 D -6 2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 6 -2 B 4 0 2 -2 8 C 10 -2 0 2 -2 D -6 2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4244: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) B D A C E (7) E A B C D (6) C E D A B (6) C D B E A (6) A E B C D (6) E C A D B (5) E C D A B (4) E A C D B (4) E A C B D (4) D C B A E (4) C D E B A (4) C D E A B (4) B D C A E (4) A E B D C (4) A B E D C (4) B A E C D (3) E A B D C (2) D C B E A (2) A B E C D (2) E D C A B (1) D E C A B (1) D C E B A (1) D C E A B (1) D B C A E (1) D B A C E (1) C E D B A (1) B C D A E (1) B A D C E (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 8 2 -4 B -12 0 4 6 -10 C -8 -4 0 12 -12 D -2 -6 -12 0 -20 E 4 10 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 8 2 -4 B -12 0 4 6 -10 C -8 -4 0 12 -12 D -2 -6 -12 0 -20 E 4 10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 C=21 A=18 D=11 so D is eliminated. Round 2 votes counts: C=29 E=27 B=26 A=18 so A is eliminated. Round 3 votes counts: E=38 B=33 C=29 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:209 B:194 C:194 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 8 2 -4 B -12 0 4 6 -10 C -8 -4 0 12 -12 D -2 -6 -12 0 -20 E 4 10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 2 -4 B -12 0 4 6 -10 C -8 -4 0 12 -12 D -2 -6 -12 0 -20 E 4 10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 2 -4 B -12 0 4 6 -10 C -8 -4 0 12 -12 D -2 -6 -12 0 -20 E 4 10 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4245: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) E C B D A (6) A D B C E (6) D A B C E (5) A D C B E (5) C E D B A (4) C E A B D (4) C A E D B (4) A C D B E (4) E B D C A (3) C D A E B (3) B D E A C (3) B A D E C (3) A B E D C (3) A B D E C (3) E C B A D (2) E C A B D (2) D E B C A (2) D C B E A (2) D B E A C (2) D B A E C (2) C A D E B (2) C A D B E (2) A E B C D (2) A C E B D (2) A B E C D (2) E B C D A (1) E B C A D (1) E B A D C (1) D C A B E (1) D B C E A (1) D B A C E (1) C E D A B (1) C E B A D (1) C E A D B (1) C D E B A (1) B E A D C (1) B D E C A (1) B D A E C (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -4 6 6 B -8 0 4 -14 12 C 4 -4 0 -6 4 D -6 14 6 0 14 E -6 -12 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999999 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 6 6 B -8 0 4 -14 12 C 4 -4 0 -6 4 D -6 14 6 0 14 E -6 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=23 C=23 E=16 B=9 so B is eliminated. Round 2 votes counts: A=32 D=28 C=23 E=17 so E is eliminated. Round 3 votes counts: C=35 A=34 D=31 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:214 A:208 C:199 B:197 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -4 6 6 B -8 0 4 -14 12 C 4 -4 0 -6 4 D -6 14 6 0 14 E -6 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 6 6 B -8 0 4 -14 12 C 4 -4 0 -6 4 D -6 14 6 0 14 E -6 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 6 6 B -8 0 4 -14 12 C 4 -4 0 -6 4 D -6 14 6 0 14 E -6 -12 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4246: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) C B D A E (12) E A D B C (8) A E B C D (8) A E B D C (7) E D C A B (6) D C E B A (4) A B C E D (4) E D A B C (3) E A C D B (3) D E C B A (3) C D B E A (3) C D B A E (3) B C A D E (3) E D C B A (2) D E B C A (2) A B E C D (2) A B C D E (2) E D A C B (1) E A D C B (1) E A B D C (1) D E C A B (1) D B E C A (1) D B C E A (1) C D E B A (1) B D C A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -10 -12 -6 B 0 0 -8 -12 -6 C 10 8 0 -12 -4 D 12 12 12 0 2 E 6 6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -12 -6 B 0 0 -8 -12 -6 C 10 8 0 -12 -4 D 12 12 12 0 2 E 6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994096 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 D=24 C=19 B=6 so B is eliminated. Round 2 votes counts: A=28 E=25 D=25 C=22 so C is eliminated. Round 3 votes counts: D=44 A=31 E=25 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:207 C:201 B:187 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -12 -6 B 0 0 -8 -12 -6 C 10 8 0 -12 -4 D 12 12 12 0 2 E 6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994096 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -12 -6 B 0 0 -8 -12 -6 C 10 8 0 -12 -4 D 12 12 12 0 2 E 6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994096 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -12 -6 B 0 0 -8 -12 -6 C 10 8 0 -12 -4 D 12 12 12 0 2 E 6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994096 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4247: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (11) E C B D A (8) D A C E B (7) B E C A D (7) E C D A B (5) D A E C B (5) C E B D A (5) A D B E C (5) C B E D A (4) E B C A D (3) B E A C D (3) B C E D A (3) B C E A D (3) E D C A B (2) D A B C E (2) B A D C E (2) B A C D E (2) A D E C B (2) A D E B C (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D B A (1) E C B A D (1) E C A B D (1) D E A C B (1) D C E A B (1) D C A B E (1) D A C B E (1) C E D B A (1) B E C D A (1) B D A C E (1) B C D E A (1) B A E D C (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 2 4 -2 -4 B -2 0 6 4 8 C -4 -6 0 0 -6 D 2 -4 0 0 -2 E 4 -8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.42857142803 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 2 4 -2 -4 B -2 0 6 4 8 C -4 -6 0 0 -6 D 2 -4 0 0 -2 E 4 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.428571426215 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 E=22 D=18 C=10 so C is eliminated. Round 2 votes counts: B=30 E=28 A=24 D=18 so D is eliminated. Round 3 votes counts: A=40 E=30 B=30 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:208 E:202 A:200 D:198 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 4 -2 -4 B -2 0 6 4 8 C -4 -6 0 0 -6 D 2 -4 0 0 -2 E 4 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.428571426215 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 -4 B -2 0 6 4 8 C -4 -6 0 0 -6 D 2 -4 0 0 -2 E 4 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.428571426215 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 -4 B -2 0 6 4 8 C -4 -6 0 0 -6 D 2 -4 0 0 -2 E 4 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.428571426215 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4248: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) E C B D A (6) A C E B D (6) C E D B A (5) A E B C D (5) A D B E C (5) D B C E A (4) C E B D A (4) B D A E C (4) A D C B E (4) D C B E A (3) D B A E C (3) D B A C E (3) C E A D B (3) A E C B D (3) A C E D B (3) A B E D C (3) C E D A B (2) C E A B D (2) B D E A C (2) A C D E B (2) E D C B A (1) E C D B A (1) E C B A D (1) E C A B D (1) E B D C A (1) E B C D A (1) E B C A D (1) E B A D C (1) E A C B D (1) D C B A E (1) D C A B E (1) D B E C A (1) C D E B A (1) C D E A B (1) C D A E B (1) C A E D B (1) B E D A C (1) B A E D C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 8 14 6 12 B -8 0 -10 4 -8 C -14 10 0 4 -2 D -6 -4 -4 0 -8 E -12 8 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 6 12 B -8 0 -10 4 -8 C -14 10 0 4 -2 D -6 -4 -4 0 -8 E -12 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=20 D=16 E=15 B=8 so B is eliminated. Round 2 votes counts: A=42 D=22 C=20 E=16 so E is eliminated. Round 3 votes counts: A=44 C=31 D=25 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:203 C:199 B:189 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 6 12 B -8 0 -10 4 -8 C -14 10 0 4 -2 D -6 -4 -4 0 -8 E -12 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 6 12 B -8 0 -10 4 -8 C -14 10 0 4 -2 D -6 -4 -4 0 -8 E -12 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 6 12 B -8 0 -10 4 -8 C -14 10 0 4 -2 D -6 -4 -4 0 -8 E -12 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4249: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B C D A E (9) C B D A E (7) B C E D A (6) E A C D B (5) E A D B C (4) B D A C E (4) B C D E A (4) A D E C B (4) D A C E B (3) D A C B E (3) B D C A E (3) A E D C B (3) A D E B C (3) E C A D B (2) E B C A D (2) C E A D B (2) C B E D A (2) B E C A D (2) B E A D C (2) B C E A D (2) E C B A D (1) E C A B D (1) E B A D C (1) E B A C D (1) E A C B D (1) E A B D C (1) D C A E B (1) D C A B E (1) D B A C E (1) D A E C B (1) D A E B C (1) D A B E C (1) C E A B D (1) C D B A E (1) C D A E B (1) B E D A C (1) B E C D A (1) B D A E C (1) Total count = 100 A B C D E A 0 -2 2 -4 -4 B 2 0 0 6 2 C -2 0 0 2 2 D 4 -6 -2 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.669045 C: 0.330955 D: 0.000000 E: 0.000000 Sum of squares = 0.557152565855 Cumulative probabilities = A: 0.000000 B: 0.669045 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 -4 B 2 0 0 6 2 C -2 0 0 2 2 D 4 -6 -2 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500123 C: 0.499877 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003032 Cumulative probabilities = A: 0.000000 B: 0.500123 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=29 C=14 D=12 A=10 so A is eliminated. Round 2 votes counts: B=35 E=32 D=19 C=14 so C is eliminated. Round 3 votes counts: B=44 E=35 D=21 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:205 C:201 E:201 D:197 A:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -4 -4 B 2 0 0 6 2 C -2 0 0 2 2 D 4 -6 -2 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500123 C: 0.499877 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003032 Cumulative probabilities = A: 0.000000 B: 0.500123 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 -4 B 2 0 0 6 2 C -2 0 0 2 2 D 4 -6 -2 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500123 C: 0.499877 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003032 Cumulative probabilities = A: 0.000000 B: 0.500123 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 -4 B 2 0 0 6 2 C -2 0 0 2 2 D 4 -6 -2 0 -2 E 4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500123 C: 0.499877 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003032 Cumulative probabilities = A: 0.000000 B: 0.500123 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4250: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) A D C B E (7) D E C B A (5) C E D B A (5) B E A C D (5) E C B D A (4) D A C E B (4) B E C A D (4) B A C E D (4) A D B C E (4) D A E B C (3) C E B D A (3) C B A E D (3) A D B E C (3) A B D E C (3) D E C A B (2) D E B C A (2) D E A B C (2) D A E C B (2) C B E A D (2) C A B E D (2) B C E A D (2) A C D B E (2) A B E C D (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E B D A C (1) D E A C B (1) D C E A B (1) C D E A B (1) C D A E B (1) C A D B E (1) C A B D E (1) B A E D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 0 4 -2 B 4 0 4 4 4 C 0 -4 0 10 -4 D -4 -4 -10 0 -4 E 2 -4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 4 -2 B 4 0 4 4 4 C 0 -4 0 10 -4 D -4 -4 -10 0 -4 E 2 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=22 C=19 E=16 B=16 so E is eliminated. Round 2 votes counts: A=27 B=26 D=24 C=23 so C is eliminated. Round 3 votes counts: B=38 D=31 A=31 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:203 C:201 A:199 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 4 -2 B 4 0 4 4 4 C 0 -4 0 10 -4 D -4 -4 -10 0 -4 E 2 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 4 -2 B 4 0 4 4 4 C 0 -4 0 10 -4 D -4 -4 -10 0 -4 E 2 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 4 -2 B 4 0 4 4 4 C 0 -4 0 10 -4 D -4 -4 -10 0 -4 E 2 -4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4251: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) E B A C D (8) D A C E B (8) B E C A D (8) E D A C B (6) E B D A C (6) D E A C B (5) D C A B E (5) E D B A C (4) E B D C A (3) E A D C B (3) D A E C B (3) C A B D E (3) A C D E B (3) A C D B E (3) E D A B C (2) E A C D B (2) D B E C A (2) C A D B E (2) B E C D A (2) B C A D E (2) A D C E B (2) E B C A D (1) E B A D C (1) D E B C A (1) D A C B E (1) C D A B E (1) C B A D E (1) B D E C A (1) B C A E D (1) A E C B D (1) Total count = 100 A B C D E A 0 0 16 -18 -28 B 0 0 2 -6 -18 C -16 -2 0 -24 -36 D 18 6 24 0 -14 E 28 18 36 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 16 -18 -28 B 0 0 2 -6 -18 C -16 -2 0 -24 -36 D 18 6 24 0 -14 E 28 18 36 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=25 B=23 A=9 C=7 so C is eliminated. Round 2 votes counts: E=36 D=26 B=24 A=14 so A is eliminated. Round 3 votes counts: E=37 D=36 B=27 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:248 D:217 B:189 A:185 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 16 -18 -28 B 0 0 2 -6 -18 C -16 -2 0 -24 -36 D 18 6 24 0 -14 E 28 18 36 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 -18 -28 B 0 0 2 -6 -18 C -16 -2 0 -24 -36 D 18 6 24 0 -14 E 28 18 36 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 -18 -28 B 0 0 2 -6 -18 C -16 -2 0 -24 -36 D 18 6 24 0 -14 E 28 18 36 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4252: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (9) D C A B E (7) E B A C D (6) C D E B A (5) D A E B C (4) C D B E A (4) A D B E C (4) A B E D C (4) D C A E B (3) D A C B E (3) A B D E C (3) E A B C D (2) D C B A E (2) D A B E C (2) D A B C E (2) B E A C D (2) A E B D C (2) A D B C E (2) E D C B A (1) E B C D A (1) E B C A D (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A B C (1) D C E A B (1) D A E C B (1) D A C E B (1) C E D B A (1) C E B D A (1) C E B A D (1) C D B A E (1) C D A B E (1) C B E A D (1) C B D E A (1) C A D B E (1) B E C A D (1) B A E C D (1) A E B C D (1) A D E B C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 22 12 -6 16 B -22 0 6 -20 8 C -12 -6 0 -12 -2 D 6 20 12 0 22 E -16 -8 2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 -6 16 B -22 0 6 -20 8 C -12 -6 0 -12 -2 D 6 20 12 0 22 E -16 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=29 C=17 E=13 B=4 so B is eliminated. Round 2 votes counts: D=37 A=30 C=17 E=16 so E is eliminated. Round 3 votes counts: A=42 D=38 C=20 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:222 B:186 C:184 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 12 -6 16 B -22 0 6 -20 8 C -12 -6 0 -12 -2 D 6 20 12 0 22 E -16 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 -6 16 B -22 0 6 -20 8 C -12 -6 0 -12 -2 D 6 20 12 0 22 E -16 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 -6 16 B -22 0 6 -20 8 C -12 -6 0 -12 -2 D 6 20 12 0 22 E -16 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4253: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) E C D A B (6) A B D C E (6) A B C E D (6) C E B D A (5) B A D C E (5) A D B E C (5) D E C B A (4) D B A E C (4) C E D B A (4) C E D A B (4) A B C D E (4) C A E B D (3) A C B E D (3) E D C B A (2) D E B C A (2) C E B A D (2) B D E C A (2) B D A E C (2) A C E B D (2) A B D E C (2) E D C A B (1) E C A D B (1) D E C A B (1) D E B A C (1) D E A C B (1) D B E C A (1) D B E A C (1) D A E B C (1) D A B E C (1) C E A D B (1) C B E A D (1) B D E A C (1) B C E D A (1) B C D A E (1) B C A E D (1) B A D E C (1) B A C E D (1) A D E C B (1) A D E B C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 2 -4 4 B -4 0 2 -2 0 C -2 -2 0 6 4 D 4 2 -6 0 -2 E -4 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -4 4 B -4 0 2 -2 0 C -2 -2 0 6 4 D 4 2 -6 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=20 D=17 E=16 B=15 so B is eliminated. Round 2 votes counts: A=39 C=23 D=22 E=16 so E is eliminated. Round 3 votes counts: A=39 C=36 D=25 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:203 C:203 D:199 B:198 E:197 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 2 -4 4 B -4 0 2 -2 0 C -2 -2 0 6 4 D 4 2 -6 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -4 4 B -4 0 2 -2 0 C -2 -2 0 6 4 D 4 2 -6 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -4 4 B -4 0 2 -2 0 C -2 -2 0 6 4 D 4 2 -6 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4254: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) E B D A C (9) C A D B E (8) B E A D C (7) C A B D E (6) B E C D A (4) B C A E D (4) C D A B E (3) C B A D E (3) B C E A D (3) B A E C D (3) A B C E D (3) E D B A C (2) D E A C B (2) D C E A B (2) D C A E B (2) D A E C B (2) B E D A C (2) B E C A D (2) B E A C D (2) B A C E D (2) A D C E B (2) A C D B E (2) E D A B C (1) E B D C A (1) E B A D C (1) D E C B A (1) D E A B C (1) D A C E B (1) C D E A B (1) C D B A E (1) C B D E A (1) B E D C A (1) B C E D A (1) A E D B C (1) A D E B C (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -8 4 12 B 0 0 4 12 18 C 8 -4 0 20 12 D -4 -12 -20 0 0 E -12 -18 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.217686 B: 0.782314 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.659402183402 Cumulative probabilities = A: 0.217686 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 4 12 B 0 0 4 12 18 C 8 -4 0 20 12 D -4 -12 -20 0 0 E -12 -18 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555943217 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=31 E=14 D=11 A=11 so D is eliminated. Round 2 votes counts: C=37 B=31 E=18 A=14 so A is eliminated. Round 3 votes counts: C=43 B=35 E=22 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:218 B:217 A:204 D:182 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 4 12 B 0 0 4 12 18 C 8 -4 0 20 12 D -4 -12 -20 0 0 E -12 -18 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555943217 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 4 12 B 0 0 4 12 18 C 8 -4 0 20 12 D -4 -12 -20 0 0 E -12 -18 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555943217 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 4 12 B 0 0 4 12 18 C 8 -4 0 20 12 D -4 -12 -20 0 0 E -12 -18 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555943217 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4255: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (12) D C B A E (11) E B A C D (5) D C A B E (4) E D A B C (3) E B A D C (3) D C E A B (3) D C B E A (3) D C A E B (3) C D B A E (3) C D A B E (3) C B D A E (3) C A B E D (3) B E A C D (3) B A E C D (3) B A C E D (3) D E B A C (2) D E A C B (2) D C E B A (2) C D A E B (2) C B A D E (2) B C A E D (2) B C A D E (2) A E B C D (2) E D B A C (1) E A D B C (1) E A C D B (1) D E A B C (1) D B C A E (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E A C (1) A C E B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -2 2 10 B 6 0 -2 4 10 C 2 2 0 14 14 D -2 -4 -14 0 2 E -10 -10 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 2 10 B 6 0 -2 4 10 C 2 2 0 14 14 D -2 -4 -14 0 2 E -10 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=26 C=20 B=16 A=6 so A is eliminated. Round 2 votes counts: D=32 E=28 C=22 B=18 so B is eliminated. Round 3 votes counts: E=37 D=33 C=30 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:216 B:209 A:202 D:191 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 2 10 B 6 0 -2 4 10 C 2 2 0 14 14 D -2 -4 -14 0 2 E -10 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 10 B 6 0 -2 4 10 C 2 2 0 14 14 D -2 -4 -14 0 2 E -10 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 10 B 6 0 -2 4 10 C 2 2 0 14 14 D -2 -4 -14 0 2 E -10 -10 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4256: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) E D C B A (6) E D B A C (6) E C D B A (5) C B E A D (5) A B C D E (5) D A E C B (4) B A D E C (4) E D B C A (3) D E A B C (3) C B A E D (3) B C E A D (3) B A C D E (3) D E C A B (2) C A D E B (2) C A B E D (2) C A B D E (2) B E C D A (2) B D A E C (2) B A C E D (2) A D B E C (2) A D B C E (2) E B C D A (1) D E B A C (1) D E A C B (1) D B A E C (1) D A E B C (1) D A B E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A D B (1) C A E B D (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C A D (1) B E A C D (1) B D E A C (1) B C A E D (1) B A D C E (1) A D C E B (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -8 -6 -12 B 12 0 0 -8 0 C 8 0 0 -10 -18 D 6 8 10 0 -14 E 12 0 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.405610 C: 0.000000 D: 0.000000 E: 0.594390 Sum of squares = 0.517819015102 Cumulative probabilities = A: 0.000000 B: 0.405610 C: 0.405610 D: 0.405610 E: 1.000000 A B C D E A 0 -12 -8 -6 -12 B 12 0 0 -8 0 C 8 0 0 -10 -18 D 6 8 10 0 -14 E 12 0 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=23 C=20 D=14 A=13 so A is eliminated. Round 2 votes counts: E=30 B=29 C=21 D=20 so D is eliminated. Round 3 votes counts: E=42 B=35 C=23 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:222 D:205 B:202 C:190 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 -6 -12 B 12 0 0 -8 0 C 8 0 0 -10 -18 D 6 8 10 0 -14 E 12 0 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -6 -12 B 12 0 0 -8 0 C 8 0 0 -10 -18 D 6 8 10 0 -14 E 12 0 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -6 -12 B 12 0 0 -8 0 C 8 0 0 -10 -18 D 6 8 10 0 -14 E 12 0 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4257: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (12) A C D E B (10) D B E C A (9) B E D C A (6) D B E A C (4) C E B A D (4) C A E B D (4) B E D A C (4) B D E C A (4) A D C B E (4) E B C D A (3) E B C A D (3) A C E D B (3) E C B A D (2) E C A B D (2) E B A C D (2) D C A B E (2) D B C E A (2) D A C B E (2) D A B E C (2) C E A B D (2) B D E A C (2) E A C B D (1) D C B E A (1) D B A E C (1) D A B C E (1) C E B D A (1) C D B E A (1) C D A B E (1) C A D E B (1) A E C B D (1) A E B D C (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 2 4 8 -6 B -2 0 -8 10 -6 C -4 8 0 8 4 D -8 -10 -8 0 -4 E 6 6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775535 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 2 4 8 -6 B -2 0 -8 10 -6 C -4 8 0 8 4 D -8 -10 -8 0 -4 E 6 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=24 B=16 C=14 E=13 so E is eliminated. Round 2 votes counts: A=34 D=24 B=24 C=18 so C is eliminated. Round 3 votes counts: A=43 B=31 D=26 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:208 E:206 A:204 B:197 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 4 8 -6 B -2 0 -8 10 -6 C -4 8 0 8 4 D -8 -10 -8 0 -4 E 6 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 -6 B -2 0 -8 10 -6 C -4 8 0 8 4 D -8 -10 -8 0 -4 E 6 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 -6 B -2 0 -8 10 -6 C -4 8 0 8 4 D -8 -10 -8 0 -4 E 6 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4258: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A D C B E (8) C A D E B (7) D B E A C (6) C E B A D (5) A C D B E (5) C E A B D (4) C A E B D (4) A C B E D (4) E C B D A (3) C E D B A (3) B E D A C (3) A C B D E (3) E C B A D (2) E B C A D (2) D E C B A (2) D B A E C (2) D A C B E (2) D A B E C (2) C E A D B (2) C A E D B (2) B A E D C (2) A D B C E (2) E D C B A (1) E B C D A (1) D E B A C (1) D C E A B (1) D C A E B (1) C E D A B (1) C D E A B (1) C D A E B (1) B E D C A (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E C D (1) B A D E C (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -4 16 0 B -4 0 -26 -2 -6 C 4 26 0 8 16 D -16 2 -8 0 -4 E 0 6 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 16 0 B -4 0 -26 -2 -6 C 4 26 0 8 16 D -16 2 -8 0 -4 E 0 6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=24 E=17 D=17 B=12 so B is eliminated. Round 2 votes counts: C=30 A=28 E=23 D=19 so D is eliminated. Round 3 votes counts: A=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:227 A:208 E:197 D:187 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 16 0 B -4 0 -26 -2 -6 C 4 26 0 8 16 D -16 2 -8 0 -4 E 0 6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 16 0 B -4 0 -26 -2 -6 C 4 26 0 8 16 D -16 2 -8 0 -4 E 0 6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 16 0 B -4 0 -26 -2 -6 C 4 26 0 8 16 D -16 2 -8 0 -4 E 0 6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4259: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) C A E D B (6) B C D E A (6) A E D C B (6) D B E C A (5) B D E C A (5) B C D A E (5) E D C A B (4) C A B E D (4) B D A E C (4) D E B A C (3) C E D B A (3) C E A D B (3) C B A E D (3) C A E B D (3) E D A C B (2) D E B C A (2) D B E A C (2) B C A D E (2) A E D B C (2) A B C E D (2) E D C B A (1) E D A B C (1) E C A D B (1) E A D C B (1) E A C D B (1) D E A B C (1) D C B E A (1) D A B E C (1) C B D E A (1) C B D A E (1) B D E A C (1) B D C E A (1) B A D C E (1) A E B D C (1) A D E B C (1) A C E D B (1) A C E B D (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -14 0 10 B -6 0 -4 -12 -4 C 14 4 0 6 -8 D 0 12 -6 0 -10 E -10 4 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.312500 D: 0.000000 E: 0.437500 Sum of squares = 0.3515625 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.562500 D: 0.562500 E: 1.000000 A B C D E A 0 6 -14 0 10 B -6 0 -4 -12 -4 C 14 4 0 6 -8 D 0 12 -6 0 -10 E -10 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.312500 D: 0.000000 E: 0.437500 Sum of squares = 0.351562500002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.562500 D: 0.562500 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=25 A=25 C=24 D=15 E=11 so E is eliminated. Round 2 votes counts: A=27 C=25 B=25 D=23 so D is eliminated. Round 3 votes counts: B=37 A=32 C=31 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:208 E:206 A:201 D:198 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 0 10 B -6 0 -4 -12 -4 C 14 4 0 6 -8 D 0 12 -6 0 -10 E -10 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.312500 D: 0.000000 E: 0.437500 Sum of squares = 0.351562500002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.562500 D: 0.562500 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 0 10 B -6 0 -4 -12 -4 C 14 4 0 6 -8 D 0 12 -6 0 -10 E -10 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.312500 D: 0.000000 E: 0.437500 Sum of squares = 0.351562500002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.562500 D: 0.562500 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 0 10 B -6 0 -4 -12 -4 C 14 4 0 6 -8 D 0 12 -6 0 -10 E -10 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.312500 D: 0.000000 E: 0.437500 Sum of squares = 0.351562500002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.562500 D: 0.562500 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4260: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (6) C D B A E (6) A E D B C (6) D C A B E (5) D A B C E (4) C D A E B (4) B E A D C (4) A E C B D (4) E C B A D (3) E B A D C (3) E B A C D (3) D B C E A (3) D B A C E (3) C D B E A (3) C D A B E (3) C B D E A (3) B E C A D (3) B D C E A (3) E B C A D (2) E A B D C (2) D B A E C (2) B E A C D (2) B D E C A (2) A E C D B (2) A E B C D (2) A D E B C (2) D C B A E (1) D C A E B (1) D B E A C (1) D A C E B (1) C E B A D (1) C B E A D (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E A C (1) B C E D A (1) B C D E A (1) A E D C B (1) A E B D C (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 4 0 0 B 8 0 18 0 8 C -4 -18 0 4 -8 D 0 0 -4 0 2 E 0 -8 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.450776 C: 0.000000 D: 0.549224 E: 0.000000 Sum of squares = 0.504846022861 Cumulative probabilities = A: 0.000000 B: 0.450776 C: 0.450776 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 0 0 B 8 0 18 0 8 C -4 -18 0 4 -8 D 0 0 -4 0 2 E 0 -8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 D=21 E=19 B=19 A=19 so E is eliminated. Round 2 votes counts: B=27 A=27 C=25 D=21 so D is eliminated. Round 3 votes counts: B=36 C=32 A=32 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:199 E:199 A:198 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 0 0 B 8 0 18 0 8 C -4 -18 0 4 -8 D 0 0 -4 0 2 E 0 -8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 0 0 B 8 0 18 0 8 C -4 -18 0 4 -8 D 0 0 -4 0 2 E 0 -8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 0 0 B 8 0 18 0 8 C -4 -18 0 4 -8 D 0 0 -4 0 2 E 0 -8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4261: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (15) E A C D B (13) A C E B D (8) B D E C A (7) E D B A C (6) C A B D E (6) E D A B C (4) E A D C B (4) A E C D B (4) A C E D B (4) C A E D B (3) B D E A C (3) D B E C A (2) D B C A E (2) C D A B E (2) A E C B D (2) E D C A B (1) E D A C B (1) E C A D B (1) E A C B D (1) D E B C A (1) D B E A C (1) D B C E A (1) C B A D E (1) C A E B D (1) B D C E A (1) B C D A E (1) B C A D E (1) A E B C D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 8 4 6 B -16 0 -6 0 -10 C -8 6 0 2 -4 D -4 0 -2 0 -10 E -6 10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 4 6 B -16 0 -6 0 -10 C -8 6 0 2 -4 D -4 0 -2 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999031 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=28 A=21 C=13 D=7 so D is eliminated. Round 2 votes counts: B=34 E=32 A=21 C=13 so C is eliminated. Round 3 votes counts: B=35 A=33 E=32 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:209 C:198 D:192 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 4 6 B -16 0 -6 0 -10 C -8 6 0 2 -4 D -4 0 -2 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999031 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 4 6 B -16 0 -6 0 -10 C -8 6 0 2 -4 D -4 0 -2 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999031 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 4 6 B -16 0 -6 0 -10 C -8 6 0 2 -4 D -4 0 -2 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999031 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4262: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (12) E C B D A (6) C E B A D (5) C A B E D (5) E D C B A (4) B C E D A (4) A D E C B (4) E D A C B (3) D E A B C (3) D B A C E (3) D A B E C (3) D A B C E (3) C B E D A (3) B D A C E (3) B C A D E (3) E D B C A (2) E C D B A (2) E C B A D (2) D B E A C (2) D B C A E (2) C E B D A (2) B C D A E (2) A E C D B (2) A C B D E (2) A B D C E (2) E C A B D (1) E B D C A (1) E A D C B (1) E A C D B (1) D B E C A (1) D B C E A (1) D B A E C (1) C B E A D (1) C B A E D (1) B A C D E (1) A D E B C (1) A D C E B (1) A D B E C (1) A C E B D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 4 -2 10 B 4 0 4 -8 16 C -4 -4 0 -8 18 D 2 8 8 0 6 E -10 -16 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -2 10 B 4 0 4 -8 16 C -4 -4 0 -8 18 D 2 8 8 0 6 E -10 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 D=19 C=17 B=13 so B is eliminated. Round 2 votes counts: A=29 C=26 E=23 D=22 so D is eliminated. Round 3 votes counts: A=42 E=29 C=29 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:212 B:208 A:204 C:201 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -2 10 B 4 0 4 -8 16 C -4 -4 0 -8 18 D 2 8 8 0 6 E -10 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -2 10 B 4 0 4 -8 16 C -4 -4 0 -8 18 D 2 8 8 0 6 E -10 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -2 10 B 4 0 4 -8 16 C -4 -4 0 -8 18 D 2 8 8 0 6 E -10 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4263: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) A E B D C (8) C D E A B (5) B C D A E (5) B A E D C (5) C D E B A (4) C B D A E (4) E D A C B (3) E A D C B (3) D E C A B (3) D C E B A (3) C D B E A (3) B A E C D (3) B A C E D (3) B A C D E (3) A B E D C (3) E A B D C (2) D C B E A (2) D B E A C (2) D B C E A (2) C D B A E (2) C A E D B (2) C A B E D (2) B D C A E (2) A E C D B (2) A B C E D (2) E D A B C (1) D E A C B (1) D C E A B (1) C A E B D (1) C A D B E (1) B E D A C (1) B D E A C (1) B D C E A (1) B D A E C (1) B A D E C (1) A E D C B (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 14 6 8 B -4 0 12 2 2 C -14 -12 0 -10 -2 D -6 -2 10 0 -6 E -8 -2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 6 8 B -4 0 12 2 2 C -14 -12 0 -10 -2 D -6 -2 10 0 -6 E -8 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 A=19 E=17 D=14 so D is eliminated. Round 2 votes counts: C=30 B=30 E=21 A=19 so A is eliminated. Round 3 votes counts: B=36 E=33 C=31 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:206 E:199 D:198 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 6 8 B -4 0 12 2 2 C -14 -12 0 -10 -2 D -6 -2 10 0 -6 E -8 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 6 8 B -4 0 12 2 2 C -14 -12 0 -10 -2 D -6 -2 10 0 -6 E -8 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 6 8 B -4 0 12 2 2 C -14 -12 0 -10 -2 D -6 -2 10 0 -6 E -8 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4264: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) C D E B A (6) B A E D C (6) E A C D B (5) D C B A E (5) E A B C D (4) C D E A B (4) B D A E C (4) B D A C E (4) B A E C D (4) E A C B D (3) D C B E A (3) D A E B C (3) C D B E A (3) B C D A E (3) E C A D B (2) E C A B D (2) E A D C B (2) D C E A B (2) D B A E C (2) C E B A D (2) C E A D B (2) C E A B D (2) C B E D A (2) A E B C D (2) E C D A B (1) E C B A D (1) D C E B A (1) D B C A E (1) D B A C E (1) D A E C B (1) C E D A B (1) C B E A D (1) B E A C D (1) B D C A E (1) B C A D E (1) A E D B C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 8 4 0 B 20 0 -2 6 4 C -8 2 0 4 -10 D -4 -6 -4 0 10 E 0 -4 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000014 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -20 8 4 0 B 20 0 -2 6 4 C -8 2 0 4 -10 D -4 -6 -4 0 10 E 0 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000301 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=23 E=20 D=19 A=6 so A is eliminated. Round 2 votes counts: B=34 E=23 C=23 D=20 so D is eliminated. Round 3 votes counts: B=38 C=34 E=28 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:214 D:198 E:198 A:196 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 8 4 0 B 20 0 -2 6 4 C -8 2 0 4 -10 D -4 -6 -4 0 10 E 0 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000301 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 8 4 0 B 20 0 -2 6 4 C -8 2 0 4 -10 D -4 -6 -4 0 10 E 0 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000301 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 8 4 0 B 20 0 -2 6 4 C -8 2 0 4 -10 D -4 -6 -4 0 10 E 0 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000301 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4265: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) E B D C A (12) C D A E B (6) A C D E B (6) A C D B E (6) B E A D C (5) E D C B A (4) E B A D C (3) D C B E A (3) D C B A E (3) C D A B E (3) B D C E A (3) A B C D E (3) E A C D B (2) E A B C D (2) C A D E B (2) B A E C D (2) A E C D B (2) A E B C D (2) E D C A B (1) E D B C A (1) E C D A B (1) E C A D B (1) E A D C B (1) D C E B A (1) D C E A B (1) D C A B E (1) C D E A B (1) B E D A C (1) B D C A E (1) B D A C E (1) B A E D C (1) B A D C E (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -18 -16 -14 B 12 0 6 6 0 C 18 -6 0 -16 -12 D 16 -6 16 0 -14 E 14 0 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.738013 C: 0.000000 D: 0.000000 E: 0.261987 Sum of squares = 0.613300327198 Cumulative probabilities = A: 0.000000 B: 0.738013 C: 0.738013 D: 0.738013 E: 1.000000 A B C D E A 0 -12 -18 -16 -14 B 12 0 6 6 0 C 18 -6 0 -16 -12 D 16 -6 16 0 -14 E 14 0 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=28 A=22 C=12 D=9 so D is eliminated. Round 2 votes counts: B=29 E=28 A=22 C=21 so C is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:220 B:212 D:206 C:192 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -18 -16 -14 B 12 0 6 6 0 C 18 -6 0 -16 -12 D 16 -6 16 0 -14 E 14 0 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -16 -14 B 12 0 6 6 0 C 18 -6 0 -16 -12 D 16 -6 16 0 -14 E 14 0 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -16 -14 B 12 0 6 6 0 C 18 -6 0 -16 -12 D 16 -6 16 0 -14 E 14 0 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4266: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (13) E A B D C (11) C D B E A (8) B D A C E (8) D B C A E (6) D B A C E (6) E A C B D (5) E C A D B (4) D C B A E (4) C E D B A (4) C E D A B (3) C D E B A (3) A E B D C (3) A B D E C (3) E C B A D (2) E A B C D (2) C D A B E (2) B D C A E (2) B A D E C (2) E C B D A (1) E C A B D (1) E A C D B (1) C E A D B (1) C B D E A (1) C A E D B (1) B E D A C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -12 -24 4 B 22 0 -8 -14 16 C 12 8 0 4 28 D 24 14 -4 0 20 E -4 -16 -28 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -12 -24 4 B 22 0 -8 -14 16 C 12 8 0 4 28 D 24 14 -4 0 20 E -4 -16 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=27 D=16 B=13 A=8 so A is eliminated. Round 2 votes counts: C=36 E=30 D=17 B=17 so D is eliminated. Round 3 votes counts: C=40 E=30 B=30 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:227 C:226 B:208 A:173 E:166 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -12 -24 4 B 22 0 -8 -14 16 C 12 8 0 4 28 D 24 14 -4 0 20 E -4 -16 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -12 -24 4 B 22 0 -8 -14 16 C 12 8 0 4 28 D 24 14 -4 0 20 E -4 -16 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -12 -24 4 B 22 0 -8 -14 16 C 12 8 0 4 28 D 24 14 -4 0 20 E -4 -16 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998229 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4267: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) A B D E C (8) B A E D C (6) C E D B A (5) D E C A B (4) C E B D A (4) B A E C D (4) E D C B A (3) E C D B A (3) D C E A B (3) C D E B A (3) B E C A D (3) B A C E D (3) A C D B E (3) A B C D E (3) E D B C A (2) C D A E B (2) C B E D A (2) B E A D C (2) B E A C D (2) A D E C B (2) A D C E B (2) A D B C E (2) A B E D C (2) A B D C E (2) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) D E B A C (1) D E A C B (1) D E A B C (1) D A E C B (1) D A C E B (1) C B E A D (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) A D E B C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 2 -2 -10 B 2 0 -2 0 -2 C -2 2 0 2 -8 D 2 0 -2 0 2 E 10 2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 1.000000 A B C D E A 0 -2 2 -2 -10 B 2 0 -2 0 -2 C -2 2 0 2 -8 D 2 0 -2 0 2 E 10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 B=23 E=12 D=12 so E is eliminated. Round 2 votes counts: C=30 A=27 B=26 D=17 so D is eliminated. Round 3 votes counts: C=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:209 D:201 B:199 C:197 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 -10 B 2 0 -2 0 -2 C -2 2 0 2 -8 D 2 0 -2 0 2 E 10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 -10 B 2 0 -2 0 -2 C -2 2 0 2 -8 D 2 0 -2 0 2 E 10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 -10 B 2 0 -2 0 -2 C -2 2 0 2 -8 D 2 0 -2 0 2 E 10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4268: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A E C B D (7) A C D B E (7) E D B A C (6) D E B C A (5) D B E C A (5) D C A B E (4) D A C B E (4) E D B C A (3) C A D B E (3) C A B D E (3) B C D E A (3) B C A E D (3) A C B D E (3) E B C D A (2) E B A C D (2) E A D B C (2) D E A B C (2) B E C A D (2) A C E B D (2) E B D A C (1) E B A D C (1) E A B D C (1) E A B C D (1) D E B A C (1) D E A C B (1) D C E B A (1) D C B A E (1) D B C E A (1) D A C E B (1) C B D A E (1) C B A E D (1) C B A D E (1) B E D C A (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A E B C D (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -2 -8 -4 B 4 0 10 -2 -2 C 2 -10 0 -4 -10 D 8 2 4 0 2 E 4 2 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 -4 B 4 0 10 -2 -2 C 2 -10 0 -4 -10 D 8 2 4 0 2 E 4 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 A=26 B=12 C=9 so C is eliminated. Round 2 votes counts: A=32 E=27 D=26 B=15 so B is eliminated. Round 3 votes counts: A=37 D=32 E=31 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:207 B:205 A:191 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 -4 B 4 0 10 -2 -2 C 2 -10 0 -4 -10 D 8 2 4 0 2 E 4 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 -4 B 4 0 10 -2 -2 C 2 -10 0 -4 -10 D 8 2 4 0 2 E 4 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 -4 B 4 0 10 -2 -2 C 2 -10 0 -4 -10 D 8 2 4 0 2 E 4 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4269: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (20) D E A B C (8) C A B E D (7) D C A E B (5) C B E A D (5) D E B C A (4) C B A E D (4) E B D A C (3) D A E B C (3) C D A E B (3) B E A C D (3) A C B E D (3) E D A B C (2) D E C B A (2) D E A C B (2) C B E D A (2) C A B D E (2) B E C A D (2) B C E A D (2) A D E B C (2) D C E B A (1) D C E A B (1) D C B E A (1) D A E C B (1) D A C E B (1) C D B E A (1) C D A B E (1) C B D E A (1) C B A D E (1) C A D E B (1) C A D B E (1) B E C D A (1) B C A E D (1) A E B D C (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 2 -26 -22 B 8 0 4 -24 -22 C -2 -4 0 -16 -8 D 26 24 16 0 28 E 22 22 8 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -26 -22 B 8 0 4 -24 -22 C -2 -4 0 -16 -8 D 26 24 16 0 28 E 22 22 8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=49 C=29 B=9 A=8 E=5 so E is eliminated. Round 2 votes counts: D=51 C=29 B=12 A=8 so A is eliminated. Round 3 votes counts: D=54 C=33 B=13 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:247 E:212 C:185 B:183 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -26 -22 B 8 0 4 -24 -22 C -2 -4 0 -16 -8 D 26 24 16 0 28 E 22 22 8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -26 -22 B 8 0 4 -24 -22 C -2 -4 0 -16 -8 D 26 24 16 0 28 E 22 22 8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -26 -22 B 8 0 4 -24 -22 C -2 -4 0 -16 -8 D 26 24 16 0 28 E 22 22 8 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4270: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) E B D C A (8) D B C A E (7) B D C E A (6) A C D E B (6) E A C D B (5) B E D C A (5) A C D B E (5) E B D A C (4) E A B C D (4) B E D A C (4) A E C D B (4) A C E D B (4) E A C B D (3) D C A B E (3) C D A B E (3) B D E C A (3) E B C D A (2) E B A D C (2) E B A C D (2) C A D E B (2) C A D B E (2) B D E A C (2) E C A D B (1) C E A D B (1) B D A C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -6 -16 -4 B 12 0 20 14 4 C 6 -20 0 -10 2 D 16 -14 10 0 0 E 4 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -16 -4 B 12 0 20 14 4 C 6 -20 0 -10 2 D 16 -14 10 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997371 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=31 B=31 A=20 D=10 C=8 so C is eliminated. Round 2 votes counts: E=32 B=31 A=24 D=13 so D is eliminated. Round 3 votes counts: B=38 E=32 A=30 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:206 E:199 C:189 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -16 -4 B 12 0 20 14 4 C 6 -20 0 -10 2 D 16 -14 10 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997371 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -16 -4 B 12 0 20 14 4 C 6 -20 0 -10 2 D 16 -14 10 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997371 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -16 -4 B 12 0 20 14 4 C 6 -20 0 -10 2 D 16 -14 10 0 0 E 4 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997371 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4271: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (14) B E C A D (10) E B D A C (8) B E A D C (7) C A D B E (6) A D C B E (6) E D A B C (4) E B C D A (4) C A D E B (4) B C E A D (4) B C A D E (3) E D C A B (2) E D A C B (2) E C D B A (2) D A E C B (2) D A B C E (2) C B A D E (2) B A D C E (2) B A C D E (2) A C D B E (2) E D B C A (1) E D B A C (1) E C D A B (1) E B D C A (1) D E A C B (1) D A C B E (1) D A B E C (1) C E D A B (1) C D E A B (1) B E D A C (1) B E A C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 16 0 -4 B -2 0 6 -8 2 C -16 -6 0 -14 2 D 0 8 14 0 0 E 4 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.471392 E: 0.528608 Sum of squares = 0.501636793625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.471392 E: 1.000000 A B C D E A 0 2 16 0 -4 B -2 0 6 -8 2 C -16 -6 0 -14 2 D 0 8 14 0 0 E 4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 D=21 C=14 A=9 so A is eliminated. Round 2 votes counts: B=31 D=27 E=26 C=16 so C is eliminated. Round 3 votes counts: D=40 B=33 E=27 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:207 E:200 B:199 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 16 0 -4 B -2 0 6 -8 2 C -16 -6 0 -14 2 D 0 8 14 0 0 E 4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 0 -4 B -2 0 6 -8 2 C -16 -6 0 -14 2 D 0 8 14 0 0 E 4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 0 -4 B -2 0 6 -8 2 C -16 -6 0 -14 2 D 0 8 14 0 0 E 4 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4272: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) D A E B C (9) D A B C E (9) B C E D A (9) E C B A D (8) B C E A D (7) B C D E A (6) C B E A D (5) D A E C B (4) E B C A D (3) D A B E C (3) B E C A D (3) A D E B C (3) E C A B D (2) D B C A E (2) C E B A D (2) C B E D A (2) E A C D B (1) E A C B D (1) E A B C D (1) D C B A E (1) D A C B E (1) C E B D A (1) C D E B A (1) B D C A E (1) A E D C B (1) A E B D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -6 2 -4 B 2 0 18 6 2 C 6 -18 0 4 -6 D -2 -6 -4 0 4 E 4 -2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 2 -4 B 2 0 18 6 2 C 6 -18 0 4 -6 D -2 -6 -4 0 4 E 4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 A=18 E=16 C=11 so C is eliminated. Round 2 votes counts: B=33 D=30 E=19 A=18 so A is eliminated. Round 3 votes counts: D=45 B=34 E=21 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:202 D:196 A:195 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 2 -4 B 2 0 18 6 2 C 6 -18 0 4 -6 D -2 -6 -4 0 4 E 4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 -4 B 2 0 18 6 2 C 6 -18 0 4 -6 D -2 -6 -4 0 4 E 4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 -4 B 2 0 18 6 2 C 6 -18 0 4 -6 D -2 -6 -4 0 4 E 4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4273: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) C B A E D (8) D E C A B (7) B A C E D (7) A B C E D (7) D E C B A (6) D C E B A (4) C B A D E (4) E D A C B (3) E D A B C (3) C D E B A (3) C D B E A (3) B C A D E (3) B A C D E (3) A E B D C (3) E D C A B (2) D E B C A (2) D E A C B (2) C B D A E (2) B C A E D (2) A D B E C (2) E C A B D (1) E A D C B (1) E A B D C (1) E A B C D (1) D C B E A (1) D B E C A (1) D B C E A (1) D A B E C (1) C E D B A (1) C E B D A (1) C D B A E (1) C B E A D (1) C A B E D (1) B A D C E (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -10 -4 -6 B 10 0 -4 -6 0 C 10 4 0 0 8 D 4 6 0 0 10 E 6 0 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.474527 D: 0.525473 E: 0.000000 Sum of squares = 0.501297787908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.474527 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -4 -6 B 10 0 -4 -6 0 C 10 4 0 0 8 D 4 6 0 0 10 E 6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=25 B=16 A=14 E=12 so E is eliminated. Round 2 votes counts: D=41 C=26 A=17 B=16 so B is eliminated. Round 3 votes counts: D=41 C=31 A=28 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:210 B:200 E:194 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 -4 -6 B 10 0 -4 -6 0 C 10 4 0 0 8 D 4 6 0 0 10 E 6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -4 -6 B 10 0 -4 -6 0 C 10 4 0 0 8 D 4 6 0 0 10 E 6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -4 -6 B 10 0 -4 -6 0 C 10 4 0 0 8 D 4 6 0 0 10 E 6 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4274: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D B C A E (10) A E D B C (10) C B D A E (8) C B D E A (7) D B A C E (6) C B E D A (5) E C A B D (4) E A D C B (4) D B A E C (4) D A B E C (4) E A D B C (3) C E A B D (3) B D C A E (3) E A C D B (2) D A E B C (2) A E C B D (2) D C B E A (1) C E B D A (1) C B E A D (1) C B A D E (1) A E B C D (1) A D E B C (1) A C B E D (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 12 -2 12 B -6 0 -6 6 8 C -12 6 0 0 -4 D 2 -6 0 0 -2 E -12 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 -2 12 B -6 0 -6 6 8 C -12 6 0 0 -4 D 2 -6 0 0 -2 E -12 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=26 C=26 A=18 B=3 so B is eliminated. Round 2 votes counts: D=30 E=26 C=26 A=18 so A is eliminated. Round 3 votes counts: E=40 D=33 C=27 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:214 B:201 D:197 C:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 -2 12 B -6 0 -6 6 8 C -12 6 0 0 -4 D 2 -6 0 0 -2 E -12 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 -2 12 B -6 0 -6 6 8 C -12 6 0 0 -4 D 2 -6 0 0 -2 E -12 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 -2 12 B -6 0 -6 6 8 C -12 6 0 0 -4 D 2 -6 0 0 -2 E -12 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.428571 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4275: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) C E A D B (7) A E C D B (7) A E B D C (7) D B C E A (6) B D C E A (6) E A C B D (5) C D B A E (5) A E C B D (5) D B C A E (4) B D A E C (4) E A C D B (3) E A B D C (3) C D A B E (3) B D E A C (3) D C B E A (2) D C B A E (2) C A E D B (2) B D C A E (2) A E B C D (2) E C A D B (1) E C A B D (1) E A B C D (1) D A B C E (1) C E D B A (1) C B D E A (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E C A (1) B A E D C (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -8 -2 -4 B 0 0 -10 -8 8 C 8 10 0 8 4 D 2 8 -8 0 2 E 4 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -2 -4 B 0 0 -10 -8 8 C 8 10 0 8 4 D 2 8 -8 0 2 E 4 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=22 B=20 D=15 E=14 so E is eliminated. Round 2 votes counts: A=34 C=31 B=20 D=15 so D is eliminated. Round 3 votes counts: C=35 A=35 B=30 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:202 B:195 E:195 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -2 -4 B 0 0 -10 -8 8 C 8 10 0 8 4 D 2 8 -8 0 2 E 4 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 -4 B 0 0 -10 -8 8 C 8 10 0 8 4 D 2 8 -8 0 2 E 4 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 -4 B 0 0 -10 -8 8 C 8 10 0 8 4 D 2 8 -8 0 2 E 4 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4276: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) E C A B D (7) E A C B D (7) A E C D B (6) D B C A E (5) D B A C E (4) C E B D A (4) C E B A D (4) C B D E A (4) A D B E C (4) E C A D B (3) E B C A D (3) E A C D B (3) B C D E A (3) A E D B C (3) A E B D C (3) A D E C B (3) A D E B C (3) D B C E A (2) C E D B A (2) C D B E A (2) B D C A E (2) A E D C B (2) E C B A D (1) E A B C D (1) D C B E A (1) D C A E B (1) D A C B E (1) C E A D B (1) C B E D A (1) B E C A D (1) B E A C D (1) B D E A C (1) B D A E C (1) B D A C E (1) Total count = 100 A B C D E A 0 -4 -12 12 -22 B 4 0 -6 8 -14 C 12 6 0 8 -6 D -12 -8 -8 0 -6 E 22 14 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -12 12 -22 B 4 0 -6 8 -14 C 12 6 0 8 -6 D -12 -8 -8 0 -6 E 22 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=24 B=19 C=18 D=14 so D is eliminated. Round 2 votes counts: B=30 E=25 A=25 C=20 so C is eliminated. Round 3 votes counts: B=38 E=36 A=26 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 C:210 B:196 A:187 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -12 12 -22 B 4 0 -6 8 -14 C 12 6 0 8 -6 D -12 -8 -8 0 -6 E 22 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 12 -22 B 4 0 -6 8 -14 C 12 6 0 8 -6 D -12 -8 -8 0 -6 E 22 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 12 -22 B 4 0 -6 8 -14 C 12 6 0 8 -6 D -12 -8 -8 0 -6 E 22 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4277: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (12) D C A E B (11) E D B A C (9) C A B D E (9) B E A C D (6) E B A C D (5) C A D B E (5) B A C E D (5) D C A B E (4) C A B E D (4) A C B D E (4) E D C A B (3) E B D A C (3) E B C A D (3) D E B A C (3) D A C B E (3) E D B C A (2) A C B E D (2) E D C B A (1) E C A B D (1) E B A D C (1) D E B C A (1) D E A C B (1) B D E A C (1) B A C D E (1) Total count = 100 A B C D E A 0 18 -12 -8 -4 B -18 0 -20 -10 -12 C 12 20 0 -10 -4 D 8 10 10 0 10 E 4 12 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 -8 -4 B -18 0 -20 -10 -12 C 12 20 0 -10 -4 D 8 10 10 0 10 E 4 12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=28 C=18 B=13 A=6 so A is eliminated. Round 2 votes counts: D=35 E=28 C=24 B=13 so B is eliminated. Round 3 votes counts: D=36 E=34 C=30 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:209 E:205 A:197 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -12 -8 -4 B -18 0 -20 -10 -12 C 12 20 0 -10 -4 D 8 10 10 0 10 E 4 12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 -8 -4 B -18 0 -20 -10 -12 C 12 20 0 -10 -4 D 8 10 10 0 10 E 4 12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 -8 -4 B -18 0 -20 -10 -12 C 12 20 0 -10 -4 D 8 10 10 0 10 E 4 12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4278: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) E B D C A (5) B D E A C (5) A C D B E (5) D C E A B (4) D A C B E (4) A C B D E (4) E C A D B (3) D E C B A (3) D C A B E (3) C E A D B (3) C A D E B (3) B E D A C (3) B D A E C (3) B D A C E (3) B A E C D (3) E C D A B (2) E C A B D (2) E B A C D (2) D E B C A (2) D B C A E (2) C E D A B (2) B E A D C (2) A C E B D (2) E D C B A (1) E C B D A (1) E B C D A (1) E B C A D (1) E A C B D (1) E A B C D (1) D C E B A (1) D C B A E (1) D B E C A (1) D B A C E (1) D A B C E (1) C D A E B (1) C A D B E (1) B E A C D (1) B D E C A (1) B A C E D (1) A D B C E (1) A C E D B (1) A C D E B (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -4 -2 4 B -12 0 -16 -10 -2 C 4 16 0 4 12 D 2 10 -4 0 4 E -4 2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 -2 4 B -12 0 -16 -10 -2 C 4 16 0 4 12 D 2 10 -4 0 4 E -4 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 B=22 E=20 C=18 A=17 so A is eliminated. Round 2 votes counts: C=32 D=24 B=24 E=20 so E is eliminated. Round 3 votes counts: C=41 B=34 D=25 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:206 A:205 E:191 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 -2 4 B -12 0 -16 -10 -2 C 4 16 0 4 12 D 2 10 -4 0 4 E -4 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 -2 4 B -12 0 -16 -10 -2 C 4 16 0 4 12 D 2 10 -4 0 4 E -4 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 -2 4 B -12 0 -16 -10 -2 C 4 16 0 4 12 D 2 10 -4 0 4 E -4 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4279: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) C A E B D (11) D B E A C (10) A C D E B (7) C E B D A (6) A C D B E (6) E C B D A (5) B E D C A (5) A D B E C (5) D A B E C (4) A D B C E (3) D B E C A (2) C E B A D (2) C B E D A (2) C A B E D (2) A C E B D (2) E C A B D (1) E B C D A (1) D E B A C (1) D B A C E (1) C E A B D (1) C B D E A (1) C B A E D (1) C A B D E (1) B D E C A (1) B D C A E (1) B C E D A (1) B C D E A (1) A D E B C (1) A D C B E (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -12 -6 -2 B 4 0 -4 16 0 C 12 4 0 8 6 D 6 -16 -8 0 -6 E 2 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -6 -2 B 4 0 -4 16 0 C 12 4 0 8 6 D 6 -16 -8 0 -6 E 2 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=27 E=18 D=18 B=9 so B is eliminated. Round 2 votes counts: C=29 A=28 E=23 D=20 so D is eliminated. Round 3 votes counts: E=37 A=33 C=30 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:215 B:208 E:201 A:188 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -6 -2 B 4 0 -4 16 0 C 12 4 0 8 6 D 6 -16 -8 0 -6 E 2 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -6 -2 B 4 0 -4 16 0 C 12 4 0 8 6 D 6 -16 -8 0 -6 E 2 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -6 -2 B 4 0 -4 16 0 C 12 4 0 8 6 D 6 -16 -8 0 -6 E 2 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4280: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) D C A E B (7) D B E A C (6) B A E C D (6) A C B E D (6) D E C B A (5) D E B C A (5) D A B C E (4) B E A D C (4) A B C E D (4) E C B A D (3) D A C B E (3) C A D E B (3) C A D B E (3) C A B E D (3) A C D B E (3) E B C D A (2) D E B A C (2) D C E A B (2) D C A B E (2) D B A E C (2) C D A E B (2) B E D A C (2) E B D C A (1) E B C A D (1) E B A C D (1) D C E B A (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E A B (1) C A E D B (1) C A E B D (1) A D C B E (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 10 2 8 B 2 0 2 -12 20 C -10 -2 0 2 0 D -2 12 -2 0 8 E -8 -20 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000025 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 2 8 B 2 0 2 -12 20 C -10 -2 0 2 0 D -2 12 -2 0 8 E -8 -20 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000053 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=20 C=16 A=16 E=8 so E is eliminated. Round 2 votes counts: D=40 B=25 C=19 A=16 so A is eliminated. Round 3 votes counts: D=42 B=30 C=28 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:208 B:206 C:195 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 10 2 8 B 2 0 2 -12 20 C -10 -2 0 2 0 D -2 12 -2 0 8 E -8 -20 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000053 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 2 8 B 2 0 2 -12 20 C -10 -2 0 2 0 D -2 12 -2 0 8 E -8 -20 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000053 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 2 8 B 2 0 2 -12 20 C -10 -2 0 2 0 D -2 12 -2 0 8 E -8 -20 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000053 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4281: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) B D E A C (11) D B E C A (6) D B C E A (6) A C E D B (6) A E B D C (5) C D B E A (4) B D C E A (4) D C B E A (3) C D A B E (3) A E C B D (3) A E B C D (3) C D E B A (2) C D B A E (2) C A E B D (2) C A D E B (2) B E D A C (2) B E A D C (2) B D E C A (2) B A D E C (2) A C E B D (2) E D A B C (1) E B D A C (1) E A D C B (1) E A B D C (1) D E C B A (1) D E B C A (1) D B E A C (1) C E D A B (1) C E A D B (1) C B D A E (1) C A D B E (1) C A B D E (1) B D C A E (1) A E C D B (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 -6 -2 B 4 0 2 -8 10 C 10 -2 0 -4 10 D 6 8 4 0 10 E 2 -10 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -2 B 4 0 2 -8 10 C 10 -2 0 -4 10 D 6 8 4 0 10 E 2 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=24 A=23 D=18 E=4 so E is eliminated. Round 2 votes counts: C=31 B=25 A=25 D=19 so D is eliminated. Round 3 votes counts: B=39 C=35 A=26 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 C:207 B:204 A:189 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -2 B 4 0 2 -8 10 C 10 -2 0 -4 10 D 6 8 4 0 10 E 2 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -2 B 4 0 2 -8 10 C 10 -2 0 -4 10 D 6 8 4 0 10 E 2 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -2 B 4 0 2 -8 10 C 10 -2 0 -4 10 D 6 8 4 0 10 E 2 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4282: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) E D B A C (10) E D B C A (9) A C D E B (8) C A D E B (7) B D E A C (7) C A B D E (6) A C B D E (5) C A B E D (4) E D C B A (3) D E B A C (3) B A D E C (3) B E D C A (2) B E C D A (2) B D A E C (2) A D E B C (2) A C D B E (2) A B C D E (2) E B D A C (1) D A E B C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A D B (1) C B E D A (1) C B A E D (1) C A E B D (1) B E D A C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -4 10 14 B -6 0 -8 -20 -22 C 4 8 0 10 6 D -10 20 -10 0 -2 E -14 22 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 10 14 B -6 0 -8 -20 -22 C 4 8 0 10 6 D -10 20 -10 0 -2 E -14 22 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=23 A=20 B=17 D=4 so D is eliminated. Round 2 votes counts: C=36 E=26 A=21 B=17 so B is eliminated. Round 3 votes counts: E=38 C=36 A=26 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:213 E:202 D:199 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 10 14 B -6 0 -8 -20 -22 C 4 8 0 10 6 D -10 20 -10 0 -2 E -14 22 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 10 14 B -6 0 -8 -20 -22 C 4 8 0 10 6 D -10 20 -10 0 -2 E -14 22 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 10 14 B -6 0 -8 -20 -22 C 4 8 0 10 6 D -10 20 -10 0 -2 E -14 22 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4283: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) A E C B D (6) C A B D E (5) E A C D B (4) D C B E A (4) C D B A E (4) B D C A E (4) E A D B C (3) D C E B A (3) D B E C A (3) C E D A B (3) A C B E D (3) A C B D E (3) A B E C D (3) E D B A C (2) D C B A E (2) C D E B A (2) C D B E A (2) C A D E B (2) C A D B E (2) B C D A E (2) B A D E C (2) B A D C E (2) A E B D C (2) A E B C D (2) A B E D C (2) A B C E D (2) A B C D E (2) E D C B A (1) E D A C B (1) E C D A B (1) E C A D B (1) E B A D C (1) E A D C B (1) E A C B D (1) D E B C A (1) D B C E A (1) D B C A E (1) C E A D B (1) C D A B E (1) C B D A E (1) B D A C E (1) B C A D E (1) B A C D E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 18 4 20 16 B -18 0 -10 8 10 C -4 10 0 12 14 D -20 -8 -12 0 6 E -16 -10 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 20 16 B -18 0 -10 8 10 C -4 10 0 12 14 D -20 -8 -12 0 6 E -16 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 E=22 D=15 B=13 so B is eliminated. Round 2 votes counts: A=32 C=26 E=22 D=20 so D is eliminated. Round 3 votes counts: C=41 A=33 E=26 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:216 B:195 D:183 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 20 16 B -18 0 -10 8 10 C -4 10 0 12 14 D -20 -8 -12 0 6 E -16 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 20 16 B -18 0 -10 8 10 C -4 10 0 12 14 D -20 -8 -12 0 6 E -16 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 20 16 B -18 0 -10 8 10 C -4 10 0 12 14 D -20 -8 -12 0 6 E -16 -10 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4284: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (10) D B A C E (5) C E D A B (5) C A B E D (5) E D C B A (4) E C D B A (4) E C A B D (4) D E B A C (4) B A D E C (4) D B E A C (3) C E A D B (3) C D B A E (3) C A B D E (3) A B C D E (3) E C D A B (2) E B A D C (2) E A B C D (2) D E C B A (2) D C B E A (2) C A E B D (2) B A E D C (2) E D B C A (1) E D B A C (1) E B D A C (1) E A C B D (1) D B E C A (1) D B C A E (1) C D E B A (1) C D A B E (1) C A E D B (1) B D A E C (1) B A D C E (1) A C E B D (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -10 -6 -6 B 8 0 -8 -10 -2 C 10 8 0 6 -2 D 6 10 -6 0 -6 E 6 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 -6 B 8 0 -8 -10 -2 C 10 8 0 6 -2 D 6 10 -6 0 -6 E 6 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=29 E=22 B=8 A=7 so A is eliminated. Round 2 votes counts: C=35 D=29 E=22 B=14 so B is eliminated. Round 3 votes counts: C=39 D=36 E=25 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 E:208 D:202 B:194 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 -6 B 8 0 -8 -10 -2 C 10 8 0 6 -2 D 6 10 -6 0 -6 E 6 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 -6 B 8 0 -8 -10 -2 C 10 8 0 6 -2 D 6 10 -6 0 -6 E 6 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 -6 B 8 0 -8 -10 -2 C 10 8 0 6 -2 D 6 10 -6 0 -6 E 6 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4285: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) E A D C B (7) A D E C B (6) E A D B C (5) E A B C D (5) D C A B E (5) A E D C B (5) A E C D B (5) D A C E B (4) B D C E A (4) E B A D C (3) E A C D B (3) E A C B D (3) E A B D C (3) D B C A E (3) B E C A D (3) B C D E A (3) B C D A E (3) E B C A D (2) E B A C D (2) D A E C B (2) A C D E B (2) D B E C A (1) D B C E A (1) C D B A E (1) C B D A E (1) C A E B D (1) C A B D E (1) B E C D A (1) B D C A E (1) B C E D A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 18 14 16 6 B -18 0 -18 -24 -20 C -14 18 0 -24 -14 D -16 24 24 0 0 E -6 20 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 14 16 6 B -18 0 -18 -24 -20 C -14 18 0 -24 -14 D -16 24 24 0 0 E -6 20 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=27 A=20 B=16 C=4 so C is eliminated. Round 2 votes counts: E=33 D=28 A=22 B=17 so B is eliminated. Round 3 votes counts: D=40 E=38 A=22 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:227 D:216 E:214 C:183 B:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 14 16 6 B -18 0 -18 -24 -20 C -14 18 0 -24 -14 D -16 24 24 0 0 E -6 20 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 14 16 6 B -18 0 -18 -24 -20 C -14 18 0 -24 -14 D -16 24 24 0 0 E -6 20 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 14 16 6 B -18 0 -18 -24 -20 C -14 18 0 -24 -14 D -16 24 24 0 0 E -6 20 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4286: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (14) D C A E B (11) B E A C D (11) D B C A E (8) E A C D B (5) C A E D B (5) B D C A E (5) D C B A E (4) B D E A C (4) D C A B E (3) C D A E B (3) B D E C A (3) B D C E A (3) A E C D B (3) E B A C D (2) E A B C D (2) C A D E B (2) B E D A C (2) B E A D C (2) A C E D B (2) D A C B E (1) C E A D B (1) C D B E A (1) B D A C E (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 4 -2 -2 -4 B -4 0 -10 2 0 C 2 10 0 6 4 D 2 -2 -6 0 0 E 4 0 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -2 -4 B -4 0 -10 2 0 C 2 10 0 6 4 D 2 -2 -6 0 0 E 4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=27 E=23 C=12 A=5 so A is eliminated. Round 2 votes counts: B=33 D=27 E=26 C=14 so C is eliminated. Round 3 votes counts: E=34 D=33 B=33 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 E:200 A:198 D:197 B:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -2 -4 B -4 0 -10 2 0 C 2 10 0 6 4 D 2 -2 -6 0 0 E 4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -2 -4 B -4 0 -10 2 0 C 2 10 0 6 4 D 2 -2 -6 0 0 E 4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -2 -4 B -4 0 -10 2 0 C 2 10 0 6 4 D 2 -2 -6 0 0 E 4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4287: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (16) C A E D B (13) E A D B C (11) C B D A E (9) E D A B C (8) A E C D B (7) C A B D E (3) B D C E A (3) D B E A C (2) D A E B C (2) C E A B D (2) C B E A D (2) C B D E A (2) C B A E D (2) C B A D E (2) C A E B D (2) B C D E A (2) A E D C B (2) E D B A C (1) E A C D B (1) D E A B C (1) C E B A D (1) C E A D B (1) C A D E B (1) C A D B E (1) B D E C A (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 14 6 6 -8 B -14 0 -4 -6 -10 C -6 4 0 4 -6 D -6 6 -4 0 -10 E 8 10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 6 6 -8 B -14 0 -4 -6 -10 C -6 4 0 4 -6 D -6 6 -4 0 -10 E 8 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 B=22 E=21 A=11 D=5 so D is eliminated. Round 2 votes counts: C=41 B=24 E=22 A=13 so A is eliminated. Round 3 votes counts: C=42 E=34 B=24 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:209 C:198 D:193 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 6 6 -8 B -14 0 -4 -6 -10 C -6 4 0 4 -6 D -6 6 -4 0 -10 E 8 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 6 -8 B -14 0 -4 -6 -10 C -6 4 0 4 -6 D -6 6 -4 0 -10 E 8 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 6 -8 B -14 0 -4 -6 -10 C -6 4 0 4 -6 D -6 6 -4 0 -10 E 8 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4288: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (6) C B D E A (6) D A E C B (5) B C E A D (5) B C D A E (5) A E D B C (5) E A B D C (3) D C A E B (3) D A C E B (3) D A C B E (3) C B E D A (3) B A E C D (3) A E B D C (3) A D E B C (3) E C B A D (2) E A D C B (2) D C E A B (2) D C B A E (2) D C A B E (2) D A E B C (2) C D B E A (2) B C A D E (2) B A C D E (2) A D B E C (2) E C D A B (1) E C B D A (1) E C A D B (1) E C A B D (1) E B C A D (1) E A D B C (1) E A C B D (1) E A B C D (1) D E C A B (1) D B C A E (1) D B A C E (1) C E D B A (1) C E B D A (1) C D E B A (1) C B D A E (1) B E C A D (1) B D C A E (1) B C E D A (1) B C A E D (1) B A D E C (1) B A C E D (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -10 -10 22 B 4 0 -2 -2 10 C 10 2 0 2 14 D 10 2 -2 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -10 22 B 4 0 -2 -2 10 C 10 2 0 2 14 D 10 2 -2 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=23 C=21 A=16 E=15 so E is eliminated. Round 2 votes counts: C=27 D=25 B=24 A=24 so B is eliminated. Round 3 votes counts: C=43 A=31 D=26 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:214 B:205 A:199 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -10 22 B 4 0 -2 -2 10 C 10 2 0 2 14 D 10 2 -2 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -10 22 B 4 0 -2 -2 10 C 10 2 0 2 14 D 10 2 -2 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -10 22 B 4 0 -2 -2 10 C 10 2 0 2 14 D 10 2 -2 0 18 E -22 -10 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4289: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (15) E D A B C (11) A C E D B (11) E D B A C (9) C A B D E (9) B C D E A (7) C B A D E (6) D E B A C (5) D E B C A (4) A E D C B (4) B C D A E (3) A C B E D (3) A C B D E (3) E D B C A (2) D B E C A (2) E D A C B (1) C A B E D (1) B E D C A (1) B D C E A (1) B C E D A (1) B C A D E (1) Total count = 100 A B C D E A 0 -14 -6 -24 -18 B 14 0 24 2 6 C 6 -24 0 -10 -8 D 24 -2 10 0 12 E 18 -6 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -24 -18 B 14 0 24 2 6 C 6 -24 0 -10 -8 D 24 -2 10 0 12 E 18 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995115 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 A=21 C=16 D=11 so D is eliminated. Round 2 votes counts: E=32 B=31 A=21 C=16 so C is eliminated. Round 3 votes counts: B=37 E=32 A=31 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:222 E:204 C:182 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -24 -18 B 14 0 24 2 6 C 6 -24 0 -10 -8 D 24 -2 10 0 12 E 18 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995115 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -24 -18 B 14 0 24 2 6 C 6 -24 0 -10 -8 D 24 -2 10 0 12 E 18 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995115 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -24 -18 B 14 0 24 2 6 C 6 -24 0 -10 -8 D 24 -2 10 0 12 E 18 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995115 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4290: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) D A B C E (7) B C D E A (7) E A C B D (5) C E B A D (5) A D E B C (5) E C A B D (4) E A C D B (4) D B A C E (4) C E B D A (4) E C B A D (3) E A D C B (3) C E A B D (3) B D C E A (3) A E D C B (3) A E C D B (3) A D C B E (3) A D B C E (3) E C B D A (2) E B C D A (2) D A C B E (2) C B D E A (2) C B D A E (2) C A E B D (2) E A D B C (1) D E B A C (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B E C (1) B D E C A (1) B C E D A (1) B C D A E (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -2 -2 -20 B -2 0 -18 10 -4 C 2 18 0 16 16 D 2 -10 -16 0 -8 E 20 4 -16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -2 -20 B -2 0 -18 10 -4 C 2 18 0 16 16 D 2 -10 -16 0 -8 E 20 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 A=19 D=18 B=13 so B is eliminated. Round 2 votes counts: C=35 E=24 D=22 A=19 so A is eliminated. Round 3 votes counts: C=35 D=34 E=31 so E is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:208 B:193 A:189 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 -2 -20 B -2 0 -18 10 -4 C 2 18 0 16 16 D 2 -10 -16 0 -8 E 20 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -2 -20 B -2 0 -18 10 -4 C 2 18 0 16 16 D 2 -10 -16 0 -8 E 20 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -2 -20 B -2 0 -18 10 -4 C 2 18 0 16 16 D 2 -10 -16 0 -8 E 20 4 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4291: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) C D E B A (8) E C A D B (6) D C B A E (6) B A D C E (6) E A B C D (5) D C B E A (5) B D A C E (5) E A C B D (4) D B C A E (4) C E D A B (4) B A E D C (4) B A D E C (4) A E B D C (3) E C D B A (2) E A C D B (2) E A B D C (2) A E C B D (2) E D C B A (1) E C D A B (1) E B A D C (1) D C E B A (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B A E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) A E C D B (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 8 14 4 B 6 0 0 4 2 C -8 0 0 -14 -8 D -14 -4 14 0 -6 E -4 -2 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.892961 C: 0.107039 D: 0.000000 E: 0.000000 Sum of squares = 0.808837335158 Cumulative probabilities = A: 0.000000 B: 0.892961 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 14 4 B 6 0 0 4 2 C -8 0 0 -14 -8 D -14 -4 14 0 -6 E -4 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000056176 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=23 C=19 A=18 D=16 so D is eliminated. Round 2 votes counts: C=31 B=27 E=24 A=18 so A is eliminated. Round 3 votes counts: B=38 E=31 C=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:210 B:206 E:204 D:195 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 14 4 B 6 0 0 4 2 C -8 0 0 -14 -8 D -14 -4 14 0 -6 E -4 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000056176 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 14 4 B 6 0 0 4 2 C -8 0 0 -14 -8 D -14 -4 14 0 -6 E -4 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000056176 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 14 4 B 6 0 0 4 2 C -8 0 0 -14 -8 D -14 -4 14 0 -6 E -4 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000056176 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4292: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (13) B E D C A (9) D C E B A (8) C D A E B (8) D C A E B (5) D B C E A (5) E B C D A (4) B D E C A (4) A E B C D (3) E C D B A (2) E C B D A (2) E B D C A (2) E B A C D (2) C A D E B (2) B E D A C (2) B E A C D (2) B D E A C (2) A E C B D (2) A D C B E (2) A C E D B (2) E C B A D (1) E A B C D (1) D E B C A (1) D C B E A (1) D B E C A (1) D A C B E (1) C E D B A (1) C D E B A (1) B E A D C (1) B D A E C (1) B A E D C (1) B A D E C (1) A D C E B (1) A C D B E (1) A B E D C (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -14 -20 -4 B 8 0 -6 -10 -22 C 14 6 0 -2 6 D 20 10 2 0 20 E 4 22 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -20 -4 B 8 0 -6 -10 -22 C 14 6 0 -2 6 D 20 10 2 0 20 E 4 22 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 D=22 E=14 C=12 so C is eliminated. Round 2 votes counts: D=31 A=31 B=23 E=15 so E is eliminated. Round 3 votes counts: D=34 B=34 A=32 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:212 E:200 B:185 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -14 -20 -4 B 8 0 -6 -10 -22 C 14 6 0 -2 6 D 20 10 2 0 20 E 4 22 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -20 -4 B 8 0 -6 -10 -22 C 14 6 0 -2 6 D 20 10 2 0 20 E 4 22 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -20 -4 B 8 0 -6 -10 -22 C 14 6 0 -2 6 D 20 10 2 0 20 E 4 22 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4293: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (6) B A E C D (6) B E A C D (5) E B C A D (4) D C E A B (4) D A C E B (4) D A C B E (4) C D E A B (4) B E D C A (4) D C E B A (3) D C A B E (3) C E D A B (3) A D C E B (3) E D C B A (2) E C A D B (2) E B A C D (2) E A C B D (2) E A B C D (2) D B C E A (2) C E A D B (2) C D A E B (2) C A D E B (2) B D E C A (2) B A E D C (2) A E B C D (2) A C E D B (2) A C D E B (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A B D (1) D C B E A (1) D C B A E (1) D B C A E (1) D B A C E (1) D A B C E (1) C E D B A (1) C A E D B (1) B E C A D (1) B D E A C (1) B A D E C (1) A E C B D (1) A D C B E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -10 -4 -2 B -16 0 -20 -20 -20 C 10 20 0 2 10 D 4 20 -2 0 0 E 2 20 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -10 -4 -2 B -16 0 -20 -20 -20 C 10 20 0 2 10 D 4 20 -2 0 0 E 2 20 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=22 E=18 C=15 A=14 so A is eliminated. Round 2 votes counts: D=35 B=25 E=21 C=19 so C is eliminated. Round 3 votes counts: D=45 E=30 B=25 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:221 D:211 E:206 A:200 B:162 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -10 -4 -2 B -16 0 -20 -20 -20 C 10 20 0 2 10 D 4 20 -2 0 0 E 2 20 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -10 -4 -2 B -16 0 -20 -20 -20 C 10 20 0 2 10 D 4 20 -2 0 0 E 2 20 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -10 -4 -2 B -16 0 -20 -20 -20 C 10 20 0 2 10 D 4 20 -2 0 0 E 2 20 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990433 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4294: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) B A D C E (7) E A C B D (6) E A B D C (6) D C B A E (6) D B C A E (6) A B E D C (6) E C A D B (5) E A B C D (5) C D E B A (5) B D A C E (5) C D B A E (4) A E B C D (4) A B D C E (4) E C D B A (3) C E D B A (3) C E A D B (3) C E D A B (2) C D E A B (2) C D B E A (2) E C A B D (1) E B A D C (1) D B C E A (1) C D A B E (1) B A D E C (1) A E B D C (1) A C E B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -2 6 -4 B -12 0 -2 0 -10 C 2 2 0 10 6 D -6 0 -10 0 -8 E 4 10 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 6 -4 B -12 0 -2 0 -10 C 2 2 0 10 6 D -6 0 -10 0 -8 E 4 10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=22 A=18 D=13 B=13 so D is eliminated. Round 2 votes counts: E=34 C=28 B=20 A=18 so A is eliminated. Round 3 votes counts: E=39 B=32 C=29 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:208 A:206 B:188 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 6 -4 B -12 0 -2 0 -10 C 2 2 0 10 6 D -6 0 -10 0 -8 E 4 10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 6 -4 B -12 0 -2 0 -10 C 2 2 0 10 6 D -6 0 -10 0 -8 E 4 10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 6 -4 B -12 0 -2 0 -10 C 2 2 0 10 6 D -6 0 -10 0 -8 E 4 10 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4295: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) B C D A E (6) A E C D B (6) E D A C B (5) E A D C B (4) D C E B A (4) D C B E A (4) D B C E A (4) C D B E A (4) C B A D E (4) C A E B D (4) A E D B C (4) A E C B D (4) A E B D C (4) E A C D B (3) C B D A E (3) B A C E D (3) A B E C D (3) E D A B C (2) C A B E D (2) B D C A E (2) B D A E C (2) B C D E A (2) D E B C A (1) D E A C B (1) D E A B C (1) D B E A C (1) C B D E A (1) C B A E D (1) C A E D B (1) B D C E A (1) B D A C E (1) B C A E D (1) B C A D E (1) B A D C E (1) A E D C B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 8 10 12 B -6 0 -4 -4 -2 C -8 4 0 0 2 D -10 4 0 0 -12 E -12 2 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 10 12 B -6 0 -4 -4 -2 C -8 4 0 0 2 D -10 4 0 0 -12 E -12 2 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=20 C=20 B=20 D=16 so D is eliminated. Round 2 votes counts: C=28 B=25 A=24 E=23 so E is eliminated. Round 3 votes counts: A=46 C=28 B=26 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:200 C:199 B:192 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 10 12 B -6 0 -4 -4 -2 C -8 4 0 0 2 D -10 4 0 0 -12 E -12 2 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 10 12 B -6 0 -4 -4 -2 C -8 4 0 0 2 D -10 4 0 0 -12 E -12 2 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 10 12 B -6 0 -4 -4 -2 C -8 4 0 0 2 D -10 4 0 0 -12 E -12 2 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4296: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (11) C B E A D (10) A D E B C (8) E D A C B (6) C B E D A (5) C B A E D (5) A E D C B (5) D A E B C (4) B C D E A (4) B C D A E (4) E D C A B (3) E D A B C (3) D E B A C (3) D E A B C (3) C E B D A (3) E D B C A (2) E C D B A (2) E A D C B (2) C B A D E (2) B D A C E (2) E A C D B (1) D B A E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C A B E D (1) B D C E A (1) B D C A E (1) B C E D A (1) B A C D E (1) A D E C B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 -16 4 -4 B 18 0 0 10 2 C 16 0 0 8 12 D -4 -10 -8 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.469897 C: 0.530103 D: 0.000000 E: 0.000000 Sum of squares = 0.501812347189 Cumulative probabilities = A: 0.000000 B: 0.469897 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 4 -4 B 18 0 0 10 2 C 16 0 0 8 12 D -4 -10 -8 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=25 E=19 A=16 D=11 so D is eliminated. Round 2 votes counts: C=29 B=26 E=25 A=20 so A is eliminated. Round 3 votes counts: E=43 C=30 B=27 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:215 E:197 D:187 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 4 -4 B 18 0 0 10 2 C 16 0 0 8 12 D -4 -10 -8 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 4 -4 B 18 0 0 10 2 C 16 0 0 8 12 D -4 -10 -8 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 4 -4 B 18 0 0 10 2 C 16 0 0 8 12 D -4 -10 -8 0 -4 E 4 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4297: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (14) B A E C D (13) A B E D C (12) D C E A B (9) C B A E D (5) D E A B C (4) C D B A E (4) B A C E D (4) E B A D C (3) E B A C D (3) D C A B E (3) C E B A D (3) A B D E C (3) D C A E B (2) D A E B C (2) D A B E C (2) C B A D E (2) E D A B C (1) E C D B A (1) E C B A D (1) E A B D C (1) D C E B A (1) D A B C E (1) C E D B A (1) C D A B E (1) B E A C D (1) A D B E C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 6 8 14 B 12 0 6 6 8 C -6 -6 0 8 2 D -8 -6 -8 0 0 E -14 -8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 8 14 B 12 0 6 6 8 C -6 -6 0 8 2 D -8 -6 -8 0 0 E -14 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=24 B=18 A=18 E=10 so E is eliminated. Round 2 votes counts: C=32 D=25 B=24 A=19 so A is eliminated. Round 3 votes counts: B=42 C=32 D=26 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:208 C:199 D:189 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 8 14 B 12 0 6 6 8 C -6 -6 0 8 2 D -8 -6 -8 0 0 E -14 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 8 14 B 12 0 6 6 8 C -6 -6 0 8 2 D -8 -6 -8 0 0 E -14 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 8 14 B 12 0 6 6 8 C -6 -6 0 8 2 D -8 -6 -8 0 0 E -14 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4298: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) A E C B D (7) C D B E A (6) B D A E C (6) D B C E A (4) C A B D E (4) A B E D C (4) E C A D B (3) E A B D C (3) D B E C A (3) C E D A B (3) A E C D B (3) A C E B D (3) E D A B C (2) E A D B C (2) D E C B A (2) D E B C A (2) D C B E A (2) C E D B A (2) C E A D B (2) C D B A E (2) C A E D B (2) B D E A C (2) B D C A E (2) B D A C E (2) B C D A E (2) B A D C E (2) A E B D C (2) A C B E D (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D A B (1) D E B A C (1) D B E A C (1) C D E B A (1) B C A D E (1) Total count = 100 A B C D E A 0 10 6 2 -4 B -10 0 -12 -8 -2 C -6 12 0 10 -8 D -2 8 -10 0 -2 E 4 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 6 2 -4 B -10 0 -12 -8 -2 C -6 12 0 10 -8 D -2 8 -10 0 -2 E 4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=22 E=21 B=17 D=15 so D is eliminated. Round 2 votes counts: E=26 B=25 A=25 C=24 so C is eliminated. Round 3 votes counts: B=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:208 A:207 C:204 D:197 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 2 -4 B -10 0 -12 -8 -2 C -6 12 0 10 -8 D -2 8 -10 0 -2 E 4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 -4 B -10 0 -12 -8 -2 C -6 12 0 10 -8 D -2 8 -10 0 -2 E 4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 -4 B -10 0 -12 -8 -2 C -6 12 0 10 -8 D -2 8 -10 0 -2 E 4 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4299: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) B D A C E (7) B D E C A (6) E C B D A (5) E C A D B (4) E C A B D (4) D B E A C (4) C E B A D (4) A C B D E (4) D E A B C (3) D A B E C (3) A D B C E (3) A C E D B (3) A B D C E (3) E C D A B (2) E A C D B (2) D B A C E (2) D A E B C (2) C E A D B (2) C A E B D (2) C A B E D (2) B D C E A (2) B C E D A (2) B C A E D (2) B A D C E (2) B A C D E (2) A C E B D (2) E D C A B (1) E C D B A (1) E C B A D (1) E A D C B (1) D E B C A (1) D B E C A (1) D A B C E (1) C B E D A (1) C B E A D (1) B C D A E (1) B C A D E (1) A E C D B (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -6 10 -10 B -8 0 -2 24 2 C 6 2 0 14 14 D -10 -24 -14 0 0 E 10 -2 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 10 -10 B -8 0 -2 24 2 C 6 2 0 14 14 D -10 -24 -14 0 0 E 10 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=21 C=19 A=18 D=17 so D is eliminated. Round 2 votes counts: B=32 E=25 A=24 C=19 so C is eliminated. Round 3 votes counts: E=38 B=34 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:218 B:208 A:201 E:197 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 10 -10 B -8 0 -2 24 2 C 6 2 0 14 14 D -10 -24 -14 0 0 E 10 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 10 -10 B -8 0 -2 24 2 C 6 2 0 14 14 D -10 -24 -14 0 0 E 10 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 10 -10 B -8 0 -2 24 2 C 6 2 0 14 14 D -10 -24 -14 0 0 E 10 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4300: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (14) D B E A C (10) E A C D B (8) D E A C B (8) B D C A E (8) C A E B D (7) C A B E D (5) B C D A E (5) E D A C B (4) D E B A C (4) C B A E D (4) A E C D B (4) E A D C B (3) B D C E A (3) B D E C A (2) B D E A C (2) E A C B D (1) D E A B C (1) D C B A E (1) D C A E B (1) D B E C A (1) C B A D E (1) B C A D E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 -6 0 6 B 12 0 2 8 14 C 6 -2 0 4 4 D 0 -8 -4 0 -4 E -6 -14 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 0 6 B 12 0 2 8 14 C 6 -2 0 4 4 D 0 -8 -4 0 -4 E -6 -14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979131 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=26 C=17 E=16 A=6 so A is eliminated. Round 2 votes counts: B=35 D=26 E=20 C=19 so C is eliminated. Round 3 votes counts: B=45 E=29 D=26 so D is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:206 A:194 D:192 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 0 6 B 12 0 2 8 14 C 6 -2 0 4 4 D 0 -8 -4 0 -4 E -6 -14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979131 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 0 6 B 12 0 2 8 14 C 6 -2 0 4 4 D 0 -8 -4 0 -4 E -6 -14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979131 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 0 6 B 12 0 2 8 14 C 6 -2 0 4 4 D 0 -8 -4 0 -4 E -6 -14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979131 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4301: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (18) E D C B A (9) B A C E D (8) A B C D E (8) C E D B A (7) C B E A D (7) D E A B C (6) C B A E D (6) A B D E C (5) D A E B C (4) A D B E C (4) B C A E D (3) A B D C E (3) B A E D C (2) E D B C A (1) E D B A C (1) E C D B A (1) D A B E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C D E A B (1) C A B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -14 -2 -8 B -6 0 -6 -8 -2 C 14 6 0 -8 -4 D 2 8 8 0 2 E 8 2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 -2 -8 B -6 0 -6 -8 -2 C 14 6 0 -8 -4 D 2 8 8 0 2 E 8 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 A=21 B=13 E=12 so E is eliminated. Round 2 votes counts: D=40 C=26 A=21 B=13 so B is eliminated. Round 3 votes counts: D=40 A=31 C=29 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:206 C:204 A:191 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -14 -2 -8 B -6 0 -6 -8 -2 C 14 6 0 -8 -4 D 2 8 8 0 2 E 8 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 -2 -8 B -6 0 -6 -8 -2 C 14 6 0 -8 -4 D 2 8 8 0 2 E 8 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 -2 -8 B -6 0 -6 -8 -2 C 14 6 0 -8 -4 D 2 8 8 0 2 E 8 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4302: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) D C B E A (7) A E B D C (7) A E B C D (6) E B C A D (4) E B A C D (4) D C A B E (4) A D C E B (4) E B C D A (3) E A B C D (3) D B C E A (3) D A C B E (3) C D E B A (3) C D A E B (3) B E C D A (3) B D E C A (3) A C D E B (3) E C B D A (2) E B A D C (2) D B E C A (2) C D A B E (2) A D B E C (2) A B E D C (2) D C B A E (1) D B A E C (1) C E B D A (1) C E B A D (1) C D E A B (1) C B E D A (1) B E D C A (1) B E D A C (1) B E A D C (1) B C E D A (1) B C D E A (1) A E D B C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -16 -16 -18 B 14 0 4 -4 4 C 16 -4 0 6 2 D 16 4 -6 0 12 E 18 -4 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.285714 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -16 -18 B 14 0 4 -4 4 C 16 -4 0 6 2 D 16 4 -6 0 12 E 18 -4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.285714 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 D=21 E=18 B=11 so B is eliminated. Round 2 votes counts: A=27 C=25 E=24 D=24 so E is eliminated. Round 3 votes counts: C=37 A=37 D=26 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:210 B:209 E:200 A:168 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -16 -18 B 14 0 4 -4 4 C 16 -4 0 6 2 D 16 4 -6 0 12 E 18 -4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.285714 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -16 -18 B 14 0 4 -4 4 C 16 -4 0 6 2 D 16 4 -6 0 12 E 18 -4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.285714 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -16 -18 B 14 0 4 -4 4 C 16 -4 0 6 2 D 16 4 -6 0 12 E 18 -4 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.285714 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877551 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4303: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) E C D A B (9) A B C E D (8) E D C A B (6) E C A D B (6) D E C B A (6) C E A B D (6) D B A E C (5) D E B A C (4) D B E A C (4) C E D A B (3) C A E B D (3) B A C E D (3) B A C D E (3) A C E B D (3) A C B E D (3) E C A B D (2) C E A D B (2) E A C B D (1) D E B C A (1) D E A B C (1) D B E C A (1) D B A C E (1) C B D A E (1) C B A E D (1) C B A D E (1) C A B E D (1) B D A E C (1) B A D E C (1) A E B C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 14 2 14 -4 B -14 0 -8 2 -8 C -2 8 0 16 -2 D -14 -2 -16 0 -20 E 4 8 2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 14 -4 B -14 0 -8 2 -8 C -2 8 0 16 -2 D -14 -2 -16 0 -20 E 4 8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=23 C=18 B=18 A=17 so A is eliminated. Round 2 votes counts: B=28 E=25 C=24 D=23 so D is eliminated. Round 3 votes counts: B=39 E=37 C=24 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:213 C:210 B:186 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 14 -4 B -14 0 -8 2 -8 C -2 8 0 16 -2 D -14 -2 -16 0 -20 E 4 8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 14 -4 B -14 0 -8 2 -8 C -2 8 0 16 -2 D -14 -2 -16 0 -20 E 4 8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 14 -4 B -14 0 -8 2 -8 C -2 8 0 16 -2 D -14 -2 -16 0 -20 E 4 8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4304: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) D C A E B (6) E B A D C (5) D C A B E (5) B A E D C (5) D C E A B (4) D A B C E (4) C D E A B (4) B A E C D (4) E B C A D (3) D A C B E (3) D A B E C (3) E D A B C (2) E C D B A (2) E B A C D (2) D E C A B (2) C E B D A (2) C E B A D (2) C D E B A (2) C D A E B (2) C B A D E (2) B A D E C (2) A D B E C (2) A B E D C (2) A B D E C (2) A B D C E (2) E D C B A (1) E D C A B (1) E D A C B (1) E C D A B (1) E A B D C (1) D E A B C (1) C E D B A (1) C D B A E (1) B E C A D (1) B E A C D (1) B A D C E (1) B A C E D (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 22 -8 -20 22 B -22 0 -8 -22 10 C 8 8 0 -12 12 D 20 22 12 0 24 E -22 -10 -12 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -8 -20 22 B -22 0 -8 -22 10 C 8 8 0 -12 12 D 20 22 12 0 24 E -22 -10 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 E=19 B=16 A=9 so A is eliminated. Round 2 votes counts: D=31 C=28 B=22 E=19 so E is eliminated. Round 3 votes counts: D=36 B=33 C=31 so C is eliminated. Round 4 votes counts: D=61 B=39 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:239 A:208 C:208 B:179 E:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 -8 -20 22 B -22 0 -8 -22 10 C 8 8 0 -12 12 D 20 22 12 0 24 E -22 -10 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -8 -20 22 B -22 0 -8 -22 10 C 8 8 0 -12 12 D 20 22 12 0 24 E -22 -10 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -8 -20 22 B -22 0 -8 -22 10 C 8 8 0 -12 12 D 20 22 12 0 24 E -22 -10 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4305: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (17) C B A E D (10) E D A C B (6) D E A B C (6) C A B E D (6) B C A D E (6) A C B E D (6) B C A E D (4) E A D C B (3) D E C B A (3) D E B C A (3) D B C E A (3) A B C E D (3) E A C D B (2) D E B A C (2) B C D A E (2) E D C A B (1) E C A D B (1) D E C A B (1) D B E C A (1) D B A E C (1) D A E B C (1) C E B D A (1) C E A B D (1) C B E A D (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B A C D E (1) A E D C B (1) A E C B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 16 6 -4 -8 B -16 0 -22 -4 0 C -6 22 0 -6 -4 D 4 4 6 0 4 E 8 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 -4 -8 B -16 0 -22 -4 0 C -6 22 0 -6 -4 D 4 4 6 0 4 E 8 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=19 B=17 E=13 A=13 so E is eliminated. Round 2 votes counts: D=45 C=20 A=18 B=17 so B is eliminated. Round 3 votes counts: D=48 C=33 A=19 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 A:205 E:204 C:203 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 -4 -8 B -16 0 -22 -4 0 C -6 22 0 -6 -4 D 4 4 6 0 4 E 8 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -4 -8 B -16 0 -22 -4 0 C -6 22 0 -6 -4 D 4 4 6 0 4 E 8 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -4 -8 B -16 0 -22 -4 0 C -6 22 0 -6 -4 D 4 4 6 0 4 E 8 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4306: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) B E D C A (7) B C E D A (6) E B D A C (5) C A B D E (5) B E C D A (5) E B D C A (4) D C E B A (4) C B E D A (4) C A D B E (4) A D C E B (4) E D B A C (3) D E A B C (3) D A E B C (3) A D E C B (3) E D B C A (2) E D A B C (2) C D E B A (2) C D A E B (2) C B A E D (2) A E D B C (2) A C D E B (2) D E C B A (1) D E B A C (1) D A E C B (1) D A C E B (1) C D A B E (1) C B D E A (1) C B D A E (1) C A D E B (1) C A B E D (1) B E C A D (1) B E A D C (1) B C A E D (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -10 -18 -4 B 2 0 14 -4 -10 C 10 -14 0 -12 -8 D 18 4 12 0 2 E 4 10 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -18 -4 B 2 0 14 -4 -10 C 10 -14 0 -12 -8 D 18 4 12 0 2 E 4 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=24 B=21 E=16 D=14 so D is eliminated. Round 2 votes counts: A=30 C=28 E=21 B=21 so E is eliminated. Round 3 votes counts: B=36 A=35 C=29 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:218 E:210 B:201 C:188 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -10 -18 -4 B 2 0 14 -4 -10 C 10 -14 0 -12 -8 D 18 4 12 0 2 E 4 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -18 -4 B 2 0 14 -4 -10 C 10 -14 0 -12 -8 D 18 4 12 0 2 E 4 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -18 -4 B 2 0 14 -4 -10 C 10 -14 0 -12 -8 D 18 4 12 0 2 E 4 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4307: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) A C E D B (9) B D E A C (5) D E C A B (4) D E B C A (4) B D A E C (4) A C B E D (4) E D C A B (3) D B E C A (3) D A E C B (3) B C E D A (3) B C A E D (3) A D E C B (3) A D C E B (3) E D C B A (2) E C D A B (2) D E C B A (2) D E A C B (2) C E D A B (2) C E A D B (2) C B E A D (2) C A E D B (2) C A E B D (2) B E D C A (2) B D C E A (2) B C E A D (2) B A D E C (2) A C D E B (2) E C B D A (1) D B A E C (1) C E D B A (1) C A B E D (1) B E C D A (1) B C D E A (1) B A D C E (1) B A C E D (1) B A C D E (1) A C E B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -12 -14 -10 B 6 0 -6 0 0 C 12 6 0 -12 -8 D 14 0 12 0 8 E 10 0 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.445250 C: 0.000000 D: 0.554750 E: 0.000000 Sum of squares = 0.505995077999 Cumulative probabilities = A: 0.000000 B: 0.445250 C: 0.445250 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -14 -10 B 6 0 -6 0 0 C 12 6 0 -12 -8 D 14 0 12 0 8 E 10 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=24 D=19 C=12 E=8 so E is eliminated. Round 2 votes counts: B=37 D=24 A=24 C=15 so C is eliminated. Round 3 votes counts: B=40 A=31 D=29 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:217 E:205 B:200 C:199 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -14 -10 B 6 0 -6 0 0 C 12 6 0 -12 -8 D 14 0 12 0 8 E 10 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -14 -10 B 6 0 -6 0 0 C 12 6 0 -12 -8 D 14 0 12 0 8 E 10 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -14 -10 B 6 0 -6 0 0 C 12 6 0 -12 -8 D 14 0 12 0 8 E 10 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4308: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) B A E D C (13) D E C B A (11) A C B D E (8) A B C E D (7) C D E B A (6) C D A E B (6) A B E D C (6) E B D A C (5) A B E C D (5) E D B A C (4) C A D E B (4) E D B C A (3) D C E B A (2) C A D B E (2) B E D A C (2) E D C B A (1) C A B D E (1) Total count = 100 A B C D E A 0 6 0 -8 4 B -6 0 -10 -6 -12 C 0 10 0 6 0 D 8 6 -6 0 8 E -4 12 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.319333 B: 0.000000 C: 0.680667 D: 0.000000 E: 0.000000 Sum of squares = 0.565280841217 Cumulative probabilities = A: 0.319333 B: 0.319333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -8 4 B -6 0 -10 -6 -12 C 0 10 0 6 0 D 8 6 -6 0 8 E -4 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428569 B: 0.000000 C: 0.571431 D: 0.000000 E: 0.000000 Sum of squares = 0.510204685495 Cumulative probabilities = A: 0.428569 B: 0.428569 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=26 B=15 E=13 D=13 so E is eliminated. Round 2 votes counts: C=33 A=26 D=21 B=20 so B is eliminated. Round 3 votes counts: A=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:208 D:208 A:201 E:200 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 -8 4 B -6 0 -10 -6 -12 C 0 10 0 6 0 D 8 6 -6 0 8 E -4 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428569 B: 0.000000 C: 0.571431 D: 0.000000 E: 0.000000 Sum of squares = 0.510204685495 Cumulative probabilities = A: 0.428569 B: 0.428569 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -8 4 B -6 0 -10 -6 -12 C 0 10 0 6 0 D 8 6 -6 0 8 E -4 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428569 B: 0.000000 C: 0.571431 D: 0.000000 E: 0.000000 Sum of squares = 0.510204685495 Cumulative probabilities = A: 0.428569 B: 0.428569 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -8 4 B -6 0 -10 -6 -12 C 0 10 0 6 0 D 8 6 -6 0 8 E -4 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428569 B: 0.000000 C: 0.571431 D: 0.000000 E: 0.000000 Sum of squares = 0.510204685495 Cumulative probabilities = A: 0.428569 B: 0.428569 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4309: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) C A B D E (11) E D B A C (10) D B A E C (7) D B E A C (6) E C A B D (4) D E B A C (4) C A E B D (4) C A B E D (4) B D A C E (4) D B E C A (3) D B A C E (3) E C B D A (2) E C A D B (2) D B C A E (2) C E A B D (2) B D C A E (2) A E D B C (2) A C B D E (2) E D C B A (1) E D C A B (1) E D A C B (1) E C D B A (1) E A C D B (1) D E B C A (1) C B A E D (1) B C D E A (1) B A D C E (1) A E C D B (1) A D B E C (1) A D B C E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -8 -22 -4 B 22 0 22 -20 0 C 8 -22 0 -26 -20 D 22 20 26 0 0 E 4 0 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.587702 E: 0.412298 Sum of squares = 0.515383279078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.587702 E: 1.000000 A B C D E A 0 -22 -8 -22 -4 B 22 0 22 -20 0 C 8 -22 0 -26 -20 D 22 20 26 0 0 E 4 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=26 C=22 A=9 B=8 so B is eliminated. Round 2 votes counts: E=35 D=32 C=23 A=10 so A is eliminated. Round 3 votes counts: E=38 D=36 C=26 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:234 B:212 E:212 A:172 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -8 -22 -4 B 22 0 22 -20 0 C 8 -22 0 -26 -20 D 22 20 26 0 0 E 4 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -8 -22 -4 B 22 0 22 -20 0 C 8 -22 0 -26 -20 D 22 20 26 0 0 E 4 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -8 -22 -4 B 22 0 22 -20 0 C 8 -22 0 -26 -20 D 22 20 26 0 0 E 4 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4310: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (14) B A C E D (9) A B E D C (6) E A B D C (5) B A E C D (5) A B E C D (5) E D C A B (4) E D A B C (4) C D B A E (4) D E B A C (3) D E A B C (3) D C E A B (3) C D E B A (3) C D E A B (3) C D B E A (3) C B A D E (3) B C A D E (3) E C D A B (2) D C E B A (2) D B A E C (2) C B A E D (2) A E B D C (2) E D A C B (1) D B C A E (1) C A B E D (1) B D C A E (1) B C A E D (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 0 -6 0 B -8 0 10 -4 2 C 0 -10 0 -8 -16 D 6 4 8 0 2 E 0 -2 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -6 0 B -8 0 10 -4 2 C 0 -10 0 -8 -16 D 6 4 8 0 2 E 0 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=23 C=19 E=16 A=14 so A is eliminated. Round 2 votes counts: B=35 D=28 C=19 E=18 so E is eliminated. Round 3 votes counts: B=42 D=37 C=21 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:206 A:201 B:200 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -6 0 B -8 0 10 -4 2 C 0 -10 0 -8 -16 D 6 4 8 0 2 E 0 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -6 0 B -8 0 10 -4 2 C 0 -10 0 -8 -16 D 6 4 8 0 2 E 0 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -6 0 B -8 0 10 -4 2 C 0 -10 0 -8 -16 D 6 4 8 0 2 E 0 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4311: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (6) D B C A E (6) D B A C E (6) A B D C E (6) E D C B A (5) E C A B D (5) E A C B D (5) C D B A E (4) E C D B A (3) E C D A B (3) D C B E A (3) C E A B D (3) C B D A E (3) C B A D E (3) B A C D E (3) E D A B C (2) E A B D C (2) D C B A E (2) D B A E C (2) C E D B A (2) C A B E D (2) A C B E D (2) A B E C D (2) A B C D E (2) E D C A B (1) E C B D A (1) E A D B C (1) D E C B A (1) D E A B C (1) D B E A C (1) D A B E C (1) C E B D A (1) C D E B A (1) C D B E A (1) C B A E D (1) B D A C E (1) B C A D E (1) B A D C E (1) A E B D C (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -4 -2 4 B 4 0 -6 8 8 C 4 6 0 14 12 D 2 -8 -14 0 -2 E -4 -8 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 4 B 4 0 -6 8 8 C 4 6 0 14 12 D 2 -8 -14 0 -2 E -4 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=23 C=21 A=16 B=6 so B is eliminated. Round 2 votes counts: E=34 D=24 C=22 A=20 so A is eliminated. Round 3 votes counts: E=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:207 A:197 D:189 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 4 B 4 0 -6 8 8 C 4 6 0 14 12 D 2 -8 -14 0 -2 E -4 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 4 B 4 0 -6 8 8 C 4 6 0 14 12 D 2 -8 -14 0 -2 E -4 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 4 B 4 0 -6 8 8 C 4 6 0 14 12 D 2 -8 -14 0 -2 E -4 -8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4312: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (12) E D C B A (8) D E A B C (7) C B A E D (7) B A C D E (7) E D C A B (5) E D A C B (5) A D E B C (5) D E B A C (4) A B D C E (4) D E B C A (3) C E B D A (3) A C B E D (3) E C D A B (2) C E D B A (2) C E B A D (2) A E D B C (2) A D B E C (2) A B D E C (2) E C D B A (1) E C A D B (1) E A D C B (1) E A C D B (1) D E C B A (1) D B A E C (1) C E A D B (1) C B E A D (1) C B D E A (1) C A B E D (1) B C D E A (1) B C D A E (1) B C A D E (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 16 10 0 B -12 0 6 -6 -10 C -16 -6 0 0 -4 D -10 6 0 0 4 E 0 10 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545203 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.454797 Sum of squares = 0.504086637215 Cumulative probabilities = A: 0.545203 B: 0.545203 C: 0.545203 D: 0.545203 E: 1.000000 A B C D E A 0 12 16 10 0 B -12 0 6 -6 -10 C -16 -6 0 0 -4 D -10 6 0 0 4 E 0 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=24 C=18 D=16 B=10 so B is eliminated. Round 2 votes counts: A=39 E=24 C=21 D=16 so D is eliminated. Round 3 votes counts: A=40 E=39 C=21 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:205 D:200 B:189 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 10 0 B -12 0 6 -6 -10 C -16 -6 0 0 -4 D -10 6 0 0 4 E 0 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 10 0 B -12 0 6 -6 -10 C -16 -6 0 0 -4 D -10 6 0 0 4 E 0 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 10 0 B -12 0 6 -6 -10 C -16 -6 0 0 -4 D -10 6 0 0 4 E 0 10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4313: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (12) E C B D A (11) E C D A B (5) E B C D A (5) B A E D C (5) A D C B E (5) E C D B A (4) E C A D B (4) B A D C E (4) A D C E B (4) E C B A D (3) B D A C E (3) B A D E C (3) E B C A D (2) D C B A E (2) D B C A E (2) D B A C E (2) C E D A B (2) C D E A B (2) C D A E B (2) A B D E C (2) E C A B D (1) E B A C D (1) E A B C D (1) D B C E A (1) C E D B A (1) C E A D B (1) C D B E A (1) B E C D A (1) B E A D C (1) B E A C D (1) B D C A E (1) B C D E A (1) A E B C D (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 8 2 B 10 0 4 -2 0 C 4 -4 0 0 -6 D -8 2 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.358801 C: 0.000000 D: 0.000000 E: 0.641199 Sum of squares = 0.53987419859 Cumulative probabilities = A: 0.000000 B: 0.358801 C: 0.358801 D: 0.358801 E: 1.000000 A B C D E A 0 -10 -4 8 2 B 10 0 4 -2 0 C 4 -4 0 0 -6 D -8 2 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499678 C: 0.000000 D: 0.000000 E: 0.500322 Sum of squares = 0.500000207586 Cumulative probabilities = A: 0.000000 B: 0.499678 C: 0.499678 D: 0.499678 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=27 B=20 C=9 D=7 so D is eliminated. Round 2 votes counts: E=37 A=27 B=25 C=11 so C is eliminated. Round 3 votes counts: E=43 A=29 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:206 E:203 A:198 C:197 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 8 2 B 10 0 4 -2 0 C 4 -4 0 0 -6 D -8 2 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499678 C: 0.000000 D: 0.000000 E: 0.500322 Sum of squares = 0.500000207586 Cumulative probabilities = A: 0.000000 B: 0.499678 C: 0.499678 D: 0.499678 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 8 2 B 10 0 4 -2 0 C 4 -4 0 0 -6 D -8 2 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499678 C: 0.000000 D: 0.000000 E: 0.500322 Sum of squares = 0.500000207586 Cumulative probabilities = A: 0.000000 B: 0.499678 C: 0.499678 D: 0.499678 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 8 2 B 10 0 4 -2 0 C 4 -4 0 0 -6 D -8 2 0 0 -2 E -2 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499678 C: 0.000000 D: 0.000000 E: 0.500322 Sum of squares = 0.500000207586 Cumulative probabilities = A: 0.000000 B: 0.499678 C: 0.499678 D: 0.499678 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4314: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) E B A D C (9) A D B E C (9) A B E D C (9) C E B D A (5) A B D E C (5) E C B D A (4) C D E B A (4) C D E A B (4) E C B A D (3) E B C A D (3) E B A C D (3) E A B D C (3) C D A B E (3) A D B C E (3) D A C B E (2) C D B A E (2) E C A B D (1) E B D C A (1) E B C D A (1) E A B C D (1) D C B E A (1) D C A B E (1) D B A C E (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B A D (1) C E A B D (1) C D B E A (1) C A D B E (1) B E A D C (1) B D E A C (1) B A E D C (1) B A D E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 4 12 -18 B 6 0 8 8 -10 C -4 -8 0 0 -14 D -12 -8 0 0 -16 E 18 10 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 12 -18 B 6 0 8 8 -10 C -4 -8 0 0 -14 D -12 -8 0 0 -16 E 18 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=29 A=27 D=7 B=4 so B is eliminated. Round 2 votes counts: C=33 E=30 A=29 D=8 so D is eliminated. Round 3 votes counts: C=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:229 B:206 A:196 C:187 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 12 -18 B 6 0 8 8 -10 C -4 -8 0 0 -14 D -12 -8 0 0 -16 E 18 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 12 -18 B 6 0 8 8 -10 C -4 -8 0 0 -14 D -12 -8 0 0 -16 E 18 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 12 -18 B 6 0 8 8 -10 C -4 -8 0 0 -14 D -12 -8 0 0 -16 E 18 10 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4315: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) A C E D B (7) E A C D B (6) A C D B E (6) E D B A C (5) D B E A C (5) C A D B E (5) C B A E D (4) C A B D E (4) A D C B E (4) C E A B D (3) C A E B D (3) E C B A D (2) E B D C A (2) E B D A C (2) D B C A E (2) D B A C E (2) D A E C B (2) D A C B E (2) D A B C E (2) C B A D E (2) C A B E D (2) B D E C A (2) E D A C B (1) E D A B C (1) E C A B D (1) E B C D A (1) E A D C B (1) E A D B C (1) D E B A C (1) D E A B C (1) D B A E C (1) C A E D B (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E A D (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 8 2 12 20 B -8 0 -14 -14 16 C -2 14 0 6 26 D -12 14 -6 0 6 E -20 -16 -26 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 12 20 B -8 0 -14 -14 16 C -2 14 0 6 26 D -12 14 -6 0 6 E -20 -16 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=23 D=18 A=18 B=17 so B is eliminated. Round 2 votes counts: C=29 D=28 E=25 A=18 so A is eliminated. Round 3 votes counts: C=42 D=33 E=25 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:221 D:201 B:190 E:166 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 12 20 B -8 0 -14 -14 16 C -2 14 0 6 26 D -12 14 -6 0 6 E -20 -16 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 12 20 B -8 0 -14 -14 16 C -2 14 0 6 26 D -12 14 -6 0 6 E -20 -16 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 12 20 B -8 0 -14 -14 16 C -2 14 0 6 26 D -12 14 -6 0 6 E -20 -16 -26 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4316: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) B C A D E (8) B C A E D (7) D E A C B (5) C D E B A (5) B C E D A (5) B A C E D (5) A D E B C (5) A B E D C (5) D E C A B (4) B A E C D (4) A E D B C (4) A D E C B (4) E D A C B (3) C E D A B (3) C B D E A (3) B C D E A (3) A E D C B (3) E A D C B (2) D A E C B (2) C E D B A (2) B C D A E (2) E D C A B (1) D C E A B (1) C D B E A (1) B C E A D (1) B A D E C (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 -14 -10 2 4 B 14 0 4 10 10 C 10 -4 0 18 12 D -2 -10 -18 0 -10 E -4 -10 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 2 4 B 14 0 4 10 10 C 10 -4 0 18 12 D -2 -10 -18 0 -10 E -4 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=23 A=22 D=12 E=6 so E is eliminated. Round 2 votes counts: B=37 A=24 C=23 D=16 so D is eliminated. Round 3 votes counts: B=37 A=34 C=29 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:218 E:192 A:191 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -10 2 4 B 14 0 4 10 10 C 10 -4 0 18 12 D -2 -10 -18 0 -10 E -4 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 2 4 B 14 0 4 10 10 C 10 -4 0 18 12 D -2 -10 -18 0 -10 E -4 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 2 4 B 14 0 4 10 10 C 10 -4 0 18 12 D -2 -10 -18 0 -10 E -4 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4317: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) A C D E B (11) D E A C B (8) C A D B E (8) B E D A C (7) E D A C B (6) B E D C A (6) E D B A C (5) E B D A C (4) D E B A C (4) B C A E D (4) D A C E B (3) B E C D A (3) E B A C D (2) C A B E D (2) B E A C D (2) A C E D B (2) E D A B C (1) E A D C B (1) E A B C D (1) D E A B C (1) D C A E B (1) D B C E A (1) D A E C B (1) C A B D E (1) B E C A D (1) B D C E A (1) B C A D E (1) A E C B D (1) Total count = 100 A B C D E A 0 18 20 -4 -8 B -18 0 -12 -28 -26 C -20 12 0 0 -8 D 4 28 0 0 4 E 8 26 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.098304 D: 0.901696 E: 0.000000 Sum of squares = 0.822719427491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.098304 D: 1.000000 E: 1.000000 A B C D E A 0 18 20 -4 -8 B -18 0 -12 -28 -26 C -20 12 0 0 -8 D 4 28 0 0 4 E 8 26 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 0.000000 Sum of squares = 0.722222250303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=22 E=20 D=19 A=14 so A is eliminated. Round 2 votes counts: C=35 B=25 E=21 D=19 so D is eliminated. Round 3 votes counts: C=39 E=35 B=26 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:218 A:213 C:192 B:158 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 20 -4 -8 B -18 0 -12 -28 -26 C -20 12 0 0 -8 D 4 28 0 0 4 E 8 26 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 0.000000 Sum of squares = 0.722222250303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 -4 -8 B -18 0 -12 -28 -26 C -20 12 0 0 -8 D 4 28 0 0 4 E 8 26 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 0.000000 Sum of squares = 0.722222250303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 -4 -8 B -18 0 -12 -28 -26 C -20 12 0 0 -8 D 4 28 0 0 4 E 8 26 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.833333 E: 0.000000 Sum of squares = 0.722222250303 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4318: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) C E A B D (6) E C B A D (5) E B D A C (5) E B A D C (5) D B A E C (5) C E D A B (5) C A B E D (5) B A E D C (4) E B C A D (3) D E B A C (3) D C E A B (3) B A D E C (3) A B D C E (3) E C D A B (2) E C B D A (2) E B A C D (2) D E C B A (2) D E C A B (2) D C A B E (2) C D E A B (2) C D A B E (2) C B A E D (2) B E A D C (2) B A E C D (2) A C B D E (2) A B C D E (2) E D C B A (1) E D B A C (1) E C D B A (1) D C A E B (1) D A B E C (1) C E B A D (1) C D A E B (1) C A B D E (1) B E A C D (1) B A C E D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 4 -8 B 2 0 4 16 -6 C -2 -4 0 -2 -4 D -4 -16 2 0 -14 E 8 6 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 4 -8 B 2 0 4 16 -6 C -2 -4 0 -2 -4 D -4 -16 2 0 -14 E 8 6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 C=25 B=13 A=9 so A is eliminated. Round 2 votes counts: E=27 D=27 C=27 B=19 so B is eliminated. Round 3 votes counts: E=36 D=33 C=31 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:208 A:198 C:194 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 4 -8 B 2 0 4 16 -6 C -2 -4 0 -2 -4 D -4 -16 2 0 -14 E 8 6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -8 B 2 0 4 16 -6 C -2 -4 0 -2 -4 D -4 -16 2 0 -14 E 8 6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -8 B 2 0 4 16 -6 C -2 -4 0 -2 -4 D -4 -16 2 0 -14 E 8 6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4319: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) B E D A C (8) B E D C A (7) B E A D C (6) A C E D B (6) E B A C D (4) D C A B E (4) D A C B E (4) B D E C A (4) E B D A C (3) D B C E A (3) C D E B A (3) C D A B E (3) A E C B D (3) E B C D A (2) E B C A D (2) D B C A E (2) C E A D B (2) C A D E B (2) B D A E C (2) A E B C D (2) A D C B E (2) A C D E B (2) A C D B E (2) E C B D A (1) E C A B D (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B A C E (1) C D E A B (1) C D A E B (1) C A E D B (1) B D E A C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 18 -4 -20 B 20 0 16 18 4 C -18 -16 0 -18 -16 D 4 -18 18 0 -20 E 20 -4 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 18 -4 -20 B 20 0 16 18 4 C -18 -16 0 -18 -16 D 4 -18 18 0 -20 E 20 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 A=19 D=16 C=13 so C is eliminated. Round 2 votes counts: B=28 E=26 D=24 A=22 so A is eliminated. Round 3 votes counts: E=39 D=32 B=29 so B is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:229 E:226 D:192 A:187 C:166 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 18 -4 -20 B 20 0 16 18 4 C -18 -16 0 -18 -16 D 4 -18 18 0 -20 E 20 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 18 -4 -20 B 20 0 16 18 4 C -18 -16 0 -18 -16 D 4 -18 18 0 -20 E 20 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 18 -4 -20 B 20 0 16 18 4 C -18 -16 0 -18 -16 D 4 -18 18 0 -20 E 20 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998151 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4320: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) A C B E D (7) D E B C A (5) C E A D B (5) D B E C A (4) D A B E C (4) D A B C E (4) B E C D A (4) B E C A D (4) A C E D B (4) E C B A D (3) D E C A B (3) D A C E B (3) B A C E D (3) A C D E B (3) A B C E D (3) E C B D A (2) D E C B A (2) D B A E C (2) C E B A D (2) C B A E D (2) B E D C A (2) B D E C A (2) A D C E B (2) A D B C E (2) A B D C E (2) A B C D E (2) E D C B A (1) E C D A B (1) E B C D A (1) D E A C B (1) C E A B D (1) C B E A D (1) C A E D B (1) C A E B D (1) B D E A C (1) B D A E C (1) Total count = 100 A B C D E A 0 16 6 14 10 B -16 0 -8 6 0 C -6 8 0 18 14 D -14 -6 -18 0 -14 E -10 0 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 14 10 B -16 0 -8 6 0 C -6 8 0 18 14 D -14 -6 -18 0 -14 E -10 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=28 B=17 C=13 E=8 so E is eliminated. Round 2 votes counts: A=34 D=29 C=19 B=18 so B is eliminated. Round 3 votes counts: A=37 D=35 C=28 so C is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:217 E:195 B:191 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 14 10 B -16 0 -8 6 0 C -6 8 0 18 14 D -14 -6 -18 0 -14 E -10 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 14 10 B -16 0 -8 6 0 C -6 8 0 18 14 D -14 -6 -18 0 -14 E -10 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 14 10 B -16 0 -8 6 0 C -6 8 0 18 14 D -14 -6 -18 0 -14 E -10 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4321: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) D C E B A (7) D C B A E (6) B A E D C (6) E D C B A (5) B D A C E (5) A B E D C (5) E A B C D (4) C D E B A (4) C D B A E (4) C A B D E (4) C D A B E (3) B A D E C (3) A B E C D (3) E C A D B (2) E B D A C (2) E B A D C (2) E A C B D (2) D E C B A (2) D B A C E (2) C E A D B (2) C D E A B (2) B A D C E (2) A E B C D (2) A B C D E (2) E D C A B (1) E D B A C (1) E C D A B (1) D E B C A (1) D B C A E (1) C A E B D (1) C A B E D (1) B E A D C (1) B D E A C (1) B C D A E (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 4 2 6 B 12 0 6 12 2 C -4 -6 0 -22 -4 D -2 -12 22 0 0 E -6 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 2 6 B 12 0 6 12 2 C -4 -6 0 -22 -4 D -2 -12 22 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=21 D=19 B=19 A=14 so A is eliminated. Round 2 votes counts: E=30 B=30 C=21 D=19 so D is eliminated. Round 3 votes counts: C=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:204 A:200 E:198 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 2 6 B 12 0 6 12 2 C -4 -6 0 -22 -4 D -2 -12 22 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 2 6 B 12 0 6 12 2 C -4 -6 0 -22 -4 D -2 -12 22 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 2 6 B 12 0 6 12 2 C -4 -6 0 -22 -4 D -2 -12 22 0 0 E -6 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4322: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) C B D A E (8) E A D B C (7) A E B C D (7) A B C E D (7) D E C B A (6) B C A D E (6) E D A B C (5) D C B E A (4) A E B D C (4) A B E C D (4) C D B E A (3) B A C D E (3) E A B D C (2) D C E B A (2) C B D E A (2) C A B D E (2) B C D A E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B A C (1) E B D C A (1) E B D A C (1) E A D C B (1) D E B C A (1) D C A E B (1) D B C E A (1) C D B A E (1) B E A C D (1) B C D E A (1) A E D C B (1) A E D B C (1) A C B E D (1) Total count = 100 A B C D E A 0 10 14 -2 0 B -10 0 14 8 -4 C -14 -14 0 0 -8 D 2 -8 0 0 -10 E 0 4 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.416262 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.583738 Sum of squares = 0.514024157389 Cumulative probabilities = A: 0.416262 B: 0.416262 C: 0.416262 D: 0.416262 E: 1.000000 A B C D E A 0 10 14 -2 0 B -10 0 14 8 -4 C -14 -14 0 0 -8 D 2 -8 0 0 -10 E 0 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 C=16 D=15 B=13 so B is eliminated. Round 2 votes counts: E=30 A=30 C=25 D=15 so D is eliminated. Round 3 votes counts: E=37 C=33 A=30 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:211 E:211 B:204 D:192 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 -2 0 B -10 0 14 8 -4 C -14 -14 0 0 -8 D 2 -8 0 0 -10 E 0 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -2 0 B -10 0 14 8 -4 C -14 -14 0 0 -8 D 2 -8 0 0 -10 E 0 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -2 0 B -10 0 14 8 -4 C -14 -14 0 0 -8 D 2 -8 0 0 -10 E 0 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4323: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (6) B C D A E (6) C B E A D (5) A D E B C (5) E D A C B (4) D A E C B (4) B C E A D (4) B C A E D (4) A D B E C (4) A B D E C (4) E D C A B (3) E C D A B (3) D E A C B (3) D A B E C (3) C B E D A (3) B A D C E (3) E C A B D (2) D C E B A (2) D A E B C (2) C E D B A (2) C D B E A (2) C B D E A (2) B D A C E (2) E C B A D (1) E C A D B (1) E B C A D (1) E A D C B (1) E A D B C (1) E A C B D (1) D E C A B (1) D C B E A (1) D C A E B (1) D C A B E (1) D B A C E (1) D A C B E (1) C E B A D (1) B D C A E (1) B C E D A (1) B C A D E (1) B A E C D (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 -10 -2 B 4 0 -2 6 6 C 10 2 0 -2 6 D 10 -6 2 0 4 E 2 -6 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000054 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -10 -2 B 4 0 -2 6 6 C 10 2 0 -2 6 D 10 -6 2 0 4 E 2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000078 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=21 D=20 E=18 A=16 so A is eliminated. Round 2 votes counts: B=30 D=29 C=21 E=20 so E is eliminated. Round 3 votes counts: D=39 B=32 C=29 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:207 D:205 E:193 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -10 -2 B 4 0 -2 6 6 C 10 2 0 -2 6 D 10 -6 2 0 4 E 2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000078 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -10 -2 B 4 0 -2 6 6 C 10 2 0 -2 6 D 10 -6 2 0 4 E 2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000078 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -10 -2 B 4 0 -2 6 6 C 10 2 0 -2 6 D 10 -6 2 0 4 E 2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000078 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4324: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) C B E D A (7) A D E B C (7) C D A B E (6) D A C B E (5) B C E A D (5) E B C D A (4) D A C E B (4) C B D A E (4) E A D B C (3) D A E C B (3) C D B A E (3) B E C A D (3) A D B E C (3) A D B C E (3) E D A C B (2) E D A B C (2) E C B D A (2) E A B D C (2) D C A B E (2) D A E B C (2) C B E A D (2) C B A D E (2) B E A C D (2) B C A E D (2) A D C B E (2) A B C D E (2) E D C A B (1) E B A D C (1) D E C A B (1) D E A B C (1) D C A E B (1) C E D B A (1) C A D B E (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -8 -2 8 B -8 0 2 -6 12 C 8 -2 0 8 6 D 2 6 -8 0 6 E -8 -12 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407401 Cumulative probabilities = A: 0.111111 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -2 8 B -8 0 2 -6 12 C 8 -2 0 8 6 D 2 6 -8 0 6 E -8 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407352 Cumulative probabilities = A: 0.111111 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 D=19 A=18 B=13 so B is eliminated. Round 2 votes counts: C=33 E=29 D=19 A=19 so D is eliminated. Round 3 votes counts: C=36 A=33 E=31 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:203 D:203 B:200 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -2 8 B -8 0 2 -6 12 C 8 -2 0 8 6 D 2 6 -8 0 6 E -8 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407352 Cumulative probabilities = A: 0.111111 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -2 8 B -8 0 2 -6 12 C 8 -2 0 8 6 D 2 6 -8 0 6 E -8 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407352 Cumulative probabilities = A: 0.111111 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -2 8 B -8 0 2 -6 12 C 8 -2 0 8 6 D 2 6 -8 0 6 E -8 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.444444 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407352 Cumulative probabilities = A: 0.111111 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4325: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) A D B C E (7) D C E B A (5) B D E A C (5) A B D E C (5) E C B D A (4) D B E C A (4) A C D E B (4) E B C A D (3) D C A E B (3) D A C B E (3) C D A E B (3) B E C D A (3) B E A C D (3) B A D E C (3) E B D C A (2) E B C D A (2) D E C B A (2) C E D A B (2) C E B D A (2) B E D C A (2) B D E C A (2) B D A E C (2) A C E B D (2) A B E C D (2) A B D C E (2) E C D B A (1) E A C B D (1) D E B C A (1) D B E A C (1) D B C E A (1) D A C E B (1) C E B A D (1) C D E B A (1) C D E A B (1) C A E D B (1) B E D A C (1) B E A D C (1) B A E D C (1) B A E C D (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -22 -6 -22 -16 B 22 0 12 2 2 C 6 -12 0 -10 -6 D 22 -2 10 0 14 E 16 -2 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -6 -22 -16 B 22 0 12 2 2 C 6 -12 0 -10 -6 D 22 -2 10 0 14 E 16 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 D=21 C=18 E=13 so E is eliminated. Round 2 votes counts: B=31 A=25 C=23 D=21 so D is eliminated. Round 3 votes counts: B=38 C=33 A=29 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:219 E:203 C:189 A:167 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -6 -22 -16 B 22 0 12 2 2 C 6 -12 0 -10 -6 D 22 -2 10 0 14 E 16 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -6 -22 -16 B 22 0 12 2 2 C 6 -12 0 -10 -6 D 22 -2 10 0 14 E 16 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -6 -22 -16 B 22 0 12 2 2 C 6 -12 0 -10 -6 D 22 -2 10 0 14 E 16 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4326: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A D C B E (11) E B C A D (6) C B E A D (5) C B D A E (5) B E C D A (5) B C E D A (5) A D E C B (4) E D A B C (3) E B D C A (3) D A E C B (3) C A D B E (3) C A B D E (3) A D C E B (3) E B D A C (2) D E A B C (2) D A E B C (2) C B E D A (2) C B D E A (2) C B A D E (2) B C E A D (2) B C D E A (2) A E D B C (2) A C D B E (2) E D B C A (1) E D B A C (1) E B A D C (1) E B A C D (1) E A B C D (1) D E B C A (1) D A C B E (1) C D A B E (1) C A B E D (1) B E D C A (1) Total count = 100 A B C D E A 0 -16 -22 -6 -14 B 16 0 4 20 6 C 22 -4 0 18 0 D 6 -20 -18 0 -6 E 14 -6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -22 -6 -14 B 16 0 4 20 6 C 22 -4 0 18 0 D 6 -20 -18 0 -6 E 14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=24 A=22 B=15 D=9 so D is eliminated. Round 2 votes counts: E=33 A=28 C=24 B=15 so B is eliminated. Round 3 votes counts: E=39 C=33 A=28 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:223 C:218 E:207 D:181 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -22 -6 -14 B 16 0 4 20 6 C 22 -4 0 18 0 D 6 -20 -18 0 -6 E 14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -22 -6 -14 B 16 0 4 20 6 C 22 -4 0 18 0 D 6 -20 -18 0 -6 E 14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -22 -6 -14 B 16 0 4 20 6 C 22 -4 0 18 0 D 6 -20 -18 0 -6 E 14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4327: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) C A B E D (8) A B E C D (7) E C D A B (6) B A D E C (6) D E C B A (5) C D A B E (5) E D B A C (4) E B A D C (4) C E D A B (4) C E A B D (4) D B A E C (3) D B A C E (3) C D E A B (3) B A E D C (3) A B C D E (3) E D C B A (2) E C A B D (2) D C E B A (2) C A E B D (2) C A B D E (2) B A D C E (2) A B C E D (2) E B D A C (1) E B A C D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B C A E (1) C E A D B (1) C A E D B (1) B D A E C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 0 0 2 B -4 0 -2 0 -2 C 0 2 0 6 -8 D 0 0 -6 0 -8 E -2 2 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.850304 B: 0.000000 C: 0.149696 D: 0.000000 E: 0.000000 Sum of squares = 0.745425397597 Cumulative probabilities = A: 0.850304 B: 0.850304 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 0 2 B -4 0 -2 0 -2 C 0 2 0 6 -8 D 0 0 -6 0 -8 E -2 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000001789 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=24 E=21 A=13 B=12 so B is eliminated. Round 2 votes counts: C=30 D=25 A=24 E=21 so E is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:208 A:203 C:200 B:196 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 0 2 B -4 0 -2 0 -2 C 0 2 0 6 -8 D 0 0 -6 0 -8 E -2 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000001789 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 0 2 B -4 0 -2 0 -2 C 0 2 0 6 -8 D 0 0 -6 0 -8 E -2 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000001789 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 0 2 B -4 0 -2 0 -2 C 0 2 0 6 -8 D 0 0 -6 0 -8 E -2 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000001789 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4328: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (16) D E C A B (11) D E C B A (7) E D C A B (6) D E A C B (5) E C A D B (4) B C A E D (4) A B C E D (4) D B E C A (3) B D C E A (3) B A D C E (3) B A C D E (3) D E B A C (2) C E A D B (2) C A E B D (2) B D E C A (2) A C E D B (2) A C E B D (2) A C B E D (2) E D A C B (1) E C D A B (1) D E B C A (1) D E A B C (1) D B E A C (1) D A B E C (1) C E D A B (1) C B A E D (1) C A B E D (1) B D E A C (1) B D C A E (1) B D A C E (1) B C A D E (1) B A D E C (1) A E C D B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -2 2 -4 B 2 0 2 -2 2 C 2 -2 0 -6 0 D -2 2 6 0 0 E 4 -2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.565899 E: 0.434101 Sum of squares = 0.508685291835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.565899 E: 1.000000 A B C D E A 0 -2 -2 2 -4 B 2 0 2 -2 2 C 2 -2 0 -6 0 D -2 2 6 0 0 E 4 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 0.499860 Sum of squares = 0.500000039084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=32 A=13 E=12 C=7 so C is eliminated. Round 2 votes counts: B=37 D=32 A=16 E=15 so E is eliminated. Round 3 votes counts: D=41 B=37 A=22 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:203 B:202 E:201 A:197 C:197 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 2 -4 B 2 0 2 -2 2 C 2 -2 0 -6 0 D -2 2 6 0 0 E 4 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 0.499860 Sum of squares = 0.500000039084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 -4 B 2 0 2 -2 2 C 2 -2 0 -6 0 D -2 2 6 0 0 E 4 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 0.499860 Sum of squares = 0.500000039084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 -4 B 2 0 2 -2 2 C 2 -2 0 -6 0 D -2 2 6 0 0 E 4 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 0.499860 Sum of squares = 0.500000039084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500140 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4329: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) E D A C B (8) B C A D E (8) B C D A E (7) C D A B E (6) B C E D A (6) C D A E B (5) C B D A E (5) B E A D C (5) E A D B C (3) B E C A D (3) E D C B A (2) E B C D A (2) E B A D C (2) C A D B E (2) B E C D A (2) B C D E A (2) B A E D C (2) B A C D E (2) A E D C B (2) A D E C B (2) E D B A C (1) D E C A B (1) D C A E B (1) C E D A B (1) C D E A B (1) C A D E B (1) C A B D E (1) B E A C D (1) B C E A D (1) B C A E D (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -16 0 -6 B 6 0 0 4 12 C 16 0 0 14 0 D 0 -4 -14 0 -8 E 6 -12 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.572491 C: 0.427509 D: 0.000000 E: 0.000000 Sum of squares = 0.510509951471 Cumulative probabilities = A: 0.000000 B: 0.572491 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 0 -6 B 6 0 0 4 12 C 16 0 0 14 0 D 0 -4 -14 0 -8 E 6 -12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=30 C=22 A=5 D=2 so D is eliminated. Round 2 votes counts: B=41 E=31 C=23 A=5 so A is eliminated. Round 3 votes counts: B=42 E=35 C=23 so C is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:211 E:201 D:187 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -16 0 -6 B 6 0 0 4 12 C 16 0 0 14 0 D 0 -4 -14 0 -8 E 6 -12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 0 -6 B 6 0 0 4 12 C 16 0 0 14 0 D 0 -4 -14 0 -8 E 6 -12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 0 -6 B 6 0 0 4 12 C 16 0 0 14 0 D 0 -4 -14 0 -8 E 6 -12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4330: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) D C B A E (7) C A E D B (7) C A E B D (7) D B E A C (6) A E C B D (6) E B A C D (4) D B E C A (4) B E D A C (4) E A C B D (3) C E A B D (3) C D B A E (3) C D A E B (3) C D A B E (3) E B A D C (2) E A B D C (2) E A B C D (2) D C B E A (2) D C A B E (2) D B C E A (2) B E A D C (2) B D E C A (2) B C D E A (2) A C E D B (2) D A E C B (1) D A C E B (1) D A C B E (1) C B E A D (1) B E C A D (1) B C E A D (1) A E D C B (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 0 -6 -6 B 6 0 -12 6 6 C 0 12 0 0 0 D 6 -6 0 0 2 E 6 -6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.463648 D: 0.536352 E: 0.000000 Sum of squares = 0.502642970052 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.463648 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 -6 B 6 0 -12 6 6 C 0 12 0 0 0 D 6 -6 0 0 2 E 6 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 B=22 E=13 A=12 so A is eliminated. Round 2 votes counts: C=31 D=27 B=22 E=20 so E is eliminated. Round 3 votes counts: C=40 B=32 D=28 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:203 D:201 E:199 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -6 -6 B 6 0 -12 6 6 C 0 12 0 0 0 D 6 -6 0 0 2 E 6 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 -6 B 6 0 -12 6 6 C 0 12 0 0 0 D 6 -6 0 0 2 E 6 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 -6 B 6 0 -12 6 6 C 0 12 0 0 0 D 6 -6 0 0 2 E 6 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4331: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) D C B E A (8) C D E A B (7) B D C E A (6) B A E D C (6) B D C A E (5) A B E C D (5) E C A D B (4) E A C D B (4) C E D A B (4) C E A D B (4) A E C B D (4) E A C B D (3) D C B A E (3) D B C A E (3) C D A B E (3) B A D E C (3) B A D C E (3) A E C D B (3) D C E B A (2) C D A E B (2) B D A C E (2) E C D A B (1) D C E A B (1) C A D E B (1) B E A D C (1) B A E C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 14 -8 6 10 B -14 0 -12 -2 -2 C 8 12 0 14 12 D -6 2 -14 0 -2 E -10 2 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 6 10 B -14 0 -12 -2 -2 C 8 12 0 14 12 D -6 2 -14 0 -2 E -10 2 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=23 C=21 D=17 E=12 so E is eliminated. Round 2 votes counts: A=30 B=27 C=26 D=17 so D is eliminated. Round 3 votes counts: C=40 B=30 A=30 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:211 E:191 D:190 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -8 6 10 B -14 0 -12 -2 -2 C 8 12 0 14 12 D -6 2 -14 0 -2 E -10 2 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 6 10 B -14 0 -12 -2 -2 C 8 12 0 14 12 D -6 2 -14 0 -2 E -10 2 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 6 10 B -14 0 -12 -2 -2 C 8 12 0 14 12 D -6 2 -14 0 -2 E -10 2 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4332: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) E A B D C (8) C D B A E (6) B D C A E (6) B A E D C (6) A E C D B (6) C D B E A (5) A E C B D (5) A E B C D (5) E A C B D (4) C D E A B (4) D C B E A (3) B D E A C (3) A E B D C (3) D B C E A (2) D B C A E (2) C E A D B (2) B D E C A (2) B D A E C (2) B A E C D (2) A C E D B (2) E A B C D (1) D C E A B (1) D C B A E (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C E A (1) B D A C E (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 10 24 18 4 B -10 0 -4 6 -8 C -24 4 0 12 -20 D -18 -6 -12 0 -16 E -4 8 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 24 18 4 B -10 0 -4 6 -8 C -24 4 0 12 -20 D -18 -6 -12 0 -16 E -4 8 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 A=22 C=20 D=9 so D is eliminated. Round 2 votes counts: B=30 C=25 E=23 A=22 so A is eliminated. Round 3 votes counts: E=42 B=31 C=27 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:228 E:220 B:192 C:186 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 24 18 4 B -10 0 -4 6 -8 C -24 4 0 12 -20 D -18 -6 -12 0 -16 E -4 8 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 24 18 4 B -10 0 -4 6 -8 C -24 4 0 12 -20 D -18 -6 -12 0 -16 E -4 8 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 24 18 4 B -10 0 -4 6 -8 C -24 4 0 12 -20 D -18 -6 -12 0 -16 E -4 8 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993542 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4333: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (12) D A C B E (10) E B C D A (8) A D C B E (7) A D E C B (6) D E C B A (5) E B A C D (4) E A D B C (3) C B E D A (3) B E C A D (3) B C E D A (3) A D E B C (3) E D C B A (2) E C B D A (2) E A B C D (2) D E A C B (2) D C B E A (2) D A E C B (2) B C E A D (2) A E D B C (2) A E B C D (2) A D B C E (2) E D B C A (1) E D A B C (1) D C E B A (1) D C B A E (1) D A C E B (1) C B D E A (1) C B A D E (1) B E C D A (1) B A C E D (1) A E B D C (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 8 -16 B 6 0 6 -2 -20 C -4 -6 0 -4 -24 D -8 2 4 0 -8 E 16 20 24 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 8 -16 B 6 0 6 -2 -20 C -4 -6 0 -4 -24 D -8 2 4 0 -8 E 16 20 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=26 D=24 B=10 C=5 so C is eliminated. Round 2 votes counts: E=35 A=26 D=24 B=15 so B is eliminated. Round 3 votes counts: E=47 A=28 D=25 so D is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:234 A:195 B:195 D:195 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 8 -16 B 6 0 6 -2 -20 C -4 -6 0 -4 -24 D -8 2 4 0 -8 E 16 20 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 8 -16 B 6 0 6 -2 -20 C -4 -6 0 -4 -24 D -8 2 4 0 -8 E 16 20 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 8 -16 B 6 0 6 -2 -20 C -4 -6 0 -4 -24 D -8 2 4 0 -8 E 16 20 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4334: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) E C B D A (7) C B D A E (7) E B C D A (5) E A B D C (5) B E C D A (5) E B A D C (4) C D B A E (4) B C D A E (4) E A D C B (3) E A C D B (3) C B E D A (3) B D C A E (3) B C D E A (3) A E D B C (3) A D E B C (3) A D C E B (3) A D C B E (3) D A C B E (2) C D A B E (2) A E D C B (2) A D B C E (2) E C B A D (1) E C A D B (1) E B C A D (1) E A B C D (1) D B C A E (1) D B A C E (1) D A B C E (1) C E D A B (1) C B D E A (1) B E D C A (1) B D A C E (1) B C E D A (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -2 -6 -12 B 6 0 12 8 -8 C 2 -12 0 0 -14 D 6 -8 0 0 -14 E 12 8 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 -6 -12 B 6 0 12 8 -8 C 2 -12 0 0 -14 D 6 -8 0 0 -14 E 12 8 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=18 B=18 A=18 D=5 so D is eliminated. Round 2 votes counts: E=41 A=21 B=20 C=18 so C is eliminated. Round 3 votes counts: E=42 B=35 A=23 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:209 D:192 C:188 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -6 -12 B 6 0 12 8 -8 C 2 -12 0 0 -14 D 6 -8 0 0 -14 E 12 8 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -6 -12 B 6 0 12 8 -8 C 2 -12 0 0 -14 D 6 -8 0 0 -14 E 12 8 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -6 -12 B 6 0 12 8 -8 C 2 -12 0 0 -14 D 6 -8 0 0 -14 E 12 8 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4335: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) B C A D E (7) B A E D C (7) D E B A C (6) B A C E D (6) D E A C B (5) E D A C B (4) E A D C B (4) C A B E D (4) B C A E D (4) D E B C A (3) D B C E A (3) D E C B A (2) D C E A B (2) D B E A C (2) C A D E B (2) B D E A C (2) B D C E A (2) B C D E A (2) B C D A E (2) A E D C B (2) A E C D B (2) E D C A B (1) E A D B C (1) D E A B C (1) D B E C A (1) C D B E A (1) C B D E A (1) C B D A E (1) C B A E D (1) C A E D B (1) C A E B D (1) B D A E C (1) A E D B C (1) A E B D C (1) A E B C D (1) A C E B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -2 -4 -6 B 8 0 10 -8 -2 C 2 -10 0 -24 -18 D 4 8 24 0 12 E 6 2 18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -4 -6 B 8 0 10 -8 -2 C 2 -10 0 -24 -18 D 4 8 24 0 12 E 6 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=33 C=12 E=10 A=10 so E is eliminated. Round 2 votes counts: D=40 B=33 A=15 C=12 so C is eliminated. Round 3 votes counts: D=41 B=36 A=23 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:207 B:204 A:190 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 -6 B 8 0 10 -8 -2 C 2 -10 0 -24 -18 D 4 8 24 0 12 E 6 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 -6 B 8 0 10 -8 -2 C 2 -10 0 -24 -18 D 4 8 24 0 12 E 6 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 -6 B 8 0 10 -8 -2 C 2 -10 0 -24 -18 D 4 8 24 0 12 E 6 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4336: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) E B A C D (8) E B C A D (6) E B A D C (6) C D A B E (6) A B E D C (6) C D E B A (5) D C A B E (4) C E B A D (4) A D B E C (4) D C A E B (3) D A B C E (3) C E B D A (3) A B D E C (3) D E B A C (2) D E A B C (2) D C E B A (2) C E D B A (2) C A D B E (2) C A B E D (2) B E A D C (2) A B E C D (2) E C B A D (1) E B C D A (1) D A E B C (1) D A C E B (1) C D A E B (1) C B A E D (1) B E A C D (1) B A E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 14 2 10 B -10 0 26 -2 4 C -14 -26 0 -6 -18 D -2 2 6 0 6 E -10 -4 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 2 10 B -10 0 26 -2 4 C -14 -26 0 -6 -18 D -2 2 6 0 6 E -10 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=26 E=22 A=17 B=4 so B is eliminated. Round 2 votes counts: D=31 C=26 E=25 A=18 so A is eliminated. Round 3 votes counts: D=39 E=34 C=27 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:218 B:209 D:206 E:199 C:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 2 10 B -10 0 26 -2 4 C -14 -26 0 -6 -18 D -2 2 6 0 6 E -10 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 2 10 B -10 0 26 -2 4 C -14 -26 0 -6 -18 D -2 2 6 0 6 E -10 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 2 10 B -10 0 26 -2 4 C -14 -26 0 -6 -18 D -2 2 6 0 6 E -10 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4337: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) C A D E B (8) C D A E B (6) D E C B A (5) B A C E D (5) E B D C A (4) B A E C D (4) A C D B E (4) A C B D E (4) A B C E D (4) E D C B A (3) D C A E B (3) B E D C A (3) B E D A C (3) B E A D C (3) B E A C D (3) B A D C E (3) A C B E D (3) A B C D E (3) C D E A B (2) C A E D B (2) B D E A C (2) A C E B D (2) E D B C A (1) E C D A B (1) E B A C D (1) D E C A B (1) D C E B A (1) D B E C A (1) C A D B E (1) B D E C A (1) B A E D C (1) B A C D E (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -2 10 14 B -10 0 -12 2 0 C 2 12 0 14 24 D -10 -2 -14 0 10 E -14 0 -24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 10 14 B -10 0 -12 2 0 C 2 12 0 14 24 D -10 -2 -14 0 10 E -14 0 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 D=19 C=19 E=10 so E is eliminated. Round 2 votes counts: B=34 D=23 A=23 C=20 so C is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:226 A:216 D:192 B:190 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 10 14 B -10 0 -12 2 0 C 2 12 0 14 24 D -10 -2 -14 0 10 E -14 0 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 10 14 B -10 0 -12 2 0 C 2 12 0 14 24 D -10 -2 -14 0 10 E -14 0 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 10 14 B -10 0 -12 2 0 C 2 12 0 14 24 D -10 -2 -14 0 10 E -14 0 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4338: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (23) E A B C D (19) A B E C D (7) C D B A E (6) E D A B C (5) D C E B A (5) B A E C D (5) E A B D C (4) D C E A B (4) D E C A B (3) C B A D E (3) B A C E D (3) D C A B E (2) B A C D E (2) E C B A D (1) D E A B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C D E B A (1) B C A D E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 0 -4 12 B 2 0 2 -4 8 C 0 -2 0 0 4 D 4 4 0 0 6 E -12 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.384430 D: 0.615570 E: 0.000000 Sum of squares = 0.526712946738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.384430 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -4 12 B 2 0 2 -4 8 C 0 -2 0 0 4 D 4 4 0 0 6 E -12 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=29 C=11 B=11 A=9 so A is eliminated. Round 2 votes counts: D=40 E=31 B=18 C=11 so C is eliminated. Round 3 votes counts: D=47 E=32 B=21 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:207 B:204 A:203 C:201 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -4 12 B 2 0 2 -4 8 C 0 -2 0 0 4 D 4 4 0 0 6 E -12 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 12 B 2 0 2 -4 8 C 0 -2 0 0 4 D 4 4 0 0 6 E -12 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 12 B 2 0 2 -4 8 C 0 -2 0 0 4 D 4 4 0 0 6 E -12 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4339: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) A E B D C (11) E A D C B (10) B A C D E (9) C D E B A (7) B C D A E (7) A E D C B (7) A B E C D (7) E D C A B (6) C D B E A (5) B A C E D (4) A B E D C (3) D C E B A (2) C B D E A (2) B A E C D (2) E D C B A (1) D C E A B (1) C D B A E (1) B C A D E (1) A E D B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 10 12 8 B 6 0 16 18 8 C -10 -16 0 16 4 D -12 -18 -16 0 -4 E -8 -8 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 12 8 B 6 0 16 18 8 C -10 -16 0 16 4 D -12 -18 -16 0 -4 E -8 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=30 E=17 C=15 D=3 so D is eliminated. Round 2 votes counts: B=35 A=30 C=18 E=17 so E is eliminated. Round 3 votes counts: A=40 B=35 C=25 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:212 C:197 E:192 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 12 8 B 6 0 16 18 8 C -10 -16 0 16 4 D -12 -18 -16 0 -4 E -8 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 12 8 B 6 0 16 18 8 C -10 -16 0 16 4 D -12 -18 -16 0 -4 E -8 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 12 8 B 6 0 16 18 8 C -10 -16 0 16 4 D -12 -18 -16 0 -4 E -8 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4340: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) B A C E D (8) A C D B E (8) A B C D E (7) C A D E B (6) D C E A B (5) D C A E B (5) B E D A C (5) B E A C D (5) D E C A B (4) E D C A B (3) B E A D C (3) B A E C D (3) E D C B A (2) E D B C A (2) D B A C E (2) C D E A B (2) B A E D C (2) B A D C E (2) A C B D E (2) A B C E D (2) E C D B A (1) E C D A B (1) E C B A D (1) E B C D A (1) D E C B A (1) D E B C A (1) D B E A C (1) D A B C E (1) C D A E B (1) B E D C A (1) B A C D E (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 10 6 6 B 0 0 10 6 8 C -10 -10 0 4 10 D -6 -6 -4 0 0 E -6 -8 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.546628 B: 0.453372 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.504348252862 Cumulative probabilities = A: 0.546628 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 6 6 B 0 0 10 6 8 C -10 -10 0 4 10 D -6 -6 -4 0 0 E -6 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=22 D=20 E=19 C=9 so C is eliminated. Round 2 votes counts: B=30 A=28 D=23 E=19 so E is eliminated. Round 3 votes counts: B=40 D=32 A=28 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:211 C:197 D:192 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 6 6 B 0 0 10 6 8 C -10 -10 0 4 10 D -6 -6 -4 0 0 E -6 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 6 6 B 0 0 10 6 8 C -10 -10 0 4 10 D -6 -6 -4 0 0 E -6 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 6 6 B 0 0 10 6 8 C -10 -10 0 4 10 D -6 -6 -4 0 0 E -6 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4341: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) D B A C E (6) D B C E A (5) C D E B A (5) C D E A B (5) A E B C D (5) B D A E C (4) A E C B D (4) E A C B D (3) D C E B A (3) D C E A B (3) D A C B E (3) C E A D B (3) B E A C D (3) B A E D C (3) B A E C D (3) A E C D B (3) E C A B D (2) D C A E B (2) D B C A E (2) D A C E B (2) B D C E A (2) B D A C E (2) B A D E C (2) A E D C B (2) A E B D C (2) A D B E C (2) E C A D B (1) E A C D B (1) C E D B A (1) C E D A B (1) C E B D A (1) C D B E A (1) B D E C A (1) B D C A E (1) A D C E B (1) A C E D B (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 10 -12 6 B 2 0 -8 -18 -2 C -10 8 0 -12 12 D 12 18 12 0 18 E -6 2 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -12 6 B 2 0 -8 -18 -2 C -10 8 0 -12 12 D 12 18 12 0 18 E -6 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=23 B=21 C=17 E=7 so E is eliminated. Round 2 votes counts: D=32 A=27 B=21 C=20 so C is eliminated. Round 3 votes counts: D=45 A=33 B=22 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:201 C:199 B:187 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -12 6 B 2 0 -8 -18 -2 C -10 8 0 -12 12 D 12 18 12 0 18 E -6 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -12 6 B 2 0 -8 -18 -2 C -10 8 0 -12 12 D 12 18 12 0 18 E -6 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -12 6 B 2 0 -8 -18 -2 C -10 8 0 -12 12 D 12 18 12 0 18 E -6 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4342: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) A D C B E (7) E B C D A (6) C A E D B (6) D A C E B (5) E C B A D (4) E B D C A (4) D A E C B (4) B E C A D (4) E C B D A (3) C E A D B (3) B E D A C (3) A D C E B (3) A D B C E (3) A C D E B (3) A C D B E (3) E C D A B (2) D B A E C (2) D A E B C (2) D A B C E (2) C E A B D (2) C B E A D (2) C A E B D (2) C A D B E (2) B E C D A (2) B A D C E (2) E D B A C (1) E B D A C (1) D E B A C (1) D E A B C (1) D B E A C (1) D A C B E (1) D A B E C (1) C B A E D (1) C A D E B (1) B D E A C (1) B C E A D (1) Total count = 100 A B C D E A 0 6 -6 14 0 B -6 0 -24 -8 -24 C 6 24 0 10 14 D -14 8 -10 0 -10 E 0 24 -14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 14 0 B -6 0 -24 -8 -24 C 6 24 0 10 14 D -14 8 -10 0 -10 E 0 24 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=21 D=20 A=19 B=13 so B is eliminated. Round 2 votes counts: E=30 C=28 D=21 A=21 so D is eliminated. Round 3 votes counts: A=38 E=34 C=28 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:227 E:210 A:207 D:187 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 14 0 B -6 0 -24 -8 -24 C 6 24 0 10 14 D -14 8 -10 0 -10 E 0 24 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 14 0 B -6 0 -24 -8 -24 C 6 24 0 10 14 D -14 8 -10 0 -10 E 0 24 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 14 0 B -6 0 -24 -8 -24 C 6 24 0 10 14 D -14 8 -10 0 -10 E 0 24 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4343: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) B C E D A (7) E A C D B (6) D A B E C (6) C B E D A (6) D B A E C (4) B D C A E (4) A E D C B (4) A D E C B (4) A D E B C (4) E C A D B (3) D B A C E (3) C E A B D (3) C B E A D (3) C B A D E (3) B D A C E (3) E C B D A (2) E A D C B (2) D A B C E (2) B D C E A (2) E D B A C (1) E D A B C (1) E C B A D (1) E C A B D (1) E B C D A (1) D E A B C (1) D B E A C (1) D A E B C (1) C E B D A (1) C B A E D (1) C A E D B (1) B E D C A (1) B E C D A (1) B D E A C (1) B C D E A (1) B C D A E (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -2 -2 -8 B 12 0 -6 2 4 C 2 6 0 6 8 D 2 -2 -6 0 -10 E 8 -4 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -2 -8 B 12 0 -6 2 4 C 2 6 0 6 8 D 2 -2 -6 0 -10 E 8 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=21 E=18 D=18 A=17 so A is eliminated. Round 2 votes counts: C=29 D=28 E=22 B=21 so B is eliminated. Round 3 votes counts: D=38 C=38 E=24 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:206 E:203 D:192 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 -2 -8 B 12 0 -6 2 4 C 2 6 0 6 8 D 2 -2 -6 0 -10 E 8 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -2 -8 B 12 0 -6 2 4 C 2 6 0 6 8 D 2 -2 -6 0 -10 E 8 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -2 -8 B 12 0 -6 2 4 C 2 6 0 6 8 D 2 -2 -6 0 -10 E 8 -4 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4344: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) C D A E B (8) B C E A D (7) D C A E B (6) B E C D A (6) E B D A C (5) D A C E B (5) C D A B E (5) C D B E A (4) B E A C D (4) B C D E A (4) E D A B C (3) C D B A E (3) D A E C B (2) B E D C A (2) B E C A D (2) A E B D C (2) A D E C B (2) E D B C A (1) E B D C A (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C E A B (1) C B A D E (1) C A D B E (1) C A B D E (1) B C E D A (1) B C A E D (1) B C A D E (1) A E D B C (1) A E B C D (1) A D C E B (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -14 -18 -12 B 8 0 14 6 12 C 14 -14 0 6 4 D 18 -6 -6 0 -4 E 12 -12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -18 -12 B 8 0 14 6 12 C 14 -14 0 6 4 D 18 -6 -6 0 -4 E 12 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 C=23 D=16 E=13 A=10 so A is eliminated. Round 2 votes counts: B=40 C=24 D=19 E=17 so E is eliminated. Round 3 votes counts: B=51 D=25 C=24 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:205 D:201 E:200 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -14 -18 -12 B 8 0 14 6 12 C 14 -14 0 6 4 D 18 -6 -6 0 -4 E 12 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -18 -12 B 8 0 14 6 12 C 14 -14 0 6 4 D 18 -6 -6 0 -4 E 12 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -18 -12 B 8 0 14 6 12 C 14 -14 0 6 4 D 18 -6 -6 0 -4 E 12 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4345: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) D C B A E (6) D C A B E (6) D B C A E (6) E D C A B (5) E B A C D (4) E A C B D (4) B D C A E (4) A C E B D (4) E A C D B (3) C D A B E (3) C A B D E (3) B D E A C (3) B D A C E (3) E D A C B (2) E A B C D (2) D E B C A (2) D B E C A (2) C D B A E (2) B A C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A B C (1) E C D A B (1) E C A D B (1) E B D A C (1) E B A D C (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A E B (1) C D A E B (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A D C (1) B E A C D (1) B C A D E (1) B A D C E (1) A E C B D (1) A C E D B (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 -12 18 B 6 0 -4 -2 18 C 2 4 0 -4 20 D 12 2 4 0 16 E -18 -18 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -12 18 B 6 0 -4 -2 18 C 2 4 0 -4 20 D 12 2 4 0 16 E -18 -18 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 B=24 C=11 A=11 so C is eliminated. Round 2 votes counts: D=32 E=28 B=24 A=16 so A is eliminated. Round 3 votes counts: E=34 D=34 B=32 so B is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:211 B:209 A:199 E:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 -12 18 B 6 0 -4 -2 18 C 2 4 0 -4 20 D 12 2 4 0 16 E -18 -18 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -12 18 B 6 0 -4 -2 18 C 2 4 0 -4 20 D 12 2 4 0 16 E -18 -18 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -12 18 B 6 0 -4 -2 18 C 2 4 0 -4 20 D 12 2 4 0 16 E -18 -18 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4346: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (15) C B A E D (9) B C A E D (9) E A D C B (8) D B C A E (5) B C D A E (5) E A C B D (4) D E A C B (4) C B D E A (4) E D A C B (3) C B D A E (3) A E B C D (3) E A D B C (2) E A C D B (2) D B C E A (2) D B A C E (2) D A E B C (2) D A B E C (2) A E D B C (2) A D E B C (2) E C D B A (1) E C A B D (1) D E B C A (1) D C B E A (1) C B E A D (1) B D C A E (1) B C A D E (1) B A D C E (1) A E C B D (1) A E B D C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 12 -2 2 B -8 0 14 -8 -6 C -12 -14 0 -8 -8 D 2 8 8 0 2 E -2 6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -2 2 B -8 0 14 -8 -6 C -12 -14 0 -8 -8 D 2 8 8 0 2 E -2 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=21 C=17 B=17 A=11 so A is eliminated. Round 2 votes counts: D=36 E=28 C=18 B=18 so C is eliminated. Round 3 votes counts: D=36 B=35 E=29 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:210 E:205 B:196 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -2 2 B -8 0 14 -8 -6 C -12 -14 0 -8 -8 D 2 8 8 0 2 E -2 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -2 2 B -8 0 14 -8 -6 C -12 -14 0 -8 -8 D 2 8 8 0 2 E -2 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -2 2 B -8 0 14 -8 -6 C -12 -14 0 -8 -8 D 2 8 8 0 2 E -2 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4347: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) D C A E B (7) D A C E B (6) C D B E A (6) C D A B E (6) B E C A D (6) D C E B A (5) A B E C D (4) D E C B A (3) D C E A B (3) C B E D A (3) C B A E D (3) B E C D A (3) B E A C D (3) B A C E D (3) A D E B C (3) A B E D C (3) E B C D A (2) E B A D C (2) E B A C D (2) D E C A B (2) D E A C B (2) C B D E A (2) B A E C D (2) A B C E D (2) E D A B C (1) E A B D C (1) D C A B E (1) C E B D A (1) B C E D A (1) B C E A D (1) B C A E D (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -12 -8 2 B -2 0 -2 6 0 C 12 2 0 2 4 D 8 -6 -2 0 -6 E -2 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -8 2 B -2 0 -2 6 0 C 12 2 0 2 4 D 8 -6 -2 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=22 C=21 B=20 E=8 so E is eliminated. Round 2 votes counts: D=30 B=26 A=23 C=21 so C is eliminated. Round 3 votes counts: D=42 B=35 A=23 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:210 B:201 E:200 D:197 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -8 2 B -2 0 -2 6 0 C 12 2 0 2 4 D 8 -6 -2 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -8 2 B -2 0 -2 6 0 C 12 2 0 2 4 D 8 -6 -2 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -8 2 B -2 0 -2 6 0 C 12 2 0 2 4 D 8 -6 -2 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4348: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) B A D E C (6) C E D A B (5) C D E A B (5) B A E D C (5) A E B C D (5) D C E A B (4) D B A E C (4) B A E C D (4) E C A D B (3) E C A B D (3) D C B E A (3) D B C A E (3) D B A C E (3) C E A D B (3) C D E B A (3) B C E A D (3) B A C E D (3) A E B D C (3) E A C B D (2) D A E C B (2) D A B E C (2) C E D B A (2) C E A B D (2) B D A C E (2) B C D E A (2) A D B E C (2) E A C D B (1) E A B C D (1) D C E B A (1) D A E B C (1) C B E A D (1) B D A E C (1) B A D C E (1) A E C B D (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 2 14 0 B 6 0 6 4 -8 C -2 -6 0 10 4 D -14 -4 -10 0 -8 E 0 8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.333333 Sum of squares = 0.358024691295 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -6 2 14 0 B 6 0 6 4 -8 C -2 -6 0 10 4 D -14 -4 -10 0 -8 E 0 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.333333 Sum of squares = 0.358024688867 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=27 B=27 D=23 A=13 E=10 so E is eliminated. Round 2 votes counts: C=33 B=27 D=23 A=17 so A is eliminated. Round 3 votes counts: C=37 B=37 D=26 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:206 A:205 B:204 C:203 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 2 14 0 B 6 0 6 4 -8 C -2 -6 0 10 4 D -14 -4 -10 0 -8 E 0 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.333333 Sum of squares = 0.358024688867 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 14 0 B 6 0 6 4 -8 C -2 -6 0 10 4 D -14 -4 -10 0 -8 E 0 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.333333 Sum of squares = 0.358024688867 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 14 0 B 6 0 6 4 -8 C -2 -6 0 10 4 D -14 -4 -10 0 -8 E 0 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.000000 E: 0.333333 Sum of squares = 0.358024688867 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4349: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C A E D B (7) B E D C A (7) B C A D E (6) A D C E B (6) B E C D A (5) E B C A D (4) D B A E C (4) B D E A C (4) B C E A D (4) E C A D B (3) D B E A C (3) A D E C B (3) A C E D B (3) E B D A C (2) D A E B C (2) D A B E C (2) C B E A D (2) C A E B D (2) B E C A D (2) B D E C A (2) B D A C E (2) B C E D A (2) A C D E B (2) E A D C B (1) D B A C E (1) D A C B E (1) D A B C E (1) C E B A D (1) C A B E D (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D E A (1) B C D A E (1) A E D C B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -4 -2 12 B 12 0 16 2 10 C 4 -16 0 -6 -8 D 2 -2 6 0 6 E -12 -10 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -2 12 B 12 0 16 2 10 C 4 -16 0 -6 -8 D 2 -2 6 0 6 E -12 -10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=22 A=16 C=13 E=10 so E is eliminated. Round 2 votes counts: B=45 D=22 A=17 C=16 so C is eliminated. Round 3 votes counts: B=48 A=30 D=22 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:206 A:197 E:190 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -2 12 B 12 0 16 2 10 C 4 -16 0 -6 -8 D 2 -2 6 0 6 E -12 -10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -2 12 B 12 0 16 2 10 C 4 -16 0 -6 -8 D 2 -2 6 0 6 E -12 -10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -2 12 B 12 0 16 2 10 C 4 -16 0 -6 -8 D 2 -2 6 0 6 E -12 -10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4350: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) D B C A E (8) B D A E C (8) E A C B D (7) C D E A B (7) E A B C D (5) D C B A E (5) B E A D C (5) D B A C E (3) C E A D B (3) B D C A E (3) E C A D B (2) E C A B D (2) E B C A D (2) D C A B E (2) C D B E A (2) C A E D B (2) B D E A C (2) B A E D C (2) B A D E C (2) A E B D C (2) A D E C B (2) E A C D B (1) D C A E B (1) D B C E A (1) C E D A B (1) C D E B A (1) B E D C A (1) B E D A C (1) B E C D A (1) B D E C A (1) B D C E A (1) A E C D B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -12 -18 8 B -2 0 -4 -10 -4 C 12 4 0 -2 6 D 18 10 2 0 22 E -8 4 -6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -18 8 B -2 0 -4 -10 -4 C 12 4 0 -2 6 D 18 10 2 0 22 E -8 4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999941723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 D=20 E=19 A=8 so A is eliminated. Round 2 votes counts: C=28 B=27 D=23 E=22 so E is eliminated. Round 3 votes counts: C=41 B=36 D=23 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:226 C:210 A:190 B:190 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -18 8 B -2 0 -4 -10 -4 C 12 4 0 -2 6 D 18 10 2 0 22 E -8 4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999941723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -18 8 B -2 0 -4 -10 -4 C 12 4 0 -2 6 D 18 10 2 0 22 E -8 4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999941723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -18 8 B -2 0 -4 -10 -4 C 12 4 0 -2 6 D 18 10 2 0 22 E -8 4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999941723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4351: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) C A E D B (9) D B E C A (6) C A B E D (6) D C B E A (5) B E D A C (5) A C E B D (5) D C E B A (4) D E B A C (3) D B E A C (3) C D E A B (3) C D A E B (3) B E A D C (3) B D E A C (3) B D C E A (3) E A B D C (2) C D B E A (2) C B D A E (2) C A E B D (2) C A D B E (2) A E C B D (2) A E B C D (2) E D B A C (1) E A D B C (1) E A C D B (1) D E C A B (1) D E B C A (1) D C E A B (1) D B C E A (1) C D E B A (1) C A D E B (1) C A B D E (1) B C A D E (1) B A E C D (1) A E C D B (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -10 4 0 B 10 0 -4 0 10 C 10 4 0 -8 4 D -4 0 8 0 -6 E 0 -10 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.513283 C: 0.000000 D: 0.486717 E: 0.000000 Sum of squares = 0.500352835346 Cumulative probabilities = A: 0.000000 B: 0.513283 C: 0.513283 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 4 0 B 10 0 -4 0 10 C 10 4 0 -8 4 D -4 0 8 0 -6 E 0 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=26 D=25 A=12 E=5 so E is eliminated. Round 2 votes counts: C=32 D=26 B=26 A=16 so A is eliminated. Round 3 votes counts: C=41 B=32 D=27 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:208 C:205 D:199 E:196 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 4 0 B 10 0 -4 0 10 C 10 4 0 -8 4 D -4 0 8 0 -6 E 0 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 4 0 B 10 0 -4 0 10 C 10 4 0 -8 4 D -4 0 8 0 -6 E 0 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 4 0 B 10 0 -4 0 10 C 10 4 0 -8 4 D -4 0 8 0 -6 E 0 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4352: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) B E D C A (8) C E A B D (7) A C E B D (7) D A B C E (4) B E C A D (4) A D C B E (4) A C E D B (4) D B E C A (3) D B A E C (3) D B A C E (3) B E C D A (3) B D E C A (3) A D B C E (3) A C D E B (3) A B C D E (3) E C D B A (2) E B C D A (2) D C A E B (2) C E D A B (2) C A E D B (2) C A E B D (2) B E D A C (2) B D A E C (2) E C B D A (1) E C A B D (1) E B C A D (1) D C E A B (1) C E A D B (1) C D E B A (1) C D E A B (1) C A D E B (1) B E A C D (1) A D C E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -16 14 -8 B -2 0 -6 18 -4 C 16 6 0 22 6 D -14 -18 -22 0 -24 E 8 4 -6 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 14 -8 B -2 0 -6 18 -4 C 16 6 0 22 6 D -14 -18 -22 0 -24 E 8 4 -6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=23 E=17 C=17 D=16 so D is eliminated. Round 2 votes counts: B=32 A=31 C=20 E=17 so E is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:215 B:203 A:196 D:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 14 -8 B -2 0 -6 18 -4 C 16 6 0 22 6 D -14 -18 -22 0 -24 E 8 4 -6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 14 -8 B -2 0 -6 18 -4 C 16 6 0 22 6 D -14 -18 -22 0 -24 E 8 4 -6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 14 -8 B -2 0 -6 18 -4 C 16 6 0 22 6 D -14 -18 -22 0 -24 E 8 4 -6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4353: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) C E B D A (8) A D B E C (8) E C A D B (7) C E A B D (6) B D A C E (6) E C A B D (5) E A D C B (5) A D E B C (5) C B D E A (4) B D C A E (4) B C D E A (4) E C B D A (3) D B A E C (3) D A B E C (3) C E B A D (3) A E D C B (2) A D B C E (2) A C E D B (2) E D C A B (1) E C D B A (1) E C B A D (1) E A C D B (1) D B A C E (1) C B E A D (1) C B D A E (1) C B A E D (1) C A B E D (1) B C D A E (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 -22 2 -18 B 2 0 -24 16 -2 C 22 24 0 18 8 D -2 -16 -18 0 -16 E 18 2 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -22 2 -18 B 2 0 -24 16 -2 C 22 24 0 18 8 D -2 -16 -18 0 -16 E 18 2 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=24 A=20 B=15 D=7 so D is eliminated. Round 2 votes counts: C=34 E=24 A=23 B=19 so B is eliminated. Round 3 votes counts: C=43 A=33 E=24 so E is eliminated. Round 4 votes counts: C=61 A=39 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:236 E:214 B:196 A:180 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -22 2 -18 B 2 0 -24 16 -2 C 22 24 0 18 8 D -2 -16 -18 0 -16 E 18 2 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -22 2 -18 B 2 0 -24 16 -2 C 22 24 0 18 8 D -2 -16 -18 0 -16 E 18 2 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -22 2 -18 B 2 0 -24 16 -2 C 22 24 0 18 8 D -2 -16 -18 0 -16 E 18 2 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4354: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) A C B D E (9) C E A B D (8) D B A E C (7) C A D B E (6) E D B C A (5) E C D B A (5) E B D A C (5) C E A D B (5) C A E B D (4) C A B D E (4) C A B E D (3) B D E A C (3) B D A E C (3) A B D C E (3) D E B A C (2) D B E A C (2) C E D B A (2) C A E D B (2) B A D E C (2) A D B C E (2) E D C B A (1) E C B D A (1) E C B A D (1) E B D C A (1) D A B E C (1) C A D E B (1) B E D A C (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 2 2 -4 B 2 0 -6 -4 -6 C -2 6 0 4 0 D -2 4 -4 0 -8 E 4 6 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.478445 D: 0.000000 E: 0.521555 Sum of squares = 0.500929236331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.478445 D: 0.478445 E: 1.000000 A B C D E A 0 -2 2 2 -4 B 2 0 -6 -4 -6 C -2 6 0 4 0 D -2 4 -4 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=29 A=15 D=12 B=9 so B is eliminated. Round 2 votes counts: C=35 E=30 D=18 A=17 so A is eliminated. Round 3 votes counts: C=45 E=30 D=25 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:204 A:199 D:195 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 2 -4 B 2 0 -6 -4 -6 C -2 6 0 4 0 D -2 4 -4 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 -4 B 2 0 -6 -4 -6 C -2 6 0 4 0 D -2 4 -4 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 -4 B 2 0 -6 -4 -6 C -2 6 0 4 0 D -2 4 -4 0 -8 E 4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4355: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) C A D B E (11) C D A B E (10) D C E A B (7) B A E C D (7) E D B C A (6) B E A C D (6) A B C D E (6) E D C B A (5) E B A C D (5) A C B D E (5) E D B A C (2) E B D A C (2) D E C A B (2) D C A E B (2) B E A D C (2) E D C A B (1) E B D C A (1) E B C A D (1) D E C B A (1) D C A B E (1) D A C B E (1) D A B C E (1) C A B D E (1) B A C E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 2 16 -4 B 0 0 6 0 8 C -2 -6 0 10 -6 D -16 0 -10 0 -4 E 4 -8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.347040 B: 0.652960 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.546793589686 Cumulative probabilities = A: 0.347040 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 16 -4 B 0 0 6 0 8 C -2 -6 0 10 -6 D -16 0 -10 0 -4 E 4 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=22 B=16 D=15 A=13 so A is eliminated. Round 2 votes counts: E=34 C=27 B=24 D=15 so D is eliminated. Round 3 votes counts: C=38 E=37 B=25 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:207 B:207 E:203 C:198 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 16 -4 B 0 0 6 0 8 C -2 -6 0 10 -6 D -16 0 -10 0 -4 E 4 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 16 -4 B 0 0 6 0 8 C -2 -6 0 10 -6 D -16 0 -10 0 -4 E 4 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 16 -4 B 0 0 6 0 8 C -2 -6 0 10 -6 D -16 0 -10 0 -4 E 4 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4356: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (12) B E C D A (7) D A C E B (6) B C E D A (5) D C A E B (4) C D B E A (4) B A E D C (4) B E A C D (3) A E D C B (3) A D C E B (3) A B E D C (3) A B D C E (3) C E D B A (2) C E B D A (2) C B E D A (2) C B D E A (2) B C D E A (2) A E D B C (2) A E B D C (2) A D C B E (2) A D B E C (2) A D B C E (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B D A (1) E B C D A (1) E B A C D (1) E A D C B (1) E A B C D (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A B E (1) D A C B E (1) C D E A B (1) C D B A E (1) B E A D C (1) B D C A E (1) B C E A D (1) B A D E C (1) B A D C E (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -8 2 -10 B 10 0 14 14 24 C 8 -14 0 0 -6 D -2 -14 0 0 -14 E 10 -24 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 2 -10 B 10 0 14 14 24 C 8 -14 0 0 -6 D -2 -14 0 0 -14 E 10 -24 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=25 D=15 C=14 E=8 so E is eliminated. Round 2 votes counts: B=40 A=27 C=17 D=16 so D is eliminated. Round 3 votes counts: B=40 A=35 C=25 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:203 C:194 A:187 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 2 -10 B 10 0 14 14 24 C 8 -14 0 0 -6 D -2 -14 0 0 -14 E 10 -24 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 2 -10 B 10 0 14 14 24 C 8 -14 0 0 -6 D -2 -14 0 0 -14 E 10 -24 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 2 -10 B 10 0 14 14 24 C 8 -14 0 0 -6 D -2 -14 0 0 -14 E 10 -24 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4357: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (6) B D C A E (6) E D B C A (5) E A C D B (5) C D B A E (5) E A B C D (4) B C A D E (4) A E C D B (4) A E C B D (4) A C B D E (4) E D B A C (3) E A B D C (3) D B E C A (3) D B C E A (3) E D C B A (2) E D C A B (2) D E C B A (2) D C B E A (2) C D E A B (2) C A D E B (2) C A D B E (2) B D E C A (2) B D E A C (2) B C D A E (2) A C E B D (2) A B E C D (2) E D A B C (1) E B D A C (1) E B A D C (1) D E B C A (1) D C E B A (1) D C B A E (1) D B C A E (1) C D A E B (1) C B A D E (1) C A E D B (1) B E D A C (1) B D A E C (1) B D A C E (1) B A D E C (1) B A C D E (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -14 -2 8 B 6 0 2 2 4 C 14 -2 0 8 -2 D 2 -2 -8 0 14 E -8 -4 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -2 8 B 6 0 2 2 4 C 14 -2 0 8 -2 D 2 -2 -8 0 14 E -8 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=21 C=20 A=18 D=14 so D is eliminated. Round 2 votes counts: E=30 B=28 C=24 A=18 so A is eliminated. Round 3 votes counts: E=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:207 D:203 A:193 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 -2 8 B 6 0 2 2 4 C 14 -2 0 8 -2 D 2 -2 -8 0 14 E -8 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -2 8 B 6 0 2 2 4 C 14 -2 0 8 -2 D 2 -2 -8 0 14 E -8 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -2 8 B 6 0 2 2 4 C 14 -2 0 8 -2 D 2 -2 -8 0 14 E -8 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4358: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) B D C A E (6) A D B E C (6) A E D B C (5) E C A D B (4) C B E D A (4) B A D E C (4) A D E B C (4) A B E D C (4) E C B A D (3) E C A B D (3) E A D C B (3) E A C D B (3) D A B E C (3) D A B C E (3) B C D A E (3) B A D C E (3) E A C B D (2) D B A C E (2) D A E C B (2) C E D B A (2) C E B D A (2) C B D E A (2) B C D E A (2) B A E D C (2) A D E C B (2) E C D A B (1) D E C A B (1) D C B A E (1) D B C A E (1) D A E B C (1) C E B A D (1) B C E A D (1) B A E C D (1) A E D C B (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 26 8 32 B 0 0 26 10 18 C -26 -26 0 -32 -14 D -8 -10 32 0 14 E -32 -18 14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307412 B: 0.692588 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.574180310596 Cumulative probabilities = A: 0.307412 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 26 8 32 B 0 0 26 10 18 C -26 -26 0 -32 -14 D -8 -10 32 0 14 E -32 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=24 E=19 D=14 C=11 so C is eliminated. Round 2 votes counts: B=38 E=24 A=24 D=14 so D is eliminated. Round 3 votes counts: B=42 A=33 E=25 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 B:227 D:214 E:175 C:151 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 26 8 32 B 0 0 26 10 18 C -26 -26 0 -32 -14 D -8 -10 32 0 14 E -32 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 26 8 32 B 0 0 26 10 18 C -26 -26 0 -32 -14 D -8 -10 32 0 14 E -32 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 26 8 32 B 0 0 26 10 18 C -26 -26 0 -32 -14 D -8 -10 32 0 14 E -32 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4359: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) B D C E A (11) A D B E C (8) E C B A D (7) C E B D A (7) D B A C E (5) A E C D B (5) C E B A D (4) C E A B D (4) E C A D B (3) D A B E C (3) C E A D B (3) B D C A E (3) A D E B C (3) E C B D A (2) D A B C E (2) B D A E C (2) B D A C E (2) A E D C B (2) A D E C B (2) A D B C E (2) A C E D B (2) E B C D A (1) E A C B D (1) D B C A E (1) B E D C A (1) B D E A C (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 2 -20 14 -16 B -2 0 -6 18 -14 C 20 6 0 4 -4 D -14 -18 -4 0 -8 E 16 14 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -20 14 -16 B -2 0 -6 18 -14 C 20 6 0 4 -4 D -14 -18 -4 0 -8 E 16 14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=24 B=22 C=18 D=11 so D is eliminated. Round 2 votes counts: A=29 B=28 E=25 C=18 so C is eliminated. Round 3 votes counts: E=43 A=29 B=28 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:213 B:198 A:190 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -20 14 -16 B -2 0 -6 18 -14 C 20 6 0 4 -4 D -14 -18 -4 0 -8 E 16 14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -20 14 -16 B -2 0 -6 18 -14 C 20 6 0 4 -4 D -14 -18 -4 0 -8 E 16 14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -20 14 -16 B -2 0 -6 18 -14 C 20 6 0 4 -4 D -14 -18 -4 0 -8 E 16 14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4360: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) C D E B A (9) A C B E D (7) D C E B A (6) A B E D C (5) A B D E C (5) A B C E D (5) D B E C A (4) C E D B A (4) D B E A C (3) C E A D B (3) C D E A B (3) E B C A D (2) C E D A B (2) C E B D A (2) C D A E B (2) C A E D B (2) C A E B D (2) C A D E B (2) B E D A C (2) B A D E C (2) A D B E C (2) A C E B D (2) A C D B E (2) A B E C D (2) E D C B A (1) D E C B A (1) D C E A B (1) D A C B E (1) D A B E C (1) C E A B D (1) B E D C A (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E D C (1) Total count = 100 A B C D E A 0 0 -14 -8 -12 B 0 0 -6 -16 -8 C 14 6 0 4 12 D 8 16 -4 0 10 E 12 8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -8 -12 B 0 0 -6 -16 -8 C 14 6 0 4 12 D 8 16 -4 0 10 E 12 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=30 D=26 B=9 E=3 so E is eliminated. Round 2 votes counts: C=32 A=30 D=27 B=11 so B is eliminated. Round 3 votes counts: C=34 A=34 D=32 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:215 E:199 B:185 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 -8 -12 B 0 0 -6 -16 -8 C 14 6 0 4 12 D 8 16 -4 0 10 E 12 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -8 -12 B 0 0 -6 -16 -8 C 14 6 0 4 12 D 8 16 -4 0 10 E 12 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -8 -12 B 0 0 -6 -16 -8 C 14 6 0 4 12 D 8 16 -4 0 10 E 12 8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4361: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) A E D B C (8) A D E B C (8) A C B E D (7) D E B C A (6) C B D E A (5) E D B C A (4) B E D C A (4) B C E D A (4) A E D C B (4) A C B D E (4) D E B A C (3) D E A B C (3) C A B E D (3) B E C D A (3) A D E C B (3) A C D B E (3) E B D C A (2) A C E B D (2) E D B A C (1) E D A B C (1) E B C A D (1) E B A C D (1) E A D B C (1) E A B D C (1) D C B E A (1) D A C E B (1) C B E A D (1) C B A E D (1) C B A D E (1) C A B D E (1) B D E C A (1) B C D E A (1) A C E D B (1) Total count = 100 A B C D E A 0 2 4 2 -6 B -2 0 6 4 -2 C -4 -6 0 -4 -10 D -2 -4 4 0 -18 E 6 2 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 2 -6 B -2 0 6 4 -2 C -4 -6 0 -4 -10 D -2 -4 4 0 -18 E 6 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=21 D=14 B=13 E=12 so E is eliminated. Round 2 votes counts: A=42 C=21 D=20 B=17 so B is eliminated. Round 3 votes counts: A=43 C=30 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:218 B:203 A:201 D:190 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 2 -6 B -2 0 6 4 -2 C -4 -6 0 -4 -10 D -2 -4 4 0 -18 E 6 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 -6 B -2 0 6 4 -2 C -4 -6 0 -4 -10 D -2 -4 4 0 -18 E 6 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 -6 B -2 0 6 4 -2 C -4 -6 0 -4 -10 D -2 -4 4 0 -18 E 6 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4362: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (7) E D C A B (6) B C A D E (6) A D E B C (6) E D A B C (5) B A C D E (5) A B C D E (5) E D C B A (4) B A E D C (4) A B D C E (4) E D B C A (3) E D A C B (3) C E B D A (3) B C A E D (3) E C D B A (2) D E C A B (2) D A E C B (2) C D E A B (2) C B A D E (2) C A D B E (2) C A B D E (2) B E D A C (2) B C E A D (2) A B D E C (2) E B D C A (1) E B D A C (1) D E A B C (1) D C E A B (1) D C A E B (1) D A E B C (1) C E D B A (1) C B E D A (1) C B E A D (1) C B A E D (1) B A E C D (1) B A D C E (1) B A C E D (1) A D B E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 8 2 4 B -10 0 10 -2 -4 C -8 -10 0 -16 -8 D -2 2 16 0 10 E -4 4 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 2 4 B -10 0 10 -2 -4 C -8 -10 0 -16 -8 D -2 2 16 0 10 E -4 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=25 B=25 A=20 D=15 C=15 so D is eliminated. Round 2 votes counts: E=35 B=25 A=23 C=17 so C is eliminated. Round 3 votes counts: E=42 B=30 A=28 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:213 A:212 E:199 B:197 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 2 4 B -10 0 10 -2 -4 C -8 -10 0 -16 -8 D -2 2 16 0 10 E -4 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 2 4 B -10 0 10 -2 -4 C -8 -10 0 -16 -8 D -2 2 16 0 10 E -4 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 2 4 B -10 0 10 -2 -4 C -8 -10 0 -16 -8 D -2 2 16 0 10 E -4 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4363: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) C B E A D (8) D A E B C (6) A D E C B (5) A D C E B (5) B C D E A (4) A D E B C (4) E D A B C (3) E C B A D (3) E B D A C (3) C E B A D (3) C B E D A (3) C B D A E (3) C A B D E (3) B D C E A (3) A D C B E (3) E D B A C (2) E B D C A (2) D A B C E (2) C A E B D (2) B E D C A (2) B E C D A (2) B D E C A (2) A C E D B (2) E C A B D (1) E B C D A (1) E B A D C (1) E A C D B (1) D E B A C (1) D E A B C (1) D B E A C (1) D A C B E (1) C B A E D (1) C B A D E (1) B E D A C (1) B D C A E (1) A E D C B (1) A C D B E (1) Total count = 100 A B C D E A 0 -18 -12 -10 -18 B 18 0 6 22 6 C 12 -6 0 0 14 D 10 -22 0 0 -6 E 18 -6 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -12 -10 -18 B 18 0 6 22 6 C 12 -6 0 0 14 D 10 -22 0 0 -6 E 18 -6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 A=21 E=17 D=12 so D is eliminated. Round 2 votes counts: A=30 B=27 C=24 E=19 so E is eliminated. Round 3 votes counts: B=37 A=35 C=28 so C is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:210 E:202 D:191 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -12 -10 -18 B 18 0 6 22 6 C 12 -6 0 0 14 D 10 -22 0 0 -6 E 18 -6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -10 -18 B 18 0 6 22 6 C 12 -6 0 0 14 D 10 -22 0 0 -6 E 18 -6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -10 -18 B 18 0 6 22 6 C 12 -6 0 0 14 D 10 -22 0 0 -6 E 18 -6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4364: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) E A C D B (8) D B C E A (8) C A E B D (5) E C A D B (4) D B E C A (4) D B E A C (4) B D A C E (4) C A B E D (3) B D C A E (3) A C E B D (3) A B C E D (3) E D C B A (2) E D A C B (2) E A D B C (2) C E A B D (2) C D B E A (2) C B A D E (2) B D C E A (2) B C D A E (2) A E D B C (2) A E C D B (2) E D C A B (1) E C D A B (1) E C A B D (1) E A D C B (1) D E B A C (1) D E A B C (1) D B C A E (1) D B A E C (1) C E D B A (1) C E A D B (1) C B E D A (1) C B D E A (1) C B D A E (1) B C A D E (1) B A D C E (1) A E B D C (1) A E B C D (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 16 2 14 0 B -16 0 -12 2 -6 C -2 12 0 16 -2 D -14 -2 -16 0 -18 E 0 6 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.538342 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.461658 Sum of squares = 0.502940189336 Cumulative probabilities = A: 0.538342 B: 0.538342 C: 0.538342 D: 0.538342 E: 1.000000 A B C D E A 0 16 2 14 0 B -16 0 -12 2 -6 C -2 12 0 16 -2 D -14 -2 -16 0 -18 E 0 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997744 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 D=20 C=19 B=13 so B is eliminated. Round 2 votes counts: D=29 A=27 E=22 C=22 so E is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:213 C:212 B:184 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 14 0 B -16 0 -12 2 -6 C -2 12 0 16 -2 D -14 -2 -16 0 -18 E 0 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997744 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 14 0 B -16 0 -12 2 -6 C -2 12 0 16 -2 D -14 -2 -16 0 -18 E 0 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997744 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 14 0 B -16 0 -12 2 -6 C -2 12 0 16 -2 D -14 -2 -16 0 -18 E 0 6 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999997744 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4365: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C A D B E (7) C D A B E (4) B E A D C (4) A D B C E (4) E D B C A (3) E B D C A (3) E B C A D (3) E B A D C (3) C E D B A (3) C D A E B (3) C A E B D (3) C A B E D (3) E C B D A (2) E C B A D (2) E B C D A (2) D E B C A (2) D E B A C (2) D B E A C (2) D A B E C (2) D A B C E (2) C E A B D (2) C A D E B (2) C A B D E (2) B A E D C (2) B A E C D (2) A D C B E (2) A D B E C (2) A C D B E (2) A B D E C (2) E D B A C (1) E B A C D (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A E B (1) D C A B E (1) D B A E C (1) D A C B E (1) C E A D B (1) B E D A C (1) B A D E C (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -4 6 4 B 0 0 10 -2 0 C 4 -10 0 -8 -8 D -6 2 8 0 -4 E -4 0 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.537803 B: 0.462197 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50285811018 Cumulative probabilities = A: 0.537803 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 6 4 B 0 0 10 -2 0 C 4 -10 0 -8 -8 D -6 2 8 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=28 D=17 A=15 B=10 so B is eliminated. Round 2 votes counts: E=33 C=30 A=20 D=17 so D is eliminated. Round 3 votes counts: E=40 C=34 A=26 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:204 E:204 A:203 D:200 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 6 4 B 0 0 10 -2 0 C 4 -10 0 -8 -8 D -6 2 8 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 6 4 B 0 0 10 -2 0 C 4 -10 0 -8 -8 D -6 2 8 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 6 4 B 0 0 10 -2 0 C 4 -10 0 -8 -8 D -6 2 8 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4366: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A C B E D (8) A E C B D (7) A C B D E (6) E D A B C (5) D E B C A (5) D B E C A (5) D B C E A (5) D B C A E (5) C A B D E (5) B C D A E (5) E D B A C (4) E A C D B (4) E A C B D (4) A C E B D (3) E A D C B (2) C B A D E (2) B D C E A (2) A E D C B (2) E B C A D (1) E A D B C (1) E A B C D (1) D E B A C (1) D C B A E (1) C B D A E (1) C A D B E (1) B E C D A (1) B D E C A (1) B D C A E (1) B C E D A (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 0 -2 -2 B -2 0 4 -2 2 C 0 -4 0 4 -4 D 2 2 -4 0 -4 E 2 -2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.149170 B: 0.425415 C: 0.000000 D: 0.138122 E: 0.287293 Sum of squares = 0.304844405312 Cumulative probabilities = A: 0.149170 B: 0.574585 C: 0.574585 D: 0.712707 E: 1.000000 A B C D E A 0 2 0 -2 -2 B -2 0 4 -2 2 C 0 -4 0 4 -4 D 2 2 -4 0 -4 E 2 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.299999999999 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=28 D=22 B=11 C=9 so C is eliminated. Round 2 votes counts: A=34 E=30 D=22 B=14 so B is eliminated. Round 3 votes counts: A=36 E=32 D=32 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:204 B:201 A:199 C:198 D:198 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 -2 -2 B -2 0 4 -2 2 C 0 -4 0 4 -4 D 2 2 -4 0 -4 E 2 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.299999999999 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 -2 B -2 0 4 -2 2 C 0 -4 0 4 -4 D 2 2 -4 0 -4 E 2 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.299999999999 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 -2 B -2 0 4 -2 2 C 0 -4 0 4 -4 D 2 2 -4 0 -4 E 2 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.100000 E: 0.300000 Sum of squares = 0.299999999999 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4367: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) D B C E A (6) A E C B D (5) D E B A C (4) D B E A C (4) C D B A E (4) B E D C A (4) E A B C D (3) D C B A E (3) C A E B D (3) B D E C A (3) B C D E A (3) A E B D C (3) A C D B E (3) E C B A D (2) E B D C A (2) E B C A D (2) E A D B C (2) E A B D C (2) D E A B C (2) D C B E A (2) D A E B C (2) C B E A D (2) B C E D A (2) A E B C D (2) A D E C B (2) A C E B D (2) A C D E B (2) E D B C A (1) E B C D A (1) E B A C D (1) E A C B D (1) D E B C A (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C B E (1) C E B A D (1) C B D E A (1) C A B E D (1) A E D C B (1) A E D B C (1) A E C D B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 -4 -12 -18 B 16 0 20 -8 0 C 4 -20 0 -14 -20 D 12 8 14 0 8 E 18 0 20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -12 -18 B 16 0 20 -8 0 C 4 -20 0 -14 -20 D 12 8 14 0 8 E 18 0 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=24 E=17 C=12 B=12 so C is eliminated. Round 2 votes counts: D=39 A=28 E=18 B=15 so B is eliminated. Round 3 votes counts: D=46 A=28 E=26 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:215 B:214 A:175 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -4 -12 -18 B 16 0 20 -8 0 C 4 -20 0 -14 -20 D 12 8 14 0 8 E 18 0 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -12 -18 B 16 0 20 -8 0 C 4 -20 0 -14 -20 D 12 8 14 0 8 E 18 0 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -12 -18 B 16 0 20 -8 0 C 4 -20 0 -14 -20 D 12 8 14 0 8 E 18 0 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4368: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (16) A D C B E (9) E B D C A (8) B E D A C (7) D A C E B (6) B E A C D (5) C D A E B (4) B E C D A (4) A C D B E (4) B A E C D (3) E B D A C (2) E B C D A (2) C E B A D (2) B E D C A (2) B D E A C (2) B A C E D (2) A D C E B (2) E D B C A (1) D E B A C (1) D E A B C (1) D C A E B (1) D B E A C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A D B (1) C A E B D (1) C A D B E (1) C A B E D (1) B E C A D (1) B E A D C (1) A D B E C (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 6 12 B -8 0 -4 -10 -6 C -4 4 0 4 12 D -6 10 -4 0 8 E -12 6 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 6 12 B -8 0 -4 -10 -6 C -4 4 0 4 12 D -6 10 -4 0 8 E -12 6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=27 A=19 E=13 D=12 so D is eliminated. Round 2 votes counts: C=30 B=28 A=27 E=15 so E is eliminated. Round 3 votes counts: B=42 C=30 A=28 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:215 C:208 D:204 E:187 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 6 12 B -8 0 -4 -10 -6 C -4 4 0 4 12 D -6 10 -4 0 8 E -12 6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 6 12 B -8 0 -4 -10 -6 C -4 4 0 4 12 D -6 10 -4 0 8 E -12 6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 6 12 B -8 0 -4 -10 -6 C -4 4 0 4 12 D -6 10 -4 0 8 E -12 6 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4369: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (6) C E B D A (6) C E B A D (6) C A D E B (6) D A C E B (5) D A B E C (5) A D C B E (5) A D B E C (5) A C B E D (5) A B E D C (4) E B D C A (3) D E B C A (3) D C A E B (3) C D A E B (3) B E C A D (3) B E A C D (3) D E C B A (2) D E B A C (2) C E A B D (2) B E D A C (2) B E A D C (2) A C D E B (2) A B E C D (2) A B D E C (2) E D B C A (1) E B C A D (1) D C E B A (1) D B E A C (1) D B A E C (1) D A E B C (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E B A (1) C A E B D (1) B E D C A (1) A C D B E (1) Total count = 100 A B C D E A 0 8 0 0 6 B -8 0 -2 -2 -12 C 0 2 0 -2 0 D 0 2 2 0 2 E -6 12 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.407876 B: 0.000000 C: 0.000000 D: 0.592124 E: 0.000000 Sum of squares = 0.51697358073 Cumulative probabilities = A: 0.407876 B: 0.407876 C: 0.407876 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 0 6 B -8 0 -2 -2 -12 C 0 2 0 -2 0 D 0 2 2 0 2 E -6 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 A=26 E=11 B=11 so E is eliminated. Round 2 votes counts: D=27 C=26 A=26 B=21 so B is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:203 E:202 C:200 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 0 6 B -8 0 -2 -2 -12 C 0 2 0 -2 0 D 0 2 2 0 2 E -6 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 0 6 B -8 0 -2 -2 -12 C 0 2 0 -2 0 D 0 2 2 0 2 E -6 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 0 6 B -8 0 -2 -2 -12 C 0 2 0 -2 0 D 0 2 2 0 2 E -6 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4370: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (9) B C D E A (8) E C A B D (7) D A E B C (7) A D E C B (6) E A C D B (5) D B A C E (5) B D C A E (5) B C E D A (5) C E B A D (4) C B E A D (4) B C E A D (4) D B A E C (3) D A E C B (3) D A B E C (3) E C B A D (2) E A D B C (2) C E A B D (2) C B D A E (2) C A E D B (2) B D C E A (2) A E C D B (2) E B C A D (1) E A D C B (1) D C A B E (1) D A B C E (1) C E A D B (1) C D A B E (1) B E C D A (1) A E D C B (1) Total count = 100 A B C D E A 0 -10 -22 -12 2 B 10 0 12 12 8 C 22 -12 0 20 12 D 12 -12 -20 0 12 E -2 -8 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 -12 2 B 10 0 12 12 8 C 22 -12 0 20 12 D 12 -12 -20 0 12 E -2 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997432 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=23 E=18 C=16 A=9 so A is eliminated. Round 2 votes counts: B=34 D=29 E=21 C=16 so C is eliminated. Round 3 votes counts: B=40 E=30 D=30 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:221 D:196 E:183 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -22 -12 2 B 10 0 12 12 8 C 22 -12 0 20 12 D 12 -12 -20 0 12 E -2 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997432 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 -12 2 B 10 0 12 12 8 C 22 -12 0 20 12 D 12 -12 -20 0 12 E -2 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997432 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 -12 2 B 10 0 12 12 8 C 22 -12 0 20 12 D 12 -12 -20 0 12 E -2 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997432 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4371: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) D B E A C (6) D A B E C (6) B E A D C (6) E B A D C (5) D C B E A (5) D B A E C (5) A E B D C (5) E B A C D (4) C D E B A (4) B E D A C (4) C A E D B (3) C A E B D (3) C A D E B (3) B D E A C (3) A E B C D (3) E C B A D (2) E A B C D (2) D C B A E (2) D B C E A (2) D A C B E (2) C E A D B (2) C D B E A (2) C D A E B (2) E C A B D (1) E B C A D (1) E A C B D (1) D C A B E (1) D B C A E (1) D B A C E (1) C E B D A (1) C E B A D (1) C D E A B (1) C D B A E (1) C D A B E (1) A E C B D (1) Total count = 100 A B C D E A 0 -12 8 0 -20 B 12 0 8 0 -4 C -8 -8 0 -8 -10 D 0 0 8 0 -4 E 20 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 8 0 -20 B 12 0 8 0 -4 C -8 -8 0 -8 -10 D 0 0 8 0 -4 E 20 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=31 C=31 E=16 B=13 A=9 so A is eliminated. Round 2 votes counts: D=31 C=31 E=25 B=13 so B is eliminated. Round 3 votes counts: E=35 D=34 C=31 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:208 D:202 A:188 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 8 0 -20 B 12 0 8 0 -4 C -8 -8 0 -8 -10 D 0 0 8 0 -4 E 20 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 8 0 -20 B 12 0 8 0 -4 C -8 -8 0 -8 -10 D 0 0 8 0 -4 E 20 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 8 0 -20 B 12 0 8 0 -4 C -8 -8 0 -8 -10 D 0 0 8 0 -4 E 20 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4372: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (12) E A C B D (8) C A B D E (8) A B C D E (7) D B A C E (6) E C A B D (5) D C B A E (5) D B C A E (4) A C B D E (4) E D C B A (3) E D C A B (3) E C D A B (3) D B A E C (3) B D A C E (3) A B C E D (3) E D B A C (2) E A B D C (2) C E A B D (2) C D B A E (2) E D A B C (1) E C A D B (1) E A D B C (1) D E B C A (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E A C (1) C B D A E (1) C A B E D (1) B C A D E (1) B A D E C (1) B A D C E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 26 16 18 4 B -26 0 2 22 6 C -16 -2 0 18 4 D -18 -22 -18 0 4 E -4 -6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 16 18 4 B -26 0 2 22 6 C -16 -2 0 18 4 D -18 -22 -18 0 4 E -4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 D=23 A=16 C=14 B=6 so B is eliminated. Round 2 votes counts: E=41 D=26 A=18 C=15 so C is eliminated. Round 3 votes counts: E=43 D=29 A=28 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:232 B:202 C:202 E:191 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 16 18 4 B -26 0 2 22 6 C -16 -2 0 18 4 D -18 -22 -18 0 4 E -4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 16 18 4 B -26 0 2 22 6 C -16 -2 0 18 4 D -18 -22 -18 0 4 E -4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 16 18 4 B -26 0 2 22 6 C -16 -2 0 18 4 D -18 -22 -18 0 4 E -4 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4373: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (13) D C E A B (12) E D C B A (9) B E D C A (9) A C D B E (9) B E A D C (8) B A E C D (7) E D B C A (6) E B D C A (6) A B C D E (6) A C D E B (5) D E C B A (2) B A E D C (2) C D E A B (1) C A D E B (1) B E D A C (1) B C D A E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -20 -20 -8 B 4 0 -4 -16 -10 C 20 4 0 -10 -2 D 20 16 10 0 0 E 8 10 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.201526 E: 0.798474 Sum of squares = 0.678173412858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.201526 E: 1.000000 A B C D E A 0 -4 -20 -20 -8 B 4 0 -4 -16 -10 C 20 4 0 -10 -2 D 20 16 10 0 0 E 8 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=21 A=21 C=15 D=14 so D is eliminated. Round 2 votes counts: B=29 C=27 E=23 A=21 so A is eliminated. Round 3 votes counts: C=41 B=36 E=23 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:223 E:210 C:206 B:187 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -20 -20 -8 B 4 0 -4 -16 -10 C 20 4 0 -10 -2 D 20 16 10 0 0 E 8 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -20 -20 -8 B 4 0 -4 -16 -10 C 20 4 0 -10 -2 D 20 16 10 0 0 E 8 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -20 -20 -8 B 4 0 -4 -16 -10 C 20 4 0 -10 -2 D 20 16 10 0 0 E 8 10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4374: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E A B D C (6) B A E C D (6) B A D C E (6) E A B C D (5) D E C A B (5) C D B A E (5) A B E D C (5) C D E B A (4) C B D A E (4) D A E B C (3) B C A D E (3) A B D E C (3) E C D A B (2) E A C B D (2) D E A C B (2) D C B A E (2) D C A B E (2) B A C D E (2) A B D C E (2) E D C A B (1) E D A C B (1) E D A B C (1) E C B A D (1) E B A C D (1) E A D C B (1) E A D B C (1) D E A B C (1) D C E B A (1) D A B C E (1) C E D B A (1) C E D A B (1) C D B E A (1) C B E A D (1) C B A E D (1) C B A D E (1) B C A E D (1) B A E D C (1) A E D B C (1) A E B D C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 8 6 4 B -16 0 4 6 -2 C -8 -4 0 -14 -4 D -6 -6 14 0 16 E -4 2 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 6 4 B -16 0 4 6 -2 C -8 -4 0 -14 -4 D -6 -6 14 0 16 E -4 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=22 C=19 B=19 A=14 so A is eliminated. Round 2 votes counts: B=30 D=27 E=24 C=19 so C is eliminated. Round 3 votes counts: D=37 B=37 E=26 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:217 D:209 B:196 E:193 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 6 4 B -16 0 4 6 -2 C -8 -4 0 -14 -4 D -6 -6 14 0 16 E -4 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 6 4 B -16 0 4 6 -2 C -8 -4 0 -14 -4 D -6 -6 14 0 16 E -4 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 6 4 B -16 0 4 6 -2 C -8 -4 0 -14 -4 D -6 -6 14 0 16 E -4 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4375: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (6) A E B C D (6) B A E D C (5) D B C A E (4) D B A C E (4) C E A D B (4) A B E D C (4) E C A B D (3) E A C B D (3) D C B E A (3) D C B A E (3) D C A B E (3) D A B C E (3) C D E B A (3) B D C E A (3) B A D E C (3) A B D E C (3) E C B A D (2) E B C A D (2) E A B C D (2) D C E B A (2) D C E A B (2) D A C E B (2) C E D B A (2) C E D A B (2) B E A D C (2) B D A E C (2) B A E C D (2) A D B E C (2) A C E D B (2) E B A C D (1) D C A E B (1) D B C E A (1) C E B A D (1) C D A E B (1) B E D A C (1) B E C A D (1) B D A C E (1) B C E D A (1) A E C D B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 0 0 6 B -2 0 8 -2 4 C 0 -8 0 -8 8 D 0 2 8 0 4 E -6 -4 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.420899 B: 0.000000 C: 0.000000 D: 0.579101 E: 0.000000 Sum of squares = 0.512513830707 Cumulative probabilities = A: 0.420899 B: 0.420899 C: 0.420899 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 6 B -2 0 8 -2 4 C 0 -8 0 -8 8 D 0 2 8 0 4 E -6 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=21 C=19 A=19 E=13 so E is eliminated. Round 2 votes counts: D=28 C=24 B=24 A=24 so C is eliminated. Round 3 votes counts: D=42 A=31 B=27 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:207 A:204 B:204 C:196 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 0 6 B -2 0 8 -2 4 C 0 -8 0 -8 8 D 0 2 8 0 4 E -6 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 6 B -2 0 8 -2 4 C 0 -8 0 -8 8 D 0 2 8 0 4 E -6 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 6 B -2 0 8 -2 4 C 0 -8 0 -8 8 D 0 2 8 0 4 E -6 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4376: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) C A E B D (9) C A D E B (9) D A C B E (8) D B E A C (7) D B A E C (6) B E D A C (6) A C D E B (6) B E C A D (5) D A B C E (4) B D E A C (4) E B D C A (3) E B C D A (3) C E A B D (3) B E D C A (3) A D C B E (3) E C B A D (2) D C A E B (2) C A E D B (2) E C B D A (1) D E B C A (1) C D A E B (1) B E A C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -6 2 4 B 2 0 6 2 -2 C 6 -6 0 6 -2 D -2 -2 -6 0 2 E -4 2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 -6 2 4 B 2 0 6 2 -2 C 6 -6 0 6 -2 D -2 -2 -6 0 2 E -4 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999996 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 B=19 E=18 A=11 so A is eliminated. Round 2 votes counts: D=31 C=31 B=20 E=18 so E is eliminated. Round 3 votes counts: B=35 C=34 D=31 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:204 C:202 A:199 E:199 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 2 4 B 2 0 6 2 -2 C 6 -6 0 6 -2 D -2 -2 -6 0 2 E -4 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999996 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 4 B 2 0 6 2 -2 C 6 -6 0 6 -2 D -2 -2 -6 0 2 E -4 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999996 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 4 B 2 0 6 2 -2 C 6 -6 0 6 -2 D -2 -2 -6 0 2 E -4 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999996 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4377: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) D A B C E (8) A D C B E (6) D E A B C (5) E B D C A (4) E B D A C (4) E B C D A (4) D B A E C (4) C B E A D (4) E C A B D (3) D B A C E (3) C A B D E (3) B D C A E (3) B D A C E (3) A C D E B (3) E A D C B (2) D A E B C (2) C E A B D (2) C B A D E (2) C A E B D (2) C A D B E (2) B C D A E (2) A D C E B (2) A C D B E (2) E D B A C (1) E C A D B (1) E A D B C (1) E A C D B (1) D A C B E (1) D A B E C (1) C E B A D (1) C B D A E (1) B E C D A (1) B C E D A (1) A E D C B (1) A E D B C (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 4 8 B -2 0 -4 2 -6 C -6 4 0 -4 4 D -4 -2 4 0 6 E -8 6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 4 8 B -2 0 -4 2 -6 C -6 4 0 -4 4 D -4 -2 4 0 6 E -8 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=24 C=17 A=17 B=10 so B is eliminated. Round 2 votes counts: E=33 D=30 C=20 A=17 so A is eliminated. Round 3 votes counts: D=38 E=36 C=26 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:202 C:199 B:195 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 8 B -2 0 -4 2 -6 C -6 4 0 -4 4 D -4 -2 4 0 6 E -8 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 8 B -2 0 -4 2 -6 C -6 4 0 -4 4 D -4 -2 4 0 6 E -8 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 8 B -2 0 -4 2 -6 C -6 4 0 -4 4 D -4 -2 4 0 6 E -8 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4378: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) D B A E C (9) B D A C E (7) B C E D A (7) D A B E C (6) A E D C B (6) A D E C B (6) B D C E A (5) B D C A E (5) E A C D B (4) C B E D A (4) B C D E A (4) C E B A D (3) C E A B D (3) C B E A D (3) E C A B D (2) A E C D B (2) A D E B C (2) E D C A B (1) E C D A B (1) D E A C B (1) D B E C A (1) D B A C E (1) D A B C E (1) C E A D B (1) B C E A D (1) B A D C E (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -4 -6 -4 B 2 0 2 -8 12 C 4 -2 0 -6 -6 D 6 8 6 0 0 E 4 -12 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.726545 E: 0.273455 Sum of squares = 0.602645473033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.726545 E: 1.000000 A B C D E A 0 -2 -4 -6 -4 B 2 0 2 -8 12 C 4 -2 0 -6 -6 D 6 8 6 0 0 E 4 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000001338 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=19 D=19 A=18 C=14 so C is eliminated. Round 2 votes counts: B=37 E=26 D=19 A=18 so A is eliminated. Round 3 votes counts: B=37 E=35 D=28 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:210 B:204 E:199 C:195 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 -4 B 2 0 2 -8 12 C 4 -2 0 -6 -6 D 6 8 6 0 0 E 4 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000001338 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 -4 B 2 0 2 -8 12 C 4 -2 0 -6 -6 D 6 8 6 0 0 E 4 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000001338 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 -4 B 2 0 2 -8 12 C 4 -2 0 -6 -6 D 6 8 6 0 0 E 4 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000001338 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4379: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) C E A D B (7) B D C A E (7) A E D C B (6) D B C A E (5) B C D E A (5) A E D B C (5) E B A D C (4) E A C B D (4) E A B D C (4) D B A E C (4) C E A B D (4) C B E A D (3) C B D E A (3) B D E A C (3) E C A B D (2) E B A C D (2) E A D B C (2) E A B C D (2) C D B A E (2) C B D A E (2) C A E D B (2) B E D A C (2) A D E B C (2) D B A C E (1) D A B E C (1) C E B A D (1) C B E D A (1) B E C A D (1) B D C E A (1) B D A E C (1) B D A C E (1) B C D A E (1) A E C D B (1) Total count = 100 A B C D E A 0 0 6 20 -18 B 0 0 8 8 -12 C -6 -8 0 2 -8 D -20 -8 -2 0 -22 E 18 12 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 6 20 -18 B 0 0 8 8 -12 C -6 -8 0 2 -8 D -20 -8 -2 0 -22 E 18 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=25 B=22 A=14 D=11 so D is eliminated. Round 2 votes counts: B=32 E=28 C=25 A=15 so A is eliminated. Round 3 votes counts: E=42 B=33 C=25 so C is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 A:204 B:202 C:190 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 20 -18 B 0 0 8 8 -12 C -6 -8 0 2 -8 D -20 -8 -2 0 -22 E 18 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 20 -18 B 0 0 8 8 -12 C -6 -8 0 2 -8 D -20 -8 -2 0 -22 E 18 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 20 -18 B 0 0 8 8 -12 C -6 -8 0 2 -8 D -20 -8 -2 0 -22 E 18 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4380: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) B D E C A (9) A C D B E (7) D B E C A (6) B E D C A (6) A C E B D (6) C E D B A (4) A C E D B (4) E D B C A (3) C D E B A (3) C D B E A (3) C A D B E (3) B E D A C (3) B D E A C (3) A C D E B (3) A B E D C (3) D E B C A (2) D C B E A (2) C E A D B (2) A E B D C (2) A E B C D (2) A B E C D (2) A B D E C (2) E C B A D (1) E B D A C (1) C E D A B (1) C D A B E (1) C A E D B (1) C A D E B (1) B E A D C (1) A D C B E (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 -12 -18 B 12 0 10 6 10 C 14 -10 0 -8 -10 D 12 -6 8 0 -4 E 18 -10 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -12 -18 B 12 0 10 6 10 C 14 -10 0 -8 -10 D 12 -6 8 0 -4 E 18 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=22 C=19 E=14 D=10 so D is eliminated. Round 2 votes counts: A=35 B=28 C=21 E=16 so E is eliminated. Round 3 votes counts: B=43 A=35 C=22 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:211 D:205 C:193 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 -12 -18 B 12 0 10 6 10 C 14 -10 0 -8 -10 D 12 -6 8 0 -4 E 18 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -12 -18 B 12 0 10 6 10 C 14 -10 0 -8 -10 D 12 -6 8 0 -4 E 18 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -12 -18 B 12 0 10 6 10 C 14 -10 0 -8 -10 D 12 -6 8 0 -4 E 18 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4381: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) C A B D E (7) C B A E D (6) B A C D E (6) E D C A B (5) D A B E C (5) A B D E C (5) A B D C E (5) A B C D E (5) D E A B C (4) B A D E C (4) D E C A B (3) C E D B A (3) C B E A D (3) B A E D C (3) A D B E C (3) E D C B A (2) E C D A B (2) D C E A B (2) C D E A B (2) C A D B E (2) B A D C E (2) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E C B A D (1) E B D A C (1) E B C A D (1) E B A D C (1) E B A C D (1) D A B C E (1) C E D A B (1) C D A B E (1) C B A D E (1) B A C E D (1) Total count = 100 A B C D E A 0 6 -4 28 14 B -6 0 0 22 20 C 4 0 0 4 10 D -28 -22 -4 0 16 E -14 -20 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.292134 C: 0.707866 D: 0.000000 E: 0.000000 Sum of squares = 0.586416664087 Cumulative probabilities = A: 0.000000 B: 0.292134 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 28 14 B -6 0 0 22 20 C 4 0 0 4 10 D -28 -22 -4 0 16 E -14 -20 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.399997 C: 0.600003 D: 0.000000 E: 0.000000 Sum of squares = 0.520001170478 Cumulative probabilities = A: 0.000000 B: 0.399997 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=18 A=18 B=16 D=15 so D is eliminated. Round 2 votes counts: C=35 E=25 A=24 B=16 so B is eliminated. Round 3 votes counts: A=40 C=35 E=25 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:222 B:218 C:209 D:181 E:170 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 28 14 B -6 0 0 22 20 C 4 0 0 4 10 D -28 -22 -4 0 16 E -14 -20 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.399997 C: 0.600003 D: 0.000000 E: 0.000000 Sum of squares = 0.520001170478 Cumulative probabilities = A: 0.000000 B: 0.399997 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 28 14 B -6 0 0 22 20 C 4 0 0 4 10 D -28 -22 -4 0 16 E -14 -20 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.399997 C: 0.600003 D: 0.000000 E: 0.000000 Sum of squares = 0.520001170478 Cumulative probabilities = A: 0.000000 B: 0.399997 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 28 14 B -6 0 0 22 20 C 4 0 0 4 10 D -28 -22 -4 0 16 E -14 -20 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.399997 C: 0.600003 D: 0.000000 E: 0.000000 Sum of squares = 0.520001170478 Cumulative probabilities = A: 0.000000 B: 0.399997 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4382: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) C D E B A (4) C A D B E (4) A E B D C (4) A B C E D (4) E D A B C (3) D E B C A (3) D A E C B (3) D A C E B (3) C B A D E (3) C A B D E (3) B C D E A (3) B A E C D (3) A B E D C (3) A B E C D (3) E B D A C (2) D E C B A (2) D E A C B (2) D C E B A (2) C D B E A (2) C D A E B (2) C A D E B (2) B E D C A (2) B E D A C (2) B E A D C (2) B D E C A (2) B C A E D (2) A E D B C (2) A D E C B (2) A C E B D (2) D E B A C (1) D C B E A (1) D C A E B (1) C D E A B (1) C D B A E (1) C B D E A (1) C B A E D (1) C A B E D (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C E A D (1) A E B C D (1) A C E D B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 12 6 -4 6 B -12 0 8 0 -2 C -6 -8 0 2 -4 D 4 0 -2 0 12 E -6 2 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.231835 C: 0.000000 D: 0.768165 E: 0.000000 Sum of squares = 0.643825421825 Cumulative probabilities = A: 0.000000 B: 0.231835 C: 0.231835 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -4 6 B -12 0 8 0 -2 C -6 -8 0 2 -4 D 4 0 -2 0 12 E -6 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000014108 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 A=24 B=21 E=5 so E is eliminated. Round 2 votes counts: D=28 C=25 A=24 B=23 so B is eliminated. Round 3 votes counts: D=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:210 D:207 B:197 E:194 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -4 6 B -12 0 8 0 -2 C -6 -8 0 2 -4 D 4 0 -2 0 12 E -6 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000014108 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 6 B -12 0 8 0 -2 C -6 -8 0 2 -4 D 4 0 -2 0 12 E -6 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000014108 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 6 B -12 0 8 0 -2 C -6 -8 0 2 -4 D 4 0 -2 0 12 E -6 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000014108 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4383: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C B E D A (7) A D E B C (7) B C E A D (6) A D B C E (6) B C A E D (5) E D A C B (4) C B E A D (4) B C A D E (4) A B C D E (4) E D C A B (3) D A E C B (3) C B A D E (3) B A C D E (3) A D B E C (3) E D C B A (2) E D B C A (2) E C D B A (2) E C B D A (2) E B C D A (2) C E B D A (2) B C E D A (2) A D C B E (2) A B D C E (2) E B A C D (1) E A B D C (1) D E A C B (1) D C A B E (1) D A E B C (1) D A C E B (1) C B D A E (1) B E C A D (1) B A C E D (1) A D C E B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 2 12 0 B 0 0 20 4 14 C -2 -20 0 2 12 D -12 -4 -2 0 -10 E 0 -14 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.501498 B: 0.498502 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500004486874 Cumulative probabilities = A: 0.501498 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 12 0 B 0 0 20 4 14 C -2 -20 0 2 12 D -12 -4 -2 0 -10 E 0 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 B=22 C=17 D=7 so D is eliminated. Round 2 votes counts: A=32 E=28 B=22 C=18 so C is eliminated. Round 3 votes counts: B=37 A=33 E=30 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:219 A:207 C:196 E:192 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 12 0 B 0 0 20 4 14 C -2 -20 0 2 12 D -12 -4 -2 0 -10 E 0 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 12 0 B 0 0 20 4 14 C -2 -20 0 2 12 D -12 -4 -2 0 -10 E 0 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 12 0 B 0 0 20 4 14 C -2 -20 0 2 12 D -12 -4 -2 0 -10 E 0 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4384: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) B A D C E (9) E C D A B (7) C E D A B (7) B A D E C (6) D E B C A (5) C E A B D (5) A B D C E (5) A B C E D (5) E D C B A (4) D B A E C (4) D E C B A (3) D E B A C (3) D A B C E (3) C A E B D (3) C A B E D (3) E C D B A (2) D B A C E (2) B D A E C (2) A C B E D (2) A B C D E (2) E C A B D (1) D C A E B (1) C E A D B (1) C A E D B (1) C A D E B (1) B E D A C (1) B A C E D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 12 -6 4 B 2 0 16 -8 12 C -12 -16 0 -16 6 D 6 8 16 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 -6 4 B 2 0 16 -8 12 C -12 -16 0 -16 6 D 6 8 16 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=21 B=19 A=16 E=14 so E is eliminated. Round 2 votes counts: D=34 C=31 B=19 A=16 so A is eliminated. Round 3 votes counts: D=35 C=34 B=31 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:211 A:204 E:182 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 -6 4 B 2 0 16 -8 12 C -12 -16 0 -16 6 D 6 8 16 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -6 4 B 2 0 16 -8 12 C -12 -16 0 -16 6 D 6 8 16 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -6 4 B 2 0 16 -8 12 C -12 -16 0 -16 6 D 6 8 16 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4385: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) E C A B D (7) D B E A C (7) A C B E D (6) D E B C A (5) D B E C A (5) C E A B D (5) D E C A B (4) C A E B D (4) A C E B D (4) E C B A D (3) E B D C A (3) D A B C E (3) B D A C E (3) A C B D E (3) E D B C A (2) D E A C B (2) D A C E B (2) C A B E D (2) B E D C A (2) B C A E D (2) A B D C E (2) E D C B A (1) E D C A B (1) E C D A B (1) E C A D B (1) D E C B A (1) D B A E C (1) D A B E C (1) B E C A D (1) B D E C A (1) B D E A C (1) B D A E C (1) B A D C E (1) B A C E D (1) A D C B E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 -10 -6 B -2 0 4 6 8 C 2 -4 0 -18 -2 D 10 -6 18 0 8 E 6 -8 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765429 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -10 -6 B -2 0 4 6 8 C 2 -4 0 -18 -2 D 10 -6 18 0 8 E 6 -8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=19 A=18 B=13 C=11 so C is eliminated. Round 2 votes counts: D=39 E=24 A=24 B=13 so B is eliminated. Round 3 votes counts: D=45 A=28 E=27 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:208 E:196 A:192 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 -10 -6 B -2 0 4 6 8 C 2 -4 0 -18 -2 D 10 -6 18 0 8 E 6 -8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -10 -6 B -2 0 4 6 8 C 2 -4 0 -18 -2 D 10 -6 18 0 8 E 6 -8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -10 -6 B -2 0 4 6 8 C 2 -4 0 -18 -2 D 10 -6 18 0 8 E 6 -8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4386: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) B C E D A (8) B E C D A (7) A D E C B (6) E C B D A (5) C E A B D (5) A C E D B (5) D A B C E (4) A D B C E (4) A C E B D (4) E C A B D (3) D B E C A (3) D B A E C (3) D A E B C (3) B D E C A (3) A D C B E (3) D E A C B (2) D A B E C (2) C E B D A (2) B E D C A (2) B D C E A (2) A D C E B (2) E D C B A (1) E A C D B (1) D B E A C (1) D B A C E (1) C B E D A (1) C B A E D (1) C A E B D (1) B D C A E (1) B C E A D (1) B C D A E (1) B C A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 -10 -16 -4 -14 B 10 0 -4 18 0 C 16 4 0 12 16 D 4 -18 -12 0 -16 E 14 0 -16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -4 -14 B 10 0 -4 18 0 C 16 4 0 12 16 D 4 -18 -12 0 -16 E 14 0 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 C=20 D=19 E=10 so E is eliminated. Round 2 votes counts: C=28 B=27 A=25 D=20 so D is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:224 B:212 E:207 D:179 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -16 -4 -14 B 10 0 -4 18 0 C 16 4 0 12 16 D 4 -18 -12 0 -16 E 14 0 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -4 -14 B 10 0 -4 18 0 C 16 4 0 12 16 D 4 -18 -12 0 -16 E 14 0 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -4 -14 B 10 0 -4 18 0 C 16 4 0 12 16 D 4 -18 -12 0 -16 E 14 0 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4387: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) A C D B E (10) B A C D E (9) B E C D A (5) B C A D E (5) A B C D E (5) E D C A B (4) E B D C A (4) E A D C B (4) A E D C B (4) E D A C B (3) D C E B A (3) C D B E A (3) B C D A E (3) A C D E B (3) E D C B A (2) E D B C A (2) E A B D C (2) D C E A B (2) D C A E B (2) C D B A E (2) B C D E A (2) A D C E B (2) A C B D E (2) E A D B C (1) C A D B E (1) B E A C D (1) B D C E A (1) B C A E D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -4 4 2 B 6 0 4 2 22 C 4 -4 0 8 14 D -4 -2 -8 0 10 E -2 -22 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 4 2 B 6 0 4 2 22 C 4 -4 0 8 14 D -4 -2 -8 0 10 E -2 -22 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973793 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=28 E=22 D=7 C=6 so C is eliminated. Round 2 votes counts: B=37 A=29 E=22 D=12 so D is eliminated. Round 3 votes counts: B=42 A=31 E=27 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:211 A:198 D:198 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 4 2 B 6 0 4 2 22 C 4 -4 0 8 14 D -4 -2 -8 0 10 E -2 -22 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973793 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 4 2 B 6 0 4 2 22 C 4 -4 0 8 14 D -4 -2 -8 0 10 E -2 -22 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973793 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 4 2 B 6 0 4 2 22 C 4 -4 0 8 14 D -4 -2 -8 0 10 E -2 -22 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973793 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4388: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) E D A B C (10) C B A D E (9) D C E A B (7) B A C E D (7) E B A D C (6) D E C A B (6) B A C D E (6) A B E D C (6) A B C D E (6) E A B D C (3) D E A B C (3) C D B A E (3) E C D B A (2) B A E C D (2) A B C E D (2) E D C A B (1) D E C B A (1) D E A C B (1) D A B E C (1) C D E A B (1) C D A B E (1) C B A E D (1) C A B D E (1) B C A E D (1) B A E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 10 4 -4 B 0 0 10 4 -4 C -10 -10 0 6 12 D -4 -4 -6 0 16 E 4 4 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.277777785989 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 0 10 4 -4 B 0 0 10 4 -4 C -10 -10 0 6 12 D -4 -4 -6 0 16 E 4 4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.277777777776 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 D=19 B=17 A=15 so A is eliminated. Round 2 votes counts: B=32 C=27 E=22 D=19 so D is eliminated. Round 3 votes counts: C=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:205 B:205 D:201 C:199 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 4 -4 B 0 0 10 4 -4 C -10 -10 0 6 12 D -4 -4 -6 0 16 E 4 4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.277777777776 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 4 -4 B 0 0 10 4 -4 C -10 -10 0 6 12 D -4 -4 -6 0 16 E 4 4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.277777777776 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 4 -4 B 0 0 10 4 -4 C -10 -10 0 6 12 D -4 -4 -6 0 16 E 4 4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.277777777776 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4389: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) B E D A C (8) D E A C B (6) B D E C A (6) A E D C B (6) A C E D B (6) C B A D E (5) C A B D E (4) B E D C A (4) B C D E A (4) E D A B C (3) E A D B C (3) D E B A C (3) C A D E B (3) C A B E D (3) B C A E D (3) A C B E D (3) E D A C B (2) D E C A B (2) C B D A E (2) B D C E A (2) E A B D C (1) D C E A B (1) D B C E A (1) D A C E B (1) C D B A E (1) C B D E A (1) C A E D B (1) B D E A C (1) B C A D E (1) B A E C D (1) B A C E D (1) A E D B C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 12 -12 -12 B 4 0 2 2 2 C -12 -2 0 -18 -10 D 12 -2 18 0 -10 E 12 -2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 -12 -12 B 4 0 2 2 2 C -12 -2 0 -18 -10 D 12 -2 18 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=20 A=18 E=17 D=14 so D is eliminated. Round 2 votes counts: B=32 E=28 C=21 A=19 so A is eliminated. Round 3 votes counts: E=35 C=33 B=32 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:209 B:205 A:192 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 -12 -12 B 4 0 2 2 2 C -12 -2 0 -18 -10 D 12 -2 18 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -12 -12 B 4 0 2 2 2 C -12 -2 0 -18 -10 D 12 -2 18 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -12 -12 B 4 0 2 2 2 C -12 -2 0 -18 -10 D 12 -2 18 0 -10 E 12 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4390: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) E C A D B (5) E C A B D (5) D B A E C (5) D A B E C (5) C D E A B (5) C E B A D (4) A E D C B (4) E A C D B (3) D C B A E (3) D A E B C (3) C E A D B (3) C B E A D (3) C B D E A (3) B C E A D (3) D C E A B (2) D B C A E (2) C D B E A (2) C B E D A (2) B D A C E (2) B C D E A (2) B A E D C (2) B A D E C (2) A D E B C (2) E D A C B (1) E C D A B (1) E C B A D (1) E A B C D (1) D E A C B (1) D A C E B (1) D A B C E (1) C E B D A (1) B E C A D (1) B E A C D (1) B D C A E (1) B D A E C (1) B C E D A (1) B C A E D (1) B A E C D (1) A E B D C (1) A E B C D (1) A D E C B (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -2 -4 -2 B -12 0 -16 -16 -6 C 2 16 0 2 -16 D 4 16 -2 0 6 E 2 6 16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.666667 E: 0.083333 Sum of squares = 0.513888888889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.916667 E: 1.000000 A B C D E A 0 12 -2 -4 -2 B -12 0 -16 -16 -6 C 2 16 0 2 -16 D 4 16 -2 0 6 E 2 6 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.666667 E: 0.083333 Sum of squares = 0.513888889087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.916667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=23 B=18 E=17 A=12 so A is eliminated. Round 2 votes counts: D=34 E=23 C=23 B=20 so B is eliminated. Round 3 votes counts: D=41 C=30 E=29 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:212 E:209 A:202 C:202 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -2 -4 -2 B -12 0 -16 -16 -6 C 2 16 0 2 -16 D 4 16 -2 0 6 E 2 6 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.666667 E: 0.083333 Sum of squares = 0.513888889087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.916667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -4 -2 B -12 0 -16 -16 -6 C 2 16 0 2 -16 D 4 16 -2 0 6 E 2 6 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.666667 E: 0.083333 Sum of squares = 0.513888889087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.916667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -4 -2 B -12 0 -16 -16 -6 C 2 16 0 2 -16 D 4 16 -2 0 6 E 2 6 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.666667 E: 0.083333 Sum of squares = 0.513888889087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.916667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4391: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (14) E A B C D (12) D C B A E (8) E A D B C (7) E A C B D (7) E A B D C (7) C B D A E (5) B C D E A (5) A D E C B (4) E A D C B (3) E A C D B (3) E B A C D (2) D A C E B (2) C D B A E (2) C B D E A (2) B E C D A (2) B E C A D (2) B D C A E (2) B C E D A (2) A E D B C (2) D C A B E (1) D B C A E (1) D A C B E (1) C D A B E (1) C B E A D (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 32 32 32 -10 B -32 0 -10 -2 -30 C -32 10 0 -6 -34 D -32 2 6 0 -30 E 10 30 34 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 32 32 32 -10 B -32 0 -10 -2 -30 C -32 10 0 -6 -34 D -32 2 6 0 -30 E 10 30 34 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 A=22 D=13 B=13 C=11 so C is eliminated. Round 2 votes counts: E=41 A=22 B=21 D=16 so D is eliminated. Round 3 votes counts: E=41 B=32 A=27 so A is eliminated. Round 4 votes counts: E=65 B=35 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:252 A:243 D:173 C:169 B:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 32 32 32 -10 B -32 0 -10 -2 -30 C -32 10 0 -6 -34 D -32 2 6 0 -30 E 10 30 34 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 32 32 32 -10 B -32 0 -10 -2 -30 C -32 10 0 -6 -34 D -32 2 6 0 -30 E 10 30 34 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 32 32 32 -10 B -32 0 -10 -2 -30 C -32 10 0 -6 -34 D -32 2 6 0 -30 E 10 30 34 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4392: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) C D B A E (10) A E D C B (9) A E C D B (9) B C D E A (8) D C B A E (7) E A B D C (5) E B A C D (4) B E D C A (4) B D C E A (4) C B D E A (3) A E D B C (3) D C A E B (2) D C A B E (2) C A E D B (2) B D E C A (2) A D C E B (2) A C D E B (2) E C A B D (1) E B A D C (1) E A C D B (1) D B C A E (1) D A C B E (1) C D B E A (1) C D A E B (1) B E C D A (1) B E A C D (1) B D C A E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 0 4 8 B -4 0 -6 -6 -8 C 0 6 0 10 -6 D -4 6 -10 0 -6 E -8 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.650964 B: 0.000000 C: 0.349036 D: 0.000000 E: 0.000000 Sum of squares = 0.545580179102 Cumulative probabilities = A: 0.650964 B: 0.650964 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 4 8 B -4 0 -6 -6 -8 C 0 6 0 10 -6 D -4 6 -10 0 -6 E -8 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 B=21 C=17 D=13 so D is eliminated. Round 2 votes counts: C=28 A=28 E=22 B=22 so E is eliminated. Round 3 votes counts: A=44 C=29 B=27 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:206 C:205 D:193 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 4 8 B -4 0 -6 -6 -8 C 0 6 0 10 -6 D -4 6 -10 0 -6 E -8 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 4 8 B -4 0 -6 -6 -8 C 0 6 0 10 -6 D -4 6 -10 0 -6 E -8 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 4 8 B -4 0 -6 -6 -8 C 0 6 0 10 -6 D -4 6 -10 0 -6 E -8 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4393: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) D E C B A (8) B C A D E (7) A E D B C (7) A D E B C (7) A B C D E (7) E D C B A (6) E D A C B (5) E A D C B (3) D E A B C (3) C B E D A (3) C B E A D (3) C B A E D (3) A B D C E (3) E D C A B (2) E D A B C (2) C E B D A (2) C B D A E (2) B C A E D (2) B A C D E (2) A E D C B (2) A D B E C (2) E A C D B (1) D E B A C (1) D E A C B (1) D A E B C (1) D A B E C (1) B D A C E (1) B C D E A (1) B C D A E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 0 0 -2 B 4 0 -2 -4 -2 C 0 2 0 -10 -4 D 0 4 10 0 16 E 2 2 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.370122 B: 0.000000 C: 0.000000 D: 0.629878 E: 0.000000 Sum of squares = 0.533736357503 Cumulative probabilities = A: 0.370122 B: 0.370122 C: 0.370122 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 0 -2 B 4 0 -2 -4 -2 C 0 2 0 -10 -4 D 0 4 10 0 16 E 2 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499520 B: 0.000000 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461503 Cumulative probabilities = A: 0.499520 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 E=19 D=15 B=14 so B is eliminated. Round 2 votes counts: C=34 A=31 E=19 D=16 so D is eliminated. Round 3 votes counts: C=34 A=34 E=32 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 B:198 A:197 E:196 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 0 -2 B 4 0 -2 -4 -2 C 0 2 0 -10 -4 D 0 4 10 0 16 E 2 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499520 B: 0.000000 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461503 Cumulative probabilities = A: 0.499520 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 0 -2 B 4 0 -2 -4 -2 C 0 2 0 -10 -4 D 0 4 10 0 16 E 2 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499520 B: 0.000000 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461503 Cumulative probabilities = A: 0.499520 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 0 -2 B 4 0 -2 -4 -2 C 0 2 0 -10 -4 D 0 4 10 0 16 E 2 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499520 B: 0.000000 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461503 Cumulative probabilities = A: 0.499520 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4394: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C B A D E (8) A B C E D (7) C B D A E (6) B C A E D (6) E A B C D (5) D C B E A (5) D C B A E (5) D E A C B (4) E B C A D (3) E B A C D (3) B A C E D (3) A E D B C (3) A C B D E (3) E B C D A (2) D E C B A (2) C D B A E (2) C B E D A (2) C B D E A (2) B C E A D (2) A E B C D (2) A D E B C (2) A C D B E (2) A B E C D (2) E D B C A (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E B A (1) D A E B C (1) D A C B E (1) C B A E D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 2 12 12 B 8 0 6 16 14 C -2 -6 0 22 14 D -12 -16 -22 0 -6 E -12 -14 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 12 12 B 8 0 6 16 14 C -2 -6 0 22 14 D -12 -16 -22 0 -6 E -12 -14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 C=21 D=20 B=11 so B is eliminated. Round 2 votes counts: C=29 A=26 E=25 D=20 so D is eliminated. Round 3 votes counts: C=40 E=32 A=28 so A is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:222 C:214 A:209 E:183 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 12 12 B 8 0 6 16 14 C -2 -6 0 22 14 D -12 -16 -22 0 -6 E -12 -14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 12 12 B 8 0 6 16 14 C -2 -6 0 22 14 D -12 -16 -22 0 -6 E -12 -14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 12 12 B 8 0 6 16 14 C -2 -6 0 22 14 D -12 -16 -22 0 -6 E -12 -14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4395: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) E D A C B (6) E D A B C (5) E C D A B (5) E A D C B (5) C B A E D (5) C E A D B (4) C D B E A (4) C B D E A (4) C B D A E (4) B A D E C (4) D E B A C (3) C E D B A (3) C B E D A (3) B C A D E (3) E D C A B (2) E A D B C (2) D E A B C (2) C B A D E (2) C A B E D (2) B C D A E (2) A E B D C (2) A C E B D (2) A B E D C (2) A B D E C (2) A B C E D (2) E A C D B (1) D B E A C (1) D A E B C (1) C E D A B (1) C E A B D (1) C D E B A (1) C B E A D (1) C A E B D (1) B D A E C (1) B A D C E (1) B A C D E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 14 4 4 -8 B -14 0 -18 -6 -10 C -4 18 0 8 -4 D -4 6 -8 0 -28 E 8 10 4 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 4 4 -8 B -14 0 -18 -6 -10 C -4 18 0 8 -4 D -4 6 -8 0 -28 E 8 10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=26 A=19 B=12 D=7 so D is eliminated. Round 2 votes counts: C=36 E=31 A=20 B=13 so B is eliminated. Round 3 votes counts: C=41 E=32 A=27 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:209 A:207 D:183 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 4 4 -8 B -14 0 -18 -6 -10 C -4 18 0 8 -4 D -4 6 -8 0 -28 E 8 10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 4 -8 B -14 0 -18 -6 -10 C -4 18 0 8 -4 D -4 6 -8 0 -28 E 8 10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 4 -8 B -14 0 -18 -6 -10 C -4 18 0 8 -4 D -4 6 -8 0 -28 E 8 10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4396: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) B D E A C (8) A C D E B (8) A C E B D (6) B E D C A (4) B A D C E (4) A C E D B (4) D E B C A (3) D B E C A (3) D B E A C (3) D B A C E (3) B E D A C (3) B D A E C (3) E D C B A (2) E C D A B (2) E C A B D (2) D E C B A (2) D E C A B (2) C E A D B (2) C A E D B (2) C A E B D (2) B D A C E (2) B A E C D (2) A C B E D (2) A B C E D (2) A B C D E (2) E B D C A (1) E B C D A (1) E B C A D (1) D C E A B (1) D C A E B (1) D B A E C (1) D A B C E (1) C E A B D (1) C D E A B (1) C D A E B (1) B A C D E (1) A D C E B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 16 -12 0 B 10 0 14 14 8 C -16 -14 0 -14 -2 D 12 -14 14 0 22 E 0 -8 2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 16 -12 0 B 10 0 14 14 8 C -16 -14 0 -14 -2 D 12 -14 14 0 22 E 0 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=27 D=20 E=9 C=9 so E is eliminated. Round 2 votes counts: B=38 A=27 D=22 C=13 so C is eliminated. Round 3 votes counts: B=38 A=36 D=26 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:217 A:197 E:186 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 16 -12 0 B 10 0 14 14 8 C -16 -14 0 -14 -2 D 12 -14 14 0 22 E 0 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 -12 0 B 10 0 14 14 8 C -16 -14 0 -14 -2 D 12 -14 14 0 22 E 0 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 -12 0 B 10 0 14 14 8 C -16 -14 0 -14 -2 D 12 -14 14 0 22 E 0 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4397: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (15) E B A D C (14) C D A B E (10) E B C A D (9) D A C B E (8) C B E D A (7) D A C E B (5) A D E B C (4) A D C E B (4) A D C B E (4) C B D A E (3) E A D B C (2) D A E C B (2) E D A B C (1) E B D A C (1) E B C D A (1) E B A C D (1) D C A E B (1) D C A B E (1) D A E B C (1) C D B A E (1) C B E A D (1) C B A D E (1) B E C D A (1) B C E A D (1) B C A D E (1) Total count = 100 A B C D E A 0 -14 -6 14 -8 B 14 0 4 12 8 C 6 -4 0 4 -4 D -14 -12 -4 0 -8 E 8 -8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 14 -8 B 14 0 4 12 8 C 6 -4 0 4 -4 D -14 -12 -4 0 -8 E 8 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=23 D=18 B=18 A=12 so A is eliminated. Round 2 votes counts: D=30 E=29 C=23 B=18 so B is eliminated. Round 3 votes counts: E=45 D=30 C=25 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:206 C:201 A:193 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 14 -8 B 14 0 4 12 8 C 6 -4 0 4 -4 D -14 -12 -4 0 -8 E 8 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 14 -8 B 14 0 4 12 8 C 6 -4 0 4 -4 D -14 -12 -4 0 -8 E 8 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 14 -8 B 14 0 4 12 8 C 6 -4 0 4 -4 D -14 -12 -4 0 -8 E 8 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4398: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (12) B C D E A (11) A E D C B (11) C D E A B (8) B A D E C (8) C E D A B (7) B C E D A (6) A D E C B (6) B C D A E (4) E A D C B (3) B A D C E (3) A B E D C (3) E C D A B (2) E C A D B (2) B A E C D (2) B A C D E (2) E A C D B (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C E B (1) B E C A D (1) B D C A E (1) B D A C E (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 10 12 14 B 4 0 10 10 10 C -10 -10 0 -6 -4 D -12 -10 6 0 -2 E -14 -10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 12 14 B 4 0 10 10 10 C -10 -10 0 -6 -4 D -12 -10 6 0 -2 E -14 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=52 A=21 C=15 E=8 D=4 so D is eliminated. Round 2 votes counts: B=52 A=23 C=17 E=8 so E is eliminated. Round 3 votes counts: B=52 A=27 C=21 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:216 D:191 E:191 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 12 14 B 4 0 10 10 10 C -10 -10 0 -6 -4 D -12 -10 6 0 -2 E -14 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 12 14 B 4 0 10 10 10 C -10 -10 0 -6 -4 D -12 -10 6 0 -2 E -14 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 12 14 B 4 0 10 10 10 C -10 -10 0 -6 -4 D -12 -10 6 0 -2 E -14 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4399: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) C A B E D (9) D E B C A (6) C A D B E (6) D A C E B (5) E B D A C (4) A B C E D (4) E D B A C (3) E B D C A (3) C A D E B (3) C A B D E (3) B E D A C (3) B E C A D (3) A C B E D (3) E D B C A (2) D C A E B (2) D A E B C (2) C D A E B (2) B E A C D (2) B A C E D (2) A C B D E (2) A B E D C (2) E B C D A (1) D E C B A (1) D E C A B (1) D E A C B (1) D C E B A (1) D C E A B (1) D A E C B (1) D A B E C (1) C B E A D (1) C B A E D (1) B E A D C (1) B A E D C (1) B A E C D (1) A D B E C (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 -4 8 B -4 0 12 -4 -2 C -8 -12 0 -10 -6 D 4 4 10 0 6 E -8 2 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -4 8 B -4 0 12 -4 -2 C -8 -12 0 -10 -6 D 4 4 10 0 6 E -8 2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=25 A=15 E=13 B=13 so E is eliminated. Round 2 votes counts: D=39 C=25 B=21 A=15 so A is eliminated. Round 3 votes counts: D=40 C=31 B=29 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:208 B:201 E:197 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -4 8 B -4 0 12 -4 -2 C -8 -12 0 -10 -6 D 4 4 10 0 6 E -8 2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -4 8 B -4 0 12 -4 -2 C -8 -12 0 -10 -6 D 4 4 10 0 6 E -8 2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -4 8 B -4 0 12 -4 -2 C -8 -12 0 -10 -6 D 4 4 10 0 6 E -8 2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4400: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (14) E D A B C (7) E D B C A (5) B A C E D (5) A C B D E (5) A B C E D (5) D E A C B (4) D E A B C (4) B C E A D (4) E B A D C (3) C B E A D (3) C A B D E (3) B A E C D (3) D E C B A (2) D E C A B (2) C D B A E (2) C B A E D (2) B C A E D (2) A D E B C (2) A D C B E (2) A D B C E (2) A C D B E (2) E D C B A (1) E D B A C (1) E C D B A (1) E B D C A (1) E A D B C (1) D C E B A (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) D A C E B (1) C B E D A (1) C B D E A (1) C A D B E (1) B E C A D (1) B E A C D (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 0 24 10 B 8 0 -4 8 20 C 0 4 0 12 18 D -24 -8 -12 0 4 E -10 -20 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.201468 B: 0.000000 C: 0.798532 D: 0.000000 E: 0.000000 Sum of squares = 0.678242963182 Cumulative probabilities = A: 0.201468 B: 0.201468 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 24 10 B 8 0 -4 8 20 C 0 4 0 12 18 D -24 -8 -12 0 4 E -10 -20 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555784958 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=20 A=19 D=18 B=16 so B is eliminated. Round 2 votes counts: C=33 A=27 E=22 D=18 so D is eliminated. Round 3 votes counts: C=37 E=34 A=29 so A is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:216 A:213 D:180 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 0 24 10 B 8 0 -4 8 20 C 0 4 0 12 18 D -24 -8 -12 0 4 E -10 -20 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555784958 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 24 10 B 8 0 -4 8 20 C 0 4 0 12 18 D -24 -8 -12 0 4 E -10 -20 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555784958 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 24 10 B 8 0 -4 8 20 C 0 4 0 12 18 D -24 -8 -12 0 4 E -10 -20 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555784958 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4401: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (15) C D B E A (13) D C E B A (8) A B E C D (7) C B D E A (6) E A B D C (4) D C E A B (4) A E D B C (4) C D B A E (3) B C E D A (3) B C E A D (3) D E A C B (2) D C A E B (2) D A E C B (2) D A C E B (2) C B E D A (2) B A E C D (2) A D E C B (2) A D E B C (2) E D A B C (1) E B D A C (1) D E C B A (1) D C B E A (1) C D A B E (1) C B D A E (1) C A B E D (1) B E D C A (1) B E A D C (1) B E A C D (1) A D C E B (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 0 -8 -4 B -6 0 -8 0 -2 C 0 8 0 -8 8 D 8 0 8 0 6 E 4 2 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.365741 C: 0.000000 D: 0.634259 E: 0.000000 Sum of squares = 0.536050760352 Cumulative probabilities = A: 0.000000 B: 0.365741 C: 0.365741 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -8 -4 B -6 0 -8 0 -2 C 0 8 0 -8 8 D 8 0 8 0 6 E 4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499838 C: 0.000000 D: 0.500162 E: 0.000000 Sum of squares = 0.500000052503 Cumulative probabilities = A: 0.000000 B: 0.499838 C: 0.499838 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=27 D=22 B=11 E=6 so E is eliminated. Round 2 votes counts: A=38 C=27 D=23 B=12 so B is eliminated. Round 3 votes counts: A=42 C=33 D=25 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:211 C:204 A:197 E:196 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -8 -4 B -6 0 -8 0 -2 C 0 8 0 -8 8 D 8 0 8 0 6 E 4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499838 C: 0.000000 D: 0.500162 E: 0.000000 Sum of squares = 0.500000052503 Cumulative probabilities = A: 0.000000 B: 0.499838 C: 0.499838 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -8 -4 B -6 0 -8 0 -2 C 0 8 0 -8 8 D 8 0 8 0 6 E 4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499838 C: 0.000000 D: 0.500162 E: 0.000000 Sum of squares = 0.500000052503 Cumulative probabilities = A: 0.000000 B: 0.499838 C: 0.499838 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -8 -4 B -6 0 -8 0 -2 C 0 8 0 -8 8 D 8 0 8 0 6 E 4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499838 C: 0.000000 D: 0.500162 E: 0.000000 Sum of squares = 0.500000052503 Cumulative probabilities = A: 0.000000 B: 0.499838 C: 0.499838 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4402: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (6) E B A D C (5) C A D B E (5) B C A E D (5) D C A E B (4) C A B D E (4) E D A B C (3) E B C A D (3) D C A B E (3) C D E B A (3) C B E D A (3) C B A D E (3) B E C A D (3) B A C E D (3) A C D B E (3) A B E D C (3) E D B C A (2) E C D B A (2) E C B D A (2) E B D A C (2) E B C D A (2) E B A C D (2) D E C A B (2) D C E B A (2) D A E C B (2) C D B A E (2) B E A C D (2) A D B C E (2) A B E C D (2) E D C B A (1) E D B A C (1) E B D C A (1) E A B D C (1) D E C B A (1) D E A B C (1) D C E A B (1) D A C E B (1) D A C B E (1) C D B E A (1) C B A E D (1) B A E C D (1) A E B D C (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -24 2 8 B 6 0 -6 0 8 C 24 6 0 18 8 D -2 0 -18 0 -2 E -8 -8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -24 2 8 B 6 0 -6 0 8 C 24 6 0 18 8 D -2 0 -18 0 -2 E -8 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 D=18 B=14 A=13 so A is eliminated. Round 2 votes counts: C=31 E=28 D=21 B=20 so B is eliminated. Round 3 votes counts: C=40 E=39 D=21 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:204 A:190 D:189 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -24 2 8 B 6 0 -6 0 8 C 24 6 0 18 8 D -2 0 -18 0 -2 E -8 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -24 2 8 B 6 0 -6 0 8 C 24 6 0 18 8 D -2 0 -18 0 -2 E -8 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -24 2 8 B 6 0 -6 0 8 C 24 6 0 18 8 D -2 0 -18 0 -2 E -8 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4403: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) A B D C E (6) E B D C A (5) E B A D C (4) D B C A E (4) B D A E C (4) B D A C E (4) B A D E C (4) E C A D B (3) E C A B D (3) E A B C D (3) C D E A B (3) C D A B E (3) C A D E B (3) D C B A E (2) D B C E A (2) D A C B E (2) D A B C E (2) C E D B A (2) C D A E B (2) B E A D C (2) B A D C E (2) A E C B D (2) A D C B E (2) A C D B E (2) E D B C A (1) E C B A D (1) E B D A C (1) E B C D A (1) E B C A D (1) E B A C D (1) D C E B A (1) D C B E A (1) D C A B E (1) D B E C A (1) C D E B A (1) C A E D B (1) C A D B E (1) B E D C A (1) B E D A C (1) B D E C A (1) A E B C D (1) A D B C E (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -6 -8 4 B 12 0 10 2 4 C 6 -10 0 -12 -2 D 8 -2 12 0 14 E -4 -4 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -8 4 B 12 0 10 2 4 C 6 -10 0 -12 -2 D 8 -2 12 0 14 E -4 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=19 A=17 D=16 C=16 so D is eliminated. Round 2 votes counts: E=32 B=26 C=21 A=21 so C is eliminated. Round 3 votes counts: E=39 A=32 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:216 B:214 C:191 E:190 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -8 4 B 12 0 10 2 4 C 6 -10 0 -12 -2 D 8 -2 12 0 14 E -4 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -8 4 B 12 0 10 2 4 C 6 -10 0 -12 -2 D 8 -2 12 0 14 E -4 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -8 4 B 12 0 10 2 4 C 6 -10 0 -12 -2 D 8 -2 12 0 14 E -4 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4404: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) B E D A C (8) D B A E C (7) D A B E C (7) C A E D B (7) C E A B D (6) A D C B E (6) E C B D A (5) C E B A D (5) C A D E B (5) B E D C A (5) B D E A C (5) A D B C E (5) B E C D A (3) A C D E B (3) A C D B E (3) E C B A D (2) D A B C E (2) C E A D B (2) B D A E C (2) E C D A B (1) E B D C A (1) E B D A C (1) C A D B E (1) Total count = 100 A B C D E A 0 -4 -2 -10 -4 B 4 0 8 2 8 C 2 -8 0 2 -10 D 10 -2 -2 0 -8 E 4 -8 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -10 -4 B 4 0 8 2 8 C 2 -8 0 2 -10 D 10 -2 -2 0 -8 E 4 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=23 E=18 A=17 D=16 so D is eliminated. Round 2 votes counts: B=30 C=26 A=26 E=18 so E is eliminated. Round 3 votes counts: B=40 C=34 A=26 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:207 D:199 C:193 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -10 -4 B 4 0 8 2 8 C 2 -8 0 2 -10 D 10 -2 -2 0 -8 E 4 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -10 -4 B 4 0 8 2 8 C 2 -8 0 2 -10 D 10 -2 -2 0 -8 E 4 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -10 -4 B 4 0 8 2 8 C 2 -8 0 2 -10 D 10 -2 -2 0 -8 E 4 -8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4405: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E B D A C (7) D E C B A (7) A B C E D (7) E B A D C (5) C D E A B (5) B E A C D (5) B A E C D (5) E D B A C (4) D C E B A (4) D C A B E (4) B E A D C (4) B A E D C (4) A B E C D (4) C A D B E (3) A B C D E (3) E D C B A (2) C D A B E (2) C A B D E (2) A C B E D (2) E D B C A (1) E C D B A (1) E C B A D (1) E B A C D (1) C E B A D (1) C D E B A (1) C A E B D (1) C A B E D (1) B E D A C (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 10 4 -18 B 8 0 6 12 0 C -10 -6 0 -8 -4 D -4 -12 8 0 -14 E 18 0 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.502869 C: 0.000000 D: 0.000000 E: 0.497131 Sum of squares = 0.500016467214 Cumulative probabilities = A: 0.000000 B: 0.502869 C: 0.502869 D: 0.502869 E: 1.000000 A B C D E A 0 -8 10 4 -18 B 8 0 6 12 0 C -10 -6 0 -8 -4 D -4 -12 8 0 -14 E 18 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999795 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=22 B=19 A=19 C=16 so C is eliminated. Round 2 votes counts: D=32 A=26 E=23 B=19 so B is eliminated. Round 3 votes counts: A=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 B:213 A:194 D:189 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 4 -18 B 8 0 6 12 0 C -10 -6 0 -8 -4 D -4 -12 8 0 -14 E 18 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999795 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 4 -18 B 8 0 6 12 0 C -10 -6 0 -8 -4 D -4 -12 8 0 -14 E 18 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999795 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 4 -18 B 8 0 6 12 0 C -10 -6 0 -8 -4 D -4 -12 8 0 -14 E 18 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999795 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4406: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) D E C A B (8) D E B A C (8) B A C E D (7) D C E B A (6) C D E A B (5) C B A E D (5) B A E D C (5) D E C B A (4) C D E B A (4) E D A C B (3) E D A B C (3) C A B E D (3) B C A E D (3) A B C E D (3) E A D B C (2) D E A B C (2) C D B E A (2) A E B D C (2) A B E D C (2) E D B A C (1) E C D A B (1) D E A C B (1) C E D A B (1) C B D E A (1) C B D A E (1) C A E D B (1) B D C E A (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -22 -12 -4 -6 B 22 0 -12 -4 -6 C 12 12 0 2 10 D 4 4 -2 0 10 E 6 6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -12 -4 -6 B 22 0 -12 -4 -6 C 12 12 0 2 10 D 4 4 -2 0 10 E 6 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=29 B=20 E=10 A=9 so A is eliminated. Round 2 votes counts: C=33 D=29 B=26 E=12 so E is eliminated. Round 3 votes counts: D=38 C=34 B=28 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:208 B:200 E:196 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -12 -4 -6 B 22 0 -12 -4 -6 C 12 12 0 2 10 D 4 4 -2 0 10 E 6 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -12 -4 -6 B 22 0 -12 -4 -6 C 12 12 0 2 10 D 4 4 -2 0 10 E 6 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -12 -4 -6 B 22 0 -12 -4 -6 C 12 12 0 2 10 D 4 4 -2 0 10 E 6 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4407: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) B E D C A (7) D A E C B (5) C E A B D (5) B C E A D (4) B C A E D (4) B C A D E (4) E D A C B (3) E C A D B (3) D B A E C (3) D A E B C (3) C B E A D (3) C B A E D (3) C A E D B (3) C A B E D (3) B D E C A (3) B D C A E (3) A D E C B (3) E C B A D (2) E A D C B (2) E A C D B (2) D E A B C (2) D A C E B (2) D A B C E (2) B E C A D (2) A C E D B (2) E D A B C (1) D E A C B (1) D B A C E (1) D A B E C (1) C E A D B (1) C A E B D (1) C A B D E (1) B E D A C (1) B D A C E (1) B C D A E (1) A E C D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -24 2 -2 B 2 0 4 14 14 C 24 -4 0 12 -8 D -2 -14 -12 0 -24 E 2 -14 8 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -24 2 -2 B 2 0 4 14 14 C 24 -4 0 12 -8 D -2 -14 -12 0 -24 E 2 -14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=20 C=20 E=13 A=8 so A is eliminated. Round 2 votes counts: B=39 C=24 D=23 E=14 so E is eliminated. Round 3 votes counts: B=39 C=32 D=29 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:212 E:210 A:187 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -24 2 -2 B 2 0 4 14 14 C 24 -4 0 12 -8 D -2 -14 -12 0 -24 E 2 -14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -24 2 -2 B 2 0 4 14 14 C 24 -4 0 12 -8 D -2 -14 -12 0 -24 E 2 -14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -24 2 -2 B 2 0 4 14 14 C 24 -4 0 12 -8 D -2 -14 -12 0 -24 E 2 -14 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4408: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) E D A C B (8) D E B C A (7) E A C D B (6) D B E C A (6) A C E B D (6) E D A B C (5) E A D C B (5) D E B A C (5) E A C B D (4) D B C A E (4) C B A D E (3) C A B E D (3) B C D A E (3) A E C B D (3) E D B A C (2) D E C A B (2) D B C E A (2) C D B A E (2) B D C A E (2) D E A C B (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B D E (1) B D E A C (1) B A C E D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -2 -8 -12 B -2 0 -6 -24 -18 C 2 6 0 -8 -10 D 8 24 8 0 6 E 12 18 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -8 -12 B -2 0 -6 -24 -18 C 2 6 0 -8 -10 D 8 24 8 0 6 E 12 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999261 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=30 B=16 C=12 A=11 so A is eliminated. Round 2 votes counts: E=33 D=31 C=20 B=16 so B is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:217 C:195 A:190 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -8 -12 B -2 0 -6 -24 -18 C 2 6 0 -8 -10 D 8 24 8 0 6 E 12 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999261 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -8 -12 B -2 0 -6 -24 -18 C 2 6 0 -8 -10 D 8 24 8 0 6 E 12 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999261 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -8 -12 B -2 0 -6 -24 -18 C 2 6 0 -8 -10 D 8 24 8 0 6 E 12 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999261 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4409: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C D A E B (8) B E D A C (7) E B A D C (6) C B D E A (6) E B A C D (5) E A B D C (5) C D B A E (5) B D A E C (5) E A B C D (3) D C B A E (3) D C A B E (3) D A C E B (3) C D A B E (3) C A D E B (3) D A E B C (2) C E A B D (2) B E C A D (2) B D E C A (2) A E D B C (2) E A D B C (1) E A C D B (1) E A C B D (1) D B A C E (1) D A E C B (1) D A B E C (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D A E (1) C A E D B (1) B D C A E (1) B C D E A (1) B C D A E (1) Total count = 100 A B C D E A 0 -20 10 -10 -12 B 20 0 12 24 10 C -10 -12 0 -8 -10 D 10 -24 8 0 0 E 12 -10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 10 -10 -12 B 20 0 12 24 10 C -10 -12 0 -8 -10 D 10 -24 8 0 0 E 12 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=30 E=22 D=14 A=2 so A is eliminated. Round 2 votes counts: C=32 B=30 E=24 D=14 so D is eliminated. Round 3 votes counts: C=41 B=32 E=27 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:206 D:197 A:184 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 10 -10 -12 B 20 0 12 24 10 C -10 -12 0 -8 -10 D 10 -24 8 0 0 E 12 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 10 -10 -12 B 20 0 12 24 10 C -10 -12 0 -8 -10 D 10 -24 8 0 0 E 12 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 10 -10 -12 B 20 0 12 24 10 C -10 -12 0 -8 -10 D 10 -24 8 0 0 E 12 -10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4410: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (12) C E D A B (7) C D E A B (7) D E C A B (5) D E A B C (5) C E A D B (5) B C A D E (4) B A D E C (4) E A D C B (3) D E A C B (3) C B D E A (3) C B A E D (3) B D A E C (3) A E D B C (3) A B E C D (3) E A D B C (2) D C B E A (2) D B C E A (2) D B A E C (2) C E A B D (2) C B D A E (2) B D A C E (2) B C A E D (2) B A C E D (2) A E B D C (2) E A C D B (1) D C E A B (1) D B E A C (1) C D E B A (1) C D B E A (1) C A B E D (1) B D C A E (1) B C D A E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 4 0 2 -2 B -4 0 4 -2 2 C 0 -4 0 -6 -2 D -2 2 6 0 0 E 2 -2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.193188 B: 0.193188 C: 0.000000 D: 0.113623 E: 0.500000 Sum of squares = 0.337553712171 Cumulative probabilities = A: 0.193188 B: 0.386377 C: 0.386377 D: 0.500000 E: 1.000000 A B C D E A 0 4 0 2 -2 B -4 0 4 -2 2 C 0 -4 0 -6 -2 D -2 2 6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.33333333334 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=31 D=21 A=10 E=6 so E is eliminated. Round 2 votes counts: C=32 B=31 D=21 A=16 so A is eliminated. Round 3 votes counts: B=37 C=34 D=29 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:203 A:202 E:201 B:200 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 -2 B -4 0 4 -2 2 C 0 -4 0 -6 -2 D -2 2 6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.33333333334 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 -2 B -4 0 4 -2 2 C 0 -4 0 -6 -2 D -2 2 6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.33333333334 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 -2 B -4 0 4 -2 2 C 0 -4 0 -6 -2 D -2 2 6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.166667 E: 0.500000 Sum of squares = 0.33333333334 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4411: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) B C E A D (9) D E B A C (8) D E B C A (5) B E C A D (5) D B E C A (4) D A C E B (4) C B E A D (4) A C B E D (4) E B D C A (3) D A E C B (3) A C D E B (3) A C D B E (3) E B A C D (2) E A B C D (2) D E A B C (2) D C B A E (2) D C A B E (2) D B C E A (2) D A C B E (2) B E D C A (2) B E C D A (2) B D E C A (2) A D E C B (2) A C E B D (2) E D B A C (1) E B C D A (1) E B A D C (1) D A E B C (1) C D A B E (1) C B A D E (1) C A D B E (1) C A B D E (1) B D C E A (1) A E B D C (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -14 4 -12 B 10 0 8 4 16 C 14 -8 0 2 6 D -4 -4 -2 0 2 E 12 -16 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 4 -12 B 10 0 8 4 16 C 14 -8 0 2 6 D -4 -4 -2 0 2 E 12 -16 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998223 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=21 C=17 A=17 E=10 so E is eliminated. Round 2 votes counts: D=36 B=28 A=19 C=17 so C is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:207 D:196 E:194 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 4 -12 B 10 0 8 4 16 C 14 -8 0 2 6 D -4 -4 -2 0 2 E 12 -16 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998223 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 4 -12 B 10 0 8 4 16 C 14 -8 0 2 6 D -4 -4 -2 0 2 E 12 -16 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998223 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 4 -12 B 10 0 8 4 16 C 14 -8 0 2 6 D -4 -4 -2 0 2 E 12 -16 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998223 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4412: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) D C E A B (11) B A D C E (10) B A E C D (9) E C A D B (7) D C A E B (4) B E A C D (4) E C D B A (3) D B C A E (3) B D A C E (3) E B C A D (2) D A C B E (2) B E A D C (2) B A E D C (2) A E C B D (2) A B D C E (2) E D C B A (1) E C B D A (1) E C A B D (1) E B A C D (1) E A C B D (1) E A B C D (1) D E C B A (1) D C B A E (1) D C A B E (1) D B A C E (1) D A B C E (1) C E D A B (1) C D E A B (1) C D A E B (1) B E D C A (1) B D E C A (1) B D E A C (1) B D C E A (1) B A D E C (1) B A C E D (1) A E B C D (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -6 -2 -6 B 0 0 -2 -2 -2 C 6 2 0 -2 -8 D 2 2 2 0 -4 E 6 2 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -6 -2 -6 B 0 0 -2 -2 -2 C 6 2 0 -2 -8 D 2 2 2 0 -4 E 6 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=29 D=25 A=7 C=3 so C is eliminated. Round 2 votes counts: B=36 E=30 D=27 A=7 so A is eliminated. Round 3 votes counts: B=39 E=33 D=28 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 D:201 C:199 B:197 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 -2 -6 B 0 0 -2 -2 -2 C 6 2 0 -2 -8 D 2 2 2 0 -4 E 6 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -2 -6 B 0 0 -2 -2 -2 C 6 2 0 -2 -8 D 2 2 2 0 -4 E 6 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -2 -6 B 0 0 -2 -2 -2 C 6 2 0 -2 -8 D 2 2 2 0 -4 E 6 2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4413: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) E D C B A (7) E D C A B (6) E D A C B (5) A B C D E (5) D E C A B (4) B C D E A (4) B C A D E (4) A E D C B (4) E D B C A (3) E A D C B (3) D C E A B (3) B C D A E (3) A C D E B (3) A B C E D (3) E B D C A (2) E B A D C (2) D C E B A (2) C D B E A (2) B E A D C (2) B D C E A (2) A E C D B (2) A C D B E (2) A C B D E (2) E D B A C (1) E D A B C (1) E B D A C (1) E A D B C (1) E A B D C (1) C D E A B (1) C D A B E (1) C B D A E (1) C A D E B (1) B E D C A (1) B A E C D (1) B A C E D (1) A C E D B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 6 0 -8 B -2 0 0 -6 -8 C -6 0 0 -4 2 D 0 6 4 0 0 E 8 8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.649603 E: 0.350397 Sum of squares = 0.544762040069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.649603 E: 1.000000 A B C D E A 0 2 6 0 -8 B -2 0 0 -6 -8 C -6 0 0 -4 2 D 0 6 4 0 0 E 8 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=28 A=24 D=9 C=6 so C is eliminated. Round 2 votes counts: E=33 B=29 A=25 D=13 so D is eliminated. Round 3 votes counts: E=43 B=31 A=26 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:207 D:205 A:200 C:196 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 0 -8 B -2 0 0 -6 -8 C -6 0 0 -4 2 D 0 6 4 0 0 E 8 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 -8 B -2 0 0 -6 -8 C -6 0 0 -4 2 D 0 6 4 0 0 E 8 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 -8 B -2 0 0 -6 -8 C -6 0 0 -4 2 D 0 6 4 0 0 E 8 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4414: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) A D B C E (7) B C E A D (6) E C D B A (5) E C B D A (5) D A C E B (5) C D A E B (5) B E C D A (4) B A D E C (4) A D C B E (4) E C D A B (3) C E D A B (3) C E B D A (3) B E D A C (3) B E C A D (3) B C E D A (3) B A D C E (3) A D E C B (3) D A E C B (2) B E A D C (2) B A E D C (2) B A C D E (2) E D C A B (1) E B D A C (1) E B C D A (1) C E D B A (1) C B E D A (1) C B E A D (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B D E (1) B C A E D (1) B C A D E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -2 8 8 B 6 0 -10 -2 4 C 2 10 0 4 18 D -8 2 -4 0 2 E -8 -4 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 8 8 B 6 0 -10 -2 4 C 2 10 0 4 18 D -8 2 -4 0 2 E -8 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=25 C=18 E=16 D=7 so D is eliminated. Round 2 votes counts: B=34 A=32 C=18 E=16 so E is eliminated. Round 3 votes counts: B=36 C=32 A=32 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:217 A:204 B:199 D:196 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 8 8 B 6 0 -10 -2 4 C 2 10 0 4 18 D -8 2 -4 0 2 E -8 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 8 8 B 6 0 -10 -2 4 C 2 10 0 4 18 D -8 2 -4 0 2 E -8 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 8 8 B 6 0 -10 -2 4 C 2 10 0 4 18 D -8 2 -4 0 2 E -8 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4415: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) C E D B A (7) A B D E C (7) B A D C E (6) C E D A B (5) C B A D E (5) E D B A C (4) D E B A C (4) C A E D B (4) C A B E D (4) C A B D E (4) A B D C E (4) E D A B C (3) D B E A C (3) B C A D E (3) B A D E C (3) E D C B A (2) E D B C A (2) D B A E C (2) C E A D B (2) B D E A C (2) A B C D E (2) E C D B A (1) E C D A B (1) E A D B C (1) D E B C A (1) D B E C A (1) C E B D A (1) C B D A E (1) C B A E D (1) B D C A E (1) B C D A E (1) A E B D C (1) A D E B C (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -18 6 0 20 B 18 0 20 10 18 C -6 -20 0 -12 6 D 0 -10 12 0 20 E -20 -18 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 0 20 B 18 0 20 10 18 C -6 -20 0 -12 6 D 0 -10 12 0 20 E -20 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=24 A=17 E=14 D=11 so D is eliminated. Round 2 votes counts: C=34 B=30 E=19 A=17 so A is eliminated. Round 3 votes counts: B=43 C=36 E=21 so E is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:233 D:211 A:204 C:184 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 0 20 B 18 0 20 10 18 C -6 -20 0 -12 6 D 0 -10 12 0 20 E -20 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 0 20 B 18 0 20 10 18 C -6 -20 0 -12 6 D 0 -10 12 0 20 E -20 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 0 20 B 18 0 20 10 18 C -6 -20 0 -12 6 D 0 -10 12 0 20 E -20 -18 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4416: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (12) C E B A D (7) E D A C B (6) C B A E D (6) B A C D E (6) A B D C E (6) E C B D A (5) D A B E C (5) D E A B C (4) D A E B C (3) B C E A D (3) B C A D E (3) A D C B E (3) A B C D E (3) E D C B A (2) E B D C A (2) D A E C B (2) C B E A D (2) A C B D E (2) E D C A B (1) E D B C A (1) E D B A C (1) E C D B A (1) E C D A B (1) E C B A D (1) E C A D B (1) D E A C B (1) D A B C E (1) C B A D E (1) C A B D E (1) B E D C A (1) B E C D A (1) B D A C E (1) B C E D A (1) B C A E D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 16 20 16 B -8 0 10 6 18 C -16 -10 0 -6 22 D -20 -6 6 0 12 E -16 -18 -22 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 20 16 B -8 0 10 6 18 C -16 -10 0 -6 22 D -20 -6 6 0 12 E -16 -18 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 C=17 B=17 D=16 so D is eliminated. Round 2 votes counts: A=39 E=27 C=17 B=17 so C is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 B:213 D:196 C:195 E:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 20 16 B -8 0 10 6 18 C -16 -10 0 -6 22 D -20 -6 6 0 12 E -16 -18 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 20 16 B -8 0 10 6 18 C -16 -10 0 -6 22 D -20 -6 6 0 12 E -16 -18 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 20 16 B -8 0 10 6 18 C -16 -10 0 -6 22 D -20 -6 6 0 12 E -16 -18 -22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4417: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) D E C A B (8) A B D E C (8) C E D B A (7) C E D A B (7) E C D B A (5) D A E C B (5) B A C E D (5) A D B E C (5) C E B D A (4) B A E C D (4) A B D C E (3) E D C A B (2) E C B D A (2) D C A E B (2) C D E A B (2) B C E A D (2) A D B C E (2) A B C D E (2) E D C B A (1) E C D A B (1) E C B A D (1) D E A B C (1) D C E A B (1) D A E B C (1) D A C E B (1) C E B A D (1) C E A B D (1) C D A E B (1) C B E A D (1) C A B D E (1) B E C A D (1) B C A E D (1) B A D C E (1) A D C B E (1) Total count = 100 A B C D E A 0 10 -4 -2 4 B -10 0 -10 -6 -8 C 4 10 0 -2 -8 D 2 6 2 0 8 E -4 8 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -2 4 B -10 0 -10 -6 -8 C 4 10 0 -2 -8 D 2 6 2 0 8 E -4 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 A=21 D=19 E=12 so E is eliminated. Round 2 votes counts: C=34 B=23 D=22 A=21 so A is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:209 A:204 C:202 E:202 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 -2 4 B -10 0 -10 -6 -8 C 4 10 0 -2 -8 D 2 6 2 0 8 E -4 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -2 4 B -10 0 -10 -6 -8 C 4 10 0 -2 -8 D 2 6 2 0 8 E -4 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -2 4 B -10 0 -10 -6 -8 C 4 10 0 -2 -8 D 2 6 2 0 8 E -4 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4418: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) E D B A C (9) D E B A C (7) C E A B D (7) D B E A C (5) A B C D E (5) C A E B D (4) B A D C E (4) E D C B A (3) E C D A B (3) C E D A B (3) C A B E D (3) A B D C E (3) E C D B A (2) E A D B C (2) E A B D C (2) D B C A E (2) C A B D E (2) B A D E C (2) B A C D E (2) A E B D C (2) A B C E D (2) E D B C A (1) E D A B C (1) E C A B D (1) D E C B A (1) D C E B A (1) D C B A E (1) D B A C E (1) C E D B A (1) C E A D B (1) C B A D E (1) B C A D E (1) A E B C D (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 24 -2 0 B 8 0 26 -8 -6 C -24 -26 0 -16 -6 D 2 8 16 0 -2 E 0 6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.187521 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.812479 Sum of squares = 0.695286178411 Cumulative probabilities = A: 0.187521 B: 0.187521 C: 0.187521 D: 0.187521 E: 1.000000 A B C D E A 0 -8 24 -2 0 B 8 0 26 -8 -6 C -24 -26 0 -16 -6 D 2 8 16 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428570 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.571430 Sum of squares = 0.510204536918 Cumulative probabilities = A: 0.428570 B: 0.428570 C: 0.428570 D: 0.428570 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 C=22 A=17 B=9 so B is eliminated. Round 2 votes counts: D=28 A=25 E=24 C=23 so C is eliminated. Round 3 votes counts: E=36 A=36 D=28 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:212 B:210 A:207 E:207 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 24 -2 0 B 8 0 26 -8 -6 C -24 -26 0 -16 -6 D 2 8 16 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428570 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.571430 Sum of squares = 0.510204536918 Cumulative probabilities = A: 0.428570 B: 0.428570 C: 0.428570 D: 0.428570 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 24 -2 0 B 8 0 26 -8 -6 C -24 -26 0 -16 -6 D 2 8 16 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428570 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.571430 Sum of squares = 0.510204536918 Cumulative probabilities = A: 0.428570 B: 0.428570 C: 0.428570 D: 0.428570 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 24 -2 0 B 8 0 26 -8 -6 C -24 -26 0 -16 -6 D 2 8 16 0 -2 E 0 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428570 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.571430 Sum of squares = 0.510204536918 Cumulative probabilities = A: 0.428570 B: 0.428570 C: 0.428570 D: 0.428570 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4419: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (13) B C E A D (13) E A B D C (9) C B D A E (8) C B A E D (6) B E A D C (6) D A E C B (5) C D A B E (5) B E A C D (5) D C A E B (4) D A E B C (4) C B A D E (4) C D B A E (3) E D A B C (2) E B A D C (2) C B E A D (2) E D A C B (1) E B A C D (1) E A D C B (1) D E A C B (1) D E A B C (1) D A C E B (1) D A B E C (1) B D A C E (1) B A E D C (1) Total count = 100 A B C D E A 0 -4 10 26 -14 B 4 0 18 16 10 C -10 -18 0 -6 -6 D -26 -16 6 0 -24 E 14 -10 6 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 26 -14 B 4 0 18 16 10 C -10 -18 0 -6 -6 D -26 -16 6 0 -24 E 14 -10 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=28 B=26 D=17 so A is eliminated. Round 2 votes counts: E=29 C=28 B=26 D=17 so D is eliminated. Round 3 votes counts: E=40 C=33 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:224 E:217 A:209 C:180 D:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 26 -14 B 4 0 18 16 10 C -10 -18 0 -6 -6 D -26 -16 6 0 -24 E 14 -10 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 26 -14 B 4 0 18 16 10 C -10 -18 0 -6 -6 D -26 -16 6 0 -24 E 14 -10 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 26 -14 B 4 0 18 16 10 C -10 -18 0 -6 -6 D -26 -16 6 0 -24 E 14 -10 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4420: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) B E C A D (10) D A C E B (9) B C E A D (7) D A E C B (6) C B E A D (6) C B D A E (6) E A D B C (5) B E A D C (5) C A D E B (4) B C E D A (4) C D A E B (3) A D E C B (3) C D A B E (2) B E D A C (2) A E D B C (2) A D E B C (2) A D C E B (2) E B D A C (1) E B A D C (1) D C A B E (1) D B A E C (1) C E B A D (1) C E A B D (1) C B A D E (1) C A E D B (1) B E C D A (1) B E A C D (1) B D C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 4 2 6 8 B -4 0 6 -4 -4 C -2 -6 0 -4 -2 D -6 4 4 0 2 E -8 4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 6 8 B -4 0 6 -4 -4 C -2 -6 0 -4 -2 D -6 4 4 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=27 C=25 A=10 E=7 so E is eliminated. Round 2 votes counts: B=33 D=27 C=25 A=15 so A is eliminated. Round 3 votes counts: D=42 B=33 C=25 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:202 E:198 B:197 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 6 8 B -4 0 6 -4 -4 C -2 -6 0 -4 -2 D -6 4 4 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 6 8 B -4 0 6 -4 -4 C -2 -6 0 -4 -2 D -6 4 4 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 6 8 B -4 0 6 -4 -4 C -2 -6 0 -4 -2 D -6 4 4 0 2 E -8 4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4421: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) E C B A D (7) E C A B D (5) E A C D B (5) D A B C E (5) C E B A D (5) B D C A E (5) C D B A E (4) B C D E A (4) E B A D C (3) C E B D A (3) B E C D A (3) B D A C E (3) E C A D B (2) E B C A D (2) E A D C B (2) E A C B D (2) D B A C E (2) C E A B D (2) C A D B E (2) B D C E A (2) B D A E C (2) B C D A E (2) A D E B C (2) A D B C E (2) E C B D A (1) E B C D A (1) E B A C D (1) E A B C D (1) D B A E C (1) D A C B E (1) C B D E A (1) C B D A E (1) B E D C A (1) B D E A C (1) A E D C B (1) A E D B C (1) A D E C B (1) A D C B E (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -22 14 -24 B 10 0 -10 16 -12 C 22 10 0 26 8 D -14 -16 -26 0 -14 E 24 12 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 14 -24 B 10 0 -10 16 -12 C 22 10 0 26 8 D -14 -16 -26 0 -14 E 24 12 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=26 B=23 A=10 D=9 so D is eliminated. Round 2 votes counts: E=32 C=26 B=26 A=16 so A is eliminated. Round 3 votes counts: E=37 B=34 C=29 so C is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:233 E:221 B:202 A:179 D:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -22 14 -24 B 10 0 -10 16 -12 C 22 10 0 26 8 D -14 -16 -26 0 -14 E 24 12 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 14 -24 B 10 0 -10 16 -12 C 22 10 0 26 8 D -14 -16 -26 0 -14 E 24 12 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 14 -24 B 10 0 -10 16 -12 C 22 10 0 26 8 D -14 -16 -26 0 -14 E 24 12 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4422: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) A B C D E (8) B A C D E (7) C A B D E (6) C D A B E (5) E B A C D (4) D C A B E (4) C A D B E (4) E D C B A (3) E C D A B (3) D E C B A (3) D E B A C (3) B A D C E (3) B A C E D (3) E D B C A (2) E C A B D (2) E B D A C (2) D C A E B (2) D B A C E (2) C E A B D (2) B E A C D (2) B A E D C (2) A B C E D (2) E D B A C (1) E C D B A (1) E B A D C (1) D E B C A (1) D C E A B (1) D B A E C (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E A B (1) C A E B D (1) C A B E D (1) B A E C D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 16 -8 2 12 B -16 0 -8 -2 10 C 8 8 0 12 14 D -2 2 -12 0 8 E -12 -10 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -8 2 12 B -16 0 -8 -2 10 C 8 8 0 12 14 D -2 2 -12 0 8 E -12 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=21 D=19 B=18 A=12 so A is eliminated. Round 2 votes counts: E=30 B=28 C=23 D=19 so D is eliminated. Round 3 votes counts: E=37 B=32 C=31 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:221 A:211 D:198 B:192 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -8 2 12 B -16 0 -8 -2 10 C 8 8 0 12 14 D -2 2 -12 0 8 E -12 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 2 12 B -16 0 -8 -2 10 C 8 8 0 12 14 D -2 2 -12 0 8 E -12 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 2 12 B -16 0 -8 -2 10 C 8 8 0 12 14 D -2 2 -12 0 8 E -12 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4423: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) A D B E C (9) C E D A B (8) B A D E C (6) A D C E B (6) C E D B A (4) C A E D B (4) D E A C B (3) B E C D A (3) E C B D A (2) E B C D A (2) D A E B C (2) C E B A D (2) C E A D B (2) C D E A B (2) C A D E B (2) C A B E D (2) B E D C A (2) B E D A C (2) B C E D A (2) B C A E D (2) B A C E D (2) A C B D E (2) E D C B A (1) E D B A C (1) E B D C A (1) D E B A C (1) D E A B C (1) D B E A C (1) D A C E B (1) D A B E C (1) C E A B D (1) C B E A D (1) C A B D E (1) B E C A D (1) B D E A C (1) B D A E C (1) B C E A D (1) B A E C D (1) A D B C E (1) A C D E B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -10 -2 -8 B -2 0 -8 -2 -12 C 10 8 0 16 14 D 2 2 -16 0 -12 E 8 12 -14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -2 -8 B -2 0 -8 -2 -12 C 10 8 0 16 14 D 2 2 -16 0 -12 E 8 12 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=24 A=21 D=10 E=7 so E is eliminated. Round 2 votes counts: C=40 B=27 A=21 D=12 so D is eliminated. Round 3 votes counts: C=41 B=30 A=29 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:209 A:191 B:188 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -2 -8 B -2 0 -8 -2 -12 C 10 8 0 16 14 D 2 2 -16 0 -12 E 8 12 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -2 -8 B -2 0 -8 -2 -12 C 10 8 0 16 14 D 2 2 -16 0 -12 E 8 12 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -2 -8 B -2 0 -8 -2 -12 C 10 8 0 16 14 D 2 2 -16 0 -12 E 8 12 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4424: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (13) B E A D C (10) D C A E B (9) C D B A E (8) C D A E B (6) C D A B E (5) A E B D C (5) E A B D C (4) D A E C B (4) B C E A D (4) E B A D C (3) D A C E B (3) C B D E A (3) B C E D A (3) A E D B C (3) D E A C B (2) C D B E A (2) B C D E A (2) A E B C D (2) D C E B A (1) D C E A B (1) D C B E A (1) D A E B C (1) C B E A D (1) C B A D E (1) B E D A C (1) B E C A D (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 4 -4 -4 B 8 0 4 6 10 C -4 -4 0 2 0 D 4 -6 -2 0 0 E 4 -10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -4 -4 B 8 0 4 6 10 C -4 -4 0 2 0 D 4 -6 -2 0 0 E 4 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=26 D=22 A=11 E=7 so E is eliminated. Round 2 votes counts: B=37 C=26 D=22 A=15 so A is eliminated. Round 3 votes counts: B=48 D=26 C=26 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:198 C:197 E:197 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -4 -4 B 8 0 4 6 10 C -4 -4 0 2 0 D 4 -6 -2 0 0 E 4 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -4 -4 B 8 0 4 6 10 C -4 -4 0 2 0 D 4 -6 -2 0 0 E 4 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -4 -4 B 8 0 4 6 10 C -4 -4 0 2 0 D 4 -6 -2 0 0 E 4 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4425: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) A B C E D (9) E D C B A (7) E A D B C (6) D E C B A (6) D C B A E (6) E D A C B (5) A B C D E (5) C B D A E (4) B C A E D (4) A E B C D (4) E D A B C (3) D C B E A (3) B C A D E (3) D E C A B (2) D E A C B (2) D C E B A (2) D C A B E (2) C D B A E (2) B A C E D (2) A C B D E (2) E D C A B (1) E D B C A (1) E B C D A (1) E B C A D (1) E B A C D (1) E A D C B (1) E A B D C (1) C B D E A (1) C A B D E (1) B A C D E (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 6 4 -6 B -10 0 2 -2 -8 C -6 -2 0 2 -4 D -4 2 -2 0 -12 E 6 8 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 6 4 -6 B -10 0 2 -2 -8 C -6 -2 0 2 -4 D -4 2 -2 0 -12 E 6 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=23 A=22 B=10 C=8 so C is eliminated. Round 2 votes counts: E=37 D=25 A=23 B=15 so B is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:207 C:195 D:192 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 4 -6 B -10 0 2 -2 -8 C -6 -2 0 2 -4 D -4 2 -2 0 -12 E 6 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 -6 B -10 0 2 -2 -8 C -6 -2 0 2 -4 D -4 2 -2 0 -12 E 6 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 -6 B -10 0 2 -2 -8 C -6 -2 0 2 -4 D -4 2 -2 0 -12 E 6 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4426: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) C A D E B (7) A C E B D (7) E B D C A (6) B E D A C (6) A B E C D (5) E B A C D (4) D C E A B (4) D B E C A (4) B E A C D (4) D C E B A (3) B E D C A (3) A C D B E (3) A C B E D (3) A C B D E (3) A B C E D (3) E B D A C (2) D C B E A (2) D C A E B (2) D B E A C (2) C D A E B (2) A E B C D (2) E D C B A (1) E D B C A (1) E B C A D (1) D E C B A (1) D E B C A (1) D C B A E (1) D B C E A (1) D B A C E (1) C E D A B (1) C D E A B (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) B D A C E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 0 -12 -2 B -2 0 2 8 6 C 0 -2 0 -6 6 D 12 -8 6 0 -2 E 2 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363636 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.438016528921 Cumulative probabilities = A: 0.363636 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -12 -2 B -2 0 2 8 6 C 0 -2 0 -6 6 D 12 -8 6 0 -2 E 2 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.438016528932 Cumulative probabilities = A: 0.363636 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 B=18 E=15 C=11 so C is eliminated. Round 2 votes counts: A=34 D=32 B=18 E=16 so E is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:207 D:204 C:199 E:196 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 0 -12 -2 B -2 0 2 8 6 C 0 -2 0 -6 6 D 12 -8 6 0 -2 E 2 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.438016528932 Cumulative probabilities = A: 0.363636 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -12 -2 B -2 0 2 8 6 C 0 -2 0 -6 6 D 12 -8 6 0 -2 E 2 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.438016528932 Cumulative probabilities = A: 0.363636 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -12 -2 B -2 0 2 8 6 C 0 -2 0 -6 6 D 12 -8 6 0 -2 E 2 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.000000 Sum of squares = 0.438016528932 Cumulative probabilities = A: 0.363636 B: 0.909091 C: 0.909091 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4427: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) C B A E D (8) C A B D E (6) E D B C A (5) B C E D A (5) E D B A C (4) D E A C B (4) D E C A B (3) B E D C A (3) B E C D A (3) B C A E D (3) B A C E D (3) A D E C B (3) A C D E B (3) A C D B E (3) A C B D E (3) D E B C A (2) D E B A C (2) D A E C B (2) D A E B C (2) A E D B C (2) A D E B C (2) A D C E B (2) E D C B A (1) E C D B A (1) E B D C A (1) D E C B A (1) D A C E B (1) C E D B A (1) C D E A B (1) C D A B E (1) C B E D A (1) C B E A D (1) C B D E A (1) C A D E B (1) C A D B E (1) B E D A C (1) B C E A D (1) B A E D C (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 -8 0 B -2 0 2 -12 -4 C 2 -2 0 -4 -6 D 8 12 4 0 8 E 0 4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -8 0 B -2 0 2 -12 -4 C 2 -2 0 -4 -6 D 8 12 4 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 A=21 B=20 E=12 so E is eliminated. Round 2 votes counts: D=35 C=23 B=21 A=21 so B is eliminated. Round 3 votes counts: D=40 C=35 A=25 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:201 A:196 C:195 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -8 0 B -2 0 2 -12 -4 C 2 -2 0 -4 -6 D 8 12 4 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -8 0 B -2 0 2 -12 -4 C 2 -2 0 -4 -6 D 8 12 4 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -8 0 B -2 0 2 -12 -4 C 2 -2 0 -4 -6 D 8 12 4 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4428: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) B C D A E (6) D E C B A (5) C A B D E (4) B D C A E (4) B A C D E (4) A E C B D (4) A C B E D (4) E A D C B (3) E A C B D (3) C B A D E (3) A E B C D (3) A C E B D (3) A C B D E (3) E D B A C (2) E C A D B (2) E A B D C (2) E A B C D (2) D E B C A (2) D C B E A (2) D B E C A (2) D B E A C (2) D B C E A (2) D B C A E (2) C D B A E (2) B A E D C (2) A B E C D (2) A B C E D (2) E D C A B (1) E D A C B (1) E D A B C (1) D E B A C (1) D C E B A (1) C E A D B (1) C D E A B (1) C D A B E (1) C B D A E (1) C A E B D (1) B E D A C (1) B D A E C (1) B D A C E (1) B C A D E (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 12 16 12 B -4 0 -6 18 8 C -12 6 0 20 -2 D -16 -18 -20 0 4 E -12 -8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 16 12 B -4 0 -6 18 8 C -12 6 0 20 -2 D -16 -18 -20 0 4 E -12 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=22 B=21 D=19 C=14 so C is eliminated. Round 2 votes counts: A=27 E=25 B=25 D=23 so D is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:222 B:208 C:206 E:189 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 16 12 B -4 0 -6 18 8 C -12 6 0 20 -2 D -16 -18 -20 0 4 E -12 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 16 12 B -4 0 -6 18 8 C -12 6 0 20 -2 D -16 -18 -20 0 4 E -12 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 16 12 B -4 0 -6 18 8 C -12 6 0 20 -2 D -16 -18 -20 0 4 E -12 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4429: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) E A B C D (7) D B C A E (7) C D A E B (6) A E B D C (5) B E A D C (4) B D C E A (4) A D C E B (4) E A C D B (3) D C A B E (3) B E A C D (3) A E D C B (3) E C A D B (2) E B C D A (2) E A C B D (2) E A B D C (2) D C B A E (2) D A C B E (2) C E D A B (2) C D E A B (2) C D B E A (2) B E C D A (2) A D E C B (2) E C D A B (1) E C A B D (1) E B C A D (1) E B A C D (1) D C B E A (1) D A B C E (1) C D A B E (1) C B E D A (1) C B D E A (1) C A D E B (1) B D E A C (1) B D C A E (1) B D A E C (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D B C (1) A E B C D (1) A C D E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 26 10 10 6 B -26 0 -2 -10 -16 C -10 2 0 6 -12 D -10 10 -6 0 -10 E -6 16 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 10 10 6 B -26 0 -2 -10 -16 C -10 2 0 6 -12 D -10 10 -6 0 -10 E -6 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 B=19 D=16 C=16 so D is eliminated. Round 2 votes counts: A=30 B=26 E=22 C=22 so E is eliminated. Round 3 votes counts: A=44 B=30 C=26 so C is eliminated. Round 4 votes counts: A=63 B=37 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:216 C:193 D:192 B:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 10 10 6 B -26 0 -2 -10 -16 C -10 2 0 6 -12 D -10 10 -6 0 -10 E -6 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 10 10 6 B -26 0 -2 -10 -16 C -10 2 0 6 -12 D -10 10 -6 0 -10 E -6 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 10 10 6 B -26 0 -2 -10 -16 C -10 2 0 6 -12 D -10 10 -6 0 -10 E -6 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4430: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (14) E A D C B (12) B C D A E (10) D E B C A (9) C B A D E (9) A C E B D (5) A C B E D (5) E A D B C (4) A E C B D (4) E D B C A (3) C A B D E (3) B D C E A (3) E D B A C (2) E A C D B (2) D B E C A (2) C B A E D (2) A C B D E (2) E D C B A (1) E C A B D (1) E A C B D (1) D B C A E (1) C B D A E (1) B D C A E (1) B C D E A (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 10 6 4 -10 B -10 0 0 -4 -20 C -6 0 0 -6 -14 D -4 4 6 0 -16 E 10 20 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 6 4 -10 B -10 0 0 -4 -20 C -6 0 0 -6 -14 D -4 4 6 0 -16 E 10 20 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=18 C=15 B=15 D=12 so D is eliminated. Round 2 votes counts: E=49 B=18 A=18 C=15 so C is eliminated. Round 3 votes counts: E=49 B=30 A=21 so A is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 A:205 D:195 C:187 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 4 -10 B -10 0 0 -4 -20 C -6 0 0 -6 -14 D -4 4 6 0 -16 E 10 20 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 -10 B -10 0 0 -4 -20 C -6 0 0 -6 -14 D -4 4 6 0 -16 E 10 20 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 -10 B -10 0 0 -4 -20 C -6 0 0 -6 -14 D -4 4 6 0 -16 E 10 20 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4431: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) E B D C A (7) D E B A C (7) D A C E B (7) C A B E D (6) A C B D E (5) D E A B C (4) B E C A D (4) E D C A B (3) E B C D A (3) D E A C B (3) D B E A C (3) D A C B E (3) D A B C E (3) C A E D B (3) D B A C E (2) D A B E C (2) C B A E D (2) C A E B D (2) C A D E B (2) B E D A C (2) B A C E D (2) B A C D E (2) A D C B E (2) A C D B E (2) A B C D E (2) E D B A C (1) E C D B A (1) E C B A D (1) E C A B D (1) E B C A D (1) D E C A B (1) C E B A D (1) C E A B D (1) B C A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 4 6 -18 -2 B -4 0 8 -12 -12 C -6 -8 0 -16 -2 D 18 12 16 0 2 E 2 12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -18 -2 B -4 0 8 -12 -12 C -6 -8 0 -16 -2 D 18 12 16 0 2 E 2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999976214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=25 C=17 B=12 A=11 so A is eliminated. Round 2 votes counts: D=37 E=25 C=24 B=14 so B is eliminated. Round 3 votes counts: D=38 E=31 C=31 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:207 A:195 B:190 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -18 -2 B -4 0 8 -12 -12 C -6 -8 0 -16 -2 D 18 12 16 0 2 E 2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999976214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -18 -2 B -4 0 8 -12 -12 C -6 -8 0 -16 -2 D 18 12 16 0 2 E 2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999976214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -18 -2 B -4 0 8 -12 -12 C -6 -8 0 -16 -2 D 18 12 16 0 2 E 2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999976214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4432: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) D B A C E (6) D B C E A (5) C E D B A (5) B D E C A (5) B D A E C (5) A B D C E (5) C E A D B (4) A B D E C (4) E C D B A (3) E A C B D (3) D B E C A (3) D B C A E (3) B A D E C (3) B A D C E (3) A D B C E (3) A C E B D (3) A C D B E (3) D C A B E (2) C E D A B (2) B E D A C (2) B D E A C (2) B A E D C (2) E C D A B (1) E C A D B (1) E B D C A (1) E B A D C (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B A E (1) C E A B D (1) C D E B A (1) C D A B E (1) C A E D B (1) A E B D C (1) A D C B E (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 2 -2 0 B 8 0 16 2 24 C -2 -16 0 -24 4 D 2 -2 24 0 18 E 0 -24 -4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -2 0 B 8 0 16 2 24 C -2 -16 0 -24 4 D 2 -2 24 0 18 E 0 -24 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980206 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 A=23 B=22 E=17 C=15 so C is eliminated. Round 2 votes counts: E=29 D=25 A=24 B=22 so B is eliminated. Round 3 votes counts: D=37 A=32 E=31 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:225 D:221 A:196 C:181 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -2 0 B 8 0 16 2 24 C -2 -16 0 -24 4 D 2 -2 24 0 18 E 0 -24 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980206 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 0 B 8 0 16 2 24 C -2 -16 0 -24 4 D 2 -2 24 0 18 E 0 -24 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980206 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 0 B 8 0 16 2 24 C -2 -16 0 -24 4 D 2 -2 24 0 18 E 0 -24 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980206 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4433: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) C E D A B (11) B D A C E (11) D B C E A (6) E C A D B (5) E C A B D (5) C E A D B (5) A E C B D (5) D C E A B (4) B A E C D (4) A E C D B (4) A B E C D (4) D A B E C (3) B D A E C (3) D C E B A (2) C E D B A (2) A D E C B (2) E A C B D (1) D C B E A (1) D B C A E (1) D B A C E (1) D A E C B (1) D A C B E (1) C E A B D (1) C D E A B (1) B D C E A (1) B C E A D (1) B C D E A (1) A D B E C (1) Total count = 100 A B C D E A 0 8 6 0 6 B -8 0 -2 -2 2 C -6 2 0 0 0 D 0 2 0 0 4 E -6 -2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.336792 B: 0.000000 C: 0.000000 D: 0.663208 E: 0.000000 Sum of squares = 0.553273894573 Cumulative probabilities = A: 0.336792 B: 0.336792 C: 0.336792 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 0 6 B -8 0 -2 -2 2 C -6 2 0 0 0 D 0 2 0 0 4 E -6 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=20 C=20 A=16 E=11 so E is eliminated. Round 2 votes counts: B=33 C=30 D=20 A=17 so A is eliminated. Round 3 votes counts: C=40 B=37 D=23 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:210 D:203 C:198 B:195 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 0 6 B -8 0 -2 -2 2 C -6 2 0 0 0 D 0 2 0 0 4 E -6 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 0 6 B -8 0 -2 -2 2 C -6 2 0 0 0 D 0 2 0 0 4 E -6 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 0 6 B -8 0 -2 -2 2 C -6 2 0 0 0 D 0 2 0 0 4 E -6 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4434: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) D B E A C (9) E A D C B (8) C A E B D (8) B D C A E (8) B C D E A (6) C B E A D (5) A E C D B (5) C E A B D (4) A E D C B (4) E A D B C (3) D E A B C (3) C B D A E (3) C B A E D (3) B D C E A (3) D B A E C (2) D A E B C (2) C A E D B (2) B C D A E (2) E D A B C (1) D E B A C (1) D B A C E (1) C E B A D (1) C B D E A (1) C B A D E (1) B D E C A (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 4 14 -12 B -6 0 -14 -8 -10 C -4 14 0 6 0 D -14 8 -6 0 -10 E 12 10 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.457723 D: 0.000000 E: 0.542277 Sum of squares = 0.503574635934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.457723 D: 0.457723 E: 1.000000 A B C D E A 0 6 4 14 -12 B -6 0 -14 -8 -10 C -4 14 0 6 0 D -14 8 -6 0 -10 E 12 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=22 B=20 D=18 A=12 so A is eliminated. Round 2 votes counts: E=31 C=30 B=20 D=19 so D is eliminated. Round 3 votes counts: E=38 B=32 C=30 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:208 A:206 D:189 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 14 -12 B -6 0 -14 -8 -10 C -4 14 0 6 0 D -14 8 -6 0 -10 E 12 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 14 -12 B -6 0 -14 -8 -10 C -4 14 0 6 0 D -14 8 -6 0 -10 E 12 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 14 -12 B -6 0 -14 -8 -10 C -4 14 0 6 0 D -14 8 -6 0 -10 E 12 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4435: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) C B E A D (6) D A E C B (5) A D E C B (5) A D C E B (5) B E C D A (4) B C E A D (4) A C E D B (4) E D C A B (3) E B D C A (3) D E A C B (3) D B A E C (3) C A E B D (3) B D E C A (3) B A D C E (3) A C B E D (3) E B C D A (2) D E A B C (2) D B E A C (2) D A B E C (2) C E A D B (2) C A B E D (2) A B D C E (2) E D B C A (1) E C D B A (1) E C B D A (1) E A C D B (1) D E B C A (1) D A E B C (1) C E B D A (1) C E B A D (1) C E A B D (1) C B A E D (1) C A E D B (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D E A (1) A E C D B (1) A D C B E (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -4 -4 -6 B 2 0 -4 10 4 C 4 4 0 0 6 D 4 -10 0 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.806911 D: 0.193089 E: 0.000000 Sum of squares = 0.68838849723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.806911 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -6 B 2 0 -4 10 4 C 4 4 0 0 6 D 4 -10 0 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735587 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=23 D=19 C=18 E=12 so E is eliminated. Round 2 votes counts: B=33 A=24 D=23 C=20 so C is eliminated. Round 3 votes counts: B=43 A=33 D=24 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:207 B:206 E:203 A:192 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -6 B 2 0 -4 10 4 C 4 4 0 0 6 D 4 -10 0 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735587 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -6 B 2 0 -4 10 4 C 4 4 0 0 6 D 4 -10 0 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735587 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -6 B 2 0 -4 10 4 C 4 4 0 0 6 D 4 -10 0 0 -10 E 6 -4 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836735587 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4436: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) D C B E A (6) A E B D C (6) D C B A E (4) D C A B E (4) D B C E A (4) A D B C E (4) E C B D A (3) E A C B D (3) D B C A E (3) C E D B A (3) B D C E A (3) A E B C D (3) A B E D C (3) E C B A D (2) D B A C E (2) C E D A B (2) C D E A B (2) C D A E B (2) B E C D A (2) B D C A E (2) B C D E A (2) A E C B D (2) E C D B A (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A C D (1) E A B D C (1) E A B C D (1) D C A E B (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E B A (1) C B E D A (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A C E (1) B A E D C (1) B A D E C (1) A E D C B (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -14 -20 -24 -6 B 14 0 -2 -10 14 C 20 2 0 -12 18 D 24 10 12 0 14 E 6 -14 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -24 -6 B 14 0 -2 -10 14 C 20 2 0 -12 18 D 24 10 12 0 14 E 6 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=24 C=20 E=15 B=15 so E is eliminated. Round 2 votes counts: A=29 C=27 D=26 B=18 so B is eliminated. Round 3 votes counts: A=34 D=33 C=33 so D is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:230 C:214 B:208 E:180 A:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -20 -24 -6 B 14 0 -2 -10 14 C 20 2 0 -12 18 D 24 10 12 0 14 E 6 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -24 -6 B 14 0 -2 -10 14 C 20 2 0 -12 18 D 24 10 12 0 14 E 6 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -24 -6 B 14 0 -2 -10 14 C 20 2 0 -12 18 D 24 10 12 0 14 E 6 -14 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4437: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (14) D B C A E (8) E A D C B (7) E A C B D (7) E B C A D (5) C B A E D (5) A E D C B (4) E C B A D (3) D E A B C (3) D B C E A (3) C B A D E (3) A D E C B (3) E A D B C (2) E A C D B (2) D A C B E (2) B D C A E (2) B C D E A (2) A E C D B (2) A E C B D (2) A C E B D (2) E D B A C (1) E D A C B (1) E B A C D (1) E A B C D (1) D E B A C (1) D C B A E (1) D B E C A (1) D A E C B (1) D A E B C (1) D A C E B (1) C D B A E (1) C B E A D (1) C A B E D (1) C A B D E (1) B D C E A (1) B C E D A (1) B C E A D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -8 10 12 B 10 0 -4 10 0 C 8 4 0 14 4 D -10 -10 -14 0 0 E -12 0 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 10 12 B 10 0 -4 10 0 C 8 4 0 14 4 D -10 -10 -14 0 0 E -12 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=22 B=21 A=15 C=12 so C is eliminated. Round 2 votes counts: E=30 B=30 D=23 A=17 so A is eliminated. Round 3 votes counts: E=40 B=34 D=26 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:208 A:202 E:192 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 10 12 B 10 0 -4 10 0 C 8 4 0 14 4 D -10 -10 -14 0 0 E -12 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 10 12 B 10 0 -4 10 0 C 8 4 0 14 4 D -10 -10 -14 0 0 E -12 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 10 12 B 10 0 -4 10 0 C 8 4 0 14 4 D -10 -10 -14 0 0 E -12 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4438: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) C A B E D (10) D E B A C (8) E D B C A (7) B C A D E (6) B A C D E (6) D B E A C (4) C A E D B (4) E D A C B (3) D E A B C (3) C B A E D (3) B D E A C (3) B C A E D (3) C E A D B (2) C A E B D (2) B D E C A (2) B A D C E (2) A D C E B (2) A C B E D (2) A C B D E (2) E D C B A (1) E D A B C (1) E A D C B (1) D E B C A (1) D E A C B (1) D B A E C (1) C A B D E (1) B D A E C (1) B C D E A (1) A D B E C (1) A C E D B (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 4 2 B -2 0 6 -6 4 C 8 -6 0 -10 0 D -4 6 10 0 -2 E -2 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.181818 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825726 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.636364 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 4 2 B -2 0 6 -6 4 C 8 -6 0 -10 0 D -4 6 10 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.181818 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825305 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.636364 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 C=22 D=18 A=12 so A is eliminated. Round 2 votes counts: C=28 B=27 E=24 D=21 so D is eliminated. Round 3 votes counts: E=37 B=33 C=30 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:205 B:201 A:200 E:198 C:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 4 2 B -2 0 6 -6 4 C 8 -6 0 -10 0 D -4 6 10 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.181818 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825305 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.636364 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 2 B -2 0 6 -6 4 C 8 -6 0 -10 0 D -4 6 10 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.181818 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825305 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.636364 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 2 B -2 0 6 -6 4 C 8 -6 0 -10 0 D -4 6 10 0 -2 E -2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.181818 D: 0.363636 E: 0.000000 Sum of squares = 0.371900825305 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.636364 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4439: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) D B C E A (5) C B D E A (5) A E D B C (5) A E C B D (5) A E B D C (5) E C D A B (4) E A D C B (4) B D C E A (4) B D C A E (4) B C D E A (4) A E D C B (4) C E D B A (3) B A C D E (3) A B E D C (3) E D C A B (2) E D A C B (2) E C A D B (2) C B A E D (2) B C A E D (2) A E C D B (2) A B E C D (2) A B D C E (2) E D C B A (1) E C D B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D C B E A (1) D B A E C (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E D A (1) C A B E D (1) B D A C E (1) B A D C E (1) A D E B C (1) A D B E C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -10 -4 10 B 0 0 10 12 8 C 10 -10 0 -2 4 D 4 -12 2 0 -6 E -10 -8 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.370815 B: 0.629185 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.533377423696 Cumulative probabilities = A: 0.370815 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -4 10 B 0 0 10 12 8 C 10 -10 0 -2 4 D 4 -12 2 0 -6 E -10 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499631 B: 0.500369 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272483 Cumulative probabilities = A: 0.499631 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=27 E=16 C=15 D=10 so D is eliminated. Round 2 votes counts: B=33 A=32 E=18 C=17 so C is eliminated. Round 3 votes counts: B=43 A=33 E=24 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:215 C:201 A:198 D:194 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -10 -4 10 B 0 0 10 12 8 C 10 -10 0 -2 4 D 4 -12 2 0 -6 E -10 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499631 B: 0.500369 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272483 Cumulative probabilities = A: 0.499631 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 10 B 0 0 10 12 8 C 10 -10 0 -2 4 D 4 -12 2 0 -6 E -10 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499631 B: 0.500369 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272483 Cumulative probabilities = A: 0.499631 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 10 B 0 0 10 12 8 C 10 -10 0 -2 4 D 4 -12 2 0 -6 E -10 -8 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499631 B: 0.500369 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272483 Cumulative probabilities = A: 0.499631 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4440: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) E B D A C (7) C D A B E (7) D B E C A (5) C A B E D (5) E B A C D (4) D E B A C (4) D C A E B (4) A C E B D (4) D E B C A (3) A E C B D (3) A E B C D (3) E B A D C (2) E A B C D (2) D E A B C (2) D B C E A (2) D A C E B (2) C D A E B (2) C A B D E (2) A E D B C (2) A C D E B (2) A C B E D (2) E D B A C (1) E A B D C (1) D E A C B (1) D C E A B (1) D C B A E (1) D C A B E (1) D B E A C (1) C D B A E (1) C B D A E (1) C B A E D (1) C A E B D (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E C A (1) B C E D A (1) B C A E D (1) A D E C B (1) A D C E B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 12 0 2 B -6 0 8 6 -10 C -12 -8 0 10 -10 D 0 -6 -10 0 -6 E -2 10 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.783531 B: 0.000000 C: 0.000000 D: 0.216469 E: 0.000000 Sum of squares = 0.660780059252 Cumulative probabilities = A: 0.783531 B: 0.783531 C: 0.783531 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 0 2 B -6 0 8 6 -10 C -12 -8 0 10 -10 D 0 -6 -10 0 -6 E -2 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000020324 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=22 A=20 E=17 B=14 so B is eliminated. Round 2 votes counts: E=28 D=28 C=24 A=20 so A is eliminated. Round 3 votes counts: E=37 C=33 D=30 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:210 B:199 C:190 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 0 2 B -6 0 8 6 -10 C -12 -8 0 10 -10 D 0 -6 -10 0 -6 E -2 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000020324 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 0 2 B -6 0 8 6 -10 C -12 -8 0 10 -10 D 0 -6 -10 0 -6 E -2 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000020324 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 0 2 B -6 0 8 6 -10 C -12 -8 0 10 -10 D 0 -6 -10 0 -6 E -2 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000020324 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4441: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) B A C D E (6) E D C A B (5) A C B E D (5) A B C D E (5) C A B E D (4) B A D C E (4) A B D E C (4) A B C E D (4) E C D A B (3) D E C B A (3) C E D B A (3) A E C D B (3) E D B A C (2) E D A B C (2) E A C D B (2) D E B A C (2) D C B E A (2) D B E A C (2) C D E B A (2) C B A D E (2) C A E B D (2) B D C E A (2) B D C A E (2) B C A D E (2) A B E D C (2) E D B C A (1) E D A C B (1) D C E B A (1) C E D A B (1) C E A D B (1) C D B E A (1) C B D E A (1) C A E D B (1) C A B D E (1) B D E A C (1) B D A E C (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B D C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 4 8 8 B -2 0 -8 6 6 C -4 8 0 4 6 D -8 -6 -4 0 -10 E -8 -6 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 8 B -2 0 -8 6 6 C -4 8 0 4 6 D -8 -6 -4 0 -10 E -8 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 C=19 B=19 D=10 so D is eliminated. Round 2 votes counts: E=29 A=28 C=22 B=21 so B is eliminated. Round 3 votes counts: A=40 E=32 C=28 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:207 B:201 E:195 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 8 B -2 0 -8 6 6 C -4 8 0 4 6 D -8 -6 -4 0 -10 E -8 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 8 B -2 0 -8 6 6 C -4 8 0 4 6 D -8 -6 -4 0 -10 E -8 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 8 B -2 0 -8 6 6 C -4 8 0 4 6 D -8 -6 -4 0 -10 E -8 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4442: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) E A C D B (7) D B E C A (6) D B A C E (5) B D C A E (5) A E C B D (5) E A D C B (4) E A C B D (4) C B A E D (4) B C A E D (4) B C A D E (4) E D A C B (3) E C A B D (3) D E B C A (3) D B E A C (3) D B C E A (3) D B A E C (3) C A B E D (3) E C B A D (2) D A B E C (2) D A B C E (2) B C E A D (2) A E D C B (2) A C E B D (2) E B D C A (1) D E A C B (1) D E A B C (1) B D C E A (1) B C D A E (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -8 -4 12 B 18 0 18 -16 22 C 8 -18 0 -14 -2 D 4 16 14 0 6 E -12 -22 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 -4 12 B 18 0 18 -16 22 C 8 -18 0 -14 -2 D 4 16 14 0 6 E -12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=24 B=17 A=11 C=7 so C is eliminated. Round 2 votes counts: D=41 E=24 B=21 A=14 so A is eliminated. Round 3 votes counts: D=41 E=34 B=25 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:221 D:220 A:191 C:187 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -8 -4 12 B 18 0 18 -16 22 C 8 -18 0 -14 -2 D 4 16 14 0 6 E -12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -4 12 B 18 0 18 -16 22 C 8 -18 0 -14 -2 D 4 16 14 0 6 E -12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -4 12 B 18 0 18 -16 22 C 8 -18 0 -14 -2 D 4 16 14 0 6 E -12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4443: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) D A E B C (6) C A B E D (5) A D B E C (5) E B C D A (4) C E D B A (4) D E C B A (3) C E B D A (3) C D E B A (3) C D E A B (3) C D A E B (3) C A D E B (3) B E D A C (3) B E A D C (3) A C D B E (3) A C B E D (3) A B E D C (3) A B D E C (3) E D B C A (2) E C D B A (2) D E B A C (2) B E D C A (2) B A E C D (2) A C D E B (2) A B C E D (2) E D B A C (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E B A (1) D C E A B (1) D A E C B (1) C B A E D (1) C A E B D (1) B E C D A (1) B D E A C (1) B C E A D (1) B C A E D (1) B A E D C (1) A D E B C (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 2 -2 B 0 0 -6 0 2 C 8 6 0 14 2 D -2 0 -14 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 2 -2 B 0 0 -6 0 2 C 8 6 0 14 2 D -2 0 -14 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=25 D=17 B=15 E=9 so E is eliminated. Round 2 votes counts: C=36 A=25 D=20 B=19 so B is eliminated. Round 3 votes counts: C=43 A=31 D=26 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:202 B:198 A:196 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 2 -2 B 0 0 -6 0 2 C 8 6 0 14 2 D -2 0 -14 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 -2 B 0 0 -6 0 2 C 8 6 0 14 2 D -2 0 -14 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 -2 B 0 0 -6 0 2 C 8 6 0 14 2 D -2 0 -14 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4444: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) D E C A B (6) D E A C B (6) D C B A E (5) C B A D E (5) B C A E D (5) A C B D E (5) E B A C D (4) E A B C D (4) B C A D E (4) A B C E D (4) E D B C A (3) E D A C B (3) E B D C A (3) E B C A D (3) B A C E D (3) E D B A C (2) E A B D C (2) D E C B A (2) D C E B A (2) C D B A E (2) C B D A E (2) C A B D E (2) B C E A D (2) B A E C D (2) E D C A B (1) E D A B C (1) E A D B C (1) D C B E A (1) D A E C B (1) D A C B E (1) C D B E A (1) B E A C D (1) A E B C D (1) A D E C B (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -12 2 4 B 4 0 -8 6 8 C 12 8 0 2 4 D -2 -6 -2 0 8 E -4 -8 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 2 4 B 4 0 -8 6 8 C 12 8 0 2 4 D -2 -6 -2 0 8 E -4 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=27 B=17 A=13 C=12 so C is eliminated. Round 2 votes counts: D=34 E=27 B=24 A=15 so A is eliminated. Round 3 votes counts: D=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:205 D:199 A:195 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 2 4 B 4 0 -8 6 8 C 12 8 0 2 4 D -2 -6 -2 0 8 E -4 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 2 4 B 4 0 -8 6 8 C 12 8 0 2 4 D -2 -6 -2 0 8 E -4 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 2 4 B 4 0 -8 6 8 C 12 8 0 2 4 D -2 -6 -2 0 8 E -4 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4445: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (18) C D A E B (11) B C D E A (6) A E B D C (6) C D B A E (5) B E A D C (5) A E D B C (5) A D C E B (4) E A B D C (3) D C E A B (3) A C D E B (3) D B C E A (2) C B D E A (2) B E C D A (2) B A E D C (2) B A E C D (2) A E D C B (2) A B E C D (2) E B D C A (1) E B D A C (1) E B A D C (1) E A D B C (1) D C B E A (1) D A E C B (1) C D E B A (1) C B D A E (1) C A D E B (1) C A D B E (1) B E A C D (1) B C E D A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -18 -14 2 B 12 0 -10 -18 12 C 18 10 0 24 30 D 14 18 -24 0 26 E -2 -12 -30 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -14 2 B 12 0 -10 -18 12 C 18 10 0 24 30 D 14 18 -24 0 26 E -2 -12 -30 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=23 A=23 E=7 D=7 so E is eliminated. Round 2 votes counts: C=40 A=27 B=26 D=7 so D is eliminated. Round 3 votes counts: C=44 B=28 A=28 so B is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:241 D:217 B:198 A:179 E:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -18 -14 2 B 12 0 -10 -18 12 C 18 10 0 24 30 D 14 18 -24 0 26 E -2 -12 -30 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -14 2 B 12 0 -10 -18 12 C 18 10 0 24 30 D 14 18 -24 0 26 E -2 -12 -30 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -14 2 B 12 0 -10 -18 12 C 18 10 0 24 30 D 14 18 -24 0 26 E -2 -12 -30 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4446: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) D E B A C (8) B C A E D (8) E D A C B (7) D E B C A (7) B C A D E (7) A C E D B (6) E D B C A (5) C B A E D (3) B E C D A (3) B C E A D (3) A C B E D (3) A C B D E (3) E A C D B (2) D E A B C (2) B E D C A (2) B D E C A (2) B C E D A (2) A E D C B (2) A C E B D (2) A C D E B (2) E D B A C (1) E C B D A (1) D B E C A (1) D A E C B (1) C E B A D (1) C A E B D (1) C A B E D (1) C A B D E (1) B D C E A (1) B A C D E (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 2 -4 -14 B 12 0 6 -10 -18 C -2 -6 0 2 -10 D 4 10 -2 0 -8 E 14 18 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 2 -4 -14 B 12 0 6 -10 -18 C -2 -6 0 2 -10 D 4 10 -2 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=28 A=20 E=16 C=7 so C is eliminated. Round 2 votes counts: B=32 D=28 A=23 E=17 so E is eliminated. Round 3 votes counts: D=41 B=34 A=25 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:225 D:202 B:195 C:192 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 2 -4 -14 B 12 0 6 -10 -18 C -2 -6 0 2 -10 D 4 10 -2 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -4 -14 B 12 0 6 -10 -18 C -2 -6 0 2 -10 D 4 10 -2 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -4 -14 B 12 0 6 -10 -18 C -2 -6 0 2 -10 D 4 10 -2 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4447: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (13) D C A E B (9) D A C B E (7) D C E A B (6) E B C A D (5) C D E A B (5) C D A E B (5) B E A D C (5) A D C B E (5) E B C D A (4) B E C A D (4) A B D C E (4) E C D B A (2) E B D C A (2) D A C E B (2) B A E C D (2) B A D E C (2) E D C B A (1) E C D A B (1) E C B D A (1) E C B A D (1) D E C B A (1) D E B C A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B D A E C (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -2 0 -8 B -6 0 -4 2 2 C 2 4 0 2 4 D 0 -2 -2 0 8 E 8 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 0 -8 B -6 0 -4 2 2 C 2 4 0 2 4 D 0 -2 -2 0 8 E 8 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 E=17 A=16 C=12 so C is eliminated. Round 2 votes counts: D=36 B=29 A=18 E=17 so E is eliminated. Round 3 votes counts: B=42 D=40 A=18 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:206 D:202 A:198 B:197 E:197 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 0 -8 B -6 0 -4 2 2 C 2 4 0 2 4 D 0 -2 -2 0 8 E 8 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 0 -8 B -6 0 -4 2 2 C 2 4 0 2 4 D 0 -2 -2 0 8 E 8 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 0 -8 B -6 0 -4 2 2 C 2 4 0 2 4 D 0 -2 -2 0 8 E 8 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4448: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) D E A C B (7) B A C E D (7) A D E C B (6) E C D B A (5) D E C B A (5) D B E C A (4) D A E C B (4) B C E A D (4) A B D C E (4) E D C A B (3) D B A C E (3) D A B E C (3) C E B A D (3) B C A E D (3) A D B C E (3) E C D A B (2) E C B D A (2) E C A D B (2) D A E B C (2) B D C E A (2) B C E D A (2) A B C E D (2) A B C D E (2) E D C B A (1) E C B A D (1) D E A B C (1) D B A E C (1) D A B C E (1) C E A B D (1) C B E A D (1) B A D C E (1) B A C D E (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 8 0 -14 -10 B -8 0 -6 -26 -10 C 0 6 0 -22 -18 D 14 26 22 0 20 E 10 10 18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -14 -10 B -8 0 -6 -26 -10 C 0 6 0 -22 -18 D 14 26 22 0 20 E 10 10 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=20 A=19 E=16 C=5 so C is eliminated. Round 2 votes counts: D=40 B=21 E=20 A=19 so A is eliminated. Round 3 votes counts: D=50 B=29 E=21 so E is eliminated. Round 4 votes counts: D=63 B=37 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:241 E:209 A:192 C:183 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -14 -10 B -8 0 -6 -26 -10 C 0 6 0 -22 -18 D 14 26 22 0 20 E 10 10 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -14 -10 B -8 0 -6 -26 -10 C 0 6 0 -22 -18 D 14 26 22 0 20 E 10 10 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -14 -10 B -8 0 -6 -26 -10 C 0 6 0 -22 -18 D 14 26 22 0 20 E 10 10 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4449: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) C B A E D (11) B C A D E (6) E D A C B (5) C A B E D (5) B C D A E (5) E A C D B (4) D B C E A (4) C B E A D (4) E A D C B (3) D E B C A (3) D E A C B (3) D A E B C (3) A E D C B (3) A E C B D (3) A C E B D (3) E D C A B (2) B D C A E (2) A D E B C (2) A C B E D (2) A B C E D (2) E D C B A (1) E C D A B (1) E C B A D (1) D E C B A (1) D E B A C (1) D B E C A (1) D B C A E (1) D B A E C (1) D A B E C (1) C E B A D (1) C B E D A (1) A E D B C (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 0 6 6 B -12 0 -10 -6 -6 C 0 10 0 2 -4 D -6 6 -2 0 -8 E -6 6 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.675565 B: 0.000000 C: 0.324435 D: 0.000000 E: 0.000000 Sum of squares = 0.56164588934 Cumulative probabilities = A: 0.675565 B: 0.675565 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 6 6 B -12 0 -10 -6 -6 C 0 10 0 2 -4 D -6 6 -2 0 -8 E -6 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=22 A=18 E=17 B=13 so B is eliminated. Round 2 votes counts: C=33 D=32 A=18 E=17 so E is eliminated. Round 3 votes counts: D=40 C=35 A=25 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:212 E:206 C:204 D:195 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 6 6 B -12 0 -10 -6 -6 C 0 10 0 2 -4 D -6 6 -2 0 -8 E -6 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 6 6 B -12 0 -10 -6 -6 C 0 10 0 2 -4 D -6 6 -2 0 -8 E -6 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 6 6 B -12 0 -10 -6 -6 C 0 10 0 2 -4 D -6 6 -2 0 -8 E -6 6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4450: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (12) A B C D E (7) C B E D A (6) D E A C B (5) B C A D E (5) D E C B A (4) D E C A B (4) C B D E A (4) E D A C B (3) E D A B C (3) E C D B A (3) E A D B C (3) C B A D E (3) B C E A D (3) B C A E D (3) E D C A B (2) D C E B A (2) D A C B E (2) B A C E D (2) A D B C E (2) A B E D C (2) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C A D (1) E B A C D (1) E A B D C (1) D C E A B (1) D C A E B (1) D C A B E (1) C E D B A (1) C D E B A (1) C D B A E (1) C B D A E (1) B E C A D (1) B C E D A (1) A D E B C (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -28 -24 -32 B 18 0 -20 -10 -6 C 28 20 0 -6 -2 D 24 10 6 0 -6 E 32 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -28 -24 -32 B 18 0 -20 -10 -6 C 28 20 0 -6 -2 D 24 10 6 0 -6 E 32 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=20 C=17 B=15 A=15 so B is eliminated. Round 2 votes counts: E=34 C=29 D=20 A=17 so A is eliminated. Round 3 votes counts: C=39 E=37 D=24 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:220 D:217 B:191 A:149 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -28 -24 -32 B 18 0 -20 -10 -6 C 28 20 0 -6 -2 D 24 10 6 0 -6 E 32 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -28 -24 -32 B 18 0 -20 -10 -6 C 28 20 0 -6 -2 D 24 10 6 0 -6 E 32 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -28 -24 -32 B 18 0 -20 -10 -6 C 28 20 0 -6 -2 D 24 10 6 0 -6 E 32 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4451: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) B D E C A (5) B A C D E (5) E D B A C (4) D E B A C (4) B C A D E (4) E D B C A (3) E D A C B (3) E B D C A (3) D E A C B (3) C B E A D (3) B E D C A (3) B D E A C (3) B C E D A (3) B C A E D (3) A D C E B (3) A C D B E (3) D E A B C (2) D B E A C (2) D A E C B (2) C B A E D (2) C A B D E (2) B D C A E (2) B D A C E (2) B C E A D (2) A E C D B (2) A C E D B (2) A C D E B (2) A C B D E (2) E D A B C (1) E C B A D (1) E A D C B (1) D B E C A (1) D B A E C (1) C E A D B (1) C A E B D (1) B E C D A (1) B C D E A (1) B C D A E (1) B A D C E (1) A D E C B (1) Total count = 100 A B C D E A 0 -20 -2 0 0 B 20 0 14 18 22 C 2 -14 0 0 8 D 0 -18 0 0 4 E 0 -22 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -2 0 0 B 20 0 14 18 22 C 2 -14 0 0 8 D 0 -18 0 0 4 E 0 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=18 E=16 D=15 A=15 so D is eliminated. Round 2 votes counts: B=40 E=25 C=18 A=17 so A is eliminated. Round 3 votes counts: B=40 E=30 C=30 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:237 C:198 D:193 A:189 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -2 0 0 B 20 0 14 18 22 C 2 -14 0 0 8 D 0 -18 0 0 4 E 0 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -2 0 0 B 20 0 14 18 22 C 2 -14 0 0 8 D 0 -18 0 0 4 E 0 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -2 0 0 B 20 0 14 18 22 C 2 -14 0 0 8 D 0 -18 0 0 4 E 0 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4452: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (15) C A D E B (11) A D C E B (7) A C D E B (6) D A E B C (5) C A D B E (4) E B D A C (3) D A C E B (3) C B A E D (3) C A B E D (3) B E C D A (3) B E C A D (3) E D A B C (2) E A D B C (2) D E B A C (2) D B E A C (2) D A B E C (2) C B E A D (2) C A E B D (2) B E D C A (2) B C E A D (2) E C B A D (1) E B A D C (1) E B A C D (1) E A D C B (1) E A B D C (1) D E A B C (1) D C A B E (1) D A E C B (1) C E B A D (1) C D A B E (1) C B A D E (1) C A E D B (1) C A B D E (1) B D E A C (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 12 14 10 6 B -12 0 -2 -6 -6 C -14 2 0 -6 0 D -10 6 6 0 -2 E -6 6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 10 6 B -12 0 -2 -6 -6 C -14 2 0 -6 0 D -10 6 6 0 -2 E -6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=27 D=17 A=14 E=12 so E is eliminated. Round 2 votes counts: B=32 C=31 D=19 A=18 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:221 E:201 D:200 C:191 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 10 6 B -12 0 -2 -6 -6 C -14 2 0 -6 0 D -10 6 6 0 -2 E -6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 10 6 B -12 0 -2 -6 -6 C -14 2 0 -6 0 D -10 6 6 0 -2 E -6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 10 6 B -12 0 -2 -6 -6 C -14 2 0 -6 0 D -10 6 6 0 -2 E -6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4453: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) E A B C D (8) E D B A C (7) C A B D E (5) A B C D E (5) E B D A C (4) E B A D C (4) D E B C A (4) A B C E D (4) E D C B A (3) D E C B A (3) D C B A E (3) D B C A E (3) C A D B E (3) D E C A B (2) D C A B E (2) C D A B E (2) C A E B D (2) B E D A C (2) A E B C D (2) A C B E D (2) E D C A B (1) E D A B C (1) E B A C D (1) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E A C (1) C D A E B (1) C A E D B (1) C A D E B (1) B D E A C (1) B D A C E (1) B A D C E (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 0 -12 -20 B 6 0 26 -8 -24 C 0 -26 0 -20 -18 D 12 8 20 0 -14 E 20 24 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 0 -12 -20 B 6 0 26 -8 -24 C 0 -26 0 -20 -18 D 12 8 20 0 -14 E 20 24 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 D=22 C=15 A=14 B=6 so B is eliminated. Round 2 votes counts: E=45 D=24 A=16 C=15 so C is eliminated. Round 3 votes counts: E=45 A=28 D=27 so D is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:238 D:213 B:200 A:181 C:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 -12 -20 B 6 0 26 -8 -24 C 0 -26 0 -20 -18 D 12 8 20 0 -14 E 20 24 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -12 -20 B 6 0 26 -8 -24 C 0 -26 0 -20 -18 D 12 8 20 0 -14 E 20 24 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -12 -20 B 6 0 26 -8 -24 C 0 -26 0 -20 -18 D 12 8 20 0 -14 E 20 24 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4454: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) E A C D B (5) D B A C E (5) B E D A C (5) A E C D B (5) E B D A C (4) E A C B D (4) C A E D B (4) B D E C A (4) E B C A D (3) D A C B E (3) C A E B D (3) C A D B E (3) B E D C A (3) B D C E A (3) A C D E B (3) E B A C D (2) D B C A E (2) D B A E C (2) D A C E B (2) D A B E C (2) C B A D E (2) C A D E B (2) B E C D A (2) B E C A D (2) B D E A C (2) B D C A E (2) B C E D A (2) E C B A D (1) E A B C D (1) D E A B C (1) D C B A E (1) D A E B C (1) D A B C E (1) C E A B D (1) C D B A E (1) C D A B E (1) C B E A D (1) B C E A D (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 10 -12 -6 B 14 0 14 -2 14 C -10 -14 0 2 -10 D 12 2 -2 0 0 E 6 -14 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.777778 E: 0.000000 Sum of squares = 0.629629629642 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 -12 -6 B 14 0 14 -2 14 C -10 -14 0 2 -10 D 12 2 -2 0 0 E 6 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.777778 E: 0.000000 Sum of squares = 0.629629629737 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 E=20 C=18 A=9 so A is eliminated. Round 2 votes counts: B=27 D=26 E=25 C=22 so C is eliminated. Round 3 votes counts: D=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:220 D:206 E:201 A:189 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 -12 -6 B 14 0 14 -2 14 C -10 -14 0 2 -10 D 12 2 -2 0 0 E 6 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.777778 E: 0.000000 Sum of squares = 0.629629629737 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 -12 -6 B 14 0 14 -2 14 C -10 -14 0 2 -10 D 12 2 -2 0 0 E 6 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.777778 E: 0.000000 Sum of squares = 0.629629629737 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 -12 -6 B 14 0 14 -2 14 C -10 -14 0 2 -10 D 12 2 -2 0 0 E 6 -14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.777778 E: 0.000000 Sum of squares = 0.629629629737 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4455: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (15) E C D A B (11) B E A D C (7) B E D A C (6) A B C D E (5) E D C B A (4) C D E A B (4) A C D B E (4) E D C A B (3) E C D B A (3) E B D C A (3) E B A C D (3) C D A E B (3) E D B C A (2) E C A D B (2) C E D A B (2) C A D B E (2) B D A C E (2) B A C E D (2) A D C B E (2) E C B A D (1) D E C B A (1) D C E A B (1) D C B A E (1) D B C E A (1) D B A C E (1) D A C B E (1) D A B C E (1) C A E D B (1) C A D E B (1) B A E D C (1) B A E C D (1) B A D E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 8 0 -8 B 10 0 4 -2 10 C -8 -4 0 -8 2 D 0 2 8 0 -4 E 8 -10 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999994 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 A B C D E A 0 -10 8 0 -8 B 10 0 4 -2 10 C -8 -4 0 -8 2 D 0 2 8 0 -4 E 8 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999904 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=32 C=13 A=13 D=7 so D is eliminated. Round 2 votes counts: B=37 E=33 C=15 A=15 so C is eliminated. Round 3 votes counts: E=40 B=38 A=22 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:203 E:200 A:195 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 0 -8 B 10 0 4 -2 10 C -8 -4 0 -8 2 D 0 2 8 0 -4 E 8 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999904 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 0 -8 B 10 0 4 -2 10 C -8 -4 0 -8 2 D 0 2 8 0 -4 E 8 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999904 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 0 -8 B 10 0 4 -2 10 C -8 -4 0 -8 2 D 0 2 8 0 -4 E 8 -10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999904 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4456: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) D E C A B (8) A B C E D (8) A E B D C (7) A B E C D (7) D E C B A (5) C B A D E (5) B A C E D (5) C D B A E (4) A B E D C (4) A B C D E (3) E A D B C (2) E A B D C (2) C D E B A (2) C D B E A (2) C B E D A (2) B C A E D (2) A E D B C (2) A B D C E (2) E D C B A (1) E D A C B (1) E D A B C (1) E C B A D (1) E B A D C (1) D E A C B (1) D E A B C (1) D C E A B (1) D C B E A (1) D C B A E (1) C E D B A (1) C D A B E (1) C B E A D (1) C B D A E (1) B E A C D (1) B C E A D (1) B C A D E (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -2 12 8 B -4 0 2 8 6 C 2 -2 0 -4 8 D -12 -8 4 0 0 E -8 -6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999979 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 12 8 B -4 0 2 8 6 C 2 -2 0 -4 8 D -12 -8 4 0 0 E -8 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=28 C=19 B=10 E=9 so E is eliminated. Round 2 votes counts: A=38 D=31 C=20 B=11 so B is eliminated. Round 3 votes counts: A=45 D=31 C=24 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:206 C:202 D:192 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 12 8 B -4 0 2 8 6 C 2 -2 0 -4 8 D -12 -8 4 0 0 E -8 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 12 8 B -4 0 2 8 6 C 2 -2 0 -4 8 D -12 -8 4 0 0 E -8 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 12 8 B -4 0 2 8 6 C 2 -2 0 -4 8 D -12 -8 4 0 0 E -8 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4457: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (12) A D B C E (9) B E C D A (8) C E D B A (7) C D A E B (7) B E A D C (6) A D C E B (6) A D C B E (6) A B D E C (5) D C A E B (4) E C B D A (3) C E D A B (3) B E A C D (3) B A E D C (3) A D B E C (3) D A C E B (2) C D E A B (2) B E D C A (2) B A D E C (2) E B C A D (1) D B C A E (1) D A C B E (1) C E B D A (1) C D E B A (1) B E C A D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -6 -8 0 B 2 0 14 -4 2 C 6 -14 0 0 0 D 8 4 0 0 -2 E 0 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999993 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -2 -6 -8 0 B 2 0 14 -4 2 C 6 -14 0 0 0 D 8 4 0 0 -2 E 0 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000075 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=25 C=21 E=16 D=8 so D is eliminated. Round 2 votes counts: A=33 B=26 C=25 E=16 so E is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:207 D:205 E:200 C:196 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 0 B 2 0 14 -4 2 C 6 -14 0 0 0 D 8 4 0 0 -2 E 0 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000075 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 0 B 2 0 14 -4 2 C 6 -14 0 0 0 D 8 4 0 0 -2 E 0 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000075 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 0 B 2 0 14 -4 2 C 6 -14 0 0 0 D 8 4 0 0 -2 E 0 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000075 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4458: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) D C E A B (7) C B D E A (5) A B E C D (5) C D B E A (4) B C A D E (4) B A C E D (4) A D E C B (4) E D C B A (3) E B A C D (3) E A B D C (3) D C E B A (3) D A C E B (3) C D B A E (3) C D A B E (3) B A E C D (3) A E B D C (3) A B E D C (3) E B D A C (2) E B C D A (2) D E C B A (2) C D E B A (2) B E C A D (2) B E A C D (2) B C A E D (2) A B C D E (2) E D A C B (1) E B A D C (1) D C A B E (1) C E D B A (1) C B E D A (1) C B A D E (1) B C E D A (1) B C D E A (1) A D C E B (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -14 -8 8 B 4 0 -10 4 0 C 14 10 0 6 22 D 8 -4 -6 0 14 E -8 0 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -8 8 B 4 0 -10 4 0 C 14 10 0 6 22 D 8 -4 -6 0 14 E -8 0 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=21 C=20 B=19 E=15 so E is eliminated. Round 2 votes counts: D=29 B=27 A=24 C=20 so C is eliminated. Round 3 votes counts: D=42 B=34 A=24 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:226 D:206 B:199 A:191 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -8 8 B 4 0 -10 4 0 C 14 10 0 6 22 D 8 -4 -6 0 14 E -8 0 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -8 8 B 4 0 -10 4 0 C 14 10 0 6 22 D 8 -4 -6 0 14 E -8 0 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -8 8 B 4 0 -10 4 0 C 14 10 0 6 22 D 8 -4 -6 0 14 E -8 0 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4459: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (15) D B C E A (9) E A B D C (7) D B E C A (6) C D B A E (6) C A D B E (6) C D B E A (5) C A E D B (5) C A D E B (5) D C B E A (4) A C E D B (4) E B D A C (3) C D A B E (3) B E D A C (3) B D E C A (3) B D E A C (3) A C E B D (3) E B A D C (2) A E B D C (2) A E B C D (2) D B E A C (1) C B D E A (1) C A E B D (1) B D C E A (1) Total count = 100 A B C D E A 0 6 -10 4 4 B -6 0 -16 -8 2 C 10 16 0 12 6 D -4 8 -12 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 4 4 B -6 0 -16 -8 2 C 10 16 0 12 6 D -4 8 -12 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=26 D=20 E=12 B=10 so B is eliminated. Round 2 votes counts: C=32 D=27 A=26 E=15 so E is eliminated. Round 3 votes counts: A=35 D=33 C=32 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:222 A:202 D:199 E:191 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 4 4 B -6 0 -16 -8 2 C 10 16 0 12 6 D -4 8 -12 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 4 4 B -6 0 -16 -8 2 C 10 16 0 12 6 D -4 8 -12 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 4 4 B -6 0 -16 -8 2 C 10 16 0 12 6 D -4 8 -12 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4460: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) C D A E B (5) C D A B E (5) C A E D B (4) B E A C D (4) B D E A C (4) B A E C D (4) E C A D B (3) E B D A C (3) E A B C D (3) D E B C A (3) D C E B A (3) D C E A B (3) D B C A E (3) C D E A B (3) C A D B E (3) A C E B D (3) E D C B A (2) E C A B D (2) E B A D C (2) E A C B D (2) D C B A E (2) D C A E B (2) D C A B E (2) C E A D B (2) B E A D C (2) A E C B D (2) E B D C A (1) D E C B A (1) D B E C A (1) D B C E A (1) D B A C E (1) C E D A B (1) C A E B D (1) B D E C A (1) B D A E C (1) B D A C E (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B C D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -8 2 -4 B 0 0 -4 -2 -14 C 8 4 0 16 0 D -2 2 -16 0 -2 E 4 14 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.464733 D: 0.000000 E: 0.535267 Sum of squares = 0.502487530836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.464733 D: 0.464733 E: 1.000000 A B C D E A 0 0 -8 2 -4 B 0 0 -4 -2 -14 C 8 4 0 16 0 D -2 2 -16 0 -2 E 4 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=23 D=22 B=22 A=9 so A is eliminated. Round 2 votes counts: C=28 E=26 B=24 D=22 so D is eliminated. Round 3 votes counts: C=40 E=30 B=30 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:210 A:195 D:191 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 2 -4 B 0 0 -4 -2 -14 C 8 4 0 16 0 D -2 2 -16 0 -2 E 4 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 -4 B 0 0 -4 -2 -14 C 8 4 0 16 0 D -2 2 -16 0 -2 E 4 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 -4 B 0 0 -4 -2 -14 C 8 4 0 16 0 D -2 2 -16 0 -2 E 4 14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4461: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (17) A B E D C (14) C D E A B (6) D C E B A (5) A E B D C (5) E B D A C (4) D E B A C (4) C A B E D (4) B A E D C (4) A B C E D (4) C D A E B (3) B E A D C (3) A C E B D (3) A B E C D (3) D E B C A (2) C D B E A (2) C A D E B (2) C A B D E (2) B E D A C (2) E D B A C (1) D E C B A (1) D B E C A (1) D B C E A (1) C D A B E (1) C B D A E (1) C B A D E (1) C A E D B (1) B D E C A (1) B D E A C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 -2 -6 -2 B 2 0 2 6 -10 C 2 -2 0 2 6 D 6 -6 -2 0 2 E 2 10 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.111111 Sum of squares = 0.432098765436 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 -2 -2 -6 -2 B 2 0 2 6 -10 C 2 -2 0 2 6 D 6 -6 -2 0 2 E 2 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.111111 Sum of squares = 0.432098765438 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 A=30 D=14 B=11 E=5 so E is eliminated. Round 2 votes counts: C=40 A=30 D=15 B=15 so D is eliminated. Round 3 votes counts: C=46 A=30 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:204 E:202 B:200 D:200 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 -2 B 2 0 2 6 -10 C 2 -2 0 2 6 D 6 -6 -2 0 2 E 2 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.111111 Sum of squares = 0.432098765438 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 -2 B 2 0 2 6 -10 C 2 -2 0 2 6 D 6 -6 -2 0 2 E 2 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.111111 Sum of squares = 0.432098765438 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 -2 B 2 0 2 6 -10 C 2 -2 0 2 6 D 6 -6 -2 0 2 E 2 10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.111111 Sum of squares = 0.432098765438 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4462: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (16) C E B D A (11) D B E A C (6) E B C D A (5) E B D A C (4) C E A B D (4) C A E D B (4) C A D B E (4) A D C B E (4) A D B C E (4) E B D C A (3) D B A E C (3) D A B E C (3) C E B A D (3) C D A B E (3) A C E D B (3) E C B D A (2) C B D E A (2) A C D B E (2) E C B A D (1) E B A D C (1) E A C B D (1) D C A B E (1) D B C E A (1) D B C A E (1) D A B C E (1) C D B E A (1) C A E B D (1) B E D A C (1) B D E A C (1) A D E B C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 6 2 6 B -8 0 2 -18 8 C -6 -2 0 -2 4 D -2 18 2 0 10 E -6 -8 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 2 6 B -8 0 2 -18 8 C -6 -2 0 -2 4 D -2 18 2 0 10 E -6 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=32 E=17 D=16 B=2 so B is eliminated. Round 2 votes counts: C=33 A=32 E=18 D=17 so D is eliminated. Round 3 votes counts: A=39 C=36 E=25 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:211 C:197 B:192 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 2 6 B -8 0 2 -18 8 C -6 -2 0 -2 4 D -2 18 2 0 10 E -6 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 2 6 B -8 0 2 -18 8 C -6 -2 0 -2 4 D -2 18 2 0 10 E -6 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 2 6 B -8 0 2 -18 8 C -6 -2 0 -2 4 D -2 18 2 0 10 E -6 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4463: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) C D B A E (7) E D B A C (5) E A B D C (5) C D A B E (5) A E B C D (5) E B D A C (4) C A D B E (4) A C E D B (4) E B A D C (3) D E B C A (3) C D A E B (3) C B A D E (3) A E C B D (3) A C E B D (3) E A D B C (2) E A C D B (2) D E C B A (2) D C B A E (2) C B D A E (2) C A E D B (2) B D E C A (2) B A C D E (2) A C B E D (2) A C B D E (2) E A B C D (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E C A (1) C E D A B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C E A (1) B D C A E (1) B A E C D (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 -6 -6 6 B 6 0 -4 -16 -4 C 6 4 0 8 16 D 6 16 -8 0 8 E -6 4 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -6 6 B 6 0 -4 -16 -4 C 6 4 0 8 16 D 6 16 -8 0 8 E -6 4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=22 D=21 A=20 B=9 so B is eliminated. Round 2 votes counts: C=28 D=25 E=24 A=23 so A is eliminated. Round 3 votes counts: C=41 E=34 D=25 so D is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:211 A:194 B:191 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -6 6 B 6 0 -4 -16 -4 C 6 4 0 8 16 D 6 16 -8 0 8 E -6 4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -6 6 B 6 0 -4 -16 -4 C 6 4 0 8 16 D 6 16 -8 0 8 E -6 4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -6 6 B 6 0 -4 -16 -4 C 6 4 0 8 16 D 6 16 -8 0 8 E -6 4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4464: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) E B A D C (7) C D A B E (5) B E C D A (5) B E A C D (5) B A E C D (5) E B D C A (4) D C A E B (4) C D A E B (4) B C E D A (4) A C D B E (4) E A B D C (3) B E C A D (3) B A C E D (3) A E B D C (3) A D C E B (3) E D C B A (2) D A C E B (2) C B D E A (2) C B D A E (2) C A B D E (2) B C D E A (2) A E D C B (2) E D C A B (1) E D A B C (1) E B D A C (1) D E C B A (1) D C E B A (1) D C E A B (1) C D E B A (1) C D B A E (1) C B A D E (1) C A D B E (1) B C E A D (1) B A C D E (1) A E D B C (1) A D E C B (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 4 12 -6 B 20 0 18 28 14 C -4 -18 0 8 -8 D -12 -28 -8 0 -20 E 6 -14 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 4 12 -6 B 20 0 18 28 14 C -4 -18 0 8 -8 D -12 -28 -8 0 -20 E 6 -14 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=19 C=19 A=16 D=9 so D is eliminated. Round 2 votes counts: B=37 C=25 E=20 A=18 so A is eliminated. Round 3 votes counts: B=39 C=34 E=27 so E is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:240 E:210 A:195 C:189 D:166 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 4 12 -6 B 20 0 18 28 14 C -4 -18 0 8 -8 D -12 -28 -8 0 -20 E 6 -14 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 4 12 -6 B 20 0 18 28 14 C -4 -18 0 8 -8 D -12 -28 -8 0 -20 E 6 -14 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 4 12 -6 B 20 0 18 28 14 C -4 -18 0 8 -8 D -12 -28 -8 0 -20 E 6 -14 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4465: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) D B E A C (7) C A E B D (7) C A E D B (6) B D E A C (6) D B C E A (5) D B C A E (5) B D E C A (4) E B D A C (3) D B E C A (3) B E A D C (3) B D C E A (3) A E C D B (3) A E C B D (3) A C E D B (3) E C A B D (2) E A C D B (2) E A B D C (2) D E B A C (2) D C B A E (2) D B A E C (2) C D A B E (2) C B A D E (2) C A D E B (2) B E A C D (2) E D A B C (1) E B A C D (1) E A D B C (1) E A B C D (1) D C A B E (1) D A E C B (1) C E A B D (1) C D B A E (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C A D (1) Total count = 100 A B C D E A 0 -6 2 2 -16 B 6 0 6 0 4 C -2 -6 0 -4 -16 D -2 0 4 0 0 E 16 -4 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.694513 C: 0.000000 D: 0.305487 E: 0.000000 Sum of squares = 0.575670513927 Cumulative probabilities = A: 0.000000 B: 0.694513 C: 0.694513 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 2 -16 B 6 0 6 0 4 C -2 -6 0 -4 -16 D -2 0 4 0 0 E 16 -4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 E=20 B=20 A=9 so A is eliminated. Round 2 votes counts: D=28 E=26 C=26 B=20 so B is eliminated. Round 3 votes counts: D=41 E=33 C=26 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:214 B:208 D:201 A:191 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 2 -16 B 6 0 6 0 4 C -2 -6 0 -4 -16 D -2 0 4 0 0 E 16 -4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 -16 B 6 0 6 0 4 C -2 -6 0 -4 -16 D -2 0 4 0 0 E 16 -4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 -16 B 6 0 6 0 4 C -2 -6 0 -4 -16 D -2 0 4 0 0 E 16 -4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4466: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D C A B E (7) D A C B E (7) D C E A B (5) B E A C D (5) B A E C D (5) D A B C E (4) A C B D E (4) D E C A B (3) C D A E B (3) A B D C E (3) E D C B A (2) E C D B A (2) E B D C A (2) E B D A C (2) E B A D C (2) E B A C D (2) D E C B A (2) D C A E B (2) D B A E C (2) C E D A B (2) C E B A D (2) C D E A B (2) B E A D C (2) B A D E C (2) B A C E D (2) A D B C E (2) A B C D E (2) E C B A D (1) E B C D A (1) D E A B C (1) C D A B E (1) C A E D B (1) C A B E D (1) C A B D E (1) B A E D C (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 0 0 4 B -6 0 2 2 6 C 0 -2 0 -4 6 D 0 -2 4 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.518776 B: 0.000000 C: 0.000000 D: 0.481224 E: 0.000000 Sum of squares = 0.500705074556 Cumulative probabilities = A: 0.518776 B: 0.518776 C: 0.518776 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 4 B -6 0 2 2 6 C 0 -2 0 -4 6 D 0 -2 4 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=24 B=17 C=13 A=13 so C is eliminated. Round 2 votes counts: D=39 E=28 B=17 A=16 so A is eliminated. Round 3 votes counts: D=42 E=29 B=29 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:205 D:205 B:202 C:200 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 0 4 B -6 0 2 2 6 C 0 -2 0 -4 6 D 0 -2 4 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 4 B -6 0 2 2 6 C 0 -2 0 -4 6 D 0 -2 4 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 4 B -6 0 2 2 6 C 0 -2 0 -4 6 D 0 -2 4 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4467: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (11) C A E D B (7) D B E A C (6) C A D E B (6) E B D A C (5) E A C B D (5) D B E C A (5) A C E D B (5) C E A B D (4) C A E B D (4) E C A B D (3) C D B A E (3) B E D A C (3) E C B D A (2) E C B A D (2) E B D C A (2) D B A E C (2) C D B E A (2) A D B E C (2) A C E B D (2) E A B C D (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A C E (1) D A B C E (1) C E B D A (1) C E B A D (1) C D A B E (1) C B D E A (1) C A D B E (1) B E D C A (1) B D E C A (1) B D C E A (1) A E D B C (1) A E C B D (1) A E B D C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -2 -4 -14 B 4 0 -10 4 -8 C 2 10 0 8 -8 D 4 -4 -8 0 -2 E 14 8 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 -14 B 4 0 -10 4 -8 C 2 10 0 8 -8 D 4 -4 -8 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=20 D=18 B=17 A=14 so A is eliminated. Round 2 votes counts: C=40 E=23 D=20 B=17 so B is eliminated. Round 3 votes counts: C=40 D=33 E=27 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:216 C:206 B:195 D:195 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 -14 B 4 0 -10 4 -8 C 2 10 0 8 -8 D 4 -4 -8 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 -14 B 4 0 -10 4 -8 C 2 10 0 8 -8 D 4 -4 -8 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 -14 B 4 0 -10 4 -8 C 2 10 0 8 -8 D 4 -4 -8 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4468: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) B D E A C (7) A C D B E (7) E C B D A (5) C E A D B (5) E A C B D (4) C A E B D (4) C A D B E (4) A C E D B (4) E C A B D (3) E B D A C (3) E B A D C (3) D B C E A (3) D B A C E (3) C E D B A (3) C A E D B (3) B D E C A (3) A D C B E (3) A C E B D (3) A B D E C (3) C A D E B (2) B D A E C (2) A D B C E (2) A C B D E (2) A B D C E (2) E C D B A (1) E C B A D (1) E A B D C (1) D B C A E (1) C E A B D (1) C D B E A (1) C D A B E (1) B E D A C (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 12 4 18 -4 B -12 0 -18 12 -10 C -4 18 0 12 10 D -18 -12 -12 0 -6 E 4 10 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407431 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 12 4 18 -4 B -12 0 -18 12 -10 C -4 18 0 12 10 D -18 -12 -12 0 -6 E 4 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.40740740733 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 C=24 B=13 D=7 so D is eliminated. Round 2 votes counts: E=28 A=28 C=24 B=20 so B is eliminated. Round 3 votes counts: E=39 A=33 C=28 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:218 A:215 E:205 B:186 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 18 -4 B -12 0 -18 12 -10 C -4 18 0 12 10 D -18 -12 -12 0 -6 E 4 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.40740740733 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 18 -4 B -12 0 -18 12 -10 C -4 18 0 12 10 D -18 -12 -12 0 -6 E 4 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.40740740733 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 18 -4 B -12 0 -18 12 -10 C -4 18 0 12 10 D -18 -12 -12 0 -6 E 4 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.222222 D: 0.000000 E: 0.222222 Sum of squares = 0.40740740733 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4469: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) E A D C B (6) E D A C B (5) E A B D C (5) D E C A B (5) C D B A E (5) B A E C D (5) B A C E D (5) A B E C D (5) D C B E A (4) B C A D E (4) A E B C D (4) E D B A C (3) C D A E B (3) A C E D B (3) A C B E D (3) E B A D C (2) D E B C A (2) D C B A E (2) D C A E B (2) D B C E A (2) C A B D E (2) B E A C D (2) A B C E D (2) E D C A B (1) E D A B C (1) E B A C D (1) E A C D B (1) E A C B D (1) D E C B A (1) D C E B A (1) D C E A B (1) C D A B E (1) C B A D E (1) B D C A E (1) B A C D E (1) Total count = 100 A B C D E A 0 16 24 20 -2 B -16 0 4 2 -10 C -24 -4 0 12 -14 D -20 -2 -12 0 -24 E 2 10 14 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 24 20 -2 B -16 0 4 2 -10 C -24 -4 0 12 -14 D -20 -2 -12 0 -24 E 2 10 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=20 B=18 A=17 C=12 so C is eliminated. Round 2 votes counts: E=33 D=29 B=19 A=19 so B is eliminated. Round 3 votes counts: E=35 A=35 D=30 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:229 E:225 B:190 C:185 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 24 20 -2 B -16 0 4 2 -10 C -24 -4 0 12 -14 D -20 -2 -12 0 -24 E 2 10 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 24 20 -2 B -16 0 4 2 -10 C -24 -4 0 12 -14 D -20 -2 -12 0 -24 E 2 10 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 24 20 -2 B -16 0 4 2 -10 C -24 -4 0 12 -14 D -20 -2 -12 0 -24 E 2 10 14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4470: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) E A B D C (9) E A D B C (8) D E A C B (7) C D B A E (7) B A E C D (6) D E C A B (5) B A C E D (5) A B E C D (5) D C E A B (4) B C A E D (4) D C E B A (3) D C B A E (3) C D B E A (3) B C D A E (3) C B D A E (2) B E A C D (2) B C A D E (2) A E D C B (2) A E B C D (2) E D A C B (1) E A B C D (1) C B D E A (1) C B A D E (1) A E D B C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 2 0 -10 B 6 0 0 -10 12 C -2 0 0 -10 0 D 0 10 10 0 4 E 10 -12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.171813 B: 0.000000 C: 0.000000 D: 0.828187 E: 0.000000 Sum of squares = 0.715413908347 Cumulative probabilities = A: 0.171813 B: 0.171813 C: 0.171813 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 0 -10 B 6 0 0 -10 12 C -2 0 0 -10 0 D 0 10 10 0 4 E 10 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.59183673512 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=22 E=19 C=14 A=12 so A is eliminated. Round 2 votes counts: D=33 B=28 E=25 C=14 so C is eliminated. Round 3 votes counts: D=43 B=32 E=25 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:204 E:197 C:194 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 0 -10 B 6 0 0 -10 12 C -2 0 0 -10 0 D 0 10 10 0 4 E 10 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.59183673512 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 -10 B 6 0 0 -10 12 C -2 0 0 -10 0 D 0 10 10 0 4 E 10 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.59183673512 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 -10 B 6 0 0 -10 12 C -2 0 0 -10 0 D 0 10 10 0 4 E 10 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.59183673512 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4471: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (16) A E C B D (15) B C E A D (11) D A E C B (10) A E C D B (10) B D C E A (6) D C E B A (5) E C A D B (2) E C A B D (2) E A C B D (2) D B A C E (2) B D C A E (2) B D A C E (2) B C E D A (2) A D E C B (2) A B E C D (2) D E C A B (1) D E A C B (1) D C E A B (1) C E D B A (1) C E A B D (1) B A E D C (1) B A D C E (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 0 2 -2 B 0 0 -6 -4 -6 C 0 6 0 -2 2 D -2 4 2 0 0 E 2 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300391 B: 0.000000 C: 0.300391 D: 0.199609 E: 0.199609 Sum of squares = 0.260156943791 Cumulative probabilities = A: 0.300391 B: 0.300391 C: 0.600782 D: 0.800391 E: 1.000000 A B C D E A 0 0 0 2 -2 B 0 0 -6 -4 -6 C 0 6 0 -2 2 D -2 4 2 0 0 E 2 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=30 B=26 E=6 C=2 so C is eliminated. Round 2 votes counts: D=36 A=30 B=26 E=8 so E is eliminated. Round 3 votes counts: D=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:203 E:203 D:202 A:200 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 2 -2 B 0 0 -6 -4 -6 C 0 6 0 -2 2 D -2 4 2 0 0 E 2 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -2 B 0 0 -6 -4 -6 C 0 6 0 -2 2 D -2 4 2 0 0 E 2 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -2 B 0 0 -6 -4 -6 C 0 6 0 -2 2 D -2 4 2 0 0 E 2 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.250000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4472: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) E C A B D (8) B D A C E (6) E D B C A (5) C A E B D (5) D B A C E (4) C A E D B (4) B D E A C (4) E D C A B (3) E B C A D (3) D E B A C (3) D B E C A (3) D B E A C (3) A C D E B (3) A C B D E (3) E D A C B (2) E C D A B (2) D E B C A (2) D E A C B (2) B E D C A (2) B E C A D (2) B D E C A (2) B D C A E (2) B A D C E (2) A D C B E (2) A C D B E (2) E A D C B (1) D B A E C (1) D A C E B (1) D A C B E (1) D A B C E (1) B E C D A (1) B C A E D (1) B A C D E (1) A D C E B (1) A C E D B (1) A C E B D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -6 0 -12 B -6 0 -2 -10 -10 C 6 2 0 -8 -14 D 0 10 8 0 0 E 12 10 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.421231 E: 0.578769 Sum of squares = 0.512409076387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.421231 E: 1.000000 A B C D E A 0 6 -6 0 -12 B -6 0 -2 -10 -10 C 6 2 0 -8 -14 D 0 10 8 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=23 D=21 A=15 C=9 so C is eliminated. Round 2 votes counts: E=32 A=24 B=23 D=21 so D is eliminated. Round 3 votes counts: E=39 B=34 A=27 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:209 A:194 C:193 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 0 -12 B -6 0 -2 -10 -10 C 6 2 0 -8 -14 D 0 10 8 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 0 -12 B -6 0 -2 -10 -10 C 6 2 0 -8 -14 D 0 10 8 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 0 -12 B -6 0 -2 -10 -10 C 6 2 0 -8 -14 D 0 10 8 0 0 E 12 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4473: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) E C B D A (7) C B E D A (6) A D E B C (6) E C B A D (5) D B C E A (5) C B D E A (5) D B A C E (4) A D B E C (4) E C D B A (3) E A D C B (3) C E B D A (3) A E C B D (3) E D C A B (2) E D A C B (2) E D A B C (2) E A C D B (2) E A C B D (2) D E B C A (2) D E B A C (2) D A B C E (2) C B A E D (2) B D C A E (2) B C D E A (2) B C D A E (2) A E D B C (2) E D C B A (1) E C D A B (1) D B E A C (1) D B C A E (1) D A E B C (1) D A B E C (1) B D A C E (1) A E D C B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 -16 -12 B 8 0 2 -16 0 C -2 -2 0 -10 -6 D 16 16 10 0 4 E 12 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -16 -12 B 8 0 2 -16 0 C -2 -2 0 -10 -6 D 16 16 10 0 4 E 12 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=28 D=19 C=16 B=7 so B is eliminated. Round 2 votes counts: E=30 A=28 D=22 C=20 so C is eliminated. Round 3 votes counts: E=39 D=31 A=30 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:207 B:197 C:190 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -16 -12 B 8 0 2 -16 0 C -2 -2 0 -10 -6 D 16 16 10 0 4 E 12 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -16 -12 B 8 0 2 -16 0 C -2 -2 0 -10 -6 D 16 16 10 0 4 E 12 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -16 -12 B 8 0 2 -16 0 C -2 -2 0 -10 -6 D 16 16 10 0 4 E 12 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4474: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) D E B A C (6) D B E C A (6) B D C A E (6) E D A C B (5) E A C D B (5) D E A C B (5) D E A B C (5) C A E B D (5) C A B E D (5) B C A E D (5) B D C E A (4) B C D A E (4) E A D C B (3) D B C E A (3) D B C A E (3) B C A D E (3) A E C B D (3) E C A B D (2) D B A C E (2) B D E C A (2) A E C D B (2) D B E A C (1) D A E C B (1) C E A B D (1) C B A E D (1) B C E A D (1) B C D E A (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -4 -8 0 B -4 0 4 2 -6 C 4 -4 0 -8 6 D 8 -2 8 0 6 E 0 6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.142857 Sum of squares = 0.387755102034 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.857143 E: 1.000000 A B C D E A 0 4 -4 -8 0 B -4 0 4 2 -6 C 4 -4 0 -8 6 D 8 -2 8 0 6 E 0 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.142857 Sum of squares = 0.387755102035 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=26 E=15 A=15 C=12 so C is eliminated. Round 2 votes counts: D=32 B=27 A=25 E=16 so E is eliminated. Round 3 votes counts: D=37 A=36 B=27 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:199 B:198 E:197 A:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -8 0 B -4 0 4 2 -6 C 4 -4 0 -8 6 D 8 -2 8 0 6 E 0 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.142857 Sum of squares = 0.387755102035 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.857143 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 0 B -4 0 4 2 -6 C 4 -4 0 -8 6 D 8 -2 8 0 6 E 0 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.142857 Sum of squares = 0.387755102035 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 0 B -4 0 4 2 -6 C 4 -4 0 -8 6 D 8 -2 8 0 6 E 0 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.428571 E: 0.142857 Sum of squares = 0.387755102035 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4475: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) B C E A D (9) A D C E B (8) C E B D A (7) D A E C B (5) A D E B C (5) C E D B A (4) C B E D A (4) D E A C B (3) D A C E B (3) C D E A B (3) B C E D A (3) A D C B E (3) D C E A B (2) C B E A D (2) C A D E B (2) B E A D C (2) A D E C B (2) A D B E C (2) A C B D E (2) A B D E C (2) E D C B A (1) E D B C A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D C A E B (1) C E D A B (1) C D A E B (1) C A B E D (1) B E D A C (1) B E C A D (1) B C A E D (1) B A E D C (1) B A D E C (1) B A C E D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -14 -6 -14 B 2 0 -14 2 -6 C 14 14 0 6 20 D 6 -2 -6 0 -4 E 14 6 -20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 -6 -14 B 2 0 -14 2 -6 C 14 14 0 6 20 D 6 -2 -6 0 -4 E 14 6 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=26 C=25 D=15 E=5 so E is eliminated. Round 2 votes counts: B=31 C=26 A=26 D=17 so D is eliminated. Round 3 votes counts: A=37 B=32 C=31 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:227 E:202 D:197 B:192 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 -6 -14 B 2 0 -14 2 -6 C 14 14 0 6 20 D 6 -2 -6 0 -4 E 14 6 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -6 -14 B 2 0 -14 2 -6 C 14 14 0 6 20 D 6 -2 -6 0 -4 E 14 6 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -6 -14 B 2 0 -14 2 -6 C 14 14 0 6 20 D 6 -2 -6 0 -4 E 14 6 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4476: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) E D B C A (8) C A E D B (7) E D C B A (6) B E D A C (5) B A D E C (5) A B D C E (5) E C D B A (4) C E D A B (4) A C D B E (4) A C B E D (4) A B C D E (4) D E B A C (3) C A D E B (3) C A B E D (3) A C D E B (3) E B D C A (2) D E B C A (2) B A E D C (2) A C B D E (2) E C D A B (1) D E A C B (1) D B E A C (1) D A C E B (1) D A B E C (1) C E D B A (1) C E A D B (1) C B A E D (1) B D A E C (1) B C E A D (1) B A D C E (1) B A C E D (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 12 -2 0 B 10 0 8 -2 6 C -12 -8 0 -8 -4 D 2 2 8 0 -4 E 0 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.500000 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.833333 E: 1.000000 A B C D E A 0 -10 12 -2 0 B 10 0 8 -2 6 C -12 -8 0 -8 -4 D 2 2 8 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.500000 E: 0.166667 Sum of squares = 0.388888888845 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=23 E=21 C=20 D=9 so D is eliminated. Round 2 votes counts: B=28 E=27 A=25 C=20 so C is eliminated. Round 3 votes counts: A=38 E=33 B=29 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:211 D:204 E:201 A:200 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 12 -2 0 B 10 0 8 -2 6 C -12 -8 0 -8 -4 D 2 2 8 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.500000 E: 0.166667 Sum of squares = 0.388888888845 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.833333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 12 -2 0 B 10 0 8 -2 6 C -12 -8 0 -8 -4 D 2 2 8 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.500000 E: 0.166667 Sum of squares = 0.388888888845 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 12 -2 0 B 10 0 8 -2 6 C -12 -8 0 -8 -4 D 2 2 8 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.500000 E: 0.166667 Sum of squares = 0.388888888845 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4477: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) B A E D C (9) A B D C E (8) E D C B A (5) E B D C A (5) A B C D E (5) C D A E B (4) B E D C A (4) B E A D C (4) B A E C D (4) A C D E B (4) E C D B A (3) B E D A C (3) B E A C D (3) A D B C E (3) E B C D A (2) D C A E B (2) C A D E B (2) B A D E C (2) A D C E B (2) A C B D E (2) A B D E C (2) A B C E D (2) E D B C A (1) E C B D A (1) D E C B A (1) C E D B A (1) C E D A B (1) C A E B D (1) B E C D A (1) B E C A D (1) B A C E D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 12 14 10 B 2 0 20 20 12 C -12 -20 0 -4 -2 D -14 -20 4 0 -4 E -10 -12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999465 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 14 10 B 2 0 20 20 12 C -12 -20 0 -4 -2 D -14 -20 4 0 -4 E -10 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961784 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=30 C=18 E=17 D=3 so D is eliminated. Round 2 votes counts: B=32 A=30 C=20 E=18 so E is eliminated. Round 3 votes counts: B=40 C=30 A=30 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:217 E:192 D:183 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 12 14 10 B 2 0 20 20 12 C -12 -20 0 -4 -2 D -14 -20 4 0 -4 E -10 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961784 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 14 10 B 2 0 20 20 12 C -12 -20 0 -4 -2 D -14 -20 4 0 -4 E -10 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961784 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 14 10 B 2 0 20 20 12 C -12 -20 0 -4 -2 D -14 -20 4 0 -4 E -10 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961784 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4478: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) D B E A C (5) A D C B E (5) D A B E C (4) C A E D B (4) C A B E D (4) B E D C A (4) E C B D A (3) E C B A D (3) E B C D A (3) D B A E C (3) D A B C E (3) C E A B D (3) C A E B D (3) B D E C A (3) B D E A C (3) B D A C E (3) A D C E B (3) A C D B E (3) E D A C B (2) E C A D B (2) E B D C A (2) D E A B C (2) C B E A D (2) A D B C E (2) A C D E B (2) A C B D E (2) E D C A B (1) E C D B A (1) E C D A B (1) E B C A D (1) D E B A C (1) D E A C B (1) D B A C E (1) D A E C B (1) D A C B E (1) C E B A D (1) C E A D B (1) C B A E D (1) B E C A D (1) B C A D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 -8 -4 B -4 0 0 -10 6 C 2 0 0 -14 -6 D 8 10 14 0 2 E 4 -6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -8 -4 B -4 0 0 -10 6 C 2 0 0 -14 -6 D 8 10 14 0 2 E 4 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 C=19 A=19 B=15 so B is eliminated. Round 2 votes counts: D=31 E=30 C=20 A=19 so A is eliminated. Round 3 votes counts: D=42 E=30 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:201 B:196 A:195 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -8 -4 B -4 0 0 -10 6 C 2 0 0 -14 -6 D 8 10 14 0 2 E 4 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -8 -4 B -4 0 0 -10 6 C 2 0 0 -14 -6 D 8 10 14 0 2 E 4 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -8 -4 B -4 0 0 -10 6 C 2 0 0 -14 -6 D 8 10 14 0 2 E 4 -6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4479: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E C D B A (11) A B D E C (10) A E D B C (7) A D B E C (7) C E D B A (6) C E B D A (5) E D B C A (4) A E C D B (4) E D B A C (3) C B D E A (3) B D C E A (3) A C E B D (3) A C B D E (3) E C A D B (2) D B E C A (2) D B E A C (2) D B A E C (2) C E A B D (2) C B D A E (2) C A B D E (2) E D C B A (1) E C D A B (1) B D A C E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 12 8 10 B -10 0 8 -6 0 C -12 -8 0 -8 -12 D -8 6 8 0 0 E -10 0 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 8 10 B -10 0 8 -6 0 C -12 -8 0 -8 -12 D -8 6 8 0 0 E -10 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=48 E=22 C=20 D=6 B=4 so B is eliminated. Round 2 votes counts: A=48 E=22 C=20 D=10 so D is eliminated. Round 3 votes counts: A=51 E=26 C=23 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:203 E:201 B:196 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 8 10 B -10 0 8 -6 0 C -12 -8 0 -8 -12 D -8 6 8 0 0 E -10 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 8 10 B -10 0 8 -6 0 C -12 -8 0 -8 -12 D -8 6 8 0 0 E -10 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 8 10 B -10 0 8 -6 0 C -12 -8 0 -8 -12 D -8 6 8 0 0 E -10 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4480: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) C E B A D (7) D A B E C (5) B C D A E (5) D A E B C (4) C B D E A (4) A B C E D (4) D E A C B (3) D B C A E (3) C B E A D (3) C B A E D (3) B D C A E (3) A E C B D (3) A D B E C (3) E D A C B (2) E C D A B (2) E C A B D (2) E A D C B (2) D B A C E (2) C E B D A (2) B C A D E (2) B A D C E (2) B A C D E (2) A E D C B (2) E D C A B (1) E C D B A (1) E C B D A (1) E C B A D (1) E A C D B (1) E A C B D (1) D E C A B (1) D B E C A (1) D B C E A (1) D B A E C (1) D A B C E (1) C A E B D (1) B C A E D (1) B A C E D (1) A E D B C (1) A D E B C (1) A D B C E (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -8 -4 10 B 10 0 -6 22 22 C 8 6 0 16 20 D 4 -22 -16 0 -4 E -10 -22 -20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -4 10 B 10 0 -6 22 22 C 8 6 0 16 20 D 4 -22 -16 0 -4 E -10 -22 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=22 A=19 B=16 E=14 so E is eliminated. Round 2 votes counts: C=36 D=25 A=23 B=16 so B is eliminated. Round 3 votes counts: C=44 D=28 A=28 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:224 A:194 D:181 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -4 10 B 10 0 -6 22 22 C 8 6 0 16 20 D 4 -22 -16 0 -4 E -10 -22 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -4 10 B 10 0 -6 22 22 C 8 6 0 16 20 D 4 -22 -16 0 -4 E -10 -22 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -4 10 B 10 0 -6 22 22 C 8 6 0 16 20 D 4 -22 -16 0 -4 E -10 -22 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4481: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (12) B E D A C (9) A C B E D (7) A C D E B (6) D E B C A (5) C D E B A (5) C A B D E (4) B A E D C (4) A B E D C (4) A B C E D (4) E D B C A (3) D C E B A (3) C D E A B (3) C D A E B (3) A D E C B (3) A D C E B (3) E B D C A (2) D E B A C (2) A C D B E (2) A C B D E (2) A B E C D (2) D E C B A (1) D E A B C (1) C B E D A (1) C B A E D (1) C A B E D (1) B E D C A (1) B E A D C (1) B C A E D (1) B A E C D (1) B A C E D (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 18 8 22 26 B -18 0 -14 -8 -8 C -8 14 0 12 18 D -22 8 -12 0 12 E -26 8 -18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 8 22 26 B -18 0 -14 -8 -8 C -8 14 0 12 18 D -22 8 -12 0 12 E -26 8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=30 B=18 D=12 E=5 so E is eliminated. Round 2 votes counts: A=35 C=30 B=20 D=15 so D is eliminated. Round 3 votes counts: A=36 C=34 B=30 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:237 C:218 D:193 B:176 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 8 22 26 B -18 0 -14 -8 -8 C -8 14 0 12 18 D -22 8 -12 0 12 E -26 8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 8 22 26 B -18 0 -14 -8 -8 C -8 14 0 12 18 D -22 8 -12 0 12 E -26 8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 8 22 26 B -18 0 -14 -8 -8 C -8 14 0 12 18 D -22 8 -12 0 12 E -26 8 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4482: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) B A E D C (7) A B D E C (7) B A E C D (6) A B E D C (6) E C D B A (5) A D B C E (5) E C B D A (4) D C A E B (4) D A C E B (4) C D E A B (4) B E C A D (4) D C E A B (3) C D E B A (3) A D E C B (3) D C A B E (2) D A E C B (2) C E D B A (2) C D B E A (2) B E C D A (2) A D C B E (2) A B D C E (2) E D C A B (1) E C D A B (1) E B C D A (1) E B A D C (1) E A D C B (1) D E C A B (1) C D B A E (1) B C E D A (1) B C E A D (1) B C A E D (1) B A D E C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 0 14 14 10 B 0 0 8 4 18 C -14 -8 0 -8 -24 D -14 -4 8 0 -4 E -10 -18 24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.418303 B: 0.581697 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513348687122 Cumulative probabilities = A: 0.418303 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 14 10 B 0 0 8 4 18 C -14 -8 0 -8 -24 D -14 -4 8 0 -4 E -10 -18 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=27 D=16 E=14 C=12 so C is eliminated. Round 2 votes counts: B=31 A=27 D=26 E=16 so E is eliminated. Round 3 votes counts: B=37 D=35 A=28 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:215 E:200 D:193 C:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 14 10 B 0 0 8 4 18 C -14 -8 0 -8 -24 D -14 -4 8 0 -4 E -10 -18 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 14 10 B 0 0 8 4 18 C -14 -8 0 -8 -24 D -14 -4 8 0 -4 E -10 -18 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 14 10 B 0 0 8 4 18 C -14 -8 0 -8 -24 D -14 -4 8 0 -4 E -10 -18 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4483: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C A B (8) B A D E C (8) A B C D E (7) B A C E D (6) E D C B A (5) D E C A B (5) D C E A B (5) C E D A B (5) A C B D E (4) E C D A B (3) B E D A C (3) B A E C D (3) A B D C E (3) C D E A B (2) C A D E B (2) B C A E D (2) B A E D C (2) E B C D A (1) D A E C B (1) D A C E B (1) C E D B A (1) C E A D B (1) C D A E B (1) C B A E D (1) C A B D E (1) B E C A D (1) B E A C D (1) B C E A D (1) B A D C E (1) A D C E B (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 10 18 16 B -10 0 2 10 16 C -10 -2 0 6 14 D -18 -10 -6 0 12 E -16 -16 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 18 16 B -10 0 2 10 16 C -10 -2 0 6 14 D -18 -10 -6 0 12 E -16 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=20 E=17 C=14 D=12 so D is eliminated. Round 2 votes counts: B=37 E=22 A=22 C=19 so C is eliminated. Round 3 votes counts: B=38 E=36 A=26 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:227 B:209 C:204 D:189 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 18 16 B -10 0 2 10 16 C -10 -2 0 6 14 D -18 -10 -6 0 12 E -16 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 18 16 B -10 0 2 10 16 C -10 -2 0 6 14 D -18 -10 -6 0 12 E -16 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 18 16 B -10 0 2 10 16 C -10 -2 0 6 14 D -18 -10 -6 0 12 E -16 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4484: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (7) C E A D B (7) C A E D B (7) B D E C A (7) E D C B A (6) E C D A B (6) A C B E D (5) D E C B A (4) D E B C A (4) B A D E C (4) A B D E C (4) D E C A B (3) B E C D A (3) A C E D B (3) D B E A C (2) D A E C B (2) B E D C A (2) B D E A C (2) B D A E C (2) B C E D A (2) B C A E D (2) E D C A B (1) E C B D A (1) D E A C B (1) C E B A D (1) C B E A D (1) B E D A C (1) B C E A D (1) B A D C E (1) B A C E D (1) B A C D E (1) A D E B C (1) A D C E B (1) A D B E C (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -30 -12 -24 B -4 0 -14 -12 -12 C 30 14 0 0 -14 D 12 12 0 0 -16 E 24 12 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -30 -12 -24 B -4 0 -14 -12 -12 C 30 14 0 0 -14 D 12 12 0 0 -16 E 24 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=23 A=18 D=16 E=14 so E is eliminated. Round 2 votes counts: C=30 B=29 D=23 A=18 so A is eliminated. Round 3 votes counts: C=39 B=35 D=26 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:233 C:215 D:204 B:179 A:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -30 -12 -24 B -4 0 -14 -12 -12 C 30 14 0 0 -14 D 12 12 0 0 -16 E 24 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -30 -12 -24 B -4 0 -14 -12 -12 C 30 14 0 0 -14 D 12 12 0 0 -16 E 24 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -30 -12 -24 B -4 0 -14 -12 -12 C 30 14 0 0 -14 D 12 12 0 0 -16 E 24 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4485: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (16) A D E B C (12) C A D E B (8) A D B E C (8) B E D C A (5) C E B D A (4) A B D E C (4) C D A E B (3) C B E A D (3) B C E D A (3) A D C E B (3) E D B A C (2) D E A B C (2) C E D A B (2) C B E D A (2) C A D B E (2) B E C D A (2) B A D E C (2) A C D E B (2) E D B C A (1) E B D C A (1) E B C D A (1) D E B A C (1) D C A E B (1) D A E C B (1) C E D B A (1) C E B A D (1) C B A E D (1) C B A D E (1) C A E D B (1) B C E A D (1) A D E C B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 12 4 4 B -4 0 22 -6 4 C -12 -22 0 -22 -18 D -4 6 22 0 6 E -4 -4 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 4 4 B -4 0 22 -6 4 C -12 -22 0 -22 -18 D -4 6 22 0 6 E -4 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=29 B=29 E=5 D=5 so E is eliminated. Round 2 votes counts: A=32 B=31 C=29 D=8 so D is eliminated. Round 3 votes counts: B=35 A=35 C=30 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:212 B:208 E:202 C:163 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 4 4 B -4 0 22 -6 4 C -12 -22 0 -22 -18 D -4 6 22 0 6 E -4 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 4 4 B -4 0 22 -6 4 C -12 -22 0 -22 -18 D -4 6 22 0 6 E -4 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 4 4 B -4 0 22 -6 4 C -12 -22 0 -22 -18 D -4 6 22 0 6 E -4 -4 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4486: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) D B A C E (7) B D C E A (7) A C E D B (6) E A C B D (5) B D A E C (4) B D A C E (4) E C B D A (3) E C B A D (3) E C A D B (3) E C A B D (3) E B C A D (3) D A B C E (3) C E A D B (3) A E B C D (3) A D C E B (3) D A C B E (2) B E C A D (2) B E A C D (2) B D E C A (2) B D C A E (2) A E C B D (2) E B A C D (1) E A C D B (1) E A B C D (1) D C B E A (1) D C B A E (1) D B C A E (1) C E D B A (1) C E B D A (1) C A E D B (1) B E D C A (1) B E D A C (1) B E C D A (1) B A D E C (1) A E D C B (1) A D B E C (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 22 16 10 B -2 0 -4 6 -12 C -22 4 0 14 -12 D -16 -6 -14 0 -20 E -10 12 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998063 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 22 16 10 B -2 0 -4 6 -12 C -22 4 0 14 -12 D -16 -6 -14 0 -20 E -10 12 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988122 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 E=23 D=15 C=6 so C is eliminated. Round 2 votes counts: A=30 E=28 B=27 D=15 so D is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:217 B:194 C:192 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 22 16 10 B -2 0 -4 6 -12 C -22 4 0 14 -12 D -16 -6 -14 0 -20 E -10 12 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988122 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 22 16 10 B -2 0 -4 6 -12 C -22 4 0 14 -12 D -16 -6 -14 0 -20 E -10 12 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988122 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 22 16 10 B -2 0 -4 6 -12 C -22 4 0 14 -12 D -16 -6 -14 0 -20 E -10 12 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988122 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4487: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) B E D C A (7) D E C B A (6) B C E D A (6) C B E D A (5) A D E B C (5) A C D E B (5) A B E D C (4) D E B C A (3) C A B D E (3) B E D A C (3) B C A E D (3) A D C E B (3) A C B D E (3) A B C E D (3) E D B C A (2) E D A B C (2) D E A B C (2) C D A E B (2) B E C D A (2) B A E D C (2) B A E C D (2) A E D B C (2) E D C B A (1) E D B A C (1) D E C A B (1) D E A C B (1) D C E B A (1) D C E A B (1) D A E C B (1) C D E B A (1) C D E A B (1) C A D E B (1) B C E A D (1) B A C E D (1) A D B E C (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 2 6 B -6 0 8 -4 -2 C -6 -8 0 -18 -16 D -2 4 18 0 4 E -6 2 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 2 6 B -6 0 8 -4 -2 C -6 -8 0 -18 -16 D -2 4 18 0 4 E -6 2 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=27 D=16 C=13 E=6 so E is eliminated. Round 2 votes counts: A=38 B=27 D=22 C=13 so C is eliminated. Round 3 votes counts: A=42 B=32 D=26 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:210 E:204 B:198 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 2 6 B -6 0 8 -4 -2 C -6 -8 0 -18 -16 D -2 4 18 0 4 E -6 2 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 2 6 B -6 0 8 -4 -2 C -6 -8 0 -18 -16 D -2 4 18 0 4 E -6 2 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 2 6 B -6 0 8 -4 -2 C -6 -8 0 -18 -16 D -2 4 18 0 4 E -6 2 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998603 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4488: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) C E D B A (6) E B C D A (5) D C A E B (5) A C D B E (5) D A C E B (4) C E B D A (4) B E A D C (4) A D C B E (4) E C B A D (3) D C E B A (3) D A B E C (3) C A E B D (3) B E A C D (3) A D B E C (3) A D B C E (3) A B E D C (3) A B E C D (3) E D B C A (2) E C B D A (2) D C E A B (2) A B D E C (2) A B C E D (2) E D C B A (1) E C D B A (1) E B D C A (1) E B C A D (1) D E B A C (1) D B E A C (1) D B A E C (1) D A E B C (1) C E B A D (1) C D E A B (1) C A D E B (1) B E D C A (1) B E C D A (1) B D E A C (1) B A E C D (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -4 6 -6 B 4 0 2 2 2 C 4 -2 0 8 -6 D -6 -2 -8 0 -14 E 6 -2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 6 -6 B 4 0 2 2 2 C 4 -2 0 8 -6 D -6 -2 -8 0 -14 E 6 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=21 B=19 E=16 C=16 so E is eliminated. Round 2 votes counts: A=28 B=26 D=24 C=22 so C is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:212 B:205 C:202 A:196 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 6 -6 B 4 0 2 2 2 C 4 -2 0 8 -6 D -6 -2 -8 0 -14 E 6 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 6 -6 B 4 0 2 2 2 C 4 -2 0 8 -6 D -6 -2 -8 0 -14 E 6 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 6 -6 B 4 0 2 2 2 C 4 -2 0 8 -6 D -6 -2 -8 0 -14 E 6 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4489: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (14) C D B E A (8) E A B C D (6) D C A B E (6) B E A D C (6) D C B A E (4) C B D E A (4) E B C A D (3) E B A C D (3) C E B D A (3) C D A E B (3) B C D E A (3) E C B A D (2) E A B D C (2) D A C B E (2) C D E A B (2) B C E D A (2) A E D B C (2) A E C D B (2) A E B C D (2) E C A B D (1) E B A D C (1) E A C B D (1) D B C A E (1) C D E B A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A E D B (1) C A D E B (1) B E C D A (1) B D E C A (1) B D C E A (1) B A E D C (1) A E C B D (1) A D E B C (1) A D C E B (1) A D B E C (1) A C D E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -2 10 -4 B -6 0 6 22 -8 C 2 -6 0 8 -6 D -10 -22 -8 0 -12 E 4 8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -2 10 -4 B -6 0 6 22 -8 C 2 -6 0 8 -6 D -10 -22 -8 0 -12 E 4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 E=19 B=15 D=13 so D is eliminated. Round 2 votes counts: C=36 A=29 E=19 B=16 so B is eliminated. Round 3 votes counts: C=43 A=30 E=27 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:215 B:207 A:205 C:199 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 10 -4 B -6 0 6 22 -8 C 2 -6 0 8 -6 D -10 -22 -8 0 -12 E 4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 10 -4 B -6 0 6 22 -8 C 2 -6 0 8 -6 D -10 -22 -8 0 -12 E 4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 10 -4 B -6 0 6 22 -8 C 2 -6 0 8 -6 D -10 -22 -8 0 -12 E 4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4490: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) E A C B D (6) E C A B D (5) C B E D A (5) A D E B C (5) E C D B A (4) E C B A D (4) C E B A D (4) A D B E C (4) E C A D B (3) E A C D B (3) D A E B C (3) C E B D A (3) B D C A E (3) A E D B C (3) A E C B D (3) A B C D E (3) E D C A B (2) E C B D A (2) E A D C B (2) D B C A E (2) D B A E C (2) C B D E A (2) B D A C E (2) E D C B A (1) E C D A B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C E A (1) D A B E C (1) D A B C E (1) C E A B D (1) C B E A D (1) C B A E D (1) C B A D E (1) B C D E A (1) B C D A E (1) B A D C E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -4 4 -10 B 4 0 -12 0 -14 C 4 12 0 8 -12 D -4 0 -8 0 -8 E 10 14 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 4 -10 B 4 0 -12 0 -14 C 4 12 0 8 -12 D -4 0 -8 0 -8 E 10 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=21 A=20 C=18 B=8 so B is eliminated. Round 2 votes counts: E=33 D=26 A=21 C=20 so C is eliminated. Round 3 votes counts: E=47 D=30 A=23 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:206 A:193 D:190 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 4 -10 B 4 0 -12 0 -14 C 4 12 0 8 -12 D -4 0 -8 0 -8 E 10 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 4 -10 B 4 0 -12 0 -14 C 4 12 0 8 -12 D -4 0 -8 0 -8 E 10 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 4 -10 B 4 0 -12 0 -14 C 4 12 0 8 -12 D -4 0 -8 0 -8 E 10 14 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4491: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) D A C B E (8) E B C A D (6) D C A B E (6) B E D A C (6) B E A D C (6) A D B C E (5) E B A C D (4) C A D E B (4) B E D C A (4) A D C B E (4) E B C D A (3) B E C D A (3) A C D E B (3) E C B D A (2) E C B A D (2) E C A D B (2) C E A D B (2) B A E D C (2) A E B C D (2) E A C D B (1) D C B A E (1) D C A E B (1) D B A C E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D E A B (1) C D A B E (1) B D E A C (1) B D A E C (1) B A D E C (1) A E C D B (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 10 -4 -6 8 B -10 0 -8 -12 2 C 4 8 0 2 4 D 6 12 -2 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.999709 D: 0.000000 E: 0.000291 Sum of squares = 0.999418172076 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.999709 D: 0.999709 E: 1.000000 A B C D E A 0 10 -4 -6 8 B -10 0 -8 -12 2 C 4 8 0 2 4 D 6 12 -2 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=22 E=20 D=17 A=17 so D is eliminated. Round 2 votes counts: C=30 B=25 A=25 E=20 so E is eliminated. Round 3 votes counts: B=38 C=36 A=26 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:209 A:204 E:191 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -6 8 B -10 0 -8 -12 2 C 4 8 0 2 4 D 6 12 -2 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -6 8 B -10 0 -8 -12 2 C 4 8 0 2 4 D 6 12 -2 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -6 8 B -10 0 -8 -12 2 C 4 8 0 2 4 D 6 12 -2 0 4 E -8 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4492: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A C B E D (8) E B D A C (5) A C D E B (5) E D B C A (4) D C A E B (4) B C A E D (4) A C D B E (4) E B A D C (3) D A C E B (3) C D A B E (3) C B A D E (3) C A D B E (3) E D B A C (2) E B D C A (2) E A D B C (2) E A B D C (2) D E C B A (2) D C B E A (2) C A D E B (2) C A B D E (2) B E A C D (2) A E B C D (2) E D A C B (1) E D A B C (1) E B A C D (1) D E C A B (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C E A (1) C B D E A (1) C B D A E (1) C A B E D (1) B E C A D (1) B D E C A (1) B C E D A (1) B C E A D (1) B A C E D (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 8 6 B -2 0 -6 6 2 C 2 6 0 0 12 D -8 -6 0 0 -12 E -6 -2 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.867822 D: 0.132178 E: 0.000000 Sum of squares = 0.770586598431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.867822 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 8 6 B -2 0 -6 6 2 C 2 6 0 0 12 D -8 -6 0 0 -12 E -6 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 B=19 D=16 C=16 so D is eliminated. Round 2 votes counts: A=29 E=27 C=23 B=21 so B is eliminated. Round 3 votes counts: E=40 C=30 A=30 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:210 A:207 B:200 E:196 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 8 6 B -2 0 -6 6 2 C 2 6 0 0 12 D -8 -6 0 0 -12 E -6 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 6 B -2 0 -6 6 2 C 2 6 0 0 12 D -8 -6 0 0 -12 E -6 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 6 B -2 0 -6 6 2 C 2 6 0 0 12 D -8 -6 0 0 -12 E -6 -2 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4493: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) C E A D B (10) A D C E B (7) E B C D A (6) C A E D B (4) B E C D A (4) B E C A D (4) A D B C E (4) E C B D A (3) E C B A D (3) D A C B E (3) C E D A B (3) C E B A D (3) C E A B D (3) B E D C A (3) A C D E B (3) E C D B A (2) E C D A B (2) D A C E B (2) B D E C A (2) B D E A C (2) B D A E C (2) B A D E C (2) A C B E D (2) E D C A B (1) E B D C A (1) D B E A C (1) D B A E C (1) D A B E C (1) B E A D C (1) B E A C D (1) B A E D C (1) B A C E D (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 14 -8 2 -10 B -14 0 -6 -10 -8 C 8 6 0 10 14 D -2 10 -10 0 -18 E 10 8 -14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 2 -10 B -14 0 -6 -10 -8 C 8 6 0 10 14 D -2 10 -10 0 -18 E 10 8 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 E=18 D=18 A=18 so E is eliminated. Round 2 votes counts: C=33 B=30 D=19 A=18 so A is eliminated. Round 3 votes counts: C=39 D=31 B=30 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:211 A:199 D:190 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -8 2 -10 B -14 0 -6 -10 -8 C 8 6 0 10 14 D -2 10 -10 0 -18 E 10 8 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 2 -10 B -14 0 -6 -10 -8 C 8 6 0 10 14 D -2 10 -10 0 -18 E 10 8 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 2 -10 B -14 0 -6 -10 -8 C 8 6 0 10 14 D -2 10 -10 0 -18 E 10 8 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4494: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (5) C D B E A (5) C D B A E (5) E D A B C (4) E A D B C (4) C B D A E (4) B D A C E (4) B A C E D (4) A B E D C (4) E D C A B (3) E A B C D (3) D E C B A (3) D E B A C (3) D B C A E (3) A E B D C (3) A E B C D (3) E D A C B (2) E C A D B (2) E A D C B (2) E A C D B (2) E A C B D (2) D E C A B (2) D E A B C (2) D B A E C (2) C B A D E (2) B A D E C (2) A C B E D (2) E C D A B (1) E A B D C (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C E A (1) C E D A B (1) C E A B D (1) C D E B A (1) C B A E D (1) C A B E D (1) B D C A E (1) B D A E C (1) B C A D E (1) B A D C E (1) A E C B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -12 -6 B 4 0 2 -12 4 C -8 -2 0 -12 -10 D 12 12 12 0 2 E 6 -4 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -12 -6 B 4 0 2 -12 4 C -8 -2 0 -12 -10 D 12 12 12 0 2 E 6 -4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 C=21 A=15 B=14 so B is eliminated. Round 2 votes counts: D=30 E=26 C=22 A=22 so C is eliminated. Round 3 votes counts: D=45 E=28 A=27 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:205 B:199 A:193 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -12 -6 B 4 0 2 -12 4 C -8 -2 0 -12 -10 D 12 12 12 0 2 E 6 -4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -12 -6 B 4 0 2 -12 4 C -8 -2 0 -12 -10 D 12 12 12 0 2 E 6 -4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -12 -6 B 4 0 2 -12 4 C -8 -2 0 -12 -10 D 12 12 12 0 2 E 6 -4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4495: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (13) D C B E A (10) D B C E A (5) C D B E A (5) C D B A E (5) A C E B D (5) E B A D C (4) C D A B E (4) C A E B D (4) B E D A C (4) A E B C D (4) C A E D B (3) A E B D C (3) E B D A C (2) E A B D C (2) E A B C D (2) D C B A E (2) D B E C A (2) C B D E A (2) B E D C A (2) B D E C A (2) B D E A C (2) A E D B C (2) E A D B C (1) D C A B E (1) D B E A C (1) C E A B D (1) C B A E D (1) C A D E B (1) B E C D A (1) B D C E A (1) B C D E A (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -6 -4 0 B 4 0 -18 12 2 C 6 18 0 6 4 D 4 -12 -6 0 -12 E 0 -2 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 0 B 4 0 -18 12 2 C 6 18 0 6 4 D 4 -12 -6 0 -12 E 0 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=26 D=21 B=13 E=11 so E is eliminated. Round 2 votes counts: A=34 C=26 D=21 B=19 so B is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:217 E:203 B:200 A:193 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 0 B 4 0 -18 12 2 C 6 18 0 6 4 D 4 -12 -6 0 -12 E 0 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 0 B 4 0 -18 12 2 C 6 18 0 6 4 D 4 -12 -6 0 -12 E 0 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 0 B 4 0 -18 12 2 C 6 18 0 6 4 D 4 -12 -6 0 -12 E 0 -2 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4496: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) B D A E C (7) D A C B E (6) B E C D A (6) A D B E C (6) C E B D A (5) C D A B E (5) B E A D C (5) C E A D B (4) A D C E B (4) A D C B E (4) E C B D A (3) D B A C E (3) B D A C E (3) A D E B C (3) D A B C E (2) C B E D A (2) C A D E B (2) A E D C B (2) A E D B C (2) A C D E B (2) E C A B D (1) E B A D C (1) E B A C D (1) E A D B C (1) D C A B E (1) D A B E C (1) C E D A B (1) C E B A D (1) C D E A B (1) C B D E A (1) B E D C A (1) B E D A C (1) B C E D A (1) B C D E A (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 10 -2 8 B 2 0 -8 -2 14 C -10 8 0 -8 0 D 2 2 8 0 6 E -8 -14 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -2 8 B 2 0 -8 -2 14 C -10 8 0 -8 0 D 2 2 8 0 6 E -8 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=24 C=22 E=16 D=13 so D is eliminated. Round 2 votes counts: A=33 B=28 C=23 E=16 so E is eliminated. Round 3 votes counts: C=36 A=34 B=30 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:209 A:207 B:203 C:195 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -2 8 B 2 0 -8 -2 14 C -10 8 0 -8 0 D 2 2 8 0 6 E -8 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -2 8 B 2 0 -8 -2 14 C -10 8 0 -8 0 D 2 2 8 0 6 E -8 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -2 8 B 2 0 -8 -2 14 C -10 8 0 -8 0 D 2 2 8 0 6 E -8 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4497: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) D B C E A (8) E A B D C (6) C D B A E (6) C A D B E (6) B D C E A (6) E B D A C (5) A E C D B (5) E B D C A (4) A E B D C (4) E D B C A (3) C D A B E (3) B D C A E (3) A E C B D (3) A E B C D (3) A C E D B (3) A C D B E (3) D C B E A (2) C D B E A (2) A B E D C (2) E B A D C (1) E A D C B (1) D B E C A (1) C E D B A (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B C D A E (1) A C E B D (1) A C D E B (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -16 -14 -4 B 10 0 20 6 14 C 16 -20 0 -18 -2 D 14 -6 18 0 12 E 4 -14 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -14 -4 B 10 0 20 6 14 C 16 -20 0 -18 -2 D 14 -6 18 0 12 E 4 -14 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=22 E=20 C=20 D=11 so D is eliminated. Round 2 votes counts: B=31 A=27 C=22 E=20 so E is eliminated. Round 3 votes counts: B=44 A=34 C=22 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:219 E:190 C:188 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -16 -14 -4 B 10 0 20 6 14 C 16 -20 0 -18 -2 D 14 -6 18 0 12 E 4 -14 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -14 -4 B 10 0 20 6 14 C 16 -20 0 -18 -2 D 14 -6 18 0 12 E 4 -14 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -14 -4 B 10 0 20 6 14 C 16 -20 0 -18 -2 D 14 -6 18 0 12 E 4 -14 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4498: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E D B C A (7) D E B C A (7) E A D B C (6) C A B D E (5) A C B E D (5) E D B A C (4) D B E C A (4) C B A D E (4) B C D E A (4) A E D B C (4) A E B D C (4) A C E D B (4) D B C E A (3) C D B E A (3) C B D E A (3) A E C D B (3) A C E B D (3) E D A B C (2) D C B E A (2) A C D E B (2) E B A D C (1) E A B D C (1) D C E B A (1) C A D B E (1) B E C D A (1) B D E C A (1) B C A E D (1) B A C E D (1) A E D C B (1) A E C B D (1) A E B C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -10 0 0 B 10 0 6 -8 -4 C 10 -6 0 4 4 D 0 8 -4 0 -2 E 0 4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.333333 E: 0.000000 Sum of squares = 0.358024690556 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 0 0 B 10 0 6 -8 -4 C 10 -6 0 4 4 D 0 8 -4 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691071 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=24 E=21 D=17 B=8 so B is eliminated. Round 2 votes counts: A=31 C=29 E=22 D=18 so D is eliminated. Round 3 votes counts: C=35 E=34 A=31 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:202 D:201 E:201 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 0 0 B 10 0 6 -8 -4 C 10 -6 0 4 4 D 0 8 -4 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691071 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 0 0 B 10 0 6 -8 -4 C 10 -6 0 4 4 D 0 8 -4 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691071 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 0 0 B 10 0 6 -8 -4 C 10 -6 0 4 4 D 0 8 -4 0 -2 E 0 4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.444444 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691071 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4499: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (14) E B C D A (12) D E B C A (11) A C B E D (9) D A E C B (7) C B E A D (7) B E C A D (7) C B A E D (5) E D B C A (3) E B D C A (3) D A E B C (3) A C D B E (3) E B C A D (2) D E A B C (2) D A C E B (2) D A C B E (2) B C E A D (2) A C B D E (2) D E B A C (1) D A B E C (1) C E D B A (1) C A B E D (1) Total count = 100 A B C D E A 0 -8 -8 4 -2 B 8 0 -6 0 6 C 8 6 0 2 -4 D -4 0 -2 0 -4 E 2 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 -8 -8 4 -2 B 8 0 -6 0 6 C 8 6 0 2 -4 D -4 0 -2 0 -4 E 2 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999978 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=28 E=20 C=14 B=9 so B is eliminated. Round 2 votes counts: D=29 A=28 E=27 C=16 so C is eliminated. Round 3 votes counts: E=37 A=34 D=29 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:206 B:204 E:202 D:195 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 4 -2 B 8 0 -6 0 6 C 8 6 0 2 -4 D -4 0 -2 0 -4 E 2 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999978 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 4 -2 B 8 0 -6 0 6 C 8 6 0 2 -4 D -4 0 -2 0 -4 E 2 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999978 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 4 -2 B 8 0 -6 0 6 C 8 6 0 2 -4 D -4 0 -2 0 -4 E 2 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999978 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4500: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E D A C B (7) E D C A B (5) B C A E D (5) B A D C E (5) E C D A B (4) D A E B C (4) B C E A D (4) D E A C B (3) C E B D A (3) C E B A D (3) B C A D E (3) A E D B C (3) A D B E C (3) E C B A D (2) E C A B D (2) E A D C B (2) D E C A B (2) D C E B A (2) D A E C B (2) C B E D A (2) C B E A D (2) C B D A E (2) B C D A E (2) A D E B C (2) A B D C E (2) E C D B A (1) E A D B C (1) E A C B D (1) E A B D C (1) D C B E A (1) D B C A E (1) D B A C E (1) D A B E C (1) C D B E A (1) C B D E A (1) B D C A E (1) B A E C D (1) A E B C D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 2 -10 0 B -14 0 4 -12 -2 C -2 -4 0 -20 2 D 10 12 20 0 0 E 0 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.415064 E: 0.584936 Sum of squares = 0.514428412222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.415064 E: 1.000000 A B C D E A 0 14 2 -10 0 B -14 0 4 -12 -2 C -2 -4 0 -20 2 D 10 12 20 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=26 D=26 B=21 C=14 A=13 so A is eliminated. Round 2 votes counts: D=32 E=30 B=24 C=14 so C is eliminated. Round 3 votes counts: E=36 D=33 B=31 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:203 E:200 B:188 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 2 -10 0 B -14 0 4 -12 -2 C -2 -4 0 -20 2 D 10 12 20 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 -10 0 B -14 0 4 -12 -2 C -2 -4 0 -20 2 D 10 12 20 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 -10 0 B -14 0 4 -12 -2 C -2 -4 0 -20 2 D 10 12 20 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4501: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) D A E B C (8) C B D E A (7) C B E A D (6) E A C B D (4) D A E C B (4) C B E D A (4) B C E A D (4) B C A E D (4) A E D B C (4) D E A C B (3) B D A C E (3) B C D E A (3) B C A D E (3) B A C E D (3) A E D C B (3) E A C D B (2) C D B E A (2) B C D A E (2) B A D E C (2) A D E B C (2) E D A C B (1) E C A D B (1) D E C A B (1) D C B E A (1) D B C A E (1) D B A E C (1) D B A C E (1) C E B A D (1) C E A B D (1) B D C A E (1) B A E D C (1) B A E C D (1) A E C B D (1) A E B D C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 16 14 -4 B 2 0 -6 8 2 C -16 6 0 0 -6 D -14 -8 0 0 -10 E 4 -2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101895 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 -2 16 14 -4 B 2 0 -6 8 2 C -16 6 0 0 -6 D -14 -8 0 0 -10 E 4 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101823 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=21 D=20 E=19 A=13 so A is eliminated. Round 2 votes counts: E=29 B=28 D=22 C=21 so C is eliminated. Round 3 votes counts: B=45 E=31 D=24 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:212 E:209 B:203 C:192 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 16 14 -4 B 2 0 -6 8 2 C -16 6 0 0 -6 D -14 -8 0 0 -10 E 4 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101823 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 14 -4 B 2 0 -6 8 2 C -16 6 0 0 -6 D -14 -8 0 0 -10 E 4 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101823 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 14 -4 B 2 0 -6 8 2 C -16 6 0 0 -6 D -14 -8 0 0 -10 E 4 -2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101823 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4502: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (17) D B C E A (8) E A C B D (7) C B E A D (7) B C D E A (6) B C E D A (5) A E C D B (5) E C A B D (4) D B C A E (4) D A E C B (4) C E B A D (4) B C E A D (4) A E D C B (4) D B A C E (3) D A B E C (3) B D C E A (3) C E A B D (2) C B E D A (2) E D C A B (1) D B A E C (1) D A E B C (1) B D C A E (1) B C D A E (1) A E D B C (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -4 14 -6 B -2 0 -16 28 -2 C 4 16 0 30 0 D -14 -28 -30 0 -28 E 6 2 0 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.380944 D: 0.000000 E: 0.619056 Sum of squares = 0.528348501078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.380944 D: 0.380944 E: 1.000000 A B C D E A 0 2 -4 14 -6 B -2 0 -16 28 -2 C 4 16 0 30 0 D -14 -28 -30 0 -28 E 6 2 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=24 B=20 C=15 E=12 so E is eliminated. Round 2 votes counts: A=36 D=25 B=20 C=19 so C is eliminated. Round 3 votes counts: A=42 B=33 D=25 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:225 E:218 B:204 A:203 D:150 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 14 -6 B -2 0 -16 28 -2 C 4 16 0 30 0 D -14 -28 -30 0 -28 E 6 2 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 14 -6 B -2 0 -16 28 -2 C 4 16 0 30 0 D -14 -28 -30 0 -28 E 6 2 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 14 -6 B -2 0 -16 28 -2 C 4 16 0 30 0 D -14 -28 -30 0 -28 E 6 2 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4503: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E C A D B (8) C E A D B (7) D B A C E (6) B D A E C (6) C D A E B (5) B D C E A (5) E C A B D (4) C E B D A (3) B A D E C (3) A E C D B (3) A D C E B (3) E C B D A (2) E B C A D (2) E A C D B (2) D C A B E (2) C D B E A (2) B E D A C (2) B E A D C (2) B C E D A (2) E B C D A (1) E B A D C (1) E A C B D (1) D A B E C (1) D A B C E (1) C E D A B (1) C E A B D (1) C D E A B (1) C B E D A (1) B E A C D (1) B D C A E (1) B C D E A (1) B A E D C (1) A E D C B (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 2 -8 0 B 6 0 0 2 2 C -2 0 0 -2 12 D 8 -2 2 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.736350 C: 0.263650 D: 0.000000 E: 0.000000 Sum of squares = 0.61172231505 Cumulative probabilities = A: 0.000000 B: 0.736350 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -8 0 B 6 0 0 2 2 C -2 0 0 -2 12 D 8 -2 2 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500170 C: 0.499830 D: 0.000000 E: 0.000000 Sum of squares = 0.500000058132 Cumulative probabilities = A: 0.000000 B: 0.500170 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=21 C=21 A=13 D=10 so D is eliminated. Round 2 votes counts: B=41 C=23 E=21 A=15 so A is eliminated. Round 3 votes counts: B=45 C=28 E=27 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:207 B:205 C:204 A:194 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -8 0 B 6 0 0 2 2 C -2 0 0 -2 12 D 8 -2 2 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500170 C: 0.499830 D: 0.000000 E: 0.000000 Sum of squares = 0.500000058132 Cumulative probabilities = A: 0.000000 B: 0.500170 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -8 0 B 6 0 0 2 2 C -2 0 0 -2 12 D 8 -2 2 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500170 C: 0.499830 D: 0.000000 E: 0.000000 Sum of squares = 0.500000058132 Cumulative probabilities = A: 0.000000 B: 0.500170 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -8 0 B 6 0 0 2 2 C -2 0 0 -2 12 D 8 -2 2 0 6 E 0 -2 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500170 C: 0.499830 D: 0.000000 E: 0.000000 Sum of squares = 0.500000058132 Cumulative probabilities = A: 0.000000 B: 0.500170 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4504: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) C A D E B (7) B E D A C (6) E B D C A (5) D A C E B (5) D E C A B (4) B A C D E (4) A C D B E (4) E C D A B (3) D B E A C (3) A C B D E (3) E C D B A (2) E C B D A (2) E B C D A (2) D B A E C (2) D B A C E (2) D A E B C (2) C E A D B (2) C E A B D (2) C A E B D (2) C A B E D (2) B E A C D (2) B A E D C (2) A D C B E (2) A B C D E (2) E D C A B (1) E D B C A (1) E B D A C (1) E B C A D (1) D E A B C (1) D C E A B (1) D C A E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C D A E B (1) C A E D B (1) B E D C A (1) B E A D C (1) B D A E C (1) B C A E D (1) B A D C E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -4 0 -2 B -2 0 4 4 4 C 4 -4 0 8 -4 D 0 -4 -8 0 -2 E 2 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.455380 B: 0.316930 C: 0.089240 D: 0.000000 E: 0.138450 Sum of squares = 0.3349477722 Cumulative probabilities = A: 0.455380 B: 0.772310 C: 0.861550 D: 0.861550 E: 1.000000 A B C D E A 0 2 -4 0 -2 B -2 0 4 4 4 C 4 -4 0 8 -4 D 0 -4 -8 0 -2 E 2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.444444 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 E=18 C=18 A=13 so A is eliminated. Round 2 votes counts: B=31 C=26 D=25 E=18 so E is eliminated. Round 3 votes counts: B=40 C=33 D=27 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:205 C:202 E:202 A:198 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -4 0 -2 B -2 0 4 4 4 C 4 -4 0 8 -4 D 0 -4 -8 0 -2 E 2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.444444 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 -2 B -2 0 4 4 4 C 4 -4 0 8 -4 D 0 -4 -8 0 -2 E 2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.444444 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 -2 B -2 0 4 4 4 C 4 -4 0 8 -4 D 0 -4 -8 0 -2 E 2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.444444 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4505: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (12) B C D A E (10) E D A C B (9) A B C E D (8) D C B E A (7) E A D C B (5) B C A D E (5) A E C B D (5) A C B E D (4) A B C D E (4) E D A B C (3) D E B C A (3) C B D E A (3) E D C B A (2) B A C D E (2) A D E B C (2) E D C A B (1) E C B D A (1) E C A B D (1) E A D B C (1) E A C D B (1) E A C B D (1) D B C A E (1) D A B C E (1) C E B D A (1) C B E A D (1) C B A E D (1) C B A D E (1) A E D C B (1) A E B D C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 0 -8 -4 B 0 0 -14 0 -2 C 0 14 0 0 -2 D 8 0 0 0 4 E 4 2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.329186 D: 0.670814 E: 0.000000 Sum of squares = 0.558355029687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.329186 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -8 -4 B 0 0 -14 0 -2 C 0 14 0 0 -2 D 8 0 0 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 D=24 B=17 C=7 so C is eliminated. Round 2 votes counts: A=27 E=26 D=24 B=23 so B is eliminated. Round 3 votes counts: D=37 A=36 E=27 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:206 D:206 E:202 A:194 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -8 -4 B 0 0 -14 0 -2 C 0 14 0 0 -2 D 8 0 0 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 -4 B 0 0 -14 0 -2 C 0 14 0 0 -2 D 8 0 0 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 -4 B 0 0 -14 0 -2 C 0 14 0 0 -2 D 8 0 0 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4506: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (14) D E B A C (7) C B E D A (7) C A B E D (7) A D E B C (6) D A E B C (5) B E C D A (5) B C E D A (5) A C E B D (5) A D E C B (4) E D B A C (3) C B E A D (3) A C B E D (3) D E B C A (2) D E A B C (2) D B E C A (2) B D E C A (2) A D C E B (2) A C D B E (2) E D B C A (1) E C B A D (1) E B D C A (1) E A B D C (1) D C B A E (1) C B D A E (1) C A B D E (1) B E D C A (1) A E D B C (1) A E C B D (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -8 10 14 B 12 0 -12 18 12 C 8 12 0 16 10 D -10 -18 -16 0 -20 E -14 -12 -10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 10 14 B 12 0 -12 18 12 C 8 12 0 16 10 D -10 -18 -16 0 -20 E -14 -12 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=28 D=19 B=13 E=7 so E is eliminated. Round 2 votes counts: C=34 A=29 D=23 B=14 so B is eliminated. Round 3 votes counts: C=44 A=29 D=27 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 B:215 A:202 E:192 D:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 10 14 B 12 0 -12 18 12 C 8 12 0 16 10 D -10 -18 -16 0 -20 E -14 -12 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 10 14 B 12 0 -12 18 12 C 8 12 0 16 10 D -10 -18 -16 0 -20 E -14 -12 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 10 14 B 12 0 -12 18 12 C 8 12 0 16 10 D -10 -18 -16 0 -20 E -14 -12 -10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4507: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) C D E B A (7) B D E A C (7) C E D A B (6) E D C A B (5) D E C B A (5) D E B C A (5) C D E A B (5) C B D E A (5) A B E D C (4) C A E D B (3) B D E C A (3) B D C E A (3) A C E D B (3) A B C E D (3) E D B A C (2) E C D A B (2) D E B A C (2) C B A D E (2) A E D C B (2) A E D B C (2) A E B D C (2) A C B E D (2) E D A B C (1) E A D C B (1) E A D B C (1) D B E C A (1) C E A D B (1) C D B A E (1) B D A E C (1) B C D E A (1) B A E D C (1) B A D C E (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -10 -24 -26 B 10 0 -4 -12 -14 C 10 4 0 -14 -12 D 24 12 14 0 14 E 26 14 12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -24 -26 B 10 0 -4 -12 -14 C 10 4 0 -14 -12 D 24 12 14 0 14 E 26 14 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=25 A=20 D=13 E=12 so E is eliminated. Round 2 votes counts: C=32 B=25 A=22 D=21 so D is eliminated. Round 3 votes counts: C=42 B=35 A=23 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:232 E:219 C:194 B:190 A:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 -24 -26 B 10 0 -4 -12 -14 C 10 4 0 -14 -12 D 24 12 14 0 14 E 26 14 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -24 -26 B 10 0 -4 -12 -14 C 10 4 0 -14 -12 D 24 12 14 0 14 E 26 14 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -24 -26 B 10 0 -4 -12 -14 C 10 4 0 -14 -12 D 24 12 14 0 14 E 26 14 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4508: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) D C E A B (6) B E A C D (6) E D C B A (5) D C A E B (5) C D E A B (5) C D A B E (5) C D E B A (4) B E C A D (4) B A E C D (4) B A E D C (3) A D C B E (3) A B D C E (3) E C D B A (2) E C B D A (2) E B A D C (2) E B A C D (2) D A C B E (2) C D A E B (2) C B E D A (2) B E A D C (2) B C A E D (2) B A C E D (2) A B D E C (2) A B C D E (2) E D B A C (1) E D A B C (1) E B C D A (1) E A B D C (1) D E C A B (1) D C A B E (1) D A C E B (1) C E D B A (1) C D B E A (1) C D B A E (1) C B A D E (1) C A D B E (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -4 2 2 B -4 0 -4 2 14 C 4 4 0 2 2 D -2 -2 -2 0 -6 E -2 -14 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 2 2 B -4 0 -4 2 14 C 4 4 0 2 2 D -2 -2 -2 0 -6 E -2 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 A=21 E=17 D=16 so D is eliminated. Round 2 votes counts: C=35 A=24 B=23 E=18 so E is eliminated. Round 3 votes counts: C=45 B=29 A=26 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:204 A:202 D:194 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 2 2 B -4 0 -4 2 14 C 4 4 0 2 2 D -2 -2 -2 0 -6 E -2 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 2 2 B -4 0 -4 2 14 C 4 4 0 2 2 D -2 -2 -2 0 -6 E -2 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 2 2 B -4 0 -4 2 14 C 4 4 0 2 2 D -2 -2 -2 0 -6 E -2 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4509: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (13) B A D E C (11) C E D B A (6) B D A E C (6) B D C E A (5) B D A C E (5) C E D A B (4) E C D A B (3) C D E B A (3) A E C D B (3) A B D E C (3) D B E C A (2) D B A E C (2) C B D E A (2) B D E C A (2) B C D E A (2) B C A D E (2) A E B D C (2) A B E D C (2) E D C A B (1) E D A C B (1) E C A D B (1) E A D C B (1) E A C D B (1) D E B C A (1) D B E A C (1) D B C E A (1) C E A B D (1) C B E D A (1) C B A E D (1) C A E B D (1) B D E A C (1) B D C A E (1) B A C E D (1) B A C D E (1) A E D C B (1) A E D B C (1) A E C B D (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -6 0 -6 B 14 0 8 8 8 C 6 -8 0 0 4 D 0 -8 0 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 0 -6 B 14 0 8 8 8 C 6 -8 0 0 4 D 0 -8 0 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=32 A=16 E=8 D=7 so D is eliminated. Round 2 votes counts: B=43 C=32 A=16 E=9 so E is eliminated. Round 3 votes counts: B=44 C=37 A=19 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:201 D:197 E:196 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 0 -6 B 14 0 8 8 8 C 6 -8 0 0 4 D 0 -8 0 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 0 -6 B 14 0 8 8 8 C 6 -8 0 0 4 D 0 -8 0 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 0 -6 B 14 0 8 8 8 C 6 -8 0 0 4 D 0 -8 0 0 2 E 6 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4510: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D C A B E (6) D B E A C (6) D A C E B (6) C A E B D (5) B C E A D (5) E A C B D (4) D A E C B (4) C B A E D (4) A C E B D (4) E B C A D (3) E B A C D (3) B C E D A (3) E A D B C (2) D E A B C (2) D C B A E (2) D A C B E (2) C B D A E (2) C A D E B (2) C A D B E (2) B E D A C (2) B D C E A (2) E C A B D (1) E B D A C (1) E B A D C (1) E A B D C (1) D E B A C (1) D C A E B (1) D B C E A (1) D B C A E (1) D A E B C (1) C E A B D (1) C D A B E (1) C B E A D (1) C B A D E (1) C A E D B (1) C A B E D (1) B E D C A (1) B E A D C (1) B D E C A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 -14 8 -4 B 2 0 -6 16 8 C 14 6 0 8 10 D -8 -16 -8 0 -10 E 4 -8 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 8 -4 B 2 0 -6 16 8 C 14 6 0 8 10 D -8 -16 -8 0 -10 E 4 -8 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=24 C=21 E=16 A=6 so A is eliminated. Round 2 votes counts: D=34 C=25 B=24 E=17 so E is eliminated. Round 3 votes counts: D=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:219 B:210 E:198 A:194 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 8 -4 B 2 0 -6 16 8 C 14 6 0 8 10 D -8 -16 -8 0 -10 E 4 -8 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 8 -4 B 2 0 -6 16 8 C 14 6 0 8 10 D -8 -16 -8 0 -10 E 4 -8 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 8 -4 B 2 0 -6 16 8 C 14 6 0 8 10 D -8 -16 -8 0 -10 E 4 -8 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4511: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D C B E A (8) A E C B D (6) A E B C D (5) D C E B A (4) D C A E B (4) D B C E A (4) B C E D A (4) A D E C B (4) A D B E C (4) D B C A E (3) D A C E B (3) C E B A D (3) C B E D A (3) A E D C B (3) A B E C D (3) E A C B D (2) D C E A B (2) C D E B A (2) A E C D B (2) A E B D C (2) E C B A D (1) D C B A E (1) D B A C E (1) D A E B C (1) D A C B E (1) D A B E C (1) C E A B D (1) C D E A B (1) C D B E A (1) C B E A D (1) B E A D C (1) B D C E A (1) B D A E C (1) B D A C E (1) B A E D C (1) B A E C D (1) B A D E C (1) A D E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -6 6 4 B 4 0 -10 -6 2 C 6 10 0 -8 2 D -6 6 8 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.300000 E: 0.000000 Sum of squares = 0.339999999988 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 6 4 B 4 0 -10 -6 2 C 6 10 0 -8 2 D -6 6 8 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=32 B=20 C=12 E=3 so E is eliminated. Round 2 votes counts: A=34 D=33 B=20 C=13 so C is eliminated. Round 3 votes counts: D=37 A=35 B=28 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:205 D:205 A:200 B:195 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 -6 6 4 B 4 0 -10 -6 2 C 6 10 0 -8 2 D -6 6 8 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 4 B 4 0 -10 -6 2 C 6 10 0 -8 2 D -6 6 8 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 4 B 4 0 -10 -6 2 C 6 10 0 -8 2 D -6 6 8 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.300000 E: 0.000000 Sum of squares = 0.34 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4512: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (15) E A B C D (14) D E C A B (9) D C E B A (6) D E A B C (5) A B E C D (5) E A D B C (4) B C A E D (4) C B A D E (3) A E B C D (3) E A B D C (2) D C E A B (2) C D E B A (2) C D B A E (2) C B D A E (2) B C A D E (2) B A E C D (2) B A C E D (2) A B E D C (2) E D A C B (1) E D A B C (1) E C D A B (1) E C A B D (1) E A C B D (1) D E A C B (1) D C B E A (1) D A B E C (1) C E B A D (1) C D B E A (1) C B D E A (1) C B A E D (1) B A E D C (1) B A C D E (1) Total count = 100 A B C D E A 0 6 -8 -2 -8 B -6 0 -2 -4 -8 C 8 2 0 -2 -8 D 2 4 2 0 8 E 8 8 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -2 -8 B -6 0 -2 -4 -8 C 8 2 0 -2 -8 D 2 4 2 0 8 E 8 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=25 C=13 B=12 A=10 so A is eliminated. Round 2 votes counts: D=40 E=28 B=19 C=13 so C is eliminated. Round 3 votes counts: D=45 E=29 B=26 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:208 C:200 A:194 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -2 -8 B -6 0 -2 -4 -8 C 8 2 0 -2 -8 D 2 4 2 0 8 E 8 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -2 -8 B -6 0 -2 -4 -8 C 8 2 0 -2 -8 D 2 4 2 0 8 E 8 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -2 -8 B -6 0 -2 -4 -8 C 8 2 0 -2 -8 D 2 4 2 0 8 E 8 8 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4513: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) E C A B D (6) D A C B E (6) B C E D A (5) E B C A D (4) D B A C E (4) D A B E C (4) C E A D B (4) C A E D B (4) B D E A C (4) A D C E B (4) D A B C E (3) B E C D A (3) B E C A D (3) B D E C A (3) B D C E A (3) E B A D C (2) E A C D B (2) E A C B D (2) D C A B E (2) D B C A E (2) D B A E C (2) D A C E B (2) B E D C A (2) A E C D B (2) A D E C B (2) A C E D B (2) A C D E B (2) E A B D C (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) C A D E B (1) B E D A C (1) B E A D C (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 0 -6 0 -12 B 0 0 -4 2 6 C 6 4 0 2 -2 D 0 -2 -2 0 -6 E 12 -6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 0 -6 0 -12 B 0 0 -4 2 6 C 6 4 0 2 -2 D 0 -2 -2 0 -6 E 12 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888893 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 E=23 C=13 A=13 so C is eliminated. Round 2 votes counts: B=30 E=27 D=25 A=18 so A is eliminated. Round 3 votes counts: E=36 D=34 B=30 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 C:205 B:202 D:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 0 -12 B 0 0 -4 2 6 C 6 4 0 2 -2 D 0 -2 -2 0 -6 E 12 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888893 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 0 -12 B 0 0 -4 2 6 C 6 4 0 2 -2 D 0 -2 -2 0 -6 E 12 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888893 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 0 -12 B 0 0 -4 2 6 C 6 4 0 2 -2 D 0 -2 -2 0 -6 E 12 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888893 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4514: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) C A E D B (8) B A D C E (8) C E D A B (6) C D E B A (5) C A E B D (5) B D E A C (5) B D A E C (5) B A D E C (4) A C E B D (4) A B C E D (4) E C A D B (3) D C E B A (3) B D C A E (3) E D C A B (2) D C B E A (2) D B E C A (2) D B E A C (2) D B C E A (2) A C E D B (2) A C B E D (2) A B E C D (2) E D B A C (1) E D A B C (1) E C D A B (1) E A C D B (1) D E B C A (1) C E A D B (1) C D E A B (1) B D A C E (1) B A E D C (1) A E C B D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -8 -4 4 B 8 0 -12 -6 -12 C 8 12 0 -8 14 D 4 6 8 0 6 E -4 12 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -4 4 B 8 0 -12 -6 -12 C 8 12 0 -8 14 D 4 6 8 0 6 E -4 12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 D=21 A=17 E=9 so E is eliminated. Round 2 votes counts: C=30 B=27 D=25 A=18 so A is eliminated. Round 3 votes counts: C=40 B=35 D=25 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:212 E:194 A:192 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 4 B 8 0 -12 -6 -12 C 8 12 0 -8 14 D 4 6 8 0 6 E -4 12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 4 B 8 0 -12 -6 -12 C 8 12 0 -8 14 D 4 6 8 0 6 E -4 12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 4 B 8 0 -12 -6 -12 C 8 12 0 -8 14 D 4 6 8 0 6 E -4 12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4515: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (14) D E C A B (8) C B A E D (6) B A C E D (5) A C B E D (5) A B C E D (5) E D A B C (4) C D E A B (4) D E A B C (3) C D E B A (3) C D B E A (3) C A E D B (3) C A B E D (3) D E B A C (2) D E A C B (2) D C E B A (2) C B D E A (2) C B D A E (2) B E D A C (2) B C D A E (2) B C A E D (2) B A E D C (2) A E D B C (2) A B E D C (2) D E C B A (1) D C E A B (1) D C B E A (1) C A D E B (1) B D E A C (1) B C D E A (1) A E D C B (1) A E C B D (1) A E B D C (1) A C E D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -18 12 20 B 2 0 -30 16 18 C 18 30 0 30 34 D -12 -16 -30 0 6 E -20 -18 -34 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -18 12 20 B 2 0 -30 16 18 C 18 30 0 30 34 D -12 -16 -30 0 6 E -20 -18 -34 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 D=20 A=20 B=15 E=4 so E is eliminated. Round 2 votes counts: C=41 D=24 A=20 B=15 so B is eliminated. Round 3 votes counts: C=46 D=27 A=27 so D is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:256 A:206 B:203 D:174 E:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -18 12 20 B 2 0 -30 16 18 C 18 30 0 30 34 D -12 -16 -30 0 6 E -20 -18 -34 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -18 12 20 B 2 0 -30 16 18 C 18 30 0 30 34 D -12 -16 -30 0 6 E -20 -18 -34 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -18 12 20 B 2 0 -30 16 18 C 18 30 0 30 34 D -12 -16 -30 0 6 E -20 -18 -34 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4516: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) B D A C E (8) E C A D B (5) D B E C A (5) C A E B D (5) A C E B D (5) E C D B A (4) B D C E A (4) B A C D E (4) D E A B C (3) B D C A E (3) A C B E D (3) A B C D E (3) E D C A B (2) E A C D B (2) D E B C A (2) D E B A C (2) D B A E C (2) D A E B C (2) C E B A D (2) C E A B D (2) A D B E C (2) A B D C E (2) A B C E D (2) E D C B A (1) E D B C A (1) E D A C B (1) E C D A B (1) E A D C B (1) D A E C B (1) D A B E C (1) C E B D A (1) C E A D B (1) C B E D A (1) C B E A D (1) C B A E D (1) C A B E D (1) B D A E C (1) B C D E A (1) B C D A E (1) B A D C E (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 10 -12 -2 B 8 0 16 4 10 C -10 -16 0 -8 4 D 12 -4 8 0 12 E 2 -10 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 -12 -2 B 8 0 16 4 10 C -10 -16 0 -8 4 D 12 -4 8 0 12 E 2 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=23 E=18 A=18 C=15 so C is eliminated. Round 2 votes counts: D=26 B=26 E=24 A=24 so E is eliminated. Round 3 votes counts: D=36 A=35 B=29 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 D:214 A:194 E:188 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 -12 -2 B 8 0 16 4 10 C -10 -16 0 -8 4 D 12 -4 8 0 12 E 2 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 -12 -2 B 8 0 16 4 10 C -10 -16 0 -8 4 D 12 -4 8 0 12 E 2 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 -12 -2 B 8 0 16 4 10 C -10 -16 0 -8 4 D 12 -4 8 0 12 E 2 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4517: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) B A E D C (8) A D C B E (7) E B C D A (6) D C A E B (6) D A C E B (6) C D E A B (6) B E A C D (6) B A D E C (6) A D B C E (6) A B D C E (6) E C B D A (4) A D C E B (4) E C D A B (3) B A E C D (3) B E C A D (2) B A D C E (2) E D A C B (1) D C E A B (1) D A E C B (1) C E D B A (1) C E D A B (1) C D B A E (1) B E C D A (1) B E A D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 16 4 14 B 2 0 -4 -8 0 C -16 4 0 -12 -6 D -4 8 12 0 6 E -14 0 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428632 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 4 14 B 2 0 -4 -8 0 C -16 4 0 -12 -6 D -4 8 12 0 6 E -14 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428349 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=24 A=24 D=14 C=9 so C is eliminated. Round 2 votes counts: B=29 E=26 A=24 D=21 so D is eliminated. Round 3 votes counts: A=37 E=33 B=30 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:211 B:195 E:193 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 16 4 14 B 2 0 -4 -8 0 C -16 4 0 -12 -6 D -4 8 12 0 6 E -14 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428349 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 4 14 B 2 0 -4 -8 0 C -16 4 0 -12 -6 D -4 8 12 0 6 E -14 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428349 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 4 14 B 2 0 -4 -8 0 C -16 4 0 -12 -6 D -4 8 12 0 6 E -14 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428349 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4518: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) A B C D E (7) E C B D A (6) D A C E B (6) A D B C E (6) D E C B A (5) E B A C D (4) A D C B E (4) E D B C A (3) E C D B A (3) E B C D A (3) D A E C B (3) D A C B E (3) B C A E D (3) A B C E D (3) E D C B A (2) E B C A D (2) D E A C B (2) D C E A B (2) D C A B E (2) D A E B C (2) C E B D A (2) C B A D E (2) B E C A D (2) B A C E D (2) D C A E B (1) C E D B A (1) C D B E A (1) C B E A D (1) C B A E D (1) C A D B E (1) C A B D E (1) B E A C D (1) A D E B C (1) A D B E C (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 12 6 20 B 4 0 0 2 4 C -12 0 0 12 2 D -6 -2 -12 0 4 E -20 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.837258 C: 0.162742 D: 0.000000 E: 0.000000 Sum of squares = 0.727485972699 Cumulative probabilities = A: 0.000000 B: 0.837258 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 6 20 B 4 0 0 2 4 C -12 0 0 12 2 D -6 -2 -12 0 4 E -20 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.62500014777 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 E=23 B=16 C=10 so C is eliminated. Round 2 votes counts: D=27 A=27 E=26 B=20 so B is eliminated. Round 3 votes counts: A=43 E=30 D=27 so D is eliminated. Round 4 votes counts: A=60 E=40 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:205 C:201 D:192 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 6 20 B 4 0 0 2 4 C -12 0 0 12 2 D -6 -2 -12 0 4 E -20 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.62500014777 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 6 20 B 4 0 0 2 4 C -12 0 0 12 2 D -6 -2 -12 0 4 E -20 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.62500014777 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 6 20 B 4 0 0 2 4 C -12 0 0 12 2 D -6 -2 -12 0 4 E -20 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.62500014777 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4519: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) A C D E B (7) E B D C A (5) E B D A C (5) C A D B E (5) B E D C A (5) B D E C A (5) A C E B D (5) D A C B E (4) A C D B E (4) E C A B D (3) E B C D A (3) D B E A C (3) D B A E C (3) C A E D B (3) B E D A C (3) E C B D A (2) E B C A D (2) E A C B D (2) D C B A E (2) D B E C A (2) D B C A E (2) D B A C E (2) C A D E B (2) E C B A D (1) E B A D C (1) E A B C D (1) D B C E A (1) D A B C E (1) C E B A D (1) C E A B D (1) C D B A E (1) C D A B E (1) B C D E A (1) A E D B C (1) A E C B D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 4 -16 -8 B 14 0 6 8 6 C -4 -6 0 -6 -10 D 16 -8 6 0 8 E 8 -6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -16 -8 B 14 0 6 8 6 C -4 -6 0 -6 -10 D 16 -8 6 0 8 E 8 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=21 D=20 A=20 C=14 so C is eliminated. Round 2 votes counts: A=30 E=27 D=22 B=21 so B is eliminated. Round 3 votes counts: E=35 D=35 A=30 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:217 D:211 E:202 C:187 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 -16 -8 B 14 0 6 8 6 C -4 -6 0 -6 -10 D 16 -8 6 0 8 E 8 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -16 -8 B 14 0 6 8 6 C -4 -6 0 -6 -10 D 16 -8 6 0 8 E 8 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -16 -8 B 14 0 6 8 6 C -4 -6 0 -6 -10 D 16 -8 6 0 8 E 8 -6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4520: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (16) C A B D E (12) B A C E D (11) D E C A B (10) D C A B E (10) E D B A C (9) E B D A C (6) B E A C D (5) D C A E B (4) D E B C A (3) E B A D C (2) D E C B A (2) C A D B E (2) A C B E D (2) E D C A B (1) E D B C A (1) E A B C D (1) B A E C D (1) B A D C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 10 8 -12 B 14 0 12 16 -10 C -10 -12 0 2 -14 D -8 -16 -2 0 -10 E 12 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 10 8 -12 B 14 0 12 16 -10 C -10 -12 0 2 -14 D -8 -16 -2 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=29 B=18 C=14 A=3 so A is eliminated. Round 2 votes counts: E=36 D=29 B=18 C=17 so C is eliminated. Round 3 votes counts: E=36 B=33 D=31 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:216 A:196 C:183 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 10 8 -12 B 14 0 12 16 -10 C -10 -12 0 2 -14 D -8 -16 -2 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 8 -12 B 14 0 12 16 -10 C -10 -12 0 2 -14 D -8 -16 -2 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 8 -12 B 14 0 12 16 -10 C -10 -12 0 2 -14 D -8 -16 -2 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4521: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) E D B C A (6) A C B D E (6) C E B D A (5) C B A E D (5) C A B E D (5) D E B A C (4) D A E B C (4) B C A E D (4) E D C A B (3) D E A C B (3) D E A B C (3) D A E C B (3) B D E A C (3) A D C E B (3) A B C D E (3) E C D B A (2) E C B D A (2) E B D C A (2) D B A E C (2) C E D A B (2) C B E D A (2) C B E A D (2) B E D C A (2) B A D E C (2) A D E C B (2) A B D C E (2) D E C A B (1) C E A D B (1) C E A B D (1) C A E D B (1) C A E B D (1) C A D E B (1) B D A E C (1) B C E D A (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 2 -2 10 B 4 0 -4 14 -2 C -2 4 0 4 6 D 2 -14 -4 0 6 E -10 2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -2 10 B 4 0 -4 14 -2 C -2 4 0 4 6 D 2 -14 -4 0 6 E -10 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000005 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=20 B=20 A=19 E=15 so E is eliminated. Round 2 votes counts: C=30 D=29 B=22 A=19 so A is eliminated. Round 3 votes counts: C=37 D=35 B=28 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:206 C:206 A:203 D:195 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 -2 10 B 4 0 -4 14 -2 C -2 4 0 4 6 D 2 -14 -4 0 6 E -10 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000005 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 10 B 4 0 -4 14 -2 C -2 4 0 4 6 D 2 -14 -4 0 6 E -10 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000005 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 10 B 4 0 -4 14 -2 C -2 4 0 4 6 D 2 -14 -4 0 6 E -10 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000005 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4522: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) D A E C B (7) B C E A D (7) E D A B C (6) B E C A D (6) A D C E B (5) E A D C B (4) E A D B C (4) D A E B C (4) C B D A E (4) A D E C B (4) E B D A C (3) D A C E B (3) B E D C A (3) D A B E C (2) C B E A D (2) B E C D A (2) B D E A C (2) B C D A E (2) A E D C B (2) E D A C B (1) E C A B D (1) E B A D C (1) E A C D B (1) D B E A C (1) D A C B E (1) C E B A D (1) C D A B E (1) C B A E D (1) C A D E B (1) C A D B E (1) B E D A C (1) B D E C A (1) B D C E A (1) B C E D A (1) B C D E A (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 6 6 0 B 2 0 -4 2 2 C -6 4 0 -12 -14 D -6 -2 12 0 4 E 0 -2 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.071429 B: 0.642857 C: 0.119048 D: 0.095238 E: 0.071429 Sum of squares = 0.446712018141 Cumulative probabilities = A: 0.071429 B: 0.714286 C: 0.833333 D: 0.928571 E: 1.000000 A B C D E A 0 -2 6 6 0 B 2 0 -4 2 2 C -6 4 0 -12 -14 D -6 -2 12 0 4 E 0 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.642857 C: 0.119048 D: 0.095238 E: 0.071429 Sum of squares = 0.446712018164 Cumulative probabilities = A: 0.071429 B: 0.714286 C: 0.833333 D: 0.928571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=22 E=21 D=18 A=12 so A is eliminated. Round 2 votes counts: D=27 B=27 E=24 C=22 so C is eliminated. Round 3 votes counts: B=45 D=30 E=25 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:205 D:204 E:204 B:201 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 6 0 B 2 0 -4 2 2 C -6 4 0 -12 -14 D -6 -2 12 0 4 E 0 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.642857 C: 0.119048 D: 0.095238 E: 0.071429 Sum of squares = 0.446712018164 Cumulative probabilities = A: 0.071429 B: 0.714286 C: 0.833333 D: 0.928571 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 0 B 2 0 -4 2 2 C -6 4 0 -12 -14 D -6 -2 12 0 4 E 0 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.642857 C: 0.119048 D: 0.095238 E: 0.071429 Sum of squares = 0.446712018164 Cumulative probabilities = A: 0.071429 B: 0.714286 C: 0.833333 D: 0.928571 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 0 B 2 0 -4 2 2 C -6 4 0 -12 -14 D -6 -2 12 0 4 E 0 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.642857 C: 0.119048 D: 0.095238 E: 0.071429 Sum of squares = 0.446712018164 Cumulative probabilities = A: 0.071429 B: 0.714286 C: 0.833333 D: 0.928571 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4523: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (17) B D C A E (15) B D C E A (10) A E C D B (5) D B C A E (4) C E A D B (4) E A C B D (3) D C B A E (3) D A C E B (3) B E C A D (3) B E A D C (3) B D E C A (3) B D A E C (3) A E D C B (3) E B A C D (2) E A B C D (2) D B A C E (2) B E A C D (2) B D A C E (2) B C D E A (2) E C A D B (1) E B C A D (1) D C A B E (1) D A C B E (1) C E D A B (1) C B D E A (1) B C E D A (1) B A E D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 0 -4 -12 B 16 0 12 8 14 C 0 -12 0 -10 0 D 4 -8 10 0 2 E 12 -14 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -4 -12 B 16 0 12 8 14 C 0 -12 0 -10 0 D 4 -8 10 0 2 E 12 -14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=45 E=26 D=14 A=9 C=6 so C is eliminated. Round 2 votes counts: B=46 E=31 D=14 A=9 so A is eliminated. Round 3 votes counts: B=46 E=39 D=15 so D is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:204 E:198 C:189 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -4 -12 B 16 0 12 8 14 C 0 -12 0 -10 0 D 4 -8 10 0 2 E 12 -14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -4 -12 B 16 0 12 8 14 C 0 -12 0 -10 0 D 4 -8 10 0 2 E 12 -14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -4 -12 B 16 0 12 8 14 C 0 -12 0 -10 0 D 4 -8 10 0 2 E 12 -14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4524: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (15) A D B C E (12) D B A C E (8) E C B A D (5) A B D E C (5) E C B D A (4) A E B D C (4) C D B E A (3) C A D E B (3) B A D E C (3) A D C B E (3) E C A B D (2) E B C D A (2) E B C A D (2) E A B D C (2) D A C B E (2) D A B C E (2) C E D A B (2) C E B D A (2) C E A D B (2) C D E B A (2) C D E A B (2) C D A E B (2) B D A E C (2) A D B E C (2) E C D A B (1) E B A C D (1) D C B A E (1) D B C A E (1) B E A D C (1) B D A C E (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -2 -4 4 B 6 0 -2 -26 -8 C 2 2 0 2 26 D 4 26 -2 0 8 E -4 8 -26 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -4 4 B 6 0 -2 -26 -8 C 2 2 0 2 26 D 4 26 -2 0 8 E -4 8 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=27 E=19 D=14 B=7 so B is eliminated. Round 2 votes counts: C=33 A=30 E=20 D=17 so D is eliminated. Round 3 votes counts: A=45 C=35 E=20 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:216 A:196 B:185 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 4 B 6 0 -2 -26 -8 C 2 2 0 2 26 D 4 26 -2 0 8 E -4 8 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 4 B 6 0 -2 -26 -8 C 2 2 0 2 26 D 4 26 -2 0 8 E -4 8 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 4 B 6 0 -2 -26 -8 C 2 2 0 2 26 D 4 26 -2 0 8 E -4 8 -26 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4525: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) E A C D B (5) E A C B D (5) D E C A B (5) D B A E C (5) C E A B D (5) A E C B D (5) E C A B D (4) D C E A B (4) D C B E A (4) D B C A E (4) B D A C E (4) D B A C E (3) C E A D B (3) B A C E D (3) E C D A B (2) E C A D B (2) D E B C A (2) D E A B C (2) C E D A B (2) B D A E C (2) B C A E D (2) B C A D E (2) B A E C D (2) B A D E C (2) E D C A B (1) E A D C B (1) D E A C B (1) D B E C A (1) D B E A C (1) C E B D A (1) C B D A E (1) C B A E D (1) B C D A E (1) B A E D C (1) B A D C E (1) A E B D C (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 -4 -14 B -2 0 -6 -8 -2 C 6 6 0 -2 -2 D 4 8 2 0 2 E 14 2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -4 -14 B -2 0 -6 -8 -2 C 6 6 0 -2 -2 D 4 8 2 0 2 E 14 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=20 B=20 C=13 A=9 so A is eliminated. Round 2 votes counts: D=38 E=26 B=22 C=14 so C is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:208 C:204 B:191 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -14 B -2 0 -6 -8 -2 C 6 6 0 -2 -2 D 4 8 2 0 2 E 14 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -14 B -2 0 -6 -8 -2 C 6 6 0 -2 -2 D 4 8 2 0 2 E 14 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -14 B -2 0 -6 -8 -2 C 6 6 0 -2 -2 D 4 8 2 0 2 E 14 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4526: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) A E C D B (7) B D C A E (6) B A E D C (6) D C B E A (5) D B C E A (5) C E D A B (5) B D C E A (5) A B E D C (5) B A D C E (4) D E C A B (3) D C E A B (3) C D E B A (3) C D E A B (3) B C D A E (3) B A E C D (3) A E B D C (3) A E B C D (3) E C D A B (2) E C A D B (2) E A D C B (2) C D B E A (2) B C D E A (2) D E A C B (1) C E A D B (1) B D A E C (1) B D A C E (1) B C A E D (1) B C A D E (1) B A D E C (1) B A C E D (1) A E D C B (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -4 0 -2 B 0 0 4 -4 6 C 4 -4 0 -4 2 D 0 4 4 0 -2 E 2 -6 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 A B C D E A 0 0 -4 0 -2 B 0 0 4 -4 6 C 4 -4 0 -4 2 D 0 4 4 0 -2 E 2 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=21 D=17 C=14 E=13 so E is eliminated. Round 2 votes counts: B=35 A=30 C=18 D=17 so D is eliminated. Round 3 votes counts: B=40 A=31 C=29 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:203 D:203 C:199 E:198 A:197 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 0 -2 B 0 0 4 -4 6 C 4 -4 0 -4 2 D 0 4 4 0 -2 E 2 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 0 -2 B 0 0 4 -4 6 C 4 -4 0 -4 2 D 0 4 4 0 -2 E 2 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 0 -2 B 0 0 4 -4 6 C 4 -4 0 -4 2 D 0 4 4 0 -2 E 2 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4527: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) B C E A D (6) E D A B C (5) C B A D E (5) D E A C B (4) D A C B E (4) C B E A D (4) B A E C D (4) E C D B A (3) E B A D C (3) D C E A B (3) D A E B C (3) C B A E D (3) B C A E D (3) A D B E C (3) A D B C E (3) A B D C E (3) E C B D A (2) D A C E B (2) C E D B A (2) C E B D A (2) C D E A B (2) C D A B E (2) C B E D A (2) B E C A D (2) B A C D E (2) E D C B A (1) E D C A B (1) E D A C B (1) E A D B C (1) E A B D C (1) D E A B C (1) D C A B E (1) D A E C B (1) C D E B A (1) B E A C D (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -4 14 -10 B 8 0 8 8 2 C 4 -8 0 8 2 D -14 -8 -8 0 -16 E 10 -2 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 14 -10 B 8 0 8 8 2 C 4 -8 0 8 2 D -14 -8 -8 0 -16 E 10 -2 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 D=19 B=19 A=14 so A is eliminated. Round 2 votes counts: E=27 D=26 B=24 C=23 so C is eliminated. Round 3 votes counts: B=38 E=31 D=31 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:211 C:203 A:196 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 14 -10 B 8 0 8 8 2 C 4 -8 0 8 2 D -14 -8 -8 0 -16 E 10 -2 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 14 -10 B 8 0 8 8 2 C 4 -8 0 8 2 D -14 -8 -8 0 -16 E 10 -2 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 14 -10 B 8 0 8 8 2 C 4 -8 0 8 2 D -14 -8 -8 0 -16 E 10 -2 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4528: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) E A B C D (7) D E B C A (7) E D B C A (6) D B C E A (6) D C B E A (5) E B D C A (4) A E C B D (4) A E B C D (4) A C D B E (4) A C B E D (4) E D A B C (3) D C B A E (3) C B D A E (3) B C E A D (3) E A D B C (2) D B E C A (2) D A C B E (2) C B D E A (2) C B A D E (2) B C D E A (2) E D B A C (1) E B D A C (1) E B C A D (1) E B A D C (1) D E A B C (1) D C A B E (1) C A B E D (1) B E C D A (1) B D C E A (1) A E C D B (1) A D C E B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 -2 -12 B 2 0 8 10 10 C 0 -8 0 6 8 D 2 -10 -6 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -2 -12 B 2 0 8 10 10 C 0 -8 0 6 8 D 2 -10 -6 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=27 E=26 C=8 B=7 so B is eliminated. Round 2 votes counts: A=32 D=28 E=27 C=13 so C is eliminated. Round 3 votes counts: D=35 A=35 E=30 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:215 C:203 D:197 E:193 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -2 -12 B 2 0 8 10 10 C 0 -8 0 6 8 D 2 -10 -6 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -2 -12 B 2 0 8 10 10 C 0 -8 0 6 8 D 2 -10 -6 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -2 -12 B 2 0 8 10 10 C 0 -8 0 6 8 D 2 -10 -6 0 8 E 12 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4529: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D C A (6) E B D A C (6) E B C A D (6) B E C D A (6) A D C E B (5) E B A D C (4) D C B A E (4) A C D E B (4) E B A C D (3) E A B C D (3) C D A B E (3) A E C B D (3) A D C B E (3) E B C D A (2) D B E C A (2) D B C E A (2) D A C E B (2) D A C B E (2) C D B A E (2) C B D A E (2) B D C E A (2) B C D E A (2) A C E D B (2) E D B A C (1) E A D B C (1) E A B D C (1) D E B C A (1) D C A B E (1) D B C A E (1) C A D B E (1) C A B D E (1) B E C A D (1) A E D C B (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 10 6 2 B 6 0 0 -2 -4 C -10 0 0 10 4 D -6 2 -10 0 6 E -2 4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.100000 D: 0.100000 E: 0.250000 Sum of squares = 0.244999999999 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.650000 D: 0.750000 E: 1.000000 A B C D E A 0 -6 10 6 2 B 6 0 0 -2 -4 C -10 0 0 10 4 D -6 2 -10 0 6 E -2 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.100000 D: 0.100000 E: 0.250000 Sum of squares = 0.245 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.650000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=32 D=15 B=11 C=9 so C is eliminated. Round 2 votes counts: A=34 E=33 D=20 B=13 so B is eliminated. Round 3 votes counts: E=40 A=34 D=26 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 C:202 B:200 D:196 E:196 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 6 2 B 6 0 0 -2 -4 C -10 0 0 10 4 D -6 2 -10 0 6 E -2 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.100000 D: 0.100000 E: 0.250000 Sum of squares = 0.245 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.650000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 6 2 B 6 0 0 -2 -4 C -10 0 0 10 4 D -6 2 -10 0 6 E -2 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.100000 D: 0.100000 E: 0.250000 Sum of squares = 0.245 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.650000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 6 2 B 6 0 0 -2 -4 C -10 0 0 10 4 D -6 2 -10 0 6 E -2 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.100000 D: 0.100000 E: 0.250000 Sum of squares = 0.245 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.650000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4530: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) A D B E C (8) E D A C B (6) D A E B C (5) C B E D A (5) B C D E A (5) C B E A D (4) A D E C B (4) E A D C B (3) C E D A B (3) B A D C E (3) A B D C E (3) E D C B A (2) E D B C A (2) E D A B C (2) C E B D A (2) C E B A D (2) C E A D B (2) C E A B D (2) C B A E D (2) B D E C A (2) A D B C E (2) A C E D B (2) A C D E B (2) E D C A B (1) E C B D A (1) D E B A C (1) D E A B C (1) D B E A C (1) D B A E C (1) D A B E C (1) C B A D E (1) C A B E D (1) C A B D E (1) B D C E A (1) B C E D A (1) B C D A E (1) B C A D E (1) B A D E C (1) B A C D E (1) A E D C B (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 20 16 12 2 B -20 0 4 -16 -4 C -16 -4 0 -20 -4 D -12 16 20 0 12 E -2 4 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 16 12 2 B -20 0 4 -16 -4 C -16 -4 0 -20 -4 D -12 16 20 0 12 E -2 4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996362 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=25 E=17 B=16 D=10 so D is eliminated. Round 2 votes counts: A=38 C=25 E=19 B=18 so B is eliminated. Round 3 votes counts: A=44 C=34 E=22 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:218 E:197 B:182 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 16 12 2 B -20 0 4 -16 -4 C -16 -4 0 -20 -4 D -12 16 20 0 12 E -2 4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996362 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 12 2 B -20 0 4 -16 -4 C -16 -4 0 -20 -4 D -12 16 20 0 12 E -2 4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996362 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 12 2 B -20 0 4 -16 -4 C -16 -4 0 -20 -4 D -12 16 20 0 12 E -2 4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996362 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4531: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) D B A C E (8) B D A C E (7) D A B C E (6) E C D A B (5) C B E A D (4) A C E B D (4) E C B A D (3) D E A C B (3) D B E C A (3) D A E C B (3) B C A E D (3) E D A C B (2) E C A D B (2) E A C D B (2) D E B C A (2) D B A E C (2) C E B A D (2) C E A B D (2) B D C E A (2) B A D C E (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D B A (1) E C B D A (1) E A C B D (1) D E B A C (1) D E A B C (1) D B E A C (1) D A E B C (1) D A B E C (1) C A E B D (1) C A B E D (1) B E D C A (1) B D C A E (1) B C E A D (1) B A C E D (1) B A C D E (1) A E C D B (1) A D E C B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 2 -10 -8 B -2 0 -6 2 -6 C -2 6 0 -2 -4 D 10 -2 2 0 -6 E 8 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 -10 -8 B -2 0 -6 2 -6 C -2 6 0 -2 -4 D 10 -2 2 0 -6 E 8 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=31 B=19 C=10 A=8 so A is eliminated. Round 2 votes counts: D=33 E=32 B=20 C=15 so C is eliminated. Round 3 votes counts: E=41 D=33 B=26 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:202 C:199 B:194 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 -10 -8 B -2 0 -6 2 -6 C -2 6 0 -2 -4 D 10 -2 2 0 -6 E 8 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -10 -8 B -2 0 -6 2 -6 C -2 6 0 -2 -4 D 10 -2 2 0 -6 E 8 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -10 -8 B -2 0 -6 2 -6 C -2 6 0 -2 -4 D 10 -2 2 0 -6 E 8 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4532: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (15) D B C A E (12) B D C E A (12) E A C B D (7) D B C E A (5) E C A B D (4) A E C B D (4) B C D E A (3) A E D C B (3) E A C D B (2) D A B C E (2) C D B E A (2) B E D C A (2) B E C D A (2) B D C A E (2) A D C B E (2) E C B D A (1) E C A D B (1) E B C A D (1) E A B C D (1) D C B A E (1) D B A C E (1) D A C B E (1) C B E D A (1) B E D A C (1) B E C A D (1) B E A D C (1) B D A E C (1) A E B D C (1) A E B C D (1) A D E C B (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 2 6 B -2 0 4 -4 10 C 0 -4 0 -4 -4 D -2 4 4 0 -2 E -6 -10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.802931 B: 0.000000 C: 0.197069 D: 0.000000 E: 0.000000 Sum of squares = 0.683533783371 Cumulative probabilities = A: 0.802931 B: 0.802931 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 6 B -2 0 4 -4 10 C 0 -4 0 -4 -4 D -2 4 4 0 -2 E -6 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555623961 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=25 D=22 E=17 C=3 so C is eliminated. Round 2 votes counts: A=33 B=26 D=24 E=17 so E is eliminated. Round 3 votes counts: A=48 B=28 D=24 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:205 B:204 D:202 E:195 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 2 6 B -2 0 4 -4 10 C 0 -4 0 -4 -4 D -2 4 4 0 -2 E -6 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555623961 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 6 B -2 0 4 -4 10 C 0 -4 0 -4 -4 D -2 4 4 0 -2 E -6 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555623961 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 6 B -2 0 4 -4 10 C 0 -4 0 -4 -4 D -2 4 4 0 -2 E -6 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555623961 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4533: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) E D A B C (6) D A B C E (6) E A B D C (5) C D B A E (5) D A B E C (4) C E B A D (4) B A D C E (4) D E A B C (3) D A C B E (3) C D E B A (3) C B D A E (3) A B E D C (3) E C A D B (2) E B C A D (2) E B A C D (2) E A D B C (2) D E C A B (2) D E A C B (2) D C B A E (2) D A E B C (2) C B E A D (2) B C A E D (2) B C A D E (2) A D B E C (2) E D C A B (1) E C D A B (1) E C A B D (1) E A B C D (1) D C E A B (1) D C A E B (1) C E D B A (1) C D B E A (1) C B D E A (1) C B A D E (1) B C D A E (1) B A E D C (1) B A E C D (1) B A D E C (1) B A C D E (1) A E B D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 4 2 -4 B -2 0 8 0 -4 C -4 -8 0 -8 -10 D -2 0 8 0 6 E 4 4 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888794 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 2 4 2 -4 B -2 0 8 0 -4 C -4 -8 0 -8 -10 D -2 0 8 0 6 E 4 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=26 C=21 B=13 A=8 so A is eliminated. Round 2 votes counts: E=33 D=28 C=21 B=18 so B is eliminated. Round 3 votes counts: E=38 D=35 C=27 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 E:206 A:202 B:201 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 2 -4 B -2 0 8 0 -4 C -4 -8 0 -8 -10 D -2 0 8 0 6 E 4 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 -4 B -2 0 8 0 -4 C -4 -8 0 -8 -10 D -2 0 8 0 6 E 4 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 -4 B -2 0 8 0 -4 C -4 -8 0 -8 -10 D -2 0 8 0 6 E 4 4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4534: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) A B D C E (10) E B A C D (8) C D E A B (7) B A D C E (6) D A C B E (5) E C B A D (4) B A E C D (4) B A D E C (4) E B A D C (3) C E D A B (3) B E A D C (3) B E A C D (3) A D B C E (3) E C D A B (2) E B D A C (2) E B C A D (2) D C A B E (2) D A B C E (2) C D A E B (2) C D A B E (2) C A D B E (2) B A C D E (2) A C B D E (2) E D C B A (1) D C E A B (1) C B A E D (1) B D E A C (1) B A E D C (1) B A C E D (1) Total count = 100 A B C D E A 0 -14 20 18 -2 B 14 0 10 14 8 C -20 -10 0 12 2 D -18 -14 -12 0 2 E 2 -8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999454 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 20 18 -2 B 14 0 10 14 8 C -20 -10 0 12 2 D -18 -14 -12 0 2 E 2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=25 C=17 A=15 D=10 so D is eliminated. Round 2 votes counts: E=33 B=25 A=22 C=20 so C is eliminated. Round 3 votes counts: E=44 A=30 B=26 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:223 A:211 E:195 C:192 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 20 18 -2 B 14 0 10 14 8 C -20 -10 0 12 2 D -18 -14 -12 0 2 E 2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 20 18 -2 B 14 0 10 14 8 C -20 -10 0 12 2 D -18 -14 -12 0 2 E 2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 20 18 -2 B 14 0 10 14 8 C -20 -10 0 12 2 D -18 -14 -12 0 2 E 2 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4535: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) E A D C B (7) A B E D C (7) D E A C B (5) B A D E C (5) A B E C D (5) E A C D B (4) D E C A B (4) D A E B C (4) C E D B A (4) B C A E D (4) B A C E D (4) C D E B A (3) C D B E A (3) B C D A E (3) B A C D E (3) D C E B A (2) D C E A B (2) C B A E D (2) B A E C D (2) A D E B C (2) A B D E C (2) E D C A B (1) E D A B C (1) E C A D B (1) D E B C A (1) D B C A E (1) D B A E C (1) C E D A B (1) C E B D A (1) C B E D A (1) C B E A D (1) B D A C E (1) B C A D E (1) B A D C E (1) A E D C B (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 12 28 8 2 B -12 0 -2 -12 -6 C -28 2 0 -12 -24 D -8 12 12 0 -12 E -2 6 24 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999621 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 28 8 2 B -12 0 -2 -12 -6 C -28 2 0 -12 -24 D -8 12 12 0 -12 E -2 6 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979008 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=21 D=20 A=19 C=16 so C is eliminated. Round 2 votes counts: B=28 E=27 D=26 A=19 so A is eliminated. Round 3 votes counts: B=42 E=30 D=28 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:225 E:220 D:202 B:184 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 28 8 2 B -12 0 -2 -12 -6 C -28 2 0 -12 -24 D -8 12 12 0 -12 E -2 6 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979008 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 28 8 2 B -12 0 -2 -12 -6 C -28 2 0 -12 -24 D -8 12 12 0 -12 E -2 6 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979008 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 28 8 2 B -12 0 -2 -12 -6 C -28 2 0 -12 -24 D -8 12 12 0 -12 E -2 6 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979008 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4536: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) D C A E B (6) E B A C D (5) D C E B A (5) A B E C D (5) C D B E A (4) C B D A E (4) B C A D E (4) E D C B A (3) E D A C B (3) E B A D C (3) E A D B C (3) D E C A B (3) D C B E A (3) D C A B E (3) C B D E A (3) B A E C D (3) A E B D C (3) D E C B A (2) C D A B E (2) B C E D A (2) B A C E D (2) A D E C B (2) A B C E D (2) A B C D E (2) E D C A B (1) E D A B C (1) E A D C B (1) E A B C D (1) D C E A B (1) D C B A E (1) C D B A E (1) B E C D A (1) B E A C D (1) B C E A D (1) B C A E D (1) B A C D E (1) A E D C B (1) Total count = 100 A B C D E A 0 0 -2 2 -14 B 0 0 2 8 -8 C 2 -2 0 -10 -4 D -2 -8 10 0 -6 E 14 8 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 2 -14 B 0 0 2 8 -8 C 2 -2 0 -10 -4 D -2 -8 10 0 -6 E 14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=24 B=16 A=15 C=14 so C is eliminated. Round 2 votes counts: E=31 D=31 B=23 A=15 so A is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:201 D:197 A:193 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 2 -14 B 0 0 2 8 -8 C 2 -2 0 -10 -4 D -2 -8 10 0 -6 E 14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 -14 B 0 0 2 8 -8 C 2 -2 0 -10 -4 D -2 -8 10 0 -6 E 14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 -14 B 0 0 2 8 -8 C 2 -2 0 -10 -4 D -2 -8 10 0 -6 E 14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4537: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) C E D B A (6) A B E D C (6) E D C A B (5) E C D B A (5) B A C E D (5) A B D E C (5) C D E B A (4) B A D C E (4) E B C A D (3) D A C E B (3) C D B E A (3) B E A C D (3) B A E C D (3) A E B D C (3) E A B D C (2) D E C A B (2) D A C B E (2) C D B A E (2) C B E D A (2) B E C A D (2) A E D B C (2) A D E B C (2) A D B E C (2) E D A C B (1) E B A C D (1) D E A C B (1) D C E A B (1) D A E C B (1) C D E A B (1) C B D E A (1) C B D A E (1) B C E D A (1) B C E A D (1) B C D A E (1) B C A D E (1) B A E D C (1) B A D E C (1) B A C D E (1) A D E C B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 4 2 -2 B 4 0 4 -2 0 C -4 -4 0 6 -16 D -2 2 -6 0 -16 E 2 0 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.545047 C: 0.000000 D: 0.000000 E: 0.454953 Sum of squares = 0.504058397972 Cumulative probabilities = A: 0.000000 B: 0.545047 C: 0.545047 D: 0.545047 E: 1.000000 A B C D E A 0 -4 4 2 -2 B 4 0 4 -2 0 C -4 -4 0 6 -16 D -2 2 -6 0 -16 E 2 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 A=23 C=20 D=10 so D is eliminated. Round 2 votes counts: A=29 E=26 B=24 C=21 so C is eliminated. Round 3 votes counts: E=38 B=33 A=29 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:217 B:203 A:200 C:191 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 2 -2 B 4 0 4 -2 0 C -4 -4 0 6 -16 D -2 2 -6 0 -16 E 2 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 2 -2 B 4 0 4 -2 0 C -4 -4 0 6 -16 D -2 2 -6 0 -16 E 2 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 2 -2 B 4 0 4 -2 0 C -4 -4 0 6 -16 D -2 2 -6 0 -16 E 2 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4538: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) B A E D C (10) D C E A B (9) B A E C D (9) E A B D C (4) D E A B C (4) D A E B C (4) A B E D C (4) E D A B C (3) E A D B C (3) B A D E C (3) E C D A B (2) E A B C D (2) D E C A B (2) D E A C B (2) C E D A B (2) C E B A D (2) C D B E A (2) C B D A E (2) C B A E D (2) B C A E D (2) B A C E D (2) B A C D E (2) E A C D B (1) D C B A E (1) D C A B E (1) D B A E C (1) D B A C E (1) D A B E C (1) D A B C E (1) C E A B D (1) C D B A E (1) C B E D A (1) C B A D E (1) B C A D E (1) A E B D C (1) Total count = 100 A B C D E A 0 14 16 0 0 B -14 0 16 -2 -4 C -16 -16 0 -10 -12 D 0 2 10 0 -2 E 0 4 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.261527 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.738473 Sum of squares = 0.61373914567 Cumulative probabilities = A: 0.261527 B: 0.261527 C: 0.261527 D: 0.261527 E: 1.000000 A B C D E A 0 14 16 0 0 B -14 0 16 -2 -4 C -16 -16 0 -10 -12 D 0 2 10 0 -2 E 0 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 C=24 E=15 A=5 so A is eliminated. Round 2 votes counts: B=33 D=27 C=24 E=16 so E is eliminated. Round 3 votes counts: B=40 D=33 C=27 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:215 E:209 D:205 B:198 C:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 16 0 0 B -14 0 16 -2 -4 C -16 -16 0 -10 -12 D 0 2 10 0 -2 E 0 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 0 0 B -14 0 16 -2 -4 C -16 -16 0 -10 -12 D 0 2 10 0 -2 E 0 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 0 0 B -14 0 16 -2 -4 C -16 -16 0 -10 -12 D 0 2 10 0 -2 E 0 4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4539: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (15) C D A B E (8) C D E B A (5) C D E A B (5) A B E D C (5) D C E A B (4) D A C B E (4) B E A C D (4) D C A E B (3) C E D B A (3) C D B A E (3) B E A D C (3) B A E D C (3) A D B E C (3) E C D B A (2) E B C A D (2) E A D B C (2) D C A B E (2) C E B D A (2) B A E C D (2) B A C D E (2) A B D E C (2) E D C A B (1) E C B D A (1) E C B A D (1) D E C A B (1) D A E B C (1) C D A E B (1) C B E A D (1) C B D E A (1) C B A D E (1) C A B D E (1) B E C A D (1) B C E A D (1) B A C E D (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 0 6 -10 B 8 0 0 2 2 C 0 0 0 -2 2 D -6 -2 2 0 0 E 10 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.601072 C: 0.398928 D: 0.000000 E: 0.000000 Sum of squares = 0.520430990876 Cumulative probabilities = A: 0.000000 B: 0.601072 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 6 -10 B 8 0 0 2 2 C 0 0 0 -2 2 D -6 -2 2 0 0 E 10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500456 C: 0.499544 D: 0.000000 E: 0.000000 Sum of squares = 0.500000415565 Cumulative probabilities = A: 0.000000 B: 0.500456 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=24 B=17 D=15 A=13 so A is eliminated. Round 2 votes counts: C=31 B=26 E=24 D=19 so D is eliminated. Round 3 votes counts: C=44 B=30 E=26 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:206 E:203 C:200 D:197 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 6 -10 B 8 0 0 2 2 C 0 0 0 -2 2 D -6 -2 2 0 0 E 10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500456 C: 0.499544 D: 0.000000 E: 0.000000 Sum of squares = 0.500000415565 Cumulative probabilities = A: 0.000000 B: 0.500456 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 6 -10 B 8 0 0 2 2 C 0 0 0 -2 2 D -6 -2 2 0 0 E 10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500456 C: 0.499544 D: 0.000000 E: 0.000000 Sum of squares = 0.500000415565 Cumulative probabilities = A: 0.000000 B: 0.500456 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 6 -10 B 8 0 0 2 2 C 0 0 0 -2 2 D -6 -2 2 0 0 E 10 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500456 C: 0.499544 D: 0.000000 E: 0.000000 Sum of squares = 0.500000415565 Cumulative probabilities = A: 0.000000 B: 0.500456 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4540: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (14) E D A B C (7) D E B A C (7) C A B E D (7) D E C A B (6) D C E A B (6) B A E D C (6) B A C E D (6) C D A B E (5) E A B D C (4) D C E B A (4) C D E A B (4) A B E C D (4) E B A D C (3) D E A B C (3) D E A C B (2) C A B D E (2) A B E D C (2) D C B E A (1) C D B E A (1) C B A D E (1) B E A D C (1) B D E A C (1) B D A E C (1) B C A E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 24 4 0 B -6 0 22 8 8 C -24 -22 0 -8 -22 D -4 -8 8 0 -12 E 0 -8 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.798968 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.201032 Sum of squares = 0.678763657142 Cumulative probabilities = A: 0.798968 B: 0.798968 C: 0.798968 D: 0.798968 E: 1.000000 A B C D E A 0 6 24 4 0 B -6 0 22 8 8 C -24 -22 0 -8 -22 D -4 -8 8 0 -12 E 0 -8 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571433 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.428567 Sum of squares = 0.510205454671 Cumulative probabilities = A: 0.571433 B: 0.571433 C: 0.571433 D: 0.571433 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=29 C=20 E=14 A=7 so A is eliminated. Round 2 votes counts: B=37 D=29 C=20 E=14 so E is eliminated. Round 3 votes counts: B=44 D=36 C=20 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:216 E:213 D:192 C:162 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 24 4 0 B -6 0 22 8 8 C -24 -22 0 -8 -22 D -4 -8 8 0 -12 E 0 -8 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571433 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.428567 Sum of squares = 0.510205454671 Cumulative probabilities = A: 0.571433 B: 0.571433 C: 0.571433 D: 0.571433 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 24 4 0 B -6 0 22 8 8 C -24 -22 0 -8 -22 D -4 -8 8 0 -12 E 0 -8 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571433 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.428567 Sum of squares = 0.510205454671 Cumulative probabilities = A: 0.571433 B: 0.571433 C: 0.571433 D: 0.571433 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 24 4 0 B -6 0 22 8 8 C -24 -22 0 -8 -22 D -4 -8 8 0 -12 E 0 -8 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571433 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.428567 Sum of squares = 0.510205454671 Cumulative probabilities = A: 0.571433 B: 0.571433 C: 0.571433 D: 0.571433 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4541: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) B A D C E (8) B D A C E (6) A B C D E (6) E D C B A (4) E C D A B (4) E A C B D (4) D B E C A (4) B D A E C (4) B A C D E (4) A C B E D (4) A B C E D (4) D E B C A (3) C D E A B (3) B D E A C (3) A C B D E (3) E D B C A (2) E B D A C (2) D E C B A (2) C E D A B (2) C E A D B (2) C A E D B (2) A C E B D (2) E D C A B (1) E B A C D (1) D C B A E (1) D B C E A (1) C D A E B (1) C D A B E (1) C A D E B (1) B E A D C (1) B D E C A (1) B A E D C (1) B A E C D (1) B A C E D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 12 10 2 B 0 0 6 12 8 C -12 -6 0 12 6 D -10 -12 -12 0 4 E -2 -8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.510989 B: 0.489011 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500241512781 Cumulative probabilities = A: 0.510989 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 10 2 B 0 0 6 12 8 C -12 -6 0 12 6 D -10 -12 -12 0 4 E -2 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 A=20 C=12 D=11 so D is eliminated. Round 2 votes counts: B=35 E=32 A=20 C=13 so C is eliminated. Round 3 votes counts: E=39 B=36 A=25 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:212 C:200 E:190 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 10 2 B 0 0 6 12 8 C -12 -6 0 12 6 D -10 -12 -12 0 4 E -2 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 10 2 B 0 0 6 12 8 C -12 -6 0 12 6 D -10 -12 -12 0 4 E -2 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 10 2 B 0 0 6 12 8 C -12 -6 0 12 6 D -10 -12 -12 0 4 E -2 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4542: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (6) A B E D C (6) C D E B A (5) C D E A B (5) C D A E B (5) C B E D A (5) C B A E D (5) C A D E B (4) C A B E D (4) C A B D E (4) B A C E D (4) A D E B C (4) D E C A B (3) D E A B C (3) B C A E D (3) A E B D C (3) A D E C B (3) E D B A C (2) E B D A C (2) D E B A C (2) D C E A B (2) C D B E A (2) A B C E D (2) E D B C A (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) D A E C B (1) D A E B C (1) C B E A D (1) C A D B E (1) B E D A C (1) B E A D C (1) B C E D A (1) B A E C D (1) A E D B C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -8 10 20 B -10 0 -8 2 -2 C 8 8 0 8 10 D -10 -2 -8 0 -2 E -20 2 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 10 20 B -10 0 -8 2 -2 C 8 8 0 8 10 D -10 -2 -8 0 -2 E -20 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=21 B=17 D=16 E=5 so E is eliminated. Round 2 votes counts: C=41 A=21 D=19 B=19 so D is eliminated. Round 3 votes counts: C=48 A=27 B=25 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:216 B:191 D:189 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 10 20 B -10 0 -8 2 -2 C 8 8 0 8 10 D -10 -2 -8 0 -2 E -20 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 10 20 B -10 0 -8 2 -2 C 8 8 0 8 10 D -10 -2 -8 0 -2 E -20 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 10 20 B -10 0 -8 2 -2 C 8 8 0 8 10 D -10 -2 -8 0 -2 E -20 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4543: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) D B A C E (9) B A E C D (9) A B E C D (9) D C E A B (8) D B E C A (5) A E C B D (5) E B C A D (3) C E A D B (3) B D E C A (3) B D A E C (3) B A E D C (3) A C E D B (3) E C B A D (2) C E D A B (2) C D E A B (2) B E D C A (2) B E C A D (2) B A D E C (2) A C E B D (2) A B C E D (2) E C B D A (1) D C B E A (1) D C A E B (1) D C A B E (1) D B A E C (1) D A C E B (1) D A C B E (1) B E A C D (1) A C D E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 8 -2 10 B 14 0 12 2 12 C -8 -12 0 -6 -4 D 2 -2 6 0 2 E -10 -12 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 -2 10 B 14 0 12 2 12 C -8 -12 0 -6 -4 D 2 -2 6 0 2 E -10 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=25 A=24 C=7 E=6 so E is eliminated. Round 2 votes counts: D=38 B=28 A=24 C=10 so C is eliminated. Round 3 votes counts: D=42 B=31 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:204 A:201 E:190 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 -2 10 B 14 0 12 2 12 C -8 -12 0 -6 -4 D 2 -2 6 0 2 E -10 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -2 10 B 14 0 12 2 12 C -8 -12 0 -6 -4 D 2 -2 6 0 2 E -10 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -2 10 B 14 0 12 2 12 C -8 -12 0 -6 -4 D 2 -2 6 0 2 E -10 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999973612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4544: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (13) C D B E A (12) E B A D C (8) E A B D C (8) C D A B E (6) C A D B E (6) E B D C A (5) C D B A E (5) A C B D E (5) A C D B E (4) A B D E C (4) E B D A C (3) A E B C D (3) A E C B D (2) A B D C E (2) E D B C A (1) E C D B A (1) E C B D A (1) E A B C D (1) D E B C A (1) D B C A E (1) C E D A B (1) C D A E B (1) C A D E B (1) B D E A C (1) B D A C E (1) B A E D C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 18 16 20 14 B -18 0 8 18 -2 C -16 -8 0 0 -8 D -20 -18 0 0 0 E -14 2 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 20 14 B -18 0 8 18 -2 C -16 -8 0 0 -8 D -20 -18 0 0 0 E -14 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=32 E=28 B=3 D=2 so D is eliminated. Round 2 votes counts: A=35 C=32 E=29 B=4 so B is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:234 B:203 E:198 C:184 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 20 14 B -18 0 8 18 -2 C -16 -8 0 0 -8 D -20 -18 0 0 0 E -14 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 20 14 B -18 0 8 18 -2 C -16 -8 0 0 -8 D -20 -18 0 0 0 E -14 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 20 14 B -18 0 8 18 -2 C -16 -8 0 0 -8 D -20 -18 0 0 0 E -14 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4545: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (13) A E B C D (12) B C D A E (10) D C B E A (7) C B D A E (7) D E A C B (6) B C A E D (5) E D A C B (4) D A E C B (4) B C D E A (4) B C A D E (4) E A D B C (3) A E B D C (3) D C B A E (2) B C E A D (2) B A E C D (2) A E D C B (2) A E C B D (2) E A B D C (1) E A B C D (1) D E C A B (1) D C A E B (1) C D B E A (1) C D B A E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 10 4 14 B -8 0 -2 10 -6 C -10 2 0 6 -10 D -4 -10 -6 0 -4 E -14 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 4 14 B -8 0 -2 10 -6 C -10 2 0 6 -10 D -4 -10 -6 0 -4 E -14 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=22 D=21 A=20 C=9 so C is eliminated. Round 2 votes counts: B=35 D=23 E=22 A=20 so A is eliminated. Round 3 votes counts: E=41 B=36 D=23 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:218 E:203 B:197 C:194 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 4 14 B -8 0 -2 10 -6 C -10 2 0 6 -10 D -4 -10 -6 0 -4 E -14 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 4 14 B -8 0 -2 10 -6 C -10 2 0 6 -10 D -4 -10 -6 0 -4 E -14 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 4 14 B -8 0 -2 10 -6 C -10 2 0 6 -10 D -4 -10 -6 0 -4 E -14 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4546: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (7) B C A E D (6) E D A C B (5) C E D A B (5) A E D C B (5) A E D B C (5) C B D E A (4) C B A E D (4) D E C A B (3) D E A C B (3) D E A B C (3) D B E A C (3) B D C A E (3) B C D E A (3) B C D A E (3) E A D C B (2) C D E B A (2) C B E A D (2) B D A E C (2) A E C D B (2) A C B E D (2) E D C A B (1) E C D A B (1) D E B A C (1) D B C E A (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B A D (1) C E A B D (1) C D B E A (1) C B E D A (1) C A B E D (1) B D C E A (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B D C (1) A E B C D (1) A D E B C (1) A D B E C (1) A B E D C (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 10 2 10 B 2 0 6 4 10 C -10 -6 0 -8 -8 D -2 -4 8 0 -6 E -10 -10 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 2 10 B 2 0 6 4 10 C -10 -6 0 -8 -8 D -2 -4 8 0 -6 E -10 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=23 A=22 D=16 E=9 so E is eliminated. Round 2 votes counts: B=30 C=24 A=24 D=22 so D is eliminated. Round 3 votes counts: A=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:210 D:198 E:197 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 2 10 B 2 0 6 4 10 C -10 -6 0 -8 -8 D -2 -4 8 0 -6 E -10 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 2 10 B 2 0 6 4 10 C -10 -6 0 -8 -8 D -2 -4 8 0 -6 E -10 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 2 10 B 2 0 6 4 10 C -10 -6 0 -8 -8 D -2 -4 8 0 -6 E -10 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4547: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B A D C E (7) E D C B A (4) E D C A B (4) E D B C A (4) D B A E C (4) C E A D B (4) C A B E D (4) B D A E C (4) A D B C E (4) A C B D E (4) A B C D E (4) E C B A D (3) D E C A B (3) C E D A B (3) B A C D E (3) D E B A C (2) D C A E B (2) D A C B E (2) D A B C E (2) C A E D B (2) C A B D E (2) B D E A C (2) B A E D C (2) E D B A C (1) E C D B A (1) E C B D A (1) E C A B D (1) E B D C A (1) E B D A C (1) E B C A D (1) D B E A C (1) C E A B D (1) C B A E D (1) C A D B E (1) B E D A C (1) B E A D C (1) B A D E C (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 -4 2 B -6 0 -4 -6 4 C 4 4 0 -8 -4 D 4 6 8 0 -2 E -2 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999927 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 6 -4 -4 2 B -6 0 -4 -6 4 C 4 4 0 -8 -4 D 4 6 8 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=22 C=18 D=16 A=13 so A is eliminated. Round 2 votes counts: E=31 B=27 C=22 D=20 so D is eliminated. Round 3 votes counts: B=38 E=36 C=26 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:208 A:200 E:200 C:198 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -4 2 B -6 0 -4 -6 4 C 4 4 0 -8 -4 D 4 6 8 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 2 B -6 0 -4 -6 4 C 4 4 0 -8 -4 D 4 6 8 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 2 B -6 0 -4 -6 4 C 4 4 0 -8 -4 D 4 6 8 0 -2 E -2 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4548: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (16) C B D A E (12) E A D B C (10) D C B E A (9) B C A E D (9) A E B C D (7) D C B A E (5) E A B C D (4) E D A C B (3) E D A B C (3) E A D C B (3) B C D A E (3) B C A D E (3) D E C A B (2) D A E C B (2) B C E A D (2) E A B D C (1) D C E B A (1) D C E A B (1) D C A B E (1) B C D E A (1) B A C E D (1) A E D C B (1) Total count = 100 A B C D E A 0 8 2 -18 -12 B -8 0 -12 -14 -8 C -2 12 0 -16 -4 D 18 14 16 0 12 E 12 8 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -18 -12 B -8 0 -12 -14 -8 C -2 12 0 -16 -4 D 18 14 16 0 12 E 12 8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=24 B=19 C=12 A=8 so A is eliminated. Round 2 votes counts: D=37 E=32 B=19 C=12 so C is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:230 E:206 C:195 A:190 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -18 -12 B -8 0 -12 -14 -8 C -2 12 0 -16 -4 D 18 14 16 0 12 E 12 8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -18 -12 B -8 0 -12 -14 -8 C -2 12 0 -16 -4 D 18 14 16 0 12 E 12 8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -18 -12 B -8 0 -12 -14 -8 C -2 12 0 -16 -4 D 18 14 16 0 12 E 12 8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4549: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (14) C E D B A (8) E C B D A (7) B E A C D (7) E B C D A (5) E B C A D (5) B E C A D (5) C E D A B (4) B A D E C (4) C D E B A (3) C D A E B (3) B E A D C (3) A D C B E (3) E C B A D (2) D C A E B (2) D B A C E (2) D A C E B (2) D A C B E (2) D A B C E (2) C E B D A (2) B A E D C (2) A D C E B (2) A D B E C (2) A D B C E (2) E B A C D (1) E A C B D (1) D C E A B (1) D C A B E (1) C D E A B (1) B E C D A (1) A E C B D (1) Total count = 100 A B C D E A 0 -14 0 8 -12 B 14 0 10 20 0 C 0 -10 0 12 -20 D -8 -20 -12 0 -8 E 12 0 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.516201 C: 0.000000 D: 0.000000 E: 0.483799 Sum of squares = 0.500524955074 Cumulative probabilities = A: 0.000000 B: 0.516201 C: 0.516201 D: 0.516201 E: 1.000000 A B C D E A 0 -14 0 8 -12 B 14 0 10 20 0 C 0 -10 0 12 -20 D -8 -20 -12 0 -8 E 12 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=22 E=21 C=21 D=12 so D is eliminated. Round 2 votes counts: A=30 C=25 B=24 E=21 so E is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:220 A:191 C:191 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 8 -12 B 14 0 10 20 0 C 0 -10 0 12 -20 D -8 -20 -12 0 -8 E 12 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 8 -12 B 14 0 10 20 0 C 0 -10 0 12 -20 D -8 -20 -12 0 -8 E 12 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 8 -12 B 14 0 10 20 0 C 0 -10 0 12 -20 D -8 -20 -12 0 -8 E 12 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4550: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) A B D C E (10) B A E C D (7) A B E D C (7) E C D B A (5) E C B D A (5) C E D B A (5) A B D E C (4) E C D A B (3) B A D C E (3) A E B D C (3) A D E B C (3) A D B C E (3) E C B A D (2) E B C A D (2) E B A C D (2) D E C A B (2) D A C B E (2) C D E B A (2) A D B E C (2) E D C A B (1) E A C B D (1) D E A C B (1) D C B A E (1) D C A E B (1) D C A B E (1) C E B D A (1) C D E A B (1) C B E D A (1) B E C A D (1) B E A C D (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C E D (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 12 4 12 6 B -12 0 10 14 -2 C -4 -10 0 -8 -6 D -12 -14 8 0 0 E -6 2 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 12 6 B -12 0 10 14 -2 C -4 -10 0 -8 -6 D -12 -14 8 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=21 D=18 B=18 C=10 so C is eliminated. Round 2 votes counts: A=33 E=27 D=21 B=19 so B is eliminated. Round 3 votes counts: A=47 E=30 D=23 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:205 E:201 D:191 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 12 6 B -12 0 10 14 -2 C -4 -10 0 -8 -6 D -12 -14 8 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 12 6 B -12 0 10 14 -2 C -4 -10 0 -8 -6 D -12 -14 8 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 12 6 B -12 0 10 14 -2 C -4 -10 0 -8 -6 D -12 -14 8 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996301 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4551: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A E D B (8) D E B C A (7) E D B A C (6) B D E C A (6) B D E A C (6) B A C D E (4) C A B E D (3) C A B D E (3) B E D A C (3) A E C D B (3) A C E D B (3) D B E C A (2) D B C E A (2) C D E A B (2) C A E B D (2) B D C E A (2) B C D E A (2) A E D C B (2) A E D B C (2) A C E B D (2) E D C A B (1) E D A C B (1) E D A B C (1) E A D C B (1) E A D B C (1) D E C B A (1) D B E A C (1) C E A D B (1) C D B E A (1) C D B A E (1) C D A E B (1) C B D E A (1) C B A D E (1) C A D E B (1) B D A C E (1) B C D A E (1) B C A D E (1) B A E D C (1) A E B D C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 0 -16 -14 B 18 0 20 -16 -12 C 0 -20 0 -14 -12 D 16 16 14 0 12 E 14 12 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 0 -16 -14 B 18 0 20 -16 -12 C 0 -20 0 -14 -12 D 16 16 14 0 12 E 14 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=25 D=22 A=15 E=11 so E is eliminated. Round 2 votes counts: D=31 B=27 C=25 A=17 so A is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:213 B:205 C:177 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 0 -16 -14 B 18 0 20 -16 -12 C 0 -20 0 -14 -12 D 16 16 14 0 12 E 14 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -16 -14 B 18 0 20 -16 -12 C 0 -20 0 -14 -12 D 16 16 14 0 12 E 14 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -16 -14 B 18 0 20 -16 -12 C 0 -20 0 -14 -12 D 16 16 14 0 12 E 14 12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4552: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) A E C D B (11) B D C E A (9) D B C E A (6) C B D E A (6) B D C A E (6) B D A E C (6) A B D E C (6) D B A E C (5) E C A D B (4) C E D B A (3) C D B E A (3) B D A C E (3) A E D B C (3) A E C B D (3) C E A B D (2) B C D E A (2) E A D C B (1) D C B E A (1) D B E A C (1) C E B D A (1) C E A D B (1) C A E B D (1) B A D E C (1) A E D C B (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 10 -4 -4 B 6 0 0 -6 12 C -10 0 0 -2 -12 D 4 6 2 0 12 E 4 -12 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 -4 -4 B 6 0 0 -6 12 C -10 0 0 -2 -12 D 4 6 2 0 12 E 4 -12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 E=17 C=17 D=13 so D is eliminated. Round 2 votes counts: B=39 A=26 C=18 E=17 so E is eliminated. Round 3 votes counts: B=39 A=39 C=22 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:212 B:206 A:198 E:196 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 10 -4 -4 B 6 0 0 -6 12 C -10 0 0 -2 -12 D 4 6 2 0 12 E 4 -12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 -4 -4 B 6 0 0 -6 12 C -10 0 0 -2 -12 D 4 6 2 0 12 E 4 -12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 -4 -4 B 6 0 0 -6 12 C -10 0 0 -2 -12 D 4 6 2 0 12 E 4 -12 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4553: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) E A D C B (6) C D E B A (6) B C D A E (6) E C B D A (5) C B D E A (5) B A D C E (4) E A B D C (3) C E D B A (3) C D E A B (3) B C E D A (3) B A D E C (3) B A C D E (3) A E B D C (3) A D E C B (3) E C D B A (2) E A D B C (2) D C B A E (2) D C A E B (2) C D B E A (2) B E C A D (2) B C D E A (2) A D B C E (2) A B D E C (2) E C A D B (1) E B C D A (1) E A C D B (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B D A (1) C B E D A (1) B E C D A (1) B D A C E (1) B C A D E (1) B A E D C (1) A E D C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -16 -16 -18 B 10 0 -14 -2 -10 C 16 14 0 10 2 D 16 2 -10 0 12 E 18 10 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -16 -18 B 10 0 -14 -2 -10 C 16 14 0 10 2 D 16 2 -10 0 12 E 18 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=27 B=27 C=21 A=15 D=10 so D is eliminated. Round 2 votes counts: E=29 C=27 B=27 A=17 so A is eliminated. Round 3 votes counts: E=38 B=33 C=29 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:221 D:210 E:207 B:192 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -16 -16 -18 B 10 0 -14 -2 -10 C 16 14 0 10 2 D 16 2 -10 0 12 E 18 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -16 -18 B 10 0 -14 -2 -10 C 16 14 0 10 2 D 16 2 -10 0 12 E 18 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -16 -18 B 10 0 -14 -2 -10 C 16 14 0 10 2 D 16 2 -10 0 12 E 18 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4554: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (15) A E D B C (13) C E D B A (8) A E D C B (8) A B D E C (6) C E D A B (5) B C D E A (5) C D E B A (4) B D E C A (4) B D E A C (4) B A D E C (4) C A E D B (3) E D A C B (2) B C A D E (2) A E C D B (2) A C E D B (2) A B C D E (2) E D C B A (1) E D B A C (1) E A D C B (1) C D B E A (1) C B A D E (1) B D C E A (1) B D A E C (1) B A C D E (1) A E B C D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 0 -4 -4 B 6 0 -8 -2 -2 C 0 8 0 6 2 D 4 2 -6 0 2 E 4 2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.245563 B: 0.000000 C: 0.754437 D: 0.000000 E: 0.000000 Sum of squares = 0.629476077068 Cumulative probabilities = A: 0.245563 B: 0.245563 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -4 -4 B 6 0 -8 -2 -2 C 0 8 0 6 2 D 4 2 -6 0 2 E 4 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555565099 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=36 B=22 E=5 so D is eliminated. Round 2 votes counts: C=37 A=36 B=22 E=5 so E is eliminated. Round 3 votes counts: A=39 C=38 B=23 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:208 D:201 E:201 B:197 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -4 -4 B 6 0 -8 -2 -2 C 0 8 0 6 2 D 4 2 -6 0 2 E 4 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555565099 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -4 -4 B 6 0 -8 -2 -2 C 0 8 0 6 2 D 4 2 -6 0 2 E 4 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555565099 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -4 -4 B 6 0 -8 -2 -2 C 0 8 0 6 2 D 4 2 -6 0 2 E 4 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.55555565099 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4555: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (5) D A C B E (5) C D B A E (5) B C E D A (5) A D E C B (5) E D C A B (4) E A D C B (4) C B D A E (4) A D C B E (4) E A B D C (3) D C A B E (3) C D A B E (3) B E A C D (3) E B C D A (2) E B C A D (2) E B A D C (2) E B A C D (2) D C A E B (2) D A C E B (2) C B E D A (2) C B D E A (2) B E C D A (2) B E C A D (2) B C D A E (2) B C A D E (2) B A D C E (2) B A C D E (2) A E D B C (2) A D E B C (2) A D B E C (2) A D B C E (2) E D C B A (1) E D A C B (1) E C D B A (1) C E D B A (1) C D E A B (1) C D B E A (1) B C E A D (1) B C D E A (1) B C A E D (1) B A E C D (1) A D C E B (1) Total count = 100 A B C D E A 0 2 0 0 4 B -2 0 -4 -14 14 C 0 4 0 -4 8 D 0 14 4 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.538434 B: 0.000000 C: 0.000000 D: 0.461566 E: 0.000000 Sum of squares = 0.502954320108 Cumulative probabilities = A: 0.538434 B: 0.538434 C: 0.538434 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 4 B -2 0 -4 -14 14 C 0 4 0 -4 8 D 0 14 4 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=24 C=19 A=18 D=12 so D is eliminated. Round 2 votes counts: E=27 A=25 C=24 B=24 so C is eliminated. Round 3 votes counts: B=38 A=33 E=29 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:212 C:204 A:203 B:197 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 0 4 B -2 0 -4 -14 14 C 0 4 0 -4 8 D 0 14 4 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 4 B -2 0 -4 -14 14 C 0 4 0 -4 8 D 0 14 4 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 4 B -2 0 -4 -14 14 C 0 4 0 -4 8 D 0 14 4 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4556: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (14) D C B E A (11) E A B C D (8) D C B A E (8) D E C B A (7) A C B D E (5) A B C E D (5) C B D A E (4) E B A C D (3) D C A B E (3) D A C B E (3) A D C B E (3) E D B C A (2) C B D E A (2) B C E A D (2) B C D E A (2) A C B E D (2) A B E C D (2) E D A C B (1) E B D C A (1) E B C D A (1) E B C A D (1) E A D C B (1) E A D B C (1) E A B D C (1) D E B C A (1) D A C E B (1) B C A E D (1) B C A D E (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 6 6 6 10 B -6 0 -8 10 8 C -6 8 0 8 6 D -6 -10 -8 0 4 E -10 -8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 10 B -6 0 -8 10 8 C -6 8 0 8 6 D -6 -10 -8 0 4 E -10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=34 A=34 E=20 C=6 B=6 so C is eliminated. Round 2 votes counts: D=34 A=34 E=20 B=12 so B is eliminated. Round 3 votes counts: D=42 A=36 E=22 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:208 B:202 D:190 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 10 B -6 0 -8 10 8 C -6 8 0 8 6 D -6 -10 -8 0 4 E -10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 10 B -6 0 -8 10 8 C -6 8 0 8 6 D -6 -10 -8 0 4 E -10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 10 B -6 0 -8 10 8 C -6 8 0 8 6 D -6 -10 -8 0 4 E -10 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4557: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (16) A C E B D (15) B E C D A (6) D E B C A (5) D A E C B (5) B D C E A (4) B C E A D (4) A B C E D (4) E C B D A (3) E C B A D (3) D B C E A (3) C E A B D (3) A E C B D (3) A C B E D (3) E C A B D (2) D E C B A (2) D E C A B (2) C E B A D (2) A E C D B (2) A D B C E (2) E C A D B (1) E B C D A (1) D B A E C (1) D B A C E (1) D A C E B (1) D A B E C (1) C B E A D (1) B A C E D (1) A D E C B (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -16 -2 -16 B 6 0 0 12 -4 C 16 0 0 12 -8 D 2 -12 -12 0 -10 E 16 4 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -16 -2 -16 B 6 0 0 12 -4 C 16 0 0 12 -8 D 2 -12 -12 0 -10 E 16 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=32 B=15 E=10 C=6 so C is eliminated. Round 2 votes counts: D=37 A=32 B=16 E=15 so E is eliminated. Round 3 votes counts: A=38 D=37 B=25 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:219 C:210 B:207 D:184 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -16 -2 -16 B 6 0 0 12 -4 C 16 0 0 12 -8 D 2 -12 -12 0 -10 E 16 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -2 -16 B 6 0 0 12 -4 C 16 0 0 12 -8 D 2 -12 -12 0 -10 E 16 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -2 -16 B 6 0 0 12 -4 C 16 0 0 12 -8 D 2 -12 -12 0 -10 E 16 4 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4558: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) C B D E A (8) A E C D B (8) E A C D B (7) E C A B D (6) C B E D A (6) D B C A E (5) B C D E A (5) A D E B C (5) A D B E C (5) E A C B D (4) D B A C E (4) C E A B D (4) B D C E A (4) B D C A E (3) A E D C B (3) A E D B C (3) D B A E C (2) C E B D A (2) C E B A D (2) E C A D B (1) D A B C E (1) C E A D B (1) B D A C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 6 2 0 B -16 0 -4 -6 8 C -6 4 0 8 -8 D -2 6 -8 0 4 E 0 -8 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750963 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.249037 Sum of squares = 0.625964834244 Cumulative probabilities = A: 0.750963 B: 0.750963 C: 0.750963 D: 0.750963 E: 1.000000 A B C D E A 0 16 6 2 0 B -16 0 -4 -6 8 C -6 4 0 8 -8 D -2 6 -8 0 4 E 0 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555862719 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=23 D=20 E=18 B=13 so B is eliminated. Round 2 votes counts: D=28 C=28 A=26 E=18 so E is eliminated. Round 3 votes counts: A=37 C=35 D=28 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:200 C:199 E:198 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 2 0 B -16 0 -4 -6 8 C -6 4 0 8 -8 D -2 6 -8 0 4 E 0 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555862719 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 2 0 B -16 0 -4 -6 8 C -6 4 0 8 -8 D -2 6 -8 0 4 E 0 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555862719 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 2 0 B -16 0 -4 -6 8 C -6 4 0 8 -8 D -2 6 -8 0 4 E 0 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555862719 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4559: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) A D B E C (7) E C B A D (6) C E B D A (5) A E D C B (5) E A C D B (4) C E A D B (4) C D B A E (4) B D C A E (4) E C A B D (3) D A B C E (3) B D A E C (3) B C E D A (3) A D E B C (3) E C A D B (2) E B A D C (2) E B A C D (2) D B A C E (2) D A C B E (2) C E D A B (2) B D A C E (2) B C D E A (2) A D C E B (2) E B C D A (1) E A D B C (1) E A B D C (1) E A B C D (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E A B (1) C D A E B (1) C D A B E (1) C B E D A (1) B D E A C (1) B C D A E (1) B A D E C (1) A E D B C (1) A D E C B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -6 -2 -8 B 4 0 -16 0 -2 C 6 16 0 14 8 D 2 0 -14 0 6 E 8 2 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -2 -8 B 4 0 -16 0 -2 C 6 16 0 14 8 D 2 0 -14 0 6 E 8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 A=21 B=17 D=8 so D is eliminated. Round 2 votes counts: C=31 A=27 E=23 B=19 so B is eliminated. Round 3 votes counts: C=41 A=35 E=24 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:198 D:197 B:193 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 -8 B 4 0 -16 0 -2 C 6 16 0 14 8 D 2 0 -14 0 6 E 8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 -8 B 4 0 -16 0 -2 C 6 16 0 14 8 D 2 0 -14 0 6 E 8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 -8 B 4 0 -16 0 -2 C 6 16 0 14 8 D 2 0 -14 0 6 E 8 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4560: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) B C E A D (6) D A C E B (5) E D C A B (4) E C D A B (4) E C B D A (4) B A D E C (4) A C D B E (4) D A E C B (3) D A E B C (3) D A B E C (3) C E D A B (3) B E C D A (3) B E C A D (3) A B D C E (3) E C D B A (2) D E C A B (2) B E D A C (2) B D E A C (2) B D A E C (2) B C A E D (2) B A E D C (2) B A E C D (2) A D B E C (2) A D B C E (2) E D B C A (1) E B C D A (1) D E A C B (1) D C E A B (1) C E B D A (1) C E B A D (1) C E A D B (1) C E A B D (1) C B A E D (1) C A E D B (1) C A D E B (1) C A B E D (1) B E D C A (1) B A D C E (1) B A C E D (1) B A C D E (1) A D C B E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 14 10 4 12 B -14 0 -6 -8 0 C -10 6 0 -8 -2 D -4 8 8 0 4 E -12 0 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 4 12 B -14 0 -6 -8 0 C -10 6 0 -8 -2 D -4 8 8 0 4 E -12 0 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 D=18 E=16 C=11 so C is eliminated. Round 2 votes counts: B=33 A=26 E=23 D=18 so D is eliminated. Round 3 votes counts: A=40 B=33 E=27 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:208 C:193 E:193 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 4 12 B -14 0 -6 -8 0 C -10 6 0 -8 -2 D -4 8 8 0 4 E -12 0 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 4 12 B -14 0 -6 -8 0 C -10 6 0 -8 -2 D -4 8 8 0 4 E -12 0 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 4 12 B -14 0 -6 -8 0 C -10 6 0 -8 -2 D -4 8 8 0 4 E -12 0 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4561: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) D C B A E (6) A E B D C (6) A E D B C (5) A E B C D (5) D A E C B (4) C D B E A (4) B C A E D (4) E A D C B (3) E A B C D (3) C B E D A (3) B A E C D (3) A D E B C (3) A B E C D (3) D E C A B (2) D C E A B (2) D C A B E (2) D A C B E (2) B C E A D (2) B C A D E (2) B A C D E (2) A D B E C (2) A D B C E (2) A B D C E (2) E D C A B (1) E D A C B (1) E C D B A (1) E C B D A (1) E C B A D (1) E B A C D (1) E A C D B (1) E A B D C (1) D C A E B (1) D A C E B (1) D A B C E (1) C E B D A (1) C D E B A (1) C D B A E (1) C B D E A (1) B E A C D (1) B A D C E (1) B A C E D (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 14 14 24 B -12 0 6 -6 10 C -14 -6 0 -14 0 D -14 6 14 0 2 E -24 -10 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 14 24 B -12 0 6 -6 10 C -14 -6 0 -14 0 D -14 6 14 0 2 E -24 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=28 B=16 E=14 C=11 so C is eliminated. Round 2 votes counts: D=34 A=31 B=20 E=15 so E is eliminated. Round 3 votes counts: A=39 D=37 B=24 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:232 D:204 B:199 C:183 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 14 24 B -12 0 6 -6 10 C -14 -6 0 -14 0 D -14 6 14 0 2 E -24 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 14 24 B -12 0 6 -6 10 C -14 -6 0 -14 0 D -14 6 14 0 2 E -24 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 14 24 B -12 0 6 -6 10 C -14 -6 0 -14 0 D -14 6 14 0 2 E -24 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4562: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) B A C D E (7) E D C A B (6) E D B A C (6) E B A D C (4) D C B A E (4) D B C A E (4) C D E A B (4) C A B D E (4) E A B D C (3) E A B C D (3) D E C B A (3) C D B A E (3) C A E B D (3) C A B E D (3) B A D E C (3) B A D C E (3) A B C E D (3) A B C D E (3) D E B C A (2) D C E A B (2) D B E A C (2) D B A C E (2) C E A B D (2) A C B E D (2) E D A B C (1) E C A D B (1) E C A B D (1) E A C B D (1) D E C A B (1) D C E B A (1) C D A B E (1) C B A D E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 8 -2 -4 B 10 0 14 -4 -6 C -8 -14 0 -12 6 D 2 4 12 0 18 E 4 6 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -2 -4 B 10 0 14 -4 -6 C -8 -14 0 -12 6 D 2 4 12 0 18 E 4 6 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=26 C=21 B=14 A=9 so A is eliminated. Round 2 votes counts: D=30 E=26 C=23 B=21 so B is eliminated. Round 3 votes counts: C=37 D=36 E=27 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:207 A:196 E:193 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 -2 -4 B 10 0 14 -4 -6 C -8 -14 0 -12 6 D 2 4 12 0 18 E 4 6 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -2 -4 B 10 0 14 -4 -6 C -8 -14 0 -12 6 D 2 4 12 0 18 E 4 6 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -2 -4 B 10 0 14 -4 -6 C -8 -14 0 -12 6 D 2 4 12 0 18 E 4 6 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4563: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) B E A D C (9) B E A C D (6) C A D B E (5) B E C D A (5) B C A D E (5) C D E A B (4) C D A E B (4) B C E D A (4) B A C E D (4) D C A E B (3) C B A D E (3) B E C A D (3) A D C B E (3) E D C B A (2) E D C A B (2) E B D C A (2) E B D A C (2) E B A D C (2) D E C A B (2) D C E A B (2) B A E D C (2) B A C D E (2) A D E C B (2) A C D B E (2) E C B D A (1) E B C D A (1) E A B D C (1) D E A C B (1) C D A B E (1) C B E D A (1) C B D A E (1) C A B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 24 0 B 10 0 -2 12 16 C 4 2 0 6 14 D -24 -12 -6 0 6 E 0 -16 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 24 0 B 10 0 -2 12 16 C 4 2 0 6 14 D -24 -12 -6 0 6 E 0 -16 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 C=20 A=19 E=13 D=8 so D is eliminated. Round 2 votes counts: B=40 C=25 A=19 E=16 so E is eliminated. Round 3 votes counts: B=47 C=32 A=21 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:213 A:205 D:182 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 24 0 B 10 0 -2 12 16 C 4 2 0 6 14 D -24 -12 -6 0 6 E 0 -16 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 24 0 B 10 0 -2 12 16 C 4 2 0 6 14 D -24 -12 -6 0 6 E 0 -16 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 24 0 B 10 0 -2 12 16 C 4 2 0 6 14 D -24 -12 -6 0 6 E 0 -16 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997345 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4564: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) A C D B E (6) C D A E B (5) C A D E B (5) D E C B A (4) B E D A C (4) E D C B A (3) D E C A B (3) D C A E B (3) B A E D C (3) A D C B E (3) A B D E C (3) A B C D E (3) E D B C A (2) E C D B A (2) E B D C A (2) E B D A C (2) E B C D A (2) D C E A B (2) D A C B E (2) D A B E C (2) C E D B A (2) C E B A D (2) C D E A B (2) C A E B D (2) B E C A D (2) B E A D C (2) B E A C D (2) B D E A C (2) B D A E C (2) B A D E C (2) A C B D E (2) E C B D A (1) D B E A C (1) D A C E B (1) C E D A B (1) C E A B D (1) C B E A D (1) A D B E C (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 6 0 10 B 0 0 -8 0 6 C -6 8 0 0 -8 D 0 0 0 0 10 E -10 -6 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.508510 B: 0.188382 C: 0.000000 D: 0.303108 E: 0.000000 Sum of squares = 0.385944584943 Cumulative probabilities = A: 0.508510 B: 0.696892 C: 0.696892 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 0 10 B 0 0 -8 0 6 C -6 8 0 0 -8 D 0 0 0 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378379 B: 0.283784 C: 0.000000 D: 0.337838 E: 0.000000 Sum of squares = 0.337837865186 Cumulative probabilities = A: 0.378379 B: 0.662162 C: 0.662162 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=21 A=21 D=18 E=14 so E is eliminated. Round 2 votes counts: B=32 C=24 D=23 A=21 so A is eliminated. Round 3 votes counts: B=40 C=33 D=27 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:208 D:205 B:199 C:197 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 0 10 B 0 0 -8 0 6 C -6 8 0 0 -8 D 0 0 0 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378379 B: 0.283784 C: 0.000000 D: 0.337838 E: 0.000000 Sum of squares = 0.337837865186 Cumulative probabilities = A: 0.378379 B: 0.662162 C: 0.662162 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 10 B 0 0 -8 0 6 C -6 8 0 0 -8 D 0 0 0 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378379 B: 0.283784 C: 0.000000 D: 0.337838 E: 0.000000 Sum of squares = 0.337837865186 Cumulative probabilities = A: 0.378379 B: 0.662162 C: 0.662162 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 10 B 0 0 -8 0 6 C -6 8 0 0 -8 D 0 0 0 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378379 B: 0.283784 C: 0.000000 D: 0.337838 E: 0.000000 Sum of squares = 0.337837865186 Cumulative probabilities = A: 0.378379 B: 0.662162 C: 0.662162 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4565: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (6) A E B D C (6) B C A E D (5) E D A C B (4) E A D B C (4) D A E C B (4) C D B E A (4) C B D A E (4) B C D E A (4) A E B C D (4) A D E C B (4) E D C B A (3) E A B D C (3) D C E B A (3) B A C E D (3) A D C E B (3) A B E C D (3) A B C E D (3) E B A C D (2) D E C B A (2) D C B E A (2) D C B A E (2) D C A E B (2) B E C D A (2) B E A C D (2) B C A D E (2) E D B C A (1) E B C D A (1) E B A D C (1) E A B C D (1) D E A C B (1) D C A B E (1) D A C E B (1) C B D E A (1) C A B D E (1) B E C A D (1) B C E A D (1) B A E C D (1) A E D C B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 16 16 14 B -6 0 12 2 -14 C -16 -12 0 -8 -14 D -16 -2 8 0 -16 E -14 14 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 16 14 B -6 0 12 2 -14 C -16 -12 0 -8 -14 D -16 -2 8 0 -16 E -14 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=21 E=20 D=18 C=10 so C is eliminated. Round 2 votes counts: A=32 B=26 D=22 E=20 so E is eliminated. Round 3 votes counts: A=40 D=30 B=30 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:215 B:197 D:187 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 16 14 B -6 0 12 2 -14 C -16 -12 0 -8 -14 D -16 -2 8 0 -16 E -14 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 16 14 B -6 0 12 2 -14 C -16 -12 0 -8 -14 D -16 -2 8 0 -16 E -14 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 16 14 B -6 0 12 2 -14 C -16 -12 0 -8 -14 D -16 -2 8 0 -16 E -14 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4566: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) D B E A C (6) C A E B D (6) E C D A B (5) D E B C A (5) A C B D E (5) D B A E C (4) C E B A D (4) B E C A D (4) A C B E D (4) E D B C A (3) E C D B A (3) E B D C A (3) E B C D A (3) D A B C E (3) A B D C E (3) D E C B A (2) D E C A B (2) C A B E D (2) B A C D E (2) A D C B E (2) A C E B D (2) A C D E B (2) E D C B A (1) E C B D A (1) E B C A D (1) D E B A C (1) D E A C B (1) D A C E B (1) D A C B E (1) C E A D B (1) C E A B D (1) C A E D B (1) C A D E B (1) B D E A C (1) B D A E C (1) B C A E D (1) B A D C E (1) B A C E D (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -12 6 -8 B 8 0 -8 8 -12 C 12 8 0 16 -6 D -6 -8 -16 0 -8 E 8 12 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 6 -8 B 8 0 -8 8 -12 C 12 8 0 16 -6 D -6 -8 -16 0 -8 E 8 12 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=26 D=26 A=21 C=16 B=11 so B is eliminated. Round 2 votes counts: E=30 D=28 A=25 C=17 so C is eliminated. Round 3 votes counts: E=36 A=36 D=28 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:215 B:198 A:189 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 6 -8 B 8 0 -8 8 -12 C 12 8 0 16 -6 D -6 -8 -16 0 -8 E 8 12 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 6 -8 B 8 0 -8 8 -12 C 12 8 0 16 -6 D -6 -8 -16 0 -8 E 8 12 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 6 -8 B 8 0 -8 8 -12 C 12 8 0 16 -6 D -6 -8 -16 0 -8 E 8 12 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4567: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) D B A C E (8) B D C E A (7) B D C A E (7) E C A B D (6) C A E D B (6) D B A E C (5) A E C D B (5) E A C D B (4) D B C A E (4) C E A D B (4) C B E A D (4) A C E D B (4) E B C A D (3) E A C B D (3) B D A E C (3) D C B A E (2) C E A B D (2) B E D C A (2) E C B A D (1) E C A D B (1) D A C B E (1) C D A B E (1) B E D A C (1) B E C A D (1) B D E C A (1) B C E D A (1) A E D B C (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -20 -6 -6 -2 B 20 0 8 4 16 C 6 -8 0 -6 4 D 6 -4 6 0 2 E 2 -16 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -6 -2 B 20 0 8 4 16 C 6 -8 0 -6 4 D 6 -4 6 0 2 E 2 -16 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=20 E=18 C=17 A=12 so A is eliminated. Round 2 votes counts: B=33 E=24 C=22 D=21 so D is eliminated. Round 3 votes counts: B=50 E=25 C=25 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:224 D:205 C:198 E:190 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -6 -6 -2 B 20 0 8 4 16 C 6 -8 0 -6 4 D 6 -4 6 0 2 E 2 -16 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -6 -2 B 20 0 8 4 16 C 6 -8 0 -6 4 D 6 -4 6 0 2 E 2 -16 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -6 -2 B 20 0 8 4 16 C 6 -8 0 -6 4 D 6 -4 6 0 2 E 2 -16 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994436 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4568: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) B E D A C (7) B D A E C (6) E B C D A (5) E B A C D (5) D C A B E (5) C D A E B (5) E C A D B (4) D A C B E (3) C E D B A (3) C E A D B (3) A B E D C (3) E A B C D (2) D A B C E (2) C A E D B (2) B E A D C (2) B D E A C (2) B D C E A (2) B D A C E (2) A E C D B (2) A E B D C (2) A D C B E (2) A D B C E (2) A C D E B (2) A B D E C (2) E C B D A (1) E C B A D (1) E C A B D (1) E B A D C (1) E A C D B (1) E A C B D (1) D C B A E (1) D B C A E (1) C E D A B (1) C D E A B (1) C A D E B (1) B E D C A (1) B E C D A (1) B D C A E (1) B A E D C (1) B A D E C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 6 0 -4 B 2 0 16 14 -6 C -6 -16 0 0 -18 D 0 -14 0 0 -16 E 4 6 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 0 -4 B 2 0 16 14 -6 C -6 -16 0 0 -18 D 0 -14 0 0 -16 E 4 6 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=26 A=17 C=16 D=12 so D is eliminated. Round 2 votes counts: E=29 B=27 C=22 A=22 so C is eliminated. Round 3 votes counts: E=37 A=35 B=28 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:213 A:200 D:185 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 0 -4 B 2 0 16 14 -6 C -6 -16 0 0 -18 D 0 -14 0 0 -16 E 4 6 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 -4 B 2 0 16 14 -6 C -6 -16 0 0 -18 D 0 -14 0 0 -16 E 4 6 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 -4 B 2 0 16 14 -6 C -6 -16 0 0 -18 D 0 -14 0 0 -16 E 4 6 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4569: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) A B D E C (9) C E B A D (7) B A D E C (6) E D A C B (5) C B E D A (5) E C D A B (4) A E D B C (4) A D B E C (4) D E A C B (3) D A E B C (3) C E D B A (3) C E D A B (3) B C A D E (3) B A C D E (3) A D E B C (3) E A D C B (2) D A B E C (2) C B D E A (2) B C A E D (2) B A C E D (2) E D C A B (1) E C A D B (1) E C A B D (1) E A D B C (1) D E C A B (1) D E A B C (1) D C E A B (1) D B A E C (1) C E A B D (1) C D E B A (1) C D B E A (1) C B D A E (1) C B A E D (1) B D A C E (1) B C D A E (1) A E B D C (1) Total count = 100 A B C D E A 0 2 2 20 -6 B -2 0 -6 10 6 C -2 6 0 2 -6 D -20 -10 -2 0 -6 E 6 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333253 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 2 2 20 -6 B -2 0 -6 10 6 C -2 6 0 2 -6 D -20 -10 -2 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=21 B=18 E=15 D=12 so D is eliminated. Round 2 votes counts: C=35 A=26 E=20 B=19 so B is eliminated. Round 3 votes counts: C=41 A=39 E=20 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:206 B:204 C:200 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 2 20 -6 B -2 0 -6 10 6 C -2 6 0 2 -6 D -20 -10 -2 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 20 -6 B -2 0 -6 10 6 C -2 6 0 2 -6 D -20 -10 -2 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 20 -6 B -2 0 -6 10 6 C -2 6 0 2 -6 D -20 -10 -2 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4570: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (21) C E D A B (14) C E D B A (12) A B D E C (11) D E C B A (10) A B C E D (9) C E A D B (3) E D C B A (2) E C D B A (2) D E C A B (2) D B E C A (2) C A B E D (2) A B C D E (2) E C D A B (1) E C A D B (1) D E B C A (1) C E B A D (1) B D A E C (1) B A C E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -6 6 -2 B 8 0 0 0 2 C 6 0 0 0 -10 D -6 0 0 0 2 E 2 -2 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.451095 C: 0.119753 D: 0.429151 E: 0.000000 Sum of squares = 0.401998799353 Cumulative probabilities = A: 0.000000 B: 0.451095 C: 0.570849 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 6 -2 B 8 0 0 0 2 C 6 0 0 0 -10 D -6 0 0 0 2 E 2 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.416667 C: 0.166667 D: 0.416667 E: 0.000000 Sum of squares = 0.375000002992 Cumulative probabilities = A: 0.000000 B: 0.416667 C: 0.583333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=24 A=23 D=15 E=6 so E is eliminated. Round 2 votes counts: C=36 B=24 A=23 D=17 so D is eliminated. Round 3 votes counts: C=50 B=27 A=23 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:205 E:204 C:198 D:198 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 6 -2 B 8 0 0 0 2 C 6 0 0 0 -10 D -6 0 0 0 2 E 2 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.416667 C: 0.166667 D: 0.416667 E: 0.000000 Sum of squares = 0.375000002992 Cumulative probabilities = A: 0.000000 B: 0.416667 C: 0.583333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 6 -2 B 8 0 0 0 2 C 6 0 0 0 -10 D -6 0 0 0 2 E 2 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.416667 C: 0.166667 D: 0.416667 E: 0.000000 Sum of squares = 0.375000002992 Cumulative probabilities = A: 0.000000 B: 0.416667 C: 0.583333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 6 -2 B 8 0 0 0 2 C 6 0 0 0 -10 D -6 0 0 0 2 E 2 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.416667 C: 0.166667 D: 0.416667 E: 0.000000 Sum of squares = 0.375000002992 Cumulative probabilities = A: 0.000000 B: 0.416667 C: 0.583333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4571: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) A E B C D (8) E A B C D (7) D C B A E (7) D C A E B (7) C D B E A (7) A E B D C (7) C D B A E (6) B A E C D (6) C B D E A (5) D C B E A (3) D A E C B (3) D A E B C (3) C D E A B (3) C B E A D (3) A E D B C (3) E A C B D (2) D E A C B (2) B E A C D (2) B C A E D (2) E C A B D (1) E A B D C (1) D A B E C (1) C E A B D (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 16 -6 -10 10 B -16 0 -16 -8 -12 C 6 16 0 6 4 D 10 8 -6 0 12 E -10 12 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -6 -10 10 B -16 0 -16 -8 -12 C 6 16 0 6 4 D 10 8 -6 0 12 E -10 12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=25 A=19 E=11 B=11 so E is eliminated. Round 2 votes counts: D=34 A=29 C=26 B=11 so B is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:212 A:205 E:193 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -6 -10 10 B -16 0 -16 -8 -12 C 6 16 0 6 4 D 10 8 -6 0 12 E -10 12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 -10 10 B -16 0 -16 -8 -12 C 6 16 0 6 4 D 10 8 -6 0 12 E -10 12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 -10 10 B -16 0 -16 -8 -12 C 6 16 0 6 4 D 10 8 -6 0 12 E -10 12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4572: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) E B D A C (8) D A C E B (7) C A D B E (7) C A D E B (6) B E D A C (6) D E A B C (5) C A B E D (4) C A B D E (4) A D C E B (4) C B A E D (3) B E C D A (3) A C D E B (3) E D B A C (2) E D A B C (2) D E B A C (2) D A E C B (2) B E C A D (2) B C E D A (2) B C E A D (2) B C D E A (2) A D E C B (2) E B A D C (1) E A C B D (1) D C A E B (1) D C A B E (1) D B E A C (1) C B D A E (1) C B A D E (1) C A E D B (1) B D E C A (1) A E D C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -2 -12 0 B -6 0 -2 4 0 C 2 2 0 -12 2 D 12 -4 12 0 0 E 0 0 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.146683 B: 0.440048 C: 0.000000 D: 0.220024 E: 0.193245 Sum of squares = 0.300912407015 Cumulative probabilities = A: 0.146683 B: 0.586731 C: 0.586731 D: 0.806755 E: 1.000000 A B C D E A 0 6 -2 -12 0 B -6 0 -2 4 0 C 2 2 0 -12 2 D 12 -4 12 0 0 E 0 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.129412 B: 0.388235 C: 0.000000 D: 0.194118 E: 0.288235 Sum of squares = 0.288235294111 Cumulative probabilities = A: 0.129412 B: 0.517647 C: 0.517647 D: 0.711765 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=27 D=19 E=14 A=12 so A is eliminated. Round 2 votes counts: C=32 B=28 D=25 E=15 so E is eliminated. Round 3 votes counts: B=37 C=33 D=30 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:210 E:199 B:198 C:197 A:196 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 -12 0 B -6 0 -2 4 0 C 2 2 0 -12 2 D 12 -4 12 0 0 E 0 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.129412 B: 0.388235 C: 0.000000 D: 0.194118 E: 0.288235 Sum of squares = 0.288235294111 Cumulative probabilities = A: 0.129412 B: 0.517647 C: 0.517647 D: 0.711765 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -12 0 B -6 0 -2 4 0 C 2 2 0 -12 2 D 12 -4 12 0 0 E 0 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.129412 B: 0.388235 C: 0.000000 D: 0.194118 E: 0.288235 Sum of squares = 0.288235294111 Cumulative probabilities = A: 0.129412 B: 0.517647 C: 0.517647 D: 0.711765 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -12 0 B -6 0 -2 4 0 C 2 2 0 -12 2 D 12 -4 12 0 0 E 0 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.129412 B: 0.388235 C: 0.000000 D: 0.194118 E: 0.288235 Sum of squares = 0.288235294111 Cumulative probabilities = A: 0.129412 B: 0.517647 C: 0.517647 D: 0.711765 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4573: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (16) C E A D B (7) C A E D B (7) D B A C E (6) B D E A C (6) A C D E B (6) E B D C A (5) E B C D A (5) D A B C E (5) A C E D B (5) A C D B E (5) B E D C A (4) E C B D A (3) D B A E C (2) D A C B E (2) C E A B D (2) B D E C A (2) B D A E C (2) B D A C E (2) E C A D B (1) D C A B E (1) D B E C A (1) A D C B E (1) A D B C E (1) A C E B D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 24 -8 8 -4 B -24 0 -16 0 -16 C 8 16 0 18 6 D -8 0 -18 0 -14 E 4 16 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 -8 8 -4 B -24 0 -16 0 -16 C 8 16 0 18 6 D -8 0 -18 0 -14 E 4 16 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=21 D=17 C=16 B=16 so C is eliminated. Round 2 votes counts: E=39 A=28 D=17 B=16 so B is eliminated. Round 3 votes counts: E=43 D=29 A=28 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:224 E:214 A:210 D:180 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -8 8 -4 B -24 0 -16 0 -16 C 8 16 0 18 6 D -8 0 -18 0 -14 E 4 16 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -8 8 -4 B -24 0 -16 0 -16 C 8 16 0 18 6 D -8 0 -18 0 -14 E 4 16 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -8 8 -4 B -24 0 -16 0 -16 C 8 16 0 18 6 D -8 0 -18 0 -14 E 4 16 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4574: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) A D E C B (7) E A D C B (6) D A C B E (6) B C E D A (6) D C B A E (4) B C D A E (4) E B C D A (3) B C E A D (3) B C D E A (3) A E D C B (3) E D C A B (2) E C D B A (2) E A D B C (2) E A B C D (2) D E A C B (2) D A E C B (2) D A C E B (2) C E B D A (2) C B E D A (2) C B D E A (2) B C A D E (2) A D E B C (2) A D C B E (2) E D C B A (1) E C B D A (1) E B A C D (1) E A C D B (1) E A B D C (1) D E C B A (1) D C E B A (1) C B D A E (1) B E C D A (1) B E C A D (1) B E A C D (1) B D C A E (1) B A C D E (1) A E D B C (1) A D C E B (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -4 2 -12 B 6 0 -2 0 -10 C 4 2 0 -2 -8 D -2 0 2 0 -4 E 12 10 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 2 -12 B 6 0 -2 0 -10 C 4 2 0 -2 -8 D -2 0 2 0 -4 E 12 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=23 A=21 D=18 C=7 so C is eliminated. Round 2 votes counts: E=33 B=28 A=21 D=18 so D is eliminated. Round 3 votes counts: E=37 B=32 A=31 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:198 D:198 B:197 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 2 -12 B 6 0 -2 0 -10 C 4 2 0 -2 -8 D -2 0 2 0 -4 E 12 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 2 -12 B 6 0 -2 0 -10 C 4 2 0 -2 -8 D -2 0 2 0 -4 E 12 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 2 -12 B 6 0 -2 0 -10 C 4 2 0 -2 -8 D -2 0 2 0 -4 E 12 10 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4575: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) C E D A B (8) B A E D C (8) B A E C D (7) B A D E C (7) E B C A D (6) D A C E B (5) E C D B A (4) D C E A B (4) B E A C D (4) B A D C E (4) E C D A B (3) C D E A B (3) A D B C E (3) E C B D A (2) E C A D B (2) D C A E B (2) C E D B A (2) B E C A D (2) A D C B E (2) E C A B D (1) E A C D B (1) D C E B A (1) D C A B E (1) C D E B A (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C A E (1) B D A C E (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 8 18 6 B 6 0 14 14 8 C -8 -14 0 -4 -4 D -18 -14 4 0 -6 E -6 -8 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 18 6 B 6 0 14 14 8 C -8 -14 0 -4 -4 D -18 -14 4 0 -6 E -6 -8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=19 A=17 C=14 D=13 so D is eliminated. Round 2 votes counts: B=37 C=22 A=22 E=19 so E is eliminated. Round 3 votes counts: B=43 C=34 A=23 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:213 E:198 C:185 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 18 6 B 6 0 14 14 8 C -8 -14 0 -4 -4 D -18 -14 4 0 -6 E -6 -8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 18 6 B 6 0 14 14 8 C -8 -14 0 -4 -4 D -18 -14 4 0 -6 E -6 -8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 18 6 B 6 0 14 14 8 C -8 -14 0 -4 -4 D -18 -14 4 0 -6 E -6 -8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4576: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (11) E A D C B (10) A E D C B (10) B C A D E (7) C B D E A (6) B C D E A (6) E D C A B (5) A E D B C (4) A E B D C (4) A B E D C (4) E D A C B (3) E B A C D (3) D C A E B (3) C D E B A (3) D C E B A (2) C D B A E (2) B A C E D (2) B A C D E (2) A B E C D (2) E C B D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E A B (1) D C A B E (1) C D B E A (1) B A E C D (1) B A D C E (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 0 8 12 B -4 0 -2 2 -6 C 0 2 0 -6 0 D -8 -2 6 0 -2 E -12 6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.582078 B: 0.000000 C: 0.417922 D: 0.000000 E: 0.000000 Sum of squares = 0.513473732863 Cumulative probabilities = A: 0.582078 B: 0.582078 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 8 12 B -4 0 -2 2 -6 C 0 2 0 -6 0 D -8 -2 6 0 -2 E -12 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=26 E=24 C=12 D=8 so D is eliminated. Round 2 votes counts: B=30 A=26 E=25 C=19 so C is eliminated. Round 3 votes counts: B=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 C:198 E:198 D:197 B:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 8 12 B -4 0 -2 2 -6 C 0 2 0 -6 0 D -8 -2 6 0 -2 E -12 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 8 12 B -4 0 -2 2 -6 C 0 2 0 -6 0 D -8 -2 6 0 -2 E -12 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 8 12 B -4 0 -2 2 -6 C 0 2 0 -6 0 D -8 -2 6 0 -2 E -12 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4577: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (10) C E A B D (9) D B A E C (8) B D A C E (8) E C D A B (5) C A E B D (5) E D C A B (4) E C A D B (3) D B E A C (3) B A D C E (3) A C B E D (3) A B D E C (3) E C D B A (2) E A C D B (2) D E B C A (2) B D C A E (2) A C E B D (2) E D A C B (1) E D A B C (1) D E C B A (1) C E D B A (1) C E D A B (1) C D E B A (1) C B D E A (1) C B A E D (1) B D C E A (1) B C A D E (1) A E C D B (1) A E B C D (1) A D B E C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -2 -4 2 B -10 0 -8 4 -6 C 2 8 0 2 2 D 4 -4 -2 0 -10 E -2 6 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -4 2 B -10 0 -8 4 -6 C 2 8 0 2 2 D 4 -4 -2 0 -10 E -2 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=25 E=18 D=14 A=13 so A is eliminated. Round 2 votes counts: C=35 B=30 E=20 D=15 so D is eliminated. Round 3 votes counts: B=42 C=35 E=23 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 E:206 A:203 D:194 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 -4 2 B -10 0 -8 4 -6 C 2 8 0 2 2 D 4 -4 -2 0 -10 E -2 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -4 2 B -10 0 -8 4 -6 C 2 8 0 2 2 D 4 -4 -2 0 -10 E -2 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -4 2 B -10 0 -8 4 -6 C 2 8 0 2 2 D 4 -4 -2 0 -10 E -2 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4578: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) B D A C E (8) B C D A E (7) E C A D B (6) D A E B C (6) E A D C B (5) C E A D B (5) C B E A D (5) D B A E C (4) D A B E C (4) C E B A D (4) C E A B D (4) E A C D B (3) D E A B C (3) C B E D A (3) C B A E D (3) B D C A E (3) B C E D A (3) C A E D B (2) B C A D E (2) E C D B A (1) E C D A B (1) D E A C B (1) D B E A C (1) C A E B D (1) C A B D E (1) B D C E A (1) B A D C E (1) A D E C B (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -4 -10 8 B 10 0 8 10 12 C 4 -8 0 2 8 D 10 -10 -2 0 8 E -8 -12 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -10 8 B 10 0 8 10 12 C 4 -8 0 2 8 D 10 -10 -2 0 8 E -8 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=28 D=19 E=16 A=3 so A is eliminated. Round 2 votes counts: B=34 C=28 D=22 E=16 so E is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:203 D:203 A:192 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -10 8 B 10 0 8 10 12 C 4 -8 0 2 8 D 10 -10 -2 0 8 E -8 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -10 8 B 10 0 8 10 12 C 4 -8 0 2 8 D 10 -10 -2 0 8 E -8 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -10 8 B 10 0 8 10 12 C 4 -8 0 2 8 D 10 -10 -2 0 8 E -8 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4579: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) C B D A E (7) C D B E A (6) C E A B D (5) B A E C D (5) E A B C D (4) D E A B C (4) D C B A E (4) C D B A E (4) E A D C B (3) E A B D C (3) D B C A E (3) D B A E C (3) C E A D B (3) A B E D C (3) D E A C B (2) D B A C E (2) D A E B C (2) C E D A B (2) C E B A D (2) C D E A B (2) C B E A D (2) C B A E D (2) C B A D E (2) B D C A E (2) B A C E D (2) A E B D C (2) E D A C B (1) E C A D B (1) E A C D B (1) E A C B D (1) D E C A B (1) C E B D A (1) B D A C E (1) B A D E C (1) A E D B C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 6 -4 B -2 0 -4 -6 0 C -2 4 0 6 4 D -6 6 -6 0 -6 E 4 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999991 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 2 2 6 -4 B -2 0 -4 -6 0 C -2 4 0 6 4 D -6 6 -6 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=22 D=21 B=11 A=8 so A is eliminated. Round 2 votes counts: C=38 E=26 D=21 B=15 so B is eliminated. Round 3 votes counts: C=40 E=34 D=26 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 A:203 E:203 B:194 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 2 6 -4 B -2 0 -4 -6 0 C -2 4 0 6 4 D -6 6 -6 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 6 -4 B -2 0 -4 -6 0 C -2 4 0 6 4 D -6 6 -6 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 6 -4 B -2 0 -4 -6 0 C -2 4 0 6 4 D -6 6 -6 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4580: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) C E A D B (9) E A C B D (8) A E C D B (7) E C A B D (6) D B A C E (5) D C B A E (4) C E A B D (4) C D B E A (3) C A D E B (3) B D A E C (3) A E B D C (3) E A B C D (2) D A B E C (2) D A B C E (2) C D A E B (2) B E D A C (2) B D C E A (2) B D C A E (2) B C D E A (2) E B C A D (1) E B A D C (1) E A B D C (1) D B A E C (1) C E D A B (1) C E B A D (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A B E (1) B E D C A (1) B D A C E (1) A E D B C (1) A D C E B (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -14 2 8 B -14 0 -10 -18 -10 C 14 10 0 10 20 D -2 18 -10 0 2 E -8 10 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -14 2 8 B -14 0 -10 -18 -10 C 14 10 0 10 20 D -2 18 -10 0 2 E -8 10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 E=19 A=15 B=13 so B is eliminated. Round 2 votes counts: D=34 C=29 E=22 A=15 so A is eliminated. Round 3 votes counts: D=36 E=33 C=31 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:227 A:205 D:204 E:190 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -14 2 8 B -14 0 -10 -18 -10 C 14 10 0 10 20 D -2 18 -10 0 2 E -8 10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -14 2 8 B -14 0 -10 -18 -10 C 14 10 0 10 20 D -2 18 -10 0 2 E -8 10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -14 2 8 B -14 0 -10 -18 -10 C 14 10 0 10 20 D -2 18 -10 0 2 E -8 10 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4581: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (16) C E A D B (8) A B D E C (8) E C A B D (7) D B A E C (6) B D A E C (6) E C D B A (4) D B A C E (4) C D B E A (4) E C B D A (3) D B E A C (3) C D B A E (3) A B D C E (3) E B A D C (2) E A C B D (2) D B E C A (2) C E D A B (2) C D E B A (2) A E C B D (2) A D B C E (2) A B E D C (2) E C B A D (1) E B D A C (1) D B C E A (1) D B C A E (1) C E A B D (1) C A E D B (1) B D E A C (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -12 -18 -20 B 20 0 -14 -20 -6 C 12 14 0 14 -2 D 18 20 -14 0 -6 E 20 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999129 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -12 -18 -20 B 20 0 -14 -20 -6 C 12 14 0 14 -2 D 18 20 -14 0 -6 E 20 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=20 A=19 D=17 B=7 so B is eliminated. Round 2 votes counts: C=37 D=24 E=20 A=19 so A is eliminated. Round 3 votes counts: C=38 D=37 E=25 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:217 D:209 B:190 A:165 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -12 -18 -20 B 20 0 -14 -20 -6 C 12 14 0 14 -2 D 18 20 -14 0 -6 E 20 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -12 -18 -20 B 20 0 -14 -20 -6 C 12 14 0 14 -2 D 18 20 -14 0 -6 E 20 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -12 -18 -20 B 20 0 -14 -20 -6 C 12 14 0 14 -2 D 18 20 -14 0 -6 E 20 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4582: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (15) B A D E C (8) A B C E D (6) B D A E C (5) B A C E D (5) D E C B A (4) A C E D B (4) A B E D C (4) C E D B A (3) C D E B A (3) B A C D E (3) A B E C D (3) A B D E C (3) D E C A B (2) D C E B A (2) D C E A B (2) C E A D B (2) B D E C A (2) B A D C E (2) A E C D B (2) A D E C B (2) E C D A B (1) D E B C A (1) D E A B C (1) D B E C A (1) C E B D A (1) C D B E A (1) C B E D A (1) C A E D B (1) B D E A C (1) B D C E A (1) B C E D A (1) B C D E A (1) B C A D E (1) B A E D C (1) A E D C B (1) A E C B D (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 8 2 8 B -4 0 2 2 2 C -8 -2 0 12 12 D -2 -2 -12 0 -6 E -8 -2 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 2 8 B -4 0 2 2 2 C -8 -2 0 12 12 D -2 -2 -12 0 -6 E -8 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=28 C=27 D=13 E=1 so E is eliminated. Round 2 votes counts: B=31 C=28 A=28 D=13 so D is eliminated. Round 3 votes counts: C=38 B=33 A=29 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 C:207 B:201 E:192 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 2 8 B -4 0 2 2 2 C -8 -2 0 12 12 D -2 -2 -12 0 -6 E -8 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 2 8 B -4 0 2 2 2 C -8 -2 0 12 12 D -2 -2 -12 0 -6 E -8 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 2 8 B -4 0 2 2 2 C -8 -2 0 12 12 D -2 -2 -12 0 -6 E -8 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4583: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) B C A D E (7) B A C D E (6) E D C A B (5) B C E D A (5) A D E C B (5) E B D C A (3) C E D A B (3) C D E A B (3) C A D E B (3) C A B D E (3) B E A D C (3) B C A E D (3) B A E D C (3) B A D E C (3) A D C E B (3) E D A B C (2) D A E C B (2) B E D A C (2) B E C D A (2) B C E A D (2) B A D C E (2) A D E B C (2) A D B E C (2) A D B C E (2) A B D E C (2) E D C B A (1) E D B C A (1) E B D A C (1) E B C D A (1) E A D B C (1) C E D B A (1) C B E D A (1) C B D E A (1) B E D C A (1) B E C A D (1) A E D B C (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 6 14 4 B 0 0 18 8 8 C -6 -18 0 -14 -4 D -14 -8 14 0 -2 E -4 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.416858 B: 0.583142 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513825151509 Cumulative probabilities = A: 0.416858 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 14 4 B 0 0 18 8 8 C -6 -18 0 -14 -4 D -14 -8 14 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=23 A=20 C=15 D=2 so D is eliminated. Round 2 votes counts: B=40 E=23 A=22 C=15 so C is eliminated. Round 3 votes counts: B=42 E=30 A=28 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:212 E:197 D:195 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 14 4 B 0 0 18 8 8 C -6 -18 0 -14 -4 D -14 -8 14 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 14 4 B 0 0 18 8 8 C -6 -18 0 -14 -4 D -14 -8 14 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 14 4 B 0 0 18 8 8 C -6 -18 0 -14 -4 D -14 -8 14 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4584: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (13) C B E A D (13) B C E D A (12) B C D E A (9) D A B C E (7) A E D C B (7) D A B E C (6) E A D C B (5) E A C B D (4) A D E C B (4) E C B A D (3) E C A B D (3) C E B A D (3) D B C A E (2) D B A C E (2) C B D E A (2) D E A B C (1) D B E A C (1) B C D A E (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 4 4 -12 -12 B -4 0 10 0 10 C -4 -10 0 0 4 D 12 0 0 0 0 E 12 -10 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.359256 C: 0.000000 D: 0.640744 E: 0.000000 Sum of squares = 0.539617793946 Cumulative probabilities = A: 0.000000 B: 0.359256 C: 0.359256 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -12 -12 B -4 0 10 0 10 C -4 -10 0 0 4 D 12 0 0 0 0 E 12 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=22 C=18 E=15 A=13 so A is eliminated. Round 2 votes counts: D=38 E=22 B=22 C=18 so C is eliminated. Round 3 votes counts: D=38 B=37 E=25 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:206 E:199 C:195 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -12 -12 B -4 0 10 0 10 C -4 -10 0 0 4 D 12 0 0 0 0 E 12 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -12 -12 B -4 0 10 0 10 C -4 -10 0 0 4 D 12 0 0 0 0 E 12 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -12 -12 B -4 0 10 0 10 C -4 -10 0 0 4 D 12 0 0 0 0 E 12 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4585: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (11) C E D B A (10) D A B E C (9) B A E C D (8) A B D E C (5) E C B D A (4) E C B A D (4) D A B C E (4) A B D C E (4) E C D B A (3) D A C B E (3) C E D A B (3) C E B D A (3) C E B A D (3) C D E A B (3) B A E D C (3) B E A C D (2) B A D E C (2) A D B E C (2) A D B C E (2) E D B A C (1) E B C A D (1) D E C A B (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C E B (1) C D A E B (1) B E C A D (1) B E A D C (1) B C A E D (1) B A C E D (1) Total count = 100 A B C D E A 0 4 0 -20 -2 B -4 0 -4 -14 -4 C 0 4 0 -4 2 D 20 14 4 0 2 E 2 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -20 -2 B -4 0 -4 -14 -4 C 0 4 0 -4 2 D 20 14 4 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=23 B=19 E=13 A=13 so E is eliminated. Round 2 votes counts: C=34 D=33 B=20 A=13 so A is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:201 E:201 A:191 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -20 -2 B -4 0 -4 -14 -4 C 0 4 0 -4 2 D 20 14 4 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -20 -2 B -4 0 -4 -14 -4 C 0 4 0 -4 2 D 20 14 4 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -20 -2 B -4 0 -4 -14 -4 C 0 4 0 -4 2 D 20 14 4 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4586: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) C A D B E (8) C A E B D (6) E B D A C (5) A C B D E (5) A B D C E (5) E C B D A (4) D E B C A (4) C E D B A (4) C E A D B (4) C A E D B (3) B D E A C (3) A C D B E (3) A C B E D (3) E D B C A (2) E C B A D (2) E C A B D (2) E B C D A (2) E A C B D (2) D B E A C (2) D B A E C (2) C E D A B (2) C A D E B (2) E C D B A (1) D B C E A (1) D B C A E (1) D A C B E (1) D A B C E (1) C E A B D (1) C D E B A (1) C D E A B (1) B D A E C (1) B A D E C (1) A E B C D (1) A D B E C (1) A D B C E (1) A C E B D (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -18 8 -2 B -12 0 -12 10 -16 C 18 12 0 18 10 D -8 -10 -18 0 -8 E 2 16 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -18 8 -2 B -12 0 -12 10 -16 C 18 12 0 18 10 D -8 -10 -18 0 -8 E 2 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=28 A=23 D=12 B=5 so B is eliminated. Round 2 votes counts: C=32 E=28 A=24 D=16 so D is eliminated. Round 3 votes counts: E=37 C=34 A=29 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:229 E:208 A:200 B:185 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -18 8 -2 B -12 0 -12 10 -16 C 18 12 0 18 10 D -8 -10 -18 0 -8 E 2 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -18 8 -2 B -12 0 -12 10 -16 C 18 12 0 18 10 D -8 -10 -18 0 -8 E 2 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -18 8 -2 B -12 0 -12 10 -16 C 18 12 0 18 10 D -8 -10 -18 0 -8 E 2 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4587: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) B D A E C (10) C E B A D (8) C E A D B (8) B E C D A (8) E C B D A (7) B D E A C (7) A D B C E (6) D B A E C (4) C E B D A (3) C E A B D (3) C A E D B (3) A D C E B (3) E B C D A (2) B C E D A (2) A D C B E (2) E D A C B (1) E D A B C (1) E C D A B (1) E C B A D (1) D A E B C (1) B E D C A (1) A D E C B (1) A D B E C (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 6 -22 -6 B 6 0 12 4 10 C -6 -12 0 -2 -18 D 22 -4 2 0 0 E 6 -10 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -22 -6 B 6 0 12 4 10 C -6 -12 0 -2 -18 D 22 -4 2 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999175 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 D=18 A=16 E=13 so E is eliminated. Round 2 votes counts: C=34 B=30 D=20 A=16 so A is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:210 E:207 A:186 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -22 -6 B 6 0 12 4 10 C -6 -12 0 -2 -18 D 22 -4 2 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999175 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -22 -6 B 6 0 12 4 10 C -6 -12 0 -2 -18 D 22 -4 2 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999175 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -22 -6 B 6 0 12 4 10 C -6 -12 0 -2 -18 D 22 -4 2 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999175 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4588: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (10) C D E B A (9) A B E D C (7) A B D C E (6) D C A E B (5) E C D B A (4) A B E C D (4) E B C D A (3) D C E B A (3) D C E A B (3) D C B E A (3) D A C B E (3) B E C D A (3) B A E C D (3) A D C E B (3) A D B C E (3) A B D E C (3) E C B D A (2) E B A C D (2) E A B C D (2) D C B A E (2) B E C A D (2) B E A C D (2) B C D E A (2) A E B D C (2) A E B C D (2) E B C A D (1) D C A B E (1) C E D B A (1) C D E A B (1) C D B E A (1) B A D C E (1) A E C D B (1) Total count = 100 A B C D E A 0 12 8 8 12 B -12 0 -4 -6 12 C -8 4 0 -10 14 D -8 6 10 0 18 E -12 -12 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 8 12 B -12 0 -4 -6 12 C -8 4 0 -10 14 D -8 6 10 0 18 E -12 -12 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 D=20 E=14 B=13 C=12 so C is eliminated. Round 2 votes counts: A=41 D=31 E=15 B=13 so B is eliminated. Round 3 votes counts: A=45 D=33 E=22 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:213 C:200 B:195 E:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 8 12 B -12 0 -4 -6 12 C -8 4 0 -10 14 D -8 6 10 0 18 E -12 -12 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 8 12 B -12 0 -4 -6 12 C -8 4 0 -10 14 D -8 6 10 0 18 E -12 -12 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 8 12 B -12 0 -4 -6 12 C -8 4 0 -10 14 D -8 6 10 0 18 E -12 -12 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4589: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) D B A C E (11) B D A C E (11) E C A B D (10) E B D C A (4) E B A C D (4) D C A B E (4) C A E D B (4) C A D E B (4) B E D A C (4) A C D B E (4) C A D B E (3) E D B C A (2) E B A D C (2) D A C B E (2) B D E A C (2) A C E B D (2) E B D A C (1) E B C D A (1) E B C A D (1) D E C B A (1) D C A E B (1) D B E C A (1) D B C A E (1) C E A D B (1) C D A E B (1) B E A C D (1) B D E C A (1) B A D C E (1) B A C D E (1) A D C B E (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -2 4 6 B 0 0 0 -8 -2 C 2 0 0 -2 8 D -4 8 2 0 4 E -6 2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 4 6 B 0 0 0 -8 -2 C 2 0 0 -2 8 D -4 8 2 0 4 E -6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000175 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=21 B=21 C=13 A=9 so A is eliminated. Round 2 votes counts: E=36 D=23 B=21 C=20 so C is eliminated. Round 3 votes counts: E=43 D=36 B=21 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:205 A:204 C:204 B:195 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 4 6 B 0 0 0 -8 -2 C 2 0 0 -2 8 D -4 8 2 0 4 E -6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000175 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 6 B 0 0 0 -8 -2 C 2 0 0 -2 8 D -4 8 2 0 4 E -6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000175 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 6 B 0 0 0 -8 -2 C 2 0 0 -2 8 D -4 8 2 0 4 E -6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000175 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4590: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (13) E A C B D (11) E A D C B (5) D B E C A (5) B D C A E (5) E D A B C (4) D A C B E (3) A E C B D (3) E D B A C (2) E B D C A (2) E B C D A (2) D E B C A (2) D E B A C (2) D B E A C (2) D B A E C (2) C B D A E (2) C B A D E (2) C A B D E (2) B D C E A (2) B C D A E (2) A C E D B (2) A C E B D (2) E D B C A (1) E C B A D (1) E C A B D (1) E A D B C (1) E A C D B (1) E A B C D (1) D C A B E (1) D B C E A (1) D B A C E (1) D A E C B (1) D A E B C (1) D A B C E (1) C E A B D (1) C B E A D (1) B D E C A (1) B C D E A (1) B C A D E (1) A E D C B (1) A E C D B (1) A D C B E (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 2 -18 0 B 6 0 10 -10 2 C -2 -10 0 -20 -6 D 18 10 20 0 12 E 0 -2 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -18 0 B 6 0 10 -10 2 C -2 -10 0 -20 -6 D 18 10 20 0 12 E 0 -2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=32 A=13 B=12 C=8 so C is eliminated. Round 2 votes counts: D=35 E=33 B=17 A=15 so A is eliminated. Round 3 votes counts: E=42 D=37 B=21 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:230 B:204 E:196 A:189 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -18 0 B 6 0 10 -10 2 C -2 -10 0 -20 -6 D 18 10 20 0 12 E 0 -2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -18 0 B 6 0 10 -10 2 C -2 -10 0 -20 -6 D 18 10 20 0 12 E 0 -2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -18 0 B 6 0 10 -10 2 C -2 -10 0 -20 -6 D 18 10 20 0 12 E 0 -2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4591: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) A E B D C (8) B E C A D (6) C D B A E (5) E C B A D (4) E B A C D (4) E A B C D (4) D C B A E (4) E B C A D (3) D C A E B (3) C E B A D (3) C D B E A (3) C B E D A (3) C B D E A (3) A D B E C (3) E A B D C (2) D A C E B (2) D A C B E (2) D A B E C (2) D A B C E (2) C D E A B (2) C D A E B (2) B A E D C (2) A E D B C (2) A D E C B (2) E C A B D (1) E B A D C (1) E A C B D (1) D B A C E (1) D A E C B (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E B A (1) B E A C D (1) B A D E C (1) A E D C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 10 -12 6 10 B -10 0 -12 -2 -2 C 12 12 0 -2 -2 D -6 2 2 0 0 E -10 2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999843 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 6 10 B -10 0 -12 -2 -2 C 12 12 0 -2 -2 D -6 2 2 0 0 E -10 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999992 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 E=20 A=18 B=10 so B is eliminated. Round 2 votes counts: E=27 D=27 C=25 A=21 so A is eliminated. Round 3 votes counts: E=41 D=34 C=25 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 A:207 D:199 E:197 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 6 10 B -10 0 -12 -2 -2 C 12 12 0 -2 -2 D -6 2 2 0 0 E -10 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999992 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 6 10 B -10 0 -12 -2 -2 C 12 12 0 -2 -2 D -6 2 2 0 0 E -10 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999992 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 6 10 B -10 0 -12 -2 -2 C 12 12 0 -2 -2 D -6 2 2 0 0 E -10 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999992 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4592: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) A E D C B (8) D B E C A (5) D E A C B (4) B D E A C (4) E A D B C (3) D E C A B (3) D C E A B (3) C B D A E (3) C A D E B (3) C A B E D (3) B E A D C (3) B A E D C (3) A E C D B (3) A C E B D (3) E D A B C (2) C D E A B (2) C D B E A (2) C D A E B (2) C B A D E (2) C A E D B (2) B E D A C (2) B C A E D (2) B A E C D (2) B A C E D (2) A E D B C (2) A E B D C (2) A C E D B (2) E A D C B (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A B C (1) D C E B A (1) D C B E A (1) D B C E A (1) C D E B A (1) C D B A E (1) C B D E A (1) C A B D E (1) B D E C A (1) B C D A E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 -4 -4 B -2 0 -6 -12 -2 C 2 6 0 -6 -4 D 4 12 6 0 8 E 4 2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -4 -4 B -2 0 -6 -12 -2 C 2 6 0 -6 -4 D 4 12 6 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 D=22 A=21 E=6 so E is eliminated. Round 2 votes counts: B=28 A=25 D=24 C=23 so C is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:215 E:201 C:199 A:196 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -4 -4 B -2 0 -6 -12 -2 C 2 6 0 -6 -4 D 4 12 6 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 -4 B -2 0 -6 -12 -2 C 2 6 0 -6 -4 D 4 12 6 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 -4 B -2 0 -6 -12 -2 C 2 6 0 -6 -4 D 4 12 6 0 8 E 4 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4593: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) E D A B C (6) E A B C D (6) C B D A E (6) E A D B C (5) D C B E A (5) C B D E A (5) A C B E D (5) C B A D E (4) E B D C A (3) E B A C D (3) E A B D C (3) D E B C A (3) D B C E A (3) A E C B D (3) E D B C A (2) D E A B C (2) D C B A E (2) D A E C B (2) C D B A E (2) C A B E D (2) C A B D E (2) B C D E A (2) E D B A C (1) E B D A C (1) E B C A D (1) D E C B A (1) D E C A B (1) D C E A B (1) C B A E D (1) B E C A D (1) B C E D A (1) B C A E D (1) B A C E D (1) A E D C B (1) A E B D C (1) A C E B D (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 2 2 -12 B -2 0 10 26 -8 C -2 -10 0 14 -8 D -2 -26 -14 0 -16 E 12 8 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 2 2 -12 B -2 0 10 26 -8 C -2 -10 0 14 -8 D -2 -26 -14 0 -16 E 12 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 A=21 D=20 B=6 so B is eliminated. Round 2 votes counts: E=32 C=26 A=22 D=20 so D is eliminated. Round 3 votes counts: E=39 C=37 A=24 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:213 A:197 C:197 D:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 2 -12 B -2 0 10 26 -8 C -2 -10 0 14 -8 D -2 -26 -14 0 -16 E 12 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 -12 B -2 0 10 26 -8 C -2 -10 0 14 -8 D -2 -26 -14 0 -16 E 12 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 -12 B -2 0 10 26 -8 C -2 -10 0 14 -8 D -2 -26 -14 0 -16 E 12 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4594: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) C B D E A (9) A E D B C (8) A B E C D (8) E D A B C (7) D E A C B (7) D C E B A (7) C D B E A (5) B C A E D (5) A B C E D (5) D E A B C (3) C B A D E (3) A E B D C (3) C B A E D (2) B A C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E B A D C (1) D E C A B (1) C D E B A (1) C D B A E (1) C B D A E (1) B C E D A (1) A D E C B (1) A D E B C (1) A D C E B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 0 -16 -14 B 6 0 -4 -16 -12 C 0 4 0 -10 -12 D 16 16 10 0 6 E 14 12 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -16 -14 B 6 0 -4 -16 -12 C 0 4 0 -10 -12 D 16 16 10 0 6 E 14 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 C=22 E=12 B=8 so B is eliminated. Round 2 votes counts: A=31 D=29 C=28 E=12 so E is eliminated. Round 3 votes counts: D=39 A=32 C=29 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:216 C:191 B:187 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -16 -14 B 6 0 -4 -16 -12 C 0 4 0 -10 -12 D 16 16 10 0 6 E 14 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -16 -14 B 6 0 -4 -16 -12 C 0 4 0 -10 -12 D 16 16 10 0 6 E 14 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -16 -14 B 6 0 -4 -16 -12 C 0 4 0 -10 -12 D 16 16 10 0 6 E 14 12 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4595: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (7) D E A C B (5) A C B E D (5) E B C D A (4) D A C B E (4) C B A E D (4) B E C A D (4) B C A E D (4) A D C B E (4) E D B C A (3) E D B A C (3) D E B C A (3) D A E C B (3) C B A D E (3) A D E C B (3) A C B D E (3) E D A B C (2) E B D C A (2) E B A C D (2) D C A B E (2) C D B A E (2) C B E D A (2) C B D E A (2) C B D A E (2) B C E D A (2) B C E A D (2) A E D C B (2) A E B C D (2) A D E B C (2) A B C E D (2) E B C A D (1) E A D B C (1) E A B C D (1) D E A B C (1) D C B E A (1) C A D B E (1) C A B D E (1) B E C D A (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 6 8 12 16 B -6 0 -14 2 16 C -8 14 0 16 6 D -12 -2 -16 0 -2 E -16 -16 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 12 16 B -6 0 -14 2 16 C -8 14 0 16 6 D -12 -2 -16 0 -2 E -16 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=19 D=19 C=17 B=13 so B is eliminated. Round 2 votes counts: A=32 C=25 E=24 D=19 so D is eliminated. Round 3 votes counts: A=39 E=33 C=28 so C is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:214 B:199 D:184 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 12 16 B -6 0 -14 2 16 C -8 14 0 16 6 D -12 -2 -16 0 -2 E -16 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 12 16 B -6 0 -14 2 16 C -8 14 0 16 6 D -12 -2 -16 0 -2 E -16 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 12 16 B -6 0 -14 2 16 C -8 14 0 16 6 D -12 -2 -16 0 -2 E -16 -16 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4596: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) A E B D C (6) A B D E C (6) E C B D A (5) E A C D B (5) E A C B D (4) D B C E A (4) C B D E A (4) A E B C D (4) D C B E A (3) D C B A E (3) D B A C E (3) B D C E A (3) B D A C E (3) A E C B D (3) E C A B D (2) E B A C D (2) E A B C D (2) D B C A E (2) D A C B E (2) D A B C E (2) C E D B A (2) C B E D A (2) B D C A E (2) B C D E A (2) A D B E C (2) A D B C E (2) E C D A B (1) E C B A D (1) C E B D A (1) C D E A B (1) C A E D B (1) B E C D A (1) B A D E C (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 4 -8 -6 B 4 0 -2 8 10 C -4 2 0 4 0 D 8 -8 -4 0 10 E 6 -10 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -8 -6 B 4 0 -2 8 10 C -4 2 0 4 0 D 8 -8 -4 0 10 E 6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999871 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 D=19 C=19 B=12 so B is eliminated. Round 2 votes counts: A=29 D=27 E=23 C=21 so C is eliminated. Round 3 votes counts: D=42 A=30 E=28 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:203 C:201 A:193 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 -8 -6 B 4 0 -2 8 10 C -4 2 0 4 0 D 8 -8 -4 0 10 E 6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999871 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -8 -6 B 4 0 -2 8 10 C -4 2 0 4 0 D 8 -8 -4 0 10 E 6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999871 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -8 -6 B 4 0 -2 8 10 C -4 2 0 4 0 D 8 -8 -4 0 10 E 6 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999871 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4597: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (13) D A E C B (6) B C A D E (6) E D C A B (5) B A C D E (5) D E A C B (4) B C E A D (4) B C A E D (4) A D E C B (4) A D C E B (4) E C B D A (3) E B C D A (3) C B A D E (3) B A D E C (3) E D B C A (2) E D A C B (2) C E D B A (2) C E B D A (2) C D E A B (2) C B E D A (2) B E C D A (2) A D B C E (2) A B D E C (2) E D A B C (1) E C D B A (1) E A D B C (1) D E A B C (1) D A E B C (1) D A C E B (1) C D A B E (1) C B E A D (1) C B D E A (1) C A B D E (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -16 -10 -4 B 14 0 6 12 8 C 16 -6 0 16 14 D 10 -12 -16 0 4 E 4 -8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -10 -4 B 14 0 6 12 8 C 16 -6 0 16 14 D 10 -12 -16 0 4 E 4 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=18 A=17 C=15 D=13 so D is eliminated. Round 2 votes counts: B=37 A=25 E=23 C=15 so C is eliminated. Round 3 votes counts: B=44 E=29 A=27 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:220 D:193 E:189 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -10 -4 B 14 0 6 12 8 C 16 -6 0 16 14 D 10 -12 -16 0 4 E 4 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -10 -4 B 14 0 6 12 8 C 16 -6 0 16 14 D 10 -12 -16 0 4 E 4 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -10 -4 B 14 0 6 12 8 C 16 -6 0 16 14 D 10 -12 -16 0 4 E 4 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4598: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) B E A C D (10) B E C A D (7) E A C B D (5) C D E A B (4) B D C E A (4) B A E D C (4) E B A C D (3) D B C E A (3) B D C A E (3) B D A E C (3) A E C D B (3) E C A D B (2) E C A B D (2) E A B C D (2) D C B A E (2) D C A B E (2) C D B E A (2) C D A E B (2) B E A D C (2) B D E A C (2) B D A C E (2) B A D E C (2) A C E D B (2) E C B A D (1) E A C D B (1) D B C A E (1) D B A C E (1) D A C E B (1) D A B C E (1) C E D A B (1) C E B A D (1) C E A D B (1) C A E D B (1) B D E C A (1) B C D E A (1) A E C B D (1) A E B C D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -4 6 -10 B 10 0 8 16 8 C 4 -8 0 8 -8 D -6 -16 -8 0 -2 E 10 -8 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 6 -10 B 10 0 8 16 8 C 4 -8 0 8 -8 D -6 -16 -8 0 -2 E 10 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=22 E=16 C=12 A=9 so A is eliminated. Round 2 votes counts: B=42 D=23 E=21 C=14 so C is eliminated. Round 3 votes counts: B=42 D=31 E=27 so E is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:206 C:198 A:191 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 6 -10 B 10 0 8 16 8 C 4 -8 0 8 -8 D -6 -16 -8 0 -2 E 10 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 6 -10 B 10 0 8 16 8 C 4 -8 0 8 -8 D -6 -16 -8 0 -2 E 10 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 6 -10 B 10 0 8 16 8 C 4 -8 0 8 -8 D -6 -16 -8 0 -2 E 10 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4599: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (14) D A C E B (12) D C E B A (9) B E C A D (8) D A B C E (7) A B E C D (7) A D B E C (6) E C B A D (4) E B C A D (4) D A B E C (3) A D B C E (3) A B D E C (3) D C E A B (2) D A C B E (2) C D E B A (2) B A E C D (2) A D C E B (2) A B E D C (2) D C B E A (1) D C A E B (1) D B C E A (1) D B A E C (1) C E D B A (1) C E B A D (1) B A D E C (1) A E B C D (1) Total count = 100 A B C D E A 0 2 4 -12 6 B -2 0 -2 -6 -6 C -4 2 0 -12 16 D 12 6 12 0 12 E -6 6 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -12 6 B -2 0 -2 -6 -6 C -4 2 0 -12 16 D 12 6 12 0 12 E -6 6 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=24 C=18 B=11 E=8 so E is eliminated. Round 2 votes counts: D=39 A=24 C=22 B=15 so B is eliminated. Round 3 votes counts: D=39 C=34 A=27 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:201 A:200 B:192 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -12 6 B -2 0 -2 -6 -6 C -4 2 0 -12 16 D 12 6 12 0 12 E -6 6 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -12 6 B -2 0 -2 -6 -6 C -4 2 0 -12 16 D 12 6 12 0 12 E -6 6 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -12 6 B -2 0 -2 -6 -6 C -4 2 0 -12 16 D 12 6 12 0 12 E -6 6 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4600: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (6) D E A C B (5) B E C A D (5) B D A C E (5) B C A E D (5) E C D A B (4) E C B A D (4) E C A B D (4) E D C A B (3) E C A D B (3) D E C A B (3) D A C B E (3) C A E B D (3) B D E C A (3) B D E A C (3) B A C E D (3) B A C D E (3) E D B C A (2) E B C D A (2) D B A C E (2) D A E C B (2) D A B C E (2) C E A B D (2) B E D C A (2) B C E A D (2) A C D E B (2) A C B E D (2) A C B D E (2) E B D C A (1) E B C A D (1) D E B C A (1) D E B A C (1) D B A E C (1) C E A D B (1) C A E D B (1) B D A E C (1) B A D C E (1) A D C B E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -4 -4 -4 B -4 0 -4 10 -2 C 4 4 0 0 -2 D 4 -10 0 0 0 E 4 2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.105765 E: 0.894235 Sum of squares = 0.810841971974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.105765 E: 1.000000 A B C D E A 0 4 -4 -4 -4 B -4 0 -4 10 -2 C 4 4 0 0 -2 D 4 -10 0 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222247021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=26 E=24 A=10 C=7 so C is eliminated. Round 2 votes counts: B=33 E=27 D=26 A=14 so A is eliminated. Round 3 votes counts: B=38 E=31 D=31 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:204 C:203 B:200 D:197 A:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 -4 -4 B -4 0 -4 10 -2 C 4 4 0 0 -2 D 4 -10 0 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222247021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -4 -4 B -4 0 -4 10 -2 C 4 4 0 0 -2 D 4 -10 0 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222247021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -4 -4 B -4 0 -4 10 -2 C 4 4 0 0 -2 D 4 -10 0 0 0 E 4 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222247021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4601: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (8) E A C B D (7) E A B C D (6) D C A E B (6) A C E D B (6) D B C A E (5) C D A E B (5) B E A D C (5) D C B A E (4) D B C E A (4) A E C B D (4) E B A C D (3) D C A B E (3) C A E D B (3) B E A C D (3) B D A E C (3) B D A C E (3) D C B E A (2) D B E C A (2) C E A D B (2) C A D E B (2) B E D A C (2) B D E C A (2) A E B C D (2) E C A D B (1) E C A B D (1) D E C B A (1) D C E B A (1) D C E A B (1) D B A C E (1) C D E A B (1) B A E C D (1) Total count = 100 A B C D E A 0 0 8 -8 -4 B 0 0 0 0 -4 C -8 0 0 -6 -2 D 8 0 6 0 8 E 4 4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.487652 C: 0.000000 D: 0.512348 E: 0.000000 Sum of squares = 0.500304925019 Cumulative probabilities = A: 0.000000 B: 0.487652 C: 0.487652 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -8 -4 B 0 0 0 0 -4 C -8 0 0 -6 -2 D 8 0 6 0 8 E 4 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=27 E=18 C=13 A=12 so A is eliminated. Round 2 votes counts: D=30 B=27 E=24 C=19 so C is eliminated. Round 3 votes counts: D=38 E=35 B=27 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:201 A:198 B:198 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 8 -8 -4 B 0 0 0 0 -4 C -8 0 0 -6 -2 D 8 0 6 0 8 E 4 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -8 -4 B 0 0 0 0 -4 C -8 0 0 -6 -2 D 8 0 6 0 8 E 4 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -8 -4 B 0 0 0 0 -4 C -8 0 0 -6 -2 D 8 0 6 0 8 E 4 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4602: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (20) B D A C E (19) B E C A D (9) C E A D B (8) B D A E C (8) D B A C E (4) C A E D B (4) D A E C B (3) B D E A C (3) A C D E B (3) E C A B D (2) D A C E B (2) D A B C E (2) B C E A D (2) E D A C B (1) E C D A B (1) D A E B C (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C A E (1) B C A E D (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -4 4 2 B 2 0 8 0 8 C 4 -8 0 8 -2 D -4 0 -8 0 -4 E -2 -8 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.741411 C: 0.000000 D: 0.258589 E: 0.000000 Sum of squares = 0.616558777373 Cumulative probabilities = A: 0.000000 B: 0.741411 C: 0.741411 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 4 2 B 2 0 8 0 8 C 4 -8 0 8 -2 D -4 0 -8 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.55555555595 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=47 E=24 C=13 D=12 A=4 so A is eliminated. Round 2 votes counts: B=47 E=24 C=17 D=12 so D is eliminated. Round 3 votes counts: B=53 E=28 C=19 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:201 A:200 E:198 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 4 2 B 2 0 8 0 8 C 4 -8 0 8 -2 D -4 0 -8 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.55555555595 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 2 B 2 0 8 0 8 C 4 -8 0 8 -2 D -4 0 -8 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.55555555595 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 2 B 2 0 8 0 8 C 4 -8 0 8 -2 D -4 0 -8 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.55555555595 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4603: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C B A D E (8) C B A E D (6) B A D C E (6) C B D A E (5) E A D B C (4) E D A C B (3) E C D B A (3) E C D A B (3) E C A B D (3) D E A B C (3) D B A C E (3) D A B E C (3) C E D B A (3) C E B D A (3) C B E A D (3) B A C D E (3) A E D B C (3) E D C A B (2) E A B C D (2) D A E B C (2) C E B A D (2) B D A C E (2) A B D E C (2) E C B A D (1) E C A D B (1) E A D C B (1) E A C D B (1) E A C B D (1) E A B D C (1) D C E B A (1) D C B A E (1) D B C A E (1) D A B C E (1) C B E D A (1) B C D A E (1) B A C E D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 4 2 0 B 8 0 -4 6 -4 C -4 4 0 4 2 D -2 -6 -4 0 -16 E 0 4 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.267070 B: 0.198790 C: 0.397579 D: 0.000000 E: 0.136561 Sum of squares = 0.287561935104 Cumulative probabilities = A: 0.267070 B: 0.465860 C: 0.863439 D: 0.863439 E: 1.000000 A B C D E A 0 -8 4 2 0 B 8 0 -4 6 -4 C -4 4 0 4 2 D -2 -6 -4 0 -16 E 0 4 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.281818 B: 0.154545 C: 0.309091 D: 0.000000 E: 0.254545 Sum of squares = 0.263636363636 Cumulative probabilities = A: 0.281818 B: 0.436364 C: 0.745455 D: 0.745455 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=31 D=15 B=13 A=7 so A is eliminated. Round 2 votes counts: E=38 C=31 B=16 D=15 so D is eliminated. Round 3 votes counts: E=43 C=33 B=24 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:209 B:203 C:203 A:199 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 2 0 B 8 0 -4 6 -4 C -4 4 0 4 2 D -2 -6 -4 0 -16 E 0 4 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.281818 B: 0.154545 C: 0.309091 D: 0.000000 E: 0.254545 Sum of squares = 0.263636363636 Cumulative probabilities = A: 0.281818 B: 0.436364 C: 0.745455 D: 0.745455 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 2 0 B 8 0 -4 6 -4 C -4 4 0 4 2 D -2 -6 -4 0 -16 E 0 4 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.281818 B: 0.154545 C: 0.309091 D: 0.000000 E: 0.254545 Sum of squares = 0.263636363636 Cumulative probabilities = A: 0.281818 B: 0.436364 C: 0.745455 D: 0.745455 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 2 0 B 8 0 -4 6 -4 C -4 4 0 4 2 D -2 -6 -4 0 -16 E 0 4 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.281818 B: 0.154545 C: 0.309091 D: 0.000000 E: 0.254545 Sum of squares = 0.263636363636 Cumulative probabilities = A: 0.281818 B: 0.436364 C: 0.745455 D: 0.745455 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4604: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (11) A D C B E (10) C B A D E (9) B C E D A (9) A D E C B (8) E B C D A (6) D E A B C (6) A C B D E (6) A D E B C (5) E D A B C (3) D A E B C (3) C B D E A (3) C B A E D (3) E D B A C (2) E B C A D (2) C B E A D (2) C B D A E (2) B E C D A (2) B C E A D (2) A C D B E (2) E B A C D (1) E A D B C (1) D E B C A (1) C A B D E (1) Total count = 100 A B C D E A 0 -10 -6 4 -2 B 10 0 -14 18 24 C 6 14 0 22 20 D -4 -18 -22 0 12 E 2 -24 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 4 -2 B 10 0 -14 18 24 C 6 14 0 22 20 D -4 -18 -22 0 12 E 2 -24 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=31 A=31 E=15 B=13 D=10 so D is eliminated. Round 2 votes counts: A=34 C=31 E=22 B=13 so B is eliminated. Round 3 votes counts: C=42 A=34 E=24 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:231 B:219 A:193 D:184 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 4 -2 B 10 0 -14 18 24 C 6 14 0 22 20 D -4 -18 -22 0 12 E 2 -24 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 4 -2 B 10 0 -14 18 24 C 6 14 0 22 20 D -4 -18 -22 0 12 E 2 -24 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 4 -2 B 10 0 -14 18 24 C 6 14 0 22 20 D -4 -18 -22 0 12 E 2 -24 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4605: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) E B C D A (11) B D A E C (11) C A D E B (10) D A B E C (9) A D C B E (9) B E D A C (8) E C B A D (7) D A B C E (6) C E A B D (4) E C B D A (3) C E B A D (3) B E D C A (2) A D C E B (2) A D B C E (2) A C D B E (1) Total count = 100 A B C D E A 0 10 -4 0 0 B -10 0 -2 -2 -4 C 4 2 0 2 -2 D 0 2 -2 0 0 E 0 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.114194 B: 0.000000 C: 0.000000 D: 0.280882 E: 0.604924 Sum of squares = 0.457867898302 Cumulative probabilities = A: 0.114194 B: 0.114194 C: 0.114194 D: 0.395076 E: 1.000000 A B C D E A 0 10 -4 0 0 B -10 0 -2 -2 -4 C 4 2 0 2 -2 D 0 2 -2 0 0 E 0 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.571429 Sum of squares = 0.428571484351 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.428571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=21 B=21 D=15 A=14 so A is eliminated. Round 2 votes counts: C=30 D=28 E=21 B=21 so E is eliminated. Round 3 votes counts: C=40 B=32 D=28 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:203 C:203 E:203 D:200 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -4 0 0 B -10 0 -2 -2 -4 C 4 2 0 2 -2 D 0 2 -2 0 0 E 0 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.571429 Sum of squares = 0.428571484351 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.428571 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 0 0 B -10 0 -2 -2 -4 C 4 2 0 2 -2 D 0 2 -2 0 0 E 0 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.571429 Sum of squares = 0.428571484351 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 0 0 B -10 0 -2 -2 -4 C 4 2 0 2 -2 D 0 2 -2 0 0 E 0 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.571429 Sum of squares = 0.428571484351 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4606: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) A C B E D (7) C A E B D (6) E C D A B (5) A B C E D (5) D E B C A (4) D C B A E (4) D B E A C (4) B A C D E (4) E C A D B (3) E A C B D (3) D B C A E (3) D B A C E (3) B D A C E (3) B A D C E (3) D B A E C (2) C E D A B (2) C D E A B (2) C A B E D (2) B A E D C (2) B A D E C (2) A E B C D (2) A C B D E (2) A B E C D (2) E D C A B (1) E D B A C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E B A C (1) D C B E A (1) D B E C A (1) C E A D B (1) C E A B D (1) C A B D E (1) B C A D E (1) B A E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 18 0 24 12 B -18 0 -6 22 8 C 0 6 0 26 4 D -24 -22 -26 0 -14 E -12 -8 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.751252 B: 0.000000 C: 0.248748 D: 0.000000 E: 0.000000 Sum of squares = 0.62625539062 Cumulative probabilities = A: 0.751252 B: 0.751252 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 24 12 B -18 0 -6 22 8 C 0 6 0 26 4 D -24 -22 -26 0 -14 E -12 -8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 A=19 B=16 C=15 so C is eliminated. Round 2 votes counts: E=30 A=28 D=26 B=16 so B is eliminated. Round 3 votes counts: A=41 E=30 D=29 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 C:218 B:203 E:195 D:157 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 24 12 B -18 0 -6 22 8 C 0 6 0 26 4 D -24 -22 -26 0 -14 E -12 -8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 24 12 B -18 0 -6 22 8 C 0 6 0 26 4 D -24 -22 -26 0 -14 E -12 -8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 24 12 B -18 0 -6 22 8 C 0 6 0 26 4 D -24 -22 -26 0 -14 E -12 -8 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4607: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) C E D B A (7) B A D C E (7) A D B E C (7) E C D A B (6) E C D B A (5) E A C D B (5) A B D C E (5) C E B D A (3) C D B E A (3) B C D A E (3) A E D B C (3) E D C A B (2) E C B A D (2) E C A D B (2) E A D C B (2) C D E B A (2) C B D E A (2) A E B D C (2) A D E B C (2) A D B C E (2) E C B D A (1) E C A B D (1) E B C A D (1) D E C A B (1) D E A C B (1) D C E B A (1) D C E A B (1) D A E C B (1) D A B C E (1) C E B A D (1) C B E D A (1) B E C A D (1) B D A C E (1) B C E A D (1) B C A D E (1) B A E C D (1) B A C D E (1) A D E C B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 4 16 -4 B -10 0 -2 -10 -6 C -4 2 0 2 -14 D -16 10 -2 0 2 E 4 6 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.000000 D: 0.181818 E: 0.727273 Sum of squares = 0.570247933883 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.090909 D: 0.272727 E: 1.000000 A B C D E A 0 10 4 16 -4 B -10 0 -2 -10 -6 C -4 2 0 2 -14 D -16 10 -2 0 2 E 4 6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.000000 D: 0.181818 E: 0.727273 Sum of squares = 0.570247933848 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.090909 D: 0.272727 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=27 C=19 B=16 D=6 so D is eliminated. Round 2 votes counts: A=34 E=29 C=21 B=16 so B is eliminated. Round 3 votes counts: A=44 E=30 C=26 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:213 E:211 D:197 C:193 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 4 16 -4 B -10 0 -2 -10 -6 C -4 2 0 2 -14 D -16 10 -2 0 2 E 4 6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.000000 D: 0.181818 E: 0.727273 Sum of squares = 0.570247933848 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.090909 D: 0.272727 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 16 -4 B -10 0 -2 -10 -6 C -4 2 0 2 -14 D -16 10 -2 0 2 E 4 6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.000000 D: 0.181818 E: 0.727273 Sum of squares = 0.570247933848 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.090909 D: 0.272727 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 16 -4 B -10 0 -2 -10 -6 C -4 2 0 2 -14 D -16 10 -2 0 2 E 4 6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.000000 D: 0.181818 E: 0.727273 Sum of squares = 0.570247933848 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.090909 D: 0.272727 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4608: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) A E C D B (11) D B C A E (10) E A C D B (8) E A B C D (6) C B D A E (5) B D C A E (5) B C D E A (5) E A D B C (4) E A C B D (4) D C B A E (3) E B A D C (2) C D B A E (2) C A D E B (2) B D E A C (2) B C D A E (2) A E D B C (2) E C A B D (1) E A D C B (1) E A B D C (1) D C A E B (1) D B A E C (1) D A E C B (1) D A C E B (1) D A B C E (1) C B E A D (1) C B A E D (1) C A E B D (1) B E D A C (1) A E D C B (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 0 -2 6 B 2 0 6 -4 0 C 0 -6 0 2 6 D 2 4 -2 0 8 E -6 0 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -2 6 B 2 0 6 -4 0 C 0 -6 0 2 6 D 2 4 -2 0 8 E -6 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888827 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 D=18 A=17 C=12 so C is eliminated. Round 2 votes counts: B=33 E=27 D=20 A=20 so D is eliminated. Round 3 votes counts: B=49 E=27 A=24 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:206 B:202 A:201 C:201 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -2 6 B 2 0 6 -4 0 C 0 -6 0 2 6 D 2 4 -2 0 8 E -6 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888827 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -2 6 B 2 0 6 -4 0 C 0 -6 0 2 6 D 2 4 -2 0 8 E -6 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888827 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -2 6 B 2 0 6 -4 0 C 0 -6 0 2 6 D 2 4 -2 0 8 E -6 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888827 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4609: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (6) E C B A D (5) D B A E C (5) C E A B D (5) B C E A D (5) B A D C E (5) E C A D B (4) D E A C B (4) B D E C A (4) A C E B D (4) E D C A B (3) D E C A B (3) D E B C A (3) D A E C B (3) D A B C E (3) C A E B D (3) D E C B A (2) D B E C A (2) D A B E C (2) B E C A D (2) B D A C E (2) B C A E D (2) A D C E B (2) A C B E D (2) E D C B A (1) E D B C A (1) E C D B A (1) E C D A B (1) E C A B D (1) D E A B C (1) D B E A C (1) D B A C E (1) D A E B C (1) D A C B E (1) C E B A D (1) C B E A D (1) C A E D B (1) B E D C A (1) B D E A C (1) B D A E C (1) B A C D E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 4 -6 B 8 0 4 6 0 C 4 -4 0 -8 -6 D -4 -6 8 0 0 E 6 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.551576 C: 0.000000 D: 0.000000 E: 0.448424 Sum of squares = 0.505320174512 Cumulative probabilities = A: 0.000000 B: 0.551576 C: 0.551576 D: 0.551576 E: 1.000000 A B C D E A 0 -8 -4 4 -6 B 8 0 4 6 0 C 4 -4 0 -8 -6 D -4 -6 8 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=30 E=17 C=11 A=10 so A is eliminated. Round 2 votes counts: D=35 B=31 E=17 C=17 so E is eliminated. Round 3 votes counts: D=40 B=31 C=29 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:206 D:199 A:193 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 4 -6 B 8 0 4 6 0 C 4 -4 0 -8 -6 D -4 -6 8 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 4 -6 B 8 0 4 6 0 C 4 -4 0 -8 -6 D -4 -6 8 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 4 -6 B 8 0 4 6 0 C 4 -4 0 -8 -6 D -4 -6 8 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4610: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) E B C A D (8) B E C A D (7) C A D E B (6) B E D A C (6) D A C B E (5) C A E D B (5) B D E A C (4) A D C E B (4) A C D E B (4) E C B A D (3) C E A D B (3) B E D C A (3) E C A B D (2) E B A C D (2) C E B A D (2) C D A E B (2) C D A B E (2) C B E A D (2) B E C D A (2) B E A D C (2) E D B A C (1) E B A D C (1) E A C B D (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A B E C (1) D A B C E (1) B D C A E (1) B C E D A (1) B C D A E (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -6 6 0 B 0 0 -10 -4 -14 C 6 10 0 6 6 D -6 4 -6 0 -2 E 0 14 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 6 0 B 0 0 -10 -4 -14 C 6 10 0 6 6 D -6 4 -6 0 -2 E 0 14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=24 C=22 E=18 A=9 so A is eliminated. Round 2 votes counts: D=29 B=27 C=26 E=18 so E is eliminated. Round 3 votes counts: B=38 C=32 D=30 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:205 A:200 D:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 6 0 B 0 0 -10 -4 -14 C 6 10 0 6 6 D -6 4 -6 0 -2 E 0 14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 6 0 B 0 0 -10 -4 -14 C 6 10 0 6 6 D -6 4 -6 0 -2 E 0 14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 6 0 B 0 0 -10 -4 -14 C 6 10 0 6 6 D -6 4 -6 0 -2 E 0 14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4611: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) A B D C E (8) A D B E C (7) D A B E C (5) C A B D E (5) E D B A C (4) E D A B C (4) A C B D E (4) E C D B A (3) C B E D A (3) B D E A C (3) B D A E C (3) A B D E C (3) E C D A B (2) E B D C A (2) C E D A B (2) C E A D B (2) C B E A D (2) C A E D B (2) B E D C A (2) B A D E C (2) A D B C E (2) E D C B A (1) E D B C A (1) E C B D A (1) E B D A C (1) D E B A C (1) D E A B C (1) D B A E C (1) D A E B C (1) C E D B A (1) C E B A D (1) C B D E A (1) C B A E D (1) C A E B D (1) C A D B E (1) B E C D A (1) B C E D A (1) B C D E A (1) B C D A E (1) A E C D B (1) A D E B C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 8 -10 2 B -8 0 14 12 16 C -8 -14 0 -6 -2 D 10 -12 6 0 6 E -2 -16 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.333333 C: 0.000000 D: 0.266667 E: 0.000000 Sum of squares = 0.342222222223 Cumulative probabilities = A: 0.400000 B: 0.733333 C: 0.733333 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 -10 2 B -8 0 14 12 16 C -8 -14 0 -6 -2 D 10 -12 6 0 6 E -2 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.333333 C: 0.000000 D: 0.266667 E: 0.000000 Sum of squares = 0.342222222208 Cumulative probabilities = A: 0.400000 B: 0.733333 C: 0.733333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=28 E=19 B=14 D=9 so D is eliminated. Round 2 votes counts: A=34 C=30 E=21 B=15 so B is eliminated. Round 3 votes counts: A=40 C=33 E=27 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:217 D:205 A:204 E:189 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 8 8 -10 2 B -8 0 14 12 16 C -8 -14 0 -6 -2 D 10 -12 6 0 6 E -2 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.333333 C: 0.000000 D: 0.266667 E: 0.000000 Sum of squares = 0.342222222208 Cumulative probabilities = A: 0.400000 B: 0.733333 C: 0.733333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -10 2 B -8 0 14 12 16 C -8 -14 0 -6 -2 D 10 -12 6 0 6 E -2 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.333333 C: 0.000000 D: 0.266667 E: 0.000000 Sum of squares = 0.342222222208 Cumulative probabilities = A: 0.400000 B: 0.733333 C: 0.733333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -10 2 B -8 0 14 12 16 C -8 -14 0 -6 -2 D 10 -12 6 0 6 E -2 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.333333 C: 0.000000 D: 0.266667 E: 0.000000 Sum of squares = 0.342222222208 Cumulative probabilities = A: 0.400000 B: 0.733333 C: 0.733333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4612: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (14) A D C E B (10) B E A D C (8) C B E D A (7) C A D B E (4) B E D A C (4) E B D A C (3) B C E D A (3) A D E B C (3) A C D B E (3) E D B A C (2) E B D C A (2) D E A B C (2) D C A E B (2) D A E B C (2) C E D B A (2) C D E A B (2) C D A E B (2) C A B D E (2) B C E A D (2) B A E C D (2) A D E C B (2) A D B E C (2) A C B D E (2) A B D E C (2) A B C D E (2) D C E A B (1) D A C E B (1) C E B D A (1) C B A E D (1) C A D E B (1) B E D C A (1) B A E D C (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 6 -2 -8 B 6 0 12 16 22 C -6 -12 0 2 -2 D 2 -16 -2 0 -8 E 8 -22 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -2 -8 B 6 0 12 16 22 C -6 -12 0 2 -2 D 2 -16 -2 0 -8 E 8 -22 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=28 C=22 D=8 E=7 so E is eliminated. Round 2 votes counts: B=40 A=28 C=22 D=10 so D is eliminated. Round 3 votes counts: B=42 A=33 C=25 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:228 E:198 A:195 C:191 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -2 -8 B 6 0 12 16 22 C -6 -12 0 2 -2 D 2 -16 -2 0 -8 E 8 -22 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -2 -8 B 6 0 12 16 22 C -6 -12 0 2 -2 D 2 -16 -2 0 -8 E 8 -22 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -2 -8 B 6 0 12 16 22 C -6 -12 0 2 -2 D 2 -16 -2 0 -8 E 8 -22 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4613: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (16) E C D A B (9) A B D E C (7) A B E C D (6) B D C E A (5) A B E D C (5) B C D E A (4) A E C D B (4) D C E B A (3) D C B E A (3) C E D B A (3) B D A C E (3) A E D C B (3) A E C B D (3) E D C A B (2) E C D B A (2) C D E B A (2) B A C D E (2) A E B C D (2) A B D C E (2) A B C E D (2) A B C D E (2) E A D C B (1) E A C D B (1) D E C B A (1) D E B C A (1) C E D A B (1) C B D E A (1) C A B E D (1) B D C A E (1) B A D E C (1) A E D B C (1) Total count = 100 A B C D E A 0 4 22 18 22 B -4 0 20 26 22 C -22 -20 0 -10 2 D -18 -26 10 0 8 E -22 -22 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 22 18 22 B -4 0 20 26 22 C -22 -20 0 -10 2 D -18 -26 10 0 8 E -22 -22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=32 E=15 D=8 C=8 so D is eliminated. Round 2 votes counts: A=37 B=32 E=17 C=14 so C is eliminated. Round 3 votes counts: A=38 B=36 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 B:232 D:187 C:175 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 22 18 22 B -4 0 20 26 22 C -22 -20 0 -10 2 D -18 -26 10 0 8 E -22 -22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 22 18 22 B -4 0 20 26 22 C -22 -20 0 -10 2 D -18 -26 10 0 8 E -22 -22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 22 18 22 B -4 0 20 26 22 C -22 -20 0 -10 2 D -18 -26 10 0 8 E -22 -22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4614: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) A D B C E (7) C A E D B (6) C E A D B (5) B D C A E (5) E D B A C (4) E B D C A (4) E B D A C (4) B D A C E (4) E C A B D (3) E A D B C (3) E A C D B (3) D B A E C (3) C B D A E (3) C A D B E (3) B D E A C (3) A C D B E (3) E C B D A (2) E B C D A (2) D B A C E (2) D A B C E (2) C E A B D (2) C A E B D (2) B D E C A (2) B D A E C (2) A E C D B (2) E C B A D (1) E A D C B (1) C B A D E (1) B E D A C (1) B C D E A (1) A E D B C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 12 -6 12 -4 B -12 0 2 -16 -14 C 6 -2 0 2 -6 D -12 16 -2 0 -14 E 4 14 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -6 12 -4 B -12 0 2 -16 -14 C 6 -2 0 2 -6 D -12 16 -2 0 -14 E 4 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=22 B=18 A=15 D=7 so D is eliminated. Round 2 votes counts: E=38 B=23 C=22 A=17 so A is eliminated. Round 3 votes counts: E=41 B=33 C=26 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:207 C:200 D:194 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -6 12 -4 B -12 0 2 -16 -14 C 6 -2 0 2 -6 D -12 16 -2 0 -14 E 4 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 12 -4 B -12 0 2 -16 -14 C 6 -2 0 2 -6 D -12 16 -2 0 -14 E 4 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 12 -4 B -12 0 2 -16 -14 C 6 -2 0 2 -6 D -12 16 -2 0 -14 E 4 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4615: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (17) B C D A E (11) B A E D C (9) C D B E A (7) A E B D C (7) C D E A B (6) D C E A B (4) B A E C D (4) B E A C D (3) E D A C B (2) E A B D C (2) D C A E B (2) C D B A E (2) C B D E A (2) B A C D E (2) A E D B C (2) A B E D C (2) E D C A B (1) E A B C D (1) D E C A B (1) D E A C B (1) D B A C E (1) D A C E B (1) C D E B A (1) C D A B E (1) C B D A E (1) B E A D C (1) B C E A D (1) B A D E C (1) B A D C E (1) B A C E D (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 4 20 12 0 B -4 0 -2 -2 0 C -20 2 0 -14 -12 D -12 2 14 0 -8 E 0 0 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.210051 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.789949 Sum of squares = 0.668140956706 Cumulative probabilities = A: 0.210051 B: 0.210051 C: 0.210051 D: 0.210051 E: 1.000000 A B C D E A 0 4 20 12 0 B -4 0 -2 -2 0 C -20 2 0 -14 -12 D -12 2 14 0 -8 E 0 0 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=23 C=20 A=13 D=10 so D is eliminated. Round 2 votes counts: B=35 C=26 E=25 A=14 so A is eliminated. Round 3 votes counts: B=37 E=36 C=27 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:218 E:210 D:198 B:196 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 12 0 B -4 0 -2 -2 0 C -20 2 0 -14 -12 D -12 2 14 0 -8 E 0 0 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 12 0 B -4 0 -2 -2 0 C -20 2 0 -14 -12 D -12 2 14 0 -8 E 0 0 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 12 0 B -4 0 -2 -2 0 C -20 2 0 -14 -12 D -12 2 14 0 -8 E 0 0 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4616: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) E C A B D (12) D B A C E (12) B A D E C (7) E B A D C (6) E C D B A (5) E A B C D (4) C D E A B (4) C D B A E (4) E C B A D (3) E A B D C (3) C E A B D (3) B A E D C (3) B A D C E (3) E C D A B (2) D A B C E (2) C D A B E (2) A B E D C (2) E B A C D (1) E A C B D (1) D E C B A (1) D C E B A (1) D C B A E (1) D C A B E (1) D B A E C (1) C E D B A (1) C D E B A (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -2 -22 B 0 0 -10 -2 -22 C 8 10 0 14 -4 D 2 2 -14 0 -20 E 22 22 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -8 -2 -22 B 0 0 -10 -2 -22 C 8 10 0 14 -4 D 2 2 -14 0 -20 E 22 22 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=28 D=19 B=13 A=3 so A is eliminated. Round 2 votes counts: E=37 C=28 D=19 B=16 so B is eliminated. Round 3 votes counts: E=43 D=29 C=28 so C is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:234 C:214 D:185 A:184 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -2 -22 B 0 0 -10 -2 -22 C 8 10 0 14 -4 D 2 2 -14 0 -20 E 22 22 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 -22 B 0 0 -10 -2 -22 C 8 10 0 14 -4 D 2 2 -14 0 -20 E 22 22 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 -22 B 0 0 -10 -2 -22 C 8 10 0 14 -4 D 2 2 -14 0 -20 E 22 22 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4617: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) E A C D B (10) E A C B D (9) E A D C B (7) B D C A E (7) B C D A E (7) E D A C B (5) B C A E D (5) D B E A C (4) D B C E A (4) C A E B D (4) A E C B D (4) D B E C A (3) C B A E D (3) D E A C B (2) D B A C E (2) D A E B C (2) C A B E D (2) A C E B D (2) E C A B D (1) D E B A C (1) D E A B C (1) C E A B D (1) B C A D E (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 2 2 4 B -4 0 -4 -6 0 C -2 4 0 2 0 D -2 6 -2 0 -10 E -4 0 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 4 B -4 0 -4 -6 0 C -2 4 0 2 0 D -2 6 -2 0 -10 E -4 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=30 B=20 C=10 A=8 so A is eliminated. Round 2 votes counts: E=37 D=30 B=20 C=13 so C is eliminated. Round 3 votes counts: E=44 D=30 B=26 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:206 E:203 C:202 D:196 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 4 B -4 0 -4 -6 0 C -2 4 0 2 0 D -2 6 -2 0 -10 E -4 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 4 B -4 0 -4 -6 0 C -2 4 0 2 0 D -2 6 -2 0 -10 E -4 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 4 B -4 0 -4 -6 0 C -2 4 0 2 0 D -2 6 -2 0 -10 E -4 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4618: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) B E A D C (10) C D B A E (6) C D A E B (6) B D C E A (6) B E A C D (5) E A D B C (4) A E D B C (4) E A C B D (3) C B D A E (3) C A E D B (3) A E C D B (3) E A B C D (2) D C B A E (2) D C A E B (2) D B C A E (2) D A E B C (2) C B E A D (2) C B D E A (2) C B A E D (2) B E D A C (2) B D E A C (2) A E D C B (2) E B A D C (1) D E A B C (1) D B E A C (1) D B C E A (1) D B A E C (1) D A E C B (1) D A C E B (1) C A E B D (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D E A (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 16 12 -14 B 4 0 18 14 2 C -16 -18 0 -16 -18 D -12 -14 16 0 -16 E 14 -2 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999054 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 12 -14 B 4 0 18 14 2 C -16 -18 0 -16 -18 D -12 -14 16 0 -16 E 14 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=25 E=21 D=14 A=11 so A is eliminated. Round 2 votes counts: E=31 B=29 C=25 D=15 so D is eliminated. Round 3 votes counts: E=36 B=34 C=30 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:223 B:219 A:205 D:187 C:166 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 16 12 -14 B 4 0 18 14 2 C -16 -18 0 -16 -18 D -12 -14 16 0 -16 E 14 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 12 -14 B 4 0 18 14 2 C -16 -18 0 -16 -18 D -12 -14 16 0 -16 E 14 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 12 -14 B 4 0 18 14 2 C -16 -18 0 -16 -18 D -12 -14 16 0 -16 E 14 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4619: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (8) E D B C A (7) E D B A C (6) C A B D E (6) E B A D C (5) A C B D E (5) A B C D E (5) E D C B A (4) B E A D C (4) B A C E D (4) A B C E D (4) E B D A C (3) D E C B A (3) D C A E B (3) C D E B A (3) C D A B E (3) D E C A B (2) C D A E B (2) B C A E D (2) E B C D A (1) E B C A D (1) E B A C D (1) D A C E B (1) C E B D A (1) C B E A D (1) C A D B E (1) B E C A D (1) B E A C D (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -2 2 -8 B 12 0 8 8 -6 C 2 -8 0 2 4 D -2 -8 -2 0 -8 E 8 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691358 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 A B C D E A 0 -12 -2 2 -8 B 12 0 8 8 -6 C 2 -8 0 2 4 D -2 -8 -2 0 -8 E 8 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=20 D=19 C=17 A=16 so A is eliminated. Round 2 votes counts: B=30 E=28 C=23 D=19 so D is eliminated. Round 3 votes counts: C=37 E=33 B=30 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:211 E:209 C:200 A:190 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -2 2 -8 B 12 0 8 8 -6 C 2 -8 0 2 4 D -2 -8 -2 0 -8 E 8 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 2 -8 B 12 0 8 8 -6 C 2 -8 0 2 4 D -2 -8 -2 0 -8 E 8 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 2 -8 B 12 0 8 8 -6 C 2 -8 0 2 4 D -2 -8 -2 0 -8 E 8 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691355 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4620: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (16) D B C A E (11) B D E A C (8) C A E D B (6) A E C D B (6) A C E D B (6) E A B C D (5) B D C A E (5) E B A D C (4) D C B A E (4) C A D E B (4) B D C E A (4) E C A B D (3) E A C D B (3) E B A C D (2) C E A D B (2) C D A B E (2) E C A D B (1) E B C A D (1) D B A C E (1) C D A E B (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E C A (1) A D C E B (1) Total count = 100 A B C D E A 0 12 8 22 -6 B -12 0 -10 4 -22 C -8 10 0 16 -6 D -22 -4 -16 0 -16 E 6 22 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999518 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 8 22 -6 B -12 0 -10 4 -22 C -8 10 0 16 -6 D -22 -4 -16 0 -16 E 6 22 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999122 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=21 D=16 C=15 A=13 so A is eliminated. Round 2 votes counts: E=41 C=21 B=21 D=17 so D is eliminated. Round 3 votes counts: E=41 B=33 C=26 so C is eliminated. Round 4 votes counts: E=61 B=39 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:218 C:206 B:180 D:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 8 22 -6 B -12 0 -10 4 -22 C -8 10 0 16 -6 D -22 -4 -16 0 -16 E 6 22 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999122 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 22 -6 B -12 0 -10 4 -22 C -8 10 0 16 -6 D -22 -4 -16 0 -16 E 6 22 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999122 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 22 -6 B -12 0 -10 4 -22 C -8 10 0 16 -6 D -22 -4 -16 0 -16 E 6 22 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999122 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4621: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) E B C D A (5) E B A D C (4) C D A B E (4) C B E D A (4) E C B A D (3) D C B E A (3) D B A C E (3) C E A D B (3) C D B E A (3) C D B A E (3) B D E C A (3) B D E A C (3) A D C B E (3) E C A B D (2) E A B C D (2) D C B A E (2) D C A B E (2) D B C E A (2) D B C A E (2) D B A E C (2) D A B C E (2) C D E A B (2) C A E D B (2) C A D E B (2) B E D A C (2) B D A E C (2) A E B D C (2) A E B C D (2) A B D E C (2) E C B D A (1) E B C A D (1) D A B E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C A D B E (1) B E D C A (1) B D C E A (1) B A D E C (1) A D E B C (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 2 12 C 24 6 0 6 16 D 10 -2 -6 0 8 E 10 -12 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 2 12 C 24 6 0 6 16 D 10 -2 -6 0 8 E 10 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=19 E=18 A=16 B=13 so B is eliminated. Round 2 votes counts: C=34 D=28 E=21 A=17 so A is eliminated. Round 3 votes counts: D=37 C=37 E=26 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:207 D:205 E:187 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 2 12 C 24 6 0 6 16 D 10 -2 -6 0 8 E 10 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 2 12 C 24 6 0 6 16 D 10 -2 -6 0 8 E 10 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 2 12 C 24 6 0 6 16 D 10 -2 -6 0 8 E 10 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4622: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) D B E A C (7) D C A B E (6) C D A B E (6) C A D E B (6) B E D A C (5) A C D E B (5) E B C A D (4) D B A E C (4) D A C B E (4) B E D C A (4) E B C D A (3) D C B A E (3) C A E B D (3) A D C B E (3) E C A B D (2) E A C B D (2) D B A C E (2) C A E D B (2) B D E A C (2) A D C E B (2) A C E B D (2) E B A D C (1) E B A C D (1) E A B D C (1) D B E C A (1) D B C E A (1) D B C A E (1) D A B E C (1) C E B A D (1) C B E D A (1) B E C D A (1) B D E C A (1) A E C D B (1) A E C B D (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 14 8 -6 24 B -14 0 -18 -28 8 C -8 18 0 2 14 D 6 28 -2 0 12 E -24 -8 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999976 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 -6 24 B -14 0 -18 -28 8 C -8 18 0 2 14 D 6 28 -2 0 12 E -24 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000003 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=24 C=19 E=14 B=13 so B is eliminated. Round 2 votes counts: D=33 E=24 A=24 C=19 so C is eliminated. Round 3 votes counts: D=39 A=35 E=26 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:220 C:213 B:174 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 8 -6 24 B -14 0 -18 -28 8 C -8 18 0 2 14 D 6 28 -2 0 12 E -24 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000003 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -6 24 B -14 0 -18 -28 8 C -8 18 0 2 14 D 6 28 -2 0 12 E -24 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000003 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -6 24 B -14 0 -18 -28 8 C -8 18 0 2 14 D 6 28 -2 0 12 E -24 -8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000003 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4623: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) E C B A D (5) D E A C B (5) A C B E D (5) A C B D E (5) E D B C A (4) D E B A C (4) A D C B E (4) E B C D A (3) D B E C A (3) D A C B E (3) D A B C E (3) C A B E D (3) B C E A D (3) B C A E D (3) A C D B E (3) A B C D E (3) E B D C A (2) D E A B C (2) D B A E C (2) D A E C B (2) D A E B C (2) D A B E C (2) C B A E D (2) B A C D E (2) E A D C B (1) E A C D B (1) D B A C E (1) C E A B D (1) C B E A D (1) C A E B D (1) B E C A D (1) B D E C A (1) B D A C E (1) A E C B D (1) A D E C B (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 10 -4 2 B 0 0 10 -10 4 C -10 -10 0 -12 -8 D 4 10 12 0 24 E -2 -4 8 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 -4 2 B 0 0 10 -10 4 C -10 -10 0 -12 -8 D 4 10 12 0 24 E -2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=24 E=16 B=11 C=8 so C is eliminated. Round 2 votes counts: D=41 A=28 E=17 B=14 so B is eliminated. Round 3 votes counts: D=43 A=35 E=22 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:204 B:202 E:189 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 10 -4 2 B 0 0 10 -10 4 C -10 -10 0 -12 -8 D 4 10 12 0 24 E -2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -4 2 B 0 0 10 -10 4 C -10 -10 0 -12 -8 D 4 10 12 0 24 E -2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -4 2 B 0 0 10 -10 4 C -10 -10 0 -12 -8 D 4 10 12 0 24 E -2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4624: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) A D E B C (7) D A E C B (6) C B A D E (5) B C E A D (5) E A B D C (4) B C A E D (4) B C A D E (4) A D B C E (4) E D C B A (3) E D A B C (3) E A D B C (3) D E C A B (3) C B E D A (3) C B D A E (3) B A C E D (3) A E B D C (3) E D C A B (2) C E B D A (2) C B E A D (2) C B D E A (2) A E D B C (2) E D A C B (1) E C D B A (1) E B C A D (1) E B A C D (1) E A B C D (1) D C E B A (1) D C A B E (1) D A E B C (1) D A C E B (1) D A C B E (1) C E D B A (1) C D B E A (1) B E A C D (1) B A C D E (1) A D C B E (1) A D B E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 12 12 2 B -12 0 4 -4 -10 C -12 -4 0 -14 -4 D -12 4 14 0 6 E -2 10 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 12 2 B -12 0 4 -4 -10 C -12 -4 0 -14 -4 D -12 4 14 0 6 E -2 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=22 A=21 E=20 C=19 B=18 so B is eliminated. Round 2 votes counts: C=32 A=25 D=22 E=21 so E is eliminated. Round 3 votes counts: A=35 C=34 D=31 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:206 E:203 B:189 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 12 2 B -12 0 4 -4 -10 C -12 -4 0 -14 -4 D -12 4 14 0 6 E -2 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 12 2 B -12 0 4 -4 -10 C -12 -4 0 -14 -4 D -12 4 14 0 6 E -2 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 12 2 B -12 0 4 -4 -10 C -12 -4 0 -14 -4 D -12 4 14 0 6 E -2 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4625: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (13) C E A D B (11) D B A C E (8) D A B C E (6) B D A C E (6) A E C D B (5) E C B A D (4) E C A B D (3) E A C B D (3) C E B D A (3) E A C D B (2) D B A E C (2) C E B A D (2) C D A E B (2) B E C D A (2) A D E C B (2) A C E D B (2) A C D E B (2) E C B D A (1) E C A D B (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B A E (1) D A C B E (1) D A B E C (1) C E A B D (1) C B E D A (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A D C (1) B D E A C (1) B D C E A (1) B A D E C (1) A E D B C (1) A D E B C (1) A D C E B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 24 -2 18 B 0 0 0 -4 -4 C -24 0 0 0 0 D 2 4 0 0 2 E -18 4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.054544 D: 0.945456 E: 0.000000 Sum of squares = 0.896862880454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.054544 D: 1.000000 E: 1.000000 A B C D E A 0 0 24 -2 18 B 0 0 0 -4 -4 C -24 0 0 0 0 D 2 4 0 0 2 E -18 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.076923 D: 0.923077 E: 0.000000 Sum of squares = 0.857988340486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.076923 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=22 D=19 E=17 A=16 so A is eliminated. Round 2 votes counts: B=27 C=26 D=24 E=23 so E is eliminated. Round 3 votes counts: C=45 B=30 D=25 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:220 D:204 B:196 E:192 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 24 -2 18 B 0 0 0 -4 -4 C -24 0 0 0 0 D 2 4 0 0 2 E -18 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.076923 D: 0.923077 E: 0.000000 Sum of squares = 0.857988340486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.076923 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 24 -2 18 B 0 0 0 -4 -4 C -24 0 0 0 0 D 2 4 0 0 2 E -18 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.076923 D: 0.923077 E: 0.000000 Sum of squares = 0.857988340486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.076923 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 24 -2 18 B 0 0 0 -4 -4 C -24 0 0 0 0 D 2 4 0 0 2 E -18 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.076923 D: 0.923077 E: 0.000000 Sum of squares = 0.857988340486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.076923 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4626: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) E D B C A (8) A C B E D (8) D E B C A (7) C A D E B (5) E B D C A (4) D C A E B (4) B E A D C (4) A C B D E (4) E B D A C (3) D B E A C (3) C D A E B (3) C A D B E (3) B D E A C (3) B A E C D (3) E B C A D (2) D E C B A (2) D C E A B (2) D B A E C (2) D B A C E (2) C A E D B (2) B E A C D (2) A C D B E (2) A B C E D (2) A B C D E (2) E D C B A (1) E C B A D (1) E B A C D (1) D E C A B (1) D A C B E (1) C A E B D (1) C A B E D (1) B D A E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -16 6 -12 -6 B 16 0 18 2 6 C -6 -18 0 -16 -14 D 12 -2 16 0 -4 E 6 -6 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 -12 -6 B 16 0 18 2 6 C -6 -18 0 -16 -14 D 12 -2 16 0 -4 E 6 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986333 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 E=20 A=19 C=15 so C is eliminated. Round 2 votes counts: A=31 D=27 B=22 E=20 so E is eliminated. Round 3 votes counts: D=36 B=33 A=31 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:211 E:209 A:186 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 -12 -6 B 16 0 18 2 6 C -6 -18 0 -16 -14 D 12 -2 16 0 -4 E 6 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986333 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -12 -6 B 16 0 18 2 6 C -6 -18 0 -16 -14 D 12 -2 16 0 -4 E 6 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986333 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -12 -6 B 16 0 18 2 6 C -6 -18 0 -16 -14 D 12 -2 16 0 -4 E 6 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986333 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4627: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) A D C E B (7) D E A C B (6) D A E C B (6) E D C B A (5) E C D B A (5) B E C D A (5) A D E C B (5) D E C A B (4) B C E D A (4) B A C E D (4) A B D C E (4) A B C D E (4) C E D A B (3) B E D C A (3) A D E B C (3) B C A E D (2) A D C B E (2) A B D E C (2) E D C A B (1) E B D C A (1) D E B A C (1) D E A B C (1) D C E A B (1) D A E B C (1) D A C E B (1) C D E A B (1) C A E D B (1) C A D E B (1) B E C A D (1) B A E D C (1) B A E C D (1) B A D E C (1) A D B E C (1) A D B C E (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 16 8 2 0 B -16 0 0 -16 -10 C -8 0 0 -16 -8 D -2 16 16 0 8 E 0 10 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.857775 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.142225 Sum of squares = 0.756005233243 Cumulative probabilities = A: 0.857775 B: 0.857775 C: 0.857775 D: 0.857775 E: 1.000000 A B C D E A 0 16 8 2 0 B -16 0 0 -16 -10 C -8 0 0 -16 -8 D -2 16 16 0 8 E 0 10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000035067 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=30 D=21 E=12 C=6 so C is eliminated. Round 2 votes counts: A=33 B=30 D=22 E=15 so E is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:219 A:213 E:205 C:184 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 2 0 B -16 0 0 -16 -10 C -8 0 0 -16 -8 D -2 16 16 0 8 E 0 10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000035067 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 2 0 B -16 0 0 -16 -10 C -8 0 0 -16 -8 D -2 16 16 0 8 E 0 10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000035067 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 2 0 B -16 0 0 -16 -10 C -8 0 0 -16 -8 D -2 16 16 0 8 E 0 10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000035067 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4628: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) E C D A B (9) A B D E C (8) D A B E C (7) C E D B A (7) C E B A D (7) E D A B C (6) B D A C E (6) E C A D B (5) A D B E C (5) C E B D A (4) C B A D E (4) D B A E C (3) D B A C E (3) C E A B D (3) B A D E C (2) E D C A B (1) E C A B D (1) E A D B C (1) C D E B A (1) C B E A D (1) C B D E A (1) C B D A E (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 10 2 6 B 8 0 10 4 10 C -10 -10 0 -10 4 D -2 -4 10 0 10 E -6 -10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 2 6 B 8 0 10 4 10 C -10 -10 0 -10 4 D -2 -4 10 0 10 E -6 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=23 B=22 D=13 A=13 so D is eliminated. Round 2 votes counts: C=29 B=28 E=23 A=20 so A is eliminated. Round 3 votes counts: B=48 C=29 E=23 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:207 A:205 C:187 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 2 6 B 8 0 10 4 10 C -10 -10 0 -10 4 D -2 -4 10 0 10 E -6 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 2 6 B 8 0 10 4 10 C -10 -10 0 -10 4 D -2 -4 10 0 10 E -6 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 2 6 B 8 0 10 4 10 C -10 -10 0 -10 4 D -2 -4 10 0 10 E -6 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4629: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) B E C A D (9) B D E A C (8) D A C E B (5) D A C B E (5) B E D A C (5) C A D E B (4) E C B A D (3) E B D A C (3) D B A C E (3) C A D B E (3) B C E A D (3) A C D E B (3) E D A C B (2) E C A B D (2) E B C D A (2) D E A C B (2) D B E A C (2) D A E B C (2) B E D C A (2) B D C A E (2) B D A E C (2) B D A C E (2) B C A D E (2) A D C E B (2) E D B A C (1) E A C D B (1) D E B A C (1) D A E C B (1) C B E A D (1) C B A E D (1) C A E D B (1) C A B E D (1) B E C D A (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 -30 2 -4 -18 B 30 0 24 22 8 C -2 -24 0 -2 -20 D 4 -22 2 0 2 E 18 -8 20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 2 -4 -18 B 30 0 24 22 8 C -2 -24 0 -2 -20 D 4 -22 2 0 2 E 18 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=25 D=21 C=11 A=6 so A is eliminated. Round 2 votes counts: B=37 E=25 D=24 C=14 so C is eliminated. Round 3 votes counts: B=40 D=34 E=26 so E is eliminated. Round 4 votes counts: B=61 D=39 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:242 E:214 D:193 C:176 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 2 -4 -18 B 30 0 24 22 8 C -2 -24 0 -2 -20 D 4 -22 2 0 2 E 18 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 2 -4 -18 B 30 0 24 22 8 C -2 -24 0 -2 -20 D 4 -22 2 0 2 E 18 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 2 -4 -18 B 30 0 24 22 8 C -2 -24 0 -2 -20 D 4 -22 2 0 2 E 18 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998399 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4630: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C B D A E (8) B C D E A (8) E A B C D (7) B C E A D (6) E A D C B (5) E A D B C (5) A E D C B (5) D E A B C (4) C B A E D (4) B C D A E (4) A E C B D (4) A E C D B (3) E A C D B (2) E A C B D (2) D C B A E (2) D C A E B (2) D B C E A (2) D A E B C (2) B D C E A (2) A D E C B (2) E D A B C (1) E B A D C (1) E A B D C (1) D C A B E (1) D B E C A (1) D B E A C (1) D B C A E (1) C B E A D (1) C B A D E (1) C A D E B (1) B E D A C (1) B D E A C (1) A C E D B (1) Total count = 100 A B C D E A 0 12 12 2 -2 B -12 0 -4 2 -12 C -12 4 0 4 -12 D -2 -2 -4 0 2 E 2 12 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 12 12 2 -2 B -12 0 -4 2 -12 C -12 4 0 4 -12 D -2 -2 -4 0 2 E 2 12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 B=22 C=15 A=15 so C is eliminated. Round 2 votes counts: B=36 E=24 D=24 A=16 so A is eliminated. Round 3 votes counts: E=37 B=36 D=27 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:212 D:197 C:192 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 2 -2 B -12 0 -4 2 -12 C -12 4 0 4 -12 D -2 -2 -4 0 2 E 2 12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 2 -2 B -12 0 -4 2 -12 C -12 4 0 4 -12 D -2 -2 -4 0 2 E 2 12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 2 -2 B -12 0 -4 2 -12 C -12 4 0 4 -12 D -2 -2 -4 0 2 E 2 12 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4631: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) E A B C D (8) E A C B D (7) D B C E A (7) D B C A E (6) A E C B D (6) E D B C A (5) D B E C A (5) E A D B C (4) C B A D E (4) C A B D E (4) A C E D B (4) E D B A C (3) E A D C B (3) C B D A E (3) E D A C B (2) E B A D C (2) E A C D B (2) B C D A E (2) A E C D B (2) A C E B D (2) A C B D E (2) E D A B C (1) D E B C A (1) C D B A E (1) B D E C A (1) B D C E A (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 4 -4 B 2 0 14 8 -4 C 0 -14 0 -2 -6 D -4 -8 2 0 -6 E 4 4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 4 -4 B 2 0 14 8 -4 C 0 -14 0 -2 -6 D -4 -8 2 0 -6 E 4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=19 A=18 B=14 C=12 so C is eliminated. Round 2 votes counts: E=37 A=22 B=21 D=20 so D is eliminated. Round 3 votes counts: B=40 E=38 A=22 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:210 A:199 D:192 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 4 -4 B 2 0 14 8 -4 C 0 -14 0 -2 -6 D -4 -8 2 0 -6 E 4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 -4 B 2 0 14 8 -4 C 0 -14 0 -2 -6 D -4 -8 2 0 -6 E 4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 -4 B 2 0 14 8 -4 C 0 -14 0 -2 -6 D -4 -8 2 0 -6 E 4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4632: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (8) A B E C D (7) D C E B A (6) D C B E A (6) B C A D E (6) A E B C D (5) E D A C B (4) E D A B C (4) E A B D C (4) D C E A B (4) D C B A E (4) E B D C A (3) E A D B C (3) B E A C D (3) A B C E D (3) D C A E B (2) C B D A E (2) C A D B E (2) B C D E A (2) B C D A E (2) B A E C D (2) B A C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E B A C D (1) E A D C B (1) D E C B A (1) D E A C B (1) D C A B E (1) D A E C B (1) C D A B E (1) C B D E A (1) C A B D E (1) B E C A D (1) B C A E D (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -12 -12 4 B 8 0 4 -4 12 C 12 -4 0 2 10 D 12 4 -2 0 4 E -4 -12 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -12 4 B 8 0 4 -4 12 C 12 -4 0 2 10 D 12 4 -2 0 4 E -4 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000005 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 B=19 A=17 C=15 so C is eliminated. Round 2 votes counts: D=35 E=23 B=22 A=20 so A is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:210 C:210 D:209 A:186 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 -12 4 B 8 0 4 -4 12 C 12 -4 0 2 10 D 12 4 -2 0 4 E -4 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000005 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -12 4 B 8 0 4 -4 12 C 12 -4 0 2 10 D 12 4 -2 0 4 E -4 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000005 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -12 4 B 8 0 4 -4 12 C 12 -4 0 2 10 D 12 4 -2 0 4 E -4 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000005 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4633: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) C E B A D (10) E C D A B (7) C E A B D (6) D B A E C (5) B D A C E (5) E C B D A (4) E C A D B (4) A D B C E (4) E D C A B (3) E D A C B (3) E C D B A (3) C E B D A (3) C A E B D (3) B A D C E (3) A B D C E (3) E D C B A (2) E D A B C (2) D E A B C (2) B C A D E (2) B A C D E (2) E C A B D (1) E B C D A (1) D B E A C (1) D B A C E (1) C E A D B (1) C B E A D (1) C B A D E (1) C A E D B (1) C A B E D (1) B D E A C (1) B D C A E (1) B C D A E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -12 -10 -10 B -6 0 -12 0 -12 C 12 12 0 8 2 D 10 0 -8 0 -12 E 10 12 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -10 -10 B -6 0 -12 0 -12 C 12 12 0 8 2 D 10 0 -8 0 -12 E 10 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=27 D=19 B=15 A=9 so A is eliminated. Round 2 votes counts: E=30 C=29 D=23 B=18 so B is eliminated. Round 3 votes counts: D=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:216 D:195 A:187 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -10 -10 B -6 0 -12 0 -12 C 12 12 0 8 2 D 10 0 -8 0 -12 E 10 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -10 -10 B -6 0 -12 0 -12 C 12 12 0 8 2 D 10 0 -8 0 -12 E 10 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -10 -10 B -6 0 -12 0 -12 C 12 12 0 8 2 D 10 0 -8 0 -12 E 10 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4634: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (18) C D E A B (12) B A E C D (12) D C E B A (6) A E B C D (6) E C A D B (5) B A D E C (5) D B C E A (4) B D A E C (4) B A E D C (4) A B E C D (4) D C B E A (3) A E C D B (3) C E D A B (2) B D C A E (2) B D A C E (2) A E C B D (2) E A C D B (1) E A C B D (1) E A B C D (1) D B A C E (1) B D C E A (1) A C E D B (1) Total count = 100 A B C D E A 0 12 -6 -10 -8 B -12 0 -8 -12 -16 C 6 8 0 0 4 D 10 12 0 0 16 E 8 16 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.639096 D: 0.360904 E: 0.000000 Sum of squares = 0.538695242691 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.639096 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 -10 -8 B -12 0 -8 -12 -16 C 6 8 0 0 4 D 10 12 0 0 16 E 8 16 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=30 A=16 C=14 E=8 so E is eliminated. Round 2 votes counts: D=32 B=30 C=19 A=19 so C is eliminated. Round 3 votes counts: D=46 B=30 A=24 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:209 E:202 A:194 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 -10 -8 B -12 0 -8 -12 -16 C 6 8 0 0 4 D 10 12 0 0 16 E 8 16 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -10 -8 B -12 0 -8 -12 -16 C 6 8 0 0 4 D 10 12 0 0 16 E 8 16 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -10 -8 B -12 0 -8 -12 -16 C 6 8 0 0 4 D 10 12 0 0 16 E 8 16 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4635: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) B D A C E (9) D B C A E (8) B D C A E (8) E C A D B (7) E A C B D (5) C D E B A (5) B D C E A (5) B A D E C (4) C D E A B (3) A B E D C (3) E C B D A (2) E C A B D (2) E B C D A (2) E A C D B (2) E A B C D (2) D C B E A (2) D C B A E (2) C D B E A (2) B D A E C (2) A B D E C (2) E A B D C (1) D A B C E (1) C E A D B (1) C D B A E (1) C A E D B (1) B E A D C (1) B A E D C (1) B A D C E (1) A E C D B (1) A E C B D (1) A E B D C (1) A D B C E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -20 -22 -2 B 10 0 6 6 8 C 20 -6 0 -6 22 D 22 -6 6 0 14 E 2 -8 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 -22 -2 B 10 0 6 6 8 C 20 -6 0 -6 22 D 22 -6 6 0 14 E 2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=23 C=22 D=13 A=11 so A is eliminated. Round 2 votes counts: B=37 E=26 C=23 D=14 so D is eliminated. Round 3 votes counts: B=47 C=27 E=26 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:218 B:215 C:215 E:179 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -20 -22 -2 B 10 0 6 6 8 C 20 -6 0 -6 22 D 22 -6 6 0 14 E 2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 -22 -2 B 10 0 6 6 8 C 20 -6 0 -6 22 D 22 -6 6 0 14 E 2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 -22 -2 B 10 0 6 6 8 C 20 -6 0 -6 22 D 22 -6 6 0 14 E 2 -8 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4636: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) B E C A D (7) D A B E C (6) D A B C E (6) A D B C E (6) D A C B E (5) A D B E C (5) D C E A B (4) C B E A D (4) E C B D A (3) D A E B C (3) C E D B A (3) B E A D C (3) E D B C A (2) E C D B A (2) E B C D A (2) E B A D C (2) D E C A B (2) D E A B C (2) D C A E B (2) D A E C B (2) B A E C D (2) A B C E D (2) E D C B A (1) E D B A C (1) E C B A D (1) E B D C A (1) E B A C D (1) D E A C B (1) D C A B E (1) C E B D A (1) C E B A D (1) C D E A B (1) C B A E D (1) C A D B E (1) B E A C D (1) B C A E D (1) B A E D C (1) A D C B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 -2 -6 B -4 0 26 -14 10 C -4 -26 0 -16 -18 D 2 14 16 0 0 E 6 -10 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.585988 E: 0.414012 Sum of squares = 0.514787831724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.585988 E: 1.000000 A B C D E A 0 4 4 -2 -6 B -4 0 26 -14 10 C -4 -26 0 -16 -18 D 2 14 16 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=23 A=16 B=15 C=12 so C is eliminated. Round 2 votes counts: D=35 E=28 B=20 A=17 so A is eliminated. Round 3 votes counts: D=48 E=28 B=24 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:209 E:207 A:200 C:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -2 -6 B -4 0 26 -14 10 C -4 -26 0 -16 -18 D 2 14 16 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -2 -6 B -4 0 26 -14 10 C -4 -26 0 -16 -18 D 2 14 16 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -2 -6 B -4 0 26 -14 10 C -4 -26 0 -16 -18 D 2 14 16 0 0 E 6 -10 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4637: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) A E D B C (8) C B E D A (7) C A B E D (7) C B E A D (5) C B D E A (5) B E C D A (5) E B D A C (4) C B D A E (3) B C E D A (3) A D E C B (3) E D B A C (2) E D A B C (2) E A D B C (2) E A B D C (2) D A E B C (2) C D A B E (2) C B A E D (2) C A D B E (2) B E D C A (2) B E C A D (2) B D E C A (2) A E D C B (2) A E C D B (2) A D E B C (2) A C B E D (2) E B A D C (1) D E B A C (1) D C B E A (1) D B C E A (1) C D B A E (1) C A D E B (1) C A B D E (1) B C E A D (1) B C A E D (1) A E B D C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -8 -2 -12 B -2 0 6 14 10 C 8 -6 0 8 -8 D 2 -14 -8 0 -28 E 12 -10 8 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.416667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.083333 Sum of squares = 0.430555555547 Cumulative probabilities = A: 0.416667 B: 0.916667 C: 0.916667 D: 0.916667 E: 1.000000 A B C D E A 0 2 -8 -2 -12 B -2 0 6 14 10 C 8 -6 0 8 -8 D 2 -14 -8 0 -28 E 12 -10 8 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.083333 Sum of squares = 0.430555555455 Cumulative probabilities = A: 0.416667 B: 0.916667 C: 0.916667 D: 0.916667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=22 B=16 E=13 D=13 so E is eliminated. Round 2 votes counts: C=36 A=26 B=21 D=17 so D is eliminated. Round 3 votes counts: A=38 C=37 B=25 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:219 B:214 C:201 A:190 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -8 -2 -12 B -2 0 6 14 10 C 8 -6 0 8 -8 D 2 -14 -8 0 -28 E 12 -10 8 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.083333 Sum of squares = 0.430555555455 Cumulative probabilities = A: 0.416667 B: 0.916667 C: 0.916667 D: 0.916667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 -12 B -2 0 6 14 10 C 8 -6 0 8 -8 D 2 -14 -8 0 -28 E 12 -10 8 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.083333 Sum of squares = 0.430555555455 Cumulative probabilities = A: 0.416667 B: 0.916667 C: 0.916667 D: 0.916667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 -12 B -2 0 6 14 10 C 8 -6 0 8 -8 D 2 -14 -8 0 -28 E 12 -10 8 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.416667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.083333 Sum of squares = 0.430555555455 Cumulative probabilities = A: 0.416667 B: 0.916667 C: 0.916667 D: 0.916667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4638: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A B C D E (8) E D C B A (7) A C B E D (7) C A E D B (6) B D E A C (6) B A D E C (6) C E D A B (5) A B D E C (5) B D A E C (4) A C E D B (4) A B E D C (4) E D C A B (2) E C D B A (2) D B E C A (2) C E D B A (2) C E A D B (2) C B D E A (2) B D E C A (2) A C B D E (2) A B D C E (2) E D B C A (1) E C D A B (1) D E C B A (1) D C B E A (1) C D E B A (1) C A E B D (1) C A B D E (1) B A D C E (1) B A C D E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 4 8 B -4 0 4 8 10 C -4 -4 0 -6 -4 D -4 -8 6 0 8 E -8 -10 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 4 8 B -4 0 4 8 10 C -4 -4 0 -6 -4 D -4 -8 6 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=20 B=20 E=13 D=13 so E is eliminated. Round 2 votes counts: A=34 D=23 C=23 B=20 so B is eliminated. Round 3 votes counts: A=42 D=35 C=23 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:209 D:201 C:191 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 4 8 B -4 0 4 8 10 C -4 -4 0 -6 -4 D -4 -8 6 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 8 B -4 0 4 8 10 C -4 -4 0 -6 -4 D -4 -8 6 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 8 B -4 0 4 8 10 C -4 -4 0 -6 -4 D -4 -8 6 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4639: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) D C B E A (6) D C B A E (6) C B A D E (6) E D A B C (5) E A D B C (5) E A B C D (5) C B D A E (5) B C A E D (5) B C A D E (5) E D B C A (4) D E B C A (4) D E A C B (4) E D A C B (3) A B C E D (3) D E C B A (2) D B C E A (2) C B A E D (2) B C D E A (2) B C D A E (2) B A C E D (2) A E C B D (2) E B A D C (1) E A D C B (1) D E A B C (1) D C E B A (1) D C A B E (1) D A C E B (1) C A B D E (1) B E C A D (1) B C E A D (1) A E B D C (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 -12 2 4 B 14 0 12 8 4 C 12 -12 0 4 8 D -2 -8 -4 0 0 E -4 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 2 4 B 14 0 12 8 4 C 12 -12 0 4 8 D -2 -8 -4 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998321 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 B=18 A=16 C=14 so C is eliminated. Round 2 votes counts: B=31 D=28 E=24 A=17 so A is eliminated. Round 3 votes counts: B=37 E=35 D=28 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:206 D:193 E:192 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 2 4 B 14 0 12 8 4 C 12 -12 0 4 8 D -2 -8 -4 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998321 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 2 4 B 14 0 12 8 4 C 12 -12 0 4 8 D -2 -8 -4 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998321 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 2 4 B 14 0 12 8 4 C 12 -12 0 4 8 D -2 -8 -4 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998321 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4640: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) B C A D E (9) D E C A B (8) E D A C B (5) E D A B C (4) D E C B A (4) D E A B C (4) D C B E A (4) C D B E A (4) A E B D C (4) A B C E D (4) E A D C B (3) C B D E A (3) C B D A E (3) A B E D C (3) D E A C B (2) D C E B A (2) D B C E A (2) C D E B A (2) C B A E D (2) B D C E A (2) B A C E D (2) A C B E D (2) A B E C D (2) E A D B C (1) D E B A C (1) D B E A C (1) C D E A B (1) B C D E A (1) B C D A E (1) B C A E D (1) B A E D C (1) A E D C B (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 -16 -8 -8 B 8 0 -12 -2 12 C 16 12 0 -6 8 D 8 2 6 0 26 E 8 -12 -8 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -8 -8 B 8 0 -12 -2 12 C 16 12 0 -6 8 D 8 2 6 0 26 E 8 -12 -8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 A=18 B=17 E=13 so E is eliminated. Round 2 votes counts: D=37 C=24 A=22 B=17 so B is eliminated. Round 3 votes counts: D=39 C=36 A=25 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:215 B:203 E:181 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 -8 B 8 0 -12 -2 12 C 16 12 0 -6 8 D 8 2 6 0 26 E 8 -12 -8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 -8 B 8 0 -12 -2 12 C 16 12 0 -6 8 D 8 2 6 0 26 E 8 -12 -8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 -8 B 8 0 -12 -2 12 C 16 12 0 -6 8 D 8 2 6 0 26 E 8 -12 -8 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993256 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4641: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) D C E A B (7) A B C D E (7) E B A C D (6) D E C A B (5) C E D B A (5) B A E C D (5) B A C E D (5) D C A B E (4) E C D B A (3) C D E A B (3) C A D B E (3) C A B D E (3) A B D E C (3) E C B D A (2) E C B A D (2) E B A D C (2) D E A B C (2) D A C B E (2) C E B D A (2) B A E D C (2) A D B C E (2) A B E D C (2) A B C E D (2) E D B A C (1) E D A B C (1) E B C A D (1) E A B D C (1) D C A E B (1) D A E B C (1) D A B E C (1) D A B C E (1) C D A E B (1) C B A E D (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 26 -4 -2 14 B -26 0 -8 -6 8 C 4 8 0 22 18 D 2 6 -22 0 14 E -14 -8 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 -4 -2 14 B -26 0 -8 -6 8 C 4 8 0 22 18 D 2 6 -22 0 14 E -14 -8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=24 E=19 A=18 B=12 so B is eliminated. Round 2 votes counts: A=30 C=27 D=24 E=19 so E is eliminated. Round 3 votes counts: A=39 C=35 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:217 D:200 B:184 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 26 -4 -2 14 B -26 0 -8 -6 8 C 4 8 0 22 18 D 2 6 -22 0 14 E -14 -8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 -4 -2 14 B -26 0 -8 -6 8 C 4 8 0 22 18 D 2 6 -22 0 14 E -14 -8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 -4 -2 14 B -26 0 -8 -6 8 C 4 8 0 22 18 D 2 6 -22 0 14 E -14 -8 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4642: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) E C B A D (7) C E B D A (5) B E C A D (5) B A D E C (5) A D B E C (5) D C A E B (4) B E A C D (4) B A E D C (4) D A C B E (3) D A B C E (3) C B E D A (3) B C D A E (3) B A D C E (3) E C B D A (2) E B A C D (2) E A B D C (2) D B A C E (2) D A E C B (2) C E D B A (2) C E D A B (2) C D E A B (2) B C D E A (2) A E D B C (2) A D E C B (2) A D E B C (2) E C D B A (1) E C D A B (1) E C A B D (1) E B C A D (1) E A D C B (1) E A C D B (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A E B (1) B A E C D (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 10 2 8 B 10 0 -2 4 -4 C -10 2 0 -2 -6 D -2 -4 2 0 4 E -8 4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.074449 B: 0.345741 C: 0.000000 D: 0.196844 E: 0.382966 Sum of squares = 0.310490136702 Cumulative probabilities = A: 0.074449 B: 0.420190 C: 0.420190 D: 0.617034 E: 1.000000 A B C D E A 0 -10 10 2 8 B 10 0 -2 4 -4 C -10 2 0 -2 -6 D -2 -4 2 0 4 E -8 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068966 B: 0.344828 C: 0.000000 D: 0.206897 E: 0.379310 Sum of squares = 0.310344827587 Cumulative probabilities = A: 0.068966 B: 0.413793 C: 0.413793 D: 0.620690 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 E=19 C=18 A=14 so A is eliminated. Round 2 votes counts: D=32 B=29 E=21 C=18 so C is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:205 B:204 D:200 E:199 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 2 8 B 10 0 -2 4 -4 C -10 2 0 -2 -6 D -2 -4 2 0 4 E -8 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068966 B: 0.344828 C: 0.000000 D: 0.206897 E: 0.379310 Sum of squares = 0.310344827587 Cumulative probabilities = A: 0.068966 B: 0.413793 C: 0.413793 D: 0.620690 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 2 8 B 10 0 -2 4 -4 C -10 2 0 -2 -6 D -2 -4 2 0 4 E -8 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068966 B: 0.344828 C: 0.000000 D: 0.206897 E: 0.379310 Sum of squares = 0.310344827587 Cumulative probabilities = A: 0.068966 B: 0.413793 C: 0.413793 D: 0.620690 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 2 8 B 10 0 -2 4 -4 C -10 2 0 -2 -6 D -2 -4 2 0 4 E -8 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.068966 B: 0.344828 C: 0.000000 D: 0.206897 E: 0.379310 Sum of squares = 0.310344827587 Cumulative probabilities = A: 0.068966 B: 0.413793 C: 0.413793 D: 0.620690 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4643: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) A C E D B (9) B D E C A (8) B D A C E (8) D B A E C (7) C E A B D (6) A C E B D (5) E C A B D (4) D B E C A (4) D B E A C (3) D A E C B (3) C E A D B (3) C A E B D (3) B C E A D (3) D B A C E (2) D A B C E (2) B E C A D (2) B D A E C (2) A C B E D (2) E D C A B (1) E B C D A (1) E A C D B (1) D E C B A (1) D E C A B (1) D E A C B (1) D A E B C (1) D A C E B (1) B E D C A (1) B E C D A (1) B D C E A (1) A E C D B (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 0 2 -4 B -12 0 -6 -4 -6 C 0 6 0 4 -6 D -2 4 -4 0 -8 E 4 6 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 0 2 -4 B -12 0 -6 -4 -6 C 0 6 0 4 -6 D -2 4 -4 0 -8 E 4 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 A=19 E=17 C=12 so C is eliminated. Round 2 votes counts: E=26 D=26 B=26 A=22 so A is eliminated. Round 3 votes counts: E=44 B=29 D=27 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:205 C:202 D:195 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 0 2 -4 B -12 0 -6 -4 -6 C 0 6 0 4 -6 D -2 4 -4 0 -8 E 4 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 2 -4 B -12 0 -6 -4 -6 C 0 6 0 4 -6 D -2 4 -4 0 -8 E 4 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 2 -4 B -12 0 -6 -4 -6 C 0 6 0 4 -6 D -2 4 -4 0 -8 E 4 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4644: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) C B A E D (8) B D C A E (8) D B C E A (7) A E C B D (7) D E A B C (6) D B C A E (6) C B A D E (6) B C D A E (6) D E B A C (5) D B E C A (5) D B E A C (5) E A D C B (4) C A B E D (4) A C E B D (4) B C A D E (3) E D A C B (2) E A C B D (2) E D A B C (1) B C A E D (1) Total count = 100 A B C D E A 0 -20 -8 -2 6 B 20 0 6 -2 18 C 8 -6 0 2 6 D 2 2 -2 0 14 E -6 -18 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.44000000003 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 -2 6 B 20 0 6 -2 18 C 8 -6 0 2 6 D 2 2 -2 0 14 E -6 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000041 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=19 C=18 B=18 A=11 so A is eliminated. Round 2 votes counts: D=34 E=26 C=22 B=18 so B is eliminated. Round 3 votes counts: D=42 C=32 E=26 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:221 D:208 C:205 A:188 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -8 -2 6 B 20 0 6 -2 18 C 8 -6 0 2 6 D 2 2 -2 0 14 E -6 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000041 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -2 6 B 20 0 6 -2 18 C 8 -6 0 2 6 D 2 2 -2 0 14 E -6 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000041 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -2 6 B 20 0 6 -2 18 C 8 -6 0 2 6 D 2 2 -2 0 14 E -6 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000041 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4645: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) C A E D B (7) D E C A B (6) D E B C A (6) B D E A C (6) B A C D E (6) A C B E D (6) A B C E D (6) D E C B A (5) B A C E D (5) D B E C A (4) B D A E C (4) B A D C E (4) E C D A B (3) B D E C A (3) A C E B D (3) D B E A C (2) B A D E C (2) D E B A C (1) C E D A B (1) C E A D B (1) C B A D E (1) C A E B D (1) C A B D E (1) B E D A C (1) B D C A E (1) B D A C E (1) A E C B D (1) A C E D B (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 2 -6 4 B 4 0 8 8 10 C -2 -8 0 -10 -8 D 6 -8 10 0 8 E -4 -10 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -6 4 B 4 0 8 8 10 C -2 -8 0 -10 -8 D 6 -8 10 0 8 E -4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=24 A=19 E=12 C=12 so E is eliminated. Round 2 votes counts: D=33 B=33 A=19 C=15 so C is eliminated. Round 3 votes counts: D=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:208 A:198 E:193 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -6 4 B 4 0 8 8 10 C -2 -8 0 -10 -8 D 6 -8 10 0 8 E -4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -6 4 B 4 0 8 8 10 C -2 -8 0 -10 -8 D 6 -8 10 0 8 E -4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -6 4 B 4 0 8 8 10 C -2 -8 0 -10 -8 D 6 -8 10 0 8 E -4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4646: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (13) B E A C D (12) A E C D B (7) D C A E B (6) B D C E A (6) E A C D B (5) D C B A E (5) B E A D C (5) B E D A C (4) B D C A E (4) E A D C B (3) E A B C D (3) D B C A E (2) C D B A E (2) C D A B E (2) B D E A C (2) B C D A E (2) B C A E D (2) B A E C D (2) E D A C B (1) E B A D C (1) E A C B D (1) D C E A B (1) D B E C A (1) D B C E A (1) C A E D B (1) C A D E B (1) B D E C A (1) B A C E D (1) A E D C B (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 0 -6 6 B 6 0 -2 -4 8 C 0 2 0 12 0 D 6 4 -12 0 -2 E -6 -8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.189154 B: 0.000000 C: 0.810846 D: 0.000000 E: 0.000000 Sum of squares = 0.693250683291 Cumulative probabilities = A: 0.189154 B: 0.189154 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 6 B 6 0 -2 -4 8 C 0 2 0 12 0 D 6 4 -12 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000203637 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=19 D=16 E=14 A=10 so A is eliminated. Round 2 votes counts: B=41 E=23 C=20 D=16 so D is eliminated. Round 3 votes counts: B=45 C=32 E=23 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:204 D:198 A:197 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -6 6 B 6 0 -2 -4 8 C 0 2 0 12 0 D 6 4 -12 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000203637 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 6 B 6 0 -2 -4 8 C 0 2 0 12 0 D 6 4 -12 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000203637 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 6 B 6 0 -2 -4 8 C 0 2 0 12 0 D 6 4 -12 0 -2 E -6 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000203637 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4647: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (15) D E A B C (12) E D C B A (8) C B A E D (8) A B C D E (8) C B E D A (6) C E D B A (5) B C A D E (5) B A C D E (5) A B D E C (5) E D A B C (4) E D A C B (3) E D C A B (2) D A E B C (2) C B A D E (2) B C A E D (2) A B D C E (2) E D B A C (1) E C D B A (1) C E B D A (1) B C E D A (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 8 18 8 12 B -8 0 26 -6 -8 C -18 -26 0 -10 -8 D -8 6 10 0 16 E -12 8 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 8 12 B -8 0 26 -6 -8 C -18 -26 0 -10 -8 D -8 6 10 0 16 E -12 8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=22 E=19 D=14 B=14 so D is eliminated. Round 2 votes counts: A=33 E=31 C=22 B=14 so B is eliminated. Round 3 votes counts: A=39 E=31 C=30 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 D:212 B:202 E:194 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 8 12 B -8 0 26 -6 -8 C -18 -26 0 -10 -8 D -8 6 10 0 16 E -12 8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 8 12 B -8 0 26 -6 -8 C -18 -26 0 -10 -8 D -8 6 10 0 16 E -12 8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 8 12 B -8 0 26 -6 -8 C -18 -26 0 -10 -8 D -8 6 10 0 16 E -12 8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4648: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) E A B C D (7) D E B C A (7) E D B A C (5) D C B A E (5) D C A E B (4) D C A B E (4) A E C D B (4) E A D C B (3) D E A C B (3) C A D B E (3) C A B D E (3) A C E D B (3) A C E B D (3) E D B C A (2) E B A D C (2) E B A C D (2) D B C E A (2) C B D A E (2) B E D C A (2) B D E C A (2) B C A D E (2) B A C E D (2) A E C B D (2) A C D E B (2) E D A C B (1) E D A B C (1) E B D A C (1) E A D B C (1) E A C B D (1) E A B D C (1) D E C A B (1) D E A B C (1) D C E A B (1) C D A E B (1) B D C A E (1) B C D A E (1) B C A E D (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 12 6 -8 B -6 0 4 -10 -18 C -12 -4 0 0 -14 D -6 10 0 0 -6 E 8 18 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 12 6 -8 B -6 0 4 -10 -18 C -12 -4 0 0 -14 D -6 10 0 0 -6 E 8 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 B=19 A=17 C=9 so C is eliminated. Round 2 votes counts: D=29 E=27 A=23 B=21 so B is eliminated. Round 3 votes counts: E=37 D=35 A=28 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:208 D:199 B:185 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 6 -8 B -6 0 4 -10 -18 C -12 -4 0 0 -14 D -6 10 0 0 -6 E 8 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 6 -8 B -6 0 4 -10 -18 C -12 -4 0 0 -14 D -6 10 0 0 -6 E 8 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 6 -8 B -6 0 4 -10 -18 C -12 -4 0 0 -14 D -6 10 0 0 -6 E 8 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4649: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) D C B A E (7) C D A B E (7) B D E C A (7) D B E C A (6) C A B D E (6) A C E D B (6) E B A D C (5) A C E B D (5) E B D A C (4) D B C E A (3) A C D E B (3) E B A C D (2) E A B C D (2) D C A E B (2) D B C A E (2) C A D E B (2) C A B E D (2) B E D A C (2) B D C E A (2) A E C D B (2) E D B A C (1) E D A B C (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E C A B (1) D C A B E (1) C D A E B (1) C B A D E (1) B C D A E (1) A E D C B (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 14 -20 4 20 B -14 0 -20 -14 14 C 20 20 0 6 22 D -4 14 -6 0 22 E -20 -14 -22 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -20 4 20 B -14 0 -20 -14 14 C 20 20 0 6 22 D -4 14 -6 0 22 E -20 -14 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=22 E=19 A=19 B=12 so B is eliminated. Round 2 votes counts: D=31 C=29 E=21 A=19 so A is eliminated. Round 3 votes counts: C=44 D=31 E=25 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:234 D:213 A:209 B:183 E:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -20 4 20 B -14 0 -20 -14 14 C 20 20 0 6 22 D -4 14 -6 0 22 E -20 -14 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -20 4 20 B -14 0 -20 -14 14 C 20 20 0 6 22 D -4 14 -6 0 22 E -20 -14 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -20 4 20 B -14 0 -20 -14 14 C 20 20 0 6 22 D -4 14 -6 0 22 E -20 -14 -22 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4650: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) C E D A B (8) C E B A D (7) D A B E C (6) E C B D A (5) A B D C E (5) E C D B A (4) C E A B D (4) A D C B E (4) C E A D B (3) C A E D B (3) B E D A C (3) A C D B E (3) A B D E C (3) E B D C A (2) D B A E C (2) C D A E B (2) C A E B D (2) B E C A D (2) B E A D C (2) A D B E C (2) A D B C E (2) A B C D E (2) E C B A D (1) E B C D A (1) E B C A D (1) D E C A B (1) D E B A C (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C A D E B (1) C A B E D (1) B E A C D (1) B D A E C (1) B A E D C (1) A C D E B (1) Total count = 100 A B C D E A 0 10 0 20 4 B -10 0 -6 8 2 C 0 6 0 8 4 D -20 -8 -8 0 -6 E -4 -2 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.190340 B: 0.000000 C: 0.809660 D: 0.000000 E: 0.000000 Sum of squares = 0.691778344957 Cumulative probabilities = A: 0.190340 B: 0.190340 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 20 4 B -10 0 -6 8 2 C 0 6 0 8 4 D -20 -8 -8 0 -6 E -4 -2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=22 B=19 E=14 D=12 so D is eliminated. Round 2 votes counts: C=33 A=30 B=21 E=16 so E is eliminated. Round 3 votes counts: C=44 A=30 B=26 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:209 E:198 B:197 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 20 4 B -10 0 -6 8 2 C 0 6 0 8 4 D -20 -8 -8 0 -6 E -4 -2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 20 4 B -10 0 -6 8 2 C 0 6 0 8 4 D -20 -8 -8 0 -6 E -4 -2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 20 4 B -10 0 -6 8 2 C 0 6 0 8 4 D -20 -8 -8 0 -6 E -4 -2 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4651: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D B C A E (6) A B C D E (6) E D C B A (5) E A C B D (5) D B C E A (5) D A B C E (5) B C A E D (5) A E D C B (4) E C D B A (3) E A D C B (3) D E C B A (3) C B D E A (3) A E B C D (3) A B C E D (3) E C B A D (2) D E A C B (2) D C E B A (2) D C B E A (2) D B A C E (2) C E B D A (2) B D C A E (2) B C D E A (2) B A C D E (2) A E D B C (2) A B E C D (2) E D C A B (1) E D A C B (1) E C A B D (1) E A C D B (1) D A B E C (1) C B E D A (1) B C D A E (1) B A D C E (1) B A C E D (1) A E C B D (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -4 -10 0 B 12 0 4 2 6 C 4 -4 0 0 4 D 10 -2 0 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -10 0 B 12 0 4 2 6 C 4 -4 0 0 4 D 10 -2 0 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994184 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 A=24 B=14 C=6 so C is eliminated. Round 2 votes counts: E=30 D=28 A=24 B=18 so B is eliminated. Round 3 votes counts: D=36 A=33 E=31 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:212 C:202 D:202 E:197 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -10 0 B 12 0 4 2 6 C 4 -4 0 0 4 D 10 -2 0 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994184 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -10 0 B 12 0 4 2 6 C 4 -4 0 0 4 D 10 -2 0 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994184 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -10 0 B 12 0 4 2 6 C 4 -4 0 0 4 D 10 -2 0 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994184 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4652: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) B D A C E (10) C E A D B (8) D A B C E (7) E C A D B (6) A D C B E (6) E C B A D (5) B E D A C (5) E C B D A (4) D B A C E (4) A C D E B (4) E B C D A (3) E C A B D (2) E B C A D (2) D A C B E (2) C A E D B (2) C A D E B (2) B E C D A (2) B D E C A (2) B D E A C (2) B A D E C (2) E C D A B (1) C E D B A (1) C E D A B (1) B E D C A (1) B E A D C (1) B A D C E (1) A D C E B (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 16 -10 8 B 10 0 10 8 16 C -16 -10 0 -14 2 D 10 -8 14 0 10 E -8 -16 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 16 -10 8 B 10 0 10 8 16 C -16 -10 0 -14 2 D 10 -8 14 0 10 E -8 -16 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=23 C=14 A=14 D=13 so D is eliminated. Round 2 votes counts: B=40 E=23 A=23 C=14 so C is eliminated. Round 3 votes counts: B=40 E=33 A=27 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:213 A:202 E:182 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 16 -10 8 B 10 0 10 8 16 C -16 -10 0 -14 2 D 10 -8 14 0 10 E -8 -16 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 -10 8 B 10 0 10 8 16 C -16 -10 0 -14 2 D 10 -8 14 0 10 E -8 -16 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 -10 8 B 10 0 10 8 16 C -16 -10 0 -14 2 D 10 -8 14 0 10 E -8 -16 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4653: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) B D E A C (8) C A B D E (6) A D E B C (5) D B E A C (4) A B D C E (4) E B D C A (3) D E B A C (3) C E B D A (3) C A D E B (3) B E D C A (3) A C D E B (3) A C D B E (3) E D B A C (2) E D A C B (2) E C A D B (2) E B D A C (2) E B C D A (2) C E B A D (2) C A E B D (2) C A D B E (2) B E D A C (2) B E C D A (2) B D C A E (2) B D A E C (2) A D B C E (2) E D C B A (1) E D C A B (1) E D A B C (1) E A D C B (1) D E A B C (1) C E A B D (1) C B E D A (1) C B E A D (1) C B A D E (1) C A B E D (1) B C E D A (1) B C D A E (1) A D C E B (1) A D C B E (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 6 2 B -8 0 2 4 -2 C 0 -2 0 -2 6 D -6 -4 2 0 8 E -2 2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.471544 B: 0.000000 C: 0.528456 D: 0.000000 E: 0.000000 Sum of squares = 0.501619484651 Cumulative probabilities = A: 0.471544 B: 0.471544 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 6 2 B -8 0 2 4 -2 C 0 -2 0 -2 6 D -6 -4 2 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=22 B=21 E=17 D=8 so D is eliminated. Round 2 votes counts: C=32 B=25 A=22 E=21 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:208 C:201 D:200 B:198 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 6 2 B -8 0 2 4 -2 C 0 -2 0 -2 6 D -6 -4 2 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 6 2 B -8 0 2 4 -2 C 0 -2 0 -2 6 D -6 -4 2 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 6 2 B -8 0 2 4 -2 C 0 -2 0 -2 6 D -6 -4 2 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4654: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) E D B C A (7) B A C E D (7) A C B D E (7) A C B E D (6) D E A C B (5) A B C D E (5) A C D B E (4) C A D E B (3) B A C D E (3) A C D E B (3) E D C B A (2) E B D C A (2) D E C B A (2) D E B C A (2) D E A B C (2) C A B E D (2) B E C A D (2) B D E A C (2) B C A E D (2) B A D E C (2) A D C B E (2) A B D C E (2) E D C A B (1) E C D B A (1) E C B D A (1) E B C D A (1) D A E C B (1) C E B A D (1) C B E A D (1) C B A E D (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B A E D C (1) A D E B C (1) A D B E C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 14 14 18 12 B -14 0 -6 6 12 C -14 6 0 6 2 D -18 -6 -6 0 16 E -12 -12 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 18 12 B -14 0 -6 6 12 C -14 6 0 6 2 D -18 -6 -6 0 16 E -12 -12 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=22 D=21 E=15 C=9 so C is eliminated. Round 2 votes counts: A=39 B=24 D=21 E=16 so E is eliminated. Round 3 votes counts: A=39 D=32 B=29 so B is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:200 B:199 D:193 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 18 12 B -14 0 -6 6 12 C -14 6 0 6 2 D -18 -6 -6 0 16 E -12 -12 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 18 12 B -14 0 -6 6 12 C -14 6 0 6 2 D -18 -6 -6 0 16 E -12 -12 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 18 12 B -14 0 -6 6 12 C -14 6 0 6 2 D -18 -6 -6 0 16 E -12 -12 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4655: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (12) D E B C A (10) C A B E D (8) A D B C E (6) C B A E D (5) C B E A D (4) B E C D A (4) B C E D A (4) E D B C A (3) E C B D A (3) D A E B C (3) A D C E B (3) A D C B E (3) A C B D E (3) D E A B C (2) C A E B D (2) A C D B E (2) E D C B A (1) E C A B D (1) E B D C A (1) E B C D A (1) D E A C B (1) D B E C A (1) D B E A C (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A B D (1) B E D C A (1) B D E C A (1) B C E A D (1) B C D E A (1) B C A E D (1) B C A D E (1) A E C D B (1) A D E C B (1) A D E B C (1) A C E D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -12 20 14 B -8 0 -8 16 24 C 12 8 0 18 22 D -20 -16 -18 0 -14 E -14 -24 -22 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 20 14 B -8 0 -8 16 24 C 12 8 0 18 22 D -20 -16 -18 0 -14 E -14 -24 -22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=21 D=20 B=14 E=10 so E is eliminated. Round 2 votes counts: A=35 C=25 D=24 B=16 so B is eliminated. Round 3 votes counts: C=38 A=35 D=27 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:230 A:215 B:212 E:177 D:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 20 14 B -8 0 -8 16 24 C 12 8 0 18 22 D -20 -16 -18 0 -14 E -14 -24 -22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 20 14 B -8 0 -8 16 24 C 12 8 0 18 22 D -20 -16 -18 0 -14 E -14 -24 -22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 20 14 B -8 0 -8 16 24 C 12 8 0 18 22 D -20 -16 -18 0 -14 E -14 -24 -22 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4656: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) A B E C D (8) E C A D B (5) C D E B A (5) E A C B D (4) E A B C D (4) D E C A B (4) D C E B A (4) D C B E A (4) D B C A E (4) C B D A E (4) D B A E C (3) C B A D E (3) B C A D E (3) B A D C E (3) B A C D E (3) E D A C B (2) D C B A E (2) D B A C E (2) C E D A B (2) B D A C E (2) B C D A E (2) B A C E D (2) A E B D C (2) E D C A B (1) E C D A B (1) E C A B D (1) E A C D B (1) E A B D C (1) D E B C A (1) D C E A B (1) D B E A C (1) C E A B D (1) C D B E A (1) C B E A D (1) B D C A E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 6 10 B 2 0 6 12 2 C 2 -6 0 22 2 D -6 -12 -22 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 6 10 B 2 0 6 12 2 C 2 -6 0 22 2 D -6 -12 -22 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=21 E=20 C=17 B=16 so B is eliminated. Round 2 votes counts: D=29 A=29 C=22 E=20 so E is eliminated. Round 3 votes counts: A=39 D=32 C=29 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:211 C:210 A:206 E:190 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 6 10 B 2 0 6 12 2 C 2 -6 0 22 2 D -6 -12 -22 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 6 10 B 2 0 6 12 2 C 2 -6 0 22 2 D -6 -12 -22 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 6 10 B 2 0 6 12 2 C 2 -6 0 22 2 D -6 -12 -22 0 6 E -10 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4657: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) D C B A E (8) C D E B A (8) A B D E C (7) E C D A B (6) C E D B A (6) A E B D C (6) E A B C D (5) D B C A E (5) A B D C E (5) E C A D B (4) E A C B D (4) A B E D C (4) E C D B A (3) C D B E A (3) B D C A E (3) B A D C E (3) E C A B D (2) E A C D B (2) B A D E C (2) D C B E A (1) D B A C E (1) B E A D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 2 -8 10 B 8 0 6 4 8 C -2 -6 0 -14 8 D 8 -4 14 0 14 E -10 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -8 10 B 8 0 6 4 8 C -2 -6 0 -14 8 D 8 -4 14 0 14 E -10 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=23 B=19 C=17 D=15 so D is eliminated. Round 2 votes counts: E=26 C=26 B=25 A=23 so A is eliminated. Round 3 votes counts: B=42 E=32 C=26 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:213 A:198 C:193 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -8 10 B 8 0 6 4 8 C -2 -6 0 -14 8 D 8 -4 14 0 14 E -10 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -8 10 B 8 0 6 4 8 C -2 -6 0 -14 8 D 8 -4 14 0 14 E -10 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -8 10 B 8 0 6 4 8 C -2 -6 0 -14 8 D 8 -4 14 0 14 E -10 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4658: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) D E C A B (6) B A C D E (5) D A E B C (4) A D B E C (4) A B D E C (4) E C D A B (3) D C E B A (3) D A B C E (3) C E D B A (3) C B D A E (3) C B A E D (3) A E D B C (3) A B D C E (3) E C A B D (2) E A D C B (2) E A B C D (2) D E A C B (2) D C B A E (2) D A B E C (2) C E B A D (2) C D E B A (2) B D A C E (2) A B E D C (2) A B E C D (2) A B C E D (2) E D A C B (1) E D A B C (1) E C D B A (1) E C B A D (1) E A D B C (1) E A C D B (1) E A C B D (1) D C B E A (1) D C A E B (1) D B C A E (1) D B A C E (1) C E B D A (1) C D B E A (1) C B E A D (1) B C D A E (1) B C A E D (1) B C A D E (1) B A D C E (1) B A C E D (1) A E B D C (1) A E B C D (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 24 8 -2 12 B -24 0 2 -12 -4 C -8 -2 0 -16 -8 D 2 12 16 0 10 E -12 4 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 8 -2 12 B -24 0 2 -12 -4 C -8 -2 0 -16 -8 D 2 12 16 0 10 E -12 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=24 E=22 C=16 B=12 so B is eliminated. Round 2 votes counts: A=31 D=28 E=22 C=19 so C is eliminated. Round 3 votes counts: A=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:221 D:220 E:195 C:183 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 8 -2 12 B -24 0 2 -12 -4 C -8 -2 0 -16 -8 D 2 12 16 0 10 E -12 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 8 -2 12 B -24 0 2 -12 -4 C -8 -2 0 -16 -8 D 2 12 16 0 10 E -12 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 8 -2 12 B -24 0 2 -12 -4 C -8 -2 0 -16 -8 D 2 12 16 0 10 E -12 4 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4659: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) C D A B E (7) E A B D C (6) D E C B A (5) A B E C D (5) D C B E A (4) C D B A E (4) B A E C D (4) A E B C D (4) A B C E D (4) E D B C A (3) D E C A B (3) D C E A B (3) D C A E B (3) C B D A E (3) B E A D C (3) B E A C D (3) B C D A E (3) B A C E D (3) E D B A C (2) D C E B A (2) B C A D E (2) B A C D E (2) A C D B E (2) E D C B A (1) E D C A B (1) E D A C B (1) E D A B C (1) E B D A C (1) E A D C B (1) E A D B C (1) D A C E B (1) C B A D E (1) C A D B E (1) B E D C A (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 6 2 2 B 8 0 10 6 4 C -6 -10 0 0 -8 D -2 -6 0 0 -6 E -2 -4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 2 2 B 8 0 10 6 4 C -6 -10 0 0 -8 D -2 -6 0 0 -6 E -2 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=21 B=21 A=17 C=16 so C is eliminated. Round 2 votes counts: D=32 E=25 B=25 A=18 so A is eliminated. Round 3 votes counts: D=36 B=34 E=30 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:204 A:201 D:193 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 2 2 B 8 0 10 6 4 C -6 -10 0 0 -8 D -2 -6 0 0 -6 E -2 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 2 2 B 8 0 10 6 4 C -6 -10 0 0 -8 D -2 -6 0 0 -6 E -2 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 2 2 B 8 0 10 6 4 C -6 -10 0 0 -8 D -2 -6 0 0 -6 E -2 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4660: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (15) A D E B C (11) A E D C B (10) C B E D A (8) C B A E D (6) B C A D E (5) B D E A C (4) D E A B C (3) C B E A D (3) C B D E A (3) A C E D B (3) D A E B C (2) C E D B A (2) C E A D B (2) C A E D B (2) C A B E D (2) B D C E A (2) B A D E C (2) A E D B C (2) E D C A B (1) E D A C B (1) E D A B C (1) E C A D B (1) E A D C B (1) D A B E C (1) C E D A B (1) B D E C A (1) B D A E C (1) A E C D B (1) A D E C B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -8 8 2 B 4 0 2 8 10 C 8 -2 0 10 12 D -8 -8 -10 0 4 E -2 -10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 8 2 B 4 0 2 8 10 C 8 -2 0 10 12 D -8 -8 -10 0 4 E -2 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=30 A=30 C=29 D=6 E=5 so E is eliminated. Round 2 votes counts: A=31 C=30 B=30 D=9 so D is eliminated. Round 3 votes counts: A=39 C=31 B=30 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:212 A:199 D:189 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 8 2 B 4 0 2 8 10 C 8 -2 0 10 12 D -8 -8 -10 0 4 E -2 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 8 2 B 4 0 2 8 10 C 8 -2 0 10 12 D -8 -8 -10 0 4 E -2 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 8 2 B 4 0 2 8 10 C 8 -2 0 10 12 D -8 -8 -10 0 4 E -2 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4661: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) E D C B A (7) E D A B C (6) E B A C D (6) B A C D E (6) E A B D C (5) C B A D E (5) B C A D E (4) B A E C D (4) E D A C B (3) D E C A B (3) D C B A E (3) D C A B E (3) D A C B E (3) C B D A E (3) A B C D E (3) E C B D A (2) C D B A E (2) B C A E D (2) B A C E D (2) A B C E D (2) E C D B A (1) E B D C A (1) D E A C B (1) D C E B A (1) D C E A B (1) D C A E B (1) D A E C B (1) C E B D A (1) C D A B E (1) C A B D E (1) B E C A D (1) A E D B C (1) A D C B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -8 -10 0 B 2 0 -12 0 -4 C 8 12 0 -6 -8 D 10 0 6 0 -12 E 0 4 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.352157 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.647843 Sum of squares = 0.543714930357 Cumulative probabilities = A: 0.352157 B: 0.352157 C: 0.352157 D: 0.352157 E: 1.000000 A B C D E A 0 -2 -8 -10 0 B 2 0 -12 0 -4 C 8 12 0 -6 -8 D 10 0 6 0 -12 E 0 4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500258 Sum of squares = 0.500000132629 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 0.499742 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 B=19 D=17 C=13 A=9 so A is eliminated. Round 2 votes counts: E=43 B=25 D=18 C=14 so C is eliminated. Round 3 votes counts: E=44 B=35 D=21 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:203 D:202 B:193 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 -10 0 B 2 0 -12 0 -4 C 8 12 0 -6 -8 D 10 0 6 0 -12 E 0 4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500258 Sum of squares = 0.500000132629 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 0.499742 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -10 0 B 2 0 -12 0 -4 C 8 12 0 -6 -8 D 10 0 6 0 -12 E 0 4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500258 Sum of squares = 0.500000132629 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 0.499742 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -10 0 B 2 0 -12 0 -4 C 8 12 0 -6 -8 D 10 0 6 0 -12 E 0 4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500258 Sum of squares = 0.500000132629 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 0.499742 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4662: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) B D E A C (7) C A E B D (6) D B A C E (5) C A E D B (5) E C A B D (4) C A D E B (4) B E D A C (4) A C B D E (4) D E B C A (3) C A D B E (3) E D B C A (2) E C A D B (2) E B C A D (2) E B A C D (2) D E C B A (2) D E C A B (2) D C A E B (2) D C A B E (2) D A C B E (2) B A D C E (2) A C E B D (2) A C D B E (2) A C B E D (2) E D C B A (1) E D C A B (1) E C B D A (1) E C B A D (1) E B D C A (1) E B C D A (1) E A B C D (1) D C E A B (1) D B E C A (1) D B A E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C D A E B (1) B E A D C (1) B E A C D (1) B D A E C (1) B D A C E (1) B A E D C (1) B A E C D (1) B A C E D (1) A E C B D (1) Total count = 100 A B C D E A 0 0 0 0 0 B 0 0 -6 -2 0 C 0 6 0 -2 -4 D 0 2 2 0 8 E 0 0 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.598911 B: 0.000000 C: 0.000000 D: 0.401089 E: 0.000000 Sum of squares = 0.519566684222 Cumulative probabilities = A: 0.598911 B: 0.598911 C: 0.598911 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 0 0 B 0 0 -6 -2 0 C 0 6 0 -2 -4 D 0 2 2 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=21 B=20 E=19 A=11 so A is eliminated. Round 2 votes counts: C=31 D=29 E=20 B=20 so E is eliminated. Round 3 votes counts: C=40 D=33 B=27 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:206 A:200 C:200 E:198 B:196 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 0 0 B 0 0 -6 -2 0 C 0 6 0 -2 -4 D 0 2 2 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 0 B 0 0 -6 -2 0 C 0 6 0 -2 -4 D 0 2 2 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 0 B 0 0 -6 -2 0 C 0 6 0 -2 -4 D 0 2 2 0 8 E 0 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4663: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) D A C B E (12) A C D B E (8) A D C B E (7) E A B C D (5) A C B E D (5) D C B A E (4) A E C B D (4) A C B D E (4) E B D C A (3) D C A B E (3) E D B C A (2) E D B A C (2) E B A C D (2) E A D C B (2) E A C B D (2) D B C A E (2) B C D A E (2) A C E B D (2) E B C D A (1) D E A C B (1) D E A B C (1) D B C E A (1) D A E C B (1) D A C E B (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B C E D A (1) B C E A D (1) B C A E D (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 22 20 24 20 B -22 0 -18 2 12 C -20 18 0 14 16 D -24 -2 -14 0 2 E -20 -12 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 20 24 20 B -22 0 -18 2 12 C -20 18 0 14 16 D -24 -2 -14 0 2 E -20 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=31 D=26 B=8 C=2 so C is eliminated. Round 2 votes counts: E=33 A=33 D=26 B=8 so B is eliminated. Round 3 votes counts: E=37 A=35 D=28 so D is eliminated. Round 4 votes counts: A=60 E=40 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:243 C:214 B:187 D:181 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 20 24 20 B -22 0 -18 2 12 C -20 18 0 14 16 D -24 -2 -14 0 2 E -20 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 20 24 20 B -22 0 -18 2 12 C -20 18 0 14 16 D -24 -2 -14 0 2 E -20 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 20 24 20 B -22 0 -18 2 12 C -20 18 0 14 16 D -24 -2 -14 0 2 E -20 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4664: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (14) D C B E A (11) E A B C D (8) E A B D C (7) D E C B A (6) A B C D E (6) A B E C D (5) E D C B A (4) E A D C B (4) D C B A E (4) C B D A E (4) B C D A E (4) E D A C B (3) E A D B C (3) D C E B A (3) A B C E D (3) E D C A B (2) C D B A E (2) E D B A C (1) E D A B C (1) E B A C D (1) D C A B E (1) C B A D E (1) B C A D E (1) A C B D E (1) Total count = 100 A B C D E A 0 16 14 8 -8 B -16 0 8 10 -14 C -14 -8 0 0 -18 D -8 -10 0 0 -12 E 8 14 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 14 8 -8 B -16 0 8 10 -14 C -14 -8 0 0 -18 D -8 -10 0 0 -12 E 8 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=29 D=25 C=7 B=5 so B is eliminated. Round 2 votes counts: E=34 A=29 D=25 C=12 so C is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:215 B:194 D:185 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 14 8 -8 B -16 0 8 10 -14 C -14 -8 0 0 -18 D -8 -10 0 0 -12 E 8 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 8 -8 B -16 0 8 10 -14 C -14 -8 0 0 -18 D -8 -10 0 0 -12 E 8 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 8 -8 B -16 0 8 10 -14 C -14 -8 0 0 -18 D -8 -10 0 0 -12 E 8 14 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4665: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) E D C B A (7) B A E C D (7) D C E A B (5) A B C E D (5) D C A E B (4) C D E A B (4) C D A E B (4) C A B E D (4) B E A D C (4) E B D A C (3) E B A C D (3) B E A C D (3) A C B E D (3) E C B A D (2) D C A B E (2) D A C B E (2) B A E D C (2) A D C B E (2) E D C A B (1) E D B A C (1) E C B D A (1) E C A B D (1) E B D C A (1) E B C D A (1) E B C A D (1) D E C A B (1) D E B C A (1) D E B A C (1) D B A E C (1) D B A C E (1) D A B C E (1) C E A B D (1) C D A B E (1) C A E B D (1) C A D E B (1) C A D B E (1) B D A E C (1) B A C E D (1) A D B C E (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -10 -8 -4 B 4 0 -18 -4 -10 C 10 18 0 -4 -8 D 8 4 4 0 -8 E 4 10 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 -8 -4 B 4 0 -18 -4 -10 C 10 18 0 -4 -8 D 8 4 4 0 -8 E 4 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=22 B=18 C=17 A=14 so A is eliminated. Round 2 votes counts: D=32 B=25 E=22 C=21 so C is eliminated. Round 3 votes counts: D=43 B=33 E=24 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:215 C:208 D:204 A:187 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 -8 -4 B 4 0 -18 -4 -10 C 10 18 0 -4 -8 D 8 4 4 0 -8 E 4 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -8 -4 B 4 0 -18 -4 -10 C 10 18 0 -4 -8 D 8 4 4 0 -8 E 4 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -8 -4 B 4 0 -18 -4 -10 C 10 18 0 -4 -8 D 8 4 4 0 -8 E 4 10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4666: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) D C E B A (6) A B E C D (6) E D C A B (5) C D E A B (5) B A E C D (5) D E C A B (4) B C D A E (4) B A C D E (4) E A D C B (3) D C E A B (3) C E D A B (3) C D E B A (3) B D C A E (3) B C A D E (3) A B E D C (3) E D A C B (2) E C D A B (2) E A C D B (2) D E C B A (2) D C B E A (2) B A D E C (2) A E D B C (2) A E C D B (2) E D A B C (1) D E A C B (1) D B C E A (1) C D B E A (1) C B D A E (1) C B A E D (1) C B A D E (1) B D C E A (1) B D A E C (1) B C D E A (1) B A D C E (1) A E D C B (1) A E C B D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -4 -4 4 B 4 0 -2 -2 0 C 4 2 0 10 6 D 4 2 -10 0 0 E -4 0 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -4 4 B 4 0 -2 -2 0 C 4 2 0 10 6 D 4 2 -10 0 0 E -4 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=19 A=17 E=15 C=15 so E is eliminated. Round 2 votes counts: B=34 D=27 A=22 C=17 so C is eliminated. Round 3 votes counts: D=41 B=37 A=22 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:211 B:200 D:198 A:196 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -4 4 B 4 0 -2 -2 0 C 4 2 0 10 6 D 4 2 -10 0 0 E -4 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -4 4 B 4 0 -2 -2 0 C 4 2 0 10 6 D 4 2 -10 0 0 E -4 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -4 4 B 4 0 -2 -2 0 C 4 2 0 10 6 D 4 2 -10 0 0 E -4 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4667: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (13) D A C E B (11) B E C A D (10) B C A D E (10) E B D A C (7) C A D B E (5) E D A B C (4) C B A D E (4) A C D B E (4) E B C D A (3) D A C B E (3) C D A B E (3) C A B D E (3) B C E A D (3) B C A E D (3) A D C B E (3) E D B A C (2) E B D C A (2) E B C A D (2) E D C A B (1) D E A C B (1) D C E A B (1) D A E C B (1) B A C D E (1) Total count = 100 A B C D E A 0 6 0 -4 2 B -6 0 -6 -4 4 C 0 6 0 2 8 D 4 4 -2 0 0 E -2 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.182499 B: 0.000000 C: 0.817501 D: 0.000000 E: 0.000000 Sum of squares = 0.701613356821 Cumulative probabilities = A: 0.182499 B: 0.182499 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -4 2 B -6 0 -6 -4 4 C 0 6 0 2 8 D 4 4 -2 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559661 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=27 D=17 C=15 A=7 so A is eliminated. Round 2 votes counts: E=34 B=27 D=20 C=19 so C is eliminated. Round 3 votes counts: E=34 B=34 D=32 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:208 D:203 A:202 B:194 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 -4 2 B -6 0 -6 -4 4 C 0 6 0 2 8 D 4 4 -2 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559661 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -4 2 B -6 0 -6 -4 4 C 0 6 0 2 8 D 4 4 -2 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559661 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -4 2 B -6 0 -6 -4 4 C 0 6 0 2 8 D 4 4 -2 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559661 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4668: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) D E B C A (7) D E A C B (6) C B D E A (5) A E B C D (5) E D B C A (4) E D B A C (4) E D A B C (4) E A B D C (4) C B D A E (4) B E C D A (4) B C D E A (4) E D A C B (3) E A D B C (3) D E C B A (3) B A C E D (3) A C B E D (3) C B A D E (2) A B C E D (2) E B D C A (1) E B A C D (1) D C B E A (1) D C B A E (1) D C A E B (1) D B C E A (1) D A E C B (1) D A C E B (1) C D B E A (1) C A D B E (1) B D E C A (1) B C E D A (1) A E D C B (1) A E D B C (1) A E C B D (1) A D E C B (1) A C E B D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -6 -16 -16 B 18 0 24 8 -8 C 6 -24 0 2 -12 D 16 -8 -2 0 -16 E 16 8 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -6 -16 -16 B 18 0 24 8 -8 C 6 -24 0 2 -12 D 16 -8 -2 0 -16 E 16 8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 D=22 A=17 C=13 so C is eliminated. Round 2 votes counts: B=35 E=24 D=23 A=18 so A is eliminated. Round 3 votes counts: B=41 E=33 D=26 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:221 D:195 C:186 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -6 -16 -16 B 18 0 24 8 -8 C 6 -24 0 2 -12 D 16 -8 -2 0 -16 E 16 8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -16 -16 B 18 0 24 8 -8 C 6 -24 0 2 -12 D 16 -8 -2 0 -16 E 16 8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -16 -16 B 18 0 24 8 -8 C 6 -24 0 2 -12 D 16 -8 -2 0 -16 E 16 8 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4669: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (15) D E B C A (8) D A C E B (8) A C D B E (7) B E C A D (6) D A E C B (5) A C B E D (5) E B A C D (4) D C A B E (4) C A B E D (4) E D B C A (2) E B C A D (2) D E B A C (2) D E A C B (2) C A D B E (2) C A B D E (2) B C A E D (2) A E D B C (2) A C B D E (2) E B D A C (1) E B A D C (1) E A B C D (1) D E C A B (1) D C E B A (1) D C A E B (1) D B E C A (1) D B C E A (1) D A C B E (1) C B A D E (1) B E D C A (1) B C E A D (1) B C D E A (1) A D C E B (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -12 -10 -2 B 0 0 4 0 -14 C 12 -4 0 -18 -8 D 10 0 18 0 4 E 2 14 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.163887 C: 0.000000 D: 0.836113 E: 0.000000 Sum of squares = 0.725944444414 Cumulative probabilities = A: 0.000000 B: 0.163887 C: 0.163887 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -10 -2 B 0 0 4 0 -14 C 12 -4 0 -18 -8 D 10 0 18 0 4 E 2 14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321018202 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=26 A=19 B=11 C=9 so C is eliminated. Round 2 votes counts: D=35 A=27 E=26 B=12 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:210 B:195 C:191 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -12 -10 -2 B 0 0 4 0 -14 C 12 -4 0 -18 -8 D 10 0 18 0 4 E 2 14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321018202 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -10 -2 B 0 0 4 0 -14 C 12 -4 0 -18 -8 D 10 0 18 0 4 E 2 14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321018202 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -10 -2 B 0 0 4 0 -14 C 12 -4 0 -18 -8 D 10 0 18 0 4 E 2 14 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.777778 E: 0.000000 Sum of squares = 0.654321018202 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4670: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) A D B E C (6) A D C B E (5) D A C E B (4) C D E B A (4) E B C D A (3) D E C B A (3) D A E B C (3) D A C B E (3) C E D B A (3) C D A E B (3) C B E A D (3) C A B E D (3) E C B D A (2) E B D C A (2) E B C A D (2) E B A D C (2) D E B A C (2) D C E A B (2) D C A B E (2) C E B D A (2) C D E A B (2) C D A B E (2) C B A E D (2) C A B D E (2) B E A C D (2) A C D B E (2) A B D E C (2) A B D C E (2) A B C E D (2) E B D A C (1) E B A C D (1) D E C A B (1) D E B C A (1) D E A B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C A D B E (1) B C E A D (1) B A E D C (1) B A E C D (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 22 -10 -12 18 B -22 0 -22 -20 -6 C 10 22 0 -8 22 D 12 20 8 0 28 E -18 6 -22 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -10 -12 18 B -22 0 -22 -20 -6 C 10 22 0 -8 22 D 12 20 8 0 28 E -18 6 -22 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=28 A=22 E=13 B=5 so B is eliminated. Round 2 votes counts: D=32 C=29 A=24 E=15 so E is eliminated. Round 3 votes counts: C=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:234 C:223 A:209 E:169 B:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 -10 -12 18 B -22 0 -22 -20 -6 C 10 22 0 -8 22 D 12 20 8 0 28 E -18 6 -22 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -10 -12 18 B -22 0 -22 -20 -6 C 10 22 0 -8 22 D 12 20 8 0 28 E -18 6 -22 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -10 -12 18 B -22 0 -22 -20 -6 C 10 22 0 -8 22 D 12 20 8 0 28 E -18 6 -22 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4671: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) D C B E A (6) A E B C D (6) E B A C D (5) E A B C D (4) D E C B A (4) D C B A E (4) A E C B D (4) E D B A C (3) E A B D C (3) D E B C A (3) D C A B E (3) A C B E D (3) A C B D E (3) A B E C D (3) E D B C A (2) E B A D C (2) D C E B A (2) D B C E A (2) C D B A E (2) C D A B E (2) C A B D E (2) B A C E D (2) A D E C B (2) E D A C B (1) E D A B C (1) E B D C A (1) E B D A C (1) E B C A D (1) E A D C B (1) E A D B C (1) D E A C B (1) D B E C A (1) D A C B E (1) C B D A E (1) C A D B E (1) B E D C A (1) B E C A D (1) B E A C D (1) B C D E A (1) B C A E D (1) B C A D E (1) B A E C D (1) A E D C B (1) Total count = 100 A B C D E A 0 2 16 14 2 B -2 0 12 12 2 C -16 -12 0 6 -10 D -14 -12 -6 0 -16 E -2 -2 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 14 2 B -2 0 12 12 2 C -16 -12 0 6 -10 D -14 -12 -6 0 -16 E -2 -2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=27 E=26 B=9 C=8 so C is eliminated. Round 2 votes counts: A=33 D=31 E=26 B=10 so B is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:212 E:211 C:184 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 14 2 B -2 0 12 12 2 C -16 -12 0 6 -10 D -14 -12 -6 0 -16 E -2 -2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 14 2 B -2 0 12 12 2 C -16 -12 0 6 -10 D -14 -12 -6 0 -16 E -2 -2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 14 2 B -2 0 12 12 2 C -16 -12 0 6 -10 D -14 -12 -6 0 -16 E -2 -2 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4672: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (6) D A B E C (6) C E B A D (6) A D C E B (6) E B C D A (5) D A B C E (5) C B E A D (5) C A D E B (5) B C E D A (5) A D C B E (5) E C B A D (4) B D A E C (4) E D A B C (3) D A E B C (3) B D A C E (3) B C A D E (3) A D B C E (3) E C B D A (2) E A D C B (2) D B A C E (2) C E A D B (2) C B A D E (2) C A D B E (2) E D A C B (1) E B D C A (1) E B D A C (1) D B A E C (1) D A E C B (1) C B A E D (1) B D E A C (1) B C D A E (1) A D E C B (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 6 0 10 12 B -6 0 -4 -12 0 C 0 4 0 0 14 D -10 12 0 0 10 E -12 0 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.793441 B: 0.000000 C: 0.206559 D: 0.000000 E: 0.000000 Sum of squares = 0.672214655482 Cumulative probabilities = A: 0.793441 B: 0.793441 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 10 12 B -6 0 -4 -12 0 C 0 4 0 0 14 D -10 12 0 0 10 E -12 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 D=18 B=17 A=17 so B is eliminated. Round 2 votes counts: C=32 D=26 E=25 A=17 so A is eliminated. Round 3 votes counts: D=42 C=33 E=25 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:209 D:206 B:189 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 10 12 B -6 0 -4 -12 0 C 0 4 0 0 14 D -10 12 0 0 10 E -12 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 10 12 B -6 0 -4 -12 0 C 0 4 0 0 14 D -10 12 0 0 10 E -12 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 10 12 B -6 0 -4 -12 0 C 0 4 0 0 14 D -10 12 0 0 10 E -12 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4673: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) D C E A B (10) A B E C D (8) E C A B D (7) D C E B A (6) B A D E C (6) C E D A B (5) C E A B D (5) D B A E C (4) B A E D C (4) B A E C D (4) E C B A D (3) D A B C E (3) C E D B A (3) E B A C D (2) D C B E A (2) D B C A E (2) D A C B E (2) B D A E C (2) A B D E C (2) D C A B E (1) D A C E B (1) C E B A D (1) C E A D B (1) B E C D A (1) A D B C E (1) A C E B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 6 -6 8 B 2 0 4 -4 10 C -6 -4 0 -18 12 D 6 4 18 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -6 8 B 2 0 4 -4 10 C -6 -4 0 -18 12 D 6 4 18 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 B=17 C=15 A=14 E=12 so E is eliminated. Round 2 votes counts: D=42 C=25 B=19 A=14 so A is eliminated. Round 3 votes counts: D=43 B=31 C=26 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:206 A:203 C:192 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -6 8 B 2 0 4 -4 10 C -6 -4 0 -18 12 D 6 4 18 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -6 8 B 2 0 4 -4 10 C -6 -4 0 -18 12 D 6 4 18 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -6 8 B 2 0 4 -4 10 C -6 -4 0 -18 12 D 6 4 18 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4674: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (5) D A C B E (5) A E B D C (5) A D B C E (5) E C B D A (4) E B C A D (4) D C A B E (4) E D A B C (3) D E C B A (3) D E A B C (3) D A E B C (3) D A C E B (3) C D B E A (3) C A B D E (3) A B C E D (3) E D B A C (2) E B A C D (2) D E C A B (2) D E A C B (2) D A E C B (2) C E D B A (2) C E B D A (2) C B E D A (2) C B D E A (2) B C E A D (2) A D C B E (2) E C B A D (1) E B D A C (1) E B A D C (1) E A B C D (1) D C E B A (1) D C B A E (1) D C A E B (1) C D B A E (1) C D A B E (1) C B E A D (1) C B D A E (1) C B A E D (1) C A D B E (1) B E C A D (1) B E A C D (1) B C A E D (1) B A C E D (1) A E D B C (1) A C B D E (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 0 -18 -2 B -8 0 -4 -2 -8 C 0 4 0 -6 2 D 18 2 6 0 4 E 2 8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -18 -2 B -8 0 -4 -2 -8 C 0 4 0 -6 2 D 18 2 6 0 4 E 2 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=24 C=20 A=20 B=6 so B is eliminated. Round 2 votes counts: D=30 E=26 C=23 A=21 so A is eliminated. Round 3 votes counts: D=39 E=33 C=28 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:202 C:200 A:194 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -18 -2 B -8 0 -4 -2 -8 C 0 4 0 -6 2 D 18 2 6 0 4 E 2 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -18 -2 B -8 0 -4 -2 -8 C 0 4 0 -6 2 D 18 2 6 0 4 E 2 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -18 -2 B -8 0 -4 -2 -8 C 0 4 0 -6 2 D 18 2 6 0 4 E 2 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4675: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) E C B A D (6) B D A C E (5) D B A C E (4) D A B C E (4) C B A D E (4) E C A D B (3) E B D A C (3) C E B A D (3) B D E C A (3) B D C A E (3) A D C B E (3) A C D E B (3) E C A B D (2) E B D C A (2) E B C D A (2) E A D C B (2) E A C D B (2) D B A E C (2) D A B E C (2) C E A D B (2) C A E D B (2) C A E B D (2) C A D E B (2) C A D B E (2) B E D C A (2) B E D A C (2) B D E A C (2) B C D E A (2) A D E C B (2) A C D B E (2) E D A B C (1) E B C A D (1) D B E A C (1) D A E C B (1) D A C B E (1) C E A B D (1) B D C E A (1) B D A E C (1) B C A D E (1) A E D C B (1) A E C D B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -6 -2 14 B 14 0 2 10 8 C 6 -2 0 2 12 D 2 -10 -2 0 18 E -14 -8 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -2 14 B 14 0 2 10 8 C 6 -2 0 2 12 D 2 -10 -2 0 18 E -14 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=24 C=18 D=15 A=14 so A is eliminated. Round 2 votes counts: B=29 E=26 C=24 D=21 so D is eliminated. Round 3 votes counts: B=42 E=29 C=29 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:209 D:204 A:196 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -2 14 B 14 0 2 10 8 C 6 -2 0 2 12 D 2 -10 -2 0 18 E -14 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -2 14 B 14 0 2 10 8 C 6 -2 0 2 12 D 2 -10 -2 0 18 E -14 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -2 14 B 14 0 2 10 8 C 6 -2 0 2 12 D 2 -10 -2 0 18 E -14 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4676: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (6) A B E D C (6) E A C B D (5) D B C A E (5) C D E B A (5) B A D E C (5) A E B C D (5) E C A B D (4) E A B C D (3) D C B E A (3) D C B A E (3) C E D B A (3) C D B E A (3) E C A D B (2) E B A D C (2) E A B D C (2) D B A C E (2) C E D A B (2) C D E A B (2) C A D B E (2) B E A D C (2) B D E A C (2) B D A E C (2) B A E D C (2) A E C B D (2) A E B D C (2) A B D C E (2) E D B C A (1) E C B D A (1) E C B A D (1) E B C D A (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C E A (1) D B A E C (1) C E A B D (1) C D B A E (1) C D A B E (1) C A D E B (1) B E D A C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -4 14 -12 B 0 0 -2 4 -6 C 4 2 0 6 -8 D -14 -4 -6 0 -8 E 12 6 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -4 14 -12 B 0 0 -2 4 -6 C 4 2 0 6 -8 D -14 -4 -6 0 -8 E 12 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 A=19 D=18 B=14 so B is eliminated. Round 2 votes counts: C=27 A=26 E=25 D=22 so D is eliminated. Round 3 votes counts: C=40 A=31 E=29 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:217 C:202 A:199 B:198 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 14 -12 B 0 0 -2 4 -6 C 4 2 0 6 -8 D -14 -4 -6 0 -8 E 12 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 14 -12 B 0 0 -2 4 -6 C 4 2 0 6 -8 D -14 -4 -6 0 -8 E 12 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 14 -12 B 0 0 -2 4 -6 C 4 2 0 6 -8 D -14 -4 -6 0 -8 E 12 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4677: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (11) A B E D C (10) B A C D E (9) E D C A B (8) C D E B A (8) A E D B C (7) E D A C B (5) D E C B A (5) C B D E A (4) B A C E D (4) A E B D C (4) D C E B A (3) E D C B A (2) E A D C B (2) D C E A B (2) C D B E A (2) B C D A E (2) A B C D E (2) E D B A C (1) E A D B C (1) D E C A B (1) C B D A E (1) B C E D A (1) B C A E D (1) A E D C B (1) A D E C B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -2 10 10 B 8 0 8 0 -4 C 2 -8 0 -6 4 D -10 0 6 0 4 E -10 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.454545 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.371900826109 Cumulative probabilities = A: 0.181818 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 A B C D E A 0 -8 -2 10 10 B 8 0 8 0 -4 C 2 -8 0 -6 4 D -10 0 6 0 4 E -10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.454545 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.371900825986 Cumulative probabilities = A: 0.181818 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=27 E=19 C=15 D=11 so D is eliminated. Round 2 votes counts: B=28 A=27 E=25 C=20 so C is eliminated. Round 3 votes counts: E=38 B=35 A=27 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:206 A:205 D:200 C:196 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 10 10 B 8 0 8 0 -4 C 2 -8 0 -6 4 D -10 0 6 0 4 E -10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.454545 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.371900825986 Cumulative probabilities = A: 0.181818 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 10 10 B 8 0 8 0 -4 C 2 -8 0 -6 4 D -10 0 6 0 4 E -10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.454545 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.371900825986 Cumulative probabilities = A: 0.181818 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 10 10 B 8 0 8 0 -4 C 2 -8 0 -6 4 D -10 0 6 0 4 E -10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.454545 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.371900825986 Cumulative probabilities = A: 0.181818 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4678: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (14) A B C D E (11) E D C B A (9) C E D B A (7) D E C A B (6) A C B E D (5) D E A C B (4) D E A B C (4) C B E D A (4) C B A E D (4) A D B E C (4) A B C E D (4) E C D B A (3) A D E C B (3) A D E B C (3) C E D A B (2) B A C D E (2) E D B C A (1) D E C B A (1) D E B C A (1) D E B A C (1) D B A E C (1) D A E B C (1) C E B D A (1) C A B E D (1) B D E A C (1) B C A E D (1) A B D C E (1) Total count = 100 A B C D E A 0 26 18 6 10 B -26 0 0 -2 6 C -18 0 0 -10 -14 D -6 2 10 0 16 E -10 -6 14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 18 6 10 B -26 0 0 -2 6 C -18 0 0 -10 -14 D -6 2 10 0 16 E -10 -6 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 D=19 C=19 E=13 B=4 so B is eliminated. Round 2 votes counts: A=47 D=20 C=20 E=13 so E is eliminated. Round 3 votes counts: A=47 D=30 C=23 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:230 D:211 E:191 B:189 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 18 6 10 B -26 0 0 -2 6 C -18 0 0 -10 -14 D -6 2 10 0 16 E -10 -6 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 18 6 10 B -26 0 0 -2 6 C -18 0 0 -10 -14 D -6 2 10 0 16 E -10 -6 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 18 6 10 B -26 0 0 -2 6 C -18 0 0 -10 -14 D -6 2 10 0 16 E -10 -6 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4679: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) D C B E A (6) E A B C D (5) D C B A E (5) B C A D E (5) B A E C D (5) A B E C D (5) E A D C B (4) D E A C B (4) D C E B A (4) C D B E A (4) B C A E D (4) A E B C D (4) E D A C B (3) E A C D B (3) C D B A E (3) E A D B C (2) C D E B A (2) C B D A E (2) B C D A E (2) B A C E D (2) B A C D E (2) E A B D C (1) D C E A B (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C B E (1) C E D B A (1) C E B A D (1) C B E D A (1) C B D E A (1) B D C A E (1) B C E A D (1) B A D C E (1) A E D B C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 10 12 12 B 6 0 4 4 6 C -10 -4 0 6 2 D -12 -4 -6 0 -4 E -12 -6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 12 12 B 6 0 4 4 6 C -10 -4 0 6 2 D -12 -4 -6 0 -4 E -12 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998423 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=23 A=20 E=18 C=15 so C is eliminated. Round 2 votes counts: D=33 B=27 E=20 A=20 so E is eliminated. Round 3 votes counts: D=37 A=35 B=28 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:210 C:197 E:192 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 12 12 B 6 0 4 4 6 C -10 -4 0 6 2 D -12 -4 -6 0 -4 E -12 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998423 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 12 12 B 6 0 4 4 6 C -10 -4 0 6 2 D -12 -4 -6 0 -4 E -12 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998423 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 12 12 B 6 0 4 4 6 C -10 -4 0 6 2 D -12 -4 -6 0 -4 E -12 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998423 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4680: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) E A C B D (9) E A B D C (8) B D E C A (8) B D C E A (8) C A D B E (6) A C E D B (6) D B C A E (5) A E C B D (4) E B A D C (3) C D B A E (3) C B D A E (3) C A E B D (3) C A B D E (3) E B D A C (2) E A C D B (2) D C B A E (2) D B C E A (2) C D A B E (2) C A D E B (2) B E D A C (2) B D C A E (2) E D B A C (1) D B E A C (1) C B A E D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 14 0 18 8 B -14 0 -16 14 -4 C 0 16 0 12 0 D -18 -14 -12 0 -4 E -8 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.562533 B: 0.000000 C: 0.437467 D: 0.000000 E: 0.000000 Sum of squares = 0.507820668477 Cumulative probabilities = A: 0.562533 B: 0.562533 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 18 8 B -14 0 -16 14 -4 C 0 16 0 12 0 D -18 -14 -12 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 A=22 B=20 D=10 so D is eliminated. Round 2 votes counts: B=28 E=25 C=25 A=22 so A is eliminated. Round 3 votes counts: E=39 C=33 B=28 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:220 C:214 E:200 B:190 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 18 8 B -14 0 -16 14 -4 C 0 16 0 12 0 D -18 -14 -12 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 18 8 B -14 0 -16 14 -4 C 0 16 0 12 0 D -18 -14 -12 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 18 8 B -14 0 -16 14 -4 C 0 16 0 12 0 D -18 -14 -12 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4681: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) D E B C A (7) A B D E C (7) D E C A B (5) A B C E D (5) A D B E C (4) D A B E C (3) B A C E D (3) E D B C A (2) D E B A C (2) D C E A B (2) D C A E B (2) C E D B A (2) C B E D A (2) C A B E D (2) B D A E C (2) B C A E D (2) B A D E C (2) A D C E B (2) A C D E B (2) A C D B E (2) A B E C D (2) A B D C E (2) A B C D E (2) E D C B A (1) E C D B A (1) E B C D A (1) D E A C B (1) D B E A C (1) D A E C B (1) D A C E B (1) C E D A B (1) C E B D A (1) C E B A D (1) C D A E B (1) C B A E D (1) C A E D B (1) C A E B D (1) C A D E B (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B A E D C (1) B A E C D (1) A D E C B (1) A D B C E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 8 2 2 14 B -8 0 8 -14 0 C -2 -8 0 -20 -14 D -2 14 20 0 26 E -14 0 14 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 2 14 B -8 0 8 -14 0 C -2 -8 0 -20 -14 D -2 14 20 0 26 E -14 0 14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996416 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=32 B=15 C=14 E=5 so E is eliminated. Round 2 votes counts: D=37 A=32 B=16 C=15 so C is eliminated. Round 3 votes counts: D=42 A=37 B=21 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:229 A:213 B:193 E:187 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 2 14 B -8 0 8 -14 0 C -2 -8 0 -20 -14 D -2 14 20 0 26 E -14 0 14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996416 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 14 B -8 0 8 -14 0 C -2 -8 0 -20 -14 D -2 14 20 0 26 E -14 0 14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996416 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 14 B -8 0 8 -14 0 C -2 -8 0 -20 -14 D -2 14 20 0 26 E -14 0 14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996416 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4682: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A B E D C (7) D E C A B (4) D E A C B (4) A B E C D (4) A B C E D (4) D C E B A (3) C D A E B (3) B A C E D (3) A E D B C (3) A D C E B (3) A C D B E (3) A C B D E (3) D C E A B (2) C D E A B (2) C D B E A (2) C D A B E (2) C B D E A (2) C B D A E (2) C B A D E (2) B A E C D (2) A E B D C (2) A D E B C (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) E B D C A (1) E B D A C (1) E B A D C (1) E A D B C (1) D E C B A (1) D A C E B (1) C D B A E (1) C A D E B (1) C A B D E (1) B E A D C (1) B E A C D (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E D C (1) A D E C B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 20 10 4 12 B -20 0 -14 -12 -6 C -10 14 0 8 12 D -4 12 -8 0 18 E -12 6 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 4 12 B -20 0 -14 -12 -6 C -10 14 0 8 12 D -4 12 -8 0 18 E -12 6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=28 D=15 B=11 E=10 so E is eliminated. Round 2 votes counts: A=37 C=28 D=21 B=14 so B is eliminated. Round 3 votes counts: A=46 C=31 D=23 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:212 D:209 E:182 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 4 12 B -20 0 -14 -12 -6 C -10 14 0 8 12 D -4 12 -8 0 18 E -12 6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 4 12 B -20 0 -14 -12 -6 C -10 14 0 8 12 D -4 12 -8 0 18 E -12 6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 4 12 B -20 0 -14 -12 -6 C -10 14 0 8 12 D -4 12 -8 0 18 E -12 6 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999141 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4683: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) A D B C E (11) E C D B A (6) D A B C E (6) A B D E C (6) B A D E C (5) B A D C E (5) E C D A B (4) D A C B E (4) A B D C E (4) E B C A D (3) D C A B E (3) C E D A B (3) C E B D A (3) E C A B D (2) D B A C E (2) C E D B A (2) C D E A B (2) C D B A E (2) C B D E A (2) B A E D C (2) E C B A D (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D B C (1) C D B E A (1) C D A E B (1) C D A B E (1) B E A D C (1) B C D A E (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 0 2 -10 10 B 0 0 4 0 14 C -2 -4 0 -6 6 D 10 0 6 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.450965 C: 0.000000 D: 0.549035 E: 0.000000 Sum of squares = 0.504808786465 Cumulative probabilities = A: 0.000000 B: 0.450965 C: 0.450965 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -10 10 B 0 0 4 0 14 C -2 -4 0 -6 6 D 10 0 6 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=23 C=17 D=15 B=14 so B is eliminated. Round 2 votes counts: A=35 E=32 C=18 D=15 so D is eliminated. Round 3 votes counts: A=47 E=32 C=21 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 B:209 A:201 C:197 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -10 10 B 0 0 4 0 14 C -2 -4 0 -6 6 D 10 0 6 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -10 10 B 0 0 4 0 14 C -2 -4 0 -6 6 D 10 0 6 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -10 10 B 0 0 4 0 14 C -2 -4 0 -6 6 D 10 0 6 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4684: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) D A C E B (6) C B E A D (5) B C D E A (5) E B A C D (4) D B C A E (4) B D C E A (4) A E C D B (4) A D E C B (4) E A C B D (3) E A B C D (3) C D A B E (3) C B D E A (3) B C E A D (3) A C D E B (3) D C B A E (2) D B A E C (2) D A E B C (2) C E A B D (2) C D B A E (2) C D A E B (2) C A E B D (2) B E D C A (2) B D E A C (2) E C A B D (1) E B A D C (1) E A D B C (1) E A C D B (1) D E A B C (1) D C A E B (1) D C A B E (1) D B A C E (1) D A E C B (1) D A C B E (1) C E B A D (1) C B D A E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E C A (1) B C E D A (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 6 2 2 -6 B -6 0 -8 6 -6 C -2 8 0 6 14 D -2 -6 -6 0 10 E 6 6 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.48760330575 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 6 2 2 -6 B -6 0 -8 6 -6 C -2 8 0 6 14 D -2 -6 -6 0 10 E 6 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305759 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=22 B=21 E=20 A=13 so A is eliminated. Round 2 votes counts: D=27 C=27 E=25 B=21 so B is eliminated. Round 3 votes counts: C=36 D=34 E=30 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:202 D:198 E:194 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 2 2 -6 B -6 0 -8 6 -6 C -2 8 0 6 14 D -2 -6 -6 0 10 E 6 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305759 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 -6 B -6 0 -8 6 -6 C -2 8 0 6 14 D -2 -6 -6 0 10 E 6 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305759 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 -6 B -6 0 -8 6 -6 C -2 8 0 6 14 D -2 -6 -6 0 10 E 6 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.272727 D: 0.000000 E: 0.090909 Sum of squares = 0.487603305759 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4685: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) E A B D C (6) B C E A D (6) C B D A E (5) B C E D A (5) E B A C D (4) E A D C B (4) D A C E B (4) C D B A E (4) B C A D E (4) B E C A D (3) B E A C D (3) B C D E A (3) B C D A E (3) A E D C B (3) E D C A B (2) D C E A B (2) C D A B E (2) A D E C B (2) A D E B C (2) A B E C D (2) E D C B A (1) E D A C B (1) E B C A D (1) E B A D C (1) D E A C B (1) D C A E B (1) D C A B E (1) D A C B E (1) C E D B A (1) C D E B A (1) C B D E A (1) A E D B C (1) A E B D C (1) A D C E B (1) A D C B E (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 8 22 -14 B -8 0 22 2 -2 C -8 -22 0 -4 -4 D -22 -2 4 0 -14 E 14 2 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 22 -14 B -8 0 22 2 -2 C -8 -22 0 -4 -4 D -22 -2 4 0 -14 E 14 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=27 A=18 C=14 D=10 so D is eliminated. Round 2 votes counts: E=32 B=27 A=23 C=18 so C is eliminated. Round 3 votes counts: B=37 E=36 A=27 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:212 B:207 D:183 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 22 -14 B -8 0 22 2 -2 C -8 -22 0 -4 -4 D -22 -2 4 0 -14 E 14 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 22 -14 B -8 0 22 2 -2 C -8 -22 0 -4 -4 D -22 -2 4 0 -14 E 14 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 22 -14 B -8 0 22 2 -2 C -8 -22 0 -4 -4 D -22 -2 4 0 -14 E 14 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4686: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (7) E B A D C (6) D C B E A (5) D C A E B (5) C D A B E (5) A E B C D (5) E D B C A (4) E B D C A (4) D C E B A (4) C D A E B (4) B A E C D (4) A B E C D (4) E B D A C (3) D C E A B (3) C A D B E (3) B E D A C (3) A B C E D (3) D E C B A (2) D E C A B (2) C A D E B (2) B D E C A (2) B D C E A (2) A C D B E (2) A C B D E (2) E D C A B (1) E D A C B (1) D E B C A (1) D C B A E (1) D B E C A (1) C A B D E (1) B E D C A (1) B E A C D (1) B A C D E (1) A E C D B (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -6 -8 -6 B 4 0 4 2 -2 C 6 -4 0 -16 -6 D 8 -2 16 0 -2 E 6 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -6 -8 -6 B 4 0 4 2 -2 C 6 -4 0 -16 -6 D 8 -2 16 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=21 A=21 E=19 C=15 so C is eliminated. Round 2 votes counts: D=33 A=27 B=21 E=19 so E is eliminated. Round 3 votes counts: D=39 B=34 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:210 E:208 B:204 C:190 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -6 -8 -6 B 4 0 4 2 -2 C 6 -4 0 -16 -6 D 8 -2 16 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -8 -6 B 4 0 4 2 -2 C 6 -4 0 -16 -6 D 8 -2 16 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -8 -6 B 4 0 4 2 -2 C 6 -4 0 -16 -6 D 8 -2 16 0 -2 E 6 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4687: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) A D E B C (11) A E B C D (9) A E D B C (8) D A E C B (6) C D B E A (5) C B E D A (5) E B C D A (4) E B C A D (4) B C E A D (4) A E B D C (4) A D E C B (4) D A C B E (3) D E C B A (2) D A E B C (2) B E C D A (2) B E C A D (2) A B E C D (2) E D B C A (1) E B A C D (1) D C E B A (1) D C B A E (1) D C A B E (1) D A C E B (1) C B D E A (1) C B A D E (1) B C E D A (1) A D C E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 8 8 6 12 B -8 0 10 -18 -18 C -8 -10 0 -14 -24 D -6 18 14 0 4 E -12 18 24 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 6 12 B -8 0 10 -18 -18 C -8 -10 0 -14 -24 D -6 18 14 0 4 E -12 18 24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 D=28 C=12 E=10 B=9 so B is eliminated. Round 2 votes counts: A=41 D=28 C=17 E=14 so E is eliminated. Round 3 votes counts: A=42 D=29 C=29 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:215 E:213 B:183 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 6 12 B -8 0 10 -18 -18 C -8 -10 0 -14 -24 D -6 18 14 0 4 E -12 18 24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 12 B -8 0 10 -18 -18 C -8 -10 0 -14 -24 D -6 18 14 0 4 E -12 18 24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 12 B -8 0 10 -18 -18 C -8 -10 0 -14 -24 D -6 18 14 0 4 E -12 18 24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4688: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) B C A D E (9) E D B C A (8) A C D B E (8) E D A C B (5) D E C A B (5) D A C E B (5) E B A D C (4) B A C E D (4) A C B D E (4) A B C D E (4) D C E A B (3) B C A E D (3) E D C B A (2) E D C A B (2) E D B A C (2) E B D A C (2) B E D C A (2) B E A C D (2) B C D E A (2) B A E C D (2) A C D E B (2) E A B D C (1) D E C B A (1) D E A C B (1) D A E C B (1) B E C D A (1) B E C A D (1) B A C D E (1) A D C E B (1) A C E D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 4 -2 -6 B 10 0 16 6 -10 C -4 -16 0 -8 -2 D 2 -6 8 0 -6 E 6 10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 4 -2 -6 B 10 0 16 6 -10 C -4 -16 0 -8 -2 D 2 -6 8 0 -6 E 6 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=27 A=22 D=16 so C is eliminated. Round 2 votes counts: E=35 B=27 A=22 D=16 so D is eliminated. Round 3 votes counts: E=45 A=28 B=27 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:211 D:199 A:193 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 4 -2 -6 B 10 0 16 6 -10 C -4 -16 0 -8 -2 D 2 -6 8 0 -6 E 6 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -2 -6 B 10 0 16 6 -10 C -4 -16 0 -8 -2 D 2 -6 8 0 -6 E 6 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -2 -6 B 10 0 16 6 -10 C -4 -16 0 -8 -2 D 2 -6 8 0 -6 E 6 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4689: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (18) D A B E C (15) C E A B D (7) D C A B E (4) E C B A D (3) C D A E B (3) C A E B D (3) B E D A C (3) B E A C D (3) A D B E C (3) D B E A C (2) D B A E C (2) D A C E B (2) C E B D A (2) C D B E A (2) C B E D A (2) C A D E B (2) B E C A D (2) B D E A C (2) A D E B C (2) A D C E B (2) E B C A D (1) E B A C D (1) D B E C A (1) D B C E A (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D E A B (1) B E A D C (1) B D E C A (1) A E C B D (1) A E B C D (1) A C E D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -10 6 -10 B -2 0 -12 6 -4 C 10 12 0 10 8 D -6 -6 -10 0 -2 E 10 4 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 6 -10 B -2 0 -12 6 -4 C 10 12 0 10 8 D -6 -6 -10 0 -2 E 10 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 D=29 B=12 A=12 E=5 so E is eliminated. Round 2 votes counts: C=45 D=29 B=14 A=12 so A is eliminated. Round 3 votes counts: C=47 D=36 B=17 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:204 A:194 B:194 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 6 -10 B -2 0 -12 6 -4 C 10 12 0 10 8 D -6 -6 -10 0 -2 E 10 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 6 -10 B -2 0 -12 6 -4 C 10 12 0 10 8 D -6 -6 -10 0 -2 E 10 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 6 -10 B -2 0 -12 6 -4 C 10 12 0 10 8 D -6 -6 -10 0 -2 E 10 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4690: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (10) A E D C B (9) D E C B A (6) D C B E A (6) C B D E A (6) E D C A B (5) C D B E A (5) B C D E A (5) A E B C D (5) E D C B A (4) B C A D E (4) A B C D E (4) E D A C B (3) C B E D A (2) B A C D E (2) A E B D C (2) A B E C D (2) A B C E D (2) E C D B A (1) D C E B A (1) D B C E A (1) C D E B A (1) B A C E D (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -14 -12 -14 B 10 0 -20 -8 2 C 14 20 0 0 0 D 12 8 0 0 2 E 14 -2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.194843 D: 0.805157 E: 0.000000 Sum of squares = 0.686241956365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.194843 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -12 -14 B 10 0 -20 -8 2 C 14 20 0 0 0 D 12 8 0 0 2 E 14 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999996153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 B=22 D=14 C=14 so D is eliminated. Round 2 votes counts: E=30 A=26 B=23 C=21 so C is eliminated. Round 3 votes counts: B=42 E=32 A=26 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:217 D:211 E:205 B:192 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -12 -14 B 10 0 -20 -8 2 C 14 20 0 0 0 D 12 8 0 0 2 E 14 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999996153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -12 -14 B 10 0 -20 -8 2 C 14 20 0 0 0 D 12 8 0 0 2 E 14 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999996153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -12 -14 B 10 0 -20 -8 2 C 14 20 0 0 0 D 12 8 0 0 2 E 14 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999996153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4691: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) D C B E A (7) A C D B E (7) C D A E B (6) A B E D C (6) C D E B A (5) B E A D C (5) E B A D C (4) A C B D E (4) C D E A B (3) C D A B E (3) C A D E B (3) C A D B E (3) B E D A C (3) E D C B A (2) E D B C A (2) E C D B A (2) E A B C D (2) D E C B A (2) D E B C A (2) D C B A E (2) B E D C A (2) B A D E C (2) A C E D B (2) A B E C D (2) A B C D E (2) E B D A C (1) C D B A E (1) B D A C E (1) B A E D C (1) A E C B D (1) A E B D C (1) A C E B D (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 -2 2 B 2 0 -12 -6 6 C 4 12 0 0 6 D 2 6 0 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.447192 D: 0.552808 E: 0.000000 Sum of squares = 0.505577400718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.447192 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -2 2 B 2 0 -12 -6 6 C 4 12 0 0 6 D 2 6 0 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=24 E=20 B=14 D=13 so D is eliminated. Round 2 votes counts: C=33 A=29 E=24 B=14 so B is eliminated. Round 3 votes counts: E=34 C=33 A=33 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:208 A:197 B:195 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 2 B 2 0 -12 -6 6 C 4 12 0 0 6 D 2 6 0 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 2 B 2 0 -12 -6 6 C 4 12 0 0 6 D 2 6 0 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 2 B 2 0 -12 -6 6 C 4 12 0 0 6 D 2 6 0 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4692: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) C D E B A (8) E B D A C (7) D E B A C (7) C A B E D (7) A C B E D (6) D C E B A (5) A B C E D (5) E B A D C (4) C A D B E (4) A B E C D (4) D A C E B (3) A B E D C (3) E B D C A (2) D C E A B (2) D C A E B (2) B E C A D (2) B E A D C (2) B E A C D (2) A C D B E (2) A C B D E (2) E D B C A (1) E C D B A (1) D E C B A (1) D E C A B (1) D A E B C (1) C E D B A (1) C E B A D (1) C D A E B (1) C A B D E (1) B E D A C (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 -10 2 -6 -14 B 10 0 2 0 -14 C -2 -2 0 -4 2 D 6 0 4 0 0 E 14 14 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.673616 E: 0.326384 Sum of squares = 0.560284898371 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.673616 E: 1.000000 A B C D E A 0 -10 2 -6 -14 B 10 0 2 0 -14 C -2 -2 0 -4 2 D 6 0 4 0 0 E 14 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=23 A=23 E=15 B=8 so B is eliminated. Round 2 votes counts: D=31 A=24 C=23 E=22 so E is eliminated. Round 3 votes counts: D=42 A=32 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:205 B:199 C:197 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 -6 -14 B 10 0 2 0 -14 C -2 -2 0 -4 2 D 6 0 4 0 0 E 14 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -6 -14 B 10 0 2 0 -14 C -2 -2 0 -4 2 D 6 0 4 0 0 E 14 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -6 -14 B 10 0 2 0 -14 C -2 -2 0 -4 2 D 6 0 4 0 0 E 14 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4693: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (13) D B E A C (8) C A E B D (7) C D A B E (6) B E A D C (6) E B A D C (5) E B D A C (4) C A D E B (4) E B C D A (3) E B C A D (3) E B A C D (3) D C A B E (3) D B A E C (3) A C D B E (3) E B D C A (2) C E B A D (2) C E A B D (2) C D E B A (2) B E D A C (2) B A E D C (2) A B D E C (2) E C B D A (1) E A B C D (1) D E B C A (1) D C E B A (1) D C B E A (1) D C B A E (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C A E D B (1) A E B D C (1) A D B E C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -10 16 2 B 4 0 0 -2 10 C 10 0 0 10 2 D -16 2 -10 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.353339 C: 0.646661 D: 0.000000 E: 0.000000 Sum of squares = 0.543019153353 Cumulative probabilities = A: 0.000000 B: 0.353339 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 16 2 B 4 0 0 -2 10 C 10 0 0 10 2 D -16 2 -10 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=22 D=20 B=10 A=9 so A is eliminated. Round 2 votes counts: C=42 E=23 D=21 B=14 so B is eliminated. Round 3 votes counts: C=43 E=34 D=23 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:206 A:202 E:191 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 16 2 B 4 0 0 -2 10 C 10 0 0 10 2 D -16 2 -10 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 16 2 B 4 0 0 -2 10 C 10 0 0 10 2 D -16 2 -10 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 16 2 B 4 0 0 -2 10 C 10 0 0 10 2 D -16 2 -10 0 4 E -2 -10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4694: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) D B A C E (12) C A E D B (9) E B C A D (7) B D E A C (7) D B C A E (6) D A C B E (4) B E C A D (4) B D A E C (4) A C E D B (4) E A C B D (3) C A D E B (3) D C A B E (2) D A C E B (2) C E A B D (2) B E D A C (2) B D E C A (2) E C B A D (1) D C B A E (1) D C A E B (1) D B A E C (1) D A B C E (1) C E A D B (1) C D A E B (1) C A E B D (1) B E A D C (1) B E A C D (1) A D C E B (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -10 8 10 B -2 0 -2 -2 -2 C 10 2 0 4 4 D -8 2 -4 0 0 E -10 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 8 10 B -2 0 -2 -2 -2 C 10 2 0 4 4 D -8 2 -4 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=25 B=21 C=17 A=7 so A is eliminated. Round 2 votes counts: D=32 E=25 C=22 B=21 so B is eliminated. Round 3 votes counts: D=45 E=33 C=22 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 A:205 B:196 D:195 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 8 10 B -2 0 -2 -2 -2 C 10 2 0 4 4 D -8 2 -4 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 8 10 B -2 0 -2 -2 -2 C 10 2 0 4 4 D -8 2 -4 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 8 10 B -2 0 -2 -2 -2 C 10 2 0 4 4 D -8 2 -4 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4695: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) B D E C A (10) C A E B D (9) D B E A C (8) B E C D A (5) D B A E C (4) A D C B E (4) E B D C A (3) C E B A D (3) B D C E A (3) B C D E A (3) A C D E B (3) A C D B E (3) E D A B C (2) E C A B D (2) E B C D A (2) D A B E C (2) C A B D E (2) B E D C A (2) B C E D A (2) A E C D B (2) A D E C B (2) A D E B C (2) E D B A C (1) E C B D A (1) E C B A D (1) E A C D B (1) D E B A C (1) D E A B C (1) D A E B C (1) C E A B D (1) C B E D A (1) C A B E D (1) A D C E B (1) Total count = 100 A B C D E A 0 0 -2 -4 -6 B 0 0 4 2 0 C 2 -4 0 6 -6 D 4 -2 -6 0 0 E 6 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.655520 C: 0.000000 D: 0.000000 E: 0.344480 Sum of squares = 0.548372777952 Cumulative probabilities = A: 0.000000 B: 0.655520 C: 0.655520 D: 0.655520 E: 1.000000 A B C D E A 0 0 -2 -4 -6 B 0 0 4 2 0 C 2 -4 0 6 -6 D 4 -2 -6 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999859 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 D=17 C=17 E=13 so E is eliminated. Round 2 votes counts: B=30 A=29 C=21 D=20 so D is eliminated. Round 3 votes counts: B=44 A=35 C=21 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 B:203 C:199 D:198 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -4 -6 B 0 0 4 2 0 C 2 -4 0 6 -6 D 4 -2 -6 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999859 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -4 -6 B 0 0 4 2 0 C 2 -4 0 6 -6 D 4 -2 -6 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999859 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -4 -6 B 0 0 4 2 0 C 2 -4 0 6 -6 D 4 -2 -6 0 0 E 6 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999859 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4696: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) D E B A C (9) C A B E D (8) B D E C A (8) A C E B D (8) E D B A C (5) B C A D E (5) D B E A C (4) C A B D E (4) B C D A E (3) B C A E D (3) A E C D B (3) E D A C B (2) D E B C A (2) C B A D E (2) C A E D B (2) B E D A C (2) B D C E A (2) B D C A E (2) A C B E D (2) E D A B C (1) E B D A C (1) E B A D C (1) E A D C B (1) E A C D B (1) E A C B D (1) E A B C D (1) D C A E B (1) D B E C A (1) C B D A E (1) C A E B D (1) B D E A C (1) A E C B D (1) Total count = 100 A B C D E A 0 -4 10 10 14 B 4 0 2 14 -4 C -10 -2 0 14 10 D -10 -14 -14 0 -10 E -14 4 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380164 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 A B C D E A 0 -4 10 10 14 B 4 0 2 14 -4 C -10 -2 0 14 10 D -10 -14 -14 0 -10 E -14 4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379279 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=25 C=18 D=17 E=14 so E is eliminated. Round 2 votes counts: A=29 B=28 D=25 C=18 so C is eliminated. Round 3 votes counts: A=44 B=31 D=25 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:208 C:206 E:195 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 10 10 14 B 4 0 2 14 -4 C -10 -2 0 14 10 D -10 -14 -14 0 -10 E -14 4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379279 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 10 14 B 4 0 2 14 -4 C -10 -2 0 14 10 D -10 -14 -14 0 -10 E -14 4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379279 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 10 14 B 4 0 2 14 -4 C -10 -2 0 14 10 D -10 -14 -14 0 -10 E -14 4 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.000000 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379279 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 0.818182 D: 0.818182 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4697: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (19) B A D C E (14) B E C D A (8) A D C B E (7) A D C E B (6) E C D B A (4) C D A E B (4) B E A D C (4) E C B D A (3) E B C D A (3) B E C A D (3) A D B C E (3) E B C A D (2) D C A E B (2) D A C E B (2) C E D A B (2) C D E A B (2) C A D E B (2) B D A C E (2) B A D E C (2) E C A D B (1) B E D A C (1) B D A E C (1) B A E D C (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -10 -6 -4 B -4 0 -10 -10 -6 C 10 10 0 8 -4 D 6 10 -8 0 -2 E 4 6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -10 -6 -4 B -4 0 -10 -10 -6 C 10 10 0 8 -4 D 6 10 -8 0 -2 E 4 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=32 A=18 C=10 D=4 so D is eliminated. Round 2 votes counts: B=36 E=32 A=20 C=12 so C is eliminated. Round 3 votes counts: E=36 B=36 A=28 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:208 D:203 A:192 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -10 -6 -4 B -4 0 -10 -10 -6 C 10 10 0 8 -4 D 6 10 -8 0 -2 E 4 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -6 -4 B -4 0 -10 -10 -6 C 10 10 0 8 -4 D 6 10 -8 0 -2 E 4 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -6 -4 B -4 0 -10 -10 -6 C 10 10 0 8 -4 D 6 10 -8 0 -2 E 4 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) C B E D A (7) B E D A C (7) C E B D A (6) B A D E C (6) B E C D A (4) A D E B C (4) A B C D E (4) E D B C A (3) C E D B A (3) C E A D B (3) C A B D E (3) B C E D A (3) A B D E C (3) C E D A B (2) C D E A B (2) C B E A D (2) C A E D B (2) B D E A C (2) B C A E D (2) B A E D C (2) A D C E B (2) A C D E B (2) E D B A C (1) E D A C B (1) E D A B C (1) E C D B A (1) E B D C A (1) D E A B C (1) D A E B C (1) C E B A D (1) C D A E B (1) C A E B D (1) A D E C B (1) A D C B E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -16 6 -2 B 2 0 -6 10 -2 C 16 6 0 22 22 D -6 -10 -22 0 -6 E 2 2 -22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 6 -2 B 2 0 -6 10 -2 C 16 6 0 22 22 D -6 -10 -22 0 -6 E 2 2 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 B=26 A=20 E=8 D=2 so D is eliminated. Round 2 votes counts: C=44 B=26 A=21 E=9 so E is eliminated. Round 3 votes counts: C=45 B=31 A=24 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:233 B:202 E:194 A:193 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 6 -2 B 2 0 -6 10 -2 C 16 6 0 22 22 D -6 -10 -22 0 -6 E 2 2 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 6 -2 B 2 0 -6 10 -2 C 16 6 0 22 22 D -6 -10 -22 0 -6 E 2 2 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 6 -2 B 2 0 -6 10 -2 C 16 6 0 22 22 D -6 -10 -22 0 -6 E 2 2 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4699: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (12) E B D A C (9) B E D C A (7) A C D B E (7) A C D E B (6) C A B D E (5) B D C A E (5) E B C A D (3) E A C D B (3) D A C B E (3) B E C D A (3) B C D A E (3) E D B A C (2) E B A C D (2) E A D C B (2) C A E D B (2) A C E D B (2) E D A C B (1) E D A B C (1) E C A B D (1) E B D C A (1) E B A D C (1) D E B A C (1) D E A B C (1) D B E A C (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A B D (1) C B A E D (1) C B A D E (1) C A B E D (1) B E D A C (1) B D E C A (1) B D C E A (1) B C A D E (1) A E C D B (1) A D E C B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 10 2 10 14 B -10 0 -8 -4 12 C -2 8 0 10 12 D -10 4 -10 0 10 E -14 -12 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 10 14 B -10 0 -8 -4 12 C -2 8 0 10 12 D -10 4 -10 0 10 E -14 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=23 B=22 A=19 D=10 so D is eliminated. Round 2 votes counts: E=28 A=25 B=24 C=23 so C is eliminated. Round 3 votes counts: A=45 E=29 B=26 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:214 D:197 B:195 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 10 14 B -10 0 -8 -4 12 C -2 8 0 10 12 D -10 4 -10 0 10 E -14 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 10 14 B -10 0 -8 -4 12 C -2 8 0 10 12 D -10 4 -10 0 10 E -14 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 10 14 B -10 0 -8 -4 12 C -2 8 0 10 12 D -10 4 -10 0 10 E -14 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4700: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (15) C B A E D (12) A B C D E (10) D A B E C (8) E C D B A (7) D E A B C (7) A B D C E (7) B A C D E (5) E D C B A (4) E D C A B (3) E C B D A (3) E D A B C (2) D E C A B (2) D E A C B (2) D A E B C (2) B A C E D (2) A D B C E (2) E D B A C (1) D C E A B (1) C E B D A (1) C E A D B (1) C E A B D (1) B C A E D (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -2 14 0 B 2 0 -4 14 -4 C 2 4 0 16 16 D -14 -14 -16 0 -6 E 0 4 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 14 0 B 2 0 -4 14 -4 C 2 4 0 16 16 D -14 -14 -16 0 -6 E 0 4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 E=20 A=20 B=8 so B is eliminated. Round 2 votes counts: C=31 A=27 D=22 E=20 so E is eliminated. Round 3 votes counts: C=41 D=32 A=27 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:205 B:204 E:197 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 14 0 B 2 0 -4 14 -4 C 2 4 0 16 16 D -14 -14 -16 0 -6 E 0 4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 14 0 B 2 0 -4 14 -4 C 2 4 0 16 16 D -14 -14 -16 0 -6 E 0 4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 14 0 B 2 0 -4 14 -4 C 2 4 0 16 16 D -14 -14 -16 0 -6 E 0 4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4701: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (16) E B D C A (13) D B A C E (10) D B E A C (6) C A E D B (6) C A E B D (6) B E D A C (5) B D E A C (5) A C D B E (5) D A B C E (4) E B C D A (3) D A C B E (3) C A D E B (3) E B D A C (2) E B C A D (2) D B A E C (2) A C B D E (2) E D B A C (1) E C B D A (1) E C A D B (1) C E A B D (1) C A D B E (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -6 -10 -12 B 2 0 8 14 -10 C 6 -8 0 -4 -16 D 10 -14 4 0 -16 E 12 10 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 -10 -12 B 2 0 8 14 -10 C 6 -8 0 -4 -16 D 10 -14 4 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=25 C=17 B=11 A=8 so A is eliminated. Round 2 votes counts: E=39 D=26 C=24 B=11 so B is eliminated. Round 3 votes counts: E=45 D=31 C=24 so C is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:207 D:192 C:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 -10 -12 B 2 0 8 14 -10 C 6 -8 0 -4 -16 D 10 -14 4 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -10 -12 B 2 0 8 14 -10 C 6 -8 0 -4 -16 D 10 -14 4 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -10 -12 B 2 0 8 14 -10 C 6 -8 0 -4 -16 D 10 -14 4 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4702: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (11) A C E B D (10) D C B E A (7) D B C E A (6) D B E A C (5) C A E D B (5) A E C B D (5) A C E D B (5) A E B C D (4) A C D E B (4) D C E B A (3) C D E B A (3) C D A E B (3) C A E B D (3) D C A E B (2) D B A E C (2) C D E A B (2) B E A D C (2) B D E A C (2) E C B A D (1) E C A B D (1) E A C B D (1) D C B A E (1) D C A B E (1) D A C E B (1) D A C B E (1) D A B E C (1) C E A B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B D E C A (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -8 -10 2 B -4 0 -22 -28 -10 C 8 22 0 2 18 D 10 28 -2 0 14 E -2 10 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -10 2 B -4 0 -22 -28 -10 C 8 22 0 2 18 D 10 28 -2 0 14 E -2 10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=29 C=18 B=9 E=3 so E is eliminated. Round 2 votes counts: D=41 A=30 C=20 B=9 so B is eliminated. Round 3 votes counts: D=46 A=33 C=21 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:225 D:225 A:194 E:188 B:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -10 2 B -4 0 -22 -28 -10 C 8 22 0 2 18 D 10 28 -2 0 14 E -2 10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 2 B -4 0 -22 -28 -10 C 8 22 0 2 18 D 10 28 -2 0 14 E -2 10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 2 B -4 0 -22 -28 -10 C 8 22 0 2 18 D 10 28 -2 0 14 E -2 10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4703: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (10) C D A B E (9) E B A D C (8) A D E C B (8) D A C E B (7) C B D A E (6) B E C A D (6) C D B A E (5) E D A B C (4) D A E C B (4) B E A D C (4) E B D A C (3) E A B D C (3) B E C D A (3) B E A C D (3) B C E A D (3) C B D E A (2) C B A D E (2) A E D B C (2) A D C E B (2) E D A C B (1) E A D B C (1) D E C B A (1) C D A E B (1) B A C E D (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 4 -12 -4 B 14 0 4 8 8 C -4 -4 0 2 -4 D 12 -8 -2 0 -4 E 4 -8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -12 -4 B 14 0 4 8 8 C -4 -4 0 2 -4 D 12 -8 -2 0 -4 E 4 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=25 E=20 A=13 D=12 so D is eliminated. Round 2 votes counts: B=30 C=25 A=24 E=21 so E is eliminated. Round 3 votes counts: B=41 A=33 C=26 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:202 D:199 C:195 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 -12 -4 B 14 0 4 8 8 C -4 -4 0 2 -4 D 12 -8 -2 0 -4 E 4 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -12 -4 B 14 0 4 8 8 C -4 -4 0 2 -4 D 12 -8 -2 0 -4 E 4 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -12 -4 B 14 0 4 8 8 C -4 -4 0 2 -4 D 12 -8 -2 0 -4 E 4 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4704: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) E A B C D (7) D B E A C (6) D E A C B (5) D B C A E (5) C B D A E (5) E A C B D (4) D C B A E (4) D C A E B (4) C B A E D (4) C A E B D (4) C A B E D (4) D C E A B (3) D C B E A (3) B E A C D (3) B A E C D (3) A E C B D (3) D E C A B (2) D E B A C (2) D B C E A (2) B D E A C (2) B D C A E (2) B A C E D (2) E D A B C (1) E A D B C (1) E A B D C (1) D B E C A (1) C E A D B (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B E A D C (1) B C D A E (1) B C A E D (1) B C A D E (1) Total count = 100 A B C D E A 0 4 -2 -18 -8 B -4 0 0 -4 2 C 2 0 0 -6 0 D 18 4 6 0 18 E 8 -2 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -18 -8 B -4 0 0 -4 2 C 2 0 0 -6 0 D 18 4 6 0 18 E 8 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 C=22 B=16 E=14 A=3 so A is eliminated. Round 2 votes counts: D=45 C=22 E=17 B=16 so B is eliminated. Round 3 votes counts: D=49 C=27 E=24 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:198 B:197 E:194 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -18 -8 B -4 0 0 -4 2 C 2 0 0 -6 0 D 18 4 6 0 18 E 8 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -18 -8 B -4 0 0 -4 2 C 2 0 0 -6 0 D 18 4 6 0 18 E 8 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -18 -8 B -4 0 0 -4 2 C 2 0 0 -6 0 D 18 4 6 0 18 E 8 -2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4705: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) E D B C A (8) A B C E D (8) B E A D C (7) E B D A C (5) D C E B A (5) B A E D C (5) E B D C A (4) B A E C D (4) A B E C D (4) D C E A B (3) C D E A B (3) B E D A C (3) D E C A B (2) C E D B A (2) C A D E B (2) C A D B E (2) C A B D E (2) A C B D E (2) A B E D C (2) A B D C E (2) E D C B A (1) E D B A C (1) D E B C A (1) D E B A C (1) D C A E B (1) D A E B C (1) D A C E B (1) C D E B A (1) C D A E B (1) B A C E D (1) A D C B E (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 2 -10 -16 B 20 0 16 2 -8 C -2 -16 0 -30 -20 D 10 -2 30 0 -12 E 16 8 20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 2 -10 -16 B 20 0 16 2 -8 C -2 -16 0 -30 -20 D 10 -2 30 0 -12 E 16 8 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=22 B=20 E=19 C=13 so C is eliminated. Round 2 votes counts: D=31 A=28 E=21 B=20 so B is eliminated. Round 3 votes counts: A=38 E=31 D=31 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:228 B:215 D:213 A:178 C:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 2 -10 -16 B 20 0 16 2 -8 C -2 -16 0 -30 -20 D 10 -2 30 0 -12 E 16 8 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 2 -10 -16 B 20 0 16 2 -8 C -2 -16 0 -30 -20 D 10 -2 30 0 -12 E 16 8 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 2 -10 -16 B 20 0 16 2 -8 C -2 -16 0 -30 -20 D 10 -2 30 0 -12 E 16 8 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4706: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) E C A D B (8) E A C D B (7) C E A B D (6) D B E C A (5) B D A C E (5) A E C B D (5) E C A B D (4) B D C A E (3) A E C D B (3) A B C E D (3) E D A C B (2) D B C E A (2) D B A C E (2) D A B E C (2) C E B D A (2) C B E D A (2) C A E B D (2) B D C E A (2) A E D B C (2) E D C A B (1) E C D A B (1) E A D C B (1) E A C B D (1) D E B C A (1) D E B A C (1) D B E A C (1) C E D B A (1) C E B A D (1) C B D E A (1) C B A E D (1) B C D A E (1) B A D C E (1) B A C E D (1) A E D C B (1) A E B D C (1) A D E B C (1) A C B E D (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 12 6 0 B -10 0 -2 -8 -4 C -12 2 0 4 -24 D -6 8 -4 0 -16 E 0 4 24 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.497873 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.502127 Sum of squares = 0.500009051513 Cumulative probabilities = A: 0.497873 B: 0.497873 C: 0.497873 D: 0.497873 E: 1.000000 A B C D E A 0 10 12 6 0 B -10 0 -2 -8 -4 C -12 2 0 4 -24 D -6 8 -4 0 -16 E 0 4 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 A=20 C=16 B=13 so B is eliminated. Round 2 votes counts: D=36 E=25 A=22 C=17 so C is eliminated. Round 3 votes counts: D=38 E=37 A=25 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:214 D:191 B:188 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 6 0 B -10 0 -2 -8 -4 C -12 2 0 4 -24 D -6 8 -4 0 -16 E 0 4 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 6 0 B -10 0 -2 -8 -4 C -12 2 0 4 -24 D -6 8 -4 0 -16 E 0 4 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 6 0 B -10 0 -2 -8 -4 C -12 2 0 4 -24 D -6 8 -4 0 -16 E 0 4 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4707: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) E B D C A (8) D C A E B (8) E D B C A (7) A C D B E (7) A B C E D (7) D E C B A (6) E B D A C (4) C D A E B (4) C A D E B (4) C A D B E (4) B E A D C (4) D E C A B (3) D E B C A (3) D A C E B (3) B E D A C (3) B A E C D (3) A C B D E (3) E D B A C (2) D C E B A (2) D C E A B (2) E D C B A (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -4 -12 -12 B 8 0 4 -14 -16 C 4 -4 0 -12 -10 D 12 14 12 0 0 E 12 16 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.553924 E: 0.446076 Sum of squares = 0.505815599431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.553924 E: 1.000000 A B C D E A 0 -8 -4 -12 -12 B 8 0 4 -14 -16 C 4 -4 0 -12 -10 D 12 14 12 0 0 E 12 16 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=22 B=21 A=18 C=12 so C is eliminated. Round 2 votes counts: D=31 A=26 E=22 B=21 so B is eliminated. Round 3 votes counts: E=40 D=31 A=29 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:219 B:191 C:189 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 -12 -12 B 8 0 4 -14 -16 C 4 -4 0 -12 -10 D 12 14 12 0 0 E 12 16 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -12 -12 B 8 0 4 -14 -16 C 4 -4 0 -12 -10 D 12 14 12 0 0 E 12 16 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -12 -12 B 8 0 4 -14 -16 C 4 -4 0 -12 -10 D 12 14 12 0 0 E 12 16 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4708: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (12) B E C D A (9) A D C E B (7) A D B C E (7) D A B E C (6) C E A D B (6) B D A E C (5) C E B A D (4) E C D B A (3) E C D A B (3) D B A E C (3) D A E C B (3) B E D C A (3) A B D C E (3) E C A D B (2) C E A B D (2) C B E A D (2) C A E D B (2) B C E A D (2) B A D C E (2) A D C B E (2) A C E D B (2) A C D E B (2) E D C B A (1) E C B A D (1) E B C D A (1) D E A C B (1) D A B C E (1) B D E C A (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -8 -4 -6 B 0 0 -12 -2 -6 C 8 12 0 10 -8 D 4 2 -10 0 -12 E 6 6 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -8 -4 -6 B 0 0 -12 -2 -6 C 8 12 0 10 -8 D 4 2 -10 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 B=23 C=16 D=14 so D is eliminated. Round 2 votes counts: A=34 B=26 E=24 C=16 so C is eliminated. Round 3 votes counts: E=36 A=36 B=28 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:211 D:192 A:191 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -4 -6 B 0 0 -12 -2 -6 C 8 12 0 10 -8 D 4 2 -10 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 -6 B 0 0 -12 -2 -6 C 8 12 0 10 -8 D 4 2 -10 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 -6 B 0 0 -12 -2 -6 C 8 12 0 10 -8 D 4 2 -10 0 -12 E 6 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4709: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) A D C B E (7) A C D E B (7) C A E D B (6) A C E D B (6) E B C D A (5) D C A E B (4) D A C B E (4) E C D B A (3) E C D A B (3) E B D C A (3) E C B D A (2) E C A B D (2) E B C A D (2) D B A C E (2) D A B C E (2) C E A D B (2) C D A E B (2) C A D E B (2) B E D A C (2) B E A C D (2) B D E A C (2) B A E D C (2) B A D C E (2) A B D C E (2) E D C B A (1) D C E B A (1) D C E A B (1) D B E C A (1) D B C A E (1) D A C E B (1) C E D A B (1) B E C A D (1) B E A D C (1) B D E C A (1) B A E C D (1) B A D E C (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -2 2 14 B -14 0 -16 -16 -10 C 2 16 0 4 14 D -2 16 -4 0 -8 E -14 10 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 2 14 B -14 0 -16 -16 -10 C 2 16 0 4 14 D -2 16 -4 0 -8 E -14 10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=22 E=21 D=17 C=13 so C is eliminated. Round 2 votes counts: A=35 E=24 B=22 D=19 so D is eliminated. Round 3 votes counts: A=48 E=26 B=26 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:214 D:201 E:195 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 2 14 B -14 0 -16 -16 -10 C 2 16 0 4 14 D -2 16 -4 0 -8 E -14 10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 2 14 B -14 0 -16 -16 -10 C 2 16 0 4 14 D -2 16 -4 0 -8 E -14 10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 2 14 B -14 0 -16 -16 -10 C 2 16 0 4 14 D -2 16 -4 0 -8 E -14 10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4710: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) E C D A B (5) D E C A B (5) E D C B A (4) E D C A B (4) D B A E C (4) B D A E C (4) D E B C A (3) D A B C E (3) C E A B D (3) C A B E D (3) B E D C A (3) A B C D E (3) E D B C A (2) E C B D A (2) E C B A D (2) D E C B A (2) D C A E B (2) B E A C D (2) B D E A C (2) B A D C E (2) B A C D E (2) A B D C E (2) A B C E D (2) E C D B A (1) E C A D B (1) E C A B D (1) E B D C A (1) D E A C B (1) D B E A C (1) D B A C E (1) D A C E B (1) D A B E C (1) C E B A D (1) C A E D B (1) C A E B D (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A D C (1) B D A C E (1) B A E C D (1) A D B C E (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 2 -10 0 B 14 0 14 12 10 C -2 -14 0 -4 -12 D 10 -12 4 0 -16 E 0 -10 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -10 0 B 14 0 14 12 10 C -2 -14 0 -4 -12 D 10 -12 4 0 -16 E 0 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=24 E=23 A=11 C=9 so C is eliminated. Round 2 votes counts: B=33 E=27 D=24 A=16 so A is eliminated. Round 3 votes counts: B=44 E=30 D=26 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:209 D:193 A:189 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 -10 0 B 14 0 14 12 10 C -2 -14 0 -4 -12 D 10 -12 4 0 -16 E 0 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -10 0 B 14 0 14 12 10 C -2 -14 0 -4 -12 D 10 -12 4 0 -16 E 0 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -10 0 B 14 0 14 12 10 C -2 -14 0 -4 -12 D 10 -12 4 0 -16 E 0 -10 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4711: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) E D C B A (7) E C D B A (7) A B C E D (7) E A B C D (5) D E C B A (5) D C B A E (5) A B E C D (4) E C B A D (3) D C E B A (3) B C A D E (3) B A C E D (3) A E B C D (3) A D B C E (3) A B D C E (3) E C B D A (2) E B C A D (2) E A D B C (2) D E C A B (2) D C E A B (2) D A C B E (2) C B E D A (2) B C A E D (2) A E D B C (2) A E B D C (2) D E A C B (1) D C B E A (1) D C A E B (1) D A C E B (1) D A B C E (1) C E D B A (1) C D B E A (1) C B A E D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 0 12 8 B -4 0 6 4 -4 C 0 -6 0 10 2 D -12 -4 -10 0 -10 E -8 4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.858586 B: 0.000000 C: 0.141414 D: 0.000000 E: 0.000000 Sum of squares = 0.757167581558 Cumulative probabilities = A: 0.858586 B: 0.858586 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 12 8 B -4 0 6 4 -4 C 0 -6 0 10 2 D -12 -4 -10 0 -10 E -8 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000012366 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=28 D=24 B=8 C=5 so C is eliminated. Round 2 votes counts: A=35 E=29 D=25 B=11 so B is eliminated. Round 3 votes counts: A=44 E=31 D=25 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:203 E:202 B:201 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 12 8 B -4 0 6 4 -4 C 0 -6 0 10 2 D -12 -4 -10 0 -10 E -8 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000012366 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 12 8 B -4 0 6 4 -4 C 0 -6 0 10 2 D -12 -4 -10 0 -10 E -8 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000012366 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 12 8 B -4 0 6 4 -4 C 0 -6 0 10 2 D -12 -4 -10 0 -10 E -8 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000012366 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4712: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E D A (10) C B E A D (7) E B A D C (6) A D E B C (6) C D A B E (5) C D B E A (4) C D B A E (4) C B D E A (4) C A E B D (4) B E A C D (3) A E D B C (3) E A B D C (2) D B E A C (2) D B C E A (2) D A C E B (2) C E B A D (2) C A D B E (2) B E C A D (2) B C E D A (2) A E B C D (2) A D C E B (2) E B A C D (1) D C B E A (1) D C B A E (1) D C A B E (1) D B A E C (1) D A C B E (1) C D A E B (1) C A E D B (1) C A D E B (1) C A B E D (1) B E D A C (1) B E A D C (1) B D E A C (1) A E B D C (1) Total count = 100 A B C D E A 0 -10 -10 -6 -2 B 10 0 -8 0 12 C 10 8 0 12 16 D 6 0 -12 0 2 E 2 -12 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -6 -2 B 10 0 -8 0 12 C 10 8 0 12 16 D 6 0 -12 0 2 E 2 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 D=21 A=14 B=10 E=9 so E is eliminated. Round 2 votes counts: C=46 D=21 B=17 A=16 so A is eliminated. Round 3 votes counts: C=46 D=32 B=22 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 B:207 D:198 A:186 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 -6 -2 B 10 0 -8 0 12 C 10 8 0 12 16 D 6 0 -12 0 2 E 2 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -6 -2 B 10 0 -8 0 12 C 10 8 0 12 16 D 6 0 -12 0 2 E 2 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -6 -2 B 10 0 -8 0 12 C 10 8 0 12 16 D 6 0 -12 0 2 E 2 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4713: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) B A C E D (8) B A C D E (8) D E A C B (6) E D C A B (5) C A E B D (5) B C A E D (5) A C B E D (5) A B C E D (4) E D C B A (3) E C D A B (3) D E B C A (3) C E A D B (3) C A E D B (3) D E B A C (2) C E A B D (2) C A B E D (2) B D E C A (2) B D E A C (2) B C D A E (2) A D E C B (2) E D A C B (1) E C B D A (1) D E A B C (1) D B E C A (1) D B E A C (1) D A E C B (1) D A B E C (1) C E D A B (1) B E D C A (1) B E C D A (1) B D A E C (1) B C E D A (1) B C E A D (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -10 0 -2 B -14 0 -8 4 -6 C 10 8 0 12 4 D 0 -4 -12 0 -12 E 2 6 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 0 -2 B -14 0 -8 4 -6 C 10 8 0 12 4 D 0 -4 -12 0 -12 E 2 6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=26 C=16 E=13 A=13 so E is eliminated. Round 2 votes counts: D=35 B=32 C=20 A=13 so A is eliminated. Round 3 votes counts: D=37 B=37 C=26 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:217 E:208 A:201 B:188 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 0 -2 B -14 0 -8 4 -6 C 10 8 0 12 4 D 0 -4 -12 0 -12 E 2 6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 0 -2 B -14 0 -8 4 -6 C 10 8 0 12 4 D 0 -4 -12 0 -12 E 2 6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 0 -2 B -14 0 -8 4 -6 C 10 8 0 12 4 D 0 -4 -12 0 -12 E 2 6 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4714: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C E D B A (11) A B D E C (11) E C D A B (6) A B D C E (6) E C D B A (5) D E A B C (4) C A B E D (4) E D C A B (3) C B A E D (3) C B A D E (3) B A C D E (3) A B C D E (3) E A D B C (2) D E B A C (2) D B A E C (2) D A B E C (2) C E B A D (2) C D E B A (2) C B D A E (2) B A D C E (2) E C A D B (1) E A C B D (1) D E B C A (1) D B E A C (1) C E B D A (1) C E A B D (1) C B E D A (1) C B E A D (1) B C D A E (1) A D B E C (1) Total count = 100 A B C D E A 0 14 4 -12 -14 B -14 0 6 -10 -8 C -4 -6 0 2 -8 D 12 10 -2 0 -8 E 14 8 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 4 -12 -14 B -14 0 6 -10 -8 C -4 -6 0 2 -8 D 12 10 -2 0 -8 E 14 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=30 A=21 D=12 B=6 so B is eliminated. Round 2 votes counts: C=32 E=30 A=26 D=12 so D is eliminated. Round 3 votes counts: E=38 C=32 A=30 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:206 A:196 C:192 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 4 -12 -14 B -14 0 6 -10 -8 C -4 -6 0 2 -8 D 12 10 -2 0 -8 E 14 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -12 -14 B -14 0 6 -10 -8 C -4 -6 0 2 -8 D 12 10 -2 0 -8 E 14 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -12 -14 B -14 0 6 -10 -8 C -4 -6 0 2 -8 D 12 10 -2 0 -8 E 14 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4715: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) C A B E D (9) A C D B E (7) E B D C A (6) D E B A C (5) C E B A D (5) E D B C A (4) E C B A D (4) D E A B C (4) D B E A C (4) D A B E C (4) A C D E B (4) E B C D A (3) B E C D A (3) D A E C B (2) C B A E D (2) C A E D B (2) B C E A D (2) B C A E D (2) A D B C E (2) A C B D E (2) D E A C B (1) D A C E B (1) D A B C E (1) C E A D B (1) C E A B D (1) C B E A D (1) C A E B D (1) B E C A D (1) B D E A C (1) B D A E C (1) B A C D E (1) A D C E B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -14 0 -12 B 10 0 10 12 10 C 14 -10 0 4 -6 D 0 -12 -4 0 -14 E 12 -10 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 0 -12 B 10 0 10 12 10 C 14 -10 0 4 -6 D 0 -12 -4 0 -14 E 12 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=22 C=22 B=21 A=18 E=17 so E is eliminated. Round 2 votes counts: B=30 D=26 C=26 A=18 so A is eliminated. Round 3 votes counts: C=39 B=31 D=30 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:211 C:201 D:185 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 0 -12 B 10 0 10 12 10 C 14 -10 0 4 -6 D 0 -12 -4 0 -14 E 12 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 0 -12 B 10 0 10 12 10 C 14 -10 0 4 -6 D 0 -12 -4 0 -14 E 12 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 0 -12 B 10 0 10 12 10 C 14 -10 0 4 -6 D 0 -12 -4 0 -14 E 12 -10 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4716: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (12) D B A C E (9) D A C B E (9) E C B A D (8) A D C E B (7) E C A B D (6) A C E D B (6) A C D E B (6) E B C A D (5) C E A B D (5) C A E D B (4) B E D C A (4) D A B C E (3) B D E C A (3) B D E A C (3) E A C B D (1) D C A B E (1) D B C A E (1) D B A E C (1) D A C E B (1) C E B A D (1) C E A D B (1) C A D E B (1) B D A E C (1) A D C B E (1) Total count = 100 A B C D E A 0 4 -4 4 2 B -4 0 -16 -2 -4 C 4 16 0 12 12 D -4 2 -12 0 -6 E -2 4 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 4 2 B -4 0 -16 -2 -4 C 4 16 0 12 12 D -4 2 -12 0 -6 E -2 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999518 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=23 E=20 A=20 C=12 so C is eliminated. Round 2 votes counts: E=27 D=25 A=25 B=23 so B is eliminated. Round 3 votes counts: E=43 D=32 A=25 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:222 A:203 E:198 D:190 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 2 B -4 0 -16 -2 -4 C 4 16 0 12 12 D -4 2 -12 0 -6 E -2 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999518 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 2 B -4 0 -16 -2 -4 C 4 16 0 12 12 D -4 2 -12 0 -6 E -2 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999518 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 2 B -4 0 -16 -2 -4 C 4 16 0 12 12 D -4 2 -12 0 -6 E -2 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999518 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4717: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) B A D E C (10) E B D C A (5) C E D A B (5) A D B C E (5) A C D B E (5) A B D C E (5) D B A E C (4) E C B D A (3) D A B C E (3) C D A E B (3) C A D E B (3) B D A E C (3) E B D A C (2) E B C A D (2) C E A D B (2) C E A B D (2) C A E B D (2) B E D A C (2) B E A D C (2) B E A C D (2) A D C B E (2) E D B C A (1) E C D A B (1) E B C D A (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E A B (1) D C A E B (1) D B E A C (1) D A C B E (1) D A B E C (1) C D E A B (1) C D A B E (1) C A E D B (1) B D E A C (1) B A E D C (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 6 -8 6 B 8 0 8 -10 2 C -6 -8 0 -8 -10 D 8 10 8 0 10 E -6 -2 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 -8 6 B 8 0 8 -10 2 C -6 -8 0 -8 -10 D 8 10 8 0 10 E -6 -2 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=22 C=20 A=18 D=15 so D is eliminated. Round 2 votes counts: E=28 B=27 A=23 C=22 so C is eliminated. Round 3 votes counts: E=39 A=34 B=27 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:218 B:204 A:198 E:196 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 6 -8 6 B 8 0 8 -10 2 C -6 -8 0 -8 -10 D 8 10 8 0 10 E -6 -2 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -8 6 B 8 0 8 -10 2 C -6 -8 0 -8 -10 D 8 10 8 0 10 E -6 -2 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -8 6 B 8 0 8 -10 2 C -6 -8 0 -8 -10 D 8 10 8 0 10 E -6 -2 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4718: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) E B D C A (9) B C A E D (5) A C D E B (5) A C B E D (5) D E B C A (4) D C A B E (4) E D B C A (3) E D B A C (3) C A D B E (3) C A B D E (3) B E D C A (3) B E C D A (3) A E C D B (3) A D E C B (3) A D C E B (3) E D A B C (2) E B D A C (2) E B A C D (2) D A C E B (2) B C D E A (2) A E D C B (2) A C B D E (2) E D A C B (1) E B A D C (1) E A D C B (1) E A B D C (1) E A B C D (1) D E A B C (1) D B E C A (1) D A E C B (1) C D B A E (1) C B D A E (1) C B A E D (1) C B A D E (1) B C E A D (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 10 14 18 B -14 0 -10 -10 -6 C -10 10 0 6 2 D -14 10 -6 0 -4 E -18 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 14 18 B -14 0 -10 -10 -6 C -10 10 0 6 2 D -14 10 -6 0 -4 E -18 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=26 B=14 D=13 C=10 so C is eliminated. Round 2 votes counts: A=43 E=26 B=17 D=14 so D is eliminated. Round 3 votes counts: A=50 E=31 B=19 so B is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:204 E:195 D:193 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 14 18 B -14 0 -10 -10 -6 C -10 10 0 6 2 D -14 10 -6 0 -4 E -18 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 14 18 B -14 0 -10 -10 -6 C -10 10 0 6 2 D -14 10 -6 0 -4 E -18 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 14 18 B -14 0 -10 -10 -6 C -10 10 0 6 2 D -14 10 -6 0 -4 E -18 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4719: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) A D E C B (7) B C E D A (6) E D C B A (5) A D C E B (5) A B C D E (5) D E A C B (4) C E D B A (4) B C E A D (4) B A E D C (4) B A C E D (4) A B D E C (4) C B E D A (3) B C A E D (3) B A E C D (3) E D B C A (2) E C B D A (2) D E C B A (2) D A E C B (2) C D E A B (2) A D E B C (2) A C B D E (2) A B D C E (2) E C D B A (1) E B D C A (1) E B D A C (1) D C E A B (1) D C A E B (1) C B A E D (1) C B A D E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) A D C B E (1) A D B E C (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 4 0 B 2 0 -6 2 -2 C -2 6 0 -12 -6 D -4 -2 12 0 2 E 0 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.492230 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.507770 Sum of squares = 0.500120728679 Cumulative probabilities = A: 0.492230 B: 0.492230 C: 0.492230 D: 0.492230 E: 1.000000 A B C D E A 0 -2 2 4 0 B 2 0 -6 2 -2 C -2 6 0 -12 -6 D -4 -2 12 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499590 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500410 Sum of squares = 0.500000336063 Cumulative probabilities = A: 0.499590 B: 0.499590 C: 0.499590 D: 0.499590 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=28 D=18 E=12 C=11 so C is eliminated. Round 2 votes counts: B=33 A=31 D=20 E=16 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:204 E:203 A:202 B:198 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 4 0 B 2 0 -6 2 -2 C -2 6 0 -12 -6 D -4 -2 12 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499590 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500410 Sum of squares = 0.500000336063 Cumulative probabilities = A: 0.499590 B: 0.499590 C: 0.499590 D: 0.499590 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 0 B 2 0 -6 2 -2 C -2 6 0 -12 -6 D -4 -2 12 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499590 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500410 Sum of squares = 0.500000336063 Cumulative probabilities = A: 0.499590 B: 0.499590 C: 0.499590 D: 0.499590 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 0 B 2 0 -6 2 -2 C -2 6 0 -12 -6 D -4 -2 12 0 2 E 0 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499590 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500410 Sum of squares = 0.500000336063 Cumulative probabilities = A: 0.499590 B: 0.499590 C: 0.499590 D: 0.499590 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4720: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) A E D C B (7) E A D C B (6) C B E D A (6) C E D B A (5) B C D E A (5) C E D A B (4) C B D E A (4) A D B E C (4) A B D E C (4) E A C D B (3) C E B D A (3) B D A C E (3) B C A E D (3) A E D B C (3) E D C A B (2) E C A D B (2) D B A C E (2) C D E B A (2) B D C E A (2) B C E D A (2) B C D A E (2) B C A D E (2) A D E C B (2) A D E B C (2) E D A C B (1) E C A B D (1) D E C A B (1) D E A C B (1) D C E B A (1) D A E C B (1) C E B A D (1) C D B E A (1) B A D C E (1) B A C E D (1) B A C D E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 6 -12 -10 -20 B -6 0 -24 -14 -14 C 12 24 0 14 2 D 10 14 -14 0 -18 E 20 14 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -10 -20 B -6 0 -24 -14 -14 C 12 24 0 14 2 D 10 14 -14 0 -18 E 20 14 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 E=22 B=22 D=6 so D is eliminated. Round 2 votes counts: C=27 A=25 E=24 B=24 so E is eliminated. Round 3 votes counts: C=40 A=36 B=24 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:225 D:196 A:182 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -10 -20 B -6 0 -24 -14 -14 C 12 24 0 14 2 D 10 14 -14 0 -18 E 20 14 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -10 -20 B -6 0 -24 -14 -14 C 12 24 0 14 2 D 10 14 -14 0 -18 E 20 14 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -10 -20 B -6 0 -24 -14 -14 C 12 24 0 14 2 D 10 14 -14 0 -18 E 20 14 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4721: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) C A D E B (8) B E D A C (7) C D A B E (5) E A D B C (4) D E C A B (4) B E A D C (4) B C E D A (4) A D C E B (4) A C D E B (4) E B D A C (3) C A D B E (3) B E D C A (3) B E C A D (3) B C E A D (3) B A E C D (3) E D B A C (2) E D A B C (2) E B A D C (2) D C A E B (2) C B D A E (2) C B A D E (2) A E D B C (2) E A B D C (1) D E B C A (1) D A E C B (1) C D E B A (1) C D B E A (1) C D B A E (1) C B A E D (1) C A B D E (1) B E C D A (1) B C A E D (1) B C A D E (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -16 0 8 B -8 0 -2 -10 -2 C 16 2 0 14 12 D 0 10 -14 0 2 E -8 2 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -16 0 8 B -8 0 -2 -10 -2 C 16 2 0 14 12 D 0 10 -14 0 2 E -8 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=30 E=14 A=13 D=8 so D is eliminated. Round 2 votes counts: C=37 B=30 E=19 A=14 so A is eliminated. Round 3 votes counts: C=46 B=32 E=22 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:200 D:199 E:190 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 0 8 B -8 0 -2 -10 -2 C 16 2 0 14 12 D 0 10 -14 0 2 E -8 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 0 8 B -8 0 -2 -10 -2 C 16 2 0 14 12 D 0 10 -14 0 2 E -8 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 0 8 B -8 0 -2 -10 -2 C 16 2 0 14 12 D 0 10 -14 0 2 E -8 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999962306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4722: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) B A C E D (8) A B C D E (8) D C E A B (5) C A D B E (5) B A E D C (5) E D B A C (4) E C D B A (4) C D A E B (4) A B D C E (4) E D C B A (3) E B D A C (3) D E C A B (3) C E B D A (3) A C B D E (3) C E D B A (2) C D A B E (2) C A B D E (2) B E A C D (2) E D C A B (1) E D B C A (1) E D A B C (1) E B A D C (1) D E A B C (1) D A C E B (1) C B A E D (1) C B A D E (1) B E C A D (1) B C A E D (1) B A D E C (1) B A D C E (1) B A C D E (1) A D B E C (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -6 -4 2 B -14 0 -10 -6 -2 C 6 10 0 26 34 D 4 6 -26 0 18 E -2 2 -34 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 -4 2 B -14 0 -10 -6 -2 C 6 10 0 26 34 D 4 6 -26 0 18 E -2 2 -34 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=20 E=18 A=18 D=10 so D is eliminated. Round 2 votes counts: C=39 E=22 B=20 A=19 so A is eliminated. Round 3 votes counts: C=44 B=34 E=22 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:238 A:203 D:201 B:184 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 -4 2 B -14 0 -10 -6 -2 C 6 10 0 26 34 D 4 6 -26 0 18 E -2 2 -34 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 -4 2 B -14 0 -10 -6 -2 C 6 10 0 26 34 D 4 6 -26 0 18 E -2 2 -34 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 -4 2 B -14 0 -10 -6 -2 C 6 10 0 26 34 D 4 6 -26 0 18 E -2 2 -34 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4723: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (13) C D A B E (8) A C D E B (8) B E D C A (7) B D C E A (7) A E C D B (7) E A B D C (6) B D C A E (6) E B A D C (5) A C D B E (5) E B D C A (4) E B D A C (4) D C B A E (4) C D B A E (4) E A B C D (3) B D E C A (2) E B C D A (1) E A C B D (1) C A D B E (1) B E C D A (1) A E D C B (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 10 10 4 -8 B -10 0 -6 -4 -10 C -10 6 0 6 -12 D -4 4 -6 0 -10 E 8 10 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 10 4 -8 B -10 0 -6 -4 -10 C -10 6 0 6 -12 D -4 4 -6 0 -10 E 8 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=23 A=23 C=13 D=4 so D is eliminated. Round 2 votes counts: E=37 B=23 A=23 C=17 so C is eliminated. Round 3 votes counts: E=37 A=32 B=31 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:208 C:195 D:192 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 10 4 -8 B -10 0 -6 -4 -10 C -10 6 0 6 -12 D -4 4 -6 0 -10 E 8 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 4 -8 B -10 0 -6 -4 -10 C -10 6 0 6 -12 D -4 4 -6 0 -10 E 8 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 4 -8 B -10 0 -6 -4 -10 C -10 6 0 6 -12 D -4 4 -6 0 -10 E 8 10 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4724: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (5) B D C E A (5) B C E D A (5) E B C A D (4) E B A C D (4) D A E C B (4) D A C E B (4) C B A E D (4) A E C B D (4) E A B C D (3) B E C D A (3) B C D E A (3) E D B A C (2) E D A B C (2) E A D B C (2) E A B D C (2) D E B A C (2) D E A B C (2) D C B A E (2) D C A B E (2) C B D A E (2) C A D B E (2) B C E A D (2) A E D C B (2) A E C D B (2) A D E C B (2) A D C B E (2) A C B E D (2) E B D C A (1) E B D A C (1) E B A D C (1) E A C B D (1) D E B C A (1) D B E C A (1) D B C E A (1) D B C A E (1) D A E B C (1) C E B A D (1) C D B A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C A D (1) A D C E B (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 8 -4 -2 B 0 0 2 10 -2 C -8 -2 0 0 2 D 4 -10 0 0 -4 E 2 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 0 8 -4 -2 B 0 0 2 10 -2 C -8 -2 0 0 2 D 4 -10 0 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 B=20 A=18 C=13 so C is eliminated. Round 2 votes counts: D=27 B=27 E=24 A=22 so A is eliminated. Round 3 votes counts: D=35 E=33 B=32 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:205 E:203 A:201 C:196 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 8 -4 -2 B 0 0 2 10 -2 C -8 -2 0 0 2 D 4 -10 0 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -4 -2 B 0 0 2 10 -2 C -8 -2 0 0 2 D 4 -10 0 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -4 -2 B 0 0 2 10 -2 C -8 -2 0 0 2 D 4 -10 0 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999872 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4725: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) E C B D A (6) C E A B D (6) E C D B A (5) D E C B A (5) A C E D B (5) E C A D B (4) B C E D A (4) B A C D E (4) A B D C E (4) E C D A B (3) D A B E C (3) C E B A D (3) B D A C E (3) E D C B A (2) D B E C A (2) D B E A C (2) D B A C E (2) D A B C E (2) C A E B D (2) B D E C A (2) B D A E C (2) B C E A D (2) B A C E D (2) A D C E B (2) A D B C E (2) A B C E D (2) E B C D A (1) D E C A B (1) D E A B C (1) C E A D B (1) C B A E D (1) B E C D A (1) B C A E D (1) B A D C E (1) A D E C B (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 -4 -10 -2 B 18 0 4 -4 2 C 4 -4 0 8 0 D 10 4 -8 0 -4 E 2 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.425905 C: 0.064763 D: 0.212953 E: 0.296380 Sum of squares = 0.318779022709 Cumulative probabilities = A: 0.000000 B: 0.425905 C: 0.490668 D: 0.703620 E: 1.000000 A B C D E A 0 -18 -4 -10 -2 B 18 0 4 -4 2 C 4 -4 0 8 0 D 10 4 -8 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.446809 C: 0.117021 D: 0.223404 E: 0.212766 Sum of squares = 0.308510638313 Cumulative probabilities = A: 0.000000 B: 0.446809 C: 0.563830 D: 0.787234 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=22 E=21 A=18 C=13 so C is eliminated. Round 2 votes counts: E=31 D=26 B=23 A=20 so A is eliminated. Round 3 votes counts: E=39 D=32 B=29 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:210 C:204 E:202 D:201 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -4 -10 -2 B 18 0 4 -4 2 C 4 -4 0 8 0 D 10 4 -8 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.446809 C: 0.117021 D: 0.223404 E: 0.212766 Sum of squares = 0.308510638313 Cumulative probabilities = A: 0.000000 B: 0.446809 C: 0.563830 D: 0.787234 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -10 -2 B 18 0 4 -4 2 C 4 -4 0 8 0 D 10 4 -8 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.446809 C: 0.117021 D: 0.223404 E: 0.212766 Sum of squares = 0.308510638313 Cumulative probabilities = A: 0.000000 B: 0.446809 C: 0.563830 D: 0.787234 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -10 -2 B 18 0 4 -4 2 C 4 -4 0 8 0 D 10 4 -8 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.446809 C: 0.117021 D: 0.223404 E: 0.212766 Sum of squares = 0.308510638313 Cumulative probabilities = A: 0.000000 B: 0.446809 C: 0.563830 D: 0.787234 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4726: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) A E C B D (8) C D B A E (6) D C B A E (5) D B C A E (5) C A E D B (5) C A E B D (5) B D E C A (4) B D C E A (4) D B E A C (3) B D E A C (3) A E C D B (3) A C E B D (3) E C A B D (2) D C B E A (2) D B E C A (2) D B A E C (2) C E A B D (2) C D B E A (2) C A D E B (2) B E D A C (2) B E A D C (2) A E B D C (2) A C E D B (2) E B A D C (1) E B A C D (1) E A C B D (1) E A B C D (1) D C A B E (1) D A C E B (1) C E B A D (1) C D A E B (1) C B E A D (1) C B D E A (1) B C D E A (1) A E D C B (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 -24 -10 8 B 16 0 -12 -10 12 C 24 12 0 -4 20 D 10 10 4 0 12 E -8 -12 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -24 -10 8 B 16 0 -12 -10 12 C 24 12 0 -4 20 D 10 10 4 0 12 E -8 -12 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=26 A=21 B=16 E=6 so E is eliminated. Round 2 votes counts: D=31 C=28 A=23 B=18 so B is eliminated. Round 3 votes counts: D=44 C=29 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:226 D:218 B:203 A:179 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -24 -10 8 B 16 0 -12 -10 12 C 24 12 0 -4 20 D 10 10 4 0 12 E -8 -12 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -24 -10 8 B 16 0 -12 -10 12 C 24 12 0 -4 20 D 10 10 4 0 12 E -8 -12 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -24 -10 8 B 16 0 -12 -10 12 C 24 12 0 -4 20 D 10 10 4 0 12 E -8 -12 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4727: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) C D E B A (6) A B E D C (6) D C E A B (5) C D E A B (5) B A E D C (5) E D C A B (4) E C D B A (4) E B A C D (4) C B E D A (4) E A B D C (3) B E A C D (3) B A C E D (3) B A C D E (3) A B D C E (3) E D A C B (2) E C D A B (2) E A D B C (2) D E C A B (2) D C A E B (2) B E C A D (2) B C E D A (2) A E D B C (2) A E B D C (2) E C B D A (1) E C A D B (1) E A D C B (1) D C B A E (1) D A C E B (1) C E D B A (1) C D B E A (1) C D B A E (1) B E C D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A D E C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 0 6 -16 B 10 0 10 12 -2 C 0 -10 0 12 -14 D -6 -12 -12 0 -28 E 16 2 14 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 0 6 -16 B 10 0 10 12 -2 C 0 -10 0 12 -14 D -6 -12 -12 0 -28 E 16 2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999946541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=24 C=18 A=15 D=11 so D is eliminated. Round 2 votes counts: B=32 E=26 C=26 A=16 so A is eliminated. Round 3 votes counts: B=42 E=31 C=27 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:230 B:215 C:194 A:190 D:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 0 6 -16 B 10 0 10 12 -2 C 0 -10 0 12 -14 D -6 -12 -12 0 -28 E 16 2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999946541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 -16 B 10 0 10 12 -2 C 0 -10 0 12 -14 D -6 -12 -12 0 -28 E 16 2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999946541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 -16 B 10 0 10 12 -2 C 0 -10 0 12 -14 D -6 -12 -12 0 -28 E 16 2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999946541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4728: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) D B C A E (7) E A C B D (6) A E B C D (5) A D B E C (5) A B C D E (5) E C D B A (4) E C B D A (4) D C B E A (4) D A B C E (4) A D E B C (4) A D B C E (4) D B A C E (3) C E B D A (3) B C D E A (3) B A C D E (3) E D C A B (2) E C D A B (2) E A D C B (2) D B C E A (2) C B D E A (2) B C D A E (2) A B D C E (2) E D C B A (1) E D A C B (1) E C B A D (1) D E C B A (1) C B E A D (1) B D A C E (1) B C E D A (1) B C A E D (1) B A D C E (1) A E D C B (1) A E D B C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -12 2 B 8 0 12 4 22 C 0 -12 0 6 18 D 12 -4 -6 0 6 E -2 -22 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -12 2 B 8 0 12 4 22 C 0 -12 0 6 18 D 12 -4 -6 0 6 E -2 -22 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=23 D=21 C=15 B=12 so B is eliminated. Round 2 votes counts: A=33 E=23 D=22 C=22 so D is eliminated. Round 3 votes counts: A=41 C=35 E=24 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:223 C:206 D:204 A:191 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -12 2 B 8 0 12 4 22 C 0 -12 0 6 18 D 12 -4 -6 0 6 E -2 -22 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -12 2 B 8 0 12 4 22 C 0 -12 0 6 18 D 12 -4 -6 0 6 E -2 -22 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -12 2 B 8 0 12 4 22 C 0 -12 0 6 18 D 12 -4 -6 0 6 E -2 -22 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4729: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) D B A E C (7) A C E D B (7) E C A B D (6) B C E D A (6) A E C D B (6) D B A C E (5) B D E C A (5) C A E D B (4) B D E A C (4) B D C E A (4) A D C E B (4) A C D E B (4) D B C A E (3) C E B A D (3) E A C B D (2) C E A B D (2) C B E A D (2) A E D C B (2) A D E B C (2) E C B A D (1) E A C D B (1) E A B C D (1) D C B A E (1) D A E C B (1) D A E B C (1) D A B C E (1) C A E B D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 8 16 0 16 B -8 0 4 -16 2 C -16 -4 0 -2 -4 D 0 16 2 0 4 E -16 -2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.547931 B: 0.000000 C: 0.000000 D: 0.452069 E: 0.000000 Sum of squares = 0.504594785896 Cumulative probabilities = A: 0.547931 B: 0.547931 C: 0.547931 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 0 16 B -8 0 4 -16 2 C -16 -4 0 -2 -4 D 0 16 2 0 4 E -16 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 B=24 C=12 E=11 so E is eliminated. Round 2 votes counts: A=30 D=27 B=24 C=19 so C is eliminated. Round 3 votes counts: A=43 B=30 D=27 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:211 B:191 E:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 0 16 B -8 0 4 -16 2 C -16 -4 0 -2 -4 D 0 16 2 0 4 E -16 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 0 16 B -8 0 4 -16 2 C -16 -4 0 -2 -4 D 0 16 2 0 4 E -16 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 0 16 B -8 0 4 -16 2 C -16 -4 0 -2 -4 D 0 16 2 0 4 E -16 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4730: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C A D (9) D A E B C (6) E B C A D (4) D B C E A (4) D B C A E (4) D A B C E (4) A C E B D (4) E C B A D (3) D C A B E (3) B D C E A (3) A E C B D (3) E B A D C (2) E B A C D (2) E A C B D (2) D B E C A (2) D B E A C (2) D A E C B (2) D A C B E (2) C B E A D (2) C B A E D (2) C A E B D (2) B E D A C (2) B D E C A (2) E C A B D (1) E B D A C (1) E A B C D (1) D C A E B (1) D B A E C (1) D B A C E (1) C E B A D (1) C D A B E (1) C A B E D (1) C A B D E (1) B E A C D (1) B D E A C (1) B C E D A (1) B C E A D (1) A E C D B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 4 -8 6 B 2 0 8 4 0 C -4 -8 0 -10 4 D 8 -4 10 0 6 E -6 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.847826 C: 0.000000 D: 0.000000 E: 0.152174 Sum of squares = 0.741966536773 Cumulative probabilities = A: 0.000000 B: 0.847826 C: 0.847826 D: 0.847826 E: 1.000000 A B C D E A 0 -2 4 -8 6 B 2 0 8 4 0 C -4 -8 0 -10 4 D 8 -4 10 0 6 E -6 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000029 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 B=20 E=16 A=11 C=10 so C is eliminated. Round 2 votes counts: D=44 B=24 E=17 A=15 so A is eliminated. Round 3 votes counts: D=46 E=28 B=26 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:207 A:200 E:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 -8 6 B 2 0 8 4 0 C -4 -8 0 -10 4 D 8 -4 10 0 6 E -6 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000029 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -8 6 B 2 0 8 4 0 C -4 -8 0 -10 4 D 8 -4 10 0 6 E -6 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000029 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -8 6 B 2 0 8 4 0 C -4 -8 0 -10 4 D 8 -4 10 0 6 E -6 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000029 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4731: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E C A B D (7) A D B C E (6) C E A D B (5) C E B D A (4) B D E A C (4) B D A E C (4) A E D B C (4) A D B E C (4) E B C D A (3) E A C B D (3) E A B D C (3) D A B C E (3) C D B A E (3) B D C E A (3) A E C D B (3) E B D A C (2) D B C A E (2) D B A C E (2) C D A B E (2) B E D C A (2) B D A C E (2) A D E B C (2) A C E D B (2) A C D B E (2) E C A D B (1) E B A D C (1) E A C D B (1) D A B E C (1) C E D B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D B E (1) B E D A C (1) B D E C A (1) A E B D C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 6 -2 -4 B -10 0 4 2 -4 C -6 -4 0 0 -12 D 2 -2 0 0 -6 E 4 4 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 6 -2 -4 B -10 0 4 2 -4 C -6 -4 0 0 -12 D 2 -2 0 0 -6 E 4 4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 C=21 B=17 D=8 so D is eliminated. Round 2 votes counts: A=30 E=28 C=21 B=21 so C is eliminated. Round 3 votes counts: E=38 A=35 B=27 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:205 D:197 B:196 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 -2 -4 B -10 0 4 2 -4 C -6 -4 0 0 -12 D 2 -2 0 0 -6 E 4 4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -2 -4 B -10 0 4 2 -4 C -6 -4 0 0 -12 D 2 -2 0 0 -6 E 4 4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -2 -4 B -10 0 4 2 -4 C -6 -4 0 0 -12 D 2 -2 0 0 -6 E 4 4 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4732: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) A D E C B (8) C B E D A (7) E D A C B (6) B E C A D (5) B C E D A (5) B C A E D (5) E B C D A (4) E A D B C (4) C B D A E (4) A D E B C (4) E D A B C (3) E B A C D (3) D A C B E (3) C D B A E (3) C B A D E (3) B C E A D (3) E B D A C (2) D E A C B (2) D A E C B (2) A C B D E (2) E B C A D (1) E B A D C (1) D A C E B (1) C D E B A (1) C D A B E (1) B E C D A (1) A E D B C (1) A D C E B (1) A D B E C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 14 10 4 B -4 0 -12 -6 12 C -14 12 0 -2 4 D -10 6 2 0 -2 E -4 -12 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 10 4 B -4 0 -12 -6 12 C -14 12 0 -2 4 D -10 6 2 0 -2 E -4 -12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=24 C=19 B=19 D=8 so D is eliminated. Round 2 votes counts: A=36 E=26 C=19 B=19 so C is eliminated. Round 3 votes counts: A=37 B=36 E=27 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:200 D:198 B:195 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 4 B -4 0 -12 -6 12 C -14 12 0 -2 4 D -10 6 2 0 -2 E -4 -12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 4 B -4 0 -12 -6 12 C -14 12 0 -2 4 D -10 6 2 0 -2 E -4 -12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 4 B -4 0 -12 -6 12 C -14 12 0 -2 4 D -10 6 2 0 -2 E -4 -12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4733: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) C E B D A (5) B E A C D (5) A E C B D (5) D C B E A (4) D B A C E (4) D A C E B (4) A D B E C (4) E C B A D (3) D A B C E (3) C E D B A (3) B E C A D (3) B D C E A (3) B C E D A (3) A D E C B (3) A D C E B (3) A B E C D (3) E C A B D (2) E B C A D (2) D B A E C (2) C D B E A (2) C B E D A (2) B E C D A (2) B D A E C (2) B C D E A (2) B A E D C (2) B A E C D (2) A E C D B (2) A C E D B (2) E B A C D (1) E A C B D (1) D C A E B (1) D A C B E (1) C E D A B (1) C E B A D (1) C E A B D (1) C D E B A (1) C A E D B (1) B E D A C (1) B D E C A (1) A B D E C (1) Total count = 100 A B C D E A 0 -24 2 -6 -10 B 24 0 4 6 16 C -2 -4 0 10 6 D 6 -6 -10 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 2 -6 -10 B 24 0 4 6 16 C -2 -4 0 10 6 D 6 -6 -10 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993339 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 A=23 C=17 E=9 so E is eliminated. Round 2 votes counts: B=29 D=25 A=24 C=22 so C is eliminated. Round 3 votes counts: B=40 D=32 A=28 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:205 E:197 D:192 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 2 -6 -10 B 24 0 4 6 16 C -2 -4 0 10 6 D 6 -6 -10 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993339 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 2 -6 -10 B 24 0 4 6 16 C -2 -4 0 10 6 D 6 -6 -10 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993339 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 2 -6 -10 B 24 0 4 6 16 C -2 -4 0 10 6 D 6 -6 -10 0 -6 E 10 -16 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993339 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4734: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) E A D B C (8) C B D E A (7) C B D A E (7) E D A B C (6) E A C D B (5) C E D B A (5) C E A D B (5) C B A D E (5) D B A E C (4) C E A B D (4) B D C E A (4) E A D C B (3) C B E D A (3) C A E B D (3) D E B A C (2) C E B A D (2) B C D A E (2) E C A D B (1) D E B C A (1) D E A B C (1) D B E C A (1) D B E A C (1) C D B E A (1) C A B E D (1) B D C A E (1) B D A E C (1) B C D E A (1) B A D C E (1) A E D C B (1) A C E B D (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 4 -22 B -2 0 -10 -8 -14 C 8 10 0 10 10 D -4 8 -10 0 -16 E 22 14 -10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 4 -22 B -2 0 -10 -8 -14 C 8 10 0 10 10 D -4 8 -10 0 -16 E 22 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=43 E=23 A=14 D=10 B=10 so D is eliminated. Round 2 votes counts: C=43 E=27 B=16 A=14 so A is eliminated. Round 3 votes counts: C=45 E=37 B=18 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:219 D:189 A:188 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 4 -22 B -2 0 -10 -8 -14 C 8 10 0 10 10 D -4 8 -10 0 -16 E 22 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 -22 B -2 0 -10 -8 -14 C 8 10 0 10 10 D -4 8 -10 0 -16 E 22 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 -22 B -2 0 -10 -8 -14 C 8 10 0 10 10 D -4 8 -10 0 -16 E 22 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4735: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) C A D B E (9) C A B D E (7) B E D A C (7) E D A C B (6) E B D A C (6) A C D E B (6) D E A B C (5) C A D E B (4) E D A B C (3) D E A C B (3) C B A D E (3) C A E D B (3) B E C D A (3) B C A D E (3) D E B A C (2) C B A E D (2) B E D C A (2) B C E A D (2) A D E C B (2) A D C E B (2) E C D A B (1) D B E A C (1) D A E C B (1) D A E B C (1) B D E A C (1) B C E D A (1) A E C D B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 12 20 -8 -8 B -12 0 -4 -26 -16 C -20 4 0 -6 -12 D 8 26 6 0 2 E 8 16 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 20 -8 -8 B -12 0 -4 -26 -16 C -20 4 0 -6 -12 D 8 26 6 0 2 E 8 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 B=19 D=13 A=13 so D is eliminated. Round 2 votes counts: E=37 C=28 B=20 A=15 so A is eliminated. Round 3 votes counts: E=42 C=38 B=20 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:217 A:208 C:183 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 20 -8 -8 B -12 0 -4 -26 -16 C -20 4 0 -6 -12 D 8 26 6 0 2 E 8 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 20 -8 -8 B -12 0 -4 -26 -16 C -20 4 0 -6 -12 D 8 26 6 0 2 E 8 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 20 -8 -8 B -12 0 -4 -26 -16 C -20 4 0 -6 -12 D 8 26 6 0 2 E 8 16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4736: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) B D A C E (8) E C A D B (6) D C A B E (6) E C D A B (5) D B A C E (5) D E B C A (4) B A D C E (4) E D C B A (3) D E C B A (3) D B A E C (3) C A E D B (3) A C B E D (3) E D C A B (2) E C B A D (2) E B A C D (2) D C E A B (2) D C A E B (2) C A B D E (2) B A E C D (2) A B D C E (2) A B C D E (2) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) D E C A B (1) D E B A C (1) D A B C E (1) C A E B D (1) C A D E B (1) C A D B E (1) B E A C D (1) B D E A C (1) B D A E C (1) B A C D E (1) A D C B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -16 -10 2 B -2 0 -10 -6 -8 C 16 10 0 -8 -6 D 10 6 8 0 6 E -2 8 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 -10 2 B -2 0 -10 -6 -8 C 16 10 0 -8 -6 D 10 6 8 0 6 E -2 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=28 B=18 A=10 C=8 so C is eliminated. Round 2 votes counts: E=36 D=28 B=18 A=18 so B is eliminated. Round 3 votes counts: D=38 E=37 A=25 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:206 E:203 A:189 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -16 -10 2 B -2 0 -10 -6 -8 C 16 10 0 -8 -6 D 10 6 8 0 6 E -2 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -10 2 B -2 0 -10 -6 -8 C 16 10 0 -8 -6 D 10 6 8 0 6 E -2 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -10 2 B -2 0 -10 -6 -8 C 16 10 0 -8 -6 D 10 6 8 0 6 E -2 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4737: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) D B A E C (6) D A B E C (6) D A B C E (5) C E B D A (5) E C B A D (4) E C A D B (4) E C A B D (4) C E A D B (4) C B E D A (4) B D A C E (4) E A C D B (3) C E B A D (3) A E D C B (3) A D E C B (3) A D B E C (3) A B D E C (3) D C B A E (2) D A C E B (2) C D B E A (2) B C E D A (2) B A D E C (2) A E C D B (2) A D E B C (2) E A D B C (1) E A C B D (1) D B A C E (1) C E D A B (1) C D E A B (1) C A D E B (1) B D E C A (1) B D E A C (1) B D C A E (1) B A E D C (1) A E B D C (1) A D C E B (1) Total count = 100 A B C D E A 0 2 22 -8 18 B -2 0 0 -6 8 C -22 0 0 -18 -22 D 8 6 18 0 14 E -18 -8 22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 22 -8 18 B -2 0 0 -6 8 C -22 0 0 -18 -22 D 8 6 18 0 14 E -18 -8 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=22 B=22 C=21 A=18 E=17 so E is eliminated. Round 2 votes counts: C=33 A=23 D=22 B=22 so D is eliminated. Round 3 votes counts: A=36 C=35 B=29 so B is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:223 A:217 B:200 E:191 C:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 22 -8 18 B -2 0 0 -6 8 C -22 0 0 -18 -22 D 8 6 18 0 14 E -18 -8 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 22 -8 18 B -2 0 0 -6 8 C -22 0 0 -18 -22 D 8 6 18 0 14 E -18 -8 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 22 -8 18 B -2 0 0 -6 8 C -22 0 0 -18 -22 D 8 6 18 0 14 E -18 -8 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4738: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) E A D B C (6) B D C E A (6) D E B C A (5) B C D A E (5) E A D C B (4) B C A D E (4) A E C D B (4) A E C B D (4) E D A C B (3) D E C B A (3) B C D E A (3) B A C D E (3) A E D C B (3) A E B C D (3) A C B E D (3) A C B D E (3) E D C A B (2) D C B E A (2) D B E C A (2) C A B D E (2) A E B D C (2) A B E C D (2) A B C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A B C (1) E C A D B (1) D B C E A (1) C D E B A (1) C D B E A (1) C B D E A (1) C A D B E (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -8 -2 10 B 4 0 -4 12 6 C 8 4 0 12 2 D 2 -12 -12 0 10 E -10 -6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -2 10 B 4 0 -4 12 6 C 8 4 0 12 2 D 2 -12 -12 0 10 E -10 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=21 E=20 C=18 D=13 so D is eliminated. Round 2 votes counts: E=28 A=28 B=24 C=20 so C is eliminated. Round 3 votes counts: B=40 A=31 E=29 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:209 A:198 D:194 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -2 10 B 4 0 -4 12 6 C 8 4 0 12 2 D 2 -12 -12 0 10 E -10 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -2 10 B 4 0 -4 12 6 C 8 4 0 12 2 D 2 -12 -12 0 10 E -10 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -2 10 B 4 0 -4 12 6 C 8 4 0 12 2 D 2 -12 -12 0 10 E -10 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4739: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (13) A C E D B (13) B A D E C (11) A B C D E (10) C E D A B (8) E C D A B (7) B A D C E (6) A B C E D (6) B E D C A (4) B D E C A (4) D E C B A (3) A C B E D (3) E C D B A (2) C E A D B (2) E D B C A (1) B D E A C (1) B D A E C (1) B A E D C (1) A E C B D (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 12 12 10 B -6 0 -8 0 -2 C -12 8 0 10 0 D -12 0 -10 0 -24 E -10 2 0 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 12 10 B -6 0 -8 0 -2 C -12 8 0 10 0 D -12 0 -10 0 -24 E -10 2 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=28 E=23 C=10 D=3 so D is eliminated. Round 2 votes counts: A=36 B=28 E=26 C=10 so C is eliminated. Round 3 votes counts: E=36 A=36 B=28 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:208 C:203 B:192 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 12 10 B -6 0 -8 0 -2 C -12 8 0 10 0 D -12 0 -10 0 -24 E -10 2 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 12 10 B -6 0 -8 0 -2 C -12 8 0 10 0 D -12 0 -10 0 -24 E -10 2 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 12 10 B -6 0 -8 0 -2 C -12 8 0 10 0 D -12 0 -10 0 -24 E -10 2 0 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4740: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) E A D B C (6) C A B D E (6) A E C B D (5) D B E C A (4) D B E A C (4) D B C E A (4) C B D A E (4) C A E B D (4) A C E B D (4) E D A C B (3) E D A B C (3) E A C D B (3) D E B C A (3) C D B E A (3) B D E A C (3) B D C A E (3) B A D E C (3) A C B E D (3) E D B C A (2) E A D C B (2) C B A D E (2) C A E D B (2) C A B E D (2) B C D A E (2) A E C D B (2) E C D B A (1) D E C B A (1) D E B A C (1) D C B E A (1) C E D A B (1) C D E B A (1) B D E C A (1) B D C E A (1) B A D C E (1) A E B C D (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 4 -4 -8 B 2 0 -2 -8 -4 C -4 2 0 -6 -8 D 4 8 6 0 -2 E 8 4 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 -4 -8 B 2 0 -2 -8 -4 C -4 2 0 -6 -8 D 4 8 6 0 -2 E 8 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 D=18 A=17 B=14 so B is eliminated. Round 2 votes counts: C=27 E=26 D=26 A=21 so A is eliminated. Round 3 votes counts: C=35 E=34 D=31 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:208 A:195 B:194 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 -4 -8 B 2 0 -2 -8 -4 C -4 2 0 -6 -8 D 4 8 6 0 -2 E 8 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -4 -8 B 2 0 -2 -8 -4 C -4 2 0 -6 -8 D 4 8 6 0 -2 E 8 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -4 -8 B 2 0 -2 -8 -4 C -4 2 0 -6 -8 D 4 8 6 0 -2 E 8 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4741: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (15) B E C D A (14) D A B E C (12) A C D E B (6) D A C E B (4) A D C E B (4) A D B C E (4) E D C B A (2) E B C D A (2) D E B A C (2) D A E C B (2) D A E B C (2) C E B A D (2) C E A D B (2) C B E A D (2) C A E D B (2) C A E B D (2) B D E A C (2) A D C B E (2) A C D B E (2) E C B D A (1) E C B A D (1) D E C A B (1) D E A B C (1) D A C B E (1) C B A E D (1) C A D E B (1) C A B E D (1) B E D A C (1) B E C A D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -2 -4 -26 0 B 2 0 22 0 26 C 4 -22 0 -14 -20 D 26 0 14 0 -4 E 0 -26 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.486544 C: 0.000000 D: 0.513456 E: 0.000000 Sum of squares = 0.500362107105 Cumulative probabilities = A: 0.000000 B: 0.486544 C: 0.486544 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -26 0 B 2 0 22 0 26 C 4 -22 0 -14 -20 D 26 0 14 0 -4 E 0 -26 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=25 A=18 C=13 E=6 so E is eliminated. Round 2 votes counts: B=40 D=27 A=18 C=15 so C is eliminated. Round 3 votes counts: B=47 D=27 A=26 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:218 E:199 A:184 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -26 0 B 2 0 22 0 26 C 4 -22 0 -14 -20 D 26 0 14 0 -4 E 0 -26 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -26 0 B 2 0 22 0 26 C 4 -22 0 -14 -20 D 26 0 14 0 -4 E 0 -26 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -26 0 B 2 0 22 0 26 C 4 -22 0 -14 -20 D 26 0 14 0 -4 E 0 -26 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4742: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A D E B (8) D E B A C (4) C D A E B (4) C B A E D (4) C A B E D (4) B E D C A (4) B E A D C (4) C B D E A (3) C A B D E (3) B D E C A (3) A E D B C (3) A C E D B (3) E D B A C (2) E D A B C (2) E B D A C (2) E B A D C (2) D C A E B (2) D B E C A (2) C A D B E (2) B E C D A (2) B C D E A (2) A C E B D (2) E A B D C (1) D E C A B (1) D E A B C (1) D C E B A (1) D C E A B (1) D A E C B (1) D A C E B (1) C D E B A (1) C D B E A (1) C D A B E (1) C B D A E (1) C B A D E (1) B E A C D (1) B D C E A (1) B A E C D (1) B A C E D (1) A E D C B (1) A E B C D (1) A D E C B (1) A D E B C (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -4 -8 -4 B 8 0 4 10 6 C 4 -4 0 -4 -4 D 8 -10 4 0 -4 E 4 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -8 -4 B 8 0 4 10 6 C 4 -4 0 -4 -4 D 8 -10 4 0 -4 E 4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=30 D=14 A=14 E=9 so E is eliminated. Round 2 votes counts: B=34 C=33 D=18 A=15 so A is eliminated. Round 3 votes counts: C=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:203 D:199 C:196 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -8 -4 B 8 0 4 10 6 C 4 -4 0 -4 -4 D 8 -10 4 0 -4 E 4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -8 -4 B 8 0 4 10 6 C 4 -4 0 -4 -4 D 8 -10 4 0 -4 E 4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -8 -4 B 8 0 4 10 6 C 4 -4 0 -4 -4 D 8 -10 4 0 -4 E 4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4743: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) E A B C D (5) D C B A E (5) E C A B D (4) D C E A B (4) C E D B A (4) C D E B A (4) B D C A E (4) A E D B C (4) A E B D C (4) A B E D C (4) E D C A B (3) D C B E A (3) C E B A D (3) C D B E A (3) B C A D E (3) B A D C E (3) A D B E C (3) A B D E C (3) E C A D B (2) C E D A B (2) C E B D A (2) B D A C E (2) B A C E D (2) E C D A B (1) E A D B C (1) E A C B D (1) D E C A B (1) D B A C E (1) D A B E C (1) C D B A E (1) C B E D A (1) C B D E A (1) C B D A E (1) B E A C D (1) B A E D C (1) B A E C D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -22 -6 8 B 10 0 8 -4 8 C 22 -8 0 -12 16 D 6 4 12 0 4 E -8 -8 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 -6 8 B 10 0 8 -4 8 C 22 -8 0 -12 16 D 6 4 12 0 4 E -8 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 A=20 E=17 B=17 so E is eliminated. Round 2 votes counts: C=29 D=27 A=27 B=17 so B is eliminated. Round 3 votes counts: A=35 D=33 C=32 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:211 C:209 A:185 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -22 -6 8 B 10 0 8 -4 8 C 22 -8 0 -12 16 D 6 4 12 0 4 E -8 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 -6 8 B 10 0 8 -4 8 C 22 -8 0 -12 16 D 6 4 12 0 4 E -8 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 -6 8 B 10 0 8 -4 8 C 22 -8 0 -12 16 D 6 4 12 0 4 E -8 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4744: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) C E B A D (6) B E C A D (6) D A C E B (5) A D B E C (5) D E A C B (4) C E D A B (4) B A D E C (4) E C D B A (3) E C B A D (3) E B D A C (3) E B C D A (3) D A E C B (3) C B E A D (3) A D C B E (3) E D B A C (2) E C B D A (2) D E B A C (2) D E A B C (2) C B A E D (2) C A D B E (2) C A B D E (2) B E D A C (2) B E A D C (2) B A E D C (2) A D B C E (2) A C D B E (2) E D B C A (1) E B D C A (1) E B C A D (1) D E C A B (1) D C A E B (1) D A E B C (1) D A C B E (1) C D A E B (1) C A D E B (1) B E A C D (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 -6 10 2 -6 B 6 0 2 -6 0 C -10 -2 0 -10 -22 D -2 6 10 0 2 E 6 0 22 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.439999999989 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 A B C D E A 0 -6 10 2 -6 B 6 0 2 -6 0 C -10 -2 0 -10 -22 D -2 6 10 0 2 E 6 0 22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000101 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=21 B=21 E=19 A=12 so A is eliminated. Round 2 votes counts: D=37 C=23 B=21 E=19 so E is eliminated. Round 3 votes counts: D=40 C=31 B=29 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:208 B:201 A:200 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 10 2 -6 B 6 0 2 -6 0 C -10 -2 0 -10 -22 D -2 6 10 0 2 E 6 0 22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000101 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 2 -6 B 6 0 2 -6 0 C -10 -2 0 -10 -22 D -2 6 10 0 2 E 6 0 22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000101 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 2 -6 B 6 0 2 -6 0 C -10 -2 0 -10 -22 D -2 6 10 0 2 E 6 0 22 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000101 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4745: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (12) E B C D A (10) C E B A D (10) A D C B E (10) A D C E B (7) D A B C E (6) E C B A D (5) D A B E C (5) B E C D A (5) A C D E B (5) B C E D A (4) D A C B E (3) C B E D A (3) A D E C B (3) B E D A C (2) A D E B C (2) E B D C A (1) D E A B C (1) D B E A C (1) D B C A E (1) D A E B C (1) C B E A D (1) B E D C A (1) B C D E A (1) Total count = 100 A B C D E A 0 -14 -8 10 -14 B 14 0 6 10 -14 C 8 -6 0 12 2 D -10 -10 -12 0 -8 E 14 14 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.636364 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305776 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.727273 D: 0.727273 E: 1.000000 A B C D E A 0 -14 -8 10 -14 B 14 0 6 10 -14 C 8 -6 0 12 2 D -10 -10 -12 0 -8 E 14 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.636364 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305721 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.727273 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 D=18 C=14 B=13 so B is eliminated. Round 2 votes counts: E=36 A=27 C=19 D=18 so D is eliminated. Round 3 votes counts: A=42 E=38 C=20 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:208 C:208 A:187 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -8 10 -14 B 14 0 6 10 -14 C 8 -6 0 12 2 D -10 -10 -12 0 -8 E 14 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.636364 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305721 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.727273 D: 0.727273 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 10 -14 B 14 0 6 10 -14 C 8 -6 0 12 2 D -10 -10 -12 0 -8 E 14 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.636364 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305721 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.727273 D: 0.727273 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 10 -14 B 14 0 6 10 -14 C 8 -6 0 12 2 D -10 -10 -12 0 -8 E 14 14 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.636364 D: 0.000000 E: 0.272727 Sum of squares = 0.487603305721 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.727273 D: 0.727273 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4746: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) C D A B E (6) B E A C D (6) B A E D C (6) E B A D C (4) D A E B C (4) D A B C E (4) E C D B A (3) E B C A D (3) D C A E B (3) C E D B A (3) C D E B A (3) B E A D C (3) A B D C E (3) E D A B C (2) E B A C D (2) D E C A B (2) D C E A B (2) D A C E B (2) C D E A B (2) C B E A D (2) B A E C D (2) B A C D E (2) A D B E C (2) A D B C E (2) E D B C A (1) E D B A C (1) E C B D A (1) E C B A D (1) D E A C B (1) D E A B C (1) D C A B E (1) D A C B E (1) C E B D A (1) C D B E A (1) C D B A E (1) C D A E B (1) C B D A E (1) C B A E D (1) C B A D E (1) C A B D E (1) B A C E D (1) A E B D C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 18 4 10 B 0 0 18 2 12 C -18 -18 0 -8 -8 D -4 -2 8 0 12 E -10 -12 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.587634 B: 0.412366 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.515359592479 Cumulative probabilities = A: 0.587634 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 4 10 B 0 0 18 2 12 C -18 -18 0 -8 -8 D -4 -2 8 0 12 E -10 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999728 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=21 B=20 E=18 A=17 so A is eliminated. Round 2 votes counts: B=32 D=25 C=24 E=19 so E is eliminated. Round 3 votes counts: B=42 D=29 C=29 so D is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:216 D:207 E:187 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 4 10 B 0 0 18 2 12 C -18 -18 0 -8 -8 D -4 -2 8 0 12 E -10 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999728 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 4 10 B 0 0 18 2 12 C -18 -18 0 -8 -8 D -4 -2 8 0 12 E -10 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999728 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 4 10 B 0 0 18 2 12 C -18 -18 0 -8 -8 D -4 -2 8 0 12 E -10 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999728 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4747: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) C A D E B (6) B E D A C (6) A C B E D (6) D E B C A (5) C A D B E (5) B E D C A (5) E B D A C (4) D E C A B (4) C A B D E (4) B A E C D (4) A C D E B (4) D E C B A (3) B A C E D (3) A D E C B (3) A B C E D (3) E D A B C (2) D E A C B (2) D A E C B (2) C B D E A (2) C B A D E (2) B C D E A (2) B C A E D (2) A C B D E (2) E D B C A (1) E D A C B (1) D E A B C (1) D C E A B (1) C D E B A (1) C D A E B (1) B E A C D (1) B C E D A (1) B A E D C (1) A E D C B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 10 -2 2 B 0 0 -2 -2 2 C -10 2 0 2 -8 D 2 2 -2 0 2 E -2 -2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408121 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 -2 2 B 0 0 -2 -2 2 C -10 2 0 2 -8 D 2 2 -2 0 2 E -2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408167 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=21 A=21 D=18 E=15 so E is eliminated. Round 2 votes counts: D=29 B=29 C=21 A=21 so C is eliminated. Round 3 votes counts: A=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:205 D:202 E:201 B:199 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 -2 2 B 0 0 -2 -2 2 C -10 2 0 2 -8 D 2 2 -2 0 2 E -2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408167 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -2 2 B 0 0 -2 -2 2 C -10 2 0 2 -8 D 2 2 -2 0 2 E -2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408167 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -2 2 B 0 0 -2 -2 2 C -10 2 0 2 -8 D 2 2 -2 0 2 E -2 -2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408167 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4748: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E D C B A (6) C B A D E (6) A B E D C (6) E D C A B (5) B A C D E (5) E D A C B (4) D E C A B (4) B C A E D (4) A D E B C (4) A B C D E (4) D E A C B (3) C D B E A (3) C B E D A (3) B C A D E (3) A B E C D (3) E D A B C (2) E A B D C (2) D A C B E (2) C D E B A (2) B A C E D (2) A B D E C (2) E B A C D (1) E A D B C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A B E (1) D A E C B (1) C E D B A (1) C D B A E (1) B E C A D (1) B C E D A (1) B C E A D (1) A E D B C (1) A E B D C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -8 -2 14 B 2 0 -10 10 18 C 8 10 0 2 4 D 2 -10 -2 0 10 E -14 -18 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -2 14 B 2 0 -10 10 18 C 8 10 0 2 4 D 2 -10 -2 0 10 E -14 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 E=21 B=17 D=14 so D is eliminated. Round 2 votes counts: E=29 C=28 A=26 B=17 so B is eliminated. Round 3 votes counts: C=37 A=33 E=30 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:210 A:201 D:200 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -2 14 B 2 0 -10 10 18 C 8 10 0 2 4 D 2 -10 -2 0 10 E -14 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -2 14 B 2 0 -10 10 18 C 8 10 0 2 4 D 2 -10 -2 0 10 E -14 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -2 14 B 2 0 -10 10 18 C 8 10 0 2 4 D 2 -10 -2 0 10 E -14 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4749: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) B D C A E (7) D A E B C (6) C B E D A (6) D B A E C (5) C E A B D (5) A E C D B (5) B C D E A (4) A D E B C (4) E C A D B (3) E A D C B (3) C A E D B (3) B D A E C (3) A E D C B (3) E A C D B (2) E A C B D (2) D B A C E (2) C B D E A (2) C A E B D (2) B D E A C (2) B D C E A (2) B C D A E (2) A D E C B (2) E D B A C (1) E C A B D (1) E A D B C (1) D E B A C (1) D B C A E (1) D A B E C (1) C D B A E (1) C B E A D (1) C B D A E (1) C A D E B (1) B E C D A (1) B D E C A (1) B D A C E (1) A E D B C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -6 0 6 B 6 0 -6 4 -12 C 6 6 0 4 4 D 0 -4 -4 0 0 E -6 12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 0 6 B 6 0 -6 4 -12 C 6 6 0 4 4 D 0 -4 -4 0 0 E -6 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=23 A=17 D=16 E=13 so E is eliminated. Round 2 votes counts: C=35 A=25 B=23 D=17 so D is eliminated. Round 3 votes counts: C=35 B=33 A=32 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 E:201 A:197 B:196 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 0 6 B 6 0 -6 4 -12 C 6 6 0 4 4 D 0 -4 -4 0 0 E -6 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 0 6 B 6 0 -6 4 -12 C 6 6 0 4 4 D 0 -4 -4 0 0 E -6 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 0 6 B 6 0 -6 4 -12 C 6 6 0 4 4 D 0 -4 -4 0 0 E -6 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4750: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) E C B A D (10) D A B C E (9) C E A D B (9) A D C B E (9) B E D A C (8) B D A E C (8) C A D E B (7) E C A D B (5) B D A C E (5) E B D A C (3) C B E A D (2) C A E D B (2) C A D B E (2) A D C E B (2) E B C A D (1) E A D C B (1) D A C B E (1) D A B E C (1) C E B A D (1) B E C D A (1) B C E D A (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -4 4 -6 B 2 0 -4 2 -6 C 4 4 0 6 2 D -4 -2 -6 0 -10 E 6 6 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 4 -6 B 2 0 -4 2 -6 C 4 4 0 6 2 D -4 -2 -6 0 -10 E 6 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=23 B=23 A=12 D=11 so D is eliminated. Round 2 votes counts: E=31 C=23 B=23 A=23 so C is eliminated. Round 3 votes counts: E=41 A=34 B=25 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:208 B:197 A:196 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 4 -6 B 2 0 -4 2 -6 C 4 4 0 6 2 D -4 -2 -6 0 -10 E 6 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 4 -6 B 2 0 -4 2 -6 C 4 4 0 6 2 D -4 -2 -6 0 -10 E 6 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 4 -6 B 2 0 -4 2 -6 C 4 4 0 6 2 D -4 -2 -6 0 -10 E 6 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4751: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) B D E A C (7) B A D E C (7) E D C B A (4) E D A B C (4) C E D A B (4) B D E C A (4) B D A E C (4) E D B A C (3) C E D B A (3) B A D C E (3) B A C D E (3) A C E D B (3) A C E B D (3) A B C D E (3) E D C A B (2) E D B C A (2) E D A C B (2) E C D A B (2) D E C B A (2) D E B C A (2) D E B A C (2) D B E C A (2) C B D E A (2) C A B E D (2) A C B E D (2) A C B D E (2) A B C E D (2) E C D B A (1) E A D C B (1) E A D B C (1) C B A D E (1) C A B D E (1) B D C E A (1) A E D C B (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 10 -6 -2 B 6 0 4 -4 -8 C -10 -4 0 -12 -10 D 6 4 12 0 -8 E 2 8 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 10 -6 -2 B 6 0 4 -4 -8 C -10 -4 0 -12 -10 D 6 4 12 0 -8 E 2 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=23 E=22 A=18 D=8 so D is eliminated. Round 2 votes counts: B=31 E=28 C=23 A=18 so A is eliminated. Round 3 votes counts: B=36 C=33 E=31 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:214 D:207 B:199 A:198 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 10 -6 -2 B 6 0 4 -4 -8 C -10 -4 0 -12 -10 D 6 4 12 0 -8 E 2 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 -6 -2 B 6 0 4 -4 -8 C -10 -4 0 -12 -10 D 6 4 12 0 -8 E 2 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 -6 -2 B 6 0 4 -4 -8 C -10 -4 0 -12 -10 D 6 4 12 0 -8 E 2 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4752: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) D E B A C (8) D E B C A (6) C A E B D (6) C A B E D (6) B D A E C (5) E D A B C (4) D B E A C (4) C E A D B (4) B A D E C (4) B A C D E (4) E D C A B (3) E D A C B (3) B D A C E (3) B A D C E (3) E A D B C (2) D E C B A (2) C E D A B (2) C D E B A (2) C A B D E (2) B C D A E (2) B C A D E (2) A B E D C (2) A B C D E (2) E D C B A (1) E C A D B (1) E A D C B (1) D C E B A (1) C B D A E (1) B D E A C (1) B D C A E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -20 -4 2 10 B 20 0 8 12 4 C 4 -8 0 -8 4 D -2 -12 8 0 26 E -10 -4 -4 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 2 10 B 20 0 8 12 4 C 4 -8 0 -8 4 D -2 -12 8 0 26 E -10 -4 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993113 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=25 D=21 E=15 A=6 so A is eliminated. Round 2 votes counts: C=34 B=29 D=21 E=16 so E is eliminated. Round 3 votes counts: D=35 C=35 B=30 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:222 D:210 C:196 A:194 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -4 2 10 B 20 0 8 12 4 C 4 -8 0 -8 4 D -2 -12 8 0 26 E -10 -4 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993113 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 2 10 B 20 0 8 12 4 C 4 -8 0 -8 4 D -2 -12 8 0 26 E -10 -4 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993113 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 2 10 B 20 0 8 12 4 C 4 -8 0 -8 4 D -2 -12 8 0 26 E -10 -4 -4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993113 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4753: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (13) E C A D B (12) C D A E B (7) B E A D C (6) E C B A D (5) E B C A D (5) C D A B E (5) D A B C E (4) C E A D B (4) B A D E C (4) E C D A B (3) D B A C E (3) C E D A B (3) B D A E C (3) B A D C E (3) E C B D A (2) E C A B D (2) E B A C D (2) D A C B E (2) B A E D C (2) E C D B A (1) E B D A C (1) E B A D C (1) D C A B E (1) C A E D B (1) C A D B E (1) B E D A C (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 2 4 B 4 0 0 2 2 C 4 0 0 10 0 D -2 -2 -10 0 -2 E -4 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.724249 C: 0.275751 D: 0.000000 E: 0.000000 Sum of squares = 0.600574899565 Cumulative probabilities = A: 0.000000 B: 0.724249 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 2 4 B 4 0 0 2 2 C 4 0 0 10 0 D -2 -2 -10 0 -2 E -4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999695 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=32 C=21 D=10 A=3 so A is eliminated. Round 2 votes counts: E=34 B=33 C=22 D=11 so D is eliminated. Round 3 votes counts: B=41 E=34 C=25 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:207 B:204 A:199 E:198 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 2 4 B 4 0 0 2 2 C 4 0 0 10 0 D -2 -2 -10 0 -2 E -4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999695 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 2 4 B 4 0 0 2 2 C 4 0 0 10 0 D -2 -2 -10 0 -2 E -4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999695 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 2 4 B 4 0 0 2 2 C 4 0 0 10 0 D -2 -2 -10 0 -2 E -4 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999695 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4754: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) D C E A B (9) E A B D C (8) D C B E A (7) B A E D C (7) E A D B C (5) C D B E A (4) B A E C D (4) A E B C D (4) E A C B D (3) D C E B A (3) C D E A B (3) B C D A E (3) E D A B C (2) E A D C B (2) E A B C D (2) D E C A B (2) D C B A E (2) D B C E A (2) B D E A C (2) B D C A E (2) B C A E D (2) A B E D C (2) E D C A B (1) E D A C B (1) E C A D B (1) D E B C A (1) D E A C B (1) C B D A E (1) C B A E D (1) C A E B D (1) C A B E D (1) B C A D E (1) Total count = 100 A B C D E A 0 -4 -14 -12 -18 B 4 0 -6 -12 2 C 14 6 0 -18 4 D 12 12 18 0 6 E 18 -2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -12 -18 B 4 0 -6 -12 2 C 14 6 0 -18 4 D 12 12 18 0 6 E 18 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 C=21 B=21 A=6 so A is eliminated. Round 2 votes counts: E=29 D=27 B=23 C=21 so C is eliminated. Round 3 votes counts: D=44 E=30 B=26 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:203 E:203 B:194 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -14 -12 -18 B 4 0 -6 -12 2 C 14 6 0 -18 4 D 12 12 18 0 6 E 18 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -12 -18 B 4 0 -6 -12 2 C 14 6 0 -18 4 D 12 12 18 0 6 E 18 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -12 -18 B 4 0 -6 -12 2 C 14 6 0 -18 4 D 12 12 18 0 6 E 18 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4755: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) A E D B C (10) D E A C B (8) E A D C B (6) D C B E A (6) D B C E A (5) B C D A E (5) A E C B D (5) E A C D B (4) E A C B D (4) D A E B C (4) C B E A D (4) C B D E A (4) A E B C D (4) C B A E D (3) B C A E D (3) D E A B C (2) B D C A E (2) B C D E A (2) A E B D C (2) D E C A B (1) D C E B A (1) C B E D A (1) B A C E D (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 -4 4 B -4 0 6 -18 -4 C -4 -6 0 -18 -4 D 4 18 18 0 4 E -4 4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -4 4 B -4 0 6 -18 -4 C -4 -6 0 -18 -4 D 4 18 18 0 4 E -4 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=23 E=14 B=13 C=12 so C is eliminated. Round 2 votes counts: D=38 B=25 A=23 E=14 so E is eliminated. Round 3 votes counts: D=38 A=37 B=25 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:204 E:200 B:190 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -4 4 B -4 0 6 -18 -4 C -4 -6 0 -18 -4 D 4 18 18 0 4 E -4 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -4 4 B -4 0 6 -18 -4 C -4 -6 0 -18 -4 D 4 18 18 0 4 E -4 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -4 4 B -4 0 6 -18 -4 C -4 -6 0 -18 -4 D 4 18 18 0 4 E -4 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4756: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) B D A E C (8) C E A D B (7) E C B A D (6) D A C B E (5) C E D A B (5) C A D E B (5) D C A E B (4) D A C E B (4) C D A E B (4) B E A D C (4) B E A C D (4) E B C A D (3) D C E A B (3) B E C D A (3) E C B D A (2) E C A B D (2) D B A C E (2) B E D C A (2) B D E A C (2) B A E C D (2) B A D E C (2) E B C D A (1) D C E B A (1) C E D B A (1) C E A B D (1) C A E D B (1) B E C A D (1) B A E D C (1) A D C E B (1) A D B C E (1) A C D E B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -4 -12 4 B -10 0 -6 -8 -4 C 4 6 0 0 12 D 12 8 0 0 6 E -4 4 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.418223 D: 0.581777 E: 0.000000 Sum of squares = 0.513374910982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.418223 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -12 4 B -10 0 -6 -8 -4 C 4 6 0 0 12 D 12 8 0 0 6 E -4 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=28 C=24 E=14 A=5 so A is eliminated. Round 2 votes counts: B=31 D=30 C=25 E=14 so E is eliminated. Round 3 votes counts: C=35 B=35 D=30 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:211 A:199 E:191 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -12 4 B -10 0 -6 -8 -4 C 4 6 0 0 12 D 12 8 0 0 6 E -4 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -12 4 B -10 0 -6 -8 -4 C 4 6 0 0 12 D 12 8 0 0 6 E -4 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -12 4 B -10 0 -6 -8 -4 C 4 6 0 0 12 D 12 8 0 0 6 E -4 4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4757: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) B E C D A (7) B E C A D (7) E C B A D (5) B D A E C (5) E C A B D (4) D A B C E (4) A D B E C (4) D A C E B (3) C E B A D (3) C E A B D (3) A E D C B (3) A E C D B (3) A E B D C (3) A D C E B (3) E B C A D (2) E A B C D (2) D C B E A (2) D A C B E (2) C B E D A (2) B D C E A (2) A E C B D (2) A E B C D (2) A D E C B (2) E B A C D (1) E A C B D (1) D B C E A (1) D B C A E (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E A B (1) C D B E A (1) C A E D B (1) B E D A C (1) B D E C A (1) B D A C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -8 8 -14 B 4 0 8 26 2 C 8 -8 0 20 -14 D -8 -26 -20 0 -28 E 14 -2 14 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 8 -14 B 4 0 8 26 2 C 8 -8 0 20 -14 D -8 -26 -20 0 -28 E 14 -2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997707 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=24 E=15 D=15 C=14 so C is eliminated. Round 2 votes counts: B=34 A=25 E=24 D=17 so D is eliminated. Round 3 votes counts: B=40 A=35 E=25 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:227 B:220 C:203 A:191 D:159 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 8 -14 B 4 0 8 26 2 C 8 -8 0 20 -14 D -8 -26 -20 0 -28 E 14 -2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997707 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 8 -14 B 4 0 8 26 2 C 8 -8 0 20 -14 D -8 -26 -20 0 -28 E 14 -2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997707 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 8 -14 B 4 0 8 26 2 C 8 -8 0 20 -14 D -8 -26 -20 0 -28 E 14 -2 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997707 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4758: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (12) E B D A C (10) D E B C A (6) C A D B E (6) B E A D C (6) A B E C D (5) E D B A C (4) E B A D C (4) A C B E D (4) C D B E A (3) C A B D E (3) D E C B A (2) D E B A C (2) D C A E B (2) C D E A B (2) C A B E D (2) B E D C A (2) B E A C D (2) B A C E D (2) A D C E B (2) A B E D C (2) E B D C A (1) D E A B C (1) D C E B A (1) D C E A B (1) D A E C B (1) D A E B C (1) C D B A E (1) C D A B E (1) C B E D A (1) C B A E D (1) C B A D E (1) C A D E B (1) B E D A C (1) B C A E D (1) B A E C D (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 0 -10 2 B 4 0 4 -2 -10 C 0 -4 0 -2 -6 D 10 2 2 0 0 E -2 10 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.517150 E: 0.482850 Sum of squares = 0.5005882395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.517150 E: 1.000000 A B C D E A 0 -4 0 -10 2 B 4 0 4 -2 -10 C 0 -4 0 -2 -6 D 10 2 2 0 0 E -2 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=19 D=17 B=15 A=15 so B is eliminated. Round 2 votes counts: C=35 E=30 A=18 D=17 so D is eliminated. Round 3 votes counts: E=41 C=39 A=20 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:207 E:207 B:198 A:194 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -10 2 B 4 0 4 -2 -10 C 0 -4 0 -2 -6 D 10 2 2 0 0 E -2 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -10 2 B 4 0 4 -2 -10 C 0 -4 0 -2 -6 D 10 2 2 0 0 E -2 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -10 2 B 4 0 4 -2 -10 C 0 -4 0 -2 -6 D 10 2 2 0 0 E -2 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4759: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) E D A B C (7) C B A D E (7) B A C E D (7) E A B D C (6) D E C A B (6) D E A B C (4) D C E B A (4) D A B C E (4) C D E B A (4) A B D C E (4) D C E A B (3) C A B D E (3) B A C D E (3) A B C D E (3) E D B A C (2) E B A C D (2) D C A B E (2) C D B A E (2) B A E C D (2) A B D E C (2) E D C A B (1) E C B A D (1) E B A D C (1) D E C B A (1) D A E B C (1) D A C B E (1) D A B E C (1) C E D B A (1) C E B A D (1) C B E A D (1) C B A E D (1) A D B E C (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 8 -4 -6 B -4 0 6 -6 -6 C -8 -6 0 -20 4 D 4 6 20 0 12 E 6 6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -4 -6 B -4 0 6 -6 -6 C -8 -6 0 -20 4 D 4 6 20 0 12 E 6 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 C=20 A=13 B=12 so B is eliminated. Round 2 votes counts: E=28 D=27 A=25 C=20 so C is eliminated. Round 3 votes counts: A=36 D=33 E=31 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:201 E:198 B:195 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -4 -6 B -4 0 6 -6 -6 C -8 -6 0 -20 4 D 4 6 20 0 12 E 6 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -4 -6 B -4 0 6 -6 -6 C -8 -6 0 -20 4 D 4 6 20 0 12 E 6 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -4 -6 B -4 0 6 -6 -6 C -8 -6 0 -20 4 D 4 6 20 0 12 E 6 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4760: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (19) B E A C D (8) C A B E D (7) D E B A C (6) C A D B E (6) E B D A C (5) D E C B A (5) D C E A B (5) D E B C A (4) D C E B A (3) D C A E B (3) C D A E B (3) A B C E D (3) E D B C A (2) E B D C A (2) D A E B C (2) B E A D C (2) A C D B E (2) E D B A C (1) D B E A C (1) D A C E B (1) C D E B A (1) C A D E B (1) B E C A D (1) B C E A D (1) B A E D C (1) B A E C D (1) A C B D E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 12 12 6 B -12 0 -14 8 12 C -12 14 0 12 14 D -12 -8 -12 0 -8 E -6 -12 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 12 6 B -12 0 -14 8 12 C -12 14 0 12 14 D -12 -8 -12 0 -8 E -6 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=28 C=18 B=14 E=10 so E is eliminated. Round 2 votes counts: D=33 A=28 B=21 C=18 so C is eliminated. Round 3 votes counts: A=42 D=37 B=21 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:214 B:197 E:188 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 12 6 B -12 0 -14 8 12 C -12 14 0 12 14 D -12 -8 -12 0 -8 E -6 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 12 6 B -12 0 -14 8 12 C -12 14 0 12 14 D -12 -8 -12 0 -8 E -6 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 12 6 B -12 0 -14 8 12 C -12 14 0 12 14 D -12 -8 -12 0 -8 E -6 -12 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4761: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (16) A E B D C (6) E D C A B (5) E C D B A (5) C D E B A (5) E A B C D (4) D C E B A (4) C D B E A (4) B A C D E (4) A B E D C (4) A B D C E (4) D C B A E (3) B A C E D (3) A D B C E (3) E A D C B (2) D E C A B (2) D C B E A (2) C B D E A (2) B C D A E (2) B A E C D (2) A E B C D (2) A B D E C (2) E C D A B (1) E B A C D (1) E A D B C (1) D C E A B (1) D C A B E (1) D B C A E (1) B D C A E (1) B C E D A (1) B C E A D (1) A E D B C (1) A D E C B (1) A D E B C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 18 18 20 18 B -18 0 24 14 16 C -18 -24 0 10 -12 D -20 -14 -10 0 -12 E -18 -16 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 18 20 18 B -18 0 24 14 16 C -18 -24 0 10 -12 D -20 -14 -10 0 -12 E -18 -16 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=19 D=14 B=14 C=11 so C is eliminated. Round 2 votes counts: A=42 D=23 E=19 B=16 so B is eliminated. Round 3 votes counts: A=51 D=28 E=21 so E is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:237 B:218 E:195 C:178 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 18 20 18 B -18 0 24 14 16 C -18 -24 0 10 -12 D -20 -14 -10 0 -12 E -18 -16 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 20 18 B -18 0 24 14 16 C -18 -24 0 10 -12 D -20 -14 -10 0 -12 E -18 -16 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 20 18 B -18 0 24 14 16 C -18 -24 0 10 -12 D -20 -14 -10 0 -12 E -18 -16 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4762: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) E A C D B (7) C E D B A (7) E C D A B (5) B D C A E (5) B D A C E (5) A E D B C (5) D B C E A (4) A E C D B (4) A B E D C (4) A B D E C (4) C D E B A (3) B D C E A (3) B C D E A (3) A E B D C (3) E A C B D (2) D C E B A (2) D C B E A (2) C E D A B (2) C E B D A (2) C B E D A (2) B A D E C (2) B A D C E (2) A E C B D (2) A D B E C (2) A B E C D (2) E D A C B (1) E C A B D (1) E A D C B (1) D B A E C (1) D A E B C (1) C D B E A (1) B C D A E (1) A E D C B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 0 0 -10 B -10 0 -6 -12 -12 C 0 6 0 4 -10 D 0 12 -4 0 -18 E 10 12 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 0 0 -10 B -10 0 -6 -12 -12 C 0 6 0 4 -10 D 0 12 -4 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 B=21 C=17 D=10 so D is eliminated. Round 2 votes counts: A=29 B=26 E=24 C=21 so C is eliminated. Round 3 votes counts: E=40 B=31 A=29 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:200 C:200 D:195 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 0 0 -10 B -10 0 -6 -12 -12 C 0 6 0 4 -10 D 0 12 -4 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 0 -10 B -10 0 -6 -12 -12 C 0 6 0 4 -10 D 0 12 -4 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 0 -10 B -10 0 -6 -12 -12 C 0 6 0 4 -10 D 0 12 -4 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4763: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) A D E C B (9) E A D B C (6) B C A D E (6) E D A C B (5) B C D E A (5) B C E A D (4) B C D A E (4) A E D B C (4) E B A D C (3) D E A C B (3) B C E D A (3) E A D C B (2) D A E C B (2) C B E D A (2) C B A D E (2) B C A E D (2) A D E B C (2) A D B E C (2) E D C B A (1) E D C A B (1) E D A B C (1) E B D A C (1) E B C D A (1) E A B D C (1) D E C A B (1) D C E A B (1) D C A E B (1) D A C E B (1) C D E B A (1) C D E A B (1) C D A E B (1) C B D A E (1) C A B D E (1) B E C D A (1) B E A D C (1) B E A C D (1) B A E D C (1) B A E C D (1) B A C D E (1) A E B D C (1) A D C E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 2 6 -10 B 2 0 8 8 -2 C -2 -8 0 -6 -6 D -6 -8 6 0 12 E 10 2 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.43801652899 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 A B C D E A 0 -2 2 6 -10 B 2 0 8 8 -2 C -2 -8 0 -6 -6 D -6 -8 6 0 12 E 10 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016529117 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=22 A=21 C=18 D=9 so D is eliminated. Round 2 votes counts: B=30 E=26 A=24 C=20 so C is eliminated. Round 3 votes counts: B=44 E=29 A=27 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:208 E:203 D:202 A:198 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 6 -10 B 2 0 8 8 -2 C -2 -8 0 -6 -6 D -6 -8 6 0 12 E 10 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016529117 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 6 -10 B 2 0 8 8 -2 C -2 -8 0 -6 -6 D -6 -8 6 0 12 E 10 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016529117 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 6 -10 B 2 0 8 8 -2 C -2 -8 0 -6 -6 D -6 -8 6 0 12 E 10 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.090909 E: 0.363636 Sum of squares = 0.438016529117 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.636364 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4764: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A D C B E (9) E B C A D (8) D C A B E (8) E B D C A (6) B E C D A (6) E B A C D (5) A D C E B (5) D A C E B (4) D A C B E (4) A E B C D (4) A C D B E (4) A C B E D (4) E B A D C (2) D C E B A (2) D C B E A (2) C D A B E (2) A E B D C (2) A D E B C (2) D E B C A (1) D C A E B (1) D A E C B (1) D A E B C (1) C D B E A (1) C D B A E (1) C B A E D (1) C A D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -2 -2 12 B -8 0 -2 2 -10 C 2 2 0 0 0 D 2 -2 0 0 0 E -12 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.549120 D: 0.339076 E: 0.111804 Sum of squares = 0.429005589772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.549120 D: 0.888196 E: 1.000000 A B C D E A 0 8 -2 -2 12 B -8 0 -2 2 -10 C 2 2 0 0 0 D 2 -2 0 0 0 E -12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.428571 E: 0.142857 Sum of squares = 0.387755198734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.857143 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=32 A=32 D=24 C=6 B=6 so C is eliminated. Round 2 votes counts: A=33 E=32 D=28 B=7 so B is eliminated. Round 3 votes counts: E=38 A=34 D=28 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:202 D:200 E:199 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 -2 12 B -8 0 -2 2 -10 C 2 2 0 0 0 D 2 -2 0 0 0 E -12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.428571 E: 0.142857 Sum of squares = 0.387755198734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.857143 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -2 12 B -8 0 -2 2 -10 C 2 2 0 0 0 D 2 -2 0 0 0 E -12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.428571 E: 0.142857 Sum of squares = 0.387755198734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -2 12 B -8 0 -2 2 -10 C 2 2 0 0 0 D 2 -2 0 0 0 E -12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.428571 E: 0.142857 Sum of squares = 0.387755198734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428572 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4765: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (15) A E B C D (14) E A D C B (12) B C D A E (10) C D B E A (6) D E C B A (5) D C E B A (4) B A C D E (4) A B C E D (4) E D C A B (3) E D A C B (3) A B E C D (3) E A D B C (2) D C B A E (2) C B D A E (2) A E D B C (2) A E B D C (2) A B C D E (2) E D C B A (1) E A B D C (1) C B D E A (1) B C A E D (1) B C A D E (1) Total count = 100 A B C D E A 0 -4 -2 -4 -6 B 4 0 -8 -10 2 C 2 8 0 -4 4 D 4 10 4 0 4 E 6 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 -6 B 4 0 -8 -10 2 C 2 8 0 -4 4 D 4 10 4 0 4 E 6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 E=22 B=16 C=9 so C is eliminated. Round 2 votes counts: D=32 A=27 E=22 B=19 so B is eliminated. Round 3 votes counts: D=45 A=33 E=22 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:205 E:198 B:194 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 -6 B 4 0 -8 -10 2 C 2 8 0 -4 4 D 4 10 4 0 4 E 6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 -6 B 4 0 -8 -10 2 C 2 8 0 -4 4 D 4 10 4 0 4 E 6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 -6 B 4 0 -8 -10 2 C 2 8 0 -4 4 D 4 10 4 0 4 E 6 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4766: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) C B D E A (6) A E B D C (6) E C D B A (5) A E B C D (5) E D A C B (4) D C B E A (4) B A C D E (4) A B D C E (4) A B C D E (4) E A C B D (3) D C E B A (3) C D B E A (3) E D C B A (2) E D C A B (2) E C B D A (2) E C B A D (2) E A D C B (2) E A D B C (2) E A C D B (2) D E A C B (2) C B D A E (2) B C A D E (2) B A D C E (2) A B D E C (2) E C A D B (1) E A B C D (1) D E C B A (1) D B C A E (1) C E D B A (1) C E B D A (1) C D B A E (1) C B E D A (1) C B E A D (1) B D A C E (1) B A C E D (1) A E D B C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 0 -2 B 8 0 -2 22 2 C -2 2 0 16 2 D 0 -22 -16 0 4 E 2 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 0 -2 B 8 0 -2 22 2 C -2 2 0 16 2 D 0 -22 -16 0 4 E 2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 B=18 C=16 D=11 so D is eliminated. Round 2 votes counts: E=31 A=27 C=23 B=19 so B is eliminated. Round 3 votes counts: A=35 C=34 E=31 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:215 C:209 E:197 A:196 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 0 -2 B 8 0 -2 22 2 C -2 2 0 16 2 D 0 -22 -16 0 4 E 2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 0 -2 B 8 0 -2 22 2 C -2 2 0 16 2 D 0 -22 -16 0 4 E 2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 0 -2 B 8 0 -2 22 2 C -2 2 0 16 2 D 0 -22 -16 0 4 E 2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999913 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4767: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) A B C E D (9) C B A E D (8) D E A B C (6) E D C B A (4) D C B A E (4) D A E B C (4) B C A D E (4) E D A B C (3) E C B A D (3) D C B E A (3) D A B C E (3) C B A D E (3) E D C A B (2) E A D B C (2) E A C B D (2) E A B C D (2) D E C A B (2) D C E B A (2) C E B A D (2) C D B A E (2) C B E A D (2) C B D A E (2) B C A E D (2) A D B C E (2) E D A C B (1) D E B C A (1) D E A C B (1) D B C A E (1) D A B E C (1) C E D B A (1) C D E B A (1) C D B E A (1) C A B E D (1) B A C E D (1) A E B C D (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -20 -8 0 B 12 0 -12 -14 0 C 20 12 0 -4 10 D 8 14 4 0 8 E 0 0 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -20 -8 0 B 12 0 -12 -14 0 C 20 12 0 -4 10 D 8 14 4 0 8 E 0 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=23 E=19 A=14 B=7 so B is eliminated. Round 2 votes counts: D=37 C=29 E=19 A=15 so A is eliminated. Round 3 votes counts: D=40 C=40 E=20 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:219 D:217 B:193 E:191 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -20 -8 0 B 12 0 -12 -14 0 C 20 12 0 -4 10 D 8 14 4 0 8 E 0 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -8 0 B 12 0 -12 -14 0 C 20 12 0 -4 10 D 8 14 4 0 8 E 0 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -8 0 B 12 0 -12 -14 0 C 20 12 0 -4 10 D 8 14 4 0 8 E 0 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4768: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) B A C E D (7) A B D C E (7) B A C D E (6) E D C A B (5) D E C A B (4) D E A C B (4) D A E B C (4) E C D B A (3) E C A B D (3) D C E B A (3) D A B C E (3) A B C E D (3) E D C B A (2) E C D A B (2) E C B A D (2) E A C B D (2) D B A C E (2) D A B E C (2) C E B D A (2) C B E A D (2) B A D C E (2) A E B C D (2) A D B E C (2) A B D E C (2) E D A C B (1) E D A B C (1) E C B D A (1) E A D B C (1) D C B A E (1) C E D B A (1) C E B A D (1) C D E B A (1) C B D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) A E B D C (1) A D B C E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 10 -4 0 B -4 0 2 -2 -8 C -10 -2 0 -12 -6 D 4 2 12 0 12 E 0 8 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 -4 0 B -4 0 2 -2 -8 C -10 -2 0 -12 -6 D 4 2 12 0 12 E 0 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=23 A=20 B=18 C=8 so C is eliminated. Round 2 votes counts: D=32 E=27 B=21 A=20 so A is eliminated. Round 3 votes counts: D=35 B=35 E=30 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:205 E:201 B:194 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 10 -4 0 B -4 0 2 -2 -8 C -10 -2 0 -12 -6 D 4 2 12 0 12 E 0 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -4 0 B -4 0 2 -2 -8 C -10 -2 0 -12 -6 D 4 2 12 0 12 E 0 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -4 0 B -4 0 2 -2 -8 C -10 -2 0 -12 -6 D 4 2 12 0 12 E 0 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4769: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) C A D B E (7) B D A C E (7) B A D C E (7) E B D A C (6) D B C A E (4) E D C A B (3) E C A B D (3) D C A B E (3) C A E D B (3) B A C D E (3) A C B D E (3) E D C B A (2) E C D A B (2) E B D C A (2) E B A D C (2) E A C B D (2) E A B C D (2) D E B C A (2) D C B A E (2) D C A E B (2) D B E C A (2) C D A B E (2) B D E A C (2) B D A E C (2) A C E B D (2) A B C D E (2) E D B C A (1) E B A C D (1) D C E A B (1) D C B E A (1) D B A C E (1) C E A D B (1) C D E A B (1) C A D E B (1) B E D A C (1) B A D E C (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 2 10 B -2 0 -2 0 6 C 6 2 0 -8 12 D -2 0 8 0 14 E -10 -6 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999966 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 2 10 B -2 0 -2 0 6 C 6 2 0 -8 12 D -2 0 8 0 14 E -10 -6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999436 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=23 D=18 C=15 A=10 so A is eliminated. Round 2 votes counts: E=34 B=26 C=22 D=18 so D is eliminated. Round 3 votes counts: E=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:210 C:206 A:204 B:201 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 2 10 B -2 0 -2 0 6 C 6 2 0 -8 12 D -2 0 8 0 14 E -10 -6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999436 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 2 10 B -2 0 -2 0 6 C 6 2 0 -8 12 D -2 0 8 0 14 E -10 -6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999436 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 2 10 B -2 0 -2 0 6 C 6 2 0 -8 12 D -2 0 8 0 14 E -10 -6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.125000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999436 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4770: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (15) D E C A B (9) A B C D E (8) A B E C D (7) C B A D E (6) B A C E D (5) E D C A B (4) E D A B C (4) E B A C D (4) B C A E D (4) D C E B A (3) B A E C D (3) E A B D C (2) C D B A E (2) B E A C D (2) B A C D E (2) A B C E D (2) E D B A C (1) E B C A D (1) E B A D C (1) E A D B C (1) D E A B C (1) D C E A B (1) D C A E B (1) D A E B C (1) D A C E B (1) C E B D A (1) C D E B A (1) C B E D A (1) C B E A D (1) C B D A E (1) A D E B C (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -2 6 -6 B 8 0 6 6 -6 C 2 -6 0 2 -18 D -6 -6 -2 0 -20 E 6 6 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -2 6 -6 B 8 0 6 6 -6 C 2 -6 0 2 -18 D -6 -6 -2 0 -20 E 6 6 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=21 D=17 B=16 C=13 so C is eliminated. Round 2 votes counts: E=34 B=25 A=21 D=20 so D is eliminated. Round 3 votes counts: E=49 B=27 A=24 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:207 A:195 C:190 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 6 -6 B 8 0 6 6 -6 C 2 -6 0 2 -18 D -6 -6 -2 0 -20 E 6 6 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 6 -6 B 8 0 6 6 -6 C 2 -6 0 2 -18 D -6 -6 -2 0 -20 E 6 6 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 6 -6 B 8 0 6 6 -6 C 2 -6 0 2 -18 D -6 -6 -2 0 -20 E 6 6 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4771: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (12) E B C A D (7) D A B C E (7) B E D C A (7) A C D E B (6) D A C E B (5) B E C D A (5) C A E B D (4) B D E A C (4) E B C D A (3) D B A E C (3) C E B A D (3) B E C A D (3) E D B A C (2) D B E A C (2) D A E C B (2) D A B E C (2) C E A D B (2) C E A B D (2) C A E D B (2) A D C B E (2) A C D B E (2) E D A C B (1) E C B A D (1) E C A D B (1) E C A B D (1) D A E B C (1) C B E A D (1) C A D E B (1) C A D B E (1) C A B D E (1) B D C A E (1) B C E D A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 12 4 8 6 B -12 0 -2 -10 -14 C -4 2 0 -4 8 D -8 10 4 0 6 E -6 14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 8 6 B -12 0 -2 -10 -14 C -4 2 0 -4 8 D -8 10 4 0 6 E -6 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=22 B=22 C=17 E=16 so E is eliminated. Round 2 votes counts: B=32 D=25 A=23 C=20 so C is eliminated. Round 3 votes counts: A=38 B=37 D=25 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:206 C:201 E:197 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 8 6 B -12 0 -2 -10 -14 C -4 2 0 -4 8 D -8 10 4 0 6 E -6 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 8 6 B -12 0 -2 -10 -14 C -4 2 0 -4 8 D -8 10 4 0 6 E -6 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 8 6 B -12 0 -2 -10 -14 C -4 2 0 -4 8 D -8 10 4 0 6 E -6 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998633 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4772: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) B E A D C (7) E A B D C (6) C D A B E (6) D C A B E (4) C D A E B (4) B E C A D (4) E B C A D (3) D A B C E (3) C E B A D (3) C D B E A (3) C D B A E (3) C B D E A (3) A E D B C (3) A E B D C (3) A D E B C (3) E A C B D (2) E A B C D (2) D A E C B (2) D A C E B (2) D A B E C (2) C E A D B (2) C E A B D (2) C A D E B (2) B C E D A (2) A E D C B (2) A D B E C (2) E C B A D (1) D C A E B (1) D B A C E (1) D A E B C (1) C E B D A (1) C B E D A (1) C B E A D (1) B E C D A (1) B D C A E (1) B D A E C (1) B C D E A (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 12 2 14 -4 B -12 0 6 4 -8 C -2 -6 0 10 -6 D -14 -4 -10 0 -8 E 4 8 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 14 -4 B -12 0 6 4 -8 C -2 -6 0 10 -6 D -14 -4 -10 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=21 B=17 D=16 A=15 so A is eliminated. Round 2 votes counts: C=32 E=29 D=22 B=17 so B is eliminated. Round 3 votes counts: E=41 C=35 D=24 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:212 C:198 B:195 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 14 -4 B -12 0 6 4 -8 C -2 -6 0 10 -6 D -14 -4 -10 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 14 -4 B -12 0 6 4 -8 C -2 -6 0 10 -6 D -14 -4 -10 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 14 -4 B -12 0 6 4 -8 C -2 -6 0 10 -6 D -14 -4 -10 0 -8 E 4 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4773: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) C E B D A (9) B E A C D (8) D A C E B (5) E B C D A (4) C D E B A (4) C B E D A (4) B E C A D (4) A D C B E (4) A D B C E (4) A B E D C (4) B E C D A (3) B A E C D (3) A D E B C (3) E C B D A (2) E B C A D (2) E B A C D (2) D C E A B (2) C D A B E (2) B C E A D (2) A D B E C (2) E D B C A (1) E A B D C (1) D E C B A (1) D E A B C (1) D C E B A (1) D C A B E (1) D A E C B (1) D A C B E (1) C D B E A (1) C D A E B (1) C B E A D (1) C B D E A (1) B E A D C (1) A C B E D (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -12 -10 -10 B 8 0 -4 10 0 C 12 4 0 12 10 D 10 -10 -12 0 -8 E 10 0 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -10 -10 B 8 0 -4 10 0 C 12 4 0 12 10 D 10 -10 -12 0 -8 E 10 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=23 C=23 B=21 A=21 E=12 so E is eliminated. Round 2 votes counts: B=29 C=25 D=24 A=22 so A is eliminated. Round 3 votes counts: D=37 B=37 C=26 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:219 B:207 E:204 D:190 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -10 -10 B 8 0 -4 10 0 C 12 4 0 12 10 D 10 -10 -12 0 -8 E 10 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -10 -10 B 8 0 -4 10 0 C 12 4 0 12 10 D 10 -10 -12 0 -8 E 10 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -10 -10 B 8 0 -4 10 0 C 12 4 0 12 10 D 10 -10 -12 0 -8 E 10 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4774: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) A E C B D (9) E A C B D (5) D B C E A (5) E D A B C (4) C D B E A (4) B D C A E (4) B D A C E (4) A E B C D (4) E C A B D (3) E A D B C (3) D B C A E (3) C E D B A (3) C E A D B (3) C D E B A (3) E C D A B (2) E A D C B (2) D B E C A (2) D B E A C (2) D B A E C (2) C B D A E (2) B D A E C (2) B C D A E (2) A E B D C (2) A C E B D (2) A B E D C (2) A B D E C (2) E D A C B (1) E C D B A (1) E A B C D (1) D C B E A (1) D B A C E (1) C E A B D (1) C A E B D (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 14 0 4 -12 B -14 0 -6 -4 -20 C 0 6 0 12 -20 D -4 4 -12 0 -18 E 12 20 20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 0 4 -12 B -14 0 -6 -4 -20 C 0 6 0 12 -20 D -4 4 -12 0 -18 E 12 20 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=21 C=17 D=16 B=14 so B is eliminated. Round 2 votes counts: E=32 D=26 A=23 C=19 so C is eliminated. Round 3 votes counts: E=39 D=37 A=24 so A is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:235 A:203 C:199 D:185 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 4 -12 B -14 0 -6 -4 -20 C 0 6 0 12 -20 D -4 4 -12 0 -18 E 12 20 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 4 -12 B -14 0 -6 -4 -20 C 0 6 0 12 -20 D -4 4 -12 0 -18 E 12 20 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 4 -12 B -14 0 -6 -4 -20 C 0 6 0 12 -20 D -4 4 -12 0 -18 E 12 20 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4775: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) D C E A B (7) E A C B D (5) B A D C E (5) E D C A B (4) C E A D B (4) A C E B D (4) A B E C D (4) E C A D B (3) E B A C D (3) C E D A B (3) C A E D B (3) C A D E B (3) B E A D C (3) B D E A C (3) B D A E C (3) A E C B D (3) E D C B A (2) D E C B A (2) D C E B A (2) D B E C A (2) D B C E A (2) D B C A E (2) D A C B E (2) B E A C D (2) B D A C E (2) A B C D E (2) E A C D B (1) E A B C D (1) D C A E B (1) C D A E B (1) B E D C A (1) B D E C A (1) B A E D C (1) B A D E C (1) B A C E D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 14 20 -2 B -6 0 -4 8 -8 C -14 4 0 8 -8 D -20 -8 -8 0 -16 E 2 8 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 14 20 -2 B -6 0 -4 8 -8 C -14 4 0 8 -8 D -20 -8 -8 0 -16 E 2 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=20 E=19 A=15 C=14 so C is eliminated. Round 2 votes counts: B=32 E=26 D=21 A=21 so D is eliminated. Round 3 votes counts: B=38 E=37 A=25 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:219 E:217 B:195 C:195 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 20 -2 B -6 0 -4 8 -8 C -14 4 0 8 -8 D -20 -8 -8 0 -16 E 2 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 20 -2 B -6 0 -4 8 -8 C -14 4 0 8 -8 D -20 -8 -8 0 -16 E 2 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 20 -2 B -6 0 -4 8 -8 C -14 4 0 8 -8 D -20 -8 -8 0 -16 E 2 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4776: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) A E D B C (8) E A D B C (6) D B A E C (6) C E B A D (4) B D A E C (4) E C A D B (3) E A C D B (3) C E A D B (3) C E A B D (3) C B E D A (3) C B D E A (3) C A D B E (3) B D E A C (3) B D C A E (3) B D A C E (3) A E C D B (3) A D E B C (3) A D B E C (3) E C A B D (2) E A D C B (2) D B A C E (2) C E B D A (2) B E D A C (2) E B D C A (1) E A B D C (1) E A B C D (1) D B C A E (1) D A E B C (1) C D B A E (1) C A E D B (1) B D C E A (1) B C D A E (1) A E D C B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 8 4 14 B 2 0 0 -4 2 C -8 0 0 -2 -6 D -4 4 2 0 2 E -14 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 4 14 B 2 0 0 -4 2 C -8 0 0 -2 -6 D -4 4 2 0 2 E -14 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=20 E=19 B=17 D=10 so D is eliminated. Round 2 votes counts: C=34 B=26 A=21 E=19 so E is eliminated. Round 3 votes counts: C=39 A=34 B=27 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:202 B:200 E:194 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 4 14 B 2 0 0 -4 2 C -8 0 0 -2 -6 D -4 4 2 0 2 E -14 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 4 14 B 2 0 0 -4 2 C -8 0 0 -2 -6 D -4 4 2 0 2 E -14 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 4 14 B 2 0 0 -4 2 C -8 0 0 -2 -6 D -4 4 2 0 2 E -14 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4777: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B C A E D (10) C A B D E (8) C A D B E (7) B E D A C (7) E D B A C (6) E D A B C (5) D E A C B (5) E B D C A (4) E B D A C (4) D A C E B (4) B E D C A (4) B E A C D (3) A C D E B (3) E D B C A (2) B E C D A (2) B E C A D (2) A D C E B (2) A C D B E (2) D E A B C (1) D C A E B (1) D A E C B (1) C D A E B (1) B E A D C (1) B C E D A (1) B C E A D (1) B C A D E (1) B A E C D (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -8 4 4 B -2 0 10 -2 0 C 8 -10 0 4 2 D -4 2 -4 0 -6 E -4 0 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.400000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999996 Cumulative probabilities = A: 0.500000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 4 4 B -2 0 10 -2 0 C 8 -10 0 4 2 D -4 2 -4 0 -6 E -4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.400000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000007 Cumulative probabilities = A: 0.500000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=26 E=21 D=12 A=8 so A is eliminated. Round 2 votes counts: B=33 C=31 E=21 D=15 so D is eliminated. Round 3 votes counts: C=38 B=33 E=29 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:203 C:202 A:201 E:200 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -8 4 4 B -2 0 10 -2 0 C 8 -10 0 4 2 D -4 2 -4 0 -6 E -4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.400000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000007 Cumulative probabilities = A: 0.500000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 4 B -2 0 10 -2 0 C 8 -10 0 4 2 D -4 2 -4 0 -6 E -4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.400000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000007 Cumulative probabilities = A: 0.500000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 4 B -2 0 10 -2 0 C 8 -10 0 4 2 D -4 2 -4 0 -6 E -4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.400000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.420000000007 Cumulative probabilities = A: 0.500000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4778: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) C D A B E (10) E B A D C (8) B E D A C (8) E B A C D (7) A C D E B (6) C A D E B (4) B E D C A (4) B D E A C (4) E A C B D (3) E A B C D (3) D B C E A (3) D B C A E (3) C D A E B (3) A C E D B (3) E B C A D (2) D C B E A (2) B D E C A (2) A E C B D (2) E C B A D (1) E B D A C (1) D B E C A (1) D B E A C (1) D B A C E (1) D A B C E (1) C E A B D (1) C D B E A (1) C D B A E (1) A E B C D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 2 -14 -4 B 0 0 0 -6 10 C -2 0 0 -2 4 D 14 6 2 0 12 E 4 -10 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -14 -4 B 0 0 0 -6 10 C -2 0 0 -2 4 D 14 6 2 0 12 E 4 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 C=20 B=18 A=14 so A is eliminated. Round 2 votes counts: C=30 E=28 D=24 B=18 so B is eliminated. Round 3 votes counts: E=40 D=30 C=30 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:217 B:202 C:200 A:192 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -14 -4 B 0 0 0 -6 10 C -2 0 0 -2 4 D 14 6 2 0 12 E 4 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -14 -4 B 0 0 0 -6 10 C -2 0 0 -2 4 D 14 6 2 0 12 E 4 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -14 -4 B 0 0 0 -6 10 C -2 0 0 -2 4 D 14 6 2 0 12 E 4 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4779: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) E D A C B (8) B C E D A (8) D A E C B (7) B D A E C (6) B C E A D (6) E C A D B (5) C E B A D (5) A D B E C (5) E C D A B (4) B D A C E (4) B A D C E (4) D A E B C (3) A D E C B (3) E C B D A (2) E A D C B (2) D A B E C (2) C A D B E (2) B C D A E (2) B C A D E (2) A C D E B (2) E D C A B (1) E C D B A (1) C E A D B (1) C E A B D (1) B D C A E (1) B C A E D (1) B A C D E (1) A E D C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -2 2 -6 B 4 0 -8 6 8 C 2 8 0 4 0 D -2 -6 -4 0 -10 E 6 -8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666111 D: 0.000000 E: 0.333889 Sum of squares = 0.555185606592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666111 D: 0.666111 E: 1.000000 A B C D E A 0 -4 -2 2 -6 B 4 0 -8 6 8 C 2 8 0 4 0 D -2 -6 -4 0 -10 E 6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.000000 E: 0.499776 Sum of squares = 0.500000100332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.500224 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=23 C=18 D=12 A=12 so D is eliminated. Round 2 votes counts: B=35 A=24 E=23 C=18 so C is eliminated. Round 3 votes counts: B=44 E=30 A=26 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:207 B:205 E:204 A:195 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 2 -6 B 4 0 -8 6 8 C 2 8 0 4 0 D -2 -6 -4 0 -10 E 6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.000000 E: 0.499776 Sum of squares = 0.500000100332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.500224 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 2 -6 B 4 0 -8 6 8 C 2 8 0 4 0 D -2 -6 -4 0 -10 E 6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.000000 E: 0.499776 Sum of squares = 0.500000100332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.500224 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 2 -6 B 4 0 -8 6 8 C 2 8 0 4 0 D -2 -6 -4 0 -10 E 6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.000000 E: 0.499776 Sum of squares = 0.500000100332 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500224 D: 0.500224 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4780: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) B D C A E (6) A E C D B (6) E C B D A (4) E A C D B (4) D C B E A (4) B E C D A (4) B D C E A (4) B D A C E (4) B A E D C (4) A E D C B (4) A D B E C (4) D B C A E (3) C D B E A (3) B E A C D (3) E C D A B (2) E C A D B (2) D A C B E (2) C E D B A (2) C D E B A (2) C D E A B (2) C B E D A (2) E C A B D (1) E B C D A (1) E B A C D (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A E B (1) D A C E B (1) D A B C E (1) C E B D A (1) B E C A D (1) B C D E A (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B D C (1) A E B C D (1) A D B C E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 6 -2 6 B 2 0 -10 -10 0 C -6 10 0 -6 0 D 2 10 6 0 0 E -6 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.776066 E: 0.223934 Sum of squares = 0.65242532523 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.776066 E: 1.000000 A B C D E A 0 -2 6 -2 6 B 2 0 -10 -10 0 C -6 10 0 -6 0 D 2 10 6 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000025399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=28 E=17 D=13 C=12 so C is eliminated. Round 2 votes counts: B=30 A=30 E=20 D=20 so E is eliminated. Round 3 votes counts: A=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:209 A:204 C:199 E:197 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -2 6 B 2 0 -10 -10 0 C -6 10 0 -6 0 D 2 10 6 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000025399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 6 B 2 0 -10 -10 0 C -6 10 0 -6 0 D 2 10 6 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000025399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 6 B 2 0 -10 -10 0 C -6 10 0 -6 0 D 2 10 6 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000025399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4781: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) A D E B C (6) C E D B A (5) C D E A B (5) C D A E B (5) D A C E B (4) C D E B A (4) C B E D A (4) B E A C D (4) B A E D C (4) C B D A E (3) A B E D C (3) E D A B C (2) E C D B A (2) E B A D C (2) D E C A B (2) D A E C B (2) C E B D A (2) C D B A E (2) C D A B E (2) C B A D E (2) B E C D A (2) B E C A D (2) B E A D C (2) B C E A D (2) A E D B C (2) A D C B E (2) A B D E C (2) E D C A B (1) E B D A C (1) E B C D A (1) D E A C B (1) D C A E B (1) B C E D A (1) B C A E D (1) B C A D E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E B D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 -4 10 B 0 0 -12 -12 -12 C 0 12 0 2 -2 D 4 12 -2 0 12 E -10 12 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.289529 B: 0.000000 C: 0.710471 D: 0.000000 E: 0.000000 Sum of squares = 0.588596464724 Cumulative probabilities = A: 0.289529 B: 0.289529 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 10 B 0 0 -12 -12 -12 C 0 12 0 2 -2 D 4 12 -2 0 12 E -10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555574131 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=25 B=22 D=10 E=9 so E is eliminated. Round 2 votes counts: C=36 B=26 A=25 D=13 so D is eliminated. Round 3 votes counts: C=40 A=34 B=26 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:213 C:206 A:203 E:196 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -4 10 B 0 0 -12 -12 -12 C 0 12 0 2 -2 D 4 12 -2 0 12 E -10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555574131 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 10 B 0 0 -12 -12 -12 C 0 12 0 2 -2 D 4 12 -2 0 12 E -10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555574131 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 10 B 0 0 -12 -12 -12 C 0 12 0 2 -2 D 4 12 -2 0 12 E -10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555574131 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4782: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) E A B D C (9) C D E A B (9) D C B E A (8) B A E C D (6) D C E A B (5) C A B E D (5) B E A D C (5) C D A B E (4) A E B C D (4) D E A B C (3) D E C A B (2) D E B A C (2) D C E B A (2) D B E A C (2) C D B A E (2) C A E B D (2) B D E A C (2) B C A E D (2) E B A D C (1) E A D C B (1) E A D B C (1) D B E C A (1) D B C E A (1) D B C A E (1) C D B E A (1) C B D A E (1) C B A E D (1) C B A D E (1) B D E C A (1) B D C A E (1) B A C D E (1) A E C D B (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 0 4 -12 B 4 0 8 8 14 C 0 -8 0 -16 -6 D -4 -8 16 0 0 E 12 -14 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 4 -12 B 4 0 8 8 14 C 0 -8 0 -16 -6 D -4 -8 16 0 0 E 12 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 C=26 E=12 A=7 so A is eliminated. Round 2 votes counts: B=29 D=27 C=26 E=18 so E is eliminated. Round 3 votes counts: B=43 D=29 C=28 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:202 E:202 A:194 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 4 -12 B 4 0 8 8 14 C 0 -8 0 -16 -6 D -4 -8 16 0 0 E 12 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 4 -12 B 4 0 8 8 14 C 0 -8 0 -16 -6 D -4 -8 16 0 0 E 12 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 4 -12 B 4 0 8 8 14 C 0 -8 0 -16 -6 D -4 -8 16 0 0 E 12 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4783: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) C D E A B (8) E D C B A (6) D C E B A (5) C D E B A (5) E D B C A (4) E B D C A (3) C E D A B (3) C D B E A (3) B E D A C (3) B D C E A (3) A C D E B (3) A B D E C (3) D E C B A (2) D E B C A (2) C B D E A (2) C A B D E (2) B D E A C (2) B C D E A (2) B A C D E (2) A E C D B (2) A E B D C (2) A B E D C (2) A B E C D (2) A B C E D (2) E D A C B (1) E D A B C (1) E C D A B (1) C E A D B (1) C D B A E (1) C D A E B (1) C A E D B (1) C A D E B (1) C A D B E (1) B E A D C (1) B D A E C (1) B A E D C (1) A E D C B (1) A E C B D (1) A C E D B (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -32 -36 -34 B 14 0 -8 -8 -12 C 32 8 0 -4 0 D 36 8 4 0 20 E 34 12 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -32 -36 -34 B 14 0 -8 -8 -12 C 32 8 0 -4 0 D 36 8 4 0 20 E 34 12 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=24 A=22 E=16 D=9 so D is eliminated. Round 2 votes counts: C=34 B=24 A=22 E=20 so E is eliminated. Round 3 votes counts: C=43 B=33 A=24 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:234 C:218 E:213 B:193 A:142 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -32 -36 -34 B 14 0 -8 -8 -12 C 32 8 0 -4 0 D 36 8 4 0 20 E 34 12 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -32 -36 -34 B 14 0 -8 -8 -12 C 32 8 0 -4 0 D 36 8 4 0 20 E 34 12 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -32 -36 -34 B 14 0 -8 -8 -12 C 32 8 0 -4 0 D 36 8 4 0 20 E 34 12 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4784: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (12) E B A D C (10) D C A E B (10) A B E C D (8) D E B C A (7) C D A B E (7) E B D A C (6) B E A C D (5) E D B A C (4) C A B D E (4) D E B A C (3) D C E B A (3) D E C A B (2) C A B E D (2) B A C E D (2) A B C E D (2) D E C B A (1) D C E A B (1) C D B E A (1) C D B A E (1) C B A E D (1) C B A D E (1) C A D B E (1) B E D A C (1) A E B D C (1) A E B C D (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 16 6 12 B -8 0 0 14 -4 C -16 0 0 0 2 D -6 -14 0 0 -12 E -12 4 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 6 12 B -8 0 0 14 -4 C -16 0 0 0 2 D -6 -14 0 0 -12 E -12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 E=20 C=18 B=8 so B is eliminated. Round 2 votes counts: A=29 D=27 E=26 C=18 so C is eliminated. Round 3 votes counts: A=38 D=36 E=26 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:201 E:201 C:193 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 6 12 B -8 0 0 14 -4 C -16 0 0 0 2 D -6 -14 0 0 -12 E -12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 6 12 B -8 0 0 14 -4 C -16 0 0 0 2 D -6 -14 0 0 -12 E -12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 6 12 B -8 0 0 14 -4 C -16 0 0 0 2 D -6 -14 0 0 -12 E -12 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4785: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) B C A D E (9) A B D C E (7) A B C D E (7) E D C A B (6) E D A C B (6) A D B E C (5) E D C B A (4) C E D B A (4) C B E A D (3) B C A E D (3) A D B C E (3) E C D B A (2) E A D C B (2) E A D B C (2) E A B C D (2) D E A C B (2) D A B C E (2) C D B E A (2) C B E D A (2) B A C D E (2) A B C E D (2) E C B D A (1) D E C B A (1) D E A B C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E B D A (1) C B D E A (1) C B D A E (1) C B A D E (1) B A C E D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 24 18 22 16 B -24 0 20 -10 6 C -18 -20 0 -12 2 D -22 10 12 0 14 E -16 -6 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 18 22 16 B -24 0 20 -10 6 C -18 -20 0 -12 2 D -22 10 12 0 14 E -16 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=25 C=15 B=15 D=9 so D is eliminated. Round 2 votes counts: A=41 E=29 C=15 B=15 so C is eliminated. Round 3 votes counts: A=41 E=34 B=25 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:240 D:207 B:196 E:181 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 18 22 16 B -24 0 20 -10 6 C -18 -20 0 -12 2 D -22 10 12 0 14 E -16 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 18 22 16 B -24 0 20 -10 6 C -18 -20 0 -12 2 D -22 10 12 0 14 E -16 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 18 22 16 B -24 0 20 -10 6 C -18 -20 0 -12 2 D -22 10 12 0 14 E -16 -6 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4786: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (6) D E C A B (5) A B D E C (5) A B D C E (5) D C E A B (4) D C A E B (4) D A C E B (4) C D E B A (4) B A C E D (4) A D C B E (4) A B E C D (4) A B C E D (4) E D C B A (3) D A E B C (3) D A C B E (3) A D B C E (3) A B C D E (3) E C D B A (2) E B C D A (2) E B A D C (2) E A B D C (2) C E D B A (2) C E B D A (2) C B A D E (2) B E A C D (2) A D E B C (2) A B E D C (2) E D A C B (1) E D A B C (1) E B A C D (1) E A D B C (1) D E A B C (1) D C E B A (1) C D B A E (1) C D A B E (1) C B E D A (1) C B E A D (1) B C E A D (1) B A C D E (1) Total count = 100 A B C D E A 0 24 28 10 22 B -24 0 10 0 6 C -28 -10 0 -12 10 D -10 0 12 0 12 E -22 -6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 28 10 22 B -24 0 10 0 6 C -28 -10 0 -12 10 D -10 0 12 0 12 E -22 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=25 E=15 C=14 B=14 so C is eliminated. Round 2 votes counts: A=32 D=31 E=19 B=18 so B is eliminated. Round 3 votes counts: A=45 D=31 E=24 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:242 D:207 B:196 C:180 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 28 10 22 B -24 0 10 0 6 C -28 -10 0 -12 10 D -10 0 12 0 12 E -22 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 28 10 22 B -24 0 10 0 6 C -28 -10 0 -12 10 D -10 0 12 0 12 E -22 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 28 10 22 B -24 0 10 0 6 C -28 -10 0 -12 10 D -10 0 12 0 12 E -22 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4787: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) B A C D E (6) B A C E D (5) D C B E A (4) B D A C E (4) E D A C B (3) E D A B C (3) D E C B A (3) D E C A B (3) D E B A C (3) C D E A B (3) C B A D E (3) C A E B D (3) B D C A E (3) B C A D E (3) B A E C D (3) A E B C D (3) E D C A B (2) E A D C B (2) E A B C D (2) D B C E A (2) A E C B D (2) A C B E D (2) A B E C D (2) A B C E D (2) E D B A C (1) E C D A B (1) E C A D B (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E B C A (1) D C E A B (1) D B C A E (1) C E D A B (1) C E A D B (1) C D B E A (1) C D B A E (1) C B D A E (1) C A E D B (1) C A D B E (1) B E D A C (1) B D A E C (1) B C D A E (1) B A E D C (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 -12 4 -2 2 B 12 0 4 4 -2 C -4 -4 0 2 14 D 2 -4 -2 0 6 E -2 2 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.100000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000034 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -12 4 -2 2 B 12 0 4 4 -2 C -4 -4 0 2 14 D 2 -4 -2 0 6 E -2 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.100000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000032 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=24 E=19 C=16 A=12 so A is eliminated. Round 2 votes counts: B=33 E=25 D=24 C=18 so C is eliminated. Round 3 votes counts: B=39 E=31 D=30 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:209 C:204 D:201 A:196 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 -2 2 B 12 0 4 4 -2 C -4 -4 0 2 14 D 2 -4 -2 0 6 E -2 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.100000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000032 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -2 2 B 12 0 4 4 -2 C -4 -4 0 2 14 D 2 -4 -2 0 6 E -2 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.100000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000032 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -2 2 B 12 0 4 4 -2 C -4 -4 0 2 14 D 2 -4 -2 0 6 E -2 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.700000 C: 0.100000 D: 0.000000 E: 0.200000 Sum of squares = 0.540000000032 Cumulative probabilities = A: 0.000000 B: 0.700000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4788: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) B A C E D (8) A B D E C (7) E C D A B (6) D C E A B (6) B A E C D (6) E D C A B (5) D E C A B (5) C E D B A (5) E C D B A (4) E C B A D (4) D E A C B (4) C D E B A (4) D A B E C (3) A D B E C (3) A B E D C (3) D A B C E (2) C B E A D (2) B C A E D (2) A B D C E (2) E D A B C (1) E A C B D (1) D C E B A (1) D C B A E (1) C E B A D (1) C B E D A (1) B C E A D (1) B C A D E (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 -4 -16 -14 -20 B 4 0 -18 0 -12 C 16 18 0 12 -4 D 14 0 -12 0 -18 E 20 12 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -16 -14 -20 B 4 0 -18 0 -12 C 16 18 0 12 -4 D 14 0 -12 0 -18 E 20 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=22 C=22 E=21 B=20 A=15 so A is eliminated. Round 2 votes counts: B=32 D=25 C=22 E=21 so E is eliminated. Round 3 votes counts: C=37 B=32 D=31 so D is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:227 C:221 D:192 B:187 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -16 -14 -20 B 4 0 -18 0 -12 C 16 18 0 12 -4 D 14 0 -12 0 -18 E 20 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -14 -20 B 4 0 -18 0 -12 C 16 18 0 12 -4 D 14 0 -12 0 -18 E 20 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -14 -20 B 4 0 -18 0 -12 C 16 18 0 12 -4 D 14 0 -12 0 -18 E 20 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4789: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (7) D C E A B (6) D B C E A (5) C D A E B (5) B A E C D (5) A C E D B (5) E A B C D (4) D C A B E (4) A C D E B (4) D C A E B (3) D B C A E (3) B E D C A (3) B E D A C (3) B E A C D (3) B D E C A (3) B D C A E (3) E B A C D (2) D C E B A (2) D C B E A (2) D C B A E (2) C A D E B (2) C A D B E (2) A E C B D (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D A B (1) E B D C A (1) E B A D C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E C B A (1) C D E A B (1) C D A B E (1) C A E D B (1) B D C E A (1) B D A E C (1) B D A C E (1) B A E D C (1) B A D C E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -10 -10 -4 B 6 0 2 -8 4 C 10 -2 0 -16 10 D 10 8 16 0 6 E 4 -4 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -10 -4 B 6 0 2 -8 4 C 10 -2 0 -16 10 D 10 8 16 0 6 E 4 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=28 E=15 A=13 C=12 so C is eliminated. Round 2 votes counts: D=35 B=32 A=18 E=15 so E is eliminated. Round 3 votes counts: D=39 B=36 A=25 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:202 C:201 E:192 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -10 -10 -4 B 6 0 2 -8 4 C 10 -2 0 -16 10 D 10 8 16 0 6 E 4 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -10 -4 B 6 0 2 -8 4 C 10 -2 0 -16 10 D 10 8 16 0 6 E 4 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -10 -4 B 6 0 2 -8 4 C 10 -2 0 -16 10 D 10 8 16 0 6 E 4 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4790: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E A B D (7) B D E C A (7) C A E D B (6) A C E D B (6) D B A C E (5) A C E B D (5) A C D B E (5) B D E A C (4) A C B D E (4) A B D C E (4) E C D B A (3) E C A B D (3) D B E A C (3) C E A D B (3) B D A E C (3) A D B C E (3) E D C A B (2) E C A D B (2) D A B C E (2) C A E B D (2) B D A C E (2) A D C B E (2) A C D E B (2) E C D A B (1) E B D C A (1) E B C D A (1) D B E C A (1) B E D C A (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 18 20 12 22 B -18 0 -6 -8 12 C -20 6 0 2 20 D -12 8 -2 0 14 E -22 -12 -20 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 20 12 22 B -18 0 -6 -8 12 C -20 6 0 2 20 D -12 8 -2 0 14 E -22 -12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=19 B=19 C=18 E=13 so E is eliminated. Round 2 votes counts: A=31 C=27 D=21 B=21 so D is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:236 C:204 D:204 B:190 E:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 20 12 22 B -18 0 -6 -8 12 C -20 6 0 2 20 D -12 8 -2 0 14 E -22 -12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 12 22 B -18 0 -6 -8 12 C -20 6 0 2 20 D -12 8 -2 0 14 E -22 -12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 12 22 B -18 0 -6 -8 12 C -20 6 0 2 20 D -12 8 -2 0 14 E -22 -12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4791: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) D A B C E (7) B E C D A (6) B D C E A (5) A D E C B (5) E A C B D (4) D B C E A (4) D B C A E (4) C D B E A (4) E C B A D (3) E C A B D (3) C E B D A (3) B C E D A (3) A D B E C (3) E C B D A (2) E B C D A (2) E B C A D (2) D C B E A (2) D A C B E (2) C D E A B (2) B E C A D (2) A E D C B (2) A E C D B (2) A E B D C (2) A D C E B (2) A C E D B (2) E B A C D (1) D C A B E (1) C E D A B (1) C B E D A (1) C A D E B (1) B D E C A (1) B D A E C (1) B C D E A (1) A E B C D (1) A D C B E (1) A D B C E (1) A C D E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -6 -4 -4 B -6 0 -4 6 2 C 6 4 0 10 -4 D 4 -6 -10 0 -2 E 4 -2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999831 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 6 -6 -4 -4 B -6 0 -4 6 2 C 6 4 0 10 -4 D 4 -6 -10 0 -2 E 4 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999998984 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=20 B=19 E=17 C=12 so C is eliminated. Round 2 votes counts: A=33 D=26 E=21 B=20 so B is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:208 E:204 B:199 A:196 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 -4 -4 B -6 0 -4 6 2 C 6 4 0 10 -4 D 4 -6 -10 0 -2 E 4 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999998984 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -4 -4 B -6 0 -4 6 2 C 6 4 0 10 -4 D 4 -6 -10 0 -2 E 4 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999998984 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -4 -4 B -6 0 -4 6 2 C 6 4 0 10 -4 D 4 -6 -10 0 -2 E 4 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999998984 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4792: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) B C D E A (6) C B A E D (5) C A B D E (5) E A D B C (4) C B D A E (4) B D C E A (4) A D C E B (4) E D A B C (3) E B C D A (3) D B C E A (3) C B A D E (3) B C E D A (3) A E C D B (3) A D E C B (3) A D C B E (3) A C D B E (3) E D B C A (2) E B D C A (2) E A D C B (2) D E B A C (2) D A E B C (2) C B E A D (2) B E D C A (2) A E D C B (2) E C B A D (1) E A C B D (1) D C B A E (1) D C A B E (1) D B E C A (1) D B E A C (1) D A C B E (1) D A B C E (1) C D B A E (1) C B E D A (1) C B D E A (1) C A E B D (1) C A D B E (1) B E C D A (1) A E C B D (1) A D E B C (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -8 -6 -4 B 12 0 -4 -6 10 C 8 4 0 -4 12 D 6 6 4 0 4 E 4 -10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -6 -4 B 12 0 -4 -6 10 C 8 4 0 -4 12 D 6 6 4 0 4 E 4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 A=22 B=16 D=13 so D is eliminated. Round 2 votes counts: E=27 C=26 A=26 B=21 so B is eliminated. Round 3 votes counts: C=42 E=32 A=26 so A is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:210 B:206 E:189 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -8 -6 -4 B 12 0 -4 -6 10 C 8 4 0 -4 12 D 6 6 4 0 4 E 4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -6 -4 B 12 0 -4 -6 10 C 8 4 0 -4 12 D 6 6 4 0 4 E 4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -6 -4 B 12 0 -4 -6 10 C 8 4 0 -4 12 D 6 6 4 0 4 E 4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4793: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) D C E A B (7) D C B E A (6) B D A E C (6) C D E A B (5) A E B C D (5) D C E B A (4) D B C E A (4) C E D A B (4) C E A D B (4) C E A B D (4) E C A B D (3) B D A C E (3) B C E A D (3) B A D E C (3) A E C B D (3) A B E C D (3) E C A D B (2) E A C D B (2) D B A C E (2) B D C A E (2) B A E D C (2) E D C A B (1) E A C B D (1) D E A C B (1) D B A E C (1) D A E C B (1) C E B A D (1) C D E B A (1) C D B E A (1) C B E A D (1) B D C E A (1) B A C E D (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -8 0 -12 B 2 0 -6 4 -4 C 8 6 0 8 8 D 0 -4 -8 0 -2 E 12 4 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 0 -12 B 2 0 -6 4 -4 C 8 6 0 8 8 D 0 -4 -8 0 -2 E 12 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=26 C=21 A=14 E=9 so E is eliminated. Round 2 votes counts: B=30 D=27 C=26 A=17 so A is eliminated. Round 3 votes counts: B=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:205 B:198 D:193 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 0 -12 B 2 0 -6 4 -4 C 8 6 0 8 8 D 0 -4 -8 0 -2 E 12 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 0 -12 B 2 0 -6 4 -4 C 8 6 0 8 8 D 0 -4 -8 0 -2 E 12 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 0 -12 B 2 0 -6 4 -4 C 8 6 0 8 8 D 0 -4 -8 0 -2 E 12 4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4794: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) A B C D E (8) B A E D C (7) E B A D C (6) D E C A B (6) D E A B C (6) E D C B A (5) D C E A B (5) C D E A B (5) B A C E D (5) A B E D C (4) A B C E D (4) E D B A C (3) D E C B A (3) A B D E C (3) E D A B C (2) D C E B A (2) C E D B A (2) C D E B A (2) C D A B E (2) C B A E D (2) B A E C D (2) E D B C A (1) C E B A D (1) C D A E B (1) C A D B E (1) B E A C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 16 6 10 0 B -16 0 8 8 0 C -6 -8 0 -8 0 D -10 -8 8 0 8 E 0 0 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.672233 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.327767 Sum of squares = 0.559328357826 Cumulative probabilities = A: 0.672233 B: 0.672233 C: 0.672233 D: 0.672233 E: 1.000000 A B C D E A 0 16 6 10 0 B -16 0 8 8 0 C -6 -8 0 -8 0 D -10 -8 8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=22 A=21 E=17 B=15 so B is eliminated. Round 2 votes counts: A=35 C=25 D=22 E=18 so E is eliminated. Round 3 votes counts: A=42 D=33 C=25 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:200 D:199 E:196 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 10 0 B -16 0 8 8 0 C -6 -8 0 -8 0 D -10 -8 8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 10 0 B -16 0 8 8 0 C -6 -8 0 -8 0 D -10 -8 8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 10 0 B -16 0 8 8 0 C -6 -8 0 -8 0 D -10 -8 8 0 8 E 0 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4795: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) D C B A E (6) C A D B E (6) E D C A B (5) E A B C D (5) E B D A C (4) B E A D C (4) B D C A E (4) A E C B D (4) A C B D E (4) E A B D C (3) D C B E A (3) D B C E A (3) D B C A E (3) C B D A E (3) E D B C A (2) E B A D C (2) D C E B A (2) C D B A E (2) C D A B E (2) B D E C A (2) B C A D E (2) B A C D E (2) A C B E D (2) E D B A C (1) E D A C B (1) E C A D B (1) E A C B D (1) D E C B A (1) D E B C A (1) D B E C A (1) C D A E B (1) C A D E B (1) C A B D E (1) B D C E A (1) B D A E C (1) B C D A E (1) B A E C D (1) B A D C E (1) A E B C D (1) A C E D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 0 2 B 6 0 -8 2 12 C 8 8 0 -2 6 D 0 -2 2 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 0.000000 Sum of squares = 0.50000000002 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 0 2 B 6 0 -8 2 12 C 8 8 0 -2 6 D 0 -2 2 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000017 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=20 B=19 C=16 A=14 so A is eliminated. Round 2 votes counts: E=36 C=24 D=20 B=20 so D is eliminated. Round 3 votes counts: E=38 C=35 B=27 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 B:206 D:205 A:194 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 0 2 B 6 0 -8 2 12 C 8 8 0 -2 6 D 0 -2 2 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000017 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 2 B 6 0 -8 2 12 C 8 8 0 -2 6 D 0 -2 2 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000017 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 2 B 6 0 -8 2 12 C 8 8 0 -2 6 D 0 -2 2 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000017 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4796: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) C B A E D (5) B C D A E (5) E D A C B (4) D E C B A (4) D B C E A (4) C B D E A (4) C B A D E (4) B D C A E (4) A E C D B (4) E A C D B (3) D B E C A (3) D B E A C (3) A C B E D (3) E D C A B (2) E D A B C (2) E C A D B (2) C E D A B (2) C D E B A (2) C A B E D (2) B D A E C (2) B A D C E (2) A E D B C (2) A E B D C (2) A C E B D (2) A B C E D (2) E C D A B (1) E A D B C (1) D E C A B (1) D A B E C (1) C E A D B (1) C E A B D (1) C B D A E (1) C A E B D (1) B D A C E (1) B C D E A (1) B C A E D (1) B C A D E (1) B A C E D (1) B A C D E (1) A E D C B (1) A E B C D (1) A D E B C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -4 6 6 B -2 0 -12 0 8 C 4 12 0 4 2 D -6 0 -4 0 -8 E -6 -8 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 6 6 B -2 0 -12 0 8 C 4 12 0 4 2 D -6 0 -4 0 -8 E -6 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 E=21 A=21 B=19 D=16 so D is eliminated. Round 2 votes counts: B=29 E=26 C=23 A=22 so A is eliminated. Round 3 votes counts: E=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 A:205 B:197 E:196 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 6 6 B -2 0 -12 0 8 C 4 12 0 4 2 D -6 0 -4 0 -8 E -6 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 6 6 B -2 0 -12 0 8 C 4 12 0 4 2 D -6 0 -4 0 -8 E -6 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 6 6 B -2 0 -12 0 8 C 4 12 0 4 2 D -6 0 -4 0 -8 E -6 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4797: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (21) E D C A B (10) C A B D E (9) B A C D E (7) C A B E D (5) E D B A C (4) E D A B C (4) E C A B D (4) B D A C E (4) D E B C A (3) C E A B D (3) E D C B A (2) E D A C B (2) E C A D B (2) E A C B D (2) D B A C E (2) C B A D E (2) C A E B D (2) B C A D E (2) E A C D B (1) E A B C D (1) D E C B A (1) D B E C A (1) D B E A C (1) D B A E C (1) C A D B E (1) B A D C E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 6 -12 -24 B 4 0 6 -12 -24 C -6 -6 0 -14 -20 D 12 12 14 0 12 E 24 24 20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -12 -24 B 4 0 6 -12 -24 C -6 -6 0 -14 -20 D 12 12 14 0 12 E 24 24 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=30 C=22 B=14 A=2 so A is eliminated. Round 2 votes counts: E=32 D=30 C=23 B=15 so B is eliminated. Round 3 votes counts: D=35 C=33 E=32 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:228 D:225 B:187 A:183 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -12 -24 B 4 0 6 -12 -24 C -6 -6 0 -14 -20 D 12 12 14 0 12 E 24 24 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -12 -24 B 4 0 6 -12 -24 C -6 -6 0 -14 -20 D 12 12 14 0 12 E 24 24 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -12 -24 B 4 0 6 -12 -24 C -6 -6 0 -14 -20 D 12 12 14 0 12 E 24 24 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4798: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) B D A E C (5) A B D E C (5) A B C E D (5) D C E B A (4) D B E A C (4) C A E B D (4) E D C A B (3) D E B A C (3) C B A D E (3) B D C A E (3) B A D C E (3) A C B E D (3) A B E C D (3) E D A B C (2) E C A D B (2) D E C B A (2) D E A B C (2) D B C E A (2) C E D A B (2) C E A D B (2) C D B E A (2) C B D A E (2) C B A E D (2) B D A C E (2) B C D A E (2) B C A D E (2) B A D E C (2) A E C B D (2) E D A C B (1) E C D A B (1) E A D C B (1) D E B C A (1) D C B E A (1) D B A E C (1) D A E B C (1) C E A B D (1) C B E A D (1) C B D E A (1) C A B E D (1) B A C D E (1) A E D B C (1) A E B C D (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 12 6 12 B -2 0 6 10 16 C -12 -6 0 0 0 D -6 -10 0 0 12 E -12 -16 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 6 12 B -2 0 6 10 16 C -12 -6 0 0 0 D -6 -10 0 0 12 E -12 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=22 D=21 C=21 B=20 E=16 so E is eliminated. Round 2 votes counts: A=29 D=27 C=24 B=20 so B is eliminated. Round 3 votes counts: D=37 A=35 C=28 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:215 D:198 C:191 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 6 12 B -2 0 6 10 16 C -12 -6 0 0 0 D -6 -10 0 0 12 E -12 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 6 12 B -2 0 6 10 16 C -12 -6 0 0 0 D -6 -10 0 0 12 E -12 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 6 12 B -2 0 6 10 16 C -12 -6 0 0 0 D -6 -10 0 0 12 E -12 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4799: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) D C B E A (5) D E C A B (4) C D E A B (4) C D A B E (4) B D E C A (4) E D C A B (3) E D B A C (3) D C E B A (3) D C E A B (3) C A E D B (3) B D E A C (3) B D C E A (3) B A E C D (3) A B E C D (3) E A B D C (2) D E C B A (2) C D B A E (2) C B A D E (2) C A D B E (2) C A B D E (2) B D C A E (2) B D A E C (2) B A C E D (2) A E C B D (2) A E B C D (2) A C E B D (2) A B C E D (2) E D C B A (1) E D A C B (1) E A D B C (1) E A C D B (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) C A D E B (1) B E D A C (1) B E A D C (1) B C D A E (1) B C A D E (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C D E (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -24 -24 4 B 0 0 -12 -10 10 C 24 12 0 -4 12 D 24 10 4 0 28 E -4 -10 -12 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -24 -24 4 B 0 0 -12 -10 10 C 24 12 0 -4 12 D 24 10 4 0 28 E -4 -10 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 D=22 A=13 E=12 so E is eliminated. Round 2 votes counts: D=30 B=27 C=26 A=17 so A is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:233 C:222 B:194 A:178 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -24 -24 4 B 0 0 -12 -10 10 C 24 12 0 -4 12 D 24 10 4 0 28 E -4 -10 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -24 -24 4 B 0 0 -12 -10 10 C 24 12 0 -4 12 D 24 10 4 0 28 E -4 -10 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -24 -24 4 B 0 0 -12 -10 10 C 24 12 0 -4 12 D 24 10 4 0 28 E -4 -10 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996712 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4800: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) C B A D E (7) E D A B C (6) C B A E D (6) C B D E A (5) B C A D E (5) A D E B C (5) E D A C B (4) D E A C B (4) C E B D A (4) C B E D A (4) A B E D C (4) E C D B A (3) D E A B C (3) C B E A D (3) A B D E C (3) A B C E D (3) D E C B A (2) C E D B A (2) C B D A E (2) B C A E D (2) A E D B C (2) A B C D E (2) E D C B A (1) D C E B A (1) D A B C E (1) B A C E D (1) B A C D E (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -18 -8 -8 B -2 0 -20 6 2 C 18 20 0 2 -2 D 8 -6 -2 0 -16 E 8 -2 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.000000 E: 0.833333 Sum of squares = 0.708333333293 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.166667 E: 1.000000 A B C D E A 0 2 -18 -8 -8 B -2 0 -20 6 2 C 18 20 0 2 -2 D 8 -6 -2 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.000000 E: 0.833333 Sum of squares = 0.70833333301 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=26 A=21 D=11 B=9 so B is eliminated. Round 2 votes counts: C=40 E=26 A=23 D=11 so D is eliminated. Round 3 votes counts: C=41 E=35 A=24 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:212 B:193 D:192 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -18 -8 -8 B -2 0 -20 6 2 C 18 20 0 2 -2 D 8 -6 -2 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.000000 E: 0.833333 Sum of squares = 0.70833333301 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -8 -8 B -2 0 -20 6 2 C 18 20 0 2 -2 D 8 -6 -2 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.000000 E: 0.833333 Sum of squares = 0.70833333301 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -8 -8 B -2 0 -20 6 2 C 18 20 0 2 -2 D 8 -6 -2 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.000000 E: 0.833333 Sum of squares = 0.70833333301 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4801: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) D E B A C (6) C A B E D (6) E D C A B (4) D B C A E (4) C E A D B (4) C D B A E (4) B A C D E (4) E C A D B (3) D E B C A (3) C E D A B (3) C D E A B (3) C B A D E (3) E D B A C (2) E D A B C (2) D E C B A (2) D B E A C (2) C D E B A (2) C D A B E (2) C A B D E (2) B D A E C (2) B C D A E (2) B C A D E (2) A C B E D (2) A B E D C (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A B D (1) E B A D C (1) E A D B C (1) E A C D B (1) E A B D C (1) D C E B A (1) D B E C A (1) D B C E A (1) D B A E C (1) C B D A E (1) C A E D B (1) B D E A C (1) B A E C D (1) B A D E C (1) B A D C E (1) A E B D C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -26 -2 6 B -4 0 -10 -10 -6 C 26 10 0 18 12 D 2 10 -18 0 2 E -6 6 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -26 -2 6 B -4 0 -10 -10 -6 C 26 10 0 18 12 D 2 10 -18 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=21 E=18 B=14 A=9 so A is eliminated. Round 2 votes counts: C=41 D=21 E=20 B=18 so B is eliminated. Round 3 votes counts: C=49 D=26 E=25 so E is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:233 D:198 E:193 A:191 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -26 -2 6 B -4 0 -10 -10 -6 C 26 10 0 18 12 D 2 10 -18 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -26 -2 6 B -4 0 -10 -10 -6 C 26 10 0 18 12 D 2 10 -18 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -26 -2 6 B -4 0 -10 -10 -6 C 26 10 0 18 12 D 2 10 -18 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4802: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (18) A B E C D (13) D C E A B (8) A C E D B (8) B D E C A (5) B A E C D (5) A E C B D (5) A B D C E (5) E C B D A (3) A D C E B (3) E C D B A (2) D B C E A (2) B E C D A (2) B D C E A (2) B A D E C (2) A D C B E (2) A C D E B (2) E C A D B (1) E B C A D (1) D A C E B (1) C D E B A (1) C D A E B (1) B E D C A (1) B E C A D (1) B E A C D (1) B A E D C (1) A E C D B (1) A D B C E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 4 8 4 B -6 0 -12 -2 -10 C -4 12 0 -4 10 D -8 2 4 0 8 E -4 10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999405 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 8 4 B -6 0 -12 -2 -10 C -4 12 0 -4 10 D -8 2 4 0 8 E -4 10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 D=29 B=20 E=7 C=2 so C is eliminated. Round 2 votes counts: A=42 D=31 B=20 E=7 so E is eliminated. Round 3 votes counts: A=43 D=33 B=24 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:207 D:203 E:194 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 8 4 B -6 0 -12 -2 -10 C -4 12 0 -4 10 D -8 2 4 0 8 E -4 10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 8 4 B -6 0 -12 -2 -10 C -4 12 0 -4 10 D -8 2 4 0 8 E -4 10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 8 4 B -6 0 -12 -2 -10 C -4 12 0 -4 10 D -8 2 4 0 8 E -4 10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4803: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (15) A D C E B (10) B A D E C (5) D A C E B (4) C E D B A (4) B E D C A (4) A D C B E (4) E C D B A (3) E C B D A (3) E B C D A (3) D A E C B (3) C E D A B (3) B C A E D (3) A D B C E (3) E D C B A (2) E B D C A (2) D E C A B (2) D C E A B (2) D A B E C (2) B E C A D (2) B E A D C (2) B E A C D (2) B C E A D (2) B A C E D (2) A C D E B (2) A B D C E (2) A B C D E (2) D E C B A (1) D C A E B (1) C E B D A (1) C D E A B (1) A D B E C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -8 -12 -8 B 12 0 6 4 6 C 8 -6 0 -2 -6 D 12 -4 2 0 -6 E 8 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -12 -8 B 12 0 6 4 6 C 8 -6 0 -2 -6 D 12 -4 2 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=26 D=15 E=13 C=9 so C is eliminated. Round 2 votes counts: B=37 A=26 E=21 D=16 so D is eliminated. Round 3 votes counts: B=37 A=36 E=27 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:207 D:202 C:197 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -12 -8 B 12 0 6 4 6 C 8 -6 0 -2 -6 D 12 -4 2 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -12 -8 B 12 0 6 4 6 C 8 -6 0 -2 -6 D 12 -4 2 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -12 -8 B 12 0 6 4 6 C 8 -6 0 -2 -6 D 12 -4 2 0 -6 E 8 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4804: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) D B E A C (5) C E A D B (5) C D B E A (5) A E B C D (5) D B C A E (4) C D B A E (4) C A E B D (4) E A C B D (3) E A B D C (3) E A B C D (3) D C B E A (3) D C B A E (3) D B A E C (3) C E D A B (3) C E A B D (3) A E C B D (3) A B E D C (3) E D B C A (2) E B D A C (2) E B A D C (2) D B E C A (2) C D A B E (2) C A E D B (2) C A D E B (2) B D A E C (2) B A D E C (2) A E B D C (2) D E B A C (1) D C E B A (1) D B A C E (1) C D E A B (1) C D A E B (1) C A D B E (1) B E A D C (1) A C E B D (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -10 -4 -4 B 0 0 -10 -16 -12 C 10 10 0 12 10 D 4 16 -12 0 6 E 4 12 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -4 -4 B 0 0 -10 -16 -12 C 10 10 0 12 10 D 4 16 -12 0 6 E 4 12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 D=23 A=17 E=15 B=5 so B is eliminated. Round 2 votes counts: C=40 D=25 A=19 E=16 so E is eliminated. Round 3 votes counts: C=40 A=31 D=29 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:207 E:200 A:191 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -4 -4 B 0 0 -10 -16 -12 C 10 10 0 12 10 D 4 16 -12 0 6 E 4 12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 -4 B 0 0 -10 -16 -12 C 10 10 0 12 10 D 4 16 -12 0 6 E 4 12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 -4 B 0 0 -10 -16 -12 C 10 10 0 12 10 D 4 16 -12 0 6 E 4 12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4805: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) D E A B C (7) B E A C D (7) B C A E D (7) E A C D B (4) D C B A E (4) D C A E B (4) D C A B E (4) D B C A E (4) B E C A D (4) B C E A D (4) E A C B D (3) E A B C D (3) D A E C B (3) C A B E D (3) E A D C B (2) C D A B E (2) C B A D E (2) C A E B D (2) B D E C A (2) B D C A E (2) B C D A E (2) A E C D B (2) A C E B D (2) E D A B C (1) E B A D C (1) C D B A E (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A D C (1) B D C E A (1) B C D E A (1) B A E C D (1) A E D C B (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 12 -4 6 2 B -12 0 -4 0 6 C 4 4 0 10 -2 D -6 0 -10 0 -2 E -2 -6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999886 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 12 -4 6 2 B -12 0 -4 0 6 C 4 4 0 10 -2 D -6 0 -10 0 -2 E -2 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=32 E=14 C=13 A=7 so A is eliminated. Round 2 votes counts: D=34 B=32 E=18 C=16 so C is eliminated. Round 3 votes counts: D=39 B=38 E=23 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:208 C:208 E:198 B:195 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 6 2 B -12 0 -4 0 6 C 4 4 0 10 -2 D -6 0 -10 0 -2 E -2 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 6 2 B -12 0 -4 0 6 C 4 4 0 10 -2 D -6 0 -10 0 -2 E -2 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 6 2 B -12 0 -4 0 6 C 4 4 0 10 -2 D -6 0 -10 0 -2 E -2 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4806: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) D C E B A (9) A E B D C (8) D C B E A (7) A E B C D (7) E B A C D (4) E A B C D (4) C D A B E (4) D E A B C (3) D C B A E (3) D A E B C (3) C B A E D (3) A D E B C (3) E A B D C (2) D C A B E (2) C D B E A (2) C B D A E (2) A E D B C (2) A D C B E (2) A C B E D (2) A B C E D (2) E D B A C (1) E D A B C (1) E B A D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E A B (1) C D B A E (1) C B E D A (1) C B E A D (1) C B A D E (1) C A B E D (1) C A B D E (1) B E A C D (1) A C B D E (1) Total count = 100 A B C D E A 0 22 16 14 18 B -22 0 8 6 -4 C -16 -8 0 -2 -8 D -14 -6 2 0 -4 E -18 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 16 14 18 B -22 0 8 6 -4 C -16 -8 0 -2 -8 D -14 -6 2 0 -4 E -18 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=31 C=17 E=13 B=1 so B is eliminated. Round 2 votes counts: A=38 D=31 C=17 E=14 so E is eliminated. Round 3 votes counts: A=50 D=33 C=17 so C is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:235 E:199 B:194 D:189 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 16 14 18 B -22 0 8 6 -4 C -16 -8 0 -2 -8 D -14 -6 2 0 -4 E -18 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 14 18 B -22 0 8 6 -4 C -16 -8 0 -2 -8 D -14 -6 2 0 -4 E -18 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 14 18 B -22 0 8 6 -4 C -16 -8 0 -2 -8 D -14 -6 2 0 -4 E -18 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4807: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) B E A D C (11) D C B E A (10) B E A C D (10) E B A D C (7) D C A E B (5) E B A C D (4) C D A E B (4) C A D E B (4) A C D E B (4) D C A B E (3) C D B E A (2) C D A B E (2) B D E C A (2) A E B D C (2) A C E B D (2) E B D A C (1) E A B D C (1) E A B C D (1) D C E B A (1) D C E A B (1) D A E B C (1) D A C E B (1) C B E A D (1) C A E B D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) A E C B D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 22 28 -12 B 6 0 14 20 -10 C -22 -14 0 0 -16 D -28 -20 0 0 -16 E 12 10 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 22 28 -12 B 6 0 14 20 -10 C -22 -14 0 0 -16 D -28 -20 0 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=23 D=22 E=14 C=14 so E is eliminated. Round 2 votes counts: B=39 A=25 D=22 C=14 so C is eliminated. Round 3 votes counts: B=40 D=30 A=30 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:227 A:216 B:215 C:174 D:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 22 28 -12 B 6 0 14 20 -10 C -22 -14 0 0 -16 D -28 -20 0 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 22 28 -12 B 6 0 14 20 -10 C -22 -14 0 0 -16 D -28 -20 0 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 22 28 -12 B 6 0 14 20 -10 C -22 -14 0 0 -16 D -28 -20 0 0 -16 E 12 10 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4808: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (14) A B C D E (12) B A D C E (9) D B E C A (8) B D A E C (8) E D C B A (7) E C D A B (6) C A E B D (6) A C E B D (6) D E B C A (4) D B E A C (3) A C B E D (3) A B D C E (3) D E C B A (2) A B C E D (2) E C D B A (1) E C A D B (1) C E A B D (1) B D E C A (1) B D A C E (1) B A D E C (1) A C E D B (1) Total count = 100 A B C D E A 0 10 -2 18 4 B -10 0 4 6 2 C 2 -4 0 6 16 D -18 -6 -6 0 4 E -4 -2 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 18 4 B -10 0 4 6 2 C 2 -4 0 6 16 D -18 -6 -6 0 4 E -4 -2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999991 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=21 B=20 D=17 E=15 so E is eliminated. Round 2 votes counts: C=29 A=27 D=24 B=20 so B is eliminated. Round 3 votes counts: A=37 D=34 C=29 so C is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:210 B:201 D:187 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 18 4 B -10 0 4 6 2 C 2 -4 0 6 16 D -18 -6 -6 0 4 E -4 -2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999991 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 18 4 B -10 0 4 6 2 C 2 -4 0 6 16 D -18 -6 -6 0 4 E -4 -2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999991 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 18 4 B -10 0 4 6 2 C 2 -4 0 6 16 D -18 -6 -6 0 4 E -4 -2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999991 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4809: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) D C A B E (10) B E A C D (7) D E B C A (6) E B D A C (5) D C A E B (4) B E D C A (4) B A C E D (4) A C D B E (4) A C B D E (4) E B D C A (3) D E C A B (3) A C E D B (3) A C E B D (3) A C B E D (3) C D A B E (2) C A D E B (2) B C A D E (2) A C D E B (2) E D B C A (1) E D B A C (1) E D A C B (1) E B A C D (1) E A C B D (1) E A B C D (1) D C B A E (1) D B E C A (1) D B C E A (1) D B C A E (1) C B A D E (1) B E D A C (1) B D E C A (1) B D C A E (1) B A E C D (1) A E C B D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -10 6 24 B -14 0 -12 -8 24 C 10 12 0 10 20 D -6 8 -10 0 16 E -24 -24 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 6 24 B -14 0 -12 -8 24 C 10 12 0 10 20 D -6 8 -10 0 16 E -24 -24 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=22 B=21 C=16 E=14 so E is eliminated. Round 2 votes counts: D=30 B=30 A=24 C=16 so C is eliminated. Round 3 votes counts: A=37 D=32 B=31 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:226 A:217 D:204 B:195 E:158 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 6 24 B -14 0 -12 -8 24 C 10 12 0 10 20 D -6 8 -10 0 16 E -24 -24 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 6 24 B -14 0 -12 -8 24 C 10 12 0 10 20 D -6 8 -10 0 16 E -24 -24 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 6 24 B -14 0 -12 -8 24 C 10 12 0 10 20 D -6 8 -10 0 16 E -24 -24 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4810: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) D E B A C (7) B E D A C (6) A C D B E (6) B E D C A (5) D A E B C (4) C B E A D (4) A C D E B (4) A C B D E (4) B E C D A (3) B C A E D (3) E D B C A (2) E D B A C (2) E B D C A (2) E B C D A (2) D E A B C (2) C E A B D (2) C B A E D (2) C A E B D (2) C A D E B (2) C A B D E (2) B E C A D (2) B E A C D (2) B D E A C (2) B C E A D (2) B A C E D (2) A D C E B (2) D E A C B (1) D B A E C (1) D A E C B (1) D A B E C (1) C E D B A (1) C E B D A (1) C E A D B (1) C A E D B (1) C A D B E (1) B A D E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -2 14 2 B 4 0 4 20 22 C 2 -4 0 20 8 D -14 -20 -20 0 -16 E -2 -22 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 14 2 B 4 0 4 20 22 C 2 -4 0 20 8 D -14 -20 -20 0 -16 E -2 -22 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=28 D=17 A=17 E=8 so E is eliminated. Round 2 votes counts: B=32 C=30 D=21 A=17 so A is eliminated. Round 3 votes counts: C=44 B=32 D=24 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:213 A:205 E:192 D:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 14 2 B 4 0 4 20 22 C 2 -4 0 20 8 D -14 -20 -20 0 -16 E -2 -22 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 14 2 B 4 0 4 20 22 C 2 -4 0 20 8 D -14 -20 -20 0 -16 E -2 -22 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 14 2 B 4 0 4 20 22 C 2 -4 0 20 8 D -14 -20 -20 0 -16 E -2 -22 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4811: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) A D C E B (8) D A E B C (7) E B D A C (6) E B A D C (6) E A D B C (5) C B D A E (5) C B A D E (5) B E C D A (5) B C E D A (5) A D E C B (5) A D E B C (5) D A C E B (3) D A C B E (3) C B E A D (3) D A E C B (2) B E D C A (2) A E D B C (2) E D B A C (1) E B D C A (1) E B C A D (1) E A B D C (1) D C A B E (1) D B C A E (1) C E B A D (1) C D A B E (1) C B D E A (1) C B A E D (1) B E C A D (1) B C E A D (1) A E D C B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 14 12 20 B -8 0 0 -10 -12 C -14 0 0 -22 -2 D -12 10 22 0 14 E -20 12 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 12 20 B -8 0 0 -10 -12 C -14 0 0 -22 -2 D -12 10 22 0 14 E -20 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 E=21 D=17 B=14 so B is eliminated. Round 2 votes counts: C=31 E=29 A=23 D=17 so D is eliminated. Round 3 votes counts: A=38 C=33 E=29 so E is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:227 D:217 E:190 B:185 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 12 20 B -8 0 0 -10 -12 C -14 0 0 -22 -2 D -12 10 22 0 14 E -20 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 12 20 B -8 0 0 -10 -12 C -14 0 0 -22 -2 D -12 10 22 0 14 E -20 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 12 20 B -8 0 0 -10 -12 C -14 0 0 -22 -2 D -12 10 22 0 14 E -20 12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4812: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) D B A C E (8) E C A B D (6) C E A B D (5) C A B E D (5) A B C D E (5) E D C B A (4) E D B A C (4) C B D A E (4) E C D B A (3) D E B C A (3) D B C A E (3) D B A E C (3) C A B D E (3) A C B D E (3) E D B C A (2) E C B D A (2) E C A D B (2) E A D B C (2) E A C B D (2) C E B D A (2) C E B A D (2) C A E B D (2) B D A C E (2) E D C A B (1) E D A B C (1) E A D C B (1) D E C B A (1) D E B A C (1) D E A B C (1) D B E C A (1) D B C E A (1) D A B E C (1) D A B C E (1) A D E B C (1) A D B E C (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -4 2 6 B -8 0 2 6 4 C 4 -2 0 -6 14 D -2 -6 6 0 6 E -6 -4 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 2 6 B -8 0 2 6 4 C 4 -2 0 -6 14 D -2 -6 6 0 6 E -6 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=24 C=23 A=21 B=2 so B is eliminated. Round 2 votes counts: E=30 D=26 C=23 A=21 so A is eliminated. Round 3 votes counts: D=38 C=32 E=30 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:206 C:205 B:202 D:202 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -4 2 6 B -8 0 2 6 4 C 4 -2 0 -6 14 D -2 -6 6 0 6 E -6 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 2 6 B -8 0 2 6 4 C 4 -2 0 -6 14 D -2 -6 6 0 6 E -6 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 2 6 B -8 0 2 6 4 C 4 -2 0 -6 14 D -2 -6 6 0 6 E -6 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4813: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) B E D C A (7) B C E D A (6) E D B A C (5) A D C E B (5) C A D B E (4) B E C D A (4) E D A B C (3) E A D C B (3) D E A B C (3) B D E C A (3) B D C A E (3) E B D C A (2) E B D A C (2) E B C D A (2) C B E A D (2) C B A E D (2) C A D E B (2) C A B E D (2) B D C E A (2) B C D E A (2) B C D A E (2) A D E C B (2) A D C B E (2) E A D B C (1) E A C B D (1) D E B A C (1) D C A B E (1) D B E A C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C E B (1) C E B A D (1) C E A B D (1) C B D A E (1) C B A D E (1) C A E D B (1) C A E B D (1) C A B D E (1) B E D A C (1) B C E A D (1) A E D C B (1) A E C D B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -8 -10 -8 B 4 0 6 0 0 C 8 -6 0 -4 10 D 10 0 4 0 -2 E 8 0 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.729062 C: 0.000000 D: 0.000000 E: 0.270938 Sum of squares = 0.604938737978 Cumulative probabilities = A: 0.000000 B: 0.729062 C: 0.729062 D: 0.729062 E: 1.000000 A B C D E A 0 -4 -8 -10 -8 B 4 0 6 0 0 C 8 -6 0 -4 10 D 10 0 4 0 -2 E 8 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625001 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250345189 Cumulative probabilities = A: 0.000000 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=21 E=19 C=19 D=10 so D is eliminated. Round 2 votes counts: B=33 A=24 E=23 C=20 so C is eliminated. Round 3 votes counts: B=39 A=36 E=25 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:206 B:205 C:204 E:200 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -10 -8 B 4 0 6 0 0 C 8 -6 0 -4 10 D 10 0 4 0 -2 E 8 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625001 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250345189 Cumulative probabilities = A: 0.000000 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -10 -8 B 4 0 6 0 0 C 8 -6 0 -4 10 D 10 0 4 0 -2 E 8 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625001 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250345189 Cumulative probabilities = A: 0.000000 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -10 -8 B 4 0 6 0 0 C 8 -6 0 -4 10 D 10 0 4 0 -2 E 8 0 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625001 C: 0.000000 D: 0.000000 E: 0.374999 Sum of squares = 0.531250345189 Cumulative probabilities = A: 0.000000 B: 0.625001 C: 0.625001 D: 0.625001 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4814: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (12) E C D B A (11) A B D C E (8) B A C E D (7) C E B D A (6) C E B A D (6) B C E A D (6) C E D B A (5) A D B C E (5) C B E A D (4) A D B E C (4) D A B E C (3) B A E C D (3) E C B D A (2) D A E C B (2) D A E B C (2) B C A E D (2) A B D E C (2) E D C A B (1) D E A C B (1) D E A B C (1) D C E A B (1) D C A E B (1) D A C E B (1) B A C D E (1) A D C B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -14 2 -12 B 6 0 -8 -2 -4 C 14 8 0 10 12 D -2 2 -10 0 -8 E 12 4 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 2 -12 B 6 0 -8 -2 -4 C 14 8 0 10 12 D -2 2 -10 0 -8 E 12 4 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 C=21 B=19 E=14 so E is eliminated. Round 2 votes counts: C=34 D=25 A=22 B=19 so B is eliminated. Round 3 votes counts: C=42 A=33 D=25 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:206 B:196 D:191 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 2 -12 B 6 0 -8 -2 -4 C 14 8 0 10 12 D -2 2 -10 0 -8 E 12 4 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 2 -12 B 6 0 -8 -2 -4 C 14 8 0 10 12 D -2 2 -10 0 -8 E 12 4 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 2 -12 B 6 0 -8 -2 -4 C 14 8 0 10 12 D -2 2 -10 0 -8 E 12 4 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4815: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (13) D B E C A (11) B D C A E (8) E D B A C (7) C A B D E (7) E D A B C (6) A C E B D (6) E A C B D (5) C B A D E (5) E A D C B (4) D B C E A (4) C A B E D (4) B C D A E (4) D B C A E (3) E D A C B (2) D E B A C (2) D B E A C (2) C A E B D (2) B C A D E (2) A E C B D (2) D E B C A (1) Total count = 100 A B C D E A 0 2 -2 0 -14 B -2 0 0 -10 0 C 2 0 0 0 -10 D 0 10 0 0 -2 E 14 0 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.067586 C: 0.000000 D: 0.000000 E: 0.932414 Sum of squares = 0.873963870106 Cumulative probabilities = A: 0.000000 B: 0.067586 C: 0.067586 D: 0.067586 E: 1.000000 A B C D E A 0 2 -2 0 -14 B -2 0 0 -10 0 C 2 0 0 0 -10 D 0 10 0 0 -2 E 14 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222312445 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=23 C=18 B=14 A=8 so A is eliminated. Round 2 votes counts: E=39 C=24 D=23 B=14 so B is eliminated. Round 3 votes counts: E=39 D=31 C=30 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:204 C:196 B:194 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 0 -14 B -2 0 0 -10 0 C 2 0 0 0 -10 D 0 10 0 0 -2 E 14 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222312445 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -14 B -2 0 0 -10 0 C 2 0 0 0 -10 D 0 10 0 0 -2 E 14 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222312445 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -14 B -2 0 0 -10 0 C 2 0 0 0 -10 D 0 10 0 0 -2 E 14 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222312445 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4816: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (12) D B C A E (12) D B C E A (7) E A C D B (5) C B D A E (5) B D C A E (5) D B A E C (4) B C D A E (4) E A D C B (3) E A D B C (3) D B E A C (3) C B A D E (3) E A B D C (2) D E B C A (2) C E A B D (2) C D B E A (2) C A E B D (2) C A B E D (2) A E C B D (2) A E B D C (2) A E B C D (2) E D A B C (1) E C A D B (1) E A B C D (1) D E C B A (1) D E B A C (1) D C E B A (1) D C B E A (1) D C B A E (1) D B E C A (1) C E D B A (1) C D B A E (1) C B A E D (1) B D A E C (1) B C A D E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -12 -8 0 B 16 0 6 -2 12 C 12 -6 0 -4 4 D 8 2 4 0 12 E 0 -12 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 -8 0 B 16 0 6 -2 12 C 12 -6 0 -4 4 D 8 2 4 0 12 E 0 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=28 C=19 B=11 A=8 so A is eliminated. Round 2 votes counts: E=34 D=34 C=20 B=12 so B is eliminated. Round 3 votes counts: D=40 E=35 C=25 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:213 C:203 E:186 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -12 -8 0 B 16 0 6 -2 12 C 12 -6 0 -4 4 D 8 2 4 0 12 E 0 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -8 0 B 16 0 6 -2 12 C 12 -6 0 -4 4 D 8 2 4 0 12 E 0 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -8 0 B 16 0 6 -2 12 C 12 -6 0 -4 4 D 8 2 4 0 12 E 0 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4817: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) D A C B E (10) A D E C B (9) B C E D A (8) E B C D A (5) E A D B C (5) B C D E A (5) E A B D C (4) B E C D A (4) E A D C B (3) A D E B C (3) A D C E B (3) E C B D A (2) E B A C D (2) D C B A E (2) D C A B E (2) D A C E B (2) C D B A E (2) C B E D A (2) C B D E A (2) C B D A E (2) A E D B C (2) A D B C E (2) E C D A B (1) E C A D B (1) E A C D B (1) D A E C B (1) C D A B E (1) B E A D C (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 4 0 -2 -14 B -4 0 6 -4 -12 C 0 -6 0 -2 -14 D 2 4 2 0 -6 E 14 12 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 -2 -14 B -4 0 6 -4 -12 C 0 -6 0 -2 -14 D 2 4 2 0 -6 E 14 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=21 B=18 D=17 C=9 so C is eliminated. Round 2 votes counts: E=35 B=24 A=21 D=20 so D is eliminated. Round 3 votes counts: A=37 E=35 B=28 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:201 A:194 B:193 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -2 -14 B -4 0 6 -4 -12 C 0 -6 0 -2 -14 D 2 4 2 0 -6 E 14 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -2 -14 B -4 0 6 -4 -12 C 0 -6 0 -2 -14 D 2 4 2 0 -6 E 14 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -2 -14 B -4 0 6 -4 -12 C 0 -6 0 -2 -14 D 2 4 2 0 -6 E 14 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4818: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (12) E D A B C (10) C B A D E (7) C B E A D (6) C B A E D (6) A D E C B (6) A D E B C (5) D E A B C (4) D A E C B (4) D A E B C (4) B E C D A (4) C B E D A (3) C A D B E (3) C A B D E (3) A D C E B (3) E B D A C (2) B C A E D (2) A D C B E (2) E D C B A (1) E D B A C (1) E D A C B (1) E B D C A (1) E B C D A (1) C E D B A (1) B C E A D (1) B C A D E (1) B A E D C (1) A E D B C (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -4 2 4 B 0 0 2 4 8 C 4 -2 0 6 6 D -2 -4 -6 0 -8 E -4 -8 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.184835 B: 0.815165 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.698657701084 Cumulative probabilities = A: 0.184835 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 4 B 0 0 2 4 8 C 4 -2 0 6 6 D -2 -4 -6 0 -8 E -4 -8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555579709 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=21 A=21 E=17 D=12 so D is eliminated. Round 2 votes counts: C=29 A=29 E=21 B=21 so E is eliminated. Round 3 votes counts: A=44 C=30 B=26 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:207 C:207 A:201 E:195 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 2 4 B 0 0 2 4 8 C 4 -2 0 6 6 D -2 -4 -6 0 -8 E -4 -8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555579709 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 4 B 0 0 2 4 8 C 4 -2 0 6 6 D -2 -4 -6 0 -8 E -4 -8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555579709 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 4 B 0 0 2 4 8 C 4 -2 0 6 6 D -2 -4 -6 0 -8 E -4 -8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555579709 Cumulative probabilities = A: 0.333333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4819: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) D E B A C (6) D E A C B (6) B D E C A (6) B C D E A (6) A C E D B (6) C B A D E (5) C A B E D (5) C B D E A (4) B D C E A (4) D E B C A (3) D B E C A (3) C A B D E (3) E D B A C (2) D E A B C (2) C D B E A (2) C B A E D (2) B E D A C (2) B C A E D (2) A E D C B (2) A E D B C (2) A E C D B (2) A C B E D (2) E B D A C (1) E A D B C (1) D E C B A (1) D C B E A (1) D B C E A (1) C D B A E (1) C B D A E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D C A (1) B E A D C (1) B C E D A (1) B A E C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -10 -22 -22 B 14 0 4 -4 10 C 10 -4 0 -6 0 D 22 4 6 0 16 E 22 -10 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -22 -22 B 14 0 4 -4 10 C 10 -4 0 -6 0 D 22 4 6 0 16 E 22 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 D=23 A=16 E=11 so E is eliminated. Round 2 votes counts: D=32 C=26 B=25 A=17 so A is eliminated. Round 3 votes counts: D=38 C=37 B=25 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:212 C:200 E:198 A:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 -22 -22 B 14 0 4 -4 10 C 10 -4 0 -6 0 D 22 4 6 0 16 E 22 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -22 -22 B 14 0 4 -4 10 C 10 -4 0 -6 0 D 22 4 6 0 16 E 22 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -22 -22 B 14 0 4 -4 10 C 10 -4 0 -6 0 D 22 4 6 0 16 E 22 -10 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4820: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) E D C B A (8) D E C B A (8) D C B A E (7) E A B C D (6) A B C D E (6) E D A B C (5) C B D A E (5) E A D B C (4) B C D A E (4) B A C D E (4) E D C A B (3) E D A C B (3) C D B A E (3) C B A D E (3) E D B C A (2) D C E B A (2) D B C A E (2) C A B D E (2) B C A D E (2) A C B E D (2) E D B A C (1) E B A C D (1) E A D C B (1) D C B E A (1) D B C E A (1) B D C A E (1) Total count = 100 A B C D E A 0 -10 -8 -12 8 B 10 0 4 -2 12 C 8 -4 0 2 16 D 12 2 -2 0 2 E -8 -12 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375000000092 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -12 8 B 10 0 4 -2 12 C 8 -4 0 2 16 D 12 2 -2 0 2 E -8 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.3750000011 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=21 A=21 C=13 B=11 so B is eliminated. Round 2 votes counts: E=34 A=25 D=22 C=19 so C is eliminated. Round 3 votes counts: E=34 D=34 A=32 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:212 C:211 D:207 A:189 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -12 8 B 10 0 4 -2 12 C 8 -4 0 2 16 D 12 2 -2 0 2 E -8 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.3750000011 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -12 8 B 10 0 4 -2 12 C 8 -4 0 2 16 D 12 2 -2 0 2 E -8 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.3750000011 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -12 8 B 10 0 4 -2 12 C 8 -4 0 2 16 D 12 2 -2 0 2 E -8 -12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.3750000011 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4821: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (12) A D C B E (8) D A C B E (7) C B A D E (7) B C E A D (7) E B C D A (6) E B C A D (6) E D A B C (5) E D B C A (4) D A E B C (4) D E A B C (3) C B E A D (3) C A B D E (3) E A C B D (2) D B C A E (2) C B A E D (2) B E C D A (2) B E C A D (2) A D E C B (2) A C D B E (2) E D B A C (1) E C B A D (1) E C A B D (1) E B D C A (1) D A C E B (1) D A B E C (1) B C E D A (1) B C D E A (1) B C D A E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 0 -4 8 B -6 0 -6 -4 2 C 0 6 0 -2 -6 D 4 4 2 0 10 E -8 -2 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -4 8 B -6 0 -6 -4 2 C 0 6 0 -2 -6 D 4 4 2 0 10 E -8 -2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 C=15 B=14 A=14 so B is eliminated. Round 2 votes counts: E=31 D=30 C=25 A=14 so A is eliminated. Round 3 votes counts: D=40 E=31 C=29 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:205 C:199 B:193 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -4 8 B -6 0 -6 -4 2 C 0 6 0 -2 -6 D 4 4 2 0 10 E -8 -2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -4 8 B -6 0 -6 -4 2 C 0 6 0 -2 -6 D 4 4 2 0 10 E -8 -2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -4 8 B -6 0 -6 -4 2 C 0 6 0 -2 -6 D 4 4 2 0 10 E -8 -2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4822: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A B E D (8) E D B A C (7) C A D E B (6) C A B D E (6) A C D E B (6) D E A C B (4) D E A B C (4) B E D C A (4) A D E C B (4) D E B A C (3) D A E C B (3) C B A D E (3) B C A E D (3) D C A E B (2) C A D B E (2) B E A D C (2) B D E C A (2) B C D E A (2) B A E C D (2) A C B E D (2) E B D A C (1) E B A D C (1) D E C A B (1) D C E A B (1) D A C E B (1) C D B E A (1) C B D E A (1) C B A E D (1) B E C D A (1) B E A C D (1) B A C E D (1) A E D C B (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 12 2 6 B -6 0 -6 8 8 C -12 6 0 -6 -8 D -2 -8 6 0 4 E -6 -8 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 2 6 B -6 0 -6 8 8 C -12 6 0 -6 -8 D -2 -8 6 0 4 E -6 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=28 D=19 A=15 E=9 so E is eliminated. Round 2 votes counts: B=31 C=28 D=26 A=15 so A is eliminated. Round 3 votes counts: C=36 B=33 D=31 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 B:202 D:200 E:195 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 2 6 B -6 0 -6 8 8 C -12 6 0 -6 -8 D -2 -8 6 0 4 E -6 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 2 6 B -6 0 -6 8 8 C -12 6 0 -6 -8 D -2 -8 6 0 4 E -6 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 2 6 B -6 0 -6 8 8 C -12 6 0 -6 -8 D -2 -8 6 0 4 E -6 -8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4823: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (11) E C A B D (7) E C D B A (6) E C D A B (6) C E B A D (6) A D B E C (6) D E C A B (5) B A D C E (5) E D C B A (3) C E B D A (3) C E A B D (3) C B E A D (3) B D A C E (3) B A C E D (3) A B D C E (3) E D C A B (2) E C A D B (2) D B C E A (2) D B A C E (2) D A E B C (2) C E D B A (2) B C A E D (2) A B C E D (2) E A C B D (1) D E A C B (1) D B C A E (1) D B A E C (1) B D C A E (1) A E D C B (1) A E C B D (1) A E B C D (1) A C E B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -8 -2 -4 B -14 0 -6 -6 -6 C 8 6 0 0 -14 D 2 6 0 0 -10 E 4 6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -8 -2 -4 B -14 0 -6 -6 -6 C 8 6 0 0 -14 D 2 6 0 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 C=17 A=17 B=14 so B is eliminated. Round 2 votes counts: D=29 E=27 A=25 C=19 so C is eliminated. Round 3 votes counts: E=44 D=29 A=27 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:200 C:200 D:199 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -8 -2 -4 B -14 0 -6 -6 -6 C 8 6 0 0 -14 D 2 6 0 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 -2 -4 B -14 0 -6 -6 -6 C 8 6 0 0 -14 D 2 6 0 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 -2 -4 B -14 0 -6 -6 -6 C 8 6 0 0 -14 D 2 6 0 0 -10 E 4 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4824: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) B E A C D (7) E C D A B (6) E A D C B (5) B E C A D (5) A D C B E (5) E B A C D (4) C D E A B (4) A E D C B (4) A D C E B (4) E B C D A (3) A B D E C (3) D C A B E (2) D A C E B (2) D A C B E (2) C E D B A (2) C E D A B (2) C D B E A (2) C D A E B (2) B E C D A (2) B C D E A (2) B C D A E (2) A D B C E (2) A B D C E (2) E C D B A (1) E C B D A (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D C B A E (1) D C A E B (1) D A B C E (1) C D E B A (1) C D B A E (1) C D A B E (1) B E A D C (1) B D A C E (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 2 14 14 -6 B -2 0 0 -6 6 C -14 0 0 0 -14 D -14 6 0 0 -16 E 6 -6 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102045 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 2 14 14 -6 B -2 0 0 -6 6 C -14 0 0 0 -14 D -14 6 0 0 -16 E 6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102258 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=24 A=21 C=15 D=9 so D is eliminated. Round 2 votes counts: B=31 A=26 E=24 C=19 so C is eliminated. Round 3 votes counts: B=35 E=33 A=32 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:215 A:212 B:199 D:188 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 14 -6 B -2 0 0 -6 6 C -14 0 0 0 -14 D -14 6 0 0 -16 E 6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102258 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 14 -6 B -2 0 0 -6 6 C -14 0 0 0 -14 D -14 6 0 0 -16 E 6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102258 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 14 -6 B -2 0 0 -6 6 C -14 0 0 0 -14 D -14 6 0 0 -16 E 6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102258 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4825: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) E B D C A (6) D B C E A (6) A E C B D (5) E B A D C (4) E A C B D (4) D C A B E (4) B D A E C (4) B A E D C (4) A E B C D (4) A B D C E (4) D C B E A (3) C D A B E (3) A C E D B (3) E C D B A (2) E C B D A (2) E B C D A (2) D A C B E (2) C E A D B (2) C D E B A (2) C D A E B (2) B E D A C (2) B D E C A (2) A C E B D (2) A C D B E (2) E C A B D (1) E B A C D (1) E A B C D (1) D B C A E (1) C E D A B (1) C D E A B (1) C D B E A (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E A C (1) B D A C E (1) B A D E C (1) A E B D C (1) A C D E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 14 4 10 B -8 0 8 24 2 C -14 -8 0 -8 -10 D -4 -24 8 0 -14 E -10 -2 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999217 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 4 10 B -8 0 8 24 2 C -14 -8 0 -8 -10 D -4 -24 8 0 -14 E -10 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=23 D=16 B=16 C=14 so C is eliminated. Round 2 votes counts: A=33 E=26 D=25 B=16 so B is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:213 E:206 D:183 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 4 10 B -8 0 8 24 2 C -14 -8 0 -8 -10 D -4 -24 8 0 -14 E -10 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 4 10 B -8 0 8 24 2 C -14 -8 0 -8 -10 D -4 -24 8 0 -14 E -10 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 4 10 B -8 0 8 24 2 C -14 -8 0 -8 -10 D -4 -24 8 0 -14 E -10 -2 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4826: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (12) C E B A D (12) E C D A B (10) B A D C E (9) E D A B C (8) B A D E C (6) B A C D E (5) D B A E C (4) C E D B A (4) A D B E C (4) C E D A B (3) C B E A D (3) C B A D E (3) E D A C B (2) D E A B C (2) B D A C E (2) B C A D E (2) A B D E C (2) E D C A B (1) E C A B D (1) D B A C E (1) D A E B C (1) C E B D A (1) C E A D B (1) C E A B D (1) Total count = 100 A B C D E A 0 -4 16 -2 2 B 4 0 16 -6 6 C -16 -16 0 -8 -6 D 2 6 8 0 6 E -2 -6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 -2 2 B 4 0 16 -6 6 C -16 -16 0 -8 -6 D 2 6 8 0 6 E -2 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=24 E=22 D=20 A=6 so A is eliminated. Round 2 votes counts: C=28 B=26 D=24 E=22 so E is eliminated. Round 3 votes counts: C=39 D=35 B=26 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:210 A:206 E:196 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 16 -2 2 B 4 0 16 -6 6 C -16 -16 0 -8 -6 D 2 6 8 0 6 E -2 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 -2 2 B 4 0 16 -6 6 C -16 -16 0 -8 -6 D 2 6 8 0 6 E -2 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 -2 2 B 4 0 16 -6 6 C -16 -16 0 -8 -6 D 2 6 8 0 6 E -2 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4827: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (17) D B E A C (13) B D C A E (8) C B A E D (7) E A D C B (6) D E A B C (6) B D E A C (6) B D C E A (6) E A C D B (4) D E B A C (4) B C D A E (4) E D A C B (3) C A B E D (3) B C A D E (3) A E C B D (3) E C A D B (1) D E C A B (1) D B E C A (1) D B C E A (1) B D E C A (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -6 -8 -6 B 8 0 6 16 6 C 6 -6 0 -12 0 D 8 -16 12 0 8 E 6 -6 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -8 -6 B 8 0 6 16 6 C 6 -6 0 -12 0 D 8 -16 12 0 8 E 6 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=27 D=26 E=14 A=5 so A is eliminated. Round 2 votes counts: C=28 B=28 D=26 E=18 so E is eliminated. Round 3 votes counts: C=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:206 E:196 C:194 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -8 -6 B 8 0 6 16 6 C 6 -6 0 -12 0 D 8 -16 12 0 8 E 6 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -8 -6 B 8 0 6 16 6 C 6 -6 0 -12 0 D 8 -16 12 0 8 E 6 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -8 -6 B 8 0 6 16 6 C 6 -6 0 -12 0 D 8 -16 12 0 8 E 6 -6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4828: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) D E B A C (7) C A E B D (7) D B E A C (5) C A B E D (5) C A B D E (4) A B D C E (4) A B C D E (4) E D C B A (3) E D B A C (3) E B A D C (3) E B A C D (3) D E C B A (3) D E B C A (3) D C E B A (3) D A B C E (3) E C A B D (2) E A B C D (2) D C B A E (2) D B A C E (2) C E D A B (2) C E A D B (2) C D A B E (2) B D E A C (2) A B E C D (2) E C D B A (1) D C B E A (1) D C A B E (1) D B C A E (1) D A C B E (1) C D E A B (1) C A E D B (1) C A D B E (1) B E A D C (1) B A E D C (1) B A D E C (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 10 -6 8 -10 B -10 0 -4 4 -8 C 6 4 0 -2 14 D -8 -4 2 0 4 E 10 8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.000000 Sum of squares = 0.40624999996 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 8 -10 B -10 0 -4 4 -8 C 6 4 0 -2 14 D -8 -4 2 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000012 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=32 E=17 A=13 B=5 so B is eliminated. Round 2 votes counts: D=34 C=33 E=18 A=15 so A is eliminated. Round 3 votes counts: D=39 C=39 E=22 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:211 A:201 E:200 D:197 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 8 -10 B -10 0 -4 4 -8 C 6 4 0 -2 14 D -8 -4 2 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000012 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 8 -10 B -10 0 -4 4 -8 C 6 4 0 -2 14 D -8 -4 2 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000012 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 8 -10 B -10 0 -4 4 -8 C 6 4 0 -2 14 D -8 -4 2 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.000000 Sum of squares = 0.406250000012 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4829: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) B D C A E (7) D A E B C (6) D B E A C (5) B D E C A (5) A C E B D (5) E C A D B (4) A E C D B (4) E A C D B (3) D B E C A (3) C E B A D (3) B D C E A (3) B D A C E (3) A D E C B (3) E C B D A (2) D B A E C (2) D A B E C (2) D A B C E (2) C E A B D (2) B C E D A (2) B C E A D (2) B C A E D (2) B C A D E (2) A C D E B (2) E D B C A (1) E D A C B (1) E C B A D (1) E B D C A (1) E B C D A (1) D E B A C (1) D E A B C (1) D B A C E (1) C B E A D (1) C A E B D (1) B E D C A (1) B E C D A (1) B C D A E (1) A D E B C (1) A D C B E (1) A C E D B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -8 -4 -4 B 2 0 12 10 -4 C 8 -12 0 0 -14 D 4 -10 0 0 6 E 4 4 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999986 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.500000 E: 1.000000 A B C D E A 0 -2 -8 -4 -4 B 2 0 12 10 -4 C 8 -12 0 0 -14 D 4 -10 0 0 6 E 4 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999991 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 E=22 A=19 C=7 so C is eliminated. Round 2 votes counts: B=30 E=27 D=23 A=20 so A is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:208 D:200 A:191 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 -4 B 2 0 12 10 -4 C 8 -12 0 0 -14 D 4 -10 0 0 6 E 4 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999991 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 -4 B 2 0 12 10 -4 C 8 -12 0 0 -14 D 4 -10 0 0 6 E 4 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999991 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 -4 B 2 0 12 10 -4 C 8 -12 0 0 -14 D 4 -10 0 0 6 E 4 4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.200000 E: 0.500000 Sum of squares = 0.379999999991 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4830: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (7) E D C B A (5) E D B C A (5) B A C E D (5) E B D C A (4) D A C E B (4) A D C B E (4) D E C A B (3) D C E B A (3) B C E A D (3) A C B D E (3) A B E C D (3) E D B A C (2) E B C D A (2) E B A D C (2) E B A C D (2) D E C B A (2) D C A E B (2) C D E B A (2) C D A B E (2) C B A D E (2) B E A C D (2) B C A D E (2) B A E C D (2) A D E C B (2) A D C E B (2) A C D B E (2) A B E D C (2) A B C E D (2) E D A C B (1) E D A B C (1) E C D B A (1) E B D A C (1) D E A C B (1) D C E A B (1) D C A B E (1) D A E C B (1) C D B E A (1) C A D B E (1) C A B D E (1) B E C D A (1) B C E D A (1) B C A E D (1) A E D B C (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -6 6 -6 B 16 0 6 -2 -2 C 6 -6 0 -2 -8 D -6 2 2 0 -12 E 6 2 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -6 6 -6 B 16 0 6 -2 -2 C 6 -6 0 -2 -8 D -6 2 2 0 -12 E 6 2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 A=23 D=18 C=9 so C is eliminated. Round 2 votes counts: E=26 B=26 A=25 D=23 so D is eliminated. Round 3 votes counts: E=38 A=35 B=27 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:209 C:195 D:193 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -6 6 -6 B 16 0 6 -2 -2 C 6 -6 0 -2 -8 D -6 2 2 0 -12 E 6 2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 6 -6 B 16 0 6 -2 -2 C 6 -6 0 -2 -8 D -6 2 2 0 -12 E 6 2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 6 -6 B 16 0 6 -2 -2 C 6 -6 0 -2 -8 D -6 2 2 0 -12 E 6 2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4831: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) E B D A C (7) E B A D C (6) D C B E A (5) A C D B E (5) E D B C A (4) E B D C A (4) E B A C D (4) B E A C D (4) A C D E B (4) D C E B A (3) D C E A B (3) D C A B E (3) C A D B E (3) B D E C A (3) B A E C D (3) D E B C A (2) D C A E B (2) A E C D B (2) A E C B D (2) A E B C D (2) A C B E D (2) A C B D E (2) E D A B C (1) E A B C D (1) D E C B A (1) C D A E B (1) C B D A E (1) C B A D E (1) B E D C A (1) B E A D C (1) B D C E A (1) A D E C B (1) A C E D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 -8 -2 B 2 0 -10 -6 -4 C 0 10 0 4 0 D 8 6 -4 0 6 E 2 4 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.720494 D: 0.000000 E: 0.279506 Sum of squares = 0.597235350622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.720494 D: 0.720494 E: 1.000000 A B C D E A 0 -2 0 -8 -2 B 2 0 -10 -6 -4 C 0 10 0 4 0 D 8 6 -4 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000503896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=23 D=19 C=18 B=13 so B is eliminated. Round 2 votes counts: E=33 A=26 D=23 C=18 so C is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 C:207 E:200 A:194 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 -8 -2 B 2 0 -10 -6 -4 C 0 10 0 4 0 D 8 6 -4 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000503896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -8 -2 B 2 0 -10 -6 -4 C 0 10 0 4 0 D 8 6 -4 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000503896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -8 -2 B 2 0 -10 -6 -4 C 0 10 0 4 0 D 8 6 -4 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000503896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4832: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) E D C A B (9) A B E D C (9) E D A C B (7) D C E A B (6) B A C E D (6) A E B D C (6) B C A D E (5) C B D E A (4) C B D A E (4) B A C D E (4) D E A C B (3) C D B E A (3) B C A E D (3) A E D B C (3) E C D B A (2) E A D B C (2) D E C A B (2) B A E C D (2) C E D B A (1) C E B D A (1) C D E A B (1) C D B A E (1) B C E A D (1) B A E D C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -10 -12 -8 B 0 0 -12 -8 -12 C 10 12 0 0 4 D 12 8 0 0 -6 E 8 12 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.767192 D: 0.232808 E: 0.000000 Sum of squares = 0.642783531585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.767192 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -12 -8 B 0 0 -12 -8 -12 C 10 12 0 0 4 D 12 8 0 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000180329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=22 E=20 A=20 D=11 so D is eliminated. Round 2 votes counts: C=33 E=25 B=22 A=20 so A is eliminated. Round 3 votes counts: E=35 C=33 B=32 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:211 D:207 A:185 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -12 -8 B 0 0 -12 -8 -12 C 10 12 0 0 4 D 12 8 0 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000180329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -12 -8 B 0 0 -12 -8 -12 C 10 12 0 0 4 D 12 8 0 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000180329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -12 -8 B 0 0 -12 -8 -12 C 10 12 0 0 4 D 12 8 0 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000180329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4833: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) D C B E A (12) A E B C D (9) D C A B E (6) B E C D A (5) A E B D C (5) E B C A D (4) E B A C D (4) C D B E A (4) A D C E B (4) A C D B E (4) E B C D A (3) A D C B E (3) C D A B E (2) C B E D A (2) A E D B C (2) A D E B C (2) A C B D E (2) A B E C D (2) E D B A C (1) E B A D C (1) D C E B A (1) D A E B C (1) C B D E A (1) C B A E D (1) B E C A D (1) B C E D A (1) A E C B D (1) A D E C B (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -10 -2 -4 B 6 0 6 12 -6 C 10 -6 0 -2 -8 D 2 -12 2 0 -14 E 4 6 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -10 -2 -4 B 6 0 6 12 -6 C 10 -6 0 -2 -8 D 2 -12 2 0 -14 E 4 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=25 D=20 C=10 B=7 so B is eliminated. Round 2 votes counts: A=38 E=31 D=20 C=11 so C is eliminated. Round 3 votes counts: A=39 E=34 D=27 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:209 C:197 A:189 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -10 -2 -4 B 6 0 6 12 -6 C 10 -6 0 -2 -8 D 2 -12 2 0 -14 E 4 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -2 -4 B 6 0 6 12 -6 C 10 -6 0 -2 -8 D 2 -12 2 0 -14 E 4 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -2 -4 B 6 0 6 12 -6 C 10 -6 0 -2 -8 D 2 -12 2 0 -14 E 4 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4834: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) E B C D A (8) E B C A D (8) D A C B E (7) A D C B E (7) D E A B C (6) B C E A D (6) A D E C B (6) A C B D E (6) E D B C A (5) C B A D E (5) D A E C B (4) E D A B C (3) E A D B C (3) C B A E D (3) E A B C D (2) D E B C A (2) D C B A E (2) D B C E A (2) A E D C B (2) A C B E D (2) B C A E D (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 -2 4 -8 B 2 0 10 2 0 C 2 -10 0 2 0 D -4 -2 -2 0 -6 E 8 0 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.431977 C: 0.000000 D: 0.000000 E: 0.568023 Sum of squares = 0.509254207037 Cumulative probabilities = A: 0.000000 B: 0.431977 C: 0.431977 D: 0.431977 E: 1.000000 A B C D E A 0 -2 -2 4 -8 B 2 0 10 2 0 C 2 -10 0 2 0 D -4 -2 -2 0 -6 E 8 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=24 D=23 B=16 C=8 so C is eliminated. Round 2 votes counts: E=29 B=24 A=24 D=23 so D is eliminated. Round 3 votes counts: E=37 A=35 B=28 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:207 E:207 C:197 A:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -8 B 2 0 10 2 0 C 2 -10 0 2 0 D -4 -2 -2 0 -6 E 8 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -8 B 2 0 10 2 0 C 2 -10 0 2 0 D -4 -2 -2 0 -6 E 8 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -8 B 2 0 10 2 0 C 2 -10 0 2 0 D -4 -2 -2 0 -6 E 8 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4835: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) D B E A C (7) E A C D B (6) C B D A E (6) E C D B A (5) E A D B C (5) C E A B D (5) C A B D E (5) A C B D E (5) A E C D B (4) A C E B D (4) E D B C A (3) E C A D B (3) C B D E A (3) C A E B D (3) B D C A E (3) A B D E C (3) D B A E C (2) C E B D A (2) C E A D B (2) B D E C A (2) B D C E A (2) A E D B C (2) A D B E C (2) A B D C E (2) E D C B A (1) E D A B C (1) D B E C A (1) C D B E A (1) B D A E C (1) B D A C E (1) A D E B C (1) Total count = 100 A B C D E A 0 6 6 4 -12 B -6 0 -10 -6 -8 C -6 10 0 8 -12 D -4 6 -8 0 -6 E 12 8 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 4 -12 B -6 0 -10 -6 -8 C -6 10 0 8 -12 D -4 6 -8 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=27 A=23 D=10 B=9 so B is eliminated. Round 2 votes counts: E=31 C=27 A=23 D=19 so D is eliminated. Round 3 votes counts: E=41 C=32 A=27 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:202 C:200 D:194 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 4 -12 B -6 0 -10 -6 -8 C -6 10 0 8 -12 D -4 6 -8 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 -12 B -6 0 -10 -6 -8 C -6 10 0 8 -12 D -4 6 -8 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 -12 B -6 0 -10 -6 -8 C -6 10 0 8 -12 D -4 6 -8 0 -6 E 12 8 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4836: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (13) D B A E C (10) A C E D B (8) A D C E B (7) D A B E C (6) E C A B D (5) E C B A D (4) D B A C E (4) D A B C E (4) C A E B D (4) E B C A D (3) D B E A C (3) C E B A D (3) D A C B E (2) C E A B D (2) B E C A D (2) B D E C A (2) B D C E A (2) B C D E A (2) A C D E B (2) E D A B C (1) E B A D C (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C E B (1) C A E D B (1) C A D E B (1) B E D C A (1) B C E D A (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 6 -10 10 B 4 0 14 -10 6 C -6 -14 0 4 -10 D 10 10 -4 0 0 E -10 -6 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.596596 E: 0.403404 Sum of squares = 0.518661651652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.596596 E: 1.000000 A B C D E A 0 -4 6 -10 10 B 4 0 14 -10 6 C -6 -14 0 4 -10 D 10 10 -4 0 0 E -10 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 0.499571 Sum of squares = 0.500000368629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=23 A=19 E=14 C=11 so C is eliminated. Round 2 votes counts: D=33 A=25 B=23 E=19 so E is eliminated. Round 3 votes counts: D=34 B=34 A=32 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:207 A:201 E:197 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -10 10 B 4 0 14 -10 6 C -6 -14 0 4 -10 D 10 10 -4 0 0 E -10 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 0.499571 Sum of squares = 0.500000368629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -10 10 B 4 0 14 -10 6 C -6 -14 0 4 -10 D 10 10 -4 0 0 E -10 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 0.499571 Sum of squares = 0.500000368629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -10 10 B 4 0 14 -10 6 C -6 -14 0 4 -10 D 10 10 -4 0 0 E -10 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 0.499571 Sum of squares = 0.500000368629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4837: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (8) E D A C B (7) E C B A D (7) C B E A D (7) E C D B A (5) E A D B C (5) E B C A D (4) D E A C B (4) B C A D E (4) A B D C E (4) C B E D A (3) A D E B C (3) E D C A B (2) E D A B C (2) E C B D A (2) E A B C D (2) D B C A E (2) D A E C B (2) B A C E D (2) B A C D E (2) A E D B C (2) A D B C E (2) A B C E D (2) A B C D E (2) E D C B A (1) E B A C D (1) D E C B A (1) D E C A B (1) D C B A E (1) D C A B E (1) D A C B E (1) D A B E C (1) C E B D A (1) C D B E A (1) C B D A E (1) C B A E D (1) B C E A D (1) B C A E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 6 6 -14 B -4 0 2 -4 -4 C -6 -2 0 0 -6 D -6 4 0 0 -18 E 14 4 6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 6 -14 B -4 0 2 -4 -4 C -6 -2 0 0 -6 D -6 4 0 0 -18 E 14 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998374 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=22 A=16 C=14 B=10 so B is eliminated. Round 2 votes counts: E=38 D=22 C=20 A=20 so C is eliminated. Round 3 votes counts: E=50 A=26 D=24 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:201 B:195 C:193 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 6 -14 B -4 0 2 -4 -4 C -6 -2 0 0 -6 D -6 4 0 0 -18 E 14 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998374 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 6 -14 B -4 0 2 -4 -4 C -6 -2 0 0 -6 D -6 4 0 0 -18 E 14 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998374 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 6 -14 B -4 0 2 -4 -4 C -6 -2 0 0 -6 D -6 4 0 0 -18 E 14 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998374 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4838: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C D B A E (7) E A D C B (5) E B A C D (4) D A C E B (4) C D A B E (4) B E A D C (4) A D B E C (4) E C B D A (3) E C A D B (3) D A B C E (3) C B D E A (3) B D A C E (3) B C E D A (3) B C D A E (3) B A E D C (3) A D E C B (3) E B A D C (2) E A D B C (2) D C B A E (2) D C A B E (2) C D E B A (2) C D E A B (2) C D A E B (2) C B E D A (2) C B D A E (2) B A D E C (2) A D E B C (2) E C D A B (1) E C B A D (1) E C A B D (1) E B C A D (1) D A C B E (1) C E D A B (1) C E B D A (1) B E C A D (1) B E A C D (1) B A D C E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 6 -2 0 B 2 0 -4 -2 6 C -6 4 0 -4 -2 D 2 2 4 0 6 E 0 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -2 0 B 2 0 -4 -2 6 C -6 4 0 -4 -2 D 2 2 4 0 6 E 0 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 B=21 D=12 A=11 so A is eliminated. Round 2 votes counts: E=30 C=26 D=22 B=22 so D is eliminated. Round 3 votes counts: E=35 C=35 B=30 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:207 A:201 B:201 C:196 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -2 0 B 2 0 -4 -2 6 C -6 4 0 -4 -2 D 2 2 4 0 6 E 0 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 0 B 2 0 -4 -2 6 C -6 4 0 -4 -2 D 2 2 4 0 6 E 0 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 0 B 2 0 -4 -2 6 C -6 4 0 -4 -2 D 2 2 4 0 6 E 0 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4839: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) D A E B C (7) A D E B C (7) A D B E C (6) C B E D A (5) E B C A D (4) E D A C B (3) E B A D C (3) E A D B C (3) D A C E B (3) D A C B E (3) D A B E C (3) C E B D A (3) C E B A D (3) C D E A B (3) C D A B E (3) B E C A D (3) E C B A D (2) D C A E B (2) C D A E B (2) C B E A D (2) C B D A E (2) B E A D C (2) B C A D E (2) B A D E C (2) A E D B C (2) E C D A B (1) E A B D C (1) D E A C B (1) D C A B E (1) D A B C E (1) C D B A E (1) C B A D E (1) B E A C D (1) B C E A D (1) B A C D E (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 24 18 -4 18 B -24 0 2 -20 -18 C -18 -2 0 -20 -22 D 4 20 20 0 20 E -18 18 22 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 18 -4 18 B -24 0 2 -20 -18 C -18 -2 0 -20 -22 D 4 20 20 0 20 E -18 18 22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 E=17 A=17 B=12 so B is eliminated. Round 2 votes counts: D=29 C=28 E=23 A=20 so A is eliminated. Round 3 votes counts: D=45 C=29 E=26 so E is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:232 A:228 E:201 B:170 C:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 18 -4 18 B -24 0 2 -20 -18 C -18 -2 0 -20 -22 D 4 20 20 0 20 E -18 18 22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 18 -4 18 B -24 0 2 -20 -18 C -18 -2 0 -20 -22 D 4 20 20 0 20 E -18 18 22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 18 -4 18 B -24 0 2 -20 -18 C -18 -2 0 -20 -22 D 4 20 20 0 20 E -18 18 22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4840: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (22) C A B D E (14) B D E C A (13) A C E D B (6) E D A B C (5) D B E A C (4) B C D A E (4) E B D C A (3) C A E B D (3) A E C D B (3) E A D C B (2) C B A D E (2) B E D C A (2) B D E A C (2) B D C E A (2) E C A B D (1) E A D B C (1) D E B A C (1) D B A E C (1) C E B A D (1) C E A B D (1) C A E D B (1) C A D B E (1) C A B E D (1) B D C A E (1) B C E D A (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 -20 -4 -24 -24 B 20 0 28 4 -2 C 4 -28 0 -20 -22 D 24 -4 20 0 -6 E 24 2 22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -4 -24 -24 B 20 0 28 4 -2 C 4 -28 0 -20 -22 D 24 -4 20 0 -6 E 24 2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=26 C=24 A=10 D=6 so D is eliminated. Round 2 votes counts: E=35 B=31 C=24 A=10 so A is eliminated. Round 3 votes counts: E=39 B=31 C=30 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:225 D:217 C:167 A:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -4 -24 -24 B 20 0 28 4 -2 C 4 -28 0 -20 -22 D 24 -4 20 0 -6 E 24 2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -24 -24 B 20 0 28 4 -2 C 4 -28 0 -20 -22 D 24 -4 20 0 -6 E 24 2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -24 -24 B 20 0 28 4 -2 C 4 -28 0 -20 -22 D 24 -4 20 0 -6 E 24 2 22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4841: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) D B C A E (7) C D B E A (7) A E B C D (7) D A B E C (5) C B D E A (5) A E D B C (5) D C B A E (4) A E B D C (4) E A C B D (3) C E B D A (3) B E A C D (3) A E C B D (3) D B C E A (2) D B A E C (2) D B A C E (2) D A E C B (2) D A B C E (2) C E B A D (2) C E A B D (2) B C E D A (2) A D E B C (2) A B E D C (2) A B E C D (2) E C B A D (1) E C A B D (1) C D E B A (1) C D E A B (1) B D C E A (1) B C D E A (1) B A E D C (1) B A D E C (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -14 -4 -18 6 B 14 0 2 -12 22 C 4 -2 0 -10 8 D 18 12 10 0 16 E -6 -22 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -18 6 B 14 0 2 -12 22 C 4 -2 0 -10 8 D 18 12 10 0 16 E -6 -22 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=27 C=21 B=9 E=5 so E is eliminated. Round 2 votes counts: D=38 A=30 C=23 B=9 so B is eliminated. Round 3 votes counts: D=39 A=35 C=26 so C is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:213 C:200 A:185 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -4 -18 6 B 14 0 2 -12 22 C 4 -2 0 -10 8 D 18 12 10 0 16 E -6 -22 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -18 6 B 14 0 2 -12 22 C 4 -2 0 -10 8 D 18 12 10 0 16 E -6 -22 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -18 6 B 14 0 2 -12 22 C 4 -2 0 -10 8 D 18 12 10 0 16 E -6 -22 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4842: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) D A C E B (7) D C A E B (6) B C E A D (6) E B A D C (5) C B D E A (5) C B D A E (5) B E A C D (5) C D A B E (4) C B A D E (4) A D E C B (4) D C A B E (3) C D B A E (3) C B A E D (3) B E C A D (3) A E D B C (3) E B D A C (2) C A B D E (2) B E D A C (2) B C E D A (2) A E D C B (2) A D E B C (2) A D C E B (2) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) D C B A E (1) D A E C B (1) D A E B C (1) C D B E A (1) C A D E B (1) C A D B E (1) B E C D A (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 4 12 B 2 0 -18 8 4 C 8 18 0 -6 16 D -4 -8 6 0 14 E -12 -4 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.250000 D: 0.562500 E: 0.000000 Sum of squares = 0.414062494606 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.437500 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 4 12 B 2 0 -18 8 4 C 8 18 0 -6 16 D -4 -8 6 0 14 E -12 -4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.250000 D: 0.562500 E: 0.000000 Sum of squares = 0.414062499249 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.437500 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=23 B=19 E=15 A=14 so A is eliminated. Round 2 votes counts: D=31 C=30 E=20 B=19 so B is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:204 A:203 B:198 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 4 12 B 2 0 -18 8 4 C 8 18 0 -6 16 D -4 -8 6 0 14 E -12 -4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.250000 D: 0.562500 E: 0.000000 Sum of squares = 0.414062499249 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.437500 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 4 12 B 2 0 -18 8 4 C 8 18 0 -6 16 D -4 -8 6 0 14 E -12 -4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.250000 D: 0.562500 E: 0.000000 Sum of squares = 0.414062499249 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.437500 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 4 12 B 2 0 -18 8 4 C 8 18 0 -6 16 D -4 -8 6 0 14 E -12 -4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.250000 D: 0.562500 E: 0.000000 Sum of squares = 0.414062499249 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.437500 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4843: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) A E C B D (7) D B E C A (6) D B C E A (6) A E C D B (6) B D E A C (5) E A C B D (4) E A B D C (4) D C B A E (4) E A D C B (3) C D B A E (3) C A E D B (3) C A D E B (3) B D C E A (3) E A C D B (2) D E B A C (2) D E A B C (2) D C A E B (2) C B D A E (2) C A E B D (2) C A D B E (2) C A B E D (2) B C D A E (2) A C E B D (2) E D A B C (1) E B D A C (1) E B A D C (1) E A D B C (1) D E A C B (1) D B E A C (1) D B C A E (1) C D A B E (1) C A B D E (1) B E D A C (1) B E A C D (1) B D E C A (1) B D C A E (1) B C D E A (1) B C A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 6 6 -8 B -14 0 -2 -2 -10 C -6 2 0 6 -14 D -6 2 -6 0 0 E 8 10 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.405419 E: 0.594581 Sum of squares = 0.517891064328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.405419 E: 1.000000 A B C D E A 0 14 6 6 -8 B -14 0 -2 -2 -10 C -6 2 0 6 -14 D -6 2 -6 0 0 E 8 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=24 C=19 B=16 A=16 so B is eliminated. Round 2 votes counts: D=35 E=26 C=23 A=16 so A is eliminated. Round 3 votes counts: E=39 D=35 C=26 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:216 A:209 D:195 C:194 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 6 6 -8 B -14 0 -2 -2 -10 C -6 2 0 6 -14 D -6 2 -6 0 0 E 8 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 6 -8 B -14 0 -2 -2 -10 C -6 2 0 6 -14 D -6 2 -6 0 0 E 8 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 6 -8 B -14 0 -2 -2 -10 C -6 2 0 6 -14 D -6 2 -6 0 0 E 8 10 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4844: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) D B A C E (13) B A D E C (10) D C E B A (8) A B E C D (7) C E A B D (6) C E D A B (5) D C B E A (4) D B C A E (4) D B A E C (4) B A D C E (4) E A C B D (3) B A C E D (3) E C D A B (2) C E D B A (2) B A E C D (2) A E B C D (2) E C A D B (1) D E C A B (1) D C E A B (1) C B E A D (1) B D A C E (1) B A E D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 4 10 6 B 14 0 6 8 12 C -4 -6 0 -6 4 D -10 -8 6 0 2 E -6 -12 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 10 6 B 14 0 6 8 12 C -4 -6 0 -6 4 D -10 -8 6 0 2 E -6 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=21 E=19 C=14 A=11 so A is eliminated. Round 2 votes counts: D=36 B=29 E=21 C=14 so C is eliminated. Round 3 votes counts: D=36 E=34 B=30 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:220 A:203 D:195 C:194 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 10 6 B 14 0 6 8 12 C -4 -6 0 -6 4 D -10 -8 6 0 2 E -6 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 10 6 B 14 0 6 8 12 C -4 -6 0 -6 4 D -10 -8 6 0 2 E -6 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 10 6 B 14 0 6 8 12 C -4 -6 0 -6 4 D -10 -8 6 0 2 E -6 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4845: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) E A C B D (9) E A C D B (7) E C A B D (6) C B A D E (5) E D A C B (4) E D A B C (4) D E B A C (4) B D C A E (4) B C A D E (4) D E C B A (3) D E B C A (3) D B C E A (3) C B A E D (3) C A E B D (3) C A B E D (3) A C B E D (3) C E A B D (2) C D B E A (2) A E C B D (2) E D C A B (1) E A D C B (1) E A D B C (1) D E A B C (1) D B E A C (1) D B A E C (1) D B A C E (1) C B E A D (1) B C D A E (1) B A C D E (1) A E B C D (1) A D E B C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -12 10 -6 B -2 0 -10 0 -6 C 12 10 0 10 0 D -10 0 -10 0 -4 E 6 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.361001 D: 0.000000 E: 0.638999 Sum of squares = 0.538641193171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.361001 D: 0.361001 E: 1.000000 A B C D E A 0 2 -12 10 -6 B -2 0 -10 0 -6 C 12 10 0 10 0 D -10 0 -10 0 -4 E 6 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=29 C=19 B=10 A=9 so A is eliminated. Round 2 votes counts: E=36 D=30 C=22 B=12 so B is eliminated. Round 3 votes counts: E=36 D=34 C=30 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:216 E:208 A:197 B:191 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 10 -6 B -2 0 -10 0 -6 C 12 10 0 10 0 D -10 0 -10 0 -4 E 6 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 10 -6 B -2 0 -10 0 -6 C 12 10 0 10 0 D -10 0 -10 0 -4 E 6 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 10 -6 B -2 0 -10 0 -6 C 12 10 0 10 0 D -10 0 -10 0 -4 E 6 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4846: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (7) D A B E C (6) C B E A D (6) D E A C B (5) D B A C E (5) D A E B C (5) A E B C D (5) E C A B D (3) D E C A B (3) D C B E A (3) D B A E C (3) D A E C B (3) C E A B D (3) A E C B D (3) A B D E C (3) D B C E A (2) D B C A E (2) C E B A D (2) C E A D B (2) C D E B A (2) C B E D A (2) B C E A D (2) A E D C B (2) A E C D B (2) A D E C B (2) E C A D B (1) E A C D B (1) E A C B D (1) D C E A B (1) C E D B A (1) C E D A B (1) C D B E A (1) C B D E A (1) B D C A E (1) B D A C E (1) B A D E C (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 8 -10 0 B -16 0 -2 -12 -2 C -8 2 0 -6 -10 D 10 12 6 0 18 E 0 2 10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 -10 0 B -16 0 -2 -12 -2 C -8 2 0 -6 -10 D 10 12 6 0 18 E 0 2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=23 C=21 B=12 E=6 so E is eliminated. Round 2 votes counts: D=38 C=25 A=25 B=12 so B is eliminated. Round 3 votes counts: D=40 C=34 A=26 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:207 E:197 C:189 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 8 -10 0 B -16 0 -2 -12 -2 C -8 2 0 -6 -10 D 10 12 6 0 18 E 0 2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 -10 0 B -16 0 -2 -12 -2 C -8 2 0 -6 -10 D 10 12 6 0 18 E 0 2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 -10 0 B -16 0 -2 -12 -2 C -8 2 0 -6 -10 D 10 12 6 0 18 E 0 2 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4847: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) A B D C E (6) D E C A B (5) B A D C E (5) D A B C E (4) A B C D E (4) E C B A D (3) E B D A C (3) D E B C A (3) D E B A C (3) D A C B E (3) C E D A B (3) B A E C D (3) E D C B A (2) E C D B A (2) E B C D A (2) D E C B A (2) D E A B C (2) D C E A B (2) D C A E B (2) D B A E C (2) C E B A D (2) B D A E C (2) B A E D C (2) B A D E C (2) E D C A B (1) E D B C A (1) E C B D A (1) E B C A D (1) D A E B C (1) C D E A B (1) C D A B E (1) C A E D B (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D A C (1) B E A D C (1) B C A E D (1) B A C D E (1) A D B C E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 22 2 18 B 16 0 30 14 14 C -22 -30 0 -12 10 D -2 -14 12 0 10 E -18 -14 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 22 2 18 B 16 0 30 14 14 C -22 -30 0 -12 10 D -2 -14 12 0 10 E -18 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 E=16 A=13 C=11 so C is eliminated. Round 2 votes counts: D=31 B=31 E=21 A=17 so A is eliminated. Round 3 votes counts: B=45 D=33 E=22 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:237 A:213 D:203 E:174 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 22 2 18 B 16 0 30 14 14 C -22 -30 0 -12 10 D -2 -14 12 0 10 E -18 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 22 2 18 B 16 0 30 14 14 C -22 -30 0 -12 10 D -2 -14 12 0 10 E -18 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 22 2 18 B 16 0 30 14 14 C -22 -30 0 -12 10 D -2 -14 12 0 10 E -18 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4848: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) D A C B E (8) C E A B D (7) E C B A D (6) C E A D B (6) B E A D C (6) D A B C E (5) A D C B E (5) B A D E C (4) C E D A B (3) C E B A D (3) C D A E B (3) C A D E B (3) B E D A C (3) B E A C D (3) B D A E C (3) E C B D A (2) C D E B A (2) B E D C A (2) B A E D C (2) A D C E B (2) A D B C E (2) E B C D A (1) D B A E C (1) D A C E B (1) C A E D B (1) C A E B D (1) B E C A D (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -4 32 -12 B 0 0 -8 12 -6 C 4 8 0 8 6 D -32 -12 -8 0 -16 E 12 6 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 32 -12 B 0 0 -8 12 -6 C 4 8 0 8 6 D -32 -12 -8 0 -16 E 12 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=24 E=20 D=15 A=12 so A is eliminated. Round 2 votes counts: C=30 D=25 B=25 E=20 so E is eliminated. Round 3 votes counts: C=38 B=37 D=25 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:214 C:213 A:208 B:199 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 32 -12 B 0 0 -8 12 -6 C 4 8 0 8 6 D -32 -12 -8 0 -16 E 12 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 32 -12 B 0 0 -8 12 -6 C 4 8 0 8 6 D -32 -12 -8 0 -16 E 12 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 32 -12 B 0 0 -8 12 -6 C 4 8 0 8 6 D -32 -12 -8 0 -16 E 12 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4849: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) B D A C E (5) E D C B A (4) D B E A C (4) B D C A E (4) B A D C E (4) A E D B C (4) A B C D E (4) E D C A B (3) E C D A B (3) E C A D B (3) E A D C B (3) C E A D B (3) C A B E D (3) B C A D E (3) A E C D B (3) A C B E D (3) E A C D B (2) C E D B A (2) C B A D E (2) B D A E C (2) B A C D E (2) A E D C B (2) A E C B D (2) A B E D C (2) A B D E C (2) E D A C B (1) E D A B C (1) E C D B A (1) E A D B C (1) D E C B A (1) D E B A C (1) D B C E A (1) C D E B A (1) C D B E A (1) C B E D A (1) C B D A E (1) C B A E D (1) C A E B D (1) C A B D E (1) B C D A E (1) A E B D C (1) A D E B C (1) A D B E C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 6 12 14 B -4 0 -12 6 10 C -6 12 0 4 4 D -12 -6 -4 0 -2 E -14 -10 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 12 14 B -4 0 -12 6 10 C -6 12 0 4 4 D -12 -6 -4 0 -2 E -14 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997158 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 E=22 B=21 D=7 so D is eliminated. Round 2 votes counts: A=27 B=26 E=24 C=23 so C is eliminated. Round 3 votes counts: B=38 A=32 E=30 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:207 B:200 D:188 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 12 14 B -4 0 -12 6 10 C -6 12 0 4 4 D -12 -6 -4 0 -2 E -14 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997158 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 12 14 B -4 0 -12 6 10 C -6 12 0 4 4 D -12 -6 -4 0 -2 E -14 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997158 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 12 14 B -4 0 -12 6 10 C -6 12 0 4 4 D -12 -6 -4 0 -2 E -14 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997158 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4850: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (13) D B E C A (9) D E B C A (8) E D C A B (6) D B E A C (6) C A E B D (6) A C E B D (6) E C A D B (5) B A C D E (5) A C B E D (5) B D E A C (3) B A D C E (3) D E C B A (2) D E C A B (2) D E B A C (2) C E A D B (2) C A E D B (2) B D E C A (2) B C A E D (2) A C E D B (2) E D C B A (1) E D A C B (1) E A D C B (1) D E A B C (1) C A B E D (1) B D C A E (1) B A C E D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 2 -14 -2 B 16 0 14 -2 4 C -2 -14 0 -22 0 D 14 2 22 0 14 E 2 -4 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 2 -14 -2 B 16 0 14 -2 4 C -2 -14 0 -22 0 D 14 2 22 0 14 E 2 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=30 B=30 A=15 E=14 C=11 so C is eliminated. Round 2 votes counts: D=30 B=30 A=24 E=16 so E is eliminated. Round 3 votes counts: D=38 A=32 B=30 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:216 E:192 A:185 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 2 -14 -2 B 16 0 14 -2 4 C -2 -14 0 -22 0 D 14 2 22 0 14 E 2 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 -14 -2 B 16 0 14 -2 4 C -2 -14 0 -22 0 D 14 2 22 0 14 E 2 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 -14 -2 B 16 0 14 -2 4 C -2 -14 0 -22 0 D 14 2 22 0 14 E 2 -4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4851: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) B E D A C (7) A C D B E (6) E D C A B (5) D E C A B (5) B E C A D (5) B C A E D (5) E D B A C (4) D E B A C (4) D C A E B (4) D A C B E (4) C A D E B (4) C A B E D (4) B E A C D (4) A C B D E (4) B A C E D (3) E B D C A (2) D A C E B (2) C A B D E (2) B A D C E (2) E D C B A (1) E D B C A (1) E B D A C (1) E B C A D (1) D C E A B (1) C D A E B (1) C B A E D (1) C A D B E (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E D C (1) B A E C D (1) B A C D E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 6 14 -4 -2 B -6 0 -8 -4 10 C -14 8 0 -10 -6 D 4 4 10 0 6 E 2 -10 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 -4 -2 B -6 0 -8 -4 10 C -14 8 0 -10 -6 D 4 4 10 0 6 E 2 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=28 E=15 C=13 A=12 so A is eliminated. Round 2 votes counts: B=33 D=28 C=24 E=15 so E is eliminated. Round 3 votes counts: D=39 B=37 C=24 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 B:196 E:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 14 -4 -2 B -6 0 -8 -4 10 C -14 8 0 -10 -6 D 4 4 10 0 6 E 2 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -4 -2 B -6 0 -8 -4 10 C -14 8 0 -10 -6 D 4 4 10 0 6 E 2 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -4 -2 B -6 0 -8 -4 10 C -14 8 0 -10 -6 D 4 4 10 0 6 E 2 -10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4852: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) A D B C E (7) E D A B C (5) E C B D A (5) E C B A D (5) D E B A C (5) C E B A D (5) E C A B D (4) C B A D E (4) B C A D E (4) C E B D A (3) C B E D A (3) C B A E D (3) A D B E C (3) A C B D E (3) A B C D E (3) E D C B A (2) D E A B C (2) D B C E A (2) D B A C E (2) C B E A D (2) A B D C E (2) E D B A C (1) E D A C B (1) E C A D B (1) E A D C B (1) E A C B D (1) D E C B A (1) D E B C A (1) D C B E A (1) D A E B C (1) D A B E C (1) D A B C E (1) C E A B D (1) C B D E A (1) C A B E D (1) C A B D E (1) B A D C E (1) B A C D E (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -14 10 -20 B 18 0 -14 8 -8 C 14 14 0 18 2 D -10 -8 -18 0 -4 E 20 8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 10 -20 B 18 0 -14 8 -8 C 14 14 0 18 2 D -10 -8 -18 0 -4 E 20 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=24 A=20 D=17 B=6 so B is eliminated. Round 2 votes counts: E=33 C=28 A=22 D=17 so D is eliminated. Round 3 votes counts: E=42 C=31 A=27 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:215 B:202 D:180 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -14 10 -20 B 18 0 -14 8 -8 C 14 14 0 18 2 D -10 -8 -18 0 -4 E 20 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 10 -20 B 18 0 -14 8 -8 C 14 14 0 18 2 D -10 -8 -18 0 -4 E 20 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 10 -20 B 18 0 -14 8 -8 C 14 14 0 18 2 D -10 -8 -18 0 -4 E 20 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4853: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) E A D B C (6) D E A B C (6) D B C A E (6) C A E B D (6) E A B D C (5) D C B E A (5) C B A E D (5) A E C B D (5) D C B A E (4) C D B A E (4) B C D A E (4) E A D C B (3) E A C B D (3) C E A B D (3) B D A E C (3) E A C D B (2) E A B C D (2) D B C E A (2) C D E A B (2) C B D A E (2) B C A E D (2) B A E D C (2) A E B C D (2) E C A D B (1) D E B A C (1) D C E A B (1) C E D A B (1) C E A D B (1) C D B E A (1) B A E C D (1) B A C E D (1) A E B D C (1) Total count = 100 A B C D E A 0 0 0 2 -4 B 0 0 2 -6 -2 C 0 -2 0 -4 0 D -2 6 4 0 -4 E 4 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.306179 D: 0.000000 E: 0.693821 Sum of squares = 0.575133427792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.306179 D: 0.306179 E: 1.000000 A B C D E A 0 0 0 2 -4 B 0 0 2 -6 -2 C 0 -2 0 -4 0 D -2 6 4 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.000000 E: 0.500042 Sum of squares = 0.500000003462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.499958 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=25 E=22 B=13 A=8 so A is eliminated. Round 2 votes counts: D=32 E=30 C=25 B=13 so B is eliminated. Round 3 votes counts: D=35 E=33 C=32 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:205 D:202 A:199 B:197 C:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 2 -4 B 0 0 2 -6 -2 C 0 -2 0 -4 0 D -2 6 4 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.000000 E: 0.500042 Sum of squares = 0.500000003462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.499958 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -4 B 0 0 2 -6 -2 C 0 -2 0 -4 0 D -2 6 4 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.000000 E: 0.500042 Sum of squares = 0.500000003462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.499958 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -4 B 0 0 2 -6 -2 C 0 -2 0 -4 0 D -2 6 4 0 -4 E 4 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.000000 E: 0.500042 Sum of squares = 0.500000003462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499958 D: 0.499958 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4854: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) A E B D C (7) A E B C D (7) D C B A E (6) D B C E A (6) A E C B D (6) C A E D B (5) B E A D C (5) C D B E A (4) E A B D C (3) D B C A E (3) C E A D B (3) C D B A E (3) C D A E B (3) C A D E B (3) B D E A C (3) B A E D C (3) A C E B D (3) D B E C A (2) C E D A B (2) C D E A B (2) C D A B E (2) B A D E C (2) E C A D B (1) E B D C A (1) E B A D C (1) E A B C D (1) C D E B A (1) B E D A C (1) B D E C A (1) B D A E C (1) B D A C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -10 2 12 B 2 0 -2 -6 2 C 10 2 0 -8 8 D -2 6 8 0 0 E -12 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.100000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999996 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 2 12 B 2 0 -2 -6 2 C 10 2 0 -8 8 D -2 6 8 0 0 E -12 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.100000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999963 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 A=24 B=17 E=7 so E is eliminated. Round 2 votes counts: C=29 A=28 D=24 B=19 so B is eliminated. Round 3 votes counts: A=39 D=32 C=29 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:206 D:206 A:201 B:198 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -10 2 12 B 2 0 -2 -6 2 C 10 2 0 -8 8 D -2 6 8 0 0 E -12 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.100000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999963 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 2 12 B 2 0 -2 -6 2 C 10 2 0 -8 8 D -2 6 8 0 0 E -12 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.100000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999963 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 2 12 B 2 0 -2 -6 2 C 10 2 0 -8 8 D -2 6 8 0 0 E -12 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.100000 D: 0.500000 E: 0.000000 Sum of squares = 0.419999999963 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4855: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) E C B A D (6) D A E B C (6) E C B D A (5) A D B E C (5) E A D B C (4) C B D A E (4) A B D C E (4) E B C A D (3) D E A C B (3) D C A B E (3) D A B C E (3) C D B A E (3) C B A D E (3) B C A D E (3) A D B C E (3) E D A C B (2) E D A B C (2) D A B E C (2) C B E D A (2) C B E A D (2) C B D E A (2) C B A E D (2) B C A E D (2) B A C D E (2) A D E B C (2) E D C A B (1) E C D B A (1) E C D A B (1) E B A C D (1) E A D C B (1) E A B C D (1) D C B A E (1) D C A E B (1) D B C A E (1) D A C B E (1) C E D B A (1) C D B E A (1) B E A C D (1) B C E A D (1) A E D B C (1) Total count = 100 A B C D E A 0 6 2 -6 18 B -6 0 -6 -12 2 C -2 6 0 -6 -10 D 6 12 6 0 20 E -18 -2 10 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -6 18 B -6 0 -6 -12 2 C -2 6 0 -6 -10 D 6 12 6 0 20 E -18 -2 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 C=20 A=15 B=9 so B is eliminated. Round 2 votes counts: E=29 D=28 C=26 A=17 so A is eliminated. Round 3 votes counts: D=42 E=30 C=28 so C is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:210 C:194 B:189 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -6 18 B -6 0 -6 -12 2 C -2 6 0 -6 -10 D 6 12 6 0 20 E -18 -2 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -6 18 B -6 0 -6 -12 2 C -2 6 0 -6 -10 D 6 12 6 0 20 E -18 -2 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -6 18 B -6 0 -6 -12 2 C -2 6 0 -6 -10 D 6 12 6 0 20 E -18 -2 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4856: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) E D B C A (11) E C D B A (6) D E B A C (6) C E A D B (6) C A E B D (6) C A B E D (6) A B D C E (5) C E D A B (4) C A E D B (4) C E D B A (3) B D E A C (3) B D A E C (3) A B C D E (3) E D C B A (2) D B A E C (2) C E B D A (2) E D C A B (1) E D B A C (1) E C B D A (1) D E A B C (1) D B E C A (1) D A B E C (1) C A B D E (1) B E D C A (1) B D E C A (1) B A D E C (1) B A D C E (1) B A C D E (1) A D B E C (1) A C E B D (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -12 0 0 B -8 0 -12 -2 -12 C 12 12 0 14 12 D 0 2 -14 0 -10 E 0 12 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 0 0 B -8 0 -12 -2 -12 C 12 12 0 14 12 D 0 2 -14 0 -10 E 0 12 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=24 E=22 D=11 B=11 so D is eliminated. Round 2 votes counts: C=32 E=29 A=25 B=14 so B is eliminated. Round 3 votes counts: E=35 A=33 C=32 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:225 E:205 A:198 D:189 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 0 0 B -8 0 -12 -2 -12 C 12 12 0 14 12 D 0 2 -14 0 -10 E 0 12 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 0 0 B -8 0 -12 -2 -12 C 12 12 0 14 12 D 0 2 -14 0 -10 E 0 12 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 0 0 B -8 0 -12 -2 -12 C 12 12 0 14 12 D 0 2 -14 0 -10 E 0 12 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4857: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) C E D A B (6) C E D B A (5) E C D B A (4) E C A B D (4) D B A C E (4) C E A D B (4) A E B C D (4) E B A C D (3) E A B C D (3) D C E B A (3) D C B A E (3) D B E A C (3) A B E C D (3) A B D E C (3) A B D C E (3) E D C B A (2) E D B C A (2) E B A D C (2) D A B C E (2) C E A B D (2) C D A B E (2) C A E B D (2) B D A E C (2) A C E B D (2) A C B E D (2) A C B D E (2) E B D C A (1) D E C B A (1) D C B E A (1) D C A B E (1) D B E C A (1) D B A E C (1) C D E B A (1) C D E A B (1) C A E D B (1) C A D B E (1) A C D B E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 8 2 B -4 0 -2 0 -6 C -4 2 0 10 2 D -8 0 -10 0 -8 E -2 6 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 8 2 B -4 0 -2 0 -6 C -4 2 0 10 2 D -8 0 -10 0 -8 E -2 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 E=21 D=20 B=11 so B is eliminated. Round 2 votes counts: A=32 C=25 D=22 E=21 so E is eliminated. Round 3 votes counts: A=40 C=33 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:205 E:205 B:194 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 8 2 B -4 0 -2 0 -6 C -4 2 0 10 2 D -8 0 -10 0 -8 E -2 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 8 2 B -4 0 -2 0 -6 C -4 2 0 10 2 D -8 0 -10 0 -8 E -2 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 8 2 B -4 0 -2 0 -6 C -4 2 0 10 2 D -8 0 -10 0 -8 E -2 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4858: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (18) B C E D A (14) A D C E B (9) C D A E B (8) B E C D A (8) E D A C B (5) A E D B C (4) C B A D E (3) E B D A C (2) E A D B C (2) D E A C B (2) C D E A B (2) C B D E A (2) B C A D E (2) B A E D C (2) A D E B C (2) E B D C A (1) D A E C B (1) D A C E B (1) C D E B A (1) C B E D A (1) C B D A E (1) C A D B E (1) B E C A D (1) B E A D C (1) B C D E A (1) B C D A E (1) B A E C D (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 16 6 -2 14 B -16 0 -14 -16 -18 C -6 14 0 -2 -2 D 2 16 2 0 14 E -14 18 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 -2 14 B -16 0 -14 -16 -18 C -6 14 0 -2 -2 D 2 16 2 0 14 E -14 18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=31 C=19 E=10 D=4 so D is eliminated. Round 2 votes counts: A=38 B=31 C=19 E=12 so E is eliminated. Round 3 votes counts: A=47 B=34 C=19 so C is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:217 C:202 E:196 B:168 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 -2 14 B -16 0 -14 -16 -18 C -6 14 0 -2 -2 D 2 16 2 0 14 E -14 18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -2 14 B -16 0 -14 -16 -18 C -6 14 0 -2 -2 D 2 16 2 0 14 E -14 18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -2 14 B -16 0 -14 -16 -18 C -6 14 0 -2 -2 D 2 16 2 0 14 E -14 18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4859: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (12) D B C A E (11) E C A B D (7) C E B D A (7) A D B C E (7) C B D E A (5) E C B D A (4) E C B A D (4) D B A C E (4) B D C A E (4) A E C B D (4) A E B D C (4) E A C D B (3) E A C B D (3) A E D C B (3) E C A D B (2) A E D B C (2) A D E B C (2) E C D A B (1) D C B A E (1) D B C E A (1) D A B C E (1) C D B E A (1) C B E D A (1) B D C E A (1) B D A C E (1) A E C D B (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 0 14 20 B -10 0 6 -4 6 C 0 -6 0 -14 -8 D -14 4 14 0 6 E -20 -6 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.662151 B: 0.000000 C: 0.337849 D: 0.000000 E: 0.000000 Sum of squares = 0.55258575698 Cumulative probabilities = A: 0.662151 B: 0.662151 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 14 20 B -10 0 6 -4 6 C 0 -6 0 -14 -8 D -14 4 14 0 6 E -20 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500542 B: 0.000000 C: 0.499458 D: 0.000000 E: 0.000000 Sum of squares = 0.500000587607 Cumulative probabilities = A: 0.500542 B: 0.500542 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=24 D=18 C=14 B=6 so B is eliminated. Round 2 votes counts: A=38 E=24 D=24 C=14 so C is eliminated. Round 3 votes counts: A=38 E=32 D=30 so D is eliminated. Round 4 votes counts: A=60 E=40 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:205 B:199 E:188 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 14 20 B -10 0 6 -4 6 C 0 -6 0 -14 -8 D -14 4 14 0 6 E -20 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500542 B: 0.000000 C: 0.499458 D: 0.000000 E: 0.000000 Sum of squares = 0.500000587607 Cumulative probabilities = A: 0.500542 B: 0.500542 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 14 20 B -10 0 6 -4 6 C 0 -6 0 -14 -8 D -14 4 14 0 6 E -20 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500542 B: 0.000000 C: 0.499458 D: 0.000000 E: 0.000000 Sum of squares = 0.500000587607 Cumulative probabilities = A: 0.500542 B: 0.500542 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 14 20 B -10 0 6 -4 6 C 0 -6 0 -14 -8 D -14 4 14 0 6 E -20 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500542 B: 0.000000 C: 0.499458 D: 0.000000 E: 0.000000 Sum of squares = 0.500000587607 Cumulative probabilities = A: 0.500542 B: 0.500542 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4860: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (5) A C B E D (5) E A B C D (4) D E B A C (4) D C B A E (4) C A B E D (4) A C E B D (4) A C D E B (4) D E A B C (3) D A C E B (3) C B A E D (3) C B A D E (3) B E C D A (3) A E C B D (3) A E B C D (3) E D B A C (2) E B D A C (2) E B A D C (2) E A D B C (2) E A B D C (2) D C E B A (2) D B C E A (2) D A E C B (2) C A B D E (2) B E D C A (2) B E C A D (2) B E A C D (2) B C E A D (2) B C A E D (2) A E C D B (2) E B A C D (1) D E B C A (1) D E A C B (1) D C E A B (1) D C B E A (1) D C A B E (1) C D A B E (1) C B D A E (1) C A D B E (1) B D C E A (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 8 12 2 B -2 0 0 12 0 C -8 0 0 10 -2 D -12 -12 -10 0 -10 E -2 0 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 12 2 B -2 0 0 12 0 C -8 0 0 10 -2 D -12 -12 -10 0 -10 E -2 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=24 B=16 E=15 C=15 so E is eliminated. Round 2 votes counts: D=32 A=32 B=21 C=15 so C is eliminated. Round 3 votes counts: A=39 D=33 B=28 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:205 E:205 C:200 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 12 2 B -2 0 0 12 0 C -8 0 0 10 -2 D -12 -12 -10 0 -10 E -2 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 12 2 B -2 0 0 12 0 C -8 0 0 10 -2 D -12 -12 -10 0 -10 E -2 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 12 2 B -2 0 0 12 0 C -8 0 0 10 -2 D -12 -12 -10 0 -10 E -2 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4861: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (7) A C E B D (7) A E D B C (6) E D B C A (5) A C B D E (5) E C D B A (4) E A D B C (4) D E B C A (4) C B D A E (4) A E C D B (4) A B D C E (4) E D A B C (3) E A C D B (3) C E D B A (3) C E B D A (3) A B D E C (3) C A E B D (2) C A B D E (2) B D C E A (2) B C D A E (2) A D E B C (2) A B C D E (2) E D C B A (1) E D B A C (1) E D A C B (1) E C D A B (1) E C A D B (1) D E C B A (1) D B C E A (1) D B A E C (1) C E A D B (1) C E A B D (1) C B A E D (1) C B A D E (1) B D C A E (1) B D A C E (1) B C D E A (1) B C A D E (1) B A D C E (1) B A C D E (1) A E C B D (1) Total count = 100 A B C D E A 0 6 0 6 4 B -6 0 -8 6 -18 C 0 8 0 16 8 D -6 -6 -16 0 -6 E -4 18 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.835925 B: 0.000000 C: 0.164075 D: 0.000000 E: 0.000000 Sum of squares = 0.725691504495 Cumulative probabilities = A: 0.835925 B: 0.835925 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 6 4 B -6 0 -8 6 -18 C 0 8 0 16 8 D -6 -6 -16 0 -6 E -4 18 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=25 E=24 B=10 D=7 so D is eliminated. Round 2 votes counts: A=34 E=29 C=25 B=12 so B is eliminated. Round 3 votes counts: A=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:216 A:208 E:206 B:187 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 6 4 B -6 0 -8 6 -18 C 0 8 0 16 8 D -6 -6 -16 0 -6 E -4 18 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 6 4 B -6 0 -8 6 -18 C 0 8 0 16 8 D -6 -6 -16 0 -6 E -4 18 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 6 4 B -6 0 -8 6 -18 C 0 8 0 16 8 D -6 -6 -16 0 -6 E -4 18 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4862: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) D E C B A (5) B D C E A (5) D C E B A (4) C A D E B (4) A E B C D (4) A B C E D (4) E D B A C (3) E A C D B (3) E A B D C (3) D E B C A (3) C A D B E (3) B E D A C (3) B D E C A (3) B C A D E (3) A C B D E (3) E D A C B (2) E D A B C (2) D C B E A (2) C D B E A (2) C A B D E (2) B E A D C (2) B C D A E (2) B A C D E (2) A E B D C (2) A C E B D (2) E D C B A (1) E D B C A (1) E B D A C (1) E A D C B (1) E A D B C (1) D C E A B (1) D B C E A (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) C A E D B (1) B D E A C (1) B D C A E (1) B A E D C (1) B A E C D (1) A E C D B (1) A E C B D (1) A C E D B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -4 -12 B 0 0 2 0 -6 C 8 -2 0 -12 -6 D 4 0 12 0 2 E 12 6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.112775 C: 0.000000 D: 0.887225 E: 0.000000 Sum of squares = 0.799886585807 Cumulative probabilities = A: 0.000000 B: 0.112775 C: 0.112775 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -4 -12 B 0 0 2 0 -6 C 8 -2 0 -12 -6 D 4 0 12 0 2 E 12 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000081038 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 A=20 D=16 C=16 so D is eliminated. Round 2 votes counts: E=32 B=25 C=23 A=20 so A is eliminated. Round 3 votes counts: E=40 B=31 C=29 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:209 B:198 C:194 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -4 -12 B 0 0 2 0 -6 C 8 -2 0 -12 -6 D 4 0 12 0 2 E 12 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000081038 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 -12 B 0 0 2 0 -6 C 8 -2 0 -12 -6 D 4 0 12 0 2 E 12 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000081038 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 -12 B 0 0 2 0 -6 C 8 -2 0 -12 -6 D 4 0 12 0 2 E 12 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000081038 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4863: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) C A B E D (7) B E A C D (7) D C A E B (6) C A D E B (6) A E C B D (6) E B D A C (5) D B E C A (5) D C A B E (4) B E D A C (4) E B A D C (3) E A B C D (3) D C B A E (3) E D A B C (2) D E B A C (2) D E A B C (2) D B E A C (2) C D A B E (2) C A E D B (2) C A E B D (2) B C A E D (2) A E C D B (2) A C E D B (2) E A D C B (1) E A B D C (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E A B (1) D B C E A (1) D B C A E (1) D A C E B (1) C D A E B (1) B E A D C (1) B D C A E (1) B C D A E (1) B A C E D (1) Total count = 100 A B C D E A 0 6 6 6 0 B -6 0 4 2 -16 C -6 -4 0 2 -12 D -6 -2 -2 0 -16 E 0 16 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.841916 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.158084 Sum of squares = 0.733813564304 Cumulative probabilities = A: 0.841916 B: 0.841916 C: 0.841916 D: 0.841916 E: 1.000000 A B C D E A 0 6 6 6 0 B -6 0 4 2 -16 C -6 -4 0 2 -12 D -6 -2 -2 0 -16 E 0 16 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 C=20 B=17 A=10 so A is eliminated. Round 2 votes counts: D=31 E=30 C=22 B=17 so B is eliminated. Round 3 votes counts: E=42 D=32 C=26 so C is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:209 B:192 C:190 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 0 B -6 0 4 2 -16 C -6 -4 0 2 -12 D -6 -2 -2 0 -16 E 0 16 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 0 B -6 0 4 2 -16 C -6 -4 0 2 -12 D -6 -2 -2 0 -16 E 0 16 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 0 B -6 0 4 2 -16 C -6 -4 0 2 -12 D -6 -2 -2 0 -16 E 0 16 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4864: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) B E A D C (6) E B A C D (5) D C A B E (5) C D E A B (5) C D A E B (5) B A E D C (5) B A D C E (5) A D B C E (5) E C D A B (4) E C B D A (4) E B C D A (4) C E D A B (4) E B C A D (3) D C A E B (3) C D B A E (3) B A D E C (3) D A B C E (2) C D A B E (2) A D C B E (2) A B D E C (2) E C D B A (1) E C A D B (1) E B A D C (1) E A B C D (1) D A C E B (1) C E D B A (1) C D E B A (1) C B E D A (1) C B D A E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C D A E (1) A D E C B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 -12 10 B -4 0 -4 -6 10 C 2 4 0 0 10 D 12 6 0 0 10 E -10 -10 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.667815 D: 0.332185 E: 0.000000 Sum of squares = 0.556323537698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.667815 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -12 10 B -4 0 -4 -6 10 C 2 4 0 0 10 D 12 6 0 0 10 E -10 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=23 B=23 D=18 A=12 so A is eliminated. Round 2 votes counts: B=27 D=26 E=24 C=23 so C is eliminated. Round 3 votes counts: D=42 E=29 B=29 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:208 A:200 B:198 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -12 10 B -4 0 -4 -6 10 C 2 4 0 0 10 D 12 6 0 0 10 E -10 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -12 10 B -4 0 -4 -6 10 C 2 4 0 0 10 D 12 6 0 0 10 E -10 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -12 10 B -4 0 -4 -6 10 C 2 4 0 0 10 D 12 6 0 0 10 E -10 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4865: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) C D B A E (6) C D A B E (6) A D C B E (6) C B D E A (5) B E C D A (5) E B A C D (4) D C A B E (4) D A C B E (4) A E B D C (4) E C B D A (3) E B C A D (3) E B A D C (3) E A B D C (3) A D E C B (3) A D C E B (3) A D B C E (3) E C B A D (2) C E D B A (2) B E A D C (2) B C E D A (2) A E D B C (2) A D E B C (2) A D B E C (2) E A C D B (1) E A C B D (1) C E D A B (1) C E B D A (1) C E A D B (1) C D B E A (1) C D A E B (1) B D C A E (1) B C D E A (1) A E D C B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -8 -4 0 B 0 0 -4 2 0 C 8 4 0 10 -2 D 4 -2 -10 0 0 E 0 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.042844 B: 0.213046 C: 0.000000 D: 0.000000 E: 0.744110 Sum of squares = 0.600924213546 Cumulative probabilities = A: 0.042844 B: 0.255890 C: 0.255890 D: 0.255890 E: 1.000000 A B C D E A 0 0 -8 -4 0 B 0 0 -4 2 0 C 8 4 0 10 -2 D 4 -2 -10 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.026316 B: 0.289473 C: 0.000000 D: 0.000000 E: 0.684211 Sum of squares = 0.552631629044 Cumulative probabilities = A: 0.026316 B: 0.315789 C: 0.315789 D: 0.315789 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=28 C=24 B=11 D=8 so D is eliminated. Round 2 votes counts: A=32 E=29 C=28 B=11 so B is eliminated. Round 3 votes counts: E=36 C=32 A=32 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:210 E:201 B:199 D:196 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -4 0 B 0 0 -4 2 0 C 8 4 0 10 -2 D 4 -2 -10 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.026316 B: 0.289473 C: 0.000000 D: 0.000000 E: 0.684211 Sum of squares = 0.552631629044 Cumulative probabilities = A: 0.026316 B: 0.315789 C: 0.315789 D: 0.315789 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 0 B 0 0 -4 2 0 C 8 4 0 10 -2 D 4 -2 -10 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.026316 B: 0.289473 C: 0.000000 D: 0.000000 E: 0.684211 Sum of squares = 0.552631629044 Cumulative probabilities = A: 0.026316 B: 0.315789 C: 0.315789 D: 0.315789 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 0 B 0 0 -4 2 0 C 8 4 0 10 -2 D 4 -2 -10 0 0 E 0 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.026316 B: 0.289473 C: 0.000000 D: 0.000000 E: 0.684211 Sum of squares = 0.552631629044 Cumulative probabilities = A: 0.026316 B: 0.315789 C: 0.315789 D: 0.315789 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4866: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) C E B A D (10) A D B C E (7) E C B A D (6) D A B C E (6) E C B D A (5) E D C A B (4) D A B E C (3) C B E A D (3) B D A C E (3) A D B E C (3) E D C B A (2) E C D A B (2) E C A D B (2) E C A B D (2) E A C B D (2) D E C B A (2) D A E B C (2) B C A D E (2) A D E B C (2) A B D C E (2) A B C D E (2) D E C A B (1) D E B A C (1) D E A C B (1) D B C A E (1) D B A C E (1) D A E C B (1) C E B D A (1) C E A B D (1) C B E D A (1) C B D E A (1) C B A E D (1) B C E D A (1) B C A E D (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -18 2 -16 B 10 0 -16 -2 -16 C 18 16 0 14 -4 D -2 2 -14 0 -14 E 16 16 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -18 2 -16 B 10 0 -16 -2 -16 C 18 16 0 14 -4 D -2 2 -14 0 -14 E 16 16 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=19 C=18 A=18 B=10 so B is eliminated. Round 2 votes counts: E=35 D=22 C=22 A=21 so A is eliminated. Round 3 votes counts: D=37 E=36 C=27 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:222 B:188 D:186 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -18 2 -16 B 10 0 -16 -2 -16 C 18 16 0 14 -4 D -2 2 -14 0 -14 E 16 16 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 2 -16 B 10 0 -16 -2 -16 C 18 16 0 14 -4 D -2 2 -14 0 -14 E 16 16 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 2 -16 B 10 0 -16 -2 -16 C 18 16 0 14 -4 D -2 2 -14 0 -14 E 16 16 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4867: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) C B D E A (6) B C E A D (6) E A D C B (5) C B E D A (5) B C D A E (5) B C A E D (5) D C E A B (4) D A C E B (4) A E D B C (4) A D E C B (4) D E C A B (3) D A E C B (3) D A B C E (3) B E A C D (3) B A E C D (3) A D E B C (3) E D A C B (2) E C B A D (2) D C B A E (2) B C A D E (2) A B E D C (2) E D C A B (1) E C A B D (1) E A C D B (1) E A B C D (1) D E A C B (1) D C A E B (1) D C A B E (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B D A (1) C D B E A (1) B A D C E (1) A E B C D (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 6 8 12 B -14 0 2 6 -2 C -6 -2 0 -12 -2 D -8 -6 12 0 -2 E -12 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 8 12 B -14 0 2 6 -2 C -6 -2 0 -12 -2 D -8 -6 12 0 -2 E -12 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 A=24 C=14 E=13 so E is eliminated. Round 2 votes counts: A=31 D=27 B=25 C=17 so C is eliminated. Round 3 votes counts: B=39 A=32 D=29 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:198 E:197 B:196 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 8 12 B -14 0 2 6 -2 C -6 -2 0 -12 -2 D -8 -6 12 0 -2 E -12 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 8 12 B -14 0 2 6 -2 C -6 -2 0 -12 -2 D -8 -6 12 0 -2 E -12 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 8 12 B -14 0 2 6 -2 C -6 -2 0 -12 -2 D -8 -6 12 0 -2 E -12 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4868: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (13) A D B E C (10) B E C A D (7) E C B A D (6) E C B D A (5) A D B C E (5) D A C E B (4) C E D B A (4) C D E A B (4) B E C D A (4) B A D E C (4) A D C E B (4) E B C A D (3) D A C B E (3) D A B C E (3) B E A C D (3) A D C B E (3) D A B E C (2) C E D A B (2) D B A C E (1) C E A D B (1) C D E B A (1) C D A E B (1) C A D E B (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E D C (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -4 2 -10 B 10 0 -6 2 0 C 4 6 0 14 0 D -2 -2 -14 0 -4 E 10 0 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.495690 D: 0.000000 E: 0.504310 Sum of squares = 0.500037145883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.495690 D: 0.495690 E: 1.000000 A B C D E A 0 -10 -4 2 -10 B 10 0 -6 2 0 C 4 6 0 14 0 D -2 -2 -14 0 -4 E 10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 B=22 E=14 D=13 so D is eliminated. Round 2 votes counts: A=36 C=27 B=23 E=14 so E is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:207 B:203 A:189 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 2 -10 B 10 0 -6 2 0 C 4 6 0 14 0 D -2 -2 -14 0 -4 E 10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 2 -10 B 10 0 -6 2 0 C 4 6 0 14 0 D -2 -2 -14 0 -4 E 10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 2 -10 B 10 0 -6 2 0 C 4 6 0 14 0 D -2 -2 -14 0 -4 E 10 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4869: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) A C B E D (8) D E B C A (7) D E B A C (6) D C A B E (4) C D A B E (4) C A B E D (4) B E C A D (4) A C E B D (4) E B D C A (3) E B D A C (3) E B A C D (3) D B E C A (3) C A D B E (3) B E C D A (3) A E B C D (3) E B A D C (2) D E A B C (2) D A C E B (2) C A B D E (2) B E A C D (2) A E D B C (2) A E B D C (2) A D C E B (2) A C D E B (2) A C D B E (2) D C E A B (1) D C A E B (1) D A E C B (1) C D B A E (1) C B E A D (1) C B A E D (1) B E D C A (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -4 -2 0 B -2 0 -6 -8 4 C 4 6 0 -4 4 D 2 8 4 0 6 E 0 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -2 0 B -2 0 -6 -8 4 C 4 6 0 -4 4 D 2 8 4 0 6 E 0 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=27 C=16 E=11 B=10 so B is eliminated. Round 2 votes counts: D=36 A=27 E=21 C=16 so C is eliminated. Round 3 votes counts: D=41 A=37 E=22 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:205 A:198 B:194 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -2 0 B -2 0 -6 -8 4 C 4 6 0 -4 4 D 2 8 4 0 6 E 0 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 0 B -2 0 -6 -8 4 C 4 6 0 -4 4 D 2 8 4 0 6 E 0 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 0 B -2 0 -6 -8 4 C 4 6 0 -4 4 D 2 8 4 0 6 E 0 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4870: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) D C A E B (5) B E C A D (5) C D A E B (4) B D C A E (4) B C D E A (4) A D C E B (4) E C D A B (3) E A C D B (3) D C A B E (3) C E D B A (3) B E A C D (3) B C E D A (3) B A D C E (3) A E D C B (3) E C B D A (2) E C A D B (2) E B A D C (2) E A D C B (2) C D B E A (2) C D A B E (2) C B D E A (2) B E C D A (2) B E A D C (2) B D A C E (2) B A E D C (2) A E D B C (2) A D E C B (2) A D C B E (2) A D B C E (2) E C B A D (1) E B C A D (1) E A D B C (1) E A C B D (1) E A B C D (1) D A C E B (1) C E B D A (1) C D E B A (1) B C D A E (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -18 -6 -10 B -8 0 -14 -10 -6 C 18 14 0 10 16 D 6 10 -10 0 8 E 10 6 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 -6 -10 B -8 0 -14 -10 -6 C 18 14 0 10 16 D 6 10 -10 0 8 E 10 6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=23 E=19 A=18 D=9 so D is eliminated. Round 2 votes counts: C=31 B=31 E=19 A=19 so E is eliminated. Round 3 votes counts: C=39 B=34 A=27 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:207 E:196 A:187 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 -6 -10 B -8 0 -14 -10 -6 C 18 14 0 10 16 D 6 10 -10 0 8 E 10 6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -6 -10 B -8 0 -14 -10 -6 C 18 14 0 10 16 D 6 10 -10 0 8 E 10 6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -6 -10 B -8 0 -14 -10 -6 C 18 14 0 10 16 D 6 10 -10 0 8 E 10 6 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4871: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) D A B E C (7) D E A B C (6) E A D C B (5) E C A B D (4) D A E B C (4) C E B A D (4) B A D C E (4) A D B E C (4) A B D C E (4) E D A C B (3) E C D B A (3) E A D B C (3) C E D B A (3) C B A E D (3) B D C A E (3) B C A E D (3) B C A D E (3) E D A B C (2) E C D A B (2) D E C B A (2) C B E D A (2) B D A C E (2) A B D E C (2) E D C A B (1) E C B A D (1) E C A D B (1) E A C B D (1) E A B C D (1) D B C A E (1) D B A C E (1) D A B C E (1) C E B D A (1) C B D A E (1) B C D A E (1) B A C D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 6 8 -6 B -6 0 10 2 6 C -6 -10 0 -12 -6 D -8 -2 12 0 -6 E 6 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 6 6 8 -6 B -6 0 10 2 6 C -6 -10 0 -12 -6 D -8 -2 12 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=22 C=22 B=17 A=12 so A is eliminated. Round 2 votes counts: E=27 D=26 B=25 C=22 so C is eliminated. Round 3 votes counts: B=39 E=35 D=26 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:207 B:206 E:206 D:198 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 8 -6 B -6 0 10 2 6 C -6 -10 0 -12 -6 D -8 -2 12 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 -6 B -6 0 10 2 6 C -6 -10 0 -12 -6 D -8 -2 12 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 -6 B -6 0 10 2 6 C -6 -10 0 -12 -6 D -8 -2 12 0 -6 E 6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4872: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) B D A C E (8) A E C D B (8) E C A D B (6) A E C B D (5) E A C D B (4) C E D B A (4) B A D E C (4) E C A B D (3) E A C B D (3) D C B E A (3) A E B D C (3) A D B E C (3) A B D E C (3) D B C A E (2) D B A C E (2) C D E B A (2) C D E A B (2) C B E D A (2) B D C A E (2) B A E D C (2) E C D A B (1) E C B D A (1) E B C A D (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C E A (1) D A C E B (1) C E D A B (1) C E B D A (1) C D B E A (1) C B D E A (1) B E C D A (1) B C E D A (1) B C D E A (1) B A E C D (1) A E D C B (1) A D E C B (1) A D C E B (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 4 2 4 B 0 0 -10 4 0 C -4 10 0 2 -4 D -2 -4 -2 0 0 E -4 0 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.803550 B: 0.196450 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.684284876959 Cumulative probabilities = A: 0.803550 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 2 4 B 0 0 -10 4 0 C -4 10 0 2 -4 D -2 -4 -2 0 0 E -4 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738715 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=27 E=19 C=14 D=12 so D is eliminated. Round 2 votes counts: B=33 A=28 C=20 E=19 so E is eliminated. Round 3 votes counts: A=35 B=34 C=31 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:205 C:202 E:200 B:197 D:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 2 4 B 0 0 -10 4 0 C -4 10 0 2 -4 D -2 -4 -2 0 0 E -4 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738715 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 2 4 B 0 0 -10 4 0 C -4 10 0 2 -4 D -2 -4 -2 0 0 E -4 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738715 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 2 4 B 0 0 -10 4 0 C -4 10 0 2 -4 D -2 -4 -2 0 0 E -4 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738715 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4873: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) A E B C D (8) B D C E A (7) A B E D C (6) A E C D B (5) D B C E A (4) C E D A B (4) C D E B A (4) C D E A B (4) D C E B A (3) D C B A E (3) B A E D C (3) E C A D B (2) E B A C D (2) E A C B D (2) C E D B A (2) B E D C A (2) B E C D A (2) B E A C D (2) B A D E C (2) A E C B D (2) A C E D B (2) A B E C D (2) A B D E C (2) E C B D A (1) E C B A D (1) E C A B D (1) E A C D B (1) E A B C D (1) D C E A B (1) D A B C E (1) C E A D B (1) C A E D B (1) C A D E B (1) B E D A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B A E C D (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -8 -2 -14 B 4 0 -2 2 0 C 8 2 0 4 0 D 2 -2 -4 0 -10 E 14 0 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.864037 D: 0.000000 E: 0.135963 Sum of squares = 0.765045675802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.864037 D: 0.864037 E: 1.000000 A B C D E A 0 -4 -8 -2 -14 B 4 0 -2 2 0 C 8 2 0 4 0 D 2 -2 -4 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=23 D=21 C=17 E=11 so E is eliminated. Round 2 votes counts: A=32 B=25 C=22 D=21 so D is eliminated. Round 3 votes counts: C=38 A=33 B=29 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:207 B:202 D:193 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -2 -14 B 4 0 -2 2 0 C 8 2 0 4 0 D 2 -2 -4 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -2 -14 B 4 0 -2 2 0 C 8 2 0 4 0 D 2 -2 -4 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -2 -14 B 4 0 -2 2 0 C 8 2 0 4 0 D 2 -2 -4 0 -10 E 14 0 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4874: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (8) B A D E C (7) B A E D C (6) C E D A B (5) A C E B D (5) C D E A B (4) D E C B A (3) D C E A B (3) D B C E A (3) C E A D B (3) B E A D C (3) B D A C E (3) A B C E D (3) E D C B A (2) E A B C D (2) D E B C A (2) D C B E A (2) D C B A E (2) D B A C E (2) B D A E C (2) B A E C D (2) A E B C D (2) E C D B A (1) E C D A B (1) E C A B D (1) E B D C A (1) E B A C D (1) E A C B D (1) D B A E C (1) C D A B E (1) C A D E B (1) C A B D E (1) B D E A C (1) B A D C E (1) A E C B D (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 12 6 6 B 6 0 6 12 6 C -12 -6 0 -6 0 D -6 -12 6 0 -2 E -6 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 6 6 B 6 0 6 12 6 C -12 -6 0 -6 0 D -6 -12 6 0 -2 E -6 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 A=24 C=15 E=10 so E is eliminated. Round 2 votes counts: D=28 B=27 A=27 C=18 so C is eliminated. Round 3 votes counts: D=40 A=33 B=27 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:209 E:195 D:193 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 6 6 B 6 0 6 12 6 C -12 -6 0 -6 0 D -6 -12 6 0 -2 E -6 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 6 6 B 6 0 6 12 6 C -12 -6 0 -6 0 D -6 -12 6 0 -2 E -6 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 6 6 B 6 0 6 12 6 C -12 -6 0 -6 0 D -6 -12 6 0 -2 E -6 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4875: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) D E C B A (5) A B C E D (5) A B C D E (5) E D C B A (4) A E B D C (4) A D C B E (4) E D A C B (3) E A B D C (3) D E C A B (3) D C E B A (3) D C E A B (3) D A C E B (3) C D A B E (3) B A C E D (3) A D E C B (3) A B E C D (3) E D C A B (2) E C B D A (2) E B A D C (2) C D B E A (2) B C A D E (2) A B E D C (2) E D B C A (1) E D A B C (1) E B D C A (1) E B C D A (1) E B A C D (1) E A D C B (1) E A D B C (1) D E A C B (1) D C A E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C D B A E (1) C B A D E (1) C A D B E (1) B E A C D (1) B C E A D (1) B C D E A (1) B C A E D (1) B A C D E (1) A E D C B (1) A E D B C (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 16 6 0 -2 B -16 0 -16 -2 -6 C -6 16 0 -10 6 D 0 2 10 0 6 E 2 6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.362721 B: 0.000000 C: 0.000000 D: 0.637279 E: 0.000000 Sum of squares = 0.537690929527 Cumulative probabilities = A: 0.362721 B: 0.362721 C: 0.362721 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 0 -2 B -16 0 -16 -2 -6 C -6 16 0 -10 6 D 0 2 10 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=23 D=20 C=16 B=10 so B is eliminated. Round 2 votes counts: A=35 E=24 C=21 D=20 so D is eliminated. Round 3 votes counts: A=39 E=33 C=28 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:210 D:209 C:203 E:198 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 0 -2 B -16 0 -16 -2 -6 C -6 16 0 -10 6 D 0 2 10 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 0 -2 B -16 0 -16 -2 -6 C -6 16 0 -10 6 D 0 2 10 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 0 -2 B -16 0 -16 -2 -6 C -6 16 0 -10 6 D 0 2 10 0 6 E 2 6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4876: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) D C B E A (7) E A C D B (6) B D C E A (6) D C B A E (5) D B C E A (5) B D C A E (5) E C D A B (4) B A D C E (4) A E C D B (4) A E C B D (4) D B C A E (3) C D A E B (3) B E A D C (3) B D E C A (3) B A E D C (3) A B E C D (3) E A C B D (2) E A B C D (2) D C E B A (2) C E A D B (2) B E D C A (2) B A D E C (2) A B C D E (2) E C A D B (1) E B A D C (1) E B A C D (1) C D E A B (1) C A D E B (1) B E D A C (1) B D A E C (1) B D A C E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 0 2 2 B 10 0 12 10 12 C 0 -12 0 -8 -2 D -2 -10 8 0 4 E -2 -12 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 2 2 B 10 0 12 10 12 C 0 -12 0 -8 -2 D -2 -10 8 0 4 E -2 -12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=23 D=22 E=17 C=7 so C is eliminated. Round 2 votes counts: B=31 D=26 A=24 E=19 so E is eliminated. Round 3 votes counts: A=37 B=33 D=30 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:200 A:197 E:192 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 2 2 B 10 0 12 10 12 C 0 -12 0 -8 -2 D -2 -10 8 0 4 E -2 -12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 2 2 B 10 0 12 10 12 C 0 -12 0 -8 -2 D -2 -10 8 0 4 E -2 -12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 2 2 B 10 0 12 10 12 C 0 -12 0 -8 -2 D -2 -10 8 0 4 E -2 -12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4877: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (15) D E B A C (13) C A E B D (13) C A B E D (11) D B E C A (6) E B A D C (5) D E B C A (5) C D A B E (4) D C E B A (2) C D A E B (2) C A B D E (2) A E B C D (2) A C E B D (2) A B E D C (2) E D B A C (1) E B D A C (1) E A B D C (1) E A B C D (1) D E C B A (1) D C B A E (1) D B C E A (1) C D E A B (1) C D B A E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A D C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 -10 -10 B 8 0 12 -12 -4 C 6 -12 0 -12 -10 D 10 12 12 0 12 E 10 4 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -10 -10 B 8 0 12 -12 -4 C 6 -12 0 -12 -10 D 10 12 12 0 12 E 10 4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 C=37 E=9 A=8 B=2 so B is eliminated. Round 2 votes counts: D=44 C=37 E=11 A=8 so A is eliminated. Round 3 votes counts: D=44 C=41 E=15 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:206 B:202 C:186 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 -10 B 8 0 12 -12 -4 C 6 -12 0 -12 -10 D 10 12 12 0 12 E 10 4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 -10 B 8 0 12 -12 -4 C 6 -12 0 -12 -10 D 10 12 12 0 12 E 10 4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 -10 B 8 0 12 -12 -4 C 6 -12 0 -12 -10 D 10 12 12 0 12 E 10 4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4878: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) E D A C B (7) C B A D E (7) A D E C B (6) C B E A D (5) C E A B D (4) A E D C B (4) A C E D B (4) E D B A C (3) D E A B C (3) D A E B C (3) C E B A D (3) C B A E D (3) B E D C A (3) A C D B E (3) E C A D B (2) E A C D B (2) D E B A C (2) D B A E C (2) D B A C E (2) C A B E D (2) B D E A C (2) B D C A E (2) B C E A D (2) B C D A E (2) B C A D E (2) A C D E B (2) E C B D A (1) E C B A D (1) E A D C B (1) D A E C B (1) C A E D B (1) B E C D A (1) B D C E A (1) B D A E C (1) B C A E D (1) Total count = 100 A B C D E A 0 -10 -4 10 -4 B 10 0 -18 4 0 C 4 18 0 14 10 D -10 -4 -14 0 -18 E 4 0 -10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 10 -4 B 10 0 -18 4 0 C 4 18 0 14 10 D -10 -4 -14 0 -18 E 4 0 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 A=19 E=17 D=13 so D is eliminated. Round 2 votes counts: B=30 C=25 A=23 E=22 so E is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:223 E:206 B:198 A:196 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 10 -4 B 10 0 -18 4 0 C 4 18 0 14 10 D -10 -4 -14 0 -18 E 4 0 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 10 -4 B 10 0 -18 4 0 C 4 18 0 14 10 D -10 -4 -14 0 -18 E 4 0 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 10 -4 B 10 0 -18 4 0 C 4 18 0 14 10 D -10 -4 -14 0 -18 E 4 0 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999308 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4879: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) C E A D B (9) D E A C B (4) B D A C E (4) B C E A D (4) B C A D E (4) B A C D E (4) E D A C B (3) E C D B A (3) E C D A B (3) E C A D B (3) B D E A C (3) B D A E C (3) A D B C E (3) D E A B C (2) D B A E C (2) D A B E C (2) C E B A D (2) C B A E D (2) C A B E D (2) B C A E D (2) B A D C E (2) E D C B A (1) E D B C A (1) E D A B C (1) E C B D A (1) E C B A D (1) E B D C A (1) E B C D A (1) D B A C E (1) D A E C B (1) D A E B C (1) C B E A D (1) C B A D E (1) C A E D B (1) C A E B D (1) C A B D E (1) B E D C A (1) B C E D A (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -16 -2 -14 B -8 0 -14 -14 -8 C 16 14 0 0 0 D 2 14 0 0 -14 E 14 8 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.311028 D: 0.000000 E: 0.688972 Sum of squares = 0.571420911012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.311028 D: 0.311028 E: 1.000000 A B C D E A 0 8 -16 -2 -14 B -8 0 -14 -14 -8 C 16 14 0 0 0 D 2 14 0 0 -14 E 14 8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=28 C=20 D=13 A=9 so A is eliminated. Round 2 votes counts: E=30 B=28 C=23 D=19 so D is eliminated. Round 3 votes counts: E=39 B=36 C=25 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:215 D:201 A:188 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 -2 -14 B -8 0 -14 -14 -8 C 16 14 0 0 0 D 2 14 0 0 -14 E 14 8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 -2 -14 B -8 0 -14 -14 -8 C 16 14 0 0 0 D 2 14 0 0 -14 E 14 8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 -2 -14 B -8 0 -14 -14 -8 C 16 14 0 0 0 D 2 14 0 0 -14 E 14 8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4880: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) D C B E A (9) C D B E A (6) A E D C B (6) A E B C D (6) C D E B A (5) B D C A E (5) A E B D C (5) E A B C D (4) D C B A E (3) D B C A E (3) C B D E A (3) B C D E A (3) A B E D C (3) E C D A B (2) E A C B D (2) D C E A B (2) C E D A B (2) B A E C D (2) B A D C E (2) A D B E C (2) E D A C B (1) E C A D B (1) E C A B D (1) E B A C D (1) D E C A B (1) D C A E B (1) C B E D A (1) B D A C E (1) B C E A D (1) B C D A E (1) B C A D E (1) B A E D C (1) B A C E D (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -2 2 -10 B -2 0 -14 -10 -2 C 2 14 0 6 0 D -2 10 -6 0 0 E 10 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.505277 D: 0.000000 E: 0.494723 Sum of squares = 0.500055689911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.505277 D: 0.505277 E: 1.000000 A B C D E A 0 2 -2 2 -10 B -2 0 -14 -10 -2 C 2 14 0 6 0 D -2 10 -6 0 0 E 10 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=22 D=19 B=18 C=17 so C is eliminated. Round 2 votes counts: D=30 E=24 A=24 B=22 so B is eliminated. Round 3 votes counts: D=43 A=31 E=26 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:211 E:206 D:201 A:196 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 2 -10 B -2 0 -14 -10 -2 C 2 14 0 6 0 D -2 10 -6 0 0 E 10 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 -10 B -2 0 -14 -10 -2 C 2 14 0 6 0 D -2 10 -6 0 0 E 10 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 -10 B -2 0 -14 -10 -2 C 2 14 0 6 0 D -2 10 -6 0 0 E 10 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4881: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) E B D A C (8) A C D E B (8) C A D B E (5) C A B E D (5) B E D C A (5) D A C B E (4) A D C E B (4) E B A D C (3) D B E C A (3) C D B A E (3) A C E B D (3) E B C A D (2) E B A C D (2) D E B A C (2) D C A B E (2) C D A B E (2) C B D A E (2) A E D B C (2) A E C B D (2) A D E B C (2) E B D C A (1) E B C D A (1) E A D B C (1) E A B D C (1) E A B C D (1) D E A B C (1) D B C E A (1) D A E C B (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D E A (1) C A E B D (1) B D E C A (1) B D C E A (1) B C E D A (1) B C D E A (1) A E B D C (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -2 B 4 0 4 14 2 C 4 -4 0 12 -2 D 6 -14 -12 0 -10 E 2 -2 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -2 B 4 0 4 14 2 C 4 -4 0 12 -2 D 6 -14 -12 0 -10 E 2 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999598 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 E=20 B=20 D=14 so D is eliminated. Round 2 votes counts: A=29 C=24 B=24 E=23 so E is eliminated. Round 3 votes counts: B=43 A=33 C=24 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:206 C:205 A:192 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -2 B 4 0 4 14 2 C 4 -4 0 12 -2 D 6 -14 -12 0 -10 E 2 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999598 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 4 14 2 C 4 -4 0 12 -2 D 6 -14 -12 0 -10 E 2 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999598 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 4 14 2 C 4 -4 0 12 -2 D 6 -14 -12 0 -10 E 2 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999598 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4882: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (11) D B A C E (8) E C B A D (7) A C E B D (7) E C A B D (6) E A C B D (4) E D B C A (3) E A C D B (3) D E B C A (3) D B A E C (3) D A B C E (3) B D C A E (3) E C A D B (2) D B E A C (2) D B C E A (2) D B C A E (2) D A C B E (2) B E C A D (2) B D A C E (2) B C A E D (2) A E C D B (2) A C B E D (2) E D C B A (1) E C D B A (1) E C D A B (1) E B C D A (1) D E C B A (1) D E A C B (1) D A E C B (1) C A E B D (1) B D E C A (1) B D C E A (1) B A C D E (1) A E D C B (1) A D C E B (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -2 -6 -6 B 14 0 6 -14 4 C 2 -6 0 -8 -14 D 6 14 8 0 8 E 6 -4 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -6 -6 B 14 0 6 -14 4 C 2 -6 0 -8 -14 D 6 14 8 0 8 E 6 -4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=29 A=19 B=12 C=1 so C is eliminated. Round 2 votes counts: D=39 E=29 A=20 B=12 so B is eliminated. Round 3 votes counts: D=46 E=31 A=23 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:205 E:204 C:187 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -2 -6 -6 B 14 0 6 -14 4 C 2 -6 0 -8 -14 D 6 14 8 0 8 E 6 -4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -6 -6 B 14 0 6 -14 4 C 2 -6 0 -8 -14 D 6 14 8 0 8 E 6 -4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -6 -6 B 14 0 6 -14 4 C 2 -6 0 -8 -14 D 6 14 8 0 8 E 6 -4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4883: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (21) A D E C B (11) B C A E D (10) E D C A B (8) A D E B C (5) B C E A D (4) D A E C B (3) C E D A B (3) C B E D A (3) B E C D A (3) B A D C E (3) A D C E B (3) C E B D A (2) A D B E C (2) A C D E B (2) A B C D E (2) E D C B A (1) E D B C A (1) E D A C B (1) E C B D A (1) E B D C A (1) D E A C B (1) D B A E C (1) D A E B C (1) C A D E B (1) C A B E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -22 -2 0 B 8 0 16 12 10 C 22 -16 0 14 18 D 2 -12 -14 0 -22 E 0 -10 -18 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -22 -2 0 B 8 0 16 12 10 C 22 -16 0 14 18 D 2 -12 -14 0 -22 E 0 -10 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 A=27 E=13 C=10 D=6 so D is eliminated. Round 2 votes counts: B=45 A=31 E=14 C=10 so C is eliminated. Round 3 votes counts: B=48 A=33 E=19 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:219 E:197 A:184 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -22 -2 0 B 8 0 16 12 10 C 22 -16 0 14 18 D 2 -12 -14 0 -22 E 0 -10 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -22 -2 0 B 8 0 16 12 10 C 22 -16 0 14 18 D 2 -12 -14 0 -22 E 0 -10 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -22 -2 0 B 8 0 16 12 10 C 22 -16 0 14 18 D 2 -12 -14 0 -22 E 0 -10 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4884: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) C A E B D (7) A B D E C (7) D B E C A (5) C D E B A (5) D C E B A (4) C E A B D (4) A C E B D (4) A B E D C (4) A B E C D (4) E B A C D (3) D A B C E (3) B A D E C (3) A B C E D (3) D B C E A (2) D A B E C (2) C E D B A (2) C E D A B (2) C D A B E (2) C A E D B (2) B E D A C (2) B E A D C (2) B A E D C (2) A E C B D (2) A D B E C (2) A C D B E (2) E C D B A (1) E C B A D (1) D E C B A (1) D C B E A (1) D B A E C (1) C E B A D (1) C D E A B (1) C A D E B (1) C A D B E (1) B D E A C (1) B A E C D (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 14 16 10 B -10 0 10 4 18 C -14 -10 0 0 -2 D -16 -4 0 0 4 E -10 -18 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 16 10 B -10 0 10 4 18 C -14 -10 0 0 -2 D -16 -4 0 0 4 E -10 -18 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=28 D=26 B=11 E=5 so E is eliminated. Round 2 votes counts: C=30 A=30 D=26 B=14 so B is eliminated. Round 3 votes counts: A=41 C=30 D=29 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 B:211 D:192 C:187 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 16 10 B -10 0 10 4 18 C -14 -10 0 0 -2 D -16 -4 0 0 4 E -10 -18 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 16 10 B -10 0 10 4 18 C -14 -10 0 0 -2 D -16 -4 0 0 4 E -10 -18 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 16 10 B -10 0 10 4 18 C -14 -10 0 0 -2 D -16 -4 0 0 4 E -10 -18 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4885: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) B E A C D (12) A C D B E (12) D C A E B (8) C D A B E (6) C A D B E (5) B E A D C (5) A C D E B (5) E B A D C (4) B E D C A (4) B E C D A (4) D A C E B (3) A D C E B (3) A C B D E (3) E D B C A (2) D C E B A (2) C D B E A (2) C D A E B (2) D E C B A (1) D C E A B (1) C B D E A (1) B A E C D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 0 4 0 B 0 0 -8 -4 14 C 0 8 0 10 8 D -4 4 -10 0 8 E 0 -14 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.542813 B: 0.000000 C: 0.457187 D: 0.000000 E: 0.000000 Sum of squares = 0.503665851983 Cumulative probabilities = A: 0.542813 B: 0.542813 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 0 B 0 0 -8 -4 14 C 0 8 0 10 8 D -4 4 -10 0 8 E 0 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=25 E=18 C=16 D=15 so D is eliminated. Round 2 votes counts: A=28 C=27 B=26 E=19 so E is eliminated. Round 3 votes counts: B=44 C=28 A=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:202 B:201 D:199 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 0 B 0 0 -8 -4 14 C 0 8 0 10 8 D -4 4 -10 0 8 E 0 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 0 B 0 0 -8 -4 14 C 0 8 0 10 8 D -4 4 -10 0 8 E 0 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 0 B 0 0 -8 -4 14 C 0 8 0 10 8 D -4 4 -10 0 8 E 0 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4886: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) C E A B D (9) B A D C E (9) C A B E D (8) E C A D B (7) E D C A B (6) B D A C E (6) D B E A C (5) D E B C A (4) D E B A C (4) D B A E C (4) C A E B D (4) A C B E D (4) A B C D E (4) B D A E C (3) E D C B A (2) C B A E D (2) E D A C B (1) E C D A B (1) C E A D B (1) B C A D E (1) A E C D B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 10 28 20 B 4 0 8 28 20 C -10 -8 0 10 24 D -28 -28 -10 0 6 E -20 -20 -24 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999253 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 28 20 B 4 0 8 28 20 C -10 -8 0 10 24 D -28 -28 -10 0 6 E -20 -20 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=24 E=17 D=17 A=11 so A is eliminated. Round 2 votes counts: B=37 C=28 E=18 D=17 so D is eliminated. Round 3 votes counts: B=46 C=28 E=26 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:230 A:227 C:208 D:170 E:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 28 20 B 4 0 8 28 20 C -10 -8 0 10 24 D -28 -28 -10 0 6 E -20 -20 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 28 20 B 4 0 8 28 20 C -10 -8 0 10 24 D -28 -28 -10 0 6 E -20 -20 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 28 20 B 4 0 8 28 20 C -10 -8 0 10 24 D -28 -28 -10 0 6 E -20 -20 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4887: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (8) A B E D C (6) D E C B A (5) C E D B A (5) C D E A B (5) D A E C B (4) C E B D A (4) B C E A D (4) D E A B C (3) C B E D A (3) B A E C D (3) A D C E B (3) A B C E D (3) E D C B A (2) E B D C A (2) D E C A B (2) D E B A C (2) D C E A B (2) D C A E B (2) C B E A D (2) B E A D C (2) B A E D C (2) B A C E D (2) A D E C B (2) A D C B E (2) A C B D E (2) E C D B A (1) E B C D A (1) D C E B A (1) D A E B C (1) D A C E B (1) C E A D B (1) C D E B A (1) C B A E D (1) C A B E D (1) B E D A C (1) B E C A D (1) B E A C D (1) A D B E C (1) A D B C E (1) A C D E B (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 8 4 -2 B -8 0 -8 2 -2 C -8 8 0 -12 0 D -4 -2 12 0 4 E 2 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 A B C D E A 0 8 8 4 -2 B -8 0 -8 2 -2 C -8 8 0 -12 0 D -4 -2 12 0 4 E 2 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 C=23 B=16 E=6 so E is eliminated. Round 2 votes counts: A=32 D=25 C=24 B=19 so B is eliminated. Round 3 votes counts: A=42 C=30 D=28 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 D:205 E:200 C:194 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 4 -2 B -8 0 -8 2 -2 C -8 8 0 -12 0 D -4 -2 12 0 4 E 2 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 4 -2 B -8 0 -8 2 -2 C -8 8 0 -12 0 D -4 -2 12 0 4 E 2 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 4 -2 B -8 0 -8 2 -2 C -8 8 0 -12 0 D -4 -2 12 0 4 E 2 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4888: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) D B A C E (9) B D E A C (8) E B C A D (5) D A B E C (5) D A B C E (5) B E C D A (5) E C B A D (4) A E C D B (4) E A C B D (3) C E A D B (3) C E A B D (3) B D E C A (3) A D C E B (3) D B C A E (2) D B A E C (2) D A C B E (2) C E B A D (2) B E C A D (2) B D A E C (2) A D E C B (2) A C E D B (2) A C D E B (2) E C A B D (1) E A C D B (1) D A E C B (1) D A E B C (1) C B E A D (1) B E D C A (1) B E D A C (1) B D C E A (1) B C E D A (1) B C E A D (1) A E D C B (1) Total count = 100 A B C D E A 0 0 8 2 8 B 0 0 8 -12 2 C -8 -8 0 2 -4 D -2 12 -2 0 -4 E -8 -2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.900429 B: 0.099571 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.820686447048 Cumulative probabilities = A: 0.900429 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 8 B 0 0 8 -12 2 C -8 -8 0 2 -4 D -2 12 -2 0 -4 E -8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.755102043729 Cumulative probabilities = A: 0.857143 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 C=20 E=14 A=14 so E is eliminated. Round 2 votes counts: B=30 D=27 C=25 A=18 so A is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:209 D:202 B:199 E:199 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 8 B 0 0 8 -12 2 C -8 -8 0 2 -4 D -2 12 -2 0 -4 E -8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.755102043729 Cumulative probabilities = A: 0.857143 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 8 B 0 0 8 -12 2 C -8 -8 0 2 -4 D -2 12 -2 0 -4 E -8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.755102043729 Cumulative probabilities = A: 0.857143 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 8 B 0 0 8 -12 2 C -8 -8 0 2 -4 D -2 12 -2 0 -4 E -8 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.755102043729 Cumulative probabilities = A: 0.857143 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4889: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) D A E B C (5) A D B E C (5) D C A E B (4) C E B D A (4) C E B A D (4) B C A E D (4) A B D E C (4) E B C A D (3) E B A C D (3) D A B C E (3) C B A D E (3) B A E C D (3) E D C B A (2) D A E C B (2) D A C B E (2) D A B E C (2) C E D B A (2) C D E B A (2) C B E D A (2) C B D A E (2) B E A C D (2) B C E A D (2) B C A D E (2) A B D C E (2) A B C D E (2) E D C A B (1) E D B A C (1) E D A B C (1) E C D B A (1) E C B D A (1) E C B A D (1) E B A D C (1) E A B D C (1) D E C B A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C E B A (1) D C A B E (1) C D B A E (1) B E C A D (1) A E B D C (1) A D E B C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -10 12 2 B 16 0 4 16 8 C 10 -4 0 10 6 D -12 -16 -10 0 -2 E -2 -8 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 12 2 B 16 0 4 16 8 C 10 -4 0 10 6 D -12 -16 -10 0 -2 E -2 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 A=17 E=16 B=14 so B is eliminated. Round 2 votes counts: C=37 D=24 A=20 E=19 so E is eliminated. Round 3 votes counts: C=44 D=29 A=27 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:222 C:211 A:194 E:193 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 12 2 B 16 0 4 16 8 C 10 -4 0 10 6 D -12 -16 -10 0 -2 E -2 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 12 2 B 16 0 4 16 8 C 10 -4 0 10 6 D -12 -16 -10 0 -2 E -2 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 12 2 B 16 0 4 16 8 C 10 -4 0 10 6 D -12 -16 -10 0 -2 E -2 -8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4890: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) C D B A E (12) B A E D C (5) A E B C D (5) E A C B D (4) C E D A B (4) E B A D C (3) E A C D B (3) D B C E A (3) C D B E A (3) B D E A C (3) A C E B D (3) E D B A C (2) D B E A C (2) D B C A E (2) C D A B E (2) C B D A E (2) C A B D E (2) B D A C E (2) B A C D E (2) A E B D C (2) A B C D E (2) E D C A B (1) E D B C A (1) E A D B C (1) E A B C D (1) D E B C A (1) D C E B A (1) D C B E A (1) D C B A E (1) C D E B A (1) C D E A B (1) C D A E B (1) B E D A C (1) B E A D C (1) B D E C A (1) B D C A E (1) B C A D E (1) B A D E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 16 2 -4 B 6 0 16 14 4 C -16 -16 0 0 -6 D -2 -14 0 0 -4 E 4 -4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 16 2 -4 B 6 0 16 14 4 C -16 -16 0 0 -6 D -2 -14 0 0 -4 E 4 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=28 B=18 A=14 D=11 so D is eliminated. Round 2 votes counts: C=31 E=30 B=25 A=14 so A is eliminated. Round 3 votes counts: E=37 C=34 B=29 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:205 A:204 D:190 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 16 2 -4 B 6 0 16 14 4 C -16 -16 0 0 -6 D -2 -14 0 0 -4 E 4 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 2 -4 B 6 0 16 14 4 C -16 -16 0 0 -6 D -2 -14 0 0 -4 E 4 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 2 -4 B 6 0 16 14 4 C -16 -16 0 0 -6 D -2 -14 0 0 -4 E 4 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4891: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) D B A C E (6) A D E C B (6) E C A D B (5) E A C D B (5) B C E D A (5) B C D E A (5) A E C D B (5) C E A B D (4) B D C A E (4) E C B A D (3) D B A E C (3) D A B C E (3) C E A D B (3) C A E D B (3) B D A C E (3) A C E D B (3) D A E B C (2) D A B E C (2) C E B A D (2) B D E C A (2) B D C E A (2) B D A E C (2) A E D C B (2) E B C A D (1) E A D C B (1) C D A B E (1) C B E A D (1) C B D A E (1) C A D B E (1) B E D A C (1) B E C D A (1) B E C A D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 -8 14 -2 B -14 0 -14 -6 -12 C 8 14 0 20 -2 D -14 6 -20 0 -10 E 2 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -8 14 -2 B -14 0 -14 -6 -12 C 8 14 0 20 -2 D -14 6 -20 0 -10 E 2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=24 A=18 D=16 C=16 so D is eliminated. Round 2 votes counts: B=35 A=25 E=24 C=16 so C is eliminated. Round 3 votes counts: B=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:220 E:213 A:209 D:181 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -8 14 -2 B -14 0 -14 -6 -12 C 8 14 0 20 -2 D -14 6 -20 0 -10 E 2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 14 -2 B -14 0 -14 -6 -12 C 8 14 0 20 -2 D -14 6 -20 0 -10 E 2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 14 -2 B -14 0 -14 -6 -12 C 8 14 0 20 -2 D -14 6 -20 0 -10 E 2 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4892: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) A D C E B (6) D A B C E (5) B E A C D (5) B D E A C (5) A D C B E (5) E C B D A (4) D C A E B (4) D B A C E (4) C A D E B (4) B E D C A (4) A C E D B (4) B E C A D (3) A E C B D (3) E C B A D (2) E B C D A (2) E B C A D (2) D B E C A (2) D A C E B (2) C E A D B (2) C D A E B (2) C A E D B (2) B D A E C (2) B A E C D (2) A C E B D (2) E C D B A (1) E C A B D (1) D E B C A (1) D C E A B (1) C E D B A (1) C E D A B (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E C A (1) B A E D C (1) B A D C E (1) A D B C E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 18 12 18 B -8 0 -8 -10 -8 C -18 8 0 4 8 D -12 10 -4 0 8 E -18 8 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 12 18 B -8 0 -8 -10 -8 C -18 8 0 4 8 D -12 10 -4 0 8 E -18 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=27 D=19 E=12 C=12 so E is eliminated. Round 2 votes counts: B=31 A=30 C=20 D=19 so D is eliminated. Round 3 votes counts: B=38 A=37 C=25 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:201 D:201 E:187 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 12 18 B -8 0 -8 -10 -8 C -18 8 0 4 8 D -12 10 -4 0 8 E -18 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 12 18 B -8 0 -8 -10 -8 C -18 8 0 4 8 D -12 10 -4 0 8 E -18 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 12 18 B -8 0 -8 -10 -8 C -18 8 0 4 8 D -12 10 -4 0 8 E -18 8 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4893: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) B C E A D (7) A B D C E (7) A D E C B (6) A B C D E (6) E D C B A (5) E C B D A (5) D E C B A (5) D E C A B (5) B C A E D (5) D A E C B (4) C E B D A (4) C B E D A (4) E C D B A (3) D A E B C (3) B A C E D (3) A D E B C (3) A B C E D (3) D E A C B (2) A D C B E (2) A D B C E (2) A C E B D (2) D E B C A (1) C E B A D (1) C B E A D (1) C A E D B (1) B E C D A (1) B A C D E (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -10 2 -2 B 6 0 -2 14 -2 C 10 2 0 8 14 D -2 -14 -8 0 -4 E 2 2 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 2 -2 B 6 0 -2 14 -2 C 10 2 0 8 14 D -2 -14 -8 0 -4 E 2 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=24 D=20 E=13 C=11 so C is eliminated. Round 2 votes counts: A=33 B=29 D=20 E=18 so E is eliminated. Round 3 votes counts: B=39 A=33 D=28 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:217 B:208 E:197 A:192 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 2 -2 B 6 0 -2 14 -2 C 10 2 0 8 14 D -2 -14 -8 0 -4 E 2 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 2 -2 B 6 0 -2 14 -2 C 10 2 0 8 14 D -2 -14 -8 0 -4 E 2 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 2 -2 B 6 0 -2 14 -2 C 10 2 0 8 14 D -2 -14 -8 0 -4 E 2 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4894: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) D A E B C (7) C B E D A (6) B C E A D (6) B A C E D (6) D A E C B (4) C B D E A (4) B C A E D (4) E A B C D (3) D C E A B (3) A E D B C (3) A D B E C (3) E D A C B (2) E C A B D (2) D E C A B (2) D B A C E (2) D A B E C (2) C E D B A (2) C E B D A (2) C B E A D (2) B C D E A (2) B A D E C (2) A B E D C (2) A B D E C (2) E C D A B (1) E C A D B (1) E B C A D (1) E A C D B (1) D C B A E (1) D C A E B (1) D C A B E (1) D B A E C (1) D A C E B (1) C E D A B (1) C D E B A (1) C D B E A (1) C B D A E (1) B E C A D (1) B D C A E (1) B C D A E (1) B A E D C (1) B A E C D (1) B A D C E (1) Total count = 100 A B C D E A 0 0 4 0 12 B 0 0 20 2 8 C -4 -20 0 0 0 D 0 -2 0 0 4 E -12 -8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500482 B: 0.499518 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000462703 Cumulative probabilities = A: 0.500482 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 0 12 B 0 0 20 2 8 C -4 -20 0 0 0 D 0 -2 0 0 4 E -12 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999808 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 C=20 A=18 E=11 so E is eliminated. Round 2 votes counts: D=27 B=27 C=24 A=22 so A is eliminated. Round 3 votes counts: D=41 B=34 C=25 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:208 D:201 C:188 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 12 B 0 0 20 2 8 C -4 -20 0 0 0 D 0 -2 0 0 4 E -12 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999808 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 12 B 0 0 20 2 8 C -4 -20 0 0 0 D 0 -2 0 0 4 E -12 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999808 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 12 B 0 0 20 2 8 C -4 -20 0 0 0 D 0 -2 0 0 4 E -12 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999808 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4895: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) D B A C E (9) E A C B D (7) B D E C A (7) D B C A E (6) D B A E C (6) D A C B E (5) C A E D B (5) B E C A D (4) B D E A C (4) A C E D B (4) D A C E B (3) C E A B D (3) B E C D A (3) B E D C A (2) B D C E A (2) A D C E B (2) E C B A D (1) E B C A D (1) D C B A E (1) D C A E B (1) D A E C B (1) D A B E C (1) D A B C E (1) C D A E B (1) C A E B D (1) B E D A C (1) B C E A D (1) B C D E A (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -8 -10 -2 B -2 0 -2 4 8 C 8 2 0 -4 -6 D 10 -4 4 0 4 E 2 -8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -10 -2 B -2 0 -2 4 8 C 8 2 0 -4 -6 D 10 -4 4 0 4 E 2 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=25 E=23 C=10 A=8 so A is eliminated. Round 2 votes counts: D=36 B=25 E=24 C=15 so C is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:207 B:204 C:200 E:198 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -8 -10 -2 B -2 0 -2 4 8 C 8 2 0 -4 -6 D 10 -4 4 0 4 E 2 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -10 -2 B -2 0 -2 4 8 C 8 2 0 -4 -6 D 10 -4 4 0 4 E 2 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -10 -2 B -2 0 -2 4 8 C 8 2 0 -4 -6 D 10 -4 4 0 4 E 2 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4896: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (17) B C D E A (12) B C A E D (11) C B A E D (6) A E D C B (6) C D E A B (5) B D E A C (4) B D C E A (4) E A D C B (3) D E A B C (3) C B D E A (3) B A E D C (3) A E D B C (3) E A D B C (2) D E B A C (2) C B A D E (2) C A E D B (2) B C D A E (2) B C A D E (2) E D A C B (1) E D A B C (1) D C E B A (1) D C E A B (1) D B E A C (1) C D B A E (1) B D E C A (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -6 -18 -22 B 10 0 2 2 4 C 6 -2 0 -6 6 D 18 -2 6 0 22 E 22 -4 -6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -18 -22 B 10 0 2 2 4 C 6 -2 0 -6 6 D 18 -2 6 0 22 E 22 -4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=25 C=19 A=10 E=7 so E is eliminated. Round 2 votes counts: B=39 D=27 C=19 A=15 so A is eliminated. Round 3 votes counts: D=41 B=39 C=20 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:209 C:202 E:195 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -18 -22 B 10 0 2 2 4 C 6 -2 0 -6 6 D 18 -2 6 0 22 E 22 -4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -18 -22 B 10 0 2 2 4 C 6 -2 0 -6 6 D 18 -2 6 0 22 E 22 -4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -18 -22 B 10 0 2 2 4 C 6 -2 0 -6 6 D 18 -2 6 0 22 E 22 -4 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4897: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) D B C E A (6) D B A E C (6) C E D A B (6) E C A B D (5) D C E B A (5) B D A E C (5) B A E C D (5) A E C B D (5) A B E C D (5) D B A C E (4) C E A D B (4) C E A B D (4) D B C A E (3) C E D B A (3) E A C B D (2) D C B E A (2) D C A E B (2) C D E A B (2) B E A C D (2) B A D E C (2) A E B C D (2) E B A C D (1) D B E C A (1) D A C E B (1) D A B C E (1) B E C D A (1) B D E C A (1) B D E A C (1) B A E D C (1) A C E D B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -8 -18 -10 B -2 0 -4 -12 -6 C 8 4 0 -2 8 D 18 12 2 0 2 E 10 6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -18 -10 B -2 0 -4 -12 -6 C 8 4 0 -2 8 D 18 12 2 0 2 E 10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 C=19 B=18 A=15 E=8 so E is eliminated. Round 2 votes counts: D=40 C=24 B=19 A=17 so A is eliminated. Round 3 votes counts: D=40 C=33 B=27 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:209 E:203 B:188 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -18 -10 B -2 0 -4 -12 -6 C 8 4 0 -2 8 D 18 12 2 0 2 E 10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -18 -10 B -2 0 -4 -12 -6 C 8 4 0 -2 8 D 18 12 2 0 2 E 10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -18 -10 B -2 0 -4 -12 -6 C 8 4 0 -2 8 D 18 12 2 0 2 E 10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4898: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (11) C A E D B (6) D B A E C (5) C E A D B (5) B C D A E (5) E D A C B (4) B E D A C (4) E C A D B (3) E A C D B (3) C A D B E (3) B D A E C (3) A E C D B (3) D E B A C (2) D B E A C (2) C E A B D (2) C B A E D (2) C B A D E (2) C A D E B (2) B E D C A (2) B E C D A (2) B D E C A (2) B D C A E (2) B C E A D (2) B C A D E (2) A E D C B (2) A D C E B (2) A C E D B (2) E D B A C (1) E D A B C (1) E A D C B (1) D E A B C (1) D B A C E (1) D A E C B (1) D A E B C (1) D A B E C (1) D A B C E (1) C A E B D (1) C A B E D (1) C A B D E (1) B D A C E (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 10 -6 4 B 2 0 4 -10 10 C -10 -4 0 -4 -12 D 6 10 4 0 6 E -4 -10 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -6 4 B 2 0 4 -10 10 C -10 -4 0 -4 -12 D 6 10 4 0 6 E -4 -10 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=25 D=15 E=13 A=11 so A is eliminated. Round 2 votes counts: B=36 C=28 E=18 D=18 so E is eliminated. Round 3 votes counts: C=37 B=36 D=27 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:213 A:203 B:203 E:196 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -6 4 B 2 0 4 -10 10 C -10 -4 0 -4 -12 D 6 10 4 0 6 E -4 -10 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -6 4 B 2 0 4 -10 10 C -10 -4 0 -4 -12 D 6 10 4 0 6 E -4 -10 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -6 4 B 2 0 4 -10 10 C -10 -4 0 -4 -12 D 6 10 4 0 6 E -4 -10 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4899: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E B D C (8) C D A E B (7) B C D E A (7) D C A E B (6) C D B A E (5) D A E C B (4) C B D E A (4) A E D B C (4) A D E C B (4) D C E B A (3) C B D A E (3) B C E D A (3) B C E A D (3) A E D C B (3) A B E C D (3) D E A C B (2) D A C E B (2) C D A B E (2) B E D A C (2) B E A C D (2) E B D C A (1) E A B D C (1) D E C A B (1) D E B C A (1) D E A B C (1) D C E A B (1) C D E B A (1) C A D B E (1) B E D C A (1) B C D A E (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -24 -38 10 B -2 0 -20 -18 0 C 24 20 0 8 22 D 38 18 -8 0 34 E -10 0 -22 -34 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -24 -38 10 B -2 0 -20 -18 0 C 24 20 0 8 22 D 38 18 -8 0 34 E -10 0 -22 -34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=23 D=21 B=20 E=2 so E is eliminated. Round 2 votes counts: C=34 A=24 D=21 B=21 so D is eliminated. Round 3 votes counts: C=45 A=33 B=22 so B is eliminated. Round 4 votes counts: C=62 A=38 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:241 C:237 B:180 A:175 E:167 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -24 -38 10 B -2 0 -20 -18 0 C 24 20 0 8 22 D 38 18 -8 0 34 E -10 0 -22 -34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -24 -38 10 B -2 0 -20 -18 0 C 24 20 0 8 22 D 38 18 -8 0 34 E -10 0 -22 -34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -24 -38 10 B -2 0 -20 -18 0 C 24 20 0 8 22 D 38 18 -8 0 34 E -10 0 -22 -34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4900: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) C D A B E (11) C D E A B (10) E B A C D (9) B A E D C (9) B A D C E (9) E C D A B (5) A B D C E (4) E B A D C (3) C E D A B (3) C D E B A (3) B E A D C (3) E C D B A (2) E C A B D (2) D C B A E (2) B A D E C (2) E D B C A (1) E C B D A (1) E C A D B (1) E A B C D (1) D C E B A (1) D B A C E (1) D A B C E (1) C E D B A (1) C D B A E (1) C D A E B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -12 -10 8 B -4 0 -10 -10 12 C 12 10 0 4 20 D 10 10 -4 0 16 E -8 -12 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 -10 8 B -4 0 -10 -10 12 C 12 10 0 4 20 D 10 10 -4 0 16 E -8 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=25 B=23 D=16 A=6 so A is eliminated. Round 2 votes counts: C=30 B=29 E=25 D=16 so D is eliminated. Round 3 votes counts: C=44 B=31 E=25 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:216 A:195 B:194 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -10 8 B -4 0 -10 -10 12 C 12 10 0 4 20 D 10 10 -4 0 16 E -8 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -10 8 B -4 0 -10 -10 12 C 12 10 0 4 20 D 10 10 -4 0 16 E -8 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -10 8 B -4 0 -10 -10 12 C 12 10 0 4 20 D 10 10 -4 0 16 E -8 -12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4901: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) A C B D E (8) E A C D B (5) A E C B D (5) E D C A B (4) E A B C D (4) B D E C A (4) E C D A B (3) E B D C A (3) E A B D C (3) D C B A E (3) E D C B A (2) E C A D B (2) D E B C A (2) D B C E A (2) C D B A E (2) C A D B E (2) B E D C A (2) B E D A C (2) B D C A E (2) B A D C E (2) A E B C D (2) A C E D B (2) A C B E D (2) E D B A C (1) E B D A C (1) E B A D C (1) D E C B A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) C E D A B (1) C E A D B (1) C D E A B (1) C A B D E (1) B D A E C (1) B D A C E (1) B C A D E (1) B A E D C (1) B A C D E (1) A C E B D (1) A C D E B (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -6 -4 -16 B -2 0 2 2 -14 C 6 -2 0 -4 -22 D 4 -2 4 0 -20 E 16 14 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -6 -4 -16 B -2 0 2 2 -14 C 6 -2 0 -4 -22 D 4 -2 4 0 -20 E 16 14 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=24 B=17 D=12 C=8 so C is eliminated. Round 2 votes counts: E=41 A=27 B=17 D=15 so D is eliminated. Round 3 votes counts: E=46 B=27 A=27 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:236 B:194 D:193 C:189 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -16 B -2 0 2 2 -14 C 6 -2 0 -4 -22 D 4 -2 4 0 -20 E 16 14 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -16 B -2 0 2 2 -14 C 6 -2 0 -4 -22 D 4 -2 4 0 -20 E 16 14 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -16 B -2 0 2 2 -14 C 6 -2 0 -4 -22 D 4 -2 4 0 -20 E 16 14 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4902: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) D E C A B (9) D E A C B (8) B C A E D (6) B A C E D (6) C B A D E (5) B C A D E (5) E D A B C (4) D E C B A (4) D E B C A (4) C B D A E (4) E D B A C (3) E A D B C (3) D C B E A (3) B C D A E (3) E A D C B (2) D E A B C (2) B C D E A (2) A B C E D (2) E B D A C (1) D C E B A (1) D B C E A (1) C D B A E (1) C D A B E (1) C B D E A (1) C A B D E (1) B E D A C (1) B D E C A (1) A E D B C (1) A E C D B (1) A E B D C (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -4 -28 -20 B 4 0 -8 -18 -12 C 4 8 0 -18 -10 D 28 18 18 0 12 E 20 12 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -28 -20 B 4 0 -8 -18 -12 C 4 8 0 -18 -10 D 28 18 18 0 12 E 20 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=24 E=23 C=13 A=8 so A is eliminated. Round 2 votes counts: D=32 E=26 B=26 C=16 so C is eliminated. Round 3 votes counts: B=38 D=34 E=28 so E is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:238 E:215 C:192 B:183 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -28 -20 B 4 0 -8 -18 -12 C 4 8 0 -18 -10 D 28 18 18 0 12 E 20 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -28 -20 B 4 0 -8 -18 -12 C 4 8 0 -18 -10 D 28 18 18 0 12 E 20 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -28 -20 B 4 0 -8 -18 -12 C 4 8 0 -18 -10 D 28 18 18 0 12 E 20 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (26) D A B C E (16) E D A C B (6) D A E B C (6) B C A D E (6) E A D C B (5) E A C B D (5) D E A B C (5) C B A E D (5) D B C A E (4) C B E A D (4) E C B D A (2) D B A C E (2) B D C A E (2) E C A B D (1) E B C D A (1) B C D A E (1) B C A E D (1) A E D C B (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -6 10 -10 B 8 0 -10 8 -16 C 6 10 0 4 -16 D -10 -8 -4 0 -14 E 10 16 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 10 -10 B 8 0 -10 8 -16 C 6 10 0 4 -16 D -10 -8 -4 0 -14 E 10 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=46 D=33 B=10 C=9 A=2 so A is eliminated. Round 2 votes counts: E=47 D=34 B=10 C=9 so C is eliminated. Round 3 votes counts: E=47 D=34 B=19 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:202 B:195 A:193 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 10 -10 B 8 0 -10 8 -16 C 6 10 0 4 -16 D -10 -8 -4 0 -14 E 10 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 10 -10 B 8 0 -10 8 -16 C 6 10 0 4 -16 D -10 -8 -4 0 -14 E 10 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 10 -10 B 8 0 -10 8 -16 C 6 10 0 4 -16 D -10 -8 -4 0 -14 E 10 16 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4904: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) C B E A D (8) E D A C B (7) B C A E D (7) A B C D E (7) D E A C B (6) D A E B C (6) C B A E D (6) A D B C E (6) E D C A B (5) D E A B C (5) B C A D E (5) A D E B C (5) A D B E C (3) D A B E C (2) C E B D A (2) C B E D A (2) B A C D E (2) A B D C E (2) E C D B A (1) E C B D A (1) D B A E C (1) D A B C E (1) C E B A D (1) A C B D E (1) Total count = 100 A B C D E A 0 12 8 6 8 B -12 0 4 -12 6 C -8 -4 0 -14 0 D -6 12 14 0 4 E -8 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 6 8 B -12 0 4 -12 6 C -8 -4 0 -14 0 D -6 12 14 0 4 E -8 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=22 D=21 C=19 B=14 so B is eliminated. Round 2 votes counts: C=31 A=26 E=22 D=21 so D is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:212 B:193 E:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 6 8 B -12 0 4 -12 6 C -8 -4 0 -14 0 D -6 12 14 0 4 E -8 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 6 8 B -12 0 4 -12 6 C -8 -4 0 -14 0 D -6 12 14 0 4 E -8 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 6 8 B -12 0 4 -12 6 C -8 -4 0 -14 0 D -6 12 14 0 4 E -8 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4905: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (14) B C D A E (11) E A D C B (9) B C D E A (8) E D A C B (7) D E C B A (5) C B D E A (5) A E D C B (4) A E D B C (4) A E B C D (4) D C E B A (3) B C A D E (3) B A C D E (3) E D C B A (2) E D C A B (2) D C B A E (2) C D B E A (2) A E B D C (2) A B C E D (2) A B C D E (2) D E A C B (1) C E B D A (1) C B E D A (1) B C A E D (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -22 -20 -28 -20 B 22 0 -16 -12 12 C 20 16 0 -12 18 D 28 12 12 0 20 E 20 -12 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -20 -28 -20 B 22 0 -16 -12 12 C 20 16 0 -12 18 D 28 12 12 0 20 E 20 -12 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 E=20 A=20 C=9 so C is eliminated. Round 2 votes counts: B=32 D=27 E=21 A=20 so A is eliminated. Round 3 votes counts: B=37 E=35 D=28 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:236 C:221 B:203 E:185 A:155 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -20 -28 -20 B 22 0 -16 -12 12 C 20 16 0 -12 18 D 28 12 12 0 20 E 20 -12 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -20 -28 -20 B 22 0 -16 -12 12 C 20 16 0 -12 18 D 28 12 12 0 20 E 20 -12 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -20 -28 -20 B 22 0 -16 -12 12 C 20 16 0 -12 18 D 28 12 12 0 20 E 20 -12 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4906: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D A E B C (9) D A C B E (7) B C E D A (7) E B C D A (6) A D E C B (5) A D C B E (5) E B D C A (4) B E C D A (4) A C B E D (4) E C A B D (3) D E A B C (3) C E B A D (3) A D C E B (3) E D B C A (2) E C B A D (2) E A C B D (2) D E B A C (2) D B C E A (2) D B C A E (2) D A E C B (2) D A B E C (2) C B E A D (2) E D A B C (1) D E B C A (1) D B E C A (1) D A C E B (1) C E A B D (1) C B A E D (1) C A B D E (1) B D E C A (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -4 -14 -12 B -2 0 12 4 -20 C 4 -12 0 -6 -18 D 14 -4 6 0 -4 E 12 20 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -4 -14 -12 B -2 0 12 4 -20 C 4 -12 0 -6 -18 D 14 -4 6 0 -4 E 12 20 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=29 A=19 B=12 C=8 so C is eliminated. Round 2 votes counts: E=33 D=32 A=20 B=15 so B is eliminated. Round 3 votes counts: E=46 D=33 A=21 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:206 B:197 A:186 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 -14 -12 B -2 0 12 4 -20 C 4 -12 0 -6 -18 D 14 -4 6 0 -4 E 12 20 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -14 -12 B -2 0 12 4 -20 C 4 -12 0 -6 -18 D 14 -4 6 0 -4 E 12 20 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -14 -12 B -2 0 12 4 -20 C 4 -12 0 -6 -18 D 14 -4 6 0 -4 E 12 20 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999609 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4907: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) A C B D E (10) E D A B C (8) C B A E D (8) C B E D A (7) B C A D E (6) E D B C A (5) E D A C B (4) D E B C A (3) D E A B C (3) D B E C A (3) B C D E A (3) A D E B C (3) A C B E D (3) A B C D E (3) E C B A D (2) E A D C B (2) D B C E A (2) A E D C B (2) A D E C B (2) A D B C E (2) E D C B A (1) E A D B C (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D A E (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -10 14 8 B 10 0 -8 16 24 C 10 8 0 14 20 D -14 -16 -14 0 8 E -8 -24 -20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 14 8 B 10 0 -8 16 24 C 10 8 0 14 20 D -14 -16 -14 0 8 E -8 -24 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 E=23 B=13 D=11 so D is eliminated. Round 2 votes counts: E=29 A=27 C=26 B=18 so B is eliminated. Round 3 votes counts: C=41 E=32 A=27 so A is eliminated. Round 4 votes counts: C=60 E=40 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:221 A:201 D:182 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 14 8 B 10 0 -8 16 24 C 10 8 0 14 20 D -14 -16 -14 0 8 E -8 -24 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 14 8 B 10 0 -8 16 24 C 10 8 0 14 20 D -14 -16 -14 0 8 E -8 -24 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 14 8 B 10 0 -8 16 24 C 10 8 0 14 20 D -14 -16 -14 0 8 E -8 -24 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4908: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) B C E D A (10) B C A D E (8) B C E A D (5) A D B C E (5) E D C A B (4) E C D A B (4) D E A C B (4) B C A E D (4) A D E C B (4) A D B E C (4) C E B D A (3) C B E A D (3) A C B D E (3) D E A B C (2) D A E C B (2) C E B A D (2) C B E D A (2) C B A E D (2) B A C D E (2) A D E B C (2) A B D C E (2) E C B D A (1) E C A D B (1) E B C D A (1) D A E B C (1) C E D B A (1) C E A D B (1) C B A D E (1) C A E B D (1) C A B E D (1) B E C D A (1) B D A E C (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -12 6 -10 B -4 0 0 10 12 C 12 0 0 14 14 D -6 -10 -14 0 -14 E 10 -12 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.290634 C: 0.709366 D: 0.000000 E: 0.000000 Sum of squares = 0.58766828419 Cumulative probabilities = A: 0.000000 B: 0.290634 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 6 -10 B -4 0 0 10 12 C 12 0 0 14 14 D -6 -10 -14 0 -14 E 10 -12 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=21 A=21 C=17 D=9 so D is eliminated. Round 2 votes counts: B=32 E=27 A=24 C=17 so C is eliminated. Round 3 votes counts: B=40 E=34 A=26 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:209 E:199 A:194 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 6 -10 B -4 0 0 10 12 C 12 0 0 14 14 D -6 -10 -14 0 -14 E 10 -12 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 6 -10 B -4 0 0 10 12 C 12 0 0 14 14 D -6 -10 -14 0 -14 E 10 -12 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 6 -10 B -4 0 0 10 12 C 12 0 0 14 14 D -6 -10 -14 0 -14 E 10 -12 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4909: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) A B E D C (8) A D C E B (7) D C E B A (6) A D C B E (6) A D B C E (6) A B E C D (6) A B D E C (5) E B C A D (4) C D E B A (3) B A E D C (3) A E B C D (3) E C B D A (2) E B C D A (2) D C E A B (2) D C A E B (2) D B A C E (2) D A C E B (2) D A C B E (2) D A B C E (2) C E D A B (2) B E D A C (2) B E C D A (2) E C B A D (1) E C A B D (1) E B A C D (1) D B C E A (1) C D E A B (1) B E A C D (1) B D E C A (1) B D E A C (1) B A D E C (1) A E C B D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 16 22 12 16 B -16 0 2 -10 -2 C -22 -2 0 -18 10 D -12 10 18 0 2 E -16 2 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 22 12 16 B -16 0 2 -10 -2 C -22 -2 0 -18 10 D -12 10 18 0 2 E -16 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 D=19 C=15 E=11 B=11 so E is eliminated. Round 2 votes counts: A=44 D=19 C=19 B=18 so B is eliminated. Round 3 votes counts: A=50 C=27 D=23 so D is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:233 D:209 B:187 E:187 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 22 12 16 B -16 0 2 -10 -2 C -22 -2 0 -18 10 D -12 10 18 0 2 E -16 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 22 12 16 B -16 0 2 -10 -2 C -22 -2 0 -18 10 D -12 10 18 0 2 E -16 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 22 12 16 B -16 0 2 -10 -2 C -22 -2 0 -18 10 D -12 10 18 0 2 E -16 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4910: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) E A C D B (8) D A C B E (8) B E C D A (7) A D C E B (6) B E D A C (4) B C D A E (4) A D C B E (4) E C A D B (3) E C A B D (3) E A D C B (3) B C D E A (3) A E D C B (3) A D E C B (3) E B C D A (2) E B A D C (2) C E A D B (2) B D C A E (2) B D A E C (2) B D A C E (2) B C E D A (2) B A D E C (2) A D B C E (2) E C B D A (1) E B A C D (1) E A C B D (1) E A B D C (1) D C A B E (1) D B C A E (1) D B A C E (1) D A B C E (1) C D A B E (1) C A D E B (1) C A D B E (1) B E D C A (1) B D E C A (1) A D B E C (1) Total count = 100 A B C D E A 0 6 10 12 -8 B -6 0 2 0 2 C -10 -2 0 -2 -16 D -12 0 2 0 -6 E 8 -2 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999998 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 6 10 12 -8 B -6 0 2 0 2 C -10 -2 0 -2 -16 D -12 0 2 0 -6 E 8 -2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999723 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=30 A=19 D=12 C=5 so C is eliminated. Round 2 votes counts: E=36 B=30 A=21 D=13 so D is eliminated. Round 3 votes counts: E=36 B=32 A=32 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:210 B:199 D:192 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 10 12 -8 B -6 0 2 0 2 C -10 -2 0 -2 -16 D -12 0 2 0 -6 E 8 -2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999723 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 12 -8 B -6 0 2 0 2 C -10 -2 0 -2 -16 D -12 0 2 0 -6 E 8 -2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999723 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 12 -8 B -6 0 2 0 2 C -10 -2 0 -2 -16 D -12 0 2 0 -6 E 8 -2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999723 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4911: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) C D B A E (6) B E D C A (6) E B A D C (4) D C B E A (4) C D B E A (4) B E A D C (4) A E B D C (4) D E C B A (3) D C E B A (3) C B D E A (3) B E A C D (3) B A C E D (3) A C B E D (3) E B D C A (2) D C E A B (2) C A D B E (2) B C E D A (2) A E D B C (2) A E C D B (2) A D C E B (2) A C D E B (2) A C D B E (2) A C B D E (2) A B E C D (2) E D B C A (1) E D B A C (1) E D A B C (1) E B D A C (1) E A D B C (1) E A B D C (1) D E C A B (1) D B E C A (1) C D A E B (1) C B D A E (1) C A D E B (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D E A (1) B C A E D (1) A E C B D (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -14 -12 -4 B 14 0 -10 -4 26 C 14 10 0 6 12 D 12 4 -6 0 6 E 4 -26 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 -12 -4 B 14 0 -10 -4 26 C 14 10 0 6 12 D 12 4 -6 0 6 E 4 -26 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 B=23 D=14 E=12 so E is eliminated. Round 2 votes counts: B=30 C=27 A=26 D=17 so D is eliminated. Round 3 votes counts: C=40 B=33 A=27 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:213 D:208 E:180 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 -12 -4 B 14 0 -10 -4 26 C 14 10 0 6 12 D 12 4 -6 0 6 E 4 -26 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -12 -4 B 14 0 -10 -4 26 C 14 10 0 6 12 D 12 4 -6 0 6 E 4 -26 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -12 -4 B 14 0 -10 -4 26 C 14 10 0 6 12 D 12 4 -6 0 6 E 4 -26 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4912: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (15) D B C E A (14) B C D E A (8) B D C E A (7) C B E A D (6) E A C B D (5) A E C D B (5) B C E D A (4) D B C A E (3) D A E B C (3) D A B C E (3) B C E A D (3) A E D C B (3) E C A B D (2) D B A E C (2) D B A C E (2) D A B E C (2) C E B A D (2) B D C A E (2) A C E B D (2) D B E C A (1) D B E A C (1) C E A B D (1) C B A E D (1) B C D A E (1) B C A D E (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 -12 -6 -8 B 16 0 14 20 22 C 12 -14 0 12 20 D 6 -20 -12 0 2 E 8 -22 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 -6 -8 B 16 0 14 20 22 C 12 -14 0 12 20 D 6 -20 -12 0 2 E 8 -22 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 A=26 C=10 E=7 so E is eliminated. Round 2 votes counts: D=31 A=31 B=26 C=12 so C is eliminated. Round 3 votes counts: B=35 A=34 D=31 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:236 C:215 D:188 E:182 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 -6 -8 B 16 0 14 20 22 C 12 -14 0 12 20 D 6 -20 -12 0 2 E 8 -22 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -6 -8 B 16 0 14 20 22 C 12 -14 0 12 20 D 6 -20 -12 0 2 E 8 -22 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -6 -8 B 16 0 14 20 22 C 12 -14 0 12 20 D 6 -20 -12 0 2 E 8 -22 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4913: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) B C A D E (6) A B E C D (6) E D C B A (5) E D A C B (5) D C E B A (5) E A B D C (4) C D B E A (4) E D C A B (3) E A D C B (3) D E C A B (3) C D B A E (3) C B D A E (3) A E D B C (3) A E B D C (3) D C E A B (2) C B D E A (2) B E C D A (2) B C E A D (2) B A E C D (2) B A C E D (2) A D E C B (2) A B E D C (2) E B D C A (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D B C (1) D E C B A (1) D C B A E (1) D C A E B (1) D C A B E (1) D A E C B (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A E D (1) A E D C B (1) A D E B C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 0 6 4 B 10 0 8 8 4 C 0 -8 0 -2 -4 D -6 -8 2 0 -2 E -4 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 6 4 B 10 0 8 8 4 C 0 -8 0 -2 -4 D -6 -8 2 0 -2 E -4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 A=21 D=15 C=12 so C is eliminated. Round 2 votes counts: B=32 E=25 D=22 A=21 so A is eliminated. Round 3 votes counts: B=43 E=32 D=25 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:200 E:199 C:193 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 6 4 B 10 0 8 8 4 C 0 -8 0 -2 -4 D -6 -8 2 0 -2 E -4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 4 B 10 0 8 8 4 C 0 -8 0 -2 -4 D -6 -8 2 0 -2 E -4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 4 B 10 0 8 8 4 C 0 -8 0 -2 -4 D -6 -8 2 0 -2 E -4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4914: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) C D E A B (7) B A E D C (7) A E D C B (6) A B E D C (6) E D A C B (5) B A C E D (5) E D C A B (4) E A D C B (4) D E C A B (4) D C E A B (4) B C D A E (4) B C A D E (4) A E B D C (4) C B D A E (3) B E A D C (3) B A C D E (3) A E D B C (3) A B E C D (3) C B D E A (2) B E D C A (2) A C D E B (2) A B C D E (2) E B A D C (1) D E C B A (1) D C E B A (1) C D A E B (1) B C D E A (1) B A E C D (1) Total count = 100 A B C D E A 0 10 10 8 8 B -10 0 -2 2 -8 C -10 2 0 -10 -8 D -8 -2 10 0 -8 E -8 8 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 8 8 B -10 0 -2 2 -8 C -10 2 0 -10 -8 D -8 -2 10 0 -8 E -8 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=26 C=20 E=14 D=10 so D is eliminated. Round 2 votes counts: B=30 A=26 C=25 E=19 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:208 D:196 B:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 8 8 B -10 0 -2 2 -8 C -10 2 0 -10 -8 D -8 -2 10 0 -8 E -8 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 8 8 B -10 0 -2 2 -8 C -10 2 0 -10 -8 D -8 -2 10 0 -8 E -8 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 8 8 B -10 0 -2 2 -8 C -10 2 0 -10 -8 D -8 -2 10 0 -8 E -8 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4915: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) C A B E D (7) C A E B D (6) B C A D E (5) B A C D E (5) C B A E D (4) B E D C A (4) B D E C A (4) B C A E D (4) A C E D B (4) E D C B A (3) E D B C A (3) E D A C B (3) A C B D E (3) E C D A B (2) D E B C A (2) D B E A C (2) D A E C B (2) B D C E A (2) B D A C E (2) B C E D A (2) A C D E B (2) A C B E D (2) E D B A C (1) E B D C A (1) E B C D A (1) E B C A D (1) D E A C B (1) D B A E C (1) C E B A D (1) C E A B D (1) C B E A D (1) C A E D B (1) B E C D A (1) B D E A C (1) B D C A E (1) B D A E C (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -14 4 10 B 22 0 6 24 10 C 14 -6 0 12 14 D -4 -24 -12 0 -8 E -10 -10 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -14 4 10 B 22 0 6 24 10 C 14 -6 0 12 14 D -4 -24 -12 0 -8 E -10 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=21 D=16 A=16 E=15 so E is eliminated. Round 2 votes counts: B=35 D=26 C=23 A=16 so A is eliminated. Round 3 votes counts: C=36 B=36 D=28 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:231 C:217 A:189 E:187 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -14 4 10 B 22 0 6 24 10 C 14 -6 0 12 14 D -4 -24 -12 0 -8 E -10 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 4 10 B 22 0 6 24 10 C 14 -6 0 12 14 D -4 -24 -12 0 -8 E -10 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 4 10 B 22 0 6 24 10 C 14 -6 0 12 14 D -4 -24 -12 0 -8 E -10 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4916: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) B A E D C (10) B E D A C (7) B E A D C (6) C B D E A (5) C A D E B (5) C D E B A (4) B E D C A (4) B C E D A (4) A D E C B (4) C A D B E (3) B C A E D (3) A E D B C (3) A D C E B (3) E D A B C (2) E B D A C (2) D E C B A (2) D E C A B (2) D C E A B (2) C D A E B (2) B E C D A (2) B A C E D (2) A C B D E (2) E D A C B (1) E A D B C (1) E A B D C (1) D E A C B (1) D A E C B (1) C D B E A (1) C D A B E (1) C B D A E (1) C B A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 -6 -10 -16 B 8 0 -6 0 4 C 6 6 0 -6 0 D 10 0 6 0 2 E 16 -4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.341243 C: 0.000000 D: 0.658757 E: 0.000000 Sum of squares = 0.550407714253 Cumulative probabilities = A: 0.000000 B: 0.341243 C: 0.341243 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -10 -16 B 8 0 -6 0 4 C 6 6 0 -6 0 D 10 0 6 0 2 E 16 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499520 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461017 Cumulative probabilities = A: 0.000000 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 C=34 A=13 D=8 E=7 so E is eliminated. Round 2 votes counts: B=40 C=34 A=15 D=11 so D is eliminated. Round 3 votes counts: C=40 B=40 A=20 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:209 E:205 B:203 C:203 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 -16 B 8 0 -6 0 4 C 6 6 0 -6 0 D 10 0 6 0 2 E 16 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499520 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461017 Cumulative probabilities = A: 0.000000 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 -16 B 8 0 -6 0 4 C 6 6 0 -6 0 D 10 0 6 0 2 E 16 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499520 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461017 Cumulative probabilities = A: 0.000000 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 -16 B 8 0 -6 0 4 C 6 6 0 -6 0 D 10 0 6 0 2 E 16 -4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499520 C: 0.000000 D: 0.500480 E: 0.000000 Sum of squares = 0.500000461017 Cumulative probabilities = A: 0.000000 B: 0.499520 C: 0.499520 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4917: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) E D C A B (5) D E B A C (5) C E A D B (5) B A D E C (5) B A C D E (5) A B C D E (5) E D C B A (4) E C D A B (4) E C D B A (3) D E A C B (3) C E D A B (3) C A E D B (3) C A B E D (3) B A D C E (3) B A C E D (3) E D B C A (2) D C E A B (2) C B A E D (2) C A D E B (2) B E D A C (2) B D E A C (2) B D A E C (2) B A E D C (2) A C B D E (2) A B D C E (2) D E C B A (1) D E B C A (1) D A E B C (1) C E D B A (1) C E B D A (1) C D E A B (1) C A B D E (1) B E D C A (1) B C E A D (1) B C A E D (1) A D B E C (1) A C D E B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -8 -2 -8 B -6 0 -10 -12 -10 C 8 10 0 -4 -4 D 2 12 4 0 8 E 8 10 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -2 -8 B -6 0 -10 -12 -10 C 8 10 0 -4 -4 D 2 12 4 0 8 E 8 10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=22 D=20 E=18 A=13 so A is eliminated. Round 2 votes counts: B=35 C=26 D=21 E=18 so E is eliminated. Round 3 votes counts: B=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 E:207 C:205 A:194 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -2 -8 B -6 0 -10 -12 -10 C 8 10 0 -4 -4 D 2 12 4 0 8 E 8 10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -2 -8 B -6 0 -10 -12 -10 C 8 10 0 -4 -4 D 2 12 4 0 8 E 8 10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -2 -8 B -6 0 -10 -12 -10 C 8 10 0 -4 -4 D 2 12 4 0 8 E 8 10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4918: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) A D E C B (8) D A E B C (7) D A B E C (6) C B E A D (6) B C E A D (6) E C A B D (5) C E B A D (5) B D C A E (4) B C D A E (4) A E D C B (4) E A D C B (3) D B A E C (3) C E A B D (3) B D A C E (3) E C A D B (2) D B A C E (2) D A B C E (2) B C D E A (2) E C B A D (1) E A D B C (1) D A E C B (1) D A C B E (1) C B D A E (1) C A E B D (1) B E D C A (1) B E C D A (1) B D E C A (1) B D E A C (1) B D C E A (1) A E C D B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 16 12 18 0 B -16 0 -10 -8 -10 C -12 10 0 0 -14 D -18 8 0 0 -4 E 0 10 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.481668 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.518332 Sum of squares = 0.500672139539 Cumulative probabilities = A: 0.481668 B: 0.481668 C: 0.481668 D: 0.481668 E: 1.000000 A B C D E A 0 16 12 18 0 B -16 0 -10 -8 -10 C -12 10 0 0 -14 D -18 8 0 0 -4 E 0 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 D=22 C=16 A=15 so A is eliminated. Round 2 votes counts: D=31 E=28 B=24 C=17 so C is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:223 E:214 D:193 C:192 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 18 0 B -16 0 -10 -8 -10 C -12 10 0 0 -14 D -18 8 0 0 -4 E 0 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 18 0 B -16 0 -10 -8 -10 C -12 10 0 0 -14 D -18 8 0 0 -4 E 0 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 18 0 B -16 0 -10 -8 -10 C -12 10 0 0 -14 D -18 8 0 0 -4 E 0 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4919: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (6) C E B A D (6) D A B E C (5) C E D A B (5) A D B C E (5) E C B A D (4) D A C E B (4) B E C A D (4) A B C D E (4) E C B D A (3) E B C A D (3) D C A E B (3) B D E A C (3) A D C B E (3) A B D C E (3) E B D C A (2) E B C D A (2) D E C A B (2) D E A B C (2) D B A E C (2) C E A B D (2) C D E A B (2) C A D E B (2) C A B E D (2) B E A D C (2) A C D B E (2) E C D B A (1) D E B A C (1) D B E A C (1) D A E C B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D A E B (1) C B E A D (1) C A E D B (1) C A E B D (1) B E D A C (1) B C E A D (1) B A D E C (1) B A C E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 18 0 0 0 B -18 0 -8 -2 0 C 0 8 0 2 18 D 0 2 -2 0 10 E 0 0 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.264832 B: 0.000000 C: 0.735168 D: 0.000000 E: 0.000000 Sum of squares = 0.61060760523 Cumulative probabilities = A: 0.264832 B: 0.264832 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 0 0 B -18 0 -8 -2 0 C 0 8 0 2 18 D 0 2 -2 0 10 E 0 0 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 A=19 E=15 B=13 so B is eliminated. Round 2 votes counts: D=31 C=26 E=22 A=21 so A is eliminated. Round 3 votes counts: D=44 C=34 E=22 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:209 D:205 B:186 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 0 0 B -18 0 -8 -2 0 C 0 8 0 2 18 D 0 2 -2 0 10 E 0 0 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 0 0 B -18 0 -8 -2 0 C 0 8 0 2 18 D 0 2 -2 0 10 E 0 0 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 0 0 B -18 0 -8 -2 0 C 0 8 0 2 18 D 0 2 -2 0 10 E 0 0 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4920: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) C B E A D (6) C B A D E (6) A D E C B (6) A C D E B (5) D E A B C (4) D A E B C (4) C A B D E (4) B C D A E (4) E A D C B (3) B E D C A (3) B C E D A (3) A D B E C (3) A C D B E (3) E D A B C (2) E C D A B (2) D B E A C (2) D A B E C (2) C B A E D (2) C A E D B (2) C A B E D (2) B D E C A (2) B D E A C (2) B D C E A (2) A E C D B (2) E D B C A (1) E D B A C (1) E D A C B (1) E C A D B (1) E B D C A (1) E A C D B (1) C B E D A (1) B E C D A (1) B D C A E (1) B D A E C (1) B D A C E (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E B C (1) A D C E B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 12 6 10 B -4 0 -2 -10 8 C -12 2 0 -4 -6 D -6 10 4 0 26 E -10 -8 6 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 6 10 B -4 0 -2 -10 8 C -12 2 0 -4 -6 D -6 10 4 0 26 E -10 -8 6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 B=22 D=18 E=13 so E is eliminated. Round 2 votes counts: A=28 C=26 D=23 B=23 so D is eliminated. Round 3 votes counts: A=41 B=33 C=26 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:216 B:196 C:190 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 6 10 B -4 0 -2 -10 8 C -12 2 0 -4 -6 D -6 10 4 0 26 E -10 -8 6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 6 10 B -4 0 -2 -10 8 C -12 2 0 -4 -6 D -6 10 4 0 26 E -10 -8 6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 6 10 B -4 0 -2 -10 8 C -12 2 0 -4 -6 D -6 10 4 0 26 E -10 -8 6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4921: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) A B D E C (10) A B C D E (6) C E B D A (5) C D E A B (5) B C A E D (5) B A C E D (5) A D E C B (5) A D B E C (5) B E C D A (4) E D C B A (3) B C E A D (3) B A E D C (3) B A E C D (3) A C B D E (3) D E A C B (2) C E D B A (2) C B E D A (2) A D E B C (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B D A (1) D E A B C (1) D C E A B (1) D A E C B (1) D A C E B (1) C E D A B (1) B E D C A (1) B E D A C (1) B A D E C (1) A D C E B (1) A D C B E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 20 6 10 8 B -20 0 2 8 10 C -6 -2 0 -4 -14 D -10 -8 4 0 16 E -8 -10 14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 10 8 B -20 0 2 8 10 C -6 -2 0 -4 -14 D -10 -8 4 0 16 E -8 -10 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=26 D=17 C=15 E=7 so E is eliminated. Round 2 votes counts: A=35 B=26 D=21 C=18 so C is eliminated. Round 3 votes counts: A=35 B=34 D=31 so D is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:201 B:200 E:190 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 10 8 B -20 0 2 8 10 C -6 -2 0 -4 -14 D -10 -8 4 0 16 E -8 -10 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 10 8 B -20 0 2 8 10 C -6 -2 0 -4 -14 D -10 -8 4 0 16 E -8 -10 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 10 8 B -20 0 2 8 10 C -6 -2 0 -4 -14 D -10 -8 4 0 16 E -8 -10 14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4922: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C B A (7) B C E D A (7) A D E C B (6) A B C D E (6) D E C A B (5) B C A E D (5) B A C E D (4) A B D C E (4) E C D B A (3) D E A C B (3) C E B D A (3) C B E D A (3) B C D E A (3) A B D E C (3) D E C B A (2) D B C E A (2) C E D B A (2) B C A D E (2) A E D C B (2) A D E B C (2) A D B E C (2) A B C E D (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E A D C B (1) D E B C A (1) D B E C A (1) D B A E C (1) C E A B D (1) C B A E D (1) B C E A D (1) B A D C E (1) A E D B C (1) Total count = 100 A B C D E A 0 -16 -4 8 2 B 16 0 14 10 14 C 4 -14 0 8 12 D -8 -10 -8 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 8 2 B 16 0 14 10 14 C 4 -14 0 8 12 D -8 -10 -8 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=28 E=15 D=15 C=10 so C is eliminated. Round 2 votes counts: B=36 A=28 E=21 D=15 so D is eliminated. Round 3 votes counts: B=40 E=32 A=28 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:205 A:195 D:190 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 8 2 B 16 0 14 10 14 C 4 -14 0 8 12 D -8 -10 -8 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 8 2 B 16 0 14 10 14 C 4 -14 0 8 12 D -8 -10 -8 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 8 2 B 16 0 14 10 14 C 4 -14 0 8 12 D -8 -10 -8 0 6 E -2 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4923: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) C E D B A (7) D E A B C (6) E D C A B (5) E C D A B (5) B A D C E (4) A D E B C (4) E D A C B (3) D E C B A (3) D E C A B (3) D E A C B (3) C B E A D (3) C B A E D (3) B A C D E (3) E D C B A (2) E C D B A (2) E A D C B (2) C E D A B (2) C E B D A (2) C B E D A (2) B C D E A (2) A B C E D (2) E A C D B (1) D E B C A (1) D E B A C (1) D A E B C (1) C E B A D (1) C E A B D (1) C D B E A (1) C B D E A (1) B D C E A (1) B D A E C (1) B A D E C (1) B A C E D (1) A E D C B (1) A E D B C (1) A E C B D (1) A E B D C (1) A D B E C (1) A C E B D (1) A C B E D (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 16 6 -8 -20 B -16 0 -12 -10 -20 C -6 12 0 -12 -20 D 8 10 12 0 -4 E 20 20 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 6 -8 -20 B -16 0 -12 -10 -20 C -6 12 0 -12 -20 D 8 10 12 0 -4 E 20 20 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=23 E=20 D=18 B=13 so B is eliminated. Round 2 votes counts: A=35 C=25 E=20 D=20 so E is eliminated. Round 3 votes counts: A=38 C=32 D=30 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:232 D:213 A:197 C:187 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 6 -8 -20 B -16 0 -12 -10 -20 C -6 12 0 -12 -20 D 8 10 12 0 -4 E 20 20 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -8 -20 B -16 0 -12 -10 -20 C -6 12 0 -12 -20 D 8 10 12 0 -4 E 20 20 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -8 -20 B -16 0 -12 -10 -20 C -6 12 0 -12 -20 D 8 10 12 0 -4 E 20 20 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4924: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) B E D C A (7) E B D C A (5) D B C E A (5) C D B A E (5) C A D B E (5) E B D A C (4) D B E C A (4) A C D B E (4) E A B C D (3) C B D E A (3) E B C D A (2) E B A D C (2) E B A C D (2) E A D B C (2) C D A B E (2) C A B D E (2) B E C A D (2) A E D C B (2) A E C B D (2) A E B C D (2) A D E C B (2) A D C E B (2) A C E B D (2) A C B E D (2) E D B C A (1) E D B A C (1) E B C A D (1) E A B D C (1) D E B C A (1) D E B A C (1) D C A B E (1) D A C E B (1) D A C B E (1) C A B E D (1) B E C D A (1) B D E C A (1) B D C E A (1) B C E D A (1) A E D B C (1) A E C D B (1) A E B D C (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -2 4 -2 B 0 0 6 -2 -2 C 2 -6 0 4 -6 D -4 2 -4 0 2 E 2 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000023 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 0 -2 4 -2 B 0 0 6 -2 -2 C 2 -6 0 4 -6 D -4 2 -4 0 2 E 2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 C=18 D=14 B=13 so B is eliminated. Round 2 votes counts: E=34 A=31 C=19 D=16 so D is eliminated. Round 3 votes counts: E=41 A=33 C=26 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:204 B:201 A:200 D:198 C:197 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 4 -2 B 0 0 6 -2 -2 C 2 -6 0 4 -6 D -4 2 -4 0 2 E 2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 -2 B 0 0 6 -2 -2 C 2 -6 0 4 -6 D -4 2 -4 0 2 E 2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 -2 B 0 0 6 -2 -2 C 2 -6 0 4 -6 D -4 2 -4 0 2 E 2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4925: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) C A E D B (9) B E D A C (8) C A D E B (5) A D C B E (5) E C B A D (4) B D E A C (4) B D A E C (4) A D C E B (4) E B C A D (3) D B A C E (3) D A C B E (3) C E B D A (3) C E B A D (3) C E A D B (3) A D B E C (3) A C D E B (3) E C B D A (2) E C A B D (2) D B A E C (2) D A B E C (2) D A B C E (2) C E A B D (2) B E C D A (2) B E A D C (2) E B A D C (1) E A B D C (1) C E D B A (1) C D E A B (1) B E D C A (1) B D E C A (1) A E C B D (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -2 4 -6 B 6 0 -2 6 -14 C 2 2 0 6 -4 D -4 -6 -6 0 -14 E 6 14 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 4 -6 B 6 0 -2 6 -14 C 2 2 0 6 -4 D -4 -6 -6 0 -14 E 6 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 B=22 A=17 D=12 so D is eliminated. Round 2 votes counts: C=27 B=27 A=24 E=22 so E is eliminated. Round 3 votes counts: B=40 C=35 A=25 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:219 C:203 B:198 A:195 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 4 -6 B 6 0 -2 6 -14 C 2 2 0 6 -4 D -4 -6 -6 0 -14 E 6 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 4 -6 B 6 0 -2 6 -14 C 2 2 0 6 -4 D -4 -6 -6 0 -14 E 6 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 4 -6 B 6 0 -2 6 -14 C 2 2 0 6 -4 D -4 -6 -6 0 -14 E 6 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4926: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) D B C A E (7) A C E B D (7) D B E A C (5) B D E A C (5) E B D A C (4) E A B C D (4) D E B A C (4) D B E C A (4) D B C E A (4) A E C B D (4) E B A D C (3) E B A C D (3) D C B A E (3) C D A B E (3) B D C A E (3) E A C D B (2) E A C B D (2) C D B A E (2) C A D E B (2) C A D B E (2) B E D A C (2) B E A D C (2) B D E C A (2) E A D C B (1) D E B C A (1) D C E A B (1) D C A B E (1) C D A E B (1) C B A D E (1) C A E D B (1) C A B D E (1) B E A C D (1) B C D A E (1) B C A E D (1) B C A D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -18 0 -6 0 B 18 0 16 12 2 C 0 -16 0 -4 0 D 6 -12 4 0 8 E 0 -2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999475 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 0 -6 0 B 18 0 16 12 2 C 0 -16 0 -4 0 D 6 -12 4 0 8 E 0 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=21 E=19 B=18 A=12 so A is eliminated. Round 2 votes counts: D=30 C=28 E=24 B=18 so B is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:224 D:203 E:195 C:190 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 0 -6 0 B 18 0 16 12 2 C 0 -16 0 -4 0 D 6 -12 4 0 8 E 0 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -6 0 B 18 0 16 12 2 C 0 -16 0 -4 0 D 6 -12 4 0 8 E 0 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -6 0 B 18 0 16 12 2 C 0 -16 0 -4 0 D 6 -12 4 0 8 E 0 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4927: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) A C D B E (9) E D C A B (8) D C E A B (7) B A C D E (7) E D C B A (6) D C A E B (6) B A E C D (6) A C B D E (5) B E A C D (4) A B C D E (4) D E C A B (3) E D B C A (2) E B D A C (2) E B A D C (2) C D E A B (2) C D A E B (2) C D A B E (2) A D C B E (2) D A C E B (1) C D E B A (1) C A D B E (1) B E C D A (1) B E C A D (1) B C A D E (1) B A E D C (1) B A C E D (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 -8 -8 0 B -8 0 -14 -8 -10 C 8 14 0 -6 4 D 8 8 6 0 8 E 0 10 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -8 0 B -8 0 -14 -8 -10 C 8 14 0 -6 4 D 8 8 6 0 8 E 0 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=22 A=22 D=17 C=8 so C is eliminated. Round 2 votes counts: E=31 D=24 A=23 B=22 so B is eliminated. Round 3 votes counts: A=39 E=37 D=24 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 C:210 E:199 A:196 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -8 0 B -8 0 -14 -8 -10 C 8 14 0 -6 4 D 8 8 6 0 8 E 0 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -8 0 B -8 0 -14 -8 -10 C 8 14 0 -6 4 D 8 8 6 0 8 E 0 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -8 0 B -8 0 -14 -8 -10 C 8 14 0 -6 4 D 8 8 6 0 8 E 0 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4928: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) E A D B C (6) C B D A E (6) B D C E A (6) C B D E A (5) A E D B C (5) E D B C A (4) B C D A E (4) E D B A C (3) E D A B C (3) D B C E A (3) C B A D E (3) C A B D E (3) B D E C A (3) B C D E A (3) A E C D B (3) A C B E D (3) E A C B D (2) D E B C A (2) D E B A C (2) D B E C A (2) C E A B D (2) A D E B C (2) A C E B D (2) E B D C A (1) E B C D A (1) E A D C B (1) D E A B C (1) D B C A E (1) C A E B D (1) C A B E D (1) B D C A E (1) A E D C B (1) A E C B D (1) A D C B E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -4 -2 0 B 0 0 8 16 14 C 4 -8 0 2 14 D 2 -16 -2 0 20 E 0 -14 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.404818 B: 0.595182 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.518119359008 Cumulative probabilities = A: 0.404818 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -2 0 B 0 0 8 16 14 C 4 -8 0 2 14 D 2 -16 -2 0 20 E 0 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=21 C=21 B=17 D=11 so D is eliminated. Round 2 votes counts: A=30 E=26 B=23 C=21 so C is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:219 C:206 D:202 A:197 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -2 0 B 0 0 8 16 14 C 4 -8 0 2 14 D 2 -16 -2 0 20 E 0 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 0 B 0 0 8 16 14 C 4 -8 0 2 14 D 2 -16 -2 0 20 E 0 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 0 B 0 0 8 16 14 C 4 -8 0 2 14 D 2 -16 -2 0 20 E 0 -14 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4929: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) E C D B A (6) D E C A B (5) E B D C A (4) D E C B A (4) D E B C A (4) A D B C E (4) E D C B A (3) E C B D A (3) D E B A C (3) D A E B C (3) C E B A D (3) C E A B D (3) C A B E D (3) B E C A D (3) B D A E C (3) B A D E C (3) B A C E D (3) A C B E D (3) A B C D E (3) D A C E B (2) C E A D B (2) C A E B D (2) B E D A C (2) B D E A C (2) A D C B E (2) A B D C E (2) D C E A B (1) D B E A C (1) D A E C B (1) C E D A B (1) C E B D A (1) C D A E B (1) C B A E D (1) B C E A D (1) B C A E D (1) B A E D C (1) B A D C E (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -18 -14 -18 B 20 0 4 -4 -18 C 18 -4 0 -18 -18 D 14 4 18 0 -6 E 18 18 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -18 -14 -18 B 20 0 4 -4 -18 C 18 -4 0 -18 -18 D 14 4 18 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 B=20 C=17 A=16 so A is eliminated. Round 2 votes counts: D=31 B=25 E=23 C=21 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:230 D:215 B:201 C:189 A:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -18 -14 -18 B 20 0 4 -4 -18 C 18 -4 0 -18 -18 D 14 4 18 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 -14 -18 B 20 0 4 -4 -18 C 18 -4 0 -18 -18 D 14 4 18 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 -14 -18 B 20 0 4 -4 -18 C 18 -4 0 -18 -18 D 14 4 18 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4930: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) C D A B E (7) B C D E A (7) E A D C B (6) D C A E B (5) B E C D A (5) A D C E B (5) A C D B E (5) E A B D C (4) B E A C D (4) A E D C B (4) E D C B A (3) E B D C A (3) D C B A E (3) D C A B E (3) A E B C D (3) E B A C D (2) C B D A E (2) B E C A D (2) A D E C B (2) A C D E B (2) A B E C D (2) E D C A B (1) E D A C B (1) E B D A C (1) E A D B C (1) D C E B A (1) C D B E A (1) C D B A E (1) C A D B E (1) B E D C A (1) B C D A E (1) B C A D E (1) A E B D C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 8 -2 B -8 0 -6 -4 -6 C -4 6 0 -6 -8 D -8 4 6 0 -4 E 2 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 4 8 -2 B -8 0 -6 -4 -6 C -4 6 0 -6 -8 D -8 4 6 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=25 B=21 D=12 C=12 so D is eliminated. Round 2 votes counts: E=30 A=25 C=24 B=21 so B is eliminated. Round 3 votes counts: E=42 C=33 A=25 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:209 D:199 C:194 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 4 8 -2 B -8 0 -6 -4 -6 C -4 6 0 -6 -8 D -8 4 6 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 8 -2 B -8 0 -6 -4 -6 C -4 6 0 -6 -8 D -8 4 6 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 8 -2 B -8 0 -6 -4 -6 C -4 6 0 -6 -8 D -8 4 6 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4931: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (27) C E A B D (13) E C A B D (7) E B A C D (4) E A B C D (3) D B A C E (3) C E D A B (3) B A D E C (3) E C B A D (2) D C E B A (2) D C B E A (2) D C A B E (2) C E B A D (2) C D E A B (2) C A D B E (2) B D E A C (2) B D A E C (2) B A E D C (2) E B A D C (1) E A C B D (1) D E C B A (1) D E B C A (1) D C E A B (1) D C B A E (1) D B E A C (1) D B C A E (1) D A B C E (1) C E D B A (1) C D A E B (1) C A E B D (1) C A D E B (1) B E D A C (1) B E A D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -20 8 -10 -2 B 20 0 10 -6 6 C -8 -10 0 -12 -22 D 10 6 12 0 12 E 2 -6 22 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 8 -10 -2 B 20 0 10 -6 6 C -8 -10 0 -12 -22 D 10 6 12 0 12 E 2 -6 22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 C=26 E=18 B=11 A=2 so A is eliminated. Round 2 votes counts: D=43 C=26 E=18 B=13 so B is eliminated. Round 3 votes counts: D=50 C=26 E=24 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:215 E:203 A:188 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 8 -10 -2 B 20 0 10 -6 6 C -8 -10 0 -12 -22 D 10 6 12 0 12 E 2 -6 22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 8 -10 -2 B 20 0 10 -6 6 C -8 -10 0 -12 -22 D 10 6 12 0 12 E 2 -6 22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 8 -10 -2 B 20 0 10 -6 6 C -8 -10 0 -12 -22 D 10 6 12 0 12 E 2 -6 22 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4932: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (18) E A C B D (11) B C D A E (7) B C D E A (6) E C A B D (5) A D E B C (5) D B A C E (4) D A E B C (4) A E D C B (4) A E D B C (4) C B E A D (3) E A D C B (2) E A D B C (2) E A C D B (2) D B C E A (2) D A B C E (2) C E B A D (2) C B D A E (2) B D C E A (2) B D C A E (2) A E C B D (2) A D B C E (2) E C B D A (1) D B A E C (1) D A B E C (1) C E A B D (1) C B D E A (1) C B A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -6 -6 20 B 4 0 24 -8 8 C 6 -24 0 -10 12 D 6 8 10 0 18 E -20 -8 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -6 20 B 4 0 24 -8 8 C 6 -24 0 -10 12 D 6 8 10 0 18 E -20 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=23 A=18 B=17 C=10 so C is eliminated. Round 2 votes counts: D=32 E=26 B=24 A=18 so A is eliminated. Round 3 votes counts: D=39 E=37 B=24 so B is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:214 A:202 C:192 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -6 20 B 4 0 24 -8 8 C 6 -24 0 -10 12 D 6 8 10 0 18 E -20 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -6 20 B 4 0 24 -8 8 C 6 -24 0 -10 12 D 6 8 10 0 18 E -20 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -6 20 B 4 0 24 -8 8 C 6 -24 0 -10 12 D 6 8 10 0 18 E -20 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4933: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) B A D C E (11) D C E A B (6) C E D A B (6) E C B A D (4) D E C A B (4) B A D E C (4) B A C E D (4) E D C A B (3) E C D B A (3) D B A E C (3) D A B C E (3) C E A D B (3) B A E C D (3) A B D C E (3) A B C D E (3) E C B D A (2) E C A B D (2) E B C D A (2) C E A B D (2) C D E A B (2) A D B C E (2) E D B C A (1) E B D C A (1) E B C A D (1) D E B A C (1) D C A E B (1) D B A C E (1) D A E C B (1) D A C B E (1) C A B E D (1) B E A C D (1) B A E D C (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 -12 -6 -12 B -12 0 -8 -6 -14 C 12 8 0 6 2 D 6 6 -6 0 -4 E 12 14 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 -6 -12 B -12 0 -8 -6 -14 C 12 8 0 6 2 D 6 6 -6 0 -4 E 12 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=25 D=21 C=14 A=9 so A is eliminated. Round 2 votes counts: E=31 B=31 D=23 C=15 so C is eliminated. Round 3 votes counts: E=42 B=33 D=25 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:214 D:201 A:191 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -12 -6 -12 B -12 0 -8 -6 -14 C 12 8 0 6 2 D 6 6 -6 0 -4 E 12 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 -6 -12 B -12 0 -8 -6 -14 C 12 8 0 6 2 D 6 6 -6 0 -4 E 12 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 -6 -12 B -12 0 -8 -6 -14 C 12 8 0 6 2 D 6 6 -6 0 -4 E 12 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4934: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) B C E D A (8) A D C E B (7) C D B A E (6) C B D E A (6) E B A C D (5) D C A E B (5) B E C A D (5) D A C E B (4) B E A C D (4) B C D E A (4) E B A D C (3) E A D B C (3) B E A D C (3) A E D C B (3) A E D B C (3) E A D C B (2) D C A B E (2) C D A B E (2) B C D A E (2) A E B D C (2) A D E C B (2) E C D A B (1) E A B C D (1) C B E D A (1) C B D A E (1) B E C D A (1) B A E D C (1) Total count = 100 A B C D E A 0 0 12 14 -20 B 0 0 16 20 -8 C -12 -16 0 -6 -4 D -14 -20 6 0 -18 E 20 8 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 12 14 -20 B 0 0 16 20 -8 C -12 -16 0 -6 -4 D -14 -20 6 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=28 B=28 A=17 C=16 D=11 so D is eliminated. Round 2 votes counts: E=28 B=28 C=23 A=21 so A is eliminated. Round 3 votes counts: E=38 C=34 B=28 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:214 A:203 C:181 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 14 -20 B 0 0 16 20 -8 C -12 -16 0 -6 -4 D -14 -20 6 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 14 -20 B 0 0 16 20 -8 C -12 -16 0 -6 -4 D -14 -20 6 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 14 -20 B 0 0 16 20 -8 C -12 -16 0 -6 -4 D -14 -20 6 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4935: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E B C A D (6) C A B E D (6) A E C B D (6) A C E B D (6) A D C B E (5) D A C B E (4) E C B A D (3) E C A B D (3) D B C A E (3) D A E B C (3) D A B C E (3) B E C D A (3) A D E C B (3) A C D B E (3) E D B C A (2) E B D C A (2) D B E C A (2) D B A C E (2) D A E C B (2) C E B A D (2) B E C A D (2) B D E C A (2) A D C E B (2) A C E D B (2) E D B A C (1) E D A B C (1) E A D C B (1) E A C B D (1) D E B C A (1) D E B A C (1) D B E A C (1) D B C E A (1) D B A E C (1) D A B E C (1) C B A E D (1) B E D C A (1) B C E A D (1) B C D E A (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 2 10 10 B -8 0 -4 8 -12 C -2 4 0 10 -14 D -10 -8 -10 0 -16 E -10 12 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 10 10 B -8 0 -4 8 -12 C -2 4 0 10 -14 D -10 -8 -10 0 -16 E -10 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998254 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 D=25 B=10 C=9 so C is eliminated. Round 2 votes counts: A=35 E=29 D=25 B=11 so B is eliminated. Round 3 votes counts: E=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:215 C:199 B:192 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 10 10 B -8 0 -4 8 -12 C -2 4 0 10 -14 D -10 -8 -10 0 -16 E -10 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998254 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 10 10 B -8 0 -4 8 -12 C -2 4 0 10 -14 D -10 -8 -10 0 -16 E -10 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998254 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 10 10 B -8 0 -4 8 -12 C -2 4 0 10 -14 D -10 -8 -10 0 -16 E -10 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998254 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4936: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) D C B E A (8) D B C E A (7) D C B A E (6) E A B C D (5) C A D E B (5) B E D A C (5) B D E C A (5) D C A E B (4) C A D B E (4) A E C B D (4) A C E B D (4) E B A D C (3) E A B D C (3) D B E C A (3) C D B A E (3) C D A B E (3) E D B A C (2) C A B D E (2) B C D E A (2) A E C D B (2) A C D E B (2) E A D C B (1) E A D B C (1) D E B A C (1) B E D C A (1) B D C E A (1) B C A D E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -10 -2 4 B -4 0 0 -6 4 C 10 0 0 -2 8 D 2 6 2 0 14 E -4 -4 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999213 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -2 4 B -4 0 0 -6 4 C 10 0 0 -2 8 D 2 6 2 0 14 E -4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 C=17 E=15 B=15 so E is eliminated. Round 2 votes counts: A=34 D=31 B=18 C=17 so C is eliminated. Round 3 votes counts: A=45 D=37 B=18 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:208 A:198 B:197 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -10 -2 4 B -4 0 0 -6 4 C 10 0 0 -2 8 D 2 6 2 0 14 E -4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -2 4 B -4 0 0 -6 4 C 10 0 0 -2 8 D 2 6 2 0 14 E -4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -2 4 B -4 0 0 -6 4 C 10 0 0 -2 8 D 2 6 2 0 14 E -4 -4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4937: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) D C A B E (8) B D A C E (8) B A E D C (7) C D E A B (5) C D A E B (5) B D C A E (4) E C D B A (3) E B A C D (3) E A C D B (3) E A C B D (3) E A B C D (3) D C B A E (3) C D E B A (3) B E D C A (3) E C A D B (2) C E D A B (2) B D C E A (2) B A D C E (2) A E B C D (2) A B E D C (2) A B D C E (2) E C D A B (1) E C B D A (1) E B D C A (1) E B C D A (1) E B A D C (1) D C E B A (1) D C B E A (1) D B C A E (1) C E D B A (1) C D A B E (1) B D E A C (1) B A D E C (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -18 2 -12 -4 B 18 0 10 18 14 C -2 -10 0 -18 0 D 12 -18 18 0 -2 E 4 -14 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 2 -12 -4 B 18 0 10 18 14 C -2 -10 0 -18 0 D 12 -18 18 0 -2 E 4 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=22 C=17 D=14 A=8 so A is eliminated. Round 2 votes counts: B=43 E=25 C=18 D=14 so D is eliminated. Round 3 votes counts: B=44 C=31 E=25 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:230 D:205 E:196 C:185 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 2 -12 -4 B 18 0 10 18 14 C -2 -10 0 -18 0 D 12 -18 18 0 -2 E 4 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 2 -12 -4 B 18 0 10 18 14 C -2 -10 0 -18 0 D 12 -18 18 0 -2 E 4 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 2 -12 -4 B 18 0 10 18 14 C -2 -10 0 -18 0 D 12 -18 18 0 -2 E 4 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4938: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) C A D E B (6) B E D A C (6) A C D E B (6) C A E D B (5) B D E C A (5) D B A C E (4) B E D C A (4) B D A E C (4) E C A B D (3) E B C A D (3) E B A C D (3) D B C A E (3) D A C B E (3) C E A D B (3) E C B D A (2) E B C D A (2) D A B C E (2) C E D A B (2) B D A C E (2) A E C B D (2) A D C B E (2) A C E D B (2) E C B A D (1) E C A D B (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B C E A (1) C D B A E (1) C D A E B (1) C D A B E (1) B E A D C (1) B A E D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 8 -12 -4 B 14 0 12 8 8 C -8 -12 0 -8 -6 D 12 -8 8 0 8 E 4 -8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 -12 -4 B 14 0 12 8 8 C -8 -12 0 -8 -6 D 12 -8 8 0 8 E 4 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=19 C=19 D=15 A=14 so A is eliminated. Round 2 votes counts: B=34 C=27 E=21 D=18 so D is eliminated. Round 3 votes counts: B=45 C=34 E=21 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:210 E:197 A:189 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 -12 -4 B 14 0 12 8 8 C -8 -12 0 -8 -6 D 12 -8 8 0 8 E 4 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -12 -4 B 14 0 12 8 8 C -8 -12 0 -8 -6 D 12 -8 8 0 8 E 4 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -12 -4 B 14 0 12 8 8 C -8 -12 0 -8 -6 D 12 -8 8 0 8 E 4 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4939: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (14) E C A B D (13) D B E C A (13) A B D C E (9) D B A C E (7) B D A C E (7) E C A D B (6) C E A D B (4) B D A E C (4) E C D B A (3) C A E D B (3) A C B D E (3) D B E A C (2) C E A B D (2) C A E B D (2) A E C B D (2) E D C B A (1) D B C A E (1) B D E A C (1) B A D E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 4 22 10 B -20 0 -6 18 0 C -4 6 0 6 8 D -22 -18 -6 0 0 E -10 0 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 4 22 10 B -20 0 -6 18 0 C -4 6 0 6 8 D -22 -18 -6 0 0 E -10 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 D=23 B=13 C=11 so C is eliminated. Round 2 votes counts: A=35 E=29 D=23 B=13 so B is eliminated. Round 3 votes counts: A=36 D=35 E=29 so E is eliminated. Round 4 votes counts: A=61 D=39 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:208 B:196 E:191 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 4 22 10 B -20 0 -6 18 0 C -4 6 0 6 8 D -22 -18 -6 0 0 E -10 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 4 22 10 B -20 0 -6 18 0 C -4 6 0 6 8 D -22 -18 -6 0 0 E -10 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 4 22 10 B -20 0 -6 18 0 C -4 6 0 6 8 D -22 -18 -6 0 0 E -10 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4940: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E A D C B (7) D E A C B (7) B D E C A (7) B C A D E (7) B C D A E (6) A C E D B (6) B D E A C (5) E A C D B (4) C A D E B (4) E D B A C (3) D C A E B (3) C A B E D (3) B D C E A (3) B D C A E (3) B C A E D (3) E D A B C (2) D B E C A (2) C B A D E (2) C A E D B (2) B E D A C (2) B E A C D (2) B A C E D (2) A E C D B (2) A C E B D (2) E B D A C (1) Total count = 100 A B C D E A 0 4 10 -8 -10 B -4 0 -4 -4 -6 C -10 4 0 -10 -8 D 8 4 10 0 -2 E 10 6 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 -8 -10 B -4 0 -4 -4 -6 C -10 4 0 -10 -8 D 8 4 10 0 -2 E 10 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=27 D=12 C=11 A=10 so A is eliminated. Round 2 votes counts: B=40 E=29 C=19 D=12 so D is eliminated. Round 3 votes counts: B=42 E=36 C=22 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:210 A:198 B:191 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 -8 -10 B -4 0 -4 -4 -6 C -10 4 0 -10 -8 D 8 4 10 0 -2 E 10 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -8 -10 B -4 0 -4 -4 -6 C -10 4 0 -10 -8 D 8 4 10 0 -2 E 10 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -8 -10 B -4 0 -4 -4 -6 C -10 4 0 -10 -8 D 8 4 10 0 -2 E 10 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4941: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) A D E B C (7) C E D B A (6) B C E D A (6) B A D C E (5) C B E D A (4) B C E A D (4) D A E C B (3) D A B E C (3) B D C A E (3) B A C E D (3) A D E C B (3) E D C B A (2) E C D B A (2) E C D A B (2) E C A D B (2) E A D C B (2) D E C B A (2) C E B D A (2) C E B A D (2) B D C E A (2) B C D E A (2) B C A E D (2) A E C D B (2) A E C B D (2) A B E C D (2) A B C E D (2) E D C A B (1) E D A C B (1) D E C A B (1) D E A C B (1) D C E B A (1) D B A E C (1) C E A B D (1) C D E B A (1) B D A C E (1) B C A D E (1) B A C D E (1) A E D C B (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 2 6 6 B 6 0 10 -4 4 C -2 -10 0 0 2 D -6 4 0 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999652 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 6 6 B 6 0 10 -4 4 C -2 -10 0 0 2 D -6 4 0 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999975 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=30 A=30 C=16 E=12 D=12 so E is eliminated. Round 2 votes counts: A=32 B=30 C=22 D=16 so D is eliminated. Round 3 votes counts: A=40 B=31 C=29 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:204 D:197 E:196 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 6 6 B 6 0 10 -4 4 C -2 -10 0 0 2 D -6 4 0 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999975 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 6 6 B 6 0 10 -4 4 C -2 -10 0 0 2 D -6 4 0 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999975 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 6 6 B 6 0 10 -4 4 C -2 -10 0 0 2 D -6 4 0 0 -4 E -6 -4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999975 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4942: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) D B E C A (6) C D B A E (6) C A D B E (6) E A B C D (5) C A B E D (5) A E C B D (5) A C E B D (5) D B C E A (4) C B D E A (4) D E A B C (3) B D E C A (3) B C E D A (3) E A C B D (2) E A B D C (2) D C B A E (2) C D B E A (2) C B D A E (2) C B A E D (2) C A E B D (2) A E D B C (2) A C E D B (2) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) E B C A D (1) D B C A E (1) D B A E C (1) D A E B C (1) D A C B E (1) D A B E C (1) C B E A D (1) C B A D E (1) C A B D E (1) B D C E A (1) B C D E A (1) A E D C B (1) A E B C D (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 -8 0 B 4 0 -2 -2 8 C 10 2 0 16 6 D 8 2 -16 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -8 0 B 4 0 -2 -2 8 C 10 2 0 16 6 D 8 2 -16 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=28 A=18 E=14 B=8 so B is eliminated. Round 2 votes counts: C=36 D=32 A=18 E=14 so E is eliminated. Round 3 votes counts: C=37 D=36 A=27 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:204 D:204 A:189 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -8 0 B 4 0 -2 -2 8 C 10 2 0 16 6 D 8 2 -16 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -8 0 B 4 0 -2 -2 8 C 10 2 0 16 6 D 8 2 -16 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -8 0 B 4 0 -2 -2 8 C 10 2 0 16 6 D 8 2 -16 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4943: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (14) C B D A E (10) A E B C D (10) C D B A E (7) E A D B C (6) E A B D C (6) D E A C B (6) D C E A B (6) D C B A E (5) B C A D E (5) B C A E D (4) B A E C D (4) D C E B A (3) E A B C D (2) D E C A B (2) B C D A E (2) E D A C B (1) E A D C B (1) D C A E B (1) C B D E A (1) C B A D E (1) B A C E D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -22 -16 4 B 14 0 -18 -6 10 C 22 18 0 -4 20 D 16 6 4 0 28 E -4 -10 -20 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -22 -16 4 B 14 0 -18 -6 10 C 22 18 0 -4 20 D 16 6 4 0 28 E -4 -10 -20 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997106 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=19 E=16 B=16 A=12 so A is eliminated. Round 2 votes counts: D=38 E=26 C=19 B=17 so B is eliminated. Round 3 votes counts: D=38 E=31 C=31 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:228 D:227 B:200 A:176 E:169 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -22 -16 4 B 14 0 -18 -6 10 C 22 18 0 -4 20 D 16 6 4 0 28 E -4 -10 -20 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997106 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -22 -16 4 B 14 0 -18 -6 10 C 22 18 0 -4 20 D 16 6 4 0 28 E -4 -10 -20 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997106 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -22 -16 4 B 14 0 -18 -6 10 C 22 18 0 -4 20 D 16 6 4 0 28 E -4 -10 -20 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997106 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4944: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) D B A C E (6) D A B C E (6) C E B A D (6) C E B D A (5) B D A C E (5) B A D E C (5) C E D A B (4) A D B E C (4) E C A D B (3) D A C B E (3) C B E D A (3) B D A E C (3) A B D E C (3) E C B A D (2) E C A B D (2) E A C D B (2) D B C A E (2) D A E B C (2) C E A D B (2) C D E A B (2) B E A C D (2) B C D A E (2) A D E B C (2) E D A C B (1) E B A D C (1) E B A C D (1) E A D C B (1) E A D B C (1) E A C B D (1) D C A B E (1) D B A E C (1) D A E C B (1) C E D B A (1) C E A B D (1) C D B E A (1) C D B A E (1) B D C A E (1) B C A E D (1) B C A D E (1) A E D B C (1) Total count = 100 A B C D E A 0 0 18 -16 16 B 0 0 14 -10 16 C -18 -14 0 -14 8 D 16 10 14 0 18 E -16 -16 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 -16 16 B 0 0 14 -10 16 C -18 -14 0 -14 8 D 16 10 14 0 18 E -16 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 B=20 E=15 A=10 so A is eliminated. Round 2 votes counts: D=35 C=26 B=23 E=16 so E is eliminated. Round 3 votes counts: D=39 C=36 B=25 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:229 B:210 A:209 C:181 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 18 -16 16 B 0 0 14 -10 16 C -18 -14 0 -14 8 D 16 10 14 0 18 E -16 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 -16 16 B 0 0 14 -10 16 C -18 -14 0 -14 8 D 16 10 14 0 18 E -16 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 -16 16 B 0 0 14 -10 16 C -18 -14 0 -14 8 D 16 10 14 0 18 E -16 -16 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4945: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (18) A C D B E (15) C D A E B (11) E B D C A (9) B E A D C (7) E D C B A (5) D C E B A (5) B A E C D (5) B E D C A (4) A B C D E (4) D C A B E (3) C D E A B (3) A B E C D (3) E C D A B (2) B E A C D (2) E B A C D (1) B E D A C (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 18 -20 -22 -14 B -18 0 -24 -24 -8 C 20 24 0 -4 22 D 22 24 4 0 20 E 14 8 -22 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -20 -22 -14 B -18 0 -24 -24 -8 C 20 24 0 -4 22 D 22 24 4 0 20 E 14 8 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=22 B=21 E=17 C=14 so C is eliminated. Round 2 votes counts: D=40 A=22 B=21 E=17 so E is eliminated. Round 3 votes counts: D=47 B=31 A=22 so A is eliminated. Round 4 votes counts: D=62 B=38 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:235 C:231 E:190 A:181 B:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -20 -22 -14 B -18 0 -24 -24 -8 C 20 24 0 -4 22 D 22 24 4 0 20 E 14 8 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -20 -22 -14 B -18 0 -24 -24 -8 C 20 24 0 -4 22 D 22 24 4 0 20 E 14 8 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -20 -22 -14 B -18 0 -24 -24 -8 C 20 24 0 -4 22 D 22 24 4 0 20 E 14 8 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998554 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4946: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) A B D C E (11) E C A B D (9) C A B D E (6) E D C B A (5) D B A C E (5) A B E D C (5) A B D E C (5) E D B A C (4) E B A D C (4) C E D B A (4) A C B D E (4) D B A E C (3) C E A B D (3) E C D A B (2) E A B D C (2) A B C D E (2) E D B C A (1) E B D A C (1) D E B A C (1) D C B A E (1) D B E A C (1) C E D A B (1) C D E B A (1) C D B A E (1) C D A B E (1) C A E B D (1) B D A E C (1) B A E D C (1) B A D E C (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 12 0 B -8 0 0 16 -2 C -6 0 0 -4 -16 D -12 -16 4 0 -12 E 0 2 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.602526 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.397474 Sum of squares = 0.521023051004 Cumulative probabilities = A: 0.602526 B: 0.602526 C: 0.602526 D: 0.602526 E: 1.000000 A B C D E A 0 8 6 12 0 B -8 0 0 16 -2 C -6 0 0 -4 -16 D -12 -16 4 0 -12 E 0 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=29 C=18 D=11 B=3 so B is eliminated. Round 2 votes counts: E=39 A=31 C=18 D=12 so D is eliminated. Round 3 votes counts: E=41 A=40 C=19 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:213 B:203 C:187 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 12 0 B -8 0 0 16 -2 C -6 0 0 -4 -16 D -12 -16 4 0 -12 E 0 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 12 0 B -8 0 0 16 -2 C -6 0 0 -4 -16 D -12 -16 4 0 -12 E 0 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 12 0 B -8 0 0 16 -2 C -6 0 0 -4 -16 D -12 -16 4 0 -12 E 0 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4947: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) B C E D A (8) C A B E D (7) A C B E D (6) E D B A C (5) A D C B E (5) C A B D E (4) A D E C B (4) A D C E B (4) E D B C A (3) E D A B C (3) E B D C A (3) D E A B C (3) C B A E D (3) E B D A C (2) E B C D A (2) E B C A D (2) E A D B C (2) D E B C A (2) D B E C A (2) D A E C B (2) D A C B E (2) A C E B D (2) E A C B D (1) D E B A C (1) D C B A E (1) D C A B E (1) D B A E C (1) C B D A E (1) C A D B E (1) B E D C A (1) B E C D A (1) B C D E A (1) A E C D B (1) A E C B D (1) A D E B C (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 22 14 10 16 B -22 0 -14 -8 10 C -14 14 0 4 14 D -10 8 -4 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 14 10 16 B -22 0 -14 -8 10 C -14 14 0 4 14 D -10 8 -4 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=23 C=16 D=15 B=11 so B is eliminated. Round 2 votes counts: A=35 E=25 C=25 D=15 so D is eliminated. Round 3 votes counts: A=40 E=33 C=27 so C is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 C:209 D:194 B:183 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 14 10 16 B -22 0 -14 -8 10 C -14 14 0 4 14 D -10 8 -4 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 14 10 16 B -22 0 -14 -8 10 C -14 14 0 4 14 D -10 8 -4 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 14 10 16 B -22 0 -14 -8 10 C -14 14 0 4 14 D -10 8 -4 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4948: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) C E B A D (7) A C E D B (7) A D B C E (6) A C E B D (6) B D E C A (5) D A B E C (4) C E D B A (4) C E A D B (4) B D A E C (4) A D C E B (4) D B E C A (3) C E A B D (3) B E D C A (3) B D E A C (3) A B C E D (3) E B C D A (2) D E C B A (2) D B A E C (2) C E D A B (2) B A E C D (2) B A D E C (2) A D B E C (2) A C D E B (2) E C D B A (1) E B C A D (1) C E B D A (1) C A E B D (1) B E C D A (1) B E C A D (1) A D C B E (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 12 0 B 2 0 -6 12 -8 C -4 6 0 14 8 D -12 -12 -14 0 -14 E 0 8 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.726017 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.273983 Sum of squares = 0.602167676447 Cumulative probabilities = A: 0.726017 B: 0.726017 C: 0.726017 D: 0.726017 E: 1.000000 A B C D E A 0 -2 4 12 0 B 2 0 -6 12 -8 C -4 6 0 14 8 D -12 -12 -14 0 -14 E 0 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555776239 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=22 B=21 E=11 D=11 so E is eliminated. Round 2 votes counts: A=35 C=30 B=24 D=11 so D is eliminated. Round 3 votes counts: A=39 C=32 B=29 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:207 E:207 B:200 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 12 0 B 2 0 -6 12 -8 C -4 6 0 14 8 D -12 -12 -14 0 -14 E 0 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555776239 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 12 0 B 2 0 -6 12 -8 C -4 6 0 14 8 D -12 -12 -14 0 -14 E 0 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555776239 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 12 0 B 2 0 -6 12 -8 C -4 6 0 14 8 D -12 -12 -14 0 -14 E 0 8 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555776239 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4949: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) A D B E C (9) E C B A D (5) E B A D C (5) C D A B E (5) C E B A D (4) E B C D A (3) E A B D C (3) D B A E C (3) D A B E C (3) D A B C E (3) C E D A B (3) C E A D B (3) C E A B D (3) C D B A E (3) C A D E B (3) B E A D C (3) B A E D C (3) B A D E C (3) E B C A D (2) D C A B E (2) D B A C E (2) D A C B E (2) C E D B A (2) A B D E C (2) E C B D A (1) E C A B D (1) E B A C D (1) C D E B A (1) C D E A B (1) C A D B E (1) B E D A C (1) B D E A C (1) B D A E C (1) A E B D C (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -4 8 -4 B 6 0 -2 4 -4 C 4 2 0 4 -2 D -8 -4 -4 0 -8 E 4 4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 8 -4 B 6 0 -2 4 -4 C 4 2 0 4 -2 D -8 -4 -4 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999244 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=21 D=15 A=14 B=12 so B is eliminated. Round 2 votes counts: C=38 E=25 A=20 D=17 so D is eliminated. Round 3 votes counts: C=40 A=34 E=26 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:204 B:202 A:197 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 8 -4 B 6 0 -2 4 -4 C 4 2 0 4 -2 D -8 -4 -4 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999244 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 8 -4 B 6 0 -2 4 -4 C 4 2 0 4 -2 D -8 -4 -4 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999244 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 8 -4 B 6 0 -2 4 -4 C 4 2 0 4 -2 D -8 -4 -4 0 -8 E 4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999244 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4950: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) A B D E C (9) B A E D C (8) B A E C D (7) A D B C E (7) A B D C E (7) D A C B E (6) E C B D A (5) D C E A B (5) B E A C D (5) E C D B A (4) D A C E B (4) D A B C E (4) E C B A D (3) D C A E B (3) C E D A B (3) C D E A B (3) B E C A D (2) B A D E C (2) E B C D A (1) E B C A D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 22 6 18 B -6 0 10 2 18 C -22 -10 0 -14 2 D -6 -2 14 0 2 E -18 -18 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 22 6 18 B -6 0 10 2 18 C -22 -10 0 -14 2 D -6 -2 14 0 2 E -18 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=24 D=22 C=15 E=14 so E is eliminated. Round 2 votes counts: C=27 B=26 A=25 D=22 so D is eliminated. Round 3 votes counts: A=39 C=35 B=26 so B is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:212 D:204 E:180 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 22 6 18 B -6 0 10 2 18 C -22 -10 0 -14 2 D -6 -2 14 0 2 E -18 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 22 6 18 B -6 0 10 2 18 C -22 -10 0 -14 2 D -6 -2 14 0 2 E -18 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 22 6 18 B -6 0 10 2 18 C -22 -10 0 -14 2 D -6 -2 14 0 2 E -18 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4951: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (13) D E C B A (10) A C E D B (10) D B E C A (9) A B C E D (9) D E C A B (6) B D E C A (6) E C D B A (5) B C E D A (4) B A C E D (4) D A E C B (3) A D E C B (3) A D C E B (3) D E B C A (2) C E B D A (2) B C E A D (2) D A B E C (1) C E D B A (1) C E D A B (1) C E A D B (1) C E A B D (1) B A D C E (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 0 0 0 B -8 0 -22 -12 -24 C 0 22 0 10 10 D 0 12 -10 0 -8 E 0 24 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.476355 B: 0.000000 C: 0.523645 D: 0.000000 E: 0.000000 Sum of squares = 0.501118192949 Cumulative probabilities = A: 0.476355 B: 0.476355 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 0 0 B -8 0 -22 -12 -24 C 0 22 0 10 10 D 0 12 -10 0 -8 E 0 24 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 D=31 B=17 C=6 E=5 so E is eliminated. Round 2 votes counts: A=41 D=31 B=17 C=11 so C is eliminated. Round 3 votes counts: A=43 D=38 B=19 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:221 E:211 A:204 D:197 B:167 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 0 0 B -8 0 -22 -12 -24 C 0 22 0 10 10 D 0 12 -10 0 -8 E 0 24 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 0 0 B -8 0 -22 -12 -24 C 0 22 0 10 10 D 0 12 -10 0 -8 E 0 24 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 0 0 B -8 0 -22 -12 -24 C 0 22 0 10 10 D 0 12 -10 0 -8 E 0 24 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999892 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4952: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) A B E D C (9) E B C D A (7) C D E B A (7) E C D B A (5) C D E A B (5) C D A E B (5) E B A C D (4) B A E D C (4) A D C B E (4) A C D E B (4) E C D A B (3) D C B A E (3) A D B C E (3) D C B E A (2) C E D B A (2) B E A D C (2) B D C E A (2) B D A C E (2) A E C D B (2) A B E C D (2) E A B C D (1) D C E B A (1) D C A E B (1) D C A B E (1) C E D A B (1) C D B E A (1) C A D E B (1) B E D C A (1) B E C D A (1) B D C A E (1) B A D C E (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 0 -2 10 B -8 0 4 -2 0 C 0 -4 0 4 14 D 2 2 -4 0 10 E -10 0 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.470541 B: 0.000000 C: 0.529459 D: 0.000000 E: 0.000000 Sum of squares = 0.501735710361 Cumulative probabilities = A: 0.470541 B: 0.470541 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -2 10 B -8 0 4 -2 0 C 0 -4 0 4 14 D 2 2 -4 0 10 E -10 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=22 E=20 B=14 D=8 so D is eliminated. Round 2 votes counts: A=36 C=30 E=20 B=14 so B is eliminated. Round 3 votes counts: A=43 C=33 E=24 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:207 D:205 B:197 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 -2 10 B -8 0 4 -2 0 C 0 -4 0 4 14 D 2 2 -4 0 10 E -10 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -2 10 B -8 0 4 -2 0 C 0 -4 0 4 14 D 2 2 -4 0 10 E -10 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -2 10 B -8 0 4 -2 0 C 0 -4 0 4 14 D 2 2 -4 0 10 E -10 0 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4953: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) D B A E C (6) C E B A D (6) D C E B A (5) E D C A B (4) D E C A B (4) D B A C E (4) B A C E D (4) A B D E C (4) A B C E D (4) E A D C B (3) E A C B D (3) D E C B A (3) D E A B C (3) D C B E A (3) C B A E D (3) B A C D E (3) D E A C B (2) D B C A E (2) C E D B A (2) B C A D E (2) A E D B C (2) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) D E B C A (1) D B C E A (1) D A B E C (1) C E A B D (1) C D E B A (1) C B E A D (1) B D A C E (1) B C D A E (1) B C A E D (1) B A D E C (1) A E C B D (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 12 8 6 B 18 0 8 -2 6 C -12 -8 0 -24 6 D -8 2 24 0 14 E -6 -6 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.071429 B: 0.285714 C: 0.000000 D: 0.642857 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.071429 B: 0.357143 C: 0.357143 D: 1.000000 E: 1.000000 A B C D E A 0 -18 12 8 6 B 18 0 8 -2 6 C -12 -8 0 -24 6 D -8 2 24 0 14 E -6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.285714 C: 0.000000 D: 0.642857 E: 0.000000 Sum of squares = 0.500000000069 Cumulative probabilities = A: 0.071429 B: 0.357143 C: 0.357143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=21 A=16 E=14 C=14 so E is eliminated. Round 2 votes counts: D=40 A=22 B=21 C=17 so C is eliminated. Round 3 votes counts: D=44 B=31 A=25 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:215 A:204 E:184 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 12 8 6 B 18 0 8 -2 6 C -12 -8 0 -24 6 D -8 2 24 0 14 E -6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.285714 C: 0.000000 D: 0.642857 E: 0.000000 Sum of squares = 0.500000000069 Cumulative probabilities = A: 0.071429 B: 0.357143 C: 0.357143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 12 8 6 B 18 0 8 -2 6 C -12 -8 0 -24 6 D -8 2 24 0 14 E -6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.285714 C: 0.000000 D: 0.642857 E: 0.000000 Sum of squares = 0.500000000069 Cumulative probabilities = A: 0.071429 B: 0.357143 C: 0.357143 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 12 8 6 B 18 0 8 -2 6 C -12 -8 0 -24 6 D -8 2 24 0 14 E -6 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.071429 B: 0.285714 C: 0.000000 D: 0.642857 E: 0.000000 Sum of squares = 0.500000000069 Cumulative probabilities = A: 0.071429 B: 0.357143 C: 0.357143 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4954: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) E C B A D (5) D A B C E (5) B A D C E (5) D E B A C (4) D B A E C (4) C A E B D (4) E D C B A (3) E C D A B (3) E B A D C (3) D C A B E (3) D B A C E (3) D A C B E (3) C A D B E (3) B E A D C (3) B A E C D (3) A B C D E (3) E D C A B (2) E D B C A (2) E C A B D (2) E B D A C (2) C E A B D (2) C A B D E (2) B E A C D (2) B D A E C (2) B A C E D (2) E D B A C (1) E C D B A (1) E C B D A (1) E B A C D (1) D E C A B (1) D E B C A (1) D C E A B (1) D C A E B (1) C E D A B (1) C D E A B (1) C D A E B (1) C A E D B (1) B E D A C (1) B D E A C (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 4 12 B 0 0 -2 8 12 C -2 2 0 -6 4 D -4 -8 6 0 -6 E -12 -12 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.542470 B: 0.457530 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.503607391698 Cumulative probabilities = A: 0.542470 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 12 B 0 0 -2 8 12 C -2 2 0 -6 4 D -4 -8 6 0 -6 E -12 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500102 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000020664 Cumulative probabilities = A: 0.500102 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=26 D=26 C=23 B=19 A=6 so A is eliminated. Round 2 votes counts: D=27 E=26 C=24 B=23 so B is eliminated. Round 3 votes counts: D=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:209 B:209 C:199 D:194 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 4 12 B 0 0 -2 8 12 C -2 2 0 -6 4 D -4 -8 6 0 -6 E -12 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500102 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000020664 Cumulative probabilities = A: 0.500102 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 12 B 0 0 -2 8 12 C -2 2 0 -6 4 D -4 -8 6 0 -6 E -12 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500102 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000020664 Cumulative probabilities = A: 0.500102 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 12 B 0 0 -2 8 12 C -2 2 0 -6 4 D -4 -8 6 0 -6 E -12 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500102 B: 0.499898 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000020664 Cumulative probabilities = A: 0.500102 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4955: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) E D C A B (8) B A C D E (8) A B D E C (7) D E A C B (6) C E D A B (5) C D E A B (5) B C A D E (5) B A E D C (5) E D A C B (4) C B E D A (4) C B A D E (4) D E C A B (3) C E D B A (3) C D E B A (2) C B D E A (2) C B D A E (2) C A B D E (2) B C A E D (2) A D E B C (2) E D C B A (1) E A D B C (1) C A D E B (1) C A D B E (1) B E A D C (1) B A C E D (1) A E D B C (1) A D E C B (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 20 0 -10 -10 B -20 0 -10 -8 -6 C 0 10 0 -2 -2 D 10 8 2 0 6 E 10 6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 0 -10 -10 B -20 0 -10 -8 -6 C 0 10 0 -2 -2 D 10 8 2 0 6 E 10 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=24 B=22 A=14 D=9 so D is eliminated. Round 2 votes counts: E=33 C=31 B=22 A=14 so A is eliminated. Round 3 votes counts: E=37 C=32 B=31 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:206 C:203 A:200 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 0 -10 -10 B -20 0 -10 -8 -6 C 0 10 0 -2 -2 D 10 8 2 0 6 E 10 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 -10 -10 B -20 0 -10 -8 -6 C 0 10 0 -2 -2 D 10 8 2 0 6 E 10 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 -10 -10 B -20 0 -10 -8 -6 C 0 10 0 -2 -2 D 10 8 2 0 6 E 10 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4956: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) D C E A B (8) B E A D C (8) D E C A B (7) B A C E D (7) E D B A C (6) C A B D E (6) B A E C D (6) E B D A C (5) C A D B E (5) E D B C A (4) C D A E B (4) E D C B A (3) D E C B A (3) D C A E B (3) C D A B E (3) A B C E D (3) A B C D E (2) E B A D C (1) E A D C B (1) D C E B A (1) C A D E B (1) B C D E A (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 2 4 4 B -10 0 -14 -2 4 C -2 14 0 0 12 D -4 2 0 0 12 E -4 -4 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 4 4 B -10 0 -14 -2 4 C -2 14 0 0 12 D -4 2 0 0 12 E -4 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=22 B=22 E=20 C=19 A=17 so A is eliminated. Round 2 votes counts: C=31 B=27 D=22 E=20 so E is eliminated. Round 3 votes counts: D=36 B=33 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:212 A:210 D:205 B:189 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 4 4 B -10 0 -14 -2 4 C -2 14 0 0 12 D -4 2 0 0 12 E -4 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 4 4 B -10 0 -14 -2 4 C -2 14 0 0 12 D -4 2 0 0 12 E -4 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 4 4 B -10 0 -14 -2 4 C -2 14 0 0 12 D -4 2 0 0 12 E -4 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4957: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) C A E B D (8) C A B E D (8) E D B A C (7) A C E D B (6) D E B A C (4) D B E C A (4) C A D E B (4) D B C A E (3) C B A D E (3) C A D B E (3) C A B D E (3) A E C B D (3) E B A D C (2) E A C B D (2) E A B C D (2) D C B A E (2) D B E A C (2) C B A E D (2) B D E C A (2) B C D A E (2) E B D A C (1) E B A C D (1) E A D C B (1) E A C D B (1) D E A C B (1) D C E B A (1) D C A E B (1) D A E C B (1) D A C E B (1) C D B A E (1) C D A E B (1) C A E D B (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E A C (1) B D C A E (1) B C A D E (1) Total count = 100 A B C D E A 0 14 -2 26 30 B -14 0 -28 10 -18 C 2 28 0 26 24 D -26 -10 -26 0 -16 E -30 18 -24 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 26 30 B -14 0 -28 10 -18 C 2 28 0 26 24 D -26 -10 -26 0 -16 E -30 18 -24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=20 A=19 E=17 B=10 so B is eliminated. Round 2 votes counts: C=37 D=24 E=20 A=19 so A is eliminated. Round 3 votes counts: C=53 D=24 E=23 so E is eliminated. Round 4 votes counts: C=63 D=37 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:240 A:234 E:190 B:175 D:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 26 30 B -14 0 -28 10 -18 C 2 28 0 26 24 D -26 -10 -26 0 -16 E -30 18 -24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 26 30 B -14 0 -28 10 -18 C 2 28 0 26 24 D -26 -10 -26 0 -16 E -30 18 -24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 26 30 B -14 0 -28 10 -18 C 2 28 0 26 24 D -26 -10 -26 0 -16 E -30 18 -24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4958: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) D C A E B (8) B D E C A (7) B E D A C (5) B E A C D (5) B C A E D (5) D E B A C (3) D B C E A (3) C D A B E (3) C A E B D (3) B C D A E (3) A C E B D (3) E A D C B (2) D E A C B (2) D C B E A (2) D C A B E (2) C D A E B (2) C A B D E (2) B D C E A (2) A E D C B (2) E D B A C (1) E D A C B (1) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D E C B A (1) D E B C A (1) D E A B C (1) D C B A E (1) D A C E B (1) C D B A E (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B E D (1) B E D C A (1) B C E A D (1) B C A D E (1) B A C E D (1) A E C D B (1) A E C B D (1) A D C E B (1) A C E D B (1) A C D E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -24 -8 16 B -4 0 -10 -4 2 C 24 10 0 0 22 D 8 4 0 0 20 E -16 -2 -22 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454406 D: 0.545594 E: 0.000000 Sum of squares = 0.504157608797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454406 D: 1.000000 E: 1.000000 A B C D E A 0 4 -24 -8 16 B -4 0 -10 -4 2 C 24 10 0 0 22 D 8 4 0 0 20 E -16 -2 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=25 C=24 A=12 E=8 so E is eliminated. Round 2 votes counts: B=33 D=28 C=24 A=15 so A is eliminated. Round 3 votes counts: B=35 D=33 C=32 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:228 D:216 A:194 B:192 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -24 -8 16 B -4 0 -10 -4 2 C 24 10 0 0 22 D 8 4 0 0 20 E -16 -2 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -24 -8 16 B -4 0 -10 -4 2 C 24 10 0 0 22 D 8 4 0 0 20 E -16 -2 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -24 -8 16 B -4 0 -10 -4 2 C 24 10 0 0 22 D 8 4 0 0 20 E -16 -2 -22 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4959: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (6) E C A B D (5) E C A D B (4) D C E B A (4) D B C E A (4) D B A C E (4) D A B E C (4) C B E D A (4) E A C D B (3) C E D B A (3) C E B A D (3) B D C A E (3) B C E D A (3) B C E A D (3) B C D E A (3) A E D C B (3) A E D B C (3) A D E C B (3) D E C B A (2) D C B E A (2) D B C A E (2) D A E B C (2) B D C E A (2) B C A E D (2) B C A D E (2) A D E B C (2) A D B E C (2) A B E C D (2) A B D E C (2) E D C A B (1) E A D C B (1) D E A C B (1) C E D A B (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E A D (1) C B D E A (1) B D A C E (1) B A C E D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -16 2 -8 B 4 0 -4 -8 0 C 16 4 0 4 6 D -2 8 -4 0 -4 E 8 0 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 2 -8 B 4 0 -4 -8 0 C 16 4 0 4 6 D -2 8 -4 0 -4 E 8 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 B=20 C=16 E=14 so E is eliminated. Round 2 votes counts: A=29 D=26 C=25 B=20 so B is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:203 D:199 B:196 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 2 -8 B 4 0 -4 -8 0 C 16 4 0 4 6 D -2 8 -4 0 -4 E 8 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 2 -8 B 4 0 -4 -8 0 C 16 4 0 4 6 D -2 8 -4 0 -4 E 8 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 2 -8 B 4 0 -4 -8 0 C 16 4 0 4 6 D -2 8 -4 0 -4 E 8 0 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4960: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (13) D C A B E (13) B E D C A (9) A C D B E (7) E B A D C (5) D C B A E (5) A C E D B (5) E B D C A (4) B E C A D (3) A E C B D (3) A C E B D (3) D A C B E (2) C A D B E (2) B E A C D (2) B D E C A (2) A D C E B (2) A C D E B (2) E D B C A (1) E D B A C (1) E B D A C (1) E A D C B (1) E A D B C (1) E A C B D (1) E A B C D (1) D E B C A (1) D B E C A (1) D B C E A (1) D B C A E (1) C A B D E (1) B E C D A (1) B D C E A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C E D (1) Total count = 100 A B C D E A 0 -12 2 8 -2 B 12 0 6 8 10 C -2 -6 0 -4 -2 D -8 -8 4 0 -12 E 2 -10 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 8 -2 B 12 0 6 8 10 C -2 -6 0 -4 -2 D -8 -8 4 0 -12 E 2 -10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=24 B=22 A=22 C=3 so C is eliminated. Round 2 votes counts: E=29 A=25 D=24 B=22 so B is eliminated. Round 3 votes counts: E=44 D=29 A=27 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:218 E:203 A:198 C:193 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 8 -2 B 12 0 6 8 10 C -2 -6 0 -4 -2 D -8 -8 4 0 -12 E 2 -10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 8 -2 B 12 0 6 8 10 C -2 -6 0 -4 -2 D -8 -8 4 0 -12 E 2 -10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 8 -2 B 12 0 6 8 10 C -2 -6 0 -4 -2 D -8 -8 4 0 -12 E 2 -10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4961: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) B E A C D (7) B A C E D (7) D C E A B (6) A B C D E (6) E D B C A (5) D C A E B (5) E C D A B (4) D A C B E (4) B E D A C (4) E D C B A (3) E B D C A (3) E B C A D (3) D E C A B (3) C D A E B (3) B A E C D (3) C E A D B (2) B E C A D (2) B A D E C (2) A D C B E (2) A C B D E (2) E D C A B (1) E D B A C (1) E C D B A (1) E C B A D (1) E C A B D (1) E B D A C (1) D A B C E (1) C E B A D (1) C A E B D (1) C A D E B (1) C A D B E (1) B E A D C (1) B D A C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 6 8 0 B 12 0 14 12 8 C -6 -14 0 14 8 D -8 -12 -14 0 -4 E 0 -8 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 8 0 B 12 0 14 12 8 C -6 -14 0 14 8 D -8 -12 -14 0 -4 E 0 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=24 D=19 A=11 C=9 so C is eliminated. Round 2 votes counts: B=37 E=27 D=22 A=14 so A is eliminated. Round 3 votes counts: B=45 E=28 D=27 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:201 C:201 E:194 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 8 0 B 12 0 14 12 8 C -6 -14 0 14 8 D -8 -12 -14 0 -4 E 0 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 8 0 B 12 0 14 12 8 C -6 -14 0 14 8 D -8 -12 -14 0 -4 E 0 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 8 0 B 12 0 14 12 8 C -6 -14 0 14 8 D -8 -12 -14 0 -4 E 0 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4962: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) D A C B E (7) B D A C E (7) A D C E B (5) A D B C E (5) D C A B E (4) B C E D A (4) E C B A D (3) E C A D B (3) E A C B D (3) D A B C E (3) C E D A B (3) B D C A E (3) A B D E C (3) E C A B D (2) E B C D A (2) E B C A D (2) C D B E A (2) C D A B E (2) C B D E A (2) A D C B E (2) E C B D A (1) E B A C D (1) E A C D B (1) D C B A E (1) D B A C E (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A E B (1) C B E D A (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E A C (1) B D A E C (1) B A D E C (1) A E D C B (1) A D E C B (1) A D E B C (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -2 -20 6 B -4 0 -20 0 12 C 2 20 0 0 36 D 20 0 0 0 14 E -6 -12 -36 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.417515 D: 0.582485 E: 0.000000 Sum of squares = 0.513607672769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.417515 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -20 6 B -4 0 -20 0 12 C 2 20 0 0 36 D 20 0 0 0 14 E -6 -12 -36 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=21 A=21 E=18 D=16 so D is eliminated. Round 2 votes counts: A=31 C=29 B=22 E=18 so E is eliminated. Round 3 votes counts: C=38 A=35 B=27 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:217 A:194 B:194 E:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -20 6 B -4 0 -20 0 12 C 2 20 0 0 36 D 20 0 0 0 14 E -6 -12 -36 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -20 6 B -4 0 -20 0 12 C 2 20 0 0 36 D 20 0 0 0 14 E -6 -12 -36 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -20 6 B -4 0 -20 0 12 C 2 20 0 0 36 D 20 0 0 0 14 E -6 -12 -36 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4963: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) D A E C B (8) E B C D A (6) E B C A D (6) C B A D E (6) B E C D A (6) B C E A D (6) A D C B E (6) D A E B C (5) E D A C B (4) D A C E B (4) E B D A C (3) D A C B E (3) D A B C E (3) A D E C B (3) E D A B C (2) C E B A D (2) C B E A D (2) C A D E B (2) B C D A E (2) E D B A C (1) E C B A D (1) D E A B C (1) D B A C E (1) C E A B D (1) C B A E D (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C A D (1) B D A C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -6 -2 14 B 10 0 8 10 2 C 6 -8 0 8 4 D 2 -10 -8 0 14 E -14 -2 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -2 14 B 10 0 8 10 2 C 6 -8 0 8 4 D 2 -10 -8 0 14 E -14 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 E=23 C=16 A=10 so A is eliminated. Round 2 votes counts: D=34 B=26 E=23 C=17 so C is eliminated. Round 3 votes counts: D=38 B=36 E=26 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:205 D:199 A:198 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -2 14 B 10 0 8 10 2 C 6 -8 0 8 4 D 2 -10 -8 0 14 E -14 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -2 14 B 10 0 8 10 2 C 6 -8 0 8 4 D 2 -10 -8 0 14 E -14 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -2 14 B 10 0 8 10 2 C 6 -8 0 8 4 D 2 -10 -8 0 14 E -14 -2 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4964: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) A D E B C (7) E B C A D (6) D B C E A (6) A D C E B (6) E C B A D (5) D A C B E (5) C E B A D (4) D B A E C (3) B C E D A (3) B C D E A (3) A E C B D (3) A D E C B (3) E B A C D (2) E A B C D (2) D A C E B (2) C E B D A (2) C E A B D (2) C B E D A (2) C A E B D (2) B E D A C (2) B E C D A (2) B D E C A (2) B D C E A (2) A E C D B (2) E C A B D (1) E A C B D (1) D C A B E (1) D B E C A (1) D B C A E (1) C E A D B (1) C D B E A (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A D C (1) A E D C B (1) A E D B C (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 2 0 8 -4 B -2 0 8 0 -14 C 0 -8 0 -6 4 D -8 0 6 0 2 E 4 14 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.026667 C: 0.386667 D: 0.080000 E: 0.173333 Sum of squares = 0.297777777797 Cumulative probabilities = A: 0.333333 B: 0.360000 C: 0.746667 D: 0.826667 E: 1.000000 A B C D E A 0 2 0 8 -4 B -2 0 8 0 -14 C 0 -8 0 -6 4 D -8 0 6 0 2 E 4 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.026667 C: 0.386667 D: 0.080000 E: 0.173333 Sum of squares = 0.297777777735 Cumulative probabilities = A: 0.333333 B: 0.360000 C: 0.746667 D: 0.826667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 E=17 C=16 B=16 so C is eliminated. Round 2 votes counts: A=29 D=27 E=26 B=18 so B is eliminated. Round 3 votes counts: E=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:206 A:203 D:200 B:196 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 8 -4 B -2 0 8 0 -14 C 0 -8 0 -6 4 D -8 0 6 0 2 E 4 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.026667 C: 0.386667 D: 0.080000 E: 0.173333 Sum of squares = 0.297777777735 Cumulative probabilities = A: 0.333333 B: 0.360000 C: 0.746667 D: 0.826667 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 8 -4 B -2 0 8 0 -14 C 0 -8 0 -6 4 D -8 0 6 0 2 E 4 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.026667 C: 0.386667 D: 0.080000 E: 0.173333 Sum of squares = 0.297777777735 Cumulative probabilities = A: 0.333333 B: 0.360000 C: 0.746667 D: 0.826667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 8 -4 B -2 0 8 0 -14 C 0 -8 0 -6 4 D -8 0 6 0 2 E 4 14 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.026667 C: 0.386667 D: 0.080000 E: 0.173333 Sum of squares = 0.297777777735 Cumulative probabilities = A: 0.333333 B: 0.360000 C: 0.746667 D: 0.826667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4965: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) B C A E D (6) B A E C D (6) D C B A E (5) B C D A E (5) C D B E A (4) C B E D A (4) C B D E A (4) A B E D C (4) E B C A D (3) E A D C B (3) E A C B D (3) E A B C D (3) D C E A B (3) D C A E B (3) D A E C B (3) C D E B A (3) E D A C B (2) E A D B C (2) D C B E A (2) C B E A D (2) A E B D C (2) A D E B C (2) A B E C D (2) E C A B D (1) E A B D C (1) D C A B E (1) D B C A E (1) D A E B C (1) D A B C E (1) C E B D A (1) C E B A D (1) C D B A E (1) B E C A D (1) B E A C D (1) B C E A D (1) B C D E A (1) B C A D E (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -8 10 8 B 6 0 8 8 8 C 8 -8 0 10 2 D -10 -8 -10 0 -16 E -8 -8 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 10 8 B 6 0 8 8 8 C 8 -8 0 10 2 D -10 -8 -10 0 -16 E -8 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 D=20 C=20 A=20 E=18 so E is eliminated. Round 2 votes counts: A=32 B=25 D=22 C=21 so C is eliminated. Round 3 votes counts: B=37 A=33 D=30 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:206 A:202 E:199 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 10 8 B 6 0 8 8 8 C 8 -8 0 10 2 D -10 -8 -10 0 -16 E -8 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 10 8 B 6 0 8 8 8 C 8 -8 0 10 2 D -10 -8 -10 0 -16 E -8 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 10 8 B 6 0 8 8 8 C 8 -8 0 10 2 D -10 -8 -10 0 -16 E -8 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4966: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) E A D C B (9) D C B A E (9) B C D A E (9) E A B C D (8) A E B D C (6) B C D E A (4) A E D C B (4) D A E C B (3) D A C E B (3) B A E C D (3) A E D B C (3) E D C A B (2) E D A C B (2) D C E A B (2) C D B A E (2) C B D E A (2) B D C A E (2) B C A D E (2) A E B C D (2) A B E D C (2) E C D A B (1) E A C B D (1) E A B D C (1) D E C A B (1) D E A C B (1) D C A E B (1) D B C A E (1) B E C A D (1) B E A C D (1) B C E D A (1) A D E B C (1) Total count = 100 A B C D E A 0 6 0 -12 6 B -6 0 -6 -10 -2 C 0 6 0 -6 -4 D 12 10 6 0 6 E -6 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -12 6 B -6 0 -6 -10 -2 C 0 6 0 -6 -4 D 12 10 6 0 6 E -6 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=23 D=21 A=18 C=14 so C is eliminated. Round 2 votes counts: D=33 B=25 E=24 A=18 so A is eliminated. Round 3 votes counts: E=39 D=34 B=27 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:200 C:198 E:197 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -12 6 B -6 0 -6 -10 -2 C 0 6 0 -6 -4 D 12 10 6 0 6 E -6 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -12 6 B -6 0 -6 -10 -2 C 0 6 0 -6 -4 D 12 10 6 0 6 E -6 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -12 6 B -6 0 -6 -10 -2 C 0 6 0 -6 -4 D 12 10 6 0 6 E -6 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4967: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) C E D B A (6) B A D E C (6) E C B A D (5) D A B C E (5) C E D A B (5) D A B E C (4) C B E A D (4) E C B D A (3) D C A E B (3) D A E B C (3) C E B D A (3) C E B A D (3) B E A C D (3) B A D C E (3) B A C D E (3) A D B E C (3) E D C A B (2) E D A C B (2) E C D B A (2) E B A C D (2) D A C B E (2) C B A D E (2) B C A D E (2) A B D E C (2) E B C A D (1) E B A D C (1) E A B D C (1) D C E A B (1) D C A B E (1) D A E C B (1) C D E A B (1) C B D A E (1) C B A E D (1) B E C A D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 -10 -12 B 4 0 -16 -4 -10 C 14 16 0 16 -4 D 10 4 -16 0 -10 E 12 10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -14 -10 -12 B 4 0 -16 -4 -10 C 14 16 0 16 -4 D 10 4 -16 0 -10 E 12 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 D=20 B=18 A=7 so A is eliminated. Round 2 votes counts: E=29 C=26 D=24 B=21 so B is eliminated. Round 3 votes counts: D=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:221 E:218 D:194 B:187 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -14 -10 -12 B 4 0 -16 -4 -10 C 14 16 0 16 -4 D 10 4 -16 0 -10 E 12 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -10 -12 B 4 0 -16 -4 -10 C 14 16 0 16 -4 D 10 4 -16 0 -10 E 12 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -10 -12 B 4 0 -16 -4 -10 C 14 16 0 16 -4 D 10 4 -16 0 -10 E 12 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4968: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) D A E B C (7) D C E B A (6) A B C E D (6) D E A B C (5) D A B E C (5) E D C B A (4) E B C A D (4) D A C B E (4) C B A E D (4) A B E C D (4) E C D B A (3) E B A C D (3) C E B A D (3) A D B C E (3) E C B D A (2) D E C A B (2) D A B C E (2) C B E A D (2) B C A E D (2) B A C E D (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B A D (1) E A B D C (1) D E A C B (1) D C B A E (1) D C A B E (1) D A E C B (1) C E B D A (1) C D B E A (1) C B E D A (1) C B A D E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -22 -8 B 8 0 0 -20 -16 C 2 0 0 -18 -16 D 22 20 18 0 6 E 8 16 16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -22 -8 B 8 0 0 -20 -16 C 2 0 0 -18 -16 D 22 20 18 0 6 E 8 16 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=46 E=21 A=16 C=13 B=4 so B is eliminated. Round 2 votes counts: D=46 E=21 A=18 C=15 so C is eliminated. Round 3 votes counts: D=47 E=28 A=25 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:233 E:217 B:186 C:184 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -22 -8 B 8 0 0 -20 -16 C 2 0 0 -18 -16 D 22 20 18 0 6 E 8 16 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -22 -8 B 8 0 0 -20 -16 C 2 0 0 -18 -16 D 22 20 18 0 6 E 8 16 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -22 -8 B 8 0 0 -20 -16 C 2 0 0 -18 -16 D 22 20 18 0 6 E 8 16 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4969: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (20) A E B C D (14) D B C E A (12) D C B E A (9) D B A E C (5) E C A B D (3) D B C A E (2) D B A C E (2) C E A D B (2) C E A B D (2) A E C D B (2) A B E D C (2) A B D E C (2) E C B A D (1) E A C B D (1) D B E C A (1) D B E A C (1) D A B E C (1) C E B D A (1) C E B A D (1) C D E B A (1) C D E A B (1) C D B E A (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E A C (1) B D C E A (1) B D A E C (1) B C E D A (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 14 20 18 16 B -14 0 4 8 -10 C -20 -4 0 6 -24 D -18 -8 -6 0 -12 E -16 10 24 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 18 16 B -14 0 4 8 -10 C -20 -4 0 6 -24 D -18 -8 -6 0 -12 E -16 10 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 D=33 C=10 B=6 E=5 so E is eliminated. Round 2 votes counts: A=47 D=33 C=14 B=6 so B is eliminated. Round 3 votes counts: A=49 D=36 C=15 so C is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:215 B:194 C:179 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 18 16 B -14 0 4 8 -10 C -20 -4 0 6 -24 D -18 -8 -6 0 -12 E -16 10 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 18 16 B -14 0 4 8 -10 C -20 -4 0 6 -24 D -18 -8 -6 0 -12 E -16 10 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 18 16 B -14 0 4 8 -10 C -20 -4 0 6 -24 D -18 -8 -6 0 -12 E -16 10 24 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4970: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) C D A E B (5) C D A B E (5) B E C A D (5) E B C D A (4) B E A C D (4) E B D A C (3) D C A E B (3) C E B D A (3) C D E A B (3) C B E D A (3) C A B D E (3) B A E D C (3) A D C B E (3) A B E D C (3) A B D E C (3) E B D C A (2) E B A D C (2) D E C A B (2) D E A C B (2) D A E C B (2) C B E A D (2) C B D E A (2) C A D B E (2) B E A D C (2) B C E A D (2) B A C D E (2) A D E B C (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) E A B D C (1) D E A B C (1) D C A B E (1) D A E B C (1) C D B A E (1) C B A D E (1) B E C D A (1) B C A D E (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -8 -12 2 B -8 0 -6 10 4 C 8 6 0 2 2 D 12 -10 -2 0 10 E -2 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -12 2 B -8 0 -6 10 4 C 8 6 0 2 2 D 12 -10 -2 0 10 E -2 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=20 D=19 E=17 A=14 so A is eliminated. Round 2 votes counts: C=31 B=27 D=25 E=17 so E is eliminated. Round 3 votes counts: B=39 C=32 D=29 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:205 B:200 A:195 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -12 2 B -8 0 -6 10 4 C 8 6 0 2 2 D 12 -10 -2 0 10 E -2 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -12 2 B -8 0 -6 10 4 C 8 6 0 2 2 D 12 -10 -2 0 10 E -2 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -12 2 B -8 0 -6 10 4 C 8 6 0 2 2 D 12 -10 -2 0 10 E -2 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4971: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) D B C E A (9) B D A E C (6) A E C B D (6) C D B E A (5) B D E A C (5) B D C A E (5) A E B D C (5) D C B E A (4) E C A D B (3) E A D B C (3) D E B A C (3) D C E B A (3) D B E A C (3) C A E B D (3) A E B C D (3) D E A B C (2) C D E A B (2) C B D A E (2) C A B E D (2) E D A C B (1) E D A B C (1) E A D C B (1) E A C D B (1) D E C A B (1) D B E C A (1) C E D A B (1) C D B A E (1) B C D A E (1) B C A D E (1) B A E D C (1) B A D E C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -12 -12 -22 B 2 0 2 -14 -4 C 12 -2 0 -12 4 D 12 14 12 0 10 E 22 4 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -12 -22 B 2 0 2 -14 -4 C 12 -2 0 -12 4 D 12 14 12 0 10 E 22 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 B=20 A=16 E=10 so E is eliminated. Round 2 votes counts: C=31 D=28 A=21 B=20 so B is eliminated. Round 3 votes counts: D=44 C=33 A=23 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:206 C:201 B:193 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 -12 -22 B 2 0 2 -14 -4 C 12 -2 0 -12 4 D 12 14 12 0 10 E 22 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -12 -22 B 2 0 2 -14 -4 C 12 -2 0 -12 4 D 12 14 12 0 10 E 22 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -12 -22 B 2 0 2 -14 -4 C 12 -2 0 -12 4 D 12 14 12 0 10 E 22 4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4972: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (12) A D E C B (8) A D B E C (8) E C B A D (7) A D E B C (7) E B C A D (6) C E B D A (6) B D C A E (4) E C A D B (3) E C A B D (3) D A C B E (3) B C E D A (3) A E D C B (3) A E D B C (3) D B C A E (2) D B A C E (2) C B E D A (2) E C B D A (1) E A D B C (1) E A C D B (1) E A C B D (1) E A B C D (1) D C B A E (1) D A C E B (1) C E D A B (1) C D E A B (1) C D B E A (1) C B D E A (1) B E C D A (1) B E A C D (1) B D A C E (1) B C D E A (1) B A E D C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 18 12 12 16 B -18 0 12 -18 -8 C -12 -12 0 -18 -14 D -12 18 18 0 10 E -16 8 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 12 12 16 B -18 0 12 -18 -8 C -12 -12 0 -18 -14 D -12 18 18 0 10 E -16 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 D=21 C=12 B=12 so C is eliminated. Round 2 votes counts: E=31 A=31 D=23 B=15 so B is eliminated. Round 3 votes counts: E=38 A=32 D=30 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 D:217 E:198 B:184 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 12 12 16 B -18 0 12 -18 -8 C -12 -12 0 -18 -14 D -12 18 18 0 10 E -16 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 12 16 B -18 0 12 -18 -8 C -12 -12 0 -18 -14 D -12 18 18 0 10 E -16 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 12 16 B -18 0 12 -18 -8 C -12 -12 0 -18 -14 D -12 18 18 0 10 E -16 8 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4973: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) D C E B A (7) D E C B A (6) B A D C E (6) A B E D C (6) A B E C D (5) B D C A E (4) B D A C E (4) B C D E A (4) B A C E D (4) A E C B D (4) A E B C D (4) E D C A B (3) E D A C B (3) D C B E A (3) D B C E A (3) B D C E A (3) E C A D B (2) C E D B A (2) C D E B A (2) A E D C B (2) A E C D B (2) A B C E D (2) E A C D B (1) D E C A B (1) D B A E C (1) C E A D B (1) C B E D A (1) C B D E A (1) B C E D A (1) B C D A E (1) B A D E C (1) A E D B C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -4 -14 -2 B 8 0 4 2 2 C 4 -4 0 -12 -2 D 14 -2 12 0 -2 E 2 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -14 -2 B 8 0 4 2 2 C 4 -4 0 -12 -2 D 14 -2 12 0 -2 E 2 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=28 A=28 D=21 E=16 C=7 so C is eliminated. Round 2 votes counts: B=30 A=28 D=23 E=19 so E is eliminated. Round 3 votes counts: D=38 A=32 B=30 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:208 E:202 C:193 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -14 -2 B 8 0 4 2 2 C 4 -4 0 -12 -2 D 14 -2 12 0 -2 E 2 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -14 -2 B 8 0 4 2 2 C 4 -4 0 -12 -2 D 14 -2 12 0 -2 E 2 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -14 -2 B 8 0 4 2 2 C 4 -4 0 -12 -2 D 14 -2 12 0 -2 E 2 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4974: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (17) A E C D B (12) C E A D B (9) B D A E C (8) B D A C E (6) D B A E C (5) E C A B D (4) B A D E C (4) E C A D B (3) E A C D B (3) C E A B D (3) E A C B D (2) D A B E C (2) C E D A B (2) C E B A D (2) B D C A E (2) A E D C B (2) A E C B D (2) D B C E A (1) D B A C E (1) D A E B C (1) D A C E B (1) C D B E A (1) C B E D A (1) C A D E B (1) B E C A D (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 6 4 2 B -2 0 4 10 2 C -6 -4 0 -6 -6 D -4 -10 6 0 2 E -2 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 4 2 B -2 0 4 10 2 C -6 -4 0 -6 -6 D -4 -10 6 0 2 E -2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=20 C=19 E=12 D=11 so D is eliminated. Round 2 votes counts: B=45 A=24 C=19 E=12 so E is eliminated. Round 3 votes counts: B=45 A=29 C=26 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:207 B:207 E:200 D:197 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 2 B -2 0 4 10 2 C -6 -4 0 -6 -6 D -4 -10 6 0 2 E -2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 2 B -2 0 4 10 2 C -6 -4 0 -6 -6 D -4 -10 6 0 2 E -2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 2 B -2 0 4 10 2 C -6 -4 0 -6 -6 D -4 -10 6 0 2 E -2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4975: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (6) E D C B A (5) D C E B A (5) C D E B A (5) A B C E D (5) E D B C A (4) C A E B D (4) B A D E C (4) A B E C D (4) E B D A C (3) D E C B A (3) D B E A C (3) C E D A B (3) B D A E C (3) B A E D C (3) A C B E D (3) A C B D E (3) E D B A C (2) E C D B A (2) E B A D C (2) D E B A C (2) D B A C E (2) C E D B A (2) C E A D B (2) C D E A B (2) C D A E B (2) C A D B E (2) A B D C E (2) A B C D E (2) D E B C A (1) D C A B E (1) C D A B E (1) C A E D B (1) C A D E B (1) C A B D E (1) B E A D C (1) B A D C E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 6 -14 -6 B 18 0 4 -2 -4 C -6 -4 0 -10 0 D 14 2 10 0 4 E 6 4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 -14 -6 B 18 0 4 -2 -4 C -6 -4 0 -10 0 D 14 2 10 0 4 E 6 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=21 E=18 B=18 D=17 so D is eliminated. Round 2 votes counts: C=32 E=24 B=23 A=21 so A is eliminated. Round 3 votes counts: C=38 B=37 E=25 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:215 B:208 E:203 C:190 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 6 -14 -6 B 18 0 4 -2 -4 C -6 -4 0 -10 0 D 14 2 10 0 4 E 6 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 -14 -6 B 18 0 4 -2 -4 C -6 -4 0 -10 0 D 14 2 10 0 4 E 6 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 -14 -6 B 18 0 4 -2 -4 C -6 -4 0 -10 0 D 14 2 10 0 4 E 6 4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4976: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) E D A C B (6) B E A C D (6) C A D B E (5) E B C D A (4) E B A D C (4) E A D B C (4) D E C A B (4) D A C E B (4) B C A E D (4) C B A D E (3) B A E C D (3) A B C D E (3) E D C B A (2) E B A C D (2) D E A C B (2) D A C B E (2) C D A B E (2) B C E A D (2) B A C E D (2) B A C D E (2) A C D B E (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) E B D A C (1) E B C A D (1) E A B D C (1) D E C B A (1) D C E A B (1) D C A E B (1) D A E C B (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B C E D A (1) A E D B C (1) A E B D C (1) A D E C B (1) A D C B E (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 8 20 2 B 6 0 12 10 6 C -8 -12 0 14 -6 D -20 -10 -14 0 -6 E -2 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 20 2 B 6 0 12 10 6 C -8 -12 0 14 -6 D -20 -10 -14 0 -6 E -2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=29 D=16 C=13 A=12 so A is eliminated. Round 2 votes counts: B=34 E=32 D=18 C=16 so C is eliminated. Round 3 votes counts: B=40 E=32 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:212 E:202 C:194 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 20 2 B 6 0 12 10 6 C -8 -12 0 14 -6 D -20 -10 -14 0 -6 E -2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 20 2 B 6 0 12 10 6 C -8 -12 0 14 -6 D -20 -10 -14 0 -6 E -2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 20 2 B 6 0 12 10 6 C -8 -12 0 14 -6 D -20 -10 -14 0 -6 E -2 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4977: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) D E C B A (7) D B A E C (7) A B C E D (7) C E A B D (6) E D C B A (5) C E D A B (5) C A B E D (5) B A D E C (5) D E B A C (4) B D A E C (4) A B D C E (4) C A E B D (3) B A D C E (3) E C D B A (2) E C A B D (2) D B A C E (2) C E A D B (2) A C B E D (2) E D B A C (1) E C D A B (1) E C B A D (1) E B D A C (1) E B A D C (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E A C (1) D A B C E (1) C D E A B (1) C A D E B (1) C A B D E (1) B D A C E (1) B A E D C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 10 8 14 B -2 0 6 14 10 C -10 -6 0 -2 10 D -8 -14 2 0 8 E -14 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 8 14 B -2 0 6 14 10 C -10 -6 0 -2 10 D -8 -14 2 0 8 E -14 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 A=23 E=14 B=14 so E is eliminated. Round 2 votes counts: D=31 C=30 A=23 B=16 so B is eliminated. Round 3 votes counts: D=37 A=33 C=30 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:214 C:196 D:194 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 14 B -2 0 6 14 10 C -10 -6 0 -2 10 D -8 -14 2 0 8 E -14 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 14 B -2 0 6 14 10 C -10 -6 0 -2 10 D -8 -14 2 0 8 E -14 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 14 B -2 0 6 14 10 C -10 -6 0 -2 10 D -8 -14 2 0 8 E -14 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4978: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) A B D C E (8) E C D B A (7) B A D C E (7) E C B D A (6) B D A C E (5) A D B C E (5) E C B A D (4) B D C E A (3) B A E C D (3) A D E C B (3) A D C E B (3) E C D A B (2) E C A D B (2) E A C D B (2) D B A C E (2) D A B C E (2) C E D B A (2) C E D A B (2) C D E B A (2) B E C D A (2) B E C A D (2) B A D E C (2) A E D C B (2) A B E C D (2) E C A B D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C B A E (1) D A C E B (1) D A C B E (1) C D E A B (1) B C E D A (1) A E C D B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 20 16 16 B 0 0 10 14 12 C -20 -10 0 -14 -4 D -16 -14 14 0 14 E -16 -12 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.328348 B: 0.671652 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.558928775031 Cumulative probabilities = A: 0.328348 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 20 16 16 B 0 0 10 14 12 C -20 -10 0 -14 -4 D -16 -14 14 0 14 E -16 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=25 B=25 D=9 C=7 so C is eliminated. Round 2 votes counts: A=34 E=29 B=25 D=12 so D is eliminated. Round 3 votes counts: A=38 E=34 B=28 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:218 D:199 E:181 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 20 16 16 B 0 0 10 14 12 C -20 -10 0 -14 -4 D -16 -14 14 0 14 E -16 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 20 16 16 B 0 0 10 14 12 C -20 -10 0 -14 -4 D -16 -14 14 0 14 E -16 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 20 16 16 B 0 0 10 14 12 C -20 -10 0 -14 -4 D -16 -14 14 0 14 E -16 -12 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4979: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) A E D C B (10) A C B E D (9) E D A B C (8) B C A D E (8) C B A D E (7) D E A C B (6) D E A B C (6) D E B C A (5) E D B A C (3) B C A E D (3) A E C D B (3) D E C B A (2) C B A E D (2) B D E C A (2) B D C E A (2) B C D E A (2) A B C E D (2) E A D C B (1) D E B A C (1) D B C E A (1) C D A B E (1) C A B E D (1) B C D A E (1) B A C E D (1) A E D B C (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 20 26 0 2 B -20 0 -8 -20 -16 C -26 8 0 -18 -18 D 0 20 18 0 -10 E -2 16 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.892438 B: 0.000000 C: 0.000000 D: 0.107562 E: 0.000000 Sum of squares = 0.808015227416 Cumulative probabilities = A: 0.892438 B: 0.892438 C: 0.892438 D: 1.000000 E: 1.000000 A B C D E A 0 20 26 0 2 B -20 0 -8 -20 -16 C -26 8 0 -18 -18 D 0 20 18 0 -10 E -2 16 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833334 B: 0.000000 C: 0.000000 D: 0.166666 E: 0.000000 Sum of squares = 0.722222598504 Cumulative probabilities = A: 0.833334 B: 0.833334 C: 0.833334 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 D=21 B=19 C=11 so C is eliminated. Round 2 votes counts: B=28 A=28 E=22 D=22 so E is eliminated. Round 3 votes counts: D=43 A=29 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:224 E:221 D:214 C:173 B:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 26 0 2 B -20 0 -8 -20 -16 C -26 8 0 -18 -18 D 0 20 18 0 -10 E -2 16 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833334 B: 0.000000 C: 0.000000 D: 0.166666 E: 0.000000 Sum of squares = 0.722222598504 Cumulative probabilities = A: 0.833334 B: 0.833334 C: 0.833334 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 26 0 2 B -20 0 -8 -20 -16 C -26 8 0 -18 -18 D 0 20 18 0 -10 E -2 16 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833334 B: 0.000000 C: 0.000000 D: 0.166666 E: 0.000000 Sum of squares = 0.722222598504 Cumulative probabilities = A: 0.833334 B: 0.833334 C: 0.833334 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 26 0 2 B -20 0 -8 -20 -16 C -26 8 0 -18 -18 D 0 20 18 0 -10 E -2 16 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833334 B: 0.000000 C: 0.000000 D: 0.166666 E: 0.000000 Sum of squares = 0.722222598504 Cumulative probabilities = A: 0.833334 B: 0.833334 C: 0.833334 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4980: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (16) E A B D C (14) A E C D B (7) E B D A C (5) C A D B E (5) A C E D B (5) E A C B D (4) D B C E A (4) B D C E A (4) E A D B C (3) D B C A E (3) C D A B E (3) C B D E A (3) B D E C A (3) B D E A C (3) E B A D C (2) D B E A C (2) C E B A D (2) C A E D B (2) A E C B D (2) E B D C A (1) E B C D A (1) E B A C D (1) D B E C A (1) D B A C E (1) C A E B D (1) C A B D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -4 0 0 -6 B 4 0 -2 -6 -2 C 0 2 0 6 0 D 0 6 -6 0 -2 E 6 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.616518 D: 0.000000 E: 0.383482 Sum of squares = 0.527152691503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.616518 D: 0.616518 E: 1.000000 A B C D E A 0 -4 0 0 -6 B 4 0 -2 -6 -2 C 0 2 0 6 0 D 0 6 -6 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=31 A=15 D=11 B=10 so B is eliminated. Round 2 votes counts: C=33 E=31 D=21 A=15 so A is eliminated. Round 3 votes counts: E=41 C=38 D=21 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:205 C:204 D:199 B:197 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 0 -6 B 4 0 -2 -6 -2 C 0 2 0 6 0 D 0 6 -6 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 0 -6 B 4 0 -2 -6 -2 C 0 2 0 6 0 D 0 6 -6 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 0 -6 B 4 0 -2 -6 -2 C 0 2 0 6 0 D 0 6 -6 0 -2 E 6 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4981: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) C B D E A (7) D C B A E (6) A E C D B (6) A E B D C (6) B C D E A (5) A E D C B (5) A E D B C (5) E A B C D (4) D B C A E (4) B C E D A (4) E B A C D (3) C D B E A (3) C B E D A (3) A D E C B (3) D C B E A (2) D C A E B (2) D B A C E (2) C E B A D (2) A D B E C (2) E C B A D (1) E B C A D (1) D C A B E (1) D B C E A (1) D A E C B (1) D A B E C (1) D A B C E (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E D B (1) B E A C D (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A E C (1) B C E A D (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 0 0 2 2 B 0 0 -6 0 -4 C 0 6 0 6 0 D -2 0 -6 0 -4 E -2 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.683195 B: 0.000000 C: 0.316805 D: 0.000000 E: 0.000000 Sum of squares = 0.567121104641 Cumulative probabilities = A: 0.683195 B: 0.683195 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 2 2 B 0 0 -6 0 -4 C 0 6 0 6 0 D -2 0 -6 0 -4 E -2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=21 C=19 E=16 B=15 so B is eliminated. Round 2 votes counts: C=29 A=29 D=25 E=17 so E is eliminated. Round 3 votes counts: A=44 C=31 D=25 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:206 E:203 A:202 B:195 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 2 2 B 0 0 -6 0 -4 C 0 6 0 6 0 D -2 0 -6 0 -4 E -2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 2 B 0 0 -6 0 -4 C 0 6 0 6 0 D -2 0 -6 0 -4 E -2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 2 B 0 0 -6 0 -4 C 0 6 0 6 0 D -2 0 -6 0 -4 E -2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4982: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) A E D C B (7) D C E B A (6) D C E A B (6) B E C D A (6) B A E C D (6) A D C E B (6) C D E B A (5) B C D E A (5) A B E D C (5) C D B E A (4) A B C D E (4) E D C A B (3) B E A C D (3) E B D C A (2) D C A E B (2) B C E D A (2) B A E D C (2) A E B D C (2) A C D B E (2) A B E C D (2) E B A D C (1) E A D B C (1) D A C E B (1) C D B A E (1) C B D E A (1) B E A D C (1) B C D A E (1) B C A D E (1) B A C E D (1) B A C D E (1) A D E C B (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -4 -4 -6 B 12 0 -6 -6 0 C 4 6 0 -10 2 D 4 6 10 0 -2 E 6 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 A B C D E A 0 -12 -4 -4 -6 B 12 0 -6 -6 0 C 4 6 0 -10 2 D 4 6 10 0 -2 E 6 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=29 D=15 E=14 C=11 so C is eliminated. Round 2 votes counts: A=31 B=30 D=25 E=14 so E is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:209 E:203 C:201 B:200 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -4 -4 -6 B 12 0 -6 -6 0 C 4 6 0 -10 2 D 4 6 10 0 -2 E 6 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -4 -6 B 12 0 -6 -6 0 C 4 6 0 -10 2 D 4 6 10 0 -2 E 6 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -4 -6 B 12 0 -6 -6 0 C 4 6 0 -10 2 D 4 6 10 0 -2 E 6 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4983: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) B E A D C (11) E C B D A (8) E B A D C (8) E B C A D (7) C D A B E (6) B A D E C (5) E B C D A (4) E B A C D (4) D A C B E (4) C D E A B (4) A D B C E (4) A B D E C (4) D C A B E (3) C E D A B (3) A D C B E (3) C E D B A (2) C E B D A (2) A D B E C (2) C E B A D (1) B A E D C (1) Total count = 100 A B C D E A 0 -6 -8 0 -8 B 6 0 0 10 -14 C 8 0 0 10 -8 D 0 -10 -10 0 -2 E 8 14 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 0 -8 B 6 0 0 10 -14 C 8 0 0 10 -8 D 0 -10 -10 0 -2 E 8 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=31 B=17 A=13 D=7 so D is eliminated. Round 2 votes counts: C=35 E=31 B=17 A=17 so B is eliminated. Round 3 votes counts: E=42 C=35 A=23 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:205 B:201 A:189 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 0 -8 B 6 0 0 10 -14 C 8 0 0 10 -8 D 0 -10 -10 0 -2 E 8 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 -8 B 6 0 0 10 -14 C 8 0 0 10 -8 D 0 -10 -10 0 -2 E 8 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 -8 B 6 0 0 10 -14 C 8 0 0 10 -8 D 0 -10 -10 0 -2 E 8 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4984: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (14) E B D A C (12) B E A D C (9) D A C E B (7) C D A E B (7) C A D E B (6) D A E C B (4) B E C A D (4) C B E A D (3) C B A E D (3) C A B D E (3) B E D A C (3) B E C D A (3) B C E A D (3) E D B A C (2) D E A B C (2) D C A E B (2) D A E B C (2) B A E D C (2) A D E B C (2) A C D B E (2) C D E B A (1) C A B E D (1) B E A C D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 8 0 10 14 B -8 0 -10 -6 4 C 0 10 0 2 8 D -10 6 -2 0 8 E -14 -4 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.446246 B: 0.000000 C: 0.553754 D: 0.000000 E: 0.000000 Sum of squares = 0.505778912513 Cumulative probabilities = A: 0.446246 B: 0.446246 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 10 14 B -8 0 -10 -6 4 C 0 10 0 2 8 D -10 6 -2 0 8 E -14 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=25 D=17 E=14 A=6 so A is eliminated. Round 2 votes counts: C=40 B=25 D=21 E=14 so E is eliminated. Round 3 votes counts: C=40 B=37 D=23 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:216 C:210 D:201 B:190 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 10 14 B -8 0 -10 -6 4 C 0 10 0 2 8 D -10 6 -2 0 8 E -14 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 10 14 B -8 0 -10 -6 4 C 0 10 0 2 8 D -10 6 -2 0 8 E -14 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 10 14 B -8 0 -10 -6 4 C 0 10 0 2 8 D -10 6 -2 0 8 E -14 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4985: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E A B C D (6) B E C A D (6) E A B D C (5) C B D E A (5) B E C D A (5) B C E D A (5) E B A D C (4) D C B E A (4) C D B E A (4) A E B C D (4) D C A E B (3) C D A B E (3) A E C B D (3) E B A C D (2) D C B A E (2) C D B A E (2) C B E D A (2) B E D C A (2) B D E C A (2) B D C E A (2) A E B D C (2) A D E C B (2) A C D E B (2) D B C E A (1) D A E C B (1) D A C E B (1) D A C B E (1) C E B A D (1) C D A E B (1) C A E D B (1) B E A D C (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -22 -12 -14 B 0 0 -2 14 14 C 22 2 0 8 2 D 12 -14 -8 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -22 -12 -14 B 0 0 -2 14 14 C 22 2 0 8 2 D 12 -14 -8 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 C=19 A=18 E=17 so E is eliminated. Round 2 votes counts: B=29 A=29 D=23 C=19 so C is eliminated. Round 3 votes counts: B=37 D=33 A=30 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:217 B:213 E:202 D:192 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -22 -12 -14 B 0 0 -2 14 14 C 22 2 0 8 2 D 12 -14 -8 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -22 -12 -14 B 0 0 -2 14 14 C 22 2 0 8 2 D 12 -14 -8 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -22 -12 -14 B 0 0 -2 14 14 C 22 2 0 8 2 D 12 -14 -8 0 -6 E 14 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4986: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) D B E A C (9) B D A E C (7) A B D C E (7) E C D B A (6) C A E B D (6) D B A E C (5) C E A B D (5) E D B C A (4) C E D A B (4) C E A D B (4) A B C D E (4) E D C B A (3) A C B D E (3) A B D E C (3) D C A B E (2) C A B E D (2) E B D A C (1) E B C A D (1) D E C B A (1) D E B C A (1) D E B A C (1) C E D B A (1) C D E A B (1) C A E D B (1) C A B D E (1) B D E A C (1) B A E D C (1) A E C B D (1) A C D B E (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 14 6 14 B 4 0 14 12 18 C -14 -14 0 -14 -14 D -6 -12 14 0 14 E -14 -18 14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 6 14 B 4 0 14 12 18 C -14 -14 0 -14 -14 D -6 -12 14 0 14 E -14 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999724 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=22 D=19 B=19 E=15 so E is eliminated. Round 2 votes counts: C=31 D=26 A=22 B=21 so B is eliminated. Round 3 votes counts: D=35 A=33 C=32 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:224 A:215 D:205 E:184 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 6 14 B 4 0 14 12 18 C -14 -14 0 -14 -14 D -6 -12 14 0 14 E -14 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999724 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 6 14 B 4 0 14 12 18 C -14 -14 0 -14 -14 D -6 -12 14 0 14 E -14 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999724 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 6 14 B 4 0 14 12 18 C -14 -14 0 -14 -14 D -6 -12 14 0 14 E -14 -18 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999724 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4987: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) A E D C B (11) A D E B C (9) E C A B D (8) E A C B D (8) B D C E A (8) A E C B D (7) D B C A E (5) D A B E C (5) B C D E A (5) D B A C E (4) C E B A D (3) A E C D B (3) A D B E C (3) C B E D A (2) A E D B C (2) E C B A D (1) D B A E C (1) D A B C E (1) C E B D A (1) Total count = 100 A B C D E A 0 14 8 10 2 B -14 0 12 -14 -6 C -8 -12 0 -24 -16 D -10 14 24 0 8 E -2 6 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 10 2 B -14 0 12 -14 -6 C -8 -12 0 -24 -16 D -10 14 24 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=29 E=17 B=13 C=6 so C is eliminated. Round 2 votes counts: A=35 D=29 E=21 B=15 so B is eliminated. Round 3 votes counts: D=42 A=35 E=23 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:218 A:217 E:206 B:189 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 10 2 B -14 0 12 -14 -6 C -8 -12 0 -24 -16 D -10 14 24 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 10 2 B -14 0 12 -14 -6 C -8 -12 0 -24 -16 D -10 14 24 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 10 2 B -14 0 12 -14 -6 C -8 -12 0 -24 -16 D -10 14 24 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999779 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 4988: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (18) D A C E B (15) A D E C B (8) B C E D A (6) A D B E C (5) C E D A B (4) C B E D A (4) D A E C B (3) B D C A E (3) B A D E C (3) E B A D C (2) D C A E B (2) D A C B E (2) D A B C E (2) C E B D A (2) C D A E B (2) B E A C D (2) A D E B C (2) A B D E C (2) E C B A D (1) E C A D B (1) E B C A D (1) E A B C D (1) D A B E C (1) C E D B A (1) C D E A B (1) C B D A E (1) B E A D C (1) B D A C E (1) B C E A D (1) B A E D C (1) A E D C B (1) Total count = 100 A B C D E A 0 4 4 0 8 B -4 0 4 0 6 C -4 -4 0 -8 -6 D 0 0 8 0 6 E -8 -6 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.506237 B: 0.000000 C: 0.000000 D: 0.493763 E: 0.000000 Sum of squares = 0.500077799848 Cumulative probabilities = A: 0.506237 B: 0.506237 C: 0.506237 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 8 B -4 0 4 0 6 C -4 -4 0 -8 -6 D 0 0 8 0 6 E -8 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=25 A=18 C=15 E=6 so E is eliminated. Round 2 votes counts: B=39 D=25 A=19 C=17 so C is eliminated. Round 3 votes counts: B=47 D=33 A=20 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:208 D:207 B:203 E:193 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 8 B -4 0 4 0 6 C -4 -4 0 -8 -6 D 0 0 8 0 6 E -8 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 8 B -4 0 4 0 6 C -4 -4 0 -8 -6 D 0 0 8 0 6 E -8 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 8 B -4 0 4 0 6 C -4 -4 0 -8 -6 D 0 0 8 0 6 E -8 -6 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4989: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) A E C B D (7) E A C D B (6) D E C B A (5) A B C D E (5) A B E C D (4) E B A D C (3) E A D B C (3) E A B D C (3) D C B E A (3) B E D A C (3) A C B E D (3) E D C B A (2) E D C A B (2) E D B C A (2) E A D C B (2) E A C B D (2) D C E B A (2) B E A D C (2) B C A D E (2) A E B C D (2) A C E D B (2) A C E B D (2) E D A B C (1) E C D A B (1) E C A D B (1) E B D A C (1) D E B C A (1) D C B A E (1) D B C E A (1) D B C A E (1) C D E B A (1) C D E A B (1) C D B E A (1) C D A B E (1) C B D A E (1) C B A D E (1) C A E D B (1) B D E C A (1) B D C E A (1) B D A C E (1) B C D A E (1) B A E D C (1) B A E C D (1) B A D C E (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 16 14 -2 B -4 0 2 18 -6 C -16 -2 0 -4 -14 D -14 -18 4 0 -18 E 2 6 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 16 14 -2 B -4 0 2 18 -6 C -16 -2 0 -4 -14 D -14 -18 4 0 -18 E 2 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=28 B=22 D=14 C=7 so C is eliminated. Round 2 votes counts: E=29 A=29 B=24 D=18 so D is eliminated. Round 3 votes counts: E=39 B=31 A=30 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:216 B:205 C:182 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 16 14 -2 B -4 0 2 18 -6 C -16 -2 0 -4 -14 D -14 -18 4 0 -18 E 2 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 14 -2 B -4 0 2 18 -6 C -16 -2 0 -4 -14 D -14 -18 4 0 -18 E 2 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 14 -2 B -4 0 2 18 -6 C -16 -2 0 -4 -14 D -14 -18 4 0 -18 E 2 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 4990: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) A E D B C (6) E A B D C (5) B D C E A (5) E D B A C (4) E A D B C (4) D B E C A (4) C B D E A (4) B D E A C (4) A E C D B (4) A E B D C (4) A C E B D (4) D E B A C (3) C D B E A (3) C A D B E (3) C A B E D (3) B D E C A (3) A C E D B (3) E B D A C (2) D E A B C (2) D C B E A (2) C D B A E (2) C D A E B (2) C B D A E (2) C A E D B (2) C A E B D (2) C A D E B (2) C A B D E (2) A E C B D (2) A E B C D (2) E D A B C (1) C B A D E (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 8 14 0 -6 B -8 0 12 -8 -10 C -14 -12 0 -12 -14 D 0 8 12 0 2 E 6 10 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.170799 B: 0.000000 C: 0.000000 D: 0.829201 E: 0.000000 Sum of squares = 0.716746254129 Cumulative probabilities = A: 0.170799 B: 0.170799 C: 0.170799 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 0 -6 B -8 0 12 -8 -10 C -14 -12 0 -12 -14 D 0 8 12 0 2 E 6 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000134354 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=26 D=17 E=16 B=13 so B is eliminated. Round 2 votes counts: D=29 C=29 A=26 E=16 so E is eliminated. Round 3 votes counts: D=36 A=35 C=29 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:214 D:211 A:208 B:193 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 14 0 -6 B -8 0 12 -8 -10 C -14 -12 0 -12 -14 D 0 8 12 0 2 E 6 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000134354 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 0 -6 B -8 0 12 -8 -10 C -14 -12 0 -12 -14 D 0 8 12 0 2 E 6 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000134354 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 0 -6 B -8 0 12 -8 -10 C -14 -12 0 -12 -14 D 0 8 12 0 2 E 6 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000134354 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4991: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (5) E C D A B (4) E B D A C (4) C E B A D (4) C A D B E (4) C A B D E (4) B A D E C (4) E D C A B (3) E D A B C (3) D A C E B (3) C E A D B (3) C B A E D (3) C A D E B (3) B E D A C (3) B E C A D (3) B C A E D (3) A D C B E (3) A D B C E (3) E C B A D (2) D E B A C (2) D C A E B (2) D B E A C (2) C A B E D (2) B A C D E (2) A C D B E (2) A B D C E (2) E D B C A (1) E D B A C (1) E D A C B (1) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C E A B (1) D A E B C (1) D A C B E (1) D A B E C (1) C E D A B (1) C E A B D (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E A D (1) B A C E D (1) A C D E B (1) Total count = 100 A B C D E A 0 4 0 14 2 B -4 0 -4 0 4 C 0 4 0 -4 10 D -14 0 4 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.515800 B: 0.000000 C: 0.484200 D: 0.000000 E: 0.000000 Sum of squares = 0.500499274096 Cumulative probabilities = A: 0.515800 B: 0.515800 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 14 2 B -4 0 -4 0 4 C 0 4 0 -4 10 D -14 0 4 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=25 B=25 E=23 D=16 A=11 so A is eliminated. Round 2 votes counts: C=28 B=27 E=23 D=22 so D is eliminated. Round 3 votes counts: C=38 B=33 E=29 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:210 C:205 B:198 D:196 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 14 2 B -4 0 -4 0 4 C 0 4 0 -4 10 D -14 0 4 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 14 2 B -4 0 -4 0 4 C 0 4 0 -4 10 D -14 0 4 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 14 2 B -4 0 -4 0 4 C 0 4 0 -4 10 D -14 0 4 0 2 E -2 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4992: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (8) E D C B A (5) D E C A B (5) B A C E D (5) D C E A B (4) C E B D A (4) A D B C E (4) E B A D C (3) D C A E B (3) C B E A D (3) E D B C A (2) E D A B C (2) E C D B A (2) E B C A D (2) E A D B C (2) D C A B E (2) D A C B E (2) C E D B A (2) C D E B A (2) C D A E B (2) C D A B E (2) C B A E D (2) C A B D E (2) B E C A D (2) B C E A D (2) B A E C D (2) A B E D C (2) E D A C B (1) E C B D A (1) E B C D A (1) E B A C D (1) E A B D C (1) D E A C B (1) D A E C B (1) D A E B C (1) D A C E B (1) C D E A B (1) C D B A E (1) C B E D A (1) C B D E A (1) C B D A E (1) C A D B E (1) B C A E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A D C B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -30 -2 -2 B 8 0 -24 -4 -6 C 30 24 0 4 18 D 2 4 -4 0 -2 E 2 6 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -30 -2 -2 B 8 0 -24 -4 -6 C 30 24 0 4 18 D 2 4 -4 0 -2 E 2 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=23 D=20 B=12 A=12 so B is eliminated. Round 2 votes counts: C=36 E=25 D=20 A=19 so A is eliminated. Round 3 votes counts: C=41 E=31 D=28 so D is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:238 D:200 E:196 B:187 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -30 -2 -2 B 8 0 -24 -4 -6 C 30 24 0 4 18 D 2 4 -4 0 -2 E 2 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -30 -2 -2 B 8 0 -24 -4 -6 C 30 24 0 4 18 D 2 4 -4 0 -2 E 2 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -30 -2 -2 B 8 0 -24 -4 -6 C 30 24 0 4 18 D 2 4 -4 0 -2 E 2 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 4993: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) D B C A E (7) B D C E A (6) B D A E C (6) D B C E A (5) D B A E C (5) A E C B D (5) D C B E A (4) C E A D B (4) A E C D B (4) E A C B D (3) C D B E A (3) C B E D A (3) C B D E A (3) B C D E A (3) D C A E B (2) D B A C E (2) B A E D C (2) A E D B C (2) A D B E C (2) A B E D C (2) E A C D B (1) E A B C D (1) D A B E C (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C A D (1) B E A D C (1) B D E A C (1) B D A C E (1) A E D C B (1) A E B C D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -12 -8 -4 B 8 0 2 4 18 C 12 -2 0 -4 18 D 8 -4 4 0 8 E 4 -18 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -8 -4 B 8 0 2 4 18 C 12 -2 0 -4 18 D 8 -4 4 0 8 E 4 -18 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996441 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 B=22 A=19 E=5 so E is eliminated. Round 2 votes counts: C=28 D=26 A=24 B=22 so B is eliminated. Round 3 votes counts: D=41 C=32 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 C:212 D:208 A:184 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -8 -4 B 8 0 2 4 18 C 12 -2 0 -4 18 D 8 -4 4 0 8 E 4 -18 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996441 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -8 -4 B 8 0 2 4 18 C 12 -2 0 -4 18 D 8 -4 4 0 8 E 4 -18 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996441 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -8 -4 B 8 0 2 4 18 C 12 -2 0 -4 18 D 8 -4 4 0 8 E 4 -18 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996441 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 4994: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (15) E B D A C (8) E B A C D (5) B E D A C (5) D A C B E (4) C A D E B (4) C A D B E (4) B D E A C (4) A C D B E (4) E D C A B (3) E C B A D (3) E C A B D (3) D C A B E (3) C D A E B (3) C A E D B (3) B D A C E (3) E B D C A (2) D B E C A (2) C E A D B (2) B E A C D (2) B A D E C (2) A C B D E (2) E C D B A (1) E C D A B (1) E C A D B (1) E B C A D (1) E B A D C (1) D E B C A (1) D C B A E (1) D B C E A (1) D B C A E (1) C E A B D (1) B D A E C (1) B A E C D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -22 18 -18 6 B 22 0 12 -8 14 C -18 -12 0 -14 6 D 18 8 14 0 12 E -6 -14 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 18 -18 6 B 22 0 12 -8 14 C -18 -12 0 -14 6 D 18 8 14 0 12 E -6 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=28 B=19 C=17 A=7 so A is eliminated. Round 2 votes counts: E=29 D=28 C=24 B=19 so B is eliminated. Round 3 votes counts: D=38 E=37 C=25 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:220 A:192 C:181 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 18 -18 6 B 22 0 12 -8 14 C -18 -12 0 -14 6 D 18 8 14 0 12 E -6 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 18 -18 6 B 22 0 12 -8 14 C -18 -12 0 -14 6 D 18 8 14 0 12 E -6 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 18 -18 6 B 22 0 12 -8 14 C -18 -12 0 -14 6 D 18 8 14 0 12 E -6 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4995: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) A D E C B (8) C D A E B (7) B C D A E (7) D A E C B (6) B C E A D (6) E A D C B (5) B C E D A (5) E A D B C (4) D A C E B (4) C D B A E (4) C D A B E (4) B E C A D (4) C B D E A (3) D A C B E (2) C B D A E (2) B D C A E (2) B D A E C (2) A E D B C (2) E C A B D (1) E B A D C (1) E A B D C (1) D C B A E (1) D C A E B (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B C E (1) C D E A B (1) C B E D A (1) B E A C D (1) B D A C E (1) B A E D C (1) A E D C B (1) Total count = 100 A B C D E A 0 -2 0 -14 18 B 2 0 -2 -8 14 C 0 2 0 -8 8 D 14 8 8 0 18 E -18 -14 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -14 18 B 2 0 -2 -8 14 C 0 2 0 -8 8 D 14 8 8 0 18 E -18 -14 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=22 D=18 E=12 A=11 so A is eliminated. Round 2 votes counts: B=37 D=26 C=22 E=15 so E is eliminated. Round 3 votes counts: B=39 D=38 C=23 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:203 A:201 C:201 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -14 18 B 2 0 -2 -8 14 C 0 2 0 -8 8 D 14 8 8 0 18 E -18 -14 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -14 18 B 2 0 -2 -8 14 C 0 2 0 -8 8 D 14 8 8 0 18 E -18 -14 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -14 18 B 2 0 -2 -8 14 C 0 2 0 -8 8 D 14 8 8 0 18 E -18 -14 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4996: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) C A D E B (6) C B A D E (5) B E A D C (5) E D B A C (4) E D A B C (4) C B D E A (4) B C A E D (4) A E D C B (4) B E D C A (3) B C D E A (3) A E B D C (3) E B D A C (2) E A D B C (2) D E C A B (2) D A E C B (2) C D A E B (2) C A D B E (2) B E D A C (2) B C E D A (2) B C A D E (2) B A E D C (2) B A C E D (2) A E D B C (2) A C D E B (2) E D A C B (1) E B A D C (1) E A B D C (1) D C E A B (1) D C A E B (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D A E (1) C A B D E (1) B E A C D (1) B D E C A (1) B D C E A (1) B C E A D (1) B A E C D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 8 -2 -8 B -2 0 -2 -4 -8 C -8 2 0 -14 -10 D 2 4 14 0 6 E 8 8 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -2 -8 B -2 0 -2 -4 -8 C -8 2 0 -14 -10 D 2 4 14 0 6 E 8 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=25 D=17 E=15 A=13 so A is eliminated. Round 2 votes counts: B=30 C=27 E=24 D=19 so D is eliminated. Round 3 votes counts: E=40 C=30 B=30 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:210 A:200 B:192 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -2 -8 B -2 0 -2 -4 -8 C -8 2 0 -14 -10 D 2 4 14 0 6 E 8 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -2 -8 B -2 0 -2 -4 -8 C -8 2 0 -14 -10 D 2 4 14 0 6 E 8 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -2 -8 B -2 0 -2 -4 -8 C -8 2 0 -14 -10 D 2 4 14 0 6 E 8 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4997: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) E C D A B (6) D E C B A (6) E D C B A (5) E D B A C (5) D C B A E (5) D B A C E (5) A B C D E (5) E C A B D (4) E A B C D (4) C D B A E (4) B A C D E (4) E D C A B (3) D E B A C (3) D C E B A (3) D B C A E (3) C A B D E (3) B A D C E (3) A C B E D (3) E A B D C (2) C D A B E (2) C B A D E (2) C A B E D (2) B A D E C (2) A E B C D (2) A B E C D (2) E C A D B (1) D B E A C (1) D B A E C (1) B D A E C (1) Total count = 100 A B C D E A 0 -6 2 -6 14 B 6 0 2 -6 12 C -2 -2 0 4 4 D 6 6 -4 0 6 E -14 -12 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -6 14 B 6 0 2 -6 12 C -2 -2 0 4 4 D 6 6 -4 0 6 E -14 -12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888937 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 A=20 C=13 B=10 so B is eliminated. Round 2 votes counts: E=30 A=29 D=28 C=13 so C is eliminated. Round 3 votes counts: A=36 D=34 E=30 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:207 D:207 A:202 C:202 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 -6 14 B 6 0 2 -6 12 C -2 -2 0 4 4 D 6 6 -4 0 6 E -14 -12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888937 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -6 14 B 6 0 2 -6 12 C -2 -2 0 4 4 D 6 6 -4 0 6 E -14 -12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888937 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -6 14 B 6 0 2 -6 12 C -2 -2 0 4 4 D 6 6 -4 0 6 E -14 -12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888937 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4998: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) D E B A C (11) B A C D E (11) E C A B D (10) C A B E D (10) D B A C E (8) E D B A C (7) C A B D E (7) D B A E C (6) D E C B A (3) E A B C D (2) C E D A B (2) C E A B D (2) E D C B A (1) E C D A B (1) E A B D C (1) D E C A B (1) D E B C A (1) D C E A B (1) B D A C E (1) B A D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -8 -8 B 0 0 2 -6 -8 C 0 -2 0 -6 -10 D 8 6 6 0 4 E 8 8 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -8 -8 B 0 0 2 -6 -8 C 0 -2 0 -6 -10 D 8 6 6 0 4 E 8 8 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=31 C=21 B=13 A=2 so A is eliminated. Round 2 votes counts: E=33 D=31 C=21 B=15 so B is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:211 B:194 A:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -8 -8 B 0 0 2 -6 -8 C 0 -2 0 -6 -10 D 8 6 6 0 4 E 8 8 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 -8 B 0 0 2 -6 -8 C 0 -2 0 -6 -10 D 8 6 6 0 4 E 8 8 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 -8 B 0 0 2 -6 -8 C 0 -2 0 -6 -10 D 8 6 6 0 4 E 8 8 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 4999: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) B A E C D (9) C A B E D (8) A B C E D (7) D E B A C (6) E B A C D (4) D E B C A (4) D C E A B (4) E D B A C (3) D B E A C (3) D B A E C (3) C A E B D (3) C A B D E (3) B E A D C (3) A C B E D (3) E D B C A (2) D C A B E (2) C E A B D (2) C D A E B (2) B E A C D (2) B D E A C (2) E D C A B (1) E C D B A (1) E C B A D (1) E B A D C (1) D A C B E (1) C E D A B (1) C D E A B (1) C A D E B (1) C A D B E (1) B E D A C (1) B A E D C (1) B A D E C (1) B A D C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 6 6 -4 B 16 0 8 8 6 C -6 -8 0 2 -16 D -6 -8 -2 0 -6 E 4 -6 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 6 -4 B 16 0 8 8 6 C -6 -8 0 2 -16 D -6 -8 -2 0 -6 E 4 -6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=22 B=20 E=13 A=12 so A is eliminated. Round 2 votes counts: D=33 B=28 C=26 E=13 so E is eliminated. Round 3 votes counts: D=39 B=33 C=28 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:210 A:196 D:189 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 6 -4 B 16 0 8 8 6 C -6 -8 0 2 -16 D -6 -8 -2 0 -6 E 4 -6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 6 -4 B 16 0 8 8 6 C -6 -8 0 2 -16 D -6 -8 -2 0 -6 E 4 -6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 6 -4 B 16 0 8 8 6 C -6 -8 0 2 -16 D -6 -8 -2 0 -6 E 4 -6 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5000: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) E D B C A (8) C A B E D (8) B A C D E (8) E D C A B (6) A B C D E (6) E D C B A (5) E D A C B (5) B A D C E (5) A C B D E (5) D B E A C (4) C B A E D (4) E C D A B (3) C E D B A (2) C E A D B (2) C E A B D (2) C A E B D (2) C A B D E (2) E D A B C (1) E C B D A (1) E C A D B (1) D E A B C (1) C E B D A (1) C B A D E (1) C A E D B (1) B E C D A (1) B D E C A (1) B D E A C (1) B D A C E (1) A D E C B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -2 0 -8 B 4 0 -8 0 -4 C 2 8 0 4 4 D 0 0 -4 0 -8 E 8 4 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 0 -8 B 4 0 -8 0 -4 C 2 8 0 4 4 D 0 0 -4 0 -8 E 8 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=25 B=17 D=14 A=14 so D is eliminated. Round 2 votes counts: E=40 C=25 B=21 A=14 so A is eliminated. Round 3 votes counts: E=41 C=32 B=27 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:209 E:208 B:196 D:194 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 0 -8 B 4 0 -8 0 -4 C 2 8 0 4 4 D 0 0 -4 0 -8 E 8 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 0 -8 B 4 0 -8 0 -4 C 2 8 0 4 4 D 0 0 -4 0 -8 E 8 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 0 -8 B 4 0 -8 0 -4 C 2 8 0 4 4 D 0 0 -4 0 -8 E 8 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5001: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) E C D B A (10) E B A C D (7) B A E C D (6) A B E D C (6) E B C A D (5) A D B C E (5) D C A E B (4) D A C E B (4) C D E B A (4) B E A C D (4) A B D C E (4) D C A B E (3) D A C B E (3) E D C A B (2) E B A D C (2) D E C A B (2) C D B A E (2) B E C A D (2) A D C B E (2) E D C B A (1) E B D A C (1) E A B D C (1) D C E A B (1) D A E C B (1) C D A B E (1) C B D A E (1) B C E A D (1) B C A E D (1) B C A D E (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -18 -4 -2 -8 B 18 0 -4 6 -14 C 4 4 0 12 -26 D 2 -6 -12 0 -20 E 8 14 26 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -4 -2 -8 B 18 0 -4 6 -14 C 4 4 0 12 -26 D 2 -6 -12 0 -20 E 8 14 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=19 D=18 B=15 C=8 so C is eliminated. Round 2 votes counts: E=40 D=25 A=19 B=16 so B is eliminated. Round 3 votes counts: E=47 A=27 D=26 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:234 B:203 C:197 A:184 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -4 -2 -8 B 18 0 -4 6 -14 C 4 4 0 12 -26 D 2 -6 -12 0 -20 E 8 14 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -2 -8 B 18 0 -4 6 -14 C 4 4 0 12 -26 D 2 -6 -12 0 -20 E 8 14 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -2 -8 B 18 0 -4 6 -14 C 4 4 0 12 -26 D 2 -6 -12 0 -20 E 8 14 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5002: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) E B D C A (5) C E A D B (5) C D B E A (5) A C E B D (5) A B D E C (5) E B D A C (4) C E D B A (4) B D A E C (4) A E B D C (4) A C D B E (4) A B E D C (4) E C B D A (3) E A B D C (3) C A E D B (3) B A D E C (3) A E C B D (3) A B D C E (3) E C D B A (2) E C A B D (2) E B A D C (2) E A B C D (2) D B A C E (2) C D E B A (2) B D E A C (2) E D B C A (1) E C B A D (1) D C B E A (1) D B E C A (1) D B C E A (1) D B C A E (1) C E D A B (1) C D B A E (1) C D A E B (1) C A D E B (1) B E D A C (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 6 6 16 4 B -6 0 0 14 -8 C -6 0 0 2 -6 D -16 -14 -2 0 -12 E -4 8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999585 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 16 4 B -6 0 0 14 -8 C -6 0 0 2 -6 D -16 -14 -2 0 -12 E -4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=29 A=29 E=25 B=11 D=6 so D is eliminated. Round 2 votes counts: C=30 A=29 E=25 B=16 so B is eliminated. Round 3 votes counts: A=39 C=32 E=29 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:211 B:200 C:195 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 16 4 B -6 0 0 14 -8 C -6 0 0 2 -6 D -16 -14 -2 0 -12 E -4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 16 4 B -6 0 0 14 -8 C -6 0 0 2 -6 D -16 -14 -2 0 -12 E -4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 16 4 B -6 0 0 14 -8 C -6 0 0 2 -6 D -16 -14 -2 0 -12 E -4 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5003: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) D B C A E (6) D B A E C (6) C E B A D (5) C D B E A (5) E A C B D (4) C E A B D (4) C B D E A (4) E C A B D (3) E A C D B (3) C E A D B (3) C D E A B (3) C B E D A (3) A E D B C (3) A E B D C (3) A E B C D (3) D C B E A (2) C E D A B (2) B C E A D (2) B A E D C (2) B A E C D (2) B A D E C (2) A E D C B (2) A E C B D (2) E A D C B (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A B E (1) D B C E A (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C D E B A (1) C B E A D (1) B E A C D (1) B C D E A (1) A E C D B (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -14 2 -6 B 8 0 -10 6 2 C 14 10 0 12 12 D -2 -6 -12 0 -8 E 6 -2 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 2 -6 B 8 0 -10 6 2 C 14 10 0 12 12 D -2 -6 -12 0 -8 E 6 -2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=22 B=17 A=17 E=12 so E is eliminated. Round 2 votes counts: C=35 A=26 D=22 B=17 so B is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:203 E:200 A:187 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 2 -6 B 8 0 -10 6 2 C 14 10 0 12 12 D -2 -6 -12 0 -8 E 6 -2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 2 -6 B 8 0 -10 6 2 C 14 10 0 12 12 D -2 -6 -12 0 -8 E 6 -2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 2 -6 B 8 0 -10 6 2 C 14 10 0 12 12 D -2 -6 -12 0 -8 E 6 -2 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5004: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) E B A C D (11) B E D A C (10) B E D C A (7) B D E C A (7) D C B A E (6) C A D E B (5) B E A D C (5) D B E C A (4) A C E D B (4) A C D E B (4) C D A E B (3) A E C B D (3) E B C A D (2) E A B C D (2) B E C D A (2) B E A C D (2) A D C E B (2) A C E B D (2) E B A D C (1) E A C B D (1) D C A E B (1) D B C E A (1) D B C A E (1) C B D E A (1) Total count = 100 A B C D E A 0 -20 -6 -12 -12 B 20 0 10 12 18 C 6 -10 0 -16 -14 D 12 -12 16 0 -4 E 12 -18 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -12 -12 B 20 0 10 12 18 C 6 -10 0 -16 -14 D 12 -12 16 0 -4 E 12 -18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=26 E=17 A=15 C=9 so C is eliminated. Round 2 votes counts: B=34 D=29 A=20 E=17 so E is eliminated. Round 3 votes counts: B=48 D=29 A=23 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:230 D:206 E:206 C:183 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -6 -12 -12 B 20 0 10 12 18 C 6 -10 0 -16 -14 D 12 -12 16 0 -4 E 12 -18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -12 -12 B 20 0 10 12 18 C 6 -10 0 -16 -14 D 12 -12 16 0 -4 E 12 -18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -12 -12 B 20 0 10 12 18 C 6 -10 0 -16 -14 D 12 -12 16 0 -4 E 12 -18 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5005: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) B C A E D (12) A C B D E (12) D E A C B (11) B E D C A (11) C A B D E (8) A C D E B (5) E D A B C (4) A D C E B (4) E D B A C (3) D E A B C (3) B C E A D (3) A D E C B (3) C B A D E (2) B C E D A (2) E B D C A (1) C B A E D (1) B C A D E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -8 4 -2 B -2 0 6 6 6 C 8 -6 0 -6 2 D -4 -6 6 0 0 E 2 -6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 4 -2 B -2 0 6 6 6 C 8 -6 0 -6 2 D -4 -6 6 0 0 E 2 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000071 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=25 E=21 D=14 C=11 so C is eliminated. Round 2 votes counts: A=33 B=32 E=21 D=14 so D is eliminated. Round 3 votes counts: E=35 A=33 B=32 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:208 C:199 A:198 D:198 E:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -8 4 -2 B -2 0 6 6 6 C 8 -6 0 -6 2 D -4 -6 6 0 0 E 2 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000071 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 -2 B -2 0 6 6 6 C 8 -6 0 -6 2 D -4 -6 6 0 0 E 2 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000071 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 -2 B -2 0 6 6 6 C 8 -6 0 -6 2 D -4 -6 6 0 0 E 2 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.406250000071 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5006: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (10) C E D B A (9) A D B C E (7) A D C E B (6) D C A E B (5) C E D A B (4) C D A E B (4) B E C A D (4) A C D E B (4) E B C D A (3) C D E B A (3) C D E A B (3) B E C D A (3) B E A C D (3) E C B D A (2) D C E B A (2) D C E A B (2) B E A D C (2) B A E D C (2) A E B C D (2) A B E C D (2) A B D E C (2) E C D B A (1) E C B A D (1) E A B C D (1) D C A B E (1) D B C E A (1) D B C A E (1) D B A C E (1) D A C B E (1) D A B C E (1) C A E B D (1) C A D E B (1) B E D C A (1) B A E C D (1) B A D E C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 18 -4 4 10 B -18 0 -24 -36 -12 C 4 24 0 8 38 D -4 36 -8 0 12 E -10 12 -38 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -4 4 10 B -18 0 -24 -36 -12 C 4 24 0 8 38 D -4 36 -8 0 12 E -10 12 -38 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=25 B=17 D=15 E=8 so E is eliminated. Round 2 votes counts: A=36 C=29 B=20 D=15 so D is eliminated. Round 3 votes counts: C=39 A=38 B=23 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:237 D:218 A:214 E:176 B:155 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -4 4 10 B -18 0 -24 -36 -12 C 4 24 0 8 38 D -4 36 -8 0 12 E -10 12 -38 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 4 10 B -18 0 -24 -36 -12 C 4 24 0 8 38 D -4 36 -8 0 12 E -10 12 -38 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 4 10 B -18 0 -24 -36 -12 C 4 24 0 8 38 D -4 36 -8 0 12 E -10 12 -38 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999957696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5007: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) E B A C D (6) E C D B A (5) C D E B A (5) B A E C D (5) A D B C E (5) D C A E B (4) B E A C D (4) B A D C E (4) A B E D C (4) E C D A B (3) E C B D A (3) D C A B E (3) C E D B A (3) C D B E A (3) A D C E B (3) E C A D B (2) E A B C D (2) D A C E B (2) D A C B E (2) C D E A B (2) B A E D C (2) A E D C B (2) A E B D C (2) E B C D A (1) E A C D B (1) D C E A B (1) D C B E A (1) D C B A E (1) D A B C E (1) B E C A D (1) B D C A E (1) B D A C E (1) B C D E A (1) A E B C D (1) A D C B E (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 20 14 12 B -6 0 6 -2 4 C -20 -6 0 -2 6 D -14 2 2 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 20 14 12 B -6 0 6 -2 4 C -20 -6 0 -2 6 D -14 2 2 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 B=19 D=15 C=13 so C is eliminated. Round 2 votes counts: A=30 E=26 D=25 B=19 so B is eliminated. Round 3 votes counts: A=41 E=31 D=28 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:201 D:197 C:189 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 20 14 12 B -6 0 6 -2 4 C -20 -6 0 -2 6 D -14 2 2 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 20 14 12 B -6 0 6 -2 4 C -20 -6 0 -2 6 D -14 2 2 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 20 14 12 B -6 0 6 -2 4 C -20 -6 0 -2 6 D -14 2 2 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999003 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5008: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) C E D B A (5) B A C E D (5) A B D C E (5) E D C B A (4) E D C A B (4) E B C A D (4) C B A E D (4) C B A D E (4) E D B C A (3) D E A B C (3) D A B E C (3) C E B A D (3) B C A E D (3) A D B E C (3) A B C D E (3) E C B D A (2) E B D A C (2) D E A C B (2) D C E A B (2) C E B D A (2) C D E A B (2) C B E A D (2) C A D B E (2) B E A C D (2) B C E A D (2) A C B D E (2) E D A C B (1) E B A D C (1) E B A C D (1) D C A E B (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C B E (1) C D E B A (1) C D A B E (1) C B E D A (1) B E A D C (1) B A E D C (1) B A C D E (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -18 2 -12 B 20 0 -8 6 -4 C 18 8 0 16 6 D -2 -6 -16 0 -20 E 12 4 -6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -18 2 -12 B 20 0 -8 6 -4 C 18 8 0 16 6 D -2 -6 -16 0 -20 E 12 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 D=15 B=15 A=15 so D is eliminated. Round 2 votes counts: E=33 C=31 A=21 B=15 so B is eliminated. Round 3 votes counts: E=36 C=36 A=28 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:215 B:207 D:178 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -18 2 -12 B 20 0 -8 6 -4 C 18 8 0 16 6 D -2 -6 -16 0 -20 E 12 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 2 -12 B 20 0 -8 6 -4 C 18 8 0 16 6 D -2 -6 -16 0 -20 E 12 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 2 -12 B 20 0 -8 6 -4 C 18 8 0 16 6 D -2 -6 -16 0 -20 E 12 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5009: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) B E D C A (7) E B A D C (6) B E A C D (6) B D C E A (6) A C D E B (6) E A B C D (5) D C B E A (5) C D A E B (5) A E B C D (5) D C E A B (4) B C D A E (4) E A B D C (3) B E A D C (3) A B E C D (3) E D C A B (2) E A D C B (2) D C B A E (2) C D B A E (2) C D A B E (2) B A E C D (2) A E C B D (2) A C D B E (2) E D C B A (1) E B D C A (1) E A C D B (1) D C A E B (1) D C A B E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 14 14 -4 B -10 0 4 8 -8 C -14 -4 0 12 -18 D -14 -8 -12 0 -18 E 4 8 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 14 -4 B -10 0 4 8 -8 C -14 -4 0 12 -18 D -14 -8 -12 0 -18 E 4 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 E=21 D=13 C=9 so C is eliminated. Round 2 votes counts: A=29 B=28 D=22 E=21 so E is eliminated. Round 3 votes counts: A=40 B=35 D=25 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:224 A:217 B:197 C:188 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 14 -4 B -10 0 4 8 -8 C -14 -4 0 12 -18 D -14 -8 -12 0 -18 E 4 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 14 -4 B -10 0 4 8 -8 C -14 -4 0 12 -18 D -14 -8 -12 0 -18 E 4 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 14 -4 B -10 0 4 8 -8 C -14 -4 0 12 -18 D -14 -8 -12 0 -18 E 4 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5010: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (6) B A E D C (6) E C B D A (5) E B C A D (5) D A E B C (5) D A C B E (5) A B D E C (5) E D C B A (4) D E C B A (4) E C D B A (3) E B A D C (3) B E C A D (3) E D B A C (2) E B D A C (2) E B A C D (2) D A B E C (2) C E B A D (2) C D E A B (2) C D A E B (2) C A B D E (2) B E A D C (2) B A E C D (2) A D C B E (2) A D B E C (2) A D B C E (2) A C D B E (2) E D B C A (1) E C B A D (1) E B D C A (1) D E C A B (1) D E A B C (1) D C A E B (1) D C A B E (1) C E B D A (1) C B A E D (1) C A D B E (1) C A B E D (1) B E A C D (1) B C E A D (1) B C A E D (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 0 2 -6 B 18 0 4 2 -8 C 0 -4 0 -6 -26 D -2 -2 6 0 -14 E 6 8 26 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 0 2 -6 B 18 0 4 2 -8 C 0 -4 0 -6 -26 D -2 -2 6 0 -14 E 6 8 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=20 C=18 A=17 B=16 so B is eliminated. Round 2 votes counts: E=35 A=25 D=20 C=20 so D is eliminated. Round 3 votes counts: E=41 A=37 C=22 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:208 D:194 A:189 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 0 2 -6 B 18 0 4 2 -8 C 0 -4 0 -6 -26 D -2 -2 6 0 -14 E 6 8 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 2 -6 B 18 0 4 2 -8 C 0 -4 0 -6 -26 D -2 -2 6 0 -14 E 6 8 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 2 -6 B 18 0 4 2 -8 C 0 -4 0 -6 -26 D -2 -2 6 0 -14 E 6 8 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5011: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (10) D C E A B (9) B A E C D (9) D C A E B (6) E B A D C (5) E A B D C (4) E A B C D (4) C D A B E (4) B E A D C (4) D E B A C (3) D C B A E (3) C D A E B (3) C A B D E (3) E D B A C (2) D C A B E (2) C D B A E (2) C A D B E (2) C A B E D (2) B C A E D (2) A E B C D (2) A C B E D (2) A B E C D (2) E B D A C (1) E B A C D (1) E A D C B (1) E A D B C (1) D E A B C (1) D C E B A (1) D C B E A (1) C B D A E (1) C B A D E (1) C A E B D (1) C A D E B (1) B C E D A (1) A E D C B (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 10 20 2 B -6 0 6 14 4 C -10 -6 0 10 -4 D -20 -14 -10 0 -14 E -2 -4 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997761 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 20 2 B -6 0 6 14 4 C -10 -6 0 10 -4 D -20 -14 -10 0 -14 E -2 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 C=20 E=19 A=9 so A is eliminated. Round 2 votes counts: B=29 D=26 E=23 C=22 so C is eliminated. Round 3 votes counts: D=38 B=38 E=24 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:209 E:206 C:195 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 20 2 B -6 0 6 14 4 C -10 -6 0 10 -4 D -20 -14 -10 0 -14 E -2 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 20 2 B -6 0 6 14 4 C -10 -6 0 10 -4 D -20 -14 -10 0 -14 E -2 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 20 2 B -6 0 6 14 4 C -10 -6 0 10 -4 D -20 -14 -10 0 -14 E -2 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972283 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5012: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (7) A E C D B (7) E A D C B (6) C A D E B (6) B D C E A (6) B C D A E (5) B E D A C (4) B D C A E (4) E D B A C (3) E A C D B (3) D B E C A (3) C A B D E (3) B A E C D (3) A E C B D (3) E B D A C (2) E A C B D (2) D C B A E (2) D B C E A (2) C D B A E (2) C D A E B (2) C B D A E (2) C A D B E (2) A C E D B (2) E D A C B (1) E B A D C (1) E A D B C (1) E A B D C (1) E A B C D (1) D E C A B (1) D E B A C (1) D C A B E (1) D B C A E (1) C B A D E (1) C A E D B (1) C A B E D (1) B E A D C (1) B C A E D (1) B A C E D (1) A C E B D (1) A C D E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 2 8 B 4 0 0 4 8 C 6 0 0 4 -4 D -2 -4 -4 0 4 E -8 -8 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.557582 C: 0.442418 D: 0.000000 E: 0.000000 Sum of squares = 0.506631420486 Cumulative probabilities = A: 0.000000 B: 0.557582 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 8 B 4 0 0 4 8 C 6 0 0 4 -4 D -2 -4 -4 0 4 E -8 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=21 C=20 A=16 D=11 so D is eliminated. Round 2 votes counts: B=38 E=23 C=23 A=16 so A is eliminated. Round 3 votes counts: B=40 E=33 C=27 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 C:203 A:200 D:197 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 2 8 B 4 0 0 4 8 C 6 0 0 4 -4 D -2 -4 -4 0 4 E -8 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 8 B 4 0 0 4 8 C 6 0 0 4 -4 D -2 -4 -4 0 4 E -8 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 8 B 4 0 0 4 8 C 6 0 0 4 -4 D -2 -4 -4 0 4 E -8 -8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5013: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) C E B A D (8) B D E C A (7) A C E D B (7) E C B A D (6) D B E A C (6) B E D C A (6) B E C D A (6) D B A E C (5) D A B E C (5) C A E D B (5) A D C E B (4) D A B C E (3) C A E B D (3) B D E A C (3) D B C E A (2) A C D E B (2) D C A B E (1) D B E C A (1) D B C A E (1) D A C B E (1) C E B D A (1) C A D E B (1) B E D A C (1) B D C E A (1) B C E D A (1) A E D C B (1) A E C B D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -18 -2 -16 B 10 0 0 8 4 C 18 0 0 0 2 D 2 -8 0 0 -12 E 16 -4 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.708289 C: 0.291711 D: 0.000000 E: 0.000000 Sum of squares = 0.586768291899 Cumulative probabilities = A: 0.000000 B: 0.708289 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -2 -16 B 10 0 0 8 4 C 18 0 0 0 2 D 2 -8 0 0 -12 E 16 -4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 B=25 A=17 E=6 so E is eliminated. Round 2 votes counts: C=33 D=25 B=25 A=17 so A is eliminated. Round 3 votes counts: C=43 D=31 B=26 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:211 E:211 C:210 D:191 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -18 -2 -16 B 10 0 0 8 4 C 18 0 0 0 2 D 2 -8 0 0 -12 E 16 -4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -2 -16 B 10 0 0 8 4 C 18 0 0 0 2 D 2 -8 0 0 -12 E 16 -4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -2 -16 B 10 0 0 8 4 C 18 0 0 0 2 D 2 -8 0 0 -12 E 16 -4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5014: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) D A C B E (6) C E B D A (6) C E A B D (5) B D E A C (5) E C B A D (4) E B C D A (4) E B A C D (4) C A E D B (4) C A E B D (4) C A D E B (4) A E B C D (4) D C A B E (3) D B E C A (3) D B C E A (3) D B C A E (3) D B A C E (2) C E B A D (2) B E D A C (2) B E A D C (2) B D A E C (2) A C D E B (2) E B A D C (1) E A C B D (1) E A B C D (1) D B E A C (1) C E D B A (1) C E A D B (1) C D E B A (1) C D A B E (1) C A D B E (1) B E C D A (1) A E C B D (1) A E B D C (1) A D B E C (1) A D B C E (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -2 -8 4 B 14 0 4 4 -4 C 2 -4 0 6 2 D 8 -4 -6 0 0 E -4 4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -14 -2 -8 4 B 14 0 4 4 -4 C 2 -4 0 6 2 D 8 -4 -6 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=30 E=15 B=12 A=12 so B is eliminated. Round 2 votes counts: D=38 C=30 E=20 A=12 so A is eliminated. Round 3 votes counts: D=41 C=33 E=26 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:209 C:203 D:199 E:199 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -8 4 B 14 0 4 4 -4 C 2 -4 0 6 2 D 8 -4 -6 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -8 4 B 14 0 4 4 -4 C 2 -4 0 6 2 D 8 -4 -6 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -8 4 B 14 0 4 4 -4 C 2 -4 0 6 2 D 8 -4 -6 0 0 E -4 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5015: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) A C B D E (8) C A B E D (7) D E B A C (5) B D C E A (5) C A B D E (4) B D E C A (4) A C E D B (4) E D A B C (3) C B A D E (3) B C D E A (3) A E C D B (3) A D E B C (3) E D C B A (2) E A D B C (2) C B E D A (2) C A E B D (2) B D E A C (2) B C A D E (2) A E D C B (2) A E D B C (2) A C E B D (2) E D B A C (1) E D A C B (1) E C A D B (1) E A D C B (1) D E B C A (1) D E A B C (1) D B E C A (1) D B E A C (1) C E B D A (1) C B D E A (1) C B A E D (1) C A E D B (1) B D C A E (1) B D A E C (1) B A D C E (1) B A C D E (1) A D E C B (1) A D B E C (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 10 6 B -4 0 2 6 4 C 2 -2 0 -2 4 D -10 -6 2 0 4 E -6 -4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999847 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 10 6 B -4 0 2 6 4 C 2 -2 0 -2 4 D -10 -6 2 0 4 E -6 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999992 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=22 E=20 B=20 D=9 so D is eliminated. Round 2 votes counts: A=29 E=27 C=22 B=22 so C is eliminated. Round 3 votes counts: A=43 B=29 E=28 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:204 C:201 D:195 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 10 6 B -4 0 2 6 4 C 2 -2 0 -2 4 D -10 -6 2 0 4 E -6 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999992 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 10 6 B -4 0 2 6 4 C 2 -2 0 -2 4 D -10 -6 2 0 4 E -6 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999992 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 10 6 B -4 0 2 6 4 C 2 -2 0 -2 4 D -10 -6 2 0 4 E -6 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999992 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5016: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) E A B D C (9) D A C E B (7) E B A D C (6) C D A B E (6) E B A C D (5) C B D E A (5) B C E D A (4) B E C D A (3) B E A D C (3) B E A C D (3) A E D B C (3) D C B A E (2) D A E B C (2) C E A D B (2) C E A B D (2) C B E D A (2) A D E C B (2) A D E B C (2) E A D B C (1) E A C D B (1) E A B C D (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A E C B (1) D A B C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C D B A E (1) C D A E B (1) C B E A D (1) C A D E B (1) B E D C A (1) B E D A C (1) B D C A E (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 -12 -2 10 -30 B 12 0 22 24 0 C 2 -22 0 8 -10 D -10 -24 -8 0 -26 E 30 0 10 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.569947 C: 0.000000 D: 0.000000 E: 0.430053 Sum of squares = 0.509785251915 Cumulative probabilities = A: 0.000000 B: 0.569947 C: 0.569947 D: 0.569947 E: 1.000000 A B C D E A 0 -12 -2 10 -30 B 12 0 22 24 0 C 2 -22 0 8 -10 D -10 -24 -8 0 -26 E 30 0 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=24 E=23 D=17 A=7 so A is eliminated. Round 2 votes counts: B=29 E=26 C=24 D=21 so D is eliminated. Round 3 votes counts: C=35 E=33 B=32 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:233 B:229 C:189 A:183 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 10 -30 B 12 0 22 24 0 C 2 -22 0 8 -10 D -10 -24 -8 0 -26 E 30 0 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 10 -30 B 12 0 22 24 0 C 2 -22 0 8 -10 D -10 -24 -8 0 -26 E 30 0 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 10 -30 B 12 0 22 24 0 C 2 -22 0 8 -10 D -10 -24 -8 0 -26 E 30 0 10 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5017: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (18) B A D E C (13) C E D A B (12) B A D C E (9) D A C B E (7) E C B D A (6) B A E D C (6) A B D C E (5) D A B C E (4) E C D B A (3) E B C A D (3) C D A E B (3) C D E A B (2) B E A D C (2) B E A C D (2) E C B A D (1) D E A C B (1) D C A E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 10 2 -14 0 B -10 0 -10 -6 0 C -2 10 0 0 -10 D 14 6 0 0 -6 E 0 0 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.228500 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.771500 Sum of squares = 0.647424021655 Cumulative probabilities = A: 0.228500 B: 0.228500 C: 0.228500 D: 0.228500 E: 1.000000 A B C D E A 0 10 2 -14 0 B -10 0 -10 -6 0 C -2 10 0 0 -10 D 14 6 0 0 -6 E 0 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000116006 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=31 C=17 D=13 A=7 so A is eliminated. Round 2 votes counts: B=37 E=31 C=17 D=15 so D is eliminated. Round 3 votes counts: B=42 E=32 C=26 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:208 D:207 A:199 C:199 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 -14 0 B -10 0 -10 -6 0 C -2 10 0 0 -10 D 14 6 0 0 -6 E 0 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000116006 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -14 0 B -10 0 -10 -6 0 C -2 10 0 0 -10 D 14 6 0 0 -6 E 0 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000116006 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -14 0 B -10 0 -10 -6 0 C -2 10 0 0 -10 D 14 6 0 0 -6 E 0 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.700000 Sum of squares = 0.580000116006 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.300000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5018: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) E C B D A (9) E C D B A (7) B A E C D (7) D A C B E (6) E B C A D (5) B A E D C (5) A B D E C (5) E C B A D (4) D A C E B (4) D A B C E (4) C D E A B (4) D C E A B (3) D C A E B (3) B E A C D (3) B A D E C (3) A B D C E (3) E C D A B (2) C E D A B (2) B C E A D (2) A B E D C (2) E D A C B (1) E B A C D (1) C E B D A (1) C D B A E (1) C B E D A (1) B A D C E (1) A D B E C (1) Total count = 100 A B C D E A 0 -20 -8 -16 -10 B 20 0 -16 4 -12 C 8 16 0 18 -10 D 16 -4 -18 0 -24 E 10 12 10 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -8 -16 -10 B 20 0 -16 4 -12 C 8 16 0 18 -10 D 16 -4 -18 0 -24 E 10 12 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=21 D=20 C=19 A=11 so A is eliminated. Round 2 votes counts: B=31 E=29 D=21 C=19 so C is eliminated. Round 3 votes counts: E=42 B=32 D=26 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:216 B:198 D:185 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -8 -16 -10 B 20 0 -16 4 -12 C 8 16 0 18 -10 D 16 -4 -18 0 -24 E 10 12 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -16 -10 B 20 0 -16 4 -12 C 8 16 0 18 -10 D 16 -4 -18 0 -24 E 10 12 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -16 -10 B 20 0 -16 4 -12 C 8 16 0 18 -10 D 16 -4 -18 0 -24 E 10 12 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5019: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (6) D B A E C (5) A C B E D (5) A B D C E (5) C E A D B (4) C A E D B (4) D E C B A (3) D E B C A (3) D B E A C (3) C E D B A (3) C A E B D (3) B D E A C (3) B A E C D (3) A D B C E (3) A C E B D (3) A B C D E (3) E D C B A (2) E C B D A (2) E B C D A (2) D E C A B (2) C E B A D (2) C D E A B (2) B E D C A (2) B D A E C (2) B A D E C (2) A C D E B (2) E C D B A (1) E B D C A (1) D E B A C (1) D C E A B (1) D B E C A (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D A B (1) C E A B D (1) C D A E B (1) B E D A C (1) B E C D A (1) B E C A D (1) B D E C A (1) B C E A D (1) B C A E D (1) B A E D C (1) B A C E D (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 2 8 6 10 B -2 0 10 6 8 C -8 -10 0 6 8 D -6 -6 -6 0 -4 E -10 -8 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 6 10 B -2 0 10 6 8 C -8 -10 0 6 8 D -6 -6 -6 0 -4 E -10 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 C=21 B=20 E=8 so E is eliminated. Round 2 votes counts: A=29 D=24 C=24 B=23 so B is eliminated. Round 3 votes counts: A=36 D=34 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:211 C:198 D:189 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 6 10 B -2 0 10 6 8 C -8 -10 0 6 8 D -6 -6 -6 0 -4 E -10 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 6 10 B -2 0 10 6 8 C -8 -10 0 6 8 D -6 -6 -6 0 -4 E -10 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 6 10 B -2 0 10 6 8 C -8 -10 0 6 8 D -6 -6 -6 0 -4 E -10 -8 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5020: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (16) A B C D E (10) A D C E B (7) D E C A B (6) D C E A B (5) B A C D E (4) A E B D C (4) A B E D C (4) D C E B A (3) B E C D A (3) B E A C D (3) B A C E D (3) A C D B E (3) A B C E D (3) E D C A B (2) E B D C A (2) D A C E B (2) C D E B A (2) B C D E A (2) A D E C B (2) E D B C A (1) E C D B A (1) E B C D A (1) D E C B A (1) D C A E B (1) D A E C B (1) C D B E A (1) C D B A E (1) C B D E A (1) B E A D C (1) B A E C D (1) A E D C B (1) A E D B C (1) A C B D E (1) Total count = 100 A B C D E A 0 6 2 -4 -2 B -6 0 -14 -14 -18 C -2 14 0 -20 -2 D 4 14 20 0 6 E 2 18 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -4 -2 B -6 0 -14 -14 -18 C -2 14 0 -20 -2 D 4 14 20 0 6 E 2 18 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=23 D=19 B=17 C=5 so C is eliminated. Round 2 votes counts: A=36 E=23 D=23 B=18 so B is eliminated. Round 3 votes counts: A=44 E=30 D=26 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:222 E:208 A:201 C:195 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -4 -2 B -6 0 -14 -14 -18 C -2 14 0 -20 -2 D 4 14 20 0 6 E 2 18 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -4 -2 B -6 0 -14 -14 -18 C -2 14 0 -20 -2 D 4 14 20 0 6 E 2 18 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -4 -2 B -6 0 -14 -14 -18 C -2 14 0 -20 -2 D 4 14 20 0 6 E 2 18 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5021: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) A C D B E (10) E B D C A (9) B E C D A (8) A D C E B (7) D E A B C (6) B C E D A (5) E D B A C (4) D A E B C (4) C A B E D (4) A C D E B (4) E D B C A (3) C B A E D (3) A D E B C (3) A C B D E (3) C B E D A (2) C B E A D (2) C A B D E (2) B E D C A (2) A D E C B (2) E B D A C (1) D E C B A (1) D E B C A (1) D A E C B (1) D A C E B (1) B C E A D (1) A E D B C (1) Total count = 100 A B C D E A 0 -4 14 -16 -10 B 4 0 16 -16 -16 C -14 -16 0 -12 -12 D 16 16 12 0 10 E 10 16 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 -16 -10 B 4 0 16 -16 -16 C -14 -16 0 -12 -12 D 16 16 12 0 10 E 10 16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 E=17 B=16 C=13 so C is eliminated. Round 2 votes counts: A=36 D=24 B=23 E=17 so E is eliminated. Round 3 votes counts: A=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:227 E:214 B:194 A:192 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 14 -16 -10 B 4 0 16 -16 -16 C -14 -16 0 -12 -12 D 16 16 12 0 10 E 10 16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 -16 -10 B 4 0 16 -16 -16 C -14 -16 0 -12 -12 D 16 16 12 0 10 E 10 16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 -16 -10 B 4 0 16 -16 -16 C -14 -16 0 -12 -12 D 16 16 12 0 10 E 10 16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5022: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (12) D A B E C (9) C E B D A (8) B E C A D (7) D A C B E (6) A D B E C (5) E C B D A (4) E B C D A (4) D A C E B (4) C A D E B (4) A D B C E (4) D A B C E (3) A D C B E (3) E B C A D (2) D A E B C (2) B E D C A (2) B E A D C (2) B E A C D (2) B C E A D (2) A D C E B (2) A B D E C (2) E D B A C (1) E B D C A (1) C E D B A (1) C E D A B (1) C D E A B (1) C D A E B (1) C B E A D (1) B E D A C (1) B E C D A (1) B A E D C (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -4 0 -6 B 4 0 2 4 4 C 4 -2 0 4 8 D 0 -4 -4 0 -6 E 6 -4 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 0 -6 B 4 0 2 4 4 C 4 -2 0 4 8 D 0 -4 -4 0 -6 E 6 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 B=18 A=17 E=12 so E is eliminated. Round 2 votes counts: C=33 D=25 B=25 A=17 so A is eliminated. Round 3 votes counts: D=39 C=34 B=27 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:207 C:207 E:200 A:193 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 0 -6 B 4 0 2 4 4 C 4 -2 0 4 8 D 0 -4 -4 0 -6 E 6 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 -6 B 4 0 2 4 4 C 4 -2 0 4 8 D 0 -4 -4 0 -6 E 6 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 -6 B 4 0 2 4 4 C 4 -2 0 4 8 D 0 -4 -4 0 -6 E 6 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5023: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) E C A D B (8) D B C E A (8) C E D B A (8) A E C B D (8) B A D C E (6) B D A C E (5) A B D C E (5) E C D A B (4) B D C A E (4) E C D B A (3) C E A D B (3) D C B E A (2) C E D A B (2) C D E B A (2) C D B E A (2) E D C B A (1) E C A B D (1) E A D C B (1) E A C B D (1) D C E B A (1) D B C A E (1) D B A E C (1) D B A C E (1) C B D E A (1) C B D A E (1) C A E B D (1) B A D E C (1) A E D B C (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 6 4 B -4 0 0 0 6 C 6 0 0 -4 8 D -6 0 4 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999974 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 6 4 B -4 0 0 0 6 C 6 0 0 -4 8 D -6 0 4 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=20 E=19 B=16 D=14 so D is eliminated. Round 2 votes counts: A=31 B=27 C=23 E=19 so E is eliminated. Round 3 votes counts: C=40 A=33 B=27 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:205 A:204 D:204 B:201 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 6 4 B -4 0 0 0 6 C 6 0 0 -4 8 D -6 0 4 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 6 4 B -4 0 0 0 6 C 6 0 0 -4 8 D -6 0 4 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 6 4 B -4 0 0 0 6 C 6 0 0 -4 8 D -6 0 4 0 10 E -4 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.375000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5024: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (15) E A D B C (10) E A C D B (6) E D A B C (5) C B D E A (5) B C D E A (5) A E C D B (5) A D E B C (4) E A D C B (3) C A E D B (3) C A D B E (3) B D E C A (3) B D C A E (3) A E D C B (3) D B E A C (2) D B A C E (2) B C D A E (2) A C E D B (2) E D B A C (1) E C B A D (1) E B D A C (1) E B C D A (1) E A C B D (1) E A B D C (1) D E A B C (1) D B C A E (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C B A E D (1) C A B E D (1) C A B D E (1) B D E A C (1) B D C E A (1) A D E C B (1) Total count = 100 A B C D E A 0 8 4 -4 0 B -8 0 -6 -10 -4 C -4 6 0 8 -4 D 4 10 -8 0 6 E 0 4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000025 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -4 0 B -8 0 -6 -10 -4 C -4 6 0 8 -4 D 4 10 -8 0 6 E 0 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999994925 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=30 B=15 A=15 D=9 so D is eliminated. Round 2 votes counts: E=31 C=31 B=20 A=18 so A is eliminated. Round 3 votes counts: E=45 C=33 B=22 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:206 A:204 C:203 E:201 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 -4 0 B -8 0 -6 -10 -4 C -4 6 0 8 -4 D 4 10 -8 0 6 E 0 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999994925 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -4 0 B -8 0 -6 -10 -4 C -4 6 0 8 -4 D 4 10 -8 0 6 E 0 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999994925 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -4 0 B -8 0 -6 -10 -4 C -4 6 0 8 -4 D 4 10 -8 0 6 E 0 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999994925 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5025: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) A B D E C (8) C E D A B (7) C A D E B (7) C E B D A (6) B A D E C (6) A D B C E (6) E C B D A (5) E B D A C (4) D A B E C (3) C E A B D (3) C A D B E (3) C A B D E (3) E D B A C (2) C E D B A (2) C E A D B (2) C B E A D (2) C A E B D (2) C A B E D (2) B E C D A (2) E D C B A (1) E D A C B (1) E C D A B (1) E B C D A (1) D B A E C (1) D A E B C (1) C B A E D (1) B E D A C (1) B E A D C (1) B D E A C (1) B D A E C (1) B C E A D (1) B A D C E (1) A D C B E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 22 -2 20 14 B -22 0 0 4 10 C 2 0 0 2 2 D -20 -4 -2 0 6 E -14 -10 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.060764 C: 0.939236 D: 0.000000 E: 0.000000 Sum of squares = 0.885855727968 Cumulative probabilities = A: 0.000000 B: 0.060764 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -2 20 14 B -22 0 0 4 10 C 2 0 0 2 2 D -20 -4 -2 0 6 E -14 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.916667 D: 0.000000 E: 0.000000 Sum of squares = 0.847222246338 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 A=26 E=15 B=14 D=5 so D is eliminated. Round 2 votes counts: C=40 A=30 E=15 B=15 so E is eliminated. Round 3 votes counts: C=47 A=31 B=22 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:227 C:203 B:196 D:190 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -2 20 14 B -22 0 0 4 10 C 2 0 0 2 2 D -20 -4 -2 0 6 E -14 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.916667 D: 0.000000 E: 0.000000 Sum of squares = 0.847222246338 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -2 20 14 B -22 0 0 4 10 C 2 0 0 2 2 D -20 -4 -2 0 6 E -14 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.916667 D: 0.000000 E: 0.000000 Sum of squares = 0.847222246338 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -2 20 14 B -22 0 0 4 10 C 2 0 0 2 2 D -20 -4 -2 0 6 E -14 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.916667 D: 0.000000 E: 0.000000 Sum of squares = 0.847222246338 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5026: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) B A E D C (14) D C E A B (10) C D B E A (9) C B D A E (8) E A D B C (7) D E A C B (7) D E A B C (5) A E B D C (4) D E C A B (3) B A E C D (3) C D E B A (2) C B A E D (2) C B A D E (2) A B E D C (2) E A B D C (1) C E D A B (1) C E A D B (1) B D A E C (1) B C A E D (1) B A D E C (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 14 -6 -20 -20 B -14 0 -18 -20 -14 C 6 18 0 -12 0 D 20 20 12 0 26 E 20 14 0 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 -20 -20 B -14 0 -18 -20 -14 C 6 18 0 -12 0 D 20 20 12 0 26 E 20 14 0 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=25 B=20 E=8 A=8 so E is eliminated. Round 2 votes counts: C=39 D=25 B=20 A=16 so A is eliminated. Round 3 votes counts: C=39 D=33 B=28 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:239 C:206 E:204 A:184 B:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -6 -20 -20 B -14 0 -18 -20 -14 C 6 18 0 -12 0 D 20 20 12 0 26 E 20 14 0 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 -20 -20 B -14 0 -18 -20 -14 C 6 18 0 -12 0 D 20 20 12 0 26 E 20 14 0 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 -20 -20 B -14 0 -18 -20 -14 C 6 18 0 -12 0 D 20 20 12 0 26 E 20 14 0 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5027: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) B D A E C (12) A B D C E (9) C E D B A (8) B A D E C (7) E C D B A (6) A B D E C (6) D B E C A (5) A B C D E (5) A C B E D (4) D E B C A (3) D B E A C (3) C E D A B (3) C A E B D (3) B D E C A (3) E D C B A (2) E D B C A (1) E D A B C (1) D B A E C (1) C E B D A (1) C A E D B (1) B C A D E (1) A D E B C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 2 2 4 B 6 0 16 6 14 C -2 -16 0 -8 -2 D -2 -6 8 0 14 E -4 -14 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 2 4 B 6 0 16 6 14 C -2 -16 0 -8 -2 D -2 -6 8 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 B=23 D=12 E=10 so E is eliminated. Round 2 votes counts: C=34 A=27 B=23 D=16 so D is eliminated. Round 3 votes counts: C=36 B=36 A=28 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:207 A:201 C:186 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 2 4 B 6 0 16 6 14 C -2 -16 0 -8 -2 D -2 -6 8 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 4 B 6 0 16 6 14 C -2 -16 0 -8 -2 D -2 -6 8 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 4 B 6 0 16 6 14 C -2 -16 0 -8 -2 D -2 -6 8 0 14 E -4 -14 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5028: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) D C E B A (9) B A E D C (9) D B A C E (8) B A D E C (8) E A B C D (7) C E D A B (7) D B A E C (6) A B E C D (6) D C B A E (5) E C A B D (4) C E A D B (4) E A C B D (3) D C E A B (2) D C B E A (2) D B C A E (2) C D E B A (2) C E A B D (1) C D B A E (1) B D A C E (1) B A E C D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 -10 -2 B 8 0 0 -16 0 C -2 0 0 -6 8 D 10 16 6 0 12 E 2 0 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -10 -2 B 8 0 0 -16 0 C -2 0 0 -6 8 D 10 16 6 0 12 E 2 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=25 B=19 E=14 A=8 so A is eliminated. Round 2 votes counts: D=34 B=26 C=25 E=15 so E is eliminated. Round 3 votes counts: D=34 B=34 C=32 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:200 B:196 A:191 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -10 -2 B 8 0 0 -16 0 C -2 0 0 -6 8 D 10 16 6 0 12 E 2 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -10 -2 B 8 0 0 -16 0 C -2 0 0 -6 8 D 10 16 6 0 12 E 2 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -10 -2 B 8 0 0 -16 0 C -2 0 0 -6 8 D 10 16 6 0 12 E 2 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5029: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) C A E B D (8) B D E C A (8) A D C B E (8) E C B A D (7) E B C D A (7) C E A B D (7) B E D C A (7) A C D E B (6) D B A E C (4) D A B C E (4) A C E D B (4) E B C A D (3) E B D C A (2) D A C B E (2) C A E D B (2) E C B D A (1) D B A C E (1) D A B E C (1) C B D A E (1) C A D B E (1) B D E A C (1) B C E D A (1) A E C D B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -10 -2 -10 B 8 0 2 6 4 C 10 -2 0 0 -6 D 2 -6 0 0 0 E 10 -4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998797 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -2 -10 B 8 0 2 6 4 C 10 -2 0 0 -6 D 2 -6 0 0 0 E 10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=21 E=20 C=19 B=17 so B is eliminated. Round 2 votes counts: D=32 E=27 A=21 C=20 so C is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 E:206 C:201 D:198 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -2 -10 B 8 0 2 6 4 C 10 -2 0 0 -6 D 2 -6 0 0 0 E 10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -2 -10 B 8 0 2 6 4 C 10 -2 0 0 -6 D 2 -6 0 0 0 E 10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -2 -10 B 8 0 2 6 4 C 10 -2 0 0 -6 D 2 -6 0 0 0 E 10 -4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5030: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (11) C A D B E (7) E A D B C (6) B C D E A (6) B E D C A (5) E B C A D (4) C D A B E (4) A D E C B (4) E B D A C (3) A E D C B (3) E B C D A (2) E A C B D (2) D B C A E (2) D A C B E (2) D A B E C (2) D A B C E (2) C A B D E (2) B D C E A (2) B C E D A (2) A E D B C (2) A D C E B (2) A C D E B (2) E C A B D (1) E B A D C (1) E A D C B (1) E A B D C (1) E A B C D (1) D C B A E (1) D C A B E (1) D B E A C (1) D B A E C (1) D A E B C (1) C D B A E (1) C B E A D (1) C B D E A (1) C B A D E (1) C A E D B (1) B E D A C (1) B E C D A (1) B D E C A (1) B D E A C (1) B D C A E (1) A D E B C (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -16 -10 12 B 0 0 0 2 22 C 16 0 0 2 8 D 10 -2 -2 0 22 E -12 -22 -8 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.451598 C: 0.548402 D: 0.000000 E: 0.000000 Sum of squares = 0.504685535933 Cumulative probabilities = A: 0.000000 B: 0.451598 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -10 12 B 0 0 0 2 22 C 16 0 0 2 8 D 10 -2 -2 0 22 E -12 -22 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999431 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=22 B=20 A=16 D=13 so D is eliminated. Round 2 votes counts: C=31 B=24 A=23 E=22 so E is eliminated. Round 3 votes counts: B=34 A=34 C=32 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:214 C:213 B:212 A:193 E:168 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -16 -10 12 B 0 0 0 2 22 C 16 0 0 2 8 D 10 -2 -2 0 22 E -12 -22 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999431 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -10 12 B 0 0 0 2 22 C 16 0 0 2 8 D 10 -2 -2 0 22 E -12 -22 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999431 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -10 12 B 0 0 0 2 22 C 16 0 0 2 8 D 10 -2 -2 0 22 E -12 -22 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999431 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5031: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) C D A B E (12) D C E B A (6) C A B E D (6) D E B A C (5) D C B A E (5) C A B D E (4) E A B C D (3) D C B E A (3) C D E A B (3) C D A E B (3) B A E D C (3) A B E C D (3) E D B A C (2) E B D A C (2) E A C B D (2) D E C B A (2) D C E A B (2) D B E A C (2) C D B A E (2) C A E B D (2) C A D B E (2) A C B E D (2) A B C E D (2) E A B D C (1) D E B C A (1) D C A B E (1) C E D A B (1) C E A D B (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E C D (1) A E B C D (1) Total count = 100 A B C D E A 0 2 -12 -8 0 B -2 0 -18 -6 2 C 12 18 0 0 14 D 8 6 0 0 10 E 0 -2 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.332524 D: 0.667476 E: 0.000000 Sum of squares = 0.556096627271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.332524 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -8 0 B -2 0 -18 -6 2 C 12 18 0 0 14 D 8 6 0 0 10 E 0 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=27 E=22 A=8 B=7 so B is eliminated. Round 2 votes counts: C=36 D=29 E=23 A=12 so A is eliminated. Round 3 votes counts: C=40 E=31 D=29 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:212 A:191 B:188 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -8 0 B -2 0 -18 -6 2 C 12 18 0 0 14 D 8 6 0 0 10 E 0 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -8 0 B -2 0 -18 -6 2 C 12 18 0 0 14 D 8 6 0 0 10 E 0 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -8 0 B -2 0 -18 -6 2 C 12 18 0 0 14 D 8 6 0 0 10 E 0 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5032: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (11) B E A D C (10) A C D B E (10) D C E B A (9) C D E A B (9) B A E D C (9) D E C B A (6) E D B C A (4) E B D C A (4) C D A E B (4) B E D A C (4) C A D E B (3) A C B D E (3) A B E C D (3) A B C D E (3) E D C B A (1) E B D A C (1) D C E A B (1) C A B D E (1) B E D C A (1) B A E C D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 14 12 0 B 0 0 4 4 14 C -14 -4 0 -2 10 D -12 -4 2 0 0 E 0 -14 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.378456 B: 0.621544 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.529546002269 Cumulative probabilities = A: 0.378456 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 12 0 B 0 0 4 4 14 C -14 -4 0 -2 10 D -12 -4 2 0 0 E 0 -14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=25 C=17 D=16 E=10 so E is eliminated. Round 2 votes counts: A=32 B=30 D=21 C=17 so C is eliminated. Round 3 votes counts: A=36 D=34 B=30 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:211 C:195 D:193 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 12 0 B 0 0 4 4 14 C -14 -4 0 -2 10 D -12 -4 2 0 0 E 0 -14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 12 0 B 0 0 4 4 14 C -14 -4 0 -2 10 D -12 -4 2 0 0 E 0 -14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 12 0 B 0 0 4 4 14 C -14 -4 0 -2 10 D -12 -4 2 0 0 E 0 -14 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5033: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) A C B E D (9) E D B C A (8) D E A B C (7) C B A E D (7) E D A C B (6) B C A D E (5) E A C B D (4) D E B A C (3) D B C A E (3) C A B D E (3) B C D A E (3) A B C D E (3) E D C A B (2) E C A B D (2) E A D C B (2) D B C E A (2) D B A C E (2) C B A D E (2) C A B E D (2) B D C A E (2) A E C B D (2) A C E B D (2) A C B D E (2) E D C B A (1) E D A B C (1) E A C D B (1) C B E A D (1) B C D E A (1) A E C D B (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -8 -2 -2 B 0 0 2 0 -4 C 8 -2 0 0 0 D 2 0 0 0 -2 E 2 4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.331808 D: 0.000000 E: 0.668192 Sum of squares = 0.556577266226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.331808 D: 0.331808 E: 1.000000 A B C D E A 0 0 -8 -2 -2 B 0 0 2 0 -4 C 8 -2 0 0 0 D 2 0 0 0 -2 E 2 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=27 D=27 A=20 C=15 B=11 so B is eliminated. Round 2 votes counts: D=29 E=27 C=24 A=20 so A is eliminated. Round 3 votes counts: C=40 E=30 D=30 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:204 C:203 D:200 B:199 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -2 -2 B 0 0 2 0 -4 C 8 -2 0 0 0 D 2 0 0 0 -2 E 2 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 -2 B 0 0 2 0 -4 C 8 -2 0 0 0 D 2 0 0 0 -2 E 2 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 -2 B 0 0 2 0 -4 C 8 -2 0 0 0 D 2 0 0 0 -2 E 2 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5034: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (11) B C D A E (9) B C A D E (9) B A C E D (9) A B E D C (9) E D A C B (7) C D B E A (7) A E D B C (7) C D E A B (6) E A D C B (5) E D C A B (4) D C E A B (3) A E B D C (3) D E C A B (2) B C A E D (2) B A C D E (2) E A C D B (1) D E A C B (1) C D E B A (1) A E D C B (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -2 4 B 0 0 2 10 18 C 8 -2 0 16 18 D 2 -10 -16 0 2 E -4 -18 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.115832 B: 0.884168 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.795169766699 Cumulative probabilities = A: 0.115832 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -2 4 B 0 0 2 10 18 C 8 -2 0 16 18 D 2 -10 -16 0 2 E -4 -18 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005877 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=25 A=21 E=17 D=6 so D is eliminated. Round 2 votes counts: B=31 C=28 A=21 E=20 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:215 A:197 D:189 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -2 4 B 0 0 2 10 18 C 8 -2 0 16 18 D 2 -10 -16 0 2 E -4 -18 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005877 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 4 B 0 0 2 10 18 C 8 -2 0 16 18 D 2 -10 -16 0 2 E -4 -18 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005877 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 4 B 0 0 2 10 18 C 8 -2 0 16 18 D 2 -10 -16 0 2 E -4 -18 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005877 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5035: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (10) B E C D A (8) A D E B C (6) E B C A D (5) B E C A D (5) D B C A E (4) E B A C D (3) E A B C D (3) D C B E A (3) D A C E B (3) C D B E A (3) C A E D B (3) B E A C D (3) A D E C B (3) A D B E C (3) D B A C E (2) D A C B E (2) D A B E C (2) D A B C E (2) B E A D C (2) B D C E A (2) A E B D C (2) E C A B D (1) E B C D A (1) E A C B D (1) D C A B E (1) D B E C A (1) D B E A C (1) C E D B A (1) C E B D A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D A E B (1) C B E D A (1) C B D E A (1) B E D C A (1) B D E A C (1) A E D C B (1) A E D B C (1) A E C D B (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 8 12 -2 B 0 0 18 -14 -2 C -8 -18 0 -8 -12 D -12 14 8 0 6 E 2 2 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.600000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.400000 E: 1.000000 A B C D E A 0 0 8 12 -2 B 0 0 18 -14 -2 C -8 -18 0 -8 -12 D -12 14 8 0 6 E 2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.600000 Sum of squares = 0.459999999864 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=22 D=21 E=14 C=14 so E is eliminated. Round 2 votes counts: A=33 B=31 D=21 C=15 so C is eliminated. Round 3 votes counts: A=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 D:208 E:205 B:201 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 12 -2 B 0 0 18 -14 -2 C -8 -18 0 -8 -12 D -12 14 8 0 6 E 2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.600000 Sum of squares = 0.459999999864 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 12 -2 B 0 0 18 -14 -2 C -8 -18 0 -8 -12 D -12 14 8 0 6 E 2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.600000 Sum of squares = 0.459999999864 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 12 -2 B 0 0 18 -14 -2 C -8 -18 0 -8 -12 D -12 14 8 0 6 E 2 2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.000000 D: 0.100000 E: 0.600000 Sum of squares = 0.459999999864 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.300000 D: 0.400000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5036: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) B A C E D (9) D E C A B (8) A B D E C (8) B C A E D (6) E D C B A (5) D E A C B (5) B A C D E (5) A D E C B (5) C E D B A (4) A C D E B (4) C B E D A (3) E C D B A (2) E B D C A (2) D A E C B (2) C E D A B (2) C B E A D (2) A D E B C (2) E D B C A (1) D E B C A (1) D E A B C (1) D A E B C (1) C E B D A (1) C B A E D (1) C A B E D (1) B E D C A (1) B E D A C (1) B D E A C (1) B C E A D (1) B A D E C (1) A D C E B (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 18 18 18 B -6 0 8 10 6 C -18 -8 0 4 4 D -18 -10 -4 0 14 E -18 -6 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 18 18 B -6 0 8 10 6 C -18 -8 0 4 4 D -18 -10 -4 0 14 E -18 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=25 D=18 C=14 E=10 so E is eliminated. Round 2 votes counts: A=33 B=27 D=24 C=16 so C is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 B:209 C:191 D:191 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 18 18 B -6 0 8 10 6 C -18 -8 0 4 4 D -18 -10 -4 0 14 E -18 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 18 18 B -6 0 8 10 6 C -18 -8 0 4 4 D -18 -10 -4 0 14 E -18 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 18 18 B -6 0 8 10 6 C -18 -8 0 4 4 D -18 -10 -4 0 14 E -18 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999529 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5037: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) C A E B D (9) A E B D C (8) E A D B C (7) C D B A E (6) D B E A C (5) C E A D B (4) C D E B A (4) C A B E D (4) E D B A C (3) D E B A C (3) C A B D E (3) E C A D B (2) E A B D C (2) D B C E A (2) D B C A E (2) C E D A B (2) C B A D E (2) B D A E C (2) B D A C E (2) B A D C E (2) A B E D C (2) E D A B C (1) E C D A B (1) D E B C A (1) D B E C A (1) C D A E B (1) C B D A E (1) C A E D B (1) B D E A C (1) B D C A E (1) B C A D E (1) B A D E C (1) A E C B D (1) A E B C D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -14 4 4 B -2 0 -4 -10 -4 C 14 4 0 8 16 D -4 10 -8 0 0 E -4 4 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 4 4 B -2 0 -4 -10 -4 C 14 4 0 8 16 D -4 10 -8 0 0 E -4 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 E=16 D=14 A=14 B=10 so B is eliminated. Round 2 votes counts: C=47 D=20 A=17 E=16 so E is eliminated. Round 3 votes counts: C=50 A=26 D=24 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:199 A:198 E:192 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 4 4 B -2 0 -4 -10 -4 C 14 4 0 8 16 D -4 10 -8 0 0 E -4 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 4 4 B -2 0 -4 -10 -4 C 14 4 0 8 16 D -4 10 -8 0 0 E -4 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 4 4 B -2 0 -4 -10 -4 C 14 4 0 8 16 D -4 10 -8 0 0 E -4 4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5038: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) D C A E B (6) B E D C A (5) B D E C A (5) A B C D E (5) D C E A B (4) B A C E D (4) A D C B E (4) A C B D E (4) E C D A B (3) E B D C A (3) C A D E B (3) B E D A C (3) B A E C D (3) B A D C E (3) A C D E B (3) E B C D A (2) E B C A D (2) D B A C E (2) C E D A B (2) C D E A B (2) C D A E B (2) C A E D B (2) B D E A C (2) B A E D C (2) B A C D E (2) A C D B E (2) E D C B A (1) E D C A B (1) E D B C A (1) D E C B A (1) D E B C A (1) D B C E A (1) D A C E B (1) D A C B E (1) C E A D B (1) B E A D C (1) B D A C E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 4 0 4 B 4 0 10 12 14 C -4 -10 0 2 12 D 0 -12 -2 0 10 E -4 -14 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 0 4 B 4 0 10 12 14 C -4 -10 0 2 12 D 0 -12 -2 0 10 E -4 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997762 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=20 D=17 E=13 C=12 so C is eliminated. Round 2 votes counts: B=38 A=25 D=21 E=16 so E is eliminated. Round 3 votes counts: B=45 D=29 A=26 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:202 C:200 D:198 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 0 4 B 4 0 10 12 14 C -4 -10 0 2 12 D 0 -12 -2 0 10 E -4 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997762 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 0 4 B 4 0 10 12 14 C -4 -10 0 2 12 D 0 -12 -2 0 10 E -4 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997762 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 0 4 B 4 0 10 12 14 C -4 -10 0 2 12 D 0 -12 -2 0 10 E -4 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997762 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5039: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) D C A E B (8) A C D E B (8) A D C B E (7) E B C D A (5) B E C D A (4) A D C E B (4) E B C A D (3) E B A C D (3) D C A B E (3) C E D B A (3) C A D E B (3) B E C A D (3) B E A C D (3) A C E D B (3) A B E D C (3) E C D B A (2) E C B D A (2) E A B C D (2) D C E B A (2) C D E A B (2) B D C E A (2) A D B C E (2) A B E C D (2) A B D C E (2) D C E A B (1) D C B A E (1) D B C A E (1) C E D A B (1) C D E B A (1) B E A D C (1) B D E A C (1) B A E D C (1) Total count = 100 A B C D E A 0 2 -16 0 -4 B -2 0 -2 -4 -6 C 16 2 0 0 8 D 0 4 0 0 -4 E 4 6 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.512919 D: 0.487081 E: 0.000000 Sum of squares = 0.500333798749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.512919 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 0 -4 B -2 0 -2 -4 -6 C 16 2 0 0 8 D 0 4 0 0 -4 E 4 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=26 E=17 D=16 C=10 so C is eliminated. Round 2 votes counts: A=34 B=26 E=21 D=19 so D is eliminated. Round 3 votes counts: A=45 B=28 E=27 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 E:203 D:200 B:193 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 0 -4 B -2 0 -2 -4 -6 C 16 2 0 0 8 D 0 4 0 0 -4 E 4 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 0 -4 B -2 0 -2 -4 -6 C 16 2 0 0 8 D 0 4 0 0 -4 E 4 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 0 -4 B -2 0 -2 -4 -6 C 16 2 0 0 8 D 0 4 0 0 -4 E 4 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5040: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (11) E C B D A (8) D A E B C (5) B C A D E (5) D B A C E (4) D A B C E (4) C E B A D (4) C B A E D (4) E D A B C (3) E C A D B (3) E C A B D (3) D E A B C (3) C B E D A (3) A D E B C (3) E D B C A (2) E C D B A (2) E C D A B (2) E C B A D (2) E B D C A (2) E A D C B (2) D B E C A (2) C B A D E (2) C A E B D (2) B C E D A (2) B C D A E (2) A D E C B (2) A B D C E (2) E D C B A (1) E D A C B (1) E B C D A (1) D E B C A (1) D B C E A (1) C A B E D (1) B D C A E (1) B A D C E (1) B A C D E (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -12 0 4 B 2 0 12 -6 -6 C 12 -12 0 -2 4 D 0 6 2 0 2 E -4 6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.084160 B: 0.000000 C: 0.000000 D: 0.915840 E: 0.000000 Sum of squares = 0.845845965548 Cumulative probabilities = A: 0.084160 B: 0.084160 C: 0.084160 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 0 4 B 2 0 12 -6 -6 C 12 -12 0 -2 4 D 0 6 2 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102042627 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=20 A=20 C=16 B=12 so B is eliminated. Round 2 votes counts: E=32 C=25 A=22 D=21 so D is eliminated. Round 3 votes counts: E=38 A=35 C=27 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:205 B:201 C:201 E:198 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 0 4 B 2 0 12 -6 -6 C 12 -12 0 -2 4 D 0 6 2 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102042627 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 0 4 B 2 0 12 -6 -6 C 12 -12 0 -2 4 D 0 6 2 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102042627 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 0 4 B 2 0 12 -6 -6 C 12 -12 0 -2 4 D 0 6 2 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.857143 E: 0.000000 Sum of squares = 0.755102042627 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5041: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) E B C D A (7) C A B E D (6) D A B E C (5) A D B C E (5) E C B D A (4) C E B D A (4) C A B D E (4) D E B A C (3) D B E A C (3) C E D A B (3) C E B A D (3) C D A E B (3) C B E A D (3) A D C B E (3) A C B D E (3) E D C B A (2) E B D A C (2) D C A E B (2) D A E C B (2) D A C E B (2) C A D E B (2) C A D B E (2) B E D A C (2) B E C D A (2) A D C E B (2) E C D B A (1) E C D A B (1) D E A C B (1) D E A B C (1) D A E B C (1) C D E A B (1) C A E B D (1) B E C A D (1) B E A C D (1) B D A E C (1) B C E A D (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -22 -22 -8 B -2 0 -12 8 -12 C 22 12 0 10 4 D 22 -8 -10 0 -6 E 8 12 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -22 -22 -8 B -2 0 -12 8 -12 C 22 12 0 10 4 D 22 -8 -10 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=25 D=20 A=14 B=9 so B is eliminated. Round 2 votes counts: C=33 E=31 D=21 A=15 so A is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:211 D:199 B:191 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -22 -22 -8 B -2 0 -12 8 -12 C 22 12 0 10 4 D 22 -8 -10 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -22 -22 -8 B -2 0 -12 8 -12 C 22 12 0 10 4 D 22 -8 -10 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -22 -22 -8 B -2 0 -12 8 -12 C 22 12 0 10 4 D 22 -8 -10 0 -6 E 8 12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5042: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (14) B D E A C (9) D B E A C (7) C A D E B (5) A C D E B (5) C A E B D (4) B E D A C (4) E D A B C (3) E B D A C (3) D B A E C (3) A D C E B (3) E C B A D (2) E C A B D (2) D A B E C (2) D A B C E (2) C E B A D (2) C B E A D (2) C B D A E (2) C A D B E (2) B E D C A (2) A E C D B (2) A C E D B (2) E B A C D (1) E A C B D (1) D E B A C (1) D C A B E (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C B E (1) C E A B D (1) C D B A E (1) C B A E D (1) B E C A D (1) B D E C A (1) B D C A E (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 8 6 4 12 B -8 0 -12 -18 -10 C -6 12 0 4 4 D -4 18 -4 0 2 E -12 10 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 4 12 B -8 0 -12 -18 -10 C -6 12 0 4 4 D -4 18 -4 0 2 E -12 10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=20 B=20 A=14 E=12 so E is eliminated. Round 2 votes counts: C=38 B=24 D=23 A=15 so A is eliminated. Round 3 votes counts: C=48 D=28 B=24 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:215 C:207 D:206 E:196 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 4 12 B -8 0 -12 -18 -10 C -6 12 0 4 4 D -4 18 -4 0 2 E -12 10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 4 12 B -8 0 -12 -18 -10 C -6 12 0 4 4 D -4 18 -4 0 2 E -12 10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 4 12 B -8 0 -12 -18 -10 C -6 12 0 4 4 D -4 18 -4 0 2 E -12 10 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5043: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) E B A C D (5) E B C A D (4) C E B D A (4) C E B A D (4) A D C E B (4) A D C B E (4) A C E D B (4) E C B A D (3) E B C D A (3) D C A B E (3) D B A C E (3) C D B E A (3) C A E D B (3) A E C B D (3) A C D E B (3) A C D B E (3) E B D A C (2) D B E C A (2) D A C B E (2) C D A B E (2) B D E C A (2) A E D C B (2) A E D B C (2) A E C D B (2) E C A B D (1) E B A D C (1) E A B D C (1) E A B C D (1) D B E A C (1) D B A E C (1) D A B E C (1) D A B C E (1) C D B A E (1) C B E D A (1) C A E B D (1) C A D E B (1) C A D B E (1) B E C D A (1) B D E A C (1) B D C E A (1) B C E D A (1) A E B D C (1) A D E C B (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 2 14 2 B 2 0 -12 -4 -14 C -2 12 0 10 2 D -14 4 -10 0 -14 E -2 14 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.754912 B: 0.122544 C: 0.102898 D: 0.000000 E: 0.019646 Sum of squares = 0.595882505026 Cumulative probabilities = A: 0.754912 B: 0.877456 C: 0.980354 D: 0.980354 E: 1.000000 A B C D E A 0 -2 2 14 2 B 2 0 -12 -4 -14 C -2 12 0 10 2 D -14 4 -10 0 -14 E -2 14 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.124999 D: 0.000000 E: 0.000001 Sum of squares = 0.59375008244 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.999999 D: 0.999999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=21 C=21 D=14 B=13 so B is eliminated. Round 2 votes counts: A=31 E=29 C=22 D=18 so D is eliminated. Round 3 votes counts: A=39 E=35 C=26 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:212 C:211 A:208 B:186 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 14 2 B 2 0 -12 -4 -14 C -2 12 0 10 2 D -14 4 -10 0 -14 E -2 14 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.124999 D: 0.000000 E: 0.000001 Sum of squares = 0.59375008244 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.999999 D: 0.999999 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 14 2 B 2 0 -12 -4 -14 C -2 12 0 10 2 D -14 4 -10 0 -14 E -2 14 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.124999 D: 0.000000 E: 0.000001 Sum of squares = 0.59375008244 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.999999 D: 0.999999 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 14 2 B 2 0 -12 -4 -14 C -2 12 0 10 2 D -14 4 -10 0 -14 E -2 14 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.124999 D: 0.000000 E: 0.000001 Sum of squares = 0.59375008244 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.999999 D: 0.999999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5044: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C E A D B (6) D A B E C (5) C E A B D (5) C D B A E (5) D B A C E (4) C D B E A (4) A E D B C (4) E C A B D (3) E A B C D (3) C E B D A (3) B C D A E (3) B A D E C (3) E C A D B (2) E A C D B (2) E A C B D (2) D B C A E (2) D B A E C (2) D A B C E (2) C E D B A (2) C E D A B (2) C D E B A (2) C B E D A (2) C B D E A (2) C B D A E (2) B D A E C (2) B A E D C (2) A D E B C (2) A D B E C (2) A B D E C (2) E B C A D (1) E A D C B (1) E A D B C (1) D C B A E (1) D A E B C (1) C E B A D (1) C D A E B (1) B D C A E (1) B D A C E (1) B C E A D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -2 2 -4 B -8 0 4 -6 -2 C 2 -4 0 8 4 D -2 6 -8 0 -2 E 4 2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428565 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 2 -4 B -8 0 4 -6 -2 C 2 -4 0 8 4 D -2 6 -8 0 -2 E 4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428594 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=22 D=17 B=13 A=11 so A is eliminated. Round 2 votes counts: C=37 E=26 D=21 B=16 so B is eliminated. Round 3 votes counts: C=41 D=30 E=29 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:205 A:202 E:202 D:197 B:194 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 2 -4 B -8 0 4 -6 -2 C 2 -4 0 8 4 D -2 6 -8 0 -2 E 4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428594 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 2 -4 B -8 0 4 -6 -2 C 2 -4 0 8 4 D -2 6 -8 0 -2 E 4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428594 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 2 -4 B -8 0 4 -6 -2 C 2 -4 0 8 4 D -2 6 -8 0 -2 E 4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428594 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5045: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) E A D C B (8) B C A E D (8) D E A C B (7) D C A E B (7) B C D A E (4) A E C D B (4) E D A C B (3) E A B C D (3) D C B A E (3) C B D A E (3) E A D B C (2) E A C D B (2) E A C B D (2) D E C A B (2) D C E A B (2) D C A B E (2) D A E C B (2) C D B A E (2) C B A E D (2) C A B E D (2) B E A D C (2) B D C E A (2) B D C A E (2) B A E C D (2) A C E D B (2) E B A C D (1) E A B D C (1) D E A B C (1) D C B E A (1) D B E C A (1) D B E A C (1) B E D C A (1) B E D A C (1) B D E A C (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 8 12 4 -6 B -8 0 -16 -4 -2 C -12 16 0 -4 -14 D -4 4 4 0 -14 E 6 2 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 12 4 -6 B -8 0 -16 -4 -2 C -12 16 0 -4 -14 D -4 4 4 0 -14 E 6 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=29 E=22 C=9 A=8 so A is eliminated. Round 2 votes counts: B=32 D=29 E=27 C=12 so C is eliminated. Round 3 votes counts: B=39 D=31 E=30 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:218 A:209 D:195 C:193 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 4 -6 B -8 0 -16 -4 -2 C -12 16 0 -4 -14 D -4 4 4 0 -14 E 6 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 4 -6 B -8 0 -16 -4 -2 C -12 16 0 -4 -14 D -4 4 4 0 -14 E 6 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 4 -6 B -8 0 -16 -4 -2 C -12 16 0 -4 -14 D -4 4 4 0 -14 E 6 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999980507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5046: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) B C A D E (6) C A E B D (5) B A C E D (5) D C E B A (4) C B D A E (4) B C A E D (4) A E B D C (4) A B E C D (4) E D A C B (3) E C A D B (3) E A D B C (3) C D E A B (3) C D B E A (3) B A E D C (3) B A C D E (3) E A D C B (2) E A C D B (2) D C E A B (2) D B E C A (2) D B E A C (2) D B C E A (2) C B D E A (2) C B A E D (2) C A B E D (2) A E B C D (2) E D C A B (1) E D A B C (1) E C D A B (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A C B (1) D E A B C (1) D C B E A (1) C E D A B (1) C E A D B (1) B D C A E (1) B D A E C (1) B C D A E (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -20 6 -2 B 0 0 -2 2 0 C 20 2 0 8 4 D -6 -2 -8 0 0 E 2 0 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 6 -2 B 0 0 -2 2 0 C 20 2 0 8 4 D -6 -2 -8 0 0 E 2 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 C=23 E=16 A=11 so A is eliminated. Round 2 votes counts: B=30 D=25 C=23 E=22 so E is eliminated. Round 3 votes counts: B=36 D=35 C=29 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:217 B:200 E:199 A:192 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -20 6 -2 B 0 0 -2 2 0 C 20 2 0 8 4 D -6 -2 -8 0 0 E 2 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 6 -2 B 0 0 -2 2 0 C 20 2 0 8 4 D -6 -2 -8 0 0 E 2 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 6 -2 B 0 0 -2 2 0 C 20 2 0 8 4 D -6 -2 -8 0 0 E 2 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5047: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) B E D A C (9) C A B E D (6) A B C E D (6) D E C B A (5) C A D E B (5) A C B E D (5) A C B D E (5) E D B C A (4) B A C E D (4) E D B A C (3) E B D C A (3) D E B A C (3) C A B D E (3) B E A D C (3) A C D B E (3) D C A E B (2) C D E A B (2) C A E D B (2) B D E A C (2) E C B D A (1) D E C A B (1) D C E A B (1) D A C E B (1) D A B E C (1) C E D A B (1) C D A E B (1) C B E A D (1) C A E B D (1) B E D C A (1) B E A C D (1) B D A E C (1) B A E D C (1) B A E C D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 2 -2 0 B 4 0 4 8 8 C -2 -4 0 -4 2 D 2 -8 4 0 -6 E 0 -8 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -2 0 B 4 0 4 8 8 C -2 -4 0 -4 2 D 2 -8 4 0 -6 E 0 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 C=22 A=21 E=11 so E is eliminated. Round 2 votes counts: D=30 B=26 C=23 A=21 so A is eliminated. Round 3 votes counts: C=36 D=32 B=32 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:198 E:198 C:196 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -2 0 B 4 0 4 8 8 C -2 -4 0 -4 2 D 2 -8 4 0 -6 E 0 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 0 B 4 0 4 8 8 C -2 -4 0 -4 2 D 2 -8 4 0 -6 E 0 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 0 B 4 0 4 8 8 C -2 -4 0 -4 2 D 2 -8 4 0 -6 E 0 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5048: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) E D B C A (7) B E A C D (7) A C B D E (7) B A C E D (5) A C D B E (5) E D C B A (4) B A E C D (4) A B C E D (4) E B D A C (3) D C A E B (3) C D A E B (3) B E C A D (3) B C E A D (3) A D C E B (3) A D B C E (3) D C E A B (2) D A E C B (2) D A C E B (2) C A D E B (2) C A B D E (2) B E C D A (2) A D C B E (2) E D B A C (1) D E C B A (1) D E C A B (1) D E A C B (1) D E A B C (1) C E D B A (1) C A D B E (1) B E D A C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 6 6 0 B 8 0 14 8 0 C -6 -14 0 2 0 D -6 -8 -2 0 -14 E 0 0 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.568531 C: 0.000000 D: 0.000000 E: 0.431469 Sum of squares = 0.509393123802 Cumulative probabilities = A: 0.000000 B: 0.568531 C: 0.568531 D: 0.568531 E: 1.000000 A B C D E A 0 -8 6 6 0 B 8 0 14 8 0 C -6 -14 0 2 0 D -6 -8 -2 0 -14 E 0 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 B=25 D=13 C=9 so C is eliminated. Round 2 votes counts: A=31 E=28 B=25 D=16 so D is eliminated. Round 3 votes counts: A=41 E=34 B=25 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 E:207 A:202 C:191 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 6 0 B 8 0 14 8 0 C -6 -14 0 2 0 D -6 -8 -2 0 -14 E 0 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 6 0 B 8 0 14 8 0 C -6 -14 0 2 0 D -6 -8 -2 0 -14 E 0 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 6 0 B 8 0 14 8 0 C -6 -14 0 2 0 D -6 -8 -2 0 -14 E 0 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5049: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) C B D A E (10) E C A B D (7) D A B E C (5) B D A C E (5) A D E B C (5) D B C A E (4) D B A C E (4) C E B D A (4) C D B A E (4) A D B E C (4) E A C D B (3) C E B A D (3) B D C A E (3) B C D A E (3) A E D B C (3) E A D B C (2) D A E B C (2) C B E D A (2) C B D E A (2) B A D C E (2) A B D E C (2) E C D A B (1) E C B A D (1) E A D C B (1) E A C B D (1) E A B C D (1) D E C A B (1) C E D B A (1) C E D A B (1) C B E A D (1) B A D E C (1) A E B D C (1) Total count = 100 A B C D E A 0 0 -16 -4 16 B 0 0 -6 -2 4 C 16 6 0 10 -2 D 4 2 -10 0 14 E -16 -4 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.615385 E: 1.000000 A B C D E A 0 0 -16 -4 16 B 0 0 -6 -2 4 C 16 6 0 10 -2 D 4 2 -10 0 14 E -16 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.615385 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 D=16 A=15 B=14 so B is eliminated. Round 2 votes counts: C=31 E=27 D=24 A=18 so A is eliminated. Round 3 votes counts: D=38 E=31 C=31 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:205 A:198 B:198 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -4 16 B 0 0 -6 -2 4 C 16 6 0 10 -2 D 4 2 -10 0 14 E -16 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.615385 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -4 16 B 0 0 -6 -2 4 C 16 6 0 10 -2 D 4 2 -10 0 14 E -16 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.615385 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -4 16 B 0 0 -6 -2 4 C 16 6 0 10 -2 D 4 2 -10 0 14 E -16 -4 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.538462 D: 0.615385 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5050: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (15) E C B A D (12) D A B E C (12) C E B A D (11) D A B C E (9) B A D C E (6) C B E A D (4) A B D E C (4) E D C A B (3) D A E B C (3) A B D C E (3) E C B D A (2) C E D B A (2) B C A D E (2) A D B C E (2) E C D B A (1) D E C A B (1) D E A C B (1) D C B A E (1) D B A C E (1) C E B D A (1) B C E A D (1) B A C E D (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 8 -12 -4 -8 B -8 0 -8 -4 -4 C 12 8 0 6 -10 D 4 4 -6 0 -6 E 8 4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -12 -4 -8 B -8 0 -8 -4 -4 C 12 8 0 6 -10 D 4 4 -6 0 -6 E 8 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=28 C=18 B=11 A=10 so A is eliminated. Round 2 votes counts: E=33 D=31 C=18 B=18 so C is eliminated. Round 3 votes counts: E=47 D=31 B=22 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:208 D:198 A:192 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -12 -4 -8 B -8 0 -8 -4 -4 C 12 8 0 6 -10 D 4 4 -6 0 -6 E 8 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 -4 -8 B -8 0 -8 -4 -4 C 12 8 0 6 -10 D 4 4 -6 0 -6 E 8 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 -4 -8 B -8 0 -8 -4 -4 C 12 8 0 6 -10 D 4 4 -6 0 -6 E 8 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5051: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C A E B D (7) D E C A B (6) A B C E D (6) D E B C A (5) C A B E D (5) E A C B D (4) D B E A C (4) C E A B D (4) B A C E D (4) E D C A B (3) E C A D B (3) D E C B A (3) C E D A B (3) B D A C E (3) D E A B C (2) D B A E C (2) C E A D B (2) C D E B A (2) B D C A E (2) B D A E C (2) B C D A E (2) B A D E C (2) B A C D E (2) E D A C B (1) E D A B C (1) E A C D B (1) E A B D C (1) D B C E A (1) D B C A E (1) C D E A B (1) C B A D E (1) B C A D E (1) B A E D C (1) B A D C E (1) A E C B D (1) A E B D C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -4 -2 -8 B -8 0 4 4 -18 C 4 -4 0 2 -2 D 2 -4 -2 0 0 E 8 18 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.418144 E: 0.581856 Sum of squares = 0.513400760251 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.418144 E: 1.000000 A B C D E A 0 8 -4 -2 -8 B -8 0 4 4 -18 C 4 -4 0 2 -2 D 2 -4 -2 0 0 E 8 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 0.500690 Sum of squares = 0.5000009535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=25 B=20 E=14 A=10 so A is eliminated. Round 2 votes counts: D=31 B=27 C=26 E=16 so E is eliminated. Round 3 votes counts: D=36 C=35 B=29 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:214 C:200 D:198 A:197 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -4 -2 -8 B -8 0 4 4 -18 C 4 -4 0 2 -2 D 2 -4 -2 0 0 E 8 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 0.500690 Sum of squares = 0.5000009535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -2 -8 B -8 0 4 4 -18 C 4 -4 0 2 -2 D 2 -4 -2 0 0 E 8 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 0.500690 Sum of squares = 0.5000009535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -2 -8 B -8 0 4 4 -18 C 4 -4 0 2 -2 D 2 -4 -2 0 0 E 8 18 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 0.500690 Sum of squares = 0.5000009535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499310 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5052: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (12) A E B D C (12) D B C A E (10) C D B E A (10) E A C D B (7) B D C A E (6) A E D B C (6) D B A E C (4) C E A D B (4) C E A B D (4) C B D E A (4) A E B C D (4) E A D C B (2) C D E B A (2) B D A E C (2) B C D A E (2) E C A D B (1) D C B E A (1) D B A C E (1) C B E A D (1) C A E B D (1) B A E D C (1) B A D E C (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 8 16 4 B -10 0 0 4 -12 C -8 0 0 6 -8 D -16 -4 -6 0 -12 E -4 12 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 16 4 B -10 0 0 4 -12 C -8 0 0 6 -8 D -16 -4 -6 0 -12 E -4 12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 E=22 D=16 B=12 so B is eliminated. Round 2 votes counts: C=28 A=26 D=24 E=22 so E is eliminated. Round 3 votes counts: A=47 C=29 D=24 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:214 C:195 B:191 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 16 4 B -10 0 0 4 -12 C -8 0 0 6 -8 D -16 -4 -6 0 -12 E -4 12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 16 4 B -10 0 0 4 -12 C -8 0 0 6 -8 D -16 -4 -6 0 -12 E -4 12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 16 4 B -10 0 0 4 -12 C -8 0 0 6 -8 D -16 -4 -6 0 -12 E -4 12 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5053: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (13) C B D E A (7) D C A B E (6) C D B E A (6) A E B D C (6) A D C E B (6) E A B C D (4) D A C E B (4) D A C B E (4) C D B A E (4) B C E D A (4) A D C B E (4) D C E B A (3) B E C D A (3) E B C D A (2) E B C A D (2) E A B D C (2) D C A E B (2) D A E C B (2) B E C A D (2) A E D B C (2) E B A D C (1) E A D B C (1) D C B A E (1) C D A B E (1) C B E D A (1) C B D A E (1) C B A D E (1) B E A C D (1) A E B C D (1) A D E C B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -2 -4 B 4 0 -10 4 -4 C -8 10 0 10 14 D 2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.147727 C: 0.159091 D: 0.045455 E: 0.147727 Sum of squares = 0.321022727256 Cumulative probabilities = A: 0.500000 B: 0.647727 C: 0.806818 D: 0.852273 E: 1.000000 A B C D E A 0 -4 8 -2 -4 B 4 0 -10 4 -4 C -8 10 0 10 14 D 2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.147727 C: 0.159091 D: 0.045455 E: 0.147727 Sum of squares = 0.321022727267 Cumulative probabilities = A: 0.500000 B: 0.647727 C: 0.806818 D: 0.852273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 A=22 C=21 B=10 so B is eliminated. Round 2 votes counts: E=31 C=25 D=22 A=22 so D is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:199 D:198 B:197 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 8 -2 -4 B 4 0 -10 4 -4 C -8 10 0 10 14 D 2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.147727 C: 0.159091 D: 0.045455 E: 0.147727 Sum of squares = 0.321022727267 Cumulative probabilities = A: 0.500000 B: 0.647727 C: 0.806818 D: 0.852273 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -2 -4 B 4 0 -10 4 -4 C -8 10 0 10 14 D 2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.147727 C: 0.159091 D: 0.045455 E: 0.147727 Sum of squares = 0.321022727267 Cumulative probabilities = A: 0.500000 B: 0.647727 C: 0.806818 D: 0.852273 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -2 -4 B 4 0 -10 4 -4 C -8 10 0 10 14 D 2 -4 -10 0 8 E 4 4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.147727 C: 0.159091 D: 0.045455 E: 0.147727 Sum of squares = 0.321022727267 Cumulative probabilities = A: 0.500000 B: 0.647727 C: 0.806818 D: 0.852273 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5054: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) D E C B A (10) D B A E C (10) C E A B D (9) B A D C E (7) B A C E D (6) E D C A B (5) D E B A C (5) B A C D E (4) A B C E D (4) E C A D B (3) D E C A B (3) C B A E D (3) C A B E D (3) D E A C B (2) C E D A B (2) B D A E C (2) B D A C E (2) A C B E D (2) E C A B D (1) E A C B D (1) D E B C A (1) D B E C A (1) D B E A C (1) D B A C E (1) C E A D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 -4 -10 -10 B 6 0 -12 -10 -8 C 4 12 0 0 -12 D 10 10 0 0 -2 E 10 8 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -10 -10 B 6 0 -12 -10 -8 C 4 12 0 0 -12 D 10 10 0 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=21 E=20 C=18 A=7 so A is eliminated. Round 2 votes counts: D=34 B=25 E=21 C=20 so C is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:209 C:202 B:188 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 -10 B 6 0 -12 -10 -8 C 4 12 0 0 -12 D 10 10 0 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 -10 B 6 0 -12 -10 -8 C 4 12 0 0 -12 D 10 10 0 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 -10 B 6 0 -12 -10 -8 C 4 12 0 0 -12 D 10 10 0 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5055: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) C D E A B (7) C E D A B (6) B A E D C (6) E A B D C (5) B E A D C (5) C D A B E (4) B E C A D (4) B C D A E (4) B A D C E (4) E C D A B (3) E B C A D (3) D C A E B (3) D C A B E (3) A D E B C (3) A D B C E (3) E C B A D (2) E A D B C (2) D A C E B (2) C E D B A (2) B E A C D (2) E C A D B (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D C B (1) E A C D B (1) D C B A E (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E B A (1) B D A C E (1) B C E D A (1) B A D E C (1) A E D B C (1) A D C B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 18 -12 -2 0 B -18 0 2 -12 -12 C 12 -2 0 6 12 D 2 12 -6 0 0 E 0 12 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.460000000117 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 -2 0 B -18 0 2 -12 -12 C 12 -2 0 6 12 D 2 12 -6 0 0 E 0 12 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.460000000001 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=28 E=21 D=11 A=10 so A is eliminated. Round 2 votes counts: C=30 B=30 E=22 D=18 so D is eliminated. Round 3 votes counts: C=41 B=34 E=25 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:214 D:204 A:202 E:200 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -12 -2 0 B -18 0 2 -12 -12 C 12 -2 0 6 12 D 2 12 -6 0 0 E 0 12 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.460000000001 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 -2 0 B -18 0 2 -12 -12 C 12 -2 0 6 12 D 2 12 -6 0 0 E 0 12 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.460000000001 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 -2 0 B -18 0 2 -12 -12 C 12 -2 0 6 12 D 2 12 -6 0 0 E 0 12 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.600000 D: 0.100000 E: 0.000000 Sum of squares = 0.460000000001 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5056: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) A E C B D (7) D E A B C (6) D A C E B (4) C B A E D (4) C B A D E (4) B C E A D (4) A E C D B (4) A D E C B (4) E A D B C (3) E A B C D (3) D C B A E (3) D C A E B (3) D C A B E (3) D B E A C (3) D B C E A (3) C D A E B (3) C A E B D (3) B C D E A (3) C D A B E (2) C A E D B (2) C A B E D (2) B E D A C (2) B E A C D (2) A C E D B (2) A C E B D (2) E D A B C (1) E B A D C (1) E A B D C (1) D B C A E (1) C B E A D (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E C A (1) B D C E A (1) Total count = 100 A B C D E A 0 28 6 6 24 B -28 0 -24 -10 -12 C -6 24 0 4 4 D -6 10 -4 0 6 E -24 12 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 6 6 24 B -28 0 -24 -10 -12 C -6 24 0 4 4 D -6 10 -4 0 6 E -24 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991197 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 A=19 B=16 E=9 so E is eliminated. Round 2 votes counts: D=34 A=26 C=23 B=17 so B is eliminated. Round 3 votes counts: D=38 C=32 A=30 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:232 C:213 D:203 E:189 B:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 6 6 24 B -28 0 -24 -10 -12 C -6 24 0 4 4 D -6 10 -4 0 6 E -24 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991197 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 6 6 24 B -28 0 -24 -10 -12 C -6 24 0 4 4 D -6 10 -4 0 6 E -24 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991197 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 6 6 24 B -28 0 -24 -10 -12 C -6 24 0 4 4 D -6 10 -4 0 6 E -24 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991197 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5057: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (15) D B E C A (10) B E D A C (10) D B E A C (7) E B D A C (6) C A E B D (5) C A D E B (5) D C A B E (4) C A D B E (4) A E B C D (4) E B A C D (3) D B C E A (3) A C D E B (3) E B D C A (2) E B C D A (2) E B A D C (2) D C B E A (2) D A C B E (2) A E C B D (2) C D E B A (1) C D B E A (1) C D A E B (1) C D A B E (1) C A E D B (1) A D B E C (1) A C D B E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 16 -4 2 B -2 0 2 6 -4 C -16 -2 0 0 0 D 4 -6 0 0 -8 E -2 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428564 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.714286 E: 1.000000 A B C D E A 0 2 16 -4 2 B -2 0 2 6 -4 C -16 -2 0 0 0 D 4 -6 0 0 -8 E -2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428081 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=28 A=28 C=19 E=15 B=10 so B is eliminated. Round 2 votes counts: D=28 A=28 E=25 C=19 so C is eliminated. Round 3 votes counts: A=43 D=32 E=25 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:208 E:205 B:201 D:195 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 16 -4 2 B -2 0 2 6 -4 C -16 -2 0 0 0 D 4 -6 0 0 -8 E -2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428081 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.714286 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 -4 2 B -2 0 2 6 -4 C -16 -2 0 0 0 D 4 -6 0 0 -8 E -2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428081 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.714286 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 -4 2 B -2 0 2 6 -4 C -16 -2 0 0 0 D 4 -6 0 0 -8 E -2 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428081 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.714286 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5058: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) C D B A E (7) A E D C B (7) B C D E A (6) E B D A C (5) B E C D A (5) A C D E B (5) E A D B C (4) E A B D C (4) D C A B E (4) B D C E A (4) A E B C D (4) E D B A C (3) C B D A E (3) B E D C A (3) A E C D B (3) A E C B D (3) A C D B E (3) E B D C A (2) E B A D C (2) E A D C B (2) D C B E A (2) A D C E B (2) E D B C A (1) E A B C D (1) D E C A B (1) D C E B A (1) C B A D E (1) C A D B E (1) B E C A D (1) B E A C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 0 -10 4 B 10 0 -8 -8 0 C 0 8 0 -10 -4 D 10 8 10 0 -4 E -4 0 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.306152 C: 0.000000 D: 0.000000 E: 0.693848 Sum of squares = 0.57515377994 Cumulative probabilities = A: 0.000000 B: 0.306152 C: 0.306152 D: 0.306152 E: 1.000000 A B C D E A 0 -10 0 -10 4 B 10 0 -8 -8 0 C 0 8 0 -10 -4 D 10 8 10 0 -4 E -4 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555581597 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 B=20 D=16 C=12 so C is eliminated. Round 2 votes counts: A=29 E=24 B=24 D=23 so D is eliminated. Round 3 votes counts: B=41 A=33 E=26 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:212 E:202 B:197 C:197 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 0 -10 4 B 10 0 -8 -8 0 C 0 8 0 -10 -4 D 10 8 10 0 -4 E -4 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555581597 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -10 4 B 10 0 -8 -8 0 C 0 8 0 -10 -4 D 10 8 10 0 -4 E -4 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555581597 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -10 4 B 10 0 -8 -8 0 C 0 8 0 -10 -4 D 10 8 10 0 -4 E -4 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555581597 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5059: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (6) E A C D B (5) D B C A E (5) E C A B D (4) D B C E A (4) C B D E A (4) B C D A E (4) A E D B C (4) A D E B C (4) E A C B D (3) D E A C B (3) D C B E A (3) D B A C E (3) D A E B C (3) B A C E D (3) A E B C D (3) D E C B A (2) D E C A B (2) D A B E C (2) D A B C E (2) C E B D A (2) C B E D A (2) C B E A D (2) B D C A E (2) B C D E A (2) A E D C B (2) A D B E C (2) A B C E D (2) E D C A B (1) E D A C B (1) E C D B A (1) E C D A B (1) E C B A D (1) E C A D B (1) C E B A D (1) B D C E A (1) B C E D A (1) B C A D E (1) B A D C E (1) B A C D E (1) A E B D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 6 -2 6 B -8 0 6 -2 -2 C -6 -6 0 2 -6 D 2 2 -2 0 4 E -6 2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000003 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -2 6 B -8 0 6 -2 -2 C -6 -6 0 2 -6 D 2 2 -2 0 4 E -6 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=26 E=18 B=16 C=11 so C is eliminated. Round 2 votes counts: D=29 A=26 B=24 E=21 so E is eliminated. Round 3 votes counts: A=39 D=33 B=28 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:203 E:199 B:197 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 -2 6 B -8 0 6 -2 -2 C -6 -6 0 2 -6 D 2 2 -2 0 4 E -6 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -2 6 B -8 0 6 -2 -2 C -6 -6 0 2 -6 D 2 2 -2 0 4 E -6 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -2 6 B -8 0 6 -2 -2 C -6 -6 0 2 -6 D 2 2 -2 0 4 E -6 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999876 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5060: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (15) B D E C A (12) C A D B E (10) E B D A C (9) E A B D C (8) A C E D B (8) B D C E A (7) D B C E A (5) A E C B D (5) B E D C A (4) A C D E B (4) D C B E A (3) C D A B E (3) A E C D B (3) C D B E A (2) A E B C D (2) Total count = 100 A B C D E A 0 -14 -22 -20 0 B 14 0 -6 -6 22 C 22 6 0 4 14 D 20 6 -4 0 22 E 0 -22 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -22 -20 0 B 14 0 -6 -6 22 C 22 6 0 4 14 D 20 6 -4 0 22 E 0 -22 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=23 A=22 E=17 D=8 so D is eliminated. Round 2 votes counts: C=33 B=28 A=22 E=17 so E is eliminated. Round 3 votes counts: B=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:222 B:212 A:172 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -22 -20 0 B 14 0 -6 -6 22 C 22 6 0 4 14 D 20 6 -4 0 22 E 0 -22 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -22 -20 0 B 14 0 -6 -6 22 C 22 6 0 4 14 D 20 6 -4 0 22 E 0 -22 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -22 -20 0 B 14 0 -6 -6 22 C 22 6 0 4 14 D 20 6 -4 0 22 E 0 -22 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5061: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C A E B D (9) C D A E B (7) B E A C D (6) D C B E A (5) B E A D C (5) D E A B C (4) C D B A E (4) C B A E D (4) B D E A C (4) A E B C D (4) A C E B D (4) D B E C A (3) D B C E A (3) C A E D B (3) C A B E D (3) D E B A C (2) C D A B E (2) C B D A E (2) B E C A D (2) B A E C D (2) A E C D B (2) A B E C D (2) E B D A C (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E A B (1) D C A E B (1) B E D A C (1) B C D E A (1) A E C B D (1) Total count = 100 A B C D E A 0 -8 0 -2 0 B 8 0 2 4 16 C 0 -2 0 16 -2 D 2 -4 -16 0 -2 E 0 -16 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -2 0 B 8 0 2 4 16 C 0 -2 0 16 -2 D 2 -4 -16 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=29 B=21 A=13 E=3 so E is eliminated. Round 2 votes counts: C=34 D=29 B=22 A=15 so A is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:206 A:195 E:194 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -2 0 B 8 0 2 4 16 C 0 -2 0 16 -2 D 2 -4 -16 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -2 0 B 8 0 2 4 16 C 0 -2 0 16 -2 D 2 -4 -16 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -2 0 B 8 0 2 4 16 C 0 -2 0 16 -2 D 2 -4 -16 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5062: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) D E C A B (7) C E D B A (7) A B D E C (6) A B E D C (5) A B E C D (5) E C D A B (4) E A B C D (4) C E B D A (4) B A C E D (4) D A E C B (3) D A B C E (3) B A E C D (3) A D B E C (3) E D C A B (2) D C E B A (2) D A C B E (2) D A B E C (2) B C E A D (2) B A D C E (2) B A C D E (2) A B D C E (2) E C D B A (1) E C B A D (1) E C A B D (1) E A C B D (1) D E A C B (1) D C A E B (1) D C A B E (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E D A (1) B D A C E (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 30 4 -10 0 B -30 0 0 -4 -4 C -4 0 0 -10 -4 D 10 4 10 0 4 E 0 4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 30 4 -10 0 B -30 0 0 -4 -4 C -4 0 0 -10 -4 D 10 4 10 0 4 E 0 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=23 B=17 E=14 C=14 so E is eliminated. Round 2 votes counts: D=34 A=28 C=21 B=17 so B is eliminated. Round 3 votes counts: A=39 D=35 C=26 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:212 E:202 C:191 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 30 4 -10 0 B -30 0 0 -4 -4 C -4 0 0 -10 -4 D 10 4 10 0 4 E 0 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 4 -10 0 B -30 0 0 -4 -4 C -4 0 0 -10 -4 D 10 4 10 0 4 E 0 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 4 -10 0 B -30 0 0 -4 -4 C -4 0 0 -10 -4 D 10 4 10 0 4 E 0 4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5063: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) C D B A E (7) B E C A D (7) A E B D C (6) E B A C D (5) E A B D C (5) D C A B E (5) C E B D A (5) C D A E B (5) E B A D C (4) E A B C D (4) D C A E B (3) C B D E A (3) A E D C B (3) A E D B C (3) E C B A D (2) D C B A E (2) D A C B E (2) C D A B E (2) A D E B C (2) E B C A D (1) E A C B D (1) D B A C E (1) D A E B C (1) C E D B A (1) C E D A B (1) C E B A D (1) C B E D A (1) C A D E B (1) B E A D C (1) B E A C D (1) B D A E C (1) B C E D A (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -16 -2 -8 B 8 0 -12 0 -10 C 16 12 0 18 2 D 2 0 -18 0 -6 E 8 10 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -2 -8 B 8 0 -12 0 -10 C 16 12 0 18 2 D 2 0 -18 0 -6 E 8 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=22 A=16 D=14 B=11 so B is eliminated. Round 2 votes counts: C=38 E=31 A=16 D=15 so D is eliminated. Round 3 votes counts: C=48 E=31 A=21 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:211 B:193 D:189 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -16 -2 -8 B 8 0 -12 0 -10 C 16 12 0 18 2 D 2 0 -18 0 -6 E 8 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -2 -8 B 8 0 -12 0 -10 C 16 12 0 18 2 D 2 0 -18 0 -6 E 8 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -2 -8 B 8 0 -12 0 -10 C 16 12 0 18 2 D 2 0 -18 0 -6 E 8 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5064: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) C B E A D (8) A E D C B (8) B C D E A (6) D E A C B (5) A E C D B (5) A E C B D (5) D B C E A (4) C E A B D (4) E A C D B (3) D B A E C (3) C B A E D (3) B D C E A (3) B C E A D (3) E D A C B (2) E C A D B (2) E A D C B (2) D B A C E (2) C B E D A (2) B A C E D (2) A E D B C (2) A D E B C (2) A B D E C (2) E D C A B (1) E C A B D (1) E A C B D (1) D E B C A (1) D E A B C (1) D C E B A (1) D B E A C (1) D A E C B (1) D A E B C (1) C A E B D (1) B D A C E (1) B C E D A (1) B C A E D (1) Total count = 100 A B C D E A 0 -2 -2 10 -4 B 2 0 -10 6 2 C 2 10 0 -4 2 D -10 -6 4 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 A B C D E A 0 -2 -2 10 -4 B 2 0 -10 6 2 C 2 10 0 -4 2 D -10 -6 4 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 D=20 C=18 E=12 so E is eliminated. Round 2 votes counts: A=30 B=26 D=23 C=21 so C is eliminated. Round 3 votes counts: B=39 A=38 D=23 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:207 C:205 A:201 B:200 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 10 -4 B 2 0 -10 6 2 C 2 10 0 -4 2 D -10 -6 4 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 10 -4 B 2 0 -10 6 2 C 2 10 0 -4 2 D -10 -6 4 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 10 -4 B 2 0 -10 6 2 C 2 10 0 -4 2 D -10 -6 4 0 -14 E 4 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999403 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5065: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) A B D C E (9) E C D A B (8) B A D C E (8) D C E A B (7) A D B C E (7) E C D B A (6) B E C A D (6) B A E C D (5) C E D A B (4) E C B D A (3) E B C D A (3) D A C E B (3) C D E A B (3) B A E D C (3) E B D C A (1) D E C A B (1) D E B C A (1) D C A E B (1) D A B C E (1) C A E D B (1) B E C D A (1) B D E A C (1) B C A E D (1) B A C E D (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 6 12 10 B 2 0 20 6 12 C -6 -20 0 -14 -2 D -12 -6 14 0 14 E -10 -12 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 12 10 B 2 0 20 6 12 C -6 -20 0 -14 -2 D -12 -6 14 0 14 E -10 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=21 A=20 D=14 C=8 so C is eliminated. Round 2 votes counts: B=37 E=25 A=21 D=17 so D is eliminated. Round 3 votes counts: E=37 B=37 A=26 so A is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:213 D:205 E:183 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 12 10 B 2 0 20 6 12 C -6 -20 0 -14 -2 D -12 -6 14 0 14 E -10 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 12 10 B 2 0 20 6 12 C -6 -20 0 -14 -2 D -12 -6 14 0 14 E -10 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 12 10 B 2 0 20 6 12 C -6 -20 0 -14 -2 D -12 -6 14 0 14 E -10 -12 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5066: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) D A B E C (10) A D C B E (6) A D B C E (6) B A D E C (5) E B D A C (4) C A D B E (4) D E A B C (3) D B A E C (3) C E D A B (3) C A B D E (3) A D B E C (3) E D B A C (2) E C D B A (2) E C B D A (2) D E B A C (2) D A E C B (2) C E A D B (2) C B A D E (2) C A E B D (2) B E A D C (2) B A E D C (2) B A D C E (2) A C D B E (2) E D C B A (1) E C B A D (1) E B C A D (1) E B A D C (1) D E C A B (1) D B E A C (1) D A C B E (1) D A B C E (1) C D E A B (1) C B E A D (1) C A D E B (1) B E D A C (1) B C E A D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 22 20 14 B -4 0 4 -14 16 C -22 -4 0 -18 2 D -20 14 18 0 22 E -14 -16 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999342 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 22 20 14 B -4 0 4 -14 16 C -22 -4 0 -18 2 D -20 14 18 0 22 E -14 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=24 A=18 E=14 B=14 so E is eliminated. Round 2 votes counts: C=35 D=27 B=20 A=18 so A is eliminated. Round 3 votes counts: D=42 C=37 B=21 so B is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:230 D:217 B:201 C:179 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 22 20 14 B -4 0 4 -14 16 C -22 -4 0 -18 2 D -20 14 18 0 22 E -14 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 22 20 14 B -4 0 4 -14 16 C -22 -4 0 -18 2 D -20 14 18 0 22 E -14 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 22 20 14 B -4 0 4 -14 16 C -22 -4 0 -18 2 D -20 14 18 0 22 E -14 -16 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5067: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) B E C D A (6) D A C E B (5) D A C B E (5) D A B C E (5) E C B A D (4) E B C A D (4) C B E D A (4) D C B A E (3) D B A E C (3) C E B D A (3) B E A C D (3) A B E D C (3) C E D A B (2) C E B A D (2) C E A D B (2) B E C A D (2) B D A E C (2) B C E D A (2) B A E D C (2) A E C D B (2) A E C B D (2) A D E B C (2) A D C E B (2) A C E D B (2) E C B D A (1) E C A B D (1) E B C D A (1) D C A B E (1) D A B E C (1) C E D B A (1) C E A B D (1) C D E B A (1) C D B E A (1) C A D E B (1) B D E C A (1) B D E A C (1) B D C E A (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 0 10 0 12 B 0 0 8 -4 22 C -10 -8 0 -2 -12 D 0 4 2 0 -2 E -12 -22 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.560130 B: 0.000000 C: 0.000000 D: 0.439870 E: 0.000000 Sum of squares = 0.507231209982 Cumulative probabilities = A: 0.560130 B: 0.560130 C: 0.560130 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 0 12 B 0 0 8 -4 22 C -10 -8 0 -2 -12 D 0 4 2 0 -2 E -12 -22 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 B=22 C=18 E=11 so E is eliminated. Round 2 votes counts: B=27 A=26 C=24 D=23 so D is eliminated. Round 3 votes counts: A=42 B=30 C=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:211 D:202 E:190 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 0 12 B 0 0 8 -4 22 C -10 -8 0 -2 -12 D 0 4 2 0 -2 E -12 -22 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 0 12 B 0 0 8 -4 22 C -10 -8 0 -2 -12 D 0 4 2 0 -2 E -12 -22 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 0 12 B 0 0 8 -4 22 C -10 -8 0 -2 -12 D 0 4 2 0 -2 E -12 -22 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5068: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) B E C A D (8) D C A B E (7) C B E D A (7) A E B C D (7) A E D B C (5) C D B E A (4) A E B D C (4) E B A C D (3) E A B D C (3) C B D E A (3) B C E A D (3) A D C E B (3) E B A D C (2) D E A B C (2) D C B E A (2) D C B A E (2) D A E C B (2) C B E A D (2) A D E C B (2) E B D A C (1) E B C A D (1) D E B C A (1) D E B A C (1) D C E B A (1) D B C E A (1) D A C B E (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B E D (1) B E C D A (1) B E A C D (1) B C E D A (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 2 4 B -2 0 -6 6 6 C 2 6 0 2 10 D -2 -6 -2 0 -6 E -4 -6 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 2 4 B -2 0 -6 6 6 C 2 6 0 2 10 D -2 -6 -2 0 -6 E -4 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 C=23 B=14 E=10 so E is eliminated. Round 2 votes counts: D=29 A=27 C=23 B=21 so B is eliminated. Round 3 votes counts: C=37 A=33 D=30 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:203 B:202 E:193 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 2 4 B -2 0 -6 6 6 C 2 6 0 2 10 D -2 -6 -2 0 -6 E -4 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 4 B -2 0 -6 6 6 C 2 6 0 2 10 D -2 -6 -2 0 -6 E -4 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 4 B -2 0 -6 6 6 C 2 6 0 2 10 D -2 -6 -2 0 -6 E -4 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5069: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) D A B C E (7) B A E C D (6) E D C B A (5) D E C A B (5) B A D E C (5) A B D C E (5) B E C A D (4) E C D A B (3) E C B A D (3) E B C A D (3) D E C B A (3) D C A E B (3) C E A D B (3) B A C E D (3) A D C B E (3) E C B D A (2) D C E A B (2) D B E A C (2) D A C B E (2) C E D A B (2) C A D E B (2) B E D A C (2) B E A C D (2) A D B C E (2) A C B D E (2) A B C D E (2) E D C A B (1) E B D C A (1) D B A E C (1) D A E B C (1) D A C E B (1) D A B E C (1) C E B A D (1) C A E B D (1) B D A E C (1) B A D C E (1) Total count = 100 A B C D E A 0 -4 -2 -4 -2 B 4 0 -2 -12 2 C 2 2 0 -8 -16 D 4 12 8 0 2 E 2 -2 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 -2 B 4 0 -2 -12 2 C 2 2 0 -8 -16 D 4 12 8 0 2 E 2 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=25 B=24 A=14 C=9 so C is eliminated. Round 2 votes counts: E=31 D=28 B=24 A=17 so A is eliminated. Round 3 votes counts: D=35 B=33 E=32 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:207 B:196 A:194 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 -2 B 4 0 -2 -12 2 C 2 2 0 -8 -16 D 4 12 8 0 2 E 2 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 -2 B 4 0 -2 -12 2 C 2 2 0 -8 -16 D 4 12 8 0 2 E 2 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 -2 B 4 0 -2 -12 2 C 2 2 0 -8 -16 D 4 12 8 0 2 E 2 -2 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5070: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) C D A B E (10) A C D E B (9) C D B E A (8) E B A D C (7) A E B D C (7) A D C E B (7) A C D B E (7) B E C D A (5) E B D C A (4) A E D C B (4) C D B A E (3) B C D E A (3) D C B E A (2) B D C E A (2) A E B C D (2) A B E C D (2) D C E B A (1) D C A B E (1) D B C E A (1) B E D A C (1) B E A C D (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 2 -2 -4 8 B -2 0 -6 -10 14 C 2 6 0 2 8 D 4 10 -2 0 8 E -8 -14 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -4 8 B -2 0 -6 -10 14 C 2 6 0 2 8 D 4 10 -2 0 8 E -8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=23 C=21 E=11 D=5 so D is eliminated. Round 2 votes counts: A=40 C=25 B=24 E=11 so E is eliminated. Round 3 votes counts: A=40 B=35 C=25 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:210 C:209 A:202 B:198 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 -4 8 B -2 0 -6 -10 14 C 2 6 0 2 8 D 4 10 -2 0 8 E -8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 8 B -2 0 -6 -10 14 C 2 6 0 2 8 D 4 10 -2 0 8 E -8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 8 B -2 0 -6 -10 14 C 2 6 0 2 8 D 4 10 -2 0 8 E -8 -14 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5071: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) D A C E B (9) E B C A D (8) B E D C A (6) D B A C E (5) D B A E C (4) B E C A D (4) A C D E B (4) E C B A D (3) E C A B D (3) C A E D B (3) C A E B D (3) B E C D A (3) B D E A C (3) B D C E A (3) B D C A E (3) A C E D B (3) E A C D B (2) E A C B D (2) D B C A E (2) C E A B D (2) B D E C A (2) E D A B C (1) E B D A C (1) E B A C D (1) E A D C B (1) D E B A C (1) D B E A C (1) D A E C B (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 2 -12 2 B 4 0 0 0 0 C -2 0 0 -10 4 D 12 0 10 0 2 E -2 0 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.649975 C: 0.000000 D: 0.350025 E: 0.000000 Sum of squares = 0.544984896605 Cumulative probabilities = A: 0.000000 B: 0.649975 C: 0.649975 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -12 2 B 4 0 0 0 0 C -2 0 0 -10 4 D 12 0 10 0 2 E -2 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=26 E=22 C=11 A=8 so A is eliminated. Round 2 votes counts: D=34 B=26 E=22 C=18 so C is eliminated. Round 3 votes counts: D=40 E=33 B=27 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:202 C:196 E:196 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -12 2 B 4 0 0 0 0 C -2 0 0 -10 4 D 12 0 10 0 2 E -2 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -12 2 B 4 0 0 0 0 C -2 0 0 -10 4 D 12 0 10 0 2 E -2 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -12 2 B 4 0 0 0 0 C -2 0 0 -10 4 D 12 0 10 0 2 E -2 0 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5072: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) D A E B C (8) D A B E C (5) C B E A D (5) E C A D B (4) D E A C B (4) E D C A B (3) E D A C B (3) E A D C B (3) D B A C E (3) C E B D A (3) C E A B D (3) C B E D A (3) B C E D A (3) E D C B A (2) E C B D A (2) E C B A D (2) D E B C A (2) D E A B C (2) D B A E C (2) B D C E A (2) B C D A E (2) B C A E D (2) A E C B D (2) A D E C B (2) A D B C E (2) A B C D E (2) E C A B D (1) D E C B A (1) D E C A B (1) D B E C A (1) D B C E A (1) D A E C B (1) C E B A D (1) C B A E D (1) C A B E D (1) B D A C E (1) B C D E A (1) B C A D E (1) B A C E D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -14 -12 -24 B 4 0 2 0 -2 C 14 -2 0 2 -2 D 12 0 -2 0 -10 E 24 2 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -14 -12 -24 B 4 0 2 0 -2 C 14 -2 0 2 -2 D 12 0 -2 0 -10 E 24 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=23 E=20 C=17 A=9 so A is eliminated. Round 2 votes counts: D=35 B=25 E=22 C=18 so C is eliminated. Round 3 votes counts: D=35 B=35 E=30 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:219 C:206 B:202 D:200 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -14 -12 -24 B 4 0 2 0 -2 C 14 -2 0 2 -2 D 12 0 -2 0 -10 E 24 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -12 -24 B 4 0 2 0 -2 C 14 -2 0 2 -2 D 12 0 -2 0 -10 E 24 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -12 -24 B 4 0 2 0 -2 C 14 -2 0 2 -2 D 12 0 -2 0 -10 E 24 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5073: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (13) A D C E B (8) D A C B E (6) E B C A D (5) D C A B E (5) B D C E A (4) A E D B C (4) E B A C D (3) C D B A E (3) C B D E A (3) B E D A C (3) A E D C B (3) A E C D B (3) A E B D C (3) A D E B C (3) E B A D C (2) E A B D C (2) D A B C E (2) C D B E A (2) C A E D B (2) B E D C A (2) A D C B E (2) E C B A D (1) E C A B D (1) E B C D A (1) D C B A E (1) D B C A E (1) D B A E C (1) D B A C E (1) C E A D B (1) C E A B D (1) C D A B E (1) C B E D A (1) C A D E B (1) B D E A C (1) B C E D A (1) A E B C D (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 0 -4 6 B -2 0 6 -4 6 C 0 -6 0 -10 -6 D 4 4 10 0 -6 E -6 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999985 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 A B C D E A 0 2 0 -4 6 B -2 0 6 -4 6 C 0 -6 0 -10 -6 D 4 4 10 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999933 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 D=17 E=15 C=15 so E is eliminated. Round 2 votes counts: B=35 A=31 D=17 C=17 so D is eliminated. Round 3 votes counts: A=39 B=38 C=23 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:206 B:203 A:202 E:200 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 -4 6 B -2 0 6 -4 6 C 0 -6 0 -10 -6 D 4 4 10 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999933 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 6 B -2 0 6 -4 6 C 0 -6 0 -10 -6 D 4 4 10 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999933 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 6 B -2 0 6 -4 6 C 0 -6 0 -10 -6 D 4 4 10 0 -6 E -6 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999933 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5074: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) A E D B C (9) E A B C D (7) B C D E A (7) C B D E A (6) B E C A D (6) B C E D A (6) A E D C B (6) D C B A E (5) D C A E B (5) A D E C B (5) E B A C D (4) E A B D C (4) D C A B E (4) C D B A E (4) D A E C B (3) C B D A E (2) B C E A D (2) E A D C B (1) E A D B C (1) D A C B E (1) B E A C D (1) A D E B C (1) Total count = 100 A B C D E A 0 14 6 -6 10 B -14 0 -4 -10 -12 C -6 4 0 -10 4 D 6 10 10 0 6 E -10 12 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -6 10 B -14 0 -4 -10 -12 C -6 4 0 -10 4 D 6 10 10 0 6 E -10 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=22 A=21 E=17 C=12 so C is eliminated. Round 2 votes counts: D=32 B=30 A=21 E=17 so E is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:212 C:196 E:196 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -6 10 B -14 0 -4 -10 -12 C -6 4 0 -10 4 D 6 10 10 0 6 E -10 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -6 10 B -14 0 -4 -10 -12 C -6 4 0 -10 4 D 6 10 10 0 6 E -10 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -6 10 B -14 0 -4 -10 -12 C -6 4 0 -10 4 D 6 10 10 0 6 E -10 12 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5075: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) E C B D A (8) D A B E C (7) D A E B C (6) A D B C E (6) C E B A D (5) B A D C E (5) E C D B A (4) E C D A B (4) D E A B C (4) C E B D A (4) C B E A D (4) B A C D E (4) E D C A B (3) E D A C B (3) D A B C E (3) B C A D E (3) A B C D E (3) E C B A D (2) E C A D B (2) C B A E D (2) C B A D E (2) E D C B A (1) E C A B D (1) D E B A C (1) C E A B D (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 10 8 4 6 B -10 0 6 10 0 C -8 -6 0 0 6 D -4 -10 0 0 10 E -6 0 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 4 6 B -10 0 6 10 0 C -8 -6 0 0 6 D -4 -10 0 0 10 E -6 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=21 A=21 C=18 B=12 so B is eliminated. Round 2 votes counts: A=30 E=28 D=21 C=21 so D is eliminated. Round 3 votes counts: A=46 E=33 C=21 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:203 D:198 C:196 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 4 6 B -10 0 6 10 0 C -8 -6 0 0 6 D -4 -10 0 0 10 E -6 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 4 6 B -10 0 6 10 0 C -8 -6 0 0 6 D -4 -10 0 0 10 E -6 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 4 6 B -10 0 6 10 0 C -8 -6 0 0 6 D -4 -10 0 0 10 E -6 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5076: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) A E D B C (9) E B D A C (7) C B D E A (6) C A E D B (6) E A D B C (5) C A D E B (5) C A D B E (5) E B A D C (4) E A B D C (4) C D B A E (4) B D C E A (4) A E C D B (4) C E A B D (3) A C D E B (3) C D A B E (2) C B D A E (2) B E C D A (2) A D C E B (2) A C E D B (2) E D A B C (1) E C A B D (1) E A C D B (1) E A C B D (1) D B E A C (1) C B E D A (1) C A E B D (1) B E D C A (1) B E D A C (1) B D E C A (1) A D E B C (1) Total count = 100 A B C D E A 0 12 12 14 -8 B -12 0 2 -2 -20 C -12 -2 0 -2 -8 D -14 2 2 0 -8 E 8 20 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 12 14 -8 B -12 0 2 -2 -20 C -12 -2 0 -2 -8 D -14 2 2 0 -8 E 8 20 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=24 A=21 B=19 D=1 so D is eliminated. Round 2 votes counts: C=35 E=24 A=21 B=20 so B is eliminated. Round 3 votes counts: E=40 C=39 A=21 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:215 D:191 C:188 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 12 14 -8 B -12 0 2 -2 -20 C -12 -2 0 -2 -8 D -14 2 2 0 -8 E 8 20 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 14 -8 B -12 0 2 -2 -20 C -12 -2 0 -2 -8 D -14 2 2 0 -8 E 8 20 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 14 -8 B -12 0 2 -2 -20 C -12 -2 0 -2 -8 D -14 2 2 0 -8 E 8 20 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5077: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) E C A B D (9) D B A C E (9) C E B D A (7) D B A E C (6) C B D E A (6) B D C A E (5) A E D B C (5) E C B D A (4) C B D A E (4) A D B E C (4) E A D B C (3) D B C A E (3) C E A B D (3) B D A C E (3) A B D C E (3) E C D B A (2) A E C D B (2) E D A B C (1) E C A D B (1) E A D C B (1) D B E A C (1) C E B A D (1) C B E D A (1) B D C E A (1) B C D A E (1) A E C B D (1) A E B C D (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 4 -8 -2 B 8 0 -6 4 -4 C -4 6 0 10 -2 D 8 -4 -10 0 -6 E 2 4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 4 -8 -2 B 8 0 -6 4 -4 C -4 6 0 10 -2 D 8 -4 -10 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 D=19 A=18 B=10 so B is eliminated. Round 2 votes counts: E=31 D=28 C=23 A=18 so A is eliminated. Round 3 votes counts: E=40 D=35 C=25 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 C:205 B:201 D:194 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 -8 -2 B 8 0 -6 4 -4 C -4 6 0 10 -2 D 8 -4 -10 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -8 -2 B 8 0 -6 4 -4 C -4 6 0 10 -2 D 8 -4 -10 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -8 -2 B 8 0 -6 4 -4 C -4 6 0 10 -2 D 8 -4 -10 0 -6 E 2 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5078: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (15) C B E D A (9) C D B E A (6) D A E B C (5) A D C E B (5) A C B E D (4) A B E C D (4) D E B C A (3) C B E A D (3) B E D C A (3) B E C D A (3) A E D B C (3) A E B D C (3) A D E C B (3) E B D C A (2) D E B A C (2) D C E B A (2) C D E B A (2) C D B A E (2) C B D E A (2) C A D B E (2) A C D E B (2) E B D A C (1) E B A D C (1) D E C B A (1) D C A E B (1) D B E C A (1) D A E C B (1) D A C E B (1) C D A E B (1) C B A E D (1) C A B D E (1) B E C A D (1) B E A D C (1) B E A C D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 8 4 12 B -6 0 0 -18 -8 C -8 0 0 -10 -10 D -4 18 10 0 18 E -12 8 10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 4 12 B -6 0 0 -18 -8 C -8 0 0 -10 -10 D -4 18 10 0 18 E -12 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=29 D=17 B=9 E=4 so E is eliminated. Round 2 votes counts: A=41 C=29 D=17 B=13 so B is eliminated. Round 3 votes counts: A=44 C=33 D=23 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:215 E:194 C:186 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 4 12 B -6 0 0 -18 -8 C -8 0 0 -10 -10 D -4 18 10 0 18 E -12 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 4 12 B -6 0 0 -18 -8 C -8 0 0 -10 -10 D -4 18 10 0 18 E -12 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 4 12 B -6 0 0 -18 -8 C -8 0 0 -10 -10 D -4 18 10 0 18 E -12 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5079: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) B E D A C (9) A C B E D (8) A B C E D (7) C A E D B (6) A B C D E (6) B A E D C (5) A C B D E (5) D E C B A (4) C A D B E (4) B E A D C (4) D B E A C (3) C D E A B (3) C A D E B (3) B D E A C (3) B A E C D (3) E D B C A (2) E D B A C (2) D C E A B (2) C E A D B (2) E D C B A (1) E B D C A (1) D C A E B (1) C D A E B (1) C A B D E (1) B D A E C (1) B A D E C (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 18 14 8 B 0 0 16 10 24 C -18 -16 0 2 2 D -14 -10 -2 0 -2 E -8 -24 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.506994 B: 0.493006 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500097842685 Cumulative probabilities = A: 0.506994 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 14 8 B 0 0 16 10 24 C -18 -16 0 2 2 D -14 -10 -2 0 -2 E -8 -24 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 D=20 C=20 E=6 so E is eliminated. Round 2 votes counts: B=28 A=27 D=25 C=20 so C is eliminated. Round 3 votes counts: A=43 D=29 B=28 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:225 A:220 D:186 C:185 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 14 8 B 0 0 16 10 24 C -18 -16 0 2 2 D -14 -10 -2 0 -2 E -8 -24 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 14 8 B 0 0 16 10 24 C -18 -16 0 2 2 D -14 -10 -2 0 -2 E -8 -24 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 14 8 B 0 0 16 10 24 C -18 -16 0 2 2 D -14 -10 -2 0 -2 E -8 -24 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5080: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) E D C A B (6) B D C E A (6) A E C D B (6) A E B D C (5) E A D B C (4) D C B E A (4) A E D C B (4) E D B C A (3) D C E B A (3) C D B E A (3) B D E C A (3) B A C D E (3) E D A C B (2) E B D A C (2) E A D C B (2) D E C B A (2) D E B C A (2) D B E C A (2) D B C E A (2) C B D E A (2) C B D A E (2) B C D E A (2) B C A D E (2) A E B C D (2) A B E C D (2) E D C B A (1) E D B A C (1) E B A D C (1) E A C D B (1) C B A D E (1) C A D E B (1) B E D C A (1) B E A D C (1) B A D E C (1) A E D B C (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -10 -12 -12 B 14 0 10 -4 -4 C 10 -10 0 -18 -10 D 12 4 18 0 2 E 12 4 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -12 -12 B 14 0 10 -4 -4 C 10 -10 0 -18 -10 D 12 4 18 0 2 E 12 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 E=23 D=15 C=9 so C is eliminated. Round 2 votes counts: B=31 A=28 E=23 D=18 so D is eliminated. Round 3 votes counts: B=42 E=30 A=28 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:212 B:208 C:186 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 -12 -12 B 14 0 10 -4 -4 C 10 -10 0 -18 -10 D 12 4 18 0 2 E 12 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -12 -12 B 14 0 10 -4 -4 C 10 -10 0 -18 -10 D 12 4 18 0 2 E 12 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -12 -12 B 14 0 10 -4 -4 C 10 -10 0 -18 -10 D 12 4 18 0 2 E 12 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5081: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (12) D A C E B (8) E B C A D (7) A D E C B (7) B E C A D (5) B C E D A (5) E A D C B (4) C B E A D (4) B C D A E (4) D A B C E (3) B E C D A (3) B D C A E (3) A D C E B (3) E C B A D (2) E A C D B (2) D A E C B (2) C E A D B (2) B E D C A (2) B C D E A (2) E D A B C (1) E C A D B (1) E C A B D (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B D C (1) D C A B E (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) C D A B E (1) C B D A E (1) C A D E B (1) C A D B E (1) B C E A D (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 14 2 -2 4 B -14 0 -12 -10 0 C -2 12 0 -8 10 D 2 10 8 0 6 E -4 0 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 -2 4 B -14 0 -12 -10 0 C -2 12 0 -8 10 D 2 10 8 0 6 E -4 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=25 E=22 C=12 A=12 so C is eliminated. Round 2 votes counts: D=30 B=30 E=26 A=14 so A is eliminated. Round 3 votes counts: D=43 B=30 E=27 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:209 C:206 E:190 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 2 -2 4 B -14 0 -12 -10 0 C -2 12 0 -8 10 D 2 10 8 0 6 E -4 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 -2 4 B -14 0 -12 -10 0 C -2 12 0 -8 10 D 2 10 8 0 6 E -4 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 -2 4 B -14 0 -12 -10 0 C -2 12 0 -8 10 D 2 10 8 0 6 E -4 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5082: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) D A B C E (11) C D A B E (9) E C B A D (8) B A D E C (8) B E A D C (7) E C D A B (5) C E D A B (5) C E B D A (5) E A D B C (3) E B C A D (2) E B A C D (2) D C A B E (2) D A C B E (2) D A B E C (2) C D E A B (2) C D A E B (2) B A E D C (2) A D B E C (2) D A E B C (1) C E D B A (1) C E B A D (1) C D B A E (1) C B D A E (1) B D C A E (1) B C D A E (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 8 -2 -6 B 6 0 12 4 2 C -8 -12 0 -10 -10 D 2 -4 10 0 -6 E 6 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998485 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 -2 -6 B 6 0 12 4 2 C -8 -12 0 -10 -10 D 2 -4 10 0 -6 E 6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=27 B=20 D=18 A=3 so A is eliminated. Round 2 votes counts: E=32 C=27 B=21 D=20 so D is eliminated. Round 3 votes counts: B=36 E=33 C=31 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:210 D:201 A:197 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 -2 -6 B 6 0 12 4 2 C -8 -12 0 -10 -10 D 2 -4 10 0 -6 E 6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 -2 -6 B 6 0 12 4 2 C -8 -12 0 -10 -10 D 2 -4 10 0 -6 E 6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 -2 -6 B 6 0 12 4 2 C -8 -12 0 -10 -10 D 2 -4 10 0 -6 E 6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5083: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) C D B E A (7) C D E B A (6) E A C D B (5) D B C E A (5) A E C B D (4) E A D C B (3) E A D B C (3) C E A D B (3) C B D A E (3) B D E A C (3) B A C D E (3) A E C D B (3) A E B D C (3) A C E B D (3) E D B A C (2) E D A C B (2) E D A B C (2) D C B E A (2) C E D A B (2) C B A D E (2) C A B D E (2) B D C E A (2) A B E D C (2) E C A D B (1) E A B D C (1) D E C B A (1) D E B C A (1) D E B A C (1) D B E C A (1) D B E A C (1) C D E A B (1) C D B A E (1) C A E B D (1) C A B E D (1) B D E C A (1) B D A C E (1) B C D A E (1) A E D B C (1) A E B C D (1) A C B E D (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -8 -12 B 4 0 -8 -8 0 C 4 8 0 8 14 D 8 8 -8 0 8 E 12 0 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -8 -12 B 4 0 -8 -8 0 C 4 8 0 8 14 D 8 8 -8 0 8 E 12 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=21 E=19 B=19 D=12 so D is eliminated. Round 2 votes counts: C=31 B=26 E=22 A=21 so A is eliminated. Round 3 votes counts: C=35 E=34 B=31 so B is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:208 E:195 B:194 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -8 -12 B 4 0 -8 -8 0 C 4 8 0 8 14 D 8 8 -8 0 8 E 12 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -8 -12 B 4 0 -8 -8 0 C 4 8 0 8 14 D 8 8 -8 0 8 E 12 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -8 -12 B 4 0 -8 -8 0 C 4 8 0 8 14 D 8 8 -8 0 8 E 12 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5084: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) C B A D E (7) A B C E D (7) A B E C D (6) E D A B C (5) D C E B A (5) C D B E A (5) E A B D C (4) D E C A B (4) C B D A E (4) A E D B C (4) D E A C B (3) D E A B C (3) E B D C A (2) E A D B C (2) D A C B E (2) C D B A E (2) C A B D E (2) B C E A D (2) B C A E D (2) A D C B E (2) A C B D E (2) E D B A C (1) E B A C D (1) E A B C D (1) D C B A E (1) D C A E B (1) D A E C B (1) C D A B E (1) C B E D A (1) C B A E D (1) B C E D A (1) B A C E D (1) A E B D C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -4 -6 -2 B -6 0 -10 -6 0 C 4 10 0 -6 -2 D 6 6 6 0 12 E 2 0 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -6 -2 B -6 0 -10 -6 0 C 4 10 0 -6 -2 D 6 6 6 0 12 E 2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=24 C=23 E=16 B=6 so B is eliminated. Round 2 votes counts: D=31 C=28 A=25 E=16 so E is eliminated. Round 3 votes counts: D=39 A=33 C=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:203 A:197 E:196 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -6 -2 B -6 0 -10 -6 0 C 4 10 0 -6 -2 D 6 6 6 0 12 E 2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -6 -2 B -6 0 -10 -6 0 C 4 10 0 -6 -2 D 6 6 6 0 12 E 2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -6 -2 B -6 0 -10 -6 0 C 4 10 0 -6 -2 D 6 6 6 0 12 E 2 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5085: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) D B C A E (8) B D C E A (6) D A E C B (5) B E D C A (5) B E C A D (5) E A C B D (4) D A C E B (4) B D E C A (4) A E C D B (4) E A C D B (3) D E A B C (3) D B A C E (3) B C E A D (3) E A D C B (2) E A B C D (2) C D A B E (2) C B D A E (2) C A D E B (2) B E C D A (2) A E C B D (2) A D C E B (2) A C D E B (2) E B C A D (1) E B A C D (1) D E B A C (1) D E A C B (1) D C A B E (1) D B A E C (1) D A C B E (1) C E A B D (1) C A E B D (1) C A D B E (1) B E D A C (1) B D E A C (1) B D C A E (1) B C A D E (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 8 -22 -10 B 10 0 16 -14 14 C -8 -16 0 -20 -16 D 22 14 20 0 22 E 10 -14 16 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -22 -10 B 10 0 16 -14 14 C -8 -16 0 -20 -16 D 22 14 20 0 22 E 10 -14 16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=29 E=13 A=12 C=9 so C is eliminated. Round 2 votes counts: D=39 B=31 A=16 E=14 so E is eliminated. Round 3 votes counts: D=39 B=33 A=28 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:239 B:213 E:195 A:183 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 -22 -10 B 10 0 16 -14 14 C -8 -16 0 -20 -16 D 22 14 20 0 22 E 10 -14 16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -22 -10 B 10 0 16 -14 14 C -8 -16 0 -20 -16 D 22 14 20 0 22 E 10 -14 16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -22 -10 B 10 0 16 -14 14 C -8 -16 0 -20 -16 D 22 14 20 0 22 E 10 -14 16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5086: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) C E B A D (7) B E C D A (7) A D C E B (6) A E B D C (5) A D E B C (5) E B C A D (4) D B E C A (4) C D B E A (4) E B A C D (3) D C B E A (3) D C A B E (3) D B E A C (3) C B E D A (3) C A E B D (3) C A D E B (3) B E D C A (3) B E D A C (3) A C D E B (3) C D A B E (2) C B E A D (2) A E C B D (2) A E B C D (2) A C E B D (2) E C B A D (1) E B D A C (1) E B C D A (1) E A B D C (1) D C B A E (1) D B A E C (1) D A B E C (1) C A D B E (1) B D E A C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -4 2 -2 B 4 0 -10 2 2 C 4 10 0 0 4 D -2 -2 0 0 0 E 2 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.542462 D: 0.457538 E: 0.000000 Sum of squares = 0.503606004021 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.542462 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 2 -2 B 4 0 -10 2 2 C 4 10 0 0 4 D -2 -2 0 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 D=24 B=14 E=11 so E is eliminated. Round 2 votes counts: A=27 C=26 D=24 B=23 so B is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:199 D:198 E:198 A:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 2 -2 B 4 0 -10 2 2 C 4 10 0 0 4 D -2 -2 0 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 2 -2 B 4 0 -10 2 2 C 4 10 0 0 4 D -2 -2 0 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 2 -2 B 4 0 -10 2 2 C 4 10 0 0 4 D -2 -2 0 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5087: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) D E C B A (11) E A D B C (8) D C B E A (8) E D A C B (6) C B D A E (6) A B C E D (6) A B C D E (6) D C E B A (5) C D B A E (5) B C A D E (5) B A C D E (5) E D C A B (4) D C B A E (4) E A D C B (3) E A B C D (3) E D C B A (1) E D B C A (1) B C D E A (1) Total count = 100 A B C D E A 0 -4 -2 -4 -2 B 4 0 -6 -12 -8 C 2 6 0 -2 2 D 4 12 2 0 12 E 2 8 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -4 -2 B 4 0 -6 -12 -8 C 2 6 0 -2 2 D 4 12 2 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 A=24 C=11 B=11 so C is eliminated. Round 2 votes counts: D=33 E=26 A=24 B=17 so B is eliminated. Round 3 votes counts: D=40 A=34 E=26 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:204 E:198 A:194 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 -2 B 4 0 -6 -12 -8 C 2 6 0 -2 2 D 4 12 2 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 -2 B 4 0 -6 -12 -8 C 2 6 0 -2 2 D 4 12 2 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 -2 B 4 0 -6 -12 -8 C 2 6 0 -2 2 D 4 12 2 0 12 E 2 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5088: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) A D E B C (7) E C D B A (5) A D B E C (5) E D A C B (4) C E D B A (4) A B C E D (4) E D C B A (3) E D C A B (3) E C D A B (3) D E C B A (3) D B E C A (3) C B E A D (3) B C D E A (3) A E D C B (3) A B D E C (3) D A E B C (2) C E B A D (2) B D A E C (2) B D A C E (2) B C A E D (2) B A D C E (2) B A C E D (2) B A C D E (2) A B D C E (2) A B C D E (2) E C A D B (1) E A C D B (1) D E C A B (1) D E B A C (1) D E A C B (1) D E A B C (1) D B E A C (1) D B A E C (1) C E D A B (1) C E A B D (1) C A E B D (1) C A B E D (1) B D C E A (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 0 -8 -10 B 2 0 -4 -10 -16 C 0 4 0 -2 -10 D 8 10 2 0 -10 E 10 16 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 -8 -10 B 2 0 -4 -10 -16 C 0 4 0 -2 -10 D 8 10 2 0 -10 E 10 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 E=20 B=16 D=14 so D is eliminated. Round 2 votes counts: A=30 E=27 C=22 B=21 so B is eliminated. Round 3 votes counts: A=41 E=31 C=28 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:205 C:196 A:190 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 -8 -10 B 2 0 -4 -10 -16 C 0 4 0 -2 -10 D 8 10 2 0 -10 E 10 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -8 -10 B 2 0 -4 -10 -16 C 0 4 0 -2 -10 D 8 10 2 0 -10 E 10 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -8 -10 B 2 0 -4 -10 -16 C 0 4 0 -2 -10 D 8 10 2 0 -10 E 10 16 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5089: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) D C E A B (9) C D A E B (8) A C D B E (7) E D C A B (5) C A D B E (5) B E A D C (5) E B D C A (4) D E C A B (4) B A E C D (4) A B C D E (4) E D B C A (3) E B D A C (3) B E D A C (3) B A C E D (3) A C B D E (3) C A D E B (2) B E A C D (2) E D B A C (1) D E A C B (1) D C A E B (1) D A E C B (1) D A C E B (1) C D E A B (1) C D A B E (1) C B A D E (1) B E D C A (1) B E C A D (1) B C E D A (1) B A C D E (1) A E B D C (1) A C D E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -14 -16 -8 B -14 0 -22 -22 -12 C 14 22 0 -8 0 D 16 22 8 0 2 E 8 12 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999114 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -14 -16 -8 B -14 0 -22 -22 -12 C 14 22 0 -8 0 D 16 22 8 0 2 E 8 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=21 C=18 A=18 D=17 so D is eliminated. Round 2 votes counts: E=31 C=28 B=21 A=20 so A is eliminated. Round 3 votes counts: C=40 E=33 B=27 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:224 C:214 E:209 A:188 B:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -14 -16 -8 B -14 0 -22 -22 -12 C 14 22 0 -8 0 D 16 22 8 0 2 E 8 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -14 -16 -8 B -14 0 -22 -22 -12 C 14 22 0 -8 0 D 16 22 8 0 2 E 8 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -14 -16 -8 B -14 0 -22 -22 -12 C 14 22 0 -8 0 D 16 22 8 0 2 E 8 12 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5090: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (13) D C B E A (9) E A B D C (7) D C B A E (7) E D C B A (6) B C D A E (6) A B C D E (5) A C D B E (4) E D C A B (3) E B A C D (3) E A D C B (3) E A B C D (3) C D B A E (3) B A C D E (3) E B D C A (2) D C E A B (2) B D C E A (2) B A E C D (2) A E C D B (2) A C D E B (2) E D A C B (1) E A D B C (1) D E C B A (1) D C A E B (1) D C A B E (1) D B C E A (1) B E D C A (1) B C D E A (1) B C A D E (1) B A C E D (1) A D C E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 6 6 8 B -2 0 8 4 -2 C -6 -8 0 2 2 D -6 -4 -2 0 0 E -8 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 6 8 B -2 0 8 4 -2 C -6 -8 0 2 2 D -6 -4 -2 0 0 E -8 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=29 A=29 D=22 B=17 C=3 so C is eliminated. Round 2 votes counts: E=29 A=29 D=25 B=17 so B is eliminated. Round 3 votes counts: A=36 D=34 E=30 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:204 E:196 C:195 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 6 8 B -2 0 8 4 -2 C -6 -8 0 2 2 D -6 -4 -2 0 0 E -8 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 6 8 B -2 0 8 4 -2 C -6 -8 0 2 2 D -6 -4 -2 0 0 E -8 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 6 8 B -2 0 8 4 -2 C -6 -8 0 2 2 D -6 -4 -2 0 0 E -8 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5091: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) C A B D E (9) E D C B A (6) C A E B D (6) A B C D E (6) D B E A C (5) C E D A B (5) E C D B A (4) D B A E C (4) C A D B E (4) E D B C A (3) D E B A C (3) C A D E B (3) B D A E C (3) B A D E C (3) E B D A C (2) C E D B A (2) C D E B A (2) C A B E D (2) B D E A C (2) B A E D C (2) A B D C E (2) E A C B D (1) D E C B A (1) D C E B A (1) D B A C E (1) C E A B D (1) C D E A B (1) C A E D B (1) B E A D C (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -2 -8 2 B 8 0 -2 -10 -2 C 2 2 0 0 -4 D 8 10 0 0 4 E -2 2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.398897 D: 0.601103 E: 0.000000 Sum of squares = 0.520443576532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.398897 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -8 2 B 8 0 -2 -10 -2 C 2 2 0 0 -4 D 8 10 0 0 4 E -2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=25 D=15 A=13 B=11 so B is eliminated. Round 2 votes counts: C=36 E=26 D=20 A=18 so A is eliminated. Round 3 votes counts: C=44 E=30 D=26 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:211 C:200 E:200 B:197 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -8 2 B 8 0 -2 -10 -2 C 2 2 0 0 -4 D 8 10 0 0 4 E -2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -8 2 B 8 0 -2 -10 -2 C 2 2 0 0 -4 D 8 10 0 0 4 E -2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -8 2 B 8 0 -2 -10 -2 C 2 2 0 0 -4 D 8 10 0 0 4 E -2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499995 D: 0.500005 E: 0.000000 Sum of squares = 0.500000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499995 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5092: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) A D C E B (6) C D E A B (5) C A B D E (5) A D E C B (5) B E C D A (4) B C A E D (4) B A E D C (4) D E C A B (3) C E D B A (3) C A D E B (3) B E D C A (3) B E D A C (3) B E A D C (3) B C E D A (3) B A C E D (3) A B E D C (3) E C D B A (2) D E A C B (2) D C E A B (2) D A E C B (2) C D A E B (2) B C E A D (2) B A E C D (2) A C D B E (2) A B D E C (2) E D B C A (1) E D A B C (1) E B D A C (1) E B C D A (1) D C A E B (1) C E B D A (1) C B E D A (1) B E C A D (1) B A C D E (1) A D E B C (1) A D C B E (1) A D B C E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 4 2 B 0 0 -8 0 2 C 8 8 0 -4 -2 D -4 0 4 0 -8 E -2 -2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.058190 B: 0.108477 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.487375512862 Cumulative probabilities = A: 0.058190 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 0 -8 4 2 B 0 0 -8 0 2 C 8 8 0 -4 -2 D -4 0 4 0 -8 E -2 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.486111111347 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=24 C=20 E=13 D=10 so D is eliminated. Round 2 votes counts: B=33 A=26 C=23 E=18 so E is eliminated. Round 3 votes counts: B=36 C=35 A=29 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:205 E:203 A:199 B:197 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 4 2 B 0 0 -8 0 2 C 8 8 0 -4 -2 D -4 0 4 0 -8 E -2 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.486111111347 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 4 2 B 0 0 -8 0 2 C 8 8 0 -4 -2 D -4 0 4 0 -8 E -2 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.486111111347 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 4 2 B 0 0 -8 0 2 C 8 8 0 -4 -2 D -4 0 4 0 -8 E -2 -2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.486111111347 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5093: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (6) B A E D C (6) A B E C D (6) D C E B A (5) D C E A B (5) D B E A C (5) C E A D B (5) C D E A B (5) B A E C D (5) A E C B D (5) D E C A B (4) B D E A C (4) D B E C A (3) D B C E A (3) A C E B D (3) A B C E D (3) D C B E A (2) C E D A B (2) C B D A E (2) C A E B D (2) B D A C E (2) B A D E C (2) E C D A B (1) E A D C B (1) E A C D B (1) E A B D C (1) D E A B C (1) C D E B A (1) C A E D B (1) C A B E D (1) B E A D C (1) B C A E D (1) B C A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 12 -2 0 B 4 0 6 10 14 C -12 -6 0 -4 -4 D 2 -10 4 0 6 E 0 -14 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 -2 0 B 4 0 6 10 14 C -12 -6 0 -4 -4 D 2 -10 4 0 6 E 0 -14 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=28 C=19 A=18 E=4 so E is eliminated. Round 2 votes counts: B=31 D=28 A=21 C=20 so C is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:203 D:201 E:192 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 -2 0 B 4 0 6 10 14 C -12 -6 0 -4 -4 D 2 -10 4 0 6 E 0 -14 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -2 0 B 4 0 6 10 14 C -12 -6 0 -4 -4 D 2 -10 4 0 6 E 0 -14 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -2 0 B 4 0 6 10 14 C -12 -6 0 -4 -4 D 2 -10 4 0 6 E 0 -14 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5094: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) C A B D E (6) B E C D A (5) A D E C B (5) A D C E B (5) A C B E D (5) E A D B C (4) D E A B C (4) D C A B E (4) D A E C B (4) E D A B C (3) D C B E A (3) C B A D E (3) B C E D A (3) A C D B E (3) A C B D E (3) E D B A C (2) E B C A D (2) C B D A E (2) C B A E D (2) B C E A D (2) A E B C D (2) E D B C A (1) E B D A C (1) E A B D C (1) E A B C D (1) D E A C B (1) D A E B C (1) C D B E A (1) C B E A D (1) C B D E A (1) C A D B E (1) B E D C A (1) B E C A D (1) B C D E A (1) B C A E D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 18 2 8 6 B -18 0 -8 6 0 C -2 8 0 -4 -2 D -8 -6 4 0 0 E -6 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 8 6 B -18 0 -8 6 0 C -2 8 0 -4 -2 D -8 -6 4 0 0 E -6 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971451 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=23 D=17 C=17 B=14 so B is eliminated. Round 2 votes counts: E=30 A=29 C=24 D=17 so D is eliminated. Round 3 votes counts: E=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:200 E:198 D:195 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 8 6 B -18 0 -8 6 0 C -2 8 0 -4 -2 D -8 -6 4 0 0 E -6 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971451 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 8 6 B -18 0 -8 6 0 C -2 8 0 -4 -2 D -8 -6 4 0 0 E -6 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971451 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 8 6 B -18 0 -8 6 0 C -2 8 0 -4 -2 D -8 -6 4 0 0 E -6 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971451 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5095: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) A C D B E (7) E D B C A (6) D E B A C (6) D A C B E (6) C A B E D (6) C A D E B (5) B E D A C (5) E B D C A (4) A C B E D (4) E B C A D (3) D A C E B (3) C A E B D (3) B E A C D (3) D B A E C (2) B E D C A (2) B E C A D (2) B D E A C (2) A D C B E (2) A C D E B (2) A C B D E (2) E D C B A (1) E C B D A (1) E C B A D (1) E B C D A (1) D E C A B (1) D C A E B (1) D B E A C (1) D A B C E (1) C E B A D (1) C D E A B (1) C A D B E (1) C A B D E (1) B D A E C (1) B A D E C (1) B A D C E (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 -4 4 B 2 0 2 -8 6 C -4 -2 0 -10 0 D 4 8 10 0 14 E -4 -6 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -4 4 B 2 0 2 -8 6 C -4 -2 0 -10 0 D 4 8 10 0 14 E -4 -6 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=20 C=18 E=17 B=17 so E is eliminated. Round 2 votes counts: D=35 B=25 C=20 A=20 so C is eliminated. Round 3 votes counts: D=36 A=36 B=28 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:201 B:201 C:192 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -4 4 B 2 0 2 -8 6 C -4 -2 0 -10 0 D 4 8 10 0 14 E -4 -6 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -4 4 B 2 0 2 -8 6 C -4 -2 0 -10 0 D 4 8 10 0 14 E -4 -6 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -4 4 B 2 0 2 -8 6 C -4 -2 0 -10 0 D 4 8 10 0 14 E -4 -6 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5096: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (13) C A D B E (13) A D E B C (8) C B E D A (6) B E D A C (5) E B C D A (4) C A D E B (4) B D E A C (4) D B A E C (3) D A B E C (3) C E A B D (3) E B D C A (2) D A E B C (2) C E B D A (2) C E B A D (2) C A E D B (2) C A E B D (2) C A B D E (2) B E D C A (2) A D C B E (2) A C D E B (2) A C D B E (2) E C B A D (1) E B C A D (1) E A C D B (1) D E B A C (1) D E A B C (1) D B E A C (1) D A C B E (1) C B E A D (1) C B A E D (1) B C E D A (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 2 -2 -2 B 0 0 4 4 -6 C -2 -4 0 2 -6 D 2 -4 -2 0 -2 E 2 6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 -2 -2 B 0 0 4 4 -6 C -2 -4 0 2 -6 D 2 -4 -2 0 -2 E 2 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=22 A=16 D=12 B=12 so D is eliminated. Round 2 votes counts: C=38 E=24 A=22 B=16 so B is eliminated. Round 3 votes counts: C=39 E=36 A=25 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:208 B:201 A:199 D:197 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -2 -2 B 0 0 4 4 -6 C -2 -4 0 2 -6 D 2 -4 -2 0 -2 E 2 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -2 B 0 0 4 4 -6 C -2 -4 0 2 -6 D 2 -4 -2 0 -2 E 2 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -2 B 0 0 4 4 -6 C -2 -4 0 2 -6 D 2 -4 -2 0 -2 E 2 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5097: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) D C B E A (8) D B C E A (6) C D A B E (6) B E D A C (6) A E C B D (4) A E B D C (4) A E B C D (4) A C E D B (4) A C E B D (4) E A B D C (3) D B E C A (3) C D A E B (3) C A D E B (3) B E A D C (3) B D E C A (3) A C D E B (3) E B A C D (2) D C B A E (2) D C A E B (2) D C A B E (2) C D B A E (2) B E A C D (2) E D A B C (1) E A D B C (1) C D B E A (1) C A D B E (1) B E D C A (1) B E C A D (1) B D C E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 10 6 -8 B 6 0 6 0 -4 C -10 -6 0 -20 -4 D -6 0 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 10 6 -8 B 6 0 6 0 -4 C -10 -6 0 -20 -4 D -6 0 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 E=19 B=17 C=16 so C is eliminated. Round 2 votes counts: D=35 A=29 E=19 B=17 so B is eliminated. Round 3 votes counts: D=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 B:204 D:204 A:201 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 10 6 -8 B 6 0 6 0 -4 C -10 -6 0 -20 -4 D -6 0 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 6 -8 B 6 0 6 0 -4 C -10 -6 0 -20 -4 D -6 0 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 6 -8 B 6 0 6 0 -4 C -10 -6 0 -20 -4 D -6 0 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5098: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) B A E C D (7) E A D C B (6) D C B E A (6) C D B A E (6) D C E A B (5) A E C B D (5) D C B A E (4) C D A E B (4) B E A D C (4) B C D A E (4) D C E B A (3) D B C E A (3) C B A D E (3) B E D A C (3) B D C A E (3) A E B C D (3) E A C D B (2) E A B D C (2) D E C A B (2) C D A B E (2) B A C E D (2) A C E D B (2) E D A C B (1) E B A D C (1) E A D B C (1) D E A C B (1) D B C A E (1) C A E D B (1) B E A C D (1) B D C E A (1) B C A D E (1) B A E D C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 2 2 2 B 8 0 -8 0 8 C -2 8 0 4 4 D -2 0 -4 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407384 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 2 2 B 8 0 -8 0 8 C -2 8 0 4 4 D -2 0 -4 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407398 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 E=20 C=16 A=12 so A is eliminated. Round 2 votes counts: E=28 B=28 D=25 C=19 so C is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:207 B:204 A:199 D:196 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 2 2 2 B 8 0 -8 0 8 C -2 8 0 4 4 D -2 0 -4 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407398 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 2 2 B 8 0 -8 0 8 C -2 8 0 4 4 D -2 0 -4 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407398 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 2 2 B 8 0 -8 0 8 C -2 8 0 4 4 D -2 0 -4 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.111111 C: 0.444444 D: 0.000000 E: 0.000000 Sum of squares = 0.407407407398 Cumulative probabilities = A: 0.444444 B: 0.555556 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5099: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) A E C B D (9) D B C E A (8) E A C B D (6) C B D A E (6) A C E B D (6) D B C A E (5) E D B A C (4) E A B D C (3) E A B C D (3) C A B D E (3) B D C E A (3) A E D B C (3) A C D B E (3) E B D A C (2) E A D B C (2) D B E A C (2) C D B A E (2) A C E D B (2) E B D C A (1) E B C D A (1) E B C A D (1) D C B A E (1) D B A E C (1) D A B E C (1) C E A B D (1) C D A B E (1) C B E D A (1) C B A D E (1) C A E B D (1) C A D B E (1) B D E C A (1) B C E D A (1) B C D E A (1) A E D C B (1) A E C D B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 2 -2 0 B 2 0 4 4 4 C -2 -4 0 6 -2 D 2 -4 -6 0 0 E 0 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 0 B 2 0 4 4 4 C -2 -4 0 6 -2 D 2 -4 -6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 E=23 C=17 B=6 so B is eliminated. Round 2 votes counts: D=31 A=27 E=23 C=19 so C is eliminated. Round 3 votes counts: D=41 A=33 E=26 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:207 A:199 C:199 E:199 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 0 B 2 0 4 4 4 C -2 -4 0 6 -2 D 2 -4 -6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 0 B 2 0 4 4 4 C -2 -4 0 6 -2 D 2 -4 -6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 0 B 2 0 4 4 4 C -2 -4 0 6 -2 D 2 -4 -6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998641 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5100: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) E D A C B (5) D A E B C (5) C B E D A (5) B A C D E (5) E C B D A (4) E C B A D (4) E A D B C (4) D A B C E (4) B C A E D (4) D E A C B (3) C E B D A (3) C E B A D (3) B C A D E (3) B A D C E (3) A D B C E (3) A B D C E (3) D E C A B (2) D A E C B (2) D A C E B (2) D A C B E (2) D A B E C (2) B C E A D (2) A D B E C (2) E C D B A (1) E C D A B (1) E B A C D (1) D C A E B (1) C E D B A (1) C D E B A (1) C B D A E (1) C B A E D (1) C B A D E (1) B E C A D (1) B E A C D (1) B C D A E (1) A D E B C (1) Total count = 100 A B C D E A 0 -16 -4 8 -8 B 16 0 -10 16 12 C 4 10 0 12 22 D -8 -16 -12 0 -6 E 8 -12 -22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 8 -8 B 16 0 -10 16 12 C 4 10 0 12 22 D -8 -16 -12 0 -6 E 8 -12 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 E=20 B=20 A=9 so A is eliminated. Round 2 votes counts: D=29 C=28 B=23 E=20 so E is eliminated. Round 3 votes counts: D=38 C=38 B=24 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:217 A:190 E:190 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -4 8 -8 B 16 0 -10 16 12 C 4 10 0 12 22 D -8 -16 -12 0 -6 E 8 -12 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 8 -8 B 16 0 -10 16 12 C 4 10 0 12 22 D -8 -16 -12 0 -6 E 8 -12 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 8 -8 B 16 0 -10 16 12 C 4 10 0 12 22 D -8 -16 -12 0 -6 E 8 -12 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5101: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) A B D C E (7) D C E B A (6) C D E B A (6) C D E A B (6) A B E C D (6) E C D B A (4) A B C E D (4) A B C D E (4) E A B C D (3) B A D E C (3) B A D C E (3) E D B C A (2) E B A D C (2) D E B C A (2) D C A B E (2) C E D B A (2) C E D A B (2) B E A D C (2) B A E D C (2) A B E D C (2) A B D E C (2) E D B A C (1) E C A B D (1) E B D A C (1) E B A C D (1) E A C B D (1) D E C B A (1) D C B E A (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A C E (1) C E A D B (1) C D A B E (1) C A E B D (1) C A D B E (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -2 -2 -12 B 6 0 4 -4 -8 C 2 -4 0 -6 2 D 2 4 6 0 0 E 12 8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.635898 E: 0.364102 Sum of squares = 0.53693653924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.635898 E: 1.000000 A B C D E A 0 -6 -2 -2 -12 B 6 0 4 -4 -8 C 2 -4 0 -6 2 D 2 4 6 0 0 E 12 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=25 C=20 D=17 B=10 so B is eliminated. Round 2 votes counts: A=36 E=27 C=20 D=17 so D is eliminated. Round 3 votes counts: A=37 E=32 C=31 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:209 D:206 B:199 C:197 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 -2 -12 B 6 0 4 -4 -8 C 2 -4 0 -6 2 D 2 4 6 0 0 E 12 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -2 -12 B 6 0 4 -4 -8 C 2 -4 0 -6 2 D 2 4 6 0 0 E 12 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -2 -12 B 6 0 4 -4 -8 C 2 -4 0 -6 2 D 2 4 6 0 0 E 12 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5102: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (14) D C A B E (9) B C A D E (7) E D B C A (6) B A C E D (6) A C B D E (6) E B D C A (5) E D C A B (4) D E C B A (4) D E C A B (4) A C D B E (4) D E A C B (3) E D B A C (2) D C B A E (2) D C A E B (2) D B E C A (2) D A C E B (2) B C D A E (2) B C A E D (2) E D A C B (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D C B (1) C D B A E (1) C B A D E (1) C A B D E (1) B E C A D (1) B D E C A (1) B A E C D (1) B A C D E (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -20 -8 -4 -2 B 20 0 6 0 -4 C 8 -6 0 -4 -4 D 4 0 4 0 8 E 2 4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.337970 C: 0.000000 D: 0.662030 E: 0.000000 Sum of squares = 0.552507419094 Cumulative probabilities = A: 0.000000 B: 0.337970 C: 0.337970 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 -4 -2 B 20 0 6 0 -4 C 8 -6 0 -4 -4 D 4 0 4 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=28 B=21 A=12 C=3 so C is eliminated. Round 2 votes counts: E=36 D=29 B=22 A=13 so A is eliminated. Round 3 votes counts: E=36 D=35 B=29 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:211 D:208 E:201 C:197 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -8 -4 -2 B 20 0 6 0 -4 C 8 -6 0 -4 -4 D 4 0 4 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -4 -2 B 20 0 6 0 -4 C 8 -6 0 -4 -4 D 4 0 4 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -4 -2 B 20 0 6 0 -4 C 8 -6 0 -4 -4 D 4 0 4 0 8 E 2 4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5103: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (16) A D B C E (14) C E A D B (13) B D A E C (13) D B A E C (5) C E B D A (5) B D E A C (5) A C E D B (4) A C D B E (4) E C A D B (3) E B D C A (3) C E A B D (3) B D E C A (3) C A E D B (2) A D B E C (2) E B D A C (1) E B C D A (1) B D A C E (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 2 -6 -6 B 6 0 -4 2 -4 C -2 4 0 4 -4 D 6 -2 -4 0 -2 E 6 4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 2 -6 -6 B 6 0 -4 2 -4 C -2 4 0 4 -4 D 6 -2 -4 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 C=23 B=22 D=5 so D is eliminated. Round 2 votes counts: B=27 A=26 E=24 C=23 so C is eliminated. Round 3 votes counts: E=45 A=28 B=27 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 C:201 B:200 D:199 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 2 -6 -6 B 6 0 -4 2 -4 C -2 4 0 4 -4 D 6 -2 -4 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -6 -6 B 6 0 -4 2 -4 C -2 4 0 4 -4 D 6 -2 -4 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -6 -6 B 6 0 -4 2 -4 C -2 4 0 4 -4 D 6 -2 -4 0 -2 E 6 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5104: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) B D E C A (7) C A E B D (5) B C A D E (5) A C B E D (5) E D C B A (4) D E A B C (4) C B A E D (4) C A B E D (4) A C E B D (4) E D B C A (3) E C D B A (3) D E B C A (3) D B E A C (3) C A E D B (3) B C E D A (3) B C D A E (3) B A C D E (3) A E D C B (3) E D A C B (2) E C A D B (2) D B A E C (2) C E A B D (2) A C B D E (2) E D C A B (1) E A D C B (1) D E B A C (1) D B E C A (1) C E A D B (1) B D E A C (1) B D C A E (1) B A D C E (1) A D E B C (1) A D C B E (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -10 16 16 B -4 0 -10 4 -2 C 10 10 0 16 14 D -16 -4 -16 0 -16 E -16 2 -14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 16 16 B -4 0 -10 4 -2 C 10 10 0 16 14 D -16 -4 -16 0 -16 E -16 2 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=24 C=19 E=16 D=14 so D is eliminated. Round 2 votes counts: B=30 A=27 E=24 C=19 so C is eliminated. Round 3 votes counts: A=39 B=34 E=27 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:225 A:213 B:194 E:194 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 16 16 B -4 0 -10 4 -2 C 10 10 0 16 14 D -16 -4 -16 0 -16 E -16 2 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 16 16 B -4 0 -10 4 -2 C 10 10 0 16 14 D -16 -4 -16 0 -16 E -16 2 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 16 16 B -4 0 -10 4 -2 C 10 10 0 16 14 D -16 -4 -16 0 -16 E -16 2 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5105: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (5) C A B D E (5) A E D B C (5) A C D B E (5) C B E A D (4) C B A E D (4) C B A D E (4) C A B E D (4) B C D E A (4) A E D C B (4) A D E B C (4) A C D E B (4) D E B A C (3) C B D E A (3) A D E C B (3) A C E D B (3) A C E B D (3) E A D B C (2) E A B D C (2) D E B C A (2) D B E C A (2) C B E D A (2) C B D A E (2) C A D B E (2) B E D C A (2) B E C D A (2) B D E C A (2) B C E D A (2) A C B E D (2) E D B C A (1) E B D C A (1) D E A B C (1) D C A B E (1) D A E B C (1) C D B E A (1) C D B A E (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 6 -2 24 18 B -6 0 -16 -2 8 C 2 16 0 18 12 D -24 2 -18 0 0 E -18 -8 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 24 18 B -6 0 -16 -2 8 C 2 16 0 18 12 D -24 2 -18 0 0 E -18 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=32 B=12 E=11 D=10 so D is eliminated. Round 2 votes counts: A=36 C=33 E=17 B=14 so B is eliminated. Round 3 votes counts: C=39 A=36 E=25 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:223 B:192 E:181 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 24 18 B -6 0 -16 -2 8 C 2 16 0 18 12 D -24 2 -18 0 0 E -18 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 24 18 B -6 0 -16 -2 8 C 2 16 0 18 12 D -24 2 -18 0 0 E -18 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 24 18 B -6 0 -16 -2 8 C 2 16 0 18 12 D -24 2 -18 0 0 E -18 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5106: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) E A D B C (8) C D B A E (8) E D A B C (7) C D E B A (7) A B E C D (6) D C E B A (5) C B D A E (5) B A E C D (5) B A E D C (4) B A C E D (4) C D B E A (3) B C A E D (3) B A C D E (3) A B E D C (3) E D A C B (2) D E C A B (2) D E A B C (2) D C E A B (2) C B A D E (2) D E C B A (1) D E A C B (1) D C B E A (1) D B A E C (1) C E D A B (1) C B A E D (1) C A B E D (1) B C A D E (1) B A D E C (1) A E B D C (1) Total count = 100 A B C D E A 0 -10 14 4 -2 B 10 0 16 -2 4 C -14 -16 0 0 -6 D -4 2 0 0 -10 E 2 -4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000033 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 A B C D E A 0 -10 14 4 -2 B 10 0 16 -2 4 C -14 -16 0 0 -6 D -4 2 0 0 -10 E 2 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468749999995 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=21 D=15 A=10 so A is eliminated. Round 2 votes counts: B=30 C=28 E=27 D=15 so D is eliminated. Round 3 votes counts: C=36 E=33 B=31 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:207 A:203 D:194 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 4 -2 B 10 0 16 -2 4 C -14 -16 0 0 -6 D -4 2 0 0 -10 E 2 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468749999995 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 4 -2 B 10 0 16 -2 4 C -14 -16 0 0 -6 D -4 2 0 0 -10 E 2 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468749999995 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 4 -2 B 10 0 16 -2 4 C -14 -16 0 0 -6 D -4 2 0 0 -10 E 2 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.000000 D: 0.250000 E: 0.125000 Sum of squares = 0.468749999995 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.625000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5107: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) D B E C A (7) E B A D C (4) D B C E A (4) C A B E D (4) B E D A C (4) B E A C D (4) A E C B D (4) A C E D B (4) E B A C D (3) E A B C D (3) D C B E A (3) C A D B E (3) C A B D E (3) B D E C A (3) A E C D B (3) A C E B D (3) D E B C A (2) D E B A C (2) D A E C B (2) C D B A E (2) C A D E B (2) B E D C A (2) B D E A C (2) E D B A C (1) E B D A C (1) E A D C B (1) E A D B C (1) D E A C B (1) D E A B C (1) D C B A E (1) D C A B E (1) C D A E B (1) C B A E D (1) C B A D E (1) B E C A D (1) B E A D C (1) B D C E A (1) B C A E D (1) B A E C D (1) A E B C D (1) A D E C B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 0 4 -4 B 4 0 0 -2 2 C 0 0 0 -6 -12 D -4 2 6 0 2 E 4 -2 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999961 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 4 -4 B 4 0 0 -2 2 C 0 0 0 -6 -12 D -4 2 6 0 2 E 4 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.35999999987 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=20 A=18 C=17 E=14 so E is eliminated. Round 2 votes counts: D=32 B=28 A=23 C=17 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:203 B:202 A:198 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 4 -4 B 4 0 0 -2 2 C 0 0 0 -6 -12 D -4 2 6 0 2 E 4 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.35999999987 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 4 -4 B 4 0 0 -2 2 C 0 0 0 -6 -12 D -4 2 6 0 2 E 4 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.35999999987 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 4 -4 B 4 0 0 -2 2 C 0 0 0 -6 -12 D -4 2 6 0 2 E 4 -2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.35999999987 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5108: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) E D C A B (5) E A D B C (4) E C B D A (3) E B C A D (3) E B A C D (3) D A E C B (3) D A C B E (3) C D E B A (3) C D B A E (3) C D A B E (3) C B A D E (3) B E C A D (3) B E A C D (3) B A C D E (3) E D C B A (2) E D B A C (2) E D A C B (2) E D A B C (2) E B C D A (2) E A B D C (2) D E C A B (2) D C E A B (2) D C A E B (2) D C A B E (2) C B E D A (2) B C A D E (2) B A E C D (2) B A C E D (2) A E D B C (2) A E B D C (2) A C D B E (2) A C B D E (2) A B D E C (2) E B D C A (1) E B A D C (1) D E C B A (1) D E A C B (1) D A C E B (1) C A D B E (1) B C E A D (1) A D C B E (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -6 -12 B 2 0 -10 -12 -14 C 4 10 0 6 -18 D 6 12 -6 0 -10 E 12 14 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -6 -12 B 2 0 -10 -12 -14 C 4 10 0 6 -18 D 6 12 -6 0 -10 E 12 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=17 B=16 C=15 A=14 so A is eliminated. Round 2 votes counts: E=42 B=20 D=19 C=19 so D is eliminated. Round 3 votes counts: E=49 C=30 B=21 so B is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:227 C:201 D:201 A:188 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 -12 B 2 0 -10 -12 -14 C 4 10 0 6 -18 D 6 12 -6 0 -10 E 12 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 -12 B 2 0 -10 -12 -14 C 4 10 0 6 -18 D 6 12 -6 0 -10 E 12 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 -12 B 2 0 -10 -12 -14 C 4 10 0 6 -18 D 6 12 -6 0 -10 E 12 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5109: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) D E B A C (10) D C E B A (8) D E C B A (7) D A B E C (7) E B A D C (5) A C B E D (4) E B A C D (3) D E B C A (3) D C A B E (3) D A C B E (3) C E B A D (3) E D B C A (2) E D B A C (2) E B D A C (2) E B C A D (2) D C E A B (2) C D E A B (2) C D A E B (2) C D A B E (2) A B D E C (2) A B C E D (2) E C B D A (1) E C B A D (1) C E B D A (1) C B A E D (1) C A D B E (1) C A B D E (1) B E A C D (1) B A E C D (1) A D B C E (1) A C D B E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -6 -14 -10 B 6 0 -8 -12 -12 C 6 8 0 -16 -2 D 14 12 16 0 12 E 10 12 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -14 -10 B 6 0 -8 -12 -12 C 6 8 0 -16 -2 D 14 12 16 0 12 E 10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 C=24 E=18 A=13 B=2 so B is eliminated. Round 2 votes counts: D=43 C=24 E=19 A=14 so A is eliminated. Round 3 votes counts: D=46 C=32 E=22 so E is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:206 C:198 B:187 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -14 -10 B 6 0 -8 -12 -12 C 6 8 0 -16 -2 D 14 12 16 0 12 E 10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -14 -10 B 6 0 -8 -12 -12 C 6 8 0 -16 -2 D 14 12 16 0 12 E 10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -14 -10 B 6 0 -8 -12 -12 C 6 8 0 -16 -2 D 14 12 16 0 12 E 10 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5110: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) B E D A C (8) E B A C D (7) E B C A D (6) B E A C D (6) A C D E B (6) D B A C E (5) C A D E B (5) E C A B D (4) D C A E B (4) D A C B E (4) C E A D B (4) C D A E B (4) E C A D B (3) C A E D B (3) A C E D B (3) D B C A E (2) B E D C A (2) B E A D C (2) B D A E C (2) A D C B E (2) E D C B A (1) E A C B D (1) E A B C D (1) D C E B A (1) D A B C E (1) C E D A B (1) B D E A C (1) B D A C E (1) Total count = 100 A B C D E A 0 12 0 6 4 B -12 0 -12 -18 -8 C 0 12 0 8 12 D -6 18 -8 0 -4 E -4 8 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.763447 B: 0.000000 C: 0.236553 D: 0.000000 E: 0.000000 Sum of squares = 0.638809098771 Cumulative probabilities = A: 0.763447 B: 0.763447 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 6 4 B -12 0 -12 -18 -8 C 0 12 0 8 12 D -6 18 -8 0 -4 E -4 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=23 B=22 C=17 A=11 so A is eliminated. Round 2 votes counts: D=29 C=26 E=23 B=22 so B is eliminated. Round 3 votes counts: E=41 D=33 C=26 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:216 A:211 D:200 E:198 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 6 4 B -12 0 -12 -18 -8 C 0 12 0 8 12 D -6 18 -8 0 -4 E -4 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 6 4 B -12 0 -12 -18 -8 C 0 12 0 8 12 D -6 18 -8 0 -4 E -4 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 6 4 B -12 0 -12 -18 -8 C 0 12 0 8 12 D -6 18 -8 0 -4 E -4 8 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5111: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) E C A D B (8) B A D E C (8) E B A C D (6) C E D A B (6) C D B A E (6) E B A D C (4) D A B C E (4) B E A D C (4) E C A B D (3) D C A B E (3) D A C B E (3) C D A E B (3) B D A C E (3) A B D E C (3) E B C D A (2) B A E D C (2) E C D A B (1) E C B A D (1) E B C A D (1) E A D B C (1) E A C D B (1) E A B D C (1) D C B A E (1) C E D B A (1) C E B D A (1) C B E D A (1) C B D E A (1) C A D E B (1) B E D A C (1) B E C D A (1) B C D E A (1) B A D C E (1) A E D B C (1) A E B D C (1) A D E C B (1) A D B E C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -4 2 8 B -8 0 -6 -8 10 C 4 6 0 12 -4 D -2 8 -12 0 4 E -8 -10 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 8 -4 2 8 B -8 0 -6 -8 10 C 4 6 0 12 -4 D -2 8 -12 0 4 E -8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=29 B=21 D=11 A=9 so A is eliminated. Round 2 votes counts: E=31 C=31 B=24 D=14 so D is eliminated. Round 3 votes counts: C=38 E=32 B=30 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:209 A:207 D:199 B:194 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -4 2 8 B -8 0 -6 -8 10 C 4 6 0 12 -4 D -2 8 -12 0 4 E -8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 2 8 B -8 0 -6 -8 10 C 4 6 0 12 -4 D -2 8 -12 0 4 E -8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 2 8 B -8 0 -6 -8 10 C 4 6 0 12 -4 D -2 8 -12 0 4 E -8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999964 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5112: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) C A B E D (11) D E B A C (9) C B E A D (9) B E C A D (9) B E C D A (8) A C D B E (7) E B D C A (6) D E A B C (4) D A E B C (4) C A D B E (4) D A E C B (3) A D C E B (3) E B D A C (2) B C E A D (2) A D C B E (2) E B C D A (1) D E B C A (1) C B A E D (1) B E D A C (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -4 0 -4 B -2 0 -6 2 10 C 4 6 0 8 4 D 0 -2 -8 0 -2 E 4 -10 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 0 -4 B -2 0 -6 2 10 C 4 6 0 8 4 D 0 -2 -8 0 -2 E 4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=25 B=20 A=14 E=9 so E is eliminated. Round 2 votes counts: D=32 B=29 C=25 A=14 so A is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:202 A:197 E:196 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 0 -4 B -2 0 -6 2 10 C 4 6 0 8 4 D 0 -2 -8 0 -2 E 4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 -4 B -2 0 -6 2 10 C 4 6 0 8 4 D 0 -2 -8 0 -2 E 4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 -4 B -2 0 -6 2 10 C 4 6 0 8 4 D 0 -2 -8 0 -2 E 4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5113: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) B D C A E (10) E C A D B (8) D A B E C (7) A E D B C (6) E A C D B (5) D B A E C (5) C E A B D (5) C B E D A (5) B D A C E (5) B C D A E (5) E C A B D (4) D B A C E (4) A D E B C (4) A D B E C (3) C E B A D (2) C B D E A (2) A E D C B (2) E C D A B (1) E A D C B (1) D B E C A (1) D A E B C (1) C E D B A (1) B C D E A (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 -10 -16 8 B 2 0 4 0 -4 C 10 -4 0 2 0 D 16 0 -2 0 -4 E -8 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.480453 D: 0.000000 E: 0.519547 Sum of squares = 0.500764138197 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.480453 D: 0.480453 E: 1.000000 A B C D E A 0 -2 -10 -16 8 B 2 0 4 0 -4 C 10 -4 0 2 0 D 16 0 -2 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.000000 E: 0.502870 Sum of squares = 0.500016470954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.497130 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=21 E=19 D=18 A=17 so A is eliminated. Round 2 votes counts: E=29 D=25 C=25 B=21 so B is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:205 C:204 B:201 E:200 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -10 -16 8 B 2 0 4 0 -4 C 10 -4 0 2 0 D 16 0 -2 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.000000 E: 0.502870 Sum of squares = 0.500016470954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.497130 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -16 8 B 2 0 4 0 -4 C 10 -4 0 2 0 D 16 0 -2 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.000000 E: 0.502870 Sum of squares = 0.500016470954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.497130 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -16 8 B 2 0 4 0 -4 C 10 -4 0 2 0 D 16 0 -2 0 -4 E -8 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.000000 E: 0.502870 Sum of squares = 0.500016470954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.497130 D: 0.497130 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5114: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) E C D A B (8) D C B E A (7) B A D C E (7) B D C A E (6) A B E C D (6) D B C E A (5) C D E B A (5) A B E D C (5) E C A D B (4) E A C D B (4) B D C E A (4) E C D B A (3) D C E B A (3) A E B C D (3) E C B D A (2) E A C B D (2) E A B C D (2) C E D B A (2) B D A C E (2) B A E D C (2) A E C D B (2) A B D E C (2) D C B A E (1) D A C B E (1) B E A C D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 0 0 2 -4 B 0 0 8 6 18 C 0 -8 0 -12 8 D -2 -6 12 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.535504 B: 0.464496 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50252113635 Cumulative probabilities = A: 0.535504 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 2 -4 B 0 0 8 6 18 C 0 -8 0 -12 8 D -2 -6 12 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=25 B=22 D=17 C=7 so C is eliminated. Round 2 votes counts: A=29 E=27 D=22 B=22 so D is eliminated. Round 3 votes counts: E=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:206 A:199 C:194 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 2 -4 B 0 0 8 6 18 C 0 -8 0 -12 8 D -2 -6 12 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -4 B 0 0 8 6 18 C 0 -8 0 -12 8 D -2 -6 12 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -4 B 0 0 8 6 18 C 0 -8 0 -12 8 D -2 -6 12 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5115: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) B D C E A (9) D A C E B (5) B C E D A (5) A E B C D (5) D C E B A (4) D B A C E (4) A D B E C (4) A D B C E (4) D B C E A (3) D A B C E (3) C E B D A (3) B E C D A (3) A E C D B (3) A E C B D (3) A D E C B (3) A D E B C (3) E C B D A (2) E A C B D (2) D C E A B (2) D A C B E (2) C B E D A (2) B A E C D (2) E C B A D (1) E C A B D (1) E B C A D (1) E A C D B (1) E A B C D (1) D C A B E (1) C D E B A (1) B E C A D (1) B E A C D (1) A E D C B (1) A D C E B (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 -18 -6 B 4 0 4 -10 14 C -2 -4 0 -24 20 D 18 10 24 0 22 E 6 -14 -20 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -18 -6 B 4 0 4 -10 14 C -2 -4 0 -24 20 D 18 10 24 0 22 E 6 -14 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=30 B=21 E=9 C=6 so C is eliminated. Round 2 votes counts: D=35 A=30 B=23 E=12 so E is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:237 B:206 C:195 A:187 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -18 -6 B 4 0 4 -10 14 C -2 -4 0 -24 20 D 18 10 24 0 22 E 6 -14 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -18 -6 B 4 0 4 -10 14 C -2 -4 0 -24 20 D 18 10 24 0 22 E 6 -14 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -18 -6 B 4 0 4 -10 14 C -2 -4 0 -24 20 D 18 10 24 0 22 E 6 -14 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5116: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) D E C B A (6) D B A E C (6) B A D C E (6) D E B C A (5) C E D A B (5) B D A E C (5) A B C D E (5) E C A B D (4) D B A C E (4) E C D A B (3) D E B A C (3) D C E A B (3) D B E A C (3) C A E B D (3) C A B E D (3) B A E C D (3) A C B E D (3) E D B C A (2) A B D C E (2) A B C E D (2) E D B A C (1) E C D B A (1) E C B A D (1) E B D A C (1) E B C A D (1) E B A C D (1) D C E B A (1) D B C E A (1) D A B C E (1) C E A D B (1) C E A B D (1) C D E A B (1) C A B D E (1) B D A C E (1) B A D E C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 0 -20 -4 B 20 0 8 -12 -2 C 0 -8 0 -20 -8 D 20 12 20 0 14 E 4 2 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 0 -20 -4 B 20 0 8 -12 -2 C 0 -8 0 -20 -8 D 20 12 20 0 14 E 4 2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=22 B=16 C=15 A=14 so A is eliminated. Round 2 votes counts: D=34 B=25 E=22 C=19 so C is eliminated. Round 3 votes counts: D=36 E=32 B=32 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:233 B:207 E:200 C:182 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 0 -20 -4 B 20 0 8 -12 -2 C 0 -8 0 -20 -8 D 20 12 20 0 14 E 4 2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 0 -20 -4 B 20 0 8 -12 -2 C 0 -8 0 -20 -8 D 20 12 20 0 14 E 4 2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 0 -20 -4 B 20 0 8 -12 -2 C 0 -8 0 -20 -8 D 20 12 20 0 14 E 4 2 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5117: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (12) E D A B C (9) C B D A E (9) A C B D E (7) D B C E A (6) A E B C D (6) E D B C A (5) D C B E A (5) A B C D E (5) E D C B A (4) D E B C A (4) A E C B D (4) C A B D E (3) A C E B D (3) E D B A C (2) D B E C A (2) C B A D E (2) B D C A E (2) A C B E D (2) E D C A B (1) D E C B A (1) D E B A C (1) D C E B A (1) D B C A E (1) C D E B A (1) B C D A E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 -10 -8 B -6 0 14 -10 -8 C -4 -14 0 -14 -2 D 10 10 14 0 4 E 8 8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -10 -8 B -6 0 14 -10 -8 C -4 -14 0 -14 -2 D 10 10 14 0 4 E 8 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=28 D=21 C=15 B=3 so B is eliminated. Round 2 votes counts: E=33 A=28 D=23 C=16 so C is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:207 A:196 B:195 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -10 -8 B -6 0 14 -10 -8 C -4 -14 0 -14 -2 D 10 10 14 0 4 E 8 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -10 -8 B -6 0 14 -10 -8 C -4 -14 0 -14 -2 D 10 10 14 0 4 E 8 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -10 -8 B -6 0 14 -10 -8 C -4 -14 0 -14 -2 D 10 10 14 0 4 E 8 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5118: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) D C E B A (6) B A E D C (6) E A B C D (5) A B E C D (5) E B A D C (4) C D A E B (4) B E A D C (4) B A E C D (4) A E B C D (4) A C E B D (4) D C B E A (3) C A D B E (3) A E C B D (3) E D C B A (2) E D B C A (2) E B D A C (2) E B A C D (2) D C E A B (2) D C B A E (2) D C A E B (2) C E A D B (2) C D E A B (2) C A D E B (2) B E D A C (2) B D E A C (2) A B C E D (2) E A C B D (1) D E C B A (1) D C A B E (1) D B C E A (1) D B C A E (1) C E D A B (1) C A E D B (1) B D A E C (1) A C E D B (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 12 12 B -10 0 -4 6 -6 C -10 4 0 12 0 D -12 -6 -12 0 -14 E -12 6 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 12 12 B -10 0 -4 6 -6 C -10 4 0 12 0 D -12 -6 -12 0 -14 E -12 6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=22 A=22 D=19 B=19 E=18 so E is eliminated. Round 2 votes counts: A=28 B=27 D=23 C=22 so C is eliminated. Round 3 votes counts: D=37 A=36 B=27 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:204 C:203 B:193 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 12 12 B -10 0 -4 6 -6 C -10 4 0 12 0 D -12 -6 -12 0 -14 E -12 6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 12 12 B -10 0 -4 6 -6 C -10 4 0 12 0 D -12 -6 -12 0 -14 E -12 6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 12 12 B -10 0 -4 6 -6 C -10 4 0 12 0 D -12 -6 -12 0 -14 E -12 6 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5119: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C A D B E (6) E A B D C (5) C A D E B (5) B E D A C (5) B D E A C (5) E A D B C (4) D B E A C (4) B E D C A (4) B D C E A (4) C B E A D (3) C B D A E (3) C B A D E (3) B D E C A (3) A E D B C (3) A C E D B (3) E D B A C (2) E B C A D (2) D B E C A (2) D A E B C (2) C B E D A (2) B C D E A (2) A E D C B (2) A E C D B (2) E D A B C (1) E C A B D (1) E B D C A (1) E A C B D (1) D E B A C (1) D E A B C (1) D B C E A (1) C B A E D (1) C A E D B (1) C A E B D (1) C A B E D (1) C A B D E (1) B E C D A (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -18 2 -8 -30 B 18 0 26 18 4 C -2 -26 0 -20 -22 D 8 -18 20 0 -12 E 30 -4 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 2 -8 -30 B 18 0 26 18 4 C -2 -26 0 -20 -22 D 8 -18 20 0 -12 E 30 -4 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 B=25 D=11 A=11 so D is eliminated. Round 2 votes counts: B=32 E=28 C=27 A=13 so A is eliminated. Round 3 votes counts: E=37 B=32 C=31 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:230 D:199 A:173 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 2 -8 -30 B 18 0 26 18 4 C -2 -26 0 -20 -22 D 8 -18 20 0 -12 E 30 -4 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 2 -8 -30 B 18 0 26 18 4 C -2 -26 0 -20 -22 D 8 -18 20 0 -12 E 30 -4 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 2 -8 -30 B 18 0 26 18 4 C -2 -26 0 -20 -22 D 8 -18 20 0 -12 E 30 -4 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5120: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (18) A C E B D (11) E B D C A (8) E B D A C (6) D B E C A (6) C A D B E (6) A E B D C (6) C D B E A (5) D C B E A (4) E B A D C (3) E B C D A (2) E A B D C (2) C D B A E (2) C D A B E (2) B D E C A (2) A D B E C (2) A C E D B (2) A C D E B (2) E C B D A (1) D C A B E (1) D B C E A (1) C E D B A (1) C E A B D (1) C A E D B (1) B E D C A (1) A E C B D (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 12 16 14 B -16 0 -16 -6 4 C -12 16 0 12 14 D -16 6 -12 0 4 E -14 -4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 16 14 B -16 0 -16 -6 4 C -12 16 0 12 14 D -16 6 -12 0 4 E -14 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 E=22 C=18 D=12 B=3 so B is eliminated. Round 2 votes counts: A=45 E=23 C=18 D=14 so D is eliminated. Round 3 votes counts: A=45 E=31 C=24 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:215 D:191 B:183 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 16 14 B -16 0 -16 -6 4 C -12 16 0 12 14 D -16 6 -12 0 4 E -14 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 16 14 B -16 0 -16 -6 4 C -12 16 0 12 14 D -16 6 -12 0 4 E -14 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 16 14 B -16 0 -16 -6 4 C -12 16 0 12 14 D -16 6 -12 0 4 E -14 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5121: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (19) D C E A B (16) B A E D C (8) A E B C D (8) D C B E A (6) B D C E A (5) D B C E A (4) C D E A B (4) B D A E C (4) E A C B D (3) D C E B A (3) A E C B D (3) D C A E B (2) C E A D B (2) B E A C D (2) A B E C D (2) E A B C D (1) D C B A E (1) C D A E B (1) C A E D B (1) B E A D C (1) B D C A E (1) B A D E C (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 8 6 6 B 10 0 12 16 8 C -8 -12 0 -4 -6 D -6 -16 4 0 -4 E -6 -8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 6 6 B 10 0 12 16 8 C -8 -12 0 -4 -6 D -6 -16 4 0 -4 E -6 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=32 A=15 C=8 E=4 so E is eliminated. Round 2 votes counts: B=41 D=32 A=19 C=8 so C is eliminated. Round 3 votes counts: B=41 D=37 A=22 so A is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:205 E:198 D:189 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 6 6 B 10 0 12 16 8 C -8 -12 0 -4 -6 D -6 -16 4 0 -4 E -6 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 6 6 B 10 0 12 16 8 C -8 -12 0 -4 -6 D -6 -16 4 0 -4 E -6 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 6 6 B 10 0 12 16 8 C -8 -12 0 -4 -6 D -6 -16 4 0 -4 E -6 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5122: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) D E C A B (10) E D C A B (8) B A C D E (7) B A E C D (5) D E C B A (4) D C E A B (4) A B C E D (4) D C B A E (3) C D E A B (3) C D A B E (3) A C B E D (3) E C D A B (2) D E B C A (2) D C B E A (2) C D A E B (2) B E A D C (2) B D E C A (2) B D C A E (2) B A E D C (2) B A D C E (2) E D B A C (1) E D A C B (1) E C A D B (1) E A B D C (1) D C E B A (1) C B D A E (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E A C (1) B D C E A (1) B D A C E (1) A E C D B (1) A E C B D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -6 -10 6 B 4 0 -6 2 12 C 6 6 0 -2 8 D 10 -2 2 0 4 E -6 -12 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000052 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -10 6 B 4 0 -6 2 12 C 6 6 0 -2 8 D 10 -2 2 0 4 E -6 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=26 E=14 C=11 A=11 so C is eliminated. Round 2 votes counts: B=39 D=34 E=14 A=13 so A is eliminated. Round 3 votes counts: B=48 D=34 E=18 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:209 D:207 B:206 A:193 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -10 6 B 4 0 -6 2 12 C 6 6 0 -2 8 D 10 -2 2 0 4 E -6 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -10 6 B 4 0 -6 2 12 C 6 6 0 -2 8 D 10 -2 2 0 4 E -6 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -10 6 B 4 0 -6 2 12 C 6 6 0 -2 8 D 10 -2 2 0 4 E -6 -12 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.43999999952 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5123: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) A E B D C (9) C D B E A (8) E B A C D (7) D A E B C (7) B E A C D (7) A E D B C (7) D C B A E (6) B C E A D (6) D C A B E (5) B E C A D (5) E A B C D (4) D C A E B (3) C B D E A (3) A E B C D (3) D A C E B (2) C B E D A (2) A D E B C (2) D A E C B (1) C D A B E (1) C B E A D (1) C B A E D (1) Total count = 100 A B C D E A 0 -12 -2 4 -6 B 12 0 14 -4 10 C 2 -14 0 -4 -4 D -4 4 4 0 -4 E 6 -10 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.222222 Sum of squares = 0.407407407382 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.777778 E: 1.000000 A B C D E A 0 -12 -2 4 -6 B 12 0 14 -4 10 C 2 -14 0 -4 -4 D -4 4 4 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.222222 Sum of squares = 0.407407407385 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=21 B=18 C=16 E=11 so E is eliminated. Round 2 votes counts: D=34 B=25 A=25 C=16 so C is eliminated. Round 3 votes counts: D=43 B=32 A=25 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:216 E:202 D:200 A:192 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 4 -6 B 12 0 14 -4 10 C 2 -14 0 -4 -4 D -4 4 4 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.222222 Sum of squares = 0.407407407385 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.777778 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 4 -6 B 12 0 14 -4 10 C 2 -14 0 -4 -4 D -4 4 4 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.222222 Sum of squares = 0.407407407385 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.777778 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 4 -6 B 12 0 14 -4 10 C 2 -14 0 -4 -4 D -4 4 4 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.555556 E: 0.222222 Sum of squares = 0.407407407385 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.777778 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5124: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) B A C E D (7) E D A B C (6) D E C A B (6) E D C B A (5) D E C B A (5) D C E A B (5) C A B D E (5) C B A D E (4) A B C E D (4) E D B C A (3) E A B D C (3) C D E B A (3) C D A B E (3) B A E D C (3) B A E C D (3) A B E D C (3) E B A D C (2) E A D B C (2) D C E B A (2) C D B E A (2) A B C D E (2) E D C A B (1) E B D C A (1) D E A C B (1) D A E C B (1) C E D B A (1) C D E A B (1) C D A E B (1) C B D A E (1) B E A D C (1) B E A C D (1) B C A E D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 0 -10 -16 B 4 0 2 -12 -12 C 0 -2 0 -16 -12 D 10 12 16 0 -10 E 16 12 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 -10 -16 B 4 0 2 -12 -12 C 0 -2 0 -16 -12 D 10 12 16 0 -10 E 16 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=21 D=20 B=16 A=13 so A is eliminated. Round 2 votes counts: E=30 B=27 C=23 D=20 so D is eliminated. Round 3 votes counts: E=43 C=30 B=27 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:214 B:191 A:185 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -10 -16 B 4 0 2 -12 -12 C 0 -2 0 -16 -12 D 10 12 16 0 -10 E 16 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -10 -16 B 4 0 2 -12 -12 C 0 -2 0 -16 -12 D 10 12 16 0 -10 E 16 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -10 -16 B 4 0 2 -12 -12 C 0 -2 0 -16 -12 D 10 12 16 0 -10 E 16 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5125: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) C B A E D (7) C B A D E (7) A B C E D (7) D E C B A (6) D C B E A (6) B C A D E (6) E A D B C (4) A E B C D (4) D E A B C (3) D C E B A (3) B C D A E (3) E D C B A (2) E D A B C (2) D E C A B (2) D A E B C (2) C D B E A (2) C B D A E (2) B A C D E (2) A B C D E (2) E D C A B (1) E C D A B (1) E C A B D (1) E A D C B (1) E A B D C (1) D E B C A (1) D E A C B (1) D B C A E (1) D B A C E (1) C E B A D (1) C D E B A (1) C B E D A (1) C B D E A (1) B C A E D (1) A E D B C (1) A E C B D (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -12 2 4 B 8 0 -8 0 4 C 12 8 0 0 10 D -2 0 0 0 10 E -4 -4 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.534618 D: 0.465382 E: 0.000000 Sum of squares = 0.502396812247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.534618 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 2 4 B 8 0 -8 0 4 C 12 8 0 0 10 D -2 0 0 0 10 E -4 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=22 E=20 A=20 B=12 so B is eliminated. Round 2 votes counts: C=32 D=26 A=22 E=20 so E is eliminated. Round 3 votes counts: D=38 C=34 A=28 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:204 B:202 A:193 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 2 4 B 8 0 -8 0 4 C 12 8 0 0 10 D -2 0 0 0 10 E -4 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 2 4 B 8 0 -8 0 4 C 12 8 0 0 10 D -2 0 0 0 10 E -4 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 2 4 B 8 0 -8 0 4 C 12 8 0 0 10 D -2 0 0 0 10 E -4 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5126: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) D A C E B (8) E B D A C (7) C B E A D (7) E B C A D (6) C B A E D (6) E D B A C (5) D A C B E (5) B E A D C (4) E B A D C (3) C A B D E (3) A D B C E (3) E B D C A (2) D E A B C (2) D A B E C (2) C E B D A (2) B E C A D (2) B E A C D (2) B C A E D (2) B A E D C (2) B A C E D (2) A D C B E (2) E D C B A (1) E C B D A (1) E B C D A (1) D C A B E (1) D A E C B (1) D A E B C (1) C E B A D (1) C D A E B (1) C D A B E (1) C B A D E (1) B A E C D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 4 18 8 B 16 0 -2 12 16 C -4 2 0 -2 12 D -18 -12 2 0 -14 E -8 -16 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.570247933606 Cumulative probabilities = A: 0.090909 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 18 8 B 16 0 -2 12 16 C -4 2 0 -2 12 D -18 -12 2 0 -14 E -8 -16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.570247929215 Cumulative probabilities = A: 0.090909 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=26 D=20 B=15 A=7 so A is eliminated. Round 2 votes counts: C=32 E=26 D=26 B=16 so B is eliminated. Round 3 votes counts: E=37 C=36 D=27 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:221 A:207 C:204 E:189 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 18 8 B 16 0 -2 12 16 C -4 2 0 -2 12 D -18 -12 2 0 -14 E -8 -16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.570247929215 Cumulative probabilities = A: 0.090909 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 18 8 B 16 0 -2 12 16 C -4 2 0 -2 12 D -18 -12 2 0 -14 E -8 -16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.570247929215 Cumulative probabilities = A: 0.090909 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 18 8 B 16 0 -2 12 16 C -4 2 0 -2 12 D -18 -12 2 0 -14 E -8 -16 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.570247929215 Cumulative probabilities = A: 0.090909 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5127: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (11) C D E B A (11) D C E A B (9) B A E C D (7) B A C D E (7) A B E D C (7) A B E C D (7) C D B E A (6) A E B D C (6) E A D B C (4) D E C A B (4) B A C E D (4) E D A C B (3) C D B A E (3) B C D A E (3) C B D A E (2) B C A D E (2) E A D C B (1) E A B D C (1) C D E A B (1) B A E D C (1) Total count = 100 A B C D E A 0 8 -4 -6 -2 B -8 0 -2 -6 -2 C 4 2 0 6 -4 D 6 6 -6 0 -4 E 2 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -4 -6 -2 B -8 0 -2 -6 -2 C 4 2 0 6 -4 D 6 6 -6 0 -4 E 2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 E=20 A=20 D=13 so D is eliminated. Round 2 votes counts: C=32 E=24 B=24 A=20 so A is eliminated. Round 3 votes counts: B=38 C=32 E=30 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:206 C:204 D:201 A:198 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -4 -6 -2 B -8 0 -2 -6 -2 C 4 2 0 6 -4 D 6 6 -6 0 -4 E 2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -6 -2 B -8 0 -2 -6 -2 C 4 2 0 6 -4 D 6 6 -6 0 -4 E 2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -6 -2 B -8 0 -2 -6 -2 C 4 2 0 6 -4 D 6 6 -6 0 -4 E 2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5128: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) C D B E A (8) B E A D C (7) D C A B E (6) D C B A E (5) D C A E B (5) E A C B D (4) C B D E A (4) C A D E B (3) B E A C D (3) E B A D C (2) E A B D C (2) E A B C D (2) D B C A E (2) C E B A D (2) C D B A E (2) C B E D A (2) B E C A D (2) B D A E C (2) A E D B C (2) A E B D C (2) A B D E C (2) E C A B D (1) E B C A D (1) E B A C D (1) D B C E A (1) D B A E C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E A B D (1) C D E A B (1) C D A B E (1) C A E D B (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D E A (1) A E D C B (1) A D E C B (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -24 -14 4 B -2 0 -20 -12 10 C 24 20 0 2 18 D 14 12 -2 0 26 E -4 -10 -18 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -24 -14 4 B -2 0 -20 -12 10 C 24 20 0 2 18 D 14 12 -2 0 26 E -4 -10 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=23 B=18 E=13 A=11 so A is eliminated. Round 2 votes counts: C=35 D=27 B=20 E=18 so E is eliminated. Round 3 votes counts: C=40 D=30 B=30 so D is eliminated. Round 4 votes counts: C=60 B=40 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 D:225 B:188 A:184 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -24 -14 4 B -2 0 -20 -12 10 C 24 20 0 2 18 D 14 12 -2 0 26 E -4 -10 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -24 -14 4 B -2 0 -20 -12 10 C 24 20 0 2 18 D 14 12 -2 0 26 E -4 -10 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -24 -14 4 B -2 0 -20 -12 10 C 24 20 0 2 18 D 14 12 -2 0 26 E -4 -10 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5129: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) D C B A E (5) C A E B D (5) B D A E C (5) E A C B D (4) D B E A C (4) A E B C D (4) E A B C D (3) D C B E A (3) D B C E A (3) B A E D C (3) B A D E C (3) A B E C D (3) E D C A B (2) E C D A B (2) E A D C B (2) D B E C A (2) C D E A B (2) C A B E D (2) B D E A C (2) B D C A E (2) B D A C E (2) B A D C E (2) B A C E D (2) A E C B D (2) A E B D C (2) A B E D C (2) E D B A C (1) E D A B C (1) E C A D B (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E B C A (1) D C E A B (1) D B C A E (1) C E A D B (1) C E A B D (1) C D B A E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E A D C (1) B A C D E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 10 2 4 B 4 0 10 14 8 C -10 -10 0 -22 -10 D -2 -14 22 0 0 E -4 -8 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 2 4 B 4 0 10 14 8 C -10 -10 0 -22 -10 D -2 -14 22 0 0 E -4 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 E=20 C=15 A=15 so C is eliminated. Round 2 votes counts: D=31 A=24 B=23 E=22 so E is eliminated. Round 3 votes counts: A=38 D=37 B=25 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:218 A:206 D:203 E:199 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 2 4 B 4 0 10 14 8 C -10 -10 0 -22 -10 D -2 -14 22 0 0 E -4 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 2 4 B 4 0 10 14 8 C -10 -10 0 -22 -10 D -2 -14 22 0 0 E -4 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 2 4 B 4 0 10 14 8 C -10 -10 0 -22 -10 D -2 -14 22 0 0 E -4 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5130: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (18) B E C D A (12) B E C A D (8) D C E A B (6) C E B D A (5) C E D B A (4) C D E A B (4) B E D C A (3) A C D E B (3) A B D E C (3) A B C E D (3) E C B D A (2) E B C D A (2) D A E C B (2) C E D A B (2) B D E A C (2) A D B E C (2) A D B C E (2) E D B C A (1) D E C B A (1) D C A E B (1) D A C E B (1) C E B A D (1) C B E A D (1) B E A C D (1) B D E C A (1) B C E A D (1) B A E D C (1) B A E C D (1) B A D E C (1) B A C E D (1) A D C B E (1) A C B E D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -10 2 -14 B -2 0 -6 4 -6 C 10 6 0 6 12 D -2 -4 -6 0 -2 E 14 6 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 2 -14 B -2 0 -6 4 -6 C 10 6 0 6 12 D -2 -4 -6 0 -2 E 14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=32 C=17 D=11 E=5 so E is eliminated. Round 2 votes counts: A=35 B=34 C=19 D=12 so D is eliminated. Round 3 votes counts: A=38 B=35 C=27 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 E:205 B:195 D:193 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 2 -14 B -2 0 -6 4 -6 C 10 6 0 6 12 D -2 -4 -6 0 -2 E 14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 2 -14 B -2 0 -6 4 -6 C 10 6 0 6 12 D -2 -4 -6 0 -2 E 14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 2 -14 B -2 0 -6 4 -6 C 10 6 0 6 12 D -2 -4 -6 0 -2 E 14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5131: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) D A B E C (7) A E B D C (7) C B D E A (6) C E B A D (4) C D B E A (4) C B E D A (4) A E D C B (4) A E B C D (4) A D E B C (4) E A C B D (3) E A B C D (3) D C E A B (3) D B C A E (3) D B A C E (3) D A E C B (3) D A E B C (3) A E D B C (3) E C B A D (2) E B A C D (2) D C B E A (2) D C A B E (2) D A B C E (2) C E B D A (2) C B E A D (2) A B D E C (2) E C A D B (1) E C A B D (1) E A D C B (1) D C E B A (1) D A C E B (1) C D E B A (1) B E A C D (1) B C E A D (1) B A E D C (1) Total count = 100 A B C D E A 0 8 8 -8 12 B -8 0 -8 -10 -6 C -8 8 0 -18 -4 D 8 10 18 0 8 E -12 6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 -8 12 B -8 0 -8 -10 -6 C -8 8 0 -18 -4 D 8 10 18 0 8 E -12 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=24 C=23 E=13 B=3 so B is eliminated. Round 2 votes counts: D=37 A=25 C=24 E=14 so E is eliminated. Round 3 votes counts: D=37 A=35 C=28 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:210 E:195 C:189 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 8 -8 12 B -8 0 -8 -10 -6 C -8 8 0 -18 -4 D 8 10 18 0 8 E -12 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -8 12 B -8 0 -8 -10 -6 C -8 8 0 -18 -4 D 8 10 18 0 8 E -12 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -8 12 B -8 0 -8 -10 -6 C -8 8 0 -18 -4 D 8 10 18 0 8 E -12 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5132: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) B D C A E (8) E A D C B (7) D E B C A (7) C B A D E (7) A E C B D (7) E D B A C (5) D B E C A (5) E D A B C (4) E A D B C (3) D E B A C (3) C A E B D (3) B C D A E (3) A C E B D (3) A C B E D (3) A C B D E (3) E C A D B (2) D B E A C (2) D B C E A (2) C B D A E (2) C A B E D (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E D A C B (1) E C D A B (1) D E C B A (1) C A B D E (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 4 6 6 -8 B -4 0 -8 -10 -18 C -6 8 0 -2 -20 D -6 10 2 0 -6 E 8 18 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 6 -8 B -4 0 -8 -10 -18 C -6 8 0 -2 -20 D -6 10 2 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=20 A=19 C=15 B=12 so B is eliminated. Round 2 votes counts: E=34 D=28 A=20 C=18 so C is eliminated. Round 3 votes counts: E=34 D=33 A=33 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:204 D:200 C:190 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 6 -8 B -4 0 -8 -10 -18 C -6 8 0 -2 -20 D -6 10 2 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 6 -8 B -4 0 -8 -10 -18 C -6 8 0 -2 -20 D -6 10 2 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 6 -8 B -4 0 -8 -10 -18 C -6 8 0 -2 -20 D -6 10 2 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5133: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) A B E C D (8) B A E D C (7) E C D A B (6) A B E D C (6) C E D A B (5) C D E A B (5) E D C A B (4) D C B E A (4) C D E B A (4) A E B D C (4) A E B C D (4) D B C A E (3) B A D C E (3) E D A C B (2) E C A D B (2) E A D C B (2) E A D B C (2) E A C D B (2) E A B C D (2) C D B A E (2) B A C D E (2) D E C B A (1) D E B C A (1) D E B A C (1) D B E C A (1) D B C E A (1) C E A D B (1) C D B E A (1) C B D A E (1) C A B D E (1) B D C A E (1) B C D A E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 16 -6 -4 -10 B -16 0 -4 -16 -12 C 6 4 0 -2 -10 D 4 16 2 0 -18 E 10 12 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -6 -4 -10 B -16 0 -4 -16 -12 C 6 4 0 -2 -10 D 4 16 2 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=22 D=20 C=20 B=14 so B is eliminated. Round 2 votes counts: A=36 E=22 D=21 C=21 so D is eliminated. Round 3 votes counts: C=38 A=36 E=26 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:225 D:202 C:199 A:198 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -6 -4 -10 B -16 0 -4 -16 -12 C 6 4 0 -2 -10 D 4 16 2 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 -4 -10 B -16 0 -4 -16 -12 C 6 4 0 -2 -10 D 4 16 2 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 -4 -10 B -16 0 -4 -16 -12 C 6 4 0 -2 -10 D 4 16 2 0 -18 E 10 12 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5134: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) A D C B E (7) D A C E B (5) B E D C A (5) E C D A B (4) C E A D B (4) B E C A D (4) B A C D E (4) A C D E B (4) E D A C B (3) D E A C B (3) D A E C B (3) C A E D B (3) C A D E B (3) B D E A C (3) A D C E B (3) E D C B A (2) E D C A B (2) E C A D B (2) E B D A C (2) E B C D A (2) B E C D A (2) B C E A D (2) A C D B E (2) A B C D E (2) E D B C A (1) E D A B C (1) E C D B A (1) E C B D A (1) E B C A D (1) D B E A C (1) D A E B C (1) C E A B D (1) C B A E D (1) C A B E D (1) B E D A C (1) B E A C D (1) B D A E C (1) B C A E D (1) B A C E D (1) Total count = 100 A B C D E A 0 8 14 12 2 B -8 0 -10 -10 -4 C -14 10 0 -6 6 D -12 10 6 0 2 E -2 4 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 12 2 B -8 0 -10 -10 -4 C -14 10 0 -6 6 D -12 10 6 0 2 E -2 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=22 A=18 D=13 C=13 so D is eliminated. Round 2 votes counts: B=35 A=27 E=25 C=13 so C is eliminated. Round 3 votes counts: B=36 A=34 E=30 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:203 C:198 E:197 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 12 2 B -8 0 -10 -10 -4 C -14 10 0 -6 6 D -12 10 6 0 2 E -2 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 12 2 B -8 0 -10 -10 -4 C -14 10 0 -6 6 D -12 10 6 0 2 E -2 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 12 2 B -8 0 -10 -10 -4 C -14 10 0 -6 6 D -12 10 6 0 2 E -2 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5135: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) C D B E A (8) A C E B D (8) A E C B D (6) E B C D A (5) D B C E A (5) C E B A D (5) A E B C D (5) C E B D A (4) C B D E A (3) E B D C A (2) E B D A C (2) E B C A D (2) E A B D C (2) D C B E A (2) D C B A E (2) D B E A C (2) D B A E C (2) C D A B E (2) C A E B D (2) C A D B E (2) B E D C A (2) B D E C A (2) A E B D C (2) A D C B E (2) E C B D A (1) E C A B D (1) E A C B D (1) E A B C D (1) D B C A E (1) D B A C E (1) D A B E C (1) C A E D B (1) B E C D A (1) A D E B C (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -22 -12 -18 B 20 0 -6 14 -6 C 22 6 0 22 2 D 12 -14 -22 0 -8 E 18 6 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -22 -12 -18 B 20 0 -6 14 -6 C 22 6 0 22 2 D 12 -14 -22 0 -8 E 18 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=27 A=27 D=24 E=17 B=5 so B is eliminated. Round 2 votes counts: C=27 A=27 D=26 E=20 so E is eliminated. Round 3 votes counts: C=37 D=32 A=31 so A is eliminated. Round 4 votes counts: C=61 D=39 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:215 B:211 D:184 A:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -22 -12 -18 B 20 0 -6 14 -6 C 22 6 0 22 2 D 12 -14 -22 0 -8 E 18 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -22 -12 -18 B 20 0 -6 14 -6 C 22 6 0 22 2 D 12 -14 -22 0 -8 E 18 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -22 -12 -18 B 20 0 -6 14 -6 C 22 6 0 22 2 D 12 -14 -22 0 -8 E 18 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999961203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5136: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) B E A C D (8) D C B E A (6) D C A E B (6) A D C E B (5) E B A C D (3) E A C B D (3) D A C E B (3) C D E A B (3) B E C A D (3) B C E D A (3) A E C D B (3) A D E C B (3) A B E D C (3) A B E C D (3) E A B C D (2) D B C E A (2) D B C A E (2) C D B E A (2) B E C D A (2) B C D E A (2) B A E D C (2) B A E C D (2) A E D C B (2) A E D B C (2) A E B D C (2) E C A D B (1) E B C A D (1) E A C D B (1) D C E B A (1) D C E A B (1) D C B A E (1) D A B C E (1) C E D B A (1) C E D A B (1) C D E B A (1) B E A D C (1) B D E A C (1) B D C A E (1) A C E D B (1) Total count = 100 A B C D E A 0 10 20 20 2 B -10 0 10 2 -10 C -20 -10 0 10 -14 D -20 -2 -10 0 -18 E -2 10 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 20 20 2 B -10 0 10 2 -10 C -20 -10 0 10 -14 D -20 -2 -10 0 -18 E -2 10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=25 D=23 E=11 C=8 so C is eliminated. Round 2 votes counts: A=33 D=29 B=25 E=13 so E is eliminated. Round 3 votes counts: A=40 D=31 B=29 so B is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:220 B:196 C:183 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 20 20 2 B -10 0 10 2 -10 C -20 -10 0 10 -14 D -20 -2 -10 0 -18 E -2 10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 20 20 2 B -10 0 10 2 -10 C -20 -10 0 10 -14 D -20 -2 -10 0 -18 E -2 10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 20 20 2 B -10 0 10 2 -10 C -20 -10 0 10 -14 D -20 -2 -10 0 -18 E -2 10 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999971402 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5137: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) A D B C E (11) E C B D A (9) E B D C A (6) B D A E C (6) A C D B E (6) C E B A D (5) C A E D B (5) B A D E C (5) D A B E C (4) C E A D B (4) E B C D A (3) C E A B D (3) D B A E C (2) D A B C E (2) C E D A B (2) C E B D A (2) B D E A C (2) A D C B E (2) E D C A B (1) E C D B A (1) D E B A C (1) D A C E B (1) D A C B E (1) C E D B A (1) B E D A C (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -6 10 18 B -12 0 -10 -14 -12 C 6 10 0 6 14 D -10 14 -6 0 14 E -18 12 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 10 18 B -12 0 -10 -14 -12 C 6 10 0 6 14 D -10 14 -6 0 14 E -18 12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=22 E=20 B=14 D=11 so D is eliminated. Round 2 votes counts: C=33 A=30 E=21 B=16 so B is eliminated. Round 3 votes counts: A=43 C=33 E=24 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:217 D:206 E:183 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 10 18 B -12 0 -10 -14 -12 C 6 10 0 6 14 D -10 14 -6 0 14 E -18 12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 10 18 B -12 0 -10 -14 -12 C 6 10 0 6 14 D -10 14 -6 0 14 E -18 12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 10 18 B -12 0 -10 -14 -12 C 6 10 0 6 14 D -10 14 -6 0 14 E -18 12 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5138: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) B A C D E (6) D A C E B (5) B A C E D (5) E B C A D (4) B C A E D (4) A B C D E (4) E D C A B (3) E D B C A (3) E C D A B (3) D E C A B (3) D E B A C (3) D E A C B (3) D A B C E (3) C B E A D (3) B E C A D (3) D E A B C (2) D C A E B (2) D B E A C (2) D A C B E (2) D A B E C (2) C E A D B (2) C E A B D (2) C A D E B (2) B A D C E (2) A C B D E (2) E D C B A (1) E C D B A (1) E C B D A (1) E C B A D (1) E B C D A (1) D B A E C (1) D B A C E (1) C E B A D (1) C A E D B (1) C A E B D (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A C E (1) B C E A D (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 -2 12 8 B -2 0 4 8 10 C 2 -4 0 16 18 D -12 -8 -16 0 -4 E -8 -10 -18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 12 8 B -2 0 4 8 10 C 2 -4 0 16 18 D -12 -8 -16 0 -4 E -8 -10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 C=20 E=18 A=7 so A is eliminated. Round 2 votes counts: D=30 B=30 C=22 E=18 so E is eliminated. Round 3 votes counts: D=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:216 A:210 B:210 E:184 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 12 8 B -2 0 4 8 10 C 2 -4 0 16 18 D -12 -8 -16 0 -4 E -8 -10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 12 8 B -2 0 4 8 10 C 2 -4 0 16 18 D -12 -8 -16 0 -4 E -8 -10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 12 8 B -2 0 4 8 10 C 2 -4 0 16 18 D -12 -8 -16 0 -4 E -8 -10 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5139: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) E A B D C (7) C D A E B (7) C E D A B (6) C D E A B (6) B A D E C (6) C B D A E (4) B A E C D (4) E B A D C (3) C D B A E (3) B C E A D (3) B C A D E (3) E D C A B (2) E A D B C (2) D E A C B (2) D C A E B (2) D C A B E (2) D A E C B (2) D A B C E (2) C E D B A (2) C E B A D (2) C B E D A (2) E C D A B (1) E C A B D (1) E B A C D (1) E A D C B (1) E A B C D (1) D B C A E (1) C E B D A (1) C D E B A (1) C D B E A (1) C B E A D (1) C B D E A (1) B E A D C (1) B E A C D (1) B D A E C (1) B C D A E (1) B C A E D (1) A E D B C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -8 0 2 B 6 0 0 12 -4 C 8 0 0 8 4 D 0 -12 -8 0 -8 E -2 4 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.229975 C: 0.770025 D: 0.000000 E: 0.000000 Sum of squares = 0.645827412457 Cumulative probabilities = A: 0.000000 B: 0.229975 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 0 2 B 6 0 0 12 -4 C 8 0 0 8 4 D 0 -12 -8 0 -8 E -2 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499593 C: 0.500407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000330645 Cumulative probabilities = A: 0.000000 B: 0.499593 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=30 E=19 D=11 A=3 so A is eliminated. Round 2 votes counts: C=37 B=31 E=21 D=11 so D is eliminated. Round 3 votes counts: C=41 B=34 E=25 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:210 B:207 E:203 A:194 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 0 2 B 6 0 0 12 -4 C 8 0 0 8 4 D 0 -12 -8 0 -8 E -2 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499593 C: 0.500407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000330645 Cumulative probabilities = A: 0.000000 B: 0.499593 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 2 B 6 0 0 12 -4 C 8 0 0 8 4 D 0 -12 -8 0 -8 E -2 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499593 C: 0.500407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000330645 Cumulative probabilities = A: 0.000000 B: 0.499593 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 2 B 6 0 0 12 -4 C 8 0 0 8 4 D 0 -12 -8 0 -8 E -2 4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499593 C: 0.500407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000330645 Cumulative probabilities = A: 0.000000 B: 0.499593 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5140: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) A D B C E (8) B E C D A (7) E B C A D (6) E C B A D (5) C E A D B (5) B A D E C (5) A E C D B (4) E C B D A (3) E C A D B (3) E C A B D (3) D B A C E (3) D A C E B (3) B E D C A (3) A D C B E (3) E B C D A (2) E B A C D (2) D A C B E (2) D A B C E (2) C E D B A (2) C E D A B (2) B E C A D (2) B D A C E (2) B C D E A (2) B A E D C (2) A D B E C (2) E A B C D (1) C E B D A (1) C D E A B (1) C D A E B (1) B E D A C (1) B D C E A (1) B C E D A (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 0 0 22 -6 B 0 0 6 -2 -8 C 0 -6 0 6 -6 D -22 2 -6 0 -12 E 6 8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 22 -6 B 0 0 6 -2 -8 C 0 -6 0 6 -6 D -22 2 -6 0 -12 E 6 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 E=25 C=12 D=10 so D is eliminated. Round 2 votes counts: A=34 B=29 E=25 C=12 so C is eliminated. Round 3 votes counts: E=36 A=35 B=29 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:208 B:198 C:197 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 22 -6 B 0 0 6 -2 -8 C 0 -6 0 6 -6 D -22 2 -6 0 -12 E 6 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 22 -6 B 0 0 6 -2 -8 C 0 -6 0 6 -6 D -22 2 -6 0 -12 E 6 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 22 -6 B 0 0 6 -2 -8 C 0 -6 0 6 -6 D -22 2 -6 0 -12 E 6 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5141: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) E D C B A (6) D E C A B (6) B A D C E (6) B A C E D (6) A B C D E (6) E C D A B (5) D E C B A (5) D E B A C (5) B A C D E (5) C A B E D (4) A B D C E (4) E C D B A (3) D B E A C (2) D B A E C (2) D A B C E (2) C E D A B (2) C E B A D (2) C E A D B (2) C E A B D (2) C A E B D (2) A C B E D (2) A B C E D (2) E D B C A (1) E C B D A (1) E B C D A (1) D E B C A (1) D E A C B (1) D C E A B (1) D A E B C (1) D A B E C (1) C B E A D (1) B A D E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -4 -6 -8 B -4 0 -8 -8 -8 C 4 8 0 -4 2 D 6 8 4 0 2 E 8 8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -6 -8 B -4 0 -8 -8 -8 C 4 8 0 -4 2 D 6 8 4 0 2 E 8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 B=18 A=16 C=15 so C is eliminated. Round 2 votes counts: E=32 D=27 A=22 B=19 so B is eliminated. Round 3 votes counts: A=40 E=33 D=27 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 E:206 C:205 A:193 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -6 -8 B -4 0 -8 -8 -8 C 4 8 0 -4 2 D 6 8 4 0 2 E 8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 -8 B -4 0 -8 -8 -8 C 4 8 0 -4 2 D 6 8 4 0 2 E 8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 -8 B -4 0 -8 -8 -8 C 4 8 0 -4 2 D 6 8 4 0 2 E 8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5142: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) E C B A D (5) D A E C B (5) C E B A D (5) B C A D E (5) D A C E B (4) D A C B E (4) D A B C E (4) B C A E D (4) B A D C E (4) B A C D E (4) A D B C E (4) E D A C B (3) E D A B C (3) E C D A B (3) D A E B C (3) A D B E C (3) E B C D A (2) E B C A D (2) D A B E C (2) C D A E B (2) C D A B E (2) C B E A D (2) B A E D C (2) E C D B A (1) E B D A C (1) E B A D C (1) D C A E B (1) D C A B E (1) C E D B A (1) C B A D E (1) B E C A D (1) B C E A D (1) A D E B C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 0 -6 16 B 6 0 -4 4 -8 C 0 4 0 4 2 D 6 -4 -4 0 4 E -16 8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.223731 B: 0.000000 C: 0.776269 D: 0.000000 E: 0.000000 Sum of squares = 0.652649254159 Cumulative probabilities = A: 0.223731 B: 0.223731 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 16 B 6 0 -4 4 -8 C 0 4 0 4 2 D 6 -4 -4 0 4 E -16 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000007238 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=24 B=21 C=13 A=10 so A is eliminated. Round 2 votes counts: D=33 E=32 B=22 C=13 so C is eliminated. Round 3 votes counts: E=38 D=37 B=25 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:205 A:202 D:201 B:199 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -6 16 B 6 0 -4 4 -8 C 0 4 0 4 2 D 6 -4 -4 0 4 E -16 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000007238 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 16 B 6 0 -4 4 -8 C 0 4 0 4 2 D 6 -4 -4 0 4 E -16 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000007238 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 16 B 6 0 -4 4 -8 C 0 4 0 4 2 D 6 -4 -4 0 4 E -16 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.52000007238 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5143: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) C D E B A (7) D E C A B (6) C B D E A (6) A B E C D (6) B A C D E (5) E C D A B (4) E A D C B (4) C E D A B (4) E D C A B (3) B C A D E (3) B A C E D (3) A E D C B (3) A E D B C (3) A E B D C (3) D E C B A (2) D E B C A (2) D C E B A (2) D B C E A (2) C B A E D (2) B D A C E (2) B C D E A (2) A E C B D (2) E D A C B (1) E A D B C (1) D E A B C (1) D C B E A (1) D B E C A (1) D B E A C (1) C D B E A (1) C B D A E (1) B D A E C (1) B A E D C (1) B A D C E (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 0 -2 B -8 0 -2 2 2 C -2 2 0 -2 -10 D 0 -2 2 0 -6 E 2 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000096 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 8 2 0 -2 B -8 0 -2 2 2 C -2 2 0 -2 -10 D 0 -2 2 0 -6 E 2 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000165 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=21 D=18 B=18 E=13 so E is eliminated. Round 2 votes counts: A=35 C=25 D=22 B=18 so B is eliminated. Round 3 votes counts: A=45 C=30 D=25 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:204 B:197 D:197 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 0 -2 B -8 0 -2 2 2 C -2 2 0 -2 -10 D 0 -2 2 0 -6 E 2 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000165 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 0 -2 B -8 0 -2 2 2 C -2 2 0 -2 -10 D 0 -2 2 0 -6 E 2 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000165 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 0 -2 B -8 0 -2 2 2 C -2 2 0 -2 -10 D 0 -2 2 0 -6 E 2 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000165 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5144: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A E C B (6) D A C B E (6) C B D A E (6) B C E A D (6) A D C E B (5) E A D B C (4) A D E C B (4) E D B A C (3) E D A B C (3) D A C E B (3) B E C A D (3) A E D C B (3) E B A C D (2) D E A B C (2) D C A B E (2) D B E C A (2) C B E A D (2) C B A D E (2) B E C D A (2) B C D E A (2) A E C D B (2) A D C B E (2) A C D B E (2) E B D C A (1) E B D A C (1) E B C D A (1) E A C B D (1) D E B C A (1) D C B A E (1) D B C A E (1) D A E B C (1) C E A B D (1) C D B A E (1) C A B D E (1) B C E D A (1) B C D A E (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 6 6 4 B -2 0 -6 -8 -12 C -6 6 0 -2 -6 D -6 8 2 0 2 E -4 12 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999482 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 6 4 B -2 0 -6 -8 -12 C -6 6 0 -2 -6 D -6 8 2 0 2 E -4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=25 A=21 B=15 C=13 so C is eliminated. Round 2 votes counts: E=27 D=26 B=25 A=22 so A is eliminated. Round 3 votes counts: D=39 E=34 B=27 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:209 E:206 D:203 C:196 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 6 4 B -2 0 -6 -8 -12 C -6 6 0 -2 -6 D -6 8 2 0 2 E -4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 6 4 B -2 0 -6 -8 -12 C -6 6 0 -2 -6 D -6 8 2 0 2 E -4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 6 4 B -2 0 -6 -8 -12 C -6 6 0 -2 -6 D -6 8 2 0 2 E -4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5145: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (21) C A D E B (10) A C E B D (8) C D B E A (7) A E B D C (7) D B E C A (6) E B D A C (4) A C E D B (4) D E B A C (3) A B E D C (3) A B E C D (3) D B C E A (2) C D E B A (2) C A E B D (2) C A D B E (2) B E A D C (2) B D E C A (2) B D E A C (2) E D B A C (1) E B A D C (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) C A B E D (1) B A E D C (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 24 -6 -8 B 8 0 16 16 8 C -24 -16 0 -10 -12 D 6 -16 10 0 -20 E 8 -8 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 24 -6 -8 B 8 0 16 16 8 C -24 -16 0 -10 -12 D 6 -16 10 0 -20 E 8 -8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=28 B=28 A=27 D=11 E=6 so E is eliminated. Round 2 votes counts: B=33 C=28 A=27 D=12 so D is eliminated. Round 3 votes counts: B=45 C=28 A=27 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:216 A:201 D:190 C:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 24 -6 -8 B 8 0 16 16 8 C -24 -16 0 -10 -12 D 6 -16 10 0 -20 E 8 -8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 24 -6 -8 B 8 0 16 16 8 C -24 -16 0 -10 -12 D 6 -16 10 0 -20 E 8 -8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 24 -6 -8 B 8 0 16 16 8 C -24 -16 0 -10 -12 D 6 -16 10 0 -20 E 8 -8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5146: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) D B C A E (9) B D A E C (8) E A C B D (7) D B A C E (7) C D E B A (7) A E C B D (7) C E A D B (6) B A D E C (6) E C A B D (4) D B C E A (4) B D A C E (4) A E B C D (4) E C A D B (3) D C E B A (3) D C B E A (3) A B E C D (3) C D E A B (2) A B E D C (2) B A E D C (1) Total count = 100 A B C D E A 0 -4 -2 -14 2 B 4 0 -4 -8 -6 C 2 4 0 6 10 D 14 8 -6 0 6 E -2 6 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -14 2 B 4 0 -4 -8 -6 C 2 4 0 6 10 D 14 8 -6 0 6 E -2 6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=25 B=19 A=16 E=14 so E is eliminated. Round 2 votes counts: C=32 D=26 A=23 B=19 so B is eliminated. Round 3 votes counts: D=38 C=32 A=30 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:211 E:194 B:193 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -14 2 B 4 0 -4 -8 -6 C 2 4 0 6 10 D 14 8 -6 0 6 E -2 6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -14 2 B 4 0 -4 -8 -6 C 2 4 0 6 10 D 14 8 -6 0 6 E -2 6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -14 2 B 4 0 -4 -8 -6 C 2 4 0 6 10 D 14 8 -6 0 6 E -2 6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5147: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (15) C B D E A (12) A E D C B (10) B C D E A (7) B C A D E (7) A B C E D (7) E D A C B (6) D E C B A (5) B A C E D (5) A E B D C (4) C D B E A (3) D E B C A (2) D E A C B (2) C B D A E (2) B C D A E (2) A B E C D (2) E D B C A (1) E A D B C (1) D C B E A (1) C B A D E (1) B C A E D (1) A E B C D (1) A D E C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 12 14 20 B -2 0 10 6 2 C -12 -10 0 4 0 D -14 -6 -4 0 -10 E -20 -2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 14 20 B -2 0 10 6 2 C -12 -10 0 4 0 D -14 -6 -4 0 -10 E -20 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 B=22 C=18 D=10 E=8 so E is eliminated. Round 2 votes counts: A=43 B=22 C=18 D=17 so D is eliminated. Round 3 votes counts: A=51 B=25 C=24 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:208 E:194 C:191 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 14 20 B -2 0 10 6 2 C -12 -10 0 4 0 D -14 -6 -4 0 -10 E -20 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 14 20 B -2 0 10 6 2 C -12 -10 0 4 0 D -14 -6 -4 0 -10 E -20 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 14 20 B -2 0 10 6 2 C -12 -10 0 4 0 D -14 -6 -4 0 -10 E -20 -2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5148: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) A B D C E (7) D E A B C (6) A B D E C (5) D E C A B (4) C E D B A (4) A B C D E (4) E C D B A (3) E C A B D (3) D E C B A (3) D B A E C (3) D A B E C (3) E D C B A (2) E D C A B (2) E D A C B (2) D E B A C (2) D A E B C (2) C E B A D (2) C B E A D (2) C B D A E (2) B C A E D (2) B A D C E (2) B A C E D (2) A D B E C (2) A B E C D (2) E A D B C (1) E A C D B (1) D E B C A (1) D E A C B (1) D C E B A (1) D C B E A (1) D B A C E (1) C E B D A (1) C D E B A (1) C D B E A (1) C B E D A (1) C B A E D (1) C B A D E (1) B D A C E (1) A E D B C (1) A E B C D (1) A D E B C (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 20 6 -14 -8 B -20 0 2 -16 -8 C -6 -2 0 -10 -22 D 14 16 10 0 10 E 8 8 22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 -14 -8 B -20 0 2 -16 -8 C -6 -2 0 -10 -22 D 14 16 10 0 10 E 8 8 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=26 E=23 C=16 B=7 so B is eliminated. Round 2 votes counts: A=30 D=29 E=23 C=18 so C is eliminated. Round 3 votes counts: A=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:214 A:202 C:180 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 6 -14 -8 B -20 0 2 -16 -8 C -6 -2 0 -10 -22 D 14 16 10 0 10 E 8 8 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 -14 -8 B -20 0 2 -16 -8 C -6 -2 0 -10 -22 D 14 16 10 0 10 E 8 8 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 -14 -8 B -20 0 2 -16 -8 C -6 -2 0 -10 -22 D 14 16 10 0 10 E 8 8 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5149: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) D C E B A (10) B C D A E (10) A B E C D (10) E D C A B (9) E D A C B (6) E A D B C (6) B C A D E (6) B A C D E (6) C B D A E (4) E A D C B (3) C D B E A (3) B A C E D (3) A B E D C (3) A B C E D (2) E D C B A (1) E B A D C (1) E A B D C (1) C D E B A (1) C D A B E (1) A E D B C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 10 18 B -10 0 24 18 -2 C -10 -24 0 -6 -6 D -10 -18 6 0 -16 E -18 2 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 10 18 B -10 0 24 18 -2 C -10 -24 0 -6 -6 D -10 -18 6 0 -16 E -18 2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 B=25 D=10 C=9 so C is eliminated. Round 2 votes counts: B=29 A=29 E=27 D=15 so D is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:224 B:215 E:203 D:181 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 10 18 B -10 0 24 18 -2 C -10 -24 0 -6 -6 D -10 -18 6 0 -16 E -18 2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 10 18 B -10 0 24 18 -2 C -10 -24 0 -6 -6 D -10 -18 6 0 -16 E -18 2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 10 18 B -10 0 24 18 -2 C -10 -24 0 -6 -6 D -10 -18 6 0 -16 E -18 2 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5150: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) D E B A C (7) E D C A B (6) E D B A C (5) D B E A C (5) C B A D E (5) C A B D E (5) A C E B D (5) E C A D B (4) D B C A E (4) B A C D E (4) A C B E D (4) E D B C A (3) E A C B D (3) D E C B A (3) C A B E D (3) B C A D E (3) E D A B C (2) E A C D B (2) D B C E A (2) C A E B D (2) B D A C E (2) E D A C B (1) E C D A B (1) E C A B D (1) E A B C D (1) D C E B A (1) D C B A E (1) D B E C A (1) C D E A B (1) B D C A E (1) B D A E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -12 -12 -16 B 14 0 4 -16 -14 C 12 -4 0 -8 -10 D 12 16 8 0 12 E 16 14 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -12 -16 B 14 0 4 -16 -14 C 12 -4 0 -8 -10 D 12 16 8 0 12 E 16 14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=29 C=16 B=11 A=11 so B is eliminated. Round 2 votes counts: D=37 E=29 C=19 A=15 so A is eliminated. Round 3 votes counts: D=37 C=34 E=29 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:214 C:195 B:194 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -12 -12 -16 B 14 0 4 -16 -14 C 12 -4 0 -8 -10 D 12 16 8 0 12 E 16 14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -12 -16 B 14 0 4 -16 -14 C 12 -4 0 -8 -10 D 12 16 8 0 12 E 16 14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -12 -16 B 14 0 4 -16 -14 C 12 -4 0 -8 -10 D 12 16 8 0 12 E 16 14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5151: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) A C E B D (12) D B E C A (11) D B E A C (7) C A E B D (6) B D E A C (5) C A E D B (4) C A D E B (4) C A D B E (4) B E D A C (4) E A C B D (3) D C A B E (3) D B C A E (3) D B A C E (3) E C A B D (2) E B D C A (2) D A C B E (2) A C D E B (2) A C D B E (2) E D B C A (1) E B C A D (1) E B A D C (1) D C B A E (1) D B A E C (1) C E A B D (1) C D A B E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 12 -12 0 B 4 0 4 2 -2 C -12 -4 0 -14 -2 D 12 -2 14 0 -2 E 0 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.085685 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.914315 Sum of squares = 0.843314039871 Cumulative probabilities = A: 0.085685 B: 0.085685 C: 0.085685 D: 0.085685 E: 1.000000 A B C D E A 0 -4 12 -12 0 B 4 0 4 2 -2 C -12 -4 0 -14 -2 D 12 -2 14 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102042129 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 C=20 A=18 B=9 so B is eliminated. Round 2 votes counts: D=36 E=26 C=20 A=18 so A is eliminated. Round 3 votes counts: C=37 D=36 E=27 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:204 E:203 A:198 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 12 -12 0 B 4 0 4 2 -2 C -12 -4 0 -14 -2 D 12 -2 14 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102042129 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -12 0 B 4 0 4 2 -2 C -12 -4 0 -14 -2 D 12 -2 14 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102042129 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -12 0 B 4 0 4 2 -2 C -12 -4 0 -14 -2 D 12 -2 14 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857143 Sum of squares = 0.755102042129 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.142857 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5152: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E C A D B (10) B A E D C (6) A C E B D (5) E C D A B (4) D C E A B (4) C E D A B (4) B A E C D (4) B A D C E (4) E D C B A (3) D C A B E (3) D B C A E (3) C E A D B (3) C D E A B (3) E A C B D (2) E A B C D (2) D E B C A (2) D C E B A (2) D C A E B (2) D B C E A (2) B D C A E (2) B A D E C (2) A E C B D (2) A D C B E (2) A B E C D (2) E D C A B (1) E C A B D (1) D B A C E (1) D A C B E (1) C D A E B (1) B D E C A (1) B D A E C (1) A E B C D (1) A C E D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -2 -2 12 B -12 0 -8 -4 -6 C 2 8 0 -6 12 D 2 4 6 0 -4 E -12 6 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839484 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 A B C D E A 0 12 -2 -2 12 B -12 0 -8 -4 -6 C 2 8 0 -6 12 D 2 4 6 0 -4 E -12 6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839434 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=23 D=20 A=15 C=11 so C is eliminated. Round 2 votes counts: B=31 E=30 D=24 A=15 so A is eliminated. Round 3 votes counts: E=39 B=35 D=26 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:210 C:208 D:204 E:193 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 -2 -2 12 B -12 0 -8 -4 -6 C 2 8 0 -6 12 D 2 4 6 0 -4 E -12 6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839434 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -2 12 B -12 0 -8 -4 -6 C 2 8 0 -6 12 D 2 4 6 0 -4 E -12 6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839434 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -2 12 B -12 0 -8 -4 -6 C 2 8 0 -6 12 D 2 4 6 0 -4 E -12 6 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839434 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5153: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) E B A C D (6) C A E B D (6) E A C B D (5) D B A C E (5) B A C E D (5) E D B C A (4) D B E A C (4) D B C A E (4) C A E D B (4) E D C A B (3) E C A B D (3) E B D A C (3) C A D B E (3) C A B D E (3) B E D A C (3) B E A C D (3) E D B A C (2) D E C A B (2) D E B C A (2) D E B A C (2) C D A B E (2) C A B E D (2) B D A C E (2) A C B E D (2) E C D A B (1) E B D C A (1) E B A D C (1) D C E A B (1) D C A E B (1) D B C E A (1) B E A D C (1) B D E A C (1) B A E C D (1) B A D C E (1) B A C D E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 0 -2 2 B 6 0 6 4 4 C 0 -6 0 -2 4 D 2 -4 2 0 -14 E -2 -4 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 2 B 6 0 6 4 4 C 0 -6 0 -2 4 D 2 -4 2 0 -14 E -2 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=29 D=29 C=20 B=18 A=4 so A is eliminated. Round 2 votes counts: E=29 D=29 C=24 B=18 so B is eliminated. Round 3 votes counts: E=37 D=33 C=30 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:202 C:198 A:197 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -2 2 B 6 0 6 4 4 C 0 -6 0 -2 4 D 2 -4 2 0 -14 E -2 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 2 B 6 0 6 4 4 C 0 -6 0 -2 4 D 2 -4 2 0 -14 E -2 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 2 B 6 0 6 4 4 C 0 -6 0 -2 4 D 2 -4 2 0 -14 E -2 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5154: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (16) A D B C E (10) B C E D A (6) E C B A D (5) E C A D B (4) E C A B D (4) E A C D B (4) D B A C E (4) A D E C B (4) B D C E A (3) B D C A E (3) A D B E C (3) A B D C E (3) D B E C A (2) D A B E C (2) C E B D A (2) C B E D A (2) B D A C E (2) B C D A E (2) A E D C B (2) A D E B C (2) E D A C B (1) E C D B A (1) E C D A B (1) E A D C B (1) D E A C B (1) D B A E C (1) D A E B C (1) D A B C E (1) C E B A D (1) B E D C A (1) B C E A D (1) B C D E A (1) B C A D E (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -12 -6 -14 B 8 0 0 8 -2 C 12 0 0 6 -14 D 6 -8 -6 0 -8 E 14 2 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 -6 -14 B 8 0 0 8 -2 C 12 0 0 6 -14 D 6 -8 -6 0 -8 E 14 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=26 B=20 D=12 C=5 so C is eliminated. Round 2 votes counts: E=40 A=26 B=22 D=12 so D is eliminated. Round 3 votes counts: E=41 A=30 B=29 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:207 C:202 D:192 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 -6 -14 B 8 0 0 8 -2 C 12 0 0 6 -14 D 6 -8 -6 0 -8 E 14 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -6 -14 B 8 0 0 8 -2 C 12 0 0 6 -14 D 6 -8 -6 0 -8 E 14 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -6 -14 B 8 0 0 8 -2 C 12 0 0 6 -14 D 6 -8 -6 0 -8 E 14 2 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5155: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) C D A E B (9) E B D C A (6) A B E D C (6) C D E B A (5) C D E A B (5) A B E C D (5) E D C B A (4) D E C B A (4) D C E B A (4) A C D B E (4) E D B C A (3) E B D A C (3) D C A E B (3) B E A C D (3) B A E D C (3) A C B D E (3) A B C E D (3) D C E A B (2) C A D B E (2) A B C D E (2) E C B D A (1) E B C D A (1) D E B C A (1) D E B A C (1) C A B E D (1) B E D A C (1) B C A E D (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 -8 -4 -6 -12 B 8 0 4 4 -4 C 4 -4 0 -8 -10 D 6 -4 8 0 -8 E 12 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -4 -6 -12 B 8 0 4 4 -4 C 4 -4 0 -8 -10 D 6 -4 8 0 -8 E 12 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 B=21 E=18 D=15 so D is eliminated. Round 2 votes counts: C=31 E=24 A=24 B=21 so B is eliminated. Round 3 votes counts: E=40 C=32 A=28 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:206 D:201 C:191 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -6 -12 B 8 0 4 4 -4 C 4 -4 0 -8 -10 D 6 -4 8 0 -8 E 12 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -6 -12 B 8 0 4 4 -4 C 4 -4 0 -8 -10 D 6 -4 8 0 -8 E 12 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -6 -12 B 8 0 4 4 -4 C 4 -4 0 -8 -10 D 6 -4 8 0 -8 E 12 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5156: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (11) A B C E D (10) E B D A C (9) D E C B A (9) E D B A C (7) D E C A B (7) E D B C A (6) B E A C D (6) C D A B E (5) C A D B E (5) C A B D E (5) B A C E D (5) E B A D C (4) D C A E B (4) A C B D E (4) D E B C A (2) D C E A B (1) Total count = 100 A B C D E A 0 -18 12 0 -2 B 18 0 20 8 2 C -12 -20 0 2 -22 D 0 -8 -2 0 -16 E 2 -2 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999121 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 12 0 -2 B 18 0 20 8 2 C -12 -20 0 2 -22 D 0 -8 -2 0 -16 E 2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=23 B=22 C=15 A=14 so A is eliminated. Round 2 votes counts: B=32 E=26 D=23 C=19 so C is eliminated. Round 3 votes counts: B=41 D=33 E=26 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:219 A:196 D:187 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 12 0 -2 B 18 0 20 8 2 C -12 -20 0 2 -22 D 0 -8 -2 0 -16 E 2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 12 0 -2 B 18 0 20 8 2 C -12 -20 0 2 -22 D 0 -8 -2 0 -16 E 2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 12 0 -2 B 18 0 20 8 2 C -12 -20 0 2 -22 D 0 -8 -2 0 -16 E 2 -2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5157: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) E B A D C (6) D C E B A (6) E D B C A (5) E C A D B (5) C E D A B (4) C D E A B (4) B A E D C (4) A B E C D (4) E C D A B (3) C A D E B (3) B E A D C (3) B A D E C (3) B A D C E (3) A E C B D (3) A E B C D (3) E A C B D (2) D E B C A (2) D B E C A (2) C E A D B (2) C D A B E (2) C A E D B (2) B D A C E (2) A C B D E (2) A B C E D (2) A B C D E (2) E D C B A (1) E C A B D (1) E B D A C (1) D C B A E (1) D B E A C (1) D B C E A (1) D B C A E (1) C D A E B (1) C A E B D (1) B E D A C (1) B E A C D (1) B D A E C (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 8 6 22 -16 B -8 0 12 8 -24 C -6 -12 0 12 -20 D -22 -8 -12 0 -26 E 16 24 20 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 22 -16 B -8 0 12 8 -24 C -6 -12 0 12 -20 D -22 -8 -12 0 -26 E 16 24 20 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=19 B=19 A=17 D=14 so D is eliminated. Round 2 votes counts: E=33 C=26 B=24 A=17 so A is eliminated. Round 3 votes counts: E=39 B=32 C=29 so C is eliminated. Round 4 votes counts: E=62 B=38 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:243 A:210 B:194 C:187 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 22 -16 B -8 0 12 8 -24 C -6 -12 0 12 -20 D -22 -8 -12 0 -26 E 16 24 20 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 22 -16 B -8 0 12 8 -24 C -6 -12 0 12 -20 D -22 -8 -12 0 -26 E 16 24 20 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 22 -16 B -8 0 12 8 -24 C -6 -12 0 12 -20 D -22 -8 -12 0 -26 E 16 24 20 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5158: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) D B A C E (11) E B D A C (7) E D B A C (6) C A D B E (6) D B E A C (5) B D A C E (5) E C A D B (4) C A E B D (4) A C B D E (4) E D C B A (3) E D B C A (3) E C D B A (3) E B A D C (3) C A B D E (3) A B D C E (3) D E B A C (2) D B A E C (2) C A B E D (2) E B A C D (1) D C B A E (1) D C A B E (1) C E A D B (1) C E A B D (1) C D A B E (1) C A E D B (1) B D E A C (1) A D B C E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 6 -2 -6 B 6 0 4 -2 -4 C -6 -4 0 -8 -8 D 2 2 8 0 -6 E 6 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 -2 -6 B 6 0 4 -2 -4 C -6 -4 0 -8 -8 D 2 2 8 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 D=22 C=19 A=10 B=6 so B is eliminated. Round 2 votes counts: E=43 D=28 C=19 A=10 so A is eliminated. Round 3 votes counts: E=44 D=32 C=24 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:203 B:202 A:196 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 -2 -6 B 6 0 4 -2 -4 C -6 -4 0 -8 -8 D 2 2 8 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -2 -6 B 6 0 4 -2 -4 C -6 -4 0 -8 -8 D 2 2 8 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -2 -6 B 6 0 4 -2 -4 C -6 -4 0 -8 -8 D 2 2 8 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5159: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (6) A C E D B (6) D E A B C (5) A C E B D (5) E B D C A (4) D B E C A (4) D A E B C (4) D A C B E (4) B C E D A (4) A E D C B (4) A D E C B (4) A C B D E (4) E D A B C (3) C B A E D (3) B C E A D (3) A C D B E (3) E D B C A (2) E D B A C (2) E A D B C (2) D E B A C (2) C B A D E (2) C A E B D (2) B E D C A (2) B E C D A (2) B C D E A (2) E C A B D (1) E B C D A (1) E A B D C (1) D E B C A (1) D C B A E (1) D B C E A (1) D B A E C (1) D A C E B (1) C B E A D (1) C B D A E (1) C A D B E (1) B D C E A (1) A E C D B (1) A E C B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 20 10 4 12 B -20 0 -6 -6 -6 C -10 6 0 2 6 D -4 6 -2 0 -14 E -12 6 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 4 12 B -20 0 -6 -6 -6 C -10 6 0 2 6 D -4 6 -2 0 -14 E -12 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 E=16 C=16 B=14 so B is eliminated. Round 2 votes counts: A=30 D=25 C=25 E=20 so E is eliminated. Round 3 votes counts: D=38 A=33 C=29 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:202 E:201 D:193 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 4 12 B -20 0 -6 -6 -6 C -10 6 0 2 6 D -4 6 -2 0 -14 E -12 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 4 12 B -20 0 -6 -6 -6 C -10 6 0 2 6 D -4 6 -2 0 -14 E -12 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 4 12 B -20 0 -6 -6 -6 C -10 6 0 2 6 D -4 6 -2 0 -14 E -12 6 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5160: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) A D E C B (7) E D A B C (6) A D C E B (6) E B C D A (5) D A E C B (5) D E A B C (4) B E C D A (4) B C E A D (4) A D E B C (4) E B D C A (3) C D B A E (3) C D A B E (3) C A B D E (3) B E C A D (3) A E D B C (3) E B D A C (2) E B C A D (2) E B A C D (2) E A D B C (2) D C A B E (2) D A C E B (2) D A C B E (2) C B E D A (2) C B D A E (2) B C E D A (2) A C D B E (2) E D B A C (1) E B A D C (1) D E A C B (1) C B D E A (1) C B A E D (1) C A D B E (1) B C A E D (1) A D C B E (1) Total count = 100 A B C D E A 0 8 2 0 10 B -8 0 -2 -10 -12 C -2 2 0 -4 -10 D 0 10 4 0 12 E -10 12 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.623285 B: 0.000000 C: 0.000000 D: 0.376715 E: 0.000000 Sum of squares = 0.530398305516 Cumulative probabilities = A: 0.623285 B: 0.623285 C: 0.623285 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 0 10 B -8 0 -2 -10 -12 C -2 2 0 -4 -10 D 0 10 4 0 12 E -10 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=23 A=23 D=16 B=14 so B is eliminated. Round 2 votes counts: E=31 C=30 A=23 D=16 so D is eliminated. Round 3 votes counts: E=36 C=32 A=32 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:210 E:200 C:193 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 0 10 B -8 0 -2 -10 -12 C -2 2 0 -4 -10 D 0 10 4 0 12 E -10 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 0 10 B -8 0 -2 -10 -12 C -2 2 0 -4 -10 D 0 10 4 0 12 E -10 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 0 10 B -8 0 -2 -10 -12 C -2 2 0 -4 -10 D 0 10 4 0 12 E -10 12 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5161: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (13) D A C B E (11) E D A C B (7) E B C D A (7) E B C A D (6) B E C A D (4) A D C B E (4) E D A B C (3) D A C E B (3) C A D B E (3) B E A D C (3) B C E A D (3) B C A E D (3) B A C D E (3) E D C B A (2) E C B D A (2) E B D A C (2) D E A C B (2) D A E C B (2) C B A D E (2) B A D C E (2) A D B C E (2) E D C A B (1) E C D A B (1) E B D C A (1) D A E B C (1) D A B E C (1) C D A B E (1) C B A E D (1) C A B D E (1) B E A C D (1) B A D E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -2 6 10 B 12 0 12 10 20 C 2 -12 0 4 6 D -6 -10 -4 0 6 E -10 -20 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 6 10 B 12 0 12 10 20 C 2 -12 0 4 6 D -6 -10 -4 0 6 E -10 -20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=32 D=20 C=8 A=7 so A is eliminated. Round 2 votes counts: B=33 E=32 D=26 C=9 so C is eliminated. Round 3 votes counts: B=37 E=32 D=31 so D is eliminated. Round 4 votes counts: B=60 E=40 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:201 C:200 D:193 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 6 10 B 12 0 12 10 20 C 2 -12 0 4 6 D -6 -10 -4 0 6 E -10 -20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 6 10 B 12 0 12 10 20 C 2 -12 0 4 6 D -6 -10 -4 0 6 E -10 -20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 6 10 B 12 0 12 10 20 C 2 -12 0 4 6 D -6 -10 -4 0 6 E -10 -20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5162: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) E D B A C (6) D B A C E (5) D A B C E (5) C A D E B (5) C A B D E (5) E B D A C (4) D A C B E (4) C A D B E (4) B E D A C (4) A C D B E (4) E D B C A (3) E C B A D (3) E C A B D (3) D B E A C (3) D B A E C (3) E C D A B (2) E C B D A (2) E C A D B (2) E B C D A (2) E B C A D (2) D E B A C (2) D A E C B (2) C E A D B (2) C A E D B (2) B A D C E (2) A D B C E (2) A B C D E (2) D E A B C (1) C E A B D (1) C A E B D (1) B E A D C (1) B D E A C (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 8 -12 -2 B 0 0 12 -16 -4 C -8 -12 0 -14 -6 D 12 16 14 0 6 E 2 4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -12 -2 B 0 0 12 -16 -4 C -8 -12 0 -14 -6 D 12 16 14 0 6 E 2 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=25 C=20 A=11 B=8 so B is eliminated. Round 2 votes counts: E=41 D=26 C=20 A=13 so A is eliminated. Round 3 votes counts: E=41 D=32 C=27 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:203 A:197 B:196 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 8 -12 -2 B 0 0 12 -16 -4 C -8 -12 0 -14 -6 D 12 16 14 0 6 E 2 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -12 -2 B 0 0 12 -16 -4 C -8 -12 0 -14 -6 D 12 16 14 0 6 E 2 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -12 -2 B 0 0 12 -16 -4 C -8 -12 0 -14 -6 D 12 16 14 0 6 E 2 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5163: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) A C E B D (7) C E A D B (6) D B E C A (5) D B C E A (5) A E C B D (5) A E B C D (5) D C E B A (4) D C B E A (4) B A D E C (4) D C B A E (3) C E D A B (3) C A E B D (3) B D A C E (3) A B E C D (3) E C D B A (2) E A C B D (2) D B E A C (2) D B C A E (2) B D E A C (2) A B E D C (2) A B C E D (2) A B C D E (2) E D C B A (1) E C A D B (1) E C A B D (1) D E C B A (1) D B A E C (1) D B A C E (1) C D E A B (1) C D A B E (1) C A E D B (1) B E A D C (1) B A E D C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 12 -4 18 B 6 0 6 12 14 C -12 -6 0 -8 0 D 4 -12 8 0 6 E -18 -14 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -4 18 B 6 0 6 12 14 C -12 -6 0 -8 0 D 4 -12 8 0 6 E -18 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=28 A=28 B=22 C=15 E=7 so E is eliminated. Round 2 votes counts: A=30 D=29 B=22 C=19 so C is eliminated. Round 3 votes counts: A=42 D=36 B=22 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 A:210 D:203 C:187 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 -4 18 B 6 0 6 12 14 C -12 -6 0 -8 0 D 4 -12 8 0 6 E -18 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -4 18 B 6 0 6 12 14 C -12 -6 0 -8 0 D 4 -12 8 0 6 E -18 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -4 18 B 6 0 6 12 14 C -12 -6 0 -8 0 D 4 -12 8 0 6 E -18 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5164: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) C A D E B (8) E B A D C (6) D C A E B (6) B E A D C (6) C D B A E (5) B C E D A (5) B E D C A (4) A E B D C (4) C D B E A (3) C D A E B (3) C A D B E (3) B E C D A (3) B E C A D (3) A D C E B (3) E D B A C (2) E A B D C (2) D A C E B (2) C B D E A (2) B E A C D (2) A E D B C (2) A E B C D (2) A D E C B (2) A C D E B (2) E B D A C (1) C B E D A (1) C B E A D (1) B E D A C (1) B D C E A (1) B C E A D (1) B C D E A (1) A E D C B (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -20 0 10 B -4 0 -6 -6 4 C 20 6 0 12 16 D 0 6 -12 0 4 E -10 -4 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -20 0 10 B -4 0 -6 -6 4 C 20 6 0 12 16 D 0 6 -12 0 4 E -10 -4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=27 A=18 E=11 D=8 so D is eliminated. Round 2 votes counts: C=42 B=27 A=20 E=11 so E is eliminated. Round 3 votes counts: C=42 B=36 A=22 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:199 A:197 B:194 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -20 0 10 B -4 0 -6 -6 4 C 20 6 0 12 16 D 0 6 -12 0 4 E -10 -4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 0 10 B -4 0 -6 -6 4 C 20 6 0 12 16 D 0 6 -12 0 4 E -10 -4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 0 10 B -4 0 -6 -6 4 C 20 6 0 12 16 D 0 6 -12 0 4 E -10 -4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5165: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) E A C B D (6) D B C E A (6) C B A E D (6) A E D C B (6) D B C A E (5) B C E A D (5) A E C D B (5) E B A C D (4) D B E C A (4) A C E B D (4) C A E B D (3) B C E D A (3) A E C B D (3) D C B A E (2) D C A B E (2) C D A B E (2) C B D A E (2) B E A C D (2) B D E C A (2) A D E C B (2) A C E D B (2) E D A B C (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E B A C (1) D E A B C (1) D B A C E (1) D A E C B (1) C E A B D (1) C B E A D (1) C A D E B (1) C A D B E (1) B E C A D (1) B D C E A (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 0 16 -2 B 12 0 -2 -4 8 C 0 2 0 10 0 D -16 4 -10 0 -14 E 2 -8 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.812470 D: 0.000000 E: 0.187530 Sum of squares = 0.69527439196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.812470 D: 0.812470 E: 1.000000 A B C D E A 0 -12 0 16 -2 B 12 0 -2 -4 8 C 0 2 0 10 0 D -16 4 -10 0 -14 E 2 -8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000237679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 C=17 E=15 B=15 so E is eliminated. Round 2 votes counts: D=31 A=31 B=21 C=17 so C is eliminated. Round 3 votes counts: A=37 D=33 B=30 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:207 C:206 E:204 A:201 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 0 16 -2 B 12 0 -2 -4 8 C 0 2 0 10 0 D -16 4 -10 0 -14 E 2 -8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000237679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 16 -2 B 12 0 -2 -4 8 C 0 2 0 10 0 D -16 4 -10 0 -14 E 2 -8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000237679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 16 -2 B 12 0 -2 -4 8 C 0 2 0 10 0 D -16 4 -10 0 -14 E 2 -8 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000237679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5166: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) B D E C A (7) D C B E A (6) A C D E B (6) E B A C D (5) B E D C A (5) E B A D C (4) E A B C D (4) D C A B E (4) D B C E A (4) A C E B D (4) E B D A C (3) E B C A D (3) D C B A E (3) C E B D A (3) C D A B E (3) D A B E C (2) C A E D B (2) B E D A C (2) A E B D C (2) A E B C D (2) A C E D B (2) E B C D A (1) E A C B D (1) D B E C A (1) D B E A C (1) D B A E C (1) D A C B E (1) D A B C E (1) C E B A D (1) C D B E A (1) C D B A E (1) C D A E B (1) C A E B D (1) C A D B E (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -14 -2 -4 B 4 0 -2 -2 -10 C 14 2 0 4 8 D 2 2 -4 0 8 E 4 10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999084 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -2 -4 B 4 0 -2 -2 -10 C 14 2 0 4 8 D 2 2 -4 0 8 E 4 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 E=21 A=18 B=14 so B is eliminated. Round 2 votes counts: D=31 E=28 C=23 A=18 so A is eliminated. Round 3 votes counts: C=35 E=33 D=32 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:204 E:199 B:195 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -2 -4 B 4 0 -2 -2 -10 C 14 2 0 4 8 D 2 2 -4 0 8 E 4 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -2 -4 B 4 0 -2 -2 -10 C 14 2 0 4 8 D 2 2 -4 0 8 E 4 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -2 -4 B 4 0 -2 -2 -10 C 14 2 0 4 8 D 2 2 -4 0 8 E 4 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5167: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) C D E A B (7) B A C E D (7) E D A C B (5) E A D C B (5) B C D A E (5) B A E C D (5) D E C A B (4) D C E A B (4) B C A D E (4) B A E D C (4) A E B D C (4) A B E D C (4) E D C A B (3) E A D B C (3) C B D E A (3) E D A B C (2) C D E B A (2) B A C D E (2) A E C D B (2) A B E C D (2) E C A D B (1) E A C D B (1) D E B A C (1) D C B E A (1) C E A D B (1) C B D A E (1) C A E B D (1) B D C E A (1) B D A C E (1) B C A E D (1) A E D C B (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 4 2 0 -8 B -4 0 -6 -8 -18 C -2 6 0 -8 2 D 0 8 8 0 -8 E 8 18 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.555556 E: 1.000000 A B C D E A 0 4 2 0 -8 B -4 0 -6 -8 -18 C -2 6 0 -8 2 D 0 8 8 0 -8 E 8 18 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.111111 E: 0.444444 Sum of squares = 0.40740740734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=20 D=20 C=15 A=15 so C is eliminated. Round 2 votes counts: B=34 D=29 E=21 A=16 so A is eliminated. Round 3 votes counts: B=40 E=31 D=29 so D is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:204 A:199 C:199 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 0 -8 B -4 0 -6 -8 -18 C -2 6 0 -8 2 D 0 8 8 0 -8 E 8 18 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.111111 E: 0.444444 Sum of squares = 0.40740740734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.555556 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 0 -8 B -4 0 -6 -8 -18 C -2 6 0 -8 2 D 0 8 8 0 -8 E 8 18 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.111111 E: 0.444444 Sum of squares = 0.40740740734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 0 -8 B -4 0 -6 -8 -18 C -2 6 0 -8 2 D 0 8 8 0 -8 E 8 18 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.111111 E: 0.444444 Sum of squares = 0.40740740734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.555556 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5168: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (12) E A D C B (9) A E D C B (9) B A E C D (8) C D E A B (6) B A E D C (6) A E D B C (6) A E B D C (6) D C E A B (5) B C D E A (5) B C D A E (5) C B D E A (4) E D A C B (3) B C A E D (3) B C A D E (3) D E C A B (2) B E A D C (2) E D C A B (1) E A D B C (1) C D E B A (1) B E A C D (1) B C E A D (1) B A C E D (1) Total count = 100 A B C D E A 0 -4 4 12 -6 B 4 0 -4 -10 2 C -4 4 0 0 -8 D -12 10 0 0 -14 E 6 -2 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.538462 C: 0.000000 D: 0.076923 E: 0.384615 Sum of squares = 0.4437869821 Cumulative probabilities = A: 0.000000 B: 0.538462 C: 0.538462 D: 0.615385 E: 1.000000 A B C D E A 0 -4 4 12 -6 B 4 0 -4 -10 2 C -4 4 0 0 -8 D -12 10 0 0 -14 E 6 -2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.538462 C: 0.000000 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982208 Cumulative probabilities = A: 0.000000 B: 0.538462 C: 0.538462 D: 0.615385 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=23 A=21 E=14 D=7 so D is eliminated. Round 2 votes counts: B=35 C=28 A=21 E=16 so E is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:213 A:203 B:196 C:196 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 12 -6 B 4 0 -4 -10 2 C -4 4 0 0 -8 D -12 10 0 0 -14 E 6 -2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.538462 C: 0.000000 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982208 Cumulative probabilities = A: 0.000000 B: 0.538462 C: 0.538462 D: 0.615385 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 12 -6 B 4 0 -4 -10 2 C -4 4 0 0 -8 D -12 10 0 0 -14 E 6 -2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.538462 C: 0.000000 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982208 Cumulative probabilities = A: 0.000000 B: 0.538462 C: 0.538462 D: 0.615385 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 12 -6 B 4 0 -4 -10 2 C -4 4 0 0 -8 D -12 10 0 0 -14 E 6 -2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.538462 C: 0.000000 D: 0.076923 E: 0.384615 Sum of squares = 0.443786982208 Cumulative probabilities = A: 0.000000 B: 0.538462 C: 0.538462 D: 0.615385 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5169: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (13) A C B D E (8) A C D B E (7) E B D C A (6) E A B C D (6) E D B C A (5) B D C A E (5) E A D C B (4) E A C B D (4) D C B A E (4) D B E C A (4) B C A D E (4) A E C B D (4) D B C E A (3) E D A C B (2) E B A C D (2) E A C D B (2) D C A B E (2) C A D B E (2) B E D C A (2) E D B A C (1) E B D A C (1) E B C A D (1) E A D B C (1) D E B C A (1) D C A E B (1) C D A B E (1) C A B D E (1) B C D A E (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -12 -4 10 B 6 0 12 -10 14 C 12 -12 0 -10 6 D 4 10 10 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -4 10 B 6 0 12 -10 14 C 12 -12 0 -10 6 D 4 10 10 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=28 A=21 B=12 C=4 so C is eliminated. Round 2 votes counts: E=35 D=29 A=24 B=12 so B is eliminated. Round 3 votes counts: E=37 D=35 A=28 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:211 C:198 A:194 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -4 10 B 6 0 12 -10 14 C 12 -12 0 -10 6 D 4 10 10 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -4 10 B 6 0 12 -10 14 C 12 -12 0 -10 6 D 4 10 10 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -4 10 B 6 0 12 -10 14 C 12 -12 0 -10 6 D 4 10 10 0 14 E -10 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5170: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) A B C D E (8) E D C B A (7) C D A E B (7) B A E D C (7) C A D B E (6) B E A D C (5) A C D B E (5) E D B C A (4) B E D A C (4) A C B D E (4) E D C A B (3) E C D A B (3) D E C B A (3) E B A C D (2) D E C A B (2) C D E A B (2) B E A C D (2) B A E C D (2) B A D E C (2) B A D C E (2) E C D B A (1) E B A D C (1) D C E A B (1) D B A C E (1) D A C B E (1) C A D E B (1) B D E A C (1) B A C E D (1) B A C D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 20 2 0 B 10 0 8 6 8 C -20 -8 0 -8 -16 D -2 -6 8 0 -4 E 0 -8 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 20 2 0 B 10 0 8 6 8 C -20 -8 0 -8 -16 D -2 -6 8 0 -4 E 0 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998475 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=27 A=19 C=16 D=8 so D is eliminated. Round 2 votes counts: E=35 B=28 A=20 C=17 so C is eliminated. Round 3 votes counts: E=38 A=34 B=28 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:216 A:206 E:206 D:198 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 20 2 0 B 10 0 8 6 8 C -20 -8 0 -8 -16 D -2 -6 8 0 -4 E 0 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998475 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 20 2 0 B 10 0 8 6 8 C -20 -8 0 -8 -16 D -2 -6 8 0 -4 E 0 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998475 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 20 2 0 B 10 0 8 6 8 C -20 -8 0 -8 -16 D -2 -6 8 0 -4 E 0 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998475 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5171: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (13) C E A B D (10) B A D C E (7) D E C B A (5) D B E C A (5) E C D A B (4) D E C A B (4) D C E B A (4) B A D E C (4) B D A E C (3) B A C D E (3) E D A C B (2) E C A D B (2) D B C A E (2) C E A D B (2) C B E A D (2) B D C A E (2) B D A C E (2) A E C B D (2) A C E B D (2) A B E C D (2) A B C E D (2) E A C D B (1) E A C B D (1) D E A C B (1) D B E A C (1) D B C E A (1) D A B E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B E A (1) C A E B D (1) C A B E D (1) B C A E D (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -2 -8 0 B 18 0 2 -2 8 C 2 -2 0 -16 -6 D 8 2 16 0 22 E 0 -8 6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -2 -8 0 B 18 0 2 -2 8 C 2 -2 0 -16 -6 D 8 2 16 0 22 E 0 -8 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=22 C=21 E=10 A=10 so E is eliminated. Round 2 votes counts: D=39 C=27 B=22 A=12 so A is eliminated. Round 3 votes counts: D=39 C=33 B=28 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:213 C:189 E:188 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -2 -8 0 B 18 0 2 -2 8 C 2 -2 0 -16 -6 D 8 2 16 0 22 E 0 -8 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 -8 0 B 18 0 2 -2 8 C 2 -2 0 -16 -6 D 8 2 16 0 22 E 0 -8 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 -8 0 B 18 0 2 -2 8 C 2 -2 0 -16 -6 D 8 2 16 0 22 E 0 -8 6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5172: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) E A D C B (5) E A C D B (5) D B C A E (5) C A B D E (5) B C D A E (5) E D B A C (4) E B D C A (4) E A D B C (4) C B D A E (4) A E C D B (4) E A B D C (3) D C B A E (3) B D C E A (3) B D C A E (3) D C A B E (2) D B C E A (2) D A C B E (2) C A D B E (2) C A B E D (2) A E C B D (2) A C D E B (2) E D A B C (1) E B D A C (1) E B A C D (1) E A B C D (1) D E A B C (1) D B E C A (1) D B E A C (1) D A B C E (1) C E B A D (1) C D B A E (1) C B E A D (1) C B A D E (1) B E D C A (1) B E C D A (1) B D E C A (1) B C D E A (1) A D E C B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 2 4 -6 B -10 0 -10 2 0 C -2 10 0 0 -4 D -4 -2 0 0 -2 E 6 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.192993 C: 0.000000 D: 0.000000 E: 0.807007 Sum of squares = 0.688506591351 Cumulative probabilities = A: 0.000000 B: 0.192993 C: 0.192993 D: 0.192993 E: 1.000000 A B C D E A 0 10 2 4 -6 B -10 0 -10 2 0 C -2 10 0 0 -4 D -4 -2 0 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735081 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=18 C=17 B=15 A=11 so A is eliminated. Round 2 votes counts: E=45 D=20 C=20 B=15 so B is eliminated. Round 3 votes counts: E=47 D=27 C=26 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:206 A:205 C:202 D:196 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 4 -6 B -10 0 -10 2 0 C -2 10 0 0 -4 D -4 -2 0 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735081 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 4 -6 B -10 0 -10 2 0 C -2 10 0 0 -4 D -4 -2 0 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735081 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 4 -6 B -10 0 -10 2 0 C -2 10 0 0 -4 D -4 -2 0 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735081 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5173: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) C A E B D (9) D B E A C (8) B D E C A (8) A C D E B (8) E C B A D (6) D A B C E (6) D B A E C (5) B E D C A (4) E B C D A (3) B E C D A (3) E C A B D (2) D E B A C (2) D B A C E (2) C E B A D (2) C B E A D (2) B D A C E (2) A E C D B (2) A D C B E (2) A C D B E (2) E B D C A (1) E B C A D (1) E A C D B (1) D B E C A (1) D A E C B (1) D A E B C (1) C E A B D (1) C B A E D (1) C A B E D (1) B D E A C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 10 4 8 B 4 0 -4 -4 -4 C -10 4 0 10 0 D -4 4 -10 0 0 E -8 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.100000 D: 0.050000 E: 0.100000 Sum of squares = 0.334999999951 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.850000 D: 0.900000 E: 1.000000 A B C D E A 0 -4 10 4 8 B 4 0 -4 -4 -4 C -10 4 0 10 0 D -4 4 -10 0 0 E -8 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.100000 D: 0.050000 E: 0.100000 Sum of squares = 0.335000000007 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.850000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 B=18 C=16 E=14 so E is eliminated. Round 2 votes counts: A=27 D=26 C=24 B=23 so B is eliminated. Round 3 votes counts: D=42 C=31 A=27 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:209 C:202 E:198 B:196 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 10 4 8 B 4 0 -4 -4 -4 C -10 4 0 10 0 D -4 4 -10 0 0 E -8 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.100000 D: 0.050000 E: 0.100000 Sum of squares = 0.335000000007 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.850000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 4 8 B 4 0 -4 -4 -4 C -10 4 0 10 0 D -4 4 -10 0 0 E -8 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.100000 D: 0.050000 E: 0.100000 Sum of squares = 0.335000000007 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.850000 D: 0.900000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 4 8 B 4 0 -4 -4 -4 C -10 4 0 10 0 D -4 4 -10 0 0 E -8 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.100000 D: 0.050000 E: 0.100000 Sum of squares = 0.335000000007 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.850000 D: 0.900000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5174: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) B D E C A (5) B D C A E (5) E B D A C (4) D C B A E (4) C A E D B (4) C A D E B (4) B A E D C (4) A C B E D (4) E D B C A (3) E C A D B (3) E A B C D (3) D B E C A (3) B A C D E (3) E D C B A (2) E B A D C (2) E A C D B (2) E A C B D (2) D E B C A (2) D C E B A (2) D B C E A (2) D B C A E (2) C D A B E (2) B E D A C (2) B D E A C (2) B D A C E (2) A C E B D (2) A C B D E (2) A B C D E (2) E C D A B (1) E A B D C (1) D C B E A (1) D C A B E (1) C E A D B (1) C D E A B (1) B D C E A (1) A E C B D (1) A E B C D (1) A C E D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 2 6 10 B 20 0 20 18 16 C -2 -20 0 -18 12 D -6 -18 18 0 10 E -10 -16 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 2 6 10 B 20 0 20 18 16 C -2 -20 0 -18 12 D -6 -18 18 0 10 E -10 -16 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=23 D=17 A=15 C=12 so C is eliminated. Round 2 votes counts: B=33 E=24 A=23 D=20 so D is eliminated. Round 3 votes counts: B=45 E=29 A=26 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:237 D:202 A:199 C:186 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 2 6 10 B 20 0 20 18 16 C -2 -20 0 -18 12 D -6 -18 18 0 10 E -10 -16 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 2 6 10 B 20 0 20 18 16 C -2 -20 0 -18 12 D -6 -18 18 0 10 E -10 -16 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 2 6 10 B 20 0 20 18 16 C -2 -20 0 -18 12 D -6 -18 18 0 10 E -10 -16 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5175: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C A B (7) C E D B A (7) A B D E C (6) E C D B A (5) B C A E D (5) D E C A B (4) B A C E D (4) E D C B A (3) E D B C A (3) D A E C B (3) C E D A B (3) C D E A B (3) C A D E B (3) C A B D E (3) B E D A C (3) E C D A B (2) C E B D A (2) C A D B E (2) B A E D C (2) B A E C D (2) B A D C E (2) A B C D E (2) E D A C B (1) E C B D A (1) E B C D A (1) D E A C B (1) D C E A B (1) C D A E B (1) C B E D A (1) B E D C A (1) B C E D A (1) B C E A D (1) B A D E C (1) A D E B C (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -20 -8 -2 B 8 0 -8 -2 -4 C 20 8 0 18 4 D 8 2 -18 0 -10 E 2 4 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -20 -8 -2 B 8 0 -8 -2 -4 C 20 8 0 18 4 D 8 2 -18 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=25 E=23 A=12 D=9 so D is eliminated. Round 2 votes counts: B=31 E=28 C=26 A=15 so A is eliminated. Round 3 votes counts: B=41 E=32 C=27 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:225 E:206 B:197 D:191 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -20 -8 -2 B 8 0 -8 -2 -4 C 20 8 0 18 4 D 8 2 -18 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 -8 -2 B 8 0 -8 -2 -4 C 20 8 0 18 4 D 8 2 -18 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 -8 -2 B 8 0 -8 -2 -4 C 20 8 0 18 4 D 8 2 -18 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5176: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) A D B E C (9) C E B D A (8) E C B A D (6) C D A B E (6) C D B A E (5) E B C A D (4) E B A D C (4) E C B D A (2) E A B D C (2) D C A B E (2) D B A C E (2) D A C B E (2) D A B C E (2) C E D A B (2) C D A E B (2) C B D A E (2) B D A E C (2) A E D B C (2) A D E B C (2) A D B C E (2) E C A D B (1) E C A B D (1) E B C D A (1) E B A C D (1) E A D B C (1) D A B E C (1) C E D B A (1) C E A D B (1) C D B E A (1) C B D E A (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B D A C E (1) B C D E A (1) B A D E C (1) A E B D C (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 0 6 0 B 10 0 4 2 8 C 0 -4 0 0 -8 D -6 -2 0 0 0 E 0 -8 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 6 0 B 10 0 4 2 8 C 0 -4 0 0 -8 D -6 -2 0 0 0 E 0 -8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=23 A=21 B=17 D=9 so D is eliminated. Round 2 votes counts: C=32 A=26 E=23 B=19 so B is eliminated. Round 3 votes counts: E=35 C=33 A=32 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:212 E:200 A:198 D:196 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 6 0 B 10 0 4 2 8 C 0 -4 0 0 -8 D -6 -2 0 0 0 E 0 -8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 0 B 10 0 4 2 8 C 0 -4 0 0 -8 D -6 -2 0 0 0 E 0 -8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 0 B 10 0 4 2 8 C 0 -4 0 0 -8 D -6 -2 0 0 0 E 0 -8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5177: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (7) A E C B D (7) D B C A E (6) B C E D A (6) A D C E B (6) A C E D B (6) D B C E A (5) E B A C D (4) D A C B E (4) E C A B D (3) D A B C E (3) C B E D A (3) B C D E A (3) A D E C B (3) E A B C D (2) D C B E A (2) D C A B E (2) D A C E B (2) D A B E C (2) C E A B D (2) C D A E B (2) C B D E A (2) B D C E A (2) A E C D B (2) A D E B C (2) E C B A D (1) E B C A D (1) E A C B D (1) D C A E B (1) D B A E C (1) C E B D A (1) C E B A D (1) B E D C A (1) B E C A D (1) B E A D C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -4 -10 2 B -4 0 -4 0 2 C 4 4 0 14 20 D 10 0 -14 0 -2 E -2 -2 -20 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -10 2 B -4 0 -4 0 2 C 4 4 0 14 20 D 10 0 -14 0 -2 E -2 -2 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=28 A=28 B=21 E=12 C=11 so C is eliminated. Round 2 votes counts: D=30 A=28 B=26 E=16 so E is eliminated. Round 3 votes counts: A=36 B=34 D=30 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:221 B:197 D:197 A:196 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -10 2 B -4 0 -4 0 2 C 4 4 0 14 20 D 10 0 -14 0 -2 E -2 -2 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -10 2 B -4 0 -4 0 2 C 4 4 0 14 20 D 10 0 -14 0 -2 E -2 -2 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -10 2 B -4 0 -4 0 2 C 4 4 0 14 20 D 10 0 -14 0 -2 E -2 -2 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5178: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) E D B C A (5) A C B E D (5) D C B A E (4) C B A E D (4) C B A D E (4) B C A E D (4) A C B D E (4) E B A C D (3) E A D B C (3) D E C B A (3) D E B C A (3) D E A C B (3) D C B E A (3) E B D C A (2) E B D A C (2) E B C D A (2) E B C A D (2) D E C A B (2) D E A B C (2) D A E C B (2) C D B A E (2) B C E D A (2) A E D C B (2) A E C B D (2) A E B C D (2) A D C B E (2) A B E C D (2) E D B A C (1) E A B D C (1) E A B C D (1) D C E A B (1) D B C E A (1) D A C E B (1) D A C B E (1) C B D A E (1) C A B D E (1) B E C A D (1) B C D E A (1) B C A D E (1) A E C D B (1) A E B D C (1) A D E C B (1) A D C E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 0 0 B 2 0 -2 -2 -10 C -2 2 0 -6 -10 D 0 2 6 0 -10 E 0 10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.576035 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.423965 Sum of squares = 0.511562763865 Cumulative probabilities = A: 0.576035 B: 0.576035 C: 0.576035 D: 0.576035 E: 1.000000 A B C D E A 0 -2 2 0 0 B 2 0 -2 -2 -10 C -2 2 0 -6 -10 D 0 2 6 0 -10 E 0 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 A=25 C=12 B=9 so B is eliminated. Round 2 votes counts: E=29 D=26 A=25 C=20 so C is eliminated. Round 3 votes counts: A=39 E=31 D=30 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:200 D:199 B:194 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 0 0 B 2 0 -2 -2 -10 C -2 2 0 -6 -10 D 0 2 6 0 -10 E 0 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 0 B 2 0 -2 -2 -10 C -2 2 0 -6 -10 D 0 2 6 0 -10 E 0 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 0 B 2 0 -2 -2 -10 C -2 2 0 -6 -10 D 0 2 6 0 -10 E 0 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5179: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) A B C D E (9) D E B C A (7) E C A B D (6) E D A B C (5) D E A B C (5) C B A E D (5) B A C D E (5) A C B E D (5) E C B A D (4) B C A D E (4) A D B C E (3) E D C A B (2) E D B C A (2) E C D B A (2) E C D A B (2) D E B A C (2) D A B E C (2) C E B A D (2) C B A D E (2) C A B E D (2) E D A C B (1) E C B D A (1) E A C D B (1) E A C B D (1) D B A C E (1) D A E B C (1) D A B C E (1) C E A B D (1) B D A C E (1) B A D C E (1) A E C B D (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -6 8 -10 B 0 0 0 4 -16 C 6 0 0 8 -12 D -8 -4 -8 0 -10 E 10 16 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -6 8 -10 B 0 0 0 4 -16 C 6 0 0 8 -12 D -8 -4 -8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=20 D=19 C=12 B=11 so B is eliminated. Round 2 votes counts: E=38 A=26 D=20 C=16 so C is eliminated. Round 3 votes counts: E=41 A=39 D=20 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 C:201 A:196 B:194 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 8 -10 B 0 0 0 4 -16 C 6 0 0 8 -12 D -8 -4 -8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 8 -10 B 0 0 0 4 -16 C 6 0 0 8 -12 D -8 -4 -8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 8 -10 B 0 0 0 4 -16 C 6 0 0 8 -12 D -8 -4 -8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5180: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) D A E B C (10) C B E A D (9) E B C D A (8) C B A E D (7) B C E A D (6) D A E C B (4) A D C B E (4) A D B C E (4) A C D B E (4) C E B D A (3) B E C D A (3) A D B E C (3) E D C B A (2) E D B C A (2) E C B D A (2) E B D C A (2) B C E D A (2) B C A E D (2) A C B D E (2) E D B A C (1) E D A B C (1) D E C A B (1) D E B A C (1) C E D B A (1) C B E D A (1) C A B D E (1) B E A C D (1) A D C E B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -4 -8 -12 B 6 0 14 2 2 C 4 -14 0 6 -4 D 8 -2 -6 0 -8 E 12 -2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -8 -12 B 6 0 14 2 2 C 4 -14 0 6 -4 D 8 -2 -6 0 -8 E 12 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995481 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=22 A=20 E=18 B=14 so B is eliminated. Round 2 votes counts: C=32 D=26 E=22 A=20 so A is eliminated. Round 3 votes counts: D=39 C=39 E=22 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:212 E:211 C:196 D:196 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 -12 B 6 0 14 2 2 C 4 -14 0 6 -4 D 8 -2 -6 0 -8 E 12 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995481 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 -12 B 6 0 14 2 2 C 4 -14 0 6 -4 D 8 -2 -6 0 -8 E 12 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995481 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 -12 B 6 0 14 2 2 C 4 -14 0 6 -4 D 8 -2 -6 0 -8 E 12 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995481 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5181: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E C B D A (5) C D A E B (5) D A C B E (4) B A D E C (4) B A D C E (4) A D C B E (4) A C D E B (4) A B E D C (4) E C D B A (3) E C D A B (3) E A C D B (3) E A B C D (3) D C A B E (3) C E D A B (3) B E A D C (3) B A E D C (3) A D B C E (3) E C A D B (2) E B A C D (2) D C B A E (2) D B C A E (2) B E D A C (2) B E A C D (2) B D C E A (2) B D C A E (2) B D A C E (2) A B D C E (2) E C A B D (1) E B C A D (1) E A C B D (1) D C A E B (1) D B A C E (1) D A B C E (1) C E D B A (1) C A E D B (1) B E D C A (1) A E B C D (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 10 0 10 B -2 0 6 6 4 C -10 -6 0 -2 -4 D 0 -6 2 0 -6 E -10 -4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.893632 B: 0.000000 C: 0.000000 D: 0.106368 E: 0.000000 Sum of squares = 0.809892236238 Cumulative probabilities = A: 0.893632 B: 0.893632 C: 0.893632 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 0 10 B -2 0 6 6 4 C -10 -6 0 -2 -4 D 0 -6 2 0 -6 E -10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000000475 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=25 A=20 D=14 C=10 so C is eliminated. Round 2 votes counts: E=35 B=25 A=21 D=19 so D is eliminated. Round 3 votes counts: E=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:207 E:198 D:195 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 0 10 B -2 0 6 6 4 C -10 -6 0 -2 -4 D 0 -6 2 0 -6 E -10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000000475 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 0 10 B -2 0 6 6 4 C -10 -6 0 -2 -4 D 0 -6 2 0 -6 E -10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000000475 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 0 10 B -2 0 6 6 4 C -10 -6 0 -2 -4 D 0 -6 2 0 -6 E -10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000000475 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5182: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) E A C B D (9) B C A E D (8) D E B C A (7) D B C A E (6) A C E B D (5) A C B E D (5) E D A C B (4) E A C D B (4) D E B A C (4) D B E C A (4) C A B E D (4) B C A D E (4) B D C A E (3) E A D C B (2) E A B C D (2) D E A B C (2) C A B D E (2) E D B C A (1) E D A B C (1) E B D C A (1) E B C A D (1) E A B D C (1) D C B A E (1) D B C E A (1) D A C B E (1) C B A E D (1) B E C A D (1) B D E C A (1) B C E A D (1) A C E D B (1) Total count = 100 A B C D E A 0 10 6 2 -18 B -10 0 -2 -2 -14 C -6 2 0 -4 -14 D -2 2 4 0 -4 E 18 14 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 6 2 -18 B -10 0 -2 -2 -14 C -6 2 0 -4 -14 D -2 2 4 0 -4 E 18 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=26 B=18 A=11 C=7 so C is eliminated. Round 2 votes counts: D=38 E=26 B=19 A=17 so A is eliminated. Round 3 votes counts: D=38 E=32 B=30 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:200 D:200 C:189 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 2 -18 B -10 0 -2 -2 -14 C -6 2 0 -4 -14 D -2 2 4 0 -4 E 18 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 -18 B -10 0 -2 -2 -14 C -6 2 0 -4 -14 D -2 2 4 0 -4 E 18 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 -18 B -10 0 -2 -2 -14 C -6 2 0 -4 -14 D -2 2 4 0 -4 E 18 14 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5183: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) E C B D A (11) B A D E C (9) E C D A B (7) E B C A D (7) B E A D C (7) C D A E B (5) A D B C E (5) D A C E B (4) D A C B E (4) E C B A D (3) B A D C E (3) E D C A B (2) E D A B C (2) C E B D A (2) C E B A D (2) C D E A B (2) C D A B E (2) B E C A D (2) E D A C B (1) E C D B A (1) E B A D C (1) D C A E B (1) C B A D E (1) B E A C D (1) B C A D E (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -24 -14 -28 B 2 0 -24 0 -28 C 24 24 0 20 -8 D 14 0 -20 0 -24 E 28 28 8 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -24 -14 -28 B 2 0 -24 0 -28 C 24 24 0 20 -8 D 14 0 -20 0 -24 E 28 28 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=27 B=23 D=9 A=6 so A is eliminated. Round 2 votes counts: E=35 C=27 B=23 D=15 so D is eliminated. Round 3 votes counts: C=37 E=35 B=28 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:244 C:230 D:185 B:175 A:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -24 -14 -28 B 2 0 -24 0 -28 C 24 24 0 20 -8 D 14 0 -20 0 -24 E 28 28 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -24 -14 -28 B 2 0 -24 0 -28 C 24 24 0 20 -8 D 14 0 -20 0 -24 E 28 28 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -24 -14 -28 B 2 0 -24 0 -28 C 24 24 0 20 -8 D 14 0 -20 0 -24 E 28 28 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5184: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (7) E B D A C (6) D E B A C (5) D A E B C (5) D A C E B (5) E B A C D (4) D C A B E (4) C B E A D (4) C A D B E (4) B E C D A (4) B E C A D (4) E D B A C (3) D E A B C (3) D A C B E (3) C D B A E (3) A D E B C (3) A C D B E (3) A C B E D (3) D A E C B (2) C D B E A (2) C D A B E (2) C B E D A (2) C A B E D (2) B E D C A (2) B C E D A (2) A E B D C (2) A C D E B (2) E B D C A (1) E B C D A (1) E A B D C (1) C B A D E (1) C A B D E (1) B D E C A (1) A E D B C (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 10 20 -12 10 B -10 0 -4 -18 -4 C -20 4 0 -12 2 D 12 18 12 0 16 E -10 4 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 20 -12 10 B -10 0 -4 -18 -4 C -20 4 0 -12 2 D 12 18 12 0 16 E -10 4 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 C=21 E=16 B=13 so B is eliminated. Round 2 votes counts: D=28 E=26 C=23 A=23 so C is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 A:214 E:188 C:187 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 20 -12 10 B -10 0 -4 -18 -4 C -20 4 0 -12 2 D 12 18 12 0 16 E -10 4 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 20 -12 10 B -10 0 -4 -18 -4 C -20 4 0 -12 2 D 12 18 12 0 16 E -10 4 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 20 -12 10 B -10 0 -4 -18 -4 C -20 4 0 -12 2 D 12 18 12 0 16 E -10 4 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5185: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (21) D B C E A (14) E C B A D (8) D B C A E (7) E C B D A (5) E A C B D (5) D A B C E (3) B D C E A (3) A C B E D (3) E C A B D (2) D E B C A (2) D A E B C (2) D A B E C (2) B C E D A (2) B C D E A (2) B C A E D (2) A D E C B (2) E D C B A (1) E A D C B (1) E A C D B (1) D E A C B (1) D B E C A (1) D B A C E (1) C E B A D (1) B C E A D (1) B C D A E (1) B C A D E (1) A E C D B (1) A D C B E (1) A D B C E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -6 6 0 B 4 0 -8 18 -8 C 6 8 0 14 -10 D -6 -18 -14 0 -10 E 0 8 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.395738 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.604261 Sum of squares = 0.521740898935 Cumulative probabilities = A: 0.395738 B: 0.395738 C: 0.395739 D: 0.395739 E: 1.000000 A B C D E A 0 -4 -6 6 0 B 4 0 -8 18 -8 C 6 8 0 14 -10 D -6 -18 -14 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=31 E=23 B=12 C=1 so C is eliminated. Round 2 votes counts: D=33 A=31 E=24 B=12 so B is eliminated. Round 3 votes counts: D=39 A=34 E=27 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:214 C:209 B:203 A:198 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -6 6 0 B 4 0 -8 18 -8 C 6 8 0 14 -10 D -6 -18 -14 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 0 B 4 0 -8 18 -8 C 6 8 0 14 -10 D -6 -18 -14 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 0 B 4 0 -8 18 -8 C 6 8 0 14 -10 D -6 -18 -14 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5186: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) B A C D E (6) E D C A B (5) D A C E B (5) B E A D C (5) B C A E D (5) E B D A C (4) B C E A D (4) B C A D E (4) A D C B E (4) E C D B A (2) E C D A B (2) E B D C A (2) D C A E B (2) C D A E B (2) C B A D E (2) C A D B E (2) C A B D E (2) B E A C D (2) B A D E C (2) A D C E B (2) A D B C E (2) A C D B E (2) A B C D E (2) E D A C B (1) E D A B C (1) E C B D A (1) E B C D A (1) D E C A B (1) D E A C B (1) D C E A B (1) D A E C B (1) C E D B A (1) C E D A B (1) C E B D A (1) C B E D A (1) C A D E B (1) B E D A C (1) B A E D C (1) B A E C D (1) A D B E C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -8 26 2 B 14 0 12 20 24 C 8 -12 0 14 6 D -26 -20 -14 0 -6 E -2 -24 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 26 2 B 14 0 12 20 24 C 8 -12 0 14 6 D -26 -20 -14 0 -6 E -2 -24 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 E=19 A=15 C=13 D=11 so D is eliminated. Round 2 votes counts: B=42 E=21 A=21 C=16 so C is eliminated. Round 3 votes counts: B=45 A=30 E=25 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:235 C:208 A:203 E:187 D:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 26 2 B 14 0 12 20 24 C 8 -12 0 14 6 D -26 -20 -14 0 -6 E -2 -24 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 26 2 B 14 0 12 20 24 C 8 -12 0 14 6 D -26 -20 -14 0 -6 E -2 -24 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 26 2 B 14 0 12 20 24 C 8 -12 0 14 6 D -26 -20 -14 0 -6 E -2 -24 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5187: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) D B C E A (7) D B C A E (7) B C D E A (6) A E C B D (6) D E A B C (5) E D A B C (4) D E B C A (4) D A E C B (4) D A C B E (4) B C E A D (4) B C E D A (3) A E D C B (3) E C B A D (2) E B C A D (2) E A D C B (2) E A C B D (2) E A B C D (2) D C B A E (2) B D C E A (2) B C D A E (2) A C D E B (2) A C B E D (2) E D A C B (1) E B A C D (1) D C A B E (1) D A E B C (1) D A C E B (1) D A B C E (1) C E B A D (1) C A B D E (1) B C A E D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -8 -10 4 B 6 0 4 -4 6 C 8 -4 0 -2 20 D 10 4 2 0 6 E -4 -6 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -10 4 B 6 0 4 -4 6 C 8 -4 0 -2 20 D 10 4 2 0 6 E -4 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=18 A=18 E=16 C=11 so C is eliminated. Round 2 votes counts: D=37 B=27 A=19 E=17 so E is eliminated. Round 3 votes counts: D=42 B=33 A=25 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:211 D:211 B:206 A:190 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -8 -10 4 B 6 0 4 -4 6 C 8 -4 0 -2 20 D 10 4 2 0 6 E -4 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -10 4 B 6 0 4 -4 6 C 8 -4 0 -2 20 D 10 4 2 0 6 E -4 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -10 4 B 6 0 4 -4 6 C 8 -4 0 -2 20 D 10 4 2 0 6 E -4 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5188: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) B D C A E (6) B D A E C (6) D E A B C (5) D C B E A (5) E A C D B (4) D E A C B (4) D B E A C (4) C E A D B (4) C A E B D (4) B C D A E (4) B C A E D (4) A E C B D (4) C B D E A (3) C B A E D (3) B A C E D (3) A E B C D (3) E A D C B (2) D E C A B (2) D C E A B (2) C D E A B (2) B A E C D (2) A E D B C (2) E D A B C (1) D C E B A (1) D B C A E (1) D B A E C (1) D A E B C (1) C E D A B (1) C E B A D (1) C D E B A (1) C D B E A (1) C B D A E (1) C A B E D (1) B A E D C (1) A E C D B (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -8 -18 0 B 10 0 6 -4 8 C 8 -6 0 -6 10 D 18 4 6 0 14 E 0 -8 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -18 0 B 10 0 6 -4 8 C 8 -6 0 -6 10 D 18 4 6 0 14 E 0 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=26 C=22 A=12 E=7 so E is eliminated. Round 2 votes counts: D=34 B=26 C=22 A=18 so A is eliminated. Round 3 votes counts: D=38 C=31 B=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:210 C:203 E:184 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -8 -18 0 B 10 0 6 -4 8 C 8 -6 0 -6 10 D 18 4 6 0 14 E 0 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -18 0 B 10 0 6 -4 8 C 8 -6 0 -6 10 D 18 4 6 0 14 E 0 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -18 0 B 10 0 6 -4 8 C 8 -6 0 -6 10 D 18 4 6 0 14 E 0 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5189: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) D B A C E (9) E B A D C (8) B A D C E (8) E D B A C (7) E C D A B (7) D A B C E (7) E C A D B (6) C D A B E (5) C A B D E (4) E C B A D (3) C A D B E (3) B D A C E (3) E B D A C (2) E B A C D (2) D C A B E (2) B A C D E (2) E D C A B (1) D E A B C (1) D B A E C (1) D A C B E (1) C E D A B (1) C E A D B (1) C E A B D (1) B D A E C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 6 4 -6 B -8 0 4 -4 -6 C -6 -4 0 -4 -4 D -4 4 4 0 -4 E 6 6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 4 -6 B -8 0 4 -4 -6 C -6 -4 0 -4 -4 D -4 4 4 0 -4 E 6 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=49 D=21 C=15 B=14 A=1 so A is eliminated. Round 2 votes counts: E=49 D=21 C=15 B=15 so C is eliminated. Round 3 votes counts: E=52 D=29 B=19 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:206 D:200 B:193 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 4 -6 B -8 0 4 -4 -6 C -6 -4 0 -4 -4 D -4 4 4 0 -4 E 6 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 4 -6 B -8 0 4 -4 -6 C -6 -4 0 -4 -4 D -4 4 4 0 -4 E 6 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 4 -6 B -8 0 4 -4 -6 C -6 -4 0 -4 -4 D -4 4 4 0 -4 E 6 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5190: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (15) E B A C D (13) C A D B E (11) D E B C A (9) D C A B E (8) E D B A C (4) B E D A C (4) E D B C A (3) E A C B D (3) A C B E D (3) D E C A B (2) D B E C A (2) C A E B D (2) C A B E D (2) C A B D E (2) B E A C D (2) A C B D E (2) A B C E D (2) E D C A B (1) E C A B D (1) E B A D C (1) E A B C D (1) D E B A C (1) D C A E B (1) C D A B E (1) C A D E B (1) B E A D C (1) B A C E D (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 8 -2 -26 B 12 0 18 12 -18 C -8 -18 0 -4 -26 D 2 -12 4 0 -20 E 26 18 26 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 8 -2 -26 B 12 0 18 12 -18 C -8 -18 0 -4 -26 D 2 -12 4 0 -20 E 26 18 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 D=23 C=19 B=8 A=8 so B is eliminated. Round 2 votes counts: E=49 D=23 C=19 A=9 so A is eliminated. Round 3 votes counts: E=49 C=28 D=23 so D is eliminated. Round 4 votes counts: E=63 C=37 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:245 B:212 D:187 A:184 C:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 8 -2 -26 B 12 0 18 12 -18 C -8 -18 0 -4 -26 D 2 -12 4 0 -20 E 26 18 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 8 -2 -26 B 12 0 18 12 -18 C -8 -18 0 -4 -26 D 2 -12 4 0 -20 E 26 18 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 8 -2 -26 B 12 0 18 12 -18 C -8 -18 0 -4 -26 D 2 -12 4 0 -20 E 26 18 26 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5191: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (6) B C A E D (6) E D B A C (5) E D A B C (5) E B D A C (5) C B A E D (5) C D A E B (4) E D B C A (3) E A D B C (3) D E C A B (3) D E A C B (3) D C E A B (3) C B A D E (3) C A D B E (3) B E A C D (3) A B E D C (3) D E C B A (2) D E A B C (2) D A E C B (2) C D E B A (2) C D A B E (2) C B E A D (2) B E C A D (2) B E A D C (2) B C E A D (2) B A E C D (2) B A C E D (2) A D E B C (2) A C B D E (2) E B D C A (1) E B A D C (1) E A B D C (1) D A C E B (1) C B D A E (1) A E D B C (1) A D B E C (1) A C D B E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 12 0 B -2 0 10 4 2 C 0 -10 0 0 -4 D -12 -4 0 0 -10 E 0 -2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.759017 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.240983 Sum of squares = 0.634179456687 Cumulative probabilities = A: 0.759017 B: 0.759017 C: 0.759017 D: 0.759017 E: 1.000000 A B C D E A 0 2 0 12 0 B -2 0 10 4 2 C 0 -10 0 0 -4 D -12 -4 0 0 -10 E 0 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499867 Sum of squares = 0.500000035383 Cumulative probabilities = A: 0.500133 B: 0.500133 C: 0.500133 D: 0.500133 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=24 B=19 D=16 A=13 so A is eliminated. Round 2 votes counts: C=31 E=25 B=25 D=19 so D is eliminated. Round 3 votes counts: E=39 C=35 B=26 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:207 B:207 E:206 C:193 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 12 0 B -2 0 10 4 2 C 0 -10 0 0 -4 D -12 -4 0 0 -10 E 0 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499867 Sum of squares = 0.500000035383 Cumulative probabilities = A: 0.500133 B: 0.500133 C: 0.500133 D: 0.500133 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 12 0 B -2 0 10 4 2 C 0 -10 0 0 -4 D -12 -4 0 0 -10 E 0 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499867 Sum of squares = 0.500000035383 Cumulative probabilities = A: 0.500133 B: 0.500133 C: 0.500133 D: 0.500133 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 12 0 B -2 0 10 4 2 C 0 -10 0 0 -4 D -12 -4 0 0 -10 E 0 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499867 Sum of squares = 0.500000035383 Cumulative probabilities = A: 0.500133 B: 0.500133 C: 0.500133 D: 0.500133 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5192: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (13) E A B D C (11) C D B A E (7) D E C A B (6) D C E A B (6) C B D A E (6) E A D B C (5) D E A C B (5) C B A D E (5) B A C E D (5) E D A B C (4) B A E C D (4) D C B A E (3) C B A E D (3) A E B C D (3) E A B C D (2) D C B E A (2) B C A E D (2) A B E C D (2) E D A C B (1) D C E B A (1) C D E B A (1) B C A D E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 20 14 -10 -14 B -20 0 6 -8 -16 C -14 -6 0 -16 -14 D 10 8 16 0 12 E 14 16 14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 14 -10 -14 B -20 0 6 -8 -16 C -14 -6 0 -16 -14 D 10 8 16 0 12 E 14 16 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=23 C=22 B=12 A=7 so A is eliminated. Round 2 votes counts: D=36 E=26 C=23 B=15 so B is eliminated. Round 3 votes counts: D=36 E=33 C=31 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:216 A:205 B:181 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 14 -10 -14 B -20 0 6 -8 -16 C -14 -6 0 -16 -14 D 10 8 16 0 12 E 14 16 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 14 -10 -14 B -20 0 6 -8 -16 C -14 -6 0 -16 -14 D 10 8 16 0 12 E 14 16 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 14 -10 -14 B -20 0 6 -8 -16 C -14 -6 0 -16 -14 D 10 8 16 0 12 E 14 16 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5193: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) A B E D C (9) C D E B A (7) C D B E A (7) D E C A B (6) E A D C B (5) C B D E A (5) B A C E D (5) A E D B C (5) E D C A B (4) D C E B A (4) B A C D E (4) A B E C D (4) E D A C B (3) B C D A E (3) B C A D E (3) A B C E D (3) D C B E A (2) C E D B A (2) A E D C B (2) A E C B D (2) E A C D B (1) D E C B A (1) D A E B C (1) B D C A E (1) B D A C E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 10 6 6 B -10 0 -2 0 -4 C -10 2 0 -8 -6 D -6 0 8 0 -8 E -6 4 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 6 6 B -10 0 -2 0 -4 C -10 2 0 -8 -6 D -6 0 8 0 -8 E -6 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=21 B=17 D=14 E=13 so E is eliminated. Round 2 votes counts: A=41 D=21 C=21 B=17 so B is eliminated. Round 3 votes counts: A=50 C=27 D=23 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:206 D:197 B:192 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 6 6 B -10 0 -2 0 -4 C -10 2 0 -8 -6 D -6 0 8 0 -8 E -6 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 6 6 B -10 0 -2 0 -4 C -10 2 0 -8 -6 D -6 0 8 0 -8 E -6 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 6 6 B -10 0 -2 0 -4 C -10 2 0 -8 -6 D -6 0 8 0 -8 E -6 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5194: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (10) A D C B E (10) C A E D B (9) E B C D A (6) C E A B D (6) D B A E C (5) C A D B E (5) B E D A C (5) A C D B E (5) E C B A D (4) C A D E B (4) B D E A C (4) E B D A C (3) C E B A D (3) C A E B D (3) A D B E C (3) E C B D A (2) D B E A C (2) D A B C E (2) A D B C E (2) E C A B D (1) E B A D C (1) D B A C E (1) D A B E C (1) C E B D A (1) B E C D A (1) B D E C A (1) Total count = 100 A B C D E A 0 2 -12 12 0 B -2 0 -6 2 -6 C 12 6 0 0 2 D -12 -2 0 0 -10 E 0 6 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.910992 D: 0.089008 E: 0.000000 Sum of squares = 0.837828875029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.910992 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 12 0 B -2 0 -6 2 -6 C 12 6 0 0 2 D -12 -2 0 0 -10 E 0 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833333 D: 0.166667 E: 0.000000 Sum of squares = 0.722222222603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=27 A=20 D=11 B=11 so D is eliminated. Round 2 votes counts: C=31 E=27 A=23 B=19 so B is eliminated. Round 3 votes counts: E=40 C=31 A=29 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 E:207 A:201 B:194 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 12 0 B -2 0 -6 2 -6 C 12 6 0 0 2 D -12 -2 0 0 -10 E 0 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833333 D: 0.166667 E: 0.000000 Sum of squares = 0.722222222603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 12 0 B -2 0 -6 2 -6 C 12 6 0 0 2 D -12 -2 0 0 -10 E 0 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833333 D: 0.166667 E: 0.000000 Sum of squares = 0.722222222603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 12 0 B -2 0 -6 2 -6 C 12 6 0 0 2 D -12 -2 0 0 -10 E 0 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833333 D: 0.166667 E: 0.000000 Sum of squares = 0.722222222603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5195: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) E D B A C (8) E B D A C (8) E B A D C (6) E B A C D (6) C A D B E (6) C A B E D (6) C D A B E (5) C A B D E (5) E B C A D (3) D A B C E (3) E D C B A (2) E B D C A (2) D E C B A (2) D E B A C (2) C D E A B (2) B A D E C (2) E D B C A (1) E C D B A (1) E B C D A (1) D E C A B (1) D C E A B (1) D C A E B (1) D B E A C (1) D B A E C (1) D A C B E (1) C E D A B (1) C E A D B (1) C D A E B (1) C A E B D (1) B E D A C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E C D (1) B A C E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 -18 -4 B 4 0 2 -4 -2 C 8 -2 0 -12 -4 D 18 4 12 0 -4 E 4 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 -18 -4 B 4 0 2 -4 -2 C 8 -2 0 -12 -4 D 18 4 12 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=28 D=24 B=8 A=2 so A is eliminated. Round 2 votes counts: E=38 C=29 D=24 B=9 so B is eliminated. Round 3 votes counts: E=41 C=31 D=28 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:207 B:200 C:195 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 -18 -4 B 4 0 2 -4 -2 C 8 -2 0 -12 -4 D 18 4 12 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -18 -4 B 4 0 2 -4 -2 C 8 -2 0 -12 -4 D 18 4 12 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -18 -4 B 4 0 2 -4 -2 C 8 -2 0 -12 -4 D 18 4 12 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5196: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) D B A E C (9) D C A E B (4) B E A D C (4) E C B A D (3) D B A C E (3) C E D B A (3) C E D A B (3) C E A D B (3) C E A B D (3) B A E D C (3) A D C B E (3) E B C D A (2) E B C A D (2) E A C B D (2) D C E B A (2) D C A B E (2) D B C A E (2) D A C E B (2) C D A E B (2) C A E D B (2) B E D A C (2) B E C A D (2) B D A E C (2) A E C B D (2) A D C E B (2) A C D E B (2) A B E D C (2) A B E C D (2) E C B D A (1) E C A B D (1) E B A C D (1) E A B C D (1) D C B A E (1) D B C E A (1) D A C B E (1) C E B D A (1) C D E B A (1) C D E A B (1) B E D C A (1) B D E A C (1) A E B C D (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 14 -14 18 B -6 0 4 -20 4 C -14 -4 0 -16 10 D 14 20 16 0 4 E -18 -4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 -14 18 B -6 0 4 -20 4 C -14 -4 0 -16 10 D 14 20 16 0 4 E -18 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=19 A=16 B=15 E=13 so E is eliminated. Round 2 votes counts: D=37 C=24 B=20 A=19 so A is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:212 B:191 C:188 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 14 -14 18 B -6 0 4 -20 4 C -14 -4 0 -16 10 D 14 20 16 0 4 E -18 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -14 18 B -6 0 4 -20 4 C -14 -4 0 -16 10 D 14 20 16 0 4 E -18 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -14 18 B -6 0 4 -20 4 C -14 -4 0 -16 10 D 14 20 16 0 4 E -18 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5197: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E A D (10) B C D E A (9) D B C A E (6) E A C B D (5) C B A E D (5) B D C E A (5) D E A B C (4) D B C E A (4) C B E D A (4) C B D E A (4) A E D C B (4) A E C B D (4) E D A B C (2) E C B A D (2) E C A B D (2) E A D C B (2) D E B C A (2) D B A C E (2) D A B C E (2) A D E B C (2) E D B C A (1) E A D B C (1) D E B A C (1) C E B A D (1) C B D A E (1) C B A D E (1) B C E D A (1) B C D A E (1) A E D B C (1) A C B E D (1) Total count = 100 A B C D E A 0 -20 -18 -18 -20 B 20 0 8 12 12 C 18 -8 0 2 14 D 18 -12 -2 0 8 E 20 -12 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -18 -18 -20 B 20 0 8 12 12 C 18 -8 0 2 14 D 18 -12 -2 0 8 E 20 -12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=26 B=16 E=15 A=12 so A is eliminated. Round 2 votes counts: D=33 C=27 E=24 B=16 so B is eliminated. Round 3 votes counts: D=38 C=38 E=24 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:226 C:213 D:206 E:193 A:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -18 -18 -20 B 20 0 8 12 12 C 18 -8 0 2 14 D 18 -12 -2 0 8 E 20 -12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 -18 -20 B 20 0 8 12 12 C 18 -8 0 2 14 D 18 -12 -2 0 8 E 20 -12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 -18 -20 B 20 0 8 12 12 C 18 -8 0 2 14 D 18 -12 -2 0 8 E 20 -12 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5198: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) E C B A D (7) A D B E C (7) C B D E A (6) D A B C E (5) E C D A B (4) C D B A E (4) C B E D A (4) E C D B A (3) E A D B C (3) C E B D A (3) B C D A E (3) B A C D E (3) A B D E C (3) E C B D A (2) E A D C B (2) E A B D C (2) D C E A B (2) C E D B A (2) C D B E A (2) C B D A E (2) B D C A E (2) A B E D C (2) A B D C E (2) E D C A B (1) E D A C B (1) E C A D B (1) E B C A D (1) E B A C D (1) E A C B D (1) D C B A E (1) D C A B E (1) D B A C E (1) D A E C B (1) D A E B C (1) D A C B E (1) C D E B A (1) B E C A D (1) B C E A D (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -16 -8 -6 -2 B 16 0 -4 10 20 C 8 4 0 4 10 D 6 -10 -4 0 14 E 2 -20 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -6 -2 B 16 0 -4 10 20 C 8 4 0 4 10 D 6 -10 -4 0 14 E 2 -20 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=24 B=18 A=16 D=13 so D is eliminated. Round 2 votes counts: E=29 C=28 A=24 B=19 so B is eliminated. Round 3 votes counts: A=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:213 D:203 A:184 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -8 -6 -2 B 16 0 -4 10 20 C 8 4 0 4 10 D 6 -10 -4 0 14 E 2 -20 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -6 -2 B 16 0 -4 10 20 C 8 4 0 4 10 D 6 -10 -4 0 14 E 2 -20 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -6 -2 B 16 0 -4 10 20 C 8 4 0 4 10 D 6 -10 -4 0 14 E 2 -20 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5199: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (12) D B A E C (9) B D A C E (7) C E A D B (6) E A C D B (5) C E A B D (5) C B E D A (5) E C A D B (4) B D A E C (4) A D E B C (4) B D C A E (3) B C D E A (3) B C D A E (3) A E D C B (3) A E D B C (3) E A D C B (2) C B D E A (2) C B D A E (2) B C E D A (2) A D E C B (2) E C A B D (1) E B A D C (1) E A D B C (1) E A B C D (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B C E (1) C E D A B (1) C E B A D (1) C D B A E (1) C D A B E (1) C A E D B (1) B E C A D (1) Total count = 100 A B C D E A 0 8 14 -18 18 B -8 0 16 -18 16 C -14 -16 0 -10 -8 D 18 18 10 0 14 E -18 -16 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 -18 18 B -8 0 16 -18 16 C -14 -16 0 -10 -8 D 18 18 10 0 14 E -18 -16 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 B=23 E=15 A=12 so A is eliminated. Round 2 votes counts: D=31 C=25 B=23 E=21 so E is eliminated. Round 3 votes counts: D=40 C=35 B=25 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:211 B:203 E:180 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 14 -18 18 B -8 0 16 -18 16 C -14 -16 0 -10 -8 D 18 18 10 0 14 E -18 -16 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 -18 18 B -8 0 16 -18 16 C -14 -16 0 -10 -8 D 18 18 10 0 14 E -18 -16 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 -18 18 B -8 0 16 -18 16 C -14 -16 0 -10 -8 D 18 18 10 0 14 E -18 -16 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5200: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) D E B C A (7) E D C A B (6) A C E B D (6) D E C A B (5) D C E A B (5) B D E A C (5) E D B C A (3) D E C B A (3) D B E C A (3) C A D E B (3) C A D B E (3) B A E C D (3) B A C D E (3) E C D A B (2) D B C E A (2) B E A D C (2) B D C A E (2) B D A C E (2) A C B E D (2) A C B D E (2) A B C E D (2) A B C D E (2) E D B A C (1) E C A D B (1) E B D A C (1) E B A C D (1) D C E B A (1) D C A B E (1) D B E A C (1) D B C A E (1) C E D A B (1) C D E A B (1) C D A E B (1) C A B D E (1) B E A C D (1) B A E D C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 -26 -8 -4 B -14 0 -10 -24 -20 C 26 10 0 -4 4 D 8 24 4 0 8 E 4 20 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -26 -8 -4 B -14 0 -10 -24 -20 C 26 10 0 -4 4 D 8 24 4 0 8 E 4 20 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=21 B=19 A=16 E=15 so E is eliminated. Round 2 votes counts: D=39 C=24 B=21 A=16 so A is eliminated. Round 3 votes counts: D=39 C=34 B=27 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:218 E:206 A:188 B:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -26 -8 -4 B -14 0 -10 -24 -20 C 26 10 0 -4 4 D 8 24 4 0 8 E 4 20 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -26 -8 -4 B -14 0 -10 -24 -20 C 26 10 0 -4 4 D 8 24 4 0 8 E 4 20 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -26 -8 -4 B -14 0 -10 -24 -20 C 26 10 0 -4 4 D 8 24 4 0 8 E 4 20 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5201: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (13) A D C E B (9) E D B C A (5) E B C D A (5) B C E D A (5) A D E C B (5) A C D B E (5) E D C B A (4) D C A E B (3) C B D E A (3) A D C B E (3) E B D C A (2) E B D A C (2) E A D C B (2) D E C B A (2) C D B E A (2) C D A B E (2) C A D B E (2) B E C A D (2) B E A C D (2) A E D C B (2) A E D B C (2) A E B D C (2) A B E C D (2) A B C D E (2) E D C A B (1) E B A D C (1) D E C A B (1) D C E B A (1) D C E A B (1) D A E C B (1) B C E A D (1) B C D E A (1) B C A D E (1) B A E C D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -14 -8 -12 B 6 0 0 -6 -2 C 14 0 0 0 -16 D 8 6 0 0 -10 E 12 2 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -14 -8 -12 B 6 0 0 -6 -2 C 14 0 0 0 -16 D 8 6 0 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=26 E=22 D=9 C=9 so D is eliminated. Round 2 votes counts: A=35 B=26 E=25 C=14 so C is eliminated. Round 3 votes counts: A=42 B=31 E=27 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:220 D:202 B:199 C:199 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -14 -8 -12 B 6 0 0 -6 -2 C 14 0 0 0 -16 D 8 6 0 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -8 -12 B 6 0 0 -6 -2 C 14 0 0 0 -16 D 8 6 0 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -8 -12 B 6 0 0 -6 -2 C 14 0 0 0 -16 D 8 6 0 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5202: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) A B D C E (11) E C D B A (9) A B C D E (9) E D C A B (6) D E B C A (4) C E B D A (4) A D E B C (4) A D B E C (4) D B A E C (3) D A E B C (3) C B E D A (3) B A D C E (3) A B D E C (3) D E C B A (2) D A B E C (2) B A C D E (2) A B C E D (2) E D A C B (1) E C D A B (1) E A C D B (1) D E B A C (1) D E A C B (1) D E A B C (1) D B E C A (1) C E D B A (1) C B E A D (1) C A E B D (1) C A B E D (1) B D C E A (1) B C A E D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 6 6 -10 2 B -6 0 12 -12 -2 C -6 -12 0 -26 -18 D 10 12 26 0 10 E -2 2 18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -10 2 B -6 0 12 -12 -2 C -6 -12 0 -26 -18 D 10 12 26 0 10 E -2 2 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=29 D=18 C=11 B=7 so B is eliminated. Round 2 votes counts: A=40 E=29 D=19 C=12 so C is eliminated. Round 3 votes counts: A=43 E=38 D=19 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:229 E:204 A:202 B:196 C:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -10 2 B -6 0 12 -12 -2 C -6 -12 0 -26 -18 D 10 12 26 0 10 E -2 2 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -10 2 B -6 0 12 -12 -2 C -6 -12 0 -26 -18 D 10 12 26 0 10 E -2 2 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -10 2 B -6 0 12 -12 -2 C -6 -12 0 -26 -18 D 10 12 26 0 10 E -2 2 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5203: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) E B D A C (9) D C A B E (9) C D A B E (7) E B A D C (5) D E B C A (5) E A B C D (4) C A D E B (4) C A D B E (3) B E A D C (3) B A E C D (3) A C E B D (3) A B C E D (3) E C A B D (2) D C B A E (2) D B C A E (2) D B A C E (2) C A E B D (2) B D E A C (2) A C B E D (2) A C B D E (2) A B E C D (2) E D C B A (1) E D B A C (1) E C D A B (1) E C A D B (1) D E C B A (1) D C A E B (1) D B C E A (1) C E D A B (1) C E A D B (1) C D A E B (1) C A E D B (1) C A B D E (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 8 4 B -6 0 8 10 -10 C -6 -8 0 12 -2 D -8 -10 -12 0 -12 E -4 10 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 8 4 B -6 0 8 10 -10 C -6 -8 0 12 -2 D -8 -10 -12 0 -12 E -4 10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=23 C=21 A=14 B=8 so B is eliminated. Round 2 votes counts: E=37 D=25 C=21 A=17 so A is eliminated. Round 3 votes counts: E=43 C=32 D=25 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:210 B:201 C:198 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 8 4 B -6 0 8 10 -10 C -6 -8 0 12 -2 D -8 -10 -12 0 -12 E -4 10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 4 B -6 0 8 10 -10 C -6 -8 0 12 -2 D -8 -10 -12 0 -12 E -4 10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 4 B -6 0 8 10 -10 C -6 -8 0 12 -2 D -8 -10 -12 0 -12 E -4 10 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5204: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) A B E D C (7) A E B D C (6) E D C A B (5) E C D A B (5) B C D A E (5) A E B C D (5) E A B C D (4) D C B A E (4) E C D B A (3) E A D C B (3) D E C A B (3) C D E B A (3) B D A C E (3) B A C E D (3) A D B C E (3) A B E C D (3) A B D C E (3) E A B D C (2) D C B E A (2) C B D A E (2) B A D C E (2) A B D E C (2) E A C B D (1) D C E B A (1) D B A C E (1) D A B C E (1) C E D B A (1) C E B D A (1) C D B A E (1) B D C A E (1) B C A D E (1) B A E C D (1) B A C D E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 10 6 -2 14 B -10 0 12 6 12 C -6 -12 0 -2 -4 D 2 -6 2 0 -2 E -14 -12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -2 14 B -10 0 12 6 12 C -6 -12 0 -2 -4 D 2 -6 2 0 -2 E -14 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.432098764783 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=23 C=17 B=17 D=12 so D is eliminated. Round 2 votes counts: A=32 E=26 C=24 B=18 so B is eliminated. Round 3 votes counts: A=43 C=31 E=26 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:210 D:198 E:190 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 -2 14 B -10 0 12 6 12 C -6 -12 0 -2 -4 D 2 -6 2 0 -2 E -14 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.432098764783 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -2 14 B -10 0 12 6 12 C -6 -12 0 -2 -4 D 2 -6 2 0 -2 E -14 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.432098764783 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -2 14 B -10 0 12 6 12 C -6 -12 0 -2 -4 D 2 -6 2 0 -2 E -14 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.111111 C: 0.000000 D: 0.555556 E: 0.000000 Sum of squares = 0.432098764783 Cumulative probabilities = A: 0.333333 B: 0.444444 C: 0.444444 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5205: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (12) B C A D E (8) E A D C B (6) A E C D B (6) B C D A E (5) C B A E D (4) B C E A D (4) E D C B A (3) E D A C B (3) D E A C B (3) D E A B C (3) D B A C E (3) B D C E A (3) B D C A E (3) D E B A C (2) D B E A C (2) D B A E C (2) D A E C B (2) D A E B C (2) C E A B D (2) C B E A D (2) C A E B D (2) C A B E D (2) A E D C B (2) A D E C B (2) E C A D B (1) E C A B D (1) D B E C A (1) D A B C E (1) C E B A D (1) B E D C A (1) B C D E A (1) B A C D E (1) A E C B D (1) A D B C E (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -12 20 22 B 16 0 12 10 14 C 12 -12 0 10 14 D -20 -10 -10 0 -8 E -22 -14 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 20 22 B 16 0 12 10 14 C 12 -12 0 10 14 D -20 -10 -10 0 -8 E -22 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=21 E=14 A=14 C=13 so C is eliminated. Round 2 votes counts: B=44 D=21 A=18 E=17 so E is eliminated. Round 3 votes counts: B=45 A=28 D=27 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:212 A:207 E:179 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 20 22 B 16 0 12 10 14 C 12 -12 0 10 14 D -20 -10 -10 0 -8 E -22 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 20 22 B 16 0 12 10 14 C 12 -12 0 10 14 D -20 -10 -10 0 -8 E -22 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 20 22 B 16 0 12 10 14 C 12 -12 0 10 14 D -20 -10 -10 0 -8 E -22 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5206: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (7) E C D B A (6) B A D E C (6) D A B C E (5) E C D A B (4) C E D A B (4) B E C A D (4) A B D C E (4) E C B A D (3) D C E A B (3) D C A E B (3) D A C E B (3) C E B A D (3) C E A D B (3) E B C A D (2) D B E A C (2) D B A E C (2) D A C B E (2) C E A B D (2) C D E A B (2) C D A E B (2) B A E D C (2) B A D C E (2) A D C B E (2) A D B C E (2) A C B D E (2) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E B C A (1) D A E C B (1) D A E B C (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E C D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 2 4 6 B -2 0 -2 -4 -2 C -2 2 0 6 12 D -4 4 -6 0 2 E -6 2 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 4 6 B -2 0 -2 -4 -2 C -2 2 0 6 12 D -4 4 -6 0 2 E -6 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=24 C=19 E=18 A=12 so A is eliminated. Round 2 votes counts: B=32 D=28 C=22 E=18 so E is eliminated. Round 3 votes counts: C=36 B=36 D=28 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:207 D:198 B:195 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 6 B -2 0 -2 -4 -2 C -2 2 0 6 12 D -4 4 -6 0 2 E -6 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 6 B -2 0 -2 -4 -2 C -2 2 0 6 12 D -4 4 -6 0 2 E -6 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 6 B -2 0 -2 -4 -2 C -2 2 0 6 12 D -4 4 -6 0 2 E -6 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5207: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) B C A D E (9) B A C E D (6) E D B A C (5) D E C A B (5) A B C E D (5) B C D E A (4) A E D C B (4) C B D A E (3) C A B D E (3) E D A B C (2) E A D C B (2) D E C B A (2) D E B C A (2) D E A C B (2) D C E B A (2) C D E A B (2) C D A E B (2) A E D B C (2) A E B D C (2) A C B D E (2) A B E D C (2) E D C B A (1) E D B C A (1) E A D B C (1) E A B D C (1) D C E A B (1) D B E C A (1) C D E B A (1) C D B E A (1) C B D E A (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E C A (1) B C D A E (1) B A E D C (1) B A E C D (1) A E B C D (1) A D E C B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 12 -6 -4 B -10 0 2 -6 -10 C -12 -2 0 -10 -10 D 6 6 10 0 -6 E 4 10 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 12 -6 -4 B -10 0 2 -6 -10 C -12 -2 0 -10 -10 D 6 6 10 0 -6 E 4 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=25 B=25 A=21 D=15 C=14 so C is eliminated. Round 2 votes counts: B=29 E=25 A=25 D=21 so D is eliminated. Round 3 votes counts: E=42 B=31 A=27 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:208 A:206 B:188 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 12 -6 -4 B -10 0 2 -6 -10 C -12 -2 0 -10 -10 D 6 6 10 0 -6 E 4 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -6 -4 B -10 0 2 -6 -10 C -12 -2 0 -10 -10 D 6 6 10 0 -6 E 4 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -6 -4 B -10 0 2 -6 -10 C -12 -2 0 -10 -10 D 6 6 10 0 -6 E 4 10 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5208: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) A D C B E (10) E C B A D (9) D A B C E (6) D A E B C (5) A C B D E (4) E B C D A (3) D A B E C (3) E D B C A (2) E D B A C (2) E C B D A (2) E C A D B (2) E C A B D (2) E B D C A (2) D E B A C (2) D B E A C (2) D B A C E (2) C E A B D (2) C B E A D (2) B D A C E (2) B C E A D (2) A D E C B (2) A D C E B (2) E D A C B (1) E C D A B (1) E A C D B (1) D E B C A (1) D E A C B (1) D E A B C (1) D B A E C (1) C B A E D (1) C A E D B (1) C A B E D (1) B E C D A (1) B D C E A (1) B C E D A (1) B C A D E (1) A D B C E (1) A C E D B (1) A C D E B (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 16 -6 B -2 0 -14 -4 -12 C -6 14 0 0 8 D -16 4 0 0 2 E 6 12 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 2 6 16 -6 B -2 0 -14 -4 -12 C -6 14 0 0 8 D -16 4 0 0 2 E 6 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 A=24 C=17 B=8 so B is eliminated. Round 2 votes counts: E=28 D=27 A=24 C=21 so C is eliminated. Round 3 votes counts: E=45 A=28 D=27 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:209 C:208 E:204 D:195 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 16 -6 B -2 0 -14 -4 -12 C -6 14 0 0 8 D -16 4 0 0 2 E 6 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 16 -6 B -2 0 -14 -4 -12 C -6 14 0 0 8 D -16 4 0 0 2 E 6 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 16 -6 B -2 0 -14 -4 -12 C -6 14 0 0 8 D -16 4 0 0 2 E 6 12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.300000 Sum of squares = 0.340000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5209: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (19) E C A D B (13) B E C A D (11) B D A C E (8) D A C B E (7) A C D E B (5) D A B C E (4) B E C D A (4) E C A B D (3) B D A E C (3) E C D A B (2) E B C A D (2) D B A C E (2) C A D E B (2) B E D A C (2) B D E A C (2) A D C E B (2) D C A E B (1) C E A D B (1) C D A E B (1) C A E D B (1) B E D C A (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 22 16 -12 18 B -22 0 -14 -20 -4 C -16 14 0 -6 10 D 12 20 6 0 16 E -18 4 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 16 -12 18 B -22 0 -14 -20 -4 C -16 14 0 -6 10 D 12 20 6 0 16 E -18 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=33 E=20 A=7 C=5 so C is eliminated. Round 2 votes counts: B=35 D=34 E=21 A=10 so A is eliminated. Round 3 votes counts: D=43 B=35 E=22 so E is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:222 C:201 E:180 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 16 -12 18 B -22 0 -14 -20 -4 C -16 14 0 -6 10 D 12 20 6 0 16 E -18 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 -12 18 B -22 0 -14 -20 -4 C -16 14 0 -6 10 D 12 20 6 0 16 E -18 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 -12 18 B -22 0 -14 -20 -4 C -16 14 0 -6 10 D 12 20 6 0 16 E -18 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5210: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) B E A C D (8) D C A E B (7) D C B E A (5) B C D E A (5) A E C B D (5) D E A C B (4) D E A B C (4) C A E D B (4) B E D A C (4) B E A D C (4) E B A D C (3) C D B A E (3) B D C E A (3) B C A E D (3) A E B C D (3) E A D B C (2) E A B C D (2) D A E C B (2) C D A B E (2) C B D A E (2) C A E B D (2) E D A B C (1) D C E A B (1) D C B A E (1) D B E C A (1) D B E A C (1) C B A E D (1) C B A D E (1) C A D E B (1) B D E A C (1) B C D A E (1) A E C D B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 14 4 -18 B -6 0 14 18 -6 C -14 -14 0 -8 -14 D -4 -18 8 0 -10 E 18 6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 14 4 -18 B -6 0 14 18 -6 C -14 -14 0 -8 -14 D -4 -18 8 0 -10 E 18 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 E=18 C=16 A=11 so A is eliminated. Round 2 votes counts: B=30 E=27 D=26 C=17 so C is eliminated. Round 3 votes counts: E=34 B=34 D=32 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:210 A:203 D:188 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 4 -18 B -6 0 14 18 -6 C -14 -14 0 -8 -14 D -4 -18 8 0 -10 E 18 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 4 -18 B -6 0 14 18 -6 C -14 -14 0 -8 -14 D -4 -18 8 0 -10 E 18 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 4 -18 B -6 0 14 18 -6 C -14 -14 0 -8 -14 D -4 -18 8 0 -10 E 18 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5211: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D A B E C (8) C E B A D (8) E C D B A (7) A B D C E (6) D E A B C (5) D A B C E (4) A B D E C (4) D A E B C (3) C E B D A (3) C D B E A (3) C B A D E (3) B A C D E (3) A D B E C (3) E D C A B (2) E D A B C (2) E C B D A (2) E C A D B (2) E C A B D (2) D C B A E (2) C E D B A (2) C B E A D (2) B C A D E (2) A D B C E (2) A B C D E (2) E D C B A (1) E D A C B (1) E C D A B (1) E A D C B (1) D E C A B (1) C D E B A (1) C B A E D (1) B C A E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -8 4 -8 B -2 0 -6 -2 -4 C 8 6 0 8 -10 D -4 2 -8 0 4 E 8 4 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826445 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.636364 E: 1.000000 A B C D E A 0 2 -8 4 -8 B -2 0 -6 -2 -4 C 8 6 0 8 -10 D -4 2 -8 0 4 E 8 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=23 C=23 A=19 B=6 so B is eliminated. Round 2 votes counts: E=29 C=26 D=23 A=22 so A is eliminated. Round 3 votes counts: D=38 E=31 C=31 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:206 D:197 A:195 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 4 -8 B -2 0 -6 -2 -4 C 8 6 0 8 -10 D -4 2 -8 0 4 E 8 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.636364 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 -8 B -2 0 -6 -2 -4 C 8 6 0 8 -10 D -4 2 -8 0 4 E 8 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.636364 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 -8 B -2 0 -6 -2 -4 C 8 6 0 8 -10 D -4 2 -8 0 4 E 8 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.181818 D: 0.636364 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5212: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) C B D E A (7) C B A E D (7) C A D E B (7) D E A B C (6) C D E B A (5) A E D B C (5) B C A E D (4) A D E C B (4) D E B A C (3) B D E C A (3) B A E D C (3) E A D B C (2) D E C B A (2) D E A C B (2) C D B E A (2) C D A E B (2) C B A D E (2) C A B E D (2) B E D C A (2) B C E D A (2) A E D C B (2) E D B A C (1) E D A B C (1) D E C A B (1) D C E A B (1) D B E C A (1) D B E A C (1) C D E A B (1) C B E A D (1) C A B D E (1) B E A D C (1) B C D E A (1) B A C E D (1) A E B D C (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -8 -4 -8 B 14 0 -6 -2 2 C 8 6 0 -2 0 D 4 2 2 0 6 E 8 -2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -4 -8 B 14 0 -6 -2 2 C 8 6 0 -2 0 D 4 2 2 0 6 E 8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=25 D=17 A=17 E=4 so E is eliminated. Round 2 votes counts: C=37 B=25 D=19 A=19 so D is eliminated. Round 3 votes counts: C=41 B=31 A=28 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:207 C:206 B:204 E:200 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -8 -4 -8 B 14 0 -6 -2 2 C 8 6 0 -2 0 D 4 2 2 0 6 E 8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -4 -8 B 14 0 -6 -2 2 C 8 6 0 -2 0 D 4 2 2 0 6 E 8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -4 -8 B 14 0 -6 -2 2 C 8 6 0 -2 0 D 4 2 2 0 6 E 8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5213: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) E A B D C (9) C B A D E (8) B A D E C (8) C E D B A (7) E D A B C (6) E A D B C (5) E C D A B (4) E C A B D (4) C E B A D (4) C D B A E (4) D B A C E (3) D A B E C (3) C E A B D (3) A B D E C (3) E D C A B (2) E A B C D (2) E D A C B (1) E C A D B (1) D E A B C (1) D C B A E (1) D B A E C (1) C E A D B (1) C D E B A (1) C B D A E (1) C B A E D (1) B C A D E (1) B A D C E (1) B A C D E (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 16 -8 8 -24 B -16 0 -8 -8 -26 C 8 8 0 8 -4 D -8 8 -8 0 -22 E 24 26 4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -8 8 -24 B -16 0 -8 -8 -26 C 8 8 0 8 -4 D -8 8 -8 0 -22 E 24 26 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 E=34 B=11 D=9 A=5 so A is eliminated. Round 2 votes counts: C=41 E=34 B=14 D=11 so D is eliminated. Round 3 votes counts: C=42 E=36 B=22 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:238 C:210 A:196 D:185 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -8 8 -24 B -16 0 -8 -8 -26 C 8 8 0 8 -4 D -8 8 -8 0 -22 E 24 26 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 8 -24 B -16 0 -8 -8 -26 C 8 8 0 8 -4 D -8 8 -8 0 -22 E 24 26 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 8 -24 B -16 0 -8 -8 -26 C 8 8 0 8 -4 D -8 8 -8 0 -22 E 24 26 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5214: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) E B C A D (8) D A C B E (8) A D C B E (8) E B C D A (7) D B E A C (7) B E D C A (7) B D E A C (7) A C D E B (5) D B A E C (4) A C D B E (4) E B D C A (3) C E A B D (3) C A E D B (3) C A D E B (3) E C B A D (2) B E D A C (2) E C A B D (1) D E C B A (1) D B A C E (1) D A E C B (1) D A B C E (1) B E C A D (1) B E A C D (1) B D A E C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 4 0 0 B 4 0 4 6 8 C -4 -4 0 -4 -6 D 0 -6 4 0 4 E 0 -8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 0 0 B 4 0 4 6 8 C -4 -4 0 -4 -6 D 0 -6 4 0 4 E 0 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=21 B=19 A=19 C=18 so C is eliminated. Round 2 votes counts: A=34 E=24 D=23 B=19 so B is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:211 D:201 A:200 E:197 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 0 0 B 4 0 4 6 8 C -4 -4 0 -4 -6 D 0 -6 4 0 4 E 0 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 0 0 B 4 0 4 6 8 C -4 -4 0 -4 -6 D 0 -6 4 0 4 E 0 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 0 0 B 4 0 4 6 8 C -4 -4 0 -4 -6 D 0 -6 4 0 4 E 0 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5215: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) A D E B C (11) C B E D A (10) B D E A C (7) E D B A C (5) C A D B E (4) A E D B C (4) E B D A C (3) C A D E B (3) A C D E B (3) E D A B C (2) E A D C B (2) D E B A C (2) D A B E C (2) C B D E A (2) C B A D E (2) B E D A C (2) B E C D A (2) B D A E C (2) A E D C B (2) A D E C B (2) A D B E C (2) E B D C A (1) E A D B C (1) D E A B C (1) C E B A D (1) C E A D B (1) C B D A E (1) C A B D E (1) B E D C A (1) B D E C A (1) B D C A E (1) B C E D A (1) B C D E A (1) A E C D B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 12 6 8 B -10 0 4 -22 -16 C -12 -4 0 -10 -12 D -6 22 10 0 -2 E -8 16 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 6 8 B -10 0 4 -22 -16 C -12 -4 0 -10 -12 D -6 22 10 0 -2 E -8 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=27 B=18 E=14 D=5 so D is eliminated. Round 2 votes counts: C=36 A=29 B=18 E=17 so E is eliminated. Round 3 votes counts: C=36 A=35 B=29 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:212 E:211 C:181 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 6 8 B -10 0 4 -22 -16 C -12 -4 0 -10 -12 D -6 22 10 0 -2 E -8 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 6 8 B -10 0 4 -22 -16 C -12 -4 0 -10 -12 D -6 22 10 0 -2 E -8 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 6 8 B -10 0 4 -22 -16 C -12 -4 0 -10 -12 D -6 22 10 0 -2 E -8 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5216: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (15) E A C D B (14) B D C A E (11) C D B E A (8) A E B C D (8) D C B E A (6) E C D A B (5) E A B D C (5) C D E B A (5) A E C B D (3) D B C E A (2) C E D A B (2) B A D C E (2) A B E D C (2) E D B C A (1) E C D B A (1) E A B C D (1) D C B A E (1) C D B A E (1) C D A E B (1) C D A B E (1) B D A E C (1) B D A C E (1) B A C D E (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 18 10 6 0 B -18 0 2 2 -24 C -10 -2 0 4 -16 D -6 -2 -4 0 -16 E 0 24 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.659399 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.340601 Sum of squares = 0.55081605845 Cumulative probabilities = A: 0.659399 B: 0.659399 C: 0.659399 D: 0.659399 E: 1.000000 A B C D E A 0 18 10 6 0 B -18 0 2 2 -24 C -10 -2 0 4 -16 D -6 -2 -4 0 -16 E 0 24 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=27 C=18 B=16 D=9 so D is eliminated. Round 2 votes counts: A=30 E=27 C=25 B=18 so B is eliminated. Round 3 votes counts: C=38 A=35 E=27 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:228 A:217 C:188 D:186 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 6 0 B -18 0 2 2 -24 C -10 -2 0 4 -16 D -6 -2 -4 0 -16 E 0 24 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 6 0 B -18 0 2 2 -24 C -10 -2 0 4 -16 D -6 -2 -4 0 -16 E 0 24 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 6 0 B -18 0 2 2 -24 C -10 -2 0 4 -16 D -6 -2 -4 0 -16 E 0 24 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5217: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) C B E A D (7) E B C D A (6) D B E A C (5) D E B A C (4) C A B E D (4) A D C E B (4) A C D E B (4) E B D C A (3) D E A B C (3) C B E D A (3) A E C B D (3) A C D B E (3) E D B A C (2) E B C A D (2) D B E C A (2) D A B C E (2) C E B A D (2) C A B D E (2) B C E D A (2) A E C D B (2) A D E C B (2) A D C B E (2) A C E D B (2) A C E B D (2) A C B E D (2) E B A C D (1) E A B D C (1) D C A B E (1) D B C E A (1) D B A E C (1) D A C B E (1) C E A B D (1) C B A D E (1) C A E B D (1) C A D B E (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D E A (1) A E D B C (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 6 12 4 0 B -6 0 -2 -2 -8 C -12 2 0 8 2 D -4 2 -8 0 2 E 0 8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.701732 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.298268 Sum of squares = 0.581391855603 Cumulative probabilities = A: 0.701732 B: 0.701732 C: 0.701732 D: 0.701732 E: 1.000000 A B C D E A 0 6 12 4 0 B -6 0 -2 -2 -8 C -12 2 0 8 2 D -4 2 -8 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=27 C=22 E=15 B=7 so B is eliminated. Round 2 votes counts: D=29 A=29 C=25 E=17 so E is eliminated. Round 3 votes counts: D=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:211 E:202 C:200 D:196 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 4 0 B -6 0 -2 -2 -8 C -12 2 0 8 2 D -4 2 -8 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 4 0 B -6 0 -2 -2 -8 C -12 2 0 8 2 D -4 2 -8 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 4 0 B -6 0 -2 -2 -8 C -12 2 0 8 2 D -4 2 -8 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5218: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (15) B A D C E (13) B D C E A (11) D C E A B (7) B A E C D (7) B D C A E (5) A E C D B (5) A D C E B (5) E B C D A (3) D C B E A (3) C D E A B (3) A D C B E (3) A C D E B (3) A B E C D (3) E A C D B (2) D C E B A (2) B E C A D (2) A E B C D (2) D C A B E (1) C D A E B (1) B E C D A (1) B E A C D (1) B D E C A (1) B D A E C (1) Total count = 100 A B C D E A 0 0 -10 -8 -2 B 0 0 0 0 4 C 10 0 0 -4 14 D 8 0 4 0 18 E 2 -4 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.503759 C: 0.000000 D: 0.496241 E: 0.000000 Sum of squares = 0.500028252831 Cumulative probabilities = A: 0.000000 B: 0.503759 C: 0.503759 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -8 -2 B 0 0 0 0 4 C 10 0 0 -4 14 D 8 0 4 0 18 E 2 -4 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=21 E=20 D=13 C=4 so C is eliminated. Round 2 votes counts: B=42 A=21 E=20 D=17 so D is eliminated. Round 3 votes counts: B=45 E=32 A=23 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:215 C:210 B:202 A:190 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -10 -8 -2 B 0 0 0 0 4 C 10 0 0 -4 14 D 8 0 4 0 18 E 2 -4 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -8 -2 B 0 0 0 0 4 C 10 0 0 -4 14 D 8 0 4 0 18 E 2 -4 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -8 -2 B 0 0 0 0 4 C 10 0 0 -4 14 D 8 0 4 0 18 E 2 -4 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5219: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A B D E C (7) C D E B A (6) E D B C A (5) A B E D C (5) E B D A C (4) D E B C A (4) D C E B A (4) C E D B A (4) A C D B E (4) D B E C A (3) C A D E B (3) C A D B E (3) B D E A C (3) A D B C E (3) A C B D E (3) E C D B A (2) D B E A C (2) C A E D B (2) B E D A C (2) B E A D C (2) E C B D A (1) E B C D A (1) E B A D C (1) E B A C D (1) D B A E C (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E B D (1) B A E D C (1) A E C B D (1) A C E B D (1) A C B E D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 -12 -14 B 10 0 14 -6 -8 C 6 -14 0 -16 -8 D 12 6 16 0 6 E 14 8 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -12 -14 B 10 0 14 -6 -8 C 6 -14 0 -16 -8 D 12 6 16 0 6 E 14 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=25 E=23 D=16 B=8 so B is eliminated. Round 2 votes counts: A=29 E=27 C=25 D=19 so D is eliminated. Round 3 votes counts: E=39 A=32 C=29 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:220 E:212 B:205 C:184 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -6 -12 -14 B 10 0 14 -6 -8 C 6 -14 0 -16 -8 D 12 6 16 0 6 E 14 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -12 -14 B 10 0 14 -6 -8 C 6 -14 0 -16 -8 D 12 6 16 0 6 E 14 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -12 -14 B 10 0 14 -6 -8 C 6 -14 0 -16 -8 D 12 6 16 0 6 E 14 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5220: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C E A B D (11) B A D E C (9) B D A E C (6) D B A C E (4) C E D A B (4) B C A E D (4) E C A B D (3) E A C B D (3) D B C A E (3) D A E B C (3) D A B E C (3) C E A D B (3) C B E A D (3) E A C D B (2) D E A C B (2) D C B A E (2) C E B A D (2) B D A C E (2) B A E C D (2) A E B D C (2) A B E D C (2) A B D E C (2) E A B C D (1) D E C A B (1) D C E B A (1) D C E A B (1) C E D B A (1) C D E B A (1) C D E A B (1) C D B E A (1) B C D E A (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 14 2 16 B 10 0 16 8 16 C -14 -16 0 -12 -8 D -2 -8 12 0 12 E -16 -16 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 2 16 B 10 0 16 8 16 C -14 -16 0 -12 -8 D -2 -8 12 0 12 E -16 -16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=27 B=25 E=9 A=7 so A is eliminated. Round 2 votes counts: D=33 B=29 C=27 E=11 so E is eliminated. Round 3 votes counts: C=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:225 A:211 D:207 E:182 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 2 16 B 10 0 16 8 16 C -14 -16 0 -12 -8 D -2 -8 12 0 12 E -16 -16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 2 16 B 10 0 16 8 16 C -14 -16 0 -12 -8 D -2 -8 12 0 12 E -16 -16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 2 16 B 10 0 16 8 16 C -14 -16 0 -12 -8 D -2 -8 12 0 12 E -16 -16 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5221: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) C D B A E (6) A C B D E (6) E B D C A (5) A C D B E (5) D B C E A (4) B D E C A (4) A B C D E (4) E B D A C (3) E A C D B (3) C D B E A (3) C A D B E (3) B E D C A (3) A E B D C (3) A E B C D (3) A B E D C (3) E A B D C (2) D B E C A (2) C A D E B (2) B D C E A (2) B A E D C (2) A C E D B (2) A B E C D (2) E D C B A (1) E B A D C (1) E A D B C (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B E A (1) C E D B A (1) C E D A B (1) C B A D E (1) C A B D E (1) B E D A C (1) B D A C E (1) B C D A E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -6 -2 0 B 8 0 16 2 12 C 6 -16 0 -4 -6 D 2 -2 4 0 2 E 0 -12 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -2 0 B 8 0 16 2 12 C 6 -16 0 -4 -6 D 2 -2 4 0 2 E 0 -12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=25 C=18 B=14 D=10 so D is eliminated. Round 2 votes counts: A=33 E=27 C=20 B=20 so C is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:203 E:196 A:192 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -2 0 B 8 0 16 2 12 C 6 -16 0 -4 -6 D 2 -2 4 0 2 E 0 -12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -2 0 B 8 0 16 2 12 C 6 -16 0 -4 -6 D 2 -2 4 0 2 E 0 -12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -2 0 B 8 0 16 2 12 C 6 -16 0 -4 -6 D 2 -2 4 0 2 E 0 -12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5222: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) A D B E C (10) B E A D C (7) A B D E C (7) E C B D A (6) B E C D A (6) A D C B E (6) C D A E B (5) C E B D A (4) E B C D A (3) D A C E B (3) C D E A B (3) B C E A D (3) A D C E B (3) A D B C E (3) E B D A C (2) D C A E B (2) C E D A B (2) C A D B E (2) E B D C A (1) D A E C B (1) D A E B C (1) C E D B A (1) C B E A D (1) C B A D E (1) B E A C D (1) B C E D A (1) B A E D C (1) B A D E C (1) B A D C E (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -2 18 -2 B 0 0 18 12 24 C 2 -18 0 0 -16 D -18 -12 0 0 2 E 2 -24 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.320575 B: 0.679425 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.564386482812 Cumulative probabilities = A: 0.320575 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 18 -2 B 0 0 18 12 24 C 2 -18 0 0 -16 D -18 -12 0 0 2 E 2 -24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=31 A=31 C=19 E=12 D=7 so D is eliminated. Round 2 votes counts: A=36 B=31 C=21 E=12 so E is eliminated. Round 3 votes counts: B=37 A=36 C=27 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:227 A:207 E:196 D:186 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 18 -2 B 0 0 18 12 24 C 2 -18 0 0 -16 D -18 -12 0 0 2 E 2 -24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 18 -2 B 0 0 18 12 24 C 2 -18 0 0 -16 D -18 -12 0 0 2 E 2 -24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 18 -2 B 0 0 18 12 24 C 2 -18 0 0 -16 D -18 -12 0 0 2 E 2 -24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5223: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (15) A D B E C (13) C A E D B (5) C A D E B (5) B D E A C (5) E B D C A (4) D E B A C (4) B E D A C (4) A C D E B (4) E D B A C (3) E B D A C (3) D B E A C (3) D B A E C (3) A D E B C (3) A C D B E (3) E B C D A (2) D A B E C (2) C E A D B (2) C A D B E (2) C A B E D (2) B E D C A (2) E C D A B (1) E C B D A (1) E A D B C (1) D A E B C (1) C E B A D (1) C A E B D (1) C A B D E (1) B E C D A (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 10 -8 -4 B 2 0 10 -16 -14 C -10 -10 0 -8 -12 D 8 16 8 0 4 E 4 14 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -8 -4 B 2 0 10 -16 -14 C -10 -10 0 -8 -12 D 8 16 8 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=26 E=15 D=13 B=12 so B is eliminated. Round 2 votes counts: C=34 A=26 E=22 D=18 so D is eliminated. Round 3 votes counts: E=34 C=34 A=32 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:213 A:198 B:191 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -8 -4 B 2 0 10 -16 -14 C -10 -10 0 -8 -12 D 8 16 8 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -8 -4 B 2 0 10 -16 -14 C -10 -10 0 -8 -12 D 8 16 8 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -8 -4 B 2 0 10 -16 -14 C -10 -10 0 -8 -12 D 8 16 8 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5224: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (16) A B C E D (8) B C A D E (7) C B D E A (6) A E D B C (6) D E C B A (5) A E B D C (5) E A D C B (4) D E C A B (4) D E A C B (3) B A C E D (3) A E B C D (3) E D C A B (2) E D A B C (2) C D B E A (2) C B A E D (2) C B A D E (2) E D C B A (1) D E B A C (1) D E A B C (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C E A (1) D A E B C (1) C E B A D (1) C D E B A (1) C B E D A (1) C B D A E (1) B C D E A (1) B C D A E (1) B C A E D (1) B A C D E (1) A E C B D (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 18 16 -6 -10 B -18 0 -8 -8 -16 C -16 8 0 -14 -18 D 6 8 14 0 -14 E 10 16 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 16 -6 -10 B -18 0 -8 -8 -16 C -16 8 0 -14 -18 D 6 8 14 0 -14 E 10 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 D=19 C=16 B=14 so B is eliminated. Round 2 votes counts: A=30 C=26 E=25 D=19 so D is eliminated. Round 3 votes counts: E=40 A=31 C=29 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 A:209 D:207 C:180 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 16 -6 -10 B -18 0 -8 -8 -16 C -16 8 0 -14 -18 D 6 8 14 0 -14 E 10 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 -6 -10 B -18 0 -8 -8 -16 C -16 8 0 -14 -18 D 6 8 14 0 -14 E 10 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 -6 -10 B -18 0 -8 -8 -16 C -16 8 0 -14 -18 D 6 8 14 0 -14 E 10 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5225: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) C E A B D (8) C E D B A (7) E C D B A (6) D B A C E (6) A B D C E (6) D E C B A (5) A B D E C (4) E D C B A (3) E C D A B (3) E C A B D (3) C E A D B (3) C D E B A (3) B A D E C (3) A B C D E (3) D E B A C (2) D B E A C (2) D B A E C (2) C D B A E (2) B D A E C (2) B D A C E (2) B A D C E (2) A B E D C (2) E D B A C (1) E C A D B (1) E A B D C (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C A E (1) C D B E A (1) C A E B D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -20 -22 -22 B 6 0 -16 -22 -18 C 20 16 0 4 14 D 22 22 -4 0 -2 E 22 18 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -22 -22 B 6 0 -16 -22 -18 C 20 16 0 4 14 D 22 22 -4 0 -2 E 22 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=21 E=18 A=17 B=9 so B is eliminated. Round 2 votes counts: C=35 D=25 A=22 E=18 so E is eliminated. Round 3 votes counts: C=48 D=29 A=23 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:219 E:214 B:175 A:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -20 -22 -22 B 6 0 -16 -22 -18 C 20 16 0 4 14 D 22 22 -4 0 -2 E 22 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -22 -22 B 6 0 -16 -22 -18 C 20 16 0 4 14 D 22 22 -4 0 -2 E 22 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -22 -22 B 6 0 -16 -22 -18 C 20 16 0 4 14 D 22 22 -4 0 -2 E 22 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5226: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) A D E C B (7) D A C B E (6) E B C A D (5) B C E A D (5) D A E C B (4) D A E B C (4) D A B C E (4) A C E B D (4) E C B A D (3) E B C D A (3) C B D A E (3) B C D E A (3) A C E D B (3) D B E C A (2) D B A C E (2) D A C E B (2) C B E A D (2) C A B E D (2) B E C A D (2) B D C E A (2) B C D A E (2) A E D C B (2) A C D B E (2) A C B E D (2) E D B C A (1) E D B A C (1) E B D C A (1) E A C B D (1) E A B C D (1) D E B C A (1) D E A B C (1) D B C E A (1) D B C A E (1) D A B E C (1) C B A E D (1) C A B D E (1) B E D C A (1) A E C D B (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 0 -8 12 B 0 0 4 6 10 C 0 -4 0 10 16 D 8 -6 -10 0 2 E -12 -10 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.231014 B: 0.768986 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.644706665411 Cumulative probabilities = A: 0.231014 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -8 12 B 0 0 4 6 10 C 0 -4 0 10 16 D 8 -6 -10 0 2 E -12 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204102942 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=23 A=23 E=16 C=9 so C is eliminated. Round 2 votes counts: D=29 B=29 A=26 E=16 so E is eliminated. Round 3 votes counts: B=41 D=31 A=28 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:210 A:202 D:197 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -8 12 B 0 0 4 6 10 C 0 -4 0 10 16 D 8 -6 -10 0 2 E -12 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204102942 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 12 B 0 0 4 6 10 C 0 -4 0 10 16 D 8 -6 -10 0 2 E -12 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204102942 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 12 B 0 0 4 6 10 C 0 -4 0 10 16 D 8 -6 -10 0 2 E -12 -10 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204102942 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5227: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) A C D E B (8) E B C D A (7) B E C A D (6) E B C A D (5) C E B D A (5) B E D C A (4) B E A D C (4) D A B E C (3) C E D B A (3) C A D E B (3) B E C D A (3) B D E A C (3) B A D E C (3) D B A E C (2) D A C E B (2) D A B C E (2) C E A D B (2) C D A E B (2) A D C B E (2) A D B C E (2) E D B C A (1) E C D B A (1) E C B A D (1) E B D C A (1) D E B C A (1) D C E B A (1) D C A E B (1) D A C B E (1) C E D A B (1) C E B A D (1) C E A B D (1) C D E B A (1) C D E A B (1) B E D A C (1) B E A C D (1) B D A E C (1) B A E D C (1) A D B E C (1) A C E B D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -4 4 -10 B 14 0 8 2 -16 C 4 -8 0 6 -4 D -4 -2 -6 0 -2 E 10 16 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -4 4 -10 B 14 0 8 2 -16 C 4 -8 0 6 -4 D -4 -2 -6 0 -2 E 10 16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 C=20 E=16 D=13 so D is eliminated. Round 2 votes counts: A=32 B=29 C=22 E=17 so E is eliminated. Round 3 votes counts: B=44 A=32 C=24 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:204 C:199 D:193 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -4 4 -10 B 14 0 8 2 -16 C 4 -8 0 6 -4 D -4 -2 -6 0 -2 E 10 16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 4 -10 B 14 0 8 2 -16 C 4 -8 0 6 -4 D -4 -2 -6 0 -2 E 10 16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 4 -10 B 14 0 8 2 -16 C 4 -8 0 6 -4 D -4 -2 -6 0 -2 E 10 16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5228: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) E B D A C (8) E B D C A (6) C A E B D (6) A B D E C (5) C E D B A (4) C E B D A (4) C A D E B (4) D E B A C (3) D B A E C (3) C E A D B (3) A D B C E (3) A C D B E (3) E D C B A (2) E D B C A (2) E D B A C (2) E C D B A (2) E C B D A (2) E B C D A (2) D A B E C (2) C E D A B (2) C E B A D (2) B E D A C (2) A C B D E (2) A B D C E (2) E B A C D (1) D C B A E (1) D B E A C (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B E A D C (1) B D E A C (1) B D A E C (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -14 -6 -2 B 0 0 -4 -6 -16 C 14 4 0 6 2 D 6 6 -6 0 -8 E 2 16 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -6 -2 B 0 0 -4 -6 -16 C 14 4 0 6 2 D 6 6 -6 0 -8 E 2 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=27 A=18 D=10 B=5 so B is eliminated. Round 2 votes counts: C=40 E=30 A=18 D=12 so D is eliminated. Round 3 votes counts: C=41 E=35 A=24 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:212 D:199 A:189 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 -6 -2 B 0 0 -4 -6 -16 C 14 4 0 6 2 D 6 6 -6 0 -8 E 2 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -6 -2 B 0 0 -4 -6 -16 C 14 4 0 6 2 D 6 6 -6 0 -8 E 2 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -6 -2 B 0 0 -4 -6 -16 C 14 4 0 6 2 D 6 6 -6 0 -8 E 2 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5229: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) D C A B E (7) D A E C B (5) C B E D A (5) D C A E B (4) D A C E B (4) C D B E A (4) B E C A D (4) A E D B C (4) A D E B C (4) E D A B C (3) E B C A D (3) E B A C D (3) E A D B C (3) C B E A D (3) C B D A E (3) B E A C D (3) B C A E D (3) B A E C D (3) E B C D A (2) C D B A E (2) C D A B E (2) C B D E A (2) A B E D C (2) E B D C A (1) E B A D C (1) D E A C B (1) D C E A B (1) D A E B C (1) D A C B E (1) C E B D A (1) C D E B A (1) C B A E D (1) C A B D E (1) B C A D E (1) A E B D C (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -18 0 2 B 8 0 0 4 14 C 18 0 0 12 12 D 0 -4 -12 0 -8 E -2 -14 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.783796 C: 0.216204 D: 0.000000 E: 0.000000 Sum of squares = 0.661080282509 Cumulative probabilities = A: 0.000000 B: 0.783796 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 0 2 B 8 0 0 4 14 C 18 0 0 12 12 D 0 -4 -12 0 -8 E -2 -14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 B=22 E=16 A=13 so A is eliminated. Round 2 votes counts: D=29 C=26 B=24 E=21 so E is eliminated. Round 3 votes counts: D=39 B=35 C=26 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:221 B:213 E:190 A:188 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -18 0 2 B 8 0 0 4 14 C 18 0 0 12 12 D 0 -4 -12 0 -8 E -2 -14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 0 2 B 8 0 0 4 14 C 18 0 0 12 12 D 0 -4 -12 0 -8 E -2 -14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 0 2 B 8 0 0 4 14 C 18 0 0 12 12 D 0 -4 -12 0 -8 E -2 -14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5230: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) C B E D A (8) B D E C A (5) D A B E C (4) C A E B D (4) C A B E D (4) E D B C A (3) E B D C A (3) D E B A C (3) D E A B C (3) C A B D E (3) B E D C A (3) A D C B E (3) A C E B D (3) A C D E B (3) A C D B E (3) E A D B C (2) D B A E C (2) D A E B C (2) C E B A D (2) C B E A D (2) C B A D E (2) B E C D A (2) A D E C B (2) A D B E C (2) A C B D E (2) E D B A C (1) E D A B C (1) E C B D A (1) E C B A D (1) E C A D B (1) E B C D A (1) D B E C A (1) D B E A C (1) C E B D A (1) C B D A E (1) C B A E D (1) B D C A E (1) B C E D A (1) B C D E A (1) B C A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 4 -6 2 6 B -4 0 4 6 6 C 6 -4 0 -4 -8 D -2 -6 4 0 8 E -6 -6 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775509 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 2 6 B -4 0 4 6 6 C 6 -4 0 -4 -8 D -2 -6 4 0 8 E -6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=28 A=28 D=16 E=14 B=14 so E is eliminated. Round 2 votes counts: C=31 A=30 D=21 B=18 so B is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:206 A:203 D:202 C:195 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -6 2 6 B -4 0 4 6 6 C 6 -4 0 -4 -8 D -2 -6 4 0 8 E -6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 2 6 B -4 0 4 6 6 C 6 -4 0 -4 -8 D -2 -6 4 0 8 E -6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 2 6 B -4 0 4 6 6 C 6 -4 0 -4 -8 D -2 -6 4 0 8 E -6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.428571 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.285714 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5231: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) B E A D C (8) B A C E D (8) E D C B A (6) C D E B A (5) A B C D E (5) E D B A C (4) D E C A B (4) D C E A B (4) C E D B A (3) C A B D E (3) B A E D C (3) A D B E C (3) A B E D C (3) E D B C A (2) E B D C A (2) E B D A C (2) C B E D A (2) C B A D E (2) C A D B E (2) B E C A D (2) B A E C D (2) A C B D E (2) E D A B C (1) E B C D A (1) D E A C B (1) D C A E B (1) D A E C B (1) D A E B C (1) C E B D A (1) C D B E A (1) C B A E D (1) B A C D E (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -4 -4 -18 B 12 0 0 0 2 C 4 0 0 4 6 D 4 0 -4 0 -2 E 18 -2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.425074 C: 0.574926 D: 0.000000 E: 0.000000 Sum of squares = 0.511227671391 Cumulative probabilities = A: 0.000000 B: 0.425074 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -4 -18 B 12 0 0 0 2 C 4 0 0 4 6 D 4 0 -4 0 -2 E 18 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=24 E=18 A=16 D=12 so D is eliminated. Round 2 votes counts: C=35 B=24 E=23 A=18 so A is eliminated. Round 3 votes counts: C=38 B=37 E=25 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:207 E:206 D:199 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -4 -18 B 12 0 0 0 2 C 4 0 0 4 6 D 4 0 -4 0 -2 E 18 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -4 -18 B 12 0 0 0 2 C 4 0 0 4 6 D 4 0 -4 0 -2 E 18 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -4 -18 B 12 0 0 0 2 C 4 0 0 4 6 D 4 0 -4 0 -2 E 18 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5232: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (14) E A B D C (7) E B A C D (6) D C A B E (6) E A D B C (5) A E B D C (5) C B D E A (4) B E A C D (4) A E D B C (4) E A D C B (3) D C A E B (3) C D B E A (3) C B E D A (3) B E C A D (3) B C E A D (3) E D A C B (2) D A C B E (2) B A E D C (2) A D E B C (2) E C B A D (1) E B C A D (1) D C E A B (1) D C B A E (1) D B A C E (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D E A B (1) C D A B E (1) C B D A E (1) B C E D A (1) B C D A E (1) B A E C D (1) A E D C B (1) A D E C B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 0 2 0 B 4 0 -4 -10 6 C 0 4 0 0 0 D -2 10 0 0 -6 E 0 -6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.216613 B: 0.000000 C: 0.636402 D: 0.000000 E: 0.146984 Sum of squares = 0.473533743879 Cumulative probabilities = A: 0.216613 B: 0.216613 C: 0.853016 D: 0.853016 E: 1.000000 A B C D E A 0 -4 0 2 0 B 4 0 -4 -10 6 C 0 4 0 0 0 D -2 10 0 0 -6 E 0 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.261905 B: 0.000000 C: 0.547619 D: 0.000000 E: 0.190476 Sum of squares = 0.40476209374 Cumulative probabilities = A: 0.261905 B: 0.261905 C: 0.809524 D: 0.809524 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=25 D=16 B=15 A=15 so B is eliminated. Round 2 votes counts: C=34 E=32 A=18 D=16 so D is eliminated. Round 3 votes counts: C=45 E=32 A=23 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:202 D:201 E:200 A:199 B:198 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 2 0 B 4 0 -4 -10 6 C 0 4 0 0 0 D -2 10 0 0 -6 E 0 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.261905 B: 0.000000 C: 0.547619 D: 0.000000 E: 0.190476 Sum of squares = 0.40476209374 Cumulative probabilities = A: 0.261905 B: 0.261905 C: 0.809524 D: 0.809524 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 0 B 4 0 -4 -10 6 C 0 4 0 0 0 D -2 10 0 0 -6 E 0 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.261905 B: 0.000000 C: 0.547619 D: 0.000000 E: 0.190476 Sum of squares = 0.40476209374 Cumulative probabilities = A: 0.261905 B: 0.261905 C: 0.809524 D: 0.809524 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 0 B 4 0 -4 -10 6 C 0 4 0 0 0 D -2 10 0 0 -6 E 0 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.261905 B: 0.000000 C: 0.547619 D: 0.000000 E: 0.190476 Sum of squares = 0.40476209374 Cumulative probabilities = A: 0.261905 B: 0.261905 C: 0.809524 D: 0.809524 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5233: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (14) C B D A E (14) B C D A E (12) C B E A D (8) D A E B C (6) E A D C B (5) B C E D A (5) E C B A D (4) A E D C B (4) A D E C B (4) C B E D A (3) E C A B D (2) E B C A D (2) E A C B D (2) D A C B E (2) A E D B C (2) E D A B C (1) E B A D C (1) D C B A E (1) D B E A C (1) D B A C E (1) D A B E C (1) D A B C E (1) C B D E A (1) C B A E D (1) C B A D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -8 2 2 B 10 0 -4 12 4 C 8 4 0 10 0 D -2 -12 -10 0 -8 E -2 -4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.748311 D: 0.000000 E: 0.251689 Sum of squares = 0.623316235086 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.748311 D: 0.748311 E: 1.000000 A B C D E A 0 -10 -8 2 2 B 10 0 -4 12 4 C 8 4 0 10 0 D -2 -12 -10 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.000000 E: 0.499990 Sum of squares = 0.500000000214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.500010 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=28 B=17 D=13 A=11 so A is eliminated. Round 2 votes counts: E=37 C=28 D=18 B=17 so B is eliminated. Round 3 votes counts: C=45 E=37 D=18 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:211 E:201 A:193 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 2 2 B 10 0 -4 12 4 C 8 4 0 10 0 D -2 -12 -10 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.000000 E: 0.499990 Sum of squares = 0.500000000214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.500010 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 2 2 B 10 0 -4 12 4 C 8 4 0 10 0 D -2 -12 -10 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.000000 E: 0.499990 Sum of squares = 0.500000000214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.500010 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 2 2 B 10 0 -4 12 4 C 8 4 0 10 0 D -2 -12 -10 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.000000 E: 0.499990 Sum of squares = 0.500000000214 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500010 D: 0.500010 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5234: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (14) E B D A C (12) E D B A C (7) C A D B E (6) E D B C A (4) A E B D C (4) A B E D C (4) D E B C A (3) D E B A C (3) D B E C A (3) D B E A C (3) C D E B A (3) C D A B E (3) A E B C D (3) E A B D C (2) D B C E A (2) C A E D B (2) C A E B D (2) B D E A C (2) A C E B D (2) A C B E D (2) A B D E C (2) E D C B A (1) E B A D C (1) D C B E A (1) C E D B A (1) C D B E A (1) C D A E B (1) B E D A C (1) B D A C E (1) B A D E C (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 6 -4 0 B 0 0 20 12 -2 C -6 -20 0 -16 -14 D 4 -12 16 0 2 E 0 2 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593750000024 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 A B C D E A 0 0 6 -4 0 B 0 0 20 12 -2 C -6 -20 0 -16 -14 D 4 -12 16 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999707 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=27 A=20 D=15 B=5 so B is eliminated. Round 2 votes counts: C=33 E=28 A=21 D=18 so D is eliminated. Round 3 votes counts: E=42 C=36 A=22 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:207 D:205 A:201 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 -4 0 B 0 0 20 12 -2 C -6 -20 0 -16 -14 D 4 -12 16 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999707 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -4 0 B 0 0 20 12 -2 C -6 -20 0 -16 -14 D 4 -12 16 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999707 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -4 0 B 0 0 20 12 -2 C -6 -20 0 -16 -14 D 4 -12 16 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593749999707 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5235: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (13) A C B D E (7) E D B C A (5) E D A C B (5) B C D A E (4) E D B A C (3) E B D C A (3) E B A C D (3) E A B C D (3) D E C A B (3) A E C B D (3) E D C A B (2) E B A D C (2) E A D C B (2) E A D B C (2) D E B C A (2) C B A D E (2) B E D C A (2) B D C E A (2) B C D E A (2) B A C E D (2) A C D E B (2) A C D B E (2) A C B E D (2) A B C D E (2) E D C B A (1) D E C B A (1) D E A C B (1) D C E B A (1) D C B E A (1) D C B A E (1) D C A E B (1) C D A B E (1) C B D A E (1) C A D B E (1) C A B D E (1) B E A C D (1) B C E D A (1) B C E A D (1) B C A E D (1) A E C D B (1) A E B C D (1) A D E C B (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -6 14 2 B 10 0 12 20 2 C 6 -12 0 16 6 D -14 -20 -16 0 6 E -2 -2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 14 2 B 10 0 12 20 2 C 6 -12 0 16 6 D -14 -20 -16 0 6 E -2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999939943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=29 A=23 D=11 C=6 so C is eliminated. Round 2 votes counts: B=32 E=31 A=25 D=12 so D is eliminated. Round 3 votes counts: E=39 B=34 A=27 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:208 A:200 E:192 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 14 2 B 10 0 12 20 2 C 6 -12 0 16 6 D -14 -20 -16 0 6 E -2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999939943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 14 2 B 10 0 12 20 2 C 6 -12 0 16 6 D -14 -20 -16 0 6 E -2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999939943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 14 2 B 10 0 12 20 2 C 6 -12 0 16 6 D -14 -20 -16 0 6 E -2 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999939943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5236: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) A C D E B (6) A C D B E (6) C D A E B (5) B C E D A (5) D C A E B (4) B A E C D (4) D E A C B (3) C D A B E (3) B E C A D (3) B C A D E (3) A E D C B (3) A E D B C (3) A D C E B (3) A C B D E (3) E D B C A (2) E D A C B (2) E B D C A (2) E B A D C (2) D E C A B (2) D A C E B (2) C D B A E (2) C A D B E (2) A B E D C (2) A B C D E (2) E A B D C (1) D C E B A (1) D C E A B (1) C D E B A (1) C B D E A (1) C B D A E (1) C B A D E (1) B E D A C (1) B E A D C (1) B E A C D (1) B C D E A (1) B C D A E (1) B A C E D (1) B A C D E (1) A E B D C (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 12 0 2 22 B -12 0 -6 -4 10 C 0 6 0 26 12 D -2 4 -26 0 12 E -22 -10 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.575365 B: 0.000000 C: 0.424635 D: 0.000000 E: 0.000000 Sum of squares = 0.511359653357 Cumulative probabilities = A: 0.575365 B: 0.575365 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 2 22 B -12 0 -6 -4 10 C 0 6 0 26 12 D -2 4 -26 0 12 E -22 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=31 A=31 C=16 D=13 E=9 so E is eliminated. Round 2 votes counts: B=35 A=32 D=17 C=16 so C is eliminated. Round 3 votes counts: B=38 A=34 D=28 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:222 A:218 B:194 D:194 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 2 22 B -12 0 -6 -4 10 C 0 6 0 26 12 D -2 4 -26 0 12 E -22 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 2 22 B -12 0 -6 -4 10 C 0 6 0 26 12 D -2 4 -26 0 12 E -22 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 2 22 B -12 0 -6 -4 10 C 0 6 0 26 12 D -2 4 -26 0 12 E -22 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5237: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B A C D E (7) D A C E B (5) A D B E C (5) D E C A B (4) D E A C B (4) C E D A B (4) E D B C A (3) E C D B A (3) E B C D A (3) C A B D E (3) B A C E D (3) A B D C E (3) A B C D E (3) E D C B A (2) E C B D A (2) D A E C B (2) D A E B C (2) D A B E C (2) C E D B A (2) C E B D A (2) C D E A B (2) C B A E D (2) C A D E B (2) C A B E D (2) B C A E D (2) A D B C E (2) E D B A C (1) E C D A B (1) E B D C A (1) D E B A C (1) D E A B C (1) D C E A B (1) C E B A D (1) C D A E B (1) C B E A D (1) C A D B E (1) B E C A D (1) B E A D C (1) B C E A D (1) B A E C D (1) B A D C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 18 -8 -12 2 B -18 0 -12 -18 -14 C 8 12 0 4 6 D 12 18 -4 0 8 E -2 14 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -8 -12 2 B -18 0 -12 -18 -14 C 8 12 0 4 6 D 12 18 -4 0 8 E -2 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=23 C=23 D=22 B=17 A=15 so A is eliminated. Round 2 votes counts: D=29 C=25 E=23 B=23 so E is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:215 A:200 E:199 B:169 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -8 -12 2 B -18 0 -12 -18 -14 C 8 12 0 4 6 D 12 18 -4 0 8 E -2 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 -12 2 B -18 0 -12 -18 -14 C 8 12 0 4 6 D 12 18 -4 0 8 E -2 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 -12 2 B -18 0 -12 -18 -14 C 8 12 0 4 6 D 12 18 -4 0 8 E -2 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5238: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (7) B D E A C (5) E C A D B (4) E A C B D (4) E A B C D (4) D C B E A (4) D B E C A (4) C A E D B (4) E B D C A (3) E B A D C (3) D B C E A (3) C A D B E (3) B E D A C (3) E C D B A (2) E B D A C (2) E B A C D (2) D B C A E (2) C D A E B (2) C D A B E (2) B D E C A (2) B D A C E (2) B A E D C (2) A D B C E (2) A C E D B (2) A B E C D (2) E C D A B (1) E C B D A (1) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) D C E B A (1) D C B A E (1) D C A B E (1) D B A C E (1) D A C B E (1) C E D B A (1) C E D A B (1) C E A D B (1) C D E A B (1) C A D E B (1) B E A D C (1) B D A E C (1) B A D E C (1) B A D C E (1) A E C B D (1) A E B C D (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 4 -14 B 2 0 0 -4 6 C -2 0 0 8 -6 D -4 4 -8 0 0 E 14 -6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.618432 C: 0.381568 D: 0.000000 E: 0.000000 Sum of squares = 0.528052504751 Cumulative probabilities = A: 0.000000 B: 0.618432 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 4 -14 B 2 0 0 -4 6 C -2 0 0 8 -6 D -4 4 -8 0 0 E 14 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500593 C: 0.499407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000704332 Cumulative probabilities = A: 0.000000 B: 0.500593 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=18 B=18 A=18 C=16 so C is eliminated. Round 2 votes counts: E=33 A=26 D=23 B=18 so B is eliminated. Round 3 votes counts: E=37 D=33 A=30 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:207 B:202 C:200 D:196 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 4 -14 B 2 0 0 -4 6 C -2 0 0 8 -6 D -4 4 -8 0 0 E 14 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500593 C: 0.499407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000704332 Cumulative probabilities = A: 0.000000 B: 0.500593 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -14 B 2 0 0 -4 6 C -2 0 0 8 -6 D -4 4 -8 0 0 E 14 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500593 C: 0.499407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000704332 Cumulative probabilities = A: 0.000000 B: 0.500593 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -14 B 2 0 0 -4 6 C -2 0 0 8 -6 D -4 4 -8 0 0 E 14 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500593 C: 0.499407 D: 0.000000 E: 0.000000 Sum of squares = 0.500000704332 Cumulative probabilities = A: 0.000000 B: 0.500593 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5239: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (12) A D C B E (9) E B C D A (7) A D C E B (7) A B C D E (6) B C D E A (5) B E C A D (4) B C E D A (4) A E D C B (4) A E B D C (4) A D E C B (4) D A C E B (3) A B E C D (3) E D C B A (2) E C D B A (2) D C E B A (2) C D E B A (2) B C A D E (2) B A C D E (2) A E B C D (2) A B D C E (2) A B C E D (2) E C B D A (1) E B A D C (1) E A B D C (1) D C B E A (1) D C B A E (1) D C A B E (1) C D B E A (1) C B D E A (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 4 10 8 B 2 0 18 22 16 C -4 -18 0 14 2 D -10 -22 -14 0 -2 E -8 -16 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 10 8 B 2 0 18 22 16 C -4 -18 0 14 2 D -10 -22 -14 0 -2 E -8 -16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 B=30 E=14 D=8 C=4 so C is eliminated. Round 2 votes counts: A=44 B=31 E=14 D=11 so D is eliminated. Round 3 votes counts: A=48 B=34 E=18 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 A:210 C:197 E:188 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 10 8 B 2 0 18 22 16 C -4 -18 0 14 2 D -10 -22 -14 0 -2 E -8 -16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 8 B 2 0 18 22 16 C -4 -18 0 14 2 D -10 -22 -14 0 -2 E -8 -16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 8 B 2 0 18 22 16 C -4 -18 0 14 2 D -10 -22 -14 0 -2 E -8 -16 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999947519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5240: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (14) C B D A E (8) C A E D B (8) B D C A E (8) E A C D B (7) B D E A C (7) C A B D E (6) E C A D B (4) D B E A C (4) B D A E C (4) B D A C E (4) E A D B C (2) C E A D B (2) C E A B D (2) B D E C A (2) A E C D B (2) A C D B E (2) E D A B C (1) E C B D A (1) E B D C A (1) E B D A C (1) E A D C B (1) D E B A C (1) C B A D E (1) B E D C A (1) B D C E A (1) B C D E A (1) B C D A E (1) A E D C B (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -20 6 -20 -6 B 20 0 8 -2 2 C -6 -8 0 -8 -10 D 20 2 8 0 2 E 6 -2 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 -20 -6 B 20 0 8 -2 2 C -6 -8 0 -8 -10 D 20 2 8 0 2 E 6 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=29 C=27 A=7 D=5 so D is eliminated. Round 2 votes counts: E=33 B=33 C=27 A=7 so A is eliminated. Round 3 votes counts: E=36 B=34 C=30 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:214 E:206 C:184 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 6 -20 -6 B 20 0 8 -2 2 C -6 -8 0 -8 -10 D 20 2 8 0 2 E 6 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -20 -6 B 20 0 8 -2 2 C -6 -8 0 -8 -10 D 20 2 8 0 2 E 6 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -20 -6 B 20 0 8 -2 2 C -6 -8 0 -8 -10 D 20 2 8 0 2 E 6 -2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5241: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) B E C D A (7) B C E A D (7) E B C D A (5) D B E A C (5) D A E C B (5) B E D C A (5) A D C B E (5) D A B C E (4) C A E B D (4) E C B A D (3) D B A E C (3) B E C A D (3) B D E C A (3) A D C E B (3) A C E D B (3) E C A B D (2) E B C A D (2) D E A B C (2) D A C B E (2) C E B A D (2) C B A D E (2) B C D A E (2) A D E C B (2) E D A C B (1) E A D C B (1) E A C D B (1) D B A C E (1) D A E B C (1) D A C E B (1) D A B E C (1) C B A E D (1) C A B D E (1) B C A E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -16 4 -14 B 20 0 4 16 24 C 16 -4 0 10 -4 D -4 -16 -10 0 -12 E 14 -24 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -16 4 -14 B 20 0 4 16 24 C 16 -4 0 10 -4 D -4 -16 -10 0 -12 E 14 -24 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=25 C=18 E=15 A=14 so A is eliminated. Round 2 votes counts: D=35 B=28 C=22 E=15 so E is eliminated. Round 3 votes counts: D=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:209 E:203 D:179 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -16 4 -14 B 20 0 4 16 24 C 16 -4 0 10 -4 D -4 -16 -10 0 -12 E 14 -24 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 4 -14 B 20 0 4 16 24 C 16 -4 0 10 -4 D -4 -16 -10 0 -12 E 14 -24 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 4 -14 B 20 0 4 16 24 C 16 -4 0 10 -4 D -4 -16 -10 0 -12 E 14 -24 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997303 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5242: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C A E B D (9) C E A B D (6) D B E A C (5) D A B C E (5) A D B C E (5) D B E C A (4) C E A D B (4) C A D E B (4) B E A C D (3) A C B E D (3) E C B A D (2) E B D C A (2) E B C D A (2) E B C A D (2) D E B C A (2) D C A E B (2) C A E D B (2) B E D A C (2) B D E A C (2) A C E B D (2) A C B D E (2) A B E C D (2) E C D B A (1) E C B D A (1) E B A C D (1) E A C B D (1) D C E B A (1) D C A B E (1) D B C E A (1) D A C B E (1) D A B E C (1) C E D B A (1) C E B A D (1) B D A E C (1) A C D B E (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 4 6 12 B -8 0 10 -6 8 C -4 -10 0 2 4 D -6 6 -2 0 2 E -12 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 6 12 B -8 0 10 -6 8 C -4 -10 0 2 4 D -6 6 -2 0 2 E -12 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=27 A=18 E=12 B=8 so B is eliminated. Round 2 votes counts: D=38 C=27 A=18 E=17 so E is eliminated. Round 3 votes counts: D=42 C=35 A=23 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:215 B:202 D:200 C:196 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 6 12 B -8 0 10 -6 8 C -4 -10 0 2 4 D -6 6 -2 0 2 E -12 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 6 12 B -8 0 10 -6 8 C -4 -10 0 2 4 D -6 6 -2 0 2 E -12 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 6 12 B -8 0 10 -6 8 C -4 -10 0 2 4 D -6 6 -2 0 2 E -12 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5243: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) B E A C D (7) E A D B C (6) A E D C B (6) A E C D B (6) E A B D C (5) D C A E B (5) C D A E B (4) E D A B C (3) D E A B C (3) D C B A E (3) C B D A E (3) C A E D B (3) B E C A D (3) B C D E A (3) A C E D B (3) E A D C B (2) D C B E A (2) D B C E A (2) D A E C B (2) C D B A E (2) C D A B E (2) B E A D C (2) B D C E A (2) B C E A D (2) A E C B D (2) E B A D C (1) E A B C D (1) D E C B A (1) D E A C B (1) D C A B E (1) C A E B D (1) B D E A C (1) B C D A E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 16 6 18 6 B -16 0 -14 -14 -14 C -6 14 0 4 -6 D -18 14 -4 0 -24 E -6 14 6 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 18 6 B -16 0 -14 -14 -14 C -6 14 0 4 -6 D -18 14 -4 0 -24 E -6 14 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 B=21 D=20 A=19 E=18 so E is eliminated. Round 2 votes counts: A=33 D=23 C=22 B=22 so C is eliminated. Round 3 votes counts: A=37 B=32 D=31 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:219 C:203 D:184 B:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 18 6 B -16 0 -14 -14 -14 C -6 14 0 4 -6 D -18 14 -4 0 -24 E -6 14 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 18 6 B -16 0 -14 -14 -14 C -6 14 0 4 -6 D -18 14 -4 0 -24 E -6 14 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 18 6 B -16 0 -14 -14 -14 C -6 14 0 4 -6 D -18 14 -4 0 -24 E -6 14 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999012 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5244: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) D A C E B (7) A B E C D (7) B E C A D (6) D C E B A (5) A D C B E (5) E B D C A (4) D C A E B (4) B E A C D (4) A C D E B (4) A C B E D (4) A B C E D (4) E B C D A (3) D E B C A (3) C A E B D (3) B E A D C (3) A D B E C (3) A B E D C (3) D C E A B (2) D A C B E (2) B E C D A (2) A D C E B (2) E D C B A (1) D E C B A (1) D B E C A (1) C E B A D (1) C D E A B (1) C D A E B (1) C A E D B (1) C A D E B (1) B D E C A (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 0 6 6 B -12 0 8 10 12 C 0 -8 0 -14 -2 D -6 -10 14 0 -10 E -6 -12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.796221 B: 0.000000 C: 0.203779 D: 0.000000 E: 0.000000 Sum of squares = 0.67549398324 Cumulative probabilities = A: 0.796221 B: 0.796221 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 6 6 B -12 0 8 10 12 C 0 -8 0 -14 -2 D -6 -10 14 0 -10 E -6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000349084 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=25 B=25 E=8 C=8 so E is eliminated. Round 2 votes counts: A=34 B=32 D=26 C=8 so C is eliminated. Round 3 votes counts: A=39 B=33 D=28 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:209 E:197 D:194 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 6 6 B -12 0 8 10 12 C 0 -8 0 -14 -2 D -6 -10 14 0 -10 E -6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000349084 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 6 6 B -12 0 8 10 12 C 0 -8 0 -14 -2 D -6 -10 14 0 -10 E -6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000349084 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 6 6 B -12 0 8 10 12 C 0 -8 0 -14 -2 D -6 -10 14 0 -10 E -6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000349084 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5245: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (7) E A D B C (6) D A E C B (5) C B D A E (5) A E D B C (5) A D E C B (5) E A B D C (4) C D B A E (4) B C D E A (4) E B D A C (3) D E A C B (3) D C A E B (3) D A C E B (3) B E A C D (3) B C E A D (3) A E B C D (3) E D A B C (2) E B A D C (2) D C B E A (2) C D A E B (2) C D A B E (2) C B A E D (2) C A D E B (2) B E A D C (2) B D E C A (2) B C A E D (2) E B A C D (1) D E A B C (1) D C B A E (1) C D B E A (1) C B D E A (1) C A D B E (1) C A B D E (1) B E D C A (1) B E D A C (1) B E C D A (1) B C E D A (1) A E B D C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 16 18 4 12 B -16 0 -4 -12 -20 C -18 4 0 -18 -16 D -4 12 18 0 -2 E -12 20 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 4 12 B -16 0 -4 -12 -20 C -18 4 0 -18 -16 D -4 12 18 0 -2 E -12 20 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 C=21 B=20 E=18 D=18 so E is eliminated. Round 2 votes counts: A=33 B=26 C=21 D=20 so D is eliminated. Round 3 votes counts: A=47 C=27 B=26 so B is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:213 D:212 C:176 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 4 12 B -16 0 -4 -12 -20 C -18 4 0 -18 -16 D -4 12 18 0 -2 E -12 20 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 4 12 B -16 0 -4 -12 -20 C -18 4 0 -18 -16 D -4 12 18 0 -2 E -12 20 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 4 12 B -16 0 -4 -12 -20 C -18 4 0 -18 -16 D -4 12 18 0 -2 E -12 20 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5246: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (13) D E B C A (9) D B E A C (5) A B D E C (5) D E B A C (4) C A E B D (4) A D B C E (4) E D B C A (3) E C B D A (3) D A E B C (3) C E B A D (3) B E D C A (3) B D E A C (3) A C D E B (3) A C B D E (3) E B D C A (2) D B E C A (2) D A E C B (2) C E D B A (2) C A B E D (2) B C A E D (2) B A D E C (2) A D C E B (2) E C D B A (1) D E C B A (1) D E C A B (1) D E A B C (1) C E D A B (1) C E A D B (1) C B E D A (1) C B A E D (1) C A E D B (1) B E C D A (1) B C E A D (1) B A C E D (1) A D E C B (1) A C E D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 10 4 4 B 0 0 6 4 2 C -10 -6 0 -6 -4 D -4 -4 6 0 4 E -4 -2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.332331 B: 0.667669 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.556225848962 Cumulative probabilities = A: 0.332331 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 4 4 B 0 0 6 4 2 C -10 -6 0 -6 -4 D -4 -4 6 0 4 E -4 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=28 C=16 B=13 E=9 so E is eliminated. Round 2 votes counts: A=34 D=31 C=20 B=15 so B is eliminated. Round 3 votes counts: D=39 A=37 C=24 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:206 D:201 E:197 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 4 4 B 0 0 6 4 2 C -10 -6 0 -6 -4 D -4 -4 6 0 4 E -4 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 4 4 B 0 0 6 4 2 C -10 -6 0 -6 -4 D -4 -4 6 0 4 E -4 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 4 4 B 0 0 6 4 2 C -10 -6 0 -6 -4 D -4 -4 6 0 4 E -4 -2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5247: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) E D B A C (8) C A B D E (6) E B D A C (5) D E C A B (5) C D A E B (5) A C B E D (4) E B A D C (3) E A B C D (3) D E C B A (3) D C A E B (3) C A D B E (3) E D C A B (2) E D B C A (2) E B A C D (2) E A C B D (2) D C A B E (2) D B E C A (2) C A D E B (2) C A B E D (2) B E A C D (2) B D A C E (2) B A E C D (2) A C E B D (2) A B E C D (2) E D A B C (1) E A C D B (1) D E B A C (1) D C B E A (1) D C B A E (1) D B C E A (1) C D A B E (1) C A E D B (1) C A E B D (1) B E D A C (1) B E A D C (1) B D E A C (1) B A C E D (1) A E C B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -6 -14 -14 B -2 0 4 -10 -26 C 6 -4 0 -10 -20 D 14 10 10 0 0 E 14 26 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.742107 E: 0.257893 Sum of squares = 0.617231130053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.742107 E: 1.000000 A B C D E A 0 2 -6 -14 -14 B -2 0 4 -10 -26 C 6 -4 0 -10 -20 D 14 10 10 0 0 E 14 26 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=29 D=29 C=21 A=11 B=10 so B is eliminated. Round 2 votes counts: E=33 D=32 C=21 A=14 so A is eliminated. Round 3 votes counts: E=38 D=32 C=30 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:230 D:217 C:186 A:184 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -14 -14 B -2 0 4 -10 -26 C 6 -4 0 -10 -20 D 14 10 10 0 0 E 14 26 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -14 -14 B -2 0 4 -10 -26 C 6 -4 0 -10 -20 D 14 10 10 0 0 E 14 26 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -14 -14 B -2 0 4 -10 -26 C 6 -4 0 -10 -20 D 14 10 10 0 0 E 14 26 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5248: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) C E A D B (7) C A E D B (7) E A C D B (6) D B E A C (6) C A E B D (6) B D E A C (5) D B C E A (4) C B D A E (4) E D B A C (3) E C A D B (3) D B E C A (3) C D B A E (3) B D A E C (3) A E C B D (3) A E B D C (3) A E B C D (2) A C E B D (2) A B E D C (2) E C D A B (1) E B A D C (1) E A D B C (1) D C B E A (1) D B C A E (1) C E D B A (1) C E D A B (1) C D B E A (1) C D A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C E A (1) B C D A E (1) B A D E C (1) B A D C E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -18 2 8 B 0 0 -4 -4 4 C 18 4 0 8 10 D -2 4 -8 0 0 E -8 -4 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 2 8 B 0 0 -4 -4 4 C 18 4 0 8 10 D -2 4 -8 0 0 E -8 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=21 E=15 D=15 A=14 so A is eliminated. Round 2 votes counts: C=38 B=24 E=23 D=15 so D is eliminated. Round 3 votes counts: C=39 B=38 E=23 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:198 D:197 A:196 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -18 2 8 B 0 0 -4 -4 4 C 18 4 0 8 10 D -2 4 -8 0 0 E -8 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 2 8 B 0 0 -4 -4 4 C 18 4 0 8 10 D -2 4 -8 0 0 E -8 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 2 8 B 0 0 -4 -4 4 C 18 4 0 8 10 D -2 4 -8 0 0 E -8 -4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5249: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (12) D E C B A (11) C B D E A (8) C D E B A (7) E D A B C (6) D E A C B (5) C B D A E (5) B A C E D (5) B C A E D (4) D E C A B (3) C B E D A (3) C B A E D (3) C B A D E (3) A E D B C (3) A B E C D (3) E D B A C (2) D E A B C (2) D C E B A (2) C D B E A (2) E D B C A (1) E B D C A (1) E A B D C (1) D A E C B (1) C B E A D (1) B C E D A (1) A E B D C (1) A D E C B (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -10 -20 -12 B 18 0 -12 8 6 C 10 12 0 18 16 D 20 -8 -18 0 4 E 12 -6 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -20 -12 B 18 0 -12 8 6 C 10 12 0 18 16 D 20 -8 -18 0 4 E 12 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 A=23 E=11 B=10 so B is eliminated. Round 2 votes counts: C=37 A=28 D=24 E=11 so E is eliminated. Round 3 votes counts: C=37 D=34 A=29 so A is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:210 D:199 E:193 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -10 -20 -12 B 18 0 -12 8 6 C 10 12 0 18 16 D 20 -8 -18 0 4 E 12 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -20 -12 B 18 0 -12 8 6 C 10 12 0 18 16 D 20 -8 -18 0 4 E 12 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -20 -12 B 18 0 -12 8 6 C 10 12 0 18 16 D 20 -8 -18 0 4 E 12 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5250: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) B E D C A (9) E B D A C (8) E B C A D (5) D E B A C (4) C A E B D (4) A D C E B (4) E D A B C (3) E B D C A (3) B E C D A (3) A C D E B (3) E A C B D (2) D C A B E (2) D B E A C (2) D B C A E (2) D B A C E (2) D A E B C (2) D A C B E (2) C B E A D (2) C B D A E (2) C B A E D (2) C A D E B (2) B D E A C (2) A C E D B (2) E D B A C (1) E B C D A (1) E A B D C (1) D E A B C (1) D B E C A (1) D B C E A (1) D A B E C (1) C E B A D (1) C A B E D (1) B E D A C (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D E A (1) A E C D B (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -8 -12 -10 B 12 0 14 4 0 C 8 -14 0 -10 -8 D 12 -4 10 0 -6 E 10 0 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.510425 C: 0.000000 D: 0.000000 E: 0.489575 Sum of squares = 0.500217368495 Cumulative probabilities = A: 0.000000 B: 0.510425 C: 0.510425 D: 0.510425 E: 1.000000 A B C D E A 0 -12 -8 -12 -10 B 12 0 14 4 0 C 8 -14 0 -10 -8 D 12 -4 10 0 -6 E 10 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=23 D=20 B=19 A=14 so A is eliminated. Round 2 votes counts: C=30 D=26 E=25 B=19 so B is eliminated. Round 3 votes counts: E=39 C=32 D=29 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:212 D:206 C:188 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -12 -10 B 12 0 14 4 0 C 8 -14 0 -10 -8 D 12 -4 10 0 -6 E 10 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -12 -10 B 12 0 14 4 0 C 8 -14 0 -10 -8 D 12 -4 10 0 -6 E 10 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -12 -10 B 12 0 14 4 0 C 8 -14 0 -10 -8 D 12 -4 10 0 -6 E 10 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5251: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (13) A D C E B (13) E B A C D (11) D C A B E (6) C D B A E (5) E B A D C (4) A E D C B (4) C D A E B (3) B C E D A (3) B C D E A (3) A E D B C (3) C B D E A (2) B E A D C (2) A D E C B (2) A D E B C (2) A D B E C (2) A C D E B (2) E B C A D (1) E A C D B (1) E A C B D (1) E A B D C (1) D A C E B (1) D A C B E (1) D A B C E (1) C E A D B (1) C D B E A (1) C D A B E (1) C B D A E (1) B E D C A (1) B E D A C (1) B E C A D (1) B E A C D (1) B D C A E (1) B C D A E (1) A E B D C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 12 10 4 B 4 0 8 -2 -2 C -12 -8 0 4 -4 D -10 2 -4 0 0 E -4 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999972 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -4 12 10 4 B 4 0 8 -2 -2 C -12 -8 0 4 -4 D -10 2 -4 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999984 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=27 E=19 C=14 D=9 so D is eliminated. Round 2 votes counts: A=34 B=27 C=20 E=19 so E is eliminated. Round 3 votes counts: B=43 A=37 C=20 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:211 B:204 E:201 D:194 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 10 4 B 4 0 8 -2 -2 C -12 -8 0 4 -4 D -10 2 -4 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999984 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 10 4 B 4 0 8 -2 -2 C -12 -8 0 4 -4 D -10 2 -4 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999984 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 10 4 B 4 0 8 -2 -2 C -12 -8 0 4 -4 D -10 2 -4 0 0 E -4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999984 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5252: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D B A C E (6) D E C A B (5) D B E C A (5) D A B C E (5) B A C E D (5) E D C A B (4) D E C B A (4) D B A E C (4) C E B A D (4) B C E A D (4) D A B E C (3) A D B C E (3) E D C B A (2) E C B A D (2) E C A D B (2) D E B C A (2) D A E B C (2) B D C E A (2) B D A C E (2) B C E D A (2) B C A E D (2) A D E C B (2) A C B E D (2) E D A C B (1) E C D B A (1) E C D A B (1) E C A B D (1) E A C D B (1) D E A C B (1) D B E A C (1) D B C A E (1) D A E C B (1) C A B E D (1) B D C A E (1) B A D C E (1) A E C D B (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 8 -12 10 B 2 0 26 -16 24 C -8 -26 0 -18 8 D 12 16 18 0 6 E -10 -24 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -12 10 B 2 0 26 -16 24 C -8 -26 0 -18 8 D 12 16 18 0 6 E -10 -24 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=21 B=19 E=15 C=5 so C is eliminated. Round 2 votes counts: D=40 A=22 E=19 B=19 so E is eliminated. Round 3 votes counts: D=49 A=26 B=25 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:218 A:202 C:178 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 -12 10 B 2 0 26 -16 24 C -8 -26 0 -18 8 D 12 16 18 0 6 E -10 -24 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -12 10 B 2 0 26 -16 24 C -8 -26 0 -18 8 D 12 16 18 0 6 E -10 -24 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -12 10 B 2 0 26 -16 24 C -8 -26 0 -18 8 D 12 16 18 0 6 E -10 -24 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5253: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) D C A E B (10) D E C A B (6) C D A E B (6) A C D E B (6) C A D E B (5) B E A D C (5) E B A D C (4) C A D B E (4) B C A D E (4) A C B D E (4) E B D A C (3) C D A B E (3) C A B D E (3) E D C B A (2) E D B A C (2) E D A C B (2) B E D A C (2) B A E C D (2) B A C E D (2) E D C A B (1) E D A B C (1) E B D C A (1) E A D C B (1) D E C B A (1) D E B C A (1) D B E C A (1) C D B A E (1) C B D A E (1) B A C D E (1) A E C D B (1) A E C B D (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 12 0 12 12 B -12 0 -18 -10 -10 C 0 18 0 12 2 D -12 10 -12 0 16 E -12 10 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.853131 B: 0.000000 C: 0.146869 D: 0.000000 E: 0.000000 Sum of squares = 0.749403270171 Cumulative probabilities = A: 0.853131 B: 0.853131 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 12 12 B -12 0 -18 -10 -10 C 0 18 0 12 2 D -12 10 -12 0 16 E -12 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 D=19 E=17 A=14 so A is eliminated. Round 2 votes counts: C=33 B=28 D=20 E=19 so E is eliminated. Round 3 votes counts: B=36 C=35 D=29 so D is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:218 C:216 D:201 E:190 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 12 12 B -12 0 -18 -10 -10 C 0 18 0 12 2 D -12 10 -12 0 16 E -12 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 12 12 B -12 0 -18 -10 -10 C 0 18 0 12 2 D -12 10 -12 0 16 E -12 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 12 12 B -12 0 -18 -10 -10 C 0 18 0 12 2 D -12 10 -12 0 16 E -12 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5254: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A C B D E (6) E D B A C (5) E B D A C (5) D A B E C (4) A D B C E (4) E D B C A (3) E C D B A (3) C A B E D (3) B E D A C (3) B A E D C (3) E B C A D (2) D E B A C (2) D E A C B (2) D E A B C (2) D B A E C (2) D A C E B (2) D A B C E (2) C E B A D (2) C D A E B (2) C B A E D (2) C A D E B (2) B E A D C (2) B C A E D (2) B A D E C (2) B A C D E (2) A C D B E (2) A B C D E (2) E D C B A (1) E D C A B (1) E C B D A (1) E C B A D (1) E B C D A (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E A B (1) C A E D B (1) C A E B D (1) C A D B E (1) B E C A D (1) B E A C D (1) B D E A C (1) B D A E C (1) B A D C E (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 16 10 14 B 0 0 10 12 14 C -16 -10 0 2 2 D -10 -12 -2 0 4 E -14 -14 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.412384 B: 0.587616 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.515353223326 Cumulative probabilities = A: 0.412384 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 10 14 B 0 0 10 12 14 C -16 -10 0 2 2 D -10 -12 -2 0 4 E -14 -14 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=20 D=16 A=15 so A is eliminated. Round 2 votes counts: C=34 E=23 B=23 D=20 so D is eliminated. Round 3 votes counts: C=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:220 B:218 D:190 C:189 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 16 10 14 B 0 0 10 12 14 C -16 -10 0 2 2 D -10 -12 -2 0 4 E -14 -14 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 10 14 B 0 0 10 12 14 C -16 -10 0 2 2 D -10 -12 -2 0 4 E -14 -14 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 10 14 B 0 0 10 12 14 C -16 -10 0 2 2 D -10 -12 -2 0 4 E -14 -14 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5255: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) D A E B C (6) C B E A D (6) C A D E B (6) B E D A C (6) B E D C A (5) B A D E C (4) A C D E B (4) A B C D E (4) E D B C A (3) E B D C A (3) D E B A C (3) D A E C B (3) B E C D A (3) A D B E C (3) E D B A C (2) E B C D A (2) D E A B C (2) D A C E B (2) C E A B D (2) B C E D A (2) B A E D C (2) A D C B E (2) A C D B E (2) D B E A C (1) D A B E C (1) C E D B A (1) C E D A B (1) C B A E D (1) C A E D B (1) C A B D E (1) B E C A D (1) B D E A C (1) B C E A D (1) A D E B C (1) A D B C E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 22 6 10 B -6 0 18 -6 0 C -22 -18 0 -22 -6 D -6 6 22 0 16 E -10 0 6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 22 6 10 B -6 0 18 -6 0 C -22 -18 0 -22 -6 D -6 6 22 0 16 E -10 0 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 C=19 D=18 E=10 so E is eliminated. Round 2 votes counts: B=30 A=28 D=23 C=19 so C is eliminated. Round 3 votes counts: A=38 B=37 D=25 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:219 B:203 E:190 C:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 22 6 10 B -6 0 18 -6 0 C -22 -18 0 -22 -6 D -6 6 22 0 16 E -10 0 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 22 6 10 B -6 0 18 -6 0 C -22 -18 0 -22 -6 D -6 6 22 0 16 E -10 0 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 22 6 10 B -6 0 18 -6 0 C -22 -18 0 -22 -6 D -6 6 22 0 16 E -10 0 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5256: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (20) C B D E A (14) A E D B C (8) B C A E D (6) E A D B C (5) E A D C B (4) C B E A D (4) B C A D E (4) E D A C B (3) D C E A B (3) C B D A E (3) B C D A E (3) B A C E D (3) A B E D C (3) D A E B C (2) C E D A B (2) C D B E A (2) E D A B C (1) E A C B D (1) D E C A B (1) C D E A B (1) C B E D A (1) C B A E D (1) B D C A E (1) B A E D C (1) B A E C D (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 8 -14 -24 B -12 0 -20 -6 -4 C -8 20 0 -6 -4 D 14 6 6 0 10 E 24 4 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -14 -24 B -12 0 -20 -6 -4 C -8 20 0 -6 -4 D 14 6 6 0 10 E 24 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 B=19 E=14 A=13 so A is eliminated. Round 2 votes counts: C=28 D=27 B=23 E=22 so E is eliminated. Round 3 votes counts: D=48 C=29 B=23 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:211 C:201 A:191 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -14 -24 B -12 0 -20 -6 -4 C -8 20 0 -6 -4 D 14 6 6 0 10 E 24 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -14 -24 B -12 0 -20 -6 -4 C -8 20 0 -6 -4 D 14 6 6 0 10 E 24 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -14 -24 B -12 0 -20 -6 -4 C -8 20 0 -6 -4 D 14 6 6 0 10 E 24 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5257: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (14) A C D E B (10) D A C E B (9) B E C A D (7) D E B A C (6) B E D C A (6) E D B C A (5) E B D C A (5) E B D A C (5) C A B E D (5) D E B C A (4) A D C E B (4) A C B D E (4) B E C D A (3) D A E C B (2) C B E A D (2) C B A E D (2) A C B E D (2) E D B A C (1) D E A B C (1) C A D E B (1) C A D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 18 6 10 B -8 0 -12 -16 -6 C -18 12 0 4 10 D -6 16 -4 0 12 E -10 6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 6 10 B -8 0 -12 -16 -6 C -18 12 0 4 10 D -6 16 -4 0 12 E -10 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=22 E=16 B=16 C=11 so C is eliminated. Round 2 votes counts: A=42 D=22 B=20 E=16 so E is eliminated. Round 3 votes counts: A=42 B=30 D=28 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:209 C:204 E:187 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 6 10 B -8 0 -12 -16 -6 C -18 12 0 4 10 D -6 16 -4 0 12 E -10 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 6 10 B -8 0 -12 -16 -6 C -18 12 0 4 10 D -6 16 -4 0 12 E -10 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 6 10 B -8 0 -12 -16 -6 C -18 12 0 4 10 D -6 16 -4 0 12 E -10 6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5258: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (12) C D E B A (9) C E D B A (8) E B A C D (7) D A C B E (7) D A B C E (7) A B E D C (6) E B C A D (4) C D A B E (4) A D B E C (4) E C B D A (3) C E B D A (3) B A E C D (3) A B D E C (3) E C B A D (2) E A B D C (2) D C E A B (2) D C A E B (2) B A E D C (2) E C D B A (1) D A E B C (1) D A B E C (1) C E B A D (1) C D B E A (1) C D B A E (1) C B E A D (1) B E A D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -8 -24 10 B -6 0 -14 -22 10 C 8 14 0 -2 18 D 24 22 2 0 10 E -10 -10 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -24 10 B -6 0 -14 -22 10 C 8 14 0 -2 18 D 24 22 2 0 10 E -10 -10 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=28 E=19 A=15 B=6 so B is eliminated. Round 2 votes counts: D=32 C=28 E=20 A=20 so E is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:229 C:219 A:192 B:184 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -24 10 B -6 0 -14 -22 10 C 8 14 0 -2 18 D 24 22 2 0 10 E -10 -10 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -24 10 B -6 0 -14 -22 10 C 8 14 0 -2 18 D 24 22 2 0 10 E -10 -10 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -24 10 B -6 0 -14 -22 10 C 8 14 0 -2 18 D 24 22 2 0 10 E -10 -10 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5259: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) A B D E C (8) E C A D B (6) C E D A B (5) B D C A E (5) A B E D C (5) E C D B A (4) E A C D B (4) D C B E A (4) C E D B A (4) C D E B A (4) B D A C E (4) B A D E C (4) E A C B D (3) D B C A E (3) D B A C E (3) B A D C E (3) D C E B A (2) D A B C E (2) B A E D C (2) A E C B D (2) A E B C D (2) E C A B D (1) D C B A E (1) D B C E A (1) C E B D A (1) C D B E A (1) B E C D A (1) B E C A D (1) B D C E A (1) A E C D B (1) A D C E B (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -12 2 C 8 6 0 0 -8 D 10 12 0 0 -4 E 4 -2 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.50617283957 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -12 2 C 8 6 0 0 -8 D 10 12 0 0 -4 E 4 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.5061728396 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=21 A=21 D=16 C=15 so C is eliminated. Round 2 votes counts: E=37 D=21 B=21 A=21 so D is eliminated. Round 3 votes counts: E=43 B=34 A=23 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:209 E:207 C:203 B:191 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -12 2 C 8 6 0 0 -8 D 10 12 0 0 -4 E 4 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.5061728396 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -12 2 C 8 6 0 0 -8 D 10 12 0 0 -4 E 4 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.5061728396 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -10 -4 B -2 0 -6 -12 2 C 8 6 0 0 -8 D 10 12 0 0 -4 E 4 -2 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.111111 E: 0.666667 Sum of squares = 0.5061728396 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.333333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5260: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) C B D E A (11) C D B A E (10) C B D A E (7) E B C D A (6) E A C B D (6) E A B D C (6) D B C A E (6) E A D B C (4) E A B C D (3) B C D E A (3) A D C B E (3) D C B A E (2) D A C B E (2) C A D B E (2) B D C E A (2) A E D C B (2) A D E B C (2) E C B D A (1) E C B A D (1) E B D A C (1) E A C D B (1) D A E B C (1) D A B C E (1) C B E D A (1) B C E D A (1) A E C D B (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -6 -10 6 B 4 0 -4 0 2 C 6 4 0 10 2 D 10 0 -10 0 6 E -6 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -10 6 B 4 0 -4 0 2 C 6 4 0 10 2 D 10 0 -10 0 6 E -6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=29 A=22 D=12 B=6 so B is eliminated. Round 2 votes counts: C=35 E=29 A=22 D=14 so D is eliminated. Round 3 votes counts: C=45 E=29 A=26 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:203 B:201 A:193 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -10 6 B 4 0 -4 0 2 C 6 4 0 10 2 D 10 0 -10 0 6 E -6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -10 6 B 4 0 -4 0 2 C 6 4 0 10 2 D 10 0 -10 0 6 E -6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -10 6 B 4 0 -4 0 2 C 6 4 0 10 2 D 10 0 -10 0 6 E -6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5261: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) C A B E D (7) A C E D B (7) D B E A C (6) B E D C A (5) E B D C A (4) D A C E B (4) C B A E D (4) C A E B D (4) A C D E B (4) D E B A C (3) D E A B C (3) B E C A D (3) A C D B E (3) E B C A D (2) E A D C B (2) D B C A E (2) D B A C E (2) D A E C B (2) D A C B E (2) D A B E C (2) C E A B D (2) C B A D E (2) B C D A E (2) A C E B D (2) E D B A C (1) E B D A C (1) E A D B C (1) D B A E C (1) D A B C E (1) C E B A D (1) C A B D E (1) B E C D A (1) B D C E A (1) B D C A E (1) B C D E A (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -4 -6 10 B 4 0 4 6 12 C 4 -4 0 -6 10 D 6 -6 6 0 6 E -10 -12 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 10 B 4 0 4 6 12 C 4 -4 0 -6 10 D 6 -6 6 0 6 E -10 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=23 C=21 A=17 E=11 so E is eliminated. Round 2 votes counts: B=30 D=29 C=21 A=20 so A is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:213 D:206 C:202 A:198 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 10 B 4 0 4 6 12 C 4 -4 0 -6 10 D 6 -6 6 0 6 E -10 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 10 B 4 0 4 6 12 C 4 -4 0 -6 10 D 6 -6 6 0 6 E -10 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 10 B 4 0 4 6 12 C 4 -4 0 -6 10 D 6 -6 6 0 6 E -10 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5262: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) E B C D A (8) D E C B A (7) E A B C D (6) D A C B E (6) D E A B C (4) D C B E A (4) D A E C B (4) C B A E D (4) A D E B C (4) E D A B C (3) A E D B C (3) A C B D E (3) E D B C A (2) E B C A D (2) E B A C D (2) D C E B A (2) C B D E A (2) A D C B E (2) A C B E D (2) E B D C A (1) E B D A C (1) E A D B C (1) E A B D C (1) D E B C A (1) D E B A C (1) D C B A E (1) D A C E B (1) C B E A D (1) C B A D E (1) B C E D A (1) B C E A D (1) B C A E D (1) B A C E D (1) A E B C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 22 2 -2 B -12 0 18 6 -10 C -22 -18 0 2 -6 D -2 -6 -2 0 -12 E 2 10 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 22 2 -2 B -12 0 18 6 -10 C -22 -18 0 2 -6 D -2 -6 -2 0 -12 E 2 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=30 E=27 C=8 B=4 so B is eliminated. Round 2 votes counts: D=31 A=31 E=27 C=11 so C is eliminated. Round 3 votes counts: A=37 D=33 E=30 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:215 B:201 D:189 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 22 2 -2 B -12 0 18 6 -10 C -22 -18 0 2 -6 D -2 -6 -2 0 -12 E 2 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 22 2 -2 B -12 0 18 6 -10 C -22 -18 0 2 -6 D -2 -6 -2 0 -12 E 2 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 22 2 -2 B -12 0 18 6 -10 C -22 -18 0 2 -6 D -2 -6 -2 0 -12 E 2 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5263: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) A D E C B (8) A C B E D (8) A C E B D (6) E C B D A (4) D E C B A (4) C B E D A (4) B C E D A (4) B C E A D (4) D E A C B (3) D A E B C (3) C E B A D (3) B C D E A (3) B C A E D (3) A E D C B (3) A B C D E (3) E D C B A (2) E C B A D (2) D B E C A (2) D B C E A (2) C B A E D (2) A E C B D (2) A D B C E (2) E D C A B (1) E D A C B (1) E C A B D (1) E A D C B (1) E A C D B (1) D E A B C (1) D B A C E (1) D A B C E (1) B C D A E (1) A E C D B (1) A D E B C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 6 -6 B 2 0 -14 2 -16 C 4 14 0 6 -4 D -6 -2 -6 0 -8 E 6 16 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 6 -6 B 2 0 -14 2 -16 C 4 14 0 6 -4 D -6 -2 -6 0 -8 E 6 16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=27 B=15 E=13 C=9 so C is eliminated. Round 2 votes counts: A=36 D=27 B=21 E=16 so E is eliminated. Round 3 votes counts: A=39 D=31 B=30 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:217 C:210 A:197 D:189 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 6 -6 B 2 0 -14 2 -16 C 4 14 0 6 -4 D -6 -2 -6 0 -8 E 6 16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 6 -6 B 2 0 -14 2 -16 C 4 14 0 6 -4 D -6 -2 -6 0 -8 E 6 16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 6 -6 B 2 0 -14 2 -16 C 4 14 0 6 -4 D -6 -2 -6 0 -8 E 6 16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5264: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) A C D E B (9) E B D C A (8) D C A E B (7) B E A C D (6) A C D B E (6) E D C A B (5) C D A B E (5) B C D A E (4) B A C D E (4) E D C B A (3) E B A D C (3) D C E A B (3) C D A E B (3) E B A C D (2) D E C B A (2) D C A B E (2) B A C E D (2) A B E C D (2) E D B C A (1) E B D A C (1) E A B D C (1) D C B E A (1) D C B A E (1) D B C E A (1) C A D B E (1) B E C D A (1) B E A D C (1) B D E C A (1) B A E C D (1) A E C D B (1) A E C B D (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -16 -16 2 B 4 0 -2 -4 -4 C 16 2 0 -2 0 D 16 4 2 0 2 E -2 4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -16 2 B 4 0 -2 -4 -4 C 16 2 0 -2 0 D 16 4 2 0 2 E -2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=24 A=21 D=17 C=9 so C is eliminated. Round 2 votes counts: B=29 D=25 E=24 A=22 so A is eliminated. Round 3 votes counts: D=42 B=31 E=27 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:208 E:200 B:197 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -16 -16 2 B 4 0 -2 -4 -4 C 16 2 0 -2 0 D 16 4 2 0 2 E -2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -16 2 B 4 0 -2 -4 -4 C 16 2 0 -2 0 D 16 4 2 0 2 E -2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -16 2 B 4 0 -2 -4 -4 C 16 2 0 -2 0 D 16 4 2 0 2 E -2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5265: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (6) C E D A B (5) B D A E C (5) B D A C E (5) D B C A E (4) C E A D B (4) C D E B A (4) A E C B D (4) A B D C E (4) E C D B A (3) E B D A C (3) D C B A E (3) C A D E B (3) B A D C E (3) A B E D C (3) A B E C D (3) A B D E C (3) E D C B A (2) E D B C A (2) E C D A B (2) E C A D B (2) E A C B D (2) E A B C D (2) D E B C A (2) D C E B A (2) D C B E A (2) D B C E A (2) C A E D B (2) B A D E C (2) A C B E D (2) A B C E D (2) A B C D E (2) D B E C A (1) C E D B A (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 8 4 18 B -8 0 8 6 -4 C -8 -8 0 4 6 D -4 -6 -4 0 -2 E -18 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 4 18 B -8 0 8 6 -4 C -8 -8 0 4 6 D -4 -6 -4 0 -2 E -18 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=19 E=18 D=16 B=15 so B is eliminated. Round 2 votes counts: A=37 D=26 C=19 E=18 so E is eliminated. Round 3 votes counts: A=41 D=33 C=26 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:201 C:197 D:192 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 4 18 B -8 0 8 6 -4 C -8 -8 0 4 6 D -4 -6 -4 0 -2 E -18 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 4 18 B -8 0 8 6 -4 C -8 -8 0 4 6 D -4 -6 -4 0 -2 E -18 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 4 18 B -8 0 8 6 -4 C -8 -8 0 4 6 D -4 -6 -4 0 -2 E -18 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5266: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (11) D E C A B (10) C D E A B (9) D C E A B (7) B A E D C (7) B A E C D (6) C D B A E (5) C B A D E (5) E D A B C (4) E A B D C (4) D E A B C (4) C D E B A (4) C B A E D (4) E D C A B (2) B A C D E (2) A B E D C (2) E D C B A (1) E D A C B (1) E C D B A (1) E B A D C (1) D C A E B (1) D C A B E (1) D A B E C (1) C D A B E (1) C B D A E (1) C A B D E (1) B C A E D (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -8 -6 4 B -2 0 -8 -6 0 C 8 8 0 2 6 D 6 6 -2 0 8 E -4 0 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -6 4 B -2 0 -8 -6 0 C 8 8 0 2 6 D 6 6 -2 0 8 E -4 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=27 D=24 E=14 A=5 so A is eliminated. Round 2 votes counts: C=30 B=30 D=25 E=15 so E is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:212 D:209 A:196 B:192 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -6 4 B -2 0 -8 -6 0 C 8 8 0 2 6 D 6 6 -2 0 8 E -4 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -6 4 B -2 0 -8 -6 0 C 8 8 0 2 6 D 6 6 -2 0 8 E -4 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -6 4 B -2 0 -8 -6 0 C 8 8 0 2 6 D 6 6 -2 0 8 E -4 0 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5267: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) C A B E D (7) B E C D A (6) D E B A C (5) B C E A D (5) A C D E B (5) D E B C A (4) C B A E D (4) C A B D E (4) B E C A D (4) A C B E D (4) E D B C A (3) E B D C A (3) E B D A C (3) D C E B A (3) C B E A D (3) A C D B E (3) E B A D C (2) C D A E B (2) C D A B E (2) B E D C A (2) B C E D A (2) E D B A C (1) E A B D C (1) D E C A B (1) D E A C B (1) C D B E A (1) C B D E A (1) C A D E B (1) B E A D C (1) B E A C D (1) B C A E D (1) A E B D C (1) A E B C D (1) A D E C B (1) A D E B C (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -32 20 -6 B 10 0 -4 16 20 C 32 4 0 34 14 D -20 -16 -34 0 -14 E 6 -20 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -32 20 -6 B 10 0 -4 16 20 C 32 4 0 34 14 D -20 -16 -34 0 -14 E 6 -20 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=22 A=19 D=14 E=13 so E is eliminated. Round 2 votes counts: C=32 B=30 A=20 D=18 so D is eliminated. Round 3 votes counts: B=43 C=36 A=21 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:242 B:221 E:193 A:186 D:158 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -32 20 -6 B 10 0 -4 16 20 C 32 4 0 34 14 D -20 -16 -34 0 -14 E 6 -20 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -32 20 -6 B 10 0 -4 16 20 C 32 4 0 34 14 D -20 -16 -34 0 -14 E 6 -20 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -32 20 -6 B 10 0 -4 16 20 C 32 4 0 34 14 D -20 -16 -34 0 -14 E 6 -20 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5268: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (6) C D E A B (6) B A E D C (5) B A D E C (5) A D E B C (5) D E A C B (4) A B E D C (4) A B D E C (4) D C E A B (3) D A C E B (3) C B E D A (3) A D C E B (3) E D C A B (2) E D A C B (2) E C D B A (2) E C B D A (2) C E D B A (2) C D E B A (2) C B A D E (2) B E A D C (2) B E A C D (2) B C E A D (2) B C A D E (2) B A C E D (2) A E B D C (2) A B C D E (2) E D C B A (1) E C D A B (1) E B D C A (1) E B D A C (1) E A D B C (1) E A B D C (1) D E C A B (1) C E D A B (1) C E B D A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D E A (1) C A D B E (1) B E C D A (1) B C E D A (1) B C A E D (1) B A D C E (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 14 16 0 8 B -14 0 -10 -2 -8 C -16 10 0 -16 -10 D 0 2 16 0 14 E -8 8 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.749994 B: 0.000000 C: 0.000000 D: 0.250006 E: 0.000000 Sum of squares = 0.624993704786 Cumulative probabilities = A: 0.749994 B: 0.749994 C: 0.749994 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 0 8 B -14 0 -10 -2 -8 C -16 10 0 -16 -10 D 0 2 16 0 14 E -8 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 C=22 D=17 E=14 so E is eliminated. Round 2 votes counts: C=27 B=26 A=25 D=22 so D is eliminated. Round 3 votes counts: A=40 C=34 B=26 so B is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:216 E:198 C:184 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 14 16 0 8 B -14 0 -10 -2 -8 C -16 10 0 -16 -10 D 0 2 16 0 14 E -8 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 0 8 B -14 0 -10 -2 -8 C -16 10 0 -16 -10 D 0 2 16 0 14 E -8 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 0 8 B -14 0 -10 -2 -8 C -16 10 0 -16 -10 D 0 2 16 0 14 E -8 8 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5269: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) C E D A B (9) C B A D E (9) B A C D E (9) E D A C B (6) E D A B C (6) C B E D A (6) B A D E C (6) E D C A B (4) B C A D E (4) C B A E D (3) C A B D E (3) A E D C B (3) A B D E C (3) C A E D B (2) A D E B C (2) E D B C A (1) E D B A C (1) E C D A B (1) D E B A C (1) D E A B C (1) D A E B C (1) C E B D A (1) C E A D B (1) C B E A D (1) C B D E A (1) C A B E D (1) B D A E C (1) B C D A E (1) B A D C E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -14 0 2 B 10 0 -20 0 0 C 14 20 0 22 22 D 0 0 -22 0 -12 E -2 0 -22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 0 2 B 10 0 -20 0 0 C 14 20 0 22 22 D 0 0 -22 0 -12 E -2 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 B=22 E=19 A=10 D=3 so D is eliminated. Round 2 votes counts: C=46 B=22 E=21 A=11 so A is eliminated. Round 3 votes counts: C=46 E=28 B=26 so B is eliminated. Round 4 votes counts: C=61 E=39 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:239 B:195 E:194 A:189 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 0 2 B 10 0 -20 0 0 C 14 20 0 22 22 D 0 0 -22 0 -12 E -2 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 0 2 B 10 0 -20 0 0 C 14 20 0 22 22 D 0 0 -22 0 -12 E -2 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 0 2 B 10 0 -20 0 0 C 14 20 0 22 22 D 0 0 -22 0 -12 E -2 0 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5270: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (7) E A C B D (5) C A E B D (5) B D E C A (5) A E C D B (5) E B D A C (4) C A E D B (4) C A D B E (4) D B E A C (3) C B D A E (3) B E D C A (3) B D C A E (3) A C D B E (3) E D B A C (2) E C B A D (2) E B D C A (2) E A B D C (2) D B E C A (2) D B A E C (2) D A E B C (2) C A D E B (2) B D E A C (2) B D C E A (2) B C D E A (2) E C A B D (1) E B C D A (1) E B C A D (1) E B A D C (1) E B A C D (1) E A D C B (1) E A D B C (1) E A C D B (1) D E B A C (1) D C B A E (1) D B C A E (1) D B A C E (1) D A C B E (1) C E B D A (1) C E B A D (1) C E A B D (1) C B E A D (1) C B D E A (1) C A B E D (1) C A B D E (1) B C E D A (1) A E D C B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -4 8 -2 B 0 0 -8 6 -14 C 4 8 0 12 -4 D -8 -6 -12 0 -12 E 2 14 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -4 8 -2 B 0 0 -8 6 -14 C 4 8 0 12 -4 D -8 -6 -12 0 -12 E 2 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=25 C=25 B=18 A=18 D=14 so D is eliminated. Round 2 votes counts: B=27 E=26 C=26 A=21 so A is eliminated. Round 3 votes counts: C=38 E=35 B=27 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:210 A:201 B:192 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 8 -2 B 0 0 -8 6 -14 C 4 8 0 12 -4 D -8 -6 -12 0 -12 E 2 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 8 -2 B 0 0 -8 6 -14 C 4 8 0 12 -4 D -8 -6 -12 0 -12 E 2 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 8 -2 B 0 0 -8 6 -14 C 4 8 0 12 -4 D -8 -6 -12 0 -12 E 2 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5271: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (13) C A E D B (11) C A E B D (11) E A C D B (6) D B E A C (6) D C B A E (5) B D C A E (5) A E C D B (5) D B C A E (4) E A D C B (3) E A C B D (3) C B A E D (3) B D C E A (3) D E A B C (2) C D B A E (2) B C E A D (2) E B D A C (1) D C A B E (1) D B C E A (1) D B A E C (1) C E A B D (1) C D A E B (1) C D A B E (1) C B E A D (1) C B A D E (1) C A D B E (1) C A B E D (1) B E D A C (1) B E A D C (1) B E A C D (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -10 6 10 B 2 0 -18 -2 8 C 10 18 0 6 12 D -6 2 -6 0 -6 E -10 -8 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 6 10 B 2 0 -18 -2 8 C 10 18 0 6 12 D -6 2 -6 0 -6 E -10 -8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=26 D=20 E=13 A=7 so A is eliminated. Round 2 votes counts: C=35 B=26 D=20 E=19 so E is eliminated. Round 3 votes counts: C=50 B=27 D=23 so D is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:202 B:195 D:192 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 6 10 B 2 0 -18 -2 8 C 10 18 0 6 12 D -6 2 -6 0 -6 E -10 -8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 6 10 B 2 0 -18 -2 8 C 10 18 0 6 12 D -6 2 -6 0 -6 E -10 -8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 6 10 B 2 0 -18 -2 8 C 10 18 0 6 12 D -6 2 -6 0 -6 E -10 -8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5272: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) B D E C A (7) D E B C A (6) A C E D B (6) E A D C B (5) C A E D B (4) C A E B D (4) C A B E D (4) B C A D E (4) A C E B D (4) E D A B C (3) C B A D E (3) B D C E A (3) B C D A E (3) B A C D E (3) A C B E D (3) E D B A C (2) E D A C B (2) E C D A B (2) E A D B C (2) D B E A C (2) A E D C B (2) A E D B C (2) A E C D B (2) A E B D C (2) E D C A B (1) E A C D B (1) D B E C A (1) C D E B A (1) C A B D E (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D E A (1) B A D E C (1) Total count = 100 A B C D E A 0 0 8 6 2 B 0 0 10 -6 -20 C -8 -10 0 -8 -8 D -6 6 8 0 -2 E -2 20 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.952874 B: 0.047126 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.910189051608 Cumulative probabilities = A: 0.952874 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 6 2 B 0 0 10 -6 -20 C -8 -10 0 -8 -8 D -6 6 8 0 -2 E -2 20 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.909091 B: 0.090909 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.834710755859 Cumulative probabilities = A: 0.909091 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=21 E=18 D=18 C=17 so C is eliminated. Round 2 votes counts: A=34 B=29 D=19 E=18 so E is eliminated. Round 3 votes counts: A=42 D=29 B=29 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:214 A:208 D:203 B:192 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 6 2 B 0 0 10 -6 -20 C -8 -10 0 -8 -8 D -6 6 8 0 -2 E -2 20 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.909091 B: 0.090909 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.834710755859 Cumulative probabilities = A: 0.909091 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 6 2 B 0 0 10 -6 -20 C -8 -10 0 -8 -8 D -6 6 8 0 -2 E -2 20 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.909091 B: 0.090909 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.834710755859 Cumulative probabilities = A: 0.909091 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 6 2 B 0 0 10 -6 -20 C -8 -10 0 -8 -8 D -6 6 8 0 -2 E -2 20 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.909091 B: 0.090909 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.834710755859 Cumulative probabilities = A: 0.909091 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5273: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E A B D (8) A D B C E (8) B A D E C (7) E C D B A (6) E C B A D (5) E C D A B (4) D A B C E (4) C E D A B (4) B D A E C (4) B A D C E (4) A C B E D (4) E D B C A (3) D E B C A (3) A B D C E (3) D B E A C (2) D B A C E (2) C A D E B (2) A D C B E (2) A C B D E (2) A B C D E (2) E D C B A (1) E C A D B (1) E C A B D (1) D C E A B (1) D C A E B (1) D A C E B (1) D A C B E (1) C A E D B (1) C A E B D (1) B E D C A (1) B E A D C (1) B D E A C (1) B A E D C (1) Total count = 100 A B C D E A 0 2 14 6 16 B -2 0 8 -10 14 C -14 -8 0 -18 2 D -6 10 18 0 16 E -16 -14 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 6 16 B -2 0 8 -10 14 C -14 -8 0 -18 2 D -6 10 18 0 16 E -16 -14 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=21 A=21 B=19 C=16 so C is eliminated. Round 2 votes counts: E=33 A=25 D=23 B=19 so B is eliminated. Round 3 votes counts: A=37 E=35 D=28 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:219 B:205 C:181 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 6 16 B -2 0 8 -10 14 C -14 -8 0 -18 2 D -6 10 18 0 16 E -16 -14 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 6 16 B -2 0 8 -10 14 C -14 -8 0 -18 2 D -6 10 18 0 16 E -16 -14 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 6 16 B -2 0 8 -10 14 C -14 -8 0 -18 2 D -6 10 18 0 16 E -16 -14 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5274: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) B A D E C (7) A B D E C (6) E C B D A (5) E B C A D (5) C E D B A (5) C E B D A (5) B E A C D (5) A D B C E (5) A B D C E (5) C D E A B (4) D A E C B (3) D A C E B (3) C D A E B (3) B A E D C (3) E B C D A (2) D C E A B (2) D C A B E (2) D A E B C (2) D A C B E (2) D A B C E (2) C E D A B (2) A D B E C (2) E D C A B (1) E D A B C (1) E C D B A (1) E C D A B (1) E B A D C (1) C E B A D (1) C D A B E (1) C B A D E (1) C A D B E (1) B E A D C (1) B A C E D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 14 2 -8 16 B -14 0 -4 -4 -8 C -2 4 0 -12 8 D 8 4 12 0 20 E -16 8 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 -8 16 B -14 0 -4 -4 -8 C -2 4 0 -12 8 D 8 4 12 0 20 E -16 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=23 C=23 A=20 E=17 B=17 so E is eliminated. Round 2 votes counts: C=30 D=25 B=25 A=20 so A is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:212 C:199 B:185 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 2 -8 16 B -14 0 -4 -4 -8 C -2 4 0 -12 8 D 8 4 12 0 20 E -16 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 -8 16 B -14 0 -4 -4 -8 C -2 4 0 -12 8 D 8 4 12 0 20 E -16 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 -8 16 B -14 0 -4 -4 -8 C -2 4 0 -12 8 D 8 4 12 0 20 E -16 8 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5275: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) C E A B D (6) C B A D E (6) E C A D B (5) D E B A C (5) D B C A E (4) D B A E C (4) C B A E D (4) E C A B D (3) D E C B A (3) C D B E A (3) B D A C E (3) E D A C B (2) E A D B C (2) E A C B D (2) D E A B C (2) D B E A C (2) D B A C E (2) C E B A D (2) C D E B A (2) C B D A E (2) C A E B D (2) C A B E D (2) B D C A E (2) B C A D E (2) A B C E D (2) A B C D E (2) E D C A B (1) E C D A B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E B C A (1) D C B A E (1) D B E C A (1) D A E B C (1) C E A D B (1) C B E D A (1) C B D E A (1) A E B C D (1) A D B E C (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -12 -2 -12 B 2 0 -8 -6 -6 C 12 8 0 10 4 D 2 6 -10 0 0 E 12 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -2 -12 B 2 0 -8 -6 -6 C 12 8 0 10 4 D 2 6 -10 0 0 E 12 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=26 D=26 A=9 B=7 so B is eliminated. Round 2 votes counts: C=34 D=31 E=26 A=9 so A is eliminated. Round 3 votes counts: C=40 D=32 E=28 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:207 D:199 B:191 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -2 -12 B 2 0 -8 -6 -6 C 12 8 0 10 4 D 2 6 -10 0 0 E 12 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -2 -12 B 2 0 -8 -6 -6 C 12 8 0 10 4 D 2 6 -10 0 0 E 12 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -2 -12 B 2 0 -8 -6 -6 C 12 8 0 10 4 D 2 6 -10 0 0 E 12 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5276: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) D C E B A (10) C D E B A (8) B A E C D (6) A E B C D (6) D A C B E (5) E B C A D (4) A E B D C (4) A B E D C (4) E C B D A (3) E C B A D (3) A D B C E (3) E B A C D (2) D E C A B (2) D C E A B (2) D C B A E (2) C E D B A (2) C E B D A (2) B E C A D (2) B E A C D (2) B C E A D (2) E D A C B (1) E C D B A (1) E A D C B (1) D C B E A (1) D C A E B (1) D A C E B (1) D A B C E (1) C D B E A (1) C B D E A (1) B C E D A (1) B C D E A (1) A D E B C (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 2 8 -4 B 8 0 6 12 -8 C -2 -6 0 16 -10 D -8 -12 -16 0 -14 E 4 8 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 2 8 -4 B 8 0 6 12 -8 C -2 -6 0 16 -10 D -8 -12 -16 0 -14 E 4 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=25 E=15 C=14 B=14 so C is eliminated. Round 2 votes counts: D=34 A=32 E=19 B=15 so B is eliminated. Round 3 votes counts: A=38 D=36 E=26 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:218 B:209 A:199 C:199 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 2 8 -4 B 8 0 6 12 -8 C -2 -6 0 16 -10 D -8 -12 -16 0 -14 E 4 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 8 -4 B 8 0 6 12 -8 C -2 -6 0 16 -10 D -8 -12 -16 0 -14 E 4 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 8 -4 B 8 0 6 12 -8 C -2 -6 0 16 -10 D -8 -12 -16 0 -14 E 4 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5277: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (20) A E C D B (14) D C E B A (6) C E D A B (6) B A D E C (6) B A E C D (5) D C E A B (4) B D C A E (4) A D C E B (4) A B E C D (4) D B C E A (3) E C D A B (2) E C A D B (2) D C A E B (2) B E C A D (2) B D A C E (2) A E C B D (2) E B C A D (1) E A C D B (1) D C B E A (1) D A C E B (1) D A B C E (1) C A D E B (1) B E A C D (1) B D E C A (1) B D A E C (1) B A D C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -10 -8 0 B 8 0 8 4 8 C 10 -8 0 -18 12 D 8 -4 18 0 18 E 0 -8 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -8 0 B 8 0 8 4 8 C 10 -8 0 -18 12 D 8 -4 18 0 18 E 0 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 A=26 D=18 C=7 E=6 so E is eliminated. Round 2 votes counts: B=44 A=27 D=18 C=11 so C is eliminated. Round 3 votes counts: B=44 A=30 D=26 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:214 C:198 A:187 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -8 0 B 8 0 8 4 8 C 10 -8 0 -18 12 D 8 -4 18 0 18 E 0 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -8 0 B 8 0 8 4 8 C 10 -8 0 -18 12 D 8 -4 18 0 18 E 0 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -8 0 B 8 0 8 4 8 C 10 -8 0 -18 12 D 8 -4 18 0 18 E 0 -8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5278: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B A E D C (8) E C D A B (7) A E B D C (7) C D E B A (6) C D B A E (6) C D E A B (5) B C D A E (5) B A D C E (5) C D B E A (4) B C D E A (4) E D C A B (3) C B D A E (3) A B E D C (3) E A C D B (2) C B D E A (2) B A C D E (2) A E D C B (2) E B A C D (1) E A D B C (1) E A B D C (1) E A B C D (1) D C E A B (1) D C A E B (1) D A C E B (1) C E D A B (1) C E B D A (1) B E C A D (1) B E A C D (1) B D A C E (1) B C A D E (1) B A E C D (1) A E D B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -2 -2 -4 B 4 0 -12 -4 -4 C 2 12 0 8 0 D 2 4 -8 0 -4 E 4 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.355485 D: 0.000000 E: 0.644515 Sum of squares = 0.541769343645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.355485 D: 0.355485 E: 1.000000 A B C D E A 0 -4 -2 -2 -4 B 4 0 -12 -4 -4 C 2 12 0 8 0 D 2 4 -8 0 -4 E 4 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=28 E=26 A=14 D=3 so D is eliminated. Round 2 votes counts: C=30 B=29 E=26 A=15 so A is eliminated. Round 3 votes counts: E=36 C=32 B=32 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:211 E:206 D:197 A:194 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 -4 B 4 0 -12 -4 -4 C 2 12 0 8 0 D 2 4 -8 0 -4 E 4 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 -4 B 4 0 -12 -4 -4 C 2 12 0 8 0 D 2 4 -8 0 -4 E 4 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 -4 B 4 0 -12 -4 -4 C 2 12 0 8 0 D 2 4 -8 0 -4 E 4 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5279: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) E B A D C (10) C D A B E (6) A D C E B (6) E B A C D (5) D C A B E (5) B E D C A (5) B E D A C (5) B E C D A (5) C D B E A (4) C B E D A (4) E A B D C (3) D A C E B (3) B E C A D (3) D A E B C (2) C B E A D (2) C A D E B (2) B E A D C (2) B C E D A (2) A E B C D (2) A D E C B (2) E B C A D (1) D C B E A (1) D B E C A (1) D B E A C (1) C D B A E (1) C D A E B (1) C B D E A (1) C A D B E (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 10 6 -10 B 6 0 20 26 -2 C -10 -20 0 -18 -20 D -6 -26 18 0 -24 E 10 2 20 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 10 6 -10 B 6 0 20 26 -2 C -10 -20 0 -18 -20 D -6 -26 18 0 -24 E 10 2 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999971597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 B=22 E=19 D=13 so D is eliminated. Round 2 votes counts: A=29 C=28 B=24 E=19 so E is eliminated. Round 3 votes counts: B=40 A=32 C=28 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:228 B:225 A:200 D:181 C:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 10 6 -10 B 6 0 20 26 -2 C -10 -20 0 -18 -20 D -6 -26 18 0 -24 E 10 2 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999971597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 6 -10 B 6 0 20 26 -2 C -10 -20 0 -18 -20 D -6 -26 18 0 -24 E 10 2 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999971597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 6 -10 B 6 0 20 26 -2 C -10 -20 0 -18 -20 D -6 -26 18 0 -24 E 10 2 20 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999971597 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5280: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) D C E A B (8) B E A D C (8) A E D C B (7) E D C A B (5) C D E A B (5) B A C D E (5) A B C D E (5) E A D C B (4) B C D E A (4) B C D A E (4) A B E D C (4) E A B D C (3) B E D C A (3) A D C E B (3) A C D E B (3) E D C B A (2) E D A C B (2) C D B E A (2) B E C D A (2) B A E D C (2) A E B D C (2) E D B C A (1) E B D C A (1) C D E B A (1) C D B A E (1) C D A B E (1) C B D E A (1) B C A D E (1) A C D B E (1) Total count = 100 A B C D E A 0 24 -2 -4 -4 B -24 0 -10 -10 -12 C 2 10 0 -10 8 D 4 10 10 0 8 E 4 12 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 -2 -4 -4 B -24 0 -10 -10 -12 C 2 10 0 -10 8 D 4 10 10 0 8 E 4 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=25 C=20 E=18 D=8 so D is eliminated. Round 2 votes counts: B=29 C=28 A=25 E=18 so E is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:216 A:207 C:205 E:200 B:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 -2 -4 -4 B -24 0 -10 -10 -12 C 2 10 0 -10 8 D 4 10 10 0 8 E 4 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -2 -4 -4 B -24 0 -10 -10 -12 C 2 10 0 -10 8 D 4 10 10 0 8 E 4 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -2 -4 -4 B -24 0 -10 -10 -12 C 2 10 0 -10 8 D 4 10 10 0 8 E 4 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5281: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) B E A C D (8) E B C A D (7) A B D E C (7) E C B D A (5) E B C D A (5) D C A E B (5) D C A B E (5) C D E A B (5) A B E D C (5) C E D B A (4) B E C A D (4) A D C B E (4) D A C B E (3) A D B C E (3) E B A C D (2) D A C E B (2) B E A D C (2) E B D C A (1) E A B D C (1) D E C A B (1) D C E A B (1) C E B D A (1) C D B E A (1) C B E D A (1) B C A E D (1) B A E D C (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -18 -4 -22 B 10 0 0 4 -6 C 18 0 0 12 -2 D 4 -4 -12 0 4 E 22 6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839461 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 1.000000 A B C D E A 0 -10 -18 -4 -22 B 10 0 0 4 -6 C 18 0 0 12 -2 D 4 -4 -12 0 4 E 22 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=22 E=21 D=17 B=16 so B is eliminated. Round 2 votes counts: E=35 C=25 A=23 D=17 so D is eliminated. Round 3 votes counts: E=36 C=36 A=28 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:213 B:204 D:196 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -18 -4 -22 B 10 0 0 4 -6 C 18 0 0 12 -2 D 4 -4 -12 0 4 E 22 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -4 -22 B 10 0 0 4 -6 C 18 0 0 12 -2 D 4 -4 -12 0 4 E 22 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -4 -22 B 10 0 0 4 -6 C 18 0 0 12 -2 D 4 -4 -12 0 4 E 22 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.666667 Sum of squares = 0.506172839486 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.222222 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5282: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E A D B C (7) A E D B C (7) D E A B C (6) C B D E A (6) B C D E A (6) A E D C B (6) B D C E A (5) C B A E D (4) B C E D A (4) E D A B C (3) D E B A C (3) C B A D E (3) B C E A D (3) A D E C B (3) E B A C D (2) D C A B E (2) D A E B C (2) C A B E D (2) B E C A D (2) A D E B C (2) E D B A C (1) E A B D C (1) E A B C D (1) D C B A E (1) D C A E B (1) D B E C A (1) D B C E A (1) D A E C B (1) C D A B E (1) B E D C A (1) B E C D A (1) B D E C A (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -6 -10 -10 B 6 0 22 2 4 C 6 -22 0 -10 -6 D 10 -2 10 0 6 E 10 -4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -10 -10 B 6 0 22 2 4 C 6 -22 0 -10 -6 D 10 -2 10 0 6 E 10 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=23 A=20 D=18 E=15 so E is eliminated. Round 2 votes counts: A=29 B=25 C=24 D=22 so D is eliminated. Round 3 votes counts: A=41 B=31 C=28 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:212 E:203 A:184 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -10 -10 B 6 0 22 2 4 C 6 -22 0 -10 -6 D 10 -2 10 0 6 E 10 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -10 -10 B 6 0 22 2 4 C 6 -22 0 -10 -6 D 10 -2 10 0 6 E 10 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -10 -10 B 6 0 22 2 4 C 6 -22 0 -10 -6 D 10 -2 10 0 6 E 10 -4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5283: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (17) A B C E D (10) C E D B A (6) D C E A B (5) D A B E C (5) C E B A D (5) C E A B D (5) B A E C D (5) B A C E D (5) A B D E C (5) A B D C E (4) E C D B A (3) E C B A D (3) D A B C E (3) D E B A C (2) D C E B A (2) D B A E C (2) A D B C E (2) E D C B A (1) E C B D A (1) E B C A D (1) D E B C A (1) D E A B C (1) D A C B E (1) C E D A B (1) C B A E D (1) B A D E C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -4 -2 -8 B 12 0 -4 -4 -8 C 4 4 0 -4 4 D 2 4 4 0 4 E 8 8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -2 -8 B 12 0 -4 -4 -8 C 4 4 0 -4 4 D 2 4 4 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=23 C=18 B=11 E=9 so E is eliminated. Round 2 votes counts: D=40 C=25 A=23 B=12 so B is eliminated. Round 3 votes counts: D=40 A=34 C=26 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:207 C:204 E:204 B:198 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -4 -2 -8 B 12 0 -4 -4 -8 C 4 4 0 -4 4 D 2 4 4 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -2 -8 B 12 0 -4 -4 -8 C 4 4 0 -4 4 D 2 4 4 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -2 -8 B 12 0 -4 -4 -8 C 4 4 0 -4 4 D 2 4 4 0 4 E 8 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5284: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) B E A D C (9) D B C E A (5) B E D A C (5) B D C A E (5) D C E A B (4) D C B E A (4) B A E C D (4) E B A D C (3) E A C D B (3) D C B A E (3) C D E A B (3) A E B C D (3) A C E D B (3) E A C B D (2) E A B C D (2) D B C A E (2) C E A D B (2) C D A B E (2) B D E C A (2) B D E A C (2) B D C E A (2) A E C D B (2) A E C B D (2) A B E C D (2) E D C B A (1) E C D A B (1) E C A D B (1) E B A C D (1) D C E B A (1) D C A B E (1) D B E C A (1) C A E D B (1) B E D C A (1) B D A E C (1) B D A C E (1) B A E D C (1) B A C D E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -2 -12 -10 B 10 0 6 2 12 C 2 -6 0 -8 2 D 12 -2 8 0 0 E 10 -12 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -12 -10 B 10 0 6 2 12 C 2 -6 0 -8 2 D 12 -2 8 0 0 E 10 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=21 C=17 E=14 A=14 so E is eliminated. Round 2 votes counts: B=38 D=22 A=21 C=19 so C is eliminated. Round 3 votes counts: B=38 D=37 A=25 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:209 E:198 C:195 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -12 -10 B 10 0 6 2 12 C 2 -6 0 -8 2 D 12 -2 8 0 0 E 10 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -12 -10 B 10 0 6 2 12 C 2 -6 0 -8 2 D 12 -2 8 0 0 E 10 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -12 -10 B 10 0 6 2 12 C 2 -6 0 -8 2 D 12 -2 8 0 0 E 10 -12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5285: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) B D E A C (9) B A C D E (8) D E B C A (6) C A E D B (6) B E D A C (6) A C B E D (6) C A D E B (4) A C E D B (4) A C B D E (4) E D B A C (3) E A C D B (3) D E C A B (3) A B C D E (3) E D C B A (2) D B E C A (2) C D A E B (2) B E A D C (2) B D E C A (2) B D C A E (2) B C A D E (2) A C E B D (2) E D B C A (1) E C D A B (1) D E C B A (1) C E A D B (1) B D A C E (1) B A E C D (1) B A C E D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 10 -2 -4 B -2 0 0 0 0 C -10 0 0 0 -6 D 2 0 0 0 -2 E 4 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.431873 C: 0.000000 D: 0.000000 E: 0.568127 Sum of squares = 0.50928255667 Cumulative probabilities = A: 0.000000 B: 0.431873 C: 0.431873 D: 0.431873 E: 1.000000 A B C D E A 0 2 10 -2 -4 B -2 0 0 0 0 C -10 0 0 0 -6 D 2 0 0 0 -2 E 4 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=21 E=20 C=13 D=12 so D is eliminated. Round 2 votes counts: B=36 E=30 A=21 C=13 so C is eliminated. Round 3 votes counts: B=36 A=33 E=31 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:203 D:200 B:199 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 10 -2 -4 B -2 0 0 0 0 C -10 0 0 0 -6 D 2 0 0 0 -2 E 4 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -2 -4 B -2 0 0 0 0 C -10 0 0 0 -6 D 2 0 0 0 -2 E 4 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -2 -4 B -2 0 0 0 0 C -10 0 0 0 -6 D 2 0 0 0 -2 E 4 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5286: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) C B E A D (8) D A E B C (7) C E B A D (6) A D E B C (6) A D B E C (5) E C B D A (4) A D B C E (4) E B D C A (3) E B C D A (3) C E B D A (3) A D E C B (3) A D C B E (3) A C D B E (3) E A D B C (2) D A B E C (2) B E D C A (2) A D C E B (2) A C D E B (2) E D B A C (1) E D A B C (1) E C B A D (1) D E B A C (1) D E A B C (1) D B E C A (1) D B E A C (1) D B C A E (1) D A B C E (1) C B D E A (1) C B A E D (1) C A E B D (1) C A B E D (1) C A B D E (1) B E C D A (1) B C E D A (1) B C D E A (1) A E D B C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -4 4 -8 B 4 0 -8 2 0 C 4 8 0 4 8 D -4 -2 -4 0 -8 E 8 0 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 4 -8 B 4 0 -8 2 0 C 4 8 0 4 8 D -4 -2 -4 0 -8 E 8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=31 E=15 D=15 B=5 so B is eliminated. Round 2 votes counts: C=36 A=31 E=18 D=15 so D is eliminated. Round 3 votes counts: A=41 C=37 E=22 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:204 B:199 A:194 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 4 -8 B 4 0 -8 2 0 C 4 8 0 4 8 D -4 -2 -4 0 -8 E 8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 4 -8 B 4 0 -8 2 0 C 4 8 0 4 8 D -4 -2 -4 0 -8 E 8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 4 -8 B 4 0 -8 2 0 C 4 8 0 4 8 D -4 -2 -4 0 -8 E 8 0 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) B E A C D (8) E B D C A (7) C A D B E (6) B E C A D (6) A C D B E (6) E B D A C (5) A D C B E (5) E B C D A (4) D C A E B (4) C A B D E (4) D C A B E (3) A D C E B (3) E D B C A (2) E B A C D (2) D E A C B (2) D C E B A (2) C D A B E (2) C B E D A (2) B E C D A (2) B C E A D (2) A D E B C (2) E D B A C (1) E D A B C (1) D E C B A (1) D E C A B (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) C D E B A (1) B A E C D (1) Total count = 100 A B C D E A 0 -12 0 10 -20 B 12 0 8 6 -4 C 0 -8 0 -8 -14 D -10 -6 8 0 -6 E 20 4 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 0 10 -20 B 12 0 8 6 -4 C 0 -8 0 -8 -14 D -10 -6 8 0 -6 E 20 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=19 D=18 A=16 C=15 so C is eliminated. Round 2 votes counts: E=32 A=26 D=21 B=21 so D is eliminated. Round 3 votes counts: E=40 A=39 B=21 so B is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:211 D:193 A:189 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 0 10 -20 B 12 0 8 6 -4 C 0 -8 0 -8 -14 D -10 -6 8 0 -6 E 20 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 10 -20 B 12 0 8 6 -4 C 0 -8 0 -8 -14 D -10 -6 8 0 -6 E 20 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 10 -20 B 12 0 8 6 -4 C 0 -8 0 -8 -14 D -10 -6 8 0 -6 E 20 4 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5288: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) E C D A B (11) B A D C E (11) D C E B A (7) E C A D B (6) B A C E D (5) A B E C D (4) A B D E C (4) B C E D A (3) B C D E A (3) B A E C D (3) A E C D B (3) E C A B D (2) E A C D B (2) D A E C B (2) B C E A D (2) B A C D E (2) A D B E C (2) A B E D C (2) E A C B D (1) D E A C B (1) D C B E A (1) D B A C E (1) D A B E C (1) C E B D A (1) C D E B A (1) B D C E A (1) B D A C E (1) B C A E D (1) A E D C B (1) A E C B D (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -2 8 -8 B 10 0 -6 -6 -6 C 2 6 0 26 4 D -8 6 -26 0 -20 E 8 6 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 8 -8 B 10 0 -6 -6 -6 C 2 6 0 26 4 D -8 6 -26 0 -20 E 8 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999950339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=22 A=19 C=14 D=13 so D is eliminated. Round 2 votes counts: B=33 E=23 C=22 A=22 so C is eliminated. Round 3 votes counts: E=44 B=34 A=22 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:215 B:196 A:194 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -2 8 -8 B 10 0 -6 -6 -6 C 2 6 0 26 4 D -8 6 -26 0 -20 E 8 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999950339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 8 -8 B 10 0 -6 -6 -6 C 2 6 0 26 4 D -8 6 -26 0 -20 E 8 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999950339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 8 -8 B 10 0 -6 -6 -6 C 2 6 0 26 4 D -8 6 -26 0 -20 E 8 6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999950339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5289: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) E B A D C (9) C A E B D (7) E B D A C (6) D C A B E (5) C E B A D (5) C D A E B (5) D A B E C (4) C D E B A (4) C A D B E (4) E C B A D (3) E B A C D (3) D B A E C (3) C E B D A (3) B E A D C (3) E B C A D (2) C E A B D (2) C A D E B (2) B D E A C (2) A D B E C (2) A B E D C (2) E B D C A (1) E B C D A (1) E A B C D (1) D C E B A (1) D B E C A (1) D A C B E (1) D A B C E (1) C E D B A (1) C D E A B (1) B E D A C (1) A E B C D (1) A D C B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -16 -2 0 B -2 0 -10 8 -16 C 16 10 0 14 8 D 2 -8 -14 0 -6 E 0 16 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 -2 0 B -2 0 -10 8 -16 C 16 10 0 14 8 D 2 -8 -14 0 -6 E 0 16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 E=26 D=16 A=8 B=6 so B is eliminated. Round 2 votes counts: C=44 E=30 D=18 A=8 so A is eliminated. Round 3 votes counts: C=45 E=34 D=21 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:207 A:192 B:190 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 -2 0 B -2 0 -10 8 -16 C 16 10 0 14 8 D 2 -8 -14 0 -6 E 0 16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -2 0 B -2 0 -10 8 -16 C 16 10 0 14 8 D 2 -8 -14 0 -6 E 0 16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -2 0 B -2 0 -10 8 -16 C 16 10 0 14 8 D 2 -8 -14 0 -6 E 0 16 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5290: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) D B E C A (7) B D C A E (6) A C E B D (6) E A C B D (5) D E B C A (5) C B A D E (5) B C A D E (5) A B C E D (5) E D C A B (4) D B C A E (4) A C B E D (4) E D B A C (3) C A B D E (3) B A C D E (3) E D C B A (2) E D A C B (2) E D A B C (2) E C D A B (2) D B C E A (2) B C D A E (2) A E C B D (2) E C A D B (1) E B D A C (1) E A D C B (1) E A B D C (1) D C E A B (1) D C B A E (1) C B D A E (1) C A E D B (1) C A E B D (1) B D E A C (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -6 8 0 B -4 0 -6 4 0 C 6 6 0 14 2 D -8 -4 -14 0 -6 E 0 0 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 8 0 B -4 0 -6 4 0 C 6 6 0 14 2 D -8 -4 -14 0 -6 E 0 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=20 A=18 B=17 C=11 so C is eliminated. Round 2 votes counts: E=34 B=23 A=23 D=20 so D is eliminated. Round 3 votes counts: E=40 B=37 A=23 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:214 A:203 E:202 B:197 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 8 0 B -4 0 -6 4 0 C 6 6 0 14 2 D -8 -4 -14 0 -6 E 0 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 8 0 B -4 0 -6 4 0 C 6 6 0 14 2 D -8 -4 -14 0 -6 E 0 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 8 0 B -4 0 -6 4 0 C 6 6 0 14 2 D -8 -4 -14 0 -6 E 0 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5291: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (18) C E A D B (11) A D E C B (11) B C E D A (7) D A B E C (5) E C A D B (4) D A E B C (4) C E B D A (4) E A D C B (3) C E B A D (3) D B A E C (2) C E A B D (2) C B E D A (2) C B E A D (2) B E D A C (2) B D A C E (2) A D E B C (2) A D C E B (2) A D B C E (2) E D C A B (1) E C D A B (1) E C B D A (1) E B C D A (1) D A E C B (1) C A E D B (1) B E D C A (1) B E C D A (1) B C A D E (1) B A D E C (1) B A D C E (1) A D B E C (1) Total count = 100 A B C D E A 0 2 14 -6 8 B -2 0 2 -2 -4 C -14 -2 0 -18 -20 D 6 2 18 0 6 E -8 4 20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 -6 8 B -2 0 2 -2 -4 C -14 -2 0 -18 -20 D 6 2 18 0 6 E -8 4 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=25 A=18 D=12 E=11 so E is eliminated. Round 2 votes counts: B=35 C=31 A=21 D=13 so D is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:216 A:209 E:205 B:197 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 14 -6 8 B -2 0 2 -2 -4 C -14 -2 0 -18 -20 D 6 2 18 0 6 E -8 4 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -6 8 B -2 0 2 -2 -4 C -14 -2 0 -18 -20 D 6 2 18 0 6 E -8 4 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -6 8 B -2 0 2 -2 -4 C -14 -2 0 -18 -20 D 6 2 18 0 6 E -8 4 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5292: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) B C A E D (8) E D B C A (6) D E B A C (6) C B A E D (6) B A C D E (6) D A B C E (4) B A D C E (4) A C B D E (4) A B C D E (4) E C A B D (3) E B D C A (3) E B C A D (3) D A B E C (3) C A E B D (3) A D C B E (3) E D C B A (2) E C D A B (2) E C A D B (2) D E A C B (2) D B E A C (2) B A C E D (2) E C D B A (1) E B C D A (1) D E C A B (1) D B A E C (1) D B A C E (1) D A E C B (1) C E A D B (1) C A E D B (1) C A B E D (1) B E C D A (1) B D A C E (1) A D B C E (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -6 8 12 B 8 0 14 2 6 C 6 -14 0 2 4 D -8 -2 -2 0 -10 E -12 -6 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 8 12 B 8 0 14 2 6 C 6 -14 0 2 4 D -8 -2 -2 0 -10 E -12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=22 D=21 A=14 C=12 so C is eliminated. Round 2 votes counts: E=32 B=28 D=21 A=19 so A is eliminated. Round 3 votes counts: B=38 E=36 D=26 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:203 C:199 E:194 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 8 12 B 8 0 14 2 6 C 6 -14 0 2 4 D -8 -2 -2 0 -10 E -12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 8 12 B 8 0 14 2 6 C 6 -14 0 2 4 D -8 -2 -2 0 -10 E -12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 8 12 B 8 0 14 2 6 C 6 -14 0 2 4 D -8 -2 -2 0 -10 E -12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5293: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) C B E A D (13) C B E D A (11) D A E B C (10) C B A E D (5) D E A B C (4) D A C E B (4) C D E B A (4) E D C B A (2) E B C D A (2) D C E B A (2) D C A E B (2) D A E C B (2) B C E A D (2) A D B E C (2) A B D E C (2) E D B C A (1) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) D E C B A (1) C E B D A (1) C D E A B (1) C B D A E (1) C A D B E (1) B E C D A (1) B C E D A (1) B C A E D (1) B A E D C (1) B A E C D (1) A E D B C (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 -6 0 B 6 0 -2 -10 -10 C 4 2 0 -10 2 D 6 10 10 0 4 E 0 10 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -6 0 B 6 0 -2 -10 -10 C 4 2 0 -10 2 D 6 10 10 0 4 E 0 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=25 A=22 E=9 B=7 so B is eliminated. Round 2 votes counts: C=41 D=25 A=24 E=10 so E is eliminated. Round 3 votes counts: C=44 D=31 A=25 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:202 C:199 A:192 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -6 0 B 6 0 -2 -10 -10 C 4 2 0 -10 2 D 6 10 10 0 4 E 0 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -6 0 B 6 0 -2 -10 -10 C 4 2 0 -10 2 D 6 10 10 0 4 E 0 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -6 0 B 6 0 -2 -10 -10 C 4 2 0 -10 2 D 6 10 10 0 4 E 0 10 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5294: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) A E B D C (9) D B E A C (8) C D B E A (8) C E B D A (7) C D B A E (7) C A E B D (6) C A D B E (5) C B D E A (4) E A B D C (3) D A B E C (3) C A E D B (3) B D E A C (3) E B D A C (2) D C B A E (2) D B A E C (2) C B E D A (2) B E D A C (2) A E D B C (2) A D E B C (2) A D C B E (2) E B C D A (1) E A C B D (1) C E A B D (1) C D A B E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 6 -4 16 B -4 0 0 -18 24 C -6 0 0 -6 -2 D 4 18 6 0 20 E -16 -24 2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -4 16 B -4 0 0 -18 24 C -6 0 0 -6 -2 D 4 18 6 0 20 E -16 -24 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=29 D=15 E=7 B=5 so B is eliminated. Round 2 votes counts: C=44 A=29 D=18 E=9 so E is eliminated. Round 3 votes counts: C=45 A=33 D=22 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:224 A:211 B:201 C:193 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -4 16 B -4 0 0 -18 24 C -6 0 0 -6 -2 D 4 18 6 0 20 E -16 -24 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -4 16 B -4 0 0 -18 24 C -6 0 0 -6 -2 D 4 18 6 0 20 E -16 -24 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -4 16 B -4 0 0 -18 24 C -6 0 0 -6 -2 D 4 18 6 0 20 E -16 -24 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5295: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) E C A D B (7) D A B E C (7) C E A D B (5) B D A E C (5) B C E A D (5) E A C D B (4) C E B A D (4) B E C A D (4) B D A C E (4) B C E D A (4) D A C E B (3) C D A E B (3) B D C A E (3) E C A B D (2) D C A E B (2) D A E B C (2) B E C D A (2) B E A D C (2) B C D E A (2) B A D E C (2) A D E B C (2) A D B E C (2) E C B A D (1) E B C A D (1) E A D C B (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B D A (1) C B E D A (1) C B E A D (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 12 0 10 2 B -12 0 -2 -12 -6 C 0 2 0 -2 -14 D -10 12 2 0 4 E -2 6 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.952882 B: 0.000000 C: 0.047118 D: 0.000000 E: 0.000000 Sum of squares = 0.910204684058 Cumulative probabilities = A: 0.952882 B: 0.952882 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 10 2 B -12 0 -2 -12 -6 C 0 2 0 -2 -14 D -10 12 2 0 4 E -2 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250036349 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=18 C=17 E=16 A=16 so E is eliminated. Round 2 votes counts: B=34 C=27 A=21 D=18 so D is eliminated. Round 3 votes counts: A=36 B=35 C=29 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:207 D:204 C:193 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 10 2 B -12 0 -2 -12 -6 C 0 2 0 -2 -14 D -10 12 2 0 4 E -2 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250036349 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 10 2 B -12 0 -2 -12 -6 C 0 2 0 -2 -14 D -10 12 2 0 4 E -2 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250036349 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 10 2 B -12 0 -2 -12 -6 C 0 2 0 -2 -14 D -10 12 2 0 4 E -2 6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.875000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.781250036349 Cumulative probabilities = A: 0.875000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5296: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) D C A E B (7) E C B D A (6) B E A C D (6) B A E D C (6) B A D E C (6) A B D E C (6) E C B A D (5) C D E A B (5) E B A C D (4) D C E A B (4) C E D A B (4) B A E C D (4) E C D B A (3) E B C A D (3) D A C B E (3) D A B C E (3) A D B C E (3) A B E D C (3) D C A B E (2) C D E B A (2) E C A B D (1) D C E B A (1) D B C E A (1) D B A C E (1) A D C B E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -4 0 -6 B 12 0 -8 2 -8 C 4 8 0 6 -6 D 0 -2 -6 0 -10 E 6 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -4 0 -6 B 12 0 -8 2 -8 C 4 8 0 6 -6 D 0 -2 -6 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 D=22 B=22 C=19 A=15 so A is eliminated. Round 2 votes counts: B=31 D=26 E=22 C=21 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:206 B:199 D:191 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -4 0 -6 B 12 0 -8 2 -8 C 4 8 0 6 -6 D 0 -2 -6 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 0 -6 B 12 0 -8 2 -8 C 4 8 0 6 -6 D 0 -2 -6 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 0 -6 B 12 0 -8 2 -8 C 4 8 0 6 -6 D 0 -2 -6 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5297: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) E D A B C (9) C B A D E (8) D E A B C (6) C D E A B (6) C B D E A (6) D E C A B (5) C E D B A (4) C B A E D (4) B A E C D (4) D A E B C (3) C E B D A (3) C D E B A (3) B C A E D (3) B A C D E (3) A B E D C (3) E D C A B (2) C B D A E (2) B A C E D (2) A B D E C (2) E C D A B (1) D E A C B (1) D C E A B (1) C E D A B (1) C E B A D (1) C D B E A (1) B C E A D (1) B C A D E (1) B A D E C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -16 -6 -8 -2 B 16 0 2 10 6 C 6 -2 0 8 0 D 8 -10 -8 0 2 E 2 -6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -8 -2 B 16 0 2 10 6 C 6 -2 0 8 0 D 8 -10 -8 0 2 E 2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=26 D=16 E=12 A=7 so A is eliminated. Round 2 votes counts: C=39 B=31 D=18 E=12 so E is eliminated. Round 3 votes counts: C=40 B=31 D=29 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:206 E:197 D:196 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -8 -2 B 16 0 2 10 6 C 6 -2 0 8 0 D 8 -10 -8 0 2 E 2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -8 -2 B 16 0 2 10 6 C 6 -2 0 8 0 D 8 -10 -8 0 2 E 2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -8 -2 B 16 0 2 10 6 C 6 -2 0 8 0 D 8 -10 -8 0 2 E 2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5298: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (15) B A D C E (12) E C D B A (7) A E C D B (7) B D C E A (6) D C E B A (5) B E A C D (5) A B D C E (5) B E D C A (4) A C D E B (4) E A C D B (3) B E C D A (3) A B C D E (3) E D C B A (2) B A E D C (2) B A E C D (2) A B E C D (2) E C A D B (1) E B D C A (1) E B C D A (1) D C B E A (1) D C A E B (1) D C A B E (1) B D C A E (1) B D A C E (1) A E B C D (1) A D C E B (1) A D C B E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 2 2 -8 B 6 0 0 -2 2 C -2 0 0 10 -12 D -2 2 -10 0 -12 E 8 -2 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593750000061 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 A B C D E A 0 -6 2 2 -8 B 6 0 0 -2 2 C -2 0 0 10 -12 D -2 2 -10 0 -12 E 8 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593750000172 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=30 A=26 D=8 so C is eliminated. Round 2 votes counts: B=36 E=30 A=26 D=8 so D is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:203 C:198 A:195 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 2 -8 B 6 0 0 -2 2 C -2 0 0 10 -12 D -2 2 -10 0 -12 E 8 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593750000172 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 -8 B 6 0 0 -2 2 C -2 0 0 10 -12 D -2 2 -10 0 -12 E 8 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593750000172 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 -8 B 6 0 0 -2 2 C -2 0 0 10 -12 D -2 2 -10 0 -12 E 8 -2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593750000172 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5299: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) A E D B C (6) A D B E C (6) D A B E C (5) C E B D A (5) A D B C E (5) E C B A D (4) D B A E C (4) D B A C E (4) C E A B D (4) C A E D B (4) C E A D B (3) C B E D A (3) B E D C A (3) B D C A E (3) A E D C B (3) A C D E B (3) E B D A C (2) E B C D A (2) E A D B C (2) E A C D B (2) C E B A D (2) B D E C A (2) A E C D B (2) A D E B C (2) A C E D B (2) E C B D A (1) E C A D B (1) E A B D C (1) D A B C E (1) C D B A E (1) C D A B E (1) C B D E A (1) C B D A E (1) B E D A C (1) B D C E A (1) A D C B E (1) Total count = 100 A B C D E A 0 20 4 18 8 B -20 0 0 -16 -14 C -4 0 0 -4 -10 D -18 16 4 0 -18 E -8 14 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 4 18 8 B -20 0 0 -16 -14 C -4 0 0 -4 -10 D -18 16 4 0 -18 E -8 14 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=25 E=21 D=14 B=10 so B is eliminated. Round 2 votes counts: A=30 E=25 C=25 D=20 so D is eliminated. Round 3 votes counts: A=44 C=29 E=27 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:217 D:192 C:191 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 4 18 8 B -20 0 0 -16 -14 C -4 0 0 -4 -10 D -18 16 4 0 -18 E -8 14 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 4 18 8 B -20 0 0 -16 -14 C -4 0 0 -4 -10 D -18 16 4 0 -18 E -8 14 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 4 18 8 B -20 0 0 -16 -14 C -4 0 0 -4 -10 D -18 16 4 0 -18 E -8 14 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5300: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) B A C E D (11) D E C A B (9) D E C B A (8) A B D C E (6) B C E A D (5) D E A C B (4) D A E C B (4) D A E B C (4) D A B E C (4) E C D A B (3) C E B D A (3) C E B A D (3) D B A E C (2) C E D B A (2) C E A D B (2) B C A E D (2) B A D C E (2) A D C E B (2) A D B E C (2) E D C B A (1) E C B D A (1) D B E C A (1) C E A B D (1) B E C D A (1) B D E C A (1) B C E D A (1) B A D E C (1) A D E C B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -10 -20 -14 B 12 0 -12 -22 -20 C 10 12 0 -4 -16 D 20 22 4 0 4 E 14 20 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -20 -14 B 12 0 -12 -22 -20 C 10 12 0 -4 -16 D 20 22 4 0 4 E 14 20 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=24 E=16 A=13 C=11 so C is eliminated. Round 2 votes counts: D=36 E=27 B=24 A=13 so A is eliminated. Round 3 votes counts: D=42 B=31 E=27 so E is eliminated. Round 4 votes counts: D=61 B=39 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:223 C:201 B:179 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -10 -20 -14 B 12 0 -12 -22 -20 C 10 12 0 -4 -16 D 20 22 4 0 4 E 14 20 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -20 -14 B 12 0 -12 -22 -20 C 10 12 0 -4 -16 D 20 22 4 0 4 E 14 20 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -20 -14 B 12 0 -12 -22 -20 C 10 12 0 -4 -16 D 20 22 4 0 4 E 14 20 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5301: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) D C E A B (8) C D A B E (8) D C E B A (7) C D E A B (7) A B C D E (6) E C D B A (5) B A E C D (5) B A D C E (5) E D C B A (4) E B A C D (4) C A B D E (4) A B E C D (4) A B D C E (4) E B A D C (3) D E C B A (3) A B C E D (3) D C A B E (2) C D A E B (2) E C D A B (1) E B C A D (1) D B A C E (1) D A C B E (1) C A D B E (1) B E A D C (1) B A D E C (1) Total count = 100 A B C D E A 0 2 -6 2 12 B -2 0 -6 0 10 C 6 6 0 2 18 D -2 0 -2 0 20 E -12 -10 -18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 2 12 B -2 0 -6 0 10 C 6 6 0 2 18 D -2 0 -2 0 20 E -12 -10 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=22 C=22 B=21 E=18 A=17 so A is eliminated. Round 2 votes counts: B=38 D=22 C=22 E=18 so E is eliminated. Round 3 votes counts: B=46 C=28 D=26 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:208 A:205 B:201 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 2 12 B -2 0 -6 0 10 C 6 6 0 2 18 D -2 0 -2 0 20 E -12 -10 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 2 12 B -2 0 -6 0 10 C 6 6 0 2 18 D -2 0 -2 0 20 E -12 -10 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 2 12 B -2 0 -6 0 10 C 6 6 0 2 18 D -2 0 -2 0 20 E -12 -10 -18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5302: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (7) C B A D E (5) C A D E B (5) B C D E A (5) A E D B C (5) E D B C A (4) D E C B A (4) B E D C A (4) A C D E B (4) A B E D C (4) E D B A C (3) E D A B C (3) D E A C B (3) C B D E A (3) B E D A C (3) B C E D A (3) B A C E D (3) A B C E D (3) E D A C B (2) C D E B A (2) C D A E B (2) A C B D E (2) A B C D E (2) E B D C A (1) E B D A C (1) D E C A B (1) D C E B A (1) D C E A B (1) C D B E A (1) C D A B E (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A E D (1) B A E D C (1) A E B D C (1) A D E C B (1) A D C E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 4 4 B -4 0 2 -4 -4 C -4 -2 0 -4 -2 D -4 4 4 0 -8 E -4 4 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 4 4 B -4 0 2 -4 -4 C -4 -2 0 -4 -2 D -4 4 4 0 -8 E -4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=23 C=21 E=14 D=10 so D is eliminated. Round 2 votes counts: A=32 C=23 B=23 E=22 so E is eliminated. Round 3 votes counts: A=40 B=32 C=28 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:205 D:198 B:195 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 4 4 B -4 0 2 -4 -4 C -4 -2 0 -4 -2 D -4 4 4 0 -8 E -4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 4 B -4 0 2 -4 -4 C -4 -2 0 -4 -2 D -4 4 4 0 -8 E -4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 4 B -4 0 2 -4 -4 C -4 -2 0 -4 -2 D -4 4 4 0 -8 E -4 4 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5303: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) C D E B A (8) B A E C D (6) A B D E C (6) D E C A B (5) C E B D A (5) B C A E D (5) A B E D C (5) E C B D A (4) A D B E C (4) A B D C E (4) E C D B A (3) D C E A B (3) B A E D C (3) A D B C E (3) E C B A D (2) E B A D C (2) D A B E C (2) C D A B E (2) B A C E D (2) E D C B A (1) E D C A B (1) E D A B C (1) D E A B C (1) D C A E B (1) D A C E B (1) D A C B E (1) D A B C E (1) C D B E A (1) C D A E B (1) C B E A D (1) C B D E A (1) C A B D E (1) B E C A D (1) B E A C D (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -10 -4 0 B 10 0 -4 0 4 C 10 4 0 8 4 D 4 0 -8 0 -4 E 0 -4 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -4 0 B 10 0 -4 0 4 C 10 4 0 8 4 D 4 0 -8 0 -4 E 0 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=24 B=18 D=15 E=14 so E is eliminated. Round 2 votes counts: C=38 A=24 B=20 D=18 so D is eliminated. Round 3 votes counts: C=49 A=31 B=20 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:205 E:198 D:196 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 -4 0 B 10 0 -4 0 4 C 10 4 0 8 4 D 4 0 -8 0 -4 E 0 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -4 0 B 10 0 -4 0 4 C 10 4 0 8 4 D 4 0 -8 0 -4 E 0 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -4 0 B 10 0 -4 0 4 C 10 4 0 8 4 D 4 0 -8 0 -4 E 0 -4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5304: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (15) A B E C D (11) D C E B A (9) E C A D B (7) D C E A B (6) A E C D B (5) D C B E A (4) D B C E A (4) A E C B D (4) A E B C D (4) B D C E A (3) A B E D C (3) E A C B D (2) D B C A E (2) D B A C E (2) C D E A B (2) B E C A D (2) B A E D C (2) A B D E C (2) E C B D A (1) E C A B D (1) D C A E B (1) D A C B E (1) C E D B A (1) C E D A B (1) B D C A E (1) B C E D A (1) B A D E C (1) B A D C E (1) A E D C B (1) Total count = 100 A B C D E A 0 2 8 22 12 B -2 0 8 8 10 C -8 -8 0 14 -22 D -22 -8 -14 0 -22 E -12 -10 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 22 12 B -2 0 8 8 10 C -8 -8 0 14 -22 D -22 -8 -14 0 -22 E -12 -10 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=29 B=26 E=11 C=4 so C is eliminated. Round 2 votes counts: D=31 A=30 B=26 E=13 so E is eliminated. Round 3 votes counts: A=40 D=33 B=27 so B is eliminated. Round 4 votes counts: A=61 D=39 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:212 E:211 C:188 D:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 22 12 B -2 0 8 8 10 C -8 -8 0 14 -22 D -22 -8 -14 0 -22 E -12 -10 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 22 12 B -2 0 8 8 10 C -8 -8 0 14 -22 D -22 -8 -14 0 -22 E -12 -10 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 22 12 B -2 0 8 8 10 C -8 -8 0 14 -22 D -22 -8 -14 0 -22 E -12 -10 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5305: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) C E D B A (5) A B E C D (5) D A C B E (4) C E A B D (4) B A E C D (4) E C B D A (3) D C A E B (3) D B E C A (3) D A C E B (3) C D E A B (3) B E D C A (3) B E C D A (3) B E C A D (3) B A D E C (3) A D B C E (3) A B E D C (3) A B D E C (3) E C D B A (2) D C E A B (2) C E A D B (2) C A E D B (2) B D E C A (2) B D A E C (2) B A E D C (2) A D C B E (2) A B D C E (2) E C B A D (1) E C A B D (1) E B C A D (1) D E C B A (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A E C (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D A E B (1) B E A D C (1) B E A C D (1) B D E A C (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -14 -10 -6 B 6 0 -2 0 12 C 14 2 0 -10 2 D 10 0 10 0 4 E 6 -12 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.340581 C: 0.000000 D: 0.659419 E: 0.000000 Sum of squares = 0.550828687671 Cumulative probabilities = A: 0.000000 B: 0.340581 C: 0.340581 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -10 -6 B 6 0 -2 0 12 C 14 2 0 -10 2 D 10 0 10 0 4 E 6 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 C=20 A=20 E=8 so E is eliminated. Round 2 votes counts: D=27 C=27 B=26 A=20 so A is eliminated. Round 3 votes counts: B=40 D=33 C=27 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:212 B:208 C:204 E:194 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -14 -10 -6 B 6 0 -2 0 12 C 14 2 0 -10 2 D 10 0 10 0 4 E 6 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -10 -6 B 6 0 -2 0 12 C 14 2 0 -10 2 D 10 0 10 0 4 E 6 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -10 -6 B 6 0 -2 0 12 C 14 2 0 -10 2 D 10 0 10 0 4 E 6 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5306: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) B C A D E (12) B C A E D (10) A C E B D (8) E D A C B (7) D E B A C (7) D E A B C (6) D B E C A (4) B D C A E (4) E A C D B (3) D E A C B (3) D B E A C (3) B D E C A (3) B D C E A (3) B C D A E (3) A C E D B (3) D E B C A (2) A E C D B (2) E D B C A (1) D B A C E (1) D A B C E (1) C A E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -10 4 16 B 6 0 20 14 14 C 10 -20 0 10 18 D -4 -14 -10 0 6 E -16 -14 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 4 16 B 6 0 20 14 14 C 10 -20 0 10 18 D -4 -14 -10 0 6 E -16 -14 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=27 A=14 C=13 E=11 so E is eliminated. Round 2 votes counts: D=35 B=35 A=17 C=13 so C is eliminated. Round 3 votes counts: D=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:209 A:202 D:189 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 4 16 B 6 0 20 14 14 C 10 -20 0 10 18 D -4 -14 -10 0 6 E -16 -14 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 4 16 B 6 0 20 14 14 C 10 -20 0 10 18 D -4 -14 -10 0 6 E -16 -14 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 4 16 B 6 0 20 14 14 C 10 -20 0 10 18 D -4 -14 -10 0 6 E -16 -14 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5307: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) E A C D B (12) D B C A E (8) B D A C E (7) E C A D B (5) E A B D C (5) C E A D B (4) C D B A E (4) B D C E A (4) E B D A C (3) C A D E B (3) E B D C A (2) E A C B D (2) C D B E A (2) C D A B E (2) C A E D B (2) B E D A C (2) B D E C A (2) B D A E C (2) A E C D B (2) A C D E B (2) E B C A D (1) E B A D C (1) D C B A E (1) C E D B A (1) C E D A B (1) C D E A B (1) C A D B E (1) B A D E C (1) A D C B E (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -14 -10 4 B 8 0 4 -8 4 C 14 -4 0 -6 20 D 10 8 6 0 12 E -4 -4 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -10 4 B 8 0 4 -8 4 C 14 -4 0 -6 20 D 10 8 6 0 12 E -4 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=31 B=31 C=21 D=9 A=8 so A is eliminated. Round 2 votes counts: E=33 B=32 C=25 D=10 so D is eliminated. Round 3 votes counts: B=40 E=33 C=27 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:218 C:212 B:204 A:186 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -14 -10 4 B 8 0 4 -8 4 C 14 -4 0 -6 20 D 10 8 6 0 12 E -4 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -10 4 B 8 0 4 -8 4 C 14 -4 0 -6 20 D 10 8 6 0 12 E -4 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -10 4 B 8 0 4 -8 4 C 14 -4 0 -6 20 D 10 8 6 0 12 E -4 -4 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5308: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) A E C D B (8) E D A B C (7) C B D A E (7) C B A D E (7) B D C E A (6) E A D B C (5) C A B E D (5) D B E C A (4) C A B D E (4) A E C B D (4) A C E B D (4) E D B A C (3) E A D C B (3) D E B C A (3) B C D E A (3) B C D A E (3) A C B E D (3) D E B A C (2) D B C E A (2) A C E D B (2) E D A C B (1) D E A C B (1) D C B A E (1) C D B A E (1) C B A E D (1) B E A C D (1) B D E C A (1) Total count = 100 A B C D E A 0 10 4 10 16 B -10 0 -20 -2 -2 C -4 20 0 6 -2 D -10 2 -6 0 -10 E -16 2 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 10 16 B -10 0 -20 -2 -2 C -4 20 0 6 -2 D -10 2 -6 0 -10 E -16 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 E=19 B=14 D=13 so D is eliminated. Round 2 votes counts: A=29 C=26 E=25 B=20 so B is eliminated. Round 3 votes counts: C=40 E=31 A=29 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:220 C:210 E:199 D:188 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 10 16 B -10 0 -20 -2 -2 C -4 20 0 6 -2 D -10 2 -6 0 -10 E -16 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 10 16 B -10 0 -20 -2 -2 C -4 20 0 6 -2 D -10 2 -6 0 -10 E -16 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 10 16 B -10 0 -20 -2 -2 C -4 20 0 6 -2 D -10 2 -6 0 -10 E -16 2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5309: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) D B C A E (6) C D E B A (6) B A D E C (6) D B A E C (5) A B D E C (5) E A C B D (4) D C E B A (4) D B A C E (4) B D A C E (4) B A D C E (4) A E B C D (4) E D C A B (3) D C B E A (3) D B C E A (3) C E D B A (3) C E A D B (3) A B E D C (3) E C A D B (2) E C A B D (2) E A C D B (2) C E A B D (2) A E C B D (2) A E B D C (2) A B E C D (2) D E C B A (1) C D E A B (1) C A E B D (1) B A C E D (1) B A C D E (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 -4 4 B 2 0 4 -12 -2 C -2 -4 0 -8 12 D 4 12 8 0 8 E -4 2 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 4 B 2 0 4 -12 -2 C -2 -4 0 -8 12 D 4 12 8 0 8 E -4 2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=25 A=20 B=16 E=13 so E is eliminated. Round 2 votes counts: D=29 C=29 A=26 B=16 so B is eliminated. Round 3 votes counts: A=38 D=33 C=29 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:200 C:199 B:196 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -4 4 B 2 0 4 -12 -2 C -2 -4 0 -8 12 D 4 12 8 0 8 E -4 2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 4 B 2 0 4 -12 -2 C -2 -4 0 -8 12 D 4 12 8 0 8 E -4 2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 4 B 2 0 4 -12 -2 C -2 -4 0 -8 12 D 4 12 8 0 8 E -4 2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5310: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (13) E B D A C (12) D C A E B (12) C A D B E (12) D E B C A (11) A B C E D (5) C D A E B (4) A C B E D (4) B E A D C (3) E D B C A (2) E B D C A (2) D C E B A (2) C D A B E (2) C A B E D (2) C A B D E (2) B A C E D (2) A C D B E (2) A C B D E (2) E D B A C (1) D E C B A (1) D C E A B (1) B E C A D (1) B C A E D (1) B A E C D (1) Total count = 100 A B C D E A 0 -4 -10 0 2 B 4 0 8 0 4 C 10 -8 0 6 6 D 0 0 -6 0 2 E -2 -4 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.626509 C: 0.000000 D: 0.373491 E: 0.000000 Sum of squares = 0.532009155199 Cumulative probabilities = A: 0.000000 B: 0.626509 C: 0.626509 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 0 2 B 4 0 8 0 4 C 10 -8 0 6 6 D 0 0 -6 0 2 E -2 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=22 B=21 E=17 A=13 so A is eliminated. Round 2 votes counts: C=30 D=27 B=26 E=17 so E is eliminated. Round 3 votes counts: B=40 D=30 C=30 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:208 C:207 D:198 A:194 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 0 2 B 4 0 8 0 4 C 10 -8 0 6 6 D 0 0 -6 0 2 E -2 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 0 2 B 4 0 8 0 4 C 10 -8 0 6 6 D 0 0 -6 0 2 E -2 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 0 2 B 4 0 8 0 4 C 10 -8 0 6 6 D 0 0 -6 0 2 E -2 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5311: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) B A D C E (9) A B C E D (9) B D A E C (8) E C D A B (7) D B E C A (7) E D C B A (5) B A D E C (5) A C B E D (5) C E A D B (4) A B C D E (4) E A C B D (3) C E A B D (3) E D B C A (2) D E C B A (2) D E B C A (2) D C E B A (2) D B A E C (2) C E D A B (2) E C D B A (1) E C A D B (1) E C A B D (1) D B E A C (1) D B A C E (1) B E D A C (1) B D E A C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 2 12 10 B -2 0 6 22 8 C -2 -6 0 2 2 D -12 -22 -2 0 -10 E -10 -8 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 12 10 B -2 0 6 22 8 C -2 -6 0 2 2 D -12 -22 -2 0 -10 E -10 -8 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=20 A=20 C=19 D=17 so D is eliminated. Round 2 votes counts: B=35 E=24 C=21 A=20 so A is eliminated. Round 3 votes counts: B=49 C=27 E=24 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:213 C:198 E:195 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 12 10 B -2 0 6 22 8 C -2 -6 0 2 2 D -12 -22 -2 0 -10 E -10 -8 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 12 10 B -2 0 6 22 8 C -2 -6 0 2 2 D -12 -22 -2 0 -10 E -10 -8 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 12 10 B -2 0 6 22 8 C -2 -6 0 2 2 D -12 -22 -2 0 -10 E -10 -8 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5312: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (17) C D B E A (16) E B A D C (8) C A E B D (8) B E D A C (8) D C B E A (5) C A D E B (5) D B E A C (3) B E A D C (3) D C B A E (2) D B E C A (2) D B C E A (2) D A B E C (2) C E B A D (2) C D A B E (2) B E D C A (2) A E B C D (2) A D E B C (2) E B A C D (1) D C A B E (1) C D B A E (1) C D A E B (1) B D E A C (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 2 4 -6 B 12 0 6 6 0 C -2 -6 0 -18 -2 D -4 -6 18 0 -6 E 6 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.304405 C: 0.000000 D: 0.000000 E: 0.695595 Sum of squares = 0.576514440401 Cumulative probabilities = A: 0.000000 B: 0.304405 C: 0.304405 D: 0.304405 E: 1.000000 A B C D E A 0 -12 2 4 -6 B 12 0 6 6 0 C -2 -6 0 -18 -2 D -4 -6 18 0 -6 E 6 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=25 D=17 B=14 E=9 so E is eliminated. Round 2 votes counts: C=35 A=25 B=23 D=17 so D is eliminated. Round 3 votes counts: C=43 B=30 A=27 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:207 D:201 A:194 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 4 -6 B 12 0 6 6 0 C -2 -6 0 -18 -2 D -4 -6 18 0 -6 E 6 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 4 -6 B 12 0 6 6 0 C -2 -6 0 -18 -2 D -4 -6 18 0 -6 E 6 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 4 -6 B 12 0 6 6 0 C -2 -6 0 -18 -2 D -4 -6 18 0 -6 E 6 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5313: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) D C B A E (9) A C B E D (8) A C B D E (6) D B C E A (5) D E B C A (4) D E A C B (4) E B C A D (3) E A D C B (3) E A B C D (3) D A C B E (3) C B A D E (3) A E C B D (3) A D C B E (3) E D A C B (2) D C A B E (2) C B D A E (2) C A B D E (2) B E C A D (2) B C E A D (2) B C D A E (2) A E D C B (2) A B C E D (2) E B D C A (1) E B D A C (1) E B C D A (1) E B A C D (1) E A D B C (1) E A C B D (1) D E C B A (1) D C E A B (1) D B C A E (1) D A C E B (1) B D C E A (1) B C A E D (1) A E B C D (1) A D E C B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -4 0 8 B -2 0 -18 -6 8 C 4 18 0 -12 12 D 0 6 12 0 4 E -8 -8 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.443939 B: 0.000000 C: 0.000000 D: 0.556060 E: 0.000000 Sum of squares = 0.506285519936 Cumulative probabilities = A: 0.443939 B: 0.443939 C: 0.443940 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 0 8 B -2 0 -18 -6 8 C 4 18 0 -12 12 D 0 6 12 0 4 E -8 -8 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=28 E=26 B=8 C=7 so C is eliminated. Round 2 votes counts: D=31 A=30 E=26 B=13 so B is eliminated. Round 3 votes counts: D=36 A=34 E=30 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:211 A:203 B:191 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 0 8 B -2 0 -18 -6 8 C 4 18 0 -12 12 D 0 6 12 0 4 E -8 -8 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 8 B -2 0 -18 -6 8 C 4 18 0 -12 12 D 0 6 12 0 4 E -8 -8 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 8 B -2 0 -18 -6 8 C 4 18 0 -12 12 D 0 6 12 0 4 E -8 -8 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5314: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (17) C A E B D (16) B C D A E (8) A C E D B (8) B D C A E (7) E A C D B (6) D E A B C (5) E D A C B (4) D E B A C (4) C B A D E (4) B D E C A (4) C A B E D (3) A C E B D (3) B D C E A (2) B C A D E (2) E D B A C (1) E A D C B (1) D B C E A (1) C B A E D (1) C A E D B (1) B D E A C (1) A E C D B (1) Total count = 100 A B C D E A 0 -4 2 -8 8 B 4 0 4 2 0 C -2 -4 0 6 12 D 8 -2 -6 0 10 E -8 0 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.901310 C: 0.000000 D: 0.000000 E: 0.098690 Sum of squares = 0.822098691538 Cumulative probabilities = A: 0.000000 B: 0.901310 C: 0.901310 D: 0.901310 E: 1.000000 A B C D E A 0 -4 2 -8 8 B 4 0 4 2 0 C -2 -4 0 6 12 D 8 -2 -6 0 10 E -8 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222224943 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 B=24 E=12 A=12 so E is eliminated. Round 2 votes counts: D=32 C=25 B=24 A=19 so A is eliminated. Round 3 votes counts: C=43 D=33 B=24 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:205 D:205 A:199 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -8 8 B 4 0 4 2 0 C -2 -4 0 6 12 D 8 -2 -6 0 10 E -8 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222224943 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -8 8 B 4 0 4 2 0 C -2 -4 0 6 12 D 8 -2 -6 0 10 E -8 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222224943 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -8 8 B 4 0 4 2 0 C -2 -4 0 6 12 D 8 -2 -6 0 10 E -8 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.833333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.722222224943 Cumulative probabilities = A: 0.000000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5315: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) C A D E B (9) C A D B E (9) A C D E B (9) E B C A D (6) B E D C A (6) E B D C A (5) D A C B E (5) D B E A C (4) E B A C D (3) D C A B E (3) C D A B E (3) B E D A C (3) B D E C A (3) E B A D C (2) D B E C A (2) C B E A D (2) C A E B D (2) A C E B D (2) E C B A D (1) D C B A E (1) D B C E A (1) D B A E C (1) D A C E B (1) C E A B D (1) C A E D B (1) B C E D A (1) A D C E B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -12 0 -2 B 4 0 -4 -4 -10 C 12 4 0 2 6 D 0 4 -2 0 6 E 2 10 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 0 -2 B 4 0 -4 -4 -10 C 12 4 0 2 6 D 0 4 -2 0 6 E 2 10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 D=18 A=14 B=13 so B is eliminated. Round 2 votes counts: E=37 C=28 D=21 A=14 so A is eliminated. Round 3 votes counts: C=41 E=37 D=22 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:204 E:200 B:193 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 0 -2 B 4 0 -4 -4 -10 C 12 4 0 2 6 D 0 4 -2 0 6 E 2 10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 0 -2 B 4 0 -4 -4 -10 C 12 4 0 2 6 D 0 4 -2 0 6 E 2 10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 0 -2 B 4 0 -4 -4 -10 C 12 4 0 2 6 D 0 4 -2 0 6 E 2 10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5316: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (11) E A D C B (9) C E B A D (8) E A C D B (7) B C D A E (6) D B A E C (5) C B E A D (5) B D A C E (5) A E D C B (5) E C A D B (4) D A E B C (4) C E A B D (4) C B E D A (4) D E A B C (3) E A D B C (2) B C D E A (2) A D E B C (2) E D C B A (1) E D C A B (1) E D A C B (1) E D A B C (1) E C D A B (1) E C A B D (1) C E B D A (1) C B D A E (1) C B A E D (1) B A D C E (1) B A C D E (1) A D E C B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 18 20 6 -10 B -18 0 -10 -20 -12 C -20 10 0 -8 -20 D -6 20 8 0 -12 E 10 12 20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 20 6 -10 B -18 0 -10 -20 -12 C -20 10 0 -8 -20 D -6 20 8 0 -12 E 10 12 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=24 D=23 B=15 A=10 so A is eliminated. Round 2 votes counts: E=33 D=28 C=24 B=15 so B is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:227 A:217 D:205 C:181 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 20 6 -10 B -18 0 -10 -20 -12 C -20 10 0 -8 -20 D -6 20 8 0 -12 E 10 12 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 6 -10 B -18 0 -10 -20 -12 C -20 10 0 -8 -20 D -6 20 8 0 -12 E 10 12 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 6 -10 B -18 0 -10 -20 -12 C -20 10 0 -8 -20 D -6 20 8 0 -12 E 10 12 20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5317: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) D A C E B (7) A C D E B (7) E B A C D (6) A E C D B (6) D C B A E (5) B D C A E (5) E A B C D (4) E B C A D (3) E A D C B (3) E A C B D (3) D B C A E (3) D A C B E (3) B E C D A (3) E B D A C (2) E B A D C (2) B E D C A (2) B E C A D (2) B D C E A (2) B C D A E (2) A D C E B (2) A C E D B (2) E D B A C (1) E B C D A (1) E A D B C (1) E A C D B (1) E A B D C (1) D E A C B (1) D B E C A (1) D B E A C (1) D B A E C (1) D B A C E (1) D A E C B (1) C D A B E (1) C A D B E (1) B D E C A (1) B C E A D (1) B C D E A (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 8 16 -6 14 B -8 0 -6 -16 -10 C -16 6 0 -8 4 D 6 16 8 0 10 E -14 10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 -6 14 B -8 0 -6 -16 -10 C -16 6 0 -8 4 D 6 16 8 0 10 E -14 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=28 B=19 A=19 C=2 so C is eliminated. Round 2 votes counts: D=33 E=28 A=20 B=19 so B is eliminated. Round 3 votes counts: D=44 E=36 A=20 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:216 C:193 E:191 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 16 -6 14 B -8 0 -6 -16 -10 C -16 6 0 -8 4 D 6 16 8 0 10 E -14 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -6 14 B -8 0 -6 -16 -10 C -16 6 0 -8 4 D 6 16 8 0 10 E -14 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -6 14 B -8 0 -6 -16 -10 C -16 6 0 -8 4 D 6 16 8 0 10 E -14 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5318: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) B D A C E (6) C D E B A (5) C D B E A (4) B D A E C (4) A E D B C (4) E A D C B (3) E A C D B (3) D B E C A (3) C B A D E (3) B D C A E (3) B C D A E (3) E D C A B (2) E C A D B (2) D E B C A (2) D E B A C (2) D B C E A (2) D B A E C (2) C E A D B (2) C E A B D (2) C B D E A (2) C A E B D (2) B A D C E (2) A E C B D (2) A E B D C (2) A E B C D (2) A C B E D (2) A B D E C (2) A B C D E (2) E D A B C (1) E C D A B (1) E C A B D (1) E A D B C (1) D E C B A (1) D C E B A (1) D C B E A (1) D B E A C (1) C E D A B (1) C A B E D (1) C A B D E (1) B D C E A (1) B C A D E (1) B A C D E (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -10 -12 -4 B 16 0 0 -6 0 C 10 0 0 4 12 D 12 6 -4 0 12 E 4 0 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.245025 C: 0.754975 D: 0.000000 E: 0.000000 Sum of squares = 0.630024295948 Cumulative probabilities = A: 0.000000 B: 0.245025 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -12 -4 B 16 0 0 -6 0 C 10 0 0 4 12 D 12 6 -4 0 12 E 4 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000063985 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=21 A=19 D=15 E=14 so E is eliminated. Round 2 votes counts: C=35 A=26 B=21 D=18 so D is eliminated. Round 3 votes counts: C=40 B=33 A=27 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:213 B:205 E:190 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -10 -12 -4 B 16 0 0 -6 0 C 10 0 0 4 12 D 12 6 -4 0 12 E 4 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000063985 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -12 -4 B 16 0 0 -6 0 C 10 0 0 4 12 D 12 6 -4 0 12 E 4 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000063985 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -12 -4 B 16 0 0 -6 0 C 10 0 0 4 12 D 12 6 -4 0 12 E 4 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000063985 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5319: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) E C D A B (10) D E C B A (9) B A D C E (9) D B A E C (7) E C A D B (6) A B C E D (5) D B E C A (4) C E A D B (4) A B C D E (4) E C D B A (3) D E B C A (3) C E A B D (3) B D A C E (3) D E C A B (2) B D A E C (2) B C A E D (2) B A C D E (2) A B D C E (2) E D C B A (1) E D C A B (1) D A B E C (1) C E B A D (1) C A E B D (1) B C E D A (1) B C E A D (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -4 6 2 B -2 0 -6 -2 -10 C 4 6 0 12 2 D -6 2 -12 0 -4 E -2 10 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 6 2 B -2 0 -6 -2 -10 C 4 6 0 12 2 D -6 2 -12 0 -4 E -2 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=23 E=21 B=21 C=9 so C is eliminated. Round 2 votes counts: E=29 D=26 A=24 B=21 so B is eliminated. Round 3 votes counts: A=38 E=31 D=31 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 E:205 A:203 B:190 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 6 2 B -2 0 -6 -2 -10 C 4 6 0 12 2 D -6 2 -12 0 -4 E -2 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 6 2 B -2 0 -6 -2 -10 C 4 6 0 12 2 D -6 2 -12 0 -4 E -2 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 6 2 B -2 0 -6 -2 -10 C 4 6 0 12 2 D -6 2 -12 0 -4 E -2 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5320: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) E A B D C (7) B E A C D (7) B C D E A (7) C D B A E (6) B E A D C (5) A E B D C (5) E B A D C (4) D C A E B (4) D A C E B (4) B C E D A (4) E A D B C (3) C D A B E (3) C B D E A (3) A D C E B (3) D E A C B (2) D A E C B (2) C D B E A (2) C B D A E (2) B E D A C (2) B C E A D (2) A E D C B (2) A E D B C (2) D C B E A (1) C B A D E (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B C D A E (1) B C A E D (1) B C A D E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -2 -8 -4 B 4 0 8 10 2 C 2 -8 0 6 12 D 8 -10 -6 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 -4 B 4 0 8 10 2 C 2 -8 0 6 12 D 8 -10 -6 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=27 E=14 D=13 A=13 so D is eliminated. Round 2 votes counts: B=33 C=32 A=19 E=16 so E is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:206 D:198 E:193 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 -4 B 4 0 8 10 2 C 2 -8 0 6 12 D 8 -10 -6 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 -4 B 4 0 8 10 2 C 2 -8 0 6 12 D 8 -10 -6 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 -4 B 4 0 8 10 2 C 2 -8 0 6 12 D 8 -10 -6 0 4 E 4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5321: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (12) D B E A C (11) B C A E D (11) B D C A E (8) D E A C B (7) D E A B C (6) C A E D B (6) A E C D B (5) E A D C B (3) E A C D B (3) C B A E D (3) C E A D B (2) B D E C A (2) B D E A C (2) B D C E A (2) B D A E C (2) A C E D B (2) A C E B D (2) D E B A C (1) D A E B C (1) C E A B D (1) C B E D A (1) B D A C E (1) B C D E A (1) B C D A E (1) B C A D E (1) B A E C D (1) B A C E D (1) A E D C B (1) Total count = 100 A B C D E A 0 2 -2 8 16 B -2 0 4 4 -4 C 2 -4 0 6 10 D -8 -4 -6 0 -8 E -16 4 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 8 16 B -2 0 4 4 -4 C 2 -4 0 6 10 D -8 -4 -6 0 -8 E -16 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=26 C=25 A=10 E=6 so E is eliminated. Round 2 votes counts: B=33 D=26 C=25 A=16 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:212 C:207 B:201 E:193 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 8 16 B -2 0 4 4 -4 C 2 -4 0 6 10 D -8 -4 -6 0 -8 E -16 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 16 B -2 0 4 4 -4 C 2 -4 0 6 10 D -8 -4 -6 0 -8 E -16 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 16 B -2 0 4 4 -4 C 2 -4 0 6 10 D -8 -4 -6 0 -8 E -16 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5322: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) D B E C A (7) A C E B D (7) A C D E B (7) E C A B D (6) D B A C E (6) A C E D B (6) D B E A C (4) D A C E B (4) C A E B D (4) D A B C E (3) C E A B D (3) A D C E B (3) E C B A D (2) E C A D B (2) E B C A D (2) B E D C A (2) B D E A C (2) B C E A D (2) B A C E D (2) A C D B E (2) E C D B A (1) E B C D A (1) D E C A B (1) D E B C A (1) D E A B C (1) D B A E C (1) D A C B E (1) C E A D B (1) C B A E D (1) B A D C E (1) A D C B E (1) A D B C E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 8 10 4 B -10 0 -8 -6 -4 C -8 8 0 2 14 D -10 6 -2 0 14 E -4 4 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 10 4 B -10 0 -8 -6 -4 C -8 8 0 2 14 D -10 6 -2 0 14 E -4 4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 B=19 E=14 C=9 so C is eliminated. Round 2 votes counts: A=33 D=29 B=20 E=18 so E is eliminated. Round 3 votes counts: A=45 D=30 B=25 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:208 D:204 B:186 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 10 4 B -10 0 -8 -6 -4 C -8 8 0 2 14 D -10 6 -2 0 14 E -4 4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 10 4 B -10 0 -8 -6 -4 C -8 8 0 2 14 D -10 6 -2 0 14 E -4 4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 10 4 B -10 0 -8 -6 -4 C -8 8 0 2 14 D -10 6 -2 0 14 E -4 4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5323: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) C D B A E (7) C B E D A (7) B E C A D (7) C B D E A (5) B E A C D (5) E A B C D (4) D C B A E (4) D C A E B (3) D A C E B (3) C B E A D (3) B C D E A (3) A E D B C (3) A D E B C (3) E B A C D (2) E A B D C (2) D B C A E (2) D A E C B (2) C D B E A (2) B C E D A (2) B C E A D (2) A E D C B (2) A E B D C (2) A D E C B (2) E B C A D (1) E A C B D (1) D C B E A (1) D B C E A (1) D B A E C (1) D B A C E (1) D A C B E (1) D A B E C (1) C D A E B (1) C D A B E (1) B E C D A (1) B E A D C (1) B D C E A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 -18 -24 -16 -2 B 18 0 -10 0 34 C 24 10 0 10 18 D 16 0 -10 0 8 E 2 -34 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -24 -16 -2 B 18 0 -10 0 34 C 24 10 0 10 18 D 16 0 -10 0 8 E 2 -34 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 B=22 A=14 E=10 so E is eliminated. Round 2 votes counts: D=28 C=26 B=25 A=21 so A is eliminated. Round 3 votes counts: D=39 B=33 C=28 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:231 B:221 D:207 E:171 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -24 -16 -2 B 18 0 -10 0 34 C 24 10 0 10 18 D 16 0 -10 0 8 E 2 -34 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -24 -16 -2 B 18 0 -10 0 34 C 24 10 0 10 18 D 16 0 -10 0 8 E 2 -34 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -24 -16 -2 B 18 0 -10 0 34 C 24 10 0 10 18 D 16 0 -10 0 8 E 2 -34 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5324: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) D C B E A (9) B D A E C (8) C D E A B (6) A E C B D (5) A E B C D (5) B D E A C (4) B A E D C (4) A B E C D (4) C E D A B (3) C D A B E (3) E C A D B (2) E A C D B (2) D B E C A (2) D B C A E (2) C E A D B (2) C D E B A (2) C D B A E (2) C D A E B (2) C A E D B (2) B D A C E (2) B A D E C (2) A B E D C (2) E D B C A (1) E D A B C (1) E C D A B (1) E A C B D (1) E A B D C (1) D E C B A (1) D E B A C (1) D C E B A (1) D C B A E (1) C A B E D (1) B E D A C (1) B D C A E (1) Total count = 100 A B C D E A 0 -14 -14 -34 -8 B 14 0 8 -18 22 C 14 -8 0 -14 4 D 34 18 14 0 24 E 8 -22 -4 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 -34 -8 B 14 0 8 -18 22 C 14 -8 0 -14 4 D 34 18 14 0 24 E 8 -22 -4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=23 B=22 A=16 E=9 so E is eliminated. Round 2 votes counts: D=32 C=26 B=22 A=20 so A is eliminated. Round 3 votes counts: C=34 B=34 D=32 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:245 B:213 C:198 E:179 A:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -14 -34 -8 B 14 0 8 -18 22 C 14 -8 0 -14 4 D 34 18 14 0 24 E 8 -22 -4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -34 -8 B 14 0 8 -18 22 C 14 -8 0 -14 4 D 34 18 14 0 24 E 8 -22 -4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -34 -8 B 14 0 8 -18 22 C 14 -8 0 -14 4 D 34 18 14 0 24 E 8 -22 -4 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5325: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (5) D E C B A (5) D E B C A (5) B A C E D (5) A B C E D (5) D E C A B (4) C E D B A (4) B C E A D (4) A D E C B (4) D E A C B (3) D E A B C (3) C A E B D (3) B C E D A (3) B C D E A (3) B C A E D (3) A C E D B (3) A B D C E (3) E C D B A (2) E C D A B (2) D B E C A (2) C B E D A (2) C B E A D (2) B D E C A (2) B A C D E (2) A E D C B (2) A E C D B (2) A D E B C (2) A C E B D (2) A C B E D (2) E A C D B (1) D B E A C (1) D A E C B (1) C E B D A (1) C E A D B (1) C E A B D (1) C B A E D (1) C A E D B (1) B D E A C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -12 2 -14 B -4 0 -8 -6 -14 C 12 8 0 12 4 D -2 6 -12 0 -14 E 14 14 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 2 -14 B -4 0 -8 -6 -14 C 12 8 0 12 4 D -2 6 -12 0 -14 E 14 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 B=23 C=16 E=10 so E is eliminated. Round 2 votes counts: D=29 A=28 B=23 C=20 so C is eliminated. Round 3 votes counts: D=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:219 C:218 A:190 D:189 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 2 -14 B -4 0 -8 -6 -14 C 12 8 0 12 4 D -2 6 -12 0 -14 E 14 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 2 -14 B -4 0 -8 -6 -14 C 12 8 0 12 4 D -2 6 -12 0 -14 E 14 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 2 -14 B -4 0 -8 -6 -14 C 12 8 0 12 4 D -2 6 -12 0 -14 E 14 14 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5326: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) D E B A C (6) C A B E D (6) D B C A E (5) C A E B D (5) B D C A E (5) A C E B D (5) B C A E D (4) B C A D E (4) E D A C B (3) E A C B D (3) D E A C B (3) D E A B C (3) A E C D B (3) E D B A C (2) E D A B C (2) E A D C B (2) E A B D C (2) D C A B E (2) D B E C A (2) D B E A C (2) C B A D E (2) C A D E B (2) C A B D E (2) B D C E A (2) B C D A E (2) A C E D B (2) E B A C D (1) E A C D B (1) E A B C D (1) D C B A E (1) D B C E A (1) D A C E B (1) B E D A C (1) B E A C D (1) B D E C A (1) B D E A C (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 -8 10 20 B 2 0 -4 14 4 C 8 4 0 6 18 D -10 -14 -6 0 -6 E -20 -4 -18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 10 20 B 2 0 -4 14 4 C 8 4 0 6 18 D -10 -14 -6 0 -6 E -20 -4 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=25 B=21 E=17 A=11 so A is eliminated. Round 2 votes counts: C=32 D=26 E=21 B=21 so E is eliminated. Round 3 votes counts: C=40 D=35 B=25 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:210 B:208 D:182 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 10 20 B 2 0 -4 14 4 C 8 4 0 6 18 D -10 -14 -6 0 -6 E -20 -4 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 10 20 B 2 0 -4 14 4 C 8 4 0 6 18 D -10 -14 -6 0 -6 E -20 -4 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 10 20 B 2 0 -4 14 4 C 8 4 0 6 18 D -10 -14 -6 0 -6 E -20 -4 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5327: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) C B A D E (11) E D A C B (10) D E C B A (10) E D A B C (7) A B E C D (6) E A D B C (5) D C B E A (5) C D B A E (4) C B D A E (4) A E B C D (4) E A B D C (3) B C A D E (3) B A C E D (3) E A B C D (2) D C E B A (2) D C B A E (2) D E C A B (1) D E A C B (1) B A C D E (1) A E B D C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 16 8 8 B -10 0 -2 6 8 C -16 2 0 6 0 D -8 -6 -6 0 -8 E -8 -8 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999545 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 8 8 B -10 0 -2 6 8 C -16 2 0 6 0 D -8 -6 -6 0 -8 E -8 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 D=21 C=19 B=7 so B is eliminated. Round 2 votes counts: A=30 E=27 C=22 D=21 so D is eliminated. Round 3 votes counts: E=39 C=31 A=30 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:221 B:201 C:196 E:196 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 8 8 B -10 0 -2 6 8 C -16 2 0 6 0 D -8 -6 -6 0 -8 E -8 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 8 8 B -10 0 -2 6 8 C -16 2 0 6 0 D -8 -6 -6 0 -8 E -8 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 8 8 B -10 0 -2 6 8 C -16 2 0 6 0 D -8 -6 -6 0 -8 E -8 -8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5328: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C B A E D (7) E D A C B (5) C E A B D (4) B D C A E (4) D E A B C (3) D B E C A (3) B D C E A (3) B D A C E (3) B C D E A (3) B C A E D (3) B C A D E (3) E D C A B (2) E C A D B (2) D E C B A (2) D E A C B (2) D B E A C (2) C E A D B (2) C B E D A (2) C A B E D (2) B C D A E (2) B A D C E (2) A E C D B (2) A D B E C (2) A B C E D (2) E C D A B (1) E A D C B (1) E A C D B (1) D E B C A (1) D E B A C (1) D B C E A (1) D B C A E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E B A D (1) C B E A D (1) C B A D E (1) C A E B D (1) B D A E C (1) B A C D E (1) A E D C B (1) A E D B C (1) A E C B D (1) A E B C D (1) A C E B D (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -10 -8 8 B 16 0 10 6 22 C 10 -10 0 -6 8 D 8 -6 6 0 4 E -8 -22 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -8 8 B 16 0 10 6 22 C 10 -10 0 -6 8 D 8 -6 6 0 4 E -8 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 C=24 A=14 E=12 so E is eliminated. Round 2 votes counts: D=32 C=27 B=25 A=16 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:227 D:206 C:201 A:187 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -8 8 B 16 0 10 6 22 C 10 -10 0 -6 8 D 8 -6 6 0 4 E -8 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -8 8 B 16 0 10 6 22 C 10 -10 0 -6 8 D 8 -6 6 0 4 E -8 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -8 8 B 16 0 10 6 22 C 10 -10 0 -6 8 D 8 -6 6 0 4 E -8 -22 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5329: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) C A D B E (11) C D B E A (7) E B D C A (6) E B D A C (6) E B A D C (6) A C E B D (6) A C D B E (6) D B E A C (4) C D A B E (3) C A E B D (3) B E D A C (3) A E B C D (3) A C D E B (3) D C B E A (2) D B E C A (2) D A C B E (2) C A D E B (2) B E D C A (2) A D B E C (2) E B C D A (1) E B A C D (1) C E B D A (1) C E B A D (1) C D B A E (1) C B E D A (1) B D E C A (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 10 12 16 12 B -10 0 -2 8 -4 C -12 2 0 2 0 D -16 -8 -2 0 -6 E -12 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 16 12 B -10 0 -2 8 -4 C -12 2 0 2 0 D -16 -8 -2 0 -6 E -12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=30 E=20 D=10 B=6 so B is eliminated. Round 2 votes counts: A=34 C=30 E=25 D=11 so D is eliminated. Round 3 votes counts: A=36 E=32 C=32 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:199 B:196 C:196 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 16 12 B -10 0 -2 8 -4 C -12 2 0 2 0 D -16 -8 -2 0 -6 E -12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 16 12 B -10 0 -2 8 -4 C -12 2 0 2 0 D -16 -8 -2 0 -6 E -12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 16 12 B -10 0 -2 8 -4 C -12 2 0 2 0 D -16 -8 -2 0 -6 E -12 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5330: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) C E A D B (6) B D E C A (6) B D A E C (6) A B D E C (6) C E D A B (5) C E D B A (4) C A E D B (4) A E C D B (4) A D E B C (4) A D B E C (4) A C E D B (4) A C B D E (4) E D B C A (3) C E B D A (3) B C D E A (3) A E D C B (3) A B D C E (3) D B E A C (2) C E B A D (2) B D E A C (2) B C D A E (2) A E D B C (2) E D A B C (1) E C D B A (1) D B E C A (1) D B A E C (1) C B E D A (1) C B A D E (1) C A B D E (1) B D C E A (1) B D C A E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 -2 2 B -4 0 -2 2 8 C 6 2 0 8 8 D 2 -2 -8 0 14 E -2 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -2 2 B -4 0 -2 2 8 C 6 2 0 8 8 D 2 -2 -8 0 14 E -2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=35 A=35 B=21 E=5 D=4 so D is eliminated. Round 2 votes counts: C=35 A=35 B=25 E=5 so E is eliminated. Round 3 votes counts: C=36 A=36 B=28 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:203 B:202 A:199 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -2 2 B -4 0 -2 2 8 C 6 2 0 8 8 D 2 -2 -8 0 14 E -2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -2 2 B -4 0 -2 2 8 C 6 2 0 8 8 D 2 -2 -8 0 14 E -2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -2 2 B -4 0 -2 2 8 C 6 2 0 8 8 D 2 -2 -8 0 14 E -2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5331: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) C D B A E (8) C D B E A (7) A E B D C (7) C A D B E (6) D C B E A (5) A C D B E (5) A E C D B (4) E B A D C (3) C E D B A (3) C D E B A (3) C D A B E (3) B E D A C (3) A E C B D (3) A E B C D (3) E D B C A (2) E B D A C (2) B E D C A (2) B D E C A (2) B D C E A (2) A B E D C (2) A B C D E (2) E C D B A (1) E C D A B (1) E A B D C (1) D B E C A (1) D B C E A (1) C A D E B (1) B D E A C (1) B D C A E (1) B A E D C (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -18 -20 -18 -2 B 18 0 -6 -6 6 C 20 6 0 6 0 D 18 6 -6 0 0 E 2 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.756839 D: 0.000000 E: 0.243161 Sum of squares = 0.631932400899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.756839 D: 0.756839 E: 1.000000 A B C D E A 0 -18 -20 -18 -2 B 18 0 -6 -6 6 C 20 6 0 6 0 D 18 6 -6 0 0 E 2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.000000 E: 0.499674 Sum of squares = 0.500000212117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.500326 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=29 E=21 B=12 D=7 so D is eliminated. Round 2 votes counts: C=36 A=29 E=21 B=14 so B is eliminated. Round 3 votes counts: C=40 E=30 A=30 so E is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:209 B:206 E:198 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -20 -18 -2 B 18 0 -6 -6 6 C 20 6 0 6 0 D 18 6 -6 0 0 E 2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.000000 E: 0.499674 Sum of squares = 0.500000212117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.500326 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -20 -18 -2 B 18 0 -6 -6 6 C 20 6 0 6 0 D 18 6 -6 0 0 E 2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.000000 E: 0.499674 Sum of squares = 0.500000212117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.500326 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -20 -18 -2 B 18 0 -6 -6 6 C 20 6 0 6 0 D 18 6 -6 0 0 E 2 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.000000 E: 0.499674 Sum of squares = 0.500000212117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500326 D: 0.500326 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5332: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (12) E A B C D (11) C B D A E (8) E B C D A (7) E A D B C (7) B C E D A (7) D C B A E (6) D A C B E (6) A D E C B (6) B C D E A (5) A E D C B (5) E A B D C (4) A D C B E (3) A E D B C (2) A D B C E (2) E C B D A (1) D C B E A (1) D C A B E (1) C D B A E (1) C B D E A (1) A D E B C (1) A D C E B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 12 -12 B -2 0 20 16 -14 C 0 -20 0 10 -12 D -12 -16 -10 0 -14 E 12 14 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 12 -12 B -2 0 20 16 -14 C 0 -20 0 10 -12 D -12 -16 -10 0 -14 E 12 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 A=22 D=14 B=12 C=10 so C is eliminated. Round 2 votes counts: E=42 A=22 B=21 D=15 so D is eliminated. Round 3 votes counts: E=42 B=29 A=29 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:210 A:201 C:189 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 12 -12 B -2 0 20 16 -14 C 0 -20 0 10 -12 D -12 -16 -10 0 -14 E 12 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 12 -12 B -2 0 20 16 -14 C 0 -20 0 10 -12 D -12 -16 -10 0 -14 E 12 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 12 -12 B -2 0 20 16 -14 C 0 -20 0 10 -12 D -12 -16 -10 0 -14 E 12 14 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5333: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) E A D B C (11) C B D A E (11) C B A E D (11) A E C B D (6) D B C E A (5) D E B C A (4) C B A D E (4) E D A B C (3) D E A C B (3) D B E C A (3) B C A E D (3) A E D C B (3) A E D B C (3) A C B E D (3) D E B A C (2) C B D E A (2) B C D E A (2) B C A D E (2) E B D A C (1) E A D C B (1) D E C B A (1) C A B E D (1) B C D A E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 0 0 0 B 4 0 6 -2 -4 C 0 -6 0 -4 -8 D 0 2 4 0 4 E 0 4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222306 B: 0.000000 C: 0.000000 D: 0.777694 E: 0.000000 Sum of squares = 0.654228332083 Cumulative probabilities = A: 0.222306 B: 0.222306 C: 0.222306 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 0 0 B 4 0 6 -2 -4 C 0 -6 0 -4 -8 D 0 2 4 0 4 E 0 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555555983 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=29 A=17 E=16 B=8 so B is eliminated. Round 2 votes counts: C=37 D=30 A=17 E=16 so E is eliminated. Round 3 votes counts: C=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:205 E:204 B:202 A:198 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 0 0 B 4 0 6 -2 -4 C 0 -6 0 -4 -8 D 0 2 4 0 4 E 0 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555555983 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 0 0 B 4 0 6 -2 -4 C 0 -6 0 -4 -8 D 0 2 4 0 4 E 0 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555555983 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 0 0 B 4 0 6 -2 -4 C 0 -6 0 -4 -8 D 0 2 4 0 4 E 0 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555555983 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5334: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) C E A B D (5) B D A C E (5) B A D C E (5) A E C B D (5) E C D A B (4) E C A D B (4) D B A E C (4) B A C E D (4) A B C E D (4) E C A B D (3) D E C A B (3) C E D B A (3) A D B E C (3) E D C A B (2) E D A C B (2) D B E C A (2) D A B E C (2) C A E B D (2) B D C E A (2) B C D E A (2) A E D B C (2) A B E C D (2) A B D E C (2) E D C B A (1) E A C D B (1) E A C B D (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) D C B E A (1) D B E A C (1) D B A C E (1) D A E C B (1) C B E A D (1) B D C A E (1) B C E D A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) A E D C B (1) A E B C D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 2 2 2 B -6 0 14 4 10 C -2 -14 0 -4 -2 D -2 -4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 2 2 B -6 0 14 4 10 C -2 -14 0 -4 -2 D -2 -4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=24 A=22 E=18 C=11 so C is eliminated. Round 2 votes counts: E=26 D=25 B=25 A=24 so A is eliminated. Round 3 votes counts: E=37 B=35 D=28 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:206 D:198 E:196 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 2 2 B -6 0 14 4 10 C -2 -14 0 -4 -2 D -2 -4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 2 B -6 0 14 4 10 C -2 -14 0 -4 -2 D -2 -4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 2 B -6 0 14 4 10 C -2 -14 0 -4 -2 D -2 -4 4 0 -2 E -2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5335: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) E A B D C (6) E C D B A (5) E C D A B (5) E C A D B (5) E A B C D (5) C A E D B (4) A B E D C (4) E D B C A (3) D B E A C (3) D B C A E (3) D B A C E (3) C D B E A (3) C D B A E (3) B D A C E (3) B A D C E (3) E D C B A (2) E A C B D (2) C D A B E (2) C A D E B (2) C A B D E (2) B A D E C (2) A E B D C (2) A E B C D (2) A B D E C (2) A B D C E (2) A B C D E (2) E D B A C (1) D C B E A (1) D C B A E (1) D B C E A (1) C E D B A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D E A B (1) C A D B E (1) B E D A C (1) B D A E C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 18 -8 10 -8 B -18 0 -2 -6 -12 C 8 2 0 12 -14 D -10 6 -12 0 -14 E 8 12 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 -8 10 -8 B -18 0 -2 -6 -12 C 8 2 0 12 -14 D -10 6 -12 0 -14 E 8 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=22 A=16 D=12 B=10 so B is eliminated. Round 2 votes counts: E=41 C=22 A=21 D=16 so D is eliminated. Round 3 votes counts: E=44 C=28 A=28 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:206 C:204 D:185 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -8 10 -8 B -18 0 -2 -6 -12 C 8 2 0 12 -14 D -10 6 -12 0 -14 E 8 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 10 -8 B -18 0 -2 -6 -12 C 8 2 0 12 -14 D -10 6 -12 0 -14 E 8 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 10 -8 B -18 0 -2 -6 -12 C 8 2 0 12 -14 D -10 6 -12 0 -14 E 8 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5336: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) C D E B A (6) C A E D B (6) A B D E C (6) E B D A C (5) C E D A B (5) C A E B D (4) C A B E D (4) B D E A C (4) B A E D C (4) E D B C A (3) E D B A C (3) C E D B A (3) B D A E C (3) B A D E C (3) A C B D E (3) A B D C E (3) A B C E D (3) A B C D E (3) E C D B A (2) D E B C A (2) D E B A C (2) B E D A C (2) A C E B D (2) E A B D C (1) D C E B A (1) D B E A C (1) C E A D B (1) C D E A B (1) C A D B E (1) B E A D C (1) A E B D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 22 14 14 B -10 0 4 26 4 C -22 -4 0 8 10 D -14 -26 -8 0 -22 E -14 -4 -10 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 22 14 14 B -10 0 4 26 4 C -22 -4 0 8 10 D -14 -26 -8 0 -22 E -14 -4 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=31 B=17 E=14 D=6 so D is eliminated. Round 2 votes counts: C=32 A=32 E=18 B=18 so E is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 B:212 E:197 C:196 D:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 22 14 14 B -10 0 4 26 4 C -22 -4 0 8 10 D -14 -26 -8 0 -22 E -14 -4 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 22 14 14 B -10 0 4 26 4 C -22 -4 0 8 10 D -14 -26 -8 0 -22 E -14 -4 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 22 14 14 B -10 0 4 26 4 C -22 -4 0 8 10 D -14 -26 -8 0 -22 E -14 -4 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5337: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) C A D B E (6) D E B A C (5) D A C B E (5) B D E C A (5) A C D B E (5) E D B A C (4) D B E C A (4) E B D A C (3) E B A D C (3) D B E A C (3) D B A C E (3) C D A B E (3) C A E B D (3) C A B E D (3) A D C E B (3) A C E D B (3) A C E B D (3) A C D E B (3) E B A C D (2) E A C B D (2) C B D A E (2) C B A E D (2) C B A D E (2) B E C D A (2) B D C E A (2) A E D C B (2) A E C D B (2) A D E C B (2) E D A B C (1) E A D B C (1) D E A B C (1) D B C A E (1) D A E B C (1) A E C B D (1) Total count = 100 A B C D E A 0 0 16 -4 10 B 0 0 -4 -16 10 C -16 4 0 -12 -2 D 4 16 12 0 12 E -10 -10 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 -4 10 B 0 0 -4 -16 10 C -16 4 0 -12 -2 D 4 16 12 0 12 E -10 -10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 C=21 E=16 B=16 so E is eliminated. Round 2 votes counts: D=28 A=27 B=24 C=21 so C is eliminated. Round 3 votes counts: A=39 D=31 B=30 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:211 B:195 C:187 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 16 -4 10 B 0 0 -4 -16 10 C -16 4 0 -12 -2 D 4 16 12 0 12 E -10 -10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 -4 10 B 0 0 -4 -16 10 C -16 4 0 -12 -2 D 4 16 12 0 12 E -10 -10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 -4 10 B 0 0 -4 -16 10 C -16 4 0 -12 -2 D 4 16 12 0 12 E -10 -10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5338: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (14) B D A C E (10) D B A C E (8) E B C A D (7) C A E D B (6) A C D B E (6) D A C B E (5) B E D A C (5) C A D E B (4) E B D C A (3) B D E A C (3) B D A E C (3) B A D C E (3) B A C D E (3) A C B D E (3) E D B C A (2) E C D A B (2) E C B A D (2) D A C E B (2) D A B C E (2) E C A B D (1) E B D A C (1) D E A C B (1) C E A D B (1) C A D B E (1) B E D C A (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 12 4 14 B 2 0 2 -10 6 C -12 -2 0 0 10 D -4 10 0 0 10 E -14 -6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.46875000019 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 4 14 B 2 0 2 -10 6 C -12 -2 0 0 10 D -4 10 0 0 10 E -14 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.468749999981 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=28 D=18 C=12 A=10 so A is eliminated. Round 2 votes counts: E=32 B=28 C=21 D=19 so D is eliminated. Round 3 votes counts: B=38 E=33 C=29 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:214 D:208 B:200 C:198 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 12 4 14 B 2 0 2 -10 6 C -12 -2 0 0 10 D -4 10 0 0 10 E -14 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.468749999981 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 4 14 B 2 0 2 -10 6 C -12 -2 0 0 10 D -4 10 0 0 10 E -14 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.468749999981 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 4 14 B 2 0 2 -10 6 C -12 -2 0 0 10 D -4 10 0 0 10 E -14 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.468749999981 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5339: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) D A E B C (8) A E D B C (8) D A C B E (6) E A B C D (5) D C B A E (5) C B E D A (5) C B D A E (5) D C B E A (4) D B C E A (4) D A E C B (4) A E D C B (4) A E C B D (4) E B C A D (3) C B D E A (3) B E C A D (3) C D B E A (2) B C E A D (2) A D E C B (2) A C E B D (2) E B A D C (1) E B A C D (1) E A C B D (1) D C A B E (1) D B E C A (1) D A C E B (1) C D A B E (1) C B A E D (1) C A D B E (1) B C E D A (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -2 -2 10 B 0 0 -24 -8 8 C 2 24 0 0 8 D 2 8 0 0 -2 E -10 -8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.481225 D: 0.518774 E: 0.000000 Sum of squares = 0.500704920623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.481226 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -2 10 B 0 0 -24 -8 8 C 2 24 0 0 8 D 2 8 0 0 -2 E -10 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=27 A=22 E=11 B=6 so B is eliminated. Round 2 votes counts: D=34 C=30 A=22 E=14 so E is eliminated. Round 3 votes counts: C=36 D=34 A=30 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:204 A:203 B:188 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 -2 10 B 0 0 -24 -8 8 C 2 24 0 0 8 D 2 8 0 0 -2 E -10 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 10 B 0 0 -24 -8 8 C 2 24 0 0 8 D 2 8 0 0 -2 E -10 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 10 B 0 0 -24 -8 8 C 2 24 0 0 8 D 2 8 0 0 -2 E -10 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999689 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5340: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) E B A D C (8) C A D E B (8) D A C B E (6) D A B E C (5) C D A B E (5) B E C D A (4) E B D A C (3) E A B D C (3) C E B A D (3) C E A B D (3) C B D E A (3) A E C D B (3) E B C A D (2) D B A E C (2) D B A C E (2) C D B A E (2) C A E D B (2) C A D B E (2) B D E A C (2) A D E C B (2) A D C E B (2) E C B A D (1) E C A B D (1) E A D B C (1) E A C B D (1) E A B C D (1) D C B A E (1) D C A B E (1) D A C E B (1) C B E D A (1) C B E A D (1) C A E B D (1) B C E D A (1) B C D E A (1) A E D B C (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 2 14 -4 -4 B -2 0 -2 4 4 C -14 2 0 -8 -6 D 4 -4 8 0 -6 E 4 -4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999962 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 2 14 -4 -4 B -2 0 -2 4 4 C -14 2 0 -8 -6 D 4 -4 8 0 -6 E 4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.35999999981 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=21 B=20 D=18 A=10 so A is eliminated. Round 2 votes counts: C=31 E=25 D=24 B=20 so B is eliminated. Round 3 votes counts: E=41 C=33 D=26 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:206 A:204 B:202 D:201 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 14 -4 -4 B -2 0 -2 4 4 C -14 2 0 -8 -6 D 4 -4 8 0 -6 E 4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.35999999981 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -4 -4 B -2 0 -2 4 4 C -14 2 0 -8 -6 D 4 -4 8 0 -6 E 4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.35999999981 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -4 -4 B -2 0 -2 4 4 C -14 2 0 -8 -6 D 4 -4 8 0 -6 E 4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.35999999981 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5341: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) C D A E B (7) A C D B E (7) B A E C D (6) D E B A C (5) C A B E D (4) A B C E D (4) D E B C A (3) D C E A B (3) C E B A D (3) C A E B D (3) C A D E B (3) C A D B E (3) C A B D E (3) E D B C A (2) E D B A C (2) E B D C A (2) E B C D A (2) D C E B A (2) D C A E B (2) D B E A C (2) C E D B A (2) B E D A C (2) B E C A D (2) B E A D C (2) B A E D C (2) A D C B E (2) D E C B A (1) D A C E B (1) C E B D A (1) C D E A B (1) C A E D B (1) B E A C D (1) B A C E D (1) A D B E C (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 6 10 B 0 0 -2 0 -6 C 0 2 0 16 14 D -6 0 -16 0 -2 E -10 6 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.888491 B: 0.000000 C: 0.111509 D: 0.000000 E: 0.000000 Sum of squares = 0.801850799309 Cumulative probabilities = A: 0.888491 B: 0.888491 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 6 10 B 0 0 -2 0 -6 C 0 2 0 16 14 D -6 0 -16 0 -2 E -10 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999043 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=19 A=19 B=16 E=15 so E is eliminated. Round 2 votes counts: C=31 B=27 D=23 A=19 so A is eliminated. Round 3 votes counts: C=40 B=34 D=26 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:208 B:196 E:192 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 6 10 B 0 0 -2 0 -6 C 0 2 0 16 14 D -6 0 -16 0 -2 E -10 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999043 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 6 10 B 0 0 -2 0 -6 C 0 2 0 16 14 D -6 0 -16 0 -2 E -10 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999043 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 6 10 B 0 0 -2 0 -6 C 0 2 0 16 14 D -6 0 -16 0 -2 E -10 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999043 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5342: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) D C A E B (7) A B C D E (7) E B A D C (6) E D C A B (5) E B C D A (5) B E A C D (5) E D C B A (4) E C D B A (3) E B D A C (3) E B C A D (3) E B A C D (3) C D A B E (3) A D C B E (3) A B E D C (3) C D E B A (2) B A E D C (2) B A E C D (2) B A C E D (2) A C B D E (2) A B D C E (2) E D B A C (1) E C B D A (1) D E C A B (1) D E A C B (1) D C E B A (1) D C A B E (1) D A E C B (1) D A C E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C D B E A (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D A E (1) C A D B E (1) B E C A D (1) B C A E D (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -8 -12 -14 B 2 0 -6 2 -18 C 8 6 0 -4 0 D 12 -2 4 0 -6 E 14 18 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.406981 D: 0.000000 E: 0.593019 Sum of squares = 0.517305005184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.406981 D: 0.406981 E: 1.000000 A B C D E A 0 -2 -8 -12 -14 B 2 0 -6 2 -18 C 8 6 0 -4 0 D 12 -2 4 0 -6 E 14 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=22 A=18 C=13 B=13 so C is eliminated. Round 2 votes counts: E=36 D=30 A=19 B=15 so B is eliminated. Round 3 votes counts: E=43 D=31 A=26 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:205 D:204 B:190 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 -12 -14 B 2 0 -6 2 -18 C 8 6 0 -4 0 D 12 -2 4 0 -6 E 14 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -12 -14 B 2 0 -6 2 -18 C 8 6 0 -4 0 D 12 -2 4 0 -6 E 14 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -12 -14 B 2 0 -6 2 -18 C 8 6 0 -4 0 D 12 -2 4 0 -6 E 14 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5343: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (13) A D E C B (7) E D B A C (6) B C E D A (6) D E B A C (4) D E A B C (4) C E A B D (4) C A E D B (4) B E D C A (4) E D A C B (3) C B A E D (3) C A B D E (3) E C A D B (2) E B D C A (2) E B D A C (2) E A D C B (2) D E A C B (2) C B E A D (2) B E C D A (2) B D E A C (2) B C D A E (2) A D C E B (2) E D C A B (1) E C D B A (1) E C D A B (1) E B C D A (1) D B A E C (1) D A E C B (1) D A E B C (1) C E B D A (1) C A B E D (1) B D C A E (1) B D A C E (1) B C E A D (1) B C D E A (1) B C A E D (1) A E D C B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -14 -14 0 -8 B 14 0 -16 6 -8 C 14 16 0 4 0 D 0 -6 -4 0 -4 E 8 8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.452458 D: 0.000000 E: 0.547542 Sum of squares = 0.504520479517 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.452458 D: 0.452458 E: 1.000000 A B C D E A 0 -14 -14 0 -8 B 14 0 -16 6 -8 C 14 16 0 4 0 D 0 -6 -4 0 -4 E 8 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=21 B=21 A=14 D=13 so D is eliminated. Round 2 votes counts: E=31 C=31 B=22 A=16 so A is eliminated. Round 3 votes counts: E=41 C=37 B=22 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:210 B:198 D:193 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 0 -8 B 14 0 -16 6 -8 C 14 16 0 4 0 D 0 -6 -4 0 -4 E 8 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 0 -8 B 14 0 -16 6 -8 C 14 16 0 4 0 D 0 -6 -4 0 -4 E 8 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 0 -8 B 14 0 -16 6 -8 C 14 16 0 4 0 D 0 -6 -4 0 -4 E 8 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5344: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (6) D B E A C (6) C A E B D (6) E D A B C (5) D B C E A (5) B D E C A (5) A E C B D (5) A C E D B (5) D E B A C (4) D B C A E (4) B C D E A (4) E A D B C (3) A C D E B (3) E D B A C (2) E B D A C (2) D E A B C (2) C B E A D (2) C B D A E (2) C B A D E (2) C A B E D (2) C A B D E (2) B C D A E (2) A E C D B (2) A C E B D (2) E C A B D (1) E B C A D (1) E A C D B (1) D C B A E (1) D B E C A (1) D A E C B (1) D A C B E (1) C D B A E (1) C D A B E (1) C B A E D (1) C A D B E (1) B E C D A (1) B D C E A (1) B C E D A (1) B C E A D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 4 -4 -8 B -2 0 0 -2 -6 C -4 0 0 10 2 D 4 2 -10 0 2 E 8 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428545 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 2 4 -4 -8 B -2 0 0 -2 -6 C -4 0 0 10 2 D 4 2 -10 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=21 C=20 A=19 B=15 so B is eliminated. Round 2 votes counts: D=31 C=28 E=22 A=19 so A is eliminated. Round 3 votes counts: C=38 D=33 E=29 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:205 C:204 D:199 A:197 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 -4 -8 B -2 0 0 -2 -6 C -4 0 0 10 2 D 4 2 -10 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -4 -8 B -2 0 0 -2 -6 C -4 0 0 10 2 D 4 2 -10 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -4 -8 B -2 0 0 -2 -6 C -4 0 0 10 2 D 4 2 -10 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5345: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (20) D C E B A (19) B E C A D (5) A D C E B (5) E B C D A (4) D B E C A (4) C E D B A (4) A D C B E (4) A C E D B (4) A C E B D (4) D C E A B (2) D C A E B (2) D A C E B (2) C E B D A (2) B E C D A (2) A C B E D (2) A B E D C (2) D E C B A (1) D B E A C (1) D B A E C (1) D A C B E (1) C E B A D (1) C D E B A (1) C D A E B (1) B E D C A (1) B E A C D (1) B A E C D (1) A D B C E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 4 2 4 4 B -4 0 -12 -8 -6 C -2 12 0 6 12 D -4 8 -6 0 -6 E -4 6 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 4 4 B -4 0 -12 -8 -6 C -2 12 0 6 12 D -4 8 -6 0 -6 E -4 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 D=33 B=10 C=9 E=4 so E is eliminated. Round 2 votes counts: A=44 D=33 B=14 C=9 so C is eliminated. Round 3 votes counts: A=44 D=39 B=17 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:214 A:207 E:198 D:196 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 4 4 B -4 0 -12 -8 -6 C -2 12 0 6 12 D -4 8 -6 0 -6 E -4 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 4 B -4 0 -12 -8 -6 C -2 12 0 6 12 D -4 8 -6 0 -6 E -4 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 4 B -4 0 -12 -8 -6 C -2 12 0 6 12 D -4 8 -6 0 -6 E -4 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5346: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B D C A (7) C A E B D (7) D B E A C (6) D B A E C (6) B D E A C (6) D A B C E (5) C A E D B (5) E C A B D (4) B E D C A (4) B D E C A (4) A C E D B (4) A C D E B (4) E C B A D (3) E B C D A (3) C E A B D (3) C A B D E (3) A D C B E (3) E D A B C (2) D B A C E (2) D A C B E (2) D A B E C (2) C A D E B (2) C E B A D (1) B E D A C (1) B E C D A (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 10 4 -4 8 B -10 0 0 -4 8 C -4 0 0 -2 2 D 4 4 2 0 8 E -8 -8 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -4 8 B -10 0 0 -4 8 C -4 0 0 -2 2 D 4 4 2 0 8 E -8 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=21 A=20 E=19 B=17 so B is eliminated. Round 2 votes counts: D=33 E=25 C=22 A=20 so A is eliminated. Round 3 votes counts: C=38 D=37 E=25 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:209 C:198 B:197 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -4 8 B -10 0 0 -4 8 C -4 0 0 -2 2 D 4 4 2 0 8 E -8 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -4 8 B -10 0 0 -4 8 C -4 0 0 -2 2 D 4 4 2 0 8 E -8 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -4 8 B -10 0 0 -4 8 C -4 0 0 -2 2 D 4 4 2 0 8 E -8 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5347: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) B E D A C (7) A B E C D (7) C D A E B (6) D E B C A (5) D B E C A (5) A C E B D (5) A C B E D (5) C A D B E (4) B E A D C (4) A C E D B (4) C D E B A (3) C D E A B (3) C D B E A (3) C D A B E (3) B A E D C (3) A E B C D (3) D C E B A (2) C D B A E (2) B D E C A (2) A B E D C (2) A B C E D (2) E D C A B (1) E B D C A (1) E B A D C (1) D E C A B (1) D C B E A (1) C A B D E (1) A E C B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 22 -8 10 22 B -22 0 -16 -10 4 C 8 16 0 30 14 D -10 10 -30 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -8 10 22 B -22 0 -16 -10 4 C 8 16 0 30 14 D -10 10 -30 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=31 B=16 D=14 E=3 so E is eliminated. Round 2 votes counts: C=36 A=31 B=18 D=15 so D is eliminated. Round 3 votes counts: C=41 A=31 B=28 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:234 A:223 D:189 B:178 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -8 10 22 B -22 0 -16 -10 4 C 8 16 0 30 14 D -10 10 -30 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -8 10 22 B -22 0 -16 -10 4 C 8 16 0 30 14 D -10 10 -30 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -8 10 22 B -22 0 -16 -10 4 C 8 16 0 30 14 D -10 10 -30 0 8 E -22 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5348: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) E C A D B (7) D B C E A (7) B C E D A (6) A E C B D (6) D B C A E (5) C E D A B (5) A E C D B (5) A E D C B (4) A E B C D (4) E C A B D (3) D C E B A (3) A D E C B (3) A B E C D (3) E A C B D (2) D C E A B (2) D C B E A (2) B E A C D (2) B A E C D (2) B A D E C (2) A D E B C (2) A B D E C (2) E C D A B (1) E A C D B (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E B D A (1) C B E D A (1) B E C A D (1) B D A C E (1) B C E A D (1) A E B D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 14 -8 6 -8 B -14 0 2 -4 -4 C 8 -2 0 2 -8 D -6 4 -2 0 -14 E 8 4 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -8 6 -8 B -14 0 2 -4 -4 C 8 -2 0 2 -8 D -6 4 -2 0 -14 E 8 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=24 D=23 E=14 C=7 so C is eliminated. Round 2 votes counts: A=32 B=25 D=23 E=20 so E is eliminated. Round 3 votes counts: A=45 D=29 B=26 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:217 A:202 C:200 D:191 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -8 6 -8 B -14 0 2 -4 -4 C 8 -2 0 2 -8 D -6 4 -2 0 -14 E 8 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 6 -8 B -14 0 2 -4 -4 C 8 -2 0 2 -8 D -6 4 -2 0 -14 E 8 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 6 -8 B -14 0 2 -4 -4 C 8 -2 0 2 -8 D -6 4 -2 0 -14 E 8 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5349: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (14) B C D A E (11) D C B A E (7) D C A E B (6) C D A B E (4) B C A D E (4) E A B D C (3) E A B C D (3) C D B A E (3) B E A C D (3) A E C D B (3) A E B C D (3) E D A C B (2) D E C A B (2) D E A C B (2) D C B E A (2) D C A B E (2) B D C E A (2) B D C A E (2) B C A E D (2) A E D C B (2) E D B A C (1) E B A D C (1) E A D B C (1) E A C D B (1) D E B C A (1) D C E B A (1) D C E A B (1) D B C A E (1) C D A E B (1) C B D A E (1) B E D C A (1) B E C D A (1) B C D E A (1) B A E C D (1) B A C E D (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -12 -10 14 B -6 0 -12 -16 2 C 12 12 0 -8 8 D 10 16 8 0 10 E -14 -2 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -10 14 B -6 0 -12 -16 2 C 12 12 0 -8 8 D 10 16 8 0 10 E -14 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=26 D=25 A=11 C=9 so C is eliminated. Round 2 votes counts: D=33 B=30 E=26 A=11 so A is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:212 A:199 B:184 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -12 -10 14 B -6 0 -12 -16 2 C 12 12 0 -8 8 D 10 16 8 0 10 E -14 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -10 14 B -6 0 -12 -16 2 C 12 12 0 -8 8 D 10 16 8 0 10 E -14 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -10 14 B -6 0 -12 -16 2 C 12 12 0 -8 8 D 10 16 8 0 10 E -14 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5350: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (13) C A D E B (9) B E D A C (9) C B E D A (7) A D C E B (7) B E C D A (5) D A E B C (4) C B A E D (4) C A D B E (4) C D A E B (3) C A B E D (3) B E D C A (3) A D E C B (3) A D E B C (3) D E B A C (2) D C A E B (2) D A C E B (2) B E A C D (2) A C D E B (2) E D B A C (1) E B A D C (1) E A B D C (1) D E A B C (1) D A E C B (1) C D B A E (1) C A E B D (1) C A B D E (1) B E A D C (1) B D E A C (1) B C E D A (1) A E D B C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 -14 14 4 B 2 0 -26 8 10 C 14 26 0 14 20 D -14 -8 -14 0 -8 E -4 -10 -20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 14 4 B 2 0 -26 8 10 C 14 26 0 14 20 D -14 -8 -14 0 -8 E -4 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 B=22 A=17 D=12 E=3 so E is eliminated. Round 2 votes counts: C=46 B=23 A=18 D=13 so D is eliminated. Round 3 votes counts: C=48 B=26 A=26 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:237 A:201 B:197 E:187 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 14 4 B 2 0 -26 8 10 C 14 26 0 14 20 D -14 -8 -14 0 -8 E -4 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 14 4 B 2 0 -26 8 10 C 14 26 0 14 20 D -14 -8 -14 0 -8 E -4 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 14 4 B 2 0 -26 8 10 C 14 26 0 14 20 D -14 -8 -14 0 -8 E -4 -10 -20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5351: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D A E C B (8) A D E B C (8) B C E A D (7) A D E C B (6) B C A D E (5) A E D C B (5) E C B D A (4) E C B A D (4) D B C A E (4) D A B C E (4) B C D E A (4) E A D C B (3) B C D A E (3) A B C D E (3) E A C B D (2) D A E B C (2) C E B D A (2) C B E A D (2) B C E D A (2) A E B C D (2) A D B C E (2) A B C E D (2) E D C B A (1) E B C A D (1) D C B E A (1) C E B A D (1) C B E D A (1) B D C A E (1) A E D B C (1) Total count = 100 A B C D E A 0 -4 -4 26 30 B 4 0 20 10 0 C 4 -20 0 8 6 D -26 -10 -8 0 2 E -30 0 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.919503 C: 0.000000 D: 0.000000 E: 0.080497 Sum of squares = 0.851965773223 Cumulative probabilities = A: 0.000000 B: 0.919503 C: 0.919503 D: 0.919503 E: 1.000000 A B C D E A 0 -4 -4 26 30 B 4 0 20 10 0 C 4 -20 0 8 6 D -26 -10 -8 0 2 E -30 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.882353 C: 0.000000 D: 0.000000 E: 0.117647 Sum of squares = 0.792387733271 Cumulative probabilities = A: 0.000000 B: 0.882353 C: 0.882353 D: 0.882353 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=29 D=19 E=15 C=6 so C is eliminated. Round 2 votes counts: B=34 A=29 D=19 E=18 so E is eliminated. Round 3 votes counts: B=46 A=34 D=20 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:224 B:217 C:199 E:181 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 26 30 B 4 0 20 10 0 C 4 -20 0 8 6 D -26 -10 -8 0 2 E -30 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.882353 C: 0.000000 D: 0.000000 E: 0.117647 Sum of squares = 0.792387733271 Cumulative probabilities = A: 0.000000 B: 0.882353 C: 0.882353 D: 0.882353 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 26 30 B 4 0 20 10 0 C 4 -20 0 8 6 D -26 -10 -8 0 2 E -30 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.882353 C: 0.000000 D: 0.000000 E: 0.117647 Sum of squares = 0.792387733271 Cumulative probabilities = A: 0.000000 B: 0.882353 C: 0.882353 D: 0.882353 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 26 30 B 4 0 20 10 0 C 4 -20 0 8 6 D -26 -10 -8 0 2 E -30 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.882353 C: 0.000000 D: 0.000000 E: 0.117647 Sum of squares = 0.792387733271 Cumulative probabilities = A: 0.000000 B: 0.882353 C: 0.882353 D: 0.882353 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5352: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) E A D B C (6) E B C A D (5) B E A D C (5) B A D E C (5) B A D C E (5) E B A D C (4) C D A E B (4) B A E D C (4) A D B C E (4) E C B A D (3) D A C E B (3) D A C B E (3) C E B D A (3) C B D A E (3) B C D A E (3) E D A C B (2) D C A E B (2) C E D A B (2) C D E A B (2) B E C A D (2) B E A C D (2) B C A D E (2) E C B D A (1) E C A D B (1) E B A C D (1) E A D C B (1) D E A C B (1) D A B C E (1) C D B E A (1) C D B A E (1) C D A B E (1) B D A C E (1) B C E A D (1) A E D C B (1) A E D B C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 4 10 -8 B 4 0 6 0 -12 C -4 -6 0 -2 -16 D -10 0 2 0 -12 E 8 12 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 10 -8 B 4 0 6 0 -12 C -4 -6 0 -2 -16 D -10 0 2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=30 C=17 D=10 A=8 so A is eliminated. Round 2 votes counts: E=37 B=30 C=17 D=16 so D is eliminated. Round 3 votes counts: E=40 B=35 C=25 so C is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:201 B:199 D:190 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 10 -8 B 4 0 6 0 -12 C -4 -6 0 -2 -16 D -10 0 2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 10 -8 B 4 0 6 0 -12 C -4 -6 0 -2 -16 D -10 0 2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 10 -8 B 4 0 6 0 -12 C -4 -6 0 -2 -16 D -10 0 2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5353: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C A B E D (8) D E C B A (6) E A D C B (5) E D A C B (4) D B E A C (4) C B A D E (4) A E D B C (4) A C B E D (4) A B C E D (4) D E B A C (3) D E A B C (3) C B D E A (3) C B D A E (3) B C A D E (3) A E B C D (3) E D C A B (2) E C D A B (2) C D E B A (2) A E C B D (2) A B E D C (2) A B E C D (2) A B D E C (2) E A D B C (1) D E C A B (1) D E B C A (1) D C E B A (1) D B C E A (1) C E D B A (1) C E D A B (1) C E A D B (1) C D B E A (1) C B A E D (1) B D E A C (1) B D A E C (1) B A D C E (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 24 16 -2 -8 B -24 0 -6 -8 -8 C -16 6 0 -6 -18 D 2 8 6 0 -16 E 8 8 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 16 -2 -8 B -24 0 -6 -8 -8 C -16 6 0 -6 -18 D 2 8 6 0 -16 E 8 8 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 A=24 D=20 B=7 so B is eliminated. Round 2 votes counts: C=28 A=26 E=24 D=22 so D is eliminated. Round 3 votes counts: E=43 C=30 A=27 so A is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:215 D:200 C:183 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 16 -2 -8 B -24 0 -6 -8 -8 C -16 6 0 -6 -18 D 2 8 6 0 -16 E 8 8 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 16 -2 -8 B -24 0 -6 -8 -8 C -16 6 0 -6 -18 D 2 8 6 0 -16 E 8 8 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 16 -2 -8 B -24 0 -6 -8 -8 C -16 6 0 -6 -18 D 2 8 6 0 -16 E 8 8 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5354: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) C D E B A (8) B E C A D (7) B E A C D (7) A D B E C (7) D C E B A (6) A B E C D (6) D A C B E (5) D C E A B (4) B A E C D (4) A B D E C (4) E B C A D (3) A D B C E (3) A B E D C (3) E B C D A (2) D C A E B (2) D A E C B (2) D A C E B (2) D A B E C (2) C E D B A (2) C B E A D (2) A B C E D (2) E D B A C (1) E C B A D (1) D E B C A (1) D C A B E (1) D A E B C (1) D A B C E (1) C E B A D (1) C A D B E (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 2 4 -6 B 6 0 8 0 12 C -2 -8 0 8 -2 D -4 0 -8 0 2 E 6 -12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.658737 C: 0.000000 D: 0.341263 E: 0.000000 Sum of squares = 0.550394857478 Cumulative probabilities = A: 0.000000 B: 0.658737 C: 0.658737 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 -6 B 6 0 8 0 12 C -2 -8 0 8 -2 D -4 0 -8 0 2 E 6 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500449 C: 0.000000 D: 0.499551 E: 0.000000 Sum of squares = 0.500000403872 Cumulative probabilities = A: 0.000000 B: 0.500449 C: 0.500449 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 C=22 B=18 E=7 so E is eliminated. Round 2 votes counts: D=28 A=26 C=23 B=23 so C is eliminated. Round 3 votes counts: D=38 B=35 A=27 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:198 A:197 E:197 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 -6 B 6 0 8 0 12 C -2 -8 0 8 -2 D -4 0 -8 0 2 E 6 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500449 C: 0.000000 D: 0.499551 E: 0.000000 Sum of squares = 0.500000403872 Cumulative probabilities = A: 0.000000 B: 0.500449 C: 0.500449 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 -6 B 6 0 8 0 12 C -2 -8 0 8 -2 D -4 0 -8 0 2 E 6 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500449 C: 0.000000 D: 0.499551 E: 0.000000 Sum of squares = 0.500000403872 Cumulative probabilities = A: 0.000000 B: 0.500449 C: 0.500449 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 -6 B 6 0 8 0 12 C -2 -8 0 8 -2 D -4 0 -8 0 2 E 6 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500449 C: 0.000000 D: 0.499551 E: 0.000000 Sum of squares = 0.500000403872 Cumulative probabilities = A: 0.000000 B: 0.500449 C: 0.500449 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5355: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) E B D C A (7) E B A D C (5) E B A C D (5) D C A B E (5) B A E C D (5) B A C E D (5) E D C A B (4) E B D A C (4) D E C B A (4) A C B D E (4) E D C B A (3) E A B C D (3) D E C A B (3) C D A B E (3) B A C D E (3) D E B C A (2) D C A E B (2) C D A E B (2) C A D B E (2) B D C E A (2) A C D B E (2) A B E C D (2) A B C D E (2) E D A C B (1) D C E B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D B C E A (1) B E A D C (1) B E A C D (1) B D E C A (1) A E C D B (1) A E B C D (1) A C E D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -4 -10 -14 B 18 0 16 6 -14 C 4 -16 0 -12 -20 D 10 -6 12 0 -16 E 14 14 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -4 -10 -14 B 18 0 16 6 -14 C 4 -16 0 -12 -20 D 10 -6 12 0 -16 E 14 14 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=21 B=18 A=15 C=7 so C is eliminated. Round 2 votes counts: E=39 D=26 B=18 A=17 so A is eliminated. Round 3 votes counts: E=42 D=30 B=28 so B is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:232 B:213 D:200 C:178 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -4 -10 -14 B 18 0 16 6 -14 C 4 -16 0 -12 -20 D 10 -6 12 0 -16 E 14 14 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -10 -14 B 18 0 16 6 -14 C 4 -16 0 -12 -20 D 10 -6 12 0 -16 E 14 14 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -10 -14 B 18 0 16 6 -14 C 4 -16 0 -12 -20 D 10 -6 12 0 -16 E 14 14 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5356: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (21) E C D B A (11) C D E B A (10) D C B A E (7) E C A B D (6) A E B D C (6) E A B C D (5) E C D A B (4) C E D B A (4) B A D C E (3) A B E D C (3) E A B D C (2) D B A C E (2) D A B C E (2) C D B E A (2) C D B A E (2) A B D E C (2) E C B A D (1) E C A D B (1) E A D C B (1) E A C B D (1) D C E B A (1) D B C A E (1) B D C A E (1) B A C D E (1) Total count = 100 A B C D E A 0 8 -2 6 2 B -8 0 -2 4 -6 C 2 2 0 -4 14 D -6 -4 4 0 10 E -2 6 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 6 2 B -8 0 -2 4 -6 C 2 2 0 -4 14 D -6 -4 4 0 10 E -2 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888891 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=32 A=32 C=18 D=13 B=5 so B is eliminated. Round 2 votes counts: A=36 E=32 C=18 D=14 so D is eliminated. Round 3 votes counts: A=40 E=32 C=28 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 C:207 D:202 B:194 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 6 2 B -8 0 -2 4 -6 C 2 2 0 -4 14 D -6 -4 4 0 10 E -2 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888891 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 2 B -8 0 -2 4 -6 C 2 2 0 -4 14 D -6 -4 4 0 10 E -2 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888891 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 2 B -8 0 -2 4 -6 C 2 2 0 -4 14 D -6 -4 4 0 10 E -2 6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.38888888891 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5357: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) B C E A D (7) E D B A C (6) E B D A C (6) D A C E B (5) C A D E B (5) A D C E B (5) B E C D A (4) B E C A D (4) B E A C D (4) C D A E B (3) B E A D C (3) A C D B E (3) E D A B C (2) E C D B A (2) D E A C B (2) D C A E B (2) D A E C B (2) C D E A B (2) C B A D E (2) C A B D E (2) B C E D A (2) B C A E D (2) B C A D E (2) E D C B A (1) E C B D A (1) E B D C A (1) D E C A B (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D E A (1) B E D C A (1) B A C E D (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -18 6 -4 B 4 0 -8 -10 6 C 18 8 0 22 18 D -6 10 -22 0 4 E 4 -6 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 6 -4 B 4 0 -8 -10 6 C 18 8 0 22 18 D -6 10 -22 0 4 E 4 -6 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=29 E=19 D=12 A=10 so A is eliminated. Round 2 votes counts: C=32 B=31 E=19 D=18 so D is eliminated. Round 3 votes counts: C=44 B=31 E=25 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:233 B:196 D:193 A:190 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -18 6 -4 B 4 0 -8 -10 6 C 18 8 0 22 18 D -6 10 -22 0 4 E 4 -6 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 6 -4 B 4 0 -8 -10 6 C 18 8 0 22 18 D -6 10 -22 0 4 E 4 -6 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 6 -4 B 4 0 -8 -10 6 C 18 8 0 22 18 D -6 10 -22 0 4 E 4 -6 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5358: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) B A C D E (9) E D C B A (7) B C D A E (7) E D C A B (6) C D B E A (6) B C A D E (6) A E B D C (5) D C B E A (4) C D E B A (4) C B D E A (4) E C D B A (3) D C E B A (3) A E D C B (3) E D A C B (2) E A C D B (2) C E D B A (2) A E D B C (2) A B C E D (2) E C D A B (1) D E C B A (1) D E B A C (1) D B C A E (1) D A B E C (1) C E B D A (1) B A D C E (1) B A C E D (1) A D E B C (1) A D B E C (1) A C E B D (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -12 -8 -12 B 22 0 -18 -20 -8 C 12 18 0 2 8 D 8 20 -2 0 4 E 12 8 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -12 -8 -12 B 22 0 -18 -20 -8 C 12 18 0 2 8 D 8 20 -2 0 4 E 12 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=24 A=18 C=17 D=11 so D is eliminated. Round 2 votes counts: E=32 B=25 C=24 A=19 so A is eliminated. Round 3 votes counts: E=43 B=32 C=25 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:220 D:215 E:204 B:188 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -12 -8 -12 B 22 0 -18 -20 -8 C 12 18 0 2 8 D 8 20 -2 0 4 E 12 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -12 -8 -12 B 22 0 -18 -20 -8 C 12 18 0 2 8 D 8 20 -2 0 4 E 12 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -12 -8 -12 B 22 0 -18 -20 -8 C 12 18 0 2 8 D 8 20 -2 0 4 E 12 8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5359: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (16) A E B C D (11) E B C A D (8) E B C D A (7) E A B C D (7) D C B E A (6) C B E D A (5) A D C B E (5) E B A C D (4) D E C B A (3) D C A B E (3) A D E B C (3) D E B C A (2) D C B A E (2) D A C E B (2) B C E A D (2) A E B D C (2) E D B C A (1) E D A B C (1) E C B D A (1) E A B D C (1) D E A B C (1) D C E B A (1) D A E C B (1) D A E B C (1) C E B D A (1) A D B E C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 16 -8 -2 B -14 0 6 0 -16 C -16 -6 0 -4 -10 D 8 0 4 0 -4 E 2 16 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 16 -8 -2 B -14 0 6 0 -16 C -16 -6 0 -4 -10 D 8 0 4 0 -4 E 2 16 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=30 A=24 C=6 B=2 so B is eliminated. Round 2 votes counts: D=38 E=30 A=24 C=8 so C is eliminated. Round 3 votes counts: E=38 D=38 A=24 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:210 D:204 B:188 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 16 -8 -2 B -14 0 6 0 -16 C -16 -6 0 -4 -10 D 8 0 4 0 -4 E 2 16 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 -8 -2 B -14 0 6 0 -16 C -16 -6 0 -4 -10 D 8 0 4 0 -4 E 2 16 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 -8 -2 B -14 0 6 0 -16 C -16 -6 0 -4 -10 D 8 0 4 0 -4 E 2 16 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5360: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) D A B C E (8) A B E D C (7) B A E C D (5) A D B E C (5) E C B A D (4) D C E B A (4) D B A C E (4) D A C B E (4) D A B E C (4) C E D B A (4) B A D E C (4) C E D A B (3) C E B A D (3) A B D E C (3) E B C A D (2) E A C B D (2) D C E A B (2) D C B E A (2) D C A E B (2) D A E C B (2) C D E B A (2) C B E A D (2) B E A C D (2) E C A B D (1) E B A C D (1) D A C E B (1) C D E A B (1) C B E D A (1) C B D E A (1) B E C A D (1) B C A D E (1) B A D C E (1) A D E B C (1) Total count = 100 A B C D E A 0 -8 8 -10 4 B 8 0 -2 2 10 C -8 2 0 -8 12 D 10 -2 8 0 4 E -4 -10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000000167 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -10 4 B 8 0 -2 2 10 C -8 2 0 -8 12 D 10 -2 8 0 4 E -4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000001091 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=27 A=16 B=14 E=10 so E is eliminated. Round 2 votes counts: D=33 C=32 A=18 B=17 so B is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:209 C:199 A:197 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -10 4 B 8 0 -2 2 10 C -8 2 0 -8 12 D 10 -2 8 0 4 E -4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000001091 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -10 4 B 8 0 -2 2 10 C -8 2 0 -8 12 D 10 -2 8 0 4 E -4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000001091 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -10 4 B 8 0 -2 2 10 C -8 2 0 -8 12 D 10 -2 8 0 4 E -4 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000001091 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5361: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) B C D E A (8) C D B A E (7) A D E C B (7) D C A B E (5) C D A E B (5) B D C A E (5) B C D A E (5) A E D B C (4) E B A C D (3) E A D B C (3) D C A E B (3) D A C E B (3) B E A D C (3) A E D C B (3) E B A D C (2) D C B A E (2) C D E A B (2) C D B E A (2) C B D E A (2) B E A C D (2) B A D E C (2) A E B D C (2) E A B C D (1) D B A C E (1) C E B D A (1) C E B A D (1) C B D A E (1) C A D E B (1) B E C A D (1) B D A E C (1) B D A C E (1) B C E D A (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -4 -10 18 B 2 0 10 2 -2 C 4 -10 0 -14 12 D 10 -2 14 0 28 E -18 2 -12 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.062500 E: 0.062500 Sum of squares = 0.773437499933 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.937500 E: 1.000000 A B C D E A 0 -2 -4 -10 18 B 2 0 10 2 -2 C 4 -10 0 -14 12 D 10 -2 14 0 28 E -18 2 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.062500 E: 0.062500 Sum of squares = 0.773437499463 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.937500 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 E=18 A=17 D=14 so D is eliminated. Round 2 votes counts: C=32 B=30 A=20 E=18 so E is eliminated. Round 3 votes counts: B=35 A=33 C=32 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:225 B:206 A:201 C:196 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -10 18 B 2 0 10 2 -2 C 4 -10 0 -14 12 D 10 -2 14 0 28 E -18 2 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.062500 E: 0.062500 Sum of squares = 0.773437499463 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.937500 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -10 18 B 2 0 10 2 -2 C 4 -10 0 -14 12 D 10 -2 14 0 28 E -18 2 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.062500 E: 0.062500 Sum of squares = 0.773437499463 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.937500 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -10 18 B 2 0 10 2 -2 C 4 -10 0 -14 12 D 10 -2 14 0 28 E -18 2 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.062500 E: 0.062500 Sum of squares = 0.773437499463 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.937500 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5362: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) D E C A B (6) A D E B C (6) A B D E C (6) E D A B C (5) D E A B C (5) B C E D A (5) E D C B A (4) C A B D E (4) A D E C B (4) D E A C B (3) C B E D A (3) B A E D C (3) A D C E B (3) A B C D E (3) E D C A B (2) E D B C A (2) E B D A C (2) C E D B A (2) C D E A B (2) C D A E B (2) C B A D E (2) B C E A D (2) B A C E D (2) A D B E C (2) A C D E B (2) A C B D E (2) E D B A C (1) E C D B A (1) E B D C A (1) D C A E B (1) D A E C B (1) C E B D A (1) B E A D C (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 20 6 2 4 B -20 0 -4 -8 -12 C -6 4 0 -18 -12 D -2 8 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 2 4 B -20 0 -4 -8 -12 C -6 4 0 -18 -12 D -2 8 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 E=18 D=16 B=14 so B is eliminated. Round 2 votes counts: A=35 C=30 E=19 D=16 so D is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:216 E:205 C:184 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 2 4 B -20 0 -4 -8 -12 C -6 4 0 -18 -12 D -2 8 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 2 4 B -20 0 -4 -8 -12 C -6 4 0 -18 -12 D -2 8 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 2 4 B -20 0 -4 -8 -12 C -6 4 0 -18 -12 D -2 8 18 0 10 E -4 12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5363: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) E D C A B (7) B C D A E (7) A E C D B (7) E A D C B (6) D C E A B (6) A B E C D (6) B A C E D (5) D E C A B (4) C D B E A (4) B D C E A (4) B A C D E (4) A E B C D (4) B A E D C (3) A E B D C (3) E D A C B (2) B C D E A (2) B A D C E (2) A E D C B (2) A B E D C (2) E C D A B (1) E A D B C (1) D E C B A (1) D B C E A (1) C D E B A (1) C D E A B (1) B D C A E (1) B D A C E (1) B C A D E (1) B A E C D (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -2 2 B -8 0 -2 -4 -10 C -2 2 0 -8 -2 D 2 4 8 0 -4 E -2 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.37500000004 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 8 2 -2 2 B -8 0 -2 -4 -10 C -2 2 0 -8 -2 D 2 4 8 0 -4 E -2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=26 D=20 E=17 C=6 so C is eliminated. Round 2 votes counts: B=31 D=26 A=26 E=17 so E is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:207 A:205 D:205 C:195 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 -2 2 B -8 0 -2 -4 -10 C -2 2 0 -8 -2 D 2 4 8 0 -4 E -2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 2 B -8 0 -2 -4 -10 C -2 2 0 -8 -2 D 2 4 8 0 -4 E -2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 2 B -8 0 -2 -4 -10 C -2 2 0 -8 -2 D 2 4 8 0 -4 E -2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5364: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (13) D C E B A (10) E B A C D (9) D C A B E (6) C E D B A (6) C D E B A (6) A B E C D (6) D A C B E (5) D C A E B (4) A B D C E (4) E C B D A (3) E B C A D (3) D C E A B (3) B E A C D (3) A D B C E (3) A B E D C (3) A B D E C (3) E C D B A (2) E B C D A (2) C D E A B (2) C E D A B (1) B E D A C (1) B E A D C (1) B A E D C (1) Total count = 100 A B C D E A 0 -20 4 -2 -4 B 20 0 4 4 -2 C -4 -4 0 12 0 D 2 -4 -12 0 -8 E 4 2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100309 D: 0.000000 E: 0.899691 Sum of squares = 0.819505287742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100309 D: 0.100309 E: 1.000000 A B C D E A 0 -20 4 -2 -4 B 20 0 4 4 -2 C -4 -4 0 12 0 D 2 -4 -12 0 -8 E 4 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555567812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=19 B=19 A=19 C=15 so C is eliminated. Round 2 votes counts: D=36 E=26 B=19 A=19 so B is eliminated. Round 3 votes counts: D=36 A=33 E=31 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:213 E:207 C:202 A:189 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 4 -2 -4 B 20 0 4 4 -2 C -4 -4 0 12 0 D 2 -4 -12 0 -8 E 4 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555567812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 4 -2 -4 B 20 0 4 4 -2 C -4 -4 0 12 0 D 2 -4 -12 0 -8 E 4 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555567812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 4 -2 -4 B 20 0 4 4 -2 C -4 -4 0 12 0 D 2 -4 -12 0 -8 E 4 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555567812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5365: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) D A B C E (8) A D C E B (8) B E D C A (6) E B C A D (5) E C B A D (4) B E D A C (4) D A C B E (3) D A B E C (3) C E B A D (3) C B E A D (3) C A D B E (3) B E C A D (3) B C E D A (3) A E D C B (3) A D E C B (3) E B A D C (2) D A E C B (2) D A E B C (2) C A E D B (2) C A E B D (2) E D A B C (1) E C A B D (1) E B D A C (1) E A D B C (1) E A C D B (1) D B A E C (1) D B A C E (1) D A C E B (1) C E A B D (1) C D A B E (1) C B D A E (1) C A D E B (1) B D E C A (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 -6 -2 B 4 0 12 8 10 C 2 -12 0 -10 -12 D 6 -8 10 0 -12 E 2 -10 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -6 -2 B 4 0 12 8 10 C 2 -12 0 -10 -12 D 6 -8 10 0 -12 E 2 -10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=21 C=17 E=16 A=15 so A is eliminated. Round 2 votes counts: D=32 B=31 E=19 C=18 so C is eliminated. Round 3 votes counts: D=38 B=35 E=27 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:208 D:198 A:193 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -6 -2 B 4 0 12 8 10 C 2 -12 0 -10 -12 D 6 -8 10 0 -12 E 2 -10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -6 -2 B 4 0 12 8 10 C 2 -12 0 -10 -12 D 6 -8 10 0 -12 E 2 -10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -6 -2 B 4 0 12 8 10 C 2 -12 0 -10 -12 D 6 -8 10 0 -12 E 2 -10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5366: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) E B C A D (6) D C A B E (6) E D A B C (5) D A C B E (5) C B D A E (5) B C D A E (5) A D E C B (5) A D C B E (5) E A D C B (4) B C E D A (4) B C D E A (4) A E D C B (4) E A D B C (3) A D C E B (3) A C D B E (3) E B D A C (2) E B A C D (2) E A C B D (2) E A B C D (2) C D B A E (2) B E C D A (2) B C E A D (2) A E C D B (2) E D A C B (1) E B D C A (1) E A C D B (1) E A B D C (1) D E B C A (1) D C B A E (1) D B C A E (1) B E D C A (1) B D C E A (1) Total count = 100 A B C D E A 0 4 0 -10 -6 B -4 0 2 -4 -6 C 0 -2 0 0 -6 D 10 4 0 0 -6 E 6 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 -10 -6 B -4 0 2 -4 -6 C 0 -2 0 0 -6 D 10 4 0 0 -6 E 6 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=22 B=19 D=14 C=7 so C is eliminated. Round 2 votes counts: E=38 B=24 A=22 D=16 so D is eliminated. Round 3 votes counts: E=39 A=33 B=28 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:204 C:196 A:194 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -10 -6 B -4 0 2 -4 -6 C 0 -2 0 0 -6 D 10 4 0 0 -6 E 6 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -10 -6 B -4 0 2 -4 -6 C 0 -2 0 0 -6 D 10 4 0 0 -6 E 6 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -10 -6 B -4 0 2 -4 -6 C 0 -2 0 0 -6 D 10 4 0 0 -6 E 6 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5367: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (13) D B E A C (13) A C E B D (12) C A E B D (8) E B D C A (7) A C D E B (7) E B D A C (6) C A D B E (6) B E D C A (3) D E B A C (2) D B C E A (2) D A C B E (2) B D E C A (2) A D C B E (2) A C D B E (2) E D B A C (1) E B C D A (1) E A C B D (1) D C B E A (1) D C B A E (1) D B A E C (1) C E B A D (1) C E A B D (1) C A B E D (1) C A B D E (1) A E C B D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 4 -10 -8 B 8 0 2 -10 0 C -4 -2 0 -14 -2 D 10 10 14 0 12 E 8 0 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -10 -8 B 8 0 2 -10 0 C -4 -2 0 -14 -2 D 10 10 14 0 12 E 8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=26 C=18 E=16 B=5 so B is eliminated. Round 2 votes counts: D=37 A=26 E=19 C=18 so C is eliminated. Round 3 votes counts: A=42 D=37 E=21 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:200 E:199 A:189 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 4 -10 -8 B 8 0 2 -10 0 C -4 -2 0 -14 -2 D 10 10 14 0 12 E 8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -10 -8 B 8 0 2 -10 0 C -4 -2 0 -14 -2 D 10 10 14 0 12 E 8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -10 -8 B 8 0 2 -10 0 C -4 -2 0 -14 -2 D 10 10 14 0 12 E 8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5368: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) B C E D A (8) D E A B C (7) A D E C B (6) E D A C B (5) D A E B C (5) E D A B C (4) D A E C B (4) C B A D E (4) B E C D A (4) A D C E B (4) E C B D A (3) E B D C A (3) C B E A D (3) B C E A D (3) A C D B E (3) A C B D E (3) E B C D A (2) C B A E D (2) C A B D E (2) B C A D E (2) A C D E B (2) E D B C A (1) D E A C B (1) C E A D B (1) C E A B D (1) C A E D B (1) C A B E D (1) B E D C A (1) A E D C B (1) A D E B C (1) A D C B E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 16 -14 -12 B -10 0 4 -14 -22 C -16 -4 0 -8 -14 D 14 14 8 0 -6 E 12 22 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 16 -14 -12 B -10 0 4 -14 -22 C -16 -4 0 -8 -14 D 14 14 8 0 -6 E 12 22 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=23 B=18 D=17 C=15 so C is eliminated. Round 2 votes counts: E=29 B=27 A=27 D=17 so D is eliminated. Round 3 votes counts: E=37 A=36 B=27 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:215 A:200 B:179 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 16 -14 -12 B -10 0 4 -14 -22 C -16 -4 0 -8 -14 D 14 14 8 0 -6 E 12 22 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 -14 -12 B -10 0 4 -14 -22 C -16 -4 0 -8 -14 D 14 14 8 0 -6 E 12 22 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 -14 -12 B -10 0 4 -14 -22 C -16 -4 0 -8 -14 D 14 14 8 0 -6 E 12 22 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5369: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) B E A C D (8) D C B E A (7) D C A E B (6) D C A B E (6) A C D E B (6) E B C D A (5) B E D C A (5) A B E D C (5) B E A D C (4) A E B C D (4) C E B D A (3) C D A E B (3) B E D A C (3) B E C D A (3) D C E B A (2) D B E C A (2) D B C E A (2) D A C E B (2) D A C B E (2) C D E B A (2) A D C E B (2) A D B E C (2) A C E B D (2) E B C A D (1) E A B C D (1) C E A B D (1) B D E C A (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 2 -8 -16 B 12 0 8 10 0 C -2 -8 0 -2 -6 D 8 -10 2 0 -10 E 16 0 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.467802 C: 0.000000 D: 0.000000 E: 0.532198 Sum of squares = 0.502073461661 Cumulative probabilities = A: 0.000000 B: 0.467802 C: 0.467802 D: 0.467802 E: 1.000000 A B C D E A 0 -12 2 -8 -16 B 12 0 8 10 0 C -2 -8 0 -2 -6 D 8 -10 2 0 -10 E 16 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=24 A=23 E=15 C=9 so C is eliminated. Round 2 votes counts: D=34 B=24 A=23 E=19 so E is eliminated. Round 3 votes counts: B=41 D=34 A=25 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:215 D:195 C:191 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 -8 -16 B 12 0 8 10 0 C -2 -8 0 -2 -6 D 8 -10 2 0 -10 E 16 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -8 -16 B 12 0 8 10 0 C -2 -8 0 -2 -6 D 8 -10 2 0 -10 E 16 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -8 -16 B 12 0 8 10 0 C -2 -8 0 -2 -6 D 8 -10 2 0 -10 E 16 0 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5370: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) D B C E A (9) C E D B A (9) A B D E C (7) A B D C E (7) E C A D B (5) E C D B A (4) C D B E A (4) A E C B D (4) D B E A C (3) C E A D B (3) B D A E C (3) E C D A B (2) D B A E C (2) A E D B C (2) A E B D C (2) A C E B D (2) E D C B A (1) E D B C A (1) E C A B D (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) C D E B A (1) C B D E A (1) C A B D E (1) B D C A E (1) B C D A E (1) B A D C E (1) B A C D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 2 -16 -4 B 16 0 14 -8 14 C -2 -14 0 -14 12 D 16 8 14 0 18 E 4 -14 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 2 -16 -4 B 16 0 14 -8 14 C -2 -14 0 -14 12 D 16 8 14 0 18 E 4 -14 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=19 C=19 E=18 B=18 so E is eliminated. Round 2 votes counts: C=31 A=30 D=21 B=18 so B is eliminated. Round 3 votes counts: D=36 C=32 A=32 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:218 C:191 A:183 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 2 -16 -4 B 16 0 14 -8 14 C -2 -14 0 -14 12 D 16 8 14 0 18 E 4 -14 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 -16 -4 B 16 0 14 -8 14 C -2 -14 0 -14 12 D 16 8 14 0 18 E 4 -14 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 -16 -4 B 16 0 14 -8 14 C -2 -14 0 -14 12 D 16 8 14 0 18 E 4 -14 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5371: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C E A B D (9) D C B A E (8) B D A E C (7) B A D E C (7) E C A B D (5) C D A B E (5) D A B C E (4) D B A C E (3) C E D B A (3) C A E D B (3) C A E B D (3) E B D A C (2) E A C B D (2) E A B C D (2) D C E B A (2) D C B E A (2) D B E A C (2) C E D A B (2) C E A D B (2) E C B A D (1) E B C A D (1) D E C B A (1) D E B C A (1) D C A B E (1) D B C E A (1) C D E B A (1) C D E A B (1) C D A E B (1) B E D A C (1) A E C B D (1) A E B C D (1) A D B C E (1) A C D B E (1) A C B E D (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 -14 18 B 4 0 -10 -8 12 C 6 10 0 -8 10 D 14 8 8 0 18 E -18 -12 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -14 18 B 4 0 -10 -8 12 C 6 10 0 -8 10 D 14 8 8 0 18 E -18 -12 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=30 B=15 E=13 A=8 so A is eliminated. Round 2 votes counts: D=35 C=32 B=18 E=15 so E is eliminated. Round 3 votes counts: C=41 D=35 B=24 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:209 B:199 A:197 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -14 18 B 4 0 -10 -8 12 C 6 10 0 -8 10 D 14 8 8 0 18 E -18 -12 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -14 18 B 4 0 -10 -8 12 C 6 10 0 -8 10 D 14 8 8 0 18 E -18 -12 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -14 18 B 4 0 -10 -8 12 C 6 10 0 -8 10 D 14 8 8 0 18 E -18 -12 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5372: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) D E A B C (8) A D C E B (6) E D B A C (5) E B D C A (5) D A E C B (5) C B A E D (5) E D A B C (4) C A B D E (4) B E C D A (4) A C D E B (4) E B C A D (3) D B E C A (3) B C E A D (3) A C D B E (3) E B A C D (2) D A E B C (2) D A C E B (2) C A B E D (2) B E D C A (2) B E C A D (2) B C E D A (2) A E D C B (2) A D E C B (2) A C E B D (2) E B D A C (1) E B C D A (1) E B A D C (1) D B C A E (1) C A D B E (1) B D E C A (1) B C D E A (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 20 -12 -14 B 2 0 20 -16 -32 C -20 -20 0 -18 -26 D 12 16 18 0 4 E 14 32 26 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 20 -12 -14 B 2 0 20 -16 -32 C -20 -20 0 -18 -26 D 12 16 18 0 4 E 14 32 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=22 A=21 B=15 C=12 so C is eliminated. Round 2 votes counts: D=30 A=28 E=22 B=20 so B is eliminated. Round 3 votes counts: E=35 A=33 D=32 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:234 D:225 A:196 B:187 C:158 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 20 -12 -14 B 2 0 20 -16 -32 C -20 -20 0 -18 -26 D 12 16 18 0 4 E 14 32 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 20 -12 -14 B 2 0 20 -16 -32 C -20 -20 0 -18 -26 D 12 16 18 0 4 E 14 32 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 20 -12 -14 B 2 0 20 -16 -32 C -20 -20 0 -18 -26 D 12 16 18 0 4 E 14 32 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5373: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) C A B E D (5) A D C B E (5) E B D A C (4) B A C E D (4) A B D C E (4) D A C B E (3) C A D B E (3) C A B D E (3) B E A D C (3) B A E D C (3) B A D E C (3) A B C D E (3) E D C B A (2) E C B D A (2) E B D C A (2) E B C D A (2) E B C A D (2) D E B A C (2) D A B C E (2) C E D A B (2) C D E A B (2) C D A E B (2) C B A E D (2) B C A E D (2) A D B E C (2) A B D E C (2) E D B C A (1) E C D B A (1) E C B A D (1) E B A D C (1) D E C A B (1) D E A B C (1) D C E A B (1) D B A E C (1) D A E C B (1) D A E B C (1) C E D B A (1) C E B D A (1) C E A D B (1) C B E A D (1) C A E B D (1) C A D E B (1) B E C A D (1) B E A C D (1) B A E C D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 14 30 28 B -12 0 -2 10 28 C -14 2 0 8 18 D -30 -10 -8 0 6 E -28 -28 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 30 28 B -12 0 -2 10 28 C -14 2 0 8 18 D -30 -10 -8 0 6 E -28 -28 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 E=18 B=18 D=13 so D is eliminated. Round 2 votes counts: A=33 C=26 E=22 B=19 so B is eliminated. Round 3 votes counts: A=45 C=28 E=27 so E is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:242 B:212 C:207 D:179 E:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 30 28 B -12 0 -2 10 28 C -14 2 0 8 18 D -30 -10 -8 0 6 E -28 -28 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 30 28 B -12 0 -2 10 28 C -14 2 0 8 18 D -30 -10 -8 0 6 E -28 -28 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 30 28 B -12 0 -2 10 28 C -14 2 0 8 18 D -30 -10 -8 0 6 E -28 -28 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5374: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) C D A E B (11) E B C D A (7) B A E D C (6) E C D B A (5) A D C B E (5) A D B C E (5) E B C A D (4) C E D B A (4) B E A C D (4) E C B D A (3) E B A D C (3) D C A B E (3) D A C B E (3) B E A D C (3) B A D E C (3) A B D C E (3) D C A E B (2) D A C E B (2) C D E B A (2) A B D E C (2) E C D A B (1) E B A C D (1) E A D B C (1) E A B D C (1) C E D A B (1) C D A B E (1) C B E D A (1) B A D C E (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -12 -14 -4 B -6 0 -10 -16 -18 C 12 10 0 12 10 D 14 16 -12 0 10 E 4 18 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -14 -4 B -6 0 -10 -16 -18 C 12 10 0 12 10 D 14 16 -12 0 10 E 4 18 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 B=17 A=16 D=10 so D is eliminated. Round 2 votes counts: C=36 E=26 A=21 B=17 so B is eliminated. Round 3 votes counts: C=36 E=33 A=31 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:214 E:201 A:188 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -14 -4 B -6 0 -10 -16 -18 C 12 10 0 12 10 D 14 16 -12 0 10 E 4 18 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -14 -4 B -6 0 -10 -16 -18 C 12 10 0 12 10 D 14 16 -12 0 10 E 4 18 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -14 -4 B -6 0 -10 -16 -18 C 12 10 0 12 10 D 14 16 -12 0 10 E 4 18 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5375: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) E C D A B (8) D B E A C (8) C E A D B (8) E D C B A (7) B A D C E (7) A C B E D (5) A B C E D (5) E C D B A (4) B D A E C (4) A B C D E (4) D E C B A (3) D E B C A (3) C E D A B (3) C E A B D (3) B A D E C (3) D B E C A (2) D B A E C (2) A B D C E (2) E C A D B (1) E C A B D (1) D E B A C (1) D B C A E (1) C E D B A (1) C A B E D (1) B D A C E (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -12 4 -6 B -6 0 -12 -4 -8 C 12 12 0 12 2 D -4 4 -12 0 -18 E 6 8 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 4 -6 B -6 0 -12 -4 -8 C 12 12 0 12 2 D -4 4 -12 0 -18 E 6 8 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=21 D=20 A=18 B=15 so B is eliminated. Round 2 votes counts: A=28 C=26 D=25 E=21 so E is eliminated. Round 3 votes counts: C=40 D=32 A=28 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:215 A:196 B:185 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 4 -6 B -6 0 -12 -4 -8 C 12 12 0 12 2 D -4 4 -12 0 -18 E 6 8 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 4 -6 B -6 0 -12 -4 -8 C 12 12 0 12 2 D -4 4 -12 0 -18 E 6 8 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 4 -6 B -6 0 -12 -4 -8 C 12 12 0 12 2 D -4 4 -12 0 -18 E 6 8 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5376: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A E B C D (8) E A C B D (7) D C B E A (7) C D E A B (7) A B E D C (7) C E A D B (5) A E C B D (4) D B A C E (3) B D A E C (3) B A E D C (3) E C A B D (2) E B C A D (2) D C B A E (2) D B C E A (2) D B C A E (2) C E D A B (2) C A D E B (2) B E A C D (2) A D C E B (2) A B E C D (2) A B D E C (2) E C B A D (1) E C A D B (1) E B A C D (1) E A B C D (1) D C A B E (1) C D A E B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D A C E (1) B A D E C (1) A E C D B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 14 2 14 -6 B -14 0 -14 0 -18 C -2 14 0 20 -4 D -14 0 -20 0 -4 E 6 18 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 14 -6 B -14 0 -14 0 -18 C -2 14 0 20 -4 D -14 0 -20 0 -4 E 6 18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=27 D=17 E=15 B=13 so B is eliminated. Round 2 votes counts: A=32 C=27 D=22 E=19 so E is eliminated. Round 3 votes counts: A=44 C=33 D=23 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:216 C:214 A:212 D:181 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 14 -6 B -14 0 -14 0 -18 C -2 14 0 20 -4 D -14 0 -20 0 -4 E 6 18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 14 -6 B -14 0 -14 0 -18 C -2 14 0 20 -4 D -14 0 -20 0 -4 E 6 18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 14 -6 B -14 0 -14 0 -18 C -2 14 0 20 -4 D -14 0 -20 0 -4 E 6 18 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5377: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) B E A C D (7) B A E D C (7) A B D C E (7) A D C B E (6) E C D A B (5) D C A E B (4) C D A E B (4) B E A D C (4) B A D C E (4) E C D B A (3) E B C D A (3) E B C A D (3) A D B C E (3) A B D E C (3) E C B D A (2) E B A D C (2) E B A C D (2) E A C D B (2) D C A B E (2) D A C B E (2) C E D A B (2) C D A B E (2) B A D E C (2) E A B D C (1) C E D B A (1) C B E D A (1) C B D A E (1) B E C D A (1) B E C A D (1) B D C A E (1) A D C E B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 10 10 14 2 B -10 0 6 6 12 C -10 -6 0 -2 0 D -14 -6 2 0 4 E -2 -12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 14 2 B -10 0 6 6 12 C -10 -6 0 -2 0 D -14 -6 2 0 4 E -2 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=23 A=22 C=20 D=8 so D is eliminated. Round 2 votes counts: B=27 C=26 A=24 E=23 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:218 B:207 D:193 C:191 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 14 2 B -10 0 6 6 12 C -10 -6 0 -2 0 D -14 -6 2 0 4 E -2 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 14 2 B -10 0 6 6 12 C -10 -6 0 -2 0 D -14 -6 2 0 4 E -2 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 14 2 B -10 0 6 6 12 C -10 -6 0 -2 0 D -14 -6 2 0 4 E -2 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5378: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (13) A D E B C (11) E B C D A (10) D A E B C (5) D A B E C (5) E B D A C (4) C E B A D (4) B E C D A (4) A D C E B (4) A D C B E (4) E C B D A (3) C A D B E (3) E C B A D (2) E B C A D (2) E B A D C (2) E A D B C (2) D A C B E (2) D A B C E (2) C B E A D (2) C B D A E (2) B E D A C (2) A D E C B (2) E C A D B (1) E B D C A (1) C B D E A (1) C B A D E (1) C A E B D (1) B D A E C (1) B C E D A (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -2 -12 -8 B 10 0 8 12 -8 C 2 -8 0 2 -16 D 12 -12 -2 0 -8 E 8 8 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -2 -12 -8 B 10 0 8 12 -8 C 2 -8 0 2 -16 D 12 -12 -2 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=27 C=27 A=24 D=14 B=8 so B is eliminated. Round 2 votes counts: E=33 C=28 A=24 D=15 so D is eliminated. Round 3 votes counts: A=39 E=33 C=28 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:211 D:195 C:190 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -2 -12 -8 B 10 0 8 12 -8 C 2 -8 0 2 -16 D 12 -12 -2 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -12 -8 B 10 0 8 12 -8 C 2 -8 0 2 -16 D 12 -12 -2 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -12 -8 B 10 0 8 12 -8 C 2 -8 0 2 -16 D 12 -12 -2 0 -8 E 8 8 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5379: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (13) D C E B A (12) A B E D C (11) E A C B D (8) C E D A B (8) B A D E C (7) A E B C D (7) E C A D B (6) C D E B A (6) E C D A B (4) D B C E A (4) A B E C D (3) E C A B D (2) D B C A E (2) D B A C E (2) D C B E A (1) C D E A B (1) C D A E B (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 -8 -4 B -6 0 0 6 -12 C -6 0 0 -6 0 D 8 -6 6 0 0 E 4 12 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.472181 E: 0.527819 Sum of squares = 0.501547742314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.472181 E: 1.000000 A B C D E A 0 6 6 -8 -4 B -6 0 0 6 -12 C -6 0 0 -6 0 D 8 -6 6 0 0 E 4 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=21 E=20 B=20 C=16 so C is eliminated. Round 2 votes counts: D=29 E=28 A=23 B=20 so B is eliminated. Round 3 votes counts: D=42 A=30 E=28 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:208 D:204 A:200 B:194 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -8 -4 B -6 0 0 6 -12 C -6 0 0 -6 0 D 8 -6 6 0 0 E 4 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -8 -4 B -6 0 0 6 -12 C -6 0 0 -6 0 D 8 -6 6 0 0 E 4 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -8 -4 B -6 0 0 6 -12 C -6 0 0 -6 0 D 8 -6 6 0 0 E 4 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5380: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) D A E C B (7) C B E A D (5) C A B E D (5) D E A B C (4) C B A E D (4) B E D A C (4) A C D B E (4) D E B A C (3) C E B D A (3) C E B A D (3) B E C A D (3) A D B E C (3) A C B D E (3) A B D C E (3) E C B D A (2) D E C B A (2) D E B C A (2) D A E B C (2) D A C E B (2) D A B E C (2) C A D E B (2) B C E A D (2) B C A E D (2) B A D E C (2) A D B C E (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E C D B A (1) D E C A B (1) D B A E C (1) C E D A B (1) C A E D B (1) B E D C A (1) B E A C D (1) B D E A C (1) B A D C E (1) B A C E D (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 2 4 4 B 6 0 4 14 10 C -2 -4 0 8 -2 D -4 -14 -8 0 0 E -4 -10 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 4 B 6 0 4 14 10 C -2 -4 0 8 -2 D -4 -14 -8 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998368 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=24 A=20 B=18 E=12 so E is eliminated. Round 2 votes counts: D=28 C=27 B=25 A=20 so A is eliminated. Round 3 votes counts: D=34 C=34 B=32 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:217 A:202 C:200 E:194 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 4 B 6 0 4 14 10 C -2 -4 0 8 -2 D -4 -14 -8 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998368 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 4 B 6 0 4 14 10 C -2 -4 0 8 -2 D -4 -14 -8 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998368 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 4 B 6 0 4 14 10 C -2 -4 0 8 -2 D -4 -14 -8 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998368 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5381: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) E A C B D (6) D B C E A (5) E A C D B (4) D E B A C (4) B D C A E (4) E D A C B (3) E A B C D (3) D E A C B (3) D C B A E (3) C B A D E (3) C A E B D (3) B C A E D (3) A C E B D (3) A C B E D (3) A B E C D (3) A B C E D (3) E A D B C (2) D E C A B (2) D C E A B (2) C D B A E (2) C D A E B (2) C A B E D (2) C A B D E (2) B E A D C (2) B C A D E (2) B A C E D (2) E D A B C (1) D E C B A (1) D E B C A (1) D C E B A (1) D C A B E (1) D B E C A (1) C D A B E (1) C A E D B (1) C A D B E (1) B E D A C (1) B D E A C (1) B D A E C (1) B C D A E (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -8 -2 14 B -4 0 0 -2 14 C 8 0 0 2 20 D 2 2 -2 0 8 E -14 -14 -20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.287908 C: 0.712092 D: 0.000000 E: 0.000000 Sum of squares = 0.589965986366 Cumulative probabilities = A: 0.000000 B: 0.287908 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -2 14 B -4 0 0 -2 14 C 8 0 0 2 20 D 2 2 -2 0 8 E -14 -14 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499996 C: 0.500004 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000027 Cumulative probabilities = A: 0.000000 B: 0.499996 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=19 C=17 B=17 A=13 so A is eliminated. Round 2 votes counts: D=34 C=23 B=23 E=20 so E is eliminated. Round 3 votes counts: D=40 C=34 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:205 A:204 B:204 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -2 14 B -4 0 0 -2 14 C 8 0 0 2 20 D 2 2 -2 0 8 E -14 -14 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499996 C: 0.500004 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000027 Cumulative probabilities = A: 0.000000 B: 0.499996 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -2 14 B -4 0 0 -2 14 C 8 0 0 2 20 D 2 2 -2 0 8 E -14 -14 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499996 C: 0.500004 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000027 Cumulative probabilities = A: 0.000000 B: 0.499996 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -2 14 B -4 0 0 -2 14 C 8 0 0 2 20 D 2 2 -2 0 8 E -14 -14 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499996 C: 0.500004 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000027 Cumulative probabilities = A: 0.000000 B: 0.499996 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5382: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) D B A C E (8) A B D C E (8) E C A B D (5) D C A B E (5) C E D A B (5) C D E A B (5) E C D B A (4) E B A C D (4) E C B A D (3) E B A D C (3) D C B A E (3) C A D B E (3) A C B D E (3) A B C D E (3) E A B C D (2) D C E B A (2) C E A B D (2) C A B D E (2) B A E D C (2) E D C B A (1) E D B A C (1) E B D A C (1) E B C D A (1) E B C A D (1) D E C B A (1) D B E A C (1) D B A E C (1) C E D B A (1) C D E B A (1) C A E B D (1) B E D A C (1) B A D E C (1) A E B C D (1) A D C B E (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 18 -12 -4 10 B -18 0 -16 -6 10 C 12 16 0 18 28 D 4 6 -18 0 16 E -10 -10 -28 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 -4 10 B -18 0 -16 -6 10 C 12 16 0 18 28 D 4 6 -18 0 16 E -10 -10 -28 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=26 D=21 A=19 B=4 so B is eliminated. Round 2 votes counts: C=30 E=27 A=22 D=21 so D is eliminated. Round 3 votes counts: C=40 A=31 E=29 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:237 A:206 D:204 B:185 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -12 -4 10 B -18 0 -16 -6 10 C 12 16 0 18 28 D 4 6 -18 0 16 E -10 -10 -28 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 -4 10 B -18 0 -16 -6 10 C 12 16 0 18 28 D 4 6 -18 0 16 E -10 -10 -28 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 -4 10 B -18 0 -16 -6 10 C 12 16 0 18 28 D 4 6 -18 0 16 E -10 -10 -28 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5383: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) C A E D B (7) B D E A C (7) A C D E B (7) B D E C A (6) B A C E D (5) A C B E D (4) D E C A B (3) D A E C B (3) C A B E D (3) B C A E D (3) A D C E B (3) A D B E C (3) E B D C A (2) D E B C A (2) D E A C B (2) D B E A C (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B C A (1) D E B A C (1) C E D A B (1) C E B A D (1) C E A D B (1) C B E D A (1) C A E B D (1) C A D E B (1) B E D A C (1) B E C D A (1) B D A E C (1) B C E D A (1) B C E A D (1) B A D E C (1) B A D C E (1) B A C D E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -4 -2 0 B 8 0 14 20 24 C 4 -14 0 -12 -4 D 2 -20 12 0 0 E 0 -24 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -2 0 B 8 0 14 20 24 C 4 -14 0 -12 -4 D 2 -20 12 0 0 E 0 -24 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998226 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 A=23 C=16 D=13 E=5 so E is eliminated. Round 2 votes counts: B=45 A=23 D=16 C=16 so D is eliminated. Round 3 votes counts: B=51 A=28 C=21 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:233 D:197 A:193 E:190 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -2 0 B 8 0 14 20 24 C 4 -14 0 -12 -4 D 2 -20 12 0 0 E 0 -24 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998226 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -2 0 B 8 0 14 20 24 C 4 -14 0 -12 -4 D 2 -20 12 0 0 E 0 -24 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998226 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -2 0 B 8 0 14 20 24 C 4 -14 0 -12 -4 D 2 -20 12 0 0 E 0 -24 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998226 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5384: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) B C D A E (8) E D A B C (6) D B C A E (5) C A B D E (5) B D C A E (5) A C B D E (5) E D B A C (4) E B D C A (4) E A D B C (4) A C E B D (4) A C B E D (4) E D B C A (3) E A C B D (3) C B A D E (3) B D E C A (3) A C D B E (3) E A C D B (2) D B E C A (2) C B D A E (2) B D C E A (2) E B C D A (1) E B C A D (1) D E B C A (1) D E B A C (1) D B C E A (1) D A E B C (1) C D A B E (1) C B E A D (1) C A B E D (1) B C E D A (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 0 -2 4 B -4 0 6 8 6 C 0 -6 0 -6 4 D 2 -8 6 0 0 E -4 -6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428712 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -2 4 B -4 0 6 8 6 C 0 -6 0 -6 4 D 2 -8 6 0 0 E -4 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428562 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=21 B=19 C=13 D=11 so D is eliminated. Round 2 votes counts: E=38 B=27 A=22 C=13 so C is eliminated. Round 3 votes counts: E=38 B=33 A=29 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:203 D:200 C:196 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 -2 4 B -4 0 6 8 6 C 0 -6 0 -6 4 D 2 -8 6 0 0 E -4 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428562 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -2 4 B -4 0 6 8 6 C 0 -6 0 -6 4 D 2 -8 6 0 0 E -4 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428562 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -2 4 B -4 0 6 8 6 C 0 -6 0 -6 4 D 2 -8 6 0 0 E -4 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428562 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5385: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) D B E C A (6) D B C E A (6) D E B C A (5) A C B E D (5) E D C B A (4) E C B D A (4) A E D C B (4) E C B A D (3) E C A B D (3) E A C D B (3) C B E A D (3) B C D E A (3) B C A D E (3) D E A C B (2) D B C A E (2) D A B C E (2) C B E D A (2) B C E D A (2) B C D A E (2) B A C D E (2) A E C D B (2) A E C B D (2) A C E B D (2) A B D C E (2) E D A C B (1) E C A D B (1) E A C B D (1) D E A B C (1) D B A E C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) C A E B D (1) B D C E A (1) B D C A E (1) B C A E D (1) A D C B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -10 -14 -6 B 22 0 6 -6 14 C 10 -6 0 -2 8 D 14 6 2 0 6 E 6 -14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999046 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -10 -14 -6 B 22 0 6 -6 14 C 10 -6 0 -2 8 D 14 6 2 0 6 E 6 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=20 A=20 B=15 C=8 so C is eliminated. Round 2 votes counts: D=37 E=22 A=21 B=20 so B is eliminated. Round 3 votes counts: D=44 E=29 A=27 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:214 C:205 E:189 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -10 -14 -6 B 22 0 6 -6 14 C 10 -6 0 -2 8 D 14 6 2 0 6 E 6 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -10 -14 -6 B 22 0 6 -6 14 C 10 -6 0 -2 8 D 14 6 2 0 6 E 6 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -10 -14 -6 B 22 0 6 -6 14 C 10 -6 0 -2 8 D 14 6 2 0 6 E 6 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5386: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (10) B D E A C (7) B E D C A (6) A C E D B (6) E B D C A (4) B D E C A (4) A C D E B (4) E C A B D (3) D B E A C (3) D B A E C (3) D B A C E (3) D A C B E (3) C A E D B (3) C A E B D (3) B A D C E (3) E C D B A (2) C E A B D (2) C D A E B (2) A C D B E (2) E D C B A (1) E D B C A (1) E C D A B (1) E C B A D (1) D E B C A (1) D C E A B (1) D C A E B (1) D A B C E (1) C E D A B (1) B E D A C (1) B E C D A (1) B E A C D (1) B A E D C (1) A D C B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 4 -14 -2 B 6 0 0 0 4 C -4 0 0 -10 -2 D 14 0 10 0 0 E 2 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.596059 C: 0.000000 D: 0.403941 E: 0.000000 Sum of squares = 0.518454655833 Cumulative probabilities = A: 0.000000 B: 0.596059 C: 0.596059 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -14 -2 B 6 0 0 0 4 C -4 0 0 -10 -2 D 14 0 10 0 0 E 2 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 B=24 E=13 C=11 so C is eliminated. Round 2 votes counts: A=32 D=28 B=24 E=16 so E is eliminated. Round 3 votes counts: A=37 D=34 B=29 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:205 E:200 C:192 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -14 -2 B 6 0 0 0 4 C -4 0 0 -10 -2 D 14 0 10 0 0 E 2 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -14 -2 B 6 0 0 0 4 C -4 0 0 -10 -2 D 14 0 10 0 0 E 2 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -14 -2 B 6 0 0 0 4 C -4 0 0 -10 -2 D 14 0 10 0 0 E 2 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5387: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (8) B E C D A (6) E B C A D (5) A E B C D (5) C D E B A (4) C A E D B (4) B E A D C (4) E C B A D (3) E B A C D (3) C A D E B (3) A C E D B (3) A C D E B (3) E A B C D (2) D B E C A (2) D B C E A (2) D A B C E (2) C E D A B (2) C E B D A (2) C E A D B (2) B E D A C (2) B A E D C (2) A D B E C (2) A B E C D (2) E C B D A (1) E B C D A (1) D C B E A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C E B (1) D A C B E (1) C E D B A (1) C E A B D (1) C D E A B (1) C D B E A (1) B E D C A (1) B E A C D (1) B D E A C (1) B D C E A (1) B D A E C (1) B C D E A (1) B A D E C (1) A E C B D (1) A E B D C (1) A D E C B (1) A D C E B (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 2 20 -2 B 0 0 18 14 -4 C -2 -18 0 16 -14 D -20 -14 -16 0 -26 E 2 4 14 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 20 -2 B 0 0 18 14 -4 C -2 -18 0 16 -14 D -20 -14 -16 0 -26 E 2 4 14 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=21 B=21 E=15 D=14 so D is eliminated. Round 2 votes counts: A=33 B=27 C=25 E=15 so E is eliminated. Round 3 votes counts: B=36 A=35 C=29 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:223 B:214 A:210 C:191 D:162 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 20 -2 B 0 0 18 14 -4 C -2 -18 0 16 -14 D -20 -14 -16 0 -26 E 2 4 14 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 20 -2 B 0 0 18 14 -4 C -2 -18 0 16 -14 D -20 -14 -16 0 -26 E 2 4 14 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 20 -2 B 0 0 18 14 -4 C -2 -18 0 16 -14 D -20 -14 -16 0 -26 E 2 4 14 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5388: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (9) E D C A B (8) A C D E B (6) B E D C A (5) E D C B A (4) B D A E C (4) B A D C E (4) E D B C A (3) E C D B A (3) E C D A B (3) E B D C A (3) C E D A B (3) C E A D B (3) C A B E D (3) A C B D E (3) A B C D E (3) D E B C A (2) D E B A C (2) D B E A C (2) D E C A B (1) D E A C B (1) D E A B C (1) D B A E C (1) D A E C B (1) C E A B D (1) C D E A B (1) C A E B D (1) B C A E D (1) B A D E C (1) B A C E D (1) A C D B E (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -14 -14 -10 B -10 0 -12 -16 -18 C 14 12 0 -6 -8 D 14 16 6 0 -12 E 10 18 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -14 -14 -10 B -10 0 -12 -16 -18 C 14 12 0 -6 -8 D 14 16 6 0 -12 E 10 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 C=24 A=16 D=11 so D is eliminated. Round 2 votes counts: E=31 B=28 C=24 A=17 so A is eliminated. Round 3 votes counts: C=35 B=33 E=32 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:224 D:212 C:206 A:186 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -14 -14 -10 B -10 0 -12 -16 -18 C 14 12 0 -6 -8 D 14 16 6 0 -12 E 10 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 -14 -10 B -10 0 -12 -16 -18 C 14 12 0 -6 -8 D 14 16 6 0 -12 E 10 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 -14 -10 B -10 0 -12 -16 -18 C 14 12 0 -6 -8 D 14 16 6 0 -12 E 10 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5389: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) A D B C E (8) E C B D A (7) E B A D C (7) E B C D A (6) C D A B E (6) E C B A D (3) E B D A C (3) C E D B A (3) C E D A B (3) C E A D B (3) C A D E B (3) C A D B E (3) B D A E C (3) E B C A D (2) E B A C D (2) D A C B E (2) C E B A D (2) C D E B A (2) C D A E B (2) B E D A C (2) B D E A C (2) B A D E C (2) A D C B E (2) A B D E C (2) E A B D C (1) D B A E C (1) D B A C E (1) C E B D A (1) C D E A B (1) C A E D B (1) A E B D C (1) A E B C D (1) A D B E C (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 4 -8 0 B -2 0 10 -4 -8 C -4 -10 0 4 6 D 8 4 -4 0 2 E 0 8 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 0.000000 Sum of squares = 0.407407407439 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.444444 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -8 0 B -2 0 10 -4 -8 C -4 -10 0 4 6 D 8 4 -4 0 2 E 0 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 0.000000 Sum of squares = 0.40740740741 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.444444 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 A=17 D=13 B=9 so B is eliminated. Round 2 votes counts: E=33 C=30 A=19 D=18 so D is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:205 E:200 A:199 B:198 C:198 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -8 0 B -2 0 10 -4 -8 C -4 -10 0 4 6 D 8 4 -4 0 2 E 0 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 0.000000 Sum of squares = 0.40740740741 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.444444 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -8 0 B -2 0 10 -4 -8 C -4 -10 0 4 6 D 8 4 -4 0 2 E 0 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 0.000000 Sum of squares = 0.40740740741 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.444444 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -8 0 B -2 0 10 -4 -8 C -4 -10 0 4 6 D 8 4 -4 0 2 E 0 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 0.000000 Sum of squares = 0.40740740741 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.444444 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5390: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (14) E D A B C (11) E A D C B (7) E B D C A (6) A C E D B (6) C A B D E (5) A C D E B (5) A C D B E (5) C B A D E (4) A E D C B (4) B C D A E (3) D B E A C (2) C A E D B (2) B D E A C (2) B D C E A (2) B D C A E (2) B D A C E (2) B C E D A (2) A E C D B (2) E D B C A (1) E C A D B (1) E C A B D (1) D E B A C (1) D E A B C (1) D B A C E (1) C E B A D (1) C B E D A (1) C B A E D (1) B E D C A (1) B D E C A (1) B C D E A (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 4 30 -8 -12 B -4 0 8 -30 -30 C -30 -8 0 -20 -12 D 8 30 20 0 -24 E 12 30 12 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 30 -8 -12 B -4 0 8 -30 -30 C -30 -8 0 -20 -12 D 8 30 20 0 -24 E 12 30 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 A=24 B=16 C=14 D=5 so D is eliminated. Round 2 votes counts: E=43 A=24 B=19 C=14 so C is eliminated. Round 3 votes counts: E=44 A=31 B=25 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:239 D:217 A:207 B:172 C:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 30 -8 -12 B -4 0 8 -30 -30 C -30 -8 0 -20 -12 D 8 30 20 0 -24 E 12 30 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 30 -8 -12 B -4 0 8 -30 -30 C -30 -8 0 -20 -12 D 8 30 20 0 -24 E 12 30 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 30 -8 -12 B -4 0 8 -30 -30 C -30 -8 0 -20 -12 D 8 30 20 0 -24 E 12 30 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5391: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (13) E C A B D (7) C E D B A (7) E C B D A (5) D B A E C (5) D A B C E (5) C D E B A (5) A E B D C (4) A D B E C (4) E C B A D (3) E B A D C (3) D B C A E (3) C E B D A (3) C E A B D (3) C D B E A (3) B D A E C (3) A E C B D (3) D B C E A (2) D B A C E (2) C E D A B (2) C E A D B (2) C A D B E (2) E B D A C (1) E B A C D (1) E A C B D (1) C D A B E (1) C A E D B (1) C A D E B (1) B D E C A (1) B D E A C (1) A D B C E (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -2 2 0 B -4 0 0 6 -6 C 2 0 0 2 -12 D -2 -6 -2 0 4 E 0 6 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.725178 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.274822 Sum of squares = 0.601410247269 Cumulative probabilities = A: 0.725178 B: 0.725178 C: 0.725178 D: 0.725178 E: 1.000000 A B C D E A 0 4 -2 2 0 B -4 0 0 6 -6 C 2 0 0 2 -12 D -2 -6 -2 0 4 E 0 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555561485 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=27 E=21 D=17 B=5 so B is eliminated. Round 2 votes counts: C=30 A=27 D=22 E=21 so E is eliminated. Round 3 votes counts: C=45 A=32 D=23 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:207 A:202 B:198 D:197 C:196 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 2 0 B -4 0 0 6 -6 C 2 0 0 2 -12 D -2 -6 -2 0 4 E 0 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555561485 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 0 B -4 0 0 6 -6 C 2 0 0 2 -12 D -2 -6 -2 0 4 E 0 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555561485 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 0 B -4 0 0 6 -6 C 2 0 0 2 -12 D -2 -6 -2 0 4 E 0 6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555561485 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5392: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) A D E B C (9) C B A D E (6) A C B D E (6) D E A B C (5) C A B D E (5) E D A B C (4) E C D B A (4) D E B A C (4) C B E D A (4) A C B E D (4) C B E A D (3) A E D C B (3) A E D B C (3) A C E D B (3) A B D C E (3) E D C B A (2) C B D E A (2) C B A E D (2) C A B E D (2) B C D E A (2) B C D A E (2) E D A C B (1) E C B D A (1) E B D C A (1) D E B C A (1) D B E C A (1) C E B D A (1) C E A B D (1) C B D A E (1) B E D C A (1) B D A E C (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -4 4 4 B 0 0 -2 -2 -6 C 4 2 0 0 -2 D -4 2 0 0 0 E -4 6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999776 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 0 -4 4 4 B 0 0 -2 -2 -6 C 4 2 0 0 -2 D -4 2 0 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=27 E=23 D=11 B=7 so B is eliminated. Round 2 votes counts: A=33 C=31 E=24 D=12 so D is eliminated. Round 3 votes counts: E=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:202 C:202 E:202 D:199 B:195 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 4 4 B 0 0 -2 -2 -6 C 4 2 0 0 -2 D -4 2 0 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 4 B 0 0 -2 -2 -6 C 4 2 0 0 -2 D -4 2 0 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 4 B 0 0 -2 -2 -6 C 4 2 0 0 -2 D -4 2 0 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5393: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) C D B E A (10) D C B E A (9) E A B C D (8) A E D C B (6) A E D B C (6) A E B D C (6) E B A C D (5) D C A B E (5) A E B C D (5) B E A C D (4) B C D E A (4) A D E C B (4) A D C E B (4) D A C E B (2) B E C A D (2) B E A D C (2) B C E D A (2) E A C D B (1) D C A E B (1) C B D E A (1) B E C D A (1) B D C E A (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 8 8 0 B 2 0 -6 -18 2 C -8 6 0 -14 -2 D -8 18 14 0 4 E 0 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.642857 B: 0.285714 C: 0.000000 D: 0.071429 E: 0.000000 Sum of squares = 0.500000000056 Cumulative probabilities = A: 0.642857 B: 0.928571 C: 0.928571 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 8 0 B 2 0 -6 -18 2 C -8 6 0 -14 -2 D -8 18 14 0 4 E 0 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.285714 C: 0.000000 D: 0.071429 E: 0.000000 Sum of squares = 0.499999999095 Cumulative probabilities = A: 0.642857 B: 0.928571 C: 0.928571 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=27 B=16 E=14 C=11 so C is eliminated. Round 2 votes counts: D=37 A=32 B=17 E=14 so E is eliminated. Round 3 votes counts: A=41 D=37 B=22 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:207 E:198 C:191 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 8 0 B 2 0 -6 -18 2 C -8 6 0 -14 -2 D -8 18 14 0 4 E 0 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.285714 C: 0.000000 D: 0.071429 E: 0.000000 Sum of squares = 0.499999999095 Cumulative probabilities = A: 0.642857 B: 0.928571 C: 0.928571 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 8 0 B 2 0 -6 -18 2 C -8 6 0 -14 -2 D -8 18 14 0 4 E 0 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.285714 C: 0.000000 D: 0.071429 E: 0.000000 Sum of squares = 0.499999999095 Cumulative probabilities = A: 0.642857 B: 0.928571 C: 0.928571 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 8 0 B 2 0 -6 -18 2 C -8 6 0 -14 -2 D -8 18 14 0 4 E 0 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.642857 B: 0.285714 C: 0.000000 D: 0.071429 E: 0.000000 Sum of squares = 0.499999999095 Cumulative probabilities = A: 0.642857 B: 0.928571 C: 0.928571 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5394: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) A C D B E (8) A B C D E (8) E D C B A (5) B E A D C (5) A D C E B (5) E B D A C (4) D E C A B (4) B A E C D (4) E D B C A (3) E D A C B (3) C D A E B (3) B E D C A (3) B A C D E (3) A C B D E (3) E D C A B (2) D C E A B (2) D C A E B (2) D A C E B (2) C B A D E (2) B E A C D (2) B C E D A (2) B C A D E (2) B A C E D (2) A D E C B (2) E C D B A (1) D A E C B (1) C D A B E (1) C A D B E (1) B E C D A (1) B E C A D (1) B A E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 8 0 2 B 4 0 6 10 0 C -8 -6 0 -12 -6 D 0 -10 12 0 0 E -2 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.677681 C: 0.000000 D: 0.000000 E: 0.322319 Sum of squares = 0.563141124221 Cumulative probabilities = A: 0.000000 B: 0.677681 C: 0.677681 D: 0.677681 E: 1.000000 A B C D E A 0 -4 8 0 2 B 4 0 6 10 0 C -8 -6 0 -12 -6 D 0 -10 12 0 0 E -2 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 B=26 D=11 C=7 so C is eliminated. Round 2 votes counts: E=29 B=28 A=28 D=15 so D is eliminated. Round 3 votes counts: A=37 E=35 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:203 E:202 D:201 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 0 2 B 4 0 6 10 0 C -8 -6 0 -12 -6 D 0 -10 12 0 0 E -2 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 0 2 B 4 0 6 10 0 C -8 -6 0 -12 -6 D 0 -10 12 0 0 E -2 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 0 2 B 4 0 6 10 0 C -8 -6 0 -12 -6 D 0 -10 12 0 0 E -2 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5395: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (16) C E B A D (11) B C E A D (10) E C A D B (8) D A B E C (8) B C E D A (7) A E C D B (6) D B A E C (5) B D A C E (5) A D E C B (5) C E A B D (4) B D A E C (2) A C E D B (2) E C B A D (1) D B E A C (1) D A B C E (1) C B E A D (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D A E (1) B A C E D (1) A E D C B (1) Total count = 100 A B C D E A 0 2 6 0 6 B -2 0 -10 -6 -8 C -6 10 0 6 -12 D 0 6 -6 0 -8 E -6 8 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.742175 B: 0.000000 C: 0.000000 D: 0.257825 E: 0.000000 Sum of squares = 0.617297084653 Cumulative probabilities = A: 0.742175 B: 0.742175 C: 0.742175 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 0 6 B -2 0 -10 -6 -8 C -6 10 0 6 -12 D 0 6 -6 0 -8 E -6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204149741 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=30 C=16 A=14 E=9 so E is eliminated. Round 2 votes counts: D=31 B=30 C=25 A=14 so A is eliminated. Round 3 votes counts: D=37 C=33 B=30 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 A:207 C:199 D:196 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 0 6 B -2 0 -10 -6 -8 C -6 10 0 6 -12 D 0 6 -6 0 -8 E -6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204149741 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 0 6 B -2 0 -10 -6 -8 C -6 10 0 6 -12 D 0 6 -6 0 -8 E -6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204149741 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 0 6 B -2 0 -10 -6 -8 C -6 10 0 6 -12 D 0 6 -6 0 -8 E -6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.000000 Sum of squares = 0.510204149741 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5396: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) E A C B D (10) B D C A E (9) C E A D B (8) D C B E A (5) D B C E A (5) B D A C E (5) E C A D B (4) C E D A B (4) B D A E C (4) A E B C D (4) B A D E C (3) A C E B D (3) E D C A B (2) E C D A B (2) E A D B C (2) E A C D B (2) D C E B A (2) D B E C A (2) C D E B A (2) C A E B D (2) B A E D C (2) D E A B C (1) D B A E C (1) C D B E A (1) B A E C D (1) A E B D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 4 10 -4 B -16 0 -16 14 -20 C -4 16 0 10 -8 D -10 -14 -10 0 -20 E 4 20 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 4 10 -4 B -16 0 -16 14 -20 C -4 16 0 10 -8 D -10 -14 -10 0 -20 E 4 20 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=22 A=21 C=17 D=16 so D is eliminated. Round 2 votes counts: B=32 C=24 E=23 A=21 so A is eliminated. Round 3 votes counts: E=39 B=34 C=27 so C is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:213 C:207 B:181 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 4 10 -4 B -16 0 -16 14 -20 C -4 16 0 10 -8 D -10 -14 -10 0 -20 E 4 20 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 10 -4 B -16 0 -16 14 -20 C -4 16 0 10 -8 D -10 -14 -10 0 -20 E 4 20 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 10 -4 B -16 0 -16 14 -20 C -4 16 0 10 -8 D -10 -14 -10 0 -20 E 4 20 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5397: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (17) E A C B D (12) C A E D B (11) C A E B D (7) B D C E A (7) B D C A E (5) A E C D B (5) D B C E A (4) B D E A C (4) A E C B D (4) E A C D B (3) E A B D C (3) D B E A C (3) B D E C A (3) E D B A C (1) E C A B D (1) E B A D C (1) E A B C D (1) D E A B C (1) C D A B E (1) C B D A E (1) C A B E D (1) B E D A C (1) B C D E A (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -18 2 8 B -4 0 4 6 -4 C 18 -4 0 0 14 D -2 -6 0 0 -6 E -8 4 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627218666 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -18 2 8 B -4 0 4 6 -4 C 18 -4 0 0 14 D -2 -6 0 0 -6 E -8 4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627218825 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=22 C=21 B=21 A=11 so A is eliminated. Round 2 votes counts: E=31 D=25 C=23 B=21 so B is eliminated. Round 3 votes counts: D=44 E=32 C=24 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:214 B:201 A:198 E:194 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -18 2 8 B -4 0 4 6 -4 C 18 -4 0 0 14 D -2 -6 0 0 -6 E -8 4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627218825 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -18 2 8 B -4 0 4 6 -4 C 18 -4 0 0 14 D -2 -6 0 0 -6 E -8 4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627218825 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -18 2 8 B -4 0 4 6 -4 C 18 -4 0 0 14 D -2 -6 0 0 -6 E -8 4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.692308 C: 0.153846 D: 0.000000 E: 0.000000 Sum of squares = 0.526627218825 Cumulative probabilities = A: 0.153846 B: 0.846154 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5398: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) B C A D E (11) A B C E D (10) E D C A B (8) E D A C B (5) B A C D E (5) B A C E D (4) E A D C B (3) E A D B C (3) C D E B A (3) C B D E A (3) C B D A E (3) A B E C D (3) D E C A B (2) D C E B A (2) C B A D E (2) B A D E C (2) A E B D C (2) A E B C D (2) A B E D C (2) E C D A B (1) E C A D B (1) E A C D B (1) D E B C A (1) D E A B C (1) D C B E A (1) D B C E A (1) C E D B A (1) C D B E A (1) B D C E A (1) B D C A E (1) B C D A E (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 -10 6 0 B 8 0 2 4 2 C 10 -2 0 6 0 D -6 -4 -6 0 4 E 0 -2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 6 0 B 8 0 2 4 2 C 10 -2 0 6 0 D -6 -4 -6 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=22 A=21 D=19 C=13 so C is eliminated. Round 2 votes counts: B=33 E=23 D=23 A=21 so A is eliminated. Round 3 votes counts: B=48 E=29 D=23 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 C:207 E:197 A:194 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 6 0 B 8 0 2 4 2 C 10 -2 0 6 0 D -6 -4 -6 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 6 0 B 8 0 2 4 2 C 10 -2 0 6 0 D -6 -4 -6 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 6 0 B 8 0 2 4 2 C 10 -2 0 6 0 D -6 -4 -6 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5399: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) E B A D C (7) D C E A B (7) D C E B A (6) C D B E A (6) E A B D C (5) B E A D C (5) A E B D C (5) A B E C D (5) C D B A E (4) B A E C D (4) D C A E B (3) E D B A C (2) E D A B C (2) E B D C A (2) D E C B A (2) C D A E B (2) C B A D E (2) C A D B E (2) A C D B E (2) A B C E D (2) E B D A C (1) D C B E A (1) C D E B A (1) C D E A B (1) C B D A E (1) C A D E B (1) C A B D E (1) B E A C D (1) B C A E D (1) B A C E D (1) A D C E B (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -8 -4 2 B -6 0 -14 -10 2 C 8 14 0 2 18 D 4 10 -2 0 12 E -2 -2 -18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -4 2 B -6 0 -14 -10 2 C 8 14 0 2 18 D 4 10 -2 0 12 E -2 -2 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=19 D=19 A=18 B=12 so B is eliminated. Round 2 votes counts: C=33 E=25 A=23 D=19 so D is eliminated. Round 3 votes counts: C=50 E=27 A=23 so A is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:212 A:198 B:186 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -4 2 B -6 0 -14 -10 2 C 8 14 0 2 18 D 4 10 -2 0 12 E -2 -2 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -4 2 B -6 0 -14 -10 2 C 8 14 0 2 18 D 4 10 -2 0 12 E -2 -2 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -4 2 B -6 0 -14 -10 2 C 8 14 0 2 18 D 4 10 -2 0 12 E -2 -2 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5400: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (15) E C A B D (6) A D B C E (6) A C E D B (6) B D E C A (5) E C B D A (4) E B C D A (4) D B E C A (4) D B A C E (4) B D E A C (4) B D A E C (4) A D C B E (4) A C E B D (4) D B C E A (3) A B D E C (3) E C D B A (2) E B D C A (2) E A C B D (2) B E D C A (2) A C D E B (2) E C B A D (1) E B D A C (1) E A B C D (1) D C B E A (1) D B E A C (1) D A C B E (1) D A B C E (1) C A E D B (1) C A D E B (1) B A D E C (1) B A D C E (1) A E C B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -2 14 -16 B -12 0 -4 -6 -6 C 2 4 0 2 4 D -14 6 -2 0 -4 E 16 6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 14 -16 B -12 0 -4 -6 -6 C 2 4 0 2 4 D -14 6 -2 0 -4 E 16 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 C=17 B=17 D=15 so D is eliminated. Round 2 votes counts: A=30 B=29 E=23 C=18 so C is eliminated. Round 3 votes counts: E=38 A=32 B=30 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:206 A:204 D:193 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 14 -16 B -12 0 -4 -6 -6 C 2 4 0 2 4 D -14 6 -2 0 -4 E 16 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 14 -16 B -12 0 -4 -6 -6 C 2 4 0 2 4 D -14 6 -2 0 -4 E 16 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 14 -16 B -12 0 -4 -6 -6 C 2 4 0 2 4 D -14 6 -2 0 -4 E 16 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5401: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (14) C B E D A (10) E C B D A (9) D A B C E (6) A D E B C (6) A D E C B (5) D A E B C (4) C E B D A (4) B C D E A (4) E C B A D (3) B C E D A (3) A E D C B (3) A D B E C (3) E C A B D (2) D E A B C (2) D B C E A (2) C B E A D (2) B C A D E (2) A C B E D (2) A B C D E (2) E D C B A (1) E D A C B (1) D B C A E (1) D B A C E (1) D A B E C (1) C E B A D (1) C B A E D (1) B D C E A (1) B A C D E (1) A E C B D (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 8 0 10 B -8 0 8 -2 16 C -8 -8 0 -6 18 D 0 2 6 0 14 E -10 -16 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.424432 B: 0.000000 C: 0.000000 D: 0.575568 E: 0.000000 Sum of squares = 0.511421129168 Cumulative probabilities = A: 0.424432 B: 0.424432 C: 0.424432 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 0 10 B -8 0 8 -2 16 C -8 -8 0 -6 18 D 0 2 6 0 14 E -10 -16 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=18 D=17 E=16 B=11 so B is eliminated. Round 2 votes counts: A=39 C=27 D=18 E=16 so E is eliminated. Round 3 votes counts: C=41 A=39 D=20 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:211 B:207 C:198 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 0 10 B -8 0 8 -2 16 C -8 -8 0 -6 18 D 0 2 6 0 14 E -10 -16 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 0 10 B -8 0 8 -2 16 C -8 -8 0 -6 18 D 0 2 6 0 14 E -10 -16 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 0 10 B -8 0 8 -2 16 C -8 -8 0 -6 18 D 0 2 6 0 14 E -10 -16 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5402: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) D B A C E (7) C E A B D (7) B E C D A (7) C E B D A (6) D B C E A (5) B D E C A (5) D B A E C (4) A D C E B (4) A D B E C (4) C E B A D (3) B D C E A (3) B C E D A (3) A E C D B (3) A E C B D (3) A D E C B (3) D A C B E (2) D A B C E (2) C B E D A (2) B D E A C (2) A D C B E (2) A C E D B (2) E C A B D (1) E B C A D (1) E A C B D (1) D B E C A (1) D B C A E (1) C E A D B (1) C D B E A (1) B C D E A (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -26 -18 -4 -22 B 26 0 -8 14 4 C 18 8 0 10 8 D 4 -14 -10 0 -4 E 22 -4 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -18 -4 -22 B 26 0 -8 14 4 C 18 8 0 10 8 D 4 -14 -10 0 -4 E 22 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=22 B=21 C=20 E=14 so E is eliminated. Round 2 votes counts: C=32 A=24 D=22 B=22 so D is eliminated. Round 3 votes counts: B=40 C=32 A=28 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:218 E:207 D:188 A:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -26 -18 -4 -22 B 26 0 -8 14 4 C 18 8 0 10 8 D 4 -14 -10 0 -4 E 22 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -18 -4 -22 B 26 0 -8 14 4 C 18 8 0 10 8 D 4 -14 -10 0 -4 E 22 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -18 -4 -22 B 26 0 -8 14 4 C 18 8 0 10 8 D 4 -14 -10 0 -4 E 22 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5403: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (15) A B C E D (13) D E C A B (7) C E D A B (6) B D A E C (5) A C E B D (5) A C B E D (5) E C D A B (4) B A D E C (4) B A C E D (4) D E B C A (3) D B E C A (3) C E A D B (3) A C E D B (3) E D C B A (2) C A E D B (2) B D E C A (2) B D E A C (2) B A E C D (2) A B D C E (2) E D C A B (1) D C E A B (1) D B E A C (1) D B A E C (1) D A B E C (1) B A D C E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 2 -8 0 B -10 0 -8 -8 -4 C -2 8 0 -6 -8 D 8 8 6 0 0 E 0 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.592262 E: 0.407738 Sum of squares = 0.517024679064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.592262 E: 1.000000 A B C D E A 0 10 2 -8 0 B -10 0 -8 -8 -4 C -2 8 0 -6 -8 D 8 8 6 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=30 B=20 C=11 E=7 so E is eliminated. Round 2 votes counts: D=35 A=30 B=20 C=15 so C is eliminated. Round 3 votes counts: D=45 A=35 B=20 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:206 A:202 C:196 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 10 2 -8 0 B -10 0 -8 -8 -4 C -2 8 0 -6 -8 D 8 8 6 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -8 0 B -10 0 -8 -8 -4 C -2 8 0 -6 -8 D 8 8 6 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -8 0 B -10 0 -8 -8 -4 C -2 8 0 -6 -8 D 8 8 6 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5404: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) A B E C D (8) E A C D B (6) B A D C E (6) A E B D C (6) A E B C D (6) A B D C E (6) E D C B A (5) D C E B A (4) C D E B A (4) C D B E A (4) B D C A E (4) D C B E A (3) B D A C E (3) B C D A E (3) A B C D E (3) E C D A B (2) E A D C B (2) B A C D E (2) A B C E D (2) E D C A B (1) E C A D B (1) E C A B D (1) E A C B D (1) D E C A B (1) C B D E A (1) B D C E A (1) B C E A D (1) B C D E A (1) A E C D B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 8 8 4 B 2 0 8 12 0 C -8 -8 0 12 -4 D -8 -12 -12 0 -6 E -4 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.784131 C: 0.000000 D: 0.000000 E: 0.215869 Sum of squares = 0.661460509143 Cumulative probabilities = A: 0.000000 B: 0.784131 C: 0.784131 D: 0.784131 E: 1.000000 A B C D E A 0 -2 8 8 4 B 2 0 8 12 0 C -8 -8 0 12 -4 D -8 -12 -12 0 -6 E -4 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555886999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=28 B=21 C=9 D=8 so D is eliminated. Round 2 votes counts: A=34 E=29 B=21 C=16 so C is eliminated. Round 3 votes counts: E=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:211 A:209 E:203 C:196 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 8 4 B 2 0 8 12 0 C -8 -8 0 12 -4 D -8 -12 -12 0 -6 E -4 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555886999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 8 4 B 2 0 8 12 0 C -8 -8 0 12 -4 D -8 -12 -12 0 -6 E -4 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555886999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 8 4 B 2 0 8 12 0 C -8 -8 0 12 -4 D -8 -12 -12 0 -6 E -4 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555886999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5405: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (14) A B C D E (12) D E B A C (11) E D C B A (9) B A D E C (9) B A D C E (5) E C D A B (4) D E C B A (4) D B A E C (4) C E A B D (3) E D C A B (2) E D B A C (2) C E D A B (2) C E A D B (2) C A B D E (2) B A E D C (2) B A C D E (2) A B D C E (2) A B C E D (2) E D B C A (1) E C D B A (1) D E B C A (1) D B E A C (1) C A E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 8 16 14 B 4 0 8 12 14 C -8 -8 0 -6 -2 D -16 -12 6 0 8 E -14 -14 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 16 14 B 4 0 8 12 14 C -8 -8 0 -6 -2 D -16 -12 6 0 8 E -14 -14 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=21 E=19 B=18 A=18 so B is eliminated. Round 2 votes counts: A=36 C=24 D=21 E=19 so E is eliminated. Round 3 votes counts: A=36 D=35 C=29 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:219 A:217 D:193 C:188 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 16 14 B 4 0 8 12 14 C -8 -8 0 -6 -2 D -16 -12 6 0 8 E -14 -14 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 16 14 B 4 0 8 12 14 C -8 -8 0 -6 -2 D -16 -12 6 0 8 E -14 -14 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 16 14 B 4 0 8 12 14 C -8 -8 0 -6 -2 D -16 -12 6 0 8 E -14 -14 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5406: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (6) E B A D C (5) B A E C D (5) C D B E A (4) C D A E B (4) C D A B E (4) C B D A E (4) B A C D E (4) A B E D C (4) E D C B A (3) E B D A C (3) D E C A B (3) C D E B A (3) C A D B E (3) B E A D C (3) B E A C D (3) A B C D E (3) E B C D A (2) D C E A B (2) C D E A B (2) C D B A E (2) C B A D E (2) B C A E D (2) B A E D C (2) A B D C E (2) E D C A B (1) E D B A C (1) E D A C B (1) E D A B C (1) E B D C A (1) E A D B C (1) E A B D C (1) D E C B A (1) D E A C B (1) D A E C B (1) D A C E B (1) D A C B E (1) C E D B A (1) B E C D A (1) B E C A D (1) B C A D E (1) B A C E D (1) A D C B E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -6 -8 10 B 10 0 -4 2 10 C 6 4 0 6 8 D 8 -2 -6 0 14 E -10 -10 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -8 10 B 10 0 -4 2 10 C 6 4 0 6 8 D 8 -2 -6 0 14 E -10 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=23 E=20 D=16 A=12 so A is eliminated. Round 2 votes counts: B=33 C=30 E=20 D=17 so D is eliminated. Round 3 votes counts: C=41 B=33 E=26 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:209 D:207 A:193 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 -8 10 B 10 0 -4 2 10 C 6 4 0 6 8 D 8 -2 -6 0 14 E -10 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -8 10 B 10 0 -4 2 10 C 6 4 0 6 8 D 8 -2 -6 0 14 E -10 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -8 10 B 10 0 -4 2 10 C 6 4 0 6 8 D 8 -2 -6 0 14 E -10 -10 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5407: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) B E C A D (7) E C B A D (6) B D E C A (5) B C E A D (5) D A E C B (4) B E C D A (4) E C A D B (3) E C A B D (3) D E B C A (3) C A E B D (3) B C A E D (3) B A D C E (3) B A C E D (3) A C B E D (3) E B C A D (2) D E A C B (2) D B A E C (2) D B A C E (2) B E D C A (2) A D C E B (2) A D C B E (2) A C E D B (2) A C D E B (2) A C B D E (2) E D C A B (1) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E B A C (1) D B E C A (1) D B E A C (1) D A C B E (1) D A B C E (1) C A E D B (1) C A B E D (1) B D E A C (1) B D A C E (1) A D B C E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -10 12 -4 B 12 0 2 18 4 C 10 -2 0 10 -6 D -12 -18 -10 0 -10 E 4 -4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 12 -4 B 12 0 2 18 4 C 10 -2 0 10 -6 D -12 -18 -10 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=26 E=19 A=16 C=5 so C is eliminated. Round 2 votes counts: B=34 D=26 A=21 E=19 so E is eliminated. Round 3 votes counts: B=45 D=28 A=27 so A is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:208 C:206 A:193 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 12 -4 B 12 0 2 18 4 C 10 -2 0 10 -6 D -12 -18 -10 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 12 -4 B 12 0 2 18 4 C 10 -2 0 10 -6 D -12 -18 -10 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 12 -4 B 12 0 2 18 4 C 10 -2 0 10 -6 D -12 -18 -10 0 -10 E 4 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986421 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5408: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) B C A E D (6) C E B A D (5) C E A B D (5) D A B E C (4) B A D C E (4) E D C A B (3) E C D A B (3) C E D B A (3) B D C E A (3) B D A C E (3) B C D E A (3) A E C D B (3) A B C E D (3) D B C E A (2) D A E B C (2) C A E B D (2) B C E A D (2) B C A D E (2) B A C E D (2) A E D C B (2) A E C B D (2) A D E B C (2) A C E B D (2) E D A C B (1) E C D B A (1) E C A D B (1) E C A B D (1) E A C D B (1) D E C B A (1) D E A C B (1) D C B E A (1) D B A C E (1) D A E C B (1) C E D A B (1) C E A D B (1) C B E A D (1) C B D E A (1) C B A E D (1) B D C A E (1) B C E D A (1) B C D A E (1) B A C D E (1) A E D B C (1) A D E C B (1) A D B E C (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -4 10 16 B 6 0 8 10 8 C 4 -8 0 14 20 D -10 -10 -14 0 -12 E -16 -8 -20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 10 16 B 6 0 8 10 8 C 4 -8 0 14 20 D -10 -10 -14 0 -12 E -16 -8 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=20 C=20 A=20 E=11 so E is eliminated. Round 2 votes counts: B=29 C=26 D=24 A=21 so A is eliminated. Round 3 votes counts: C=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:215 A:208 E:184 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 10 16 B 6 0 8 10 8 C 4 -8 0 14 20 D -10 -10 -14 0 -12 E -16 -8 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 10 16 B 6 0 8 10 8 C 4 -8 0 14 20 D -10 -10 -14 0 -12 E -16 -8 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 10 16 B 6 0 8 10 8 C 4 -8 0 14 20 D -10 -10 -14 0 -12 E -16 -8 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5409: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) D E C A B (10) D E C B A (9) A B C E D (9) B A C E D (7) E D C B A (5) A B D C E (5) A B C D E (5) D A E B C (4) B C A E D (4) A D B C E (4) E C D B A (3) D E A B C (3) E D C A B (2) D E B C A (2) D A B E C (2) C E B A D (2) E C B D A (1) E C B A D (1) E C A B D (1) D B E C A (1) D B A E C (1) D B A C E (1) D A B C E (1) C B E A D (1) C A E B D (1) B C E A D (1) A D E C B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 22 12 -12 -6 B -22 0 0 -22 -14 C -12 0 0 -24 -14 D 12 22 24 0 20 E 6 14 14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 -12 -6 B -22 0 0 -22 -14 C -12 0 0 -24 -14 D 12 22 24 0 20 E 6 14 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 A=26 E=13 B=12 C=4 so C is eliminated. Round 2 votes counts: D=45 A=27 E=15 B=13 so B is eliminated. Round 3 votes counts: D=45 A=38 E=17 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:239 A:208 E:207 C:175 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 12 -12 -6 B -22 0 0 -22 -14 C -12 0 0 -24 -14 D 12 22 24 0 20 E 6 14 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 -12 -6 B -22 0 0 -22 -14 C -12 0 0 -24 -14 D 12 22 24 0 20 E 6 14 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 -12 -6 B -22 0 0 -22 -14 C -12 0 0 -24 -14 D 12 22 24 0 20 E 6 14 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5410: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) C A B D E (6) B E D C A (6) B E C D A (6) D A B C E (5) B C E A D (5) E B D C A (4) E B D A C (4) A D C B E (4) E B C D A (3) D B E A C (3) C E A B D (3) C A E B D (3) C A D B E (3) D A E C B (2) C A B E D (2) B E D A C (2) B D C A E (2) B C E D A (2) B C D A E (2) A E D C B (2) A D C E B (2) A C D E B (2) A C D B E (2) E C B A D (1) E B C A D (1) E A D C B (1) E A C D B (1) D E B A C (1) D E A B C (1) D B A C E (1) D A C E B (1) D A C B E (1) C E B A D (1) C B A E D (1) C B A D E (1) C A E D B (1) B D E A C (1) B D A C E (1) B C D E A (1) A E C D B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -8 -10 -6 B 10 0 14 16 14 C 8 -14 0 -2 6 D 10 -16 2 0 -14 E 6 -14 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -10 -6 B 10 0 14 16 14 C 8 -14 0 -2 6 D 10 -16 2 0 -14 E 6 -14 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=21 C=21 D=15 A=15 so D is eliminated. Round 2 votes counts: B=32 A=24 E=23 C=21 so C is eliminated. Round 3 votes counts: A=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:200 C:199 D:191 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -10 -6 B 10 0 14 16 14 C 8 -14 0 -2 6 D 10 -16 2 0 -14 E 6 -14 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -10 -6 B 10 0 14 16 14 C 8 -14 0 -2 6 D 10 -16 2 0 -14 E 6 -14 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -10 -6 B 10 0 14 16 14 C 8 -14 0 -2 6 D 10 -16 2 0 -14 E 6 -14 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5411: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (13) C B D A E (11) C E D A B (7) E A D C B (6) E C A D B (5) E A B D C (4) C D B A E (4) C B E D A (4) B D A C E (4) B C D A E (4) E B A D C (3) D C A B E (3) B E A D C (3) B D A E C (3) E C A B D (2) E A C D B (2) D B A C E (2) D A B E C (2) C E B A D (2) B D C A E (2) B C E A D (2) E C D A B (1) D B A E C (1) D A B C E (1) C E A D B (1) C D E A B (1) C D A E B (1) C B D E A (1) B A D E C (1) B A D C E (1) A E D B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -2 -4 -14 B -4 0 -2 -4 2 C 2 2 0 -4 2 D 4 4 4 0 -14 E 14 -2 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 A B C D E A 0 4 -2 -4 -14 B -4 0 -2 -4 2 C 2 2 0 -4 2 D 4 4 4 0 -14 E 14 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=32 B=20 D=9 A=3 so A is eliminated. Round 2 votes counts: E=37 C=32 B=21 D=10 so D is eliminated. Round 3 votes counts: E=37 C=35 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:201 D:199 B:196 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 -4 -14 B -4 0 -2 -4 2 C 2 2 0 -4 2 D 4 4 4 0 -14 E 14 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -4 -14 B -4 0 -2 -4 2 C 2 2 0 -4 2 D 4 4 4 0 -14 E 14 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -4 -14 B -4 0 -2 -4 2 C 2 2 0 -4 2 D 4 4 4 0 -14 E 14 -2 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.100000 E: 0.200000 Sum of squares = 0.539999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5412: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (5) A B E D C (5) C D E B A (4) C B D E A (4) B A E D C (4) B A C E D (4) A E B D C (4) A B E C D (4) E B A D C (3) D E A B C (3) D C E A B (3) C B A E D (3) B E A D C (3) B E A C D (3) A C B D E (3) D E C B A (2) D C E B A (2) D C A E B (2) D A E C B (2) C D B A E (2) C D A E B (2) C D A B E (2) C B E D A (2) C A D B E (2) C A B E D (2) B C E D A (2) B C E A D (2) A D E B C (2) A D C E B (2) E D B C A (1) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) E B C D A (1) E A B D C (1) D E C A B (1) D E A C B (1) D A C E B (1) C D B E A (1) C B D A E (1) C B A D E (1) C A B D E (1) A D E C B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 12 14 14 B -4 0 4 12 6 C -12 -4 0 -4 0 D -14 -12 4 0 -6 E -14 -6 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 14 14 B -4 0 4 12 6 C -12 -4 0 -4 0 D -14 -12 4 0 -6 E -14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=27 B=18 D=17 E=10 so E is eliminated. Round 2 votes counts: A=29 C=27 B=24 D=20 so D is eliminated. Round 3 votes counts: C=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:209 E:193 C:190 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 14 14 B -4 0 4 12 6 C -12 -4 0 -4 0 D -14 -12 4 0 -6 E -14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 14 14 B -4 0 4 12 6 C -12 -4 0 -4 0 D -14 -12 4 0 -6 E -14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 14 14 B -4 0 4 12 6 C -12 -4 0 -4 0 D -14 -12 4 0 -6 E -14 -6 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5413: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) B E A C D (6) D B E C A (5) C A D E B (5) E B D A C (3) E B A D C (3) E A B D C (3) D E B A C (3) D C B E A (3) D C A E B (3) C D B A E (3) A C E B D (3) E D A B C (2) D E A B C (2) D C E B A (2) D A C E B (2) C D A E B (2) C D A B E (2) C B D A E (2) B E D C A (2) B E D A C (2) B E C D A (2) B D E C A (2) B D C E A (2) B C D E A (2) B C A E D (2) A E C D B (2) A C D E B (2) A C B E D (2) E D B A C (1) E A D B C (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E A B (1) D C B A E (1) C B A E D (1) C B A D E (1) C A E D B (1) C A D B E (1) B E C A D (1) B E A D C (1) B C E D A (1) B C E A D (1) A E C B D (1) A D E C B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 2 6 C 14 2 0 4 6 D 6 -2 -4 0 0 E 8 -6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 2 6 C 14 2 0 4 6 D 6 -2 -4 0 0 E 8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 B=24 E=13 A=13 so E is eliminated. Round 2 votes counts: B=30 D=28 C=25 A=17 so A is eliminated. Round 3 votes counts: C=36 B=34 D=30 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:207 D:200 E:198 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 2 6 C 14 2 0 4 6 D 6 -2 -4 0 0 E 8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 2 6 C 14 2 0 4 6 D 6 -2 -4 0 0 E 8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 2 6 C 14 2 0 4 6 D 6 -2 -4 0 0 E 8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5414: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) E B A C D (6) D A C B E (6) E B C D A (5) B E D A C (5) D C A B E (4) C E D B A (4) C D A E B (4) E B C A D (3) D C E B A (3) D B A E C (3) C D E B A (3) C A E B D (3) B E A D C (3) B E A C D (3) A D B C E (3) A B E C D (3) E C B D A (2) D E B C A (2) D C A E B (2) D A B E C (2) D A B C E (2) C E B A D (2) C D E A B (2) C A E D B (2) C A D E B (2) B A E D C (2) A B E D C (2) E D B C A (1) D C B E A (1) D B E C A (1) D B E A C (1) C E D A B (1) C E B D A (1) C E A D B (1) B D E A C (1) A D C B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -10 -12 -14 B 16 0 0 -2 -10 C 10 0 0 6 -4 D 12 2 -6 0 -10 E 14 10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -10 -12 -14 B 16 0 0 -2 -10 C 10 0 0 6 -4 D 12 2 -6 0 -10 E 14 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 E=23 B=14 A=11 so A is eliminated. Round 2 votes counts: D=31 C=25 E=23 B=21 so B is eliminated. Round 3 votes counts: E=41 D=34 C=25 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:206 B:202 D:199 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -10 -12 -14 B 16 0 0 -2 -10 C 10 0 0 6 -4 D 12 2 -6 0 -10 E 14 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -12 -14 B 16 0 0 -2 -10 C 10 0 0 6 -4 D 12 2 -6 0 -10 E 14 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -12 -14 B 16 0 0 -2 -10 C 10 0 0 6 -4 D 12 2 -6 0 -10 E 14 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5415: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) C B D E A (7) A E C D B (7) C E A B D (6) B C D E A (6) D E A B C (5) C B A E D (5) D A E B C (4) C B D A E (4) B D C E A (4) B D A E C (4) A E D C B (4) D B A E C (3) E D A C B (2) E A D B C (2) D B E A C (2) D A B E C (2) C B E D A (2) B D C A E (2) A B D E C (2) E D C A B (1) E D A B C (1) E C D A B (1) E C A D B (1) E A D C B (1) D E B A C (1) D B E C A (1) D B C E A (1) C E B A D (1) C D E B A (1) C B E A D (1) C A E B D (1) C A B E D (1) B C A D E (1) A E B C D (1) Total count = 100 A B C D E A 0 8 6 -8 6 B -8 0 8 -4 -4 C -6 -8 0 -8 -14 D 8 4 8 0 0 E -6 4 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.682464 E: 0.317536 Sum of squares = 0.566586067347 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.682464 E: 1.000000 A B C D E A 0 8 6 -8 6 B -8 0 8 -4 -4 C -6 -8 0 -8 -14 D 8 4 8 0 0 E -6 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 D=19 B=17 E=9 so E is eliminated. Round 2 votes counts: C=31 A=29 D=23 B=17 so B is eliminated. Round 3 votes counts: C=38 D=33 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:206 E:206 B:196 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -8 6 B -8 0 8 -4 -4 C -6 -8 0 -8 -14 D 8 4 8 0 0 E -6 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -8 6 B -8 0 8 -4 -4 C -6 -8 0 -8 -14 D 8 4 8 0 0 E -6 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -8 6 B -8 0 8 -4 -4 C -6 -8 0 -8 -14 D 8 4 8 0 0 E -6 4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 0.499998 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500002 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5416: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C D E B A (6) A D E B C (6) A B E D C (6) B C A E D (5) C B E D A (4) C B D E A (4) B A E C D (4) D E C A B (3) D E A B C (3) B C E D A (3) B A E D C (3) A E D B C (3) A E B D C (3) E D B A C (2) E B D C A (2) E B D A C (2) E A D B C (2) D E C B A (2) D C E A B (2) C D E A B (2) C B A E D (2) C A D E B (2) B E C A D (2) B E A D C (2) B C E A D (2) A B C E D (2) E D B C A (1) D E A C B (1) D C A E B (1) C D B E A (1) C B E A D (1) C B D A E (1) C A D B E (1) B E D C A (1) B A C E D (1) A D E C B (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 4 2 -12 B 2 0 28 4 -6 C -4 -28 0 -8 -16 D -2 -4 8 0 -24 E 12 6 16 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 2 -12 B 2 0 28 4 -6 C -4 -28 0 -8 -16 D -2 -4 8 0 -24 E 12 6 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998608 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 B=23 E=17 D=12 so D is eliminated. Round 2 votes counts: C=27 E=26 A=24 B=23 so B is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:229 B:214 A:196 D:189 C:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 2 -12 B 2 0 28 4 -6 C -4 -28 0 -8 -16 D -2 -4 8 0 -24 E 12 6 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998608 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 2 -12 B 2 0 28 4 -6 C -4 -28 0 -8 -16 D -2 -4 8 0 -24 E 12 6 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998608 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 2 -12 B 2 0 28 4 -6 C -4 -28 0 -8 -16 D -2 -4 8 0 -24 E 12 6 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998608 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5417: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) D B A C E (8) E C A B D (7) D E C B A (6) E C D B A (5) E C A D B (5) E C D A B (4) C E D B A (4) E D C B A (3) D B C E A (3) C E A B D (3) A B E C D (3) A B D E C (3) D C E B A (2) D B A E C (2) C E B A D (2) C D B E A (2) B A D C E (2) A D B E C (2) A C E B D (2) A B C D E (2) E D C A B (1) E D A C B (1) E A C D B (1) D E B C A (1) D B E C A (1) D B C A E (1) D A E B C (1) D A B E C (1) C E B D A (1) C D E B A (1) C B D A E (1) C B A E D (1) B D A C E (1) B A C D E (1) A E D B C (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -8 0 -6 B -4 0 -8 -12 -6 C 8 8 0 -4 0 D 0 12 4 0 6 E 6 6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.175238 B: 0.000000 C: 0.000000 D: 0.824762 E: 0.000000 Sum of squares = 0.710940625935 Cumulative probabilities = A: 0.175238 B: 0.175238 C: 0.175238 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 0 -6 B -4 0 -8 -12 -6 C 8 8 0 -4 0 D 0 12 4 0 6 E 6 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555635 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 D=26 C=15 B=4 so B is eliminated. Round 2 votes counts: A=31 E=27 D=27 C=15 so C is eliminated. Round 3 votes counts: E=37 A=32 D=31 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:211 C:206 E:203 A:195 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 0 -6 B -4 0 -8 -12 -6 C 8 8 0 -4 0 D 0 12 4 0 6 E 6 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555635 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 0 -6 B -4 0 -8 -12 -6 C 8 8 0 -4 0 D 0 12 4 0 6 E 6 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555635 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 0 -6 B -4 0 -8 -12 -6 C 8 8 0 -4 0 D 0 12 4 0 6 E 6 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555635 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5418: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) E B A C D (5) D C E A B (5) D C A E B (5) C D E A B (5) A D E C B (5) E B C A D (4) E C D A B (3) D A C B E (3) B E C A D (3) B C E D A (3) B A D C E (3) A E D C B (3) E D A C B (2) E A D C B (2) C D E B A (2) C D B E A (2) C D B A E (2) B E A C D (2) B C E A D (2) B C A D E (2) B A E D C (2) B A D E C (2) A E D B C (2) A D C E B (2) A D C B E (2) A B E D C (2) A B D E C (2) E D C A B (1) E C D B A (1) E B C D A (1) D E C A B (1) D A E C B (1) C B E D A (1) C B D E A (1) B C D E A (1) B C A E D (1) B A E C D (1) B A C E D (1) B A C D E (1) A E B D C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 14 8 4 6 B -14 0 -14 -18 -20 C -8 14 0 -12 6 D -4 18 12 0 14 E -6 20 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 4 6 B -14 0 -14 -18 -20 C -8 14 0 -12 6 D -4 18 12 0 14 E -6 20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=23 A=21 E=19 C=13 so C is eliminated. Round 2 votes counts: D=34 B=26 A=21 E=19 so E is eliminated. Round 3 votes counts: D=41 B=36 A=23 so A is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:216 C:200 E:197 B:167 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 4 6 B -14 0 -14 -18 -20 C -8 14 0 -12 6 D -4 18 12 0 14 E -6 20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 4 6 B -14 0 -14 -18 -20 C -8 14 0 -12 6 D -4 18 12 0 14 E -6 20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 4 6 B -14 0 -14 -18 -20 C -8 14 0 -12 6 D -4 18 12 0 14 E -6 20 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5419: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) E A C D B (7) A E D B C (6) B D C A E (5) E C A D B (4) E A D C B (4) D B C E A (4) C B D A E (4) B C D A E (4) A E C B D (4) A D B E C (4) C E A B D (3) C B E D A (3) B D A C E (3) E A D B C (2) D B A E C (2) C E B A D (2) C E A D B (2) C D B E A (2) C B A E D (2) C A E B D (2) B D C E A (2) A D E B C (2) D B E C A (1) D B C A E (1) D A E B C (1) D A B E C (1) C E D A B (1) C E B D A (1) C A B E D (1) B D A E C (1) B C A D E (1) A E C D B (1) A E B D C (1) A C B E D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -14 4 0 B 0 0 -12 10 14 C 14 12 0 16 12 D -4 -10 -16 0 2 E 0 -14 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 4 0 B 0 0 -12 10 14 C 14 12 0 16 12 D -4 -10 -16 0 2 E 0 -14 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=22 E=17 B=16 D=10 so D is eliminated. Round 2 votes counts: C=35 B=24 A=24 E=17 so E is eliminated. Round 3 votes counts: C=39 A=37 B=24 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:206 A:195 D:186 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 4 0 B 0 0 -12 10 14 C 14 12 0 16 12 D -4 -10 -16 0 2 E 0 -14 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 4 0 B 0 0 -12 10 14 C 14 12 0 16 12 D -4 -10 -16 0 2 E 0 -14 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 4 0 B 0 0 -12 10 14 C 14 12 0 16 12 D -4 -10 -16 0 2 E 0 -14 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5420: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) D C B A E (7) E D C A B (6) D C A B E (6) C D A B E (6) E B A D C (5) E A C B D (5) B A C D E (5) A B C E D (5) E D C B A (4) C D B A E (4) E B A C D (3) A C B D E (3) A B C D E (3) E D B C A (2) E C A D B (2) D E C B A (2) B D A C E (2) B C A D E (2) E D B A C (1) E A B D C (1) D E C A B (1) D E B C A (1) D C E A B (1) D C B E A (1) D B C A E (1) C D A E B (1) C B A D E (1) C A B D E (1) B E A D C (1) B E A C D (1) B D C A E (1) B A E D C (1) B A D C E (1) B A C E D (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 0 6 6 B -6 0 -4 8 6 C 0 4 0 10 6 D -6 -8 -10 0 0 E -6 -6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.538695 B: 0.000000 C: 0.461305 D: 0.000000 E: 0.000000 Sum of squares = 0.502994568522 Cumulative probabilities = A: 0.538695 B: 0.538695 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 6 6 B -6 0 -4 8 6 C 0 4 0 10 6 D -6 -8 -10 0 0 E -6 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=20 B=15 C=13 A=13 so C is eliminated. Round 2 votes counts: E=39 D=31 B=16 A=14 so A is eliminated. Round 3 votes counts: E=41 D=31 B=28 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 A:209 B:202 E:191 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 6 6 B -6 0 -4 8 6 C 0 4 0 10 6 D -6 -8 -10 0 0 E -6 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 6 6 B -6 0 -4 8 6 C 0 4 0 10 6 D -6 -8 -10 0 0 E -6 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 6 6 B -6 0 -4 8 6 C 0 4 0 10 6 D -6 -8 -10 0 0 E -6 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5421: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) E C A B D (7) C B D E A (7) A E D B C (7) D B A C E (6) E A C D B (4) E A C B D (4) C E B D A (4) C D B E A (4) B C D E A (4) A D E B C (4) A D B E C (4) E A B C D (3) A E D C B (3) A E B D C (3) E C A D B (2) C D B A E (2) C B E D A (2) B D C E A (2) B D C A E (2) A E C D B (2) E C B D A (1) E C B A D (1) E A B D C (1) D C B A E (1) D A B C E (1) C E A D B (1) C D A B E (1) B E C D A (1) B C E D A (1) A D C E B (1) A D C B E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -8 0 2 B -2 0 2 -12 2 C 8 -2 0 4 4 D 0 12 -4 0 4 E -2 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839526 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 0 2 B -2 0 2 -12 2 C 8 -2 0 4 4 D 0 12 -4 0 4 E -2 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839473 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=23 C=21 D=19 B=10 so B is eliminated. Round 2 votes counts: A=27 C=26 E=24 D=23 so D is eliminated. Round 3 votes counts: C=42 A=34 E=24 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 D:206 A:198 B:195 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 0 2 B -2 0 2 -12 2 C 8 -2 0 4 4 D 0 12 -4 0 4 E -2 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839473 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 0 2 B -2 0 2 -12 2 C 8 -2 0 4 4 D 0 12 -4 0 4 E -2 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839473 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 0 2 B -2 0 2 -12 2 C 8 -2 0 4 4 D 0 12 -4 0 4 E -2 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839473 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5422: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) E D A B C (7) B E C D A (6) B C A D E (5) D A C E B (4) C A D E B (4) C A D B E (4) C A B D E (4) E C B D A (3) E B D A C (3) D E A C B (3) D A E C B (3) C D A E B (3) B E D C A (3) B A C D E (3) A D C E B (3) E D C A B (2) E B D C A (2) E B C D A (2) C B A D E (2) B E D A C (2) B E A D C (2) B E A C D (2) B C E A D (2) A D C B E (2) A C D E B (2) E D C B A (1) E D B A C (1) E C D B A (1) D E A B C (1) D C A E B (1) C E B D A (1) C D E A B (1) B E C A D (1) B A E C D (1) A D E C B (1) A D B C E (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 4 -16 -8 B -14 0 -10 -8 -14 C -4 10 0 0 -10 D 16 8 0 0 0 E 8 14 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.324723 E: 0.675277 Sum of squares = 0.561444276843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.324723 E: 1.000000 A B C D E A 0 14 4 -16 -8 B -14 0 -10 -8 -14 C -4 10 0 0 -10 D 16 8 0 0 0 E 8 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=27 C=19 D=12 A=12 so D is eliminated. Round 2 votes counts: E=34 B=27 C=20 A=19 so A is eliminated. Round 3 votes counts: E=38 C=33 B=29 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:212 C:198 A:197 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 14 4 -16 -8 B -14 0 -10 -8 -14 C -4 10 0 0 -10 D 16 8 0 0 0 E 8 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -16 -8 B -14 0 -10 -8 -14 C -4 10 0 0 -10 D 16 8 0 0 0 E 8 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -16 -8 B -14 0 -10 -8 -14 C -4 10 0 0 -10 D 16 8 0 0 0 E 8 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5423: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) A D B E C (9) C E B A D (8) D A B E C (7) B D A E C (7) A D C E B (7) A D B C E (7) B C E D A (6) A D E C B (6) B E C D A (5) E C B D A (4) E C A D B (3) C E A D B (3) A D E B C (3) C E A B D (2) B D A C E (2) E D B C A (1) E D B A C (1) E B C D A (1) E A C D B (1) D A E B C (1) D A B C E (1) C B E D A (1) C B E A D (1) B D E C A (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 4 4 B -4 0 6 -2 -2 C -8 -6 0 -8 0 D -4 2 8 0 6 E -4 2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 4 4 B -4 0 6 -2 -2 C -8 -6 0 -8 0 D -4 2 8 0 6 E -4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=25 B=21 E=11 D=9 so D is eliminated. Round 2 votes counts: A=43 C=25 B=21 E=11 so E is eliminated. Round 3 votes counts: A=44 C=32 B=24 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:206 B:199 E:196 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 4 B -4 0 6 -2 -2 C -8 -6 0 -8 0 D -4 2 8 0 6 E -4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 4 B -4 0 6 -2 -2 C -8 -6 0 -8 0 D -4 2 8 0 6 E -4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 4 B -4 0 6 -2 -2 C -8 -6 0 -8 0 D -4 2 8 0 6 E -4 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5424: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (14) A D C B E (12) C E A D B (10) C A D E B (10) A D B C E (9) E B C D A (7) D A B C E (7) E C B D A (4) B E D A C (4) E C B A D (3) C E B D A (2) C E B A D (2) B E A D C (2) B D A C E (2) A C D E B (2) E C A D B (1) E B A D C (1) D C A B E (1) D B A C E (1) D A B E C (1) C E D A B (1) C A E D B (1) B C D A E (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 14 14 10 26 B -14 0 -2 -16 10 C -14 2 0 -10 26 D -10 16 10 0 24 E -26 -10 -26 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 10 26 B -14 0 -2 -16 10 C -14 2 0 -10 26 D -10 16 10 0 24 E -26 -10 -26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=25 B=23 E=16 D=10 so D is eliminated. Round 2 votes counts: A=33 C=27 B=24 E=16 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:232 D:220 C:202 B:189 E:157 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 10 26 B -14 0 -2 -16 10 C -14 2 0 -10 26 D -10 16 10 0 24 E -26 -10 -26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 10 26 B -14 0 -2 -16 10 C -14 2 0 -10 26 D -10 16 10 0 24 E -26 -10 -26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 10 26 B -14 0 -2 -16 10 C -14 2 0 -10 26 D -10 16 10 0 24 E -26 -10 -26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5425: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) C E B A D (6) C D A E B (6) E B A D C (5) D A E B C (5) C D A B E (5) B E A D C (5) B A D E C (5) E A D B C (4) E A B D C (4) D C A B E (4) C D E A B (4) C D B A E (4) E B C A D (3) D A C B E (3) C E B D A (3) E B A C D (2) D C A E B (2) C B E D A (2) B E C A D (2) A E D B C (2) A B E D C (2) E C B A D (1) E A C B D (1) D E A C B (1) D C E A B (1) D A E C B (1) D A C E B (1) C D E B A (1) C D B E A (1) C B D E A (1) C B D A E (1) C B A D E (1) B E A C D (1) B C D A E (1) B C A D E (1) B A E D C (1) Total count = 100 A B C D E A 0 6 0 -8 4 B -6 0 0 -4 -6 C 0 0 0 -6 -4 D 8 4 6 0 12 E -4 6 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -8 4 B -6 0 0 -4 -6 C 0 0 0 -6 -4 D 8 4 6 0 12 E -4 6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=25 E=20 B=16 A=4 so A is eliminated. Round 2 votes counts: C=35 D=25 E=22 B=18 so B is eliminated. Round 3 votes counts: C=37 E=33 D=30 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:215 A:201 E:197 C:195 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -8 4 B -6 0 0 -4 -6 C 0 0 0 -6 -4 D 8 4 6 0 12 E -4 6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -8 4 B -6 0 0 -4 -6 C 0 0 0 -6 -4 D 8 4 6 0 12 E -4 6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -8 4 B -6 0 0 -4 -6 C 0 0 0 -6 -4 D 8 4 6 0 12 E -4 6 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5426: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) B A C E D (7) A B E D C (7) E D C A B (5) D E C A B (5) D C E B A (5) D C E A B (5) C D B A E (5) E A B D C (4) D E A B C (4) C E D B A (4) C E B A D (4) A B D E C (4) E D A C B (3) E C B A D (3) D A B E C (3) D A B C E (3) C D E B A (3) E D A B C (2) B A C D E (2) A B E C D (2) A B C D E (2) E C D B A (1) E A B C D (1) D C B A E (1) D A E B C (1) C B D A E (1) C B A D E (1) B A E C D (1) Total count = 100 A B C D E A 0 2 -8 -2 2 B -2 0 -14 0 0 C 8 14 0 -4 8 D 2 0 4 0 -10 E -2 0 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.363636 E: 0.181818 Sum of squares = 0.371900826447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.818182 E: 1.000000 A B C D E A 0 2 -8 -2 2 B -2 0 -14 0 0 C 8 14 0 -4 8 D 2 0 4 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.363636 E: 0.181818 Sum of squares = 0.37190082641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=27 E=19 A=15 B=10 so B is eliminated. Round 2 votes counts: C=29 D=27 A=25 E=19 so E is eliminated. Round 3 votes counts: D=37 C=33 A=30 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:213 E:200 D:198 A:197 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -2 2 B -2 0 -14 0 0 C 8 14 0 -4 8 D 2 0 4 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.363636 E: 0.181818 Sum of squares = 0.37190082641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.818182 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 2 B -2 0 -14 0 0 C 8 14 0 -4 8 D 2 0 4 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.363636 E: 0.181818 Sum of squares = 0.37190082641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.818182 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 2 B -2 0 -14 0 0 C 8 14 0 -4 8 D 2 0 4 0 -10 E -2 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.363636 E: 0.181818 Sum of squares = 0.37190082641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.818182 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5427: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) C E A D B (7) A D C E B (7) A D C B E (6) A D B C E (6) E C B D A (5) D A B C E (5) C E A B D (5) D A B E C (4) C B E A D (4) B E C D A (4) C E B A D (3) C A E D B (3) B D E A C (3) B D A E C (3) E C D A B (2) E C B A D (2) C A D E B (2) C A D B E (2) B E D A C (2) B D A C E (2) A D E C B (2) E D B A C (1) E C A D B (1) E B D C A (1) D E A B C (1) D B E A C (1) D B A E C (1) D A E B C (1) C B A D E (1) B E D C A (1) B C E D A (1) B C D A E (1) B C A D E (1) A E C D B (1) Total count = 100 A B C D E A 0 10 -8 6 -4 B -10 0 -6 -6 -4 C 8 6 0 6 12 D -6 6 -6 0 -2 E 4 4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 6 -4 B -10 0 -6 -6 -4 C 8 6 0 6 12 D -6 6 -6 0 -2 E 4 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=22 E=20 B=18 D=13 so D is eliminated. Round 2 votes counts: A=32 C=27 E=21 B=20 so B is eliminated. Round 3 votes counts: A=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:216 A:202 E:199 D:196 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 6 -4 B -10 0 -6 -6 -4 C 8 6 0 6 12 D -6 6 -6 0 -2 E 4 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 6 -4 B -10 0 -6 -6 -4 C 8 6 0 6 12 D -6 6 -6 0 -2 E 4 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 6 -4 B -10 0 -6 -6 -4 C 8 6 0 6 12 D -6 6 -6 0 -2 E 4 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5428: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) D A E C B (8) E D A B C (7) C B A D E (6) D A C E B (5) C A D B E (5) C B D A E (4) C B A E D (4) C A B D E (4) B C E A D (4) E D B A C (3) B E C A D (3) B C A E D (3) A C D E B (3) E B D A C (2) E B A D C (2) E A D C B (2) D E A B C (2) D C A E B (2) B E C D A (2) B C D A E (2) A E D C B (2) A C D B E (2) E D A C B (1) E B D C A (1) E B A C D (1) E A D B C (1) D E B C A (1) D B E C A (1) C D B A E (1) C D A B E (1) C A B E D (1) B E D C A (1) B D C E A (1) A D E C B (1) A D C E B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 16 6 -6 14 B -16 0 -24 -14 -8 C -6 24 0 -4 2 D 6 14 4 0 16 E -14 8 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 -6 14 B -16 0 -24 -14 -8 C -6 24 0 -4 2 D 6 14 4 0 16 E -14 8 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=26 E=20 B=16 A=11 so A is eliminated. Round 2 votes counts: C=32 D=29 E=22 B=17 so B is eliminated. Round 3 votes counts: C=42 D=30 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:215 C:208 E:188 B:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 6 -6 14 B -16 0 -24 -14 -8 C -6 24 0 -4 2 D 6 14 4 0 16 E -14 8 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 -6 14 B -16 0 -24 -14 -8 C -6 24 0 -4 2 D 6 14 4 0 16 E -14 8 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 -6 14 B -16 0 -24 -14 -8 C -6 24 0 -4 2 D 6 14 4 0 16 E -14 8 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5429: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) B A E D C (10) A E B C D (8) C D E A B (6) D C E B A (5) C D E B A (5) B E A D C (5) E A B C D (4) D B C E A (4) C E A D B (3) C D A E B (3) B D E A C (3) B D A E C (3) A B E D C (3) B A D E C (2) A E C B D (2) A B C E D (2) E D C B A (1) E C D A B (1) E B D C A (1) E B D A C (1) E B A D C (1) E A B D C (1) D E C B A (1) D B C A E (1) C E D A B (1) C D A B E (1) C A E D B (1) C A D B E (1) B E D A C (1) B D E C A (1) B D C E A (1) B D A C E (1) B A D C E (1) A C E D B (1) A C D E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 4 -4 -14 B 18 0 10 6 6 C -4 -10 0 -16 0 D 4 -6 16 0 2 E 14 -6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 4 -4 -14 B 18 0 10 6 6 C -4 -10 0 -16 0 D 4 -6 16 0 2 E 14 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=22 C=21 A=19 E=10 so E is eliminated. Round 2 votes counts: B=31 A=24 D=23 C=22 so C is eliminated. Round 3 votes counts: D=40 B=31 A=29 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:208 E:203 C:185 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 4 -4 -14 B 18 0 10 6 6 C -4 -10 0 -16 0 D 4 -6 16 0 2 E 14 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 -4 -14 B 18 0 10 6 6 C -4 -10 0 -16 0 D 4 -6 16 0 2 E 14 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 -4 -14 B 18 0 10 6 6 C -4 -10 0 -16 0 D 4 -6 16 0 2 E 14 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5430: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (13) C D E A B (8) D C B E A (7) B A E D C (7) D C B A E (5) A B E C D (5) E A B C D (4) E B A D C (3) D B C E A (3) C D E B A (3) B E A D C (3) E B D A C (2) D B E A C (2) D B A C E (2) C E D A B (2) C D A B E (2) C A E D B (2) C A E B D (2) C A D E B (2) B D A E C (2) B A D C E (2) A E B C D (2) A B E D C (2) E C D A B (1) E A C B D (1) D E B C A (1) D C A B E (1) D B E C A (1) D B C A E (1) D B A E C (1) C E D B A (1) C E A D B (1) C E A B D (1) B E D A C (1) B D E A C (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -14 -20 -18 B 22 0 -8 -18 0 C 14 8 0 -20 22 D 20 18 20 0 18 E 18 0 -22 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -14 -20 -18 B 22 0 -8 -18 0 C 14 8 0 -20 22 D 20 18 20 0 18 E 18 0 -22 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=24 B=16 A=12 E=11 so E is eliminated. Round 2 votes counts: D=37 C=25 B=21 A=17 so A is eliminated. Round 3 votes counts: D=37 B=35 C=28 so C is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:238 C:212 B:198 E:189 A:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -14 -20 -18 B 22 0 -8 -18 0 C 14 8 0 -20 22 D 20 18 20 0 18 E 18 0 -22 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -14 -20 -18 B 22 0 -8 -18 0 C 14 8 0 -20 22 D 20 18 20 0 18 E 18 0 -22 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -14 -20 -18 B 22 0 -8 -18 0 C 14 8 0 -20 22 D 20 18 20 0 18 E 18 0 -22 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5431: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) D B C E A (10) E B A D C (6) D B E C A (6) E A B C D (5) C A D E B (5) C A D B E (5) A E C B D (5) B E D A C (4) A C E B D (4) D C B A E (3) A C B E D (3) E B A C D (2) E A C B D (2) D C E B A (2) D C A B E (2) C A E D B (2) B E A D C (2) A C B D E (2) E D B C A (1) E D B A C (1) E B D A C (1) E A D C B (1) E A C D B (1) D E C A B (1) D E B C A (1) D C B E A (1) D B E A C (1) D B C A E (1) C D A E B (1) C A B D E (1) B E A C D (1) B D A E C (1) B D A C E (1) B C D A E (1) B A E C D (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -8 0 2 B -4 0 -4 -12 18 C 8 4 0 8 14 D 0 12 -8 0 12 E -2 -18 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 0 2 B -4 0 -4 -12 18 C 8 4 0 8 14 D 0 12 -8 0 12 E -2 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 E=20 A=15 B=12 so B is eliminated. Round 2 votes counts: D=30 E=27 C=26 A=17 so A is eliminated. Round 3 votes counts: C=37 E=33 D=30 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:208 A:199 B:199 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 0 2 B -4 0 -4 -12 18 C 8 4 0 8 14 D 0 12 -8 0 12 E -2 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 0 2 B -4 0 -4 -12 18 C 8 4 0 8 14 D 0 12 -8 0 12 E -2 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 0 2 B -4 0 -4 -12 18 C 8 4 0 8 14 D 0 12 -8 0 12 E -2 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5432: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (14) E B A D C (7) C D B E A (7) B E C D A (7) A E B D C (6) C D E B A (5) D C E B A (4) B E A C D (4) A D C E B (4) A D C B E (4) A B E C D (4) E B D C A (3) C D A B E (3) B E C A D (3) A D E B C (3) A C D B E (3) A B E D C (3) E B C D A (2) D A C E B (2) C B E D A (2) E B A C D (1) D C E A B (1) D A E C B (1) C D A E B (1) C B E A D (1) B A E C D (1) A D E C B (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -6 -4 6 B -6 0 -10 -6 -10 C 6 10 0 -6 8 D 4 6 6 0 8 E -6 10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -4 6 B -6 0 -10 -6 -10 C 6 10 0 -6 8 D 4 6 6 0 8 E -6 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=22 C=19 B=15 E=13 so E is eliminated. Round 2 votes counts: A=31 B=28 D=22 C=19 so C is eliminated. Round 3 votes counts: D=38 B=31 A=31 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:209 A:201 E:194 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 -4 6 B -6 0 -10 -6 -10 C 6 10 0 -6 8 D 4 6 6 0 8 E -6 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -4 6 B -6 0 -10 -6 -10 C 6 10 0 -6 8 D 4 6 6 0 8 E -6 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -4 6 B -6 0 -10 -6 -10 C 6 10 0 -6 8 D 4 6 6 0 8 E -6 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5433: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (13) E C A D B (8) B D A E C (7) C B D A E (6) A D B E C (6) C E A D B (5) E C B D A (4) E A D B C (4) E A C D B (4) C B D E A (4) B D A C E (4) E C B A D (3) C E A B D (3) E C A B D (2) E B D A C (2) D B A E C (2) C E B A D (2) C B E D A (2) B E D A C (2) B D C A E (2) B C D A E (2) A E D B C (2) E B A D C (1) E A D C B (1) E A B D C (1) D C B A E (1) D C A B E (1) D A B C E (1) C D B A E (1) C D A B E (1) C A D B E (1) B D E C A (1) A D C B E (1) Total count = 100 A B C D E A 0 -18 -24 -12 -24 B 18 0 -26 22 -10 C 24 26 0 22 0 D 12 -22 -22 0 -18 E 24 10 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.317075 D: 0.000000 E: 0.682925 Sum of squares = 0.566923127202 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.317075 D: 0.317075 E: 1.000000 A B C D E A 0 -18 -24 -12 -24 B 18 0 -26 22 -10 C 24 26 0 22 0 D 12 -22 -22 0 -18 E 24 10 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=30 B=18 A=9 D=5 so D is eliminated. Round 2 votes counts: C=40 E=30 B=20 A=10 so A is eliminated. Round 3 votes counts: C=41 E=32 B=27 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:236 E:226 B:202 D:175 A:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -24 -12 -24 B 18 0 -26 22 -10 C 24 26 0 22 0 D 12 -22 -22 0 -18 E 24 10 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -24 -12 -24 B 18 0 -26 22 -10 C 24 26 0 22 0 D 12 -22 -22 0 -18 E 24 10 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -24 -12 -24 B 18 0 -26 22 -10 C 24 26 0 22 0 D 12 -22 -22 0 -18 E 24 10 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5434: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (10) D E C B A (9) D B E C A (7) B D A C E (6) A B C E D (6) A B D C E (5) E C D B A (4) D A B E C (4) D B A E C (3) C E B A D (3) B C E D A (3) B A C E D (3) E D C B A (2) E C D A B (2) E C A B D (2) D E A C B (2) D B C E A (2) D A E C B (2) C E B D A (2) C E A B D (2) B A D C E (2) B A C D E (2) A E C D B (2) A D E C B (2) A D B E C (2) A C E B D (2) E C B A D (1) E C A D B (1) D A E B C (1) C A E B D (1) C A B E D (1) B C E A D (1) B C A E D (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -8 -18 -6 B 2 0 0 -12 -2 C 8 0 0 -20 -14 D 18 12 20 0 22 E 6 2 14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -18 -6 B 2 0 0 -12 -2 C 8 0 0 -20 -14 D 18 12 20 0 22 E 6 2 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=21 B=18 E=12 C=9 so C is eliminated. Round 2 votes counts: D=40 A=23 E=19 B=18 so B is eliminated. Round 3 votes counts: D=46 A=31 E=23 so E is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:236 E:200 B:194 C:187 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -18 -6 B 2 0 0 -12 -2 C 8 0 0 -20 -14 D 18 12 20 0 22 E 6 2 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -18 -6 B 2 0 0 -12 -2 C 8 0 0 -20 -14 D 18 12 20 0 22 E 6 2 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -18 -6 B 2 0 0 -12 -2 C 8 0 0 -20 -14 D 18 12 20 0 22 E 6 2 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5435: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (14) D E C A B (10) B A C E D (10) D B E C A (7) A C E B D (7) C E A D B (6) B D E C A (6) C E D A B (5) D E B C A (4) A B C E D (4) E D C A B (3) E C D A B (3) B A D E C (3) A C E D B (3) A C B E D (3) C A E D B (2) B D A E C (2) B A D C E (2) D B A E C (1) D A E B C (1) D A B C E (1) B A C D E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -20 -14 -16 B 0 0 -12 -22 -16 C 20 12 0 -12 -8 D 14 22 12 0 8 E 16 16 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 -14 -16 B 0 0 -12 -22 -16 C 20 12 0 -12 -8 D 14 22 12 0 8 E 16 16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=24 A=19 C=13 E=6 so E is eliminated. Round 2 votes counts: D=41 B=24 A=19 C=16 so C is eliminated. Round 3 votes counts: D=49 A=27 B=24 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:216 C:206 A:175 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -20 -14 -16 B 0 0 -12 -22 -16 C 20 12 0 -12 -8 D 14 22 12 0 8 E 16 16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 -14 -16 B 0 0 -12 -22 -16 C 20 12 0 -12 -8 D 14 22 12 0 8 E 16 16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 -14 -16 B 0 0 -12 -22 -16 C 20 12 0 -12 -8 D 14 22 12 0 8 E 16 16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5436: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) B E D C A (11) E D B A C (9) D E A C B (9) B E C A D (7) B C A D E (7) E B D A C (6) C A B D E (6) B C A E D (6) D A C E B (5) B E D A C (4) E D A C B (3) D A E C B (3) C A D E B (3) B D E A C (2) A C D E B (2) E A C D B (1) B E C D A (1) B D C A E (1) B D A C E (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 -8 -10 -8 B 12 0 12 6 16 C 8 -12 0 -10 -12 D 10 -6 10 0 2 E 8 -16 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -10 -8 B 12 0 12 6 16 C 8 -12 0 -10 -12 D 10 -6 10 0 2 E 8 -16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=20 E=19 D=17 A=3 so A is eliminated. Round 2 votes counts: B=41 C=22 E=19 D=18 so D is eliminated. Round 3 votes counts: B=41 E=31 C=28 so C is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:208 E:201 C:187 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -10 -8 B 12 0 12 6 16 C 8 -12 0 -10 -12 D 10 -6 10 0 2 E 8 -16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -10 -8 B 12 0 12 6 16 C 8 -12 0 -10 -12 D 10 -6 10 0 2 E 8 -16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -10 -8 B 12 0 12 6 16 C 8 -12 0 -10 -12 D 10 -6 10 0 2 E 8 -16 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5437: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (9) E D B C A (8) A D C E B (6) C A B D E (5) A C D B E (5) D E C A B (4) D E A C B (4) B E C D A (4) B C A E D (4) A B C E D (4) D E C B A (3) C A D E B (3) B C E D A (3) A B C D E (3) E D B A C (2) E B D C A (2) D A E C B (2) C D A E B (2) C B E D A (2) B E D A C (2) B E A D C (2) B A C E D (2) A D E C B (2) A C D E B (2) A B E D C (2) E D C B A (1) E B D A C (1) D C E A B (1) D C A E B (1) D A C E B (1) C E B D A (1) C D A B E (1) C B D E A (1) C B A D E (1) B E D C A (1) A E B D C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 20 4 6 16 B -20 0 -14 2 4 C -4 14 0 4 14 D -6 -2 -4 0 16 E -16 -4 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 4 6 16 B -20 0 -14 2 4 C -4 14 0 4 14 D -6 -2 -4 0 16 E -16 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=18 D=16 C=16 E=14 so E is eliminated. Round 2 votes counts: A=36 D=27 B=21 C=16 so C is eliminated. Round 3 votes counts: A=44 D=30 B=26 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:214 D:202 B:186 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 4 6 16 B -20 0 -14 2 4 C -4 14 0 4 14 D -6 -2 -4 0 16 E -16 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 4 6 16 B -20 0 -14 2 4 C -4 14 0 4 14 D -6 -2 -4 0 16 E -16 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 4 6 16 B -20 0 -14 2 4 C -4 14 0 4 14 D -6 -2 -4 0 16 E -16 -4 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998464 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5438: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) E A B C D (6) E A C B D (5) D A B E C (5) C B E D A (5) C B D E A (5) A D E B C (5) D B C A E (4) A E D C B (4) E B C A D (3) E A D B C (3) E A B D C (3) D C B A E (3) B C E D A (3) E C B A D (2) E B C D A (2) D B A E C (2) D A B C E (2) C E B A D (2) C D B A E (2) C A D B E (2) B E C D A (2) B C D E A (2) A E C B D (2) A D E C B (2) A C E D B (2) E B D A C (1) E B A D C (1) E B A C D (1) D B E A C (1) D B C E A (1) D A E B C (1) D A C B E (1) C D A B E (1) C A E B D (1) B D C E A (1) A E C D B (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 14 18 12 2 B -14 0 14 -6 -14 C -18 -14 0 0 -20 D -12 6 0 0 -14 E -2 14 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 12 2 B -14 0 14 -6 -14 C -18 -14 0 0 -20 D -12 6 0 0 -14 E -2 14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 D=20 C=18 B=8 so B is eliminated. Round 2 votes counts: E=29 A=27 C=23 D=21 so D is eliminated. Round 3 votes counts: A=38 C=32 E=30 so E is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:223 B:190 D:190 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 12 2 B -14 0 14 -6 -14 C -18 -14 0 0 -20 D -12 6 0 0 -14 E -2 14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 12 2 B -14 0 14 -6 -14 C -18 -14 0 0 -20 D -12 6 0 0 -14 E -2 14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 12 2 B -14 0 14 -6 -14 C -18 -14 0 0 -20 D -12 6 0 0 -14 E -2 14 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998419 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5439: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) C D A B E (8) A C D E B (7) A C D B E (6) D C B A E (5) B E D A C (4) E A B C D (3) D C A B E (3) C D B E A (3) C A D E B (3) A D C B E (3) E C D B A (2) E B D C A (2) E B D A C (2) E B C D A (2) E A C B D (2) D B C E A (2) C D A E B (2) B E D C A (2) B E A D C (2) B D E C A (2) A E C D B (2) A E B C D (2) A D B C E (2) A B E D C (2) A B D E C (2) E C B D A (1) E C A D B (1) E C A B D (1) E A C D B (1) D C B E A (1) D B C A E (1) D B A C E (1) D A C B E (1) C D E B A (1) C D E A B (1) C D B A E (1) B D A C E (1) B A E D C (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 8 12 4 10 B -8 0 -12 -14 6 C -12 12 0 0 4 D -4 14 0 0 12 E -10 -6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 4 10 B -8 0 -12 -14 6 C -12 12 0 0 4 D -4 14 0 0 12 E -10 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 C=19 D=14 B=12 so B is eliminated. Round 2 votes counts: E=35 A=29 C=19 D=17 so D is eliminated. Round 3 votes counts: E=37 A=32 C=31 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:211 C:202 B:186 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 4 10 B -8 0 -12 -14 6 C -12 12 0 0 4 D -4 14 0 0 12 E -10 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 4 10 B -8 0 -12 -14 6 C -12 12 0 0 4 D -4 14 0 0 12 E -10 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 4 10 B -8 0 -12 -14 6 C -12 12 0 0 4 D -4 14 0 0 12 E -10 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5440: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) E C B D A (9) B D A E C (9) A D B C E (9) E C A D B (6) D A B C E (6) E C B A D (4) E C A B D (4) D B A C E (4) B D E C A (4) B D E A C (4) A D C B E (4) B E D C A (3) A C E D B (3) E B C D A (2) C E D A B (2) B E C D A (2) B D A C E (2) A E C D B (2) A C D E B (2) E B A C D (1) D C B A E (1) D C A E B (1) C D A E B (1) C A E D B (1) B A E D C (1) B A D E C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 0 0 -2 B -6 0 0 -6 4 C 0 0 0 -2 -8 D 0 6 2 0 0 E 2 -4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.629865 E: 0.370135 Sum of squares = 0.533730006492 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.629865 E: 1.000000 A B C D E A 0 6 0 0 -2 B -6 0 0 -6 4 C 0 0 0 -2 -8 D 0 6 2 0 0 E 2 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=26 B=26 A=22 C=14 D=12 so D is eliminated. Round 2 votes counts: B=30 A=28 E=26 C=16 so C is eliminated. Round 3 votes counts: E=38 B=31 A=31 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:204 E:203 A:202 B:196 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 0 -2 B -6 0 0 -6 4 C 0 0 0 -2 -8 D 0 6 2 0 0 E 2 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 -2 B -6 0 0 -6 4 C 0 0 0 -2 -8 D 0 6 2 0 0 E 2 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 -2 B -6 0 0 -6 4 C 0 0 0 -2 -8 D 0 6 2 0 0 E 2 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5441: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) D B A C E (7) B A E D C (7) B A E C D (7) C E D A B (6) D C E A B (5) C D E B A (5) E A C B D (4) E A B C D (4) D C B A E (4) C D E A B (4) A E B C D (4) E C A D B (3) B D A E C (3) D C B E A (2) D B C A E (2) D A B C E (2) C E B A D (2) C D B E A (2) A B E D C (2) A B E C D (2) E C A B D (1) E B A C D (1) D C E B A (1) D A B E C (1) C E D B A (1) C E A D B (1) C E A B D (1) B D C A E (1) B D A C E (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -16 18 6 14 B 16 0 16 6 12 C -18 -16 0 -4 -6 D -6 -6 4 0 6 E -14 -12 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 18 6 14 B 16 0 16 6 12 C -18 -16 0 -4 -6 D -6 -6 4 0 6 E -14 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=24 C=22 E=13 A=10 so A is eliminated. Round 2 votes counts: B=35 D=25 C=22 E=18 so E is eliminated. Round 3 votes counts: B=45 C=30 D=25 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:211 D:199 E:187 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 18 6 14 B 16 0 16 6 12 C -18 -16 0 -4 -6 D -6 -6 4 0 6 E -14 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 18 6 14 B 16 0 16 6 12 C -18 -16 0 -4 -6 D -6 -6 4 0 6 E -14 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 18 6 14 B 16 0 16 6 12 C -18 -16 0 -4 -6 D -6 -6 4 0 6 E -14 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5442: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C B E A D (7) B A C D E (7) C B A E D (6) C B A D E (6) E C B D A (5) D A E B C (4) A D B E C (4) E D C A B (3) D E A C B (3) D E A B C (3) D A C E B (3) B E A C D (3) A D B C E (3) E C D B A (2) E B A D C (2) C E B D A (2) C E B A D (2) C D A E B (2) C B E D A (2) A C B D E (2) A B D C E (2) E D B C A (1) E D A C B (1) E B D C A (1) E B D A C (1) E B C A D (1) D E C A B (1) D A E C B (1) D A C B E (1) D A B E C (1) D A B C E (1) C E D B A (1) C D E A B (1) C D A B E (1) C A B D E (1) B E C A D (1) A D E B C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 8 0 -4 B 0 0 -6 6 0 C -8 6 0 6 2 D 0 -6 -6 0 0 E 4 0 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428572 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 A B C D E A 0 0 8 0 -4 B 0 0 -6 6 0 C -8 6 0 6 2 D 0 -6 -6 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428608 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 D=18 A=14 B=11 so B is eliminated. Round 2 votes counts: C=31 E=30 A=21 D=18 so D is eliminated. Round 3 votes counts: E=37 A=32 C=31 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:203 A:202 E:201 B:200 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 0 -4 B 0 0 -6 6 0 C -8 6 0 6 2 D 0 -6 -6 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428608 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 0 -4 B 0 0 -6 6 0 C -8 6 0 6 2 D 0 -6 -6 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428608 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 0 -4 B 0 0 -6 6 0 C -8 6 0 6 2 D 0 -6 -6 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428608 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5443: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) C E B A D (6) B C E D A (6) D B E A C (5) D A B E C (5) A C E D B (5) D E A B C (4) D B A E C (4) C A E B D (4) B D E C A (4) E C A D B (3) C E A B D (3) C B E D A (3) C B E A D (3) B C D E A (3) E A D C B (2) E A C D B (2) D A E B C (2) C B A E D (2) A D C E B (2) A D B C E (2) A C D E B (2) E D C B A (1) E D A C B (1) E D A B C (1) E C D A B (1) E C B D A (1) E B C D A (1) D E B C A (1) D E B A C (1) D A B C E (1) C B A D E (1) C A E D B (1) B E C D A (1) B D E A C (1) B D C E A (1) B D A C E (1) B C D A E (1) B C A D E (1) A E D C B (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 4 4 0 -10 B -4 0 -6 -14 -10 C -4 6 0 2 -4 D 0 14 -2 0 2 E 10 10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 4 4 0 -10 B -4 0 -6 -14 -10 C -4 6 0 2 -4 D 0 14 -2 0 2 E 10 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=23 C=23 A=22 B=19 E=13 so E is eliminated. Round 2 votes counts: C=28 D=26 A=26 B=20 so B is eliminated. Round 3 votes counts: C=41 D=33 A=26 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 D:207 C:200 A:199 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 0 -10 B -4 0 -6 -14 -10 C -4 6 0 2 -4 D 0 14 -2 0 2 E 10 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 -10 B -4 0 -6 -14 -10 C -4 6 0 2 -4 D 0 14 -2 0 2 E 10 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 -10 B -4 0 -6 -14 -10 C -4 6 0 2 -4 D 0 14 -2 0 2 E 10 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5444: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (12) E D B C A (4) E D B A C (4) D E C B A (4) C D B E A (4) C D A E B (4) C B A D E (4) A C B D E (4) E B D C A (3) D C E A B (3) C D E B A (3) C D A B E (3) B E A C D (3) B A E C D (3) A D E C B (3) A C D B E (3) A B E D C (3) A B E C D (3) D E C A B (2) D C E B A (2) D A C E B (2) C D B A E (2) C B D E A (2) C A D B E (2) B E C D A (2) B E A D C (2) B C A E D (2) A E B D C (2) A C D E B (2) A B C D E (2) E D A B C (1) E B D A C (1) D E A C B (1) C A B D E (1) B C E D A (1) A E D B C (1) Total count = 100 A B C D E A 0 8 4 4 16 B -8 0 -2 0 16 C -4 2 0 24 16 D -4 0 -24 0 6 E -16 -16 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 4 16 B -8 0 -2 0 16 C -4 2 0 24 16 D -4 0 -24 0 6 E -16 -16 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996109 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=25 D=14 E=13 B=13 so E is eliminated. Round 2 votes counts: A=35 C=25 D=23 B=17 so B is eliminated. Round 3 votes counts: A=43 C=30 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:219 A:216 B:203 D:189 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 4 16 B -8 0 -2 0 16 C -4 2 0 24 16 D -4 0 -24 0 6 E -16 -16 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996109 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 4 16 B -8 0 -2 0 16 C -4 2 0 24 16 D -4 0 -24 0 6 E -16 -16 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996109 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 4 16 B -8 0 -2 0 16 C -4 2 0 24 16 D -4 0 -24 0 6 E -16 -16 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996109 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5445: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) A E D B C (6) E A B C D (5) D C B E A (5) E B C D A (4) D C E B A (4) C D B A E (4) B C E D A (4) A D C B E (4) E B C A D (3) E A D B C (3) D A C E B (3) D A C B E (3) A D E C B (3) A B C E D (3) A B C D E (3) E D C B A (2) E A B D C (2) D C B A E (2) D A E C B (2) C B D E A (2) B E C A D (2) B C A E D (2) A D C E B (2) A C B D E (2) A B E C D (2) E D A B C (1) E C D B A (1) E B D A C (1) E B A D C (1) E B A C D (1) D C A E B (1) D C A B E (1) C D B E A (1) C D A B E (1) C B D A E (1) B E A C D (1) B C E A D (1) B A E C D (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 14 18 14 12 B -14 0 10 0 -8 C -18 -10 0 6 0 D -14 0 -6 0 -10 E -12 8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 14 12 B -14 0 10 0 -8 C -18 -10 0 6 0 D -14 0 -6 0 -10 E -12 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=24 D=21 B=11 C=9 so C is eliminated. Round 2 votes counts: A=35 D=27 E=24 B=14 so B is eliminated. Round 3 votes counts: A=38 E=32 D=30 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:203 B:194 C:189 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 14 12 B -14 0 10 0 -8 C -18 -10 0 6 0 D -14 0 -6 0 -10 E -12 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 14 12 B -14 0 10 0 -8 C -18 -10 0 6 0 D -14 0 -6 0 -10 E -12 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 14 12 B -14 0 10 0 -8 C -18 -10 0 6 0 D -14 0 -6 0 -10 E -12 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5446: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) E D C B A (7) D E A C B (7) A B D E C (6) C B A E D (5) B A C E D (5) A D B E C (5) A B C D E (5) C E D B A (4) C B E D A (4) C B E A D (4) B C A E D (4) A D E B C (4) A B D C E (4) D E A B C (3) B A C D E (3) E D A B C (2) C A B D E (2) B C E A D (2) E D C A B (1) E D B C A (1) E C D B A (1) D E C B A (1) D A C E B (1) C E B D A (1) C D E B A (1) C D A E B (1) C B A D E (1) C A D B E (1) B E C D A (1) B E A D C (1) B A E C D (1) A D E C B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 0 12 2 B -6 0 -4 2 12 C 0 4 0 -6 -2 D -12 -2 6 0 10 E -2 -12 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.678457 B: 0.000000 C: 0.321543 D: 0.000000 E: 0.000000 Sum of squares = 0.563693763406 Cumulative probabilities = A: 0.678457 B: 0.678457 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 12 2 B -6 0 -4 2 12 C 0 4 0 -6 -2 D -12 -2 6 0 10 E -2 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500154 B: 0.000000 C: 0.499846 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047474 Cumulative probabilities = A: 0.500154 B: 0.500154 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=24 D=20 B=17 E=12 so E is eliminated. Round 2 votes counts: D=31 A=27 C=25 B=17 so B is eliminated. Round 3 votes counts: A=37 C=32 D=31 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:202 D:201 C:198 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 12 2 B -6 0 -4 2 12 C 0 4 0 -6 -2 D -12 -2 6 0 10 E -2 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500154 B: 0.000000 C: 0.499846 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047474 Cumulative probabilities = A: 0.500154 B: 0.500154 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 12 2 B -6 0 -4 2 12 C 0 4 0 -6 -2 D -12 -2 6 0 10 E -2 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500154 B: 0.000000 C: 0.499846 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047474 Cumulative probabilities = A: 0.500154 B: 0.500154 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 12 2 B -6 0 -4 2 12 C 0 4 0 -6 -2 D -12 -2 6 0 10 E -2 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500154 B: 0.000000 C: 0.499846 D: 0.000000 E: 0.000000 Sum of squares = 0.500000047474 Cumulative probabilities = A: 0.500154 B: 0.500154 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5447: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D B A C E (8) B D C A E (8) A D B E C (8) E B D C A (5) E A C D B (5) D B A E C (5) C E B D A (4) C B D A E (4) B D C E A (4) E C B D A (3) A D B C E (3) E B D A C (2) E B C D A (2) E A D B C (2) C E A B D (2) C B D E A (2) C A D B E (2) B C D E A (2) A C E D B (2) E D B A C (1) E C B A D (1) E A D C B (1) D B E A C (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E D B (1) C A E B D (1) C A B D E (1) B E D C A (1) B D E A C (1) B D A C E (1) B C E D A (1) B C D A E (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -16 -14 -14 -4 B 16 0 16 14 8 C 14 -16 0 -6 -2 D 14 -14 6 0 4 E 4 -8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -14 -4 B 16 0 16 14 8 C 14 -16 0 -6 -2 D 14 -14 6 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=19 B=19 D=15 A=15 so D is eliminated. Round 2 votes counts: B=33 E=32 C=19 A=16 so A is eliminated. Round 3 votes counts: B=45 E=34 C=21 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 D:205 E:197 C:195 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -14 -14 -4 B 16 0 16 14 8 C 14 -16 0 -6 -2 D 14 -14 6 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -14 -4 B 16 0 16 14 8 C 14 -16 0 -6 -2 D 14 -14 6 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -14 -4 B 16 0 16 14 8 C 14 -16 0 -6 -2 D 14 -14 6 0 4 E 4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5448: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C B A D E (9) E A D C B (5) B C D A E (5) A C B D E (5) E C A B D (4) B D C E A (4) B D C A E (4) E D B A C (3) D E B A C (3) C B D A E (3) A C E B D (3) E D B C A (2) E B D C A (2) E B C D A (2) E A C D B (2) D A E B C (2) C B E A D (2) C A E B D (2) C A B D E (2) B C A D E (2) A C D B E (2) E D A C B (1) E C A D B (1) E A D B C (1) D E A B C (1) D B E A C (1) D A B E C (1) C E B A D (1) C E A B D (1) C A B E D (1) B D E C A (1) B C E D A (1) B C D E A (1) B A C D E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C B E (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 6 0 2 -2 B -6 0 2 12 -4 C 0 -2 0 4 4 D -2 -12 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.550525 B: 0.000000 C: 0.449475 D: 0.000000 E: 0.000000 Sum of squares = 0.505105422286 Cumulative probabilities = A: 0.550525 B: 0.550525 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 2 -2 B -6 0 2 12 -4 C 0 -2 0 4 4 D -2 -12 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=21 B=19 A=17 D=8 so D is eliminated. Round 2 votes counts: E=39 C=21 B=20 A=20 so B is eliminated. Round 3 votes counts: E=41 C=38 A=21 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:203 C:203 B:202 E:200 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 2 -2 B -6 0 2 12 -4 C 0 -2 0 4 4 D -2 -12 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 2 -2 B -6 0 2 12 -4 C 0 -2 0 4 4 D -2 -12 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 2 -2 B -6 0 2 12 -4 C 0 -2 0 4 4 D -2 -12 -4 0 2 E 2 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5449: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) E A C B D (9) E C A D B (8) B D A C E (7) E C A B D (6) E C D A B (5) D E C B A (5) D B C E A (4) D B C A E (4) B A D C E (4) A B C E D (4) D C E B A (3) D C B E A (3) C E A D B (3) A E B C D (3) E D C A B (2) D B A C E (2) B D A E C (2) B A C D E (2) A B E C D (2) E D A C B (1) E A C D B (1) E A B C D (1) D E B A C (1) D B E A C (1) C E A B D (1) C D B A E (1) C A B E D (1) B C A D E (1) B A E D C (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 16 6 18 -8 B -16 0 -20 12 -18 C -6 20 0 20 -16 D -18 -12 -20 0 -20 E 8 18 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 6 18 -8 B -16 0 -20 12 -18 C -6 20 0 20 -16 D -18 -12 -20 0 -20 E 8 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=23 A=20 B=18 C=6 so C is eliminated. Round 2 votes counts: E=37 D=24 A=21 B=18 so B is eliminated. Round 3 votes counts: E=37 D=33 A=30 so A is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:231 A:216 C:209 B:179 D:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 6 18 -8 B -16 0 -20 12 -18 C -6 20 0 20 -16 D -18 -12 -20 0 -20 E 8 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 18 -8 B -16 0 -20 12 -18 C -6 20 0 20 -16 D -18 -12 -20 0 -20 E 8 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 18 -8 B -16 0 -20 12 -18 C -6 20 0 20 -16 D -18 -12 -20 0 -20 E 8 18 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5450: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C A D E B (6) C A D B E (5) B E D C A (5) B E C D A (5) E B C A D (4) B E D A C (4) A D C E B (4) E D B A C (3) E A D B C (3) D B E A C (3) D A E B C (3) D A C B E (3) C B A D E (3) C A B D E (3) A E D C B (3) A D C B E (3) E B D C A (2) E A C D B (2) D B A C E (2) C B A E D (2) C A E D B (2) B C E D A (2) A C D E B (2) E D A B C (1) E C A B D (1) D E A B C (1) D C A B E (1) D B A E C (1) D A C E B (1) D A B C E (1) C E A B D (1) C D A B E (1) C B E A D (1) C B D A E (1) B E C A D (1) B D E C A (1) B D E A C (1) B D A C E (1) A E C D B (1) Total count = 100 A B C D E A 0 -4 6 -6 -2 B 4 0 8 -4 0 C -6 -8 0 -14 -10 D 6 4 14 0 -6 E 2 0 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.385436 C: 0.000000 D: 0.000000 E: 0.614564 Sum of squares = 0.526250035323 Cumulative probabilities = A: 0.000000 B: 0.385436 C: 0.385436 D: 0.385436 E: 1.000000 A B C D E A 0 -4 6 -6 -2 B 4 0 8 -4 0 C -6 -8 0 -14 -10 D 6 4 14 0 -6 E 2 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 B=20 D=16 A=13 so A is eliminated. Round 2 votes counts: E=30 C=27 D=23 B=20 so B is eliminated. Round 3 votes counts: E=45 C=29 D=26 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:209 E:209 B:204 A:197 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 -6 -2 B 4 0 8 -4 0 C -6 -8 0 -14 -10 D 6 4 14 0 -6 E 2 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -6 -2 B 4 0 8 -4 0 C -6 -8 0 -14 -10 D 6 4 14 0 -6 E 2 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -6 -2 B 4 0 8 -4 0 C -6 -8 0 -14 -10 D 6 4 14 0 -6 E 2 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5451: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) B C A E D (8) D E B C A (7) D B C E A (6) C B A E D (6) E A D C B (5) B C A D E (5) E D A C B (4) D E A B C (4) D C B A E (4) D B E C A (4) B D C E A (4) A C E B D (4) C A B D E (3) A E C D B (3) E D B A C (2) D E C B A (2) B C D A E (2) A C B E D (2) E D A B C (1) E B A D C (1) E A B C D (1) D E C A B (1) D C A E B (1) D A E C B (1) C D B A E (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D C A (1) B C E D A (1) B C D E A (1) A E D C B (1) A E C B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -20 -12 -6 B 14 0 -2 -14 2 C 20 2 0 -14 6 D 12 14 14 0 16 E 6 -2 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -12 -6 B 14 0 -2 -14 2 C 20 2 0 -14 6 D 12 14 14 0 16 E 6 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=22 E=14 C=13 A=13 so C is eliminated. Round 2 votes counts: D=39 B=30 A=17 E=14 so E is eliminated. Round 3 votes counts: D=46 B=31 A=23 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:228 C:207 B:200 E:191 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -20 -12 -6 B 14 0 -2 -14 2 C 20 2 0 -14 6 D 12 14 14 0 16 E 6 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -12 -6 B 14 0 -2 -14 2 C 20 2 0 -14 6 D 12 14 14 0 16 E 6 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -12 -6 B 14 0 -2 -14 2 C 20 2 0 -14 6 D 12 14 14 0 16 E 6 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5452: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) B C E D A (8) C E B A D (6) C B E A D (6) B C E A D (6) A D B C E (6) E D A C B (5) B C A D E (5) E C B D A (4) B A D C E (4) D A E B C (3) D A B E C (3) C E A D B (3) C B A D E (3) C A E D B (3) B C A E D (3) E C D A B (2) E C A D B (2) E A D C B (2) D E A C B (2) A D C E B (2) E C D B A (1) E B C D A (1) C E A B D (1) C A D B E (1) B E D C A (1) B E C D A (1) B D E A C (1) B D A E C (1) B A C D E (1) A D E C B (1) A D E B C (1) A C D E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -14 18 -4 B 4 0 -8 6 4 C 14 8 0 20 22 D -18 -6 -20 0 -10 E 4 -4 -22 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 18 -4 B 4 0 -8 6 4 C 14 8 0 20 22 D -18 -6 -20 0 -10 E 4 -4 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=23 E=17 D=16 A=13 so A is eliminated. Round 2 votes counts: B=32 D=26 C=25 E=17 so E is eliminated. Round 3 votes counts: C=34 D=33 B=33 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 B:203 A:198 E:194 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 18 -4 B 4 0 -8 6 4 C 14 8 0 20 22 D -18 -6 -20 0 -10 E 4 -4 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 18 -4 B 4 0 -8 6 4 C 14 8 0 20 22 D -18 -6 -20 0 -10 E 4 -4 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 18 -4 B 4 0 -8 6 4 C 14 8 0 20 22 D -18 -6 -20 0 -10 E 4 -4 -22 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5453: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (6) C E B D A (5) C A D E B (5) B E C D A (5) A D B E C (4) A C B D E (4) E D C B A (3) C E D B A (3) B E D C A (3) A D C B E (3) A D B C E (3) A C D E B (3) E B D C A (2) E B C D A (2) D E C A B (2) D E B A C (2) D A E C B (2) D A E B C (2) D A C E B (2) D A B E C (2) C D A E B (2) C B E A D (2) B D E A C (2) B C E D A (2) B C E A D (2) B A D E C (2) B A C E D (2) A D E C B (2) A B D E C (2) E D B C A (1) E C D B A (1) E C B D A (1) D E A B C (1) D B A E C (1) C E D A B (1) C E B A D (1) C E A B D (1) C D E A B (1) C B A E D (1) C A E D B (1) B E D A C (1) B E A D C (1) B D A E C (1) B C A E D (1) B A E C D (1) A D E B C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 0 10 B -4 0 -6 -10 -6 C -4 6 0 -2 6 D 0 10 2 0 14 E -10 6 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.626952 B: 0.000000 C: 0.000000 D: 0.373048 E: 0.000000 Sum of squares = 0.532233749619 Cumulative probabilities = A: 0.626952 B: 0.626952 C: 0.626952 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 10 B -4 0 -6 -10 -6 C -4 6 0 -2 6 D 0 10 2 0 14 E -10 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=23 B=23 D=14 E=10 so E is eliminated. Round 2 votes counts: A=30 B=27 C=25 D=18 so D is eliminated. Round 3 votes counts: A=39 B=31 C=30 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:209 C:203 E:188 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 10 B -4 0 -6 -10 -6 C -4 6 0 -2 6 D 0 10 2 0 14 E -10 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 10 B -4 0 -6 -10 -6 C -4 6 0 -2 6 D 0 10 2 0 14 E -10 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 10 B -4 0 -6 -10 -6 C -4 6 0 -2 6 D 0 10 2 0 14 E -10 6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5454: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) D A E B C (7) C B E D A (6) E B C D A (5) B E C A D (5) A D C B E (5) D C E B A (4) A B E C D (4) E D B C A (3) C E B D A (3) C B E A D (3) C B A E D (3) C A B D E (3) B C E A D (3) A B C E D (3) D E C B A (2) D E B C A (2) D E A B C (2) D C A E B (2) D A E C B (2) D A C E B (2) C D A B E (2) B E A C D (2) B A E C D (2) A E D B C (2) A D B E C (2) E D B A C (1) E B D C A (1) E B D A C (1) D C A B E (1) D A C B E (1) C B D E A (1) C A B E D (1) B C E D A (1) A E B D C (1) A D C E B (1) A D B C E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -2 2 10 B -4 0 14 0 2 C 2 -14 0 -4 -6 D -2 0 4 0 -2 E -10 -2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999641 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 2 10 B -4 0 14 0 2 C 2 -14 0 -4 -6 D -2 0 4 0 -2 E -10 -2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999338 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=25 C=22 B=13 E=11 so E is eliminated. Round 2 votes counts: D=29 A=29 C=22 B=20 so B is eliminated. Round 3 votes counts: C=36 A=33 D=31 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:207 B:206 D:200 E:198 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 2 10 B -4 0 14 0 2 C 2 -14 0 -4 -6 D -2 0 4 0 -2 E -10 -2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999338 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 10 B -4 0 14 0 2 C 2 -14 0 -4 -6 D -2 0 4 0 -2 E -10 -2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999338 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 10 B -4 0 14 0 2 C 2 -14 0 -4 -6 D -2 0 4 0 -2 E -10 -2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999338 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5455: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) D A E C B (6) E A B D C (5) B C D E A (5) B C A E D (5) C B D A E (4) B E A C D (4) B C E A D (4) A E D B C (4) A D E C B (4) E B D A C (3) D C B E A (3) D C A E B (3) C D B A E (3) C B D E A (3) A E B C D (3) D E A C B (2) D C E B A (2) D C E A B (2) C B A D E (2) C A B D E (2) B E C D A (2) B C E D A (2) A E D C B (2) A E B D C (2) E D A B C (1) E B A D C (1) E A B C D (1) D E A B C (1) D C A B E (1) D B E C A (1) D A C E B (1) C D B E A (1) C D A E B (1) C D A B E (1) C B A E D (1) C A B E D (1) B A E C D (1) B A C E D (1) A E C B D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 2 4 2 B -4 0 6 8 -4 C -2 -6 0 0 -2 D -4 -8 0 0 -2 E -2 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 4 2 B -4 0 6 8 -4 C -2 -6 0 0 -2 D -4 -8 0 0 -2 E -2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=22 C=19 A=18 E=17 so E is eliminated. Round 2 votes counts: A=30 B=28 D=23 C=19 so C is eliminated. Round 3 votes counts: B=38 A=33 D=29 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:203 E:203 C:195 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 4 2 B -4 0 6 8 -4 C -2 -6 0 0 -2 D -4 -8 0 0 -2 E -2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 2 B -4 0 6 8 -4 C -2 -6 0 0 -2 D -4 -8 0 0 -2 E -2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 2 B -4 0 6 8 -4 C -2 -6 0 0 -2 D -4 -8 0 0 -2 E -2 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5456: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) A C D B E (6) E D B C A (5) E B D C A (5) C A D E B (5) D E C B A (4) B E D C A (4) A C D E B (4) E D C B A (3) D E C A B (3) C E D A B (3) C E A D B (3) C D A E B (3) B E A D C (3) B A E D C (3) A C B E D (3) A B D E C (3) E C D B A (2) C E B D A (2) C D E A B (2) A D C E B (2) A B D C E (2) A B C E D (2) A B C D E (2) E C B D A (1) E B C D A (1) D E B C A (1) D C E A B (1) D C A E B (1) D A C E B (1) C E D B A (1) C B A E D (1) C A B E D (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B C A E D (1) B A E C D (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -10 10 4 B 2 0 -8 -2 -6 C 10 8 0 14 14 D -10 2 -14 0 -16 E -4 6 -14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 10 4 B 2 0 -8 -2 -6 C 10 8 0 14 14 D -10 2 -14 0 -16 E -4 6 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 C=21 E=17 D=11 so D is eliminated. Round 2 votes counts: A=27 E=25 B=25 C=23 so C is eliminated. Round 3 votes counts: E=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:223 E:202 A:201 B:193 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 10 4 B 2 0 -8 -2 -6 C 10 8 0 14 14 D -10 2 -14 0 -16 E -4 6 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 10 4 B 2 0 -8 -2 -6 C 10 8 0 14 14 D -10 2 -14 0 -16 E -4 6 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 10 4 B 2 0 -8 -2 -6 C 10 8 0 14 14 D -10 2 -14 0 -16 E -4 6 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5457: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (5) C A E D B (5) A B E D C (5) A B C E D (5) D E C B A (4) D E B C A (4) D B E C A (4) C E D A B (4) C A D E B (4) B A E D C (4) A C E B D (4) A C B E D (4) A C B D E (4) E D B C A (3) E B D A C (3) E A D C B (3) C D E B A (3) B D E C A (3) A B E C D (3) E D C B A (2) C D B E A (2) B E D A C (2) B D C E A (2) B D A E C (2) B A D E C (2) A C E D B (2) E D B A C (1) E C D A B (1) E A D B C (1) E A C D B (1) D C E B A (1) C E A D B (1) C D A E B (1) C B D E A (1) C A D B E (1) C A B D E (1) B D E A C (1) B A E C D (1) Total count = 100 A B C D E A 0 10 -4 2 -4 B -10 0 -8 -6 -6 C 4 8 0 6 0 D -2 6 -6 0 -10 E 4 6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.674058 D: 0.000000 E: 0.325942 Sum of squares = 0.560592717421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.674058 D: 0.674058 E: 1.000000 A B C D E A 0 10 -4 2 -4 B -10 0 -8 -6 -6 C 4 8 0 6 0 D -2 6 -6 0 -10 E 4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 B=17 E=15 D=13 so D is eliminated. Round 2 votes counts: C=29 A=27 E=23 B=21 so B is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:209 A:202 D:194 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 2 -4 B -10 0 -8 -6 -6 C 4 8 0 6 0 D -2 6 -6 0 -10 E 4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 2 -4 B -10 0 -8 -6 -6 C 4 8 0 6 0 D -2 6 -6 0 -10 E 4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 2 -4 B -10 0 -8 -6 -6 C 4 8 0 6 0 D -2 6 -6 0 -10 E 4 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5458: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) A B C D E (8) A D C B E (7) E B C D A (5) C B D E A (5) B C A D E (5) B C E D A (4) A D E C B (4) A B C E D (4) E A B C D (3) D E C B A (3) D C B E A (3) D C B A E (3) C B D A E (3) E D C A B (2) E B A C D (2) E A D B C (2) D C E B A (2) C B A D E (2) B C D A E (2) A E D C B (2) A E D B C (2) A B E C D (2) E D A C B (1) E B C A D (1) E A D C B (1) E A B D C (1) D E A C B (1) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A C B E (1) C D B E A (1) B A C E D (1) A E B C D (1) A D E B C (1) A D C E B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 4 8 B 0 0 -8 0 8 C 2 8 0 -2 14 D -4 0 2 0 16 E -8 -8 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999951 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 4 8 B 0 0 -8 0 8 C 2 8 0 -2 14 D -4 0 2 0 16 E -8 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000119 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=26 D=17 B=12 C=11 so C is eliminated. Round 2 votes counts: A=34 E=26 B=22 D=18 so D is eliminated. Round 3 votes counts: A=38 E=33 B=29 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:207 A:205 B:200 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 4 8 B 0 0 -8 0 8 C 2 8 0 -2 14 D -4 0 2 0 16 E -8 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000119 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 8 B 0 0 -8 0 8 C 2 8 0 -2 14 D -4 0 2 0 16 E -8 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000119 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 8 B 0 0 -8 0 8 C 2 8 0 -2 14 D -4 0 2 0 16 E -8 -8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000119 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5459: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C E A B D (7) B A D C E (6) E C B D A (5) E C B A D (5) D A B C E (5) A B D C E (5) A B C D E (5) E C D B A (4) E C D A B (4) C E A D B (4) C A E B D (4) C E B A D (3) A D B C E (3) E B D C A (2) E B D A C (2) C A B E D (2) B D A E C (2) A B C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E B C D A (1) E B C A D (1) D E C B A (1) D E B A C (1) D E A B C (1) D C E A B (1) D B E A C (1) D B A C E (1) D A B E C (1) C D E A B (1) C B E A D (1) C A E D B (1) B E A D C (1) B E A C D (1) B A E C D (1) B A D E C (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 2 10 0 B 6 0 10 16 -2 C -2 -10 0 8 6 D -10 -16 -8 0 -10 E 0 2 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765423 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 A B C D E A 0 -6 2 10 0 B 6 0 10 16 -2 C -2 -10 0 8 6 D -10 -16 -8 0 -10 E 0 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=23 D=21 A=16 B=13 so B is eliminated. Round 2 votes counts: E=29 A=25 D=23 C=23 so D is eliminated. Round 3 votes counts: A=43 E=33 C=24 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:203 E:203 C:201 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 10 0 B 6 0 10 16 -2 C -2 -10 0 8 6 D -10 -16 -8 0 -10 E 0 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 10 0 B 6 0 10 16 -2 C -2 -10 0 8 6 D -10 -16 -8 0 -10 E 0 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 10 0 B 6 0 10 16 -2 C -2 -10 0 8 6 D -10 -16 -8 0 -10 E 0 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5460: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) A E B D C (9) C D E A B (6) B C D A E (6) D C B E A (5) C D B E A (5) A B E D C (5) B A E D C (4) E D A C B (3) E A D C B (3) C D E B A (3) B C D E A (3) A B E C D (3) E D C A B (2) D C E A B (2) D B C E A (2) B E D A C (2) B D C E A (2) A E D C B (2) A E D B C (2) E D B A C (1) E C D A B (1) E B D A C (1) E B A D C (1) E A B D C (1) D E C B A (1) D E B C A (1) D C E B A (1) D B E C A (1) C E A D B (1) C B D A E (1) C A E D B (1) C A D E B (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C A E (1) B C A D E (1) B A D E C (1) B A C D E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 -2 2 B -10 0 0 -10 -6 C 0 0 0 -8 -8 D 2 10 8 0 -6 E -2 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.44000000008 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 10 0 -2 2 B -10 0 0 -10 -6 C 0 0 0 -8 -8 D 2 10 8 0 -6 E -2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=22 C=20 E=13 D=13 so E is eliminated. Round 2 votes counts: A=36 B=24 C=21 D=19 so D is eliminated. Round 3 votes counts: A=39 C=32 B=29 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:209 D:207 A:205 C:192 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 -2 2 B -10 0 0 -10 -6 C 0 0 0 -8 -8 D 2 10 8 0 -6 E -2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -2 2 B -10 0 0 -10 -6 C 0 0 0 -8 -8 D 2 10 8 0 -6 E -2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -2 2 B -10 0 0 -10 -6 C 0 0 0 -8 -8 D 2 10 8 0 -6 E -2 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5461: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) D A B C E (6) D B C E A (5) D B C A E (4) D B A C E (4) C B E D A (4) A E C B D (4) A D B E C (4) A B E C D (4) C E B D A (3) B C E A D (3) A E B C D (3) A D E C B (3) A D E B C (3) E C B A D (2) E C A B D (2) E B C A D (2) E A C D B (2) E A C B D (2) D C B A E (2) D A C E B (2) D A C B E (2) C E B A D (2) B C E D A (2) A E D C B (2) A E B D C (2) E D C A B (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A E B (1) D A E C B (1) C E D B A (1) C D E B A (1) C D B E A (1) C B D E A (1) B D C E A (1) B D C A E (1) B D A C E (1) B A E C D (1) B A D E C (1) A E D B C (1) A E C D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 4 -6 10 B 0 0 2 -12 12 C -4 -2 0 -16 12 D 6 12 16 0 8 E -10 -12 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -6 10 B 0 0 2 -12 12 C -4 -2 0 -16 12 D 6 12 16 0 8 E -10 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=29 C=13 E=12 B=10 so B is eliminated. Round 2 votes counts: D=39 A=31 C=18 E=12 so E is eliminated. Round 3 votes counts: D=40 A=36 C=24 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:204 B:201 C:195 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -6 10 B 0 0 2 -12 12 C -4 -2 0 -16 12 D 6 12 16 0 8 E -10 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -6 10 B 0 0 2 -12 12 C -4 -2 0 -16 12 D 6 12 16 0 8 E -10 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -6 10 B 0 0 2 -12 12 C -4 -2 0 -16 12 D 6 12 16 0 8 E -10 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5462: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (17) B C E A D (16) E A D B C (6) D A C E B (6) D C A E B (5) C B D A E (5) E A D C B (4) B E A D C (4) B E A C D (4) C B E A D (3) B E C A D (3) B C D A E (3) E B A D C (2) E A B D C (2) D A E B C (2) C E A D B (2) C D A E B (2) B D C A E (2) B C D E A (2) A E D C B (2) A D E C B (2) D C B A E (1) D B C A E (1) C D B A E (1) B D A E C (1) B C E D A (1) B A E D C (1) Total count = 100 A B C D E A 0 0 6 2 2 B 0 0 0 -2 -4 C -6 0 0 -16 0 D -2 2 16 0 0 E -2 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.761335 B: 0.238665 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.636591751166 Cumulative probabilities = A: 0.761335 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 2 2 B 0 0 0 -2 -4 C -6 0 0 -16 0 D -2 2 16 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555608127 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=32 E=14 C=13 A=4 so A is eliminated. Round 2 votes counts: B=37 D=34 E=16 C=13 so C is eliminated. Round 3 votes counts: B=45 D=37 E=18 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:208 A:205 E:201 B:197 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 2 2 B 0 0 0 -2 -4 C -6 0 0 -16 0 D -2 2 16 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555608127 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 2 2 B 0 0 0 -2 -4 C -6 0 0 -16 0 D -2 2 16 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555608127 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 2 2 B 0 0 0 -2 -4 C -6 0 0 -16 0 D -2 2 16 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555608127 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5463: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (12) B C A E D (12) D E A C B (8) D E A B C (8) B A C E D (8) C B D A E (7) D E C B A (6) E D A B C (4) C D E B A (4) C B A E D (4) E A D B C (3) C D B E A (3) A B E D C (3) A E B D C (2) A B E C D (2) E D B A C (1) D E C A B (1) D E B A C (1) D C E B A (1) D C E A B (1) D C B E A (1) C D B A E (1) C D A B E (1) C B D E A (1) C A B D E (1) B E D A C (1) B E A D C (1) B C A D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -30 -14 0 10 B 30 0 -4 10 18 C 14 4 0 14 16 D 0 -10 -14 0 16 E -10 -18 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -14 0 10 B 30 0 -4 10 18 C 14 4 0 14 16 D 0 -10 -14 0 16 E -10 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=27 B=23 E=8 A=8 so E is eliminated. Round 2 votes counts: C=34 D=32 B=23 A=11 so A is eliminated. Round 3 votes counts: D=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:227 C:224 D:196 A:183 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -30 -14 0 10 B 30 0 -4 10 18 C 14 4 0 14 16 D 0 -10 -14 0 16 E -10 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -14 0 10 B 30 0 -4 10 18 C 14 4 0 14 16 D 0 -10 -14 0 16 E -10 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -14 0 10 B 30 0 -4 10 18 C 14 4 0 14 16 D 0 -10 -14 0 16 E -10 -18 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5464: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) A E D B C (7) D C E A B (6) C B D A E (5) C B A E D (5) C D A B E (4) C B E D A (4) B C A E D (4) B A E C D (4) D E C A B (3) D E A C B (3) D C E B A (3) C D E B A (3) B C E A D (3) A B E D C (3) E A D B C (2) D E A B C (2) D A E C B (2) D A C E B (2) C D B E A (2) C B E A D (2) C B D E A (2) C A B D E (2) A B E C D (2) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) D E B A C (1) D C A E B (1) D A E B C (1) C D B A E (1) C D A E B (1) C B A D E (1) C A D B E (1) B E C A D (1) B E A C D (1) B C E D A (1) A E B D C (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -10 0 14 B -8 0 -8 -12 0 C 10 8 0 0 10 D 0 12 0 0 10 E -14 0 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.512075 D: 0.487925 E: 0.000000 Sum of squares = 0.500291602756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.512075 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 0 14 B -8 0 -8 -12 0 C 10 8 0 0 10 D 0 12 0 0 10 E -14 0 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=24 A=23 B=14 E=6 so E is eliminated. Round 2 votes counts: C=33 D=26 A=25 B=16 so B is eliminated. Round 3 votes counts: C=42 A=31 D=27 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:211 A:206 B:186 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 0 14 B -8 0 -8 -12 0 C 10 8 0 0 10 D 0 12 0 0 10 E -14 0 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 0 14 B -8 0 -8 -12 0 C 10 8 0 0 10 D 0 12 0 0 10 E -14 0 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 0 14 B -8 0 -8 -12 0 C 10 8 0 0 10 D 0 12 0 0 10 E -14 0 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5465: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) A C E D B (9) A C E B D (9) B D E C A (7) E C A D B (6) D E B C A (5) D B E C A (5) D B A C E (5) B D A C E (5) A C B E D (4) E C A B D (3) D E C B A (3) C E A B D (3) B A D C E (3) E D C A B (2) E D B C A (2) C E A D B (2) C A E D B (2) C A E B D (2) B E D C A (2) D E C A B (1) D B E A C (1) D B A E C (1) D A C E B (1) B E C A D (1) B D E A C (1) B A E D C (1) B A E C D (1) B A C E D (1) A D C E B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -10 0 -6 B -12 0 -16 -12 -22 C 10 16 0 8 -2 D 0 12 -8 0 -20 E 6 22 2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -10 0 -6 B -12 0 -16 -12 -22 C 10 16 0 8 -2 D 0 12 -8 0 -20 E 6 22 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=22 D=22 B=22 C=9 so C is eliminated. Round 2 votes counts: A=29 E=27 D=22 B=22 so D is eliminated. Round 3 votes counts: E=36 B=34 A=30 so A is eliminated. Round 4 votes counts: E=61 B=39 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:216 A:198 D:192 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -10 0 -6 B -12 0 -16 -12 -22 C 10 16 0 8 -2 D 0 12 -8 0 -20 E 6 22 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 0 -6 B -12 0 -16 -12 -22 C 10 16 0 8 -2 D 0 12 -8 0 -20 E 6 22 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 0 -6 B -12 0 -16 -12 -22 C 10 16 0 8 -2 D 0 12 -8 0 -20 E 6 22 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5466: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) B E C A D (9) A B E C D (8) D C E A B (7) D A C E B (7) C D E B A (6) B E A C D (5) A B E D C (5) E B C A D (4) C E D B A (3) C E B D A (3) B C E D A (3) B A E C D (3) A D E B C (3) A D B E C (3) A B D E C (3) E C B D A (2) D C A E B (2) D C A B E (2) D A C B E (2) B C E A D (2) A D B C E (2) E C D B A (1) E C D A B (1) B C D A E (1) B A C D E (1) A E D B C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 2 -10 B 4 0 8 0 2 C 10 -8 0 4 4 D -2 0 -4 0 0 E 10 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.644118 C: 0.000000 D: 0.355882 E: 0.000000 Sum of squares = 0.541540162288 Cumulative probabilities = A: 0.000000 B: 0.644118 C: 0.644118 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 2 -10 B 4 0 8 0 2 C 10 -8 0 4 4 D -2 0 -4 0 0 E 10 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 B=24 C=12 E=8 so E is eliminated. Round 2 votes counts: D=29 B=28 A=27 C=16 so C is eliminated. Round 3 votes counts: D=40 B=33 A=27 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:205 E:202 D:197 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 2 -10 B 4 0 8 0 2 C 10 -8 0 4 4 D -2 0 -4 0 0 E 10 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 2 -10 B 4 0 8 0 2 C 10 -8 0 4 4 D -2 0 -4 0 0 E 10 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 2 -10 B 4 0 8 0 2 C 10 -8 0 4 4 D -2 0 -4 0 0 E 10 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5467: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) E C A D B (9) B D A C E (9) A B E D C (9) D B C A E (8) A B D E C (6) E A B C D (5) C E D B A (5) D B C E A (3) D B A C E (3) B A D E C (3) A E B C D (3) A B D C E (3) E C D A B (2) E A C B D (2) D C B E A (2) C D B E A (2) E C D B A (1) E C B D A (1) E C A B D (1) E A C D B (1) E A B D C (1) D C B A E (1) C E D A B (1) C D E A B (1) C D A E B (1) B D C A E (1) B A E D C (1) B A D C E (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -2 -6 2 B 6 0 14 -4 4 C 2 -14 0 -4 6 D 6 4 4 0 12 E -2 -4 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -6 2 B 6 0 14 -4 4 C 2 -14 0 -4 6 D 6 4 4 0 12 E -2 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 A=23 C=22 D=17 B=15 so B is eliminated. Round 2 votes counts: A=28 D=27 E=23 C=22 so C is eliminated. Round 3 votes counts: D=43 E=29 A=28 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:210 C:195 A:194 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 -6 2 B 6 0 14 -4 4 C 2 -14 0 -4 6 D 6 4 4 0 12 E -2 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -6 2 B 6 0 14 -4 4 C 2 -14 0 -4 6 D 6 4 4 0 12 E -2 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -6 2 B 6 0 14 -4 4 C 2 -14 0 -4 6 D 6 4 4 0 12 E -2 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5468: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) E B D C A (5) D A B C E (5) B D A E C (5) A D C B E (5) E C B D A (4) E C B A D (4) D C A E B (4) D B A E C (4) A D B C E (4) A C D B E (4) E B C D A (3) C E A B D (3) C A E D B (3) B D E A C (3) E D B C A (2) D B E C A (2) D B A C E (2) C E D B A (2) C A E B D (2) B E D C A (2) B E D A C (2) B A E D C (2) B A D E C (2) A C E D B (2) A C D E B (2) E C A B D (1) E B C A D (1) D E C B A (1) D C E A B (1) D B E A C (1) D B C E A (1) D A C B E (1) C D A E B (1) C A D E B (1) B E A D C (1) A D C E B (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 -2 6 B -2 0 -2 -14 -4 C 2 2 0 -16 8 D 2 14 16 0 4 E -6 4 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -2 6 B -2 0 -2 -14 -4 C 2 2 0 -16 8 D 2 14 16 0 4 E -6 4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=22 A=21 E=20 C=20 B=17 so B is eliminated. Round 2 votes counts: D=30 E=25 A=25 C=20 so C is eliminated. Round 3 votes counts: E=38 D=31 A=31 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:218 A:202 C:198 E:193 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -2 6 B -2 0 -2 -14 -4 C 2 2 0 -16 8 D 2 14 16 0 4 E -6 4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -2 6 B -2 0 -2 -14 -4 C 2 2 0 -16 8 D 2 14 16 0 4 E -6 4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -2 6 B -2 0 -2 -14 -4 C 2 2 0 -16 8 D 2 14 16 0 4 E -6 4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5469: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) E A B D C (5) E D A B C (4) E A D B C (4) C A D E B (4) E D A C B (3) E A D C B (3) D E B C A (3) D C B A E (3) C D B A E (3) C B A D E (3) B C A D E (3) B A E C D (3) A C E D B (3) A C B E D (3) E D B A C (2) E A B C D (2) D E A C B (2) D B E C A (2) D B C E A (2) C A B D E (2) B E D C A (2) B E D A C (2) B E C A D (2) B C D E A (2) B C D A E (2) B C A E D (2) A E C B D (2) A C E B D (2) E B D A C (1) E B A D C (1) E A C D B (1) D E C B A (1) D C E A B (1) D C B E A (1) D C A E B (1) D A E C B (1) C D A E B (1) C D A B E (1) C A D B E (1) B E A D C (1) B D E C A (1) A D C E B (1) A C D E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 16 14 18 4 B -16 0 -4 -14 -14 C -14 4 0 6 -14 D -18 14 -6 0 -16 E -4 14 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 18 4 B -16 0 -4 -14 -14 C -14 4 0 6 -14 D -18 14 -6 0 -16 E -4 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 B=20 D=17 C=15 so C is eliminated. Round 2 votes counts: A=29 E=26 B=23 D=22 so D is eliminated. Round 3 votes counts: B=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:220 C:191 D:187 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 18 4 B -16 0 -4 -14 -14 C -14 4 0 6 -14 D -18 14 -6 0 -16 E -4 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 18 4 B -16 0 -4 -14 -14 C -14 4 0 6 -14 D -18 14 -6 0 -16 E -4 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 18 4 B -16 0 -4 -14 -14 C -14 4 0 6 -14 D -18 14 -6 0 -16 E -4 14 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5470: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) C D A E B (8) E C D B A (5) E B A C D (4) C E D A B (4) C A D E B (4) B A D C E (4) A D C B E (4) A B D C E (4) E B C A D (3) D C A E B (3) D C A B E (3) C D E A B (3) B E A D C (3) B A E D C (3) B A D E C (3) A C D B E (3) A C B D E (3) A B C D E (3) E C B A D (2) D C E A B (2) B E A C D (2) A D B C E (2) A C B E D (2) E C B D A (1) E C A B D (1) E B D C A (1) E B C D A (1) D C E B A (1) D A B C E (1) C D A B E (1) B E D C A (1) B E D A C (1) B D E A C (1) B D A E C (1) B D A C E (1) B A E C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 22 -4 6 12 B -22 0 -18 -4 -4 C 4 18 0 22 14 D -6 4 -22 0 10 E -12 4 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -4 6 12 B -22 0 -18 -4 -4 C 4 18 0 22 14 D -6 4 -22 0 10 E -12 4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=23 B=21 C=20 D=10 so D is eliminated. Round 2 votes counts: C=29 E=26 A=24 B=21 so B is eliminated. Round 3 votes counts: A=37 E=34 C=29 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:229 A:218 D:193 E:184 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -4 6 12 B -22 0 -18 -4 -4 C 4 18 0 22 14 D -6 4 -22 0 10 E -12 4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -4 6 12 B -22 0 -18 -4 -4 C 4 18 0 22 14 D -6 4 -22 0 10 E -12 4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -4 6 12 B -22 0 -18 -4 -4 C 4 18 0 22 14 D -6 4 -22 0 10 E -12 4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5471: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) C E D A B (8) B A E D C (7) B A D E C (7) C D E A B (6) D A C E B (5) D B A C E (4) D A B C E (4) B A E C D (4) A B D E C (4) E C D A B (3) E C B A D (3) E C A D B (3) E B C A D (3) C D E B A (3) B E C A D (3) B E A C D (3) A D B C E (3) E B A C D (2) D C E A B (2) C E D B A (2) B C E D A (2) B A D C E (2) A E B C D (2) D C B E A (1) D A C B E (1) B D C A E (1) B D A C E (1) A E C D B (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 -2 12 B -4 0 6 -10 -4 C -4 -6 0 -2 8 D 2 10 2 0 6 E -12 4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -2 12 B -4 0 6 -10 -4 C -4 -6 0 -2 8 D 2 10 2 0 6 E -12 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=25 C=19 E=14 A=12 so A is eliminated. Round 2 votes counts: B=35 D=29 C=19 E=17 so E is eliminated. Round 3 votes counts: B=42 D=29 C=29 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:209 C:198 B:194 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -2 12 B -4 0 6 -10 -4 C -4 -6 0 -2 8 D 2 10 2 0 6 E -12 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -2 12 B -4 0 6 -10 -4 C -4 -6 0 -2 8 D 2 10 2 0 6 E -12 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -2 12 B -4 0 6 -10 -4 C -4 -6 0 -2 8 D 2 10 2 0 6 E -12 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5472: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) A C E D B (9) E C B A D (8) E C A B D (8) D A B C E (7) C E A D B (5) B E C D A (5) B D E C A (5) A D C E B (5) A C D E B (5) D A C B E (4) C A E D B (4) B E D C A (4) B D A E C (4) E B C D A (3) E B C A D (3) C E A B D (2) B D E A C (2) A D C B E (2) D B A E C (1) C D A E B (1) B D A C E (1) B C D E A (1) Total count = 100 A B C D E A 0 4 2 2 8 B -4 0 -6 -8 -6 C -2 6 0 8 14 D -2 8 -8 0 -2 E -8 6 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 8 B -4 0 -6 -8 -6 C -2 6 0 8 14 D -2 8 -8 0 -2 E -8 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=22 B=22 A=21 C=12 so C is eliminated. Round 2 votes counts: E=29 A=25 D=24 B=22 so B is eliminated. Round 3 votes counts: E=38 D=37 A=25 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:213 A:208 D:198 E:193 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 8 B -4 0 -6 -8 -6 C -2 6 0 8 14 D -2 8 -8 0 -2 E -8 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 8 B -4 0 -6 -8 -6 C -2 6 0 8 14 D -2 8 -8 0 -2 E -8 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 8 B -4 0 -6 -8 -6 C -2 6 0 8 14 D -2 8 -8 0 -2 E -8 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5473: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) C E D B A (7) E C D B A (6) C D E A B (6) B A E C D (6) A B D C E (6) D C E A B (5) B A E D C (5) E B C A D (4) B E A C D (4) D E C A B (3) D A B E C (3) C E B A D (3) B A C E D (3) A D B C E (3) E C B A D (2) D E C B A (2) C E B D A (2) A C D B E (2) A C B D E (2) A B C E D (2) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) D E B A C (1) D E A B C (1) D B A E C (1) D A B C E (1) C E D A B (1) C B E A D (1) C B A E D (1) C A D B E (1) B E C A D (1) B A D E C (1) A D C B E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -10 0 -20 B 22 0 -6 -6 -8 C 10 6 0 12 -4 D 0 6 -12 0 -18 E 20 8 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -22 -10 0 -20 B 22 0 -6 -6 -8 C 10 6 0 12 -4 D 0 6 -12 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 C=22 B=20 A=18 D=17 so D is eliminated. Round 2 votes counts: E=30 C=27 A=22 B=21 so B is eliminated. Round 3 votes counts: A=38 E=35 C=27 so C is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:212 B:201 D:188 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -22 -10 0 -20 B 22 0 -6 -6 -8 C 10 6 0 12 -4 D 0 6 -12 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -10 0 -20 B 22 0 -6 -6 -8 C 10 6 0 12 -4 D 0 6 -12 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -10 0 -20 B 22 0 -6 -6 -8 C 10 6 0 12 -4 D 0 6 -12 0 -18 E 20 8 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5474: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) B A C E D (10) D E C A B (8) D B E C A (8) C A E B D (7) D E C B A (5) C E A D B (5) C A E D B (5) B A D C E (5) A C B E D (5) D B E A C (4) A C E B D (4) E C A D B (3) D C E A B (3) B A C D E (3) D C A E B (2) B D A E C (2) E D C A B (1) E C D A B (1) E A C B D (1) D E B A C (1) C D E A B (1) C D A E B (1) C A B D E (1) B D A C E (1) B A E C D (1) B A D E C (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 -2 4 -2 B 2 0 -8 4 2 C 2 8 0 -2 6 D -4 -4 2 0 12 E 2 -2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 4 -2 B 2 0 -8 4 2 C 2 8 0 -2 6 D -4 -4 2 0 12 E 2 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=31 C=20 A=10 E=6 so E is eliminated. Round 2 votes counts: B=33 D=32 C=24 A=11 so A is eliminated. Round 3 votes counts: C=35 B=33 D=32 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 D:203 B:200 A:199 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -2 B 2 0 -8 4 2 C 2 8 0 -2 6 D -4 -4 2 0 12 E 2 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -2 B 2 0 -8 4 2 C 2 8 0 -2 6 D -4 -4 2 0 12 E 2 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -2 B 2 0 -8 4 2 C 2 8 0 -2 6 D -4 -4 2 0 12 E 2 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5475: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (14) E B D C A (8) B E C D A (8) D E B C A (6) A C B D E (5) E B A C D (4) A D C E B (4) E D B C A (3) D A C E B (3) C D A B E (3) A E B C D (3) A B E C D (3) E D B A C (2) E B D A C (2) E A B D C (2) D E C B A (2) D E A B C (2) D C A B E (2) C B D E A (2) C A D B E (2) B E C A D (2) A B C E D (2) E B C A D (1) E B A D C (1) D E C A B (1) D E A C B (1) D C B E A (1) D C B A E (1) D B E C A (1) C D B A E (1) C B D A E (1) C B A D E (1) C A B D E (1) B C E A D (1) A E D B C (1) A D E C B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 4 0 0 B -4 0 4 -4 6 C -4 -4 0 10 -8 D 0 4 -10 0 12 E 0 -6 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.785384 B: 0.000000 C: 0.000000 D: 0.214616 E: 0.000000 Sum of squares = 0.662887772537 Cumulative probabilities = A: 0.785384 B: 0.785384 C: 0.785384 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 0 0 B -4 0 4 -4 6 C -4 -4 0 10 -8 D 0 4 -10 0 12 E 0 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836760267 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=23 D=20 C=11 B=11 so C is eliminated. Round 2 votes counts: A=38 D=24 E=23 B=15 so B is eliminated. Round 3 votes counts: A=39 E=34 D=27 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:204 D:203 B:201 C:197 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 0 B -4 0 4 -4 6 C -4 -4 0 10 -8 D 0 4 -10 0 12 E 0 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836760267 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 0 B -4 0 4 -4 6 C -4 -4 0 10 -8 D 0 4 -10 0 12 E 0 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836760267 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 0 B -4 0 4 -4 6 C -4 -4 0 10 -8 D 0 4 -10 0 12 E 0 -6 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836760267 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5476: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) D C A B E (5) C A E D B (5) B D A C E (5) B A E C D (5) A C E B D (5) D B E C A (4) D B E A C (4) B E A C D (4) B A C E D (4) B A C D E (4) E D B A C (3) E C A D B (3) E A C B D (3) D E C A B (3) D C A E B (3) C A E B D (3) E D C A B (2) E C D A B (2) E A B C D (2) D E C B A (2) D C E A B (2) C E A D B (2) C A D B E (2) C A B E D (2) A C B E D (2) E C A B D (1) D E B C A (1) D E B A C (1) D C B A E (1) C D E A B (1) C A D E B (1) C A B D E (1) B E A D C (1) B D E A C (1) B D A E C (1) B A D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 4 16 B -2 0 -2 -8 10 C 6 2 0 6 14 D -4 8 -6 0 0 E -16 -10 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 4 16 B -2 0 -2 -8 10 C 6 2 0 6 14 D -4 8 -6 0 0 E -16 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=26 C=17 E=16 A=8 so A is eliminated. Round 2 votes counts: D=33 B=27 C=24 E=16 so E is eliminated. Round 3 votes counts: D=38 C=33 B=29 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:208 B:199 D:199 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 4 16 B -2 0 -2 -8 10 C 6 2 0 6 14 D -4 8 -6 0 0 E -16 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 4 16 B -2 0 -2 -8 10 C 6 2 0 6 14 D -4 8 -6 0 0 E -16 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 4 16 B -2 0 -2 -8 10 C 6 2 0 6 14 D -4 8 -6 0 0 E -16 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5477: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) D A C E B (7) B E C A D (7) B E D C A (6) D B E C A (5) A B C E D (5) D E C B A (4) D A B E C (4) D E C A B (3) D E B C A (3) D B E A C (3) B E C D A (3) B D E A C (3) A D C E B (3) A C B E D (3) E B D C A (2) D C E A B (2) D A C B E (2) C A E B D (2) C A B E D (2) B D E C A (2) B D A E C (2) B A E C D (2) A C E D B (2) A C E B D (2) E D B C A (1) D C A E B (1) D A E B C (1) C E B A D (1) C A E D B (1) B E A D C (1) B E A C D (1) B C E A D (1) B C A E D (1) B A C E D (1) A D B C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 6 -8 4 B -2 0 10 -4 14 C -6 -10 0 -12 -6 D 8 4 12 0 12 E -4 -14 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -8 4 B -2 0 10 -4 14 C -6 -10 0 -12 -6 D 8 4 12 0 12 E -4 -14 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=30 A=26 C=6 E=3 so E is eliminated. Round 2 votes counts: D=36 B=32 A=26 C=6 so C is eliminated. Round 3 votes counts: D=36 B=33 A=31 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:209 A:202 E:188 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -8 4 B -2 0 10 -4 14 C -6 -10 0 -12 -6 D 8 4 12 0 12 E -4 -14 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -8 4 B -2 0 10 -4 14 C -6 -10 0 -12 -6 D 8 4 12 0 12 E -4 -14 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -8 4 B -2 0 10 -4 14 C -6 -10 0 -12 -6 D 8 4 12 0 12 E -4 -14 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5478: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) A C D E B (6) D C B E A (5) B E D C A (5) D C A E B (4) B E C D A (4) A D C E B (4) A B D E C (4) E C B D A (3) E C B A D (3) E C A B D (3) C E D B A (3) B E A C D (3) B D C E A (3) E B C D A (2) E A C B D (2) D B C E A (2) D B A C E (2) D A C B E (2) C E A D B (2) C A D E B (2) B E A D C (2) B D A E C (2) A E C B D (2) A D B C E (2) A C E D B (2) E B A C D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A B E (1) D A C E B (1) D A B C E (1) C E B D A (1) C A E D B (1) B E D A C (1) B D E C A (1) B D E A C (1) B D A C E (1) B A E D C (1) B A D E C (1) A E D C B (1) A E C D B (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -8 4 -12 B 10 0 -6 8 -6 C 8 6 0 -2 -2 D -4 -8 2 0 -2 E 12 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -8 4 -12 B 10 0 -6 8 -6 C 8 6 0 -2 -2 D -4 -8 2 0 -2 E 12 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=24 E=21 D=21 C=9 so C is eliminated. Round 2 votes counts: E=27 A=27 B=25 D=21 so D is eliminated. Round 3 votes counts: A=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:211 C:205 B:203 D:194 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -8 4 -12 B 10 0 -6 8 -6 C 8 6 0 -2 -2 D -4 -8 2 0 -2 E 12 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 4 -12 B 10 0 -6 8 -6 C 8 6 0 -2 -2 D -4 -8 2 0 -2 E 12 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 4 -12 B 10 0 -6 8 -6 C 8 6 0 -2 -2 D -4 -8 2 0 -2 E 12 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5479: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) C B D A E (7) A E D B C (7) E A D B C (6) E A C D B (5) E C A D B (4) D B A E C (4) C E A D B (4) B D E C A (4) B D C E A (4) B D A E C (4) E D B A C (3) C B D E A (3) C A E B D (3) B D E A C (3) B D C A E (3) A E C D B (3) E B D A C (2) D B E A C (2) C E A B D (2) C A B E D (2) B D A C E (2) A E D C B (2) A D B E C (2) A C E D B (2) E C B D A (1) E C A B D (1) E A D C B (1) D E B A C (1) C B E D A (1) C A D B E (1) B C D A E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 2 10 6 B -10 0 -2 -14 -10 C -2 2 0 -2 -10 D -10 14 2 0 -14 E -6 10 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 10 6 B -10 0 -2 -14 -10 C -2 2 0 -2 -10 D -10 14 2 0 -14 E -6 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 B=21 A=18 D=7 so D is eliminated. Round 2 votes counts: C=31 B=27 E=24 A=18 so A is eliminated. Round 3 votes counts: E=36 C=34 B=30 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:214 E:214 D:196 C:194 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 10 6 B -10 0 -2 -14 -10 C -2 2 0 -2 -10 D -10 14 2 0 -14 E -6 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 10 6 B -10 0 -2 -14 -10 C -2 2 0 -2 -10 D -10 14 2 0 -14 E -6 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 10 6 B -10 0 -2 -14 -10 C -2 2 0 -2 -10 D -10 14 2 0 -14 E -6 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5480: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) C D A B E (7) A C B E D (6) A B E C D (5) E B D A C (4) E A D B C (4) E A B D C (4) C A D B E (4) A E B D C (4) D C A E B (3) D B C E A (3) C B A D E (3) C A B D E (3) B D E C A (3) B C D E A (3) A E D B C (3) A C E D B (3) E B A D C (2) D E B C A (2) C D A E B (2) B E D C A (2) B E D A C (2) B D C E A (2) B A E C D (2) A E D C B (2) A E B C D (2) A C D B E (2) E D B C A (1) E D B A C (1) E B D C A (1) D E A C B (1) D C E B A (1) D C B E A (1) D A E C B (1) B E C D A (1) B A C E D (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 16 2 4 10 B -16 0 4 2 14 C -2 -4 0 6 4 D -4 -2 -6 0 -2 E -10 -14 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 4 10 B -16 0 4 2 14 C -2 -4 0 6 4 D -4 -2 -6 0 -2 E -10 -14 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=26 E=17 B=16 D=12 so D is eliminated. Round 2 votes counts: C=31 A=30 E=20 B=19 so B is eliminated. Round 3 votes counts: C=39 A=33 E=28 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:202 C:202 D:193 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 4 10 B -16 0 4 2 14 C -2 -4 0 6 4 D -4 -2 -6 0 -2 E -10 -14 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 4 10 B -16 0 4 2 14 C -2 -4 0 6 4 D -4 -2 -6 0 -2 E -10 -14 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 4 10 B -16 0 4 2 14 C -2 -4 0 6 4 D -4 -2 -6 0 -2 E -10 -14 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5481: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) A D B E C (8) E C B A D (6) D B A C E (6) D A B C E (6) A E C D B (6) D A C E B (5) B E C A D (5) C E B D A (4) C E A B D (4) B E C D A (4) B D C E A (4) A E C B D (4) B C E D A (3) E C A B D (2) D C E B A (2) D C A E B (2) D A B E C (2) C E D B A (2) C E B A D (2) B D E C A (2) A D C E B (2) A B E C D (2) E B C A D (1) D C E A B (1) D B A E C (1) C E A D B (1) B A E C D (1) B A D E C (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -10 -8 -6 B 8 0 12 -10 10 C 10 -12 0 -6 8 D 8 10 6 0 6 E 6 -10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -8 -6 B 8 0 12 -10 10 C 10 -12 0 -6 8 D 8 10 6 0 6 E 6 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=23 B=20 C=13 E=9 so E is eliminated. Round 2 votes counts: D=35 A=23 C=21 B=21 so C is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:210 C:200 E:191 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -8 -6 B 8 0 12 -10 10 C 10 -12 0 -6 8 D 8 10 6 0 6 E 6 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -8 -6 B 8 0 12 -10 10 C 10 -12 0 -6 8 D 8 10 6 0 6 E 6 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -8 -6 B 8 0 12 -10 10 C 10 -12 0 -6 8 D 8 10 6 0 6 E 6 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5482: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) D C E A B (6) D C B A E (6) B A E C D (6) D E C A B (5) C D A B E (5) E D A B C (4) B E A C D (4) B C D A E (4) D C B E A (3) C D B A E (3) C D A E B (3) C B D A E (3) B E A D C (3) B A C D E (3) E D B C A (2) E B A D C (2) E A D C B (2) D C A E B (2) D B C E A (2) B D C E A (2) B C A D E (2) A E C B D (2) A B C D E (2) E D C B A (1) E D C A B (1) E D A C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D E C B A (1) D C E B A (1) D B C A E (1) C A D E B (1) C A D B E (1) B A C E D (1) A C D E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -10 -12 -4 B 0 0 2 -8 6 C 10 -2 0 6 6 D 12 8 -6 0 14 E 4 -6 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000003 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -12 -4 B 0 0 2 -8 6 C 10 -2 0 6 6 D 12 8 -6 0 14 E 4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.125000 E: 0.000000 Sum of squares = 0.40624999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 B=25 C=16 A=7 so A is eliminated. Round 2 votes counts: B=29 E=27 D=27 C=17 so C is eliminated. Round 3 votes counts: D=41 B=32 E=27 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:210 B:200 E:189 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -12 -4 B 0 0 2 -8 6 C 10 -2 0 6 6 D 12 8 -6 0 14 E 4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.125000 E: 0.000000 Sum of squares = 0.40624999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -12 -4 B 0 0 2 -8 6 C 10 -2 0 6 6 D 12 8 -6 0 14 E 4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.125000 E: 0.000000 Sum of squares = 0.40624999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -12 -4 B 0 0 2 -8 6 C 10 -2 0 6 6 D 12 8 -6 0 14 E 4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.500000 D: 0.125000 E: 0.000000 Sum of squares = 0.40624999999 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5483: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) E B C A D (10) D A E B C (8) D A C B E (6) A D E B C (6) C B E A D (4) C B D A E (4) A D E C B (4) D A B C E (3) C B E D A (3) B E C A D (3) B C D E A (3) E B A D C (2) E A B C D (2) D B A C E (2) D A C E B (2) D A B E C (2) A E D B C (2) E C B A D (1) E C A B D (1) E B A C D (1) E A D B C (1) E A B D C (1) D C A B E (1) D B A E C (1) D A E C B (1) C E B A D (1) C E A B D (1) C D B A E (1) C B D E A (1) C B A E D (1) C A E B D (1) C A D E B (1) C A B D E (1) B E D A C (1) B E C D A (1) B C E A D (1) B C D A E (1) A E D C B (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -4 -4 2 B 6 0 24 12 2 C 4 -24 0 10 0 D 4 -12 -10 0 -2 E -2 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -4 2 B 6 0 24 12 2 C 4 -24 0 10 0 D 4 -12 -10 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=21 E=19 C=19 A=15 so A is eliminated. Round 2 votes counts: D=37 E=23 B=21 C=19 so C is eliminated. Round 3 votes counts: D=39 B=35 E=26 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:199 C:195 A:194 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -4 2 B 6 0 24 12 2 C 4 -24 0 10 0 D 4 -12 -10 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -4 2 B 6 0 24 12 2 C 4 -24 0 10 0 D 4 -12 -10 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -4 2 B 6 0 24 12 2 C 4 -24 0 10 0 D 4 -12 -10 0 -2 E -2 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5484: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (19) B E D C A (15) D E B C A (6) A C B E D (6) A D C E B (5) D E C B A (3) C B A E D (3) A C E D B (3) A B C E D (3) D B E C A (2) D A E C B (2) C D E B A (2) C B E D A (2) C B E A D (2) B E C D A (2) B D E C A (2) A D E C B (2) A B D E C (2) E D C B A (1) E D B C A (1) E B D C A (1) D C A E B (1) C E B D A (1) C D E A B (1) C A E D B (1) B E D A C (1) B D E A C (1) B C E D A (1) B A E D C (1) B A C E D (1) A D E B C (1) A D B E C (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 6 10 12 B -4 0 -14 -4 -2 C -6 14 0 2 8 D -10 4 -2 0 4 E -12 2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 10 12 B -4 0 -14 -4 -2 C -6 14 0 2 8 D -10 4 -2 0 4 E -12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=47 B=24 D=14 C=12 E=3 so E is eliminated. Round 2 votes counts: A=47 B=25 D=16 C=12 so C is eliminated. Round 3 votes counts: A=48 B=33 D=19 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:209 D:198 E:189 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 10 12 B -4 0 -14 -4 -2 C -6 14 0 2 8 D -10 4 -2 0 4 E -12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 10 12 B -4 0 -14 -4 -2 C -6 14 0 2 8 D -10 4 -2 0 4 E -12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 10 12 B -4 0 -14 -4 -2 C -6 14 0 2 8 D -10 4 -2 0 4 E -12 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5485: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (16) E B C D A (12) D A C B E (10) A D C B E (10) A D E C B (7) D C A B E (6) E A B D C (4) C B D A E (4) E C B D A (3) E B A C D (3) E A D B C (3) D A C E B (3) B E C A D (3) E B A D C (2) C B E D A (2) E D C B A (1) E D B A C (1) D E C A B (1) D A E C B (1) C D B A E (1) C B D E A (1) B E A C D (1) B C E A D (1) B C A D E (1) A E D B C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -4 8 -8 B 4 0 0 6 -18 C 4 0 0 -4 -20 D -8 -6 4 0 -6 E 8 18 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 8 -8 B 4 0 0 6 -18 C 4 0 0 -4 -20 D -8 -6 4 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=45 D=21 A=20 C=8 B=6 so B is eliminated. Round 2 votes counts: E=49 D=21 A=20 C=10 so C is eliminated. Round 3 votes counts: E=52 D=27 A=21 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:196 B:196 D:192 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 8 -8 B 4 0 0 6 -18 C 4 0 0 -4 -20 D -8 -6 4 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 8 -8 B 4 0 0 6 -18 C 4 0 0 -4 -20 D -8 -6 4 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 8 -8 B 4 0 0 6 -18 C 4 0 0 -4 -20 D -8 -6 4 0 -6 E 8 18 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5486: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (13) D B A C E (9) D A B E C (8) C E B A D (7) E D A B C (6) E C B A D (4) E C A B D (4) D A B C E (4) B C A D E (4) B A D C E (4) E C D A B (3) E A D B C (3) B D A C E (3) E D A C B (2) E A D C B (2) B C D A E (2) A D B C E (2) A B D C E (2) E C D B A (1) E A C B D (1) E A B D C (1) D E B C A (1) D E B A C (1) D E A B C (1) D B C A E (1) D A E B C (1) C E B D A (1) C D E B A (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A E D (1) B D C A E (1) A D E B C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 6 4 18 B 14 0 14 4 18 C -6 -14 0 -8 18 D -4 -4 8 0 24 E -18 -18 -18 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 4 18 B 14 0 14 4 18 C -6 -14 0 -8 18 D -4 -4 8 0 24 E -18 -18 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 C=26 B=14 A=7 so A is eliminated. Round 2 votes counts: D=30 E=27 C=26 B=17 so B is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:225 D:212 A:207 C:195 E:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 6 4 18 B 14 0 14 4 18 C -6 -14 0 -8 18 D -4 -4 8 0 24 E -18 -18 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 4 18 B 14 0 14 4 18 C -6 -14 0 -8 18 D -4 -4 8 0 24 E -18 -18 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 4 18 B 14 0 14 4 18 C -6 -14 0 -8 18 D -4 -4 8 0 24 E -18 -18 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5487: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (19) D A E B C (18) C B D A E (6) A D E C B (5) A D E B C (4) D A B C E (3) B E C D A (3) B C E D A (3) E B A D C (2) D A C E B (2) B E D A C (2) B C E A D (2) B C D E A (2) A E D C B (2) A E D B C (2) A D C E B (2) E D B A C (1) E D A B C (1) E B D A C (1) E B C A D (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D E B A C (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B E C (1) C E B A D (1) C D A B E (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D E B (1) B E C A D (1) B E A D C (1) B D E A C (1) B C D A E (1) Total count = 100 A B C D E A 0 -2 12 -4 4 B 2 0 8 -2 0 C -12 -8 0 -10 -6 D 4 2 10 0 4 E -4 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 -4 4 B 2 0 8 -2 0 C -12 -8 0 -10 -6 D 4 2 10 0 4 E -4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=28 B=16 A=15 E=10 so E is eliminated. Round 2 votes counts: C=31 D=30 B=20 A=19 so A is eliminated. Round 3 votes counts: D=47 C=32 B=21 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:205 B:204 E:199 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 -4 4 B 2 0 8 -2 0 C -12 -8 0 -10 -6 D 4 2 10 0 4 E -4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 -4 4 B 2 0 8 -2 0 C -12 -8 0 -10 -6 D 4 2 10 0 4 E -4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 -4 4 B 2 0 8 -2 0 C -12 -8 0 -10 -6 D 4 2 10 0 4 E -4 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5488: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (14) C A E D B (10) B D E C A (8) B D E A C (8) D E B C A (5) C E D A B (4) B C D A E (4) B C A D E (4) B A D E C (4) E D C B A (3) E D A C B (3) D E B A C (3) C E D B A (3) B D C E A (3) A C B D E (3) E D B C A (2) D B E C A (2) C A B D E (2) A E D C B (2) A B D E C (2) E D C A B (1) E D B A C (1) D B C E A (1) C A E B D (1) B D C A E (1) B D A E C (1) B C D E A (1) B A C D E (1) A E C D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -10 -8 4 B 10 0 2 -10 -8 C 10 -2 0 0 8 D 8 10 0 0 6 E -4 8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.472681 D: 0.527319 E: 0.000000 Sum of squares = 0.501492693888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.472681 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -8 4 B 10 0 2 -10 -8 C 10 -2 0 0 8 D 8 10 0 0 6 E -4 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=24 C=20 D=11 E=10 so E is eliminated. Round 2 votes counts: B=35 A=24 D=21 C=20 so C is eliminated. Round 3 votes counts: A=37 B=35 D=28 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:212 C:208 B:197 E:195 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 -8 4 B 10 0 2 -10 -8 C 10 -2 0 0 8 D 8 10 0 0 6 E -4 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -8 4 B 10 0 2 -10 -8 C 10 -2 0 0 8 D 8 10 0 0 6 E -4 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -8 4 B 10 0 2 -10 -8 C 10 -2 0 0 8 D 8 10 0 0 6 E -4 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5489: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) C A B E D (8) D E A B C (7) D E B A C (5) C A E D B (5) A D E C B (5) B E D A C (4) B C E A D (4) A C E D B (4) D B E A C (3) C A D E B (3) B C E D A (3) B C A E D (3) A E D C B (3) E D A B C (2) D C B E A (2) D A C E B (2) C A E B D (2) C A D B E (2) B E A C D (2) B D E C A (2) B D E A C (2) B C D E A (2) A E D B C (2) E D B A C (1) E B D A C (1) E A B D C (1) D E A C B (1) D B E C A (1) C D B E A (1) C D A E B (1) C D A B E (1) C B D E A (1) C A B D E (1) B E C D A (1) B E C A D (1) B D C E A (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -6 12 4 B -4 0 -4 -6 6 C 6 4 0 10 10 D -12 6 -10 0 -12 E -4 -6 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 12 4 B -4 0 -4 -6 6 C 6 4 0 10 10 D -12 6 -10 0 -12 E -4 -6 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=25 D=21 A=16 E=5 so E is eliminated. Round 2 votes counts: C=33 B=26 D=24 A=17 so A is eliminated. Round 3 votes counts: C=39 D=34 B=27 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:207 B:196 E:196 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 12 4 B -4 0 -4 -6 6 C 6 4 0 10 10 D -12 6 -10 0 -12 E -4 -6 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 12 4 B -4 0 -4 -6 6 C 6 4 0 10 10 D -12 6 -10 0 -12 E -4 -6 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 12 4 B -4 0 -4 -6 6 C 6 4 0 10 10 D -12 6 -10 0 -12 E -4 -6 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5490: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) C A E B D (6) B D E A C (6) A C B D E (6) E D B C A (5) E A C B D (4) C A E D B (4) C A D E B (4) B D A C E (4) A C D B E (4) E D C A B (3) E B D C A (3) D B E C A (3) D B A C E (3) C A D B E (3) E C A D B (2) E C A B D (2) E B A C D (2) D E B C A (2) B E D A C (2) A C E B D (2) A C B E D (2) A B C D E (2) E C B A D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B C E A (1) D B C A E (1) C D A B E (1) B E A D C (1) B E A C D (1) B A E C D (1) B A C D E (1) A E C B D (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 14 4 -2 B 0 0 4 22 -8 C -14 -4 0 2 -6 D -4 -22 -2 0 -12 E 2 8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 14 4 -2 B 0 0 4 22 -8 C -14 -4 0 2 -6 D -4 -22 -2 0 -12 E 2 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=19 C=18 B=16 D=12 so D is eliminated. Round 2 votes counts: E=37 B=24 C=20 A=19 so A is eliminated. Round 3 votes counts: E=38 C=35 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:209 A:208 C:189 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 14 4 -2 B 0 0 4 22 -8 C -14 -4 0 2 -6 D -4 -22 -2 0 -12 E 2 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 4 -2 B 0 0 4 22 -8 C -14 -4 0 2 -6 D -4 -22 -2 0 -12 E 2 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 4 -2 B 0 0 4 22 -8 C -14 -4 0 2 -6 D -4 -22 -2 0 -12 E 2 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5491: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) C B E A D (10) B C E A D (8) D A E C B (6) D A E B C (5) D A B E C (4) B E C A D (4) D A C E B (3) C E D B A (3) C E B D A (3) C B E D A (3) C B D E A (3) B E A C D (3) E D C A B (2) E C A B D (2) D C E A B (2) C D E B A (2) B D A C E (2) B A E C D (2) A E D B C (2) E C B A D (1) E B C A D (1) E A D C B (1) E A C B D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C E B A (1) D A C B E (1) D A B C E (1) C D E A B (1) C D B E A (1) B D C A E (1) B C D A E (1) B A E D C (1) B A D E C (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -24 -22 6 -32 B 24 0 -18 22 -4 C 22 18 0 22 14 D -6 -22 -22 0 -22 E 32 4 -14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -22 6 -32 B 24 0 -18 22 -4 C 22 18 0 22 14 D -6 -22 -22 0 -22 E 32 4 -14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=25 B=23 E=9 A=6 so A is eliminated. Round 2 votes counts: C=37 D=27 B=24 E=12 so E is eliminated. Round 3 votes counts: C=41 D=32 B=27 so B is eliminated. Round 4 votes counts: C=61 D=39 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:238 E:222 B:212 A:164 D:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -24 -22 6 -32 B 24 0 -18 22 -4 C 22 18 0 22 14 D -6 -22 -22 0 -22 E 32 4 -14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -22 6 -32 B 24 0 -18 22 -4 C 22 18 0 22 14 D -6 -22 -22 0 -22 E 32 4 -14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -22 6 -32 B 24 0 -18 22 -4 C 22 18 0 22 14 D -6 -22 -22 0 -22 E 32 4 -14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5492: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (16) E C A D B (11) C E D B A (11) E A C B D (9) B D A C E (9) A E C B D (9) D B C A E (7) C D B E A (7) A E B D C (5) D B A C E (3) D B C E A (2) E C D B A (1) E A C D B (1) D C B E A (1) C E D A B (1) C E A D B (1) C A D B E (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A E C (1) A C E D B (1) Total count = 100 A B C D E A 0 10 10 8 4 B -10 0 -8 4 0 C -10 8 0 6 -10 D -8 -4 -6 0 0 E -4 0 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 8 4 B -10 0 -8 4 0 C -10 8 0 6 -10 D -8 -4 -6 0 0 E -4 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=22 C=21 D=13 B=13 so D is eliminated. Round 2 votes counts: A=31 B=25 E=22 C=22 so E is eliminated. Round 3 votes counts: A=41 C=34 B=25 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:203 C:197 B:193 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 8 4 B -10 0 -8 4 0 C -10 8 0 6 -10 D -8 -4 -6 0 0 E -4 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 8 4 B -10 0 -8 4 0 C -10 8 0 6 -10 D -8 -4 -6 0 0 E -4 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 8 4 B -10 0 -8 4 0 C -10 8 0 6 -10 D -8 -4 -6 0 0 E -4 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5493: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (13) B E D A C (11) A C B E D (11) D E B C A (10) C A B E D (10) C A D E B (7) C D E B A (6) B A E D C (5) C D E A B (4) A B E D C (4) E B D A C (2) D E C B A (2) D C E B A (2) C A D B E (2) C A B D E (2) B E A D C (2) D E A B C (1) C D B E A (1) C A E D B (1) C A E B D (1) B D E A C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 4 -6 -10 B 10 0 0 0 0 C -4 0 0 -6 -4 D 6 0 6 0 2 E 10 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.425742 C: 0.000000 D: 0.574258 E: 0.000000 Sum of squares = 0.511028492931 Cumulative probabilities = A: 0.000000 B: 0.425742 C: 0.425742 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -6 -10 B 10 0 0 0 0 C -4 0 0 -6 -4 D 6 0 6 0 2 E 10 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=28 B=19 A=17 E=2 so E is eliminated. Round 2 votes counts: C=34 D=28 B=21 A=17 so A is eliminated. Round 3 votes counts: C=46 D=28 B=26 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:207 E:206 B:205 C:193 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 -6 -10 B 10 0 0 0 0 C -4 0 0 -6 -4 D 6 0 6 0 2 E 10 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -6 -10 B 10 0 0 0 0 C -4 0 0 -6 -4 D 6 0 6 0 2 E 10 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -6 -10 B 10 0 0 0 0 C -4 0 0 -6 -4 D 6 0 6 0 2 E 10 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5494: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) C B D E A (6) D E C A B (5) D C E A B (5) C D E B A (5) C D B E A (5) B C A D E (5) E D A C B (4) B C D E A (4) B A E D C (4) A E D C B (4) A E D B C (4) A B E D C (4) C D A E B (3) B A C E D (3) D E C B A (2) D E A C B (2) D C E B A (2) C B D A E (2) B E D C A (2) B E A D C (2) B C D A E (2) B C A E D (2) B A E C D (2) A E B D C (2) E D C A B (1) E A D C B (1) E A B D C (1) C D E A B (1) C D B A E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D A C (1) B D C E A (1) A E C D B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -14 -8 -12 B 4 0 -10 -10 -2 C 14 10 0 -8 4 D 8 10 8 0 12 E 12 2 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -8 -12 B 4 0 -10 -10 -2 C 14 10 0 -8 4 D 8 10 8 0 12 E 12 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=26 A=17 D=16 E=13 so E is eliminated. Round 2 votes counts: B=28 C=26 A=25 D=21 so D is eliminated. Round 3 votes counts: C=41 A=31 B=28 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:210 E:199 B:191 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -14 -8 -12 B 4 0 -10 -10 -2 C 14 10 0 -8 4 D 8 10 8 0 12 E 12 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -8 -12 B 4 0 -10 -10 -2 C 14 10 0 -8 4 D 8 10 8 0 12 E 12 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -8 -12 B 4 0 -10 -10 -2 C 14 10 0 -8 4 D 8 10 8 0 12 E 12 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5495: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) B C D E A (6) C B E A D (5) B D A C E (5) A E D B C (5) A E C D B (5) E A D C B (4) D B A E C (4) C A E B D (4) B D C E A (4) A E D C B (4) E A C D B (3) C E A D B (3) C B D E A (3) C B A E D (3) B D C A E (3) B C A D E (3) E D C A B (2) D E A B C (2) C E B D A (2) C B E D A (2) C B A D E (2) B D A E C (2) B C D A E (2) B A D C E (2) E D A C B (1) E D A B C (1) E C D A B (1) D E B A C (1) D A B E C (1) C E D B A (1) C E A B D (1) B A D E C (1) B A C D E (1) A D B E C (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 14 16 B 4 0 6 8 4 C -6 -6 0 -4 8 D -14 -8 4 0 4 E -16 -4 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 14 16 B 4 0 6 8 4 C -6 -6 0 -4 8 D -14 -8 4 0 4 E -16 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=26 A=25 E=12 D=8 so D is eliminated. Round 2 votes counts: B=33 C=26 A=26 E=15 so E is eliminated. Round 3 votes counts: A=37 B=34 C=29 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:211 C:196 D:193 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 14 16 B 4 0 6 8 4 C -6 -6 0 -4 8 D -14 -8 4 0 4 E -16 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 14 16 B 4 0 6 8 4 C -6 -6 0 -4 8 D -14 -8 4 0 4 E -16 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 14 16 B 4 0 6 8 4 C -6 -6 0 -4 8 D -14 -8 4 0 4 E -16 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5496: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (11) A C E D B (8) A E C B D (7) D B C E A (6) C A E D B (5) B D C E A (5) E A C D B (4) C D B A E (4) A E B D C (4) E B D A C (3) B E D A C (3) E C A D B (2) E A B D C (2) D B E C A (2) C E A D B (2) C D B E A (2) C A D B E (2) B D C A E (2) B D A C E (2) A E B C D (2) A C B E D (2) A C B D E (2) A B D E C (2) E D C B A (1) E C D B A (1) E C D A B (1) E B A D C (1) E A C B D (1) C D E B A (1) C A D E B (1) B D E A C (1) B D A E C (1) B A E D C (1) B A D C E (1) A E C D B (1) A C E B D (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 4 8 2 B -4 0 0 10 2 C -4 0 0 2 -4 D -8 -10 -2 0 -6 E -2 -2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 8 2 B -4 0 0 10 2 C -4 0 0 2 -4 D -8 -10 -2 0 -6 E -2 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=27 C=17 E=16 D=8 so D is eliminated. Round 2 votes counts: B=35 A=32 C=17 E=16 so E is eliminated. Round 3 votes counts: B=39 A=39 C=22 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:204 E:203 C:197 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 8 2 B -4 0 0 10 2 C -4 0 0 2 -4 D -8 -10 -2 0 -6 E -2 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 8 2 B -4 0 0 10 2 C -4 0 0 2 -4 D -8 -10 -2 0 -6 E -2 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 8 2 B -4 0 0 10 2 C -4 0 0 2 -4 D -8 -10 -2 0 -6 E -2 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5497: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) C A E B D (6) A C E B D (6) D B A C E (5) C E A B D (5) A C E D B (5) E C B A D (4) D B E A C (4) B D C E A (4) B D C A E (4) D B C A E (3) D A C B E (3) D A B C E (3) C A B E D (3) A D C E B (3) A C D E B (3) B C E D A (2) A E D C B (2) A E C B D (2) A D C B E (2) A C D B E (2) E D A B C (1) E C A B D (1) E B D C A (1) E B C D A (1) E B C A D (1) E A C D B (1) E A C B D (1) D E B C A (1) D E B A C (1) D B C E A (1) D B A E C (1) C E B A D (1) C B E A D (1) C B A E D (1) B E D C A (1) B D E C A (1) B C E A D (1) B C D A E (1) A E C D B (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -4 8 16 B -4 0 -10 -2 4 C 4 10 0 0 32 D -8 2 0 0 2 E -16 -4 -32 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.775950 D: 0.224050 E: 0.000000 Sum of squares = 0.652296941453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.775950 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 8 16 B -4 0 -10 -2 4 C 4 10 0 0 32 D -8 2 0 0 2 E -16 -4 -32 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555672072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=28 C=17 B=14 E=11 so E is eliminated. Round 2 votes counts: D=31 A=30 C=22 B=17 so B is eliminated. Round 3 votes counts: D=42 A=30 C=28 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:223 A:212 D:198 B:194 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 8 16 B -4 0 -10 -2 4 C 4 10 0 0 32 D -8 2 0 0 2 E -16 -4 -32 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555672072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 8 16 B -4 0 -10 -2 4 C 4 10 0 0 32 D -8 2 0 0 2 E -16 -4 -32 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555672072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 8 16 B -4 0 -10 -2 4 C 4 10 0 0 32 D -8 2 0 0 2 E -16 -4 -32 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555672072 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5498: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) D A C E B (10) B E A C D (8) B E D A C (6) B E C A D (6) C A D E B (5) B E A D C (5) E A D C B (4) E D A C B (3) D E A C B (3) C D A E B (3) C B A D E (3) B E D C A (3) B C A E D (3) E D B A C (2) E A B C D (2) D C A B E (2) B D C E A (2) B C D A E (2) A C D E B (2) E B D A C (1) E B A D C (1) E B A C D (1) E A B D C (1) D E B A C (1) D B C A E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E C D A (1) B D C A E (1) B C D E A (1) B C A D E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 2 4 -8 -2 B -2 0 -2 0 -4 C -4 2 0 -14 2 D 8 0 14 0 2 E 2 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.280438 C: 0.000000 D: 0.719562 E: 0.000000 Sum of squares = 0.596415284686 Cumulative probabilities = A: 0.000000 B: 0.280438 C: 0.280438 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -8 -2 B -2 0 -2 0 -4 C -4 2 0 -14 2 D 8 0 14 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555573007 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=28 E=15 C=14 A=4 so A is eliminated. Round 2 votes counts: B=39 D=28 C=17 E=16 so E is eliminated. Round 3 votes counts: B=45 D=37 C=18 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:212 E:201 A:198 B:196 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -8 -2 B -2 0 -2 0 -4 C -4 2 0 -14 2 D 8 0 14 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555573007 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -8 -2 B -2 0 -2 0 -4 C -4 2 0 -14 2 D 8 0 14 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555573007 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -8 -2 B -2 0 -2 0 -4 C -4 2 0 -14 2 D 8 0 14 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555573007 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5499: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (12) D B C A E (11) E A C D B (7) E A C B D (7) D A E C B (7) B D C A E (6) B C E A D (5) D B A E C (4) C E A B D (4) C B D E A (4) A E D B C (4) C B D A E (3) B C D E A (3) B C D A E (3) E A D C B (2) D A E B C (2) C D B A E (2) E A B D C (1) E A B C D (1) D C B A E (1) D C A E B (1) D C A B E (1) C E A D B (1) C D A E B (1) C D A B E (1) C B E A D (1) B E A C D (1) B D E C A (1) B D C E A (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 8 0 -4 22 B -8 0 -12 -18 -4 C 0 12 0 -10 -2 D 4 18 10 0 6 E -22 4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -4 22 B -8 0 -12 -18 -4 C 0 12 0 -10 -2 D 4 18 10 0 6 E -22 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=20 E=18 A=18 C=17 so C is eliminated. Round 2 votes counts: D=31 B=28 E=23 A=18 so A is eliminated. Round 3 votes counts: E=40 D=32 B=28 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:213 C:200 E:189 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -4 22 B -8 0 -12 -18 -4 C 0 12 0 -10 -2 D 4 18 10 0 6 E -22 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -4 22 B -8 0 -12 -18 -4 C 0 12 0 -10 -2 D 4 18 10 0 6 E -22 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -4 22 B -8 0 -12 -18 -4 C 0 12 0 -10 -2 D 4 18 10 0 6 E -22 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5500: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) B A D E C (11) E C B A D (9) A B D C E (8) E C D B A (7) D C E A B (7) D C A E B (7) C D E A B (5) B E A C D (5) B A E C D (5) C E D A B (4) E B A C D (3) D A C B E (3) A B D E C (3) B A D C E (2) A D B C E (2) E D C A B (1) E D A B C (1) E C D A B (1) E C B D A (1) E B C A D (1) D E C A B (1) C E D B A (1) B A E D C (1) Total count = 100 A B C D E A 0 8 10 0 6 B -8 0 6 -2 2 C -10 -6 0 -16 0 D 0 2 16 0 20 E -6 -2 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.626349 B: 0.000000 C: 0.000000 D: 0.373651 E: 0.000000 Sum of squares = 0.531928348195 Cumulative probabilities = A: 0.626349 B: 0.626349 C: 0.626349 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 0 6 B -8 0 6 -2 2 C -10 -6 0 -16 0 D 0 2 16 0 20 E -6 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 B=24 A=13 C=10 so C is eliminated. Round 2 votes counts: D=34 E=29 B=24 A=13 so A is eliminated. Round 3 votes counts: D=36 B=35 E=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:212 B:199 E:186 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 0 6 B -8 0 6 -2 2 C -10 -6 0 -16 0 D 0 2 16 0 20 E -6 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 0 6 B -8 0 6 -2 2 C -10 -6 0 -16 0 D 0 2 16 0 20 E -6 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 0 6 B -8 0 6 -2 2 C -10 -6 0 -16 0 D 0 2 16 0 20 E -6 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5501: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) C D E A B (8) B A C E D (8) E D C A B (7) D E C B A (5) A B C E D (4) D C E B A (3) C D E B A (3) B D E A C (3) B C A D E (3) B A E D C (3) B A C D E (3) A C B E D (3) E D B A C (2) D E B A C (2) D E A B C (2) C E D A B (2) C D B E A (2) C A B E D (2) C A B D E (2) B D A C E (2) B A E C D (2) B A D E C (2) B A D C E (2) A E B D C (2) E D A C B (1) E A D C B (1) D E B C A (1) D C E A B (1) D C B E A (1) C E A D B (1) C B D A E (1) C A E D B (1) B E A D C (1) B D A E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -16 -14 B 0 0 -12 -8 -8 C 8 12 0 -6 6 D 16 8 6 0 16 E 14 8 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -16 -14 B 0 0 -12 -8 -8 C 8 12 0 -6 6 D 16 8 6 0 16 E 14 8 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=26 C=22 E=11 A=11 so E is eliminated. Round 2 votes counts: D=36 B=30 C=22 A=12 so A is eliminated. Round 3 votes counts: D=37 B=37 C=26 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:210 E:200 B:186 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -16 -14 B 0 0 -12 -8 -8 C 8 12 0 -6 6 D 16 8 6 0 16 E 14 8 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -16 -14 B 0 0 -12 -8 -8 C 8 12 0 -6 6 D 16 8 6 0 16 E 14 8 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -16 -14 B 0 0 -12 -8 -8 C 8 12 0 -6 6 D 16 8 6 0 16 E 14 8 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5502: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E C B A D (7) E B A C D (7) B A E C D (7) E D C B A (6) E C D B A (6) D E A B C (6) E D A B C (5) D A B C E (5) C D B A E (5) B A C E D (4) A B E D C (4) D E C A B (3) D C E A B (3) B A C D E (3) E A B D C (2) C B A D E (2) A B D E C (2) A B D C E (2) E C B D A (1) E B A D C (1) D C E B A (1) D C A E B (1) D A B E C (1) C E B A D (1) C D E B A (1) C D A B E (1) C B D A E (1) C B A E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -12 0 B 8 0 0 -8 -2 C 0 0 0 -4 -16 D 12 8 4 0 -6 E 0 2 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.109457 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.890543 Sum of squares = 0.805047822053 Cumulative probabilities = A: 0.109457 B: 0.109457 C: 0.109457 D: 0.109457 E: 1.000000 A B C D E A 0 -8 0 -12 0 B 8 0 0 -8 -2 C 0 0 0 -4 -16 D 12 8 4 0 -6 E 0 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000004064 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=30 B=14 C=12 A=9 so A is eliminated. Round 2 votes counts: E=35 D=30 B=23 C=12 so C is eliminated. Round 3 votes counts: D=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:209 B:199 A:190 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -12 0 B 8 0 0 -8 -2 C 0 0 0 -4 -16 D 12 8 4 0 -6 E 0 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000004064 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -12 0 B 8 0 0 -8 -2 C 0 0 0 -4 -16 D 12 8 4 0 -6 E 0 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000004064 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -12 0 B 8 0 0 -8 -2 C 0 0 0 -4 -16 D 12 8 4 0 -6 E 0 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000004064 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5503: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) D E B A C (7) D A E C B (7) B C E A D (7) D E A B C (6) E D B C A (5) E B C D A (5) D A C E B (5) C B A E D (5) C A B E D (5) E B D C A (4) D E B C A (4) B E C A D (4) A D C E B (4) A C B D E (4) D E A C B (3) C A B D E (3) B C A E D (3) A D C B E (3) E D B A C (2) C B E A D (1) C B A D E (1) B E C D A (1) B A E C D (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 0 4 2 2 B 0 0 0 -8 -6 C -4 0 0 -2 0 D -2 8 2 0 10 E -2 6 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.887264 B: 0.112736 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.799946870181 Cumulative probabilities = A: 0.887264 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 2 2 B 0 0 0 -8 -6 C -4 0 0 -2 0 D -2 8 2 0 10 E -2 6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000003 Cumulative probabilities = A: 0.800000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=21 E=16 B=16 C=15 so C is eliminated. Round 2 votes counts: D=32 A=29 B=23 E=16 so E is eliminated. Round 3 votes counts: D=39 B=32 A=29 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:209 A:204 C:197 E:197 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 2 2 B 0 0 0 -8 -6 C -4 0 0 -2 0 D -2 8 2 0 10 E -2 6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000003 Cumulative probabilities = A: 0.800000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 2 2 B 0 0 0 -8 -6 C -4 0 0 -2 0 D -2 8 2 0 10 E -2 6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000003 Cumulative probabilities = A: 0.800000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 2 2 B 0 0 0 -8 -6 C -4 0 0 -2 0 D -2 8 2 0 10 E -2 6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000000003 Cumulative probabilities = A: 0.800000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5504: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) A C B E D (7) D E B C A (6) E D B A C (5) E A C D B (5) C A B D E (5) B D E C A (5) E D A C B (4) D B C E A (4) A E C D B (4) A E C B D (4) D B C A E (3) B C D A E (3) E D A B C (2) E A B D C (2) D E C A B (2) D C E A B (2) D C B A E (2) D B E C A (2) C A D B E (2) B D C E A (2) B C A D E (2) A C E D B (2) A C E B D (2) E D C A B (1) E B D A C (1) E B A D C (1) E A D C B (1) D E C B A (1) D E B A C (1) D C B E A (1) D C A E B (1) C D E A B (1) C B A D E (1) C A B E D (1) B D E A C (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -12 -18 0 B 2 0 2 -4 4 C 12 -2 0 -18 4 D 18 4 18 0 14 E 0 -4 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -18 0 B 2 0 2 -4 4 C 12 -2 0 -18 4 D 18 4 18 0 14 E 0 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=23 E=22 A=20 C=10 so C is eliminated. Round 2 votes counts: A=28 D=26 B=24 E=22 so E is eliminated. Round 3 votes counts: D=38 A=36 B=26 so B is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 B:202 C:198 E:189 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 -18 0 B 2 0 2 -4 4 C 12 -2 0 -18 4 D 18 4 18 0 14 E 0 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -18 0 B 2 0 2 -4 4 C 12 -2 0 -18 4 D 18 4 18 0 14 E 0 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -18 0 B 2 0 2 -4 4 C 12 -2 0 -18 4 D 18 4 18 0 14 E 0 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5505: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (13) C B E D A (12) D A C B E (11) E B C A D (10) D A C E B (9) B E C A D (6) B C E D A (6) E C B D A (3) D A E C B (3) C E B D A (3) A D E C B (3) A D B E C (3) A D B C E (3) E A D C B (2) D C A B E (2) C D A B E (2) E B A C D (1) E A D B C (1) E A B D C (1) D E C A B (1) C D B A E (1) C B D E A (1) B C E A D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 12 4 -8 4 B -12 0 -6 -10 -4 C -4 6 0 -8 2 D 8 10 8 0 4 E -4 4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -8 4 B -12 0 -6 -10 -4 C -4 6 0 -8 2 D 8 10 8 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=24 C=19 E=18 B=13 so B is eliminated. Round 2 votes counts: D=26 C=26 E=24 A=24 so E is eliminated. Round 3 votes counts: C=45 A=29 D=26 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:206 C:198 E:197 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 4 -8 4 B -12 0 -6 -10 -4 C -4 6 0 -8 2 D 8 10 8 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -8 4 B -12 0 -6 -10 -4 C -4 6 0 -8 2 D 8 10 8 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -8 4 B -12 0 -6 -10 -4 C -4 6 0 -8 2 D 8 10 8 0 4 E -4 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5506: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) C D B E A (7) C B D E A (7) B C A E D (5) C A B E D (4) A E D B C (4) D E A B C (3) D C E B A (3) D C E A B (3) D C B E A (3) C B A D E (3) C A E D B (3) B D C E A (3) B C D E A (3) A E C B D (3) E D A C B (2) E D A B C (2) E A D C B (2) E A D B C (2) D E C A B (2) D E B A C (2) D E A C B (2) D B E C A (2) D B E A C (2) C B D A E (2) C A D E B (2) A E D C B (2) E B A D C (1) C B A E D (1) C A E B D (1) B D E C A (1) B D E A C (1) B C D A E (1) B C A D E (1) B A E D C (1) B A E C D (1) B A C E D (1) A E C D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -14 -2 -6 B 2 0 -10 2 0 C 14 10 0 -4 10 D 2 -2 4 0 6 E 6 0 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000022 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 -2 -6 B 2 0 -10 2 0 C 14 10 0 -4 10 D 2 -2 4 0 6 E 6 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999984 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 A=21 B=18 E=9 so E is eliminated. Round 2 votes counts: C=30 D=26 A=25 B=19 so B is eliminated. Round 3 votes counts: C=40 D=31 A=29 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:205 B:197 E:195 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -14 -2 -6 B 2 0 -10 2 0 C 14 10 0 -4 10 D 2 -2 4 0 6 E 6 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999984 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -2 -6 B 2 0 -10 2 0 C 14 10 0 -4 10 D 2 -2 4 0 6 E 6 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999984 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -2 -6 B 2 0 -10 2 0 C 14 10 0 -4 10 D 2 -2 4 0 6 E 6 0 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999984 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5507: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (16) D E A B C (10) C B A E D (7) B C A D E (6) A B E C D (6) D E A C B (5) B A C E D (5) D E C A B (4) E D A C B (3) E A D C B (3) D B C E A (3) A B C E D (3) D E B A C (2) C B D E A (2) B C D A E (2) A E D B C (2) E A D B C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E C A (1) C E D A B (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E D A (1) C B A D E (1) C A B E D (1) B D A C E (1) B C D E A (1) B A D E C (1) A E D C B (1) A E B D C (1) A E B C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -6 16 12 B 8 0 26 16 20 C 6 -26 0 14 14 D -16 -16 -14 0 -10 E -12 -20 -14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 16 12 B 8 0 26 16 20 C 6 -26 0 14 14 D -16 -16 -14 0 -10 E -12 -20 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=29 C=16 A=16 E=7 so E is eliminated. Round 2 votes counts: D=32 B=32 A=20 C=16 so C is eliminated. Round 3 votes counts: B=43 D=35 A=22 so A is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:235 A:207 C:204 E:182 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 16 12 B 8 0 26 16 20 C 6 -26 0 14 14 D -16 -16 -14 0 -10 E -12 -20 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 16 12 B 8 0 26 16 20 C 6 -26 0 14 14 D -16 -16 -14 0 -10 E -12 -20 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 16 12 B 8 0 26 16 20 C 6 -26 0 14 14 D -16 -16 -14 0 -10 E -12 -20 -14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5508: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) B A C E D (9) B A E C D (8) B A D C E (8) D C E A B (7) D E C A B (4) B D E C A (4) B D A C E (4) A B C E D (4) B A C D E (3) A C E D B (3) E D C B A (2) D E C B A (2) C E D A B (2) B E D C A (2) B E C D A (2) B E C A D (2) B D C E A (2) B D A E C (2) A C E B D (2) A B E C D (2) E C B D A (1) E C B A D (1) E C A D B (1) E C A B D (1) D C E B A (1) D B E C A (1) D B C E A (1) C E A D B (1) B E A C D (1) B A D E C (1) A D C E B (1) A D B C E (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 4 4 2 B 14 0 18 22 18 C -4 -18 0 12 4 D -4 -22 -12 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 4 2 B 14 0 18 22 18 C -4 -18 0 12 4 D -4 -22 -12 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=48 E=17 D=16 A=16 C=3 so C is eliminated. Round 2 votes counts: B=48 E=20 D=16 A=16 so D is eliminated. Round 3 votes counts: B=50 E=34 A=16 so A is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:236 A:198 C:197 E:193 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 4 2 B 14 0 18 22 18 C -4 -18 0 12 4 D -4 -22 -12 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 4 2 B 14 0 18 22 18 C -4 -18 0 12 4 D -4 -22 -12 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 4 2 B 14 0 18 22 18 C -4 -18 0 12 4 D -4 -22 -12 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5509: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) E B D A C (6) E D B A C (5) B C E D A (5) D E A C B (4) C B A D E (4) B E D C A (4) A C D E B (4) D E C B A (3) D A E C B (3) B E C D A (3) B E A D C (3) A D C E B (3) A C B D E (3) A B C E D (3) E D B C A (2) E D A B C (2) E B D C A (2) E A D B C (2) D E C A B (2) C D B E A (2) B E D A C (2) B E C A D (2) B C E A D (2) A E D B C (2) A D E C B (2) A C B E D (2) D E B C A (1) D E A B C (1) D C E B A (1) D C E A B (1) C D E B A (1) C D A B E (1) C B D E A (1) C B A E D (1) C A D E B (1) C A D B E (1) B C D E A (1) B C A E D (1) B A E D C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 0 -6 -16 B 6 0 2 10 4 C 0 -2 0 -4 -6 D 6 -10 4 0 -2 E 16 -4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 -16 B 6 0 2 10 4 C 0 -2 0 -4 -6 D 6 -10 4 0 -2 E 16 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=21 C=20 E=19 D=16 so D is eliminated. Round 2 votes counts: E=30 B=24 A=24 C=22 so C is eliminated. Round 3 votes counts: A=35 E=33 B=32 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:211 E:210 D:199 C:194 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -6 -16 B 6 0 2 10 4 C 0 -2 0 -4 -6 D 6 -10 4 0 -2 E 16 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 -16 B 6 0 2 10 4 C 0 -2 0 -4 -6 D 6 -10 4 0 -2 E 16 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 -16 B 6 0 2 10 4 C 0 -2 0 -4 -6 D 6 -10 4 0 -2 E 16 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5510: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) D A B C E (9) C B E D A (7) A D E B C (6) E C B D A (5) C E B D A (5) C E B A D (5) A D B E C (5) B C E D A (4) D B E A C (3) C E A B D (3) A D E C B (3) E C A D B (2) E B C D A (2) E A D C B (2) D B A C E (2) D A B E C (2) C A B D E (2) B E C D A (2) B C D E A (2) A E C D B (2) A C E D B (2) E D A B C (1) E C D A B (1) E C B A D (1) D B A E C (1) D A E B C (1) C B D E A (1) C B A D E (1) C A E D B (1) B E D C A (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 6 2 -8 2 B -6 0 12 -8 16 C -2 -12 0 -2 18 D 8 8 2 0 8 E -2 -16 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -8 2 B -6 0 12 -8 16 C -2 -12 0 -2 18 D 8 8 2 0 8 E -2 -16 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 D=18 E=14 B=14 so E is eliminated. Round 2 votes counts: C=34 A=31 D=19 B=16 so B is eliminated. Round 3 votes counts: C=45 A=31 D=24 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:213 B:207 A:201 C:201 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -8 2 B -6 0 12 -8 16 C -2 -12 0 -2 18 D 8 8 2 0 8 E -2 -16 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -8 2 B -6 0 12 -8 16 C -2 -12 0 -2 18 D 8 8 2 0 8 E -2 -16 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -8 2 B -6 0 12 -8 16 C -2 -12 0 -2 18 D 8 8 2 0 8 E -2 -16 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5511: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) D B A E C (7) C E A B D (7) C A E B D (7) E C D A B (5) D E B C A (5) A B C E D (5) B D A E C (4) E C D B A (3) E C A D B (3) D E B A C (3) D B E C A (3) C E A D B (3) C A B E D (3) B A C D E (3) A C B E D (3) E D C B A (2) D E C B A (2) C B A E D (2) B D C A E (2) B D A C E (2) B C A D E (2) B A D E C (2) B A D C E (2) E D C A B (1) E A C D B (1) D E C A B (1) D B C E A (1) D A B E C (1) C E D A B (1) C E B A D (1) B C D A E (1) B A C E D (1) A E C D B (1) A C E B D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -10 -2 2 B 10 0 6 0 6 C 10 -6 0 6 -4 D 2 0 -6 0 0 E -2 -6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.630439 C: 0.000000 D: 0.369561 E: 0.000000 Sum of squares = 0.534028846072 Cumulative probabilities = A: 0.000000 B: 0.630439 C: 0.630439 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -2 2 B 10 0 6 0 6 C 10 -6 0 6 -4 D 2 0 -6 0 0 E -2 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500289 C: 0.000000 D: 0.499711 E: 0.000000 Sum of squares = 0.500000167061 Cumulative probabilities = A: 0.000000 B: 0.500289 C: 0.500289 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=24 B=19 E=15 A=12 so A is eliminated. Round 2 votes counts: D=30 C=28 B=26 E=16 so E is eliminated. Round 3 votes counts: C=41 D=33 B=26 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:203 D:198 E:198 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 -2 2 B 10 0 6 0 6 C 10 -6 0 6 -4 D 2 0 -6 0 0 E -2 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500289 C: 0.000000 D: 0.499711 E: 0.000000 Sum of squares = 0.500000167061 Cumulative probabilities = A: 0.000000 B: 0.500289 C: 0.500289 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -2 2 B 10 0 6 0 6 C 10 -6 0 6 -4 D 2 0 -6 0 0 E -2 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500289 C: 0.000000 D: 0.499711 E: 0.000000 Sum of squares = 0.500000167061 Cumulative probabilities = A: 0.000000 B: 0.500289 C: 0.500289 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -2 2 B 10 0 6 0 6 C 10 -6 0 6 -4 D 2 0 -6 0 0 E -2 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500289 C: 0.000000 D: 0.499711 E: 0.000000 Sum of squares = 0.500000167061 Cumulative probabilities = A: 0.000000 B: 0.500289 C: 0.500289 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5512: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A C B D E (7) D B E C A (6) E D B C A (5) E B C D A (5) D E B C A (4) D B E A C (4) D A B C E (4) C A B E D (4) B C A D E (4) E C B A D (3) C E A B D (3) C A E B D (3) B D C A E (3) A C E D B (3) A C E B D (3) D E B A C (2) C E B A D (2) B E D C A (2) B D E C A (2) B D C E A (2) A C B E D (2) E D C A B (1) E D B A C (1) E C A D B (1) E A D C B (1) D E A B C (1) D B A E C (1) D B A C E (1) D A E B C (1) D A B E C (1) C B E A D (1) C B A E D (1) C A B D E (1) B E C D A (1) B C E D A (1) A E D C B (1) A E C D B (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -18 -26 -12 -12 B 18 0 20 16 0 C 26 -20 0 -8 -4 D 12 -16 8 0 -6 E 12 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.376467 C: 0.000000 D: 0.000000 E: 0.623533 Sum of squares = 0.53052104307 Cumulative probabilities = A: 0.000000 B: 0.376467 C: 0.376467 D: 0.376467 E: 1.000000 A B C D E A 0 -18 -26 -12 -12 B 18 0 20 16 0 C 26 -20 0 -8 -4 D 12 -16 8 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 A=20 C=15 B=15 so C is eliminated. Round 2 votes counts: E=30 A=28 D=25 B=17 so B is eliminated. Round 3 votes counts: E=35 A=33 D=32 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:227 E:211 D:199 C:197 A:166 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -26 -12 -12 B 18 0 20 16 0 C 26 -20 0 -8 -4 D 12 -16 8 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -26 -12 -12 B 18 0 20 16 0 C 26 -20 0 -8 -4 D 12 -16 8 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -26 -12 -12 B 18 0 20 16 0 C 26 -20 0 -8 -4 D 12 -16 8 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5513: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) A C E B D (9) E D B C A (8) D B E C A (7) D B A C E (6) A E C B D (6) A C B E D (5) E C B A D (4) A C B D E (4) E D B A C (3) E B D C A (3) D A B C E (3) B D C E A (3) A D C B E (3) E A D B C (2) E A C D B (2) D A E B C (2) C E A B D (2) C A B D E (2) B D E C A (2) A E C D B (2) D E B C A (1) D B E A C (1) D B C A E (1) C B E A D (1) C B D A E (1) C A E B D (1) C A B E D (1) B D C A E (1) B C D E A (1) B C D A E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 12 20 12 2 B -12 0 -10 12 -12 C -20 10 0 8 -4 D -12 -12 -8 0 -18 E -2 12 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 20 12 2 B -12 0 -10 12 -12 C -20 10 0 8 -4 D -12 -12 -8 0 -18 E -2 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995552 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=31 D=21 C=8 B=8 so C is eliminated. Round 2 votes counts: A=36 E=33 D=21 B=10 so B is eliminated. Round 3 votes counts: A=36 E=34 D=30 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:216 C:197 B:189 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 20 12 2 B -12 0 -10 12 -12 C -20 10 0 8 -4 D -12 -12 -8 0 -18 E -2 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995552 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 20 12 2 B -12 0 -10 12 -12 C -20 10 0 8 -4 D -12 -12 -8 0 -18 E -2 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995552 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 20 12 2 B -12 0 -10 12 -12 C -20 10 0 8 -4 D -12 -12 -8 0 -18 E -2 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995552 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5514: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) D B A C E (8) C E A B D (6) B D C A E (6) B C E D A (6) A D E C B (6) D A E B C (5) B D A C E (5) A E D C B (5) A E C D B (5) E A D C B (4) D B C A E (4) D A B E C (4) C E B A D (4) E C A B D (3) E A C D B (3) C B E A D (3) B D C E A (3) A D E B C (3) D E A C B (1) D B A E C (1) D A E C B (1) C B E D A (1) C A E B D (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 -6 2 -12 10 B 6 0 14 0 6 C -2 -14 0 -12 18 D 12 0 12 0 16 E -10 -6 -18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.578739 C: 0.000000 D: 0.421261 E: 0.000000 Sum of squares = 0.51239950811 Cumulative probabilities = A: 0.000000 B: 0.578739 C: 0.578739 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -12 10 B 6 0 14 0 6 C -2 -14 0 -12 18 D 12 0 12 0 16 E -10 -6 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=24 A=19 C=15 E=10 so E is eliminated. Round 2 votes counts: B=32 A=26 D=24 C=18 so C is eliminated. Round 3 votes counts: B=40 A=36 D=24 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:213 A:197 C:195 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -12 10 B 6 0 14 0 6 C -2 -14 0 -12 18 D 12 0 12 0 16 E -10 -6 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -12 10 B 6 0 14 0 6 C -2 -14 0 -12 18 D 12 0 12 0 16 E -10 -6 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -12 10 B 6 0 14 0 6 C -2 -14 0 -12 18 D 12 0 12 0 16 E -10 -6 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5515: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B E D A C (7) B D E C A (7) B A E D C (7) A C E D B (6) B C A D E (4) B A C E D (4) A E D C B (4) A B E D C (4) E D A C B (3) E D A B C (3) C B D E A (3) C A D E B (3) A C B E D (3) E D B A C (2) E A D C B (2) D E C B A (2) D E B C A (2) D E A C B (2) C D E A B (2) C B D A E (2) C A B E D (2) B D E A C (2) A E D B C (2) E B D A C (1) D C E B A (1) D B E C A (1) C D E B A (1) C B A D E (1) C A E D B (1) B E A D C (1) B C D E A (1) B C D A E (1) B A E C D (1) A E C D B (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 16 -2 -2 B 2 0 4 8 6 C -16 -4 0 -22 -26 D 2 -8 22 0 -14 E 2 -6 26 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 -2 -2 B 2 0 4 8 6 C -16 -4 0 -22 -26 D 2 -8 22 0 -14 E 2 -6 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=23 D=16 C=15 E=11 so E is eliminated. Round 2 votes counts: B=36 A=25 D=24 C=15 so C is eliminated. Round 3 votes counts: B=42 A=31 D=27 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:210 A:205 D:201 C:166 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 -2 -2 B 2 0 4 8 6 C -16 -4 0 -22 -26 D 2 -8 22 0 -14 E 2 -6 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 -2 -2 B 2 0 4 8 6 C -16 -4 0 -22 -26 D 2 -8 22 0 -14 E 2 -6 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 -2 -2 B 2 0 4 8 6 C -16 -4 0 -22 -26 D 2 -8 22 0 -14 E 2 -6 26 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5516: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D E A C B (8) B A C E D (8) E D C B A (5) D E C A B (5) C E D B A (5) C B E D A (5) E D C A B (4) B C E D A (4) A D E B C (4) D E B C A (3) C B E A D (3) A D E C B (3) A B D E C (3) A B C D E (3) E C D B A (2) D E C B A (2) D E A B C (2) C E B D A (2) C B A E D (2) B D E C A (2) B A C D E (2) A C E D B (2) A B D C E (2) E C D A B (1) D A E C B (1) D A E B C (1) C E D A B (1) C E B A D (1) C A E D B (1) C A E B D (1) B D A E C (1) B A D C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -16 -8 -10 B 14 0 -10 -2 -10 C 16 10 0 6 6 D 8 2 -6 0 -12 E 10 10 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -8 -10 B 14 0 -10 -2 -10 C 16 10 0 6 6 D 8 2 -6 0 -12 E 10 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 C=21 A=18 E=12 so E is eliminated. Round 2 votes counts: D=31 B=27 C=24 A=18 so A is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:219 E:213 B:196 D:196 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -16 -8 -10 B 14 0 -10 -2 -10 C 16 10 0 6 6 D 8 2 -6 0 -12 E 10 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -8 -10 B 14 0 -10 -2 -10 C 16 10 0 6 6 D 8 2 -6 0 -12 E 10 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -8 -10 B 14 0 -10 -2 -10 C 16 10 0 6 6 D 8 2 -6 0 -12 E 10 10 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5517: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E B A C D (8) E B D A C (7) C A D B E (6) C A B D E (6) D C E A B (5) B A C E D (5) D C A E B (4) B A C D E (4) A B E C D (4) E D B C A (3) D E C A B (3) A E B C D (3) A C B D E (3) E D B A C (2) D E C B A (2) D E B C A (2) D C B A E (2) C D A B E (2) C A D E B (2) B E D A C (2) B E A C D (2) B C D A E (2) B A E C D (2) A B C E D (2) E D A C B (1) E B A D C (1) E A D B C (1) D C E B A (1) D C B E A (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -2 2 18 B -8 0 0 6 8 C 2 0 0 6 12 D -2 -6 -6 0 10 E -18 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.118535 C: 0.881465 D: 0.000000 E: 0.000000 Sum of squares = 0.791031044155 Cumulative probabilities = A: 0.000000 B: 0.118535 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 2 18 B -8 0 0 6 8 C 2 0 0 6 12 D -2 -6 -6 0 10 E -18 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.199999 C: 0.800001 D: 0.000000 E: 0.000000 Sum of squares = 0.680000670448 Cumulative probabilities = A: 0.000000 B: 0.199999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=23 B=17 C=16 A=14 so A is eliminated. Round 2 votes counts: D=30 E=27 B=23 C=20 so C is eliminated. Round 3 votes counts: D=40 B=33 E=27 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 C:210 B:203 D:198 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 2 18 B -8 0 0 6 8 C 2 0 0 6 12 D -2 -6 -6 0 10 E -18 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.199999 C: 0.800001 D: 0.000000 E: 0.000000 Sum of squares = 0.680000670448 Cumulative probabilities = A: 0.000000 B: 0.199999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 2 18 B -8 0 0 6 8 C 2 0 0 6 12 D -2 -6 -6 0 10 E -18 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.199999 C: 0.800001 D: 0.000000 E: 0.000000 Sum of squares = 0.680000670448 Cumulative probabilities = A: 0.000000 B: 0.199999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 2 18 B -8 0 0 6 8 C 2 0 0 6 12 D -2 -6 -6 0 10 E -18 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.199999 C: 0.800001 D: 0.000000 E: 0.000000 Sum of squares = 0.680000670448 Cumulative probabilities = A: 0.000000 B: 0.199999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5518: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) D B A E C (7) D E B A C (6) A C B E D (6) C A E B D (5) E B D A C (4) E B A D C (4) A B E D C (4) E A B C D (3) D E C B A (3) C E D A B (3) C E A B D (3) C D E B A (3) C D A B E (3) C A B E D (3) B A E D C (3) A B D C E (3) E D B A C (2) D C E B A (2) D C B A E (2) D B C E A (2) D B A C E (2) C D E A B (2) C A B D E (2) B D A E C (2) E D B C A (1) E C D B A (1) E A B D C (1) D E B C A (1) D C B E A (1) C E A D B (1) C D A E B (1) C A E D B (1) C A D B E (1) B E D A C (1) A E C B D (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 18 -12 -2 B 8 0 10 -4 2 C -18 -10 0 -18 -6 D 12 4 18 0 2 E 2 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 18 -12 -2 B 8 0 10 -4 2 C -18 -10 0 -18 -6 D 12 4 18 0 2 E 2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=28 A=17 E=16 B=6 so B is eliminated. Round 2 votes counts: D=35 C=28 A=20 E=17 so E is eliminated. Round 3 votes counts: D=43 C=29 A=28 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:208 E:202 A:198 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 18 -12 -2 B 8 0 10 -4 2 C -18 -10 0 -18 -6 D 12 4 18 0 2 E 2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 18 -12 -2 B 8 0 10 -4 2 C -18 -10 0 -18 -6 D 12 4 18 0 2 E 2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 18 -12 -2 B 8 0 10 -4 2 C -18 -10 0 -18 -6 D 12 4 18 0 2 E 2 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5519: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) E D C A B (8) A B C D E (8) C D E B A (7) B A C E D (7) C B D E A (6) E D C B A (5) C E D B A (5) A E D B C (5) D E C B A (4) D E C A B (4) E D A B C (3) B C A E D (3) B A C D E (3) D E A C B (2) C B E D A (2) B C A D E (2) A D E C B (2) A D E B C (2) A B C E D (2) E D B A C (1) E D A C B (1) E C D B A (1) E B D C A (1) C D E A B (1) B C E D A (1) B A E D C (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 -4 -4 B -2 0 2 -2 -4 C 0 -2 0 -2 -2 D 4 2 2 0 -12 E 4 4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 -4 -4 B -2 0 2 -2 -4 C 0 -2 0 -2 -2 D 4 2 2 0 -12 E 4 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=21 E=20 B=17 D=10 so D is eliminated. Round 2 votes counts: A=32 E=30 C=21 B=17 so B is eliminated. Round 3 votes counts: A=43 E=30 C=27 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:198 A:197 B:197 C:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -4 -4 B -2 0 2 -2 -4 C 0 -2 0 -2 -2 D 4 2 2 0 -12 E 4 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 -4 B -2 0 2 -2 -4 C 0 -2 0 -2 -2 D 4 2 2 0 -12 E 4 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 -4 B -2 0 2 -2 -4 C 0 -2 0 -2 -2 D 4 2 2 0 -12 E 4 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5520: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) E D B A C (6) B A C E D (6) B A E C D (5) A C B D E (5) A B E C D (5) D E C A B (4) C A D B E (4) B E A D C (4) A E B C D (4) D C E A B (3) B E A C D (3) A E C D B (3) A C D E B (3) E D A C B (2) E A D C B (2) D E C B A (2) D E B C A (2) D B E C A (2) C D A B E (2) C A D E B (2) C A B D E (2) B E D A C (2) B C D A E (2) A E B D C (2) A C B E D (2) E D B C A (1) E D A B C (1) E B D A C (1) E B A D C (1) C D B A E (1) C B D E A (1) C B D A E (1) B E D C A (1) B C E A D (1) B C D E A (1) B C A D E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 18 14 20 B -12 0 4 2 4 C -18 -4 0 26 -8 D -14 -2 -26 0 -8 E -20 -4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 18 14 20 B -12 0 4 2 4 C -18 -4 0 26 -8 D -14 -2 -26 0 -8 E -20 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 C=21 E=14 D=13 so D is eliminated. Round 2 votes counts: B=28 A=26 C=24 E=22 so E is eliminated. Round 3 votes counts: B=39 A=31 C=30 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:232 B:199 C:198 E:196 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 18 14 20 B -12 0 4 2 4 C -18 -4 0 26 -8 D -14 -2 -26 0 -8 E -20 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 18 14 20 B -12 0 4 2 4 C -18 -4 0 26 -8 D -14 -2 -26 0 -8 E -20 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 18 14 20 B -12 0 4 2 4 C -18 -4 0 26 -8 D -14 -2 -26 0 -8 E -20 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5521: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (8) C D B E A (6) E B C D A (5) A D B E C (5) E B D C A (4) E A B D C (4) A C D E B (4) E B C A D (3) D A C B E (3) C E D B A (3) C A D B E (3) A D C B E (3) E B D A C (2) D C A B E (2) C D B A E (2) C D A E B (2) B E D C A (2) B E D A C (2) B D A E C (2) A E B D C (2) A D B C E (2) A C E D B (2) A C D B E (2) A B D E C (2) E C B D A (1) E C B A D (1) E C A B D (1) E B A C D (1) E A B C D (1) D C B A E (1) D B C E A (1) D A B C E (1) C E B D A (1) C E A D B (1) B D E C A (1) B D E A C (1) B A E D C (1) A E C B D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 4 0 0 B -2 0 6 -2 2 C -4 -6 0 -2 -6 D 0 2 2 0 2 E 0 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.354816 B: 0.000000 C: 0.000000 D: 0.645184 E: 0.000000 Sum of squares = 0.542156627756 Cumulative probabilities = A: 0.354816 B: 0.354816 C: 0.354816 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 0 0 B -2 0 6 -2 2 C -4 -6 0 -2 -6 D 0 2 2 0 2 E 0 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=26 A=25 B=9 D=8 so D is eliminated. Round 2 votes counts: E=32 C=29 A=29 B=10 so B is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:203 D:203 B:202 E:201 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 0 0 B -2 0 6 -2 2 C -4 -6 0 -2 -6 D 0 2 2 0 2 E 0 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 0 0 B -2 0 6 -2 2 C -4 -6 0 -2 -6 D 0 2 2 0 2 E 0 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 0 0 B -2 0 6 -2 2 C -4 -6 0 -2 -6 D 0 2 2 0 2 E 0 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5522: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (5) B E C A D (5) B E A C D (5) E D A C B (4) E D A B C (4) D A E C B (4) C B E D A (4) C A D B E (4) B E A D C (4) B C A D E (4) A B D E C (4) E D B A C (3) E B A D C (3) C D E A B (3) C B A D E (3) B E C D A (3) B C E A D (3) B C A E D (3) A D E B C (3) E D C A B (2) D C A E B (2) C E D A B (2) C E B D A (2) C D A B E (2) C A B D E (2) A E D B C (2) A D E C B (2) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) D A C E B (1) C E D B A (1) B A E C D (1) B A D E C (1) B A D C E (1) A D C E B (1) A D B C E (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -6 8 -4 B 0 0 6 4 4 C 6 -6 0 10 -10 D -8 -4 -10 0 -12 E 4 -4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.241135 B: 0.758865 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.634021828781 Cumulative probabilities = A: 0.241135 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 8 -4 B 0 0 6 4 4 C 6 -6 0 10 -10 D -8 -4 -10 0 -12 E 4 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499860 B: 0.500140 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000038936 Cumulative probabilities = A: 0.499860 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=28 E=20 A=15 D=7 so D is eliminated. Round 2 votes counts: C=30 B=30 E=20 A=20 so E is eliminated. Round 3 votes counts: B=38 C=34 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:211 B:207 C:200 A:199 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 8 -4 B 0 0 6 4 4 C 6 -6 0 10 -10 D -8 -4 -10 0 -12 E 4 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499860 B: 0.500140 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000038936 Cumulative probabilities = A: 0.499860 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 8 -4 B 0 0 6 4 4 C 6 -6 0 10 -10 D -8 -4 -10 0 -12 E 4 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499860 B: 0.500140 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000038936 Cumulative probabilities = A: 0.499860 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 8 -4 B 0 0 6 4 4 C 6 -6 0 10 -10 D -8 -4 -10 0 -12 E 4 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499860 B: 0.500140 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000038936 Cumulative probabilities = A: 0.499860 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5523: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (17) E A B C D (11) D C B A E (10) E A B D C (9) E A C D B (8) B D C A E (8) A E B D C (7) C D E B A (4) C D E A B (4) C D B A E (4) A B E D C (4) B A D C E (3) B E A D C (2) E C D B A (1) E C D A B (1) C E D A B (1) C D A E B (1) B E C D A (1) B D C E A (1) B C D E A (1) B A E D C (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 -8 -8 -22 B 6 0 -4 -4 4 C 8 4 0 10 8 D 8 4 -10 0 6 E 22 -4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -8 -22 B 6 0 -4 -4 4 C 8 4 0 10 8 D 8 4 -10 0 6 E 22 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=30 B=17 A=12 D=10 so D is eliminated. Round 2 votes counts: C=41 E=30 B=17 A=12 so A is eliminated. Round 3 votes counts: C=41 E=38 B=21 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:204 E:202 B:201 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -8 -22 B 6 0 -4 -4 4 C 8 4 0 10 8 D 8 4 -10 0 6 E 22 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -8 -22 B 6 0 -4 -4 4 C 8 4 0 10 8 D 8 4 -10 0 6 E 22 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -8 -22 B 6 0 -4 -4 4 C 8 4 0 10 8 D 8 4 -10 0 6 E 22 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5524: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) D A B E C (5) C E B D A (5) A D E B C (5) E C A B D (4) C E D B A (4) B D A C E (4) A D B E C (4) A B D E C (4) E C B A D (3) D C A B E (3) D A C B E (3) C E B A D (3) B D C A E (3) B C E D A (3) A E D C B (3) E B C A D (2) E B A C D (2) E A C D B (2) E A B C D (2) D C B A E (2) D B A E C (2) C D E B A (2) C B D E A (2) B A E D C (2) A E B D C (2) A B E D C (2) E C A D B (1) E A C B D (1) E A B D C (1) D B C A E (1) D A E C B (1) D A B C E (1) C E D A B (1) C E A B D (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E D A (1) B E A C D (1) B C E A D (1) B A D E C (1) Total count = 100 A B C D E A 0 -4 10 -6 16 B 4 0 10 0 8 C -10 -10 0 -12 0 D 6 0 12 0 6 E -16 -8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.611880 C: 0.000000 D: 0.388120 E: 0.000000 Sum of squares = 0.525034070205 Cumulative probabilities = A: 0.000000 B: 0.611880 C: 0.611880 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -6 16 B 4 0 10 0 8 C -10 -10 0 -12 0 D 6 0 12 0 6 E -16 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 A=20 E=18 B=15 so B is eliminated. Round 2 votes counts: D=32 C=26 A=23 E=19 so E is eliminated. Round 3 votes counts: C=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:212 B:211 A:208 E:185 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 -6 16 B 4 0 10 0 8 C -10 -10 0 -12 0 D 6 0 12 0 6 E -16 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -6 16 B 4 0 10 0 8 C -10 -10 0 -12 0 D 6 0 12 0 6 E -16 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -6 16 B 4 0 10 0 8 C -10 -10 0 -12 0 D 6 0 12 0 6 E -16 -8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5525: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (14) B C A E D (14) D E A C B (13) D A E B C (9) C B D E A (6) B C D A E (6) D A B E C (3) C D B E A (3) E C A B D (2) E A D C B (2) E A C B D (2) D C E A B (2) D C B E A (2) C B E D A (2) B C E A D (2) B A E C D (2) A E D B C (2) A B E D C (2) A B E C D (2) E D A C B (1) E A D B C (1) D E C A B (1) D C B A E (1) D B C A E (1) B C A D E (1) B A D E C (1) B A C E D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -12 -14 0 -6 B 12 0 -2 18 26 C 14 2 0 16 10 D 0 -18 -16 0 -2 E 6 -26 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 0 -6 B 12 0 -2 18 26 C 14 2 0 16 10 D 0 -18 -16 0 -2 E 6 -26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=27 C=25 E=8 A=8 so E is eliminated. Round 2 votes counts: D=33 C=27 B=27 A=13 so A is eliminated. Round 3 votes counts: D=38 B=33 C=29 so C is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:221 E:186 A:184 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 0 -6 B 12 0 -2 18 26 C 14 2 0 16 10 D 0 -18 -16 0 -2 E 6 -26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 0 -6 B 12 0 -2 18 26 C 14 2 0 16 10 D 0 -18 -16 0 -2 E 6 -26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 0 -6 B 12 0 -2 18 26 C 14 2 0 16 10 D 0 -18 -16 0 -2 E 6 -26 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5526: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (13) D C E B A (9) B E C A D (8) A B E C D (8) A D B C E (7) B E C D A (6) A D C B E (6) B E A C D (5) A B D E C (5) E B C D A (4) D A C E B (4) C E D B A (3) C D E B A (3) D E C B A (2) C E B D A (2) B E D C A (2) A C B E D (2) A B E D C (2) A B C E D (2) E C B D A (1) D C E A B (1) D C A E B (1) C A E B D (1) A D B E C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 14 24 8 B -10 0 2 -2 10 C -14 -2 0 -6 12 D -24 2 6 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 24 8 B -10 0 2 -2 10 C -14 -2 0 -6 12 D -24 2 6 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=48 B=21 D=17 C=9 E=5 so E is eliminated. Round 2 votes counts: A=48 B=25 D=17 C=10 so C is eliminated. Round 3 votes counts: A=49 B=28 D=23 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:200 D:196 C:195 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 24 8 B -10 0 2 -2 10 C -14 -2 0 -6 12 D -24 2 6 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 24 8 B -10 0 2 -2 10 C -14 -2 0 -6 12 D -24 2 6 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 24 8 B -10 0 2 -2 10 C -14 -2 0 -6 12 D -24 2 6 0 8 E -8 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5527: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) C E D A B (11) C E D B A (8) C E B D A (5) C B E D A (4) A D B E C (4) E D C A B (3) C A D B E (3) B C A E D (3) A B D E C (3) E D B C A (2) E D A B C (2) C B A E D (2) B E D A C (2) B C E A D (2) B A C D E (2) A D E B C (2) A C D B E (2) E D B A C (1) E D A C B (1) E C D B A (1) E C D A B (1) E C B D A (1) E B D A C (1) D E A C B (1) D E A B C (1) D A E C B (1) D A E B C (1) C E B A D (1) C D A E B (1) C B E A D (1) C B A D E (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B D A E C (1) B C E D A (1) B A D C E (1) A D C E B (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 -4 -8 B 12 0 -6 2 6 C 14 6 0 12 10 D 4 -2 -12 0 -16 E 8 -6 -10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -4 -8 B 12 0 -6 2 6 C 14 6 0 12 10 D 4 -2 -12 0 -16 E 8 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=28 A=15 E=13 D=4 so D is eliminated. Round 2 votes counts: C=40 B=28 A=17 E=15 so E is eliminated. Round 3 votes counts: C=46 B=32 A=22 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:207 E:204 D:187 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -4 -8 B 12 0 -6 2 6 C 14 6 0 12 10 D 4 -2 -12 0 -16 E 8 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -4 -8 B 12 0 -6 2 6 C 14 6 0 12 10 D 4 -2 -12 0 -16 E 8 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -4 -8 B 12 0 -6 2 6 C 14 6 0 12 10 D 4 -2 -12 0 -16 E 8 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5528: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) C A D E B (7) A C D B E (7) E C B A D (6) E B C D A (6) B D E A C (6) A D C B E (6) B E D C A (5) E B D C A (4) D A B C E (4) C A E D B (4) A C D E B (4) E C A B D (3) D A C B E (3) E B C A D (2) D B A E C (2) C E D B A (2) C A E B D (2) A C E B D (2) A C B E D (2) E C D B A (1) E B A C D (1) D B E A C (1) D B A C E (1) C E A B D (1) C A D B E (1) B D A E C (1) B A D E C (1) A E C B D (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 12 4 0 B 2 0 -6 12 6 C -12 6 0 6 -4 D -4 -12 -6 0 -10 E 0 -6 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000007 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 12 4 0 B 2 0 -6 12 6 C -12 6 0 6 -4 D -4 -12 -6 0 -10 E 0 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=24 E=23 C=17 D=11 so D is eliminated. Round 2 votes counts: A=31 B=29 E=23 C=17 so C is eliminated. Round 3 votes counts: A=45 B=29 E=26 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:207 B:207 E:204 C:198 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 12 4 0 B 2 0 -6 12 6 C -12 6 0 6 -4 D -4 -12 -6 0 -10 E 0 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 4 0 B 2 0 -6 12 6 C -12 6 0 6 -4 D -4 -12 -6 0 -10 E 0 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 4 0 B 2 0 -6 12 6 C -12 6 0 6 -4 D -4 -12 -6 0 -10 E 0 -6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5529: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (14) D C E B A (12) D E C B A (6) D C A B E (6) B A E C D (6) C D E B A (5) A B E D C (5) E B A D C (4) E C B D A (3) E B A C D (3) D C E A B (3) C D B E A (3) A D B E C (3) E C D B A (2) E B D A C (2) E B C A D (2) C D A B E (2) C B E A D (2) C B A D E (2) B A C E D (2) A B D E C (2) E D C B A (1) D C B A E (1) D C A E B (1) D A E C B (1) D A E B C (1) C E D B A (1) C A D B E (1) B E C A D (1) A E B D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -8 0 0 B 16 0 -4 2 4 C 8 4 0 2 -14 D 0 -2 -2 0 0 E 0 -4 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380142 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 A B C D E A 0 -16 -8 0 0 B 16 0 -4 2 4 C 8 4 0 2 -14 D 0 -2 -2 0 0 E 0 -4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379808 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=27 E=17 C=16 B=9 so B is eliminated. Round 2 votes counts: A=35 D=31 E=18 C=16 so C is eliminated. Round 3 votes counts: D=41 A=38 E=21 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:209 E:205 C:200 D:198 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -8 0 0 B 16 0 -4 2 4 C 8 4 0 2 -14 D 0 -2 -2 0 0 E 0 -4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379808 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 0 0 B 16 0 -4 2 4 C 8 4 0 2 -14 D 0 -2 -2 0 0 E 0 -4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379808 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 0 0 B 16 0 -4 2 4 C 8 4 0 2 -14 D 0 -2 -2 0 0 E 0 -4 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074379808 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5530: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (13) E A B C D (11) A B E C D (6) C B D A E (5) A E B D C (5) B C A E D (4) E A D B C (3) D C E A B (3) C D B E A (3) C D B A E (3) A B E D C (3) E B C A D (2) E A B D C (2) D C B E A (2) D B A C E (2) D A B C E (2) C D E B A (2) C B D E A (2) B C A D E (2) B A E C D (2) B A D C E (2) B A C E D (2) B A C D E (2) A E B C D (2) A D E B C (2) E C D A B (1) E B A C D (1) E A C D B (1) D E C A B (1) D E A C B (1) D A B E C (1) C E D B A (1) C E B A D (1) B E A C D (1) B D C A E (1) A E D B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 8 14 24 B 6 0 22 14 20 C -8 -22 0 8 4 D -14 -14 -8 0 2 E -24 -20 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 14 24 B 6 0 22 14 20 C -8 -22 0 8 4 D -14 -14 -8 0 2 E -24 -20 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=21 A=21 C=17 B=16 so B is eliminated. Round 2 votes counts: A=29 D=26 C=23 E=22 so E is eliminated. Round 3 votes counts: A=48 D=26 C=26 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:231 A:220 C:191 D:183 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 14 24 B 6 0 22 14 20 C -8 -22 0 8 4 D -14 -14 -8 0 2 E -24 -20 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 14 24 B 6 0 22 14 20 C -8 -22 0 8 4 D -14 -14 -8 0 2 E -24 -20 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 14 24 B 6 0 22 14 20 C -8 -22 0 8 4 D -14 -14 -8 0 2 E -24 -20 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5531: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) C B A D E (12) A C B D E (8) D E A B C (7) E D A B C (6) C A B D E (5) B C E D A (5) E A D C B (4) A E D C B (4) A D E C B (4) A C D B E (4) E D B A C (3) A D C B E (3) A C B E D (3) E D A C B (2) C B A E D (2) C A B E D (2) B C D E A (2) B C D A E (2) B C A D E (2) E B D C A (1) D E B C A (1) D E B A C (1) B E D C A (1) B E C D A (1) B C E A D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 2 12 6 B -8 0 -10 -6 6 C -2 10 0 0 6 D -12 6 0 0 6 E -6 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 12 6 B -8 0 -10 -6 6 C -2 10 0 0 6 D -12 6 0 0 6 E -6 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 C=21 B=14 D=9 so D is eliminated. Round 2 votes counts: E=37 A=28 C=21 B=14 so B is eliminated. Round 3 votes counts: E=39 C=33 A=28 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:207 D:200 B:191 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 12 6 B -8 0 -10 -6 6 C -2 10 0 0 6 D -12 6 0 0 6 E -6 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 12 6 B -8 0 -10 -6 6 C -2 10 0 0 6 D -12 6 0 0 6 E -6 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 12 6 B -8 0 -10 -6 6 C -2 10 0 0 6 D -12 6 0 0 6 E -6 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5532: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) A E D B C (9) C B D E A (8) D A C E B (7) B C E A D (7) C B E D A (6) D C B A E (5) D A E C B (5) A D E B C (5) E B C A D (4) D A C B E (4) A E B C D (4) E C B A D (3) D C B E A (3) D C A B E (3) D A E B C (2) C D B E A (2) C B E A D (2) B E C A D (2) E B A C D (1) E A D C B (1) D B C A E (1) C E B D A (1) B E A C D (1) B C E D A (1) B A E C D (1) A E D C B (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 4 -2 B -4 0 -2 4 -6 C -4 2 0 6 0 D -4 -4 -6 0 -10 E 2 6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.137284 D: 0.000000 E: 0.862716 Sum of squares = 0.763125477127 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.137284 D: 0.137284 E: 1.000000 A B C D E A 0 4 4 4 -2 B -4 0 -2 4 -6 C -4 2 0 6 0 D -4 -4 -6 0 -10 E 2 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555559352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=21 C=19 E=18 B=12 so B is eliminated. Round 2 votes counts: D=30 C=27 A=22 E=21 so E is eliminated. Round 3 votes counts: C=36 A=34 D=30 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:205 C:202 B:196 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 4 -2 B -4 0 -2 4 -6 C -4 2 0 6 0 D -4 -4 -6 0 -10 E 2 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555559352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 -2 B -4 0 -2 4 -6 C -4 2 0 6 0 D -4 -4 -6 0 -10 E 2 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555559352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 -2 B -4 0 -2 4 -6 C -4 2 0 6 0 D -4 -4 -6 0 -10 E 2 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555559352 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5533: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) A C D E B (9) C A D E B (8) E B D C A (6) B D E C A (6) A E B C D (6) D C E B A (4) B E A D C (4) A C E D B (4) E A B D C (3) D B C E A (3) C D A E B (3) C D A B E (3) C A D B E (3) B E D A C (3) B D C E A (3) A E C D B (3) A C B E D (3) E B D A C (2) B A C D E (2) A C D B E (2) A B C E D (2) E D C B A (1) E B A D C (1) D C E A B (1) D B E C A (1) A E C B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -6 6 2 B -6 0 8 10 -4 C 6 -8 0 2 2 D -6 -10 -2 0 -2 E -2 4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999202 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 6 2 B -6 0 8 10 -4 C 6 -8 0 2 2 D -6 -10 -2 0 -2 E -2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999705 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=29 C=17 E=13 D=9 so D is eliminated. Round 2 votes counts: B=33 A=32 C=22 E=13 so E is eliminated. Round 3 votes counts: B=42 A=35 C=23 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:204 B:204 C:201 E:201 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 6 2 B -6 0 8 10 -4 C 6 -8 0 2 2 D -6 -10 -2 0 -2 E -2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999705 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 6 2 B -6 0 8 10 -4 C 6 -8 0 2 2 D -6 -10 -2 0 -2 E -2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999705 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 6 2 B -6 0 8 10 -4 C 6 -8 0 2 2 D -6 -10 -2 0 -2 E -2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.300000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999705 Cumulative probabilities = A: 0.400000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5534: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) E D B C A (6) E B C D A (6) C A B D E (6) D E A B C (5) A C D B E (5) E B D C A (4) E B C A D (4) D E C A B (4) D A E C B (4) B E C A D (4) E D C B A (3) E B D A C (3) D E A C B (3) C B A E D (3) C A B E D (3) B C E A D (3) A D C B E (3) A C B D E (3) A B C D E (3) E D B A C (2) D E B A C (2) E C B D A (1) D C E A B (1) D C A E B (1) D A E B C (1) D A C B E (1) D A B E C (1) C B E A D (1) C A D B E (1) B E A C D (1) B C A E D (1) B A C E D (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 -4 -12 -6 B -10 0 -4 -4 -16 C 4 4 0 -6 -8 D 12 4 6 0 6 E 6 16 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -12 -6 B -10 0 -4 -4 -16 C 4 4 0 -6 -8 D 12 4 6 0 6 E 6 16 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=29 A=16 C=14 B=10 so B is eliminated. Round 2 votes counts: E=34 D=31 C=18 A=17 so A is eliminated. Round 3 votes counts: D=35 E=34 C=31 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:212 C:197 A:194 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 -12 -6 B -10 0 -4 -4 -16 C 4 4 0 -6 -8 D 12 4 6 0 6 E 6 16 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -12 -6 B -10 0 -4 -4 -16 C 4 4 0 -6 -8 D 12 4 6 0 6 E 6 16 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -12 -6 B -10 0 -4 -4 -16 C 4 4 0 -6 -8 D 12 4 6 0 6 E 6 16 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5535: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) E B C D A (7) B E C A D (7) E C D B A (5) A D B C E (5) D C E A B (4) D A C E B (4) B C E A D (4) E C B D A (3) C B E A D (3) C A D B E (3) B A E D C (3) E D C B A (2) E B D C A (2) E B D A C (2) D A E C B (2) D A E B C (2) D A C B E (2) C E D B A (2) C E B D A (2) C D E B A (2) C D E A B (2) C D A E B (2) C D A B E (2) B E A C D (2) A D B E C (2) E D B C A (1) E B A D C (1) E B A C D (1) D E A C B (1) D E A B C (1) D C A E B (1) C D B E A (1) C B A E D (1) B E A D C (1) B C A E D (1) B A C E D (1) A E D B C (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -14 -4 -12 B 8 0 -12 -16 0 C 14 12 0 4 8 D 4 16 -4 0 -4 E 12 0 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -4 -12 B 8 0 -12 -16 0 C 14 12 0 4 8 D 4 16 -4 0 -4 E 12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=20 A=20 B=19 D=17 so D is eliminated. Round 2 votes counts: A=30 E=26 C=25 B=19 so B is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:219 D:206 E:204 B:190 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -4 -12 B 8 0 -12 -16 0 C 14 12 0 4 8 D 4 16 -4 0 -4 E 12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -4 -12 B 8 0 -12 -16 0 C 14 12 0 4 8 D 4 16 -4 0 -4 E 12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -4 -12 B 8 0 -12 -16 0 C 14 12 0 4 8 D 4 16 -4 0 -4 E 12 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5536: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) E D A C B (6) B C A D E (5) B A C D E (5) A B C D E (5) E D C B A (4) E D C A B (4) C B D A E (4) C B A D E (4) A B D C E (4) E C D B A (3) E C B D A (3) D C E A B (3) B C E A D (3) B A E C D (3) E B C A D (2) D E C A B (2) D C A B E (2) D A E B C (2) D A C B E (2) C E D B A (2) C D B A E (2) C B E A D (2) B A C E D (2) A E B D C (2) A D B C E (2) A B E C D (2) E C B A D (1) E B A C D (1) E A D B C (1) E A B D C (1) D E A C B (1) D A B C E (1) C E B D A (1) C D E B A (1) C B A E D (1) A E B C D (1) A D B E C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 2 0 4 B -2 0 4 8 4 C -2 -4 0 8 4 D 0 -8 -8 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.895247 B: 0.000000 C: 0.000000 D: 0.104753 E: 0.000000 Sum of squares = 0.812440009359 Cumulative probabilities = A: 0.895247 B: 0.895247 C: 0.895247 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 4 B -2 0 4 8 4 C -2 -4 0 8 4 D 0 -8 -8 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000449 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=19 B=18 C=17 D=13 so D is eliminated. Round 2 votes counts: E=36 A=24 C=22 B=18 so B is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:207 A:204 C:203 E:198 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 4 B -2 0 4 8 4 C -2 -4 0 8 4 D 0 -8 -8 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000449 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 4 B -2 0 4 8 4 C -2 -4 0 8 4 D 0 -8 -8 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000449 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 4 B -2 0 4 8 4 C -2 -4 0 8 4 D 0 -8 -8 0 -8 E -4 -4 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000000449 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5537: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (20) E B C D A (13) E B C A D (11) D A C B E (7) B E C D A (6) A D C E B (5) D B E A C (4) A C D B E (3) E B D C A (2) E B A C D (2) D A B E C (2) C E B A D (2) C B E D A (2) A E B D C (2) A D E B C (2) A D B E C (2) A C D E B (2) E C B A D (1) E B D A C (1) D C A B E (1) D B E C A (1) D B A C E (1) D A B C E (1) C E A B D (1) C D B A E (1) C D A B E (1) C A D B E (1) B D E C A (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 12 12 6 B -4 0 4 -12 10 C -12 -4 0 -6 0 D -12 12 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 12 6 B -4 0 4 -12 10 C -12 -4 0 -6 0 D -12 12 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=30 D=17 C=8 B=7 so B is eliminated. Round 2 votes counts: A=38 E=36 D=18 C=8 so C is eliminated. Round 3 votes counts: E=41 A=39 D=20 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:209 B:199 C:189 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 12 6 B -4 0 4 -12 10 C -12 -4 0 -6 0 D -12 12 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 12 6 B -4 0 4 -12 10 C -12 -4 0 -6 0 D -12 12 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 12 6 B -4 0 4 -12 10 C -12 -4 0 -6 0 D -12 12 6 0 12 E -6 -10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5538: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (13) C E B A D (12) E B C A D (11) D A B E C (10) D A C B E (7) A D C E B (7) C B E D A (6) E B A D C (5) B E C D A (5) A D E B C (4) C A D E B (3) B E A D C (3) C D B E A (2) C D A B E (2) C A E B D (2) B E D C A (2) D B E A C (1) D A E B C (1) D A C E B (1) C D A E B (1) B E C A D (1) A E B D C (1) Total count = 100 A B C D E A 0 -22 6 -2 -22 B 22 0 14 22 4 C -6 -14 0 -10 -14 D 2 -22 10 0 -22 E 22 -4 14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997696 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 6 -2 -22 B 22 0 14 22 4 C -6 -14 0 -10 -14 D 2 -22 10 0 -22 E 22 -4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=24 D=20 E=16 A=12 so A is eliminated. Round 2 votes counts: D=31 C=28 B=24 E=17 so E is eliminated. Round 3 votes counts: B=41 D=31 C=28 so C is eliminated. Round 4 votes counts: B=61 D=39 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:227 D:184 A:180 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 6 -2 -22 B 22 0 14 22 4 C -6 -14 0 -10 -14 D 2 -22 10 0 -22 E 22 -4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 6 -2 -22 B 22 0 14 22 4 C -6 -14 0 -10 -14 D 2 -22 10 0 -22 E 22 -4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 6 -2 -22 B 22 0 14 22 4 C -6 -14 0 -10 -14 D 2 -22 10 0 -22 E 22 -4 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5539: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E A B D C (7) E C B A D (6) B D A C E (6) E C A D B (5) D A C B E (4) C D B A E (4) E B C A D (3) E B A C D (3) D B C A E (3) C E D B A (3) C E B D A (3) B A D E C (3) A D B C E (3) E A D C B (2) E A D B C (2) D B A C E (2) D A B C E (2) C D B E A (2) C D A B E (2) C B D E A (2) B D C A E (2) B C E D A (2) B C D A E (2) A E D B C (2) A D C B E (2) E C D A B (1) E C A B D (1) E A C B D (1) E A B C D (1) D C A B E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) C B E D A (1) B E A D C (1) A E B D C (1) A D E C B (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -8 -4 -12 B 10 0 -4 6 -8 C 8 4 0 6 0 D 4 -6 -6 0 -8 E 12 8 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.604297 D: 0.000000 E: 0.395703 Sum of squares = 0.521755841353 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.604297 D: 0.604297 E: 1.000000 A B C D E A 0 -10 -8 -4 -12 B 10 0 -4 6 -8 C 8 4 0 6 0 D 4 -6 -6 0 -8 E 12 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=21 B=16 D=12 A=12 so D is eliminated. Round 2 votes counts: E=39 C=22 B=21 A=18 so A is eliminated. Round 3 votes counts: E=44 C=28 B=28 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:209 B:202 D:192 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -4 -12 B 10 0 -4 6 -8 C 8 4 0 6 0 D 4 -6 -6 0 -8 E 12 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -4 -12 B 10 0 -4 6 -8 C 8 4 0 6 0 D 4 -6 -6 0 -8 E 12 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -4 -12 B 10 0 -4 6 -8 C 8 4 0 6 0 D 4 -6 -6 0 -8 E 12 8 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5540: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) B A E D C (9) E D A C B (5) C D E B A (5) D C E A B (4) C B E D A (4) C B D E A (4) C B D A E (4) E D C A B (3) C E D B A (3) C D B E A (3) C D A E B (3) B C E A D (3) B A E C D (3) A D E C B (3) A D E B C (3) A B D E C (3) E A D B C (2) D A E C B (2) C E D A B (2) B C E D A (2) B C A E D (2) B A C D E (2) A E D B C (2) E C D B A (1) E C D A B (1) E C B D A (1) E A D C B (1) D E A C B (1) D C A E B (1) C E B D A (1) C A D B E (1) B E A D C (1) B C D A E (1) B C A D E (1) B A D E C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -20 -22 -14 B 2 0 -26 -12 -10 C 20 26 0 14 12 D 22 12 -14 0 4 E 14 10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -20 -22 -14 B 2 0 -26 -12 -10 C 20 26 0 14 12 D 22 12 -14 0 4 E 14 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=25 E=14 A=13 D=8 so D is eliminated. Round 2 votes counts: C=45 B=25 E=15 A=15 so E is eliminated. Round 3 votes counts: C=51 B=25 A=24 so A is eliminated. Round 4 votes counts: C=63 B=37 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:236 D:212 E:204 B:177 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -20 -22 -14 B 2 0 -26 -12 -10 C 20 26 0 14 12 D 22 12 -14 0 4 E 14 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -20 -22 -14 B 2 0 -26 -12 -10 C 20 26 0 14 12 D 22 12 -14 0 4 E 14 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -20 -22 -14 B 2 0 -26 -12 -10 C 20 26 0 14 12 D 22 12 -14 0 4 E 14 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5541: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) B A E C D (9) D A B C E (8) A D B E C (7) B C E A D (6) A B D E C (6) D C E A B (5) D A C E B (4) D A E C B (3) C E B D A (3) B A C D E (3) E D C A B (2) E D A C B (2) E C D A B (2) D E C A B (2) C E D A B (2) C D E A B (2) C B E A D (2) B E C A D (2) B C E D A (2) B A C E D (2) A D E B C (2) E C B A D (1) E B C A D (1) E A C D B (1) D E A C B (1) D C B A E (1) D B A C E (1) D A E B C (1) C D B E A (1) B A D E C (1) B A D C E (1) A E D C B (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 12 -4 6 B -8 0 10 -16 10 C -12 -10 0 0 8 D 4 16 0 0 0 E -6 -10 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.205100 D: 0.794900 E: 0.000000 Sum of squares = 0.673931723068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.205100 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -4 6 B -8 0 10 -16 10 C -12 -10 0 0 8 D 4 16 0 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000020269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 C=20 A=19 E=9 so E is eliminated. Round 2 votes counts: D=30 B=27 C=23 A=20 so A is eliminated. Round 3 votes counts: D=40 B=36 C=24 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:211 D:210 B:198 C:193 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -4 6 B -8 0 10 -16 10 C -12 -10 0 0 8 D 4 16 0 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000020269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -4 6 B -8 0 10 -16 10 C -12 -10 0 0 8 D 4 16 0 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000020269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -4 6 B -8 0 10 -16 10 C -12 -10 0 0 8 D 4 16 0 0 0 E -6 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000020269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5542: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) C E D B A (10) A B D E C (10) B A E C D (9) E C B A D (8) D C E A B (7) C D E A B (5) B A E D C (5) D A C B E (4) C E D A B (4) B E A C D (4) E C D B A (3) E C B D A (3) E B A C D (2) B A D E C (2) A D B E C (2) A D B C E (2) A B D C E (2) E B C A D (1) D E C A B (1) D A C E B (1) D A B E C (1) B E C A D (1) Total count = 100 A B C D E A 0 4 14 -4 2 B -4 0 8 -6 10 C -14 -8 0 0 -4 D 4 6 0 0 0 E -2 -10 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.721046 E: 0.278954 Sum of squares = 0.597722670054 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.721046 E: 1.000000 A B C D E A 0 4 14 -4 2 B -4 0 8 -6 10 C -14 -8 0 0 -4 D 4 6 0 0 0 E -2 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.375000 Sum of squares = 0.531250178333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=21 C=19 E=17 A=16 so A is eliminated. Round 2 votes counts: B=33 D=31 C=19 E=17 so E is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:208 D:205 B:204 E:196 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 14 -4 2 B -4 0 8 -6 10 C -14 -8 0 0 -4 D 4 6 0 0 0 E -2 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.375000 Sum of squares = 0.531250178333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -4 2 B -4 0 8 -6 10 C -14 -8 0 0 -4 D 4 6 0 0 0 E -2 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.375000 Sum of squares = 0.531250178333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -4 2 B -4 0 8 -6 10 C -14 -8 0 0 -4 D 4 6 0 0 0 E -2 -10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.375000 Sum of squares = 0.531250178333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.625000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5543: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (10) D A E C B (9) C B A E D (8) E B C D A (5) D A E B C (5) D A C B E (5) B C E D A (5) A D E C B (5) E B C A D (4) C B E A D (4) A E D C B (4) E C A B D (3) D B C A E (3) B E C D A (3) E B D C A (2) D A B C E (2) C A B D E (2) B D C E A (2) B C D E A (2) A E C B D (2) E C B A D (1) E A C B D (1) D E B C A (1) D E B A C (1) D E A B C (1) D B C E A (1) C E B A D (1) C A B E D (1) B E C A D (1) B C A E D (1) A D C B E (1) A C E B D (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -20 6 4 B 10 0 -2 22 8 C 20 2 0 16 4 D -6 -22 -16 0 -16 E -4 -8 -4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 6 4 B 10 0 -2 22 8 C 20 2 0 16 4 D -6 -22 -16 0 -16 E -4 -8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=24 E=16 C=16 A=16 so E is eliminated. Round 2 votes counts: B=35 D=28 C=20 A=17 so A is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: B=61 D=39 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:221 B:219 E:200 A:190 D:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -20 6 4 B 10 0 -2 22 8 C 20 2 0 16 4 D -6 -22 -16 0 -16 E -4 -8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 6 4 B 10 0 -2 22 8 C 20 2 0 16 4 D -6 -22 -16 0 -16 E -4 -8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 6 4 B 10 0 -2 22 8 C 20 2 0 16 4 D -6 -22 -16 0 -16 E -4 -8 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994044 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5544: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (12) A E D C B (11) C B E D A (10) D B C A E (9) D A B C E (8) B C D E A (8) E C B A D (7) D B A C E (6) E C A B D (5) A D E B C (5) D A B E C (4) E A C D B (3) B D C E A (3) C E B A D (2) B C E D A (2) A D E C B (2) B D C A E (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 4 6 -2 -4 B -4 0 -6 0 4 C -6 6 0 0 -2 D 2 0 0 0 -6 E 4 -4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333321 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 6 -2 -4 B -4 0 -6 0 4 C -6 6 0 0 -2 D 2 0 0 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333314 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=27 D=27 A=20 B=14 C=12 so C is eliminated. Round 2 votes counts: E=29 D=27 B=24 A=20 so A is eliminated. Round 3 votes counts: E=41 D=35 B=24 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:204 A:202 C:199 D:198 B:197 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 -2 -4 B -4 0 -6 0 4 C -6 6 0 0 -2 D 2 0 0 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333314 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -2 -4 B -4 0 -6 0 4 C -6 6 0 0 -2 D 2 0 0 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333314 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -2 -4 B -4 0 -6 0 4 C -6 6 0 0 -2 D 2 0 0 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333314 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5545: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) A E D C B (6) D B C A E (5) C B A D E (5) E D A B C (4) C B D A E (4) B C E D A (4) A E C B D (4) A C D B E (4) E A D B C (3) E A C D B (3) E A C B D (3) B C D E A (3) B C D A E (3) E C B A D (2) E C A B D (2) D E A B C (2) D B E C A (2) D A C B E (2) D A B E C (2) C B E A D (2) B D C E A (2) B D C A E (2) A E C D B (2) A D E C B (2) A D C E B (2) A D C B E (2) A C E B D (2) E D B C A (1) E B D C A (1) E B C A D (1) E A D C B (1) E A B C D (1) D E B A C (1) D B E A C (1) D B A C E (1) D A E B C (1) C E B A D (1) C B A E D (1) B E D C A (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 14 16 2 18 B -14 0 -4 -10 8 C -16 4 0 -2 8 D -2 10 2 0 6 E -18 -8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 2 18 B -14 0 -4 -10 8 C -16 4 0 -2 8 D -2 10 2 0 6 E -18 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963063 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 E=22 B=15 C=13 so C is eliminated. Round 2 votes counts: B=27 A=26 D=24 E=23 so E is eliminated. Round 3 votes counts: A=39 B=32 D=29 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:208 C:197 B:190 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 16 2 18 B -14 0 -4 -10 8 C -16 4 0 -2 8 D -2 10 2 0 6 E -18 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963063 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 2 18 B -14 0 -4 -10 8 C -16 4 0 -2 8 D -2 10 2 0 6 E -18 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963063 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 2 18 B -14 0 -4 -10 8 C -16 4 0 -2 8 D -2 10 2 0 6 E -18 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963063 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5546: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) B E D C A (5) B C A E D (5) A C B D E (5) E D C A B (4) D A E C B (4) B D E C A (4) A C D E B (4) E C A D B (3) D E A C B (3) D B A C E (3) C A B E D (3) A C E D B (3) E D B C A (2) E D A C B (2) E C D B A (2) E C B D A (2) E C B A D (2) E C A B D (2) D E B A C (2) D B A E C (2) D A B C E (2) C E A B D (2) C B E A D (2) B C E D A (2) B C E A D (2) A D C B E (2) E D C B A (1) E B D C A (1) E B C D A (1) E A D C B (1) D E A B C (1) D B E C A (1) D A C E B (1) D A C B E (1) C E B A D (1) C A E B D (1) B E C D A (1) B D E A C (1) B C A D E (1) B A C D E (1) A E D C B (1) A D E C B (1) A C E B D (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -8 -12 B 0 0 -12 -6 4 C 0 12 0 -2 -10 D 8 6 2 0 -6 E 12 -4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.384615 C: 0.153846 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384151 Cumulative probabilities = A: 0.000000 B: 0.384615 C: 0.538462 D: 0.538462 E: 1.000000 A B C D E A 0 0 0 -8 -12 B 0 0 -12 -6 4 C 0 12 0 -2 -10 D 8 6 2 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.384615 C: 0.153846 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384271 Cumulative probabilities = A: 0.000000 B: 0.384615 C: 0.538462 D: 0.538462 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 B=22 A=20 C=9 so C is eliminated. Round 2 votes counts: E=26 D=26 B=24 A=24 so B is eliminated. Round 3 votes counts: E=38 D=31 A=31 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:205 C:200 B:193 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -8 -12 B 0 0 -12 -6 4 C 0 12 0 -2 -10 D 8 6 2 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.384615 C: 0.153846 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384271 Cumulative probabilities = A: 0.000000 B: 0.384615 C: 0.538462 D: 0.538462 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 -12 B 0 0 -12 -6 4 C 0 12 0 -2 -10 D 8 6 2 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.384615 C: 0.153846 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384271 Cumulative probabilities = A: 0.000000 B: 0.384615 C: 0.538462 D: 0.538462 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 -12 B 0 0 -12 -6 4 C 0 12 0 -2 -10 D 8 6 2 0 -6 E 12 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.384615 C: 0.153846 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384271 Cumulative probabilities = A: 0.000000 B: 0.384615 C: 0.538462 D: 0.538462 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5547: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) C B E D A (7) A D E B C (7) C B E A D (6) B C A D E (6) C B A E D (4) B D C E A (4) B C D A E (4) A D B E C (4) E D A C B (3) D E A B C (3) D B C E A (3) C B D E A (3) A E D C B (3) A E C B D (3) A B D C E (3) E C D B A (2) E A D C B (2) D A E B C (2) B A D C E (2) B A C D E (2) E D C B A (1) E A C D B (1) D E C A B (1) D E B C A (1) D B E A C (1) D B A C E (1) C E B D A (1) C E B A D (1) B D C A E (1) B D A C E (1) A E D B C (1) A C E B D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -26 -14 -2 -4 B 26 0 20 28 34 C 14 -20 0 12 30 D 2 -28 -12 0 24 E 4 -34 -30 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -14 -2 -4 B 26 0 20 28 34 C 14 -20 0 12 30 D 2 -28 -12 0 24 E 4 -34 -30 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=25 C=22 D=12 E=9 so E is eliminated. Round 2 votes counts: B=32 A=28 C=24 D=16 so D is eliminated. Round 3 votes counts: B=38 A=36 C=26 so C is eliminated. Round 4 votes counts: B=63 A=37 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:254 C:218 D:193 A:177 E:158 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -14 -2 -4 B 26 0 20 28 34 C 14 -20 0 12 30 D 2 -28 -12 0 24 E 4 -34 -30 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -14 -2 -4 B 26 0 20 28 34 C 14 -20 0 12 30 D 2 -28 -12 0 24 E 4 -34 -30 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -14 -2 -4 B 26 0 20 28 34 C 14 -20 0 12 30 D 2 -28 -12 0 24 E 4 -34 -30 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5548: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (5) D A E B C (5) D A C B E (5) D A B C E (4) C E D B A (4) C B E A D (4) C B A D E (4) A D E B C (4) E D A C B (3) E C D B A (3) E C B A D (3) D C A E B (3) C B E D A (3) A D B C E (3) A B D C E (3) C D E B A (2) C D E A B (2) C D B E A (2) C D A B E (2) B E C A D (2) B C A E D (2) B A E D C (2) B A C D E (2) A E D B C (2) E D C A B (1) E D A B C (1) E C D A B (1) E C B D A (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B D C (1) D E A C B (1) D E A B C (1) D C E A B (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B D A (1) C E B A D (1) C D B A E (1) C B D E A (1) C B D A E (1) B E A D C (1) B C E A D (1) B C A D E (1) B A E C D (1) B A C E D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -6 -4 2 B 2 0 -6 -12 -2 C 6 6 0 4 12 D 4 12 -4 0 4 E -2 2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -4 2 B 2 0 -6 -12 -2 C 6 6 0 4 12 D 4 12 -4 0 4 E -2 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 E=22 A=14 B=13 so B is eliminated. Round 2 votes counts: C=32 E=25 D=23 A=20 so A is eliminated. Round 3 votes counts: C=35 D=34 E=31 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:208 A:195 E:192 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 2 B 2 0 -6 -12 -2 C 6 6 0 4 12 D 4 12 -4 0 4 E -2 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 2 B 2 0 -6 -12 -2 C 6 6 0 4 12 D 4 12 -4 0 4 E -2 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 2 B 2 0 -6 -12 -2 C 6 6 0 4 12 D 4 12 -4 0 4 E -2 2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5549: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) B A C E D (8) A B C D E (7) E C D B A (5) C A B D E (5) E D C B A (4) E D B A C (4) D E C A B (4) D E A B C (4) A B D C E (4) E C B A D (3) D C E A B (3) D A B C E (3) C E D B A (3) C E B A D (3) C D E B A (3) E B A C D (2) D E C B A (2) D A B E C (2) B A E C D (2) A D B C E (2) A B D E C (2) A B C E D (2) E D B C A (1) E C B D A (1) E B A D C (1) D C A E B (1) D C A B E (1) D A E B C (1) C E B D A (1) C D A B E (1) C B E A D (1) C B A D E (1) C A D B E (1) B E A C D (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -8 12 8 B 12 0 -4 10 8 C 8 4 0 20 20 D -12 -10 -20 0 -6 E -8 -8 -20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 12 8 B 12 0 -4 10 8 C 8 4 0 20 20 D -12 -10 -20 0 -6 E -8 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=21 D=21 A=18 B=12 so B is eliminated. Round 2 votes counts: C=29 A=28 E=22 D=21 so D is eliminated. Round 3 votes counts: C=34 A=34 E=32 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:213 A:200 E:185 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 12 8 B 12 0 -4 10 8 C 8 4 0 20 20 D -12 -10 -20 0 -6 E -8 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 12 8 B 12 0 -4 10 8 C 8 4 0 20 20 D -12 -10 -20 0 -6 E -8 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 12 8 B 12 0 -4 10 8 C 8 4 0 20 20 D -12 -10 -20 0 -6 E -8 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5550: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (6) C E B D A (5) C D E B A (5) B E C D A (5) A B E C D (5) E B A C D (4) D A C B E (4) A D C E B (4) A D B C E (4) C E D B A (3) C D A E B (3) B E D A C (3) B E A D C (3) B D E C A (3) B A E D C (3) A B E D C (3) E C B D A (2) D C A B E (2) D B C E A (2) D A B C E (2) C E A B D (2) B E D C A (2) A D B E C (2) A C E D B (2) A C D E B (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C A D (1) E A C B D (1) D C B E A (1) D B E C A (1) D B A E C (1) D A C E B (1) C E D A B (1) C E A D B (1) B E C A D (1) B D E A C (1) B D A E C (1) A E C B D (1) A E B C D (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 16 4 -10 B 8 0 12 16 16 C -16 -12 0 8 -8 D -4 -16 -8 0 -16 E 10 -16 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 16 4 -10 B 8 0 12 16 16 C -16 -12 0 8 -8 D -4 -16 -8 0 -16 E 10 -16 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=28 A=28 C=20 D=14 E=10 so E is eliminated. Round 2 votes counts: B=33 A=29 C=24 D=14 so D is eliminated. Round 3 votes counts: B=37 A=36 C=27 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:209 A:201 C:186 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 16 4 -10 B 8 0 12 16 16 C -16 -12 0 8 -8 D -4 -16 -8 0 -16 E 10 -16 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 4 -10 B 8 0 12 16 16 C -16 -12 0 8 -8 D -4 -16 -8 0 -16 E 10 -16 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 4 -10 B 8 0 12 16 16 C -16 -12 0 8 -8 D -4 -16 -8 0 -16 E 10 -16 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5551: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C A B (7) D E A C B (7) D E C B A (6) B C A E D (6) B A C E D (6) B A C D E (5) B C D E A (4) A C B E D (4) E C D B A (3) E A C B D (3) A E C B D (3) A B D C E (3) E D C B A (2) E D C A B (2) D E B C A (2) D A E B C (2) C B E A D (2) C B A E D (2) B C A D E (2) E D A C B (1) E C B D A (1) E A C D B (1) D E B A C (1) D B E C A (1) D B C E A (1) D B A E C (1) D A B E C (1) D A B C E (1) C E B D A (1) C E B A D (1) C B E D A (1) C B D E A (1) C A B E D (1) B D A C E (1) B C E A D (1) A E D B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 6 6 2 B 2 0 2 20 12 C -6 -2 0 16 6 D -6 -20 -16 0 -6 E -2 -12 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 6 2 B 2 0 2 20 12 C -6 -2 0 16 6 D -6 -20 -16 0 -6 E -2 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=25 A=23 E=13 C=9 so C is eliminated. Round 2 votes counts: B=31 D=30 A=24 E=15 so E is eliminated. Round 3 votes counts: D=38 B=34 A=28 so A is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:207 A:206 E:193 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 6 2 B 2 0 2 20 12 C -6 -2 0 16 6 D -6 -20 -16 0 -6 E -2 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 2 B 2 0 2 20 12 C -6 -2 0 16 6 D -6 -20 -16 0 -6 E -2 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 2 B 2 0 2 20 12 C -6 -2 0 16 6 D -6 -20 -16 0 -6 E -2 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5552: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) C A E B D (9) B D A E C (8) A E D C B (8) A E C D B (8) D A E B C (5) C E A B D (4) C B E D A (4) C B E A D (4) B D C E A (4) B D C A E (4) B C D E A (3) A D E B C (3) E C A D B (2) E A C D B (2) D B E A C (2) D B A E C (2) C B A E D (2) C A B E D (2) E C D A B (1) E A D C B (1) D E B C A (1) D E A C B (1) D E A B C (1) C E B D A (1) C E B A D (1) C A E D B (1) B C E A D (1) B C A E D (1) A E B D C (1) A D E C B (1) A D B E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 24 -8 26 16 B -24 0 -24 0 -22 C 8 24 0 12 2 D -26 0 -12 0 -26 E -16 22 -2 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 -8 26 16 B -24 0 -24 0 -22 C 8 24 0 12 2 D -26 0 -12 0 -26 E -16 22 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=24 B=21 D=12 E=6 so E is eliminated. Round 2 votes counts: C=40 A=27 B=21 D=12 so D is eliminated. Round 3 votes counts: C=40 A=34 B=26 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:229 C:223 E:215 D:168 B:165 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -8 26 16 B -24 0 -24 0 -22 C 8 24 0 12 2 D -26 0 -12 0 -26 E -16 22 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -8 26 16 B -24 0 -24 0 -22 C 8 24 0 12 2 D -26 0 -12 0 -26 E -16 22 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -8 26 16 B -24 0 -24 0 -22 C 8 24 0 12 2 D -26 0 -12 0 -26 E -16 22 -2 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5553: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (15) D B C A E (12) E A C D B (6) C A E D B (6) E A B C D (4) B D C A E (4) D B E A C (3) D B C E A (3) C A E B D (3) B E D A C (3) A E C B D (3) E B A D C (2) D E B A C (2) D C B A E (2) D B E C A (2) C D A B E (2) B E A D C (2) B E A C D (2) B A E C D (2) A C E B D (2) E D C A B (1) E D A C B (1) E C D A B (1) E B A C D (1) D E A C B (1) D C B E A (1) D C A E B (1) D C A B E (1) C E A D B (1) C D A E B (1) C A D E B (1) C A B E D (1) C A B D E (1) B D E A C (1) B C D A E (1) B C A D E (1) B A E D C (1) B A C E D (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 6 8 14 -4 B -6 0 -2 4 -6 C -8 2 0 14 -8 D -14 -4 -14 0 -18 E 4 6 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 14 -4 B -6 0 -2 4 -6 C -8 2 0 14 -8 D -14 -4 -14 0 -18 E 4 6 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=28 B=19 C=16 A=6 so A is eliminated. Round 2 votes counts: E=35 D=28 B=19 C=18 so C is eliminated. Round 3 votes counts: E=47 D=32 B=21 so B is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:212 C:200 B:195 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 14 -4 B -6 0 -2 4 -6 C -8 2 0 14 -8 D -14 -4 -14 0 -18 E 4 6 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 14 -4 B -6 0 -2 4 -6 C -8 2 0 14 -8 D -14 -4 -14 0 -18 E 4 6 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 14 -4 B -6 0 -2 4 -6 C -8 2 0 14 -8 D -14 -4 -14 0 -18 E 4 6 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5554: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (22) C B E D A (14) E D A B C (6) A D B E C (5) C A E D B (4) B C D E A (4) E D B A C (3) C E A D B (3) A C D E B (3) C E D B A (2) C E B D A (2) C B E A D (2) C B A D E (2) C A B D E (2) B C E D A (2) B A D E C (2) A B D E C (2) E D C B A (1) E D A C B (1) E B C D A (1) D E A B C (1) D B A E C (1) D A E B C (1) C E D A B (1) C E B A D (1) C B D E A (1) C B A E D (1) B E D C A (1) B E C D A (1) B D E A C (1) B D A E C (1) B A D C E (1) A D E C B (1) A D C E B (1) A D B C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 10 10 4 B -12 0 12 -16 -8 C -10 -12 0 -4 -2 D -10 16 4 0 8 E -4 8 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 10 4 B -12 0 12 -16 -8 C -10 -12 0 -4 -2 D -10 16 4 0 8 E -4 8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=35 B=13 E=12 D=3 so D is eliminated. Round 2 votes counts: A=38 C=35 B=14 E=13 so E is eliminated. Round 3 votes counts: A=46 C=36 B=18 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:209 E:199 B:188 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 10 4 B -12 0 12 -16 -8 C -10 -12 0 -4 -2 D -10 16 4 0 8 E -4 8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 10 4 B -12 0 12 -16 -8 C -10 -12 0 -4 -2 D -10 16 4 0 8 E -4 8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 10 4 B -12 0 12 -16 -8 C -10 -12 0 -4 -2 D -10 16 4 0 8 E -4 8 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5555: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) B D A E C (8) A C E D B (6) B A D E C (5) E D A C B (4) D B E A C (4) C E A B D (4) C A E B D (4) B D C E A (4) E C D A B (3) D E B C A (3) D B E C A (3) C A E D B (3) B D E C A (3) B D A C E (3) B A D C E (3) B A C D E (3) A C E B D (3) C E D B A (2) B D E A C (2) B C A D E (2) A E D C B (2) A B D E C (2) E D C B A (1) E D C A B (1) E C A D B (1) E A C D B (1) D E C B A (1) D A B E C (1) C E D A B (1) C D E B A (1) C B D E A (1) C A B E D (1) A E D B C (1) A E C D B (1) A D B E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 4 8 2 B -2 0 -2 0 -4 C -4 2 0 -4 4 D -8 0 4 0 0 E -2 4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 2 B -2 0 -2 0 -4 C -4 2 0 -4 4 D -8 0 4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=26 A=18 D=12 E=11 so E is eliminated. Round 2 votes counts: B=33 C=30 A=19 D=18 so D is eliminated. Round 3 votes counts: B=43 C=33 A=24 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:208 C:199 E:199 D:198 B:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 2 B -2 0 -2 0 -4 C -4 2 0 -4 4 D -8 0 4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 2 B -2 0 -2 0 -4 C -4 2 0 -4 4 D -8 0 4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 2 B -2 0 -2 0 -4 C -4 2 0 -4 4 D -8 0 4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5556: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) E B C A D (8) A D C B E (6) E B C D A (5) D A C B E (5) B E C D A (5) A C D B E (5) E B A C D (4) B E D C A (4) D C B A E (3) D C A B E (3) B D E C A (3) B D C E A (3) B C E A D (3) B C D A E (3) A D E C B (3) E A D C B (2) E A B D C (2) E A B C D (2) C D A B E (2) C A D B E (2) B C E D A (2) A D C E B (2) E D A B C (1) E A D B C (1) E A C D B (1) E A C B D (1) D B E C A (1) D B C A E (1) D A C E B (1) C D B A E (1) C B A D E (1) B D C A E (1) A E C D B (1) A E C B D (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -18 -4 -12 B 12 0 14 16 12 C 18 -14 0 0 -6 D 4 -16 0 0 -4 E 12 -12 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -4 -12 B 12 0 14 16 12 C 18 -14 0 0 -6 D 4 -16 0 0 -4 E 12 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=24 A=21 D=14 C=6 so C is eliminated. Round 2 votes counts: E=35 B=25 A=23 D=17 so D is eliminated. Round 3 votes counts: E=35 A=34 B=31 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:227 E:205 C:199 D:192 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -18 -4 -12 B 12 0 14 16 12 C 18 -14 0 0 -6 D 4 -16 0 0 -4 E 12 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -4 -12 B 12 0 14 16 12 C 18 -14 0 0 -6 D 4 -16 0 0 -4 E 12 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -4 -12 B 12 0 14 16 12 C 18 -14 0 0 -6 D 4 -16 0 0 -4 E 12 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5557: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) D B A E C (6) E C A D B (5) C E A B D (5) C D E B A (5) B A D E C (5) D B E A C (4) A B E D C (4) A B D E C (4) D B E C A (3) C E D B A (3) C E D A B (3) C B D E A (3) B D A C E (3) B A D C E (3) A D E B C (3) E C D A B (2) E A C D B (2) D E A B C (2) C B A E D (2) B D A E C (2) A E D B C (2) A E C B D (2) A E B D C (2) A E B C D (2) A D B E C (2) A B C E D (2) E D C A B (1) E A D C B (1) E A D B C (1) D E C B A (1) D B C E A (1) D B A C E (1) C E B A D (1) C A E B D (1) C A B E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 10 20 -2 B -12 0 8 -12 -4 C -10 -8 0 -2 -14 D -20 12 2 0 -2 E 2 4 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 10 20 -2 B -12 0 8 -12 -4 C -10 -8 0 -2 -14 D -20 12 2 0 -2 E 2 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=24 D=18 B=14 E=12 so E is eliminated. Round 2 votes counts: C=39 A=28 D=19 B=14 so B is eliminated. Round 3 votes counts: C=39 A=37 D=24 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:211 D:196 B:190 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 10 20 -2 B -12 0 8 -12 -4 C -10 -8 0 -2 -14 D -20 12 2 0 -2 E 2 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 20 -2 B -12 0 8 -12 -4 C -10 -8 0 -2 -14 D -20 12 2 0 -2 E 2 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 20 -2 B -12 0 8 -12 -4 C -10 -8 0 -2 -14 D -20 12 2 0 -2 E 2 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999069 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5558: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (17) D B E C A (10) A C B E D (10) C B E A D (8) D A E B C (6) A D E B C (5) A C E B D (5) A D E C B (4) C E B A D (3) C B E D A (3) B E C D A (3) B C E D A (3) E D C B A (2) E D B C A (2) D E B A C (2) C A B E D (2) B D E C A (2) A D C E B (2) A C D E B (2) E B C D A (1) C E A B D (1) B E D C A (1) A E C D B (1) A E C B D (1) A D C B E (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -16 -4 -16 B 14 0 8 -10 -8 C 16 -8 0 -12 -16 D 4 10 12 0 8 E 16 8 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -4 -16 B 14 0 8 -10 -8 C 16 -8 0 -12 -16 D 4 10 12 0 8 E 16 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=34 C=17 B=9 E=5 so E is eliminated. Round 2 votes counts: D=39 A=34 C=17 B=10 so B is eliminated. Round 3 votes counts: D=42 A=34 C=24 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:216 B:202 C:190 A:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -16 -4 -16 B 14 0 8 -10 -8 C 16 -8 0 -12 -16 D 4 10 12 0 8 E 16 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -4 -16 B 14 0 8 -10 -8 C 16 -8 0 -12 -16 D 4 10 12 0 8 E 16 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -4 -16 B 14 0 8 -10 -8 C 16 -8 0 -12 -16 D 4 10 12 0 8 E 16 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995031 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5559: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (24) D A E C B (17) C B E A D (8) D A E B C (7) D A B E C (5) B D A C E (4) B C E D A (4) C E B A D (3) B C D A E (3) A D E C B (3) E C A D B (2) E C A B D (2) D B A C E (2) C E B D A (2) A D B E C (2) E C B A D (1) E B C A D (1) E A D C B (1) E A C D B (1) D E C A B (1) D C E A B (1) D A B C E (1) C B D E A (1) B C A E D (1) B C A D E (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -10 4 -4 B 10 0 12 10 12 C 10 -12 0 8 10 D -4 -10 -8 0 -2 E 4 -12 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 4 -4 B 10 0 12 10 12 C 10 -12 0 8 10 D -4 -10 -8 0 -2 E 4 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=34 C=14 E=8 A=7 so A is eliminated. Round 2 votes counts: D=40 B=37 C=14 E=9 so E is eliminated. Round 3 votes counts: D=42 B=38 C=20 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:208 E:192 A:190 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 4 -4 B 10 0 12 10 12 C 10 -12 0 8 10 D -4 -10 -8 0 -2 E 4 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 4 -4 B 10 0 12 10 12 C 10 -12 0 8 10 D -4 -10 -8 0 -2 E 4 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 4 -4 B 10 0 12 10 12 C 10 -12 0 8 10 D -4 -10 -8 0 -2 E 4 -12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5560: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) E B C D A (8) C B A E D (8) B C E A D (8) A D C B E (8) D A B C E (6) A C B D E (6) E B C A D (5) D A C B E (5) D E B C A (4) E D B C A (2) E C A B D (2) D E A B C (2) D B C A E (2) D A E C B (2) D A E B C (2) C B E A D (2) B C A E D (2) A D B C E (2) A C B E D (2) E D C A B (1) E C B D A (1) E A D C B (1) E A C B D (1) D E B A C (1) D E A C B (1) D B A C E (1) C A B E D (1) B D C E A (1) B C E D A (1) B A C D E (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -14 20 0 B 12 0 -2 18 14 C 14 2 0 16 14 D -20 -18 -16 0 -8 E 0 -14 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 20 0 B 12 0 -2 18 14 C 14 2 0 16 14 D -20 -18 -16 0 -8 E 0 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 A=20 B=13 C=11 so C is eliminated. Round 2 votes counts: E=30 D=26 B=23 A=21 so A is eliminated. Round 3 votes counts: D=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:223 B:221 A:197 E:190 D:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 20 0 B 12 0 -2 18 14 C 14 2 0 16 14 D -20 -18 -16 0 -8 E 0 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 20 0 B 12 0 -2 18 14 C 14 2 0 16 14 D -20 -18 -16 0 -8 E 0 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 20 0 B 12 0 -2 18 14 C 14 2 0 16 14 D -20 -18 -16 0 -8 E 0 -14 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5561: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (20) B A C E D (8) D C E B A (5) C B E A D (5) A B E C D (5) B C A E D (4) B A D E C (4) E D C A B (3) E C A D B (3) D E C B A (3) D E A C B (3) B C D E A (3) E C D A B (2) D C E A B (2) D B E C A (2) C E D B A (2) B D A E C (2) B D A C E (2) A D E C B (2) A D E B C (2) A C E B D (2) E A C D B (1) D B C E A (1) D B A E C (1) D A E C B (1) C E D A B (1) C E B A D (1) C D E B A (1) C B E D A (1) B D C A E (1) B C E A D (1) B A D C E (1) B A C D E (1) A E C B D (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -22 -12 -20 B -2 0 -18 -12 -10 C 22 18 0 -16 -16 D 12 12 16 0 18 E 20 10 16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -22 -12 -20 B -2 0 -18 -12 -10 C 22 18 0 -16 -16 D 12 12 16 0 18 E 20 10 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=27 A=15 C=11 E=9 so E is eliminated. Round 2 votes counts: D=41 B=27 C=16 A=16 so C is eliminated. Round 3 votes counts: D=47 B=34 A=19 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:214 C:204 B:179 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -22 -12 -20 B -2 0 -18 -12 -10 C 22 18 0 -16 -16 D 12 12 16 0 18 E 20 10 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -22 -12 -20 B -2 0 -18 -12 -10 C 22 18 0 -16 -16 D 12 12 16 0 18 E 20 10 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -22 -12 -20 B -2 0 -18 -12 -10 C 22 18 0 -16 -16 D 12 12 16 0 18 E 20 10 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5562: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (10) C B D A E (8) A D E C B (8) A D C B E (8) E B C D A (7) E A D B C (6) D A E C B (6) D A C B E (4) C D A B E (4) E B A C D (3) E A B D C (3) D E A C B (3) D A C E B (3) C B A D E (3) B E C A D (3) E D A C B (2) E B A D C (2) C D B A E (2) B C A D E (2) A D C E B (2) E C D B A (1) E B C A D (1) E A D C B (1) D C A B E (1) C E B D A (1) C D E B A (1) C B E D A (1) B E C D A (1) A E D B C (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 8 8 -10 8 B -8 0 -20 -8 -4 C -8 20 0 -2 2 D 10 8 2 0 14 E -8 4 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 -10 8 B -8 0 -20 -8 -4 C -8 20 0 -2 2 D 10 8 2 0 14 E -8 4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=21 C=20 D=17 B=16 so B is eliminated. Round 2 votes counts: C=32 E=30 A=21 D=17 so D is eliminated. Round 3 votes counts: A=34 E=33 C=33 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:207 C:206 E:190 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 8 -10 8 B -8 0 -20 -8 -4 C -8 20 0 -2 2 D 10 8 2 0 14 E -8 4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -10 8 B -8 0 -20 -8 -4 C -8 20 0 -2 2 D 10 8 2 0 14 E -8 4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -10 8 B -8 0 -20 -8 -4 C -8 20 0 -2 2 D 10 8 2 0 14 E -8 4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5563: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (17) D B A C E (10) E C A D B (8) E C A B D (7) D B C E A (7) B D A C E (7) C E D B A (6) A E C B D (6) C E A D B (5) C E D A B (4) A E B C D (4) E C D B A (3) B D A E C (3) D B C A E (2) C D E B A (2) A B E D C (2) E C D A B (1) E A C B D (1) D C E B A (1) D C B E A (1) D B E C A (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 12 4 4 6 B -12 0 8 -2 2 C -4 -8 0 -4 -8 D -4 2 4 0 4 E -6 -2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 4 6 B -12 0 8 -2 2 C -4 -8 0 -4 -8 D -4 2 4 0 4 E -6 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 E=20 C=17 B=11 so B is eliminated. Round 2 votes counts: D=32 A=31 E=20 C=17 so C is eliminated. Round 3 votes counts: E=35 D=34 A=31 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:213 D:203 B:198 E:198 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 4 6 B -12 0 8 -2 2 C -4 -8 0 -4 -8 D -4 2 4 0 4 E -6 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 4 6 B -12 0 8 -2 2 C -4 -8 0 -4 -8 D -4 2 4 0 4 E -6 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 4 6 B -12 0 8 -2 2 C -4 -8 0 -4 -8 D -4 2 4 0 4 E -6 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5564: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E C D B A (6) D C A B E (6) B A D C E (6) E B C D A (5) B A E C D (5) A B E D C (5) E B A C D (4) D C B A E (4) C D E A B (4) B E A C D (4) B A E D C (4) A D C B E (4) D C E A B (3) A E D C B (3) A B D E C (3) A B D C E (3) E C B D A (2) D C A E B (2) C D E B A (2) C D B E A (2) B E C D A (2) B D C A E (2) E B C A D (1) E A B C D (1) D A C E B (1) B E C A D (1) B C D E A (1) B A D E C (1) A D E C B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -6 -4 4 B 4 0 0 0 10 C 6 0 0 0 -18 D 4 0 0 0 -6 E -4 -10 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.626653 C: 0.179504 D: 0.193843 E: 0.000000 Sum of squares = 0.462490869974 Cumulative probabilities = A: 0.000000 B: 0.626653 C: 0.806157 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 4 B 4 0 0 0 10 C 6 0 0 0 -18 D 4 0 0 0 -6 E -4 -10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.506757 C: 0.175676 D: 0.317568 E: 0.000000 Sum of squares = 0.388513545936 Cumulative probabilities = A: 0.000000 B: 0.506757 C: 0.682432 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=26 A=21 D=16 C=8 so C is eliminated. Round 2 votes counts: E=29 B=26 D=24 A=21 so A is eliminated. Round 3 votes counts: B=37 E=32 D=31 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:207 E:205 D:199 A:195 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 4 B 4 0 0 0 10 C 6 0 0 0 -18 D 4 0 0 0 -6 E -4 -10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.506757 C: 0.175676 D: 0.317568 E: 0.000000 Sum of squares = 0.388513545936 Cumulative probabilities = A: 0.000000 B: 0.506757 C: 0.682432 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 4 B 4 0 0 0 10 C 6 0 0 0 -18 D 4 0 0 0 -6 E -4 -10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.506757 C: 0.175676 D: 0.317568 E: 0.000000 Sum of squares = 0.388513545936 Cumulative probabilities = A: 0.000000 B: 0.506757 C: 0.682432 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 4 B 4 0 0 0 10 C 6 0 0 0 -18 D 4 0 0 0 -6 E -4 -10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.506757 C: 0.175676 D: 0.317568 E: 0.000000 Sum of squares = 0.388513545936 Cumulative probabilities = A: 0.000000 B: 0.506757 C: 0.682432 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5565: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) A B E C D (8) E D B C A (7) D E C B A (7) B A E D C (7) E B D A C (5) C D A E B (5) C D A B E (5) E D C B A (4) D C E B A (4) C A D E B (4) A C B E D (4) C D E A B (3) C A D B E (3) A C B D E (3) A B C E D (3) E B D C A (2) E B A D C (2) D E B C A (2) C E A D B (2) B E D A C (2) A C D B E (2) A B E D C (2) A B C D E (2) E C D B A (1) D B E A C (1) C E D A B (1) B D E A C (1) Total count = 100 A B C D E A 0 -6 0 0 -4 B 6 0 4 -2 2 C 0 -4 0 -8 -18 D 0 2 8 0 -16 E 4 -2 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.660000000023 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 A B C D E A 0 -6 0 0 -4 B 6 0 4 -2 2 C 0 -4 0 -8 -18 D 0 2 8 0 -16 E 4 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999998683 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 E=21 B=18 D=14 so D is eliminated. Round 2 votes counts: E=30 C=27 A=24 B=19 so B is eliminated. Round 3 votes counts: E=42 A=31 C=27 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 B:205 D:197 A:195 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 0 -4 B 6 0 4 -2 2 C 0 -4 0 -8 -18 D 0 2 8 0 -16 E 4 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999998683 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 0 -4 B 6 0 4 -2 2 C 0 -4 0 -8 -18 D 0 2 8 0 -16 E 4 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999998683 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 0 -4 B 6 0 4 -2 2 C 0 -4 0 -8 -18 D 0 2 8 0 -16 E 4 -2 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.100000 E: 0.100000 Sum of squares = 0.659999998683 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.900000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5566: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) D B E C A (6) C D A B E (6) E B A D C (5) D E B C A (5) C A D E B (5) A C E B D (5) E B D A C (4) D B C E A (4) C D B E A (4) C D B A E (4) C A D B E (4) B E D A C (4) B D E C A (4) D C B E A (3) C D E B A (3) C A B D E (3) A E C D B (3) E A B D C (2) C A B E D (2) B E A D C (2) A E B D C (2) A C E D B (2) D E C B A (1) D C E A B (1) C D E A B (1) C D A E B (1) B D E A C (1) B A E D C (1) B A E C D (1) A E C B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -14 -4 0 B 4 0 0 -6 4 C 14 0 0 10 0 D 4 6 -10 0 12 E 0 -4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.430118 C: 0.569882 D: 0.000000 E: 0.000000 Sum of squares = 0.509767091101 Cumulative probabilities = A: 0.000000 B: 0.430118 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -4 0 B 4 0 0 -6 4 C 14 0 0 10 0 D 4 6 -10 0 12 E 0 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=23 D=20 B=13 E=11 so E is eliminated. Round 2 votes counts: C=33 A=25 B=22 D=20 so D is eliminated. Round 3 votes counts: C=38 B=37 A=25 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 D:206 B:201 E:192 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -4 0 B 4 0 0 -6 4 C 14 0 0 10 0 D 4 6 -10 0 12 E 0 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -4 0 B 4 0 0 -6 4 C 14 0 0 10 0 D 4 6 -10 0 12 E 0 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -4 0 B 4 0 0 -6 4 C 14 0 0 10 0 D 4 6 -10 0 12 E 0 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5567: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) C B D E A (10) E A C D B (9) C B D A E (7) B D C A E (6) E C A B D (5) D B A E C (5) A D B E C (5) D B A C E (4) C B E D A (4) B C D E A (4) A D E B C (4) A E D C B (3) E A D C B (2) E A D B C (2) D A B C E (2) B D C E A (2) B C D A E (2) E D B A C (1) E C B A D (1) E A C B D (1) D B C A E (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) C E A B D (1) B D E C A (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 8 -4 10 B 0 0 6 -6 8 C -8 -6 0 -4 -8 D 4 6 4 0 10 E -10 -8 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -4 10 B 0 0 6 -6 8 C -8 -6 0 -4 -8 D 4 6 4 0 10 E -10 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 E=21 B=15 D=14 so D is eliminated. Round 2 votes counts: A=30 B=25 C=24 E=21 so E is eliminated. Round 3 votes counts: A=44 C=30 B=26 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:207 B:204 E:190 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 8 -4 10 B 0 0 6 -6 8 C -8 -6 0 -4 -8 D 4 6 4 0 10 E -10 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -4 10 B 0 0 6 -6 8 C -8 -6 0 -4 -8 D 4 6 4 0 10 E -10 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -4 10 B 0 0 6 -6 8 C -8 -6 0 -4 -8 D 4 6 4 0 10 E -10 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) A C E B D (7) E B C A D (6) D B E C A (6) C A E B D (6) B E C D A (6) B E C A D (6) B D E C A (5) E C B A D (4) A C E D B (4) E C A B D (3) E A C B D (3) D B E A C (3) D B A C E (3) B E D C A (3) A C D E B (3) D E B A C (2) D A E C B (2) C E A B D (2) A D C E B (2) A C D B E (2) E D B A C (1) E B D C A (1) D E A B C (1) D B C E A (1) D A E B C (1) D A C B E (1) D A B E C (1) D A B C E (1) B D C E A (1) B D C A E (1) B C E A D (1) B C A D E (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 -6 2 -10 B 2 0 2 12 -16 C 6 -2 0 10 -10 D -2 -12 -10 0 -8 E 10 16 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 2 -10 B 2 0 2 12 -16 C 6 -2 0 10 -10 D -2 -12 -10 0 -8 E 10 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=24 A=19 E=18 C=8 so C is eliminated. Round 2 votes counts: D=31 A=25 B=24 E=20 so E is eliminated. Round 3 votes counts: B=35 A=33 D=32 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:222 C:202 B:200 A:192 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 2 -10 B 2 0 2 12 -16 C 6 -2 0 10 -10 D -2 -12 -10 0 -8 E 10 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 -10 B 2 0 2 12 -16 C 6 -2 0 10 -10 D -2 -12 -10 0 -8 E 10 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 -10 B 2 0 2 12 -16 C 6 -2 0 10 -10 D -2 -12 -10 0 -8 E 10 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5569: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) D E B A C (9) C A B D E (7) E D C B A (5) C A D B E (5) A B C E D (5) D E C B A (4) D C A B E (4) E B D A C (3) E B A D C (3) D A C B E (3) B A C E D (3) E B C A D (2) D E C A B (2) D E B C A (2) D C E A B (2) D A B C E (2) C D A B E (2) B E A D C (2) B A E D C (2) B A E C D (2) A C B E D (2) A B C D E (2) E D B C A (1) E D B A C (1) E C B A D (1) E B A C D (1) D C A E B (1) C E A B D (1) C D A E B (1) C B A E D (1) C A E B D (1) B E D A C (1) B E A C D (1) B D A E C (1) B A D E C (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -8 12 18 B -8 0 -12 10 20 C 8 12 0 0 12 D -12 -10 0 0 0 E -18 -20 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.756431 D: 0.243569 E: 0.000000 Sum of squares = 0.631514068137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.756431 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 12 18 B -8 0 -12 10 20 C 8 12 0 0 12 D -12 -10 0 0 0 E -18 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 E=17 B=13 A=11 so A is eliminated. Round 2 votes counts: C=33 D=30 B=20 E=17 so E is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:215 B:205 D:189 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 12 18 B -8 0 -12 10 20 C 8 12 0 0 12 D -12 -10 0 0 0 E -18 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 12 18 B -8 0 -12 10 20 C 8 12 0 0 12 D -12 -10 0 0 0 E -18 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 12 18 B -8 0 -12 10 20 C 8 12 0 0 12 D -12 -10 0 0 0 E -18 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000008094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5570: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) D E C A B (6) C B D E A (6) A B E D C (6) E D A C B (4) C D E B A (4) E A D C B (3) E A C D B (3) B D C A E (3) B C D A E (3) B C A D E (3) A E D B C (3) A E B D C (3) A E B C D (3) A B E C D (3) A B D E C (3) E D C A B (2) E C D A B (2) D E C B A (2) D C E B A (2) D B C E A (2) D B C A E (2) D A E B C (2) C E D A B (2) B D A C E (2) B A C E D (2) B A C D E (2) A B C E D (2) E C A D B (1) D E A C B (1) D E A B C (1) C E D B A (1) C E A B D (1) C B E D A (1) C B E A D (1) C B A E D (1) B D A E C (1) B C A E D (1) B A D E C (1) B A D C E (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -6 -12 -4 B -4 0 -2 0 6 C 6 2 0 0 -2 D 12 0 0 0 8 E 4 -6 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.532279 D: 0.467721 E: 0.000000 Sum of squares = 0.502083855692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.532279 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -12 -4 B -4 0 -2 0 6 C 6 2 0 0 -2 D 12 0 0 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 B=19 D=18 E=15 so E is eliminated. Round 2 votes counts: A=30 C=27 D=24 B=19 so B is eliminated. Round 3 votes counts: A=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:203 B:200 E:196 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 -12 -4 B -4 0 -2 0 6 C 6 2 0 0 -2 D 12 0 0 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -12 -4 B -4 0 -2 0 6 C 6 2 0 0 -2 D 12 0 0 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -12 -4 B -4 0 -2 0 6 C 6 2 0 0 -2 D 12 0 0 0 8 E 4 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5571: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) C A D E B (9) C B E D A (7) E B D A C (6) C D A B E (6) C A D B E (6) A D C E B (5) C D B E A (4) A E D B C (4) A D E B C (4) D C A E B (3) C B D E A (3) B E C D A (3) B E A D C (3) A D E C B (3) D E B A C (2) C D A E B (2) B E D A C (2) A E B D C (2) A C D E B (2) E B A D C (1) E A B D C (1) D E A B C (1) D C B E A (1) D B E C A (1) D A C E B (1) C B E A D (1) C B D A E (1) C B A E D (1) C A B E D (1) C A B D E (1) B E C A D (1) B C E D A (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -24 -8 4 B -4 0 -16 -8 8 C 24 16 0 0 12 D 8 8 0 0 10 E -4 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.259878 D: 0.740122 E: 0.000000 Sum of squares = 0.615317543856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.259878 D: 1.000000 E: 1.000000 A B C D E A 0 4 -24 -8 4 B -4 0 -16 -8 8 C 24 16 0 0 12 D 8 8 0 0 10 E -4 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 A=21 B=20 D=9 E=8 so E is eliminated. Round 2 votes counts: C=42 B=27 A=22 D=9 so D is eliminated. Round 3 votes counts: C=46 B=30 A=24 so A is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 D:213 B:190 A:188 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -24 -8 4 B -4 0 -16 -8 8 C 24 16 0 0 12 D 8 8 0 0 10 E -4 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -24 -8 4 B -4 0 -16 -8 8 C 24 16 0 0 12 D 8 8 0 0 10 E -4 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -24 -8 4 B -4 0 -16 -8 8 C 24 16 0 0 12 D 8 8 0 0 10 E -4 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5572: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) B C E A D (7) D A E C B (6) A D E C B (6) C E B A D (5) D A E B C (4) B D A C E (4) B C E D A (4) B C A D E (4) E C D A B (3) D B E A C (3) C B E A D (3) B A D C E (3) B A C D E (3) A D B C E (3) E B D C A (2) D E A C B (2) D B A E C (2) D A B E C (2) C E B D A (2) C A B D E (2) B D A E C (2) B C A E D (2) A D C E B (2) A D C B E (2) A C B D E (2) E D C B A (1) E D A C B (1) E C A D B (1) E A D C B (1) D E B A C (1) D E A B C (1) C E A D B (1) C B A E D (1) C A E B D (1) C A B E D (1) B E C D A (1) B D E A C (1) A D B E C (1) Total count = 100 A B C D E A 0 -16 4 2 6 B 16 0 0 14 6 C -4 0 0 0 4 D -2 -14 0 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.442344 C: 0.557656 D: 0.000000 E: 0.000000 Sum of squares = 0.506648479965 Cumulative probabilities = A: 0.000000 B: 0.442344 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 2 6 B 16 0 0 14 6 C -4 0 0 0 4 D -2 -14 0 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=21 E=16 C=16 A=16 so E is eliminated. Round 2 votes counts: B=33 C=27 D=23 A=17 so A is eliminated. Round 3 votes counts: D=38 B=33 C=29 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:200 A:198 D:198 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 2 6 B 16 0 0 14 6 C -4 0 0 0 4 D -2 -14 0 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 2 6 B 16 0 0 14 6 C -4 0 0 0 4 D -2 -14 0 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 2 6 B 16 0 0 14 6 C -4 0 0 0 4 D -2 -14 0 0 12 E -6 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5573: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) C A B E D (7) D B E A C (6) C D A B E (6) E B A C D (5) E A C B D (5) E C A B D (4) E A B C D (4) D B A C E (4) C A E B D (4) E D B A C (3) D C E A B (3) D C A B E (3) D B C A E (3) C B A D E (3) A C E B D (3) D E C B A (2) D E B A C (2) C A D B E (2) C A B D E (2) B E A C D (2) B D E A C (2) B D A C E (2) B C A D E (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B D C (1) D E B C A (1) D C B E A (1) C E D A B (1) C E A B D (1) C D E A B (1) C A D E B (1) B E D A C (1) B A D C E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -8 0 4 B -2 0 -16 6 10 C 8 16 0 10 16 D 0 -6 -10 0 8 E -4 -10 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 0 4 B -2 0 -16 6 10 C 8 16 0 10 16 D 0 -6 -10 0 8 E -4 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=28 E=25 B=10 A=5 so A is eliminated. Round 2 votes counts: D=32 C=32 E=26 B=10 so B is eliminated. Round 3 votes counts: D=37 C=34 E=29 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 A:199 B:199 D:196 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 0 4 B -2 0 -16 6 10 C 8 16 0 10 16 D 0 -6 -10 0 8 E -4 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 0 4 B -2 0 -16 6 10 C 8 16 0 10 16 D 0 -6 -10 0 8 E -4 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 0 4 B -2 0 -16 6 10 C 8 16 0 10 16 D 0 -6 -10 0 8 E -4 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5574: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) E A C B D (7) B D A C E (5) B A E D C (5) B A D E C (5) A B D E C (5) D B C A E (4) D B A C E (4) C E D A B (4) C D E A B (4) A B E C D (4) E C B D A (3) E C B A D (3) E C A D B (3) D C E B A (3) D C B E A (3) C E D B A (3) B D A E C (3) E C A B D (2) C E A D B (2) E A B C D (1) D C B A E (1) D C A E B (1) D B C E A (1) D A C E B (1) D A B C E (1) C E B D A (1) C D E B A (1) C D A E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B D C E A (1) B D C A E (1) B A E C D (1) A E C D B (1) A E C B D (1) A D C E B (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 12 2 10 B 2 0 6 20 -2 C -12 -6 0 2 -10 D -2 -20 -2 0 -2 E -10 2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408148 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -2 12 2 10 B 2 0 6 20 -2 C -12 -6 0 2 -10 D -2 -20 -2 0 -2 E -10 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408195 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=22 E=19 D=19 C=16 so C is eliminated. Round 2 votes counts: E=29 D=25 B=24 A=22 so A is eliminated. Round 3 votes counts: E=39 B=34 D=27 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:213 A:211 E:202 C:187 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 12 2 10 B 2 0 6 20 -2 C -12 -6 0 2 -10 D -2 -20 -2 0 -2 E -10 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408195 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 2 10 B 2 0 6 20 -2 C -12 -6 0 2 -10 D -2 -20 -2 0 -2 E -10 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408195 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 2 10 B 2 0 6 20 -2 C -12 -6 0 2 -10 D -2 -20 -2 0 -2 E -10 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408195 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5575: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) E D A C B (6) E A D C B (6) B C D A E (6) B C A D E (5) B C A E D (4) A E C B D (4) E A D B C (3) C D B A E (3) C B A D E (3) C A D B E (3) C A B E D (3) B D E C A (3) B D E A C (3) B A E C D (3) A E C D B (3) A E B C D (3) A C E D B (3) E A B D C (2) D E C A B (2) D E B A C (2) D E A C B (2) D B C E A (2) C B D A E (2) C A E D B (2) B A C E D (2) E D B A C (1) E D A B C (1) D E B C A (1) D C E B A (1) D C E A B (1) C D A E B (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B A C D E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 10 10 B 2 0 4 12 8 C 2 -4 0 10 6 D -10 -12 -10 0 0 E -10 -8 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 10 10 B 2 0 4 12 8 C 2 -4 0 10 6 D -10 -12 -10 0 0 E -10 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=19 C=19 A=15 D=11 so D is eliminated. Round 2 votes counts: B=38 E=26 C=21 A=15 so A is eliminated. Round 3 votes counts: B=39 E=36 C=25 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:208 C:207 E:188 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 10 10 B 2 0 4 12 8 C 2 -4 0 10 6 D -10 -12 -10 0 0 E -10 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 10 10 B 2 0 4 12 8 C 2 -4 0 10 6 D -10 -12 -10 0 0 E -10 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 10 10 B 2 0 4 12 8 C 2 -4 0 10 6 D -10 -12 -10 0 0 E -10 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5576: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (13) B D E C A (7) D B E C A (5) D B A E C (5) C E A D B (5) A B D C E (5) E C D B A (4) C E A B D (4) B D A E C (4) E C A D B (3) D E C B A (3) C A E B D (3) B A D C E (3) A C E D B (3) A C B E D (3) A B C D E (3) E C D A B (2) D B A C E (2) D A B E C (2) B D A C E (2) A B C E D (2) E D C B A (1) E C B A D (1) E C A B D (1) E A C D B (1) D E C A B (1) D E B A C (1) D E A C B (1) D B E A C (1) D A E B C (1) D A B C E (1) C E B A D (1) C A E D B (1) C A B E D (1) B E C D A (1) A E C D B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 18 12 12 14 B -18 0 -8 8 -2 C -12 8 0 6 8 D -12 -8 -6 0 -2 E -14 2 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 12 12 14 B -18 0 -8 8 -2 C -12 8 0 6 8 D -12 -8 -6 0 -2 E -14 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 B=17 C=15 E=13 so E is eliminated. Round 2 votes counts: A=33 C=26 D=24 B=17 so B is eliminated. Round 3 votes counts: D=37 A=36 C=27 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:205 E:191 B:190 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 12 12 14 B -18 0 -8 8 -2 C -12 8 0 6 8 D -12 -8 -6 0 -2 E -14 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 12 14 B -18 0 -8 8 -2 C -12 8 0 6 8 D -12 -8 -6 0 -2 E -14 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 12 14 B -18 0 -8 8 -2 C -12 8 0 6 8 D -12 -8 -6 0 -2 E -14 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5577: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (14) C D A B E (13) E B A D C (10) C D A E B (7) B D A E C (7) D A B E C (6) E C B A D (5) C E A B D (4) B A E D C (4) D C A B E (3) C E D A B (3) C E B A D (3) E C A B D (2) E B A C D (2) D B A E C (2) D A C E B (2) D A C B E (2) C E B D A (2) C E A D B (2) E C A D B (1) E B C A D (1) E A B D C (1) D B A C E (1) D A B C E (1) C D E A B (1) B A D E C (1) Total count = 100 A B C D E A 0 -4 6 0 -2 B 4 0 0 12 8 C -6 0 0 -8 -12 D 0 -12 8 0 -8 E 2 -8 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.760673 C: 0.239327 D: 0.000000 E: 0.000000 Sum of squares = 0.635900589249 Cumulative probabilities = A: 0.000000 B: 0.760673 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 0 -2 B 4 0 0 12 8 C -6 0 0 -8 -12 D 0 -12 8 0 -8 E 2 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000146686 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=26 E=22 D=17 so A is eliminated. Round 2 votes counts: C=35 B=26 E=22 D=17 so D is eliminated. Round 3 votes counts: C=42 B=36 E=22 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:207 A:200 D:194 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 0 -2 B 4 0 0 12 8 C -6 0 0 -8 -12 D 0 -12 8 0 -8 E 2 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000146686 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 0 -2 B 4 0 0 12 8 C -6 0 0 -8 -12 D 0 -12 8 0 -8 E 2 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000146686 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 0 -2 B 4 0 0 12 8 C -6 0 0 -8 -12 D 0 -12 8 0 -8 E 2 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000146686 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5578: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) E D B A C (10) D A C B E (9) A D C B E (9) C A B D E (7) C A D B E (6) D A E C B (5) E D A C B (4) E B C A D (4) A C D B E (4) E D A B C (3) E A D C B (3) D A C E B (3) E B D A C (2) D E A B C (2) D A E B C (2) C B A D E (2) B E C D A (2) B C E A D (2) B C A D E (2) A D C E B (2) E B A D C (1) E A D B C (1) E A C D B (1) D E B A C (1) C B E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 22 24 -10 4 B -22 0 -12 -30 -10 C -24 12 0 -14 -4 D 10 30 14 0 10 E -4 10 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 24 -10 4 B -22 0 -12 -30 -10 C -24 12 0 -14 -4 D 10 30 14 0 10 E -4 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=22 C=16 A=15 B=7 so B is eliminated. Round 2 votes counts: E=42 D=22 C=21 A=15 so A is eliminated. Round 3 votes counts: E=42 D=33 C=25 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:232 A:220 E:200 C:185 B:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 24 -10 4 B -22 0 -12 -30 -10 C -24 12 0 -14 -4 D 10 30 14 0 10 E -4 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 24 -10 4 B -22 0 -12 -30 -10 C -24 12 0 -14 -4 D 10 30 14 0 10 E -4 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 24 -10 4 B -22 0 -12 -30 -10 C -24 12 0 -14 -4 D 10 30 14 0 10 E -4 10 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5579: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) D B E A C (5) C D E B A (5) C A B E D (5) B A C D E (5) A B D E C (5) E A D B C (4) D E B A C (4) C E A D B (4) E D C A B (3) E D A C B (3) C E A B D (3) B D A E C (3) B A D E C (3) A E B D C (3) E D A B C (2) E C D A B (2) E C A D B (2) D E C B A (2) D E A B C (2) D B E C A (2) D B A E C (2) A E B C D (2) A C E B D (2) A B C E D (2) E A D C B (1) D E C A B (1) D E B C A (1) D B C E A (1) C E D B A (1) C D B E A (1) C B D A E (1) C B A E D (1) C B A D E (1) C A E B D (1) B D C A E (1) B D A C E (1) A E D B C (1) A D E B C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 20 6 -4 -16 B -20 0 4 -18 -18 C -6 -4 0 -4 -12 D 4 18 4 0 -6 E 16 18 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 6 -4 -16 B -20 0 4 -18 -18 C -6 -4 0 -4 -12 D 4 18 4 0 -6 E 16 18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=20 A=18 E=17 B=13 so B is eliminated. Round 2 votes counts: C=32 A=26 D=25 E=17 so E is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:226 D:210 A:203 C:187 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 6 -4 -16 B -20 0 4 -18 -18 C -6 -4 0 -4 -12 D 4 18 4 0 -6 E 16 18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 -4 -16 B -20 0 4 -18 -18 C -6 -4 0 -4 -12 D 4 18 4 0 -6 E 16 18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 -4 -16 B -20 0 4 -18 -18 C -6 -4 0 -4 -12 D 4 18 4 0 -6 E 16 18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5580: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) E D B C A (7) E D B A C (7) E B C A D (5) E B D A C (4) C D A E B (4) C A B D E (4) A C D B E (4) E D C B A (3) E B D C A (3) D E A C B (3) D E A B C (3) D C A E B (3) C A D B E (3) E D C A B (2) D E C A B (2) D E B A C (2) D C E A B (2) D A E C B (2) D A E B C (2) D A C E B (2) C A B E D (2) B E C A D (2) B C A E D (2) B A E C D (2) B A C E D (2) B A C D E (2) E C B A D (1) E C A D B (1) E B A C D (1) D B E A C (1) D B A E C (1) D A B C E (1) C B E A D (1) C A E D B (1) C A E B D (1) B E A C D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 2 -26 -2 B -6 0 -2 -30 -22 C -2 2 0 -18 -10 D 26 30 18 0 4 E 2 22 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -26 -2 B -6 0 -2 -30 -22 C -2 2 0 -18 -10 D 26 30 18 0 4 E 2 22 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983138 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=33 C=16 B=11 A=6 so A is eliminated. Round 2 votes counts: E=34 D=33 C=21 B=12 so B is eliminated. Round 3 votes counts: E=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:239 E:215 A:190 C:186 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -26 -2 B -6 0 -2 -30 -22 C -2 2 0 -18 -10 D 26 30 18 0 4 E 2 22 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983138 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -26 -2 B -6 0 -2 -30 -22 C -2 2 0 -18 -10 D 26 30 18 0 4 E 2 22 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983138 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -26 -2 B -6 0 -2 -30 -22 C -2 2 0 -18 -10 D 26 30 18 0 4 E 2 22 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983138 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5581: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) E D B A C (7) D E C A B (7) D E A B C (6) A B C E D (6) E B A D C (5) D E C B A (5) C B A E D (5) B A C E D (5) A B C D E (5) C B A D E (4) E C B A D (3) E B A C D (3) D E A C B (3) D A B C E (3) C D A B E (3) C A B D E (3) E D A B C (2) D A C B E (2) C D B A E (2) E D C B A (1) E B D A C (1) D E B A C (1) D C E A B (1) D C A E B (1) C E B A D (1) B E C A D (1) B C E A D (1) B C A E D (1) B A E C D (1) A E B D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 -6 4 B -6 0 2 -4 4 C -6 -2 0 -10 2 D 6 4 10 0 10 E -4 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -6 4 B -6 0 2 -4 4 C -6 -2 0 -10 2 D 6 4 10 0 10 E -4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=22 C=18 A=14 B=9 so B is eliminated. Round 2 votes counts: D=37 E=23 C=20 A=20 so C is eliminated. Round 3 votes counts: D=42 A=33 E=25 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:205 B:198 C:192 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -6 4 B -6 0 2 -4 4 C -6 -2 0 -10 2 D 6 4 10 0 10 E -4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -6 4 B -6 0 2 -4 4 C -6 -2 0 -10 2 D 6 4 10 0 10 E -4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -6 4 B -6 0 2 -4 4 C -6 -2 0 -10 2 D 6 4 10 0 10 E -4 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5582: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) D B E C A (8) A E C B D (7) D B C E A (6) A E B C D (6) B D E A C (5) E A B D C (4) E A B C D (4) D B C A E (4) E A C B D (3) B D A E C (3) B A E D C (3) A E B D C (3) E B A D C (2) D C B A E (2) C D E B A (2) C D E A B (2) C D B A E (2) C D A E B (2) C D A B E (2) C A D E B (2) B E D A C (2) B D A C E (2) A C E B D (2) E C A D B (1) D E C B A (1) D E B C A (1) D C B E A (1) C E D A B (1) C A D B E (1) B A D E C (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 8 14 B -10 0 14 0 -10 C 0 -14 0 2 -12 D -8 0 -2 0 -6 E -14 10 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.723293 B: 0.000000 C: 0.276707 D: 0.000000 E: 0.000000 Sum of squares = 0.599719755281 Cumulative probabilities = A: 0.723293 B: 0.723293 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 8 14 B -10 0 14 0 -10 C 0 -14 0 2 -12 D -8 0 -2 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.416667 D: 0.000000 E: 0.000000 Sum of squares = 0.513888889451 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=23 A=21 B=16 E=14 so E is eliminated. Round 2 votes counts: A=32 C=27 D=23 B=18 so B is eliminated. Round 3 votes counts: A=38 D=35 C=27 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:207 B:197 D:192 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 8 14 B -10 0 14 0 -10 C 0 -14 0 2 -12 D -8 0 -2 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.416667 D: 0.000000 E: 0.000000 Sum of squares = 0.513888889451 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 8 14 B -10 0 14 0 -10 C 0 -14 0 2 -12 D -8 0 -2 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.416667 D: 0.000000 E: 0.000000 Sum of squares = 0.513888889451 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 8 14 B -10 0 14 0 -10 C 0 -14 0 2 -12 D -8 0 -2 0 -6 E -14 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.000000 C: 0.416667 D: 0.000000 E: 0.000000 Sum of squares = 0.513888889451 Cumulative probabilities = A: 0.583333 B: 0.583333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5583: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) B C A D E (8) C B D A E (6) E D A B C (5) B E C A D (4) B C A E D (4) B A C D E (4) A D C E B (4) E B A D C (3) D C A E B (3) C D E B A (3) C B D E A (3) C B A D E (3) B C E A D (3) A D E C B (3) E D C A B (2) D E A C B (2) D A C E B (2) C D B A E (2) B E C D A (2) B E A C D (2) A E D C B (2) E C D B A (1) E B D C A (1) E B D A C (1) E B C D A (1) E A D B C (1) E A B D C (1) D C E A B (1) C D E A B (1) C A D B E (1) C A B D E (1) B C E D A (1) B A E D C (1) B A E C D (1) B A C E D (1) A E D B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -2 6 4 B 10 0 -2 6 4 C 2 2 0 6 10 D -6 -6 -6 0 4 E -4 -4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 6 4 B 10 0 -2 6 4 C 2 2 0 6 10 D -6 -6 -6 0 4 E -4 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=26 C=20 A=15 D=8 so D is eliminated. Round 2 votes counts: B=31 E=28 C=24 A=17 so A is eliminated. Round 3 votes counts: B=35 E=34 C=31 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:210 B:209 A:199 D:193 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -2 6 4 B 10 0 -2 6 4 C 2 2 0 6 10 D -6 -6 -6 0 4 E -4 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 6 4 B 10 0 -2 6 4 C 2 2 0 6 10 D -6 -6 -6 0 4 E -4 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 6 4 B 10 0 -2 6 4 C 2 2 0 6 10 D -6 -6 -6 0 4 E -4 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5584: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E C D B A (8) B C A E D (8) D E A C B (7) A D E B C (7) E D C A B (6) D A E B C (6) C B E D A (5) C E D B A (4) B C A D E (4) E C B D A (3) A D B E C (3) D E A B C (2) C E B D A (2) C B E A D (2) C B D E A (2) C B A E D (2) C B A D E (2) B A C D E (2) A E D B C (2) A B D C E (2) E D C B A (1) E A B C D (1) D E C A B (1) D C B A E (1) D B C A E (1) D B A C E (1) C E B A D (1) C B D A E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 -22 -10 B 0 0 -16 -20 -22 C 8 16 0 0 -16 D 22 20 0 0 -14 E 10 22 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -8 -22 -10 B 0 0 -16 -20 -22 C 8 16 0 0 -16 D 22 20 0 0 -14 E 10 22 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=21 D=19 A=17 B=14 so B is eliminated. Round 2 votes counts: C=33 E=29 D=19 A=19 so D is eliminated. Round 3 votes counts: E=39 C=35 A=26 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 D:214 C:204 A:180 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -22 -10 B 0 0 -16 -20 -22 C 8 16 0 0 -16 D 22 20 0 0 -14 E 10 22 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -22 -10 B 0 0 -16 -20 -22 C 8 16 0 0 -16 D 22 20 0 0 -14 E 10 22 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -22 -10 B 0 0 -16 -20 -22 C 8 16 0 0 -16 D 22 20 0 0 -14 E 10 22 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5585: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) C A D E B (6) E B A C D (5) D C A B E (5) B E A C D (5) E A B D C (4) C D B E A (4) B E D A C (4) E B A D C (3) D C A E B (3) D B E A C (3) D B C E A (3) C D B A E (3) C D A E B (3) C B D E A (3) B D E C A (3) A C E D B (3) E A B C D (2) D C B A E (2) D B A E C (2) D A E B C (2) B E D C A (2) B E C A D (2) B E A D C (2) A E B C D (2) A C D E B (2) D A C E B (1) C E B A D (1) C D A B E (1) B E C D A (1) B D E A C (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E B C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -20 -6 -14 -18 B 20 0 -2 -16 14 C 6 2 0 -12 6 D 14 16 12 0 20 E 18 -14 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -14 -18 B 20 0 -2 -16 14 C 6 2 0 -12 6 D 14 16 12 0 20 E 18 -14 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=21 B=20 E=14 A=13 so A is eliminated. Round 2 votes counts: D=34 C=27 B=20 E=19 so E is eliminated. Round 3 votes counts: B=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:231 B:208 C:201 E:189 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -6 -14 -18 B 20 0 -2 -16 14 C 6 2 0 -12 6 D 14 16 12 0 20 E 18 -14 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -14 -18 B 20 0 -2 -16 14 C 6 2 0 -12 6 D 14 16 12 0 20 E 18 -14 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -14 -18 B 20 0 -2 -16 14 C 6 2 0 -12 6 D 14 16 12 0 20 E 18 -14 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5586: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (13) A B D C E (12) A D B C E (10) E A C B D (9) E C B D A (7) C D B E A (7) A E B D C (6) D C B A E (4) C E D B A (4) D B C A E (3) E C A B D (2) E A D C B (2) E A B C D (2) C D B A E (2) C B D A E (2) B D C A E (2) B A D C E (2) A D B E C (2) A B D E C (2) E C B A D (1) E A C D B (1) D E C B A (1) C D E B A (1) C B E D A (1) C B D E A (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 -2 4 -4 B 2 0 -16 0 0 C 2 16 0 8 2 D -4 0 -8 0 2 E 4 0 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 4 -4 B 2 0 -16 0 0 C 2 16 0 8 2 D -4 0 -8 0 2 E 4 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=33 C=18 D=8 B=4 so B is eliminated. Round 2 votes counts: E=37 A=35 C=18 D=10 so D is eliminated. Round 3 votes counts: E=38 A=35 C=27 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:200 A:198 D:195 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -4 B 2 0 -16 0 0 C 2 16 0 8 2 D -4 0 -8 0 2 E 4 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -4 B 2 0 -16 0 0 C 2 16 0 8 2 D -4 0 -8 0 2 E 4 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -4 B 2 0 -16 0 0 C 2 16 0 8 2 D -4 0 -8 0 2 E 4 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5587: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (18) E D A C B (17) D A C B E (9) C B A D E (6) E B C A D (5) D A C E B (4) A C D B E (4) E D A B C (3) D E A C B (3) B C A E D (3) E D B A C (2) E B D C A (2) E B C D A (2) E A D C B (2) E A C B D (2) C A B D E (2) B E D C A (2) B E C A D (2) B C D A E (2) A D C B E (2) A C B D E (2) E D B C A (1) E A D B C (1) E A C D B (1) D E B C A (1) D A E C B (1) A E C D B (1) Total count = 100 A B C D E A 0 8 8 2 8 B -8 0 -12 -4 4 C -8 12 0 0 4 D -2 4 0 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 2 8 B -8 0 -12 -4 4 C -8 12 0 0 4 D -2 4 0 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996115 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=27 D=18 A=9 C=8 so C is eliminated. Round 2 votes counts: E=38 B=33 D=18 A=11 so A is eliminated. Round 3 votes counts: E=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:213 D:205 C:204 B:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 2 8 B -8 0 -12 -4 4 C -8 12 0 0 4 D -2 4 0 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996115 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 2 8 B -8 0 -12 -4 4 C -8 12 0 0 4 D -2 4 0 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996115 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 2 8 B -8 0 -12 -4 4 C -8 12 0 0 4 D -2 4 0 0 8 E -8 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996115 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5588: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (6) D C E B A (6) D C B E A (6) A E B C D (6) B E C A D (5) A E B D C (5) D C B A E (4) D C A B E (4) C D A B E (4) C D B A E (3) C B A E D (3) A B E C D (3) E B A D C (2) D E B C A (2) D E A B C (2) D A C E B (2) C D B E A (2) C B D E A (2) A E D B C (2) A D E B C (2) A D C B E (2) A C D B E (2) E D B C A (1) E D B A C (1) E B D C A (1) E B C D A (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D E B A C (1) D C E A B (1) D C A E B (1) D B E C A (1) D B C E A (1) D A E C B (1) D A E B C (1) C B E D A (1) C B E A D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E A C D (1) B C E D A (1) B C E A D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -14 -4 0 B 8 0 -4 -10 2 C 14 4 0 -4 4 D 4 10 4 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -4 0 B 8 0 -4 -10 2 C 14 4 0 -4 4 D 4 10 4 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=24 C=19 E=14 B=8 so B is eliminated. Round 2 votes counts: D=35 A=24 C=21 E=20 so E is eliminated. Round 3 votes counts: D=38 A=35 C=27 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 C:209 B:198 E:193 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -14 -4 0 B 8 0 -4 -10 2 C 14 4 0 -4 4 D 4 10 4 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -4 0 B 8 0 -4 -10 2 C 14 4 0 -4 4 D 4 10 4 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -4 0 B 8 0 -4 -10 2 C 14 4 0 -4 4 D 4 10 4 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5589: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) C E B D A (8) A B E C D (8) A B D E C (8) A D B E C (7) D C E B A (6) D A B E C (5) C E B A D (5) D A C E B (4) B E C D A (4) A B E D C (4) E B C D A (2) D E C B A (2) C D E B A (2) C D E A B (2) C A E B D (2) A B C E D (2) E C B D A (1) D C E A B (1) D B E A C (1) D B A E C (1) D A E B C (1) C E A D B (1) C E A B D (1) C B E A D (1) C A E D B (1) C A B E D (1) B E C A D (1) B E A D C (1) B D E A C (1) B C E A D (1) B A E C D (1) A D C E B (1) A D C B E (1) A C D E B (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 0 2 2 B -8 0 -2 8 2 C 0 2 0 10 4 D -2 -8 -10 0 -8 E -2 -2 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.482298 B: 0.000000 C: 0.517702 D: 0.000000 E: 0.000000 Sum of squares = 0.500626737668 Cumulative probabilities = A: 0.482298 B: 0.482298 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 2 2 B -8 0 -2 8 2 C 0 2 0 10 4 D -2 -8 -10 0 -8 E -2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=32 D=21 B=9 E=3 so E is eliminated. Round 2 votes counts: A=35 C=33 D=21 B=11 so B is eliminated. Round 3 votes counts: C=41 A=37 D=22 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:208 A:206 B:200 E:200 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 2 2 B -8 0 -2 8 2 C 0 2 0 10 4 D -2 -8 -10 0 -8 E -2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 2 2 B -8 0 -2 8 2 C 0 2 0 10 4 D -2 -8 -10 0 -8 E -2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 2 2 B -8 0 -2 8 2 C 0 2 0 10 4 D -2 -8 -10 0 -8 E -2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5590: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) D A C E B (9) C B E A D (8) B C E A D (8) A D E B C (8) C B D E A (7) D A E C B (6) B E C A D (6) E B C A D (5) D C B A E (5) D C A B E (5) E B A C D (3) E A B D C (3) D A C B E (3) A E D B C (3) C D B E A (2) A E B D C (2) E C B D A (1) E A D B C (1) D A E B C (1) C D B A E (1) B E A C D (1) Total count = 100 A B C D E A 0 -18 -20 -4 -14 B 18 0 -18 12 16 C 20 18 0 8 20 D 4 -12 -8 0 -6 E 14 -16 -20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -20 -4 -14 B 18 0 -18 12 16 C 20 18 0 8 20 D 4 -12 -8 0 -6 E 14 -16 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 B=15 E=13 A=13 so E is eliminated. Round 2 votes counts: C=31 D=29 B=23 A=17 so A is eliminated. Round 3 votes counts: D=41 C=31 B=28 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:233 B:214 E:192 D:189 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -20 -4 -14 B 18 0 -18 12 16 C 20 18 0 8 20 D 4 -12 -8 0 -6 E 14 -16 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -20 -4 -14 B 18 0 -18 12 16 C 20 18 0 8 20 D 4 -12 -8 0 -6 E 14 -16 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -20 -4 -14 B 18 0 -18 12 16 C 20 18 0 8 20 D 4 -12 -8 0 -6 E 14 -16 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5591: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (11) D B C A E (11) D E B C A (10) E A C D B (7) A C E B D (7) B C A D E (6) A E C B D (6) D B E C A (5) D B C E A (5) A C B E D (4) E D A C B (3) C A B E D (3) B D C A E (3) E D C B A (2) E A D C B (2) D E B A C (2) B D A C E (2) B C D A E (2) E D B C A (1) E D A B C (1) D E C B A (1) C B A E D (1) C B A D E (1) A E D B C (1) A E B D C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 4 0 B 4 0 2 -2 -10 C 2 -2 0 0 -6 D -4 2 0 0 0 E 0 10 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.507465 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.492535 Sum of squares = 0.500111419922 Cumulative probabilities = A: 0.507465 B: 0.507465 C: 0.507465 D: 0.507465 E: 1.000000 A B C D E A 0 -4 -2 4 0 B 4 0 2 -2 -10 C 2 -2 0 0 -6 D -4 2 0 0 0 E 0 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=27 A=21 B=13 C=5 so C is eliminated. Round 2 votes counts: D=34 E=27 A=24 B=15 so B is eliminated. Round 3 votes counts: D=41 A=32 E=27 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:199 D:199 B:197 C:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 4 0 B 4 0 2 -2 -10 C 2 -2 0 0 -6 D -4 2 0 0 0 E 0 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 4 0 B 4 0 2 -2 -10 C 2 -2 0 0 -6 D -4 2 0 0 0 E 0 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 4 0 B 4 0 2 -2 -10 C 2 -2 0 0 -6 D -4 2 0 0 0 E 0 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5592: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E A C D B (6) B D C A E (6) B A D C E (6) E C A D B (5) A E C D B (5) A B E D C (5) A B C D E (5) E D C B A (4) C D E B A (4) D C E B A (3) D C B E A (3) C E D A B (3) A E B C D (3) E C D B A (2) E A B D C (2) C D E A B (2) B D E C A (2) A B D C E (2) E D B C A (1) E A C B D (1) D E C B A (1) D B E C A (1) D B C E A (1) C E D B A (1) C D B E A (1) B E D A C (1) B D E A C (1) B D C E A (1) B D A C E (1) B C D A E (1) B A D E C (1) B A C D E (1) A E C B D (1) A E B D C (1) A C E D B (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -4 0 -12 B -14 0 -10 -8 -12 C 4 10 0 12 -10 D 0 8 -12 0 -10 E 12 12 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -4 0 -12 B -14 0 -10 -8 -12 C 4 10 0 12 -10 D 0 8 -12 0 -10 E 12 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=28 B=21 C=11 D=9 so D is eliminated. Round 2 votes counts: E=32 A=28 B=23 C=17 so C is eliminated. Round 3 votes counts: E=45 A=28 B=27 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:208 A:199 D:193 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 0 -12 B -14 0 -10 -8 -12 C 4 10 0 12 -10 D 0 8 -12 0 -10 E 12 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 0 -12 B -14 0 -10 -8 -12 C 4 10 0 12 -10 D 0 8 -12 0 -10 E 12 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 0 -12 B -14 0 -10 -8 -12 C 4 10 0 12 -10 D 0 8 -12 0 -10 E 12 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5593: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) E A B D C (9) C D E B A (7) E D A B C (6) A B E D C (6) E D C A B (5) C B A D E (5) E D C B A (4) B A D E C (4) E D B A C (3) D E C B A (3) D C E B A (3) C E D A B (3) C A B E D (3) A E B D C (3) E C D A B (2) D C B E A (2) C D B E A (2) B A D C E (2) B A C D E (2) A B E C D (2) E A D B C (1) E A C B D (1) D E B C A (1) D E B A C (1) C E A B D (1) C B D A E (1) C A B D E (1) B C A D E (1) A E B C D (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -12 -10 -8 B 6 0 -10 -10 -8 C 12 10 0 -8 -6 D 10 10 8 0 -2 E 8 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -12 -10 -8 B 6 0 -10 -10 -8 C 12 10 0 -8 -6 D 10 10 8 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=31 A=15 D=10 B=9 so B is eliminated. Round 2 votes counts: C=36 E=31 A=23 D=10 so D is eliminated. Round 3 votes counts: C=41 E=36 A=23 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:212 C:204 B:189 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -12 -10 -8 B 6 0 -10 -10 -8 C 12 10 0 -8 -6 D 10 10 8 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -10 -8 B 6 0 -10 -10 -8 C 12 10 0 -8 -6 D 10 10 8 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -10 -8 B 6 0 -10 -10 -8 C 12 10 0 -8 -6 D 10 10 8 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5594: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) D A E C B (6) E B A D C (4) D C B E A (4) D A C E B (4) C B D E A (4) B E A C D (4) B C E A D (4) A E D B C (4) A E B C D (4) A D E C B (4) E B A C D (3) D C A B E (3) C D B E A (3) C B A E D (3) C A D B E (3) A E D C B (3) A D C E B (3) E A B D C (2) D E A B C (2) D C E B A (2) D B C E A (2) D A E B C (2) C A B E D (2) A E B D C (2) E D B A C (1) E B D A C (1) D E A C B (1) D C A E B (1) D B E C A (1) C D B A E (1) C B E A D (1) C B D A E (1) C B A D E (1) B E C D A (1) B C E D A (1) B C A E D (1) B A E C D (1) B A C E D (1) A E C D B (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 22 6 6 B 2 0 -4 -16 -16 C -22 4 0 -18 -10 D -6 16 18 0 10 E -6 16 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.000000 D: 0.083333 E: 0.000000 Sum of squares = 0.513888888903 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 0.916667 D: 1.000000 E: 1.000000 A B C D E A 0 -2 22 6 6 B 2 0 -4 -16 -16 C -22 4 0 -18 -10 D -6 16 18 0 10 E -6 16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.000000 D: 0.083333 E: 0.000000 Sum of squares = 0.51388889065 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 0.916667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=23 C=19 B=13 E=11 so E is eliminated. Round 2 votes counts: D=35 A=25 B=21 C=19 so C is eliminated. Round 3 votes counts: D=39 B=31 A=30 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:216 E:205 B:183 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 22 6 6 B 2 0 -4 -16 -16 C -22 4 0 -18 -10 D -6 16 18 0 10 E -6 16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.000000 D: 0.083333 E: 0.000000 Sum of squares = 0.51388889065 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 0.916667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 22 6 6 B 2 0 -4 -16 -16 C -22 4 0 -18 -10 D -6 16 18 0 10 E -6 16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.000000 D: 0.083333 E: 0.000000 Sum of squares = 0.51388889065 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 0.916667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 22 6 6 B 2 0 -4 -16 -16 C -22 4 0 -18 -10 D -6 16 18 0 10 E -6 16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.000000 D: 0.083333 E: 0.000000 Sum of squares = 0.51388889065 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 0.916667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5595: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) E C A B D (6) C A E D B (6) B D E A C (6) E C B D A (5) E C B A D (5) C A D E B (5) B D A E C (5) A C D E B (5) E B C D A (4) D B A E C (4) D A B C E (4) A D C B E (4) D B A C E (3) D A C B E (3) B E D A C (3) A C D B E (3) E B D C A (2) E B C A D (2) E B A C D (2) D C A B E (2) C E A D B (2) C E A B D (2) A C E D B (2) E B A D C (1) D C B A E (1) C A E B D (1) B E A D C (1) B D E C A (1) B A D E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -4 -2 0 B 8 0 -4 10 0 C 4 4 0 0 -12 D 2 -10 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.456085 C: 0.000000 D: 0.000000 E: 0.543915 Sum of squares = 0.50385704982 Cumulative probabilities = A: 0.000000 B: 0.456085 C: 0.456085 D: 0.456085 E: 1.000000 A B C D E A 0 -8 -4 -2 0 B 8 0 -4 10 0 C 4 4 0 0 -12 D 2 -10 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 D=17 C=16 A=15 so A is eliminated. Round 2 votes counts: E=27 C=26 B=25 D=22 so D is eliminated. Round 3 votes counts: B=37 C=36 E=27 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:208 B:207 C:198 D:194 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -2 0 B 8 0 -4 10 0 C 4 4 0 0 -12 D 2 -10 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -2 0 B 8 0 -4 10 0 C 4 4 0 0 -12 D 2 -10 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -2 0 B 8 0 -4 10 0 C 4 4 0 0 -12 D 2 -10 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5596: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) B A C E D (7) D C B A E (6) E A B C D (5) A B C E D (5) E D A B C (4) C B E A D (4) E D A C B (3) E C B A D (3) E A D B C (3) D E C B A (3) D E C A B (3) D E A B C (3) C B D A E (3) C B A E D (3) B C A E D (3) A E B C D (3) E C D B A (2) E A C B D (2) E A B D C (2) D E A C B (2) D B A C E (2) D A B E C (2) D A B C E (2) A E B D C (2) E D C B A (1) E C A B D (1) E A D C B (1) D C E B A (1) D B C A E (1) D A E B C (1) C E B D A (1) C D B E A (1) C D B A E (1) B C E A D (1) B C D A E (1) B C A D E (1) A D E B C (1) A D B E C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 6 0 -6 B 4 0 4 -2 6 C -6 -4 0 -2 2 D 0 2 2 0 -14 E 6 -6 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.090909 Sum of squares = 0.48760330582 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.909091 E: 1.000000 A B C D E A 0 -4 6 0 -6 B 4 0 4 -2 6 C -6 -4 0 -2 2 D 0 2 2 0 -14 E 6 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.090909 Sum of squares = 0.487603305786 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=27 A=14 C=13 B=13 so C is eliminated. Round 2 votes counts: D=35 E=28 B=23 A=14 so A is eliminated. Round 3 votes counts: D=37 E=33 B=30 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:206 E:206 A:198 C:195 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 0 -6 B 4 0 4 -2 6 C -6 -4 0 -2 2 D 0 2 2 0 -14 E 6 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.090909 Sum of squares = 0.487603305786 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.909091 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 0 -6 B 4 0 4 -2 6 C -6 -4 0 -2 2 D 0 2 2 0 -14 E 6 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.090909 Sum of squares = 0.487603305786 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.909091 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 0 -6 B 4 0 4 -2 6 C -6 -4 0 -2 2 D 0 2 2 0 -14 E 6 -6 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.090909 Sum of squares = 0.487603305786 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.909091 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5597: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (7) E B A D C (6) E B A C D (6) C D A B E (6) B E D C A (5) B E C D A (5) A E B C D (5) D C B E A (4) D C A B E (4) A C D E B (4) D B E C A (3) D A E B C (3) C A D B E (3) B E C A D (3) A E B D C (3) E B C A D (2) D C A E B (2) D A C E B (2) D A C B E (2) C D B E A (2) C A E B D (2) C A D E B (2) B E D A C (2) B C E D A (2) E B D A C (1) E A C B D (1) E A B D C (1) E A B C D (1) D E B A C (1) D C B A E (1) D B E A C (1) C D B A E (1) C D A E B (1) C B E D A (1) C A B E D (1) B E A C D (1) A E D C B (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 6 0 2 4 B -6 0 4 -2 -6 C 0 -4 0 0 -6 D -2 2 0 0 -2 E -4 6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.710855 B: 0.000000 C: 0.289145 D: 0.000000 E: 0.000000 Sum of squares = 0.588919957329 Cumulative probabilities = A: 0.710855 B: 0.710855 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 2 4 B -6 0 4 -2 -6 C 0 -4 0 0 -6 D -2 2 0 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000919 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=22 C=19 E=18 B=18 so E is eliminated. Round 2 votes counts: B=33 A=25 D=23 C=19 so C is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:205 D:199 B:195 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 2 4 B -6 0 4 -2 -6 C 0 -4 0 0 -6 D -2 2 0 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000919 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 2 4 B -6 0 4 -2 -6 C 0 -4 0 0 -6 D -2 2 0 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000919 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 2 4 B -6 0 4 -2 -6 C 0 -4 0 0 -6 D -2 2 0 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000000919 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5598: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) C A E D B (10) D E B C A (8) A C E B D (6) B D E C A (4) A C E D B (4) A C B E D (4) E D C A B (3) C E D A B (3) B E D A C (3) E D C B A (2) E D B C A (2) E D B A C (2) C E A D B (2) C B D A E (2) C A D E B (2) C A B D E (2) B D A E C (2) B A E D C (2) B A C D E (2) A E C B D (2) A E B C D (2) E D A C B (1) E D A B C (1) E C A D B (1) E B D A C (1) E A D B C (1) E A C D B (1) D E C B A (1) D C E B A (1) D B E C A (1) D B E A C (1) C D A E B (1) C B D E A (1) C A E B D (1) C A D B E (1) B D C E A (1) A E C D B (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 -6 -6 B -4 0 -4 0 -18 C -2 4 0 2 -10 D 6 0 -2 0 -14 E 6 18 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 -6 -6 B -4 0 -4 0 -18 C -2 4 0 2 -10 D 6 0 -2 0 -14 E 6 18 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 A=22 E=15 D=12 so D is eliminated. Round 2 votes counts: B=28 C=26 E=24 A=22 so A is eliminated. Round 3 votes counts: C=40 B=31 E=29 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:224 A:197 C:197 D:195 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 -6 -6 B -4 0 -4 0 -18 C -2 4 0 2 -10 D 6 0 -2 0 -14 E 6 18 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -6 -6 B -4 0 -4 0 -18 C -2 4 0 2 -10 D 6 0 -2 0 -14 E 6 18 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -6 -6 B -4 0 -4 0 -18 C -2 4 0 2 -10 D 6 0 -2 0 -14 E 6 18 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5599: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (15) B E D C A (11) A D C E B (10) D A C E B (9) B E C D A (6) E B C A D (5) A C E B D (5) A C D E B (5) D B C E A (4) E B A C D (3) D C B E A (3) D A C B E (3) B E A C D (3) D C A E B (2) D B E C A (2) D A B C E (2) B D E A C (2) E C B D A (1) E C B A D (1) D C A B E (1) D B A E C (1) C A E D B (1) B D E C A (1) B A E C D (1) A E C B D (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -18 -6 4 -14 B 18 0 16 12 14 C 6 -16 0 -4 -8 D -4 -12 4 0 -8 E 14 -14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 4 -14 B 18 0 16 12 14 C 6 -16 0 -4 -8 D -4 -12 4 0 -8 E 14 -14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=27 A=23 E=10 C=1 so C is eliminated. Round 2 votes counts: B=39 D=27 A=24 E=10 so E is eliminated. Round 3 votes counts: B=49 D=27 A=24 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:230 E:208 D:190 C:189 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -6 4 -14 B 18 0 16 12 14 C 6 -16 0 -4 -8 D -4 -12 4 0 -8 E 14 -14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 4 -14 B 18 0 16 12 14 C 6 -16 0 -4 -8 D -4 -12 4 0 -8 E 14 -14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 4 -14 B 18 0 16 12 14 C 6 -16 0 -4 -8 D -4 -12 4 0 -8 E 14 -14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5600: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) E A D C B (9) B C D A E (9) D B C A E (7) D A B C E (7) E A D B C (5) C B A D E (5) E B C D A (4) C B E D A (4) C B D A E (4) A D C B E (4) E D A B C (3) E A C B D (3) E D B C A (2) D B A C E (2) C B E A D (2) B D C A E (2) B C E D A (2) B C D E A (2) A D B C E (2) A C D B E (2) E C B D A (1) E C A B D (1) D E A B C (1) D B E A C (1) D A C B E (1) C E B A D (1) C B D E A (1) C B A E D (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -20 -16 -6 -4 B 20 0 -2 4 16 C 16 2 0 4 16 D 6 -4 -4 0 2 E 4 -16 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -16 -6 -4 B 20 0 -2 4 16 C 16 2 0 4 16 D 6 -4 -4 0 2 E 4 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=19 C=18 B=15 A=10 so A is eliminated. Round 2 votes counts: E=39 D=26 C=20 B=15 so B is eliminated. Round 3 votes counts: E=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:219 C:219 D:200 E:185 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -16 -6 -4 B 20 0 -2 4 16 C 16 2 0 4 16 D 6 -4 -4 0 2 E 4 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -16 -6 -4 B 20 0 -2 4 16 C 16 2 0 4 16 D 6 -4 -4 0 2 E 4 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -16 -6 -4 B 20 0 -2 4 16 C 16 2 0 4 16 D 6 -4 -4 0 2 E 4 -16 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5601: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (15) D B E C A (10) D B C E A (7) B C E A D (6) E B C A D (5) D E B C A (4) B E C D A (4) D B E A C (3) B D C E A (3) B C E D A (3) A C D E B (3) E C B A D (2) E C A B D (2) D E A B C (2) D B C A E (2) C B A E D (2) B D E C A (2) A E C D B (2) A D C E B (2) A C D B E (2) A C B E D (2) E D B A C (1) E B D C A (1) E B C D A (1) E A C B D (1) D E B A C (1) D E A C B (1) D A E C B (1) D A B C E (1) C E B A D (1) C B E A D (1) C A B E D (1) B E C A D (1) B D C A E (1) B C D A E (1) B C A D E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -26 -22 2 -24 B 26 0 20 14 6 C 22 -20 0 16 10 D -2 -14 -16 0 -6 E 24 -6 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -22 2 -24 B 26 0 20 14 6 C 22 -20 0 16 10 D -2 -14 -16 0 -6 E 24 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=28 B=22 E=13 C=5 so C is eliminated. Round 2 votes counts: D=32 A=29 B=25 E=14 so E is eliminated. Round 3 votes counts: B=35 D=33 A=32 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:233 C:214 E:207 D:181 A:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -22 2 -24 B 26 0 20 14 6 C 22 -20 0 16 10 D -2 -14 -16 0 -6 E 24 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -22 2 -24 B 26 0 20 14 6 C 22 -20 0 16 10 D -2 -14 -16 0 -6 E 24 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -22 2 -24 B 26 0 20 14 6 C 22 -20 0 16 10 D -2 -14 -16 0 -6 E 24 -6 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999352 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5602: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (13) A D C E B (10) A B E C D (9) D C E B A (8) D C A E B (6) C E B D A (5) B E D C A (4) A E C B D (4) A B D E C (4) B E A C D (3) A E B C D (3) E B C D A (2) E B C A D (2) D C B E A (2) D A C B E (2) B D E C A (2) A D B E C (2) A D B C E (2) A C E D B (2) E C B D A (1) E C B A D (1) D B E C A (1) D B C E A (1) D A C E B (1) D A B C E (1) C E D B A (1) C D E A B (1) B E D A C (1) B E C A D (1) B C E D A (1) A E C D B (1) A D C B E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -4 -6 0 B -2 0 6 14 2 C 4 -6 0 2 -10 D 6 -14 -2 0 -10 E 0 -2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.534801 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.465199 Sum of squares = 0.502422199715 Cumulative probabilities = A: 0.534801 B: 0.534801 C: 0.534801 D: 0.534801 E: 1.000000 A B C D E A 0 2 -4 -6 0 B -2 0 6 14 2 C 4 -6 0 2 -10 D 6 -14 -2 0 -10 E 0 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500271 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499729 Sum of squares = 0.500000146936 Cumulative probabilities = A: 0.500271 B: 0.500271 C: 0.500271 D: 0.500271 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=25 D=22 C=7 E=6 so E is eliminated. Round 2 votes counts: A=40 B=29 D=22 C=9 so C is eliminated. Round 3 votes counts: A=40 B=36 D=24 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:210 E:209 A:196 C:195 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -4 -6 0 B -2 0 6 14 2 C 4 -6 0 2 -10 D 6 -14 -2 0 -10 E 0 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500271 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499729 Sum of squares = 0.500000146936 Cumulative probabilities = A: 0.500271 B: 0.500271 C: 0.500271 D: 0.500271 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -6 0 B -2 0 6 14 2 C 4 -6 0 2 -10 D 6 -14 -2 0 -10 E 0 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500271 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499729 Sum of squares = 0.500000146936 Cumulative probabilities = A: 0.500271 B: 0.500271 C: 0.500271 D: 0.500271 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -6 0 B -2 0 6 14 2 C 4 -6 0 2 -10 D 6 -14 -2 0 -10 E 0 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500271 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499729 Sum of squares = 0.500000146936 Cumulative probabilities = A: 0.500271 B: 0.500271 C: 0.500271 D: 0.500271 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5603: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) D E C B A (7) D E C A B (7) C E D B A (7) C A D E B (5) B A C E D (4) A D E B C (4) A B C D E (4) E D C B A (3) D E B A C (3) B E D C A (3) A D B E C (3) D E A C B (2) D E A B C (2) D C E A B (2) C D E A B (2) C B A E D (2) B A E D C (2) A D C B E (2) A C D E B (2) A C B D E (2) A B E D C (2) E B D C A (1) D A E C B (1) D A E B C (1) C E B D A (1) C D E B A (1) C B E D A (1) C A B E D (1) C A B D E (1) B E D A C (1) B E C D A (1) B E C A D (1) B C E D A (1) B C E A D (1) B A E C D (1) A D E C B (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 18 4 6 6 B -18 0 -2 -10 -4 C -4 2 0 -20 -16 D -6 10 20 0 30 E -6 4 16 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998075 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 6 6 B -18 0 -2 -10 -4 C -4 2 0 -20 -16 D -6 10 20 0 30 E -6 4 16 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=25 C=21 B=15 E=4 so E is eliminated. Round 2 votes counts: A=35 D=28 C=21 B=16 so B is eliminated. Round 3 votes counts: A=42 D=33 C=25 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:227 A:217 E:192 B:183 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 4 6 6 B -18 0 -2 -10 -4 C -4 2 0 -20 -16 D -6 10 20 0 30 E -6 4 16 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 6 6 B -18 0 -2 -10 -4 C -4 2 0 -20 -16 D -6 10 20 0 30 E -6 4 16 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 6 6 B -18 0 -2 -10 -4 C -4 2 0 -20 -16 D -6 10 20 0 30 E -6 4 16 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5604: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (12) E C B D A (11) A D C E B (9) B D C E A (8) A E C D B (7) A B E C D (7) D C E B A (5) A B D C E (5) B D E C A (4) A D B C E (4) A C E D B (4) C E D B A (3) B E C D A (3) B D A E C (3) C E D A B (2) A B D E C (2) E C D A B (1) E C A D B (1) E B C D A (1) E B C A D (1) D C E A B (1) D B C E A (1) D A C E B (1) B A D E C (1) A E C B D (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -8 -12 -8 B 6 0 -18 -4 -24 C 8 18 0 12 -12 D 12 4 -12 0 -10 E 8 24 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -12 -8 B 6 0 -18 -4 -24 C 8 18 0 12 -12 D 12 4 -12 0 -10 E 8 24 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 E=27 B=19 D=8 C=5 so C is eliminated. Round 2 votes counts: A=41 E=32 B=19 D=8 so D is eliminated. Round 3 votes counts: A=42 E=38 B=20 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 C:213 D:197 A:183 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -12 -8 B 6 0 -18 -4 -24 C 8 18 0 12 -12 D 12 4 -12 0 -10 E 8 24 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -12 -8 B 6 0 -18 -4 -24 C 8 18 0 12 -12 D 12 4 -12 0 -10 E 8 24 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -12 -8 B 6 0 -18 -4 -24 C 8 18 0 12 -12 D 12 4 -12 0 -10 E 8 24 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5605: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (14) A D B E C (12) E B C D A (7) D A B E C (7) C E B A D (5) B E D C A (5) A D C E B (5) D B E A C (4) C A E D B (4) C A E B D (4) B E C D A (4) A D C B E (4) E C B D A (3) D B A E C (3) C E A B D (3) A C D E B (3) C B E D A (2) B D E A C (2) A D E B C (2) E B D C A (1) D A E B C (1) C E A D B (1) C B E A D (1) A D E C B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -8 -6 -4 B 2 0 -2 2 -8 C 8 2 0 4 -4 D 6 -2 -4 0 -8 E 4 8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -8 -6 -4 B 2 0 -2 2 -8 C 8 2 0 4 -4 D 6 -2 -4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=29 D=15 E=11 B=11 so E is eliminated. Round 2 votes counts: C=37 A=29 B=19 D=15 so D is eliminated. Round 3 votes counts: C=37 A=37 B=26 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:205 B:197 D:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 -6 -4 B 2 0 -2 2 -8 C 8 2 0 4 -4 D 6 -2 -4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -6 -4 B 2 0 -2 2 -8 C 8 2 0 4 -4 D 6 -2 -4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -6 -4 B 2 0 -2 2 -8 C 8 2 0 4 -4 D 6 -2 -4 0 -8 E 4 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5606: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) C D B E A (7) E A C D B (6) D B C A E (6) E B A C D (5) A E C D B (5) D C B A E (4) C D E A B (4) C D A E B (4) B D C E A (4) B D C A E (4) E A B C D (3) C E D A B (3) B C D E A (3) B A D E C (3) A B E D C (3) E A C B D (2) C E D B A (2) C D E B A (2) C B D E A (2) B E C D A (2) B E C A D (2) A E D C B (2) A D E C B (2) E C B A D (1) D C A B E (1) D B A C E (1) D A B C E (1) C D A B E (1) C A E D B (1) B E A C D (1) B D A C E (1) B A E D C (1) A E D B C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -6 -4 2 B 2 0 0 -8 -6 C 6 0 0 16 4 D 4 8 -16 0 2 E -2 6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.295260 C: 0.704740 D: 0.000000 E: 0.000000 Sum of squares = 0.583836760544 Cumulative probabilities = A: 0.000000 B: 0.295260 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -4 2 B 2 0 0 -8 -6 C 6 0 0 16 4 D 4 8 -16 0 2 E -2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044892 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 B=21 E=17 D=13 so D is eliminated. Round 2 votes counts: C=31 B=28 A=24 E=17 so E is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:199 E:199 A:195 B:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 2 B 2 0 0 -8 -6 C 6 0 0 16 4 D 4 8 -16 0 2 E -2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044892 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 2 B 2 0 0 -8 -6 C 6 0 0 16 4 D 4 8 -16 0 2 E -2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044892 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 2 B 2 0 0 -8 -6 C 6 0 0 16 4 D 4 8 -16 0 2 E -2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044892 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5607: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) B A D C E (9) D C A E B (8) B E A C D (6) E B C D A (5) A D C B E (5) E C D B A (4) D C E A B (4) C D E A B (4) B A E C D (4) C D A E B (3) B E D C A (3) B A E D C (3) E D C B A (2) E C B D A (2) D A C E B (2) D A C B E (2) B E A D C (2) B D A C E (2) B A C D E (2) A B D C E (2) A B C D E (2) E D C A B (1) E C A D B (1) E B D C A (1) E B C A D (1) D E C A B (1) C E D A B (1) C E A D B (1) B A D E C (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -8 -14 -2 B -6 0 -12 -10 -12 C 8 12 0 0 2 D 14 10 0 0 0 E 2 12 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.676729 D: 0.323271 E: 0.000000 Sum of squares = 0.562466566807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.676729 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -14 -2 B -6 0 -12 -10 -12 C 8 12 0 0 2 D 14 10 0 0 0 E 2 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=29 D=17 A=13 C=9 so C is eliminated. Round 2 votes counts: B=32 E=31 D=24 A=13 so A is eliminated. Round 3 votes counts: B=36 E=32 D=32 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:211 E:206 A:191 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -14 -2 B -6 0 -12 -10 -12 C 8 12 0 0 2 D 14 10 0 0 0 E 2 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -14 -2 B -6 0 -12 -10 -12 C 8 12 0 0 2 D 14 10 0 0 0 E 2 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -14 -2 B -6 0 -12 -10 -12 C 8 12 0 0 2 D 14 10 0 0 0 E 2 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5608: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (12) D C A B E (10) D A C E B (7) B E C D A (7) A E B D C (5) E B A D C (4) D A C B E (4) C B E D A (4) B E C A D (4) A D E B C (4) A E D B C (3) A E C B D (3) A D C E B (3) E B C A D (2) D A E C B (2) C D B A E (2) C D A B E (2) C B E A D (2) C B D E A (2) C A D B E (2) B E D C A (2) A E B C D (2) A C D E B (2) E A B D C (1) D E B A C (1) D C B E A (1) D C B A E (1) D B E A C (1) D A E B C (1) B C E D A (1) B C E A D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 14 4 10 B -6 0 2 6 -8 C -14 -2 0 -2 -10 D -4 -6 2 0 -8 E -10 8 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 4 10 B -6 0 2 6 -8 C -14 -2 0 -2 -10 D -4 -6 2 0 -8 E -10 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=24 E=19 B=15 C=14 so C is eliminated. Round 2 votes counts: D=32 A=26 B=23 E=19 so E is eliminated. Round 3 votes counts: B=41 D=32 A=27 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:217 E:208 B:197 D:192 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 4 10 B -6 0 2 6 -8 C -14 -2 0 -2 -10 D -4 -6 2 0 -8 E -10 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 4 10 B -6 0 2 6 -8 C -14 -2 0 -2 -10 D -4 -6 2 0 -8 E -10 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 4 10 B -6 0 2 6 -8 C -14 -2 0 -2 -10 D -4 -6 2 0 -8 E -10 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5609: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C B A D E (5) B A C E D (5) E D C A B (4) D E C A B (4) D C E B A (4) B C A E D (4) B A C D E (4) E D A C B (3) E C D B A (3) E A B D C (3) D E A B C (3) C B A E D (3) B C A D E (3) A D B E C (3) A B D C E (3) E A D B C (2) E A B C D (2) D E A C B (2) D B A C E (2) D A E B C (2) D A B C E (2) C E D B A (2) C D E B A (2) C B D E A (2) B A E C D (2) A B E D C (2) A B D E C (2) E C B D A (1) E C A B D (1) D C E A B (1) D C B A E (1) C E B D A (1) C E B A D (1) C D B E A (1) C D B A E (1) C B D A E (1) B E A C D (1) B A D C E (1) A E D B C (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 10 0 -2 B 0 0 14 -2 -2 C -10 -14 0 -8 -2 D 0 2 8 0 -2 E 2 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 10 0 -2 B 0 0 14 -2 -2 C -10 -14 0 -8 -2 D 0 2 8 0 -2 E 2 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=21 B=20 C=19 A=13 so A is eliminated. Round 2 votes counts: E=29 B=28 D=24 C=19 so C is eliminated. Round 3 votes counts: B=39 E=33 D=28 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:205 A:204 D:204 E:204 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 0 -2 B 0 0 14 -2 -2 C -10 -14 0 -8 -2 D 0 2 8 0 -2 E 2 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 0 -2 B 0 0 14 -2 -2 C -10 -14 0 -8 -2 D 0 2 8 0 -2 E 2 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 0 -2 B 0 0 14 -2 -2 C -10 -14 0 -8 -2 D 0 2 8 0 -2 E 2 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5610: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) A B D E C (7) D E C B A (6) C E D B A (6) E D C B A (5) D E C A B (4) D C E A B (4) B A E C D (4) A D B C E (4) A B C D E (4) E B D A C (3) C B A E D (3) A D B E C (3) D C E B A (2) D A E B C (2) C E B A D (2) C D A B E (2) C B E A D (2) B C E A D (2) E D B A C (1) E C D B A (1) E C B D A (1) E B D C A (1) E B A D C (1) D E B A C (1) D E A C B (1) D C A E B (1) D A E C B (1) D A B E C (1) D A B C E (1) C E B D A (1) C D E B A (1) C D E A B (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A D C (1) B E A C D (1) B A C E D (1) A C B D E (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 4 6 2 B 16 0 6 2 8 C -4 -6 0 -26 -16 D -6 -2 26 0 0 E -2 -8 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999101 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 6 2 B 16 0 6 2 8 C -4 -6 0 -26 -16 D -6 -2 26 0 0 E -2 -8 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 B=21 C=20 E=13 so E is eliminated. Round 2 votes counts: D=30 B=26 C=22 A=22 so C is eliminated. Round 3 votes counts: D=41 B=35 A=24 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:209 E:203 A:198 C:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 6 2 B 16 0 6 2 8 C -4 -6 0 -26 -16 D -6 -2 26 0 0 E -2 -8 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 6 2 B 16 0 6 2 8 C -4 -6 0 -26 -16 D -6 -2 26 0 0 E -2 -8 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 6 2 B 16 0 6 2 8 C -4 -6 0 -26 -16 D -6 -2 26 0 0 E -2 -8 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5611: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (11) E B C D A (10) C B E D A (9) A D C B E (8) C B D A E (7) E A D B C (6) A D E C B (6) A D B E C (6) A D B C E (6) E C B A D (5) C A D B E (5) C E B D A (3) E C A B D (2) D A C B E (2) D A B C E (2) C B A D E (2) E C A D B (1) E B D A C (1) E B A D C (1) E A C D B (1) C B E A D (1) C B D E A (1) B D A C E (1) B C E D A (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 16 6 26 16 B -16 0 -10 -12 4 C -6 10 0 0 0 D -26 12 0 0 18 E -16 -4 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 26 16 B -16 0 -10 -12 4 C -6 10 0 0 0 D -26 12 0 0 18 E -16 -4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=28 E=27 D=4 B=2 so B is eliminated. Round 2 votes counts: A=39 C=29 E=27 D=5 so D is eliminated. Round 3 votes counts: A=44 C=29 E=27 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:232 C:202 D:202 B:183 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 26 16 B -16 0 -10 -12 4 C -6 10 0 0 0 D -26 12 0 0 18 E -16 -4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 26 16 B -16 0 -10 -12 4 C -6 10 0 0 0 D -26 12 0 0 18 E -16 -4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 26 16 B -16 0 -10 -12 4 C -6 10 0 0 0 D -26 12 0 0 18 E -16 -4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997547 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5612: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (5) A E D B C (5) E D C A B (4) E D A C B (4) C E D A B (4) C E A D B (4) A C E B D (4) E C D A B (3) D B C E A (3) C D E B A (3) C B D E A (3) C A B E D (3) B D C A E (3) E C A D B (2) E A D C B (2) E A C D B (2) D E C A B (2) D E A B C (2) D B E A C (2) C B D A E (2) C B A E D (2) B D E A C (2) B D C E A (2) B D A E C (2) B C D A E (2) B C A D E (2) B A D E C (2) B A D C E (2) A E C D B (2) E D A B C (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A C B (1) D C E B A (1) D C B E A (1) C D B E A (1) C B A D E (1) C A E B D (1) B D A C E (1) B A C D E (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E B C (1) A C E D B (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -2 -4 -4 B -8 0 -12 -8 -10 C 2 12 0 4 6 D 4 8 -4 0 -10 E 4 10 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -4 -4 B -8 0 -12 -8 -10 C 2 12 0 4 6 D 4 8 -4 0 -10 E 4 10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 A=19 E=18 D=15 so D is eliminated. Round 2 votes counts: B=29 E=26 C=26 A=19 so A is eliminated. Round 3 votes counts: E=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:209 A:199 D:199 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 -4 -4 B -8 0 -12 -8 -10 C 2 12 0 4 6 D 4 8 -4 0 -10 E 4 10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -4 -4 B -8 0 -12 -8 -10 C 2 12 0 4 6 D 4 8 -4 0 -10 E 4 10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -4 -4 B -8 0 -12 -8 -10 C 2 12 0 4 6 D 4 8 -4 0 -10 E 4 10 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5613: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) E C A D B (5) D C E B A (5) B A C E D (5) A B E D C (5) E D C A B (4) E C A B D (4) D B A C E (4) C E B A D (4) C D E B A (4) A E B C D (4) A B D E C (4) E C D A B (3) E A C B D (3) D E C A B (3) D E A C B (3) D A B E C (3) C B A E D (3) E A D C B (2) D C B A E (2) D B C A E (2) D B A E C (2) D A E B C (2) C E D B A (2) E D A C B (1) E A B C D (1) D E A B C (1) D C B E A (1) C E B D A (1) C D B A E (1) C B E A D (1) C B D A E (1) C B A D E (1) B A D E C (1) B A C D E (1) A E B D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 6 10 4 B -2 0 -8 0 -6 C -6 8 0 -8 -6 D -10 0 8 0 -2 E -4 6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 10 4 B -2 0 -8 0 -6 C -6 8 0 -8 -6 D -10 0 8 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999203 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=23 C=18 A=16 B=15 so B is eliminated. Round 2 votes counts: A=31 D=28 E=23 C=18 so C is eliminated. Round 3 votes counts: A=35 D=34 E=31 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:205 D:198 C:194 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 10 4 B -2 0 -8 0 -6 C -6 8 0 -8 -6 D -10 0 8 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999203 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 10 4 B -2 0 -8 0 -6 C -6 8 0 -8 -6 D -10 0 8 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999203 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 10 4 B -2 0 -8 0 -6 C -6 8 0 -8 -6 D -10 0 8 0 -2 E -4 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999203 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5614: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) C E A B D (7) B A C D E (6) C A B E D (5) B C A D E (5) B A D C E (5) D E A B C (4) C B A E D (4) E D C A B (3) E D A C B (3) E D A B C (3) C E D B A (3) C E D A B (3) C E B A D (3) A D E B C (3) A B D E C (3) E C D B A (2) E C A D B (2) D B E A C (2) D B A E C (2) D A B E C (2) C E A D B (2) C B E A D (2) C A E D B (2) B D A E C (2) B A D E C (2) E D C B A (1) D E B A C (1) D A E B C (1) C B E D A (1) B D E C A (1) B D E A C (1) B C D E A (1) B C D A E (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -14 8 -8 B -10 0 -6 2 -4 C 14 6 0 18 4 D -8 -2 -18 0 -10 E 8 4 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -14 8 -8 B -10 0 -6 2 -4 C 14 6 0 18 4 D -8 -2 -18 0 -10 E 8 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=24 E=23 D=12 A=9 so A is eliminated. Round 2 votes counts: C=33 B=28 E=23 D=16 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 E:209 A:198 B:191 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -14 8 -8 B -10 0 -6 2 -4 C 14 6 0 18 4 D -8 -2 -18 0 -10 E 8 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 8 -8 B -10 0 -6 2 -4 C 14 6 0 18 4 D -8 -2 -18 0 -10 E 8 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 8 -8 B -10 0 -6 2 -4 C 14 6 0 18 4 D -8 -2 -18 0 -10 E 8 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5615: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) B A D E C (9) C D E A B (8) B D A E C (8) E C A B D (7) B A E D C (7) A E B C D (7) D C E B A (5) D C E A B (5) D B A C E (5) A B E C D (5) E A B C D (4) E A C B D (3) D C B E A (3) D B C A E (3) C E A D B (3) D C B A E (1) D C A B E (1) D B C E A (1) C D E B A (1) B A E C D (1) A D C E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 4 -2 0 B -12 0 4 6 -8 C -4 -4 0 -2 -6 D 2 -6 2 0 4 E 0 8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.459999999995 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -2 0 B -12 0 4 6 -8 C -4 -4 0 -2 -6 D 2 -6 2 0 4 E 0 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.460000000025 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 C=22 A=15 E=14 so E is eliminated. Round 2 votes counts: C=29 B=25 D=24 A=22 so A is eliminated. Round 3 votes counts: B=43 C=32 D=25 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:207 E:205 D:201 B:195 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 -2 0 B -12 0 4 6 -8 C -4 -4 0 -2 -6 D 2 -6 2 0 4 E 0 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.460000000025 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -2 0 B -12 0 4 6 -8 C -4 -4 0 -2 -6 D 2 -6 2 0 4 E 0 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.460000000025 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -2 0 B -12 0 4 6 -8 C -4 -4 0 -2 -6 D 2 -6 2 0 4 E 0 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.460000000025 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5616: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E C A B D (8) C D E A B (8) B A E D C (7) D C A B E (5) D B A C E (5) C E D A B (5) E B A C D (4) B E A C D (4) D C E B A (3) D C A E B (3) A B E C D (3) E C D B A (2) E B C A D (2) E A C B D (2) E A B C D (2) C E A D B (2) B D A E C (2) B A E C D (2) B A D E C (2) A E C B D (2) A C E B D (2) A B D C E (2) E C B A D (1) E C A D B (1) E B D C A (1) D E C B A (1) D C B E A (1) D C B A E (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B C E (1) C A E D B (1) B D A C E (1) B A D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -12 -2 -16 B -14 0 -14 -2 -18 C 12 14 0 4 6 D 2 2 -4 0 -4 E 16 18 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -12 -2 -16 B -14 0 -14 -2 -18 C 12 14 0 4 6 D 2 2 -4 0 -4 E 16 18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=23 B=19 C=16 A=10 so A is eliminated. Round 2 votes counts: D=32 E=25 B=25 C=18 so C is eliminated. Round 3 votes counts: D=40 E=35 B=25 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:218 E:216 D:198 A:192 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -12 -2 -16 B -14 0 -14 -2 -18 C 12 14 0 4 6 D 2 2 -4 0 -4 E 16 18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -12 -2 -16 B -14 0 -14 -2 -18 C 12 14 0 4 6 D 2 2 -4 0 -4 E 16 18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -12 -2 -16 B -14 0 -14 -2 -18 C 12 14 0 4 6 D 2 2 -4 0 -4 E 16 18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5617: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) E C D A B (6) A D B C E (6) E C B D A (5) E C B A D (5) D A B C E (5) B A D C E (5) E B C A D (4) E B A C D (3) D A C B E (3) C E D A B (3) C A B D E (3) B E A D C (3) B A C D E (3) A D C B E (3) E D C B A (2) E C D B A (2) E B D A C (2) E B C D A (2) E B A D C (2) D C A B E (2) C B E A D (2) B E A C D (2) A B D C E (2) E D B A C (1) E D A C B (1) D E A C B (1) D C A E B (1) D A E C B (1) D A E B C (1) D A B E C (1) C E B A D (1) C D A B E (1) C A D B E (1) B E C A D (1) B D A E C (1) B C E A D (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 6 -10 B 2 0 0 6 2 C 0 0 0 2 -6 D -6 -6 -2 0 -14 E 10 -2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.816963 C: 0.183037 D: 0.000000 E: 0.000000 Sum of squares = 0.700931256561 Cumulative probabilities = A: 0.000000 B: 0.816963 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 6 -10 B 2 0 0 6 2 C 0 0 0 2 -6 D -6 -6 -2 0 -14 E 10 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000137415 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=20 D=15 A=13 C=11 so C is eliminated. Round 2 votes counts: E=45 B=22 A=17 D=16 so D is eliminated. Round 3 votes counts: E=46 A=32 B=22 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:205 C:198 A:197 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 6 -10 B 2 0 0 6 2 C 0 0 0 2 -6 D -6 -6 -2 0 -14 E 10 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000137415 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 6 -10 B 2 0 0 6 2 C 0 0 0 2 -6 D -6 -6 -2 0 -14 E 10 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000137415 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 6 -10 B 2 0 0 6 2 C 0 0 0 2 -6 D -6 -6 -2 0 -14 E 10 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000137415 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5618: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) B A C D E (8) D B E A C (6) C A E D B (6) C A B E D (6) B D A C E (6) A C E D B (6) C A E B D (5) C E A D B (4) B A D C E (4) E D C A B (3) D E C A B (3) B C A E D (3) B A C E D (3) A C B E D (3) D E A C B (2) D B A C E (2) C E B A D (2) B D E C A (2) B D E A C (2) E C D B A (1) E C D A B (1) E C B A D (1) E B D C A (1) D E B C A (1) D E B A C (1) D B A E C (1) D A C E B (1) D A C B E (1) C E A B D (1) B E C D A (1) B C A D E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 30 16 B -8 0 -14 0 2 C 4 14 0 28 28 D -30 0 -28 0 -14 E -16 -2 -28 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 30 16 B -8 0 -14 0 2 C 4 14 0 28 28 D -30 0 -28 0 -14 E -16 -2 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=24 D=18 E=17 A=11 so A is eliminated. Round 2 votes counts: C=34 B=31 D=18 E=17 so E is eliminated. Round 3 votes counts: C=47 B=32 D=21 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:237 A:225 B:190 E:184 D:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 30 16 B -8 0 -14 0 2 C 4 14 0 28 28 D -30 0 -28 0 -14 E -16 -2 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 30 16 B -8 0 -14 0 2 C 4 14 0 28 28 D -30 0 -28 0 -14 E -16 -2 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 30 16 B -8 0 -14 0 2 C 4 14 0 28 28 D -30 0 -28 0 -14 E -16 -2 -28 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5619: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) A D B E C (10) D A B C E (9) C E B D A (9) D A B E C (8) E B C A D (5) D C A E B (5) D A C B E (5) C E B A D (5) B E C A D (4) E C B A D (3) A B E D C (3) D C B E A (2) D C A B E (2) C D E B A (2) C D E A B (2) C D A E B (2) B E A D C (2) A D E C B (2) E B A C D (1) D C B A E (1) D B A E C (1) C E D B A (1) C E A B D (1) B E C D A (1) B E A C D (1) A E B C D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 24 10 -20 22 B -24 0 -4 -26 2 C -10 4 0 -24 14 D 20 26 24 0 26 E -22 -2 -14 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 10 -20 22 B -24 0 -4 -26 2 C -10 4 0 -24 14 D 20 26 24 0 26 E -22 -2 -14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 C=22 A=18 E=9 B=8 so B is eliminated. Round 2 votes counts: D=43 C=22 A=18 E=17 so E is eliminated. Round 3 votes counts: D=43 C=35 A=22 so A is eliminated. Round 4 votes counts: D=62 C=38 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:248 A:218 C:192 B:174 E:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 10 -20 22 B -24 0 -4 -26 2 C -10 4 0 -24 14 D 20 26 24 0 26 E -22 -2 -14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 10 -20 22 B -24 0 -4 -26 2 C -10 4 0 -24 14 D 20 26 24 0 26 E -22 -2 -14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 10 -20 22 B -24 0 -4 -26 2 C -10 4 0 -24 14 D 20 26 24 0 26 E -22 -2 -14 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5620: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) A C D B E (9) E B D C A (8) E D B A C (7) D A C B E (6) D A B C E (6) E C A B D (5) E B D A C (4) B D A C E (4) E B C A D (3) D B E A C (3) C A E D B (3) C A E B D (3) C A B D E (3) B D E A C (3) E C B A D (2) E C A D B (2) D E B A C (2) D B A C E (2) C E A B D (2) C A B E D (2) B E D A C (2) A D C B E (2) E D A C B (1) E B C D A (1) D B A E C (1) C B A D E (1) C A D E B (1) B E C A D (1) A C B D E (1) Total count = 100 A B C D E A 0 12 6 0 8 B -12 0 -6 -10 12 C -6 6 0 -2 10 D 0 10 2 0 8 E -8 -12 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.567569 B: 0.000000 C: 0.000000 D: 0.432431 E: 0.000000 Sum of squares = 0.509131097215 Cumulative probabilities = A: 0.567569 B: 0.567569 C: 0.567569 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 0 8 B -12 0 -6 -10 12 C -6 6 0 -2 10 D 0 10 2 0 8 E -8 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=25 D=20 A=12 B=10 so B is eliminated. Round 2 votes counts: E=36 D=27 C=25 A=12 so A is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 D:210 C:204 B:192 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 0 8 B -12 0 -6 -10 12 C -6 6 0 -2 10 D 0 10 2 0 8 E -8 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 0 8 B -12 0 -6 -10 12 C -6 6 0 -2 10 D 0 10 2 0 8 E -8 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 0 8 B -12 0 -6 -10 12 C -6 6 0 -2 10 D 0 10 2 0 8 E -8 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5621: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (15) C D E B A (8) B E A D C (7) C D E A B (6) C D A E B (5) A E B D C (5) B A E D C (4) A B C D E (4) D C E A B (3) C D A B E (3) C B D E A (3) B A E C D (3) A E D B C (3) E D C B A (2) D E C B A (2) C B D A E (2) B E A C D (2) B C A E D (2) A C D B E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) E B D C A (1) E B D A C (1) E B A D C (1) D C E B A (1) D C A E B (1) C D B E A (1) C D B A E (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D C A (1) A D E C B (1) A D C E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 10 14 B -8 0 8 10 8 C -6 -8 0 -4 -6 D -10 -10 4 0 -6 E -14 -8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 10 14 B -8 0 8 10 8 C -6 -8 0 -4 -6 D -10 -10 4 0 -6 E -14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=32 B=19 E=9 D=7 so D is eliminated. Round 2 votes counts: C=37 A=33 B=19 E=11 so E is eliminated. Round 3 votes counts: C=42 A=34 B=24 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:209 E:195 D:189 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 10 14 B -8 0 8 10 8 C -6 -8 0 -4 -6 D -10 -10 4 0 -6 E -14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 10 14 B -8 0 8 10 8 C -6 -8 0 -4 -6 D -10 -10 4 0 -6 E -14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 10 14 B -8 0 8 10 8 C -6 -8 0 -4 -6 D -10 -10 4 0 -6 E -14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5622: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) B A E D C (8) E D A B C (6) B E D A C (6) B A C E D (6) C D E B A (5) B E D C A (5) A C D E B (5) C B D E A (3) B C A D E (3) A E D B C (3) A E B D C (3) A B E D C (3) E D A C B (2) D E C A B (2) D E A C B (2) D C E A B (2) C D B E A (2) C A D E B (2) B C E D A (2) B C D E A (2) A D E C B (2) E D C A B (1) E D B C A (1) E D B A C (1) E B D A C (1) E A D B C (1) D E C B A (1) D A E C B (1) C D A E B (1) C B A D E (1) C A B D E (1) B E C D A (1) B E A D C (1) B A E C D (1) A E D C B (1) A D C E B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 12 -12 -12 B 0 0 12 -2 -6 C -12 -12 0 -8 -6 D 12 2 8 0 -8 E 12 6 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 12 -12 -12 B 0 0 12 -2 -6 C -12 -12 0 -8 -6 D 12 2 8 0 -8 E 12 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=24 A=20 E=13 D=8 so D is eliminated. Round 2 votes counts: B=35 C=26 A=21 E=18 so E is eliminated. Round 3 votes counts: B=38 A=32 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:216 D:207 B:202 A:194 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 -12 -12 B 0 0 12 -2 -6 C -12 -12 0 -8 -6 D 12 2 8 0 -8 E 12 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -12 -12 B 0 0 12 -2 -6 C -12 -12 0 -8 -6 D 12 2 8 0 -8 E 12 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -12 -12 B 0 0 12 -2 -6 C -12 -12 0 -8 -6 D 12 2 8 0 -8 E 12 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5623: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (14) D E C B A (12) B A D E C (11) E D C B A (9) D E B A C (8) C E D A B (7) C A B E D (6) D E C A B (5) B A C E D (5) A B C D E (4) D E B C A (3) A C B E D (3) E C D A B (2) D B E A C (2) A B D E C (2) C E A D B (1) C E A B D (1) C B A E D (1) B E A D C (1) B A C D E (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 6 4 -2 B 6 0 6 0 2 C -6 -6 0 -10 -12 D -4 0 10 0 0 E 2 -2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.691153 C: 0.000000 D: 0.308847 E: 0.000000 Sum of squares = 0.573079285839 Cumulative probabilities = A: 0.000000 B: 0.691153 C: 0.691153 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 4 -2 B 6 0 6 0 2 C -6 -6 0 -10 -12 D -4 0 10 0 0 E 2 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=25 B=18 C=16 E=11 so E is eliminated. Round 2 votes counts: D=39 A=25 C=18 B=18 so C is eliminated. Round 3 votes counts: D=48 A=33 B=19 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:207 E:206 D:203 A:201 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 4 -2 B 6 0 6 0 2 C -6 -6 0 -10 -12 D -4 0 10 0 0 E 2 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 4 -2 B 6 0 6 0 2 C -6 -6 0 -10 -12 D -4 0 10 0 0 E 2 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 4 -2 B 6 0 6 0 2 C -6 -6 0 -10 -12 D -4 0 10 0 0 E 2 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5624: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (15) D E C B A (9) C B A D E (9) E D C A B (8) D E C A B (7) B A C E D (7) B A C D E (7) A B C E D (7) C D E B A (6) A B E D C (4) D E A B C (3) C D B A E (3) B A D E C (3) E A B D C (2) C E D A B (2) C A B E D (2) D E B A C (1) D C E B A (1) D A B E C (1) C E A B D (1) C B A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 4 2 -12 -10 B -4 0 2 -12 -10 C -2 -2 0 -10 -6 D 12 12 10 0 2 E 10 10 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -12 -10 B -4 0 2 -12 -10 C -2 -2 0 -10 -6 D 12 12 10 0 2 E 10 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 D=22 B=18 A=11 so A is eliminated. Round 2 votes counts: B=29 E=25 C=24 D=22 so D is eliminated. Round 3 votes counts: E=45 B=30 C=25 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:212 A:192 C:190 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -12 -10 B -4 0 2 -12 -10 C -2 -2 0 -10 -6 D 12 12 10 0 2 E 10 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -12 -10 B -4 0 2 -12 -10 C -2 -2 0 -10 -6 D 12 12 10 0 2 E 10 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -12 -10 B -4 0 2 -12 -10 C -2 -2 0 -10 -6 D 12 12 10 0 2 E 10 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5625: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (7) C A E B D (7) C A D B E (7) B D E A C (7) E B D A C (6) C A E D B (5) E B A D C (4) D B E C A (4) D B E A C (4) C D A B E (4) C A D E B (4) B E D A C (4) A E C B D (4) A E B D C (4) E A B D C (3) D C B A E (3) D B C E A (3) C D B E A (3) B D E C A (3) C E A B D (2) A C E B D (2) E B C D A (1) E B A C D (1) E A C B D (1) E A B C D (1) D B A E C (1) B E D C A (1) A E B C D (1) A D C B E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -8 -2 4 B 4 0 -4 4 6 C 8 4 0 4 0 D 2 -4 -4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.739671 D: 0.000000 E: 0.260329 Sum of squares = 0.614884496643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.739671 D: 0.739671 E: 1.000000 A B C D E A 0 -4 -8 -2 4 B 4 0 -4 4 6 C 8 4 0 4 0 D 2 -4 -4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000003532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=17 D=15 B=15 A=14 so A is eliminated. Round 2 votes counts: C=43 E=26 D=16 B=15 so B is eliminated. Round 3 votes counts: C=43 E=31 D=26 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:205 D:199 A:195 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -2 4 B 4 0 -4 4 6 C 8 4 0 4 0 D 2 -4 -4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000003532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -2 4 B 4 0 -4 4 6 C 8 4 0 4 0 D 2 -4 -4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000003532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -2 4 B 4 0 -4 4 6 C 8 4 0 4 0 D 2 -4 -4 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000003532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5626: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) A B E D C (6) A D E B C (5) E D B C A (4) E B A D C (4) C D E B A (4) E D B A C (3) E B D A C (3) D E A C B (3) D A C E B (3) C D A E B (3) C B A E D (3) B E A C D (3) B C E D A (3) B C A E D (3) B A E C D (3) A B E C D (3) A B C E D (3) E B D C A (2) D E C B A (2) D E C A B (2) D E A B C (2) D C E A B (2) C D E A B (2) C D A B E (2) C B E D A (2) E D C B A (1) E A D B C (1) D E B A C (1) D A E B C (1) C B E A D (1) C B D E A (1) C A B D E (1) B E C D A (1) B C E A D (1) B A E D C (1) B A C E D (1) A E D B C (1) A E B D C (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 6 6 4 B -6 0 14 -4 -2 C -6 -14 0 -2 -6 D -6 4 2 0 -8 E -4 2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 4 B -6 0 14 -4 -2 C -6 -14 0 -2 -6 D -6 4 2 0 -8 E -4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 E=18 D=16 B=16 so D is eliminated. Round 2 votes counts: E=28 C=28 A=28 B=16 so B is eliminated. Round 3 votes counts: C=35 A=33 E=32 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:206 B:201 D:196 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 4 B -6 0 14 -4 -2 C -6 -14 0 -2 -6 D -6 4 2 0 -8 E -4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 4 B -6 0 14 -4 -2 C -6 -14 0 -2 -6 D -6 4 2 0 -8 E -4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 4 B -6 0 14 -4 -2 C -6 -14 0 -2 -6 D -6 4 2 0 -8 E -4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5627: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) B D A C E (6) E A C D B (5) D A E B C (5) C E A B D (5) B D E A C (5) E A D C B (4) D A B E C (4) D A B C E (4) E D B A C (3) E D A B C (3) E C B A D (3) E B D A C (3) D B A E C (3) C E B A D (3) C B A D E (3) C A B D E (3) B D A E C (3) E B D C A (2) E B C D A (2) D B A C E (2) C E A D B (2) C A E D B (2) B D E C A (2) B C D A E (2) E D A C B (1) E C A B D (1) D B E A C (1) C B E A D (1) C A E B D (1) C A D B E (1) B E D C A (1) B E D A C (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 12 -6 -16 B -8 0 4 -6 -16 C -12 -4 0 -8 -26 D 6 6 8 0 -8 E 16 16 26 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 12 -6 -16 B -8 0 4 -6 -16 C -12 -4 0 -8 -26 D 6 6 8 0 -8 E 16 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=21 B=20 D=19 A=3 so A is eliminated. Round 2 votes counts: E=37 C=23 D=20 B=20 so D is eliminated. Round 3 votes counts: E=43 B=34 C=23 so C is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:233 D:206 A:199 B:187 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 -6 -16 B -8 0 4 -6 -16 C -12 -4 0 -8 -26 D 6 6 8 0 -8 E 16 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -6 -16 B -8 0 4 -6 -16 C -12 -4 0 -8 -26 D 6 6 8 0 -8 E 16 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -6 -16 B -8 0 4 -6 -16 C -12 -4 0 -8 -26 D 6 6 8 0 -8 E 16 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5628: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) B D C E A (6) D B E C A (5) A C B E D (5) C D B E A (4) B C D A E (4) A E C D B (4) E D C B A (3) E D B A C (3) E A D B C (3) D E B C A (3) D B E A C (3) A C E B D (3) A B E D C (3) A B C D E (3) E D C A B (2) E D B C A (2) E A D C B (2) E A C D B (2) D B C E A (2) C A E B D (2) B D C A E (2) B C A D E (2) B A D C E (2) A E D C B (2) A E C B D (2) E D A C B (1) E D A B C (1) E C A D B (1) D E C B A (1) D E B A C (1) D C E B A (1) D C B E A (1) C B A D E (1) C A B D E (1) B D E A C (1) B A D E C (1) B A C D E (1) A E D B C (1) A C E D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -6 -12 4 B 18 0 2 2 18 C 6 -2 0 -4 4 D 12 -2 4 0 10 E -4 -18 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -12 4 B 18 0 2 2 18 C 6 -2 0 -4 4 D 12 -2 4 0 10 E -4 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=20 B=19 C=18 D=17 so D is eliminated. Round 2 votes counts: B=29 A=26 E=25 C=20 so C is eliminated. Round 3 votes counts: B=45 A=29 E=26 so E is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:212 C:202 A:184 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -6 -12 4 B 18 0 2 2 18 C 6 -2 0 -4 4 D 12 -2 4 0 10 E -4 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -12 4 B 18 0 2 2 18 C 6 -2 0 -4 4 D 12 -2 4 0 10 E -4 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -12 4 B 18 0 2 2 18 C 6 -2 0 -4 4 D 12 -2 4 0 10 E -4 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5629: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (6) B E C D A (6) E A D B C (5) D A E C B (5) C B E A D (5) A D C E B (5) C A D E B (4) A E D C B (4) E A C D B (3) C A D B E (3) B E C A D (3) B D A C E (3) B C D A E (3) E B C A D (2) E B A C D (2) D B A C E (2) D A E B C (2) D A C E B (2) D A B E C (2) C E B A D (2) C E A B D (2) C D A B E (2) B E D A C (2) B D C A E (2) B C E D A (2) B C D E A (2) A D E C B (2) E D A B C (1) E C B A D (1) E C A D B (1) E C A B D (1) E A B C D (1) D C A B E (1) D B C A E (1) D A B C E (1) C D B A E (1) C B E D A (1) C B D A E (1) C B A E D (1) B E D C A (1) B D E C A (1) B C E A D (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 10 0 0 10 B -10 0 -8 -10 6 C 0 8 0 2 8 D 0 10 -2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.357247 B: 0.000000 C: 0.642753 D: 0.000000 E: 0.000000 Sum of squares = 0.540757054953 Cumulative probabilities = A: 0.357247 B: 0.357247 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 0 10 B -10 0 -8 -10 6 C 0 8 0 2 8 D 0 10 -2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=22 C=22 E=17 A=13 so A is eliminated. Round 2 votes counts: D=29 B=26 C=23 E=22 so E is eliminated. Round 3 votes counts: D=40 B=31 C=29 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:210 C:209 D:206 B:189 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 0 10 B -10 0 -8 -10 6 C 0 8 0 2 8 D 0 10 -2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 0 10 B -10 0 -8 -10 6 C 0 8 0 2 8 D 0 10 -2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 0 10 B -10 0 -8 -10 6 C 0 8 0 2 8 D 0 10 -2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5630: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) E C A B D (6) D B C A E (6) D B A C E (6) A E D C B (6) E A C D B (5) C E A B D (5) C B A E D (5) D B E A C (4) E A C B D (3) D A B C E (3) C A E B D (3) A E C D B (3) E D C B A (2) E C B D A (2) D E A B C (2) D B E C A (2) D B A E C (2) C B E A D (2) B D C A E (2) B C D A E (2) A E C B D (2) A C E B D (2) E D B C A (1) E D A C B (1) E C B A D (1) E C A D B (1) E B C D A (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) B D C E A (1) B C E D A (1) B C D E A (1) B C A D E (1) A D E B C (1) A D C B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 0 -2 B 4 0 -6 -12 0 C 10 6 0 -2 6 D 0 12 2 0 -10 E 2 0 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.333333 E: 0.111111 Sum of squares = 0.432098765438 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.888889 E: 1.000000 A B C D E A 0 -4 -10 0 -2 B 4 0 -6 -12 0 C 10 6 0 -2 6 D 0 12 2 0 -10 E 2 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.333333 E: 0.111111 Sum of squares = 0.432098765695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=23 C=17 A=17 B=8 so B is eliminated. Round 2 votes counts: D=38 E=23 C=22 A=17 so A is eliminated. Round 3 votes counts: D=41 E=34 C=25 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:203 D:202 B:193 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 0 -2 B 4 0 -6 -12 0 C 10 6 0 -2 6 D 0 12 2 0 -10 E 2 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.333333 E: 0.111111 Sum of squares = 0.432098765695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.888889 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 0 -2 B 4 0 -6 -12 0 C 10 6 0 -2 6 D 0 12 2 0 -10 E 2 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.333333 E: 0.111111 Sum of squares = 0.432098765695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.888889 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 0 -2 B 4 0 -6 -12 0 C 10 6 0 -2 6 D 0 12 2 0 -10 E 2 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.333333 E: 0.111111 Sum of squares = 0.432098765695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.888889 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5631: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) B C E D A (7) D B E C A (6) C B E A D (5) B D C E A (5) B D A C E (5) B C D E A (5) D B A E C (4) D A E B C (4) C E A B D (4) A D E C B (4) E D C A B (3) E C D A B (3) E C A D B (3) E A C D B (3) C E B A D (3) D A B E C (2) C B E D A (2) B D C A E (2) B D A E C (2) B C A D E (2) A E D C B (2) A D E B C (2) A C E B D (2) E D C B A (1) E C D B A (1) E A D C B (1) D E A B C (1) D B E A C (1) D A E C B (1) C E B D A (1) B C E A D (1) B C D A E (1) B A D C E (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -10 -14 -12 B 10 0 4 -4 4 C 10 -4 0 4 -6 D 14 4 -4 0 -2 E 12 -4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 -10 -10 -14 -12 B 10 0 4 -4 4 C 10 -4 0 4 -6 D 14 4 -4 0 -2 E 12 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=20 D=19 E=15 C=15 so E is eliminated. Round 2 votes counts: B=31 A=24 D=23 C=22 so C is eliminated. Round 3 votes counts: B=42 A=31 D=27 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:208 B:207 D:206 C:202 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 -14 -12 B 10 0 4 -4 4 C 10 -4 0 4 -6 D 14 4 -4 0 -2 E 12 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -14 -12 B 10 0 4 -4 4 C 10 -4 0 4 -6 D 14 4 -4 0 -2 E 12 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -14 -12 B 10 0 4 -4 4 C 10 -4 0 4 -6 D 14 4 -4 0 -2 E 12 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5632: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) A B E D C (9) D C A E B (7) C E D B A (6) B E C A D (6) B E A C D (6) A B D E C (6) D A C B E (5) E C B D A (4) D C E A B (4) C E B D A (4) B A E C D (4) E B C D A (3) C D E A B (3) A D C B E (3) A D B E C (3) D C A B E (2) D A C E B (2) B A E D C (2) A D C E B (2) A D B C E (2) E B C A D (1) E B A C D (1) C D B E A (1) B C E D A (1) A E D B C (1) A E B D C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 -2 2 B -4 0 -4 -2 2 C 2 4 0 -2 4 D 2 2 2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -2 2 B -4 0 -4 -2 2 C 2 4 0 -2 4 D 2 2 2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 D=20 B=19 E=9 so E is eliminated. Round 2 votes counts: A=29 C=27 B=24 D=20 so D is eliminated. Round 3 votes counts: C=40 A=36 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:204 D:204 A:201 B:196 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -2 2 B -4 0 -4 -2 2 C 2 4 0 -2 4 D 2 2 2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -2 2 B -4 0 -4 -2 2 C 2 4 0 -2 4 D 2 2 2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -2 2 B -4 0 -4 -2 2 C 2 4 0 -2 4 D 2 2 2 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5633: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) D C B A E (8) C B E A D (8) A E D B C (8) D C A B E (7) E B A C D (6) E A B C D (5) E A B D C (4) D A E B C (4) D A C E B (4) C D B A E (4) C B E D A (4) C B D E A (4) B E A C D (4) D C A E B (3) A E B D C (3) C D B E A (2) C B D A E (2) B E C A D (2) B C E A D (2) A E B C D (2) D C E B A (1) D C B E A (1) D A C B E (1) C D E B A (1) A D E B C (1) Total count = 100 A B C D E A 0 2 2 -10 12 B -2 0 -18 -8 -2 C -2 18 0 -8 4 D 10 8 8 0 4 E -12 2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -10 12 B -2 0 -18 -8 -2 C -2 18 0 -8 4 D 10 8 8 0 4 E -12 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=25 E=15 A=14 B=8 so B is eliminated. Round 2 votes counts: D=38 C=27 E=21 A=14 so A is eliminated. Round 3 votes counts: D=39 E=34 C=27 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:206 A:203 E:191 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 2 -10 12 B -2 0 -18 -8 -2 C -2 18 0 -8 4 D 10 8 8 0 4 E -12 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -10 12 B -2 0 -18 -8 -2 C -2 18 0 -8 4 D 10 8 8 0 4 E -12 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -10 12 B -2 0 -18 -8 -2 C -2 18 0 -8 4 D 10 8 8 0 4 E -12 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5634: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) C A D B E (7) C A B D E (7) D E B C A (6) C A D E B (6) D E C A B (5) D C A E B (5) B E A C D (5) A C E D B (5) E B D A C (4) B E D A C (4) A C B E D (4) E D B A C (3) D E A C B (3) B D E C A (3) B C A D E (3) B A C E D (3) A C E B D (3) D B E C A (2) B E D C A (2) A C D E B (2) E D A C B (1) E B A D C (1) E B A C D (1) E A B C D (1) D E B A C (1) D A C E B (1) C A B E D (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -12 20 16 B -4 0 0 6 4 C 12 0 0 18 14 D -20 -6 -18 0 2 E -16 -4 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.430753 C: 0.569247 D: 0.000000 E: 0.000000 Sum of squares = 0.50959028496 Cumulative probabilities = A: 0.000000 B: 0.430753 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 20 16 B -4 0 0 6 4 C 12 0 0 18 14 D -20 -6 -18 0 2 E -16 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 C=21 A=15 E=11 so E is eliminated. Round 2 votes counts: B=36 D=27 C=21 A=16 so A is eliminated. Round 3 votes counts: B=38 C=35 D=27 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:222 A:214 B:203 E:182 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 20 16 B -4 0 0 6 4 C 12 0 0 18 14 D -20 -6 -18 0 2 E -16 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 20 16 B -4 0 0 6 4 C 12 0 0 18 14 D -20 -6 -18 0 2 E -16 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 20 16 B -4 0 0 6 4 C 12 0 0 18 14 D -20 -6 -18 0 2 E -16 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5635: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) E A C B D (8) D B C E A (8) B E D C A (8) B D E C A (8) D B C A E (7) A E C B D (6) E B D A C (4) E A B C D (4) A C D E B (4) E B A D C (2) E B A C D (2) D C B A E (2) D B E A C (2) C A E D B (2) C A D B E (2) B D C E A (2) A E C D B (2) A D C B E (2) A C E B D (2) E C A B D (1) E B D C A (1) E B C D A (1) E A B D C (1) D C B E A (1) D B A E C (1) C E B A D (1) C E A B D (1) C D A B E (1) C A E B D (1) C A D E B (1) B E D A C (1) B D E A C (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 4 4 -14 B 4 0 6 10 -8 C -4 -6 0 -4 -6 D -4 -10 4 0 -14 E 14 8 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 4 -14 B 4 0 6 10 -8 C -4 -6 0 -4 -6 D -4 -10 4 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 D=21 B=20 C=9 so C is eliminated. Round 2 votes counts: A=32 E=26 D=22 B=20 so B is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:206 A:195 C:190 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 4 -14 B 4 0 6 10 -8 C -4 -6 0 -4 -6 D -4 -10 4 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 -14 B 4 0 6 10 -8 C -4 -6 0 -4 -6 D -4 -10 4 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 -14 B 4 0 6 10 -8 C -4 -6 0 -4 -6 D -4 -10 4 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5636: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (24) B A E C D (13) C E A D B (7) C D E A B (7) B D C E A (7) D B C E A (6) E A C D B (5) A E B C D (5) A E C B D (4) B D C A E (3) D C E B A (2) B D A E C (2) B C D E A (2) B A D E C (2) A E C D B (2) E C A D B (1) E A C B D (1) D C B E A (1) D B C A E (1) C E D A B (1) C A E D B (1) B E A C D (1) B C E A D (1) A B E C D (1) Total count = 100 A B C D E A 0 18 -28 -12 -32 B -18 0 -12 -16 -20 C 28 12 0 4 26 D 12 16 -4 0 14 E 32 20 -26 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -28 -12 -32 B -18 0 -12 -16 -20 C 28 12 0 4 26 D 12 16 -4 0 14 E 32 20 -26 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999511 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=31 C=16 A=12 E=7 so E is eliminated. Round 2 votes counts: D=34 B=31 A=18 C=17 so C is eliminated. Round 3 votes counts: D=42 B=31 A=27 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:235 D:219 E:206 A:173 B:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -28 -12 -32 B -18 0 -12 -16 -20 C 28 12 0 4 26 D 12 16 -4 0 14 E 32 20 -26 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999511 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -28 -12 -32 B -18 0 -12 -16 -20 C 28 12 0 4 26 D 12 16 -4 0 14 E 32 20 -26 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999511 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -28 -12 -32 B -18 0 -12 -16 -20 C 28 12 0 4 26 D 12 16 -4 0 14 E 32 20 -26 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999511 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5637: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) B D A E C (7) B A D C E (7) E C B D A (6) D A E C B (6) E C D A B (5) C E A D B (5) A D E C B (5) A D C E B (5) D A E B C (4) C E B A D (4) B E C D A (4) B C A D E (4) C E B D A (3) C B E A D (3) B E D A C (3) B C E D A (3) B C E A D (3) E B C D A (2) E C D B A (1) E C A D B (1) D E A B C (1) C E A B D (1) B D C A E (1) B D A C E (1) B A C D E (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 8 -14 10 B 6 0 6 6 0 C -8 -6 0 -6 -12 D 14 -6 6 0 12 E -10 0 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.828246 C: 0.000000 D: 0.000000 E: 0.171754 Sum of squares = 0.715491041956 Cumulative probabilities = A: 0.000000 B: 0.828246 C: 0.828246 D: 0.828246 E: 1.000000 A B C D E A 0 -6 8 -14 10 B 6 0 6 6 0 C -8 -6 0 -6 -12 D 14 -6 6 0 12 E -10 0 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560782 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=21 C=16 E=15 A=14 so A is eliminated. Round 2 votes counts: D=34 B=34 C=17 E=15 so E is eliminated. Round 3 votes counts: B=36 D=34 C=30 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:209 A:199 E:195 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 -14 10 B 6 0 6 6 0 C -8 -6 0 -6 -12 D 14 -6 6 0 12 E -10 0 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560782 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 -14 10 B 6 0 6 6 0 C -8 -6 0 -6 -12 D 14 -6 6 0 12 E -10 0 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560782 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 -14 10 B 6 0 6 6 0 C -8 -6 0 -6 -12 D 14 -6 6 0 12 E -10 0 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560782 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5638: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) B E D A C (6) E B C D A (5) E A B C D (4) C E A D B (4) C D A E B (4) C A E D B (4) A D C B E (4) E C A D B (3) E C A B D (3) D C B A E (3) C E D A B (3) C A D E B (3) B E A D C (3) B D E C A (3) B D A E C (3) A C E D B (3) E C D B A (2) E C B D A (2) E C B A D (2) E B C A D (2) E B A D C (2) E A C B D (2) B E D C A (2) B D E A C (2) B D C E A (2) B D C A E (2) E C D A B (1) E B A C D (1) D C A B E (1) D B C A E (1) D B A C E (1) C D A B E (1) B A E D C (1) B A D E C (1) B A D C E (1) A E C D B (1) A E B D C (1) A E B C D (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -6 -2 -8 B 8 0 6 18 -8 C 6 -6 0 4 -6 D 2 -18 -4 0 -16 E 8 8 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 -2 -8 B 8 0 6 18 -8 C 6 -6 0 4 -6 D 2 -18 -4 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=29 C=19 A=13 D=6 so D is eliminated. Round 2 votes counts: B=35 E=29 C=23 A=13 so A is eliminated. Round 3 votes counts: B=37 E=32 C=31 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:212 C:199 A:188 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -2 -8 B 8 0 6 18 -8 C 6 -6 0 4 -6 D 2 -18 -4 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -2 -8 B 8 0 6 18 -8 C 6 -6 0 4 -6 D 2 -18 -4 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -2 -8 B 8 0 6 18 -8 C 6 -6 0 4 -6 D 2 -18 -4 0 -16 E 8 8 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5639: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) E D C A B (7) E D A C B (7) A D E C B (5) B E C D A (4) B C E D A (4) E B C D A (3) B A C D E (3) E D B A C (2) E D A B C (2) E B D C A (2) E A D B C (2) D E C A B (2) D A E C B (2) C D E A B (2) C D A E B (2) C D A B E (2) B C E A D (2) B C A E D (2) B A E D C (2) A E D B C (2) A C D B E (2) A B D E C (2) E D B C A (1) E C D A B (1) E C B D A (1) E B D A C (1) E B A D C (1) E A D C B (1) D E A C B (1) D C E A B (1) D C A E B (1) D A C E B (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A C D (1) B C D A E (1) B A D C E (1) A E D C B (1) A D E B C (1) A D C E B (1) A D C B E (1) A D B E C (1) A D B C E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -6 -8 -2 B -12 0 4 -10 -6 C 6 -4 0 -6 -12 D 8 10 6 0 0 E 2 6 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.524697 E: 0.475303 Sum of squares = 0.501219893771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.524697 E: 1.000000 A B C D E A 0 12 -6 -8 -2 B -12 0 4 -10 -6 C 6 -4 0 -6 -12 D 8 10 6 0 0 E 2 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=29 A=20 C=12 D=8 so D is eliminated. Round 2 votes counts: E=34 B=29 A=23 C=14 so C is eliminated. Round 3 votes counts: E=37 B=33 A=30 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:210 A:198 C:192 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -6 -8 -2 B -12 0 4 -10 -6 C 6 -4 0 -6 -12 D 8 10 6 0 0 E 2 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -8 -2 B -12 0 4 -10 -6 C 6 -4 0 -6 -12 D 8 10 6 0 0 E 2 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -8 -2 B -12 0 4 -10 -6 C 6 -4 0 -6 -12 D 8 10 6 0 0 E 2 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5640: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (6) B C E A D (6) B C A D E (6) A C B D E (6) E D B C A (5) C A B E D (5) E D A C B (4) D B E C A (4) D A E C B (4) B C E D A (4) D A B C E (3) C B A E D (3) B C A E D (3) A C E B D (3) E B D C A (2) E A C D B (2) D E A C B (2) B E C D A (2) B D E C A (2) B D C E A (2) A E D C B (2) A D E C B (2) A D C B E (2) A C B E D (2) E D B A C (1) E C B A D (1) E B C D A (1) E A D C B (1) E A C B D (1) D E B C A (1) D E B A C (1) D B C E A (1) D B C A E (1) D B A C E (1) D A E B C (1) D A C B E (1) C B E A D (1) C A B D E (1) B D A C E (1) A D C E B (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -2 0 0 B -2 0 10 4 16 C 2 -10 0 -4 10 D 0 -4 4 0 0 E 0 -16 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.697465 B: 0.092396 C: 0.092396 D: 0.117743 E: 0.000000 Sum of squares = 0.517395167241 Cumulative probabilities = A: 0.697465 B: 0.789861 C: 0.882257 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 0 0 B -2 0 10 4 16 C 2 -10 0 -4 10 D 0 -4 4 0 0 E 0 -16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.691177 B: 0.073530 C: 0.073530 D: 0.161764 E: 0.000000 Sum of squares = 0.514705882352 Cumulative probabilities = A: 0.691177 B: 0.764706 C: 0.838236 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 A=20 E=18 C=10 so C is eliminated. Round 2 votes counts: B=30 D=26 A=26 E=18 so E is eliminated. Round 3 votes counts: D=36 B=34 A=30 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:200 D:200 C:199 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 0 0 B -2 0 10 4 16 C 2 -10 0 -4 10 D 0 -4 4 0 0 E 0 -16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.691177 B: 0.073530 C: 0.073530 D: 0.161764 E: 0.000000 Sum of squares = 0.514705882352 Cumulative probabilities = A: 0.691177 B: 0.764706 C: 0.838236 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 0 B -2 0 10 4 16 C 2 -10 0 -4 10 D 0 -4 4 0 0 E 0 -16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.691177 B: 0.073530 C: 0.073530 D: 0.161764 E: 0.000000 Sum of squares = 0.514705882352 Cumulative probabilities = A: 0.691177 B: 0.764706 C: 0.838236 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 0 B -2 0 10 4 16 C 2 -10 0 -4 10 D 0 -4 4 0 0 E 0 -16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.691177 B: 0.073530 C: 0.073530 D: 0.161764 E: 0.000000 Sum of squares = 0.514705882352 Cumulative probabilities = A: 0.691177 B: 0.764706 C: 0.838236 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5641: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (14) B D C A E (12) E A D C B (11) C B D E A (6) B D A E C (6) B C D A E (6) C E A D B (5) A E D B C (4) E C A D B (3) B C D E A (3) E A B D C (2) D C B A E (2) D C A E B (2) D B C A E (2) D A E B C (2) C D E A B (2) B C E D A (2) B A E D C (2) A E D C B (2) E A D B C (1) E A C B D (1) E A B C D (1) D A E C B (1) C D B A E (1) C B E D A (1) C B E A D (1) B E C A D (1) B E A D C (1) B A D E C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 2 4 -10 B -6 0 -4 -6 -4 C -2 4 0 -6 -10 D -4 6 6 0 -4 E 10 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 2 4 -10 B -6 0 -4 -6 -4 C -2 4 0 -6 -10 D -4 6 6 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=33 C=16 D=9 A=8 so A is eliminated. Round 2 votes counts: E=39 B=35 C=16 D=10 so D is eliminated. Round 3 votes counts: E=43 B=37 C=20 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:202 A:201 C:193 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 4 -10 B -6 0 -4 -6 -4 C -2 4 0 -6 -10 D -4 6 6 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 4 -10 B -6 0 -4 -6 -4 C -2 4 0 -6 -10 D -4 6 6 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 4 -10 B -6 0 -4 -6 -4 C -2 4 0 -6 -10 D -4 6 6 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5642: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) D B A E C (6) C E D A B (6) C B E A D (6) E A C B D (5) C D B A E (5) C B D A E (5) E C A B D (4) D E A C B (4) C E B A D (4) E B A C D (3) E A B D C (3) E A B C D (3) D A B E C (3) C D B E A (3) A B E D C (3) E D A C B (2) D C B A E (2) D B C A E (2) D B A C E (2) E D A B C (1) E C D A B (1) E C A D B (1) E A D C B (1) E A D B C (1) D E A B C (1) D A E C B (1) C E D B A (1) C D E A B (1) C B A D E (1) B D A E C (1) B D A C E (1) B C E A D (1) B C D A E (1) B C A E D (1) B C A D E (1) B A D E C (1) B A D C E (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 8 -12 -4 B -4 0 -6 -8 -6 C -8 6 0 6 -12 D 12 8 -6 0 2 E 4 6 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.600000 E: 0.300000 Sum of squares = 0.460000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.700000 E: 1.000000 A B C D E A 0 4 8 -12 -4 B -4 0 -6 -8 -6 C -8 6 0 6 -12 D 12 8 -6 0 2 E 4 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.600000 E: 0.300000 Sum of squares = 0.459999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=28 E=25 B=8 A=7 so A is eliminated. Round 2 votes counts: C=32 D=30 E=27 B=11 so B is eliminated. Round 3 votes counts: C=36 D=34 E=30 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:210 D:208 A:198 C:196 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 -12 -4 B -4 0 -6 -8 -6 C -8 6 0 6 -12 D 12 8 -6 0 2 E 4 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.600000 E: 0.300000 Sum of squares = 0.459999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.700000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -12 -4 B -4 0 -6 -8 -6 C -8 6 0 6 -12 D 12 8 -6 0 2 E 4 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.600000 E: 0.300000 Sum of squares = 0.459999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.700000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -12 -4 B -4 0 -6 -8 -6 C -8 6 0 6 -12 D 12 8 -6 0 2 E 4 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.600000 E: 0.300000 Sum of squares = 0.459999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5643: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (9) E C A B D (8) B A C E D (8) D B A C E (6) D B E A C (5) C A E B D (5) B D A C E (5) D E C A B (4) D C A E B (4) B A C D E (4) A C B E D (4) E D C A B (3) E B C A D (3) D E B C A (3) D A C B E (3) C A E D B (3) E C A D B (2) B E A C D (2) D C E A B (1) D B A E C (1) C E A B D (1) C D A E B (1) B E D C A (1) B E D A C (1) B D E A C (1) B A E C D (1) Total count = 100 A B C D E A 0 0 4 4 12 B 0 0 0 10 2 C -4 0 0 6 12 D -4 -10 -6 0 -6 E -12 -2 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.532645 B: 0.467355 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50213141633 Cumulative probabilities = A: 0.532645 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 4 12 B 0 0 0 10 2 C -4 0 0 6 12 D -4 -10 -6 0 -6 E -12 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=23 E=16 A=15 C=10 so C is eliminated. Round 2 votes counts: D=37 B=23 A=23 E=17 so E is eliminated. Round 3 votes counts: D=40 A=34 B=26 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:207 B:206 E:190 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 4 12 B 0 0 0 10 2 C -4 0 0 6 12 D -4 -10 -6 0 -6 E -12 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 12 B 0 0 0 10 2 C -4 0 0 6 12 D -4 -10 -6 0 -6 E -12 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 12 B 0 0 0 10 2 C -4 0 0 6 12 D -4 -10 -6 0 -6 E -12 -2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5644: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (29) C B A D E (26) B A D C E (10) D E A B C (6) C E B A D (6) C B A E D (5) A B D E C (3) E C D A B (2) D B A C E (2) D A B E C (2) B A C D E (2) E D C A B (1) E C D B A (1) D A B C E (1) C E D B A (1) C B E A D (1) B C A D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 12 10 6 B 10 0 14 10 8 C -12 -14 0 -10 12 D -10 -10 10 0 8 E -6 -8 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 12 10 6 B 10 0 14 10 8 C -12 -14 0 -10 12 D -10 -10 10 0 8 E -6 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=33 B=13 D=11 A=4 so A is eliminated. Round 2 votes counts: C=39 E=33 B=17 D=11 so D is eliminated. Round 3 votes counts: E=39 C=39 B=22 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:221 A:209 D:199 C:188 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 12 10 6 B 10 0 14 10 8 C -12 -14 0 -10 12 D -10 -10 10 0 8 E -6 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 12 10 6 B 10 0 14 10 8 C -12 -14 0 -10 12 D -10 -10 10 0 8 E -6 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 12 10 6 B 10 0 14 10 8 C -12 -14 0 -10 12 D -10 -10 10 0 8 E -6 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5645: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (14) C E A D B (12) A D C B E (10) E C B A D (8) D A B C E (8) D A B E C (5) A D C E B (5) D B A E C (4) C A E D B (4) C A D E B (4) C E B A D (3) B E D C A (3) B E D A C (3) B D E A C (3) A D B C E (3) A C D E B (3) C E A B D (2) B E C D A (2) B D A E C (2) E C B D A (1) B D A C E (1) Total count = 100 A B C D E A 0 12 -6 8 -2 B -12 0 -4 -16 -12 C 6 4 0 6 10 D -8 16 -6 0 -4 E 2 12 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 8 -2 B -12 0 -4 -16 -12 C 6 4 0 6 10 D -8 16 -6 0 -4 E 2 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=23 A=21 D=17 B=14 so B is eliminated. Round 2 votes counts: E=31 C=25 D=23 A=21 so A is eliminated. Round 3 votes counts: D=41 E=31 C=28 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:213 A:206 E:204 D:199 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 8 -2 B -12 0 -4 -16 -12 C 6 4 0 6 10 D -8 16 -6 0 -4 E 2 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 8 -2 B -12 0 -4 -16 -12 C 6 4 0 6 10 D -8 16 -6 0 -4 E 2 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 8 -2 B -12 0 -4 -16 -12 C 6 4 0 6 10 D -8 16 -6 0 -4 E 2 12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5646: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) E B A D C (7) C D A B E (7) B E A C D (7) C D E B A (6) A C D B E (6) A B E C D (6) E B D C A (5) C A D B E (5) B E A D C (5) D C E B A (3) D C E A B (3) B E C A D (3) A D C B E (3) E D B C A (2) E B C D A (2) B A E D C (2) B A E C D (2) A E B D C (2) A D C E B (2) A B E D C (2) E D B A C (1) D E C B A (1) D E A C B (1) D A C E B (1) C D B E A (1) C D A E B (1) C B E D A (1) B E C D A (1) B C E A D (1) B C A E D (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -4 10 0 B 2 0 0 -6 6 C 4 0 0 0 0 D -10 6 0 0 0 E 0 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.658478 D: 0.204016 E: 0.137506 Sum of squares = 0.494123870166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.658478 D: 0.862494 E: 1.000000 A B C D E A 0 -2 -4 10 0 B 2 0 0 -6 6 C 4 0 0 0 0 D -10 6 0 0 0 E 0 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.222222 Sum of squares = 0.407407499993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.777778 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=22 A=22 C=21 D=18 E=17 so E is eliminated. Round 2 votes counts: B=36 A=22 D=21 C=21 so D is eliminated. Round 3 votes counts: B=39 C=37 A=24 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:202 C:202 B:201 D:198 E:197 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 10 0 B 2 0 0 -6 6 C 4 0 0 0 0 D -10 6 0 0 0 E 0 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.222222 Sum of squares = 0.407407499993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.777778 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 0 B 2 0 0 -6 6 C 4 0 0 0 0 D -10 6 0 0 0 E 0 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.222222 Sum of squares = 0.407407499993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 0 B 2 0 0 -6 6 C 4 0 0 0 0 D -10 6 0 0 0 E 0 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.222222 Sum of squares = 0.407407499993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.555556 D: 0.777778 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5647: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B E D A C (7) E B C D A (6) C E A B D (5) A C D B E (5) A C B D E (5) E C B D A (4) A D B E C (4) A B D C E (4) E B D C A (3) D B E A C (3) C E D B A (3) C A E B D (3) A D C B E (3) A C D E B (3) A C B E D (3) E B C A D (2) D E B C A (2) D A C B E (2) D A B E C (2) C D A E B (2) C A E D B (2) B E D C A (2) B E A D C (2) A D B C E (2) A B D E C (2) E D B C A (1) D C E A B (1) C E B A D (1) B E A C D (1) B D E A C (1) B A E D C (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 22 6 22 12 B -22 0 -4 10 4 C -6 4 0 16 12 D -22 -10 -16 0 4 E -12 -4 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 6 22 12 B -22 0 -4 10 4 C -6 4 0 16 12 D -22 -10 -16 0 4 E -12 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=26 E=16 B=14 D=10 so D is eliminated. Round 2 votes counts: A=38 C=27 E=18 B=17 so B is eliminated. Round 3 votes counts: A=39 E=34 C=27 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 C:213 B:194 E:184 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 6 22 12 B -22 0 -4 10 4 C -6 4 0 16 12 D -22 -10 -16 0 4 E -12 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 6 22 12 B -22 0 -4 10 4 C -6 4 0 16 12 D -22 -10 -16 0 4 E -12 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 6 22 12 B -22 0 -4 10 4 C -6 4 0 16 12 D -22 -10 -16 0 4 E -12 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5648: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) A C B D E (9) E D B C A (7) D E C B A (7) D E A B C (5) D A E C B (5) A B C E D (5) E D B A C (4) C B A E D (4) A C D B E (4) E B D C A (3) C B A D E (3) C A B D E (3) B A C E D (3) E B C D A (2) D A C E B (2) C D A B E (2) B E C A D (2) A D C B E (2) D E C A B (1) D E B C A (1) D E A C B (1) C D E A B (1) C D B E A (1) C B E A D (1) C B D A E (1) B E A D C (1) B E A C D (1) B C E A D (1) B A E D C (1) A D B E C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -6 14 22 B 12 0 6 12 22 C 6 -6 0 16 14 D -14 -12 -16 0 2 E -22 -22 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 14 22 B 12 0 6 12 22 C 6 -6 0 16 14 D -14 -12 -16 0 2 E -22 -22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=22 B=22 E=16 C=16 so E is eliminated. Round 2 votes counts: D=33 B=27 A=24 C=16 so C is eliminated. Round 3 votes counts: D=37 B=36 A=27 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:215 A:209 D:180 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 14 22 B 12 0 6 12 22 C 6 -6 0 16 14 D -14 -12 -16 0 2 E -22 -22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 14 22 B 12 0 6 12 22 C 6 -6 0 16 14 D -14 -12 -16 0 2 E -22 -22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 14 22 B 12 0 6 12 22 C 6 -6 0 16 14 D -14 -12 -16 0 2 E -22 -22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5649: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B A E C D (9) C D A E B (8) E D C B A (6) A C D B E (5) A B C D E (5) E B D C A (4) E B A D C (4) D E C B A (4) C A D B E (4) A B D C E (4) D C E B A (3) C D E A B (3) B A E D C (3) A B C E D (3) E D B C A (2) E C D B A (2) E C A B D (2) E B D A C (2) D C A E B (2) C D A B E (2) B E A D C (2) B A D E C (2) E C B D A (1) E B C D A (1) E B A C D (1) D B A C E (1) D A C B E (1) C E A D B (1) B E A C D (1) A D C B E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -8 -2 4 B -4 0 -8 -8 -10 C 8 8 0 -2 4 D 2 8 2 0 10 E -4 10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -2 4 B -4 0 -8 -8 -10 C 8 8 0 -2 4 D 2 8 2 0 10 E -4 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=20 A=20 C=18 B=17 so B is eliminated. Round 2 votes counts: A=34 E=28 D=20 C=18 so C is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:209 A:199 E:196 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -2 4 B -4 0 -8 -8 -10 C 8 8 0 -2 4 D 2 8 2 0 10 E -4 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -2 4 B -4 0 -8 -8 -10 C 8 8 0 -2 4 D 2 8 2 0 10 E -4 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -2 4 B -4 0 -8 -8 -10 C 8 8 0 -2 4 D 2 8 2 0 10 E -4 10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5650: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) B D A C E (7) B D A E C (6) A D B C E (6) E C D B A (4) B A D E C (4) A D C E B (4) A C E D B (4) E C B D A (3) E C B A D (3) E C A B D (3) D B C A E (3) D A B C E (3) C E D B A (3) B D E C A (3) A E B C D (3) A B D E C (3) E B C D A (2) E A C B D (2) C E A D B (2) B E C D A (2) B A D C E (2) A C D E B (2) A B D C E (2) E C D A B (1) E B C A D (1) D C E B A (1) D C E A B (1) D C B A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E B A (1) C D A E B (1) C A E D B (1) B E D C A (1) B E C A D (1) B E A D C (1) B D C E A (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 6 6 12 B 2 0 6 4 -2 C -6 -6 0 -4 -4 D -6 -4 4 0 6 E -12 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000143 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -2 6 6 12 B 2 0 6 4 -2 C -6 -6 0 -4 -4 D -6 -4 4 0 6 E -12 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000408 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=26 A=26 D=11 C=9 so C is eliminated. Round 2 votes counts: E=32 B=28 A=27 D=13 so D is eliminated. Round 3 votes counts: E=35 B=33 A=32 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:211 B:205 D:200 E:194 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 6 12 B 2 0 6 4 -2 C -6 -6 0 -4 -4 D -6 -4 4 0 6 E -12 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000408 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 12 B 2 0 6 4 -2 C -6 -6 0 -4 -4 D -6 -4 4 0 6 E -12 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000408 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 12 B 2 0 6 4 -2 C -6 -6 0 -4 -4 D -6 -4 4 0 6 E -12 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000408 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5651: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) E B D A C (8) C B E A D (8) C A D B E (8) D A C E B (7) D A C B E (6) E D B A C (5) B C E A D (5) A D C B E (5) D A E B C (4) B E C D A (4) E B D C A (3) D A E C B (3) C B A E D (3) B E C A D (3) A D C E B (3) E B A C D (2) C A B E D (2) A C D B E (2) E B C D A (1) E B A D C (1) C D B A E (1) C B A D E (1) B C E D A (1) Total count = 100 A B C D E A 0 -20 -8 14 -10 B 20 0 2 12 -2 C 8 -2 0 10 4 D -14 -12 -10 0 -20 E 10 2 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000049 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -20 -8 14 -10 B 20 0 2 12 -2 C 8 -2 0 10 4 D -14 -12 -10 0 -20 E 10 2 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000068 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=23 D=20 B=13 A=10 so A is eliminated. Round 2 votes counts: E=34 D=28 C=25 B=13 so B is eliminated. Round 3 votes counts: E=41 C=31 D=28 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:216 E:214 C:210 A:188 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -8 14 -10 B 20 0 2 12 -2 C 8 -2 0 10 4 D -14 -12 -10 0 -20 E 10 2 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000068 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 14 -10 B 20 0 2 12 -2 C 8 -2 0 10 4 D -14 -12 -10 0 -20 E 10 2 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000068 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 14 -10 B 20 0 2 12 -2 C 8 -2 0 10 4 D -14 -12 -10 0 -20 E 10 2 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000068 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5652: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (12) B D A E C (7) A E B C D (7) C A E D B (6) E C A D B (5) D C B E A (5) D B C E A (5) B D E A C (4) E A B C D (3) D C B A E (3) C E A D B (3) B E D A C (3) B A E D C (3) E C D A B (2) E A C D B (2) E A C B D (2) D C E A B (2) C D E A B (2) C D A E B (2) C D A B E (2) C A D E B (2) B E A D C (2) B D C A E (2) A C E D B (2) E D C A B (1) E B A D C (1) D C E B A (1) C D B A E (1) C A B D E (1) B D E C A (1) B D A C E (1) B A E C D (1) B A D E C (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 18 8 12 12 B -18 0 -16 8 -12 C -8 16 0 16 -14 D -12 -8 -16 0 -16 E -12 12 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 8 12 12 B -18 0 -16 8 -12 C -8 16 0 16 -14 D -12 -8 -16 0 -16 E -12 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=24 C=19 E=16 D=16 so E is eliminated. Round 2 votes counts: A=31 C=26 B=26 D=17 so D is eliminated. Round 3 votes counts: C=38 B=31 A=31 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:215 C:205 B:181 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 8 12 12 B -18 0 -16 8 -12 C -8 16 0 16 -14 D -12 -8 -16 0 -16 E -12 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 8 12 12 B -18 0 -16 8 -12 C -8 16 0 16 -14 D -12 -8 -16 0 -16 E -12 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 8 12 12 B -18 0 -16 8 -12 C -8 16 0 16 -14 D -12 -8 -16 0 -16 E -12 12 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5653: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (10) B E A D C (6) E C A D B (5) A C D B E (5) E B A C D (4) D B A C E (4) C A D B E (4) B E A C D (4) B A D C E (4) D E B C A (3) D B E C A (3) C A D E B (3) B E D A C (3) A C B D E (3) E D B C A (2) E C D B A (2) E B D A C (2) E B C A D (2) D E C B A (2) D C A E B (2) D C A B E (2) C D A E B (2) C A E D B (2) C A E B D (2) B D A C E (2) B A C D E (2) A B C E D (2) A B C D E (2) E D C B A (1) E C B A D (1) E B C D A (1) E B A D C (1) D C E B A (1) D C E A B (1) C D A B E (1) B A E C D (1) B A C E D (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 -4 12 -8 B 24 0 20 8 0 C 4 -20 0 0 -6 D -12 -8 0 0 -4 E 8 0 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.174591 C: 0.000000 D: 0.000000 E: 0.825409 Sum of squares = 0.711781698637 Cumulative probabilities = A: 0.000000 B: 0.174591 C: 0.174591 D: 0.174591 E: 1.000000 A B C D E A 0 -24 -4 12 -8 B 24 0 20 8 0 C 4 -20 0 0 -6 D -12 -8 0 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=23 D=18 C=14 A=14 so C is eliminated. Round 2 votes counts: E=31 A=25 B=23 D=21 so D is eliminated. Round 3 votes counts: E=38 A=32 B=30 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:226 E:209 C:189 A:188 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -4 12 -8 B 24 0 20 8 0 C 4 -20 0 0 -6 D -12 -8 0 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -4 12 -8 B 24 0 20 8 0 C 4 -20 0 0 -6 D -12 -8 0 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -4 12 -8 B 24 0 20 8 0 C 4 -20 0 0 -6 D -12 -8 0 0 -4 E 8 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5654: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) B A C D E (8) E D B A C (6) E A D B C (6) E D C A B (5) D E C B A (5) C D B A E (4) C B A D E (4) A B E D C (4) A B E C D (4) A B C E D (4) E D C B A (3) E D A C B (3) E D A B C (3) E A B D C (3) C D E B A (3) A E B D C (3) C B D A E (2) B C A D E (2) A E B C D (2) E A D C B (1) E A B C D (1) D C B A E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A B E (1) C A D B E (1) C A B D E (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D A E (1) B A D C E (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 10 -2 -2 B 4 0 6 -6 -12 C -10 -6 0 -12 -2 D 2 6 12 0 -2 E 2 12 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 10 -2 -2 B 4 0 6 -6 -12 C -10 -6 0 -12 -2 D 2 6 12 0 -2 E 2 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=20 C=19 B=16 D=14 so D is eliminated. Round 2 votes counts: E=36 C=28 A=20 B=16 so B is eliminated. Round 3 votes counts: E=37 C=32 A=31 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:209 E:209 A:201 B:196 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 10 -2 -2 B 4 0 6 -6 -12 C -10 -6 0 -12 -2 D 2 6 12 0 -2 E 2 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -2 -2 B 4 0 6 -6 -12 C -10 -6 0 -12 -2 D 2 6 12 0 -2 E 2 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -2 -2 B 4 0 6 -6 -12 C -10 -6 0 -12 -2 D 2 6 12 0 -2 E 2 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5655: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) C E D B A (7) B A E C D (7) A B D E C (7) D C E A B (6) D A B C E (6) D A C B E (5) C E B D A (4) B A D E C (4) E B C A D (3) D C A E B (3) C E D A B (3) C E B A D (3) C D E B A (3) A D B E C (3) A B E D C (3) E C A D B (2) E A C B D (2) D C B A E (2) D C A B E (2) D B C A E (2) B C E A D (2) A E B C D (2) A B E C D (2) E B A C D (1) E A B C D (1) D C E B A (1) D C B E A (1) D B A C E (1) D A E B C (1) D A C E B (1) C D E A B (1) C B E D A (1) B D C A E (1) Total count = 100 A B C D E A 0 0 -8 -2 4 B 0 0 -8 0 -2 C 8 8 0 2 10 D 2 0 -2 0 0 E -4 2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -2 4 B 0 0 -8 0 -2 C 8 8 0 2 10 D 2 0 -2 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=22 A=17 E=16 B=14 so B is eliminated. Round 2 votes counts: D=32 A=28 C=24 E=16 so E is eliminated. Round 3 votes counts: C=36 D=32 A=32 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:200 A:197 B:195 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -2 4 B 0 0 -8 0 -2 C 8 8 0 2 10 D 2 0 -2 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 4 B 0 0 -8 0 -2 C 8 8 0 2 10 D 2 0 -2 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 4 B 0 0 -8 0 -2 C 8 8 0 2 10 D 2 0 -2 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5656: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) B C E D A (9) B D A C E (8) C E B D A (6) D A C E B (5) B A D E C (5) D A C B E (4) D A B C E (4) B E C A D (4) B C D A E (4) A D E C B (4) E C A D B (3) E A D C B (3) C E D A B (3) A D B E C (3) C D E A B (2) C D A E B (2) B E A D C (2) A D E B C (2) A B D E C (2) E B C A D (1) D B A C E (1) C D B E A (1) C B E D A (1) C B D E A (1) B E A C D (1) B D C A E (1) B C E A D (1) B C D E A (1) B A E D C (1) B A D C E (1) A E D C B (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -20 -2 -6 0 B 20 0 6 22 10 C 2 -6 0 2 10 D 6 -22 -2 0 2 E 0 -10 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -2 -6 0 B 20 0 6 22 10 C 2 -6 0 2 10 D 6 -22 -2 0 2 E 0 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999148 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=18 C=16 D=14 A=14 so D is eliminated. Round 2 votes counts: B=39 A=27 E=18 C=16 so C is eliminated. Round 3 votes counts: B=42 E=29 A=29 so E is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 C:204 D:192 E:189 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -2 -6 0 B 20 0 6 22 10 C 2 -6 0 2 10 D 6 -22 -2 0 2 E 0 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999148 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -2 -6 0 B 20 0 6 22 10 C 2 -6 0 2 10 D 6 -22 -2 0 2 E 0 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999148 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -2 -6 0 B 20 0 6 22 10 C 2 -6 0 2 10 D 6 -22 -2 0 2 E 0 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999148 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5657: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) B D C A E (6) B D A E C (6) D A C B E (4) D A B E C (4) C E B A D (4) C B E D A (4) A D E C B (4) E C A B D (3) E B A C D (3) C D B A E (3) C D A B E (3) B E C A D (3) B C E D A (3) A E D C B (3) E C B A D (2) E C A D B (2) E A C D B (2) D A B C E (2) C A E D B (2) B E C D A (2) B E A D C (2) B C D E A (2) A D E B C (2) A D C E B (2) A C E D B (2) E B A D C (1) E A C B D (1) E A B D C (1) E A B C D (1) D C A E B (1) D B A E C (1) C E D B A (1) C E B D A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D B E A (1) C A D E B (1) B E D C A (1) B D E C A (1) B D C E A (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -12 0 -12 10 B 12 0 2 0 12 C 0 -2 0 -2 6 D 12 0 2 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.276361 C: 0.000000 D: 0.723639 E: 0.000000 Sum of squares = 0.600028792846 Cumulative probabilities = A: 0.000000 B: 0.276361 C: 0.276361 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -12 10 B 12 0 2 0 12 C 0 -2 0 -2 6 D 12 0 2 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 D=19 E=16 A=15 so A is eliminated. Round 2 votes counts: D=27 B=27 C=25 E=21 so E is eliminated. Round 3 votes counts: C=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:210 C:201 A:193 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -12 10 B 12 0 2 0 12 C 0 -2 0 -2 6 D 12 0 2 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -12 10 B 12 0 2 0 12 C 0 -2 0 -2 6 D 12 0 2 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -12 10 B 12 0 2 0 12 C 0 -2 0 -2 6 D 12 0 2 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5658: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) A E C D B (7) A E B C D (6) E B A D C (5) E B D A C (4) D C B A E (4) B E D C A (4) E A D C B (3) E A B C D (3) C D B A E (3) B D C E A (3) A C E B D (3) A C D E B (3) E D C B A (2) E D A C B (2) D E C B A (2) D C E A B (2) D C A E B (2) C D A B E (2) B D E C A (2) B C A D E (2) B A E C D (2) A C D B E (2) A B E C D (2) A B C E D (2) A B C D E (2) E D B C A (1) E D A B C (1) E B A C D (1) E A D B C (1) E A C D B (1) D E C A B (1) D C E B A (1) D B C E A (1) C D A E B (1) C B A D E (1) C A D B E (1) B E D A C (1) B E A D C (1) B D C A E (1) B C D A E (1) B A C E D (1) B A C D E (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 12 4 2 B 2 0 -4 -2 -6 C -12 4 0 -4 -6 D -4 2 4 0 -8 E -2 6 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000008 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -2 12 4 2 B 2 0 -4 -2 -6 C -12 4 0 -4 -6 D -4 2 4 0 -8 E -2 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 D=20 B=19 C=8 so C is eliminated. Round 2 votes counts: A=30 D=26 E=24 B=20 so B is eliminated. Round 3 votes counts: A=37 D=33 E=30 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:208 D:197 B:195 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 12 4 2 B 2 0 -4 -2 -6 C -12 4 0 -4 -6 D -4 2 4 0 -8 E -2 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 4 2 B 2 0 -4 -2 -6 C -12 4 0 -4 -6 D -4 2 4 0 -8 E -2 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 4 2 B 2 0 -4 -2 -6 C -12 4 0 -4 -6 D -4 2 4 0 -8 E -2 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5659: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) C D A E B (7) A C D B E (7) E B D C A (5) D C A B E (5) B E A D C (5) E C D B A (4) D C E B A (4) C D E A B (4) E B A D C (3) C A D E B (3) B E D C A (3) B A E D C (3) E D B C A (2) E B D A C (2) E A B C D (2) D C B A E (2) C D E B A (2) C D A B E (2) A E B C D (2) A D C B E (2) A C D E B (2) E D C B A (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C D B (1) D E C B A (1) D C E A B (1) D C B E A (1) D B E C A (1) D B C A E (1) C E D A B (1) B E D A C (1) B D A C E (1) B A D C E (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 -4 6 B -8 0 -2 -8 0 C 4 2 0 8 0 D 4 8 -8 0 0 E -6 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.766120 D: 0.000000 E: 0.233880 Sum of squares = 0.641639327538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.766120 D: 0.766120 E: 1.000000 A B C D E A 0 8 -4 -4 6 B -8 0 -2 -8 0 C 4 2 0 8 0 D 4 8 -8 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.52000000295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 C=19 D=16 B=14 so B is eliminated. Round 2 votes counts: E=32 A=32 C=19 D=17 so D is eliminated. Round 3 votes counts: E=34 C=33 A=33 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:203 D:202 E:197 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -4 6 B -8 0 -2 -8 0 C 4 2 0 8 0 D 4 8 -8 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.52000000295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -4 6 B -8 0 -2 -8 0 C 4 2 0 8 0 D 4 8 -8 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.52000000295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -4 6 B -8 0 -2 -8 0 C 4 2 0 8 0 D 4 8 -8 0 0 E -6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.400000 Sum of squares = 0.52000000295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5660: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) E B A D C (7) D C E A B (5) C D A B E (5) B A E C D (5) A B D C E (5) A B C D E (5) E D C B A (4) E C D B A (4) D C A E B (4) B A E D C (4) B A C E D (4) E B D A C (3) D E C A B (3) D C A B E (3) C D E A B (3) C A D B E (3) B E A C D (3) E B D C A (2) C E D A B (2) C A B D E (2) B E A D C (2) E D C A B (1) E D B C A (1) E D B A C (1) E C B D A (1) E B C A D (1) D E C B A (1) D A C B E (1) B A D E C (1) B A D C E (1) B A C D E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 10 12 -8 B 12 0 14 16 -6 C -10 -14 0 0 -8 D -12 -16 0 0 -10 E 8 6 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 10 12 -8 B 12 0 14 16 -6 C -10 -14 0 0 -8 D -12 -16 0 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=21 D=17 C=15 A=12 so A is eliminated. Round 2 votes counts: E=35 B=31 D=18 C=16 so C is eliminated. Round 3 votes counts: E=37 B=34 D=29 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:218 E:216 A:201 C:184 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 10 12 -8 B 12 0 14 16 -6 C -10 -14 0 0 -8 D -12 -16 0 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 12 -8 B 12 0 14 16 -6 C -10 -14 0 0 -8 D -12 -16 0 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 12 -8 B 12 0 14 16 -6 C -10 -14 0 0 -8 D -12 -16 0 0 -10 E 8 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5661: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) A D C E B (9) C D A B E (8) B E C D A (8) B E C A D (6) C B E D A (5) B E A D C (5) D C A E B (4) C D B E A (4) C D A E B (4) B E A C D (4) A E B D C (4) E B C A D (3) D C A B E (3) B C E D A (3) A D E B C (3) E A B D C (2) D A C E B (2) D A C B E (2) C D E B A (2) C D B A E (2) C B D E A (2) A D B E C (2) E B A C D (1) D C B A E (1) C E B D A (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -12 -2 -10 B 12 0 0 6 10 C 12 0 0 6 4 D 2 -6 -6 0 -2 E 10 -10 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.307127 C: 0.692873 D: 0.000000 E: 0.000000 Sum of squares = 0.57439967988 Cumulative probabilities = A: 0.000000 B: 0.307127 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -2 -10 B 12 0 0 6 10 C 12 0 0 6 4 D 2 -6 -6 0 -2 E 10 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=26 A=19 E=15 D=12 so D is eliminated. Round 2 votes counts: C=36 B=26 A=23 E=15 so E is eliminated. Round 3 votes counts: B=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:211 E:199 D:194 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -2 -10 B 12 0 0 6 10 C 12 0 0 6 4 D 2 -6 -6 0 -2 E 10 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -2 -10 B 12 0 0 6 10 C 12 0 0 6 4 D 2 -6 -6 0 -2 E 10 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -2 -10 B 12 0 0 6 10 C 12 0 0 6 4 D 2 -6 -6 0 -2 E 10 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5662: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) A D B C E (11) D B A E C (9) C E B D A (9) C E A B D (7) A C E D B (6) A D B E C (5) E C A D B (4) C E B A D (4) A E C D B (3) E C B A D (2) D B E A C (2) D B A C E (2) D A B E C (2) B D E C A (2) B D C E A (2) B D A E C (2) A C D B E (2) E D C B A (1) E D B A C (1) E C D B A (1) E C A B D (1) E B D C A (1) E A C D B (1) C B D E A (1) C B A E D (1) C A E B D (1) B D C A E (1) B D A C E (1) A D E C B (1) A C E B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 2 4 0 B 6 0 -16 -2 -10 C -2 16 0 12 2 D -4 2 -12 0 -10 E 0 10 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.560266 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.439734 Sum of squares = 0.507264004468 Cumulative probabilities = A: 0.560266 B: 0.560266 C: 0.560266 D: 0.560266 E: 1.000000 A B C D E A 0 -6 2 4 0 B 6 0 -16 -2 -10 C -2 16 0 12 2 D -4 2 -12 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500375 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499625 Sum of squares = 0.500000281672 Cumulative probabilities = A: 0.500375 B: 0.500375 C: 0.500375 D: 0.500375 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=23 C=23 D=15 B=8 so B is eliminated. Round 2 votes counts: A=31 E=23 D=23 C=23 so E is eliminated. Round 3 votes counts: C=42 A=32 D=26 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:214 E:209 A:200 B:189 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 4 0 B 6 0 -16 -2 -10 C -2 16 0 12 2 D -4 2 -12 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500375 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499625 Sum of squares = 0.500000281672 Cumulative probabilities = A: 0.500375 B: 0.500375 C: 0.500375 D: 0.500375 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 0 B 6 0 -16 -2 -10 C -2 16 0 12 2 D -4 2 -12 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500375 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499625 Sum of squares = 0.500000281672 Cumulative probabilities = A: 0.500375 B: 0.500375 C: 0.500375 D: 0.500375 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 0 B 6 0 -16 -2 -10 C -2 16 0 12 2 D -4 2 -12 0 -10 E 0 10 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500375 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499625 Sum of squares = 0.500000281672 Cumulative probabilities = A: 0.500375 B: 0.500375 C: 0.500375 D: 0.500375 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5663: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) C D B E A (6) A C D E B (5) C D A E B (4) C D A B E (4) C B E A D (4) C B D E A (4) C A D E B (4) B E D A C (4) B E C A D (4) A E D B C (4) E A C B D (3) D C A E B (3) D A B E C (3) C D B A E (3) B E A D C (3) B C E A D (3) E B A C D (2) E A B D C (2) E A B C D (2) D C A B E (2) D A E B C (2) C A E D B (2) B E A C D (2) B D E A C (2) B C E D A (2) A D C E B (2) D C B A E (1) D B C E A (1) D A C E B (1) D A C B E (1) C E B A D (1) C B E D A (1) B D E C A (1) B D C E A (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 8 -8 B 2 0 -10 -2 4 C 2 10 0 18 12 D -8 2 -18 0 2 E 8 -4 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 8 -8 B 2 0 -10 -2 4 C 2 10 0 18 12 D -8 2 -18 0 2 E 8 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=22 A=16 E=15 D=14 so D is eliminated. Round 2 votes counts: C=39 B=23 A=23 E=15 so E is eliminated. Round 3 votes counts: C=39 B=31 A=30 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 A:198 B:197 E:195 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 8 -8 B 2 0 -10 -2 4 C 2 10 0 18 12 D -8 2 -18 0 2 E 8 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 8 -8 B 2 0 -10 -2 4 C 2 10 0 18 12 D -8 2 -18 0 2 E 8 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 8 -8 B 2 0 -10 -2 4 C 2 10 0 18 12 D -8 2 -18 0 2 E 8 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5664: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) E D A C B (8) B C A E D (6) D B A C E (5) C E B A D (4) A C B E D (4) D E A C B (3) D E A B C (3) C B E A D (3) A B C E D (3) E D C B A (2) E C D B A (2) E C D A B (2) E C A B D (2) D E C B A (2) D A B E C (2) C E A B D (2) C B E D A (2) B C D E A (2) B C A D E (2) B A D C E (2) B A C D E (2) A E D C B (2) A D E C B (2) E D C A B (1) E C A D B (1) D E B C A (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) D A E C B (1) C E B D A (1) C A B E D (1) B D C A E (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D A E (1) A E C D B (1) A E C B D (1) A D E B C (1) A D B E C (1) A D B C E (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 10 8 B 10 0 -16 10 14 C 6 16 0 12 20 D -10 -10 -12 0 -22 E -8 -14 -20 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 10 8 B 10 0 -16 10 14 C 6 16 0 12 20 D -10 -10 -12 0 -22 E -8 -14 -20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 D=21 B=19 A=19 E=18 so E is eliminated. Round 2 votes counts: D=32 C=30 B=19 A=19 so B is eliminated. Round 3 votes counts: C=43 D=34 A=23 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:209 A:201 E:190 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 10 8 B 10 0 -16 10 14 C 6 16 0 12 20 D -10 -10 -12 0 -22 E -8 -14 -20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 10 8 B 10 0 -16 10 14 C 6 16 0 12 20 D -10 -10 -12 0 -22 E -8 -14 -20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 10 8 B 10 0 -16 10 14 C 6 16 0 12 20 D -10 -10 -12 0 -22 E -8 -14 -20 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5665: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (17) C B D A E (8) B C A D E (6) A E D B C (6) E A D C B (5) D A C E B (4) E D A C B (3) E A B D C (3) D C B A E (3) D A E C B (3) D A E B C (3) C B E D A (3) A D E B C (3) E C B A D (2) E B C A D (2) E A B C D (2) D C E A B (2) D B C A E (2) C B D E A (2) B A C D E (2) A B D E C (2) E C D A B (1) E B A C D (1) E A C D B (1) D E A C B (1) D C A E B (1) D B A C E (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B D A (1) C D E B A (1) C D B A E (1) B E C A D (1) B C E A D (1) B C D A E (1) B A E C D (1) A D B E C (1) Total count = 100 A B C D E A 0 22 24 12 2 B -22 0 14 -24 -24 C -24 -14 0 -26 -18 D -12 24 26 0 0 E -2 24 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 24 12 2 B -22 0 14 -24 -24 C -24 -14 0 -26 -18 D -12 24 26 0 0 E -2 24 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980465 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=23 C=16 B=12 A=12 so B is eliminated. Round 2 votes counts: E=38 C=24 D=23 A=15 so A is eliminated. Round 3 votes counts: E=45 D=29 C=26 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:230 E:220 D:219 B:172 C:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 24 12 2 B -22 0 14 -24 -24 C -24 -14 0 -26 -18 D -12 24 26 0 0 E -2 24 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980465 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 24 12 2 B -22 0 14 -24 -24 C -24 -14 0 -26 -18 D -12 24 26 0 0 E -2 24 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980465 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 24 12 2 B -22 0 14 -24 -24 C -24 -14 0 -26 -18 D -12 24 26 0 0 E -2 24 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980465 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5666: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) C E D A B (7) B C E D A (7) B C E A D (7) C E B D A (6) B A C E D (6) E C D A B (5) D E C A B (4) B A D C E (4) B A C D E (4) A D B E C (4) E D C A B (3) D A E B C (3) B D A E C (3) B C A E D (3) A B D C E (3) D E A C B (2) C E B A D (2) A D E C B (2) E D C B A (1) E C A D B (1) E A D C B (1) E A C D B (1) D B A E C (1) D A B E C (1) C E D B A (1) C E A D B (1) C B E D A (1) C B E A D (1) B D C E A (1) B A D E C (1) A D E B C (1) A C E D B (1) A C E B D (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 -6 -4 B -2 0 4 6 0 C 2 -4 0 14 16 D 6 -6 -14 0 -14 E 4 0 -16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999922 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -6 -4 B -2 0 4 6 0 C 2 -4 0 14 16 D 6 -6 -14 0 -14 E 4 0 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=19 D=18 A=15 E=12 so E is eliminated. Round 2 votes counts: B=36 C=25 D=22 A=17 so A is eliminated. Round 3 votes counts: B=42 D=30 C=28 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:214 B:204 E:201 A:195 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 -6 -4 B -2 0 4 6 0 C 2 -4 0 14 16 D 6 -6 -14 0 -14 E 4 0 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -6 -4 B -2 0 4 6 0 C 2 -4 0 14 16 D 6 -6 -14 0 -14 E 4 0 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -6 -4 B -2 0 4 6 0 C 2 -4 0 14 16 D 6 -6 -14 0 -14 E 4 0 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5667: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (14) D A E B C (10) C B E A D (10) E B C D A (9) D A E C B (6) B E C A D (6) E D B A C (4) E B C A D (4) D E A B C (4) D A C B E (4) A D E B C (4) E B D C A (3) B C E A D (3) A D B E C (3) D A C E B (2) C B A E D (2) A C D B E (2) E C B D A (1) E B A C D (1) E A D B C (1) C D B A E (1) C B E D A (1) C B A D E (1) C A D B E (1) B A E C D (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 16 10 6 B -6 0 10 -14 2 C -16 -10 0 -12 -16 D -10 14 12 0 6 E -6 -2 16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 10 6 B -6 0 10 -14 2 C -16 -10 0 -12 -16 D -10 14 12 0 6 E -6 -2 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 E=23 C=16 B=10 so B is eliminated. Round 2 votes counts: E=29 D=26 A=26 C=19 so C is eliminated. Round 3 votes counts: E=43 A=30 D=27 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:211 E:201 B:196 C:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 10 6 B -6 0 10 -14 2 C -16 -10 0 -12 -16 D -10 14 12 0 6 E -6 -2 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 10 6 B -6 0 10 -14 2 C -16 -10 0 -12 -16 D -10 14 12 0 6 E -6 -2 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 10 6 B -6 0 10 -14 2 C -16 -10 0 -12 -16 D -10 14 12 0 6 E -6 -2 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5668: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) B C D A E (5) E A D C B (4) D E A B C (4) B A D E C (4) B A C E D (4) A E B C D (4) A B E D C (4) A B E C D (4) C E D A B (3) B D C A E (3) A B C E D (3) E D C A B (2) E D A C B (2) E C A D B (2) E A D B C (2) D E B A C (2) D B C E A (2) C D B E A (2) C B D E A (2) C B D A E (2) C B A E D (2) B D A C E (2) B C A D E (2) B A D C E (2) E D A B C (1) E C D A B (1) E A C D B (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E A C (1) C E A D B (1) C E A B D (1) C B E A D (1) C B A D E (1) C A E B D (1) C A B E D (1) B A E D C (1) B A E C D (1) B A C D E (1) A E D B C (1) A E B D C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 2 2 2 B 6 0 12 8 4 C -2 -12 0 10 8 D -2 -8 -10 0 2 E -2 -4 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 2 2 B 6 0 12 8 4 C -2 -12 0 10 8 D -2 -8 -10 0 2 E -2 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 A=19 E=15 D=15 so E is eliminated. Round 2 votes counts: C=29 A=26 B=25 D=20 so D is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:215 C:202 A:200 E:192 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 2 2 B 6 0 12 8 4 C -2 -12 0 10 8 D -2 -8 -10 0 2 E -2 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 2 B 6 0 12 8 4 C -2 -12 0 10 8 D -2 -8 -10 0 2 E -2 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 2 B 6 0 12 8 4 C -2 -12 0 10 8 D -2 -8 -10 0 2 E -2 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5669: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) B D E A C (8) A C B D E (7) E B D C A (6) C A D E B (6) A C B E D (6) D C A E B (5) A C D B E (5) D E B C A (4) E D B C A (3) D B E A C (3) B A C E D (3) B A C D E (3) E D C A B (2) D A C B E (2) C A E D B (2) B E A C D (2) B A E C D (2) A C D E B (2) E D C B A (1) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) D E C A B (1) D C E A B (1) D B E C A (1) D A B C E (1) C E A B D (1) C D A E B (1) C A E B D (1) B E D C A (1) B D A E C (1) B D A C E (1) B A D C E (1) A D C B E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 20 -6 4 B 6 0 6 18 18 C -20 -6 0 -6 2 D 6 -18 6 0 8 E -4 -18 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 20 -6 4 B 6 0 6 18 18 C -20 -6 0 -6 2 D 6 -18 6 0 8 E -4 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 D=18 E=16 C=11 so C is eliminated. Round 2 votes counts: B=32 A=32 D=19 E=17 so E is eliminated. Round 3 votes counts: B=41 A=34 D=25 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:206 D:201 C:185 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 20 -6 4 B 6 0 6 18 18 C -20 -6 0 -6 2 D 6 -18 6 0 8 E -4 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 20 -6 4 B 6 0 6 18 18 C -20 -6 0 -6 2 D 6 -18 6 0 8 E -4 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 20 -6 4 B 6 0 6 18 18 C -20 -6 0 -6 2 D 6 -18 6 0 8 E -4 -18 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5670: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (7) A B E D C (6) B A C E D (5) E D B A C (4) E B D A C (4) D E C B A (4) D C E B A (4) D C E A B (4) E D C B A (3) E B A D C (3) D E C A B (3) D E A C B (3) C D B A E (3) C D A E B (3) B A E C D (3) A C D B E (3) A B C E D (3) E D B C A (2) E D A B C (2) E B D C A (2) D A C E B (2) C D B E A (2) C D A B E (2) C B A D E (2) C A D B E (2) C A B D E (2) B E D C A (2) B C A D E (2) A E D C B (2) A C B D E (2) E D A C B (1) E A D B C (1) C D E B A (1) B E A C D (1) B C D E A (1) B A E D C (1) B A C D E (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 12 -4 6 B 0 0 2 -4 0 C -12 -2 0 -8 -10 D 4 4 8 0 -6 E -6 0 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999995 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 A B C D E A 0 0 12 -4 6 B 0 0 2 -4 0 C -12 -2 0 -8 -10 D 4 4 8 0 -6 E -6 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998648 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=22 D=20 C=17 B=16 so B is eliminated. Round 2 votes counts: A=35 E=25 D=20 C=20 so D is eliminated. Round 3 votes counts: A=37 E=35 C=28 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 D:205 E:205 B:199 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 -4 6 B 0 0 2 -4 0 C -12 -2 0 -8 -10 D 4 4 8 0 -6 E -6 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998648 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -4 6 B 0 0 2 -4 0 C -12 -2 0 -8 -10 D 4 4 8 0 -6 E -6 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998648 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -4 6 B 0 0 2 -4 0 C -12 -2 0 -8 -10 D 4 4 8 0 -6 E -6 0 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998648 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5671: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) A E D C B (7) C B A E D (6) A C B E D (6) E A D B C (5) D E A B C (5) C B A D E (5) B C D E A (5) A C E B D (5) E D A B C (3) E A D C B (3) D E B C A (3) C A B D E (3) B C E D A (3) B C D A E (3) A D E C B (3) D E B A C (2) D C A B E (2) D B E C A (2) D A E C B (2) B D E C A (2) B C E A D (2) A D C E B (2) E D A C B (1) E B A D C (1) E B A C D (1) E A B D C (1) E A B C D (1) D E A C B (1) D C B E A (1) D A C B E (1) C D B A E (1) C B D A E (1) C A B E D (1) B E C A D (1) A E C D B (1) Total count = 100 A B C D E A 0 22 18 24 14 B -22 0 -20 10 -10 C -18 20 0 6 -6 D -24 -10 -6 0 -12 E -14 10 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 18 24 14 B -22 0 -20 10 -10 C -18 20 0 6 -6 D -24 -10 -6 0 -12 E -14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=19 C=17 E=16 B=16 so E is eliminated. Round 2 votes counts: A=42 D=23 B=18 C=17 so C is eliminated. Round 3 votes counts: A=46 B=30 D=24 so D is eliminated. Round 4 votes counts: A=61 B=39 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:239 E:207 C:201 B:179 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 18 24 14 B -22 0 -20 10 -10 C -18 20 0 6 -6 D -24 -10 -6 0 -12 E -14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 24 14 B -22 0 -20 10 -10 C -18 20 0 6 -6 D -24 -10 -6 0 -12 E -14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 24 14 B -22 0 -20 10 -10 C -18 20 0 6 -6 D -24 -10 -6 0 -12 E -14 10 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5672: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (13) A E B C D (11) E B C A D (10) C B D E A (9) E A B C D (7) B E C A D (6) A D E B C (6) C B E D A (5) D C B A E (4) C D B E A (4) B C E A D (4) A E D B C (4) E B A C D (3) B C E D A (3) D A C E B (2) D A C B E (2) E A B D C (1) D C E B A (1) D C A B E (1) D A E C B (1) A E B D C (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -24 -20 10 -32 B 24 0 12 20 4 C 20 -12 0 24 -2 D -10 -20 -24 0 -10 E 32 -4 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999619 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -20 10 -32 B 24 0 12 20 4 C 20 -12 0 24 -2 D -10 -20 -24 0 -10 E 32 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=21 C=18 B=13 so B is eliminated. Round 2 votes counts: E=27 C=25 D=24 A=24 so D is eliminated. Round 3 votes counts: C=44 A=29 E=27 so E is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:230 E:220 C:215 D:168 A:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -20 10 -32 B 24 0 12 20 4 C 20 -12 0 24 -2 D -10 -20 -24 0 -10 E 32 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -20 10 -32 B 24 0 12 20 4 C 20 -12 0 24 -2 D -10 -20 -24 0 -10 E 32 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -20 10 -32 B 24 0 12 20 4 C 20 -12 0 24 -2 D -10 -20 -24 0 -10 E 32 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5673: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (17) E D B C A (14) B D E A C (13) A C B D E (10) C E D B A (5) A B D E C (5) D B E C A (4) A C E B D (4) A C B E D (4) A B D C E (4) A B C D E (4) E C D B A (3) C A D B E (3) B D A E C (2) E D C B A (1) E B D A C (1) D E B C A (1) D B E A C (1) C E D A B (1) C E A D B (1) C A D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 10 -2 8 10 B -10 0 0 -4 2 C 2 0 0 6 8 D -8 4 -6 0 -4 E -10 -2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.101153 C: 0.898847 D: 0.000000 E: 0.000000 Sum of squares = 0.818157529877 Cumulative probabilities = A: 0.000000 B: 0.101153 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 8 10 B -10 0 0 -4 2 C 2 0 0 6 8 D -8 4 -6 0 -4 E -10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222223813 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 E=19 B=15 D=6 so D is eliminated. Round 2 votes counts: A=32 C=28 E=20 B=20 so E is eliminated. Round 3 votes counts: B=36 C=32 A=32 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:208 B:194 D:193 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 8 10 B -10 0 0 -4 2 C 2 0 0 6 8 D -8 4 -6 0 -4 E -10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222223813 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 8 10 B -10 0 0 -4 2 C 2 0 0 6 8 D -8 4 -6 0 -4 E -10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222223813 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 8 10 B -10 0 0 -4 2 C 2 0 0 6 8 D -8 4 -6 0 -4 E -10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222223813 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5674: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (6) E B C A D (5) E B A D C (5) C B E A D (5) B E C A D (4) A D B E C (4) E D A B C (3) E A D B C (3) D A E C B (3) D A E B C (3) C E D B A (3) B C A D E (3) B A C D E (3) A D E B C (3) A D B C E (3) E D C A B (2) E C B D A (2) D E C A B (2) D E A C B (2) D C A E B (2) D A C E B (2) D A C B E (2) C D E B A (2) C D B E A (2) C D B A E (2) B E A C D (2) B A C E D (2) A B D E C (2) E D A C B (1) E C D B A (1) E B A C D (1) E A B D C (1) D C A B E (1) D A B C E (1) C E D A B (1) C E B A D (1) C D E A B (1) C D A E B (1) C D A B E (1) C B E D A (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D B E (1) B E A D C (1) B A E D C (1) Total count = 100 A B C D E A 0 -10 -4 2 -16 B 10 0 0 -4 -12 C 4 0 0 6 -2 D -2 4 -6 0 -2 E 16 12 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 2 -16 B 10 0 0 -4 -12 C 4 0 0 6 -2 D -2 4 -6 0 -2 E 16 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=24 D=18 B=16 A=12 so A is eliminated. Round 2 votes counts: C=30 D=28 E=24 B=18 so B is eliminated. Round 3 votes counts: C=38 E=32 D=30 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:204 B:197 D:197 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 2 -16 B 10 0 0 -4 -12 C 4 0 0 6 -2 D -2 4 -6 0 -2 E 16 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 2 -16 B 10 0 0 -4 -12 C 4 0 0 6 -2 D -2 4 -6 0 -2 E 16 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 2 -16 B 10 0 0 -4 -12 C 4 0 0 6 -2 D -2 4 -6 0 -2 E 16 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5675: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (14) B C A D E (7) E D B A C (6) D E A B C (6) D E A C B (5) A C B E D (5) A C B D E (5) C B A E D (4) C A B E D (4) B C A E D (4) B C E A D (3) A C E D B (3) A C D B E (3) E D B C A (2) E D A B C (2) E C A B D (2) C B A D E (2) C A B D E (2) B C E D A (2) A D C B E (2) E C B A D (1) E B D C A (1) E A D C B (1) D E B C A (1) D E B A C (1) D B E C A (1) D B A C E (1) D A E C B (1) D A B E C (1) C E B A D (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D E A (1) B C D A E (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 16 16 2 -4 B -16 0 -14 -2 2 C -16 14 0 2 8 D -2 2 -2 0 -12 E 4 -2 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.42857142858 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 A B C D E A 0 16 16 2 -4 B -16 0 -14 -2 2 C -16 14 0 2 8 D -2 2 -2 0 -12 E 4 -2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428547 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=21 A=20 D=17 C=13 so C is eliminated. Round 2 votes counts: E=30 B=27 A=26 D=17 so D is eliminated. Round 3 votes counts: E=43 B=29 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:215 C:204 E:203 D:193 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 2 -4 B -16 0 -14 -2 2 C -16 14 0 2 8 D -2 2 -2 0 -12 E 4 -2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428547 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 2 -4 B -16 0 -14 -2 2 C -16 14 0 2 8 D -2 2 -2 0 -12 E 4 -2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428547 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 2 -4 B -16 0 -14 -2 2 C -16 14 0 2 8 D -2 2 -2 0 -12 E 4 -2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428547 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5676: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) A E D C B (6) C B E A D (5) B C D A E (5) C B D A E (4) A C D B E (4) E D B A C (3) E C A B D (3) E B D C A (3) E A D B C (3) D B A C E (3) D A E B C (3) C E B A D (3) C B E D A (3) B D C A E (3) E D B C A (2) E A D C B (2) E A C B D (2) D E A B C (2) D B E A C (2) D B A E C (2) C B A D E (2) C A E B D (2) B D C E A (2) B C D E A (2) A E C D B (2) A D E C B (2) A D E B C (2) A C E D B (2) E D A B C (1) E C B A D (1) D C B A E (1) D B C A E (1) D A B C E (1) C E A B D (1) C B D E A (1) C B A E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B C E D A (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 -12 -8 -2 B 18 0 0 10 -6 C 12 0 0 8 0 D 8 -10 -8 0 -12 E 2 6 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.479814 D: 0.000000 E: 0.520186 Sum of squares = 0.500814959936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.479814 D: 0.479814 E: 1.000000 A B C D E A 0 -18 -12 -8 -2 B 18 0 0 10 -6 C 12 0 0 8 0 D 8 -10 -8 0 -12 E 2 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=23 A=20 D=15 B=15 so D is eliminated. Round 2 votes counts: E=29 C=24 A=24 B=23 so B is eliminated. Round 3 votes counts: C=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:210 E:210 D:189 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -12 -8 -2 B 18 0 0 10 -6 C 12 0 0 8 0 D 8 -10 -8 0 -12 E 2 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -12 -8 -2 B 18 0 0 10 -6 C 12 0 0 8 0 D 8 -10 -8 0 -12 E 2 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -12 -8 -2 B 18 0 0 10 -6 C 12 0 0 8 0 D 8 -10 -8 0 -12 E 2 6 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5677: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (12) B E C D A (9) B C E D A (8) B C E A D (7) C B E A D (6) E A D C B (5) C B A D E (5) D A E C B (4) D A E B C (4) B C D A E (4) E D B A C (3) E D A B C (3) D E A B C (3) C B A E D (3) A E D C B (3) A C D E B (3) E A D B C (2) C A D B E (2) A D C E B (2) E C A D B (1) E B D C A (1) E B D A C (1) D A C E B (1) D A C B E (1) C B D A E (1) C A B E D (1) B D E A C (1) B C D E A (1) B C A E D (1) B C A D E (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -2 10 -2 B 4 0 -2 0 4 C 2 2 0 6 -4 D -10 0 -6 0 -8 E 2 -4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -4 -2 10 -2 B 4 0 -2 0 4 C 2 2 0 6 -4 D -10 0 -6 0 -8 E 2 -4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.360000000026 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=21 C=18 E=16 D=13 so D is eliminated. Round 2 votes counts: B=32 A=31 E=19 C=18 so C is eliminated. Round 3 votes counts: B=47 A=34 E=19 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:205 B:203 C:203 A:201 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 10 -2 B 4 0 -2 0 4 C 2 2 0 6 -4 D -10 0 -6 0 -8 E 2 -4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.360000000026 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 10 -2 B 4 0 -2 0 4 C 2 2 0 6 -4 D -10 0 -6 0 -8 E 2 -4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.360000000026 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 10 -2 B 4 0 -2 0 4 C 2 2 0 6 -4 D -10 0 -6 0 -8 E 2 -4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.200000 Sum of squares = 0.360000000026 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5678: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D A E C B (8) C B A E D (5) A C B D E (5) E B D C A (4) D E A B C (4) D A C B E (4) B C E A D (4) A C D B E (4) E D B C A (3) E A D C B (3) D C A B E (3) C A B D E (3) B C A E D (3) A D E C B (3) A D C B E (3) E B D A C (2) E B C D A (2) D E A C B (2) D C B A E (2) D B E C A (2) D A C E B (2) C B A D E (2) B E C D A (2) A D C E B (2) A C B E D (2) E D B A C (1) E A C B D (1) D E B C A (1) C B D A E (1) C A B E D (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D A E (1) B C A D E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -4 8 14 B -4 0 -16 6 2 C 4 16 0 0 0 D -8 -6 0 0 8 E -14 -2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.736182 D: 0.263818 E: 0.000000 Sum of squares = 0.611563681102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.736182 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 8 14 B -4 0 -16 6 2 C 4 16 0 0 0 D -8 -6 0 0 8 E -14 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555647353 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=25 A=21 B=14 C=12 so C is eliminated. Round 2 votes counts: D=28 E=25 A=25 B=22 so B is eliminated. Round 3 votes counts: A=36 E=33 D=31 so D is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:210 D:197 B:194 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 8 14 B -4 0 -16 6 2 C 4 16 0 0 0 D -8 -6 0 0 8 E -14 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555647353 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 8 14 B -4 0 -16 6 2 C 4 16 0 0 0 D -8 -6 0 0 8 E -14 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555647353 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 8 14 B -4 0 -16 6 2 C 4 16 0 0 0 D -8 -6 0 0 8 E -14 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555647353 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5679: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) C E A B D (7) A E D C B (7) C E B A D (6) A E C D B (6) A E C B D (5) D B A E C (4) D B A C E (4) A D E B C (4) A C E B D (4) D B C A E (3) C B E D A (3) B D E C A (3) A D E C B (3) E A C B D (2) D B E C A (2) D B C E A (2) D A B E C (2) D A B C E (2) C A E B D (2) B C D E A (2) A D C B E (2) A C E D B (2) E C B A D (1) E A C D B (1) D E B A C (1) D B E A C (1) D A E B C (1) C E B D A (1) C B E A D (1) C B A D E (1) B E C D A (1) B D C A E (1) B C E D A (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 6 12 10 B -4 0 -8 2 -6 C -6 8 0 -8 10 D -12 -2 8 0 0 E -10 6 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 12 10 B -4 0 -8 2 -6 C -6 8 0 -8 10 D -12 -2 8 0 0 E -10 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=22 C=21 B=18 E=4 so E is eliminated. Round 2 votes counts: A=38 D=22 C=22 B=18 so B is eliminated. Round 3 votes counts: A=38 D=36 C=26 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:202 D:197 E:193 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 12 10 B -4 0 -8 2 -6 C -6 8 0 -8 10 D -12 -2 8 0 0 E -10 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 12 10 B -4 0 -8 2 -6 C -6 8 0 -8 10 D -12 -2 8 0 0 E -10 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 12 10 B -4 0 -8 2 -6 C -6 8 0 -8 10 D -12 -2 8 0 0 E -10 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5680: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (12) E C D A B (11) B A D C E (9) E C D B A (7) E C B D A (7) E B C D A (6) D C E A B (6) A B D C E (5) B A E C D (4) A D B C E (4) D C A E B (3) B E C A D (3) B A E D C (3) A D C B E (3) B E C D A (2) B A D E C (2) A D C E B (2) E D C B A (1) E B C A D (1) D E C A B (1) D A C B E (1) D A B C E (1) C E A B D (1) C A D E B (1) B E D C A (1) B E A C D (1) B A C D E (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -2 -18 4 B -4 0 -14 -8 -20 C 2 14 0 -8 0 D 18 8 8 0 4 E -4 20 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -18 4 B -4 0 -14 -8 -20 C 2 14 0 -8 0 D 18 8 8 0 4 E -4 20 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=26 D=24 A=15 C=2 so C is eliminated. Round 2 votes counts: E=34 B=26 D=24 A=16 so A is eliminated. Round 3 votes counts: D=35 E=34 B=31 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:206 C:204 A:194 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -18 4 B -4 0 -14 -8 -20 C 2 14 0 -8 0 D 18 8 8 0 4 E -4 20 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -18 4 B -4 0 -14 -8 -20 C 2 14 0 -8 0 D 18 8 8 0 4 E -4 20 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -18 4 B -4 0 -14 -8 -20 C 2 14 0 -8 0 D 18 8 8 0 4 E -4 20 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5681: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) A B C D E (10) D B A E C (8) C E A B D (8) E C A B D (6) D B A C E (6) E D C B A (5) D E C B A (5) D E B A C (5) B A D E C (5) B A D C E (4) C A B D E (3) E D B C A (2) E C A D B (2) D B E A C (2) C E A D B (2) C A B E D (2) B D A C E (2) A C B E D (2) A B C E D (2) D E B C A (1) D C E B A (1) D B C A E (1) C D B A E (1) C D A B E (1) C A E B D (1) B A E D C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 0 2 B -2 0 0 -4 4 C 2 0 0 2 -4 D 0 4 -2 0 14 E -2 -4 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.100000 Sum of squares = 0.539999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.900000 E: 1.000000 A B C D E A 0 2 -2 0 2 B -2 0 0 -4 4 C 2 0 0 2 -4 D 0 4 -2 0 14 E -2 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.100000 Sum of squares = 0.539999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 C=18 A=16 B=12 so B is eliminated. Round 2 votes counts: D=31 A=26 E=25 C=18 so C is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 A:201 C:200 B:199 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 0 2 B -2 0 0 -4 4 C 2 0 0 2 -4 D 0 4 -2 0 14 E -2 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.100000 Sum of squares = 0.539999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.900000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 2 B -2 0 0 -4 4 C 2 0 0 2 -4 D 0 4 -2 0 14 E -2 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.100000 Sum of squares = 0.539999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.900000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 2 B -2 0 0 -4 4 C 2 0 0 2 -4 D 0 4 -2 0 14 E -2 -4 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.200000 E: 0.100000 Sum of squares = 0.539999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.700000 D: 0.900000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5682: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) E D C A B (6) E C D A B (6) B A C E D (5) B A C D E (5) C E D B A (4) B C E D A (4) B A D E C (4) A D E B C (4) A B D E C (4) D E A C B (3) C B E D A (3) C B E A D (3) C B A E D (3) A D E C B (3) E D C B A (2) D E C A B (2) D E B A C (2) D A E C B (2) D A B E C (2) C A E B D (2) B C A E D (2) A E D C B (2) A B D C E (2) E D A C B (1) E C A D B (1) D E A B C (1) D B E A C (1) D B A E C (1) D A E B C (1) C E D A B (1) C E B A D (1) C A B E D (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D A E (1) B A D C E (1) A C E D B (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 -6 -2 B 4 0 -10 8 -4 C 2 10 0 6 0 D 6 -8 -6 0 -14 E 2 4 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.497899 D: 0.000000 E: 0.502101 Sum of squares = 0.50000882745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.497899 D: 0.497899 E: 1.000000 A B C D E A 0 -4 -2 -6 -2 B 4 0 -10 8 -4 C 2 10 0 6 0 D 6 -8 -6 0 -14 E 2 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=25 B=25 A=19 E=16 D=15 so D is eliminated. Round 2 votes counts: B=27 C=25 E=24 A=24 so E is eliminated. Round 3 votes counts: C=42 B=29 A=29 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:210 C:209 B:199 A:193 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -6 -2 B 4 0 -10 8 -4 C 2 10 0 6 0 D 6 -8 -6 0 -14 E 2 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -6 -2 B 4 0 -10 8 -4 C 2 10 0 6 0 D 6 -8 -6 0 -14 E 2 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -6 -2 B 4 0 -10 8 -4 C 2 10 0 6 0 D 6 -8 -6 0 -14 E 2 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5683: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) D E A C B (8) E D C B A (7) E C B D A (6) E B C D A (6) B C E A D (6) A B D C E (6) E D B C A (4) D E C A B (4) D A E B C (4) B A C E D (4) A C B D E (4) E D B A C (3) D A E C B (3) D A C B E (3) C B A E D (3) C A B D E (3) A D C B E (3) A D B E C (3) D A B E C (2) C B E A D (2) B C A E D (2) E B D C A (1) D E A B C (1) B E C A D (1) B E A C D (1) A C D B E (1) Total count = 100 A B C D E A 0 8 10 -4 0 B -8 0 6 8 6 C -10 -6 0 -4 -8 D 4 -8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 -4 0 B -8 0 6 8 6 C -10 -6 0 -4 -8 D 4 -8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 D=25 B=14 C=8 so C is eliminated. Round 2 votes counts: A=29 E=27 D=25 B=19 so B is eliminated. Round 3 votes counts: A=38 E=37 D=25 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 B:206 D:204 E:197 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 -4 0 B -8 0 6 8 6 C -10 -6 0 -4 -8 D 4 -8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 -4 0 B -8 0 6 8 6 C -10 -6 0 -4 -8 D 4 -8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 -4 0 B -8 0 6 8 6 C -10 -6 0 -4 -8 D 4 -8 4 0 8 E 0 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5684: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (13) B E A D C (11) B E D A C (8) A B E C D (8) D E B C A (7) C A D E B (6) A B E D C (6) A C B E D (5) E B D A C (4) D C E B A (4) A B C E D (4) C D E A B (3) C A B D E (3) E D B A C (2) C D E B A (2) B A E C D (2) E D A B C (1) E B A D C (1) E A B D C (1) D E C A B (1) D C E A B (1) C D B E A (1) C D A B E (1) C A D B E (1) B E D C A (1) B C A D E (1) B A E D C (1) A C D E B (1) Total count = 100 A B C D E A 0 10 10 2 4 B -10 0 16 12 6 C -10 -16 0 2 -8 D -2 -12 -2 0 -10 E -4 -6 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 2 4 B -10 0 16 12 6 C -10 -16 0 2 -8 D -2 -12 -2 0 -10 E -4 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986319 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=24 A=24 D=13 E=9 so E is eliminated. Round 2 votes counts: C=30 B=29 A=25 D=16 so D is eliminated. Round 3 votes counts: B=38 C=36 A=26 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:212 E:204 D:187 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 2 4 B -10 0 16 12 6 C -10 -16 0 2 -8 D -2 -12 -2 0 -10 E -4 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986319 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 2 4 B -10 0 16 12 6 C -10 -16 0 2 -8 D -2 -12 -2 0 -10 E -4 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986319 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 2 4 B -10 0 16 12 6 C -10 -16 0 2 -8 D -2 -12 -2 0 -10 E -4 -6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986319 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5685: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) A B D E C (7) D C E A B (6) A D B E C (6) A D C E B (5) D E C A B (4) C D E A B (4) C B E A D (4) B A E C D (4) E D C B A (3) E C D B A (3) B E C A D (3) B E A C D (3) A B D C E (3) A B C D E (3) D E A B C (2) D C A E B (2) D A E C B (2) C E D B A (2) C E D A B (2) C B A E D (2) B E D C A (2) B E A D C (2) B C E A D (2) B A E D C (2) B A C E D (2) E B D C A (1) D E C B A (1) D E B A C (1) D A E B C (1) D A B E C (1) C A D B E (1) B D E A C (1) B C A E D (1) A C D E B (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -2 8 -8 B -6 0 -4 4 4 C 2 4 0 -4 2 D -8 -4 4 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.212766 B: 0.042553 C: 0.212766 D: 0.276596 E: 0.255319 Sum of squares = 0.234042553188 Cumulative probabilities = A: 0.212766 B: 0.255319 C: 0.468085 D: 0.744681 E: 1.000000 A B C D E A 0 6 -2 8 -8 B -6 0 -4 4 4 C 2 4 0 -4 2 D -8 -4 4 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.212766 B: 0.042553 C: 0.212766 D: 0.276596 E: 0.255319 Sum of squares = 0.23404255319 Cumulative probabilities = A: 0.212766 B: 0.255319 C: 0.468085 D: 0.744681 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=23 B=22 D=20 E=7 so E is eliminated. Round 2 votes counts: A=28 C=26 D=23 B=23 so D is eliminated. Round 3 votes counts: C=42 A=34 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:202 C:202 B:199 E:199 D:198 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 8 -8 B -6 0 -4 4 4 C 2 4 0 -4 2 D -8 -4 4 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.212766 B: 0.042553 C: 0.212766 D: 0.276596 E: 0.255319 Sum of squares = 0.23404255319 Cumulative probabilities = A: 0.212766 B: 0.255319 C: 0.468085 D: 0.744681 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 8 -8 B -6 0 -4 4 4 C 2 4 0 -4 2 D -8 -4 4 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.212766 B: 0.042553 C: 0.212766 D: 0.276596 E: 0.255319 Sum of squares = 0.23404255319 Cumulative probabilities = A: 0.212766 B: 0.255319 C: 0.468085 D: 0.744681 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 8 -8 B -6 0 -4 4 4 C 2 4 0 -4 2 D -8 -4 4 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.212766 B: 0.042553 C: 0.212766 D: 0.276596 E: 0.255319 Sum of squares = 0.23404255319 Cumulative probabilities = A: 0.212766 B: 0.255319 C: 0.468085 D: 0.744681 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5686: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) B D E A C (8) A C D E B (8) C A E D B (7) E D C A B (5) B E D C A (5) D E B A C (4) D A E B C (4) A D C E B (4) A C B D E (4) E B D C A (3) B E C D A (3) B C A D E (3) E D B A C (2) E B C D A (2) D B A E C (2) C B A E D (2) C A B D E (2) A D B C E (2) E D B C A (1) E D A C B (1) E C D A B (1) E C B D A (1) E C A D B (1) D E A C B (1) D A E C B (1) C E A D B (1) C E A B D (1) C B E A D (1) C A D E B (1) B D E C A (1) B D A C E (1) B C E A D (1) B C A E D (1) B A D C E (1) A E C D B (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 16 -4 8 14 B -16 0 -10 -2 -4 C 4 10 0 2 2 D -8 2 -2 0 2 E -14 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 8 14 B -16 0 -10 -2 -4 C 4 10 0 2 2 D -8 2 -2 0 2 E -14 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 A=23 E=17 D=12 so D is eliminated. Round 2 votes counts: A=28 B=26 C=24 E=22 so E is eliminated. Round 3 votes counts: B=38 C=32 A=30 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:217 C:209 D:197 E:193 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 8 14 B -16 0 -10 -2 -4 C 4 10 0 2 2 D -8 2 -2 0 2 E -14 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 8 14 B -16 0 -10 -2 -4 C 4 10 0 2 2 D -8 2 -2 0 2 E -14 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 8 14 B -16 0 -10 -2 -4 C 4 10 0 2 2 D -8 2 -2 0 2 E -14 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5687: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (18) C B A D E (9) A B D E C (8) C A B D E (6) C E D B A (5) C B D E A (4) A D E B C (4) E D C A B (3) C E D A B (3) C A B E D (3) A D B E C (3) E D B C A (2) E D B A C (2) E C D B A (2) E C D A B (2) C E B D A (2) C E A D B (2) B C A D E (2) B A D E C (2) B A D C E (2) B A C D E (2) A B C D E (2) E D A C B (1) E A D C B (1) D E A B C (1) D A E B C (1) C E A B D (1) C B E D A (1) C B A E D (1) C A E D B (1) C A E B D (1) B D E A C (1) B D C E A (1) A E D B C (1) Total count = 100 A B C D E A 0 24 -2 2 -4 B -24 0 4 -4 -6 C 2 -4 0 -2 -4 D -2 4 2 0 -4 E 4 6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 -2 2 -4 B -24 0 4 -4 -6 C 2 -4 0 -2 -4 D -2 4 2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=31 A=18 B=10 D=2 so D is eliminated. Round 2 votes counts: C=39 E=32 A=19 B=10 so B is eliminated. Round 3 votes counts: C=42 E=33 A=25 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:210 E:209 D:200 C:196 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 -2 2 -4 B -24 0 4 -4 -6 C 2 -4 0 -2 -4 D -2 4 2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -2 2 -4 B -24 0 4 -4 -6 C 2 -4 0 -2 -4 D -2 4 2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -2 2 -4 B -24 0 4 -4 -6 C 2 -4 0 -2 -4 D -2 4 2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5688: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) C A D B E (7) E B D A C (5) E D B A C (4) E B C A D (4) D E C B A (4) D E B A C (4) C D A E B (4) D A B E C (3) C E D A B (3) B E A D C (3) B E A C D (3) A B C E D (3) E B D C A (2) E B A D C (2) D C A B E (2) D B E A C (2) D A C B E (2) C E B A D (2) C D E A B (2) C A B E D (2) C A B D E (2) B A E D C (2) B A E C D (2) A D B C E (2) A C B D E (2) A B D C E (2) E D C B A (1) E C B D A (1) E B A C D (1) D E B C A (1) D C E A B (1) D B A E C (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D A C (1) A D C B E (1) A C D B E (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -2 0 4 B -6 0 -2 -10 -4 C 2 2 0 -6 6 D 0 10 6 0 6 E -4 4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.432562 B: 0.000000 C: 0.000000 D: 0.567438 E: 0.000000 Sum of squares = 0.509095670841 Cumulative probabilities = A: 0.432562 B: 0.432562 C: 0.432562 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 0 4 B -6 0 -2 -10 -4 C 2 2 0 -6 6 D 0 10 6 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=27 E=20 A=14 B=11 so B is eliminated. Round 2 votes counts: C=28 E=27 D=27 A=18 so A is eliminated. Round 3 votes counts: C=36 D=33 E=31 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:204 C:202 E:194 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 0 4 B -6 0 -2 -10 -4 C 2 2 0 -6 6 D 0 10 6 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 0 4 B -6 0 -2 -10 -4 C 2 2 0 -6 6 D 0 10 6 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 0 4 B -6 0 -2 -10 -4 C 2 2 0 -6 6 D 0 10 6 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5689: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E D A B C (10) C E D A B (10) C B E A D (8) C B A D E (6) B C A D E (6) D A E B C (5) C E B D A (5) B E A D C (4) A D B E C (4) E A D B C (3) B C E A D (3) B A D E C (3) B A D C E (3) E C D A B (2) C D A E B (2) C D A B E (2) C B E D A (2) C B D A E (2) A E D B C (2) E B A D C (1) E A B D C (1) D A C E B (1) D A C B E (1) C D B A E (1) B E A C D (1) B C A E D (1) B A E D C (1) Total count = 100 A B C D E A 0 6 0 -6 -20 B -6 0 -4 -6 -4 C 0 4 0 2 6 D 6 6 -2 0 -28 E 20 4 -6 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.121041 B: 0.000000 C: 0.878959 D: 0.000000 E: 0.000000 Sum of squares = 0.787220484643 Cumulative probabilities = A: 0.121041 B: 0.121041 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -6 -20 B -6 0 -4 -6 -4 C 0 4 0 2 6 D 6 6 -2 0 -28 E 20 4 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.644970469395 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=27 B=22 D=7 A=6 so A is eliminated. Round 2 votes counts: C=38 E=29 B=22 D=11 so D is eliminated. Round 3 votes counts: C=40 E=34 B=26 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:206 D:191 A:190 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 -6 -20 B -6 0 -4 -6 -4 C 0 4 0 2 6 D 6 6 -2 0 -28 E 20 4 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.644970469395 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -6 -20 B -6 0 -4 -6 -4 C 0 4 0 2 6 D 6 6 -2 0 -28 E 20 4 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.644970469395 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -6 -20 B -6 0 -4 -6 -4 C 0 4 0 2 6 D 6 6 -2 0 -28 E 20 4 -6 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.769231 D: 0.000000 E: 0.000000 Sum of squares = 0.644970469395 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5690: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) E C A B D (7) D A B C E (7) A D B E C (7) B E C D A (5) B D A E C (5) A D B C E (5) E B C A D (4) C E B D A (4) B E A C D (4) A B D E C (4) E C B D A (3) C E A D B (3) B D A C E (3) D A C B E (2) C A E D B (2) B D C E A (2) A E B C D (2) A D E B C (2) A D C E B (2) E C B A D (1) E C A D B (1) E B A C D (1) D C E B A (1) D C E A B (1) D C B A E (1) D B A C E (1) C B D E A (1) B E A D C (1) B C E D A (1) B C D E A (1) B A E D C (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B D C (1) A D C B E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 8 8 2 B -2 0 22 10 10 C -8 -22 0 0 -8 D -8 -10 0 0 -6 E -2 -10 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 8 2 B -2 0 22 10 10 C -8 -22 0 0 -8 D -8 -10 0 0 -6 E -2 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=24 C=18 E=17 D=13 so D is eliminated. Round 2 votes counts: A=37 B=25 C=21 E=17 so E is eliminated. Round 3 votes counts: A=37 C=33 B=30 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:210 E:201 D:188 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 8 2 B -2 0 22 10 10 C -8 -22 0 0 -8 D -8 -10 0 0 -6 E -2 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 8 2 B -2 0 22 10 10 C -8 -22 0 0 -8 D -8 -10 0 0 -6 E -2 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 8 2 B -2 0 22 10 10 C -8 -22 0 0 -8 D -8 -10 0 0 -6 E -2 -10 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5691: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (13) A D E C B (12) C B E D A (8) B C A E D (8) B C E D A (6) A D B E C (5) A B D C E (5) C E B D A (4) B A D C E (4) E D C A B (3) D E A C B (3) D B E A C (2) C B E A D (2) C B A E D (2) C A B D E (2) B A D E C (2) B A C D E (2) A D C E B (2) A B D E C (2) E D C B A (1) E C D B A (1) D E A B C (1) D A E C B (1) C E D B A (1) C E D A B (1) C A E D B (1) C A D E B (1) B E D C A (1) B D E A C (1) B C E A D (1) B C A D E (1) A D B C E (1) Total count = 100 A B C D E A 0 6 12 32 28 B -6 0 10 2 10 C -12 -10 0 -18 4 D -32 -2 18 0 20 E -28 -10 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 32 28 B -6 0 10 2 10 C -12 -10 0 -18 4 D -32 -2 18 0 20 E -28 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=26 C=22 D=7 E=5 so E is eliminated. Round 2 votes counts: A=40 B=26 C=23 D=11 so D is eliminated. Round 3 votes counts: A=45 B=28 C=27 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:239 B:208 D:202 C:182 E:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 32 28 B -6 0 10 2 10 C -12 -10 0 -18 4 D -32 -2 18 0 20 E -28 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 32 28 B -6 0 10 2 10 C -12 -10 0 -18 4 D -32 -2 18 0 20 E -28 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 32 28 B -6 0 10 2 10 C -12 -10 0 -18 4 D -32 -2 18 0 20 E -28 -10 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5692: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) E A B D C (6) D C B A E (6) C D E A B (6) C B D E A (5) C D B E A (4) E A B C D (3) D B A C E (3) C D B A E (3) C B E A D (3) C B D A E (3) A E D B C (3) E C A D B (2) E A D C B (2) E A D B C (2) E A C D B (2) D C A E B (2) D C A B E (2) D B C A E (2) D A E C B (2) D A E B C (2) C E B D A (2) B D C A E (2) B D A C E (2) E C A B D (1) E B A C D (1) D E A C B (1) D A C E B (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B A D (1) C E A D B (1) C E A B D (1) C D E B A (1) C B E D A (1) B E C A D (1) B E A D C (1) B E A C D (1) B D A E C (1) B C E A D (1) B C D E A (1) B C A E D (1) B A E D C (1) B A E C D (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -24 -26 -4 B 16 0 -8 0 12 C 24 8 0 12 30 D 26 0 -12 0 20 E 4 -12 -30 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -24 -26 -4 B 16 0 -8 0 12 C 24 8 0 12 30 D 26 0 -12 0 20 E 4 -12 -30 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 B=22 E=19 A=4 so A is eliminated. Round 2 votes counts: C=33 B=23 E=22 D=22 so E is eliminated. Round 3 votes counts: C=38 B=33 D=29 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:237 D:217 B:210 E:171 A:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -24 -26 -4 B 16 0 -8 0 12 C 24 8 0 12 30 D 26 0 -12 0 20 E 4 -12 -30 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -24 -26 -4 B 16 0 -8 0 12 C 24 8 0 12 30 D 26 0 -12 0 20 E 4 -12 -30 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -24 -26 -4 B 16 0 -8 0 12 C 24 8 0 12 30 D 26 0 -12 0 20 E 4 -12 -30 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5693: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) C D B A E (10) D C B A E (9) E B A D C (6) C D A B E (5) E A B C D (4) C A E D B (4) B D E A C (4) A E B D C (4) D C B E A (3) C E D B A (3) C E D A B (3) C D E B A (3) C A D B E (3) E B D A C (2) E B A C D (2) D B C E A (2) D B C A E (2) C A D E B (2) B E A D C (2) A E B C D (2) A C B D E (2) A B E D C (2) A B D C E (2) E D C B A (1) E C D B A (1) E B D C A (1) E A C B D (1) D B E C A (1) D A B C E (1) C D E A B (1) C D B E A (1) B A D E C (1) Total count = 100 A B C D E A 0 -8 -10 -6 -2 B 8 0 -4 -10 0 C 10 4 0 -6 12 D 6 10 6 0 4 E 2 0 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 -2 B 8 0 -4 -10 0 C 10 4 0 -6 12 D 6 10 6 0 4 E 2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=28 D=18 A=12 B=7 so B is eliminated. Round 2 votes counts: C=35 E=30 D=22 A=13 so A is eliminated. Round 3 votes counts: E=38 C=37 D=25 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:210 B:197 E:193 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 -2 B 8 0 -4 -10 0 C 10 4 0 -6 12 D 6 10 6 0 4 E 2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 -2 B 8 0 -4 -10 0 C 10 4 0 -6 12 D 6 10 6 0 4 E 2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 -2 B 8 0 -4 -10 0 C 10 4 0 -6 12 D 6 10 6 0 4 E 2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5694: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C D A B E (7) C A D B E (6) E A B C D (5) D E B C A (5) A C B E D (5) D B E C A (4) C A B D E (4) A B C E D (4) E D B C A (3) D C B A E (3) D B E A C (3) D B A E C (3) C A D E B (3) A C E B D (3) E B A D C (2) E B A C D (2) D E C B A (2) D C A B E (2) C E D A B (2) C E A D B (2) C D E A B (2) C A E D B (2) C A E B D (2) B E A D C (2) B A E D C (2) E D C B A (1) E C A D B (1) D C E B A (1) D C E A B (1) D B C E A (1) C D A E B (1) B E D A C (1) B D E A C (1) B D A E C (1) A E B C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -10 -4 2 B -10 0 0 -10 2 C 10 0 0 6 4 D 4 10 -6 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.263869 C: 0.736131 D: 0.000000 E: 0.000000 Sum of squares = 0.61151540469 Cumulative probabilities = A: 0.000000 B: 0.263869 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -10 -4 2 B -10 0 0 -10 2 C 10 0 0 6 4 D 4 10 -6 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250002262 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 E=22 A=15 B=7 so B is eliminated. Round 2 votes counts: C=31 D=27 E=25 A=17 so A is eliminated. Round 3 votes counts: C=43 E=29 D=28 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:205 A:199 E:195 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 -4 2 B -10 0 0 -10 2 C 10 0 0 6 4 D 4 10 -6 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250002262 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 -4 2 B -10 0 0 -10 2 C 10 0 0 6 4 D 4 10 -6 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250002262 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 -4 2 B -10 0 0 -10 2 C 10 0 0 6 4 D 4 10 -6 0 2 E -2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250002262 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5695: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) B A C E D (9) E D C B A (8) E D B C A (5) C A D B E (5) B E A C D (5) B A E C D (5) C A B E D (4) E D B A C (3) E B D A C (3) D E C B A (3) D C E A B (3) C A B D E (3) A C B D E (3) A B C E D (3) E B A D C (2) D E B C A (2) D E B A C (2) D C A B E (2) E C D B A (1) E B D C A (1) E B C D A (1) E B C A D (1) D E A B C (1) D C A E B (1) D A C B E (1) C E D A B (1) C D A E B (1) C B E A D (1) B E D A C (1) B E A D C (1) B C E A D (1) B A E D C (1) B A D E C (1) A D C B E (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -10 -2 -14 B 14 0 0 -4 0 C 10 0 0 -8 -16 D 2 4 8 0 -14 E 14 0 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.491227 C: 0.000000 D: 0.000000 E: 0.508773 Sum of squares = 0.500153935242 Cumulative probabilities = A: 0.000000 B: 0.491227 C: 0.491227 D: 0.491227 E: 1.000000 A B C D E A 0 -14 -10 -2 -14 B 14 0 0 -4 0 C 10 0 0 -8 -16 D 2 4 8 0 -14 E 14 0 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 B=24 C=15 A=10 so A is eliminated. Round 2 votes counts: B=29 D=27 E=25 C=19 so C is eliminated. Round 3 votes counts: B=40 D=34 E=26 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:222 B:205 D:200 C:193 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -10 -2 -14 B 14 0 0 -4 0 C 10 0 0 -8 -16 D 2 4 8 0 -14 E 14 0 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -2 -14 B 14 0 0 -4 0 C 10 0 0 -8 -16 D 2 4 8 0 -14 E 14 0 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -2 -14 B 14 0 0 -4 0 C 10 0 0 -8 -16 D 2 4 8 0 -14 E 14 0 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5696: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) E D C A B (5) A E B C D (5) D C E B A (4) C D B E A (4) C B D E A (4) A B E C D (4) E A D C B (3) D E C A B (3) D C B E A (3) D C B A E (3) B E C A D (3) B C E D A (3) B C D A E (3) B C A D E (3) A E D C B (3) A E D B C (3) A D E C B (3) A B E D C (3) A B D C E (3) E D C B A (2) E D A C B (2) D A C E B (2) B C D E A (2) B A C D E (2) A D E B C (2) E C D B A (1) E B C A D (1) E A C D B (1) D C E A B (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) C E D B A (1) C D E B A (1) C B E D A (1) B E A C D (1) B C E A D (1) B C A E D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -6 2 -8 B -8 0 2 -4 -2 C 6 -2 0 0 -4 D -2 4 0 0 -4 E 8 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -6 2 -8 B -8 0 2 -4 -2 C 6 -2 0 0 -4 D -2 4 0 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 D=20 B=19 C=11 so C is eliminated. Round 2 votes counts: A=28 D=25 B=24 E=23 so E is eliminated. Round 3 votes counts: A=39 D=36 B=25 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:209 C:200 D:199 A:198 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 2 -8 B -8 0 2 -4 -2 C 6 -2 0 0 -4 D -2 4 0 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 2 -8 B -8 0 2 -4 -2 C 6 -2 0 0 -4 D -2 4 0 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 2 -8 B -8 0 2 -4 -2 C 6 -2 0 0 -4 D -2 4 0 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5697: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (12) C B E D A (8) D A B C E (7) B C E D A (5) A D E C B (5) E A C B D (4) D B A C E (4) D A C B E (4) B C D A E (4) A E D C B (4) A D E B C (4) E C A B D (3) D C B A E (3) C E B A D (3) C B D E A (3) B D C A E (3) B C D E A (3) A E D B C (3) E A D C B (2) D B C A E (2) D A B E C (2) C E B D A (2) C B D A E (2) E B C A D (1) E B A D C (1) E A D B C (1) C D B A E (1) C A D E B (1) B E C D A (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -16 -14 -8 2 B 16 0 -18 10 6 C 14 18 0 6 14 D 8 -10 -6 0 0 E -2 -6 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -8 2 B 16 0 -18 10 6 C 14 18 0 6 14 D 8 -10 -6 0 0 E -2 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 C=20 A=18 B=16 so B is eliminated. Round 2 votes counts: C=32 E=25 D=25 A=18 so A is eliminated. Round 3 votes counts: D=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:207 D:196 E:189 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -14 -8 2 B 16 0 -18 10 6 C 14 18 0 6 14 D 8 -10 -6 0 0 E -2 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -8 2 B 16 0 -18 10 6 C 14 18 0 6 14 D 8 -10 -6 0 0 E -2 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -8 2 B 16 0 -18 10 6 C 14 18 0 6 14 D 8 -10 -6 0 0 E -2 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) D C A B E (11) B E A C D (11) E B A C D (10) E D C A B (7) A B C D E (7) D C A E B (5) B A E C D (5) E D C B A (4) E B A D C (4) D C E A B (4) A C D B E (4) B A C D E (3) E B D A C (2) C D B A E (2) C A D B E (2) E B D C A (1) E B C D A (1) D E C A B (1) D C E B A (1) B C A D E (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -4 -2 8 B -8 0 -8 -6 20 C 4 8 0 20 8 D 2 6 -20 0 8 E -8 -20 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -2 8 B -8 0 -8 -6 20 C 4 8 0 20 8 D 2 6 -20 0 8 E -8 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=22 B=21 C=16 A=12 so A is eliminated. Round 2 votes counts: E=29 B=28 D=22 C=21 so C is eliminated. Round 3 votes counts: D=42 E=29 B=29 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:220 A:205 B:199 D:198 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -2 8 B -8 0 -8 -6 20 C 4 8 0 20 8 D 2 6 -20 0 8 E -8 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -2 8 B -8 0 -8 -6 20 C 4 8 0 20 8 D 2 6 -20 0 8 E -8 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -2 8 B -8 0 -8 -6 20 C 4 8 0 20 8 D 2 6 -20 0 8 E -8 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5699: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) D C A B E (9) A B E D C (7) C D A B E (5) B E A D C (5) E C B D A (4) E B A C D (4) D C B A E (4) C E D B A (4) B A E D C (4) A D B C E (4) E C D B A (3) E A B C D (3) C D E B A (3) A E B C D (3) E B C D A (2) D B C A E (2) C D E A B (2) C A D E B (2) A E B D C (2) A D C B E (2) E C A D B (1) E C A B D (1) E B D C A (1) E B C A D (1) E A C B D (1) E A B D C (1) D C B E A (1) D A C B E (1) C E D A B (1) C D B E A (1) C A E D B (1) B D A E C (1) B D A C E (1) B A D E C (1) A E C B D (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 4 10 4 B 2 0 4 6 -2 C -4 -4 0 -12 -10 D -10 -6 12 0 -18 E -4 2 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 4 10 4 B 2 0 4 6 -2 C -4 -4 0 -12 -10 D -10 -6 12 0 -18 E -4 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999935 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=21 C=19 D=17 B=12 so B is eliminated. Round 2 votes counts: E=36 A=26 D=19 C=19 so D is eliminated. Round 3 votes counts: E=36 C=35 A=29 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:208 B:205 D:189 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 10 4 B 2 0 4 6 -2 C -4 -4 0 -12 -10 D -10 -6 12 0 -18 E -4 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999935 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 4 B 2 0 4 6 -2 C -4 -4 0 -12 -10 D -10 -6 12 0 -18 E -4 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999935 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 4 B 2 0 4 6 -2 C -4 -4 0 -12 -10 D -10 -6 12 0 -18 E -4 2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999935 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5700: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (6) C D E A B (6) C D B E A (6) E A B D C (5) D A E C B (5) D A C E B (5) C B D E A (5) E B A C D (4) D A B E C (4) C D B A E (4) B E A D C (4) D C A B E (3) C E B A D (3) B E C A D (3) B E A C D (3) B A E D C (3) E B A D C (2) C E B D A (2) C D A B E (2) C B E A D (2) B A D E C (2) A E B D C (2) A B D E C (2) E C B A D (1) E A D C B (1) E A D B C (1) E A C D B (1) D C B A E (1) D B A E C (1) D A C B E (1) C E D B A (1) C E D A B (1) C E A B D (1) C D A E B (1) C B D A E (1) B D A E C (1) B D A C E (1) B C D E A (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 0 -16 -6 B 2 0 -20 -4 0 C 0 20 0 -4 6 D 16 4 4 0 20 E 6 0 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -16 -6 B 2 0 -20 -4 0 C 0 20 0 -4 6 D 16 4 4 0 20 E 6 0 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=26 B=18 E=15 A=6 so A is eliminated. Round 2 votes counts: C=35 D=28 B=20 E=17 so E is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: C=60 B=40 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:222 C:211 E:190 B:189 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -16 -6 B 2 0 -20 -4 0 C 0 20 0 -4 6 D 16 4 4 0 20 E 6 0 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -16 -6 B 2 0 -20 -4 0 C 0 20 0 -4 6 D 16 4 4 0 20 E 6 0 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -16 -6 B 2 0 -20 -4 0 C 0 20 0 -4 6 D 16 4 4 0 20 E 6 0 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5701: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (11) D A E C B (9) A D E B C (8) E C B D A (7) D E C B A (7) B C A E D (7) D E A C B (6) A D B C E (5) A B C E D (5) A D B E C (4) A B C D E (4) C E B D A (3) C B E D A (3) E D C B A (2) E C D B A (2) C B E A D (2) A B E C D (2) E D C A B (1) D E C A B (1) D C E B A (1) D C B E A (1) D A E B C (1) D A B E C (1) B E C A D (1) B C E D A (1) B C A D E (1) B A C E D (1) A E B C D (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 8 2 B 0 0 8 0 0 C 2 -8 0 2 -8 D -8 0 -2 0 2 E -2 0 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.471132 B: 0.528868 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501666722105 Cumulative probabilities = A: 0.471132 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 8 2 B 0 0 8 0 0 C 2 -8 0 2 -8 D -8 0 -2 0 2 E -2 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=27 B=22 E=12 C=8 so C is eliminated. Round 2 votes counts: A=31 D=27 B=27 E=15 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:204 B:204 E:202 D:196 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 8 2 B 0 0 8 0 0 C 2 -8 0 2 -8 D -8 0 -2 0 2 E -2 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 8 2 B 0 0 8 0 0 C 2 -8 0 2 -8 D -8 0 -2 0 2 E -2 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 8 2 B 0 0 8 0 0 C 2 -8 0 2 -8 D -8 0 -2 0 2 E -2 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5702: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) E A B C D (7) A E B C D (7) E A D C B (6) D C B E A (5) C D B E A (4) B C D A E (4) E D C B A (3) E C D B A (3) D C E B A (3) C B E D A (3) C B D E A (3) B A C D E (3) A D B E C (3) E D C A B (2) E A C D B (2) D E C B A (2) D E A C B (2) D C E A B (2) D B A C E (2) B D C A E (2) B C A D E (2) B A C E D (2) A B E D C (2) A B D C E (2) E C D A B (1) E C A B D (1) E A C B D (1) D E C A B (1) D C B A E (1) D B C A E (1) D A E B C (1) C E D B A (1) C D E B A (1) B E A C D (1) B A E C D (1) B A D C E (1) A E D B C (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 10 6 -8 B -4 0 6 6 6 C -10 -6 0 14 -12 D -6 -6 -14 0 -6 E 8 -6 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 4 10 6 -8 B -4 0 6 6 6 C -10 -6 0 14 -12 D -6 -6 -14 0 -6 E 8 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 A=26 D=20 B=16 C=12 so C is eliminated. Round 2 votes counts: E=27 A=26 D=25 B=22 so B is eliminated. Round 3 votes counts: A=35 D=34 E=31 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:210 B:207 A:206 C:193 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 10 6 -8 B -4 0 6 6 6 C -10 -6 0 14 -12 D -6 -6 -14 0 -6 E 8 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 6 -8 B -4 0 6 6 6 C -10 -6 0 14 -12 D -6 -6 -14 0 -6 E 8 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 6 -8 B -4 0 6 6 6 C -10 -6 0 14 -12 D -6 -6 -14 0 -6 E 8 -6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5703: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (13) D E C A B (9) B A E C D (9) D C A B E (8) B A C E D (6) E B A D C (5) C A B D E (5) A B C D E (5) E D B A C (4) D C A E B (4) E D C B A (3) D E C B A (3) C D A B E (3) B E A C D (3) A C B D E (3) D C E A B (2) C E B A D (2) E D B C A (1) E B D A C (1) E B C A D (1) D E B C A (1) D B A E C (1) D A C B E (1) D A B E C (1) C D E A B (1) C B A E D (1) C A D B E (1) B A E D C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 10 14 2 B 10 0 8 14 0 C -10 -8 0 8 -14 D -14 -14 -8 0 -4 E -2 0 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.585500 C: 0.000000 D: 0.000000 E: 0.414500 Sum of squares = 0.51462057971 Cumulative probabilities = A: 0.000000 B: 0.585500 C: 0.585500 D: 0.585500 E: 1.000000 A B C D E A 0 -10 10 14 2 B 10 0 8 14 0 C -10 -8 0 8 -14 D -14 -14 -8 0 -4 E -2 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=28 B=19 C=13 A=10 so A is eliminated. Round 2 votes counts: D=30 E=28 B=26 C=16 so C is eliminated. Round 3 votes counts: D=35 B=35 E=30 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:208 E:208 C:188 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 14 2 B 10 0 8 14 0 C -10 -8 0 8 -14 D -14 -14 -8 0 -4 E -2 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 14 2 B 10 0 8 14 0 C -10 -8 0 8 -14 D -14 -14 -8 0 -4 E -2 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 14 2 B 10 0 8 14 0 C -10 -8 0 8 -14 D -14 -14 -8 0 -4 E -2 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5704: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) A E D B C (6) C D E A B (5) C B D A E (5) B C E A D (5) B C D A E (5) E D A C B (4) B E A C D (4) A D E B C (4) E D C A B (3) E A D C B (3) B C A D E (3) A E B D C (3) A D B E C (3) A B E D C (3) E C D A B (2) E C B A D (2) D E A C B (2) D A E C B (2) D A E B C (2) C E D A B (2) C D B A E (2) B C E D A (2) B A D E C (2) B A C D E (2) E B A C D (1) E A D B C (1) E A B C D (1) D C B A E (1) D C A E B (1) D A C E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C D B E A (1) C D A E B (1) C B D E A (1) B E C A D (1) B D A C E (1) B C D E A (1) B C A E D (1) B A E D C (1) B A C E D (1) Total count = 100 A B C D E A 0 0 -4 -6 2 B 0 0 6 4 4 C 4 -6 0 12 0 D 6 -4 -12 0 -8 E -2 -4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.241050 B: 0.758950 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.634110575923 Cumulative probabilities = A: 0.241050 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -6 2 B 0 0 6 4 4 C 4 -6 0 12 0 D 6 -4 -12 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000018339 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=25 A=19 E=17 D=10 so D is eliminated. Round 2 votes counts: B=29 C=27 A=25 E=19 so E is eliminated. Round 3 votes counts: A=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:207 C:205 E:201 A:196 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -6 2 B 0 0 6 4 4 C 4 -6 0 12 0 D 6 -4 -12 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000018339 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -6 2 B 0 0 6 4 4 C 4 -6 0 12 0 D 6 -4 -12 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000018339 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -6 2 B 0 0 6 4 4 C 4 -6 0 12 0 D 6 -4 -12 0 -8 E -2 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000018339 Cumulative probabilities = A: 0.400000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5705: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (5) B D C E A (5) B C D E A (5) A D C E B (5) E B C D A (4) E A C B D (4) D A C B E (4) C B E D A (4) B E C D A (4) B D E C A (4) E C B A D (3) E B C A D (3) D B C A E (3) D A B C E (3) B C E D A (3) A E C D B (3) A D E C B (3) A D C B E (3) A C E D B (3) E C A B D (2) E B D C A (2) E A B D C (2) D B A E C (2) D B A C E (2) C E B A D (2) C E A B D (2) C A D E B (2) B E D C A (2) E B A D C (1) E B A C D (1) E A C D B (1) D C B A E (1) D A E B C (1) C B E A D (1) C A D B E (1) B E D A C (1) A D E B C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -6 -10 -12 B 6 0 8 10 8 C 6 -8 0 0 2 D 10 -10 0 0 4 E 12 -8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -10 -12 B 6 0 8 10 8 C 6 -8 0 0 2 D 10 -10 0 0 4 E 12 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 D=21 A=20 C=12 so C is eliminated. Round 2 votes counts: B=29 E=27 A=23 D=21 so D is eliminated. Round 3 votes counts: B=37 A=36 E=27 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:202 C:200 E:199 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -10 -12 B 6 0 8 10 8 C 6 -8 0 0 2 D 10 -10 0 0 4 E 12 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -10 -12 B 6 0 8 10 8 C 6 -8 0 0 2 D 10 -10 0 0 4 E 12 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -10 -12 B 6 0 8 10 8 C 6 -8 0 0 2 D 10 -10 0 0 4 E 12 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5706: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) A B C D E (7) A B C E D (6) D B E C A (5) C B D E A (4) C B A D E (4) B A C D E (4) A C B E D (4) A B D E C (4) E D A B C (3) E C D A B (3) D C E B A (3) B C D E A (3) A E D B C (3) E D C A B (2) D E C B A (2) C E D A B (2) C A B E D (2) B C D A E (2) A C E D B (2) E D C B A (1) E D A C B (1) E A D C B (1) D E B C A (1) D E A B C (1) D B E A C (1) C E B D A (1) C E A D B (1) C D E B A (1) C D B E A (1) C B D A E (1) C B A E D (1) C A E D B (1) C A E B D (1) C A B D E (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B C A D E (1) B A D E C (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E B C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -8 0 6 B -2 0 -4 10 16 C 8 4 0 28 28 D 0 -10 -28 0 8 E -6 -16 -28 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 0 6 B -2 0 -4 10 16 C 8 4 0 28 28 D 0 -10 -28 0 8 E -6 -16 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 B=16 D=13 E=11 so E is eliminated. Round 2 votes counts: A=33 C=31 D=20 B=16 so B is eliminated. Round 3 votes counts: A=38 C=37 D=25 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:234 B:210 A:200 D:185 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 0 6 B -2 0 -4 10 16 C 8 4 0 28 28 D 0 -10 -28 0 8 E -6 -16 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 0 6 B -2 0 -4 10 16 C 8 4 0 28 28 D 0 -10 -28 0 8 E -6 -16 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 0 6 B -2 0 -4 10 16 C 8 4 0 28 28 D 0 -10 -28 0 8 E -6 -16 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5707: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (16) D E C B A (12) A B C E D (12) A B C D E (10) E D B C A (9) E A B C D (9) E D C B A (8) E D A C B (4) D C B A E (3) E A D B C (2) C B A D E (2) E D C A B (1) E D B A C (1) D C E B A (1) D C B E A (1) D B C A E (1) C D B A E (1) C B D A E (1) B A C E D (1) B A C D E (1) A E B C D (1) A D C B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 16 20 -18 -28 B -16 0 26 -22 -28 C -20 -26 0 -20 -26 D 18 22 20 0 -30 E 28 28 26 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 20 -18 -28 B -16 0 26 -22 -28 C -20 -26 0 -20 -26 D 18 22 20 0 -30 E 28 28 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=50 A=26 D=18 C=4 B=2 so B is eliminated. Round 2 votes counts: E=50 A=28 D=18 C=4 so C is eliminated. Round 3 votes counts: E=50 A=30 D=20 so D is eliminated. Round 4 votes counts: E=64 A=36 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:256 D:215 A:195 B:180 C:154 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 20 -18 -28 B -16 0 26 -22 -28 C -20 -26 0 -20 -26 D 18 22 20 0 -30 E 28 28 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 20 -18 -28 B -16 0 26 -22 -28 C -20 -26 0 -20 -26 D 18 22 20 0 -30 E 28 28 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 20 -18 -28 B -16 0 26 -22 -28 C -20 -26 0 -20 -26 D 18 22 20 0 -30 E 28 28 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5708: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (14) B C A E D (12) E D A B C (8) E A D B C (8) C B A D E (5) A E D B C (5) C B D A E (4) D E C A B (3) B C E D A (3) B C E A D (3) B C A D E (3) E D C B A (2) C D E B A (2) C B E D A (2) C B D E A (2) C A B D E (2) A D E C B (2) A B E C D (2) E B D A C (1) E A B D C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C B E A (1) D A E C B (1) C D B E A (1) C D B A E (1) B E C A D (1) B E A C D (1) B A E C D (1) B A C E D (1) A E B D C (1) A D E B C (1) A D C E B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 0 4 -12 B -4 0 8 -6 -6 C 0 -8 0 -6 -8 D -4 6 6 0 -6 E 12 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 4 -12 B -4 0 8 -6 -6 C 0 -8 0 -6 -8 D -4 6 6 0 -6 E 12 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=22 E=20 C=19 A=14 so A is eliminated. Round 2 votes counts: B=29 E=26 D=26 C=19 so C is eliminated. Round 3 votes counts: B=44 D=30 E=26 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:201 A:198 B:196 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 4 -12 B -4 0 8 -6 -6 C 0 -8 0 -6 -8 D -4 6 6 0 -6 E 12 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 4 -12 B -4 0 8 -6 -6 C 0 -8 0 -6 -8 D -4 6 6 0 -6 E 12 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 4 -12 B -4 0 8 -6 -6 C 0 -8 0 -6 -8 D -4 6 6 0 -6 E 12 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5709: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) E D C B A (8) D C E A B (7) D E C A B (5) B A C E D (5) A B D C E (5) A B C D E (5) E D B A C (4) E C B A D (4) C D A B E (4) E C D B A (3) E B A C D (3) D E C B A (3) D A B C E (3) C E D B A (3) E B C A D (2) D C E B A (2) D C A E B (2) D C A B E (2) D A C B E (2) C E B A D (2) C D E B A (2) C B A E D (2) A B D E C (2) E C B D A (1) E B A D C (1) D E A B C (1) D A B E C (1) C A B D E (1) B C A E D (1) B A E D C (1) A D B C E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -8 -6 -2 B 14 0 -6 -6 -6 C 8 6 0 -2 0 D 6 6 2 0 -4 E 2 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.409152 D: 0.000000 E: 0.590848 Sum of squares = 0.516506756864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.409152 D: 0.409152 E: 1.000000 A B C D E A 0 -14 -8 -6 -2 B 14 0 -6 -6 -6 C 8 6 0 -2 0 D 6 6 2 0 -4 E 2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=17 A=15 C=14 so C is eliminated. Round 2 votes counts: D=34 E=31 B=19 A=16 so A is eliminated. Round 3 votes counts: D=35 B=34 E=31 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:206 E:206 D:205 B:198 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -8 -6 -2 B 14 0 -6 -6 -6 C 8 6 0 -2 0 D 6 6 2 0 -4 E 2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -6 -2 B 14 0 -6 -6 -6 C 8 6 0 -2 0 D 6 6 2 0 -4 E 2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -6 -2 B 14 0 -6 -6 -6 C 8 6 0 -2 0 D 6 6 2 0 -4 E 2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5710: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) D E C A B (8) E C B A D (7) D A B C E (7) C E B A D (6) A D B C E (6) E C B D A (4) D E C B A (4) D A C E B (4) D A B E C (4) B A C E D (4) A B D C E (4) A B C E D (4) D A E B C (3) A B C D E (3) D E A B C (2) A C B E D (2) A C B D E (2) E D B C A (1) E B D C A (1) E B C A D (1) D E A C B (1) D B A E C (1) D A E C B (1) C E A D B (1) C D E A B (1) C B E A D (1) C B A E D (1) C A B E D (1) B E A C D (1) B D E A C (1) B C A E D (1) B A E D C (1) B A D E C (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 12 8 -2 4 B -12 0 -6 -8 -6 C -8 6 0 -2 -2 D 2 8 2 0 10 E -4 6 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -2 4 B -12 0 -6 -8 -6 C -8 6 0 -2 -2 D 2 8 2 0 10 E -4 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=23 E=22 C=11 B=9 so B is eliminated. Round 2 votes counts: D=36 A=29 E=23 C=12 so C is eliminated. Round 3 votes counts: D=37 A=32 E=31 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:211 D:211 C:197 E:197 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -2 4 B -12 0 -6 -8 -6 C -8 6 0 -2 -2 D 2 8 2 0 10 E -4 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -2 4 B -12 0 -6 -8 -6 C -8 6 0 -2 -2 D 2 8 2 0 10 E -4 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -2 4 B -12 0 -6 -8 -6 C -8 6 0 -2 -2 D 2 8 2 0 10 E -4 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5711: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) A E C D B (9) B D E C A (7) E C A D B (5) E C A B D (4) D B C A E (4) C E A D B (4) C D B E A (4) A C E D B (4) E C B D A (3) E B C D A (3) E A C D B (3) D B C E A (3) C D B A E (3) B D A E C (3) A D B C E (3) E C D B A (2) D A B C E (2) B E D C A (2) B D A C E (2) A E C B D (2) A C D B E (2) E C D A B (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B A E (1) D B A C E (1) C D A B E (1) C A E D B (1) C A D B E (1) B D E A C (1) A E B C D (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -22 -12 -16 B 6 0 -4 -10 6 C 22 4 0 12 0 D 12 10 -12 0 4 E 16 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.913317 D: 0.000000 E: 0.086683 Sum of squares = 0.841662674912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.913317 D: 0.913317 E: 1.000000 A B C D E A 0 -6 -22 -12 -16 B 6 0 -4 -10 6 C 22 4 0 12 0 D 12 10 -12 0 4 E 16 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000361365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 A=23 C=14 D=11 so D is eliminated. Round 2 votes counts: B=35 E=25 A=25 C=15 so C is eliminated. Round 3 votes counts: B=43 E=29 A=28 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:219 D:207 E:203 B:199 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -22 -12 -16 B 6 0 -4 -10 6 C 22 4 0 12 0 D 12 10 -12 0 4 E 16 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000361365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -22 -12 -16 B 6 0 -4 -10 6 C 22 4 0 12 0 D 12 10 -12 0 4 E 16 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000361365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -22 -12 -16 B 6 0 -4 -10 6 C 22 4 0 12 0 D 12 10 -12 0 4 E 16 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.399999 Sum of squares = 0.520000361365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600001 D: 0.600001 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5712: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (15) B C E D A (9) E B C D A (4) E A D B C (4) D A E C B (4) C B A D E (4) B C A E D (4) D A C B E (3) C B D E A (3) B E C D A (3) A D E C B (3) A B C D E (3) E D B C A (2) E D A C B (2) E D A B C (2) E B A C D (2) E A B D C (2) D E C A B (2) D C B E A (2) D C A B E (2) D A C E B (2) C D B E A (2) C B D A E (2) A E D C B (2) A E D B C (2) E D C A B (1) E A D C B (1) D E C B A (1) D E A C B (1) D C E B A (1) C D A B E (1) C B E D A (1) B E C A D (1) B C E A D (1) A E B D C (1) A D E B C (1) A D C E B (1) A D B C E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 16 8 0 6 B -16 0 -14 -16 18 C -8 14 0 -16 16 D 0 16 16 0 10 E -6 -18 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.678946 B: 0.000000 C: 0.000000 D: 0.321054 E: 0.000000 Sum of squares = 0.564043161017 Cumulative probabilities = A: 0.678946 B: 0.678946 C: 0.678946 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 0 6 B -16 0 -14 -16 18 C -8 14 0 -16 16 D 0 16 16 0 10 E -6 -18 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999735 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=20 D=18 B=18 C=13 so C is eliminated. Round 2 votes counts: A=31 B=28 D=21 E=20 so E is eliminated. Round 3 votes counts: A=38 B=34 D=28 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:215 C:203 B:186 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 0 6 B -16 0 -14 -16 18 C -8 14 0 -16 16 D 0 16 16 0 10 E -6 -18 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999735 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 0 6 B -16 0 -14 -16 18 C -8 14 0 -16 16 D 0 16 16 0 10 E -6 -18 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999735 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 0 6 B -16 0 -14 -16 18 C -8 14 0 -16 16 D 0 16 16 0 10 E -6 -18 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999735 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5713: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) A C B E D (7) E C B A D (6) D A B C E (6) A C B D E (4) E D C B A (3) E C D B A (3) E C B D A (3) D E C A B (3) D E B C A (3) D E B A C (3) D B A E C (3) B D E A C (3) B A C D E (3) A C D B E (3) A B C D E (3) E D B C A (2) D E C B A (2) D A E C B (2) D A E B C (2) D A C E B (2) C E A B D (2) C B A E D (2) B D E C A (2) B A C E D (2) A C E B D (2) E C A D B (1) E B D C A (1) E B C D A (1) D C A E B (1) D B E C A (1) D B E A C (1) D B A C E (1) C E A D B (1) C A E D B (1) C A B E D (1) B E C A D (1) B A D C E (1) A D C E B (1) A D C B E (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 4 4 16 B -6 0 -18 6 -6 C -4 18 0 10 8 D -4 -6 -10 0 4 E -16 6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 4 16 B -6 0 -18 6 -6 C -4 18 0 10 8 D -4 -6 -10 0 4 E -16 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999488 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 E=20 C=15 B=12 so B is eliminated. Round 2 votes counts: D=35 A=29 E=21 C=15 so C is eliminated. Round 3 votes counts: A=41 D=35 E=24 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:216 A:215 D:192 E:189 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 4 16 B -6 0 -18 6 -6 C -4 18 0 10 8 D -4 -6 -10 0 4 E -16 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999488 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 4 16 B -6 0 -18 6 -6 C -4 18 0 10 8 D -4 -6 -10 0 4 E -16 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999488 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 4 16 B -6 0 -18 6 -6 C -4 18 0 10 8 D -4 -6 -10 0 4 E -16 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999488 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5714: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) E B C A D (5) D C A B E (5) C D A E B (5) D C E B A (4) D B E A C (4) C D E B A (4) C D A B E (4) B A E D C (4) E B C D A (3) E B A D C (3) D E B A C (3) D A C B E (3) D A B E C (3) D A B C E (3) C A E B D (3) B E A C D (3) A B E C D (3) A B D E C (3) E B D A C (2) D E B C A (2) D C A E B (2) B E A D C (2) A C B E D (2) A B E D C (2) E D B C A (1) E C B A D (1) E B D C A (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B A D (1) C A D E B (1) B A E C D (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 12 -4 -2 B 12 0 22 4 -6 C -12 -22 0 2 -14 D 4 -4 -2 0 0 E 2 6 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.488210 E: 0.511790 Sum of squares = 0.500278012165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.488210 E: 1.000000 A B C D E A 0 -12 12 -4 -2 B 12 0 22 4 -6 C -12 -22 0 2 -14 D 4 -4 -2 0 0 E 2 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=26 C=20 A=14 B=10 so B is eliminated. Round 2 votes counts: E=31 D=30 C=20 A=19 so A is eliminated. Round 3 votes counts: E=41 D=34 C=25 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:216 E:211 D:199 A:197 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 12 -4 -2 B 12 0 22 4 -6 C -12 -22 0 2 -14 D 4 -4 -2 0 0 E 2 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 12 -4 -2 B 12 0 22 4 -6 C -12 -22 0 2 -14 D 4 -4 -2 0 0 E 2 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 12 -4 -2 B 12 0 22 4 -6 C -12 -22 0 2 -14 D 4 -4 -2 0 0 E 2 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5715: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) B D C E A (8) E C D B A (7) B D A C E (5) E C D A B (4) E C B D A (4) E A C B D (4) C E D B A (4) A B D E C (4) A B D C E (4) E A C D B (3) B D C A E (3) A E B C D (3) A B E D C (3) E C A D B (2) D C B E A (2) D B C E A (2) C D E B A (2) B A D C E (2) A E D C B (2) A D E C B (2) A D B E C (2) D C B A E (1) D C A E B (1) D B C A E (1) D A C B E (1) D A B C E (1) C D B E A (1) C B D E A (1) B E C D A (1) B E C A D (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D B C (1) A E C B D (1) A E B D C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 4 6 -2 4 B -4 0 -8 -4 -6 C -6 8 0 2 -14 D 2 4 -2 0 -8 E -4 6 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428584 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 A B C D E A 0 4 6 -2 4 B -4 0 -8 -4 -6 C -6 8 0 2 -14 D 2 4 -2 0 -8 E -4 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=24 B=23 D=9 C=8 so C is eliminated. Round 2 votes counts: A=36 E=28 B=24 D=12 so D is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:212 A:206 D:198 C:195 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 -2 4 B -4 0 -8 -4 -6 C -6 8 0 2 -14 D 2 4 -2 0 -8 E -4 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -2 4 B -4 0 -8 -4 -6 C -6 8 0 2 -14 D 2 4 -2 0 -8 E -4 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -2 4 B -4 0 -8 -4 -6 C -6 8 0 2 -14 D 2 4 -2 0 -8 E -4 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428569 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5716: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) B A D C E (6) E D C A B (5) D B A E C (5) B D A E C (5) E C B D A (4) D B E C A (4) A C B E D (4) E C D B A (3) E C D A B (3) D E A C B (3) C A E D B (3) A D C E B (3) A C E B D (3) E C A D B (2) D E B C A (2) D B E A C (2) C E A B D (2) B E D C A (2) B E C D A (2) B D E A C (2) A E D C B (2) A C E D B (2) E D C B A (1) D E B A C (1) D E A B C (1) D A E C B (1) D A B E C (1) C E B D A (1) C E B A D (1) C B E D A (1) C B E A D (1) C B A E D (1) C A E B D (1) C A B E D (1) B D E C A (1) B D C E A (1) B A C D E (1) A D E C B (1) A D C B E (1) A D B E C (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 -2 -10 B -6 0 -24 -18 -12 C 4 24 0 -4 -8 D 2 18 4 0 -10 E 10 12 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 -2 -10 B -6 0 -24 -18 -12 C 4 24 0 -4 -8 D 2 18 4 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 D=20 B=20 A=20 E=18 so E is eliminated. Round 2 votes counts: C=34 D=26 B=20 A=20 so B is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:220 C:208 D:207 A:195 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 -2 -10 B -6 0 -24 -18 -12 C 4 24 0 -4 -8 D 2 18 4 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -2 -10 B -6 0 -24 -18 -12 C 4 24 0 -4 -8 D 2 18 4 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -2 -10 B -6 0 -24 -18 -12 C 4 24 0 -4 -8 D 2 18 4 0 -10 E 10 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5717: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (14) B A C D E (10) E C D A B (8) B A D C E (8) B A E C D (5) D E C A B (4) D A C E B (4) C E D A B (4) B E A D C (4) A B C D E (4) E C D B A (3) D C A E B (3) B A C E D (3) A B D C E (3) E D B C A (2) D C E A B (2) B A D E C (2) A C B D E (2) E D C B A (1) E C B D A (1) E B D A C (1) E B C D A (1) E B C A D (1) D E A C B (1) D E A B C (1) D A B C E (1) C E A B D (1) C D E A B (1) C A D E B (1) B C E A D (1) B C A E D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 12 2 -4 -2 B -12 0 -4 -4 -8 C -2 4 0 -4 2 D 4 4 4 0 -2 E 2 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 12 2 -4 -2 B -12 0 -4 -4 -8 C -2 4 0 -4 2 D 4 4 4 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=32 D=16 A=11 C=7 so C is eliminated. Round 2 votes counts: E=37 B=34 D=17 A=12 so A is eliminated. Round 3 votes counts: B=43 E=37 D=20 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:205 E:205 A:204 C:200 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 2 -4 -2 B -12 0 -4 -4 -8 C -2 4 0 -4 2 D 4 4 4 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -4 -2 B -12 0 -4 -4 -8 C -2 4 0 -4 2 D 4 4 4 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -4 -2 B -12 0 -4 -4 -8 C -2 4 0 -4 2 D 4 4 4 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5718: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (11) C D E B A (9) D C E A B (8) E C D B A (6) D C A E B (6) A B E D C (6) B E A C D (5) B E C D A (4) A D E C B (4) A D C E B (4) D A C E B (3) E C B D A (2) E B C D A (2) E B A C D (2) D E C A B (2) D C A B E (2) C E D B A (2) C E B D A (2) B E C A D (2) B C E D A (2) B C E A D (2) B A C D E (2) A E B D C (2) A D B E C (2) E D C A B (1) E A B D C (1) D C E B A (1) D A C B E (1) C B E D A (1) A D C B E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -8 -8 -8 B 10 0 -10 -6 -14 C 8 10 0 8 -6 D 8 6 -8 0 -6 E 8 14 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -8 -8 -8 B 10 0 -10 -6 -14 C 8 10 0 8 -6 D 8 6 -8 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 A=21 E=14 C=14 so E is eliminated. Round 2 votes counts: B=32 D=24 C=22 A=22 so C is eliminated. Round 3 votes counts: D=41 B=37 A=22 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:217 C:210 D:200 B:190 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -8 -8 -8 B 10 0 -10 -6 -14 C 8 10 0 8 -6 D 8 6 -8 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -8 -8 B 10 0 -10 -6 -14 C 8 10 0 8 -6 D 8 6 -8 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -8 -8 B 10 0 -10 -6 -14 C 8 10 0 8 -6 D 8 6 -8 0 -6 E 8 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5719: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A C D E B (8) A D C E B (7) A C D B E (7) C A D E B (6) B E D A C (6) C A B E D (5) E D B C A (4) D E B A C (3) C B A E D (3) A B C D E (3) D E C A B (2) D E A C B (2) C E B D A (2) C B E A D (2) B E A D C (2) B C A E D (2) B A C E D (2) E D C B A (1) E D B A C (1) E B D C A (1) D C E A B (1) D B E A C (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E A B (1) C B E D A (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B D E (1) B E A C D (1) B D E A C (1) B C E A D (1) B A D E C (1) A D E B C (1) A D C B E (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 6 22 12 B -10 0 -20 -6 6 C -6 20 0 6 24 D -22 6 -6 0 4 E -12 -6 -24 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 22 12 B -10 0 -20 -6 6 C -6 20 0 6 24 D -22 6 -6 0 4 E -12 -6 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=27 B=24 D=12 E=7 so E is eliminated. Round 2 votes counts: A=30 C=27 B=25 D=18 so D is eliminated. Round 3 votes counts: A=35 B=34 C=31 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:222 D:191 B:185 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 22 12 B -10 0 -20 -6 6 C -6 20 0 6 24 D -22 6 -6 0 4 E -12 -6 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 22 12 B -10 0 -20 -6 6 C -6 20 0 6 24 D -22 6 -6 0 4 E -12 -6 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 22 12 B -10 0 -20 -6 6 C -6 20 0 6 24 D -22 6 -6 0 4 E -12 -6 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5720: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) E B D C A (6) A E B D C (6) A C D E B (6) E D B C A (5) D E C B A (5) B E A D C (5) A B E C D (5) E B A D C (4) D C E B A (4) D C B E A (3) C D B E A (3) B E D A C (3) E B D A C (2) E A B D C (2) C A D E B (2) B E A C D (2) B A E C D (2) A E C D B (2) A E B C D (2) E D B A C (1) E A D B C (1) D C E A B (1) D B C E A (1) D A C E B (1) C D E A B (1) C D B A E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D C A (1) B E C A D (1) B C E D A (1) B C A E D (1) B A E D C (1) B A C E D (1) A E D C B (1) A E D B C (1) A D C E B (1) A C D B E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 6 4 -4 B 6 0 12 -2 -24 C -6 -12 0 -10 -16 D -4 2 10 0 -14 E 4 24 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 4 -4 B 6 0 12 -2 -24 C -6 -12 0 -10 -16 D -4 2 10 0 -14 E 4 24 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=21 C=18 B=18 D=15 so D is eliminated. Round 2 votes counts: A=29 E=26 C=26 B=19 so B is eliminated. Round 3 votes counts: E=38 A=33 C=29 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 A:200 D:197 B:196 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 4 -4 B 6 0 12 -2 -24 C -6 -12 0 -10 -16 D -4 2 10 0 -14 E 4 24 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 4 -4 B 6 0 12 -2 -24 C -6 -12 0 -10 -16 D -4 2 10 0 -14 E 4 24 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 4 -4 B 6 0 12 -2 -24 C -6 -12 0 -10 -16 D -4 2 10 0 -14 E 4 24 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5721: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) C D A E B (7) E B A C D (6) E B D C A (5) B E A D C (5) B E A C D (5) E B C A D (4) D C B A E (4) B E D C A (4) B A D C E (4) D C A E B (3) C A D E B (3) B A C D E (3) A C D B E (3) E D C B A (2) E A C B D (2) E A B C D (2) C D A B E (2) B E D A C (2) B D C A E (2) A D C B E (2) A C D E B (2) E C D B A (1) E C D A B (1) D C E B A (1) D C E A B (1) D B C E A (1) D B C A E (1) D B A C E (1) C E D A B (1) C D E A B (1) C A D B E (1) B D E C A (1) B D A C E (1) B A E C D (1) A C E D B (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -12 -4 10 B 8 0 0 2 14 C 12 0 0 0 18 D 4 -2 0 0 12 E -10 -14 -18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.841923 C: 0.158077 D: 0.000000 E: 0.000000 Sum of squares = 0.733822702153 Cumulative probabilities = A: 0.000000 B: 0.841923 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -4 10 B 8 0 0 2 14 C 12 0 0 0 18 D 4 -2 0 0 12 E -10 -14 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=23 D=22 C=15 A=12 so A is eliminated. Round 2 votes counts: B=30 D=24 E=23 C=23 so E is eliminated. Round 3 votes counts: B=47 C=27 D=26 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:212 D:207 A:193 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -4 10 B 8 0 0 2 14 C 12 0 0 0 18 D 4 -2 0 0 12 E -10 -14 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -4 10 B 8 0 0 2 14 C 12 0 0 0 18 D 4 -2 0 0 12 E -10 -14 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -4 10 B 8 0 0 2 14 C 12 0 0 0 18 D 4 -2 0 0 12 E -10 -14 -18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5722: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (15) D B C E A (7) C D B A E (7) A E C B D (7) C A E D B (6) A E B D C (5) E D B A C (4) D B E C A (4) A E B C D (4) D E B A C (3) D C B E A (3) D B E A C (3) C D B E A (3) A C E B D (3) E B D A C (2) E B A D C (2) C A D E B (2) B D E C A (2) A C E D B (2) E D A B C (1) E A D C B (1) E A D B C (1) D E C B A (1) C D E A B (1) C D A B E (1) C B D A E (1) C A E B D (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E A C (1) B D C E A (1) B A E C D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 18 8 -14 B -6 0 18 -2 -24 C -18 -18 0 -16 -20 D -8 2 16 0 -18 E 14 24 20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 18 8 -14 B -6 0 18 -2 -24 C -18 -18 0 -16 -20 D -8 2 16 0 -18 E 14 24 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=23 A=23 D=21 B=7 so B is eliminated. Round 2 votes counts: E=28 D=25 A=24 C=23 so C is eliminated. Round 3 votes counts: D=38 A=34 E=28 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:238 A:209 D:196 B:193 C:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 18 8 -14 B -6 0 18 -2 -24 C -18 -18 0 -16 -20 D -8 2 16 0 -18 E 14 24 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 8 -14 B -6 0 18 -2 -24 C -18 -18 0 -16 -20 D -8 2 16 0 -18 E 14 24 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 8 -14 B -6 0 18 -2 -24 C -18 -18 0 -16 -20 D -8 2 16 0 -18 E 14 24 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5723: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) C E D B A (6) C D E B A (6) E D C B A (5) B A E C D (5) D C E B A (4) D C E A B (4) A B D E C (4) A B D C E (4) E D C A B (3) E D A C B (3) D E C A B (3) D A C E B (3) C E B D A (3) C B E D A (3) C B D E A (3) B E C A D (3) B A C E D (3) A B E D C (3) E C D B A (2) D C A B E (2) B E A C D (2) B C A E D (2) A D B E C (2) A B C D E (2) E D A B C (1) E B A C D (1) E A B D C (1) D C A E B (1) C D B A E (1) B C E A D (1) B C A D E (1) A E D B C (1) A E B D C (1) A D E C B (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -6 -6 -8 B 2 0 -8 -14 -14 C 6 8 0 -10 2 D 6 14 10 0 0 E 8 14 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.627531 E: 0.372469 Sum of squares = 0.53252851146 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.627531 E: 1.000000 A B C D E A 0 -2 -6 -6 -8 B 2 0 -8 -14 -14 C 6 8 0 -10 2 D 6 14 10 0 0 E 8 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 D=17 B=17 E=16 so E is eliminated. Round 2 votes counts: D=29 A=29 C=24 B=18 so B is eliminated. Round 3 votes counts: A=40 C=31 D=29 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:215 E:210 C:203 A:189 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -8 B 2 0 -8 -14 -14 C 6 8 0 -10 2 D 6 14 10 0 0 E 8 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -8 B 2 0 -8 -14 -14 C 6 8 0 -10 2 D 6 14 10 0 0 E 8 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -8 B 2 0 -8 -14 -14 C 6 8 0 -10 2 D 6 14 10 0 0 E 8 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5724: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) A E D C B (11) E A D C B (8) D C A E B (8) B C D A E (7) B E A C D (6) E A B C D (5) E B A C D (3) E A B D C (3) D C B A E (3) C D E A B (3) B E C D A (3) B E C A D (3) B D C A E (3) B A E D C (3) E A C D B (2) C D B E A (2) C D B A E (2) A D C E B (2) E C D A B (1) E C B D A (1) D C A B E (1) C D A E B (1) C B D E A (1) B E A D C (1) B A D C E (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -6 0 -12 B 6 0 6 10 2 C 6 -6 0 8 -4 D 0 -10 -8 0 -2 E 12 -2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 0 -12 B 6 0 6 10 2 C 6 -6 0 8 -4 D 0 -10 -8 0 -2 E 12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=23 A=15 D=12 C=9 so C is eliminated. Round 2 votes counts: B=42 E=23 D=20 A=15 so A is eliminated. Round 3 votes counts: B=43 E=34 D=23 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:208 C:202 D:190 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 0 -12 B 6 0 6 10 2 C 6 -6 0 8 -4 D 0 -10 -8 0 -2 E 12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 0 -12 B 6 0 6 10 2 C 6 -6 0 8 -4 D 0 -10 -8 0 -2 E 12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 0 -12 B 6 0 6 10 2 C 6 -6 0 8 -4 D 0 -10 -8 0 -2 E 12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5725: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) D A B E C (9) C A D B E (6) E B D A C (4) B E D A C (4) A D C B E (4) A D B E C (4) E C B D A (3) E B D C A (3) C E D B A (3) C E B A D (3) C D E A B (3) C A D E B (3) B E A D C (3) B A E D C (3) E B C D A (2) D E B A C (2) D B E A C (2) C E A D B (2) C E A B D (2) C D A B E (2) B A D E C (2) A C D B E (2) A B C E D (2) E D B C A (1) E C B A D (1) E B C A D (1) D C E B A (1) D C E A B (1) D B A E C (1) D A E B C (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E B A (1) C D A E B (1) B D E A C (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -2 -18 -10 B 4 0 -2 -6 0 C 2 2 0 0 4 D 18 6 0 0 0 E 10 0 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.653962 D: 0.346038 E: 0.000000 Sum of squares = 0.547408835212 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.653962 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -18 -10 B 4 0 -2 -6 0 C 2 2 0 0 4 D 18 6 0 0 0 E 10 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=19 E=15 A=15 B=13 so B is eliminated. Round 2 votes counts: C=38 E=22 D=20 A=20 so D is eliminated. Round 3 votes counts: C=40 A=33 E=27 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:204 E:203 B:198 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -18 -10 B 4 0 -2 -6 0 C 2 2 0 0 4 D 18 6 0 0 0 E 10 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -18 -10 B 4 0 -2 -6 0 C 2 2 0 0 4 D 18 6 0 0 0 E 10 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -18 -10 B 4 0 -2 -6 0 C 2 2 0 0 4 D 18 6 0 0 0 E 10 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5726: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (14) D E A B C (8) E D C B A (6) D E C B A (6) C D E B A (6) B A C E D (6) A B D E C (6) D E C A B (5) B C A E D (5) A B C D E (4) E D A B C (3) D E A C B (3) C B A D E (3) A B C E D (3) D A E B C (2) C B E D A (2) C B E A D (2) A D B C E (2) A B E D C (2) E D B A C (1) E C D B A (1) E B D A C (1) D C E A B (1) C E B D A (1) C A B D E (1) B A E C D (1) A E D B C (1) A D E B C (1) A D C B E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -6 8 8 B 10 0 -4 4 8 C 6 4 0 -2 4 D -8 -4 2 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999981 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 8 8 B 10 0 -4 4 8 C 6 4 0 -2 4 D -8 -4 2 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 A=22 E=12 B=12 so E is eliminated. Round 2 votes counts: D=35 C=30 A=22 B=13 so B is eliminated. Round 3 votes counts: D=36 C=35 A=29 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:209 C:206 A:200 D:196 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 8 8 B 10 0 -4 4 8 C 6 4 0 -2 4 D -8 -4 2 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 8 8 B 10 0 -4 4 8 C 6 4 0 -2 4 D -8 -4 2 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 8 8 B 10 0 -4 4 8 C 6 4 0 -2 4 D -8 -4 2 0 2 E -8 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5727: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) C E B D A (6) A E C B D (5) E C A B D (4) D E A C B (4) D C B E A (4) D B C E A (4) A E D B C (4) E A C D B (3) D B A E C (3) D B A C E (3) C B E A D (3) C B D E A (3) B D C E A (3) B D A C E (3) B A D C E (3) E C A D B (2) E A D C B (2) E A C B D (2) D E C B A (2) D B E C A (2) B C D E A (2) A E C D B (2) A E B C D (2) A D E B C (2) A D B E C (2) A B D E C (2) E D C A B (1) E D A C B (1) D E A B C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B A D (1) C E A B D (1) C D B E A (1) C B E D A (1) B C A E D (1) B C A D E (1) B A C E D (1) B A C D E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 -8 -8 B 12 0 0 6 4 C 0 0 0 -12 2 D 8 -6 12 0 10 E 8 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.777957 C: 0.222043 D: 0.000000 E: 0.000000 Sum of squares = 0.654519923771 Cumulative probabilities = A: 0.000000 B: 0.777957 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -8 -8 B 12 0 0 6 4 C 0 0 0 -12 2 D 8 -6 12 0 10 E 8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555647044 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=22 A=21 C=17 E=15 so E is eliminated. Round 2 votes counts: A=28 D=27 C=23 B=22 so B is eliminated. Round 3 votes counts: D=40 A=33 C=27 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:211 E:196 C:195 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -8 -8 B 12 0 0 6 4 C 0 0 0 -12 2 D 8 -6 12 0 10 E 8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555647044 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -8 -8 B 12 0 0 6 4 C 0 0 0 -12 2 D 8 -6 12 0 10 E 8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555647044 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -8 -8 B 12 0 0 6 4 C 0 0 0 -12 2 D 8 -6 12 0 10 E 8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555647044 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5728: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (5) B C A E D (5) B A C E D (5) E C D A B (4) B D A C E (4) A E D C B (4) E C A D B (3) E A D C B (3) E A C D B (3) D E C B A (3) C E A B D (3) C B E A D (3) B C D A E (3) A E C B D (3) A B E C D (3) A B D E C (3) A B C E D (3) D E A B C (2) D B A E C (2) D A E B C (2) C E D B A (2) C E D A B (2) C B D E A (2) B D C E A (2) B C D E A (2) E C D B A (1) E C A B D (1) D E C A B (1) D E A C B (1) D C E B A (1) D B E A C (1) D B C E A (1) D A B E C (1) C E B D A (1) C E A D B (1) C D E B A (1) C B E D A (1) C B A E D (1) B D A E C (1) B C A D E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E D B C (1) A E C D B (1) A E B C D (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 0 8 10 B 2 0 8 14 8 C 0 -8 0 14 2 D -8 -14 -14 0 -12 E -10 -8 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 8 10 B 2 0 8 14 8 C 0 -8 0 14 2 D -8 -14 -14 0 -12 E -10 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=22 C=17 E=15 D=15 so E is eliminated. Round 2 votes counts: B=31 A=28 C=26 D=15 so D is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:208 C:204 E:196 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 8 10 B 2 0 8 14 8 C 0 -8 0 14 2 D -8 -14 -14 0 -12 E -10 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 8 10 B 2 0 8 14 8 C 0 -8 0 14 2 D -8 -14 -14 0 -12 E -10 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 8 10 B 2 0 8 14 8 C 0 -8 0 14 2 D -8 -14 -14 0 -12 E -10 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5729: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) D C B E A (5) D B C A E (5) E B D C A (4) E B A D C (4) E A B C D (4) D C B A E (4) B A E D C (4) E A C B D (3) C D A E B (3) C A E D B (3) B E A D C (3) B D E A C (3) A C E B D (3) A B E D C (3) E C A B D (2) E B A C D (2) D B A C E (2) C E A D B (2) C D E B A (2) C D E A B (2) C D A B E (2) B D A E C (2) B D A C E (2) B A D E C (2) A E B C D (2) A B E C D (2) A B D C E (2) E D C B A (1) E C D B A (1) E C B D A (1) E C B A D (1) D B E C A (1) D B C E A (1) D A B C E (1) C E D B A (1) C E D A B (1) C A D E B (1) C A D B E (1) B D E C A (1) B A D C E (1) A E C B D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -4 -4 10 B 6 0 6 8 12 C 4 -6 0 -18 4 D 4 -8 18 0 2 E -10 -12 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -4 10 B 6 0 6 8 12 C 4 -6 0 -18 4 D 4 -8 18 0 2 E -10 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 C=18 B=18 A=15 so A is eliminated. Round 2 votes counts: B=27 E=26 D=26 C=21 so C is eliminated. Round 3 votes counts: D=37 E=36 B=27 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:208 A:198 C:192 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -4 10 B 6 0 6 8 12 C 4 -6 0 -18 4 D 4 -8 18 0 2 E -10 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -4 10 B 6 0 6 8 12 C 4 -6 0 -18 4 D 4 -8 18 0 2 E -10 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -4 10 B 6 0 6 8 12 C 4 -6 0 -18 4 D 4 -8 18 0 2 E -10 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5730: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (23) A C B E D (23) A C B D E (8) C B E D A (6) C A B E D (4) A D E B C (4) C B E A D (3) A D B E C (3) E D B C A (2) D E B A C (2) D B E C A (2) D A E B C (2) B D E C A (2) B C E D A (2) A E C D B (2) A C E D B (2) A B C D E (2) E D C B A (1) D E A B C (1) D A B E C (1) B E D C A (1) B E C D A (1) B C D E A (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 8 4 6 6 B -8 0 0 8 20 C -4 0 0 8 4 D -6 -8 -8 0 6 E -6 -20 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 6 6 B -8 0 0 8 20 C -4 0 0 8 4 D -6 -8 -8 0 6 E -6 -20 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 D=31 C=13 B=7 E=3 so E is eliminated. Round 2 votes counts: A=46 D=34 C=13 B=7 so B is eliminated. Round 3 votes counts: A=46 D=37 C=17 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:210 C:204 D:192 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 6 6 B -8 0 0 8 20 C -4 0 0 8 4 D -6 -8 -8 0 6 E -6 -20 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 6 6 B -8 0 0 8 20 C -4 0 0 8 4 D -6 -8 -8 0 6 E -6 -20 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 6 6 B -8 0 0 8 20 C -4 0 0 8 4 D -6 -8 -8 0 6 E -6 -20 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5731: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (13) A C D E B (12) A C D B E (11) D C A E B (9) B E D C A (8) B E A C D (7) E D C B A (5) B A E C D (5) A B C E D (4) A B C D E (4) D C E A B (3) C D A E B (3) B E C D A (3) D A C E B (2) B E A D C (2) B A C E D (2) A C B D E (2) E D B C A (1) D E C B A (1) D E C A B (1) C A D E B (1) A D C E B (1) Total count = 100 A B C D E A 0 6 4 2 12 B -6 0 -2 0 -4 C -4 2 0 8 8 D -2 0 -8 0 0 E -12 4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 12 B -6 0 -2 0 -4 C -4 2 0 8 8 D -2 0 -8 0 0 E -12 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=27 E=19 D=16 C=4 so C is eliminated. Round 2 votes counts: A=35 B=27 E=19 D=19 so E is eliminated. Round 3 votes counts: B=40 A=35 D=25 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:207 D:195 B:194 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 12 B -6 0 -2 0 -4 C -4 2 0 8 8 D -2 0 -8 0 0 E -12 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 12 B -6 0 -2 0 -4 C -4 2 0 8 8 D -2 0 -8 0 0 E -12 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 12 B -6 0 -2 0 -4 C -4 2 0 8 8 D -2 0 -8 0 0 E -12 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5732: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) A B C D E (10) B A C D E (9) E D C A B (8) D E B A C (7) E D C B A (6) E D B A C (5) E C D A B (4) C B A E D (4) B A D E C (4) D B A E C (3) D A B E C (3) C E D A B (3) C E A B D (3) B A D C E (3) E D B C A (2) D B E A C (2) C A E B D (2) B A C E D (2) A B D C E (2) D E C A B (1) D E A B C (1) C A B D E (1) B D A E C (1) B C A E D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 8 8 16 B -2 0 14 8 16 C -8 -14 0 0 4 D -8 -8 0 0 -2 E -16 -16 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 8 16 B -2 0 14 8 16 C -8 -14 0 0 4 D -8 -8 0 0 -2 E -16 -16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 B=20 D=17 A=14 so A is eliminated. Round 2 votes counts: B=33 E=25 C=24 D=18 so D is eliminated. Round 3 votes counts: B=42 E=34 C=24 so C is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:217 C:191 D:191 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 8 16 B -2 0 14 8 16 C -8 -14 0 0 4 D -8 -8 0 0 -2 E -16 -16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 8 16 B -2 0 14 8 16 C -8 -14 0 0 4 D -8 -8 0 0 -2 E -16 -16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 8 16 B -2 0 14 8 16 C -8 -14 0 0 4 D -8 -8 0 0 -2 E -16 -16 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5733: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) A D B C E (6) E C D A B (5) E A D C B (5) B C D A E (5) A D E B C (5) E C D B A (4) E C A B D (4) C E B D A (4) C B D E A (4) A D B E C (4) E C B A D (3) B C E A D (3) B A D C E (3) E D A C B (2) E C B D A (2) E C A D B (2) D E A C B (2) D C B A E (2) D A E C B (2) D A C E B (2) D A C B E (2) C B E D A (2) B D A C E (2) B C A D E (2) A B D E C (2) E B C A D (1) E A C B D (1) D A E B C (1) C E B A D (1) C D E B A (1) C D B E A (1) B C D E A (1) B C A E D (1) B A E C D (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 4 -8 4 B -14 0 -2 -12 4 C -4 2 0 -4 6 D 8 12 4 0 16 E -4 -4 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 -8 4 B -14 0 -2 -12 4 C -4 2 0 -4 6 D 8 12 4 0 16 E -4 -4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=21 A=19 B=18 C=13 so C is eliminated. Round 2 votes counts: E=34 B=24 D=23 A=19 so A is eliminated. Round 3 votes counts: D=38 E=35 B=27 so B is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:207 C:200 B:188 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 4 -8 4 B -14 0 -2 -12 4 C -4 2 0 -4 6 D 8 12 4 0 16 E -4 -4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -8 4 B -14 0 -2 -12 4 C -4 2 0 -4 6 D 8 12 4 0 16 E -4 -4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -8 4 B -14 0 -2 -12 4 C -4 2 0 -4 6 D 8 12 4 0 16 E -4 -4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5734: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (7) B A E D C (7) D C B E A (6) E B A D C (5) D C B A E (5) B A E C D (5) C D A E B (4) C D A B E (4) B E A D C (4) E B D A C (3) D C E A B (3) C A D E B (3) A C B E D (3) A C B D E (3) A B C E D (3) D E C B A (2) C E D A B (2) C D E A B (2) C A D B E (2) A E C B D (2) A E B C D (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B A C (1) E A B C D (1) D E B C A (1) D B C A E (1) B D E C A (1) B D C A E (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 8 10 20 B 0 0 -6 6 14 C -8 6 0 2 6 D -10 -6 -2 0 -6 E -20 -14 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.610959 B: 0.389041 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.524623796635 Cumulative probabilities = A: 0.610959 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 10 20 B 0 0 -6 6 14 C -8 6 0 2 6 D -10 -6 -2 0 -6 E -20 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 B=18 C=17 E=12 so E is eliminated. Round 2 votes counts: A=29 D=28 B=26 C=17 so C is eliminated. Round 3 votes counts: D=40 A=34 B=26 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:207 C:203 D:188 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 10 20 B 0 0 -6 6 14 C -8 6 0 2 6 D -10 -6 -2 0 -6 E -20 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 10 20 B 0 0 -6 6 14 C -8 6 0 2 6 D -10 -6 -2 0 -6 E -20 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 10 20 B 0 0 -6 6 14 C -8 6 0 2 6 D -10 -6 -2 0 -6 E -20 -14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5735: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (5) C A D B E (5) B E C A D (5) B C E A D (5) A D C B E (5) E D C A B (4) E C B D A (4) D C A E B (4) C D E A B (4) B E A C D (4) C E B D A (3) B E A D C (3) B A C D E (3) E C D B A (2) E B C D A (2) D E C A B (2) D A E B C (2) D A C E B (2) C E D B A (2) C E D A B (2) C B A D E (2) B C A D E (2) A D C E B (2) A C B D E (2) E D C B A (1) E D A B C (1) E C D A B (1) E B D C A (1) E B A D C (1) D E A C B (1) D E A B C (1) D A E C B (1) C D A E B (1) C B E D A (1) C B A E D (1) C A D E B (1) C A B D E (1) B E C D A (1) B C A E D (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C E D (1) A D E B C (1) A D B E C (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -16 4 -12 B 6 0 -10 4 -2 C 16 10 0 18 8 D -4 -4 -18 0 -6 E 12 2 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 4 -12 B 6 0 -10 4 -2 C 16 10 0 18 8 D -4 -4 -18 0 -6 E 12 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 E=22 A=14 D=13 so D is eliminated. Round 2 votes counts: B=28 C=27 E=26 A=19 so A is eliminated. Round 3 votes counts: C=39 B=31 E=30 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:206 B:199 A:185 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 4 -12 B 6 0 -10 4 -2 C 16 10 0 18 8 D -4 -4 -18 0 -6 E 12 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 4 -12 B 6 0 -10 4 -2 C 16 10 0 18 8 D -4 -4 -18 0 -6 E 12 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 4 -12 B 6 0 -10 4 -2 C 16 10 0 18 8 D -4 -4 -18 0 -6 E 12 2 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5736: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) A D E B C (6) E A B D C (4) E A B C D (4) D C A E B (4) C B E D A (4) C B E A D (4) A E D C B (4) A D E C B (4) D C A B E (3) D A B E C (3) C E B A D (3) B E C A D (3) B E A C D (3) B D A C E (3) B C E D A (3) B C E A D (3) E C B A D (2) E B A C D (2) E A C B D (2) D B A C E (2) D A E C B (2) D A E B C (2) D A C E B (2) D A B C E (2) C D E A B (2) C D B E A (2) C D B A E (2) C D A E B (2) A E D B C (2) E C A D B (1) E A C D B (1) D B C A E (1) D B A E C (1) C E A D B (1) B D C A E (1) B D A E C (1) B C D A E (1) B A D E C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 4 2 0 B -4 0 -2 0 0 C -4 2 0 2 2 D -2 0 -2 0 8 E 0 0 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.872756 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.127244 Sum of squares = 0.777894468716 Cumulative probabilities = A: 0.872756 B: 0.872756 C: 0.872756 D: 0.872756 E: 1.000000 A B C D E A 0 4 4 2 0 B -4 0 -2 0 0 C -4 2 0 2 2 D -2 0 -2 0 8 E 0 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000489 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=22 B=19 A=17 E=16 so E is eliminated. Round 2 votes counts: C=29 A=28 D=22 B=21 so B is eliminated. Round 3 votes counts: C=39 A=34 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:205 D:202 C:201 B:197 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 0 B -4 0 -2 0 0 C -4 2 0 2 2 D -2 0 -2 0 8 E 0 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000489 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 0 B -4 0 -2 0 0 C -4 2 0 2 2 D -2 0 -2 0 8 E 0 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000489 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 0 B -4 0 -2 0 0 C -4 2 0 2 2 D -2 0 -2 0 8 E 0 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000489 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5737: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (12) E B C A D (9) D E A B C (7) A D C B E (7) E B C D A (6) D E B C A (5) C B E A D (5) A C B E D (5) D B E C A (4) B C E A D (4) A C D B E (4) D A E C B (3) D A E B C (3) C B A E D (3) A D C E B (3) E D B C A (2) D C A B E (2) D A C E B (2) D A B C E (2) B E C A D (2) A C E B D (2) E B D C A (1) D B A E C (1) C A B E D (1) B E C D A (1) B C E D A (1) B C D E A (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 6 -6 4 B -10 0 -2 -18 10 C -6 2 0 -10 10 D 6 18 10 0 14 E -4 -10 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -6 4 B -10 0 -2 -18 10 C -6 2 0 -10 10 D 6 18 10 0 14 E -4 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=23 E=18 C=9 B=9 so C is eliminated. Round 2 votes counts: D=41 A=24 E=18 B=17 so B is eliminated. Round 3 votes counts: D=42 E=31 A=27 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 A:207 C:198 B:190 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 -6 4 B -10 0 -2 -18 10 C -6 2 0 -10 10 D 6 18 10 0 14 E -4 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -6 4 B -10 0 -2 -18 10 C -6 2 0 -10 10 D 6 18 10 0 14 E -4 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -6 4 B -10 0 -2 -18 10 C -6 2 0 -10 10 D 6 18 10 0 14 E -4 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5738: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) C D B A E (8) D B C A E (7) E C A B D (6) E A C B D (6) E A B D C (5) C D A B E (5) E A B C D (4) D B A E C (4) D B C E A (3) D B A C E (3) C D B E A (3) B E D A C (3) B D E A C (3) A D B C E (3) A B D E C (3) E B D A C (2) E B A D C (2) C A E D B (2) B D A E C (2) A D C B E (2) A D B E C (2) A C D B E (2) A B E D C (2) E B C D A (1) D C B E A (1) D A B C E (1) C E B D A (1) C D E B A (1) C A D E B (1) C A D B E (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 12 2 4 0 B -12 0 2 2 16 C -2 -2 0 2 6 D -4 -2 -2 0 10 E 0 -16 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.849470 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.150530 Sum of squares = 0.744258522357 Cumulative probabilities = A: 0.849470 B: 0.849470 C: 0.849470 D: 0.849470 E: 1.000000 A B C D E A 0 12 2 4 0 B -12 0 2 2 16 C -2 -2 0 2 6 D -4 -2 -2 0 10 E 0 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000007128 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 D=19 A=16 B=8 so B is eliminated. Round 2 votes counts: C=31 E=29 D=24 A=16 so A is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:209 B:204 C:202 D:201 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 4 0 B -12 0 2 2 16 C -2 -2 0 2 6 D -4 -2 -2 0 10 E 0 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000007128 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 4 0 B -12 0 2 2 16 C -2 -2 0 2 6 D -4 -2 -2 0 10 E 0 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000007128 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 4 0 B -12 0 2 2 16 C -2 -2 0 2 6 D -4 -2 -2 0 10 E 0 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000007128 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5739: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) E D B C A (6) C B A D E (6) C A B D E (6) A C B D E (5) E D B A C (4) D B E C A (4) A E D B C (4) E D A C B (3) D E B C A (3) B C D A E (3) A C E B D (3) A B C D E (3) E C A D B (2) E A D C B (2) E A C D B (2) C B D E A (2) C A E B D (2) B D C E A (2) B D A E C (2) B D A C E (2) A E C D B (2) A E B D C (2) A C B E D (2) E D C B A (1) E D C A B (1) E C D A B (1) D E C B A (1) D B E A C (1) D B C E A (1) C E D B A (1) C E B D A (1) C D B E A (1) C B D A E (1) C B A E D (1) C A B E D (1) B D C A E (1) B C A D E (1) B A C D E (1) A E D C B (1) A E C B D (1) A D B E C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 2 2 8 B -8 0 2 2 0 C -2 -2 0 -2 -8 D -2 -2 2 0 -4 E -8 0 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 2 8 B -8 0 2 2 0 C -2 -2 0 -2 -8 D -2 -2 2 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999725 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 C=22 B=12 D=10 so D is eliminated. Round 2 votes counts: E=33 A=27 C=22 B=18 so B is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 E:202 B:198 D:197 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 2 8 B -8 0 2 2 0 C -2 -2 0 -2 -8 D -2 -2 2 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999725 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 8 B -8 0 2 2 0 C -2 -2 0 -2 -8 D -2 -2 2 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999725 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 8 B -8 0 2 2 0 C -2 -2 0 -2 -8 D -2 -2 2 0 -4 E -8 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999725 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5740: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (18) E A B D C (7) B A E C D (6) A E B D C (6) D C B A E (5) B C A D E (5) D C E A B (4) C B D A E (4) B C D A E (4) E D A C B (3) E A D C B (3) E A D B C (3) B A E D C (3) A B E D C (3) E C D A B (2) E A B C D (2) D E A C B (2) C D E B A (2) C D B E A (2) B D C A E (2) B A C E D (2) B A C D E (2) E D C A B (1) E A C B D (1) D E C A B (1) D A B C E (1) C D E A B (1) C B A D E (1) B D A C E (1) B A D E C (1) B A D C E (1) A E D B C (1) Total count = 100 A B C D E A 0 -18 -4 -6 32 B 18 0 0 2 22 C 4 0 0 4 10 D 6 -2 -4 0 14 E -32 -22 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.508940 C: 0.491060 D: 0.000000 E: 0.000000 Sum of squares = 0.50015980169 Cumulative probabilities = A: 0.000000 B: 0.508940 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -4 -6 32 B 18 0 0 2 22 C 4 0 0 4 10 D 6 -2 -4 0 14 E -32 -22 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999838 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=27 E=22 D=13 A=10 so A is eliminated. Round 2 votes counts: B=30 E=29 C=28 D=13 so D is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:209 D:207 A:202 E:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -4 -6 32 B 18 0 0 2 22 C 4 0 0 4 10 D 6 -2 -4 0 14 E -32 -22 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999838 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -6 32 B 18 0 0 2 22 C 4 0 0 4 10 D 6 -2 -4 0 14 E -32 -22 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999838 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -6 32 B 18 0 0 2 22 C 4 0 0 4 10 D 6 -2 -4 0 14 E -32 -22 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999838 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5741: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) C A E B D (11) D B E A C (10) D B C E A (9) B D E A C (9) C A D E B (7) A E C B D (7) E B A D C (6) E A B C D (4) E A B D C (3) C D A E B (3) D B E C A (2) D B C A E (2) C D A B E (2) B E A D C (2) A C E B D (2) E D B A C (1) E B D A C (1) E A C B D (1) D E B A C (1) D C B E A (1) D C B A E (1) C D B A E (1) B E D A C (1) B A E D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 0 12 -2 B -4 0 6 -2 -18 C 0 -6 0 0 0 D -12 2 0 0 -4 E 2 18 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.512624 D: 0.000000 E: 0.487376 Sum of squares = 0.500318730047 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.512624 D: 0.512624 E: 1.000000 A B C D E A 0 4 0 12 -2 B -4 0 6 -2 -18 C 0 -6 0 0 0 D -12 2 0 0 -4 E 2 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=26 E=16 B=13 A=10 so A is eliminated. Round 2 votes counts: C=37 D=26 E=24 B=13 so B is eliminated. Round 3 votes counts: C=37 D=35 E=28 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:212 A:207 C:197 D:193 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 12 -2 B -4 0 6 -2 -18 C 0 -6 0 0 0 D -12 2 0 0 -4 E 2 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 12 -2 B -4 0 6 -2 -18 C 0 -6 0 0 0 D -12 2 0 0 -4 E 2 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 12 -2 B -4 0 6 -2 -18 C 0 -6 0 0 0 D -12 2 0 0 -4 E 2 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5742: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A B E D (8) E D B C A (6) B E D A C (6) A C D B E (5) A D C B E (4) E C D B A (3) E B D C A (3) E B D A C (3) D E B C A (3) D B E A C (3) D A C E B (3) C D A E B (3) E D C B A (2) D E A B C (2) D B A E C (2) D A E B C (2) D A C B E (2) C A D E B (2) C A D B E (2) C A B D E (2) B A E C D (2) A C B D E (2) A B C E D (2) E D B A C (1) E C B A D (1) D E C B A (1) D E C A B (1) D E A C B (1) D C A E B (1) D A B E C (1) C E D A B (1) C B A E D (1) C A E D B (1) C A E B D (1) B E C A D (1) B D E A C (1) B A E D C (1) B A D C E (1) B A C D E (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 14 -20 4 B 2 0 6 -24 0 C -14 -6 0 -22 -10 D 20 24 22 0 14 E -4 0 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 -20 4 B 2 0 6 -24 0 C -14 -6 0 -22 -10 D 20 24 22 0 14 E -4 0 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=21 E=19 A=16 B=13 so B is eliminated. Round 2 votes counts: D=32 E=26 C=21 A=21 so C is eliminated. Round 3 votes counts: A=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:240 A:198 E:196 B:192 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 14 -20 4 B 2 0 6 -24 0 C -14 -6 0 -22 -10 D 20 24 22 0 14 E -4 0 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 -20 4 B 2 0 6 -24 0 C -14 -6 0 -22 -10 D 20 24 22 0 14 E -4 0 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 -20 4 B 2 0 6 -24 0 C -14 -6 0 -22 -10 D 20 24 22 0 14 E -4 0 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5743: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) D A B E C (9) C E B A D (9) E B A C D (8) E A B D C (8) D C A B E (6) C D E A B (6) C D B A E (6) D C A E B (4) E A B C D (3) D C E A B (3) B E A C D (3) A B E D C (3) D A C B E (2) A E B D C (2) A B D E C (2) E D A C B (1) E C B A D (1) E A D B C (1) D E A C B (1) D C B A E (1) D B A E C (1) D A E B C (1) D A B C E (1) C E B D A (1) C E A B D (1) C D E B A (1) C D B E A (1) C D A B E (1) B E A D C (1) B D A E C (1) B A E C D (1) Total count = 100 A B C D E A 0 10 18 6 2 B -10 0 10 8 -2 C -18 -10 0 -16 -14 D -6 -8 16 0 -6 E -2 2 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 18 6 2 B -10 0 10 8 -2 C -18 -10 0 -16 -14 D -6 -8 16 0 -6 E -2 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 E=22 B=16 A=7 so A is eliminated. Round 2 votes counts: D=29 C=26 E=24 B=21 so B is eliminated. Round 3 votes counts: E=42 D=32 C=26 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:218 E:210 B:203 D:198 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 18 6 2 B -10 0 10 8 -2 C -18 -10 0 -16 -14 D -6 -8 16 0 -6 E -2 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 6 2 B -10 0 10 8 -2 C -18 -10 0 -16 -14 D -6 -8 16 0 -6 E -2 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 6 2 B -10 0 10 8 -2 C -18 -10 0 -16 -14 D -6 -8 16 0 -6 E -2 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5744: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) B C E A D (9) D A E C B (8) D A E B C (8) C B D E A (8) A E D B C (7) E A B D C (6) D C A E B (6) B E A C D (5) E A B C D (3) D A C E B (3) B C D E A (3) A E D C B (3) A D E B C (3) E B A C D (2) E A D B C (2) C D B A E (2) B E C A D (2) B D C E A (2) E B A D C (1) D E A B C (1) D C B A E (1) D B E A C (1) D B C A E (1) C D A E B (1) C B D A E (1) B C E D A (1) Total count = 100 A B C D E A 0 2 6 6 -12 B -2 0 14 6 -8 C -6 -14 0 -6 -4 D -6 -6 6 0 -2 E 12 8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 6 6 -12 B -2 0 14 6 -8 C -6 -14 0 -6 -4 D -6 -6 6 0 -2 E 12 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=22 B=22 E=14 A=13 so A is eliminated. Round 2 votes counts: D=32 E=24 C=22 B=22 so C is eliminated. Round 3 votes counts: B=41 D=35 E=24 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:213 B:205 A:201 D:196 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 6 -12 B -2 0 14 6 -8 C -6 -14 0 -6 -4 D -6 -6 6 0 -2 E 12 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 6 -12 B -2 0 14 6 -8 C -6 -14 0 -6 -4 D -6 -6 6 0 -2 E 12 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 6 -12 B -2 0 14 6 -8 C -6 -14 0 -6 -4 D -6 -6 6 0 -2 E 12 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5745: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) D B A C E (12) C E D B A (6) C E A D B (6) C E A B D (5) A B D C E (5) C A E D B (4) E B D C A (3) A D B C E (3) E C D A B (2) E C B D A (2) E C B A D (2) E B D A C (2) C D B E A (2) C D A B E (2) C A D E B (2) C A D B E (2) B D E A C (2) B A D E C (2) A C E B D (2) A C D B E (2) A B D E C (2) A B C E D (2) E C A D B (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C A E (1) D B A E C (1) C D B A E (1) C A E B D (1) B E D A C (1) B D A C E (1) A E B C D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 20 -14 22 -2 B -20 0 -18 4 -12 C 14 18 0 24 22 D -22 -4 -24 0 -18 E 2 12 -22 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -14 22 -2 B -20 0 -18 4 -12 C 14 18 0 24 22 D -22 -4 -24 0 -18 E 2 12 -22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=29 A=19 D=15 B=6 so B is eliminated. Round 2 votes counts: C=31 E=30 A=21 D=18 so D is eliminated. Round 3 votes counts: A=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:239 A:213 E:205 B:177 D:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -14 22 -2 B -20 0 -18 4 -12 C 14 18 0 24 22 D -22 -4 -24 0 -18 E 2 12 -22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -14 22 -2 B -20 0 -18 4 -12 C 14 18 0 24 22 D -22 -4 -24 0 -18 E 2 12 -22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -14 22 -2 B -20 0 -18 4 -12 C 14 18 0 24 22 D -22 -4 -24 0 -18 E 2 12 -22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5746: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) B D E C A (6) C D E A B (5) C B A D E (5) B E A D C (5) C D B E A (4) C A D E B (4) B A E D C (4) A B C E D (4) E D A B C (3) E A D C B (3) E A B D C (3) D C E B A (3) B E D A C (3) A E D C B (3) A C E B D (3) A C B E D (3) E D B A C (2) E D A C B (2) D E C B A (2) C A B D E (2) B D C E A (2) B A C E D (2) A E D B C (2) A B E D C (2) E D C A B (1) E A D B C (1) D E B C A (1) D C E A B (1) C D A E B (1) C D A B E (1) C B D E A (1) B C D A E (1) B C A D E (1) B A E C D (1) A E C D B (1) A E B D C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 18 22 4 B -14 0 -10 2 -2 C -18 10 0 0 6 D -22 -2 0 0 -18 E -4 2 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 22 4 B -14 0 -10 2 -2 C -18 10 0 0 6 D -22 -2 0 0 -18 E -4 2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=25 C=23 E=15 D=7 so D is eliminated. Round 2 votes counts: A=30 C=27 B=25 E=18 so E is eliminated. Round 3 votes counts: A=42 C=30 B=28 so B is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:205 C:199 B:188 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 22 4 B -14 0 -10 2 -2 C -18 10 0 0 6 D -22 -2 0 0 -18 E -4 2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 22 4 B -14 0 -10 2 -2 C -18 10 0 0 6 D -22 -2 0 0 -18 E -4 2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 22 4 B -14 0 -10 2 -2 C -18 10 0 0 6 D -22 -2 0 0 -18 E -4 2 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998527 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5747: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) D E C A B (6) B A E C D (6) C A B D E (5) B A C D E (5) A B E C D (5) E D C A B (4) E D B A C (4) D E B C A (4) D C E A B (4) B A C E D (4) A B C E D (4) C A B E D (3) B A E D C (3) E D A C B (2) E B A D C (2) E A B C D (2) D E B A C (2) D C E B A (2) D C B A E (2) C D A E B (2) C D A B E (2) C A D B E (2) B E D A C (2) A C B E D (2) E D B C A (1) E B D A C (1) C D E A B (1) C B D A E (1) C B A D E (1) C A E B D (1) C A D E B (1) B E A D C (1) B D E A C (1) B D A C E (1) B C D A E (1) B A D C E (1) A E B C D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 2 10 B 2 0 4 10 8 C -2 -4 0 2 -6 D -2 -10 -2 0 4 E -10 -8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 2 10 B 2 0 4 10 8 C -2 -4 0 2 -6 D -2 -10 -2 0 4 E -10 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 C=19 E=16 A=14 so A is eliminated. Round 2 votes counts: B=35 D=26 C=22 E=17 so E is eliminated. Round 3 votes counts: B=41 D=37 C=22 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:206 C:195 D:195 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 2 10 B 2 0 4 10 8 C -2 -4 0 2 -6 D -2 -10 -2 0 4 E -10 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 10 B 2 0 4 10 8 C -2 -4 0 2 -6 D -2 -10 -2 0 4 E -10 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 10 B 2 0 4 10 8 C -2 -4 0 2 -6 D -2 -10 -2 0 4 E -10 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5748: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) D E C A B (6) E B C D A (5) E D C B A (4) E B C A D (4) E B A C D (4) D C A E B (4) C A D B E (4) B E C A D (4) A D C B E (4) A C B D E (4) E D B C A (3) C B A D E (3) B C E A D (3) A B C E D (3) E D A B C (2) E B D A C (2) E A B D C (2) D E C B A (2) D C B E A (2) D C A B E (2) D A E C B (2) B E A C D (2) B A C E D (2) A D C E B (2) A C D B E (2) E D B A C (1) E D A C B (1) E B D C A (1) E B A D C (1) E A D B C (1) E A B C D (1) D E A C B (1) D C E B A (1) D A C B E (1) C D B A E (1) C B D A E (1) B E C D A (1) B C A D E (1) A D E C B (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 0 -8 B -4 0 -10 -8 -16 C 4 10 0 -6 -4 D 0 8 6 0 4 E 8 16 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.173462 B: 0.000000 C: 0.000000 D: 0.826538 E: 0.000000 Sum of squares = 0.713253698969 Cumulative probabilities = A: 0.173462 B: 0.173462 C: 0.173462 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 0 -8 B -4 0 -10 -8 -16 C 4 10 0 -6 -4 D 0 8 6 0 4 E 8 16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555626781 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=28 A=18 B=13 C=9 so C is eliminated. Round 2 votes counts: E=32 D=29 A=22 B=17 so B is eliminated. Round 3 votes counts: E=42 D=30 A=28 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:212 D:209 C:202 A:196 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 0 -8 B -4 0 -10 -8 -16 C 4 10 0 -6 -4 D 0 8 6 0 4 E 8 16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555626781 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 -8 B -4 0 -10 -8 -16 C 4 10 0 -6 -4 D 0 8 6 0 4 E 8 16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555626781 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 -8 B -4 0 -10 -8 -16 C 4 10 0 -6 -4 D 0 8 6 0 4 E 8 16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555626781 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5749: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) E B C A D (6) D A C B E (6) D C A E B (5) C D A E B (5) B E A D C (5) A B D E C (5) E C B D A (4) C D E A B (4) B E A C D (4) A D B E C (4) E B A C D (3) D A B C E (3) C E D B A (3) C E B A D (3) C D E B A (3) A D C B E (3) E C D B A (2) E B C D A (2) D C E B A (2) C E B D A (2) B A E D C (2) B A E C D (2) E B D C A (1) D E C B A (1) D E B A C (1) D C E A B (1) D C A B E (1) D B A E C (1) D A B E C (1) C A D B E (1) C A B E D (1) B E D A C (1) B D A E C (1) B A D E C (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 6 0 4 B 0 0 4 -8 4 C -6 -4 0 -6 4 D 0 8 6 0 14 E -4 -4 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.529775 B: 0.000000 C: 0.000000 D: 0.470225 E: 0.000000 Sum of squares = 0.501773054076 Cumulative probabilities = A: 0.529775 B: 0.529775 C: 0.529775 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 0 4 B 0 0 4 -8 4 C -6 -4 0 -6 4 D 0 8 6 0 14 E -4 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=22 C=22 A=22 E=18 B=16 so B is eliminated. Round 2 votes counts: E=28 A=27 D=23 C=22 so C is eliminated. Round 3 votes counts: E=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:205 B:200 C:194 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 0 4 B 0 0 4 -8 4 C -6 -4 0 -6 4 D 0 8 6 0 14 E -4 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 4 B 0 0 4 -8 4 C -6 -4 0 -6 4 D 0 8 6 0 14 E -4 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 4 B 0 0 4 -8 4 C -6 -4 0 -6 4 D 0 8 6 0 14 E -4 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5750: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) D E C A B (10) B D C E A (6) A E C B D (6) B A D E C (4) D B C E A (3) C E D A B (3) B D A E C (3) B A E C D (3) B A D C E (3) A E C D B (3) A B E C D (3) A B C E D (3) E C D A B (2) E C A D B (2) D E C B A (2) D C E B A (2) D B A E C (2) C D E B A (2) B A E D C (2) E D C A B (1) E A C D B (1) D E A C B (1) D C E A B (1) D C B E A (1) D B E C A (1) D A B E C (1) C E D B A (1) C E A D B (1) C D B E A (1) C B D E A (1) C A E D B (1) C A E B D (1) C A B E D (1) B D C A E (1) B D A C E (1) B C D E A (1) B C A E D (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 8 6 14 B 10 0 10 10 12 C -8 -10 0 2 -2 D -6 -10 -2 0 -2 E -14 -12 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 6 14 B 10 0 10 10 12 C -8 -10 0 2 -2 D -6 -10 -2 0 -2 E -14 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=24 A=19 C=12 E=6 so E is eliminated. Round 2 votes counts: B=39 D=25 A=20 C=16 so C is eliminated. Round 3 votes counts: B=40 D=34 A=26 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:209 C:191 D:190 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 6 14 B 10 0 10 10 12 C -8 -10 0 2 -2 D -6 -10 -2 0 -2 E -14 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 6 14 B 10 0 10 10 12 C -8 -10 0 2 -2 D -6 -10 -2 0 -2 E -14 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 6 14 B 10 0 10 10 12 C -8 -10 0 2 -2 D -6 -10 -2 0 -2 E -14 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5751: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) B E D C A (8) E B C D A (5) C D A B E (5) A D C B E (5) E B A C D (4) C D B E A (4) C A D E B (4) B E D A C (4) E A B C D (3) C A D B E (3) A C D E B (3) E C B A D (2) E B D C A (2) E B C A D (2) E A B D C (2) D C A B E (2) C E D B A (2) C D E B A (2) C A E D B (2) B E A D C (2) E C B D A (1) E A C B D (1) D C B A E (1) D B C A E (1) D A C B E (1) D A B E C (1) C E B D A (1) C E A B D (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D E A (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A E C (1) B C E D A (1) A E C D B (1) A E C B D (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -10 2 -22 B 16 0 4 14 0 C 10 -4 0 8 -8 D -2 -14 -8 0 -14 E 22 0 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.585709 C: 0.000000 D: 0.000000 E: 0.414291 Sum of squares = 0.5146921356 Cumulative probabilities = A: 0.000000 B: 0.585709 C: 0.585709 D: 0.585709 E: 1.000000 A B C D E A 0 -16 -10 2 -22 B 16 0 4 14 0 C 10 -4 0 8 -8 D -2 -14 -8 0 -14 E 22 0 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=28 B=19 A=16 D=6 so D is eliminated. Round 2 votes counts: E=31 C=31 B=20 A=18 so A is eliminated. Round 3 votes counts: C=42 E=33 B=25 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:217 C:203 D:181 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 2 -22 B 16 0 4 14 0 C 10 -4 0 8 -8 D -2 -14 -8 0 -14 E 22 0 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 2 -22 B 16 0 4 14 0 C 10 -4 0 8 -8 D -2 -14 -8 0 -14 E 22 0 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 2 -22 B 16 0 4 14 0 C 10 -4 0 8 -8 D -2 -14 -8 0 -14 E 22 0 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5752: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (21) A D E C B (20) B C A D E (6) C B A D E (5) B C E A D (5) A D E B C (5) C B D E A (4) E D B C A (3) E D A C B (3) D E A C B (2) C B E D A (2) C B D A E (2) B E C D A (2) A E D B C (2) A D C B E (2) E D C A B (1) E D A B C (1) E C D B A (1) E B D C A (1) E B C D A (1) E B A D C (1) D A E C B (1) D A C E B (1) C E B D A (1) C D A B E (1) C A B D E (1) B E A C D (1) B A E D C (1) A D C E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -14 4 0 B 14 0 2 10 10 C 14 -2 0 10 8 D -4 -10 -10 0 6 E 0 -10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 4 0 B 14 0 2 10 10 C 14 -2 0 10 8 D -4 -10 -10 0 6 E 0 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=32 C=16 E=12 D=4 so D is eliminated. Round 2 votes counts: B=36 A=34 C=16 E=14 so E is eliminated. Round 3 votes counts: B=42 A=40 C=18 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:215 D:191 A:188 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -14 4 0 B 14 0 2 10 10 C 14 -2 0 10 8 D -4 -10 -10 0 6 E 0 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 4 0 B 14 0 2 10 10 C 14 -2 0 10 8 D -4 -10 -10 0 6 E 0 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 4 0 B 14 0 2 10 10 C 14 -2 0 10 8 D -4 -10 -10 0 6 E 0 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994557 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5753: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (8) A D B E C (8) D A B E C (6) B A D C E (6) E C B A D (5) B D A C E (5) C E D B A (4) C E B D A (4) C E B A D (4) E D A C B (3) E C D A B (3) E C A D B (3) D A E B C (3) E C A B D (2) E A D C B (2) E A C D B (2) D C B A E (2) D A E C B (2) C B E D A (2) C B E A D (2) B C D A E (2) B C A E D (2) B C A D E (2) B A E C D (2) B A D E C (2) B A C D E (2) A D E B C (2) A B D E C (2) E C D B A (1) C D E B A (1) C D B E A (1) C B D E A (1) B D C A E (1) B A C E D (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 16 2 20 B 4 0 12 -4 16 C -16 -12 0 -8 2 D -2 4 8 0 14 E -20 -16 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000014 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 2 20 B 4 0 12 -4 16 C -16 -12 0 -8 2 D -2 4 8 0 14 E -20 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999997 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=21 D=21 C=19 A=14 so A is eliminated. Round 2 votes counts: D=31 B=28 E=22 C=19 so C is eliminated. Round 3 votes counts: E=34 D=33 B=33 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:214 D:212 C:183 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 16 2 20 B 4 0 12 -4 16 C -16 -12 0 -8 2 D -2 4 8 0 14 E -20 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999997 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 2 20 B 4 0 12 -4 16 C -16 -12 0 -8 2 D -2 4 8 0 14 E -20 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999997 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 2 20 B 4 0 12 -4 16 C -16 -12 0 -8 2 D -2 4 8 0 14 E -20 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999997 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5754: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) A E C B D (7) E A C D B (6) D B E A C (6) D B C E A (6) C A E D B (5) C A E B D (5) B D C A E (4) B A C E D (4) A C E B D (4) E B D A C (3) D B C A E (3) E D A B C (2) E B A D C (2) E A D C B (2) D C E A B (2) D B E C A (2) C A D B E (2) B E D A C (2) B D E C A (2) B A E C D (2) A C B E D (2) A B E C D (2) E D B A C (1) E A C B D (1) E A B D C (1) D E B A C (1) D C B E A (1) C D E A B (1) C D B A E (1) C B D A E (1) C B A D E (1) C A D E B (1) C A B E D (1) C A B D E (1) B D A E C (1) B D A C E (1) B A C D E (1) A E B C D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 22 6 4 B 4 0 10 14 8 C -22 -10 0 2 -2 D -6 -14 -2 0 -10 E -4 -8 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 22 6 4 B 4 0 10 14 8 C -22 -10 0 2 -2 D -6 -14 -2 0 -10 E -4 -8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=21 C=19 E=18 A=18 so E is eliminated. Round 2 votes counts: B=29 A=28 D=24 C=19 so C is eliminated. Round 3 votes counts: A=43 B=31 D=26 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:214 E:200 C:184 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 22 6 4 B 4 0 10 14 8 C -22 -10 0 2 -2 D -6 -14 -2 0 -10 E -4 -8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 6 4 B 4 0 10 14 8 C -22 -10 0 2 -2 D -6 -14 -2 0 -10 E -4 -8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 6 4 B 4 0 10 14 8 C -22 -10 0 2 -2 D -6 -14 -2 0 -10 E -4 -8 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5755: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) D A C B E (8) C A D B E (8) A D C B E (7) E B D A C (5) E B C A D (5) D A E C B (4) B C E A D (4) E D A B C (3) E C D A B (3) C D A E B (3) E D B A C (2) E C D B A (2) E B C D A (2) E B A D C (2) D C A E B (2) D C A B E (2) D A E B C (2) D A B E C (2) B E C A D (2) B E A D C (2) B C A D E (2) B A D E C (2) A D B C E (2) A C D B E (2) E D C A B (1) E C B D A (1) E C B A D (1) E B D C A (1) C E D A B (1) C D E A B (1) C B E A D (1) C B A D E (1) C A D E B (1) B E D A C (1) B A E D C (1) B A D C E (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 22 12 -8 20 B -22 0 -14 -28 0 C -12 14 0 -18 12 D 8 28 18 0 20 E -20 0 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 -8 20 B -22 0 -14 -28 0 C -12 14 0 -18 12 D 8 28 18 0 20 E -20 0 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 C=16 B=16 A=12 so A is eliminated. Round 2 votes counts: D=37 E=28 C=18 B=17 so B is eliminated. Round 3 votes counts: D=41 E=34 C=25 so C is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:237 A:223 C:198 E:174 B:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 12 -8 20 B -22 0 -14 -28 0 C -12 14 0 -18 12 D 8 28 18 0 20 E -20 0 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 -8 20 B -22 0 -14 -28 0 C -12 14 0 -18 12 D 8 28 18 0 20 E -20 0 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 -8 20 B -22 0 -14 -28 0 C -12 14 0 -18 12 D 8 28 18 0 20 E -20 0 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5756: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) E B A C D (10) D C A E B (8) C D A E B (6) E B A D C (5) D A C E B (5) B E D C A (5) B E A C D (5) A C D E B (5) C D A B E (4) C A D E B (4) B E D A C (4) B E C D A (4) B E C A D (4) A E B C D (4) D B E C A (3) B E A D C (2) A D C E B (2) E B D A C (1) D E B A C (1) D C B E A (1) C D B E A (1) C B E A D (1) C A D B E (1) C A B E D (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -8 -8 6 B -6 0 -2 -4 -6 C 8 2 0 2 2 D 8 4 -2 0 4 E -6 6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -8 6 B -6 0 -2 -4 -6 C 8 2 0 2 2 D 8 4 -2 0 4 E -6 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=24 C=18 E=16 A=13 so A is eliminated. Round 2 votes counts: D=31 C=24 B=24 E=21 so E is eliminated. Round 3 votes counts: B=45 D=31 C=24 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:207 A:198 E:197 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -8 6 B -6 0 -2 -4 -6 C 8 2 0 2 2 D 8 4 -2 0 4 E -6 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -8 6 B -6 0 -2 -4 -6 C 8 2 0 2 2 D 8 4 -2 0 4 E -6 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -8 6 B -6 0 -2 -4 -6 C 8 2 0 2 2 D 8 4 -2 0 4 E -6 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5757: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) E B C A D (6) B A D E C (6) E C B A D (4) E B A C D (4) C E D A B (4) C E A D B (4) C D A E B (4) B E A D C (4) E C D B A (3) E B C D A (3) D C A B E (3) D A C B E (3) C A D E B (3) A D B C E (3) A B E D C (3) A B D C E (3) E C B D A (2) E B D C A (2) D C E B A (2) C E D B A (2) C D E B A (2) B E D C A (2) A D C B E (2) A C D B E (2) A B D E C (2) E D C B A (1) E B D A C (1) E A C B D (1) E A B C D (1) D E C B A (1) D C A E B (1) C A E D B (1) B E D A C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E D C (1) A E B C D (1) A C E B D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -12 22 -16 B 0 0 -6 16 -22 C 12 6 0 14 -18 D -22 -16 -14 0 -20 E 16 22 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -12 22 -16 B 0 0 -6 16 -22 C 12 6 0 14 -18 D -22 -16 -14 0 -20 E 16 22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=20 A=19 B=17 D=10 so D is eliminated. Round 2 votes counts: E=35 C=26 A=22 B=17 so B is eliminated. Round 3 votes counts: E=44 A=30 C=26 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:238 C:207 A:197 B:194 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -12 22 -16 B 0 0 -6 16 -22 C 12 6 0 14 -18 D -22 -16 -14 0 -20 E 16 22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 22 -16 B 0 0 -6 16 -22 C 12 6 0 14 -18 D -22 -16 -14 0 -20 E 16 22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 22 -16 B 0 0 -6 16 -22 C 12 6 0 14 -18 D -22 -16 -14 0 -20 E 16 22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5758: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) D E B C A (10) C A B D E (8) A C D E B (8) C A D E B (6) B E D A C (6) C A B E D (5) A C E D B (5) E D B A C (4) D E B A C (4) A C B E D (4) E D A B C (2) D E C B A (2) D E C A B (2) C D E A B (2) C B A E D (2) C B A D E (2) B C E D A (2) A E D B C (2) A B E C D (2) E D B C A (1) D E A B C (1) C B D E A (1) C A D B E (1) B A E C D (1) A E D C B (1) A E B D C (1) A D E C B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -12 2 2 B -6 0 -2 -4 -4 C 12 2 0 0 -6 D -2 4 0 0 -2 E -2 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000201 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 6 -12 2 2 B -6 0 -2 -4 -4 C 12 2 0 0 -6 D -2 4 0 0 -2 E -2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000012 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 B=21 D=19 E=7 so E is eliminated. Round 2 votes counts: C=27 D=26 A=26 B=21 so B is eliminated. Round 3 votes counts: D=44 C=29 A=27 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:205 C:204 D:200 A:199 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -12 2 2 B -6 0 -2 -4 -4 C 12 2 0 0 -6 D -2 4 0 0 -2 E -2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000012 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 2 2 B -6 0 -2 -4 -4 C 12 2 0 0 -6 D -2 4 0 0 -2 E -2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000012 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 2 2 B -6 0 -2 -4 -4 C 12 2 0 0 -6 D -2 4 0 0 -2 E -2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000012 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5759: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) C B A E D (7) E D C B A (6) E D A B C (6) E C D B A (6) C B A D E (6) B A C D E (6) D E A B C (5) B A C E D (5) D E C B A (4) D C B A E (4) E D C A B (3) D E C A B (3) D C E B A (3) E C B A D (2) E A B C D (2) D A E B C (2) C B D E A (2) B C A D E (2) E D A C B (1) E B C A D (1) E B A C D (1) E A C B D (1) E A B D C (1) D C A B E (1) D B C A E (1) D A C B E (1) D A B E C (1) C E B A D (1) C D B E A (1) C D B A E (1) B D C A E (1) B C A E D (1) B A E C D (1) A E B D C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -24 -12 -18 2 B 24 0 -6 -12 2 C 12 6 0 -8 2 D 18 12 8 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -12 -18 2 B 24 0 -6 -12 2 C 12 6 0 -8 2 D 18 12 8 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=30 C=18 B=16 A=4 so A is eliminated. Round 2 votes counts: D=32 E=31 B=19 C=18 so C is eliminated. Round 3 votes counts: D=34 B=34 E=32 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:206 B:204 E:194 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 -12 -18 2 B 24 0 -6 -12 2 C 12 6 0 -8 2 D 18 12 8 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -12 -18 2 B 24 0 -6 -12 2 C 12 6 0 -8 2 D 18 12 8 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -12 -18 2 B 24 0 -6 -12 2 C 12 6 0 -8 2 D 18 12 8 0 6 E -2 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996032 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5760: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) A C B D E (7) E D C B A (6) C D E B A (5) A B E D C (5) C D E A B (4) C A D E B (4) A B C D E (4) E B D A C (3) C D B E A (3) C A D B E (3) B A C D E (3) A C B E D (3) E B D C A (2) D E B C A (2) D C E B A (2) D C B E A (2) B E D A C (2) B A D C E (2) A E B D C (2) A E B C D (2) A C D B E (2) E D C A B (1) E D B A C (1) E C D B A (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E C B A (1) C E D A B (1) C D A E B (1) C D A B E (1) C A E D B (1) C A B D E (1) B E D C A (1) B E A D C (1) B D E C A (1) B D C E A (1) B D C A E (1) B A E D C (1) A C D E B (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -8 -2 -6 B 2 0 -4 -8 -6 C 8 4 0 -2 8 D 2 8 2 0 6 E 6 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -2 -6 B 2 0 -4 -8 -6 C 8 4 0 -2 8 D 2 8 2 0 6 E 6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 C=24 B=13 D=7 so D is eliminated. Round 2 votes counts: E=30 A=29 C=28 B=13 so B is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:209 D:209 E:199 B:192 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -2 -6 B 2 0 -4 -8 -6 C 8 4 0 -2 8 D 2 8 2 0 6 E 6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -2 -6 B 2 0 -4 -8 -6 C 8 4 0 -2 8 D 2 8 2 0 6 E 6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -2 -6 B 2 0 -4 -8 -6 C 8 4 0 -2 8 D 2 8 2 0 6 E 6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5761: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) B D C E A (7) B A D C E (7) B D A C E (6) C E B D A (5) A B D E C (5) E C A B D (4) E A C D B (4) A E C D B (4) E C D A B (3) E C B A D (3) D C E B A (3) B D C A E (3) A D E C B (3) A D B C E (3) E C D B A (2) E C B D A (2) B C D E A (2) A E C B D (2) A D B E C (2) A B E D C (2) A B D C E (2) E A C B D (1) D B C E A (1) D B C A E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D B E A (1) C B E D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C E A D (1) B A D E C (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B C D (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 0 18 -4 B 4 0 2 18 2 C 0 -2 0 -2 -2 D -18 -18 2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 18 -4 B 4 0 2 18 2 C 0 -2 0 -2 -2 D -18 -18 2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=28 E=26 C=9 D=6 so D is eliminated. Round 2 votes counts: B=33 A=29 E=26 C=12 so C is eliminated. Round 3 votes counts: E=36 B=35 A=29 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:205 E:200 C:197 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 18 -4 B 4 0 2 18 2 C 0 -2 0 -2 -2 D -18 -18 2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 18 -4 B 4 0 2 18 2 C 0 -2 0 -2 -2 D -18 -18 2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 18 -4 B 4 0 2 18 2 C 0 -2 0 -2 -2 D -18 -18 2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5762: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) B C E D A (8) E B C A D (6) E A B D C (5) E A B C D (5) A D C E B (5) E B A C D (4) C E A D B (4) B E A D C (4) B C D E A (4) D A C B E (3) C D A B E (3) C B E D A (3) C B D E A (3) B E C D A (3) B D C E A (3) D C B A E (2) D B C A E (2) D A B E C (2) B E C A D (2) A E D B C (2) A E B D C (2) A D E C B (2) A D E B C (2) E C B A D (1) E A C B D (1) D B A E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C D B A E (1) C D A E B (1) B E D C A (1) B D E A C (1) A E C D B (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -14 0 -20 B 0 0 18 18 12 C 14 -18 0 6 6 D 0 -18 -6 0 -10 E 20 -12 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.261390 B: 0.738610 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.613869131907 Cumulative probabilities = A: 0.261390 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 0 -20 B 0 0 18 18 12 C 14 -18 0 6 6 D 0 -18 -6 0 -10 E 20 -12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250068946 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 D=19 C=17 A=16 so A is eliminated. Round 2 votes counts: D=29 E=28 B=26 C=17 so C is eliminated. Round 3 votes counts: E=34 D=34 B=32 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:224 E:206 C:204 A:183 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -14 0 -20 B 0 0 18 18 12 C 14 -18 0 6 6 D 0 -18 -6 0 -10 E 20 -12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250068946 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 0 -20 B 0 0 18 18 12 C 14 -18 0 6 6 D 0 -18 -6 0 -10 E 20 -12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250068946 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 0 -20 B 0 0 18 18 12 C 14 -18 0 6 6 D 0 -18 -6 0 -10 E 20 -12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250068946 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5763: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A D C E B (7) E B C A D (6) D A C E B (6) B E A D C (6) D A C B E (5) C E B D A (4) B E C A D (4) A D B E C (4) E C B A D (3) E B A C D (3) C A E D B (3) B A E D C (3) C E A D B (2) C D A E B (2) C A D E B (2) B E A C D (2) B D A E C (2) A E C D B (2) A D E B C (2) A D C B E (2) A C D E B (2) E C B D A (1) E C A B D (1) E A C B D (1) D C A E B (1) D B C A E (1) D B A C E (1) D A B C E (1) C E D B A (1) C E A B D (1) C D E B A (1) C D E A B (1) C B E D A (1) B E D C A (1) B D C E A (1) B C E D A (1) B C D E A (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 0 14 -6 B 8 0 2 6 -4 C 0 -2 0 12 -4 D -14 -6 -12 0 -14 E 6 4 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 14 -6 B 8 0 2 6 -4 C 0 -2 0 12 -4 D -14 -6 -12 0 -14 E 6 4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=20 C=18 E=15 D=15 so E is eliminated. Round 2 votes counts: B=41 C=23 A=21 D=15 so D is eliminated. Round 3 votes counts: B=43 A=33 C=24 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:206 C:203 A:200 D:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 14 -6 B 8 0 2 6 -4 C 0 -2 0 12 -4 D -14 -6 -12 0 -14 E 6 4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 14 -6 B 8 0 2 6 -4 C 0 -2 0 12 -4 D -14 -6 -12 0 -14 E 6 4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 14 -6 B 8 0 2 6 -4 C 0 -2 0 12 -4 D -14 -6 -12 0 -14 E 6 4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5764: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) A E D B C (9) E A B D C (7) D C A E B (7) C D A E B (7) C B D E A (7) C D A B E (6) D A E B C (5) B C E A D (5) A D E B C (5) E B A D C (4) C D B A E (4) C B E A D (4) B E A D C (4) D A C E B (3) B E C A D (3) B E A C D (3) A D E C B (2) D B E A C (1) C E B A D (1) C B E D A (1) C A D E B (1) B C E D A (1) A E B D C (1) Total count = 100 A B C D E A 0 24 6 -2 18 B -24 0 -4 -18 -22 C -6 4 0 -14 -6 D 2 18 14 0 14 E -18 22 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 6 -2 18 B -24 0 -4 -18 -22 C -6 4 0 -14 -6 D 2 18 14 0 14 E -18 22 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 A=17 B=16 E=11 so E is eliminated. Round 2 votes counts: C=31 D=25 A=24 B=20 so B is eliminated. Round 3 votes counts: C=40 A=35 D=25 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:224 A:223 E:198 C:189 B:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 6 -2 18 B -24 0 -4 -18 -22 C -6 4 0 -14 -6 D 2 18 14 0 14 E -18 22 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 6 -2 18 B -24 0 -4 -18 -22 C -6 4 0 -14 -6 D 2 18 14 0 14 E -18 22 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 6 -2 18 B -24 0 -4 -18 -22 C -6 4 0 -14 -6 D 2 18 14 0 14 E -18 22 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5765: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D A E C B (6) E B A D C (5) D A C B E (5) A D C B E (5) D A E B C (4) C D A B E (4) C B E D A (4) E B D A C (3) E B C D A (3) D E A B C (3) C B E A D (3) C B D E A (3) C B D A E (3) C B A E D (3) C A D B E (3) B E C A D (3) A D E B C (3) E B D C A (2) E B A C D (2) C D B E A (2) B C E A D (2) B C A E D (2) A D B E C (2) A B C D E (2) E D B C A (1) E D A B C (1) E A D B C (1) D E C B A (1) D E A C B (1) D C E B A (1) D C A E B (1) D A C E B (1) C D B A E (1) C A B D E (1) B E A C D (1) B A E C D (1) A E D B C (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -2 0 0 B 8 0 2 4 2 C 2 -2 0 4 -6 D 0 -4 -4 0 6 E 0 -2 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 0 0 B 8 0 2 4 2 C 2 -2 0 4 -6 D 0 -4 -4 0 6 E 0 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 D=23 A=15 B=9 so B is eliminated. Round 2 votes counts: C=31 E=30 D=23 A=16 so A is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:208 C:199 D:199 E:199 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 0 0 B 8 0 2 4 2 C 2 -2 0 4 -6 D 0 -4 -4 0 6 E 0 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 0 B 8 0 2 4 2 C 2 -2 0 4 -6 D 0 -4 -4 0 6 E 0 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 0 B 8 0 2 4 2 C 2 -2 0 4 -6 D 0 -4 -4 0 6 E 0 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5766: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (13) B D E C A (8) A C E D B (7) B D C E A (6) D B E A C (5) B E D A C (5) E A D B C (4) D B E C A (4) C A E D B (4) A E C B D (4) E B D A C (3) D B C E A (3) D B C A E (3) C A D E B (3) E B A D C (2) E A C B D (2) D C B A E (2) C D B A E (2) C A D B E (2) C A B D E (2) B D E A C (2) E D B A C (1) E D A B C (1) E B A C D (1) E A D C B (1) C D A E B (1) C D A B E (1) C B A D E (1) C A E B D (1) B D C A E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 6 -2 0 B 4 0 4 -18 0 C -6 -4 0 -4 -14 D 2 18 4 0 -2 E 0 0 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.092035 C: 0.000000 D: 0.000000 E: 0.907965 Sum of squares = 0.832871529786 Cumulative probabilities = A: 0.000000 B: 0.092035 C: 0.092035 D: 0.092035 E: 1.000000 A B C D E A 0 -4 6 -2 0 B 4 0 4 -18 0 C -6 -4 0 -4 -14 D 2 18 4 0 -2 E 0 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.900000 Sum of squares = 0.82000011069 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.100000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 D=17 C=17 E=15 so E is eliminated. Round 2 votes counts: A=33 B=31 D=19 C=17 so C is eliminated. Round 3 votes counts: A=45 B=32 D=23 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 E:208 A:200 B:195 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 -2 0 B 4 0 4 -18 0 C -6 -4 0 -4 -14 D 2 18 4 0 -2 E 0 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.900000 Sum of squares = 0.82000011069 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.100000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -2 0 B 4 0 4 -18 0 C -6 -4 0 -4 -14 D 2 18 4 0 -2 E 0 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.900000 Sum of squares = 0.82000011069 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.100000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -2 0 B 4 0 4 -18 0 C -6 -4 0 -4 -14 D 2 18 4 0 -2 E 0 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.900000 Sum of squares = 0.82000011069 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.100000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5767: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) E A B D C (7) C D A E B (7) E B A D C (6) D C A B E (5) E B C A D (4) E B A C D (4) C D A B E (4) D A B E C (3) C E B D A (3) C D B A E (3) B E A D C (3) B A E D C (3) A E B D C (3) A D B E C (3) A B E D C (3) D B A C E (2) D A E B C (2) D A C B E (2) C E D A B (2) C D E B A (2) C D E A B (2) C D B E A (2) B E C A D (2) B E A C D (2) B A D E C (2) E C B A D (1) D C B A E (1) D C A E B (1) C E D B A (1) C B E D A (1) C B E A D (1) B C A E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -8 14 -8 B 10 0 2 16 -12 C 8 -2 0 6 2 D -14 -16 -6 0 -16 E 8 12 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000116 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -10 -8 14 -8 B 10 0 2 16 -12 C 8 -2 0 6 2 D -14 -16 -6 0 -16 E 8 12 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.59374999996 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=22 D=16 B=13 A=10 so A is eliminated. Round 2 votes counts: C=39 E=25 D=19 B=17 so B is eliminated. Round 3 votes counts: C=40 E=38 D=22 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:217 B:208 C:207 A:194 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 14 -8 B 10 0 2 16 -12 C 8 -2 0 6 2 D -14 -16 -6 0 -16 E 8 12 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.59374999996 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 14 -8 B 10 0 2 16 -12 C 8 -2 0 6 2 D -14 -16 -6 0 -16 E 8 12 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.59374999996 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 14 -8 B 10 0 2 16 -12 C 8 -2 0 6 2 D -14 -16 -6 0 -16 E 8 12 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.59374999996 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5768: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) E A B D C (8) D B C E A (8) A E C B D (7) E A C D B (6) A E C D B (6) D C B E A (5) A E B C D (5) C D A E B (4) B E A D C (4) B A E D C (4) C D E A B (3) B D E C A (3) C D A B E (2) C A E D B (2) C A D E B (2) B D C A E (2) B C D A E (2) A E B D C (2) E D C A B (1) E D B A C (1) E C A D B (1) E A D C B (1) E A D B C (1) D C B A E (1) D B E C A (1) C D B A E (1) C B D A E (1) C A B D E (1) B D E A C (1) B A E C D (1) A C E D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 8 -8 B -10 0 8 6 -4 C 0 -8 0 -6 -6 D -8 -6 6 0 -6 E 8 4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 0 8 -8 B -10 0 8 6 -4 C 0 -8 0 -6 -6 D -8 -6 6 0 -6 E 8 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=23 E=19 C=16 D=15 so D is eliminated. Round 2 votes counts: B=36 A=23 C=22 E=19 so E is eliminated. Round 3 votes counts: A=39 B=37 C=24 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:212 A:205 B:200 D:193 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 0 8 -8 B -10 0 8 6 -4 C 0 -8 0 -6 -6 D -8 -6 6 0 -6 E 8 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 8 -8 B -10 0 8 6 -4 C 0 -8 0 -6 -6 D -8 -6 6 0 -6 E 8 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 8 -8 B -10 0 8 6 -4 C 0 -8 0 -6 -6 D -8 -6 6 0 -6 E 8 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5769: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) A E C B D (6) D B C E A (5) E A D B C (4) E A B D C (4) C B D A E (4) B C D A E (4) D E B A C (3) D C B E A (3) C D B E A (3) C A E D B (3) A C E B D (3) E D A C B (2) D E C B A (2) D E A B C (2) D C E A B (2) D B E A C (2) C D A E B (2) C B A D E (2) B D E A C (2) B D C A E (2) B C A E D (2) B A E D C (2) B A E C D (2) B A C E D (2) A E C D B (2) A E B D C (2) A E B C D (2) A C E D B (2) E D A B C (1) D E C A B (1) D E A C B (1) D B E C A (1) C E A D B (1) C D E A B (1) C B A E D (1) C A E B D (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D A C (1) B D C E A (1) B D A C E (1) B C A D E (1) B A C D E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 10 8 4 B -6 0 -4 0 -8 C -10 4 0 0 4 D -8 0 0 0 -4 E -4 8 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 8 4 B -6 0 -4 0 -8 C -10 4 0 0 4 D -8 0 0 0 -4 E -4 8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=22 C=21 B=21 A=19 E=17 so E is eliminated. Round 2 votes counts: A=33 D=25 C=21 B=21 so C is eliminated. Round 3 votes counts: A=41 D=31 B=28 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:202 C:199 D:194 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 8 4 B -6 0 -4 0 -8 C -10 4 0 0 4 D -8 0 0 0 -4 E -4 8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 8 4 B -6 0 -4 0 -8 C -10 4 0 0 4 D -8 0 0 0 -4 E -4 8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 8 4 B -6 0 -4 0 -8 C -10 4 0 0 4 D -8 0 0 0 -4 E -4 8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5770: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) B A E C D (8) E A B C D (4) D C B A E (4) B C E A D (4) A E B D C (4) A E B C D (4) A B E C D (4) E A D C B (3) D E C A B (3) D C E B A (3) D A B E C (3) C D E B A (3) C D B E A (3) C B D E A (3) B D A C E (3) B C A D E (3) E D C A B (2) D C B E A (2) D A E B C (2) C B E D A (2) B C D A E (2) B C A E D (2) A D E B C (2) A B E D C (2) A B D E C (2) E C A D B (1) E A D B C (1) E A C D B (1) E A C B D (1) D E A C B (1) D C E A B (1) D C A E B (1) D B C A E (1) C E B A D (1) C D B A E (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 -16 -6 20 0 B 16 0 6 24 24 C 6 -6 0 20 4 D -20 -24 -20 0 -12 E 0 -24 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 20 0 B 16 0 6 24 24 C 6 -6 0 20 4 D -20 -24 -20 0 -12 E 0 -24 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 D=21 A=18 E=13 so E is eliminated. Round 2 votes counts: A=28 C=25 B=24 D=23 so D is eliminated. Round 3 votes counts: C=41 A=34 B=25 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:235 C:212 A:199 E:192 D:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 20 0 B 16 0 6 24 24 C 6 -6 0 20 4 D -20 -24 -20 0 -12 E 0 -24 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 20 0 B 16 0 6 24 24 C 6 -6 0 20 4 D -20 -24 -20 0 -12 E 0 -24 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 20 0 B 16 0 6 24 24 C 6 -6 0 20 4 D -20 -24 -20 0 -12 E 0 -24 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5771: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) E D C A B (9) D E A B C (8) A B D C E (8) E C D B A (7) C E B D A (7) E D C B A (5) D A E B C (5) D E A C B (4) A D B E C (4) A B C D E (4) D A B E C (3) B C A E D (3) B C A D E (3) E D A C B (2) C E B A D (2) C B A E D (2) A D E B C (2) A D B C E (2) E C D A B (1) E C B D A (1) C A B E D (1) B D A C E (1) B C E D A (1) B A C D E (1) A D E C B (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 14 -4 -8 -14 B -14 0 -10 -8 -12 C 4 10 0 -8 -4 D 8 8 8 0 -8 E 14 12 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -4 -8 -14 B -14 0 -10 -8 -12 C 4 10 0 -8 -4 D 8 8 8 0 -8 E 14 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=24 C=22 D=20 B=9 so B is eliminated. Round 2 votes counts: C=29 E=25 A=25 D=21 so D is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:208 C:201 A:194 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 -8 -14 B -14 0 -10 -8 -12 C 4 10 0 -8 -4 D 8 8 8 0 -8 E 14 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -8 -14 B -14 0 -10 -8 -12 C 4 10 0 -8 -4 D 8 8 8 0 -8 E 14 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -8 -14 B -14 0 -10 -8 -12 C 4 10 0 -8 -4 D 8 8 8 0 -8 E 14 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5772: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (14) C D A E B (13) B E A D C (12) D A E C B (9) B E A C D (9) B C E A D (7) C D B A E (5) C D E A B (4) E A D B C (2) E A B D C (2) D A C E B (2) C B E D A (2) C B D E A (2) B A E D C (2) A E D C B (2) A D E B C (2) D C B A E (1) D A E B C (1) C E D A B (1) C E A B D (1) C D B E A (1) C B E A D (1) C B D A E (1) B D A E C (1) B C D E A (1) B C D A E (1) A E D B C (1) Total count = 100 A B C D E A 0 8 -10 -18 10 B -8 0 -18 -16 -8 C 10 18 0 -2 14 D 18 16 2 0 16 E -10 8 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 -18 10 B -8 0 -18 -16 -8 C 10 18 0 -2 14 D 18 16 2 0 16 E -10 8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=31 D=27 A=5 E=4 so E is eliminated. Round 2 votes counts: B=33 C=31 D=27 A=9 so A is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:220 A:195 E:184 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -10 -18 10 B -8 0 -18 -16 -8 C 10 18 0 -2 14 D 18 16 2 0 16 E -10 8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 -18 10 B -8 0 -18 -16 -8 C 10 18 0 -2 14 D 18 16 2 0 16 E -10 8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 -18 10 B -8 0 -18 -16 -8 C 10 18 0 -2 14 D 18 16 2 0 16 E -10 8 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5773: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) B C E D A (8) D E B C A (7) C B E D A (6) A C B E D (6) B C A E D (5) E D C B A (4) D E C B A (4) C E B D A (4) A B C D E (4) E C D B A (3) C B E A D (3) B C D E A (3) B A C E D (3) D B E C A (2) B D E C A (2) B A C D E (2) A E D C B (2) A D E B C (2) A C E D B (2) E D C A B (1) D E B A C (1) D E A B C (1) D B E A C (1) D A E C B (1) C E D B A (1) C E D A B (1) C B A E D (1) B C E A D (1) B C A D E (1) B A D E C (1) A E C D B (1) A D B C E (1) A C E B D (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -26 -14 0 -6 B 26 0 -4 8 8 C 14 4 0 16 12 D 0 -8 -16 0 -8 E 6 -8 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -14 0 -6 B 26 0 -4 8 8 C 14 4 0 16 12 D 0 -8 -16 0 -8 E 6 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=26 D=17 C=16 E=8 so E is eliminated. Round 2 votes counts: A=33 B=26 D=22 C=19 so C is eliminated. Round 3 votes counts: B=40 A=33 D=27 so D is eliminated. Round 4 votes counts: B=63 A=37 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:223 B:219 E:197 D:184 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -26 -14 0 -6 B 26 0 -4 8 8 C 14 4 0 16 12 D 0 -8 -16 0 -8 E 6 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -14 0 -6 B 26 0 -4 8 8 C 14 4 0 16 12 D 0 -8 -16 0 -8 E 6 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -14 0 -6 B 26 0 -4 8 8 C 14 4 0 16 12 D 0 -8 -16 0 -8 E 6 -8 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5774: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (12) B A C D E (12) A E D C B (10) A B C D E (9) C D E B A (8) E D C A B (6) A B E D C (6) E A D C B (4) C D E A B (4) B C D E A (4) B A E D C (4) A E B D C (4) B A E C D (3) B E C D A (2) B E A D C (2) A B E C D (2) D E C A B (1) D C E B A (1) D C E A B (1) C D B E A (1) C B D E A (1) C B D A E (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 14 14 6 B 4 0 -2 2 -2 C -14 2 0 -2 -12 D -14 -2 2 0 -10 E -6 2 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888894 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -4 14 14 6 B 4 0 -2 2 -2 C -14 2 0 -2 -12 D -14 -2 2 0 -10 E -6 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=28 E=22 C=15 D=3 so D is eliminated. Round 2 votes counts: A=32 B=28 E=23 C=17 so C is eliminated. Round 3 votes counts: E=37 A=32 B=31 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:209 B:201 D:188 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 14 6 B 4 0 -2 2 -2 C -14 2 0 -2 -12 D -14 -2 2 0 -10 E -6 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 14 6 B 4 0 -2 2 -2 C -14 2 0 -2 -12 D -14 -2 2 0 -10 E -6 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 14 6 B 4 0 -2 2 -2 C -14 2 0 -2 -12 D -14 -2 2 0 -10 E -6 2 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5775: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) E D C A B (7) E B C A D (5) B E C A D (5) B A C E D (5) B A C D E (5) A B D E C (5) D E A C B (4) C E D B A (4) B A E D C (4) E D A B C (3) E C D A B (3) D C E A B (3) C E B D A (3) E C B D A (2) D A E B C (2) D A C B E (2) C E D A B (2) C D E A B (2) C B E A D (2) B E A C D (2) B C E A D (2) A D B E C (2) A D B C E (2) A B D C E (2) E D A C B (1) E C B A D (1) D E C A B (1) D A E C B (1) C D A E B (1) C D A B E (1) B E A D C (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D E C (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -10 -2 -24 B 8 0 2 -2 -10 C 10 -2 0 16 -22 D 2 2 -16 0 -26 E 24 10 22 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -10 -2 -24 B 8 0 2 -2 -10 C 10 -2 0 16 -22 D 2 2 -16 0 -26 E 24 10 22 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=28 C=15 D=13 A=13 so D is eliminated. Round 2 votes counts: E=36 B=28 C=18 A=18 so C is eliminated. Round 3 votes counts: E=50 B=30 A=20 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:241 C:201 B:199 D:181 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -2 -24 B 8 0 2 -2 -10 C 10 -2 0 16 -22 D 2 2 -16 0 -26 E 24 10 22 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -2 -24 B 8 0 2 -2 -10 C 10 -2 0 16 -22 D 2 2 -16 0 -26 E 24 10 22 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -2 -24 B 8 0 2 -2 -10 C 10 -2 0 16 -22 D 2 2 -16 0 -26 E 24 10 22 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5776: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) A D E B C (7) C B E D A (5) B C A D E (5) A D B E C (5) E D A C B (4) D A E C B (4) C E B D A (4) C B D E A (4) B C E D A (4) E C D B A (3) D E A C B (3) B C D E A (3) B C A E D (3) A D E C B (3) A B D C E (3) E C D A B (2) E C B D A (2) D C E A B (2) D A C E B (2) D A B C E (2) B A C D E (2) A E D B C (2) A D B C E (2) A B E D C (2) E D C A B (1) E C B A D (1) E A D C B (1) E A B C D (1) D C A B E (1) D A C B E (1) C D E B A (1) C D B E A (1) C D B A E (1) B C D A E (1) B A D C E (1) B A C E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -4 -2 0 B 0 0 8 4 14 C 4 -8 0 6 14 D 2 -4 -6 0 10 E 0 -14 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.409572 B: 0.590428 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.516354563899 Cumulative probabilities = A: 0.409572 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -2 0 B 0 0 8 4 14 C 4 -8 0 6 14 D 2 -4 -6 0 10 E 0 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=26 C=16 E=15 D=15 so E is eliminated. Round 2 votes counts: B=28 A=28 C=24 D=20 so D is eliminated. Round 3 votes counts: A=44 C=28 B=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 C:208 D:201 A:197 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -2 0 B 0 0 8 4 14 C 4 -8 0 6 14 D 2 -4 -6 0 10 E 0 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 0 B 0 0 8 4 14 C 4 -8 0 6 14 D 2 -4 -6 0 10 E 0 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 0 B 0 0 8 4 14 C 4 -8 0 6 14 D 2 -4 -6 0 10 E 0 -14 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5777: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) B E D C A (6) E C D B A (5) D E B C A (5) D E B A C (5) E B D C A (4) C A E D B (4) C A B E D (4) B E D A C (4) A C B E D (4) A C B D E (4) E D C B A (3) D B E A C (3) C A E B D (3) C A D E B (3) A D C E B (3) A D B C E (3) E D B C A (2) C E D A B (2) C D E A B (2) C B E A D (2) B E C D A (2) B A D E C (2) A C D B E (2) A B C E D (2) A B C D E (2) E C B D A (1) E B C D A (1) D E C B A (1) D B A E C (1) D A B E C (1) C E A D B (1) C D A E B (1) B D A E C (1) B C E A D (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -6 -14 -14 B 12 0 10 4 8 C 6 -10 0 -8 -12 D 14 -4 8 0 -2 E 14 -8 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -14 -14 B 12 0 10 4 8 C 6 -10 0 -8 -12 D 14 -4 8 0 -2 E 14 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=23 A=23 C=22 E=16 D=16 so E is eliminated. Round 2 votes counts: C=28 B=28 A=23 D=21 so D is eliminated. Round 3 votes counts: B=44 C=32 A=24 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:210 D:208 C:188 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -14 -14 B 12 0 10 4 8 C 6 -10 0 -8 -12 D 14 -4 8 0 -2 E 14 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -14 -14 B 12 0 10 4 8 C 6 -10 0 -8 -12 D 14 -4 8 0 -2 E 14 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -14 -14 B 12 0 10 4 8 C 6 -10 0 -8 -12 D 14 -4 8 0 -2 E 14 -8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5778: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) D A C B E (9) D C E B A (8) E B C A D (6) D C E A B (6) B E A C D (6) A B E C D (6) C E D B A (5) D A B E C (4) E C B A D (3) E B C D A (3) C E B A D (3) C D E B A (3) A D B E C (3) A B E D C (3) D A B C E (2) C E B D A (2) B A E D C (2) B A E C D (2) A B D E C (2) E C D B A (1) E C B D A (1) D E C B A (1) D E B C A (1) D B E A C (1) D A C E B (1) C E A B D (1) C D E A B (1) C A D E B (1) B E D C A (1) A D B C E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -12 -18 -6 B -2 0 -14 -16 -14 C 12 14 0 -8 8 D 18 16 8 0 10 E 6 14 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -18 -6 B -2 0 -14 -16 -14 C 12 14 0 -8 8 D 18 16 8 0 10 E 6 14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 A=17 C=16 E=14 B=11 so B is eliminated. Round 2 votes counts: D=42 E=21 A=21 C=16 so C is eliminated. Round 3 votes counts: D=46 E=32 A=22 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:213 E:201 A:183 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -18 -6 B -2 0 -14 -16 -14 C 12 14 0 -8 8 D 18 16 8 0 10 E 6 14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -18 -6 B -2 0 -14 -16 -14 C 12 14 0 -8 8 D 18 16 8 0 10 E 6 14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -18 -6 B -2 0 -14 -16 -14 C 12 14 0 -8 8 D 18 16 8 0 10 E 6 14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5779: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) D C B E A (10) D C B A E (8) B A E D C (8) E B A D C (6) C D E A B (6) C D A E B (6) B E A D C (6) B D C E A (5) A C D E B (4) E A B C D (3) A E C D B (3) E A C B D (2) C D E B A (2) B D E C A (2) A C E D B (2) A B E D C (2) E C D A B (1) E C A D B (1) E B D C A (1) E B A C D (1) E A B D C (1) D B C E A (1) D B C A E (1) C E D A B (1) C D B E A (1) C D A B E (1) B E D C A (1) A E C B D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 4 6 -2 B 6 0 0 2 -4 C -4 0 0 -4 -2 D -6 -2 4 0 -4 E 2 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 6 -2 B 6 0 0 2 -4 C -4 0 0 -4 -2 D -6 -2 4 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=22 D=20 C=17 E=16 so E is eliminated. Round 2 votes counts: A=31 B=30 D=20 C=19 so C is eliminated. Round 3 votes counts: D=38 A=32 B=30 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 B:202 A:201 D:196 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 6 -2 B 6 0 0 2 -4 C -4 0 0 -4 -2 D -6 -2 4 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 6 -2 B 6 0 0 2 -4 C -4 0 0 -4 -2 D -6 -2 4 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 6 -2 B 6 0 0 2 -4 C -4 0 0 -4 -2 D -6 -2 4 0 -4 E 2 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5780: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) B A D C E (5) A E C D B (5) E C A D B (4) D B A E C (4) B D C E A (4) B C D E A (4) B C A E D (4) A E C B D (4) E D C A B (3) D A E C B (3) D A B E C (3) C E A B D (3) A E D C B (3) A D E B C (3) A C E B D (3) A B C E D (3) E A C D B (2) D E A C B (2) D A E B C (2) C E B A D (2) C B E A D (2) C A E B D (2) B D C A E (2) B C A D E (2) B A C E D (2) B A C D E (2) E C D A B (1) D E C B A (1) D E B A C (1) D E A B C (1) D C B E A (1) D B C E A (1) C D E B A (1) C B E D A (1) B D A C E (1) B C E D A (1) B C D A E (1) A D E C B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 8 14 B 0 0 10 0 6 C -4 -10 0 2 -2 D -8 0 -2 0 8 E -14 -6 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.492143 B: 0.507857 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500123479024 Cumulative probabilities = A: 0.492143 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 8 14 B 0 0 10 0 6 C -4 -10 0 2 -2 D -8 0 -2 0 8 E -14 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 A=24 C=11 E=10 so E is eliminated. Round 2 votes counts: D=30 B=28 A=26 C=16 so C is eliminated. Round 3 votes counts: A=35 B=33 D=32 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:208 D:199 C:193 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 8 14 B 0 0 10 0 6 C -4 -10 0 2 -2 D -8 0 -2 0 8 E -14 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 8 14 B 0 0 10 0 6 C -4 -10 0 2 -2 D -8 0 -2 0 8 E -14 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 8 14 B 0 0 10 0 6 C -4 -10 0 2 -2 D -8 0 -2 0 8 E -14 -6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5781: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) A B D C E (7) E D C B A (6) C A B E D (6) D B A E C (5) C E B A D (5) C E A B D (5) E C B D A (4) C D E A B (4) D E B A C (3) C A E B D (3) B A D E C (3) A C B D E (3) A B C E D (3) E D B C A (2) E C D B A (2) D B E A C (2) C E D B A (2) C E D A B (2) C E A D B (2) B A E D C (2) A D B E C (2) A D B C E (2) A C B E D (2) A B C D E (2) E D B A C (1) E C B A D (1) E B C A D (1) D E A B C (1) D A C E B (1) C E B D A (1) C D A E B (1) C B E A D (1) C A D E B (1) B E A C D (1) B D A E C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 2 8 8 B -14 0 -6 4 4 C -2 6 0 4 8 D -8 -4 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 8 8 B -14 0 -6 4 4 C -2 6 0 4 8 D -8 -4 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=23 D=20 E=17 B=7 so B is eliminated. Round 2 votes counts: C=33 A=28 D=21 E=18 so E is eliminated. Round 3 votes counts: C=41 D=30 A=29 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:216 C:208 B:194 E:192 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 8 8 B -14 0 -6 4 4 C -2 6 0 4 8 D -8 -4 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 8 8 B -14 0 -6 4 4 C -2 6 0 4 8 D -8 -4 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 8 8 B -14 0 -6 4 4 C -2 6 0 4 8 D -8 -4 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5782: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) A D C E B (10) B E C D A (9) D A C E B (8) B E C A D (8) A D C B E (8) E B C D A (7) C D A E B (6) B E A D C (5) B A E D C (5) D A C B E (4) B E A C D (4) E C B D A (3) A B D E C (3) C D E A B (2) A D B E C (2) C E D B A (1) B E D C A (1) B D E A C (1) B A D E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 10 6 -4 18 B -10 0 -4 -4 6 C -6 4 0 -20 0 D 4 4 20 0 12 E -18 -6 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -4 18 B -10 0 -4 -4 6 C -6 4 0 -20 0 D 4 4 20 0 12 E -18 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=25 D=22 E=10 C=9 so C is eliminated. Round 2 votes counts: B=34 D=30 A=25 E=11 so E is eliminated. Round 3 votes counts: B=44 D=31 A=25 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:215 B:194 C:189 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 -4 18 B -10 0 -4 -4 6 C -6 4 0 -20 0 D 4 4 20 0 12 E -18 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -4 18 B -10 0 -4 -4 6 C -6 4 0 -20 0 D 4 4 20 0 12 E -18 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -4 18 B -10 0 -4 -4 6 C -6 4 0 -20 0 D 4 4 20 0 12 E -18 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998169 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5783: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) D A B C E (6) A B D C E (6) A D B E C (5) A B E C D (5) A B D E C (5) E C D A B (4) E C B D A (4) D E C A B (4) E C D B A (3) E A B C D (3) D C B A E (3) D B A C E (3) B C A D E (3) B A C E D (3) E C A D B (2) E C A B D (2) D C E A B (2) D B C A E (2) D A B E C (2) C E D B A (2) C D E B A (2) C B D A E (2) A D B C E (2) E D C B A (1) E D C A B (1) E D A C B (1) E B C A D (1) E B A C D (1) E A D B C (1) E A C B D (1) D E C B A (1) D E A C B (1) D C E B A (1) D A E B C (1) D A C B E (1) C B E D A (1) C B E A D (1) B D C A E (1) B C E A D (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -2 2 4 B -14 0 6 -4 6 C 2 -6 0 -2 -14 D -2 4 2 0 10 E -4 -6 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.525719 B: 0.000000 C: 0.263375 D: 0.158438 E: 0.052469 Sum of squares = 0.373601934409 Cumulative probabilities = A: 0.525719 B: 0.525719 C: 0.789094 D: 0.947531 E: 1.000000 A B C D E A 0 14 -2 2 4 B -14 0 6 -4 6 C 2 -6 0 -2 -14 D -2 4 2 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378057 B: 0.000000 C: 0.317070 D: 0.292676 E: 0.012197 Sum of squares = 0.329268292796 Cumulative probabilities = A: 0.378057 B: 0.378057 C: 0.695127 D: 0.987803 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=27 A=25 C=8 B=8 so C is eliminated. Round 2 votes counts: E=34 D=29 A=25 B=12 so B is eliminated. Round 3 votes counts: E=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:207 B:197 E:197 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 14 -2 2 4 B -14 0 6 -4 6 C 2 -6 0 -2 -14 D -2 4 2 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378057 B: 0.000000 C: 0.317070 D: 0.292676 E: 0.012197 Sum of squares = 0.329268292796 Cumulative probabilities = A: 0.378057 B: 0.378057 C: 0.695127 D: 0.987803 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 2 4 B -14 0 6 -4 6 C 2 -6 0 -2 -14 D -2 4 2 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378057 B: 0.000000 C: 0.317070 D: 0.292676 E: 0.012197 Sum of squares = 0.329268292796 Cumulative probabilities = A: 0.378057 B: 0.378057 C: 0.695127 D: 0.987803 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 2 4 B -14 0 6 -4 6 C 2 -6 0 -2 -14 D -2 4 2 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.378057 B: 0.000000 C: 0.317070 D: 0.292676 E: 0.012197 Sum of squares = 0.329268292796 Cumulative probabilities = A: 0.378057 B: 0.378057 C: 0.695127 D: 0.987803 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5784: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) A C B E D (9) D E B C A (8) C A B D E (7) C B A D E (5) E D B C A (4) B E D A C (4) A B C E D (4) E D C A B (3) E D A B C (3) D E C B A (3) C A E D B (3) B D E A C (3) A E B D C (3) C D E B A (2) C D E A B (2) C B D A E (2) C A D E B (2) B A E D C (2) B A C D E (2) A E D C B (2) A E D B C (2) A C E D B (2) A B E D C (2) E C D A B (1) C D B E A (1) C B D E A (1) C A D B E (1) B E A D C (1) B D E C A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A D E C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 2 2 6 B 4 0 4 4 0 C -2 -4 0 -2 -4 D -2 -4 2 0 -10 E -6 0 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.757770 C: 0.000000 D: 0.000000 E: 0.242230 Sum of squares = 0.63289123308 Cumulative probabilities = A: 0.000000 B: 0.757770 C: 0.757770 D: 0.757770 E: 1.000000 A B C D E A 0 -4 2 2 6 B 4 0 4 4 0 C -2 -4 0 -2 -4 D -2 -4 2 0 -10 E -6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000041096 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 E=20 B=17 D=11 so D is eliminated. Round 2 votes counts: E=31 C=26 A=26 B=17 so B is eliminated. Round 3 votes counts: E=40 A=31 C=29 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:206 E:204 A:203 C:194 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 2 6 B 4 0 4 4 0 C -2 -4 0 -2 -4 D -2 -4 2 0 -10 E -6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000041096 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 2 6 B 4 0 4 4 0 C -2 -4 0 -2 -4 D -2 -4 2 0 -10 E -6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000041096 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 2 6 B 4 0 4 4 0 C -2 -4 0 -2 -4 D -2 -4 2 0 -10 E -6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000041096 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5785: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) D B C A E (7) D B A C E (7) C E A D B (6) E A C B D (5) D B C E A (5) B E D A C (5) B D E A C (5) A C E D B (5) E A B C D (3) D C B E A (3) C A D E B (3) B D E C A (3) A C E B D (3) E C A B D (2) E B A C D (2) C D A B E (2) C A D B E (2) B E D C A (2) A E C B D (2) E C D B A (1) E C A D B (1) E B D C A (1) D C B A E (1) D C A B E (1) D B E C A (1) D B E A C (1) C E D B A (1) C E D A B (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A E B (1) B E A D C (1) B D A E C (1) A E B C D (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -14 -4 -2 B -2 0 -8 -26 0 C 14 8 0 10 24 D 4 26 -10 0 -4 E 2 0 -24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 -4 -2 B -2 0 -8 -26 0 C 14 8 0 10 24 D 4 26 -10 0 -4 E 2 0 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 B=17 E=15 A=13 so A is eliminated. Round 2 votes counts: C=38 D=26 E=18 B=18 so E is eliminated. Round 3 votes counts: C=49 D=26 B=25 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 D:208 A:191 E:191 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 -4 -2 B -2 0 -8 -26 0 C 14 8 0 10 24 D 4 26 -10 0 -4 E 2 0 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -4 -2 B -2 0 -8 -26 0 C 14 8 0 10 24 D 4 26 -10 0 -4 E 2 0 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -4 -2 B -2 0 -8 -26 0 C 14 8 0 10 24 D 4 26 -10 0 -4 E 2 0 -24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5786: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (13) B C A E D (12) B E D C A (9) E D B A C (6) D E B C A (5) D E B A C (5) C A B D E (5) B E C A D (4) E D B C A (3) E B D C A (3) D A C E B (3) C A B E D (3) B C A D E (3) A C D B E (3) A C B E D (3) D E A B C (2) C B A E D (2) B E C D A (2) A D C E B (2) A C D E B (2) E D A B C (1) E B A D C (1) E A B C D (1) C D A B E (1) C B A D E (1) C A D B E (1) B D E C A (1) B C E A D (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 -20 -12 -8 -14 B 20 0 22 6 6 C 12 -22 0 -8 -14 D 8 -6 8 0 -6 E 14 -6 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999813 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -12 -8 -14 B 20 0 22 6 6 C 12 -22 0 -8 -14 D 8 -6 8 0 -6 E 14 -6 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993574 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=28 E=15 C=13 A=10 so A is eliminated. Round 2 votes counts: B=34 D=30 C=21 E=15 so E is eliminated. Round 3 votes counts: D=40 B=39 C=21 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:214 D:202 C:184 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -12 -8 -14 B 20 0 22 6 6 C 12 -22 0 -8 -14 D 8 -6 8 0 -6 E 14 -6 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993574 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -12 -8 -14 B 20 0 22 6 6 C 12 -22 0 -8 -14 D 8 -6 8 0 -6 E 14 -6 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993574 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -12 -8 -14 B 20 0 22 6 6 C 12 -22 0 -8 -14 D 8 -6 8 0 -6 E 14 -6 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993574 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5787: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) D C B A E (6) B D E C A (6) D C A E B (5) B E A C D (5) E A C D B (4) D C E A B (4) C A D E B (4) B E D A C (4) A E C D B (4) E A D C B (3) B D C E A (3) B D C A E (3) B C D A E (3) A C E D B (3) E B D A C (2) E B A C D (2) C D A B E (2) C B D A E (2) C A D B E (2) B E D C A (2) B E A D C (2) B A E C D (2) A E C B D (2) A C D E B (2) E D C A B (1) E D A C B (1) E B A D C (1) E A C B D (1) E A B D C (1) E A B C D (1) D E C A B (1) D E B C A (1) D C E B A (1) D C A B E (1) D B E C A (1) D B C E A (1) B C A D E (1) B A C D E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -14 -14 4 B -2 0 -14 -8 -2 C 14 14 0 0 4 D 14 8 0 0 16 E -4 2 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444003 D: 0.555997 E: 0.000000 Sum of squares = 0.506271381753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444003 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 -14 4 B -2 0 -14 -8 -2 C 14 14 0 0 4 D 14 8 0 0 16 E -4 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=21 E=17 C=17 A=13 so A is eliminated. Round 2 votes counts: B=33 E=23 C=23 D=21 so D is eliminated. Round 3 votes counts: C=40 B=35 E=25 so E is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:216 A:189 E:189 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 -14 4 B -2 0 -14 -8 -2 C 14 14 0 0 4 D 14 8 0 0 16 E -4 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -14 4 B -2 0 -14 -8 -2 C 14 14 0 0 4 D 14 8 0 0 16 E -4 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -14 4 B -2 0 -14 -8 -2 C 14 14 0 0 4 D 14 8 0 0 16 E -4 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5788: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) D C B E A (7) A E B C D (6) A B E D C (6) E A B C D (5) D C B A E (4) C E D B A (4) E C D A B (3) D B C A E (3) C D E B A (3) C D B E A (3) A E D C B (3) A B E C D (3) A B D E C (3) E C B A D (2) E C A D B (2) D C A B E (2) D A B C E (2) C D E A B (2) C B E D A (2) B E C A D (2) B E A C D (2) B C E D A (2) B A E C D (2) A E C D B (2) E C D B A (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C D B (1) E A C B D (1) D C E B A (1) D C A E B (1) D A C B E (1) C B D E A (1) B E C D A (1) B C D E A (1) B A E D C (1) B A D C E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -16 -6 -16 B 8 0 -10 8 2 C 16 10 0 26 -18 D 6 -8 -26 0 -26 E 16 -2 18 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.475555555564 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -8 -16 -6 -16 B 8 0 -10 8 2 C 16 10 0 26 -18 D 6 -8 -26 0 -26 E 16 -2 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.47555555554 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=25 D=21 C=15 B=12 so B is eliminated. Round 2 votes counts: E=32 A=29 D=21 C=18 so C is eliminated. Round 3 votes counts: E=40 D=31 A=29 so A is eliminated. Round 4 votes counts: E=63 D=37 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:217 B:204 A:177 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -16 -6 -16 B 8 0 -10 8 2 C 16 10 0 26 -18 D 6 -8 -26 0 -26 E 16 -2 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.47555555554 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -6 -16 B 8 0 -10 8 2 C 16 10 0 26 -18 D 6 -8 -26 0 -26 E 16 -2 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.47555555554 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -6 -16 B 8 0 -10 8 2 C 16 10 0 26 -18 D 6 -8 -26 0 -26 E 16 -2 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.47555555554 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5789: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) A C D E B (6) E D C B A (5) D E A C B (5) D E B C A (4) C B A E D (4) B E D C A (4) A D C E B (4) A C B D E (4) A B D C E (4) D A E C B (3) C B E A D (3) C A E D B (3) C A B E D (3) B E C D A (3) E C D B A (2) D E B A C (2) D B E A C (2) D A E B C (2) C E A D B (2) B C E A D (2) B A C E D (2) B A C D E (2) A D B C E (2) A C D B E (2) E D C A B (1) E C D A B (1) E C B D A (1) E B D C A (1) E B C D A (1) D E C A B (1) C E D A B (1) C E B D A (1) C E B A D (1) B D E A C (1) B D A E C (1) B C E D A (1) B C A E D (1) B A D E C (1) A D E C B (1) A D C B E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 0 -4 B 4 0 -10 -16 -10 C 6 10 0 -6 0 D 0 16 6 0 0 E 4 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.450578 E: 0.549422 Sum of squares = 0.504885134512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.450578 E: 1.000000 A B C D E A 0 -4 -6 0 -4 B 4 0 -10 -16 -10 C 6 10 0 -6 0 D 0 16 6 0 0 E 4 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=19 D=19 C=18 B=18 so C is eliminated. Round 2 votes counts: A=32 B=25 E=24 D=19 so D is eliminated. Round 3 votes counts: A=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:207 C:205 A:193 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 0 -4 B 4 0 -10 -16 -10 C 6 10 0 -6 0 D 0 16 6 0 0 E 4 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 -4 B 4 0 -10 -16 -10 C 6 10 0 -6 0 D 0 16 6 0 0 E 4 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 -4 B 4 0 -10 -16 -10 C 6 10 0 -6 0 D 0 16 6 0 0 E 4 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5790: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) E B D C A (10) C A E B D (7) A C D B E (7) D B E C A (6) E B D A C (5) B E D C A (5) B E D A C (5) A D B E C (4) A C E B D (4) E B A D C (3) C E B D A (3) B D E A C (3) A D C B E (3) E C B D A (2) D B A E C (2) D A C B E (2) C E B A D (2) A E B C D (2) E C B A D (1) E B C D A (1) E B A C D (1) E A B C D (1) D C B E A (1) C E A B D (1) C D A B E (1) C A D E B (1) B E A D C (1) B D E C A (1) A E B D C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -28 16 -18 -28 B 28 0 28 18 8 C -16 -28 0 -30 -32 D 18 -18 30 0 -10 E 28 -8 32 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 16 -18 -28 B 28 0 28 18 8 C -16 -28 0 -30 -32 D 18 -18 30 0 -10 E 28 -8 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=23 A=23 C=15 B=15 so C is eliminated. Round 2 votes counts: A=31 E=30 D=24 B=15 so B is eliminated. Round 3 votes counts: E=41 A=31 D=28 so D is eliminated. Round 4 votes counts: E=64 A=36 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:241 E:231 D:210 A:171 C:147 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 16 -18 -28 B 28 0 28 18 8 C -16 -28 0 -30 -32 D 18 -18 30 0 -10 E 28 -8 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 16 -18 -28 B 28 0 28 18 8 C -16 -28 0 -30 -32 D 18 -18 30 0 -10 E 28 -8 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 16 -18 -28 B 28 0 28 18 8 C -16 -28 0 -30 -32 D 18 -18 30 0 -10 E 28 -8 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5791: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) C D A E B (9) D C E B A (6) E B C D A (5) C D E A B (5) A C D B E (5) E B A C D (4) D C A B E (4) D B E C A (4) C A D E B (4) A C B E D (4) E B C A D (3) B E D A C (3) B E A C D (3) D C E A B (2) D C A E B (2) D B E A C (2) D A B C E (2) B A E D C (2) A C E B D (2) A B E C D (2) A B D C E (2) A B C E D (2) E D B C A (1) E C D B A (1) E C B D A (1) C E D A B (1) C E A D B (1) C D E B A (1) C A E D B (1) A E B C D (1) A D C B E (1) A D B C E (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -2 2 -6 B -8 0 -2 -6 0 C 2 2 0 14 14 D -2 6 -14 0 4 E 6 0 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 2 -6 B -8 0 -2 -6 0 C 2 2 0 14 14 D -2 6 -14 0 4 E 6 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=22 C=22 B=18 E=15 so E is eliminated. Round 2 votes counts: B=30 C=24 D=23 A=23 so D is eliminated. Round 3 votes counts: C=38 B=37 A=25 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:201 D:197 E:194 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 2 -6 B -8 0 -2 -6 0 C 2 2 0 14 14 D -2 6 -14 0 4 E 6 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 2 -6 B -8 0 -2 -6 0 C 2 2 0 14 14 D -2 6 -14 0 4 E 6 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 2 -6 B -8 0 -2 -6 0 C 2 2 0 14 14 D -2 6 -14 0 4 E 6 0 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5792: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) E D B C A (10) A C B D E (8) D E B A C (6) C A E B D (4) B E D C A (4) B A C D E (4) E D C B A (3) E B D C A (3) D E A B C (3) C E A D B (3) C B A E D (3) C A E D B (3) A D C E B (3) E C D B A (2) D E A C B (2) D B E A C (2) D A E C B (2) C A B D E (2) B C A D E (2) A D B C E (2) A C D B E (2) E D C A B (1) E D B A C (1) E C A D B (1) E B C D A (1) D B A E C (1) D A E B C (1) C E B A D (1) C E A B D (1) B D E A C (1) B D C E A (1) B C E A D (1) B C A E D (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -18 12 6 B -6 0 -8 2 -4 C 18 8 0 6 12 D -12 -2 -6 0 -10 E -6 4 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -18 12 6 B -6 0 -8 2 -4 C 18 8 0 6 12 D -12 -2 -6 0 -10 E -6 4 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=22 A=18 D=17 B=14 so B is eliminated. Round 2 votes counts: C=33 E=26 A=22 D=19 so D is eliminated. Round 3 votes counts: E=40 C=34 A=26 so A is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:203 E:198 B:192 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -18 12 6 B -6 0 -8 2 -4 C 18 8 0 6 12 D -12 -2 -6 0 -10 E -6 4 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 12 6 B -6 0 -8 2 -4 C 18 8 0 6 12 D -12 -2 -6 0 -10 E -6 4 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 12 6 B -6 0 -8 2 -4 C 18 8 0 6 12 D -12 -2 -6 0 -10 E -6 4 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5793: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) E C D A B (9) A D B C E (7) B E A C D (5) D A C E B (4) B E C A D (4) E C B D A (3) E C A D B (3) E B C A D (3) D C A E B (3) C E D B A (3) B C E D A (3) B C D A E (3) E C D B A (2) E A D C B (2) E A C D B (2) D C E A B (2) D C A B E (2) C D A E B (2) B D A C E (2) B C A E D (2) A D E C B (2) A D E B C (2) A D C E B (2) A B E D C (2) E D C A B (1) E B C D A (1) E B A D C (1) E A B C D (1) C E D A B (1) C E B D A (1) C B E D A (1) C B D A E (1) B C E A D (1) B C D E A (1) B A E D C (1) A E D C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -4 10 0 B 2 0 4 -2 -2 C 4 -4 0 4 10 D -10 2 -4 0 -6 E 0 2 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.46874999979 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 -4 10 0 B 2 0 4 -2 -2 C 4 -4 0 4 10 D -10 2 -4 0 -6 E 0 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.46875 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=28 A=17 D=11 C=9 so C is eliminated. Round 2 votes counts: B=37 E=33 A=17 D=13 so D is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:207 A:202 B:201 E:199 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 10 0 B 2 0 4 -2 -2 C 4 -4 0 4 10 D -10 2 -4 0 -6 E 0 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.46875 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 0 B 2 0 4 -2 -2 C 4 -4 0 4 10 D -10 2 -4 0 -6 E 0 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.46875 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 0 B 2 0 4 -2 -2 C 4 -4 0 4 10 D -10 2 -4 0 -6 E 0 2 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.46875 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5794: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) C D E B A (10) E B A C D (9) C D A B E (8) D C A B E (6) A B E D C (6) C D A E B (5) A B D E C (5) E B C A D (4) B A E D C (4) A D C B E (4) D A C B E (3) C D E A B (3) E C B D A (2) E C B A D (2) E B C D A (2) C E D B A (2) B E A D C (2) B A E C D (2) A D B C E (2) E D B C A (1) E B D A C (1) D A B E C (1) C A D B E (1) A D B E C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 8 12 0 B 6 0 8 6 -6 C -8 -8 0 4 -10 D -12 -6 -4 0 -2 E 0 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.302326 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.697674 Sum of squares = 0.578149907894 Cumulative probabilities = A: 0.302326 B: 0.302326 C: 0.302326 D: 0.302326 E: 1.000000 A B C D E A 0 -6 8 12 0 B 6 0 8 6 -6 C -8 -8 0 4 -10 D -12 -6 -4 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499839 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500161 Sum of squares = 0.500000051842 Cumulative probabilities = A: 0.499839 B: 0.499839 C: 0.499839 D: 0.499839 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=29 A=20 D=10 B=8 so B is eliminated. Round 2 votes counts: E=35 C=29 A=26 D=10 so D is eliminated. Round 3 votes counts: E=35 C=35 A=30 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 A:207 B:207 C:189 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 8 12 0 B 6 0 8 6 -6 C -8 -8 0 4 -10 D -12 -6 -4 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499839 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500161 Sum of squares = 0.500000051842 Cumulative probabilities = A: 0.499839 B: 0.499839 C: 0.499839 D: 0.499839 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 12 0 B 6 0 8 6 -6 C -8 -8 0 4 -10 D -12 -6 -4 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499839 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500161 Sum of squares = 0.500000051842 Cumulative probabilities = A: 0.499839 B: 0.499839 C: 0.499839 D: 0.499839 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 12 0 B 6 0 8 6 -6 C -8 -8 0 4 -10 D -12 -6 -4 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499839 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500161 Sum of squares = 0.500000051842 Cumulative probabilities = A: 0.499839 B: 0.499839 C: 0.499839 D: 0.499839 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5795: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) E C A D B (7) B D A C E (7) E C A B D (6) E A C D B (5) C E A B D (5) D B C E A (4) D B A E C (4) D B A C E (4) A B D C E (4) C A E B D (3) A C E B D (3) A C B E D (3) E D C B A (2) E C D B A (2) D B E A C (2) D A B E C (2) C E B D A (2) C A B D E (2) B D C E A (2) B C D A E (2) A E C D B (2) E D B C A (1) E D B A C (1) E D A B C (1) E A D C B (1) E A D B C (1) E A C B D (1) D E A B C (1) D B E C A (1) D B C A E (1) D A E B C (1) C B A D E (1) C A B E D (1) A E D B C (1) A E C B D (1) A D B E C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 0 10 B -8 0 4 10 6 C 4 -4 0 -4 12 D 0 -10 4 0 2 E -10 -6 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999971 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 0 10 B -8 0 4 10 6 C 4 -4 0 -4 12 D 0 -10 4 0 2 E -10 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999932 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=21 D=20 A=17 C=14 so C is eliminated. Round 2 votes counts: E=35 A=23 B=22 D=20 so D is eliminated. Round 3 votes counts: B=38 E=36 A=26 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:207 B:206 C:204 D:198 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -4 0 10 B -8 0 4 10 6 C 4 -4 0 -4 12 D 0 -10 4 0 2 E -10 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999932 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 0 10 B -8 0 4 10 6 C 4 -4 0 -4 12 D 0 -10 4 0 2 E -10 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999932 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 0 10 B -8 0 4 10 6 C 4 -4 0 -4 12 D 0 -10 4 0 2 E -10 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999932 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5796: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) B D E A C (8) C A E D B (7) E C A D B (5) D C A E B (5) C E A D B (5) B D A C E (5) D E B C A (4) B E A C D (4) E B A C D (3) E A C B D (3) D E C A B (3) D C E A B (3) B A C E D (3) E D B C A (2) D E C B A (2) D B E C A (2) D B C A E (2) D B A C E (2) B E D A C (2) B E A D C (2) A E C B D (2) A C B E D (2) E D C A B (1) E B C A D (1) D C E B A (1) D C A B E (1) C E A B D (1) C D A B E (1) C A E B D (1) C A D E B (1) B D A E C (1) B A D C E (1) B A C D E (1) A C E D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -16 10 -24 B -8 0 -12 4 -24 C 16 12 0 6 -10 D -10 -4 -6 0 -12 E 24 24 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -16 10 -24 B -8 0 -12 4 -24 C 16 12 0 6 -10 D -10 -4 -6 0 -12 E 24 24 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 D=25 C=16 A=7 so A is eliminated. Round 2 votes counts: B=28 E=27 D=25 C=20 so C is eliminated. Round 3 votes counts: E=43 B=30 D=27 so D is eliminated. Round 4 votes counts: E=62 B=38 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:235 C:212 A:189 D:184 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -16 10 -24 B -8 0 -12 4 -24 C 16 12 0 6 -10 D -10 -4 -6 0 -12 E 24 24 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 10 -24 B -8 0 -12 4 -24 C 16 12 0 6 -10 D -10 -4 -6 0 -12 E 24 24 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 10 -24 B -8 0 -12 4 -24 C 16 12 0 6 -10 D -10 -4 -6 0 -12 E 24 24 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5797: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) C A D E B (6) E D B C A (5) D E C A B (5) C D E A B (5) A B E D C (5) A B C E D (5) B A E D C (4) D C E B A (3) C D B E A (3) C B D E A (3) B E D A C (3) B E A D C (3) E D B A C (2) E D A C B (2) E D A B C (2) E B D A C (2) D E B C A (2) D C E A B (2) C D E B A (2) C B A D E (2) C A B D E (2) B E D C A (2) B D E C A (2) A E B D C (2) A C E D B (2) A C B E D (2) A C B D E (2) E A D B C (1) D E C B A (1) C D A E B (1) C D A B E (1) C A D B E (1) B D C E A (1) B C E D A (1) B C D E A (1) B C A D E (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -4 -2 -6 B -12 0 -10 -12 -8 C 4 10 0 2 14 D 2 12 -2 0 12 E 6 8 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 -2 -6 B -12 0 -10 -12 -8 C 4 10 0 2 14 D 2 12 -2 0 12 E 6 8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=26 B=19 E=14 D=13 so D is eliminated. Round 2 votes counts: C=31 A=28 E=22 B=19 so B is eliminated. Round 3 votes counts: C=35 A=33 E=32 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:212 A:200 E:194 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 -2 -6 B -12 0 -10 -12 -8 C 4 10 0 2 14 D 2 12 -2 0 12 E 6 8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 -2 -6 B -12 0 -10 -12 -8 C 4 10 0 2 14 D 2 12 -2 0 12 E 6 8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 -2 -6 B -12 0 -10 -12 -8 C 4 10 0 2 14 D 2 12 -2 0 12 E 6 8 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5798: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) E C D A B (7) C E A B D (7) A D B C E (7) C E A D B (5) C A B E D (5) E C D B A (4) D A B C E (4) B D A E C (4) A B C D E (4) E C B A D (3) E C A D B (3) D B E A C (3) B E C D A (3) A D C B E (3) A C D B E (3) E D C B A (2) E B D C A (2) D E B A C (2) D B A E C (2) C A E D B (2) C A E B D (2) A C D E B (2) E D C A B (1) E D B C A (1) E C B D A (1) D E C A B (1) D E A C B (1) D A B E C (1) C E B A D (1) C B A E D (1) B E D A C (1) B E C A D (1) B D E A C (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 16 -8 18 -2 B -16 0 -8 -8 6 C 8 8 0 12 12 D -18 8 -12 0 -8 E 2 -6 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -8 18 -2 B -16 0 -8 -8 6 C 8 8 0 12 12 D -18 8 -12 0 -8 E 2 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=23 B=20 A=19 D=14 so D is eliminated. Round 2 votes counts: E=28 B=25 A=24 C=23 so C is eliminated. Round 3 votes counts: E=41 A=33 B=26 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:220 A:212 E:196 B:187 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -8 18 -2 B -16 0 -8 -8 6 C 8 8 0 12 12 D -18 8 -12 0 -8 E 2 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 18 -2 B -16 0 -8 -8 6 C 8 8 0 12 12 D -18 8 -12 0 -8 E 2 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 18 -2 B -16 0 -8 -8 6 C 8 8 0 12 12 D -18 8 -12 0 -8 E 2 -6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5799: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) D E C B A (8) B C A D E (8) E D A C B (7) A B E C D (7) E A D B C (6) A B C E D (6) B A C D E (5) A B C D E (5) E D C B A (4) E D C A B (4) E A D C B (3) D C E B A (3) B C D A E (3) A E B D C (3) D E A C B (2) D E A B C (2) C D E B A (2) C B D A E (2) B A C E D (2) A E D B C (2) A E B C D (2) A B E D C (2) A B D C E (2) D A E B C (1) Total count = 100 A B C D E A 0 8 14 6 0 B -8 0 12 12 2 C -14 -12 0 2 -6 D -6 -12 -2 0 4 E 0 -2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.758296 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.241704 Sum of squares = 0.633433596688 Cumulative probabilities = A: 0.758296 B: 0.758296 C: 0.758296 D: 0.758296 E: 1.000000 A B C D E A 0 8 14 6 0 B -8 0 12 12 2 C -14 -12 0 2 -6 D -6 -12 -2 0 4 E 0 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 B=18 D=16 C=13 so C is eliminated. Round 2 votes counts: B=29 A=29 E=24 D=18 so D is eliminated. Round 3 votes counts: E=41 A=30 B=29 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:209 E:200 D:192 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 6 0 B -8 0 12 12 2 C -14 -12 0 2 -6 D -6 -12 -2 0 4 E 0 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 6 0 B -8 0 12 12 2 C -14 -12 0 2 -6 D -6 -12 -2 0 4 E 0 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 6 0 B -8 0 12 12 2 C -14 -12 0 2 -6 D -6 -12 -2 0 4 E 0 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5800: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D B A C E (6) A B E D C (6) D C B A E (5) D C E B A (4) D B C A E (4) B C E A D (4) A B E C D (4) E C B A D (3) D C E A B (3) D C B E A (3) D A E B C (3) D A B E C (3) C E D B A (3) B A E C D (3) E D C A B (2) E A C B D (2) E A B C D (2) D A E C B (2) D A B C E (2) C E B D A (2) C D E B A (2) C B E A D (2) C B D A E (2) B A D C E (2) A E D B C (2) A E B D C (2) A D B E C (2) A B D E C (2) E A D C B (1) D C A E B (1) C B E D A (1) C B D E A (1) B D C A E (1) B D A C E (1) B C D A E (1) B A C D E (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -18 -4 -4 14 B 18 0 6 2 12 C 4 -6 0 -16 18 D 4 -2 16 0 4 E -14 -12 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -4 -4 14 B 18 0 6 2 12 C 4 -6 0 -16 18 D 4 -2 16 0 4 E -14 -12 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=21 A=20 B=13 E=10 so E is eliminated. Round 2 votes counts: D=38 A=25 C=24 B=13 so B is eliminated. Round 3 votes counts: D=40 A=31 C=29 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 D:211 C:200 A:194 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -4 -4 14 B 18 0 6 2 12 C 4 -6 0 -16 18 D 4 -2 16 0 4 E -14 -12 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -4 14 B 18 0 6 2 12 C 4 -6 0 -16 18 D 4 -2 16 0 4 E -14 -12 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -4 14 B 18 0 6 2 12 C 4 -6 0 -16 18 D 4 -2 16 0 4 E -14 -12 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5801: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) E D B A C (8) B E D C A (8) C B A E D (7) D E A B C (6) B C E D A (6) D E A C B (5) B C A E D (5) A D E B C (5) A C D E B (5) A B C D E (4) E D C B A (2) D E C B A (2) A D E C B (2) A C B D E (2) A B D E C (2) E D A B C (1) E B D C A (1) E B D A C (1) D E C A B (1) D E B C A (1) D A E C B (1) C D E B A (1) C B E D A (1) C B D E A (1) C B A D E (1) C A B E D (1) C A B D E (1) B E D A C (1) B E C D A (1) B E A D C (1) B A E D C (1) A E D B C (1) A D C E B (1) A C D B E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 0 -16 -16 B 18 0 30 -6 -8 C 0 -30 0 -24 -24 D 16 6 24 0 -16 E 16 8 24 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 0 -16 -16 B 18 0 30 -6 -8 C 0 -30 0 -24 -24 D 16 6 24 0 -16 E 16 8 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 B=23 D=16 C=13 so C is eliminated. Round 2 votes counts: B=33 A=27 E=23 D=17 so D is eliminated. Round 3 votes counts: E=39 B=33 A=28 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:232 B:217 D:215 A:175 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 0 -16 -16 B 18 0 30 -6 -8 C 0 -30 0 -24 -24 D 16 6 24 0 -16 E 16 8 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -16 -16 B 18 0 30 -6 -8 C 0 -30 0 -24 -24 D 16 6 24 0 -16 E 16 8 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -16 -16 B 18 0 30 -6 -8 C 0 -30 0 -24 -24 D 16 6 24 0 -16 E 16 8 24 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5802: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (9) C E A B D (8) D B A E C (7) B D C E A (7) E C A B D (5) A D B E C (4) A D B C E (4) E C B D A (3) E B D C A (3) D B E A C (3) D A B E C (3) B D E C A (3) A E D C B (3) A D E C B (3) D B C E A (2) D B A C E (2) C E B D A (2) C E B A D (2) C A E B D (2) B E C D A (2) E D B C A (1) E D A B C (1) E B C D A (1) E A D C B (1) D A B C E (1) C B D A E (1) C A B E D (1) B C E D A (1) A E C B D (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -6 -4 -8 B -2 0 8 6 2 C 6 -8 0 -14 -6 D 4 -6 14 0 2 E 8 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000106 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 2 -6 -4 -8 B -2 0 8 6 2 C 6 -8 0 -14 -6 D 4 -6 14 0 2 E 8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=27 C=16 E=15 B=13 so B is eliminated. Round 2 votes counts: D=37 A=29 E=17 C=17 so E is eliminated. Round 3 votes counts: D=42 A=30 C=28 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:207 D:207 E:205 A:192 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -8 B -2 0 8 6 2 C 6 -8 0 -14 -6 D 4 -6 14 0 2 E 8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -8 B -2 0 8 6 2 C 6 -8 0 -14 -6 D 4 -6 14 0 2 E 8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -8 B -2 0 8 6 2 C 6 -8 0 -14 -6 D 4 -6 14 0 2 E 8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5803: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) A B E C D (11) D C E B A (7) B A E C D (7) C D B A E (6) D C B E A (5) C D B E A (5) A E B D C (5) E D C B A (4) B C D A E (4) A E B C D (4) D C B A E (3) E B A D C (2) D C E A B (2) B C E D A (2) B C A D E (2) E D C A B (1) E D A C B (1) E B C D A (1) E B A C D (1) E A B C D (1) D E C A B (1) D C A B E (1) C D A B E (1) C B D E A (1) B E C D A (1) B E C A D (1) B E A C D (1) B A C E D (1) A E D C B (1) A E D B C (1) A D C B E (1) A C D B E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 4 8 4 B 8 0 18 18 14 C -4 -18 0 6 -12 D -8 -18 -6 0 -18 E -4 -14 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 8 4 B 8 0 18 18 14 C -4 -18 0 6 -12 D -8 -18 -6 0 -18 E -4 -14 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 D=19 B=19 C=13 so C is eliminated. Round 2 votes counts: D=31 A=27 E=22 B=20 so B is eliminated. Round 3 votes counts: A=37 D=36 E=27 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:229 E:206 A:204 C:186 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 8 4 B 8 0 18 18 14 C -4 -18 0 6 -12 D -8 -18 -6 0 -18 E -4 -14 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 8 4 B 8 0 18 18 14 C -4 -18 0 6 -12 D -8 -18 -6 0 -18 E -4 -14 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 8 4 B 8 0 18 18 14 C -4 -18 0 6 -12 D -8 -18 -6 0 -18 E -4 -14 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5804: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) A C B D E (6) E B A C D (4) B A E C D (4) E B D A C (3) D E C B A (3) D C E B A (3) D C A B E (3) D B E A C (3) C E A D B (3) C A D B E (3) B A E D C (3) B A D E C (3) A B C E D (3) A B C D E (3) E C D A B (2) E C A B D (2) E B D C A (2) D C B A E (2) D B A C E (2) D A B C E (2) C D E A B (2) C D A E B (2) B E D A C (2) B E A D C (2) B E A C D (2) B A D C E (2) A C B E D (2) A B E C D (2) A B D C E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B D A (1) E C B A D (1) E B A D C (1) D C A E B (1) D B C A E (1) C E D A B (1) C E A B D (1) C A E D B (1) C A D E B (1) B A C E D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 8 8 2 B -4 0 -4 4 10 C -8 4 0 -2 14 D -8 -4 2 0 8 E -2 -10 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999284 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 2 B -4 0 -4 4 10 C -8 4 0 -2 14 D -8 -4 2 0 8 E -2 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=20 E=19 B=19 C=14 so C is eliminated. Round 2 votes counts: D=32 A=25 E=24 B=19 so B is eliminated. Round 3 votes counts: A=38 D=32 E=30 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:204 B:203 D:199 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 2 B -4 0 -4 4 10 C -8 4 0 -2 14 D -8 -4 2 0 8 E -2 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 2 B -4 0 -4 4 10 C -8 4 0 -2 14 D -8 -4 2 0 8 E -2 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 2 B -4 0 -4 4 10 C -8 4 0 -2 14 D -8 -4 2 0 8 E -2 -10 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996646 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5805: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) B A D C E (9) E C A D B (8) C E D B A (7) A B D E C (7) C D B E A (6) B D C A E (6) B D A C E (5) E A C D B (4) D C B A E (3) B C D A E (3) A E D C B (3) A B D C E (3) E A B D C (2) E A B C D (2) C E D A B (2) C B D E A (2) A E D B C (2) A D B C E (2) E C B D A (1) E C B A D (1) E A D C B (1) E A D B C (1) D C A B E (1) D B C A E (1) D B A C E (1) C E B D A (1) C D E B A (1) C D B A E (1) C B D A E (1) A E B D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -10 -4 2 B -2 0 -6 -10 6 C 10 6 0 0 10 D 4 10 0 0 6 E -2 -6 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.387862 D: 0.612138 E: 0.000000 Sum of squares = 0.525149958405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.387862 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -4 2 B -2 0 -6 -10 6 C 10 6 0 0 10 D 4 10 0 0 6 E -2 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=23 C=21 A=20 D=6 so D is eliminated. Round 2 votes counts: E=30 C=25 B=25 A=20 so A is eliminated. Round 3 votes counts: B=39 E=36 C=25 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:210 A:195 B:194 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -4 2 B -2 0 -6 -10 6 C 10 6 0 0 10 D 4 10 0 0 6 E -2 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -4 2 B -2 0 -6 -10 6 C 10 6 0 0 10 D 4 10 0 0 6 E -2 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -4 2 B -2 0 -6 -10 6 C 10 6 0 0 10 D 4 10 0 0 6 E -2 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5806: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (14) A D E C B (14) E D A B C (9) C B A D E (8) E D B A C (7) C A B D E (7) C B E D A (6) C B A E D (6) B E D C A (5) A D E B C (4) E D A C B (3) D E A B C (3) A C D E B (3) E D B C A (1) D A E B C (1) C E D B A (1) C E B D A (1) B E C D A (1) B D E A C (1) B C A D E (1) B A C D E (1) A D B E C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 -6 -4 B 6 0 0 4 6 C 2 0 0 2 0 D 6 -4 -2 0 -8 E 4 -6 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.341916 C: 0.658084 D: 0.000000 E: 0.000000 Sum of squares = 0.549981163639 Cumulative probabilities = A: 0.000000 B: 0.341916 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -6 -4 B 6 0 0 4 6 C 2 0 0 2 0 D 6 -4 -2 0 -8 E 4 -6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=24 B=23 E=20 D=4 so D is eliminated. Round 2 votes counts: C=29 A=25 E=23 B=23 so E is eliminated. Round 3 votes counts: A=40 B=31 C=29 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:203 C:202 D:196 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -6 -4 B 6 0 0 4 6 C 2 0 0 2 0 D 6 -4 -2 0 -8 E 4 -6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -6 -4 B 6 0 0 4 6 C 2 0 0 2 0 D 6 -4 -2 0 -8 E 4 -6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -6 -4 B 6 0 0 4 6 C 2 0 0 2 0 D 6 -4 -2 0 -8 E 4 -6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5807: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) D E B A C (8) D A B E C (6) C D A B E (6) C A B E D (6) C E B A D (5) C A D B E (5) C A B D E (5) E D B A C (4) C D E B A (4) A B E D C (4) A B E C D (4) E B A C D (3) C E D B A (3) B E A D C (3) E B D A C (2) D C A B E (2) D B A E C (2) C D E A B (2) A B D E C (2) A B C E D (2) E C D B A (1) E C B D A (1) E C B A D (1) D E A B C (1) D C E B A (1) D C E A B (1) D A B C E (1) C E B D A (1) C E A B D (1) B A E D C (1) A D B E C (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 10 8 -2 B -2 0 8 2 4 C -10 -8 0 2 -6 D -8 -2 -2 0 -4 E 2 -4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000045 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 2 10 8 -2 B -2 0 8 2 4 C -10 -8 0 2 -6 D -8 -2 -2 0 -4 E 2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000158 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=22 E=21 A=15 B=4 so B is eliminated. Round 2 votes counts: C=38 E=24 D=22 A=16 so A is eliminated. Round 3 votes counts: C=41 E=33 D=26 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:209 B:206 E:204 D:192 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 -2 B -2 0 8 2 4 C -10 -8 0 2 -6 D -8 -2 -2 0 -4 E 2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000158 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 -2 B -2 0 8 2 4 C -10 -8 0 2 -6 D -8 -2 -2 0 -4 E 2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000158 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 -2 B -2 0 8 2 4 C -10 -8 0 2 -6 D -8 -2 -2 0 -4 E 2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000158 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5808: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) B D E C A (8) A C D E B (8) E A C B D (7) D C A B E (7) D A C E B (6) A C E D B (6) D C A E B (5) B E D C A (5) D A C B E (4) B E C A D (4) E C A B D (3) E B A C D (3) A E C B D (3) E B C A D (2) E A B C D (2) D B A C E (2) C A E D B (2) B E A C D (2) E C B A D (1) D C B A E (1) D B E A C (1) D B C E A (1) D B A E C (1) C D A E B (1) C A D E B (1) B E D A C (1) B D C E A (1) B C D E A (1) A E C D B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 16 -2 -6 16 B -16 0 -16 -12 -6 C 2 16 0 -4 12 D 6 12 4 0 14 E -16 6 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -6 16 B -16 0 -16 -12 -6 C 2 16 0 -4 12 D 6 12 4 0 14 E -16 6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=22 A=20 E=18 C=4 so C is eliminated. Round 2 votes counts: D=37 A=23 B=22 E=18 so E is eliminated. Round 3 votes counts: D=37 A=35 B=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:213 A:212 E:182 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -2 -6 16 B -16 0 -16 -12 -6 C 2 16 0 -4 12 D 6 12 4 0 14 E -16 6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -6 16 B -16 0 -16 -12 -6 C 2 16 0 -4 12 D 6 12 4 0 14 E -16 6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -6 16 B -16 0 -16 -12 -6 C 2 16 0 -4 12 D 6 12 4 0 14 E -16 6 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5809: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) A C E B D (7) B D E C A (6) A C B D E (6) B C D E A (5) A C E D B (5) D B E C A (4) C A E B D (4) A E C D B (4) E D B C A (3) E C B D A (3) D E B C A (3) D E B A C (3) C E B D A (3) C E B A D (3) C E A B D (3) A C B E D (3) E D C B A (2) E D A C B (2) D B E A C (2) D B A E C (2) C B E D A (2) C B A E D (2) E D B A C (1) E C D B A (1) E A D C B (1) E A C D B (1) D E A B C (1) D A E B C (1) C A B E D (1) C A B D E (1) B E C D A (1) B C D A E (1) B C A D E (1) A E D C B (1) A D E C B (1) A D B E C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -12 -6 -14 B 10 0 -14 18 -8 C 12 14 0 16 12 D 6 -18 -16 0 -6 E 14 8 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -6 -14 B 10 0 -14 18 -8 C 12 14 0 16 12 D 6 -18 -16 0 -6 E 14 8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=21 C=19 D=16 E=14 so E is eliminated. Round 2 votes counts: A=32 D=24 C=23 B=21 so B is eliminated. Round 3 votes counts: D=37 A=32 C=31 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:227 E:208 B:203 D:183 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -6 -14 B 10 0 -14 18 -8 C 12 14 0 16 12 D 6 -18 -16 0 -6 E 14 8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -6 -14 B 10 0 -14 18 -8 C 12 14 0 16 12 D 6 -18 -16 0 -6 E 14 8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -6 -14 B 10 0 -14 18 -8 C 12 14 0 16 12 D 6 -18 -16 0 -6 E 14 8 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5810: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) D A E B C (8) C E A D B (7) B A D C E (7) C E B D A (6) E C A D B (4) C B E A D (4) A D B E C (4) E D A C B (3) D E A B C (3) D A B E C (3) C E B A D (3) B C E D A (3) A B D E C (3) E D C A B (2) E D A B C (2) E C D B A (2) E A D C B (2) C B E D A (2) C B A E D (2) B D A E C (2) A D E C B (2) A D C B E (2) E D C B A (1) E D B C A (1) E C B D A (1) D B A E C (1) D A E C B (1) C E D B A (1) C E D A B (1) C E A B D (1) C B A D E (1) B E C D A (1) B D E A C (1) B D A C E (1) B C A D E (1) B A D E C (1) B A C D E (1) A D C E B (1) Total count = 100 A B C D E A 0 14 -4 -8 -18 B -14 0 -14 -18 -20 C 4 14 0 -2 -12 D 8 18 2 0 -14 E 18 20 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -4 -8 -18 B -14 0 -14 -18 -20 C 4 14 0 -2 -12 D 8 18 2 0 -14 E 18 20 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=18 D=16 A=12 so A is eliminated. Round 2 votes counts: C=28 E=26 D=25 B=21 so B is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:232 D:207 C:202 A:192 B:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 -8 -18 B -14 0 -14 -18 -20 C 4 14 0 -2 -12 D 8 18 2 0 -14 E 18 20 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -8 -18 B -14 0 -14 -18 -20 C 4 14 0 -2 -12 D 8 18 2 0 -14 E 18 20 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -8 -18 B -14 0 -14 -18 -20 C 4 14 0 -2 -12 D 8 18 2 0 -14 E 18 20 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5811: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) B D E A C (8) C E A B D (7) C A E D B (7) E B D C A (6) D B E A C (6) C A D E B (6) C A E B D (5) A C D B E (5) E B D A C (4) D B A C E (4) D A B C E (4) E C B A D (3) B D E C A (3) E C A B D (2) E B A C D (2) D B C A E (2) D B A E C (2) B E D C A (2) B E D A C (2) A C E D B (2) E B C D A (1) E A D B C (1) D A C B E (1) C E B A D (1) C D A B E (1) C B D E A (1) B D C E A (1) A E D B C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -12 4 0 B -4 0 -2 -4 0 C 12 2 0 6 12 D -4 4 -6 0 6 E 0 0 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 4 0 B -4 0 -2 -4 0 C 12 2 0 6 12 D -4 4 -6 0 6 E 0 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=19 D=19 B=16 A=10 so A is eliminated. Round 2 votes counts: C=44 E=21 D=19 B=16 so B is eliminated. Round 3 votes counts: C=44 D=31 E=25 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:200 A:198 B:195 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 4 0 B -4 0 -2 -4 0 C 12 2 0 6 12 D -4 4 -6 0 6 E 0 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 4 0 B -4 0 -2 -4 0 C 12 2 0 6 12 D -4 4 -6 0 6 E 0 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 4 0 B -4 0 -2 -4 0 C 12 2 0 6 12 D -4 4 -6 0 6 E 0 0 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5812: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (17) A C B E D (15) C B E D A (7) C A B E D (6) B E D C A (6) A C D E B (6) E D B C A (5) D E B A C (4) D E C B A (3) C B A E D (3) B D E C A (3) A D E B C (3) A D B E C (3) D E A B C (2) D B E A C (2) C E D B A (2) C A D E B (2) A C E D B (2) C E B D A (1) C E A D B (1) C A E D B (1) B C E D A (1) A D E C B (1) A C B D E (1) A B E D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -16 -6 -8 B 8 0 -2 -8 0 C 16 2 0 -2 -2 D 6 8 2 0 -2 E 8 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.156608 C: 0.000000 D: 0.000000 E: 0.843392 Sum of squares = 0.735836370045 Cumulative probabilities = A: 0.000000 B: 0.156608 C: 0.156608 D: 0.156608 E: 1.000000 A B C D E A 0 -8 -16 -6 -8 B 8 0 -2 -8 0 C 16 2 0 -2 -2 D 6 8 2 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000022924 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=28 C=23 B=10 E=5 so E is eliminated. Round 2 votes counts: A=34 D=33 C=23 B=10 so B is eliminated. Round 3 votes counts: D=42 A=34 C=24 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:207 E:206 B:199 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -16 -6 -8 B 8 0 -2 -8 0 C 16 2 0 -2 -2 D 6 8 2 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000022924 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -6 -8 B 8 0 -2 -8 0 C 16 2 0 -2 -2 D 6 8 2 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000022924 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -6 -8 B 8 0 -2 -8 0 C 16 2 0 -2 -2 D 6 8 2 0 -2 E 8 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000022924 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5813: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C A B D E (6) D E C B A (5) D C E A B (5) C D E A B (5) C A D E B (5) B A C E D (5) E D B A C (4) C D A E B (4) B A E D C (4) B A E C D (4) A B E D C (4) B E D A C (3) E A D B C (2) D E B C A (2) D E A C B (2) C D E B A (2) C D B E A (2) B E A D C (2) B D E C A (2) B C A E D (2) A E D C B (2) A E D B C (2) A B E C D (2) A B C E D (2) E B D A C (1) D E C A B (1) D E B A C (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B E D (1) B D C E A (1) B C A D E (1) A E B D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 12 2 -2 4 B -12 0 6 -12 -8 C -2 -6 0 -4 -4 D 2 12 4 0 -2 E -4 8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 12 2 -2 4 B -12 0 6 -12 -8 C -2 -6 0 -4 -4 D 2 12 4 0 -2 E -4 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=24 D=16 E=15 A=15 so E is eliminated. Round 2 votes counts: C=30 D=28 B=25 A=17 so A is eliminated. Round 3 votes counts: D=34 B=34 C=32 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:208 D:208 E:205 C:192 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 -2 4 B -12 0 6 -12 -8 C -2 -6 0 -4 -4 D 2 12 4 0 -2 E -4 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -2 4 B -12 0 6 -12 -8 C -2 -6 0 -4 -4 D 2 12 4 0 -2 E -4 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -2 4 B -12 0 6 -12 -8 C -2 -6 0 -4 -4 D 2 12 4 0 -2 E -4 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5814: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (17) B E C D A (16) D C A E B (6) A E C D B (5) B E A C D (4) E B C A D (3) D A C B E (3) B E C A D (3) B D C E A (3) A D C B E (3) E C B D A (2) E B C D A (2) D C B A E (2) D A C E B (2) B A E D C (2) A E B C D (2) A D B C E (2) A B E D C (2) E C D A B (1) E C A D B (1) E B A C D (1) E A C B D (1) D C B E A (1) D C A B E (1) D B C A E (1) C E D B A (1) C E D A B (1) C D E B A (1) C B D E A (1) B E D C A (1) B E A D C (1) B D A C E (1) B C E D A (1) B C D E A (1) B A D C E (1) A E D B C (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 4 8 B -2 0 0 -2 4 C -2 0 0 -4 0 D -4 2 4 0 -2 E -8 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 4 8 B -2 0 0 -2 4 C -2 0 0 -4 0 D -4 2 4 0 -2 E -8 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=34 D=16 E=11 C=4 so C is eliminated. Round 2 votes counts: B=35 A=35 D=17 E=13 so E is eliminated. Round 3 votes counts: B=43 A=37 D=20 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:200 D:200 C:197 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 8 B -2 0 0 -2 4 C -2 0 0 -4 0 D -4 2 4 0 -2 E -8 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 8 B -2 0 0 -2 4 C -2 0 0 -4 0 D -4 2 4 0 -2 E -8 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 8 B -2 0 0 -2 4 C -2 0 0 -4 0 D -4 2 4 0 -2 E -8 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5815: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) A B D E C (13) C B E D A (11) B A D E C (10) C E D B A (9) C A D E B (8) B C E D A (6) C E D A B (5) E D B A C (4) C A E D B (3) B D E A C (3) A D E C B (3) C A B D E (2) B E D A C (2) A C D E B (2) D E A B C (1) C B A E D (1) B C A E D (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 2 6 18 18 B -2 0 12 2 2 C -6 -12 0 0 0 D -18 -2 0 0 16 E -18 -2 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999608 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 18 18 B -2 0 12 2 2 C -6 -12 0 0 0 D -18 -2 0 0 16 E -18 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=32 B=24 E=4 D=1 so D is eliminated. Round 2 votes counts: C=39 A=32 B=24 E=5 so E is eliminated. Round 3 votes counts: C=39 A=33 B=28 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:207 D:198 C:191 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 18 18 B -2 0 12 2 2 C -6 -12 0 0 0 D -18 -2 0 0 16 E -18 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 18 18 B -2 0 12 2 2 C -6 -12 0 0 0 D -18 -2 0 0 16 E -18 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 18 18 B -2 0 12 2 2 C -6 -12 0 0 0 D -18 -2 0 0 16 E -18 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5816: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (6) E A D B C (5) C B E A D (5) C B D E A (5) D A E B C (4) C E B D A (4) B C A D E (4) A B D E C (4) E D A C B (3) E C B A D (3) E A B D C (3) D E A C B (3) D C B A E (3) B C A E D (3) B A E C D (3) B A D C E (3) B A C E D (3) A E D B C (3) A D E B C (3) A D B E C (3) E C A B D (2) E A B C D (2) C E B A D (2) C D E B A (2) C B E D A (2) C B D A E (2) B A D E C (2) E B A C D (1) E A C B D (1) D E C A B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C D B E A (1) C D B A E (1) B C E A D (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 2 16 -6 B 4 0 0 14 -4 C -2 0 0 -4 0 D -16 -14 4 0 2 E 6 4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999682 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 A B C D E A 0 -4 2 16 -6 B 4 0 0 14 -4 C -2 0 0 -4 0 D -16 -14 4 0 2 E 6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999833 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=21 E=20 B=20 A=14 so A is eliminated. Round 2 votes counts: D=27 C=25 E=24 B=24 so E is eliminated. Round 3 votes counts: D=38 C=31 B=31 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:207 A:204 E:204 C:197 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 16 -6 B 4 0 0 14 -4 C -2 0 0 -4 0 D -16 -14 4 0 2 E 6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999833 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 16 -6 B 4 0 0 14 -4 C -2 0 0 -4 0 D -16 -14 4 0 2 E 6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999833 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 16 -6 B 4 0 0 14 -4 C -2 0 0 -4 0 D -16 -14 4 0 2 E 6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999833 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5817: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) A B D C E (7) C E B D A (6) C B E D A (5) C B D E A (5) B D C A E (5) E C D B A (4) E C D A B (4) A E D B C (4) A D B C E (4) E C B D A (3) E C A D B (3) E A C B D (3) B D A C E (3) B C D E A (3) B C D A E (3) A E D C B (3) A D B E C (3) E C A B D (2) C E D B A (2) B A D C E (2) E D A C B (1) E C B A D (1) E A D C B (1) E A B C D (1) D C E B A (1) D B C A E (1) D B A C E (1) D A B C E (1) C D E B A (1) B E C A D (1) B D C E A (1) B C E D A (1) B C E A D (1) B A C E D (1) B A C D E (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -6 -2 -16 B 4 0 -8 12 0 C 6 8 0 18 10 D 2 -12 -18 0 -10 E 16 0 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -2 -16 B 4 0 -8 12 0 C 6 8 0 18 10 D 2 -12 -18 0 -10 E 16 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=24 B=22 C=19 D=4 so D is eliminated. Round 2 votes counts: E=31 A=25 B=24 C=20 so C is eliminated. Round 3 votes counts: E=41 B=34 A=25 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:221 E:208 B:204 A:186 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 -16 B 4 0 -8 12 0 C 6 8 0 18 10 D 2 -12 -18 0 -10 E 16 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 -16 B 4 0 -8 12 0 C 6 8 0 18 10 D 2 -12 -18 0 -10 E 16 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 -16 B 4 0 -8 12 0 C 6 8 0 18 10 D 2 -12 -18 0 -10 E 16 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5818: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (14) B C D E A (13) D B C A E (6) B C E D A (6) E A C B D (5) B A E D C (5) E C A B D (4) E A C D B (4) B D C A E (4) B C E A D (4) D C B A E (3) A E D B C (3) A E B D C (3) E C A D B (2) D B A C E (2) D A B C E (2) C D B E A (2) C B D E A (2) B D C E A (2) E C B A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D B C E A (1) D A C E B (1) C E D A B (1) C E B A D (1) C B E D A (1) B A D E C (1) A E C D B (1) A E C B D (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -8 6 -4 B 12 0 10 12 10 C 8 -10 0 2 4 D -6 -12 -2 0 -18 E 4 -10 -4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 6 -4 B 12 0 10 12 10 C 8 -10 0 2 4 D -6 -12 -2 0 -18 E 4 -10 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=24 E=18 D=16 C=7 so C is eliminated. Round 2 votes counts: B=38 A=24 E=20 D=18 so D is eliminated. Round 3 votes counts: B=53 A=27 E=20 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:204 C:202 A:191 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 6 -4 B 12 0 10 12 10 C 8 -10 0 2 4 D -6 -12 -2 0 -18 E 4 -10 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 6 -4 B 12 0 10 12 10 C 8 -10 0 2 4 D -6 -12 -2 0 -18 E 4 -10 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 6 -4 B 12 0 10 12 10 C 8 -10 0 2 4 D -6 -12 -2 0 -18 E 4 -10 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5819: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) C B D A E (6) A E D C B (6) D C B E A (5) B C D E A (5) B C D A E (5) A E B C D (5) D E C A B (4) B A C E D (4) A E D B C (4) E A D C B (3) E A B D C (3) D C E A B (3) C D B A E (3) C B A D E (3) B C A E D (3) B C A D E (3) D E C B A (2) D C E B A (2) D C B A E (2) B E C D A (2) A E C D B (2) A E B D C (2) A C D E B (2) A B C E D (2) E D B C A (1) E D A B C (1) D E A C B (1) D C A E B (1) D A E C B (1) C B D E A (1) B E D C A (1) B E C A D (1) B C E D A (1) B A E C D (1) A D C E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -8 6 16 B 2 0 2 0 0 C 8 -2 0 2 6 D -6 0 -2 0 0 E -16 0 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.873834 C: 0.000000 D: 0.083518 E: 0.042648 Sum of squares = 0.772379804051 Cumulative probabilities = A: 0.000000 B: 0.873834 C: 0.873834 D: 0.957352 E: 1.000000 A B C D E A 0 -2 -8 6 16 B 2 0 2 0 0 C 8 -2 0 2 6 D -6 0 -2 0 0 E -16 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000017448 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 D=21 E=14 C=13 so C is eliminated. Round 2 votes counts: B=36 A=26 D=24 E=14 so E is eliminated. Round 3 votes counts: A=38 B=36 D=26 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:207 A:206 B:202 D:196 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 6 16 B 2 0 2 0 0 C 8 -2 0 2 6 D -6 0 -2 0 0 E -16 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000017448 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 6 16 B 2 0 2 0 0 C 8 -2 0 2 6 D -6 0 -2 0 0 E -16 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000017448 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 6 16 B 2 0 2 0 0 C 8 -2 0 2 6 D -6 0 -2 0 0 E -16 0 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000017448 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5820: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (14) C A B D E (11) E D B A C (10) D E B C A (7) E D A B C (6) C A B E D (5) A B C E D (5) D E B A C (4) C A D E B (4) E D A C B (3) B E D A C (3) B D E C A (3) B A E D C (3) D E C B A (2) C B A D E (2) B A C E D (2) A C E D B (2) E A D B C (1) D E C A B (1) C D E B A (1) C D E A B (1) C B D E A (1) C B D A E (1) B C D E A (1) B C A D E (1) A E D C B (1) A E C D B (1) A E C B D (1) A E B D C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 18 18 12 12 B -18 0 -4 12 6 C -18 4 0 8 4 D -12 -12 -8 0 -20 E -12 -6 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 18 12 12 B -18 0 -4 12 6 C -18 4 0 8 4 D -12 -12 -8 0 -20 E -12 -6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997039 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 E=20 D=14 B=13 so B is eliminated. Round 2 votes counts: A=32 C=28 E=23 D=17 so D is eliminated. Round 3 votes counts: E=40 A=32 C=28 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:199 E:199 B:198 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 18 12 12 B -18 0 -4 12 6 C -18 4 0 8 4 D -12 -12 -8 0 -20 E -12 -6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997039 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 12 12 B -18 0 -4 12 6 C -18 4 0 8 4 D -12 -12 -8 0 -20 E -12 -6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997039 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 12 12 B -18 0 -4 12 6 C -18 4 0 8 4 D -12 -12 -8 0 -20 E -12 -6 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997039 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5821: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) E A D B C (7) D E C B A (7) A B E C D (6) D C E B A (5) A B C E D (5) C B D A E (4) E D C A B (3) C E D B A (3) C B D E A (3) A E B D C (3) A E B C D (3) A B D C E (3) E D A C B (2) D E A C B (2) D A E B C (2) D A B C E (2) C D B A E (2) C B E A D (2) B C A E D (2) A E D B C (2) A B D E C (2) A B C D E (2) E D C B A (1) E D A B C (1) E C D B A (1) E A D C B (1) E A C B D (1) E A B C D (1) D E C A B (1) D C B A E (1) D B A C E (1) D A B E C (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B E A (1) C B A E D (1) B A C D E (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 20 10 10 B -6 0 12 2 0 C -20 -12 0 0 0 D -10 -2 0 0 -14 E -10 0 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 20 10 10 B -6 0 12 2 0 C -20 -12 0 0 0 D -10 -2 0 0 -14 E -10 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 C=19 E=18 B=12 so B is eliminated. Round 2 votes counts: A=39 D=22 C=21 E=18 so E is eliminated. Round 3 votes counts: A=49 D=29 C=22 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 B:204 E:202 D:187 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 20 10 10 B -6 0 12 2 0 C -20 -12 0 0 0 D -10 -2 0 0 -14 E -10 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 20 10 10 B -6 0 12 2 0 C -20 -12 0 0 0 D -10 -2 0 0 -14 E -10 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 20 10 10 B -6 0 12 2 0 C -20 -12 0 0 0 D -10 -2 0 0 -14 E -10 0 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5822: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D E B C A (8) A E B C D (6) C B E A D (5) C A B E D (5) A C B E D (5) E B D C A (4) E B C A D (4) D E B A C (4) A D E B C (4) E B A C D (3) D E A B C (3) D B E C A (3) C B D E A (3) A C D B E (3) E B A D C (2) D A E C B (2) D A C E B (2) C B E D A (2) C B A E D (2) C A B D E (2) B E C A D (2) E D B A C (1) E B D A C (1) E B C D A (1) E A B C D (1) D C B A E (1) D C A B E (1) D A E B C (1) C D B E A (1) B E C D A (1) B C E D A (1) A E D B C (1) A E C B D (1) A D E C B (1) A D C E B (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -16 -10 2 -18 B 16 0 2 6 -6 C 10 -2 0 2 -8 D -2 -6 -2 0 2 E 18 6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 A B C D E A 0 -16 -10 2 -18 B 16 0 2 6 -6 C 10 -2 0 2 -8 D -2 -6 -2 0 2 E 18 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=25 C=20 E=17 B=4 so B is eliminated. Round 2 votes counts: D=34 A=25 C=21 E=20 so E is eliminated. Round 3 votes counts: D=40 A=31 C=29 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:215 B:209 C:201 D:196 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -10 2 -18 B 16 0 2 6 -6 C 10 -2 0 2 -8 D -2 -6 -2 0 2 E 18 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 2 -18 B 16 0 2 6 -6 C 10 -2 0 2 -8 D -2 -6 -2 0 2 E 18 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 2 -18 B 16 0 2 6 -6 C 10 -2 0 2 -8 D -2 -6 -2 0 2 E 18 6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.000000 D: 0.428571 E: 0.428571 Sum of squares = 0.387755101997 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.571429 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5823: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (12) D B C A E (9) B C A D E (8) A C B E D (7) A C B D E (5) E D B C A (4) E D B A C (4) D E B C A (4) D C A B E (4) E D A C B (3) E B A C D (3) D B C E A (3) C B A D E (3) A E C B D (3) E D A B C (2) E A D C B (2) E A C D B (2) E A B C D (2) D E C A B (2) D C B A E (2) B D E C A (2) B D C A E (2) B C D A E (2) A C E B D (2) E D C A B (1) D E A C B (1) D C B E A (1) C A B D E (1) B E D C A (1) B A C E D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 2 6 2 B -2 0 -6 8 4 C -2 6 0 6 4 D -6 -8 -6 0 2 E -2 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 6 2 B -2 0 -6 8 4 C -2 6 0 6 4 D -6 -8 -6 0 2 E -2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=26 A=19 B=16 C=4 so C is eliminated. Round 2 votes counts: E=35 D=26 A=20 B=19 so B is eliminated. Round 3 votes counts: E=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:206 B:202 E:194 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 6 2 B -2 0 -6 8 4 C -2 6 0 6 4 D -6 -8 -6 0 2 E -2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 6 2 B -2 0 -6 8 4 C -2 6 0 6 4 D -6 -8 -6 0 2 E -2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 6 2 B -2 0 -6 8 4 C -2 6 0 6 4 D -6 -8 -6 0 2 E -2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5824: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) E B A C D (9) C D A B E (9) C D E B A (7) A B E D C (6) D C A E B (5) D C A B E (5) C E B A D (4) D C E A B (3) D A E B C (3) C B E A D (3) A E B D C (3) E B C A D (2) D C E B A (2) C E B D A (2) C D B E A (2) B E A C D (2) B A E C D (2) A B E C D (2) E C B A D (1) E A B D C (1) D E C B A (1) D E B A C (1) D E A C B (1) D E A B C (1) D A E C B (1) D A C B E (1) D A B E C (1) C E D B A (1) C D B A E (1) C D A E B (1) C B A E D (1) C A B E D (1) B E A D C (1) A D B E C (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -2 4 -8 B 4 0 -6 4 -18 C 2 6 0 4 0 D -4 -4 -4 0 -4 E 8 18 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.777738 D: 0.000000 E: 0.222262 Sum of squares = 0.654276722973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.777738 D: 0.777738 E: 1.000000 A B C D E A 0 -4 -2 4 -8 B 4 0 -6 4 -18 C 2 6 0 4 0 D -4 -4 -4 0 -4 E 8 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=25 E=23 A=15 B=5 so B is eliminated. Round 2 votes counts: C=32 E=26 D=25 A=17 so A is eliminated. Round 3 votes counts: E=39 C=34 D=27 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:206 A:195 B:192 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 4 -8 B 4 0 -6 4 -18 C 2 6 0 4 0 D -4 -4 -4 0 -4 E 8 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 4 -8 B 4 0 -6 4 -18 C 2 6 0 4 0 D -4 -4 -4 0 -4 E 8 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 4 -8 B 4 0 -6 4 -18 C 2 6 0 4 0 D -4 -4 -4 0 -4 E 8 18 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5825: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) E D A C B (5) D E C B A (5) D E C A B (5) C A B D E (5) B C A D E (5) A C B E D (5) E D A B C (4) C B A D E (4) B C D A E (4) B A C D E (4) A E C D B (4) C D E A B (3) B E D A C (3) B D C E A (3) B A C E D (3) A C E B D (3) E D B A C (2) E A D C B (2) D B E C A (2) C A D B E (2) A E B D C (2) A C E D B (2) E D C A B (1) D E B C A (1) D C E B A (1) D C B E A (1) D B C E A (1) C E D A B (1) C D B A E (1) C A D E B (1) B E A D C (1) B D E C A (1) B A E C D (1) A E D B C (1) A E C B D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 6 12 16 B -14 0 -4 10 10 C -6 4 0 18 14 D -12 -10 -18 0 -2 E -16 -10 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 12 16 B -14 0 -4 10 10 C -6 4 0 18 14 D -12 -10 -18 0 -2 E -16 -10 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 C=17 D=16 E=14 so E is eliminated. Round 2 votes counts: A=30 D=28 B=25 C=17 so C is eliminated. Round 3 votes counts: A=38 D=33 B=29 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:215 B:201 E:181 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 12 16 B -14 0 -4 10 10 C -6 4 0 18 14 D -12 -10 -18 0 -2 E -16 -10 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 12 16 B -14 0 -4 10 10 C -6 4 0 18 14 D -12 -10 -18 0 -2 E -16 -10 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 12 16 B -14 0 -4 10 10 C -6 4 0 18 14 D -12 -10 -18 0 -2 E -16 -10 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5826: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (13) C D A B E (9) D C A B E (7) E D C A B (6) D A C B E (6) C B A D E (5) B E A C D (5) E D B A C (3) E D A B C (3) E B C A D (3) D E C A B (3) C A B D E (3) E D C B A (2) E C D B A (2) E C B A D (2) E B C D A (2) E B A C D (2) D C A E B (2) A C B D E (2) A B D C E (2) E D A C B (1) E C D A B (1) E B D C A (1) D E A C B (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B E C (1) C E D B A (1) C E D A B (1) C E B A D (1) C A D B E (1) B E A D C (1) B A E C D (1) B A D E C (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 8 -6 -12 -10 B -8 0 -18 -10 -8 C 6 18 0 -16 -12 D 12 10 16 0 -2 E 10 8 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -6 -12 -10 B -8 0 -18 -10 -8 C 6 18 0 -16 -12 D 12 10 16 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 D=24 C=21 B=9 A=5 so A is eliminated. Round 2 votes counts: E=41 D=25 C=23 B=11 so B is eliminated. Round 3 votes counts: E=48 D=28 C=24 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:218 E:216 C:198 A:190 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 -12 -10 B -8 0 -18 -10 -8 C 6 18 0 -16 -12 D 12 10 16 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -12 -10 B -8 0 -18 -10 -8 C 6 18 0 -16 -12 D 12 10 16 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -12 -10 B -8 0 -18 -10 -8 C 6 18 0 -16 -12 D 12 10 16 0 -2 E 10 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5827: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (17) E C A D B (10) D E B C A (9) A C B E D (9) E D C A B (6) B D E A C (6) B D A C E (6) C A B E D (5) D E B A C (4) D B E A C (4) C A E B D (4) A C E D B (4) C A E D B (3) B D E C A (3) E A C D B (2) B C A D E (2) A C E B D (2) E D B C A (1) E D A C B (1) B D A E C (1) B C A E D (1) Total count = 100 A B C D E A 0 -8 12 18 8 B 8 0 8 12 8 C -12 -8 0 18 6 D -18 -12 -18 0 4 E -8 -8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999585 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 18 8 B 8 0 8 12 8 C -12 -8 0 18 6 D -18 -12 -18 0 4 E -8 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=20 D=17 A=15 C=12 so C is eliminated. Round 2 votes counts: B=36 A=27 E=20 D=17 so D is eliminated. Round 3 votes counts: B=40 E=33 A=27 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:215 C:202 E:187 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 18 8 B 8 0 8 12 8 C -12 -8 0 18 6 D -18 -12 -18 0 4 E -8 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 18 8 B 8 0 8 12 8 C -12 -8 0 18 6 D -18 -12 -18 0 4 E -8 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 18 8 B 8 0 8 12 8 C -12 -8 0 18 6 D -18 -12 -18 0 4 E -8 -8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5828: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (7) A B D C E (6) A B C E D (6) E D C B A (5) D E C B A (5) C E B A D (5) E C D B A (4) E C B D A (4) C B E A D (4) B A C D E (4) A D E B C (4) D E B C A (3) D A B E C (3) C B A E D (3) B C A E D (3) B C A D E (3) B A D C E (3) E D C A B (2) E D A C B (2) E C A B D (2) D E A C B (2) D B A C E (2) D A E B C (2) E C B A D (1) E A D C B (1) D E C A B (1) D E A B C (1) D B C E A (1) D A B C E (1) C E B D A (1) B D C A E (1) B C E A D (1) B C D A E (1) B A C E D (1) A E C B D (1) A D E C B (1) A D B E C (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 0 18 10 B 10 0 10 16 6 C 0 -10 0 4 10 D -18 -16 -4 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 18 10 B 10 0 10 16 6 C 0 -10 0 4 10 D -18 -16 -4 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=21 D=21 B=17 C=13 so C is eliminated. Round 2 votes counts: A=28 E=27 B=24 D=21 so D is eliminated. Round 3 votes counts: E=39 A=34 B=27 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:221 A:209 C:202 D:184 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 18 10 B 10 0 10 16 6 C 0 -10 0 4 10 D -18 -16 -4 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 18 10 B 10 0 10 16 6 C 0 -10 0 4 10 D -18 -16 -4 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 18 10 B 10 0 10 16 6 C 0 -10 0 4 10 D -18 -16 -4 0 6 E -10 -6 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5829: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) C B E A D (10) D B C A E (9) D A E B C (8) C B A E D (7) E A D B C (6) E A C B D (4) D B A C E (4) C B D A E (4) E C A B D (3) D E A B C (3) D C B A E (3) D E A C B (2) D A B E C (2) C D B E A (2) E D C A B (1) E D A C B (1) E C D A B (1) E C B A D (1) E C A D B (1) E A C D B (1) E A B C D (1) D C E A B (1) D A B C E (1) C E A B D (1) C B E D A (1) C B D E A (1) C B A D E (1) B D C A E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -20 10 -2 B 16 0 -12 6 6 C 20 12 0 10 18 D -10 -6 -10 0 -10 E 2 -6 -18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -20 10 -2 B 16 0 -12 6 6 C 20 12 0 10 18 D -10 -6 -10 0 -10 E 2 -6 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=33 E=20 B=5 A=5 so B is eliminated. Round 2 votes counts: C=39 D=34 E=20 A=7 so A is eliminated. Round 3 votes counts: C=40 D=35 E=25 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:208 E:194 A:186 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -20 10 -2 B 16 0 -12 6 6 C 20 12 0 10 18 D -10 -6 -10 0 -10 E 2 -6 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -20 10 -2 B 16 0 -12 6 6 C 20 12 0 10 18 D -10 -6 -10 0 -10 E 2 -6 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -20 10 -2 B 16 0 -12 6 6 C 20 12 0 10 18 D -10 -6 -10 0 -10 E 2 -6 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5830: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) E B C A D (4) C E D A B (4) C D E B A (4) B E A D C (4) A D B E C (4) A B E D C (4) E A B C D (3) D C E B A (3) D B E C A (3) D B A E C (3) D A C B E (3) A C E D B (3) E C B A D (2) E C A B D (2) E B C D A (2) D C B E A (2) D C B A E (2) D B C E A (2) D A B C E (2) C E D B A (2) C E A D B (2) C E A B D (2) C D E A B (2) C A E D B (2) B E D C A (2) B D E A C (2) B A D E C (2) A D C E B (2) A C D E B (2) A B D E C (2) E C B D A (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C E B (1) C E B D A (1) C E B A D (1) C D A E B (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B A E D C (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -2 4 -16 B 6 0 4 -10 -8 C 2 -4 0 -2 -2 D -4 10 2 0 -2 E 16 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 4 -16 B 6 0 4 -10 -8 C 2 -4 0 -2 -2 D -4 10 2 0 -2 E 16 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 A=21 E=19 B=14 so B is eliminated. Round 2 votes counts: E=27 D=27 A=24 C=22 so C is eliminated. Round 3 votes counts: E=39 D=34 A=27 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:203 C:197 B:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 4 -16 B 6 0 4 -10 -8 C 2 -4 0 -2 -2 D -4 10 2 0 -2 E 16 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 4 -16 B 6 0 4 -10 -8 C 2 -4 0 -2 -2 D -4 10 2 0 -2 E 16 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 4 -16 B 6 0 4 -10 -8 C 2 -4 0 -2 -2 D -4 10 2 0 -2 E 16 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5831: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) C A D E B (8) B E D A C (8) E C A D B (7) C A D B E (7) C A E D B (6) E B D A C (5) B D A C E (5) D B A C E (4) E B D C A (3) E B C A D (3) B D A E C (3) A C D E B (3) E D B A C (2) E C B A D (2) E C A B D (2) D B A E C (2) D A C B E (2) D A B C E (2) C E A D B (2) A D C B E (2) A C D B E (2) E B C D A (1) E A D C B (1) C E A B D (1) C A B D E (1) B E D C A (1) B D C A E (1) B C E A D (1) B C D A E (1) B C A D E (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 4 2 4 B 4 0 4 -4 4 C -4 -4 0 -4 0 D -2 4 4 0 8 E -4 -4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000059 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 2 4 B 4 0 4 -4 4 C -4 -4 0 -4 0 D -2 4 4 0 8 E -4 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 C=25 D=10 A=9 so A is eliminated. Round 2 votes counts: C=30 B=30 E=27 D=13 so D is eliminated. Round 3 votes counts: B=38 C=35 E=27 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:207 B:204 A:203 C:194 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 2 4 B 4 0 4 -4 4 C -4 -4 0 -4 0 D -2 4 4 0 8 E -4 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 2 4 B 4 0 4 -4 4 C -4 -4 0 -4 0 D -2 4 4 0 8 E -4 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 2 4 B 4 0 4 -4 4 C -4 -4 0 -4 0 D -2 4 4 0 8 E -4 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5832: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) B A E D C (7) C D E A B (5) D C E B A (4) A C B E D (4) A B D E C (4) E C D B A (3) D E B C A (3) D C E A B (3) D A B C E (3) C E D A B (3) C E B A D (3) C E A B D (3) B A D E C (3) E D C B A (2) E D B C A (2) E C B D A (2) E C B A D (2) E B A C D (2) D E C B A (2) D B A E C (2) C D A E B (2) C A E B D (2) A D B C E (2) A B D C E (2) A B C E D (2) E B D C A (1) E B D A C (1) D E B A C (1) D C A E B (1) D C A B E (1) D B E A C (1) D A B E C (1) C E B D A (1) C D E B A (1) C A E D B (1) C A D E B (1) C A D B E (1) C A B E D (1) B E A D C (1) B A E C D (1) A D B E C (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -18 -8 -10 B 8 0 -16 -10 -20 C 18 16 0 4 14 D 8 10 -4 0 -8 E 10 20 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 -8 -10 B 8 0 -16 -10 -20 C 18 16 0 4 14 D 8 10 -4 0 -8 E 10 20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 A=18 E=15 B=12 so B is eliminated. Round 2 votes counts: C=33 A=29 D=22 E=16 so E is eliminated. Round 3 votes counts: C=40 A=32 D=28 so D is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:212 D:203 B:181 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -18 -8 -10 B 8 0 -16 -10 -20 C 18 16 0 4 14 D 8 10 -4 0 -8 E 10 20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -8 -10 B 8 0 -16 -10 -20 C 18 16 0 4 14 D 8 10 -4 0 -8 E 10 20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -8 -10 B 8 0 -16 -10 -20 C 18 16 0 4 14 D 8 10 -4 0 -8 E 10 20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5833: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) C E B A D (9) D A B E C (7) D A E B C (6) B A E D C (5) D B A E C (4) D A E C B (4) C D B E A (4) E C A B D (3) E A B C D (3) D C B E A (3) C E D A B (3) C E A B D (3) C D E B A (3) C D E A B (3) B E A C D (3) A B E D C (3) E B A C D (2) D C E A B (2) C E B D A (2) C B E A D (2) A E B D C (2) A D E B C (2) E D C A B (1) E D A C B (1) E C B A D (1) D E C A B (1) D E A C B (1) D B A C E (1) C E D B A (1) C E A D B (1) C B A E D (1) B D A E C (1) B A D E C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -4 -14 -4 B 4 0 -16 -16 -10 C 4 16 0 -10 -6 D 14 16 10 0 6 E 4 10 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -14 -4 B 4 0 -16 -16 -10 C 4 16 0 -10 -6 D 14 16 10 0 6 E 4 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=32 E=11 B=10 A=9 so A is eliminated. Round 2 votes counts: D=41 C=32 E=14 B=13 so B is eliminated. Round 3 votes counts: D=43 C=32 E=25 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:207 C:202 A:187 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -14 -4 B 4 0 -16 -16 -10 C 4 16 0 -10 -6 D 14 16 10 0 6 E 4 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -14 -4 B 4 0 -16 -16 -10 C 4 16 0 -10 -6 D 14 16 10 0 6 E 4 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -14 -4 B 4 0 -16 -16 -10 C 4 16 0 -10 -6 D 14 16 10 0 6 E 4 10 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5834: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) E A D B C (7) C A E D B (6) B C E D A (6) C E A B D (5) A E D B C (5) D B A E C (4) A D E B C (4) D B C A E (3) B D C E A (3) B D A E C (3) A E D C B (3) E C A B D (2) E A D C B (2) E A B D C (2) D A E B C (2) D A B E C (2) D A B C E (2) C E B A D (2) C E A D B (2) C D B A E (2) C B E A D (2) B E D A C (2) B D E A C (2) B D C A E (2) B C D A E (2) A E C D B (2) E B A C D (1) D E A B C (1) D C A B E (1) D A C B E (1) C D A E B (1) C B E D A (1) C B D A E (1) B E C A D (1) B C D E A (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -6 -4 -4 B -4 0 10 -4 2 C 6 -10 0 -6 8 D 4 4 6 0 -2 E 4 -2 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142861 C: 0.053570 D: 0.357139 E: 0.446430 Sum of squares = 0.350127414932 Cumulative probabilities = A: 0.000000 B: 0.142861 C: 0.196430 D: 0.553570 E: 1.000000 A B C D E A 0 4 -6 -4 -4 B -4 0 10 -4 2 C 6 -10 0 -6 8 D 4 4 6 0 -2 E 4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.150000 C: 0.050000 D: 0.350000 E: 0.450000 Sum of squares = 0.349999999998 Cumulative probabilities = A: 0.000000 B: 0.150000 C: 0.200000 D: 0.550000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=22 D=16 A=16 E=14 so E is eliminated. Round 2 votes counts: C=34 A=27 B=23 D=16 so D is eliminated. Round 3 votes counts: C=35 A=35 B=30 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:206 B:202 C:199 E:198 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 -4 -4 B -4 0 10 -4 2 C 6 -10 0 -6 8 D 4 4 6 0 -2 E 4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.150000 C: 0.050000 D: 0.350000 E: 0.450000 Sum of squares = 0.349999999998 Cumulative probabilities = A: 0.000000 B: 0.150000 C: 0.200000 D: 0.550000 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -4 -4 B -4 0 10 -4 2 C 6 -10 0 -6 8 D 4 4 6 0 -2 E 4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.150000 C: 0.050000 D: 0.350000 E: 0.450000 Sum of squares = 0.349999999998 Cumulative probabilities = A: 0.000000 B: 0.150000 C: 0.200000 D: 0.550000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -4 -4 B -4 0 10 -4 2 C 6 -10 0 -6 8 D 4 4 6 0 -2 E 4 -2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.150000 C: 0.050000 D: 0.350000 E: 0.450000 Sum of squares = 0.349999999998 Cumulative probabilities = A: 0.000000 B: 0.150000 C: 0.200000 D: 0.550000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5835: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) C E B A D (6) C E A D B (6) B D A E C (6) B A D C E (5) E C D A B (4) C E B D A (4) C E D B A (3) C E A B D (3) A E C D B (3) A D B E C (3) E D C A B (2) E C D B A (2) E C A D B (2) D E C B A (2) D E A C B (2) D E A B C (2) D B A E C (2) D A E C B (2) D A B E C (2) C A E B D (2) B C D E A (2) B C D A E (2) B C A D E (2) B A C D E (2) A D E C B (2) A C E D B (2) A C E B D (2) A B D E C (2) A B D C E (2) E A C D B (1) D B E C A (1) D B E A C (1) D A E B C (1) C E D A B (1) C B E D A (1) C B E A D (1) C A E D B (1) B D C E A (1) A D E B C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 4 0 6 B 0 0 -10 4 -12 C -4 10 0 8 14 D 0 -4 -8 0 4 E -6 12 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.804173 B: 0.195827 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.685041873251 Cumulative probabilities = A: 0.804173 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 0 6 B 0 0 -10 4 -12 C -4 10 0 8 14 D 0 -4 -8 0 4 E -6 12 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738608 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=27 A=19 D=15 E=11 so E is eliminated. Round 2 votes counts: C=36 B=27 A=20 D=17 so D is eliminated. Round 3 votes counts: C=40 B=31 A=29 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:205 D:196 E:194 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 6 B 0 0 -10 4 -12 C -4 10 0 8 14 D 0 -4 -8 0 4 E -6 12 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738608 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 6 B 0 0 -10 4 -12 C -4 10 0 8 14 D 0 -4 -8 0 4 E -6 12 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738608 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 6 B 0 0 -10 4 -12 C -4 10 0 8 14 D 0 -4 -8 0 4 E -6 12 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836738608 Cumulative probabilities = A: 0.714286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5836: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (7) B C D E A (6) E A D C B (4) E A C D B (4) C B D E A (4) B C E D A (4) A D E C B (4) D C A E B (3) D B A C E (3) D A C B E (3) C E D B A (3) C D B A E (3) C B E D A (3) B E C A D (3) E A C B D (2) E A B D C (2) E A B C D (2) D C B A E (2) D B C A E (2) C E B D A (2) C E A D B (2) C D B E A (2) B D C A E (2) B D A E C (2) B D A C E (2) B C D A E (2) A E D B C (2) A E B D C (2) E C B A D (1) E C A B D (1) D B A E C (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D A B (1) C E B A D (1) C E A B D (1) C D A E B (1) C D A B E (1) B D E A C (1) B D C E A (1) B C E A D (1) A E C D B (1) A D E B C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -4 -14 -2 B 2 0 -18 -10 0 C 4 18 0 4 14 D 14 10 -4 0 2 E 2 0 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -14 -2 B 2 0 -18 -10 0 C 4 18 0 4 14 D 14 10 -4 0 2 E 2 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 A=19 D=17 E=16 so E is eliminated. Round 2 votes counts: A=33 C=26 B=24 D=17 so D is eliminated. Round 3 votes counts: A=39 C=31 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:211 E:193 A:189 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -14 -2 B 2 0 -18 -10 0 C 4 18 0 4 14 D 14 10 -4 0 2 E 2 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -14 -2 B 2 0 -18 -10 0 C 4 18 0 4 14 D 14 10 -4 0 2 E 2 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -14 -2 B 2 0 -18 -10 0 C 4 18 0 4 14 D 14 10 -4 0 2 E 2 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5837: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) E C D A B (7) E B C D A (5) E B A C D (5) D C A B E (5) C D A E B (5) B E A D C (5) B A D C E (5) D C A E B (4) C D E A B (4) B E A C D (4) A B D C E (4) E C D B A (3) B E D C A (3) B A E C D (3) A B E C D (3) E B D C A (2) E B C A D (2) D C E A B (2) B E D A C (2) B A E D C (2) B A D E C (2) A D B C E (2) A C D E B (2) A C D B E (2) E D C B A (1) E C B D A (1) E C A B D (1) D A C B E (1) C E D A B (1) B D C A E (1) B D A C E (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 4 4 B -6 0 6 6 8 C -6 -6 0 0 -4 D -4 -6 0 0 -2 E -4 -8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 4 4 B -6 0 6 6 8 C -6 -6 0 0 -4 D -4 -6 0 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=27 A=23 D=12 C=10 so C is eliminated. Round 2 votes counts: E=28 B=28 A=23 D=21 so D is eliminated. Round 3 votes counts: A=38 E=34 B=28 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:207 E:197 D:194 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 4 4 B -6 0 6 6 8 C -6 -6 0 0 -4 D -4 -6 0 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 4 B -6 0 6 6 8 C -6 -6 0 0 -4 D -4 -6 0 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 4 B -6 0 6 6 8 C -6 -6 0 0 -4 D -4 -6 0 0 -2 E -4 -8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5838: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (12) B E A D C (9) B D A C E (7) A D E C B (5) A D B E C (5) D A C E B (4) B E C A D (4) B D A E C (4) E C A D B (3) E B C A D (3) E A D C B (3) D A B C E (3) C D E A B (3) C B E D A (3) A D E B C (3) E B A D C (2) E A B D C (2) D C A B E (2) D B A C E (2) D A C B E (2) C E B D A (2) B C D A E (2) E C B A D (1) E B A C D (1) E A C B D (1) D C A E B (1) D A B E C (1) C E D A B (1) C E B A D (1) C D A B E (1) C B D A E (1) B E A C D (1) B C E D A (1) B C E A D (1) B C D E A (1) B A D E C (1) A E D C B (1) Total count = 100 A B C D E A 0 6 14 -6 14 B -6 0 6 -4 2 C -14 -6 0 -14 0 D 6 4 14 0 20 E -14 -2 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 -6 14 B -6 0 6 -4 2 C -14 -6 0 -14 0 D 6 4 14 0 20 E -14 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=24 E=16 D=15 A=14 so A is eliminated. Round 2 votes counts: B=31 D=28 C=24 E=17 so E is eliminated. Round 3 votes counts: B=39 D=32 C=29 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:214 B:199 C:183 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 14 -6 14 B -6 0 6 -4 2 C -14 -6 0 -14 0 D 6 4 14 0 20 E -14 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -6 14 B -6 0 6 -4 2 C -14 -6 0 -14 0 D 6 4 14 0 20 E -14 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -6 14 B -6 0 6 -4 2 C -14 -6 0 -14 0 D 6 4 14 0 20 E -14 -2 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5839: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) E D B C A (11) C A D B E (11) E B D A C (10) D B E C A (8) A E C B D (5) E B A D C (4) A C E B D (4) D E B C A (3) B E D A C (3) A B C D E (3) E A C D B (2) D C B A E (2) C D B A E (2) B D C A E (2) A C D B E (2) A B C E D (2) E D B A C (1) E C D A B (1) E A B C D (1) D E C B A (1) D C B E A (1) D B C E A (1) D B C A E (1) C D E B A (1) C D A B E (1) C A E D B (1) C A B D E (1) B C D A E (1) B A E D C (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 2 0 4 B 6 0 4 0 10 C -2 -4 0 2 -2 D 0 0 -2 0 6 E -4 -10 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.618441 C: 0.000000 D: 0.381559 E: 0.000000 Sum of squares = 0.528056498216 Cumulative probabilities = A: 0.000000 B: 0.618441 C: 0.618441 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 0 4 B 6 0 4 0 10 C -2 -4 0 2 -2 D 0 0 -2 0 6 E -4 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=29 D=17 C=17 B=7 so B is eliminated. Round 2 votes counts: E=33 A=30 D=19 C=18 so C is eliminated. Round 3 votes counts: A=43 E=33 D=24 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:210 D:202 A:200 C:197 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 0 4 B 6 0 4 0 10 C -2 -4 0 2 -2 D 0 0 -2 0 6 E -4 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 4 B 6 0 4 0 10 C -2 -4 0 2 -2 D 0 0 -2 0 6 E -4 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 4 B 6 0 4 0 10 C -2 -4 0 2 -2 D 0 0 -2 0 6 E -4 -10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5840: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) E D C B A (8) A C D B E (8) E B D C A (7) D C E A B (6) B A C D E (6) D E C A B (5) B E D A C (5) E D C A B (4) E D B C A (4) B E A D C (4) B E A C D (4) D C A E B (3) B A C E D (3) A B C E D (3) A B C D E (3) C D A E B (2) C A D E B (2) B A E D C (2) A C D E B (2) A C B D E (2) E C D A B (1) E B A C D (1) D E C B A (1) D E B C A (1) B E D C A (1) B D E C A (1) B D C A E (1) B A D C E (1) Total count = 100 A B C D E A 0 -18 6 0 -6 B 18 0 12 6 6 C -6 -12 0 -8 -16 D 0 -6 8 0 -12 E 6 -6 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 0 -6 B 18 0 12 6 6 C -6 -12 0 -8 -16 D 0 -6 8 0 -12 E 6 -6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=25 A=18 D=16 C=4 so C is eliminated. Round 2 votes counts: B=37 E=25 A=20 D=18 so D is eliminated. Round 3 votes counts: E=38 B=37 A=25 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:214 D:195 A:191 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 0 -6 B 18 0 12 6 6 C -6 -12 0 -8 -16 D 0 -6 8 0 -12 E 6 -6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 0 -6 B 18 0 12 6 6 C -6 -12 0 -8 -16 D 0 -6 8 0 -12 E 6 -6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 0 -6 B 18 0 12 6 6 C -6 -12 0 -8 -16 D 0 -6 8 0 -12 E 6 -6 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5841: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) E D B C A (6) B A D C E (6) A B D C E (6) E C D B A (5) D B E A C (5) C E A D B (5) B A D E C (5) A B C D E (5) C E D B A (4) C E D A B (4) B D A E C (4) D E B C A (3) C D B A E (3) C A B E D (3) A B E D C (3) E C D A B (2) D B E C A (2) C A B D E (2) A C B E D (2) A C B D E (2) A B D E C (2) A B C E D (2) E C A D B (1) E A D B C (1) D E C B A (1) D B C E A (1) D B A E C (1) C D E B A (1) C D B E A (1) C A E B D (1) B D A C E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -6 -4 0 B 14 0 10 -8 14 C 6 -10 0 -10 -2 D 4 8 10 0 2 E 0 -14 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -4 0 B 14 0 10 -8 14 C 6 -10 0 -10 -2 D 4 8 10 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 E=23 B=16 D=13 so D is eliminated. Round 2 votes counts: E=27 B=25 C=24 A=24 so C is eliminated. Round 3 votes counts: E=41 A=30 B=29 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 D:212 E:193 C:192 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 0 B 14 0 10 -8 14 C 6 -10 0 -10 -2 D 4 8 10 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 0 B 14 0 10 -8 14 C 6 -10 0 -10 -2 D 4 8 10 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 0 B 14 0 10 -8 14 C 6 -10 0 -10 -2 D 4 8 10 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5842: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) C E D B A (9) C E D A B (8) A B D E C (7) C E A D B (5) B D C E A (5) B D E C A (4) A D E B C (4) E C D A B (3) C E B A D (3) B D A E C (3) B C E D A (3) D E C A B (2) D B A E C (2) C E B D A (2) C B E A D (2) B D E A C (2) B A D C E (2) A E C D B (2) A D E C B (2) E D C B A (1) E C D B A (1) E C A D B (1) E A D C B (1) D E C B A (1) D E A C B (1) D B E C A (1) D B E A C (1) C E A B D (1) B C D E A (1) B A C D E (1) A E D C B (1) A D B E C (1) A C E D B (1) A C E B D (1) A C B E D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 0 -16 B 10 0 2 4 0 C 6 -2 0 -6 -4 D 0 -4 6 0 6 E 16 0 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.778021 C: 0.000000 D: 0.000000 E: 0.221979 Sum of squares = 0.654591740219 Cumulative probabilities = A: 0.000000 B: 0.778021 C: 0.778021 D: 0.778021 E: 1.000000 A B C D E A 0 -10 -6 0 -16 B 10 0 2 4 0 C 6 -2 0 -6 -4 D 0 -4 6 0 6 E 16 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000580664 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=30 A=23 D=8 E=7 so E is eliminated. Round 2 votes counts: C=35 B=32 A=24 D=9 so D is eliminated. Round 3 votes counts: C=39 B=36 A=25 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:207 D:204 C:197 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 0 -16 B 10 0 2 4 0 C 6 -2 0 -6 -4 D 0 -4 6 0 6 E 16 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000580664 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 0 -16 B 10 0 2 4 0 C 6 -2 0 -6 -4 D 0 -4 6 0 6 E 16 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000580664 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 0 -16 B 10 0 2 4 0 C 6 -2 0 -6 -4 D 0 -4 6 0 6 E 16 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000580664 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5843: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C D E A B (7) B A E D C (7) E B C A D (5) A D B C E (5) E B D C A (4) D A C E B (4) D A C B E (4) C E D B A (4) B A D E C (4) A B D E C (4) E C B D A (3) E B D A C (3) C E D A B (3) B E A C D (3) E D C B A (2) E B C D A (2) D E C A B (2) D C A E B (2) C D A E B (2) C A D B E (2) B E C A D (2) B E A D C (2) A C D B E (2) A B C D E (2) E D B A C (1) E C B A D (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C A E D B (1) C A D E B (1) C A B E D (1) B E D A C (1) B A E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -4 -8 -6 B 6 0 2 -4 -12 C 4 -2 0 2 -12 D 8 4 -2 0 -10 E 6 12 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -8 -6 B 6 0 2 -4 -12 C 4 -2 0 2 -12 D 8 4 -2 0 -10 E 6 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 B=20 D=15 A=15 so D is eliminated. Round 2 votes counts: E=30 A=26 C=24 B=20 so B is eliminated. Round 3 votes counts: E=38 A=38 C=24 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:200 B:196 C:196 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 -6 B 6 0 2 -4 -12 C 4 -2 0 2 -12 D 8 4 -2 0 -10 E 6 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 -6 B 6 0 2 -4 -12 C 4 -2 0 2 -12 D 8 4 -2 0 -10 E 6 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 -6 B 6 0 2 -4 -12 C 4 -2 0 2 -12 D 8 4 -2 0 -10 E 6 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5844: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) E D C B A (6) C B E D A (6) C B A D E (6) A E D B C (5) A D E B C (5) A D B E C (5) C E B D A (4) C B D E A (4) A B D C E (4) E D B A C (3) D A E B C (3) C A B E D (3) A B C D E (3) E C D B A (2) E C D A B (2) E A D C B (2) C E D B A (2) C E A B D (2) B D C E A (2) B C A D E (2) B A C D E (2) A C B D E (2) E D B C A (1) E A D B C (1) E A C D B (1) D E B C A (1) D E A B C (1) D A B E C (1) C B A E D (1) C A E B D (1) B D A C E (1) B C D A E (1) B A D E C (1) B A D C E (1) A E C D B (1) Total count = 100 A B C D E A 0 8 8 -4 -4 B -8 0 10 -8 -10 C -8 -10 0 -10 -6 D 4 8 10 0 -10 E 4 10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 -4 -4 B -8 0 10 -8 -10 C -8 -10 0 -10 -6 D 4 8 10 0 -10 E 4 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=29 A=25 B=10 D=6 so D is eliminated. Round 2 votes counts: E=32 C=29 A=29 B=10 so B is eliminated. Round 3 votes counts: C=34 A=34 E=32 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:215 D:206 A:204 B:192 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 -4 -4 B -8 0 10 -8 -10 C -8 -10 0 -10 -6 D 4 8 10 0 -10 E 4 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -4 -4 B -8 0 10 -8 -10 C -8 -10 0 -10 -6 D 4 8 10 0 -10 E 4 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -4 -4 B -8 0 10 -8 -10 C -8 -10 0 -10 -6 D 4 8 10 0 -10 E 4 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5845: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (15) E B D C A (11) E C A B D (6) C A D B E (6) B E D C A (6) A C E D B (5) E B D A C (4) B E D A C (4) E B A D C (3) E A B C D (3) D B E C A (3) C D A B E (3) C A E D B (3) B D E C A (3) A C D E B (3) E C B A D (2) D C B E A (2) D B A C E (2) A E C B D (2) E B C D A (1) E B A C D (1) E A C B D (1) D E B C A (1) D C E B A (1) D C B A E (1) D C A B E (1) D B C E A (1) C E D B A (1) C D B E A (1) C A D E B (1) B D E A C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -8 6 -12 B -2 0 -10 0 -2 C 8 10 0 12 -6 D -6 0 -12 0 -10 E 12 2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -8 6 -12 B -2 0 -10 0 -2 C 8 10 0 12 -6 D -6 0 -12 0 -10 E 12 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=27 C=15 B=14 D=12 so D is eliminated. Round 2 votes counts: E=33 A=27 C=20 B=20 so C is eliminated. Round 3 votes counts: A=41 E=35 B=24 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:212 A:194 B:193 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 6 -12 B -2 0 -10 0 -2 C 8 10 0 12 -6 D -6 0 -12 0 -10 E 12 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 6 -12 B -2 0 -10 0 -2 C 8 10 0 12 -6 D -6 0 -12 0 -10 E 12 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 6 -12 B -2 0 -10 0 -2 C 8 10 0 12 -6 D -6 0 -12 0 -10 E 12 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5846: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) E C A B D (7) D B A C E (7) C A E D B (7) B D A E C (7) D B E C A (6) D C E B A (4) D B A E C (4) B D E A C (4) B A D E C (4) C E A B D (3) A E C B D (3) A C E B D (3) E A B C D (2) D B C E A (2) D B C A E (2) C D E A B (2) B E A C D (2) B A E D C (2) E D B C A (1) E C B A D (1) E B D C A (1) E B A C D (1) E A C B D (1) D E C B A (1) D E B C A (1) D B E A C (1) D A C B E (1) C E D B A (1) C E D A B (1) C D A E B (1) B D E C A (1) B A E C D (1) A E B C D (1) A D B C E (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 4 -6 B 8 0 6 -6 -4 C 4 -6 0 0 -6 D -4 6 0 0 0 E 6 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.423393 E: 0.576607 Sum of squares = 0.511737322311 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.423393 E: 1.000000 A B C D E A 0 -8 -4 4 -6 B 8 0 6 -6 -4 C 4 -6 0 0 -6 D -4 6 0 0 0 E 6 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 B=21 E=14 A=11 so A is eliminated. Round 2 votes counts: D=30 C=29 B=23 E=18 so E is eliminated. Round 3 votes counts: C=41 D=31 B=28 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:208 B:202 D:201 C:196 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 4 -6 B 8 0 6 -6 -4 C 4 -6 0 0 -6 D -4 6 0 0 0 E 6 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 4 -6 B 8 0 6 -6 -4 C 4 -6 0 0 -6 D -4 6 0 0 0 E 6 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 4 -6 B 8 0 6 -6 -4 C 4 -6 0 0 -6 D -4 6 0 0 0 E 6 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5847: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) C A B E D (6) A C D E B (6) B C E A D (5) E B D C A (4) C B A E D (4) C A D B E (4) B E D C A (4) B E C D A (4) D A C E B (3) D A C B E (3) C A B D E (3) A D C E B (3) E D A B C (2) E B D A C (2) E B C D A (2) E A C B D (2) D E B A C (2) D C B A E (2) D C A B E (2) D B E C A (2) D B C A E (2) D A E C B (2) B E C A D (2) B D C E A (2) A C E B D (2) E D B A C (1) E C B A D (1) E B A D C (1) E A C D B (1) E A B D C (1) D E A B C (1) D A E B C (1) D A B E C (1) C B D A E (1) C A E B D (1) B D E C A (1) B C A E D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -28 12 -2 B 8 0 4 20 0 C 28 -4 0 16 4 D -12 -20 -16 0 -16 E 2 0 -4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.651824 C: 0.000000 D: 0.000000 E: 0.348176 Sum of squares = 0.546101196278 Cumulative probabilities = A: 0.000000 B: 0.651824 C: 0.651824 D: 0.651824 E: 1.000000 A B C D E A 0 -8 -28 12 -2 B 8 0 4 20 0 C 28 -4 0 16 4 D -12 -20 -16 0 -16 E 2 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091248 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=21 C=19 B=19 A=13 so A is eliminated. Round 2 votes counts: C=29 E=28 D=24 B=19 so B is eliminated. Round 3 votes counts: E=38 C=35 D=27 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:216 E:207 A:187 D:168 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -28 12 -2 B 8 0 4 20 0 C 28 -4 0 16 4 D -12 -20 -16 0 -16 E 2 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091248 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -28 12 -2 B 8 0 4 20 0 C 28 -4 0 16 4 D -12 -20 -16 0 -16 E 2 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091248 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -28 12 -2 B 8 0 4 20 0 C 28 -4 0 16 4 D -12 -20 -16 0 -16 E 2 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500214 C: 0.000000 D: 0.000000 E: 0.499786 Sum of squares = 0.500000091248 Cumulative probabilities = A: 0.000000 B: 0.500214 C: 0.500214 D: 0.500214 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5848: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (12) A C B D E (9) C E D B A (7) E D B C A (6) E C D B A (6) D B E A C (6) C E A D B (6) C A E B D (6) A B D C E (5) D B E C A (4) C E A B D (4) A C E B D (4) A B D E C (4) E C A B D (3) D B A E C (3) B D A C E (3) E B D A C (2) C A E D B (2) E A C B D (1) D B C A E (1) C D B E A (1) C A D E B (1) C A D B E (1) A E C B D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 4 -2 8 B 2 0 -4 12 0 C -4 4 0 4 0 D 2 -12 -4 0 0 E -8 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -2 8 B 2 0 -4 12 0 C -4 4 0 4 0 D 2 -12 -4 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=25 E=18 B=15 D=14 so D is eliminated. Round 2 votes counts: B=29 C=28 A=25 E=18 so E is eliminated. Round 3 votes counts: C=37 B=37 A=26 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:205 A:204 C:202 E:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 -2 8 B 2 0 -4 12 0 C -4 4 0 4 0 D 2 -12 -4 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -2 8 B 2 0 -4 12 0 C -4 4 0 4 0 D 2 -12 -4 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -2 8 B 2 0 -4 12 0 C -4 4 0 4 0 D 2 -12 -4 0 0 E -8 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.400000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000034 Cumulative probabilities = A: 0.400000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5849: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) C B A E D (6) C A E D B (6) B C A E D (6) C A B D E (5) B E D C A (5) E D A C B (4) D E A B C (4) C E A D B (4) C A B E D (4) B D A E C (4) B C A D E (4) B A D C E (4) E D B C A (3) E D B A C (3) E C D A B (3) D E A C B (3) C A D E B (3) D E B A C (2) C B A D E (2) C A E B D (2) B E C D A (2) B D E A C (2) B C E A D (2) B C D A E (2) B A C D E (2) D A E C B (1) C E D A B (1) B E D A C (1) A D E C B (1) A C D E B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -32 8 10 B 0 0 -4 10 6 C 32 4 0 12 10 D -8 -10 -12 0 -16 E -10 -6 -10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -32 8 10 B 0 0 -4 10 6 C 32 4 0 12 10 D -8 -10 -12 0 -16 E -10 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=33 E=19 D=10 A=4 so A is eliminated. Round 2 votes counts: B=36 C=34 E=19 D=11 so D is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:206 E:195 A:193 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -32 8 10 B 0 0 -4 10 6 C 32 4 0 12 10 D -8 -10 -12 0 -16 E -10 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -32 8 10 B 0 0 -4 10 6 C 32 4 0 12 10 D -8 -10 -12 0 -16 E -10 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -32 8 10 B 0 0 -4 10 6 C 32 4 0 12 10 D -8 -10 -12 0 -16 E -10 -6 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5850: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) D A C B E (14) C B E D A (12) A D E B C (8) A D C B E (6) E B C D A (4) E A B C D (4) A E B C D (4) E B A C D (3) D C B E A (3) C D B E A (3) E C B D A (2) D C B A E (2) D A E C B (2) D A E B C (2) C D B A E (2) C B D E A (2) A E B D C (2) A D B C E (2) A B E C D (2) E D C B A (1) D E A B C (1) D C E B A (1) D C A B E (1) D A C E B (1) C E B D A (1) C B E A D (1) Total count = 100 A B C D E A 0 -2 2 -8 -4 B 2 0 -8 2 0 C -2 8 0 8 2 D 8 -2 -8 0 0 E 4 0 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 2 -8 -4 B 2 0 -8 2 0 C -2 8 0 8 2 D 8 -2 -8 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 A=24 C=21 so B is eliminated. Round 2 votes counts: E=28 D=27 A=24 C=21 so C is eliminated. Round 3 votes counts: E=42 D=34 A=24 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:208 E:201 D:199 B:198 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 2 -8 -4 B 2 0 -8 2 0 C -2 8 0 8 2 D 8 -2 -8 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -8 -4 B 2 0 -8 2 0 C -2 8 0 8 2 D 8 -2 -8 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -8 -4 B 2 0 -8 2 0 C -2 8 0 8 2 D 8 -2 -8 0 0 E 4 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5851: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (12) D E B C A (7) C B A D E (7) E D A B C (6) E D B A C (5) D E B A C (5) E D A C B (4) C A B D E (4) B C A D E (4) E A D B C (3) D E C B A (3) D B C E A (3) B C A E D (3) A B C E D (3) D E A C B (2) C D B A E (2) C A B E D (2) B A C E D (2) A E B C D (2) A C E D B (2) A C D E B (2) E B A D C (1) E A B C D (1) D E C A B (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) D B E C A (1) D A E C B (1) C D A B E (1) C B D A E (1) C B A E D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 12 8 12 B -6 0 -6 -8 -2 C -12 6 0 6 8 D -8 8 -6 0 -4 E -12 2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 8 12 B -6 0 -6 -8 -2 C -12 6 0 6 8 D -8 8 -6 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 E=20 C=18 B=9 so B is eliminated. Round 2 votes counts: A=28 D=27 C=25 E=20 so E is eliminated. Round 3 votes counts: D=42 A=33 C=25 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:204 D:195 E:193 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 8 12 B -6 0 -6 -8 -2 C -12 6 0 6 8 D -8 8 -6 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 8 12 B -6 0 -6 -8 -2 C -12 6 0 6 8 D -8 8 -6 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 8 12 B -6 0 -6 -8 -2 C -12 6 0 6 8 D -8 8 -6 0 -4 E -12 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5852: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (13) C E B A D (7) D B A E C (6) E A B C D (5) D C B A E (5) D B A C E (5) B A E C D (5) E C A B D (4) E A C B D (4) D B C A E (4) A E B D C (4) A B E C D (4) D C E A B (3) D C B E A (3) C D E B A (3) B D A E C (3) A B E D C (3) B A E D C (2) B A D E C (2) E B A C D (1) E A C D B (1) D E C A B (1) D A B E C (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E A B (1) C B D E A (1) B D C A E (1) B C E A D (1) A E D B C (1) A E B C D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 0 22 -2 B 0 0 2 26 -4 C 0 -2 0 8 0 D -22 -26 -8 0 -18 E 2 4 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.297425 D: 0.000000 E: 0.702575 Sum of squares = 0.582073081833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.297425 D: 0.297425 E: 1.000000 A B C D E A 0 0 0 22 -2 B 0 0 2 26 -4 C 0 -2 0 8 0 D -22 -26 -8 0 -18 E 2 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.499999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 E=15 A=15 B=14 so B is eliminated. Round 2 votes counts: D=32 C=29 A=24 E=15 so E is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:212 A:210 C:203 D:163 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 22 -2 B 0 0 2 26 -4 C 0 -2 0 8 0 D -22 -26 -8 0 -18 E 2 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.499999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 22 -2 B 0 0 2 26 -4 C 0 -2 0 8 0 D -22 -26 -8 0 -18 E 2 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.499999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 22 -2 B 0 0 2 26 -4 C 0 -2 0 8 0 D -22 -26 -8 0 -18 E 2 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.499999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5853: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) D B A E C (9) C A E D B (7) C E A B D (6) D B E A C (5) A C E D B (5) E C B A D (3) D B E C A (3) D A B C E (3) C A E B D (3) B D E C A (3) A E B D C (3) A C D B E (3) E C A B D (2) E B D C A (2) E B C D A (2) E B A D C (2) E A C B D (2) E A B C D (2) D C B A E (2) D B C A E (2) C E B A D (2) C A D E B (2) C A D B E (2) B E D A C (2) E B C A D (1) D B C E A (1) D B A C E (1) D A B E C (1) C E B D A (1) C D E B A (1) C D B E A (1) C D A B E (1) B E D C A (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 4 -2 -4 B 8 0 10 -6 4 C -4 -10 0 -6 -10 D 2 6 6 0 8 E 4 -4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -2 -4 B 8 0 10 -6 4 C -4 -10 0 -6 -10 D 2 6 6 0 8 E 4 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=26 E=16 B=16 A=15 so A is eliminated. Round 2 votes counts: C=35 D=30 E=19 B=16 so B is eliminated. Round 3 votes counts: D=43 C=35 E=22 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:208 E:201 A:195 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 4 -2 -4 B 8 0 10 -6 4 C -4 -10 0 -6 -10 D 2 6 6 0 8 E 4 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -2 -4 B 8 0 10 -6 4 C -4 -10 0 -6 -10 D 2 6 6 0 8 E 4 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -2 -4 B 8 0 10 -6 4 C -4 -10 0 -6 -10 D 2 6 6 0 8 E 4 -4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5854: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (13) C A D E B (10) D A C E B (8) D B E A C (6) C A E D B (6) B E C A D (6) E B A C D (5) B E D C A (4) D C A E B (3) D A C B E (3) C D A E B (3) C A E B D (3) B E C D A (3) A C D E B (3) E B D A C (2) D C A B E (2) D B A C E (2) D A B E C (2) C E A B D (2) C D A B E (2) A D C E B (2) E B C A D (1) E A D C B (1) D E B A C (1) D B C A E (1) D A E B C (1) C A B D E (1) B E A D C (1) B D C A E (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 8 4 -14 10 B -8 0 -2 -16 -6 C -4 2 0 -6 6 D 14 16 6 0 2 E -10 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -14 10 B -8 0 -2 -16 -6 C -4 2 0 -6 6 D 14 16 6 0 2 E -10 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=28 C=27 E=9 A=7 so A is eliminated. Round 2 votes counts: D=31 C=31 B=28 E=10 so E is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:219 A:204 C:199 E:194 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -14 10 B -8 0 -2 -16 -6 C -4 2 0 -6 6 D 14 16 6 0 2 E -10 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -14 10 B -8 0 -2 -16 -6 C -4 2 0 -6 6 D 14 16 6 0 2 E -10 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -14 10 B -8 0 -2 -16 -6 C -4 2 0 -6 6 D 14 16 6 0 2 E -10 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5855: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) A B C E D (7) E D A C B (6) C B A E D (6) B C A E D (6) B C A D E (5) E D C B A (4) E A D C B (4) A B D C E (4) E C D B A (3) D E C B A (3) D A E B C (3) C B E D A (3) A D E B C (3) A D B E C (3) A B E C D (3) E A C B D (2) D E A C B (2) D C B E A (2) D B C A E (2) C B E A D (2) B A C E D (2) A E D B C (2) A D B C E (2) E D C A B (1) E C B D A (1) E C B A D (1) E C A B D (1) D E C A B (1) D C E B A (1) D A B E C (1) C E B D A (1) C D E B A (1) C B D A E (1) B C D A E (1) B A C D E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 8 8 12 8 B -8 0 8 -2 2 C -8 -8 0 -4 -6 D -12 2 4 0 -14 E -8 -2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 12 8 B -8 0 8 -2 2 C -8 -8 0 -4 -6 D -12 2 4 0 -14 E -8 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 D=22 B=15 C=14 so C is eliminated. Round 2 votes counts: B=27 A=26 E=24 D=23 so D is eliminated. Round 3 votes counts: E=39 B=31 A=30 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:218 E:205 B:200 D:190 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 12 8 B -8 0 8 -2 2 C -8 -8 0 -4 -6 D -12 2 4 0 -14 E -8 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 12 8 B -8 0 8 -2 2 C -8 -8 0 -4 -6 D -12 2 4 0 -14 E -8 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 12 8 B -8 0 8 -2 2 C -8 -8 0 -4 -6 D -12 2 4 0 -14 E -8 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5856: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) C E B A D (7) B E C D A (7) D A B E C (6) A D C E B (6) E B C A D (4) D B E C A (4) B E D C A (4) B D E C A (4) A D B E C (4) A C E D B (4) D C A E B (3) D B A E C (3) B D E A C (3) A C E B D (3) E B C D A (2) E B A C D (2) D A B C E (2) C E D B A (2) C E A B D (2) C D E B A (2) C A E B D (2) C A D E B (2) A D B C E (2) A C D E B (2) E C B A D (1) D B C E A (1) D B C A E (1) C E D A B (1) C A E D B (1) B E D A C (1) B E A D C (1) B E A C D (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 -16 -16 -8 -14 B 16 0 4 6 -10 C 16 -4 0 8 4 D 8 -6 -8 0 -8 E 14 10 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407462 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 -16 -16 -8 -14 B 16 0 4 6 -10 C 16 -4 0 8 4 D 8 -6 -8 0 -8 E 14 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407321 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=23 B=21 D=20 E=9 so E is eliminated. Round 2 votes counts: B=29 C=28 A=23 D=20 so D is eliminated. Round 3 votes counts: B=38 C=31 A=31 so C is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 C:212 B:208 D:193 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -16 -8 -14 B 16 0 4 6 -10 C 16 -4 0 8 4 D 8 -6 -8 0 -8 E 14 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407321 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -8 -14 B 16 0 4 6 -10 C 16 -4 0 8 4 D 8 -6 -8 0 -8 E 14 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407321 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -8 -14 B 16 0 4 6 -10 C 16 -4 0 8 4 D 8 -6 -8 0 -8 E 14 10 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407321 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5857: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (11) A C E B D (9) C E A D B (7) A B C E D (7) B D E A C (6) B D A E C (6) D E B C A (5) D C E A B (5) D B A E C (5) E C A B D (3) D E C B A (3) D E C A B (3) B A D C E (3) B A C E D (3) E C D A B (2) D B E A C (2) C A E B D (2) B E A C D (2) B A E C D (2) E D C A B (1) E C D B A (1) E A C B D (1) D C E B A (1) D C A E B (1) C E D A B (1) B E D C A (1) B E D A C (1) B D E C A (1) B A E D C (1) B A D E C (1) A C E D B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 4 -12 -14 B 10 0 16 2 8 C -4 -16 0 -14 -16 D 12 -2 14 0 8 E 14 -8 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -12 -14 B 10 0 16 2 8 C -4 -16 0 -14 -16 D 12 -2 14 0 8 E 14 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=27 A=19 C=10 E=8 so E is eliminated. Round 2 votes counts: D=37 B=27 A=20 C=16 so C is eliminated. Round 3 votes counts: D=41 A=32 B=27 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:216 E:207 A:184 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 -12 -14 B 10 0 16 2 8 C -4 -16 0 -14 -16 D 12 -2 14 0 8 E 14 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -12 -14 B 10 0 16 2 8 C -4 -16 0 -14 -16 D 12 -2 14 0 8 E 14 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -12 -14 B 10 0 16 2 8 C -4 -16 0 -14 -16 D 12 -2 14 0 8 E 14 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998394 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5858: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) A B D E C (6) A B C E D (5) D E C B A (4) D C E A B (4) C D A E B (4) C B E A D (4) E C D B A (3) D E A B C (3) D A E B C (3) C E B D A (3) C D E A B (3) B C E A D (3) B A E D C (3) B A C E D (3) E D B A C (2) D E C A B (2) D E B A C (2) C B A E D (2) C A B E D (2) C A B D E (2) B C A E D (2) A D B E C (2) E D C B A (1) E D B C A (1) E B D A C (1) D E B C A (1) D C E B A (1) D B A E C (1) D A E C B (1) C E D A B (1) C E B A D (1) C D E B A (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C A D (1) B E A D C (1) B E A C D (1) B D A E C (1) B A E C D (1) A D B C E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 -2 -2 C 14 2 0 12 12 D 6 2 -12 0 -6 E 8 2 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 -2 -2 C 14 2 0 12 12 D 6 2 -12 0 -6 E 8 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 A=19 B=18 E=8 so E is eliminated. Round 2 votes counts: C=36 D=26 B=19 A=19 so B is eliminated. Round 3 votes counts: C=42 D=30 A=28 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:202 B:201 D:195 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 -2 -2 C 14 2 0 12 12 D 6 2 -12 0 -6 E 8 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 -2 -2 C 14 2 0 12 12 D 6 2 -12 0 -6 E 8 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -6 -8 B 8 0 -2 -2 -2 C 14 2 0 12 12 D 6 2 -12 0 -6 E 8 2 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5859: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) C E A B D (8) D B A E C (7) B D A C E (6) E C D A B (5) D A B E C (5) B D A E C (5) D B E A C (4) E D A C B (3) E A D C B (3) E A C D B (3) C B A D E (3) C A E B D (3) B C D A E (3) B C A D E (3) B A D C E (3) E D C B A (2) C E B A D (2) C B E A D (2) C B A E D (2) C A B E D (2) B D C A E (2) A B D C E (2) E D A B C (1) D E B A C (1) D B E C A (1) D A E B C (1) C E B D A (1) C B D E A (1) C A B D E (1) B D C E A (1) B A C D E (1) A E D B C (1) A E C B D (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -2 2 6 B 0 0 -4 6 12 C 2 4 0 2 12 D -2 -6 -2 0 2 E -6 -12 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 2 6 B 0 0 -4 6 12 C 2 4 0 2 12 D -2 -6 -2 0 2 E -6 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=24 D=19 E=17 A=6 so A is eliminated. Round 2 votes counts: C=35 B=26 D=20 E=19 so E is eliminated. Round 3 votes counts: C=44 D=30 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:210 B:207 A:203 D:196 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 2 6 B 0 0 -4 6 12 C 2 4 0 2 12 D -2 -6 -2 0 2 E -6 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 6 B 0 0 -4 6 12 C 2 4 0 2 12 D -2 -6 -2 0 2 E -6 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 6 B 0 0 -4 6 12 C 2 4 0 2 12 D -2 -6 -2 0 2 E -6 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5860: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (11) D E C B A (10) E D C A B (9) A B C E D (8) D B C A E (6) D C E B A (4) A B C D E (4) E D A B C (3) D E B A C (3) C E A B D (3) C B D A E (3) B D C A E (3) E D C B A (2) E D A C B (2) E C D A B (2) E A B D C (2) D B A C E (2) C B A E D (2) C A B E D (2) B D A C E (2) A E C B D (2) A C B E D (2) E C A B D (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D E B C A (1) D C B E A (1) D B E A C (1) D A B E C (1) C D E B A (1) B C A D E (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 -8 -2 -12 2 B 8 0 2 -2 0 C 2 -2 0 -10 12 D 12 2 10 0 10 E -2 0 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -12 2 B 8 0 2 -2 0 C 2 -2 0 -10 12 D 12 2 10 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 B=18 A=17 C=11 so C is eliminated. Round 2 votes counts: D=30 E=28 B=23 A=19 so A is eliminated. Round 3 votes counts: B=39 E=31 D=30 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:217 B:204 C:201 A:190 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -12 2 B 8 0 2 -2 0 C 2 -2 0 -10 12 D 12 2 10 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -12 2 B 8 0 2 -2 0 C 2 -2 0 -10 12 D 12 2 10 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -12 2 B 8 0 2 -2 0 C 2 -2 0 -10 12 D 12 2 10 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999979385 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5861: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) E A C B D (7) D B C E A (7) D B C A E (7) D B A C E (7) C A E D B (6) B D E A C (6) A C E D B (5) A C D B E (5) E C A D B (3) C A D B E (3) B E A D C (3) B D E C A (3) B D A E C (3) B D A C E (3) A E C B D (3) A D B C E (3) E B D C A (2) E B A D C (2) E C D B A (1) E C D A B (1) E B D A C (1) D A C B E (1) C E D A B (1) C E A D B (1) C D E B A (1) B E D A C (1) B D C E A (1) B A D E C (1) A E B C D (1) A D C B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 10 8 2 B -2 0 6 -6 14 C -10 -6 0 -8 4 D -8 6 8 0 6 E -2 -14 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 8 2 B -2 0 6 -6 14 C -10 -6 0 -8 4 D -8 6 8 0 6 E -2 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 B=21 A=20 C=12 so C is eliminated. Round 2 votes counts: A=29 E=27 D=23 B=21 so B is eliminated. Round 3 votes counts: D=39 E=31 A=30 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:211 B:206 D:206 C:190 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 8 2 B -2 0 6 -6 14 C -10 -6 0 -8 4 D -8 6 8 0 6 E -2 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 8 2 B -2 0 6 -6 14 C -10 -6 0 -8 4 D -8 6 8 0 6 E -2 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 8 2 B -2 0 6 -6 14 C -10 -6 0 -8 4 D -8 6 8 0 6 E -2 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998073 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5862: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) B A D E C (6) A B C D E (6) E D C B A (5) B A C D E (5) E D B C A (4) D C E A B (4) D B A E C (4) C A B D E (4) E D B A C (3) E C D B A (3) E B A C D (3) D C A B E (3) C E D A B (3) C D A E B (3) B A E D C (3) A C B D E (3) E B D A C (2) D E B A C (2) D B E A C (2) D A B C E (2) C E A D B (2) C E A B D (2) C D E A B (2) C A E B D (2) B A E C D (2) B A C E D (2) E C B D A (1) E C B A D (1) E B C D A (1) D E B C A (1) D C A E B (1) D B A C E (1) C D A B E (1) C A D B E (1) B E D A C (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -2 -12 2 B 6 0 4 -8 -4 C 2 -4 0 8 -2 D 12 8 -8 0 6 E -2 4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.500000 Sum of squares = 0.406249999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.500000 E: 1.000000 A B C D E A 0 -6 -2 -12 2 B 6 0 4 -8 -4 C 2 -4 0 8 -2 D 12 8 -8 0 6 E -2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.500000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=20 C=20 B=20 A=10 so A is eliminated. Round 2 votes counts: E=30 B=27 C=23 D=20 so D is eliminated. Round 3 votes counts: B=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:209 C:202 B:199 E:199 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -12 2 B 6 0 4 -8 -4 C 2 -4 0 8 -2 D 12 8 -8 0 6 E -2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.500000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -12 2 B 6 0 4 -8 -4 C 2 -4 0 8 -2 D 12 8 -8 0 6 E -2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.500000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -12 2 B 6 0 4 -8 -4 C 2 -4 0 8 -2 D 12 8 -8 0 6 E -2 4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.125000 E: 0.500000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5863: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) C D A B E (6) A E B D C (6) D C B E A (5) D B E C A (4) D B E A C (4) C A D E B (4) A E B C D (4) A C E B D (4) E B A D C (3) E B A C D (3) E A B C D (3) D C A B E (3) A E C B D (3) D C B A E (2) D B C E A (2) D B A E C (2) D A B E C (2) C E B D A (2) C D B A E (2) C D A E B (2) C A E B D (2) B E D A C (2) A B E D C (2) E C B A D (1) E B C D A (1) E B C A D (1) E A C B D (1) D A C B E (1) D A B C E (1) C E B A D (1) C E A B D (1) C D E B A (1) C B E D A (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E A C (1) B A E D C (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -4 -8 4 B 0 0 -10 -10 6 C 4 10 0 10 6 D 8 10 -10 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -8 4 B 0 0 -10 -10 6 C 4 10 0 10 6 D 8 10 -10 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=26 A=23 E=13 B=6 so B is eliminated. Round 2 votes counts: C=32 D=27 A=24 E=17 so E is eliminated. Round 3 votes counts: C=35 A=35 D=30 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:208 A:196 B:193 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -8 4 B 0 0 -10 -10 6 C 4 10 0 10 6 D 8 10 -10 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -8 4 B 0 0 -10 -10 6 C 4 10 0 10 6 D 8 10 -10 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -8 4 B 0 0 -10 -10 6 C 4 10 0 10 6 D 8 10 -10 0 8 E -4 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5864: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (19) C A B D E (12) D E B A C (8) D B A E C (5) D A B E C (5) C A B E D (5) A B C D E (5) E D C B A (4) E D B C A (4) A B D E C (4) A B D C E (3) E C D B A (2) C E D A B (2) C E B D A (2) C E B A D (2) C A E B D (2) C A D B E (2) B D A E C (2) E B D C A (1) D E C A B (1) D E A B C (1) D B E A C (1) D A C E B (1) C E D B A (1) C E A D B (1) C E A B D (1) C B E A D (1) B E D A C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 14 -20 -4 B 6 0 20 -16 -4 C -14 -20 0 -22 -16 D 20 16 22 0 4 E 4 4 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 14 -20 -4 B 6 0 20 -16 -4 C -14 -20 0 -22 -16 D 20 16 22 0 4 E 4 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=30 D=22 A=14 B=3 so B is eliminated. Round 2 votes counts: E=31 C=31 D=24 A=14 so A is eliminated. Round 3 votes counts: C=37 D=32 E=31 so E is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:231 E:210 B:203 A:192 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 14 -20 -4 B 6 0 20 -16 -4 C -14 -20 0 -22 -16 D 20 16 22 0 4 E 4 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 -20 -4 B 6 0 20 -16 -4 C -14 -20 0 -22 -16 D 20 16 22 0 4 E 4 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 -20 -4 B 6 0 20 -16 -4 C -14 -20 0 -22 -16 D 20 16 22 0 4 E 4 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5865: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) A B D E C (7) C D E B A (6) B A E C D (6) C E B D A (5) B E C A D (5) B E A C D (5) D C A E B (4) D A C B E (4) B C E A D (4) A B E D C (4) D C E A B (3) D A E B C (3) D A C E B (3) C B E A D (3) E C D B A (2) E C B D A (2) E B C D A (2) D E C A B (2) D C A B E (2) D A E C B (2) D A B C E (2) C E D B A (2) A E B D C (2) A D E B C (2) A D B C E (2) E D C B A (1) E D A B C (1) E B D C A (1) E B A D C (1) E A B D C (1) C D E A B (1) C D B E A (1) C B E D A (1) A C D B E (1) Total count = 100 A B C D E A 0 6 6 0 2 B -6 0 10 -2 8 C -6 -10 0 -8 -12 D 0 2 8 0 4 E -2 -8 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.367720 B: 0.000000 C: 0.000000 D: 0.632280 E: 0.000000 Sum of squares = 0.534995915068 Cumulative probabilities = A: 0.367720 B: 0.367720 C: 0.367720 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 0 2 B -6 0 10 -2 8 C -6 -10 0 -8 -12 D 0 2 8 0 4 E -2 -8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 B=20 C=19 E=11 so E is eliminated. Round 2 votes counts: D=27 A=26 B=24 C=23 so C is eliminated. Round 3 votes counts: D=39 B=35 A=26 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:207 B:205 E:199 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 0 2 B -6 0 10 -2 8 C -6 -10 0 -8 -12 D 0 2 8 0 4 E -2 -8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 0 2 B -6 0 10 -2 8 C -6 -10 0 -8 -12 D 0 2 8 0 4 E -2 -8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 0 2 B -6 0 10 -2 8 C -6 -10 0 -8 -12 D 0 2 8 0 4 E -2 -8 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999918 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5866: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) B C E D A (8) A D E C B (8) C E B D A (7) B A D C E (6) D A B E C (5) D B A E C (4) C E B A D (4) B D A C E (4) A D E B C (4) C E A D B (3) B D C A E (3) A B D C E (3) E C B D A (2) D A E B C (2) C E A B D (2) B D A E C (2) B C D A E (2) A D B C E (2) E D A C B (1) E C D B A (1) E C D A B (1) E C B A D (1) E A D C B (1) D E B A C (1) D E A B C (1) D B A C E (1) D A E C B (1) C B E D A (1) C B E A D (1) C A E B D (1) B D E C A (1) B C D E A (1) B C A E D (1) A E C D B (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 2 4 B 2 0 4 0 -8 C 2 -4 0 -2 2 D -2 0 2 0 4 E -4 8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.421053 B: 0.263158 C: 0.105263 D: 0.052632 E: 0.157895 Sum of squares = 0.285318559608 Cumulative probabilities = A: 0.421053 B: 0.684211 C: 0.789474 D: 0.842105 E: 1.000000 A B C D E A 0 -2 -2 2 4 B 2 0 4 0 -8 C 2 -4 0 -2 2 D -2 0 2 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.421053 B: 0.263158 C: 0.105263 D: 0.052632 E: 0.157895 Sum of squares = 0.285318559609 Cumulative probabilities = A: 0.421053 B: 0.684211 C: 0.789474 D: 0.842105 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=20 C=19 E=18 D=15 so D is eliminated. Round 2 votes counts: B=33 A=28 E=20 C=19 so C is eliminated. Round 3 votes counts: E=36 B=35 A=29 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:202 A:201 B:199 C:199 E:199 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 2 4 B 2 0 4 0 -8 C 2 -4 0 -2 2 D -2 0 2 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.421053 B: 0.263158 C: 0.105263 D: 0.052632 E: 0.157895 Sum of squares = 0.285318559609 Cumulative probabilities = A: 0.421053 B: 0.684211 C: 0.789474 D: 0.842105 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 4 B 2 0 4 0 -8 C 2 -4 0 -2 2 D -2 0 2 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.421053 B: 0.263158 C: 0.105263 D: 0.052632 E: 0.157895 Sum of squares = 0.285318559609 Cumulative probabilities = A: 0.421053 B: 0.684211 C: 0.789474 D: 0.842105 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 4 B 2 0 4 0 -8 C 2 -4 0 -2 2 D -2 0 2 0 4 E -4 8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.421053 B: 0.263158 C: 0.105263 D: 0.052632 E: 0.157895 Sum of squares = 0.285318559609 Cumulative probabilities = A: 0.421053 B: 0.684211 C: 0.789474 D: 0.842105 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5867: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) E D B C A (8) B A D E C (8) C E D B A (7) A B D C E (7) C E A D B (6) B D E A C (6) A C B D E (6) E D C B A (5) D E B C A (5) C A E D B (5) D B E A C (4) B D A E C (4) A B D E C (4) E C D B A (3) A C B E D (3) A B C D E (3) D B E C A (2) C A E B D (2) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -6 -8 -12 B 4 0 2 -14 -4 C 6 -2 0 -6 2 D 8 14 6 0 0 E 12 4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.815591 E: 0.184409 Sum of squares = 0.699195820038 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.815591 E: 1.000000 A B C D E A 0 -4 -6 -8 -12 B 4 0 2 -14 -4 C 6 -2 0 -6 2 D 8 14 6 0 0 E 12 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=25 B=18 E=16 D=11 so D is eliminated. Round 2 votes counts: C=30 A=25 B=24 E=21 so E is eliminated. Round 3 votes counts: C=38 B=37 A=25 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 E:207 C:200 B:194 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -8 -12 B 4 0 2 -14 -4 C 6 -2 0 -6 2 D 8 14 6 0 0 E 12 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -8 -12 B 4 0 2 -14 -4 C 6 -2 0 -6 2 D 8 14 6 0 0 E 12 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -8 -12 B 4 0 2 -14 -4 C 6 -2 0 -6 2 D 8 14 6 0 0 E 12 4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5868: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) A D E B C (11) C B A D E (9) B E D A C (7) D E A B C (6) C B E D A (5) C B A E D (5) E D B A C (4) C A B D E (4) B C E D A (4) E D B C A (3) C E D A B (3) B E D C A (3) A D E C B (3) C A D E B (2) C A D B E (2) B E C D A (2) B A D E C (2) A D C E B (2) A B C D E (2) E D C A B (1) E D A C B (1) E B D C A (1) D E A C B (1) D A E B C (1) B A C D E (1) A D C B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 12 -8 -6 B -8 0 20 -6 -2 C -12 -20 0 -20 -16 D 8 6 20 0 -2 E 6 2 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 12 -8 -6 B -8 0 20 -6 -2 C -12 -20 0 -20 -16 D 8 6 20 0 -2 E 6 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=22 A=21 B=19 D=8 so D is eliminated. Round 2 votes counts: C=30 E=29 A=22 B=19 so B is eliminated. Round 3 votes counts: E=41 C=34 A=25 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:213 A:203 B:202 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 -8 -6 B -8 0 20 -6 -2 C -12 -20 0 -20 -16 D 8 6 20 0 -2 E 6 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -8 -6 B -8 0 20 -6 -2 C -12 -20 0 -20 -16 D 8 6 20 0 -2 E 6 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -8 -6 B -8 0 20 -6 -2 C -12 -20 0 -20 -16 D 8 6 20 0 -2 E 6 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5869: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (15) A C E D B (8) D B A C E (7) B D E C A (7) E C B A D (5) B E D C A (5) A C D E B (5) D B C A E (4) D B A E C (4) B E C D A (4) E B C A D (3) A D C E B (3) E B C D A (2) C E A B D (2) C A E D B (2) B D E A C (2) B D A E C (2) A E C D B (2) A D C B E (2) A D B E C (2) A D B C E (2) E C B D A (1) E A B C D (1) D C B A E (1) D A B E C (1) C E B D A (1) C E B A D (1) C E A D B (1) C D B E A (1) C D B A E (1) C B E D A (1) B D C E A (1) A E D B C (1) Total count = 100 A B C D E A 0 -6 -16 10 -6 B 6 0 -4 6 -6 C 16 4 0 12 -14 D -10 -6 -12 0 -10 E 6 6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -16 10 -6 B 6 0 -4 6 -6 C 16 4 0 12 -14 D -10 -6 -12 0 -10 E 6 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=25 B=21 D=17 C=10 so C is eliminated. Round 2 votes counts: E=32 A=27 B=22 D=19 so D is eliminated. Round 3 votes counts: B=40 E=32 A=28 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:209 B:201 A:191 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -16 10 -6 B 6 0 -4 6 -6 C 16 4 0 12 -14 D -10 -6 -12 0 -10 E 6 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 10 -6 B 6 0 -4 6 -6 C 16 4 0 12 -14 D -10 -6 -12 0 -10 E 6 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 10 -6 B 6 0 -4 6 -6 C 16 4 0 12 -14 D -10 -6 -12 0 -10 E 6 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5870: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) A D C B E (7) D A B C E (5) C E B A D (5) B D C A E (5) D B A E C (4) E C A B D (3) E B C D A (3) D A E C B (3) B C E A D (3) B C A E D (3) A D E C B (3) A C D E B (3) A C D B E (3) E D B C A (2) E D A C B (2) E C A D B (2) E A C D B (2) D E B A C (2) D A B E C (2) C B E A D (2) C A E B D (2) C A B E D (2) B E C D A (2) B C D A E (2) B C A D E (2) A E D C B (2) E D C A B (1) E C B D A (1) E C B A D (1) E B C A D (1) E A D C B (1) D E A B C (1) D B A C E (1) D A C E B (1) D A C B E (1) C B A E D (1) B D C E A (1) B C E D A (1) B C D E A (1) A D C E B (1) A D B C E (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 14 8 4 26 B -14 0 -2 -16 0 C -8 2 0 -6 10 D -4 16 6 0 14 E -26 0 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 4 26 B -14 0 -2 -16 0 C -8 2 0 -6 10 D -4 16 6 0 14 E -26 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998065 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=22 B=20 E=19 C=12 so C is eliminated. Round 2 votes counts: D=27 A=26 E=24 B=23 so B is eliminated. Round 3 votes counts: D=36 E=32 A=32 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:216 C:199 B:184 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 4 26 B -14 0 -2 -16 0 C -8 2 0 -6 10 D -4 16 6 0 14 E -26 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998065 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 4 26 B -14 0 -2 -16 0 C -8 2 0 -6 10 D -4 16 6 0 14 E -26 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998065 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 4 26 B -14 0 -2 -16 0 C -8 2 0 -6 10 D -4 16 6 0 14 E -26 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998065 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5871: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) C E D A B (9) B A E C D (7) A C E D B (7) A B D C E (7) D E C B A (6) C E D B A (5) E C D B A (4) B D E C A (4) B A D E C (4) A B D E C (4) D B E C A (3) B E D C A (3) B E C A D (3) C E B A D (2) C E A D B (2) B D E A C (2) A B E C D (2) E B C D A (1) D E B C A (1) D B A E C (1) C D E A B (1) C A E D B (1) B E C D A (1) B E A C D (1) B D A E C (1) B A C E D (1) A D B C E (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 8 16 4 B 0 0 20 16 18 C -8 -20 0 26 4 D -16 -16 -26 0 -28 E -4 -18 -4 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.512898 B: 0.487102 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500332702921 Cumulative probabilities = A: 0.512898 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 16 4 B 0 0 20 16 18 C -8 -20 0 26 4 D -16 -16 -26 0 -28 E -4 -18 -4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=27 C=20 D=11 E=5 so E is eliminated. Round 2 votes counts: A=37 B=28 C=24 D=11 so D is eliminated. Round 3 votes counts: A=37 B=33 C=30 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:227 A:214 C:201 E:201 D:157 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 16 4 B 0 0 20 16 18 C -8 -20 0 26 4 D -16 -16 -26 0 -28 E -4 -18 -4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 16 4 B 0 0 20 16 18 C -8 -20 0 26 4 D -16 -16 -26 0 -28 E -4 -18 -4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 16 4 B 0 0 20 16 18 C -8 -20 0 26 4 D -16 -16 -26 0 -28 E -4 -18 -4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5872: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (13) E A B D C (10) E A D C B (8) D C A E B (6) D A C E B (6) B E A C D (6) B C D A E (6) E A D B C (5) E B A C D (4) C B D A E (4) A E D C B (4) D C B A E (3) C B D E A (3) A D C E B (3) C D A E B (2) B D C A E (2) A E D B C (2) E A B C D (1) D C A B E (1) D A C B E (1) D A B C E (1) C D E A B (1) C D B E A (1) C D A B E (1) B E C D A (1) B C E D A (1) B C D E A (1) B A E D C (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 8 -8 16 B -6 0 -14 -16 -4 C -8 14 0 -10 14 D 8 16 10 0 12 E -16 4 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -8 16 B -6 0 -14 -16 -4 C -8 14 0 -10 14 D 8 16 10 0 12 E -16 4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=25 B=19 D=18 A=10 so A is eliminated. Round 2 votes counts: E=34 C=25 D=21 B=20 so B is eliminated. Round 3 votes counts: E=43 C=33 D=24 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:223 A:211 C:205 E:181 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -8 16 B -6 0 -14 -16 -4 C -8 14 0 -10 14 D 8 16 10 0 12 E -16 4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 16 B -6 0 -14 -16 -4 C -8 14 0 -10 14 D 8 16 10 0 12 E -16 4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 16 B -6 0 -14 -16 -4 C -8 14 0 -10 14 D 8 16 10 0 12 E -16 4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5873: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) D A B E C (5) C E D A B (5) C E A D B (5) C E A B D (5) A C B D E (5) B D E A C (4) A B D C E (4) E B D C A (3) D B A E C (3) C A E B D (3) C A D E B (3) C A D B E (3) B E D A C (3) B A D E C (3) A C D B E (3) E D C B A (2) E C B D A (2) E C B A D (2) E B C D A (2) D E B A C (2) D A B C E (2) C D A E B (2) C A B E D (2) B E A C D (2) A D C B E (2) A D B C E (2) E D B C A (1) E C D A B (1) E B D A C (1) E B C A D (1) D C E A B (1) D C A E B (1) D A C E B (1) D A C B E (1) C E B A D (1) B E D C A (1) B D A E C (1) B A E D C (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -6 -2 -2 B -14 0 -14 -4 -2 C 6 14 0 12 4 D 2 4 -12 0 -4 E 2 2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 -2 -2 B -14 0 -14 -4 -2 C 6 14 0 12 4 D 2 4 -12 0 -4 E 2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=22 A=17 D=16 B=16 so D is eliminated. Round 2 votes counts: C=31 A=26 E=24 B=19 so B is eliminated. Round 3 votes counts: A=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:218 A:202 E:202 D:195 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 -2 -2 B -14 0 -14 -4 -2 C 6 14 0 12 4 D 2 4 -12 0 -4 E 2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 -2 -2 B -14 0 -14 -4 -2 C 6 14 0 12 4 D 2 4 -12 0 -4 E 2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 -2 -2 B -14 0 -14 -4 -2 C 6 14 0 12 4 D 2 4 -12 0 -4 E 2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5874: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) B C D E A (7) B C D A E (7) E D A B C (5) E A D B C (5) D B A E C (5) C E B A D (4) C B A D E (4) A E C D B (4) E C A B D (3) E A C D B (3) D B A C E (3) D A E B C (3) C B D A E (3) C A E B D (3) B C E D A (3) E C B A D (2) E B D A C (2) E A D C B (2) D B C A E (2) D A B E C (2) D A B C E (2) C E A B D (2) C A E D B (2) A D E B C (2) E D B A C (1) D E A B C (1) D B E A C (1) C B D E A (1) C B A E D (1) B E D C A (1) B E C D A (1) B D C E A (1) A E D B C (1) A D E C B (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -14 -10 -18 10 B 14 0 26 6 6 C 10 -26 0 2 10 D 18 -6 -2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -18 10 B 14 0 26 6 6 C 10 -26 0 2 10 D 18 -6 -2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993572 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=23 C=20 D=19 A=10 so A is eliminated. Round 2 votes counts: E=28 B=28 D=23 C=21 so C is eliminated. Round 3 votes counts: E=39 B=37 D=24 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:226 D:210 C:198 A:184 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -10 -18 10 B 14 0 26 6 6 C 10 -26 0 2 10 D 18 -6 -2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993572 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -18 10 B 14 0 26 6 6 C 10 -26 0 2 10 D 18 -6 -2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993572 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -18 10 B 14 0 26 6 6 C 10 -26 0 2 10 D 18 -6 -2 0 10 E -10 -6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993572 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5875: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (17) E C B A D (16) B C E D A (15) D A B C E (12) D A E C B (5) E A D C B (4) B C D E A (3) E C A D B (2) E C A B D (2) C E B A D (2) C B E A D (2) A E D C B (2) A D E B C (2) A D B C E (2) E C B D A (1) E A C D B (1) D B A C E (1) D A E B C (1) D A C E B (1) D A B E C (1) C E B D A (1) C B E D A (1) B D C A E (1) B D A C E (1) B C E A D (1) B C D A E (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 6 4 10 -2 B -6 0 -14 -4 -14 C -4 14 0 -2 -10 D -10 4 2 0 0 E 2 14 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.086611 E: 0.913389 Sum of squares = 0.841781600844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.086611 E: 1.000000 A B C D E A 0 6 4 10 -2 B -6 0 -14 -4 -14 C -4 14 0 -2 -10 D -10 4 2 0 0 E 2 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222231176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=24 B=23 D=21 C=6 so C is eliminated. Round 2 votes counts: E=29 B=26 A=24 D=21 so D is eliminated. Round 3 votes counts: A=44 E=29 B=27 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:209 C:199 D:198 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 10 -2 B -6 0 -14 -4 -14 C -4 14 0 -2 -10 D -10 4 2 0 0 E 2 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222231176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 10 -2 B -6 0 -14 -4 -14 C -4 14 0 -2 -10 D -10 4 2 0 0 E 2 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222231176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 10 -2 B -6 0 -14 -4 -14 C -4 14 0 -2 -10 D -10 4 2 0 0 E 2 14 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222231176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5876: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) D E C A B (7) B A D C E (7) E C B A D (5) D A C E B (5) B A C D E (5) E D C A B (4) E C D B A (4) B E A D C (3) B E A C D (3) A D C B E (3) A C D B E (3) E D B C A (2) E C B D A (2) D A E C B (2) D A C B E (2) D A B C E (2) C E D A B (2) B E C A D (2) B D E A C (2) B C A E D (2) B A E D C (2) B A D E C (2) A D B C E (2) E B C D A (1) D E A B C (1) D C A E B (1) D B A E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E A D (1) C A D E B (1) C A B E D (1) B E D A C (1) B E C D A (1) B D A E C (1) B C E A D (1) A D C E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -2 -10 B -2 0 -14 -8 -2 C 0 14 0 -4 -12 D 2 8 4 0 0 E 10 2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.661358 E: 0.338642 Sum of squares = 0.552072623415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.661358 E: 1.000000 A B C D E A 0 2 0 -2 -10 B -2 0 -14 -8 -2 C 0 14 0 -4 -12 D 2 8 4 0 0 E 10 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=28 D=21 A=11 C=8 so C is eliminated. Round 2 votes counts: E=33 B=33 D=21 A=13 so A is eliminated. Round 3 votes counts: B=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:207 C:199 A:195 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -2 -10 B -2 0 -14 -8 -2 C 0 14 0 -4 -12 D 2 8 4 0 0 E 10 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 -10 B -2 0 -14 -8 -2 C 0 14 0 -4 -12 D 2 8 4 0 0 E 10 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 -10 B -2 0 -14 -8 -2 C 0 14 0 -4 -12 D 2 8 4 0 0 E 10 2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5877: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (4) C E B D A (4) B E C A D (4) B D E A C (4) A D B E C (4) A B D C E (4) A B C D E (4) E C B D A (3) D E B C A (3) C A E D B (3) C A D E B (3) B E D C A (3) B E C D A (3) B C E A D (3) A D C E B (3) A D C B E (3) E D B C A (2) E B C D A (2) D E C B A (2) D C E A B (2) D A E B C (2) D A C E B (2) C E D B A (2) C E B A D (2) C B E A D (2) C A E B D (2) C A B E D (2) B E D A C (2) B D A E C (2) A C B E D (2) A C B D E (2) E D C B A (1) E C D B A (1) D E A C B (1) D E A B C (1) D C A E B (1) D A E C B (1) D A B E C (1) C E A D B (1) C D A E B (1) C B A E D (1) B A D E C (1) A D B C E (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -14 0 -4 B -6 0 -4 8 2 C 14 4 0 6 12 D 0 -8 -6 0 0 E 4 -2 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 0 -4 B -6 0 -4 8 2 C 14 4 0 6 12 D 0 -8 -6 0 0 E 4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 B=22 D=16 E=9 so E is eliminated. Round 2 votes counts: C=31 A=26 B=24 D=19 so D is eliminated. Round 3 votes counts: C=37 A=34 B=29 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:200 E:195 A:194 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 0 -4 B -6 0 -4 8 2 C 14 4 0 6 12 D 0 -8 -6 0 0 E 4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 0 -4 B -6 0 -4 8 2 C 14 4 0 6 12 D 0 -8 -6 0 0 E 4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 0 -4 B -6 0 -4 8 2 C 14 4 0 6 12 D 0 -8 -6 0 0 E 4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5878: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) E D B A C (11) D E B A C (9) E D C A B (6) D B E A C (6) A C B D E (6) C A E B D (5) E C A D B (4) C A B E D (4) C A B D E (3) E C D A B (2) E B D A C (2) D E C A B (2) D E B C A (2) D C B A E (2) B D A E C (2) B D A C E (2) B A C E D (2) A C B E D (2) E D B C A (1) E C A B D (1) E B A C D (1) D E C B A (1) D C E A B (1) D B E C A (1) D B C A E (1) D B A C E (1) C E D A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B A E C D (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 20 -10 -2 B 18 0 14 -8 -2 C -20 -14 0 -2 -6 D 10 8 2 0 10 E 2 2 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 20 -10 -2 B 18 0 14 -8 -2 C -20 -14 0 -2 -6 D 10 8 2 0 10 E 2 2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 B=21 C=16 A=9 so A is eliminated. Round 2 votes counts: E=28 D=26 C=24 B=22 so B is eliminated. Round 3 votes counts: C=40 D=31 E=29 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:211 E:200 A:195 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 20 -10 -2 B 18 0 14 -8 -2 C -20 -14 0 -2 -6 D 10 8 2 0 10 E 2 2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 20 -10 -2 B 18 0 14 -8 -2 C -20 -14 0 -2 -6 D 10 8 2 0 10 E 2 2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 20 -10 -2 B 18 0 14 -8 -2 C -20 -14 0 -2 -6 D 10 8 2 0 10 E 2 2 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5879: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (15) D C A B E (10) D B E C A (8) B E D C A (7) A C E B D (6) A C D E B (6) D B E A C (5) B E C A D (5) C A E B D (4) C A D E B (4) B D E C A (4) E B C A D (3) E A C B D (3) D B C E A (3) D A C E B (3) B E D A C (2) A D C E B (2) E C A B D (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C A E (1) D B A C E (1) D A C B E (1) D A B E C (1) C A E D B (1) B D E A C (1) Total count = 100 A B C D E A 0 -14 -4 2 -18 B 14 0 16 4 0 C 4 -16 0 -2 -14 D -2 -4 2 0 4 E 18 0 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.581584 C: 0.000000 D: 0.000000 E: 0.418416 Sum of squares = 0.51331195152 Cumulative probabilities = A: 0.000000 B: 0.581584 C: 0.581584 D: 0.581584 E: 1.000000 A B C D E A 0 -14 -4 2 -18 B 14 0 16 4 0 C 4 -16 0 -2 -14 D -2 -4 2 0 4 E 18 0 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500481 C: 0.000000 D: 0.000000 E: 0.499519 Sum of squares = 0.500000463182 Cumulative probabilities = A: 0.000000 B: 0.500481 C: 0.500481 D: 0.500481 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=23 B=19 A=14 C=9 so C is eliminated. Round 2 votes counts: D=35 E=23 A=23 B=19 so B is eliminated. Round 3 votes counts: D=40 E=37 A=23 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:217 E:214 D:200 C:186 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 2 -18 B 14 0 16 4 0 C 4 -16 0 -2 -14 D -2 -4 2 0 4 E 18 0 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500481 C: 0.000000 D: 0.000000 E: 0.499519 Sum of squares = 0.500000463182 Cumulative probabilities = A: 0.000000 B: 0.500481 C: 0.500481 D: 0.500481 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 2 -18 B 14 0 16 4 0 C 4 -16 0 -2 -14 D -2 -4 2 0 4 E 18 0 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500481 C: 0.000000 D: 0.000000 E: 0.499519 Sum of squares = 0.500000463182 Cumulative probabilities = A: 0.000000 B: 0.500481 C: 0.500481 D: 0.500481 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 2 -18 B 14 0 16 4 0 C 4 -16 0 -2 -14 D -2 -4 2 0 4 E 18 0 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500481 C: 0.000000 D: 0.000000 E: 0.499519 Sum of squares = 0.500000463182 Cumulative probabilities = A: 0.000000 B: 0.500481 C: 0.500481 D: 0.500481 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5880: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (6) E A C D B (5) E A B D C (5) D B A C E (5) B A E D C (5) E A D B C (4) C E B D A (4) B D A E C (4) A D B E C (4) E B A D C (3) E B A C D (3) D A B E C (3) D A B C E (3) C E A D B (3) C B D E A (3) B D A C E (3) E C B A D (2) E C A B D (2) E B C A D (2) E A C B D (2) D C B A E (2) C D B A E (2) B E A D C (2) B D C E A (2) B D C A E (2) B C D E A (2) A D E B C (2) E A D C B (1) E A B C D (1) D C A B E (1) C E B A D (1) C D E A B (1) C D B E A (1) C D A E B (1) C A E D B (1) B C D A E (1) B A D E C (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 22 20 -10 B 0 0 20 4 -6 C -22 -20 0 -12 -24 D -20 -4 12 0 -14 E 10 6 24 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 22 20 -10 B 0 0 20 4 -6 C -22 -20 0 -12 -24 D -20 -4 12 0 -14 E 10 6 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=22 C=17 D=14 A=11 so A is eliminated. Round 2 votes counts: E=40 B=23 D=20 C=17 so C is eliminated. Round 3 votes counts: E=49 B=26 D=25 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:216 B:209 D:187 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 22 20 -10 B 0 0 20 4 -6 C -22 -20 0 -12 -24 D -20 -4 12 0 -14 E 10 6 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 22 20 -10 B 0 0 20 4 -6 C -22 -20 0 -12 -24 D -20 -4 12 0 -14 E 10 6 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 22 20 -10 B 0 0 20 4 -6 C -22 -20 0 -12 -24 D -20 -4 12 0 -14 E 10 6 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5881: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) B A D E C (8) A B D E C (8) C B E D A (7) A D B E C (7) B A C D E (6) D E A C B (5) C E D B A (5) C E B D A (4) B A D C E (4) E C D A B (3) B C E A D (3) B A E D C (3) E D C A B (2) E C D B A (2) D A E C B (2) B C E D A (2) B C A E D (2) A D E B C (2) A B D C E (2) E C B D A (1) D E C A B (1) D C E A B (1) D A E B C (1) C D E A B (1) C D A E B (1) C B E A D (1) C B A D E (1) C A D E B (1) B E C D A (1) B E A D C (1) B C A D E (1) B A E C D (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 2 4 2 B 6 0 4 12 16 C -2 -4 0 4 4 D -4 -12 -4 0 6 E -2 -16 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 2 B 6 0 4 12 16 C -2 -4 0 4 4 D -4 -12 -4 0 6 E -2 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=30 A=20 D=10 E=8 so E is eliminated. Round 2 votes counts: C=36 B=32 A=20 D=12 so D is eliminated. Round 3 votes counts: C=40 B=32 A=28 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:201 C:201 D:193 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 2 B 6 0 4 12 16 C -2 -4 0 4 4 D -4 -12 -4 0 6 E -2 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 2 B 6 0 4 12 16 C -2 -4 0 4 4 D -4 -12 -4 0 6 E -2 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 2 B 6 0 4 12 16 C -2 -4 0 4 4 D -4 -12 -4 0 6 E -2 -16 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5882: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) D C B E A (5) D E B C A (4) D E B A C (4) C A E B D (4) E C D A B (3) E C A D B (3) E C A B D (3) E B A D C (3) E A B C D (3) D E C B A (3) C E D A B (3) C E A D B (3) C D A B E (3) C A B D E (3) B A E D C (3) B A D E C (3) A E B C D (3) E D C B A (2) E A C B D (2) D C E B A (2) D B C A E (2) C E A B D (2) C D E A B (2) C D B A E (2) A C B E D (2) A B E C D (2) A B C E D (2) A B C D E (2) E D C A B (1) E D A B C (1) E B D A C (1) D C B A E (1) D B A E C (1) C D E B A (1) C D A E B (1) C A E D B (1) C A D B E (1) C A B E D (1) B E A D C (1) B D A E C (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -14 -4 -20 B -4 0 -10 -14 -12 C 14 10 0 8 -10 D 4 14 -8 0 0 E 20 12 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.320256 E: 0.679744 Sum of squares = 0.564615766165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.320256 E: 1.000000 A B C D E A 0 4 -14 -4 -20 B -4 0 -10 -14 -12 C 14 10 0 8 -10 D 4 14 -8 0 0 E 20 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=27 E=22 A=12 B=9 so B is eliminated. Round 2 votes counts: D=31 C=28 E=23 A=18 so A is eliminated. Round 3 votes counts: C=35 D=34 E=31 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:211 D:205 A:183 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -14 -4 -20 B -4 0 -10 -14 -12 C 14 10 0 8 -10 D 4 14 -8 0 0 E 20 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 -4 -20 B -4 0 -10 -14 -12 C 14 10 0 8 -10 D 4 14 -8 0 0 E 20 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 -4 -20 B -4 0 -10 -14 -12 C 14 10 0 8 -10 D 4 14 -8 0 0 E 20 12 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5883: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) A B C D E (9) B E A D C (8) A C D E B (7) A B E C D (7) C D E A B (6) C D A E B (6) B A E D C (6) A C D B E (6) E D C B A (5) E B D C A (5) B E D C A (5) C D A B E (2) E D C A B (1) E C D A B (1) E B A D C (1) D C E A B (1) D C B E A (1) D C B A E (1) C E D A B (1) C A D E B (1) B E D A C (1) B D E C A (1) B D C E A (1) B D C A E (1) B A E C D (1) A E C D B (1) A E C B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 0 0 2 B -4 0 -6 -2 4 C 0 6 0 2 12 D 0 2 -2 0 10 E -2 -4 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.387184 B: 0.000000 C: 0.612816 D: 0.000000 E: 0.000000 Sum of squares = 0.525454685287 Cumulative probabilities = A: 0.387184 B: 0.387184 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 0 2 B -4 0 -6 -2 4 C 0 6 0 2 12 D 0 2 -2 0 10 E -2 -4 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=24 C=16 D=14 E=13 so E is eliminated. Round 2 votes counts: A=33 B=30 D=20 C=17 so C is eliminated. Round 3 votes counts: D=36 A=34 B=30 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:210 D:205 A:203 B:196 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 0 2 B -4 0 -6 -2 4 C 0 6 0 2 12 D 0 2 -2 0 10 E -2 -4 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 0 2 B -4 0 -6 -2 4 C 0 6 0 2 12 D 0 2 -2 0 10 E -2 -4 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 0 2 B -4 0 -6 -2 4 C 0 6 0 2 12 D 0 2 -2 0 10 E -2 -4 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5884: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) D E C A B (5) A B C D E (5) E D C B A (4) E D B C A (4) D E B A C (4) C E D A B (4) C E A B D (4) B A D E C (4) A C B D E (4) E C D A B (3) D E B C A (3) D B E A C (3) D B A E C (3) C A E B D (3) B A D C E (3) A D B C E (3) E C B D A (2) E C B A D (2) D A E B C (2) D A B C E (2) C E A D B (2) B E C A D (2) B D E A C (2) B D A E C (2) B A C D E (2) A C B E D (2) E D C A B (1) E C D B A (1) E B C D A (1) D E C B A (1) D A C B E (1) C E B A D (1) C D E A B (1) C A E D B (1) B E A C D (1) B C A E D (1) B A E C D (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -8 2 -2 B -4 0 0 4 2 C 8 0 0 6 -2 D -2 -4 -6 0 0 E 2 -2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.548528 C: 0.451472 D: 0.000000 E: 0.000000 Sum of squares = 0.504710015587 Cumulative probabilities = A: 0.000000 B: 0.548528 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 2 -2 B -4 0 0 4 2 C 8 0 0 6 -2 D -2 -4 -6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500219 C: 0.499781 D: 0.000000 E: 0.000000 Sum of squares = 0.500000095669 Cumulative probabilities = A: 0.000000 B: 0.500219 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 B=19 E=18 A=15 so A is eliminated. Round 2 votes counts: C=30 D=27 B=25 E=18 so E is eliminated. Round 3 votes counts: C=38 D=36 B=26 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:201 E:201 A:198 D:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 2 -2 B -4 0 0 4 2 C 8 0 0 6 -2 D -2 -4 -6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500219 C: 0.499781 D: 0.000000 E: 0.000000 Sum of squares = 0.500000095669 Cumulative probabilities = A: 0.000000 B: 0.500219 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 2 -2 B -4 0 0 4 2 C 8 0 0 6 -2 D -2 -4 -6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500219 C: 0.499781 D: 0.000000 E: 0.000000 Sum of squares = 0.500000095669 Cumulative probabilities = A: 0.000000 B: 0.500219 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 2 -2 B -4 0 0 4 2 C 8 0 0 6 -2 D -2 -4 -6 0 0 E 2 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500219 C: 0.499781 D: 0.000000 E: 0.000000 Sum of squares = 0.500000095669 Cumulative probabilities = A: 0.000000 B: 0.500219 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5885: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (15) C D A E B (9) E D B A C (6) D E A C B (6) E B D A C (5) C B A E D (4) C B A D E (4) D C A E B (3) D A E C B (3) C A D E B (3) C A D B E (3) B E D A C (3) B E A C D (3) E D A B C (2) D E C A B (2) D C E A B (2) D A C E B (2) C D E B A (2) C D B E A (2) C D A B E (2) B C E D A (2) B C E A D (2) B C A E D (2) B A E D C (2) A D E B C (2) E D B C A (1) D E C B A (1) D E A B C (1) D A E B C (1) B E D C A (1) B E C D A (1) B E C A D (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 6 -14 -16 B 14 0 2 -8 -4 C -6 -2 0 -18 -14 D 14 8 18 0 -2 E 16 4 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 6 -14 -16 B 14 0 2 -8 -4 C -6 -2 0 -18 -14 D 14 8 18 0 -2 E 16 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=29 D=21 E=14 A=4 so A is eliminated. Round 2 votes counts: B=33 C=29 D=24 E=14 so E is eliminated. Round 3 votes counts: B=38 D=33 C=29 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:218 B:202 A:181 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 6 -14 -16 B 14 0 2 -8 -4 C -6 -2 0 -18 -14 D 14 8 18 0 -2 E 16 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 -14 -16 B 14 0 2 -8 -4 C -6 -2 0 -18 -14 D 14 8 18 0 -2 E 16 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 -14 -16 B 14 0 2 -8 -4 C -6 -2 0 -18 -14 D 14 8 18 0 -2 E 16 4 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996194 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5886: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) B C D A E (9) D B E A C (7) D B C E A (7) E A D C B (5) A C E B D (5) E A D B C (4) E A C D B (4) A E C D B (4) E D A B C (3) D E B A C (3) D E A B C (3) C B D E A (3) C B A E D (3) C B A D E (3) B D C A E (3) A E C B D (3) D B A E C (2) C B D A E (2) C A B E D (2) B D C E A (2) B D A E C (2) B A D E C (2) A E D C B (2) D E B C A (1) C E D A B (1) C E A D B (1) B D A C E (1) B C A D E (1) B A D C E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 6 2 12 B 4 0 6 6 2 C -6 -6 0 4 6 D -2 -6 -4 0 4 E -12 -2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 12 B 4 0 6 6 2 C -6 -6 0 4 6 D -2 -6 -4 0 4 E -12 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998508 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=21 E=16 A=16 so E is eliminated. Round 2 votes counts: A=29 D=26 C=24 B=21 so B is eliminated. Round 3 votes counts: D=34 C=34 A=32 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:209 A:208 C:199 D:196 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 12 B 4 0 6 6 2 C -6 -6 0 4 6 D -2 -6 -4 0 4 E -12 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998508 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 12 B 4 0 6 6 2 C -6 -6 0 4 6 D -2 -6 -4 0 4 E -12 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998508 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 12 B 4 0 6 6 2 C -6 -6 0 4 6 D -2 -6 -4 0 4 E -12 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998508 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5887: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C B A D E (6) C A B E D (5) E D C B A (4) C E D B A (4) C B D E A (4) C B D A E (4) C B A E D (4) A B C E D (4) D E B A C (3) E D A C B (2) E A D B C (2) D E A B C (2) D B C E A (2) B D A C E (2) B C D A E (2) B C A D E (2) B A D C E (2) A E D B C (2) A E B D C (2) A C E B D (2) A C B E D (2) A B D E C (2) A B D C E (2) E C D A B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E B C A (1) D C E B A (1) D B E A C (1) D A B E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A D B (1) C D E B A (1) C D B E A (1) C B E D A (1) C A E B D (1) B D A E C (1) B A C D E (1) A E C B D (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 -6 2 B 0 0 -2 10 4 C -2 2 0 6 16 D 6 -10 -6 0 -14 E -2 -4 -16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.536014 B: 0.463986 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502594065147 Cumulative probabilities = A: 0.536014 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -6 2 B 0 0 -2 10 4 C -2 2 0 6 16 D 6 -10 -6 0 -14 E -2 -4 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035169 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=23 A=20 D=12 B=10 so B is eliminated. Round 2 votes counts: C=39 E=23 A=23 D=15 so D is eliminated. Round 3 votes counts: C=42 E=31 A=27 so A is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:206 A:199 E:196 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -6 2 B 0 0 -2 10 4 C -2 2 0 6 16 D 6 -10 -6 0 -14 E -2 -4 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035169 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -6 2 B 0 0 -2 10 4 C -2 2 0 6 16 D 6 -10 -6 0 -14 E -2 -4 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035169 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -6 2 B 0 0 -2 10 4 C -2 2 0 6 16 D 6 -10 -6 0 -14 E -2 -4 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500133 B: 0.499867 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000035169 Cumulative probabilities = A: 0.500133 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5888: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) E C A D B (6) A B D C E (6) D E B C A (5) D E B A C (5) E D B A C (4) E C D A B (4) A C B E D (4) A B C D E (4) E D C B A (3) D B A E C (3) C E A B D (3) B A D C E (3) A C B D E (3) E D B C A (2) D B E C A (2) C B D A E (2) C B A D E (2) C A B E D (2) B D C A E (2) B A C D E (2) A E C D B (2) A D E B C (2) A D B E C (2) E D A C B (1) E D A B C (1) E C D B A (1) E A D C B (1) E A D B C (1) C E B D A (1) C E A D B (1) C B D E A (1) C A E B D (1) C A B D E (1) B D E C A (1) B D C E A (1) B D A E C (1) B D A C E (1) B C D E A (1) B C D A E (1) B C A D E (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 12 -2 -6 B 6 0 18 -10 8 C -12 -18 0 -12 -12 D 2 10 12 0 20 E 6 -8 12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -2 -6 B 6 0 18 -10 8 C -12 -18 0 -12 -12 D 2 10 12 0 20 E 6 -8 12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 D=23 C=14 B=14 so C is eliminated. Round 2 votes counts: E=29 A=29 D=23 B=19 so B is eliminated. Round 3 votes counts: A=37 D=34 E=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:211 A:199 E:195 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 12 -2 -6 B 6 0 18 -10 8 C -12 -18 0 -12 -12 D 2 10 12 0 20 E 6 -8 12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -2 -6 B 6 0 18 -10 8 C -12 -18 0 -12 -12 D 2 10 12 0 20 E 6 -8 12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -2 -6 B 6 0 18 -10 8 C -12 -18 0 -12 -12 D 2 10 12 0 20 E 6 -8 12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5889: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) A E D B C (8) D A E B C (6) E A D B C (5) E A B C D (5) D E A B C (5) D C E A B (4) C B E A D (4) C B D A E (4) B A E C D (4) A E B D C (4) E A B D C (3) D A E C B (3) C D B E A (3) C D B A E (3) C B E D A (3) B C A E D (3) A D E B C (3) E D A B C (2) D C A E B (2) C D E B A (2) B C E A D (2) A B E D C (2) E C A B D (1) E B A C D (1) D E C A B (1) D C E B A (1) D C B A E (1) D B A C E (1) D A B E C (1) C E B A D (1) C B D E A (1) C B A E D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 30 26 -2 -4 B -30 0 14 -20 -30 C -26 -14 0 -22 -28 D 2 20 22 0 0 E 4 30 28 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.684112 E: 0.315888 Sum of squares = 0.567794090536 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.684112 E: 1.000000 A B C D E A 0 30 26 -2 -4 B -30 0 14 -20 -30 C -26 -14 0 -22 -28 D 2 20 22 0 0 E 4 30 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=22 A=19 E=17 B=9 so B is eliminated. Round 2 votes counts: D=33 C=27 A=23 E=17 so E is eliminated. Round 3 votes counts: A=37 D=35 C=28 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:231 A:225 D:222 B:167 C:155 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 30 26 -2 -4 B -30 0 14 -20 -30 C -26 -14 0 -22 -28 D 2 20 22 0 0 E 4 30 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 26 -2 -4 B -30 0 14 -20 -30 C -26 -14 0 -22 -28 D 2 20 22 0 0 E 4 30 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 26 -2 -4 B -30 0 14 -20 -30 C -26 -14 0 -22 -28 D 2 20 22 0 0 E 4 30 28 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5890: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) D A C B E (6) E B A C D (5) D C A B E (5) E D B A C (4) E A C B D (4) D C B A E (4) D A C E B (4) B C E A D (4) A C D B E (4) E B D A C (3) E B C A D (3) E A B C D (3) D B E C A (3) A C E D B (3) A C E B D (3) A C D E B (3) E A D C B (2) D B C E A (2) C B A E D (2) B E C D A (2) B E C A D (2) B D C A E (2) A E D C B (2) A E C D B (2) E D A C B (1) D E A C B (1) D E A B C (1) C D A B E (1) C B D A E (1) C A B E D (1) C A B D E (1) B E D C A (1) B D E C A (1) B C D E A (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 6 -8 14 B 2 0 -4 -20 10 C -6 4 0 -8 18 D 8 20 8 0 6 E -14 -10 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -8 14 B 2 0 -4 -20 10 C -6 4 0 -8 18 D 8 20 8 0 6 E -14 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=25 A=19 B=13 C=6 so C is eliminated. Round 2 votes counts: D=38 E=25 A=21 B=16 so B is eliminated. Round 3 votes counts: D=43 E=34 A=23 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:205 C:204 B:194 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -8 14 B 2 0 -4 -20 10 C -6 4 0 -8 18 D 8 20 8 0 6 E -14 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -8 14 B 2 0 -4 -20 10 C -6 4 0 -8 18 D 8 20 8 0 6 E -14 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -8 14 B 2 0 -4 -20 10 C -6 4 0 -8 18 D 8 20 8 0 6 E -14 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5891: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (12) D A E C B (9) D A B E C (7) C E A D B (7) B C E D A (6) A D E C B (5) C E B A D (4) B D E A C (4) D E A C B (3) D B A E C (3) D A E B C (3) B D E C A (3) B D A C E (3) B C E A D (3) B A D C E (3) E C B D A (2) C E A B D (2) B D C E A (2) B C A E D (2) A D C E B (2) A C E D B (2) A C D E B (2) E C D A B (1) E C A D B (1) D E C A B (1) D E B A C (1) C B E D A (1) C A E D B (1) B E C D A (1) B A C D E (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 26 -24 16 B 2 0 12 2 8 C -26 -12 0 -24 -12 D 24 -2 24 0 32 E -16 -8 12 -32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 26 -24 16 B 2 0 12 2 8 C -26 -12 0 -24 -12 D 24 -2 24 0 32 E -16 -8 12 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 D=27 C=15 A=14 E=4 so E is eliminated. Round 2 votes counts: B=40 D=27 C=19 A=14 so A is eliminated. Round 3 votes counts: B=41 D=35 C=24 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:239 B:212 A:208 E:178 C:163 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 26 -24 16 B 2 0 12 2 8 C -26 -12 0 -24 -12 D 24 -2 24 0 32 E -16 -8 12 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 26 -24 16 B 2 0 12 2 8 C -26 -12 0 -24 -12 D 24 -2 24 0 32 E -16 -8 12 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 26 -24 16 B 2 0 12 2 8 C -26 -12 0 -24 -12 D 24 -2 24 0 32 E -16 -8 12 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5892: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (14) E B D A C (9) C A D E B (8) E D B A C (7) D E B A C (6) D A C E B (6) C A B E D (6) B E D A C (5) C B A E D (3) A D B C E (3) A C B D E (3) E D B C A (2) E C B D A (2) C D E A B (2) C D A E B (2) B E C A D (2) B E A D C (2) E C D B A (1) E B D C A (1) E B C D A (1) D E C B A (1) D E C A B (1) D E B C A (1) D C A E B (1) D B E A C (1) D A B E C (1) C E B D A (1) C A E D B (1) C A B D E (1) B E A C D (1) B A E D C (1) A D C B E (1) A D B E C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -2 -2 8 B -6 0 -10 -22 -6 C 2 10 0 0 8 D 2 22 0 0 8 E -8 6 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.637405 D: 0.362595 E: 0.000000 Sum of squares = 0.537760410884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.637405 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -2 8 B -6 0 -10 -22 -6 C 2 10 0 0 8 D 2 22 0 0 8 E -8 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=23 D=18 B=11 A=10 so A is eliminated. Round 2 votes counts: C=42 E=23 D=23 B=12 so B is eliminated. Round 3 votes counts: C=43 E=34 D=23 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:216 C:210 A:205 E:191 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -2 8 B -6 0 -10 -22 -6 C 2 10 0 0 8 D 2 22 0 0 8 E -8 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 8 B -6 0 -10 -22 -6 C 2 10 0 0 8 D 2 22 0 0 8 E -8 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 8 B -6 0 -10 -22 -6 C 2 10 0 0 8 D 2 22 0 0 8 E -8 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5893: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) D C B A E (8) D C A B E (8) E B D A C (7) C D A B E (7) E B A D C (6) D E B C A (5) E A B C D (4) C A D B E (4) A E B C D (4) E D B C A (3) D C E A B (3) D B C E A (3) D B C A E (3) E D B A C (2) D C E B A (2) C A D E B (2) B A C E D (2) A E C B D (2) A C E B D (2) A C B E D (2) A C B D E (2) E D C A B (1) D C A E B (1) D B E C A (1) C A B D E (1) B E A C D (1) B D E C A (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -6 -10 2 B 8 0 10 -6 -8 C 6 -10 0 -8 4 D 10 6 8 0 4 E -2 8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -10 2 B 8 0 10 -6 -8 C 6 -10 0 -8 4 D 10 6 8 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=33 A=15 C=14 B=4 so B is eliminated. Round 2 votes counts: D=35 E=34 A=17 C=14 so C is eliminated. Round 3 votes counts: D=42 E=34 A=24 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:202 E:199 C:196 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 2 B 8 0 10 -6 -8 C 6 -10 0 -8 4 D 10 6 8 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 2 B 8 0 10 -6 -8 C 6 -10 0 -8 4 D 10 6 8 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 2 B 8 0 10 -6 -8 C 6 -10 0 -8 4 D 10 6 8 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5894: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) C D B A E (7) A E C D B (7) E A B D C (6) D B C E A (6) C D B E A (5) B D C E A (5) E A C D B (4) C E D B A (4) C A D B E (4) A C D B E (4) E C B D A (3) E A C B D (3) D B C A E (3) B D E C A (3) B D A C E (3) A E C B D (3) A B D C E (3) E C D B A (2) E C A D B (2) C E A D B (2) C A E D B (2) B D E A C (2) B D C A E (2) A C E D B (2) E B D C A (1) E B C D A (1) D B A C E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 4 2 B -4 0 -8 -12 0 C 4 8 0 10 8 D -4 12 -10 0 0 E -2 0 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 4 2 B -4 0 -8 -12 0 C 4 8 0 10 8 D -4 12 -10 0 0 E -2 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=24 E=22 B=15 D=10 so D is eliminated. Round 2 votes counts: A=29 B=25 C=24 E=22 so E is eliminated. Round 3 votes counts: A=42 C=31 B=27 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:203 D:199 E:195 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 4 2 B -4 0 -8 -12 0 C 4 8 0 10 8 D -4 12 -10 0 0 E -2 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 2 B -4 0 -8 -12 0 C 4 8 0 10 8 D -4 12 -10 0 0 E -2 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 2 B -4 0 -8 -12 0 C 4 8 0 10 8 D -4 12 -10 0 0 E -2 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5895: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) E C A B D (7) E C A D B (6) D B E A C (6) B D A C E (6) D B E C A (5) B D A E C (5) A C E B D (5) E C D A B (4) C E A D B (4) E A C B D (3) D E B C A (3) D B C E A (3) D B A E C (3) C E A B D (3) B D E A C (3) B A D E C (3) B A D C E (3) B A E D C (2) B A C D E (2) A B C E D (2) E D C B A (1) E D C A B (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A B E (1) C E D A B (1) C D A E B (1) C A E D B (1) C A D E B (1) C A B D E (1) B A E C D (1) B A C E D (1) A E C B D (1) Total count = 100 A B C D E A 0 -14 12 -4 -6 B 14 0 16 0 12 C -12 -16 0 -8 -12 D 4 0 8 0 10 E 6 -12 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.482558 C: 0.000000 D: 0.517442 E: 0.000000 Sum of squares = 0.500608439157 Cumulative probabilities = A: 0.000000 B: 0.482558 C: 0.482558 D: 1.000000 E: 1.000000 A B C D E A 0 -14 12 -4 -6 B 14 0 16 0 12 C -12 -16 0 -8 -12 D 4 0 8 0 10 E 6 -12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=26 E=24 C=12 A=8 so A is eliminated. Round 2 votes counts: D=30 B=28 E=25 C=17 so C is eliminated. Round 3 votes counts: E=39 D=32 B=29 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:221 D:211 E:198 A:194 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 12 -4 -6 B 14 0 16 0 12 C -12 -16 0 -8 -12 D 4 0 8 0 10 E 6 -12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 12 -4 -6 B 14 0 16 0 12 C -12 -16 0 -8 -12 D 4 0 8 0 10 E 6 -12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 12 -4 -6 B 14 0 16 0 12 C -12 -16 0 -8 -12 D 4 0 8 0 10 E 6 -12 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5896: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) C A E D B (6) E C A B D (5) D B E A C (5) D B C A E (5) C E A D B (5) B D A E C (5) A E C B D (5) C D A E B (4) B E D A C (4) A E B C D (3) A C E B D (3) E D C B A (2) E D B C A (2) E B D A C (2) E B A D C (2) E A B C D (2) D E B C A (2) D C B A E (2) D B C E A (2) D B A C E (2) C E A B D (2) C D A B E (2) C A D E B (2) B D E A C (2) B D A C E (2) B A E D C (2) B A D E C (2) A C B E D (2) D E C B A (1) D B E C A (1) D B A E C (1) C A E B D (1) C A D B E (1) A D C B E (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 10 8 8 B -8 0 -4 6 -12 C -10 4 0 6 -12 D -8 -6 -6 0 -14 E -8 12 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 8 8 B -8 0 -4 6 -12 C -10 4 0 6 -12 D -8 -6 -6 0 -14 E -8 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 E=22 D=21 B=17 A=17 so B is eliminated. Round 2 votes counts: D=30 E=26 C=23 A=21 so A is eliminated. Round 3 votes counts: E=37 D=33 C=30 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:217 E:215 C:194 B:191 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 8 8 B -8 0 -4 6 -12 C -10 4 0 6 -12 D -8 -6 -6 0 -14 E -8 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 8 8 B -8 0 -4 6 -12 C -10 4 0 6 -12 D -8 -6 -6 0 -14 E -8 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 8 8 B -8 0 -4 6 -12 C -10 4 0 6 -12 D -8 -6 -6 0 -14 E -8 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5897: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) C B E A D (7) D A E B C (6) C B D A E (5) B C E A D (5) B A D E C (5) C D B A E (4) E A D B C (3) C E B D A (3) C D E A B (3) C D A E B (3) C D A B E (3) B D A C E (3) E D A C B (2) E C B A D (2) E C A D B (2) E B C A D (2) E B A C D (2) D A E C B (2) D A B C E (2) C E B A D (2) C B E D A (2) B C D A E (2) B C A D E (2) B A E D C (2) A E D B C (2) A D B E C (2) A B D E C (2) E D A B C (1) E B A D C (1) E A B D C (1) D A C B E (1) C E D B A (1) C E D A B (1) C D B E A (1) C B D E A (1) B E C A D (1) B E A D C (1) B C A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 -12 -6 -8 12 B 12 0 10 6 22 C 6 -10 0 10 6 D 8 -6 -10 0 12 E -12 -22 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -8 12 B 12 0 10 6 22 C 6 -10 0 10 6 D 8 -6 -10 0 12 E -12 -22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=23 D=19 E=16 A=6 so A is eliminated. Round 2 votes counts: C=36 B=25 D=21 E=18 so E is eliminated. Round 3 votes counts: C=40 B=31 D=29 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:206 D:202 A:193 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -8 12 B 12 0 10 6 22 C 6 -10 0 10 6 D 8 -6 -10 0 12 E -12 -22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -8 12 B 12 0 10 6 22 C 6 -10 0 10 6 D 8 -6 -10 0 12 E -12 -22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -8 12 B 12 0 10 6 22 C 6 -10 0 10 6 D 8 -6 -10 0 12 E -12 -22 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5898: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) A D C B E (6) E C B D A (5) C E A B D (5) B D A E C (5) E C B A D (4) D B A E C (4) C A E D B (3) B E D C A (3) A D B C E (3) A C B E D (3) A B D C E (3) E D B C A (2) E B C D A (2) E B C A D (2) D E B C A (2) D A E C B (2) C E D A B (2) C A D E B (2) B E D A C (2) B E C D A (2) B E A C D (2) B D E A C (2) A D C E B (2) A C D B E (2) A B D E C (2) A B C D E (2) E B D C A (1) D E C A B (1) D C E A B (1) D B E C A (1) D B E A C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E D B A (1) C E B A D (1) C D E A B (1) B E C A D (1) B C E A D (1) B A E D C (1) B A D E C (1) B A D C E (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 14 14 10 B -6 0 -4 4 2 C -14 4 0 2 2 D -14 -4 -2 0 12 E -10 -2 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 14 10 B -6 0 -4 4 2 C -14 4 0 2 2 D -14 -4 -2 0 12 E -10 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=21 E=16 D=15 C=15 so D is eliminated. Round 2 votes counts: A=38 B=27 E=19 C=16 so C is eliminated. Round 3 votes counts: A=43 E=30 B=27 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:198 C:197 D:196 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 14 10 B -6 0 -4 4 2 C -14 4 0 2 2 D -14 -4 -2 0 12 E -10 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 14 10 B -6 0 -4 4 2 C -14 4 0 2 2 D -14 -4 -2 0 12 E -10 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 14 10 B -6 0 -4 4 2 C -14 4 0 2 2 D -14 -4 -2 0 12 E -10 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5899: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (15) B D A C E (14) A C D B E (7) C A E D B (5) B A C D E (5) E B D C A (4) D B E A C (4) B D E A C (4) B A D C E (4) D B A C E (3) C E A D B (3) E D C A B (2) E B C A D (2) D E A C B (2) D A C B E (2) D A B C E (2) C A D E B (2) B E D C A (2) B D A E C (2) B C A E D (2) E D B C A (1) E D B A C (1) E C A B D (1) E B C D A (1) D B A E C (1) D A E C B (1) D A C E B (1) C E A B D (1) C A E B D (1) C A D B E (1) B E D A C (1) B E C A D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 12 4 10 B 4 0 10 -8 14 C -12 -10 0 -4 10 D -4 8 4 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000002 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 4 10 B 4 0 10 -8 14 C -12 -10 0 -4 10 D -4 8 4 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=27 D=16 C=13 A=9 so A is eliminated. Round 2 votes counts: B=35 E=27 C=21 D=17 so D is eliminated. Round 3 votes counts: B=46 E=30 C=24 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:211 D:211 B:210 C:192 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 12 4 10 B 4 0 10 -8 14 C -12 -10 0 -4 10 D -4 8 4 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 4 10 B 4 0 10 -8 14 C -12 -10 0 -4 10 D -4 8 4 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 4 10 B 4 0 10 -8 14 C -12 -10 0 -4 10 D -4 8 4 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5900: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) D A E C B (7) D A E B C (6) E C B D A (5) C B A E D (5) A D C B E (5) E C B A D (4) B C E A D (4) A C B D E (4) E B D C A (3) D A B E C (3) A D B C E (3) A C D B E (3) E D C A B (2) E D B A C (2) E C D A B (2) E C A D B (2) E B C D A (2) D E B A C (2) D A C B E (2) D A B C E (2) C E B A D (2) C B E A D (2) C A B D E (2) B A C D E (2) E D C B A (1) E D A C B (1) E D A B C (1) E C A B D (1) E A D C B (1) D E A B C (1) D B A E C (1) C E A B D (1) C A B E D (1) B E D C A (1) B C E D A (1) B C A E D (1) B C A D E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 -8 0 B -4 0 -8 -12 -10 C 4 8 0 -6 -14 D 8 12 6 0 -10 E 0 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.358778 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.641222 Sum of squares = 0.53988742354 Cumulative probabilities = A: 0.358778 B: 0.358778 C: 0.358778 D: 0.358778 E: 1.000000 A B C D E A 0 4 -4 -8 0 B -4 0 -8 -12 -10 C 4 8 0 -6 -14 D 8 12 6 0 -10 E 0 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=24 A=17 C=13 B=10 so B is eliminated. Round 2 votes counts: E=37 D=24 C=20 A=19 so A is eliminated. Round 3 votes counts: E=37 D=32 C=31 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:208 A:196 C:196 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 -8 0 B -4 0 -8 -12 -10 C 4 8 0 -6 -14 D 8 12 6 0 -10 E 0 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 0 B -4 0 -8 -12 -10 C 4 8 0 -6 -14 D 8 12 6 0 -10 E 0 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 0 B -4 0 -8 -12 -10 C 4 8 0 -6 -14 D 8 12 6 0 -10 E 0 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5901: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (14) E B D A C (13) E B A D C (8) E B C A D (7) D A C B E (5) B E D A C (5) C A D E B (4) E C A B D (3) D C A B E (3) D B A C E (3) D A B C E (3) C A E D B (3) E C B A D (2) C E A D B (2) C E A B D (2) C D A B E (2) C A E B D (2) B E A D C (2) A C D B E (2) E D B C A (1) E D B A C (1) E C D B A (1) E C A D B (1) E B A C D (1) D E C B A (1) D B E A C (1) D B A E C (1) B D A E C (1) B A E D C (1) B A D C E (1) A D C B E (1) A D B C E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 4 18 -2 B 0 0 2 0 -4 C -4 -2 0 -4 0 D -18 0 4 0 -12 E 2 4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.248228 D: 0.000000 E: 0.751772 Sum of squares = 0.626777845138 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.248228 D: 0.248228 E: 1.000000 A B C D E A 0 0 4 18 -2 B 0 0 2 0 -4 C -4 -2 0 -4 0 D -18 0 4 0 -12 E 2 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555973341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=29 D=17 B=10 A=6 so A is eliminated. Round 2 votes counts: E=38 C=32 D=19 B=11 so B is eliminated. Round 3 votes counts: E=46 C=33 D=21 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:210 E:209 B:199 C:195 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 18 -2 B 0 0 2 0 -4 C -4 -2 0 -4 0 D -18 0 4 0 -12 E 2 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555973341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 18 -2 B 0 0 2 0 -4 C -4 -2 0 -4 0 D -18 0 4 0 -12 E 2 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555973341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 18 -2 B 0 0 2 0 -4 C -4 -2 0 -4 0 D -18 0 4 0 -12 E 2 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555973341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5902: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) A E C D B (9) C B E A D (8) C A E B D (8) B D C E A (7) A E D C B (6) D B E A C (5) D B C A E (5) A E C B D (5) D B C E A (4) C E A B D (4) B C D E A (4) D B E C A (3) B C D A E (3) E D A B C (2) E A C D B (2) E A C B D (2) D E A B C (2) D A E C B (2) B C E D A (2) E D B A C (1) E A D C B (1) D A B E C (1) C E B A D (1) B D C A E (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 12 0 -4 4 B -12 0 0 -8 -14 C 0 0 0 -2 -4 D 4 8 2 0 -4 E -4 14 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333328 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 12 0 -4 4 B -12 0 0 -8 -14 C 0 0 0 -2 -4 D 4 8 2 0 -4 E -4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332998 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=22 C=21 B=17 E=8 so E is eliminated. Round 2 votes counts: D=35 A=27 C=21 B=17 so B is eliminated. Round 3 votes counts: D=43 C=30 A=27 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:209 A:206 D:205 C:197 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 -4 4 B -12 0 0 -8 -14 C 0 0 0 -2 -4 D 4 8 2 0 -4 E -4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332998 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -4 4 B -12 0 0 -8 -14 C 0 0 0 -2 -4 D 4 8 2 0 -4 E -4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332998 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -4 4 B -12 0 0 -8 -14 C 0 0 0 -2 -4 D 4 8 2 0 -4 E -4 14 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332998 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) A C E B D (10) C B A D E (7) E D B C A (6) E A D C B (5) C A B E D (5) B D C E A (5) E A C D B (4) D E B A C (4) D B E C A (4) C A B D E (4) B C D A E (4) E D A B C (3) D B E A C (3) A E C D B (3) A C E D B (3) A C B E D (3) D E B C A (2) B D E C A (2) B D C A E (2) A D B C E (2) E D A C B (1) E C D B A (1) E C A D B (1) E A D B C (1) C E B D A (1) B C D E A (1) B C A D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 8 0 -10 B 8 0 4 -10 -12 C -8 -4 0 -4 -4 D 0 10 4 0 -16 E 10 12 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 8 0 -10 B 8 0 4 -10 -12 C -8 -4 0 -4 -4 D 0 10 4 0 -16 E 10 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=22 C=17 B=15 D=13 so D is eliminated. Round 2 votes counts: E=39 B=22 A=22 C=17 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:221 D:199 A:195 B:195 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 8 0 -10 B 8 0 4 -10 -12 C -8 -4 0 -4 -4 D 0 10 4 0 -16 E 10 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 0 -10 B 8 0 4 -10 -12 C -8 -4 0 -4 -4 D 0 10 4 0 -16 E 10 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 0 -10 B 8 0 4 -10 -12 C -8 -4 0 -4 -4 D 0 10 4 0 -16 E 10 12 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5904: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) D C E B A (6) D A E B C (6) D E A C B (5) A E D B C (5) D A E C B (4) C D B E A (4) B C A E D (4) B A E C D (4) A E B D C (4) D E C A B (3) C E B D A (3) C B E A D (3) B C E A D (3) A D E B C (3) E C B A D (2) E A B C D (2) D A B E C (2) D A B C E (2) C B D E A (2) B E C A D (2) B A C D E (2) A E B C D (2) A B D E C (2) E D C A B (1) E D A C B (1) E C D A B (1) E C B D A (1) D C E A B (1) D C B A E (1) D C A B E (1) D B A C E (1) C E D B A (1) C E B A D (1) C B E D A (1) B A C E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 16 18 6 14 B -16 0 16 4 -4 C -18 -16 0 2 -24 D -6 -4 -2 0 -10 E -14 4 24 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 6 14 B -16 0 16 4 -4 C -18 -16 0 2 -24 D -6 -4 -2 0 -10 E -14 4 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=29 B=16 C=15 E=8 so E is eliminated. Round 2 votes counts: D=34 A=31 C=19 B=16 so B is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:212 B:200 D:189 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 6 14 B -16 0 16 4 -4 C -18 -16 0 2 -24 D -6 -4 -2 0 -10 E -14 4 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 6 14 B -16 0 16 4 -4 C -18 -16 0 2 -24 D -6 -4 -2 0 -10 E -14 4 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 6 14 B -16 0 16 4 -4 C -18 -16 0 2 -24 D -6 -4 -2 0 -10 E -14 4 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994456 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5905: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) B D E C A (7) D B E C A (6) B D E A C (6) A B D C E (6) C E A D B (5) C A E D B (5) B A D E C (5) A C E D B (5) C E D A B (4) A C E B D (4) A C B E D (4) A B C E D (4) E C D A B (3) D E B C A (3) C A E B D (3) B D A C E (3) B A D C E (3) E D C B A (2) E C D B A (2) C E D B A (2) B D A E C (2) A E C D B (2) E D C A B (1) E C A D B (1) D E A B C (1) C E B D A (1) B D C E A (1) A B D E C (1) Total count = 100 A B C D E A 0 14 8 12 10 B -14 0 12 16 12 C -8 -12 0 6 16 D -12 -16 -6 0 4 E -10 -12 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 12 10 B -14 0 12 16 12 C -8 -12 0 6 16 D -12 -16 -6 0 4 E -10 -12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=27 C=20 D=10 E=9 so E is eliminated. Round 2 votes counts: A=34 B=27 C=26 D=13 so D is eliminated. Round 3 votes counts: B=36 A=35 C=29 so C is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:213 C:201 D:185 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 12 10 B -14 0 12 16 12 C -8 -12 0 6 16 D -12 -16 -6 0 4 E -10 -12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 12 10 B -14 0 12 16 12 C -8 -12 0 6 16 D -12 -16 -6 0 4 E -10 -12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 12 10 B -14 0 12 16 12 C -8 -12 0 6 16 D -12 -16 -6 0 4 E -10 -12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5906: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) A C B E D (8) E B D C A (7) D E B C A (6) D A E C B (6) C B A E D (5) B C E D A (5) A D C E B (5) E B C D A (4) C A B D E (4) E D A B C (3) D E A C B (3) C B A D E (3) B C E A D (3) A C B D E (3) E D B C A (2) B E C D A (2) A D C B E (2) E D B A C (1) E B C A D (1) E A D B C (1) D E B A C (1) D E A B C (1) D B E C A (1) D B C E A (1) C A B E D (1) B E C A D (1) B D E C A (1) B D C E A (1) B C D E A (1) B C A E D (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E C B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 0 8 8 B -6 0 -12 6 6 C 0 12 0 12 10 D -8 -6 -12 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.552992 B: 0.000000 C: 0.447008 D: 0.000000 E: 0.000000 Sum of squares = 0.50561633001 Cumulative probabilities = A: 0.552992 B: 0.552992 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 8 8 B -6 0 -12 6 6 C 0 12 0 12 10 D -8 -6 -12 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=19 D=19 B=15 C=13 so C is eliminated. Round 2 votes counts: A=39 B=23 E=19 D=19 so E is eliminated. Round 3 votes counts: A=40 B=35 D=25 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 A:211 B:197 D:188 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 8 8 B -6 0 -12 6 6 C 0 12 0 12 10 D -8 -6 -12 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 8 8 B -6 0 -12 6 6 C 0 12 0 12 10 D -8 -6 -12 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 8 8 B -6 0 -12 6 6 C 0 12 0 12 10 D -8 -6 -12 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5907: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (15) C A E B D (6) A D C E B (6) A C E B D (6) A D E B C (5) E B C D A (4) D E B A C (4) D A E B C (4) C A B E D (4) E B C A D (3) D C B A E (3) D B E A C (3) C E B A D (3) A C D E B (3) E C B A D (2) D A B E C (2) C B E A D (2) B E D C A (2) A E C B D (2) A D C B E (2) A C D B E (2) E B D C A (1) E B D A C (1) E A B C D (1) D E B C A (1) D C B E A (1) D B C E A (1) D B A E C (1) D A C B E (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C D A (1) B D E C A (1) A E B D C (1) A E B C D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -6 2 2 B 4 0 4 -14 -12 C 6 -4 0 -12 -12 D -2 14 12 0 16 E -2 12 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999957 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 2 B 4 0 4 -14 -12 C 6 -4 0 -12 -12 D -2 14 12 0 16 E -2 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999404 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=30 C=18 E=12 B=4 so B is eliminated. Round 2 votes counts: D=37 A=30 C=18 E=15 so E is eliminated. Round 3 votes counts: D=41 A=31 C=28 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:220 E:203 A:197 B:191 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 2 2 B 4 0 4 -14 -12 C 6 -4 0 -12 -12 D -2 14 12 0 16 E -2 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999404 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 2 B 4 0 4 -14 -12 C 6 -4 0 -12 -12 D -2 14 12 0 16 E -2 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999404 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 2 B 4 0 4 -14 -12 C 6 -4 0 -12 -12 D -2 14 12 0 16 E -2 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999404 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5908: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) E D C A B (7) E A C D B (6) B D C A E (5) A C E D B (5) E D B C A (4) A E C D B (4) A C B D E (4) E D C B A (3) D B C E A (3) C D A E B (3) E B A D C (2) E A D C B (2) E A B C D (2) D E C A B (2) D C E B A (2) C D B A E (2) B C D A E (2) B C A D E (2) B A E C D (2) A E B C D (2) A C B E D (2) A B C E D (2) E D A C B (1) E C A D B (1) E B D C A (1) E B D A C (1) E A C B D (1) D E B C A (1) D C E A B (1) D C B A E (1) D C A E B (1) C A E D B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E C A (1) B D C E A (1) B D A C E (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 8 10 10 B 0 0 -2 -2 -8 C -8 2 0 14 4 D -10 2 -14 0 -8 E -10 8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.643226 B: 0.356774 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.541027219904 Cumulative probabilities = A: 0.643226 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 10 10 B 0 0 -2 -2 -8 C -8 2 0 14 4 D -10 2 -14 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=30 A=21 D=11 C=7 so C is eliminated. Round 2 votes counts: E=31 B=30 A=23 D=16 so D is eliminated. Round 3 votes counts: E=37 B=36 A=27 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:214 C:206 E:201 B:194 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 10 10 B 0 0 -2 -2 -8 C -8 2 0 14 4 D -10 2 -14 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 10 10 B 0 0 -2 -2 -8 C -8 2 0 14 4 D -10 2 -14 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 10 10 B 0 0 -2 -2 -8 C -8 2 0 14 4 D -10 2 -14 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5909: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) C E A D B (8) B E C D A (8) B D A E C (8) E C D A B (4) D E A C B (4) A D E C B (4) A D C E B (4) A D B E C (4) D A E C B (3) C E D B A (3) C E B D A (3) B E D C A (3) B D E C A (3) B C E D A (3) A C E D B (3) A B D E C (3) B C E A D (2) B A D E C (2) B A C E D (2) E C D B A (1) D E C A B (1) D E B A C (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A B D (1) C B E D A (1) C A E D B (1) B D E A C (1) A D E B C (1) A D C B E (1) A D B C E (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -6 -18 -16 B -14 0 -10 -14 -8 C 6 10 0 6 -6 D 18 14 -6 0 -10 E 16 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -6 -18 -16 B -14 0 -10 -14 -8 C 6 10 0 6 -6 D 18 14 -6 0 -10 E 16 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=28 A=24 D=11 E=5 so E is eliminated. Round 2 votes counts: C=33 B=32 A=24 D=11 so D is eliminated. Round 3 votes counts: C=34 B=34 A=32 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:220 C:208 D:208 A:187 B:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -6 -18 -16 B -14 0 -10 -14 -8 C 6 10 0 6 -6 D 18 14 -6 0 -10 E 16 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 -18 -16 B -14 0 -10 -14 -8 C 6 10 0 6 -6 D 18 14 -6 0 -10 E 16 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 -18 -16 B -14 0 -10 -14 -8 C 6 10 0 6 -6 D 18 14 -6 0 -10 E 16 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5910: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (12) C B E A D (10) B E C D A (9) D E B A C (6) C A B E D (6) A D E B C (6) D B E C A (5) D B E A C (5) A D C E B (5) C A E B D (4) A D E C B (4) A C E D B (3) E B D C A (2) E B C A D (2) D A E C B (2) D A B E C (2) B C E D A (2) B C E A D (2) B C D E A (2) A C E B D (2) A C D E B (2) D E A B C (1) D B A C E (1) C A B D E (1) B E D C A (1) B E C A D (1) B D C E A (1) A E C D B (1) Total count = 100 A B C D E A 0 2 4 -2 2 B -2 0 20 -10 -4 C -4 -20 0 -6 -18 D 2 10 6 0 10 E -2 4 18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -2 2 B -2 0 20 -10 -4 C -4 -20 0 -6 -18 D 2 10 6 0 10 E -2 4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=23 C=21 B=18 E=4 so E is eliminated. Round 2 votes counts: D=34 A=23 B=22 C=21 so C is eliminated. Round 3 votes counts: D=34 A=34 B=32 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:205 A:203 B:202 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -2 2 B -2 0 20 -10 -4 C -4 -20 0 -6 -18 D 2 10 6 0 10 E -2 4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 2 B -2 0 20 -10 -4 C -4 -20 0 -6 -18 D 2 10 6 0 10 E -2 4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 2 B -2 0 20 -10 -4 C -4 -20 0 -6 -18 D 2 10 6 0 10 E -2 4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5911: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) D A B E C (7) C E B D A (6) C E B A D (5) A D B C E (5) E C B D A (4) C E D B A (4) A D C B E (4) D E B C A (3) D B E A C (3) C E A B D (3) A B D E C (3) A B C E D (3) E D C B A (2) E B C D A (2) D E C A B (2) D E B A C (2) D B A E C (2) C E D A B (2) C D A E B (2) C B E A D (2) C A E D B (2) A D C E B (2) A C B E D (2) A B C D E (2) E D B C A (1) D E C B A (1) D E A B C (1) D A E C B (1) D A E B C (1) D A C E B (1) C E A D B (1) C A E B D (1) C A D E B (1) C A B E D (1) B E D A C (1) B E C A D (1) B A D E C (1) B A C E D (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 18 8 4 8 B -18 0 -2 -20 -2 C -8 2 0 -4 4 D -4 20 4 0 10 E -8 2 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999464 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 8 4 8 B -18 0 -2 -20 -2 C -8 2 0 -4 4 D -4 20 4 0 10 E -8 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=30 D=24 E=9 B=4 so B is eliminated. Round 2 votes counts: A=35 C=30 D=24 E=11 so E is eliminated. Round 3 votes counts: C=37 A=35 D=28 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:215 C:197 E:190 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 8 4 8 B -18 0 -2 -20 -2 C -8 2 0 -4 4 D -4 20 4 0 10 E -8 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 8 4 8 B -18 0 -2 -20 -2 C -8 2 0 -4 4 D -4 20 4 0 10 E -8 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 8 4 8 B -18 0 -2 -20 -2 C -8 2 0 -4 4 D -4 20 4 0 10 E -8 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5912: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (8) E D A B C (6) B C E D A (6) C B E A D (5) C B A D E (5) A D E C B (5) C B D A E (4) A D C E B (4) D B A E C (3) D A B E C (3) C B A E D (3) C A B E D (3) A E C D B (3) E C B A D (2) E B D A C (2) E B C D A (2) E A D B C (2) D B E A C (2) C E A B D (2) C A E B D (2) B D E A C (2) B D C A E (2) A E D C B (2) E D B A C (1) E B C A D (1) E A C D B (1) D E B A C (1) C E B A D (1) C A E D B (1) B E D C A (1) B E C A D (1) B D E C A (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 0 12 0 B 6 0 -12 6 2 C 0 12 0 2 -4 D -12 -6 -2 0 -16 E 0 -2 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839517 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 -6 0 12 0 B 6 0 -12 6 2 C 0 12 0 2 -4 D -12 -6 -2 0 -16 E 0 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839494 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 B=21 A=16 D=9 so D is eliminated. Round 2 votes counts: E=29 C=26 B=26 A=19 so A is eliminated. Round 3 votes counts: E=39 C=32 B=29 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:205 A:203 B:201 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 12 0 B 6 0 -12 6 2 C 0 12 0 2 -4 D -12 -6 -2 0 -16 E 0 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839494 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 12 0 B 6 0 -12 6 2 C 0 12 0 2 -4 D -12 -6 -2 0 -16 E 0 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839494 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 12 0 B 6 0 -12 6 2 C 0 12 0 2 -4 D -12 -6 -2 0 -16 E 0 -2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839494 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5913: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) A C D E B (9) B E C D A (8) A D C B E (5) C E B A D (4) C E A D B (4) B E C A D (4) A D C E B (4) A D B C E (4) E C B D A (3) E B C D A (3) B D A E C (3) B A C E D (3) A B C E D (3) E C B A D (2) D C E A B (2) D A B E C (2) C D E A B (2) B E D C A (2) B E A D C (2) A C D B E (2) E C D B A (1) E B C A D (1) D E C B A (1) D E C A B (1) D E B A C (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A B D (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E A C (1) B A E D C (1) B A E C D (1) B A D E C (1) B A D C E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 12 14 12 10 B -12 0 -12 -8 -6 C -14 12 0 12 22 D -12 8 -12 0 6 E -10 6 -22 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 12 10 B -12 0 -12 -8 -6 C -14 12 0 12 22 D -12 8 -12 0 6 E -10 6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 D=18 C=15 E=10 so E is eliminated. Round 2 votes counts: B=32 A=29 C=21 D=18 so D is eliminated. Round 3 votes counts: A=42 B=33 C=25 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:216 D:195 E:184 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 12 10 B -12 0 -12 -8 -6 C -14 12 0 12 22 D -12 8 -12 0 6 E -10 6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 12 10 B -12 0 -12 -8 -6 C -14 12 0 12 22 D -12 8 -12 0 6 E -10 6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 12 10 B -12 0 -12 -8 -6 C -14 12 0 12 22 D -12 8 -12 0 6 E -10 6 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5914: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) B E A D C (11) E B C A D (9) B A D E C (9) C D A E B (6) B A E D C (6) D C A E B (5) D A C B E (5) C D E A B (5) E C B D A (4) B E A C D (4) B A D C E (4) A B D E C (3) E C D A B (2) C E D B A (2) A D B E C (2) A D B C E (2) A B D C E (2) E B C D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D C A B E (1) C E B D A (1) B E C A D (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 2 12 -8 B 4 0 12 12 0 C -2 -12 0 -8 -10 D -12 -12 8 0 -8 E 8 0 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500651 C: 0.000000 D: 0.000000 E: 0.499349 Sum of squares = 0.500000846398 Cumulative probabilities = A: 0.000000 B: 0.500651 C: 0.500651 D: 0.500651 E: 1.000000 A B C D E A 0 -4 2 12 -8 B 4 0 12 12 0 C -2 -12 0 -8 -10 D -12 -12 8 0 -8 E 8 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=25 E=18 D=12 A=10 so A is eliminated. Round 2 votes counts: B=40 C=25 E=18 D=17 so D is eliminated. Round 3 votes counts: B=44 C=37 E=19 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:213 A:201 D:188 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 12 -8 B 4 0 12 12 0 C -2 -12 0 -8 -10 D -12 -12 8 0 -8 E 8 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 12 -8 B 4 0 12 12 0 C -2 -12 0 -8 -10 D -12 -12 8 0 -8 E 8 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 12 -8 B 4 0 12 12 0 C -2 -12 0 -8 -10 D -12 -12 8 0 -8 E 8 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5915: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) D B A C E (8) D C B E A (7) D A B E C (7) A B D E C (6) D B C A E (5) A E B C D (5) C E B D A (4) C E B A D (4) C E D A B (3) B A D C E (3) A B E C D (3) E A C B D (2) D C E B A (2) D C E A B (2) C E D B A (2) C E A B D (2) C D E B A (2) B D A E C (2) B D A C E (2) B A E C D (2) B A D E C (2) A D B E C (2) A B E D C (2) E C A D B (1) D C A B E (1) D A E C B (1) D A B C E (1) B D C E A (1) B C E A D (1) B C A E D (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 2 0 0 10 B -2 0 10 10 14 C 0 -10 0 -10 4 D 0 -10 10 0 8 E -10 -14 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.883225 B: 0.000000 C: 0.000000 D: 0.116775 E: 0.000000 Sum of squares = 0.793723279422 Cumulative probabilities = A: 0.883225 B: 0.883225 C: 0.883225 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 0 10 B -2 0 10 10 14 C 0 -10 0 -10 4 D 0 -10 10 0 8 E -10 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222369509 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=19 C=17 E=15 B=15 so E is eliminated. Round 2 votes counts: D=34 C=30 A=21 B=15 so B is eliminated. Round 3 votes counts: D=39 C=32 A=29 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 A:206 D:204 C:192 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 0 10 B -2 0 10 10 14 C 0 -10 0 -10 4 D 0 -10 10 0 8 E -10 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222369509 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 10 B -2 0 10 10 14 C 0 -10 0 -10 4 D 0 -10 10 0 8 E -10 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222369509 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 10 B -2 0 10 10 14 C 0 -10 0 -10 4 D 0 -10 10 0 8 E -10 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222369509 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5916: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) A D E C B (7) D E C B A (6) B C A E D (6) A B C E D (6) D E A C B (5) A B C D E (5) C B E D A (4) C B A E D (4) B C E D A (4) B C E A D (4) A D E B C (4) E D C A B (3) D E B C A (3) A C B E D (3) E D A C B (2) D E C A B (2) C E B D A (2) B A C E D (2) B A C D E (2) A E D C B (2) A D B E C (2) E D B C A (1) E C D B A (1) E C B D A (1) E A D C B (1) D E A B C (1) D B E C A (1) D A E B C (1) C E D A B (1) C A B E D (1) B C D E A (1) B C D A E (1) A E C D B (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -8 4 -2 B 2 0 -10 -6 -6 C 8 10 0 0 -4 D -4 6 0 0 -16 E 2 6 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999349 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -8 4 -2 B 2 0 -10 -6 -6 C 8 10 0 0 -4 D -4 6 0 0 -16 E 2 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=20 D=19 E=17 C=12 so C is eliminated. Round 2 votes counts: A=33 B=28 E=20 D=19 so D is eliminated. Round 3 votes counts: E=37 A=34 B=29 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:207 A:196 D:193 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 4 -2 B 2 0 -10 -6 -6 C 8 10 0 0 -4 D -4 6 0 0 -16 E 2 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 4 -2 B 2 0 -10 -6 -6 C 8 10 0 0 -4 D -4 6 0 0 -16 E 2 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 4 -2 B 2 0 -10 -6 -6 C 8 10 0 0 -4 D -4 6 0 0 -16 E 2 6 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5917: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (18) D C E A B (15) E A C B D (8) C E A D B (8) D B C A E (6) E A B C D (5) D B E A C (4) B D A E C (4) E A C D B (3) D B C E A (3) C D E A B (3) E C A D B (2) D C B A E (2) C A E B D (2) B D A C E (2) B A E D C (2) D E C A B (1) D C E B A (1) D C B E A (1) D B A C E (1) C D A E B (1) C B A E D (1) B E A C D (1) B D C A E (1) B C D A E (1) B C A E D (1) A E B C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 2 8 -10 B -2 0 2 -2 -2 C -2 -2 0 14 0 D -8 2 -14 0 -8 E 10 2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.366729 D: 0.000000 E: 0.633271 Sum of squares = 0.535522070545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.366729 D: 0.366729 E: 1.000000 A B C D E A 0 2 2 8 -10 B -2 0 2 -2 -2 C -2 -2 0 14 0 D -8 2 -14 0 -8 E 10 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.000000 E: 0.500519 Sum of squares = 0.500000539235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.499481 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=30 E=18 C=15 A=3 so A is eliminated. Round 2 votes counts: D=34 B=31 E=19 C=16 so C is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:210 C:205 A:201 B:198 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 8 -10 B -2 0 2 -2 -2 C -2 -2 0 14 0 D -8 2 -14 0 -8 E 10 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.000000 E: 0.500519 Sum of squares = 0.500000539235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.499481 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 8 -10 B -2 0 2 -2 -2 C -2 -2 0 14 0 D -8 2 -14 0 -8 E 10 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.000000 E: 0.500519 Sum of squares = 0.500000539235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.499481 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 8 -10 B -2 0 2 -2 -2 C -2 -2 0 14 0 D -8 2 -14 0 -8 E 10 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.000000 E: 0.500519 Sum of squares = 0.500000539235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499481 D: 0.499481 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5918: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) D E A C B (8) E D A B C (7) B A C E D (6) D E C A B (5) C A B D E (5) B C A E D (4) E D B A C (3) E B D C A (3) B A E C D (3) E D C B A (2) E D B C A (2) E B D A C (2) D C E A B (2) D A C E B (2) C D E B A (2) C D A B E (2) B E A C D (2) B C E A D (2) B C A D E (2) A E D B C (2) A D E B C (2) A C D B E (2) A B E D C (2) A B C E D (2) A B C D E (2) E D C A B (1) E D A C B (1) E B C D A (1) E A B D C (1) C E D B A (1) C D E A B (1) C D A E B (1) C B E D A (1) C B D E A (1) C A D B E (1) B E C A D (1) B E A D C (1) A D C B E (1) Total count = 100 A B C D E A 0 0 -2 4 0 B 0 0 0 4 2 C 2 0 0 6 2 D -4 -4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.505643 C: 0.494357 D: 0.000000 E: 0.000000 Sum of squares = 0.500063685181 Cumulative probabilities = A: 0.000000 B: 0.505643 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 4 0 B 0 0 0 4 2 C 2 0 0 6 2 D -4 -4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=21 D=17 A=13 so A is eliminated. Round 2 votes counts: C=28 B=27 E=25 D=20 so D is eliminated. Round 3 votes counts: E=40 C=33 B=27 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:205 B:203 A:201 E:198 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 4 0 B 0 0 0 4 2 C 2 0 0 6 2 D -4 -4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 0 B 0 0 0 4 2 C 2 0 0 6 2 D -4 -4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 0 B 0 0 0 4 2 C 2 0 0 6 2 D -4 -4 -6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999921 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5919: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) C D B E A (6) C D E B A (5) A B E D C (5) D C E A B (4) D C B A E (4) C E D B A (4) B C D A E (4) A E D B C (4) A E B D C (4) E A D B C (3) D E A C B (3) D B C A E (3) D A B C E (3) C B D E A (3) B A C E D (3) E A C B D (2) E A B C D (2) D E C A B (2) D C B E A (2) D A E B C (2) C B D A E (2) B D C A E (2) B A D C E (2) A E B C D (2) A B E C D (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E A B D C (1) D C E B A (1) D B A C E (1) C D B A E (1) C B E D A (1) B E C A D (1) B D A C E (1) B C E A D (1) B C A D E (1) B A C D E (1) A D E B C (1) Total count = 100 A B C D E A 0 2 0 -14 -4 B -2 0 -2 -20 -2 C 0 2 0 -14 10 D 14 20 14 0 8 E 4 2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -14 -4 B -2 0 -2 -20 -2 C 0 2 0 -14 10 D 14 20 14 0 8 E 4 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 E=19 A=18 B=16 so B is eliminated. Round 2 votes counts: D=28 C=28 A=24 E=20 so E is eliminated. Round 3 votes counts: A=39 C=31 D=30 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:228 C:199 E:194 A:192 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -14 -4 B -2 0 -2 -20 -2 C 0 2 0 -14 10 D 14 20 14 0 8 E 4 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -14 -4 B -2 0 -2 -20 -2 C 0 2 0 -14 10 D 14 20 14 0 8 E 4 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -14 -4 B -2 0 -2 -20 -2 C 0 2 0 -14 10 D 14 20 14 0 8 E 4 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5920: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (14) B E C D A (7) E D A C B (5) B C A E D (5) E B D A C (4) E B A C D (4) D E A C B (4) D C A B E (4) D A E C B (4) C A D B E (4) B E C A D (4) B E A C D (4) A C D B E (4) D C E A B (3) D C A E B (3) E D B C A (2) E D B A C (2) E B D C A (2) D A C E B (2) C D A B E (2) C A B D E (2) B C D A E (2) A D E C B (2) A D C E B (2) E A D C B (1) E A B C D (1) D E C A B (1) C B D A E (1) C B A D E (1) B E D C A (1) B C D E A (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -18 0 8 B 8 0 6 8 14 C 18 -6 0 16 4 D 0 -8 -16 0 16 E -8 -14 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 0 8 B 8 0 6 8 14 C 18 -6 0 16 4 D 0 -8 -16 0 16 E -8 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=21 D=21 C=10 A=10 so C is eliminated. Round 2 votes counts: B=40 D=23 E=21 A=16 so A is eliminated. Round 3 votes counts: B=43 D=36 E=21 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:216 D:196 A:191 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -18 0 8 B 8 0 6 8 14 C 18 -6 0 16 4 D 0 -8 -16 0 16 E -8 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 0 8 B 8 0 6 8 14 C 18 -6 0 16 4 D 0 -8 -16 0 16 E -8 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 0 8 B 8 0 6 8 14 C 18 -6 0 16 4 D 0 -8 -16 0 16 E -8 -14 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5921: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (16) A C B D E (15) B C A D E (10) E D B A C (7) E D B C A (6) C A B D E (6) E A D C B (4) E D A B C (3) E A C D B (3) B D C A E (3) A E C B D (3) A C B E D (3) D E B C A (2) D B E C A (2) B D C E A (2) B C A E D (2) A E C D B (2) A C E B D (2) E B D C A (1) E A B C D (1) D C A B E (1) D B C A E (1) C B A D E (1) B D E C A (1) B C E A D (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 20 20 8 2 B -20 0 -14 4 0 C -20 14 0 2 -2 D -8 -4 -2 0 -8 E -2 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 20 8 2 B -20 0 -14 4 0 C -20 14 0 2 -2 D -8 -4 -2 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 A=26 B=20 C=7 D=6 so D is eliminated. Round 2 votes counts: E=43 A=26 B=23 C=8 so C is eliminated. Round 3 votes counts: E=43 A=33 B=24 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:204 C:197 D:189 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 20 8 2 B -20 0 -14 4 0 C -20 14 0 2 -2 D -8 -4 -2 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 20 8 2 B -20 0 -14 4 0 C -20 14 0 2 -2 D -8 -4 -2 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 20 8 2 B -20 0 -14 4 0 C -20 14 0 2 -2 D -8 -4 -2 0 -8 E -2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5922: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E A B D (8) B D E A C (5) A E B C D (5) E A B C D (4) D C B E A (4) D C B A E (4) B E A D C (4) B E A C D (4) D B C E A (3) D A C E B (3) C D E A B (3) C D A E B (3) C A E B D (3) B A E D C (3) A E B D C (3) E B A C D (2) E A C B D (2) D C A E B (2) D B E A C (2) D A E C B (2) C E B A D (2) C D B E A (2) C A E D B (2) B A D E C (2) A E C B D (2) A B E D C (2) E C A B D (1) D B C A E (1) D A B E C (1) C E A D B (1) C D E B A (1) C B E A D (1) C A D E B (1) B D A E C (1) B C D E A (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 0 14 8 0 B 0 0 4 10 -4 C -14 -4 0 -4 -8 D -8 -10 4 0 0 E 0 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.370037 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.629963 Sum of squares = 0.533780805983 Cumulative probabilities = A: 0.370037 B: 0.370037 C: 0.370037 D: 0.370037 E: 1.000000 A B C D E A 0 0 14 8 0 B 0 0 4 10 -4 C -14 -4 0 -4 -8 D -8 -10 4 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=27 B=20 A=14 E=9 so E is eliminated. Round 2 votes counts: D=30 C=28 B=22 A=20 so A is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 E:206 B:205 D:193 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 8 0 B 0 0 4 10 -4 C -14 -4 0 -4 -8 D -8 -10 4 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 8 0 B 0 0 4 10 -4 C -14 -4 0 -4 -8 D -8 -10 4 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 8 0 B 0 0 4 10 -4 C -14 -4 0 -4 -8 D -8 -10 4 0 0 E 0 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5923: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (11) B A E C D (11) B A C E D (9) E D C A B (6) D C E B A (6) B C A D E (6) B A C D E (6) A E B D C (6) C D E B A (5) D E C A B (4) C D B A E (4) A B E C D (4) E D A C B (3) C D B E A (3) B A E D C (3) C D E A B (2) A B E D C (2) E C A D B (1) E A D B C (1) E A B D C (1) D E A B C (1) D C B A E (1) C B D A E (1) C B A D E (1) B C A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 -16 -4 6 12 B 16 0 4 4 6 C 4 -4 0 8 14 D -6 -4 -8 0 4 E -12 -6 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 6 12 B 16 0 4 4 6 C 4 -4 0 8 14 D -6 -4 -8 0 4 E -12 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=23 C=16 E=12 A=12 so E is eliminated. Round 2 votes counts: B=37 D=32 C=17 A=14 so A is eliminated. Round 3 votes counts: B=50 D=33 C=17 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:211 A:199 D:193 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 6 12 B 16 0 4 4 6 C 4 -4 0 8 14 D -6 -4 -8 0 4 E -12 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 6 12 B 16 0 4 4 6 C 4 -4 0 8 14 D -6 -4 -8 0 4 E -12 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 6 12 B 16 0 4 4 6 C 4 -4 0 8 14 D -6 -4 -8 0 4 E -12 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5924: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) B D E C A (6) E B C A D (5) E A C B D (5) A C E D B (5) E C A B D (4) A C E B D (4) D B E A C (3) D B C E A (3) D B C A E (3) B E D C A (3) A C D E B (3) E B D A C (2) D B E C A (2) D B A E C (2) D A E B C (2) C E A B D (2) C B E A D (2) C B D A E (2) C A D B E (2) C A B E D (2) B E C D A (2) B C E D A (2) A E C D B (2) A E C B D (2) A D E C B (2) A D C E B (2) E D A B C (1) E C B A D (1) E A C D B (1) E A B D C (1) E A B C D (1) D C B A E (1) D A C E B (1) D A C B E (1) D A B C E (1) C E B A D (1) C D B A E (1) C B A E D (1) C A E D B (1) B E C A D (1) B D E A C (1) B D C E A (1) B C E A D (1) B C D E A (1) Total count = 100 A B C D E A 0 6 -16 18 -4 B -6 0 -12 22 -12 C 16 12 0 24 2 D -18 -22 -24 0 -20 E 4 12 -2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -16 18 -4 B -6 0 -12 22 -12 C 16 12 0 24 2 D -18 -22 -24 0 -20 E 4 12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 E=21 A=20 D=19 B=18 so B is eliminated. Round 2 votes counts: E=27 D=27 C=26 A=20 so A is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: C=62 D=38 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:217 A:202 B:196 D:158 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -16 18 -4 B -6 0 -12 22 -12 C 16 12 0 24 2 D -18 -22 -24 0 -20 E 4 12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -16 18 -4 B -6 0 -12 22 -12 C 16 12 0 24 2 D -18 -22 -24 0 -20 E 4 12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -16 18 -4 B -6 0 -12 22 -12 C 16 12 0 24 2 D -18 -22 -24 0 -20 E 4 12 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5925: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) E B C A D (8) D A C B E (8) D E B A C (6) D E A B C (5) B E C A D (5) E B D C A (4) D B E C A (4) B E C D A (4) A D C E B (4) A D C B E (4) A C D E B (4) E D B A C (3) E A C B D (3) D A E C B (3) C A E B D (3) A C D B E (3) E B D A C (2) D A C E B (2) C B A E D (2) B E D C A (2) B D E C A (2) B C D A E (2) E D B C A (1) D E A C B (1) D A E B C (1) C E B A D (1) C B E A D (1) C B A D E (1) C A B D E (1) B C E A D (1) Total count = 100 A B C D E A 0 2 -2 0 -6 B -2 0 0 2 -2 C 2 0 0 -4 -8 D 0 -2 4 0 2 E 6 2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333217 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 2 -2 0 -6 B -2 0 0 2 -2 C 2 0 0 -4 -8 D 0 -2 4 0 2 E 6 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=21 C=18 B=16 A=15 so A is eliminated. Round 2 votes counts: D=38 C=25 E=21 B=16 so B is eliminated. Round 3 votes counts: D=40 E=32 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:207 D:202 B:199 A:197 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -2 0 -6 B -2 0 0 2 -2 C 2 0 0 -4 -8 D 0 -2 4 0 2 E 6 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -6 B -2 0 0 2 -2 C 2 0 0 -4 -8 D 0 -2 4 0 2 E 6 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -6 B -2 0 0 2 -2 C 2 0 0 -4 -8 D 0 -2 4 0 2 E 6 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5926: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (13) C D B E A (11) D C B E A (9) B E A D C (9) A E B D C (9) C D A E B (7) E B A C D (4) E B A D C (3) D C A B E (3) C D B A E (3) E A B C D (2) D C B A E (2) D C A E B (2) D B C E A (2) D A C E B (2) C D A B E (2) C A D E B (2) B E D C A (2) B E C D A (2) B C D E A (2) E A B D C (1) C B D E A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E D A (1) A E C D B (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -8 -6 -8 -4 B 8 0 8 6 4 C 6 -8 0 4 0 D 8 -6 -4 0 4 E 4 -4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -8 -4 B 8 0 8 6 4 C 6 -8 0 4 0 D 8 -6 -4 0 4 E 4 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=25 D=20 B=19 E=10 so E is eliminated. Round 2 votes counts: A=28 C=26 B=26 D=20 so D is eliminated. Round 3 votes counts: C=42 A=30 B=28 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:213 C:201 D:201 E:198 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -8 -4 B 8 0 8 6 4 C 6 -8 0 4 0 D 8 -6 -4 0 4 E 4 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -8 -4 B 8 0 8 6 4 C 6 -8 0 4 0 D 8 -6 -4 0 4 E 4 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -8 -4 B 8 0 8 6 4 C 6 -8 0 4 0 D 8 -6 -4 0 4 E 4 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5927: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) E D C B A (7) D E C A B (7) A B E D C (7) B A E D C (5) B A E C D (5) A B C D E (5) D C E A B (4) C A B D E (4) C D A E B (3) B E A D C (3) E D C A B (2) E B C D A (2) E A B D C (2) D A E C B (2) C B E D A (2) B E A C D (2) B C A E D (2) B A C E D (2) A E D B C (2) A C B D E (2) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) E B A D C (1) E A D B C (1) D E C B A (1) D E A C B (1) D C E B A (1) D A C E B (1) C E B D A (1) C D E A B (1) C D B A E (1) C A D B E (1) A E B D C (1) A D E C B (1) A D C B E (1) A D B E C (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 2 -2 B -6 0 -6 0 -8 C -2 6 0 -12 -14 D -2 0 12 0 -4 E 2 8 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 2 2 -2 B -6 0 -6 0 -8 C -2 6 0 -12 -14 D -2 0 12 0 -4 E 2 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 C=22 E=19 B=19 D=17 so D is eliminated. Round 2 votes counts: E=28 C=27 A=26 B=19 so B is eliminated. Round 3 votes counts: A=38 E=33 C=29 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:204 D:203 B:190 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 2 -2 B -6 0 -6 0 -8 C -2 6 0 -12 -14 D -2 0 12 0 -4 E 2 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 -2 B -6 0 -6 0 -8 C -2 6 0 -12 -14 D -2 0 12 0 -4 E 2 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 -2 B -6 0 -6 0 -8 C -2 6 0 -12 -14 D -2 0 12 0 -4 E 2 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5928: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) D B E A C (5) E D C B A (4) E C A D B (4) C D B A E (4) C A D B E (4) B A D E C (4) E D B C A (3) E C D B A (3) E A C B D (3) D C B A E (3) D B E C A (3) D B C E A (3) C E D A B (3) C E A D B (3) C D E A B (3) B D A E C (3) A B C E D (3) E D B A C (2) E B D A C (2) D C E B A (2) D B C A E (2) C D B E A (2) C A B E D (2) E C D A B (1) E C A B D (1) E B A D C (1) E A B D C (1) D E B C A (1) D E B A C (1) D C B E A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E B A (1) C D A E B (1) C A E D B (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E A C (1) B A E D C (1) B A D C E (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -30 -14 -14 B 14 0 -16 -24 -2 C 30 16 0 4 4 D 14 24 -4 0 -4 E 14 2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -30 -14 -14 B 14 0 -16 -24 -2 C 30 16 0 4 4 D 14 24 -4 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=25 D=23 B=12 A=6 so A is eliminated. Round 2 votes counts: C=36 E=25 D=23 B=16 so B is eliminated. Round 3 votes counts: C=39 D=32 E=29 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:215 E:208 B:186 A:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -30 -14 -14 B 14 0 -16 -24 -2 C 30 16 0 4 4 D 14 24 -4 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -30 -14 -14 B 14 0 -16 -24 -2 C 30 16 0 4 4 D 14 24 -4 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -30 -14 -14 B 14 0 -16 -24 -2 C 30 16 0 4 4 D 14 24 -4 0 -4 E 14 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5929: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) D C B A E (6) B E A C D (6) B D C E A (6) D B C E A (5) B C A E D (5) E A D C B (4) B D E C A (4) A E C D B (4) E B A D C (3) E A B C D (3) D C A E B (3) C D B A E (3) C B D A E (3) A C E D B (3) E D B A C (2) E D A C B (2) D E A C B (2) C A B D E (2) B E D A C (2) B D C A E (2) E D A B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D E A B C (1) D C A B E (1) D B E C A (1) C D A B E (1) C B A E D (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D C A (1) B E A D C (1) B D E A C (1) B A E C D (1) A E D C B (1) A E C B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -26 -14 -14 4 B 26 0 12 10 28 C 14 -12 0 0 12 D 14 -10 0 0 8 E -4 -28 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -14 -14 4 B 26 0 12 10 28 C 14 -12 0 0 12 D 14 -10 0 0 8 E -4 -28 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=19 E=18 C=13 A=11 so A is eliminated. Round 2 votes counts: B=39 E=24 D=19 C=18 so C is eliminated. Round 3 votes counts: B=46 E=28 D=26 so D is eliminated. Round 4 votes counts: B=64 E=36 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:238 C:207 D:206 A:175 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -14 -14 4 B 26 0 12 10 28 C 14 -12 0 0 12 D 14 -10 0 0 8 E -4 -28 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -14 -14 4 B 26 0 12 10 28 C 14 -12 0 0 12 D 14 -10 0 0 8 E -4 -28 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -14 -14 4 B 26 0 12 10 28 C 14 -12 0 0 12 D 14 -10 0 0 8 E -4 -28 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5930: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) E A C D B (7) E A C B D (7) D B C A E (7) B D E C A (5) E A B C D (4) E B A C D (3) D C B A E (3) B E D C A (3) B E D A C (3) B D C A E (3) B A C D E (3) A E C B D (3) A C E D B (3) A C B D E (3) E B D C A (2) E B A D C (2) D B E C A (2) D B C E A (2) C D A B E (2) C A D E B (2) C A B D E (2) B E A C D (2) A C D B E (2) E D C B A (1) E D C A B (1) E D B C A (1) E D B A C (1) E A D C B (1) D C A B E (1) C D A E B (1) C B D A E (1) C A E D B (1) C A D B E (1) B E A D C (1) B A E C D (1) B A C E D (1) A E C D B (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 18 6 -12 B 10 0 8 18 -2 C -18 -8 0 6 -18 D -6 -18 -6 0 -18 E 12 2 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 18 6 -12 B 10 0 8 18 -2 C -18 -8 0 6 -18 D -6 -18 -6 0 -18 E 12 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=22 D=15 A=15 C=10 so C is eliminated. Round 2 votes counts: E=38 B=23 A=21 D=18 so D is eliminated. Round 3 votes counts: E=38 B=37 A=25 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:217 A:201 C:181 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 18 6 -12 B 10 0 8 18 -2 C -18 -8 0 6 -18 D -6 -18 -6 0 -18 E 12 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 18 6 -12 B 10 0 8 18 -2 C -18 -8 0 6 -18 D -6 -18 -6 0 -18 E 12 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 18 6 -12 B 10 0 8 18 -2 C -18 -8 0 6 -18 D -6 -18 -6 0 -18 E 12 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999977655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5931: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) B D E C A (7) C D A B E (6) E B C A D (5) E B D A C (4) D B A C E (4) D A C B E (4) B D C A E (4) A C E D B (4) A C D E B (4) D B C A E (3) C A D E B (3) E A D B C (2) E A C D B (2) D C A B E (2) D B A E C (2) C E B A D (2) C E A D B (2) C E A B D (2) C B D A E (2) B E D A C (2) B D E A C (2) A D C E B (2) A D C B E (2) E C A D B (1) E C A B D (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D C B (1) E A C B D (1) E A B D C (1) D C B A E (1) D B E A C (1) D A B C E (1) C D B A E (1) C B E A D (1) C A E D B (1) B D C E A (1) B D A E C (1) B D A C E (1) B C E D A (1) Total count = 100 A B C D E A 0 -16 -14 -22 -4 B 16 0 10 2 18 C 14 -10 0 -18 8 D 22 -2 18 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999332 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -22 -4 B 16 0 10 2 18 C 14 -10 0 -18 8 D 22 -2 18 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=21 C=20 D=18 A=12 so A is eliminated. Round 2 votes counts: B=29 C=28 D=22 E=21 so E is eliminated. Round 3 votes counts: B=42 C=33 D=25 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:223 C:197 E:185 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -14 -22 -4 B 16 0 10 2 18 C 14 -10 0 -18 8 D 22 -2 18 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -22 -4 B 16 0 10 2 18 C 14 -10 0 -18 8 D 22 -2 18 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -22 -4 B 16 0 10 2 18 C 14 -10 0 -18 8 D 22 -2 18 0 8 E 4 -18 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5932: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E B A D C (8) A B D C E (8) C D A B E (7) E B A C D (6) B E A C D (6) E B C D A (5) E D C A B (4) E A D C B (4) D C A B E (4) A D C B E (4) D C E A B (3) C D E B A (3) B A E C D (3) B A C D E (3) A B E D C (3) E C D B A (2) E A B D C (2) D C A E B (2) C D E A B (2) B E A D C (2) E D A C B (1) E C B D A (1) E B C A D (1) D A C E B (1) C E D B A (1) C B E D A (1) B E C A D (1) A E B D C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 8 8 -6 B 4 0 0 6 6 C -8 0 0 4 -2 D -8 -6 -4 0 -4 E 6 -6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.802690 C: 0.197310 D: 0.000000 E: 0.000000 Sum of squares = 0.683242856908 Cumulative probabilities = A: 0.000000 B: 0.802690 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 8 -6 B 4 0 0 6 6 C -8 0 0 4 -2 D -8 -6 -4 0 -4 E 6 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559735 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=23 A=18 B=15 D=10 so D is eliminated. Round 2 votes counts: E=34 C=32 A=19 B=15 so B is eliminated. Round 3 votes counts: E=43 C=32 A=25 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:208 A:203 E:203 C:197 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 8 -6 B 4 0 0 6 6 C -8 0 0 4 -2 D -8 -6 -4 0 -4 E 6 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559735 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 8 -6 B 4 0 0 6 6 C -8 0 0 4 -2 D -8 -6 -4 0 -4 E 6 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559735 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 8 -6 B 4 0 0 6 6 C -8 0 0 4 -2 D -8 -6 -4 0 -4 E 6 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555559735 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5933: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A B E D C (10) A B D E C (7) C E D A B (6) B A D E C (6) C E D B A (5) E C D B A (4) D C E B A (4) D C B E A (4) E D C B A (3) D B E C A (3) C E A D B (3) B A D C E (3) A E B C D (3) E C D A B (2) B D E A C (2) B D A E C (2) B D A C E (2) A E C B D (2) A E B D C (2) A C E B D (2) A B C D E (2) E D B C A (1) E A C B D (1) E A B D C (1) E A B C D (1) D B C E A (1) C D E A B (1) C D B E A (1) C A E D B (1) C A D B E (1) A C E D B (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 -2 -6 B 2 0 -6 -2 -6 C 0 6 0 -4 0 D 2 2 4 0 2 E 6 6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -2 -6 B 2 0 -6 -2 -6 C 0 6 0 -4 0 D 2 2 4 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 B=15 E=13 D=12 so D is eliminated. Round 2 votes counts: C=36 A=32 B=19 E=13 so E is eliminated. Round 3 votes counts: C=45 A=35 B=20 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:205 E:205 C:201 A:195 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -2 -6 B 2 0 -6 -2 -6 C 0 6 0 -4 0 D 2 2 4 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -2 -6 B 2 0 -6 -2 -6 C 0 6 0 -4 0 D 2 2 4 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -2 -6 B 2 0 -6 -2 -6 C 0 6 0 -4 0 D 2 2 4 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998359 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5934: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) A E C D B (8) E A C D B (7) B D E A C (7) B D C E A (7) D B E C A (6) D B E A C (6) C A E D B (4) B C D A E (4) C A E B D (3) B C A D E (3) A C E D B (3) E D A B C (2) D C E B A (2) D B C E A (2) C B A D E (2) C A B E D (2) B D E C A (2) B A E C D (2) A E C B D (2) A C E B D (2) E D C A B (1) E C A D B (1) E A D C B (1) D E B C A (1) D E B A C (1) D C B E A (1) C E A D B (1) C D E A B (1) B D A E C (1) B D A C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -24 -12 -16 2 B 24 0 18 4 20 C 12 -18 0 -8 4 D 16 -4 8 0 20 E -2 -20 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999178 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -12 -16 2 B 24 0 18 4 20 C 12 -18 0 -8 4 D 16 -4 8 0 20 E -2 -20 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=19 A=15 C=13 E=12 so E is eliminated. Round 2 votes counts: B=41 A=23 D=22 C=14 so C is eliminated. Round 3 votes counts: B=43 A=34 D=23 so D is eliminated. Round 4 votes counts: B=62 A=38 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:233 D:220 C:195 E:177 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -12 -16 2 B 24 0 18 4 20 C 12 -18 0 -8 4 D 16 -4 8 0 20 E -2 -20 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -12 -16 2 B 24 0 18 4 20 C 12 -18 0 -8 4 D 16 -4 8 0 20 E -2 -20 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -12 -16 2 B 24 0 18 4 20 C 12 -18 0 -8 4 D 16 -4 8 0 20 E -2 -20 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999358 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5935: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (7) B C E D A (6) E D C A B (5) B A C D E (5) A B C D E (5) D E A C B (4) D C E A B (4) C B E D A (4) B A C E D (4) E D C B A (3) E B C D A (3) D E C A B (3) C E D B A (3) B E C D A (3) B C E A D (3) A D E C B (3) A D C B E (3) E C D B A (2) E C B D A (2) D A E C B (2) C E B D A (2) B C A E D (2) A D E B C (2) A D C E B (2) A D B C E (2) A B D E C (2) E D B A C (1) E D A C B (1) D C A E B (1) C D E A B (1) C D B E A (1) C D A E B (1) C B D E A (1) C B A D E (1) B E C A D (1) B C A D E (1) B A E D C (1) B A E C D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 0 -6 -6 -6 B 0 0 2 10 6 C 6 -2 0 4 18 D 6 -10 -4 0 2 E 6 -6 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.191908 B: 0.808092 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.689841886716 Cumulative probabilities = A: 0.191908 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -6 -6 B 0 0 2 10 6 C 6 -2 0 4 18 D 6 -10 -4 0 2 E 6 -6 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042144 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 E=17 D=14 C=14 so D is eliminated. Round 2 votes counts: A=30 B=27 E=24 C=19 so C is eliminated. Round 3 votes counts: E=34 B=34 A=32 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:209 D:197 A:191 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 -6 -6 B 0 0 2 10 6 C 6 -2 0 4 18 D 6 -10 -4 0 2 E 6 -6 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042144 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -6 -6 B 0 0 2 10 6 C 6 -2 0 4 18 D 6 -10 -4 0 2 E 6 -6 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042144 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -6 -6 B 0 0 2 10 6 C 6 -2 0 4 18 D 6 -10 -4 0 2 E 6 -6 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000042144 Cumulative probabilities = A: 0.250000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5936: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (10) A E C D B (8) C B A E D (7) A E D B C (6) C A E B D (5) D E B A C (4) C A B E D (4) B C D E A (4) A E D C B (4) A C E D B (4) E D A B C (3) D E A B C (3) C B D A E (3) B D C E A (3) B A D E C (3) D B E A C (2) B D E C A (2) E C D A B (1) E A D C B (1) D E A C B (1) C E A D B (1) C D B E A (1) C B D E A (1) C B A D E (1) B C D A E (1) B C A D E (1) B A E D C (1) A E C B D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 6 22 26 B -12 0 -12 -2 -10 C -6 12 0 12 -2 D -22 2 -12 0 -20 E -26 10 2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 22 26 B -12 0 -12 -2 -10 C -6 12 0 12 -2 D -22 2 -12 0 -20 E -26 10 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=25 A=25 D=10 E=5 so E is eliminated. Round 2 votes counts: C=36 A=26 B=25 D=13 so D is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:233 C:208 E:203 B:182 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 22 26 B -12 0 -12 -2 -10 C -6 12 0 12 -2 D -22 2 -12 0 -20 E -26 10 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 22 26 B -12 0 -12 -2 -10 C -6 12 0 12 -2 D -22 2 -12 0 -20 E -26 10 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 22 26 B -12 0 -12 -2 -10 C -6 12 0 12 -2 D -22 2 -12 0 -20 E -26 10 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999215 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5937: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) A C E D B (7) D B E A C (6) D B A E C (5) B D E C A (5) D A C E B (4) C E A B D (4) C A E B D (4) D E B C A (3) D A B C E (3) C E B A D (3) C E A D B (3) B C E A D (3) A C E B D (3) E C B D A (2) E C B A D (2) E C A B D (2) E B C D A (2) C A E D B (2) B E C A D (2) B A C E D (2) A D C B E (2) A D B C E (2) A C D E B (2) A C B E D (2) A B D C E (2) A B C E D (2) E C D A B (1) D B A C E (1) D A E C B (1) D A B E C (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E A C (1) B D A E C (1) B A D C E (1) B A C D E (1) A D C E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 4 10 4 B -2 0 6 -4 8 C -4 -6 0 6 14 D -10 4 -6 0 2 E -4 -8 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 10 4 B -2 0 6 -4 8 C -4 -6 0 6 14 D -10 4 -6 0 2 E -4 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=25 B=18 C=17 E=9 so E is eliminated. Round 2 votes counts: D=31 A=25 C=24 B=20 so B is eliminated. Round 3 votes counts: D=39 C=32 A=29 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:210 C:205 B:204 D:195 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 10 4 B -2 0 6 -4 8 C -4 -6 0 6 14 D -10 4 -6 0 2 E -4 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 10 4 B -2 0 6 -4 8 C -4 -6 0 6 14 D -10 4 -6 0 2 E -4 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 10 4 B -2 0 6 -4 8 C -4 -6 0 6 14 D -10 4 -6 0 2 E -4 -8 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999091 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5938: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E C A B D (10) D B A C E (10) C E D B A (8) A B C E D (6) D E C B A (4) D C E B A (4) A B D E C (4) E C A D B (3) C D E B A (3) B A D C E (3) A E B C D (3) A B E C D (3) E D C A B (2) E C D B A (2) D E A B C (2) D B C A E (2) B D A C E (2) A B E D C (2) A B C D E (2) E D C B A (1) E A C D B (1) D E C A B (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A E C (1) D A E B C (1) D A B E C (1) C B E A D (1) C B D A E (1) C A E B D (1) B A C E D (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -12 -16 -8 B -6 0 -8 -18 -12 C 12 8 0 12 -2 D 16 18 -12 0 -8 E 8 12 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -12 -16 -8 B -6 0 -8 -18 -12 C 12 8 0 12 -2 D 16 18 -12 0 -8 E 8 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=29 D=29 A=21 C=14 B=7 so B is eliminated. Round 2 votes counts: D=31 E=29 A=26 C=14 so C is eliminated. Round 3 votes counts: E=38 D=35 A=27 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:215 D:207 A:185 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -12 -16 -8 B -6 0 -8 -18 -12 C 12 8 0 12 -2 D 16 18 -12 0 -8 E 8 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -16 -8 B -6 0 -8 -18 -12 C 12 8 0 12 -2 D 16 18 -12 0 -8 E 8 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -16 -8 B -6 0 -8 -18 -12 C 12 8 0 12 -2 D 16 18 -12 0 -8 E 8 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5939: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) D C A E B (8) D A C E B (8) C D A E B (7) B E C A D (6) E B C A D (5) B E C D A (5) E B A C D (4) B E A C D (4) C D E A B (3) B D A E C (3) E C B A D (2) D C A B E (2) D A C B E (2) C D A B E (2) C B D E A (2) B C D E A (2) B A E D C (2) B A D E C (2) A D E B C (2) E A B D C (1) C E B D A (1) C E A D B (1) C D B E A (1) C D B A E (1) C B E D A (1) C A D E B (1) B E D C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B C E D A (1) A E D C B (1) A D E C B (1) A D C E B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -16 -8 -6 -8 B 16 0 8 14 8 C 8 -8 0 -2 -6 D 6 -14 2 0 6 E 8 -8 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -6 -8 B 16 0 8 14 8 C 8 -8 0 -2 -6 D 6 -14 2 0 6 E 8 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=20 C=20 E=12 A=7 so A is eliminated. Round 2 votes counts: B=41 D=26 C=20 E=13 so E is eliminated. Round 3 votes counts: B=51 D=27 C=22 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:200 E:200 C:196 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -8 -6 -8 B 16 0 8 14 8 C 8 -8 0 -2 -6 D 6 -14 2 0 6 E 8 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -6 -8 B 16 0 8 14 8 C 8 -8 0 -2 -6 D 6 -14 2 0 6 E 8 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -6 -8 B 16 0 8 14 8 C 8 -8 0 -2 -6 D 6 -14 2 0 6 E 8 -8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5940: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (8) B E C D A (7) A E C D B (5) E A C D B (4) B D C E A (4) A E B C D (4) A B E D C (4) E C B D A (3) E B C D A (3) E B C A D (3) C D E A B (3) C D A E B (3) B E D C A (3) B A D C E (3) A D C B E (3) E C D B A (2) E C D A B (2) E B A C D (2) E A C B D (2) D C B E A (2) D C A E B (2) C E D A B (2) A E B D C (2) A D C E B (2) A D B C E (2) E C B A D (1) E C A D B (1) E C A B D (1) E A B C D (1) D C A B E (1) D B C A E (1) D A C B E (1) C E D B A (1) C E A D B (1) C D E B A (1) B E A D C (1) B D E C A (1) B D C A E (1) B C D E A (1) B A E D C (1) A E D C B (1) A D E C B (1) A C D E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 -4 -4 B -2 0 8 16 -8 C 0 -8 0 8 -12 D 4 -16 -8 0 -14 E 4 8 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 -4 -4 B -2 0 8 16 -8 C 0 -8 0 8 -12 D 4 -16 -8 0 -14 E 4 8 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 E=25 C=11 D=7 so D is eliminated. Round 2 votes counts: B=31 A=28 E=25 C=16 so C is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 B:207 A:197 C:194 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -4 -4 B -2 0 8 16 -8 C 0 -8 0 8 -12 D 4 -16 -8 0 -14 E 4 8 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 -4 B -2 0 8 16 -8 C 0 -8 0 8 -12 D 4 -16 -8 0 -14 E 4 8 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 -4 B -2 0 8 16 -8 C 0 -8 0 8 -12 D 4 -16 -8 0 -14 E 4 8 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5941: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) D C B A E (6) B A E C D (6) D C B E A (5) A B E C D (5) D C E B A (4) D C E A B (4) E C D A B (3) E C B D A (3) E C A D B (3) E B C A D (3) D B C A E (3) C D B E A (3) A E B C D (3) A D B C E (3) A B E D C (3) D C A B E (2) C B D E A (2) B E C A D (2) B C D E A (2) B A E D C (2) A E D C B (2) E D C A B (1) E C B A D (1) E B C D A (1) E A C B D (1) E A B C D (1) D B C E A (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D B A (1) C B E D A (1) B E C D A (1) B E A C D (1) B D C E A (1) B D C A E (1) B C E D A (1) B A D E C (1) A E D B C (1) A E C D B (1) A E B D C (1) A D E C B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -26 -14 -8 B 20 0 -8 -10 12 C 26 8 0 6 4 D 14 10 -6 0 4 E 8 -12 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -26 -14 -8 B 20 0 -8 -10 12 C 26 8 0 6 4 D 14 10 -6 0 4 E 8 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=22 B=18 E=17 C=15 so C is eliminated. Round 2 votes counts: D=39 A=22 B=21 E=18 so E is eliminated. Round 3 votes counts: D=44 B=29 A=27 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:222 D:211 B:207 E:194 A:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -26 -14 -8 B 20 0 -8 -10 12 C 26 8 0 6 4 D 14 10 -6 0 4 E 8 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -26 -14 -8 B 20 0 -8 -10 12 C 26 8 0 6 4 D 14 10 -6 0 4 E 8 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -26 -14 -8 B 20 0 -8 -10 12 C 26 8 0 6 4 D 14 10 -6 0 4 E 8 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986025 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5942: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (14) A E C D B (10) B D E C A (8) A C D B E (8) B D C E A (7) E A C B D (6) D B C A E (6) C D B A E (5) C A D B E (5) B D C A E (5) A C E D B (5) E B D A C (3) E A C D B (2) E A B D C (2) C B D E A (2) C B D A E (2) A D C B E (2) E C B A D (1) E A D B C (1) E A B C D (1) D C B A E (1) D A C B E (1) C D A B E (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -14 -10 6 B 8 0 -4 4 6 C 14 4 0 -2 2 D 10 -4 2 0 8 E -6 -6 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -10 6 B 8 0 -4 4 6 C 14 4 0 -2 2 D 10 -4 2 0 8 E -6 -6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=27 B=20 C=15 D=8 so D is eliminated. Round 2 votes counts: E=30 A=28 B=26 C=16 so C is eliminated. Round 3 votes counts: B=36 A=34 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:209 D:208 B:207 E:189 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -10 6 B 8 0 -4 4 6 C 14 4 0 -2 2 D 10 -4 2 0 8 E -6 -6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -10 6 B 8 0 -4 4 6 C 14 4 0 -2 2 D 10 -4 2 0 8 E -6 -6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -10 6 B 8 0 -4 4 6 C 14 4 0 -2 2 D 10 -4 2 0 8 E -6 -6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5943: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E A D B (8) B D A E C (8) C E A B D (7) B D C A E (6) B D A C E (6) A D B E C (6) E C A D B (5) C E B D A (5) A D E B C (5) E A C D B (4) E A D B C (3) C E B A D (3) D A B E C (2) C D B A E (2) C B E D A (2) C A E D B (2) B D C E A (2) A E D B C (2) E C B D A (1) E C A B D (1) E B A D C (1) E A C B D (1) D B C A E (1) C B D E A (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E A C (1) B C D E A (1) A E D C B (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 2 4 4 B 0 0 10 -2 -4 C -2 -10 0 -8 -4 D -4 2 8 0 0 E -4 4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.707451 B: 0.292549 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.586072193771 Cumulative probabilities = A: 0.707451 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 4 B 0 0 10 -2 -4 C -2 -10 0 -8 -4 D -4 2 8 0 0 E -4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500146 B: 0.499854 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000042739 Cumulative probabilities = A: 0.500146 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 E=16 A=16 D=11 so D is eliminated. Round 2 votes counts: B=35 C=31 A=18 E=16 so E is eliminated. Round 3 votes counts: C=38 B=36 A=26 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:205 D:203 B:202 E:202 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 4 4 B 0 0 10 -2 -4 C -2 -10 0 -8 -4 D -4 2 8 0 0 E -4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500146 B: 0.499854 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000042739 Cumulative probabilities = A: 0.500146 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 4 B 0 0 10 -2 -4 C -2 -10 0 -8 -4 D -4 2 8 0 0 E -4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500146 B: 0.499854 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000042739 Cumulative probabilities = A: 0.500146 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 4 B 0 0 10 -2 -4 C -2 -10 0 -8 -4 D -4 2 8 0 0 E -4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500146 B: 0.499854 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000042739 Cumulative probabilities = A: 0.500146 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5944: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) E A C D B (8) E C D A B (6) D C B E A (6) D B C E A (6) A E B C D (6) C D E B A (5) A E B D C (5) A E C B D (4) B D A C E (3) A E C D B (3) A B E D C (3) A B D E C (3) E C D B A (2) C E D B A (2) C D B E A (2) B C D A E (2) B A D C E (2) B A C D E (2) A B E C D (2) A B D C E (2) A B C D E (2) E D C A B (1) E C A D B (1) E A D C B (1) D E C B A (1) C B D A E (1) C A B E D (1) B D C E A (1) B D A E C (1) B C A D E (1) B A D E C (1) A E D B C (1) A C E B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 2 16 B -4 0 8 10 6 C -4 -8 0 6 2 D -2 -10 -6 0 4 E -16 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 2 16 B -4 0 8 10 6 C -4 -8 0 6 2 D -2 -10 -6 0 4 E -16 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=23 E=19 D=13 C=11 so C is eliminated. Round 2 votes counts: A=35 B=24 E=21 D=20 so D is eliminated. Round 3 votes counts: B=38 A=35 E=27 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:210 C:198 D:193 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 16 B -4 0 8 10 6 C -4 -8 0 6 2 D -2 -10 -6 0 4 E -16 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 16 B -4 0 8 10 6 C -4 -8 0 6 2 D -2 -10 -6 0 4 E -16 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 16 B -4 0 8 10 6 C -4 -8 0 6 2 D -2 -10 -6 0 4 E -16 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5945: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) C E B D A (7) B A E D C (7) E B C A D (6) C D A E B (6) C D A B E (5) B E A D C (5) D A C B E (4) C D E B A (4) C D E A B (4) B E A C D (4) A D B E C (4) A B D E C (4) E B A D C (3) D A B E C (3) C E D B A (3) E C B D A (2) E C B A D (2) E B C D A (2) E B A C D (2) C E B A D (2) C A D B E (2) E D B A C (1) E B D C A (1) D C E A B (1) D A B C E (1) C B E A D (1) C B A E D (1) C A B D E (1) B E C A D (1) B A E C D (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -18 -4 -2 B 10 0 -8 4 -8 C 18 8 0 12 4 D 4 -4 -12 0 -2 E 2 8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -4 -2 B 10 0 -8 4 -8 C 18 8 0 12 4 D 4 -4 -12 0 -2 E 2 8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=19 B=18 D=17 A=10 so A is eliminated. Round 2 votes counts: C=36 D=23 B=22 E=19 so E is eliminated. Round 3 votes counts: C=40 B=36 D=24 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 E:204 B:199 D:193 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -18 -4 -2 B 10 0 -8 4 -8 C 18 8 0 12 4 D 4 -4 -12 0 -2 E 2 8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -4 -2 B 10 0 -8 4 -8 C 18 8 0 12 4 D 4 -4 -12 0 -2 E 2 8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -4 -2 B 10 0 -8 4 -8 C 18 8 0 12 4 D 4 -4 -12 0 -2 E 2 8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5946: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) A C B D E (10) A C E B D (9) E D B A C (8) C A D B E (7) E B D A C (6) C D B A E (4) A C E D B (4) E A B D C (3) D E B C A (3) C A B D E (3) E D C B A (2) E A C D B (2) D B C E A (2) B E D A C (2) B D E A C (2) A E C D B (2) A E C B D (2) A C B E D (2) E A C B D (1) D C E B A (1) D C B E A (1) D B E C A (1) C B D A E (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E C A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C A D E (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 14 0 2 B 4 0 -6 -4 -18 C -14 6 0 0 -2 D 0 4 0 0 -18 E -2 18 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.597222222239 Cumulative probabilities = A: 0.750000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 -4 14 0 2 B 4 0 -6 -4 -18 C -14 6 0 0 -2 D 0 4 0 0 -18 E -2 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.597222222146 Cumulative probabilities = A: 0.750000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=30 C=17 B=10 D=8 so D is eliminated. Round 2 votes counts: E=38 A=30 C=19 B=13 so B is eliminated. Round 3 votes counts: E=45 A=32 C=23 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:218 A:206 C:195 D:193 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 14 0 2 B 4 0 -6 -4 -18 C -14 6 0 0 -2 D 0 4 0 0 -18 E -2 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.597222222146 Cumulative probabilities = A: 0.750000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 0 2 B 4 0 -6 -4 -18 C -14 6 0 0 -2 D 0 4 0 0 -18 E -2 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.597222222146 Cumulative probabilities = A: 0.750000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 0 2 B 4 0 -6 -4 -18 C -14 6 0 0 -2 D 0 4 0 0 -18 E -2 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.083333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.597222222146 Cumulative probabilities = A: 0.750000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5947: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (15) E C D A B (9) C E A D B (7) B A C D E (7) C A D E B (5) D A C E B (4) A C D E B (4) E D C A B (3) E D B A C (3) B E C D A (3) B E C A D (3) A D C E B (3) A D B C E (3) E C D B A (2) E C A D B (2) E B D C A (2) C A D B E (2) C A B D E (2) B E A D C (2) B D E A C (2) B C E A D (2) B C A E D (2) E D B C A (1) E C B D A (1) E B C D A (1) D E B A C (1) D C A E B (1) D A B C E (1) C E B A D (1) C B E A D (1) B D A E C (1) B D A C E (1) B C A D E (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -2 28 8 B 6 0 4 -4 0 C 2 -4 0 10 26 D -28 4 -10 0 10 E -8 0 -26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.105263 B: 0.736842 C: 0.000000 D: 0.157895 E: 0.000000 Sum of squares = 0.57894736841 Cumulative probabilities = A: 0.105263 B: 0.842105 C: 0.842105 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 28 8 B 6 0 4 -4 0 C 2 -4 0 10 26 D -28 4 -10 0 10 E -8 0 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.105263 B: 0.736842 C: 0.000000 D: 0.157895 E: 0.000000 Sum of squares = 0.578947368362 Cumulative probabilities = A: 0.105263 B: 0.842105 C: 0.842105 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=24 C=18 A=11 D=7 so D is eliminated. Round 2 votes counts: B=40 E=25 C=19 A=16 so A is eliminated. Round 3 votes counts: B=44 C=31 E=25 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:217 A:214 B:203 D:188 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 28 8 B 6 0 4 -4 0 C 2 -4 0 10 26 D -28 4 -10 0 10 E -8 0 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.105263 B: 0.736842 C: 0.000000 D: 0.157895 E: 0.000000 Sum of squares = 0.578947368362 Cumulative probabilities = A: 0.105263 B: 0.842105 C: 0.842105 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 28 8 B 6 0 4 -4 0 C 2 -4 0 10 26 D -28 4 -10 0 10 E -8 0 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.105263 B: 0.736842 C: 0.000000 D: 0.157895 E: 0.000000 Sum of squares = 0.578947368362 Cumulative probabilities = A: 0.105263 B: 0.842105 C: 0.842105 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 28 8 B 6 0 4 -4 0 C 2 -4 0 10 26 D -28 4 -10 0 10 E -8 0 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.105263 B: 0.736842 C: 0.000000 D: 0.157895 E: 0.000000 Sum of squares = 0.578947368362 Cumulative probabilities = A: 0.105263 B: 0.842105 C: 0.842105 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5948: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) D E C B A (12) A B E C D (11) B A E C D (8) C E D B A (7) D A B E C (6) A B E D C (5) A B D E C (5) D C E B A (3) D C E A B (3) B A E D C (3) B A D E C (3) B A C E D (3) E D C B A (2) C E A B D (2) C D E B A (2) C D E A B (2) E C B A D (1) E B C D A (1) E B A D C (1) E B A C D (1) D A B C E (1) C E B A D (1) C A E B D (1) C A D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 24 22 24 B -4 0 26 22 22 C -24 -26 0 10 -18 D -22 -22 -10 0 -20 E -24 -22 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 24 22 24 B -4 0 26 22 22 C -24 -26 0 10 -18 D -22 -22 -10 0 -20 E -24 -22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=25 B=17 C=16 E=6 so E is eliminated. Round 2 votes counts: A=36 D=27 B=20 C=17 so C is eliminated. Round 3 votes counts: A=40 D=38 B=22 so B is eliminated. Round 4 votes counts: A=61 D=39 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:237 B:233 E:196 C:171 D:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 24 22 24 B -4 0 26 22 22 C -24 -26 0 10 -18 D -22 -22 -10 0 -20 E -24 -22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 24 22 24 B -4 0 26 22 22 C -24 -26 0 10 -18 D -22 -22 -10 0 -20 E -24 -22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 24 22 24 B -4 0 26 22 22 C -24 -26 0 10 -18 D -22 -22 -10 0 -20 E -24 -22 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997384 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5949: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (11) A E D C B (8) E A D B C (7) D C B E A (6) C B A D E (5) B C E D A (5) A E B C D (5) E D A B C (4) D E A C B (4) E D A C B (3) D B C E A (3) B C A D E (3) A C B E D (3) D E B C A (2) D A C E B (2) C B D A E (2) B E D C A (2) A E D B C (2) A D E C B (2) A D C E B (2) A B E C D (2) E D B C A (1) E D B A C (1) E B D C A (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B A E (1) D A E C B (1) C B D E A (1) C A D B E (1) C A B D E (1) B C E A D (1) B C D A E (1) A E C B D (1) A E B D C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 2 -4 -10 B -4 0 10 -2 0 C -2 -10 0 -6 0 D 4 2 6 0 0 E 10 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.766961 E: 0.233039 Sum of squares = 0.642536435459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.766961 E: 1.000000 A B C D E A 0 4 2 -4 -10 B -4 0 10 -2 0 C -2 -10 0 -6 0 D 4 2 6 0 0 E 10 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=23 E=20 D=19 C=10 so C is eliminated. Round 2 votes counts: B=31 A=30 E=20 D=19 so D is eliminated. Round 3 votes counts: B=41 A=33 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:206 E:205 B:202 A:196 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -4 -10 B -4 0 10 -2 0 C -2 -10 0 -6 0 D 4 2 6 0 0 E 10 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -4 -10 B -4 0 10 -2 0 C -2 -10 0 -6 0 D 4 2 6 0 0 E 10 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -4 -10 B -4 0 10 -2 0 C -2 -10 0 -6 0 D 4 2 6 0 0 E 10 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5950: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) A B D E C (8) C E B D A (7) C E D A B (6) B E C A D (5) B A E D C (5) B A D E C (5) A D E C B (5) A D B C E (5) C E D B A (4) B C E D A (4) B A E C D (4) E C D A B (3) E C B D A (3) E B C A D (3) D C E A B (3) E C B A D (2) D C A E B (2) D A B C E (2) A D B E C (2) E C A B D (1) D A E C B (1) B E C D A (1) B C E A D (1) B C A E D (1) B A D C E (1) A E D C B (1) A E C B D (1) A E B C D (1) A D E B C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 8 8 14 B -8 0 0 8 -8 C -8 0 0 -6 -4 D -8 -8 6 0 -6 E -14 8 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 14 B -8 0 0 8 -8 C -8 0 0 -6 -4 D -8 -8 6 0 -6 E -14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 D=18 C=17 E=12 so E is eliminated. Round 2 votes counts: B=30 C=26 A=26 D=18 so D is eliminated. Round 3 votes counts: A=39 C=31 B=30 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:202 B:196 D:192 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 14 B -8 0 0 8 -8 C -8 0 0 -6 -4 D -8 -8 6 0 -6 E -14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 14 B -8 0 0 8 -8 C -8 0 0 -6 -4 D -8 -8 6 0 -6 E -14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 14 B -8 0 0 8 -8 C -8 0 0 -6 -4 D -8 -8 6 0 -6 E -14 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5951: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D C B E A (10) A E B C D (10) D C B A E (7) D B C A E (5) B A E C D (5) E A D B C (4) E A C D B (3) D E C A B (3) D E A C B (3) D C E A B (3) B D C A E (3) A E B D C (3) A B E C D (3) E A B C D (2) D E A B C (2) C D B A E (2) C B A E D (2) B C D A E (2) E D A C B (1) E A D C B (1) E A B D C (1) D C E B A (1) D B A E C (1) C E D A B (1) C D E A B (1) C D B E A (1) C B E A D (1) C B D A E (1) C A B E D (1) B C A D E (1) B A E D C (1) B A C E D (1) A E C B D (1) Total count = 100 A B C D E A 0 12 10 6 -2 B -12 0 -12 2 -6 C -10 12 0 2 -14 D -6 -2 -2 0 -8 E 2 6 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 10 6 -2 B -12 0 -12 2 -6 C -10 12 0 2 -14 D -6 -2 -2 0 -8 E 2 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=25 A=17 B=13 C=10 so C is eliminated. Round 2 votes counts: D=39 E=26 A=18 B=17 so B is eliminated. Round 3 votes counts: D=45 A=28 E=27 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:213 C:195 D:191 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 10 6 -2 B -12 0 -12 2 -6 C -10 12 0 2 -14 D -6 -2 -2 0 -8 E 2 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 6 -2 B -12 0 -12 2 -6 C -10 12 0 2 -14 D -6 -2 -2 0 -8 E 2 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 6 -2 B -12 0 -12 2 -6 C -10 12 0 2 -14 D -6 -2 -2 0 -8 E 2 6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5952: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) E C A D B (8) C D B E A (7) E C A B D (6) E A B C D (6) D C B A E (6) D B C A E (5) C E D B A (5) C D B A E (5) C D E B A (4) B A D E C (4) A B E D C (4) A B D E C (4) E A C B D (3) D B A E C (2) C E D A B (2) C B D A E (2) A E B D C (2) E A D B C (1) E A C D B (1) D C E A B (1) D B A C E (1) D A B E C (1) D A B C E (1) C E B D A (1) C E B A D (1) C E A D B (1) C B D E A (1) B D C A E (1) B D A C E (1) B A D C E (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -12 8 -18 B -6 0 -8 -4 -6 C 12 8 0 6 -8 D -8 4 -6 0 -4 E 18 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -12 8 -18 B -6 0 -8 -4 -6 C 12 8 0 6 -8 D -8 4 -6 0 -4 E 18 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=29 D=17 A=11 B=7 so B is eliminated. Round 2 votes counts: E=36 C=29 D=19 A=16 so A is eliminated. Round 3 votes counts: E=42 D=29 C=29 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:209 D:193 A:192 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -12 8 -18 B -6 0 -8 -4 -6 C 12 8 0 6 -8 D -8 4 -6 0 -4 E 18 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 8 -18 B -6 0 -8 -4 -6 C 12 8 0 6 -8 D -8 4 -6 0 -4 E 18 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 8 -18 B -6 0 -8 -4 -6 C 12 8 0 6 -8 D -8 4 -6 0 -4 E 18 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5953: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) B D C A E (7) D B C A E (6) A E C D B (6) E D B C A (4) E D A C B (4) D B A C E (4) B C D A E (4) C B A E D (3) C A B E D (3) B C E A D (3) A C E B D (3) E D A B C (2) E C B A D (2) E C A B D (2) E B D C A (2) E A D C B (2) E A C B D (2) D E B A C (2) D E A B C (2) D B A E C (2) D A E C B (2) C B E A D (2) C A B D E (2) B D C E A (2) B C A D E (2) A D C E B (2) E D C A B (1) D E A C B (1) D B E C A (1) D B E A C (1) D B C E A (1) D A E B C (1) B E C D A (1) B D A C E (1) A E D C B (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 4 -2 2 B 0 0 -4 -14 -8 C -4 4 0 -2 -8 D 2 14 2 0 -12 E -2 8 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593750000003 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 A B C D E A 0 0 4 -2 2 B 0 0 -4 -14 -8 C -4 4 0 -2 -8 D 2 14 2 0 -12 E -2 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749999983 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=23 B=20 A=14 C=10 so C is eliminated. Round 2 votes counts: E=33 B=25 D=23 A=19 so A is eliminated. Round 3 votes counts: E=44 B=31 D=25 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:203 A:202 C:195 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 -2 2 B 0 0 -4 -14 -8 C -4 4 0 -2 -8 D 2 14 2 0 -12 E -2 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749999983 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 2 B 0 0 -4 -14 -8 C -4 4 0 -2 -8 D 2 14 2 0 -12 E -2 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749999983 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 2 B 0 0 -4 -14 -8 C -4 4 0 -2 -8 D 2 14 2 0 -12 E -2 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.125000 Sum of squares = 0.593749999983 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.875000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5954: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (14) B E D C A (9) A D C E B (6) E B D A C (5) D E B A C (5) B E C D A (5) A C D E B (5) D B E A C (4) C A B E D (4) E B A C D (3) D C A B E (3) B D E C A (3) E D B A C (2) E B D C A (2) E B C A D (2) D A E B C (2) D A C E B (2) C E A B D (2) C D A B E (2) C A E B D (2) C A D E B (2) A C D B E (2) E B C D A (1) E B A D C (1) D E A B C (1) D C B E A (1) D B E C A (1) D B C E A (1) D A C B E (1) C B E A D (1) C A B D E (1) B E D A C (1) B E C A D (1) B C E A D (1) A E B C D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 -16 -2 -4 B -2 0 2 -10 12 C 16 -2 0 -2 2 D 2 10 2 0 14 E 4 -12 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 -2 -4 B -2 0 2 -10 12 C 16 -2 0 -2 2 D 2 10 2 0 14 E 4 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=21 B=20 E=16 A=15 so A is eliminated. Round 2 votes counts: C=35 D=28 B=20 E=17 so E is eliminated. Round 3 votes counts: C=35 B=35 D=30 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 C:207 B:201 A:190 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -16 -2 -4 B -2 0 2 -10 12 C 16 -2 0 -2 2 D 2 10 2 0 14 E 4 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -2 -4 B -2 0 2 -10 12 C 16 -2 0 -2 2 D 2 10 2 0 14 E 4 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -2 -4 B -2 0 2 -10 12 C 16 -2 0 -2 2 D 2 10 2 0 14 E 4 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5955: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (6) D B A C E (6) A D B E C (6) E C B A D (5) A D E C B (5) E C A B D (4) E B C A D (4) D A B C E (4) A E D C B (4) A D C E B (4) A E D B C (3) A E C D B (3) A D C B E (3) E C B D A (2) E A C D B (2) E A C B D (2) E A B C D (2) D C B A E (2) D A C B E (2) D A B E C (2) C E B D A (2) C E A D B (2) C D B E A (2) C B E D A (2) B E C D A (2) B D C E A (2) B D C A E (2) A D B C E (2) E B C D A (1) E B A C D (1) E A B D C (1) C E B A D (1) C D B A E (1) C B D E A (1) B E D C A (1) B E A D C (1) B C E D A (1) B C D E A (1) A D E B C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 8 12 16 16 B -8 0 0 -22 0 C -12 0 0 -16 -6 D -16 22 16 0 4 E -16 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 16 16 B -8 0 0 -22 0 C -12 0 0 -16 -6 D -16 22 16 0 4 E -16 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=24 D=22 C=11 B=10 so B is eliminated. Round 2 votes counts: A=33 E=28 D=26 C=13 so C is eliminated. Round 3 votes counts: E=36 A=33 D=31 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:213 E:193 B:185 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 16 16 B -8 0 0 -22 0 C -12 0 0 -16 -6 D -16 22 16 0 4 E -16 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 16 16 B -8 0 0 -22 0 C -12 0 0 -16 -6 D -16 22 16 0 4 E -16 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 16 16 B -8 0 0 -22 0 C -12 0 0 -16 -6 D -16 22 16 0 4 E -16 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5956: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) E C A D B (7) D E C A B (7) B A E C D (6) B A C E D (6) B D E C A (5) B D A C E (5) A C E B D (5) E C D A B (4) D B E C A (4) D B A C E (3) B A C D E (3) A C D E B (3) E C B A D (2) C E A D B (2) B E D C A (2) B E C D A (2) B E C A D (2) B A D C E (2) E C D B A (1) E C A B D (1) E B C D A (1) D C A E B (1) D B C E A (1) D B C A E (1) D B A E C (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B A D E C (1) A E C B D (1) A D C E B (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 -4 -6 B 4 0 2 0 6 C 6 -2 0 4 -2 D 4 0 -4 0 4 E 6 -6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.892188 C: 0.000000 D: 0.107812 E: 0.000000 Sum of squares = 0.807623140556 Cumulative probabilities = A: 0.000000 B: 0.892188 C: 0.892188 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 -6 B 4 0 2 0 6 C 6 -2 0 4 -2 D 4 0 -4 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556005 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=29 E=16 A=13 C=4 so C is eliminated. Round 2 votes counts: B=38 D=29 E=19 A=14 so A is eliminated. Round 3 votes counts: B=40 D=34 E=26 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:206 C:203 D:202 E:199 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 -6 B 4 0 2 0 6 C 6 -2 0 4 -2 D 4 0 -4 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556005 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 -6 B 4 0 2 0 6 C 6 -2 0 4 -2 D 4 0 -4 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556005 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 -6 B 4 0 2 0 6 C 6 -2 0 4 -2 D 4 0 -4 0 4 E 6 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555556005 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5957: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (12) B E C D A (10) A D C E B (10) B E D C A (8) C A E D B (5) D E A C B (4) B E D A C (4) B C E A D (4) C A D E B (3) B D E A C (3) B C A E D (3) E D C A B (2) E D B C A (2) E D B A C (2) E D A C B (2) E C D A B (2) C E B D A (2) B A D C E (2) B A C D E (2) E B D C A (1) D E C A B (1) D E B A C (1) D A E C B (1) D A E B C (1) C E A D B (1) C D A E B (1) C B E A D (1) C B A E D (1) C A D B E (1) C A B D E (1) B C A D E (1) B A C E D (1) A D E C B (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 0 6 0 B -4 0 -6 -6 -8 C 0 6 0 8 10 D -6 6 -8 0 -2 E 0 8 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.452225 B: 0.000000 C: 0.547775 D: 0.000000 E: 0.000000 Sum of squares = 0.504564986858 Cumulative probabilities = A: 0.452225 B: 0.452225 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 6 0 B -4 0 -6 -6 -8 C 0 6 0 8 10 D -6 6 -8 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=27 C=16 E=11 D=8 so D is eliminated. Round 2 votes counts: B=38 A=29 E=17 C=16 so C is eliminated. Round 3 votes counts: B=40 A=40 E=20 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:205 E:200 D:195 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 6 0 B -4 0 -6 -6 -8 C 0 6 0 8 10 D -6 6 -8 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 0 B -4 0 -6 -6 -8 C 0 6 0 8 10 D -6 6 -8 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 0 B -4 0 -6 -6 -8 C 0 6 0 8 10 D -6 6 -8 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5958: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C E B D A (7) A D B E C (7) A B D C E (6) E C B D A (5) D B A E C (5) C E B A D (5) D A B E C (4) C E A B D (4) B D A E C (4) B A D C E (4) C E A D B (3) A D E C B (3) A D E B C (3) A D B C E (3) E C A D B (2) C E D A B (2) C B E D A (2) C B E A D (2) B E D C A (2) B D E A C (2) A E D C B (2) A B C D E (2) E C D A B (1) E B D C A (1) E B C D A (1) D E B A C (1) C E D B A (1) C A E D B (1) C A B E D (1) B D E C A (1) B C A D E (1) B A D E C (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 2 8 2 B 4 0 -4 4 0 C -2 4 0 -2 -4 D -8 -4 2 0 2 E -2 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.456658 C: 0.000000 D: 0.000000 E: 0.543342 Sum of squares = 0.503757040565 Cumulative probabilities = A: 0.000000 B: 0.456658 C: 0.456658 D: 0.456658 E: 1.000000 A B C D E A 0 -4 2 8 2 B 4 0 -4 4 0 C -2 4 0 -2 -4 D -8 -4 2 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499990 C: 0.000000 D: 0.000000 E: 0.500010 Sum of squares = 0.500000000184 Cumulative probabilities = A: 0.000000 B: 0.499990 C: 0.499990 D: 0.499990 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=28 E=17 B=15 D=10 so D is eliminated. Round 2 votes counts: A=34 C=28 B=20 E=18 so E is eliminated. Round 3 votes counts: C=43 A=34 B=23 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:204 B:202 E:200 C:198 D:196 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 8 2 B 4 0 -4 4 0 C -2 4 0 -2 -4 D -8 -4 2 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499990 C: 0.000000 D: 0.000000 E: 0.500010 Sum of squares = 0.500000000184 Cumulative probabilities = A: 0.000000 B: 0.499990 C: 0.499990 D: 0.499990 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 2 B 4 0 -4 4 0 C -2 4 0 -2 -4 D -8 -4 2 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499990 C: 0.000000 D: 0.000000 E: 0.500010 Sum of squares = 0.500000000184 Cumulative probabilities = A: 0.000000 B: 0.499990 C: 0.499990 D: 0.499990 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 2 B 4 0 -4 4 0 C -2 4 0 -2 -4 D -8 -4 2 0 2 E -2 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499990 C: 0.000000 D: 0.000000 E: 0.500010 Sum of squares = 0.500000000184 Cumulative probabilities = A: 0.000000 B: 0.499990 C: 0.499990 D: 0.499990 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5959: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) A B E C D (10) D A C B E (8) E B C A D (6) E B A C D (6) D A B C E (6) D C A B E (5) C B E A D (4) C B A E D (3) A D B C E (3) A C D B E (3) E C B D A (2) E C B A D (2) D E C B A (2) D A B E C (2) C E D B A (2) C E B D A (2) C E B A D (2) A E B D C (2) A D C B E (2) E B D C A (1) E B D A C (1) E B C D A (1) E A B C D (1) D E B C A (1) D E B A C (1) D E A B C (1) D C B E A (1) D C A E B (1) D A E B C (1) C D B E A (1) B E A C D (1) B C E A D (1) A E B C D (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 4 0 0 B 2 0 -4 -2 6 C -4 4 0 0 14 D 0 2 0 0 0 E 0 -6 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.291546 B: 0.000000 C: 0.000000 D: 0.661340 E: 0.047115 Sum of squares = 0.524588747176 Cumulative probabilities = A: 0.291546 B: 0.291546 C: 0.291546 D: 0.952885 E: 1.000000 A B C D E A 0 -2 4 0 0 B 2 0 -4 -2 6 C -4 4 0 0 14 D 0 2 0 0 0 E 0 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333326 B: 0.000000 C: 0.000000 D: 0.583337 E: 0.083337 Sum of squares = 0.458333342294 Cumulative probabilities = A: 0.333326 B: 0.333326 C: 0.333326 D: 0.916663 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=24 E=20 C=14 B=2 so B is eliminated. Round 2 votes counts: D=40 A=24 E=21 C=15 so C is eliminated. Round 3 votes counts: D=41 E=32 A=27 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:207 A:201 B:201 D:201 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 0 0 B 2 0 -4 -2 6 C -4 4 0 0 14 D 0 2 0 0 0 E 0 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333326 B: 0.000000 C: 0.000000 D: 0.583337 E: 0.083337 Sum of squares = 0.458333342294 Cumulative probabilities = A: 0.333326 B: 0.333326 C: 0.333326 D: 0.916663 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 0 0 B 2 0 -4 -2 6 C -4 4 0 0 14 D 0 2 0 0 0 E 0 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333326 B: 0.000000 C: 0.000000 D: 0.583337 E: 0.083337 Sum of squares = 0.458333342294 Cumulative probabilities = A: 0.333326 B: 0.333326 C: 0.333326 D: 0.916663 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 0 0 B 2 0 -4 -2 6 C -4 4 0 0 14 D 0 2 0 0 0 E 0 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333326 B: 0.000000 C: 0.000000 D: 0.583337 E: 0.083337 Sum of squares = 0.458333342294 Cumulative probabilities = A: 0.333326 B: 0.333326 C: 0.333326 D: 0.916663 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5960: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A C E (8) B D C E A (8) B C E D A (7) C B E D A (6) B C D E A (6) A D E C B (6) A D B E C (6) C E B D A (5) E C B A D (4) D A B E C (3) D A B C E (3) C E B A D (3) A E D C B (3) A E C B D (3) A D B C E (3) E C B D A (2) B D C A E (2) B D A C E (2) A D E B C (2) E C A D B (1) E B C D A (1) E A C D B (1) D E A C B (1) D B A E C (1) D A E B C (1) C B E A D (1) A E C D B (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -8 -12 -8 B 12 0 8 20 14 C 8 -8 0 0 12 D 12 -20 0 0 6 E 8 -14 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -12 -8 B 12 0 8 20 14 C 8 -8 0 0 12 D 12 -20 0 0 6 E 8 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 E=17 D=17 C=15 so C is eliminated. Round 2 votes counts: B=32 A=26 E=25 D=17 so D is eliminated. Round 3 votes counts: B=41 A=33 E=26 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:206 D:199 E:188 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -12 -8 B 12 0 8 20 14 C 8 -8 0 0 12 D 12 -20 0 0 6 E 8 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -12 -8 B 12 0 8 20 14 C 8 -8 0 0 12 D 12 -20 0 0 6 E 8 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -12 -8 B 12 0 8 20 14 C 8 -8 0 0 12 D 12 -20 0 0 6 E 8 -14 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5961: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) D C B A E (6) B E A C D (6) E B A C D (4) D B C A E (4) C E A B D (4) B E A D C (4) E A B C D (3) D B C E A (3) D A C E B (3) C D A B E (3) B E D C A (3) B E C A D (3) A E C B D (3) A D E C B (3) E C A B D (2) D C A E B (2) D C A B E (2) D B E A C (2) D B A C E (2) C B E D A (2) C A E D B (2) B E D A C (2) B D E C A (2) B D C E A (2) A E C D B (2) E B A D C (1) E A B D C (1) D C B E A (1) D A E B C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B A D (1) C D B E A (1) C D B A E (1) C D A E B (1) C B E A D (1) C B D E A (1) C A E B D (1) C A D E B (1) A E D B C (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -18 2 -8 -12 B 18 0 6 6 22 C -2 -6 0 -12 -2 D 8 -6 12 0 6 E 12 -22 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 2 -8 -12 B 18 0 6 6 22 C -2 -6 0 -12 -2 D 8 -6 12 0 6 E 12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=29 B=29 C=19 A=12 E=11 so E is eliminated. Round 2 votes counts: B=34 D=29 C=21 A=16 so A is eliminated. Round 3 votes counts: B=38 D=34 C=28 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 D:210 E:193 C:189 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 2 -8 -12 B 18 0 6 6 22 C -2 -6 0 -12 -2 D 8 -6 12 0 6 E 12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 2 -8 -12 B 18 0 6 6 22 C -2 -6 0 -12 -2 D 8 -6 12 0 6 E 12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 2 -8 -12 B 18 0 6 6 22 C -2 -6 0 -12 -2 D 8 -6 12 0 6 E 12 -22 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5962: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) D A E C B (6) D A B C E (6) B A C E D (6) A B E C D (6) E C B A D (5) E C D B A (4) D E A C B (4) C B E D A (4) A B D C E (4) D E C B A (3) D C B E A (3) C E B D A (3) C B D E A (3) B C E A D (3) A E B C D (3) E D C A B (2) E B C A D (2) D E C A B (2) B C E D A (2) A B C E D (2) E D C B A (1) E D A C B (1) E C B D A (1) E B A C D (1) E A C D B (1) E A C B D (1) E A B C D (1) D A C E B (1) D A C B E (1) D A B E C (1) C E D B A (1) B C D A E (1) B C A E D (1) B C A D E (1) A E D C B (1) A E B D C (1) A D E B C (1) A D B C E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 2 -14 -10 B 4 0 -10 6 -6 C -2 10 0 6 2 D 14 -6 -6 0 -6 E 10 6 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408224 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -4 2 -14 -10 B 4 0 -10 6 -6 C -2 10 0 6 2 D 14 -6 -6 0 -6 E 10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408123 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=21 E=20 B=14 C=11 so C is eliminated. Round 2 votes counts: D=34 E=24 B=21 A=21 so B is eliminated. Round 3 votes counts: D=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:208 D:198 B:197 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 -14 -10 B 4 0 -10 6 -6 C -2 10 0 6 2 D 14 -6 -6 0 -6 E 10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408123 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -14 -10 B 4 0 -10 6 -6 C -2 10 0 6 2 D 14 -6 -6 0 -6 E 10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408123 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -14 -10 B 4 0 -10 6 -6 C -2 10 0 6 2 D 14 -6 -6 0 -6 E 10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408123 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5963: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) D C B E A (10) E A B C D (8) A E B C D (7) A B C E D (7) C B D A E (6) E D B C A (5) A B C D E (4) A D C B E (3) E B C A D (2) E B A C D (2) E A D B C (2) D E C B A (2) D C A B E (2) D A C B E (2) C D B A E (2) C B D E A (2) B C A E D (2) B C A D E (2) A B E C D (2) E D C B A (1) E B D C A (1) E B C D A (1) E A D C B (1) E A B D C (1) D E C A B (1) D E A C B (1) D C E B A (1) D C A E B (1) D A C E B (1) C B A D E (1) B E C D A (1) B C E D A (1) B C E A D (1) A E D B C (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -12 -4 12 B 8 0 2 6 20 C 12 -2 0 4 20 D 4 -6 -4 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -4 12 B 8 0 2 6 20 C 12 -2 0 4 20 D 4 -6 -4 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996304 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=26 E=24 C=11 B=7 so B is eliminated. Round 2 votes counts: D=32 A=26 E=25 C=17 so C is eliminated. Round 3 votes counts: D=42 A=31 E=27 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:218 C:217 D:200 A:194 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -4 12 B 8 0 2 6 20 C 12 -2 0 4 20 D 4 -6 -4 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996304 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -4 12 B 8 0 2 6 20 C 12 -2 0 4 20 D 4 -6 -4 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996304 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -4 12 B 8 0 2 6 20 C 12 -2 0 4 20 D 4 -6 -4 0 6 E -12 -20 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996304 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5964: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) A E C D B (8) E A C B D (7) D C B A E (6) B D C A E (6) D C A B E (4) C D E B A (4) B E D C A (4) E C B D A (3) D B C E A (3) D B C A E (3) D B A C E (3) C A E D B (3) A D C E B (3) A C D E B (3) E C B A D (2) E C A D B (2) C E D A B (2) C D B E A (2) B E C D A (2) B D E C A (2) A E B D C (2) A C E D B (2) A B D E C (2) E B A C D (1) E A B C D (1) D C B E A (1) C E D B A (1) C D A E B (1) C D A B E (1) C B D E A (1) B D E A C (1) B D A E C (1) B A E D C (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -22 -18 6 B 10 0 -16 -8 6 C 22 16 0 -4 16 D 18 8 4 0 14 E -6 -6 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 -18 6 B 10 0 -16 -8 6 C 22 16 0 -4 16 D 18 8 4 0 14 E -6 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=24 D=20 E=16 C=15 so C is eliminated. Round 2 votes counts: D=28 A=27 B=26 E=19 so E is eliminated. Round 3 votes counts: A=37 B=32 D=31 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:225 D:222 B:196 E:179 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -22 -18 6 B 10 0 -16 -8 6 C 22 16 0 -4 16 D 18 8 4 0 14 E -6 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 -18 6 B 10 0 -16 -8 6 C 22 16 0 -4 16 D 18 8 4 0 14 E -6 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 -18 6 B 10 0 -16 -8 6 C 22 16 0 -4 16 D 18 8 4 0 14 E -6 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5965: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (14) D A C B E (9) A C D B E (8) D E B A C (7) D A E C B (6) D A C E B (6) E D B C A (5) C B A E D (5) A C B D E (5) D E A C B (4) C A B E D (4) B E C A D (4) E B D C A (3) A D C B E (3) E B C D A (2) B C A E D (2) A D C E B (2) E D A C B (1) E C A B D (1) E A D C B (1) D A B C E (1) C E B A D (1) C B E A D (1) B E D C A (1) B C A D E (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 12 10 10 B -8 0 -20 -10 -12 C -12 20 0 0 0 D -10 10 0 0 6 E -10 12 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 10 10 B -8 0 -20 -10 -12 C -12 20 0 0 0 D -10 10 0 0 6 E -10 12 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=27 A=21 C=11 B=8 so B is eliminated. Round 2 votes counts: D=33 E=32 A=21 C=14 so C is eliminated. Round 3 votes counts: E=34 D=33 A=33 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:204 D:203 E:198 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 10 10 B -8 0 -20 -10 -12 C -12 20 0 0 0 D -10 10 0 0 6 E -10 12 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 10 10 B -8 0 -20 -10 -12 C -12 20 0 0 0 D -10 10 0 0 6 E -10 12 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 10 10 B -8 0 -20 -10 -12 C -12 20 0 0 0 D -10 10 0 0 6 E -10 12 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5966: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) D E A B C (6) D C E A B (6) C E A B D (6) B A E C D (5) C A E B D (4) C A B E D (4) E D A C B (3) D E C A B (3) D E A C B (3) D C B E A (3) D B A E C (3) B C A E D (3) B C A D E (3) A E C B D (3) E A D B C (2) E A C B D (2) D E B A C (2) D B E A C (2) C D E A B (2) C B D A E (2) C B A D E (2) B D A E C (2) B A E D C (2) B A D E C (2) B A C E D (2) A B E C D (2) E D A B C (1) E A C D B (1) D C B A E (1) D B C E A (1) D B C A E (1) D B A C E (1) C E A D B (1) C D B A E (1) B D A C E (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -6 12 12 B -2 0 -14 14 8 C 6 14 0 10 10 D -12 -14 -10 0 -6 E -12 -8 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 12 12 B -2 0 -14 14 8 C 6 14 0 10 10 D -12 -14 -10 0 -6 E -12 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=32 C=32 B=20 E=9 A=7 so A is eliminated. Round 2 votes counts: D=32 C=32 B=23 E=13 so E is eliminated. Round 3 votes counts: D=38 C=38 B=24 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:210 B:203 E:188 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 12 12 B -2 0 -14 14 8 C 6 14 0 10 10 D -12 -14 -10 0 -6 E -12 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 12 12 B -2 0 -14 14 8 C 6 14 0 10 10 D -12 -14 -10 0 -6 E -12 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 12 12 B -2 0 -14 14 8 C 6 14 0 10 10 D -12 -14 -10 0 -6 E -12 -8 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5967: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) A C E B D (7) D B A C E (6) B D E C A (6) E A C B D (5) B D E A C (5) E C A B D (4) D B C E A (4) C D A E B (4) C A E D B (4) A C E D B (4) E A B C D (3) D C A E B (3) B E D C A (3) B E D A C (3) A E C B D (3) E B A C D (2) D B C A E (2) D B A E C (2) C A E B D (2) B E A C D (2) B D A E C (2) A C D E B (2) E C B A D (1) D C A B E (1) D B E A C (1) D A B C E (1) C E D A B (1) C E A D B (1) C E A B D (1) C D E A B (1) C A D E B (1) B E A D C (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 2 -10 -8 B 0 0 8 2 0 C -2 -8 0 0 -8 D 10 -2 0 0 4 E 8 0 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.761430 C: 0.000000 D: 0.000000 E: 0.238570 Sum of squares = 0.636690797514 Cumulative probabilities = A: 0.000000 B: 0.761430 C: 0.761430 D: 0.761430 E: 1.000000 A B C D E A 0 0 2 -10 -8 B 0 0 8 2 0 C -2 -8 0 0 -8 D 10 -2 0 0 4 E 8 0 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555563443 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=22 A=18 E=15 C=15 so E is eliminated. Round 2 votes counts: D=30 A=26 B=24 C=20 so C is eliminated. Round 3 votes counts: A=39 D=36 B=25 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:206 E:206 B:205 A:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -10 -8 B 0 0 8 2 0 C -2 -8 0 0 -8 D 10 -2 0 0 4 E 8 0 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555563443 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -10 -8 B 0 0 8 2 0 C -2 -8 0 0 -8 D 10 -2 0 0 4 E 8 0 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555563443 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -10 -8 B 0 0 8 2 0 C -2 -8 0 0 -8 D 10 -2 0 0 4 E 8 0 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555563443 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5968: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) B A C D E (8) D E B A C (7) B D E A C (7) B A D E C (6) A C B E D (6) E D A B C (5) D E C B A (5) C A E D B (5) C A B E D (5) B C D A E (4) A C E B D (4) D E B C A (3) A B C E D (3) A B C D E (3) C E D A B (2) C D B E A (2) E D B A C (1) E C D A B (1) E A D B C (1) E A C D B (1) E A B D C (1) D C E B A (1) D B E A C (1) C D E B A (1) C A E B D (1) B D C E A (1) B D A E C (1) B A E D C (1) A E C D B (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 18 -4 0 B -2 0 10 6 -2 C -18 -10 0 -6 -8 D 4 -6 6 0 0 E 0 2 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.212574 E: 0.787426 Sum of squares = 0.665227047357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.212574 E: 1.000000 A B C D E A 0 2 18 -4 0 B -2 0 10 6 -2 C -18 -10 0 -6 -8 D 4 -6 6 0 0 E 0 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000020198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=20 A=19 D=17 C=16 so C is eliminated. Round 2 votes counts: A=30 B=28 E=22 D=20 so D is eliminated. Round 3 votes counts: E=39 B=31 A=30 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:208 B:206 E:205 D:202 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 18 -4 0 B -2 0 10 6 -2 C -18 -10 0 -6 -8 D 4 -6 6 0 0 E 0 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000020198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 -4 0 B -2 0 10 6 -2 C -18 -10 0 -6 -8 D 4 -6 6 0 0 E 0 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000020198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 -4 0 B -2 0 10 6 -2 C -18 -10 0 -6 -8 D 4 -6 6 0 0 E 0 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000020198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5969: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) A D E B C (12) C A B D E (9) E D B A C (7) A C D E B (5) C B A E D (4) A D B E C (4) E B D A C (3) D E B A C (3) D E A B C (3) C A D B E (3) A E D B C (3) E D A B C (2) E A D B C (2) C A E D B (2) B E D C A (2) B E C D A (2) A E D C B (2) A D E C B (2) A D C E B (2) E B D C A (1) E B C D A (1) D A E B C (1) C E B D A (1) C B A D E (1) C A D E B (1) C A B E D (1) B E D A C (1) B D E A C (1) B C E D A (1) B C D E A (1) B A D E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 12 12 12 10 B -12 0 4 -12 -6 C -12 -4 0 -6 -6 D -12 12 6 0 2 E -10 6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 12 10 B -12 0 4 -12 -6 C -12 -4 0 -6 -6 D -12 12 6 0 2 E -10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=32 E=16 B=9 D=7 so D is eliminated. Round 2 votes counts: C=36 A=33 E=22 B=9 so B is eliminated. Round 3 votes counts: C=38 A=34 E=28 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 D:204 E:200 B:187 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 12 10 B -12 0 4 -12 -6 C -12 -4 0 -6 -6 D -12 12 6 0 2 E -10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 12 10 B -12 0 4 -12 -6 C -12 -4 0 -6 -6 D -12 12 6 0 2 E -10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 12 10 B -12 0 4 -12 -6 C -12 -4 0 -6 -6 D -12 12 6 0 2 E -10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5970: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (12) E A B D C (10) C D B E A (8) A E D C B (8) B C D A E (6) E A D C B (5) D C E A B (5) D C A E B (5) C D A B E (4) B E A C D (4) E D C A B (3) E A D B C (3) C B D A E (3) B C D E A (3) C D B A E (2) B A C D E (2) A E D B C (2) A E B D C (2) E D A C B (1) E B D C A (1) E B A D C (1) D E C A B (1) D C E B A (1) D A C E B (1) C D E B A (1) B E C D A (1) B C E A D (1) B C A E D (1) B A C E D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 4 8 8 2 B -4 0 0 -4 -4 C -8 0 0 -2 -10 D -8 4 2 0 -12 E -2 4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 2 B -4 0 0 -4 -4 C -8 0 0 -2 -10 D -8 4 2 0 -12 E -2 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=24 C=18 A=14 D=13 so D is eliminated. Round 2 votes counts: B=31 C=29 E=25 A=15 so A is eliminated. Round 3 votes counts: E=38 C=31 B=31 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:211 B:194 D:193 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 2 B -4 0 0 -4 -4 C -8 0 0 -2 -10 D -8 4 2 0 -12 E -2 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 2 B -4 0 0 -4 -4 C -8 0 0 -2 -10 D -8 4 2 0 -12 E -2 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 2 B -4 0 0 -4 -4 C -8 0 0 -2 -10 D -8 4 2 0 -12 E -2 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5971: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C A D B E (10) C D B E A (9) A E B D C (9) D B E C A (8) C B D E A (8) A C E B D (7) C A B D E (5) A E D B C (5) E A D B C (3) C D B A E (3) A C D E B (3) E A B D C (2) D E B A C (2) C B D A E (2) B D E C A (2) A E C B D (2) A E B C D (2) D C B E A (1) C D A B E (1) C B E D A (1) B E D C A (1) B D E A C (1) B D C E A (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 2 -4 0 2 B -2 0 -8 6 6 C 4 8 0 8 2 D 0 -6 -8 0 12 E -2 -6 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 0 2 B -2 0 -8 6 6 C 4 8 0 8 2 D 0 -6 -8 0 12 E -2 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=30 E=15 D=11 B=5 so B is eliminated. Round 2 votes counts: C=39 A=30 E=16 D=15 so D is eliminated. Round 3 votes counts: C=41 A=30 E=29 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:201 A:200 D:199 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 0 2 B -2 0 -8 6 6 C 4 8 0 8 2 D 0 -6 -8 0 12 E -2 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 2 B -2 0 -8 6 6 C 4 8 0 8 2 D 0 -6 -8 0 12 E -2 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 2 B -2 0 -8 6 6 C 4 8 0 8 2 D 0 -6 -8 0 12 E -2 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5972: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) A D B C E (9) E C D B A (8) D B C A E (8) C B D E A (8) E A C B D (6) A E B D C (6) C E B D A (5) E C A B D (4) E A C D B (4) C D B E A (4) B D C A E (4) D B A C E (3) E C B D A (2) D B C E A (2) B D A C E (2) A E C B D (2) A B D E C (2) A B D C E (2) E C A D B (1) E A D C B (1) C E D B A (1) C B A D E (1) B C D A E (1) A E D C B (1) A E C D B (1) A E B C D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 2 4 8 B -2 0 2 -8 -6 C -2 -2 0 -2 0 D -4 8 2 0 -4 E -8 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 4 8 B -2 0 2 -8 -6 C -2 -2 0 -2 0 D -4 8 2 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=26 C=19 D=13 B=7 so B is eliminated. Round 2 votes counts: A=35 E=26 C=20 D=19 so D is eliminated. Round 3 votes counts: A=40 C=34 E=26 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:201 E:201 C:197 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 8 B -2 0 2 -8 -6 C -2 -2 0 -2 0 D -4 8 2 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 8 B -2 0 2 -8 -6 C -2 -2 0 -2 0 D -4 8 2 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 8 B -2 0 2 -8 -6 C -2 -2 0 -2 0 D -4 8 2 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5973: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (10) D B A C E (9) C E A B D (8) B D E C A (5) E C A B D (4) E A C B D (4) D B C A E (4) C A E B D (4) B E D A C (4) A C D E B (4) E C B A D (3) C A D E B (3) B D C E A (3) E B A C D (2) D C A B E (2) B E D C A (2) A D C E B (2) E B D A C (1) E B C A D (1) E A D B C (1) E A B C D (1) D B A E C (1) D A C B E (1) C D B A E (1) C D A B E (1) C B E D A (1) C B E A D (1) C A D B E (1) B C D E A (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -14 8 -4 B -2 0 -8 10 -6 C 14 8 0 8 18 D -8 -10 -8 0 -2 E 4 6 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 8 -4 B -2 0 -8 10 -6 C 14 8 0 8 18 D -8 -10 -8 0 -2 E 4 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=25 E=17 D=17 A=9 so A is eliminated. Round 2 votes counts: C=37 B=25 D=20 E=18 so E is eliminated. Round 3 votes counts: C=49 B=30 D=21 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:197 E:197 A:196 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 8 -4 B -2 0 -8 10 -6 C 14 8 0 8 18 D -8 -10 -8 0 -2 E 4 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 8 -4 B -2 0 -8 10 -6 C 14 8 0 8 18 D -8 -10 -8 0 -2 E 4 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 8 -4 B -2 0 -8 10 -6 C 14 8 0 8 18 D -8 -10 -8 0 -2 E 4 6 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5974: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A B E D C (8) C D B E A (7) E A D B C (6) B C A D E (6) A E D B C (6) E D C A B (5) E D A B C (5) C B D A E (4) E D A C B (3) D E C A B (3) B A C D E (3) A E B D C (3) D E C B A (2) C E B A D (2) B C D A E (2) B A C E D (2) A C E B D (2) A B E C D (2) A B C E D (2) E C D A B (1) E C A D B (1) E A D C B (1) D E A C B (1) D E A B C (1) D C E B A (1) D C B E A (1) D A E B C (1) C B E A D (1) C B D E A (1) C B A E D (1) C B A D E (1) B D C A E (1) B C A E D (1) B A D E C (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -2 2 -4 B -4 0 4 -10 -10 C 2 -4 0 0 -2 D -2 10 0 0 -6 E 4 10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -2 2 -4 B -4 0 4 -10 -10 C 2 -4 0 0 -2 D -2 10 0 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 E=22 B=17 D=10 so D is eliminated. Round 2 votes counts: E=29 C=29 A=25 B=17 so B is eliminated. Round 3 votes counts: C=39 A=32 E=29 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:211 D:201 A:200 C:198 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 2 -4 B -4 0 4 -10 -10 C 2 -4 0 0 -2 D -2 10 0 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 -4 B -4 0 4 -10 -10 C 2 -4 0 0 -2 D -2 10 0 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 -4 B -4 0 4 -10 -10 C 2 -4 0 0 -2 D -2 10 0 0 -6 E 4 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5975: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) D B C A E (9) D B C E A (8) A E C B D (8) E C B D A (6) E C B A D (6) E C A B D (6) A D B C E (6) B D C E A (4) A D C B E (4) E A C B D (3) B C D E A (3) D B E C A (2) C B E D A (2) C A E B D (2) B C E D A (2) A E D C B (2) A D E C B (2) A D C E B (2) A C E D B (2) A C E B D (2) E D B A C (1) E A D B C (1) D E B A C (1) D B A C E (1) C B D E A (1) C A B E D (1) A E C D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -4 0 8 B -8 0 -4 -6 8 C 4 4 0 -6 22 D 0 6 6 0 10 E -8 -8 -22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.196652 B: 0.000000 C: 0.000000 D: 0.803348 E: 0.000000 Sum of squares = 0.684039687034 Cumulative probabilities = A: 0.196652 B: 0.196652 C: 0.196652 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 0 8 B -8 0 -4 -6 8 C 4 4 0 -6 22 D 0 6 6 0 10 E -8 -8 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=31 A=31 E=23 B=9 C=6 so C is eliminated. Round 2 votes counts: A=34 D=31 E=23 B=12 so B is eliminated. Round 3 votes counts: D=39 A=34 E=27 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 D:211 A:206 B:195 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -4 0 8 B -8 0 -4 -6 8 C 4 4 0 -6 22 D 0 6 6 0 10 E -8 -8 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 0 8 B -8 0 -4 -6 8 C 4 4 0 -6 22 D 0 6 6 0 10 E -8 -8 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 0 8 B -8 0 -4 -6 8 C 4 4 0 -6 22 D 0 6 6 0 10 E -8 -8 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5976: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) B C E A D (7) D E A B C (6) A D C B E (6) A C B E D (6) E D B C A (5) E B D C A (4) D E B C A (4) A E D C B (4) A E C B D (4) E D B A C (3) E B C A D (3) D A C B E (3) A C B D E (3) E D A B C (2) E B A C D (2) E A B C D (2) D E B A C (2) D B E C A (2) D B C E A (2) B E C D A (2) A D E C B (2) A C D B E (2) E B C D A (1) E B A D C (1) E A C B D (1) D C B A E (1) D A E C B (1) D A C E B (1) C B D A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B E D C A (1) B C D E A (1) A E C D B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 10 16 -2 B 2 0 0 0 -4 C -10 0 0 -2 -6 D -16 0 2 0 -18 E 2 4 6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 10 16 -2 B 2 0 0 0 -4 C -10 0 0 -2 -6 D -16 0 2 0 -18 E 2 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 D=22 C=12 B=11 so B is eliminated. Round 2 votes counts: A=31 E=27 D=22 C=20 so C is eliminated. Round 3 votes counts: A=42 E=34 D=24 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:211 B:199 C:191 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 10 16 -2 B 2 0 0 0 -4 C -10 0 0 -2 -6 D -16 0 2 0 -18 E 2 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 16 -2 B 2 0 0 0 -4 C -10 0 0 -2 -6 D -16 0 2 0 -18 E 2 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 16 -2 B 2 0 0 0 -4 C -10 0 0 -2 -6 D -16 0 2 0 -18 E 2 4 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5977: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) A B E D C (8) C D E B A (7) C D E A B (5) B C D A E (5) B A D E C (5) E D C A B (4) E C D A B (4) E C A D B (4) E A D C B (4) B A D C E (4) B A C D E (4) E A B D C (3) C D B A E (3) B A E D C (3) B A E C D (3) A E B D C (3) E D A C B (2) E A D B C (2) D C E A B (2) C E D B A (2) C E D A B (2) C B D E A (2) C B D A E (2) E C B A D (1) D C B A E (1) D A E B C (1) C E B D A (1) B D A C E (1) B C A D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -12 -8 -10 B 10 0 -12 -6 6 C 12 12 0 12 4 D 8 6 -12 0 8 E 10 -6 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 -10 B 10 0 -12 -6 6 C 12 12 0 12 4 D 8 6 -12 0 8 E 10 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=26 E=24 A=12 D=4 so D is eliminated. Round 2 votes counts: C=37 B=26 E=24 A=13 so A is eliminated. Round 3 votes counts: C=37 B=35 E=28 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:205 B:199 E:196 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 -10 B 10 0 -12 -6 6 C 12 12 0 12 4 D 8 6 -12 0 8 E 10 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 -10 B 10 0 -12 -6 6 C 12 12 0 12 4 D 8 6 -12 0 8 E 10 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 -10 B 10 0 -12 -6 6 C 12 12 0 12 4 D 8 6 -12 0 8 E 10 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5978: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) D A E C B (8) E A D B C (6) B C A E D (6) A D E C B (6) A D C E B (6) E D A B C (5) D E A C B (5) B E C D A (5) C B D A E (4) C B A D E (4) B C E D A (4) D A C E B (3) E D B A C (2) E B A D C (2) E B A C D (2) E A D C B (2) E A B D C (2) C D A B E (2) C A D B E (2) B E C A D (2) B C A D E (2) A E D C B (2) E D A C B (1) E B D C A (1) E B D A C (1) D E B C A (1) D C B E A (1) C D B A E (1) C A B D E (1) B C D E A (1) B C D A E (1) Total count = 100 A B C D E A 0 2 6 8 -4 B -2 0 4 -6 -10 C -6 -4 0 -8 -6 D -8 6 8 0 -4 E 4 10 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 6 8 -4 B -2 0 4 -6 -10 C -6 -4 0 -8 -6 D -8 6 8 0 -4 E 4 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=24 D=18 C=14 A=14 so C is eliminated. Round 2 votes counts: B=38 E=24 D=21 A=17 so A is eliminated. Round 3 votes counts: B=39 D=35 E=26 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:212 A:206 D:201 B:193 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 8 -4 B -2 0 4 -6 -10 C -6 -4 0 -8 -6 D -8 6 8 0 -4 E 4 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 8 -4 B -2 0 4 -6 -10 C -6 -4 0 -8 -6 D -8 6 8 0 -4 E 4 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 8 -4 B -2 0 4 -6 -10 C -6 -4 0 -8 -6 D -8 6 8 0 -4 E 4 10 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5979: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) B D A E C (8) C A E B D (7) C B A D E (6) B A C D E (6) D B E A C (5) C A B E D (5) B A D C E (5) D B E C A (4) E D C A B (3) D E B C A (3) A C E B D (3) E D A B C (2) E C D A B (2) E C A D B (2) E A C D B (2) C E D B A (2) C E A D B (2) B D C E A (2) B D A C E (2) A E B C D (2) A C B E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E D A C B (1) E C D B A (1) D B C E A (1) D B A E C (1) C B D A E (1) B C D A E (1) B C A D E (1) A E C D B (1) A E C B D (1) A C B D E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 10 0 14 B 20 0 14 14 10 C -10 -14 0 2 0 D 0 -14 -2 0 18 E -14 -10 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 10 0 14 B 20 0 14 14 10 C -10 -14 0 2 0 D 0 -14 -2 0 18 E -14 -10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 C=23 E=15 A=14 so A is eliminated. Round 2 votes counts: C=29 B=29 D=23 E=19 so E is eliminated. Round 3 votes counts: C=38 D=31 B=31 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 A:202 D:201 C:189 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 10 0 14 B 20 0 14 14 10 C -10 -14 0 2 0 D 0 -14 -2 0 18 E -14 -10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 10 0 14 B 20 0 14 14 10 C -10 -14 0 2 0 D 0 -14 -2 0 18 E -14 -10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 10 0 14 B 20 0 14 14 10 C -10 -14 0 2 0 D 0 -14 -2 0 18 E -14 -10 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5980: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (6) B A D C E (5) E C D B A (4) D C B A E (4) D A B C E (4) A E B D C (4) A E B C D (4) A B E C D (4) E C D A B (3) E C A D B (3) E A C D B (3) D C E B A (3) D C E A B (3) D B A C E (3) B C D E A (3) A D E C B (3) A D B E C (3) A B D C E (3) E A D C B (2) E A C B D (2) E A B C D (2) D C B E A (2) D B C A E (2) D A E C B (2) C E D B A (2) C B D E A (2) B E A C D (2) B D C A E (2) A D B C E (2) E C B D A (1) E C B A D (1) E C A B D (1) C D B E A (1) B E C A D (1) B C E D A (1) B C D A E (1) B A C E D (1) A E D C B (1) A E D B C (1) A E C B D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 8 2 4 B -6 0 0 -16 -6 C -8 0 0 -2 0 D -2 16 2 0 12 E -4 6 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 2 4 B -6 0 0 -16 -6 C -8 0 0 -2 0 D -2 16 2 0 12 E -4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=23 E=22 B=16 C=11 so C is eliminated. Round 2 votes counts: D=30 A=28 E=24 B=18 so B is eliminated. Round 3 votes counts: D=38 A=34 E=28 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:210 C:195 E:195 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 4 B -6 0 0 -16 -6 C -8 0 0 -2 0 D -2 16 2 0 12 E -4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 4 B -6 0 0 -16 -6 C -8 0 0 -2 0 D -2 16 2 0 12 E -4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 4 B -6 0 0 -16 -6 C -8 0 0 -2 0 D -2 16 2 0 12 E -4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5981: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) D B C A E (7) C A D E B (5) E B D A C (4) E A C D B (4) D E B C A (4) D B E C A (4) C A E D B (4) A C B D E (4) E A C B D (3) A E C B D (3) A C E D B (3) E D C A B (2) E D B A C (2) E C A D B (2) E B A D C (2) E A B C D (2) D C B E A (2) D B C E A (2) C A D B E (2) B E D A C (2) B E A D C (2) A C E B D (2) A B C E D (2) E D C B A (1) E D A C B (1) E D A B C (1) E C D A B (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E A B (1) D C B A E (1) D C A B E (1) C D E A B (1) C D A E B (1) C B D A E (1) C B A D E (1) C A B E D (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A C E (1) B C D A E (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 -2 -16 -8 2 B 2 0 2 -10 -4 C 16 -2 0 -12 10 D 8 10 12 0 8 E -2 4 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -8 2 B 2 0 2 -10 -4 C 16 -2 0 -12 10 D 8 10 12 0 8 E -2 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 B=20 C=16 A=14 so A is eliminated. Round 2 votes counts: E=30 C=25 D=23 B=22 so B is eliminated. Round 3 votes counts: D=37 E=34 C=29 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:206 B:195 E:192 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -16 -8 2 B 2 0 2 -10 -4 C 16 -2 0 -12 10 D 8 10 12 0 8 E -2 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -8 2 B 2 0 2 -10 -4 C 16 -2 0 -12 10 D 8 10 12 0 8 E -2 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -8 2 B 2 0 2 -10 -4 C 16 -2 0 -12 10 D 8 10 12 0 8 E -2 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5982: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) D A E C B (7) A E C B D (7) D B C E A (6) D E A B C (5) A C E B D (5) D E B C A (4) D B C A E (4) B C E A D (4) A C B E D (4) D E A C B (3) C B A E D (3) C A B E D (3) B C A E D (3) B C A D E (3) A E D C B (3) A E C D B (3) E B C D A (2) E A D C B (2) E A C D B (2) E A C B D (2) C B E A D (2) B D C A E (2) B C D A E (2) E D B C A (1) E D A C B (1) E B D C A (1) D B A C E (1) D A B C E (1) C E B A D (1) C B A D E (1) B E C D A (1) B E C A D (1) B D C E A (1) B C D E A (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -6 0 6 B 2 0 2 0 2 C 6 -2 0 0 -4 D 0 0 0 0 -2 E -6 -2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.672936 C: 0.000000 D: 0.327064 E: 0.000000 Sum of squares = 0.559813496249 Cumulative probabilities = A: 0.000000 B: 0.672936 C: 0.672936 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 0 6 B 2 0 2 0 2 C 6 -2 0 0 -4 D 0 0 0 0 -2 E -6 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500209 C: 0.000000 D: 0.499791 E: 0.000000 Sum of squares = 0.500000087703 Cumulative probabilities = A: 0.000000 B: 0.500209 C: 0.500209 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=23 B=18 E=11 C=10 so C is eliminated. Round 2 votes counts: D=38 A=26 B=24 E=12 so E is eliminated. Round 3 votes counts: D=40 A=32 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:203 C:200 A:199 D:199 E:199 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 0 6 B 2 0 2 0 2 C 6 -2 0 0 -4 D 0 0 0 0 -2 E -6 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500209 C: 0.000000 D: 0.499791 E: 0.000000 Sum of squares = 0.500000087703 Cumulative probabilities = A: 0.000000 B: 0.500209 C: 0.500209 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 0 6 B 2 0 2 0 2 C 6 -2 0 0 -4 D 0 0 0 0 -2 E -6 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500209 C: 0.000000 D: 0.499791 E: 0.000000 Sum of squares = 0.500000087703 Cumulative probabilities = A: 0.000000 B: 0.500209 C: 0.500209 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 0 6 B 2 0 2 0 2 C 6 -2 0 0 -4 D 0 0 0 0 -2 E -6 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500209 C: 0.000000 D: 0.499791 E: 0.000000 Sum of squares = 0.500000087703 Cumulative probabilities = A: 0.000000 B: 0.500209 C: 0.500209 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5983: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) E A B D C (8) B C D E A (8) E A D B C (7) E A D C B (6) C B D A E (6) B D C E A (5) B C D A E (5) C D B A E (4) A E D C B (3) E B A D C (2) E B A C D (2) D C A E B (2) D B E A C (2) D B C E A (2) C D A B E (2) C A D E B (2) B E D A C (2) B E A C D (2) B D E C A (2) B D E A C (2) A E C D B (2) E A C B D (1) D C A B E (1) D B C A E (1) D A E C B (1) D A C E B (1) C D A E B (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A D C (1) B C E A D (1) A E C B D (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 10 6 -28 B -4 0 26 20 -2 C -10 -26 0 2 -12 D -6 -20 -2 0 2 E 28 2 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.000000 D: 0.083333 E: 0.833333 Sum of squares = 0.708333333351 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.166667 E: 1.000000 A B C D E A 0 4 10 6 -28 B -4 0 26 20 -2 C -10 -26 0 2 -12 D -6 -20 -2 0 2 E 28 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.000000 D: 0.083333 E: 0.833333 Sum of squares = 0.708333333299 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.166667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=29 C=17 D=10 A=8 so A is eliminated. Round 2 votes counts: E=42 B=29 C=18 D=11 so D is eliminated. Round 3 votes counts: E=44 B=34 C=22 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:220 A:196 D:187 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 6 -28 B -4 0 26 20 -2 C -10 -26 0 2 -12 D -6 -20 -2 0 2 E 28 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.000000 D: 0.083333 E: 0.833333 Sum of squares = 0.708333333299 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.166667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 6 -28 B -4 0 26 20 -2 C -10 -26 0 2 -12 D -6 -20 -2 0 2 E 28 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.000000 D: 0.083333 E: 0.833333 Sum of squares = 0.708333333299 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 6 -28 B -4 0 26 20 -2 C -10 -26 0 2 -12 D -6 -20 -2 0 2 E 28 2 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.000000 D: 0.083333 E: 0.833333 Sum of squares = 0.708333333299 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.083333 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5984: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) E D C B A (7) D E B C A (7) D E A B C (7) C B A E D (7) B C A D E (7) A B C D E (7) A C B E D (6) E A D C B (4) D E B A C (4) D A B C E (4) B C A E D (3) A D B C E (3) E D C A B (2) E C B D A (2) D A E B C (2) A C B D E (2) E D B C A (1) E C D A B (1) E C B A D (1) E A C B D (1) D E A C B (1) D B C E A (1) C A E B D (1) B C D E A (1) B C D A E (1) B A C D E (1) A E C B D (1) A D E C B (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 16 -6 -4 B -14 0 2 -16 -12 C -16 -2 0 -16 -10 D 6 16 16 0 2 E 4 12 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 -6 -4 B -14 0 2 -16 -12 C -16 -2 0 -16 -10 D 6 16 16 0 2 E 4 12 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=26 A=22 B=13 C=8 so C is eliminated. Round 2 votes counts: E=31 D=26 A=23 B=20 so B is eliminated. Round 3 votes counts: A=41 E=31 D=28 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:220 E:212 A:210 B:180 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 16 -6 -4 B -14 0 2 -16 -12 C -16 -2 0 -16 -10 D 6 16 16 0 2 E 4 12 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 -6 -4 B -14 0 2 -16 -12 C -16 -2 0 -16 -10 D 6 16 16 0 2 E 4 12 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 -6 -4 B -14 0 2 -16 -12 C -16 -2 0 -16 -10 D 6 16 16 0 2 E 4 12 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5985: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C E B D A (6) C D B A E (6) A E D B C (5) E B A C D (4) E A B C D (4) C D A E B (4) C B D E A (4) B C E D A (4) A D C E B (4) E B C A D (3) E A C B D (3) D C B A E (3) D B A E C (3) C D B E A (3) C A E D B (3) C A D E B (3) A E C D B (3) A E B D C (3) E B A D C (2) C E A B D (2) C D A B E (2) B E D A C (2) B E C D A (2) B D E A C (2) A E C B D (2) A D B E C (2) D B C E A (1) D B C A E (1) C B E D A (1) B E D C A (1) B E A D C (1) B D C E A (1) B D A E C (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 0 6 -6 B 2 0 -2 10 -18 C 0 2 0 18 -2 D -6 -10 -18 0 -16 E 6 18 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 6 -6 B 2 0 -2 10 -18 C 0 2 0 18 -2 D -6 -10 -18 0 -16 E 6 18 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=23 A=21 B=14 D=8 so D is eliminated. Round 2 votes counts: C=37 E=23 A=21 B=19 so B is eliminated. Round 3 votes counts: C=44 E=31 A=25 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:209 A:199 B:196 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 6 -6 B 2 0 -2 10 -18 C 0 2 0 18 -2 D -6 -10 -18 0 -16 E 6 18 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 6 -6 B 2 0 -2 10 -18 C 0 2 0 18 -2 D -6 -10 -18 0 -16 E 6 18 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 6 -6 B 2 0 -2 10 -18 C 0 2 0 18 -2 D -6 -10 -18 0 -16 E 6 18 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 5986: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) A E C B D (8) D B C E A (7) A C E B D (7) E C A B D (6) D A C E B (6) B C E A D (6) D B A C E (4) B E C A D (4) A E C D B (4) E A C B D (3) D B C A E (3) D B A E C (3) D A E C B (3) C E B A D (3) B D C E A (3) E C B A D (2) D B E A C (2) B D E C A (2) A D C E B (2) A C E D B (2) E B C A D (1) D A C B E (1) D A B C E (1) C E A B D (1) C B E A D (1) C A E B D (1) B E D C A (1) B E C D A (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 -6 4 -8 B 8 0 -2 2 0 C 6 2 0 2 -2 D -4 -2 -2 0 -6 E 8 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.317767 C: 0.000000 D: 0.000000 E: 0.682233 Sum of squares = 0.566417920788 Cumulative probabilities = A: 0.000000 B: 0.317767 C: 0.317767 D: 0.317767 E: 1.000000 A B C D E A 0 -8 -6 4 -8 B 8 0 -2 2 0 C 6 2 0 2 -2 D -4 -2 -2 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499884 C: 0.000000 D: 0.000000 E: 0.500116 Sum of squares = 0.500000026762 Cumulative probabilities = A: 0.000000 B: 0.499884 C: 0.499884 D: 0.499884 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=24 B=18 E=12 C=6 so C is eliminated. Round 2 votes counts: D=40 A=25 B=19 E=16 so E is eliminated. Round 3 votes counts: D=40 A=35 B=25 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:208 B:204 C:204 D:193 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 4 -8 B 8 0 -2 2 0 C 6 2 0 2 -2 D -4 -2 -2 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499884 C: 0.000000 D: 0.000000 E: 0.500116 Sum of squares = 0.500000026762 Cumulative probabilities = A: 0.000000 B: 0.499884 C: 0.499884 D: 0.499884 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 4 -8 B 8 0 -2 2 0 C 6 2 0 2 -2 D -4 -2 -2 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499884 C: 0.000000 D: 0.000000 E: 0.500116 Sum of squares = 0.500000026762 Cumulative probabilities = A: 0.000000 B: 0.499884 C: 0.499884 D: 0.499884 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 4 -8 B 8 0 -2 2 0 C 6 2 0 2 -2 D -4 -2 -2 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499884 C: 0.000000 D: 0.000000 E: 0.500116 Sum of squares = 0.500000026762 Cumulative probabilities = A: 0.000000 B: 0.499884 C: 0.499884 D: 0.499884 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5987: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) E A D B C (7) C D A B E (7) B C D A E (7) A D E C B (7) C D A E B (6) B C E D A (6) E B A D C (5) E A D C B (5) D A C E B (5) B E C A D (4) B E A D C (3) B E A C D (3) D C A E B (2) D A E C B (2) C D B A E (2) C B E D A (2) A E D C B (2) A D C E B (2) D C A B E (1) D A E B C (1) D A C B E (1) C E A B D (1) C B E A D (1) C A D E B (1) B E C D A (1) B D A E C (1) B C E A D (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 2 -10 -14 20 B -2 0 -18 -4 6 C 10 18 0 10 16 D 14 4 -10 0 18 E -20 -6 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -14 20 B -2 0 -18 -4 6 C 10 18 0 10 16 D 14 4 -10 0 18 E -20 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=27 E=17 D=12 A=12 so D is eliminated. Round 2 votes counts: C=35 B=27 A=21 E=17 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:213 A:199 B:191 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -14 20 B -2 0 -18 -4 6 C 10 18 0 10 16 D 14 4 -10 0 18 E -20 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -14 20 B -2 0 -18 -4 6 C 10 18 0 10 16 D 14 4 -10 0 18 E -20 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -14 20 B -2 0 -18 -4 6 C 10 18 0 10 16 D 14 4 -10 0 18 E -20 -6 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5988: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) C A E B D (7) B D A E C (7) D E C B A (6) E C A B D (5) D E B C A (4) D C E B A (4) C E D A B (4) B D E A C (4) A B C E D (4) E D C B A (3) E C D A B (3) D B A E C (3) D B A C E (3) C E A D B (3) B A D E C (3) A B E C D (3) E C A D B (2) D B C A E (2) B A E C D (2) B A D C E (2) A C E B D (2) A C B E D (2) E D C A B (1) E D B C A (1) D C B A E (1) D B C E A (1) C E A B D (1) C A E D B (1) C A B E D (1) B D A C E (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 0 -20 -8 B 18 0 6 -8 4 C 0 -6 0 -16 -20 D 20 8 16 0 8 E 8 -4 20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 0 -20 -8 B 18 0 6 -8 4 C 0 -6 0 -16 -20 D 20 8 16 0 8 E 8 -4 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=19 C=17 E=15 A=13 so A is eliminated. Round 2 votes counts: D=36 B=27 C=21 E=16 so E is eliminated. Round 3 votes counts: D=41 C=32 B=27 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:210 E:208 C:179 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 0 -20 -8 B 18 0 6 -8 4 C 0 -6 0 -16 -20 D 20 8 16 0 8 E 8 -4 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -20 -8 B 18 0 6 -8 4 C 0 -6 0 -16 -20 D 20 8 16 0 8 E 8 -4 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -20 -8 B 18 0 6 -8 4 C 0 -6 0 -16 -20 D 20 8 16 0 8 E 8 -4 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5989: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D B E A C (8) B D C A E (8) C A B E D (6) B C D A E (6) E A D C B (5) C A E B D (5) B D C E A (5) A E C D B (5) D E A C B (4) C B A E D (4) D E B A C (3) C A E D B (3) B D E C A (3) A C E D B (3) A C E B D (3) E D A C B (2) D E A B C (2) D B E C A (2) C A B D E (2) B C A E D (2) E D A B C (1) E A D B C (1) E A C B D (1) D C A E B (1) D B C A E (1) C B D A E (1) C A D E B (1) B E D A C (1) B D E A C (1) B C A D E (1) A E C B D (1) Total count = 100 A B C D E A 0 8 -2 2 6 B -8 0 -10 0 2 C 2 10 0 4 4 D -2 0 -4 0 -2 E -6 -2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 2 6 B -8 0 -10 0 2 C 2 10 0 4 4 D -2 0 -4 0 -2 E -6 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=22 D=21 E=18 A=12 so A is eliminated. Round 2 votes counts: C=28 B=27 E=24 D=21 so D is eliminated. Round 3 votes counts: B=38 E=33 C=29 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:210 A:207 D:196 E:195 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 2 6 B -8 0 -10 0 2 C 2 10 0 4 4 D -2 0 -4 0 -2 E -6 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 2 6 B -8 0 -10 0 2 C 2 10 0 4 4 D -2 0 -4 0 -2 E -6 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 2 6 B -8 0 -10 0 2 C 2 10 0 4 4 D -2 0 -4 0 -2 E -6 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5990: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (8) E A B C D (5) D C E B A (5) D C B A E (5) B D E C A (5) E B D C A (4) C A D B E (4) B E D C A (4) A E C B D (4) A E B C D (4) A C D B E (4) E B D A C (3) D B C E A (3) C A D E B (3) B E A D C (3) A C E D B (3) A C D E B (3) D C B E A (2) D B E C A (2) B E D A C (2) B D C E A (2) A C B D E (2) E A C D B (1) E A C B D (1) E A B D C (1) C E A D B (1) C D B A E (1) C D A E B (1) B D C A E (1) B D A C E (1) B A D C E (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -2 2 -6 B 6 0 0 6 2 C 2 0 0 -6 2 D -2 -6 6 0 6 E 6 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.612890 C: 0.387110 D: 0.000000 E: 0.000000 Sum of squares = 0.525488303176 Cumulative probabilities = A: 0.000000 B: 0.612890 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 2 -6 B 6 0 0 6 2 C 2 0 0 -6 2 D -2 -6 6 0 6 E 6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500124 C: 0.499876 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003074 Cumulative probabilities = A: 0.000000 B: 0.500124 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=22 B=19 C=18 D=17 so D is eliminated. Round 2 votes counts: C=30 E=24 B=24 A=22 so A is eliminated. Round 3 votes counts: C=43 E=33 B=24 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:207 D:202 C:199 E:198 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 2 -6 B 6 0 0 6 2 C 2 0 0 -6 2 D -2 -6 6 0 6 E 6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500124 C: 0.499876 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003074 Cumulative probabilities = A: 0.000000 B: 0.500124 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 -6 B 6 0 0 6 2 C 2 0 0 -6 2 D -2 -6 6 0 6 E 6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500124 C: 0.499876 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003074 Cumulative probabilities = A: 0.000000 B: 0.500124 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 -6 B 6 0 0 6 2 C 2 0 0 -6 2 D -2 -6 6 0 6 E 6 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500124 C: 0.499876 D: 0.000000 E: 0.000000 Sum of squares = 0.50000003074 Cumulative probabilities = A: 0.000000 B: 0.500124 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5991: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (12) D C B E A (10) A E B C D (9) E B A C D (8) A E B D C (5) D C B A E (4) D A C E B (4) A D E C B (4) A C B E D (4) A C E B D (3) E B D C A (2) E B D A C (2) E B A D C (2) E A B D C (2) E A B C D (2) D E B C A (2) D A E C B (2) C D B A E (2) B E C D A (2) A E D B C (2) E B C A D (1) D C A E B (1) D B C E A (1) D A C B E (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D E A (1) C B D A E (1) C A D B E (1) C A B D E (1) B E D C A (1) B C E A D (1) B C D E A (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 14 6 -4 20 B -14 0 -14 -2 -6 C -6 14 0 -16 8 D 4 2 16 0 6 E -20 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -4 20 B -14 0 -14 -2 -6 C -6 14 0 -16 8 D 4 2 16 0 6 E -20 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=30 E=19 C=9 B=5 so B is eliminated. Round 2 votes counts: D=37 A=30 E=22 C=11 so C is eliminated. Round 3 votes counts: D=44 A=32 E=24 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:214 C:200 E:186 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -4 20 B -14 0 -14 -2 -6 C -6 14 0 -16 8 D 4 2 16 0 6 E -20 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -4 20 B -14 0 -14 -2 -6 C -6 14 0 -16 8 D 4 2 16 0 6 E -20 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -4 20 B -14 0 -14 -2 -6 C -6 14 0 -16 8 D 4 2 16 0 6 E -20 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 5992: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) E A C B D (7) C E D B A (6) E A B C D (5) D B C A E (5) A E B C D (5) A B D C E (5) E C D B A (4) B D A C E (4) A E B D C (4) A B E D C (4) E C B A D (3) D B A C E (3) D A B C E (3) C D B E A (3) A B D E C (3) E C B D A (2) E B C A D (2) D B C E A (2) C D E B A (2) B D C A E (2) B A E D C (2) A D B C E (2) E C D A B (1) D C B A E (1) D A C B E (1) C E B D A (1) C B E D A (1) B E A C D (1) B D C E A (1) B C E D A (1) B C D E A (1) B A D C E (1) A E D C B (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 8 -4 -2 B 12 0 16 10 10 C -8 -16 0 -10 8 D 4 -10 10 0 0 E 2 -10 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 8 -4 -2 B 12 0 16 10 10 C -8 -16 0 -10 8 D 4 -10 10 0 0 E 2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=24 D=23 C=13 B=13 so C is eliminated. Round 2 votes counts: E=31 D=28 A=27 B=14 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:224 D:202 A:195 E:192 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 8 -4 -2 B 12 0 16 10 10 C -8 -16 0 -10 8 D 4 -10 10 0 0 E 2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 8 -4 -2 B 12 0 16 10 10 C -8 -16 0 -10 8 D 4 -10 10 0 0 E 2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 8 -4 -2 B 12 0 16 10 10 C -8 -16 0 -10 8 D 4 -10 10 0 0 E 2 -10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5993: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (11) C D B E A (10) A E D C B (6) A B E C D (6) E D C A B (5) B C D A E (5) A E B D C (5) A B C D E (5) E D C B A (4) E D A C B (4) D C E B A (4) D E C B A (2) D E A C B (2) D C A B E (2) C D E B A (2) C D B A E (2) B C D E A (2) A E D B C (2) E D B C A (1) E C D B A (1) E B C D A (1) E B A C D (1) E A D C B (1) E A D B C (1) E A B D C (1) C B D E A (1) C B D A E (1) B E C D A (1) B C E D A (1) B C A D E (1) B A E C D (1) A D E C B (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D B E (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 -2 10 B 4 0 -2 -8 10 C -8 2 0 8 4 D 2 8 -8 0 12 E -10 -10 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.111111 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.555556 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -2 10 B 4 0 -2 -8 10 C -8 2 0 8 4 D 2 8 -8 0 12 E -10 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.111111 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407414 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.555556 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=22 E=20 C=16 D=10 so D is eliminated. Round 2 votes counts: A=32 E=24 C=22 B=22 so C is eliminated. Round 3 votes counts: B=36 A=34 E=30 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:207 A:206 C:203 B:202 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 8 -2 10 B 4 0 -2 -8 10 C -8 2 0 8 4 D 2 8 -8 0 12 E -10 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.111111 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407414 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.555556 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -2 10 B 4 0 -2 -8 10 C -8 2 0 8 4 D 2 8 -8 0 12 E -10 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.111111 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407414 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.555556 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -2 10 B 4 0 -2 -8 10 C -8 2 0 8 4 D 2 8 -8 0 12 E -10 -10 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.111111 D: 0.444444 E: 0.000000 Sum of squares = 0.407407407414 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.555556 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5994: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) E A D B C (7) D B A C E (5) C E A B D (5) C B A D E (5) B D C A E (5) C B D A E (4) E C A D B (3) E A C D B (3) D E B A C (3) D B E A C (3) D B C E A (3) D B A E C (3) C A E B D (3) A C E B D (3) E C A B D (2) E A C B D (2) C D B E A (2) C B A E D (2) C A B E D (2) B A C D E (2) A E C B D (2) A C B E D (2) E D B A C (1) E D A C B (1) E C D A B (1) E A D C B (1) D E A B C (1) D C B E A (1) D B E C A (1) D B C A E (1) D A B E C (1) C E B D A (1) C E B A D (1) C D E B A (1) C B D E A (1) C A B D E (1) B D A C E (1) B C A D E (1) B A D C E (1) A D E B C (1) A D B E C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 8 6 -2 B -4 0 2 -2 2 C -8 -2 0 2 10 D -6 2 -2 0 0 E 2 -2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.419999999975 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 4 8 6 -2 B -4 0 2 -2 2 C -8 -2 0 2 10 D -6 2 -2 0 0 E 2 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.419999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 D=22 A=12 B=10 so B is eliminated. Round 2 votes counts: C=29 E=28 D=28 A=15 so A is eliminated. Round 3 votes counts: C=37 D=32 E=31 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:208 C:201 B:199 D:197 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 -2 B -4 0 2 -2 2 C -8 -2 0 2 10 D -6 2 -2 0 0 E 2 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.419999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 -2 B -4 0 2 -2 2 C -8 -2 0 2 10 D -6 2 -2 0 0 E 2 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.419999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 -2 B -4 0 2 -2 2 C -8 -2 0 2 10 D -6 2 -2 0 0 E 2 -2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.419999999973 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5995: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) D E C B A (8) D C E B A (7) C A B D E (7) E B A D C (6) B A E D C (5) A B E C D (5) A B C E D (5) E C D B A (4) D E B A C (4) C D E A B (4) E D B A C (3) E A B C D (3) C A B E D (3) B A D E C (3) A B C D E (3) E C D A B (2) D C A B E (2) C E A B D (2) C A D B E (2) E D C B A (1) E C A B D (1) E B D A C (1) E B A C D (1) D C B A E (1) D B A E C (1) C E D A B (1) C A E B D (1) B E A D C (1) A E B C D (1) Total count = 100 A B C D E A 0 8 -16 -2 2 B -8 0 -16 -4 0 C 16 16 0 14 0 D 2 4 -14 0 8 E -2 0 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.552462 D: 0.000000 E: 0.447538 Sum of squares = 0.505504454512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.552462 D: 0.552462 E: 1.000000 A B C D E A 0 8 -16 -2 2 B -8 0 -16 -4 0 C 16 16 0 14 0 D 2 4 -14 0 8 E -2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=23 E=22 A=14 B=9 so B is eliminated. Round 2 votes counts: C=32 E=23 D=23 A=22 so A is eliminated. Round 3 votes counts: C=40 E=34 D=26 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:200 A:196 E:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 -2 2 B -8 0 -16 -4 0 C 16 16 0 14 0 D 2 4 -14 0 8 E -2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 -2 2 B -8 0 -16 -4 0 C 16 16 0 14 0 D 2 4 -14 0 8 E -2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 -2 2 B -8 0 -16 -4 0 C 16 16 0 14 0 D 2 4 -14 0 8 E -2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5996: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E A C D B (5) D E B A C (5) B D A E C (5) E D C A B (4) E D A B C (4) E A D C B (4) E A D B C (4) C E A B D (4) C A E B D (4) C A B E D (4) B D C A E (4) B A C D E (4) E C A D B (3) D B E A C (3) B D A C E (3) A E B D C (3) E D A C B (2) D C B E A (2) D B C E A (2) D B A E C (2) C E D B A (2) A E D B C (2) A E C B D (2) A C E B D (2) A B E C D (2) E D C B A (1) E D B A C (1) D E A B C (1) C E B A D (1) C E A D B (1) C B A D E (1) B C D A E (1) B A D E C (1) B A D C E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 18 2 2 B -8 0 2 4 -10 C -18 -2 0 -12 -10 D -2 -4 12 0 -12 E -2 10 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 2 2 B -8 0 2 4 -10 C -18 -2 0 -12 -10 D -2 -4 12 0 -12 E -2 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997029 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=24 B=19 D=15 A=14 so A is eliminated. Round 2 votes counts: E=35 C=26 B=24 D=15 so D is eliminated. Round 3 votes counts: E=41 B=31 C=28 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:215 E:215 D:197 B:194 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 2 2 B -8 0 2 4 -10 C -18 -2 0 -12 -10 D -2 -4 12 0 -12 E -2 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997029 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 2 2 B -8 0 2 4 -10 C -18 -2 0 -12 -10 D -2 -4 12 0 -12 E -2 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997029 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 2 2 B -8 0 2 4 -10 C -18 -2 0 -12 -10 D -2 -4 12 0 -12 E -2 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997029 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 5997: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B A D E C (6) A B C E D (6) E C A D B (4) D C E B A (4) C E D A B (4) C D B E A (4) B A D C E (4) B A C D E (4) E D C B A (3) E C D A B (3) C D E B A (3) C D E A B (3) C A E D B (3) B A E D C (3) A B E D C (3) E D A C B (2) D E C B A (2) D E B C A (2) D C B E A (2) D B C E A (2) C A B D E (2) B D E C A (2) A E C D B (2) A E C B D (2) A E B C D (2) A C E D B (2) E D B C A (1) E A D B C (1) D B E C A (1) C E A D B (1) C D A E B (1) C D A B E (1) C B D A E (1) B E A D C (1) B D E A C (1) B D C A E (1) B C D A E (1) B C A D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -18 -2 -6 B -2 0 -14 -16 -6 C 18 14 0 4 2 D 2 16 -4 0 -4 E 6 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 -2 -6 B -2 0 -14 -16 -6 C 18 14 0 4 2 D 2 16 -4 0 -4 E 6 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977465 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 E=21 A=19 D=13 so D is eliminated. Round 2 votes counts: C=29 B=27 E=25 A=19 so A is eliminated. Round 3 votes counts: B=37 C=32 E=31 so E is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:207 D:205 A:188 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -18 -2 -6 B -2 0 -14 -16 -6 C 18 14 0 4 2 D 2 16 -4 0 -4 E 6 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977465 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -2 -6 B -2 0 -14 -16 -6 C 18 14 0 4 2 D 2 16 -4 0 -4 E 6 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977465 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -2 -6 B -2 0 -14 -16 -6 C 18 14 0 4 2 D 2 16 -4 0 -4 E 6 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977465 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 5998: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) C E B D A (7) E C B D A (6) A D C B E (6) A D B C E (6) B D A E C (5) E B C D A (4) B E D A C (4) D B C A E (3) D A C B E (3) D A B C E (3) C E A B D (3) B D E C A (3) B D E A C (3) A D B E C (3) E B C A D (2) E B A C D (2) D B A E C (2) D B A C E (2) D A B E C (2) C E D B A (2) C E B A D (2) C E A D B (2) C A D E B (2) B E D C A (2) A E B D C (2) A C D E B (2) E C A B D (1) E A B C D (1) D B E C A (1) C E D A B (1) C D B E A (1) C D A E B (1) C A E D B (1) B A D E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -2 -10 -8 B 18 0 4 12 2 C 2 -4 0 -4 -4 D 10 -12 4 0 0 E 8 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -2 -10 -8 B 18 0 4 12 2 C 2 -4 0 -4 -4 D 10 -12 4 0 0 E 8 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985202 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 C=22 A=21 B=18 D=16 so D is eliminated. Round 2 votes counts: A=29 B=26 E=23 C=22 so C is eliminated. Round 3 votes counts: E=40 A=33 B=27 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:218 E:205 D:201 C:195 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -2 -10 -8 B 18 0 4 12 2 C 2 -4 0 -4 -4 D 10 -12 4 0 0 E 8 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985202 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 -10 -8 B 18 0 4 12 2 C 2 -4 0 -4 -4 D 10 -12 4 0 0 E 8 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985202 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 -10 -8 B 18 0 4 12 2 C 2 -4 0 -4 -4 D 10 -12 4 0 0 E 8 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985202 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 5999: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) A B D C E (9) E D C B A (8) E C D B A (7) D E C A B (7) D C E A B (6) A D B C E (6) A B C D E (6) E B C A D (5) E C B D A (4) D A B C E (4) D C A E B (3) D A C E B (3) D A C B E (3) B E C A D (3) B A E C D (3) D A E B C (2) B C A E D (2) E C B A D (1) D A E C B (1) D A B E C (1) C E D B A (1) B E A C D (1) B A C D E (1) A D B E C (1) A C B D E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 6 0 14 B -10 0 10 -6 4 C -6 -10 0 -10 10 D 0 6 10 0 10 E -14 -4 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.261551 B: 0.000000 C: 0.000000 D: 0.738449 E: 0.000000 Sum of squares = 0.613715512786 Cumulative probabilities = A: 0.261551 B: 0.261551 C: 0.261551 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 0 14 B -10 0 10 -6 4 C -6 -10 0 -10 10 D 0 6 10 0 10 E -14 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=25 A=25 B=19 C=1 so C is eliminated. Round 2 votes counts: D=30 E=26 A=25 B=19 so B is eliminated. Round 3 votes counts: A=40 E=30 D=30 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:213 B:199 C:192 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 0 14 B -10 0 10 -6 4 C -6 -10 0 -10 10 D 0 6 10 0 10 E -14 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 0 14 B -10 0 10 -6 4 C -6 -10 0 -10 10 D 0 6 10 0 10 E -14 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 0 14 B -10 0 10 -6 4 C -6 -10 0 -10 10 D 0 6 10 0 10 E -14 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6000: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) D E A B C (6) E C A D B (5) D E B A C (5) B D A C E (5) E D B C A (4) B C A D E (4) E D A C B (3) D A E B C (3) D A B E C (3) B E C D A (3) A D C E B (3) A D B C E (3) A C B D E (3) A B C D E (3) E D C A B (2) E C B A D (2) C B E A D (2) C B A E D (2) B C E D A (2) B C E A D (2) B C D E A (2) A D E C B (2) A D C B E (2) A C D E B (2) A B D C E (2) E C D A B (1) E C B D A (1) E C A B D (1) E B D C A (1) E A C D B (1) D E A C B (1) D B E C A (1) D A E C B (1) C E A B D (1) C B A D E (1) C A E B D (1) B D E C A (1) B C A E D (1) B A C D E (1) A E D C B (1) A E C D B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 20 6 10 6 B -20 0 4 -4 2 C -6 -4 0 2 2 D -10 4 -2 0 10 E -6 -2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 10 6 B -20 0 4 -4 2 C -6 -4 0 2 2 D -10 4 -2 0 10 E -6 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=21 B=21 D=20 C=14 so C is eliminated. Round 2 votes counts: A=32 B=26 E=22 D=20 so D is eliminated. Round 3 votes counts: A=39 E=34 B=27 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:201 C:197 B:191 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 10 6 B -20 0 4 -4 2 C -6 -4 0 2 2 D -10 4 -2 0 10 E -6 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 10 6 B -20 0 4 -4 2 C -6 -4 0 2 2 D -10 4 -2 0 10 E -6 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 10 6 B -20 0 4 -4 2 C -6 -4 0 2 2 D -10 4 -2 0 10 E -6 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6001: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (12) E C B A D (11) A D B C E (8) C E B D A (6) A D B E C (6) E C A B D (5) A E C D B (5) B D C E A (4) B C D E A (3) A E D C B (3) A D E C B (3) E A C B D (2) D C B E A (2) D B C E A (2) D B C A E (2) D B A C E (2) C B E D A (2) B E C D A (2) B C E D A (2) A E C B D (2) A D C B E (2) E B C D A (1) E B A C D (1) D A C B E (1) D A B C E (1) C E D A B (1) C D B E A (1) C B D E A (1) B D C A E (1) A E D B C (1) A E B D C (1) A E B C D (1) A D E B C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -16 8 -16 B 10 0 -20 16 -14 C 16 20 0 16 -16 D -8 -16 -16 0 -18 E 16 14 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -16 8 -16 B 10 0 -20 16 -14 C 16 20 0 16 -16 D -8 -16 -16 0 -18 E 16 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=32 B=12 C=11 D=10 so D is eliminated. Round 2 votes counts: A=37 E=32 B=18 C=13 so C is eliminated. Round 3 votes counts: E=39 A=37 B=24 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:232 C:218 B:196 A:183 D:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -16 8 -16 B 10 0 -20 16 -14 C 16 20 0 16 -16 D -8 -16 -16 0 -18 E 16 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 8 -16 B 10 0 -20 16 -14 C 16 20 0 16 -16 D -8 -16 -16 0 -18 E 16 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 8 -16 B 10 0 -20 16 -14 C 16 20 0 16 -16 D -8 -16 -16 0 -18 E 16 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6002: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (23) A C D B E (18) C A D B E (6) B D C E A (5) A E C B D (5) A C B D E (5) E B D A C (3) E A D B C (3) B E C D A (3) A E D C B (3) E D B A C (2) E D A B C (2) E A B D C (2) C D B A E (2) C B D A E (2) B E D C A (2) A E C D B (2) A C D E B (2) E D B C A (1) E B A C D (1) E A C B D (1) E A B C D (1) D B C E A (1) D A C B E (1) B D E C A (1) B C A E D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 6 4 -2 B -6 0 2 12 -6 C -6 -2 0 2 -10 D -4 -12 -2 0 -14 E 2 6 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 4 -2 B -6 0 2 12 -6 C -6 -2 0 2 -10 D -4 -12 -2 0 -14 E 2 6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=37 B=12 C=10 D=2 so D is eliminated. Round 2 votes counts: E=39 A=38 B=13 C=10 so C is eliminated. Round 3 votes counts: A=44 E=39 B=17 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:207 B:201 C:192 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 4 -2 B -6 0 2 12 -6 C -6 -2 0 2 -10 D -4 -12 -2 0 -14 E 2 6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 -2 B -6 0 2 12 -6 C -6 -2 0 2 -10 D -4 -12 -2 0 -14 E 2 6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 -2 B -6 0 2 12 -6 C -6 -2 0 2 -10 D -4 -12 -2 0 -14 E 2 6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6003: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E A D B C (8) E A C B D (8) E D B C A (5) D C B A E (5) D B C A E (5) A B C D E (5) E D C B A (4) E C D B A (4) E A B C D (4) D C B E A (4) A E B C D (4) E C B D A (3) E A B D C (3) A B D C E (3) E A D C B (2) D E C B A (2) C D B E A (2) C D B A E (2) C B D E A (2) A E B D C (2) A D B E C (2) A D B C E (2) E C A B D (1) D B C E A (1) D A B E C (1) D A B C E (1) C B A E D (1) B D C A E (1) B C A D E (1) A E C B D (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -4 -2 -6 B 2 0 -2 -2 -4 C 4 2 0 -4 -10 D 2 2 4 0 0 E 6 4 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.719454 E: 0.280546 Sum of squares = 0.596320478995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.719454 E: 1.000000 A B C D E A 0 -2 -4 -2 -6 B 2 0 -2 -2 -4 C 4 2 0 -4 -10 D 2 2 4 0 0 E 6 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 A=21 D=19 C=16 B=2 so B is eliminated. Round 2 votes counts: E=42 A=21 D=20 C=17 so C is eliminated. Round 3 votes counts: E=42 D=35 A=23 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:210 D:204 B:197 C:196 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 -6 B 2 0 -2 -2 -4 C 4 2 0 -4 -10 D 2 2 4 0 0 E 6 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 -6 B 2 0 -2 -2 -4 C 4 2 0 -4 -10 D 2 2 4 0 0 E 6 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 -6 B 2 0 -2 -2 -4 C 4 2 0 -4 -10 D 2 2 4 0 0 E 6 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6004: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (13) E D A C B (10) D E B C A (8) A C B E D (8) E D B C A (7) B D E C A (7) D E B A C (6) C A B E D (5) E D B A C (4) E A C D B (3) D B E C A (3) B D C A E (3) A C E D B (3) A C E B D (3) A C B D E (3) D E A C B (2) E D A B C (1) E C B A D (1) E C A D B (1) E C A B D (1) D A E C B (1) C A B D E (1) B D C E A (1) B C D A E (1) B C A E D (1) B A C D E (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -6 -8 -10 B 12 0 12 -2 -4 C 6 -12 0 -8 -10 D 8 2 8 0 4 E 10 4 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -8 -10 B 12 0 12 -2 -4 C 6 -12 0 -8 -10 D 8 2 8 0 4 E 10 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 D=20 A=19 C=6 so C is eliminated. Round 2 votes counts: E=28 B=27 A=25 D=20 so D is eliminated. Round 3 votes counts: E=44 B=30 A=26 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:210 B:209 C:188 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -8 -10 B 12 0 12 -2 -4 C 6 -12 0 -8 -10 D 8 2 8 0 4 E 10 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -8 -10 B 12 0 12 -2 -4 C 6 -12 0 -8 -10 D 8 2 8 0 4 E 10 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -8 -10 B 12 0 12 -2 -4 C 6 -12 0 -8 -10 D 8 2 8 0 4 E 10 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6005: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) E D A B C (7) E D B A C (6) D E A B C (6) D C E A B (5) C A B D E (5) A B E D C (5) E B A D C (4) D E C A B (4) D E A C B (4) C D A B E (4) C B A E D (4) B A C E D (4) C D E B A (3) A B C E D (3) E D B C A (2) D C A E B (2) C B D A E (2) B C A E D (2) B A E C D (2) A E B D C (2) A C D B E (2) E D C B A (1) E A D B C (1) D E C B A (1) D C E B A (1) D A E B C (1) D A C E B (1) C D B E A (1) C B E D A (1) B A C D E (1) A D C B E (1) A D B E C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 6 -4 6 B -12 0 -4 -8 -2 C -6 4 0 -10 6 D 4 8 10 0 12 E -6 2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -4 6 B -12 0 -4 -8 -2 C -6 4 0 -10 6 D 4 8 10 0 12 E -6 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 E=21 A=16 B=9 so B is eliminated. Round 2 votes counts: C=31 D=25 A=23 E=21 so E is eliminated. Round 3 votes counts: D=41 C=31 A=28 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:210 C:197 E:189 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -4 6 B -12 0 -4 -8 -2 C -6 4 0 -10 6 D 4 8 10 0 12 E -6 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 6 B -12 0 -4 -8 -2 C -6 4 0 -10 6 D 4 8 10 0 12 E -6 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 6 B -12 0 -4 -8 -2 C -6 4 0 -10 6 D 4 8 10 0 12 E -6 2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6006: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (6) E B A D C (5) E A D C B (5) D C B A E (5) B E A C D (5) C D A E B (4) C B D A E (4) B C D A E (4) E A D B C (3) E A B D C (3) D C A E B (3) C D A B E (3) B D C A E (3) B C D E A (3) A E C D B (3) A C E D B (3) E D A B C (2) E A C B D (2) D C A B E (2) D B C E A (2) C D B A E (2) C A D E B (2) C A D B E (2) B E D C A (2) B E A D C (2) B D E C A (2) B C E A D (2) A C E B D (2) A C D E B (2) E D B A C (1) E B A C D (1) E A C D B (1) E A B C D (1) D E A B C (1) D A C E B (1) B E D A C (1) B E C A D (1) B C A E D (1) B C A D E (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 -8 -2 -2 B 6 0 4 4 6 C 8 -4 0 0 14 D 2 -4 0 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -2 -2 B 6 0 4 4 6 C 8 -4 0 0 14 D 2 -4 0 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=24 C=17 D=14 A=12 so A is eliminated. Round 2 votes counts: B=33 E=29 C=24 D=14 so D is eliminated. Round 3 votes counts: C=35 B=35 E=30 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:209 D:201 A:191 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -8 -2 -2 B 6 0 4 4 6 C 8 -4 0 0 14 D 2 -4 0 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -2 -2 B 6 0 4 4 6 C 8 -4 0 0 14 D 2 -4 0 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -2 -2 B 6 0 4 4 6 C 8 -4 0 0 14 D 2 -4 0 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6007: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (14) A C E B D (9) A E C B D (8) D B E C A (6) D A C E B (6) A D C E B (6) E C B A D (5) C E B A D (4) B E C D A (4) D B A C E (3) D A B E C (3) D A B C E (3) C A E B D (3) B D E C A (3) E C A B D (2) D B C E A (2) D A C B E (2) B E D C A (2) B D C E A (2) A C D E B (2) E B C A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A B E (1) D B A E C (1) D A E C B (1) C E A B D (1) B E A C D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -2 20 0 B 0 0 -4 24 -4 C 2 4 0 16 -8 D -20 -24 -16 0 -16 E 0 4 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.470197 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.529803 Sum of squares = 0.501776416881 Cumulative probabilities = A: 0.470197 B: 0.470197 C: 0.470197 D: 0.470197 E: 1.000000 A B C D E A 0 0 -2 20 0 B 0 0 -4 24 -4 C 2 4 0 16 -8 D -20 -24 -16 0 -16 E 0 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 B=26 E=10 C=8 so C is eliminated. Round 2 votes counts: A=30 D=29 B=26 E=15 so E is eliminated. Round 3 votes counts: B=37 A=34 D=29 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:214 A:209 B:208 C:207 D:162 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 20 0 B 0 0 -4 24 -4 C 2 4 0 16 -8 D -20 -24 -16 0 -16 E 0 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 20 0 B 0 0 -4 24 -4 C 2 4 0 16 -8 D -20 -24 -16 0 -16 E 0 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 20 0 B 0 0 -4 24 -4 C 2 4 0 16 -8 D -20 -24 -16 0 -16 E 0 4 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6008: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) E C B A D (6) A D E C B (5) D A B C E (4) A E D C B (4) A D E B C (4) E A D C B (3) E A C D B (3) D C B A E (3) D A C B E (3) C E B D A (3) C D B E A (3) C B E D A (3) B D C A E (3) B C E D A (3) B C D E A (3) B A D C E (3) A D B E C (3) E D C A B (2) E C A D B (2) E B C A D (2) D B C A E (2) D B A C E (2) C E D B A (2) C B D E A (2) A D B C E (2) A B D E C (2) E D A C B (1) E C D A B (1) E C A B D (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A E B (1) C E D A B (1) C D E B A (1) B C E A D (1) B C D A E (1) B A E D C (1) B A E C D (1) B A D E C (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 -10 -6 -8 B 8 0 -20 -8 -8 C 10 20 0 -2 -6 D 6 8 2 0 -2 E 8 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 -8 B 8 0 -20 -8 -8 C 10 20 0 -2 -6 D 6 8 2 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=21 B=17 D=16 C=15 so C is eliminated. Round 2 votes counts: E=37 B=22 A=21 D=20 so D is eliminated. Round 3 votes counts: E=39 B=32 A=29 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:211 D:207 B:186 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 -8 B 8 0 -20 -8 -8 C 10 20 0 -2 -6 D 6 8 2 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 -8 B 8 0 -20 -8 -8 C 10 20 0 -2 -6 D 6 8 2 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 -8 B 8 0 -20 -8 -8 C 10 20 0 -2 -6 D 6 8 2 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6009: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (5) D C A B E (5) D B C E A (5) B D C E A (5) D A E B C (4) C D A B E (4) C B E A D (4) B D E C A (4) E B A D C (3) E B A C D (3) D C A E B (3) B E D A C (3) B E C D A (3) B E C A D (3) B C D E A (3) E A B D C (2) D B A C E (2) D A E C B (2) D A C B E (2) C D B A E (2) C A E B D (2) B E D C A (2) B E A C D (2) A E D C B (2) A E C B D (2) A C E D B (2) E D A B C (1) E C B A D (1) E A D B C (1) E A C B D (1) D E B A C (1) D E A B C (1) D C B A E (1) D B E A C (1) D B C A E (1) D A C E B (1) C D A E B (1) C B D E A (1) C B A E D (1) C B A D E (1) C A E D B (1) C A D E B (1) C A B D E (1) B C E A D (1) A E D B C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -12 -16 -12 B 6 0 14 6 14 C 12 -14 0 -10 2 D 16 -6 10 0 8 E 12 -14 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -16 -12 B 6 0 14 6 14 C 12 -14 0 -10 2 D 16 -6 10 0 8 E 12 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 C=19 E=17 A=9 so A is eliminated. Round 2 votes counts: D=31 B=26 E=22 C=21 so C is eliminated. Round 3 votes counts: D=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:214 C:195 E:194 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -12 -16 -12 B 6 0 14 6 14 C 12 -14 0 -10 2 D 16 -6 10 0 8 E 12 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -16 -12 B 6 0 14 6 14 C 12 -14 0 -10 2 D 16 -6 10 0 8 E 12 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -16 -12 B 6 0 14 6 14 C 12 -14 0 -10 2 D 16 -6 10 0 8 E 12 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6010: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) A B C D E (6) E C B A D (5) D C E A B (5) D A B C E (5) C B A E D (5) B A E D C (5) A B D E C (5) A B D C E (5) D E A B C (4) D A E B C (4) E C D B A (3) D C A E B (3) C E B D A (3) B A E C D (3) E D C B A (2) E D B C A (2) D E C A B (2) D A C B E (2) D A B E C (2) C D E A B (2) C B E A D (2) B C A E D (2) B A C E D (2) A D B E C (2) E C B D A (1) E B A D C (1) E B A C D (1) D E A C B (1) C E D A B (1) C E B A D (1) C D A B E (1) C A B D E (1) B C E A D (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 0 -2 10 B -4 0 4 0 2 C 0 -4 0 -2 14 D 2 0 2 0 4 E -10 -2 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.197737 C: 0.000000 D: 0.802263 E: 0.000000 Sum of squares = 0.682725702159 Cumulative probabilities = A: 0.000000 B: 0.197737 C: 0.197737 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -2 10 B -4 0 4 0 2 C 0 -4 0 -2 14 D 2 0 2 0 4 E -10 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555579758 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 A=19 E=15 B=14 so B is eliminated. Round 2 votes counts: A=30 D=28 C=27 E=15 so E is eliminated. Round 3 votes counts: C=36 D=32 A=32 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:206 C:204 D:204 B:201 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -2 10 B -4 0 4 0 2 C 0 -4 0 -2 14 D 2 0 2 0 4 E -10 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555579758 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -2 10 B -4 0 4 0 2 C 0 -4 0 -2 14 D 2 0 2 0 4 E -10 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555579758 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -2 10 B -4 0 4 0 2 C 0 -4 0 -2 14 D 2 0 2 0 4 E -10 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555579758 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6011: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) A B D C E (9) E B A C D (8) E C D B A (7) C D A B E (7) B A E D C (7) D C A E B (6) D C A B E (5) A D C B E (5) C D E B A (4) B E A C D (4) A B E D C (4) E B C D A (3) D C E A B (3) A B D E C (3) E C B D A (2) E B A D C (2) C E D B A (2) C D A E B (2) A D B C E (2) A B C D E (2) E D C B A (1) E B C A D (1) B E A D C (1) B A E C D (1) Total count = 100 A B C D E A 0 14 -4 -2 6 B -14 0 -6 -6 0 C 4 6 0 4 12 D 2 6 -4 0 14 E -6 0 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 -2 6 B -14 0 -6 -6 0 C 4 6 0 4 12 D 2 6 -4 0 14 E -6 0 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 C=24 D=14 B=13 so B is eliminated. Round 2 votes counts: A=33 E=29 C=24 D=14 so D is eliminated. Round 3 votes counts: C=38 A=33 E=29 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:209 A:207 B:187 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 -2 6 B -14 0 -6 -6 0 C 4 6 0 4 12 D 2 6 -4 0 14 E -6 0 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -2 6 B -14 0 -6 -6 0 C 4 6 0 4 12 D 2 6 -4 0 14 E -6 0 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -2 6 B -14 0 -6 -6 0 C 4 6 0 4 12 D 2 6 -4 0 14 E -6 0 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6012: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (18) A B C E D (17) B A C D E (7) E D C A B (5) E D A C B (4) E A D B C (4) D E A B C (3) D C E B A (3) D C B A E (3) A E B C D (3) A B D C E (3) E D A B C (2) E C A B D (2) E A C B D (2) D E C A B (2) D B C A E (2) D B A C E (2) C B A E D (2) B A D C E (2) A B E C D (2) A B C D E (2) E C D A B (1) E C A D B (1) E A B C D (1) D E B A C (1) D B E C A (1) D A E B C (1) C B A D E (1) C A B E D (1) B C A D E (1) A C B E D (1) Total count = 100 A B C D E A 0 14 14 4 0 B -14 0 8 -6 -6 C -14 -8 0 -12 -6 D -4 6 12 0 4 E 0 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.778113 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.221887 Sum of squares = 0.654693891324 Cumulative probabilities = A: 0.778113 B: 0.778113 C: 0.778113 D: 0.778113 E: 1.000000 A B C D E A 0 14 14 4 0 B -14 0 8 -6 -6 C -14 -8 0 -12 -6 D -4 6 12 0 4 E 0 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500324 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499676 Sum of squares = 0.500000209306 Cumulative probabilities = A: 0.500324 B: 0.500324 C: 0.500324 D: 0.500324 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=28 E=22 B=10 C=4 so C is eliminated. Round 2 votes counts: D=36 A=29 E=22 B=13 so B is eliminated. Round 3 votes counts: A=42 D=36 E=22 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:209 E:204 B:191 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 4 0 B -14 0 8 -6 -6 C -14 -8 0 -12 -6 D -4 6 12 0 4 E 0 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500324 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499676 Sum of squares = 0.500000209306 Cumulative probabilities = A: 0.500324 B: 0.500324 C: 0.500324 D: 0.500324 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 4 0 B -14 0 8 -6 -6 C -14 -8 0 -12 -6 D -4 6 12 0 4 E 0 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500324 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499676 Sum of squares = 0.500000209306 Cumulative probabilities = A: 0.500324 B: 0.500324 C: 0.500324 D: 0.500324 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 4 0 B -14 0 8 -6 -6 C -14 -8 0 -12 -6 D -4 6 12 0 4 E 0 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500324 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499676 Sum of squares = 0.500000209306 Cumulative probabilities = A: 0.500324 B: 0.500324 C: 0.500324 D: 0.500324 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6013: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) A E C B D (9) D E A B C (5) D C A B E (5) D A C E B (5) B E A C D (5) B C E A D (5) D B C E A (4) D B C A E (4) C B A E D (4) C A E B D (4) D C B A E (3) D B E C A (3) D B E A C (3) B E C A D (3) E A C B D (2) E A B C D (2) D C A E B (2) D B A E C (2) B D E C A (2) A E C D B (2) E D B A C (1) E B C A D (1) E B A C D (1) E A D C B (1) E A B D C (1) D E B A C (1) D A E B C (1) C D B A E (1) C B E A D (1) C B D A E (1) C A D B E (1) C A B E D (1) B D E A C (1) A E D C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 10 -8 16 B -8 0 -10 -12 -2 C -10 10 0 -10 -14 D 8 12 10 0 8 E -16 2 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 -8 16 B -8 0 -10 -12 -2 C -10 10 0 -10 -14 D 8 12 10 0 8 E -16 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=48 B=16 A=14 C=13 E=9 so E is eliminated. Round 2 votes counts: D=49 A=20 B=18 C=13 so C is eliminated. Round 3 votes counts: D=50 A=26 B=24 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:213 E:196 C:188 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 10 -8 16 B -8 0 -10 -12 -2 C -10 10 0 -10 -14 D 8 12 10 0 8 E -16 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 -8 16 B -8 0 -10 -12 -2 C -10 10 0 -10 -14 D 8 12 10 0 8 E -16 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 -8 16 B -8 0 -10 -12 -2 C -10 10 0 -10 -14 D 8 12 10 0 8 E -16 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6014: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) C E D B A (11) C E D A B (7) C D E B A (6) B A D E C (6) E C B A D (5) D C A B E (5) E B A C D (4) C E B D A (4) B A E D C (4) A B D E C (4) E C A B D (3) E A B C D (3) C D E A B (3) A B E D C (3) D B C A E (2) A E B D C (2) A D C E B (2) A D B E C (2) E C B D A (1) E B C A D (1) D C A E B (1) D B A C E (1) D A C B E (1) D A B E C (1) C A E D B (1) B E D A C (1) B E A D C (1) B D A E C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 0 -16 0 B -4 0 0 -14 -8 C 0 0 0 -2 16 D 16 14 2 0 -2 E 0 8 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.800000 E: 0.100000 Sum of squares = 0.66000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.900000 E: 1.000000 A B C D E A 0 4 0 -16 0 B -4 0 0 -14 -8 C 0 0 0 -2 16 D 16 14 2 0 -2 E 0 8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.800000 E: 0.100000 Sum of squares = 0.65999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 E=17 A=14 B=13 so B is eliminated. Round 2 votes counts: C=32 D=25 A=24 E=19 so E is eliminated. Round 3 votes counts: C=42 A=32 D=26 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 C:207 E:197 A:194 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -16 0 B -4 0 0 -14 -8 C 0 0 0 -2 16 D 16 14 2 0 -2 E 0 8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.800000 E: 0.100000 Sum of squares = 0.65999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.900000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -16 0 B -4 0 0 -14 -8 C 0 0 0 -2 16 D 16 14 2 0 -2 E 0 8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.800000 E: 0.100000 Sum of squares = 0.65999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.900000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -16 0 B -4 0 0 -14 -8 C 0 0 0 -2 16 D 16 14 2 0 -2 E 0 8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.800000 E: 0.100000 Sum of squares = 0.65999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.100000 D: 0.900000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6015: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) E D C B A (6) E D A B C (5) D B A E C (5) C E A B D (5) B A D C E (5) A B D E C (5) A B C E D (5) A B C D E (5) E D C A B (4) E C D B A (3) D E C B A (3) D E B A C (3) C A B E D (3) B A C D E (3) E C A B D (2) D E A B C (2) D C E B A (2) C E D A B (2) C B A E D (2) A C B E D (2) A B E C D (2) E D A C B (1) E C A D B (1) E A C B D (1) D E B C A (1) D B C A E (1) D A E B C (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E B A (1) C D B A E (1) C B A D E (1) B D A C E (1) B C D A E (1) B C A D E (1) B A D E C (1) A E B D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 0 -2 -4 B -14 0 0 -2 -8 C 0 0 0 2 -10 D 2 2 -2 0 -12 E 4 8 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 0 -2 -4 B -14 0 0 -2 -8 C 0 0 0 2 -10 D 2 2 -2 0 -12 E 4 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=22 D=18 C=18 B=12 so B is eliminated. Round 2 votes counts: A=31 E=30 C=20 D=19 so D is eliminated. Round 3 votes counts: E=39 A=38 C=23 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:204 C:196 D:195 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 -2 -4 B -14 0 0 -2 -8 C 0 0 0 2 -10 D 2 2 -2 0 -12 E 4 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 -2 -4 B -14 0 0 -2 -8 C 0 0 0 2 -10 D 2 2 -2 0 -12 E 4 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 -2 -4 B -14 0 0 -2 -8 C 0 0 0 2 -10 D 2 2 -2 0 -12 E 4 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6016: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (12) A C B D E (10) E D B C A (6) E D C B A (5) D E B A C (5) D E A B C (5) C A B E D (5) C A B D E (5) D A C E B (4) C B A E D (4) A D C B E (4) A C D B E (4) E D C A B (3) E B D C A (3) B E C A D (3) E B D A C (2) D E A C B (2) D A E B C (2) A B D C E (2) E C D A B (1) E C B D A (1) E B C D A (1) E B C A D (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B E C (1) C E B A D (1) C A E D B (1) C A E B D (1) B E A C D (1) B C E A D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -8 12 18 B -6 0 -6 8 10 C 8 6 0 6 12 D -12 -8 -6 0 -4 E -18 -10 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 12 18 B -6 0 -6 8 10 C 8 6 0 6 12 D -12 -8 -6 0 -4 E -18 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 D=22 A=21 C=17 B=17 so C is eliminated. Round 2 votes counts: A=33 E=24 D=22 B=21 so B is eliminated. Round 3 votes counts: A=49 E=29 D=22 so D is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:216 A:214 B:203 D:185 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 12 18 B -6 0 -6 8 10 C 8 6 0 6 12 D -12 -8 -6 0 -4 E -18 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 12 18 B -6 0 -6 8 10 C 8 6 0 6 12 D -12 -8 -6 0 -4 E -18 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 12 18 B -6 0 -6 8 10 C 8 6 0 6 12 D -12 -8 -6 0 -4 E -18 -10 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6017: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) B A C E D (9) D E A C B (8) A E D C B (7) B C E D A (6) B C D E A (5) A D E B C (5) D A E C B (4) C D B E A (4) A B C E D (4) D C E B A (3) D B C E A (3) A D E C B (3) E C D A B (2) D E C A B (2) B C A D E (2) A E D B C (2) A E C B D (2) E D A C B (1) D E C B A (1) D C B E A (1) D B A E C (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B D A (1) C D E B A (1) C B E D A (1) C B D E A (1) B D C E A (1) B C E A D (1) B A D E C (1) A E C D B (1) A E B C D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 8 2 14 B 6 0 12 -4 8 C -8 -12 0 6 8 D -2 4 -6 0 -2 E -14 -8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888883 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 2 14 B 6 0 12 -4 8 C -8 -12 0 6 8 D -2 4 -6 0 -2 E -14 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889011 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=28 D=25 C=9 E=3 so E is eliminated. Round 2 votes counts: B=35 A=28 D=26 C=11 so C is eliminated. Round 3 votes counts: B=38 D=34 A=28 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:211 A:209 C:197 D:197 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 2 14 B 6 0 12 -4 8 C -8 -12 0 6 8 D -2 4 -6 0 -2 E -14 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889011 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 2 14 B 6 0 12 -4 8 C -8 -12 0 6 8 D -2 4 -6 0 -2 E -14 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889011 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 2 14 B 6 0 12 -4 8 C -8 -12 0 6 8 D -2 4 -6 0 -2 E -14 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888889011 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6018: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) E D A B C (6) D E A C B (5) C A B D E (5) B E C D A (5) A C D E B (5) C B A D E (4) B C A E D (4) A E D C B (4) A D E C B (4) E D A C B (3) E B D A C (3) D E B A C (3) B D C E A (3) B C D E A (3) E B A D C (2) D E B C A (2) D E A B C (2) C B A E D (2) C A D E B (2) C A D B E (2) B E D A C (2) B C E D A (2) B C E A D (2) B C D A E (2) E A D C B (1) D C A E B (1) D C A B E (1) D B E C A (1) D A E C B (1) C D A B E (1) C A B E D (1) B E D C A (1) B E A C D (1) B C A D E (1) A E C D B (1) A E C B D (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 10 -10 -10 B 2 0 6 -10 -14 C -10 -6 0 -8 -12 D 10 10 8 0 -2 E 10 14 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 10 -10 -10 B 2 0 6 -10 -14 C -10 -6 0 -8 -12 D 10 10 8 0 -2 E 10 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 A=18 C=17 D=16 so D is eliminated. Round 2 votes counts: E=35 B=27 C=19 A=19 so C is eliminated. Round 3 votes counts: E=35 B=33 A=32 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:213 A:194 B:192 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 10 -10 -10 B 2 0 6 -10 -14 C -10 -6 0 -8 -12 D 10 10 8 0 -2 E 10 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -10 -10 B 2 0 6 -10 -14 C -10 -6 0 -8 -12 D 10 10 8 0 -2 E 10 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -10 -10 B 2 0 6 -10 -14 C -10 -6 0 -8 -12 D 10 10 8 0 -2 E 10 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6019: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) C A D E B (10) A D C B E (9) E B C D A (8) C A D B E (8) B E D A C (7) E B D A C (6) D A C B E (6) D A B C E (4) E C B A D (3) B E C A D (3) D A C E B (2) C E A D B (2) B D A E C (2) A C D E B (2) A C D B E (2) E D B A C (1) E C D A B (1) E C B D A (1) E C A D B (1) E C A B D (1) E B D C A (1) D E A C B (1) D B A E C (1) D B A C E (1) D A E B C (1) B E D C A (1) B D E A C (1) B D A C E (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -4 8 2 B -2 0 0 -6 -6 C 4 0 0 8 -2 D -8 6 -8 0 6 E -2 6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999898 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 2 -4 8 2 B -2 0 0 -6 -6 C 4 0 0 8 -2 D -8 6 -8 0 6 E -2 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=20 D=16 B=16 A=14 so A is eliminated. Round 2 votes counts: E=34 D=26 C=24 B=16 so B is eliminated. Round 3 votes counts: E=45 D=30 C=25 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:205 A:204 E:200 D:198 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 8 2 B -2 0 0 -6 -6 C 4 0 0 8 -2 D -8 6 -8 0 6 E -2 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 8 2 B -2 0 0 -6 -6 C 4 0 0 8 -2 D -8 6 -8 0 6 E -2 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 8 2 B -2 0 0 -6 -6 C 4 0 0 8 -2 D -8 6 -8 0 6 E -2 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6020: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E C D B A (8) A B C D E (7) E D C B A (6) C E D B A (6) B A E C D (5) D E C A B (4) D E A C B (4) D C E A B (4) C B E D A (4) E D C A B (3) B C E A D (3) B A C E D (3) B A C D E (3) E D A C B (2) B E A C D (2) B C A E D (2) A D C E B (2) A D B E C (2) A D B C E (2) E B C A D (1) E B A C D (1) E A D B C (1) D A C E B (1) C E B D A (1) C D E A B (1) C D A B E (1) C B D E A (1) B E C D A (1) B E C A D (1) B C A D E (1) B A E D C (1) A D E B C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 4 6 -8 B 0 0 4 4 8 C -4 -4 0 6 8 D -6 -4 -6 0 -6 E 8 -8 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.296238 B: 0.703762 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.583037577352 Cumulative probabilities = A: 0.296238 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 6 -8 B 0 0 4 4 8 C -4 -4 0 6 8 D -6 -4 -6 0 -6 E 8 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=22 B=22 C=14 D=13 so D is eliminated. Round 2 votes counts: E=30 A=30 B=22 C=18 so C is eliminated. Round 3 votes counts: E=42 A=31 B=27 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:208 C:203 A:201 E:199 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 6 -8 B 0 0 4 4 8 C -4 -4 0 6 8 D -6 -4 -6 0 -6 E 8 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 -8 B 0 0 4 4 8 C -4 -4 0 6 8 D -6 -4 -6 0 -6 E 8 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 -8 B 0 0 4 4 8 C -4 -4 0 6 8 D -6 -4 -6 0 -6 E 8 -8 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6021: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (15) D A C B E (10) C A D B E (7) A D C B E (7) E B D C A (5) E B C A D (5) D A B C E (5) A C D B E (5) B E C D A (4) E B D A C (3) D A E C B (3) C E B A D (3) A D C E B (3) E D B A C (2) D A E B C (2) D A B E C (2) B D E A C (2) B C E A D (2) E D A B C (1) E C B A D (1) E C A B D (1) D E A B C (1) D B A E C (1) D A C E B (1) C E A B D (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D A C (1) B D A C E (1) B C E D A (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 4 2 -22 2 B -4 0 8 -4 2 C -2 -8 0 0 2 D 22 4 0 0 10 E -2 -2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.171691 D: 0.828309 E: 0.000000 Sum of squares = 0.71557352648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.171691 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -22 2 B -4 0 8 -4 2 C -2 -8 0 0 2 D 22 4 0 0 10 E -2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555701631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=25 A=16 C=14 B=12 so B is eliminated. Round 2 votes counts: E=38 D=28 C=18 A=16 so A is eliminated. Round 3 votes counts: E=38 D=38 C=24 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:201 C:196 A:193 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -22 2 B -4 0 8 -4 2 C -2 -8 0 0 2 D 22 4 0 0 10 E -2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555701631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -22 2 B -4 0 8 -4 2 C -2 -8 0 0 2 D 22 4 0 0 10 E -2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555701631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -22 2 B -4 0 8 -4 2 C -2 -8 0 0 2 D 22 4 0 0 10 E -2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 0.000000 Sum of squares = 0.555555701631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6022: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) B D E A C (8) A C D E B (8) B E D C A (7) B D E C A (6) D B E A C (4) C A E B D (4) B A D E C (4) A C B D E (4) E B D C A (3) D E B C A (3) A D B E C (3) A C E D B (3) A C B E D (3) A B D E C (3) E C D B A (2) D E B A C (2) C E D A B (2) C E B D A (2) C E A D B (2) B E C D A (2) B D A E C (2) A C D B E (2) A B D C E (2) E D C B A (1) E D B C A (1) E B C D A (1) D B A E C (1) D A B E C (1) C E A B D (1) B E D A C (1) A D E C B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 8 2 4 B 0 0 10 10 8 C -8 -10 0 -6 -12 D -2 -10 6 0 8 E -4 -8 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.811655 B: 0.188345 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.694257464048 Cumulative probabilities = A: 0.811655 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 4 B 0 0 10 10 8 C -8 -10 0 -6 -12 D -2 -10 6 0 8 E -4 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=30 C=20 D=11 E=8 so E is eliminated. Round 2 votes counts: B=34 A=31 C=22 D=13 so D is eliminated. Round 3 votes counts: B=45 A=32 C=23 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:214 A:207 D:201 E:196 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 2 4 B 0 0 10 10 8 C -8 -10 0 -6 -12 D -2 -10 6 0 8 E -4 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 4 B 0 0 10 10 8 C -8 -10 0 -6 -12 D -2 -10 6 0 8 E -4 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 4 B 0 0 10 10 8 C -8 -10 0 -6 -12 D -2 -10 6 0 8 E -4 -8 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6023: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) E B C A D (6) E B A C D (5) B E A C D (5) B A D C E (5) E C D A B (4) E C A B D (4) E B D C A (4) D A C B E (4) C A D E B (4) E D C A B (3) E C A D B (3) D C A B E (3) B D A C E (3) B A E C D (3) D B A C E (2) D A B C E (2) C D A E B (2) C A E D B (2) B E A D C (2) B D E C A (2) B A C E D (2) A C D B E (2) A C B D E (2) E D C B A (1) E C D B A (1) E C B D A (1) E C B A D (1) E B C D A (1) E A C B D (1) D E C B A (1) D C E A B (1) C E A D B (1) C D E A B (1) B E D C A (1) B D E A C (1) B A E D C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -16 2 0 B -4 0 -10 2 -18 C 16 10 0 4 -2 D -2 -2 -4 0 -6 E 0 18 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.086249 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.913751 Sum of squares = 0.842379027055 Cumulative probabilities = A: 0.086249 B: 0.086249 C: 0.086249 D: 0.086249 E: 1.000000 A B C D E A 0 4 -16 2 0 B -4 0 -10 2 -18 C 16 10 0 4 -2 D -2 -2 -4 0 -6 E 0 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469239357 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=25 D=24 C=10 A=6 so A is eliminated. Round 2 votes counts: E=35 D=25 B=25 C=15 so C is eliminated. Round 3 votes counts: E=39 D=34 B=27 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:213 A:195 D:193 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -16 2 0 B -4 0 -10 2 -18 C 16 10 0 4 -2 D -2 -2 -4 0 -6 E 0 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469239357 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 2 0 B -4 0 -10 2 -18 C 16 10 0 4 -2 D -2 -2 -4 0 -6 E 0 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469239357 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 2 0 B -4 0 -10 2 -18 C 16 10 0 4 -2 D -2 -2 -4 0 -6 E 0 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469239357 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6024: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E D B C A (9) E A C D B (7) D E B C A (7) B C A D E (7) C A B D E (6) A C B D E (6) D B E C A (5) A C B E D (5) E A D C B (4) D E B A C (4) C B A D E (3) A C E B D (3) E D C A B (2) B D C E A (2) B D C A E (2) B A C D E (2) A B C D E (2) E D B A C (1) E D A B C (1) E C A D B (1) D B E A C (1) D B C E A (1) D B A E C (1) D B A C E (1) D A E B C (1) C E A B D (1) C A E B D (1) C A B E D (1) A E C B D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 8 4 4 -12 B -8 0 -6 -16 -8 C -4 6 0 -8 -10 D -4 16 8 0 6 E 12 8 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677779 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 A B C D E A 0 8 4 4 -12 B -8 0 -6 -16 -8 C -4 6 0 -8 -10 D -4 16 8 0 6 E 12 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677743 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=21 A=19 B=13 C=12 so C is eliminated. Round 2 votes counts: E=36 A=27 D=21 B=16 so B is eliminated. Round 3 votes counts: A=39 E=36 D=25 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:212 A:202 C:192 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 4 -12 B -8 0 -6 -16 -8 C -4 6 0 -8 -10 D -4 16 8 0 6 E 12 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677743 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 4 -12 B -8 0 -6 -16 -8 C -4 6 0 -8 -10 D -4 16 8 0 6 E 12 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677743 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 4 -12 B -8 0 -6 -16 -8 C -4 6 0 -8 -10 D -4 16 8 0 6 E 12 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.181818 Sum of squares = 0.404958677743 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.272727 D: 0.818182 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6025: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A C B E D (8) D B C E A (6) A C E B D (6) D B E C A (4) C B E D A (4) B C D A E (4) A B C D E (4) E A D C B (3) E A C B D (3) D E B A C (3) D E A B C (3) C B D E A (3) B C D E A (3) A E C D B (3) A D E B C (3) A D B C E (3) E D C B A (2) E D A B C (2) E C D B A (2) E A C D B (2) C B A E D (2) B A C D E (2) A E D C B (2) A E D B C (2) E D B C A (1) E A D B C (1) D B E A C (1) D B A E C (1) D A B E C (1) C B A D E (1) C A B E D (1) B D C E A (1) B D C A E (1) B D A C E (1) B C A D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 10 -4 -6 B 4 0 16 -10 6 C -10 -16 0 -2 2 D 4 10 2 0 12 E 6 -6 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -4 -6 B 4 0 16 -10 6 C -10 -16 0 -2 2 D 4 10 2 0 12 E 6 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=28 E=16 B=13 C=11 so C is eliminated. Round 2 votes counts: A=33 D=28 B=23 E=16 so E is eliminated. Round 3 votes counts: A=42 D=35 B=23 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:208 A:198 E:193 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 10 -4 -6 B 4 0 16 -10 6 C -10 -16 0 -2 2 D 4 10 2 0 12 E 6 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -4 -6 B 4 0 16 -10 6 C -10 -16 0 -2 2 D 4 10 2 0 12 E 6 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -4 -6 B 4 0 16 -10 6 C -10 -16 0 -2 2 D 4 10 2 0 12 E 6 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6026: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) D A C E B (8) B C E D A (8) D A C B E (6) B E C A D (5) A D C E B (5) E A B C D (4) B E C D A (4) B E A C D (4) A E D C B (4) A D E C B (4) A D E B C (4) E B C A D (3) E B A C D (3) D C B A E (2) D C A E B (2) D C A B E (2) D A B C E (2) C E D A B (2) C D E A B (2) C B D E A (2) A D C B E (2) A D B C E (2) E C B D A (1) E B C D A (1) E A C D B (1) E A C B D (1) C D B A E (1) B C D E A (1) B A E D C (1) B A D C E (1) A E D B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 26 28 12 16 B -26 0 10 -2 -10 C -28 -10 0 -14 -4 D -12 2 14 0 -6 E -16 10 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 28 12 16 B -26 0 10 -2 -10 C -28 -10 0 -14 -4 D -12 2 14 0 -6 E -16 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=24 D=22 E=14 C=7 so C is eliminated. Round 2 votes counts: A=33 B=26 D=25 E=16 so E is eliminated. Round 3 votes counts: A=39 B=34 D=27 so D is eliminated. Round 4 votes counts: A=63 B=37 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:241 E:202 D:199 B:186 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 28 12 16 B -26 0 10 -2 -10 C -28 -10 0 -14 -4 D -12 2 14 0 -6 E -16 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 28 12 16 B -26 0 10 -2 -10 C -28 -10 0 -14 -4 D -12 2 14 0 -6 E -16 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 28 12 16 B -26 0 10 -2 -10 C -28 -10 0 -14 -4 D -12 2 14 0 -6 E -16 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6027: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) D A B C E (11) D B A E C (9) E C D A B (8) C E A B D (8) D C E A B (6) D E C A B (5) D E C B A (4) C E D A B (4) A B C E D (4) D B A C E (3) C A B E D (3) B A E C D (3) B A D E C (3) B A C E D (3) A B D C E (3) B A E D C (2) B A D C E (2) E D C A B (1) E C D B A (1) E C A D B (1) E B C A D (1) D E B A C (1) D C A B E (1) D B E A C (1) C D E A B (1) Total count = 100 A B C D E A 0 12 -10 -12 -6 B -12 0 -8 -14 -4 C 10 8 0 -4 -2 D 12 14 4 0 0 E 6 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.320156 E: 0.679844 Sum of squares = 0.564687863534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.320156 E: 1.000000 A B C D E A 0 12 -10 -12 -6 B -12 0 -8 -14 -4 C 10 8 0 -4 -2 D 12 14 4 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=23 C=16 B=13 A=7 so A is eliminated. Round 2 votes counts: D=41 E=23 B=20 C=16 so C is eliminated. Round 3 votes counts: D=42 E=35 B=23 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:206 E:206 A:192 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -10 -12 -6 B -12 0 -8 -14 -4 C 10 8 0 -4 -2 D 12 14 4 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 -12 -6 B -12 0 -8 -14 -4 C 10 8 0 -4 -2 D 12 14 4 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 -12 -6 B -12 0 -8 -14 -4 C 10 8 0 -4 -2 D 12 14 4 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6028: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) C B E D A (9) A D E B C (8) C B A E D (7) B C E D A (5) A E D C B (5) A D E C B (5) A D B E C (4) E D B C A (3) E D A C B (3) D E B A C (3) D E A B C (3) D B E A C (3) C A B E D (3) B D E C A (3) A C B E D (3) A C B D E (3) C E D B A (2) B D E A C (2) A C E D B (2) E D C B A (1) E C D B A (1) D E B C A (1) D A E B C (1) C E D A B (1) C E B D A (1) C A E D B (1) B D A E C (1) B C D A E (1) B C A D E (1) B A D E C (1) B A D C E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 0 -8 -2 B 12 0 6 4 18 C 0 -6 0 0 2 D 8 -4 0 0 6 E 2 -18 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -8 -2 B 12 0 6 4 18 C 0 -6 0 0 2 D 8 -4 0 0 6 E 2 -18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=25 C=24 D=11 E=8 so E is eliminated. Round 2 votes counts: A=32 C=25 B=25 D=18 so D is eliminated. Round 3 votes counts: A=39 B=35 C=26 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:205 C:198 A:189 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -8 -2 B 12 0 6 4 18 C 0 -6 0 0 2 D 8 -4 0 0 6 E 2 -18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -8 -2 B 12 0 6 4 18 C 0 -6 0 0 2 D 8 -4 0 0 6 E 2 -18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -8 -2 B 12 0 6 4 18 C 0 -6 0 0 2 D 8 -4 0 0 6 E 2 -18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6029: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) D C E B A (9) C D A E B (8) C D E B A (7) C A E B D (6) A B E D C (6) C A D E B (5) B E A D C (5) E B D A C (4) B A E D C (4) D E B C A (3) D B E C A (3) D B E A C (3) C D A B E (3) C A B E D (3) B E D A C (3) A C B E D (3) D E B A C (2) A E B C D (2) A C E B D (2) E D B A C (1) E B A D C (1) D C B E A (1) D C B A E (1) C E D A B (1) C E A D B (1) C D B E A (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -4 0 10 B -4 0 -4 0 -6 C 4 4 0 6 4 D 0 0 -6 0 -6 E -10 6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 0 10 B -4 0 -4 0 -6 C 4 4 0 6 4 D 0 0 -6 0 -6 E -10 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=25 D=22 B=12 E=6 so E is eliminated. Round 2 votes counts: C=35 A=25 D=23 B=17 so B is eliminated. Round 3 votes counts: C=35 A=35 D=30 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:205 E:199 D:194 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 0 10 B -4 0 -4 0 -6 C 4 4 0 6 4 D 0 0 -6 0 -6 E -10 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 10 B -4 0 -4 0 -6 C 4 4 0 6 4 D 0 0 -6 0 -6 E -10 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 10 B -4 0 -4 0 -6 C 4 4 0 6 4 D 0 0 -6 0 -6 E -10 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6030: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) A B C E D (7) E C A B D (6) B A D E C (6) A C B E D (6) D C E A B (5) D B A E C (5) C E D A B (5) D B E C A (4) D B A C E (4) B A E C D (4) A B D C E (4) D E B C A (3) C E A B D (3) C A E B D (3) A B E C D (3) E C D B A (2) E C D A B (2) C E A D B (2) B D A E C (2) B A D C E (2) A C E B D (2) E D C B A (1) E C B D A (1) E B C D A (1) D C E B A (1) D C A E B (1) D B C A E (1) D A C B E (1) C D E A B (1) C D A E B (1) C A E D B (1) B E A C D (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -2 4 10 B -10 0 -6 4 2 C 2 6 0 6 2 D -4 -4 -6 0 -2 E -10 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 4 10 B -10 0 -6 4 2 C 2 6 0 6 2 D -4 -4 -6 0 -2 E -10 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=24 C=16 B=15 E=13 so E is eliminated. Round 2 votes counts: D=33 C=27 A=24 B=16 so B is eliminated. Round 3 votes counts: A=37 D=35 C=28 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:208 B:195 E:194 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 4 10 B -10 0 -6 4 2 C 2 6 0 6 2 D -4 -4 -6 0 -2 E -10 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 4 10 B -10 0 -6 4 2 C 2 6 0 6 2 D -4 -4 -6 0 -2 E -10 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 4 10 B -10 0 -6 4 2 C 2 6 0 6 2 D -4 -4 -6 0 -2 E -10 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6031: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (19) A C E B D (13) C E B A D (8) C E B D A (7) E C B D A (5) A D B E C (5) D B E A C (4) D A B E C (4) B E C D A (4) D B A E C (3) C E A B D (3) B E D C A (3) A D C E B (3) A D B C E (3) C A E B D (2) A D C B E (2) A C E D B (2) A C D E B (2) A B C E D (2) E B C D A (1) C E A D B (1) C A B E D (1) B D E C A (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -10 -2 -12 B 10 0 2 4 6 C 10 -2 0 2 0 D 2 -4 -2 0 -4 E 12 -6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -2 -12 B 10 0 2 4 6 C 10 -2 0 2 0 D 2 -4 -2 0 -4 E 12 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=30 C=22 B=8 E=6 so E is eliminated. Round 2 votes counts: A=34 D=30 C=27 B=9 so B is eliminated. Round 3 votes counts: D=34 A=34 C=32 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:211 C:205 E:205 D:196 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 -2 -12 B 10 0 2 4 6 C 10 -2 0 2 0 D 2 -4 -2 0 -4 E 12 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -2 -12 B 10 0 2 4 6 C 10 -2 0 2 0 D 2 -4 -2 0 -4 E 12 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -2 -12 B 10 0 2 4 6 C 10 -2 0 2 0 D 2 -4 -2 0 -4 E 12 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6032: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (18) C B D E A (15) C B A E D (8) D E A B C (5) D B C E A (4) C D B E A (4) C B E A D (4) A E B C D (4) D C B E A (3) D A E B C (3) C D B A E (3) B C E A D (3) B C D E A (3) A E B D C (3) D E A C B (2) D A E C B (2) C D A E B (2) B D C E A (2) A E D C B (2) A E C B D (2) E A D B C (1) D C A E B (1) D B E A C (1) C B D A E (1) C A E B D (1) B C A E D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -10 -2 6 B 4 0 -4 -4 4 C 10 4 0 4 12 D 2 4 -4 0 4 E -6 -4 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -2 6 B 4 0 -4 -4 4 C 10 4 0 4 12 D 2 4 -4 0 4 E -6 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=31 D=21 B=9 E=1 so E is eliminated. Round 2 votes counts: C=38 A=32 D=21 B=9 so B is eliminated. Round 3 votes counts: C=45 A=32 D=23 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:203 B:200 A:195 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -2 6 B 4 0 -4 -4 4 C 10 4 0 4 12 D 2 4 -4 0 4 E -6 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -2 6 B 4 0 -4 -4 4 C 10 4 0 4 12 D 2 4 -4 0 4 E -6 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -2 6 B 4 0 -4 -4 4 C 10 4 0 4 12 D 2 4 -4 0 4 E -6 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999402 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6033: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) A E B D C (9) B A C E D (8) C D E B A (6) C B A D E (6) B A E D C (5) A E D B C (5) E D A B C (4) D C E B A (4) D C E A B (4) B C A E D (4) D E A B C (3) C D B E A (3) C B D E A (3) E A D B C (2) D E C A B (2) D E A C B (2) C B D A E (2) C A B D E (2) B C E D A (2) B A E C D (2) A B E C D (2) A B C E D (2) E D B A C (1) E B D A C (1) D E C B A (1) C A D B E (1) B E D A C (1) B C E A D (1) Total count = 100 A B C D E A 0 0 18 22 20 B 0 0 28 24 12 C -18 -28 0 -12 -4 D -22 -24 12 0 -22 E -20 -12 4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.494337 B: 0.505663 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500064139092 Cumulative probabilities = A: 0.494337 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 18 22 20 B 0 0 28 24 12 C -18 -28 0 -12 -4 D -22 -24 12 0 -22 E -20 -12 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=23 B=23 D=16 E=8 so E is eliminated. Round 2 votes counts: A=32 B=24 C=23 D=21 so D is eliminated. Round 3 votes counts: A=41 C=34 B=25 so B is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:232 A:230 E:197 D:172 C:169 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 18 22 20 B 0 0 28 24 12 C -18 -28 0 -12 -4 D -22 -24 12 0 -22 E -20 -12 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 22 20 B 0 0 28 24 12 C -18 -28 0 -12 -4 D -22 -24 12 0 -22 E -20 -12 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 22 20 B 0 0 28 24 12 C -18 -28 0 -12 -4 D -22 -24 12 0 -22 E -20 -12 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6034: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (13) C D B E A (9) A E B D C (9) C B D E A (8) B C D E A (8) A E D B C (7) D E C A B (5) B C A D E (5) D C E B A (4) D E A C B (3) B E D A C (3) E D B C A (2) E D A C B (2) D C E A B (2) C D E A B (2) B C A E D (2) B A E D C (2) A E B C D (2) E D B A C (1) E A D C B (1) E A D B C (1) C D B A E (1) C B D A E (1) B E D C A (1) B A E C D (1) B A C E D (1) B A C D E (1) A E C B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 0 -4 -4 B 0 0 -4 -6 -10 C 0 4 0 -12 -10 D 4 6 12 0 -2 E 4 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 -4 -4 B 0 0 -4 -6 -10 C 0 4 0 -12 -10 D 4 6 12 0 -2 E 4 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=24 C=21 D=14 E=7 so E is eliminated. Round 2 votes counts: A=36 B=24 C=21 D=19 so D is eliminated. Round 3 votes counts: A=41 C=32 B=27 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:213 D:210 A:196 C:191 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -4 -4 B 0 0 -4 -6 -10 C 0 4 0 -12 -10 D 4 6 12 0 -2 E 4 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 -4 B 0 0 -4 -6 -10 C 0 4 0 -12 -10 D 4 6 12 0 -2 E 4 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 -4 B 0 0 -4 -6 -10 C 0 4 0 -12 -10 D 4 6 12 0 -2 E 4 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6035: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (7) C A B E D (7) D E B A C (6) A C B E D (6) E D B A C (5) C B A D E (4) B D E A C (4) D E B C A (3) C E A D B (3) C B D E A (3) C A E B D (3) C A B D E (3) B D A E C (3) A C E B D (3) A B C E D (3) E D C B A (2) E D B C A (2) E A D B C (2) B D E C A (2) B D C E A (2) B C D A E (2) B C A D E (2) B A D E C (2) A E D B C (2) A E B D C (2) A C B D E (2) E D A C B (1) E D A B C (1) E C D A B (1) D B E C A (1) D B E A C (1) D B C E A (1) C E D B A (1) C E D A B (1) C D E B A (1) C B D A E (1) B D A C E (1) B A D C E (1) B A C D E (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 10 14 B 2 0 0 16 6 C 4 0 0 12 18 D -10 -16 -12 0 -6 E -14 -6 -18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.552855 C: 0.447145 D: 0.000000 E: 0.000000 Sum of squares = 0.505587261106 Cumulative probabilities = A: 0.000000 B: 0.552855 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 10 14 B 2 0 0 16 6 C 4 0 0 12 18 D -10 -16 -12 0 -6 E -14 -6 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999579 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=20 A=20 E=14 D=12 so D is eliminated. Round 2 votes counts: C=34 E=23 B=23 A=20 so A is eliminated. Round 3 votes counts: C=45 E=28 B=27 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:212 A:209 E:184 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 10 14 B 2 0 0 16 6 C 4 0 0 12 18 D -10 -16 -12 0 -6 E -14 -6 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999579 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 14 B 2 0 0 16 6 C 4 0 0 12 18 D -10 -16 -12 0 -6 E -14 -6 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999579 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 14 B 2 0 0 16 6 C 4 0 0 12 18 D -10 -16 -12 0 -6 E -14 -6 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999579 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6036: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) B C A E D (8) E D C A B (7) D E A C B (6) D A E C B (6) B C E A D (6) C E A D B (3) B A D C E (3) A D C B E (3) E D A C B (2) E C D B A (2) E C D A B (2) E C B A D (2) D B A C E (2) D A C E B (2) D A C B E (2) D A B C E (2) C E B A D (2) C E A B D (2) B E D C A (2) B E C A D (2) B D A E C (2) B C A D E (2) A D C E B (2) A C B D E (2) E D B C A (1) E D B A C (1) E C A D B (1) E B C D A (1) D B A E C (1) C B E A D (1) C B A E D (1) C A E B D (1) B E D A C (1) B E C D A (1) B D E A C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 6 12 8 B 10 0 2 8 14 C -6 -2 0 4 18 D -12 -8 -4 0 2 E -8 -14 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 12 8 B 10 0 2 8 14 C -6 -2 0 4 18 D -12 -8 -4 0 2 E -8 -14 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=21 E=19 C=10 A=9 so A is eliminated. Round 2 votes counts: B=42 D=27 E=19 C=12 so C is eliminated. Round 3 votes counts: B=46 E=27 D=27 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:208 C:207 D:189 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 12 8 B 10 0 2 8 14 C -6 -2 0 4 18 D -12 -8 -4 0 2 E -8 -14 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 12 8 B 10 0 2 8 14 C -6 -2 0 4 18 D -12 -8 -4 0 2 E -8 -14 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 12 8 B 10 0 2 8 14 C -6 -2 0 4 18 D -12 -8 -4 0 2 E -8 -14 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6037: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) A E D C B (7) E A D C B (6) E A C D B (6) E A B C D (6) C D B A E (6) B C D E A (6) E B A D C (5) D C B A E (5) A D C E B (5) D C A E B (4) A D E C B (4) D C A B E (3) A E C D B (3) E B A C D (2) E A D B C (2) C D A E B (2) C D A B E (2) C B D E A (2) B D C A E (2) B C E D A (2) D A C E B (1) D A C B E (1) C D B E A (1) B E D C A (1) B E C D A (1) B E A D C (1) B E A C D (1) B D A C E (1) B C D A E (1) B A E D C (1) B A D C E (1) A E B D C (1) A D C B E (1) Total count = 100 A B C D E A 0 22 24 18 2 B -22 0 -18 -18 -22 C -24 18 0 -18 -8 D -18 18 18 0 -4 E -2 22 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 24 18 2 B -22 0 -18 -18 -22 C -24 18 0 -18 -8 D -18 18 18 0 -4 E -2 22 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999926693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=21 B=18 D=14 C=13 so C is eliminated. Round 2 votes counts: E=34 D=25 A=21 B=20 so B is eliminated. Round 3 votes counts: E=40 D=37 A=23 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:233 E:216 D:207 C:184 B:160 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 24 18 2 B -22 0 -18 -18 -22 C -24 18 0 -18 -8 D -18 18 18 0 -4 E -2 22 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999926693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 24 18 2 B -22 0 -18 -18 -22 C -24 18 0 -18 -8 D -18 18 18 0 -4 E -2 22 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999926693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 24 18 2 B -22 0 -18 -18 -22 C -24 18 0 -18 -8 D -18 18 18 0 -4 E -2 22 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999926693 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6038: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) E C B D A (6) A B C D E (6) D E A B C (5) E D A B C (4) E C D B A (4) D E A C B (4) D C A E B (4) D A E B C (4) B C A E D (4) A C B D E (4) A B D C E (4) C B E A D (3) C B A D E (3) B A C E D (3) E B C A D (2) D E C B A (2) D C A B E (2) D A B C E (2) C E D B A (2) C B A E D (2) E D C A B (1) E D B C A (1) E B D C A (1) E B C D A (1) E B A C D (1) D E C A B (1) D C E B A (1) D A C E B (1) D A C B E (1) C E B D A (1) C D E B A (1) C D A B E (1) C B E D A (1) C A B D E (1) B C E A D (1) A D E B C (1) A D B C E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -14 -24 -8 B 2 0 -16 -10 -18 C 14 16 0 -2 0 D 24 10 2 0 0 E 8 18 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.532152 E: 0.467848 Sum of squares = 0.50206742755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.532152 E: 1.000000 A B C D E A 0 -2 -14 -24 -8 B 2 0 -16 -10 -18 C 14 16 0 -2 0 D 24 10 2 0 0 E 8 18 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999996702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=27 A=18 C=15 B=8 so B is eliminated. Round 2 votes counts: E=32 D=27 A=21 C=20 so C is eliminated. Round 3 votes counts: E=40 A=31 D=29 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:218 C:214 E:213 B:179 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -14 -24 -8 B 2 0 -16 -10 -18 C 14 16 0 -2 0 D 24 10 2 0 0 E 8 18 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999996702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -24 -8 B 2 0 -16 -10 -18 C 14 16 0 -2 0 D 24 10 2 0 0 E 8 18 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999996702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -24 -8 B 2 0 -16 -10 -18 C 14 16 0 -2 0 D 24 10 2 0 0 E 8 18 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999996702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6039: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (13) E C A B D (10) B A D E C (9) A B D C E (9) C E A B D (8) A B E C D (8) D C E B A (7) C E D A B (6) E C D B A (5) A B D E C (5) D E C B A (4) E C D A B (2) D B A E C (2) C E D B A (2) E C B A D (1) D B E C A (1) D B C A E (1) C D E B A (1) C A E B D (1) B A D C E (1) A E C B D (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 12 6 B -6 0 2 12 0 C -2 -2 0 -4 2 D -12 -12 4 0 6 E -6 0 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 12 6 B -6 0 2 12 0 C -2 -2 0 -4 2 D -12 -12 4 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=26 E=18 C=18 B=10 so B is eliminated. Round 2 votes counts: A=36 D=28 E=18 C=18 so E is eliminated. Round 3 votes counts: C=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:204 C:197 D:193 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 12 6 B -6 0 2 12 0 C -2 -2 0 -4 2 D -12 -12 4 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 12 6 B -6 0 2 12 0 C -2 -2 0 -4 2 D -12 -12 4 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 12 6 B -6 0 2 12 0 C -2 -2 0 -4 2 D -12 -12 4 0 6 E -6 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6040: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) D B E A C (9) C A B E D (8) A C B D E (7) C A D E B (6) B E D A C (6) C A E B D (5) E B D C A (4) B A C E D (4) A C D B E (4) A C B E D (4) E D B C A (3) C A E D B (3) A D C B E (3) D E B C A (2) D E A C B (2) D A B C E (2) C B A E D (2) B D E A C (2) E D B A C (1) E B D A C (1) D E C B A (1) D E C A B (1) D A E B C (1) D A C E B (1) D A C B E (1) C E A B D (1) C A B D E (1) B E A C D (1) B D A E C (1) B A D E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 26 4 12 B -4 0 0 -2 16 C -26 0 0 -6 8 D -4 2 6 0 14 E -12 -16 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 26 4 12 B -4 0 0 -2 16 C -26 0 0 -6 8 D -4 2 6 0 14 E -12 -16 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=26 A=20 B=15 E=9 so E is eliminated. Round 2 votes counts: D=34 C=26 B=20 A=20 so B is eliminated. Round 3 votes counts: D=48 C=26 A=26 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 D:209 B:205 C:188 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 26 4 12 B -4 0 0 -2 16 C -26 0 0 -6 8 D -4 2 6 0 14 E -12 -16 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 26 4 12 B -4 0 0 -2 16 C -26 0 0 -6 8 D -4 2 6 0 14 E -12 -16 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 26 4 12 B -4 0 0 -2 16 C -26 0 0 -6 8 D -4 2 6 0 14 E -12 -16 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998761 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6041: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) E C A B D (9) B D E C A (9) B D E A C (6) D B A E C (5) C A E B D (5) A C D B E (5) E C A D B (4) E B D C A (4) D B E C A (4) A C E D B (4) A C D E B (4) E C B A D (3) D B E A C (3) C E A B D (3) B E D C A (3) D A B C E (2) C E A D B (2) C A E D B (2) B D A C E (2) A C E B D (2) A C B D E (2) E C B D A (1) E B C A D (1) B E C A D (1) B D A E C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -2 -2 -6 B 8 0 6 4 12 C 2 -6 0 -4 -8 D 2 -4 4 0 12 E 6 -12 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -2 -6 B 8 0 6 4 12 C 2 -6 0 -4 -8 D 2 -4 4 0 12 E 6 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=22 B=22 A=19 C=12 so C is eliminated. Round 2 votes counts: E=27 A=26 D=25 B=22 so B is eliminated. Round 3 votes counts: D=43 E=31 A=26 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:215 D:207 E:195 C:192 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -2 -6 B 8 0 6 4 12 C 2 -6 0 -4 -8 D 2 -4 4 0 12 E 6 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -2 -6 B 8 0 6 4 12 C 2 -6 0 -4 -8 D 2 -4 4 0 12 E 6 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -2 -6 B 8 0 6 4 12 C 2 -6 0 -4 -8 D 2 -4 4 0 12 E 6 -12 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6042: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) C A E D B (6) B D E A C (5) B D A E C (5) B A D E C (5) E D B C A (4) D E C A B (4) D E B A C (4) B A C E D (4) E D C B A (3) D E B C A (3) D B E A C (3) C A E B D (3) C A D E B (3) B E D A C (3) A C B E D (3) A C B D E (3) A B C E D (3) A B C D E (3) C E A D B (2) C E A B D (2) C D E A B (2) C D A E B (2) C A B E D (2) B E A C D (2) E C B D A (1) D C E A B (1) D B A E C (1) D A C B E (1) C E D A B (1) C A D B E (1) B E A D C (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C D E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 4 -2 2 B 12 0 0 -2 2 C -4 0 0 -8 -8 D 2 2 8 0 16 E -2 -2 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -2 2 B 12 0 0 -2 2 C -4 0 0 -8 -8 D 2 2 8 0 16 E -2 -2 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 C=24 A=14 E=8 so E is eliminated. Round 2 votes counts: D=32 B=29 C=25 A=14 so A is eliminated. Round 3 votes counts: B=35 D=33 C=32 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:206 A:196 E:194 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -2 2 B 12 0 0 -2 2 C -4 0 0 -8 -8 D 2 2 8 0 16 E -2 -2 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -2 2 B 12 0 0 -2 2 C -4 0 0 -8 -8 D 2 2 8 0 16 E -2 -2 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -2 2 B 12 0 0 -2 2 C -4 0 0 -8 -8 D 2 2 8 0 16 E -2 -2 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999395 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6043: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) A D C B E (7) C D B A E (6) E B C D A (5) E A B D C (4) B E D C A (4) B D C A E (4) E A B C D (3) C D A B E (3) C B D E A (3) B E A D C (3) B C D E A (3) A E C D B (3) A D E C B (3) A D B C E (3) E C D B A (2) E B C A D (2) E A C D B (2) D C B A E (2) D C A B E (2) C D B E A (2) C D A E B (2) B E D A C (2) B A E D C (2) B A D C E (2) A E D C B (2) A D C E B (2) A C D E B (2) E C B D A (1) E B D C A (1) D A C B E (1) C E D A B (1) B E C D A (1) B D C E A (1) B C E D A (1) B A D E C (1) A E D B C (1) A E B D C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 8 6 4 B 10 0 8 8 12 C -8 -8 0 -16 -4 D -6 -8 16 0 2 E -4 -12 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 6 4 B 10 0 8 8 12 C -8 -8 0 -16 -4 D -6 -8 16 0 2 E -4 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 B=24 C=17 D=5 so D is eliminated. Round 2 votes counts: A=28 E=27 B=24 C=21 so C is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:204 D:202 E:193 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 6 4 B 10 0 8 8 12 C -8 -8 0 -16 -4 D -6 -8 16 0 2 E -4 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 6 4 B 10 0 8 8 12 C -8 -8 0 -16 -4 D -6 -8 16 0 2 E -4 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 6 4 B 10 0 8 8 12 C -8 -8 0 -16 -4 D -6 -8 16 0 2 E -4 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6044: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) B D C A E (8) C E D A B (7) C D E A B (7) E A C D B (6) B A E D C (6) A E D C B (4) A B E D C (4) E A C B D (3) D B C A E (3) C D E B A (3) E C A D B (2) E C A B D (2) D C B E A (2) C E A B D (2) C D B E A (2) B D A E C (2) B C E D A (2) B A D E C (2) A E B D C (2) E C B A D (1) E B A C D (1) E A B C D (1) D C E B A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B A C E (1) D A C E B (1) D A B E C (1) D A B C E (1) C B E D A (1) C B E A D (1) C B D E A (1) B D C E A (1) B C E A D (1) B C D E A (1) B A E C D (1) B A C D E (1) A E D B C (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -2 -14 4 B 4 0 0 8 6 C 2 0 0 -4 18 D 14 -8 4 0 0 E -4 -6 -18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.782250 C: 0.217750 D: 0.000000 E: 0.000000 Sum of squares = 0.659330558328 Cumulative probabilities = A: 0.000000 B: 0.782250 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -14 4 B 4 0 0 8 6 C 2 0 0 -4 18 D 14 -8 4 0 0 E -4 -6 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=24 E=16 D=13 A=13 so D is eliminated. Round 2 votes counts: B=38 C=30 E=16 A=16 so E is eliminated. Round 3 votes counts: B=39 C=35 A=26 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:208 D:205 A:192 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -14 4 B 4 0 0 8 6 C 2 0 0 -4 18 D 14 -8 4 0 0 E -4 -6 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -14 4 B 4 0 0 8 6 C 2 0 0 -4 18 D 14 -8 4 0 0 E -4 -6 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -14 4 B 4 0 0 8 6 C 2 0 0 -4 18 D 14 -8 4 0 0 E -4 -6 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6045: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (17) A E B D C (12) D C A B E (7) B E C A D (7) D A C E B (6) D C B E A (5) B E A C D (5) A E B C D (5) E B A C D (4) C B D E A (4) A E D B C (4) E A B C D (3) B C E D A (3) A D E C B (3) E A B D C (2) D C E B A (2) E D A C B (1) D E C A B (1) D E A C B (1) D C B A E (1) D C A E B (1) D A E C B (1) C B E D A (1) B C A D E (1) A D E B C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -2 -12 B 0 0 -2 -2 6 C 0 2 0 4 -2 D 2 2 -4 0 4 E 12 -6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000003 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 0 0 -2 -12 B 0 0 -2 -2 6 C 0 2 0 4 -2 D 2 2 -4 0 4 E 12 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000106 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=25 C=22 B=16 E=10 so E is eliminated. Round 2 votes counts: A=32 D=26 C=22 B=20 so B is eliminated. Round 3 votes counts: A=41 C=33 D=26 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:202 D:202 E:202 B:201 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -2 -12 B 0 0 -2 -2 6 C 0 2 0 4 -2 D 2 2 -4 0 4 E 12 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000106 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -12 B 0 0 -2 -2 6 C 0 2 0 4 -2 D 2 2 -4 0 4 E 12 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000106 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -12 B 0 0 -2 -2 6 C 0 2 0 4 -2 D 2 2 -4 0 4 E 12 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000000106 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6046: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (11) D A E C B (8) A D E C B (7) E B A C D (6) D A C B E (6) B C E D A (6) B C E A D (6) E A D B C (5) D A C E B (5) B E C A D (5) E B C A D (4) C D B A E (4) C B D E A (4) A D E B C (4) E A B D C (3) E A B C D (3) D C A B E (3) D C B A E (2) C B D A E (2) C E B D A (1) C D B E A (1) B C D A E (1) B C A D E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -8 -2 -8 -10 B 8 0 -8 8 4 C 2 8 0 10 6 D 8 -8 -10 0 -4 E 10 -4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -8 -10 B 8 0 -8 8 4 C 2 8 0 10 6 D 8 -8 -10 0 -4 E 10 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 E=21 B=19 A=13 so A is eliminated. Round 2 votes counts: D=35 E=23 C=23 B=19 so B is eliminated. Round 3 votes counts: C=37 D=35 E=28 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:206 E:202 D:193 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 -8 -10 B 8 0 -8 8 4 C 2 8 0 10 6 D 8 -8 -10 0 -4 E 10 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -8 -10 B 8 0 -8 8 4 C 2 8 0 10 6 D 8 -8 -10 0 -4 E 10 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -8 -10 B 8 0 -8 8 4 C 2 8 0 10 6 D 8 -8 -10 0 -4 E 10 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998181 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6047: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) B A E D C (8) C A D B E (5) A D B E C (5) A B D E C (5) D A E B C (4) C D A E B (4) C B E D A (4) C B A D E (4) B A D E C (4) E D A B C (3) E B C D A (3) D A E C B (3) C E B D A (3) C A B D E (3) A D C B E (3) E D B A C (2) E B D C A (2) C E D A B (2) C B E A D (2) B E C A D (2) B C E A D (2) B A C E D (2) A D E B C (2) E D A C B (1) E C B D A (1) E B D A C (1) E B A D C (1) D A C E B (1) C E D B A (1) C B D A E (1) C B A E D (1) B E A C D (1) B A D C E (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 -16 20 28 16 B 16 0 22 26 32 C -20 -22 0 -16 -18 D -28 -26 16 0 -6 E -16 -32 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 20 28 16 B 16 0 22 26 32 C -20 -22 0 -16 -18 D -28 -26 16 0 -6 E -16 -32 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=30 A=16 E=14 D=8 so D is eliminated. Round 2 votes counts: B=32 C=30 A=24 E=14 so E is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: B=61 C=39 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:248 A:224 E:188 D:178 C:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 20 28 16 B 16 0 22 26 32 C -20 -22 0 -16 -18 D -28 -26 16 0 -6 E -16 -32 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 20 28 16 B 16 0 22 26 32 C -20 -22 0 -16 -18 D -28 -26 16 0 -6 E -16 -32 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 20 28 16 B 16 0 22 26 32 C -20 -22 0 -16 -18 D -28 -26 16 0 -6 E -16 -32 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6048: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (14) C D E A B (10) A C B D E (6) E D C B A (5) E D B C A (5) D E C B A (5) C A D E B (5) A B C E D (5) E B D A C (4) B E D A C (4) E D B A C (3) C E D A B (3) C D A E B (3) A B E D C (3) D E B C A (2) D C E B A (2) C A D B E (2) C A B D E (2) B E A D C (2) B D E A C (2) A C B E D (2) E C D B A (1) E B A C D (1) D C B A E (1) C E D B A (1) C D E B A (1) C D A B E (1) C A E D B (1) C A E B D (1) C A B E D (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -4 -6 -2 B 8 0 -6 -2 -6 C 4 6 0 -8 -6 D 6 2 8 0 -12 E 2 6 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -4 -6 -2 B 8 0 -6 -2 -6 C 4 6 0 -8 -6 D 6 2 8 0 -12 E 2 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=23 E=19 A=17 D=10 so D is eliminated. Round 2 votes counts: C=34 E=26 B=23 A=17 so A is eliminated. Round 3 votes counts: C=42 B=32 E=26 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 D:202 C:198 B:197 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -6 -2 B 8 0 -6 -2 -6 C 4 6 0 -8 -6 D 6 2 8 0 -12 E 2 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -6 -2 B 8 0 -6 -2 -6 C 4 6 0 -8 -6 D 6 2 8 0 -12 E 2 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -6 -2 B 8 0 -6 -2 -6 C 4 6 0 -8 -6 D 6 2 8 0 -12 E 2 6 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6049: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (10) A C B E D (10) C A B E D (8) C A E B D (6) B E D C A (6) E B D C A (5) D E B C A (5) C A D E B (5) A C D B E (5) D E B A C (4) D A C E B (4) C A E D B (4) B E C A D (4) D B E A C (3) D A B E C (3) C B E A D (2) B D E A C (2) A D C E B (2) A D C B E (2) E C B D A (1) E B C D A (1) E B C A D (1) D C E A B (1) D B A E C (1) D A E C B (1) D A E B C (1) C B A E D (1) B E C D A (1) A C E B D (1) Total count = 100 A B C D E A 0 26 -2 22 28 B -26 0 -26 -2 -4 C 2 26 0 20 22 D -22 2 -20 0 -2 E -28 4 -22 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 -2 22 28 B -26 0 -26 -2 -4 C 2 26 0 20 22 D -22 2 -20 0 -2 E -28 4 -22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 D=23 B=13 E=8 so E is eliminated. Round 2 votes counts: A=30 C=27 D=23 B=20 so B is eliminated. Round 3 votes counts: D=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:237 C:235 D:179 E:178 B:171 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 26 -2 22 28 B -26 0 -26 -2 -4 C 2 26 0 20 22 D -22 2 -20 0 -2 E -28 4 -22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 -2 22 28 B -26 0 -26 -2 -4 C 2 26 0 20 22 D -22 2 -20 0 -2 E -28 4 -22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 -2 22 28 B -26 0 -26 -2 -4 C 2 26 0 20 22 D -22 2 -20 0 -2 E -28 4 -22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6050: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (13) A B D E C (9) E B D A C (7) C E D B A (7) A C D B E (7) E C B D A (6) E B D C A (6) C E A D B (3) C A E D B (3) E C A B D (2) E B C D A (2) D B A E C (2) C E D A B (2) C E B D A (2) C D A B E (2) C A D E B (2) C A D B E (2) B E D A C (2) B D E A C (2) B D A E C (2) A B E D C (2) E B C A D (1) E B A D C (1) D C B A E (1) D B E A C (1) D B C A E (1) D B A C E (1) D A B E C (1) C D E B A (1) C D B E A (1) C A E B D (1) A E B D C (1) A D C B E (1) A D B E C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 10 2 8 B -8 0 12 -4 4 C -10 -12 0 -10 4 D -2 4 10 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 2 8 B -8 0 12 -4 4 C -10 -12 0 -10 4 D -2 4 10 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=26 E=25 D=7 B=6 so B is eliminated. Round 2 votes counts: A=36 E=27 C=26 D=11 so D is eliminated. Round 3 votes counts: A=42 E=30 C=28 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:207 B:202 E:191 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 2 8 B -8 0 12 -4 4 C -10 -12 0 -10 4 D -2 4 10 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 2 8 B -8 0 12 -4 4 C -10 -12 0 -10 4 D -2 4 10 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 2 8 B -8 0 12 -4 4 C -10 -12 0 -10 4 D -2 4 10 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999562 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6051: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) C B E A D (7) A D E C B (6) A D C B E (6) D A B C E (5) E D B C A (4) A D C E B (4) E C B D A (3) E C A B D (3) E B D C A (3) D E B A C (3) D E A B C (3) D A B E C (3) C E B A D (3) C B A D E (3) A E C D B (3) E C B A D (2) D B E C A (2) D B A C E (2) D A E B C (2) B E C D A (2) B C E D A (2) A E D C B (2) A C E B D (2) E A D C B (1) D E B C A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E B D A (1) C E A B D (1) C B E D A (1) C A B E D (1) B D C E A (1) B D C A E (1) B C D A E (1) A D E B C (1) A D B C E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -4 -4 -6 B 8 0 -2 -8 -16 C 4 2 0 -8 -10 D 4 8 8 0 -2 E 6 16 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -4 -4 -6 B 8 0 -2 -8 -16 C 4 2 0 -8 -10 D 4 8 8 0 -2 E 6 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 D=24 C=17 B=7 so B is eliminated. Round 2 votes counts: E=27 A=27 D=26 C=20 so C is eliminated. Round 3 votes counts: E=42 A=31 D=27 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:209 C:194 B:191 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -4 -6 B 8 0 -2 -8 -16 C 4 2 0 -8 -10 D 4 8 8 0 -2 E 6 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -4 -6 B 8 0 -2 -8 -16 C 4 2 0 -8 -10 D 4 8 8 0 -2 E 6 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -4 -6 B 8 0 -2 -8 -16 C 4 2 0 -8 -10 D 4 8 8 0 -2 E 6 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6052: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B D E A C (7) B E D C A (6) B E C D A (6) A C D E B (6) A B C D E (6) D A E C B (5) C E D B A (4) C E D A B (3) C D E A B (3) C B E D A (3) B A D E C (3) A D E B C (3) A D C E B (3) A D B E C (3) E D C B A (2) E C D B A (2) D E C B A (2) C A E D B (2) B C E D A (2) B A C E D (2) A D E C B (2) A C B D E (2) A B D E C (2) A B C E D (2) D E B C A (1) D E A B C (1) D C E A B (1) C E B D A (1) C D A E B (1) C B E A D (1) C B A E D (1) B E D A C (1) B E C A D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 0 -18 -10 B -10 0 -6 -4 0 C 0 6 0 -2 -10 D 18 4 2 0 20 E 10 0 10 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -18 -10 B -10 0 -6 -4 0 C 0 6 0 -2 -10 D 18 4 2 0 20 E 10 0 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=28 C=19 D=18 E=4 so E is eliminated. Round 2 votes counts: A=31 B=28 C=21 D=20 so D is eliminated. Round 3 votes counts: A=37 C=34 B=29 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:222 E:200 C:197 A:191 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -18 -10 B -10 0 -6 -4 0 C 0 6 0 -2 -10 D 18 4 2 0 20 E 10 0 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -18 -10 B -10 0 -6 -4 0 C 0 6 0 -2 -10 D 18 4 2 0 20 E 10 0 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -18 -10 B -10 0 -6 -4 0 C 0 6 0 -2 -10 D 18 4 2 0 20 E 10 0 10 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6053: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) E C B D A (6) B A D C E (6) E A C D B (5) D C A B E (5) E C D A B (4) E B C D A (4) E B A C D (4) E A D C B (4) C D E A B (4) B C D A E (4) A B D E C (4) B E C D A (3) B A E D C (3) A D B C E (3) E C A D B (2) D C B A E (2) C E D B A (2) C D E B A (2) B D C A E (2) B A D E C (2) A E B D C (2) A D E C B (2) E B A D C (1) E A D B C (1) E A B D C (1) D C A E B (1) D A C B E (1) D A B C E (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D E A (1) B E C A D (1) B E A D C (1) B C D E A (1) A E D C B (1) A D C E B (1) A D C B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -8 -6 -8 B 6 0 -8 -4 -10 C 8 8 0 6 -18 D 6 4 -6 0 -6 E 8 10 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -6 -8 B 6 0 -8 -4 -10 C 8 8 0 6 -18 D 6 4 -6 0 -6 E 8 10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=23 A=16 C=12 D=10 so D is eliminated. Round 2 votes counts: E=39 B=23 C=20 A=18 so A is eliminated. Round 3 votes counts: E=44 B=33 C=23 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:202 D:199 B:192 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -6 -8 B 6 0 -8 -4 -10 C 8 8 0 6 -18 D 6 4 -6 0 -6 E 8 10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -6 -8 B 6 0 -8 -4 -10 C 8 8 0 6 -18 D 6 4 -6 0 -6 E 8 10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -6 -8 B 6 0 -8 -4 -10 C 8 8 0 6 -18 D 6 4 -6 0 -6 E 8 10 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6054: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (13) A B D E C (12) D E C A B (10) B A C E D (9) E D C B A (5) A D E B C (5) A B C D E (5) B C A E D (4) B A E D C (4) D E A C B (3) C D E B A (3) A D E C B (3) E D C A B (2) C E B D A (2) C D E A B (2) B C E D A (2) A D B E C (2) A B E D C (2) E C D B A (1) E B C D A (1) D C E A B (1) D A C E B (1) C D B E A (1) C B E D A (1) C B A E D (1) B E D C A (1) B C E A D (1) A E B D C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 2 2 B -2 0 0 -6 -6 C 2 0 0 -8 -4 D -2 6 8 0 0 E -2 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 2 2 B -2 0 0 -6 -6 C 2 0 0 -8 -4 D -2 6 8 0 0 E -2 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=23 B=21 D=15 E=9 so E is eliminated. Round 2 votes counts: A=32 C=24 D=22 B=22 so D is eliminated. Round 3 votes counts: C=42 A=36 B=22 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:206 E:204 A:202 C:195 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 2 2 B -2 0 0 -6 -6 C 2 0 0 -8 -4 D -2 6 8 0 0 E -2 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 2 B -2 0 0 -6 -6 C 2 0 0 -8 -4 D -2 6 8 0 0 E -2 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 2 B -2 0 0 -6 -6 C 2 0 0 -8 -4 D -2 6 8 0 0 E -2 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6055: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (15) A E B D C (10) A B E C D (8) D C E B A (7) D C E A B (6) B A E C D (5) E D C B A (4) C D B A E (4) B A C D E (4) E A B D C (3) C D A B E (3) B C D E A (3) B C A D E (3) E D C A B (2) E D B C A (2) E A D B C (2) A B C E D (2) E D B A C (1) E D A C B (1) E B D C A (1) E A D C B (1) D E C A B (1) D C B E A (1) D C A E B (1) D A C E B (1) C B D E A (1) B E A C D (1) B C E D A (1) B C D A E (1) B A C E D (1) A E D C B (1) A E D B C (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -12 -12 -6 B 10 0 0 -10 8 C 12 0 0 6 10 D 12 10 -6 0 6 E 6 -8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.215054 C: 0.784946 D: 0.000000 E: 0.000000 Sum of squares = 0.662388933619 Cumulative probabilities = A: 0.000000 B: 0.215054 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -12 -6 B 10 0 0 -10 8 C 12 0 0 6 10 D 12 10 -6 0 6 E 6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250207908 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 B=19 E=17 D=17 so E is eliminated. Round 2 votes counts: A=30 D=27 C=23 B=20 so B is eliminated. Round 3 votes counts: A=41 C=31 D=28 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:211 B:204 E:191 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -12 -6 B 10 0 0 -10 8 C 12 0 0 6 10 D 12 10 -6 0 6 E 6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250207908 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -12 -6 B 10 0 0 -10 8 C 12 0 0 6 10 D 12 10 -6 0 6 E 6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250207908 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -12 -6 B 10 0 0 -10 8 C 12 0 0 6 10 D 12 10 -6 0 6 E 6 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250207908 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6056: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) C B D A E (8) B C D A E (8) E A C B D (7) E A D B C (6) D C B E A (6) C B A E D (5) A E D B C (5) A E B C D (5) D A E B C (4) C B E A D (4) C B D E A (4) D E A B C (3) A E C B D (3) A B C E D (3) E D A C B (2) D E C B A (2) D E B A C (2) D B C A E (2) C D B E A (2) E D C B A (1) E D A B C (1) E C A B D (1) E A C D B (1) D E B C A (1) D E A C B (1) D B E C A (1) D B C E A (1) D A B C E (1) C E A B D (1) B C A D E (1) Total count = 100 A B C D E A 0 4 4 0 -10 B -4 0 -12 0 -8 C -4 12 0 6 -8 D 0 0 -6 0 -6 E 10 8 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 0 -10 B -4 0 -12 0 -8 C -4 12 0 6 -8 D 0 0 -6 0 -6 E 10 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 C=24 A=16 B=9 so B is eliminated. Round 2 votes counts: C=33 E=27 D=24 A=16 so A is eliminated. Round 3 votes counts: E=40 C=36 D=24 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:203 A:199 D:194 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 0 -10 B -4 0 -12 0 -8 C -4 12 0 6 -8 D 0 0 -6 0 -6 E 10 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 -10 B -4 0 -12 0 -8 C -4 12 0 6 -8 D 0 0 -6 0 -6 E 10 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 -10 B -4 0 -12 0 -8 C -4 12 0 6 -8 D 0 0 -6 0 -6 E 10 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6057: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) B D C E A (7) C B A E D (6) B C D A E (6) C A B E D (5) B D E A C (4) E A D C B (3) D E A C B (3) D E A B C (3) D B E C A (3) C B A D E (3) B C A E D (3) A E C D B (3) A E B D C (3) E D A B C (2) E A D B C (2) D E B A C (2) D B E A C (2) C D E A B (2) C D B E A (2) C B D E A (2) C B D A E (2) C A E B D (2) B D E C A (2) B D A E C (2) B C A D E (2) A E D B C (2) A C E D B (2) E D A C B (1) D E C A B (1) D E B C A (1) D C B E A (1) D B C E A (1) B C D E A (1) A E D C B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -26 0 10 B 4 0 -2 4 10 C 26 2 0 6 18 D 0 -4 -6 0 4 E -10 -10 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -26 0 10 B 4 0 -2 4 10 C 26 2 0 6 18 D 0 -4 -6 0 4 E -10 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=27 D=17 A=13 E=8 so E is eliminated. Round 2 votes counts: C=35 B=27 D=20 A=18 so A is eliminated. Round 3 votes counts: C=41 B=31 D=28 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:208 D:197 A:190 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -26 0 10 B 4 0 -2 4 10 C 26 2 0 6 18 D 0 -4 -6 0 4 E -10 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -26 0 10 B 4 0 -2 4 10 C 26 2 0 6 18 D 0 -4 -6 0 4 E -10 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -26 0 10 B 4 0 -2 4 10 C 26 2 0 6 18 D 0 -4 -6 0 4 E -10 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6058: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) A E D C B (7) E A B C D (6) C B D A E (6) C B A E D (6) E A D B C (5) D A E C B (5) D C B E A (4) D C B A E (4) E A B D C (3) D E A C B (3) D E A B C (3) B C A E D (3) A E C D B (3) A E C B D (3) A D E C B (3) E B D A C (2) D B E C A (2) D A C E B (2) B C E A D (2) A E B C D (2) E D A B C (1) E B A D C (1) E B A C D (1) D E C A B (1) D E B C A (1) D B C E A (1) C D B A E (1) C B D E A (1) C B A D E (1) C A B E D (1) B E C D A (1) B D E C A (1) B C E D A (1) B A C E D (1) A E D B C (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 8 2 2 B -2 0 -6 4 -8 C -8 6 0 -2 -10 D -2 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 2 2 B -2 0 -6 4 -8 C -8 6 0 -2 -10 D -2 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999576 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=21 E=19 B=18 C=16 so C is eliminated. Round 2 votes counts: B=32 D=27 A=22 E=19 so E is eliminated. Round 3 votes counts: B=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:207 D:197 B:194 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 2 2 B -2 0 -6 4 -8 C -8 6 0 -2 -10 D -2 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999576 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 2 2 B -2 0 -6 4 -8 C -8 6 0 -2 -10 D -2 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999576 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 2 2 B -2 0 -6 4 -8 C -8 6 0 -2 -10 D -2 -4 2 0 -2 E -2 8 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999576 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6059: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B D E A C (7) A C B D E (7) E D B A C (6) C A E B D (6) D B E C A (5) A E C D B (5) D E B C A (4) C E A D B (4) B D A E C (4) B A D E C (4) A C E B D (4) E D B C A (3) D E B A C (3) D B E A C (3) C A B D E (3) A C E D B (3) B D E C A (2) B D A C E (2) B A D C E (2) E D C B A (1) E D A C B (1) E C A D B (1) E A D B C (1) E A C D B (1) C E D B A (1) C B D E A (1) C A B E D (1) B D C E A (1) B D C A E (1) B C A D E (1) A E D C B (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 14 10 10 B 2 0 4 -6 -8 C -14 -4 0 -8 -10 D -10 6 8 0 6 E -10 8 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.419999999984 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 A B C D E A 0 -2 14 10 10 B 2 0 4 -6 -8 C -14 -4 0 -8 -10 D -10 6 8 0 6 E -10 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.419999997893 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 A=23 D=15 E=14 so E is eliminated. Round 2 votes counts: D=26 C=25 A=25 B=24 so B is eliminated. Round 3 votes counts: D=43 A=31 C=26 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:205 E:201 B:196 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 14 10 10 B 2 0 4 -6 -8 C -14 -4 0 -8 -10 D -10 6 8 0 6 E -10 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.419999997893 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 10 10 B 2 0 4 -6 -8 C -14 -4 0 -8 -10 D -10 6 8 0 6 E -10 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.419999997893 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 10 10 B 2 0 4 -6 -8 C -14 -4 0 -8 -10 D -10 6 8 0 6 E -10 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.419999997893 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6060: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) C B D E A (8) B C A E D (8) B C A D E (7) E D A C B (6) A E D C B (6) D E A C B (5) B C D A E (5) B A C E D (5) E A D C B (4) B C D E A (4) D E C A B (3) D C E B A (3) A B E C D (3) D E A B C (2) D A E B C (2) B A D E C (2) A E B D C (2) A E B C D (2) D E C B A (1) D C B E A (1) C E D A B (1) C E A D B (1) C E A B D (1) C D B E A (1) C B A E D (1) C A E B D (1) B D A E C (1) B A E D C (1) B A D C E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 8 14 18 B -2 0 14 6 -2 C -8 -14 0 -4 -4 D -14 -6 4 0 -6 E -18 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999106 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 14 18 B -2 0 14 6 -2 C -8 -14 0 -4 -4 D -14 -6 4 0 -6 E -18 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=25 D=17 C=14 E=10 so E is eliminated. Round 2 votes counts: B=34 A=29 D=23 C=14 so C is eliminated. Round 3 votes counts: B=43 A=32 D=25 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:208 E:197 D:189 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 14 18 B -2 0 14 6 -2 C -8 -14 0 -4 -4 D -14 -6 4 0 -6 E -18 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 14 18 B -2 0 14 6 -2 C -8 -14 0 -4 -4 D -14 -6 4 0 -6 E -18 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 14 18 B -2 0 14 6 -2 C -8 -14 0 -4 -4 D -14 -6 4 0 -6 E -18 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6061: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) B C A E D (7) E C D A B (6) D E A C B (6) B D A E C (6) E D C A B (4) C E A D B (4) A C E D B (4) D B E A C (3) D B A E C (3) D A E C B (3) C E A B D (3) C A B E D (3) B E D C A (3) B D E A C (3) E D C B A (2) C A E B D (2) B D E C A (2) B A D C E (2) B A C D E (2) A D C E B (2) A D C B E (2) A D B C E (2) A C E B D (2) E C B D A (1) E C A D B (1) E B C D A (1) D E B A C (1) D E A B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C B A E D (1) B E C A D (1) B C A D E (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -4 6 -4 B -2 0 0 4 10 C 4 0 0 4 4 D -6 -4 -4 0 -10 E 4 -10 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.303715 C: 0.696285 D: 0.000000 E: 0.000000 Sum of squares = 0.577055684674 Cumulative probabilities = A: 0.000000 B: 0.303715 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 6 -4 B -2 0 0 4 10 C 4 0 0 4 4 D -6 -4 -4 0 -10 E 4 -10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=19 A=16 E=15 C=14 so C is eliminated. Round 2 votes counts: B=37 E=23 A=21 D=19 so D is eliminated. Round 3 votes counts: B=43 E=31 A=26 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:206 C:206 A:200 E:200 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 6 -4 B -2 0 0 4 10 C 4 0 0 4 4 D -6 -4 -4 0 -10 E 4 -10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 6 -4 B -2 0 0 4 10 C 4 0 0 4 4 D -6 -4 -4 0 -10 E 4 -10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 6 -4 B -2 0 0 4 10 C 4 0 0 4 4 D -6 -4 -4 0 -10 E 4 -10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6062: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) B D E C A (9) D E C B A (8) A E C D B (7) A C E D B (6) B D C E A (5) A E D C B (5) E C D A B (4) D E C A B (4) C E D B A (4) B A D E C (4) A B C E D (4) C D E B A (3) B C D E A (3) B D A E C (2) B A C D E (2) E D C A B (1) E D A C B (1) E A D C B (1) D E B C A (1) D E A B C (1) C E A D B (1) C B E D A (1) C B D E A (1) C B A E D (1) B D E A C (1) B D A C E (1) B C D A E (1) B C A D E (1) B A D C E (1) B A C E D (1) A E D B C (1) A C E B D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -16 -22 -18 B 0 0 -18 -16 -18 C 16 18 0 4 -6 D 22 16 -4 0 -2 E 18 18 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -16 -22 -18 B 0 0 -18 -16 -18 C 16 18 0 4 -6 D 22 16 -4 0 -2 E 18 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=27 C=21 D=14 E=7 so E is eliminated. Round 2 votes counts: B=31 A=28 C=25 D=16 so D is eliminated. Round 3 votes counts: C=38 B=32 A=30 so A is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:222 C:216 D:216 B:174 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -16 -22 -18 B 0 0 -18 -16 -18 C 16 18 0 4 -6 D 22 16 -4 0 -2 E 18 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -22 -18 B 0 0 -18 -16 -18 C 16 18 0 4 -6 D 22 16 -4 0 -2 E 18 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -22 -18 B 0 0 -18 -16 -18 C 16 18 0 4 -6 D 22 16 -4 0 -2 E 18 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6063: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) A D E C B (6) E A D C B (5) E A D B C (5) B C A D E (5) E D A B C (4) A E B C D (4) A C D B E (4) E D A C B (3) D C B A E (3) D A C E B (3) C D B A E (3) B E C D A (3) B C E D A (3) B C D E A (3) A C B D E (3) A B C E D (3) E D B C A (2) D C A B E (2) C B A D E (2) B E C A D (2) B C E A D (2) B A E C D (2) A E D C B (2) A D C E B (2) E B D A C (1) E B C D A (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E B A (1) D C B E A (1) D C A E B (1) D A E C B (1) C D A B E (1) C A D B E (1) B E A C D (1) B C D A E (1) B C A E D (1) A E C B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 6 8 6 B -10 0 4 -6 -4 C -6 -4 0 -4 -8 D -8 6 4 0 -8 E -6 4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 8 6 B -10 0 4 -6 -4 C -6 -4 0 -4 -8 D -8 6 4 0 -8 E -6 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 B=23 D=15 C=7 so C is eliminated. Round 2 votes counts: E=28 A=28 B=25 D=19 so D is eliminated. Round 3 votes counts: A=36 E=32 B=32 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:207 D:197 B:192 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 8 6 B -10 0 4 -6 -4 C -6 -4 0 -4 -8 D -8 6 4 0 -8 E -6 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 8 6 B -10 0 4 -6 -4 C -6 -4 0 -4 -8 D -8 6 4 0 -8 E -6 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 8 6 B -10 0 4 -6 -4 C -6 -4 0 -4 -8 D -8 6 4 0 -8 E -6 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6064: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) E C D B A (7) C E B D A (6) C E B A D (6) D E A B C (5) D A B E C (5) B A C E D (5) A B D C E (5) D E A C B (4) C B E A D (4) A D B C E (4) E D C B A (3) D E C A B (3) C B A E D (3) B C A E D (3) E D C A B (2) D E C B A (2) D E B C A (2) D A E C B (2) D A E B C (2) A C B E D (2) A B C D E (2) E C B D A (1) C E D A B (1) C E A B D (1) C A E D B (1) C A E B D (1) B E C D A (1) B C E A D (1) B A D E C (1) A D E B C (1) A D C B E (1) A C D E B (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 4 8 2 B -10 0 -2 -12 -2 C -4 2 0 -6 -4 D -8 12 6 0 2 E -2 2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 8 2 B -10 0 -2 -12 -2 C -4 2 0 -6 -4 D -8 12 6 0 2 E -2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 C=23 E=13 B=11 so B is eliminated. Round 2 votes counts: A=34 C=27 D=25 E=14 so E is eliminated. Round 3 votes counts: C=36 A=34 D=30 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:206 E:201 C:194 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 8 2 B -10 0 -2 -12 -2 C -4 2 0 -6 -4 D -8 12 6 0 2 E -2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 8 2 B -10 0 -2 -12 -2 C -4 2 0 -6 -4 D -8 12 6 0 2 E -2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 8 2 B -10 0 -2 -12 -2 C -4 2 0 -6 -4 D -8 12 6 0 2 E -2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999437 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6065: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (14) C B D A E (10) E A C D B (6) C A D B E (5) A E D B C (5) C B D E A (4) C A E D B (4) A E D C B (4) A E C D B (4) E C B A D (3) C D B A E (3) B D C A E (3) B C D A E (3) E B D A C (2) D B A C E (2) D A B E C (2) B E D A C (2) B E C D A (2) B D E C A (2) A D E B C (2) E A C B D (1) D C B A E (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B C E (1) C E B A D (1) C D A B E (1) C A D E B (1) B E D C A (1) B D E A C (1) B D C E A (1) B D A E C (1) B C D E A (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 6 8 16 B -8 0 -4 -24 0 C -6 4 0 2 -8 D -8 24 -2 0 0 E -16 0 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 8 16 B -8 0 -4 -24 0 C -6 4 0 2 -8 D -8 24 -2 0 0 E -16 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=26 A=19 B=17 D=9 so D is eliminated. Round 2 votes counts: C=30 E=26 B=22 A=22 so B is eliminated. Round 3 votes counts: C=39 E=35 A=26 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:219 D:207 C:196 E:196 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 8 16 B -8 0 -4 -24 0 C -6 4 0 2 -8 D -8 24 -2 0 0 E -16 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 8 16 B -8 0 -4 -24 0 C -6 4 0 2 -8 D -8 24 -2 0 0 E -16 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 8 16 B -8 0 -4 -24 0 C -6 4 0 2 -8 D -8 24 -2 0 0 E -16 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6066: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (6) E D C B A (5) D E C A B (5) D E B C A (5) B C E A D (5) A C B E D (5) A B C E D (5) E D B C A (4) B E C A D (4) A C B D E (4) E C B D A (3) E B C D A (3) D E C B A (3) C E A B D (3) B E D C A (3) A B D C E (3) A B C D E (3) D A B E C (2) C E B A D (2) A D C B E (2) A C D B E (2) E C D B A (1) E C D A B (1) E B D C A (1) D E B A C (1) D E A B C (1) D C E A B (1) D C A E B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A C E B (1) C E A D B (1) C D E A B (1) C B E A D (1) C A E B D (1) B D E A C (1) B D A E C (1) B C A E D (1) B A E C D (1) B A D C E (1) A D C E B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -10 6 -12 B 8 0 10 14 8 C 10 -10 0 6 2 D -6 -14 -6 0 -12 E 12 -8 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 6 -12 B 8 0 10 14 8 C 10 -10 0 6 2 D -6 -14 -6 0 -12 E 12 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 B=23 E=18 C=9 so C is eliminated. Round 2 votes counts: A=28 E=24 D=24 B=24 so E is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:207 C:204 A:188 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 6 -12 B 8 0 10 14 8 C 10 -10 0 6 2 D -6 -14 -6 0 -12 E 12 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 6 -12 B 8 0 10 14 8 C 10 -10 0 6 2 D -6 -14 -6 0 -12 E 12 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 6 -12 B 8 0 10 14 8 C 10 -10 0 6 2 D -6 -14 -6 0 -12 E 12 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6067: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (12) D E B A C (9) C B D E A (7) C A B E D (7) D E A B C (6) E D A B C (5) C A B D E (5) B C D E A (5) A C E D B (5) C B D A E (4) E D B A C (3) E A D B C (2) D E B C A (2) C B A D E (2) C A D E B (2) B E D C A (2) B E D A C (2) B D E C A (2) A E D C B (2) A C E B D (2) A C B E D (2) D C E B A (1) D B E C A (1) D A E B C (1) C D E B A (1) C A D B E (1) B D C E A (1) B C E D A (1) B C A E D (1) B A E D C (1) A E C D B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 10 -6 0 B -10 0 12 -12 -12 C -10 -12 0 -6 -4 D 6 12 6 0 4 E 0 12 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -6 0 B -10 0 12 -12 -12 C -10 -12 0 -6 -4 D 6 12 6 0 4 E 0 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 D=20 B=15 E=10 so E is eliminated. Round 2 votes counts: C=29 D=28 A=28 B=15 so B is eliminated. Round 3 votes counts: C=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:207 E:206 B:189 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -6 0 B -10 0 12 -12 -12 C -10 -12 0 -6 -4 D 6 12 6 0 4 E 0 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -6 0 B -10 0 12 -12 -12 C -10 -12 0 -6 -4 D 6 12 6 0 4 E 0 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -6 0 B -10 0 12 -12 -12 C -10 -12 0 -6 -4 D 6 12 6 0 4 E 0 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6068: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (10) D A C E B (9) D A E B C (5) A C D B E (5) E B D C A (4) A B E C D (4) E B D A C (3) D E B A C (3) D E A B C (3) D C A E B (3) A D C B E (3) A C B D E (3) E B A D C (2) D A C B E (2) C D B E A (2) C D A B E (2) C B E A D (2) B E C A D (2) B C E A D (2) B A E C D (2) A D E C B (2) A D E B C (2) E D B C A (1) E D A B C (1) E C B D A (1) E B C D A (1) E B A C D (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E B A (1) D C E A B (1) D A E C B (1) C D E A B (1) C D A E B (1) C B E D A (1) C B A E D (1) C A D B E (1) C A B E D (1) B E A D C (1) B C A E D (1) A E D B C (1) A E B D C (1) A D B E C (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 14 36 2 6 B -14 0 8 -10 -4 C -36 -8 0 -10 -12 D -2 10 10 0 10 E -6 4 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 36 2 6 B -14 0 8 -10 -4 C -36 -8 0 -10 -12 D -2 10 10 0 10 E -6 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=24 B=18 E=15 C=12 so C is eliminated. Round 2 votes counts: D=37 A=26 B=22 E=15 so E is eliminated. Round 3 votes counts: D=39 B=34 A=27 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:229 D:214 E:200 B:190 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 36 2 6 B -14 0 8 -10 -4 C -36 -8 0 -10 -12 D -2 10 10 0 10 E -6 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 36 2 6 B -14 0 8 -10 -4 C -36 -8 0 -10 -12 D -2 10 10 0 10 E -6 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 36 2 6 B -14 0 8 -10 -4 C -36 -8 0 -10 -12 D -2 10 10 0 10 E -6 4 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6069: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B E C A D (8) B A C E D (8) E B D C A (7) D E A C B (6) D A C E B (6) B A C D E (6) A C B D E (6) E B C A D (4) D A C B E (4) E D C A B (3) E D B C A (3) E B C D A (3) D A B C E (3) B C A E D (3) A B C D E (3) E D C B A (2) E C B A D (2) D E B A C (2) C A B D E (2) A D C B E (2) E B D A C (1) D C A E B (1) D B A E C (1) D A E C B (1) C A D B E (1) B E D A C (1) B D E A C (1) B C E A D (1) B A D C E (1) Total count = 100 A B C D E A 0 -8 4 -6 -4 B 8 0 12 14 2 C -4 -12 0 -6 -6 D 6 -14 6 0 8 E 4 -2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -6 -4 B 8 0 12 14 2 C -4 -12 0 -6 -6 D 6 -14 6 0 8 E 4 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 E=25 A=11 C=3 so C is eliminated. Round 2 votes counts: D=32 B=29 E=25 A=14 so A is eliminated. Round 3 votes counts: B=40 D=35 E=25 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:203 E:200 A:193 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -6 -4 B 8 0 12 14 2 C -4 -12 0 -6 -6 D 6 -14 6 0 8 E 4 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -6 -4 B 8 0 12 14 2 C -4 -12 0 -6 -6 D 6 -14 6 0 8 E 4 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -6 -4 B 8 0 12 14 2 C -4 -12 0 -6 -6 D 6 -14 6 0 8 E 4 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6070: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) A E D B C (7) E D A B C (6) B D E C A (5) A C E D B (5) C B A D E (4) B D C E A (4) A E D C B (4) A E C D B (4) E D B A C (3) C A E D B (3) C A B E D (3) B C D E A (3) A B D C E (3) E D A C B (2) E A D C B (2) E A D B C (2) D B E C A (2) C B E D A (2) C B D E A (2) C A B D E (2) B C D A E (2) B C A D E (2) A C B E D (2) E D B C A (1) E C D B A (1) E A C D B (1) D E B C A (1) D B E A C (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A B D (1) C B D A E (1) C B A E D (1) C A E B D (1) B E D C A (1) B D E A C (1) B D C A E (1) B D A E C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 8 0 -4 B -2 0 8 -8 -12 C -8 -8 0 -10 -6 D 0 8 10 0 -12 E 4 12 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 8 0 -4 B -2 0 8 -8 -12 C -8 -8 0 -10 -6 D 0 8 10 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 B=20 E=18 D=12 so D is eliminated. Round 2 votes counts: E=27 A=27 C=23 B=23 so C is eliminated. Round 3 votes counts: A=36 B=33 E=31 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 A:203 D:203 B:193 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 8 0 -4 B -2 0 8 -8 -12 C -8 -8 0 -10 -6 D 0 8 10 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 0 -4 B -2 0 8 -8 -12 C -8 -8 0 -10 -6 D 0 8 10 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 0 -4 B -2 0 8 -8 -12 C -8 -8 0 -10 -6 D 0 8 10 0 -12 E 4 12 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6071: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (15) B D A E C (8) D B A C E (7) D A C B E (7) B D A C E (7) E C A B D (5) D A B C E (5) C E A D B (5) D C E A B (4) C A E D B (4) B E A D C (4) C D A E B (3) C A D E B (3) E C D A B (2) E B C A D (2) E B A C D (2) D C A E B (2) D C A B E (2) B E D A C (2) B E A C D (2) D B C A E (1) B E C D A (1) B D E A C (1) B A D E C (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 24 2 -4 10 B -24 0 -14 -28 0 C -2 14 0 -4 10 D 4 28 4 0 8 E -10 0 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 2 -4 10 B -24 0 -14 -28 0 C -2 14 0 -4 10 D 4 28 4 0 8 E -10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=26 C=15 A=5 so A is eliminated. Round 2 votes counts: D=29 E=26 B=26 C=19 so C is eliminated. Round 3 votes counts: D=37 E=36 B=27 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:216 C:209 E:186 B:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 2 -4 10 B -24 0 -14 -28 0 C -2 14 0 -4 10 D 4 28 4 0 8 E -10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 2 -4 10 B -24 0 -14 -28 0 C -2 14 0 -4 10 D 4 28 4 0 8 E -10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 2 -4 10 B -24 0 -14 -28 0 C -2 14 0 -4 10 D 4 28 4 0 8 E -10 0 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6072: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (14) E B D C A (9) A B E C D (9) E B A D C (8) C D A B E (8) B E A D C (8) C D A E B (5) A E B C D (3) A D C B E (3) A C D B E (3) A B D E C (3) E B C D A (2) E B A C D (2) C D E A B (2) C A D E B (2) B E D A C (2) A B C E D (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B D A (1) D C B E A (1) D B C E A (1) C E D A B (1) C D E B A (1) C A D B E (1) B E A C D (1) A C D E B (1) A C B D E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 0 -12 B 6 0 8 8 -8 C 2 -8 0 -6 -4 D 0 -8 6 0 -4 E 12 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 0 -12 B 6 0 8 8 -8 C 2 -8 0 -6 -4 D 0 -8 6 0 -4 E 12 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=25 C=20 D=16 B=11 so B is eliminated. Round 2 votes counts: E=36 A=28 C=20 D=16 so D is eliminated. Round 3 votes counts: E=36 C=36 A=28 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:207 D:197 C:192 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 0 -12 B 6 0 8 8 -8 C 2 -8 0 -6 -4 D 0 -8 6 0 -4 E 12 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 0 -12 B 6 0 8 8 -8 C 2 -8 0 -6 -4 D 0 -8 6 0 -4 E 12 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 0 -12 B 6 0 8 8 -8 C 2 -8 0 -6 -4 D 0 -8 6 0 -4 E 12 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6073: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (15) B C D A E (12) E C B A D (9) B C E D A (9) D A B C E (8) E A D C B (6) D A E B C (6) B C E A D (5) A E D C B (3) E D A C B (2) E D A B C (2) E C A D B (2) C E B A D (2) C B E D A (2) B D A C E (2) A D B C E (2) E C B D A (1) E C A B D (1) E A C D B (1) D E A C B (1) D E A B C (1) D A B E C (1) C E A B D (1) C B A D E (1) B E C D A (1) B D C A E (1) B C A D E (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -22 -26 2 -22 B 22 0 4 26 20 C 26 -4 0 26 22 D -2 -26 -26 0 -24 E 22 -20 -22 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -26 2 -22 B 22 0 4 26 20 C 26 -4 0 26 22 D -2 -26 -26 0 -24 E 22 -20 -22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=24 C=21 D=17 A=7 so A is eliminated. Round 2 votes counts: B=31 E=27 D=21 C=21 so D is eliminated. Round 3 votes counts: B=42 E=37 C=21 so C is eliminated. Round 4 votes counts: B=60 E=40 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:236 C:235 E:202 A:166 D:161 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -26 2 -22 B 22 0 4 26 20 C 26 -4 0 26 22 D -2 -26 -26 0 -24 E 22 -20 -22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -26 2 -22 B 22 0 4 26 20 C 26 -4 0 26 22 D -2 -26 -26 0 -24 E 22 -20 -22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -26 2 -22 B 22 0 4 26 20 C 26 -4 0 26 22 D -2 -26 -26 0 -24 E 22 -20 -22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6074: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) B E A C D (8) B E C D A (7) A D C B E (6) E B C D A (5) E C B D A (4) E B C A D (4) B E D C A (4) B E C A D (4) E C D A B (3) C E D A B (3) B A E C D (3) A C D E B (3) E C D B A (2) E B A C D (2) D B C E A (2) B E D A C (2) B D E C A (2) B D E A C (2) B A E D C (2) A D C E B (2) A B D C E (2) E C A D B (1) E C A B D (1) E B D C A (1) E A C B D (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C A E (1) D A C E B (1) D A B C E (1) C E A D B (1) C D E A B (1) C D A E B (1) C A D E B (1) B D C A E (1) B A D E C (1) A E C D B (1) A D B C E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -20 -8 -26 B 18 0 14 16 4 C 20 -14 0 16 -22 D 8 -16 -16 0 -22 E 26 -4 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999179 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -20 -8 -26 B 18 0 14 16 4 C 20 -14 0 16 -22 D 8 -16 -16 0 -22 E 26 -4 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=24 A=17 D=16 C=7 so C is eliminated. Round 2 votes counts: B=36 E=28 D=18 A=18 so D is eliminated. Round 3 votes counts: B=40 E=31 A=29 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:233 B:226 C:200 D:177 A:164 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -20 -8 -26 B 18 0 14 16 4 C 20 -14 0 16 -22 D 8 -16 -16 0 -22 E 26 -4 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -20 -8 -26 B 18 0 14 16 4 C 20 -14 0 16 -22 D 8 -16 -16 0 -22 E 26 -4 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -20 -8 -26 B 18 0 14 16 4 C 20 -14 0 16 -22 D 8 -16 -16 0 -22 E 26 -4 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6075: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (14) D B E A C (9) D E B A C (8) C A E D B (8) D E C A B (5) B C A E D (5) D E A C B (4) C E A D B (3) C D E A B (3) C B A E D (3) B A C E D (3) A E C B D (3) E D A C B (2) E A C D B (2) D E A B C (2) D C E A B (2) C A E B D (2) B D C E A (2) B C D A E (2) B A E D C (2) B A E C D (2) A E C D B (2) E D A B C (1) E A D C B (1) E A D B C (1) E A B D C (1) D E B C A (1) D B E C A (1) C E D A B (1) C D B E A (1) C A B E D (1) B D A E C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 20 -18 -28 B 8 0 14 -14 -4 C -20 -14 0 -14 -26 D 18 14 14 0 10 E 28 4 26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 20 -18 -28 B 8 0 14 -14 -4 C -20 -14 0 -14 -26 D 18 14 14 0 10 E 28 4 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=31 C=22 E=8 A=7 so A is eliminated. Round 2 votes counts: B=33 D=32 C=22 E=13 so E is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:224 B:202 A:183 C:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 20 -18 -28 B 8 0 14 -14 -4 C -20 -14 0 -14 -26 D 18 14 14 0 10 E 28 4 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 20 -18 -28 B 8 0 14 -14 -4 C -20 -14 0 -14 -26 D 18 14 14 0 10 E 28 4 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 20 -18 -28 B 8 0 14 -14 -4 C -20 -14 0 -14 -26 D 18 14 14 0 10 E 28 4 26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6076: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) A B E C D (6) A B C E D (6) E D B C A (4) E B C A D (4) D C E B A (4) D A C B E (4) C E B D A (4) E D C B A (3) E C B D A (3) D E B C A (3) D C E A B (3) D C A B E (3) B E C A D (3) E C B A D (2) D E B A C (2) D A E B C (2) D A C E B (2) C D E B A (2) C B E A D (2) B E A C D (2) B A E C D (2) B A C E D (2) A D B C E (2) A C B D E (2) A B D E C (2) A B D C E (2) E D B A C (1) E B D C A (1) E B D A C (1) E B C D A (1) E A B D C (1) D E A C B (1) D E A B C (1) D C A E B (1) D A E C B (1) C E D B A (1) C E B A D (1) C B A E D (1) C A B E D (1) C A B D E (1) B C E A D (1) B C A E D (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -12 -8 -14 B 14 0 2 6 -10 C 12 -2 0 -4 -6 D 8 -6 4 0 -10 E 14 10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -12 -8 -14 B 14 0 2 6 -10 C 12 -2 0 -4 -6 D 8 -6 4 0 -10 E 14 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=22 E=21 C=13 B=11 so B is eliminated. Round 2 votes counts: D=33 E=26 A=26 C=15 so C is eliminated. Round 3 votes counts: E=35 D=35 A=30 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:206 C:200 D:198 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -12 -8 -14 B 14 0 2 6 -10 C 12 -2 0 -4 -6 D 8 -6 4 0 -10 E 14 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -8 -14 B 14 0 2 6 -10 C 12 -2 0 -4 -6 D 8 -6 4 0 -10 E 14 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -8 -14 B 14 0 2 6 -10 C 12 -2 0 -4 -6 D 8 -6 4 0 -10 E 14 10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6077: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) C A E B D (6) B E D C A (5) A E C D B (5) E B D A C (4) E A D B C (3) D B E C A (3) D B E A C (3) C D A B E (3) C B E D A (3) C B D E A (3) B D E C A (3) A D E B C (3) A D C B E (3) A C D E B (3) E A B D C (2) D B C E A (2) D B C A E (2) C E B A D (2) C D B A E (2) C B A E D (2) C A B E D (2) C A B D E (2) B E C D A (2) B C D E A (2) A C D B E (2) E D B A C (1) E D A B C (1) E C A B D (1) E B C A D (1) E B A D C (1) E B A C D (1) D E B A C (1) D E A B C (1) D C B A E (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B D A (1) C E A B D (1) C A D E B (1) C A D B E (1) B D C E A (1) A E D B C (1) A D C E B (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -18 -8 -10 B 8 0 8 6 0 C 18 -8 0 -6 -4 D 8 -6 6 0 -6 E 10 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.562446 C: 0.000000 D: 0.000000 E: 0.437554 Sum of squares = 0.507798975273 Cumulative probabilities = A: 0.000000 B: 0.562446 C: 0.562446 D: 0.562446 E: 1.000000 A B C D E A 0 -8 -18 -8 -10 B 8 0 8 6 0 C 18 -8 0 -6 -4 D 8 -6 6 0 -6 E 10 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=22 A=20 D=16 B=13 so B is eliminated. Round 2 votes counts: C=31 E=29 D=20 A=20 so D is eliminated. Round 3 votes counts: E=40 C=37 A=23 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:211 E:210 D:201 C:200 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -18 -8 -10 B 8 0 8 6 0 C 18 -8 0 -6 -4 D 8 -6 6 0 -6 E 10 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -8 -10 B 8 0 8 6 0 C 18 -8 0 -6 -4 D 8 -6 6 0 -6 E 10 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -8 -10 B 8 0 8 6 0 C 18 -8 0 -6 -4 D 8 -6 6 0 -6 E 10 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6078: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) A C D B E (9) A C B E D (9) D E B A C (7) A D E B C (7) D A E B C (6) A D C E B (5) E B D C A (4) C B E A D (4) E D B C A (3) D E B C A (3) D A E C B (3) A B E D C (3) D E C B A (2) D C A E B (2) C D B E A (2) C B A E D (2) B E C D A (2) B E C A D (2) B C E D A (2) A E B D C (2) A D C B E (2) A C B D E (2) A B C E D (2) E B C D A (1) D C E A B (1) C D A B E (1) C A B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 12 16 2 14 B -12 0 -10 -6 8 C -16 10 0 0 8 D -2 6 0 0 4 E -14 -8 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 2 14 B -12 0 -10 -6 8 C -16 10 0 0 8 D -2 6 0 0 4 E -14 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 D=24 C=20 E=8 B=6 so B is eliminated. Round 2 votes counts: A=42 D=24 C=22 E=12 so E is eliminated. Round 3 votes counts: A=42 D=31 C=27 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:204 C:201 B:190 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 2 14 B -12 0 -10 -6 8 C -16 10 0 0 8 D -2 6 0 0 4 E -14 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 2 14 B -12 0 -10 -6 8 C -16 10 0 0 8 D -2 6 0 0 4 E -14 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 2 14 B -12 0 -10 -6 8 C -16 10 0 0 8 D -2 6 0 0 4 E -14 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6079: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) A E C B D (8) A C E D B (8) C D A B E (7) D C B A E (6) D B C E A (6) B E D A C (6) A C D E B (6) E A B C D (5) E B D A C (4) E B A D C (4) B E D C A (4) B D C E A (4) A E C D B (4) E B A C D (3) C A D B E (3) D C A B E (2) C A E D B (2) C A D E B (2) E B C A D (1) E A C B D (1) E A B D C (1) D B C A E (1) D A C B E (1) B D E A C (1) Total count = 100 A B C D E A 0 0 4 -4 0 B 0 0 0 4 2 C -4 0 0 0 -4 D 4 -4 0 0 -2 E 0 -2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.354492 B: 0.645508 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.54234544571 Cumulative probabilities = A: 0.354492 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -4 0 B 0 0 0 4 2 C -4 0 0 0 -4 D 4 -4 0 0 -2 E 0 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 E=19 D=16 C=14 so C is eliminated. Round 2 votes counts: A=33 B=25 D=23 E=19 so E is eliminated. Round 3 votes counts: A=40 B=37 D=23 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:203 E:202 A:200 D:199 C:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 -4 0 B 0 0 0 4 2 C -4 0 0 0 -4 D 4 -4 0 0 -2 E 0 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -4 0 B 0 0 0 4 2 C -4 0 0 0 -4 D 4 -4 0 0 -2 E 0 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -4 0 B 0 0 0 4 2 C -4 0 0 0 -4 D 4 -4 0 0 -2 E 0 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000008 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6080: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) B D A E C (10) E C D B A (7) D B A E C (7) C E A D B (7) A C E D B (6) A C B E D (6) E D C B A (5) B A D C E (5) A B C D E (4) D E B C A (3) D B E A C (3) C A E D B (3) A C E B D (3) E D B C A (2) E C D A B (2) D B E C A (2) C E A B D (2) C A E B D (2) B D E C A (2) E C B D A (1) D E C B A (1) C A B E D (1) B E C D A (1) B C A E D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 16 8 24 B 0 0 6 2 12 C -16 -6 0 -6 8 D -8 -2 6 0 2 E -24 -12 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.234462 B: 0.765538 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.641020447069 Cumulative probabilities = A: 0.234462 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 8 24 B 0 0 6 2 12 C -16 -6 0 -6 8 D -8 -2 6 0 2 E -24 -12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=19 E=17 D=16 C=15 so C is eliminated. Round 2 votes counts: A=39 E=26 B=19 D=16 so D is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:210 D:199 C:190 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 16 8 24 B 0 0 6 2 12 C -16 -6 0 -6 8 D -8 -2 6 0 2 E -24 -12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 8 24 B 0 0 6 2 12 C -16 -6 0 -6 8 D -8 -2 6 0 2 E -24 -12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 8 24 B 0 0 6 2 12 C -16 -6 0 -6 8 D -8 -2 6 0 2 E -24 -12 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6081: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (16) D C A B E (8) A D C E B (7) D C B E A (5) C D B E A (5) E B A D C (4) E B A C D (4) E A B C D (4) C D A B E (4) A E D B C (4) E A B D C (3) D A C E B (3) C D B A E (3) B E A C D (3) B C E D A (3) B C D E A (3) D C A E B (2) C B D E A (2) C B D A E (2) B E C A D (2) A E D C B (2) A E B C D (2) A D E C B (2) A C D E B (2) E D B A C (1) B E C D A (1) B A E C D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 22 20 16 20 B -22 0 4 4 -14 C -20 -4 0 -14 -2 D -16 -4 14 0 -4 E -20 14 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 20 16 20 B -22 0 4 4 -14 C -20 -4 0 -14 -2 D -16 -4 14 0 -4 E -20 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=18 E=16 C=16 B=13 so B is eliminated. Round 2 votes counts: A=38 E=22 C=22 D=18 so D is eliminated. Round 3 votes counts: A=41 C=37 E=22 so E is eliminated. Round 4 votes counts: A=60 C=40 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:239 E:200 D:195 B:186 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 20 16 20 B -22 0 4 4 -14 C -20 -4 0 -14 -2 D -16 -4 14 0 -4 E -20 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 20 16 20 B -22 0 4 4 -14 C -20 -4 0 -14 -2 D -16 -4 14 0 -4 E -20 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 20 16 20 B -22 0 4 4 -14 C -20 -4 0 -14 -2 D -16 -4 14 0 -4 E -20 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6082: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (7) D E B A C (5) C D E B A (5) C A D E B (5) B E D C A (5) B E D A C (5) B A E D C (5) E D B A C (4) E B D A C (4) D E C B A (4) D E B C A (4) B E A D C (4) D C E A B (3) C A B D E (3) D E A C B (2) D C E B A (2) C D E A B (2) C D B E A (2) C D A E B (2) C B D E A (2) C B A E D (2) C A D B E (2) C A B E D (2) B A E C D (2) A E D B C (2) A D E C B (2) A B E D C (2) A B E C D (2) E D B C A (1) E B A D C (1) C B D A E (1) B C A E D (1) A E D C B (1) A C E D B (1) A C D E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 4 -6 -10 B 18 0 -4 -2 -2 C -4 4 0 -12 -10 D 6 2 12 0 -4 E 10 2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 4 -6 -10 B 18 0 -4 -2 -2 C -4 4 0 -12 -10 D 6 2 12 0 -4 E 10 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=22 D=20 A=20 E=10 so E is eliminated. Round 2 votes counts: C=28 B=27 D=25 A=20 so A is eliminated. Round 3 votes counts: C=38 B=32 D=30 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 D:208 B:205 C:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 4 -6 -10 B 18 0 -4 -2 -2 C -4 4 0 -12 -10 D 6 2 12 0 -4 E 10 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 -6 -10 B 18 0 -4 -2 -2 C -4 4 0 -12 -10 D 6 2 12 0 -4 E 10 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 -6 -10 B 18 0 -4 -2 -2 C -4 4 0 -12 -10 D 6 2 12 0 -4 E 10 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6083: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) C E B D A (5) B D A E C (5) C D B E A (4) A E B D C (4) E C A B D (3) D C B A E (3) D B A E C (3) D B A C E (3) D A B C E (3) C E A D B (3) C E A B D (3) B D C E A (3) A D C B E (3) A D B E C (3) A B D E C (3) E B C D A (2) E B C A D (2) E A C B D (2) E A B C D (2) D B C E A (2) D B C A E (2) C A E D B (2) C A D E B (2) B D E C A (2) B D E A C (2) A E C D B (2) A E B C D (2) E C B D A (1) E B A D C (1) E B A C D (1) D C B E A (1) D C A B E (1) D B E C A (1) D A C B E (1) C E D A B (1) C E B A D (1) C D E A B (1) B E D C A (1) B E A D C (1) B C E D A (1) B A D E C (1) A E D C B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 4 6 4 8 B -4 0 2 14 0 C -6 -2 0 -2 -8 D -4 -14 2 0 0 E -8 0 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 8 B -4 0 2 14 0 C -6 -2 0 -2 -8 D -4 -14 2 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 D=20 B=16 E=14 so E is eliminated. Round 2 votes counts: A=32 C=26 B=22 D=20 so D is eliminated. Round 3 votes counts: A=36 B=33 C=31 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:206 E:200 D:192 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 8 B -4 0 2 14 0 C -6 -2 0 -2 -8 D -4 -14 2 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 8 B -4 0 2 14 0 C -6 -2 0 -2 -8 D -4 -14 2 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 8 B -4 0 2 14 0 C -6 -2 0 -2 -8 D -4 -14 2 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6084: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) D C B A E (5) D B A E C (5) C E A B D (5) C B A E D (5) D E A B C (4) D C E B A (4) D B C A E (4) C D E A B (4) B A D E C (4) A B E C D (4) E A B D C (3) E A B C D (3) D E C A B (3) D C E A B (3) D B A C E (3) C D B A E (3) C B A D E (3) C A E B D (3) C A B E D (3) E D C A B (2) E C A B D (2) C E A D B (2) B D A C E (2) B A E D C (2) B A C E D (2) E D A B C (1) E A D B C (1) E A C B D (1) D E B C A (1) D B E A C (1) C E D A B (1) C D E B A (1) B D A E C (1) B A C D E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -8 -8 4 B 6 0 -2 -8 -4 C 8 2 0 -10 8 D 8 8 10 0 16 E -4 4 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -8 4 B 6 0 -2 -8 -4 C 8 2 0 -10 8 D 8 8 10 0 16 E -4 4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=30 E=13 B=12 A=6 so A is eliminated. Round 2 votes counts: D=39 C=30 B=16 E=15 so E is eliminated. Round 3 votes counts: D=43 C=34 B=23 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:204 B:196 A:191 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -8 -8 4 B 6 0 -2 -8 -4 C 8 2 0 -10 8 D 8 8 10 0 16 E -4 4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -8 4 B 6 0 -2 -8 -4 C 8 2 0 -10 8 D 8 8 10 0 16 E -4 4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -8 4 B 6 0 -2 -8 -4 C 8 2 0 -10 8 D 8 8 10 0 16 E -4 4 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6085: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (12) D C A E B (5) B E D C A (5) B C D A E (5) A C D B E (5) B E A C D (4) B A C E D (4) A E C D B (4) E A B D C (3) D E C A B (3) C D A B E (3) B D C E A (3) B A C D E (3) E D C B A (2) E D C A B (2) D C B A E (2) C A D B E (2) C A B D E (2) B E A D C (2) A E B C D (2) A D C E B (2) A C B D E (2) E D B C A (1) E D A C B (1) E D A B C (1) E B D C A (1) E B D A C (1) E B A D C (1) E A D C B (1) D E A C B (1) D C E B A (1) D C E A B (1) D A C E B (1) C D B A E (1) C D A E B (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D A C (1) B D E C A (1) B C A D E (1) B A E C D (1) A E D C B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 16 10 12 28 B -16 0 -18 -8 2 C -10 18 0 14 20 D -12 8 -14 0 20 E -28 -2 -20 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 12 28 B -16 0 -18 -8 2 C -10 18 0 14 20 D -12 8 -14 0 20 E -28 -2 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=30 A=30 E=14 D=14 C=12 so C is eliminated. Round 2 votes counts: A=35 B=32 D=19 E=14 so E is eliminated. Round 3 votes counts: A=39 B=35 D=26 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 C:221 D:201 B:180 E:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 12 28 B -16 0 -18 -8 2 C -10 18 0 14 20 D -12 8 -14 0 20 E -28 -2 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 12 28 B -16 0 -18 -8 2 C -10 18 0 14 20 D -12 8 -14 0 20 E -28 -2 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 12 28 B -16 0 -18 -8 2 C -10 18 0 14 20 D -12 8 -14 0 20 E -28 -2 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6086: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) C D A E B (8) E A B D C (6) B E D A C (6) B E A C D (6) E B A D C (5) E A B C D (4) C A D E B (4) E B A C D (3) D C B A E (3) D C A E B (3) B C E D A (3) E A C D B (2) D C A B E (2) D B C A E (2) D A E C B (2) D A E B C (2) C D B A E (2) C D A B E (2) C B D E A (2) C A E D B (2) B D E C A (2) B D E A C (2) A E D C B (2) A E C D B (2) A D E C B (2) A D E B C (2) E A D B C (1) D B A C E (1) D A C E B (1) C E A B D (1) B E D C A (1) B D C A E (1) B D A E C (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 0 24 8 -8 B 0 0 18 6 -8 C -24 -18 0 -18 -24 D -8 -6 18 0 -10 E 8 8 24 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 24 8 -8 B 0 0 18 6 -8 C -24 -18 0 -18 -24 D -8 -6 18 0 -10 E 8 8 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=21 C=21 D=16 A=10 so A is eliminated. Round 2 votes counts: B=32 E=26 D=21 C=21 so D is eliminated. Round 3 votes counts: B=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:212 B:208 D:197 C:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 24 8 -8 B 0 0 18 6 -8 C -24 -18 0 -18 -24 D -8 -6 18 0 -10 E 8 8 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 24 8 -8 B 0 0 18 6 -8 C -24 -18 0 -18 -24 D -8 -6 18 0 -10 E 8 8 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 24 8 -8 B 0 0 18 6 -8 C -24 -18 0 -18 -24 D -8 -6 18 0 -10 E 8 8 24 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6087: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) C B E A D (6) B E A D C (6) A D E B C (6) D A E C B (5) C D A E B (5) B C E A D (5) C B E D A (4) A D B E C (4) E C D B A (3) E C B D A (3) E B D C A (3) E B D A C (3) E B C D A (3) D E A B C (3) D A E B C (3) C A D B E (3) A D C E B (3) C B A D E (2) C A B D E (2) B A D E C (2) A D E C B (2) A D B C E (2) E D A C B (1) E C D A B (1) C E D B A (1) C E D A B (1) C D E A B (1) C B D A E (1) B E D A C (1) B E C D A (1) B E C A D (1) B C A E D (1) B A E D C (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 -14 -12 -4 -12 B 14 0 -8 10 -12 C 12 8 0 8 -4 D 4 -10 -8 0 -8 E 12 12 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -12 -4 -12 B 14 0 -8 10 -12 C 12 8 0 8 -4 D 4 -10 -8 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=19 A=18 E=17 D=11 so D is eliminated. Round 2 votes counts: C=35 A=26 E=20 B=19 so B is eliminated. Round 3 votes counts: C=41 A=30 E=29 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:218 C:212 B:202 D:189 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -12 -4 -12 B 14 0 -8 10 -12 C 12 8 0 8 -4 D 4 -10 -8 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -4 -12 B 14 0 -8 10 -12 C 12 8 0 8 -4 D 4 -10 -8 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -4 -12 B 14 0 -8 10 -12 C 12 8 0 8 -4 D 4 -10 -8 0 -8 E 12 12 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6088: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) B E D A C (7) E B D C A (6) D B E C A (5) C A D E B (5) B D E A C (5) A C B E D (5) A B E C D (5) D C A B E (4) B E D C A (4) A C E B D (4) D C B E A (3) A C D E B (3) A B E D C (3) E B A C D (2) C E A D B (2) C D E B A (2) C D E A B (2) C A E D B (2) B D E C A (2) B A E D C (2) A D B C E (2) A C B D E (2) E B D A C (1) E B C D A (1) E B A D C (1) E A C B D (1) D B A C E (1) C E D B A (1) C E D A B (1) C D A B E (1) C A E B D (1) B E A D C (1) A C E D B (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 16 8 6 B -12 0 2 12 28 C -16 -2 0 4 8 D -8 -12 -4 0 -4 E -6 -28 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 8 6 B -12 0 2 12 28 C -16 -2 0 4 8 D -8 -12 -4 0 -4 E -6 -28 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=21 C=17 D=13 E=12 so E is eliminated. Round 2 votes counts: A=38 B=32 C=17 D=13 so D is eliminated. Round 3 votes counts: B=38 A=38 C=24 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:215 C:197 D:186 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 8 6 B -12 0 2 12 28 C -16 -2 0 4 8 D -8 -12 -4 0 -4 E -6 -28 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 8 6 B -12 0 2 12 28 C -16 -2 0 4 8 D -8 -12 -4 0 -4 E -6 -28 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 8 6 B -12 0 2 12 28 C -16 -2 0 4 8 D -8 -12 -4 0 -4 E -6 -28 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998261 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6089: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) E B A D C (7) D C A B E (7) E D C A B (5) B E C A D (5) B E A C D (5) E B C D A (4) A D C B E (4) E B A C D (3) E A D B C (3) D C A E B (3) C B D A E (3) E D A C B (2) D E A C B (2) D A C E B (2) D A C B E (2) C D E B A (2) C D A E B (2) C B D E A (2) C B A D E (2) A E D B C (2) A B E D C (2) E D C B A (1) E C D B A (1) E C D A B (1) E C B D A (1) E B D C A (1) E B C A D (1) E A B D C (1) C E D B A (1) C D B A E (1) C B E D A (1) C A D B E (1) B E C D A (1) B E A D C (1) B C E D A (1) B C E A D (1) B C A D E (1) B A E D C (1) B A C D E (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 4 -20 -14 -6 B -4 0 -16 -10 8 C 20 16 0 4 0 D 14 10 -4 0 -4 E 6 -8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.578231 D: 0.000000 E: 0.421769 Sum of squares = 0.512240061987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.578231 D: 0.578231 E: 1.000000 A B C D E A 0 4 -20 -14 -6 B -4 0 -16 -10 8 C 20 16 0 4 0 D 14 10 -4 0 -4 E 6 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.499997 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.500003 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=26 B=17 D=16 A=10 so A is eliminated. Round 2 votes counts: E=33 C=26 D=22 B=19 so B is eliminated. Round 3 votes counts: E=48 C=30 D=22 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:208 E:201 B:189 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -20 -14 -6 B -4 0 -16 -10 8 C 20 16 0 4 0 D 14 10 -4 0 -4 E 6 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.499997 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.500003 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -14 -6 B -4 0 -16 -10 8 C 20 16 0 4 0 D 14 10 -4 0 -4 E 6 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.499997 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.500003 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -14 -6 B -4 0 -16 -10 8 C 20 16 0 4 0 D 14 10 -4 0 -4 E 6 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.000000 E: 0.499997 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500003 D: 0.500003 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6090: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) C E B D A (6) E C B D A (5) D A C E B (5) D A B C E (5) A D B E C (5) B E C A D (4) B D A C E (4) B C E D A (4) B A E C D (4) E C D A B (3) E C B A D (3) C E D B A (3) C B E D A (3) B C E A D (3) A E D C B (3) A D E B C (3) A D B C E (3) A B D E C (3) D A E C B (2) B A D C E (2) A B E D C (2) E D A C B (1) E C D B A (1) E B C A D (1) E A D C B (1) E A C D B (1) E A B C D (1) D E C A B (1) D E A C B (1) D C E A B (1) D C B E A (1) D A C B E (1) C E D A B (1) C D E A B (1) B C D E A (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 16 0 6 B -8 0 -6 -4 -6 C -16 6 0 -8 -8 D 0 4 8 0 0 E -6 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.487397 B: 0.000000 C: 0.000000 D: 0.512603 E: 0.000000 Sum of squares = 0.500317694861 Cumulative probabilities = A: 0.487397 B: 0.487397 C: 0.487397 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 0 6 B -8 0 -6 -4 -6 C -16 6 0 -8 -8 D 0 4 8 0 0 E -6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999926 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 E=17 D=17 C=14 so C is eliminated. Round 2 votes counts: A=29 E=27 B=26 D=18 so D is eliminated. Round 3 votes counts: A=42 E=31 B=27 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:206 E:204 B:188 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 0 6 B -8 0 -6 -4 -6 C -16 6 0 -8 -8 D 0 4 8 0 0 E -6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999926 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 0 6 B -8 0 -6 -4 -6 C -16 6 0 -8 -8 D 0 4 8 0 0 E -6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999926 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 0 6 B -8 0 -6 -4 -6 C -16 6 0 -8 -8 D 0 4 8 0 0 E -6 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999926 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6091: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) B C D E A (9) D E A C B (7) D B E A C (7) A E D C B (7) D E A B C (6) C A E B D (6) E A D C B (5) C A E D B (5) B C A E D (5) C A B E D (4) B D E A C (4) D E B A C (3) D A E C B (3) B D E C A (3) A C E D B (3) B C D A E (2) E D B A C (1) E D A C B (1) E A D B C (1) D B C A E (1) C B D A E (1) B E D A C (1) B D C E A (1) B C E D A (1) B C E A D (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 0 2 -2 -2 B 0 0 -8 -4 0 C -2 8 0 -4 -2 D 2 4 4 0 -4 E 2 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.131005 C: 0.000000 D: 0.000000 E: 0.868995 Sum of squares = 0.772314704186 Cumulative probabilities = A: 0.000000 B: 0.131005 C: 0.131005 D: 0.131005 E: 1.000000 A B C D E A 0 0 2 -2 -2 B 0 0 -8 -4 0 C -2 8 0 -4 -2 D 2 4 4 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000000497 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 C=26 A=12 E=8 so E is eliminated. Round 2 votes counts: D=29 B=27 C=26 A=18 so A is eliminated. Round 3 votes counts: D=43 C=30 B=27 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:204 D:203 C:200 A:199 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -2 -2 B 0 0 -8 -4 0 C -2 8 0 -4 -2 D 2 4 4 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000000497 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -2 B 0 0 -8 -4 0 C -2 8 0 -4 -2 D 2 4 4 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000000497 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -2 B 0 0 -8 -4 0 C -2 8 0 -4 -2 D 2 4 4 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000000497 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6092: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) D A B C E (6) A D E C B (6) B E C D A (5) A D B C E (5) A B E C D (5) E C D B A (4) B C E D A (4) A E D C B (4) A B D E C (4) E C B D A (3) E C B A D (3) D B C E A (3) B E C A D (3) B D C E A (3) B A E C D (3) A D C E B (3) E B C D A (2) E A B C D (2) D A C E B (2) D A C B E (2) B C D E A (2) A E C D B (2) A D B E C (2) E C D A B (1) E C A B D (1) E B C A D (1) E A C B D (1) D E C A B (1) D C E A B (1) D B C A E (1) D B A C E (1) B E A C D (1) A E C B D (1) A E B C D (1) A D C B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 2 2 B -6 0 12 -6 6 C -8 -12 0 -10 -14 D -2 6 10 0 4 E -2 -6 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 2 2 B -6 0 12 -6 6 C -8 -12 0 -10 -14 D -2 6 10 0 4 E -2 -6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=25 B=21 E=18 so C is eliminated. Round 2 votes counts: A=36 D=25 B=21 E=18 so E is eliminated. Round 3 votes counts: A=40 D=30 B=30 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 D:209 B:203 E:201 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 2 B -6 0 12 -6 6 C -8 -12 0 -10 -14 D -2 6 10 0 4 E -2 -6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 2 B -6 0 12 -6 6 C -8 -12 0 -10 -14 D -2 6 10 0 4 E -2 -6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 2 B -6 0 12 -6 6 C -8 -12 0 -10 -14 D -2 6 10 0 4 E -2 -6 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997842 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6093: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) B A E D C (8) E C D A B (7) B A D C E (7) E A B D C (6) C D B A E (6) E A D C B (5) E B A C D (4) C D E A B (4) C D A B E (3) B D C A E (3) A D E C B (3) E C B D A (2) E C A D B (2) E A D B C (2) E A C D B (2) D C A B E (2) B C E D A (2) B A D E C (2) A D C B E (2) A D B E C (2) A D B C E (2) A B D C E (2) E B C A D (1) E B A D C (1) D C E A B (1) D C B A E (1) D C A E B (1) D B A C E (1) D A E C B (1) D A B C E (1) C E D B A (1) C B D E A (1) B E C A D (1) B E A D C (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 0 8 10 14 B 0 0 12 -2 10 C -8 -12 0 -12 -4 D -10 2 12 0 8 E -14 -10 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.583504 B: 0.416496 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513945730031 Cumulative probabilities = A: 0.583504 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 10 14 B 0 0 12 -2 10 C -8 -12 0 -12 -4 D -10 2 12 0 8 E -14 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=32 B=32 C=15 A=13 D=8 so D is eliminated. Round 2 votes counts: B=33 E=32 C=20 A=15 so A is eliminated. Round 3 votes counts: B=40 E=38 C=22 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:210 D:206 E:186 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 10 14 B 0 0 12 -2 10 C -8 -12 0 -12 -4 D -10 2 12 0 8 E -14 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 10 14 B 0 0 12 -2 10 C -8 -12 0 -12 -4 D -10 2 12 0 8 E -14 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 10 14 B 0 0 12 -2 10 C -8 -12 0 -12 -4 D -10 2 12 0 8 E -14 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6094: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (6) C D E B A (6) C D A E B (5) E B A C D (4) D C A B E (4) D B C E A (4) B E A D C (4) A D B E C (4) E C B A D (3) D C B A E (3) B E C D A (3) B E A C D (3) A E B C D (3) E B C A D (2) D C B E A (2) D C A E B (2) D B A E C (2) D A B C E (2) C E D B A (2) C E B D A (2) C E B A D (2) C D B E A (2) B E D A C (2) B E C A D (2) B D E A C (2) B A E D C (2) A C D E B (2) A B D E C (2) E A C B D (1) E A B C D (1) D B E C A (1) D B E A C (1) D B A C E (1) D A C E B (1) C E D A B (1) C D E A B (1) C B D E A (1) C A E B D (1) B E D C A (1) B D A E C (1) B C E D A (1) A E C B D (1) A E B D C (1) A D C E B (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 -2 -18 -8 B 18 0 0 -8 14 C 2 0 0 0 6 D 18 8 0 0 12 E 8 -14 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.487749 D: 0.512251 E: 0.000000 Sum of squares = 0.500300161691 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.487749 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -2 -18 -8 B 18 0 0 -8 14 C 2 0 0 0 6 D 18 8 0 0 12 E 8 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=23 B=21 A=16 E=11 so E is eliminated. Round 2 votes counts: D=29 B=27 C=26 A=18 so A is eliminated. Round 3 votes counts: B=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:212 C:204 E:188 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -2 -18 -8 B 18 0 0 -8 14 C 2 0 0 0 6 D 18 8 0 0 12 E 8 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 -18 -8 B 18 0 0 -8 14 C 2 0 0 0 6 D 18 8 0 0 12 E 8 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 -18 -8 B 18 0 0 -8 14 C 2 0 0 0 6 D 18 8 0 0 12 E 8 -14 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6095: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (5) E A C B D (5) C B E D A (5) A E D B C (5) A D E B C (5) E C A D B (4) C E B D A (4) C B D E A (4) B A D C E (4) E A C D B (3) D A E B C (3) D A B E C (3) C E D B A (3) C D E B A (3) B C D E A (3) E C D A B (2) E A D C B (2) D E A C B (2) D C E A B (2) D C B E A (2) D B C A E (2) D B A C E (2) C E B A D (2) C B E A D (2) B C E A D (2) B C D A E (2) B A D E C (2) B A C D E (2) A E B C D (2) A B D E C (2) E D C A B (1) E D A C B (1) E A B C D (1) D E C A B (1) D C B A E (1) C D B E A (1) B D C A E (1) B A E C D (1) B A C E D (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 2 -4 4 -20 B -2 0 -14 0 -16 C 4 14 0 16 -4 D -4 0 -16 0 -6 E 20 16 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999023 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -4 4 -20 B -2 0 -14 0 -16 C 4 14 0 16 -4 D -4 0 -16 0 -6 E 20 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=24 C=24 D=18 B=18 A=16 so A is eliminated. Round 2 votes counts: E=33 C=24 D=23 B=20 so B is eliminated. Round 3 votes counts: E=34 C=34 D=32 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:215 A:191 D:187 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 4 -20 B -2 0 -14 0 -16 C 4 14 0 16 -4 D -4 0 -16 0 -6 E 20 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 -20 B -2 0 -14 0 -16 C 4 14 0 16 -4 D -4 0 -16 0 -6 E 20 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 -20 B -2 0 -14 0 -16 C 4 14 0 16 -4 D -4 0 -16 0 -6 E 20 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6096: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (29) E B D C A (23) E A C D B (10) B D E C A (4) B D C A E (4) A C D E B (4) A E C D B (3) E B A D C (2) C D B A E (2) C A D B E (2) B E D C A (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D A B (1) E B D A C (1) E B A C D (1) E A D C B (1) E A B D C (1) D C A B E (1) D B C E A (1) D B C A E (1) B D C E A (1) B C D A E (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 8 10 -2 B -8 0 -12 -18 -2 C -8 12 0 10 -4 D -10 18 -10 0 2 E 2 2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408264 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 A B C D E A 0 8 8 10 -2 B -8 0 -12 -18 -2 C -8 12 0 10 -4 D -10 18 -10 0 2 E 2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020407868 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 A=37 B=13 C=4 D=3 so D is eliminated. Round 2 votes counts: E=43 A=37 B=15 C=5 so C is eliminated. Round 3 votes counts: E=43 A=40 B=17 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:212 C:205 E:203 D:200 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 10 -2 B -8 0 -12 -18 -2 C -8 12 0 10 -4 D -10 18 -10 0 2 E 2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020407868 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 10 -2 B -8 0 -12 -18 -2 C -8 12 0 10 -4 D -10 18 -10 0 2 E 2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020407868 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 10 -2 B -8 0 -12 -18 -2 C -8 12 0 10 -4 D -10 18 -10 0 2 E 2 2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.714286 Sum of squares = 0.551020407868 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.285714 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6097: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) E C A B D (5) D B C A E (5) E D B A C (4) E A C B D (4) E A B C D (4) D B A E C (4) D B A C E (4) C D E B A (4) C D B A E (4) B D A C E (4) D C B E A (3) D C B A E (3) B A D E C (3) A B E D C (3) A B D C E (3) E D B C A (2) E C D A B (2) D B E A C (2) C E D B A (2) C D A E B (2) C A B D E (2) B A D C E (2) A B D E C (2) A B C E D (2) E C A D B (1) E B A D C (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E B A (1) D B E C A (1) D B C E A (1) C E A B D (1) C D A B E (1) C B D A E (1) B D C A E (1) B D A E C (1) A E B D C (1) A E B C D (1) A C E B D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 12 -10 4 B 12 0 24 2 8 C -12 -24 0 -22 -4 D 10 -2 22 0 16 E -4 -8 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 12 -10 4 B 12 0 24 2 8 C -12 -24 0 -22 -4 D 10 -2 22 0 16 E -4 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 C=17 A=15 B=11 so B is eliminated. Round 2 votes counts: D=33 E=30 A=20 C=17 so C is eliminated. Round 3 votes counts: D=45 E=33 A=22 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:223 D:223 A:197 E:188 C:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 12 -10 4 B 12 0 24 2 8 C -12 -24 0 -22 -4 D 10 -2 22 0 16 E -4 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 12 -10 4 B 12 0 24 2 8 C -12 -24 0 -22 -4 D 10 -2 22 0 16 E -4 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 12 -10 4 B 12 0 24 2 8 C -12 -24 0 -22 -4 D 10 -2 22 0 16 E -4 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6098: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) B C D E A (7) E A D B C (6) B C E D A (6) E A C D B (5) D A E C B (5) E B A D C (4) B D C E A (4) B C D A E (4) E A D C B (3) E A C B D (3) D E A B C (3) C B A E D (3) C B A D E (3) B E C D A (3) B E C A D (3) B D C A E (3) E C A B D (2) D B A C E (2) D A E B C (2) D A C E B (2) C D A B E (2) C A B E D (2) B E D A C (2) E D A B C (1) E B D A C (1) D C B A E (1) D B C A E (1) D B A E C (1) D A C B E (1) D A B E C (1) C A E D B (1) C A E B D (1) C A D E B (1) B D E C A (1) B C E A D (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -12 -20 -10 B 14 0 12 20 16 C 12 -12 0 12 6 D 20 -20 -12 0 4 E 10 -16 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -20 -10 B 14 0 12 20 16 C 12 -12 0 12 6 D 20 -20 -12 0 4 E 10 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=25 C=20 D=19 A=2 so A is eliminated. Round 2 votes counts: B=34 E=26 C=21 D=19 so D is eliminated. Round 3 votes counts: B=39 E=36 C=25 so C is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:231 C:209 D:196 E:192 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -20 -10 B 14 0 12 20 16 C 12 -12 0 12 6 D 20 -20 -12 0 4 E 10 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -20 -10 B 14 0 12 20 16 C 12 -12 0 12 6 D 20 -20 -12 0 4 E 10 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -20 -10 B 14 0 12 20 16 C 12 -12 0 12 6 D 20 -20 -12 0 4 E 10 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6099: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (13) B E C D A (12) C E A D B (9) E C B D A (7) C E B A D (7) B D A E C (5) D A E C B (4) C A D E B (4) B D E A C (4) E C D A B (3) E B C D A (3) D A B E C (3) B E D C A (3) D B A E C (2) D A C E B (2) C A E D B (2) B E D A C (2) B C E A D (2) B A D E C (2) A D C B E (2) E D C A B (1) E D B A C (1) E D A C B (1) D A E B C (1) C E B D A (1) C E A B D (1) C B E A D (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -12 -10 -16 B 4 0 -18 0 -22 C 12 18 0 6 -8 D 10 0 -6 0 -12 E 16 22 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -12 -10 -16 B 4 0 -18 0 -22 C 12 18 0 6 -8 D 10 0 -6 0 -12 E 16 22 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=25 A=17 E=16 D=12 so D is eliminated. Round 2 votes counts: B=32 A=27 C=25 E=16 so E is eliminated. Round 3 votes counts: C=36 B=36 A=28 so A is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:229 C:214 D:196 B:182 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -12 -10 -16 B 4 0 -18 0 -22 C 12 18 0 6 -8 D 10 0 -6 0 -12 E 16 22 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -10 -16 B 4 0 -18 0 -22 C 12 18 0 6 -8 D 10 0 -6 0 -12 E 16 22 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -10 -16 B 4 0 -18 0 -22 C 12 18 0 6 -8 D 10 0 -6 0 -12 E 16 22 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6100: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) C A D E B (9) D A E C B (6) C A B E D (6) D E A B C (5) D C A E B (5) B E D A C (5) B E A C D (5) E B A D C (4) D E B A C (4) B E C A D (4) B C E A D (3) A C E D B (3) E D A B C (2) E B A C D (2) E A C B D (2) D B E A C (2) C D A E B (2) C A E D B (2) C A E B D (2) C A B D E (2) B E A D C (2) B D E C A (2) A D C E B (2) D C B A E (1) D B E C A (1) D A C E B (1) C B D A E (1) C A D B E (1) B D E A C (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 4 -4 24 12 B -4 0 -16 8 -6 C 4 16 0 14 4 D -24 -8 -14 0 -10 E -12 6 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 24 12 B -4 0 -16 8 -6 C 4 16 0 14 4 D -24 -8 -14 0 -10 E -12 6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=25 B=22 E=10 A=7 so A is eliminated. Round 2 votes counts: C=40 D=27 B=22 E=11 so E is eliminated. Round 3 votes counts: C=43 D=29 B=28 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:218 E:200 B:191 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 24 12 B -4 0 -16 8 -6 C 4 16 0 14 4 D -24 -8 -14 0 -10 E -12 6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 24 12 B -4 0 -16 8 -6 C 4 16 0 14 4 D -24 -8 -14 0 -10 E -12 6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 24 12 B -4 0 -16 8 -6 C 4 16 0 14 4 D -24 -8 -14 0 -10 E -12 6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6101: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) C A D B E (8) A D B C E (7) A D C B E (6) E B D A C (4) C E A D B (4) B D A E C (4) E D B A C (3) E C B D A (3) E C B A D (3) E B C D A (3) D A B C E (3) C A E D B (3) C A D E B (3) C A B D E (3) B D A C E (3) E D A B C (2) E C D A B (2) E C A B D (2) D B A E C (2) D A B E C (2) C E B A D (2) C B A D E (2) B A C D E (2) E C D B A (1) E C A D B (1) E A D C B (1) D B A C E (1) D A E B C (1) C E A B D (1) C B E A D (1) C A B E D (1) B C E A D (1) B C D A E (1) B C A D E (1) B A D C E (1) A D E C B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 12 14 B -6 0 2 -4 20 C -8 -2 0 -2 12 D -12 4 2 0 6 E -14 -20 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 12 14 B -6 0 2 -4 20 C -8 -2 0 -2 12 D -12 4 2 0 6 E -14 -20 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=25 B=22 A=16 D=9 so D is eliminated. Round 2 votes counts: C=28 E=25 B=25 A=22 so A is eliminated. Round 3 votes counts: B=38 C=35 E=27 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:220 B:206 C:200 D:200 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 12 14 B -6 0 2 -4 20 C -8 -2 0 -2 12 D -12 4 2 0 6 E -14 -20 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 12 14 B -6 0 2 -4 20 C -8 -2 0 -2 12 D -12 4 2 0 6 E -14 -20 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 12 14 B -6 0 2 -4 20 C -8 -2 0 -2 12 D -12 4 2 0 6 E -14 -20 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6102: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (16) C A E D B (7) C E A D B (6) B D E C A (5) B C D E A (4) A C E D B (4) A C B D E (4) C E A B D (3) C B A E D (3) C B A D E (3) B C E D A (3) A E D C B (3) A E C D B (3) A D B E C (3) E B D C A (2) D B E A C (2) C B E D A (2) C A B E D (2) B E D C A (2) B D C E A (2) B D A E C (2) E D C B A (1) E D B A C (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D C B (1) D E B A C (1) D E A B C (1) C E B D A (1) C B E A D (1) C B D A E (1) C A E B D (1) B E C D A (1) B D C A E (1) B D A C E (1) A D E C B (1) A D E B C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 -8 -2 -16 B 12 0 -4 20 18 C 8 4 0 8 2 D 2 -20 -8 0 0 E 16 -18 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -2 -16 B 12 0 -4 20 18 C 8 4 0 8 2 D 2 -20 -8 0 0 E 16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=30 A=21 E=8 D=4 so D is eliminated. Round 2 votes counts: B=39 C=30 A=21 E=10 so E is eliminated. Round 3 votes counts: B=43 C=34 A=23 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:223 C:211 E:198 D:187 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 -2 -16 B 12 0 -4 20 18 C 8 4 0 8 2 D 2 -20 -8 0 0 E 16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -2 -16 B 12 0 -4 20 18 C 8 4 0 8 2 D 2 -20 -8 0 0 E 16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -2 -16 B 12 0 -4 20 18 C 8 4 0 8 2 D 2 -20 -8 0 0 E 16 -18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6103: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) D A E C B (8) B C E A D (6) B C D E A (6) E C A B D (5) D B C A E (5) B C E D A (5) A E D C B (5) E A C B D (4) A E C D B (4) D B A E C (3) D A E B C (3) C E B A D (3) B E C A D (3) E C B A D (2) E A B C D (2) D A C E B (2) D A B C E (2) B D C A E (2) A D E C B (2) A D C E B (2) A C E D B (2) E A C D B (1) D C B A E (1) D B C E A (1) D A B E C (1) C E A D B (1) C E A B D (1) C D E A B (1) C B E D A (1) B E D C A (1) B E C D A (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A C E (1) Total count = 100 A B C D E A 0 -8 -10 -10 -14 B 8 0 10 12 4 C 10 -10 0 -2 4 D 10 -12 2 0 2 E 14 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -10 -14 B 8 0 10 12 4 C 10 -10 0 -2 4 D 10 -12 2 0 2 E 14 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=26 A=15 E=14 C=7 so C is eliminated. Round 2 votes counts: B=39 D=27 E=19 A=15 so A is eliminated. Round 3 votes counts: B=39 D=31 E=30 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:202 C:201 D:201 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -10 -14 B 8 0 10 12 4 C 10 -10 0 -2 4 D 10 -12 2 0 2 E 14 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -10 -14 B 8 0 10 12 4 C 10 -10 0 -2 4 D 10 -12 2 0 2 E 14 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -10 -14 B 8 0 10 12 4 C 10 -10 0 -2 4 D 10 -12 2 0 2 E 14 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6104: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (13) D E C A B (9) B A E D C (9) B C A D E (4) B A C E D (4) E D C A B (3) E A D C B (3) E A D B C (3) D E C B A (3) D E B A C (3) D C E B A (3) C B A D E (3) C A E D B (3) A B E C D (3) E D A B C (2) D C E A B (2) C D B A E (2) C A B D E (2) B D E A C (2) B C D A E (2) A E B D C (2) E D B A C (1) E D A C B (1) D C B E A (1) D B E C A (1) D B C E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D B E A (1) C B D A E (1) B E A D C (1) B D C E A (1) B C A E D (1) B A C D E (1) A E C D B (1) A C E D B (1) A C E B D (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -18 -8 -14 B -8 0 -14 -20 -14 C 18 14 0 -4 4 D 8 20 4 0 12 E 14 14 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 -8 -14 B -8 0 -14 -20 -14 C 18 14 0 -4 4 D 8 20 4 0 12 E 14 14 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 D=23 E=13 A=11 so A is eliminated. Round 2 votes counts: C=31 B=30 D=23 E=16 so E is eliminated. Round 3 votes counts: D=36 C=32 B=32 so C is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:216 E:206 A:184 B:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -18 -8 -14 B -8 0 -14 -20 -14 C 18 14 0 -4 4 D 8 20 4 0 12 E 14 14 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -8 -14 B -8 0 -14 -20 -14 C 18 14 0 -4 4 D 8 20 4 0 12 E 14 14 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -8 -14 B -8 0 -14 -20 -14 C 18 14 0 -4 4 D 8 20 4 0 12 E 14 14 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999509 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6105: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) D B E A C (7) D E B C A (6) C A E B D (5) C E A B D (4) A C E B D (4) A C B E D (4) E C B D A (3) C E D A B (3) C A E D B (3) B D E A C (3) A D C B E (3) A D B C E (3) E B C D A (2) E B A C D (2) D E C B A (2) D C E B A (2) D A B E C (2) D A B C E (2) C E B D A (2) C E B A D (2) B D A E C (2) A D B E C (2) A C D E B (2) A B E C D (2) A B D E C (2) E C B A D (1) E B C A D (1) D C E A B (1) D A C E B (1) C E A D B (1) C D E A B (1) B E D C A (1) B E D A C (1) B D E C A (1) B A E D C (1) B A D E C (1) A E B C D (1) A C E D B (1) A C D B E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -2 -4 -12 B 0 0 8 -6 0 C 2 -8 0 -8 -8 D 4 6 8 0 8 E 12 0 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -4 -12 B 0 0 8 -6 0 C 2 -8 0 -8 -8 D 4 6 8 0 8 E 12 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=27 C=21 B=10 E=9 so E is eliminated. Round 2 votes counts: D=33 A=27 C=25 B=15 so B is eliminated. Round 3 votes counts: D=41 A=31 C=28 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:206 B:201 A:191 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -2 -4 -12 B 0 0 8 -6 0 C 2 -8 0 -8 -8 D 4 6 8 0 8 E 12 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -4 -12 B 0 0 8 -6 0 C 2 -8 0 -8 -8 D 4 6 8 0 8 E 12 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -4 -12 B 0 0 8 -6 0 C 2 -8 0 -8 -8 D 4 6 8 0 8 E 12 0 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6106: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) D C A E B (7) D A E B C (7) C B E A D (6) D B E A C (5) B E C A D (5) D A C E B (4) C D A B E (4) E B A D C (3) C D B A E (3) C D A E B (3) C A E B D (3) B E D A C (3) B E A C D (3) C B A E D (2) C A E D B (2) B E A D C (2) B D C E A (2) B C E A D (2) A E B C D (2) E C A B D (1) E A B C D (1) D C B E A (1) D C A B E (1) D B E C A (1) D B C E A (1) D B A E C (1) C E A B D (1) C B E D A (1) C B D E A (1) C A B E D (1) B D E A C (1) B C E D A (1) B C D E A (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E C B (1) A D E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 0 -16 16 B -10 0 -12 -12 -4 C 0 12 0 -8 -2 D 16 12 8 0 12 E -16 4 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -16 16 B -10 0 -12 -12 -4 C 0 12 0 -8 -2 D 16 12 8 0 12 E -16 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=27 B=20 A=10 E=5 so E is eliminated. Round 2 votes counts: D=38 C=28 B=23 A=11 so A is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 A:205 C:201 E:189 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -16 16 B -10 0 -12 -12 -4 C 0 12 0 -8 -2 D 16 12 8 0 12 E -16 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -16 16 B -10 0 -12 -12 -4 C 0 12 0 -8 -2 D 16 12 8 0 12 E -16 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -16 16 B -10 0 -12 -12 -4 C 0 12 0 -8 -2 D 16 12 8 0 12 E -16 4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6107: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) D B A E C (6) E C A D B (5) A D E C B (5) A B C E D (5) D E C B A (4) D B E C A (4) B C E D A (4) B A D C E (4) A E C D B (4) E C D B A (3) D E C A B (3) D A E C B (3) C E B D A (3) B D E C A (3) A D B E C (3) A C E B D (3) A B D C E (3) D E B C A (2) D A B E C (2) C B E A D (2) B D A E C (2) B D A C E (2) B A C D E (2) A D B C E (2) E D C B A (1) E C B D A (1) D E B A C (1) D B E A C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E D A (1) C A E B D (1) B C D E A (1) B C A E D (1) B A C E D (1) A C E D B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 -6 4 B 0 0 0 -14 2 C -4 0 0 -2 -18 D 6 14 2 0 8 E -4 -2 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999055 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -6 4 B 0 0 0 -14 2 C -4 0 0 -2 -18 D 6 14 2 0 8 E -4 -2 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 B=20 E=16 C=10 so C is eliminated. Round 2 votes counts: A=29 D=26 B=23 E=22 so E is eliminated. Round 3 votes counts: D=36 A=36 B=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:202 A:201 B:194 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -6 4 B 0 0 0 -14 2 C -4 0 0 -2 -18 D 6 14 2 0 8 E -4 -2 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -6 4 B 0 0 0 -14 2 C -4 0 0 -2 -18 D 6 14 2 0 8 E -4 -2 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -6 4 B 0 0 0 -14 2 C -4 0 0 -2 -18 D 6 14 2 0 8 E -4 -2 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6108: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) A E C D B (8) B D C A E (7) B D C E A (6) E A C D B (5) B D A C E (5) A D C B E (5) A B E D C (5) A B D C E (5) E C D B A (4) E C A D B (4) C E D B A (4) B E D C A (3) A C D E B (3) C D B E A (2) B D E A C (2) A D B C E (2) E C B A D (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B E A (1) D C B A E (1) D C A B E (1) D B C A E (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D E A B (1) C D A E B (1) B D E C A (1) B A D C E (1) A E C B D (1) A E B C D (1) A D C E B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -2 -8 4 B 2 0 -14 0 2 C 2 14 0 2 8 D 8 0 -2 0 0 E -4 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -8 4 B 2 0 -14 0 2 C 2 14 0 2 8 D 8 0 -2 0 0 E -4 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=25 B=25 C=11 D=6 so D is eliminated. Round 2 votes counts: A=35 B=26 E=25 C=14 so C is eliminated. Round 3 votes counts: A=37 E=33 B=30 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:213 D:203 A:196 B:195 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -8 4 B 2 0 -14 0 2 C 2 14 0 2 8 D 8 0 -2 0 0 E -4 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -8 4 B 2 0 -14 0 2 C 2 14 0 2 8 D 8 0 -2 0 0 E -4 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -8 4 B 2 0 -14 0 2 C 2 14 0 2 8 D 8 0 -2 0 0 E -4 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6109: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E A D (8) B C D A E (7) D A E C B (6) B C E A D (6) A E D C B (6) E D A B C (4) E A D C B (4) C A E B D (4) B D C E A (4) B C E D A (4) E C B A D (3) D B A C E (3) C B A D E (3) A D C E B (3) E D A C B (2) E A C D B (2) D E A B C (2) D B E A C (2) C B A E D (2) B D E C A (2) B C D E A (2) A E C D B (2) A D E C B (2) E D B A C (1) D B C A E (1) D B A E C (1) C A B E D (1) C A B D E (1) B D C A E (1) B C A D E (1) Total count = 100 A B C D E A 0 -2 0 -4 8 B 2 0 2 -2 -2 C 0 -2 0 -8 2 D 4 2 8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -4 8 B 2 0 2 -2 -2 C 0 -2 0 -8 2 D 4 2 8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 C=19 E=16 A=13 so A is eliminated. Round 2 votes counts: D=30 B=27 E=24 C=19 so C is eliminated. Round 3 votes counts: B=42 D=30 E=28 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:208 A:201 B:200 C:196 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -4 8 B 2 0 2 -2 -2 C 0 -2 0 -8 2 D 4 2 8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 8 B 2 0 2 -2 -2 C 0 -2 0 -8 2 D 4 2 8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 8 B 2 0 2 -2 -2 C 0 -2 0 -8 2 D 4 2 8 0 2 E -8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6110: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (11) E A C B D (6) D B C A E (6) D B A E C (6) B D C A E (6) E A D C B (5) C B D E A (4) C B D A E (4) C B A E D (4) E A C D B (3) D E A C B (3) D B A C E (3) C D B E A (3) B C D A E (3) E C A B D (2) E A D B C (2) D B E A C (2) D B C E A (2) C D E B A (2) C B E A D (2) A E B C D (2) A B E C D (2) E C A D B (1) D E C A B (1) D C E A B (1) D B E C A (1) D A E B C (1) D A B E C (1) C E D A B (1) C B E D A (1) C A E B D (1) B D A E C (1) B D A C E (1) B C A E D (1) B C A D E (1) A E D B C (1) A E C B D (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -16 -6 -6 B 6 0 -14 10 8 C 16 14 0 12 16 D 6 -10 -12 0 4 E 6 -8 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -6 -6 B 6 0 -14 10 8 C 16 14 0 12 16 D 6 -10 -12 0 4 E 6 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=27 E=19 B=13 A=8 so A is eliminated. Round 2 votes counts: C=34 D=27 E=24 B=15 so B is eliminated. Round 3 votes counts: C=39 D=35 E=26 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 B:205 D:194 E:189 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -6 -6 B 6 0 -14 10 8 C 16 14 0 12 16 D 6 -10 -12 0 4 E 6 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -6 -6 B 6 0 -14 10 8 C 16 14 0 12 16 D 6 -10 -12 0 4 E 6 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -6 -6 B 6 0 -14 10 8 C 16 14 0 12 16 D 6 -10 -12 0 4 E 6 -8 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6111: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) E B A C D (7) D C A B E (7) E D A C B (6) E B D C A (6) B C A D E (6) D C A E B (4) E D B C A (3) E A D C B (3) D E C A B (3) A C B D E (3) E D C B A (2) E D C A B (2) E D B A C (2) E B C A D (2) D E A C B (2) D C E A B (2) D C B A E (2) D A E C B (2) D A C E B (2) D A C B E (2) C D A B E (2) B E A C D (2) B C A E D (2) B A C D E (2) A D C E B (2) E D A B C (1) E B D A C (1) E B A D C (1) D E C B A (1) C A D B E (1) C A B D E (1) B E C A D (1) B C D A E (1) B A E C D (1) B A C E D (1) A E C B D (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 4 -6 6 B -14 0 -20 -22 -10 C -4 20 0 -14 2 D 6 22 14 0 12 E -6 10 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 -6 6 B -14 0 -20 -22 -10 C -4 20 0 -14 2 D 6 22 14 0 12 E -6 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=27 A=17 B=16 C=4 so C is eliminated. Round 2 votes counts: E=36 D=29 A=19 B=16 so B is eliminated. Round 3 votes counts: E=39 A=31 D=30 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:227 A:209 C:202 E:195 B:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 4 -6 6 B -14 0 -20 -22 -10 C -4 20 0 -14 2 D 6 22 14 0 12 E -6 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -6 6 B -14 0 -20 -22 -10 C -4 20 0 -14 2 D 6 22 14 0 12 E -6 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -6 6 B -14 0 -20 -22 -10 C -4 20 0 -14 2 D 6 22 14 0 12 E -6 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6112: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) C D A E B (6) B E A D C (6) B D A E C (6) D A C B E (5) D A B E C (5) D A B C E (5) B E C A D (5) C E B A D (4) B E C D A (4) E C B A D (3) E C A B D (3) D C A B E (3) C E B D A (3) C E A D B (3) B C E D A (3) A D E C B (3) E B A D C (2) D A C E B (2) C E D A B (2) C D E A B (2) B D C A E (2) A B D E C (2) E B C D A (1) E A B D C (1) C E A B D (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D E B (1) B E D A C (1) B D C E A (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -20 -6 -14 B 6 0 10 20 2 C 20 -10 0 10 -6 D 6 -20 -10 0 -8 E 14 -2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -6 -14 B 6 0 10 20 2 C 20 -10 0 10 -6 D 6 -20 -10 0 -8 E 14 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996458 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 E=20 D=20 A=7 so A is eliminated. Round 2 votes counts: B=30 C=26 D=24 E=20 so E is eliminated. Round 3 votes counts: B=44 C=32 D=24 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:213 C:207 D:184 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -20 -6 -14 B 6 0 10 20 2 C 20 -10 0 10 -6 D 6 -20 -10 0 -8 E 14 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996458 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -6 -14 B 6 0 10 20 2 C 20 -10 0 10 -6 D 6 -20 -10 0 -8 E 14 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996458 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -6 -14 B 6 0 10 20 2 C 20 -10 0 10 -6 D 6 -20 -10 0 -8 E 14 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996458 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6113: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) E B A D C (8) E A B D C (8) B E C A D (8) B E A C D (8) A D E C B (7) A E D B C (6) E A D B C (5) C D B A E (5) D A C E B (4) C D A B E (4) C B D E A (4) B C E D A (4) D A E C B (3) C B D A E (3) B C E A D (3) E B A C D (2) D C B A E (2) D E A C B (1) D E A B C (1) D C A B E (1) C D A E B (1) B E C D A (1) B E A D C (1) A E D C B (1) Total count = 100 A B C D E A 0 2 10 14 -8 B -2 0 10 0 -12 C -10 -10 0 -14 -20 D -14 0 14 0 -10 E 8 12 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 10 14 -8 B -2 0 10 0 -12 C -10 -10 0 -14 -20 D -14 0 14 0 -10 E 8 12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 D=21 C=17 A=14 so A is eliminated. Round 2 votes counts: E=30 D=28 B=25 C=17 so C is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:225 A:209 B:198 D:195 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 10 14 -8 B -2 0 10 0 -12 C -10 -10 0 -14 -20 D -14 0 14 0 -10 E 8 12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 14 -8 B -2 0 10 0 -12 C -10 -10 0 -14 -20 D -14 0 14 0 -10 E 8 12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 14 -8 B -2 0 10 0 -12 C -10 -10 0 -14 -20 D -14 0 14 0 -10 E 8 12 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6114: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) A E C B D (7) E A C D B (5) D C B E A (5) B D C A E (5) E A D C B (4) D B C E A (4) C E A D B (4) B D C E A (4) B D A C E (4) A E C D B (4) A E B D C (4) A E B C D (4) D B E C A (3) D B E A C (3) A B E D C (3) E C A D B (2) D E B C A (2) C E A B D (2) C B D A E (2) C B A E D (2) C A B E D (2) A E D B C (2) E D A C B (1) E C D A B (1) E A D B C (1) D E C B A (1) D B A E C (1) C E D B A (1) C D E B A (1) C D B E A (1) C B D E A (1) C B A D E (1) C A E B D (1) B D A E C (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 12 -12 -2 -10 B -12 0 -14 -10 -12 C 12 14 0 4 2 D 2 10 -4 0 -18 E 10 12 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 -2 -10 B -12 0 -14 -10 -12 C 12 14 0 4 2 D 2 10 -4 0 -18 E 10 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 D=19 B=16 E=14 so E is eliminated. Round 2 votes counts: A=34 C=30 D=20 B=16 so B is eliminated. Round 3 votes counts: D=34 A=34 C=32 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:219 C:216 D:195 A:194 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -12 -2 -10 B -12 0 -14 -10 -12 C 12 14 0 4 2 D 2 10 -4 0 -18 E 10 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 -2 -10 B -12 0 -14 -10 -12 C 12 14 0 4 2 D 2 10 -4 0 -18 E 10 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 -2 -10 B -12 0 -14 -10 -12 C 12 14 0 4 2 D 2 10 -4 0 -18 E 10 12 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6115: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (4) D E B C A (4) D E A B C (4) C B A D E (4) C A B D E (4) B C A E D (4) A E D C B (4) A E B C D (4) E A B D C (3) D E B A C (3) C B A E D (3) B E A C D (3) B C D E A (3) A C B E D (3) E A D B C (2) D E C B A (2) D E C A B (2) D C B E A (2) D C A E B (2) D A E C B (2) C D B A E (2) C B D A E (2) C A B E D (2) B E D C A (2) B A E C D (2) B A C E D (2) A E B D C (2) A C D E B (2) A B E C D (2) A B C E D (2) E B D A C (1) E B A D C (1) D C E B A (1) D C E A B (1) D B E C A (1) D B C E A (1) C D B E A (1) C D A B E (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A D E (1) A E C D B (1) A D E C B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 4 14 8 B 0 0 12 12 6 C -4 -12 0 6 -8 D -14 -12 -6 0 -6 E -8 -6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.473286 B: 0.526714 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50142730996 Cumulative probabilities = A: 0.473286 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 14 8 B 0 0 12 12 6 C -4 -12 0 6 -8 D -14 -12 -6 0 -6 E -8 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=23 B=22 C=19 E=11 so E is eliminated. Round 2 votes counts: D=29 A=28 B=24 C=19 so C is eliminated. Round 3 votes counts: A=34 D=33 B=33 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:213 E:200 C:191 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 14 8 B 0 0 12 12 6 C -4 -12 0 6 -8 D -14 -12 -6 0 -6 E -8 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 14 8 B 0 0 12 12 6 C -4 -12 0 6 -8 D -14 -12 -6 0 -6 E -8 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 14 8 B 0 0 12 12 6 C -4 -12 0 6 -8 D -14 -12 -6 0 -6 E -8 -6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6116: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) A D E C B (7) D A B E C (6) E B C A D (5) D A C B E (4) B E C D A (4) A C E D B (4) E C B A D (3) E B C D A (3) D B A C E (3) C E A B D (3) A C D E B (3) E A C B D (2) D B E C A (2) D B C A E (2) D A E B C (2) D A B C E (2) C B E A D (2) C A E B D (2) B E D C A (2) B D E C A (2) B C E A D (2) B C D E A (2) E D A B C (1) E B D C A (1) E B A C D (1) E A D C B (1) E A D B C (1) D E B A C (1) D E A B C (1) D B C E A (1) D B A E C (1) D A C E B (1) C E B A D (1) C B D A E (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B D E (1) B E C A D (1) B D C E A (1) B C E D A (1) A E D C B (1) A E C D B (1) A D C B E (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 12 10 12 12 B -12 0 -4 -14 -10 C -10 4 0 -6 2 D -12 14 6 0 12 E -12 10 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 12 12 B -12 0 -4 -14 -10 C -10 4 0 -6 2 D -12 14 6 0 12 E -12 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 E=18 B=15 C=13 so C is eliminated. Round 2 votes counts: A=32 D=26 E=22 B=20 so B is eliminated. Round 3 votes counts: E=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 D:210 C:195 E:192 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 12 12 B -12 0 -4 -14 -10 C -10 4 0 -6 2 D -12 14 6 0 12 E -12 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 12 12 B -12 0 -4 -14 -10 C -10 4 0 -6 2 D -12 14 6 0 12 E -12 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 12 12 B -12 0 -4 -14 -10 C -10 4 0 -6 2 D -12 14 6 0 12 E -12 10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6117: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) A C D B E (7) D B E A C (6) E B C D A (5) D E B A C (5) D A B E C (5) B E D C A (5) B E C D A (5) A C D E B (5) E B D C A (4) C A E B D (4) B D E C A (4) C E B A D (3) C E A B D (3) C B A E D (3) B C E A D (3) A C E D B (3) E C B A D (2) D B A E C (2) D A C E B (2) C B E A D (2) E D B C A (1) E D B A C (1) E D A C B (1) E B C A D (1) D E A B C (1) D A C B E (1) D A B C E (1) C A E D B (1) B E C A D (1) B D E A C (1) A D E C B (1) A D C E B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -10 0 -8 B 8 0 4 10 12 C 10 -4 0 14 -2 D 0 -10 -14 0 -12 E 8 -12 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 0 -8 B 8 0 4 10 12 C 10 -4 0 14 -2 D 0 -10 -14 0 -12 E 8 -12 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=19 A=19 E=15 so E is eliminated. Round 2 votes counts: B=29 D=26 C=26 A=19 so A is eliminated. Round 3 votes counts: C=41 B=30 D=29 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:209 E:205 A:187 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 0 -8 B 8 0 4 10 12 C 10 -4 0 14 -2 D 0 -10 -14 0 -12 E 8 -12 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 0 -8 B 8 0 4 10 12 C 10 -4 0 14 -2 D 0 -10 -14 0 -12 E 8 -12 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 0 -8 B 8 0 4 10 12 C 10 -4 0 14 -2 D 0 -10 -14 0 -12 E 8 -12 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6118: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) A E C B D (12) E A C B D (10) D B C E A (7) A E C D B (7) A E B D C (6) B D E C A (4) E A B C D (3) D B A E C (3) A E D C B (3) E C A B D (2) D A B E C (2) C E A B D (2) C D B E A (2) C A E D B (2) C A D E B (2) B E D C A (2) B D C E A (2) A E D B C (2) E B C A D (1) E A B D C (1) D C B E A (1) D B A C E (1) D A B C E (1) C E A D B (1) C D B A E (1) C D A B E (1) C B D E A (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E A C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 18 12 14 14 B -18 0 6 2 -10 C -12 -6 0 -2 -28 D -14 -2 2 0 -16 E -14 10 28 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 12 14 14 B -18 0 6 2 -10 C -12 -6 0 -2 -28 D -14 -2 2 0 -16 E -14 10 28 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=27 E=17 C=12 B=12 so C is eliminated. Round 2 votes counts: A=36 D=31 E=20 B=13 so B is eliminated. Round 3 votes counts: D=39 A=36 E=25 so E is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:220 B:190 D:185 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 12 14 14 B -18 0 6 2 -10 C -12 -6 0 -2 -28 D -14 -2 2 0 -16 E -14 10 28 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 14 14 B -18 0 6 2 -10 C -12 -6 0 -2 -28 D -14 -2 2 0 -16 E -14 10 28 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 14 14 B -18 0 6 2 -10 C -12 -6 0 -2 -28 D -14 -2 2 0 -16 E -14 10 28 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6119: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) B C A E D (8) B E C A D (7) D E A C B (6) D C A B E (6) D A E C B (5) E B C A D (4) E B A C D (4) B C E A D (4) E D B A C (3) E D A C B (3) E A D C B (3) D C B A E (3) C A B D E (3) B C D A E (3) B C A D E (3) D E B C A (2) D A C B E (2) C D B A E (2) B D E C A (2) A E C D B (2) A D C E B (2) A D C B E (2) E D A B C (1) E B D A C (1) E A B C D (1) D E A B C (1) D B E C A (1) D B C A E (1) C B A D E (1) B E C D A (1) B D C A E (1) B C D E A (1) B A E C D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -6 -6 10 B 6 0 0 -10 4 C 6 0 0 -6 4 D 6 10 6 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -6 10 B 6 0 0 -10 4 C 6 0 0 -6 4 D 6 10 6 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=31 E=20 A=8 C=6 so C is eliminated. Round 2 votes counts: D=37 B=32 E=20 A=11 so A is eliminated. Round 3 votes counts: D=42 B=35 E=23 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:202 B:200 A:196 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -6 10 B 6 0 0 -10 4 C 6 0 0 -6 4 D 6 10 6 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -6 10 B 6 0 0 -10 4 C 6 0 0 -6 4 D 6 10 6 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -6 10 B 6 0 0 -10 4 C 6 0 0 -6 4 D 6 10 6 0 12 E -10 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6120: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (16) C D B A E (14) D C E A B (9) B A E C D (8) D E C A B (7) A B E C D (6) E D A B C (5) E A B C D (4) D C B A E (4) B A C E D (4) C B A E D (3) C B A D E (3) E B A C D (2) D C E B A (2) D C A B E (2) B C A D E (2) E C D B A (1) D E A B C (1) D A C B E (1) D A B E C (1) C D E B A (1) C B D A E (1) B E A C D (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 6 0 2 2 B -6 0 4 4 4 C 0 -4 0 2 -6 D -2 -4 -2 0 -4 E -2 -4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.841270 B: 0.000000 C: 0.158730 D: 0.000000 E: 0.000000 Sum of squares = 0.732930558146 Cumulative probabilities = A: 0.841270 B: 0.841270 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 2 2 B -6 0 4 4 4 C 0 -4 0 2 -6 D -2 -4 -2 0 -4 E -2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000232 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 C=22 B=16 A=7 so A is eliminated. Round 2 votes counts: E=28 D=27 B=23 C=22 so C is eliminated. Round 3 votes counts: D=42 B=30 E=28 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:205 B:203 E:202 C:196 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 2 2 B -6 0 4 4 4 C 0 -4 0 2 -6 D -2 -4 -2 0 -4 E -2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000232 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 2 2 B -6 0 4 4 4 C 0 -4 0 2 -6 D -2 -4 -2 0 -4 E -2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000232 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 2 2 B -6 0 4 4 4 C 0 -4 0 2 -6 D -2 -4 -2 0 -4 E -2 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000232 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6121: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) D A C B E (7) A B C D E (7) B C A E D (6) E D A B C (5) E B C A D (4) D C E B A (4) C B E D A (4) C B A D E (4) A B C E D (4) E C B D A (3) D E A B C (3) A B E C D (3) E A D B C (2) D E A C B (2) D C A B E (2) D A E C B (2) C B E A D (2) C B D A E (2) A D B C E (2) E D B A C (1) E C D B A (1) E C B A D (1) E B A C D (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E A B (1) D C B A E (1) D A E B C (1) D A C E B (1) C E B D A (1) C D E B A (1) C B D E A (1) B C E A D (1) B A E C D (1) B A C E D (1) A D E B C (1) A D C B E (1) A D B E C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 -8 -2 B 0 0 -6 0 4 C 0 6 0 0 8 D 8 0 0 0 -6 E 2 -4 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.617016 D: 0.382984 E: 0.000000 Sum of squares = 0.527385529065 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.617016 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -8 -2 B 0 0 -6 0 4 C 0 6 0 0 8 D 8 0 0 0 -6 E 2 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 A=21 C=15 B=9 so B is eliminated. Round 2 votes counts: E=29 D=26 A=23 C=22 so C is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:207 D:201 B:199 E:198 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -8 -2 B 0 0 -6 0 4 C 0 6 0 0 8 D 8 0 0 0 -6 E 2 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 -2 B 0 0 -6 0 4 C 0 6 0 0 8 D 8 0 0 0 -6 E 2 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 -2 B 0 0 -6 0 4 C 0 6 0 0 8 D 8 0 0 0 -6 E 2 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6122: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C E D B A (6) E C D B A (5) C A D B E (5) A B D C E (5) E D B C A (4) E A B D C (4) C D E B A (4) C D A B E (4) A B E D C (4) A B D E C (4) E A B C D (3) D E B C A (3) D B A C E (3) B A E D C (3) A E B C D (3) D C B E A (2) C E D A B (2) C E A D B (2) C D B E A (2) C D B A E (2) B E A D C (2) A C D B E (2) E C B D A (1) E C B A D (1) E A C B D (1) D E C B A (1) D C B A E (1) D C A B E (1) D A B C E (1) C D E A B (1) C A E D B (1) C A E B D (1) C A B D E (1) B E D A C (1) B D A E C (1) A E B D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 0 -10 -10 B 4 0 8 -6 -8 C 0 -8 0 -4 -6 D 10 6 4 0 -12 E 10 8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 -10 -10 B 4 0 8 -6 -8 C 0 -8 0 -4 -6 D 10 6 4 0 -12 E 10 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=29 A=21 D=12 B=7 so B is eliminated. Round 2 votes counts: E=32 C=31 A=24 D=13 so D is eliminated. Round 3 votes counts: E=36 C=35 A=29 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:204 B:199 C:191 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -10 -10 B 4 0 8 -6 -8 C 0 -8 0 -4 -6 D 10 6 4 0 -12 E 10 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -10 -10 B 4 0 8 -6 -8 C 0 -8 0 -4 -6 D 10 6 4 0 -12 E 10 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -10 -10 B 4 0 8 -6 -8 C 0 -8 0 -4 -6 D 10 6 4 0 -12 E 10 8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6123: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (14) D B A E C (13) C E A B D (12) C E A D B (9) B D C E A (8) B D C A E (6) A E C D B (5) E A C D B (4) C B D E A (3) D B E A C (2) C A E D B (2) B D E C A (2) B D E A C (2) B D A C E (2) A E D C B (2) A E D B C (2) A D E B C (2) E C A B D (1) E A D B C (1) D E B A C (1) D B A C E (1) D A E B C (1) D A B E C (1) D A B C E (1) C B E D A (1) B C D E A (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 10 -18 6 B 12 0 20 4 14 C -10 -20 0 -22 -6 D 18 -4 22 0 20 E -6 -14 6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -18 6 B 12 0 20 4 14 C -10 -20 0 -22 -6 D 18 -4 22 0 20 E -6 -14 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=27 D=20 A=12 E=6 so E is eliminated. Round 2 votes counts: B=35 C=28 D=20 A=17 so A is eliminated. Round 3 votes counts: C=38 B=35 D=27 so D is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:228 B:225 A:193 E:183 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 -18 6 B 12 0 20 4 14 C -10 -20 0 -22 -6 D 18 -4 22 0 20 E -6 -14 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -18 6 B 12 0 20 4 14 C -10 -20 0 -22 -6 D 18 -4 22 0 20 E -6 -14 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -18 6 B 12 0 20 4 14 C -10 -20 0 -22 -6 D 18 -4 22 0 20 E -6 -14 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6124: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) C E A D B (9) D B E C A (8) D E B C A (7) A C E B D (7) E C D A B (6) B D E A C (5) B D A E C (5) B A D C E (5) A B C E D (5) C E D A B (4) B A C D E (4) A B C D E (4) A C E D B (3) E D B C A (2) D E C B A (2) B D A C E (2) A C B E D (2) A C B D E (2) E D C B A (1) E C A D B (1) D E C A B (1) C A D E B (1) B E D A C (1) B D E C A (1) B A D E C (1) B A C E D (1) Total count = 100 A B C D E A 0 10 -6 10 4 B -10 0 2 -10 -8 C 6 -2 0 18 18 D -10 10 -18 0 -4 E -4 8 -18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.432098765481 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 10 4 B -10 0 2 -10 -8 C 6 -2 0 18 18 D -10 10 -18 0 -4 E -4 8 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=24 A=23 D=18 E=10 so E is eliminated. Round 2 votes counts: C=31 B=25 A=23 D=21 so D is eliminated. Round 3 votes counts: B=42 C=35 A=23 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:220 A:209 E:195 D:189 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 10 4 B -10 0 2 -10 -8 C 6 -2 0 18 18 D -10 10 -18 0 -4 E -4 8 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 10 4 B -10 0 2 -10 -8 C 6 -2 0 18 18 D -10 10 -18 0 -4 E -4 8 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 10 4 B -10 0 2 -10 -8 C 6 -2 0 18 18 D -10 10 -18 0 -4 E -4 8 -18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.333333 C: 0.555556 D: 0.000000 E: 0.000000 Sum of squares = 0.432098765391 Cumulative probabilities = A: 0.111111 B: 0.444444 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6125: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) B C E A D (8) B E C A D (6) D A C E B (5) C A E B D (5) A D C E B (5) E A C B D (4) B E D A C (4) D C A B E (3) D A E C B (3) C A E D B (3) B E D C A (3) B D C E A (3) D E A B C (2) D B E A C (2) D B C A E (2) C E B A D (2) C D A E B (2) C D A B E (2) C B D A E (2) C A D B E (2) B E A C D (2) B D E C A (2) A D E C B (2) E C A B D (1) E B D A C (1) E B A C D (1) E A C D B (1) E A B C D (1) D C B A E (1) D B C E A (1) D B A E C (1) D A B E C (1) D A B C E (1) C E A B D (1) C B A E D (1) B D C A E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -24 10 6 B -8 0 -10 2 2 C 24 10 0 14 24 D -10 -2 -14 0 8 E -6 -2 -24 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -24 10 6 B -8 0 -10 2 2 C 24 10 0 14 24 D -10 -2 -14 0 8 E -6 -2 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=28 D=22 E=9 A=9 so E is eliminated. Round 2 votes counts: B=34 C=29 D=22 A=15 so A is eliminated. Round 3 votes counts: C=36 B=35 D=29 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:236 A:200 B:193 D:191 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -24 10 6 B -8 0 -10 2 2 C 24 10 0 14 24 D -10 -2 -14 0 8 E -6 -2 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -24 10 6 B -8 0 -10 2 2 C 24 10 0 14 24 D -10 -2 -14 0 8 E -6 -2 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -24 10 6 B -8 0 -10 2 2 C 24 10 0 14 24 D -10 -2 -14 0 8 E -6 -2 -24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6126: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) D C E B A (8) D C E A B (8) E C A B D (4) D B E C A (4) C D E A B (4) B E A D C (4) B A E D C (4) B A E C D (4) B A D E C (4) A E B C D (4) D B C E A (3) C E D A B (3) B E A C D (3) A E C B D (3) E B A C D (2) E A C B D (2) E A B C D (2) D C B E A (2) D B A C E (2) B E D A C (2) A C E B D (2) E D C B A (1) E D B C A (1) E C D B A (1) E B C A D (1) D E C B A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A E C (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B D A E C (1) B A D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 2 6 -12 B 4 0 8 8 -4 C -2 -8 0 -2 -16 D -6 -8 2 0 -10 E 12 4 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 6 -12 B 4 0 8 8 -4 C -2 -8 0 -2 -16 D -6 -8 2 0 -10 E 12 4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=23 A=19 E=14 C=11 so C is eliminated. Round 2 votes counts: D=38 B=23 A=22 E=17 so E is eliminated. Round 3 votes counts: D=44 A=30 B=26 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:221 B:208 A:196 D:189 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 6 -12 B 4 0 8 8 -4 C -2 -8 0 -2 -16 D -6 -8 2 0 -10 E 12 4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 -12 B 4 0 8 8 -4 C -2 -8 0 -2 -16 D -6 -8 2 0 -10 E 12 4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 -12 B 4 0 8 8 -4 C -2 -8 0 -2 -16 D -6 -8 2 0 -10 E 12 4 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6127: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (13) E D C A B (11) A B C E D (11) B A C D E (10) B A C E D (4) A C B E D (4) E D C B A (3) E C D A B (3) D E B A C (3) D E A B C (3) C A B E D (3) B A D C E (3) E D A C B (2) E C A B D (2) D E C A B (2) D E B C A (2) D B E A C (2) D B C E A (2) D B A E C (2) C B A E D (2) B C A D E (2) A E C B D (2) E A D C B (1) E A C D B (1) E A C B D (1) D B A C E (1) D A E B C (1) C E B A D (1) C E A B D (1) B D C A E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 -2 -6 B 2 0 -4 -4 -4 C -4 4 0 -4 -8 D 2 4 4 0 -4 E 6 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 -2 -6 B 2 0 -4 -4 -4 C -4 4 0 -4 -8 D 2 4 4 0 -4 E 6 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=24 B=20 A=18 C=7 so C is eliminated. Round 2 votes counts: D=31 E=26 B=22 A=21 so A is eliminated. Round 3 votes counts: B=41 D=31 E=28 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:203 A:197 B:195 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 -2 -6 B 2 0 -4 -4 -4 C -4 4 0 -4 -8 D 2 4 4 0 -4 E 6 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -2 -6 B 2 0 -4 -4 -4 C -4 4 0 -4 -8 D 2 4 4 0 -4 E 6 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -2 -6 B 2 0 -4 -4 -4 C -4 4 0 -4 -8 D 2 4 4 0 -4 E 6 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6128: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) C A B E D (8) D C E B A (6) D C A E B (6) E B D A C (5) E B A D C (5) D E B A C (5) C D B A E (4) C A D B E (4) A C D E B (4) A C B E D (4) C D A B E (3) A E B C D (3) E D B A C (2) E A B D C (2) D E C B A (2) C D A E B (2) C A B D E (2) B E D C A (2) B E D A C (2) B E C D A (2) B E A C D (2) A E D B C (2) A E B D C (2) A B E C D (2) D E A B C (1) D B E C A (1) C D B E A (1) C B E D A (1) C B A E D (1) B D C E A (1) B C E A D (1) B C A E D (1) B A C E D (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 -12 -8 2 B 6 0 0 -4 -14 C 12 0 0 -4 2 D 8 4 4 0 0 E -2 14 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.757197 E: 0.242803 Sum of squares = 0.632300901672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.757197 E: 1.000000 A B C D E A 0 -6 -12 -8 2 B 6 0 0 -4 -14 C 12 0 0 -4 2 D 8 4 4 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 A=19 E=14 B=12 so B is eliminated. Round 2 votes counts: D=30 C=28 E=22 A=20 so A is eliminated. Round 3 votes counts: C=38 E=32 D=30 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:205 E:205 B:194 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -8 2 B 6 0 0 -4 -14 C 12 0 0 -4 2 D 8 4 4 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -8 2 B 6 0 0 -4 -14 C 12 0 0 -4 2 D 8 4 4 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -8 2 B 6 0 0 -4 -14 C 12 0 0 -4 2 D 8 4 4 0 0 E -2 14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6129: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) A D E B C (9) D A B E C (8) C E B A D (7) C E B D A (6) A D B E C (5) D A B C E (3) C E A B D (3) C B D E A (3) B E D C A (3) A D C E B (3) E A B C D (2) D B C A E (2) D B A C E (2) C D B A E (2) B D C E A (2) A E D B C (2) A E C D B (2) A D E C B (2) A C E D B (2) E C B D A (1) E C B A D (1) E C A B D (1) E B D C A (1) E B C A D (1) E B A D C (1) E A C B D (1) D B E C A (1) D B A E C (1) D A C B E (1) C A E B D (1) C A D B E (1) C A B D E (1) B D E C A (1) B C E D A (1) B C D E A (1) A E C B D (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 -2 2 B -2 0 -8 2 4 C 4 8 0 4 12 D 2 -2 -4 0 2 E -2 -4 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -2 2 B -2 0 -8 2 4 C 4 8 0 4 12 D 2 -2 -4 0 2 E -2 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=29 D=18 E=9 B=8 so B is eliminated. Round 2 votes counts: C=38 A=29 D=21 E=12 so E is eliminated. Round 3 votes counts: C=42 A=33 D=25 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:199 D:199 B:198 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -2 2 B -2 0 -8 2 4 C 4 8 0 4 12 D 2 -2 -4 0 2 E -2 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 2 B -2 0 -8 2 4 C 4 8 0 4 12 D 2 -2 -4 0 2 E -2 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 2 B -2 0 -8 2 4 C 4 8 0 4 12 D 2 -2 -4 0 2 E -2 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6130: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) B D A C E (6) E C A D B (5) E C A B D (4) E A C B D (4) D B A C E (4) C A E D B (4) C A D E B (4) E B D A C (3) C A D B E (3) B D E A C (3) A B D C E (3) E C B D A (2) E B D C A (2) E B C D A (2) E A B D C (2) D B C A E (2) D A B C E (2) C D B E A (2) B E D C A (2) B E D A C (2) B D C E A (2) B D C A E (2) B D A E C (2) A E C D B (2) A D C B E (2) A C D B E (2) A B D E C (2) E B A D C (1) E B A C D (1) E A C D B (1) D C B A E (1) D B C E A (1) C E D B A (1) C E D A B (1) C E A D B (1) C D B A E (1) C D A E B (1) B D E C A (1) B A D E C (1) B A D C E (1) A E B D C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 10 -6 -8 14 B -10 0 -2 -2 14 C 6 2 0 2 14 D 8 2 -2 0 18 E -14 -14 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 -8 14 B -10 0 -2 -2 14 C 6 2 0 2 14 D 8 2 -2 0 18 E -14 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=27 C=27 B=22 A=14 D=10 so D is eliminated. Round 2 votes counts: B=29 C=28 E=27 A=16 so A is eliminated. Round 3 votes counts: B=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:212 A:205 B:200 E:170 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 -8 14 B -10 0 -2 -2 14 C 6 2 0 2 14 D 8 2 -2 0 18 E -14 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -8 14 B -10 0 -2 -2 14 C 6 2 0 2 14 D 8 2 -2 0 18 E -14 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -8 14 B -10 0 -2 -2 14 C 6 2 0 2 14 D 8 2 -2 0 18 E -14 -14 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6131: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) A D E B C (6) E A C D B (5) D B A E C (5) C E B A D (5) A E D C B (5) E A D C B (4) C B D A E (4) B D C A E (4) B C D E A (4) E C A D B (3) E C A B D (3) D E A B C (3) D A B E C (3) C B E A D (3) C A B D E (3) D B E A C (2) C B A D E (2) C A E B D (2) B D E A C (2) B D C E A (2) B D A C E (2) B C E D A (2) B C D A E (2) A C E D B (2) E D B A C (1) E C B D A (1) D B A C E (1) D A E B C (1) C B E D A (1) C A B E D (1) B E D C A (1) B D E C A (1) B D A E C (1) A E D B C (1) A E C D B (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 8 -6 14 -4 B -8 0 -10 10 -6 C 6 10 0 6 -2 D -14 -10 -6 0 0 E 4 6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.132397 E: 0.867603 Sum of squares = 0.770263872063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.132397 E: 1.000000 A B C D E A 0 8 -6 14 -4 B -8 0 -10 10 -6 C 6 10 0 6 -2 D -14 -10 -6 0 0 E 4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.777778 Sum of squares = 0.654320991328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=21 E=17 A=17 D=15 so D is eliminated. Round 2 votes counts: C=30 B=29 A=21 E=20 so E is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:206 E:206 B:193 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 14 -4 B -8 0 -10 10 -6 C 6 10 0 6 -2 D -14 -10 -6 0 0 E 4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.777778 Sum of squares = 0.654320991328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 14 -4 B -8 0 -10 10 -6 C 6 10 0 6 -2 D -14 -10 -6 0 0 E 4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.777778 Sum of squares = 0.654320991328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 14 -4 B -8 0 -10 10 -6 C 6 10 0 6 -2 D -14 -10 -6 0 0 E 4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.777778 Sum of squares = 0.654320991328 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.222222 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6132: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) B A E D C (7) E D C A B (6) C D E A B (6) B A E C D (5) B A D C E (5) D C E B A (4) A E B C D (4) A B E C D (4) E C D A B (3) E A C D B (3) E A B D C (3) D C B E A (3) B D C A E (3) A B E D C (3) A B C E D (3) E D B C A (2) D E C B A (2) C D B A E (2) C D A E B (2) C A D B E (2) B C D A E (2) A E C D B (2) A B C D E (2) E D B A C (1) E B D A C (1) E A D B C (1) D C E A B (1) D B E C A (1) C E D A B (1) C D B E A (1) C D A B E (1) C A D E B (1) B E D A C (1) B D C E A (1) B A C E D (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 0 12 12 18 B 0 0 18 10 10 C -12 -18 0 8 -2 D -12 -10 -8 0 -6 E -18 -10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.417912 B: 0.582088 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.513476859192 Cumulative probabilities = A: 0.417912 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 12 18 B 0 0 18 10 10 C -12 -18 0 8 -2 D -12 -10 -8 0 -6 E -18 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=20 A=20 C=16 D=11 so D is eliminated. Round 2 votes counts: B=34 C=24 E=22 A=20 so A is eliminated. Round 3 votes counts: B=46 E=30 C=24 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:221 B:219 E:190 C:188 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 12 18 B 0 0 18 10 10 C -12 -18 0 8 -2 D -12 -10 -8 0 -6 E -18 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 12 18 B 0 0 18 10 10 C -12 -18 0 8 -2 D -12 -10 -8 0 -6 E -18 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 12 18 B 0 0 18 10 10 C -12 -18 0 8 -2 D -12 -10 -8 0 -6 E -18 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6133: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (5) D E A B C (5) D A E B C (5) C B A E D (5) A D C E B (5) E B D C A (4) C A B E D (4) B C E A D (4) A C D B E (4) A C B D E (4) E C B D A (3) E C B A D (3) D A E C B (3) D A B E C (3) C E B A D (3) C B E A D (3) A D C B E (3) A D B C E (3) E D C B A (2) D E B A C (2) D B E A C (2) B E C D A (2) B D E C A (2) B D E A C (2) B D A C E (2) B C A D E (2) E D A C B (1) E C D B A (1) E A D C B (1) D E A C B (1) D B A E C (1) C A E B D (1) B E D C A (1) B D C E A (1) B D A E C (1) B A C D E (1) A E C D B (1) A D E C B (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 8 2 4 B 4 0 0 10 4 C -8 0 0 -4 -4 D -2 -10 4 0 12 E -4 -4 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.783544 C: 0.216456 D: 0.000000 E: 0.000000 Sum of squares = 0.660794471139 Cumulative probabilities = A: 0.000000 B: 0.783544 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 2 4 B 4 0 0 10 4 C -8 0 0 -4 -4 D -2 -10 4 0 12 E -4 -4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555558485 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=22 E=20 B=18 C=16 so C is eliminated. Round 2 votes counts: A=29 B=26 E=23 D=22 so D is eliminated. Round 3 votes counts: A=40 E=31 B=29 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:209 A:205 D:202 C:192 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 2 4 B 4 0 0 10 4 C -8 0 0 -4 -4 D -2 -10 4 0 12 E -4 -4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555558485 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 2 4 B 4 0 0 10 4 C -8 0 0 -4 -4 D -2 -10 4 0 12 E -4 -4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555558485 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 2 4 B 4 0 0 10 4 C -8 0 0 -4 -4 D -2 -10 4 0 12 E -4 -4 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555558485 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6134: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (16) B C E A D (9) A E B C D (9) D C E B A (5) D A E C B (5) B E C A D (5) A D E B C (5) E A B C D (3) D A E B C (3) C B D E A (3) B C E D A (3) E D A C B (2) E C B A D (2) E B C A D (2) D B C E A (2) D A C E B (2) C B E D A (2) B C A E D (2) A D E C B (2) A B E C D (2) E C B D A (1) E A C B D (1) D E A C B (1) D B C A E (1) D A B C E (1) C E B D A (1) C D B E A (1) C B E A D (1) B C D E A (1) B C A D E (1) B A C E D (1) A E D C B (1) A E D B C (1) A E C B D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -16 0 -22 B 18 0 6 2 6 C 16 -6 0 2 6 D 0 -2 -2 0 2 E 22 -6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 0 -22 B 18 0 6 2 6 C 16 -6 0 2 6 D 0 -2 -2 0 2 E 22 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975067 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=23 B=22 E=11 C=8 so C is eliminated. Round 2 votes counts: D=37 B=28 A=23 E=12 so E is eliminated. Round 3 votes counts: D=39 B=34 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:209 E:204 D:199 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 0 -22 B 18 0 6 2 6 C 16 -6 0 2 6 D 0 -2 -2 0 2 E 22 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975067 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 0 -22 B 18 0 6 2 6 C 16 -6 0 2 6 D 0 -2 -2 0 2 E 22 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975067 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 0 -22 B 18 0 6 2 6 C 16 -6 0 2 6 D 0 -2 -2 0 2 E 22 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999975067 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6135: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) A C B E D (9) E B C A D (7) A B C E D (7) D A C B E (6) C A B D E (4) A C B D E (4) E D B C A (3) E B C D A (3) D E C B A (3) D E A B C (3) D C A E B (3) D A B E C (3) A B E C D (3) E C B A D (2) E B D A C (2) D E C A B (2) D C E B A (2) D C A B E (2) D A B C E (2) C B E A D (2) C A D B E (2) C A B E D (2) E D C B A (1) E D B A C (1) E B A C D (1) D E B C A (1) D E A C B (1) D C E A B (1) D A E C B (1) D A E B C (1) C E B A D (1) B E C A D (1) B E A C D (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 18 16 -2 4 B -18 0 2 0 2 C -16 -2 0 0 0 D 2 0 0 0 8 E -4 -2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.062422 C: 0.000000 D: 0.937578 E: 0.000000 Sum of squares = 0.88294884981 Cumulative probabilities = A: 0.000000 B: 0.062422 C: 0.062422 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 -2 4 B -18 0 2 0 2 C -16 -2 0 0 0 D 2 0 0 0 8 E -4 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000242035 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=26 E=20 C=11 B=2 so B is eliminated. Round 2 votes counts: D=41 A=26 E=22 C=11 so C is eliminated. Round 3 votes counts: D=41 A=34 E=25 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:218 D:205 B:193 E:193 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 16 -2 4 B -18 0 2 0 2 C -16 -2 0 0 0 D 2 0 0 0 8 E -4 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000242035 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 -2 4 B -18 0 2 0 2 C -16 -2 0 0 0 D 2 0 0 0 8 E -4 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000242035 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 -2 4 B -18 0 2 0 2 C -16 -2 0 0 0 D 2 0 0 0 8 E -4 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000242035 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6136: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (12) B A E C D (11) E C D A B (9) D C E A B (9) E C D B A (8) C D E B A (4) B A E D C (4) A B E D C (4) E A B C D (3) D C A E B (3) C D E A B (3) B E A C D (3) A D C B E (3) E B A C D (2) D C E B A (2) B E C D A (2) B D C A E (2) B C D E A (2) A D C E B (2) A B D C E (2) E B C D A (1) E A C D B (1) D C B A E (1) C D B E A (1) B C E D A (1) B A C D E (1) A E D C B (1) A E B D C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 4 4 -2 B 14 0 6 6 0 C -4 -6 0 8 -4 D -4 -6 -8 0 -6 E 2 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.253510 C: 0.000000 D: 0.000000 E: 0.746490 Sum of squares = 0.621514752332 Cumulative probabilities = A: 0.000000 B: 0.253510 C: 0.253510 D: 0.253510 E: 1.000000 A B C D E A 0 -14 4 4 -2 B 14 0 6 6 0 C -4 -6 0 8 -4 D -4 -6 -8 0 -6 E 2 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=24 D=15 A=15 C=8 so C is eliminated. Round 2 votes counts: B=38 E=24 D=23 A=15 so A is eliminated. Round 3 votes counts: B=45 D=28 E=27 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:206 C:197 A:196 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 4 -2 B 14 0 6 6 0 C -4 -6 0 8 -4 D -4 -6 -8 0 -6 E 2 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 4 -2 B 14 0 6 6 0 C -4 -6 0 8 -4 D -4 -6 -8 0 -6 E 2 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 4 -2 B 14 0 6 6 0 C -4 -6 0 8 -4 D -4 -6 -8 0 -6 E 2 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6137: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) E C A D B (6) C B E D A (6) B D A C E (6) E A C D B (4) D B A E C (4) B C D A E (4) B A D C E (4) E A D C B (3) D B C A E (3) C E B A D (3) A E B D C (3) A D B E C (3) A B E D C (3) E A C B D (2) D E C A B (2) D E A C B (2) D B A C E (2) C E D B A (2) C D E B A (2) C D B E A (2) C B E A D (2) B A D E C (2) A E D B C (2) A B E C D (2) E C D A B (1) E C A B D (1) D C E B A (1) D C B E A (1) D A E B C (1) C E B D A (1) C E A B D (1) C B D E A (1) C B A E D (1) B D C A E (1) B C D E A (1) B C A E D (1) B C A D E (1) B A C E D (1) B A C D E (1) A E C B D (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 12 -2 12 B 6 0 10 0 20 C -12 -10 0 -4 -4 D 2 0 4 0 6 E -12 -20 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.327287 C: 0.000000 D: 0.672713 E: 0.000000 Sum of squares = 0.559659754746 Cumulative probabilities = A: 0.000000 B: 0.327287 C: 0.327287 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -2 12 B 6 0 10 0 20 C -12 -10 0 -4 -4 D 2 0 4 0 6 E -12 -20 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 C=21 E=17 A=16 so A is eliminated. Round 2 votes counts: D=28 B=27 E=24 C=21 so C is eliminated. Round 3 votes counts: B=37 D=32 E=31 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:208 D:206 C:185 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 -2 12 B 6 0 10 0 20 C -12 -10 0 -4 -4 D 2 0 4 0 6 E -12 -20 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -2 12 B 6 0 10 0 20 C -12 -10 0 -4 -4 D 2 0 4 0 6 E -12 -20 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -2 12 B 6 0 10 0 20 C -12 -10 0 -4 -4 D 2 0 4 0 6 E -12 -20 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6138: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (5) D B C A E (5) C A B E D (5) B D E C A (5) A E C D B (5) E A C B D (4) A E D C B (4) E B D A C (3) D E B A C (3) C E A B D (3) C B A E D (3) C A E B D (3) C A B D E (3) B D C A E (3) B C D E A (3) A D E C B (3) E D B A C (2) E D A B C (2) E C A B D (2) E A D C B (2) D A E B C (2) D A C B E (2) D A B E C (2) C B D A E (2) C B A D E (2) C A D B E (2) A C E D B (2) E B D C A (1) E B C D A (1) E A C D B (1) D E A C B (1) D E A B C (1) D B E C A (1) D B C E A (1) D B A E C (1) D A E C B (1) D A C E B (1) C E B A D (1) C B E A D (1) B E D A C (1) B E C A D (1) B D C E A (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 2 0 8 B -8 0 -12 -4 0 C -2 12 0 -10 -8 D 0 4 10 0 6 E -8 0 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.327126 B: 0.000000 C: 0.000000 D: 0.672874 E: 0.000000 Sum of squares = 0.559770599587 Cumulative probabilities = A: 0.327126 B: 0.327126 C: 0.327126 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 0 8 B -8 0 -12 -4 0 C -2 12 0 -10 -8 D 0 4 10 0 6 E -8 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=25 E=18 A=17 B=14 so B is eliminated. Round 2 votes counts: D=35 C=28 E=20 A=17 so A is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:209 E:197 C:196 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 0 8 B -8 0 -12 -4 0 C -2 12 0 -10 -8 D 0 4 10 0 6 E -8 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 0 8 B -8 0 -12 -4 0 C -2 12 0 -10 -8 D 0 4 10 0 6 E -8 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 0 8 B -8 0 -12 -4 0 C -2 12 0 -10 -8 D 0 4 10 0 6 E -8 0 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6139: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (15) C D B A E (10) C D B E A (9) A E C B D (8) E A D B C (7) C A E B D (5) D B C E A (4) C B D A E (4) B D C A E (4) A C E B D (4) E A C D B (3) D B E A C (3) B D A E C (3) A E B C D (3) E A D C B (2) E A C B D (2) D C B E A (2) C E A D B (2) A E B D C (2) E D B A C (1) E C A D B (1) E B D A C (1) D B E C A (1) C B A D E (1) C A E D B (1) B D E A C (1) B A C D E (1) Total count = 100 A B C D E A 0 10 12 14 -8 B -10 0 -8 8 -14 C -12 8 0 8 -6 D -14 -8 -8 0 -14 E 8 14 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 12 14 -8 B -10 0 -8 8 -14 C -12 8 0 8 -6 D -14 -8 -8 0 -14 E 8 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=32 C=32 A=17 D=10 B=9 so B is eliminated. Round 2 votes counts: E=32 C=32 D=18 A=18 so D is eliminated. Round 3 votes counts: C=42 E=37 A=21 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:214 C:199 B:188 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 12 14 -8 B -10 0 -8 8 -14 C -12 8 0 8 -6 D -14 -8 -8 0 -14 E 8 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 14 -8 B -10 0 -8 8 -14 C -12 8 0 8 -6 D -14 -8 -8 0 -14 E 8 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 14 -8 B -10 0 -8 8 -14 C -12 8 0 8 -6 D -14 -8 -8 0 -14 E 8 14 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6140: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) A D C B E (8) A C B D E (8) E B C D A (7) A C B E D (6) E D B C A (5) D E C B A (4) D E B A C (4) D E A B C (4) B C E A D (4) B C A E D (4) A C D B E (4) D E A C B (3) D A E B C (3) D A C E B (3) B E C A D (3) E B D C A (2) C D A E B (2) E D C B A (1) E D B A C (1) E C B D A (1) D A E C B (1) C E B D A (1) C D E B A (1) C B E A D (1) C B A E D (1) C A B E D (1) B E D A C (1) B E C D A (1) B A E D C (1) B A C E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 0 -12 -10 B 10 0 8 -10 -8 C 0 -8 0 -4 -8 D 12 10 4 0 12 E 10 8 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -12 -10 B 10 0 8 -10 -8 C 0 -8 0 -4 -8 D 12 10 4 0 12 E 10 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=28 E=17 B=15 C=7 so C is eliminated. Round 2 votes counts: D=36 A=29 E=18 B=17 so B is eliminated. Round 3 votes counts: D=36 A=36 E=28 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:207 B:200 C:190 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 0 -12 -10 B 10 0 8 -10 -8 C 0 -8 0 -4 -8 D 12 10 4 0 12 E 10 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -12 -10 B 10 0 8 -10 -8 C 0 -8 0 -4 -8 D 12 10 4 0 12 E 10 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -12 -10 B 10 0 8 -10 -8 C 0 -8 0 -4 -8 D 12 10 4 0 12 E 10 8 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6141: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) D B A C E (7) C D B A E (7) E A B D C (6) D C B A E (5) A E B D C (5) E C A B D (4) E A C B D (4) C E D B A (4) C D E B A (4) C D B E A (3) B D A C E (3) A E B C D (3) A B D E C (3) E D C A B (2) E D A C B (2) E C D B A (2) C B D A E (2) B C A D E (2) A B E D C (2) A B E C D (2) A B D C E (2) A B C E D (2) E D C B A (1) E D A B C (1) E C D A B (1) E A D B C (1) D E C A B (1) D E A B C (1) D C E B A (1) D C B E A (1) D A B E C (1) C E B A D (1) C B E A D (1) C B D E A (1) C B A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 12 0 2 B -2 0 4 10 -2 C -12 -4 0 8 2 D 0 -10 -8 0 -4 E -2 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.896698 B: 0.000000 C: 0.000000 D: 0.103302 E: 0.000000 Sum of squares = 0.814738353374 Cumulative probabilities = A: 0.896698 B: 0.896698 C: 0.896698 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 0 2 B -2 0 4 10 -2 C -12 -4 0 8 2 D 0 -10 -8 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222223034 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=24 A=20 D=17 B=8 so B is eliminated. Round 2 votes counts: E=31 C=26 A=23 D=20 so D is eliminated. Round 3 votes counts: A=34 E=33 C=33 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:205 E:201 C:197 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 0 2 B -2 0 4 10 -2 C -12 -4 0 8 2 D 0 -10 -8 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222223034 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 0 2 B -2 0 4 10 -2 C -12 -4 0 8 2 D 0 -10 -8 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222223034 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 0 2 B -2 0 4 10 -2 C -12 -4 0 8 2 D 0 -10 -8 0 -4 E -2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.722222223034 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6142: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (5) C D B A E (5) C D A B E (5) B D E A C (5) A C B D E (5) E A B D C (4) D C B E A (4) C A E D B (4) B D A C E (4) A C B E D (4) E D B C A (3) E A C B D (3) E A B C D (3) D B C E A (3) C A D E B (3) A E C B D (3) E B D A C (2) E B A D C (2) D E B C A (2) D C E B A (2) D B C A E (2) C E D A B (2) C A D B E (2) B E D A C (2) B D A E C (2) B A E D C (2) A E B D C (2) A E B C D (2) A C E B D (2) E D B A C (1) E C A D B (1) C E A D B (1) C D E B A (1) C A B D E (1) B E A D C (1) B A D E C (1) B A D C E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 8 0 6 B 0 0 4 8 14 C -8 -4 0 -4 2 D 0 -8 4 0 8 E -6 -14 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.501772 B: 0.498228 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500006279289 Cumulative probabilities = A: 0.501772 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 0 6 B 0 0 4 8 14 C -8 -4 0 -4 2 D 0 -8 4 0 8 E -6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=21 E=19 D=18 B=18 so D is eliminated. Round 2 votes counts: C=30 B=28 E=21 A=21 so E is eliminated. Round 3 votes counts: B=38 C=31 A=31 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:207 D:202 C:193 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 0 6 B 0 0 4 8 14 C -8 -4 0 -4 2 D 0 -8 4 0 8 E -6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 0 6 B 0 0 4 8 14 C -8 -4 0 -4 2 D 0 -8 4 0 8 E -6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 0 6 B 0 0 4 8 14 C -8 -4 0 -4 2 D 0 -8 4 0 8 E -6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999909 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6143: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) D C A E B (6) D E B C A (5) E C B D A (4) C E B D A (4) C E B A D (4) C D A E B (4) C A D E B (4) B E D A C (4) B E A D C (4) D B E A C (3) D A B E C (3) D A B C E (3) B E A C D (3) A C D B E (3) E B D A C (2) E B C D A (2) D E C B A (2) D C E A B (2) D A C B E (2) C E D B A (2) C D E A B (2) C A D B E (2) B E C A D (2) A D C B E (2) E D B C A (1) E C D B A (1) E B D C A (1) E B A C D (1) D E B A C (1) D C E B A (1) D A C E B (1) C E A B D (1) C D E B A (1) C A E D B (1) C A E B D (1) C A B E D (1) B D E A C (1) B D A E C (1) A D B E C (1) A C B E D (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -22 -18 -22 B 14 0 -4 -6 -22 C 22 4 0 4 -2 D 18 6 -4 0 2 E 22 22 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -14 -22 -18 -22 B 14 0 -4 -6 -22 C 22 4 0 4 -2 D 18 6 -4 0 2 E 22 22 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 E=19 B=15 A=10 so A is eliminated. Round 2 votes counts: D=32 C=31 E=19 B=18 so B is eliminated. Round 3 votes counts: D=35 E=33 C=32 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:222 C:214 D:211 B:191 A:162 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -22 -18 -22 B 14 0 -4 -6 -22 C 22 4 0 4 -2 D 18 6 -4 0 2 E 22 22 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -22 -18 -22 B 14 0 -4 -6 -22 C 22 4 0 4 -2 D 18 6 -4 0 2 E 22 22 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -22 -18 -22 B 14 0 -4 -6 -22 C 22 4 0 4 -2 D 18 6 -4 0 2 E 22 22 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000009 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6144: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) B E A C D (6) E B A D C (5) A E D C B (5) E A D B C (4) D A E C B (4) D A C E B (4) C A D E B (4) B E A D C (4) B C D E A (4) D E B A C (3) C D B A E (3) C B A E D (3) B E D A C (3) B C A E D (3) E B D A C (2) E A B D C (2) D E A C B (2) C D A B E (2) C A D B E (2) B E C A D (2) B D E A C (2) B C E A D (2) A D E C B (2) A C E D B (2) E D B A C (1) E D A B C (1) D E C B A (1) D E A B C (1) D C E B A (1) D C B E A (1) C D A E B (1) C B D E A (1) C A B E D (1) C A B D E (1) B E D C A (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 12 2 -4 B 2 0 -2 -10 -12 C -12 2 0 -20 -6 D -2 10 20 0 2 E 4 12 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 12 2 -4 B 2 0 -2 -10 -12 C -12 2 0 -20 -6 D -2 10 20 0 2 E 4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=25 C=18 E=15 A=12 so A is eliminated. Round 2 votes counts: B=30 D=28 E=21 C=21 so E is eliminated. Round 3 votes counts: D=39 B=39 C=22 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:210 A:204 B:189 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 12 2 -4 B 2 0 -2 -10 -12 C -12 2 0 -20 -6 D -2 10 20 0 2 E 4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 2 -4 B 2 0 -2 -10 -12 C -12 2 0 -20 -6 D -2 10 20 0 2 E 4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 2 -4 B 2 0 -2 -10 -12 C -12 2 0 -20 -6 D -2 10 20 0 2 E 4 12 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.374999999991 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6145: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) B A C D E (7) D A E C B (6) C E B A D (6) B C A E D (6) B C A D E (6) E D A C B (4) E C D B A (4) E C B D A (4) D E A C B (4) D A B E C (4) C B E A D (4) A D E B C (4) E D C A B (3) E A D C B (3) D A B C E (2) C E B D A (2) B D A C E (2) B C E A D (2) A D B C E (2) E D C B A (1) E C D A B (1) E C B A D (1) D C E B A (1) D B C E A (1) D B C A E (1) D B A C E (1) C E D B A (1) C B E D A (1) B C D A E (1) B A D C E (1) B A C E D (1) A E D C B (1) A E B C D (1) A D E C B (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 8 -2 14 B 8 0 4 -6 -10 C -8 -4 0 -4 -2 D 2 6 4 0 8 E -14 10 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -2 14 B 8 0 4 -6 -10 C -8 -4 0 -4 -2 D 2 6 4 0 8 E -14 10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 E=21 C=14 A=12 so A is eliminated. Round 2 votes counts: D=35 B=28 E=23 C=14 so C is eliminated. Round 3 votes counts: D=35 B=33 E=32 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:206 B:198 E:195 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -2 14 B 8 0 4 -6 -10 C -8 -4 0 -4 -2 D 2 6 4 0 8 E -14 10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -2 14 B 8 0 4 -6 -10 C -8 -4 0 -4 -2 D 2 6 4 0 8 E -14 10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -2 14 B 8 0 4 -6 -10 C -8 -4 0 -4 -2 D 2 6 4 0 8 E -14 10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6146: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) A E B D C (7) E A B D C (6) D C E B A (6) C D B A E (6) B A C D E (6) C D E B A (5) A B E D C (5) E D A C B (4) E D C B A (3) E B A D C (3) D E C B A (3) D E C A B (3) C D B E A (3) A B C E D (3) E D C A B (2) E A D B C (2) D C E A B (2) C D A E B (2) C B D A E (2) B C A D E (2) B A C E D (2) A B C D E (2) E D A B C (1) E A D C B (1) C B D E A (1) B C D E A (1) B C D A E (1) B A E D C (1) B A E C D (1) A E D C B (1) A E C D B (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 8 16 10 8 B -8 0 8 8 -6 C -16 -8 0 0 -10 D -10 -8 0 0 -8 E -8 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 10 8 B -8 0 8 8 -6 C -16 -8 0 0 -10 D -10 -8 0 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=22 C=19 D=14 B=14 so D is eliminated. Round 2 votes counts: A=31 E=28 C=27 B=14 so B is eliminated. Round 3 votes counts: A=41 C=31 E=28 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:208 B:201 D:187 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 10 8 B -8 0 8 8 -6 C -16 -8 0 0 -10 D -10 -8 0 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 10 8 B -8 0 8 8 -6 C -16 -8 0 0 -10 D -10 -8 0 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 10 8 B -8 0 8 8 -6 C -16 -8 0 0 -10 D -10 -8 0 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6147: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) E A D B C (6) C B D E A (5) D B C A E (4) C A D B E (4) B D C E A (4) B C D E A (4) A E D C B (4) A E D B C (4) A E C D B (4) A C D B E (4) E D B A C (3) C B E D A (3) C A E B D (3) E B D A C (2) D B A E C (2) D B A C E (2) B C E D A (2) A E C B D (2) A D E B C (2) A C E D B (2) E D A B C (1) E C B A D (1) E B D C A (1) E B A D C (1) E A D C B (1) E A B D C (1) E A B C D (1) D B E C A (1) D B C E A (1) D A E B C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E A B D (1) C D A B E (1) C B E A D (1) C B A D E (1) C A B D E (1) B E D C A (1) B D E C A (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 0 -4 16 B 0 0 -6 -6 12 C 0 6 0 4 16 D 4 6 -4 0 8 E -16 -12 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.254376 B: 0.000000 C: 0.745624 D: 0.000000 E: 0.000000 Sum of squares = 0.62066194775 Cumulative probabilities = A: 0.254376 B: 0.254376 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 16 B 0 0 -6 -6 12 C 0 6 0 4 16 D 4 6 -4 0 8 E -16 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499554 B: 0.000000 C: 0.500446 D: 0.000000 E: 0.000000 Sum of squares = 0.500000397509 Cumulative probabilities = A: 0.499554 B: 0.499554 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=26 E=18 D=14 B=12 so B is eliminated. Round 2 votes counts: C=36 A=26 E=19 D=19 so E is eliminated. Round 3 votes counts: C=37 A=36 D=27 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:213 D:207 A:206 B:200 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -4 16 B 0 0 -6 -6 12 C 0 6 0 4 16 D 4 6 -4 0 8 E -16 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499554 B: 0.000000 C: 0.500446 D: 0.000000 E: 0.000000 Sum of squares = 0.500000397509 Cumulative probabilities = A: 0.499554 B: 0.499554 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 16 B 0 0 -6 -6 12 C 0 6 0 4 16 D 4 6 -4 0 8 E -16 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499554 B: 0.000000 C: 0.500446 D: 0.000000 E: 0.000000 Sum of squares = 0.500000397509 Cumulative probabilities = A: 0.499554 B: 0.499554 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 16 B 0 0 -6 -6 12 C 0 6 0 4 16 D 4 6 -4 0 8 E -16 -12 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499554 B: 0.000000 C: 0.500446 D: 0.000000 E: 0.000000 Sum of squares = 0.500000397509 Cumulative probabilities = A: 0.499554 B: 0.499554 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6148: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D C A E (8) B A C E D (8) E A C B D (6) D E C A B (6) D B E C A (6) C A E D B (5) E D A C B (4) B D E A C (4) D B C A E (3) B D E C A (3) A E C B D (3) E C A D B (2) D C E A B (2) D B C E A (2) C D A E B (2) C A E B D (2) C A B D E (2) B C A E D (2) E D C A B (1) E C D A B (1) E B A C D (1) D E B C A (1) D E A C B (1) D C B A E (1) D C A E B (1) C E A D B (1) C B D A E (1) C B A E D (1) C A B E D (1) B E A C D (1) B D C E A (1) B D A C E (1) B C A D E (1) B A E C D (1) B A D C E (1) B A C D E (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -12 2 -4 B -4 0 -10 2 0 C 12 10 0 8 0 D -2 -2 -8 0 -4 E 4 0 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.310427 D: 0.000000 E: 0.689573 Sum of squares = 0.571875896111 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.310427 D: 0.310427 E: 1.000000 A B C D E A 0 4 -12 2 -4 B -4 0 -10 2 0 C 12 10 0 8 0 D -2 -2 -8 0 -4 E 4 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=24 D=23 C=15 A=6 so A is eliminated. Round 2 votes counts: B=32 E=27 D=23 C=18 so C is eliminated. Round 3 votes counts: B=38 E=37 D=25 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:215 E:204 A:195 B:194 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 2 -4 B -4 0 -10 2 0 C 12 10 0 8 0 D -2 -2 -8 0 -4 E 4 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 2 -4 B -4 0 -10 2 0 C 12 10 0 8 0 D -2 -2 -8 0 -4 E 4 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 2 -4 B -4 0 -10 2 0 C 12 10 0 8 0 D -2 -2 -8 0 -4 E 4 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6149: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (8) B D C E A (6) A E C B D (6) B E C D A (5) D B C E A (4) A E B C D (4) E C B D A (3) D C B E A (3) D C A E B (3) D B A C E (3) C E D B A (3) B D A E C (3) B C E D A (3) A E D C B (3) A D E C B (3) D C B A E (2) D A C E B (2) D A B C E (2) C D E B A (2) C D E A B (2) C D B E A (2) B E C A D (2) B D A C E (2) A D C E B (2) A D B E C (2) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C B D (1) E A B C D (1) D C E A B (1) D B C A E (1) D A C B E (1) C E B D A (1) C E A D B (1) C B E D A (1) B E A C D (1) B C D E A (1) B A E D C (1) B A D E C (1) A E B D C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 0 -14 4 B 8 0 -4 -4 -6 C 0 4 0 4 -2 D 14 4 -4 0 0 E -4 6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.314652 E: 0.685348 Sum of squares = 0.568707521632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.314652 E: 1.000000 A B C D E A 0 -8 0 -14 4 B 8 0 -4 -4 -6 C 0 4 0 4 -2 D 14 4 -4 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555751545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=25 D=22 C=12 E=10 so E is eliminated. Round 2 votes counts: A=33 B=28 D=22 C=17 so C is eliminated. Round 3 votes counts: A=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:207 C:203 E:202 B:197 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 0 -14 4 B 8 0 -4 -4 -6 C 0 4 0 4 -2 D 14 4 -4 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555751545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -14 4 B 8 0 -4 -4 -6 C 0 4 0 4 -2 D 14 4 -4 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555751545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -14 4 B 8 0 -4 -4 -6 C 0 4 0 4 -2 D 14 4 -4 0 0 E -4 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555751545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6150: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) E D B C A (6) B D C E A (5) B C D A E (5) B A C D E (5) E A D C B (4) B A C E D (4) E D C A B (3) E B D C A (3) E A C D B (3) D C E A B (3) D B E C A (3) C A D B E (3) B E A C D (3) B D E C A (3) B D C A E (3) E B A C D (2) D E C A B (2) D E B C A (2) D C B A E (2) D C A B E (2) B E A D C (2) B C A D E (2) A E C D B (2) A C D E B (2) E D C B A (1) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E B A (1) D C A E B (1) D B C E A (1) D B C A E (1) C B D A E (1) B E D C A (1) B E C D A (1) B A E C D (1) A C E D B (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -12 -22 -26 B 24 0 20 -6 0 C 12 -20 0 -22 -14 D 22 6 22 0 -2 E 26 0 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.190784 C: 0.000000 D: 0.000000 E: 0.809216 Sum of squares = 0.69122947723 Cumulative probabilities = A: 0.000000 B: 0.190784 C: 0.190784 D: 0.190784 E: 1.000000 A B C D E A 0 -24 -12 -22 -26 B 24 0 20 -6 0 C 12 -20 0 -22 -14 D 22 6 22 0 -2 E 26 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249998 C: 0.000000 D: 0.000000 E: 0.750002 Sum of squares = 0.625001544328 Cumulative probabilities = A: 0.000000 B: 0.249998 C: 0.249998 D: 0.249998 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=35 B=35 D=19 A=7 C=4 so C is eliminated. Round 2 votes counts: B=36 E=35 D=19 A=10 so A is eliminated. Round 3 votes counts: E=38 B=38 D=24 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:224 E:221 B:219 C:178 A:158 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 -12 -22 -26 B 24 0 20 -6 0 C 12 -20 0 -22 -14 D 22 6 22 0 -2 E 26 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249998 C: 0.000000 D: 0.000000 E: 0.750002 Sum of squares = 0.625001544328 Cumulative probabilities = A: 0.000000 B: 0.249998 C: 0.249998 D: 0.249998 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -12 -22 -26 B 24 0 20 -6 0 C 12 -20 0 -22 -14 D 22 6 22 0 -2 E 26 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249998 C: 0.000000 D: 0.000000 E: 0.750002 Sum of squares = 0.625001544328 Cumulative probabilities = A: 0.000000 B: 0.249998 C: 0.249998 D: 0.249998 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -12 -22 -26 B 24 0 20 -6 0 C 12 -20 0 -22 -14 D 22 6 22 0 -2 E 26 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249998 C: 0.000000 D: 0.000000 E: 0.750002 Sum of squares = 0.625001544328 Cumulative probabilities = A: 0.000000 B: 0.249998 C: 0.249998 D: 0.249998 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6151: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) E B C D A (6) D B A C E (6) A C E D B (6) C A B D E (5) B D C A E (5) E C B A D (4) E A D C B (4) E A C D B (4) D E B A C (4) A C D B E (4) D B E A C (3) D A B C E (3) C B A E D (3) C A E B D (3) E D B C A (2) E D B A C (2) E B D C A (2) D E A B C (2) D B A E C (2) D A E B C (2) C B A D E (2) B E D C A (2) B E C D A (2) B C D A E (2) A C D E B (2) E C A B D (1) D B C A E (1) D A B E C (1) C E B A D (1) C A B E D (1) B C E D A (1) B C D E A (1) A D C E B (1) Total count = 100 A B C D E A 0 -22 -8 -18 -2 B 22 0 18 2 8 C 8 -18 0 -4 -6 D 18 -2 4 0 12 E 2 -8 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -8 -18 -2 B 22 0 18 2 8 C 8 -18 0 -4 -6 D 18 -2 4 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=24 B=23 C=15 A=13 so A is eliminated. Round 2 votes counts: C=27 E=25 D=25 B=23 so B is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:225 D:216 E:194 C:190 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -8 -18 -2 B 22 0 18 2 8 C 8 -18 0 -4 -6 D 18 -2 4 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -8 -18 -2 B 22 0 18 2 8 C 8 -18 0 -4 -6 D 18 -2 4 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -8 -18 -2 B 22 0 18 2 8 C 8 -18 0 -4 -6 D 18 -2 4 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6152: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (15) E C B D A (9) D A C E B (5) B E C D A (5) C E D B A (4) C E D A B (4) A D C E B (4) E C A B D (3) D C E A B (3) B E A C D (3) B D C E A (3) A B E C D (3) E C D B A (2) E C B A D (2) E B C D A (2) D C E B A (2) D A C B E (2) D A B C E (2) C D E B A (2) B E C A D (2) B D A C E (2) B A D E C (2) B A D C E (2) A E C D B (2) A D C B E (2) A B D E C (2) E C A D B (1) E B C A D (1) D C B E A (1) D C A E B (1) D C A B E (1) D B C E A (1) D B C A E (1) C E B D A (1) C D E A B (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 -8 -4 B -6 0 -4 -12 0 C 4 4 0 -4 20 D 8 12 4 0 10 E 4 0 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -8 -4 B -6 0 -4 -12 0 C 4 4 0 -4 20 D 8 12 4 0 10 E 4 0 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=20 D=19 B=19 C=12 so C is eliminated. Round 2 votes counts: A=30 E=29 D=22 B=19 so B is eliminated. Round 3 votes counts: E=39 A=34 D=27 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:217 C:212 A:195 B:189 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -8 -4 B -6 0 -4 -12 0 C 4 4 0 -4 20 D 8 12 4 0 10 E 4 0 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -8 -4 B -6 0 -4 -12 0 C 4 4 0 -4 20 D 8 12 4 0 10 E 4 0 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -8 -4 B -6 0 -4 -12 0 C 4 4 0 -4 20 D 8 12 4 0 10 E 4 0 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6153: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) C B A D E (7) E D B A C (6) C A B D E (6) D E C B A (5) D C E A B (5) B C A E D (5) A C B D E (4) E D B C A (3) E D A B C (3) C D A E B (3) B C A D E (3) A C D E B (3) D E C A B (2) D E A C B (2) D C E B A (2) D C A E B (2) C A D E B (2) B A E D C (2) B A E C D (2) A E D B C (2) A D E C B (2) A C D B E (2) A B E C D (2) E D A C B (1) E B D A C (1) E B A D C (1) E A D B C (1) D A E C B (1) C D E B A (1) C D E A B (1) C D B E A (1) C D A B E (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A D C (1) B C D E A (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -8 12 18 B 6 0 -8 -4 2 C 8 8 0 12 18 D -12 4 -12 0 14 E -18 -2 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 12 18 B 6 0 -8 -4 2 C 8 8 0 12 18 D -12 4 -12 0 14 E -18 -2 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=23 D=19 E=16 A=16 so E is eliminated. Round 2 votes counts: D=32 B=28 C=23 A=17 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:208 B:198 D:197 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 12 18 B 6 0 -8 -4 2 C 8 8 0 12 18 D -12 4 -12 0 14 E -18 -2 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 12 18 B 6 0 -8 -4 2 C 8 8 0 12 18 D -12 4 -12 0 14 E -18 -2 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 12 18 B 6 0 -8 -4 2 C 8 8 0 12 18 D -12 4 -12 0 14 E -18 -2 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6154: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) B C E A D (8) D A E C B (7) B C A E D (7) A E D B C (7) E A B C D (6) D C B A E (5) C B D E A (5) A E B C D (5) E B C A D (4) D A C B E (4) C B E D A (4) B C A D E (4) E D A C B (3) D E A C B (3) D C E B A (3) A B C D E (3) E B A C D (2) D C B E A (2) B E C A D (2) A D E B C (2) E D C B A (1) E D A B C (1) E A D B C (1) E A B D C (1) D C A B E (1) C D B E A (1) Total count = 100 A B C D E A 0 -12 -10 4 6 B 12 0 6 18 8 C 10 -6 0 18 10 D -4 -18 -18 0 -4 E -6 -8 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 4 6 B 12 0 6 18 8 C 10 -6 0 18 10 D -4 -18 -18 0 -4 E -6 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=21 E=19 C=18 A=17 so A is eliminated. Round 2 votes counts: E=31 D=27 B=24 C=18 so C is eliminated. Round 3 votes counts: B=41 E=31 D=28 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:216 A:194 E:190 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -10 4 6 B 12 0 6 18 8 C 10 -6 0 18 10 D -4 -18 -18 0 -4 E -6 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 4 6 B 12 0 6 18 8 C 10 -6 0 18 10 D -4 -18 -18 0 -4 E -6 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 4 6 B 12 0 6 18 8 C 10 -6 0 18 10 D -4 -18 -18 0 -4 E -6 -8 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999473 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6155: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) A C D B E (12) A C D E B (10) A C B D E (7) C A D E B (6) B E A C D (5) E D B C A (4) D E C A B (4) D C E A B (4) B E C A D (4) B E D C A (3) A C B E D (3) E D C B A (2) E C D A B (2) E B C D A (2) D A C B E (2) B E D A C (2) B E A D C (2) B D A C E (2) B A C E D (2) B A C D E (2) A D C E B (2) D A C E B (1) C E A B D (1) B E C D A (1) B D E A C (1) B A D C E (1) A D C B E (1) Total count = 100 A B C D E A 0 10 10 16 2 B -10 0 -14 0 0 C -10 14 0 14 12 D -16 0 -14 0 10 E -2 0 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 16 2 B -10 0 -14 0 0 C -10 14 0 14 12 D -16 0 -14 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=25 E=22 D=11 C=7 so C is eliminated. Round 2 votes counts: A=41 B=25 E=23 D=11 so D is eliminated. Round 3 votes counts: A=44 E=31 B=25 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:215 D:190 B:188 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 16 2 B -10 0 -14 0 0 C -10 14 0 14 12 D -16 0 -14 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 16 2 B -10 0 -14 0 0 C -10 14 0 14 12 D -16 0 -14 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 16 2 B -10 0 -14 0 0 C -10 14 0 14 12 D -16 0 -14 0 10 E -2 0 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982429 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6156: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) A C D B E (9) D E B C A (8) B E A C D (8) A B E C D (8) E B D C A (6) D C E B A (6) E B D A C (5) D C A E B (4) D B E A C (4) B E D A C (4) B E A D C (4) D C A B E (3) A D C B E (3) E D B C A (2) D A C B E (2) C D A E B (2) C A D B E (2) E C B A D (1) E B A C D (1) D E B A C (1) D C E A B (1) C E B D A (1) C E B A D (1) C E A B D (1) C A E B D (1) C A D E B (1) C A B E D (1) Total count = 100 A B C D E A 0 -4 18 2 -8 B 4 0 2 4 16 C -18 -2 0 -6 -4 D -2 -4 6 0 -8 E 8 -16 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 18 2 -8 B 4 0 2 4 16 C -18 -2 0 -6 -4 D -2 -4 6 0 -8 E 8 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=29 B=16 E=15 C=10 so C is eliminated. Round 2 votes counts: A=35 D=31 E=18 B=16 so B is eliminated. Round 3 votes counts: A=35 E=34 D=31 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:213 A:204 E:202 D:196 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 18 2 -8 B 4 0 2 4 16 C -18 -2 0 -6 -4 D -2 -4 6 0 -8 E 8 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 18 2 -8 B 4 0 2 4 16 C -18 -2 0 -6 -4 D -2 -4 6 0 -8 E 8 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 18 2 -8 B 4 0 2 4 16 C -18 -2 0 -6 -4 D -2 -4 6 0 -8 E 8 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6157: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) D C E B A (12) A B C E D (12) C B A D E (11) D E C B A (10) E A B D C (8) B A C D E (8) E D C A B (7) C D B A E (7) A B E C D (4) E D C B A (1) E D A C B (1) D E A B C (1) C D E B A (1) B C A D E (1) B A C E D (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 0 -4 -6 B 4 0 0 -4 -8 C 0 0 0 -8 6 D 4 4 8 0 4 E 6 8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -4 -6 B 4 0 0 -4 -8 C 0 0 0 -8 6 D 4 4 8 0 4 E 6 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=23 C=19 A=19 B=10 so B is eliminated. Round 2 votes counts: E=29 A=28 D=23 C=20 so C is eliminated. Round 3 votes counts: A=40 D=31 E=29 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:202 C:199 B:196 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -4 -6 B 4 0 0 -4 -8 C 0 0 0 -8 6 D 4 4 8 0 4 E 6 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -4 -6 B 4 0 0 -4 -8 C 0 0 0 -8 6 D 4 4 8 0 4 E 6 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -4 -6 B 4 0 0 -4 -8 C 0 0 0 -8 6 D 4 4 8 0 4 E 6 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6158: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) E D C B A (7) E D C A B (7) D E A C B (7) B C A E D (7) A D C E B (7) E D B C A (4) E C D B A (4) B C E D A (4) A C D E B (4) D A E C B (3) B C E A D (3) B A C E D (3) A C D B E (3) A C B D E (3) E B D C A (2) D E C A B (2) C B A E D (2) B E D A C (2) B E C D A (2) B A D E C (2) B A C D E (2) A B C D E (2) E D B A C (1) D E B A C (1) D A C E B (1) C E D B A (1) C B E D A (1) C A D E B (1) C A B D E (1) B E D C A (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 2 0 2 B -2 0 -24 -26 -22 C -2 24 0 -14 -10 D 0 26 14 0 -2 E -2 22 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.717091 B: 0.000000 C: 0.000000 D: 0.282909 E: 0.000000 Sum of squares = 0.594257265554 Cumulative probabilities = A: 0.717091 B: 0.717091 C: 0.717091 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 2 B -2 0 -24 -26 -22 C -2 24 0 -14 -10 D 0 26 14 0 -2 E -2 22 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500343 B: 0.000000 C: 0.000000 D: 0.499657 E: 0.000000 Sum of squares = 0.500000235513 Cumulative probabilities = A: 0.500343 B: 0.500343 C: 0.500343 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=26 E=25 D=14 C=6 so C is eliminated. Round 2 votes counts: A=31 B=29 E=26 D=14 so D is eliminated. Round 3 votes counts: E=36 A=35 B=29 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:219 E:216 A:203 C:199 B:163 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 2 B -2 0 -24 -26 -22 C -2 24 0 -14 -10 D 0 26 14 0 -2 E -2 22 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500343 B: 0.000000 C: 0.000000 D: 0.499657 E: 0.000000 Sum of squares = 0.500000235513 Cumulative probabilities = A: 0.500343 B: 0.500343 C: 0.500343 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 2 B -2 0 -24 -26 -22 C -2 24 0 -14 -10 D 0 26 14 0 -2 E -2 22 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500343 B: 0.000000 C: 0.000000 D: 0.499657 E: 0.000000 Sum of squares = 0.500000235513 Cumulative probabilities = A: 0.500343 B: 0.500343 C: 0.500343 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 2 B -2 0 -24 -26 -22 C -2 24 0 -14 -10 D 0 26 14 0 -2 E -2 22 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500343 B: 0.000000 C: 0.000000 D: 0.499657 E: 0.000000 Sum of squares = 0.500000235513 Cumulative probabilities = A: 0.500343 B: 0.500343 C: 0.500343 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6159: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) E D A C B (7) E C D A B (7) B A D C E (7) A D B E C (6) B C E D A (5) C E D A B (4) A B D E C (4) C D E A B (3) B C D A E (3) B A C D E (3) A D E B C (3) A D B C E (3) A B D C E (3) D A C E B (2) C E D B A (2) C B E D A (2) C B D E A (2) B A D E C (2) A D E C B (2) E D C A B (1) E C D B A (1) E B A D C (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C B E (1) C E B D A (1) C D E B A (1) C D B A E (1) C D A E B (1) C B D A E (1) B E C D A (1) B C E A D (1) B C A E D (1) B A E D C (1) A E D B C (1) A D C B E (1) Total count = 100 A B C D E A 0 10 0 0 14 B -10 0 10 -6 14 C 0 -10 0 -2 18 D 0 6 2 0 24 E -14 -14 -18 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500764 B: 0.000000 C: 0.000000 D: 0.499236 E: 0.000000 Sum of squares = 0.500001147978 Cumulative probabilities = A: 0.500764 B: 0.500764 C: 0.500764 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 0 14 B -10 0 10 -6 14 C 0 -10 0 -2 18 D 0 6 2 0 24 E -14 -14 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999414 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=23 E=19 C=18 D=7 so D is eliminated. Round 2 votes counts: B=33 A=27 E=20 C=20 so E is eliminated. Round 3 votes counts: A=36 B=34 C=30 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:212 B:204 C:203 E:165 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 0 14 B -10 0 10 -6 14 C 0 -10 0 -2 18 D 0 6 2 0 24 E -14 -14 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999414 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 0 14 B -10 0 10 -6 14 C 0 -10 0 -2 18 D 0 6 2 0 24 E -14 -14 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999414 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 0 14 B -10 0 10 -6 14 C 0 -10 0 -2 18 D 0 6 2 0 24 E -14 -14 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999414 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6160: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) B E C D A (9) D A E C B (8) E B D A C (6) A D C E B (6) B C E A D (5) E D A B C (4) D A B C E (4) E B C D A (3) D A C E B (3) C B A D E (3) C A D B E (3) B E C A D (3) B D A E C (3) B C A D E (3) E C B A D (2) E B C A D (2) D A C B E (2) D A B E C (2) C B E A D (2) B E D A C (2) B C E D A (2) A D C B E (2) A C D B E (2) E C A D B (1) D E A B C (1) D A E B C (1) C E A D B (1) C B A E D (1) C A E D B (1) C A D E B (1) B D A C E (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 16 -20 -4 B -6 0 2 -6 -2 C -16 -2 0 -10 -14 D 20 6 10 0 -6 E 4 2 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 16 -20 -4 B -6 0 2 -6 -2 C -16 -2 0 -10 -14 D 20 6 10 0 -6 E 4 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=27 D=21 C=12 A=12 so C is eliminated. Round 2 votes counts: B=34 E=28 D=21 A=17 so A is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:213 A:199 B:194 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 16 -20 -4 B -6 0 2 -6 -2 C -16 -2 0 -10 -14 D 20 6 10 0 -6 E 4 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 -20 -4 B -6 0 2 -6 -2 C -16 -2 0 -10 -14 D 20 6 10 0 -6 E 4 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 -20 -4 B -6 0 2 -6 -2 C -16 -2 0 -10 -14 D 20 6 10 0 -6 E 4 2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6161: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) E D B A C (7) E B D A C (6) B A C E D (6) A C B D E (6) D E C A B (5) C D A E B (5) C A B D E (5) C B A D E (4) C A D E B (4) C A D B E (4) B E D A C (4) B C A D E (4) D C E A B (3) A C E D B (3) E D B C A (2) E B D C A (2) D E B C A (2) D C A E B (2) B E D C A (2) B A C D E (2) A C B E D (2) A B C E D (2) E D C A B (1) E A D C B (1) D E C B A (1) D E A C B (1) D C B E A (1) C D A B E (1) B E A D C (1) B D E C A (1) B C A E D (1) A E C D B (1) Total count = 100 A B C D E A 0 8 0 -8 4 B -8 0 -16 -4 -8 C 0 16 0 0 10 D 8 4 0 0 2 E -4 8 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.462864 D: 0.537136 E: 0.000000 Sum of squares = 0.502758112413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.462864 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -8 4 B -8 0 -16 -4 -8 C 0 16 0 0 10 D 8 4 0 0 2 E -4 8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=23 B=21 D=15 A=14 so A is eliminated. Round 2 votes counts: C=34 E=28 B=23 D=15 so D is eliminated. Round 3 votes counts: C=40 E=37 B=23 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:207 A:202 E:196 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 8 0 -8 4 B -8 0 -16 -4 -8 C 0 16 0 0 10 D 8 4 0 0 2 E -4 8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -8 4 B -8 0 -16 -4 -8 C 0 16 0 0 10 D 8 4 0 0 2 E -4 8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -8 4 B -8 0 -16 -4 -8 C 0 16 0 0 10 D 8 4 0 0 2 E -4 8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6162: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) E B D C A (8) A C D B E (8) A C D E B (7) C A D B E (6) A E C B D (6) D C B A E (5) E A B C D (4) D B E C A (4) B E D C A (4) B D C E A (4) A C E D B (4) D C A B E (3) C D A B E (3) A D C B E (3) E B C D A (2) E B A D C (2) C D B A E (2) B D E C A (2) E D B A C (1) E B D A C (1) E B C A D (1) E B A C D (1) D E B A C (1) D A C B E (1) C E B A D (1) C E A B D (1) C A E B D (1) B E C D A (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -14 -2 6 B -2 0 -10 -20 10 C 14 10 0 -2 20 D 2 20 2 0 20 E -6 -10 -20 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 -2 6 B -2 0 -10 -20 10 C 14 10 0 -2 20 D 2 20 2 0 20 E -6 -10 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 E=20 C=14 B=11 so B is eliminated. Round 2 votes counts: A=32 D=29 E=25 C=14 so C is eliminated. Round 3 votes counts: A=39 D=34 E=27 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:221 A:196 B:189 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -14 -2 6 B -2 0 -10 -20 10 C 14 10 0 -2 20 D 2 20 2 0 20 E -6 -10 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -2 6 B -2 0 -10 -20 10 C 14 10 0 -2 20 D 2 20 2 0 20 E -6 -10 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -2 6 B -2 0 -10 -20 10 C 14 10 0 -2 20 D 2 20 2 0 20 E -6 -10 -20 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6163: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) E D C A B (6) E C D A B (6) E C B D A (6) C E D A B (6) B A D E C (5) C E D B A (4) C D E A B (4) C D A B E (4) B A E D C (4) E C D B A (3) E B C A D (3) E B A D C (3) C E B D A (3) B E A C D (3) B A D C E (3) A B D C E (3) E D A B C (2) D E C A B (2) D A C B E (2) C B A D E (2) B A C E D (2) A D B C E (2) A B D E C (2) E D A C B (1) E B C D A (1) D C E A B (1) D A E B C (1) D A C E B (1) C D B A E (1) C D A E B (1) C B A E D (1) B E C A D (1) B E A D C (1) B C A D E (1) A E B D C (1) Total count = 100 A B C D E A 0 -10 -12 -10 -12 B 10 0 -8 6 -10 C 12 8 0 20 -2 D 10 -6 -20 0 -14 E 12 10 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 -10 -12 B 10 0 -8 6 -10 C 12 8 0 20 -2 D 10 -6 -20 0 -14 E 12 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=28 C=26 A=8 D=7 so D is eliminated. Round 2 votes counts: E=33 B=28 C=27 A=12 so A is eliminated. Round 3 votes counts: E=35 B=35 C=30 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:219 B:199 D:185 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 -10 -12 B 10 0 -8 6 -10 C 12 8 0 20 -2 D 10 -6 -20 0 -14 E 12 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -10 -12 B 10 0 -8 6 -10 C 12 8 0 20 -2 D 10 -6 -20 0 -14 E 12 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -10 -12 B 10 0 -8 6 -10 C 12 8 0 20 -2 D 10 -6 -20 0 -14 E 12 10 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6164: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (14) E C A B D (9) C E D B A (6) C E D A B (6) A E B C D (6) E C B A D (4) B D A E C (4) A B D E C (4) D C E B A (3) D B C E A (3) C E A D B (3) B A D E C (3) A E C B D (3) A D B C E (3) A B E D C (3) E A C B D (2) D C B A E (2) D C A B E (2) D B C A E (2) C D E A B (2) B A E D C (2) A C D E B (2) A B E C D (2) E C D B A (1) E C A D B (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A E B (1) C D E B A (1) C D A E B (1) B D E C A (1) B D E A C (1) Total count = 100 A B C D E A 0 2 2 -2 8 B -2 0 0 -8 -6 C -2 0 0 2 4 D 2 8 -2 0 0 E -8 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -2 8 B -2 0 0 -8 -6 C -2 0 0 2 4 D 2 8 -2 0 0 E -8 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=23 E=19 C=19 B=11 so B is eliminated. Round 2 votes counts: D=34 A=28 E=19 C=19 so E is eliminated. Round 3 votes counts: D=34 C=34 A=32 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:205 D:204 C:202 E:197 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 -2 8 B -2 0 0 -8 -6 C -2 0 0 2 4 D 2 8 -2 0 0 E -8 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -2 8 B -2 0 0 -8 -6 C -2 0 0 2 4 D 2 8 -2 0 0 E -8 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -2 8 B -2 0 0 -8 -6 C -2 0 0 2 4 D 2 8 -2 0 0 E -8 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6165: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) B C E A D (6) B E A C D (5) B C E D A (5) D C E A B (4) D C A E B (4) C D B E A (4) B A C E D (4) A B D E C (4) E D C A B (3) D A E C B (3) D A C E B (3) C D E B A (3) B A E C D (3) A D E B C (3) A D C B E (3) E D A B C (2) E C D B A (2) E C B D A (2) E B D A C (2) E B C D A (2) E B A D C (2) C E D B A (2) C A B D E (2) B E C A D (2) B A E D C (2) B A C D E (2) A B E D C (2) A B D C E (2) E D C B A (1) E D B A C (1) E A D B C (1) D E C A B (1) C D E A B (1) C D A B E (1) C B A D E (1) B C A D E (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -6 -4 -14 B 16 0 6 10 16 C 6 -6 0 10 12 D 4 -10 -10 0 -10 E 14 -16 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -4 -14 B 16 0 6 10 16 C 6 -6 0 10 12 D 4 -10 -10 0 -10 E 14 -16 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=20 E=18 A=17 D=15 so D is eliminated. Round 2 votes counts: B=30 C=28 A=23 E=19 so E is eliminated. Round 3 votes counts: C=37 B=37 A=26 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:224 C:211 E:198 D:187 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -4 -14 B 16 0 6 10 16 C 6 -6 0 10 12 D 4 -10 -10 0 -10 E 14 -16 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -4 -14 B 16 0 6 10 16 C 6 -6 0 10 12 D 4 -10 -10 0 -10 E 14 -16 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -4 -14 B 16 0 6 10 16 C 6 -6 0 10 12 D 4 -10 -10 0 -10 E 14 -16 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6166: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) D B E C A (11) D E B C A (10) B D C A E (8) A C B E D (8) A E C B D (7) A C E B D (5) D B C E A (4) B C A D E (4) A C B D E (4) E D C B A (2) E D A C B (2) E D A B C (2) E A D C B (2) D E B A C (2) D B C A E (2) C A B E D (2) B D C E A (2) B C D A E (2) E D B A C (1) E A C D B (1) C E A B D (1) C A B D E (1) B C D E A (1) B A C D E (1) A E C D B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -24 -22 -6 B 24 0 28 -6 4 C 24 -28 0 -22 -6 D 22 6 22 0 8 E 6 -4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -24 -22 -6 B 24 0 28 -6 4 C 24 -28 0 -22 -6 D 22 6 22 0 8 E 6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 E=22 B=18 C=4 so C is eliminated. Round 2 votes counts: A=30 D=29 E=23 B=18 so B is eliminated. Round 3 votes counts: D=42 A=35 E=23 so E is eliminated. Round 4 votes counts: D=61 A=39 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 B:225 E:200 C:184 A:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 -24 -22 -6 B 24 0 28 -6 4 C 24 -28 0 -22 -6 D 22 6 22 0 8 E 6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -24 -22 -6 B 24 0 28 -6 4 C 24 -28 0 -22 -6 D 22 6 22 0 8 E 6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -24 -22 -6 B 24 0 28 -6 4 C 24 -28 0 -22 -6 D 22 6 22 0 8 E 6 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995387 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6167: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (13) B E A D C (13) E B A D C (7) C D A B E (6) B E C A D (6) D C A B E (5) D A C E B (4) B E A C D (4) E A C D B (3) D C A E B (3) C E A D B (3) B E D A C (3) E A D C B (2) D A B E C (2) C A E D B (2) B E C D A (2) B C E D A (2) B C D E A (2) B C D A E (2) A E D C B (2) A D E C B (2) A D C E B (2) E C A D B (1) E B C A D (1) E B A C D (1) E A D B C (1) C D B A E (1) C B D A E (1) B E D C A (1) B D A C E (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -2 4 -4 B -6 0 -4 -8 2 C 2 4 0 4 -4 D -4 8 -4 0 -10 E 4 -2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.38888888868 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 6 -2 4 -4 B -6 0 -4 -8 2 C 2 4 0 4 -4 D -4 8 -4 0 -10 E 4 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888855 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=26 E=16 D=14 A=8 so A is eliminated. Round 2 votes counts: B=36 C=27 E=19 D=18 so D is eliminated. Round 3 votes counts: C=41 B=38 E=21 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:208 C:203 A:202 D:195 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 4 -4 B -6 0 -4 -8 2 C 2 4 0 4 -4 D -4 8 -4 0 -10 E 4 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888855 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 4 -4 B -6 0 -4 -8 2 C 2 4 0 4 -4 D -4 8 -4 0 -10 E 4 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888855 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 4 -4 B -6 0 -4 -8 2 C 2 4 0 4 -4 D -4 8 -4 0 -10 E 4 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888855 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6168: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) B E C A D (7) D C A E B (6) A D B E C (6) E B C A D (5) D A B E C (5) D A C E B (4) C E B A D (4) B A D E C (4) D A C B E (3) C E D B A (3) C D E B A (3) B A E D C (3) B A E C D (3) E B A C D (2) D C B A E (2) D B A C E (2) D A B C E (2) C B E D A (2) B E C D A (2) B C E D A (2) A E B C D (2) A D E C B (2) A B E D C (2) E C A D B (1) E A C B D (1) D C E B A (1) D C A B E (1) D B C E A (1) D B A E C (1) D A E B C (1) C E D A B (1) C D E A B (1) B D A E C (1) A E D C B (1) A E C D B (1) A D E B C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 16 12 10 B 14 0 26 2 20 C -16 -26 0 0 -24 D -12 -2 0 0 -4 E -10 -20 24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 16 12 10 B 14 0 26 2 20 C -16 -26 0 0 -24 D -12 -2 0 0 -4 E -10 -20 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 A=17 C=14 E=9 so E is eliminated. Round 2 votes counts: B=38 D=29 A=18 C=15 so C is eliminated. Round 3 votes counts: B=44 D=37 A=19 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:212 E:199 D:191 C:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 16 12 10 B 14 0 26 2 20 C -16 -26 0 0 -24 D -12 -2 0 0 -4 E -10 -20 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 16 12 10 B 14 0 26 2 20 C -16 -26 0 0 -24 D -12 -2 0 0 -4 E -10 -20 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 16 12 10 B 14 0 26 2 20 C -16 -26 0 0 -24 D -12 -2 0 0 -4 E -10 -20 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999849 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6169: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) A C D B E (7) E B D C A (6) D B A C E (5) C A B D E (5) A C D E B (5) E C A B D (4) E B C A D (4) D A C B E (4) C A B E D (4) E D A C B (3) E B C D A (3) D E B A C (3) D A C E B (3) C A E D B (3) B E D C A (3) E D B C A (2) E B D A C (2) D B E A C (2) B E D A C (2) B E C D A (2) B E C A D (2) B D E C A (2) B C A E D (2) A D C B E (2) E D B A C (1) E C D A B (1) E C B A D (1) E C A D B (1) D A E C B (1) D A E B C (1) B D E A C (1) B D A C E (1) B C E A D (1) A D C E B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -8 4 8 B -10 0 -10 8 -8 C 8 10 0 10 6 D -4 -8 -10 0 -12 E -8 8 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 4 8 B -10 0 -10 8 -8 C 8 10 0 10 6 D -4 -8 -10 0 -12 E -8 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=20 D=19 A=17 B=16 so B is eliminated. Round 2 votes counts: E=37 D=23 C=23 A=17 so A is eliminated. Round 3 votes counts: E=37 C=37 D=26 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:207 E:203 B:190 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 4 8 B -10 0 -10 8 -8 C 8 10 0 10 6 D -4 -8 -10 0 -12 E -8 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 4 8 B -10 0 -10 8 -8 C 8 10 0 10 6 D -4 -8 -10 0 -12 E -8 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 4 8 B -10 0 -10 8 -8 C 8 10 0 10 6 D -4 -8 -10 0 -12 E -8 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6170: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) A C D E B (7) E B D A C (6) B E D C A (5) D C A E B (4) B E A D C (4) B E A C D (4) E D B A C (3) D C E B A (3) C D A E B (3) B C A D E (3) B A E C D (3) A C B D E (3) E D A C B (2) E B D C A (2) D C E A B (2) C D B A E (2) C D A B E (2) C B D A E (2) C A D B E (2) C A B D E (2) B C A E D (2) A C D B E (2) E D B C A (1) E D A B C (1) E A D C B (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D E A C B (1) D C B E A (1) D C B A E (1) D A E C B (1) D A C E B (1) C A D E B (1) B E D A C (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C E D (1) A E D C B (1) A E C B D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 8 0 -6 B 16 0 2 10 -6 C -8 -2 0 -6 -2 D 0 -10 6 0 -4 E 6 6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 8 0 -6 B 16 0 2 10 -6 C -8 -2 0 -6 -2 D 0 -10 6 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=25 D=16 A=16 C=14 so C is eliminated. Round 2 votes counts: B=31 E=25 D=23 A=21 so A is eliminated. Round 3 votes counts: B=37 D=35 E=28 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:209 D:196 A:193 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 8 0 -6 B 16 0 2 10 -6 C -8 -2 0 -6 -2 D 0 -10 6 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 8 0 -6 B 16 0 2 10 -6 C -8 -2 0 -6 -2 D 0 -10 6 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 8 0 -6 B 16 0 2 10 -6 C -8 -2 0 -6 -2 D 0 -10 6 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6171: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B A E D C (8) C D E A B (7) C D B A E (6) C D A E B (6) E A B D C (5) D C E B A (5) B A E C D (5) A B E C D (5) A E B C D (4) E D B C A (3) E B A D C (3) D E C A B (3) D C B E A (3) B C D A E (3) B A C D E (3) E A D C B (2) C D B E A (2) C D A B E (2) C A D B E (2) B E A D C (2) B D C A E (2) E A C B D (1) D B E C A (1) C B D A E (1) C A D E B (1) B D E C A (1) A E C D B (1) A E C B D (1) A C E B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -14 -8 6 B -4 0 -6 -6 -4 C 14 6 0 6 10 D 8 6 -6 0 16 E -6 4 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 -8 6 B -4 0 -6 -6 -4 C 14 6 0 6 10 D 8 6 -6 0 16 E -6 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 D=21 E=14 A=14 so E is eliminated. Round 2 votes counts: C=27 B=27 D=24 A=22 so A is eliminated. Round 3 votes counts: B=43 C=31 D=26 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:212 A:194 B:190 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 -8 6 B -4 0 -6 -6 -4 C 14 6 0 6 10 D 8 6 -6 0 16 E -6 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 -8 6 B -4 0 -6 -6 -4 C 14 6 0 6 10 D 8 6 -6 0 16 E -6 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 -8 6 B -4 0 -6 -6 -4 C 14 6 0 6 10 D 8 6 -6 0 16 E -6 4 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6172: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (17) C B A E D (13) E D A B C (7) D C B A E (5) D A E B C (4) C B E A D (4) B C A E D (4) A B C E D (4) E A D B C (3) D E C B A (3) D E A C B (3) D A C B E (3) E D B C A (2) E B C A D (2) E B A C D (2) E A B D C (2) D C B E A (2) D C A B E (2) D A C E B (2) C D B A E (2) C B D A E (2) A D E B C (2) E B C D A (1) D E C A B (1) D E B C A (1) D A E C B (1) C B A D E (1) B A C E D (1) A E D B C (1) A E B D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 -16 0 B -10 0 12 -24 -10 C -10 -12 0 -26 -8 D 16 24 26 0 4 E 0 10 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -16 0 B -10 0 12 -24 -10 C -10 -12 0 -26 -8 D 16 24 26 0 4 E 0 10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 C=22 E=19 A=10 B=5 so B is eliminated. Round 2 votes counts: D=44 C=26 E=19 A=11 so A is eliminated. Round 3 votes counts: D=47 C=31 E=22 so E is eliminated. Round 4 votes counts: D=63 C=37 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:235 E:207 A:202 B:184 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -16 0 B -10 0 12 -24 -10 C -10 -12 0 -26 -8 D 16 24 26 0 4 E 0 10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -16 0 B -10 0 12 -24 -10 C -10 -12 0 -26 -8 D 16 24 26 0 4 E 0 10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -16 0 B -10 0 12 -24 -10 C -10 -12 0 -26 -8 D 16 24 26 0 4 E 0 10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6173: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) E C A B D (9) E A C B D (8) D E C B A (7) D B A C E (4) B A C D E (4) A E B C D (4) E C D B A (3) E A D C B (3) E A D B C (3) D C B A E (3) C B A E D (3) B D A C E (3) B A D C E (3) E A C D B (2) C E B A D (2) C A B E D (2) A B C E D (2) E D A C B (1) E D A B C (1) E C D A B (1) E C B A D (1) E A B D C (1) D E B C A (1) D E A B C (1) D C E B A (1) D C B E A (1) D B E A C (1) D B C E A (1) D B A E C (1) D A B E C (1) C E B D A (1) C E A B D (1) C D B A E (1) C B E A D (1) C B D A E (1) B C D A E (1) B C A D E (1) A E D B C (1) A E B D C (1) A D B E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -4 10 -2 B 10 0 -6 2 -8 C 4 6 0 0 -6 D -10 -2 0 0 -6 E 2 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 10 -2 B 10 0 -6 2 -8 C 4 6 0 0 -6 D -10 -2 0 0 -6 E 2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=32 C=12 B=12 A=11 so A is eliminated. Round 2 votes counts: E=39 D=33 B=15 C=13 so C is eliminated. Round 3 votes counts: E=44 D=34 B=22 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:202 B:199 A:197 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 10 -2 B 10 0 -6 2 -8 C 4 6 0 0 -6 D -10 -2 0 0 -6 E 2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 10 -2 B 10 0 -6 2 -8 C 4 6 0 0 -6 D -10 -2 0 0 -6 E 2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 10 -2 B 10 0 -6 2 -8 C 4 6 0 0 -6 D -10 -2 0 0 -6 E 2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6174: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) B A C D E (12) D E B A C (9) E D C A B (8) D B E A C (7) C A B E D (7) E C A D B (5) D E C B A (4) D E C A B (4) C E A B D (4) C A E B D (4) B D A C E (4) D E B C A (3) D E A C B (3) D B A E C (3) B A D C E (3) E A C D B (1) D E A B C (1) C E A D B (1) C B E A D (1) B D A E C (1) B C A E D (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -20 16 6 -2 B 20 0 12 2 4 C -16 -12 0 0 0 D -6 -2 0 0 8 E 2 -4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 16 6 -2 B 20 0 12 2 4 C -16 -12 0 0 0 D -6 -2 0 0 8 E 2 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=33 C=17 E=14 A=2 so A is eliminated. Round 2 votes counts: D=34 B=33 C=18 E=15 so E is eliminated. Round 3 votes counts: D=42 B=33 C=25 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:200 D:200 E:195 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 16 6 -2 B 20 0 12 2 4 C -16 -12 0 0 0 D -6 -2 0 0 8 E 2 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 16 6 -2 B 20 0 12 2 4 C -16 -12 0 0 0 D -6 -2 0 0 8 E 2 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 16 6 -2 B 20 0 12 2 4 C -16 -12 0 0 0 D -6 -2 0 0 8 E 2 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6175: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) E D B C A (11) A C B D E (11) A D C B E (10) E B C D A (8) A C B E D (6) A D E C B (5) D E B C A (4) D E A B C (3) C B A D E (3) B C D E A (3) E C B A D (2) E B C A D (2) E A D B C (2) B C E D A (2) A E D C B (2) A D C E B (2) A C E B D (2) E B D C A (1) E A C B D (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C A E (1) D A E B C (1) C B D A E (1) C B A E D (1) C A B E D (1) Total count = 100 A B C D E A 0 18 18 2 -6 B -18 0 4 -10 -20 C -18 -4 0 -12 -12 D -2 10 12 0 -8 E 6 20 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 18 2 -6 B -18 0 4 -10 -20 C -18 -4 0 -12 -12 D -2 10 12 0 -8 E 6 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=38 D=11 C=6 B=5 so B is eliminated. Round 2 votes counts: E=40 A=38 D=11 C=11 so D is eliminated. Round 3 votes counts: E=48 A=39 C=13 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:216 D:206 B:178 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 18 2 -6 B -18 0 4 -10 -20 C -18 -4 0 -12 -12 D -2 10 12 0 -8 E 6 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 2 -6 B -18 0 4 -10 -20 C -18 -4 0 -12 -12 D -2 10 12 0 -8 E 6 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 2 -6 B -18 0 4 -10 -20 C -18 -4 0 -12 -12 D -2 10 12 0 -8 E 6 20 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6176: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) E D C B A (6) E D C A B (5) E B C D A (5) D A C E B (5) C B E A D (5) B C A E D (5) B A C E D (4) A C B D E (4) E D B C A (3) E C B D A (3) D E C A B (3) D E B A C (3) D E A C B (3) E C D B A (2) E B D C A (2) E B D A C (2) D A B E C (2) C A B E D (2) B E C D A (2) B E C A D (2) B A D C E (2) A D C B E (2) A D B C E (2) A C D B E (2) E D B A C (1) D E A B C (1) D C A E B (1) C E D B A (1) C E D A B (1) C B A E D (1) C A D E B (1) C A B D E (1) B C E A D (1) B A E C D (1) B A D E C (1) B A C D E (1) A D C E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -4 -2 -2 B 6 0 2 10 2 C 4 -2 0 6 4 D 2 -10 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -2 -2 B 6 0 2 10 2 C 4 -2 0 6 4 D 2 -10 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 B=19 D=18 C=12 so C is eliminated. Round 2 votes counts: E=31 A=26 B=25 D=18 so D is eliminated. Round 3 votes counts: E=41 A=34 B=25 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:210 C:206 E:202 A:193 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 -2 B 6 0 2 10 2 C 4 -2 0 6 4 D 2 -10 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 -2 B 6 0 2 10 2 C 4 -2 0 6 4 D 2 -10 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 -2 B 6 0 2 10 2 C 4 -2 0 6 4 D 2 -10 -6 0 -8 E 2 -2 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6177: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (14) C D B A E (12) A E C D B (9) B D C E A (8) B D C A E (7) E B D C A (6) C D B E A (5) B D E C A (5) A E B D C (5) A C E D B (5) A E C B D (4) C A D B E (3) E A C D B (2) E A C B D (2) D B C A E (2) A B D C E (2) E C D B A (1) E C A D B (1) E B D A C (1) E B A D C (1) D B C E A (1) C D E B A (1) C A E D B (1) B D E A C (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -6 0 2 B 2 0 6 12 -6 C 6 -6 0 -6 -4 D 0 -12 6 0 -4 E -2 6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888777 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -2 -6 0 2 B 2 0 6 12 -6 C 6 -6 0 -6 -4 D 0 -12 6 0 -4 E -2 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 C=22 B=21 D=3 so D is eliminated. Round 2 votes counts: E=28 A=26 B=24 C=22 so C is eliminated. Round 3 votes counts: B=41 A=30 E=29 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:207 E:206 A:197 C:195 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 0 2 B 2 0 6 12 -6 C 6 -6 0 -6 -4 D 0 -12 6 0 -4 E -2 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 0 2 B 2 0 6 12 -6 C 6 -6 0 -6 -4 D 0 -12 6 0 -4 E -2 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 0 2 B 2 0 6 12 -6 C 6 -6 0 -6 -4 D 0 -12 6 0 -4 E -2 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6178: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (9) E A C D B (8) C A E B D (8) A E C B D (7) E A D B C (6) B D C A E (6) C B D E A (5) E A D C B (4) D B E A C (4) D B A E C (3) C E A B D (3) C B D A E (3) C B A E D (3) C B A D E (3) A E D B C (3) E D A C B (2) D E B C A (2) D E B A C (2) D C E B A (2) D B E C A (2) C E A D B (2) C D B E A (2) B D A C E (2) E D A B C (1) C E D A B (1) C D E B A (1) C B E A D (1) B D A E C (1) B A D E C (1) A E B D C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -6 4 4 B 4 0 -12 10 -10 C 6 12 0 16 4 D -4 -10 -16 0 -4 E -4 10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 4 4 B 4 0 -12 10 -10 C 6 12 0 16 4 D -4 -10 -16 0 -4 E -4 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=21 B=19 D=15 A=13 so A is eliminated. Round 2 votes counts: E=33 C=33 B=19 D=15 so D is eliminated. Round 3 votes counts: E=37 C=35 B=28 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:203 A:199 B:196 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 4 4 B 4 0 -12 10 -10 C 6 12 0 16 4 D -4 -10 -16 0 -4 E -4 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 4 B 4 0 -12 10 -10 C 6 12 0 16 4 D -4 -10 -16 0 -4 E -4 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 4 B 4 0 -12 10 -10 C 6 12 0 16 4 D -4 -10 -16 0 -4 E -4 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6179: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) E C A D B (9) A C E B D (8) D B E A C (6) B D A C E (6) C E A D B (5) A C E D B (5) C A E B D (4) B D E C A (4) B A D C E (4) E A C D B (3) D B A E C (3) C E A B D (3) B D E A C (3) E D C A B (2) D E A C B (2) C B E D A (2) B D C E A (2) B D A E C (2) B C A E D (2) A C B E D (2) E D C B A (1) E C D B A (1) D E C B A (1) D E B C A (1) D E B A C (1) D A E B C (1) C B E A D (1) B E D C A (1) B C D E A (1) B C A D E (1) A E D C B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 0 -18 B 6 0 0 -4 4 C 2 0 0 -4 -4 D 0 4 4 0 0 E 18 -4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.705400 E: 0.294600 Sum of squares = 0.58437854125 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.705400 E: 1.000000 A B C D E A 0 -6 -2 0 -18 B 6 0 0 -4 4 C 2 0 0 -4 -4 D 0 4 4 0 0 E 18 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 0.499842 Sum of squares = 0.500000049635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 A=18 E=16 C=15 so C is eliminated. Round 2 votes counts: B=29 D=25 E=24 A=22 so A is eliminated. Round 3 votes counts: E=42 B=33 D=25 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:209 D:204 B:203 C:197 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 0 -18 B 6 0 0 -4 4 C 2 0 0 -4 -4 D 0 4 4 0 0 E 18 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 0.499842 Sum of squares = 0.500000049635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 0 -18 B 6 0 0 -4 4 C 2 0 0 -4 -4 D 0 4 4 0 0 E 18 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 0.499842 Sum of squares = 0.500000049635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 0 -18 B 6 0 0 -4 4 C 2 0 0 -4 -4 D 0 4 4 0 0 E 18 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 0.499842 Sum of squares = 0.500000049635 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500158 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6180: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (14) A C D B E (10) B D E A C (9) D B E A C (8) D B A C E (7) E C A B D (6) C A E D B (6) C A D B E (6) C E A B D (5) C A D E B (5) E C B A D (4) E B C A D (4) D B A E C (3) D A B C E (3) E B D A C (2) B E D A C (2) A D C B E (2) E B C D A (1) C A E B D (1) C A B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -6 2 -10 B 8 0 8 0 4 C 6 -8 0 -2 -6 D -2 0 2 0 8 E 10 -4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.531596 C: 0.000000 D: 0.468404 E: 0.000000 Sum of squares = 0.501996570794 Cumulative probabilities = A: 0.000000 B: 0.531596 C: 0.531596 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 2 -10 B 8 0 8 0 4 C 6 -8 0 -2 -6 D -2 0 2 0 8 E 10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=24 D=21 A=13 B=11 so B is eliminated. Round 2 votes counts: E=33 D=30 C=24 A=13 so A is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:204 E:202 C:195 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 2 -10 B 8 0 8 0 4 C 6 -8 0 -2 -6 D -2 0 2 0 8 E 10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 2 -10 B 8 0 8 0 4 C 6 -8 0 -2 -6 D -2 0 2 0 8 E 10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 2 -10 B 8 0 8 0 4 C 6 -8 0 -2 -6 D -2 0 2 0 8 E 10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6181: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C A B (8) C D E A B (6) B A E D C (6) A D E C B (5) A D C E B (5) E C D B A (4) B C E D A (4) B C D E A (4) A B D E C (4) E D A C B (3) C D A E B (3) B E C D A (3) E D C B A (2) E C D A B (2) C E D B A (2) B C D A E (2) A E D C B (2) A E D B C (2) A B D C E (2) E D B C A (1) E B D C A (1) E A D C B (1) D E C A B (1) D C E A B (1) C E D A B (1) C E B D A (1) C B E D A (1) C B D E A (1) C B D A E (1) C A D E B (1) C A B D E (1) B E A D C (1) B C E A D (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C E D (1) A C D E B (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 -4 2 B -4 0 -8 -4 -6 C 6 8 0 6 2 D 4 4 -6 0 2 E -2 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -4 2 B -4 0 -8 -4 -6 C 6 8 0 6 2 D 4 4 -6 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=24 E=22 C=18 D=2 so D is eliminated. Round 2 votes counts: B=34 A=24 E=23 C=19 so C is eliminated. Round 3 votes counts: B=37 E=34 A=29 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:211 D:202 E:200 A:198 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -4 2 B -4 0 -8 -4 -6 C 6 8 0 6 2 D 4 4 -6 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -4 2 B -4 0 -8 -4 -6 C 6 8 0 6 2 D 4 4 -6 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -4 2 B -4 0 -8 -4 -6 C 6 8 0 6 2 D 4 4 -6 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6182: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) E A B D C (10) C A B E D (8) E C A B D (7) C D B A E (7) E D B A C (6) E A B C D (5) D B A C E (5) C E A B D (5) E D A B C (4) D C B A E (4) C B A D E (4) B A D C E (4) C A B D E (3) A B C D E (3) D E B A C (2) A B E C D (2) A B D E C (2) E C D B A (1) E A C B D (1) C E D B A (1) C E A D B (1) C D E B A (1) C D B E A (1) C B D A E (1) B A D E C (1) Total count = 100 A B C D E A 0 2 12 12 10 B -2 0 10 12 12 C -12 -10 0 2 -4 D -12 -12 -2 0 -2 E -10 -12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 12 10 B -2 0 10 12 12 C -12 -10 0 2 -4 D -12 -12 -2 0 -2 E -10 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=32 D=22 A=7 B=5 so B is eliminated. Round 2 votes counts: E=34 C=32 D=22 A=12 so A is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:218 B:216 E:192 C:188 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 12 10 B -2 0 10 12 12 C -12 -10 0 2 -4 D -12 -12 -2 0 -2 E -10 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 12 10 B -2 0 10 12 12 C -12 -10 0 2 -4 D -12 -12 -2 0 -2 E -10 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 12 10 B -2 0 10 12 12 C -12 -10 0 2 -4 D -12 -12 -2 0 -2 E -10 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6183: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (6) D B C E A (5) D B C A E (5) B A D E C (5) E A C B D (4) E A B C D (4) D C B E A (4) C E D A B (4) B D A C E (4) E A C D B (3) C E A D B (3) C D B A E (3) C A E B D (3) B D C A E (3) B A E D C (3) B A C E D (3) A E B C D (3) A B E C D (3) A B C E D (3) E C D A B (2) E A D C B (2) D B E C A (2) C D E A B (2) B A D C E (2) A E C B D (2) E D C A B (1) E D B A C (1) E C A B D (1) E B D A C (1) E B A D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E A B (1) D B A E C (1) D B A C E (1) C B A D E (1) C A B D E (1) B D A E C (1) B C A D E (1) B A E C D (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -2 12 0 B 0 0 8 2 4 C 2 -8 0 8 0 D -12 -2 -8 0 -10 E 0 -4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.391985 B: 0.608015 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.523334386402 Cumulative probabilities = A: 0.391985 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 12 0 B 0 0 8 2 4 C 2 -8 0 8 0 D -12 -2 -8 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=23 D=22 C=17 A=12 so A is eliminated. Round 2 votes counts: E=31 B=29 D=22 C=18 so C is eliminated. Round 3 votes counts: E=42 B=31 D=27 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:207 A:205 E:203 C:201 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 12 0 B 0 0 8 2 4 C 2 -8 0 8 0 D -12 -2 -8 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 12 0 B 0 0 8 2 4 C 2 -8 0 8 0 D -12 -2 -8 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 12 0 B 0 0 8 2 4 C 2 -8 0 8 0 D -12 -2 -8 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6184: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) E B D A C (8) A C B D E (7) C A D B E (5) D B A E C (4) C E A D B (4) A B D C E (4) E C A B D (3) D E B C A (3) C E D A B (3) C D A B E (3) C A D E B (3) B A D E C (3) E D C B A (2) E C D B A (2) E C B D A (2) D B E C A (2) D B E A C (2) C A E D B (2) C A E B D (2) C A B D E (2) B D E A C (2) B D A E C (2) A E C B D (2) A B E D C (2) A B D E C (2) E C A D B (1) E B A C D (1) E A C B D (1) D E C B A (1) D C E B A (1) D C B E A (1) D C B A E (1) D B C A E (1) D B A C E (1) C D E B A (1) C D A E B (1) B E D A C (1) A C E B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -10 -6 0 B 0 0 -2 -6 -6 C 10 2 0 -4 -10 D 6 6 4 0 6 E 0 6 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -6 0 B 0 0 -2 -6 -6 C 10 2 0 -4 -10 D 6 6 4 0 6 E 0 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 A=20 D=17 B=8 so B is eliminated. Round 2 votes counts: E=30 C=26 A=23 D=21 so D is eliminated. Round 3 votes counts: E=40 C=30 A=30 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:211 E:205 C:199 B:193 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -6 0 B 0 0 -2 -6 -6 C 10 2 0 -4 -10 D 6 6 4 0 6 E 0 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -6 0 B 0 0 -2 -6 -6 C 10 2 0 -4 -10 D 6 6 4 0 6 E 0 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -6 0 B 0 0 -2 -6 -6 C 10 2 0 -4 -10 D 6 6 4 0 6 E 0 6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6185: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) A D B E C (8) D A B E C (7) E B C A D (6) E B A C D (6) A D E B C (5) E A B C D (4) D A C B E (4) C E B D A (4) A E C B D (4) D C A B E (3) C D A B E (3) B E C D A (3) A E B D C (3) A D C B E (3) C D E B A (2) A C D E B (2) A B D E C (2) E C B A D (1) E B C D A (1) E B A D C (1) D C B E A (1) D C B A E (1) D A B C E (1) C E D B A (1) C E B A D (1) C E A B D (1) C D A E B (1) C B E D A (1) C B D E A (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) A E B C D (1) A D E C B (1) A D C E B (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 16 12 8 8 B -16 0 4 -12 2 C -12 -4 0 10 -12 D -8 12 -10 0 16 E -8 -2 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 8 8 B -16 0 4 -12 2 C -12 -4 0 10 -12 D -8 12 -10 0 16 E -8 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=26 E=19 D=17 B=6 so B is eliminated. Round 2 votes counts: A=32 C=26 E=24 D=18 so D is eliminated. Round 3 votes counts: A=44 C=31 E=25 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:205 E:193 C:191 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 8 8 B -16 0 4 -12 2 C -12 -4 0 10 -12 D -8 12 -10 0 16 E -8 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 8 8 B -16 0 4 -12 2 C -12 -4 0 10 -12 D -8 12 -10 0 16 E -8 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 8 8 B -16 0 4 -12 2 C -12 -4 0 10 -12 D -8 12 -10 0 16 E -8 -2 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6186: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) E A B D C (9) B E A D C (8) C D B A E (7) D A C E B (5) B E C A D (5) B E A C D (5) E B A D C (4) C D A B E (4) E A D B C (3) D C A E B (3) B C E D A (3) B C D E A (3) A D E C B (3) D A E C B (2) C B D E A (2) C A E D B (2) C A D E B (2) B E D A C (2) B D A E C (2) B C E A D (2) E B A C D (1) E A C D B (1) D A E B C (1) C E A B D (1) C D E A B (1) C B E D A (1) B E D C A (1) B D C A E (1) B A E D C (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 4 4 4 -4 B -4 0 6 2 -6 C -4 -6 0 2 -4 D -4 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 4 -4 B -4 0 6 2 -6 C -4 -6 0 2 -4 D -4 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=30 E=18 D=11 A=8 so A is eliminated. Round 2 votes counts: B=33 C=30 E=21 D=16 so D is eliminated. Round 3 votes counts: C=39 B=33 E=28 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:209 A:204 B:199 C:194 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 4 -4 B -4 0 6 2 -6 C -4 -6 0 2 -4 D -4 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 -4 B -4 0 6 2 -6 C -4 -6 0 2 -4 D -4 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 -4 B -4 0 6 2 -6 C -4 -6 0 2 -4 D -4 -2 -2 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6187: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) B D C A E (11) A C B D E (10) D B C A E (7) D B E C A (6) A E C B D (6) E C A D B (5) C D B A E (5) D B C E A (4) E A C B D (3) A C E D B (3) E D B A C (2) E A D C B (2) E A B D C (2) C A D B E (2) B D E C A (2) B D C E A (2) B A D C E (2) A B D C E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C D A B (1) E B D C A (1) E B D A C (1) E A D B C (1) D C B E A (1) C A B D E (1) B D A C E (1) A E C D B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -2 6 6 B -6 0 -10 -8 12 C 2 10 0 2 4 D -6 8 -2 0 12 E -6 -12 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 6 6 B -6 0 -10 -8 12 C 2 10 0 2 4 D -6 8 -2 0 12 E -6 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=24 D=18 B=18 C=8 so C is eliminated. Round 2 votes counts: E=32 A=27 D=23 B=18 so B is eliminated. Round 3 votes counts: D=39 E=32 A=29 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:209 A:208 D:206 B:194 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 6 6 B -6 0 -10 -8 12 C 2 10 0 2 4 D -6 8 -2 0 12 E -6 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 6 6 B -6 0 -10 -8 12 C 2 10 0 2 4 D -6 8 -2 0 12 E -6 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 6 6 B -6 0 -10 -8 12 C 2 10 0 2 4 D -6 8 -2 0 12 E -6 -12 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6188: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) C B D E A (7) A B E D C (6) C D E B A (5) C D B E A (5) B E A C D (4) A E D B C (4) A D E C B (4) A D E B C (4) A D C E B (4) D C E B A (3) D C A E B (3) D A E C B (3) B E C D A (3) B C E D A (3) A E B D C (3) E B D C A (2) D C E A B (2) C D A E B (2) C D A B E (2) C A D B E (2) B E C A D (2) B E A D C (2) B C A E D (2) A C B D E (2) E D C B A (1) E D B A C (1) E B D A C (1) D E C B A (1) D E A C B (1) D E A B C (1) C B A D E (1) C A B D E (1) B E D A C (1) B C E A D (1) B A E D C (1) B A E C D (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -10 -8 -6 B 10 0 -14 2 10 C 10 14 0 4 8 D 8 -2 -4 0 8 E 6 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -8 -6 B 10 0 -14 2 10 C 10 14 0 4 8 D 8 -2 -4 0 8 E 6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=28 B=21 D=14 E=5 so E is eliminated. Round 2 votes counts: C=32 A=28 B=24 D=16 so D is eliminated. Round 3 votes counts: C=42 A=33 B=25 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:205 B:204 E:190 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 -8 -6 B 10 0 -14 2 10 C 10 14 0 4 8 D 8 -2 -4 0 8 E 6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -8 -6 B 10 0 -14 2 10 C 10 14 0 4 8 D 8 -2 -4 0 8 E 6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -8 -6 B 10 0 -14 2 10 C 10 14 0 4 8 D 8 -2 -4 0 8 E 6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6189: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (9) B A E C D (6) D C E A B (5) C E D A B (5) C E A D B (5) B A D E C (5) A D C E B (5) E C D A B (4) E C B D A (4) B D E C A (4) A D B C E (4) D E C B A (3) B D A E C (3) B A D C E (3) D B E A C (2) C A E D B (2) B E D C A (2) B E C D A (2) B A C E D (2) A C D E B (2) E B C D A (1) D E B C A (1) D B E C A (1) D A E C B (1) D A C E B (1) C E B A D (1) C E A B D (1) B A C D E (1) A D C B E (1) A C E D B (1) A C D B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -2 2 -2 B 2 0 -4 -8 -4 C 2 4 0 0 2 D -2 8 0 0 6 E 2 4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.707660 D: 0.292340 E: 0.000000 Sum of squares = 0.586245194495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.707660 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 2 -2 B 2 0 -4 -8 -4 C 2 4 0 0 2 D -2 8 0 0 6 E 2 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500200 D: 0.499800 E: 0.000000 Sum of squares = 0.500000079718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500200 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=25 E=19 D=14 C=14 so D is eliminated. Round 2 votes counts: B=31 A=27 E=23 C=19 so C is eliminated. Round 3 votes counts: E=40 B=31 A=29 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:206 C:204 E:199 A:198 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 2 -2 B 2 0 -4 -8 -4 C 2 4 0 0 2 D -2 8 0 0 6 E 2 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500200 D: 0.499800 E: 0.000000 Sum of squares = 0.500000079718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500200 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 -2 B 2 0 -4 -8 -4 C 2 4 0 0 2 D -2 8 0 0 6 E 2 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500200 D: 0.499800 E: 0.000000 Sum of squares = 0.500000079718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500200 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 -2 B 2 0 -4 -8 -4 C 2 4 0 0 2 D -2 8 0 0 6 E 2 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500200 D: 0.499800 E: 0.000000 Sum of squares = 0.500000079718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500200 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6190: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) E B D A C (11) C A D B E (7) C A B D E (6) D E A C B (4) C D A E B (4) B A C D E (4) D E C A B (3) D E B A C (3) C B A E D (3) B E D A C (3) B E A C D (3) B A C E D (3) E C D B A (2) E B C A D (2) D E A B C (2) D A E C B (2) C E B D A (2) C B E A D (2) B E C A D (2) B C E A D (2) B A E C D (2) B A D E C (2) E D C B A (1) E D C A B (1) E D B C A (1) E B D C A (1) D C E A B (1) D A C B E (1) D A B E C (1) D A B C E (1) C D E A B (1) C B A D E (1) C A D E B (1) C A B E D (1) B C A E D (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -24 10 -12 -16 B 24 0 12 4 -6 C -10 -12 0 0 -16 D 12 -4 0 0 -8 E 16 6 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 10 -12 -16 B 24 0 12 4 -6 C -10 -12 0 0 -16 D 12 -4 0 0 -8 E 16 6 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=28 B=22 D=18 A=2 so A is eliminated. Round 2 votes counts: E=30 C=29 B=22 D=19 so D is eliminated. Round 3 votes counts: E=44 C=31 B=25 so B is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:217 D:200 C:181 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 10 -12 -16 B 24 0 12 4 -6 C -10 -12 0 0 -16 D 12 -4 0 0 -8 E 16 6 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 10 -12 -16 B 24 0 12 4 -6 C -10 -12 0 0 -16 D 12 -4 0 0 -8 E 16 6 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 10 -12 -16 B 24 0 12 4 -6 C -10 -12 0 0 -16 D 12 -4 0 0 -8 E 16 6 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998736 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6191: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) A B D C E (9) D E C A B (6) C E D B A (6) D E C B A (5) A D B E C (5) E D C B A (4) A D B C E (4) E C D B A (3) D A E C B (3) B C E A D (3) B C A E D (3) B A E C D (3) A B D E C (3) A B C E D (3) A B C D E (3) E C B D A (2) D C E A B (2) D A E B C (2) B E C A D (2) A D C B E (2) A C B D E (2) E C B A D (1) E B D C A (1) E B C D A (1) D E B C A (1) D E B A C (1) D E A B C (1) D A C E B (1) D A B E C (1) C E B D A (1) C E B A D (1) C D E B A (1) C B E A D (1) B E A C D (1) B A E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 12 16 12 B 4 0 18 4 14 C -12 -18 0 -4 4 D -16 -4 4 0 4 E -12 -14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 16 12 B 4 0 18 4 14 C -12 -18 0 -4 4 D -16 -4 4 0 4 E -12 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996501 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=23 B=23 E=12 C=10 so C is eliminated. Round 2 votes counts: A=32 D=24 B=24 E=20 so E is eliminated. Round 3 votes counts: D=37 A=32 B=31 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:218 D:194 C:185 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 16 12 B 4 0 18 4 14 C -12 -18 0 -4 4 D -16 -4 4 0 4 E -12 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996501 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 16 12 B 4 0 18 4 14 C -12 -18 0 -4 4 D -16 -4 4 0 4 E -12 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996501 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 16 12 B 4 0 18 4 14 C -12 -18 0 -4 4 D -16 -4 4 0 4 E -12 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996501 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6192: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) A D B C E (7) C E B D A (6) C B E A D (6) E C B D A (5) D A E C B (5) D A E B C (5) B C E A D (5) A D B E C (5) E D A C B (4) C B E D A (4) D A C E B (3) D A C B E (3) B C A E D (3) B C A D E (3) A B D C E (3) E D C A B (2) C B A E D (2) B E C A D (2) B E A C D (2) B A C D E (2) A D E B C (2) E D B A C (1) E C D B A (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A C B (1) D E A B C (1) C D E A B (1) C D B A E (1) C B D A E (1) C A D B E (1) B A C E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 14 -2 -2 B -8 0 8 -2 6 C -14 -8 0 -6 6 D 2 2 6 0 -8 E 2 -6 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 8 14 -2 -2 B -8 0 8 -2 6 C -14 -8 0 -6 6 D 2 2 6 0 -8 E 2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 C=22 A=19 D=18 B=18 so D is eliminated. Round 2 votes counts: A=35 E=25 C=22 B=18 so B is eliminated. Round 3 votes counts: A=38 C=33 E=29 so E is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:202 D:201 E:199 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 -2 -2 B -8 0 8 -2 6 C -14 -8 0 -6 6 D 2 2 6 0 -8 E 2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 -2 -2 B -8 0 8 -2 6 C -14 -8 0 -6 6 D 2 2 6 0 -8 E 2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 -2 -2 B -8 0 8 -2 6 C -14 -8 0 -6 6 D 2 2 6 0 -8 E 2 -6 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.375000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6193: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) C A E B D (6) A C B E D (6) D B E A C (5) B C A D E (5) B A E D C (5) D E C B A (4) D E C A B (4) C E A D B (4) C D E A B (4) B D E A C (4) B A C E D (4) E D C A B (3) B D A E C (3) B D A C E (3) B A C D E (3) E D A C B (2) E C A D B (2) E A C D B (2) D B E C A (2) C D B A E (2) C B A D E (2) C A B E D (2) B D C A E (2) E A D C B (1) D E B C A (1) D C E B A (1) D C E A B (1) D B C A E (1) C E D A B (1) C D E B A (1) C A B D E (1) B E D A C (1) B A E C D (1) B A D E C (1) A C E B D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 2 -2 2 B 14 0 0 6 12 C -2 0 0 -2 2 D 2 -6 2 0 12 E -2 -12 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.476359 C: 0.523641 D: 0.000000 E: 0.000000 Sum of squares = 0.501117837009 Cumulative probabilities = A: 0.000000 B: 0.476359 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -2 2 B 14 0 0 6 12 C -2 0 0 -2 2 D 2 -6 2 0 12 E -2 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=25 C=23 E=10 A=10 so E is eliminated. Round 2 votes counts: B=32 D=30 C=25 A=13 so A is eliminated. Round 3 votes counts: B=35 C=34 D=31 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:205 C:199 A:194 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 -2 2 B 14 0 0 6 12 C -2 0 0 -2 2 D 2 -6 2 0 12 E -2 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -2 2 B 14 0 0 6 12 C -2 0 0 -2 2 D 2 -6 2 0 12 E -2 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -2 2 B 14 0 0 6 12 C -2 0 0 -2 2 D 2 -6 2 0 12 E -2 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6194: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) B C A E D (7) A D B C E (7) D A E B C (6) D A E C B (5) A B D C E (5) E D C A B (4) E B A D C (4) A D E B C (4) E D A B C (3) E C D B A (3) D E A C B (3) C E B D A (3) C B A D E (3) C A D B E (3) B A C E D (3) D E C A B (2) C D A B E (2) C B E D A (2) C B E A D (2) C B A E D (2) A D B E C (2) A B D E C (2) A B C D E (2) E D A C B (1) E C B D A (1) E B D C A (1) E B D A C (1) D E A B C (1) C E D B A (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A E B (1) B E A D C (1) B A E C D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 12 6 14 16 B -12 0 16 -4 8 C -6 -16 0 -6 8 D -14 4 6 0 4 E -16 -8 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 14 16 B -12 0 16 -4 8 C -6 -16 0 -6 8 D -14 4 6 0 4 E -16 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 B=19 E=18 D=17 so D is eliminated. Round 2 votes counts: A=35 E=24 C=22 B=19 so B is eliminated. Round 3 votes counts: A=39 C=36 E=25 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:204 D:200 C:190 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 14 16 B -12 0 16 -4 8 C -6 -16 0 -6 8 D -14 4 6 0 4 E -16 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 14 16 B -12 0 16 -4 8 C -6 -16 0 -6 8 D -14 4 6 0 4 E -16 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 14 16 B -12 0 16 -4 8 C -6 -16 0 -6 8 D -14 4 6 0 4 E -16 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6195: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) E B C D A (7) D E B A C (7) A D C B E (7) D E B C A (6) D A C B E (6) D A C E B (5) E D B C A (4) E B D C A (4) C A B E D (4) A C B E D (4) D B A E C (3) C B E A D (3) C B A E D (3) B E D C A (3) A C D B E (3) D A E C B (2) C E B A D (2) C E A B D (2) B C E A D (2) A C B D E (2) E C D A B (1) E C B D A (1) E C B A D (1) D E C B A (1) D E A C B (1) D B E C A (1) D B E A C (1) D A B E C (1) C A E B D (1) B D E C A (1) B A C E D (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 -20 -10 -10 -12 B 20 0 0 0 -8 C 10 0 0 -6 -6 D 10 0 6 0 -4 E 12 8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -10 -10 -12 B 20 0 0 0 -8 C 10 0 0 -6 -6 D 10 0 6 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=26 A=17 C=15 B=8 so B is eliminated. Round 2 votes counts: D=35 E=29 A=19 C=17 so C is eliminated. Round 3 votes counts: E=38 D=35 A=27 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:206 D:206 C:199 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -10 -10 -12 B 20 0 0 0 -8 C 10 0 0 -6 -6 D 10 0 6 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -10 -10 -12 B 20 0 0 0 -8 C 10 0 0 -6 -6 D 10 0 6 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -10 -10 -12 B 20 0 0 0 -8 C 10 0 0 -6 -6 D 10 0 6 0 -4 E 12 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6196: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) A B C E D (9) C B D A E (8) A E B C D (6) E A D B C (5) B A C E D (5) A E B D C (5) A B E C D (5) E D C A B (4) D E C B A (4) B C A D E (4) B A C D E (4) D E C A B (3) D C B E A (3) E A B C D (2) D E A B C (2) D C E B A (2) D C B A E (2) E C D B A (1) E A C B D (1) E A B D C (1) D E A C B (1) D B A C E (1) C E D B A (1) C E B D A (1) C E B A D (1) C D B E A (1) C D B A E (1) C B D E A (1) C B A D E (1) B D A C E (1) B C A E D (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 14 22 4 10 B -14 0 28 14 -4 C -22 -28 0 6 -6 D -4 -14 -6 0 -22 E -10 4 6 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 22 4 10 B -14 0 28 14 -4 C -22 -28 0 6 -6 D -4 -14 -6 0 -22 E -10 4 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995347 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 D=18 C=15 B=15 so C is eliminated. Round 2 votes counts: E=28 A=27 B=25 D=20 so D is eliminated. Round 3 votes counts: E=40 B=33 A=27 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:225 B:212 E:211 D:177 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 22 4 10 B -14 0 28 14 -4 C -22 -28 0 6 -6 D -4 -14 -6 0 -22 E -10 4 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995347 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 22 4 10 B -14 0 28 14 -4 C -22 -28 0 6 -6 D -4 -14 -6 0 -22 E -10 4 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995347 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 22 4 10 B -14 0 28 14 -4 C -22 -28 0 6 -6 D -4 -14 -6 0 -22 E -10 4 6 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995347 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6197: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) E C B A D (9) D A B C E (7) A C D E B (7) E B C A D (5) B D E A C (5) D B E A C (4) D A C B E (4) C A E D B (4) C A E B D (4) A D C B E (4) A C D B E (4) E B D C A (3) D E B C A (3) C E A D B (3) C E A B D (3) E C D B A (2) E B C D A (2) D B A E C (2) D A C E B (2) B E C D A (2) E C B D A (1) E C A B D (1) D B E C A (1) D B A C E (1) B E D A C (1) B E C A D (1) B D E C A (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -14 -8 -18 B 12 0 2 4 2 C 14 -2 0 -2 -12 D 8 -4 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -8 -18 B 12 0 2 4 2 C 14 -2 0 -2 -12 D 8 -4 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 B=23 A=16 C=14 so C is eliminated. Round 2 votes counts: E=29 D=24 A=24 B=23 so B is eliminated. Round 3 votes counts: E=45 D=31 A=24 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:210 D:200 C:199 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 -8 -18 B 12 0 2 4 2 C 14 -2 0 -2 -12 D 8 -4 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -8 -18 B 12 0 2 4 2 C 14 -2 0 -2 -12 D 8 -4 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -8 -18 B 12 0 2 4 2 C 14 -2 0 -2 -12 D 8 -4 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6198: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (6) E A B D C (5) A E D C B (5) D C B E A (4) C B A E D (4) B E A C D (4) D E A B C (3) D C A E B (3) C D B A E (3) C D A E B (3) C D A B E (3) C B D E A (3) C A D E B (3) C A D B E (3) C A B D E (3) B D E C A (3) A E C D B (3) A E C B D (3) A C E D B (3) A C E B D (3) E D A B C (2) E B A D C (2) D E B A C (2) D C E B A (2) D B C E A (2) C D B E A (2) C B D A E (2) B C E D A (2) E D B A C (1) E B A C D (1) E A D B C (1) D C B A E (1) D C A B E (1) D B E C A (1) D A E C B (1) C B A D E (1) B E A D C (1) B C D E A (1) A E B D C (1) A D E C B (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 16 0 10 16 B -16 0 -24 -8 -10 C 0 24 0 14 8 D -10 8 -14 0 4 E -16 10 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.515185 B: 0.000000 C: 0.484815 D: 0.000000 E: 0.000000 Sum of squares = 0.500461138919 Cumulative probabilities = A: 0.515185 B: 0.515185 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 10 16 B -16 0 -24 -8 -10 C 0 24 0 14 8 D -10 8 -14 0 4 E -16 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=27 D=20 E=12 B=11 so B is eliminated. Round 2 votes counts: C=33 A=27 D=23 E=17 so E is eliminated. Round 3 votes counts: A=41 C=33 D=26 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:223 A:221 D:194 E:191 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 10 16 B -16 0 -24 -8 -10 C 0 24 0 14 8 D -10 8 -14 0 4 E -16 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 10 16 B -16 0 -24 -8 -10 C 0 24 0 14 8 D -10 8 -14 0 4 E -16 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 10 16 B -16 0 -24 -8 -10 C 0 24 0 14 8 D -10 8 -14 0 4 E -16 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6199: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) B C E D A (5) E B C A D (4) C A D B E (4) B D C E A (4) B C D E A (4) E D A B C (3) E B A D C (3) D B C A E (3) D A C E B (3) B E C D A (3) B E C A D (3) B C D A E (3) A D E C B (3) A C D E B (3) E B D A C (2) E B A C D (2) E A B D C (2) E A B C D (2) D C A B E (2) D B A E C (2) D A E B C (2) C D B A E (2) C D A B E (2) C B D A E (2) C A E D B (2) C A E B D (2) B E D C A (2) A D C E B (2) E C A B D (1) E A C B D (1) D E A B C (1) D C B A E (1) D B E A C (1) D A E C B (1) C E A B D (1) C B A D E (1) C A D E B (1) C A B E D (1) B D E C A (1) B C E A D (1) A E D B C (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -10 2 8 B -2 0 8 2 -6 C 10 -8 0 6 0 D -2 -2 -6 0 -4 E -8 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999796 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 2 8 B -2 0 8 2 -6 C 10 -8 0 6 0 D -2 -2 -6 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999705 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=20 A=20 C=18 D=16 so D is eliminated. Round 2 votes counts: B=32 A=26 E=21 C=21 so E is eliminated. Round 3 votes counts: B=43 A=35 C=22 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:204 A:201 B:201 E:201 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -10 2 8 B -2 0 8 2 -6 C 10 -8 0 6 0 D -2 -2 -6 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999705 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 2 8 B -2 0 8 2 -6 C 10 -8 0 6 0 D -2 -2 -6 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999705 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 2 8 B -2 0 8 2 -6 C 10 -8 0 6 0 D -2 -2 -6 0 -4 E -8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.500000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.419999999705 Cumulative probabilities = A: 0.400000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6200: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (15) D C E A B (13) A B E C D (7) B A E D C (6) A B E D C (6) E A B C D (5) C D E B A (5) D C E B A (4) D C B E A (4) D C A B E (4) C E D B A (4) B A D C E (4) E B A C D (3) D C B A E (3) E B C A D (2) C D E A B (2) A B D E C (2) E C D B A (1) E C D A B (1) E A C B D (1) D C A E B (1) D B A C E (1) D A C E B (1) D A C B E (1) C D B E A (1) B E A C D (1) B C A D E (1) B A C E D (1) Total count = 100 A B C D E A 0 -12 8 8 6 B 12 0 8 8 14 C -8 -8 0 0 0 D -8 -8 0 0 -6 E -6 -14 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 8 8 6 B 12 0 8 8 14 C -8 -8 0 0 0 D -8 -8 0 0 -6 E -6 -14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=15 E=13 C=12 so C is eliminated. Round 2 votes counts: D=40 B=28 E=17 A=15 so A is eliminated. Round 3 votes counts: B=43 D=40 E=17 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:205 E:193 C:192 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 8 8 6 B 12 0 8 8 14 C -8 -8 0 0 0 D -8 -8 0 0 -6 E -6 -14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 8 8 6 B 12 0 8 8 14 C -8 -8 0 0 0 D -8 -8 0 0 -6 E -6 -14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 8 8 6 B 12 0 8 8 14 C -8 -8 0 0 0 D -8 -8 0 0 -6 E -6 -14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6201: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (7) D E C A B (5) B E C D A (5) A B D C E (5) D E C B A (4) D E B C A (4) D A C E B (4) B A C E D (4) A C B E D (4) E C D B A (3) C E B A D (3) C E A B D (3) B E D C A (3) B D E C A (3) B D A E C (3) B A E C D (3) B A D E C (3) E D C B A (2) E D C A B (2) D B E A C (2) D A B C E (2) C A E D B (2) B E C A D (2) A C E B D (2) A B C E D (2) E C D A B (1) E C B D A (1) E C B A D (1) E B C A D (1) D B E C A (1) D A E B C (1) D A C B E (1) D A B E C (1) C E D A B (1) C E A D B (1) C A E B D (1) B E A D C (1) B E A C D (1) A D C B E (1) A D B C E (1) A C E D B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 2 2 0 B 14 0 12 18 14 C -2 -12 0 -12 -6 D -2 -18 12 0 0 E 0 -14 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 2 0 B 14 0 12 18 14 C -2 -12 0 -12 -6 D -2 -18 12 0 0 E 0 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=25 A=18 E=11 C=11 so E is eliminated. Round 2 votes counts: B=36 D=29 A=18 C=17 so C is eliminated. Round 3 votes counts: B=41 D=34 A=25 so A is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:196 E:196 A:195 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 2 0 B 14 0 12 18 14 C -2 -12 0 -12 -6 D -2 -18 12 0 0 E 0 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 2 0 B 14 0 12 18 14 C -2 -12 0 -12 -6 D -2 -18 12 0 0 E 0 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 2 0 B 14 0 12 18 14 C -2 -12 0 -12 -6 D -2 -18 12 0 0 E 0 -14 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6202: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (7) A E D C B (7) A E B C D (7) C B D E A (5) B D C A E (5) B A D C E (5) E A C D B (4) D C B E A (4) C B E D A (4) B C D E A (4) A E C D B (4) A E C B D (4) E A C B D (3) D E C A B (3) D C E B A (3) E D A C B (2) C E D B A (2) C E B A D (2) C D B E A (2) B C D A E (2) B A C E D (2) A E B D C (2) A D B E C (2) A B E D C (2) E C A B D (1) E A D C B (1) D B C E A (1) D B A C E (1) C E B D A (1) C D E B A (1) C B E A D (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E C D (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -2 4 8 B 12 0 -8 10 4 C 2 8 0 6 10 D -4 -10 -6 0 -8 E -8 -4 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 4 8 B 12 0 -8 10 4 C 2 8 0 6 10 D -4 -10 -6 0 -8 E -8 -4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=22 D=19 C=18 E=11 so E is eliminated. Round 2 votes counts: A=38 B=22 D=21 C=19 so C is eliminated. Round 3 votes counts: A=39 B=35 D=26 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:209 A:199 E:193 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 4 8 B 12 0 -8 10 4 C 2 8 0 6 10 D -4 -10 -6 0 -8 E -8 -4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 4 8 B 12 0 -8 10 4 C 2 8 0 6 10 D -4 -10 -6 0 -8 E -8 -4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 4 8 B 12 0 -8 10 4 C 2 8 0 6 10 D -4 -10 -6 0 -8 E -8 -4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998705 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6203: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) A E B D C (7) C D B A E (5) B D E C A (5) A E C B D (5) A B D E C (5) E A B C D (4) A E B C D (4) A D B C E (4) E A B D C (3) C D A B E (3) B E D C A (3) B D C E A (3) A E C D B (3) A D C B E (3) A C E D B (3) A C D B E (3) E B D A C (2) E B A D C (2) E A C B D (2) D C B E A (2) D B C E A (2) C E D B A (2) C A E D B (2) C A D E B (2) B D A C E (2) A B E D C (2) E C D B A (1) E C A B D (1) E B C A D (1) D C B A E (1) D C A B E (1) D B C A E (1) C E A D B (1) C A D B E (1) B E D A C (1) A B D C E (1) Total count = 100 A B C D E A 0 20 12 18 16 B -20 0 4 6 10 C -12 -4 0 0 -2 D -18 -6 0 0 2 E -16 -10 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 12 18 16 B -20 0 4 6 10 C -12 -4 0 0 -2 D -18 -6 0 0 2 E -16 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=23 E=16 B=14 D=7 so D is eliminated. Round 2 votes counts: A=40 C=27 B=17 E=16 so E is eliminated. Round 3 votes counts: A=49 C=29 B=22 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:233 B:200 C:191 D:189 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 12 18 16 B -20 0 4 6 10 C -12 -4 0 0 -2 D -18 -6 0 0 2 E -16 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 12 18 16 B -20 0 4 6 10 C -12 -4 0 0 -2 D -18 -6 0 0 2 E -16 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 12 18 16 B -20 0 4 6 10 C -12 -4 0 0 -2 D -18 -6 0 0 2 E -16 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6204: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) C E D A B (6) B D A E C (5) B A C E D (5) C E A D B (4) B D E A C (4) B C E A D (4) B A D C E (4) A D C E B (4) E D C A B (3) E C D A B (3) E C B D A (3) E B C D A (3) C A E D B (3) B E D C A (3) B A D E C (3) A B D C E (3) E B D C A (2) D B E A C (2) C E B D A (2) C B A E D (2) C A D E B (2) B E C A D (2) B C A E D (2) A C D E B (2) A C D B E (2) E D B C A (1) E C D B A (1) D E B C A (1) D C A E B (1) D B A E C (1) D A E C B (1) D A C E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B A E D C (1) A C E B D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -14 -6 -8 B 8 0 0 10 -2 C 14 0 0 2 2 D 6 -10 -2 0 -10 E 8 2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.306430 C: 0.693570 D: 0.000000 E: 0.000000 Sum of squares = 0.574938773426 Cumulative probabilities = A: 0.000000 B: 0.306430 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -6 -8 B 8 0 0 10 -2 C 14 0 0 2 2 D 6 -10 -2 0 -10 E 8 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499791 C: 0.500209 D: 0.000000 E: 0.000000 Sum of squares = 0.500000087416 Cumulative probabilities = A: 0.000000 B: 0.499791 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=19 E=16 A=15 D=14 so D is eliminated. Round 2 votes counts: B=39 E=24 C=20 A=17 so A is eliminated. Round 3 votes counts: B=43 C=32 E=25 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:209 E:209 B:208 D:192 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -6 -8 B 8 0 0 10 -2 C 14 0 0 2 2 D 6 -10 -2 0 -10 E 8 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499791 C: 0.500209 D: 0.000000 E: 0.000000 Sum of squares = 0.500000087416 Cumulative probabilities = A: 0.000000 B: 0.499791 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -6 -8 B 8 0 0 10 -2 C 14 0 0 2 2 D 6 -10 -2 0 -10 E 8 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499791 C: 0.500209 D: 0.000000 E: 0.000000 Sum of squares = 0.500000087416 Cumulative probabilities = A: 0.000000 B: 0.499791 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -6 -8 B 8 0 0 10 -2 C 14 0 0 2 2 D 6 -10 -2 0 -10 E 8 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499791 C: 0.500209 D: 0.000000 E: 0.000000 Sum of squares = 0.500000087416 Cumulative probabilities = A: 0.000000 B: 0.499791 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6205: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) C D E A B (8) D E C A B (7) B A E C D (7) D C E A B (6) B E A D C (5) E D A C B (4) E B D A C (4) C D A E B (4) C D B A E (3) C A B E D (3) E D A B C (2) D E B C A (2) D E B A C (2) D E A C B (2) C D B E A (2) C B A D E (2) C A E D B (2) C A B D E (2) B C A D E (2) B A C E D (2) E D C A B (1) E A D B C (1) E A B D C (1) D E C B A (1) D C E B A (1) D C B E A (1) D B E A C (1) C D E B A (1) C B D E A (1) C B A E D (1) C A D E B (1) B E D A C (1) B D E A C (1) A E B D C (1) A E B C D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 -10 -10 B 2 0 -8 -4 -4 C 2 8 0 -10 -12 D 10 4 10 0 0 E 10 4 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.511107 E: 0.488893 Sum of squares = 0.500246729013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.511107 E: 1.000000 A B C D E A 0 -2 -2 -10 -10 B 2 0 -8 -4 -4 C 2 8 0 -10 -12 D 10 4 10 0 0 E 10 4 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=29 D=23 E=13 A=5 so A is eliminated. Round 2 votes counts: C=31 B=31 D=23 E=15 so E is eliminated. Round 3 votes counts: B=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:213 D:212 C:194 B:193 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 -10 B 2 0 -8 -4 -4 C 2 8 0 -10 -12 D 10 4 10 0 0 E 10 4 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 -10 B 2 0 -8 -4 -4 C 2 8 0 -10 -12 D 10 4 10 0 0 E 10 4 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 -10 B 2 0 -8 -4 -4 C 2 8 0 -10 -12 D 10 4 10 0 0 E 10 4 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6206: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) A B E D C (6) E D A C B (5) C D E A B (5) C D B E A (5) C B D A E (5) B C A D E (5) A B E C D (5) E D A B C (4) B C D A E (4) B A E D C (4) B A C E D (4) A E D C B (4) E D C A B (3) E A D C B (3) A E D B C (3) A E B D C (3) A B C E D (3) D E B C A (2) C D E B A (2) C B D E A (2) C B A D E (2) A C E D B (2) E A D B C (1) E A B D C (1) D E C B A (1) D C E A B (1) C D B A E (1) C A D E B (1) C A B E D (1) C A B D E (1) B D C E A (1) B D A C E (1) B C D E A (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 18 2 2 14 B -18 0 -2 2 4 C -2 2 0 2 -2 D -2 -2 -2 0 -6 E -14 -4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 2 14 B -18 0 -2 2 4 C -2 2 0 2 -2 D -2 -2 -2 0 -6 E -14 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999007 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=25 B=21 E=17 D=10 so D is eliminated. Round 2 votes counts: A=27 E=26 C=26 B=21 so B is eliminated. Round 3 votes counts: C=37 A=37 E=26 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:200 E:195 D:194 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 2 14 B -18 0 -2 2 4 C -2 2 0 2 -2 D -2 -2 -2 0 -6 E -14 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999007 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 2 14 B -18 0 -2 2 4 C -2 2 0 2 -2 D -2 -2 -2 0 -6 E -14 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999007 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 2 14 B -18 0 -2 2 4 C -2 2 0 2 -2 D -2 -2 -2 0 -6 E -14 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999007 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6207: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) C B A E D (10) E D C A B (9) B A C D E (9) D E B A C (7) B A D C E (5) C E D A B (4) C A B E D (4) D E C A B (3) C E A D B (3) C E A B D (3) B C A E D (3) A B C E D (3) E D C B A (2) C E D B A (2) B C A D E (2) B A D E C (2) B A C E D (2) E C D B A (1) E C D A B (1) D E C B A (1) D E A C B (1) D B E A C (1) D B A E C (1) D A E B C (1) C E B D A (1) C E B A D (1) C B E A D (1) C A E B D (1) A E D C B (1) A D E B C (1) A D B E C (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 10 -2 B 2 0 0 0 -6 C 4 0 0 6 12 D -10 0 -6 0 -6 E 2 6 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.472585 C: 0.527415 D: 0.000000 E: 0.000000 Sum of squares = 0.501503200822 Cumulative probabilities = A: 0.000000 B: 0.472585 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 10 -2 B 2 0 0 0 -6 C 4 0 0 6 12 D -10 0 -6 0 -6 E 2 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 B=23 E=13 A=9 so A is eliminated. Round 2 votes counts: C=31 B=28 D=27 E=14 so E is eliminated. Round 3 votes counts: D=39 C=33 B=28 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:201 E:201 B:198 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 10 -2 B 2 0 0 0 -6 C 4 0 0 6 12 D -10 0 -6 0 -6 E 2 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 -2 B 2 0 0 0 -6 C 4 0 0 6 12 D -10 0 -6 0 -6 E 2 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 -2 B 2 0 0 0 -6 C 4 0 0 6 12 D -10 0 -6 0 -6 E 2 6 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6208: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) C D B E A (8) B A E C D (7) A B E C D (7) E A D B C (6) C B D E A (6) E B A D C (4) C B D A E (4) B E A D C (4) A E D B C (4) A B C E D (4) D E C A B (3) D E A C B (3) D C E A B (3) C D A E B (3) B C A E D (3) D C B E A (2) D C A E B (2) C B A E D (2) B C D E A (2) E B D A C (1) D E B A C (1) D C E B A (1) D B E C A (1) D A C E B (1) C D A B E (1) C B A D E (1) B E D C A (1) B E D A C (1) B E A C D (1) B C E D A (1) B C E A D (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 10 10 0 B 4 0 16 18 14 C -10 -16 0 2 -10 D -10 -18 -2 0 -14 E 0 -14 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 10 0 B 4 0 16 18 14 C -10 -16 0 2 -10 D -10 -18 -2 0 -14 E 0 -14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 B=21 D=17 E=11 so E is eliminated. Round 2 votes counts: A=32 B=26 C=25 D=17 so D is eliminated. Round 3 votes counts: C=36 A=36 B=28 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:226 A:208 E:205 C:183 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 10 0 B 4 0 16 18 14 C -10 -16 0 2 -10 D -10 -18 -2 0 -14 E 0 -14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 10 0 B 4 0 16 18 14 C -10 -16 0 2 -10 D -10 -18 -2 0 -14 E 0 -14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 10 0 B 4 0 16 18 14 C -10 -16 0 2 -10 D -10 -18 -2 0 -14 E 0 -14 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6209: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (15) C E B A D (13) C D E B A (13) A B E D C (9) E B A C D (8) D C A B E (8) D A C B E (6) C E D B A (5) D A B C E (4) C E B D A (3) B E A C D (3) B A E C D (3) A B D E C (2) E B C A D (1) E A B C D (1) D C E B A (1) C E A B D (1) C D E A B (1) B A E D C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 8 -12 0 B 2 0 -2 -8 6 C -8 2 0 6 10 D 12 8 -6 0 2 E 0 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.461538 D: 0.307692 E: 0.000000 Sum of squares = 0.360946745477 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.692308 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -12 0 B 2 0 -2 -8 6 C -8 2 0 6 10 D 12 8 -6 0 2 E 0 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.461538 D: 0.307692 E: 0.000000 Sum of squares = 0.360946745567 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.692308 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=34 A=13 E=10 B=7 so B is eliminated. Round 2 votes counts: C=36 D=34 A=17 E=13 so E is eliminated. Round 3 votes counts: C=37 D=34 A=29 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:205 B:199 A:197 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 -12 0 B 2 0 -2 -8 6 C -8 2 0 6 10 D 12 8 -6 0 2 E 0 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.461538 D: 0.307692 E: 0.000000 Sum of squares = 0.360946745567 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.692308 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -12 0 B 2 0 -2 -8 6 C -8 2 0 6 10 D 12 8 -6 0 2 E 0 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.461538 D: 0.307692 E: 0.000000 Sum of squares = 0.360946745567 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.692308 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -12 0 B 2 0 -2 -8 6 C -8 2 0 6 10 D 12 8 -6 0 2 E 0 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.461538 D: 0.307692 E: 0.000000 Sum of squares = 0.360946745567 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.692308 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6210: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) B D A C E (8) A D B C E (8) A C E D B (6) E C A D B (5) E C A B D (5) D B A C E (4) C E B A D (4) E D B C A (3) D E B C A (3) D A B E C (3) A D E C B (3) E C D B A (2) E C B D A (2) E B D C A (2) E A C D B (2) D E A B C (2) C E A B D (2) C B E A D (2) C A E B D (2) C A B E D (2) B C E D A (2) B A D C E (2) A C E B D (2) A C B E D (2) E D C A B (1) E B C D A (1) D E B A C (1) D A E C B (1) C B E D A (1) B D E C A (1) B D C E A (1) B C A E D (1) B A C D E (1) A E D C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 16 4 16 B 2 0 10 -14 -2 C -16 -10 0 -12 0 D -4 14 12 0 0 E -16 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.540000000032 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 4 16 B 2 0 10 -14 -2 C -16 -10 0 -12 0 D -4 14 12 0 0 E -16 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.53999999998 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=23 B=16 C=13 so C is eliminated. Round 2 votes counts: E=29 A=28 D=24 B=19 so B is eliminated. Round 3 votes counts: E=34 D=34 A=32 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:217 D:211 B:198 E:193 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 16 4 16 B 2 0 10 -14 -2 C -16 -10 0 -12 0 D -4 14 12 0 0 E -16 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.53999999998 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 4 16 B 2 0 10 -14 -2 C -16 -10 0 -12 0 D -4 14 12 0 0 E -16 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.53999999998 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 4 16 B 2 0 10 -14 -2 C -16 -10 0 -12 0 D -4 14 12 0 0 E -16 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.100000 E: 0.000000 Sum of squares = 0.53999999998 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6211: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) D C B E A (5) C E D A B (5) E A C D B (4) C D E A B (4) C A E D B (4) B A E D C (4) B A C D E (4) E C D A B (3) D C E B A (3) D C E A B (3) D C B A E (3) C D A E B (3) B D E A C (3) B A D C E (3) A B E C D (3) E A D C B (2) E A C B D (2) E A B C D (2) D E C A B (2) D B E C A (2) D B C E A (2) D B C A E (2) C A B D E (2) B A C E D (2) A E C D B (2) A E C B D (2) A E B C D (2) A C B E D (2) E D C A B (1) E D B A C (1) E B D A C (1) E A B D C (1) D E C B A (1) C E A D B (1) C D B A E (1) C D A B E (1) C A D B E (1) B D C A E (1) B D A E C (1) B D A C E (1) Total count = 100 A B C D E A 0 4 0 2 4 B -4 0 -14 -12 2 C 0 14 0 16 6 D -2 12 -16 0 -4 E -4 -2 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.779698 B: 0.000000 C: 0.220302 D: 0.000000 E: 0.000000 Sum of squares = 0.656461725248 Cumulative probabilities = A: 0.779698 B: 0.779698 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 2 4 B -4 0 -14 -12 2 C 0 14 0 16 6 D -2 12 -16 0 -4 E -4 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 C=22 E=17 A=11 so A is eliminated. Round 2 votes counts: B=30 C=24 E=23 D=23 so E is eliminated. Round 3 votes counts: C=37 B=36 D=27 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:205 E:196 D:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 4 B -4 0 -14 -12 2 C 0 14 0 16 6 D -2 12 -16 0 -4 E -4 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 4 B -4 0 -14 -12 2 C 0 14 0 16 6 D -2 12 -16 0 -4 E -4 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 4 B -4 0 -14 -12 2 C 0 14 0 16 6 D -2 12 -16 0 -4 E -4 -2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6212: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) D C B A E (8) C D B E A (6) B E A D C (5) A E C D B (5) A E B D C (5) E A B C D (4) D C A B E (4) C E B D A (4) B E C A D (3) B E A C D (3) B A E D C (3) D B A E C (2) D A C E B (2) D A C B E (2) C D E B A (2) C D E A B (2) B D E C A (2) B D E A C (2) B D C E A (2) B D A E C (2) B A D E C (2) A E B C D (2) A B E D C (2) E C B A D (1) E B C A D (1) E B A C D (1) E A C B D (1) D B C E A (1) D B A C E (1) D A B E C (1) D A B C E (1) C E A B D (1) C D A E B (1) C B D E A (1) B E D C A (1) B E C D A (1) B C D E A (1) A E D C B (1) A E D B C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -26 0 -14 -6 B 26 0 -2 0 28 C 0 2 0 -20 -4 D 14 0 20 0 10 E 6 -28 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.341949 C: 0.000000 D: 0.658051 E: 0.000000 Sum of squares = 0.549960310903 Cumulative probabilities = A: 0.000000 B: 0.341949 C: 0.341949 D: 1.000000 E: 1.000000 A B C D E A 0 -26 0 -14 -6 B 26 0 -2 0 28 C 0 2 0 -20 -4 D 14 0 20 0 10 E 6 -28 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=27 A=18 C=17 E=8 so E is eliminated. Round 2 votes counts: D=30 B=29 A=23 C=18 so C is eliminated. Round 3 votes counts: D=41 B=35 A=24 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 D:222 C:189 E:186 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -26 0 -14 -6 B 26 0 -2 0 28 C 0 2 0 -20 -4 D 14 0 20 0 10 E 6 -28 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 0 -14 -6 B 26 0 -2 0 28 C 0 2 0 -20 -4 D 14 0 20 0 10 E 6 -28 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 0 -14 -6 B 26 0 -2 0 28 C 0 2 0 -20 -4 D 14 0 20 0 10 E 6 -28 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6213: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) A B D E C (7) D E C A B (6) A C B E D (6) A B C E D (6) A B C D E (6) C B A E D (5) B A C E D (5) C B E D A (4) D E B A C (3) D A E B C (3) C B E A D (3) B C A E D (3) E D C B A (2) D E A C B (2) D E A B C (2) C E B D A (2) C D E B A (2) C A B E D (2) B E D C A (2) A D E C B (2) A D E B C (2) A D B E C (2) A C B D E (2) A B E D C (2) E D B C A (1) D E B C A (1) D A E C B (1) C E D B A (1) C D E A B (1) C A D E B (1) B E C D A (1) B C E D A (1) B C E A D (1) B A E D C (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 10 8 16 16 B -10 0 0 20 20 C -8 0 0 4 6 D -16 -20 -4 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 16 16 B -10 0 0 20 20 C -8 0 0 4 6 D -16 -20 -4 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=25 C=21 B=14 E=3 so E is eliminated. Round 2 votes counts: A=37 D=28 C=21 B=14 so B is eliminated. Round 3 votes counts: A=43 D=30 C=27 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:225 B:215 C:201 D:182 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 16 16 B -10 0 0 20 20 C -8 0 0 4 6 D -16 -20 -4 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 16 16 B -10 0 0 20 20 C -8 0 0 4 6 D -16 -20 -4 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 16 16 B -10 0 0 20 20 C -8 0 0 4 6 D -16 -20 -4 0 4 E -16 -20 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6214: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (12) E C B D A (9) E B C A D (8) A D C B E (8) D A C B E (7) A D B C E (6) D C B A E (4) A D E C B (4) E A B C D (3) B C D A E (3) E C D A B (2) D E C A B (2) D A B C E (2) C D E B A (2) C B E D A (2) C B D E A (2) B E C D A (2) A E B D C (2) A D E B C (2) A D C E B (2) A D B E C (2) E B A C D (1) E A D C B (1) E A B D C (1) D C B E A (1) D C A B E (1) C B D A E (1) B D C A E (1) B C E D A (1) B C D E A (1) B A D C E (1) B A C E D (1) A E D C B (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -8 -10 0 B 4 0 2 4 -6 C 8 -2 0 0 -8 D 10 -4 0 0 6 E 0 6 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999948 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 -8 -10 0 B 4 0 2 4 -6 C 8 -2 0 0 -8 D 10 -4 0 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=29 D=17 B=10 C=7 so C is eliminated. Round 2 votes counts: E=37 A=29 D=19 B=15 so B is eliminated. Round 3 votes counts: E=42 A=31 D=27 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:206 E:204 B:202 C:199 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -10 0 B 4 0 2 4 -6 C 8 -2 0 0 -8 D 10 -4 0 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -10 0 B 4 0 2 4 -6 C 8 -2 0 0 -8 D 10 -4 0 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -10 0 B 4 0 2 4 -6 C 8 -2 0 0 -8 D 10 -4 0 0 6 E 0 6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6215: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C B A D E (8) C A D E B (8) A D C E B (7) E D A B C (5) A D E C B (5) E D A C B (4) D A E B C (4) B C E A D (4) B C D A E (4) E A D C B (3) C A D B E (3) B C A D E (3) E C A D B (2) D B E A C (2) C E A D B (2) C E A B D (2) C A B E D (2) B E D C A (2) B E C D A (2) B D E A C (2) A C D E B (2) E B C D A (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B C E (1) C B E A D (1) C B A E D (1) C A E D B (1) C A B D E (1) B D C E A (1) B D A E C (1) B C E D A (1) B C A E D (1) A E D C B (1) Total count = 100 A B C D E A 0 10 0 14 12 B -10 0 -10 -6 4 C 0 10 0 -2 8 D -14 6 2 0 10 E -12 -4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.517421 B: 0.000000 C: 0.482579 D: 0.000000 E: 0.000000 Sum of squares = 0.50060695062 Cumulative probabilities = A: 0.517421 B: 0.517421 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 14 12 B -10 0 -10 -6 4 C 0 10 0 -2 8 D -14 6 2 0 10 E -12 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=29 E=15 A=15 D=10 so D is eliminated. Round 2 votes counts: B=34 C=29 A=22 E=15 so E is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:208 D:202 B:189 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 14 12 B -10 0 -10 -6 4 C 0 10 0 -2 8 D -14 6 2 0 10 E -12 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 14 12 B -10 0 -10 -6 4 C 0 10 0 -2 8 D -14 6 2 0 10 E -12 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 14 12 B -10 0 -10 -6 4 C 0 10 0 -2 8 D -14 6 2 0 10 E -12 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6216: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (24) B E D C A (16) C A D E B (8) D E B C A (7) A C B E D (7) C D A E B (5) B E A D C (5) A C D B E (5) B E D A C (4) C D E B A (3) A B E D C (3) A B E C D (3) A C E B D (2) E B D C A (1) D C E B A (1) D C B E A (1) D C B A E (1) C D B E A (1) C D A B E (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 20 10 18 22 B -20 0 -22 -14 -4 C -10 22 0 22 20 D -18 14 -22 0 16 E -22 4 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 18 22 B -20 0 -22 -14 -4 C -10 22 0 22 20 D -18 14 -22 0 16 E -22 4 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 B=25 C=18 D=10 E=1 so E is eliminated. Round 2 votes counts: A=46 B=26 C=18 D=10 so D is eliminated. Round 3 votes counts: A=46 B=33 C=21 so C is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:235 C:227 D:195 E:173 B:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 18 22 B -20 0 -22 -14 -4 C -10 22 0 22 20 D -18 14 -22 0 16 E -22 4 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 18 22 B -20 0 -22 -14 -4 C -10 22 0 22 20 D -18 14 -22 0 16 E -22 4 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 18 22 B -20 0 -22 -14 -4 C -10 22 0 22 20 D -18 14 -22 0 16 E -22 4 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6217: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) C E B A D (9) E A B C D (6) B A C E D (6) B C A E D (5) A B D C E (5) E A C B D (4) B A C D E (4) E C D A B (3) E C B A D (3) D E C A B (3) D E A B C (3) D B A C E (3) D A B E C (3) C E D B A (3) E D C A B (2) E A D B C (2) E A B D C (2) C D E B A (2) C B A E D (2) B A D C E (2) A E B C D (2) A B E D C (2) A B E C D (2) E D A C B (1) E C D B A (1) E C A D B (1) E C A B D (1) D C B A E (1) D A E B C (1) D A B C E (1) C E B D A (1) C D B A E (1) C B A D E (1) B C A D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 0 22 -14 B 10 0 2 18 -20 C 0 -2 0 16 14 D -22 -18 -16 0 -16 E 14 20 -14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.388889 C: 0.555556 D: 0.000000 E: 0.055556 Sum of squares = 0.462962963243 Cumulative probabilities = A: 0.000000 B: 0.388889 C: 0.944444 D: 0.944444 E: 1.000000 A B C D E A 0 -10 0 22 -14 B 10 0 2 18 -20 C 0 -2 0 16 14 D -22 -18 -16 0 -16 E 14 20 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.388889 C: 0.555556 D: 0.000000 E: 0.055556 Sum of squares = 0.462962963223 Cumulative probabilities = A: 0.000000 B: 0.388889 C: 0.944444 D: 0.944444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=25 C=19 B=18 A=12 so A is eliminated. Round 2 votes counts: E=28 B=28 D=25 C=19 so C is eliminated. Round 3 votes counts: E=41 B=31 D=28 so D is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:214 B:205 A:199 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 22 -14 B 10 0 2 18 -20 C 0 -2 0 16 14 D -22 -18 -16 0 -16 E 14 20 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.388889 C: 0.555556 D: 0.000000 E: 0.055556 Sum of squares = 0.462962963223 Cumulative probabilities = A: 0.000000 B: 0.388889 C: 0.944444 D: 0.944444 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 22 -14 B 10 0 2 18 -20 C 0 -2 0 16 14 D -22 -18 -16 0 -16 E 14 20 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.388889 C: 0.555556 D: 0.000000 E: 0.055556 Sum of squares = 0.462962963223 Cumulative probabilities = A: 0.000000 B: 0.388889 C: 0.944444 D: 0.944444 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 22 -14 B 10 0 2 18 -20 C 0 -2 0 16 14 D -22 -18 -16 0 -16 E 14 20 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.388889 C: 0.555556 D: 0.000000 E: 0.055556 Sum of squares = 0.462962963223 Cumulative probabilities = A: 0.000000 B: 0.388889 C: 0.944444 D: 0.944444 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6218: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) B E D C A (7) B A E C D (6) A C D E B (6) D C E A B (5) A B C E D (5) C A D E B (4) B E D A C (4) A C E B D (4) A C B D E (4) E D C B A (3) D E B C A (3) D C A E B (3) C E D A B (3) B D E C A (3) B A C E D (3) A B C D E (3) E B C D A (2) B E A D C (2) B D E A C (2) B A D C E (2) A C E D B (2) A C B E D (2) E C D A B (1) E C B D A (1) E C A B D (1) D B A C E (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E D B (1) B E A C D (1) B D A E C (1) B A E D C (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 2 0 2 B 4 0 -6 10 -2 C -2 6 0 4 4 D 0 -10 -4 0 0 E -2 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.38888888876 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 0 2 B 4 0 -6 10 -2 C -2 6 0 4 4 D 0 -10 -4 0 0 E -2 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888876 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=27 D=21 C=11 E=8 so E is eliminated. Round 2 votes counts: B=35 A=27 D=24 C=14 so C is eliminated. Round 3 votes counts: B=36 A=34 D=30 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:206 B:203 A:200 E:198 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 0 2 B 4 0 -6 10 -2 C -2 6 0 4 4 D 0 -10 -4 0 0 E -2 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888876 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 0 2 B 4 0 -6 10 -2 C -2 6 0 4 4 D 0 -10 -4 0 0 E -2 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888876 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 0 2 B 4 0 -6 10 -2 C -2 6 0 4 4 D 0 -10 -4 0 0 E -2 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888876 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6219: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E A D B C (9) E A B D C (8) E B A C D (7) D A C E B (7) B C E D A (7) A E D C B (7) E A D C B (6) B C D A E (6) B E C A D (5) C B D A E (4) A E D B C (4) D C A B E (3) B C D E A (3) A D E C B (3) E B A D C (2) D C A E B (2) A D C E B (2) D C E A B (1) D A E C B (1) B E A C D (1) B A D E C (1) B A C D E (1) Total count = 100 A B C D E A 0 6 18 12 2 B -6 0 8 -10 -18 C -18 -8 0 -12 -8 D -12 10 12 0 -12 E -2 18 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 12 2 B -6 0 8 -10 -18 C -18 -8 0 -12 -8 D -12 10 12 0 -12 E -2 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=24 A=16 D=14 C=14 so D is eliminated. Round 2 votes counts: E=32 B=24 A=24 C=20 so C is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:219 E:218 D:199 B:187 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 12 2 B -6 0 8 -10 -18 C -18 -8 0 -12 -8 D -12 10 12 0 -12 E -2 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 12 2 B -6 0 8 -10 -18 C -18 -8 0 -12 -8 D -12 10 12 0 -12 E -2 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 12 2 B -6 0 8 -10 -18 C -18 -8 0 -12 -8 D -12 10 12 0 -12 E -2 18 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6220: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) C D E B A (7) C D E A B (7) A B E D C (7) D C E A B (6) B C E D A (5) B A E D C (5) A E D B C (5) B E A D C (4) B A E C D (4) C D A B E (3) B C A E D (3) A E B D C (3) E B D A C (2) E B A D C (2) E A D B C (2) E A B D C (2) C B D E A (2) B C E A D (2) B A C E D (2) A D E B C (2) A B C E D (2) E D B A C (1) D A E B C (1) D A C E B (1) C D B A E (1) C B A E D (1) C B A D E (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) A D E C B (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 2 6 -8 B 8 0 16 4 10 C -2 -16 0 6 10 D -6 -4 -6 0 -12 E 8 -10 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 6 -8 B 8 0 16 4 10 C -2 -16 0 6 10 D -6 -4 -6 0 -12 E 8 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=29 A=23 E=9 D=8 so D is eliminated. Round 2 votes counts: C=37 B=29 A=25 E=9 so E is eliminated. Round 3 votes counts: C=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:200 C:199 A:196 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 6 -8 B 8 0 16 4 10 C -2 -16 0 6 10 D -6 -4 -6 0 -12 E 8 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 6 -8 B 8 0 16 4 10 C -2 -16 0 6 10 D -6 -4 -6 0 -12 E 8 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 6 -8 B 8 0 16 4 10 C -2 -16 0 6 10 D -6 -4 -6 0 -12 E 8 -10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6221: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) E D B C A (10) B A C D E (9) D E C A B (8) E D C B A (7) C A D E B (6) B E D A C (6) A C B D E (6) A B C D E (6) C D E A B (4) B A E D C (4) B A C E D (4) B A D E C (3) E D B A C (2) A C D E B (2) E C D B A (1) D E B A C (1) D E A C B (1) D C E A B (1) C E D A B (1) C D A E B (1) C A B D E (1) B E A D C (1) B C A E D (1) B A E C D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -6 -10 -10 B 0 0 -4 -16 -14 C 6 4 0 -12 -14 D 10 16 12 0 0 E 10 14 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.370808 E: 0.629192 Sum of squares = 0.533381375061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.370808 E: 1.000000 A B C D E A 0 0 -6 -10 -10 B 0 0 -4 -16 -14 C 6 4 0 -12 -14 D 10 16 12 0 0 E 10 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=29 A=15 C=13 D=11 so D is eliminated. Round 2 votes counts: E=42 B=29 A=15 C=14 so C is eliminated. Round 3 votes counts: E=48 B=29 A=23 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:219 E:219 C:192 A:187 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 -10 -10 B 0 0 -4 -16 -14 C 6 4 0 -12 -14 D 10 16 12 0 0 E 10 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -10 -10 B 0 0 -4 -16 -14 C 6 4 0 -12 -14 D 10 16 12 0 0 E 10 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -10 -10 B 0 0 -4 -16 -14 C 6 4 0 -12 -14 D 10 16 12 0 0 E 10 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6222: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D A C B E (7) C B A D E (6) D A E C B (5) B C E A D (4) A D B C E (4) E D A B C (3) E C B D A (3) C D B A E (3) C B E A D (3) B C A E D (3) B C A D E (3) A B D E C (3) E D C A B (2) E D A C B (2) E C D B A (2) E A B D C (2) D E A C B (2) D C A B E (2) D A E B C (2) D A C E B (2) C E B D A (2) C D B E A (2) C D A B E (2) C B D E A (2) B E C A D (2) A B E D C (2) A B D C E (2) E C B A D (1) E B A D C (1) E B A C D (1) E A D B C (1) D E C A B (1) D E A B C (1) D C A E B (1) C E D B A (1) C B E D A (1) C B D A E (1) C A D B E (1) B A C D E (1) A D E B C (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -12 2 6 B 0 0 -10 2 12 C 12 10 0 6 8 D -2 -2 -6 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 2 6 B 0 0 -10 2 12 C 12 10 0 6 8 D -2 -2 -6 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=24 D=23 A=14 B=13 so B is eliminated. Round 2 votes counts: C=34 E=28 D=23 A=15 so A is eliminated. Round 3 votes counts: C=36 D=34 E=30 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:202 D:201 A:198 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 2 6 B 0 0 -10 2 12 C 12 10 0 6 8 D -2 -2 -6 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 2 6 B 0 0 -10 2 12 C 12 10 0 6 8 D -2 -2 -6 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 2 6 B 0 0 -10 2 12 C 12 10 0 6 8 D -2 -2 -6 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6223: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) E B D C A (10) A C D B E (10) B C E D A (8) C B A D E (6) C B E A D (5) B E C D A (5) A D C B E (5) D E A B C (4) A D E C B (4) A D E B C (4) E B C D A (3) E A D B C (3) D A E B C (3) A D C E B (3) D A B E C (2) C B E D A (2) C A B D E (2) E D B C A (1) E D B A C (1) E C B D A (1) E C A B D (1) E A D C B (1) D A B C E (1) C B A E D (1) B E D C A (1) B C D A E (1) B C A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 8 4 -6 -12 B -8 0 16 -6 0 C -4 -16 0 -8 -10 D 6 6 8 0 -8 E 12 0 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.292407 C: 0.000000 D: 0.000000 E: 0.707593 Sum of squares = 0.586189903525 Cumulative probabilities = A: 0.000000 B: 0.292407 C: 0.292407 D: 0.292407 E: 1.000000 A B C D E A 0 8 4 -6 -12 B -8 0 16 -6 0 C -4 -16 0 -8 -10 D 6 6 8 0 -8 E 12 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=27 C=16 B=16 D=10 so D is eliminated. Round 2 votes counts: E=35 A=33 C=16 B=16 so C is eliminated. Round 3 votes counts: E=35 A=35 B=30 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:206 B:201 A:197 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 4 -6 -12 B -8 0 16 -6 0 C -4 -16 0 -8 -10 D 6 6 8 0 -8 E 12 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -6 -12 B -8 0 16 -6 0 C -4 -16 0 -8 -10 D 6 6 8 0 -8 E 12 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -6 -12 B -8 0 16 -6 0 C -4 -16 0 -8 -10 D 6 6 8 0 -8 E 12 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6224: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) B C D A E (9) E A D C B (8) E C A B D (6) D A B E C (6) C B E D A (6) C B D E A (6) A D E B C (6) E A C D B (5) C E B A D (5) C B D A E (5) D B A C E (4) A E D B C (4) E A D B C (3) C E B D A (3) C B E A D (2) E C B A D (1) E C A D B (1) E A C B D (1) E A B C D (1) D C A B E (1) C E D A B (1) B D A C E (1) B C E A D (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 -14 -20 -8 0 B 14 0 -2 22 10 C 20 2 0 8 16 D 8 -22 -8 0 4 E 0 -10 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998472 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -8 0 B 14 0 -2 22 10 C 20 2 0 8 16 D 8 -22 -8 0 4 E 0 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=25 D=11 A=10 so A is eliminated. Round 2 votes counts: E=30 C=28 B=25 D=17 so D is eliminated. Round 3 votes counts: E=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:223 B:222 D:191 E:185 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -20 -8 0 B 14 0 -2 22 10 C 20 2 0 8 16 D 8 -22 -8 0 4 E 0 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -8 0 B 14 0 -2 22 10 C 20 2 0 8 16 D 8 -22 -8 0 4 E 0 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -8 0 B 14 0 -2 22 10 C 20 2 0 8 16 D 8 -22 -8 0 4 E 0 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6225: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) C A D E B (6) B E C A D (6) D C A E B (5) B E D A C (5) A B D C E (5) E B C D A (4) C E B A D (4) C D A E B (4) A D C B E (4) A D B C E (4) E C D B A (3) E B C A D (3) D E B A C (3) C E D A B (3) C E A D B (3) B E A D C (3) E D C B A (2) E B D C A (2) D A C E B (2) B E A C D (2) B A E D C (2) B A D E C (2) D E C A B (1) D E A C B (1) D B A E C (1) D A C B E (1) C E D B A (1) C E A B D (1) C A E B D (1) C A D B E (1) C A B E D (1) B D A E C (1) B A E C D (1) A C D E B (1) A C D B E (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -16 8 -8 B 4 0 -10 6 -14 C 16 10 0 12 2 D -8 -6 -12 0 -10 E 8 14 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 8 -8 B 4 0 -10 6 -14 C 16 10 0 12 2 D -8 -6 -12 0 -10 E 8 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=22 E=21 A=18 D=14 so D is eliminated. Round 2 votes counts: C=30 E=26 B=23 A=21 so A is eliminated. Round 3 votes counts: C=41 B=33 E=26 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:215 B:193 A:190 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 8 -8 B 4 0 -10 6 -14 C 16 10 0 12 2 D -8 -6 -12 0 -10 E 8 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 8 -8 B 4 0 -10 6 -14 C 16 10 0 12 2 D -8 -6 -12 0 -10 E 8 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 8 -8 B 4 0 -10 6 -14 C 16 10 0 12 2 D -8 -6 -12 0 -10 E 8 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6226: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) E D C A B (8) B C D A E (8) B A C D E (7) C D B A E (6) A E B D C (6) A B D C E (6) E C D B A (5) B C D E A (5) A B E D C (5) B E C D A (4) A E D C B (4) E B C D A (3) E A B D C (3) D C A E B (2) C D E B A (2) C D B E A (2) B E A C D (2) A D C E B (2) A D C B E (2) E C D A B (1) E B A D C (1) D A C B E (1) C E D B A (1) C D E A B (1) B C A D E (1) B A E D C (1) B A E C D (1) Total count = 100 A B C D E A 0 2 2 2 4 B -2 0 6 6 2 C -2 -6 0 -2 -8 D -2 -6 2 0 -10 E -4 -2 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 2 4 B -2 0 6 6 2 C -2 -6 0 -2 -8 D -2 -6 2 0 -10 E -4 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=29 A=25 C=12 D=3 so D is eliminated. Round 2 votes counts: E=31 B=29 A=26 C=14 so C is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:206 E:206 A:205 D:192 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 2 4 B -2 0 6 6 2 C -2 -6 0 -2 -8 D -2 -6 2 0 -10 E -4 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 4 B -2 0 6 6 2 C -2 -6 0 -2 -8 D -2 -6 2 0 -10 E -4 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 4 B -2 0 6 6 2 C -2 -6 0 -2 -8 D -2 -6 2 0 -10 E -4 -2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6227: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) E C B D A (5) C E B A D (4) C B E A D (4) B C A D E (4) B A C D E (4) E C D A B (3) E B C D A (3) D E B A C (3) D A E B C (3) B D E A C (3) B A D C E (3) A D B C E (3) A B D C E (3) E D C B A (2) E D B C A (2) E D B A C (2) E D A B C (2) E C D B A (2) D B E A C (2) D A E C B (2) D A B E C (2) C E A D B (2) C B A E D (2) C A E D B (2) C A B E D (2) C A B D E (2) B C E D A (2) A D E C B (2) A D C E B (2) A C D E B (2) A C B D E (2) E D C A B (1) E C A D B (1) E A C D B (1) D E A C B (1) D B A E C (1) C A E B D (1) B E C D A (1) B C E A D (1) B C A E D (1) A D C B E (1) A D B E C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 6 4 -6 B 2 0 -6 -4 -8 C -6 6 0 6 -2 D -4 4 -6 0 -4 E 6 8 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 4 -6 B 2 0 -6 -4 -8 C -6 6 0 6 -2 D -4 4 -6 0 -4 E 6 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=19 B=19 A=18 D=14 so D is eliminated. Round 2 votes counts: E=34 A=25 B=22 C=19 so C is eliminated. Round 3 votes counts: E=40 A=32 B=28 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:202 A:201 D:195 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 4 -6 B 2 0 -6 -4 -8 C -6 6 0 6 -2 D -4 4 -6 0 -4 E 6 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 4 -6 B 2 0 -6 -4 -8 C -6 6 0 6 -2 D -4 4 -6 0 -4 E 6 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 4 -6 B 2 0 -6 -4 -8 C -6 6 0 6 -2 D -4 4 -6 0 -4 E 6 8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6228: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E B A C D (9) A B C E D (8) D C A E B (4) C A D B E (4) C A B D E (4) B A E C D (4) A C D B E (4) E D B C A (3) E D B A C (3) E B D A C (3) E B C A D (3) D E A B C (3) C B A E D (3) B E A C D (3) D E A C B (2) D A C E B (2) D A C B E (2) B E C A D (2) A C B D E (2) E D A B C (1) E B D C A (1) E B C D A (1) E B A D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D E B A C (1) D C E B A (1) D C B E A (1) D C B A E (1) D A E C B (1) C D B A E (1) C D A B E (1) C B D E A (1) C B A D E (1) C A B E D (1) B C E A D (1) A E B D C (1) A D B C E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 6 8 14 B -8 0 2 2 14 C -6 -2 0 10 8 D -8 -2 -10 0 0 E -14 -14 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 8 14 B -8 0 2 2 14 C -6 -2 0 10 8 D -8 -2 -10 0 0 E -14 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=25 A=18 C=16 B=10 so B is eliminated. Round 2 votes counts: D=31 E=30 A=22 C=17 so C is eliminated. Round 3 votes counts: A=35 D=34 E=31 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:205 C:205 D:190 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 8 14 B -8 0 2 2 14 C -6 -2 0 10 8 D -8 -2 -10 0 0 E -14 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 8 14 B -8 0 2 2 14 C -6 -2 0 10 8 D -8 -2 -10 0 0 E -14 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 8 14 B -8 0 2 2 14 C -6 -2 0 10 8 D -8 -2 -10 0 0 E -14 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6229: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) C A B E D (6) A C E D B (6) E D C B A (5) E D B C A (5) D E B A C (5) C A E D B (4) B C E D A (4) A C D E B (4) D E A C B (3) C E B D A (3) C E A D B (3) B D E C A (3) B D A E C (3) B C A E D (3) B A C D E (3) E D A C B (2) E C D B A (2) D E B C A (2) C E D A B (2) C E A B D (2) B D E A C (2) B C A D E (2) A B C D E (2) E C D A B (1) D E A B C (1) D B E A C (1) D B A E C (1) C B E A D (1) C B A E D (1) B E D C A (1) B D C A E (1) B C E A D (1) B C D E A (1) B A D E C (1) B A D C E (1) A E C D B (1) A D E C B (1) A D E B C (1) A D C E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -18 -8 -12 B 4 0 -10 -14 -20 C 18 10 0 6 6 D 8 14 -6 0 -20 E 12 20 -6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 -8 -12 B 4 0 -10 -14 -20 C 18 10 0 6 6 D 8 14 -6 0 -20 E 12 20 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=22 E=21 A=18 D=13 so D is eliminated. Round 2 votes counts: E=32 B=28 C=22 A=18 so A is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:220 D:198 B:180 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -18 -8 -12 B 4 0 -10 -14 -20 C 18 10 0 6 6 D 8 14 -6 0 -20 E 12 20 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -8 -12 B 4 0 -10 -14 -20 C 18 10 0 6 6 D 8 14 -6 0 -20 E 12 20 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -8 -12 B 4 0 -10 -14 -20 C 18 10 0 6 6 D 8 14 -6 0 -20 E 12 20 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6230: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) B D A C E (5) E C A D B (4) E B D C A (4) B D E C A (4) E D C B A (3) D C E A B (3) D B C A E (3) C D E A B (3) C A D E B (3) B E D A C (3) B E A C D (3) B D A E C (3) B A D C E (3) A C D B E (3) E C D A B (2) E A C B D (2) D E B C A (2) D C B A E (2) D C A E B (2) D B A C E (2) C D A E B (2) B E D C A (2) B A D E C (2) A C E D B (2) A C E B D (2) A C D E B (2) A B C E D (2) E D B C A (1) E C D B A (1) E B D A C (1) E B C A D (1) E A C D B (1) E A B C D (1) D E C B A (1) D C E B A (1) D C A B E (1) D B C E A (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) C A E D B (1) B E A D C (1) B D E A C (1) B A E C D (1) A D C B E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 2 -12 -6 B 12 0 10 -2 -4 C -2 -10 0 -10 0 D 12 2 10 0 8 E 6 4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -12 -6 B 12 0 10 -2 -4 C -2 -10 0 -10 0 D 12 2 10 0 8 E 6 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=26 D=20 A=15 C=11 so C is eliminated. Round 2 votes counts: E=28 B=28 D=25 A=19 so A is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:208 E:201 C:189 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -12 -6 B 12 0 10 -2 -4 C -2 -10 0 -10 0 D 12 2 10 0 8 E 6 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -12 -6 B 12 0 10 -2 -4 C -2 -10 0 -10 0 D 12 2 10 0 8 E 6 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -12 -6 B 12 0 10 -2 -4 C -2 -10 0 -10 0 D 12 2 10 0 8 E 6 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6231: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (13) C B E A D (10) C A E B D (10) D B E A C (9) D A E B C (5) B C D E A (5) D E A B C (4) D A E C B (4) C B A E D (4) B D C E A (4) A E C D B (4) D E B A C (3) B C E A D (3) A E D C B (3) C B A D E (2) B D E C A (2) B C E D A (2) A D E C B (2) E C B A D (1) D B A E C (1) D B A C E (1) C E B A D (1) C A E D B (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E A C (1) B D C A E (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 2 12 6 B 4 0 -14 -2 -4 C -2 14 0 16 18 D -12 2 -16 0 -10 E -6 4 -18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999992 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 12 6 B 4 0 -14 -2 -4 C -2 14 0 16 18 D -12 2 -16 0 -10 E -6 4 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999571 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=27 A=23 B=20 E=1 so E is eliminated. Round 2 votes counts: C=30 D=27 A=23 B=20 so B is eliminated. Round 3 votes counts: C=41 D=36 A=23 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:208 E:195 B:192 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 12 6 B 4 0 -14 -2 -4 C -2 14 0 16 18 D -12 2 -16 0 -10 E -6 4 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999571 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 12 6 B 4 0 -14 -2 -4 C -2 14 0 16 18 D -12 2 -16 0 -10 E -6 4 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999571 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 12 6 B 4 0 -14 -2 -4 C -2 14 0 16 18 D -12 2 -16 0 -10 E -6 4 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.539999999571 Cumulative probabilities = A: 0.700000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6232: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (18) E A D C B (14) D C B A E (10) D A C B E (8) D C A B E (7) C B D A E (6) E B A C D (5) A D E C B (4) E B C A D (3) E A D B C (3) D A E C B (3) A E D C B (3) A D C E B (3) C D B A E (2) B E C A D (2) B C E D A (2) B C A D E (2) E A B C D (1) D A C E B (1) B C D E A (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 -16 38 B 2 0 -22 -16 18 C 6 22 0 -12 22 D 16 16 12 0 32 E -38 -18 -22 -32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -16 38 B 2 0 -22 -16 18 C 6 22 0 -12 22 D 16 16 12 0 32 E -38 -18 -22 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=26 B=25 A=12 C=8 so C is eliminated. Round 2 votes counts: D=31 B=31 E=26 A=12 so A is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:238 C:219 A:207 B:191 E:145 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -16 38 B 2 0 -22 -16 18 C 6 22 0 -12 22 D 16 16 12 0 32 E -38 -18 -22 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -16 38 B 2 0 -22 -16 18 C 6 22 0 -12 22 D 16 16 12 0 32 E -38 -18 -22 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -16 38 B 2 0 -22 -16 18 C 6 22 0 -12 22 D 16 16 12 0 32 E -38 -18 -22 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6233: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) E B A D C (8) C E B A D (8) C D A B E (8) E C B A D (7) D B A E C (6) B E D A C (5) D A B E C (4) C E B D A (4) C E A D B (4) C A D B E (4) B D A E C (4) E B D A C (3) C A E D B (3) E C A B D (2) E B C A D (2) D B A C E (2) C E A B D (2) B D E A C (2) E C B D A (1) E A B D C (1) D C A B E (1) C E D A B (1) C D B E A (1) C B E D A (1) B D E C A (1) A E C D B (1) A D C B E (1) A D B C E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -20 12 -6 B 10 0 -20 4 -16 C 20 20 0 20 4 D -12 -4 -20 0 -6 E 6 16 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 12 -6 B 10 0 -20 4 -16 C 20 20 0 20 4 D -12 -4 -20 0 -6 E 6 16 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 E=24 D=13 B=12 A=5 so A is eliminated. Round 2 votes counts: C=47 E=25 D=15 B=13 so B is eliminated. Round 3 votes counts: C=47 E=30 D=23 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:232 E:212 B:189 A:188 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -20 12 -6 B 10 0 -20 4 -16 C 20 20 0 20 4 D -12 -4 -20 0 -6 E 6 16 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 12 -6 B 10 0 -20 4 -16 C 20 20 0 20 4 D -12 -4 -20 0 -6 E 6 16 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 12 -6 B 10 0 -20 4 -16 C 20 20 0 20 4 D -12 -4 -20 0 -6 E 6 16 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6234: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) B C D A E (8) A E D C B (7) C D B A E (6) E A B D C (5) D C A B E (5) A D E C B (5) B E A C D (4) B C E A D (4) B C D E A (4) A E B D C (4) E B C D A (3) D C A E B (3) B E C D A (3) E B A D C (2) E B A C D (2) E A D B C (2) D A E C B (2) C D A B E (2) C B D E A (2) B E C A D (2) B C E D A (2) B C A D E (2) B A E C D (2) B A C D E (2) E D A C B (1) D E A C B (1) D A C E B (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 14 2 B -4 0 4 4 -2 C -8 -4 0 -4 -14 D -14 -4 4 0 -10 E -2 2 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 14 2 B -4 0 4 4 -2 C -8 -4 0 -4 -14 D -14 -4 4 0 -10 E -2 2 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=27 A=18 D=12 C=10 so C is eliminated. Round 2 votes counts: B=35 E=27 D=20 A=18 so A is eliminated. Round 3 votes counts: E=38 B=36 D=26 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:214 E:212 B:201 D:188 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 14 2 B -4 0 4 4 -2 C -8 -4 0 -4 -14 D -14 -4 4 0 -10 E -2 2 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 14 2 B -4 0 4 4 -2 C -8 -4 0 -4 -14 D -14 -4 4 0 -10 E -2 2 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 14 2 B -4 0 4 4 -2 C -8 -4 0 -4 -14 D -14 -4 4 0 -10 E -2 2 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998061 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6235: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) C D A B E (6) B E C A D (6) E D A B C (5) B C E D A (5) E A D B C (4) C D B E A (4) C B D A E (4) B C A E D (4) D C E B A (3) D C A E B (3) D A E C B (3) C D B A E (3) B E C D A (3) B E A C D (3) A E B D C (3) A D E C B (3) A D C E B (3) E D B C A (2) E B D A C (2) E B A D C (2) E A B D C (2) D A C E B (2) C D A E B (2) A C D E B (2) E D B A C (1) D E C A B (1) D E A B C (1) D C B E A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A D B E (1) B C E A D (1) B A E C D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -4 -4 6 B -4 0 10 -16 -4 C 4 -10 0 2 -2 D 4 16 -2 0 -12 E -6 4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.317460 B: 0.103175 C: 0.317460 D: 0.071429 E: 0.190476 Sum of squares = 0.253590325019 Cumulative probabilities = A: 0.317460 B: 0.420635 C: 0.738095 D: 0.809524 E: 1.000000 A B C D E A 0 4 -4 -4 6 B -4 0 10 -16 -4 C 4 -10 0 2 -2 D 4 16 -2 0 -12 E -6 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.317460 B: 0.103175 C: 0.317460 D: 0.071429 E: 0.190476 Sum of squares = 0.253590325019 Cumulative probabilities = A: 0.317460 B: 0.420635 C: 0.738095 D: 0.809524 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 A=22 E=18 D=14 so D is eliminated. Round 2 votes counts: C=30 A=27 B=23 E=20 so E is eliminated. Round 3 votes counts: A=39 C=31 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:206 D:203 A:201 C:197 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 -4 6 B -4 0 10 -16 -4 C 4 -10 0 2 -2 D 4 16 -2 0 -12 E -6 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.317460 B: 0.103175 C: 0.317460 D: 0.071429 E: 0.190476 Sum of squares = 0.253590325019 Cumulative probabilities = A: 0.317460 B: 0.420635 C: 0.738095 D: 0.809524 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -4 6 B -4 0 10 -16 -4 C 4 -10 0 2 -2 D 4 16 -2 0 -12 E -6 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.317460 B: 0.103175 C: 0.317460 D: 0.071429 E: 0.190476 Sum of squares = 0.253590325019 Cumulative probabilities = A: 0.317460 B: 0.420635 C: 0.738095 D: 0.809524 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -4 6 B -4 0 10 -16 -4 C 4 -10 0 2 -2 D 4 16 -2 0 -12 E -6 4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.317460 B: 0.103175 C: 0.317460 D: 0.071429 E: 0.190476 Sum of squares = 0.253590325019 Cumulative probabilities = A: 0.317460 B: 0.420635 C: 0.738095 D: 0.809524 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6236: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) A B C E D (8) D E C B A (7) C D E A B (7) B A C D E (7) A C B D E (6) A B C D E (6) B A E D C (5) E D C A B (4) E D B C A (4) C A D E B (4) D C E B A (3) C D A E B (3) D E C A B (2) D E B C A (2) D C E A B (2) C E A D B (2) B E D A C (2) B E A D C (2) A C B E D (2) E D B A C (1) E B D A C (1) E B A D C (1) E A D B C (1) C A B D E (1) B C A D E (1) B A E C D (1) B A D C E (1) B A C E D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -6 2 -4 B 0 0 -10 -8 -12 C 6 10 0 2 10 D -2 8 -2 0 4 E 4 12 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 2 -4 B 0 0 -10 -8 -12 C 6 10 0 2 10 D -2 8 -2 0 4 E 4 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 B=20 C=17 D=16 so D is eliminated. Round 2 votes counts: E=34 A=24 C=22 B=20 so B is eliminated. Round 3 votes counts: A=39 E=38 C=23 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:214 D:204 E:201 A:196 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 2 -4 B 0 0 -10 -8 -12 C 6 10 0 2 10 D -2 8 -2 0 4 E 4 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 -4 B 0 0 -10 -8 -12 C 6 10 0 2 10 D -2 8 -2 0 4 E 4 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 -4 B 0 0 -10 -8 -12 C 6 10 0 2 10 D -2 8 -2 0 4 E 4 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6237: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) A B D E C (7) C E D A B (6) B A D E C (5) A C B E D (5) A B D C E (5) E D C B A (4) C A D E B (4) E B D C A (3) E B D A C (3) D E C B A (3) C D E A B (3) C A D B E (3) B A E D C (3) A C B D E (3) A B C D E (3) E D B C A (2) E C D B A (2) D A C B E (2) C A E B D (2) B E A D C (2) A C E B D (2) A C D B E (2) A B E D C (2) A B C E D (2) E C B D A (1) E A B C D (1) D E B C A (1) D C E B A (1) D C B A E (1) D B A E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C A E D B (1) B D E A C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 16 4 12 12 B -16 0 -12 4 0 C -4 12 0 4 16 D -12 -4 -4 0 -6 E -12 0 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 12 12 B -16 0 -12 4 0 C -4 12 0 4 16 D -12 -4 -4 0 -6 E -12 0 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=30 E=16 B=11 D=10 so D is eliminated. Round 2 votes counts: A=36 C=32 E=20 B=12 so B is eliminated. Round 3 votes counts: A=45 C=32 E=23 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:214 E:189 B:188 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 4 12 12 B -16 0 -12 4 0 C -4 12 0 4 16 D -12 -4 -4 0 -6 E -12 0 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 12 12 B -16 0 -12 4 0 C -4 12 0 4 16 D -12 -4 -4 0 -6 E -12 0 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 12 12 B -16 0 -12 4 0 C -4 12 0 4 16 D -12 -4 -4 0 -6 E -12 0 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6238: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) B E C D A (9) E B C D A (6) A D C E B (6) E C D B A (5) E A D C B (5) E B A C D (4) D C A B E (4) B C E D A (4) E B A D C (3) C D B E A (3) B E A C D (3) A E D C B (3) A B D C E (3) E C D A B (2) E A B D C (2) C D B A E (2) B E A D C (2) B C D E A (2) B A E D C (2) A E D B C (2) A E B D C (2) A D E C B (2) E D A C B (1) E B C A D (1) D C A E B (1) D A C E B (1) D A C B E (1) C D E A B (1) C D A B E (1) C B E D A (1) B E C A D (1) B A D C E (1) B A C D E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 14 12 -10 B 0 0 0 -4 6 C -14 0 0 -8 -12 D -12 4 8 0 -18 E 10 -6 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.343717 B: 0.656283 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.548848866016 Cumulative probabilities = A: 0.343717 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 12 -10 B 0 0 0 -4 6 C -14 0 0 -8 -12 D -12 4 8 0 -18 E 10 -6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250026876 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=29 B=25 C=8 D=7 so D is eliminated. Round 2 votes counts: A=33 E=29 B=25 C=13 so C is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 A:208 B:201 D:191 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 14 12 -10 B 0 0 0 -4 6 C -14 0 0 -8 -12 D -12 4 8 0 -18 E 10 -6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250026876 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 12 -10 B 0 0 0 -4 6 C -14 0 0 -8 -12 D -12 4 8 0 -18 E 10 -6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250026876 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 12 -10 B 0 0 0 -4 6 C -14 0 0 -8 -12 D -12 4 8 0 -18 E 10 -6 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.625000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250026876 Cumulative probabilities = A: 0.375000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6239: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (16) E D B C A (11) B D E C A (6) A E C D B (6) E D A B C (3) E A D C B (3) E A D B C (3) C B D A E (3) A C D B E (3) A C B E D (3) E D C B A (2) D C B E A (2) D B E C A (2) C D B E A (2) C D B A E (2) B C D A E (2) A E D B C (2) A E B D C (2) A C E D B (2) A C E B D (2) E B D C A (1) E B D A C (1) E B A D C (1) D C E B A (1) C D E B A (1) C D E A B (1) C D A B E (1) C B D E A (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E A D C (1) B D E A C (1) B D C E A (1) B C D E A (1) B C A D E (1) B A E D C (1) B A C D E (1) A E C B D (1) A C D E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 10 10 10 B -8 0 -14 0 10 C -10 14 0 10 4 D -10 0 -10 0 6 E -10 -10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 10 10 B -8 0 -14 0 10 C -10 14 0 10 4 D -10 0 -10 0 6 E -10 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 E=25 C=15 B=15 D=5 so D is eliminated. Round 2 votes counts: A=40 E=25 C=18 B=17 so B is eliminated. Round 3 votes counts: A=42 E=35 C=23 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:209 B:194 D:193 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 10 10 B -8 0 -14 0 10 C -10 14 0 10 4 D -10 0 -10 0 6 E -10 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 10 10 B -8 0 -14 0 10 C -10 14 0 10 4 D -10 0 -10 0 6 E -10 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 10 10 B -8 0 -14 0 10 C -10 14 0 10 4 D -10 0 -10 0 6 E -10 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6240: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) B E D A C (8) E B D A C (6) D A B C E (6) C B E A D (5) C E A D B (4) B E C D A (4) B C E D A (4) A D C E B (4) E D A B C (3) D A B E C (3) C B A D E (3) C A D B E (3) B C E A D (3) E C D B A (2) E C D A B (2) E C B A D (2) D A E B C (2) C A E D B (2) C A D E B (2) C A B D E (2) B D E A C (2) B D A E C (2) A D C B E (2) A D B C E (2) E B C D A (1) E A C D B (1) D E A B C (1) D A E C B (1) C B A E D (1) B E D C A (1) B D C A E (1) B D A C E (1) B C A D E (1) B A D C E (1) A E D C B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -4 0 -16 B 14 0 4 14 10 C 4 -4 0 4 14 D 0 -14 -4 0 -18 E 16 -10 -14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 0 -16 B 14 0 4 14 10 C 4 -4 0 4 14 D 0 -14 -4 0 -18 E 16 -10 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=28 E=17 D=13 A=11 so A is eliminated. Round 2 votes counts: C=32 B=28 D=22 E=18 so E is eliminated. Round 3 votes counts: C=39 B=35 D=26 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:209 E:205 A:183 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 0 -16 B 14 0 4 14 10 C 4 -4 0 4 14 D 0 -14 -4 0 -18 E 16 -10 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 0 -16 B 14 0 4 14 10 C 4 -4 0 4 14 D 0 -14 -4 0 -18 E 16 -10 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 0 -16 B 14 0 4 14 10 C 4 -4 0 4 14 D 0 -14 -4 0 -18 E 16 -10 -14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6241: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) E C A D B (7) C A D E B (6) D E C A B (5) B D E A C (5) E A C B D (4) E C D A B (3) E B A C D (3) D E C B A (3) D C E A B (3) D B C A E (3) B D E C A (3) B A D C E (3) B A C E D (3) B A C D E (3) A C B D E (3) E D C A B (2) E B D C A (2) E A B C D (2) D C A E B (2) D B E C A (2) C D A E B (2) C A E D B (2) B E D A C (2) B D A E C (2) A B C E D (2) D C E B A (1) D C B A E (1) D B C E A (1) C E A D B (1) C D E A B (1) C D A B E (1) B E A D C (1) B E A C D (1) B D A C E (1) B A E D C (1) B A D E C (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 10 -4 B 4 0 4 8 0 C 2 -4 0 12 -18 D -10 -8 -12 0 6 E 4 0 18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.677248 C: 0.000000 D: 0.000000 E: 0.322752 Sum of squares = 0.562833815559 Cumulative probabilities = A: 0.000000 B: 0.677248 C: 0.677248 D: 0.677248 E: 1.000000 A B C D E A 0 -4 -2 10 -4 B 4 0 4 8 0 C 2 -4 0 12 -18 D -10 -8 -12 0 6 E 4 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500006 C: 0.000000 D: 0.000000 E: 0.499994 Sum of squares = 0.500000000077 Cumulative probabilities = A: 0.000000 B: 0.500006 C: 0.500006 D: 0.500006 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=23 D=21 C=13 A=7 so A is eliminated. Round 2 votes counts: B=39 E=23 D=21 C=17 so C is eliminated. Round 3 votes counts: B=42 D=31 E=27 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:208 A:200 C:196 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 10 -4 B 4 0 4 8 0 C 2 -4 0 12 -18 D -10 -8 -12 0 6 E 4 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500006 C: 0.000000 D: 0.000000 E: 0.499994 Sum of squares = 0.500000000077 Cumulative probabilities = A: 0.000000 B: 0.500006 C: 0.500006 D: 0.500006 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 10 -4 B 4 0 4 8 0 C 2 -4 0 12 -18 D -10 -8 -12 0 6 E 4 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500006 C: 0.000000 D: 0.000000 E: 0.499994 Sum of squares = 0.500000000077 Cumulative probabilities = A: 0.000000 B: 0.500006 C: 0.500006 D: 0.500006 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 10 -4 B 4 0 4 8 0 C 2 -4 0 12 -18 D -10 -8 -12 0 6 E 4 0 18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500006 C: 0.000000 D: 0.000000 E: 0.499994 Sum of squares = 0.500000000077 Cumulative probabilities = A: 0.000000 B: 0.500006 C: 0.500006 D: 0.500006 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6242: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) C B E D A (9) D A C B E (7) A E D B C (6) A D E C B (6) D C B A E (5) D A E B C (5) E A B C D (4) D A E C B (4) C B E A D (4) C B D E A (4) B C E D A (4) E B C A D (3) E B A C D (3) C B D A E (3) E B C D A (2) D C B E A (2) D B C E A (2) D A C E B (2) B C E A D (2) A E B D C (2) E D B C A (1) D C A B E (1) C D B E A (1) C D B A E (1) C B A D E (1) C A B E D (1) B E C A D (1) B C D E A (1) A E C B D (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 4 -8 14 B -2 0 -6 -8 0 C -4 6 0 -6 2 D 8 8 6 0 12 E -14 0 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -8 14 B -2 0 -6 -8 0 C -4 6 0 -6 2 D 8 8 6 0 12 E -14 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=27 C=24 E=13 B=8 so B is eliminated. Round 2 votes counts: C=31 D=28 A=27 E=14 so E is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:217 A:206 C:199 B:192 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -8 14 B -2 0 -6 -8 0 C -4 6 0 -6 2 D 8 8 6 0 12 E -14 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -8 14 B -2 0 -6 -8 0 C -4 6 0 -6 2 D 8 8 6 0 12 E -14 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -8 14 B -2 0 -6 -8 0 C -4 6 0 -6 2 D 8 8 6 0 12 E -14 0 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6243: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (16) B E A C D (11) E B D A C (10) D C A E B (9) D E B C A (7) E B D C A (6) A C B E D (5) D E C A B (4) A C B D E (4) E D B C A (3) C A D E B (3) E B A C D (2) D E C B A (2) D C E A B (2) D C A B E (2) C D A B E (2) B E D A C (2) B A C E D (2) A C D B E (2) A B C E D (2) E B A D C (1) C D A E B (1) B E A D C (1) B A E C D (1) Total count = 100 A B C D E A 0 4 -14 0 -2 B -4 0 -4 -6 0 C 14 4 0 2 0 D 0 6 -2 0 8 E 2 0 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.929616 D: 0.000000 E: 0.070384 Sum of squares = 0.869139009547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.929616 D: 0.929616 E: 1.000000 A B C D E A 0 4 -14 0 -2 B -4 0 -4 -6 0 C 14 4 0 2 0 D 0 6 -2 0 8 E 2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=22 C=22 B=17 A=13 so A is eliminated. Round 2 votes counts: C=33 D=26 E=22 B=19 so B is eliminated. Round 3 votes counts: E=37 C=37 D=26 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:206 E:197 A:194 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 0 -2 B -4 0 -4 -6 0 C 14 4 0 2 0 D 0 6 -2 0 8 E 2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 0 -2 B -4 0 -4 -6 0 C 14 4 0 2 0 D 0 6 -2 0 8 E 2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 0 -2 B -4 0 -4 -6 0 C 14 4 0 2 0 D 0 6 -2 0 8 E 2 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000000825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6244: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) D C E B A (5) C D E A B (5) E D C B A (4) E C D A B (4) A B C D E (4) E C A D B (3) E B A D C (3) E A B D C (3) D E C B A (3) D B E C A (3) B D E A C (3) B A D E C (3) A B E D C (3) E D C A B (2) E D B C A (2) E B D A C (2) E A C D B (2) D E B C A (2) D C B E A (2) C E D A B (2) B D C A E (2) B D A C E (2) B A E D C (2) A E B C D (2) A C E D B (2) A C E B D (2) A C B E D (2) A C B D E (2) E D B A C (1) E A D C B (1) D B C E A (1) D B C A E (1) C E A D B (1) C D E B A (1) C D A B E (1) C B D A E (1) C A D B E (1) B E D A C (1) B E A D C (1) B D E C A (1) B D C E A (1) B D A E C (1) B C D A E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 2 -8 -18 B 12 0 8 2 -4 C -2 -8 0 -10 -6 D 8 -2 10 0 6 E 18 4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888879 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 -12 2 -8 -18 B 12 0 8 2 -4 C -2 -8 0 -10 -6 D 8 -2 10 0 6 E 18 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.3888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 A=19 D=17 C=12 so C is eliminated. Round 2 votes counts: E=30 B=26 D=24 A=20 so A is eliminated. Round 3 votes counts: B=39 E=36 D=25 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:211 E:211 B:209 C:187 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -8 -18 B 12 0 8 2 -4 C -2 -8 0 -10 -6 D 8 -2 10 0 6 E 18 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.3888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -8 -18 B 12 0 8 2 -4 C -2 -8 0 -10 -6 D 8 -2 10 0 6 E 18 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.3888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -8 -18 B 12 0 8 2 -4 C -2 -8 0 -10 -6 D 8 -2 10 0 6 E 18 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.3888888889 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6245: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) A E B C D (9) D C B E A (7) D B E A C (6) C D A B E (6) C A E B D (6) A E B D C (6) C A B E D (5) D E B A C (4) D B E C A (4) C A D E B (4) E B A D C (3) C D B E A (3) C D B A E (3) C D A E B (3) E B D A C (2) D C E A B (2) D B C E A (2) C A D B E (2) B E A D C (2) B A E C D (2) A E C B D (2) A C E B D (2) D C E B A (1) C A E D B (1) C A B D E (1) B D E A C (1) A B E C D (1) Total count = 100 A B C D E A 0 20 0 12 6 B -20 0 4 4 -10 C 0 -4 0 0 -4 D -12 -4 0 0 -2 E -6 10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555481 B: 0.000000 C: 0.444519 D: 0.000000 E: 0.000000 Sum of squares = 0.506156238048 Cumulative probabilities = A: 0.555481 B: 0.555481 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 0 12 6 B -20 0 4 4 -10 C 0 -4 0 0 -4 D -12 -4 0 0 -2 E -6 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999844 Cumulative probabilities = A: 0.500004 B: 0.500004 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=26 A=20 E=15 B=5 so B is eliminated. Round 2 votes counts: C=34 D=27 A=22 E=17 so E is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:205 C:196 D:191 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 0 12 6 B -20 0 4 4 -10 C 0 -4 0 0 -4 D -12 -4 0 0 -2 E -6 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999844 Cumulative probabilities = A: 0.500004 B: 0.500004 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 12 6 B -20 0 4 4 -10 C 0 -4 0 0 -4 D -12 -4 0 0 -2 E -6 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999844 Cumulative probabilities = A: 0.500004 B: 0.500004 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 12 6 B -20 0 4 4 -10 C 0 -4 0 0 -4 D -12 -4 0 0 -2 E -6 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999844 Cumulative probabilities = A: 0.500004 B: 0.500004 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6246: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (7) E A C B D (6) E A B D C (6) D C B A E (6) D B C A E (6) D B A C E (6) A B C D E (5) E D C B A (4) D E C B A (4) C B D A E (4) E A C D B (3) E A B C D (3) C B A D E (3) A E B C D (3) A C B E D (3) E D B A C (2) E C D B A (2) B A D C E (2) B A C D E (2) A B C E D (2) E D A B C (1) E C D A B (1) E C A D B (1) E A D C B (1) E A D B C (1) D E B A C (1) D C E B A (1) D C B E A (1) C E D B A (1) C E A B D (1) C D B E A (1) C D B A E (1) C B E D A (1) C B A E D (1) C A B E D (1) B D A C E (1) B C D A E (1) A D B E C (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 4 10 2 B 2 0 -10 10 2 C -4 10 0 10 2 D -10 -10 -10 0 -4 E -2 -2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999994 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 10 2 B 2 0 -10 10 2 C -4 10 0 10 2 D -10 -10 -10 0 -4 E -2 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.468750000085 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=25 A=17 C=14 B=6 so B is eliminated. Round 2 votes counts: E=38 D=26 A=21 C=15 so C is eliminated. Round 3 votes counts: E=41 D=33 A=26 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 A:207 B:202 E:199 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 10 2 B 2 0 -10 10 2 C -4 10 0 10 2 D -10 -10 -10 0 -4 E -2 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.468750000085 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 2 B 2 0 -10 10 2 C -4 10 0 10 2 D -10 -10 -10 0 -4 E -2 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.468750000085 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 2 B 2 0 -10 10 2 C -4 10 0 10 2 D -10 -10 -10 0 -4 E -2 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.468750000085 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6247: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) B C D A E (9) E A D B C (6) C B D A E (6) A E B D C (6) E A D C B (5) D E A B C (5) A E B C D (5) D B C E A (3) B D C A E (3) B A D E C (3) E A C D B (2) D E C A B (2) D C E B A (2) D B C A E (2) D B A E C (2) C E A D B (2) C B A E D (2) C B A D E (2) B C A E D (2) B A E C D (2) B A C E D (2) A E D B C (2) A E C B D (2) E D A C B (1) E D A B C (1) E A C B D (1) D E C B A (1) D E B A C (1) D E A C B (1) D C B E A (1) D B E A C (1) C E A B D (1) C A E B D (1) C A B E D (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 2 0 10 B 6 0 12 2 4 C -2 -12 0 2 0 D 0 -2 -2 0 6 E -10 -4 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 0 10 B 6 0 12 2 4 C -2 -12 0 2 0 D 0 -2 -2 0 6 E -10 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=21 B=21 A=18 E=16 so E is eliminated. Round 2 votes counts: A=32 C=24 D=23 B=21 so B is eliminated. Round 3 votes counts: A=39 C=35 D=26 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 A:203 D:201 C:194 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 0 10 B 6 0 12 2 4 C -2 -12 0 2 0 D 0 -2 -2 0 6 E -10 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 10 B 6 0 12 2 4 C -2 -12 0 2 0 D 0 -2 -2 0 6 E -10 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 10 B 6 0 12 2 4 C -2 -12 0 2 0 D 0 -2 -2 0 6 E -10 -4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6248: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (15) D A E C B (9) D B E A C (6) B E C D A (4) B C A E D (4) B C A D E (4) E D A C B (3) D A E B C (3) D A B C E (3) B D E C A (3) A D E C B (3) A D C E B (3) E C A B D (2) E B D C A (2) D E A C B (2) D E A B C (2) D A C E B (2) C B E A D (2) C B A E D (2) C A E B D (2) A E C D B (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A D B (1) E B C D A (1) E A D C B (1) D B A E C (1) D B A C E (1) D A B E C (1) C E B A D (1) C E A B D (1) C A B D E (1) B D A C E (1) B C E D A (1) B C D A E (1) B A C D E (1) A E D C B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 0 4 0 B 4 0 8 2 8 C 0 -8 0 4 0 D -4 -2 -4 0 0 E 0 -8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 4 0 B 4 0 8 2 8 C 0 -8 0 4 0 D -4 -2 -4 0 0 E 0 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=30 A=14 E=13 C=9 so C is eliminated. Round 2 votes counts: B=38 D=30 A=17 E=15 so E is eliminated. Round 3 votes counts: B=43 D=35 A=22 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:200 C:198 E:196 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 4 0 B 4 0 8 2 8 C 0 -8 0 4 0 D -4 -2 -4 0 0 E 0 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 4 0 B 4 0 8 2 8 C 0 -8 0 4 0 D -4 -2 -4 0 0 E 0 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 4 0 B 4 0 8 2 8 C 0 -8 0 4 0 D -4 -2 -4 0 0 E 0 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6249: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) A E B D C (8) C B D E A (7) A E C D B (7) B C D E A (6) C B A D E (5) E A D B C (4) C D B E A (4) A C E B D (4) E A D C B (3) B D C E A (3) A E D C B (3) A C E D B (3) E D B A C (2) E D A B C (2) D E B C A (2) D B E C A (2) D B C E A (2) C B D A E (2) C A D E B (2) C A B D E (2) B E D A C (2) B D E C A (2) E A B D C (1) D E C A B (1) D C E A B (1) D C B E A (1) C D E A B (1) B E A D C (1) B C D A E (1) B C A D E (1) B A E D C (1) B A C E D (1) A E C B D (1) A E B C D (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 10 18 6 B -10 0 4 4 -8 C -10 -4 0 0 -4 D -18 -4 0 0 -10 E -6 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 18 6 B -10 0 4 4 -8 C -10 -4 0 0 -4 D -18 -4 0 0 -10 E -6 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=23 B=18 E=12 D=9 so D is eliminated. Round 2 votes counts: A=38 C=25 B=22 E=15 so E is eliminated. Round 3 votes counts: A=48 C=26 B=26 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:208 B:195 C:191 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 18 6 B -10 0 4 4 -8 C -10 -4 0 0 -4 D -18 -4 0 0 -10 E -6 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 18 6 B -10 0 4 4 -8 C -10 -4 0 0 -4 D -18 -4 0 0 -10 E -6 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 18 6 B -10 0 4 4 -8 C -10 -4 0 0 -4 D -18 -4 0 0 -10 E -6 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6250: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) A E B D C (8) E A B D C (7) B D A C E (7) E C A D B (5) D B C A E (5) C E D B A (5) C D B E A (5) C B D E A (5) A B D E C (5) C E B D A (4) B D C A E (4) A D B C E (4) E C B D A (3) E C A B D (3) D B A C E (3) C D B A E (3) A E C D B (3) B A D E C (2) A E D B C (2) E A D C B (1) E A C B D (1) D C B A E (1) C A D E B (1) B D C E A (1) A E D C B (1) A D E C B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 10 8 4 B -4 0 0 2 -6 C -10 0 0 -8 -2 D -8 -2 8 0 -2 E -4 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 8 4 B -4 0 0 2 -6 C -10 0 0 -8 -2 D -8 -2 8 0 -2 E -4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 C=23 B=14 D=9 so D is eliminated. Round 2 votes counts: E=28 A=26 C=24 B=22 so B is eliminated. Round 3 votes counts: A=38 C=34 E=28 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:203 D:198 B:196 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 8 4 B -4 0 0 2 -6 C -10 0 0 -8 -2 D -8 -2 8 0 -2 E -4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 8 4 B -4 0 0 2 -6 C -10 0 0 -8 -2 D -8 -2 8 0 -2 E -4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 8 4 B -4 0 0 2 -6 C -10 0 0 -8 -2 D -8 -2 8 0 -2 E -4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6251: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) E A C D B (7) E C A D B (6) D B A E C (6) B D A C E (6) C E B A D (5) A E D B C (5) A D B E C (5) C E A D B (4) C B E D A (4) B D C A E (4) A D E B C (4) D A E B C (3) C E B D A (3) C E A B D (3) B D A E C (3) A E D C B (3) A E C D B (3) E A D C B (2) D A B E C (2) C B D E A (2) C B A E D (2) B D C E A (2) D C E B A (1) D B C E A (1) C E D B A (1) C E D A B (1) B C D A E (1) B A D E C (1) B A C E D (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 4 4 0 B 2 0 4 -8 -4 C -4 -4 0 4 -2 D -4 8 -4 0 -2 E 0 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.354979 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.645021 Sum of squares = 0.54206207433 Cumulative probabilities = A: 0.354979 B: 0.354979 C: 0.354979 D: 0.354979 E: 1.000000 A B C D E A 0 -2 4 4 0 B 2 0 4 -8 -4 C -4 -4 0 4 -2 D -4 8 -4 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 A=21 E=15 D=13 so D is eliminated. Round 2 votes counts: B=33 C=26 A=26 E=15 so E is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:204 A:203 D:199 B:197 C:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 4 0 B 2 0 4 -8 -4 C -4 -4 0 4 -2 D -4 8 -4 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 4 0 B 2 0 4 -8 -4 C -4 -4 0 4 -2 D -4 8 -4 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 4 0 B 2 0 4 -8 -4 C -4 -4 0 4 -2 D -4 8 -4 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6252: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C D A B E (8) D C B E A (6) C A D B E (6) A C D E B (6) E B A D C (5) A E B C D (5) D E B C A (4) C D B A E (4) A E B D C (4) A C B E D (4) E B D C A (3) D B E C A (3) C D B E A (3) A C E D B (3) E A B D C (2) D C E A B (2) C D A E B (2) C A B D E (2) B E D C A (2) B D E C A (2) B D C E A (2) A E C B D (2) A C D B E (2) A C B D E (2) A B E C D (2) E A D B C (1) D C E B A (1) D C A E B (1) B C E A D (1) A E D C B (1) Total count = 100 A B C D E A 0 10 -4 -4 8 B -10 0 -10 -6 -2 C 4 10 0 4 10 D 4 6 -4 0 12 E -8 2 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -4 8 B -10 0 -10 -6 -2 C 4 10 0 4 10 D 4 6 -4 0 12 E -8 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=25 E=20 D=17 B=7 so B is eliminated. Round 2 votes counts: A=31 C=26 E=22 D=21 so D is eliminated. Round 3 votes counts: C=38 E=31 A=31 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:209 A:205 B:186 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -4 8 B -10 0 -10 -6 -2 C 4 10 0 4 10 D 4 6 -4 0 12 E -8 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -4 8 B -10 0 -10 -6 -2 C 4 10 0 4 10 D 4 6 -4 0 12 E -8 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -4 8 B -10 0 -10 -6 -2 C 4 10 0 4 10 D 4 6 -4 0 12 E -8 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6253: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) A D B E C (8) A D B C E (8) A C B D E (8) A B D C E (8) C B D E A (7) E C D B A (5) E C B D A (5) C B E D A (5) A E D B C (5) C E B D A (4) E A D C B (3) B D A C E (3) E D A B C (2) D E B A C (2) B D C E A (2) A E D C B (2) A B C D E (2) E A C D B (1) D E A B C (1) D B E A C (1) D B C E A (1) D B A C E (1) C B E A D (1) C B A D E (1) C A E B D (1) C A B E D (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 14 2 0 B -4 0 8 -2 14 C -14 -8 0 -14 8 D -2 2 14 0 6 E 0 -14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.901528 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.098472 Sum of squares = 0.822448776791 Cumulative probabilities = A: 0.901528 B: 0.901528 C: 0.901528 D: 0.901528 E: 1.000000 A B C D E A 0 4 14 2 0 B -4 0 8 -2 14 C -14 -8 0 -14 8 D -2 2 14 0 6 E 0 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.654320996774 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 E=26 C=20 D=6 B=5 so B is eliminated. Round 2 votes counts: A=43 E=26 C=20 D=11 so D is eliminated. Round 3 votes counts: A=47 E=30 C=23 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:210 B:208 C:186 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 2 0 B -4 0 8 -2 14 C -14 -8 0 -14 8 D -2 2 14 0 6 E 0 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.654320996774 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 2 0 B -4 0 8 -2 14 C -14 -8 0 -14 8 D -2 2 14 0 6 E 0 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.654320996774 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 2 0 B -4 0 8 -2 14 C -14 -8 0 -14 8 D -2 2 14 0 6 E 0 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.654320996774 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6254: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (16) B D E C A (7) B D E A C (6) A C B E D (6) D E B C A (5) C B E D A (5) C A E D B (5) A E D C B (5) C E D A B (4) A D B E C (4) A B D E C (4) D E A B C (3) C E D B A (3) C E A D B (3) B A D E C (3) A D E C B (3) E D C B A (2) A B C D E (2) E D B C A (1) E D A C B (1) E C D B A (1) E A D C B (1) D E B A C (1) D E A C B (1) D B E A C (1) C E B D A (1) C B A E D (1) C A B E D (1) B C E D A (1) B C D E A (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 20 18 12 4 B -20 0 -18 -20 -12 C -18 18 0 2 2 D -12 20 -2 0 -14 E -4 12 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 12 4 B -20 0 -18 -20 -12 C -18 18 0 2 2 D -12 20 -2 0 -14 E -4 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=23 B=19 D=11 E=6 so E is eliminated. Round 2 votes counts: A=42 C=24 B=19 D=15 so D is eliminated. Round 3 votes counts: A=47 B=27 C=26 so C is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:210 C:202 D:196 B:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 12 4 B -20 0 -18 -20 -12 C -18 18 0 2 2 D -12 20 -2 0 -14 E -4 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 12 4 B -20 0 -18 -20 -12 C -18 18 0 2 2 D -12 20 -2 0 -14 E -4 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 12 4 B -20 0 -18 -20 -12 C -18 18 0 2 2 D -12 20 -2 0 -14 E -4 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6255: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) B E C D A (7) A C D E B (7) C B E A D (6) A D E C B (6) B E D C A (5) B D E C A (5) B C E D A (5) D E B C A (4) D A E B C (4) D E A C B (3) D B E C A (3) B D E A C (3) D E C B A (2) D E C A B (2) B C E A D (2) B C A E D (2) A D B E C (2) A D B C E (2) A C E D B (2) E D C A B (1) E D B C A (1) E C D A B (1) D E A B C (1) D B E A C (1) D A E C B (1) C E D B A (1) C E B D A (1) C E B A D (1) C E A D B (1) C B A E D (1) C A E B D (1) B D A E C (1) A D E B C (1) A C D B E (1) A C B E D (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -4 -12 B 2 0 4 -10 2 C 4 -4 0 -16 -10 D 4 10 16 0 22 E 12 -2 10 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -12 B 2 0 4 -10 2 C 4 -4 0 -16 -10 D 4 10 16 0 22 E 12 -2 10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=30 D=21 C=12 E=3 so E is eliminated. Round 2 votes counts: A=34 B=30 D=23 C=13 so C is eliminated. Round 3 votes counts: B=39 A=36 D=25 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:226 B:199 E:199 A:189 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -12 B 2 0 4 -10 2 C 4 -4 0 -16 -10 D 4 10 16 0 22 E 12 -2 10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -12 B 2 0 4 -10 2 C 4 -4 0 -16 -10 D 4 10 16 0 22 E 12 -2 10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -12 B 2 0 4 -10 2 C 4 -4 0 -16 -10 D 4 10 16 0 22 E 12 -2 10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6256: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (10) C A B D E (7) D E B C A (6) B E D C A (6) B A C E D (6) A C B E D (5) E D B A C (4) D C E A B (4) C D B E A (4) C D A E B (4) D E C A B (3) B E D A C (3) E D A B C (2) E B D C A (2) D E B A C (2) C D B A E (2) C B A D E (2) C A D E B (2) B D E C A (2) B C A E D (2) A E B D C (2) A B E C D (2) E B D A C (1) D E C B A (1) D E A C B (1) D C E B A (1) C D A B E (1) C B D E A (1) C B D A E (1) C B A E D (1) C A D B E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E A D (1) A E D B C (1) A E C B D (1) A D E C B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -12 -4 6 B 2 0 -8 -2 4 C 12 8 0 14 12 D 4 2 -14 0 14 E -6 -4 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -4 6 B 2 0 -8 -2 4 C 12 8 0 14 12 D 4 2 -14 0 14 E -6 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 B=23 D=18 E=9 so E is eliminated. Round 2 votes counts: C=26 B=26 D=24 A=24 so D is eliminated. Round 3 votes counts: B=38 C=35 A=27 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:203 B:198 A:194 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -4 6 B 2 0 -8 -2 4 C 12 8 0 14 12 D 4 2 -14 0 14 E -6 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -4 6 B 2 0 -8 -2 4 C 12 8 0 14 12 D 4 2 -14 0 14 E -6 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -4 6 B 2 0 -8 -2 4 C 12 8 0 14 12 D 4 2 -14 0 14 E -6 -4 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6257: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) A D E C B (9) B C E D A (8) A C D E B (7) B C A E D (6) B E D C A (5) E D C B A (4) E D C A B (4) D E C A B (4) A B D E C (4) C E D B A (3) C B A E D (3) B A C E D (3) B A C D E (3) A C B D E (3) A B C D E (3) D E B A C (2) D E A B C (2) D A E C B (2) B D E A C (2) E D B C A (1) D A E B C (1) C E D A B (1) C E A D B (1) C A B E D (1) B E D A C (1) B E C D A (1) B C E A D (1) B A D E C (1) B A D C E (1) A D E B C (1) A D C E B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 14 0 2 B -10 0 -8 -6 -6 C -14 8 0 -8 -6 D 0 6 8 0 10 E -2 6 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.552574 B: 0.000000 C: 0.000000 D: 0.447426 E: 0.000000 Sum of squares = 0.50552805908 Cumulative probabilities = A: 0.552574 B: 0.552574 C: 0.552574 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 0 2 B -10 0 -8 -6 -6 C -14 8 0 -8 -6 D 0 6 8 0 10 E -2 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=30 D=20 E=9 C=9 so E is eliminated. Round 2 votes counts: B=32 A=30 D=29 C=9 so C is eliminated. Round 3 votes counts: B=35 D=33 A=32 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:213 D:212 E:200 C:190 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 0 2 B -10 0 -8 -6 -6 C -14 8 0 -8 -6 D 0 6 8 0 10 E -2 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 0 2 B -10 0 -8 -6 -6 C -14 8 0 -8 -6 D 0 6 8 0 10 E -2 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 0 2 B -10 0 -8 -6 -6 C -14 8 0 -8 -6 D 0 6 8 0 10 E -2 6 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6258: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) D C B E A (6) A E D C B (6) C E B D A (5) B C E A D (5) B C D E A (5) A D B C E (5) E C B A D (4) E A C B D (4) E C B D A (3) E A C D B (3) B C E D A (3) A D E C B (3) A D B E C (3) E D C A B (2) E C D B A (2) D B C A E (2) D A E C B (2) C B E D A (2) B D A C E (2) B A D C E (2) B A C D E (2) A D E B C (2) A B D C E (2) E C A D B (1) E B C A D (1) E A D C B (1) D E C B A (1) D E C A B (1) D E A C B (1) D C E B A (1) D C E A B (1) D B C E A (1) D A C B E (1) B E A C D (1) B D C E A (1) B D C A E (1) B A E C D (1) A E B C D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 4 0 -10 B 2 0 0 -4 10 C -4 0 0 -12 8 D 0 4 12 0 6 E 10 -10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250520 B: 0.000000 C: 0.000000 D: 0.749480 E: 0.000000 Sum of squares = 0.624480354932 Cumulative probabilities = A: 0.250520 B: 0.250520 C: 0.250520 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 0 -10 B 2 0 0 -4 10 C -4 0 0 -12 8 D 0 4 12 0 6 E 10 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250000752 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=24 B=23 E=21 C=7 so C is eliminated. Round 2 votes counts: E=26 B=25 A=25 D=24 so D is eliminated. Round 3 votes counts: A=35 B=34 E=31 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:204 A:196 C:196 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 0 -10 B 2 0 0 -4 10 C -4 0 0 -12 8 D 0 4 12 0 6 E 10 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250000752 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 0 -10 B 2 0 0 -4 10 C -4 0 0 -12 8 D 0 4 12 0 6 E 10 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250000752 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 0 -10 B 2 0 0 -4 10 C -4 0 0 -12 8 D 0 4 12 0 6 E 10 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250000752 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6259: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (13) B D A E C (12) D A B C E (11) C E A D B (10) B E C D A (10) A D C E B (9) A D C B E (7) E B C D A (4) A C D E B (4) D B A C E (3) C A D E B (3) E C A D B (2) E C A B D (2) D A C B E (2) B E D A C (2) D B A E C (1) D A B E C (1) C E A B D (1) C A E D B (1) B E D C A (1) B D E A C (1) Total count = 100 A B C D E A 0 6 6 4 8 B -6 0 -8 -8 2 C -6 8 0 0 2 D -4 8 0 0 8 E -8 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 4 8 B -6 0 -8 -8 2 C -6 8 0 0 2 D -4 8 0 0 8 E -8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=21 A=20 D=18 C=15 so C is eliminated. Round 2 votes counts: E=32 B=26 A=24 D=18 so D is eliminated. Round 3 votes counts: A=38 E=32 B=30 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:206 C:202 B:190 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 4 8 B -6 0 -8 -8 2 C -6 8 0 0 2 D -4 8 0 0 8 E -8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 8 B -6 0 -8 -8 2 C -6 8 0 0 2 D -4 8 0 0 8 E -8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 8 B -6 0 -8 -8 2 C -6 8 0 0 2 D -4 8 0 0 8 E -8 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6260: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) C A B E D (9) B A C D E (8) C A E D B (6) B D E C A (6) A B C D E (6) D E B C A (5) D E B A C (5) D B E A C (5) B D E A C (5) E C A D B (4) B D A E C (4) B A D C E (4) C E A D B (3) C A E B D (3) E C D A B (2) C B A E D (2) B C A D E (2) A C E D B (2) E D B C A (1) E A D C B (1) D A B E C (1) C E A B D (1) B D A C E (1) B C A E D (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -10 10 4 B -2 0 10 10 14 C 10 -10 0 2 2 D -10 -10 -2 0 8 E -4 -14 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.090909 D: 0.000000 E: 0.000000 Sum of squares = 0.421487603302 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 10 4 B -2 0 10 10 14 C 10 -10 0 2 2 D -10 -10 -2 0 8 E -4 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.090909 D: 0.000000 E: 0.000000 Sum of squares = 0.421487603335 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=24 E=18 D=16 A=11 so A is eliminated. Round 2 votes counts: B=38 C=28 E=18 D=16 so D is eliminated. Round 3 votes counts: B=44 E=28 C=28 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:203 C:202 D:193 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -10 10 4 B -2 0 10 10 14 C 10 -10 0 2 2 D -10 -10 -2 0 8 E -4 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.090909 D: 0.000000 E: 0.000000 Sum of squares = 0.421487603335 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 10 4 B -2 0 10 10 14 C 10 -10 0 2 2 D -10 -10 -2 0 8 E -4 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.090909 D: 0.000000 E: 0.000000 Sum of squares = 0.421487603335 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 10 4 B -2 0 10 10 14 C 10 -10 0 2 2 D -10 -10 -2 0 8 E -4 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.454545 C: 0.090909 D: 0.000000 E: 0.000000 Sum of squares = 0.421487603335 Cumulative probabilities = A: 0.454545 B: 0.909091 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6261: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (14) A E B D C (10) C D A E B (9) D C B A E (7) D C A E B (7) B E A C D (7) E A B C D (6) C D B E A (4) E A B D C (3) D C B E A (3) B C E A D (3) B C D E A (3) E B A D C (2) E A C B D (2) D B C E A (2) B D E C A (2) A E D C B (2) A E C D B (2) E B A C D (1) D B C A E (1) D A E B C (1) D A B E C (1) C D E A B (1) C D B A E (1) B E C A D (1) B D C E A (1) B A E D C (1) A E D B C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 10 14 -10 B 6 0 22 16 2 C -10 -22 0 -16 -16 D -14 -16 16 0 -14 E 10 -2 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 14 -10 B 6 0 22 16 2 C -10 -22 0 -16 -16 D -14 -16 16 0 -14 E 10 -2 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997867 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=22 A=17 C=15 E=14 so E is eliminated. Round 2 votes counts: B=35 A=28 D=22 C=15 so C is eliminated. Round 3 votes counts: D=37 B=35 A=28 so A is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:219 A:204 D:186 C:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 14 -10 B 6 0 22 16 2 C -10 -22 0 -16 -16 D -14 -16 16 0 -14 E 10 -2 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997867 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 14 -10 B 6 0 22 16 2 C -10 -22 0 -16 -16 D -14 -16 16 0 -14 E 10 -2 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997867 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 14 -10 B 6 0 22 16 2 C -10 -22 0 -16 -16 D -14 -16 16 0 -14 E 10 -2 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997867 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6262: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) C D B E A (8) C D B A E (8) E A B D C (7) A D C E B (6) C B D A E (5) E B A D C (4) E B A C D (4) D C E B A (4) B E C D A (4) D C A E B (3) B E A C D (3) A E D C B (3) A C D B E (3) A B E C D (3) E D C B A (2) E D A B C (2) D C B E A (2) B C E D A (2) B A E C D (2) A B C E D (2) A B C D E (2) E D B C A (1) E D A C B (1) E B D C A (1) E A B C D (1) D E C A B (1) D C A B E (1) D A C E B (1) B E C A D (1) B C D E A (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 12 6 2 B 6 0 2 8 -2 C -12 -2 0 2 0 D -6 -8 -2 0 -6 E -2 2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000109 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 -6 12 6 2 B 6 0 2 8 -2 C -12 -2 0 2 0 D -6 -8 -2 0 -6 E -2 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999999 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 C=21 B=14 D=12 so D is eliminated. Round 2 votes counts: C=31 A=31 E=24 B=14 so B is eliminated. Round 3 votes counts: C=34 A=34 E=32 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:207 B:207 E:203 C:194 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 6 2 B 6 0 2 8 -2 C -12 -2 0 2 0 D -6 -8 -2 0 -6 E -2 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999999 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 6 2 B 6 0 2 8 -2 C -12 -2 0 2 0 D -6 -8 -2 0 -6 E -2 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999999 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 6 2 B 6 0 2 8 -2 C -12 -2 0 2 0 D -6 -8 -2 0 -6 E -2 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999999 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6263: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (15) C D A E B (12) E B A D C (8) E B A C D (4) D C A E B (4) D A C E B (4) C D A B E (4) C B E D A (4) B E C A D (4) A D E B C (4) E A B D C (3) C E B A D (3) C D B A E (3) B E A C D (3) D A C B E (2) D A B C E (2) C E D B A (2) B E C D A (2) A E D C B (2) A D B E C (2) E C B A D (1) E B C A D (1) E A D B C (1) C E D A B (1) C E B D A (1) C D E A B (1) C B E A D (1) C B D A E (1) B D A E C (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D B C (1) Total count = 100 A B C D E A 0 -14 6 8 -14 B 14 0 8 10 -6 C -6 -8 0 0 -6 D -8 -10 0 0 -18 E 14 6 6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 6 8 -14 B 14 0 8 10 -6 C -6 -8 0 0 -6 D -8 -10 0 0 -18 E 14 6 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=28 E=18 D=12 A=9 so A is eliminated. Round 2 votes counts: C=33 B=28 E=21 D=18 so D is eliminated. Round 3 votes counts: C=43 B=32 E=25 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:222 B:213 A:193 C:190 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 6 8 -14 B 14 0 8 10 -6 C -6 -8 0 0 -6 D -8 -10 0 0 -18 E 14 6 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 8 -14 B 14 0 8 10 -6 C -6 -8 0 0 -6 D -8 -10 0 0 -18 E 14 6 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 8 -14 B 14 0 8 10 -6 C -6 -8 0 0 -6 D -8 -10 0 0 -18 E 14 6 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6264: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) E B D A C (9) E D C B A (7) A C B D E (7) A C B E D (6) E B A D C (5) D C E B A (4) C D E B A (4) C A D B E (4) A B C D E (4) E D B A C (3) C D B A E (3) C A D E B (3) B A E D C (3) A B D C E (3) E C A D B (2) D E B C A (2) A E B C D (2) A B E D C (2) E C D B A (1) E A B D C (1) E A B C D (1) D E C B A (1) D C B A E (1) D B C A E (1) C E D A B (1) C D B E A (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E C A (1) B D E A C (1) B A D C E (1) A E C B D (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 4 -2 -10 B 18 0 2 0 -16 C -4 -2 0 -14 -8 D 2 0 14 0 -12 E 10 16 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 4 -2 -10 B 18 0 2 0 -16 C -4 -2 0 -14 -8 D 2 0 14 0 -12 E 10 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=27 C=18 D=9 B=7 so B is eliminated. Round 2 votes counts: E=40 A=31 C=18 D=11 so D is eliminated. Round 3 votes counts: E=45 A=31 C=24 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:202 D:202 A:187 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 4 -2 -10 B 18 0 2 0 -16 C -4 -2 0 -14 -8 D 2 0 14 0 -12 E 10 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 -2 -10 B 18 0 2 0 -16 C -4 -2 0 -14 -8 D 2 0 14 0 -12 E 10 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 -2 -10 B 18 0 2 0 -16 C -4 -2 0 -14 -8 D 2 0 14 0 -12 E 10 16 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6265: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (7) B C A E D (6) A C E D B (6) C A B E D (5) B C E D A (5) B E D C A (4) E D C B A (3) E D B C A (3) E C D A B (3) D E B C A (3) D E B A C (3) D E A C B (3) D A E C B (3) B D E C A (3) A D E C B (3) A C D E B (3) E D C A B (2) C E D A B (2) C B A E D (2) B D E A C (2) B D A E C (2) B A D E C (2) A D C E B (2) A C B E D (2) A C B D E (2) A B D C E (2) A B C D E (2) E B D C A (1) D B E A C (1) C E D B A (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E D A (1) C B E A D (1) C A E B D (1) B E C D A (1) B C E A D (1) B A D C E (1) B A C D E (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 4 -16 8 8 B -4 0 -12 -2 -8 C 16 12 0 12 12 D -8 2 -12 0 -22 E -8 8 -12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 8 8 B -4 0 -12 -2 -8 C 16 12 0 12 12 D -8 2 -12 0 -22 E -8 8 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=24 C=23 D=13 E=12 so E is eliminated. Round 2 votes counts: B=29 C=26 A=24 D=21 so D is eliminated. Round 3 votes counts: B=39 C=31 A=30 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:205 A:202 B:187 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 8 8 B -4 0 -12 -2 -8 C 16 12 0 12 12 D -8 2 -12 0 -22 E -8 8 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 8 8 B -4 0 -12 -2 -8 C 16 12 0 12 12 D -8 2 -12 0 -22 E -8 8 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 8 8 B -4 0 -12 -2 -8 C 16 12 0 12 12 D -8 2 -12 0 -22 E -8 8 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6266: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (14) E B A C D (7) A B E C D (7) D C A B E (6) B E A C D (6) A D B E C (6) C D E B A (5) A B E D C (5) E C B D A (4) D C E A B (4) C B E A D (4) E C B A D (3) E B C A D (3) D E C B A (3) C E B D A (3) A D B C E (3) A B D E C (3) D E C A B (1) D E A C B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) C E D B A (1) C D B A E (1) C A D B E (1) B C E A D (1) B C A E D (1) A E B D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -10 6 -20 B 12 0 -8 0 -6 C 10 8 0 -4 -4 D -6 0 4 0 6 E 20 6 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.067177 B: 0.221371 C: 0.100766 D: 0.610685 E: 0.000000 Sum of squares = 0.436608387888 Cumulative probabilities = A: 0.067177 B: 0.288548 C: 0.389315 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 6 -20 B 12 0 -8 0 -6 C 10 8 0 -4 -4 D -6 0 4 0 6 E 20 6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.095238 B: 0.174603 C: 0.142857 D: 0.587302 E: 0.000000 Sum of squares = 0.404887887391 Cumulative probabilities = A: 0.095238 B: 0.269841 C: 0.412698 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=27 E=17 C=15 B=8 so B is eliminated. Round 2 votes counts: D=33 A=27 E=23 C=17 so C is eliminated. Round 3 votes counts: D=39 E=32 A=29 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:212 C:205 D:202 B:199 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 6 -20 B 12 0 -8 0 -6 C 10 8 0 -4 -4 D -6 0 4 0 6 E 20 6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.095238 B: 0.174603 C: 0.142857 D: 0.587302 E: 0.000000 Sum of squares = 0.404887887391 Cumulative probabilities = A: 0.095238 B: 0.269841 C: 0.412698 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 6 -20 B 12 0 -8 0 -6 C 10 8 0 -4 -4 D -6 0 4 0 6 E 20 6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.095238 B: 0.174603 C: 0.142857 D: 0.587302 E: 0.000000 Sum of squares = 0.404887887391 Cumulative probabilities = A: 0.095238 B: 0.269841 C: 0.412698 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 6 -20 B 12 0 -8 0 -6 C 10 8 0 -4 -4 D -6 0 4 0 6 E 20 6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.095238 B: 0.174603 C: 0.142857 D: 0.587302 E: 0.000000 Sum of squares = 0.404887887391 Cumulative probabilities = A: 0.095238 B: 0.269841 C: 0.412698 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6267: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (11) C D E B A (8) C D E A B (7) E D C B A (6) B A E D C (6) D E C A B (5) B A C E D (5) A B D E C (5) A B C D E (5) C E D B A (4) B E D A C (4) A C B D E (4) E D B C A (3) B E D C A (3) A B E D C (3) E D B A C (2) E D A B C (2) D E C B A (2) D E A C B (2) D E A B C (2) C B E D A (2) C A D E B (2) A D E C B (2) A D E B C (2) E C D B A (1) C A B D E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 12 -6 -6 B -8 0 6 -2 -2 C -12 -6 0 2 0 D 6 2 -2 0 -4 E 6 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.149820 D: 0.000000 E: 0.850180 Sum of squares = 0.74525181306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.149820 D: 0.149820 E: 1.000000 A B C D E A 0 8 12 -6 -6 B -8 0 6 -2 -2 C -12 -6 0 2 0 D 6 2 -2 0 -4 E 6 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 B=18 E=14 D=11 so D is eliminated. Round 2 votes counts: A=33 E=25 C=24 B=18 so B is eliminated. Round 3 votes counts: A=44 E=32 C=24 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:206 A:204 D:201 B:197 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 -6 -6 B -8 0 6 -2 -2 C -12 -6 0 2 0 D 6 2 -2 0 -4 E 6 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -6 -6 B -8 0 6 -2 -2 C -12 -6 0 2 0 D 6 2 -2 0 -4 E 6 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -6 -6 B -8 0 6 -2 -2 C -12 -6 0 2 0 D 6 2 -2 0 -4 E 6 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6268: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) B E D A C (7) C A E B D (6) C A D E B (6) A E C B D (6) D B E A C (5) E B A D C (4) D C B A E (4) D B E C A (4) C A D B E (4) A C E D B (4) A C E B D (4) D B C E A (3) C A E D B (3) E B D A C (2) E A B D C (2) E A B C D (2) D C A B E (2) C D A B E (2) B E C A D (2) B D E C A (2) E B A C D (1) E A D B C (1) D E B A C (1) D E A B C (1) D C A E B (1) D A E B C (1) C D A E B (1) C B D A E (1) C B A D E (1) C A B E D (1) B E D C A (1) B E A C D (1) B D C E A (1) B C E D A (1) B C D E A (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 6 0 -2 B 2 0 4 10 4 C -6 -4 0 -4 -6 D 0 -10 4 0 2 E 2 -4 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 0 -2 B 2 0 4 10 4 C -6 -4 0 -4 -6 D 0 -10 4 0 2 E 2 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=25 B=25 D=22 A=16 E=12 so E is eliminated. Round 2 votes counts: B=32 C=25 D=22 A=21 so A is eliminated. Round 3 votes counts: C=40 B=36 D=24 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:201 E:201 D:198 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 0 -2 B 2 0 4 10 4 C -6 -4 0 -4 -6 D 0 -10 4 0 2 E 2 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 -2 B 2 0 4 10 4 C -6 -4 0 -4 -6 D 0 -10 4 0 2 E 2 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 -2 B 2 0 4 10 4 C -6 -4 0 -4 -6 D 0 -10 4 0 2 E 2 -4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6269: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) D E A B C (11) C E B A D (9) E D C A B (8) E C D B A (8) C B A E D (8) A B D C E (6) D A E B C (5) D E A C B (4) C E B D A (4) E C D A B (3) D A B E C (3) B C A E D (3) B A D C E (3) E D C B A (2) E C B D A (2) A D B E C (2) D E B A C (1) C E D B A (1) C B E A D (1) C A E D B (1) C A B D E (1) B C A D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 -4 -8 B 10 0 -4 2 -18 C 4 4 0 10 2 D 4 -2 -10 0 0 E 8 18 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998226 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -4 -8 B 10 0 -4 2 -18 C 4 4 0 10 2 D 4 -2 -10 0 0 E 8 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 E=23 B=19 A=9 so A is eliminated. Round 2 votes counts: D=26 B=26 C=25 E=23 so E is eliminated. Round 3 votes counts: C=38 D=36 B=26 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:210 D:196 B:195 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -4 -8 B 10 0 -4 2 -18 C 4 4 0 10 2 D 4 -2 -10 0 0 E 8 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -4 -8 B 10 0 -4 2 -18 C 4 4 0 10 2 D 4 -2 -10 0 0 E 8 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -4 -8 B 10 0 -4 2 -18 C 4 4 0 10 2 D 4 -2 -10 0 0 E 8 18 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6270: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (16) E A B C D (15) D E C B A (11) A E B C D (9) D C B E A (7) E A D C B (5) D C E B A (5) A B C E D (5) E D A C B (4) E D C B A (3) B C A D E (3) A B C D E (3) E D C A B (2) E A D B C (2) B A C D E (2) A B E C D (2) E D A B C (1) E A B D C (1) D E B C A (1) C B D A E (1) B D C A E (1) B C D A E (1) Total count = 100 A B C D E A 0 -2 -2 -6 -14 B 2 0 -8 -14 -18 C 2 8 0 -18 -12 D 6 14 18 0 2 E 14 18 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 -14 B 2 0 -8 -14 -18 C 2 8 0 -18 -12 D 6 14 18 0 2 E 14 18 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=33 A=19 B=7 C=1 so C is eliminated. Round 2 votes counts: D=40 E=33 A=19 B=8 so B is eliminated. Round 3 votes counts: D=43 E=33 A=24 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:221 D:220 C:190 A:188 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 -14 B 2 0 -8 -14 -18 C 2 8 0 -18 -12 D 6 14 18 0 2 E 14 18 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 -14 B 2 0 -8 -14 -18 C 2 8 0 -18 -12 D 6 14 18 0 2 E 14 18 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 -14 B 2 0 -8 -14 -18 C 2 8 0 -18 -12 D 6 14 18 0 2 E 14 18 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6271: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) B E D A C (7) E B D C A (6) A C D B E (6) D C A B E (5) C A E D B (5) B E A D C (5) D A C B E (4) B D E A C (4) E B A C D (3) C E A B D (3) E C D B A (2) E C B D A (2) E C A B D (2) E B D A C (2) D C E B A (2) D B E C A (2) D B C A E (2) D B A C E (2) C E D A B (2) B E D C A (2) B D E C A (2) B A D C E (2) A D C B E (2) A C E B D (2) A C B E D (2) A B C D E (2) E D C B A (1) E B C A D (1) D E B C A (1) D C E A B (1) D C B E A (1) D B E A C (1) C E A D B (1) C D A E B (1) C D A B E (1) B A E D C (1) B A E C D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -4 -6 -6 B 4 0 -8 -2 8 C 4 8 0 -10 10 D 6 2 10 0 -2 E 6 -8 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000071 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 A B C D E A 0 -4 -4 -6 -6 B 4 0 -8 -2 8 C 4 8 0 -10 10 D 6 2 10 0 -2 E 6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=21 C=20 E=19 A=16 so A is eliminated. Round 2 votes counts: C=32 B=26 D=23 E=19 so E is eliminated. Round 3 votes counts: C=38 B=38 D=24 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:206 B:201 E:195 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -6 B 4 0 -8 -2 8 C 4 8 0 -10 10 D 6 2 10 0 -2 E 6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -6 B 4 0 -8 -2 8 C 4 8 0 -10 10 D 6 2 10 0 -2 E 6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -6 B 4 0 -8 -2 8 C 4 8 0 -10 10 D 6 2 10 0 -2 E 6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6272: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (13) C A E D B (7) C A D B E (6) E C A D B (5) B D A E C (4) E D B A C (3) E C A B D (3) E B D A C (3) C E B A D (3) C E A B D (3) C B E D A (3) B E D A C (3) B D E C A (3) B D C A E (3) A C E D B (3) E D A B C (2) E C B D A (2) E B D C A (2) E A D C B (2) E A D B C (2) D A B E C (2) C B A D E (2) B E D C A (2) E C B A D (1) E A C D B (1) D E A B C (1) D B A E C (1) D A B C E (1) C E B D A (1) C E A D B (1) C B D A E (1) C B A E D (1) C A E B D (1) B C E D A (1) B C D E A (1) A E D B C (1) A E C D B (1) A D E C B (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -4 -4 -22 B 6 0 0 12 2 C 4 0 0 -4 -18 D 4 -12 4 0 -14 E 22 -2 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.942575 C: 0.057425 D: 0.000000 E: 0.000000 Sum of squares = 0.891744741396 Cumulative probabilities = A: 0.000000 B: 0.942575 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -4 -22 B 6 0 0 12 2 C 4 0 0 -4 -18 D 4 -12 4 0 -14 E 22 -2 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.900000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000025252 Cumulative probabilities = A: 0.000000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=29 E=26 A=10 D=5 so D is eliminated. Round 2 votes counts: B=31 C=29 E=27 A=13 so A is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:226 B:210 C:191 D:191 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -4 -22 B 6 0 0 12 2 C 4 0 0 -4 -18 D 4 -12 4 0 -14 E 22 -2 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.900000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000025252 Cumulative probabilities = A: 0.000000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -4 -22 B 6 0 0 12 2 C 4 0 0 -4 -18 D 4 -12 4 0 -14 E 22 -2 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.900000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000025252 Cumulative probabilities = A: 0.000000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -4 -22 B 6 0 0 12 2 C 4 0 0 -4 -18 D 4 -12 4 0 -14 E 22 -2 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.900000 C: 0.100000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000025252 Cumulative probabilities = A: 0.000000 B: 0.900000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6273: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) C E D B A (8) E B C A D (7) C E B D A (7) B E A C D (7) D A C E B (6) B E C A D (6) E C B A D (5) A B E D C (5) D C E A B (4) D A C B E (4) B A E C D (4) D C A E B (3) B C E A D (3) A B D E C (3) E C B D A (2) C D E B A (2) C B E A D (2) B A E D C (2) A D E B C (2) A D B C E (2) D E C A B (1) D A E C B (1) D A B E C (1) C D E A B (1) C B E D A (1) B E A D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -4 18 -14 B 14 0 6 12 2 C 4 -6 0 10 -12 D -18 -12 -10 0 -20 E 14 -2 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 18 -14 B 14 0 6 12 2 C 4 -6 0 10 -12 D -18 -12 -10 0 -20 E 14 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998338 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 A=22 C=21 D=20 E=14 so E is eliminated. Round 2 votes counts: B=30 C=28 A=22 D=20 so D is eliminated. Round 3 votes counts: C=36 A=34 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:222 B:217 C:198 A:193 D:170 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 18 -14 B 14 0 6 12 2 C 4 -6 0 10 -12 D -18 -12 -10 0 -20 E 14 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998338 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 18 -14 B 14 0 6 12 2 C 4 -6 0 10 -12 D -18 -12 -10 0 -20 E 14 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998338 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 18 -14 B 14 0 6 12 2 C 4 -6 0 10 -12 D -18 -12 -10 0 -20 E 14 -2 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998338 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6274: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) D A C B E (6) B E A C D (6) E B C D A (5) D C A E B (5) C D E A B (5) A B D C E (5) E C D A B (4) E B C A D (4) D C E B A (4) B E C A D (4) B A E D C (4) B A D C E (4) E C B D A (3) D C A B E (3) B A D E C (3) A D C B E (3) E C A B D (2) E B A C D (2) D B C E A (2) C E D A B (2) B E C D A (2) B D A C E (2) B A E C D (2) A C E D B (2) A B E C D (2) E C B A D (1) E C A D B (1) E A C B D (1) D C E A B (1) D B A C E (1) B D E A C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -8 -4 -12 B 12 0 -2 6 0 C 8 2 0 10 -6 D 4 -6 -10 0 -6 E 12 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.385787 C: 0.000000 D: 0.000000 E: 0.614213 Sum of squares = 0.526089324991 Cumulative probabilities = A: 0.000000 B: 0.385787 C: 0.385787 D: 0.385787 E: 1.000000 A B C D E A 0 -12 -8 -4 -12 B 12 0 -2 6 0 C 8 2 0 10 -6 D 4 -6 -10 0 -6 E 12 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=28 D=22 A=14 C=7 so C is eliminated. Round 2 votes counts: E=31 B=28 D=27 A=14 so A is eliminated. Round 3 votes counts: B=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:212 B:208 C:207 D:191 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 -4 -12 B 12 0 -2 6 0 C 8 2 0 10 -6 D 4 -6 -10 0 -6 E 12 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -4 -12 B 12 0 -2 6 0 C 8 2 0 10 -6 D 4 -6 -10 0 -6 E 12 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -4 -12 B 12 0 -2 6 0 C 8 2 0 10 -6 D 4 -6 -10 0 -6 E 12 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6275: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (21) C B E D A (11) C B D A E (9) A D B C E (9) A D E B C (7) E C B D A (6) C B D E A (4) B C D A E (4) A D C B E (4) E B C D A (3) C B A D E (3) E C B A D (2) B C D E A (2) A E D B C (2) E D A B C (1) E C A B D (1) E A D C B (1) E A C B D (1) D B C A E (1) D B A C E (1) D A E B C (1) C B E A D (1) C A B D E (1) A E D C B (1) A D E C B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 4 14 -8 B -6 0 4 -2 2 C -4 -4 0 0 4 D -14 2 0 0 -2 E 8 -2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999915 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 6 4 14 -8 B -6 0 4 -2 2 C -4 -4 0 0 4 D -14 2 0 0 -2 E 8 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999828 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=29 A=26 B=6 D=3 so D is eliminated. Round 2 votes counts: E=36 C=29 A=27 B=8 so B is eliminated. Round 3 votes counts: E=36 C=36 A=28 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:208 E:202 B:199 C:198 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 6 4 14 -8 B -6 0 4 -2 2 C -4 -4 0 0 4 D -14 2 0 0 -2 E 8 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999828 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 14 -8 B -6 0 4 -2 2 C -4 -4 0 0 4 D -14 2 0 0 -2 E 8 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999828 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 14 -8 B -6 0 4 -2 2 C -4 -4 0 0 4 D -14 2 0 0 -2 E 8 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999828 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6276: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A C D E B (7) C A D B E (5) A E D C B (5) E A D B C (4) C B D E A (4) B C E D A (4) B C D E A (4) B C A E D (4) A D E C B (4) E D A B C (3) E B D C A (3) E B D A C (3) D E A C B (3) C D B A E (3) A E D B C (3) A C B E D (3) E D B A C (2) C B A D E (2) C A B D E (2) B E C D A (2) A D C E B (2) A C D B E (2) A B E C D (2) E A B D C (1) D E C B A (1) D E C A B (1) D E B C A (1) D E A B C (1) D C E B A (1) D C B E A (1) D C A E B (1) C D B E A (1) C A D E B (1) B E A D C (1) B D E C A (1) B C E A D (1) B A C E D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -2 4 -2 B -4 0 -2 -4 4 C 2 2 0 0 2 D -4 4 0 0 -2 E 2 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.785658 D: 0.214342 E: 0.000000 Sum of squares = 0.663200653488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.785658 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 4 -2 B -4 0 -2 -4 4 C 2 2 0 0 2 D -4 4 0 0 -2 E 2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555566755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=26 C=18 E=16 D=10 so D is eliminated. Round 2 votes counts: A=30 B=26 E=23 C=21 so C is eliminated. Round 3 votes counts: A=39 B=37 E=24 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:203 A:202 D:199 E:199 B:197 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 4 -2 B -4 0 -2 -4 4 C 2 2 0 0 2 D -4 4 0 0 -2 E 2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555566755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 4 -2 B -4 0 -2 -4 4 C 2 2 0 0 2 D -4 4 0 0 -2 E 2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555566755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 4 -2 B -4 0 -2 -4 4 C 2 2 0 0 2 D -4 4 0 0 -2 E 2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555566755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6277: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) D C B E A (6) C D E A B (6) D C E B A (5) C E D A B (5) A B E C D (5) E D A B C (4) D E C B A (4) D E B A C (4) E C A D B (3) E A C B D (3) C E A B D (3) C D E B A (3) C D B E A (3) C B A D E (3) B A E D C (3) A E C B D (3) A B E D C (3) D E C A B (2) D B C E A (2) D B C A E (2) B D A E C (2) B D A C E (2) B A C D E (2) A E B C D (2) E D C A B (1) E C D A B (1) E A C D B (1) D E B C A (1) D E A B C (1) D B A E C (1) B C D A E (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -2 -12 -16 B 10 0 -6 -10 -6 C 2 6 0 -10 -12 D 12 10 10 0 22 E 16 6 12 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -12 -16 B 10 0 -6 -10 -6 C 2 6 0 -10 -12 D 12 10 10 0 22 E 16 6 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 B=21 A=15 E=13 so E is eliminated. Round 2 votes counts: D=33 C=27 B=21 A=19 so A is eliminated. Round 3 votes counts: C=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:206 B:194 C:193 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 -12 -16 B 10 0 -6 -10 -6 C 2 6 0 -10 -12 D 12 10 10 0 22 E 16 6 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -12 -16 B 10 0 -6 -10 -6 C 2 6 0 -10 -12 D 12 10 10 0 22 E 16 6 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -12 -16 B 10 0 -6 -10 -6 C 2 6 0 -10 -12 D 12 10 10 0 22 E 16 6 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6278: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) C A D E B (7) A C E D B (7) E B D A C (6) D B E C A (6) D B C E A (6) E A B C D (5) C A E D B (5) B E D A C (5) B D E C A (5) A E C B D (5) E D B A C (3) E B A D C (3) D C B A E (3) D C A B E (3) D B E A C (3) C D B A E (3) C D A B E (3) D B C A E (2) C A E B D (2) C A D B E (2) B D E A C (2) E B A C D (1) E A D B C (1) E A B D C (1) C A B E D (1) B E A C D (1) B D C E A (1) Total count = 100 A B C D E A 0 0 2 -2 2 B 0 0 2 -8 -8 C -2 -2 0 0 6 D 2 8 0 0 -8 E -2 8 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999828 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 0 2 -2 2 B 0 0 2 -8 -8 C -2 -2 0 0 6 D 2 8 0 0 -8 E -2 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999614 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=23 C=23 E=20 A=20 B=14 so B is eliminated. Round 2 votes counts: D=31 E=26 C=23 A=20 so A is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:204 A:201 C:201 D:201 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 -2 2 B 0 0 2 -8 -8 C -2 -2 0 0 6 D 2 8 0 0 -8 E -2 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999614 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 2 B 0 0 2 -8 -8 C -2 -2 0 0 6 D 2 8 0 0 -8 E -2 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999614 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 2 B 0 0 2 -8 -8 C -2 -2 0 0 6 D 2 8 0 0 -8 E -2 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999614 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6279: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) B E D C A (8) B E D A C (6) B D E C A (5) C A E D B (4) C A D E B (4) B E A C D (4) A C E D B (4) E B A C D (3) D C E A B (3) D B C A E (3) C D A E B (3) B E A D C (3) E D C B A (2) E C A D B (2) E B D C A (2) E B C A D (2) D E C B A (2) D C E B A (2) D C B A E (2) D C A B E (2) B D A C E (2) A C E B D (2) E C D B A (1) E B D A C (1) E A B C D (1) D C B E A (1) D C A E B (1) D B E C A (1) D B C E A (1) C E D A B (1) C E A D B (1) C D A B E (1) B D E A C (1) B A E C D (1) B A D E C (1) A E C B D (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -8 -2 -6 B 8 0 -6 -10 -6 C 8 6 0 -2 6 D 2 10 2 0 0 E 6 6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.868567 E: 0.131433 Sum of squares = 0.771683726874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.868567 E: 1.000000 A B C D E A 0 -8 -8 -2 -6 B 8 0 -6 -10 -6 C 8 6 0 -2 6 D 2 10 2 0 0 E 6 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000008832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=23 D=18 E=14 C=14 so E is eliminated. Round 2 votes counts: B=39 A=24 D=20 C=17 so C is eliminated. Round 3 votes counts: B=39 A=35 D=26 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:209 D:207 E:203 B:193 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -2 -6 B 8 0 -6 -10 -6 C 8 6 0 -2 6 D 2 10 2 0 0 E 6 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000008832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -2 -6 B 8 0 -6 -10 -6 C 8 6 0 -2 6 D 2 10 2 0 0 E 6 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000008832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -2 -6 B 8 0 -6 -10 -6 C 8 6 0 -2 6 D 2 10 2 0 0 E 6 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000008832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6280: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) C A E D B (6) B D E A C (6) A C E B D (5) E B D A C (4) D B C E A (4) C E A D B (4) C A B D E (4) A E C B D (4) A E B D C (4) A B E D C (4) E D B A C (3) C A D B E (3) B D A E C (3) B D A C E (3) A B D E C (3) E A C B D (2) E A B D C (2) D C B E A (2) C E D B A (2) C D B A E (2) C B D A E (2) C A E B D (2) B A D E C (2) A C B D E (2) A B C D E (2) E D C B A (1) E D B C A (1) E C D A B (1) E A D B C (1) E A C D B (1) D E B A C (1) D B E A C (1) C D E B A (1) C D B E A (1) B A D C E (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 12 8 8 B -4 0 8 14 8 C -12 -8 0 -8 -6 D -8 -14 8 0 2 E -8 -8 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 8 8 B -4 0 8 14 8 C -12 -8 0 -8 -6 D -8 -14 8 0 2 E -8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 E=16 D=16 B=15 so B is eliminated. Round 2 votes counts: A=29 D=28 C=27 E=16 so E is eliminated. Round 3 votes counts: D=37 A=35 C=28 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:213 D:194 E:194 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 8 8 B -4 0 8 14 8 C -12 -8 0 -8 -6 D -8 -14 8 0 2 E -8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 8 8 B -4 0 8 14 8 C -12 -8 0 -8 -6 D -8 -14 8 0 2 E -8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 8 8 B -4 0 8 14 8 C -12 -8 0 -8 -6 D -8 -14 8 0 2 E -8 -8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6281: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) C E A D B (6) B D A C E (6) E A C D B (5) E A B D C (5) C D B A E (5) B D C A E (5) C B D E A (4) E A C B D (3) C E B D A (3) C D A E B (3) B A D E C (3) A E D B C (3) E C A D B (2) E B C A D (2) E A B C D (2) D B C A E (2) D B A C E (2) D A C B E (2) C E D A B (2) C E B A D (2) C D E A B (2) C D B E A (2) A D E B C (2) A D B E C (2) A B D E C (2) E A D C B (1) E A D B C (1) D C B A E (1) D C A E B (1) D B A E C (1) D A B E C (1) C B D A E (1) B E C D A (1) B E A D C (1) B E A C D (1) B C E D A (1) B C D E A (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 8 -10 6 B 6 0 10 8 8 C -8 -10 0 -4 2 D 10 -8 4 0 14 E -6 -8 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 -10 6 B 6 0 10 8 8 C -8 -10 0 -4 2 D 10 -8 4 0 14 E -6 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=28 E=21 A=11 D=10 so D is eliminated. Round 2 votes counts: B=33 C=32 E=21 A=14 so A is eliminated. Round 3 votes counts: B=39 C=34 E=27 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:210 A:199 C:190 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 -10 6 B 6 0 10 8 8 C -8 -10 0 -4 2 D 10 -8 4 0 14 E -6 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 -10 6 B 6 0 10 8 8 C -8 -10 0 -4 2 D 10 -8 4 0 14 E -6 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 -10 6 B 6 0 10 8 8 C -8 -10 0 -4 2 D 10 -8 4 0 14 E -6 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6282: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) A D C E B (9) E B D A C (7) D C A E B (6) C A B D E (6) B C E A D (5) E D B C A (4) E B D C A (4) D E A B C (4) C B A D E (4) E D B A C (3) D A C E B (3) C A D B E (3) B C A E D (3) A D E C B (3) A C D B E (3) D E A C B (2) D A E C B (2) C B E A D (2) B E D C A (2) B E A C D (2) A C D E B (2) E D A B C (1) D E C B A (1) D C E A B (1) C D A B E (1) C B E D A (1) C A D E B (1) B E A D C (1) B C E D A (1) B A E C D (1) A E D B C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -8 -4 0 B 0 0 -2 0 -8 C 8 2 0 -8 6 D 4 0 8 0 4 E 0 8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.229271 C: 0.000000 D: 0.770729 E: 0.000000 Sum of squares = 0.646588520755 Cumulative probabilities = A: 0.000000 B: 0.229271 C: 0.229271 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -4 0 B 0 0 -2 0 -8 C 8 2 0 -8 6 D 4 0 8 0 4 E 0 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555594023 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=20 E=19 D=19 C=18 so C is eliminated. Round 2 votes counts: B=31 A=30 D=20 E=19 so E is eliminated. Round 3 votes counts: B=42 A=30 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:208 C:204 E:199 B:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -4 0 B 0 0 -2 0 -8 C 8 2 0 -8 6 D 4 0 8 0 4 E 0 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555594023 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 0 B 0 0 -2 0 -8 C 8 2 0 -8 6 D 4 0 8 0 4 E 0 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555594023 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 0 B 0 0 -2 0 -8 C 8 2 0 -8 6 D 4 0 8 0 4 E 0 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555594023 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6283: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) C B E A D (7) C E B A D (6) A D B E C (6) E C B A D (5) D A E B C (5) D E C A B (4) D C E A B (4) D A B C E (4) C E B D A (3) B C A E D (3) B A C E D (3) A E B D C (3) A B E C D (3) D E A C B (2) D E A B C (2) D A E C B (2) D A C B E (2) C D E B A (2) C B A E D (2) B C E A D (2) A B D E C (2) E D C B A (1) E D C A B (1) E D A B C (1) E C D B A (1) E C A B D (1) E B C A D (1) E B A C D (1) E A D B C (1) E A B D C (1) E A B C D (1) D C E B A (1) D A C E B (1) C E D B A (1) C D B A E (1) C B D A E (1) C B A D E (1) B E C A D (1) B E A C D (1) A D E B C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 2 8 -2 B -12 0 2 0 -4 C -2 -2 0 -4 -8 D -8 0 4 0 0 E 2 4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.134888 E: 0.865112 Sum of squares = 0.76661415818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.134888 E: 1.000000 A B C D E A 0 12 2 8 -2 B -12 0 2 0 -4 C -2 -2 0 -4 -8 D -8 0 4 0 0 E 2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.800000 Sum of squares = 0.680000054812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=24 A=17 E=15 B=10 so B is eliminated. Round 2 votes counts: D=34 C=29 A=20 E=17 so E is eliminated. Round 3 votes counts: C=38 D=37 A=25 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:210 E:207 D:198 B:193 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 8 -2 B -12 0 2 0 -4 C -2 -2 0 -4 -8 D -8 0 4 0 0 E 2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.800000 Sum of squares = 0.680000054812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 8 -2 B -12 0 2 0 -4 C -2 -2 0 -4 -8 D -8 0 4 0 0 E 2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.800000 Sum of squares = 0.680000054812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 8 -2 B -12 0 2 0 -4 C -2 -2 0 -4 -8 D -8 0 4 0 0 E 2 4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.800000 Sum of squares = 0.680000054812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6284: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) A E C D B (6) E A C B D (5) B D C E A (5) D B C A E (4) C D E A B (4) A E B D C (4) A B E D C (4) E C A D B (3) D C B A E (3) D A C B E (3) C D B E A (3) C D B A E (3) B E A D C (3) B D A C E (3) E B C D A (2) E B C A D (2) E B A C D (2) E A C D B (2) D A B C E (2) C D A E B (2) C A D E B (2) B E C D A (2) B D C A E (2) B A D E C (2) A D C E B (2) A C D E B (2) E C A B D (1) E B A D C (1) E A B D C (1) D C A B E (1) D B C E A (1) C E D B A (1) C E A D B (1) C D E B A (1) C B E D A (1) C B D E A (1) B E D C A (1) B D A E C (1) B C E D A (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 10 6 6 -2 B -10 0 2 2 -4 C -6 -2 0 10 2 D -6 -2 -10 0 -2 E 2 4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999999 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 10 6 6 -2 B -10 0 2 2 -4 C -6 -2 0 10 2 D -6 -2 -10 0 -2 E 2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999945 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=21 B=20 C=19 D=14 so D is eliminated. Round 2 votes counts: E=26 A=26 B=25 C=23 so C is eliminated. Round 3 votes counts: B=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:210 E:203 C:202 B:195 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 6 -2 B -10 0 2 2 -4 C -6 -2 0 10 2 D -6 -2 -10 0 -2 E 2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999945 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 6 -2 B -10 0 2 2 -4 C -6 -2 0 10 2 D -6 -2 -10 0 -2 E 2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999945 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 6 -2 B -10 0 2 2 -4 C -6 -2 0 10 2 D -6 -2 -10 0 -2 E 2 4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999945 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6285: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) B E A D C (7) C E A B D (5) D C B E A (4) A E C D B (4) E B A C D (3) D B C E A (3) D B A C E (3) D A C E B (3) C D A E B (3) B C E D A (3) B A E D C (3) A E D C B (3) A E B D C (3) A E B C D (3) A B E D C (3) E A C B D (2) E A B C D (2) D C A E B (2) D C A B E (2) D A B E C (2) C D B E A (2) C B D E A (2) E C B A D (1) E C A B D (1) E B C A D (1) D B C A E (1) D A C B E (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A D B (1) C B E D A (1) C A E D B (1) C A D E B (1) B E C D A (1) B E C A D (1) B E A C D (1) B D E A C (1) B D C E A (1) B D A E C (1) B A D E C (1) A E D B C (1) A E C B D (1) A D E C B (1) A D C E B (1) A D B E C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 6 10 8 B 0 0 -6 2 6 C -6 6 0 -12 2 D -10 -2 12 0 -12 E -8 -6 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.603609 B: 0.396391 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.521469571916 Cumulative probabilities = A: 0.603609 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 10 8 B 0 0 -6 2 6 C -6 6 0 -12 2 D -10 -2 12 0 -12 E -8 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500436 B: 0.499564 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000379604 Cumulative probabilities = A: 0.500436 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=23 B=20 C=19 E=10 so E is eliminated. Round 2 votes counts: D=28 A=27 B=24 C=21 so C is eliminated. Round 3 votes counts: A=36 D=34 B=30 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:201 E:198 C:195 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 10 8 B 0 0 -6 2 6 C -6 6 0 -12 2 D -10 -2 12 0 -12 E -8 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500436 B: 0.499564 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000379604 Cumulative probabilities = A: 0.500436 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 10 8 B 0 0 -6 2 6 C -6 6 0 -12 2 D -10 -2 12 0 -12 E -8 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500436 B: 0.499564 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000379604 Cumulative probabilities = A: 0.500436 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 10 8 B 0 0 -6 2 6 C -6 6 0 -12 2 D -10 -2 12 0 -12 E -8 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500436 B: 0.499564 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000379604 Cumulative probabilities = A: 0.500436 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6286: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (13) C B D E A (9) D B A E C (7) A E C D B (6) E A D C B (4) D E A B C (4) C E A D B (4) C B A E D (4) E A C D B (3) C A E B D (3) B D A E C (3) B C D A E (3) B C A D E (3) A E D B C (3) A C E B D (3) E C A D B (2) D A E B C (2) C B E A D (2) A D E B C (2) E D A C B (1) D E C B A (1) D C E B A (1) D B E C A (1) D B E A C (1) D B C E A (1) D A B E C (1) C E D B A (1) C E B D A (1) C E A B D (1) C B E D A (1) B D A C E (1) B C D E A (1) B A D C E (1) B A C D E (1) A E D C B (1) A E C B D (1) A D B E C (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -4 -6 -4 B 12 0 0 4 10 C 4 0 0 2 12 D 6 -4 -2 0 14 E 4 -10 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.421281 C: 0.578719 D: 0.000000 E: 0.000000 Sum of squares = 0.512393448833 Cumulative probabilities = A: 0.000000 B: 0.421281 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -6 -4 B 12 0 0 4 10 C 4 0 0 2 12 D 6 -4 -2 0 14 E 4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=26 B=26 D=19 A=19 E=10 so E is eliminated. Round 2 votes counts: C=28 B=26 A=26 D=20 so D is eliminated. Round 3 votes counts: B=36 A=34 C=30 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:209 D:207 A:187 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 -4 B 12 0 0 4 10 C 4 0 0 2 12 D 6 -4 -2 0 14 E 4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 -4 B 12 0 0 4 10 C 4 0 0 2 12 D 6 -4 -2 0 14 E 4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 -4 B 12 0 0 4 10 C 4 0 0 2 12 D 6 -4 -2 0 14 E 4 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (12) D C B A E (9) C D B E A (8) B C D E A (8) A E D C B (7) C B D E A (6) E B A C D (4) E A D C B (4) D C A B E (4) B D C A E (4) B C E D A (3) A D E C B (3) E B C A D (2) D C E A B (2) D C B E A (2) D A C B E (2) C D E B A (2) B A E C D (2) A E B D C (2) A E B C D (2) E C A D B (1) E A C D B (1) E A C B D (1) D C E B A (1) D B C A E (1) D A C E B (1) C E B D A (1) C B E D A (1) B E A C D (1) B A D C E (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 -10 -10 -20 B 12 0 -14 0 4 C 10 14 0 10 14 D 10 0 -10 0 12 E 20 -4 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -10 -20 B 12 0 -14 0 4 C 10 14 0 10 14 D 10 0 -10 0 12 E 20 -4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 B=19 C=18 A=16 so A is eliminated. Round 2 votes counts: E=36 D=27 B=19 C=18 so C is eliminated. Round 3 votes counts: E=37 D=37 B=26 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:224 D:206 B:201 E:195 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 -10 -20 B 12 0 -14 0 4 C 10 14 0 10 14 D 10 0 -10 0 12 E 20 -4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -10 -20 B 12 0 -14 0 4 C 10 14 0 10 14 D 10 0 -10 0 12 E 20 -4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -10 -20 B 12 0 -14 0 4 C 10 14 0 10 14 D 10 0 -10 0 12 E 20 -4 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6288: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) D B A C E (7) E C B A D (6) E C A D B (6) E C A B D (6) C E A B D (6) B D A E C (5) B D A C E (5) A C D E B (5) C E A D B (4) B E D C A (4) A C E D B (4) D A C B E (3) C A E B D (3) B D E A C (3) A D C E B (3) A D C B E (3) E B C A D (2) D B A E C (2) D A B C E (2) B D E C A (2) E B C D A (1) E A C D B (1) D E B A C (1) D A E B C (1) D A C E B (1) D A B E C (1) C B E A D (1) B E D A C (1) B D C A E (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 18 0 20 12 B -18 0 -24 -8 -20 C 0 24 0 8 12 D -20 8 -8 0 -8 E -12 20 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.491426 B: 0.000000 C: 0.508574 D: 0.000000 E: 0.000000 Sum of squares = 0.500147019457 Cumulative probabilities = A: 0.491426 B: 0.491426 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 20 12 B -18 0 -24 -8 -20 C 0 24 0 8 12 D -20 8 -8 0 -8 E -12 20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=22 C=22 B=21 D=18 A=17 so A is eliminated. Round 2 votes counts: C=31 D=25 E=23 B=21 so B is eliminated. Round 3 votes counts: D=41 C=31 E=28 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:225 C:222 E:202 D:186 B:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 20 12 B -18 0 -24 -8 -20 C 0 24 0 8 12 D -20 8 -8 0 -8 E -12 20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 20 12 B -18 0 -24 -8 -20 C 0 24 0 8 12 D -20 8 -8 0 -8 E -12 20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 20 12 B -18 0 -24 -8 -20 C 0 24 0 8 12 D -20 8 -8 0 -8 E -12 20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6289: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (9) C A D B E (8) B A D C E (7) E D B A C (5) E C B D A (5) C A B D E (5) E B D A C (4) D B A E C (4) E D C B A (3) E C D A B (3) C E B A D (3) C E A B D (3) B D E A C (3) A D B C E (3) E C B A D (2) D A B C E (2) C E D A B (2) B A C D E (2) A D C B E (2) A C D B E (2) A B D C E (2) E D B C A (1) E C D B A (1) D E A B C (1) D B E A C (1) D A B E C (1) C B A E D (1) C A E D B (1) C A D E B (1) C A B E D (1) B E D A C (1) Total count = 100 A B C D E A 0 -4 -2 8 2 B 4 0 -8 -4 8 C 2 8 0 2 12 D -8 4 -2 0 6 E -2 -8 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 8 2 B 4 0 -8 -4 8 C 2 8 0 2 12 D -8 4 -2 0 6 E -2 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=24 B=22 D=9 A=9 so D is eliminated. Round 2 votes counts: C=36 B=27 E=25 A=12 so A is eliminated. Round 3 votes counts: C=40 B=35 E=25 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:202 B:200 D:200 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 8 2 B 4 0 -8 -4 8 C 2 8 0 2 12 D -8 4 -2 0 6 E -2 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 8 2 B 4 0 -8 -4 8 C 2 8 0 2 12 D -8 4 -2 0 6 E -2 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 8 2 B 4 0 -8 -4 8 C 2 8 0 2 12 D -8 4 -2 0 6 E -2 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6290: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) B E A D C (8) A B E C D (8) D C E B A (7) A C D B E (6) A B E D C (6) C D A E B (5) E B D A C (4) E D B C A (3) E B D C A (3) D E B C A (3) B E A C D (3) A B C E D (3) E D B A C (2) E B C D A (2) D E B A C (2) D C A E B (2) C A D B E (2) B E C D A (2) B A E C D (2) A D C B E (2) A C B E D (2) E B A D C (1) D E C B A (1) D C E A B (1) C D E A B (1) C D B E A (1) C D A B E (1) C B E A D (1) B E C A D (1) B A C E D (1) A E B D C (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 8 0 -12 B 14 0 10 -2 0 C -8 -10 0 6 -4 D 0 2 -6 0 -6 E 12 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.510913 C: 0.000000 D: 0.000000 E: 0.489087 Sum of squares = 0.500238202211 Cumulative probabilities = A: 0.000000 B: 0.510913 C: 0.510913 D: 0.510913 E: 1.000000 A B C D E A 0 -14 8 0 -12 B 14 0 10 -2 0 C -8 -10 0 6 -4 D 0 2 -6 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=21 B=17 D=16 E=15 so E is eliminated. Round 2 votes counts: A=31 B=27 D=21 C=21 so D is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:211 D:195 C:192 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 8 0 -12 B 14 0 10 -2 0 C -8 -10 0 6 -4 D 0 2 -6 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 0 -12 B 14 0 10 -2 0 C -8 -10 0 6 -4 D 0 2 -6 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 0 -12 B 14 0 10 -2 0 C -8 -10 0 6 -4 D 0 2 -6 0 -6 E 12 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6291: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (7) D B A C E (6) B A C D E (5) E D C A B (4) D E C B A (4) E C D A B (3) D C E A B (3) D B C A E (3) C A B D E (3) B D A E C (3) B D A C E (3) B A E D C (3) A C E B D (3) A B E C D (3) A B C D E (3) E D C B A (2) E D B C A (2) E D B A C (2) E C A B D (2) D C E B A (2) D B E A C (2) C E A D B (2) C A E D B (2) C A E B D (2) C A D E B (2) C A B E D (2) A C B E D (2) E B D A C (1) E B A C D (1) E A C B D (1) E A B C D (1) D E C A B (1) D E B A C (1) D C B E A (1) D C B A E (1) D B E C A (1) D B A E C (1) C E D A B (1) C D E A B (1) C D A E B (1) B D E A C (1) B A E C D (1) B A D E C (1) B A C E D (1) A E C B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 10 0 22 B 10 0 6 4 10 C -10 -6 0 -10 16 D 0 -4 10 0 14 E -22 -10 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 0 22 B 10 0 6 4 10 C -10 -6 0 -10 16 D 0 -4 10 0 14 E -22 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 E=19 C=16 A=14 so A is eliminated. Round 2 votes counts: B=32 D=26 C=22 E=20 so E is eliminated. Round 3 votes counts: D=36 B=35 C=29 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:211 D:210 C:195 E:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 0 22 B 10 0 6 4 10 C -10 -6 0 -10 16 D 0 -4 10 0 14 E -22 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 0 22 B 10 0 6 4 10 C -10 -6 0 -10 16 D 0 -4 10 0 14 E -22 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 0 22 B 10 0 6 4 10 C -10 -6 0 -10 16 D 0 -4 10 0 14 E -22 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6292: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B A E D C (8) A B E D C (8) C D E B A (7) A C B E D (7) B E D C A (6) C D E A B (5) B E D A C (5) A C B D E (5) E D B A C (4) D C E B A (4) B D E C A (4) D E C B A (3) A C D E B (3) A B C E D (3) A E D B C (2) E D B C A (1) E A B D C (1) D E C A B (1) D E B C A (1) D E A C B (1) D E A B C (1) C D B E A (1) C D A E B (1) C B D E A (1) C A D B E (1) C A B D E (1) B E C D A (1) B A C E D (1) A E B D C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 4 6 6 B -6 0 -4 6 6 C -4 4 0 -2 2 D -6 -6 2 0 0 E -6 -6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 6 6 B -6 0 -4 6 6 C -4 4 0 -2 2 D -6 -6 2 0 0 E -6 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=27 B=25 D=11 E=6 so E is eliminated. Round 2 votes counts: A=32 C=27 B=25 D=16 so D is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:201 C:200 D:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 6 6 B -6 0 -4 6 6 C -4 4 0 -2 2 D -6 -6 2 0 0 E -6 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 6 6 B -6 0 -4 6 6 C -4 4 0 -2 2 D -6 -6 2 0 0 E -6 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 6 6 B -6 0 -4 6 6 C -4 4 0 -2 2 D -6 -6 2 0 0 E -6 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6293: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) D E A C B (7) B C A E D (7) A C B E D (7) E D A B C (6) D E B A C (6) E A D C B (5) C B A D E (5) E D A C B (4) D E C B A (4) D C B A E (4) C A B D E (3) B C A D E (3) B A C E D (3) A C E B D (3) A B C E D (3) E D B A C (2) D E C A B (2) D B C A E (2) C B A E D (2) C A B E D (2) E B A C D (1) E A D B C (1) D E A B C (1) D C E A B (1) D C A E B (1) D C A B E (1) D B C E A (1) C A D B E (1) B D C A E (1) B C D A E (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 2 -4 2 B 0 0 -8 -16 -8 C -2 8 0 -14 4 D 4 16 14 0 6 E -2 8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -4 2 B 0 0 -8 -16 -8 C -2 8 0 -14 4 D 4 16 14 0 6 E -2 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996197 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=19 B=15 A=15 C=13 so C is eliminated. Round 2 votes counts: D=38 B=22 A=21 E=19 so E is eliminated. Round 3 votes counts: D=50 A=27 B=23 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:200 C:198 E:198 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -4 2 B 0 0 -8 -16 -8 C -2 8 0 -14 4 D 4 16 14 0 6 E -2 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996197 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -4 2 B 0 0 -8 -16 -8 C -2 8 0 -14 4 D 4 16 14 0 6 E -2 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996197 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -4 2 B 0 0 -8 -16 -8 C -2 8 0 -14 4 D 4 16 14 0 6 E -2 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996197 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6294: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (6) C E B A D (5) E C B A D (4) D A B E C (4) C E B D A (4) C B E D A (4) C B D E A (4) B D A E C (4) A D B E C (4) D C A B E (3) D B C A E (3) D A C B E (3) D A B C E (3) C E A D B (3) C D B E A (3) A E D C B (3) E C A B D (2) E A B D C (2) D B A E C (2) D B A C E (2) D A C E B (2) C E D B A (2) C D B A E (2) B E C D A (2) B D C E A (2) B C E D A (2) A E D B C (2) A E B D C (2) A D E B C (2) E C A D B (1) E B C A D (1) E B A D C (1) E A B C D (1) C E D A B (1) C D E B A (1) C D A E B (1) C A E D B (1) B E C A D (1) B D C A E (1) B A E D C (1) A D E C B (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -18 -22 8 B 2 0 -14 -12 14 C 18 14 0 2 18 D 22 12 -2 0 10 E -8 -14 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -18 -22 8 B 2 0 -14 -12 14 C 18 14 0 2 18 D 22 12 -2 0 10 E -8 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=22 A=16 B=13 E=12 so E is eliminated. Round 2 votes counts: C=44 D=22 A=19 B=15 so B is eliminated. Round 3 votes counts: C=50 D=29 A=21 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 D:221 B:195 A:183 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -18 -22 8 B 2 0 -14 -12 14 C 18 14 0 2 18 D 22 12 -2 0 10 E -8 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -18 -22 8 B 2 0 -14 -12 14 C 18 14 0 2 18 D 22 12 -2 0 10 E -8 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -18 -22 8 B 2 0 -14 -12 14 C 18 14 0 2 18 D 22 12 -2 0 10 E -8 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6295: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B A D C (7) A B E C D (7) C D B E A (6) A C D E B (6) C D B A E (5) A E D C B (4) A E B D C (4) E B D C A (3) E B D A C (3) D C E B A (3) C D A B E (3) B E A C D (3) B C D E A (3) A D C E B (3) A C B D E (3) D C B E A (2) D C A E B (2) B E C D A (2) B C D A E (2) A D E C B (2) A B C E D (2) E D C B A (1) E B A C D (1) E A D C B (1) E A D B C (1) E A B C D (1) D E C A B (1) D E B C A (1) D C E A B (1) D C A B E (1) C B D A E (1) C A D B E (1) B E D C A (1) B C E D A (1) B A C D E (1) A E D B C (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 20 16 16 B -8 0 -8 -6 6 C -20 8 0 16 10 D -16 6 -16 0 12 E -16 -6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 20 16 16 B -8 0 -8 -6 6 C -20 8 0 16 10 D -16 6 -16 0 12 E -16 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=18 C=16 B=13 D=11 so D is eliminated. Round 2 votes counts: A=42 C=25 E=20 B=13 so B is eliminated. Round 3 votes counts: A=43 C=31 E=26 so E is eliminated. Round 4 votes counts: A=60 C=40 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:207 D:193 B:192 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 20 16 16 B -8 0 -8 -6 6 C -20 8 0 16 10 D -16 6 -16 0 12 E -16 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 20 16 16 B -8 0 -8 -6 6 C -20 8 0 16 10 D -16 6 -16 0 12 E -16 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 20 16 16 B -8 0 -8 -6 6 C -20 8 0 16 10 D -16 6 -16 0 12 E -16 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6296: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) C A B E D (7) A C B E D (6) A B C E D (6) D E A B C (5) D A C B E (5) A B E C D (4) E D B A C (3) E B D C A (3) E B A C D (3) D E B A C (3) B C E A D (3) B C A E D (3) A C B D E (3) E D B C A (2) E B D A C (2) E B C D A (2) E B C A D (2) E A B C D (2) D E C B A (2) D E B C A (2) D C E B A (2) D A E C B (2) C B E A D (2) E B A D C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C E A B (1) D C B E A (1) D C A E B (1) C D B E A (1) C D A B E (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A C D (1) B A E C D (1) A E B C D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 4 20 6 B -2 0 2 28 16 C -4 -2 0 22 10 D -20 -28 -22 0 -30 E -6 -16 -10 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 20 6 B -2 0 2 28 16 C -4 -2 0 22 10 D -20 -28 -22 0 -30 E -6 -16 -10 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=22 A=22 E=21 B=9 so B is eliminated. Round 2 votes counts: C=28 D=26 E=23 A=23 so E is eliminated. Round 3 votes counts: D=36 C=33 A=31 so A is eliminated. Round 4 votes counts: C=61 D=39 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:222 A:216 C:213 E:199 D:150 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 20 6 B -2 0 2 28 16 C -4 -2 0 22 10 D -20 -28 -22 0 -30 E -6 -16 -10 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 20 6 B -2 0 2 28 16 C -4 -2 0 22 10 D -20 -28 -22 0 -30 E -6 -16 -10 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 20 6 B -2 0 2 28 16 C -4 -2 0 22 10 D -20 -28 -22 0 -30 E -6 -16 -10 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6297: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) B A E D C (9) E B C A D (8) C E D B A (7) C D A E B (7) B E A D C (7) E B A C D (6) D A C B E (5) A D B C E (5) E C D B A (4) E C B D A (4) D C A B E (4) C E D A B (4) A D C B E (3) A B D E C (3) B E A C D (2) E B C D A (1) E B A D C (1) D C E A B (1) D A B C E (1) C E A D B (1) C D A B E (1) C A D B E (1) C A B E D (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -10 0 -14 B -2 0 -10 -12 -10 C 10 10 0 20 8 D 0 12 -20 0 -12 E 14 10 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 0 -14 B -2 0 -10 -12 -10 C 10 10 0 20 8 D 0 12 -20 0 -12 E 14 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=24 B=18 A=14 D=11 so D is eliminated. Round 2 votes counts: C=38 E=24 A=20 B=18 so B is eliminated. Round 3 votes counts: C=38 E=33 A=29 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:214 D:190 A:189 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 0 -14 B -2 0 -10 -12 -10 C 10 10 0 20 8 D 0 12 -20 0 -12 E 14 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 0 -14 B -2 0 -10 -12 -10 C 10 10 0 20 8 D 0 12 -20 0 -12 E 14 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 0 -14 B -2 0 -10 -12 -10 C 10 10 0 20 8 D 0 12 -20 0 -12 E 14 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6298: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (6) B C E D A (6) D A C E B (5) C A E D B (5) B E A C D (5) D B A E C (4) D A E C B (4) D A E B C (4) C D A E B (4) A E C D B (4) C E B A D (3) C D B A E (3) B E D A C (3) B E A D C (3) B C E A D (3) A E D C B (3) E B A D C (2) E A D B C (2) E A B D C (2) E A B C D (2) D C B A E (2) D C A B E (2) C B E A D (2) B D E A C (2) B D C A E (2) A D E C B (2) E C A B D (1) E B C A D (1) E A C B D (1) D B C A E (1) D B A C E (1) D A B E C (1) D A B C E (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B D C E A (1) B C D E A (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 4 6 -10 14 B -4 0 2 -14 -8 C -6 -2 0 -12 0 D 10 14 12 0 -4 E -14 8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.357143 Sum of squares = 0.397959183682 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.642857 E: 1.000000 A B C D E A 0 4 6 -10 14 B -4 0 2 -14 -8 C -6 -2 0 -12 0 D 10 14 12 0 -4 E -14 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.357143 Sum of squares = 0.397959183669 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.642857 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=29 C=18 E=11 A=11 so E is eliminated. Round 2 votes counts: B=32 D=31 C=19 A=18 so A is eliminated. Round 3 votes counts: D=40 B=36 C=24 so C is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:207 E:199 C:190 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -10 14 B -4 0 2 -14 -8 C -6 -2 0 -12 0 D 10 14 12 0 -4 E -14 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.357143 Sum of squares = 0.397959183669 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.642857 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -10 14 B -4 0 2 -14 -8 C -6 -2 0 -12 0 D 10 14 12 0 -4 E -14 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.357143 Sum of squares = 0.397959183669 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.642857 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -10 14 B -4 0 2 -14 -8 C -6 -2 0 -12 0 D 10 14 12 0 -4 E -14 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.357143 Sum of squares = 0.397959183669 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.642857 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6299: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) D E A B C (11) B C D A E (9) E A C D B (6) D B A C E (5) D A E B C (5) C B E A D (5) B C A D E (5) D B E A C (3) C B A E D (3) C B A D E (3) B D C A E (3) B C E D A (3) B C D E A (3) A D E C B (3) E D A B C (2) D B C A E (2) D B A E C (2) C A B E D (2) E D A C B (1) E C B A D (1) E C A B D (1) D E B A C (1) D E A C B (1) D B E C A (1) D A E C B (1) D A B E C (1) B D E C A (1) A E D C B (1) A E C B D (1) A D C E B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 16 -10 -2 B 0 0 14 -18 4 C -16 -14 0 -12 -8 D 10 18 12 0 24 E 2 -4 8 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 -10 -2 B 0 0 14 -18 4 C -16 -14 0 -12 -8 D 10 18 12 0 24 E 2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=24 E=22 C=13 A=8 so A is eliminated. Round 2 votes counts: D=37 E=24 B=24 C=15 so C is eliminated. Round 3 votes counts: D=38 B=37 E=25 so E is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:232 A:202 B:200 E:191 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 16 -10 -2 B 0 0 14 -18 4 C -16 -14 0 -12 -8 D 10 18 12 0 24 E 2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 -10 -2 B 0 0 14 -18 4 C -16 -14 0 -12 -8 D 10 18 12 0 24 E 2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 -10 -2 B 0 0 14 -18 4 C -16 -14 0 -12 -8 D 10 18 12 0 24 E 2 -4 8 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6300: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (13) E B D C A (11) A C E D B (9) B D E C A (8) A C D B E (6) C A D E B (4) B D A C E (4) A E C B D (4) A C D E B (4) E D B C A (3) B D A E C (3) E B D A C (2) E A C B D (2) D C B A E (2) C E A D B (2) C A E D B (2) B E D A C (2) A C B D E (2) A B C E D (2) E D C B A (1) E C D B A (1) E C D A B (1) E C B A D (1) E C A B D (1) E B C D A (1) D E B C A (1) D C B E A (1) D B A C E (1) C D A B E (1) B D E A C (1) B D C A E (1) B A D C E (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 10 -10 10 -4 B -10 0 -18 -6 -28 C 10 18 0 14 -12 D -10 6 -14 0 -18 E 4 28 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -10 10 -4 B -10 0 -18 -6 -28 C 10 18 0 14 -12 D -10 6 -14 0 -18 E 4 28 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=29 B=20 C=9 D=5 so D is eliminated. Round 2 votes counts: E=38 A=29 B=21 C=12 so C is eliminated. Round 3 votes counts: E=40 A=36 B=24 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:231 C:215 A:203 D:182 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -10 10 -4 B -10 0 -18 -6 -28 C 10 18 0 14 -12 D -10 6 -14 0 -18 E 4 28 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 10 -4 B -10 0 -18 -6 -28 C 10 18 0 14 -12 D -10 6 -14 0 -18 E 4 28 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 10 -4 B -10 0 -18 -6 -28 C 10 18 0 14 -12 D -10 6 -14 0 -18 E 4 28 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6301: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) A C B D E (9) D E A B C (8) C B A E D (8) C A B D E (7) E D B C A (6) E B C D A (6) D A E B C (6) A D E B C (6) B C E D A (5) A C D B E (4) E D A B C (3) E B D C A (3) E C B D A (2) C B E D A (2) C B E A D (2) A D C B E (2) E D C B A (1) E D A C B (1) C E B D A (1) C E A D B (1) C B A D E (1) C A E D B (1) B D A E C (1) A D E C B (1) A D B E C (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 -8 0 B -6 0 12 -4 -12 C -8 -12 0 2 -8 D 8 4 -2 0 -2 E 0 12 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.169872 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.830128 Sum of squares = 0.717969061023 Cumulative probabilities = A: 0.169872 B: 0.169872 C: 0.169872 D: 0.169872 E: 1.000000 A B C D E A 0 6 8 -8 0 B -6 0 12 -4 -12 C -8 -12 0 2 -8 D 8 4 -2 0 -2 E 0 12 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000121128 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=26 C=23 D=14 B=6 so B is eliminated. Round 2 votes counts: E=31 C=28 A=26 D=15 so D is eliminated. Round 3 votes counts: E=39 A=33 C=28 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:211 D:204 A:203 B:195 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 -8 0 B -6 0 12 -4 -12 C -8 -12 0 2 -8 D 8 4 -2 0 -2 E 0 12 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000121128 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 0 B -6 0 12 -4 -12 C -8 -12 0 2 -8 D 8 4 -2 0 -2 E 0 12 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000121128 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 0 B -6 0 12 -4 -12 C -8 -12 0 2 -8 D 8 4 -2 0 -2 E 0 12 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000121128 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6302: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) C D B E A (12) A E B D C (9) E A C D B (7) B C D E A (6) E A C B D (5) D C B E A (5) A E B C D (5) A E D B C (4) E C A D B (3) B D A E C (3) E A D C B (2) D B C A E (2) C E D A B (2) C E A D B (2) C D E A B (2) B A E D C (2) A E D C B (2) E D C A B (1) D C E A B (1) D C B A E (1) D C A B E (1) D A E C B (1) C E B A D (1) C D E B A (1) C B D E A (1) B D A C E (1) B C E A D (1) B A E C D (1) B A D E C (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 -4 -10 -6 -4 B 4 0 -2 0 2 C 10 2 0 2 6 D 6 0 -2 0 4 E 4 -2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -4 B 4 0 -2 0 2 C 10 2 0 2 6 D 6 0 -2 0 4 E 4 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=21 A=21 E=18 D=11 so D is eliminated. Round 2 votes counts: B=31 C=29 A=22 E=18 so E is eliminated. Round 3 votes counts: A=36 C=33 B=31 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:204 B:202 E:196 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -4 B 4 0 -2 0 2 C 10 2 0 2 6 D 6 0 -2 0 4 E 4 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 -2 0 2 C 10 2 0 2 6 D 6 0 -2 0 4 E 4 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -4 B 4 0 -2 0 2 C 10 2 0 2 6 D 6 0 -2 0 4 E 4 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6303: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) E B C D A (10) B E A D C (8) B E A C D (8) C E D B A (7) B E C D A (6) B A E D C (6) C D E A B (4) E B C A D (3) D C A E B (3) E C D B A (2) E C B D A (2) D A C E B (2) C D A E B (2) B E D C A (2) B E C A D (2) B A D E C (2) A D C E B (2) A D B C E (2) A B D E C (2) A B D C E (2) E B D C A (1) D E B C A (1) D C E A B (1) D B E C A (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E B A (1) C A D E B (1) B A E C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -26 0 4 -20 B 26 0 16 10 12 C 0 -16 0 2 -14 D -4 -10 -2 0 -18 E 20 -12 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 0 4 -20 B 26 0 16 10 12 C 0 -16 0 2 -14 D -4 -10 -2 0 -18 E 20 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=21 E=18 C=16 D=10 so D is eliminated. Round 2 votes counts: B=36 A=25 C=20 E=19 so E is eliminated. Round 3 votes counts: B=51 A=25 C=24 so C is eliminated. Round 4 votes counts: B=63 A=37 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:232 E:220 C:186 D:183 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 0 4 -20 B 26 0 16 10 12 C 0 -16 0 2 -14 D -4 -10 -2 0 -18 E 20 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 0 4 -20 B 26 0 16 10 12 C 0 -16 0 2 -14 D -4 -10 -2 0 -18 E 20 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 0 4 -20 B 26 0 16 10 12 C 0 -16 0 2 -14 D -4 -10 -2 0 -18 E 20 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6304: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (22) E B A C D (18) D E B A C (12) C A B E D (12) D E B C A (7) E B A D C (4) D C A E B (4) A C B E D (4) D C E B A (3) B E A C D (3) E D B A C (2) E B D A C (2) D E A B C (2) D E A C B (1) D C B E A (1) D A C E B (1) C A D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 0 -14 -10 B 4 0 2 -12 -12 C 0 -2 0 -22 -2 D 14 12 22 0 8 E 10 12 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -14 -10 B 4 0 2 -12 -12 C 0 -2 0 -22 -2 D 14 12 22 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=53 E=26 C=13 A=5 B=3 so B is eliminated. Round 2 votes counts: D=53 E=29 C=13 A=5 so A is eliminated. Round 3 votes counts: D=53 E=29 C=18 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:208 B:191 C:187 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -14 -10 B 4 0 2 -12 -12 C 0 -2 0 -22 -2 D 14 12 22 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -14 -10 B 4 0 2 -12 -12 C 0 -2 0 -22 -2 D 14 12 22 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -14 -10 B 4 0 2 -12 -12 C 0 -2 0 -22 -2 D 14 12 22 0 8 E 10 12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6305: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (12) D B C E A (11) D B E C A (9) C E A D B (9) B D A C E (7) A E C B D (6) A C E B D (6) E C A D B (4) D B E A C (4) E C D A B (3) D B C A E (3) C A E B D (3) C E D A B (2) C D E B A (2) C A E D B (2) B A D E C (2) A C B D E (2) E D C B A (1) E D B C A (1) E C A B D (1) E A C D B (1) E A C B D (1) D E B C A (1) D C B E A (1) C E D B A (1) B D E A C (1) B A D C E (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -8 -18 -6 B 14 0 8 -10 12 C 8 -8 0 -12 2 D 18 10 12 0 14 E 6 -12 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -18 -6 B 14 0 8 -10 12 C 8 -8 0 -12 2 D 18 10 12 0 14 E 6 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=23 C=19 A=17 E=12 so E is eliminated. Round 2 votes counts: D=31 C=27 B=23 A=19 so A is eliminated. Round 3 votes counts: C=44 D=31 B=25 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 B:212 C:195 E:189 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -8 -18 -6 B 14 0 8 -10 12 C 8 -8 0 -12 2 D 18 10 12 0 14 E 6 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -18 -6 B 14 0 8 -10 12 C 8 -8 0 -12 2 D 18 10 12 0 14 E 6 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -18 -6 B 14 0 8 -10 12 C 8 -8 0 -12 2 D 18 10 12 0 14 E 6 -12 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6306: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) B E A D C (7) E C B D A (6) E B D A C (6) B A E D C (6) A C D B E (6) E C D B A (3) E B D C A (3) D B A E C (3) C D A E B (3) A D C B E (3) A C B D E (3) A B D C E (3) E D C B A (2) E C B A D (2) E B C D A (2) E B A D C (2) D E C B A (2) D C A B E (2) D A C B E (2) C E D A B (2) C D E A B (2) C A D E B (2) B E D A C (2) A D B C E (2) E B C A D (1) E B A C D (1) D C A E B (1) D B E A C (1) D A B C E (1) C E D B A (1) C E B A D (1) C A E B D (1) C A B E D (1) B E A C D (1) B D E A C (1) B A D E C (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 8 10 4 B 8 0 -10 6 14 C -8 10 0 -2 -6 D -10 -6 2 0 -4 E -4 -14 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.384615 B: 0.307692 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.337278106526 Cumulative probabilities = A: 0.384615 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 10 4 B 8 0 -10 6 14 C -8 10 0 -2 -6 D -10 -6 2 0 -4 E -4 -14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.307692 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.337278106517 Cumulative probabilities = A: 0.384615 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 A=20 B=18 D=12 so D is eliminated. Round 2 votes counts: E=30 C=25 A=23 B=22 so B is eliminated. Round 3 votes counts: E=42 A=33 C=25 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:209 A:207 C:197 E:196 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -8 8 10 4 B 8 0 -10 6 14 C -8 10 0 -2 -6 D -10 -6 2 0 -4 E -4 -14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.307692 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.337278106517 Cumulative probabilities = A: 0.384615 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 10 4 B 8 0 -10 6 14 C -8 10 0 -2 -6 D -10 -6 2 0 -4 E -4 -14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.307692 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.337278106517 Cumulative probabilities = A: 0.384615 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 10 4 B 8 0 -10 6 14 C -8 10 0 -2 -6 D -10 -6 2 0 -4 E -4 -14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.307692 C: 0.307692 D: 0.000000 E: 0.000000 Sum of squares = 0.337278106517 Cumulative probabilities = A: 0.384615 B: 0.692308 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6307: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) A C D E B (9) E D B A C (8) E B D C A (8) B E D C A (8) C A B D E (6) A C D B E (6) B C A E D (5) B E C A D (4) C A E B D (3) C A D B E (3) C A B E D (3) B C E A D (3) E B D A C (2) E B C A D (2) D A C E B (2) D A C B E (2) C B A E D (2) B E C D A (2) B C D A E (2) B C A D E (2) E C B A D (1) E C A B D (1) D E A C B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A E B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -20 -10 4 0 B 20 0 16 8 0 C 10 -16 0 10 0 D -4 -8 -10 0 -6 E 0 0 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.442576 C: 0.000000 D: 0.000000 E: 0.557424 Sum of squares = 0.506595051637 Cumulative probabilities = A: 0.000000 B: 0.442576 C: 0.442576 D: 0.442576 E: 1.000000 A B C D E A 0 -20 -10 4 0 B 20 0 16 8 0 C 10 -16 0 10 0 D -4 -8 -10 0 -6 E 0 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 D=18 C=17 A=17 so C is eliminated. Round 2 votes counts: A=32 B=28 E=22 D=18 so D is eliminated. Round 3 votes counts: A=38 E=32 B=30 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:222 E:203 C:202 A:187 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -10 4 0 B 20 0 16 8 0 C 10 -16 0 10 0 D -4 -8 -10 0 -6 E 0 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -10 4 0 B 20 0 16 8 0 C 10 -16 0 10 0 D -4 -8 -10 0 -6 E 0 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -10 4 0 B 20 0 16 8 0 C 10 -16 0 10 0 D -4 -8 -10 0 -6 E 0 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6308: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) E B C A D (8) D A B C E (8) A D B E C (8) E B A C D (6) C D E B A (6) E C B A D (4) D C A B E (4) D A C B E (4) A E B D C (4) C E D B A (3) A B E D C (3) A B D E C (3) D C B A E (2) D C A E B (2) C D B E A (2) C B E D A (2) B A D E C (2) A D C E B (2) E B C D A (1) E A B C D (1) D C B E A (1) D A C E B (1) C D E A B (1) C D A E B (1) C B D E A (1) C A D E B (1) B E C A D (1) B E A D C (1) B E A C D (1) B C E D A (1) B A E D C (1) A E D C B (1) A E D B C (1) A E B C D (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 0 2 2 B 4 0 4 0 -10 C 0 -4 0 0 2 D -2 0 0 0 2 E -2 10 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.362345 B: 0.125000 C: 0.262655 D: 0.000000 E: 0.250000 Sum of squares = 0.27840663952 Cumulative probabilities = A: 0.362345 B: 0.487345 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 0 2 2 B 4 0 4 0 -10 C 0 -4 0 0 2 D -2 0 0 0 2 E -2 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.312501 B: 0.125000 C: 0.312499 D: 0.000000 E: 0.250000 Sum of squares = 0.273437499999 Cumulative probabilities = A: 0.312501 B: 0.437501 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=25 D=22 E=20 B=7 so B is eliminated. Round 2 votes counts: A=28 C=27 E=23 D=22 so D is eliminated. Round 3 votes counts: A=41 C=36 E=23 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:202 A:200 D:200 B:199 C:199 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 2 2 B 4 0 4 0 -10 C 0 -4 0 0 2 D -2 0 0 0 2 E -2 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.312501 B: 0.125000 C: 0.312499 D: 0.000000 E: 0.250000 Sum of squares = 0.273437499999 Cumulative probabilities = A: 0.312501 B: 0.437501 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 2 B 4 0 4 0 -10 C 0 -4 0 0 2 D -2 0 0 0 2 E -2 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.312501 B: 0.125000 C: 0.312499 D: 0.000000 E: 0.250000 Sum of squares = 0.273437499999 Cumulative probabilities = A: 0.312501 B: 0.437501 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 2 B 4 0 4 0 -10 C 0 -4 0 0 2 D -2 0 0 0 2 E -2 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.312501 B: 0.125000 C: 0.312499 D: 0.000000 E: 0.250000 Sum of squares = 0.273437499999 Cumulative probabilities = A: 0.312501 B: 0.437501 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6309: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) B A E C D (7) E C D B A (6) D E C B A (6) D C E A B (6) A B C E D (6) B E C D A (5) A C E B D (5) C E A D B (4) B A D E C (4) A D B C E (4) B D E C A (3) A D C E B (3) A C E D B (3) A B D C E (3) E C B D A (2) E C B A D (2) D B A E C (2) D A C E B (2) B E C A D (2) B D A E C (2) A D C B E (2) E D C B A (1) E C D A B (1) D E B C A (1) D C A E B (1) D B E C A (1) D B A C E (1) C E A B D (1) C A E D B (1) B E D C A (1) B E A C D (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -4 2 -2 B -6 0 -12 -8 -8 C 4 12 0 14 6 D -2 8 -14 0 -12 E 2 8 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 2 -2 B -6 0 -12 -8 -8 C 4 12 0 14 6 D -2 8 -14 0 -12 E 2 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=25 D=20 C=14 E=12 so E is eliminated. Round 2 votes counts: A=29 C=25 B=25 D=21 so D is eliminated. Round 3 votes counts: C=39 A=31 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:208 A:201 D:190 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 2 -2 B -6 0 -12 -8 -8 C 4 12 0 14 6 D -2 8 -14 0 -12 E 2 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 2 -2 B -6 0 -12 -8 -8 C 4 12 0 14 6 D -2 8 -14 0 -12 E 2 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 2 -2 B -6 0 -12 -8 -8 C 4 12 0 14 6 D -2 8 -14 0 -12 E 2 8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6310: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) E B A C D (5) D A B C E (5) C D E A B (5) A D B C E (5) C E D A B (4) B A E D C (4) B A D E C (4) E C D B A (3) E C A D B (3) E B C D A (3) C E D B A (3) A E C D B (3) A E B C D (3) A B D E C (3) A B D C E (3) E B C A D (2) D C B E A (2) D B C E A (2) D B A C E (2) C D E B A (2) B E C D A (2) B E C A D (2) B D C E A (2) B A D C E (2) A B E D C (2) E C D A B (1) E C B A D (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A B E (1) D B C A E (1) D A C E B (1) D A C B E (1) B E A D C (1) B D E C A (1) B D E A C (1) A D E B C (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 2 -2 -6 B 6 0 14 -4 -2 C -2 -14 0 2 -4 D 2 4 -2 0 0 E 6 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.325341 E: 0.674659 Sum of squares = 0.561011675041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.325341 E: 1.000000 A B C D E A 0 -6 2 -2 -6 B 6 0 14 -4 -2 C -2 -14 0 2 -4 D 2 4 -2 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 B=19 D=18 C=14 so C is eliminated. Round 2 votes counts: E=31 D=25 A=25 B=19 so B is eliminated. Round 3 votes counts: E=36 A=35 D=29 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:207 E:206 D:202 A:194 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 2 -2 -6 B 6 0 14 -4 -2 C -2 -14 0 2 -4 D 2 4 -2 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 -6 B 6 0 14 -4 -2 C -2 -14 0 2 -4 D 2 4 -2 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 -6 B 6 0 14 -4 -2 C -2 -14 0 2 -4 D 2 4 -2 0 0 E 6 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6311: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (7) A E C B D (7) D C B E A (4) D A C E B (4) B C E D A (4) B C D E A (4) B A E C D (4) A D B C E (4) E C B A D (3) D C E B A (3) D B C E A (3) B E C A D (3) B E A C D (3) A D E C B (3) E C D B A (2) E C D A B (2) E C A B D (2) E A C B D (2) D C E A B (2) D A B C E (2) C B E D A (2) B D C E A (2) B D C A E (2) B A D C E (2) B A C E D (2) A E C D B (2) A E B C D (2) A D C E B (2) A D B E C (2) A B E C D (2) E C B D A (1) D E C A B (1) D B A C E (1) D A C B E (1) C E B D A (1) C D E B A (1) C B D E A (1) B E C D A (1) A E B D C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 12 12 6 B -2 0 -6 6 2 C -12 6 0 2 -6 D -12 -6 -2 0 -8 E -6 -2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 12 6 B -2 0 -6 6 2 C -12 6 0 2 -6 D -12 -6 -2 0 -8 E -6 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=27 D=21 E=12 C=5 so C is eliminated. Round 2 votes counts: A=35 B=30 D=22 E=13 so E is eliminated. Round 3 votes counts: A=39 B=35 D=26 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:203 B:200 C:195 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 12 6 B -2 0 -6 6 2 C -12 6 0 2 -6 D -12 -6 -2 0 -8 E -6 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 12 6 B -2 0 -6 6 2 C -12 6 0 2 -6 D -12 -6 -2 0 -8 E -6 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 12 6 B -2 0 -6 6 2 C -12 6 0 2 -6 D -12 -6 -2 0 -8 E -6 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994724 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6312: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) D E B A C (7) D E A B C (6) D A E C B (6) C B A E D (6) C A B E D (5) A D C E B (5) E B D C A (4) B D E C A (4) A D E C B (4) A C B E D (4) D B E C A (3) B C D A E (3) A C E D B (3) A C E B D (3) E D B C A (2) D E B C A (2) D E A C B (2) C B E A D (2) B E C D A (2) A E C D B (2) A C D E B (2) E D B A C (1) E B C A D (1) D B C E A (1) D A E B C (1) D A C E B (1) C E B A D (1) C B A D E (1) C A E B D (1) B E C A D (1) B D C E A (1) B C E D A (1) B C A E D (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 0 6 2 B 4 0 -2 0 -8 C 0 2 0 -2 4 D -6 0 2 0 4 E -2 8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.304021 B: 0.000000 C: 0.695979 D: 0.000000 E: 0.000000 Sum of squares = 0.576815534182 Cumulative probabilities = A: 0.304021 B: 0.304021 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 6 2 B 4 0 -2 0 -8 C 0 2 0 -2 4 D -6 0 2 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556965 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=26 B=21 C=16 E=8 so E is eliminated. Round 2 votes counts: D=32 B=26 A=26 C=16 so C is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:202 C:202 D:200 E:199 B:197 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 6 2 B 4 0 -2 0 -8 C 0 2 0 -2 4 D -6 0 2 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556965 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 6 2 B 4 0 -2 0 -8 C 0 2 0 -2 4 D -6 0 2 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556965 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 6 2 B 4 0 -2 0 -8 C 0 2 0 -2 4 D -6 0 2 0 4 E -2 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556965 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6313: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) D B A C E (7) C B E A D (5) E C A B D (4) D C E A B (4) D A B E C (4) B A E C D (4) E A C B D (3) D C B E A (3) D B C A E (3) D A B C E (3) A B E D C (3) A B D E C (3) E D A C B (2) E C D A B (2) D E C A B (2) D E A C B (2) D C E B A (2) D A E C B (2) C E D A B (2) B D C A E (2) B A E D C (2) B A D C E (2) A E D B C (2) A E B C D (2) A B E C D (2) E C A D B (1) E B C A D (1) E A D C B (1) E A C D B (1) D C B A E (1) C E A D B (1) C E A B D (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C A D (1) B D A C E (1) B C E A D (1) B C A E D (1) B C A D E (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -4 10 -6 B 2 0 -6 4 6 C 4 6 0 -6 8 D -10 -4 6 0 -8 E 6 -6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.000000 Sum of squares = 0.379999999814 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 10 -6 B 2 0 -6 4 6 C 4 6 0 -6 8 D -10 -4 6 0 -8 E 6 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.000000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 E=15 B=15 A=14 so A is eliminated. Round 2 votes counts: D=35 C=23 B=23 E=19 so E is eliminated. Round 3 votes counts: D=40 C=34 B=26 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:206 B:203 E:200 A:199 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -4 10 -6 B 2 0 -6 4 6 C 4 6 0 -6 8 D -10 -4 6 0 -8 E 6 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.000000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 -6 B 2 0 -6 4 6 C 4 6 0 -6 8 D -10 -4 6 0 -8 E 6 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.000000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 -6 B 2 0 -6 4 6 C 4 6 0 -6 8 D -10 -4 6 0 -8 E 6 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.200000 E: 0.000000 Sum of squares = 0.379999999992 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6314: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) B E C D A (8) E C D A B (6) E B C D A (6) B A E C D (5) A D C E B (5) A D C B E (5) A B D C E (5) E A D C B (4) B C D A E (4) B A E D C (3) B A D C E (3) A E D C B (3) A D E C B (3) E C B D A (2) E B C A D (2) E A B D C (2) C E D B A (2) C D E A B (2) B E C A D (2) B C E D A (2) B C D E A (2) B A C D E (2) A E B D C (2) A B E D C (2) E D C A B (1) E A D B C (1) D E A C B (1) D C A E B (1) D C A B E (1) C D E B A (1) C B D E A (1) A E D B C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -2 4 -6 B 6 0 8 6 -8 C 2 -8 0 10 -26 D -4 -6 -10 0 -24 E 6 8 26 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 4 -6 B 6 0 8 6 -8 C 2 -8 0 10 -26 D -4 -6 -10 0 -24 E 6 8 26 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=31 A=28 C=6 D=3 so D is eliminated. Round 2 votes counts: E=33 B=31 A=28 C=8 so C is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:232 B:206 A:195 C:189 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 4 -6 B 6 0 8 6 -8 C 2 -8 0 10 -26 D -4 -6 -10 0 -24 E 6 8 26 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 4 -6 B 6 0 8 6 -8 C 2 -8 0 10 -26 D -4 -6 -10 0 -24 E 6 8 26 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 4 -6 B 6 0 8 6 -8 C 2 -8 0 10 -26 D -4 -6 -10 0 -24 E 6 8 26 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6315: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) E C A D B (10) B D A C E (9) D A B C E (8) A D E C B (8) E A D C B (7) B C E D A (7) E C B D A (6) C B E D A (6) A D B C E (5) A D E B C (4) A D B E C (4) C E B D A (3) D B A C E (2) C E B A D (2) B C D A E (2) E C A B D (1) E A C D B (1) D A E B C (1) D A B E C (1) C B E A D (1) B D C A E (1) B C D E A (1) Total count = 100 A B C D E A 0 0 0 6 -10 B 0 0 -10 -2 -6 C 0 10 0 0 -6 D -6 2 0 0 -8 E 10 6 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 6 -10 B 0 0 -10 -2 -6 C 0 10 0 0 -6 D -6 2 0 0 -8 E 10 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=21 B=20 D=12 C=12 so D is eliminated. Round 2 votes counts: E=35 A=31 B=22 C=12 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:202 A:198 D:194 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 6 -10 B 0 0 -10 -2 -6 C 0 10 0 0 -6 D -6 2 0 0 -8 E 10 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 6 -10 B 0 0 -10 -2 -6 C 0 10 0 0 -6 D -6 2 0 0 -8 E 10 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 6 -10 B 0 0 -10 -2 -6 C 0 10 0 0 -6 D -6 2 0 0 -8 E 10 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6316: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) C A E D B (6) C E B A D (5) C D A B E (5) B E D C A (5) E C B A D (4) C B E D A (4) B E C D A (4) B C E D A (4) E C A B D (3) E B C A D (3) C E A B D (3) C A D E B (3) C A D B E (3) A C E D B (3) D B C A E (2) D B A E C (2) D B A C E (2) D A B E C (2) B D E A C (2) A D E C B (2) E D B A C (1) E D A B C (1) E B D A C (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D B C (1) E A C D B (1) D C B A E (1) D C A B E (1) D B E A C (1) D A C B E (1) D A B C E (1) C B E A D (1) C B D A E (1) B E D A C (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D E A (1) A E D B C (1) A E C D B (1) A D E B C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -26 6 -2 B 2 0 -16 -4 -4 C 26 16 0 16 18 D -6 4 -16 0 -12 E 2 4 -18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -26 6 -2 B 2 0 -16 -4 -4 C 26 16 0 16 18 D -6 4 -16 0 -12 E 2 4 -18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=20 E=18 A=18 D=13 so D is eliminated. Round 2 votes counts: C=33 B=27 A=22 E=18 so E is eliminated. Round 3 votes counts: C=40 B=35 A=25 so A is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:238 E:200 B:189 A:188 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -26 6 -2 B 2 0 -16 -4 -4 C 26 16 0 16 18 D -6 4 -16 0 -12 E 2 4 -18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -26 6 -2 B 2 0 -16 -4 -4 C 26 16 0 16 18 D -6 4 -16 0 -12 E 2 4 -18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -26 6 -2 B 2 0 -16 -4 -4 C 26 16 0 16 18 D -6 4 -16 0 -12 E 2 4 -18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6317: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) C D E B A (7) B E A D C (7) A B D E C (7) C D E A B (5) B A E D C (5) E D C B A (4) E B D A C (4) E B A D C (4) A B D C E (4) E B C D A (3) D C A B E (3) C D A B E (3) A C D B E (3) A B E C D (3) E C D B A (2) E B D C A (2) D C E A B (2) D B E A C (2) A D C B E (2) A D B C E (2) A B E D C (2) E B A C D (1) D C E B A (1) D C A E B (1) C E D B A (1) C E A D B (1) C E A B D (1) C A E B D (1) C A D B E (1) B D E A C (1) B A E C D (1) A D B E C (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 4 -2 4 B -10 0 2 -2 0 C -4 -2 0 -8 2 D 2 2 8 0 14 E -4 0 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -2 4 B -10 0 2 -2 0 C -4 -2 0 -8 2 D 2 2 8 0 14 E -4 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=27 E=20 B=14 D=9 so D is eliminated. Round 2 votes counts: C=37 A=27 E=20 B=16 so B is eliminated. Round 3 votes counts: C=37 A=33 E=30 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:208 B:195 C:194 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -2 4 B -10 0 2 -2 0 C -4 -2 0 -8 2 D 2 2 8 0 14 E -4 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -2 4 B -10 0 2 -2 0 C -4 -2 0 -8 2 D 2 2 8 0 14 E -4 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -2 4 B -10 0 2 -2 0 C -4 -2 0 -8 2 D 2 2 8 0 14 E -4 0 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6318: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) D E A C B (7) E D C B A (6) B C A E D (6) D E C A B (5) D E A B C (5) C B A D E (5) A B D C E (5) E C D B A (4) E D C A B (3) E C B D A (3) D A E C B (3) C B E A D (3) A D E B C (3) A D B C E (3) A B C E D (3) E D A B C (2) D A E B C (2) C E B D A (2) B C E A D (2) B C A D E (2) A D B E C (2) A B D E C (2) E D B C A (1) E D A C B (1) E B C D A (1) D E C B A (1) D C E A B (1) C E D B A (1) C D A B E (1) C B E D A (1) C B D E A (1) C B D A E (1) C B A E D (1) B A C E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 -2 -4 0 B -14 0 0 -2 -2 C 2 0 0 -4 -4 D 4 2 4 0 16 E 0 2 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -4 0 B -14 0 0 -2 -2 C 2 0 0 -4 -4 D 4 2 4 0 16 E 0 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 E=21 C=16 B=12 so B is eliminated. Round 2 votes counts: A=29 C=26 D=24 E=21 so E is eliminated. Round 3 votes counts: D=37 C=34 A=29 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:204 C:197 E:195 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -4 0 B -14 0 0 -2 -2 C 2 0 0 -4 -4 D 4 2 4 0 16 E 0 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -4 0 B -14 0 0 -2 -2 C 2 0 0 -4 -4 D 4 2 4 0 16 E 0 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -4 0 B -14 0 0 -2 -2 C 2 0 0 -4 -4 D 4 2 4 0 16 E 0 2 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6319: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) E C B D A (9) B A E D C (7) C E D A B (6) C D E A B (6) B A D E C (6) D C A E B (5) E B C A D (4) A D C B E (4) A D B C E (4) A B D E C (4) A B D C E (4) E C D B A (3) E B C D A (3) E C B A D (2) E C A B D (2) D B A C E (2) B E C A D (2) B E A C D (2) B D A E C (2) D A C E B (1) D A C B E (1) D A B C E (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E C A (1) B D A C E (1) A D C E B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -12 -8 -8 B 16 0 -4 8 -8 C 12 4 0 6 -4 D 8 -8 -6 0 -12 E 8 8 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -12 -8 -8 B 16 0 -4 8 -8 C 12 4 0 6 -4 D 8 -8 -6 0 -12 E 8 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 C=23 A=19 D=10 so D is eliminated. Round 2 votes counts: C=28 B=27 E=23 A=22 so A is eliminated. Round 3 votes counts: B=41 C=36 E=23 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:216 C:209 B:206 D:191 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -12 -8 -8 B 16 0 -4 8 -8 C 12 4 0 6 -4 D 8 -8 -6 0 -12 E 8 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -8 -8 B 16 0 -4 8 -8 C 12 4 0 6 -4 D 8 -8 -6 0 -12 E 8 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -8 -8 B 16 0 -4 8 -8 C 12 4 0 6 -4 D 8 -8 -6 0 -12 E 8 8 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6320: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) C B A D E (9) E D A B C (7) C D E B A (7) C B D E A (7) D E A B C (6) A E D B C (6) D E C A B (5) D E C B A (4) C E D A B (3) C A B E D (3) E D A C B (2) E A D B C (2) D C E B A (2) C D B E A (2) B C A E D (2) B C A D E (2) A E B D C (2) E D C A B (1) E C D A B (1) E A D C B (1) D E B C A (1) D E B A C (1) D E A C B (1) D B E C A (1) D B E A C (1) C E A D B (1) C D E A B (1) C B D A E (1) B D E A C (1) B C D E A (1) B A E D C (1) B A C E D (1) B A C D E (1) A C E D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -8 -12 -18 B -10 0 -6 -14 -12 C 8 6 0 -10 -10 D 12 14 10 0 8 E 18 12 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 -12 -18 B -10 0 -6 -14 -12 C 8 6 0 -10 -10 D 12 14 10 0 8 E 18 12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=22 A=21 E=14 B=9 so B is eliminated. Round 2 votes counts: C=39 A=24 D=23 E=14 so E is eliminated. Round 3 votes counts: C=40 D=33 A=27 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:216 C:197 A:186 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -8 -12 -18 B -10 0 -6 -14 -12 C 8 6 0 -10 -10 D 12 14 10 0 8 E 18 12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -12 -18 B -10 0 -6 -14 -12 C 8 6 0 -10 -10 D 12 14 10 0 8 E 18 12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -12 -18 B -10 0 -6 -14 -12 C 8 6 0 -10 -10 D 12 14 10 0 8 E 18 12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6321: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) A D E B C (7) E D C A B (6) C B E A D (5) E D A C B (4) D A E B C (4) A E D C B (4) A D B E C (4) E C D B A (3) D E C B A (3) C B E D A (3) C B A E D (3) B C A D E (3) B A C E D (3) B A C D E (3) A B D C E (3) D E A C B (2) D E A B C (2) D A B E C (2) C E B D A (2) B C D E A (2) A D E C B (2) A B C E D (2) A B C D E (2) E D C B A (1) E C D A B (1) D B E C A (1) D B A E C (1) D A E C B (1) C E B A D (1) C E A B D (1) C B D E A (1) B D C E A (1) B C D A E (1) B A D C E (1) A E C D B (1) A C E B D (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 2 18 22 B -2 0 8 2 8 C -2 -8 0 0 0 D -18 -2 0 0 -6 E -22 -8 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 18 22 B -2 0 8 2 8 C -2 -8 0 0 0 D -18 -2 0 0 -6 E -22 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 D=16 C=16 E=15 so E is eliminated. Round 2 votes counts: A=28 D=27 B=25 C=20 so C is eliminated. Round 3 votes counts: B=40 D=31 A=29 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:222 B:208 C:195 E:188 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 18 22 B -2 0 8 2 8 C -2 -8 0 0 0 D -18 -2 0 0 -6 E -22 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 18 22 B -2 0 8 2 8 C -2 -8 0 0 0 D -18 -2 0 0 -6 E -22 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 18 22 B -2 0 8 2 8 C -2 -8 0 0 0 D -18 -2 0 0 -6 E -22 -8 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6322: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E C D B A (8) E C B D A (8) D A E C B (6) C E B A D (6) A D B C E (6) B C E A D (5) B C A E D (5) B A C D E (5) A D B E C (5) E D A C B (4) D E A C B (4) D A E B C (3) D A B E C (3) C E B D A (3) C B E A D (3) A B D C E (3) E D C A B (2) D A B C E (2) B C A D E (2) B A D C E (2) E C B A D (1) D E C A B (1) C E D B A (1) B C D E A (1) B C D A E (1) Total count = 100 A B C D E A 0 -2 -14 -14 -14 B 2 0 -14 -10 -14 C 14 14 0 18 -10 D 14 10 -18 0 -12 E 14 14 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -14 -14 -14 B 2 0 -14 -10 -14 C 14 14 0 18 -10 D 14 10 -18 0 -12 E 14 14 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=21 D=19 A=14 C=13 so C is eliminated. Round 2 votes counts: E=43 B=24 D=19 A=14 so A is eliminated. Round 3 votes counts: E=43 D=30 B=27 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:218 D:197 B:182 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -14 -14 -14 B 2 0 -14 -10 -14 C 14 14 0 18 -10 D 14 10 -18 0 -12 E 14 14 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -14 -14 B 2 0 -14 -10 -14 C 14 14 0 18 -10 D 14 10 -18 0 -12 E 14 14 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -14 -14 B 2 0 -14 -10 -14 C 14 14 0 18 -10 D 14 10 -18 0 -12 E 14 14 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6323: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) B A E C D (10) D C E A B (7) A B E C D (6) E A C D B (5) B A E D C (5) B A D E C (5) A E B C D (5) C E D A B (3) C D E B A (3) B D A C E (3) A B E D C (3) A B D E C (3) E A C B D (2) D B C E A (2) D B C A E (2) C D E A B (2) B D C E A (2) E D C A B (1) E C D A B (1) E C B A D (1) E C A D B (1) E C A B D (1) E B A C D (1) D E C A B (1) D C B E A (1) D C A E B (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C E B D A (1) B E C A D (1) B A D C E (1) A E D C B (1) A E D B C (1) A E C B D (1) A E B D C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 2 16 12 6 B -2 0 6 4 -10 C -16 -6 0 -10 -20 D -12 -4 10 0 -4 E -6 10 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 12 6 B -2 0 6 4 -10 C -16 -6 0 -10 -20 D -12 -4 10 0 -4 E -6 10 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 A=23 E=13 C=10 so C is eliminated. Round 2 votes counts: D=32 B=27 A=23 E=18 so E is eliminated. Round 3 votes counts: D=38 A=32 B=30 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:214 B:199 D:195 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 12 6 B -2 0 6 4 -10 C -16 -6 0 -10 -20 D -12 -4 10 0 -4 E -6 10 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 12 6 B -2 0 6 4 -10 C -16 -6 0 -10 -20 D -12 -4 10 0 -4 E -6 10 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 12 6 B -2 0 6 4 -10 C -16 -6 0 -10 -20 D -12 -4 10 0 -4 E -6 10 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6324: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (11) A D B C E (10) D B A E C (7) E C B D A (6) B D E A C (6) B D A E C (6) E C B A D (5) C E A D B (5) C A E D B (5) E B D C A (4) D A B C E (3) B E D C A (3) A B D C E (3) E D C B A (2) E C D B A (2) D E B C A (2) D C A E B (2) D B E C A (2) A B C D E (2) D C E B A (1) D A C B E (1) C E D B A (1) C A E B D (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B A E D C (1) B A D E C (1) A D C B E (1) A C E B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 -2 -6 B 4 0 8 8 0 C 10 -8 0 -14 0 D 2 -8 14 0 0 E 6 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.522296 C: 0.000000 D: 0.000000 E: 0.477704 Sum of squares = 0.500994188574 Cumulative probabilities = A: 0.000000 B: 0.522296 C: 0.522296 D: 0.522296 E: 1.000000 A B C D E A 0 -4 -10 -2 -6 B 4 0 8 8 0 C 10 -8 0 -14 0 D 2 -8 14 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=20 E=19 A=19 D=18 so D is eliminated. Round 2 votes counts: B=29 C=27 A=23 E=21 so E is eliminated. Round 3 votes counts: C=42 B=35 A=23 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:204 E:203 C:194 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -2 -6 B 4 0 8 8 0 C 10 -8 0 -14 0 D 2 -8 14 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -2 -6 B 4 0 8 8 0 C 10 -8 0 -14 0 D 2 -8 14 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -2 -6 B 4 0 8 8 0 C 10 -8 0 -14 0 D 2 -8 14 0 0 E 6 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6325: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (15) E A B D C (12) C D B A E (12) B E A C D (8) A E D C B (8) E A B C D (7) D C A B E (5) A D E C B (5) E B A C D (4) D C A E B (4) D C B A E (3) A D C E B (3) E A D C B (2) C B D E A (2) C B D A E (2) B E C D A (2) B C D A E (2) E A D B C (1) B E C A D (1) B C E D A (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 2 4 -10 B 4 0 8 14 6 C -2 -8 0 12 -2 D -4 -14 -12 0 6 E 10 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 4 -10 B 4 0 8 14 6 C -2 -8 0 12 -2 D -4 -14 -12 0 6 E 10 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=26 A=17 C=16 D=12 so D is eliminated. Round 2 votes counts: B=29 C=28 E=26 A=17 so A is eliminated. Round 3 votes counts: E=40 C=31 B=29 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:216 C:200 E:200 A:196 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 4 -10 B 4 0 8 14 6 C -2 -8 0 12 -2 D -4 -14 -12 0 6 E 10 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 4 -10 B 4 0 8 14 6 C -2 -8 0 12 -2 D -4 -14 -12 0 6 E 10 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 4 -10 B 4 0 8 14 6 C -2 -8 0 12 -2 D -4 -14 -12 0 6 E 10 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6326: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) D E A C B (8) C B E D A (8) A D E B C (8) B C A E D (6) B C A D E (5) A B C D E (5) E D C A B (4) E C D B A (4) B C D A E (4) E D C B A (3) D A E B C (3) C B E A D (3) B A C D E (3) E A D C B (2) E A C D B (2) D E C A B (2) D E A B C (2) C E B D A (2) C B D E A (2) A E D C B (2) A D B E C (2) D B C A E (1) D B A C E (1) C A E B D (1) B C E D A (1) B C E A D (1) B A D C E (1) B A C E D (1) A E D B C (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 -8 -6 B -8 0 -4 -8 -6 C -6 4 0 -2 -6 D 8 8 2 0 -2 E 6 6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 -8 -6 B -8 0 -4 -8 -6 C -6 4 0 -2 -6 D 8 8 2 0 -2 E 6 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=22 A=21 D=17 C=16 so C is eliminated. Round 2 votes counts: B=35 E=26 A=22 D=17 so D is eliminated. Round 3 votes counts: E=38 B=37 A=25 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 D:208 A:200 C:195 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 -8 -6 B -8 0 -4 -8 -6 C -6 4 0 -2 -6 D 8 8 2 0 -2 E 6 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -8 -6 B -8 0 -4 -8 -6 C -6 4 0 -2 -6 D 8 8 2 0 -2 E 6 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -8 -6 B -8 0 -4 -8 -6 C -6 4 0 -2 -6 D 8 8 2 0 -2 E 6 6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6327: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (14) E D C B A (9) A B E C D (9) B A D C E (7) E C D A B (6) D E C B A (5) C D E A B (5) A B C E D (5) B D E C A (4) A E C D B (3) A B C D E (3) E D C A B (2) E C A D B (2) C E A D B (2) B D C E A (2) B D C A E (2) B A D E C (2) B A C D E (2) E B A D C (1) E A B D C (1) D C E A B (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C E A (1) C E D A B (1) C D A E B (1) C A D E B (1) B E D C A (1) B A E D C (1) A E C B D (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -24 -14 -18 B 8 0 -14 -14 -16 C 24 14 0 -12 2 D 14 14 12 0 6 E 18 16 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -24 -14 -18 B 8 0 -14 -14 -16 C 24 14 0 -12 2 D 14 14 12 0 6 E 18 16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=21 B=21 C=10 so C is eliminated. Round 2 votes counts: D=30 A=25 E=24 B=21 so B is eliminated. Round 3 votes counts: D=38 A=37 E=25 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:214 E:213 B:182 A:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -24 -14 -18 B 8 0 -14 -14 -16 C 24 14 0 -12 2 D 14 14 12 0 6 E 18 16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -24 -14 -18 B 8 0 -14 -14 -16 C 24 14 0 -12 2 D 14 14 12 0 6 E 18 16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -24 -14 -18 B 8 0 -14 -14 -16 C 24 14 0 -12 2 D 14 14 12 0 6 E 18 16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6328: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (7) D E C A B (6) D E A B C (6) D C E B A (6) D C B A E (6) C B A E D (6) E A D B C (5) A E B C D (5) E A B D C (4) D B C A E (4) E D A B C (3) E C A D B (3) E A B C D (3) D C B E A (3) B C A E D (3) B C A D E (3) B A C E D (3) E D A C B (2) E A C B D (2) D E B C A (2) C E A B D (2) C B A D E (2) B C D A E (2) A C B E D (2) A B C E D (2) E D C A B (1) E A D C B (1) D E A C B (1) D B A E C (1) D B A C E (1) C B E A D (1) B D C A E (1) A B E D C (1) Total count = 100 A B C D E A 0 12 -2 10 -2 B -12 0 12 -2 -4 C 2 -12 0 -8 -6 D -10 2 8 0 -12 E 2 4 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -2 10 -2 B -12 0 12 -2 -4 C 2 -12 0 -8 -6 D -10 2 8 0 -12 E 2 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=24 A=17 B=12 C=11 so C is eliminated. Round 2 votes counts: D=36 E=26 B=21 A=17 so A is eliminated. Round 3 votes counts: D=36 B=33 E=31 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:212 A:209 B:197 D:194 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -2 10 -2 B -12 0 12 -2 -4 C 2 -12 0 -8 -6 D -10 2 8 0 -12 E 2 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 10 -2 B -12 0 12 -2 -4 C 2 -12 0 -8 -6 D -10 2 8 0 -12 E 2 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 10 -2 B -12 0 12 -2 -4 C 2 -12 0 -8 -6 D -10 2 8 0 -12 E 2 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998668 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6329: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) E D B A C (5) C B D E A (5) C A B D E (5) A E B D C (5) D C E B A (4) A D E C B (4) D E A B C (3) D C A E B (3) C D B E A (3) A D C E B (3) A C D E B (3) E D B C A (2) E B D A C (2) D E C B A (2) D C B E A (2) D C A B E (2) D A E C B (2) C D B A E (2) C B A D E (2) C A D B E (2) B E D C A (2) B C A E D (2) B A C E D (2) A E D B C (2) A C D B E (2) A B C E D (2) E D A B C (1) E B A D C (1) E B A C D (1) D E B C A (1) D E B A C (1) D E A C B (1) D C E A B (1) D A C E B (1) C D A B E (1) C B A E D (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C E A (1) B A E C D (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -8 -14 2 B 6 0 -8 -6 2 C 8 8 0 -4 20 D 14 6 4 0 16 E -2 -2 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -14 2 B 6 0 -8 -6 2 C 8 8 0 -4 20 D 14 6 4 0 16 E -2 -2 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 C=21 B=19 E=12 so E is eliminated. Round 2 votes counts: D=31 A=25 B=23 C=21 so C is eliminated. Round 3 votes counts: D=37 A=32 B=31 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:216 B:197 A:187 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -8 -14 2 B 6 0 -8 -6 2 C 8 8 0 -4 20 D 14 6 4 0 16 E -2 -2 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -14 2 B 6 0 -8 -6 2 C 8 8 0 -4 20 D 14 6 4 0 16 E -2 -2 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -14 2 B 6 0 -8 -6 2 C 8 8 0 -4 20 D 14 6 4 0 16 E -2 -2 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999665 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6330: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (14) E D A B C (12) B A C E D (11) A B E C D (9) D E C B A (8) D E C A B (8) E D A C B (7) E A B D C (5) D C E B A (5) B C A D E (5) C D B E A (4) C D B A E (4) C B D A E (2) A E B D C (2) A B C E D (2) E A D B C (1) C D E B A (1) Total count = 100 A B C D E A 0 -8 -2 -2 -2 B 8 0 -6 0 2 C 2 6 0 4 -4 D 2 0 -4 0 2 E 2 -2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.35999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 -8 -2 -2 -2 B 8 0 -6 0 2 C 2 6 0 4 -4 D 2 0 -4 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=25 C=25 D=21 B=16 A=13 so A is eliminated. Round 2 votes counts: E=27 B=27 C=25 D=21 so D is eliminated. Round 3 votes counts: E=43 C=30 B=27 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:204 B:202 E:201 D:200 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 -2 -2 B 8 0 -6 0 2 C 2 6 0 4 -4 D 2 0 -4 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -2 -2 B 8 0 -6 0 2 C 2 6 0 4 -4 D 2 0 -4 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -2 -2 B 8 0 -6 0 2 C 2 6 0 4 -4 D 2 0 -4 0 2 E 2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6331: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (8) B A D C E (8) E C D A B (5) D C E B A (5) A E C B D (5) A B D E C (5) D B C E A (4) C E D B A (4) A E B C D (4) A B E C D (4) E C A B D (3) E A C B D (3) C D E B A (3) B D C A E (3) A E D C B (3) A D B E C (3) E C A D B (2) D C B E A (2) B C E A D (2) B C A E D (2) B A C D E (2) A B E D C (2) E C B A D (1) E A C D B (1) D E C A B (1) D C E A B (1) D B C A E (1) D A E C B (1) D A B E C (1) D A B C E (1) C E B D A (1) C D B E A (1) C B E A D (1) B D C E A (1) B C E D A (1) B C D E A (1) B A E C D (1) B A C E D (1) A E C D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 10 10 14 B 6 0 12 20 12 C -10 -12 0 -2 8 D -10 -20 2 0 6 E -14 -12 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 10 14 B 6 0 12 20 12 C -10 -12 0 -2 8 D -10 -20 2 0 6 E -14 -12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=28 D=17 E=15 C=10 so C is eliminated. Round 2 votes counts: B=31 A=28 D=21 E=20 so E is eliminated. Round 3 votes counts: A=37 B=33 D=30 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:214 C:192 D:189 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 10 14 B 6 0 12 20 12 C -10 -12 0 -2 8 D -10 -20 2 0 6 E -14 -12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 10 14 B 6 0 12 20 12 C -10 -12 0 -2 8 D -10 -20 2 0 6 E -14 -12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 10 14 B 6 0 12 20 12 C -10 -12 0 -2 8 D -10 -20 2 0 6 E -14 -12 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6332: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) E D A B C (6) C A D B E (5) B E C D A (5) B C E A D (5) E B C A D (4) C B A E D (4) C B A D E (4) B C D E A (4) E A D B C (3) D E A B C (3) D A E B C (3) C A B E D (3) C A B D E (3) B C E D A (3) A D E C B (3) A C E D B (3) A C D B E (3) E B D A C (2) E B A C D (2) D B E A C (2) C A E B D (2) B E D C A (2) B C D A E (2) A D C E B (2) A C D E B (2) E D B A C (1) E A D C B (1) E A C B D (1) E A B C D (1) D C B A E (1) D A C E B (1) D A C B E (1) C E B A D (1) C B E A D (1) B D C A E (1) A E D C B (1) A D C B E (1) Total count = 100 A B C D E A 0 12 0 10 6 B -12 0 -2 0 0 C 0 2 0 16 4 D -10 0 -16 0 -2 E -6 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.699712 B: 0.000000 C: 0.300288 D: 0.000000 E: 0.000000 Sum of squares = 0.57976997656 Cumulative probabilities = A: 0.699712 B: 0.699712 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 10 6 B -12 0 -2 0 0 C 0 2 0 16 4 D -10 0 -16 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 B=22 E=21 D=19 A=15 so A is eliminated. Round 2 votes counts: C=31 D=25 E=22 B=22 so E is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:214 C:211 E:196 B:193 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 10 6 B -12 0 -2 0 0 C 0 2 0 16 4 D -10 0 -16 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 10 6 B -12 0 -2 0 0 C 0 2 0 16 4 D -10 0 -16 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 10 6 B -12 0 -2 0 0 C 0 2 0 16 4 D -10 0 -16 0 -2 E -6 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6333: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E B A C (6) D C A E B (5) C A B D E (5) D E B C A (4) C A B E D (4) B E A C D (4) E D A C B (3) E B D A C (3) E B A C D (3) D A C E B (3) B E D C A (3) B E C A D (3) A C E B D (3) E D B A C (2) E B A D C (2) E A C D B (2) E A B C D (2) D C B A E (2) D B E C A (2) C B A E D (2) B D C E A (2) B C A D E (2) A C E D B (2) E A D C B (1) E A D B C (1) E A C B D (1) E A B D C (1) D E C A B (1) D E A C B (1) D E A B C (1) D C A B E (1) D B C A E (1) C D A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D E C A (1) B D C A E (1) B C E A D (1) B A C E D (1) A E C B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 -8 12 -2 B 14 0 14 16 0 C 8 -14 0 4 2 D -12 -16 -4 0 -14 E 2 0 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.445319 C: 0.000000 D: 0.000000 E: 0.554681 Sum of squares = 0.505980046586 Cumulative probabilities = A: 0.000000 B: 0.445319 C: 0.445319 D: 0.445319 E: 1.000000 A B C D E A 0 -14 -8 12 -2 B 14 0 14 16 0 C 8 -14 0 4 2 D -12 -16 -4 0 -14 E 2 0 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 E=21 C=15 A=8 so A is eliminated. Round 2 votes counts: B=29 D=27 E=22 C=22 so E is eliminated. Round 3 votes counts: B=40 D=34 C=26 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:207 C:200 A:194 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 12 -2 B 14 0 14 16 0 C 8 -14 0 4 2 D -12 -16 -4 0 -14 E 2 0 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 12 -2 B 14 0 14 16 0 C 8 -14 0 4 2 D -12 -16 -4 0 -14 E 2 0 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 12 -2 B 14 0 14 16 0 C 8 -14 0 4 2 D -12 -16 -4 0 -14 E 2 0 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6334: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (15) B E D A C (10) B D E A C (8) A D B C E (8) E C B D A (7) E B D C A (7) C E A D B (6) E C A B D (3) C A E D B (3) B D A E C (3) E B C D A (2) E B A C D (2) C E A B D (2) C D A B E (2) C A D E B (2) C A D B E (2) B A D E C (2) A B D E C (2) E C B A D (1) E B D A C (1) E B C A D (1) D B A E C (1) D A C B E (1) C E D B A (1) C D E A B (1) C D B E A (1) B E D C A (1) A D C B E (1) A C E B D (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 16 8 -8 B -4 0 0 12 20 C -16 0 0 6 -4 D -8 -12 -6 0 2 E 8 -20 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.468750000055 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 4 16 8 -8 B -4 0 0 12 20 C -16 0 0 6 -4 D -8 -12 -6 0 2 E 8 -20 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999983 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=24 B=24 C=20 D=2 so D is eliminated. Round 2 votes counts: A=31 B=25 E=24 C=20 so C is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:214 A:210 E:195 C:193 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 16 8 -8 B -4 0 0 12 20 C -16 0 0 6 -4 D -8 -12 -6 0 2 E 8 -20 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999983 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 8 -8 B -4 0 0 12 20 C -16 0 0 6 -4 D -8 -12 -6 0 2 E 8 -20 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999983 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 8 -8 B -4 0 0 12 20 C -16 0 0 6 -4 D -8 -12 -6 0 2 E 8 -20 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.468749999983 Cumulative probabilities = A: 0.625000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6335: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (20) B C D A E (14) D C B E A (10) A E B C D (10) D E A C B (5) B C A E D (5) C B D E A (4) B C D E A (4) B A E C D (4) D C B A E (3) E D A C B (2) D E C A B (2) D C E A B (2) C D B E A (2) C B D A E (2) A E D C B (2) A B E C D (2) E A D B C (1) D A C B E (1) C D B A E (1) B E A C D (1) B A C E D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 2 -4 -6 B 2 0 -12 -4 8 C -2 12 0 0 -2 D 4 4 0 0 0 E 6 -8 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.822611 E: 0.177389 Sum of squares = 0.708155377913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.822611 E: 1.000000 A B C D E A 0 -2 2 -4 -6 B 2 0 -12 -4 8 C -2 12 0 0 -2 D 4 4 0 0 0 E 6 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555600302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 D=23 A=16 C=9 so C is eliminated. Round 2 votes counts: B=35 D=26 E=23 A=16 so A is eliminated. Round 3 votes counts: E=37 B=37 D=26 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:204 D:204 E:200 B:197 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -4 -6 B 2 0 -12 -4 8 C -2 12 0 0 -2 D 4 4 0 0 0 E 6 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555600302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 -6 B 2 0 -12 -4 8 C -2 12 0 0 -2 D 4 4 0 0 0 E 6 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555600302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 -6 B 2 0 -12 -4 8 C -2 12 0 0 -2 D 4 4 0 0 0 E 6 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555600302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6336: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) C E A D B (6) E A D C B (5) C A B E D (5) B D C E A (5) C D B E A (4) C B D E A (4) A E C D B (4) A E C B D (4) E D A B C (3) E A D B C (3) D B E C A (3) C A E D B (3) C A E B D (3) B D E A C (3) B D C A E (3) B A C D E (3) E D A C B (2) E A C D B (2) D E A B C (2) D B C E A (2) C B D A E (2) C B A E D (2) C B A D E (2) B C D A E (2) B C A D E (2) B A E D C (2) B A D E C (2) A B E D C (2) D E B A C (1) D C B E A (1) C E D A B (1) A E D C B (1) A E D B C (1) A E B D C (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 0 12 -6 B -2 0 -6 -2 10 C 0 6 0 4 4 D -12 2 -4 0 -6 E 6 -10 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.168033 B: 0.000000 C: 0.831967 D: 0.000000 E: 0.000000 Sum of squares = 0.720404642804 Cumulative probabilities = A: 0.168033 B: 0.168033 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 12 -6 B -2 0 -6 -2 10 C 0 6 0 4 4 D -12 2 -4 0 -6 E 6 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.000000 Sum of squares = 0.52000033765 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=22 A=16 E=15 D=15 so E is eliminated. Round 2 votes counts: C=32 A=26 B=22 D=20 so D is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:204 B:200 E:199 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 12 -6 B -2 0 -6 -2 10 C 0 6 0 4 4 D -12 2 -4 0 -6 E 6 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.000000 Sum of squares = 0.52000033765 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 12 -6 B -2 0 -6 -2 10 C 0 6 0 4 4 D -12 2 -4 0 -6 E 6 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.000000 Sum of squares = 0.52000033765 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 12 -6 B -2 0 -6 -2 10 C 0 6 0 4 4 D -12 2 -4 0 -6 E 6 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.399999 B: 0.000000 C: 0.600001 D: 0.000000 E: 0.000000 Sum of squares = 0.52000033765 Cumulative probabilities = A: 0.399999 B: 0.399999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6337: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) B A D C E (7) D C E B A (6) B A E C D (6) B A C D E (6) D C B E A (5) C D E A B (5) A B E C D (5) E A B D C (4) E C D A B (3) E A C D B (3) C E A D B (3) B D C E A (3) B D C A E (3) B A E D C (3) B A C E D (3) A B C E D (3) E D C A B (2) E D B C A (2) D E C B A (2) D B C E A (2) C E D A B (2) B E D A C (2) B C D A E (2) A E B C D (2) E C A D B (1) E A D C B (1) D E C A B (1) C D A E B (1) C B D A E (1) B D E C A (1) A E C D B (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -4 0 -10 B 8 0 8 6 6 C 4 -8 0 -2 20 D 0 -6 2 0 6 E 10 -6 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999387 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 0 -10 B 8 0 8 6 6 C 4 -8 0 -2 20 D 0 -6 2 0 6 E 10 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=23 E=16 A=13 C=12 so C is eliminated. Round 2 votes counts: B=37 D=29 E=21 A=13 so A is eliminated. Round 3 votes counts: B=46 D=29 E=25 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:207 D:201 A:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 0 -10 B 8 0 8 6 6 C 4 -8 0 -2 20 D 0 -6 2 0 6 E 10 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 0 -10 B 8 0 8 6 6 C 4 -8 0 -2 20 D 0 -6 2 0 6 E 10 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 0 -10 B 8 0 8 6 6 C 4 -8 0 -2 20 D 0 -6 2 0 6 E 10 -6 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6338: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (14) A B C E D (12) B A D E C (9) E C D A B (8) C E D A B (8) B A D C E (7) D B A E C (6) C E A B D (6) E D C A B (4) B D A E C (4) A C B E D (4) D E B A C (3) B A C D E (3) A B C D E (3) C E A D B (2) E C D B A (1) D E B C A (1) D B E A C (1) C E D B A (1) C A B E D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 8 -2 2 B 0 0 2 2 4 C -8 -2 0 -2 -4 D 2 -2 2 0 6 E -2 -4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.235007 B: 0.764993 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.6404425718 Cumulative probabilities = A: 0.235007 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -2 2 B 0 0 2 2 4 C -8 -2 0 -2 -4 D 2 -2 2 0 6 E -2 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499879 B: 0.500121 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029402 Cumulative probabilities = A: 0.499879 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=23 A=21 C=18 E=13 so E is eliminated. Round 2 votes counts: D=29 C=27 B=23 A=21 so A is eliminated. Round 3 votes counts: B=40 C=31 D=29 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:204 B:204 D:204 E:196 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 8 -2 2 B 0 0 2 2 4 C -8 -2 0 -2 -4 D 2 -2 2 0 6 E -2 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499879 B: 0.500121 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029402 Cumulative probabilities = A: 0.499879 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -2 2 B 0 0 2 2 4 C -8 -2 0 -2 -4 D 2 -2 2 0 6 E -2 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499879 B: 0.500121 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029402 Cumulative probabilities = A: 0.499879 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -2 2 B 0 0 2 2 4 C -8 -2 0 -2 -4 D 2 -2 2 0 6 E -2 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499879 B: 0.500121 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000029402 Cumulative probabilities = A: 0.499879 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6339: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (15) B A D E C (11) C E B A D (10) B A D C E (10) E C B D A (7) D A C E B (7) D A B C E (6) B D A E C (5) B E C A D (4) D A C B E (3) C E D A B (3) E C B A D (2) D A E C B (2) C B E A D (2) B C E A D (2) D E A C B (1) D A B E C (1) C E A D B (1) C E A B D (1) C A D E B (1) C A B D E (1) B E A C D (1) B A C D E (1) A D C B E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 2 0 2 B 10 0 -16 16 0 C -2 16 0 4 2 D 0 -16 -4 0 4 E -2 0 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.071429 C: 0.357143 D: 0.000000 E: 0.000000 Sum of squares = 0.459183673452 Cumulative probabilities = A: 0.571429 B: 0.642857 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 0 2 B 10 0 -16 16 0 C -2 16 0 4 2 D 0 -16 -4 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.071429 C: 0.357143 D: 0.000000 E: 0.000000 Sum of squares = 0.459183673459 Cumulative probabilities = A: 0.571429 B: 0.642857 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=24 D=20 C=19 A=3 so A is eliminated. Round 2 votes counts: B=34 E=24 D=22 C=20 so C is eliminated. Round 3 votes counts: E=39 B=38 D=23 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:210 B:205 A:197 E:196 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 2 0 2 B 10 0 -16 16 0 C -2 16 0 4 2 D 0 -16 -4 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.071429 C: 0.357143 D: 0.000000 E: 0.000000 Sum of squares = 0.459183673459 Cumulative probabilities = A: 0.571429 B: 0.642857 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 0 2 B 10 0 -16 16 0 C -2 16 0 4 2 D 0 -16 -4 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.071429 C: 0.357143 D: 0.000000 E: 0.000000 Sum of squares = 0.459183673459 Cumulative probabilities = A: 0.571429 B: 0.642857 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 0 2 B 10 0 -16 16 0 C -2 16 0 4 2 D 0 -16 -4 0 4 E -2 0 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.071429 C: 0.357143 D: 0.000000 E: 0.000000 Sum of squares = 0.459183673459 Cumulative probabilities = A: 0.571429 B: 0.642857 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6340: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D C B A (7) D E C A B (7) D E A C B (6) C D E A B (6) B A C E D (6) B C A E D (4) B A E C D (4) A C D E B (4) E D B C A (3) E D B A C (3) C E D B A (3) C B A E D (3) B E D A C (3) A D C E B (3) A B D E C (3) E B D C A (2) D C E A B (2) C D A E B (2) B C E D A (2) A D E B C (2) E D A B C (1) D E A B C (1) C E B D A (1) C D E B A (1) C B E D A (1) C B A D E (1) C A D E B (1) C A B D E (1) B E C D A (1) B E A D C (1) B A E D C (1) A D B E C (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 4 -4 -2 B -6 0 -2 -8 -10 C -4 2 0 8 8 D 4 8 -8 0 6 E 2 10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999925 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 -2 B -6 0 -2 -8 -10 C -4 2 0 8 8 D 4 8 -8 0 6 E 2 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=22 C=20 E=16 D=16 so E is eliminated. Round 2 votes counts: D=30 A=26 B=24 C=20 so C is eliminated. Round 3 votes counts: D=42 B=30 A=28 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:205 A:202 E:199 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 -4 -2 B -6 0 -2 -8 -10 C -4 2 0 8 8 D 4 8 -8 0 6 E 2 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 -2 B -6 0 -2 -8 -10 C -4 2 0 8 8 D 4 8 -8 0 6 E 2 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 -2 B -6 0 -2 -8 -10 C -4 2 0 8 8 D 4 8 -8 0 6 E 2 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6341: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) B D A E C (8) A E D B C (7) A B D E C (7) B C D E A (6) B D C E A (5) B A D E C (5) A E C D B (5) E A D C B (4) C E D B A (4) B D C A E (4) A E D C B (4) C E B D A (3) C E A D B (3) E A C D B (2) D B E C A (2) C E D A B (2) C D B E A (2) C B E D A (2) B D A C E (2) A D E B C (2) A B E D C (2) E C D A B (1) E C A D B (1) C E B A D (1) C E A B D (1) C D E B A (1) C B E A D (1) C B A E D (1) C A E B D (1) A E C B D (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 2 0 2 B 10 0 4 18 12 C -2 -4 0 -8 -6 D 0 -18 8 0 6 E -2 -12 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 0 2 B 10 0 4 18 12 C -2 -4 0 -8 -6 D 0 -18 8 0 6 E -2 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=30 B=30 A=30 E=8 D=2 so D is eliminated. Round 2 votes counts: B=32 C=30 A=30 E=8 so E is eliminated. Round 3 votes counts: A=36 C=32 B=32 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:198 A:197 E:193 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 0 2 B 10 0 4 18 12 C -2 -4 0 -8 -6 D 0 -18 8 0 6 E -2 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 0 2 B 10 0 4 18 12 C -2 -4 0 -8 -6 D 0 -18 8 0 6 E -2 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 0 2 B 10 0 4 18 12 C -2 -4 0 -8 -6 D 0 -18 8 0 6 E -2 -12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6342: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) C E D B A (9) A B D E C (7) B A D C E (6) D B A E C (5) C E A B D (5) B D A C E (4) E C A D B (3) E A D C B (3) E A C D B (3) D E C B A (3) C E B D A (3) C B A E D (3) B C D A E (3) E D A C B (2) D A B E C (2) C B E A D (2) C B D E A (2) B D C A E (2) B A D E C (2) A E B D C (2) A B D C E (2) E D A B C (1) E C A B D (1) E A D B C (1) D E B A C (1) D E A B C (1) D C B E A (1) D B C E A (1) C E A D B (1) C D B E A (1) C B A D E (1) B C A D E (1) B A C D E (1) A D E B C (1) A C E B D (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -2 -8 B 2 0 -8 2 -2 C 4 8 0 4 0 D 2 -2 -4 0 -4 E 8 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.513765 D: 0.000000 E: 0.486235 Sum of squares = 0.500378977291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.513765 D: 0.513765 E: 1.000000 A B C D E A 0 -2 -4 -2 -8 B 2 0 -8 2 -2 C 4 8 0 4 0 D 2 -2 -4 0 -4 E 8 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=24 B=19 A=16 D=14 so D is eliminated. Round 2 votes counts: E=29 C=28 B=25 A=18 so A is eliminated. Round 3 votes counts: B=39 E=32 C=29 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:208 E:207 B:197 D:196 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -2 -8 B 2 0 -8 2 -2 C 4 8 0 4 0 D 2 -2 -4 0 -4 E 8 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -2 -8 B 2 0 -8 2 -2 C 4 8 0 4 0 D 2 -2 -4 0 -4 E 8 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -2 -8 B 2 0 -8 2 -2 C 4 8 0 4 0 D 2 -2 -4 0 -4 E 8 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6343: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) A B E D C (7) C D E B A (6) C D E A B (6) C B A E D (6) E D B A C (5) D E C B A (5) C B E D A (5) C A B D E (5) A C B E D (5) E D B C A (4) C A B E D (4) D E A B C (3) C A D E B (3) C A D B E (3) A C D E B (3) D E B C A (2) D E B A C (2) B E D C A (2) A D E B C (2) D A E C B (1) C E D B A (1) C D B E A (1) C D A E B (1) C B E A D (1) B E D A C (1) B E A D C (1) B A C E D (1) A E D B C (1) A D E C B (1) A D C E B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -10 10 10 B -14 0 -18 -4 6 C 10 18 0 22 24 D -10 4 -22 0 -6 E -10 -6 -24 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 10 10 B -14 0 -18 -4 6 C 10 18 0 22 24 D -10 4 -22 0 -6 E -10 -6 -24 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 A=31 D=13 E=9 B=5 so B is eliminated. Round 2 votes counts: C=42 A=32 E=13 D=13 so E is eliminated. Round 3 votes counts: C=42 A=33 D=25 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:237 A:212 B:185 D:183 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 10 10 B -14 0 -18 -4 6 C 10 18 0 22 24 D -10 4 -22 0 -6 E -10 -6 -24 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 10 10 B -14 0 -18 -4 6 C 10 18 0 22 24 D -10 4 -22 0 -6 E -10 -6 -24 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 10 10 B -14 0 -18 -4 6 C 10 18 0 22 24 D -10 4 -22 0 -6 E -10 -6 -24 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6344: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) D E C A B (7) B A E C D (7) B A C E D (7) E D C A B (6) E B D C A (6) C A D E B (6) B E D C A (6) B E A D C (6) B E D A C (5) B A C D E (5) B E A C D (4) A C B D E (4) A B C D E (4) D C E A B (3) E D C B A (2) D C A E B (2) C D A E B (2) B A E D C (2) A D C E B (2) A C D B E (2) E D B C A (1) E B D A C (1) D A C E B (1) C D E A B (1) Total count = 100 A B C D E A 0 -4 16 14 4 B 4 0 8 14 4 C -16 -8 0 0 -6 D -14 -14 0 0 -6 E -4 -4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 14 4 B 4 0 8 14 4 C -16 -8 0 0 -6 D -14 -14 0 0 -6 E -4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999178 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=20 E=16 D=13 C=9 so C is eliminated. Round 2 votes counts: B=42 A=26 E=16 D=16 so E is eliminated. Round 3 votes counts: B=49 A=26 D=25 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:215 E:202 C:185 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 16 14 4 B 4 0 8 14 4 C -16 -8 0 0 -6 D -14 -14 0 0 -6 E -4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999178 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 14 4 B 4 0 8 14 4 C -16 -8 0 0 -6 D -14 -14 0 0 -6 E -4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999178 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 14 4 B 4 0 8 14 4 C -16 -8 0 0 -6 D -14 -14 0 0 -6 E -4 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999178 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6345: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) B D C A E (8) E C A D B (7) C E D B A (6) C D E B A (5) C D B E A (5) B D A C E (5) E C D A B (4) D C E B A (4) A E B D C (4) E A C D B (3) E A C B D (3) B A D C E (3) A B D E C (3) E A D C B (2) D B C A E (2) D B A C E (2) C E D A B (2) C E B D A (2) C B E D A (2) C B E A D (2) C B D E A (2) B C D A E (2) A D E B C (2) E A B C D (1) D C E A B (1) D C B E A (1) D A E C B (1) D A E B C (1) D A B E C (1) B C A D E (1) B A D E C (1) A E D B C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -12 -12 -4 B 6 0 -4 -2 0 C 12 4 0 -4 10 D 12 2 4 0 2 E 4 0 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -12 -4 B 6 0 -4 -2 0 C 12 4 0 -4 10 D 12 2 4 0 2 E 4 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=21 E=20 B=20 D=13 so D is eliminated. Round 2 votes counts: C=32 B=24 A=24 E=20 so E is eliminated. Round 3 votes counts: C=43 A=33 B=24 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:210 B:200 E:196 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -12 -4 B 6 0 -4 -2 0 C 12 4 0 -4 10 D 12 2 4 0 2 E 4 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -12 -4 B 6 0 -4 -2 0 C 12 4 0 -4 10 D 12 2 4 0 2 E 4 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -12 -4 B 6 0 -4 -2 0 C 12 4 0 -4 10 D 12 2 4 0 2 E 4 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6346: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) B E C D A (7) A C B D E (6) D A B E C (5) A D C B E (5) A D B C E (5) E C B A D (4) D E B C A (4) D E A B C (4) A C B E D (4) D E B A C (3) C E B A D (3) A D E C B (3) E D B C A (2) E B D C A (2) E B C D A (2) D B C E A (2) D B A C E (2) D A E B C (2) D A B C E (2) C E A B D (2) C B E A D (2) C A B E D (2) B D C E A (2) B C A D E (2) A D B E C (2) E D A B C (1) E C A B D (1) D B E C A (1) D B A E C (1) C B E D A (1) C A E B D (1) B D C A E (1) B C E D A (1) B A C D E (1) A E C B D (1) A C E B D (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 -4 -2 B 0 0 12 10 14 C -2 -12 0 0 -4 D 4 -10 0 0 12 E 2 -14 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.521697 B: 0.478303 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500941527104 Cumulative probabilities = A: 0.521697 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -4 -2 B 0 0 12 10 14 C -2 -12 0 0 -4 D 4 -10 0 0 12 E 2 -14 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=26 E=19 B=14 C=11 so C is eliminated. Round 2 votes counts: A=33 D=26 E=24 B=17 so B is eliminated. Round 3 votes counts: A=36 E=35 D=29 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:218 D:203 A:198 C:191 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -4 -2 B 0 0 12 10 14 C -2 -12 0 0 -4 D 4 -10 0 0 12 E 2 -14 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -4 -2 B 0 0 12 10 14 C -2 -12 0 0 -4 D 4 -10 0 0 12 E 2 -14 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -4 -2 B 0 0 12 10 14 C -2 -12 0 0 -4 D 4 -10 0 0 12 E 2 -14 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999917 Cumulative probabilities = A: 0.499999 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6347: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B C A E (8) A E C D B (7) A E C B D (7) D B C E A (6) E A C B D (5) A D B E C (5) B D C E A (4) D C B E A (3) C B E D A (3) B A D E C (3) A E B D C (3) A D E C B (3) A D E B C (3) D B A C E (2) D A B E C (2) C E A D B (2) C D E B A (2) C D B E A (2) C B D E A (2) B C D E A (2) A E D C B (2) E C B A D (1) E A C D B (1) D C B A E (1) D C A B E (1) D B A E C (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A B D (1) B D A E C (1) B D A C E (1) A E B C D (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 12 2 10 10 B -12 0 -12 -10 0 C -2 12 0 -6 -8 D -10 10 6 0 10 E -10 0 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 10 10 B -12 0 -12 -10 0 C -2 12 0 -6 -8 D -10 10 6 0 10 E -10 0 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=26 E=15 C=15 B=11 so B is eliminated. Round 2 votes counts: A=36 D=32 C=17 E=15 so E is eliminated. Round 3 votes counts: A=42 D=32 C=26 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:208 C:198 E:194 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 10 10 B -12 0 -12 -10 0 C -2 12 0 -6 -8 D -10 10 6 0 10 E -10 0 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 10 10 B -12 0 -12 -10 0 C -2 12 0 -6 -8 D -10 10 6 0 10 E -10 0 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 10 10 B -12 0 -12 -10 0 C -2 12 0 -6 -8 D -10 10 6 0 10 E -10 0 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6348: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) C E D A B (8) D B A C E (6) B D A E C (6) A B E C D (6) C E A D B (5) C E A B D (5) C D E B A (5) B A D E C (5) A E B C D (5) E C A D B (4) E C A B D (4) D B A E C (4) D C B E A (3) A E C B D (3) D C E A B (2) D B C E A (2) B A E D C (2) A B E D C (2) D E A B C (1) D C E B A (1) D B E A C (1) D B C A E (1) C D E A B (1) C D B E A (1) C B E A D (1) C B D E A (1) C B A E D (1) C A E B D (1) C A B E D (1) B D C A E (1) B C D A E (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 2 10 -6 B -14 0 -10 10 -6 C -2 10 0 26 -2 D -10 -10 -26 0 -16 E 6 6 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 10 -6 B -14 0 -10 10 -6 C -2 10 0 26 -2 D -10 -10 -26 0 -16 E 6 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=21 A=17 E=16 B=16 so E is eliminated. Round 2 votes counts: C=38 A=25 D=21 B=16 so B is eliminated. Round 3 votes counts: C=39 A=33 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:216 E:215 A:210 B:190 D:169 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 10 -6 B -14 0 -10 10 -6 C -2 10 0 26 -2 D -10 -10 -26 0 -16 E 6 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 10 -6 B -14 0 -10 10 -6 C -2 10 0 26 -2 D -10 -10 -26 0 -16 E 6 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 10 -6 B -14 0 -10 10 -6 C -2 10 0 26 -2 D -10 -10 -26 0 -16 E 6 6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6349: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (16) C A B D E (10) C A D E B (5) D E A B C (4) C A B E D (4) B E D A C (4) A D C E B (4) E B D C A (3) E B D A C (3) D A E B C (3) C E D A B (3) C B E A D (3) C B A E D (3) B E C D A (3) E C B D A (2) C B E D A (2) C A D B E (2) B E D C A (2) B C E D A (2) B A D E C (2) A C B D E (2) E D C B A (1) E D A B C (1) E C D B A (1) D E B A C (1) D E A C B (1) D A E C B (1) C E B D A (1) C A E B D (1) B E A D C (1) B D E A C (1) B D A E C (1) B A C E D (1) A D E C B (1) A C D E B (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 -12 -10 B 6 0 2 8 -6 C -4 -2 0 -4 -6 D 12 -8 4 0 -14 E 10 6 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 -12 -10 B 6 0 2 8 -6 C -4 -2 0 -4 -6 D 12 -8 4 0 -14 E 10 6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=27 B=17 A=12 D=10 so D is eliminated. Round 2 votes counts: C=34 E=33 B=17 A=16 so A is eliminated. Round 3 votes counts: C=42 E=38 B=20 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 B:205 D:197 C:192 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 -12 -10 B 6 0 2 8 -6 C -4 -2 0 -4 -6 D 12 -8 4 0 -14 E 10 6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -12 -10 B 6 0 2 8 -6 C -4 -2 0 -4 -6 D 12 -8 4 0 -14 E 10 6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -12 -10 B 6 0 2 8 -6 C -4 -2 0 -4 -6 D 12 -8 4 0 -14 E 10 6 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6350: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) B D C E A (9) E A B D C (8) E B A D C (6) B E D C A (6) A E C D B (6) D C B E A (5) C D B A E (5) E B D C A (4) E B D A C (3) B D E C A (3) A E D C B (3) A E C B D (3) A C D E B (3) E D C B A (2) C D A B E (2) B E A D C (2) A C E D B (2) A C D B E (2) A B E C D (2) E D C A B (1) E A D C B (1) D E C B A (1) D C E B A (1) D C E A B (1) D B C E A (1) C D A E B (1) C B D A E (1) C A D B E (1) B E D A C (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 0 12 6 -10 B 0 0 18 24 -18 C -12 -18 0 -18 -30 D -6 -24 18 0 -28 E 10 18 30 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 12 6 -10 B 0 0 18 24 -18 C -12 -18 0 -18 -30 D -6 -24 18 0 -28 E 10 18 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=25 B=21 C=10 D=9 so D is eliminated. Round 2 votes counts: A=35 E=26 B=22 C=17 so C is eliminated. Round 3 votes counts: A=39 B=33 E=28 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:243 B:212 A:204 D:180 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 6 -10 B 0 0 18 24 -18 C -12 -18 0 -18 -30 D -6 -24 18 0 -28 E 10 18 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 6 -10 B 0 0 18 24 -18 C -12 -18 0 -18 -30 D -6 -24 18 0 -28 E 10 18 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 6 -10 B 0 0 18 24 -18 C -12 -18 0 -18 -30 D -6 -24 18 0 -28 E 10 18 30 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6351: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) D C B E A (9) D C B A E (9) C D E B A (6) A E B D C (6) C D B E A (5) D A B E C (4) A E D B C (4) A B E D C (4) E A B C D (3) C B D E A (3) B E A C D (3) A E D C B (3) E C A B D (2) D C E A B (2) D C A E B (2) D C A B E (2) D B C A E (2) D A C E B (2) A D E C B (2) E C B A D (1) E B A C D (1) E A C B D (1) D A C B E (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B A D (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B A E D C (1) B A D E C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 2 -8 18 B -6 0 -6 -14 2 C -2 6 0 -18 2 D 8 14 18 0 12 E -18 -2 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -8 18 B -6 0 -6 -14 2 C -2 6 0 -18 2 D 8 14 18 0 12 E -18 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=32 C=17 B=9 E=8 so E is eliminated. Round 2 votes counts: A=36 D=34 C=20 B=10 so B is eliminated. Round 3 votes counts: A=42 D=37 C=21 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 A:209 C:194 B:188 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -8 18 B -6 0 -6 -14 2 C -2 6 0 -18 2 D 8 14 18 0 12 E -18 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -8 18 B -6 0 -6 -14 2 C -2 6 0 -18 2 D 8 14 18 0 12 E -18 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -8 18 B -6 0 -6 -14 2 C -2 6 0 -18 2 D 8 14 18 0 12 E -18 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6352: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B D C A (6) C D E A B (5) B E A D C (5) E C D B A (4) E C A D B (4) E B C D A (4) B A D C E (4) A C E D B (4) E A B C D (3) D C A B E (3) C A D E B (3) B A E D C (3) B A D E C (3) E D C B A (2) E C D A B (2) E B A C D (2) B D A E C (2) B D A C E (2) B A E C D (2) A D C B E (2) A C E B D (2) A B D C E (2) A B C D E (2) E D B C A (1) E C A B D (1) E B C A D (1) E B A D C (1) D E B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) D A C B E (1) C E D A B (1) C D A E B (1) C D A B E (1) B E D C A (1) B E D A C (1) B D E C A (1) B D E A C (1) A E C B D (1) A E B C D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 8 12 -2 B 0 0 2 6 -4 C -8 -2 0 8 -8 D -12 -6 -8 0 -6 E 2 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 12 -2 B 0 0 2 6 -4 C -8 -2 0 8 -8 D -12 -6 -8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=25 A=24 C=11 D=9 so D is eliminated. Round 2 votes counts: E=32 B=26 A=25 C=17 so C is eliminated. Round 3 votes counts: E=40 A=33 B=27 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:209 B:202 C:195 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 12 -2 B 0 0 2 6 -4 C -8 -2 0 8 -8 D -12 -6 -8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 12 -2 B 0 0 2 6 -4 C -8 -2 0 8 -8 D -12 -6 -8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 12 -2 B 0 0 2 6 -4 C -8 -2 0 8 -8 D -12 -6 -8 0 -6 E 2 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6353: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (9) C B A D E (9) C E D A B (7) E D A B C (6) B A C D E (6) E C D A B (5) B A D E C (5) B A D C E (5) C B E A D (4) B C A D E (4) A B D C E (4) E D A C B (3) D A E B C (3) C E D B A (3) B C A E D (3) A D B E C (3) E C D B A (2) E C B D A (2) D E A C B (2) C E B A D (2) C B A E D (2) B E A D C (2) A B D E C (2) E D C A B (1) E D B A C (1) C E B D A (1) C D A E B (1) C D A B E (1) C A B D E (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 4 6 0 B 2 0 6 4 4 C -4 -6 0 6 8 D -6 -4 -6 0 12 E 0 -4 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 6 0 B 2 0 6 4 4 C -4 -6 0 6 8 D -6 -4 -6 0 12 E 0 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=25 E=20 D=14 A=10 so A is eliminated. Round 2 votes counts: C=31 B=31 E=20 D=18 so D is eliminated. Round 3 votes counts: E=34 B=34 C=32 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 A:204 C:202 D:198 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 6 0 B 2 0 6 4 4 C -4 -6 0 6 8 D -6 -4 -6 0 12 E 0 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 6 0 B 2 0 6 4 4 C -4 -6 0 6 8 D -6 -4 -6 0 12 E 0 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 6 0 B 2 0 6 4 4 C -4 -6 0 6 8 D -6 -4 -6 0 12 E 0 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6354: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D E B A C (10) D E A C B (8) B C A E D (8) B C A D E (6) A E C B D (6) E D A C B (5) D E A B C (5) C A B E D (5) B C D A E (5) E A C D B (4) D B C A E (4) A C E B D (4) E A D C B (3) D B C E A (3) C B A E D (3) C A E D B (3) D B E C A (2) D C B A E (1) D B E A C (1) C B A D E (1) C A B D E (1) B D C E A (1) B D C A E (1) Total count = 100 A B C D E A 0 8 12 8 -4 B -8 0 -8 2 -16 C -12 8 0 12 -8 D -8 -2 -12 0 -2 E 4 16 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 12 8 -4 B -8 0 -8 2 -16 C -12 8 0 12 -8 D -8 -2 -12 0 -2 E 4 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=22 B=21 C=13 A=10 so A is eliminated. Round 2 votes counts: D=34 E=28 B=21 C=17 so C is eliminated. Round 3 votes counts: E=35 D=34 B=31 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:212 C:200 D:188 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 8 -4 B -8 0 -8 2 -16 C -12 8 0 12 -8 D -8 -2 -12 0 -2 E 4 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 8 -4 B -8 0 -8 2 -16 C -12 8 0 12 -8 D -8 -2 -12 0 -2 E 4 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 8 -4 B -8 0 -8 2 -16 C -12 8 0 12 -8 D -8 -2 -12 0 -2 E 4 16 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6355: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A D C E B (7) E B C D A (6) B C E A D (5) D E B C A (4) D A E C B (4) C A E B D (4) B E C D A (4) A C D E B (4) D B E A C (3) D B A E C (3) B D E A C (3) B C A E D (3) A D C B E (3) A C E D B (3) A C D B E (3) E D B C A (2) E C B D A (2) D B E C A (2) D A B E C (2) C E A B D (2) C B A E D (2) A D B C E (2) A C B E D (2) A C B D E (2) E C D B A (1) E C B A D (1) E B D C A (1) D E B A C (1) D E A C B (1) D A E B C (1) D A C E B (1) C E B A D (1) C B E A D (1) C A E D B (1) C A B E D (1) B D E C A (1) A C E B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 0 2 B 8 0 6 4 4 C 4 -6 0 2 0 D 0 -4 -2 0 -4 E -2 -4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 0 2 B 8 0 6 4 4 C 4 -6 0 2 0 D 0 -4 -2 0 -4 E -2 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 D=22 E=13 C=12 so C is eliminated. Round 2 votes counts: A=35 B=27 D=22 E=16 so E is eliminated. Round 3 votes counts: B=38 A=37 D=25 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:200 E:199 A:195 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 0 2 B 8 0 6 4 4 C 4 -6 0 2 0 D 0 -4 -2 0 -4 E -2 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 0 2 B 8 0 6 4 4 C 4 -6 0 2 0 D 0 -4 -2 0 -4 E -2 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 0 2 B 8 0 6 4 4 C 4 -6 0 2 0 D 0 -4 -2 0 -4 E -2 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6356: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) E B C A D (6) B E A C D (6) D E C B A (5) A B C D E (5) E C D B A (4) A D C B E (4) A B D C E (4) A B C E D (4) E B C D A (3) D C A E B (3) B E C A D (3) A D B C E (3) A C D B E (3) E D B A C (2) E B A C D (2) D E C A B (2) D E A C B (2) D E A B C (2) D C E B A (2) D C E A B (2) D C A B E (2) D A C E B (2) C E D B A (2) C D E B A (2) C A B D E (2) B A E C D (2) B A C E D (2) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) C E B D A (1) C D B A E (1) C D A B E (1) C A D B E (1) B C E A D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -4 4 -14 B 8 0 0 -4 -4 C 4 0 0 10 -2 D -4 4 -10 0 0 E 14 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.079185 E: 0.920815 Sum of squares = 0.854170336773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.079185 E: 1.000000 A B C D E A 0 -8 -4 4 -14 B 8 0 0 -4 -4 C 4 0 0 10 -2 D -4 4 -10 0 0 E 14 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222270835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 D=22 B=14 C=10 so C is eliminated. Round 2 votes counts: E=30 A=30 D=26 B=14 so B is eliminated. Round 3 votes counts: E=40 A=34 D=26 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:206 B:200 D:195 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 4 -14 B 8 0 0 -4 -4 C 4 0 0 10 -2 D -4 4 -10 0 0 E 14 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222270835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 4 -14 B 8 0 0 -4 -4 C 4 0 0 10 -2 D -4 4 -10 0 0 E 14 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222270835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 4 -14 B 8 0 0 -4 -4 C 4 0 0 10 -2 D -4 4 -10 0 0 E 14 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222270835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6357: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) E B A D C (8) E A B C D (7) D B A C E (6) C D B A E (6) D B A E C (5) B A D E C (4) A B D E C (4) E C A B D (3) E A C B D (3) C E D A B (3) C D B E A (3) B D A E C (3) B A E D C (3) E B A C D (2) D C B A E (2) C E A D B (2) C D E A B (2) C D A B E (2) C A E D B (2) B E A D C (2) A D C B E (2) A B E D C (2) E B D A C (1) E B C A D (1) D C B E A (1) D C A B E (1) D B C A E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D A E B (1) B D E C A (1) B D E A C (1) A E C B D (1) A E B D C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 34 16 -2 B 4 0 26 12 4 C -34 -26 0 -18 -22 D -16 -12 18 0 -2 E 2 -4 22 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 34 16 -2 B 4 0 26 12 4 C -34 -26 0 -18 -22 D -16 -12 18 0 -2 E 2 -4 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994505 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=23 D=17 B=14 A=12 so A is eliminated. Round 2 votes counts: E=36 C=24 D=20 B=20 so D is eliminated. Round 3 votes counts: E=36 B=34 C=30 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:222 E:211 D:194 C:150 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 34 16 -2 B 4 0 26 12 4 C -34 -26 0 -18 -22 D -16 -12 18 0 -2 E 2 -4 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994505 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 34 16 -2 B 4 0 26 12 4 C -34 -26 0 -18 -22 D -16 -12 18 0 -2 E 2 -4 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994505 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 34 16 -2 B 4 0 26 12 4 C -34 -26 0 -18 -22 D -16 -12 18 0 -2 E 2 -4 22 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994505 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6358: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C B E A D (6) B C E A D (6) C E B A D (5) B E C A D (5) E A C B D (4) E A B C D (4) C B E D A (4) A E D B C (4) A D E B C (4) E B A C D (3) D C A E B (3) D B A C E (3) C D B E A (3) E C B A D (2) E B C A D (2) D C A B E (2) D B C A E (2) D A C B E (2) D A B E C (2) D A B C E (2) C E A D B (2) C B D E A (2) B E A C D (2) B C E D A (2) B C D E A (2) A E D C B (2) E C A D B (1) E A C D B (1) D C B E A (1) D C B A E (1) D A E B C (1) D A C E B (1) C E D A B (1) C D E B A (1) B A E D C (1) B A E C D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -6 14 -18 B 8 0 -4 6 0 C 6 4 0 20 2 D -14 -6 -20 0 -20 E 18 0 -2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 14 -18 B 8 0 -4 6 0 C 6 4 0 20 2 D -14 -6 -20 0 -20 E 18 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 B=19 E=17 A=12 so A is eliminated. Round 2 votes counts: D=32 E=24 C=24 B=20 so B is eliminated. Round 3 votes counts: E=34 C=34 D=32 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:218 C:216 B:205 A:191 D:170 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 14 -18 B 8 0 -4 6 0 C 6 4 0 20 2 D -14 -6 -20 0 -20 E 18 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 14 -18 B 8 0 -4 6 0 C 6 4 0 20 2 D -14 -6 -20 0 -20 E 18 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 14 -18 B 8 0 -4 6 0 C 6 4 0 20 2 D -14 -6 -20 0 -20 E 18 0 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6359: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) C D A E B (7) B E C A D (7) A D C B E (7) A D B E C (6) B E A D C (5) B E A C D (5) E B C D A (4) E B A D C (4) C D E A B (3) C B E D A (3) A D E C B (3) A D C E B (3) C E B D A (2) C D E B A (2) C B D E A (2) B C E D A (2) B A E D C (2) B A C D E (2) A E D B C (2) A B D E C (2) E D A C B (1) E C D B A (1) E C B D A (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) D E A C B (1) D C A E B (1) D C A B E (1) C E D B A (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) A D B C E (1) A C D B E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 16 12 2 B -2 0 -4 -4 6 C -16 4 0 -2 2 D -12 4 2 0 10 E -2 -6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 12 2 B -2 0 -4 -4 6 C -16 4 0 -2 2 D -12 4 2 0 10 E -2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=25 B=23 E=15 D=10 so D is eliminated. Round 2 votes counts: A=34 C=27 B=23 E=16 so E is eliminated. Round 3 votes counts: A=38 B=33 C=29 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:202 B:198 C:194 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 12 2 B -2 0 -4 -4 6 C -16 4 0 -2 2 D -12 4 2 0 10 E -2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 12 2 B -2 0 -4 -4 6 C -16 4 0 -2 2 D -12 4 2 0 10 E -2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 12 2 B -2 0 -4 -4 6 C -16 4 0 -2 2 D -12 4 2 0 10 E -2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6360: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (13) D A C E B (10) A D B C E (8) E B C D A (7) A D C B E (7) E C D B A (6) E C B D A (6) B E C D A (6) A B D E C (5) C E D B A (4) B A D E C (3) A B D C E (3) D C E A B (2) D C A E B (2) C E D A B (2) C D E A B (2) B E A C D (2) B A E D C (2) B A E C D (2) E C D A B (1) D C B E A (1) D A C B E (1) C D E B A (1) B C E D A (1) B C D E A (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -10 -6 -10 B 10 0 8 4 12 C 10 -8 0 8 -8 D 6 -4 -8 0 -6 E 10 -12 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -6 -10 B 10 0 8 4 12 C 10 -8 0 8 -8 D 6 -4 -8 0 -6 E 10 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=25 E=20 D=16 C=9 so C is eliminated. Round 2 votes counts: B=30 E=26 A=25 D=19 so D is eliminated. Round 3 votes counts: A=38 E=31 B=31 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:206 C:201 D:194 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 -6 -10 B 10 0 8 4 12 C 10 -8 0 8 -8 D 6 -4 -8 0 -6 E 10 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -6 -10 B 10 0 8 4 12 C 10 -8 0 8 -8 D 6 -4 -8 0 -6 E 10 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -6 -10 B 10 0 8 4 12 C 10 -8 0 8 -8 D 6 -4 -8 0 -6 E 10 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6361: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) B D C A E (11) C B D A E (8) A E C B D (8) E A D B C (6) E A C D B (6) E A C B D (6) D B E A C (6) C A E B D (6) A C E B D (6) D B C E A (5) D B E C A (4) C A B D E (4) E D B A C (3) E A D C B (3) E D A B C (2) B D C E A (2) C B A D E (1) C A B E D (1) Total count = 100 A B C D E A 0 -4 -8 -6 14 B 4 0 2 6 8 C 8 -2 0 -8 12 D 6 -6 8 0 6 E -14 -8 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -6 14 B 4 0 2 6 8 C 8 -2 0 -8 12 D 6 -6 8 0 6 E -14 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=26 C=20 A=14 B=13 so B is eliminated. Round 2 votes counts: D=40 E=26 C=20 A=14 so A is eliminated. Round 3 votes counts: D=40 E=34 C=26 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:207 C:205 A:198 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 14 B 4 0 2 6 8 C 8 -2 0 -8 12 D 6 -6 8 0 6 E -14 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 14 B 4 0 2 6 8 C 8 -2 0 -8 12 D 6 -6 8 0 6 E -14 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 14 B 4 0 2 6 8 C 8 -2 0 -8 12 D 6 -6 8 0 6 E -14 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6362: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (13) D E A C B (9) C A B D E (8) C B A E D (6) B C A E D (6) A C D E B (6) D E A B C (5) C A D B E (5) B E D C A (5) B C E A D (4) A C B D E (4) E B D C A (3) B E D A C (3) E B D A C (2) D E C A B (2) D A C E B (2) B E C D A (2) A D E C B (2) E D B C A (1) E D A B C (1) D E C B A (1) D C E A B (1) D A E B C (1) C D E B A (1) C D A E B (1) C B E A D (1) C A D E B (1) B E C A D (1) B A C E D (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 2 -6 -10 B 0 0 -4 -6 -6 C -2 4 0 -4 -2 D 6 6 4 0 0 E 10 6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.633748 E: 0.366252 Sum of squares = 0.535777184612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.633748 E: 1.000000 A B C D E A 0 0 2 -6 -10 B 0 0 -4 -6 -6 C -2 4 0 -4 -2 D 6 6 4 0 0 E 10 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 B=22 D=21 E=20 A=14 so A is eliminated. Round 2 votes counts: C=34 D=24 B=22 E=20 so E is eliminated. Round 3 votes counts: D=39 C=34 B=27 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:209 D:208 C:198 A:193 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -6 -10 B 0 0 -4 -6 -6 C -2 4 0 -4 -2 D 6 6 4 0 0 E 10 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -6 -10 B 0 0 -4 -6 -6 C -2 4 0 -4 -2 D 6 6 4 0 0 E 10 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -6 -10 B 0 0 -4 -6 -6 C -2 4 0 -4 -2 D 6 6 4 0 0 E 10 6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6363: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) D E C A B (10) B C A D E (7) E D C A B (5) E B D A C (5) B E C A D (5) E D B C A (4) E D A C B (4) D C A E B (4) D C A B E (4) C A D B E (4) E B D C A (3) C A B D E (3) B C A E D (3) A C D B E (3) E D B A C (2) E B A C D (2) D E A C B (2) B D C A E (2) B A C E D (2) B A C D E (2) A C B D E (2) E A C B D (1) D E B C A (1) D B E C A (1) D B C A E (1) B E D C A (1) B C E A D (1) A E C B D (1) A C E D B (1) A C D E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -18 2 -16 B 6 0 8 6 8 C 18 -8 0 2 -16 D -2 -6 -2 0 -4 E 16 -8 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 2 -16 B 6 0 8 6 8 C 18 -8 0 2 -16 D -2 -6 -2 0 -4 E 16 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=26 D=23 A=10 C=7 so C is eliminated. Round 2 votes counts: B=34 E=26 D=23 A=17 so A is eliminated. Round 3 votes counts: B=41 D=31 E=28 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:214 C:198 D:193 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -18 2 -16 B 6 0 8 6 8 C 18 -8 0 2 -16 D -2 -6 -2 0 -4 E 16 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 2 -16 B 6 0 8 6 8 C 18 -8 0 2 -16 D -2 -6 -2 0 -4 E 16 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 2 -16 B 6 0 8 6 8 C 18 -8 0 2 -16 D -2 -6 -2 0 -4 E 16 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6364: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (20) C B D A E (13) A E D B C (10) C B D E A (8) E A C B D (7) D B C A E (5) E D A B C (3) D B C E A (3) B C D E A (3) A E C B D (3) E D B C A (2) C B E D A (2) C A B D E (2) A D B C E (2) E C B A D (1) E B C D A (1) E A D C B (1) D C B A E (1) D B A C E (1) D A B E C (1) C B A D E (1) C A B E D (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D C B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 6 6 -6 B -10 0 10 -4 0 C -6 -10 0 -6 -2 D -6 4 6 0 -10 E 6 0 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.249677 C: 0.000000 D: 0.000000 E: 0.750323 Sum of squares = 0.625323511083 Cumulative probabilities = A: 0.000000 B: 0.249677 C: 0.249677 D: 0.249677 E: 1.000000 A B C D E A 0 10 6 6 -6 B -10 0 10 -4 0 C -6 -10 0 -6 -2 D -6 4 6 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250115868 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=27 A=20 D=11 B=7 so B is eliminated. Round 2 votes counts: E=35 C=32 A=20 D=13 so D is eliminated. Round 3 votes counts: C=43 E=35 A=22 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 A:208 B:198 D:197 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 6 -6 B -10 0 10 -4 0 C -6 -10 0 -6 -2 D -6 4 6 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250115868 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 6 -6 B -10 0 10 -4 0 C -6 -10 0 -6 -2 D -6 4 6 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250115868 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 6 -6 B -10 0 10 -4 0 C -6 -10 0 -6 -2 D -6 4 6 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250115868 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6365: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) D C A B E (8) E D A C B (7) E B A C D (7) D E A C B (7) D C A E B (6) D A C E B (6) B C A E D (6) E B C A D (5) E D B A C (4) B C A D E (4) C A B D E (3) B A C E D (3) A C D B E (3) E D A B C (2) E B D A C (2) E B A D C (2) D A C B E (2) B E A C D (2) E D B C A (1) D E C A B (1) D C E A B (1) D C B A E (1) C A D B E (1) B D C E A (1) B D C A E (1) A E C D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 0 0 -6 B 0 0 2 -4 -6 C 0 -2 0 -4 -4 D 0 4 4 0 -6 E 6 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 0 -6 B 0 0 2 -4 -6 C 0 -2 0 -4 -4 D 0 4 4 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=30 B=28 A=6 C=4 so C is eliminated. Round 2 votes counts: D=32 E=30 B=28 A=10 so A is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:201 A:197 B:196 C:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 0 -6 B 0 0 2 -4 -6 C 0 -2 0 -4 -4 D 0 4 4 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 -6 B 0 0 2 -4 -6 C 0 -2 0 -4 -4 D 0 4 4 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 -6 B 0 0 2 -4 -6 C 0 -2 0 -4 -4 D 0 4 4 0 -6 E 6 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6366: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (6) B A E C D (6) C E D A B (5) B C D A E (5) E C A D B (4) C D B E A (4) B A D E C (4) A E D B C (4) E A D C B (3) E A C D B (3) D C E A B (3) D A E C B (3) A E B D C (3) A D E B C (3) E D A C B (2) E A B C D (2) D C B A E (2) D A C B E (2) C E B A D (2) C D E A B (2) C B E D A (2) C B D E A (2) B E A C D (2) B D A C E (2) B C D E A (2) A E D C B (2) A D E C B (2) A B E D C (2) A B D E C (2) E C A B D (1) E B C A D (1) E A C B D (1) D E C A B (1) D E A C B (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A B C E (1) C D E B A (1) C B E A D (1) B D A E C (1) B C E A D (1) Total count = 100 A B C D E A 0 8 16 10 8 B -8 0 -2 -4 0 C -16 2 0 -6 -18 D -10 4 6 0 -6 E -8 0 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 10 8 B -8 0 -2 -4 0 C -16 2 0 -6 -18 D -10 4 6 0 -6 E -8 0 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=19 A=18 E=17 D=17 so E is eliminated. Round 2 votes counts: B=30 A=27 C=24 D=19 so D is eliminated. Round 3 votes counts: A=36 C=32 B=32 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:208 D:197 B:193 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 10 8 B -8 0 -2 -4 0 C -16 2 0 -6 -18 D -10 4 6 0 -6 E -8 0 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 10 8 B -8 0 -2 -4 0 C -16 2 0 -6 -18 D -10 4 6 0 -6 E -8 0 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 10 8 B -8 0 -2 -4 0 C -16 2 0 -6 -18 D -10 4 6 0 -6 E -8 0 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6367: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) C A E D B (7) B A D E C (6) C B E D A (5) B D E A C (5) B C A D E (5) A D B E C (5) C E A D B (4) C B A E D (4) B A C D E (4) E D C A B (3) C B E A D (3) A B C D E (3) D E B A C (2) C E B D A (2) B E D C A (2) B D A E C (2) A D E C B (2) A D E B C (2) A C D E B (2) A C B D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C D B A (1) E C B D A (1) E B D A C (1) D E A B C (1) D B E A C (1) D B A E C (1) C E D B A (1) C A E B D (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D A C (1) B E C D A (1) B C E D A (1) B C E A D (1) B C D E A (1) B A D C E (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -10 14 4 B 8 0 0 10 14 C 10 0 0 20 18 D -14 -10 -20 0 -2 E -4 -14 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.765594 C: 0.234406 D: 0.000000 E: 0.000000 Sum of squares = 0.641079993103 Cumulative probabilities = A: 0.000000 B: 0.765594 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 14 4 B 8 0 0 10 14 C 10 0 0 20 18 D -14 -10 -20 0 -2 E -4 -14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=30 A=18 E=9 D=5 so D is eliminated. Round 2 votes counts: C=38 B=32 A=18 E=12 so E is eliminated. Round 3 votes counts: C=43 B=37 A=20 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:224 B:216 A:200 E:183 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 14 4 B 8 0 0 10 14 C 10 0 0 20 18 D -14 -10 -20 0 -2 E -4 -14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 14 4 B 8 0 0 10 14 C 10 0 0 20 18 D -14 -10 -20 0 -2 E -4 -14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 14 4 B 8 0 0 10 14 C 10 0 0 20 18 D -14 -10 -20 0 -2 E -4 -14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6368: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D C B A E (6) D B C A E (6) D B E C A (4) B E A C D (4) A E C D B (4) E D B A C (3) E B D A C (3) E B A D C (3) E A C B D (3) E A B C D (3) D C A E B (3) D C A B E (3) D A C E B (3) C A D E B (3) B E D A C (3) B D C E A (3) A C D E B (3) E A D C B (2) D B E A C (2) C B D A E (2) C A E D B (2) A E C B D (2) A D C E B (2) E A C D B (1) D E B A C (1) D E A B C (1) D B C E A (1) D A E C B (1) C D B A E (1) C A E B D (1) C A B E D (1) C A B D E (1) B E C A D (1) B D E C A (1) B D E A C (1) B D C A E (1) B C E D A (1) B C E A D (1) B C A E D (1) A E D C B (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 14 0 0 B 14 0 4 -10 -12 C -14 -4 0 -10 -6 D 0 10 10 0 0 E 0 12 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.281663 B: 0.000000 C: 0.000000 D: 0.205261 E: 0.513076 Sum of squares = 0.384713077856 Cumulative probabilities = A: 0.281663 B: 0.281663 C: 0.281663 D: 0.486924 E: 1.000000 A B C D E A 0 -14 14 0 0 B 14 0 4 -10 -12 C -14 -4 0 -10 -6 D 0 10 10 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333334 E: 0.333334 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666666 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=26 B=17 A=15 C=11 so C is eliminated. Round 2 votes counts: D=32 E=26 A=23 B=19 so B is eliminated. Round 3 votes counts: D=40 E=36 A=24 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:209 A:200 B:198 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 14 0 0 B 14 0 4 -10 -12 C -14 -4 0 -10 -6 D 0 10 10 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333334 E: 0.333334 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666666 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 14 0 0 B 14 0 4 -10 -12 C -14 -4 0 -10 -6 D 0 10 10 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333334 E: 0.333334 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666666 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 14 0 0 B 14 0 4 -10 -12 C -14 -4 0 -10 -6 D 0 10 10 0 0 E 0 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333334 E: 0.333334 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666666 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6369: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (9) E B C D A (8) E D B C A (6) D A C B E (6) E B C A D (5) D E B C A (5) A C B D E (5) D E C A B (4) D E A C B (4) C A B E D (4) A B C E D (4) E D C B A (3) E C B A D (3) D A B C E (3) A D B C E (3) A C B E D (3) E C B D A (2) D E C B A (2) D A E C B (2) D A B E C (2) C B E A D (2) C B A E D (2) B E C A D (2) B C A E D (2) D E B A C (1) D B E A C (1) D A E B C (1) D A C E B (1) C E B A D (1) B A C E D (1) A D C B E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -2 -20 -16 B -8 0 8 -10 -14 C 2 -8 0 -8 -20 D 20 10 8 0 4 E 16 14 20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -20 -16 B -8 0 8 -10 -14 C 2 -8 0 -8 -20 D 20 10 8 0 4 E 16 14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=27 A=18 C=9 B=5 so B is eliminated. Round 2 votes counts: D=41 E=29 A=19 C=11 so C is eliminated. Round 3 votes counts: D=41 E=32 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:223 D:221 B:188 A:185 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -2 -20 -16 B -8 0 8 -10 -14 C 2 -8 0 -8 -20 D 20 10 8 0 4 E 16 14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -20 -16 B -8 0 8 -10 -14 C 2 -8 0 -8 -20 D 20 10 8 0 4 E 16 14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -20 -16 B -8 0 8 -10 -14 C 2 -8 0 -8 -20 D 20 10 8 0 4 E 16 14 20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6370: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) C E B D A (10) E C B D A (6) C B E D A (5) C A E B D (5) E D B C A (4) A C D B E (4) E B D C A (3) C E A B D (3) C B D E A (3) B D E C A (3) B D A E C (3) A E D C B (3) A D B C E (3) E D B A C (2) E D A B C (2) D B A E C (2) B D C E A (2) A E D B C (2) A E C D B (2) A C B D E (2) E D A C B (1) E C D B A (1) E C D A B (1) E B C D A (1) D E B A C (1) D E A B C (1) D B E A C (1) D A B E C (1) C E B A D (1) C B D A E (1) C B A D E (1) B D C A E (1) B D A C E (1) B A D C E (1) A D E B C (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -2 -12 -2 B 6 0 -6 6 -6 C 2 6 0 -4 -4 D 12 -6 4 0 -6 E 2 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 -12 -2 B 6 0 -6 6 -6 C 2 6 0 -4 -4 D 12 -6 4 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=29 E=21 B=11 D=6 so D is eliminated. Round 2 votes counts: A=34 C=29 E=23 B=14 so B is eliminated. Round 3 votes counts: A=41 C=32 E=27 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:209 D:202 B:200 C:200 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -12 -2 B 6 0 -6 6 -6 C 2 6 0 -4 -4 D 12 -6 4 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -12 -2 B 6 0 -6 6 -6 C 2 6 0 -4 -4 D 12 -6 4 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -12 -2 B 6 0 -6 6 -6 C 2 6 0 -4 -4 D 12 -6 4 0 -6 E 2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6371: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B D A E C (8) C E A B D (6) C E A D B (5) B A E D C (5) B A D E C (5) D B A E C (4) C E B A D (4) E A C B D (3) D C A E B (3) D A C E B (3) D A B C E (3) C E D A B (3) C D E A B (3) B E A C D (3) B A E C D (3) A E C B D (3) E C A B D (2) D C E B A (2) D C B A E (2) D B C E A (2) D B C A E (2) D A B E C (2) B E C A D (2) A B E C D (2) E B A C D (1) D C B E A (1) D B A C E (1) C E D B A (1) C D E B A (1) A E C D B (1) A E B C D (1) A D E B C (1) A D C E B (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 4 0 4 B -6 0 -6 -2 -6 C -4 6 0 -12 4 D 0 2 12 0 10 E -4 6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.506139 B: 0.000000 C: 0.000000 D: 0.493861 E: 0.000000 Sum of squares = 0.500075360046 Cumulative probabilities = A: 0.506139 B: 0.506139 C: 0.506139 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 0 4 B -6 0 -6 -2 -6 C -4 6 0 -12 4 D 0 2 12 0 10 E -4 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=26 C=23 A=11 E=6 so E is eliminated. Round 2 votes counts: D=34 B=27 C=25 A=14 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 C:197 E:194 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 0 4 B -6 0 -6 -2 -6 C -4 6 0 -12 4 D 0 2 12 0 10 E -4 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 0 4 B -6 0 -6 -2 -6 C -4 6 0 -12 4 D 0 2 12 0 10 E -4 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 0 4 B -6 0 -6 -2 -6 C -4 6 0 -12 4 D 0 2 12 0 10 E -4 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6372: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) E D C B A (7) C B A E D (7) A B C D E (7) E D C A B (6) B A C D E (5) C B E A D (4) C A B E D (4) A B D C E (4) E D A B C (3) D E A C B (3) C B A D E (3) C A B D E (3) E D B A C (2) E D A C B (2) E C D B A (2) D A C E B (2) D A B E C (2) B C A E D (2) B C A D E (2) B A D C E (2) A D B C E (2) E D B C A (1) E C B D A (1) E B C D A (1) D E C A B (1) D A E B C (1) C E D A B (1) C E B A D (1) C D A E B (1) C A E B D (1) C A D B E (1) B C E A D (1) B A C E D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 16 0 4 4 B -16 0 -4 0 4 C 0 4 0 0 12 D -4 0 0 0 6 E -4 -4 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.686324 B: 0.000000 C: 0.313676 D: 0.000000 E: 0.000000 Sum of squares = 0.569433021009 Cumulative probabilities = A: 0.686324 B: 0.686324 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 4 4 B -16 0 -4 0 4 C 0 4 0 0 12 D -4 0 0 0 6 E -4 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 D=21 A=15 B=13 so B is eliminated. Round 2 votes counts: C=31 E=25 A=23 D=21 so D is eliminated. Round 3 votes counts: E=41 C=31 A=28 so A is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:212 C:208 D:201 B:192 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 4 4 B -16 0 -4 0 4 C 0 4 0 0 12 D -4 0 0 0 6 E -4 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 4 4 B -16 0 -4 0 4 C 0 4 0 0 12 D -4 0 0 0 6 E -4 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 4 4 B -16 0 -4 0 4 C 0 4 0 0 12 D -4 0 0 0 6 E -4 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6373: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) E B A D C (5) C D A B E (5) E A B C D (4) D C B E A (4) A E C B D (4) E C B D A (3) E B C D A (3) D B C E A (3) C D E A B (3) B E A D C (3) A C D E B (3) A B D C E (3) E B D C A (2) E B A C D (2) E A C B D (2) D C A B E (2) C E D A B (2) C D E B A (2) C D B E A (2) C A D E B (2) B D E C A (2) B D E A C (2) B D C E A (2) B A E D C (2) A E B D C (2) A E B C D (2) A D B C E (2) A C E D B (2) A B E D C (2) A B D E C (2) E C D B A (1) E C A D B (1) E B D A C (1) E A B D C (1) D C B A E (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D B A (1) B E D C A (1) B E D A C (1) B D C A E (1) B D A E C (1) B A D E C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 0 -8 -6 B -4 0 4 8 -8 C 0 -4 0 2 0 D 8 -8 -2 0 6 E 6 8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.363636 Sum of squares = 0.338842975197 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.636364 E: 1.000000 A B C D E A 0 4 0 -8 -6 B -4 0 4 8 -8 C 0 -4 0 2 0 D 8 -8 -2 0 6 E 6 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.363636 Sum of squares = 0.338842974464 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 A=23 B=16 D=13 so D is eliminated. Round 2 votes counts: C=30 E=25 A=24 B=21 so B is eliminated. Round 3 votes counts: C=37 E=34 A=29 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:204 D:202 B:200 C:199 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -8 -6 B -4 0 4 8 -8 C 0 -4 0 2 0 D 8 -8 -2 0 6 E 6 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.363636 Sum of squares = 0.338842974464 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.636364 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -8 -6 B -4 0 4 8 -8 C 0 -4 0 2 0 D 8 -8 -2 0 6 E 6 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.363636 Sum of squares = 0.338842974464 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.636364 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -8 -6 B -4 0 4 8 -8 C 0 -4 0 2 0 D 8 -8 -2 0 6 E 6 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.000000 D: 0.363636 E: 0.363636 Sum of squares = 0.338842974464 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.272727 D: 0.636364 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6374: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) A E B C D (9) D A B E C (8) C D B E A (6) A E D B C (6) D B C A E (5) D A B C E (4) C B E D A (4) A E B D C (4) E C A B D (3) E A B C D (3) D C B A E (3) C B D E A (3) E A C B D (2) D C E A B (2) D C A B E (2) D B C E A (2) C E B A D (2) C E A B D (2) A E C B D (2) A D E B C (2) E B A C D (1) D B A E C (1) D B A C E (1) D A E B C (1) D A C B E (1) C E B D A (1) C E A D B (1) C D E B A (1) B D C E A (1) B C E A D (1) B C D E A (1) B A D E C (1) A E D C B (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 10 -2 -16 6 B -10 0 4 -20 12 C 2 -4 0 -14 8 D 16 20 14 0 14 E -6 -12 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -16 6 B -10 0 4 -20 12 C 2 -4 0 -14 8 D 16 20 14 0 14 E -6 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=26 C=20 E=9 B=4 so B is eliminated. Round 2 votes counts: D=42 A=27 C=22 E=9 so E is eliminated. Round 3 votes counts: D=42 A=33 C=25 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:232 A:199 C:196 B:193 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -2 -16 6 B -10 0 4 -20 12 C 2 -4 0 -14 8 D 16 20 14 0 14 E -6 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -16 6 B -10 0 4 -20 12 C 2 -4 0 -14 8 D 16 20 14 0 14 E -6 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -16 6 B -10 0 4 -20 12 C 2 -4 0 -14 8 D 16 20 14 0 14 E -6 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6375: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (5) B E D A C (5) A C D E B (5) D C E A B (4) C D A E B (4) A E D C B (4) A C B D E (4) A B E C D (4) A B C E D (4) E B A D C (3) C D B E A (3) C A D B E (3) C A B D E (3) B A E C D (3) A E B D C (3) E D B C A (2) E D B A C (2) E D A B C (2) E A D C B (2) E A D B C (2) D C E B A (2) C B D E A (2) B E A D C (2) B E A C D (2) B A E D C (2) A E B C D (2) E D C B A (1) E D C A B (1) E B D A C (1) D E C B A (1) D E A C B (1) D C A E B (1) C D E B A (1) C D A B E (1) C B D A E (1) B E C D A (1) B C E D A (1) B C A E D (1) B C A D E (1) B A C E D (1) B A C D E (1) A E D B C (1) A D E C B (1) A C E B D (1) A C D B E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 22 16 8 B -12 0 4 10 6 C -22 -4 0 2 -8 D -16 -10 -2 0 -20 E -8 -6 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 22 16 8 B -12 0 4 10 6 C -22 -4 0 2 -8 D -16 -10 -2 0 -20 E -8 -6 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=25 C=18 E=16 D=9 so D is eliminated. Round 2 votes counts: A=32 C=25 B=25 E=18 so E is eliminated. Round 3 votes counts: A=39 B=33 C=28 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:207 B:204 C:184 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 22 16 8 B -12 0 4 10 6 C -22 -4 0 2 -8 D -16 -10 -2 0 -20 E -8 -6 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 22 16 8 B -12 0 4 10 6 C -22 -4 0 2 -8 D -16 -10 -2 0 -20 E -8 -6 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 22 16 8 B -12 0 4 10 6 C -22 -4 0 2 -8 D -16 -10 -2 0 -20 E -8 -6 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6376: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) A C B E D (9) D E B C A (7) E D C A B (6) C E A D B (6) D B E A C (5) B D A E C (5) A C E B D (5) A B C D E (5) C A E B D (4) B D E A C (4) B D A C E (4) B A D C E (4) B A C D E (4) E D C B A (3) D B E C A (3) A B C E D (3) E C D A B (2) E C A D B (2) D E C B A (2) C A B D E (2) B A D E C (2) E D B C A (1) E D B A C (1) C A B E D (1) Total count = 100 A B C D E A 0 10 2 14 16 B -10 0 -4 4 2 C -2 4 0 6 14 D -14 -4 -6 0 -6 E -16 -2 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 14 16 B -10 0 -4 4 2 C -2 4 0 6 14 D -14 -4 -6 0 -6 E -16 -2 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979497 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 A=22 D=17 E=15 so E is eliminated. Round 2 votes counts: D=28 C=27 B=23 A=22 so A is eliminated. Round 3 votes counts: C=41 B=31 D=28 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:221 C:211 B:196 E:187 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 14 16 B -10 0 -4 4 2 C -2 4 0 6 14 D -14 -4 -6 0 -6 E -16 -2 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979497 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 14 16 B -10 0 -4 4 2 C -2 4 0 6 14 D -14 -4 -6 0 -6 E -16 -2 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979497 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 14 16 B -10 0 -4 4 2 C -2 4 0 6 14 D -14 -4 -6 0 -6 E -16 -2 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979497 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6377: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) C E D B A (7) A B D C E (7) B A E D C (6) A B D E C (6) E C D B A (5) D C E A B (5) D C A E B (5) A D B C E (5) E C B D A (4) B E A C D (4) D B A E C (3) E B C A D (2) D C E B A (2) D A C B E (2) D A B C E (2) C D E A B (2) C D A E B (2) B A E C D (2) B A D E C (2) A B E C D (2) E D C B A (1) E D B C A (1) E C B A D (1) E B C D A (1) D E C B A (1) D E B C A (1) D B E C A (1) C E A D B (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C A D (1) B D A E C (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -10 -16 -4 B -4 0 0 -16 -4 C 10 0 0 -8 6 D 16 16 8 0 0 E 4 4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.561633 E: 0.438367 Sum of squares = 0.507597186655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.561633 E: 1.000000 A B C D E A 0 4 -10 -16 -4 B -4 0 0 -16 -4 C 10 0 0 -8 6 D 16 16 8 0 0 E 4 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 D=22 A=22 B=18 E=15 so E is eliminated. Round 2 votes counts: C=33 D=24 A=22 B=21 so B is eliminated. Round 3 votes counts: C=37 A=36 D=27 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:204 E:201 B:188 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -10 -16 -4 B -4 0 0 -16 -4 C 10 0 0 -8 6 D 16 16 8 0 0 E 4 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -16 -4 B -4 0 0 -16 -4 C 10 0 0 -8 6 D 16 16 8 0 0 E 4 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -16 -4 B -4 0 0 -16 -4 C 10 0 0 -8 6 D 16 16 8 0 0 E 4 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6378: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) D C A E B (9) C A B D E (9) E D B A C (8) C A D B E (7) B E A C D (7) A B C E D (5) E D B C A (4) E B D A C (4) D E C A B (4) D E B C A (4) D E B A C (4) B A C E D (4) C D A B E (3) E B A D C (2) D C E A B (2) A C B D E (2) E B A C D (1) D E C B A (1) D B E A C (1) C A D E B (1) C A B E D (1) B A E C D (1) B A C D E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 -16 -14 16 B -16 0 -6 -22 12 C 16 6 0 -12 18 D 14 22 12 0 24 E -16 -12 -18 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -16 -14 16 B -16 0 -6 -22 12 C 16 6 0 -12 18 D 14 22 12 0 24 E -16 -12 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=21 E=19 B=13 A=9 so A is eliminated. Round 2 votes counts: D=38 C=24 E=19 B=19 so E is eliminated. Round 3 votes counts: D=50 B=26 C=24 so C is eliminated. Round 4 votes counts: D=61 B=39 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:236 C:214 A:201 B:184 E:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -16 -14 16 B -16 0 -6 -22 12 C 16 6 0 -12 18 D 14 22 12 0 24 E -16 -12 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -16 -14 16 B -16 0 -6 -22 12 C 16 6 0 -12 18 D 14 22 12 0 24 E -16 -12 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -16 -14 16 B -16 0 -6 -22 12 C 16 6 0 -12 18 D 14 22 12 0 24 E -16 -12 -18 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6379: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (16) D B C A E (15) A E C B D (10) E A C D B (8) E C A B D (5) D B A E C (4) C E B D A (4) A C E B D (4) D B C E A (3) C B D A E (3) B D C A E (3) A E D B C (3) E D B C A (2) E D B A C (2) E A D B C (2) D B E C A (2) D B E A C (2) D B A C E (2) E C B D A (1) E A D C B (1) D A B E C (1) C B A D E (1) C A E B D (1) C A B E D (1) B C D A E (1) A E D C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 10 16 10 4 B -10 0 -12 0 -22 C -16 12 0 10 -24 D -10 0 -10 0 -22 E -4 22 24 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 10 4 B -10 0 -12 0 -22 C -16 12 0 10 -24 D -10 0 -10 0 -22 E -4 22 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998004 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=29 A=20 C=10 B=4 so B is eliminated. Round 2 votes counts: E=37 D=32 A=20 C=11 so C is eliminated. Round 3 votes counts: E=41 D=36 A=23 so A is eliminated. Round 4 votes counts: E=61 D=39 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:220 C:191 D:179 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 10 4 B -10 0 -12 0 -22 C -16 12 0 10 -24 D -10 0 -10 0 -22 E -4 22 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998004 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 10 4 B -10 0 -12 0 -22 C -16 12 0 10 -24 D -10 0 -10 0 -22 E -4 22 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998004 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 10 4 B -10 0 -12 0 -22 C -16 12 0 10 -24 D -10 0 -10 0 -22 E -4 22 24 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998004 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6380: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) B C D E A (8) D B C E A (6) C B D A E (6) E A D B C (5) E A B C D (5) D C B A E (5) A D C B E (4) A C B E D (4) E D A B C (3) E B C D A (3) E A B D C (3) D A E C B (3) A E B C D (3) A D E C B (3) A D C E B (3) A C D B E (3) D E A B C (2) D A C B E (2) B E C D A (2) B C E A D (2) A E C B D (2) E D B C A (1) E B C A D (1) D E A C B (1) D C A B E (1) D B E C A (1) C D B A E (1) C B A D E (1) C A B D E (1) B E C A D (1) B C E D A (1) A E D B C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 22 18 8 10 B -22 0 -4 -14 -2 C -18 4 0 -8 -2 D -8 14 8 0 4 E -10 2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 18 8 10 B -22 0 -4 -14 -2 C -18 4 0 -8 -2 D -8 14 8 0 4 E -10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=21 D=21 B=14 C=9 so C is eliminated. Round 2 votes counts: A=36 D=22 E=21 B=21 so E is eliminated. Round 3 votes counts: A=49 D=26 B=25 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:229 D:209 E:195 C:188 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 18 8 10 B -22 0 -4 -14 -2 C -18 4 0 -8 -2 D -8 14 8 0 4 E -10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 8 10 B -22 0 -4 -14 -2 C -18 4 0 -8 -2 D -8 14 8 0 4 E -10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 8 10 B -22 0 -4 -14 -2 C -18 4 0 -8 -2 D -8 14 8 0 4 E -10 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6381: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (14) E A C B D (7) E A B C D (6) D B A E C (6) E A B D C (5) D C B A E (4) E A C D B (3) D E B A C (3) D C B E A (3) D B A C E (3) C D B E A (3) C D B A E (3) C A E B D (3) E C A D B (2) E C A B D (2) D C E B A (2) D B C E A (2) C E D A B (2) C E A B D (2) C B A D E (2) B D A E C (2) A E B C D (2) E D A C B (1) E A D C B (1) D E C A B (1) D B E C A (1) D B E A C (1) C D E A B (1) C B D A E (1) C A B E D (1) B D C A E (1) B D A C E (1) B C D A E (1) B A E D C (1) B A D E C (1) B A C E D (1) B A C D E (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -2 -12 4 B 14 0 10 -12 14 C 2 -10 0 -8 6 D 12 12 8 0 14 E -4 -14 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -12 4 B 14 0 10 -12 14 C 2 -10 0 -8 6 D 12 12 8 0 14 E -4 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=27 C=18 B=9 A=6 so A is eliminated. Round 2 votes counts: D=40 E=29 C=19 B=12 so B is eliminated. Round 3 votes counts: D=45 E=32 C=23 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:213 C:195 A:188 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -2 -12 4 B 14 0 10 -12 14 C 2 -10 0 -8 6 D 12 12 8 0 14 E -4 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -12 4 B 14 0 10 -12 14 C 2 -10 0 -8 6 D 12 12 8 0 14 E -4 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -12 4 B 14 0 10 -12 14 C 2 -10 0 -8 6 D 12 12 8 0 14 E -4 -14 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6382: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (14) E A D C B (10) C D B A E (5) B A C D E (5) A E C D B (5) B C D E A (4) E D C A B (3) D C B E A (3) B D C E A (3) A E B C D (3) A C B D E (3) E D B C A (2) E D A B C (2) E A B D C (2) D C A E B (2) D C A B E (2) C B D A E (2) B C A D E (2) B A E C D (2) A E D C B (2) A C E D B (2) A B E C D (2) E D C B A (1) E D A C B (1) E B D C A (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B C D (1) D C E A B (1) D B C E A (1) C D A B E (1) C B A D E (1) B E D C A (1) B E A C D (1) B C E D A (1) B C A E D (1) B A C E D (1) A C D E B (1) A C D B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 0 18 B 6 0 8 8 16 C 2 -8 0 22 16 D 0 -8 -22 0 4 E -18 -16 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 0 18 B 6 0 8 8 16 C 2 -8 0 22 16 D 0 -8 -22 0 4 E -18 -16 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=26 A=21 D=9 C=9 so D is eliminated. Round 2 votes counts: B=36 E=26 A=21 C=17 so C is eliminated. Round 3 votes counts: B=47 E=27 A=26 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:216 A:205 D:187 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 0 18 B 6 0 8 8 16 C 2 -8 0 22 16 D 0 -8 -22 0 4 E -18 -16 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 0 18 B 6 0 8 8 16 C 2 -8 0 22 16 D 0 -8 -22 0 4 E -18 -16 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 0 18 B 6 0 8 8 16 C 2 -8 0 22 16 D 0 -8 -22 0 4 E -18 -16 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6383: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (11) C B D A E (9) E D B C A (8) E D A B C (8) B C D E A (7) A E D C B (7) A C B E D (6) E D B A C (4) D E B C A (4) D B C E A (3) A E C D B (3) A E C B D (3) A D E C B (3) C D B A E (2) C A B D E (2) A E B C D (2) A D C B E (2) E B D C A (1) E A D B C (1) D E B A C (1) D E A B C (1) D C B E A (1) D C B A E (1) D B E C A (1) D A E C B (1) C B D E A (1) C B A E D (1) C B A D E (1) B E D C A (1) B D C E A (1) B C E D A (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 10 -12 12 B -4 0 -12 -4 4 C -10 12 0 2 2 D 12 4 -2 0 6 E -12 -4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.500000 D: 0.416667 E: 0.000000 Sum of squares = 0.430555555554 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.583333 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 -12 12 B -4 0 -12 -4 4 C -10 12 0 2 2 D 12 4 -2 0 6 E -12 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.500000 D: 0.416667 E: 0.000000 Sum of squares = 0.430555555581 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.583333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=22 C=16 D=13 B=10 so B is eliminated. Round 2 votes counts: A=39 C=24 E=23 D=14 so D is eliminated. Round 3 votes counts: A=40 E=30 C=30 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:207 C:203 B:192 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 10 -12 12 B -4 0 -12 -4 4 C -10 12 0 2 2 D 12 4 -2 0 6 E -12 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.500000 D: 0.416667 E: 0.000000 Sum of squares = 0.430555555581 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.583333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -12 12 B -4 0 -12 -4 4 C -10 12 0 2 2 D 12 4 -2 0 6 E -12 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.500000 D: 0.416667 E: 0.000000 Sum of squares = 0.430555555581 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.583333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -12 12 B -4 0 -12 -4 4 C -10 12 0 2 2 D 12 4 -2 0 6 E -12 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.500000 D: 0.416667 E: 0.000000 Sum of squares = 0.430555555581 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.583333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6384: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) A E C D B (8) D B C A E (7) E A C D B (6) B C D E A (6) E C A B D (5) B D C E A (5) C B E A D (4) C B D A E (4) D B A C E (3) D A B E C (3) C B E D A (3) B D E C A (3) A E D C B (3) A D C E B (3) E C B A D (2) E A B D C (2) D A C B E (2) C A E B D (2) A E D B C (2) A D E C B (2) E D B A C (1) E B C A D (1) E B A C D (1) E A D C B (1) E A D B C (1) D E B A C (1) D C A B E (1) D B E A C (1) C E A B D (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C D A (1) B D C A E (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 10 2 14 -10 B -10 0 -18 2 -10 C -2 18 0 12 -10 D -14 -2 -12 0 -12 E 10 10 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 2 14 -10 B -10 0 -18 2 -10 C -2 18 0 12 -10 D -14 -2 -12 0 -12 E 10 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=20 D=18 B=17 C=16 so C is eliminated. Round 2 votes counts: E=30 B=28 A=24 D=18 so D is eliminated. Round 3 votes counts: B=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:209 A:208 B:182 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 14 -10 B -10 0 -18 2 -10 C -2 18 0 12 -10 D -14 -2 -12 0 -12 E 10 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 14 -10 B -10 0 -18 2 -10 C -2 18 0 12 -10 D -14 -2 -12 0 -12 E 10 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 14 -10 B -10 0 -18 2 -10 C -2 18 0 12 -10 D -14 -2 -12 0 -12 E 10 10 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6385: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (9) D B E A C (8) B D A E C (6) E D B A C (5) E C D B A (5) A B D C E (5) E D B C A (4) E C B D A (4) C E A D B (4) A C B D E (4) C E A B D (3) B D A C E (3) A C E D B (3) E C D A B (2) C A E B D (2) C A D B E (2) C A B D E (2) A D C B E (2) A C D B E (2) E D C B A (1) E D A B C (1) E C A D B (1) E B D C A (1) D B A E C (1) C B D A E (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D E A (1) B A D C E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 -10 -2 B 6 0 0 -8 4 C -4 0 0 -2 2 D 10 8 2 0 4 E 2 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -10 -2 B 6 0 0 -8 4 C -4 0 0 -2 2 D 10 8 2 0 4 E 2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 B=23 A=18 D=9 so D is eliminated. Round 2 votes counts: B=32 C=26 E=24 A=18 so A is eliminated. Round 3 votes counts: B=39 C=37 E=24 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:212 B:201 C:198 E:196 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 -10 -2 B 6 0 0 -8 4 C -4 0 0 -2 2 D 10 8 2 0 4 E 2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -10 -2 B 6 0 0 -8 4 C -4 0 0 -2 2 D 10 8 2 0 4 E 2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -10 -2 B 6 0 0 -8 4 C -4 0 0 -2 2 D 10 8 2 0 4 E 2 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6386: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (11) E C B D A (6) A D B C E (6) E B C A D (5) B E C A D (5) B C E D A (5) C B E D A (4) E A D C B (3) E A B C D (3) D A C E B (3) C E B D A (3) A E D B C (3) E C D A B (2) E A B D C (2) D C E A B (2) D C A B E (2) C B D E A (2) C B D A E (2) B E C D A (2) B E A C D (2) B C D A E (2) A D E C B (2) A D E B C (2) A D C B E (2) A B E D C (2) E C D B A (1) E C B A D (1) E B C D A (1) E B A C D (1) D C B A E (1) D C A E B (1) D B C A E (1) D A B C E (1) C E D B A (1) C D E B A (1) C D E A B (1) B A D C E (1) B A C D E (1) A E D C B (1) A E B D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 -10 -6 B -4 0 -4 4 8 C 2 4 0 2 8 D 10 -4 -2 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -10 -6 B -4 0 -4 4 8 C 2 4 0 2 8 D 10 -4 -2 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 A=21 B=18 C=14 so C is eliminated. Round 2 votes counts: E=29 B=26 D=24 A=21 so A is eliminated. Round 3 votes counts: D=37 E=34 B=29 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:208 B:202 E:199 D:198 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -10 -6 B -4 0 -4 4 8 C 2 4 0 2 8 D 10 -4 -2 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -10 -6 B -4 0 -4 4 8 C 2 4 0 2 8 D 10 -4 -2 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -10 -6 B -4 0 -4 4 8 C 2 4 0 2 8 D 10 -4 -2 0 -8 E 6 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6387: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) C D A B E (9) C B E D A (8) B E C A D (8) A D E B C (8) E B A D C (7) E B C A D (5) E B D A C (4) D A C B E (4) C D B E A (4) A D C B E (4) E B C D A (3) E B A C D (3) B E A C D (3) A B E D C (3) D C A E B (2) D A E B C (2) C E B D A (2) B E C D A (2) A E B D C (2) E C B D A (1) D C A B E (1) D A E C B (1) C D A E B (1) C B E A D (1) B E A D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 6 -8 -4 B 4 0 4 6 -2 C -6 -4 0 0 -8 D 8 -6 0 0 -6 E 4 2 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999188 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 6 -8 -4 B 4 0 4 6 -2 C -6 -4 0 0 -8 D 8 -6 0 0 -6 E 4 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=23 D=20 A=18 B=14 so B is eliminated. Round 2 votes counts: E=37 C=25 D=20 A=18 so A is eliminated. Round 3 votes counts: E=42 D=33 C=25 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:206 D:198 A:195 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 -8 -4 B 4 0 4 6 -2 C -6 -4 0 0 -8 D 8 -6 0 0 -6 E 4 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -8 -4 B 4 0 4 6 -2 C -6 -4 0 0 -8 D 8 -6 0 0 -6 E 4 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -8 -4 B 4 0 4 6 -2 C -6 -4 0 0 -8 D 8 -6 0 0 -6 E 4 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6388: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) D B A C E (9) D C E B A (6) D B A E C (6) E C A D B (5) E C A B D (5) C E A D B (5) B A D C E (5) D C B E A (4) D B C E A (4) B A D E C (3) A E C B D (3) A E B C D (3) A B E C D (3) E D C A B (2) E A C B D (2) D E C B A (2) D E C A B (2) D B C A E (2) C E D A B (2) C A E B D (2) B D A C E (2) B A E D C (2) B A C D E (2) D C E A B (1) D B E C A (1) D B E A C (1) C D E B A (1) C B A E D (1) B A C E D (1) A C E B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 10 -6 B 4 0 -8 -6 -4 C 10 8 0 -6 18 D -10 6 6 0 2 E 6 4 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.384615 D: 0.384615 E: 0.000000 Sum of squares = 0.349112426035 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.615385 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 10 -6 B 4 0 -8 -6 -4 C 10 8 0 -6 18 D -10 6 6 0 2 E 6 4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.384615 D: 0.384615 E: 0.000000 Sum of squares = 0.349112426011 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.615385 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=21 B=15 E=14 A=12 so A is eliminated. Round 2 votes counts: D=38 C=22 E=20 B=20 so E is eliminated. Round 3 votes counts: D=40 C=37 B=23 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:202 A:195 E:195 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 10 -6 B 4 0 -8 -6 -4 C 10 8 0 -6 18 D -10 6 6 0 2 E 6 4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.384615 D: 0.384615 E: 0.000000 Sum of squares = 0.349112426011 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.615385 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 10 -6 B 4 0 -8 -6 -4 C 10 8 0 -6 18 D -10 6 6 0 2 E 6 4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.384615 D: 0.384615 E: 0.000000 Sum of squares = 0.349112426011 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.615385 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 10 -6 B 4 0 -8 -6 -4 C 10 8 0 -6 18 D -10 6 6 0 2 E 6 4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.384615 D: 0.384615 E: 0.000000 Sum of squares = 0.349112426011 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.615385 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6389: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (14) E D B C A (8) E D B A C (7) D B A C E (7) D E B A C (6) D A C B E (6) C A B D E (6) E D C A B (5) E B C A D (4) D E A C B (4) B A C E D (4) E C B A D (3) B A C D E (3) A C D B E (3) E C A B D (2) E B D A C (2) C A E B D (2) E C A D B (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A B C (1) D C E A B (1) D B E A C (1) D A C E B (1) D A B C E (1) C A D B E (1) B E C A D (1) B D E A C (1) B D A C E (1) B A D C E (1) Total count = 100 A B C D E A 0 -2 -2 -10 0 B 2 0 0 -8 0 C 2 0 0 -10 2 D 10 8 10 0 -10 E 0 0 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.090909 E: 0.454545 Sum of squares = 0.421487603304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.545455 E: 1.000000 A B C D E A 0 -2 -2 -10 0 B 2 0 0 -8 0 C 2 0 0 -10 2 D 10 8 10 0 -10 E 0 0 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.090909 E: 0.454545 Sum of squares = 0.421487603293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.545455 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=29 C=23 B=11 A=3 so A is eliminated. Round 2 votes counts: E=34 D=29 C=26 B=11 so B is eliminated. Round 3 votes counts: E=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:209 E:204 B:197 C:197 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 0 B 2 0 0 -8 0 C 2 0 0 -10 2 D 10 8 10 0 -10 E 0 0 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.090909 E: 0.454545 Sum of squares = 0.421487603293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.545455 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 0 B 2 0 0 -8 0 C 2 0 0 -10 2 D 10 8 10 0 -10 E 0 0 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.090909 E: 0.454545 Sum of squares = 0.421487603293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.545455 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 0 B 2 0 0 -8 0 C 2 0 0 -10 2 D 10 8 10 0 -10 E 0 0 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.090909 E: 0.454545 Sum of squares = 0.421487603293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454545 D: 0.545455 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6390: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C A D (8) B E A C D (7) E B A C D (6) C E A B D (5) D C A E B (4) A E C B D (4) D B C E A (3) C D A E B (3) B D E C A (3) A C D E B (3) E B C A D (2) E A B C D (2) D B E A C (2) D A E B C (2) D A C B E (2) D A B E C (2) C A E D B (2) C A D E B (2) B E C D A (2) B D E A C (2) B C E D A (2) A D E B C (2) A D C E B (2) A C E B D (2) E A C B D (1) D C B A E (1) D C A B E (1) D B E C A (1) D B C A E (1) D B A E C (1) C E B A D (1) C D E A B (1) B E A D C (1) B D A E C (1) B A E D C (1) A E B C D (1) A D E C B (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 10 16 10 2 B -10 0 6 4 -16 C -16 -6 0 12 -6 D -10 -4 -12 0 2 E -2 16 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 10 2 B -10 0 6 4 -16 C -16 -6 0 12 -6 D -10 -4 -12 0 2 E -2 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=27 A=17 C=14 E=11 so E is eliminated. Round 2 votes counts: B=35 D=31 A=20 C=14 so C is eliminated. Round 3 votes counts: B=36 D=35 A=29 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:219 E:209 B:192 C:192 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 10 2 B -10 0 6 4 -16 C -16 -6 0 12 -6 D -10 -4 -12 0 2 E -2 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 10 2 B -10 0 6 4 -16 C -16 -6 0 12 -6 D -10 -4 -12 0 2 E -2 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 10 2 B -10 0 6 4 -16 C -16 -6 0 12 -6 D -10 -4 -12 0 2 E -2 16 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6391: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) B E A D C (8) E B A C D (7) C D A E B (6) D C A B E (5) C D E B A (5) E B A D C (4) C D A B E (4) A D C E B (4) E C D A B (3) E A B D C (3) D C A E B (3) D A C B E (3) C E B D A (3) B A E D C (3) E C B D A (2) E B C A D (2) D A C E B (2) C E D B A (2) B E C D A (2) B E C A D (2) B E A C D (2) B C E D A (2) B C D A E (2) A E B D C (2) A D B E C (2) A B E D C (2) E B C D A (1) E A C D B (1) C B E D A (1) B C D E A (1) A D E C B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 -8 -10 -18 B 14 0 -8 -2 -2 C 8 8 0 12 6 D 10 2 -12 0 -4 E 18 2 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -10 -18 B 14 0 -8 -2 -2 C 8 8 0 12 6 D 10 2 -12 0 -4 E 18 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=23 B=22 D=13 A=13 so D is eliminated. Round 2 votes counts: C=37 E=23 B=22 A=18 so A is eliminated. Round 3 votes counts: C=47 B=27 E=26 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:209 B:201 D:198 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -8 -10 -18 B 14 0 -8 -2 -2 C 8 8 0 12 6 D 10 2 -12 0 -4 E 18 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -10 -18 B 14 0 -8 -2 -2 C 8 8 0 12 6 D 10 2 -12 0 -4 E 18 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -10 -18 B 14 0 -8 -2 -2 C 8 8 0 12 6 D 10 2 -12 0 -4 E 18 2 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6392: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) B A E C D (8) E D B C A (6) E B D A C (6) D E C B A (6) B E A D C (6) D C E A B (5) C D A E B (5) A B C E D (5) A B C D E (5) D E C A B (4) A C B D E (4) E D B A C (3) E B A D C (3) C A D E B (3) B E A C D (3) B A C D E (3) A C B E D (3) B A D C E (2) E B A C D (1) D C B E A (1) D C A B E (1) D B E C A (1) D B C E A (1) C D A B E (1) C A B D E (1) B E D A C (1) B A E D C (1) B A C E D (1) Total count = 100 A B C D E A 0 -6 10 18 6 B 6 0 12 6 16 C -10 -12 0 6 2 D -18 -6 -6 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 18 6 B 6 0 12 6 16 C -10 -12 0 6 2 D -18 -6 -6 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=20 E=19 D=19 A=17 so A is eliminated. Round 2 votes counts: B=35 C=27 E=19 D=19 so E is eliminated. Round 3 votes counts: B=45 D=28 C=27 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:214 C:193 D:188 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 18 6 B 6 0 12 6 16 C -10 -12 0 6 2 D -18 -6 -6 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 18 6 B 6 0 12 6 16 C -10 -12 0 6 2 D -18 -6 -6 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 18 6 B 6 0 12 6 16 C -10 -12 0 6 2 D -18 -6 -6 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6393: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (12) D A B C E (9) D B C A E (8) E C B A D (6) E A C B D (5) C B E D A (5) E A D B C (4) A E D B C (4) E D A B C (3) E A D C B (3) D A E B C (3) D A B E C (3) C D B E A (3) C B D A E (3) E D A C B (2) E C D B A (2) E C B D A (2) D B A C E (2) C E B D A (2) C E B A D (2) C B E A D (2) C B D E A (2) B D C A E (2) A D E B C (2) A D B E C (2) A B C E D (2) E C A B D (1) E A C D B (1) B A D C E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 -26 10 B 8 0 18 -4 16 C 4 -18 0 2 14 D 26 4 -2 0 8 E -10 -16 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.750000 E: 0.000000 Sum of squares = 0.597222222379 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.250000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -26 10 B 8 0 18 -4 16 C 4 -18 0 2 14 D 26 4 -2 0 8 E -10 -16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.750000 E: 0.000000 Sum of squares = 0.597222221829 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=25 C=19 B=15 A=12 so A is eliminated. Round 2 votes counts: E=33 D=30 C=19 B=18 so B is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:219 D:218 C:201 A:186 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 -26 10 B 8 0 18 -4 16 C 4 -18 0 2 14 D 26 4 -2 0 8 E -10 -16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.750000 E: 0.000000 Sum of squares = 0.597222221829 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -26 10 B 8 0 18 -4 16 C 4 -18 0 2 14 D 26 4 -2 0 8 E -10 -16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.750000 E: 0.000000 Sum of squares = 0.597222221829 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -26 10 B 8 0 18 -4 16 C 4 -18 0 2 14 D 26 4 -2 0 8 E -10 -16 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.166667 D: 0.750000 E: 0.000000 Sum of squares = 0.597222221829 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6394: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (5) E B C D A (5) C E D A B (5) C E B A D (5) C A D B E (5) E D C A B (4) C A D E B (4) E D B A C (3) E C D B A (3) C E A D B (3) B E D A C (3) B E C A D (3) B D A E C (3) A C D E B (3) A C D B E (3) A B D C E (3) E C B D A (2) D E A C B (2) D C A E B (2) D A B E C (2) C A B D E (2) B E C D A (2) B D E A C (2) B C E A D (2) B A D C E (2) B A C D E (2) A D B E C (2) A D B C E (2) A C B D E (2) E D B C A (1) E B D A C (1) D E A B C (1) D B A E C (1) D A E B C (1) D A C E B (1) C E D B A (1) C E A B D (1) C B A E D (1) C A E D B (1) C A E B D (1) B A E D C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 -16 0 -8 B -4 0 -6 -4 -12 C 16 6 0 12 6 D 0 4 -12 0 -6 E 8 12 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 0 -8 B -4 0 -6 -4 -12 C 16 6 0 12 6 D 0 4 -12 0 -6 E 8 12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=24 B=20 A=17 D=10 so D is eliminated. Round 2 votes counts: C=31 E=27 B=21 A=21 so B is eliminated. Round 3 votes counts: E=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:210 D:193 A:190 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 0 -8 B -4 0 -6 -4 -12 C 16 6 0 12 6 D 0 4 -12 0 -6 E 8 12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 0 -8 B -4 0 -6 -4 -12 C 16 6 0 12 6 D 0 4 -12 0 -6 E 8 12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 0 -8 B -4 0 -6 -4 -12 C 16 6 0 12 6 D 0 4 -12 0 -6 E 8 12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6395: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) C E A D B (8) E C D B A (7) A B D C E (7) E B D C A (4) D A B C E (4) C E D A B (4) B D A E C (4) E C B A D (3) E B A C D (3) C A D E B (3) B D E A C (3) A D B C E (3) A B D E C (3) E C A B D (2) E B C D A (2) D C E B A (2) D B E A C (2) D B A C E (2) C E D B A (2) C E A B D (2) C A E D B (2) B E D A C (2) A C D B E (2) A C B D E (2) A B C D E (2) E B D A C (1) E B C A D (1) D C A B E (1) D B E C A (1) D B A E C (1) C D E A B (1) C D A E B (1) C A E B D (1) B E A D C (1) B A D E C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -10 -4 -18 B 0 0 -6 6 -16 C 10 6 0 14 2 D 4 -6 -14 0 -8 E 18 16 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -4 -18 B 0 0 -6 6 -16 C 10 6 0 14 2 D 4 -6 -14 0 -8 E 18 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=24 A=21 D=13 B=11 so B is eliminated. Round 2 votes counts: E=34 C=24 A=22 D=20 so D is eliminated. Round 3 votes counts: E=40 A=33 C=27 so C is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:216 B:192 D:188 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -4 -18 B 0 0 -6 6 -16 C 10 6 0 14 2 D 4 -6 -14 0 -8 E 18 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 -18 B 0 0 -6 6 -16 C 10 6 0 14 2 D 4 -6 -14 0 -8 E 18 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 -18 B 0 0 -6 6 -16 C 10 6 0 14 2 D 4 -6 -14 0 -8 E 18 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6396: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) D A E B C (7) D A E C B (5) D A B C E (5) E C B D A (4) E B C A D (4) D E C B A (4) D A B E C (4) B C E A D (4) B A C E D (4) A D B E C (4) A D B C E (4) D C A B E (3) D A C E B (3) D A C B E (3) C E B D A (3) C E B A D (3) E D C B A (2) D E C A B (2) D E A C B (2) D C E B A (2) B C A E D (2) A B E C D (2) E C D B A (1) E C B A D (1) D E A B C (1) D C E A B (1) C E D B A (1) C D E B A (1) C B D A E (1) C A B D E (1) B E C A D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -6 -10 2 B 0 0 -10 -10 6 C 6 10 0 -8 10 D 10 10 8 0 12 E -2 -6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -10 2 B 0 0 -10 -10 6 C 6 10 0 -8 10 D 10 10 8 0 12 E -2 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 C=22 A=13 E=12 B=11 so B is eliminated. Round 2 votes counts: D=42 C=28 A=17 E=13 so E is eliminated. Round 3 votes counts: D=44 C=39 A=17 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:209 A:193 B:193 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 -10 2 B 0 0 -10 -10 6 C 6 10 0 -8 10 D 10 10 8 0 12 E -2 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -10 2 B 0 0 -10 -10 6 C 6 10 0 -8 10 D 10 10 8 0 12 E -2 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -10 2 B 0 0 -10 -10 6 C 6 10 0 -8 10 D 10 10 8 0 12 E -2 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6397: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) A E B C D (11) B A E C D (8) B C D A E (6) D C B E A (5) C D B A E (5) E A D C B (4) C D B E A (4) E D A C B (3) D E C A B (3) D C E A B (3) C D E A B (3) B A E D C (3) E A D B C (2) E A C D B (2) D C E B A (2) D B C E A (2) C D E B A (2) C B D A E (2) B D A E C (2) B C A D E (2) A E B D C (2) A B E C D (2) E D C A B (1) D E B A C (1) D B C A E (1) C E D A B (1) C D A E B (1) C A E D B (1) C A B E D (1) B D C A E (1) B A C E D (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 6 8 4 2 B -6 0 10 6 -10 C -8 -10 0 8 -14 D -4 -6 -8 0 -10 E -2 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 4 2 B -6 0 10 6 -10 C -8 -10 0 8 -14 D -4 -6 -8 0 -10 E -2 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996205 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=23 B=23 C=20 D=17 A=17 so D is eliminated. Round 2 votes counts: C=30 E=27 B=26 A=17 so A is eliminated. Round 3 votes counts: E=42 C=30 B=28 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:210 B:200 C:188 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 4 2 B -6 0 10 6 -10 C -8 -10 0 8 -14 D -4 -6 -8 0 -10 E -2 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996205 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 4 2 B -6 0 10 6 -10 C -8 -10 0 8 -14 D -4 -6 -8 0 -10 E -2 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996205 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 4 2 B -6 0 10 6 -10 C -8 -10 0 8 -14 D -4 -6 -8 0 -10 E -2 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996205 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6398: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) E A D C B (8) C D B A E (8) D C B A E (7) B C D A E (7) E B A C D (6) E A B C D (5) B A C D E (5) E A B D C (4) E D A C B (3) E D C B A (2) D E C A B (2) D C E A B (2) D C B E A (2) D C A B E (2) C D B E A (2) C B D A E (2) B C D E A (2) B C A D E (2) B A C E D (2) A E B D C (2) A D C B E (2) A B C D E (2) E B C D A (1) E B A D C (1) D E C B A (1) D C E B A (1) D A C E B (1) C D A B E (1) B E C D A (1) B E A C D (1) B C E D A (1) B C E A D (1) B A E C D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -10 -12 -8 B 12 0 -10 -6 4 C 10 10 0 2 4 D 12 6 -2 0 4 E 8 -4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -12 -8 B 12 0 -10 -6 4 C 10 10 0 2 4 D 12 6 -2 0 4 E 8 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=23 D=18 C=13 A=8 so A is eliminated. Round 2 votes counts: E=40 B=26 D=21 C=13 so C is eliminated. Round 3 votes counts: E=40 D=32 B=28 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:210 B:200 E:198 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 -12 -8 B 12 0 -10 -6 4 C 10 10 0 2 4 D 12 6 -2 0 4 E 8 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -12 -8 B 12 0 -10 -6 4 C 10 10 0 2 4 D 12 6 -2 0 4 E 8 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -12 -8 B 12 0 -10 -6 4 C 10 10 0 2 4 D 12 6 -2 0 4 E 8 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6399: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (8) B D E C A (6) A D E C B (6) D E A C B (5) B C E D A (5) D A E C B (4) B C E A D (4) A C E D B (4) D A E B C (3) B D A E C (3) B A C D E (3) A B C E D (3) E D C B A (2) E D C A B (2) D E C A B (2) D E A B C (2) D A B E C (2) C E B D A (2) C E A D B (2) C E A B D (2) B E C D A (2) B D E A C (2) A D B E C (2) A C B E D (2) A C B D E (2) A B C D E (2) E C D A B (1) E C B D A (1) E A D C B (1) D E B A C (1) D B E C A (1) D B E A C (1) D B A E C (1) C B A E D (1) C A E B D (1) B C D A E (1) B A D C E (1) B A C E D (1) A E D C B (1) A D C E B (1) A D C B E (1) A C D E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 14 2 12 B -8 0 12 8 12 C -14 -12 0 -4 -4 D -2 -8 4 0 10 E -12 -12 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 2 12 B -8 0 12 8 12 C -14 -12 0 -4 -4 D -2 -8 4 0 10 E -12 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995252 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=27 D=22 C=8 E=7 so E is eliminated. Round 2 votes counts: B=36 A=28 D=26 C=10 so C is eliminated. Round 3 votes counts: B=40 A=33 D=27 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:212 D:202 E:185 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 2 12 B -8 0 12 8 12 C -14 -12 0 -4 -4 D -2 -8 4 0 10 E -12 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995252 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 2 12 B -8 0 12 8 12 C -14 -12 0 -4 -4 D -2 -8 4 0 10 E -12 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995252 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 2 12 B -8 0 12 8 12 C -14 -12 0 -4 -4 D -2 -8 4 0 10 E -12 -12 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995252 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6400: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (6) E D C A B (5) A B E C D (5) E A C D B (4) D E C A B (4) D C E A B (4) B D C E A (4) A E C D B (4) A B C E D (4) E A B D C (3) D E C B A (3) D C B E A (3) C D B A E (3) B D C A E (3) A E B C D (3) A C B D E (3) E D B C A (2) E D A C B (2) E A D C B (2) D C E B A (2) D B E C A (2) C B A D E (2) C A E D B (2) B C D A E (2) B A E C D (2) B A C E D (2) A C E D B (2) E D B A C (1) E C D A B (1) E B A D C (1) E A B C D (1) D E B C A (1) D B C E A (1) C E D A B (1) C D E A B (1) C D A B E (1) C A D E B (1) C A D B E (1) B E A D C (1) B D E C A (1) B C A D E (1) B A E D C (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -2 6 0 B -12 0 -4 -6 -2 C 2 4 0 8 2 D -6 6 -8 0 0 E 0 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 6 0 B -12 0 -4 -6 -2 C 2 4 0 8 2 D -6 6 -8 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=23 A=23 E=22 D=20 C=12 so C is eliminated. Round 2 votes counts: A=27 D=25 B=25 E=23 so E is eliminated. Round 3 votes counts: D=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:208 E:200 D:196 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 6 0 B -12 0 -4 -6 -2 C 2 4 0 8 2 D -6 6 -8 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 6 0 B -12 0 -4 -6 -2 C 2 4 0 8 2 D -6 6 -8 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 6 0 B -12 0 -4 -6 -2 C 2 4 0 8 2 D -6 6 -8 0 0 E 0 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6401: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (6) B C A E D (5) A D E C B (5) A B C E D (5) E C D A B (4) C E B D A (4) A E C D B (4) E C A D B (3) D E C B A (3) D A E C B (3) C E D B A (3) C E A B D (3) A E C B D (3) A D E B C (3) D E C A B (2) D E A C B (2) D B E C A (2) D B A E C (2) C E B A D (2) C B E D A (2) C B E A D (2) B D A C E (2) B C D E A (2) B A C E D (2) A B D E C (2) A B D C E (2) E D C A B (1) E D A C B (1) E A C D B (1) D E B A C (1) D C E B A (1) D B E A C (1) D B C E A (1) D A E B C (1) D A B E C (1) C E A D B (1) C D E B A (1) C A E B D (1) B D C E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D A E (1) B C A D E (1) B A D E C (1) A E D C B (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 2 12 8 B -2 0 -6 2 -8 C -2 6 0 18 2 D -12 -2 -18 0 -2 E -8 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999812 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 12 8 B -2 0 -6 2 -8 C -2 6 0 18 2 D -12 -2 -18 0 -2 E -8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=24 D=20 C=19 E=10 so E is eliminated. Round 2 votes counts: A=28 C=26 B=24 D=22 so D is eliminated. Round 3 votes counts: A=36 C=33 B=31 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:212 E:200 B:193 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 12 8 B -2 0 -6 2 -8 C -2 6 0 18 2 D -12 -2 -18 0 -2 E -8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 12 8 B -2 0 -6 2 -8 C -2 6 0 18 2 D -12 -2 -18 0 -2 E -8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 12 8 B -2 0 -6 2 -8 C -2 6 0 18 2 D -12 -2 -18 0 -2 E -8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6402: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) D C B A E (6) E B A D C (5) E A C B D (5) D B A C E (5) B D A E C (5) A E B C D (5) C D A B E (4) B A E D C (4) D C A B E (3) C E D A B (3) C A E D B (3) A B E D C (3) E C B D A (2) E C A B D (2) E B C D A (2) E A B D C (2) D C B E A (2) D B C A E (2) C D E B A (2) C D E A B (2) C D A E B (2) B E A D C (2) B A D E C (2) A B D E C (2) A B D C E (2) E C B A D (1) E B A C D (1) D A B C E (1) C E D B A (1) C E A D B (1) C D B E A (1) C D B A E (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B D E A C (1) A E C B D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 14 6 10 B -6 0 10 16 2 C -14 -10 0 2 -12 D -6 -16 -2 0 -8 E -10 -2 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 6 10 B -6 0 10 16 2 C -14 -10 0 2 -12 D -6 -16 -2 0 -8 E -10 -2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 D=19 B=16 A=15 so A is eliminated. Round 2 votes counts: E=34 B=24 C=23 D=19 so D is eliminated. Round 3 votes counts: E=34 C=34 B=32 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:218 B:211 E:204 D:184 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 6 10 B -6 0 10 16 2 C -14 -10 0 2 -12 D -6 -16 -2 0 -8 E -10 -2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 6 10 B -6 0 10 16 2 C -14 -10 0 2 -12 D -6 -16 -2 0 -8 E -10 -2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 6 10 B -6 0 10 16 2 C -14 -10 0 2 -12 D -6 -16 -2 0 -8 E -10 -2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999589 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6403: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) D B A E C (7) E B C A D (6) D A C B E (6) A C B E D (6) D A B C E (5) B E D A C (4) A B C E D (4) E C D B A (3) E C B A D (3) E C A B D (3) E B D C A (3) D C A E B (3) C A E D B (3) A C D E B (3) D E B C A (2) D B E C A (2) D B E A C (2) C E A D B (2) C E A B D (2) C A E B D (2) B A D E C (2) A D C B E (2) A C E B D (2) E D C B A (1) E C D A B (1) E B C D A (1) D E C B A (1) D E C A B (1) D C E A B (1) D C A B E (1) D B A C E (1) D A C E B (1) C E D A B (1) B E A C D (1) B D E A C (1) B A E C D (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 18 16 4 18 B -18 0 -14 -16 10 C -16 14 0 6 10 D -4 16 -6 0 2 E -18 -10 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 4 18 B -18 0 -14 -16 10 C -16 14 0 6 10 D -4 16 -6 0 2 E -18 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=27 E=21 C=10 B=9 so B is eliminated. Round 2 votes counts: D=34 A=30 E=26 C=10 so C is eliminated. Round 3 votes counts: A=35 D=34 E=31 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:207 D:204 B:181 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 4 18 B -18 0 -14 -16 10 C -16 14 0 6 10 D -4 16 -6 0 2 E -18 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 4 18 B -18 0 -14 -16 10 C -16 14 0 6 10 D -4 16 -6 0 2 E -18 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 4 18 B -18 0 -14 -16 10 C -16 14 0 6 10 D -4 16 -6 0 2 E -18 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6404: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) A E D C B (7) A B C E D (7) E D C B A (5) B C D E A (5) B D C E A (4) A D E C B (4) A B D C E (4) E C D B A (3) D E A B C (3) D B E C A (3) A E C D B (3) A E C B D (3) A D E B C (3) A B C D E (3) E D C A B (2) E D A C B (2) E C A B D (2) D E C B A (2) D E B C A (2) D A E B C (2) C E B D A (2) B C D A E (2) B C A E D (2) B A C D E (2) A D B E C (2) E C D A B (1) E A D C B (1) E A C D B (1) D E C A B (1) D E A C B (1) D B E A C (1) C B E D A (1) C B E A D (1) B D C A E (1) B C E D A (1) B C A D E (1) B A D C E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 8 4 -2 -2 B -8 0 12 -12 -2 C -4 -12 0 -16 -8 D 2 12 16 0 8 E 2 2 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -2 -2 B -8 0 12 -12 -2 C -4 -12 0 -16 -8 D 2 12 16 0 8 E 2 2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=22 B=19 E=17 C=4 so C is eliminated. Round 2 votes counts: A=38 D=22 B=21 E=19 so E is eliminated. Round 3 votes counts: A=42 D=35 B=23 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:204 E:202 B:195 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -2 -2 B -8 0 12 -12 -2 C -4 -12 0 -16 -8 D 2 12 16 0 8 E 2 2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -2 -2 B -8 0 12 -12 -2 C -4 -12 0 -16 -8 D 2 12 16 0 8 E 2 2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -2 -2 B -8 0 12 -12 -2 C -4 -12 0 -16 -8 D 2 12 16 0 8 E 2 2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6405: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) D B A E C (8) E C A B D (6) C E A B D (6) C E A D B (5) C A E D B (5) A E C D B (5) E B A C D (4) E A C B D (4) D B C A E (4) B D A E C (4) C D A E B (3) B E A C D (3) B D E C A (3) B D C E A (3) D C A B E (2) D B A C E (2) D A C E B (2) C B E A D (2) C A D E B (2) B E D A C (2) B E C A D (2) E C B A D (1) E C A D B (1) E B C A D (1) D C B A E (1) D C A E B (1) D A B C E (1) B E C D A (1) B E A D C (1) A E D B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 2 2 -14 B 8 0 4 10 2 C -2 -4 0 6 -18 D -2 -10 -6 0 -2 E 14 -2 18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 2 -14 B 8 0 4 10 2 C -2 -4 0 6 -18 D -2 -10 -6 0 -2 E 14 -2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998054 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=23 D=21 E=17 A=8 so A is eliminated. Round 2 votes counts: B=31 C=25 E=23 D=21 so D is eliminated. Round 3 votes counts: B=46 C=31 E=23 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:216 B:212 A:191 C:191 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 2 -14 B 8 0 4 10 2 C -2 -4 0 6 -18 D -2 -10 -6 0 -2 E 14 -2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998054 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 2 -14 B 8 0 4 10 2 C -2 -4 0 6 -18 D -2 -10 -6 0 -2 E 14 -2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998054 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 2 -14 B 8 0 4 10 2 C -2 -4 0 6 -18 D -2 -10 -6 0 -2 E 14 -2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998054 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6406: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) A B C E D (9) B E D A C (7) C A B D E (6) A C D E B (6) D E C B A (5) E D B A C (4) B E D C A (4) E B D A C (3) C D A E B (3) C B D E A (3) A D E C B (3) E D B C A (2) D E B C A (2) D E A B C (2) D C E A B (2) C D E B A (2) C D E A B (2) C B E D A (2) B C E D A (2) B C A E D (2) B A E D C (2) A D C E B (2) A C B E D (2) A C B D E (2) A B E D C (2) D E A C B (1) D C E B A (1) C B A E D (1) C A D B E (1) C A B E D (1) B E A D C (1) B A C E D (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -2 6 10 B -12 0 -12 0 -2 C 2 12 0 12 22 D -6 0 -12 0 10 E -10 2 -22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 6 10 B -12 0 -12 0 -2 C 2 12 0 12 22 D -6 0 -12 0 10 E -10 2 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=28 B=19 D=13 E=9 so E is eliminated. Round 2 votes counts: C=31 A=28 B=22 D=19 so D is eliminated. Round 3 votes counts: C=39 A=31 B=30 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:213 D:196 B:187 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 6 10 B -12 0 -12 0 -2 C 2 12 0 12 22 D -6 0 -12 0 10 E -10 2 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 6 10 B -12 0 -12 0 -2 C 2 12 0 12 22 D -6 0 -12 0 10 E -10 2 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 6 10 B -12 0 -12 0 -2 C 2 12 0 12 22 D -6 0 -12 0 10 E -10 2 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6407: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) E C B D A (8) E A D C B (7) D B C A E (6) A B D C E (6) A D E B C (5) A D B C E (5) B D C A E (4) A E D B C (4) A E B C D (4) D A E B C (3) C B E D A (3) B C A D E (3) A B C D E (3) E D A C B (2) E C B A D (2) E C A B D (2) E A C B D (2) D E A C B (2) D C B E A (2) C E B D A (2) B A C E D (2) E D C B A (1) E A C D B (1) D E C B A (1) D B C E A (1) D B A C E (1) D A B E C (1) D A B C E (1) C E B A D (1) B D A C E (1) B C E A D (1) B C D A E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 4 0 4 B 0 0 8 12 6 C -4 -8 0 -10 6 D 0 -12 10 0 14 E -4 -6 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.527772 B: 0.472228 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501542605399 Cumulative probabilities = A: 0.527772 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 0 4 B 0 0 8 12 6 C -4 -8 0 -10 6 D 0 -12 10 0 14 E -4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=25 D=18 C=16 B=12 so B is eliminated. Round 2 votes counts: A=31 E=25 D=23 C=21 so C is eliminated. Round 3 votes counts: D=34 A=34 E=32 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:213 D:206 A:204 C:192 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 4 B 0 0 8 12 6 C -4 -8 0 -10 6 D 0 -12 10 0 14 E -4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 4 B 0 0 8 12 6 C -4 -8 0 -10 6 D 0 -12 10 0 14 E -4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 4 B 0 0 8 12 6 C -4 -8 0 -10 6 D 0 -12 10 0 14 E -4 -6 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6408: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) C A E B D (10) B E A C D (8) D B E A C (7) D B E C A (6) A B E C D (5) D C B E A (4) B D E A C (4) D B A E C (3) C E A B D (3) B E D C A (3) B A E C D (3) A E C B D (3) E B A C D (2) D C A E B (2) C E B A D (2) C D A E B (2) C A D E B (2) B E A D C (2) B D E C A (2) A E B C D (2) A C E D B (2) E B C A D (1) D C E B A (1) D C A B E (1) D B C E A (1) D B A C E (1) D A C E B (1) D A C B E (1) C E B D A (1) C A E D B (1) B E D A C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 16 18 4 B 4 0 6 28 8 C -16 -6 0 16 -8 D -18 -28 -16 0 -20 E -4 -8 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 18 4 B 4 0 6 28 8 C -16 -6 0 16 -8 D -18 -28 -16 0 -20 E -4 -8 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990344 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=25 B=23 C=21 E=3 so E is eliminated. Round 2 votes counts: D=28 B=26 A=25 C=21 so C is eliminated. Round 3 votes counts: A=41 D=30 B=29 so B is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:223 A:217 E:208 C:193 D:159 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 16 18 4 B 4 0 6 28 8 C -16 -6 0 16 -8 D -18 -28 -16 0 -20 E -4 -8 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990344 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 18 4 B 4 0 6 28 8 C -16 -6 0 16 -8 D -18 -28 -16 0 -20 E -4 -8 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990344 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 18 4 B 4 0 6 28 8 C -16 -6 0 16 -8 D -18 -28 -16 0 -20 E -4 -8 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990344 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6409: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E C B A D (7) A D B E C (7) C E D B A (5) C E B D A (4) B C E A D (4) D C B A E (3) D A E C B (3) C E D A B (3) C D E A B (3) B A D E C (3) E C D A B (2) E C A D B (2) E C A B D (2) E B A C D (2) E A D B C (2) E A B D C (2) E A B C D (2) D B A C E (2) D A C B E (2) C E B A D (2) C D B A E (2) C B E D A (2) B C D A E (2) A E D B C (2) A B D E C (2) E B C A D (1) E A D C B (1) E A C B D (1) D C A E B (1) D C A B E (1) D B C A E (1) D A E B C (1) D A C E B (1) C D B E A (1) C D A E B (1) C B E A D (1) C B D A E (1) B E A D C (1) B E A C D (1) B A E D C (1) B A E C D (1) B A D C E (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 -2 0 -2 B -4 0 -2 -12 -4 C 2 2 0 4 4 D 0 12 -4 0 -4 E 2 4 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 0 -2 B -4 0 -2 -12 -4 C 2 2 0 4 4 D 0 12 -4 0 -4 E 2 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 D=24 B=14 A=13 so A is eliminated. Round 2 votes counts: D=32 E=27 C=25 B=16 so B is eliminated. Round 3 votes counts: D=38 E=31 C=31 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:206 E:203 D:202 A:200 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 0 -2 B -4 0 -2 -12 -4 C 2 2 0 4 4 D 0 12 -4 0 -4 E 2 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 0 -2 B -4 0 -2 -12 -4 C 2 2 0 4 4 D 0 12 -4 0 -4 E 2 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 0 -2 B -4 0 -2 -12 -4 C 2 2 0 4 4 D 0 12 -4 0 -4 E 2 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6410: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) D C A B E (9) E B D A C (8) C D A B E (6) C A D B E (6) A C D E B (5) E B A D C (4) E B A C D (4) D B C E A (3) D A C E B (3) C D B A E (3) B E C D A (3) A D C E B (3) E A D B C (2) E A B C D (2) D C B A E (2) D B E C A (2) D A E B C (2) C A B E D (2) B E C A D (2) A E D C B (2) D E B A C (1) D A E C B (1) C B D A E (1) C B A E D (1) C B A D E (1) C A E B D (1) C A B D E (1) B E D A C (1) B D E C A (1) B C E D A (1) B C E A D (1) A E D B C (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -16 -18 6 B 2 0 0 -6 16 C 16 0 0 -14 2 D 18 6 14 0 0 E -6 -16 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.830104 E: 0.169896 Sum of squares = 0.717937124156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.830104 E: 1.000000 A B C D E A 0 -2 -16 -18 6 B 2 0 0 -6 16 C 16 0 0 -14 2 D 18 6 14 0 0 E -6 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 0.272727 Sum of squares = 0.603305797868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 B=21 E=20 A=14 so A is eliminated. Round 2 votes counts: C=29 D=26 E=24 B=21 so B is eliminated. Round 3 votes counts: E=42 C=31 D=27 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:219 B:206 C:202 E:188 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -16 -18 6 B 2 0 0 -6 16 C 16 0 0 -14 2 D 18 6 14 0 0 E -6 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 0.272727 Sum of squares = 0.603305797868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -18 6 B 2 0 0 -6 16 C 16 0 0 -14 2 D 18 6 14 0 0 E -6 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 0.272727 Sum of squares = 0.603305797868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -18 6 B 2 0 0 -6 16 C 16 0 0 -14 2 D 18 6 14 0 0 E -6 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 0.272727 Sum of squares = 0.603305797868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.727273 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6411: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (7) A C B D E (5) A B E D C (5) B A D E C (4) B A D C E (4) A B D C E (4) E C D B A (3) E B D A C (3) D B E C A (3) C A D B E (3) A E B C D (3) E D C B A (2) E D B C A (2) E C D A B (2) E C A D B (2) E C A B D (2) E B A D C (2) E A C B D (2) E A B C D (2) D E C B A (2) D C E B A (2) D C B A E (2) D B C A E (2) D B A C E (2) C E D B A (2) C E A D B (2) C D B E A (2) B D A E C (2) B D A C E (2) A C E B D (2) A B C E D (2) E B D C A (1) E B C A D (1) E A B D C (1) D E B C A (1) D C B E A (1) D B C E A (1) C D E B A (1) C D B A E (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E A C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -2 0 -2 B 0 0 0 0 0 C 2 0 0 0 4 D 0 0 0 0 -2 E 2 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.317002 C: 0.380320 D: 0.302678 E: 0.000000 Sum of squares = 0.336747536785 Cumulative probabilities = A: 0.000000 B: 0.317002 C: 0.697322 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 0 -2 B 0 0 0 0 0 C 2 0 0 0 4 D 0 0 0 0 -2 E 2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333334 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 C=22 D=16 B=14 so B is eliminated. Round 2 votes counts: A=31 E=26 C=22 D=21 so D is eliminated. Round 3 votes counts: A=37 E=33 C=30 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:203 B:200 E:200 D:199 A:198 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 0 -2 B 0 0 0 0 0 C 2 0 0 0 4 D 0 0 0 0 -2 E 2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333334 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 0 -2 B 0 0 0 0 0 C 2 0 0 0 4 D 0 0 0 0 -2 E 2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333334 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 0 -2 B 0 0 0 0 0 C 2 0 0 0 4 D 0 0 0 0 -2 E 2 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333334 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6412: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (16) D B A C E (14) A C E D B (8) B D E C A (7) E C B A D (6) A D C E B (6) B E C D A (4) B E C A D (4) E C A D B (3) E A C D B (3) D A C E B (3) D A B C E (3) C E A D B (3) B E D C A (3) B D E A C (2) B D A E C (2) B D A C E (2) E B C A D (1) E A C B D (1) D C A E B (1) D B C A E (1) D A C B E (1) C E B D A (1) C A E D B (1) B E A C D (1) B D C A E (1) B C E D A (1) A E C D B (1) Total count = 100 A B C D E A 0 0 -6 8 -12 B 0 0 -8 4 -8 C 6 8 0 8 -8 D -8 -4 -8 0 -14 E 12 8 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -6 8 -12 B 0 0 -8 4 -8 C 6 8 0 8 -8 D -8 -4 -8 0 -14 E 12 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=27 D=23 A=15 C=5 so C is eliminated. Round 2 votes counts: E=34 B=27 D=23 A=16 so A is eliminated. Round 3 votes counts: E=44 D=29 B=27 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:207 A:195 B:194 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 8 -12 B 0 0 -8 4 -8 C 6 8 0 8 -8 D -8 -4 -8 0 -14 E 12 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 8 -12 B 0 0 -8 4 -8 C 6 8 0 8 -8 D -8 -4 -8 0 -14 E 12 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 8 -12 B 0 0 -8 4 -8 C 6 8 0 8 -8 D -8 -4 -8 0 -14 E 12 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6413: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) C D E A B (6) B A E D C (6) B A C D E (6) A E C D B (6) D B C E A (5) B D C A E (5) B D E C A (4) B C D A E (4) A B C E D (4) D C E B A (3) C A B D E (3) E D C A B (2) E C D A B (2) C E A D B (2) C D A E B (2) B E D A C (2) B E A D C (2) B D E A C (2) A E C B D (2) A C E B D (2) A C B E D (2) E D B A C (1) E A C D B (1) D E C A B (1) D E B C A (1) D B E C A (1) C E D A B (1) C D E B A (1) C D B E A (1) C D A B E (1) C B D A E (1) C A E D B (1) C A D E B (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C E D (1) A E B D C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -16 -10 6 B 16 0 16 20 26 C 16 -16 0 8 26 D 10 -20 -8 0 18 E -6 -26 -26 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -10 6 B 16 0 16 20 26 C 16 -16 0 8 26 D 10 -20 -8 0 18 E -6 -26 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 C=20 A=19 D=11 E=6 so E is eliminated. Round 2 votes counts: B=44 C=22 A=20 D=14 so D is eliminated. Round 3 votes counts: B=52 C=28 A=20 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:239 C:217 D:200 A:182 E:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -16 -10 6 B 16 0 16 20 26 C 16 -16 0 8 26 D 10 -20 -8 0 18 E -6 -26 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -10 6 B 16 0 16 20 26 C 16 -16 0 8 26 D 10 -20 -8 0 18 E -6 -26 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -10 6 B 16 0 16 20 26 C 16 -16 0 8 26 D 10 -20 -8 0 18 E -6 -26 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6414: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) C E A B D (9) D B A E C (7) A B D C E (7) E C D B A (6) A D B E C (6) D B E C A (5) D B E A C (5) E C B D A (4) C E B D A (4) C A E B D (4) A C E D B (4) A C E B D (4) C E A D B (3) A C D B E (3) A B D E C (3) E D C B A (2) E D B C A (1) D E B C A (1) D A B E C (1) C E B A D (1) C A E D B (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A E C (1) B A D E C (1) A D C B E (1) A D B C E (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -4 2 -6 B 0 0 -14 -12 -6 C 4 14 0 10 8 D -2 12 -10 0 -8 E 6 6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 -6 B 0 0 -14 -12 -6 C 4 14 0 10 8 D -2 12 -10 0 -8 E 6 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=31 D=19 E=13 B=5 so B is eliminated. Round 2 votes counts: A=33 C=31 D=22 E=14 so E is eliminated. Round 3 votes counts: C=41 A=33 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:206 A:196 D:196 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 2 -6 B 0 0 -14 -12 -6 C 4 14 0 10 8 D -2 12 -10 0 -8 E 6 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 -6 B 0 0 -14 -12 -6 C 4 14 0 10 8 D -2 12 -10 0 -8 E 6 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 -6 B 0 0 -14 -12 -6 C 4 14 0 10 8 D -2 12 -10 0 -8 E 6 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6415: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (15) E C A B D (12) B D A E C (9) C E D B A (7) C E D A B (5) C D B A E (5) E A B D C (4) D A B C E (4) C E A D B (4) C D B E A (4) B A D E C (4) D B C A E (3) C E B D A (3) E A B C D (2) C E A B D (2) B E A D C (2) A E B D C (2) A B E D C (2) A B D E C (2) E C A D B (1) E A C D B (1) D C B A E (1) D B A E C (1) C E B A D (1) C D E B A (1) C D E A B (1) C D A B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -2 -20 0 B 12 0 2 -10 8 C 2 -2 0 0 14 D 20 10 0 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.481514 D: 0.518486 E: 0.000000 Sum of squares = 0.500683494418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.481514 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -20 0 B 12 0 2 -10 8 C 2 -2 0 0 14 D 20 10 0 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=24 E=20 B=15 A=7 so A is eliminated. Round 2 votes counts: C=34 D=25 E=22 B=19 so B is eliminated. Round 3 votes counts: D=40 C=34 E=26 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:207 B:206 E:187 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -20 0 B 12 0 2 -10 8 C 2 -2 0 0 14 D 20 10 0 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -20 0 B 12 0 2 -10 8 C 2 -2 0 0 14 D 20 10 0 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -20 0 B 12 0 2 -10 8 C 2 -2 0 0 14 D 20 10 0 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6416: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E A D B C (6) E B C A D (5) D C B A E (5) C B E D A (5) D B C E A (4) C B D E A (4) E B C D A (3) E A B D C (3) D B E C A (3) D A C B E (3) C D B A E (3) C B E A D (3) B C D E A (3) A E D B C (3) E D B A C (2) E C B A D (2) E A C B D (2) D E B A C (2) D C A B E (2) D A B C E (2) B E D C A (2) A E D C B (2) A E C B D (2) A D E B C (2) A D C E B (2) A C D B E (2) E A B C D (1) D E B C A (1) D B C A E (1) D A E B C (1) C A E B D (1) C A B E D (1) C A B D E (1) B E C D A (1) B D E C A (1) B C E D A (1) B C E A D (1) A E C D B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -22 -24 -16 -10 B 22 0 -4 4 16 C 24 4 0 4 10 D 16 -4 -4 0 4 E 10 -16 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -24 -16 -10 B 22 0 -4 4 16 C 24 4 0 4 10 D 16 -4 -4 0 4 E 10 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=24 D=24 A=16 B=9 so B is eliminated. Round 2 votes counts: C=32 E=27 D=25 A=16 so A is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:219 D:206 E:190 A:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -24 -16 -10 B 22 0 -4 4 16 C 24 4 0 4 10 D 16 -4 -4 0 4 E 10 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -24 -16 -10 B 22 0 -4 4 16 C 24 4 0 4 10 D 16 -4 -4 0 4 E 10 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -24 -16 -10 B 22 0 -4 4 16 C 24 4 0 4 10 D 16 -4 -4 0 4 E 10 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6417: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) D B E C A (6) A C B E D (6) E B D C A (5) C B A D E (5) C A B D E (5) B C D A E (5) A C E B D (5) E D B C A (4) E D B A C (4) E D A B C (3) E A D C B (3) D E B C A (3) C B A E D (3) B C A D E (3) A E C D B (3) A C D E B (3) D E B A C (2) D E A B C (2) D B C A E (2) B D E C A (2) B D C E A (2) B C E D A (2) B C D E A (2) A E D C B (2) A E C B D (2) A C E D B (2) E A C D B (1) E A C B D (1) D B C E A (1) D A C B E (1) C D B A E (1) C B E A D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -18 6 12 B 6 0 -4 14 10 C 18 4 0 16 14 D -6 -14 -16 0 -8 E -12 -10 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 6 12 B 6 0 -4 14 10 C 18 4 0 16 14 D -6 -14 -16 0 -8 E -12 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 E=21 D=17 B=16 so B is eliminated. Round 2 votes counts: C=34 A=24 E=21 D=21 so E is eliminated. Round 3 votes counts: D=37 C=34 A=29 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:213 A:197 E:186 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -18 6 12 B 6 0 -4 14 10 C 18 4 0 16 14 D -6 -14 -16 0 -8 E -12 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 6 12 B 6 0 -4 14 10 C 18 4 0 16 14 D -6 -14 -16 0 -8 E -12 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 6 12 B 6 0 -4 14 10 C 18 4 0 16 14 D -6 -14 -16 0 -8 E -12 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6418: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) B A E C D (7) B A C E D (6) E D B A C (5) D E C B A (5) D C E A B (5) C A B E D (5) A B C E D (5) B E A D C (4) B A C D E (4) A C B E D (4) E A C B D (3) D E B C A (3) D E B A C (3) C D A E B (3) C A D B E (3) C A B D E (3) E B A D C (2) E B A C D (2) D B E A C (2) C A E B D (2) C A D E B (2) B D E A C (2) E D C B A (1) E D C A B (1) E D A C B (1) E C A B D (1) E B D A C (1) D C A E B (1) D C A B E (1) D B C A E (1) C D A B E (1) C A E D B (1) B E D A C (1) B E A C D (1) B A E D C (1) Total count = 100 A B C D E A 0 -2 8 12 0 B 2 0 0 8 2 C -8 0 0 6 -6 D -12 -8 -6 0 -8 E 0 -2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.904850 C: 0.095150 D: 0.000000 E: 0.000000 Sum of squares = 0.827806722993 Cumulative probabilities = A: 0.000000 B: 0.904850 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 12 0 B 2 0 0 8 2 C -8 0 0 6 -6 D -12 -8 -6 0 -8 E 0 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015198 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=26 C=20 E=17 A=9 so A is eliminated. Round 2 votes counts: B=31 D=28 C=24 E=17 so E is eliminated. Round 3 votes counts: D=36 B=36 C=28 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:206 E:206 C:196 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 12 0 B 2 0 0 8 2 C -8 0 0 6 -6 D -12 -8 -6 0 -8 E 0 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015198 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 12 0 B 2 0 0 8 2 C -8 0 0 6 -6 D -12 -8 -6 0 -8 E 0 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015198 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 12 0 B 2 0 0 8 2 C -8 0 0 6 -6 D -12 -8 -6 0 -8 E 0 -2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015198 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6419: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) C E D B A (11) A B D E C (8) E C A D B (7) D C E B A (7) E C D A B (6) E C B A D (5) A B E C D (5) E C D B A (4) C E B D A (4) A E C B D (4) D B A C E (3) B D A C E (3) B A D C E (3) A B D C E (3) E C B D A (2) D A B C E (2) C D E B A (2) A D B E C (2) D E C A B (1) D B C E A (1) D A B E C (1) C E B A D (1) B A C E D (1) A E C D B (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -24 6 -24 B -6 0 -34 2 -34 C 24 34 0 28 -18 D -6 -2 -28 0 -26 E 24 34 18 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -24 6 -24 B -6 0 -34 2 -34 C 24 34 0 28 -18 D -6 -2 -28 0 -26 E 24 34 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=25 C=18 D=15 B=7 so B is eliminated. Round 2 votes counts: E=35 A=29 D=18 C=18 so D is eliminated. Round 3 votes counts: A=38 E=36 C=26 so C is eliminated. Round 4 votes counts: E=62 A=38 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:251 C:234 A:182 D:169 B:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -24 6 -24 B -6 0 -34 2 -34 C 24 34 0 28 -18 D -6 -2 -28 0 -26 E 24 34 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -24 6 -24 B -6 0 -34 2 -34 C 24 34 0 28 -18 D -6 -2 -28 0 -26 E 24 34 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -24 6 -24 B -6 0 -34 2 -34 C 24 34 0 28 -18 D -6 -2 -28 0 -26 E 24 34 18 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6420: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (12) B A C D E (11) E D C A B (10) D E A C B (8) D A E B C (5) C E B D A (5) A D B E C (5) E D A C B (4) C B E D A (4) C E D B A (3) B C A D E (3) A B D C E (3) A B C D E (3) E C D B A (2) D E A B C (2) C B E A D (2) B C E D A (2) B C E A D (2) A B D E C (2) E D C B A (1) E D B A C (1) E C D A B (1) E C B D A (1) D A E C B (1) C E D A B (1) C B A D E (1) B D A E C (1) B A D C E (1) B A C E D (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 0 -4 2 B 6 0 10 8 6 C 0 -10 0 8 8 D 4 -8 -8 0 -4 E -2 -6 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -4 2 B 6 0 10 8 6 C 0 -10 0 8 8 D 4 -8 -8 0 -4 E -2 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=20 D=16 C=16 A=15 so A is eliminated. Round 2 votes counts: B=41 D=23 E=20 C=16 so C is eliminated. Round 3 votes counts: B=48 E=29 D=23 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:203 A:196 E:194 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -4 2 B 6 0 10 8 6 C 0 -10 0 8 8 D 4 -8 -8 0 -4 E -2 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -4 2 B 6 0 10 8 6 C 0 -10 0 8 8 D 4 -8 -8 0 -4 E -2 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -4 2 B 6 0 10 8 6 C 0 -10 0 8 8 D 4 -8 -8 0 -4 E -2 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6421: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (12) D A C B E (8) D B A E C (6) B E C D A (6) D A B E C (5) C B E D A (5) A C D E B (5) E C B A D (4) E B C D A (4) B E D C A (4) A D C E B (4) D B E C A (3) C E B D A (3) C E A B D (3) B D E C A (3) A D C B E (3) A C E B D (3) C A E B D (2) A C E D B (2) E B C A D (1) E B A C D (1) E A B D C (1) D B E A C (1) D B C E A (1) D A B C E (1) C D A B E (1) C A E D B (1) C A D E B (1) B E D A C (1) B C E D A (1) A E C B D (1) A E B C D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -10 -6 -8 B 12 0 -16 12 0 C 10 16 0 14 12 D 6 -12 -14 0 -12 E 8 0 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -6 -8 B 12 0 -16 12 0 C 10 16 0 14 12 D 6 -12 -14 0 -12 E 8 0 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=25 A=21 B=15 E=11 so E is eliminated. Round 2 votes counts: C=32 D=25 A=22 B=21 so B is eliminated. Round 3 votes counts: C=44 D=33 A=23 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:204 E:204 D:184 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 -6 -8 B 12 0 -16 12 0 C 10 16 0 14 12 D 6 -12 -14 0 -12 E 8 0 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -6 -8 B 12 0 -16 12 0 C 10 16 0 14 12 D 6 -12 -14 0 -12 E 8 0 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -6 -8 B 12 0 -16 12 0 C 10 16 0 14 12 D 6 -12 -14 0 -12 E 8 0 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6422: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B E C D A (8) A D C E B (8) B A E C D (6) E C D B A (5) D A C E B (5) C D E A B (5) A D C B E (5) E C B D A (4) B A E D C (4) A D B C E (4) A B D E C (4) E B C D A (3) D C A E B (3) C E D B A (3) C E B D A (3) B E C A D (3) B A D E C (3) D E C B A (2) C E D A B (2) B E D C A (2) A D B E C (2) A B D C E (2) E B D C A (1) C E B A D (1) B E A D C (1) B E A C D (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -6 -8 -4 B 2 0 -10 -6 -6 C 6 10 0 -8 2 D 8 6 8 0 4 E 4 6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -8 -4 B 2 0 -10 -6 -6 C 6 10 0 -8 2 D 8 6 8 0 4 E 4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=26 D=18 C=14 E=13 so E is eliminated. Round 2 votes counts: B=33 A=26 C=23 D=18 so D is eliminated. Round 3 votes counts: C=36 B=33 A=31 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:205 E:202 A:190 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 -4 B 2 0 -10 -6 -6 C 6 10 0 -8 2 D 8 6 8 0 4 E 4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 -4 B 2 0 -10 -6 -6 C 6 10 0 -8 2 D 8 6 8 0 4 E 4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 -4 B 2 0 -10 -6 -6 C 6 10 0 -8 2 D 8 6 8 0 4 E 4 6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6423: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (24) C E B A D (11) D C E A B (9) B A E C D (7) B A C E D (6) D A B C E (5) C E D B A (5) A B D E C (5) C E D A B (4) D B A C E (3) E D C A B (2) E C B A D (2) D E C A B (2) D C E B A (2) D A E B C (2) B A D E C (2) A D B E C (2) E B C A D (1) E A B C D (1) D C B E A (1) B C E A D (1) B A E D C (1) B A D C E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 20 -18 18 B -14 0 24 -22 18 C -20 -24 0 -22 -4 D 18 22 22 0 16 E -18 -18 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 -18 18 B -14 0 24 -22 18 C -20 -24 0 -22 -4 D 18 22 22 0 16 E -18 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=48 C=20 B=18 A=8 E=6 so E is eliminated. Round 2 votes counts: D=50 C=22 B=19 A=9 so A is eliminated. Round 3 votes counts: D=52 B=26 C=22 so C is eliminated. Round 4 votes counts: D=61 B=39 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:239 A:217 B:203 E:176 C:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 20 -18 18 B -14 0 24 -22 18 C -20 -24 0 -22 -4 D 18 22 22 0 16 E -18 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 -18 18 B -14 0 24 -22 18 C -20 -24 0 -22 -4 D 18 22 22 0 16 E -18 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 -18 18 B -14 0 24 -22 18 C -20 -24 0 -22 -4 D 18 22 22 0 16 E -18 -18 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6424: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) A E C D B (10) E A D B C (8) C B D A E (8) A E D B C (8) C A E D B (7) B D C E A (7) D B E A C (6) C A E B D (4) B C D E A (4) E D A B C (3) D E A B C (3) B D E A C (3) D E B A C (2) C E A D B (2) B D E C A (2) A E D C B (2) A E B D C (2) E A D C B (1) D C B E A (1) C D B E A (1) C B A D E (1) C A B E D (1) B D C A E (1) B D A E C (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 6 2 -4 -6 B -6 0 0 -10 -8 C -2 0 0 0 -4 D 4 10 0 0 0 E 6 8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.429033 E: 0.570967 Sum of squares = 0.510072679082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.429033 E: 1.000000 A B C D E A 0 6 2 -4 -6 B -6 0 0 -10 -8 C -2 0 0 0 -4 D 4 10 0 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=24 B=18 E=12 D=12 so E is eliminated. Round 2 votes counts: C=34 A=33 B=18 D=15 so D is eliminated. Round 3 votes counts: A=39 C=35 B=26 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:209 D:207 A:199 C:197 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -4 -6 B -6 0 0 -10 -8 C -2 0 0 0 -4 D 4 10 0 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -4 -6 B -6 0 0 -10 -8 C -2 0 0 0 -4 D 4 10 0 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -4 -6 B -6 0 0 -10 -8 C -2 0 0 0 -4 D 4 10 0 0 0 E 6 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6425: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (18) B E A C D (11) B A C E D (10) B E C A D (6) E B C A D (5) D C A E B (5) D A C B E (4) E B D C A (3) D B A C E (3) D A C E B (3) E D C A B (2) E D B C A (2) E C A D B (2) E C A B D (2) D C E A B (2) D B E A C (2) A C D B E (2) A C B E D (2) E D C B A (1) E C D A B (1) E C B A D (1) D E A C B (1) D B E C A (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D A C E (1) B A D C E (1) A C D E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -12 -4 -26 B 0 0 -2 -6 -4 C 12 2 0 -4 -22 D 4 6 4 0 -4 E 26 4 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999587 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -12 -4 -26 B 0 0 -2 -6 -4 C 12 2 0 -4 -22 D 4 6 4 0 -4 E 26 4 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=32 E=19 A=7 C=3 so C is eliminated. Round 2 votes counts: D=40 B=32 E=19 A=9 so A is eliminated. Round 3 votes counts: D=44 B=36 E=20 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:228 D:205 B:194 C:194 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -12 -4 -26 B 0 0 -2 -6 -4 C 12 2 0 -4 -22 D 4 6 4 0 -4 E 26 4 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -4 -26 B 0 0 -2 -6 -4 C 12 2 0 -4 -22 D 4 6 4 0 -4 E 26 4 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -4 -26 B 0 0 -2 -6 -4 C 12 2 0 -4 -22 D 4 6 4 0 -4 E 26 4 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999973568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6426: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (11) E D A C B (7) D E C B A (7) B C A D E (7) E D C A B (5) E D A B C (4) C B D A E (4) E D C B A (3) D C E B A (3) C B A D E (3) B C D A E (3) B A C D E (3) A C B E D (3) A C B D E (3) E D B C A (2) E D B A C (2) E A D B C (2) C A B D E (2) A E D C B (2) A E B D C (2) A B E D C (2) A B C D E (2) E A B D C (1) D E B C A (1) C D E B A (1) C D E A B (1) C D B E A (1) C B D E A (1) C A D B E (1) B E A D C (1) B D E C A (1) B C D E A (1) B A E D C (1) B A C E D (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B C D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 8 6 6 12 B -8 0 0 10 4 C -6 0 0 6 4 D -6 -10 -6 0 -10 E -12 -4 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 6 12 B -8 0 0 10 4 C -6 0 0 6 4 D -6 -10 -6 0 -10 E -12 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=26 B=18 C=14 D=11 so D is eliminated. Round 2 votes counts: E=34 A=31 B=18 C=17 so C is eliminated. Round 3 votes counts: E=39 A=34 B=27 so B is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:203 C:202 E:195 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 6 12 B -8 0 0 10 4 C -6 0 0 6 4 D -6 -10 -6 0 -10 E -12 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 6 12 B -8 0 0 10 4 C -6 0 0 6 4 D -6 -10 -6 0 -10 E -12 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 6 12 B -8 0 0 10 4 C -6 0 0 6 4 D -6 -10 -6 0 -10 E -12 -4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6427: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) C A D E B (9) B E D A C (8) C A E D B (6) D C A B E (5) B D E C A (5) E B C A D (4) E B A C D (4) E A B C D (4) C D A B E (4) B D E A C (4) B D C E A (4) A C E D B (4) E A C B D (3) D B A C E (3) E B A D C (2) D B E A C (2) D A C B E (2) C E A B D (2) C A E B D (2) C A D B E (2) A E D B C (2) A C D E B (2) E A B D C (1) D C B A E (1) C D B A E (1) B E D C A (1) B E C D A (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 0 -14 -2 10 B 0 0 12 -10 6 C 14 -12 0 -2 14 D 2 10 2 0 10 E -10 -6 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -2 10 B 0 0 12 -10 6 C 14 -12 0 -2 14 D 2 10 2 0 10 E -10 -6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=23 B=23 E=18 A=10 so A is eliminated. Round 2 votes counts: C=32 D=24 B=23 E=21 so E is eliminated. Round 3 votes counts: B=38 C=36 D=26 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:212 C:207 B:204 A:197 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -14 -2 10 B 0 0 12 -10 6 C 14 -12 0 -2 14 D 2 10 2 0 10 E -10 -6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -2 10 B 0 0 12 -10 6 C 14 -12 0 -2 14 D 2 10 2 0 10 E -10 -6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -2 10 B 0 0 12 -10 6 C 14 -12 0 -2 14 D 2 10 2 0 10 E -10 -6 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6428: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) E C B A D (6) E B C A D (6) E C B D A (5) C E B D A (5) B C D E A (5) B C D A E (5) A D E C B (5) E A D C B (4) D A C B E (4) A D B C E (4) D A B C E (3) C B E D A (3) C B D E A (3) B E C A D (3) B D A C E (3) A D B E C (3) E B A C D (2) E A B C D (2) D A C E B (2) C B D A E (2) A E B D C (2) E C A D B (1) E B C D A (1) E A C D B (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A C E (1) C D E B A (1) C D B E A (1) C D B A E (1) B D C A E (1) B C E A D (1) B A D C E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -32 -22 -14 -16 B 32 0 6 28 10 C 22 -6 0 24 14 D 14 -28 -24 0 -4 E 16 -10 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999635 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 -22 -14 -16 B 32 0 6 28 10 C 22 -6 0 24 14 D 14 -28 -24 0 -4 E 16 -10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 C=16 A=16 D=13 so D is eliminated. Round 2 votes counts: B=29 E=28 A=25 C=18 so C is eliminated. Round 3 votes counts: B=40 E=34 A=26 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:238 C:227 E:198 D:179 A:158 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -32 -22 -14 -16 B 32 0 6 28 10 C 22 -6 0 24 14 D 14 -28 -24 0 -4 E 16 -10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 -22 -14 -16 B 32 0 6 28 10 C 22 -6 0 24 14 D 14 -28 -24 0 -4 E 16 -10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 -22 -14 -16 B 32 0 6 28 10 C 22 -6 0 24 14 D 14 -28 -24 0 -4 E 16 -10 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998893 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6429: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (7) D C B A E (6) C E D A B (6) B A E D C (6) E C D A B (5) D B A E C (5) B A D E C (5) D B C A E (4) B A D C E (4) A B E C D (4) E C A D B (3) E C A B D (3) E A B D C (3) E A B C D (3) D B A C E (3) B D A C E (3) A E B C D (3) E D C A B (2) E A C B D (2) D C E B A (2) C E A D B (2) C E A B D (2) C D A B E (2) B A C D E (2) A B E D C (2) E B D A C (1) D E B A C (1) D C E A B (1) D C B E A (1) C D E B A (1) C D B A E (1) B E A D C (1) B D A E C (1) A E C B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 4 -4 8 B -6 0 4 -4 4 C -4 -4 0 -2 -2 D 4 4 2 0 0 E -8 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.774674 E: 0.225326 Sum of squares = 0.650891923175 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.774674 E: 1.000000 A B C D E A 0 6 4 -4 8 B -6 0 4 -4 4 C -4 -4 0 -2 -2 D 4 4 2 0 0 E -8 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=22 B=22 C=21 A=12 so A is eliminated. Round 2 votes counts: B=29 E=26 D=23 C=22 so C is eliminated. Round 3 votes counts: E=36 D=34 B=30 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:205 B:199 E:195 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 8 B -6 0 4 -4 4 C -4 -4 0 -2 -2 D 4 4 2 0 0 E -8 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 8 B -6 0 4 -4 4 C -4 -4 0 -2 -2 D 4 4 2 0 0 E -8 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 8 B -6 0 4 -4 4 C -4 -4 0 -2 -2 D 4 4 2 0 0 E -8 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555557057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6430: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) B D C A E (7) E A C B D (6) A E C D B (6) E B D A C (5) E B A D C (5) D B C A E (5) B E D A C (5) B D E C A (5) C A D E B (4) B E D C A (4) B D C E A (4) E A C D B (3) D B C E A (3) C D A B E (3) C A E D B (3) C A D B E (3) E A B C D (2) B E A D C (2) B D A E C (2) A E C B D (2) E C A D B (1) E A B D C (1) D C B E A (1) C D B A E (1) B D E A C (1) B D A C E (1) B A D E C (1) A E B D C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 12 6 4 B 4 0 8 8 -2 C -12 -8 0 -6 -4 D -6 -8 6 0 -14 E -4 2 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000005 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -4 12 6 4 B 4 0 8 8 -2 C -12 -8 0 -6 -4 D -6 -8 6 0 -14 E -4 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000009 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=23 A=22 C=14 D=9 so D is eliminated. Round 2 votes counts: B=40 E=23 A=22 C=15 so C is eliminated. Round 3 votes counts: B=42 A=35 E=23 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:209 E:208 D:189 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 6 4 B 4 0 8 8 -2 C -12 -8 0 -6 -4 D -6 -8 6 0 -14 E -4 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000009 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 6 4 B 4 0 8 8 -2 C -12 -8 0 -6 -4 D -6 -8 6 0 -14 E -4 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000009 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 6 4 B 4 0 8 8 -2 C -12 -8 0 -6 -4 D -6 -8 6 0 -14 E -4 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000009 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6431: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) A D B E C (8) E C B A D (7) D C A B E (7) D A C B E (6) C D E A B (6) D C A E B (5) C E D B A (5) C E B D A (5) D A B C E (4) B A E D C (4) A B D E C (4) B E C A D (3) B E A C D (3) E B A C D (2) E A B C D (2) C E D A B (2) C D E B A (2) B C E A D (2) B A E C D (2) B A D E C (2) E C D A B (1) E C B D A (1) E C A B D (1) E A B D C (1) D A C E B (1) C B E D A (1) C B D E A (1) C B D A E (1) B D C A E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -18 2 -6 B 0 0 -4 4 0 C 18 4 0 10 -2 D -2 -4 -10 0 -2 E 6 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.226911 C: 0.000000 D: 0.000000 E: 0.773089 Sum of squares = 0.649154745573 Cumulative probabilities = A: 0.000000 B: 0.226911 C: 0.226911 D: 0.226911 E: 1.000000 A B C D E A 0 0 -18 2 -6 B 0 0 -4 4 0 C 18 4 0 10 -2 D -2 -4 -10 0 -2 E 6 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555569944 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 C=23 B=17 A=14 so A is eliminated. Round 2 votes counts: D=32 E=23 C=23 B=22 so B is eliminated. Round 3 votes counts: D=39 E=36 C=25 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:205 B:200 D:191 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -18 2 -6 B 0 0 -4 4 0 C 18 4 0 10 -2 D -2 -4 -10 0 -2 E 6 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555569944 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 2 -6 B 0 0 -4 4 0 C 18 4 0 10 -2 D -2 -4 -10 0 -2 E 6 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555569944 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 2 -6 B 0 0 -4 4 0 C 18 4 0 10 -2 D -2 -4 -10 0 -2 E 6 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555569944 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6432: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (6) B D A C E (6) D B E C A (5) B D C E A (5) A C E B D (5) E A C D B (4) D B E A C (4) D B A C E (4) D B A E C (3) C E A B D (3) C A E B D (3) C A B E D (3) B D E C A (3) B C E D A (3) B C A E D (3) A E C D B (3) A C E D B (3) E C D B A (2) D E C A B (2) D E B C A (2) D E B A C (2) C B A E D (2) B C E A D (2) B C D E A (2) B A D C E (2) A D E C B (2) E D C A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E C A B D (1) D E A C B (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) B D C A E (1) B C A D E (1) A E D C B (1) A D C E B (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 4 -12 8 B 10 0 6 -2 6 C -4 -6 0 -10 6 D 12 2 10 0 12 E -8 -6 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -12 8 B 10 0 6 -2 6 C -4 -6 0 -10 6 D 12 2 10 0 12 E -8 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=17 C=12 E=11 so E is eliminated. Round 2 votes counts: D=33 B=28 A=21 C=18 so C is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:210 A:195 C:193 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 4 -12 8 B 10 0 6 -2 6 C -4 -6 0 -10 6 D 12 2 10 0 12 E -8 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -12 8 B 10 0 6 -2 6 C -4 -6 0 -10 6 D 12 2 10 0 12 E -8 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -12 8 B 10 0 6 -2 6 C -4 -6 0 -10 6 D 12 2 10 0 12 E -8 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6433: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) A C D E B (10) E B A D C (8) B D C E A (7) E A C D B (5) B E D C A (5) B D C A E (5) E A B C D (4) B A C D E (4) B E D A C (3) A E C B D (3) A C E D B (3) E B D A C (2) D C B E A (2) D C A B E (2) C D B A E (2) C D A E B (2) C A D E B (2) B E A D C (2) B E A C D (2) B D E C A (2) B A E C D (2) B A C E D (2) A C D B E (2) E D C B A (1) E B D C A (1) E A B D C (1) D E C A B (1) D E B C A (1) D B C E A (1) C D A B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 30 24 4 B 4 0 6 10 -8 C -30 -6 0 12 -6 D -24 -10 -12 0 -10 E -4 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000095 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 30 24 4 B 4 0 6 10 -8 C -30 -6 0 12 -6 D -24 -10 -12 0 -10 E -4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999967 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=30 E=22 D=7 C=7 so D is eliminated. Round 2 votes counts: B=35 A=30 E=24 C=11 so C is eliminated. Round 3 votes counts: B=39 A=37 E=24 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:227 E:210 B:206 C:185 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 30 24 4 B 4 0 6 10 -8 C -30 -6 0 12 -6 D -24 -10 -12 0 -10 E -4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999967 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 30 24 4 B 4 0 6 10 -8 C -30 -6 0 12 -6 D -24 -10 -12 0 -10 E -4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999967 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 30 24 4 B 4 0 6 10 -8 C -30 -6 0 12 -6 D -24 -10 -12 0 -10 E -4 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999967 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6434: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) A E B C D (6) A C B D E (6) C D B A E (5) B D E C A (5) E A B D C (4) D E B C A (4) D B C E A (4) C D B E A (4) E B D C A (3) E A D C B (3) C B D A E (3) A E B D C (3) A C E D B (3) E D C B A (2) D C E B A (2) C A D B E (2) B D C E A (2) B C D E A (2) B C D A E (2) A E C D B (2) A C B E D (2) A B E C D (2) A B C D E (2) E D A C B (1) E D A B C (1) E B A D C (1) E A D B C (1) E A C D B (1) D E C B A (1) C D E B A (1) C D A E B (1) C D A B E (1) C B D E A (1) C A D E B (1) B E D C A (1) B E D A C (1) B D A E C (1) B C A D E (1) B A E D C (1) B A C D E (1) A E D C B (1) A E C B D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -8 -8 -2 B 8 0 10 4 2 C 8 -10 0 2 -6 D 8 -4 -2 0 6 E 2 -2 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -8 -2 B 8 0 10 4 2 C 8 -10 0 2 -6 D 8 -4 -2 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 C=19 B=17 D=11 so D is eliminated. Round 2 votes counts: A=30 E=28 C=21 B=21 so C is eliminated. Round 3 votes counts: A=35 B=34 E=31 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:204 E:200 C:197 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -8 -2 B 8 0 10 4 2 C 8 -10 0 2 -6 D 8 -4 -2 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -8 -2 B 8 0 10 4 2 C 8 -10 0 2 -6 D 8 -4 -2 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -8 -2 B 8 0 10 4 2 C 8 -10 0 2 -6 D 8 -4 -2 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6435: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (12) A C D E B (11) B D E C A (9) B E D C A (7) E D C A B (6) E D B C A (6) B E D A C (6) B A C E D (5) A C B D E (5) C A D E B (4) B A C D E (4) D E B C A (3) B D A C E (3) A C B E D (3) D E C B A (2) D E C A B (2) C D A E B (2) E B D A C (1) E B A D C (1) E A C D B (1) D C A E B (1) C E A D B (1) C A D B E (1) B E A D C (1) B D C A E (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 10 2 8 B 0 0 -2 -4 -6 C -10 2 0 0 10 D -2 4 0 0 0 E -8 6 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.745387 B: 0.254613 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.620429565217 Cumulative probabilities = A: 0.745387 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 2 8 B 0 0 -2 -4 -6 C -10 2 0 0 10 D -2 4 0 0 0 E -8 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555557315 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=32 E=15 D=8 C=8 so D is eliminated. Round 2 votes counts: B=37 A=32 E=22 C=9 so C is eliminated. Round 3 votes counts: A=40 B=37 E=23 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:201 D:201 B:194 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 2 8 B 0 0 -2 -4 -6 C -10 2 0 0 10 D -2 4 0 0 0 E -8 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555557315 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 2 8 B 0 0 -2 -4 -6 C -10 2 0 0 10 D -2 4 0 0 0 E -8 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555557315 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 2 8 B 0 0 -2 -4 -6 C -10 2 0 0 10 D -2 4 0 0 0 E -8 6 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55555557315 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6436: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) B A D C E (10) E D C A B (8) D B E A C (7) D B A E C (5) C A E B D (5) E D C B A (4) E D B A C (4) E C D A B (4) C A B D E (4) B D A C E (4) E D B C A (3) E C A D B (3) C E A B D (3) B A C D E (3) A C B D E (3) A B C D E (3) D E B C A (2) C E D B A (2) C E A D B (2) C B A D E (2) C A B E D (2) A C B E D (2) E D A C B (1) E D A B C (1) D B E C A (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 10 -12 -10 B 16 0 10 -14 -4 C -10 -10 0 -24 -10 D 12 14 24 0 10 E 10 4 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 10 -12 -10 B 16 0 10 -14 -4 C -10 -10 0 -24 -10 D 12 14 24 0 10 E 10 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=25 C=20 B=18 A=9 so A is eliminated. Round 2 votes counts: E=28 D=25 C=25 B=22 so B is eliminated. Round 3 votes counts: D=40 C=31 E=29 so E is eliminated. Round 4 votes counts: D=62 C=38 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 E:207 B:204 A:186 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 10 -12 -10 B 16 0 10 -14 -4 C -10 -10 0 -24 -10 D 12 14 24 0 10 E 10 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 -12 -10 B 16 0 10 -14 -4 C -10 -10 0 -24 -10 D 12 14 24 0 10 E 10 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 -12 -10 B 16 0 10 -14 -4 C -10 -10 0 -24 -10 D 12 14 24 0 10 E 10 4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6437: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (23) D B A C E (17) C E A B D (8) E A B C D (7) E D B A C (4) D B A E C (4) D E B A C (3) C A B E D (3) A B E C D (3) E C A D B (2) C D A B E (2) C A B D E (2) B A E D C (2) A B E D C (2) E D C B A (1) E D B C A (1) E C D B A (1) E C D A B (1) E B A D C (1) D E C B A (1) D C E B A (1) D C B E A (1) D C B A E (1) C E D A B (1) C E A D B (1) C D E B A (1) C D B A E (1) C A E B D (1) B D A C E (1) B A D C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 16 -6 16 -16 B -16 0 -6 12 -16 C 6 6 0 18 -12 D -16 -12 -18 0 -24 E 16 16 12 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -6 16 -16 B -16 0 -6 12 -16 C 6 6 0 18 -12 D -16 -12 -18 0 -24 E 16 16 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 D=28 C=20 A=7 B=4 so B is eliminated. Round 2 votes counts: E=41 D=29 C=20 A=10 so A is eliminated. Round 3 votes counts: E=48 D=30 C=22 so C is eliminated. Round 4 votes counts: E=62 D=38 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:234 C:209 A:205 B:187 D:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -6 16 -16 B -16 0 -6 12 -16 C 6 6 0 18 -12 D -16 -12 -18 0 -24 E 16 16 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 16 -16 B -16 0 -6 12 -16 C 6 6 0 18 -12 D -16 -12 -18 0 -24 E 16 16 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 16 -16 B -16 0 -6 12 -16 C 6 6 0 18 -12 D -16 -12 -18 0 -24 E 16 16 12 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6438: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) B E C A D (7) B C A E D (7) D A C E B (6) D A C B E (6) B C E A D (6) A C B D E (6) D E A C B (5) E D A C B (4) D A E C B (4) A D C B E (4) E D B C A (3) E B D C A (3) D E A B C (3) B E D C A (3) E A C B D (2) D B A C E (2) C B A E D (2) C A B E D (2) B D E C A (2) B C A D E (2) A C D B E (2) E D B A C (1) E C A B D (1) E A D C B (1) D E B A C (1) D B E A C (1) D A E B C (1) D A B C E (1) B E C D A (1) A D E C B (1) A D C E B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 8 6 -2 B -4 0 2 4 10 C -8 -2 0 -6 -2 D -6 -4 6 0 -2 E 2 -10 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000016 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 4 8 6 -2 B -4 0 2 4 10 C -8 -2 0 -6 -2 D -6 -4 6 0 -2 E 2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999982 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=28 E=22 A=16 C=4 so C is eliminated. Round 2 votes counts: D=30 B=30 E=22 A=18 so A is eliminated. Round 3 votes counts: D=39 B=39 E=22 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:208 B:206 E:198 D:197 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 -2 B -4 0 2 4 10 C -8 -2 0 -6 -2 D -6 -4 6 0 -2 E 2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999982 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 -2 B -4 0 2 4 10 C -8 -2 0 -6 -2 D -6 -4 6 0 -2 E 2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999982 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 -2 B -4 0 2 4 10 C -8 -2 0 -6 -2 D -6 -4 6 0 -2 E 2 -10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999982 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6439: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) E B C A D (7) D A E C B (7) B C E A D (7) E B D C A (5) E A B C D (4) D E B A C (4) A C B E D (4) D E B C A (3) D C A B E (3) D A C E B (3) D A C B E (3) C B A D E (3) C A B D E (3) A E C B D (3) A D C E B (3) A C B D E (3) E D A B C (2) E B D A C (2) E B A C D (2) D B E C A (2) D B C E A (2) B E C A D (2) A C D B E (2) E D B C A (1) E B C D A (1) E A D B C (1) D E A B C (1) C B D A E (1) C B A E D (1) C A D B E (1) B D C E A (1) B C E D A (1) B C D E A (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -6 -2 2 B 8 0 0 6 -4 C 6 0 0 -2 4 D 2 -6 2 0 8 E -2 4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.396412 C: 0.603588 D: 0.000000 E: 0.000000 Sum of squares = 0.521461138548 Cumulative probabilities = A: 0.000000 B: 0.396412 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -2 2 B 8 0 0 6 -4 C 6 0 0 -2 4 D 2 -6 2 0 8 E -2 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499852 C: 0.500148 D: 0.000000 E: 0.000000 Sum of squares = 0.500000043518 Cumulative probabilities = A: 0.000000 B: 0.499852 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=25 A=18 B=12 C=9 so C is eliminated. Round 2 votes counts: D=36 E=25 A=22 B=17 so B is eliminated. Round 3 votes counts: D=39 E=35 A=26 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:205 C:204 D:203 E:195 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -2 2 B 8 0 0 6 -4 C 6 0 0 -2 4 D 2 -6 2 0 8 E -2 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499852 C: 0.500148 D: 0.000000 E: 0.000000 Sum of squares = 0.500000043518 Cumulative probabilities = A: 0.000000 B: 0.499852 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -2 2 B 8 0 0 6 -4 C 6 0 0 -2 4 D 2 -6 2 0 8 E -2 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499852 C: 0.500148 D: 0.000000 E: 0.000000 Sum of squares = 0.500000043518 Cumulative probabilities = A: 0.000000 B: 0.499852 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -2 2 B 8 0 0 6 -4 C 6 0 0 -2 4 D 2 -6 2 0 8 E -2 4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499852 C: 0.500148 D: 0.000000 E: 0.000000 Sum of squares = 0.500000043518 Cumulative probabilities = A: 0.000000 B: 0.499852 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6440: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) C B E A D (7) C A E D B (7) E C A B D (6) B D E A C (6) E A C B D (5) C A E B D (5) A E C D B (5) E B A D C (4) D A E B C (4) B E D A C (4) E A B D C (3) D E A B C (3) C B D E A (3) D B C A E (2) C B E D A (2) C B D A E (2) C A D E B (2) C A B E D (2) B D C E A (2) B C D E A (2) A C D E B (2) E B A C D (1) D E B A C (1) D B C E A (1) D B A E C (1) D A C E B (1) D A B E C (1) C E A B D (1) C D B A E (1) C B A E D (1) B E C D A (1) B D E C A (1) A E D C B (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 0 4 8 -22 B 0 0 -8 16 -6 C -4 8 0 12 -12 D -8 -16 -12 0 -14 E 22 6 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 8 -22 B 0 0 -8 16 -6 C -4 8 0 12 -12 D -8 -16 -12 0 -14 E 22 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 E=19 B=16 A=10 so A is eliminated. Round 2 votes counts: C=36 E=26 D=22 B=16 so B is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:227 C:202 B:201 A:195 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 8 -22 B 0 0 -8 16 -6 C -4 8 0 12 -12 D -8 -16 -12 0 -14 E 22 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 8 -22 B 0 0 -8 16 -6 C -4 8 0 12 -12 D -8 -16 -12 0 -14 E 22 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 8 -22 B 0 0 -8 16 -6 C -4 8 0 12 -12 D -8 -16 -12 0 -14 E 22 6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6441: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (20) D A C E B (16) C A D B E (7) A C D B E (7) D A C B E (6) E B D A C (5) E B C A D (5) E B C D A (4) C B A E D (4) E D B A C (3) E B D C A (3) D E A B C (3) D A E C B (3) B C E A D (3) A D C B E (3) E D A B C (2) C A B E D (2) C A B D E (2) D E B A C (1) C B D A E (1) Total count = 100 A B C D E A 0 2 -2 6 2 B -2 0 -2 -2 10 C 2 2 0 10 2 D -6 2 -10 0 -2 E -2 -10 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 6 2 B -2 0 -2 -2 10 C 2 2 0 10 2 D -6 2 -10 0 -2 E -2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=23 E=22 C=16 A=10 so A is eliminated. Round 2 votes counts: D=32 C=23 B=23 E=22 so E is eliminated. Round 3 votes counts: B=40 D=37 C=23 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:208 A:204 B:202 E:194 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 6 2 B -2 0 -2 -2 10 C 2 2 0 10 2 D -6 2 -10 0 -2 E -2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 6 2 B -2 0 -2 -2 10 C 2 2 0 10 2 D -6 2 -10 0 -2 E -2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 6 2 B -2 0 -2 -2 10 C 2 2 0 10 2 D -6 2 -10 0 -2 E -2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6442: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) B C D E A (6) E D A C B (5) C B E A D (5) A E D C B (5) E A C D B (4) D A B E C (4) C E B D A (4) C B E D A (4) A B D E C (4) E A D C B (3) D B C E A (3) D B A E C (3) B D C A E (3) A E C D B (3) A D E B C (3) E C D A B (2) D E A B C (2) D A E B C (2) C E B A D (2) C B A E D (2) B C D A E (2) A E C B D (2) A D B E C (2) E C A D B (1) E C A B D (1) D E A C B (1) D C E B A (1) D B E C A (1) D B A C E (1) C E D B A (1) C E A B D (1) C B D E A (1) B D A C E (1) B C A E D (1) B C A D E (1) B A D C E (1) A E D B C (1) A D E C B (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 2 -8 -10 B 0 0 0 2 10 C -2 0 0 -10 0 D 8 -2 10 0 2 E 10 -10 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.163900 B: 0.836100 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.725926700371 Cumulative probabilities = A: 0.163900 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -8 -10 B 0 0 0 2 10 C -2 0 0 -10 0 D 8 -2 10 0 2 E 10 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.68000008217 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=22 C=20 D=18 E=16 so E is eliminated. Round 2 votes counts: A=31 C=24 D=23 B=22 so B is eliminated. Round 3 votes counts: D=34 C=34 A=32 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:206 E:199 C:194 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -8 -10 B 0 0 0 2 10 C -2 0 0 -10 0 D 8 -2 10 0 2 E 10 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.68000008217 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -8 -10 B 0 0 0 2 10 C -2 0 0 -10 0 D 8 -2 10 0 2 E 10 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.68000008217 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -8 -10 B 0 0 0 2 10 C -2 0 0 -10 0 D 8 -2 10 0 2 E 10 -10 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.68000008217 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6443: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C A E D B (7) B D E A C (7) B C A D E (7) C A B E D (6) C E A D B (4) B C E D A (4) A C B D E (4) E D B A C (3) C B A D E (3) E D B C A (2) E D A C B (2) E A D C B (2) D E A B C (2) D B E A C (2) D A E B C (2) C E D A B (2) C E B D A (2) C B E A D (2) C A E B D (2) B C E A D (2) B A C D E (2) A E D C B (2) A E C D B (2) A C E D B (2) A B D E C (2) E D C A B (1) E D A B C (1) E B D C A (1) C E A B D (1) C B A E D (1) B E C D A (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D E A (1) B C A E D (1) B A D E C (1) B A D C E (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -6 12 -2 B 8 0 8 10 4 C 6 -8 0 12 12 D -12 -10 -12 0 -6 E 2 -4 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 12 -2 B 8 0 8 10 4 C 6 -8 0 12 12 D -12 -10 -12 0 -6 E 2 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=30 A=14 D=13 E=12 so E is eliminated. Round 2 votes counts: B=32 C=30 D=22 A=16 so A is eliminated. Round 3 votes counts: C=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:211 A:198 E:196 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 12 -2 B 8 0 8 10 4 C 6 -8 0 12 12 D -12 -10 -12 0 -6 E 2 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 12 -2 B 8 0 8 10 4 C 6 -8 0 12 12 D -12 -10 -12 0 -6 E 2 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 12 -2 B 8 0 8 10 4 C 6 -8 0 12 12 D -12 -10 -12 0 -6 E 2 -4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6444: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) A D C E B (7) D A C B E (5) B E D C A (5) B E C D A (5) A C D E B (5) E B C A D (4) C D A E B (4) C A D E B (4) E C A D B (3) E C A B D (3) E B A C D (3) E A B D C (3) C D A B E (3) B D C E A (3) E A C D B (2) D C A B E (2) D A C E B (2) C E A D B (2) C A E D B (2) B E A D C (2) B D E A C (2) B D C A E (2) A D E C B (2) A D B E C (2) E C B A D (1) E B C D A (1) E B A D C (1) E A C B D (1) D C B A E (1) D B A C E (1) D A B C E (1) C E B D A (1) C D E A B (1) C D B E A (1) C D B A E (1) C B E D A (1) C B D E A (1) B E A C D (1) B D A C E (1) B C D E A (1) Total count = 100 A B C D E A 0 8 -4 -4 -10 B -8 0 -10 -2 -4 C 4 10 0 2 4 D 4 2 -2 0 4 E 10 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -4 -10 B -8 0 -10 -2 -4 C 4 10 0 2 4 D 4 2 -2 0 4 E 10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=22 C=21 A=16 D=12 so D is eliminated. Round 2 votes counts: B=30 C=24 A=24 E=22 so E is eliminated. Round 3 votes counts: B=39 C=31 A=30 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:204 E:203 A:195 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -4 -10 B -8 0 -10 -2 -4 C 4 10 0 2 4 D 4 2 -2 0 4 E 10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -4 -10 B -8 0 -10 -2 -4 C 4 10 0 2 4 D 4 2 -2 0 4 E 10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -4 -10 B -8 0 -10 -2 -4 C 4 10 0 2 4 D 4 2 -2 0 4 E 10 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6445: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) C B A E D (8) A B C E D (8) A B C D E (6) D E C A B (5) C A B D E (5) B A C E D (5) A C B D E (5) E D B A C (4) D E C B A (4) D E B A C (4) D E A C B (4) A D B C E (4) E D C B A (3) C B E A D (3) B C A E D (3) D E B C A (2) A D B E C (2) E D B C A (1) E B D C A (1) E B C A D (1) D C E B A (1) D C A E B (1) D A E C B (1) D A C E B (1) C E B D A (1) C B E D A (1) C A B E D (1) B E C A D (1) B E A D C (1) B E A C D (1) A C B E D (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 16 16 8 B -10 0 10 10 16 C -16 -10 0 0 10 D -16 -10 0 0 10 E -8 -16 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 16 8 B -10 0 10 10 16 C -16 -10 0 0 10 D -16 -10 0 0 10 E -8 -16 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=29 C=19 B=11 E=10 so E is eliminated. Round 2 votes counts: D=39 A=29 C=19 B=13 so B is eliminated. Round 3 votes counts: D=40 A=36 C=24 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:225 B:213 C:192 D:192 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 16 8 B -10 0 10 10 16 C -16 -10 0 0 10 D -16 -10 0 0 10 E -8 -16 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 16 8 B -10 0 10 10 16 C -16 -10 0 0 10 D -16 -10 0 0 10 E -8 -16 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 16 8 B -10 0 10 10 16 C -16 -10 0 0 10 D -16 -10 0 0 10 E -8 -16 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6446: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) C D E B A (6) D E C B A (5) C B D E A (5) B A D E C (5) A B E D C (5) E D A C B (4) C E D A B (4) C B A D E (4) B D E C A (4) A E D C B (4) A B C E D (4) E D A B C (3) C D E A B (3) B C D A E (3) B A E D C (3) C D B E A (2) C A B D E (2) B D E A C (2) B D C E A (2) B D A E C (2) B C D E A (2) B C A D E (2) A E B D C (2) A C E D B (2) A C B E D (2) E A D C B (1) D E B C A (1) D E B A C (1) D B E A C (1) C E A D B (1) C B D A E (1) C A E D B (1) B A D C E (1) B A C D E (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 -10 -16 -8 B 6 0 -10 4 6 C 10 10 0 -8 -4 D 16 -4 8 0 10 E 8 -6 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826466 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -16 -8 B 6 0 -10 4 6 C 10 10 0 -8 -4 D 16 -4 8 0 10 E 8 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826454 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=27 A=21 E=15 D=8 so D is eliminated. Round 2 votes counts: C=29 B=28 E=22 A=21 so A is eliminated. Round 3 votes counts: B=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:215 C:204 B:203 E:198 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -10 -16 -8 B 6 0 -10 4 6 C 10 10 0 -8 -4 D 16 -4 8 0 10 E 8 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826454 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -16 -8 B 6 0 -10 4 6 C 10 10 0 -8 -4 D 16 -4 8 0 10 E 8 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826454 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -16 -8 B 6 0 -10 4 6 C 10 10 0 -8 -4 D 16 -4 8 0 10 E 8 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.454545 E: 0.000000 Sum of squares = 0.371900826454 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6447: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) A E D B C (10) C B D E A (9) E D C A B (8) E D A C B (6) B A C D E (5) A B C E D (5) D E C B A (4) A B D C E (4) E A D C B (3) D E A C B (3) C D B E A (3) A B E D C (3) A B C D E (3) D E C A B (2) C E D B A (2) C D E B A (2) C B E D A (2) B C A E D (2) A E B D C (2) A B E C D (2) E D C B A (1) E A D B C (1) D C E B A (1) C E B D A (1) B D C A E (1) B C D E A (1) B C D A E (1) B A C E D (1) Total count = 100 A B C D E A 0 4 -4 6 2 B -4 0 6 8 8 C 4 -6 0 2 10 D -6 -8 -2 0 2 E -2 -8 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 6 2 B -4 0 6 8 8 C 4 -6 0 2 10 D -6 -8 -2 0 2 E -2 -8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=23 E=19 C=19 D=10 so D is eliminated. Round 2 votes counts: A=29 E=28 B=23 C=20 so C is eliminated. Round 3 votes counts: B=37 E=34 A=29 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:205 A:204 D:193 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 6 2 B -4 0 6 8 8 C 4 -6 0 2 10 D -6 -8 -2 0 2 E -2 -8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 6 2 B -4 0 6 8 8 C 4 -6 0 2 10 D -6 -8 -2 0 2 E -2 -8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 6 2 B -4 0 6 8 8 C 4 -6 0 2 10 D -6 -8 -2 0 2 E -2 -8 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6448: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) D B E C A (7) E C B D A (6) D B A E C (6) A C E B D (5) D B A C E (4) C E B A D (4) C E A B D (4) A C B D E (4) D E B C A (3) C B E D A (3) B D C E A (3) A D B E C (3) A B D C E (3) A B C D E (3) E C D B A (2) E C B A D (2) E C A B D (2) D B E A C (2) C E B D A (2) C A E B D (2) B C E D A (2) A E D C B (2) A D E B C (2) A D C B E (2) E D C B A (1) E B C D A (1) D B C E A (1) D B C A E (1) C B E A D (1) C B A E D (1) C A B E D (1) B D A C E (1) A E C D B (1) A D C E B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 2 10 8 B 6 0 4 2 18 C -2 -4 0 -4 20 D -10 -2 4 0 12 E -8 -18 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 10 8 B 6 0 4 2 18 C -2 -4 0 -4 20 D -10 -2 4 0 12 E -8 -18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989279 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=24 C=18 E=14 B=6 so B is eliminated. Round 2 votes counts: A=38 D=28 C=20 E=14 so E is eliminated. Round 3 votes counts: A=38 C=33 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:215 A:207 C:205 D:202 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 10 8 B 6 0 4 2 18 C -2 -4 0 -4 20 D -10 -2 4 0 12 E -8 -18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989279 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 10 8 B 6 0 4 2 18 C -2 -4 0 -4 20 D -10 -2 4 0 12 E -8 -18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989279 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 10 8 B 6 0 4 2 18 C -2 -4 0 -4 20 D -10 -2 4 0 12 E -8 -18 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989279 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6449: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) C B D A E (7) D C B A E (6) E B A C D (4) A E B C D (4) E B C A D (3) E A D B C (3) E A B C D (3) D C E B A (3) D C A B E (3) D A E C B (3) D A C B E (3) C D B A E (3) B C A D E (3) B A C E D (3) A E D B C (3) A E B D C (3) A B C D E (3) E D A C B (2) D E A C B (2) C B D E A (2) B E C A D (2) B C E A D (2) B C A E D (2) A B E C D (2) E D A B C (1) E C D B A (1) E B D C A (1) E B C D A (1) E A B D C (1) D E C B A (1) D E C A B (1) D C E A B (1) D C B E A (1) D C A E B (1) C D B E A (1) C B E D A (1) C B A D E (1) B C E D A (1) B A E C D (1) A D C B E (1) A C D B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -10 -6 10 B 14 0 -4 4 2 C 10 4 0 4 2 D 6 -4 -4 0 -4 E -10 -2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -6 10 B 14 0 -4 4 2 C 10 4 0 4 2 D 6 -4 -4 0 -4 E -10 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 A=19 C=15 B=14 so B is eliminated. Round 2 votes counts: E=29 D=25 C=23 A=23 so C is eliminated. Round 3 votes counts: D=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:210 B:208 D:197 E:195 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -10 -6 10 B 14 0 -4 4 2 C 10 4 0 4 2 D 6 -4 -4 0 -4 E -10 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -6 10 B 14 0 -4 4 2 C 10 4 0 4 2 D 6 -4 -4 0 -4 E -10 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -6 10 B 14 0 -4 4 2 C 10 4 0 4 2 D 6 -4 -4 0 -4 E -10 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6450: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) B D C A E (6) A E C B D (6) E C A D B (5) E A D C B (5) D C B E A (5) E A C B D (4) D B C E A (3) D B A E C (3) D A E B C (3) C B D E A (3) B A C E D (3) E D C A B (2) D A B E C (2) C E D B A (2) C E B A D (2) C D E B A (2) C B E A D (2) C A B E D (2) B C D E A (2) B C D A E (2) B C A E D (2) B A D E C (2) A E D C B (2) A E C D B (2) A B E D C (2) A B C E D (2) E A C D B (1) D E C B A (1) D E A C B (1) D B E C A (1) D B A C E (1) C E B D A (1) C E A D B (1) C D E A B (1) C B A E D (1) B D A C E (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B C D (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -8 16 -2 B -6 0 -16 8 -2 C 8 16 0 12 6 D -16 -8 -12 0 -12 E 2 2 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 16 -2 B -6 0 -16 8 -2 C 8 16 0 12 6 D -16 -8 -12 0 -12 E 2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=20 B=20 A=19 E=17 so E is eliminated. Round 2 votes counts: C=29 A=29 D=22 B=20 so B is eliminated. Round 3 votes counts: C=36 A=35 D=29 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 A:206 E:205 B:192 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 16 -2 B -6 0 -16 8 -2 C 8 16 0 12 6 D -16 -8 -12 0 -12 E 2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 16 -2 B -6 0 -16 8 -2 C 8 16 0 12 6 D -16 -8 -12 0 -12 E 2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 16 -2 B -6 0 -16 8 -2 C 8 16 0 12 6 D -16 -8 -12 0 -12 E 2 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6451: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C E A (9) D B C A E (7) C B D A E (6) A E D C B (6) A E C B D (6) C B E A D (5) B C D E A (5) E A D B C (4) D E A B C (4) B D C E A (3) A E C D B (3) E A B C D (2) D B E C A (2) D B E A C (2) D A E B C (2) D A B E C (2) A E D B C (2) E D B A C (1) E D A B C (1) E C A B D (1) E A C D B (1) D E B A C (1) D C B A E (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A C B E (1) D A B C E (1) C B A D E (1) C A E B D (1) C A B D E (1) B E D A C (1) B E C A D (1) B C E A D (1) B C D A E (1) A D E C B (1) Total count = 100 A B C D E A 0 2 8 -8 -8 B -2 0 8 -10 6 C -8 -8 0 -10 -8 D 8 10 10 0 8 E 8 -6 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -8 -8 B -2 0 8 -10 6 C -8 -8 0 -10 -8 D 8 10 10 0 8 E 8 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=20 A=18 C=14 B=12 so B is eliminated. Round 2 votes counts: D=39 E=22 C=21 A=18 so A is eliminated. Round 3 votes counts: D=40 E=39 C=21 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:201 E:201 A:197 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -8 -8 B -2 0 8 -10 6 C -8 -8 0 -10 -8 D 8 10 10 0 8 E 8 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -8 -8 B -2 0 8 -10 6 C -8 -8 0 -10 -8 D 8 10 10 0 8 E 8 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -8 -8 B -2 0 8 -10 6 C -8 -8 0 -10 -8 D 8 10 10 0 8 E 8 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6452: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (17) C D B A E (17) E A D C B (8) B C D A E (8) D C B A E (7) C B D A E (6) E A B C D (4) D C A B E (4) E B A C D (3) E A D B C (3) D A C B E (2) C B D E A (2) B C E D A (2) A D E C B (2) A D B C E (2) E C B A D (1) E B C D A (1) E B C A D (1) E A C D B (1) D C A E B (1) C D B E A (1) B E C A D (1) B E A C D (1) B D C A E (1) A E D C B (1) A E D B C (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -6 -4 8 B 4 0 -10 -4 10 C 6 10 0 -2 10 D 4 4 2 0 10 E -8 -10 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 8 B 4 0 -10 -4 10 C 6 10 0 -2 10 D 4 4 2 0 10 E -8 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=26 D=14 B=13 A=8 so A is eliminated. Round 2 votes counts: E=41 C=26 D=20 B=13 so B is eliminated. Round 3 votes counts: E=43 C=36 D=21 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:210 B:200 A:197 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 8 B 4 0 -10 -4 10 C 6 10 0 -2 10 D 4 4 2 0 10 E -8 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 8 B 4 0 -10 -4 10 C 6 10 0 -2 10 D 4 4 2 0 10 E -8 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 8 B 4 0 -10 -4 10 C 6 10 0 -2 10 D 4 4 2 0 10 E -8 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6453: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) B C A E D (6) E A D B C (5) B C A D E (5) E A D C B (4) C B E A D (4) A B E C D (4) E D A C B (3) E A C B D (3) D E A C B (3) D C E A B (3) D C B A E (3) C E D B A (3) C D B E A (3) C B D E A (3) B C E A D (3) B C D A E (3) B A D E C (3) A E D B C (3) A E B C D (3) E A B D C (2) E A B C D (2) D E C A B (2) C D E B A (2) B D C A E (2) A E B D C (2) A B D E C (2) E C D A B (1) E A C D B (1) D C E B A (1) D C B E A (1) D C A E B (1) D A B C E (1) C E D A B (1) C E B A D (1) C B E D A (1) C B D A E (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 -8 0 24 -4 B 8 0 10 18 8 C 0 -10 0 14 -2 D -24 -18 -14 0 -20 E 4 -8 2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 24 -4 B 8 0 10 18 8 C 0 -10 0 14 -2 D -24 -18 -14 0 -20 E 4 -8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=21 C=19 D=15 A=14 so A is eliminated. Round 2 votes counts: B=37 E=29 C=19 D=15 so D is eliminated. Round 3 votes counts: B=38 E=34 C=28 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:209 A:206 C:201 D:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 24 -4 B 8 0 10 18 8 C 0 -10 0 14 -2 D -24 -18 -14 0 -20 E 4 -8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 24 -4 B 8 0 10 18 8 C 0 -10 0 14 -2 D -24 -18 -14 0 -20 E 4 -8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 24 -4 B 8 0 10 18 8 C 0 -10 0 14 -2 D -24 -18 -14 0 -20 E 4 -8 2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6454: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) C D A B E (9) E A B D C (8) A D E C B (8) B E A C D (7) E B A D C (6) B E A D C (6) B E C A D (5) C D A E B (4) C B D A E (4) A E D B C (4) E A D B C (3) C D B A E (3) C B D E A (3) B C E D A (3) E B A C D (2) D C A E B (2) D A E C B (2) B C D E A (2) A E D C B (2) E A C D B (1) C B E D A (1) B C E A D (1) B C D A E (1) B A E D C (1) B A D E C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 24 14 4 B -8 0 4 4 -4 C -24 -4 0 -8 -16 D -14 -4 8 0 0 E -4 4 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 24 14 4 B -8 0 4 4 -4 C -24 -4 0 -8 -16 D -14 -4 8 0 0 E -4 4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985189 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 E=20 A=16 D=13 so D is eliminated. Round 2 votes counts: B=27 A=27 C=26 E=20 so E is eliminated. Round 3 votes counts: A=39 B=35 C=26 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:208 B:198 D:195 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 24 14 4 B -8 0 4 4 -4 C -24 -4 0 -8 -16 D -14 -4 8 0 0 E -4 4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985189 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 24 14 4 B -8 0 4 4 -4 C -24 -4 0 -8 -16 D -14 -4 8 0 0 E -4 4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985189 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 24 14 4 B -8 0 4 4 -4 C -24 -4 0 -8 -16 D -14 -4 8 0 0 E -4 4 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985189 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6455: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (16) C E D A B (14) C E D B A (9) D E C B A (5) D E C A B (5) B A C E D (5) A B D E C (5) C E B D A (4) A D E C B (4) A B C E D (4) B C E D A (3) B A D C E (3) A D B E C (3) A B D C E (3) E C D B A (2) E C D A B (2) D A B E C (2) C E A D B (2) A C E D B (2) E D C A B (1) D B E C A (1) D A E C B (1) C E A B D (1) B D E C A (1) B D E A C (1) A D C E B (1) Total count = 100 A B C D E A 0 0 0 -2 -2 B 0 0 -6 -8 -6 C 0 6 0 -4 2 D 2 8 4 0 2 E 2 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -2 -2 B 0 0 -6 -8 -6 C 0 6 0 -4 2 D 2 8 4 0 2 E 2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=29 A=22 D=14 E=5 so E is eliminated. Round 2 votes counts: C=34 B=29 A=22 D=15 so D is eliminated. Round 3 votes counts: C=45 B=30 A=25 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:202 E:202 A:198 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -2 -2 B 0 0 -6 -8 -6 C 0 6 0 -4 2 D 2 8 4 0 2 E 2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -2 B 0 0 -6 -8 -6 C 0 6 0 -4 2 D 2 8 4 0 2 E 2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -2 B 0 0 -6 -8 -6 C 0 6 0 -4 2 D 2 8 4 0 2 E 2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6456: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) B D A C E (7) A E C D B (7) A D B C E (7) E C B D A (6) A D E C B (5) E C A D B (4) D A B C E (4) B D E C A (4) D B A E C (3) D A B E C (3) C E B A D (3) C E A B D (3) E D A C B (2) E C D A B (2) E C A B D (2) E B C D A (2) E A C D B (2) D E C A B (2) D A E B C (2) B D C E A (2) B D C A E (2) B C A E D (2) A C E D B (2) E D C B A (1) E C B A D (1) D E B C A (1) D E A C B (1) D B E C A (1) C E B D A (1) C B A E D (1) B E C D A (1) B C E A D (1) B A D C E (1) A D C E B (1) A C E B D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 0 -10 0 B -4 0 4 0 -2 C 0 -4 0 0 -4 D 10 0 0 0 -6 E 0 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.268973 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.731027 Sum of squares = 0.60674690463 Cumulative probabilities = A: 0.268973 B: 0.268973 C: 0.268973 D: 0.268973 E: 1.000000 A B C D E A 0 4 0 -10 0 B -4 0 4 0 -2 C 0 -4 0 0 -4 D 10 0 0 0 -6 E 0 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250036294 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=25 E=22 D=17 C=8 so C is eliminated. Round 2 votes counts: E=29 B=29 A=25 D=17 so D is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:202 B:199 A:197 C:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 -10 0 B -4 0 4 0 -2 C 0 -4 0 0 -4 D 10 0 0 0 -6 E 0 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250036294 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -10 0 B -4 0 4 0 -2 C 0 -4 0 0 -4 D 10 0 0 0 -6 E 0 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250036294 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -10 0 B -4 0 4 0 -2 C 0 -4 0 0 -4 D 10 0 0 0 -6 E 0 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250036294 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6457: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) D A E B C (7) B E C A D (7) E B D C A (6) E B C D A (6) D C A E B (6) B C E A D (6) D A C E B (4) C D A E B (4) C B E D A (4) C A B E D (4) A D E B C (4) A C D B E (4) D E B C A (3) C E B D A (3) A D C E B (3) A D C B E (3) D E B A C (2) C D A B E (2) C B E A D (2) C A D B E (2) B E A C D (2) E C B D A (1) E B A D C (1) D E C B A (1) D A E C B (1) C A B D E (1) B A E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -16 -16 -4 B 6 0 10 8 -20 C 16 -10 0 -2 -2 D 16 -8 2 0 -4 E 4 20 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -16 -16 -4 B 6 0 10 8 -20 C 16 -10 0 -2 -2 D 16 -8 2 0 -4 E 4 20 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=22 C=22 B=16 A=16 so B is eliminated. Round 2 votes counts: E=31 C=28 D=24 A=17 so A is eliminated. Round 3 votes counts: D=35 C=33 E=32 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:215 D:203 B:202 C:201 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -16 -16 -4 B 6 0 10 8 -20 C 16 -10 0 -2 -2 D 16 -8 2 0 -4 E 4 20 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -16 -4 B 6 0 10 8 -20 C 16 -10 0 -2 -2 D 16 -8 2 0 -4 E 4 20 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -16 -4 B 6 0 10 8 -20 C 16 -10 0 -2 -2 D 16 -8 2 0 -4 E 4 20 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6458: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) C B A D E (8) C A B E D (8) C B D A E (7) B C D E A (7) D E A B C (6) A E C D B (6) D B E C A (5) A C E B D (5) D E B A C (4) C B D E A (4) E D A B C (3) E A D B C (3) C B A E D (3) B D E C A (3) D B E A C (2) B C D A E (2) A C E D B (2) E A D C B (1) D E A C B (1) D B C E A (1) C D E A B (1) C A E D B (1) C A E B D (1) C A B D E (1) B D C E A (1) B C A E D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 4 -8 6 16 B -4 0 -20 4 6 C 8 20 0 14 6 D -6 -4 -14 0 6 E -16 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 6 16 B -4 0 -20 4 6 C 8 20 0 14 6 D -6 -4 -14 0 6 E -16 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=26 D=19 B=14 E=7 so E is eliminated. Round 2 votes counts: C=34 A=30 D=22 B=14 so B is eliminated. Round 3 votes counts: C=44 A=30 D=26 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:209 B:193 D:191 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 6 16 B -4 0 -20 4 6 C 8 20 0 14 6 D -6 -4 -14 0 6 E -16 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 6 16 B -4 0 -20 4 6 C 8 20 0 14 6 D -6 -4 -14 0 6 E -16 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 6 16 B -4 0 -20 4 6 C 8 20 0 14 6 D -6 -4 -14 0 6 E -16 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6459: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) D B C A E (6) A D B C E (6) E C D B A (5) D C B E A (5) A E B D C (5) E A C D B (4) C D B E A (4) B C D E A (4) A D C B E (4) E A C B D (3) C B D E A (3) B D C E A (3) A D B E C (3) A B D E C (3) A B D C E (3) E B C D A (2) E A B C D (2) D B C E A (2) C D E B A (2) B D C A E (2) A E D C B (2) A E B C D (2) E B C A D (1) E B A C D (1) D B A C E (1) D A C B E (1) C E D B A (1) C D E A B (1) C B E D A (1) B E D C A (1) B E C D A (1) B C E D A (1) B A E D C (1) B A D C E (1) A E D B C (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 -16 -10 -12 -14 B 16 0 4 0 12 C 10 -4 0 -2 4 D 12 0 2 0 10 E 14 -12 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.422097 C: 0.000000 D: 0.577903 E: 0.000000 Sum of squares = 0.512137870097 Cumulative probabilities = A: 0.000000 B: 0.422097 C: 0.422097 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -12 -14 B 16 0 4 0 12 C 10 -4 0 -2 4 D 12 0 2 0 10 E 14 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=28 D=15 B=14 C=12 so C is eliminated. Round 2 votes counts: A=31 E=29 D=22 B=18 so B is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:212 C:204 E:194 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -12 -14 B 16 0 4 0 12 C 10 -4 0 -2 4 D 12 0 2 0 10 E 14 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -12 -14 B 16 0 4 0 12 C 10 -4 0 -2 4 D 12 0 2 0 10 E 14 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -12 -14 B 16 0 4 0 12 C 10 -4 0 -2 4 D 12 0 2 0 10 E 14 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6460: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) B E A D C (8) C A D B E (7) C D E A B (6) C D A B E (5) B A E D C (5) E B D A C (4) C D A E B (4) C A B D E (4) B A C D E (4) A C B D E (3) A B D C E (3) E D B C A (2) E B D C A (2) D C A E B (2) D A E C B (2) B E C A D (2) B E A C D (2) A D C B E (2) A B D E C (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A C B (1) E C D B A (1) E C B D A (1) E B C D A (1) E B C A D (1) D C A B E (1) D A E B C (1) D A C E B (1) D A C B E (1) C E D A B (1) C B A D E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D E C (1) A D B E C (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 10 22 12 B -2 0 8 14 16 C -10 -8 0 -4 0 D -22 -14 4 0 10 E -12 -16 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 22 12 B -2 0 8 14 16 C -10 -8 0 -4 0 D -22 -14 4 0 10 E -12 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979561 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=25 B=25 A=14 D=8 so D is eliminated. Round 2 votes counts: C=31 E=25 B=25 A=19 so A is eliminated. Round 3 votes counts: C=39 B=33 E=28 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:223 B:218 C:189 D:189 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 22 12 B -2 0 8 14 16 C -10 -8 0 -4 0 D -22 -14 4 0 10 E -12 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979561 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 22 12 B -2 0 8 14 16 C -10 -8 0 -4 0 D -22 -14 4 0 10 E -12 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979561 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 22 12 B -2 0 8 14 16 C -10 -8 0 -4 0 D -22 -14 4 0 10 E -12 -16 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979561 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6461: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) B E D A C (13) C A D E B (12) E B C A D (8) C D A B E (8) C E A D B (7) D A B C E (6) B D A E C (6) E B C D A (3) E B A D C (3) B E C D A (3) A D C E B (3) E C B A D (2) E B A C D (2) C E B A D (2) A D C B E (2) A D B E C (2) E C A D B (1) E C A B D (1) C B E D A (1) B D E A C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 4 -8 6 B -12 0 -4 -10 12 C -4 4 0 0 10 D 8 10 0 0 8 E -6 -12 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.490504 D: 0.509496 E: 0.000000 Sum of squares = 0.500180335217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.490504 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -8 6 B -12 0 -4 -10 12 C -4 4 0 0 10 D 8 10 0 0 8 E -6 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=23 E=20 D=19 A=8 so A is eliminated. Round 2 votes counts: C=30 D=27 B=23 E=20 so E is eliminated. Round 3 votes counts: B=39 C=34 D=27 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 A:207 C:205 B:193 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 4 -8 6 B -12 0 -4 -10 12 C -4 4 0 0 10 D 8 10 0 0 8 E -6 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -8 6 B -12 0 -4 -10 12 C -4 4 0 0 10 D 8 10 0 0 8 E -6 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -8 6 B -12 0 -4 -10 12 C -4 4 0 0 10 D 8 10 0 0 8 E -6 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6462: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) D C A E B (8) B E A C D (8) E D B A C (6) C D A B E (5) D E B C A (4) C A D B E (4) C A B D E (4) B A E C D (4) D E C B A (3) D E A C B (3) C D B A E (3) C B A E D (3) E B D A C (2) D E C A B (2) D E A B C (2) D C B A E (2) D C A B E (2) C A B E D (2) B C E A D (2) A E B C D (2) E D A B C (1) E B D C A (1) E A B D C (1) D E B A C (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E C A (1) D B C E A (1) D A E C B (1) C B A D E (1) B E C D A (1) B E C A D (1) B C E D A (1) B C A E D (1) A E D C B (1) A E B D C (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -10 -6 -6 B 14 0 2 -8 0 C 10 -2 0 -10 -10 D 6 8 10 0 2 E 6 0 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 -6 -6 B 14 0 2 -8 0 C 10 -2 0 -10 -10 D 6 8 10 0 2 E 6 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=22 E=20 B=18 A=7 so A is eliminated. Round 2 votes counts: D=33 E=24 C=24 B=19 so B is eliminated. Round 3 votes counts: E=38 D=33 C=29 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:207 B:204 C:194 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 -6 -6 B 14 0 2 -8 0 C 10 -2 0 -10 -10 D 6 8 10 0 2 E 6 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -6 -6 B 14 0 2 -8 0 C 10 -2 0 -10 -10 D 6 8 10 0 2 E 6 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -6 -6 B 14 0 2 -8 0 C 10 -2 0 -10 -10 D 6 8 10 0 2 E 6 0 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6463: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (24) D B C A E (21) B C D A E (6) A E C B D (5) E A D C B (3) D B C E A (3) D A C B E (3) D A B C E (3) C B A E D (3) A E D C B (3) E D A B C (2) E C B A D (2) D E B C A (2) D A E C B (2) C B E A D (2) B C A D E (2) A D C B E (2) E D B C A (1) E B D C A (1) E B C D A (1) E B A C D (1) E A D B C (1) C B A D E (1) B E C D A (1) B D C A E (1) B C E D A (1) B C E A D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 2 4 8 B 0 0 -4 8 4 C -2 4 0 4 2 D -4 -8 -4 0 -6 E -8 -4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.807614 B: 0.192386 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.689252786833 Cumulative probabilities = A: 0.807614 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 8 B 0 0 -4 8 4 C -2 4 0 4 2 D -4 -8 -4 0 -6 E -8 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555661969 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=34 B=12 A=12 C=6 so C is eliminated. Round 2 votes counts: E=36 D=34 B=18 A=12 so A is eliminated. Round 3 votes counts: E=44 D=36 B=20 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:207 B:204 C:204 E:196 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 4 8 B 0 0 -4 8 4 C -2 4 0 4 2 D -4 -8 -4 0 -6 E -8 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555661969 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 8 B 0 0 -4 8 4 C -2 4 0 4 2 D -4 -8 -4 0 -6 E -8 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555661969 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 8 B 0 0 -4 8 4 C -2 4 0 4 2 D -4 -8 -4 0 -6 E -8 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555661969 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6464: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) C B A D E (6) B C D E A (6) A E D C B (6) E B D A C (5) D B E C A (5) B C A E D (5) D E A B C (4) B C E D A (4) A C E B D (4) E D B A C (3) E A D B C (3) C B A E D (3) C A B E D (3) C A B D E (3) A C E D B (3) A C D E B (3) E A D C B (2) D E B C A (2) C B D A E (2) C A D B E (2) B D C E A (2) B C A D E (2) A C B E D (2) E D B C A (1) E A B D C (1) D E B A C (1) D A E C B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B C D A E (1) A E D B C (1) A E C D B (1) A D E C B (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 2 8 2 B 4 0 14 8 2 C -2 -14 0 6 6 D -8 -8 -6 0 -16 E -2 -2 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 8 2 B 4 0 14 8 2 C -2 -14 0 6 6 D -8 -8 -6 0 -16 E -2 -2 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 E=21 C=19 D=13 so D is eliminated. Round 2 votes counts: B=29 E=28 A=24 C=19 so C is eliminated. Round 3 votes counts: B=40 A=32 E=28 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:204 E:203 C:198 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 8 2 B 4 0 14 8 2 C -2 -14 0 6 6 D -8 -8 -6 0 -16 E -2 -2 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 2 B 4 0 14 8 2 C -2 -14 0 6 6 D -8 -8 -6 0 -16 E -2 -2 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 2 B 4 0 14 8 2 C -2 -14 0 6 6 D -8 -8 -6 0 -16 E -2 -2 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6465: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) B D A C E (9) A B D E C (8) C E D B A (7) B D C A E (7) A E D B C (6) A E B D C (6) E C A D B (5) E A D C B (5) C D B E A (5) E D A B C (3) E C D B A (3) E A C D B (3) A E C D B (3) E C D A B (2) D C B E A (2) C E A D B (2) B A D C E (2) E D C B A (1) E A D B C (1) D B E C A (1) C E B D A (1) C D E B A (1) C B D A E (1) C B A E D (1) B C D A E (1) A E C B D (1) A D B E C (1) A C E B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -2 -2 C -2 6 0 -8 0 D 6 2 8 0 -4 E 2 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.148511 D: 0.000000 E: 0.851489 Sum of squares = 0.747088805334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.148511 D: 0.148511 E: 1.000000 A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -2 -2 C -2 6 0 -8 0 D 6 2 8 0 -4 E 2 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555555565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=27 E=23 B=19 D=3 so D is eliminated. Round 2 votes counts: C=29 A=28 E=23 B=20 so B is eliminated. Round 3 votes counts: A=39 C=37 E=24 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:206 E:204 C:198 A:196 B:196 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -2 -2 C -2 6 0 -8 0 D 6 2 8 0 -4 E 2 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555555565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -2 -2 C -2 6 0 -8 0 D 6 2 8 0 -4 E 2 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555555565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -2 -2 C -2 6 0 -8 0 D 6 2 8 0 -4 E 2 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555555565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6466: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (6) E A C D B (5) B C D A E (5) A E B C D (5) D C E A B (4) D B C E A (4) B D C A E (4) B A E D C (4) A B E C D (4) E D A B C (3) D E C A B (3) C B D A E (3) A E B D C (3) E D A C B (2) E A D C B (2) E A C B D (2) E A B D C (2) D C B E A (2) D C B A E (2) D B E C A (2) C D E B A (2) C D E A B (2) C D B E A (2) C B A E D (2) C A B E D (2) B C A D E (2) B A E C D (2) A E C B D (2) A C E B D (2) A C B E D (2) E D C A B (1) E C D A B (1) E A D B C (1) D E B C A (1) D E A B C (1) D C E B A (1) C E A D B (1) C A E B D (1) B E A D C (1) B D E A C (1) B D C E A (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 2 -8 -6 6 B -2 0 -4 4 6 C 8 4 0 10 4 D 6 -4 -10 0 -2 E -6 -6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -6 6 B -2 0 -4 4 6 C 8 4 0 10 4 D 6 -4 -10 0 -2 E -6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 C=21 D=20 E=19 A=18 so A is eliminated. Round 2 votes counts: E=29 B=26 C=25 D=20 so D is eliminated. Round 3 votes counts: E=34 C=34 B=32 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:202 A:197 D:195 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -6 6 B -2 0 -4 4 6 C 8 4 0 10 4 D 6 -4 -10 0 -2 E -6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -6 6 B -2 0 -4 4 6 C 8 4 0 10 4 D 6 -4 -10 0 -2 E -6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -6 6 B -2 0 -4 4 6 C 8 4 0 10 4 D 6 -4 -10 0 -2 E -6 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6467: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (12) A B D C E (6) A B C D E (6) A B E C D (5) E D C A B (4) C D E B A (4) C D B E A (4) C B D A E (4) E D A C B (3) E A D B C (3) D E A C B (3) C D B A E (3) B C A D E (3) B A C D E (3) A B E D C (3) E B C A D (2) E A B C D (2) D C E B A (2) D C E A B (2) D C B E A (2) C E D B A (2) C B D E A (2) A E B D C (2) A D B C E (2) E D C B A (1) E C B D A (1) E A D C B (1) E A C B D (1) D E C A B (1) D C B A E (1) D A E B C (1) C E B D A (1) B E A C D (1) B D C A E (1) B C E A D (1) B C D A E (1) B A E C D (1) B A C E D (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -8 -10 -10 B 6 0 -8 -4 2 C 8 8 0 20 2 D 10 4 -20 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -10 -10 B 6 0 -8 -4 2 C 8 8 0 20 2 D 10 4 -20 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=26 C=20 D=12 B=12 so D is eliminated. Round 2 votes counts: E=34 C=27 A=27 B=12 so B is eliminated. Round 3 votes counts: E=35 C=33 A=32 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:201 D:199 B:198 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -10 -10 B 6 0 -8 -4 2 C 8 8 0 20 2 D 10 4 -20 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -10 -10 B 6 0 -8 -4 2 C 8 8 0 20 2 D 10 4 -20 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -10 -10 B 6 0 -8 -4 2 C 8 8 0 20 2 D 10 4 -20 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6468: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) A E C B D (10) A E B C D (10) D B C E A (9) A C E D B (6) D C B E A (5) B D E C A (5) B E D C A (4) B A D E C (4) A E C D B (4) B D E A C (3) B A E D C (3) A B E D C (3) E A C D B (2) C D E B A (2) A E B D C (2) E C A D B (1) E C A B D (1) D C E B A (1) D B C A E (1) C E D B A (1) C E D A B (1) C D B E A (1) C D B A E (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C A E (1) B D A E C (1) B A D C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 4 2 4 B 10 0 20 22 12 C -4 -20 0 -10 -12 D -2 -22 10 0 -4 E -4 -12 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 2 4 B 10 0 20 22 12 C -4 -20 0 -10 -12 D -2 -22 10 0 -4 E -4 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=34 D=16 C=10 E=4 so E is eliminated. Round 2 votes counts: A=38 B=34 D=16 C=12 so C is eliminated. Round 3 votes counts: A=42 B=34 D=24 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:232 A:200 E:200 D:191 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 2 4 B 10 0 20 22 12 C -4 -20 0 -10 -12 D -2 -22 10 0 -4 E -4 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 2 4 B 10 0 20 22 12 C -4 -20 0 -10 -12 D -2 -22 10 0 -4 E -4 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 2 4 B 10 0 20 22 12 C -4 -20 0 -10 -12 D -2 -22 10 0 -4 E -4 -12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6469: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) B D E C A (8) E A B D C (7) E B D A C (6) E B A D C (6) D B C E A (6) C D B A E (6) A C E D B (6) C D A B E (4) A E C D B (4) A E C B D (4) C B D A E (3) C A D E B (3) B E D A C (3) A E B C D (3) E A D B C (2) D C B E A (2) C A E D B (2) C A D B E (2) C A B D E (2) B D E A C (2) B C D A E (2) A C E B D (2) E A D C B (1) D E B C A (1) D E B A C (1) C D B E A (1) C A B E D (1) B D C A E (1) Total count = 100 A B C D E A 0 -14 -6 -10 -10 B 14 0 14 18 4 C 6 -14 0 -10 4 D 10 -18 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -10 -10 B 14 0 14 18 4 C 6 -14 0 -10 4 D 10 -18 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=24 E=22 A=19 D=10 so D is eliminated. Round 2 votes counts: B=31 C=26 E=24 A=19 so A is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:225 D:204 E:198 C:193 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -10 -10 B 14 0 14 18 4 C 6 -14 0 -10 4 D 10 -18 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -10 -10 B 14 0 14 18 4 C 6 -14 0 -10 4 D 10 -18 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -10 -10 B 14 0 14 18 4 C 6 -14 0 -10 4 D 10 -18 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6470: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) E A D C B (6) C B A D E (6) C A D B E (6) C A B D E (6) E D A B C (5) C A E D B (5) B D E A C (4) B D C E A (4) D B E A C (3) A D E C B (3) A C E D B (3) A C D E B (3) E B D A C (2) E B C A D (2) D E A B C (2) D B A E C (2) C E A B D (2) B E D C A (2) B E D A C (2) B E C D A (2) B D E C A (2) B D C A E (2) B C A D E (2) A E D C B (2) E D B A C (1) E D A C B (1) E C A B D (1) E A D B C (1) E A C D B (1) D A E B C (1) D A B E C (1) C B D A E (1) C B A E D (1) C A E B D (1) C A D E B (1) B C E D A (1) B C E A D (1) B C D E A (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -12 8 8 B -4 0 0 4 12 C 12 0 0 6 8 D -8 -4 -6 0 16 E -8 -12 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.493632 C: 0.506368 D: 0.000000 E: 0.000000 Sum of squares = 0.500081103805 Cumulative probabilities = A: 0.000000 B: 0.493632 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 8 8 B -4 0 0 4 12 C 12 0 0 6 8 D -8 -4 -6 0 16 E -8 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=29 E=20 A=12 D=9 so D is eliminated. Round 2 votes counts: B=35 C=29 E=22 A=14 so A is eliminated. Round 3 votes counts: C=36 B=36 E=28 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:206 A:204 D:199 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 8 8 B -4 0 0 4 12 C 12 0 0 6 8 D -8 -4 -6 0 16 E -8 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 8 8 B -4 0 0 4 12 C 12 0 0 6 8 D -8 -4 -6 0 16 E -8 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 8 8 B -4 0 0 4 12 C 12 0 0 6 8 D -8 -4 -6 0 16 E -8 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6471: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) D C A E B (9) B E A C D (8) E A B C D (7) D C B A E (7) D C B E A (6) C D A E B (6) E A C B D (4) B D C E A (4) A E C D B (4) A E B C D (4) E A C D B (3) D B C A E (3) B D C A E (3) A C E D B (3) E B A C D (2) D C A B E (2) C D B E A (2) B E D C A (2) B A E D C (2) A C D E B (2) C A D E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B D A C E (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 6 4 -4 B 8 0 0 2 8 C -6 0 0 -6 0 D -4 -2 6 0 -4 E 4 -8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.824475 C: 0.175525 D: 0.000000 E: 0.000000 Sum of squares = 0.710568457089 Cumulative probabilities = A: 0.000000 B: 0.824475 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 4 -4 B 8 0 0 2 8 C -6 0 0 -6 0 D -4 -2 6 0 -4 E 4 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000002167 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=27 E=16 A=14 C=9 so C is eliminated. Round 2 votes counts: D=35 B=34 E=16 A=15 so A is eliminated. Round 3 votes counts: D=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:200 A:199 D:198 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 4 -4 B 8 0 0 2 8 C -6 0 0 -6 0 D -4 -2 6 0 -4 E 4 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000002167 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 4 -4 B 8 0 0 2 8 C -6 0 0 -6 0 D -4 -2 6 0 -4 E 4 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000002167 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 4 -4 B 8 0 0 2 8 C -6 0 0 -6 0 D -4 -2 6 0 -4 E 4 -8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000002167 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6472: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (12) D A B C E (8) A B D E C (7) D A B E C (6) C E B A D (5) B E A C D (5) A D B E C (5) B A E D C (4) E B C A D (3) E B A D C (3) C E B D A (3) C D A B E (3) D C A B E (2) D A C E B (2) D A C B E (2) C E D B A (2) C E D A B (2) C D E A B (2) C D A E B (2) B A D E C (2) E D A C B (1) E D A B C (1) E C D B A (1) E B A C D (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) D A E B C (1) C D E B A (1) C B E A D (1) C B A E D (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A E D (1) B A D C E (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 8 14 2 B -2 0 8 8 8 C -8 -8 0 -6 -14 D -14 -8 6 0 0 E -2 -8 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 14 2 B -2 0 8 8 8 C -8 -8 0 -6 -14 D -14 -8 6 0 0 E -2 -8 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=22 C=22 B=16 A=15 so A is eliminated. Round 2 votes counts: D=31 B=25 E=22 C=22 so E is eliminated. Round 3 votes counts: C=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:213 B:211 E:202 D:192 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 14 2 B -2 0 8 8 8 C -8 -8 0 -6 -14 D -14 -8 6 0 0 E -2 -8 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 14 2 B -2 0 8 8 8 C -8 -8 0 -6 -14 D -14 -8 6 0 0 E -2 -8 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 14 2 B -2 0 8 8 8 C -8 -8 0 -6 -14 D -14 -8 6 0 0 E -2 -8 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987162 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6473: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (9) B E C D A (8) E C B D A (7) D A B C E (4) A E B C D (4) A B D E C (4) E C B A D (3) E B C D A (3) C E D B A (3) B D C E A (3) A D C E B (3) A D B E C (3) A B E D C (3) E C A B D (2) E B C A D (2) E A C B D (2) E A B C D (2) D C B A E (2) D B A C E (2) D A C B E (2) C D E B A (2) C D B E A (2) B D E C A (2) A E C D B (2) E B A C D (1) E A C D B (1) D C B E A (1) D B C E A (1) D B C A E (1) D A C E B (1) C E D A B (1) C D E A B (1) C D A E B (1) C B E D A (1) C A E D B (1) B D C A E (1) B D A C E (1) B C E D A (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C B D (1) A C E D B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 0 -4 0 B -4 0 18 8 8 C 0 -18 0 8 -6 D 4 -8 -8 0 -4 E 0 -8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.37499999993 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -4 0 B -4 0 18 8 8 C 0 -18 0 8 -6 D 4 -8 -8 0 -4 E 0 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999978 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=23 B=17 D=14 C=12 so C is eliminated. Round 2 votes counts: A=35 E=27 D=20 B=18 so B is eliminated. Round 3 votes counts: E=37 A=35 D=28 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 E:201 A:200 C:192 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 -4 0 B -4 0 18 8 8 C 0 -18 0 8 -6 D 4 -8 -8 0 -4 E 0 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999978 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -4 0 B -4 0 18 8 8 C 0 -18 0 8 -6 D 4 -8 -8 0 -4 E 0 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999978 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -4 0 B -4 0 18 8 8 C 0 -18 0 8 -6 D 4 -8 -8 0 -4 E 0 -8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999978 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6474: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) E B C D A (8) A D C E B (8) B E D C A (7) C E B A D (6) C B E A D (6) D A E B C (5) A D C B E (5) B C E A D (4) D A C E B (3) D A B E C (3) E B C A D (2) D E B A C (2) D B E A C (2) D A E C B (2) D A B C E (2) C A D E B (2) C A D B E (2) C A B E D (2) B D E A C (2) A C D E B (2) E D B A C (1) E D A C B (1) E C B D A (1) E C B A D (1) E B D C A (1) E B D A C (1) E A C D B (1) D A C B E (1) C B A D E (1) C A E D B (1) C A E B D (1) B E C A D (1) B C E D A (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -16 -8 -16 B 16 0 6 12 2 C 16 -6 0 8 -2 D 8 -12 -8 0 -12 E 16 -2 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -8 -16 B 16 0 6 12 2 C 16 -6 0 8 -2 D 8 -12 -8 0 -12 E 16 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995577 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=21 D=20 E=17 A=16 so A is eliminated. Round 2 votes counts: D=33 B=26 C=24 E=17 so E is eliminated. Round 3 votes counts: B=38 D=35 C=27 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:214 C:208 D:188 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -16 -8 -16 B 16 0 6 12 2 C 16 -6 0 8 -2 D 8 -12 -8 0 -12 E 16 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995577 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -8 -16 B 16 0 6 12 2 C 16 -6 0 8 -2 D 8 -12 -8 0 -12 E 16 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995577 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -8 -16 B 16 0 6 12 2 C 16 -6 0 8 -2 D 8 -12 -8 0 -12 E 16 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995577 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6475: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (16) D C B A E (8) A B C D E (7) E B C A D (6) D E C B A (5) C B A D E (5) D E A C B (4) D A C B E (4) B C A E D (4) A E B C D (4) E D A B C (3) E A D B C (3) C D B A E (3) C B D A E (3) B A C E D (3) B E C A D (2) B C E A D (2) A C B D E (2) A B C E D (2) E D C B A (1) E D A C B (1) E B D C A (1) E B C D A (1) E B A C D (1) E A B D C (1) D E C A B (1) D C E A B (1) D C B E A (1) D C A B E (1) D A E C B (1) B C E D A (1) B C A D E (1) A D C B E (1) Total count = 100 A B C D E A 0 4 6 20 -2 B -4 0 16 24 0 C -6 -16 0 26 -2 D -20 -24 -26 0 -4 E 2 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.204607 C: 0.000000 D: 0.000000 E: 0.795393 Sum of squares = 0.674513553711 Cumulative probabilities = A: 0.000000 B: 0.204607 C: 0.204607 D: 0.204607 E: 1.000000 A B C D E A 0 4 6 20 -2 B -4 0 16 24 0 C -6 -16 0 26 -2 D -20 -24 -26 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555685691 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=26 A=16 B=13 C=11 so C is eliminated. Round 2 votes counts: E=34 D=29 B=21 A=16 so A is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:214 E:204 C:201 D:163 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 20 -2 B -4 0 16 24 0 C -6 -16 0 26 -2 D -20 -24 -26 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555685691 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 20 -2 B -4 0 16 24 0 C -6 -16 0 26 -2 D -20 -24 -26 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555685691 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 20 -2 B -4 0 16 24 0 C -6 -16 0 26 -2 D -20 -24 -26 0 -4 E 2 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555685691 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6476: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) B D C E A (6) B A E C D (5) A E D C B (5) A E C D B (5) D E C A B (4) D A E C B (4) B D C A E (4) B C E D A (4) A B E C D (4) D C B E A (3) D B A C E (3) C E D A B (3) A E B C D (3) E C A D B (2) E A C D B (2) D C E B A (2) D B C E A (2) C D E B A (2) C D E A B (2) B C D E A (2) A E C B D (2) A D E B C (2) A D B E C (2) A B E D C (2) E C D A B (1) E C A B D (1) E A D C B (1) E A C B D (1) D A B E C (1) C E D B A (1) C E B A D (1) C E A B D (1) C B E D A (1) B D A E C (1) B D A C E (1) B C E A D (1) B C A E D (1) B A E D C (1) B A D E C (1) B A C D E (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 14 -2 -8 0 B -14 0 -4 -10 -8 C 2 4 0 -8 -4 D 8 10 8 0 2 E 0 8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -8 0 B -14 0 -4 -10 -8 C 2 4 0 -8 -4 D 8 10 8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=27 D=26 C=11 E=8 so E is eliminated. Round 2 votes counts: A=31 B=28 D=26 C=15 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:205 A:202 C:197 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -8 0 B -14 0 -4 -10 -8 C 2 4 0 -8 -4 D 8 10 8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -8 0 B -14 0 -4 -10 -8 C 2 4 0 -8 -4 D 8 10 8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -8 0 B -14 0 -4 -10 -8 C 2 4 0 -8 -4 D 8 10 8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6477: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (7) C D A E B (7) B D A C E (7) E A B D C (5) C D B E A (5) B A E D C (5) E C A B D (4) E A B C D (4) C E A D B (4) B A D E C (4) E B A C D (3) D B A C E (3) C E D A B (3) C D E B A (3) B E A D C (3) B D C A E (3) E C A D B (2) E A C D B (2) E A C B D (2) D C B A E (2) D C A B E (2) D B C A E (2) C E D B A (2) C E B D A (2) B C D E A (2) A E B D C (2) D C A E B (1) D A C E B (1) D A B C E (1) B D C E A (1) B D A E C (1) A E D B C (1) A E C D B (1) A D E B C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 -10 -8 B -4 0 -2 -2 -16 C 4 2 0 6 18 D 10 2 -6 0 10 E 8 16 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -10 -8 B -4 0 -2 -2 -16 C 4 2 0 6 18 D 10 2 -6 0 10 E 8 16 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=26 E=22 D=12 A=7 so A is eliminated. Round 2 votes counts: C=33 B=27 E=26 D=14 so D is eliminated. Round 3 votes counts: C=40 B=33 E=27 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:208 E:198 A:191 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -10 -8 B -4 0 -2 -2 -16 C 4 2 0 6 18 D 10 2 -6 0 10 E 8 16 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -10 -8 B -4 0 -2 -2 -16 C 4 2 0 6 18 D 10 2 -6 0 10 E 8 16 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -10 -8 B -4 0 -2 -2 -16 C 4 2 0 6 18 D 10 2 -6 0 10 E 8 16 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999431 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6478: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (14) D A B C E (9) E B A C D (8) D C A B E (8) E C D B A (5) C E D A B (5) E C D A B (4) D C E A B (4) C A B D E (4) B A E C D (4) B A D C E (4) D E C A B (3) D E A B C (3) E B A D C (2) D A C B E (2) C E B A D (2) C E A B D (2) B A D E C (2) B A C E D (2) A B D C E (2) E D C B A (1) E D C A B (1) E C B D A (1) D C A E B (1) D B A C E (1) D A B E C (1) C D A B E (1) C A B E D (1) B A E D C (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -14 0 -10 B -4 0 -18 2 -12 C 14 18 0 10 0 D 0 -2 -10 0 -6 E 10 12 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.372311 D: 0.000000 E: 0.627689 Sum of squares = 0.532608726239 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.372311 D: 0.372311 E: 1.000000 A B C D E A 0 4 -14 0 -10 B -4 0 -18 2 -12 C 14 18 0 10 0 D 0 -2 -10 0 -6 E 10 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=32 C=15 B=14 A=3 so A is eliminated. Round 2 votes counts: E=36 D=32 B=17 C=15 so C is eliminated. Round 3 votes counts: E=45 D=33 B=22 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:221 E:214 D:191 A:190 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 0 -10 B -4 0 -18 2 -12 C 14 18 0 10 0 D 0 -2 -10 0 -6 E 10 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 0 -10 B -4 0 -18 2 -12 C 14 18 0 10 0 D 0 -2 -10 0 -6 E 10 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 0 -10 B -4 0 -18 2 -12 C 14 18 0 10 0 D 0 -2 -10 0 -6 E 10 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6479: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) C A D B E (8) E B D A C (6) C D A B E (6) C D B A E (5) A E B C D (5) A C D E B (5) E B D C A (4) E B A D C (4) E A D B C (4) D B E C A (3) C B D A E (3) B E D C A (3) E D B C A (2) E A B D C (2) E A B C D (2) D C B E A (2) D C A B E (2) D B C E A (2) A E C B D (2) A C E D B (2) E D B A C (1) D C E B A (1) D C B A E (1) C B D E A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C E A D (1) B C D E A (1) B A E C D (1) A E D B C (1) A E C D B (1) A E B D C (1) A D C E B (1) A D C B E (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 4 8 16 B -12 0 -6 -14 10 C -4 6 0 14 10 D -8 14 -14 0 8 E -16 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 8 16 B -12 0 -6 -14 10 C -4 6 0 14 10 D -8 14 -14 0 8 E -16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=25 C=23 D=11 B=9 so B is eliminated. Round 2 votes counts: A=33 E=29 C=25 D=13 so D is eliminated. Round 3 votes counts: C=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:213 D:200 B:189 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 8 16 B -12 0 -6 -14 10 C -4 6 0 14 10 D -8 14 -14 0 8 E -16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 8 16 B -12 0 -6 -14 10 C -4 6 0 14 10 D -8 14 -14 0 8 E -16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 8 16 B -12 0 -6 -14 10 C -4 6 0 14 10 D -8 14 -14 0 8 E -16 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6480: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) A D B C E (8) E D A C B (7) E D A B C (7) E C D A B (7) C B A D E (7) D A E C B (6) B C A D E (6) B A D C E (6) D A B E C (5) D A E B C (4) C E B A D (4) C B E A D (3) E D C A B (2) B E D A C (2) B C E A D (2) A D C E B (2) A D B E C (2) E D B A C (1) E C D B A (1) E B D C A (1) E B C D A (1) D E A B C (1) C E A D B (1) C B A E D (1) C A E D B (1) B A C D E (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 10 8 -8 2 B -10 0 -6 -14 -12 C -8 6 0 -12 -14 D 8 14 12 0 0 E -2 12 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.597167 E: 0.402833 Sum of squares = 0.518882782562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.597167 E: 1.000000 A B C D E A 0 10 8 -8 2 B -10 0 -6 -14 -12 C -8 6 0 -12 -14 D 8 14 12 0 0 E -2 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=17 B=17 D=16 A=14 so A is eliminated. Round 2 votes counts: E=36 D=30 C=17 B=17 so C is eliminated. Round 3 votes counts: E=42 D=30 B=28 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:212 A:206 C:186 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 8 -8 2 B -10 0 -6 -14 -12 C -8 6 0 -12 -14 D 8 14 12 0 0 E -2 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 -8 2 B -10 0 -6 -14 -12 C -8 6 0 -12 -14 D 8 14 12 0 0 E -2 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 -8 2 B -10 0 -6 -14 -12 C -8 6 0 -12 -14 D 8 14 12 0 0 E -2 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6481: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) C D A E B (10) C D B A E (8) D C B A E (6) C E A D B (6) C B E A D (6) A E D B C (6) C D B E A (4) B E A C D (4) A E B D C (4) E A C B D (3) C B E D A (3) B E A D C (3) E B A C D (2) E A D B C (2) D C A E B (2) D B C A E (2) D A E B C (2) C E A B D (2) B A E D C (2) D B A E C (1) D A E C B (1) D A B C E (1) C E B A D (1) C D A B E (1) C B D E A (1) B D E A C (1) B D A E C (1) B C E A D (1) A E C D B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -6 12 0 B -8 0 -10 -8 -8 C 6 10 0 6 8 D -12 8 -6 0 -16 E 0 8 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 12 0 B -8 0 -10 -8 -8 C 6 10 0 6 8 D -12 8 -6 0 -16 E 0 8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 E=18 D=15 A=13 B=12 so B is eliminated. Round 2 votes counts: C=43 E=25 D=17 A=15 so A is eliminated. Round 3 votes counts: C=43 E=39 D=18 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:208 A:207 D:187 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 12 0 B -8 0 -10 -8 -8 C 6 10 0 6 8 D -12 8 -6 0 -16 E 0 8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 12 0 B -8 0 -10 -8 -8 C 6 10 0 6 8 D -12 8 -6 0 -16 E 0 8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 12 0 B -8 0 -10 -8 -8 C 6 10 0 6 8 D -12 8 -6 0 -16 E 0 8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6482: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) B E C A D (7) B C E D A (7) E B A C D (6) A E D B C (6) E A D B C (4) C B D E A (4) C B D A E (4) D E A C B (3) D A E C B (3) C D B A E (3) B E A C D (3) B C E A D (3) A E B D C (3) E D A B C (2) E A B D C (2) C B E D A (2) C B A D E (2) C A B D E (2) B C A E D (2) A D C B E (2) A B E C D (2) E D B C A (1) E D A C B (1) E B C D A (1) E B C A D (1) D E C A B (1) D C B E A (1) D C B A E (1) D A C E B (1) D A C B E (1) C D A B E (1) B C D E A (1) B C A D E (1) B A E C D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 10 24 0 B 2 0 8 12 4 C -10 -8 0 10 -20 D -24 -12 -10 0 -8 E 0 -4 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 24 0 B 2 0 8 12 4 C -10 -8 0 10 -20 D -24 -12 -10 0 -8 E 0 -4 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 E=18 C=18 D=11 so D is eliminated. Round 2 votes counts: A=33 B=25 E=22 C=20 so C is eliminated. Round 3 votes counts: B=42 A=36 E=22 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:213 E:212 C:186 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 24 0 B 2 0 8 12 4 C -10 -8 0 10 -20 D -24 -12 -10 0 -8 E 0 -4 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 24 0 B 2 0 8 12 4 C -10 -8 0 10 -20 D -24 -12 -10 0 -8 E 0 -4 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 24 0 B 2 0 8 12 4 C -10 -8 0 10 -20 D -24 -12 -10 0 -8 E 0 -4 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6483: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (10) A D B C E (7) C E A B D (6) E C D B A (5) C E D B A (5) D B A C E (4) C E D A B (4) A B D C E (4) E C B D A (3) D C E B A (3) D C B E A (3) D B A E C (3) C E A D B (3) B D A E C (3) E C B A D (2) E B D C A (2) E A B C D (2) D B E C A (2) C D E A B (2) B A D E C (2) E D B C A (1) E B A C D (1) E A C B D (1) D C B A E (1) D C A B E (1) D B E A C (1) D B C E A (1) D B C A E (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E B A (1) C A E B D (1) C A D B E (1) B A E D C (1) A D C B E (1) A D B E C (1) A C E B D (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 2 2 B -8 0 2 -8 12 C -2 -2 0 -10 16 D -2 8 10 0 14 E -2 -12 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999506 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 2 2 B -8 0 2 -8 12 C -2 -2 0 -10 16 D -2 8 10 0 14 E -2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=24 D=22 E=17 B=6 so B is eliminated. Round 2 votes counts: A=34 D=25 C=24 E=17 so E is eliminated. Round 3 votes counts: A=38 C=34 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:207 C:201 B:199 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 2 2 B -8 0 2 -8 12 C -2 -2 0 -10 16 D -2 8 10 0 14 E -2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 2 B -8 0 2 -8 12 C -2 -2 0 -10 16 D -2 8 10 0 14 E -2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 2 B -8 0 2 -8 12 C -2 -2 0 -10 16 D -2 8 10 0 14 E -2 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6484: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) E D B A C (7) E D A B C (6) C A B E D (6) D E A C B (5) B E D C A (5) A C E D B (5) B E C D A (4) A C D E B (4) E A D C B (3) D B E C A (3) C B A E D (3) C A B D E (3) B D E C A (3) B C A E D (3) A C D B E (3) E D A C B (2) D E A B C (2) D A E C B (2) C B A D E (2) B C D A E (2) A C E B D (2) E B C A D (1) E A C B D (1) D B C A E (1) D A C E B (1) C A D B E (1) B D C E A (1) B D C A E (1) B C E A D (1) B C A D E (1) A E C D B (1) A E C B D (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 18 -14 -12 B 0 0 6 -18 -12 C -18 -6 0 -10 -16 D 14 18 10 0 -4 E 12 12 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 18 -14 -12 B 0 0 6 -18 -12 C -18 -6 0 -10 -16 D 14 18 10 0 -4 E 12 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=21 E=20 A=18 C=15 so C is eliminated. Round 2 votes counts: A=28 D=26 B=26 E=20 so E is eliminated. Round 3 votes counts: D=41 A=32 B=27 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:222 D:219 A:196 B:188 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 18 -14 -12 B 0 0 6 -18 -12 C -18 -6 0 -10 -16 D 14 18 10 0 -4 E 12 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 -14 -12 B 0 0 6 -18 -12 C -18 -6 0 -10 -16 D 14 18 10 0 -4 E 12 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 -14 -12 B 0 0 6 -18 -12 C -18 -6 0 -10 -16 D 14 18 10 0 -4 E 12 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6485: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (18) E C B D A (16) B D A E C (16) D A B C E (8) A D B C E (8) E C A D B (5) A D C B E (5) E C A B D (4) B D E C A (4) D B A C E (3) E C B A D (2) E B C D A (2) C A E D B (2) B E C D A (2) B D A C E (2) B E D C A (1) B D E A C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -12 -10 -10 B -2 0 -6 0 0 C 12 6 0 4 -6 D 10 0 -4 0 -4 E 10 0 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.327987 C: 0.000000 D: 0.000000 E: 0.672013 Sum of squares = 0.559177050451 Cumulative probabilities = A: 0.000000 B: 0.327987 C: 0.327987 D: 0.327987 E: 1.000000 A B C D E A 0 2 -12 -10 -10 B -2 0 -6 0 0 C 12 6 0 4 -6 D 10 0 -4 0 -4 E 10 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499649 C: 0.000000 D: 0.000000 E: 0.500351 Sum of squares = 0.500000245875 Cumulative probabilities = A: 0.000000 B: 0.499649 C: 0.499649 D: 0.499649 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=26 C=20 A=14 D=11 so D is eliminated. Round 2 votes counts: E=29 B=29 A=22 C=20 so C is eliminated. Round 3 votes counts: E=47 B=29 A=24 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:210 C:208 D:201 B:196 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -12 -10 -10 B -2 0 -6 0 0 C 12 6 0 4 -6 D 10 0 -4 0 -4 E 10 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499649 C: 0.000000 D: 0.000000 E: 0.500351 Sum of squares = 0.500000245875 Cumulative probabilities = A: 0.000000 B: 0.499649 C: 0.499649 D: 0.499649 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -10 -10 B -2 0 -6 0 0 C 12 6 0 4 -6 D 10 0 -4 0 -4 E 10 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499649 C: 0.000000 D: 0.000000 E: 0.500351 Sum of squares = 0.500000245875 Cumulative probabilities = A: 0.000000 B: 0.499649 C: 0.499649 D: 0.499649 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -10 -10 B -2 0 -6 0 0 C 12 6 0 4 -6 D 10 0 -4 0 -4 E 10 0 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499649 C: 0.000000 D: 0.000000 E: 0.500351 Sum of squares = 0.500000245875 Cumulative probabilities = A: 0.000000 B: 0.499649 C: 0.499649 D: 0.499649 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6486: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) A B D E C (7) E C D B A (6) C E D B A (6) B A D E C (6) A B D C E (6) E C A B D (5) E C B A D (4) D B A C E (4) A D B C E (4) E C D A B (3) C E A D B (3) B D A E C (3) A B E D C (3) E C B D A (2) E C A D B (2) D C A B E (2) D A B C E (2) C D E B A (2) C D E A B (2) B E A D C (2) B A E D C (2) B A D C E (2) A C D B E (2) E B C A D (1) E B A C D (1) E A B C D (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D E B (1) B D E C A (1) B D E A C (1) A E B C D (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -2 12 0 B -14 0 -4 -2 0 C 2 4 0 8 -2 D -12 2 -8 0 -2 E 0 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.259811 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.740189 Sum of squares = 0.61538098987 Cumulative probabilities = A: 0.259811 B: 0.259811 C: 0.259811 D: 0.259811 E: 1.000000 A B C D E A 0 14 -2 12 0 B -14 0 -4 -2 0 C 2 4 0 8 -2 D -12 2 -8 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499992 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500008 Sum of squares = 0.500000000137 Cumulative probabilities = A: 0.499992 B: 0.499992 C: 0.499992 D: 0.499992 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 C=25 A=25 B=17 D=8 so D is eliminated. Round 2 votes counts: C=27 A=27 E=25 B=21 so B is eliminated. Round 3 votes counts: A=44 E=29 C=27 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:206 E:202 B:190 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -2 12 0 B -14 0 -4 -2 0 C 2 4 0 8 -2 D -12 2 -8 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499992 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500008 Sum of squares = 0.500000000137 Cumulative probabilities = A: 0.499992 B: 0.499992 C: 0.499992 D: 0.499992 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 12 0 B -14 0 -4 -2 0 C 2 4 0 8 -2 D -12 2 -8 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499992 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500008 Sum of squares = 0.500000000137 Cumulative probabilities = A: 0.499992 B: 0.499992 C: 0.499992 D: 0.499992 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 12 0 B -14 0 -4 -2 0 C 2 4 0 8 -2 D -12 2 -8 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499992 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500008 Sum of squares = 0.500000000137 Cumulative probabilities = A: 0.499992 B: 0.499992 C: 0.499992 D: 0.499992 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6487: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) C A D B E (8) A B E C D (7) D C E B A (6) B A E D C (6) E B A D C (4) C A B D E (4) B E A D C (4) E B D A C (3) E A B C D (3) D C B A E (3) D B E C A (3) C D A B E (3) A E B C D (3) A C B E D (3) A C B D E (3) A B C E D (3) A B C D E (3) E D C B A (2) E A B D C (2) D C E A B (2) D B C E A (2) E D C A B (1) E D B A C (1) E C A D B (1) E A C B D (1) D E C B A (1) D E C A B (1) D E B C A (1) D C B E A (1) C D B A E (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E A C (1) B D A E C (1) B A C D E (1) Total count = 100 A B C D E A 0 16 0 16 18 B -16 0 -2 6 16 C 0 2 0 8 6 D -16 -6 -8 0 8 E -18 -16 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.455985 B: 0.000000 C: 0.544015 D: 0.000000 E: 0.000000 Sum of squares = 0.503874626363 Cumulative probabilities = A: 0.455985 B: 0.455985 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 16 18 B -16 0 -2 6 16 C 0 2 0 8 6 D -16 -6 -8 0 8 E -18 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=22 D=20 E=18 B=14 so B is eliminated. Round 2 votes counts: A=29 C=26 E=23 D=22 so D is eliminated. Round 3 votes counts: C=40 E=30 A=30 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:208 B:202 D:189 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 16 18 B -16 0 -2 6 16 C 0 2 0 8 6 D -16 -6 -8 0 8 E -18 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 16 18 B -16 0 -2 6 16 C 0 2 0 8 6 D -16 -6 -8 0 8 E -18 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 16 18 B -16 0 -2 6 16 C 0 2 0 8 6 D -16 -6 -8 0 8 E -18 -16 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6488: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) B D E C A (6) A B E D C (6) C D B E A (5) C A E D B (4) C A D B E (4) B D A E C (4) A E B D C (4) A B D E C (4) E C D B A (3) E A B D C (3) C D B A E (3) A E C B D (3) A C B D E (3) E D C B A (2) E D B C A (2) E B D A C (2) E A C D B (2) D E C B A (2) C E D B A (2) B D E A C (2) B A D E C (2) B A D C E (2) A C E D B (2) A C B E D (2) A B D C E (2) E D B A C (1) E B D C A (1) E B A D C (1) E A D B C (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C E A (1) C E D A B (1) C D A B E (1) C A D E B (1) B E A D C (1) B D A C E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 0 -2 0 B 10 0 0 2 2 C 0 0 0 -6 -12 D 2 -2 6 0 10 E 0 -2 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.913841 C: 0.086159 D: 0.000000 E: 0.000000 Sum of squares = 0.842528115448 Cumulative probabilities = A: 0.000000 B: 0.913841 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -2 0 B 10 0 0 2 2 C 0 0 0 -6 -12 D 2 -2 6 0 10 E 0 -2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041261 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=28 E=18 B=18 D=6 so D is eliminated. Round 2 votes counts: C=31 A=28 E=21 B=20 so B is eliminated. Round 3 votes counts: A=37 C=32 E=31 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:208 B:207 E:200 A:194 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -2 0 B 10 0 0 2 2 C 0 0 0 -6 -12 D 2 -2 6 0 10 E 0 -2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041261 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -2 0 B 10 0 0 2 2 C 0 0 0 -6 -12 D 2 -2 6 0 10 E 0 -2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041261 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -2 0 B 10 0 0 2 2 C 0 0 0 -6 -12 D 2 -2 6 0 10 E 0 -2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041261 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6489: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) C A D B E (8) D E C B A (7) E B D A C (6) C A B D E (6) B E A C D (6) E D B A C (5) A B E C D (5) E B A D C (4) D C E B A (4) C D A B E (4) A B C E D (4) E A B D C (3) B C A E D (3) B A C E D (3) E D A B C (2) C D B A E (2) B E A D C (2) B A E C D (2) A E B C D (2) A C B E D (2) E D B C A (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E A B (1) D C B E A (1) C B D A E (1) B E C D A (1) B E C A D (1) B C A D E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -4 6 6 B 2 0 4 6 6 C 4 -4 0 4 0 D -6 -6 -4 0 -4 E -6 -6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 6 6 B 2 0 4 6 6 C 4 -4 0 4 0 D -6 -6 -4 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=21 C=21 B=19 A=14 so A is eliminated. Round 2 votes counts: B=28 D=25 C=24 E=23 so E is eliminated. Round 3 votes counts: B=43 D=33 C=24 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:203 C:202 E:196 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 6 6 B 2 0 4 6 6 C 4 -4 0 4 0 D -6 -6 -4 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 6 6 B 2 0 4 6 6 C 4 -4 0 4 0 D -6 -6 -4 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 6 6 B 2 0 4 6 6 C 4 -4 0 4 0 D -6 -6 -4 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998702 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6490: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) D E A B C (7) C A B E D (7) C A B D E (7) B C A E D (6) E D B A C (5) D A C E B (5) B E C A D (5) A C D E B (5) C B A E D (4) A C D B E (4) E B D C A (3) D E B A C (3) D E A C B (3) B E C D A (3) A C E B D (3) A C B D E (3) E B C A D (2) E A C B D (2) B D E C A (2) A D C B E (2) E D A B C (1) E B D A C (1) E B C D A (1) E A D C B (1) D E B C A (1) D B E C A (1) D A E C B (1) D A C B E (1) C B D A E (1) C B A D E (1) A D E C B (1) Total count = 100 A B C D E A 0 6 -4 6 2 B -6 0 -2 18 10 C 4 2 0 8 -2 D -6 -18 -8 0 -4 E -2 -10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408176 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 6 -4 6 2 B -6 0 -2 18 10 C 4 2 0 8 -2 D -6 -18 -8 0 -4 E -2 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408153 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=22 C=20 A=18 E=16 so E is eliminated. Round 2 votes counts: B=31 D=28 A=21 C=20 so C is eliminated. Round 3 votes counts: B=37 A=35 D=28 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:210 C:206 A:205 E:197 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 6 2 B -6 0 -2 18 10 C 4 2 0 8 -2 D -6 -18 -8 0 -4 E -2 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408153 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 6 2 B -6 0 -2 18 10 C 4 2 0 8 -2 D -6 -18 -8 0 -4 E -2 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408153 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 6 2 B -6 0 -2 18 10 C 4 2 0 8 -2 D -6 -18 -8 0 -4 E -2 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408153 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6491: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (19) D B E A C (14) D B E C A (7) A D C B E (7) A C D B E (7) E B D C A (5) E B D A C (5) D A B E C (5) A C D E B (5) C E B A D (4) C A D B E (4) B E D C A (4) E B C D A (2) E B C A D (2) C E B D A (2) C A D E B (2) E B A D C (1) D A B C E (1) C B E A D (1) A D B E C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -4 10 6 B -6 0 -4 -8 4 C 4 4 0 -2 8 D -10 8 2 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000082 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 10 6 B -6 0 -4 -8 4 C 4 4 0 -2 8 D -10 8 2 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.46875000014 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=27 A=22 E=15 B=4 so B is eliminated. Round 2 votes counts: C=32 D=27 A=22 E=19 so E is eliminated. Round 3 votes counts: D=41 C=36 A=23 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:209 C:207 D:204 B:193 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 10 6 B -6 0 -4 -8 4 C 4 4 0 -2 8 D -10 8 2 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.46875000014 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 10 6 B -6 0 -4 -8 4 C 4 4 0 -2 8 D -10 8 2 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.46875000014 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 10 6 B -6 0 -4 -8 4 C 4 4 0 -2 8 D -10 8 2 0 8 E -6 -4 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.46875000014 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6492: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) C B E D A (9) E D A B C (8) B C A D E (8) E A D B C (5) B C A E D (5) E D C A B (4) C D E A B (4) C B A D E (4) B A D E C (4) C E D A B (3) C E B D A (3) A D E B C (3) A B D E C (3) E B C D A (2) D A E C B (2) B C E A D (2) B A E D C (2) A E D B C (2) E C D A B (1) E C B D A (1) E B D A C (1) D E A C B (1) D E A B C (1) D A E B C (1) D A C E B (1) C E D B A (1) C D E B A (1) C D B A E (1) C B D E A (1) C B D A E (1) B A D C E (1) B A C D E (1) A D E C B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 -14 -16 B -4 0 2 -2 -12 C 2 -2 0 -4 -4 D 14 2 4 0 -18 E 16 12 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -2 -14 -16 B -4 0 2 -2 -12 C 2 -2 0 -4 -4 D 14 2 4 0 -18 E 16 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=28 B=23 A=11 D=6 so D is eliminated. Round 2 votes counts: E=34 C=28 B=23 A=15 so A is eliminated. Round 3 votes counts: E=43 C=29 B=28 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:201 C:196 B:192 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 -14 -16 B -4 0 2 -2 -12 C 2 -2 0 -4 -4 D 14 2 4 0 -18 E 16 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -14 -16 B -4 0 2 -2 -12 C 2 -2 0 -4 -4 D 14 2 4 0 -18 E 16 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -14 -16 B -4 0 2 -2 -12 C 2 -2 0 -4 -4 D 14 2 4 0 -18 E 16 12 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6493: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) D A C E B (9) C A B E D (8) A C D B E (6) D E B C A (5) D E A B C (5) A D C B E (5) D A C B E (4) C B A E D (4) A C B E D (4) E B D A C (3) D A E B C (3) C B E A D (3) B E C A D (3) B C E A D (3) A D E B C (3) A D C E B (3) E D B A C (2) E B D C A (2) E B C D A (2) D A E C B (2) C D A B E (2) C A D B E (2) C A B D E (2) E B C A D (1) D C E B A (1) D C E A B (1) D C A E B (1) B E C D A (1) A D E C B (1) Total count = 100 A B C D E A 0 22 18 -4 18 B -22 0 -16 -28 -6 C -18 16 0 -18 16 D 4 28 18 0 28 E -18 6 -16 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 18 -4 18 B -22 0 -16 -28 -6 C -18 16 0 -18 16 D 4 28 18 0 28 E -18 6 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=22 C=21 E=10 B=7 so B is eliminated. Round 2 votes counts: D=40 C=24 A=22 E=14 so E is eliminated. Round 3 votes counts: D=47 C=31 A=22 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:239 A:227 C:198 E:172 B:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 18 -4 18 B -22 0 -16 -28 -6 C -18 16 0 -18 16 D 4 28 18 0 28 E -18 6 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 -4 18 B -22 0 -16 -28 -6 C -18 16 0 -18 16 D 4 28 18 0 28 E -18 6 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 -4 18 B -22 0 -16 -28 -6 C -18 16 0 -18 16 D 4 28 18 0 28 E -18 6 -16 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6494: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (13) B E C D A (11) D A C E B (9) E A D B C (8) D A C B E (7) C D A E B (7) B E A D C (7) A D E C B (5) E B C A D (4) B E C A D (4) A D C E B (4) C B E D A (3) C B D A E (3) E B A D C (2) A D E B C (2) E A D C B (1) E A B D C (1) D A B E C (1) C E D A B (1) C D B A E (1) C A D E B (1) B E D A C (1) B E A C D (1) B D A C E (1) B C E D A (1) B C D A E (1) Total count = 100 A B C D E A 0 20 0 -20 10 B -20 0 -10 -20 10 C 0 10 0 2 4 D 20 20 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.067007 B: 0.000000 C: 0.932993 D: 0.000000 E: 0.000000 Sum of squares = 0.874966161865 Cumulative probabilities = A: 0.067007 B: 0.067007 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 0 -20 10 B -20 0 -10 -20 10 C 0 10 0 2 4 D 20 20 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.909091 D: 0.000000 E: 0.000000 Sum of squares = 0.834710767455 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=27 D=17 E=16 A=11 so A is eliminated. Round 2 votes counts: C=29 D=28 B=27 E=16 so E is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:208 A:205 E:183 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 0 -20 10 B -20 0 -10 -20 10 C 0 10 0 2 4 D 20 20 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.909091 D: 0.000000 E: 0.000000 Sum of squares = 0.834710767455 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 -20 10 B -20 0 -10 -20 10 C 0 10 0 2 4 D 20 20 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.909091 D: 0.000000 E: 0.000000 Sum of squares = 0.834710767455 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 -20 10 B -20 0 -10 -20 10 C 0 10 0 2 4 D 20 20 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.909091 D: 0.000000 E: 0.000000 Sum of squares = 0.834710767455 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6495: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) A D B E C (7) C E B D A (6) C B A E D (6) C E B A D (5) D E B A C (4) D E A B C (4) D A B E C (4) B C E A D (4) A C B D E (4) E D B C A (3) A D C B E (3) E D C B A (2) E D C A B (2) E C B D A (2) E B D C A (2) E B C D A (2) D E C A B (2) D E B C A (2) D A E B C (2) C E D A B (2) C B E A D (2) C A B E D (2) B C A E D (2) B A C E D (2) A D B C E (2) E D B A C (1) E C D B A (1) E B D A C (1) D B E A C (1) D A E C B (1) C E D B A (1) C A E D B (1) B E C A D (1) B D E A C (1) B A D E C (1) A D C E B (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 0 -8 -18 B 4 0 -2 -8 -10 C 0 2 0 -10 -10 D 8 8 10 0 -2 E 18 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 -8 -18 B 4 0 -2 -8 -10 C 0 2 0 -10 -10 D 8 8 10 0 -2 E 18 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 A=20 E=16 B=11 so B is eliminated. Round 2 votes counts: C=31 D=29 A=23 E=17 so E is eliminated. Round 3 votes counts: D=40 C=37 A=23 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:212 B:192 C:191 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -8 -18 B 4 0 -2 -8 -10 C 0 2 0 -10 -10 D 8 8 10 0 -2 E 18 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -8 -18 B 4 0 -2 -8 -10 C 0 2 0 -10 -10 D 8 8 10 0 -2 E 18 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -8 -18 B 4 0 -2 -8 -10 C 0 2 0 -10 -10 D 8 8 10 0 -2 E 18 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6496: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) B C E D A (7) A D E C B (7) A D E B C (6) B C A E D (5) D E A B C (4) C A E D B (4) B D C E A (4) A C B D E (4) E D C A B (3) D E B C A (3) C A B E D (3) B E D C A (3) B D E C A (3) B C A D E (3) B A C D E (3) A C B E D (3) E D B C A (2) E B D C A (2) C A E B D (2) B C E A D (2) E D C B A (1) E D A C B (1) E C D A B (1) D E B A C (1) D E A C B (1) D B E C A (1) C E D B A (1) C E D A B (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E D A (1) C B E A D (1) B D E A C (1) B D A E C (1) B A D E C (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -26 16 6 B 8 0 4 18 8 C 26 -4 0 10 16 D -16 -18 -10 0 -10 E -6 -8 -16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -26 16 6 B 8 0 4 18 8 C 26 -4 0 10 16 D -16 -18 -10 0 -10 E -6 -8 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=24 C=23 E=10 D=10 so E is eliminated. Round 2 votes counts: B=35 C=24 A=24 D=17 so D is eliminated. Round 3 votes counts: B=42 A=30 C=28 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:224 B:219 A:194 E:190 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -26 16 6 B 8 0 4 18 8 C 26 -4 0 10 16 D -16 -18 -10 0 -10 E -6 -8 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -26 16 6 B 8 0 4 18 8 C 26 -4 0 10 16 D -16 -18 -10 0 -10 E -6 -8 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -26 16 6 B 8 0 4 18 8 C 26 -4 0 10 16 D -16 -18 -10 0 -10 E -6 -8 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999424 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6497: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D B C E A (8) D B E A C (6) A E C B D (6) E B A C D (4) D B A E C (4) C D B E A (4) B D E A C (4) A E B D C (4) A E B C D (4) E A B C D (3) C B E A D (3) C A E B D (3) B E A C D (3) D C B A E (2) D C A E B (2) D A E B C (2) C E A B D (2) C D B A E (2) C B E D A (2) C A D E B (2) B E A D C (2) B D C E A (2) A D C E B (2) E B A D C (1) D C A B E (1) D A C E B (1) C E B A D (1) C D A E B (1) C D A B E (1) C B D E A (1) C A E D B (1) B E D A C (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -20 4 -8 -14 B 20 0 0 -4 12 C -4 0 0 -6 6 D 8 4 6 0 10 E 14 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 4 -8 -14 B 20 0 0 -4 12 C -4 0 0 -6 6 D 8 4 6 0 10 E 14 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=23 A=21 B=13 E=8 so E is eliminated. Round 2 votes counts: D=35 A=24 C=23 B=18 so B is eliminated. Round 3 votes counts: D=42 A=34 C=24 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:214 D:214 C:198 E:193 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 4 -8 -14 B 20 0 0 -4 12 C -4 0 0 -6 6 D 8 4 6 0 10 E 14 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 4 -8 -14 B 20 0 0 -4 12 C -4 0 0 -6 6 D 8 4 6 0 10 E 14 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 4 -8 -14 B 20 0 0 -4 12 C -4 0 0 -6 6 D 8 4 6 0 10 E 14 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6498: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) E A B D C (5) D C B E A (5) B E D C A (5) A C D B E (5) E D B C A (4) E B A D C (4) A E B D C (4) A C D E B (4) E B D C A (3) C D A B E (3) B E A C D (3) B D C E A (3) A E D C B (3) A E C B D (3) E A D C B (2) E A D B C (2) E A B C D (2) C D B E A (2) B C D E A (2) B C A D E (2) A B E C D (2) A B C E D (2) E D A C B (1) D E C B A (1) D C A E B (1) C D B A E (1) C D A E B (1) C B D E A (1) C B D A E (1) C A D B E (1) B E D A C (1) B D E C A (1) B C E A D (1) B C D A E (1) B A C E D (1) B A C D E (1) A E D B C (1) A E C D B (1) A D C E B (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 22 26 4 B -14 0 22 22 -8 C -22 -22 0 6 -16 D -26 -22 -6 0 -22 E -4 8 16 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 22 26 4 B -14 0 22 22 -8 C -22 -22 0 6 -16 D -26 -22 -6 0 -22 E -4 8 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=23 B=21 C=10 D=7 so D is eliminated. Round 2 votes counts: A=39 E=24 B=21 C=16 so C is eliminated. Round 3 votes counts: A=45 B=31 E=24 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 E:221 B:211 C:173 D:162 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 22 26 4 B -14 0 22 22 -8 C -22 -22 0 6 -16 D -26 -22 -6 0 -22 E -4 8 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 22 26 4 B -14 0 22 22 -8 C -22 -22 0 6 -16 D -26 -22 -6 0 -22 E -4 8 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 22 26 4 B -14 0 22 22 -8 C -22 -22 0 6 -16 D -26 -22 -6 0 -22 E -4 8 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6499: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) C A B D E (7) B C D A E (7) C B A D E (6) A D E B C (6) C B D A E (5) C A B E D (4) A C E D B (4) E A D C B (3) E A D B C (3) E A C D B (3) C A E D B (3) B C D E A (3) A E C D B (3) E D B A C (2) E B D C A (2) D E A B C (2) D B A E C (2) D A E B C (2) C A E B D (2) B D A C E (2) A E D B C (2) A D E C B (2) A C D E B (2) E D B C A (1) E D A C B (1) E C A D B (1) E B C D A (1) D B E A C (1) D B A C E (1) D A B E C (1) C E A B D (1) C B D E A (1) B D E C A (1) B D C E A (1) B D C A E (1) A E D C B (1) A D C B E (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 26 6 12 32 B -26 0 -2 -10 -8 C -6 2 0 8 4 D -12 10 -8 0 12 E -32 8 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 6 12 32 B -26 0 -2 -10 -8 C -6 2 0 8 4 D -12 10 -8 0 12 E -32 8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=24 A=23 B=15 D=9 so D is eliminated. Round 2 votes counts: C=29 E=26 A=26 B=19 so B is eliminated. Round 3 votes counts: C=41 A=31 E=28 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:238 C:204 D:201 E:180 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 6 12 32 B -26 0 -2 -10 -8 C -6 2 0 8 4 D -12 10 -8 0 12 E -32 8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 6 12 32 B -26 0 -2 -10 -8 C -6 2 0 8 4 D -12 10 -8 0 12 E -32 8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 6 12 32 B -26 0 -2 -10 -8 C -6 2 0 8 4 D -12 10 -8 0 12 E -32 8 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6500: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) E A B C D (7) D C A B E (6) D A C E B (6) B E C A D (6) A E D C B (6) D C B A E (5) B C D E A (5) C D B A E (4) C B D E A (4) B E A C D (4) E B A C D (3) E A B D C (3) B E D C A (3) A D E C B (3) E B A D C (2) E A D B C (2) D A E C B (2) D A E B C (2) B E A D C (2) D E A B C (1) D C B E A (1) D C A E B (1) D A C B E (1) D A B E C (1) C D B E A (1) C D A B E (1) C B A E D (1) C A D E B (1) B D C E A (1) B C E D A (1) B C E A D (1) A E C B D (1) A E B C D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 12 16 8 6 B -12 0 10 -10 -4 C -16 -10 0 -16 -18 D -8 10 16 0 -6 E -6 4 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 8 6 B -12 0 10 -10 -4 C -16 -10 0 -16 -18 D -8 10 16 0 -6 E -6 4 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=23 A=22 E=17 C=12 so C is eliminated. Round 2 votes counts: D=32 B=28 A=23 E=17 so E is eliminated. Round 3 votes counts: A=35 B=33 D=32 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:211 D:206 B:192 C:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 8 6 B -12 0 10 -10 -4 C -16 -10 0 -16 -18 D -8 10 16 0 -6 E -6 4 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 8 6 B -12 0 10 -10 -4 C -16 -10 0 -16 -18 D -8 10 16 0 -6 E -6 4 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 8 6 B -12 0 10 -10 -4 C -16 -10 0 -16 -18 D -8 10 16 0 -6 E -6 4 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6501: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (13) E B A C D (10) A B E D C (9) C D E B A (6) E B C A D (5) D C E B A (5) D C E A B (4) D C A E B (4) A B E C D (4) D A C B E (3) C E D B A (3) B E A C D (3) A E B D C (3) A B C E D (3) E B A D C (2) E A B D C (2) D A E C B (2) C E B D A (2) C D A B E (2) B A E C D (2) E C B D A (1) E B C D A (1) D E C B A (1) D A E B C (1) D A C E B (1) C D B E A (1) C B E D A (1) C B E A D (1) C A D B E (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -2 -2 4 B -12 0 -6 -2 -6 C 2 6 0 -6 6 D 2 2 6 0 -4 E -4 6 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.359999999142 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 A B C D E A 0 12 -2 -2 4 B -12 0 -6 -2 -6 C 2 6 0 -6 6 D 2 2 6 0 -4 E -4 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=23 E=21 C=17 B=5 so B is eliminated. Round 2 votes counts: D=34 A=25 E=24 C=17 so C is eliminated. Round 3 votes counts: D=43 E=31 A=26 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:206 C:204 D:203 E:200 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 -2 -2 4 B -12 0 -6 -2 -6 C 2 6 0 -6 6 D 2 2 6 0 -4 E -4 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -2 4 B -12 0 -6 -2 -6 C 2 6 0 -6 6 D 2 2 6 0 -4 E -4 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -2 4 B -12 0 -6 -2 -6 C 2 6 0 -6 6 D 2 2 6 0 -4 E -4 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.200000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6502: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) B E A D C (8) E A B C D (7) C D A B E (6) D C B E A (5) E B A C D (4) D B E C A (4) C D E B A (4) D C B A E (3) D C A B E (3) D A C B E (3) C A E D B (3) B E D C A (3) A E C B D (3) A E B C D (3) D B A E C (2) C D E A B (2) C A D E B (2) A D C B E (2) A D B E C (2) A C D E B (2) A B E D C (2) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A D C (1) E A B D C (1) D A B C E (1) C E D B A (1) C E D A B (1) C A E B D (1) B E D A C (1) B D E C A (1) B A E D C (1) B A D E C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 16 -8 -4 6 B -16 0 -10 -16 -2 C 8 10 0 10 4 D 4 16 -10 0 10 E -6 2 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -8 -4 6 B -16 0 -10 -16 -2 C 8 10 0 10 4 D 4 16 -10 0 10 E -6 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=21 E=17 A=16 B=15 so B is eliminated. Round 2 votes counts: C=31 E=29 D=22 A=18 so A is eliminated. Round 3 votes counts: E=38 C=34 D=28 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:210 A:205 E:191 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -8 -4 6 B -16 0 -10 -16 -2 C 8 10 0 10 4 D 4 16 -10 0 10 E -6 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 -4 6 B -16 0 -10 -16 -2 C 8 10 0 10 4 D 4 16 -10 0 10 E -6 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 -4 6 B -16 0 -10 -16 -2 C 8 10 0 10 4 D 4 16 -10 0 10 E -6 2 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6503: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) A D B E C (11) C E B D A (10) A D E B C (6) E C D A B (5) B A D E C (5) D A E C B (4) D A E B C (4) B C E A D (4) C B E D A (3) B E D A C (3) B C E D A (3) B C A D E (3) E D C A B (2) E C D B A (2) C D E A B (2) B E A D C (2) A D C E B (2) A B D E C (2) E D A B C (1) E B D A C (1) D E A B C (1) D A C E B (1) C B A E D (1) C B A D E (1) C A D E B (1) C A D B E (1) B E C D A (1) B D A E C (1) B C A E D (1) B A E D C (1) B A D C E (1) B A C D E (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -2 -10 -2 B -12 0 6 -10 -8 C 2 -6 0 2 -6 D 10 10 -2 0 -2 E 2 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -2 -10 -2 B -12 0 6 -10 -8 C 2 -6 0 2 -6 D 10 10 -2 0 -2 E 2 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=26 A=23 E=11 D=10 so D is eliminated. Round 2 votes counts: A=32 C=30 B=26 E=12 so E is eliminated. Round 3 votes counts: C=39 A=34 B=27 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:209 D:208 A:199 C:196 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -2 -10 -2 B -12 0 6 -10 -8 C 2 -6 0 2 -6 D 10 10 -2 0 -2 E 2 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -10 -2 B -12 0 6 -10 -8 C 2 -6 0 2 -6 D 10 10 -2 0 -2 E 2 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -10 -2 B -12 0 6 -10 -8 C 2 -6 0 2 -6 D 10 10 -2 0 -2 E 2 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6504: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) C A D E B (7) C E D A B (6) B E D A C (6) B E C D A (6) B A D E C (5) A D B C E (5) D A C E B (4) C E A D B (4) A C D E B (4) E B C D A (3) B E C A D (3) B A D C E (3) E C D B A (2) E C D A B (2) E B D C A (2) D E C A B (2) D A E C B (2) D A B E C (2) C E B A D (2) C D A E B (2) C A E D B (2) A D C B E (2) E D C B A (1) E D B A C (1) E D A B C (1) D C E A B (1) D A B C E (1) C E B D A (1) C A E B D (1) C A B E D (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E A D (1) B C A D E (1) B A E D C (1) B A C D E (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -14 -8 -6 B -2 0 -8 -4 -16 C 14 8 0 14 2 D 8 4 -14 0 -8 E 6 16 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 -8 -6 B -2 0 -8 -4 -16 C 14 8 0 14 2 D 8 4 -14 0 -8 E 6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=26 E=19 A=13 D=12 so D is eliminated. Round 2 votes counts: B=30 C=27 A=22 E=21 so E is eliminated. Round 3 votes counts: C=41 B=36 A=23 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:214 D:195 A:187 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 -8 -6 B -2 0 -8 -4 -16 C 14 8 0 14 2 D 8 4 -14 0 -8 E 6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -8 -6 B -2 0 -8 -4 -16 C 14 8 0 14 2 D 8 4 -14 0 -8 E 6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -8 -6 B -2 0 -8 -4 -16 C 14 8 0 14 2 D 8 4 -14 0 -8 E 6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6505: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) D A E B C (8) A D E B C (6) E B C D A (5) D A B E C (5) C B E D A (5) C B E A D (5) B E C D A (5) E B C A D (4) D A C B E (3) B C E D A (3) A D C E B (3) E C B A D (2) E B D A C (2) E B A D C (2) D E B A C (2) D C A B E (2) C E B A D (2) B D C E A (2) A E C B D (2) A D E C B (2) A C E D B (2) E B D C A (1) E A C B D (1) E A B D C (1) E A B C D (1) D E A B C (1) D B E C A (1) D B C E A (1) D B C A E (1) D B A C E (1) D A B C E (1) C B D E A (1) C A E B D (1) C A B E D (1) C A B D E (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 14 0 6 B -10 0 6 -6 2 C -14 -6 0 -14 -4 D 0 6 14 0 8 E -6 -2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.433933 B: 0.000000 C: 0.000000 D: 0.566067 E: 0.000000 Sum of squares = 0.508729784325 Cumulative probabilities = A: 0.433933 B: 0.433933 C: 0.433933 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 0 6 B -10 0 6 -6 2 C -14 -6 0 -14 -4 D 0 6 14 0 8 E -6 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=26 E=19 C=16 B=10 so B is eliminated. Round 2 votes counts: A=29 D=28 E=24 C=19 so C is eliminated. Round 3 votes counts: E=39 A=32 D=29 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:214 B:196 E:194 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 0 6 B -10 0 6 -6 2 C -14 -6 0 -14 -4 D 0 6 14 0 8 E -6 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 0 6 B -10 0 6 -6 2 C -14 -6 0 -14 -4 D 0 6 14 0 8 E -6 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 0 6 B -10 0 6 -6 2 C -14 -6 0 -14 -4 D 0 6 14 0 8 E -6 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6506: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (16) C D A B E (16) E A B D C (6) D A C B E (5) C D A E B (5) A D B E C (5) E B A C D (4) C D B A E (4) C B D A E (4) B A D E C (4) B E A D C (3) E C B D A (2) C E D A B (2) C B E D A (2) B D A C E (2) B A E D C (2) A E D B C (2) A D B C E (2) A B E D C (2) E D A C B (1) E C A D B (1) D C A E B (1) D C A B E (1) D A C E B (1) C E D B A (1) C E B D A (1) C D E A B (1) B E C A D (1) B E A C D (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 16 2 16 B -4 0 4 4 12 C -16 -4 0 -10 -2 D -2 -4 10 0 6 E -16 -12 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 2 16 B -4 0 4 4 12 C -16 -4 0 -10 -2 D -2 -4 10 0 6 E -16 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984584 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=30 B=14 A=12 D=8 so D is eliminated. Round 2 votes counts: C=38 E=30 A=18 B=14 so B is eliminated. Round 3 votes counts: C=38 E=35 A=27 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:219 B:208 D:205 C:184 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 2 16 B -4 0 4 4 12 C -16 -4 0 -10 -2 D -2 -4 10 0 6 E -16 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984584 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 2 16 B -4 0 4 4 12 C -16 -4 0 -10 -2 D -2 -4 10 0 6 E -16 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984584 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 2 16 B -4 0 4 4 12 C -16 -4 0 -10 -2 D -2 -4 10 0 6 E -16 -12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984584 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6507: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) D E B A C (5) D B A E C (5) E A D B C (4) C B A E D (4) C B A D E (4) B D A E C (4) B D A C E (4) E D A B C (3) E C A D B (3) C E D B A (3) C E D A B (3) A E B D C (3) A B D E C (3) E C D A B (2) E C A B D (2) D E C A B (2) D C E B A (2) D B C A E (2) C E A D B (2) C B D E A (2) C B D A E (2) B C A D E (2) B A D C E (2) B A C E D (2) A B E D C (2) E D C A B (1) E D A C B (1) E A C D B (1) D E A B C (1) D C B E A (1) D C B A E (1) C E B D A (1) C B E D A (1) C B E A D (1) C A E B D (1) C A B E D (1) B D C A E (1) B C D A E (1) B C A E D (1) B A D E C (1) B A C D E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -12 4 -2 B 6 0 -4 16 -2 C 12 4 0 4 14 D -4 -16 -4 0 -8 E 2 2 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 4 -2 B 6 0 -4 16 -2 C 12 4 0 4 14 D -4 -16 -4 0 -8 E 2 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=19 B=19 E=17 A=10 so A is eliminated. Round 2 votes counts: C=36 B=25 E=20 D=19 so D is eliminated. Round 3 votes counts: C=40 B=32 E=28 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:208 E:199 A:192 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 4 -2 B 6 0 -4 16 -2 C 12 4 0 4 14 D -4 -16 -4 0 -8 E 2 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 4 -2 B 6 0 -4 16 -2 C 12 4 0 4 14 D -4 -16 -4 0 -8 E 2 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 4 -2 B 6 0 -4 16 -2 C 12 4 0 4 14 D -4 -16 -4 0 -8 E 2 2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6508: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) B E A C D (6) E B C D A (5) C E D B A (5) A B D E C (5) E B C A D (4) D C E B A (4) D B A E C (3) D A C B E (3) D A B E C (3) C D E A B (3) A C D E B (3) A B E C D (3) D C A E B (2) D B E C A (2) D A C E B (2) D A B C E (2) C E B D A (2) C D E B A (2) C D A E B (2) C A E D B (2) C A E B D (2) C A D E B (2) B E D A C (2) B D A E C (2) B A E C D (2) A E C B D (2) A D C B E (2) A D B E C (2) A D B C E (2) E C D B A (1) E C B D A (1) E C B A D (1) D E B C A (1) D B E A C (1) C E A D B (1) C E A B D (1) B E C A D (1) B E A D C (1) B D E C A (1) B D E A C (1) B A D E C (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 -2 -12 -4 B 10 0 12 -2 2 C 2 -12 0 2 -14 D 12 2 -2 0 4 E 4 -2 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000001 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -12 -4 B 10 0 12 -2 2 C 2 -12 0 2 -14 D 12 2 -2 0 4 E 4 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000064 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 C=22 A=20 E=12 so E is eliminated. Round 2 votes counts: B=32 C=25 D=23 A=20 so A is eliminated. Round 3 votes counts: B=40 D=30 C=30 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:208 E:206 C:189 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -12 -4 B 10 0 12 -2 2 C 2 -12 0 2 -14 D 12 2 -2 0 4 E 4 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000064 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -12 -4 B 10 0 12 -2 2 C 2 -12 0 2 -14 D 12 2 -2 0 4 E 4 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000064 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -12 -4 B 10 0 12 -2 2 C 2 -12 0 2 -14 D 12 2 -2 0 4 E 4 -2 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.125000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000064 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6509: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) D A E C B (6) D A E B C (5) C B A E D (5) B C E A D (5) C B E A D (4) B D A C E (4) B C A D E (4) A D E C B (4) A D C B E (4) E C B A D (3) E A C D B (3) D B A E C (3) C E B A D (3) A D C E B (3) E C A D B (2) E B C D A (2) D A B C E (2) C B A D E (2) C A E D B (2) C A B D E (2) B C D A E (2) E D C A B (1) E D B A C (1) E D A B C (1) E C A B D (1) E B D C A (1) E B C A D (1) E A D C B (1) D E A C B (1) D B A C E (1) D A C B E (1) D A B E C (1) B E C D A (1) B D E C A (1) B D C A E (1) B D A E C (1) B C D E A (1) B A C D E (1) A E D C B (1) A E C D B (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 12 10 18 B -6 0 -22 -8 -6 C -12 22 0 -4 0 D -10 8 4 0 4 E -18 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 10 18 B -6 0 -22 -8 -6 C -12 22 0 -4 0 D -10 8 4 0 4 E -18 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=21 D=20 C=18 A=16 so A is eliminated. Round 2 votes counts: D=31 E=27 C=21 B=21 so C is eliminated. Round 3 votes counts: B=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:223 C:203 D:203 E:192 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 10 18 B -6 0 -22 -8 -6 C -12 22 0 -4 0 D -10 8 4 0 4 E -18 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 10 18 B -6 0 -22 -8 -6 C -12 22 0 -4 0 D -10 8 4 0 4 E -18 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 10 18 B -6 0 -22 -8 -6 C -12 22 0 -4 0 D -10 8 4 0 4 E -18 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6510: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) E A C B D (6) D C A B E (6) D B E A C (6) E A B C D (4) D B E C A (4) D B C A E (4) B D E C A (4) B D E A C (4) A C D E B (4) E B D A C (3) E A D B C (3) D C B A E (3) D B A E C (3) C A E D B (3) C A D B E (3) B E D A C (3) B D C E A (3) A C E D B (3) A C E B D (3) E B A C D (2) C A D E B (2) B E D C A (2) A E C B D (2) E D B A C (1) E B A D C (1) E A B D C (1) D B C E A (1) D B A C E (1) D A C E B (1) D A B E C (1) C E A B D (1) C D A B E (1) B E C D A (1) B C D E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 6 8 -6 -2 B -6 0 6 -4 2 C -8 -6 0 -14 -6 D 6 4 14 0 6 E 2 -2 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -6 -2 B -6 0 6 -4 2 C -8 -6 0 -14 -6 D 6 4 14 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=21 B=18 C=17 A=14 so A is eliminated. Round 2 votes counts: D=31 C=27 E=24 B=18 so B is eliminated. Round 3 votes counts: D=42 E=30 C=28 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:203 E:200 B:199 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -6 -2 B -6 0 6 -4 2 C -8 -6 0 -14 -6 D 6 4 14 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -6 -2 B -6 0 6 -4 2 C -8 -6 0 -14 -6 D 6 4 14 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -6 -2 B -6 0 6 -4 2 C -8 -6 0 -14 -6 D 6 4 14 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6511: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (16) A B D E C (16) B D A C E (11) D B C E A (10) C E D B A (8) D B C A E (5) A E C B D (5) E A C D B (4) B D C E A (4) C D B E A (3) C E B D A (2) B D C A E (2) A E C D B (2) A E B D C (2) E C D B A (1) E C D A B (1) E A C B D (1) D B A C E (1) C E D A B (1) C D E B A (1) B D A E C (1) A E B C D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -8 -2 -4 B -2 0 10 -6 10 C 8 -10 0 -8 -2 D 2 6 8 0 10 E 4 -10 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -2 -4 B -2 0 10 -6 10 C 8 -10 0 -8 -2 D 2 6 8 0 10 E 4 -10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=23 B=18 D=16 C=15 so C is eliminated. Round 2 votes counts: E=34 A=28 D=20 B=18 so B is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:206 A:194 C:194 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -2 -4 B -2 0 10 -6 10 C 8 -10 0 -8 -2 D 2 6 8 0 10 E 4 -10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 -4 B -2 0 10 -6 10 C 8 -10 0 -8 -2 D 2 6 8 0 10 E 4 -10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 -4 B -2 0 10 -6 10 C 8 -10 0 -8 -2 D 2 6 8 0 10 E 4 -10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6512: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) C B D A E (10) D B C E A (7) C B D E A (6) A E D B C (6) A E C B D (6) E A D C B (5) D E A B C (5) A E C D B (5) D B E C A (4) B D C E A (4) C E B A D (3) B C D E A (3) B C D A E (3) A E D C B (3) D E B A C (2) D A E B C (2) C B A E D (2) C B A D E (2) E D B A C (1) E D A B C (1) E A C D B (1) E A C B D (1) D B E A C (1) D B A E C (1) C B E A D (1) C A E B D (1) C A B E D (1) B D C A E (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 4 -2 -12 B 2 0 4 -10 -8 C -4 -4 0 -8 -10 D 2 10 8 0 2 E 12 8 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -2 -12 B 2 0 4 -10 -8 C -4 -4 0 -8 -10 D 2 10 8 0 2 E 12 8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=22 A=21 E=20 B=11 so B is eliminated. Round 2 votes counts: C=32 D=27 A=21 E=20 so E is eliminated. Round 3 votes counts: A=39 C=32 D=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:214 D:211 A:194 B:194 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -2 -12 B 2 0 4 -10 -8 C -4 -4 0 -8 -10 D 2 10 8 0 2 E 12 8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -2 -12 B 2 0 4 -10 -8 C -4 -4 0 -8 -10 D 2 10 8 0 2 E 12 8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -2 -12 B 2 0 4 -10 -8 C -4 -4 0 -8 -10 D 2 10 8 0 2 E 12 8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6513: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) C D A B E (9) D C E B A (5) D C E A B (4) D C A E B (4) B E A C D (4) A C B D E (4) A B E C D (4) A B C E D (4) E B A D C (3) D E C A B (3) D C B E A (3) B A C E D (3) A C D B E (3) E D B C A (2) E D B A C (2) E D A B C (2) E A B C D (2) D E C B A (2) D E A C B (2) D C B A E (2) D C A B E (2) C B A D E (2) B C A E D (2) B C A D E (2) A E B D C (2) E D A C B (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) D A E C B (1) D A C E B (1) D A C B E (1) C D B E A (1) C A B D E (1) B C D E A (1) B A E C D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 8 -8 -16 18 B -8 0 -24 -24 18 C 8 24 0 6 30 D 16 24 -6 0 28 E -18 -18 -30 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -16 18 B -8 0 -24 -24 18 C 8 24 0 6 30 D 16 24 -6 0 28 E -18 -18 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=22 A=19 E=16 B=13 so B is eliminated. Round 2 votes counts: D=30 C=27 A=23 E=20 so E is eliminated. Round 3 votes counts: D=38 A=35 C=27 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:234 D:231 A:201 B:181 E:153 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -16 18 B -8 0 -24 -24 18 C 8 24 0 6 30 D 16 24 -6 0 28 E -18 -18 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -16 18 B -8 0 -24 -24 18 C 8 24 0 6 30 D 16 24 -6 0 28 E -18 -18 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -16 18 B -8 0 -24 -24 18 C 8 24 0 6 30 D 16 24 -6 0 28 E -18 -18 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6514: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) D A E C B (7) B C E A D (7) C B E A D (6) D B A E C (5) C E B A D (5) B C D E A (5) A D E B C (4) E A D C B (3) E A C D B (3) C B E D A (3) B C D A E (3) B C A E D (3) A D E C B (3) E C A D B (2) D E A C B (2) D A B E C (2) C E A D B (2) C E A B D (2) B D C A E (2) B A C D E (2) A E D C B (2) E D A C B (1) D C E B A (1) D C E A B (1) D C A B E (1) D B C A E (1) D A C E B (1) C D E A B (1) C D B E A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C A D E (1) B A E D C (1) B A E C D (1) A E D B C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 0 4 0 8 B 0 0 6 -10 -4 C -4 -6 0 -4 0 D 0 10 4 0 12 E -8 4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.409253 B: 0.000000 C: 0.000000 D: 0.590747 E: 0.000000 Sum of squares = 0.516469908066 Cumulative probabilities = A: 0.409253 B: 0.409253 C: 0.409253 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 0 8 B 0 0 6 -10 -4 C -4 -6 0 -4 0 D 0 10 4 0 12 E -8 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=28 C=20 A=12 E=9 so E is eliminated. Round 2 votes counts: D=32 B=28 C=22 A=18 so A is eliminated. Round 3 votes counts: D=46 B=29 C=25 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:206 B:196 C:193 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 8 B 0 0 6 -10 -4 C -4 -6 0 -4 0 D 0 10 4 0 12 E -8 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 8 B 0 0 6 -10 -4 C -4 -6 0 -4 0 D 0 10 4 0 12 E -8 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 8 B 0 0 6 -10 -4 C -4 -6 0 -4 0 D 0 10 4 0 12 E -8 4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6515: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) A B C D E (13) E D C B A (11) D E C B A (11) E D C A B (7) E D A C B (7) B C A D E (5) C D B E A (4) A B E C D (4) D E A C B (3) E C D B A (2) E A D B C (2) D C E B A (2) C B D E A (2) C B A D E (2) E D A B C (1) E C B A D (1) E A B C D (1) D C B E A (1) C E D B A (1) C E B D A (1) C B D A E (1) B A C E D (1) B A C D E (1) A E B D C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -2 -8 -14 B -8 0 -12 -6 -4 C 2 12 0 4 -6 D 8 6 -4 0 -8 E 14 4 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -2 -8 -14 B -8 0 -12 -6 -4 C 2 12 0 4 -6 D 8 6 -4 0 -8 E 14 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=32 D=17 C=11 B=7 so B is eliminated. Round 2 votes counts: A=35 E=32 D=17 C=16 so C is eliminated. Round 3 votes counts: A=42 E=34 D=24 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:206 D:201 A:192 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -2 -8 -14 B -8 0 -12 -6 -4 C 2 12 0 4 -6 D 8 6 -4 0 -8 E 14 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -8 -14 B -8 0 -12 -6 -4 C 2 12 0 4 -6 D 8 6 -4 0 -8 E 14 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -8 -14 B -8 0 -12 -6 -4 C 2 12 0 4 -6 D 8 6 -4 0 -8 E 14 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6516: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) A B D C E (6) D B A C E (5) C E D A B (5) C E A B D (5) E C D B A (4) D C A B E (4) D B A E C (4) D A C B E (4) B A D E C (4) D C E B A (3) D C E A B (3) D B E C A (3) D A B C E (3) C E A D B (3) C D E A B (3) E C D A B (2) D E B C A (2) C A E B D (2) B E A D C (2) B D E A C (2) B D A E C (2) A C B D E (2) A B C E D (2) E D C B A (1) E B C D A (1) E B C A D (1) E B A D C (1) E A C B D (1) D C A E B (1) D B E A C (1) D B C E A (1) C E D B A (1) B A E D C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 22 -10 -10 -10 B -22 0 -12 -12 2 C 10 12 0 -10 14 D 10 12 10 0 14 E 10 -2 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -10 -10 -10 B -22 0 -12 -12 2 C 10 12 0 -10 14 D 10 12 10 0 14 E 10 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=21 C=19 A=15 B=11 so B is eliminated. Round 2 votes counts: D=38 E=23 A=20 C=19 so C is eliminated. Round 3 votes counts: D=41 E=37 A=22 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:213 A:196 E:190 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 -10 -10 -10 B -22 0 -12 -12 2 C 10 12 0 -10 14 D 10 12 10 0 14 E 10 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -10 -10 -10 B -22 0 -12 -12 2 C 10 12 0 -10 14 D 10 12 10 0 14 E 10 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -10 -10 -10 B -22 0 -12 -12 2 C 10 12 0 -10 14 D 10 12 10 0 14 E 10 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6517: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) E C B A D (8) E C A B D (7) A D E C B (7) D B C A E (6) D B A C E (5) D A B C E (5) C B E A D (5) B D C E A (4) B D C A E (4) B C D E A (4) E A D C B (3) D B C E A (3) C E B A D (3) B C E D A (3) E A C D B (2) D A E B C (2) A E C D B (2) A E C B D (2) A C E B D (2) A C B E D (2) E C B D A (1) E B C D A (1) E A C B D (1) D B E C A (1) D A B E C (1) C E A B D (1) C B A D E (1) B D E C A (1) B C D A E (1) B C A D E (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -10 16 4 B 4 0 -16 4 -4 C 10 16 0 -6 4 D -16 -4 6 0 -4 E -4 4 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.500000 D: 0.312500 E: 0.000000 Sum of squares = 0.382812499996 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.687500 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 16 4 B 4 0 -16 4 -4 C 10 16 0 -6 4 D -16 -4 6 0 -4 E -4 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.500000 D: 0.312500 E: 0.000000 Sum of squares = 0.382812499951 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.687500 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 D=23 B=18 C=10 so C is eliminated. Round 2 votes counts: E=27 A=26 B=24 D=23 so D is eliminated. Round 3 votes counts: B=39 A=34 E=27 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:212 A:203 E:200 B:194 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 16 4 B 4 0 -16 4 -4 C 10 16 0 -6 4 D -16 -4 6 0 -4 E -4 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.500000 D: 0.312500 E: 0.000000 Sum of squares = 0.382812499951 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.687500 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 16 4 B 4 0 -16 4 -4 C 10 16 0 -6 4 D -16 -4 6 0 -4 E -4 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.500000 D: 0.312500 E: 0.000000 Sum of squares = 0.382812499951 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.687500 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 16 4 B 4 0 -16 4 -4 C 10 16 0 -6 4 D -16 -4 6 0 -4 E -4 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.187500 B: 0.000000 C: 0.500000 D: 0.312500 E: 0.000000 Sum of squares = 0.382812499951 Cumulative probabilities = A: 0.187500 B: 0.187500 C: 0.687500 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6518: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) D A E B C (7) D E A B C (6) E B C A D (5) D A C B E (5) C B E A D (5) B C E A D (4) E C B D A (3) E B C D A (3) D E A C B (3) D C A E B (3) C B E D A (3) B E C A D (3) A D C B E (3) A D B E C (3) A D B C E (3) E D B C A (2) D E C A B (2) D C E A B (2) D A C E B (2) C E B D A (2) C D A B E (2) C B A E D (2) B E A C D (2) A C D B E (2) A B D E C (2) A B C E D (2) E D B A C (1) E D A B C (1) E C D B A (1) E C B A D (1) E B D C A (1) E B D A C (1) C E B A D (1) C A B D E (1) B A E C D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 18 8 -14 -4 B -18 0 0 -10 -8 C -8 0 0 -12 -12 D 14 10 12 0 10 E 4 8 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 8 -14 -4 B -18 0 0 -10 -8 C -8 0 0 -12 -12 D 14 10 12 0 10 E 4 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=19 A=18 C=16 B=10 so B is eliminated. Round 2 votes counts: D=37 E=24 C=20 A=19 so A is eliminated. Round 3 votes counts: D=49 E=26 C=25 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:207 A:204 C:184 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 8 -14 -4 B -18 0 0 -10 -8 C -8 0 0 -12 -12 D 14 10 12 0 10 E 4 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 8 -14 -4 B -18 0 0 -10 -8 C -8 0 0 -12 -12 D 14 10 12 0 10 E 4 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 8 -14 -4 B -18 0 0 -10 -8 C -8 0 0 -12 -12 D 14 10 12 0 10 E 4 8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6519: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) B E D A C (8) E A B C D (6) D C B A E (6) B D E C A (6) B D E A C (6) E A B D C (4) C D A B E (4) A E C B D (4) E A C B D (3) D B C A E (3) C A E D B (3) C A D E B (3) E B A C D (2) E A C D B (2) D C A B E (2) D B C E A (2) C D B A E (2) C D A E B (2) C B D A E (2) B C A E D (2) A E C D B (2) E B D A C (1) E A D B C (1) D E C A B (1) D E B A C (1) D C A E B (1) D B E C A (1) D B E A C (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B E D (1) B E A C D (1) B D C E A (1) B D C A E (1) B C D E A (1) B C D A E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 4 -6 -12 B 14 0 16 22 8 C -4 -16 0 -8 -16 D 6 -22 8 0 -4 E 12 -8 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -6 -12 B 14 0 16 22 8 C -4 -16 0 -8 -16 D 6 -22 8 0 -4 E 12 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=27 B=27 C=20 D=18 A=8 so A is eliminated. Round 2 votes counts: E=33 B=28 C=21 D=18 so D is eliminated. Round 3 votes counts: E=35 B=35 C=30 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:230 E:212 D:194 A:186 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 -6 -12 B 14 0 16 22 8 C -4 -16 0 -8 -16 D 6 -22 8 0 -4 E 12 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -6 -12 B 14 0 16 22 8 C -4 -16 0 -8 -16 D 6 -22 8 0 -4 E 12 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -6 -12 B 14 0 16 22 8 C -4 -16 0 -8 -16 D 6 -22 8 0 -4 E 12 -8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6520: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (13) D A B C E (7) C B E D A (7) A D B E C (5) D B A C E (4) C E D B A (4) C B D A E (4) E C D A B (3) E A D B C (3) D C B A E (3) C B D E A (3) E D A C B (2) E A B C D (2) C E B D A (2) C D E B A (2) C D B E A (2) C D B A E (2) B D A C E (2) B C D A E (2) B C A D E (2) B A D C E (2) A E D B C (2) A B D E C (2) E C B D A (1) E C A D B (1) E C A B D (1) E B C A D (1) E B A C D (1) E A D C B (1) D E C A B (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C E B (1) C B E A D (1) B D C A E (1) B C A E D (1) B A D E C (1) B A C E D (1) B A C D E (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -28 -16 -12 -2 B 28 0 -10 6 14 C 16 10 0 14 14 D 12 -6 -14 0 4 E 2 -14 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -16 -12 -2 B 28 0 -10 6 14 C 16 10 0 14 14 D 12 -6 -14 0 4 E 2 -14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=27 D=19 B=13 A=12 so A is eliminated. Round 2 votes counts: E=31 C=27 D=25 B=17 so B is eliminated. Round 3 votes counts: D=34 C=34 E=32 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:219 D:198 E:185 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -28 -16 -12 -2 B 28 0 -10 6 14 C 16 10 0 14 14 D 12 -6 -14 0 4 E 2 -14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -16 -12 -2 B 28 0 -10 6 14 C 16 10 0 14 14 D 12 -6 -14 0 4 E 2 -14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -16 -12 -2 B 28 0 -10 6 14 C 16 10 0 14 14 D 12 -6 -14 0 4 E 2 -14 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6521: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (14) C D B E A (11) A E C D B (9) B D C E A (7) B C D A E (5) C E A D B (4) B D C A E (4) E A C D B (3) D B C E A (3) C D E B A (3) C B D A E (3) A B E D C (3) E D C B A (2) E C D A B (2) E A D C B (2) D C B E A (2) B D A C E (2) B A D C E (2) A E C B D (2) E D C A B (1) E D A C B (1) E C D B A (1) E B A D C (1) D E B C A (1) D B E C A (1) C E D B A (1) C D E A B (1) C D B A E (1) C A E D B (1) C A B D E (1) B D E C A (1) A E B C D (1) A C E D B (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -12 -6 4 B 2 0 -6 -2 -4 C 12 6 0 4 6 D 6 2 -4 0 -2 E -4 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -6 4 B 2 0 -6 -2 -4 C 12 6 0 4 6 D 6 2 -4 0 -2 E -4 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=26 B=21 E=13 D=7 so D is eliminated. Round 2 votes counts: A=33 C=28 B=25 E=14 so E is eliminated. Round 3 votes counts: A=39 C=34 B=27 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:201 E:198 B:195 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -6 4 B 2 0 -6 -2 -4 C 12 6 0 4 6 D 6 2 -4 0 -2 E -4 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -6 4 B 2 0 -6 -2 -4 C 12 6 0 4 6 D 6 2 -4 0 -2 E -4 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -6 4 B 2 0 -6 -2 -4 C 12 6 0 4 6 D 6 2 -4 0 -2 E -4 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6522: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) B C E A D (10) C B A E D (7) D A E C B (5) C B E A D (5) B C E D A (5) E B A D C (4) E A D B C (4) C A B E D (4) A D E C B (4) E A B C D (3) D E A B C (3) A E C B D (3) D B C E A (2) D A C E B (2) C D B A E (2) B E C D A (2) A E D C B (2) A E D B C (2) A C D E B (2) E B D A C (1) E B C A D (1) E B A C D (1) E A B D C (1) D E B A C (1) D C B A E (1) D C A B E (1) D B E A C (1) D B C A E (1) C D A B E (1) C B D A E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D E C A (1) B D C E A (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 6 12 4 B -2 0 14 4 -4 C -6 -14 0 -2 -6 D -12 -4 2 0 -16 E -4 4 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 12 4 B -2 0 14 4 -4 C -6 -14 0 -2 -6 D -12 -4 2 0 -16 E -4 4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=21 B=21 E=15 A=15 so E is eliminated. Round 2 votes counts: D=28 B=28 A=23 C=21 so C is eliminated. Round 3 votes counts: B=41 D=31 A=28 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:212 E:211 B:206 C:186 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 12 4 B -2 0 14 4 -4 C -6 -14 0 -2 -6 D -12 -4 2 0 -16 E -4 4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 12 4 B -2 0 14 4 -4 C -6 -14 0 -2 -6 D -12 -4 2 0 -16 E -4 4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 12 4 B -2 0 14 4 -4 C -6 -14 0 -2 -6 D -12 -4 2 0 -16 E -4 4 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6523: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) E B D A C (5) B C A D E (5) E A D C B (4) D E C A B (4) B A C E D (4) E D A B C (3) E B A D C (3) E B A C D (3) B E D C A (3) B D C A E (3) B C D A E (3) B A E C D (3) B A C D E (3) A C D E B (3) A C B D E (3) E D C A B (2) E D B A C (2) E D A C B (2) E B D C A (2) E A C B D (2) D C B A E (2) D C A B E (2) C D A B E (2) B D E C A (2) A E C D B (2) A E C B D (2) A C E D B (2) A C D B E (2) A B C D E (2) E A D B C (1) E A C D B (1) E A B C D (1) D E B C A (1) D C E B A (1) D C B E A (1) D C A E B (1) D B E C A (1) B E A C D (1) B D C E A (1) A D C E B (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 16 2 -4 B 10 0 18 8 -10 C -16 -18 0 -6 -12 D -2 -8 6 0 -14 E 4 10 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 16 2 -4 B 10 0 18 8 -10 C -16 -18 0 -6 -12 D -2 -8 6 0 -14 E 4 10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=28 A=20 D=13 C=2 so C is eliminated. Round 2 votes counts: E=37 B=28 A=20 D=15 so D is eliminated. Round 3 votes counts: E=43 B=32 A=25 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:213 A:202 D:191 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 16 2 -4 B 10 0 18 8 -10 C -16 -18 0 -6 -12 D -2 -8 6 0 -14 E 4 10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 2 -4 B 10 0 18 8 -10 C -16 -18 0 -6 -12 D -2 -8 6 0 -14 E 4 10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 2 -4 B 10 0 18 8 -10 C -16 -18 0 -6 -12 D -2 -8 6 0 -14 E 4 10 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6524: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) E D C A B (7) D E C A B (7) A E B C D (6) D C B E A (5) B A C D E (5) D C E B A (4) C D E A B (4) C B D A E (4) B C D A E (4) E A D B C (3) E A C D B (3) D B C E A (3) C E D A B (3) B A D C E (3) E D A C B (2) E A B D C (2) D C E A B (2) D B C A E (2) C D E B A (2) C D B E A (2) C A E B D (2) B D C A E (2) B D A E C (2) B D A C E (2) B A E C D (2) A B E D C (2) C E A D B (1) C D A B E (1) C A B E D (1) B A E D C (1) B A D E C (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 12 -12 -16 0 B -12 0 -2 -2 0 C 12 2 0 -2 4 D 16 2 2 0 10 E 0 0 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 -16 0 B -12 0 -2 -2 0 C 12 2 0 -2 4 D 16 2 2 0 10 E 0 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 B=22 C=20 A=18 E=17 so E is eliminated. Round 2 votes counts: D=32 A=26 B=22 C=20 so C is eliminated. Round 3 votes counts: D=44 A=30 B=26 so B is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:208 E:193 A:192 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -12 -16 0 B -12 0 -2 -2 0 C 12 2 0 -2 4 D 16 2 2 0 10 E 0 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 -16 0 B -12 0 -2 -2 0 C 12 2 0 -2 4 D 16 2 2 0 10 E 0 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 -16 0 B -12 0 -2 -2 0 C 12 2 0 -2 4 D 16 2 2 0 10 E 0 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6525: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) B E D A C (8) E B D C A (7) D E C A B (7) A C D B E (7) B A C E D (6) D C A E B (5) C A D B E (5) E D B C A (4) E B D A C (4) D A C E B (4) C D A E B (3) B E A D C (3) B E A C D (3) A C B E D (3) A C B D E (3) E B C A D (2) D E B A C (2) A B C E D (2) E D C B A (1) E D C A B (1) E C D A B (1) E B C D A (1) D E B C A (1) C E A B D (1) C D E A B (1) C B E A D (1) C A B E D (1) B E D C A (1) B C E A D (1) B A E D C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 -2 0 B -8 0 -6 -2 -8 C 4 6 0 2 6 D 2 2 -2 0 -4 E 0 8 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -2 0 B -8 0 -6 -2 -8 C 4 6 0 2 6 D 2 2 -2 0 -4 E 0 8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=21 C=20 D=19 A=17 so A is eliminated. Round 2 votes counts: C=34 B=26 E=21 D=19 so D is eliminated. Round 3 votes counts: C=43 E=31 B=26 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:209 E:203 A:201 D:199 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -2 0 B -8 0 -6 -2 -8 C 4 6 0 2 6 D 2 2 -2 0 -4 E 0 8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -2 0 B -8 0 -6 -2 -8 C 4 6 0 2 6 D 2 2 -2 0 -4 E 0 8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -2 0 B -8 0 -6 -2 -8 C 4 6 0 2 6 D 2 2 -2 0 -4 E 0 8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6526: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (12) E A C D B (7) B D C A E (7) D E B A C (6) D B E A C (6) D B C E A (6) C A E D B (6) E D A C B (5) C A B E D (4) B C D A E (4) B C A E D (4) B C A D E (4) D E A C B (3) D B E C A (3) E A C B D (2) D C A E B (2) C A B D E (2) B D E C A (2) B D E A C (2) A E C D B (2) A C E B D (2) E D B A C (1) E A B C D (1) D E A B C (1) D C A B E (1) C D B A E (1) C B A E D (1) C A D E B (1) B D C E A (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -22 -2 8 B -4 0 -4 -2 -4 C 22 4 0 8 16 D 2 2 -8 0 4 E -8 4 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -22 -2 8 B -4 0 -4 -2 -4 C 22 4 0 8 16 D 2 2 -8 0 4 E -8 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=27 B=24 E=16 A=5 so A is eliminated. Round 2 votes counts: C=29 D=28 B=24 E=19 so E is eliminated. Round 3 votes counts: C=41 D=34 B=25 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 D:200 A:194 B:193 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -22 -2 8 B -4 0 -4 -2 -4 C 22 4 0 8 16 D 2 2 -8 0 4 E -8 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -22 -2 8 B -4 0 -4 -2 -4 C 22 4 0 8 16 D 2 2 -8 0 4 E -8 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -22 -2 8 B -4 0 -4 -2 -4 C 22 4 0 8 16 D 2 2 -8 0 4 E -8 4 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6527: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (14) D C B E A (12) B A E C D (9) D B C A E (8) A E B C D (8) D C E A B (6) D E C A B (4) D E A C B (4) B C A E D (4) D C E B A (3) A B E C D (3) E D C A B (2) E A C D B (2) E A B C D (2) D C B A E (2) B A C E D (2) A E D B C (2) E D A C B (1) E A D B C (1) D B A E C (1) D B A C E (1) D A E B C (1) C E A B D (1) C D E B A (1) C D B A E (1) C B E A D (1) C B D E A (1) B C D A E (1) B C A D E (1) B A D E C (1) Total count = 100 A B C D E A 0 2 4 2 -10 B -2 0 -10 -4 -4 C -4 10 0 2 -10 D -2 4 -2 0 -4 E 10 4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 2 -10 B -2 0 -10 -4 -4 C -4 10 0 2 -10 D -2 4 -2 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 E=22 B=18 A=13 C=5 so C is eliminated. Round 2 votes counts: D=44 E=23 B=20 A=13 so A is eliminated. Round 3 votes counts: D=44 E=33 B=23 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:199 C:199 D:198 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 2 -10 B -2 0 -10 -4 -4 C -4 10 0 2 -10 D -2 4 -2 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 -10 B -2 0 -10 -4 -4 C -4 10 0 2 -10 D -2 4 -2 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 -10 B -2 0 -10 -4 -4 C -4 10 0 2 -10 D -2 4 -2 0 -4 E 10 4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6528: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) C B A E D (12) D C E B A (8) D E A C B (6) B A C E D (6) D C B E A (5) E A D B C (4) D E C B A (4) A B C E D (4) E D A B C (3) C D B E A (3) C B D A E (3) C B A D E (3) B C A E D (3) A B E C D (3) E A B D C (2) D E B A C (2) C D B A E (2) A E B D C (2) A E B C D (2) E B D A C (1) E A B C D (1) D E B C A (1) D A E C B (1) C D A B E (1) C B D E A (1) C A B E D (1) B C E D A (1) B A E C D (1) A E D B C (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 4 -8 -8 B 12 0 -2 -6 0 C -4 2 0 -4 8 D 8 6 4 0 4 E 8 0 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -8 -8 B 12 0 -2 -6 0 C -4 2 0 -4 8 D 8 6 4 0 4 E 8 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=26 A=13 E=11 B=11 so E is eliminated. Round 2 votes counts: D=42 C=26 A=20 B=12 so B is eliminated. Round 3 votes counts: D=43 C=30 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:202 C:201 E:198 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -8 -8 B 12 0 -2 -6 0 C -4 2 0 -4 8 D 8 6 4 0 4 E 8 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -8 -8 B 12 0 -2 -6 0 C -4 2 0 -4 8 D 8 6 4 0 4 E 8 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -8 -8 B 12 0 -2 -6 0 C -4 2 0 -4 8 D 8 6 4 0 4 E 8 0 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6529: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) D C E A B (7) C D E B A (7) A C D B E (6) E D B C A (5) A B E D C (5) D C E B A (4) B E C D A (4) A B E C D (4) C D B E A (3) C D A E B (3) A E D B C (3) A B C E D (3) A B C D E (3) E B D C A (2) D E C B A (2) C B D E A (2) B E C A D (2) B C D E A (2) B A C E D (2) A E B D C (2) A D E C B (2) A D C E B (2) A C B D E (2) E D B A C (1) E D A C B (1) E A D C B (1) E A D B C (1) E A B D C (1) D C A E B (1) C E D B A (1) C D A B E (1) B E D C A (1) B C E D A (1) B A E D C (1) B A E C D (1) B A C D E (1) A E D C B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -10 -10 -10 B -2 0 -8 -22 -12 C 10 8 0 -2 4 D 10 22 2 0 0 E 10 12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.931727 E: 0.068273 Sum of squares = 0.8727759259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.931727 E: 1.000000 A B C D E A 0 2 -10 -10 -10 B -2 0 -8 -22 -12 C 10 8 0 -2 4 D 10 22 2 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555699566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=19 C=17 B=15 D=14 so D is eliminated. Round 2 votes counts: A=35 C=29 E=21 B=15 so B is eliminated. Round 3 votes counts: A=40 C=32 E=28 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:210 E:209 A:186 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -10 -10 -10 B -2 0 -8 -22 -12 C 10 8 0 -2 4 D 10 22 2 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555699566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -10 -10 B -2 0 -8 -22 -12 C 10 8 0 -2 4 D 10 22 2 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555699566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -10 -10 B -2 0 -8 -22 -12 C 10 8 0 -2 4 D 10 22 2 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555699566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6530: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D C A (7) E B A D C (6) A C D E B (5) E D C B A (4) E C A D B (4) B E A D C (4) B D E C A (4) E A C D B (3) D C E B A (3) C D E A B (3) C A D E B (3) B E D A C (3) B A D C E (3) E C D B A (2) E B D A C (2) E A B C D (2) D C B A E (2) D C A B E (2) D B E C A (2) D B C E A (2) D B C A E (2) C D A B E (2) C A D B E (2) B E D C A (2) A E C B D (2) A E B C D (2) A C E D B (2) E D B C A (1) E B A C D (1) E A C B D (1) C D A E B (1) B D E A C (1) B D C A E (1) A E C D B (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 2 8 -14 B 4 0 -10 -18 -10 C -2 10 0 -4 -8 D -8 18 4 0 2 E 14 10 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.000000 D: 0.583333 E: 0.333333 Sum of squares = 0.458333333305 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.083333 D: 0.666667 E: 1.000000 A B C D E A 0 -4 2 8 -14 B 4 0 -10 -18 -10 C -2 10 0 -4 -8 D -8 18 4 0 2 E 14 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.000000 D: 0.583333 E: 0.333333 Sum of squares = 0.458333332412 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.083333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=25 B=18 D=13 C=11 so C is eliminated. Round 2 votes counts: E=33 A=30 D=19 B=18 so B is eliminated. Round 3 votes counts: E=42 A=33 D=25 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:208 C:198 A:196 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 8 -14 B 4 0 -10 -18 -10 C -2 10 0 -4 -8 D -8 18 4 0 2 E 14 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.000000 D: 0.583333 E: 0.333333 Sum of squares = 0.458333332412 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.083333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 -14 B 4 0 -10 -18 -10 C -2 10 0 -4 -8 D -8 18 4 0 2 E 14 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.000000 D: 0.583333 E: 0.333333 Sum of squares = 0.458333332412 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.083333 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 -14 B 4 0 -10 -18 -10 C -2 10 0 -4 -8 D -8 18 4 0 2 E 14 10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.000000 C: 0.000000 D: 0.583333 E: 0.333333 Sum of squares = 0.458333332412 Cumulative probabilities = A: 0.083333 B: 0.083333 C: 0.083333 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6531: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) D A E C B (6) B C E A D (6) B C A D E (5) A D B C E (5) E D C A B (4) E C D B A (4) D A C B E (4) C E D B A (4) B C A E D (4) E C B D A (3) D C E A B (3) B E C A D (3) B A C E D (3) B A C D E (3) E D C B A (2) E B C A D (2) D E A C B (2) D C E B A (2) C E B D A (2) B A E C D (2) A D B E C (2) A B D C E (2) A B C D E (2) E D A B C (1) E B C D A (1) E B A C D (1) E A D B C (1) D C A E B (1) D C A B E (1) D A C E B (1) C D E B A (1) C D B E A (1) C B E D A (1) C B D E A (1) B E A C D (1) A E D B C (1) A D E C B (1) A D C E B (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -16 -4 -6 B 4 0 -4 -10 -2 C 16 4 0 2 8 D 4 10 -2 0 4 E 6 2 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -4 -6 B 4 0 -4 -10 -2 C 16 4 0 2 8 D 4 10 -2 0 4 E 6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 E=19 A=17 C=10 so C is eliminated. Round 2 votes counts: D=29 B=29 E=25 A=17 so A is eliminated. Round 3 votes counts: D=38 B=36 E=26 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:208 E:198 B:194 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 -4 -6 B 4 0 -4 -10 -2 C 16 4 0 2 8 D 4 10 -2 0 4 E 6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -4 -6 B 4 0 -4 -10 -2 C 16 4 0 2 8 D 4 10 -2 0 4 E 6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -4 -6 B 4 0 -4 -10 -2 C 16 4 0 2 8 D 4 10 -2 0 4 E 6 2 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6532: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) D B E A C (6) A C E D B (6) E C A B D (5) D C A B E (5) B D E C A (5) D B A C E (4) C A E D B (4) C A D E B (4) B C E A D (4) A E C D B (4) E A C B D (3) C D B A E (3) B D E A C (3) E B A C D (2) D A C B E (2) C A E B D (2) B E D C A (2) B E C A D (2) B D C E A (2) B C D E A (2) E A B D C (1) E A B C D (1) D E B A C (1) D C B A E (1) D C A E B (1) D B C E A (1) D B A E C (1) D A E C B (1) D A E B C (1) C E B A D (1) C E A B D (1) C D A E B (1) C B A E D (1) B E D A C (1) B E C D A (1) B E A D C (1) B D C A E (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -14 -6 10 B 6 0 4 -18 12 C 14 -4 0 -4 12 D 6 18 4 0 14 E -10 -12 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -6 10 B 6 0 4 -18 12 C 14 -4 0 -4 12 D 6 18 4 0 14 E -10 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=24 C=17 A=15 E=12 so E is eliminated. Round 2 votes counts: D=32 B=26 C=22 A=20 so A is eliminated. Round 3 votes counts: D=36 C=36 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:209 B:202 A:192 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -14 -6 10 B 6 0 4 -18 12 C 14 -4 0 -4 12 D 6 18 4 0 14 E -10 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -6 10 B 6 0 4 -18 12 C 14 -4 0 -4 12 D 6 18 4 0 14 E -10 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -6 10 B 6 0 4 -18 12 C 14 -4 0 -4 12 D 6 18 4 0 14 E -10 -12 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6533: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (6) E A C B D (5) E C B D A (4) E C A D B (4) C A E D B (4) B E D C A (4) A C E D B (4) C D A B E (3) C A D E B (3) B D C A E (3) B D A E C (3) A D C B E (3) A C D E B (3) E C D B A (2) E B C D A (2) E B C A D (2) E B A D C (2) E B A C D (2) E A B C D (2) C E D A B (2) C E A D B (2) B D E C A (2) B D C E A (2) B D A C E (2) B A D E C (2) A E B D C (2) A E B C D (2) A D B C E (2) A C D B E (2) A B E D C (2) E C D A B (1) E C B A D (1) E C A B D (1) E A C D B (1) E A B D C (1) D C B A E (1) D C A B E (1) D B C E A (1) C E D B A (1) C D E B A (1) C D B E A (1) B E A D C (1) B D E A C (1) B A E D C (1) B A D C E (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -8 14 4 B -4 0 -2 2 -10 C 8 2 0 12 -4 D -14 -2 -12 0 -12 E -4 10 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 -8 14 4 B -4 0 -2 2 -10 C 8 2 0 12 -4 D -14 -2 -12 0 -12 E -4 10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=22 A=22 C=17 D=9 so D is eliminated. Round 2 votes counts: E=30 B=29 A=22 C=19 so C is eliminated. Round 3 votes counts: E=36 A=33 B=31 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:211 C:209 A:207 B:193 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 14 4 B -4 0 -2 2 -10 C 8 2 0 12 -4 D -14 -2 -12 0 -12 E -4 10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 14 4 B -4 0 -2 2 -10 C 8 2 0 12 -4 D -14 -2 -12 0 -12 E -4 10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 14 4 B -4 0 -2 2 -10 C 8 2 0 12 -4 D -14 -2 -12 0 -12 E -4 10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6534: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) E B C D A (8) D C A E B (7) C D E A B (7) B A E D C (7) A D C B E (7) B E C A D (6) B E A C D (6) A B D E C (6) C D A E B (5) D A C E B (4) A D B C E (4) B A E C D (3) A B D C E (3) D C E A B (2) C E D B A (2) C D A B E (2) A C D B E (2) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C A D (1) E B A D C (1) D E A B C (1) D A E C B (1) D A E B C (1) C E B D A (1) Total count = 100 A B C D E A 0 4 12 10 4 B -4 0 16 8 10 C -12 -16 0 -10 -8 D -10 -8 10 0 4 E -4 -10 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 10 4 B -4 0 16 8 10 C -12 -16 0 -10 -8 D -10 -8 10 0 4 E -4 -10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=22 C=17 D=16 E=14 so E is eliminated. Round 2 votes counts: B=43 A=22 C=19 D=16 so D is eliminated. Round 3 votes counts: B=43 A=29 C=28 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:215 D:198 E:195 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 10 4 B -4 0 16 8 10 C -12 -16 0 -10 -8 D -10 -8 10 0 4 E -4 -10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 10 4 B -4 0 16 8 10 C -12 -16 0 -10 -8 D -10 -8 10 0 4 E -4 -10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 10 4 B -4 0 16 8 10 C -12 -16 0 -10 -8 D -10 -8 10 0 4 E -4 -10 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6535: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) D B A E C (7) C A E D B (7) D B A C E (6) D B E C A (5) E C A B D (4) E A C B D (4) D B C A E (4) B D E A C (4) B D A E C (4) E B A C D (3) D B E A C (3) C E A B D (3) C D A E B (3) A E C B D (3) E C D B A (2) E B D C A (2) D C E B A (2) D C A E B (2) D A C B E (2) C E A D B (2) C D E A B (2) B E A D C (2) A B E C D (2) E D B C A (1) E C B A D (1) E B C A D (1) D E C B A (1) D C E A B (1) D C A B E (1) D B C E A (1) C A D E B (1) B E A C D (1) B A D E C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -8 -6 10 B 2 0 -2 -12 -6 C 8 2 0 -4 -2 D 6 12 4 0 8 E -10 6 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -6 10 B 2 0 -2 -12 -6 C 8 2 0 -4 -2 D 6 12 4 0 8 E -10 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=26 E=18 B=12 A=9 so A is eliminated. Round 2 votes counts: D=37 C=27 E=21 B=15 so B is eliminated. Round 3 votes counts: D=47 C=27 E=26 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:202 A:197 E:195 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -6 10 B 2 0 -2 -12 -6 C 8 2 0 -4 -2 D 6 12 4 0 8 E -10 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -6 10 B 2 0 -2 -12 -6 C 8 2 0 -4 -2 D 6 12 4 0 8 E -10 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -6 10 B 2 0 -2 -12 -6 C 8 2 0 -4 -2 D 6 12 4 0 8 E -10 6 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6536: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (15) C E D B A (11) E D B A C (8) C E D A B (6) B A E D C (6) E D C B A (5) D E B A C (5) C A B D E (5) C D E A B (4) C A B E D (4) B E D A C (4) A C B D E (4) A B E D C (4) E D B C A (2) D E C B A (2) C D E B A (2) C A E D B (2) C A D E B (2) A B C D E (2) E C D B A (1) D E B C A (1) B D E A C (1) B D A E C (1) A D E B C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 6 -6 -4 B -2 0 4 -4 -4 C -6 -4 0 -12 -12 D 6 4 12 0 -8 E 4 4 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 6 -6 -4 B -2 0 4 -4 -4 C -6 -4 0 -12 -12 D 6 4 12 0 -8 E 4 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=28 E=16 B=12 D=8 so D is eliminated. Round 2 votes counts: C=36 A=28 E=24 B=12 so B is eliminated. Round 3 votes counts: C=36 A=35 E=29 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:214 D:207 A:199 B:197 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 -6 -4 B -2 0 4 -4 -4 C -6 -4 0 -12 -12 D 6 4 12 0 -8 E 4 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -6 -4 B -2 0 4 -4 -4 C -6 -4 0 -12 -12 D 6 4 12 0 -8 E 4 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -6 -4 B -2 0 4 -4 -4 C -6 -4 0 -12 -12 D 6 4 12 0 -8 E 4 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6537: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) B D A E C (9) D A E C B (6) D A E B C (5) B E C A D (5) B C E A D (5) D B A E C (4) E C B A D (3) D A C E B (3) C B E A D (3) B E C D A (3) B D E A C (3) B D C A E (3) B C D E A (3) E C A D B (2) E B A D C (2) D A B E C (2) C E A B D (2) C D A E B (2) C A D E B (2) B E D C A (2) B E D A C (2) A E D C B (2) A E C D B (2) E C A B D (1) E B C A D (1) E B A C D (1) D C B A E (1) D C A E B (1) D A C B E (1) D A B C E (1) C A E D B (1) B E A D C (1) B D C E A (1) B D A C E (1) B C E D A (1) A E D B C (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -4 -8 -2 B 8 0 12 4 2 C 4 -12 0 -4 -16 D 8 -4 4 0 -2 E 2 -2 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -8 -2 B 8 0 12 4 2 C 4 -12 0 -4 -16 D 8 -4 4 0 -2 E 2 -2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=24 C=20 E=10 A=7 so A is eliminated. Round 2 votes counts: B=39 D=25 C=21 E=15 so E is eliminated. Round 3 votes counts: B=43 C=29 D=28 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:209 D:203 A:189 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -8 -2 B 8 0 12 4 2 C 4 -12 0 -4 -16 D 8 -4 4 0 -2 E 2 -2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -8 -2 B 8 0 12 4 2 C 4 -12 0 -4 -16 D 8 -4 4 0 -2 E 2 -2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -8 -2 B 8 0 12 4 2 C 4 -12 0 -4 -16 D 8 -4 4 0 -2 E 2 -2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6538: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) A D B E C (9) C A D B E (8) E B D A C (6) A C D B E (6) E B C D A (4) C E A B D (4) A E B D C (4) E B D C A (3) C D B E A (3) C D A B E (3) B E D C A (3) B D E C A (3) B D E A C (3) A D B C E (3) E A C B D (2) D B A E C (2) D A C B E (2) C B E D A (2) A D C B E (2) E C B D A (1) E B A D C (1) D C B A E (1) D B E A C (1) D B C A E (1) D A B E C (1) D A B C E (1) C E B A D (1) C B D E A (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D A C (1) A E B C D (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -4 -6 0 B -4 0 -6 6 12 C 4 6 0 6 10 D 6 -6 -6 0 4 E 0 -12 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -6 0 B -4 0 -6 6 12 C 4 6 0 6 10 D 6 -6 -6 0 4 E 0 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=28 E=17 B=10 D=9 so D is eliminated. Round 2 votes counts: C=37 A=32 E=17 B=14 so B is eliminated. Round 3 votes counts: C=38 A=34 E=28 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:204 D:199 A:197 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -6 0 B -4 0 -6 6 12 C 4 6 0 6 10 D 6 -6 -6 0 4 E 0 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 0 B -4 0 -6 6 12 C 4 6 0 6 10 D 6 -6 -6 0 4 E 0 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 0 B -4 0 -6 6 12 C 4 6 0 6 10 D 6 -6 -6 0 4 E 0 -12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6539: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) D C A B E (8) C D B A E (7) C B D A E (7) B E C A D (6) A E D B C (5) A D E C B (5) E A B D C (4) D A E C B (4) A B E C D (4) B C E D A (3) B C E A D (3) B C D E A (3) B A C E D (3) E B C D A (2) D E C A B (2) D C A E B (2) D A C B E (2) C B D E A (2) B E A C D (2) B C A E D (2) B A E C D (2) E D C B A (1) E A B C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D A C E B (1) C D B E A (1) C D A B E (1) C A D B E (1) B E C D A (1) A D C E B (1) Total count = 100 A B C D E A 0 6 -10 0 12 B -6 0 4 -10 20 C 10 -4 0 2 0 D 0 10 -2 0 0 E -12 -20 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999976 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 0 12 B -6 0 4 -10 20 C 10 -4 0 2 0 D 0 10 -2 0 0 E -12 -20 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=22 E=19 C=19 A=15 so A is eliminated. Round 2 votes counts: B=29 D=28 E=24 C=19 so C is eliminated. Round 3 votes counts: D=38 B=38 E=24 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:204 B:204 C:204 D:204 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -10 0 12 B -6 0 4 -10 20 C 10 -4 0 2 0 D 0 10 -2 0 0 E -12 -20 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 0 12 B -6 0 4 -10 20 C 10 -4 0 2 0 D 0 10 -2 0 0 E -12 -20 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 0 12 B -6 0 4 -10 20 C 10 -4 0 2 0 D 0 10 -2 0 0 E -12 -20 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000094 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6540: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (6) B D A E C (5) A D C E B (5) E B C D A (4) D B E C A (4) D A C E B (4) D A B E C (4) D A B C E (4) C E B A D (4) C A E B D (4) C E A D B (3) C A E D B (3) B D E C A (3) B A E C D (3) B A C E D (3) A C E D B (3) A C E B D (3) A C D E B (3) A B C E D (3) E C D B A (2) E C B D A (2) D C E A B (2) D B A E C (2) B D E A C (2) B A D E C (2) A D C B E (2) A D B C E (2) E C B A D (1) E B D C A (1) E B C A D (1) D E C A B (1) D E B C A (1) D A E C B (1) C E A B D (1) C D A E B (1) B E D C A (1) B E C A D (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 8 0 20 B -4 0 6 6 0 C -8 -6 0 6 4 D 0 -6 -6 0 0 E -20 0 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.693456 B: 0.000000 C: 0.000000 D: 0.306544 E: 0.000000 Sum of squares = 0.574850713392 Cumulative probabilities = A: 0.693456 B: 0.693456 C: 0.693456 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 0 20 B -4 0 6 6 0 C -8 -6 0 6 4 D 0 -6 -6 0 0 E -20 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.399998 E: 0.000000 Sum of squares = 0.520000872181 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 D=23 C=16 E=11 so E is eliminated. Round 2 votes counts: B=32 A=24 D=23 C=21 so C is eliminated. Round 3 votes counts: B=39 A=35 D=26 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:204 C:198 D:194 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 0 20 B -4 0 6 6 0 C -8 -6 0 6 4 D 0 -6 -6 0 0 E -20 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.399998 E: 0.000000 Sum of squares = 0.520000872181 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 0 20 B -4 0 6 6 0 C -8 -6 0 6 4 D 0 -6 -6 0 0 E -20 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.399998 E: 0.000000 Sum of squares = 0.520000872181 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 0 20 B -4 0 6 6 0 C -8 -6 0 6 4 D 0 -6 -6 0 0 E -20 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600002 B: 0.000000 C: 0.000000 D: 0.399998 E: 0.000000 Sum of squares = 0.520000872181 Cumulative probabilities = A: 0.600002 B: 0.600002 C: 0.600002 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6541: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) C D B A E (6) E A B D C (5) D E A B C (5) B E C A D (5) D C B E A (4) D C A E B (4) D A E C B (4) B C E D A (4) B E C D A (3) A E B C D (3) E D A B C (2) E B A D C (2) E B A C D (2) E A D B C (2) D E B C A (2) D E B A C (2) D B E C A (2) D A C E B (2) C D A E B (2) C D A B E (2) C B A E D (2) C B A D E (2) C A D B E (2) C A B E D (2) B E A D C (2) B C E A D (2) B C D E A (2) A D C E B (2) A C B E D (2) E D B A C (1) E A B C D (1) C D B E A (1) C B E A D (1) C B D A E (1) C A D E B (1) C A B D E (1) B E D C A (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 -4 0 -20 B 12 0 14 6 12 C 4 -14 0 14 -10 D 0 -6 -14 0 -4 E 20 -12 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 0 -20 B 12 0 14 6 12 C 4 -14 0 14 -10 D 0 -6 -14 0 -4 E 20 -12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=25 C=23 E=15 A=9 so A is eliminated. Round 2 votes counts: D=28 B=28 C=25 E=19 so E is eliminated. Round 3 votes counts: B=41 D=33 C=26 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:211 C:197 D:188 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 0 -20 B 12 0 14 6 12 C 4 -14 0 14 -10 D 0 -6 -14 0 -4 E 20 -12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 0 -20 B 12 0 14 6 12 C 4 -14 0 14 -10 D 0 -6 -14 0 -4 E 20 -12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 0 -20 B 12 0 14 6 12 C 4 -14 0 14 -10 D 0 -6 -14 0 -4 E 20 -12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6542: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (13) D C B A E (6) A E B D C (6) A B E D C (6) E A B C D (5) D C A B E (5) B E A D C (4) E B A C D (3) E A B D C (3) D B C A E (3) C D E B A (3) B D C E A (3) A E D B C (3) E B A D C (2) C D E A B (2) C D B A E (2) C A E D B (2) B E C D A (2) A E C B D (2) A D C B E (2) E C A D B (1) E A C B D (1) D C B E A (1) D B C E A (1) D B A C E (1) D A B C E (1) C E D A B (1) C D A E B (1) C D A B E (1) C B D E A (1) C A D E B (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A E C (1) B D A C E (1) B A E D C (1) A E D C B (1) A E C D B (1) A D E C B (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 -4 2 B 2 0 2 -12 20 C 2 -2 0 -14 6 D 4 12 14 0 8 E -2 -20 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -4 2 B 2 0 2 -12 20 C 2 -2 0 -14 6 D 4 12 14 0 8 E -2 -20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=25 D=18 E=15 B=15 so E is eliminated. Round 2 votes counts: A=34 C=28 B=20 D=18 so D is eliminated. Round 3 votes counts: C=40 A=35 B=25 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:219 B:206 A:197 C:196 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -4 2 B 2 0 2 -12 20 C 2 -2 0 -14 6 D 4 12 14 0 8 E -2 -20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -4 2 B 2 0 2 -12 20 C 2 -2 0 -14 6 D 4 12 14 0 8 E -2 -20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -4 2 B 2 0 2 -12 20 C 2 -2 0 -14 6 D 4 12 14 0 8 E -2 -20 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998277 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6543: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) E B A C D (6) D A C B E (6) E B C A D (5) B E C D A (5) C D A B E (4) C A D B E (4) A E D B C (4) D B E A C (3) C B E D A (3) C A E B D (3) C A D E B (3) A D E C B (3) A C D E B (3) D C B A E (2) D C A B E (2) D A B E C (2) C E B A D (2) C D B A E (2) B E D C A (2) B E D A C (2) B D E C A (2) A D E B C (2) A D C E B (2) A D C B E (2) A C E D B (2) E C B A D (1) E C A B D (1) E B C D A (1) E B A D C (1) D C B E A (1) D B C E A (1) D B A E C (1) D B A C E (1) D A B C E (1) C E A B D (1) C B E A D (1) C B D E A (1) C B D A E (1) C A B D E (1) B D E A C (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 2 -4 4 B 6 0 -4 -2 2 C -2 4 0 2 -2 D 4 2 -2 0 2 E -4 -2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -4 4 B 6 0 -4 -2 2 C -2 4 0 2 -2 D 4 2 -2 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 D=20 A=19 B=12 so B is eliminated. Round 2 votes counts: E=32 C=26 D=23 A=19 so A is eliminated. Round 3 votes counts: E=37 D=32 C=31 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:203 B:201 C:201 A:198 E:197 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 -4 4 B 6 0 -4 -2 2 C -2 4 0 2 -2 D 4 2 -2 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -4 4 B 6 0 -4 -2 2 C -2 4 0 2 -2 D 4 2 -2 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -4 4 B 6 0 -4 -2 2 C -2 4 0 2 -2 D 4 2 -2 0 2 E -4 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6544: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) E A C D B (9) D B C E A (9) B D A C E (9) C E D B A (8) E C A D B (6) A B D E C (5) A B D C E (5) C E B D A (4) A E C D B (4) E C D B A (3) E A C B D (3) A D B E C (3) C E A B D (2) C D B E A (2) C B D E A (2) B A D C E (2) A E B D C (2) E D B C A (1) D B A E C (1) D B A C E (1) C D E B A (1) C A B D E (1) B D C E A (1) B D C A E (1) A E D C B (1) A E D B C (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 18 14 -2 B -10 0 -18 0 -12 C -18 18 0 16 2 D -14 0 -16 0 -12 E 2 12 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.818182 Sum of squares = 0.685950413239 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.181818 D: 0.181818 E: 1.000000 A B C D E A 0 10 18 14 -2 B -10 0 -18 0 -12 C -18 18 0 16 2 D -14 0 -16 0 -12 E 2 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.818182 Sum of squares = 0.685950412979 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.181818 D: 0.181818 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=22 C=20 B=13 D=11 so D is eliminated. Round 2 votes counts: A=34 B=24 E=22 C=20 so C is eliminated. Round 3 votes counts: E=37 A=35 B=28 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:220 E:212 C:209 B:180 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 18 14 -2 B -10 0 -18 0 -12 C -18 18 0 16 2 D -14 0 -16 0 -12 E 2 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.818182 Sum of squares = 0.685950412979 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.181818 D: 0.181818 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 14 -2 B -10 0 -18 0 -12 C -18 18 0 16 2 D -14 0 -16 0 -12 E 2 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.818182 Sum of squares = 0.685950412979 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.181818 D: 0.181818 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 14 -2 B -10 0 -18 0 -12 C -18 18 0 16 2 D -14 0 -16 0 -12 E 2 12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.090909 D: 0.000000 E: 0.818182 Sum of squares = 0.685950412979 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.181818 D: 0.181818 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6545: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B D E C A (9) D B E C A (7) B E D A C (7) D C B A E (6) A E C B D (6) E A B C D (5) A C E B D (5) E B A D C (4) D B C E A (4) D B C A E (4) C A E D B (4) E B A C D (3) E A C B D (3) B E D C A (3) B D E A C (3) A C E D B (3) E B C A D (2) E C A B D (1) E B D A C (1) E B C D A (1) D C B E A (1) D C A B E (1) C D B E A (1) C D B A E (1) C A D B E (1) B E A D C (1) A E B D C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 -12 2 -12 B 16 0 12 12 -2 C 12 -12 0 -4 -16 D -2 -12 4 0 -2 E 12 2 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -12 2 -12 B 16 0 12 12 -2 C 12 -12 0 -4 -16 D -2 -12 4 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 E=20 C=17 A=17 so C is eliminated. Round 2 votes counts: A=32 D=25 B=23 E=20 so E is eliminated. Round 3 votes counts: A=41 B=34 D=25 so D is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:216 D:194 C:190 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -12 2 -12 B 16 0 12 12 -2 C 12 -12 0 -4 -16 D -2 -12 4 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 2 -12 B 16 0 12 12 -2 C 12 -12 0 -4 -16 D -2 -12 4 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 2 -12 B 16 0 12 12 -2 C 12 -12 0 -4 -16 D -2 -12 4 0 -2 E 12 2 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6546: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) C A E B D (7) A B E D C (6) E B A D C (5) C D A E B (5) C D A B E (5) D C B E A (4) D B E C A (4) A E B C D (4) A C E B D (4) E C B D A (3) E B D A C (3) B D E A C (3) A D B E C (3) A B E C D (3) E B D C A (2) C A E D B (2) C A D E B (2) B A D E C (2) A C D B E (2) A C B E D (2) A B C E D (2) E B C D A (1) E B A C D (1) E A B D C (1) D C E B A (1) D C B A E (1) D C A B E (1) D B E A C (1) D B A E C (1) D B A C E (1) D A C B E (1) D A B E C (1) C E D B A (1) C E A D B (1) C D E A B (1) C D B E A (1) C A D B E (1) B E D A C (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 12 -2 0 16 B -12 0 -10 2 -6 C 2 10 0 14 8 D 0 -2 -14 0 0 E -16 6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 0 16 B -12 0 -10 2 -6 C 2 10 0 14 8 D 0 -2 -14 0 0 E -16 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=28 E=16 D=16 B=6 so B is eliminated. Round 2 votes counts: C=34 A=30 D=19 E=17 so E is eliminated. Round 3 votes counts: C=38 A=37 D=25 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:213 D:192 E:191 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 0 16 B -12 0 -10 2 -6 C 2 10 0 14 8 D 0 -2 -14 0 0 E -16 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 0 16 B -12 0 -10 2 -6 C 2 10 0 14 8 D 0 -2 -14 0 0 E -16 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 0 16 B -12 0 -10 2 -6 C 2 10 0 14 8 D 0 -2 -14 0 0 E -16 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6547: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (6) E C B D A (5) E C A B D (4) E B D A C (4) D C A B E (4) D B E C A (4) D A B C E (4) C A E D B (4) A D C B E (4) E C B A D (3) E B C D A (3) E B A D C (3) C A D B E (3) B D E A C (3) A C E B D (3) A C D B E (3) E A C B D (2) E A B C D (2) D B C E A (2) D B A E C (2) D B A C E (2) C D A B E (2) B E D C A (2) B E D A C (2) A E B C D (2) A B E D C (2) A B D E C (2) E D C B A (1) E D B C A (1) E B D C A (1) E B A C D (1) D C E B A (1) D C B E A (1) D C B A E (1) D B C A E (1) D A C B E (1) C E D B A (1) C E D A B (1) C E A D B (1) C E A B D (1) B E A D C (1) B D E C A (1) B A D E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 6 -8 0 -2 B -6 0 -6 -4 -2 C 8 6 0 -4 -4 D 0 4 4 0 -2 E 2 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 0 -2 B -6 0 -6 -4 -2 C 8 6 0 -4 -4 D 0 4 4 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=23 C=19 A=18 B=10 so B is eliminated. Round 2 votes counts: E=35 D=27 C=19 A=19 so C is eliminated. Round 3 votes counts: E=39 A=32 D=29 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:205 C:203 D:203 A:198 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 0 -2 B -6 0 -6 -4 -2 C 8 6 0 -4 -4 D 0 4 4 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 0 -2 B -6 0 -6 -4 -2 C 8 6 0 -4 -4 D 0 4 4 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 0 -2 B -6 0 -6 -4 -2 C 8 6 0 -4 -4 D 0 4 4 0 -2 E 2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6548: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) D E C B A (6) C D E A B (6) C D A B E (6) B A E C D (6) C D E B A (5) A B E D C (5) A B C D E (5) B A E D C (4) A B E C D (4) E D C B A (3) E D B A C (3) E B A D C (3) D C A E B (3) A B C E D (3) E D B C A (2) E C D B A (2) E B D A C (2) E B A C D (2) D C E B A (2) C A B E D (2) C A B D E (2) B A C E D (2) E B C A D (1) D E B A C (1) D E A B C (1) D C A B E (1) D A C B E (1) D A B E C (1) C E D B A (1) C E B D A (1) C D A E B (1) C B E A D (1) C A D B E (1) B E A C D (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -6 -10 0 B -4 0 -2 -8 -4 C 6 2 0 4 4 D 10 8 -4 0 4 E 0 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -10 0 B -4 0 -2 -8 -4 C 6 2 0 4 4 D 10 8 -4 0 4 E 0 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=23 A=20 E=18 B=13 so B is eliminated. Round 2 votes counts: A=32 C=26 D=23 E=19 so E is eliminated. Round 3 votes counts: A=38 D=33 C=29 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:209 C:208 E:198 A:194 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -10 0 B -4 0 -2 -8 -4 C 6 2 0 4 4 D 10 8 -4 0 4 E 0 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -10 0 B -4 0 -2 -8 -4 C 6 2 0 4 4 D 10 8 -4 0 4 E 0 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -10 0 B -4 0 -2 -8 -4 C 6 2 0 4 4 D 10 8 -4 0 4 E 0 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6549: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E B D C A (6) B D C E A (6) B C D A E (5) E B D A C (4) C B A D E (4) B E D C A (4) A E C D B (4) A E C B D (4) A C D E B (4) E D B A C (3) E A B C D (3) D B C E A (3) C B D A E (3) A C E D B (3) E D B C A (2) E B A C D (2) E A B D C (2) D B E C A (2) D A E C B (2) C D B A E (2) C A D B E (2) C A B D E (2) B E C D A (2) E D A B C (1) E B C A D (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E A B C (1) D C B A E (1) D B C A E (1) C A B E D (1) B D E C A (1) B D C A E (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -2 -4 2 B 12 0 8 12 2 C 2 -8 0 10 0 D 4 -12 -10 0 0 E -2 -2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -4 2 B 12 0 8 12 2 C 2 -8 0 10 0 D 4 -12 -10 0 0 E -2 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 B=21 C=14 D=10 so D is eliminated. Round 2 votes counts: E=29 A=29 B=27 C=15 so C is eliminated. Round 3 votes counts: B=37 A=34 E=29 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:202 E:198 A:192 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 -4 2 B 12 0 8 12 2 C 2 -8 0 10 0 D 4 -12 -10 0 0 E -2 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -4 2 B 12 0 8 12 2 C 2 -8 0 10 0 D 4 -12 -10 0 0 E -2 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -4 2 B 12 0 8 12 2 C 2 -8 0 10 0 D 4 -12 -10 0 0 E -2 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6550: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) D E C A B (5) C D A E B (5) C A E D B (5) C D E A B (4) C B A E D (4) B E D A C (4) B D E A C (4) A E D C B (4) A E C D B (4) A C E D B (4) D E B A C (3) B D E C A (3) B C A D E (3) B A C E D (3) A C E B D (3) E D B A C (2) E D A B C (2) E A D C B (2) E A D B C (2) D E A C B (2) D E A B C (2) C B A D E (2) C A D E B (2) B D C E A (2) B A E D C (2) D E B C A (1) D C E B A (1) D C E A B (1) D B E A C (1) C D E B A (1) C D B A E (1) C B D A E (1) B E A D C (1) B C D E A (1) B C D A E (1) B C A E D (1) B A E C D (1) A E B D C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -2 6 12 B -14 0 -18 -8 -12 C 2 18 0 10 8 D -6 8 -10 0 -8 E -12 12 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 6 12 B -14 0 -18 -8 -12 C 2 18 0 10 8 D -6 8 -10 0 -8 E -12 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=26 A=18 D=16 E=8 so E is eliminated. Round 2 votes counts: C=32 B=26 A=22 D=20 so D is eliminated. Round 3 votes counts: C=39 B=33 A=28 so A is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:215 E:200 D:192 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 6 12 B -14 0 -18 -8 -12 C 2 18 0 10 8 D -6 8 -10 0 -8 E -12 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 6 12 B -14 0 -18 -8 -12 C 2 18 0 10 8 D -6 8 -10 0 -8 E -12 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 6 12 B -14 0 -18 -8 -12 C 2 18 0 10 8 D -6 8 -10 0 -8 E -12 12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6551: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (9) B D E C A (7) A E D B C (7) A E C D B (7) B D E A C (6) B D C E A (6) E D A B C (5) D E B A C (5) C B D E A (5) B C D E A (5) E A D B C (4) D B E A C (4) C B D A E (4) C A E B D (4) E D B A C (3) C B A D E (3) D B E C A (2) C A E D B (2) C A B E D (2) B D A E C (2) A E D C B (2) A B D E C (2) E D B C A (1) E A D C B (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 18 -10 -8 B 8 0 20 -4 -2 C -18 -20 0 -16 -18 D 10 4 16 0 4 E 8 2 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 18 -10 -8 B 8 0 20 -4 -2 C -18 -20 0 -16 -18 D 10 4 16 0 4 E 8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 C=20 E=14 D=11 so D is eliminated. Round 2 votes counts: B=33 A=28 C=20 E=19 so E is eliminated. Round 3 votes counts: B=42 A=38 C=20 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:217 E:212 B:211 A:196 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 18 -10 -8 B 8 0 20 -4 -2 C -18 -20 0 -16 -18 D 10 4 16 0 4 E 8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 18 -10 -8 B 8 0 20 -4 -2 C -18 -20 0 -16 -18 D 10 4 16 0 4 E 8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 18 -10 -8 B 8 0 20 -4 -2 C -18 -20 0 -16 -18 D 10 4 16 0 4 E 8 2 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6552: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) A B C D E (10) D E A C B (7) E D C B A (6) E D C A B (6) B C A E D (6) E C D B A (5) D E A B C (5) A D E B C (5) A B D C E (4) E D A C B (3) C B E D A (3) B A C D E (3) D A E B C (2) C E B D A (2) C B E A D (2) B C E D A (2) B C A D E (2) A D E C B (2) A D B E C (2) E D B C A (1) E B C D A (1) D E C B A (1) D E C A B (1) C E B A D (1) B E D C A (1) B A D E C (1) B A C E D (1) A D B C E (1) A C B E D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -2 8 6 B -2 0 -4 6 4 C 2 4 0 2 0 D -8 -6 -2 0 -4 E -6 -4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.840659 D: 0.000000 E: 0.159341 Sum of squares = 0.732097446772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.840659 D: 0.840659 E: 1.000000 A B C D E A 0 2 -2 8 6 B -2 0 -4 6 4 C 2 4 0 2 0 D -8 -6 -2 0 -4 E -6 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000001723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 C=19 D=16 B=16 so D is eliminated. Round 2 votes counts: E=36 A=29 C=19 B=16 so B is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 C:204 B:202 E:197 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 8 6 B -2 0 -4 6 4 C 2 4 0 2 0 D -8 -6 -2 0 -4 E -6 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000001723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 6 B -2 0 -4 6 4 C 2 4 0 2 0 D -8 -6 -2 0 -4 E -6 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000001723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 6 B -2 0 -4 6 4 C 2 4 0 2 0 D -8 -6 -2 0 -4 E -6 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000001723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6553: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (8) C E D A B (6) E C B A D (5) B D A E C (5) B A D E C (5) A B E C D (5) D B A E C (4) A E C B D (4) D C E B A (3) D C B E A (3) D B A C E (3) D A C E B (3) D A B E C (3) D A B C E (3) C E A D B (3) E C A B D (2) C E B D A (2) C E B A D (2) C E A B D (2) B E C A D (2) A D B E C (2) E A C B D (1) D C A E B (1) D B C E A (1) D B C A E (1) C E D B A (1) C D E B A (1) C D E A B (1) C A E D B (1) B E D C A (1) B E C D A (1) B D E C A (1) B C E D A (1) B A E D C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -2 -10 2 B 2 0 -2 0 2 C 2 2 0 -2 -4 D 10 0 2 0 2 E -2 -2 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.350281 C: 0.000000 D: 0.649719 E: 0.000000 Sum of squares = 0.54483154237 Cumulative probabilities = A: 0.000000 B: 0.350281 C: 0.350281 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -10 2 B 2 0 -2 0 2 C 2 2 0 -2 -4 D 10 0 2 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499796 C: 0.000000 D: 0.500204 E: 0.000000 Sum of squares = 0.500000083039 Cumulative probabilities = A: 0.000000 B: 0.499796 C: 0.499796 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=25 C=19 A=13 E=8 so E is eliminated. Round 2 votes counts: D=35 C=26 B=25 A=14 so A is eliminated. Round 3 votes counts: D=37 B=32 C=31 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:207 B:201 C:199 E:199 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 2 B 2 0 -2 0 2 C 2 2 0 -2 -4 D 10 0 2 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499796 C: 0.000000 D: 0.500204 E: 0.000000 Sum of squares = 0.500000083039 Cumulative probabilities = A: 0.000000 B: 0.499796 C: 0.499796 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 2 B 2 0 -2 0 2 C 2 2 0 -2 -4 D 10 0 2 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499796 C: 0.000000 D: 0.500204 E: 0.000000 Sum of squares = 0.500000083039 Cumulative probabilities = A: 0.000000 B: 0.499796 C: 0.499796 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 2 B 2 0 -2 0 2 C 2 2 0 -2 -4 D 10 0 2 0 2 E -2 -2 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499796 C: 0.000000 D: 0.500204 E: 0.000000 Sum of squares = 0.500000083039 Cumulative probabilities = A: 0.000000 B: 0.499796 C: 0.499796 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6554: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (9) B E A C D (7) A C D E B (7) E B A C D (6) E B D A C (5) D C B A E (4) D C A B E (4) E D B C A (3) E B A D C (3) E A B D C (3) C D A B E (3) B E D C A (3) B D E C A (3) A E D C B (3) D E B C A (2) C A D B E (2) B E D A C (2) A C D B E (2) A C B D E (2) E D B A C (1) E D A C B (1) E A D C B (1) E A D B C (1) E A C B D (1) E A B C D (1) D E C B A (1) D C B E A (1) D B C E A (1) D A C E B (1) C B D A E (1) C A B D E (1) B E C D A (1) B D C E A (1) B C E A D (1) B C D A E (1) B C A D E (1) B A E C D (1) B A C E D (1) A E C B D (1) A E B C D (1) A D C E B (1) A C E D B (1) A C E B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 14 4 -16 B 18 0 20 20 -8 C -14 -20 0 -8 -22 D -4 -20 8 0 -22 E 16 8 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 14 4 -16 B 18 0 20 20 -8 C -14 -20 0 -8 -22 D -4 -20 8 0 -22 E 16 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=22 A=22 D=14 C=7 so C is eliminated. Round 2 votes counts: E=35 A=25 B=23 D=17 so D is eliminated. Round 3 votes counts: E=38 A=33 B=29 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:234 B:225 A:192 D:181 C:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 14 4 -16 B 18 0 20 20 -8 C -14 -20 0 -8 -22 D -4 -20 8 0 -22 E 16 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 14 4 -16 B 18 0 20 20 -8 C -14 -20 0 -8 -22 D -4 -20 8 0 -22 E 16 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 14 4 -16 B 18 0 20 20 -8 C -14 -20 0 -8 -22 D -4 -20 8 0 -22 E 16 8 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6555: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) C D E A B (8) E C A B D (7) E A B C D (7) D C B E A (7) B A E D C (7) A E B C D (5) D B A C E (4) C E A D B (4) B D A C E (4) B A D E C (4) A B E D C (4) D C B A E (3) B D A E C (3) E A C B D (2) D C E B A (2) D C A B E (2) D B C A E (2) A B E C D (2) E C A D B (1) D C E A B (1) D C A E B (1) D B C E A (1) C D E B A (1) C D B E A (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A C D (1) B D C E A (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -8 -8 -14 B -12 0 -2 4 0 C 8 2 0 2 4 D 8 -4 -2 0 -8 E 14 0 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -8 -8 -14 B -12 0 -2 4 0 C 8 2 0 2 4 D 8 -4 -2 0 -8 E 14 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 B=23 E=17 A=13 so A is eliminated. Round 2 votes counts: B=30 C=24 E=23 D=23 so E is eliminated. Round 3 votes counts: B=42 C=35 D=23 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:208 D:197 B:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -8 -8 -14 B -12 0 -2 4 0 C 8 2 0 2 4 D 8 -4 -2 0 -8 E 14 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 -8 -14 B -12 0 -2 4 0 C 8 2 0 2 4 D 8 -4 -2 0 -8 E 14 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 -8 -14 B -12 0 -2 4 0 C 8 2 0 2 4 D 8 -4 -2 0 -8 E 14 0 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6556: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (10) C E D B A (9) C D E B A (5) C E B D A (4) A D B E C (4) A B E D C (4) A B E C D (4) E B A C D (3) D B A E C (3) D A B E C (3) D A B C E (3) C E B A D (3) C D E A B (3) B E A D C (3) E C B D A (2) E B D C A (2) E B C D A (2) D C B E A (2) D B E A C (2) C E D A B (2) C E A B D (2) C A D E B (2) B A E D C (2) B A D E C (2) E B C A D (1) D E B C A (1) D C E B A (1) D C A B E (1) D B E C A (1) D A C B E (1) C E A D B (1) C A E D B (1) C A E B D (1) B E D A C (1) B E A C D (1) B D E A C (1) A E C B D (1) A E B C D (1) A D B C E (1) A C E B D (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 8 2 -4 B 2 0 12 6 4 C -8 -12 0 2 -8 D -2 -6 -2 0 -4 E 4 -4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 2 -4 B 2 0 12 6 4 C -8 -12 0 2 -8 D -2 -6 -2 0 -4 E 4 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=29 D=18 E=10 B=10 so E is eliminated. Round 2 votes counts: C=35 A=29 D=18 B=18 so D is eliminated. Round 3 votes counts: C=39 A=36 B=25 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:206 A:202 D:193 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 2 -4 B 2 0 12 6 4 C -8 -12 0 2 -8 D -2 -6 -2 0 -4 E 4 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 2 -4 B 2 0 12 6 4 C -8 -12 0 2 -8 D -2 -6 -2 0 -4 E 4 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 2 -4 B 2 0 12 6 4 C -8 -12 0 2 -8 D -2 -6 -2 0 -4 E 4 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998776 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6557: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) A B C E D (9) A E D C B (7) B C D E A (5) A E B C D (5) D E A C B (4) D B C E A (4) E D C B A (3) E D A C B (3) E A C D B (3) D C E B A (3) B C A D E (3) E D C A B (2) E C D B A (2) E C D A B (2) E A D C B (2) D E C A B (2) D C B E A (2) D A E C B (2) C B E D A (2) B D C E A (2) B C D A E (2) B C A E D (2) A B D C E (2) A B C D E (2) E A C B D (1) C E D B A (1) C D B E A (1) B D C A E (1) B C E A D (1) B A C D E (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E B C (1) A D B E C (1) A D B C E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 12 2 -4 -8 B -12 0 -6 -14 -12 C -2 6 0 -10 -12 D 4 14 10 0 -2 E 8 12 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 -4 -8 B -12 0 -6 -14 -12 C -2 6 0 -10 -12 D 4 14 10 0 -2 E 8 12 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=26 E=18 B=17 C=4 so C is eliminated. Round 2 votes counts: A=35 D=27 E=19 B=19 so E is eliminated. Round 3 votes counts: A=41 D=40 B=19 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:217 D:213 A:201 C:191 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 -4 -8 B -12 0 -6 -14 -12 C -2 6 0 -10 -12 D 4 14 10 0 -2 E 8 12 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -4 -8 B -12 0 -6 -14 -12 C -2 6 0 -10 -12 D 4 14 10 0 -2 E 8 12 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -4 -8 B -12 0 -6 -14 -12 C -2 6 0 -10 -12 D 4 14 10 0 -2 E 8 12 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6558: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) E C B D A (6) C D A E B (6) B A E D C (6) A B E D C (6) E B C A D (5) E B A D C (5) A D C B E (5) E B C D A (4) E B A C D (4) C D E B A (4) C D A B E (4) D A B E C (3) C D E A B (3) E C D B A (2) D C A E B (2) D A C B E (2) A D B C E (2) A C D B E (2) A B D E C (2) A B D C E (2) D E C B A (1) D E B A C (1) D E A B C (1) D C E B A (1) D C E A B (1) D C A B E (1) C E D B A (1) C E B D A (1) C A D B E (1) B E C A D (1) B C E A D (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 10 12 -8 B 12 0 14 12 4 C -10 -14 0 16 -20 D -12 -12 -16 0 -10 E 8 -4 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 12 -8 B 12 0 14 12 4 C -10 -14 0 16 -20 D -12 -12 -16 0 -10 E 8 -4 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=21 C=20 A=20 D=13 so D is eliminated. Round 2 votes counts: E=29 C=25 A=25 B=21 so B is eliminated. Round 3 votes counts: E=42 A=32 C=26 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:221 E:217 A:201 C:186 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 12 -8 B 12 0 14 12 4 C -10 -14 0 16 -20 D -12 -12 -16 0 -10 E 8 -4 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 12 -8 B 12 0 14 12 4 C -10 -14 0 16 -20 D -12 -12 -16 0 -10 E 8 -4 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 12 -8 B 12 0 14 12 4 C -10 -14 0 16 -20 D -12 -12 -16 0 -10 E 8 -4 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6559: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (11) B E D C A (9) A C D E B (8) A D E C B (7) B C E D A (6) C A B D E (5) A D C E B (5) E D B A C (4) D E B A C (4) D E A B C (4) C A B E D (4) B E D A C (4) E B D C A (3) C D B E A (3) A D E B C (3) C D A E B (2) B E C D A (2) A C D B E (2) E D A B C (1) E B D A C (1) D E C B A (1) D E B C A (1) D C E B A (1) D A E B C (1) C B D E A (1) C B A E D (1) C A D B E (1) A E B D C (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -2 -18 -12 B 4 0 -10 4 4 C 2 10 0 -2 6 D 18 -4 2 0 0 E 12 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999993 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -18 -12 B 4 0 -10 4 4 C 2 10 0 -2 6 D 18 -4 2 0 0 E 12 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749998974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=28 B=21 D=12 E=9 so E is eliminated. Round 2 votes counts: A=30 C=28 B=25 D=17 so D is eliminated. Round 3 votes counts: A=36 B=34 C=30 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:208 D:208 B:201 E:201 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -18 -12 B 4 0 -10 4 4 C 2 10 0 -2 6 D 18 -4 2 0 0 E 12 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749998974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -18 -12 B 4 0 -10 4 4 C 2 10 0 -2 6 D 18 -4 2 0 0 E 12 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749998974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -18 -12 B 4 0 -10 4 4 C 2 10 0 -2 6 D 18 -4 2 0 0 E 12 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.250000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749998974 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6560: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) B E A D C (14) A B E C D (9) E B A D C (6) D C E B A (5) D E B C A (4) D B E C A (4) C A D E B (4) B E D C A (4) D E C B A (3) C A D B E (3) B E D A C (3) D C B E A (2) C D A B E (2) B E A C D (2) B A E C D (2) A E B D C (2) A C D E B (2) E B D C A (1) E B D A C (1) D E A C B (1) D C E A B (1) D C A E B (1) D B C E A (1) D A C E B (1) C D B A E (1) C A B D E (1) B E C D A (1) B D E C A (1) A E C B D (1) A E B C D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -6 -2 -8 B 10 0 12 2 2 C 6 -12 0 -10 -20 D 2 -2 10 0 2 E 8 -2 20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -2 -8 B 10 0 12 2 2 C 6 -12 0 -10 -20 D 2 -2 10 0 2 E 8 -2 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=25 D=23 A=17 E=8 so E is eliminated. Round 2 votes counts: B=35 C=25 D=23 A=17 so A is eliminated. Round 3 votes counts: B=47 C=30 D=23 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:212 D:206 A:187 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -2 -8 B 10 0 12 2 2 C 6 -12 0 -10 -20 D 2 -2 10 0 2 E 8 -2 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -2 -8 B 10 0 12 2 2 C 6 -12 0 -10 -20 D 2 -2 10 0 2 E 8 -2 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -2 -8 B 10 0 12 2 2 C 6 -12 0 -10 -20 D 2 -2 10 0 2 E 8 -2 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6561: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (17) D B C E A (6) D B C A E (6) D E A C B (5) C B D A E (5) E A D C B (4) E A B C D (4) B C A E D (4) E D A B C (3) E A B D C (3) C D B A E (3) B C D A E (3) A E C B D (3) A E B C D (3) A C E B D (3) E A D B C (2) D E B C A (2) C B A D E (2) B E A C D (2) B C A D E (2) B A C E D (2) A C B E D (2) E B A C D (1) E A C B D (1) D E A B C (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A E B (1) D C A B E (1) C B A E D (1) C A B D E (1) B D E C A (1) B C E D A (1) B A E C D (1) A E C D B (1) Total count = 100 A B C D E A 0 -22 -18 -16 22 B 22 0 -6 -10 22 C 18 6 0 -10 26 D 16 10 10 0 18 E -22 -22 -26 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -18 -16 22 B 22 0 -6 -10 22 C 18 6 0 -10 26 D 16 10 10 0 18 E -22 -22 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 E=18 B=16 C=12 A=12 so C is eliminated. Round 2 votes counts: D=45 B=24 E=18 A=13 so A is eliminated. Round 3 votes counts: D=45 E=28 B=27 so B is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:227 C:220 B:214 A:183 E:156 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 -18 -16 22 B 22 0 -6 -10 22 C 18 6 0 -10 26 D 16 10 10 0 18 E -22 -22 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -18 -16 22 B 22 0 -6 -10 22 C 18 6 0 -10 26 D 16 10 10 0 18 E -22 -22 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -18 -16 22 B 22 0 -6 -10 22 C 18 6 0 -10 26 D 16 10 10 0 18 E -22 -22 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6562: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (12) B A D E C (10) D E C B A (9) D B A E C (8) C E D A B (6) D C E B A (5) C E A B D (5) B D A E C (5) A B D E C (5) D B E A C (4) C E D B A (4) A B D C E (4) A C B E D (3) D E B C A (2) D B A C E (2) C A E B D (2) A C E B D (2) A B E D C (2) A B E C D (2) E C D B A (1) E C A B D (1) D E B A C (1) D B E C A (1) C E A D B (1) C D E A B (1) C A E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 22 2 18 B 4 0 18 8 18 C -22 -18 0 -16 -2 D -2 -8 16 0 16 E -18 -18 2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 22 2 18 B 4 0 18 8 18 C -22 -18 0 -16 -2 D -2 -8 16 0 16 E -18 -18 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=31 C=20 B=15 E=2 so E is eliminated. Round 2 votes counts: D=32 A=31 C=22 B=15 so B is eliminated. Round 3 votes counts: A=41 D=37 C=22 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:224 A:219 D:211 E:175 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 22 2 18 B 4 0 18 8 18 C -22 -18 0 -16 -2 D -2 -8 16 0 16 E -18 -18 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 2 18 B 4 0 18 8 18 C -22 -18 0 -16 -2 D -2 -8 16 0 16 E -18 -18 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 2 18 B 4 0 18 8 18 C -22 -18 0 -16 -2 D -2 -8 16 0 16 E -18 -18 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6563: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (8) C E A B D (6) E D B C A (5) E C B A D (5) E C D A B (4) E C A D B (4) B E D A C (4) E C A B D (3) E B C A D (3) D A C B E (3) C E A D B (3) B D E A C (3) A C D B E (3) A C B E D (3) E D C B A (2) E B D C A (2) D E B A C (2) D B E A C (2) D A C E B (2) C A E B D (2) C A D E B (2) C A B E D (2) B E D C A (2) B E C A D (2) B D A C E (2) B A C D E (2) A D B C E (2) A C D E B (2) E C D B A (1) E C B D A (1) D E B C A (1) D E A C B (1) D B A C E (1) D A E B C (1) C A E D B (1) B E C D A (1) B E A D C (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C E D (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -8 4 -18 B -8 0 -4 -2 -6 C 8 4 0 8 0 D -4 2 -8 0 -20 E 18 6 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.784115 D: 0.000000 E: 0.215885 Sum of squares = 0.661442190569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.784115 D: 0.784115 E: 1.000000 A B C D E A 0 8 -8 4 -18 B -8 0 -4 -2 -6 C 8 4 0 8 0 D -4 2 -8 0 -20 E 18 6 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=21 B=21 C=16 A=12 so A is eliminated. Round 2 votes counts: E=30 C=25 D=24 B=21 so B is eliminated. Round 3 votes counts: E=40 D=30 C=30 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:222 C:210 A:193 B:190 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 4 -18 B -8 0 -4 -2 -6 C 8 4 0 8 0 D -4 2 -8 0 -20 E 18 6 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 4 -18 B -8 0 -4 -2 -6 C 8 4 0 8 0 D -4 2 -8 0 -20 E 18 6 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 4 -18 B -8 0 -4 -2 -6 C 8 4 0 8 0 D -4 2 -8 0 -20 E 18 6 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6564: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) D B E A C (9) D E B A C (8) B D C E A (7) B C D A E (7) C A E B D (6) A E D C B (6) A C E D B (6) B D E C A (5) B C D E A (5) E D A B C (4) E A D C B (4) D E A B C (3) A E C D B (3) E A C B D (2) C B A E D (2) C B A D E (2) B C A D E (2) E D B A C (1) E A C D B (1) D E A C B (1) D B E C A (1) C B D A E (1) C A B D E (1) B E D C A (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -2 -6 -4 B 2 0 6 4 8 C 2 -6 0 0 2 D 6 -4 0 0 6 E 4 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 -4 B 2 0 6 4 8 C 2 -6 0 0 2 D 6 -4 0 0 6 E 4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 D=22 A=16 E=12 so E is eliminated. Round 2 votes counts: D=27 B=27 C=23 A=23 so C is eliminated. Round 3 votes counts: A=41 B=32 D=27 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:204 C:199 E:194 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 -4 B 2 0 6 4 8 C 2 -6 0 0 2 D 6 -4 0 0 6 E 4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 -4 B 2 0 6 4 8 C 2 -6 0 0 2 D 6 -4 0 0 6 E 4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 -4 B 2 0 6 4 8 C 2 -6 0 0 2 D 6 -4 0 0 6 E 4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6565: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C D B A E (7) C D A B E (7) B E A D C (7) C D A E B (6) C B D E A (5) D A E B C (4) C B E A D (4) A E D B C (4) D A C E B (3) B E D A C (3) B D C E A (3) A D E C B (3) E B A D C (2) E B A C D (2) E A D B C (2) C B D A E (2) C A E D B (2) C A D E B (2) B E A C D (2) B C E D A (2) A E D C B (2) A E C D B (2) D C A E B (1) D C A B E (1) D A B C E (1) C D B E A (1) C B E D A (1) C B A D E (1) C A E B D (1) C A B D E (1) B E D C A (1) B E C A D (1) B D E C A (1) B D E A C (1) B C D E A (1) A E B D C (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -2 -2 8 B -6 0 -8 -2 6 C 2 8 0 4 8 D 2 2 -4 0 6 E -8 -6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -2 8 B -6 0 -8 -2 6 C 2 8 0 4 8 D 2 2 -4 0 6 E -8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=22 A=15 E=13 D=10 so D is eliminated. Round 2 votes counts: C=42 A=23 B=22 E=13 so E is eliminated. Round 3 votes counts: C=42 A=32 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 A:205 D:203 B:195 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -2 8 B -6 0 -8 -2 6 C 2 8 0 4 8 D 2 2 -4 0 6 E -8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 8 B -6 0 -8 -2 6 C 2 8 0 4 8 D 2 2 -4 0 6 E -8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 8 B -6 0 -8 -2 6 C 2 8 0 4 8 D 2 2 -4 0 6 E -8 -6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6566: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) A D C B E (8) A D B C E (8) E C B D A (7) D A B E C (7) C A E D B (7) E C B A D (5) D B E A C (5) C E B A D (5) B E D C A (5) A D B E C (5) A C D E B (5) C E A B D (4) B D E A C (4) D B A E C (3) D A B C E (2) C A E B D (2) C A D E B (2) E B D A C (1) D C B A E (1) C E B D A (1) C E A D B (1) C A D B E (1) B D E C A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 8 0 10 6 B -8 0 -2 -12 0 C 0 2 0 -2 -4 D -10 12 2 0 6 E -6 0 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.614272 B: 0.000000 C: 0.385728 D: 0.000000 E: 0.000000 Sum of squares = 0.52611615389 Cumulative probabilities = A: 0.614272 B: 0.614272 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 10 6 B -8 0 -2 -12 0 C 0 2 0 -2 -4 D -10 12 2 0 6 E -6 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=23 E=21 D=18 B=10 so B is eliminated. Round 2 votes counts: A=28 E=26 D=23 C=23 so D is eliminated. Round 3 votes counts: A=40 E=36 C=24 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:205 C:198 E:196 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 10 6 B -8 0 -2 -12 0 C 0 2 0 -2 -4 D -10 12 2 0 6 E -6 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 10 6 B -8 0 -2 -12 0 C 0 2 0 -2 -4 D -10 12 2 0 6 E -6 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 10 6 B -8 0 -2 -12 0 C 0 2 0 -2 -4 D -10 12 2 0 6 E -6 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6567: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) A C B D E (9) C B A E D (8) C A B E D (5) E C B A D (4) D E B C A (4) D E B A C (4) A C B E D (4) E B C D A (3) D E A C B (3) D E A B C (3) D A E C B (3) C B E A D (3) A C D B E (3) E D C B A (2) E B D C A (2) D A E B C (2) D A B C E (2) C E A B D (2) B E C D A (2) B C D E A (2) A D C E B (2) A D C B E (2) A C E D B (2) A C E B D (2) A B C D E (2) E D A C B (1) E D A B C (1) E C A B D (1) D B C E A (1) D A C B E (1) C E B A D (1) B C A E D (1) B C A D E (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 6 -2 10 4 B -6 0 -22 4 -8 C 2 22 0 12 6 D -10 -4 -12 0 -10 E -4 8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999125 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 10 4 B -6 0 -22 4 -8 C 2 22 0 12 6 D -10 -4 -12 0 -10 E -4 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=23 D=23 C=19 B=6 so B is eliminated. Round 2 votes counts: A=29 E=25 D=23 C=23 so D is eliminated. Round 3 votes counts: E=39 A=37 C=24 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:221 A:209 E:204 B:184 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 10 4 B -6 0 -22 4 -8 C 2 22 0 12 6 D -10 -4 -12 0 -10 E -4 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 10 4 B -6 0 -22 4 -8 C 2 22 0 12 6 D -10 -4 -12 0 -10 E -4 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 10 4 B -6 0 -22 4 -8 C 2 22 0 12 6 D -10 -4 -12 0 -10 E -4 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6568: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (13) A B D E C (12) B A C E D (7) D E C A B (6) E C D B A (5) B A D C E (5) A B D C E (5) E C D A B (4) C E B D A (4) C E B A D (4) B C E A D (4) B A D E C (4) A D B E C (4) D A E C B (3) D A B E C (3) C E D A B (3) D E C B A (2) C B E A D (2) C B A E D (2) B A C D E (2) A D C E B (2) E C B D A (1) D E B A C (1) D E A C B (1) B C A E D (1) Total count = 100 A B C D E A 0 -14 -2 8 0 B 14 0 -4 6 2 C 2 4 0 4 8 D -8 -6 -4 0 0 E 0 -2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 8 0 B 14 0 -4 6 2 C 2 4 0 4 8 D -8 -6 -4 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=23 A=23 D=16 E=10 so E is eliminated. Round 2 votes counts: C=38 B=23 A=23 D=16 so D is eliminated. Round 3 votes counts: C=46 A=30 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:209 C:209 A:196 E:195 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -2 8 0 B 14 0 -4 6 2 C 2 4 0 4 8 D -8 -6 -4 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 8 0 B 14 0 -4 6 2 C 2 4 0 4 8 D -8 -6 -4 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 8 0 B 14 0 -4 6 2 C 2 4 0 4 8 D -8 -6 -4 0 0 E 0 -2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6569: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) E D B A C (8) A C D B E (7) E B D C A (6) A C B E D (6) B E D C A (5) D C A E B (4) C A D B E (4) D E B C A (3) C A D E B (3) B E C D A (3) B E C A D (3) B E A D C (3) B E A C D (3) B C A E D (3) B A E C D (3) C A B D E (2) B E D A C (2) B C E A D (2) A D C E B (2) A B C E D (2) E D B C A (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D B C (1) D E C A B (1) D E B A C (1) D E A C B (1) D C E B A (1) D A E C B (1) C B D A E (1) C B A E D (1) C A B E D (1) B C D E A (1) B A C E D (1) A E D C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 10 18 4 B 6 0 8 2 8 C -10 -8 0 12 2 D -18 -2 -12 0 -16 E -4 -8 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 18 4 B 6 0 8 2 8 C -10 -8 0 12 2 D -18 -2 -12 0 -16 E -4 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=28 E=19 D=12 C=12 so D is eliminated. Round 2 votes counts: B=29 A=29 E=25 C=17 so C is eliminated. Round 3 votes counts: A=43 B=31 E=26 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:212 E:201 C:198 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 18 4 B 6 0 8 2 8 C -10 -8 0 12 2 D -18 -2 -12 0 -16 E -4 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 18 4 B 6 0 8 2 8 C -10 -8 0 12 2 D -18 -2 -12 0 -16 E -4 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 18 4 B 6 0 8 2 8 C -10 -8 0 12 2 D -18 -2 -12 0 -16 E -4 -8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6570: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (16) D C A B E (10) C D A B E (9) E B A C D (8) C D A E B (8) D A C B E (4) C A D B E (4) B E A D C (4) D C E B A (3) C D E B A (3) A B E D C (3) E B C D A (2) E B C A D (2) D E B A C (2) D C A E B (2) D A B E C (2) B A E D C (2) A D B C E (2) E C B D A (1) E C B A D (1) E B D C A (1) D E B C A (1) D C B E A (1) D B E A C (1) C E B D A (1) C E A B D (1) C D E A B (1) A E B C D (1) A D C B E (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 -4 2 B -2 0 -2 -10 -8 C 2 2 0 -10 4 D 4 10 10 0 10 E -2 8 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -4 2 B -2 0 -2 -10 -8 C 2 2 0 -10 4 D 4 10 10 0 10 E -2 8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=27 D=26 A=10 B=6 so B is eliminated. Round 2 votes counts: E=35 C=27 D=26 A=12 so A is eliminated. Round 3 votes counts: E=42 D=29 C=29 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:217 A:199 C:199 E:196 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -4 2 B -2 0 -2 -10 -8 C 2 2 0 -10 4 D 4 10 10 0 10 E -2 8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 2 B -2 0 -2 -10 -8 C 2 2 0 -10 4 D 4 10 10 0 10 E -2 8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 2 B -2 0 -2 -10 -8 C 2 2 0 -10 4 D 4 10 10 0 10 E -2 8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998136 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6571: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) C A B D E (10) D B E A C (8) B D C E A (8) A C E D B (8) E A D B C (7) E D A B C (6) C A E B D (6) B D E C A (6) E D B A C (5) C B D A E (5) C B A D E (4) D E B A C (3) D B E C A (3) E A D C B (2) C A B E D (2) B C D E A (2) E A C D B (1) D E B C A (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 10 6 6 -4 B -10 0 -2 -12 -4 C -6 2 0 2 -6 D -6 12 -2 0 2 E 4 4 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 A B C D E A 0 10 6 6 -4 B -10 0 -2 -12 -4 C -6 2 0 2 -6 D -6 12 -2 0 2 E 4 4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=21 A=21 B=16 D=15 so D is eliminated. Round 2 votes counts: C=27 B=27 E=25 A=21 so A is eliminated. Round 3 votes counts: C=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:209 E:206 D:203 C:196 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 6 -4 B -10 0 -2 -12 -4 C -6 2 0 2 -6 D -6 12 -2 0 2 E 4 4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 6 -4 B -10 0 -2 -12 -4 C -6 2 0 2 -6 D -6 12 -2 0 2 E 4 4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 6 -4 B -10 0 -2 -12 -4 C -6 2 0 2 -6 D -6 12 -2 0 2 E 4 4 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6572: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) A D B C E (8) B C E A D (6) C E B D A (5) B A C E D (5) E C D A B (4) C B E D A (4) E D C A B (3) E C B A D (3) E C A B D (3) C B E A D (3) C B D E A (3) A D B E C (3) A B D C E (3) E D A C B (2) E C B D A (2) D E C A B (2) D E A C B (2) D A E C B (2) D A B E C (2) B C D A E (2) B A C D E (2) A B E D C (2) A B E C D (2) A B D E C (2) E C D B A (1) E B C A D (1) E B A C D (1) E A C D B (1) E A C B D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A E B (1) D B C A E (1) C E B A D (1) B E C A D (1) B D A C E (1) B A D C E (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 12 4 4 -4 B -12 0 10 12 20 C -4 -10 0 6 14 D -4 -12 -6 0 -6 E 4 -20 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380154 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 A B C D E A 0 12 4 4 -4 B -12 0 10 12 20 C -4 -10 0 6 14 D -4 -12 -6 0 -6 E 4 -20 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380156 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=22 D=21 B=18 C=16 so C is eliminated. Round 2 votes counts: E=29 B=28 A=22 D=21 so D is eliminated. Round 3 votes counts: A=36 E=35 B=29 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:215 A:208 C:203 E:188 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 4 -4 B -12 0 10 12 20 C -4 -10 0 6 14 D -4 -12 -6 0 -6 E 4 -20 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380156 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 4 -4 B -12 0 10 12 20 C -4 -10 0 6 14 D -4 -12 -6 0 -6 E 4 -20 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380156 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 4 -4 B -12 0 10 12 20 C -4 -10 0 6 14 D -4 -12 -6 0 -6 E 4 -20 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.181818 Sum of squares = 0.471074380156 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.818182 D: 0.818182 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6573: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) B A C D E (9) D E C B A (7) D E B A C (7) E D C A B (6) A B C E D (5) A B C D E (5) E D C B A (4) E C D A B (4) E C A B D (4) A C B E D (4) D E A C B (3) D B A E C (3) D B A C E (3) C E B A D (3) B D A C E (3) B A D C E (3) A B D C E (3) D E B C A (2) D A B E C (2) E C B A D (1) E C A D B (1) D E C A B (1) D E A B C (1) D B E A C (1) C E A B D (1) C B A E D (1) B C A E D (1) Total count = 100 A B C D E A 0 4 4 6 8 B -4 0 -4 10 10 C -4 4 0 2 6 D -6 -10 -2 0 6 E -8 -10 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 6 8 B -4 0 -4 10 10 C -4 4 0 2 6 D -6 -10 -2 0 6 E -8 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=20 C=17 A=17 B=16 so B is eliminated. Round 2 votes counts: D=33 A=29 E=20 C=18 so C is eliminated. Round 3 votes counts: A=43 D=33 E=24 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:206 C:204 D:194 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 6 8 B -4 0 -4 10 10 C -4 4 0 2 6 D -6 -10 -2 0 6 E -8 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 6 8 B -4 0 -4 10 10 C -4 4 0 2 6 D -6 -10 -2 0 6 E -8 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 6 8 B -4 0 -4 10 10 C -4 4 0 2 6 D -6 -10 -2 0 6 E -8 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6574: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (6) B C E D A (6) B C D E A (6) A E B D C (6) D A B C E (5) C D B E A (5) A E D C B (5) D B A C E (4) C E B D A (4) A E B C D (4) A D E B C (4) E B A C D (3) D C A E B (3) D A C B E (3) C B E D A (3) B E A C D (3) A D E C B (3) E A C D B (2) D C E A B (2) C E D B A (2) B D A C E (2) A D B E C (2) A B D E C (2) E C D A B (1) E C B A D (1) E C A D B (1) E C A B D (1) E B C A D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B C A E (1) D A C E B (1) C B D E A (1) B D C A E (1) B A C D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 6 -4 0 B 4 0 18 8 10 C -6 -18 0 6 4 D 4 -8 -6 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -4 0 B 4 0 18 8 10 C -6 -18 0 6 4 D 4 -8 -6 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=25 D=21 C=15 E=11 so E is eliminated. Round 2 votes counts: A=31 B=29 D=21 C=19 so C is eliminated. Round 3 votes counts: B=38 A=33 D=29 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:199 E:195 C:193 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 -4 0 B 4 0 18 8 10 C -6 -18 0 6 4 D 4 -8 -6 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 0 B 4 0 18 8 10 C -6 -18 0 6 4 D 4 -8 -6 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 0 B 4 0 18 8 10 C -6 -18 0 6 4 D 4 -8 -6 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996611 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6575: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (11) A D B E C (9) C E B D A (8) A D B C E (8) C E B A D (7) E C B D A (5) D B A E C (5) B E C D A (5) A C D E B (5) B E D C A (4) B D E A C (4) A D C E B (4) E B D C A (3) E B C D A (3) C E A D B (3) A D C B E (3) C A D E B (2) E D A B C (1) C E A B D (1) C B E A D (1) C A E D B (1) C A E B D (1) B E D A C (1) B D A E C (1) B C E D A (1) A D E C B (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 10 -4 6 B -4 0 14 -10 8 C -10 -14 0 -12 -8 D 4 10 12 0 10 E -6 -8 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999346 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 -4 6 B -4 0 14 -10 8 C -10 -14 0 -12 -8 D 4 10 12 0 10 E -6 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=24 D=16 B=16 E=12 so E is eliminated. Round 2 votes counts: A=32 C=29 B=22 D=17 so D is eliminated. Round 3 votes counts: A=44 C=29 B=27 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:218 A:208 B:204 E:192 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 10 -4 6 B -4 0 14 -10 8 C -10 -14 0 -12 -8 D 4 10 12 0 10 E -6 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -4 6 B -4 0 14 -10 8 C -10 -14 0 -12 -8 D 4 10 12 0 10 E -6 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -4 6 B -4 0 14 -10 8 C -10 -14 0 -12 -8 D 4 10 12 0 10 E -6 -8 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6576: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) E C B A D (8) B E C D A (7) B D A E C (6) A D C E B (6) D A B C E (5) C E D A B (5) C E B D A (5) E B C A D (4) D A C B E (4) C E A D B (4) E C A B D (3) D C B A E (2) D B A C E (2) C B E D A (2) C A D E B (2) B E A C D (2) B C D E A (2) B A E D C (2) A D E C B (2) E C B D A (1) E C A D B (1) E A C B D (1) E A B C D (1) D B C A E (1) D A C E B (1) D A B E C (1) C E D B A (1) C D A E B (1) C B D E A (1) B E D C A (1) B E C A D (1) B D E A C (1) B A D E C (1) A E D C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -4 2 -2 B 0 0 -6 -2 2 C 4 6 0 8 -8 D -2 2 -8 0 -2 E 2 -2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999944 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 0 -4 2 -2 B 0 0 -6 -2 2 C 4 6 0 8 -8 D -2 2 -8 0 -2 E 2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999906 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 C=21 A=21 E=19 D=16 so D is eliminated. Round 2 votes counts: A=32 B=26 C=23 E=19 so E is eliminated. Round 3 votes counts: C=36 A=34 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:205 E:205 A:198 B:197 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 2 -2 B 0 0 -6 -2 2 C 4 6 0 8 -8 D -2 2 -8 0 -2 E 2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999906 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 -2 B 0 0 -6 -2 2 C 4 6 0 8 -8 D -2 2 -8 0 -2 E 2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999906 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 -2 B 0 0 -6 -2 2 C 4 6 0 8 -8 D -2 2 -8 0 -2 E 2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.125000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999906 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6577: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (17) B D A C E (13) A C E B D (12) D B E C A (11) E C D A B (5) D B A E C (4) C E A B D (4) B A D C E (4) E D C B A (3) E C D B A (3) E C A B D (3) D E C B A (3) A C B E D (3) D E B C A (2) D B E A C (2) C A E B D (2) A B C D E (2) E A C D B (1) D B A C E (1) B D E C A (1) B D C E A (1) B D A E C (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -10 0 -12 B 0 0 -14 -6 -12 C 10 14 0 8 -12 D 0 6 -8 0 -8 E 12 12 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -10 0 -12 B 0 0 -14 -6 -12 C 10 14 0 8 -12 D 0 6 -8 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=23 B=21 A=18 C=6 so C is eliminated. Round 2 votes counts: E=36 D=23 B=21 A=20 so A is eliminated. Round 3 votes counts: E=51 B=26 D=23 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:210 D:195 A:189 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 0 -12 B 0 0 -14 -6 -12 C 10 14 0 8 -12 D 0 6 -8 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 0 -12 B 0 0 -14 -6 -12 C 10 14 0 8 -12 D 0 6 -8 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 0 -12 B 0 0 -14 -6 -12 C 10 14 0 8 -12 D 0 6 -8 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6578: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) D C E A B (7) D B C E A (7) B D A E C (6) D B A C E (5) C E D A B (5) A E C B D (5) E C A D B (4) D C E B A (4) B A E C D (4) A C E D B (4) E C A B D (3) B D E C A (3) E C D B A (2) E C B D A (2) E A C B D (2) D B E C A (2) D A C E B (2) C E A D B (2) B E C A D (2) B D A C E (2) B A D E C (2) B A D C E (2) A D B C E (2) E C B A D (1) D C B E A (1) D C A E B (1) D B C A E (1) D A B C E (1) C D E A B (1) B E C D A (1) B D E A C (1) B D C E A (1) A E C D B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 2 0 -10 -2 B -2 0 2 -8 6 C 0 -2 0 -4 0 D 10 8 4 0 6 E 2 -6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -10 -2 B -2 0 2 -8 6 C 0 -2 0 -4 0 D 10 8 4 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=24 A=23 E=14 C=8 so C is eliminated. Round 2 votes counts: D=32 B=24 A=23 E=21 so E is eliminated. Round 3 votes counts: D=39 A=34 B=27 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:199 C:197 A:195 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -10 -2 B -2 0 2 -8 6 C 0 -2 0 -4 0 D 10 8 4 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -10 -2 B -2 0 2 -8 6 C 0 -2 0 -4 0 D 10 8 4 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -10 -2 B -2 0 2 -8 6 C 0 -2 0 -4 0 D 10 8 4 0 6 E 2 -6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6579: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) D E A B C (6) E B A C D (4) D C B A E (4) D A E C B (4) D A C E B (4) B E D C A (4) B E C D A (4) B C E D A (4) B C E A D (4) A D E C B (4) A C D E B (4) E D B A C (3) E D A B C (3) E B D C A (3) C B D E A (3) C B A D E (3) E B D A C (2) E B A D C (2) E A B D C (2) C D B A E (2) C A B D E (2) B E C A D (2) B C D E A (2) A E D C B (2) A E C B D (2) A E B C D (2) A D C E B (2) E B C A D (1) E A D B C (1) D E B A C (1) D C B E A (1) D B C E A (1) C D A B E (1) C B D A E (1) B D E C A (1) B D C E A (1) A C D B E (1) Total count = 100 A B C D E A 0 -20 -2 -10 -10 B 20 0 6 12 -4 C 2 -6 0 -2 -6 D 10 -12 2 0 -4 E 10 4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -2 -10 -10 B 20 0 6 12 -4 C 2 -6 0 -2 -6 D 10 -12 2 0 -4 E 10 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 E=21 D=21 C=19 A=17 so A is eliminated. Round 2 votes counts: E=27 D=27 C=24 B=22 so B is eliminated. Round 3 votes counts: E=37 C=34 D=29 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:212 D:198 C:194 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -2 -10 -10 B 20 0 6 12 -4 C 2 -6 0 -2 -6 D 10 -12 2 0 -4 E 10 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -2 -10 -10 B 20 0 6 12 -4 C 2 -6 0 -2 -6 D 10 -12 2 0 -4 E 10 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -2 -10 -10 B 20 0 6 12 -4 C 2 -6 0 -2 -6 D 10 -12 2 0 -4 E 10 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6580: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) E A C B D (11) B D E A C (10) D B C A E (9) A C E D B (9) D B A E C (6) E C A B D (5) B D C E A (5) C E A B D (4) C D B A E (4) C A D B E (4) B D E C A (4) A E C D B (4) D B A C E (3) E A B D C (1) E A B C D (1) D B C E A (1) C E B D A (1) C D A B E (1) C A E B D (1) C A D E B (1) B E D C A (1) B E D A C (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -6 8 10 B -10 0 -16 -10 0 C 6 16 0 18 12 D -8 10 -18 0 -2 E -10 0 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 8 10 B -10 0 -16 -10 0 C 6 16 0 18 12 D -8 10 -18 0 -2 E -10 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=21 D=19 E=18 A=14 so A is eliminated. Round 2 votes counts: C=38 E=22 B=21 D=19 so D is eliminated. Round 3 votes counts: B=40 C=38 E=22 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:211 D:191 E:190 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 8 10 B -10 0 -16 -10 0 C 6 16 0 18 12 D -8 10 -18 0 -2 E -10 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 8 10 B -10 0 -16 -10 0 C 6 16 0 18 12 D -8 10 -18 0 -2 E -10 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 8 10 B -10 0 -16 -10 0 C 6 16 0 18 12 D -8 10 -18 0 -2 E -10 0 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6581: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (7) E A B D C (6) A E B C D (6) D C B E A (4) B E A C D (4) A E D B C (4) E D A C B (3) E A D B C (3) D C E A B (3) C D B A E (3) C B D A E (3) B E C A D (3) B A C E D (3) A C D B E (3) E B D C A (2) E B D A C (2) D E C B A (2) D C E B A (2) D C A E B (2) D C A B E (2) D A E C B (2) C D B E A (2) C D A B E (2) B E D C A (2) B C A D E (2) B A E C D (2) A D E C B (2) E D B C A (1) E D B A C (1) E A B C D (1) D E C A B (1) D E A C B (1) D C B A E (1) D B C E A (1) D A C E B (1) C B D E A (1) C A B D E (1) B E C D A (1) B C E D A (1) B C E A D (1) B C D A E (1) A E B D C (1) A D C E B (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -2 -8 -10 B 4 0 12 6 6 C 2 -12 0 0 -2 D 8 -6 0 0 2 E 10 -6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999126 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 -10 B 4 0 12 6 6 C 2 -12 0 0 -2 D 8 -6 0 0 2 E 10 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 A=20 E=19 C=12 so C is eliminated. Round 2 votes counts: B=31 D=29 A=21 E=19 so E is eliminated. Round 3 votes counts: B=35 D=34 A=31 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:202 E:202 C:194 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 -10 B 4 0 12 6 6 C 2 -12 0 0 -2 D 8 -6 0 0 2 E 10 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 -10 B 4 0 12 6 6 C 2 -12 0 0 -2 D 8 -6 0 0 2 E 10 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 -10 B 4 0 12 6 6 C 2 -12 0 0 -2 D 8 -6 0 0 2 E 10 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6582: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) B A D E C (10) C E D A B (9) E B A D C (8) D A B C E (8) C D E A B (8) B A E D C (5) E C D B A (4) E C B A D (4) E C B D A (3) E B C A D (3) D C A B E (3) E D A B C (2) E C D A B (2) E B A C D (2) E A B D C (2) D A C B E (2) B E A D C (2) B A D C E (2) A D B C E (2) E D C A B (1) C E B D A (1) C D A E B (1) C B A D E (1) B E A C D (1) B C A D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 -10 -4 B -6 0 0 -6 0 C 4 0 0 2 0 D 10 6 -2 0 2 E 4 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.120127 C: 0.723850 D: 0.000000 E: 0.156023 Sum of squares = 0.562732343205 Cumulative probabilities = A: 0.000000 B: 0.120127 C: 0.843977 D: 0.843977 E: 1.000000 A B C D E A 0 6 -4 -10 -4 B -6 0 0 -6 0 C 4 0 0 2 0 D 10 6 -2 0 2 E 4 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.000000 E: 0.333333 Sum of squares = 0.458333431252 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=31 C=31 B=21 D=13 A=4 so A is eliminated. Round 2 votes counts: E=31 C=31 B=23 D=15 so D is eliminated. Round 3 votes counts: C=36 B=33 E=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:208 C:203 E:201 A:194 B:194 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -10 -4 B -6 0 0 -6 0 C 4 0 0 2 0 D 10 6 -2 0 2 E 4 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.000000 E: 0.333333 Sum of squares = 0.458333431252 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -10 -4 B -6 0 0 -6 0 C 4 0 0 2 0 D 10 6 -2 0 2 E 4 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.000000 E: 0.333333 Sum of squares = 0.458333431252 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -10 -4 B -6 0 0 -6 0 C 4 0 0 2 0 D 10 6 -2 0 2 E 4 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.000000 E: 0.333333 Sum of squares = 0.458333431252 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6583: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (15) D A C E B (13) A C E B D (9) B E C D A (8) D B E C A (6) A D C E B (6) D C A E B (4) D A B E C (4) C E B A D (4) B E A C D (4) B D E C A (3) E C B A D (2) E B C A D (2) D C E A B (2) D A C B E (2) C E B D A (2) C E A B D (2) B A E C D (2) D B A E C (1) D A B C E (1) C E A D B (1) C D E B A (1) B E D C A (1) B E D A C (1) A D B E C (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -6 2 -8 B 4 0 0 12 0 C 6 0 0 8 -2 D -2 -12 -8 0 -8 E 8 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.320214 C: 0.000000 D: 0.000000 E: 0.679786 Sum of squares = 0.564645957138 Cumulative probabilities = A: 0.000000 B: 0.320214 C: 0.320214 D: 0.320214 E: 1.000000 A B C D E A 0 -4 -6 2 -8 B 4 0 0 12 0 C 6 0 0 8 -2 D -2 -12 -8 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999549 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=33 A=19 C=10 E=4 so E is eliminated. Round 2 votes counts: B=36 D=33 A=19 C=12 so C is eliminated. Round 3 votes counts: B=44 D=34 A=22 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:209 B:208 C:206 A:192 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 2 -8 B 4 0 0 12 0 C 6 0 0 8 -2 D -2 -12 -8 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999549 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 -8 B 4 0 0 12 0 C 6 0 0 8 -2 D -2 -12 -8 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999549 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 -8 B 4 0 0 12 0 C 6 0 0 8 -2 D -2 -12 -8 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999549 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6584: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) E C B D A (7) D A C B E (7) D A E B C (6) D E B A C (5) D A C E B (5) A D B E C (5) D C A E B (4) A C D B E (4) E B C D A (3) C B E A D (3) C A D E B (3) C A B E D (3) A D C B E (3) A D B C E (3) E B C A D (2) C A B D E (2) B E C A D (2) B E A D C (2) A B D C E (2) A B C E D (2) E C D B A (1) E B D C A (1) E B D A C (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E A B (1) D A B C E (1) C E D B A (1) C E B A D (1) C D E A B (1) B E D A C (1) B E A C D (1) B C E A D (1) B A D E C (1) B A C E D (1) A C B E D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 28 22 -12 26 B -28 0 2 -22 10 C -22 -2 0 -20 0 D 12 22 20 0 32 E -26 -10 0 -32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 22 -12 26 B -28 0 2 -22 10 C -22 -2 0 -20 0 D 12 22 20 0 32 E -26 -10 0 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=22 E=15 C=14 B=9 so B is eliminated. Round 2 votes counts: D=40 A=24 E=21 C=15 so C is eliminated. Round 3 votes counts: D=41 A=32 E=27 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:243 A:232 B:181 C:178 E:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 28 22 -12 26 B -28 0 2 -22 10 C -22 -2 0 -20 0 D 12 22 20 0 32 E -26 -10 0 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 22 -12 26 B -28 0 2 -22 10 C -22 -2 0 -20 0 D 12 22 20 0 32 E -26 -10 0 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 22 -12 26 B -28 0 2 -22 10 C -22 -2 0 -20 0 D 12 22 20 0 32 E -26 -10 0 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6585: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C A D B E (8) D E B A C (7) C A E B D (7) A B E D C (7) C A B E D (6) C E D B A (4) C D E B A (4) A B D E C (4) A B D C E (4) E D B C A (3) D E B C A (3) C E B D A (3) A D B E C (3) E C D B A (2) E B C D A (2) E B A D C (2) D B E A C (2) D B A E C (2) C E A B D (2) C A D E B (2) E D C B A (1) E D B A C (1) D E C B A (1) D C E B A (1) D A B C E (1) C E B A D (1) C D E A B (1) C D A E B (1) C D A B E (1) C A B D E (1) B E A D C (1) B A D E C (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 4 2 B -2 0 6 2 -12 C 8 -6 0 -6 0 D -4 -2 6 0 0 E -2 12 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.290323 B: 0.048387 C: 0.209677 D: 0.338710 E: 0.112903 Sum of squares = 0.258064516128 Cumulative probabilities = A: 0.290323 B: 0.338710 C: 0.548387 D: 0.887097 E: 1.000000 A B C D E A 0 2 -8 4 2 B -2 0 6 2 -12 C 8 -6 0 -6 0 D -4 -2 6 0 0 E -2 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.290323 B: 0.048387 C: 0.209677 D: 0.338710 E: 0.112903 Sum of squares = 0.258064516128 Cumulative probabilities = A: 0.290323 B: 0.338710 C: 0.548387 D: 0.887097 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=21 E=19 D=17 B=2 so B is eliminated. Round 2 votes counts: C=41 A=22 E=20 D=17 so D is eliminated. Round 3 votes counts: C=42 E=33 A=25 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:205 A:200 D:200 C:198 B:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 4 2 B -2 0 6 2 -12 C 8 -6 0 -6 0 D -4 -2 6 0 0 E -2 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.290323 B: 0.048387 C: 0.209677 D: 0.338710 E: 0.112903 Sum of squares = 0.258064516128 Cumulative probabilities = A: 0.290323 B: 0.338710 C: 0.548387 D: 0.887097 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 2 B -2 0 6 2 -12 C 8 -6 0 -6 0 D -4 -2 6 0 0 E -2 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.290323 B: 0.048387 C: 0.209677 D: 0.338710 E: 0.112903 Sum of squares = 0.258064516128 Cumulative probabilities = A: 0.290323 B: 0.338710 C: 0.548387 D: 0.887097 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 2 B -2 0 6 2 -12 C 8 -6 0 -6 0 D -4 -2 6 0 0 E -2 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.290323 B: 0.048387 C: 0.209677 D: 0.338710 E: 0.112903 Sum of squares = 0.258064516128 Cumulative probabilities = A: 0.290323 B: 0.338710 C: 0.548387 D: 0.887097 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6586: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) C B A E D (12) D C A B E (8) E B A D C (7) E A B D C (7) E B A C D (6) D E A B C (5) D A E B C (4) C D A B E (4) D A B E C (3) C E B A D (3) C B E A D (3) C B D A E (3) E D A B C (2) E B C A D (2) D E A C B (2) D C E A B (2) C D E A B (2) A E B D C (2) E C B A D (1) E A D B C (1) D C B A E (1) D A E C B (1) C E D B A (1) B E A C D (1) B C A E D (1) B A E D C (1) B A C E D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -10 0 10 B 12 0 -10 4 4 C 10 10 0 6 8 D 0 -4 -6 0 -2 E -10 -4 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 0 10 B 12 0 -10 4 4 C 10 10 0 6 8 D 0 -4 -6 0 -2 E -10 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=26 D=26 B=5 A=3 so A is eliminated. Round 2 votes counts: C=40 E=28 D=26 B=6 so B is eliminated. Round 3 votes counts: C=43 E=30 D=27 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:205 A:194 D:194 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -10 0 10 B 12 0 -10 4 4 C 10 10 0 6 8 D 0 -4 -6 0 -2 E -10 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 0 10 B 12 0 -10 4 4 C 10 10 0 6 8 D 0 -4 -6 0 -2 E -10 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 0 10 B 12 0 -10 4 4 C 10 10 0 6 8 D 0 -4 -6 0 -2 E -10 -4 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6587: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (10) E B D C A (9) C E B D A (9) E D B C A (7) E D B A C (7) A C D B E (7) A C B D E (5) E C D B A (4) E C B D A (4) C A E D B (4) A D B E C (4) D B E A C (3) C E D B A (3) B D E A C (3) A B D C E (3) E B D A C (2) D B A E C (2) C A D B E (2) B D A E C (2) A D B C E (2) D E B A C (1) D A B E C (1) C E D A B (1) C E B A D (1) C E A D B (1) C E A B D (1) C A E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -14 -16 -12 B 14 0 -6 2 -10 C 14 6 0 6 0 D 16 -2 -6 0 -8 E 12 10 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.519490 D: 0.000000 E: 0.480510 Sum of squares = 0.500759738872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.519490 D: 0.519490 E: 1.000000 A B C D E A 0 -14 -14 -16 -12 B 14 0 -6 2 -10 C 14 6 0 6 0 D 16 -2 -6 0 -8 E 12 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=33 C=33 A=22 D=7 B=5 so B is eliminated. Round 2 votes counts: E=33 C=33 A=22 D=12 so D is eliminated. Round 3 votes counts: E=40 C=33 A=27 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:213 B:200 D:200 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 -16 -12 B 14 0 -6 2 -10 C 14 6 0 6 0 D 16 -2 -6 0 -8 E 12 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -16 -12 B 14 0 -6 2 -10 C 14 6 0 6 0 D 16 -2 -6 0 -8 E 12 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -16 -12 B 14 0 -6 2 -10 C 14 6 0 6 0 D 16 -2 -6 0 -8 E 12 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6588: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) E D B A C (6) C B D A E (6) C B A D E (6) A E C D B (6) A C B E D (6) E D A B C (4) D E B A C (4) C A B E D (4) D E C B A (3) D E B C A (3) D B E C A (3) D B E A C (3) C A E B D (3) C A B D E (3) B C D A E (3) A C E B D (3) E A D C B (2) E A D B C (2) E A C D B (2) D C E B A (2) B C A D E (2) B A D E C (2) B A C D E (2) E D C A B (1) E D A C B (1) E C A D B (1) D C B E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D A E B (1) B D E A C (1) B D C E A (1) B D A E C (1) A E D B C (1) Total count = 100 A B C D E A 0 0 -8 8 14 B 0 0 -24 -14 -12 C 8 24 0 18 8 D -8 14 -18 0 -4 E -14 12 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 8 14 B 0 0 -24 -14 -12 C 8 24 0 18 8 D -8 14 -18 0 -4 E -14 12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=19 D=19 A=16 B=12 so B is eliminated. Round 2 votes counts: C=39 D=22 A=20 E=19 so E is eliminated. Round 3 votes counts: C=40 D=34 A=26 so A is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 A:207 E:197 D:192 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 8 14 B 0 0 -24 -14 -12 C 8 24 0 18 8 D -8 14 -18 0 -4 E -14 12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 8 14 B 0 0 -24 -14 -12 C 8 24 0 18 8 D -8 14 -18 0 -4 E -14 12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 8 14 B 0 0 -24 -14 -12 C 8 24 0 18 8 D -8 14 -18 0 -4 E -14 12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6589: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) D B E A C (9) C E B D A (9) A D B C E (8) E C B D A (7) A C D B E (6) D E B C A (5) C E A B D (5) A D B E C (5) A C E B D (5) E D B C A (3) C E B A D (3) B E D C A (3) A C B E D (3) E B C D A (2) D A E B C (2) D A B E C (2) B D A E C (2) A D C E B (2) A C B D E (2) E C D A B (1) E B D C A (1) C B E A D (1) B D E C A (1) A C E D B (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -2 -14 -20 B 12 0 8 -10 6 C 2 -8 0 -8 -6 D 14 10 8 0 12 E 20 -6 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -14 -20 B 12 0 8 -10 6 C 2 -8 0 -8 -6 D 14 10 8 0 12 E 20 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=28 C=18 E=14 B=6 so B is eliminated. Round 2 votes counts: A=34 D=31 C=18 E=17 so E is eliminated. Round 3 votes counts: D=38 A=34 C=28 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:208 E:204 C:190 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -14 -20 B 12 0 8 -10 6 C 2 -8 0 -8 -6 D 14 10 8 0 12 E 20 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -14 -20 B 12 0 8 -10 6 C 2 -8 0 -8 -6 D 14 10 8 0 12 E 20 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -14 -20 B 12 0 8 -10 6 C 2 -8 0 -8 -6 D 14 10 8 0 12 E 20 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6590: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) E D A B C (5) E C B D A (5) E A C D B (5) D B A E C (5) C E B A D (4) C B E D A (4) C B D A E (4) C A B D E (4) B D C E A (4) E B D C A (3) D A B C E (3) B D C A E (3) A D B E C (3) A D B C E (3) E D B C A (2) E C A D B (2) E C A B D (2) D B E A C (2) D A B E C (2) C E A B D (2) C A B E D (2) B D E C A (2) A D C B E (2) A C E D B (2) E D B A C (1) E A D C B (1) D B A C E (1) D A E B C (1) C E B D A (1) C B E A D (1) C B D E A (1) C B A D E (1) C A E D B (1) B E D C A (1) B D A C E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E B C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -2 -6 -10 B -4 0 4 -6 4 C 2 -4 0 -8 -4 D 6 6 8 0 -6 E 10 -4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999982 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 A B C D E A 0 4 -2 -6 -10 B -4 0 4 -6 4 C 2 -4 0 -8 -4 D 6 6 8 0 -6 E 10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=25 A=16 D=14 B=13 so B is eliminated. Round 2 votes counts: E=33 C=27 D=24 A=16 so A is eliminated. Round 3 votes counts: E=36 D=33 C=31 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:207 B:199 A:193 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 -6 -10 B -4 0 4 -6 4 C 2 -4 0 -8 -4 D 6 6 8 0 -6 E 10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -6 -10 B -4 0 4 -6 4 C 2 -4 0 -8 -4 D 6 6 8 0 -6 E 10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -6 -10 B -4 0 4 -6 4 C 2 -4 0 -8 -4 D 6 6 8 0 -6 E 10 -4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6591: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) C D B A E (6) A C E D B (5) E D C A B (4) C D A B E (4) B D E C A (4) A E C D B (4) E C D A B (3) D B C E A (3) C A D B E (3) C A B D E (3) B E A D C (3) B D C E A (3) B D C A E (3) A C B D E (3) E B A D C (2) E A B D C (2) D E B C A (2) D C B E A (2) C D E A B (2) C A D E B (2) B E D A C (2) B C A D E (2) B A E D C (2) B A D E C (2) A E B C D (2) E D B A C (1) E D A C B (1) E B D C A (1) E B D A C (1) E A C D B (1) D C E B A (1) D B E C A (1) C E A D B (1) C D E B A (1) C D A E B (1) B E D C A (1) B D E A C (1) B D A C E (1) B C D A E (1) A E B D C (1) A C D E B (1) A C D B E (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -22 -14 0 B 6 0 0 -14 8 C 22 0 0 -4 2 D 14 14 4 0 6 E 0 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -22 -14 0 B 6 0 0 -14 8 C 22 0 0 -4 2 D 14 14 4 0 6 E 0 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 C=23 A=20 D=9 so D is eliminated. Round 2 votes counts: B=29 C=26 E=25 A=20 so A is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:210 B:200 E:192 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -22 -14 0 B 6 0 0 -14 8 C 22 0 0 -4 2 D 14 14 4 0 6 E 0 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -22 -14 0 B 6 0 0 -14 8 C 22 0 0 -4 2 D 14 14 4 0 6 E 0 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -22 -14 0 B 6 0 0 -14 8 C 22 0 0 -4 2 D 14 14 4 0 6 E 0 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999556 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6592: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (15) E B C D A (11) B C E D A (9) E A D C B (8) A D E C B (7) B C D E A (6) A D C B E (6) E C B D A (3) B A C D E (3) E B A D C (2) E B A C D (2) C D B A E (2) C B D A E (2) B E C D A (2) B E C A D (2) B A E D C (2) B A D C E (2) A E D C B (2) A D C E B (2) A B D C E (2) E D C A B (1) E C D B A (1) E C D A B (1) D C A E B (1) D A B C E (1) C E D B A (1) C B E D A (1) A E D B C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -32 -16 -14 0 B 32 0 24 30 14 C 16 -24 0 22 8 D 14 -30 -22 0 0 E 0 -14 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999486 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 -16 -14 0 B 32 0 24 30 14 C 16 -24 0 22 8 D 14 -30 -22 0 0 E 0 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=29 A=22 C=6 D=2 so D is eliminated. Round 2 votes counts: B=41 E=29 A=23 C=7 so C is eliminated. Round 3 votes counts: B=46 E=30 A=24 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:250 C:211 E:189 D:181 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -32 -16 -14 0 B 32 0 24 30 14 C 16 -24 0 22 8 D 14 -30 -22 0 0 E 0 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 -16 -14 0 B 32 0 24 30 14 C 16 -24 0 22 8 D 14 -30 -22 0 0 E 0 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 -16 -14 0 B 32 0 24 30 14 C 16 -24 0 22 8 D 14 -30 -22 0 0 E 0 -14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6593: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (13) A D C B E (12) D A C B E (8) C B E D A (6) E B C A D (5) D C A B E (5) E B D C A (4) C D B A E (4) B C E D A (4) A E D B C (4) A D E B C (3) A D C E B (3) D A C E B (2) C D B E A (2) C B E A D (2) B C E A D (2) A E B D C (2) A E B C D (2) A C D B E (2) E A B C D (1) D E C B A (1) D E B C A (1) D E A B C (1) D C B E A (1) D C B A E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C D A (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -16 -10 10 B 0 0 -14 -2 14 C 16 14 0 4 22 D 10 2 -4 0 0 E -10 -14 -22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -10 10 B 0 0 -14 -2 14 C 16 14 0 4 22 D 10 2 -4 0 0 E -10 -14 -22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 D=20 C=20 B=7 so B is eliminated. Round 2 votes counts: A=30 C=26 E=24 D=20 so D is eliminated. Round 3 votes counts: A=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:228 D:204 B:199 A:192 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -10 10 B 0 0 -14 -2 14 C 16 14 0 4 22 D 10 2 -4 0 0 E -10 -14 -22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -10 10 B 0 0 -14 -2 14 C 16 14 0 4 22 D 10 2 -4 0 0 E -10 -14 -22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -10 10 B 0 0 -14 -2 14 C 16 14 0 4 22 D 10 2 -4 0 0 E -10 -14 -22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6594: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) C B E A D (5) E D C A B (4) E C D B A (4) C E B A D (4) B A D C E (4) B A C D E (4) A D B C E (4) A C B D E (4) E D A C B (3) D E A B C (3) D B A E C (3) D A E B C (3) C E B D A (3) A C D E B (3) E D B C A (2) D E B A C (2) D E A C B (2) C E A D B (2) C B A E D (2) C A E B D (2) C A B E D (2) B D E A C (2) B C A D E (2) A D B E C (2) A B D C E (2) E C D A B (1) E A D C B (1) D B E A C (1) D A E C B (1) D A B E C (1) C E D A B (1) B E D C A (1) B E C D A (1) B D E C A (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B C A E D (1) A D E C B (1) A D E B C (1) A D C B E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 0 -6 B 8 0 -8 -8 -2 C -2 8 0 -6 6 D 0 8 6 0 2 E 6 2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.170987 B: 0.000000 C: 0.000000 D: 0.829013 E: 0.000000 Sum of squares = 0.71649862523 Cumulative probabilities = A: 0.170987 B: 0.170987 C: 0.170987 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 0 -6 B 8 0 -8 -8 -2 C -2 8 0 -6 6 D 0 8 6 0 2 E 6 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000002792 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=22 C=21 B=21 A=20 D=16 so D is eliminated. Round 2 votes counts: E=29 B=25 A=25 C=21 so C is eliminated. Round 3 votes counts: E=39 B=32 A=29 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:208 C:203 E:200 B:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 0 -6 B 8 0 -8 -8 -2 C -2 8 0 -6 6 D 0 8 6 0 2 E 6 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000002792 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 0 -6 B 8 0 -8 -8 -2 C -2 8 0 -6 6 D 0 8 6 0 2 E 6 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000002792 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 0 -6 B 8 0 -8 -8 -2 C -2 8 0 -6 6 D 0 8 6 0 2 E 6 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000002792 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6595: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) D B A E C (6) C E A B D (6) C E B A D (5) B A D C E (5) E C A D B (4) D E C B A (4) C E B D A (4) B A C E D (4) A B D E C (4) A B D C E (4) E C A B D (3) D E C A B (3) D B C E A (3) D A E B C (3) D A B E C (3) B A C D E (3) A B C E D (3) E C D B A (2) E C D A B (2) D E B C A (2) D B A C E (2) C B E A D (2) B C A E D (2) A D B E C (2) A B E C D (2) E D C A B (1) E A C D B (1) D E A C B (1) C E D B A (1) C E A D B (1) C B A E D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 8 12 10 B 6 0 10 14 6 C -8 -10 0 -2 8 D -12 -14 2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 12 10 B 6 0 10 14 6 C -8 -10 0 -2 8 D -12 -14 2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=21 C=20 A=19 E=13 so E is eliminated. Round 2 votes counts: C=31 D=28 B=21 A=20 so A is eliminated. Round 3 votes counts: C=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:212 C:194 D:190 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 12 10 B 6 0 10 14 6 C -8 -10 0 -2 8 D -12 -14 2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 12 10 B 6 0 10 14 6 C -8 -10 0 -2 8 D -12 -14 2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 12 10 B 6 0 10 14 6 C -8 -10 0 -2 8 D -12 -14 2 0 4 E -10 -6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999561 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6596: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) B D C E A (10) E A C B D (9) D B A E C (7) A E D C B (7) D B C A E (6) B C D E A (4) D A E B C (3) C B E A D (3) A D E B C (3) E C A B D (2) E A B C D (2) C E B A D (2) B D E C A (2) A E D B C (2) A C E D B (2) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D C B A E (1) D C A B E (1) D B E A C (1) D B A C E (1) D A C E B (1) D A B E C (1) D A B C E (1) C E A B D (1) C B E D A (1) C B D E A (1) C B D A E (1) C B A E D (1) C A E B D (1) C A D E B (1) B E D C A (1) B E C A D (1) B D E A C (1) B C E D A (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 18 10 6 B -8 0 0 -8 -8 C -18 0 0 -10 -18 D -10 8 10 0 -4 E -6 8 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 10 6 B -8 0 0 -8 -8 C -18 0 0 -10 -18 D -10 8 10 0 -4 E -6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 B=20 E=18 C=12 so C is eliminated. Round 2 votes counts: A=29 B=27 D=23 E=21 so E is eliminated. Round 3 votes counts: A=47 B=30 D=23 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:212 D:202 B:188 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 10 6 B -8 0 0 -8 -8 C -18 0 0 -10 -18 D -10 8 10 0 -4 E -6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 10 6 B -8 0 0 -8 -8 C -18 0 0 -10 -18 D -10 8 10 0 -4 E -6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 10 6 B -8 0 0 -8 -8 C -18 0 0 -10 -18 D -10 8 10 0 -4 E -6 8 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6597: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) C E A D B (7) A B D C E (7) D B A E C (6) B D E C A (5) A C E D B (5) C E B D A (4) C E A B D (4) C A E D B (4) B A D E C (4) A C E B D (4) E C D B A (3) D B E C A (3) E D C A B (2) D E C A B (2) D B E A C (2) D A B E C (2) C E B A D (2) B A D C E (2) A D C E B (2) A D B E C (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E C D A B (1) E C B D A (1) D E C B A (1) C E D A B (1) B D E A C (1) B D A C E (1) B C E D A (1) B C E A D (1) B A C E D (1) A E D C B (1) A E C D B (1) A D B C E (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 12 8 14 B -4 0 4 2 4 C -12 -4 0 -8 6 D -8 -2 8 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 8 14 B -4 0 4 2 4 C -12 -4 0 -8 6 D -8 -2 8 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 C=22 D=16 E=9 so E is eliminated. Round 2 votes counts: A=29 C=27 B=24 D=20 so D is eliminated. Round 3 votes counts: B=36 C=33 A=31 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:203 D:201 C:191 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 8 14 B -4 0 4 2 4 C -12 -4 0 -8 6 D -8 -2 8 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 8 14 B -4 0 4 2 4 C -12 -4 0 -8 6 D -8 -2 8 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 8 14 B -4 0 4 2 4 C -12 -4 0 -8 6 D -8 -2 8 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6598: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) E A C D B (7) E A D C B (6) D A E C B (6) C A E D B (6) B D E A C (5) B C E A D (5) B C D A E (5) B E D A C (4) B E A D C (4) A E D C B (4) C B D A E (3) B E A C D (3) E A D B C (2) E A C B D (2) E A B C D (2) D C A E B (2) C E A B D (2) C D B A E (2) C B E A D (2) B C E D A (2) B C D E A (2) A E C D B (2) E C A B D (1) E B A D C (1) E B A C D (1) D E A C B (1) D E A B C (1) D B C A E (1) D A E B C (1) D A B E C (1) C D A E B (1) C D A B E (1) C B A E D (1) C A E B D (1) B D A C E (1) A C E D B (1) Total count = 100 A B C D E A 0 0 10 6 -6 B 0 0 -2 10 0 C -10 2 0 4 -8 D -6 -10 -4 0 -18 E 6 0 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.255508 C: 0.000000 D: 0.000000 E: 0.744492 Sum of squares = 0.619552896211 Cumulative probabilities = A: 0.000000 B: 0.255508 C: 0.255508 D: 0.255508 E: 1.000000 A B C D E A 0 0 10 6 -6 B 0 0 -2 10 0 C -10 2 0 4 -8 D -6 -10 -4 0 -18 E 6 0 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=22 C=19 D=13 A=7 so A is eliminated. Round 2 votes counts: B=39 E=28 C=20 D=13 so D is eliminated. Round 3 votes counts: B=41 E=37 C=22 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:216 A:205 B:204 C:194 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 6 -6 B 0 0 -2 10 0 C -10 2 0 4 -8 D -6 -10 -4 0 -18 E 6 0 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 6 -6 B 0 0 -2 10 0 C -10 2 0 4 -8 D -6 -10 -4 0 -18 E 6 0 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 6 -6 B 0 0 -2 10 0 C -10 2 0 4 -8 D -6 -10 -4 0 -18 E 6 0 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6599: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) E D C A B (9) D E C B A (8) B A D E C (7) E D C B A (6) A B C E D (6) A B C D E (6) A B D E C (5) D E B C A (4) D E A B C (4) D E B A C (3) E D A C B (2) C E D A B (2) C D E B A (2) B C D E A (2) B A C E D (2) A D B E C (2) A C B E D (2) E C D B A (1) D E C A B (1) D E A C B (1) C E B D A (1) C E B A D (1) C B E D A (1) C B E A D (1) C B D E A (1) C B A E D (1) C A E D B (1) C A B E D (1) B D E C A (1) B C E D A (1) B C A E D (1) B A C D E (1) A E D C B (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -12 -20 -24 B 10 0 -6 -16 -16 C 12 6 0 -12 -12 D 20 16 12 0 0 E 24 16 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.445135 E: 0.554865 Sum of squares = 0.506020306203 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.445135 E: 1.000000 A B C D E A 0 -10 -12 -20 -24 B 10 0 -6 -16 -16 C 12 6 0 -12 -12 D 20 16 12 0 0 E 24 16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 D=21 E=18 B=15 so B is eliminated. Round 2 votes counts: A=34 C=26 D=22 E=18 so E is eliminated. Round 3 votes counts: D=39 A=34 C=27 so C is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:226 D:224 C:197 B:186 A:167 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -12 -20 -24 B 10 0 -6 -16 -16 C 12 6 0 -12 -12 D 20 16 12 0 0 E 24 16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -20 -24 B 10 0 -6 -16 -16 C 12 6 0 -12 -12 D 20 16 12 0 0 E 24 16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -20 -24 B 10 0 -6 -16 -16 C 12 6 0 -12 -12 D 20 16 12 0 0 E 24 16 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6600: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) A C B D E (5) A B D C E (5) E C D B A (4) E B D A C (4) D B E A C (4) C E D B A (4) C A E B D (4) C A D B E (4) A B D E C (4) E D B C A (3) D B A C E (3) C D B A E (3) C A E D B (3) C A B D E (3) B A D E C (3) E D B A C (2) E C B D A (2) E C B A D (2) E C A B D (2) E A B D C (2) D E B A C (2) C E A B D (2) C D E B A (2) C A B E D (2) B D A E C (2) A B E D C (2) E D C B A (1) E A C B D (1) D B C A E (1) C E D A B (1) C E A D B (1) C D A B E (1) B D E A C (1) B A E D C (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 10 -4 20 B 12 0 6 -4 14 C -10 -6 0 -8 -10 D 4 4 8 0 12 E -20 -14 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -4 20 B 12 0 6 -4 14 C -10 -6 0 -8 -10 D 4 4 8 0 12 E -20 -14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=23 D=22 A=18 B=7 so B is eliminated. Round 2 votes counts: C=30 D=25 E=23 A=22 so A is eliminated. Round 3 votes counts: D=38 C=35 E=27 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:214 D:214 A:207 C:183 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 10 -4 20 B 12 0 6 -4 14 C -10 -6 0 -8 -10 D 4 4 8 0 12 E -20 -14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -4 20 B 12 0 6 -4 14 C -10 -6 0 -8 -10 D 4 4 8 0 12 E -20 -14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -4 20 B 12 0 6 -4 14 C -10 -6 0 -8 -10 D 4 4 8 0 12 E -20 -14 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6601: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) A B C D E (8) B C A E D (7) B A C D E (7) A B D C E (7) E D C B A (6) A B D E C (6) D E C A B (5) B A C E D (5) D E A C B (4) C E D B A (4) C B A E D (4) D A E B C (3) E C D B A (2) C D E A B (2) B C E A D (2) B A D E C (2) B A D C E (2) A D E B C (2) E C D A B (1) D C E A B (1) D A E C B (1) C E D A B (1) C E B D A (1) C D A E B (1) C B E D A (1) C B E A D (1) C B A D E (1) B E A D C (1) B C A D E (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 0 16 18 B -4 0 10 14 14 C 0 -10 0 -2 12 D -16 -14 2 0 8 E -18 -14 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.841750 B: 0.000000 C: 0.158250 D: 0.000000 E: 0.000000 Sum of squares = 0.733585595374 Cumulative probabilities = A: 0.841750 B: 0.841750 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 16 18 B -4 0 10 14 14 C 0 -10 0 -2 12 D -16 -14 2 0 8 E -18 -14 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836794388 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=24 E=18 C=16 D=14 so D is eliminated. Round 2 votes counts: B=28 A=28 E=27 C=17 so C is eliminated. Round 3 votes counts: E=36 B=35 A=29 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:217 C:200 D:190 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 16 18 B -4 0 10 14 14 C 0 -10 0 -2 12 D -16 -14 2 0 8 E -18 -14 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836794388 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 16 18 B -4 0 10 14 14 C 0 -10 0 -2 12 D -16 -14 2 0 8 E -18 -14 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836794388 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 16 18 B -4 0 10 14 14 C 0 -10 0 -2 12 D -16 -14 2 0 8 E -18 -14 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836794388 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6602: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) C A D E B (6) D E B C A (5) E D B C A (4) D C E B A (4) B E A C D (4) A C D B E (4) A B C E D (4) E B C D A (3) D A C B E (3) C E B D A (3) B E A D C (3) B A E D C (3) B A E C D (3) A D C B E (3) A C B D E (3) A B E C D (3) E C D B A (2) E B C A D (2) D C E A B (2) D B A E C (2) C E D B A (2) C E B A D (2) C D E A B (2) C D A E B (2) C A E B D (2) B E D A C (2) E C B A D (1) E B A C D (1) D C A E B (1) D C A B E (1) D A B E C (1) C E A B D (1) C D E B A (1) C A E D B (1) B D E A C (1) B A D E C (1) A D B E C (1) A C D E B (1) A C B E D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -8 4 -4 B 12 0 4 4 -10 C 8 -4 0 8 0 D -4 -4 -8 0 -10 E 4 10 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.438367 D: 0.000000 E: 0.561633 Sum of squares = 0.507597300475 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.438367 D: 0.438367 E: 1.000000 A B C D E A 0 -12 -8 4 -4 B 12 0 4 4 -10 C 8 -4 0 8 0 D -4 -4 -8 0 -10 E 4 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=22 A=22 E=20 D=19 B=17 so B is eliminated. Round 2 votes counts: E=29 A=29 C=22 D=20 so D is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:206 B:205 A:190 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 4 -4 B 12 0 4 4 -10 C 8 -4 0 8 0 D -4 -4 -8 0 -10 E 4 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 4 -4 B 12 0 4 4 -10 C 8 -4 0 8 0 D -4 -4 -8 0 -10 E 4 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 4 -4 B 12 0 4 4 -10 C 8 -4 0 8 0 D -4 -4 -8 0 -10 E 4 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6603: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (14) B D A E C (8) A E C B D (8) A B E C D (7) D C E B A (5) C E A D B (5) B A E C D (5) E A C B D (4) D C E A B (4) D C B E A (4) C E D A B (4) D B C A E (3) B D E A C (3) C D E A B (2) B E A C D (2) B D E C A (2) B D A C E (2) B A D E C (2) A E B C D (2) A C E B D (2) E C A D B (1) E C A B D (1) E B C A D (1) E B A C D (1) D B E C A (1) D A C B E (1) C A E D B (1) B E A D C (1) B A E D C (1) A C E D B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 4 -6 -10 B 10 0 14 8 16 C -4 -14 0 -4 0 D 6 -8 4 0 4 E 10 -16 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -6 -10 B 10 0 14 8 16 C -4 -14 0 -4 0 D 6 -8 4 0 4 E 10 -16 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=26 A=22 C=12 E=8 so E is eliminated. Round 2 votes counts: D=32 B=28 A=26 C=14 so C is eliminated. Round 3 votes counts: D=38 A=34 B=28 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:224 D:203 E:195 A:189 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 -6 -10 B 10 0 14 8 16 C -4 -14 0 -4 0 D 6 -8 4 0 4 E 10 -16 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -6 -10 B 10 0 14 8 16 C -4 -14 0 -4 0 D 6 -8 4 0 4 E 10 -16 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -6 -10 B 10 0 14 8 16 C -4 -14 0 -4 0 D 6 -8 4 0 4 E 10 -16 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6604: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) C D A E B (10) D C A B E (8) E B C A D (6) E B A C D (6) D C B A E (6) C D E A B (5) A E B D C (5) B E A C D (4) A D B E C (4) C E B D A (3) C E A D B (3) B D A E C (3) E A C B D (2) D C A E B (2) C D E B A (2) C D B A E (2) C A D E B (2) B D C E A (2) E A B D C (1) D B A C E (1) D A C E B (1) D A B E C (1) C E D A B (1) C D B E A (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E A C (1) B C E D A (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -12 -4 -6 B 6 0 0 -2 -2 C 12 0 0 -2 2 D 4 2 2 0 6 E 6 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -4 -6 B 6 0 0 -2 -2 C 12 0 0 -2 2 D 4 2 2 0 6 E 6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=26 D=19 E=15 A=11 so A is eliminated. Round 2 votes counts: C=29 B=26 D=25 E=20 so E is eliminated. Round 3 votes counts: B=44 C=31 D=25 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:207 C:206 B:201 E:200 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -12 -4 -6 B 6 0 0 -2 -2 C 12 0 0 -2 2 D 4 2 2 0 6 E 6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -4 -6 B 6 0 0 -2 -2 C 12 0 0 -2 2 D 4 2 2 0 6 E 6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -4 -6 B 6 0 0 -2 -2 C 12 0 0 -2 2 D 4 2 2 0 6 E 6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6605: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (5) E D A C B (4) E B D C A (4) E D C B A (3) E D A B C (3) E C D B A (3) E A D B C (3) D E A C B (3) D C A E B (3) C B E A D (3) C B A D E (3) C A D B E (3) B E C A D (3) B A C E D (3) A D E B C (3) A D B E C (3) E D B A C (2) E C B D A (2) E B A D C (2) D A C E B (2) C D E B A (2) C D A E B (2) C D A B E (2) C B E D A (2) B E A D C (2) B C E A D (2) B A E D C (2) B A E C D (2) A D B C E (2) A B D C E (2) E D B C A (1) E B D A C (1) E B C A D (1) D E C A B (1) D E A B C (1) D A E C B (1) C E D B A (1) C D E A B (1) C D B A E (1) C B D E A (1) C B D A E (1) B C A E D (1) B C A D E (1) A E D B C (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 6 6 -12 B 8 0 4 -8 -2 C -6 -4 0 -4 -14 D -6 8 4 0 -14 E 12 2 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 6 6 -12 B 8 0 4 -8 -2 C -6 -4 0 -4 -14 D -6 8 4 0 -14 E 12 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=22 B=21 A=17 D=11 so D is eliminated. Round 2 votes counts: E=34 C=25 B=21 A=20 so A is eliminated. Round 3 votes counts: E=39 C=31 B=30 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:201 A:196 D:196 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 6 -12 B 8 0 4 -8 -2 C -6 -4 0 -4 -14 D -6 8 4 0 -14 E 12 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 6 -12 B 8 0 4 -8 -2 C -6 -4 0 -4 -14 D -6 8 4 0 -14 E 12 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 6 -12 B 8 0 4 -8 -2 C -6 -4 0 -4 -14 D -6 8 4 0 -14 E 12 2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996426 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6606: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) E A D C B (6) D E A C B (6) A E D C B (6) A E C B D (6) D C B E A (4) D B C E A (4) C B D E A (4) B C D A E (4) A E B C D (4) D B C A E (3) C B E A D (3) B D C A E (3) B C D E A (3) B A D C E (3) A E D B C (3) D E A B C (2) D A E B C (2) C B A E D (2) B C A D E (2) E D C A B (1) E C A B D (1) E A C D B (1) E A C B D (1) D E C B A (1) D E C A B (1) D E B A C (1) D C E B A (1) D B A E C (1) C E B D A (1) C E B A D (1) C D B E A (1) B D C E A (1) B A C E D (1) B A C D E (1) A E C D B (1) A E B D C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 -4 12 12 B 12 0 2 10 6 C 4 -2 0 -2 8 D -12 -10 2 0 -4 E -12 -6 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 12 12 B 12 0 2 10 6 C 4 -2 0 -2 8 D -12 -10 2 0 -4 E -12 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996633 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 A=23 C=12 E=10 so E is eliminated. Round 2 votes counts: A=31 B=29 D=27 C=13 so C is eliminated. Round 3 votes counts: B=40 A=32 D=28 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:204 C:204 E:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 12 12 B 12 0 2 10 6 C 4 -2 0 -2 8 D -12 -10 2 0 -4 E -12 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996633 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 12 12 B 12 0 2 10 6 C 4 -2 0 -2 8 D -12 -10 2 0 -4 E -12 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996633 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 12 12 B 12 0 2 10 6 C 4 -2 0 -2 8 D -12 -10 2 0 -4 E -12 -6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996633 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6607: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) C D A E B (7) E B C A D (6) B E A D C (6) E A D B C (5) C D A B E (5) E C A D B (4) B C D A E (4) A D C E B (4) E C B D A (3) E B C D A (3) E A D C B (3) B E C D A (3) A D B C E (3) E C D A B (2) D C A B E (2) D A B C E (2) C B E D A (2) C B D A E (2) B D C A E (2) B A D E C (2) B A D C E (2) A D E B C (2) A B D E C (2) E C B A D (1) E A B D C (1) D C B A E (1) D A C B E (1) C E D B A (1) C D E A B (1) C A D E B (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D C B (1) A E D B C (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -4 14 -6 B 2 0 16 4 -12 C 4 -16 0 -6 -14 D -14 -4 6 0 -8 E 6 12 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 14 -6 B 2 0 16 4 -12 C 4 -16 0 -6 -14 D -14 -4 6 0 -8 E 6 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=22 C=19 A=15 D=6 so D is eliminated. Round 2 votes counts: E=38 C=22 B=22 A=18 so A is eliminated. Round 3 votes counts: E=42 B=30 C=28 so C is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:205 A:201 D:190 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 14 -6 B 2 0 16 4 -12 C 4 -16 0 -6 -14 D -14 -4 6 0 -8 E 6 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 14 -6 B 2 0 16 4 -12 C 4 -16 0 -6 -14 D -14 -4 6 0 -8 E 6 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 14 -6 B 2 0 16 4 -12 C 4 -16 0 -6 -14 D -14 -4 6 0 -8 E 6 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6608: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (12) D A B E C (6) B E C D A (6) A D C E B (6) E B C A D (4) C E B A D (4) A D E B C (4) E C A B D (3) D B A E C (3) D A B C E (3) C E A D B (3) B E D A C (3) B C E D A (3) A D E C B (3) E C B A D (2) C D A B E (2) C B E D A (2) B D C E A (2) B D C A E (2) A E C D B (2) A C E D B (2) A C D E B (2) E B C D A (1) E B A D C (1) E B A C D (1) E A C D B (1) E A B D C (1) D C A B E (1) D B C A E (1) C E A B D (1) C D B A E (1) C B D E A (1) C B D A E (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) A E D B C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 14 12 -8 14 B -14 0 -4 -14 12 C -12 4 0 -12 6 D 8 14 12 0 12 E -14 -12 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 12 -8 14 B -14 0 -4 -14 12 C -12 4 0 -12 6 D 8 14 12 0 12 E -14 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=22 B=21 C=17 E=14 so E is eliminated. Round 2 votes counts: B=28 D=26 A=24 C=22 so C is eliminated. Round 3 votes counts: B=38 A=33 D=29 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:223 A:216 C:193 B:190 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 12 -8 14 B -14 0 -4 -14 12 C -12 4 0 -12 6 D 8 14 12 0 12 E -14 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 -8 14 B -14 0 -4 -14 12 C -12 4 0 -12 6 D 8 14 12 0 12 E -14 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 -8 14 B -14 0 -4 -14 12 C -12 4 0 -12 6 D 8 14 12 0 12 E -14 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6609: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) D C A B E (7) E B A C D (6) B E C A D (6) D A C E B (5) A C B E D (5) E B C D A (4) E B A D C (3) E A B C D (3) D A C B E (3) C A D B E (3) A D C E B (3) A C D E B (3) A C D B E (3) E D B C A (2) E B D A C (2) E B C A D (2) D E B C A (2) D C A E B (2) D A E C B (2) C D A B E (2) C B A D E (2) A E C D B (2) A E C B D (2) A E B C D (2) A C B D E (2) E D B A C (1) E D A B C (1) E B D C A (1) E A D C B (1) E A C B D (1) D B C E A (1) C B D A E (1) B E C D A (1) B E A C D (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C E D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 8 8 20 B -8 0 -16 -2 -4 C -8 16 0 10 8 D -8 2 -10 0 2 E -20 4 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 20 B -8 0 -16 -2 -4 C -8 16 0 10 8 D -8 2 -10 0 2 E -20 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=27 A=24 B=12 C=8 so C is eliminated. Round 2 votes counts: D=31 E=27 A=27 B=15 so B is eliminated. Round 3 votes counts: E=36 D=34 A=30 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:222 C:213 D:193 E:187 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 20 B -8 0 -16 -2 -4 C -8 16 0 10 8 D -8 2 -10 0 2 E -20 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 20 B -8 0 -16 -2 -4 C -8 16 0 10 8 D -8 2 -10 0 2 E -20 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 20 B -8 0 -16 -2 -4 C -8 16 0 10 8 D -8 2 -10 0 2 E -20 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6610: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (6) B A C E D (6) C E B D A (5) A D B C E (5) E D C A B (4) D A C E B (4) B E A C D (4) A B D C E (4) E C D B A (3) E C B D A (3) D E C A B (3) D A C B E (3) C D A B E (3) B C A E D (3) A C D B E (3) E D C B A (2) E D B C A (2) E B C D A (2) E B C A D (2) E B A D C (2) D C A E B (2) C B A E D (2) B C E A D (2) B C A D E (2) B A C D E (2) A D C B E (2) A D B E C (2) A B C D E (2) E D A B C (1) D E A C B (1) D C E A B (1) D C A B E (1) C E D B A (1) C D E A B (1) C B E D A (1) C B E A D (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A D C (1) B A E D C (1) B A E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -10 14 4 B 10 0 2 10 22 C 10 -2 0 16 18 D -14 -10 -16 0 -10 E -4 -22 -18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999572 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 14 4 B 10 0 2 10 22 C 10 -2 0 16 18 D -14 -10 -16 0 -10 E -4 -22 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996208 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=21 A=19 C=17 D=15 so D is eliminated. Round 2 votes counts: B=28 A=26 E=25 C=21 so C is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:221 A:199 E:183 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 14 4 B 10 0 2 10 22 C 10 -2 0 16 18 D -14 -10 -16 0 -10 E -4 -22 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996208 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 14 4 B 10 0 2 10 22 C 10 -2 0 16 18 D -14 -10 -16 0 -10 E -4 -22 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996208 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 14 4 B 10 0 2 10 22 C 10 -2 0 16 18 D -14 -10 -16 0 -10 E -4 -22 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996208 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6611: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) E A B C D (11) E A D C B (9) A E D C B (9) E A D B C (8) D C B A E (8) B C D A E (6) E B C A D (5) D C A B E (5) E A B D C (4) C B D A E (4) A D E C B (4) D A C B E (3) C D B A E (3) B C E D A (3) B E C A D (2) E B C D A (1) B E C D A (1) A D C E B (1) Total count = 100 A B C D E A 0 8 -2 6 -14 B -8 0 8 0 -4 C 2 -8 0 -2 -8 D -6 0 2 0 -6 E 14 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -2 6 -14 B -8 0 8 0 -4 C 2 -8 0 -2 -8 D -6 0 2 0 -6 E 14 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=25 D=16 A=14 C=7 so C is eliminated. Round 2 votes counts: E=38 B=29 D=19 A=14 so A is eliminated. Round 3 votes counts: E=47 B=29 D=24 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:199 B:198 D:195 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -2 6 -14 B -8 0 8 0 -4 C 2 -8 0 -2 -8 D -6 0 2 0 -6 E 14 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 -14 B -8 0 8 0 -4 C 2 -8 0 -2 -8 D -6 0 2 0 -6 E 14 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 -14 B -8 0 8 0 -4 C 2 -8 0 -2 -8 D -6 0 2 0 -6 E 14 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6612: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) E B D A C (5) C B D E A (5) C A D E B (5) C A D B E (5) A E D B C (5) C B E D A (4) B E D A C (4) B D E C A (4) B C E D A (4) A C D E B (4) E A D B C (3) C D B A E (3) C D A B E (3) C B E A D (3) C B A E D (3) B E D C A (3) A D E B C (3) D E A B C (2) D B E C A (2) D A E B C (2) C D B E A (2) C A B E D (2) B E C D A (2) B C E A D (2) A D E C B (2) A D C E B (2) E D B A C (1) E D A B C (1) D A E C B (1) D A C E B (1) C B D A E (1) B E A D C (1) B E A C D (1) B D C E A (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 -14 -8 -2 -12 B 14 0 6 4 10 C 8 -6 0 -2 0 D 2 -4 2 0 -4 E 12 -10 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999362 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 -2 -12 B 14 0 6 4 10 C 8 -6 0 -2 0 D 2 -4 2 0 -4 E 12 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999242 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=22 A=18 E=16 D=8 so D is eliminated. Round 2 votes counts: C=36 B=24 A=22 E=18 so E is eliminated. Round 3 votes counts: C=36 B=36 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:203 C:200 D:198 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 -2 -12 B 14 0 6 4 10 C 8 -6 0 -2 0 D 2 -4 2 0 -4 E 12 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999242 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 -2 -12 B 14 0 6 4 10 C 8 -6 0 -2 0 D 2 -4 2 0 -4 E 12 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999242 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 -2 -12 B 14 0 6 4 10 C 8 -6 0 -2 0 D 2 -4 2 0 -4 E 12 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999242 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6613: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) E C D B A (6) B E A C D (6) A B D C E (6) D E C A B (5) C D E A B (5) B A C E D (5) E D C B A (4) C E B A D (4) E D B A C (3) B A E C D (3) A D B C E (3) E D A B C (2) E C B D A (2) D E A C B (2) D E A B C (2) D C E A B (2) D C A E B (2) D A B E C (2) C D A B E (2) C A B D E (2) B C A E D (2) A B D E C (2) A B C D E (2) E D B C A (1) E B C D A (1) E B C A D (1) E B A D C (1) E B A C D (1) D A E B C (1) D A B C E (1) C E D A B (1) C D A E B (1) C B A E D (1) C B A D E (1) C A D B E (1) B E C A D (1) B E A D C (1) B A E D C (1) B A D E C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -6 -6 -18 B 8 0 0 -10 -10 C 6 0 0 14 0 D 6 10 -14 0 -10 E 18 10 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.408512 D: 0.000000 E: 0.591488 Sum of squares = 0.516740139211 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.408512 D: 0.408512 E: 1.000000 A B C D E A 0 -8 -6 -6 -18 B 8 0 0 -10 -10 C 6 0 0 14 0 D 6 10 -14 0 -10 E 18 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=22 B=20 D=17 A=15 so A is eliminated. Round 2 votes counts: B=30 C=27 E=22 D=21 so D is eliminated. Round 3 votes counts: B=37 E=32 C=31 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:210 D:196 B:194 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -6 -18 B 8 0 0 -10 -10 C 6 0 0 14 0 D 6 10 -14 0 -10 E 18 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -6 -18 B 8 0 0 -10 -10 C 6 0 0 14 0 D 6 10 -14 0 -10 E 18 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -6 -18 B 8 0 0 -10 -10 C 6 0 0 14 0 D 6 10 -14 0 -10 E 18 10 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6614: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (7) E B A C D (6) E D A B C (5) E B A D C (5) E C B D A (4) D C A B E (4) C D A B E (4) E D C B A (3) E B C A D (3) D E C A B (3) D E A C B (3) D C E A B (3) D A E B C (3) D A C B E (3) C B A E D (3) C B A D E (3) E D C A B (2) E B D C A (2) E B D A C (2) E B C D A (2) C A B D E (2) B E A C D (2) A D B C E (2) A C B D E (2) E D B A C (1) E D A C B (1) E C D B A (1) E C B A D (1) E A D B C (1) D E A B C (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E A B (1) C B E A D (1) C B D A E (1) B E C A D (1) B C A E D (1) B C A D E (1) B A E D C (1) B A C E D (1) B A C D E (1) A D C B E (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 -4 -10 B -6 0 2 10 -12 C -6 -2 0 -2 -14 D 4 -10 2 0 -2 E 10 12 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 -4 -10 B -6 0 2 10 -12 C -6 -2 0 -2 -14 D 4 -10 2 0 -2 E 10 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=22 C=16 A=15 B=8 so B is eliminated. Round 2 votes counts: E=42 D=22 C=18 A=18 so C is eliminated. Round 3 votes counts: E=44 D=28 A=28 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:199 B:197 D:197 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -4 -10 B -6 0 2 10 -12 C -6 -2 0 -2 -14 D 4 -10 2 0 -2 E 10 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -4 -10 B -6 0 2 10 -12 C -6 -2 0 -2 -14 D 4 -10 2 0 -2 E 10 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -4 -10 B -6 0 2 10 -12 C -6 -2 0 -2 -14 D 4 -10 2 0 -2 E 10 12 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6615: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) E A C B D (5) A B E C D (5) A B C E D (5) D B A C E (4) C D B E A (4) E D C A B (3) D E C B A (3) D A B E C (3) C E D B A (3) C D E B A (3) B D A C E (3) B A D C E (3) A E B D C (3) E C D A B (2) E C A D B (2) E C A B D (2) E A B D C (2) D C B E A (2) D C B A E (2) D B A E C (2) C E A B D (2) C B D A E (2) B C A D E (2) B A C D E (2) A B D E C (2) A B D C E (2) A B C D E (2) E D A C B (1) E C D B A (1) E A D C B (1) E A D B C (1) D E B C A (1) D E A B C (1) D C E B A (1) C E D A B (1) C E B A D (1) C B A E D (1) C B A D E (1) C A E B D (1) B D C A E (1) B A C E D (1) A E B C D (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 2 -2 16 B 2 0 10 0 16 C -2 -10 0 0 16 D 2 0 0 0 10 E -16 -16 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.463936 C: 0.000000 D: 0.536064 E: 0.000000 Sum of squares = 0.502601207375 Cumulative probabilities = A: 0.000000 B: 0.463936 C: 0.463936 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 16 B 2 0 10 0 16 C -2 -10 0 0 16 D 2 0 0 0 10 E -16 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999443 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=22 E=20 C=19 B=12 so B is eliminated. Round 2 votes counts: D=31 A=28 C=21 E=20 so E is eliminated. Round 3 votes counts: A=37 D=35 C=28 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:214 A:207 D:206 C:202 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 16 B 2 0 10 0 16 C -2 -10 0 0 16 D 2 0 0 0 10 E -16 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999443 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 16 B 2 0 10 0 16 C -2 -10 0 0 16 D 2 0 0 0 10 E -16 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999443 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 16 B 2 0 10 0 16 C -2 -10 0 0 16 D 2 0 0 0 10 E -16 -16 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999443 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6616: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (15) C B A D E (13) C B A E D (8) C A B D E (6) B C A D E (6) E D A C B (5) E D C B A (4) E D B C A (4) E D B A C (4) D E A B C (4) D A E B C (4) D E B A C (2) C E B D A (2) C B E D A (2) C B E A D (2) B C D E A (2) A E D C B (2) A D E B C (2) E C B D A (1) D B E A C (1) C E D A B (1) C B D E A (1) B E D C A (1) B E C D A (1) B D E C A (1) B C D A E (1) A D E C B (1) A D B E C (1) A C E D B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -12 -12 -6 B 12 0 2 -2 -4 C 12 -2 0 -4 -8 D 12 2 4 0 -6 E 6 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -12 -12 -6 B 12 0 2 -2 -4 C 12 -2 0 -4 -8 D 12 2 4 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=33 B=12 D=11 A=9 so A is eliminated. Round 2 votes counts: C=36 E=35 D=15 B=14 so B is eliminated. Round 3 votes counts: C=46 E=37 D=17 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:206 B:204 C:199 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -12 -12 -6 B 12 0 2 -2 -4 C 12 -2 0 -4 -8 D 12 2 4 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -12 -6 B 12 0 2 -2 -4 C 12 -2 0 -4 -8 D 12 2 4 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -12 -6 B 12 0 2 -2 -4 C 12 -2 0 -4 -8 D 12 2 4 0 -6 E 6 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6617: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) D A C E B (8) B E C A D (8) A D C B E (8) C B E A D (7) D E C B A (6) A C B E D (6) E B C D A (5) D A E C B (5) A D C E B (5) D C E B A (3) D A E B C (3) C E B D A (3) E C D B A (2) E C B D A (2) D E B C A (2) D E A B C (2) B C E D A (2) A D E B C (2) A C D B E (2) E B D C A (1) D E B A C (1) D C E A B (1) D C A E B (1) C E D B A (1) B E A C D (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 12 4 6 B -12 0 -20 -22 -6 C -12 20 0 -18 -2 D -4 22 18 0 18 E -6 6 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 4 6 B -12 0 -20 -22 -6 C -12 20 0 -18 -2 D -4 22 18 0 18 E -6 6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=32 C=11 B=11 E=10 so E is eliminated. Round 2 votes counts: A=36 D=32 B=17 C=15 so C is eliminated. Round 3 votes counts: A=36 D=35 B=29 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:227 A:217 C:194 E:192 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 4 6 B -12 0 -20 -22 -6 C -12 20 0 -18 -2 D -4 22 18 0 18 E -6 6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 4 6 B -12 0 -20 -22 -6 C -12 20 0 -18 -2 D -4 22 18 0 18 E -6 6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 4 6 B -12 0 -20 -22 -6 C -12 20 0 -18 -2 D -4 22 18 0 18 E -6 6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998382 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6618: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) E D B A C (7) C B D A E (7) E A D B C (5) C B D E A (5) A C E B D (5) E D B C A (4) D B E C A (4) C A B D E (4) B D C E A (4) A E D C B (4) A E D B C (4) A C B D E (4) E C D B A (3) D E B A C (3) D B E A C (3) E A D C B (2) D E B C A (2) C A B E D (2) B D E C A (2) B D C A E (2) B C D E A (2) A E C D B (2) A B D C E (2) E D C B A (1) E D C A B (1) E D A B C (1) E A C D B (1) C B A D E (1) C A E D B (1) C A E B D (1) B D A E C (1) B C D A E (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 6 -6 0 B 4 0 -6 -12 -10 C -6 6 0 -4 0 D 6 12 4 0 -6 E 0 10 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.359560 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.640440 Sum of squares = 0.539446948436 Cumulative probabilities = A: 0.359560 B: 0.359560 C: 0.359560 D: 0.359560 E: 1.000000 A B C D E A 0 -4 6 -6 0 B 4 0 -6 -12 -10 C -6 6 0 -4 0 D 6 12 4 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499643 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500357 Sum of squares = 0.500000255532 Cumulative probabilities = A: 0.499643 B: 0.499643 C: 0.499643 D: 0.499643 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 C=21 D=12 B=12 so D is eliminated. Round 2 votes counts: E=30 A=30 C=21 B=19 so B is eliminated. Round 3 votes counts: E=39 A=31 C=30 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:208 E:208 A:198 C:198 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 -6 0 B 4 0 -6 -12 -10 C -6 6 0 -4 0 D 6 12 4 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499643 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500357 Sum of squares = 0.500000255532 Cumulative probabilities = A: 0.499643 B: 0.499643 C: 0.499643 D: 0.499643 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -6 0 B 4 0 -6 -12 -10 C -6 6 0 -4 0 D 6 12 4 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499643 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500357 Sum of squares = 0.500000255532 Cumulative probabilities = A: 0.499643 B: 0.499643 C: 0.499643 D: 0.499643 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -6 0 B 4 0 -6 -12 -10 C -6 6 0 -4 0 D 6 12 4 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499643 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500357 Sum of squares = 0.500000255532 Cumulative probabilities = A: 0.499643 B: 0.499643 C: 0.499643 D: 0.499643 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6619: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (18) B A E D C (16) B E A C D (6) D C E A B (5) A E B D C (5) D C A E B (4) D A E C B (3) C D B E A (3) B C D A E (3) E A D C B (2) E A C D B (2) E A B D C (2) E A B C D (2) D C B A E (2) C E A B D (2) B D A E C (2) B C E D A (2) B C E A D (2) E C A D B (1) E B A D C (1) E A D B C (1) E A C B D (1) D B C A E (1) D B A E C (1) D B A C E (1) D A C E B (1) C E D A B (1) C E B D A (1) C D E B A (1) C D B A E (1) C B E A D (1) B E C A D (1) B E A D C (1) B A D C E (1) A E D C B (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 2 0 -12 B -6 0 0 0 -10 C -2 0 0 -4 0 D 0 0 4 0 -6 E 12 10 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.430643 D: 0.000000 E: 0.569357 Sum of squares = 0.509620775615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.430643 D: 0.430643 E: 1.000000 A B C D E A 0 6 2 0 -12 B -6 0 0 0 -10 C -2 0 0 -4 0 D 0 0 4 0 -6 E 12 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=28 D=18 E=12 A=8 so A is eliminated. Round 2 votes counts: B=35 C=28 E=19 D=18 so D is eliminated. Round 3 votes counts: C=40 B=38 E=22 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:214 D:199 A:198 C:197 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 0 -12 B -6 0 0 0 -10 C -2 0 0 -4 0 D 0 0 4 0 -6 E 12 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 -12 B -6 0 0 0 -10 C -2 0 0 -4 0 D 0 0 4 0 -6 E 12 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 -12 B -6 0 0 0 -10 C -2 0 0 -4 0 D 0 0 4 0 -6 E 12 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6620: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (16) E D B C A (11) A C B E D (11) E A C B D (8) A C B D E (8) C B A D E (7) D E B C A (6) E D B A C (4) E C A B D (4) D B E C A (4) D E B A C (3) C A B D E (3) A B C D E (3) D B A C E (2) B D C A E (2) B C D A E (2) B C A D E (2) E D A C B (1) E D A B C (1) C A B E D (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 -16 -4 16 B 18 0 12 4 22 C 16 -12 0 0 16 D 4 -4 0 0 16 E -16 -22 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 -4 16 B 18 0 12 4 22 C 16 -12 0 0 16 D 4 -4 0 0 16 E -16 -22 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999735 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=29 A=23 C=11 B=6 so B is eliminated. Round 2 votes counts: D=33 E=29 A=23 C=15 so C is eliminated. Round 3 votes counts: A=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:228 C:210 D:208 A:189 E:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 -4 16 B 18 0 12 4 22 C 16 -12 0 0 16 D 4 -4 0 0 16 E -16 -22 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999735 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -4 16 B 18 0 12 4 22 C 16 -12 0 0 16 D 4 -4 0 0 16 E -16 -22 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999735 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -4 16 B 18 0 12 4 22 C 16 -12 0 0 16 D 4 -4 0 0 16 E -16 -22 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999735 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6621: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) B E D C A (6) E B D A C (5) E B A C D (5) A E C B D (5) E A C B D (4) D B E C A (4) C D B A E (4) A C E D B (4) D B C E A (3) D A C B E (3) C A E D B (3) C A E B D (3) C A D E B (3) B D E C A (3) A E C D B (3) E B A D C (2) E A B C D (2) D C B A E (2) D C A B E (2) D B E A C (2) D A B C E (2) C B D E A (2) B D E A C (2) A E D C B (2) E C A B D (1) E B C D A (1) E A B D C (1) D B A E C (1) D A E B C (1) C E A B D (1) C D B E A (1) C B E D A (1) C B E A D (1) C B D A E (1) B C E D A (1) A E B D C (1) A E B C D (1) A D E B C (1) A D C E B (1) A D C B E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 6 4 B -6 0 -12 0 -4 C -2 12 0 10 -6 D -6 0 -10 0 -8 E -4 4 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 6 4 B -6 0 -12 0 -4 C -2 12 0 10 -6 D -6 0 -10 0 -8 E -4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=21 A=21 D=20 B=12 so B is eliminated. Round 2 votes counts: E=27 C=27 D=25 A=21 so A is eliminated. Round 3 votes counts: E=39 C=33 D=28 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:209 C:207 E:207 B:189 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 6 4 B -6 0 -12 0 -4 C -2 12 0 10 -6 D -6 0 -10 0 -8 E -4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 6 4 B -6 0 -12 0 -4 C -2 12 0 10 -6 D -6 0 -10 0 -8 E -4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 6 4 B -6 0 -12 0 -4 C -2 12 0 10 -6 D -6 0 -10 0 -8 E -4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6622: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) D E A C B (7) C D A B E (7) C A B D E (7) D C E A B (6) C B D A E (5) C B A D E (5) B C A E D (5) D E C A B (4) B A C E D (4) E D A C B (3) E B A D C (3) C D B A E (3) C D A E B (3) E D B A C (2) E A D C B (2) E A D B C (2) E A B D C (2) D E B C A (2) B E C A D (2) B C A D E (2) B A E C D (2) E D B C A (1) D E C B A (1) D C A E B (1) D A C E B (1) B E D A C (1) B E A D C (1) B E A C D (1) B C D A E (1) A E B D C (1) A E B C D (1) A C D E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 18 -10 -14 2 B -18 0 -14 -10 -4 C 10 14 0 2 4 D 14 10 -2 0 12 E -2 4 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -10 -14 2 B -18 0 -14 -10 -4 C 10 14 0 2 4 D 14 10 -2 0 12 E -2 4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=24 D=22 B=19 A=5 so A is eliminated. Round 2 votes counts: C=32 E=26 D=22 B=20 so B is eliminated. Round 3 votes counts: C=44 E=34 D=22 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:215 A:198 E:193 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -10 -14 2 B -18 0 -14 -10 -4 C 10 14 0 2 4 D 14 10 -2 0 12 E -2 4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -10 -14 2 B -18 0 -14 -10 -4 C 10 14 0 2 4 D 14 10 -2 0 12 E -2 4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -10 -14 2 B -18 0 -14 -10 -4 C 10 14 0 2 4 D 14 10 -2 0 12 E -2 4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6623: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) B A D E C (9) E C B D A (7) B A D C E (7) D A B C E (6) C D A B E (5) E B C A D (4) E A D B C (4) D C A E B (3) C E D B A (3) C D E A B (3) B E A C D (3) A D B C E (3) A B D E C (3) E D C A B (2) D C A B E (2) C E B D A (2) B E A D C (2) B C E A D (2) A D B E C (2) E D A C B (1) E C D A B (1) E B A D C (1) E B A C D (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) D A B E C (1) C E D A B (1) C D B A E (1) C B E A D (1) B D A C E (1) B C A E D (1) B A E C D (1) B A C D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 2 12 2 B 14 0 8 12 4 C -2 -8 0 -6 -10 D -12 -12 6 0 6 E -2 -4 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999538 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 12 2 B 14 0 8 12 4 C -2 -8 0 -6 -10 D -12 -12 6 0 6 E -2 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=27 D=17 C=16 A=9 so A is eliminated. Round 2 votes counts: E=31 B=30 D=23 C=16 so C is eliminated. Round 3 votes counts: E=37 D=32 B=31 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:219 A:201 E:199 D:194 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 12 2 B 14 0 8 12 4 C -2 -8 0 -6 -10 D -12 -12 6 0 6 E -2 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 12 2 B 14 0 8 12 4 C -2 -8 0 -6 -10 D -12 -12 6 0 6 E -2 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 12 2 B 14 0 8 12 4 C -2 -8 0 -6 -10 D -12 -12 6 0 6 E -2 -4 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6624: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (14) B C A D E (9) A E B C D (9) D E C B A (7) A B E C D (7) E D A C B (4) C B D E A (4) A E B D C (4) E A D C B (3) D E C A B (3) D C E B A (3) B A C E D (3) B A C D E (3) A E D B C (3) E D C B A (2) D C B E A (2) C D B E A (2) B C D A E (2) E D C A B (1) E C D B A (1) E C D A B (1) E C A D B (1) E A C D B (1) D E A C B (1) D E A B C (1) D B C E A (1) D B A E C (1) C E D A B (1) C D E B A (1) C B A E D (1) C B A D E (1) B A D C E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 14 24 20 B -12 0 18 20 6 C -14 -18 0 24 -2 D -24 -20 -24 0 -16 E -20 -6 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 24 20 B -12 0 18 20 6 C -14 -18 0 24 -2 D -24 -20 -24 0 -16 E -20 -6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 D=19 B=18 E=14 C=10 so C is eliminated. Round 2 votes counts: A=39 B=24 D=22 E=15 so E is eliminated. Round 3 votes counts: A=44 D=32 B=24 so B is eliminated. Round 4 votes counts: A=62 D=38 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:235 B:216 E:196 C:195 D:158 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 24 20 B -12 0 18 20 6 C -14 -18 0 24 -2 D -24 -20 -24 0 -16 E -20 -6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 24 20 B -12 0 18 20 6 C -14 -18 0 24 -2 D -24 -20 -24 0 -16 E -20 -6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 24 20 B -12 0 18 20 6 C -14 -18 0 24 -2 D -24 -20 -24 0 -16 E -20 -6 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6625: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) E C A B D (8) E A C D B (8) D A B C E (8) C B E D A (8) C E B D A (7) A D E B C (7) E A C B D (5) B D C A E (5) C B D E A (4) B C D A E (4) A E D C B (4) A D B E C (4) E A D B C (3) B C D E A (3) E C B D A (2) E A D C B (2) A D B C E (2) E B C D A (1) D B C A E (1) D B A C E (1) C B D A E (1) B D C E A (1) A E D B C (1) A E C D B (1) Total count = 100 A B C D E A 0 6 -8 8 -22 B -6 0 -18 16 -16 C 8 18 0 22 -10 D -8 -16 -22 0 -18 E 22 16 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 8 -22 B -6 0 -18 16 -16 C 8 18 0 22 -10 D -8 -16 -22 0 -18 E 22 16 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=20 A=19 B=13 D=10 so D is eliminated. Round 2 votes counts: E=38 A=27 C=20 B=15 so B is eliminated. Round 3 votes counts: E=38 C=34 A=28 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:233 C:219 A:192 B:188 D:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 8 -22 B -6 0 -18 16 -16 C 8 18 0 22 -10 D -8 -16 -22 0 -18 E 22 16 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 8 -22 B -6 0 -18 16 -16 C 8 18 0 22 -10 D -8 -16 -22 0 -18 E 22 16 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 8 -22 B -6 0 -18 16 -16 C 8 18 0 22 -10 D -8 -16 -22 0 -18 E 22 16 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6626: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (16) E B A C D (14) A C D E B (12) E B D A C (8) D C A B E (8) E B A D C (5) A C E D B (5) C D A B E (4) A C D B E (4) E B D C A (3) D C B A E (3) A E C D B (3) E A B C D (2) D B C A E (2) B D E C A (2) B D C E A (2) A D C B E (2) E A C D B (1) E A C B D (1) C A D B E (1) B E D A C (1) B D C A E (1) Total count = 100 A B C D E A 0 -14 16 0 -10 B 14 0 12 10 -8 C -16 -12 0 -6 -12 D 0 -10 6 0 -18 E 10 8 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 16 0 -10 B 14 0 12 10 -8 C -16 -12 0 -6 -12 D 0 -10 6 0 -18 E 10 8 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=26 B=22 D=13 C=5 so C is eliminated. Round 2 votes counts: E=34 A=27 B=22 D=17 so D is eliminated. Round 3 votes counts: A=39 E=34 B=27 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:214 A:196 D:189 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 16 0 -10 B 14 0 12 10 -8 C -16 -12 0 -6 -12 D 0 -10 6 0 -18 E 10 8 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 16 0 -10 B 14 0 12 10 -8 C -16 -12 0 -6 -12 D 0 -10 6 0 -18 E 10 8 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 16 0 -10 B 14 0 12 10 -8 C -16 -12 0 -6 -12 D 0 -10 6 0 -18 E 10 8 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6627: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) D B C E A (10) C B A E D (6) B C D A E (6) D E B C A (5) D E A B C (5) D B C A E (5) A C B E D (5) E D A C B (4) E A C D B (3) E A C B D (3) D E A C B (3) B C A E D (3) B C A D E (3) E D A B C (2) D E B A C (2) D A E C B (2) B E C D A (2) B C E A D (2) A E D C B (2) A E C B D (2) E C B A D (1) D B E C A (1) D B A C E (1) D A E B C (1) D A C B E (1) D A B C E (1) C B A D E (1) C A B E D (1) B D C E A (1) B D C A E (1) A E C D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 4 -6 -12 B 0 0 2 -24 0 C -4 -2 0 -18 -2 D 6 24 18 0 0 E 12 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.464072 E: 0.535928 Sum of squares = 0.502581580042 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.464072 E: 1.000000 A B C D E A 0 0 4 -6 -12 B 0 0 2 -24 0 C -4 -2 0 -18 -2 D 6 24 18 0 0 E 12 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=25 B=18 A=12 C=8 so C is eliminated. Round 2 votes counts: D=37 E=25 B=25 A=13 so A is eliminated. Round 3 votes counts: D=38 E=31 B=31 so E is eliminated. Round 4 votes counts: D=62 B=38 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:207 A:193 B:189 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -6 -12 B 0 0 2 -24 0 C -4 -2 0 -18 -2 D 6 24 18 0 0 E 12 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -6 -12 B 0 0 2 -24 0 C -4 -2 0 -18 -2 D 6 24 18 0 0 E 12 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -6 -12 B 0 0 2 -24 0 C -4 -2 0 -18 -2 D 6 24 18 0 0 E 12 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6628: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) A D E B C (7) B C E A D (5) B A E C D (5) E A D B C (4) D C E B A (4) D A E B C (4) C B D E A (4) C B A D E (4) E D C B A (3) D E A C B (3) D A E C B (3) D A C B E (3) B C A E D (3) A E D B C (3) A D B C E (3) E A B D C (2) D E C A B (2) D C E A B (2) D C A E B (2) D C A B E (2) C D B A E (2) C B E A D (2) C B D A E (2) B E C A D (2) A B D E C (2) E D A B C (1) E C D B A (1) E B C A D (1) E B A D C (1) E B A C D (1) D E A B C (1) D A C E B (1) C E D B A (1) B A C E D (1) B A C D E (1) A E B D C (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 2 4 B 2 0 2 -4 4 C 0 -2 0 -12 4 D -2 4 12 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 4 B 2 0 2 -4 4 C 0 -2 0 -12 4 D -2 4 12 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999974 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=23 A=19 B=17 E=14 so E is eliminated. Round 2 votes counts: D=31 A=25 C=24 B=20 so B is eliminated. Round 3 votes counts: C=35 A=34 D=31 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:202 B:202 C:195 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 2 4 B 2 0 2 -4 4 C 0 -2 0 -12 4 D -2 4 12 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999974 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 4 B 2 0 2 -4 4 C 0 -2 0 -12 4 D -2 4 12 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999974 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 4 B 2 0 2 -4 4 C 0 -2 0 -12 4 D -2 4 12 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999974 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6629: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) C B D E A (9) D B C A E (7) C B E D A (7) A E D B C (7) E C B A D (6) E A C D B (6) B D C A E (5) E A D C B (4) D A B C E (4) A D E B C (4) D A B E C (3) B C E A D (3) B C D A E (3) E C A B D (2) D A E B C (2) C E B D A (2) C B E A D (2) B C D E A (2) B A D C E (2) A E D C B (2) A D E C B (2) A D B E C (2) E C D A B (1) D B A C E (1) D A E C B (1) C D E B A (1) C D B E A (1) Total count = 100 A B C D E A 0 -2 -2 2 -10 B 2 0 -10 4 2 C 2 10 0 8 -2 D -2 -4 -8 0 -2 E 10 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408147 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 0.285714 E: 1.000000 A B C D E A 0 -2 -2 2 -10 B 2 0 -10 4 2 C 2 10 0 8 -2 D -2 -4 -8 0 -2 E 10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408023 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 D=18 A=17 B=15 so B is eliminated. Round 2 votes counts: C=30 E=28 D=23 A=19 so A is eliminated. Round 3 votes counts: E=37 D=33 C=30 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:206 B:199 A:194 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 2 -10 B 2 0 -10 4 2 C 2 10 0 8 -2 D -2 -4 -8 0 -2 E 10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408023 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 2 -10 B 2 0 -10 4 2 C 2 10 0 8 -2 D -2 -4 -8 0 -2 E 10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408023 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 2 -10 B 2 0 -10 4 2 C 2 10 0 8 -2 D -2 -4 -8 0 -2 E 10 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408023 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6630: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (8) C A E D B (8) E A D C B (7) B D E A C (7) B C D A E (6) E D A B C (5) D E A B C (5) D B E A C (4) D A E B C (4) C B E A D (4) B C D E A (4) C B A E D (3) C A E B D (3) C E A B D (2) C B D A E (2) B E D A C (2) B D E C A (2) B D C A E (2) B D A E C (2) B D A C E (2) A C E D B (2) E D B A C (1) E C A D B (1) E B D A C (1) E A D B C (1) E A C D B (1) D B A E C (1) C E A D B (1) C A D B E (1) C A B E D (1) B E D C A (1) B D C E A (1) A E D C B (1) A D E C B (1) A D E B C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 0 -4 0 B 6 0 6 6 10 C 0 -6 0 -4 4 D 4 -6 4 0 10 E 0 -10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -4 0 B 6 0 6 6 10 C 0 -6 0 -4 4 D 4 -6 4 0 10 E 0 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=29 E=17 D=14 A=7 so A is eliminated. Round 2 votes counts: C=36 B=29 E=18 D=17 so D is eliminated. Round 3 votes counts: C=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:206 C:197 A:195 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -4 0 B 6 0 6 6 10 C 0 -6 0 -4 4 D 4 -6 4 0 10 E 0 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -4 0 B 6 0 6 6 10 C 0 -6 0 -4 4 D 4 -6 4 0 10 E 0 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -4 0 B 6 0 6 6 10 C 0 -6 0 -4 4 D 4 -6 4 0 10 E 0 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6631: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) D C A B E (10) C A B D E (10) E D B A C (7) A B C E D (7) E B A D C (5) C A D B E (5) B E A C D (5) E B D A C (4) D E C A B (4) B A C E D (4) D C A E B (3) C D A B E (3) E D B C A (2) D E C B A (2) D E B C A (2) D C E A B (2) D C B E A (2) C B A D E (2) C A B E D (2) A C B E D (2) A C B D E (2) D C E B A (1) B A E C D (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 0 16 6 B -4 0 -2 12 12 C 0 2 0 12 12 D -16 -12 -12 0 -2 E -6 -12 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.426381 B: 0.000000 C: 0.573619 D: 0.000000 E: 0.000000 Sum of squares = 0.510839412566 Cumulative probabilities = A: 0.426381 B: 0.426381 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 16 6 B -4 0 -2 12 12 C 0 2 0 12 12 D -16 -12 -12 0 -2 E -6 -12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999853 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 C=22 A=13 B=10 so B is eliminated. Round 2 votes counts: E=34 D=26 C=22 A=18 so A is eliminated. Round 3 votes counts: C=38 E=36 D=26 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:213 B:209 E:186 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 16 6 B -4 0 -2 12 12 C 0 2 0 12 12 D -16 -12 -12 0 -2 E -6 -12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999853 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 16 6 B -4 0 -2 12 12 C 0 2 0 12 12 D -16 -12 -12 0 -2 E -6 -12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999853 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 16 6 B -4 0 -2 12 12 C 0 2 0 12 12 D -16 -12 -12 0 -2 E -6 -12 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999853 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6632: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) C D A B E (8) B E A C D (8) E B A C D (7) E B A D C (6) D E A C B (6) E D A B C (5) C B A D E (5) B C A E D (5) A E B C D (4) D E C A B (3) C D B A E (3) C B D A E (3) C A B D E (3) B A C E D (3) E A B D C (2) D C E A B (2) D C A B E (2) C B A E D (2) A B E C D (2) E D B A C (1) E B D A C (1) E A B C D (1) D C E B A (1) D A C E B (1) C B D E A (1) B D E C A (1) B C E A D (1) B C D E A (1) B C A D E (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -2 4 6 B 0 0 0 14 -2 C 2 0 0 18 4 D -4 -14 -18 0 2 E -6 2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.196651 C: 0.803349 D: 0.000000 E: 0.000000 Sum of squares = 0.684041542643 Cumulative probabilities = A: 0.000000 B: 0.196651 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 4 6 B 0 0 0 14 -2 C 2 0 0 18 4 D -4 -14 -18 0 2 E -6 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 E=23 B=20 A=8 so A is eliminated. Round 2 votes counts: E=28 C=26 D=24 B=22 so B is eliminated. Round 3 votes counts: E=38 C=37 D=25 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:206 A:204 E:195 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 4 6 B 0 0 0 14 -2 C 2 0 0 18 4 D -4 -14 -18 0 2 E -6 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 6 B 0 0 0 14 -2 C 2 0 0 18 4 D -4 -14 -18 0 2 E -6 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 6 B 0 0 0 14 -2 C 2 0 0 18 4 D -4 -14 -18 0 2 E -6 2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6633: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) B D E A C (7) A D E C B (6) C B A E D (5) C A E D B (5) B E D C A (4) B C E D A (4) B A D E C (4) A C D E B (4) A C D B E (4) A C B D E (4) E D B C A (3) D E A C B (3) B A C D E (3) A D C E B (3) E D A C B (2) E D A B C (2) D A E B C (2) C E D B A (2) C E B D A (2) C B E D A (2) C B E A D (2) C A D E B (2) B C A E D (2) B C A D E (2) B A D C E (2) E D C A B (1) E D B A C (1) E B D C A (1) D E B A C (1) D E A B C (1) D A E C B (1) C E D A B (1) C A E B D (1) C A B E D (1) B D A E C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 20 4 8 B 10 0 -4 8 12 C -20 4 0 -8 6 D -4 -8 8 0 4 E -8 -12 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.117647 B: 0.588235 C: 0.294118 D: 0.000000 E: 0.000000 Sum of squares = 0.446366781962 Cumulative probabilities = A: 0.117647 B: 0.705882 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 20 4 8 B 10 0 -4 8 12 C -20 4 0 -8 6 D -4 -8 8 0 4 E -8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.588235 C: 0.294118 D: 0.000000 E: 0.000000 Sum of squares = 0.446366782137 Cumulative probabilities = A: 0.117647 B: 0.705882 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=23 A=23 E=10 D=8 so D is eliminated. Round 2 votes counts: B=36 A=26 C=23 E=15 so E is eliminated. Round 3 votes counts: B=42 A=34 C=24 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:211 D:200 C:191 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 20 4 8 B 10 0 -4 8 12 C -20 4 0 -8 6 D -4 -8 8 0 4 E -8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.588235 C: 0.294118 D: 0.000000 E: 0.000000 Sum of squares = 0.446366782137 Cumulative probabilities = A: 0.117647 B: 0.705882 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 20 4 8 B 10 0 -4 8 12 C -20 4 0 -8 6 D -4 -8 8 0 4 E -8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.588235 C: 0.294118 D: 0.000000 E: 0.000000 Sum of squares = 0.446366782137 Cumulative probabilities = A: 0.117647 B: 0.705882 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 20 4 8 B 10 0 -4 8 12 C -20 4 0 -8 6 D -4 -8 8 0 4 E -8 -12 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.117647 B: 0.588235 C: 0.294118 D: 0.000000 E: 0.000000 Sum of squares = 0.446366782137 Cumulative probabilities = A: 0.117647 B: 0.705882 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6634: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (12) E C A B D (8) E C D B A (7) A B D C E (6) D B A C E (5) C E D B A (5) A D B C E (5) E C D A B (4) C E B D A (4) A B D E C (4) E C B A D (3) E C A D B (3) D E C A B (3) D C E B A (3) B A D C E (3) A D B E C (3) A B E D C (3) E B A C D (2) D C A B E (2) B C E D A (2) B A E C D (2) A E D C B (2) E B C A D (1) D B C E A (1) D A E C B (1) D A B C E (1) B D A C E (1) B C E A D (1) A E B D C (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -18 -2 -18 B 4 0 -14 10 -20 C 18 14 0 12 -22 D 2 -10 -12 0 -24 E 18 20 22 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -18 -2 -18 B 4 0 -14 10 -20 C 18 14 0 12 -22 D 2 -10 -12 0 -24 E 18 20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=26 D=16 C=9 B=9 so C is eliminated. Round 2 votes counts: E=49 A=26 D=16 B=9 so B is eliminated. Round 3 votes counts: E=52 A=31 D=17 so D is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:242 C:211 B:190 A:179 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -18 -2 -18 B 4 0 -14 10 -20 C 18 14 0 12 -22 D 2 -10 -12 0 -24 E 18 20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -2 -18 B 4 0 -14 10 -20 C 18 14 0 12 -22 D 2 -10 -12 0 -24 E 18 20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -2 -18 B 4 0 -14 10 -20 C 18 14 0 12 -22 D 2 -10 -12 0 -24 E 18 20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6635: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (13) A E D B C (13) A E D C B (9) D E A B C (8) C D E A B (7) B C D E A (5) B A E D C (5) C A E D B (4) B C A E D (4) D E A C B (3) C D E B A (3) C B A E D (3) B A E C D (3) C B D A E (2) C B A D E (2) B E A D C (2) B D E A C (2) A E B D C (2) A B E D C (2) D E C A B (1) D C E A B (1) D C B E A (1) C D A E B (1) B D C E A (1) A E C D B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 8 4 4 6 B -8 0 -4 -4 -10 C -4 4 0 0 -4 D -4 4 0 0 0 E -6 10 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999433 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 4 6 B -8 0 -4 -4 -10 C -4 4 0 0 -4 D -4 4 0 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=29 B=22 D=14 so E is eliminated. Round 2 votes counts: C=35 A=29 B=22 D=14 so D is eliminated. Round 3 votes counts: A=40 C=38 B=22 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:204 D:200 C:198 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 4 6 B -8 0 -4 -4 -10 C -4 4 0 0 -4 D -4 4 0 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 4 6 B -8 0 -4 -4 -10 C -4 4 0 0 -4 D -4 4 0 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 4 6 B -8 0 -4 -4 -10 C -4 4 0 0 -4 D -4 4 0 0 0 E -6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6636: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) B D E C A (7) B A D C E (7) E C A D B (6) B D C A E (5) B D A C E (5) A C E D B (5) E C D A B (4) D B C A E (4) C A D E B (4) D C A E B (3) D B A C E (3) D A C B E (3) A C D E B (3) A C D B E (3) E C D B A (2) E A C D B (2) D B C E A (2) C D A E B (2) B E A C D (2) A B C D E (2) E D C B A (1) E C B A D (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C B D (1) D C A B E (1) C D E A B (1) B E D A C (1) B D C E A (1) B A E C D (1) B A C D E (1) A E C D B (1) A E C B D (1) A D B C E (1) A C E B D (1) A C B E D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -6 -6 18 B 6 0 8 -2 18 C 6 -8 0 -4 18 D 6 2 4 0 18 E -18 -18 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -6 18 B 6 0 8 -2 18 C 6 -8 0 -4 18 D 6 2 4 0 18 E -18 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=20 A=20 D=16 C=7 so C is eliminated. Round 2 votes counts: B=37 A=24 E=20 D=19 so D is eliminated. Round 3 votes counts: B=46 A=33 E=21 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:215 C:206 A:200 E:164 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -6 18 B 6 0 8 -2 18 C 6 -8 0 -4 18 D 6 2 4 0 18 E -18 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -6 18 B 6 0 8 -2 18 C 6 -8 0 -4 18 D 6 2 4 0 18 E -18 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -6 18 B 6 0 8 -2 18 C 6 -8 0 -4 18 D 6 2 4 0 18 E -18 -18 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6637: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) C B A E D (7) B C D E A (7) E A D C B (6) A E D C B (6) C B E D A (5) A E C D B (5) D E B C A (4) D B E C A (4) B D C E A (4) A D E B C (4) A C B E D (4) D E B A C (3) C B E A D (3) B C A D E (3) D A E B C (2) C E A B D (2) C B D E A (2) C B A D E (2) B C D A E (2) A E D B C (2) E D C B A (1) E D C A B (1) E D A B C (1) E C D B A (1) E C B D A (1) E B D C A (1) D E A B C (1) D B E A C (1) C A B E D (1) B D E C A (1) A D E C B (1) A D B E C (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -4 16 2 B 4 0 -16 10 -4 C 4 16 0 10 6 D -16 -10 -10 0 -14 E -2 4 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 16 2 B 4 0 -16 10 -4 C 4 16 0 10 6 D -16 -10 -10 0 -14 E -2 4 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=22 B=17 D=15 E=12 so E is eliminated. Round 2 votes counts: A=40 C=24 D=18 B=18 so D is eliminated. Round 3 votes counts: A=44 B=30 C=26 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:218 A:205 E:205 B:197 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 16 2 B 4 0 -16 10 -4 C 4 16 0 10 6 D -16 -10 -10 0 -14 E -2 4 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 16 2 B 4 0 -16 10 -4 C 4 16 0 10 6 D -16 -10 -10 0 -14 E -2 4 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 16 2 B 4 0 -16 10 -4 C 4 16 0 10 6 D -16 -10 -10 0 -14 E -2 4 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6638: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) A B C D E (6) E C A B D (5) D B C E A (5) E A D C B (4) D E B C A (4) A E C B D (4) A D B C E (4) A C B E D (4) A B D C E (4) E C D B A (3) E C B D A (3) E C B A D (3) D B C A E (3) C B E D A (3) A D E B C (3) A D B E C (3) A B C E D (3) E A C D B (2) E A C B D (2) D A B C E (2) C E B A D (2) C B E A D (2) C B A E D (2) B A C D E (2) A E D B C (2) E C D A B (1) E C A D B (1) D E C B A (1) D B A E C (1) D B A C E (1) D A E B C (1) C E B D A (1) C B D A E (1) B D C A E (1) B C D A E (1) B C A D E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 6 0 22 2 B -6 0 -4 4 -2 C 0 4 0 8 -2 D -22 -4 -8 0 -12 E -2 2 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.681784 B: 0.000000 C: 0.318216 D: 0.000000 E: 0.000000 Sum of squares = 0.566090788417 Cumulative probabilities = A: 0.681784 B: 0.681784 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 22 2 B -6 0 -4 4 -2 C 0 4 0 8 -2 D -22 -4 -8 0 -12 E -2 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500461 B: 0.000000 C: 0.499539 D: 0.000000 E: 0.000000 Sum of squares = 0.500000425585 Cumulative probabilities = A: 0.500461 B: 0.500461 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=31 D=18 C=11 B=5 so B is eliminated. Round 2 votes counts: A=37 E=31 D=19 C=13 so C is eliminated. Round 3 votes counts: A=40 E=39 D=21 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:207 C:205 B:196 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 22 2 B -6 0 -4 4 -2 C 0 4 0 8 -2 D -22 -4 -8 0 -12 E -2 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500461 B: 0.000000 C: 0.499539 D: 0.000000 E: 0.000000 Sum of squares = 0.500000425585 Cumulative probabilities = A: 0.500461 B: 0.500461 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 22 2 B -6 0 -4 4 -2 C 0 4 0 8 -2 D -22 -4 -8 0 -12 E -2 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500461 B: 0.000000 C: 0.499539 D: 0.000000 E: 0.000000 Sum of squares = 0.500000425585 Cumulative probabilities = A: 0.500461 B: 0.500461 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 22 2 B -6 0 -4 4 -2 C 0 4 0 8 -2 D -22 -4 -8 0 -12 E -2 2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500461 B: 0.000000 C: 0.499539 D: 0.000000 E: 0.000000 Sum of squares = 0.500000425585 Cumulative probabilities = A: 0.500461 B: 0.500461 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6639: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (15) E A D C B (14) E B D C A (6) B E C D A (6) A D C E B (5) E D B C A (4) E B C D A (4) B C D E A (4) B C A D E (4) A C D B E (4) E B A C D (3) A E D C B (3) A D E C B (3) D C B A E (2) C B D A E (2) C A D B E (2) E D C B A (1) E D A C B (1) E B D A C (1) E B A D C (1) E A B C D (1) D C E A B (1) D A E C B (1) C D B A E (1) C B A D E (1) C A B D E (1) B E D C A (1) B E C A D (1) B E A C D (1) B C E A D (1) B A C E D (1) A D C B E (1) A C D E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -14 0 -2 B 20 0 10 12 0 C 14 -10 0 10 -4 D 0 -12 -10 0 -2 E 2 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.399995 C: 0.000000 D: 0.000000 E: 0.600005 Sum of squares = 0.520001967722 Cumulative probabilities = A: 0.000000 B: 0.399995 C: 0.399995 D: 0.399995 E: 1.000000 A B C D E A 0 -20 -14 0 -2 B 20 0 10 12 0 C 14 -10 0 10 -4 D 0 -12 -10 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=34 A=19 C=7 D=4 so D is eliminated. Round 2 votes counts: E=36 B=34 A=20 C=10 so C is eliminated. Round 3 votes counts: B=40 E=37 A=23 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:205 E:204 D:188 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -14 0 -2 B 20 0 10 12 0 C 14 -10 0 10 -4 D 0 -12 -10 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 0 -2 B 20 0 10 12 0 C 14 -10 0 10 -4 D 0 -12 -10 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 0 -2 B 20 0 10 12 0 C 14 -10 0 10 -4 D 0 -12 -10 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6640: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) B E A D C (8) D C E A B (7) C D A B E (7) D C E B A (6) E D B A C (5) E B D A C (5) D E C B A (5) D E B A C (5) C A B E D (5) A B C E D (5) A B E C D (4) E B A D C (3) C A D B E (3) B A E C D (3) A C E B D (3) A C B E D (3) C A D E B (2) C A B D E (2) A E B C D (2) E A B C D (1) D E B C A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B E C A (1) C B A D E (1) B D C A E (1) A E C B D (1) Total count = 100 A B C D E A 0 10 -4 -8 6 B -10 0 -12 -6 -10 C 4 12 0 0 12 D 8 6 0 0 4 E -6 10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.550838 D: 0.449162 E: 0.000000 Sum of squares = 0.505168924284 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.550838 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -8 6 B -10 0 -12 -6 -10 C 4 12 0 0 12 D 8 6 0 0 4 E -6 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 A=18 E=14 B=12 so B is eliminated. Round 2 votes counts: D=29 C=28 E=22 A=21 so A is eliminated. Round 3 votes counts: C=39 E=32 D=29 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:209 A:202 E:194 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -8 6 B -10 0 -12 -6 -10 C 4 12 0 0 12 D 8 6 0 0 4 E -6 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -8 6 B -10 0 -12 -6 -10 C 4 12 0 0 12 D 8 6 0 0 4 E -6 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -8 6 B -10 0 -12 -6 -10 C 4 12 0 0 12 D 8 6 0 0 4 E -6 10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6641: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (16) A B C E D (14) D C E B A (7) B C A D E (5) A E B D C (5) E D A C B (4) D B C E A (4) C B D E A (4) A E D B C (4) D E C A B (3) E D C B A (2) E A D C B (2) D E A B C (2) C B E D A (2) C B A D E (2) B C A E D (2) B A C D E (2) A E C B D (2) A B E D C (2) A B C D E (2) E D A B C (1) E C A D B (1) D E B A C (1) D E A C B (1) D C B E A (1) C D B E A (1) C B D A E (1) C B A E D (1) B C D A E (1) A E B C D (1) A D E B C (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -6 -2 -4 B 4 0 0 -2 -6 C 6 0 0 -16 0 D 2 2 16 0 12 E 4 6 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -2 -4 B 4 0 0 -2 -6 C 6 0 0 -16 0 D 2 2 16 0 12 E 4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=34 C=11 E=10 B=10 so E is eliminated. Round 2 votes counts: D=42 A=36 C=12 B=10 so B is eliminated. Round 3 votes counts: D=42 A=38 C=20 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:199 B:198 C:195 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 -4 B 4 0 0 -2 -6 C 6 0 0 -16 0 D 2 2 16 0 12 E 4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 -4 B 4 0 0 -2 -6 C 6 0 0 -16 0 D 2 2 16 0 12 E 4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 -4 B 4 0 0 -2 -6 C 6 0 0 -16 0 D 2 2 16 0 12 E 4 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6642: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (10) E D C A B (9) D E B C A (9) B A C D E (9) D E C B A (7) C A B E D (7) D E C A B (6) C E A D B (6) A B C E D (6) C E D A B (5) B A D E C (5) D B E A C (4) B A D C E (4) B D A E C (3) A C B E D (3) D E B A C (2) E D A C B (1) D C E A B (1) C A E D B (1) C A E B D (1) B E A D C (1) Total count = 100 A B C D E A 0 12 -24 -14 -22 B -12 0 -14 -22 -16 C 24 14 0 -4 -14 D 14 22 4 0 0 E 22 16 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.566076 E: 0.433924 Sum of squares = 0.508732028951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.566076 E: 1.000000 A B C D E A 0 12 -24 -14 -22 B -12 0 -14 -22 -16 C 24 14 0 -4 -14 D 14 22 4 0 0 E 22 16 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=22 E=20 C=20 A=9 so A is eliminated. Round 2 votes counts: D=29 B=28 C=23 E=20 so E is eliminated. Round 3 votes counts: D=39 C=33 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:226 D:220 C:210 A:176 B:168 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -24 -14 -22 B -12 0 -14 -22 -16 C 24 14 0 -4 -14 D 14 22 4 0 0 E 22 16 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -24 -14 -22 B -12 0 -14 -22 -16 C 24 14 0 -4 -14 D 14 22 4 0 0 E 22 16 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -24 -14 -22 B -12 0 -14 -22 -16 C 24 14 0 -4 -14 D 14 22 4 0 0 E 22 16 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6643: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (13) A D B C E (12) E D A C B (8) E C B D A (8) D A E C B (8) B A D C E (8) B C E A D (6) B C A D E (5) E C D B A (4) E C D A B (3) C B E A D (3) B C A E D (3) E C B A D (2) D E A C B (2) D A B E C (2) C E B D A (2) C E B A D (2) B A C D E (2) E D C A B (1) E C A D B (1) C B A E D (1) C B A D E (1) C A B D E (1) B C D E A (1) A D B E C (1) Total count = 100 A B C D E A 0 4 12 -4 14 B -4 0 6 -10 -8 C -12 -6 0 -10 -6 D 4 10 10 0 12 E -14 8 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 -4 14 B -4 0 6 -10 -8 C -12 -6 0 -10 -6 D 4 10 10 0 12 E -14 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 B=25 A=13 C=10 so C is eliminated. Round 2 votes counts: E=31 B=30 D=25 A=14 so A is eliminated. Round 3 votes counts: D=38 E=31 B=31 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:213 E:194 B:192 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 12 -4 14 B -4 0 6 -10 -8 C -12 -6 0 -10 -6 D 4 10 10 0 12 E -14 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 -4 14 B -4 0 6 -10 -8 C -12 -6 0 -10 -6 D 4 10 10 0 12 E -14 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 -4 14 B -4 0 6 -10 -8 C -12 -6 0 -10 -6 D 4 10 10 0 12 E -14 8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6644: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) B D C E A (7) E A C B D (5) D B C A E (5) B D E A C (5) D B A C E (4) C E B A D (4) C B D E A (4) A E D C B (4) E C A B D (3) E B C A D (3) E A B D C (3) E A B C D (3) D C B A E (3) D B A E C (3) D A E B C (3) B C D E A (3) E B A D C (2) C D B A E (2) C D A B E (2) A E D B C (2) A E C D B (2) A E B D C (2) A D E C B (2) E C B A D (1) E B A C D (1) D B C E A (1) D A E C B (1) D A C E B (1) D A C B E (1) C D A E B (1) C B E D A (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C D A (1) B C E D A (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -6 0 -16 B 4 0 2 20 -12 C 6 -2 0 -2 2 D 0 -20 2 0 0 E 16 12 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.593750000007 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 -4 -6 0 -16 B 4 0 2 20 -12 C 6 -2 0 -2 2 D 0 -20 2 0 0 E 16 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999552 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=22 E=21 B=18 A=14 so A is eliminated. Round 2 votes counts: E=32 D=25 C=25 B=18 so B is eliminated. Round 3 votes counts: D=37 E=34 C=29 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:213 B:207 C:202 D:191 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 0 -16 B 4 0 2 20 -12 C 6 -2 0 -2 2 D 0 -20 2 0 0 E 16 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999552 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 -16 B 4 0 2 20 -12 C 6 -2 0 -2 2 D 0 -20 2 0 0 E 16 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999552 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 -16 B 4 0 2 20 -12 C 6 -2 0 -2 2 D 0 -20 2 0 0 E 16 12 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.000000 E: 0.125000 Sum of squares = 0.593749999552 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6645: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) C D B A E (8) B D C E A (7) E A C B D (6) E A B D C (6) A E D B C (6) D B A E C (4) C E A B D (4) A E C D B (4) D C B A E (3) D B A C E (3) C E B A D (3) C B D E A (3) A E D C B (3) A C E D B (3) E C A B D (2) E A D B C (2) D A B E C (2) C B E A D (2) B D E A C (2) B C D E A (2) E B C A D (1) E B A D C (1) E A B C D (1) D A E C B (1) D A C B E (1) C B E D A (1) C B D A E (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D A C (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 0 2 12 B 4 0 -2 -12 0 C 0 2 0 -10 12 D -2 12 10 0 4 E -12 0 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.111111 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.506172839491 Cumulative probabilities = A: 0.666667 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 2 12 B 4 0 -2 -12 0 C 0 2 0 -10 12 D -2 12 10 0 4 E -12 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.111111 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.5061728388 Cumulative probabilities = A: 0.666667 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 A=20 E=19 B=12 so B is eliminated. Round 2 votes counts: D=33 C=27 E=20 A=20 so E is eliminated. Round 3 votes counts: A=36 D=34 C=30 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:205 C:202 B:195 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 2 12 B 4 0 -2 -12 0 C 0 2 0 -10 12 D -2 12 10 0 4 E -12 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.111111 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.5061728388 Cumulative probabilities = A: 0.666667 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 12 B 4 0 -2 -12 0 C 0 2 0 -10 12 D -2 12 10 0 4 E -12 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.111111 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.5061728388 Cumulative probabilities = A: 0.666667 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 12 B 4 0 -2 -12 0 C 0 2 0 -10 12 D -2 12 10 0 4 E -12 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.111111 C: 0.000000 D: 0.222222 E: 0.000000 Sum of squares = 0.5061728388 Cumulative probabilities = A: 0.666667 B: 0.777778 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6646: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) C D E A B (6) C D B E A (6) B C D A E (6) A E B C D (6) B A E D C (4) A E C D B (4) A C E B D (4) B D C E A (3) A E C B D (3) A E B D C (3) A B E D C (3) E A C D B (2) D C E B A (2) D B E C A (2) D B C E A (2) C D E B A (2) C B D A E (2) C A D B E (2) B E A D C (2) B D E C A (2) B D C A E (2) B A C E D (2) A C B E D (2) A B E C D (2) A B C E D (2) E D C B A (1) E D C A B (1) E D B C A (1) E D B A C (1) E C D A B (1) E B D A C (1) E A D C B (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B C A (1) C E D A B (1) C B A D E (1) C A E D B (1) C A B D E (1) B D E A C (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -8 -4 4 B 4 0 -4 8 8 C 8 4 0 16 10 D 4 -8 -16 0 0 E -4 -8 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -4 4 B 4 0 -4 8 8 C 8 4 0 16 10 D 4 -8 -16 0 0 E -4 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998155 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=23 C=22 D=15 E=10 so E is eliminated. Round 2 votes counts: A=34 B=24 C=23 D=19 so D is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:219 B:208 A:194 D:190 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 4 B 4 0 -4 8 8 C 8 4 0 16 10 D 4 -8 -16 0 0 E -4 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998155 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 4 B 4 0 -4 8 8 C 8 4 0 16 10 D 4 -8 -16 0 0 E -4 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998155 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 4 B 4 0 -4 8 8 C 8 4 0 16 10 D 4 -8 -16 0 0 E -4 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998155 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6647: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (10) B A E C D (9) A B E D C (7) E C B A D (6) D C A B E (6) C D E B A (6) E B A C D (4) D A C B E (4) B A D C E (4) E C D A B (3) E B C A D (3) D C E A B (3) D C B A E (3) D A B C E (3) C E D B A (3) E C D B A (2) E C B D A (2) C D E A B (2) C D B A E (2) B E A C D (2) B C E A D (2) B C A E D (2) A D B E C (2) E A B C D (1) D A C E B (1) C E B D A (1) C B E D A (1) B C E D A (1) B A E D C (1) B A C E D (1) B A C D E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 4 14 16 B 12 0 10 18 26 C -4 -10 0 8 -4 D -14 -18 -8 0 -2 E -16 -26 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 14 16 B 12 0 10 18 26 C -4 -10 0 8 -4 D -14 -18 -8 0 -2 E -16 -26 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=21 A=21 D=20 C=15 so C is eliminated. Round 2 votes counts: D=30 E=25 B=24 A=21 so A is eliminated. Round 3 votes counts: B=42 D=33 E=25 so E is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:233 A:211 C:195 E:182 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 14 16 B 12 0 10 18 26 C -4 -10 0 8 -4 D -14 -18 -8 0 -2 E -16 -26 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 14 16 B 12 0 10 18 26 C -4 -10 0 8 -4 D -14 -18 -8 0 -2 E -16 -26 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 14 16 B 12 0 10 18 26 C -4 -10 0 8 -4 D -14 -18 -8 0 -2 E -16 -26 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6648: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (6) E A C D B (5) E A B D C (5) D B C E A (4) B E A D C (4) B A E D C (4) E A B C D (3) D C E B A (3) D C B E A (3) C D B A E (3) B C D A E (3) A C E B D (3) A C B E D (3) A B E C D (3) E D A C B (2) E A D B C (2) D B E C A (2) D B E A C (2) D B C A E (2) C B D A E (2) C B A D E (2) C A E D B (2) C A D E B (2) B D E A C (2) B D C A E (2) B C A D E (2) B A C E D (2) B A C D E (2) A E C D B (2) A E C B D (2) A B C E D (2) E D C A B (1) E B A D C (1) D E C A B (1) D E B C A (1) D E B A C (1) D C E A B (1) D C B A E (1) C A E B D (1) C A D B E (1) B D C E A (1) B D A E C (1) B D A C E (1) B A D C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 20 22 12 B 4 0 18 16 10 C -20 -18 0 4 0 D -22 -16 -4 0 -8 E -12 -10 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 20 22 12 B 4 0 18 16 10 C -20 -18 0 4 0 D -22 -16 -4 0 -8 E -12 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=22 D=21 E=19 C=13 so C is eliminated. Round 2 votes counts: B=29 A=28 D=24 E=19 so E is eliminated. Round 3 votes counts: A=43 B=30 D=27 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:225 B:224 E:193 C:183 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 20 22 12 B 4 0 18 16 10 C -20 -18 0 4 0 D -22 -16 -4 0 -8 E -12 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 20 22 12 B 4 0 18 16 10 C -20 -18 0 4 0 D -22 -16 -4 0 -8 E -12 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 20 22 12 B 4 0 18 16 10 C -20 -18 0 4 0 D -22 -16 -4 0 -8 E -12 -10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6649: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) C B E D A (6) E B C D A (5) A B E D C (5) E B A C D (4) D C A B E (4) C D E B A (4) C D B E A (4) C D A B E (4) B E C D A (4) B E A C D (4) E B C A D (3) E B A D C (3) D A C E B (3) C B D A E (3) B E C A D (3) D C A E B (2) C E B D A (2) B A E D C (2) A E B D C (2) A D E B C (2) A D C E B (2) A D C B E (2) A D B E C (2) A B D C E (2) E C B D A (1) E A B D C (1) D E A C B (1) D C E A B (1) C D E A B (1) C D B A E (1) B E A D C (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E C D (1) A E D B C (1) A D E C B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -10 0 B 8 0 0 12 22 C 2 0 0 8 8 D 10 -12 -8 0 -2 E 0 -22 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250511 C: 0.749489 D: 0.000000 E: 0.000000 Sum of squares = 0.624489877807 Cumulative probabilities = A: 0.000000 B: 0.250511 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -10 0 B 8 0 0 12 22 C 2 0 0 8 8 D 10 -12 -8 0 -2 E 0 -22 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=21 D=19 B=18 E=17 so E is eliminated. Round 2 votes counts: B=33 C=26 A=22 D=19 so D is eliminated. Round 3 votes counts: A=34 C=33 B=33 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:209 D:194 A:190 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -10 0 B 8 0 0 12 22 C 2 0 0 8 8 D 10 -12 -8 0 -2 E 0 -22 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -10 0 B 8 0 0 12 22 C 2 0 0 8 8 D 10 -12 -8 0 -2 E 0 -22 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -10 0 B 8 0 0 12 22 C 2 0 0 8 8 D 10 -12 -8 0 -2 E 0 -22 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6650: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) C D B A E (8) A C E D B (8) A C E B D (8) E D B A C (7) B D C E A (5) E B D A C (4) D E B A C (4) D B C E A (4) C B D A E (4) C A B D E (4) B D E C A (4) C D B E A (3) B D E A C (3) B C D A E (3) A E B C D (3) E A D C B (2) E A B D C (2) D B E C A (2) C A D B E (2) A E B D C (2) E A D B C (1) D C B E A (1) C D A B E (1) C A E D B (1) C A E B D (1) C A B E D (1) B E D A C (1) A E C D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 12 -8 14 B 6 0 -8 10 -6 C -12 8 0 14 8 D 8 -10 -14 0 -4 E -14 6 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307692 B: 0.461538 C: 0.230769 D: 0.000000 E: 0.000000 Sum of squares = 0.36094674556 Cumulative probabilities = A: 0.307692 B: 0.769231 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -8 14 B 6 0 -8 10 -6 C -12 8 0 14 8 D 8 -10 -14 0 -4 E -14 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.461538 C: 0.230769 D: 0.000000 E: 0.000000 Sum of squares = 0.360946745617 Cumulative probabilities = A: 0.307692 B: 0.769231 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=25 E=16 B=16 D=11 so D is eliminated. Round 2 votes counts: A=32 C=26 B=22 E=20 so E is eliminated. Round 3 votes counts: B=37 A=37 C=26 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:209 A:206 B:201 E:194 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -6 12 -8 14 B 6 0 -8 10 -6 C -12 8 0 14 8 D 8 -10 -14 0 -4 E -14 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.461538 C: 0.230769 D: 0.000000 E: 0.000000 Sum of squares = 0.360946745617 Cumulative probabilities = A: 0.307692 B: 0.769231 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -8 14 B 6 0 -8 10 -6 C -12 8 0 14 8 D 8 -10 -14 0 -4 E -14 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.461538 C: 0.230769 D: 0.000000 E: 0.000000 Sum of squares = 0.360946745617 Cumulative probabilities = A: 0.307692 B: 0.769231 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -8 14 B 6 0 -8 10 -6 C -12 8 0 14 8 D 8 -10 -14 0 -4 E -14 6 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.461538 C: 0.230769 D: 0.000000 E: 0.000000 Sum of squares = 0.360946745617 Cumulative probabilities = A: 0.307692 B: 0.769231 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6651: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) E C D A B (9) B C A D E (9) C E D A B (7) C E B D A (7) E D A C B (5) B D A E C (5) D E A B C (4) E D C A B (3) C A B D E (3) B C E A D (3) A D B E C (3) D E A C B (2) D A E C B (2) D A E B C (2) C E A D B (2) C B E D A (2) C B E A D (2) B A D C E (2) B A C D E (2) A D E C B (2) E D A B C (1) E C B D A (1) E B D C A (1) E B C D A (1) D E B A C (1) D A B E C (1) C E D B A (1) C E B A D (1) C B A E D (1) B E D A C (1) B C E D A (1) B C A E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -10 -14 -10 B 6 0 0 10 -4 C 10 0 0 6 -10 D 14 -10 -6 0 0 E 10 4 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.170892 E: 0.829108 Sum of squares = 0.716624158797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.170892 E: 1.000000 A B C D E A 0 -6 -10 -14 -10 B 6 0 0 10 -4 C 10 0 0 6 -10 D 14 -10 -6 0 0 E 10 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591836938082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=26 E=21 D=12 A=6 so A is eliminated. Round 2 votes counts: B=36 C=26 E=21 D=17 so D is eliminated. Round 3 votes counts: B=40 E=34 C=26 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:206 C:203 D:199 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -10 -14 -10 B 6 0 0 10 -4 C 10 0 0 6 -10 D 14 -10 -6 0 0 E 10 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591836938082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -14 -10 B 6 0 0 10 -4 C 10 0 0 6 -10 D 14 -10 -6 0 0 E 10 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591836938082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -14 -10 B 6 0 0 10 -4 C 10 0 0 6 -10 D 14 -10 -6 0 0 E 10 4 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591836938082 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6652: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) C A B E D (6) B E D C A (6) D E B A C (4) D C B E A (4) D A C E B (4) B E C A D (4) A D C E B (4) E D B A C (3) D E B C A (3) D E A B C (3) D C A E B (3) C A D B E (3) C A B D E (3) B E C D A (3) B C E A D (3) E B D C A (2) E B A D C (2) E B A C D (2) D A E B C (2) C D A B E (2) C B D E A (2) C B A E D (2) B E A C D (2) B D E C A (2) A E B C D (2) A C B E D (2) E A B D C (1) D C A B E (1) D A E C B (1) C D B E A (1) B C E D A (1) A E D B C (1) A C E B D (1) A C D E B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -2 -18 -20 B 16 0 18 18 -2 C 2 -18 0 -16 -12 D 18 -18 16 0 -12 E 20 2 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -2 -18 -20 B 16 0 18 18 -2 C 2 -18 0 -16 -12 D 18 -18 16 0 -12 E 20 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=22 B=21 C=19 A=13 so A is eliminated. Round 2 votes counts: D=29 E=25 C=24 B=22 so B is eliminated. Round 3 votes counts: E=41 D=31 C=28 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:225 E:223 D:202 C:178 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -2 -18 -20 B 16 0 18 18 -2 C 2 -18 0 -16 -12 D 18 -18 16 0 -12 E 20 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -18 -20 B 16 0 18 18 -2 C 2 -18 0 -16 -12 D 18 -18 16 0 -12 E 20 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -18 -20 B 16 0 18 18 -2 C 2 -18 0 -16 -12 D 18 -18 16 0 -12 E 20 2 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6653: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) D E A C B (6) E A C D B (5) D B E C A (5) E D A C B (4) D E A B C (4) D B E A C (4) C A E D B (4) B D E A C (4) B C A D E (4) E A D C B (3) D E B A C (3) B D C E A (3) B D C A E (3) B C A E D (3) E D A B C (2) D E C A B (2) D B C E A (2) C B A E D (2) C B A D E (2) C A E B D (2) B D E C A (2) B C D A E (2) A E C B D (2) A C E D B (2) A C E B D (2) A C B E D (2) E C A D B (1) E B A D C (1) E A B D C (1) E A B C D (1) D E B C A (1) D C B E A (1) C D B A E (1) C B D A E (1) B A E D C (1) B A C E D (1) B A C D E (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 6 2 0 -10 B -6 0 -2 -2 4 C -2 2 0 -4 -8 D 0 2 4 0 2 E 10 -4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.107135 B: 0.000000 C: 0.000000 D: 0.892865 E: 0.000000 Sum of squares = 0.808686071019 Cumulative probabilities = A: 0.107135 B: 0.107135 C: 0.107135 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 0 -10 B -6 0 -2 -2 4 C -2 2 0 -4 -8 D 0 2 4 0 2 E 10 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222317663 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=24 C=20 E=18 A=10 so A is eliminated. Round 2 votes counts: D=28 C=26 B=24 E=22 so E is eliminated. Round 3 votes counts: D=37 C=35 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:206 D:204 A:199 B:197 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 0 -10 B -6 0 -2 -2 4 C -2 2 0 -4 -8 D 0 2 4 0 2 E 10 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222317663 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 -10 B -6 0 -2 -2 4 C -2 2 0 -4 -8 D 0 2 4 0 2 E 10 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222317663 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 -10 B -6 0 -2 -2 4 C -2 2 0 -4 -8 D 0 2 4 0 2 E 10 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222317663 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6654: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) A D C E B (7) D E B C A (5) D B E C A (5) D A B E C (5) B E C D A (5) A C D E B (5) D B E A C (4) C E B A D (4) A C E B D (4) A C B E D (4) E C B A D (3) C E A B D (3) C A E B D (3) B D E C A (3) A D C B E (3) A B C D E (3) D E A C B (2) D A B C E (2) B E D C A (2) B E C A D (2) A D B C E (2) A C E D B (2) E D C B A (1) E D C A B (1) E C D B A (1) E B C D A (1) D E C B A (1) D E C A B (1) D B A E C (1) D A E B C (1) D A C E B (1) C B E A D (1) C B A E D (1) B C A E D (1) B A D C E (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -2 16 -4 B 0 0 2 -2 8 C 2 -2 0 4 12 D -16 2 -4 0 6 E 4 -8 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.318507 B: 0.681493 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.565879469604 Cumulative probabilities = A: 0.318507 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 16 -4 B 0 0 2 -2 8 C 2 -2 0 4 12 D -16 2 -4 0 6 E 4 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499676 B: 0.500324 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000209603 Cumulative probabilities = A: 0.499676 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=28 B=22 C=12 E=7 so E is eliminated. Round 2 votes counts: A=31 D=30 B=23 C=16 so C is eliminated. Round 3 votes counts: A=37 B=32 D=31 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:208 A:205 B:204 D:194 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 16 -4 B 0 0 2 -2 8 C 2 -2 0 4 12 D -16 2 -4 0 6 E 4 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499676 B: 0.500324 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000209603 Cumulative probabilities = A: 0.499676 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 16 -4 B 0 0 2 -2 8 C 2 -2 0 4 12 D -16 2 -4 0 6 E 4 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499676 B: 0.500324 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000209603 Cumulative probabilities = A: 0.499676 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 16 -4 B 0 0 2 -2 8 C 2 -2 0 4 12 D -16 2 -4 0 6 E 4 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499676 B: 0.500324 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000209603 Cumulative probabilities = A: 0.499676 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6655: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) A E D B C (7) E A B D C (6) D C A B E (6) B E C A D (5) D C B A E (4) C D B A E (4) B E A C D (4) A E B C D (4) E B A D C (3) E B A C D (3) D C B E A (3) D A E C B (3) C D A B E (3) B C E A D (3) A D C E B (3) D C E B A (2) D C A E B (2) D B E C A (2) C D B E A (2) C A D B E (2) B E C D A (2) B C E D A (2) A E B D C (2) E B D A C (1) E A B C D (1) D E C B A (1) D E A B C (1) D B C E A (1) C B E D A (1) C B E A D (1) C B A E D (1) C B A D E (1) C A D E B (1) C A B E D (1) B D E C A (1) B C D E A (1) A E D C B (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -4 2 8 B -4 0 -2 -12 2 C 4 2 0 -14 4 D -2 12 14 0 2 E -8 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.100000 D: 0.200000 E: 0.000000 Sum of squares = 0.540000000026 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 2 8 B -4 0 -2 -12 2 C 4 2 0 -14 4 D -2 12 14 0 2 E -8 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.100000 D: 0.200000 E: 0.000000 Sum of squares = 0.539999999546 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=19 B=18 C=17 E=14 so E is eliminated. Round 2 votes counts: D=32 A=26 B=25 C=17 so C is eliminated. Round 3 votes counts: D=41 A=30 B=29 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:205 C:198 B:192 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 2 8 B -4 0 -2 -12 2 C 4 2 0 -14 4 D -2 12 14 0 2 E -8 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.100000 D: 0.200000 E: 0.000000 Sum of squares = 0.539999999546 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 2 8 B -4 0 -2 -12 2 C 4 2 0 -14 4 D -2 12 14 0 2 E -8 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.100000 D: 0.200000 E: 0.000000 Sum of squares = 0.539999999546 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 2 8 B -4 0 -2 -12 2 C 4 2 0 -14 4 D -2 12 14 0 2 E -8 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.100000 D: 0.200000 E: 0.000000 Sum of squares = 0.539999999546 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6656: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (5) D E A C B (5) C A D E B (5) B E C D A (5) E D B A C (4) E D A C B (4) C E B D A (4) E D A B C (3) D E A B C (3) C B E A D (3) B E D C A (3) B C E D A (3) B C E A D (3) A D C E B (3) A D B C E (3) A C D B E (3) E C D B A (2) D A E C B (2) D A E B C (2) C E D B A (2) C E D A B (2) C D A E B (2) C B E D A (2) C B A E D (2) C A B D E (2) B E D A C (2) B A D E C (2) B A D C E (2) A C B D E (2) A B D C E (2) E D B C A (1) E B D A C (1) D E C A B (1) C B A D E (1) B E A D C (1) B C A E D (1) B C A D E (1) B A E D C (1) B A C E D (1) A D E C B (1) A D E B C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 0 -16 -18 B -6 0 -8 -10 -6 C 0 8 0 -8 0 D 16 10 8 0 -10 E 18 6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.330147 D: 0.000000 E: 0.669853 Sum of squares = 0.557699826184 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.330147 D: 0.330147 E: 1.000000 A B C D E A 0 6 0 -16 -18 B -6 0 -8 -10 -6 C 0 8 0 -8 0 D 16 10 8 0 -10 E 18 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=25 B=25 E=20 A=17 D=13 so D is eliminated. Round 2 votes counts: E=29 C=25 B=25 A=21 so A is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:217 D:212 C:200 A:186 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 -16 -18 B -6 0 -8 -10 -6 C 0 8 0 -8 0 D 16 10 8 0 -10 E 18 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -16 -18 B -6 0 -8 -10 -6 C 0 8 0 -8 0 D 16 10 8 0 -10 E 18 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -16 -18 B -6 0 -8 -10 -6 C 0 8 0 -8 0 D 16 10 8 0 -10 E 18 6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.000000 E: 0.500003 Sum of squares = 0.500000000006 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499997 D: 0.499997 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6657: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (19) C B D E A (15) A E B C D (6) D E C B A (5) D E A C B (5) D C B E A (5) B C A D E (5) E D A C B (4) E A D C B (4) A E B D C (4) D E C A B (3) C B D A E (3) B C D A E (3) D C E B A (2) D B C E A (2) C B A D E (2) B C D E A (2) B C A E D (2) B A C E D (2) A B E C D (2) A B C E D (2) C E A D B (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 2 0 2 4 B -2 0 0 -2 -8 C 0 0 0 -6 -6 D -2 2 6 0 6 E -4 8 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.825328 B: 0.000000 C: 0.174672 D: 0.000000 E: 0.000000 Sum of squares = 0.71167638001 Cumulative probabilities = A: 0.825328 B: 0.825328 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 4 B -2 0 0 -2 -8 C 0 0 0 -6 -6 D -2 2 6 0 6 E -4 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001325 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=22 C=21 B=15 E=8 so E is eliminated. Round 2 votes counts: A=38 D=26 C=21 B=15 so B is eliminated. Round 3 votes counts: A=41 C=33 D=26 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:206 A:204 E:202 B:194 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 2 4 B -2 0 0 -2 -8 C 0 0 0 -6 -6 D -2 2 6 0 6 E -4 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001325 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 4 B -2 0 0 -2 -8 C 0 0 0 -6 -6 D -2 2 6 0 6 E -4 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001325 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 4 B -2 0 0 -2 -8 C 0 0 0 -6 -6 D -2 2 6 0 6 E -4 8 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001325 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6658: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (12) A C E D B (7) C A E B D (6) A C B E D (6) E D B C A (4) E C A D B (4) E B D C A (4) E B C D A (4) B E D C A (4) B D E A C (4) A C B D E (4) E D C A B (3) C A E D B (3) B E C D A (3) B D A C E (3) A B C D E (3) E C B A D (2) D E B C A (2) D B E C A (2) D B E A C (2) B C E A D (2) B C A E D (2) A D C B E (2) A C D B E (2) E C D B A (1) E C B D A (1) D A C B E (1) D A B C E (1) C E A B D (1) B D A E C (1) B A C E D (1) A D C E B (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -20 -4 -10 B 8 0 10 30 12 C 20 -10 0 6 -6 D 4 -30 -6 0 -18 E 10 -12 6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -20 -4 -10 B 8 0 10 30 12 C 20 -10 0 6 -6 D 4 -30 -6 0 -18 E 10 -12 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=27 E=23 C=10 D=8 so D is eliminated. Round 2 votes counts: B=36 A=29 E=25 C=10 so C is eliminated. Round 3 votes counts: A=38 B=36 E=26 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:230 E:211 C:205 A:179 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -20 -4 -10 B 8 0 10 30 12 C 20 -10 0 6 -6 D 4 -30 -6 0 -18 E 10 -12 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -20 -4 -10 B 8 0 10 30 12 C 20 -10 0 6 -6 D 4 -30 -6 0 -18 E 10 -12 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -20 -4 -10 B 8 0 10 30 12 C 20 -10 0 6 -6 D 4 -30 -6 0 -18 E 10 -12 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6659: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) B A E D C (7) E B A C D (6) E C D A B (5) D C B A E (5) E C D B A (4) D C A B E (4) E A B C D (3) D A C B E (3) D A B C E (3) C D E A B (3) B E A D C (3) B D C A E (3) B A D E C (3) B A D C E (3) A D C B E (3) A B D C E (3) E C A D B (2) E A C D B (2) C E D A B (2) C D B E A (2) A E B D C (2) A D B C E (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E A C B D (1) D C A E B (1) D B C A E (1) C E D B A (1) C B E D A (1) B E D A C (1) B D C E A (1) B D A C E (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 10 0 -6 18 B -10 0 -6 -10 4 C 0 6 0 -6 8 D 6 10 6 0 10 E -18 -4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999587 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -6 18 B -10 0 -6 -10 4 C 0 6 0 -6 8 D 6 10 6 0 10 E -18 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=22 C=20 D=17 A=14 so A is eliminated. Round 2 votes counts: E=29 B=28 D=23 C=20 so C is eliminated. Round 3 votes counts: D=39 E=32 B=29 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:211 C:204 B:189 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -6 18 B -10 0 -6 -10 4 C 0 6 0 -6 8 D 6 10 6 0 10 E -18 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -6 18 B -10 0 -6 -10 4 C 0 6 0 -6 8 D 6 10 6 0 10 E -18 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -6 18 B -10 0 -6 -10 4 C 0 6 0 -6 8 D 6 10 6 0 10 E -18 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6660: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) A C D B E (9) E B D A C (7) A D C B E (7) A D B C E (7) E B C D A (6) E C B D A (5) E B A D C (4) E C B A D (3) C E A D B (3) C D B A E (3) A C D E B (3) E A C D B (2) C D A B E (2) C B D E A (2) C A D B E (2) B D E C A (2) A C E D B (2) E C A D B (1) E C A B D (1) E B C A D (1) E A B D C (1) E A B C D (1) D C B A E (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C D B E A (1) B E D C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A C E (1) A E D C B (1) A E C D B (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 2 2 -12 B 6 0 -4 2 -12 C -2 4 0 0 -2 D -2 -2 0 0 -8 E 12 12 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 2 2 -12 B 6 0 -4 2 -12 C -2 4 0 0 -2 D -2 -2 0 0 -8 E 12 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 A=32 C=15 B=7 D=3 so D is eliminated. Round 2 votes counts: E=43 A=34 C=16 B=7 so B is eliminated. Round 3 votes counts: E=47 A=35 C=18 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:200 B:196 D:194 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 2 2 -12 B 6 0 -4 2 -12 C -2 4 0 0 -2 D -2 -2 0 0 -8 E 12 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 2 -12 B 6 0 -4 2 -12 C -2 4 0 0 -2 D -2 -2 0 0 -8 E 12 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 2 -12 B 6 0 -4 2 -12 C -2 4 0 0 -2 D -2 -2 0 0 -8 E 12 12 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6661: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) E A D B C (6) C A D B E (6) E B D A C (5) E B A D C (5) C B D A E (5) E B A C D (4) E A B D C (4) E C A D B (3) C D A B E (3) C B E D A (3) B E D C A (3) B E D A C (3) B D C A E (3) B C D E A (3) A D C B E (3) A C D E B (3) A C D B E (3) E B C D A (2) D A C B E (2) C E A D B (2) B C E D A (2) B C D A E (2) A E D C B (2) A E C D B (2) E C A B D (1) E B C A D (1) E A C D B (1) E A B C D (1) D C A B E (1) D B A C E (1) C E B A D (1) C D B A E (1) C B A D E (1) C A D E B (1) B E C D A (1) B D C E A (1) B D A C E (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 4 8 16 -16 B -4 0 -2 4 -4 C -8 2 0 4 -4 D -16 -4 -4 0 -16 E 16 4 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 16 -16 B -4 0 -2 4 -4 C -8 2 0 4 -4 D -16 -4 -4 0 -16 E 16 4 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=23 B=19 A=15 D=4 so D is eliminated. Round 2 votes counts: E=39 C=24 B=20 A=17 so A is eliminated. Round 3 votes counts: E=45 C=35 B=20 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:206 B:197 C:197 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 16 -16 B -4 0 -2 4 -4 C -8 2 0 4 -4 D -16 -4 -4 0 -16 E 16 4 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 16 -16 B -4 0 -2 4 -4 C -8 2 0 4 -4 D -16 -4 -4 0 -16 E 16 4 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 16 -16 B -4 0 -2 4 -4 C -8 2 0 4 -4 D -16 -4 -4 0 -16 E 16 4 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6662: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) B D E C A (6) A E B D C (6) D B E C A (5) C D B E A (5) C A B E D (5) C A B D E (5) D E A B C (4) B E A D C (4) E D B A C (3) C B D E A (3) C A D E B (3) B E D C A (3) B D C E A (3) A C E D B (3) E B D A C (2) B E D A C (2) B C E D A (2) A C E B D (2) A C D E B (2) E D A B C (1) E B A D C (1) D E B A C (1) D E A C B (1) D B E A C (1) C D E B A (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E D A (1) C B D A E (1) C B A E D (1) C B A D E (1) C A D B E (1) B E C A D (1) B C D E A (1) A E C D B (1) A E C B D (1) A E B C D (1) A D E C B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -2 2 -2 B -2 0 18 8 10 C 2 -18 0 -8 -12 D -2 -8 8 0 -4 E 2 -10 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408126 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 2 -2 2 -2 B -2 0 18 8 10 C 2 -18 0 -8 -12 D -2 -8 8 0 -4 E 2 -10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408112 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=29 B=22 D=12 E=7 so E is eliminated. Round 2 votes counts: C=30 A=29 B=25 D=16 so D is eliminated. Round 3 votes counts: B=35 A=35 C=30 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:217 E:204 A:200 D:197 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 2 -2 B -2 0 18 8 10 C 2 -18 0 -8 -12 D -2 -8 8 0 -4 E 2 -10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408112 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 -2 B -2 0 18 8 10 C 2 -18 0 -8 -12 D -2 -8 8 0 -4 E 2 -10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408112 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 -2 B -2 0 18 8 10 C 2 -18 0 -8 -12 D -2 -8 8 0 -4 E 2 -10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408112 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6663: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) D E A B C (5) A C E B D (5) A C B D E (5) E D B C A (4) E D A B C (4) E A C B D (4) C B A E D (4) B C D A E (4) D B C A E (3) D B A C E (3) D A E B C (3) D A B E C (3) B D C A E (3) B C D E A (3) B C A D E (3) A E C B D (3) A C B E D (3) E D B A C (2) E C B A D (2) D E B C A (2) D B E C A (2) D B C E A (2) D A B C E (2) C B A D E (2) A E C D B (2) A D B C E (2) E D C A B (1) E D A C B (1) E C A B D (1) E B C D A (1) E A C D B (1) D E B A C (1) C B E A D (1) C A B E D (1) A E D C B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 16 22 2 8 B -16 0 6 -10 -8 C -22 -6 0 -10 -8 D -2 10 10 0 0 E -8 8 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998226 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 22 2 8 B -16 0 6 -10 -8 C -22 -6 0 -10 -8 D -2 10 10 0 0 E -8 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 A=23 B=13 C=8 so C is eliminated. Round 2 votes counts: E=30 D=26 A=24 B=20 so B is eliminated. Round 3 votes counts: D=36 A=33 E=31 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:209 E:204 B:186 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 22 2 8 B -16 0 6 -10 -8 C -22 -6 0 -10 -8 D -2 10 10 0 0 E -8 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 22 2 8 B -16 0 6 -10 -8 C -22 -6 0 -10 -8 D -2 10 10 0 0 E -8 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 22 2 8 B -16 0 6 -10 -8 C -22 -6 0 -10 -8 D -2 10 10 0 0 E -8 8 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999963059 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6664: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) D E A B C (8) D C B A E (7) D B C E A (7) A E C B D (7) B C D E A (5) D C B E A (4) D A E C B (4) D E B C A (3) D E B A C (3) B C E A D (3) B C A E D (3) A E B C D (3) A C B E D (3) E D A B C (2) E A D B C (2) E A B D C (2) D C A B E (2) D B E C A (2) D A C E B (2) C B D A E (2) C B A E D (2) C B A D E (2) C A B D E (2) E D B A C (1) E A D C B (1) D E C B A (1) D E A C B (1) D A C B E (1) C D A B E (1) C A B E D (1) B E D C A (1) B D C E A (1) A E D C B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 -16 -10 B -6 0 8 -8 -2 C -2 -8 0 -12 0 D 16 8 12 0 18 E 10 2 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -16 -10 B -6 0 8 -8 -2 C -2 -8 0 -12 0 D 16 8 12 0 18 E 10 2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 E=16 A=16 B=13 C=10 so C is eliminated. Round 2 votes counts: D=46 B=19 A=19 E=16 so E is eliminated. Round 3 votes counts: D=49 A=32 B=19 so B is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:197 B:196 A:191 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -16 -10 B -6 0 8 -8 -2 C -2 -8 0 -12 0 D 16 8 12 0 18 E 10 2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -16 -10 B -6 0 8 -8 -2 C -2 -8 0 -12 0 D 16 8 12 0 18 E 10 2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -16 -10 B -6 0 8 -8 -2 C -2 -8 0 -12 0 D 16 8 12 0 18 E 10 2 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6665: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) C E A D B (7) B C E D A (7) A D B E C (7) C E B D A (5) C A E D B (5) E C B D A (4) C B E A D (4) B D E A C (4) C E B A D (3) B E C D A (3) A D C E B (3) D A E B C (2) D A B E C (2) C E A B D (2) C B E D A (2) C A B D E (2) A D E B C (2) A D B C E (2) A C E D B (2) A C D E B (2) E D C B A (1) E D C A B (1) E D B A C (1) E C D A B (1) E B D A C (1) E A D C B (1) D E A B C (1) C E D A B (1) C A D E B (1) C A D B E (1) B E D A C (1) B D A C E (1) B C A D E (1) B A C D E (1) A E D C B (1) A D E C B (1) A D C B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -2 2 0 B 0 0 -2 8 4 C 2 2 0 10 10 D -2 -8 -10 0 -6 E 0 -4 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 2 0 B 0 0 -2 8 4 C 2 2 0 10 10 D -2 -8 -10 0 -6 E 0 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=29 A=23 E=10 D=5 so D is eliminated. Round 2 votes counts: C=33 B=29 A=27 E=11 so E is eliminated. Round 3 votes counts: C=40 B=31 A=29 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:205 A:200 E:196 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 2 0 B 0 0 -2 8 4 C 2 2 0 10 10 D -2 -8 -10 0 -6 E 0 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 0 B 0 0 -2 8 4 C 2 2 0 10 10 D -2 -8 -10 0 -6 E 0 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 0 B 0 0 -2 8 4 C 2 2 0 10 10 D -2 -8 -10 0 -6 E 0 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6666: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (18) C A D E B (11) D C A E B (9) B E D A C (7) A C E D B (7) B E A C D (6) E B A C D (5) D B E C A (5) B E A D C (5) D C B A E (4) B D E C A (4) E A B C D (3) C D A E B (3) D B C E A (2) D A C E B (2) A E C B D (2) E B A D C (1) E A C B D (1) B E D C A (1) B E C A D (1) B D E A C (1) B C E A D (1) A C E B D (1) Total count = 100 A B C D E A 0 14 -18 -12 14 B -14 0 -16 -22 10 C 18 16 0 -18 16 D 12 22 18 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -18 -12 14 B -14 0 -16 -22 10 C 18 16 0 -18 16 D 12 22 18 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 B=26 C=14 E=10 A=10 so E is eliminated. Round 2 votes counts: D=40 B=32 C=14 A=14 so C is eliminated. Round 3 votes counts: D=43 B=32 A=25 so A is eliminated. Round 4 votes counts: D=61 B=39 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:235 C:216 A:199 B:179 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -18 -12 14 B -14 0 -16 -22 10 C 18 16 0 -18 16 D 12 22 18 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -18 -12 14 B -14 0 -16 -22 10 C 18 16 0 -18 16 D 12 22 18 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -18 -12 14 B -14 0 -16 -22 10 C 18 16 0 -18 16 D 12 22 18 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6667: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (13) B C E A D (7) A D E C B (7) D B C E A (5) D A E B C (5) D A B E C (5) C E A D B (5) C E A B D (5) B D C E A (5) A E D C B (5) E C A B D (4) B D C A E (3) B C E D A (3) B C D E A (3) D B A E C (2) D B A C E (2) D A C E B (2) C B E A D (2) B A E C D (2) A E C D B (2) E C B A D (1) E A C D B (1) E A C B D (1) D C B E A (1) D B C A E (1) C E B A D (1) C B D E A (1) B E C A D (1) B D A E C (1) B D A C E (1) A E C B D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 4 -6 8 B -16 0 -4 -14 -8 C -4 4 0 -20 -6 D 6 14 20 0 18 E -8 8 6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 -6 8 B -16 0 -4 -14 -8 C -4 4 0 -20 -6 D 6 14 20 0 18 E -8 8 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=26 A=17 C=14 E=7 so E is eliminated. Round 2 votes counts: D=36 B=26 C=19 A=19 so C is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 A:211 E:194 C:187 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 4 -6 8 B -16 0 -4 -14 -8 C -4 4 0 -20 -6 D 6 14 20 0 18 E -8 8 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 -6 8 B -16 0 -4 -14 -8 C -4 4 0 -20 -6 D 6 14 20 0 18 E -8 8 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 -6 8 B -16 0 -4 -14 -8 C -4 4 0 -20 -6 D 6 14 20 0 18 E -8 8 6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6668: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (7) B A C E D (6) E D C B A (5) E C B D A (5) D A E B C (5) E D A B C (4) D A B C E (4) C B E A D (4) C B A D E (4) E C D B A (3) E B A C D (3) C E B D A (3) C B A E D (3) B E A C D (3) B C A E D (3) D E C A B (2) D E A B C (2) D C E A B (2) D A B E C (2) C D A B E (2) A D B E C (2) A D B C E (2) A C B D E (2) A B E D C (2) A B E C D (2) A B D E C (2) E D B C A (1) E C B A D (1) E B C A D (1) E B A D C (1) D E A C B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A C B E (1) C E B A D (1) C D E B A (1) C D B A E (1) C A B D E (1) A E B D C (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 12 6 14 B -2 0 10 12 10 C -12 -10 0 14 0 D -6 -12 -14 0 -6 E -14 -10 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 6 14 B -2 0 10 12 10 C -12 -10 0 14 0 D -6 -12 -14 0 -6 E -14 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 A=22 C=20 B=12 so B is eliminated. Round 2 votes counts: A=28 E=27 C=23 D=22 so D is eliminated. Round 3 votes counts: A=42 E=32 C=26 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:215 C:196 E:191 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 6 14 B -2 0 10 12 10 C -12 -10 0 14 0 D -6 -12 -14 0 -6 E -14 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 6 14 B -2 0 10 12 10 C -12 -10 0 14 0 D -6 -12 -14 0 -6 E -14 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 6 14 B -2 0 10 12 10 C -12 -10 0 14 0 D -6 -12 -14 0 -6 E -14 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6669: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) D E B C A (6) C B A D E (6) D E A B C (5) B C D E A (5) E A D B C (4) D E B A C (4) B C D A E (4) A D E C B (4) A C E B D (4) E D B A C (3) E D A C B (3) E A B D C (3) E A B C D (3) D B C E A (3) C B A E D (3) E B D A C (2) D C B A E (2) C A B E D (2) C A B D E (2) B E C A D (2) A E D C B (2) A C B D E (2) E D A B C (1) E B D C A (1) E B C A D (1) E B A C D (1) E A D C B (1) D C A B E (1) D A E C B (1) C B D A E (1) C A D B E (1) B E D C A (1) B D E C A (1) B D C E A (1) B C A E D (1) A E C D B (1) A E C B D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 12 10 -2 B -4 0 4 12 -4 C -12 -4 0 2 -2 D -10 -12 -2 0 0 E 2 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.102638 E: 0.897362 Sum of squares = 0.815793146246 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.102638 E: 1.000000 A B C D E A 0 4 12 10 -2 B -4 0 4 12 -4 C -12 -4 0 2 -2 D -10 -12 -2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222225081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 D=22 C=15 B=15 so C is eliminated. Round 2 votes counts: A=30 B=25 E=23 D=22 so D is eliminated. Round 3 votes counts: E=38 A=32 B=30 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:212 B:204 E:204 C:192 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 12 10 -2 B -4 0 4 12 -4 C -12 -4 0 2 -2 D -10 -12 -2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222225081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 10 -2 B -4 0 4 12 -4 C -12 -4 0 2 -2 D -10 -12 -2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222225081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 10 -2 B -4 0 4 12 -4 C -12 -4 0 2 -2 D -10 -12 -2 0 0 E 2 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222225081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6670: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) C E A B D (8) C E B A D (7) E A D B C (6) B D A C E (6) C E D A B (5) C B D A E (5) B A D C E (5) D A E B C (4) C B A D E (4) B A D E C (4) E C A D B (3) C B A E D (3) B C A D E (3) E D A B C (2) E C A B D (2) E A C D B (2) D E A B C (2) D B A C E (2) C B E A D (2) B A C D E (2) A E D B C (2) E D A C B (1) E C D A B (1) E A C B D (1) E A B C D (1) D E C A B (1) D B A E C (1) C D B A E (1) C B E D A (1) B D C A E (1) B D A E C (1) B C D A E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 4 14 10 B -2 0 6 16 4 C -4 -6 0 4 12 D -14 -16 -4 0 6 E -10 -4 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 14 10 B -2 0 6 16 4 C -4 -6 0 4 12 D -14 -16 -4 0 6 E -10 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=23 E=19 D=18 A=4 so A is eliminated. Round 2 votes counts: C=36 B=24 E=21 D=19 so D is eliminated. Round 3 votes counts: C=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:212 C:203 D:186 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 14 10 B -2 0 6 16 4 C -4 -6 0 4 12 D -14 -16 -4 0 6 E -10 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 14 10 B -2 0 6 16 4 C -4 -6 0 4 12 D -14 -16 -4 0 6 E -10 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 14 10 B -2 0 6 16 4 C -4 -6 0 4 12 D -14 -16 -4 0 6 E -10 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6671: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) C B A D E (10) D E C A B (7) B A C E D (7) D E A C B (6) A E B D C (5) C B D E A (4) C B A E D (4) A E D B C (4) A B E C D (4) D E A B C (3) C D E B A (3) C D E A B (3) C D B E A (3) C B D A E (3) B C A E D (3) A B E D C (3) E A D B C (2) B A E D C (2) A B C E D (2) E D A C B (1) D C E B A (1) D C E A B (1) D A C E B (1) C D B A E (1) C D A E B (1) B D E C A (1) B C E D A (1) B C D E A (1) B A E C D (1) A C D E B (1) Total count = 100 A B C D E A 0 10 6 -4 4 B -10 0 0 2 0 C -6 0 0 4 0 D 4 -2 -4 0 0 E -4 0 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775505 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -4 4 B -10 0 0 2 0 C -6 0 0 4 0 D 4 -2 -4 0 0 E -4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775465 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=19 A=19 B=16 E=14 so E is eliminated. Round 2 votes counts: C=32 D=31 A=21 B=16 so B is eliminated. Round 3 votes counts: C=37 D=32 A=31 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:208 C:199 D:199 E:198 B:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 -4 4 B -10 0 0 2 0 C -6 0 0 4 0 D 4 -2 -4 0 0 E -4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775465 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -4 4 B -10 0 0 2 0 C -6 0 0 4 0 D 4 -2 -4 0 0 E -4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775465 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -4 4 B -10 0 0 2 0 C -6 0 0 4 0 D 4 -2 -4 0 0 E -4 0 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775465 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6672: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) D E B A C (8) E B D C A (6) B D E A C (6) E D B A C (5) B E D C A (5) B C A E D (5) C A E B D (4) A C B D E (4) E D B C A (3) D E A C B (3) D B E A C (3) D A B C E (3) B E D A C (3) A C D B E (3) E D C A B (2) D A C E B (2) C E A D B (2) C A D E B (2) B A C E D (2) B A C D E (2) A C D E B (2) E C B D A (1) E C B A D (1) E C A B D (1) E B C A D (1) D E C A B (1) D E A B C (1) D B A E C (1) D A E C B (1) C E B A D (1) C D A E B (1) C A E D B (1) C A D B E (1) B D A E C (1) B A D C E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 6 -12 -6 B 10 0 16 8 2 C -6 -16 0 -14 -6 D 12 -8 14 0 -4 E 6 -2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -12 -6 B 10 0 16 8 2 C -6 -16 0 -14 -6 D 12 -8 14 0 -4 E 6 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989121 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 C=21 E=20 A=11 so A is eliminated. Round 2 votes counts: C=30 B=26 D=24 E=20 so E is eliminated. Round 3 votes counts: D=34 C=33 B=33 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:207 E:207 A:189 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 -12 -6 B 10 0 16 8 2 C -6 -16 0 -14 -6 D 12 -8 14 0 -4 E 6 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989121 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -12 -6 B 10 0 16 8 2 C -6 -16 0 -14 -6 D 12 -8 14 0 -4 E 6 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989121 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -12 -6 B 10 0 16 8 2 C -6 -16 0 -14 -6 D 12 -8 14 0 -4 E 6 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989121 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6673: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) B C D E A (5) A E C B D (5) A B E C D (5) E C D A B (4) B C A E D (4) A E D B C (4) E C A D B (3) E A C D B (3) C E D B A (3) C E A B D (3) C B E A D (3) B A C D E (3) A E D C B (3) E D A C B (2) D E A C B (2) D C E B A (2) C B E D A (2) B D C E A (2) B D C A E (2) B D A C E (2) B A D E C (2) A E C D B (2) A E B C D (2) A D E B C (2) A B E D C (2) A B D E C (2) E A D C B (1) D E C A B (1) D E A B C (1) D C B E A (1) D B E C A (1) D B C E A (1) D A B E C (1) C E B D A (1) C E B A D (1) C E A D B (1) C B A E D (1) C A E B D (1) B C E D A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E B D C (1) A D B E C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -2 18 0 B -8 0 -4 22 2 C 2 4 0 28 0 D -18 -22 -28 0 -22 E 0 -2 0 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.688402 D: 0.000000 E: 0.311598 Sum of squares = 0.570990248816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.688402 D: 0.688402 E: 1.000000 A B C D E A 0 8 -2 18 0 B -8 0 -4 22 2 C 2 4 0 28 0 D -18 -22 -28 0 -22 E 0 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=24 C=22 E=13 D=10 so D is eliminated. Round 2 votes counts: A=32 B=26 C=25 E=17 so E is eliminated. Round 3 votes counts: A=41 C=33 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:212 E:210 B:206 D:155 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 18 0 B -8 0 -4 22 2 C 2 4 0 28 0 D -18 -22 -28 0 -22 E 0 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 18 0 B -8 0 -4 22 2 C 2 4 0 28 0 D -18 -22 -28 0 -22 E 0 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 18 0 B -8 0 -4 22 2 C 2 4 0 28 0 D -18 -22 -28 0 -22 E 0 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6674: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) E C D B A (9) C E D B A (9) D B A E C (6) A B D E C (6) E D C A B (5) E C D A B (5) D E C B A (5) A B D C E (5) E D C B A (4) C E D A B (4) D E C A B (3) D E B C A (3) C E A B D (3) D E B A C (2) D A E B C (2) B A D E C (2) E A C D B (1) D E A C B (1) D E A B C (1) D B E A C (1) D A B E C (1) C E B A D (1) C B A E D (1) C A E B D (1) B D A E C (1) B C A E D (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D C B (1) A E C D B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -8 -24 -14 B -4 0 -10 -28 -22 C 8 10 0 0 -20 D 24 28 0 0 -16 E 14 22 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -8 -24 -14 B -4 0 -10 -28 -22 C 8 10 0 0 -20 D 24 28 0 0 -16 E 14 22 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 E=24 C=19 B=7 so B is eliminated. Round 2 votes counts: A=30 D=26 E=24 C=20 so C is eliminated. Round 3 votes counts: E=41 A=33 D=26 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:236 D:218 C:199 A:179 B:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 -24 -14 B -4 0 -10 -28 -22 C 8 10 0 0 -20 D 24 28 0 0 -16 E 14 22 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -24 -14 B -4 0 -10 -28 -22 C 8 10 0 0 -20 D 24 28 0 0 -16 E 14 22 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -24 -14 B -4 0 -10 -28 -22 C 8 10 0 0 -20 D 24 28 0 0 -16 E 14 22 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6675: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) B E D C A (6) A B C D E (5) E B D C A (4) B E D A C (4) B E A D C (4) B D C A E (4) A C D E B (4) E D C B A (3) D C E A B (3) C A D E B (3) C A D B E (3) B A D C E (3) B A C D E (3) E D B C A (2) E A C B D (2) D C E B A (2) C D A E B (2) C A E D B (2) B D E A C (2) B D C E A (2) B A C E D (2) A E C D B (2) A E C B D (2) A C E D B (2) A C B E D (2) A C B D E (2) E C D A B (1) E B A D C (1) E A C D B (1) D E C A B (1) D C B A E (1) D C A E B (1) D B C A E (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A B E (1) B D E C A (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 8 14 18 B -8 0 -8 4 18 C -8 8 0 10 26 D -14 -4 -10 0 12 E -18 -18 -26 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 14 18 B -8 0 -8 4 18 C -8 8 0 10 26 D -14 -4 -10 0 12 E -18 -18 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=31 E=14 C=14 D=9 so D is eliminated. Round 2 votes counts: B=32 A=32 C=21 E=15 so E is eliminated. Round 3 votes counts: B=39 A=35 C=26 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:218 B:203 D:192 E:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 14 18 B -8 0 -8 4 18 C -8 8 0 10 26 D -14 -4 -10 0 12 E -18 -18 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 14 18 B -8 0 -8 4 18 C -8 8 0 10 26 D -14 -4 -10 0 12 E -18 -18 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 14 18 B -8 0 -8 4 18 C -8 8 0 10 26 D -14 -4 -10 0 12 E -18 -18 -26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6676: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) D B E C A (10) A C D B E (8) A E B C D (7) A C D E B (6) E B D C A (5) E B C D A (5) D C B E A (5) A D C B E (5) D A C B E (4) C D B E A (4) C B E D A (4) E B C A D (3) A E B D C (3) A D E B C (3) E D B A C (1) E C B D A (1) E B D A C (1) E B A D C (1) E B A C D (1) D E B A C (1) D C B A E (1) D B C E A (1) C E B A D (1) C D A B E (1) C A E B D (1) C A B E D (1) B E C D A (1) B D E C A (1) A E D B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 6 10 8 8 B -6 0 -8 -6 -6 C -10 8 0 10 8 D -8 6 -10 0 4 E -8 6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 8 8 B -6 0 -8 -6 -6 C -10 8 0 10 8 D -8 6 -10 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 D=22 E=18 C=12 B=2 so B is eliminated. Round 2 votes counts: A=46 D=23 E=19 C=12 so C is eliminated. Round 3 votes counts: A=48 D=28 E=24 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:208 D:196 E:193 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 8 8 B -6 0 -8 -6 -6 C -10 8 0 10 8 D -8 6 -10 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 8 8 B -6 0 -8 -6 -6 C -10 8 0 10 8 D -8 6 -10 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 8 8 B -6 0 -8 -6 -6 C -10 8 0 10 8 D -8 6 -10 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6677: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (14) B C A D E (8) A B E D C (7) D C E A B (6) C D E B A (5) B C D A E (5) B A C E D (5) E D C A B (4) D E C A B (4) C B D E A (4) A E D B C (4) E D A C B (3) E A D B C (3) C D E A B (3) B C A E D (3) B A E C D (3) B A C D E (3) A E B D C (3) E C D A B (2) C D B A E (2) C B D A E (2) E A B D C (1) D C A E B (1) D A E C B (1) C E D A B (1) B C E A D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -2 14 26 B 10 0 24 20 18 C 2 -24 0 -6 -2 D -14 -20 6 0 -8 E -26 -18 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 14 26 B 10 0 24 20 18 C 2 -24 0 -6 -2 D -14 -20 6 0 -8 E -26 -18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 C=17 A=16 E=13 D=12 so D is eliminated. Round 2 votes counts: B=42 C=24 E=17 A=17 so E is eliminated. Round 3 votes counts: B=42 C=34 A=24 so A is eliminated. Round 4 votes counts: B=62 C=38 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:236 A:214 C:185 E:183 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 14 26 B 10 0 24 20 18 C 2 -24 0 -6 -2 D -14 -20 6 0 -8 E -26 -18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 14 26 B 10 0 24 20 18 C 2 -24 0 -6 -2 D -14 -20 6 0 -8 E -26 -18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 14 26 B 10 0 24 20 18 C 2 -24 0 -6 -2 D -14 -20 6 0 -8 E -26 -18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6678: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) D A B E C (7) C E B D A (7) C E B A D (7) A B E D C (7) C D E B A (6) D C A E B (5) D A C B E (5) A D B E C (5) E B C A D (4) D A B C E (4) C E D B A (4) A B D E C (4) E C B A D (3) E B A C D (3) D A E B C (3) E A B D C (2) D C E B A (2) D A E C B (2) D A C E B (2) C B E D A (2) B C E A D (2) D E C A B (1) D C A B E (1) C D B A E (1) B E C A D (1) B E A C D (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 12 -6 -18 -4 B -12 0 -10 -12 -18 C 6 10 0 -16 12 D 18 12 16 0 10 E 4 18 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 -18 -4 B -12 0 -10 -12 -18 C 6 10 0 -16 12 D 18 12 16 0 10 E 4 18 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=27 A=17 E=12 B=5 so B is eliminated. Round 2 votes counts: D=39 C=29 A=18 E=14 so E is eliminated. Round 3 votes counts: D=39 C=37 A=24 so A is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:228 C:206 E:200 A:192 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -6 -18 -4 B -12 0 -10 -12 -18 C 6 10 0 -16 12 D 18 12 16 0 10 E 4 18 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -18 -4 B -12 0 -10 -12 -18 C 6 10 0 -16 12 D 18 12 16 0 10 E 4 18 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -18 -4 B -12 0 -10 -12 -18 C 6 10 0 -16 12 D 18 12 16 0 10 E 4 18 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6679: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (15) D C B A E (15) A E B C D (14) A B C E D (11) E A B C D (7) E A D B C (5) E D B C A (4) D E C B A (4) C B D E A (3) A E D B C (3) D E A C B (2) D C A B E (2) A D E C B (2) E B C D A (1) E B C A D (1) E B A C D (1) D A C B E (1) C B D A E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C A E D (1) B C A D E (1) B A E C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -2 0 8 B 2 0 8 -8 12 C 2 -8 0 -6 8 D 0 8 6 0 -4 E -8 -12 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.505827 B: 0.000000 C: 0.000000 D: 0.494173 E: 0.000000 Sum of squares = 0.500067914658 Cumulative probabilities = A: 0.505827 B: 0.505827 C: 0.505827 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 0 8 B 2 0 8 -8 12 C 2 -8 0 -6 8 D 0 8 6 0 -4 E -8 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=32 E=19 B=6 C=4 so C is eliminated. Round 2 votes counts: D=39 A=32 E=19 B=10 so B is eliminated. Round 3 votes counts: D=44 A=35 E=21 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:207 D:205 A:202 C:198 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 0 8 B 2 0 8 -8 12 C 2 -8 0 -6 8 D 0 8 6 0 -4 E -8 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 0 8 B 2 0 8 -8 12 C 2 -8 0 -6 8 D 0 8 6 0 -4 E -8 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 0 8 B 2 0 8 -8 12 C 2 -8 0 -6 8 D 0 8 6 0 -4 E -8 -12 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6680: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) D B C A E (7) C E A D B (7) C D B E A (7) B D A E C (7) D C B E A (6) A E B D C (6) D C B A E (5) D C A E B (5) E A C D B (4) C D E A B (4) B D C A E (4) B A E D C (4) D B A E C (3) E C A B D (2) D B A C E (2) D A E B C (2) C E A B D (2) B E A C D (2) A B E D C (2) D A C E B (1) D A B E C (1) C E B A D (1) C B E D A (1) C B E A D (1) B D C E A (1) B C D E A (1) B A D E C (1) A E C D B (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -8 -14 6 B 6 0 -12 -10 10 C 8 12 0 -14 10 D 14 10 14 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -14 6 B 6 0 -12 -10 10 C 8 12 0 -14 10 D 14 10 14 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=23 B=20 E=14 A=11 so A is eliminated. Round 2 votes counts: D=32 E=23 C=23 B=22 so B is eliminated. Round 3 votes counts: D=45 E=31 C=24 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 C:208 B:197 A:189 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -8 -14 6 B 6 0 -12 -10 10 C 8 12 0 -14 10 D 14 10 14 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -14 6 B 6 0 -12 -10 10 C 8 12 0 -14 10 D 14 10 14 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -14 6 B 6 0 -12 -10 10 C 8 12 0 -14 10 D 14 10 14 0 14 E -6 -10 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6681: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) A D B E C (7) D A E C B (5) D A E B C (5) A E D C B (5) E C D B A (4) C E B D A (4) E D A C B (3) E A D C B (3) D E A C B (3) D A B E C (3) C B E D A (3) B C A D E (3) A D E B C (3) E C D A B (2) E C A B D (2) E A C D B (2) D E C A B (2) D B C A E (2) C E D B A (2) B D A C E (2) B C E D A (2) B C A E D (2) B A D C E (2) B A C E D (2) A D E C B (2) A B D E C (2) A B D C E (2) E D C B A (1) E D C A B (1) E C A D B (1) D C E B A (1) C E B A D (1) C D E B A (1) B D C E A (1) B D C A E (1) B C D E A (1) B C D A E (1) B A C D E (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 8 0 4 B -10 0 2 -18 -8 C -8 -2 0 -12 -14 D 0 18 12 0 2 E -4 8 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.900305 B: 0.000000 C: 0.000000 D: 0.099695 E: 0.000000 Sum of squares = 0.820487934321 Cumulative probabilities = A: 0.900305 B: 0.900305 C: 0.900305 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 0 4 B -10 0 2 -18 -8 C -8 -2 0 -12 -14 D 0 18 12 0 2 E -4 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=23 D=21 E=19 C=11 so C is eliminated. Round 2 votes counts: B=29 E=26 A=23 D=22 so D is eliminated. Round 3 votes counts: A=36 E=33 B=31 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:211 E:208 B:183 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 0 4 B -10 0 2 -18 -8 C -8 -2 0 -12 -14 D 0 18 12 0 2 E -4 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 0 4 B -10 0 2 -18 -8 C -8 -2 0 -12 -14 D 0 18 12 0 2 E -4 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 0 4 B -10 0 2 -18 -8 C -8 -2 0 -12 -14 D 0 18 12 0 2 E -4 8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6682: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) B D E A C (9) E B A D C (7) E A B D C (7) D C B E A (7) D B C E A (7) A E C B D (7) A C E B D (7) C D B A E (5) C A E D B (5) E A B C D (4) C D A E B (4) A E B C D (3) D C B A E (2) D B E C A (2) C D B E A (2) C A D E B (2) B E D A C (2) B E A D C (2) B D C A E (2) D B C A E (1) B D E C A (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -2 -10 0 B -2 0 -4 4 8 C 2 4 0 0 10 D 10 -4 0 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.715915 D: 0.284085 E: 0.000000 Sum of squares = 0.593238393776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.715915 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -10 0 B -2 0 -4 4 8 C 2 4 0 0 10 D 10 -4 0 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500009 D: 0.499991 E: 0.000000 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500009 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=19 E=18 A=18 B=16 so B is eliminated. Round 2 votes counts: D=31 C=29 E=22 A=18 so A is eliminated. Round 3 votes counts: C=36 E=33 D=31 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 D:208 B:203 A:195 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 -10 0 B -2 0 -4 4 8 C 2 4 0 0 10 D 10 -4 0 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500009 D: 0.499991 E: 0.000000 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500009 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -10 0 B -2 0 -4 4 8 C 2 4 0 0 10 D 10 -4 0 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500009 D: 0.499991 E: 0.000000 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500009 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -10 0 B -2 0 -4 4 8 C 2 4 0 0 10 D 10 -4 0 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500009 D: 0.499991 E: 0.000000 Sum of squares = 0.500000000158 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500009 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6683: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (16) E A D B C (8) E D A B C (7) D B C E A (6) C B D A E (6) E D B A C (5) C B A D E (5) B D C A E (5) D B C A E (4) A E D B C (4) E A C B D (3) C B D E A (3) C A B D E (3) B C D A E (3) A C E B D (3) E D B C A (2) C A B E D (2) A C B D E (2) A B C D E (2) E D C B A (1) E C B D A (1) E C B A D (1) D E B A C (1) D E A B C (1) D A E B C (1) C B E D A (1) C B A E D (1) B C D E A (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 8 10 4 18 B -8 0 0 18 -10 C -10 0 0 10 -4 D -4 -18 -10 0 -14 E -18 10 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999379 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 4 18 B -8 0 0 18 -10 C -10 0 0 10 -4 D -4 -18 -10 0 -14 E -18 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=28 C=21 D=13 B=9 so B is eliminated. Round 2 votes counts: A=29 E=28 C=25 D=18 so D is eliminated. Round 3 votes counts: C=40 E=30 A=30 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:205 B:200 C:198 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 4 18 B -8 0 0 18 -10 C -10 0 0 10 -4 D -4 -18 -10 0 -14 E -18 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 4 18 B -8 0 0 18 -10 C -10 0 0 10 -4 D -4 -18 -10 0 -14 E -18 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 4 18 B -8 0 0 18 -10 C -10 0 0 10 -4 D -4 -18 -10 0 -14 E -18 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6684: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (15) A C E B D (14) B E D A C (9) A E B C D (9) C A D E B (8) E B A D C (7) C D A B E (7) D C B E A (6) D C B A E (4) A C E D B (4) E A B C D (3) C A D B E (3) D B C E A (2) C D B A E (2) A E C B D (2) C A E B D (1) B E D C A (1) B D E C A (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 0 6 12 B -6 0 -4 -4 0 C 0 4 0 8 4 D -6 4 -8 0 -2 E -12 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.248036 B: 0.000000 C: 0.751964 D: 0.000000 E: 0.000000 Sum of squares = 0.626972173246 Cumulative probabilities = A: 0.248036 B: 0.248036 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 6 12 B -6 0 -4 -4 0 C 0 4 0 8 4 D -6 4 -8 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=27 C=21 B=11 E=10 so E is eliminated. Round 2 votes counts: A=34 D=27 C=21 B=18 so B is eliminated. Round 3 votes counts: A=41 D=38 C=21 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:208 D:194 B:193 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 6 12 B -6 0 -4 -4 0 C 0 4 0 8 4 D -6 4 -8 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 6 12 B -6 0 -4 -4 0 C 0 4 0 8 4 D -6 4 -8 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 6 12 B -6 0 -4 -4 0 C 0 4 0 8 4 D -6 4 -8 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6685: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (10) E A D C B (6) E A C B D (4) C B D E A (4) B D C A E (4) A E D B C (4) E D C A B (3) D E C A B (3) D E A C B (3) C B E A D (3) B C D A E (3) B A C E D (3) A D E B C (3) E A C D B (2) D B C E A (2) D A B E C (2) C E D A B (2) C D E B A (2) C B E D A (2) B C A E D (2) B C A D E (2) A E C B D (2) A B E D C (2) E D A C B (1) E C A D B (1) E C A B D (1) D C E B A (1) D C E A B (1) D C B E A (1) D A E C B (1) D A E B C (1) C E B A D (1) C D B E A (1) B D A C E (1) B C E A D (1) B A E C D (1) B A C D E (1) A E B D C (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 10 0 4 -10 B -10 0 -12 -2 -10 C 0 12 0 -2 -6 D -4 2 2 0 -6 E 10 10 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 0 4 -10 B -10 0 -12 -2 -10 C 0 12 0 -2 -6 D -4 2 2 0 -6 E 10 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=24 E=18 D=15 C=15 so D is eliminated. Round 2 votes counts: B=30 A=28 E=24 C=18 so C is eliminated. Round 3 votes counts: B=41 E=31 A=28 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:202 C:202 D:197 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 0 4 -10 B -10 0 -12 -2 -10 C 0 12 0 -2 -6 D -4 2 2 0 -6 E 10 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 4 -10 B -10 0 -12 -2 -10 C 0 12 0 -2 -6 D -4 2 2 0 -6 E 10 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 4 -10 B -10 0 -12 -2 -10 C 0 12 0 -2 -6 D -4 2 2 0 -6 E 10 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6686: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) E C A D B (7) C E A B D (7) D B A E C (6) D B C E A (5) D B A C E (5) B A D C E (5) E C D A B (4) D A B E C (4) A D E B C (4) A B E C D (4) A B D E C (4) A B C E D (4) D C E B A (3) C E D B A (3) A D B E C (3) D E C B A (2) B D C E A (2) B D A E C (2) B D A C E (2) B C A E D (2) B A C E D (2) E C A B D (1) E A C D B (1) D E C A B (1) D C B E A (1) D A E C B (1) C E D A B (1) C E B A D (1) C E A D B (1) B C E A D (1) B A D E C (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 14 16 16 18 B -14 0 12 -6 6 C -16 -12 0 -4 -8 D -16 6 4 0 2 E -18 -6 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 16 18 B -14 0 12 -6 6 C -16 -12 0 -4 -8 D -16 6 4 0 2 E -18 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=28 B=17 E=13 C=13 so E is eliminated. Round 2 votes counts: A=30 D=28 C=25 B=17 so B is eliminated. Round 3 votes counts: A=38 D=34 C=28 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:232 B:199 D:198 E:191 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 16 16 18 B -14 0 12 -6 6 C -16 -12 0 -4 -8 D -16 6 4 0 2 E -18 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 16 18 B -14 0 12 -6 6 C -16 -12 0 -4 -8 D -16 6 4 0 2 E -18 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 16 18 B -14 0 12 -6 6 C -16 -12 0 -4 -8 D -16 6 4 0 2 E -18 -6 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6687: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) E B A C D (7) D C E A B (6) C D E A B (5) B E D A C (5) B A E C D (4) B E A D C (3) B E A C D (3) B A D E C (3) A B C E D (3) A B C D E (3) E D C B A (2) E C D B A (2) E C A B D (2) E B D C A (2) E A B C D (2) D C E B A (2) D A C B E (2) C A D B E (2) B D A E C (2) A E B C D (2) A C B E D (2) A C B D E (2) A B E C D (2) E D B C A (1) E C D A B (1) E C B A D (1) E B C D A (1) E B C A D (1) E B A D C (1) D E B C A (1) D C B A E (1) D C A E B (1) D B E C A (1) D B E A C (1) D B A C E (1) D A B C E (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B D E A C (1) B A E D C (1) A C E B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 0 -2 B -6 0 6 12 10 C -8 -6 0 2 -4 D 0 -12 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765451 Cumulative probabilities = A: 0.555556 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 6 8 0 -2 B -6 0 6 12 10 C -8 -6 0 2 -4 D 0 -12 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.555556 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=23 B=22 A=17 C=11 so C is eliminated. Round 2 votes counts: D=33 E=23 B=22 A=22 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:211 A:206 E:199 C:192 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 0 -2 B -6 0 6 12 10 C -8 -6 0 2 -4 D 0 -12 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.555556 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 0 -2 B -6 0 6 12 10 C -8 -6 0 2 -4 D 0 -12 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.555556 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 0 -2 B -6 0 6 12 10 C -8 -6 0 2 -4 D 0 -12 -2 0 -2 E 2 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098765349 Cumulative probabilities = A: 0.555556 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6688: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) D E C A B (7) C E A D B (7) B A C E D (6) E D C B A (5) C E A B D (4) A B D C E (4) E C D B A (3) E B D C A (3) C E B A D (3) C A E B D (3) B E C A D (3) B D A E C (3) A D C B E (3) A D B C E (3) E C B D A (2) E C B A D (2) D B E A C (2) D B A E C (2) D A E C B (2) D A C E B (2) B A D C E (2) A C D B E (2) A C B E D (2) A B C D E (2) E D C A B (1) E C D A B (1) E B C A D (1) D E C B A (1) D E B C A (1) D A B E C (1) C E D A B (1) C D A E B (1) B E D C A (1) B D E A C (1) B C A E D (1) B A E C D (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 -2 2 2 B -12 0 -6 -8 0 C 2 6 0 -6 14 D -2 8 6 0 0 E -2 0 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000058 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 2 2 B -12 0 -6 -8 0 C 2 6 0 -6 14 D -2 8 6 0 0 E -2 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000042 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=19 B=19 E=18 A=17 so A is eliminated. Round 2 votes counts: D=33 B=25 C=24 E=18 so E is eliminated. Round 3 votes counts: D=39 C=32 B=29 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:208 A:207 D:206 E:192 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 -2 2 2 B -12 0 -6 -8 0 C 2 6 0 -6 14 D -2 8 6 0 0 E -2 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000042 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 2 2 B -12 0 -6 -8 0 C 2 6 0 -6 14 D -2 8 6 0 0 E -2 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000042 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 2 2 B -12 0 -6 -8 0 C 2 6 0 -6 14 D -2 8 6 0 0 E -2 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000042 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6689: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) B A E C D (6) B A D C E (6) A C D E B (5) E B D C A (4) D C E B A (4) C D A E B (4) B E A C D (4) B A E D C (4) B A C D E (4) E D C B A (3) E D C A B (3) E C D A B (3) E C A D B (3) D C E A B (3) D C A E B (3) C D E A B (3) B D C E A (3) A B C D E (3) E A C D B (2) E A B C D (2) D B C E A (2) C A D E B (2) B D C A E (2) A E C D B (2) A C D B E (2) A C B D E (2) A B E C D (2) D E C B A (1) D C B E A (1) D C B A E (1) D C A B E (1) B E A D C (1) B D E C A (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -6 4 2 B 6 0 2 2 4 C 6 -2 0 2 6 D -4 -2 -2 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 4 2 B 6 0 2 2 4 C 6 -2 0 2 6 D -4 -2 -2 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=20 A=18 D=16 C=9 so C is eliminated. Round 2 votes counts: B=37 D=23 E=20 A=20 so E is eliminated. Round 3 votes counts: B=41 D=32 A=27 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:206 D:199 A:197 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 4 2 B 6 0 2 2 4 C 6 -2 0 2 6 D -4 -2 -2 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 4 2 B 6 0 2 2 4 C 6 -2 0 2 6 D -4 -2 -2 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 4 2 B 6 0 2 2 4 C 6 -2 0 2 6 D -4 -2 -2 0 6 E -2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6690: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) D C A E B (12) B E A C D (12) E C D A B (10) E B C A D (9) D C E A B (8) D A C B E (5) E C B A D (4) C E D A B (4) B A E C D (4) A D B C E (3) E C D B A (2) E C B D A (2) B A E D C (2) B A D E C (2) A B D C E (2) E C A D B (1) E C A B D (1) D C A B E (1) D B E C A (1) C D E A B (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -12 8 -10 B 2 0 -4 2 -8 C 12 4 0 0 0 D -8 -2 0 0 -2 E 10 8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.729712 D: 0.000000 E: 0.270288 Sum of squares = 0.605535503404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.729712 D: 0.729712 E: 1.000000 A B C D E A 0 -2 -12 8 -10 B 2 0 -4 2 -8 C 12 4 0 0 0 D -8 -2 0 0 -2 E 10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=29 D=27 A=6 C=5 so C is eliminated. Round 2 votes counts: E=33 B=33 D=28 A=6 so A is eliminated. Round 3 votes counts: B=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:208 B:196 D:194 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 8 -10 B 2 0 -4 2 -8 C 12 4 0 0 0 D -8 -2 0 0 -2 E 10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 8 -10 B 2 0 -4 2 -8 C 12 4 0 0 0 D -8 -2 0 0 -2 E 10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 8 -10 B 2 0 -4 2 -8 C 12 4 0 0 0 D -8 -2 0 0 -2 E 10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6691: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C A E B D (10) D B E A C (8) C E A B D (6) B D E A C (6) C A D E B (5) D C A B E (3) D A C B E (3) D A B E C (3) C E B A D (3) B E D A C (3) B D E C A (3) E C A B D (2) E B A C D (2) E A B C D (2) D C B E A (2) D C B A E (2) D B C A E (2) D B A C E (2) C D A E B (2) C A E D B (2) B E D C A (2) A E C B D (2) A D E B C (2) E B C A D (1) E B A D C (1) E A C B D (1) D B C E A (1) D A B C E (1) C D B A E (1) C D A B E (1) C B E D A (1) A E B D C (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 2 -14 12 B 2 0 4 -4 10 C -2 -4 0 -14 -4 D 14 4 14 0 18 E -12 -10 4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -14 12 B 2 0 4 -4 10 C -2 -4 0 -14 -4 D 14 4 14 0 18 E -12 -10 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=31 B=14 E=9 A=8 so A is eliminated. Round 2 votes counts: D=41 C=32 B=14 E=13 so E is eliminated. Round 3 votes counts: D=41 C=37 B=22 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 B:206 A:199 C:188 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -14 12 B 2 0 4 -4 10 C -2 -4 0 -14 -4 D 14 4 14 0 18 E -12 -10 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -14 12 B 2 0 4 -4 10 C -2 -4 0 -14 -4 D 14 4 14 0 18 E -12 -10 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -14 12 B 2 0 4 -4 10 C -2 -4 0 -14 -4 D 14 4 14 0 18 E -12 -10 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6692: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C E A B D (11) B C E A D (10) D A E C B (8) E A C D B (6) C A E B D (6) B D C E A (6) D B A C E (5) A E D C B (5) C B E A D (4) B C D E A (4) A E C D B (4) E C A B D (2) D B C A E (2) D A E B C (2) B D C A E (2) B C A E D (2) E A D C B (1) E A C B D (1) D E A C B (1) D B C E A (1) D A B E C (1) C B A E D (1) B C E D A (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -4 10 4 B 0 0 -4 2 2 C 4 4 0 8 14 D -10 -2 -8 0 -12 E -4 -2 -14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 10 4 B 0 0 -4 2 2 C 4 4 0 8 14 D -10 -2 -8 0 -12 E -4 -2 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=25 C=22 A=11 E=10 so E is eliminated. Round 2 votes counts: D=32 B=25 C=24 A=19 so A is eliminated. Round 3 votes counts: D=38 C=37 B=25 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:205 B:200 E:196 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 10 4 B 0 0 -4 2 2 C 4 4 0 8 14 D -10 -2 -8 0 -12 E -4 -2 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 10 4 B 0 0 -4 2 2 C 4 4 0 8 14 D -10 -2 -8 0 -12 E -4 -2 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 10 4 B 0 0 -4 2 2 C 4 4 0 8 14 D -10 -2 -8 0 -12 E -4 -2 -14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6693: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) C A E B D (11) D B E A C (9) C E A D B (8) A C B D E (8) B D A E C (7) B D A C E (6) A B D C E (6) E C D B A (5) C A E D B (5) D B E C A (4) A C E B D (4) E C A D B (3) E D C B A (2) E C A B D (1) E B D A C (1) D E B C A (1) C E A B D (1) C D E B A (1) C D A B E (1) B D E A C (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -12 -2 0 B 2 0 0 -4 -12 C 12 0 0 -2 6 D 2 4 2 0 -8 E 0 12 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.125000 Sum of squares = 0.406250000004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.875000 E: 1.000000 A B C D E A 0 -2 -12 -2 0 B 2 0 0 -4 -12 C 12 0 0 -2 6 D 2 4 2 0 -8 E 0 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.125000 Sum of squares = 0.406249999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=25 A=19 B=15 D=14 so D is eliminated. Round 2 votes counts: B=28 C=27 E=26 A=19 so A is eliminated. Round 3 votes counts: C=39 B=35 E=26 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:208 E:207 D:200 B:193 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 -2 0 B 2 0 0 -4 -12 C 12 0 0 -2 6 D 2 4 2 0 -8 E 0 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.125000 Sum of squares = 0.406249999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.875000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -2 0 B 2 0 0 -4 -12 C 12 0 0 -2 6 D 2 4 2 0 -8 E 0 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.125000 Sum of squares = 0.406249999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -2 0 B 2 0 0 -4 -12 C 12 0 0 -2 6 D 2 4 2 0 -8 E 0 12 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.375000 E: 0.125000 Sum of squares = 0.406249999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6694: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) B C E D A (8) A D E C B (8) B C D A E (7) E A D C B (6) D A E B C (6) A E D C B (6) E D A C B (4) D B E C A (4) D E A B C (3) C B E A D (3) A D C E B (3) D A B C E (2) C E A B D (2) C A E B D (2) B D C A E (2) B C E A D (2) B C A E D (2) B C A D E (2) A E C D B (2) E D B C A (1) E C B D A (1) E C B A D (1) E A C D B (1) E A C B D (1) D E B A C (1) D E A C B (1) D B C A E (1) D B A C E (1) D A E C B (1) D A B E C (1) C E B A D (1) C B A E D (1) C A B E D (1) B D E C A (1) B C D E A (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 -2 8 0 B -20 0 -6 -6 -18 C 2 6 0 -6 -14 D -8 6 6 0 -6 E 0 18 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555265 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.444735 Sum of squares = 0.506108353413 Cumulative probabilities = A: 0.555265 B: 0.555265 C: 0.555265 D: 0.555265 E: 1.000000 A B C D E A 0 20 -2 8 0 B -20 0 -6 -6 -18 C 2 6 0 -6 -14 D -8 6 6 0 -6 E 0 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 D=21 A=21 C=10 so C is eliminated. Round 2 votes counts: B=29 E=26 A=24 D=21 so D is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 A:213 D:199 C:194 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 -2 8 0 B -20 0 -6 -6 -18 C 2 6 0 -6 -14 D -8 6 6 0 -6 E 0 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -2 8 0 B -20 0 -6 -6 -18 C 2 6 0 -6 -14 D -8 6 6 0 -6 E 0 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -2 8 0 B -20 0 -6 -6 -18 C 2 6 0 -6 -14 D -8 6 6 0 -6 E 0 18 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6695: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) E A B C D (8) A E D C B (8) B C E A D (7) D A E C B (6) B C D E A (6) C B D E A (5) A E D B C (5) A D E C B (5) E B A C D (4) D A C E B (4) E B C A D (3) E A C B D (3) D A E B C (3) B E C A D (3) D B C A E (2) D A C B E (2) C B E D A (2) A E B D C (2) A E B C D (2) A D E B C (2) E C D A B (1) E C B A D (1) D C B E A (1) D C A E B (1) D C A B E (1) C D B E A (1) B E A C D (1) B C E D A (1) B C D A E (1) Total count = 100 A B C D E A 0 6 10 8 6 B -6 0 0 -2 -16 C -10 0 0 -2 -14 D -8 2 2 0 -2 E -6 16 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999501 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 8 6 B -6 0 0 -2 -16 C -10 0 0 -2 -14 D -8 2 2 0 -2 E -6 16 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=24 E=20 B=19 C=8 so C is eliminated. Round 2 votes counts: D=30 B=26 A=24 E=20 so E is eliminated. Round 3 votes counts: A=35 B=34 D=31 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:213 D:197 B:188 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 8 6 B -6 0 0 -2 -16 C -10 0 0 -2 -14 D -8 2 2 0 -2 E -6 16 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 8 6 B -6 0 0 -2 -16 C -10 0 0 -2 -14 D -8 2 2 0 -2 E -6 16 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 8 6 B -6 0 0 -2 -16 C -10 0 0 -2 -14 D -8 2 2 0 -2 E -6 16 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6696: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (19) D B C E A (11) D C E B A (7) A E C D B (7) D A E C B (5) B C E D A (5) B C E A D (5) D A B E C (4) C E B A D (4) A D E C B (4) E C B A D (3) E C A B D (3) D E C A B (3) D C B E A (3) A B E C D (3) B A E C D (2) B A C E D (2) E A C B D (1) D B C A E (1) D B A C E (1) D A C E B (1) D A B C E (1) B D A C E (1) B C D E A (1) A E D C B (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 8 8 12 8 B -8 0 -22 0 -20 C -8 22 0 12 -14 D -12 0 -12 0 -12 E -8 20 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 12 8 B -8 0 -22 0 -20 C -8 22 0 12 -14 D -12 0 -12 0 -12 E -8 20 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=36 B=16 E=7 C=4 so C is eliminated. Round 2 votes counts: D=37 A=36 B=16 E=11 so E is eliminated. Round 3 votes counts: A=40 D=37 B=23 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:219 A:218 C:206 D:182 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 12 8 B -8 0 -22 0 -20 C -8 22 0 12 -14 D -12 0 -12 0 -12 E -8 20 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 12 8 B -8 0 -22 0 -20 C -8 22 0 12 -14 D -12 0 -12 0 -12 E -8 20 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 12 8 B -8 0 -22 0 -20 C -8 22 0 12 -14 D -12 0 -12 0 -12 E -8 20 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6697: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) E B C D A (10) E C D B A (7) C D E A B (7) A B E D C (6) D C E B A (5) C E D B A (5) E B D C A (4) B E D C A (4) B A E D C (4) A D B C E (4) A B D C E (4) C D E B A (3) A B E C D (3) D C A E B (2) C D A E B (2) C A D E B (2) B E A D C (2) B A D E C (2) A E C B D (2) A C D E B (2) A B D E C (2) E C A D B (1) D C B E A (1) C E D A B (1) B A E C D (1) A E B C D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -8 -2 0 B -4 0 -6 -10 -10 C 8 6 0 -2 2 D 2 10 2 0 -4 E 0 10 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000012 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 4 -8 -2 0 B -4 0 -6 -10 -10 C 8 6 0 -2 2 D 2 10 2 0 -4 E 0 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=22 C=20 B=13 D=8 so D is eliminated. Round 2 votes counts: A=37 C=28 E=22 B=13 so B is eliminated. Round 3 votes counts: A=44 E=28 C=28 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 E:206 D:205 A:197 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -2 0 B -4 0 -6 -10 -10 C 8 6 0 -2 2 D 2 10 2 0 -4 E 0 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -2 0 B -4 0 -6 -10 -10 C 8 6 0 -2 2 D 2 10 2 0 -4 E 0 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -2 0 B -4 0 -6 -10 -10 C 8 6 0 -2 2 D 2 10 2 0 -4 E 0 10 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000027 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) B A E C D (7) E D B C A (6) D C B E A (6) B E A D C (6) E A B D C (5) D C E B A (5) E B A D C (4) C D A E B (4) B A C D E (4) A C D E B (4) D C E A B (3) C D E A B (3) A E C D B (3) E D C B A (2) E B D C A (2) E A D C B (2) C D B A E (2) B E D C A (2) B D E C A (2) B D C E A (2) A E B C D (2) A C B D E (2) A B E C D (2) A B C E D (2) E D C A B (1) E D A C B (1) D E C B A (1) D E C A B (1) C B D A E (1) C A D B E (1) B C D A E (1) B A C E D (1) A E B D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -4 -4 -8 B 8 0 -2 -6 -4 C 4 2 0 -4 0 D 4 6 4 0 -2 E 8 4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.247519 D: 0.000000 E: 0.752481 Sum of squares = 0.627493589563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.247519 D: 0.247519 E: 1.000000 A B C D E A 0 -8 -4 -4 -8 B 8 0 -2 -6 -4 C 4 2 0 -4 0 D 4 6 4 0 -2 E 8 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555556401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 C=18 A=18 D=16 so D is eliminated. Round 2 votes counts: C=32 E=25 B=25 A=18 so A is eliminated. Round 3 votes counts: C=40 E=31 B=29 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:207 D:206 C:201 B:198 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -4 -4 -8 B 8 0 -2 -6 -4 C 4 2 0 -4 0 D 4 6 4 0 -2 E 8 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555556401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -4 -8 B 8 0 -2 -6 -4 C 4 2 0 -4 0 D 4 6 4 0 -2 E 8 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555556401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -4 -8 B 8 0 -2 -6 -4 C 4 2 0 -4 0 D 4 6 4 0 -2 E 8 4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.55555556401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6699: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D A B C E (5) C E B D A (5) B C E A D (5) B C D A E (5) A D B E C (5) A D B C E (5) E B C A D (4) E A D B C (4) A D E B C (4) D B A C E (3) D A E C B (3) C B E D A (3) C B D A E (3) B C A D E (3) B A D C E (3) A B D E C (3) A B D C E (3) E C D B A (2) E A D C B (2) D C B A E (2) C D B A E (2) C B D E A (2) A B E D C (2) E D C A B (1) E D A C B (1) E C B D A (1) E B A C D (1) E A C B D (1) E A B D C (1) E A B C D (1) D C E A B (1) D C A B E (1) D B C A E (1) D A C E B (1) D A C B E (1) B E A C D (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 2 14 12 B 8 0 20 10 16 C -2 -20 0 -6 8 D -14 -10 6 0 12 E -12 -16 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 14 12 B 8 0 20 10 16 C -2 -20 0 -6 8 D -14 -10 6 0 12 E -12 -16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=23 D=18 B=17 C=15 so C is eliminated. Round 2 votes counts: E=32 B=25 A=23 D=20 so D is eliminated. Round 3 votes counts: A=34 E=33 B=33 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:210 D:197 C:190 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 14 12 B 8 0 20 10 16 C -2 -20 0 -6 8 D -14 -10 6 0 12 E -12 -16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 14 12 B 8 0 20 10 16 C -2 -20 0 -6 8 D -14 -10 6 0 12 E -12 -16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 14 12 B 8 0 20 10 16 C -2 -20 0 -6 8 D -14 -10 6 0 12 E -12 -16 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6700: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) C E B D A (7) C A E B D (7) A D B E C (6) A C D B E (6) C E A B D (5) C A E D B (5) A B D C E (5) E B D C A (4) C E B A D (4) B D E A C (4) A D B C E (4) E D B C A (3) D E B A C (3) D B E A C (3) C A B D E (3) B D A E C (3) E D B A C (2) E C B D A (2) D E A B C (2) C A D B E (2) C A B E D (2) A C B D E (2) E C D B A (1) D A E B C (1) D A B E C (1) C E D A B (1) C B E D A (1) C B E A D (1) C B A D E (1) B C A D E (1) A C E D B (1) Total count = 100 A B C D E A 0 6 0 10 14 B -6 0 -2 4 4 C 0 2 0 4 16 D -10 -4 -4 0 8 E -14 -4 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.440983 B: 0.000000 C: 0.559017 D: 0.000000 E: 0.000000 Sum of squares = 0.506965848744 Cumulative probabilities = A: 0.440983 B: 0.440983 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 10 14 B -6 0 -2 4 4 C 0 2 0 4 16 D -10 -4 -4 0 8 E -14 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=24 D=17 E=12 B=8 so B is eliminated. Round 2 votes counts: C=40 D=24 A=24 E=12 so E is eliminated. Round 3 votes counts: C=43 D=33 A=24 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:215 C:211 B:200 D:195 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 10 14 B -6 0 -2 4 4 C 0 2 0 4 16 D -10 -4 -4 0 8 E -14 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 10 14 B -6 0 -2 4 4 C 0 2 0 4 16 D -10 -4 -4 0 8 E -14 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 10 14 B -6 0 -2 4 4 C 0 2 0 4 16 D -10 -4 -4 0 8 E -14 -4 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999896 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6701: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) A E B D C (8) E C A B D (7) E A B D C (6) D B A E C (5) C D B A E (5) A B E D C (5) E A C B D (4) D B A C E (4) B D A E C (4) E B A D C (3) D A B E C (3) C E B D A (3) C D A E B (3) E C B A D (2) E B A C D (2) E A B C D (2) D C B A E (2) D A B C E (2) C E D B A (2) C E B A D (2) C D E A B (2) C B D E A (2) B D C E A (2) A B D E C (2) D B C A E (1) D A C B E (1) C E A D B (1) C E A B D (1) C D E B A (1) C D B E A (1) C B E D A (1) B E A D C (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 8 -4 10 B -12 0 4 16 2 C -8 -4 0 -2 -10 D 4 -16 2 0 -2 E -10 -2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999999 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -4 10 B -12 0 4 16 2 C -8 -4 0 -2 -10 D 4 -16 2 0 -2 E -10 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999988 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=26 D=18 A=16 B=8 so B is eliminated. Round 2 votes counts: C=32 E=27 D=24 A=17 so A is eliminated. Round 3 votes counts: E=41 C=32 D=27 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:213 B:205 E:200 D:194 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 -4 10 B -12 0 4 16 2 C -8 -4 0 -2 -10 D 4 -16 2 0 -2 E -10 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999988 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -4 10 B -12 0 4 16 2 C -8 -4 0 -2 -10 D 4 -16 2 0 -2 E -10 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999988 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -4 10 B -12 0 4 16 2 C -8 -4 0 -2 -10 D 4 -16 2 0 -2 E -10 -2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.125000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.406249999988 Cumulative probabilities = A: 0.500000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6702: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (5) D A C E B (5) A D C E B (5) E A B C D (4) D C A B E (4) C D A B E (4) B E D A C (4) E B A D C (3) E B A C D (3) D C B E A (3) D A B E C (3) C D B E A (3) B E C D A (3) B E C A D (3) B E A D C (3) A E C B D (3) E A B D C (2) D B A E C (2) D A E C B (2) D A C B E (2) C E B A D (2) C D B A E (2) C B E D A (2) B E D C A (2) B E A C D (2) B D E C A (2) E A C B D (1) D C A E B (1) D B E A C (1) D B C A E (1) D A E B C (1) C D A E B (1) C B E A D (1) C B D E A (1) C A E D B (1) C A E B D (1) C A D E B (1) B D E A C (1) B D C E A (1) B C E D A (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A D E C B (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 10 -14 4 B 2 0 -10 -8 12 C -10 10 0 -16 -2 D 14 8 16 0 8 E -4 -12 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -14 4 B 2 0 -10 -8 12 C -10 10 0 -16 -2 D 14 8 16 0 8 E -4 -12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=22 C=19 A=16 E=13 so E is eliminated. Round 2 votes counts: D=30 B=28 A=23 C=19 so C is eliminated. Round 3 votes counts: D=40 B=34 A=26 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:199 B:198 C:191 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -14 4 B 2 0 -10 -8 12 C -10 10 0 -16 -2 D 14 8 16 0 8 E -4 -12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -14 4 B 2 0 -10 -8 12 C -10 10 0 -16 -2 D 14 8 16 0 8 E -4 -12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -14 4 B 2 0 -10 -8 12 C -10 10 0 -16 -2 D 14 8 16 0 8 E -4 -12 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6703: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) E B C A D (9) D A C E B (6) A D C B E (5) E B D C A (4) E B D A C (4) D A B E C (4) D E B A C (3) D C E A B (3) D C A E B (3) C D A E B (3) C D A B E (3) C A D B E (3) C A B D E (3) B E C A D (3) B E A C D (3) E C B A D (2) E B A D C (2) E B A C D (2) D A E B C (2) C D E A B (2) C B E A D (2) A C D B E (2) E D B C A (1) E D B A C (1) E C B D A (1) D B A E C (1) D A E C B (1) D A B C E (1) C E B A D (1) C A B E D (1) B E D A C (1) B E A D C (1) B A D E C (1) A C B D E (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 12 -8 10 B -16 0 -4 -8 0 C -12 4 0 -12 4 D 8 8 12 0 20 E -10 0 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 -8 10 B -16 0 -4 -8 0 C -12 4 0 -12 4 D 8 8 12 0 20 E -10 0 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=26 C=18 A=13 B=9 so B is eliminated. Round 2 votes counts: E=34 D=34 C=18 A=14 so A is eliminated. Round 3 votes counts: D=42 E=35 C=23 so C is eliminated. Round 4 votes counts: D=60 E=40 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 A:215 C:192 B:186 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 12 -8 10 B -16 0 -4 -8 0 C -12 4 0 -12 4 D 8 8 12 0 20 E -10 0 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 -8 10 B -16 0 -4 -8 0 C -12 4 0 -12 4 D 8 8 12 0 20 E -10 0 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 -8 10 B -16 0 -4 -8 0 C -12 4 0 -12 4 D 8 8 12 0 20 E -10 0 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6704: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) D B C E A (6) C E A D B (6) A D C E B (6) D A C E B (5) C E B A D (5) B D A E C (5) A E C B D (5) E C A B D (4) D B A C E (4) C E A B D (4) B E C A D (4) A B E C D (4) D B C A E (3) C E D A B (3) A D E C B (3) A C E D B (3) E A C B D (2) D C E A B (2) D A B E C (2) B D E C A (2) E B C A D (1) D C E B A (1) D C A E B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D E A B (1) B E C D A (1) B D C E A (1) B C E D A (1) B A E C D (1) A E C D B (1) A D B E C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 12 4 4 8 B -12 0 -10 -14 -12 C -4 10 0 -2 12 D -4 14 2 0 4 E -8 12 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 4 8 B -12 0 -10 -14 -12 C -4 10 0 -2 12 D -4 14 2 0 4 E -8 12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=25 C=21 B=15 E=7 so E is eliminated. Round 2 votes counts: D=32 A=27 C=25 B=16 so B is eliminated. Round 3 votes counts: D=40 C=32 A=28 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:214 C:208 D:208 E:194 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 4 8 B -12 0 -10 -14 -12 C -4 10 0 -2 12 D -4 14 2 0 4 E -8 12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 4 8 B -12 0 -10 -14 -12 C -4 10 0 -2 12 D -4 14 2 0 4 E -8 12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 4 8 B -12 0 -10 -14 -12 C -4 10 0 -2 12 D -4 14 2 0 4 E -8 12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6705: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) A B E C D (6) D A C B E (5) C D E A B (5) A B D C E (5) E C B D A (4) E C A B D (4) B A E D C (4) E B C D A (3) D E C B A (3) D C E B A (3) D B A C E (3) C E D B A (3) B E A C D (3) B D A E C (3) B A D E C (3) A C E B D (3) E C D B A (2) D C A B E (2) D B E C A (2) C E D A B (2) B A E C D (2) A C D E B (2) E D C B A (1) E C B A D (1) E C A D B (1) E B C A D (1) E B A C D (1) D C B A E (1) D C A E B (1) D B E A C (1) D B A E C (1) D A C E B (1) C E A D B (1) C E A B D (1) C D A E B (1) B E C A D (1) B E A D C (1) B A D C E (1) A D C B E (1) A D B C E (1) A C E D B (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 14 -6 12 B -4 0 0 -2 10 C -14 0 0 0 2 D 6 2 0 0 6 E -12 -10 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200401 D: 0.799599 E: 0.000000 Sum of squares = 0.679519539249 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200401 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 -6 12 B -4 0 0 -2 10 C -14 0 0 0 2 D 6 2 0 0 6 E -12 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.700000 E: 0.000000 Sum of squares = 0.580000088388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=22 E=18 B=18 C=13 so C is eliminated. Round 2 votes counts: D=35 E=25 A=22 B=18 so B is eliminated. Round 3 votes counts: D=38 A=32 E=30 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:212 D:207 B:202 C:194 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 14 -6 12 B -4 0 0 -2 10 C -14 0 0 0 2 D 6 2 0 0 6 E -12 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.700000 E: 0.000000 Sum of squares = 0.580000088388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -6 12 B -4 0 0 -2 10 C -14 0 0 0 2 D 6 2 0 0 6 E -12 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.700000 E: 0.000000 Sum of squares = 0.580000088388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -6 12 B -4 0 0 -2 10 C -14 0 0 0 2 D 6 2 0 0 6 E -12 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.300000 D: 0.700000 E: 0.000000 Sum of squares = 0.580000088388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.300000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6706: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) B C A E D (8) E D C A B (7) E C A D B (7) C A B E D (7) D E C A B (6) C A E B D (6) B D E C A (6) A C B E D (6) D E A C B (5) B D E A C (5) B D A C E (5) C A E D B (4) D B E A C (3) B C A D E (2) A C E D B (2) D E B C A (1) D E B A C (1) D E A B C (1) D B E C A (1) B E D C A (1) B E C A D (1) B D C A E (1) B A D C E (1) B A C E D (1) A E C D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -16 14 10 B -8 0 -6 20 14 C 16 6 0 12 8 D -14 -20 -12 0 -4 E -10 -14 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -16 14 10 B -8 0 -6 20 14 C 16 6 0 12 8 D -14 -20 -12 0 -4 E -10 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 D=18 C=17 E=14 A=11 so A is eliminated. Round 2 votes counts: B=40 C=27 D=18 E=15 so E is eliminated. Round 3 votes counts: B=40 C=35 D=25 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:210 A:208 E:186 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 14 10 B -8 0 -6 20 14 C 16 6 0 12 8 D -14 -20 -12 0 -4 E -10 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 14 10 B -8 0 -6 20 14 C 16 6 0 12 8 D -14 -20 -12 0 -4 E -10 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 14 10 B -8 0 -6 20 14 C 16 6 0 12 8 D -14 -20 -12 0 -4 E -10 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6707: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (6) C A D E B (6) D A C B E (5) B D E A C (5) A C D B E (5) C E A D B (4) C E A B D (4) B A D E C (4) A D B C E (4) E B C A D (3) D C A E B (3) C E D A B (3) B A E C D (3) A C D E B (3) A C B E D (3) A B D C E (3) E D C B A (2) E C D B A (2) E C B D A (2) E B D C A (2) D C E A B (2) C D E A B (2) C D A E B (2) B E D C A (2) B E D A C (2) B E A C D (2) B D A E C (2) E D B C A (1) D E B C A (1) D B A E C (1) D A C E B (1) C E D B A (1) C E B D A (1) C E B A D (1) C A E D B (1) C A E B D (1) B E C A D (1) B E A D C (1) B A E D C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -6 4 0 B -8 0 -10 0 -8 C 6 10 0 12 12 D -4 0 -12 0 2 E 0 8 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 4 0 B -8 0 -10 0 -8 C 6 10 0 12 12 D -4 0 -12 0 2 E 0 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=23 A=20 E=18 D=13 so D is eliminated. Round 2 votes counts: C=31 A=26 B=24 E=19 so E is eliminated. Round 3 votes counts: C=37 B=37 A=26 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:203 E:197 D:193 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 4 0 B -8 0 -10 0 -8 C 6 10 0 12 12 D -4 0 -12 0 2 E 0 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 4 0 B -8 0 -10 0 -8 C 6 10 0 12 12 D -4 0 -12 0 2 E 0 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 4 0 B -8 0 -10 0 -8 C 6 10 0 12 12 D -4 0 -12 0 2 E 0 8 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6708: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) B C A E D (9) A B C E D (8) D E A C B (7) C B A D E (7) A E D B C (7) D E C B A (6) B C E D A (5) A D E C B (5) A C B D E (5) E D A B C (4) C B D E A (4) B C E A D (3) B A C E D (3) E D B A C (2) E D C B A (1) E B D C A (1) D E A B C (1) D C E B A (1) D A E B C (1) C D B E A (1) C B E D A (1) C B D A E (1) B A E C D (1) A E B D C (1) A D E B C (1) A D C E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 -2 6 4 B 14 0 16 0 0 C 2 -16 0 0 2 D -6 0 0 0 -16 E -4 0 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.541572 C: 0.000000 D: 0.000000 E: 0.458428 Sum of squares = 0.503456531388 Cumulative probabilities = A: 0.000000 B: 0.541572 C: 0.541572 D: 0.541572 E: 1.000000 A B C D E A 0 -14 -2 6 4 B 14 0 16 0 0 C 2 -16 0 0 2 D -6 0 0 0 -16 E -4 0 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=21 E=19 D=16 C=14 so C is eliminated. Round 2 votes counts: B=34 A=30 E=19 D=17 so D is eliminated. Round 3 votes counts: B=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:205 A:197 C:194 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 6 4 B 14 0 16 0 0 C 2 -16 0 0 2 D -6 0 0 0 -16 E -4 0 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 6 4 B 14 0 16 0 0 C 2 -16 0 0 2 D -6 0 0 0 -16 E -4 0 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 6 4 B 14 0 16 0 0 C 2 -16 0 0 2 D -6 0 0 0 -16 E -4 0 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6709: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) C A B D E (10) D E A B C (7) C A B E D (7) B E D C A (7) A D C E B (7) B C E A D (5) A C D E B (5) B E D A C (4) B E C D A (4) B C A E D (4) A C D B E (4) E D B C A (3) E B D C A (3) C A D E B (3) D E B A C (2) D E A C B (2) D A E C B (2) E C D A B (1) D E C A B (1) D A B E C (1) C E D B A (1) C B A E D (1) C A D B E (1) B E C A D (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 2 -4 B -4 0 10 -2 4 C 4 -10 0 0 2 D -2 2 0 0 -6 E 4 -4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 -4 2 -4 B -4 0 10 -2 4 C 4 -10 0 0 2 D -2 2 0 0 -6 E 4 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333293 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 E=18 A=17 D=15 so D is eliminated. Round 2 votes counts: E=30 B=27 C=23 A=20 so A is eliminated. Round 3 votes counts: C=39 E=32 B=29 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:204 E:202 A:199 C:198 D:197 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 2 -4 B -4 0 10 -2 4 C 4 -10 0 0 2 D -2 2 0 0 -6 E 4 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333293 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 2 -4 B -4 0 10 -2 4 C 4 -10 0 0 2 D -2 2 0 0 -6 E 4 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333293 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 2 -4 B -4 0 10 -2 4 C 4 -10 0 0 2 D -2 2 0 0 -6 E 4 -4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333293 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6710: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C A D B E (7) B D E C A (6) A C E D B (6) E B A C D (5) E A B C D (5) D C B A E (4) D B E C A (4) D B C E A (4) A C B E D (4) E D B A C (3) C A B D E (3) B E D A C (3) A C E B D (3) A C D B E (3) A C B D E (3) E A B D C (2) D E B C A (2) C D A B E (2) B E A C D (2) B C D A E (2) A C D E B (2) E D B C A (1) E D A C B (1) E B A D C (1) E A C D B (1) E A C B D (1) D E C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A B E (1) C A D E B (1) B D C E A (1) B D C A E (1) B C A D E (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 10 4 -10 B 2 0 4 4 4 C -10 -4 0 6 2 D -4 -4 -6 0 4 E 10 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 4 -10 B 2 0 4 4 4 C -10 -4 0 6 2 D -4 -4 -6 0 4 E 10 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=23 D=20 B=16 C=13 so C is eliminated. Round 2 votes counts: A=34 E=28 D=22 B=16 so B is eliminated. Round 3 votes counts: A=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:207 A:201 E:200 C:197 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 4 -10 B 2 0 4 4 4 C -10 -4 0 6 2 D -4 -4 -6 0 4 E 10 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 4 -10 B 2 0 4 4 4 C -10 -4 0 6 2 D -4 -4 -6 0 4 E 10 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 4 -10 B 2 0 4 4 4 C -10 -4 0 6 2 D -4 -4 -6 0 4 E 10 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6711: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (12) C E A B D (12) A C E B D (9) A C D B E (7) C E B D A (6) D B E C A (4) D B E A C (4) D B A E C (4) C A E D B (4) B D E C A (4) A D B E C (4) A D B C E (4) E C A B D (2) E B D C A (2) E B D A C (2) C E D B A (2) C A E B D (2) B D E A C (2) A D C B E (2) A C E D B (2) E B C D A (1) C E B A D (1) C E A D B (1) C D E B A (1) C D B E A (1) B E D C A (1) B E D A C (1) A C B E D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -12 6 -18 B -4 0 -30 20 -18 C 12 30 0 28 12 D -6 -20 -28 0 -22 E 18 18 -12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 6 -18 B -4 0 -30 20 -18 C 12 30 0 28 12 D -6 -20 -28 0 -22 E 18 18 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=30 E=19 D=12 B=8 so B is eliminated. Round 2 votes counts: A=31 C=30 E=21 D=18 so D is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:241 E:223 A:190 B:184 D:162 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 6 -18 B -4 0 -30 20 -18 C 12 30 0 28 12 D -6 -20 -28 0 -22 E 18 18 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 6 -18 B -4 0 -30 20 -18 C 12 30 0 28 12 D -6 -20 -28 0 -22 E 18 18 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 6 -18 B -4 0 -30 20 -18 C 12 30 0 28 12 D -6 -20 -28 0 -22 E 18 18 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6712: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) E C D A B (8) D E A C B (8) D B A E C (6) B A C D E (6) E C A D B (4) D A B E C (4) C E A B D (4) B D A E C (4) B A D C E (4) A D B C E (4) E D C A B (3) D E C A B (3) D E B C A (3) D A E B C (3) B D A C E (3) E D C B A (2) D E C B A (2) C E A D B (2) C A E D B (2) A D C E B (2) E C D B A (1) E B D C A (1) E A D C B (1) D E B A C (1) D B E C A (1) D A E C B (1) C E B A D (1) C A E B D (1) B D E C A (1) B D C A E (1) B C A E D (1) A E D C B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 18 -12 8 B -6 0 6 -24 -10 C -18 -6 0 -18 -16 D 12 24 18 0 14 E -8 10 16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 -12 8 B -6 0 6 -24 -10 C -18 -6 0 -18 -16 D 12 24 18 0 14 E -8 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 E=20 C=10 A=9 so A is eliminated. Round 2 votes counts: D=38 B=30 E=21 C=11 so C is eliminated. Round 3 votes counts: D=38 E=32 B=30 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:234 A:210 E:202 B:183 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 18 -12 8 B -6 0 6 -24 -10 C -18 -6 0 -18 -16 D 12 24 18 0 14 E -8 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 -12 8 B -6 0 6 -24 -10 C -18 -6 0 -18 -16 D 12 24 18 0 14 E -8 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 -12 8 B -6 0 6 -24 -10 C -18 -6 0 -18 -16 D 12 24 18 0 14 E -8 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6713: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) B A C D E (7) E D C B A (6) E D C A B (6) E B D C A (6) B A D C E (6) A B C D E (6) D E C A B (5) B E A D C (5) B E A C D (4) E C D A B (3) C D A E B (3) B E D A C (3) B A E D C (3) B A C E D (3) A C B E D (3) E B D A C (2) D A C B E (2) C A D B E (2) A C B D E (2) E D B C A (1) E C B A D (1) E C A D B (1) E C A B D (1) D C E A B (1) D C A E B (1) D C A B E (1) C E D A B (1) C A D E B (1) B D A E C (1) B D A C E (1) B A E C D (1) A D C B E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 20 14 8 B 0 0 2 12 22 C -20 -2 0 -6 2 D -14 -12 6 0 0 E -8 -22 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.551028 B: 0.448972 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.505207715159 Cumulative probabilities = A: 0.551028 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 20 14 8 B 0 0 2 12 22 C -20 -2 0 -6 2 D -14 -12 6 0 0 E -8 -22 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=27 A=22 D=10 C=7 so C is eliminated. Round 2 votes counts: B=34 E=28 A=25 D=13 so D is eliminated. Round 3 votes counts: E=34 B=34 A=32 so A is eliminated. Round 4 votes counts: B=61 E=39 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:221 B:218 D:190 C:187 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 20 14 8 B 0 0 2 12 22 C -20 -2 0 -6 2 D -14 -12 6 0 0 E -8 -22 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 20 14 8 B 0 0 2 12 22 C -20 -2 0 -6 2 D -14 -12 6 0 0 E -8 -22 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 20 14 8 B 0 0 2 12 22 C -20 -2 0 -6 2 D -14 -12 6 0 0 E -8 -22 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6714: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) B E D C A (8) A D C B E (8) A C D E B (8) D B A E C (6) B D E C A (5) D B E C A (4) C A E B D (4) E C B A D (3) E C A B D (3) E B C D A (3) D A B E C (3) D A B C E (3) A C E D B (3) A C E B D (3) E B D A C (2) E B C A D (2) E B A C D (2) D B A C E (2) A D B C E (2) A C D B E (2) E B A D C (1) D C A B E (1) D B C A E (1) D A C B E (1) C E A B D (1) C D B A E (1) C D A B E (1) C B E D A (1) C A D E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B C E D A (1) A E C B D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 16 -6 8 B 4 0 12 -16 22 C -16 -12 0 -16 -8 D 6 16 16 0 20 E -8 -22 8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 -6 8 B 4 0 12 -16 22 C -16 -12 0 -16 -8 D 6 16 16 0 20 E -8 -22 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 E=16 B=16 C=10 so C is eliminated. Round 2 votes counts: A=35 D=31 E=17 B=17 so E is eliminated. Round 3 votes counts: A=39 D=31 B=30 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 B:211 A:207 E:179 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 16 -6 8 B 4 0 12 -16 22 C -16 -12 0 -16 -8 D 6 16 16 0 20 E -8 -22 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 -6 8 B 4 0 12 -16 22 C -16 -12 0 -16 -8 D 6 16 16 0 20 E -8 -22 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 -6 8 B 4 0 12 -16 22 C -16 -12 0 -16 -8 D 6 16 16 0 20 E -8 -22 8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6715: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) C D A B E (8) E A B D C (6) E A C B D (5) E A B C D (5) D C B A E (5) C D E B A (5) D C B E A (4) B D C A E (4) E C D A B (3) E B D C A (3) E B D A C (3) E B A D C (3) E A C D B (3) C D B E A (3) B E A D C (3) B A E D C (3) A E B C D (3) A E C D B (2) A C D B E (2) A B E D C (2) E D C B A (1) E C D B A (1) E C A D B (1) D B C E A (1) C D E A B (1) C D A E B (1) C A D E B (1) C A D B E (1) B D E A C (1) A E C B D (1) A E B D C (1) A C E D B (1) A B E C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -4 -6 -4 B -2 0 -16 -6 0 C 4 16 0 18 -2 D 6 6 -18 0 -4 E 4 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.078431 C: 0.000000 D: 0.000000 E: 0.921569 Sum of squares = 0.855440335954 Cumulative probabilities = A: 0.000000 B: 0.078431 C: 0.078431 D: 0.078431 E: 1.000000 A B C D E A 0 2 -4 -6 -4 B -2 0 -16 -6 0 C 4 16 0 18 -2 D 6 6 -18 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469136401 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=29 A=16 B=11 D=10 so D is eliminated. Round 2 votes counts: C=38 E=34 A=16 B=12 so B is eliminated. Round 3 votes counts: C=43 E=38 A=19 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:218 E:205 D:195 A:194 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 -6 -4 B -2 0 -16 -6 0 C 4 16 0 18 -2 D 6 6 -18 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469136401 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -6 -4 B -2 0 -16 -6 0 C 4 16 0 18 -2 D 6 6 -18 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469136401 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -6 -4 B -2 0 -16 -6 0 C 4 16 0 18 -2 D 6 6 -18 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469136401 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6716: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) D C E A B (6) E B D C A (5) B E A C D (5) B A E C D (5) E B C D A (4) E B A C D (4) D E C B A (4) D B E C A (4) A C D B E (4) E C A B D (3) E B D A C (3) D C E B A (3) D B A C E (3) C D A E B (3) C A D E B (3) B E A D C (3) B A E D C (3) E D C B A (2) E A C B D (2) D E B C A (2) D C B A E (2) D C A E B (2) C E D A B (2) C A E D B (2) A B C E D (2) E C D B A (1) E C D A B (1) E C A D B (1) D A B C E (1) C E A D B (1) C D E A B (1) C A D B E (1) B D A E C (1) B D A C E (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -20 -16 -14 B 10 0 -4 -12 -10 C 20 4 0 -6 -8 D 16 12 6 0 0 E 14 10 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.366872 E: 0.633128 Sum of squares = 0.535446254466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.366872 E: 1.000000 A B C D E A 0 -10 -20 -16 -14 B 10 0 -4 -12 -10 C 20 4 0 -6 -8 D 16 12 6 0 0 E 14 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=26 B=18 C=13 A=9 so A is eliminated. Round 2 votes counts: D=34 E=26 B=22 C=18 so C is eliminated. Round 3 votes counts: D=46 E=31 B=23 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:216 C:205 B:192 A:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -20 -16 -14 B 10 0 -4 -12 -10 C 20 4 0 -6 -8 D 16 12 6 0 0 E 14 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 -16 -14 B 10 0 -4 -12 -10 C 20 4 0 -6 -8 D 16 12 6 0 0 E 14 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 -16 -14 B 10 0 -4 -12 -10 C 20 4 0 -6 -8 D 16 12 6 0 0 E 14 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6717: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (14) A B C E D (12) D E C B A (8) A C B E D (8) A B C D E (8) C E D A B (6) C E D B A (5) C A E D B (5) E D C B A (4) B D E A C (4) B A C D E (3) A C E B D (3) D E B A C (2) C A B E D (2) A B D E C (2) E D C A B (1) E C D A B (1) D E B C A (1) D E A B C (1) D B E A C (1) C A E B D (1) B D E C A (1) B D A E C (1) B C D E A (1) B C D A E (1) B C A E D (1) B A C E D (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 24 24 28 B -4 0 12 26 20 C -24 -12 0 16 14 D -24 -26 -16 0 -2 E -28 -20 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 24 24 28 B -4 0 12 26 20 C -24 -12 0 16 14 D -24 -26 -16 0 -2 E -28 -20 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=27 C=19 D=13 E=6 so E is eliminated. Round 2 votes counts: A=35 B=27 C=20 D=18 so D is eliminated. Round 3 votes counts: A=36 C=33 B=31 so B is eliminated. Round 4 votes counts: A=62 C=38 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:240 B:227 C:197 E:170 D:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 24 24 28 B -4 0 12 26 20 C -24 -12 0 16 14 D -24 -26 -16 0 -2 E -28 -20 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 24 24 28 B -4 0 12 26 20 C -24 -12 0 16 14 D -24 -26 -16 0 -2 E -28 -20 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 24 24 28 B -4 0 12 26 20 C -24 -12 0 16 14 D -24 -26 -16 0 -2 E -28 -20 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984674 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6718: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) D B A E C (8) C E A B D (8) D C A B E (5) C D A B E (5) B D E A C (5) E C B A D (4) E A B C D (4) D B A C E (4) A E B D C (4) D A B C E (3) C E D B A (3) C E B A D (3) C D E A B (3) E C A B D (2) D C B E A (2) D B E C A (2) C E A D B (2) C A E D B (2) C A E B D (2) B E A D C (2) B D A E C (2) B A E D C (2) E C B D A (1) E B D C A (1) E B C A D (1) E B A D C (1) E A B D C (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B E C (1) C E D A B (1) C D E B A (1) C D A E B (1) B E D A C (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -18 -20 0 B 10 0 -8 -8 8 C 18 8 0 -12 12 D 20 8 12 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -20 0 B 10 0 -8 -8 8 C 18 8 0 -12 12 D 20 8 12 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=31 E=15 B=13 A=5 so A is eliminated. Round 2 votes counts: D=36 C=32 E=19 B=13 so B is eliminated. Round 3 votes counts: D=44 C=32 E=24 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:213 B:201 E:186 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -18 -20 0 B 10 0 -8 -8 8 C 18 8 0 -12 12 D 20 8 12 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -20 0 B 10 0 -8 -8 8 C 18 8 0 -12 12 D 20 8 12 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -20 0 B 10 0 -8 -8 8 C 18 8 0 -12 12 D 20 8 12 0 8 E 0 -8 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6719: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (13) E A C D B (9) D B C A E (9) A E D B C (9) C D B E A (8) E A C B D (7) D B A C E (6) C E D B A (5) E C A B D (4) C E B D A (4) B D C A E (4) A E B D C (4) A D B E C (4) C B D A E (3) A E C B D (2) E D A B C (1) E C A D B (1) E A D C B (1) E A D B C (1) D C B E A (1) D B A E C (1) B C D A E (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -6 -12 -10 B 10 0 -16 -12 2 C 6 16 0 16 8 D 12 12 -16 0 2 E 10 -2 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -12 -10 B 10 0 -16 -12 2 C 6 16 0 16 8 D 12 12 -16 0 2 E 10 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=24 A=21 D=17 B=5 so B is eliminated. Round 2 votes counts: C=34 E=24 D=21 A=21 so D is eliminated. Round 3 votes counts: C=48 A=28 E=24 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:205 E:199 B:192 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 -12 -10 B 10 0 -16 -12 2 C 6 16 0 16 8 D 12 12 -16 0 2 E 10 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -12 -10 B 10 0 -16 -12 2 C 6 16 0 16 8 D 12 12 -16 0 2 E 10 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -12 -10 B 10 0 -16 -12 2 C 6 16 0 16 8 D 12 12 -16 0 2 E 10 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6720: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) C A D E B (8) C A D B E (7) D B E C A (5) E B A C D (4) D E B C A (4) D C E A B (4) A C E B D (4) A C B E D (4) A C B D E (4) E D B C A (3) E B D C A (3) D E C B A (3) D C A E B (3) D C A B E (3) C D A E B (3) E A C B D (2) D C B A E (2) B E A D C (2) B E A C D (2) B A C E D (2) A E C B D (2) A E B C D (2) E D C A B (1) E B A D C (1) E A B C D (1) D E C A B (1) D C E B A (1) D B C A E (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E C A (1) B A E C D (1) B A C D E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -10 2 2 B -6 0 -10 -2 -22 C 10 10 0 2 2 D -2 2 -2 0 8 E -2 22 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998676 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 2 2 B -6 0 -10 -2 -22 C 10 10 0 2 2 D -2 2 -2 0 8 E -2 22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987275 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 C=20 A=18 B=10 so B is eliminated. Round 2 votes counts: E=30 D=28 A=22 C=20 so C is eliminated. Round 3 votes counts: A=39 D=31 E=30 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 E:205 D:203 A:200 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 2 2 B -6 0 -10 -2 -22 C 10 10 0 2 2 D -2 2 -2 0 8 E -2 22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987275 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 2 2 B -6 0 -10 -2 -22 C 10 10 0 2 2 D -2 2 -2 0 8 E -2 22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987275 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 2 2 B -6 0 -10 -2 -22 C 10 10 0 2 2 D -2 2 -2 0 8 E -2 22 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987275 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6721: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) E C A D B (7) E A C D B (4) D A B C E (4) C E B A D (4) B C E A D (4) B A E D C (4) E C A B D (3) D C A E B (3) D A C E B (3) C D E B A (3) B D A C E (3) A E D C B (3) A D B E C (3) E B A C D (2) E A B D C (2) D B A E C (2) D B A C E (2) C E B D A (2) C E A D B (2) C D B E A (2) C B E D A (2) B C E D A (2) B C D E A (2) B A D E C (2) A B D E C (2) E B C A D (1) E A D C B (1) E A B C D (1) D C B A E (1) D A E C B (1) D A B E C (1) C E D B A (1) C D E A B (1) C D B A E (1) C D A E B (1) B E C A D (1) B E A C D (1) B D A E C (1) B C D A E (1) A E D B C (1) A E C D B (1) A E B D C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 12 -6 4 -14 B -12 0 -10 -16 -16 C 6 10 0 16 6 D -4 16 -16 0 -20 E 14 16 -6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 4 -14 B -12 0 -10 -16 -16 C 6 10 0 16 6 D -4 16 -16 0 -20 E 14 16 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=21 B=21 D=17 A=13 so A is eliminated. Round 2 votes counts: C=28 E=27 B=24 D=21 so D is eliminated. Round 3 votes counts: B=36 C=35 E=29 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:222 C:219 A:198 D:188 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 4 -14 B -12 0 -10 -16 -16 C 6 10 0 16 6 D -4 16 -16 0 -20 E 14 16 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 4 -14 B -12 0 -10 -16 -16 C 6 10 0 16 6 D -4 16 -16 0 -20 E 14 16 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 4 -14 B -12 0 -10 -16 -16 C 6 10 0 16 6 D -4 16 -16 0 -20 E 14 16 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6722: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (13) E B A D C (9) B E A C D (8) D C E B A (7) C D A B E (6) C A B E D (6) D E C B A (4) D C A E B (4) D C A B E (4) C A D B E (4) E B D A C (3) D E B C A (3) D C E A B (3) B A E C D (3) D E B A C (2) D A C B E (2) C A B D E (2) B E A D C (2) A C D B E (2) E D C B A (1) E D B A C (1) E B D C A (1) E B C A D (1) D A C E B (1) C D E B A (1) C D E A B (1) C D A E B (1) C B E A D (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 0 10 -22 B 20 0 -2 6 -12 C 0 2 0 4 -6 D -10 -6 -4 0 -6 E 22 12 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 0 10 -22 B 20 0 -2 6 -12 C 0 2 0 4 -6 D -10 -6 -4 0 -6 E 22 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=29 C=22 B=13 A=6 so A is eliminated. Round 2 votes counts: D=30 E=29 C=25 B=16 so B is eliminated. Round 3 votes counts: E=44 D=30 C=26 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:206 C:200 D:187 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 0 10 -22 B 20 0 -2 6 -12 C 0 2 0 4 -6 D -10 -6 -4 0 -6 E 22 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 0 10 -22 B 20 0 -2 6 -12 C 0 2 0 4 -6 D -10 -6 -4 0 -6 E 22 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 0 10 -22 B 20 0 -2 6 -12 C 0 2 0 4 -6 D -10 -6 -4 0 -6 E 22 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6723: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D C A B E (8) B E D A C (8) E C A D B (7) E A C B D (7) D B C A E (5) E B D A C (4) D E B C A (4) A C B E D (4) E D C A B (3) D E C A B (3) C A D B E (3) B E A D C (3) B D E A C (3) B D A C E (3) B A C D E (3) A C B D E (3) E D B C A (2) D C A E B (2) D B E C A (2) A C E B D (2) A B C E D (2) E D C B A (1) E B A D C (1) E A B C D (1) D E C B A (1) D C B A E (1) C D A B E (1) C A E B D (1) B E A C D (1) B A D C E (1) B A C E D (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 12 -2 -18 B 4 0 6 14 4 C -12 -6 0 -10 -20 D 2 -14 10 0 -14 E 18 -4 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 -2 -18 B 4 0 6 14 4 C -12 -6 0 -10 -20 D 2 -14 10 0 -14 E 18 -4 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=26 B=23 A=12 C=5 so C is eliminated. Round 2 votes counts: E=34 D=27 B=23 A=16 so A is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:224 B:214 A:194 D:192 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 -2 -18 B 4 0 6 14 4 C -12 -6 0 -10 -20 D 2 -14 10 0 -14 E 18 -4 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -2 -18 B 4 0 6 14 4 C -12 -6 0 -10 -20 D 2 -14 10 0 -14 E 18 -4 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -2 -18 B 4 0 6 14 4 C -12 -6 0 -10 -20 D 2 -14 10 0 -14 E 18 -4 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994427 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6724: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D A C B E (7) E D B C A (5) E D A B C (5) E B C D A (5) C B A E D (5) A D E C B (5) A D C B E (5) D A E C B (4) B C E A D (4) E A B C D (3) D E A C B (3) D E A B C (3) A C B D E (3) E C B A D (2) E B D C A (2) E A C B D (2) D E B C A (2) D A E B C (2) C B A D E (2) C A B D E (2) B C D A E (2) A D C E B (2) A C D B E (2) E D B A C (1) E A D C B (1) E A D B C (1) D E B A C (1) D A C E B (1) D A B C E (1) C D B A E (1) C B D A E (1) C A B E D (1) B E C D A (1) B E C A D (1) B C E D A (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 8 4 -4 B -10 0 -2 -6 -20 C -8 2 0 -2 -18 D -4 6 2 0 -2 E 4 20 18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997691 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 8 4 -4 B -10 0 -2 -6 -20 C -8 2 0 -2 -18 D -4 6 2 0 -2 E 4 20 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948146 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=24 A=19 C=12 B=9 so B is eliminated. Round 2 votes counts: E=38 D=24 C=19 A=19 so C is eliminated. Round 3 votes counts: E=43 A=29 D=28 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:209 D:201 C:187 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 8 4 -4 B -10 0 -2 -6 -20 C -8 2 0 -2 -18 D -4 6 2 0 -2 E 4 20 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948146 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 4 -4 B -10 0 -2 -6 -20 C -8 2 0 -2 -18 D -4 6 2 0 -2 E 4 20 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948146 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 4 -4 B -10 0 -2 -6 -20 C -8 2 0 -2 -18 D -4 6 2 0 -2 E 4 20 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948146 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6725: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) C B E D A (7) B C E D A (7) D A B C E (5) E C B A D (4) E B C D A (4) C B D A E (4) E C A B D (3) E A D C B (3) C E B A D (3) C E A B D (3) C A D B E (3) B E D A C (3) B E C D A (3) A D E C B (3) A D C E B (3) E B D A C (2) D B A C E (2) C A E D B (2) B C D E A (2) B C D A E (2) A E D C B (2) A D C B E (2) A D B C E (2) E D A B C (1) E C B D A (1) E C A D B (1) E B D C A (1) E B C A D (1) E B A D C (1) E A C D B (1) D C A B E (1) D A E B C (1) D A C B E (1) C B E A D (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D C A (1) B D C E A (1) B D A C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -16 -16 -8 B 4 0 -4 14 18 C 16 4 0 12 12 D 16 -14 -12 0 -10 E 8 -18 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -16 -8 B 4 0 -4 14 18 C 16 4 0 12 12 D 16 -14 -12 0 -10 E 8 -18 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=20 D=18 A=13 so A is eliminated. Round 2 votes counts: D=28 C=27 E=25 B=20 so B is eliminated. Round 3 votes counts: C=38 E=32 D=30 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:216 E:194 D:190 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 -16 -8 B 4 0 -4 14 18 C 16 4 0 12 12 D 16 -14 -12 0 -10 E 8 -18 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -16 -8 B 4 0 -4 14 18 C 16 4 0 12 12 D 16 -14 -12 0 -10 E 8 -18 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -16 -8 B 4 0 -4 14 18 C 16 4 0 12 12 D 16 -14 -12 0 -10 E 8 -18 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6726: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (12) B C D E A (10) A E B D C (6) E A C B D (5) C D E A B (5) B E A C D (5) E A D C B (4) E A B C D (4) D C A E B (4) B D C A E (4) B C D A E (4) B A E D C (4) A E D B C (4) D C E A B (3) D A E C B (3) E A C D B (2) D C B A E (2) C D E B A (2) C D B E A (2) B E C A D (2) B A E C D (2) A E B C D (2) D C E B A (1) D C B E A (1) C E A D B (1) C B D E A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C E A D (1) A B E D C (1) Total count = 100 A B C D E A 0 12 12 10 0 B -12 0 4 8 -16 C -12 -4 0 -2 -14 D -10 -8 2 0 -12 E 0 16 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.585466 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.414534 Sum of squares = 0.514608936345 Cumulative probabilities = A: 0.585466 B: 0.585466 C: 0.585466 D: 0.585466 E: 1.000000 A B C D E A 0 12 12 10 0 B -12 0 4 8 -16 C -12 -4 0 -2 -14 D -10 -8 2 0 -12 E 0 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=25 E=15 D=14 C=11 so C is eliminated. Round 2 votes counts: B=36 A=25 D=23 E=16 so E is eliminated. Round 3 votes counts: A=41 B=36 D=23 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:221 A:217 B:192 D:186 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 10 0 B -12 0 4 8 -16 C -12 -4 0 -2 -14 D -10 -8 2 0 -12 E 0 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 10 0 B -12 0 4 8 -16 C -12 -4 0 -2 -14 D -10 -8 2 0 -12 E 0 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 10 0 B -12 0 4 8 -16 C -12 -4 0 -2 -14 D -10 -8 2 0 -12 E 0 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6727: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) B D C E A (10) D B C E A (7) E A C D B (6) D B A C E (6) A E C D B (6) D B E C A (5) D B A E C (5) C E B A D (5) C E A B D (5) A E C B D (4) B C D E A (3) A E D C B (3) A D B E C (3) A C E B D (3) E A D C B (2) B D C A E (2) B C E D A (2) A B C E D (2) E A C B D (1) D E C B A (1) D E A B C (1) D B E A C (1) D A B E C (1) B A C E D (1) A D E C B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 12 -22 B -4 0 0 4 0 C 4 0 0 4 -4 D -12 -4 -4 0 -6 E 22 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.529576 C: 0.000000 D: 0.000000 E: 0.470424 Sum of squares = 0.501749500996 Cumulative probabilities = A: 0.000000 B: 0.529576 C: 0.529576 D: 0.529576 E: 1.000000 A B C D E A 0 4 -4 12 -22 B -4 0 0 4 0 C 4 0 0 4 -4 D -12 -4 -4 0 -6 E 22 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=24 E=21 B=18 C=10 so C is eliminated. Round 2 votes counts: E=31 D=27 A=24 B=18 so B is eliminated. Round 3 votes counts: D=42 E=33 A=25 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:202 B:200 A:195 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 12 -22 B -4 0 0 4 0 C 4 0 0 4 -4 D -12 -4 -4 0 -6 E 22 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 12 -22 B -4 0 0 4 0 C 4 0 0 4 -4 D -12 -4 -4 0 -6 E 22 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 12 -22 B -4 0 0 4 0 C 4 0 0 4 -4 D -12 -4 -4 0 -6 E 22 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6728: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (9) C E B A D (8) E C B D A (7) C A E B D (6) B D E C A (5) D B E A C (4) D A B E C (4) B C E D A (4) A D B E C (4) A C E D B (4) A C E B D (4) D E B C A (3) A C B E D (3) E B D C A (2) E B C D A (2) D B A E C (2) D A E B C (2) C E A B D (2) C B E A D (2) B E D C A (2) B D A C E (2) A D E C B (2) A D C B E (2) A C D E B (2) A C D B E (2) E D C A B (1) E D B C A (1) E D A C B (1) E C A D B (1) D E A B C (1) C E B D A (1) C B A E D (1) C A B E D (1) B E C D A (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 0 10 4 B -6 0 -2 10 0 C 0 2 0 4 10 D -10 -10 -4 0 -8 E -4 0 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.614790 B: 0.000000 C: 0.385210 D: 0.000000 E: 0.000000 Sum of squares = 0.526353445741 Cumulative probabilities = A: 0.614790 B: 0.614790 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 10 4 B -6 0 -2 10 0 C 0 2 0 4 10 D -10 -10 -4 0 -8 E -4 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=21 D=16 E=15 B=14 so B is eliminated. Round 2 votes counts: A=34 C=25 D=23 E=18 so E is eliminated. Round 3 votes counts: C=36 A=34 D=30 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:208 B:201 E:197 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 10 4 B -6 0 -2 10 0 C 0 2 0 4 10 D -10 -10 -4 0 -8 E -4 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 10 4 B -6 0 -2 10 0 C 0 2 0 4 10 D -10 -10 -4 0 -8 E -4 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 10 4 B -6 0 -2 10 0 C 0 2 0 4 10 D -10 -10 -4 0 -8 E -4 0 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6729: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) A E D C B (6) C D B E A (5) B C D E A (5) A E D B C (4) E C A B D (3) E A D C B (3) D C E A B (3) D C B A E (3) D A E C B (3) D A B E C (3) C B E D A (3) B C E A D (3) B C D A E (3) A E B D C (3) A D E C B (3) A D E B C (3) D B C A E (2) D B A C E (2) C E D A B (2) C E B A D (2) C D E B A (2) C D E A B (2) C B E A D (2) B D C A E (2) B A E D C (2) A B D E C (2) E C D A B (1) E A C B D (1) D C E B A (1) D C A B E (1) D A E B C (1) D A C E B (1) C E D B A (1) C E B D A (1) C E A D B (1) C B D E A (1) B E C A D (1) B E A C D (1) B D A C E (1) B C E D A (1) B A E C D (1) B A C D E (1) Total count = 100 A B C D E A 0 8 -2 0 -6 B -8 0 -18 -22 -10 C 2 18 0 2 2 D 0 22 -2 0 0 E 6 10 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998724 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 0 -6 B -8 0 -18 -22 -10 C 2 18 0 2 2 D 0 22 -2 0 0 E 6 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 B=21 A=21 D=20 E=16 so E is eliminated. Round 2 votes counts: A=33 C=26 B=21 D=20 so D is eliminated. Round 3 votes counts: A=41 C=34 B=25 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:210 E:207 A:200 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 0 -6 B -8 0 -18 -22 -10 C 2 18 0 2 2 D 0 22 -2 0 0 E 6 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 0 -6 B -8 0 -18 -22 -10 C 2 18 0 2 2 D 0 22 -2 0 0 E 6 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 0 -6 B -8 0 -18 -22 -10 C 2 18 0 2 2 D 0 22 -2 0 0 E 6 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6730: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) E B A D C (7) D C A B E (6) D C B A E (5) D A C B E (5) A C D E B (5) E B C A D (4) E A B D C (4) D A E B C (4) B D E C A (4) A D C E B (4) C D B E A (3) E C B A D (2) E B D A C (2) E A C B D (2) D B C E A (2) D B C A E (2) C D A B E (2) B C D E A (2) A E B D C (2) A D E B C (2) A C E B D (2) E B C D A (1) E B A C D (1) E A B C D (1) D C B E A (1) D B E C A (1) D A C E B (1) D A B E C (1) D A B C E (1) C D B A E (1) C B E D A (1) C B D E A (1) C A D E B (1) C A D B E (1) B E C D A (1) B D C E A (1) B C E D A (1) A E D C B (1) A E D B C (1) A E C B D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 -14 0 B 2 0 6 -4 0 C 2 -6 0 -32 -2 D 14 4 32 0 14 E 0 0 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -14 0 B 2 0 6 -4 0 C 2 -6 0 -32 -2 D 14 4 32 0 14 E 0 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 A=20 B=17 C=10 so C is eliminated. Round 2 votes counts: D=35 E=24 A=22 B=19 so B is eliminated. Round 3 votes counts: D=43 E=35 A=22 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:232 B:202 E:194 A:191 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -14 0 B 2 0 6 -4 0 C 2 -6 0 -32 -2 D 14 4 32 0 14 E 0 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -14 0 B 2 0 6 -4 0 C 2 -6 0 -32 -2 D 14 4 32 0 14 E 0 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -14 0 B 2 0 6 -4 0 C 2 -6 0 -32 -2 D 14 4 32 0 14 E 0 0 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999983555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6731: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) B E A D C (8) D C B A E (7) D C A B E (7) C D A E B (7) B D C E A (6) A E C D B (6) C A E D B (5) B E D A C (5) E A C B D (4) D B C E A (4) C A D E B (4) B E A C D (4) E B A C D (3) D C A E B (3) D B C A E (3) B D E C A (3) A E C B D (3) C D A B E (2) A C E D B (2) E A B D C (1) D B E C A (1) D B E A C (1) B E C D A (1) B D E A C (1) B C D E A (1) Total count = 100 A B C D E A 0 4 -8 -4 -2 B -4 0 0 -4 8 C 8 0 0 0 2 D 4 4 0 0 0 E 2 -8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.522836 D: 0.477164 E: 0.000000 Sum of squares = 0.501043002584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.522836 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -4 -2 B -4 0 0 -4 8 C 8 0 0 0 2 D 4 4 0 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999998594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 C=18 E=16 A=11 so A is eliminated. Round 2 votes counts: B=29 D=26 E=25 C=20 so C is eliminated. Round 3 votes counts: D=39 E=32 B=29 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:205 D:204 B:200 E:196 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -4 -2 B -4 0 0 -4 8 C 8 0 0 0 2 D 4 4 0 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999998594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -4 -2 B -4 0 0 -4 8 C 8 0 0 0 2 D 4 4 0 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999998594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -4 -2 B -4 0 0 -4 8 C 8 0 0 0 2 D 4 4 0 0 0 E 2 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999998594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6732: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (10) E B A D C (7) E A B D C (6) C D A B E (5) E B D A C (4) C D E B A (4) C D B A E (4) C A D B E (4) A B E D C (4) E D B A C (3) E C A B D (3) C E D B A (3) B A E D C (3) A B D E C (3) E D B C A (2) E C D A B (2) C E D A B (2) C D B E A (2) C A D E B (2) C A B D E (2) A C B D E (2) E D C B A (1) E C B A D (1) E A C B D (1) D C E B A (1) D C B A E (1) D B A E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C A B E D (1) B A D E C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 8 -8 B -4 0 -8 0 -8 C 4 8 0 4 -4 D -8 0 -4 0 -10 E 8 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -4 8 -8 B -4 0 -8 0 -8 C 4 8 0 4 -4 D -8 0 -4 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=31 A=21 D=4 B=4 so D is eliminated. Round 2 votes counts: E=40 C=33 A=22 B=5 so B is eliminated. Round 3 votes counts: E=40 C=33 A=27 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:206 A:200 B:190 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 8 -8 B -4 0 -8 0 -8 C 4 8 0 4 -4 D -8 0 -4 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 8 -8 B -4 0 -8 0 -8 C 4 8 0 4 -4 D -8 0 -4 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 8 -8 B -4 0 -8 0 -8 C 4 8 0 4 -4 D -8 0 -4 0 -10 E 8 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6733: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) A B E C D (10) E A B C D (5) C D A B E (5) B A E C D (5) A B C D E (5) E D C B A (4) D C E B A (4) D C B A E (4) D C A B E (4) C D E A B (4) D E C B A (3) D C E A B (3) C D A E B (3) C A D B E (3) B D A C E (3) B A E D C (3) A C B D E (3) E C D A B (2) E B D A C (2) E B A C D (2) E A C B D (2) A E B C D (2) E D B C A (1) D E B C A (1) D C B E A (1) D B C A E (1) C E D A B (1) C A D E B (1) B A C D E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 10 8 10 B -10 0 4 10 0 C -10 -4 0 12 -4 D -8 -10 -12 0 -2 E -10 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 8 10 B -10 0 4 10 0 C -10 -4 0 12 -4 D -8 -10 -12 0 -2 E -10 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=22 D=21 C=17 B=12 so B is eliminated. Round 2 votes counts: A=31 E=28 D=24 C=17 so C is eliminated. Round 3 votes counts: D=36 A=35 E=29 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:202 E:198 C:197 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 8 10 B -10 0 4 10 0 C -10 -4 0 12 -4 D -8 -10 -12 0 -2 E -10 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 8 10 B -10 0 4 10 0 C -10 -4 0 12 -4 D -8 -10 -12 0 -2 E -10 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 8 10 B -10 0 4 10 0 C -10 -4 0 12 -4 D -8 -10 -12 0 -2 E -10 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6734: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) B E A D C (6) E B A D C (5) D A E B C (5) D A C E B (5) C D A B E (5) B E D A C (5) B E C D A (5) C B D A E (4) A D E C B (4) E B A C D (3) D A E C B (3) C B E A D (3) C A D E B (3) B E C A D (3) B C D E A (3) D C A E B (2) D C A B E (2) B E A C D (2) B D E A C (2) A E D B C (2) E C A B D (1) E A D B C (1) E A C D B (1) D C B A E (1) D B A E C (1) C E B A D (1) C D B A E (1) C A E B D (1) C A D B E (1) B E D C A (1) B C E D A (1) B C E A D (1) A E D C B (1) A E C D B (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 0 -14 12 B -4 0 -6 -6 -6 C 0 6 0 4 -6 D 14 6 -4 0 12 E -12 6 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.272727 E: 0.181818 Sum of squares = 0.404958677684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.818182 E: 1.000000 A B C D E A 0 4 0 -14 12 B -4 0 -6 -6 -6 C 0 6 0 4 -6 D 14 6 -4 0 12 E -12 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.272727 E: 0.181818 Sum of squares = 0.404958677683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=29 D=19 E=11 A=11 so E is eliminated. Round 2 votes counts: B=37 C=31 D=19 A=13 so A is eliminated. Round 3 votes counts: B=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:214 C:202 A:201 E:194 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -14 12 B -4 0 -6 -6 -6 C 0 6 0 4 -6 D 14 6 -4 0 12 E -12 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.272727 E: 0.181818 Sum of squares = 0.404958677683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.818182 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -14 12 B -4 0 -6 -6 -6 C 0 6 0 4 -6 D 14 6 -4 0 12 E -12 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.272727 E: 0.181818 Sum of squares = 0.404958677683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.818182 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -14 12 B -4 0 -6 -6 -6 C 0 6 0 4 -6 D 14 6 -4 0 12 E -12 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.272727 E: 0.181818 Sum of squares = 0.404958677683 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.818182 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6735: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (14) D C E B A (8) D C A E B (7) C D E B A (6) B E A C D (6) E B C D A (4) E B C A D (4) D C E A B (4) A D C B E (4) A B E D C (4) D C B E A (3) D A C B E (3) B A E C D (3) A D B C E (3) B D E A C (2) A E B C D (2) E C B D A (1) D C B A E (1) D C A B E (1) D B E C A (1) D B E A C (1) D A C E B (1) C E D B A (1) C E D A B (1) C E B D A (1) C E B A D (1) C D E A B (1) C D A E B (1) C A E B D (1) B E C D A (1) B E A D C (1) B A D E C (1) A D C E B (1) A D B E C (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 4 2 6 B -8 0 -2 -4 4 C -4 2 0 4 8 D -2 4 -4 0 6 E -6 -4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 2 6 B -8 0 -2 -4 4 C -4 2 0 4 8 D -2 4 -4 0 6 E -6 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=30 B=14 C=13 E=9 so E is eliminated. Round 2 votes counts: A=34 D=30 B=22 C=14 so C is eliminated. Round 3 votes counts: D=40 A=35 B=25 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:205 D:202 B:195 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 2 6 B -8 0 -2 -4 4 C -4 2 0 4 8 D -2 4 -4 0 6 E -6 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 6 B -8 0 -2 -4 4 C -4 2 0 4 8 D -2 4 -4 0 6 E -6 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 6 B -8 0 -2 -4 4 C -4 2 0 4 8 D -2 4 -4 0 6 E -6 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6736: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) C B A E D (7) A C B E D (7) E D A B C (6) D E A B C (6) B C D A E (6) B C A E D (6) D B C E A (5) C B D A E (5) E D A C B (4) E A D B C (4) D B E C A (3) A E B C D (3) E A D C B (2) D E C A B (2) D E B C A (2) D C B E A (2) C B A D E (2) B D C E A (2) A C E B D (2) A B C E D (2) E D B A C (1) E A B D C (1) D E C B A (1) D E B A C (1) D C A E B (1) D C A B E (1) B C E A D (1) B C D E A (1) B C A D E (1) A E C B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 4 -18 -8 B -8 0 4 -2 6 C -4 -4 0 -8 4 D 18 2 8 0 4 E 8 -6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -18 -8 B -8 0 4 -2 6 C -4 -4 0 -8 4 D 18 2 8 0 4 E 8 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=18 B=17 A=17 C=14 so C is eliminated. Round 2 votes counts: D=34 B=31 E=18 A=17 so A is eliminated. Round 3 votes counts: B=42 D=34 E=24 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:200 E:197 C:194 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -18 -8 B -8 0 4 -2 6 C -4 -4 0 -8 4 D 18 2 8 0 4 E 8 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -18 -8 B -8 0 4 -2 6 C -4 -4 0 -8 4 D 18 2 8 0 4 E 8 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -18 -8 B -8 0 4 -2 6 C -4 -4 0 -8 4 D 18 2 8 0 4 E 8 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6737: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (13) C A B E D (8) C B A E D (6) D E B A C (5) A D C B E (5) E B C A D (4) C B E A D (4) D A E B C (3) D A B E C (3) C E B A D (3) C A B D E (3) A C D B E (3) A C B D E (3) E B D C A (2) E B C D A (2) D A E C B (2) D A C E B (2) D A C B E (2) B A C E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E C B A D (1) E B D A C (1) E B A C D (1) D E C B A (1) D E B C A (1) D E A C B (1) D C A B E (1) D A B C E (1) C E B D A (1) B E C A D (1) B E A C D (1) B C A E D (1) B A E C D (1) A D B E C (1) A D B C E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 14 8 6 B -14 0 2 2 4 C -14 -2 0 0 0 D -8 -2 0 0 10 E -6 -4 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 8 6 B -14 0 2 2 4 C -14 -2 0 0 0 D -8 -2 0 0 10 E -6 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=25 A=18 E=16 B=6 so B is eliminated. Round 2 votes counts: D=35 C=26 A=21 E=18 so E is eliminated. Round 3 votes counts: D=41 C=36 A=23 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:221 D:200 B:197 C:192 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 8 6 B -14 0 2 2 4 C -14 -2 0 0 0 D -8 -2 0 0 10 E -6 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 8 6 B -14 0 2 2 4 C -14 -2 0 0 0 D -8 -2 0 0 10 E -6 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 8 6 B -14 0 2 2 4 C -14 -2 0 0 0 D -8 -2 0 0 10 E -6 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6738: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) C D B A E (7) B D C E A (7) A E B D C (7) A E B C D (7) E B A D C (6) D C B E A (6) C A D E B (6) C D A E B (5) A E C D B (5) C D B E A (4) B E A D C (4) A E C B D (4) B E D A C (3) D C B A E (2) B E D C A (2) B D E C A (2) B D E A C (2) B A E D C (2) E D C B A (1) E B D A C (1) E A C D B (1) E A B C D (1) D E B C A (1) D B C E A (1) C D A B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 10 10 -4 B 2 0 12 16 -14 C -10 -12 0 -14 -18 D -10 -16 14 0 -10 E 4 14 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 10 10 -4 B 2 0 12 16 -14 C -10 -12 0 -14 -18 D -10 -16 14 0 -10 E 4 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=23 B=22 E=20 D=10 so D is eliminated. Round 2 votes counts: C=31 A=25 B=23 E=21 so E is eliminated. Round 3 votes counts: A=37 C=32 B=31 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:223 B:208 A:207 D:189 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 10 10 -4 B 2 0 12 16 -14 C -10 -12 0 -14 -18 D -10 -16 14 0 -10 E 4 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 10 -4 B 2 0 12 16 -14 C -10 -12 0 -14 -18 D -10 -16 14 0 -10 E 4 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 10 -4 B 2 0 12 16 -14 C -10 -12 0 -14 -18 D -10 -16 14 0 -10 E 4 14 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6739: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D C A B E (5) B D A E C (5) B D A C E (5) E C B A D (4) E C A D B (4) E C A B D (4) D A C B E (4) C E D A B (4) C E A D B (4) B A D E C (4) A B D E C (4) B E A D C (3) E B C A D (2) E A C B D (2) E A B C D (2) D B C A E (2) C E D B A (2) B D C A E (2) A E B D C (2) A D C E B (2) A D C B E (2) A D B E C (2) E C D A B (1) E B C D A (1) E B A D C (1) E B A C D (1) D C A E B (1) D B A C E (1) C D E B A (1) C D E A B (1) C D A E B (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D E B (1) B E C D A (1) B E C A D (1) B E A C D (1) B D C E A (1) B A E D C (1) A E D C B (1) A E C D B (1) A D B C E (1) A C D E B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -4 8 2 B -6 0 -10 14 -2 C 4 10 0 -2 -10 D -8 -14 2 0 -4 E -2 2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.46875 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 6 -4 8 2 B -6 0 -10 14 -2 C 4 10 0 -2 -10 D -8 -14 2 0 -4 E -2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999996 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=24 A=18 C=17 D=13 so D is eliminated. Round 2 votes counts: E=28 B=27 C=23 A=22 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:207 A:206 C:201 B:198 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 8 2 B -6 0 -10 14 -2 C 4 10 0 -2 -10 D -8 -14 2 0 -4 E -2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999996 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 8 2 B -6 0 -10 14 -2 C 4 10 0 -2 -10 D -8 -14 2 0 -4 E -2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999996 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 8 2 B -6 0 -10 14 -2 C 4 10 0 -2 -10 D -8 -14 2 0 -4 E -2 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999996 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6740: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) E C B D A (5) C A B E D (5) E D C B A (4) C E A B D (4) C A E B D (4) A B D C E (4) E C D B A (3) E B D C A (3) D E C B A (3) D B E A C (3) C E B A D (3) B D E A C (3) A D C B E (3) A B D E C (3) E B C D A (2) D E C A B (2) D E B A C (2) D E A C B (2) D B A E C (2) D A E C B (2) C E D B A (2) C E D A B (2) B E C A D (2) B A D E C (2) A C D E B (2) A C B E D (2) A C B D E (2) A B C E D (2) E D B C A (1) D E A B C (1) D A C E B (1) D A B E C (1) C B A E D (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C D A (1) B E A C D (1) A D B E C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -16 -10 -18 B 6 0 -10 0 -18 C 16 10 0 -8 -18 D 10 0 8 0 2 E 18 18 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.051322 C: 0.000000 D: 0.948678 E: 0.000000 Sum of squares = 0.902624514514 Cumulative probabilities = A: 0.000000 B: 0.051322 C: 0.051322 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -10 -18 B 6 0 -10 0 -18 C 16 10 0 -8 -18 D 10 0 8 0 2 E 18 18 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012622 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 A=21 E=18 B=10 so B is eliminated. Round 2 votes counts: D=31 E=23 C=23 A=23 so E is eliminated. Round 3 votes counts: D=40 C=36 A=24 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:226 D:210 C:200 B:189 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -16 -10 -18 B 6 0 -10 0 -18 C 16 10 0 -8 -18 D 10 0 8 0 2 E 18 18 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012622 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -10 -18 B 6 0 -10 0 -18 C 16 10 0 -8 -18 D 10 0 8 0 2 E 18 18 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012622 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -10 -18 B 6 0 -10 0 -18 C 16 10 0 -8 -18 D 10 0 8 0 2 E 18 18 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012622 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6741: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (13) C D E A B (8) A B E C D (8) D C E B A (7) D C B A E (6) C E D A B (6) D B C A E (5) C E A D B (5) D B A C E (4) B A E D C (4) A E B C D (4) E A C B D (3) E A B C D (3) D C E A B (3) D C B E A (3) B A E C D (3) E C B A D (2) E C A D B (2) E C A B D (2) C D E B A (2) D A E B C (1) C E D B A (1) C E A B D (1) B D A E C (1) B D A C E (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -6 4 4 B 4 0 -2 -8 -2 C 6 2 0 0 4 D -4 8 0 0 12 E -4 2 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.702562 D: 0.297438 E: 0.000000 Sum of squares = 0.582062612643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.702562 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 4 4 B 4 0 -2 -8 -2 C 6 2 0 0 4 D -4 8 0 0 12 E -4 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=23 B=22 A=14 E=12 so E is eliminated. Round 2 votes counts: D=29 C=29 B=22 A=20 so A is eliminated. Round 3 votes counts: B=38 C=32 D=30 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:206 A:199 B:196 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 4 4 B 4 0 -2 -8 -2 C 6 2 0 0 4 D -4 8 0 0 12 E -4 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 4 B 4 0 -2 -8 -2 C 6 2 0 0 4 D -4 8 0 0 12 E -4 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 4 B 4 0 -2 -8 -2 C 6 2 0 0 4 D -4 8 0 0 12 E -4 2 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6742: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) B E C A D (7) D A E C B (6) A D C B E (6) A C B D E (6) D A C E B (5) C B A D E (5) B C A E D (5) E D A B C (4) E D B A C (3) E B D C A (3) D E C B A (3) D A C B E (3) C B E A D (3) A C D B E (3) A B C E D (3) E D C B A (2) E B C D A (2) D E A B C (2) C B A E D (2) C A B D E (2) B C E D A (2) B C E A D (2) E B A D C (1) D E C A B (1) D E A C B (1) D C A B E (1) C D B A E (1) C B E D A (1) C B D A E (1) B E A C D (1) B A E C D (1) B A C E D (1) A E D B C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -8 -2 2 8 B 8 0 -6 -4 12 C 2 6 0 -4 4 D -2 4 4 0 -4 E -8 -12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428528 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 2 8 B 8 0 -6 -4 12 C 2 6 0 -4 4 D -2 4 4 0 -4 E -8 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428564 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 D=22 A=21 B=19 C=15 so C is eliminated. Round 2 votes counts: B=31 E=23 D=23 A=23 so E is eliminated. Round 3 votes counts: D=40 B=37 A=23 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:205 C:204 D:201 A:200 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 2 8 B 8 0 -6 -4 12 C 2 6 0 -4 4 D -2 4 4 0 -4 E -8 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428564 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 2 8 B 8 0 -6 -4 12 C 2 6 0 -4 4 D -2 4 4 0 -4 E -8 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428564 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 2 8 B 8 0 -6 -4 12 C 2 6 0 -4 4 D -2 4 4 0 -4 E -8 -12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.142857 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428564 Cumulative probabilities = A: 0.285714 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6743: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) C D E A B (6) B A E D C (6) A B D E C (6) C E D A B (5) C B A E D (5) E D C A B (4) C E D B A (4) C A D B E (4) B E D A C (4) E D B A C (3) C E B D A (3) B A C D E (3) A D E B C (3) A D B E C (3) A B C D E (3) E C D B A (2) E B D A C (2) C D A E B (2) C B E A D (2) C A B D E (2) B C E A D (2) B C A E D (2) B C A D E (2) B A D C E (2) B A C E D (2) E D C B A (1) E D B C A (1) E D A C B (1) E C B D A (1) D C A E B (1) C B E D A (1) B E C A D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -2 18 14 B 16 0 12 18 22 C 2 -12 0 4 6 D -18 -18 -4 0 -4 E -14 -22 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 18 14 B 16 0 12 18 22 C 2 -12 0 4 6 D -18 -18 -4 0 -4 E -14 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=33 A=17 E=15 D=1 so D is eliminated. Round 2 votes counts: C=35 B=33 A=17 E=15 so E is eliminated. Round 3 votes counts: C=43 B=39 A=18 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:234 A:207 C:200 E:181 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 18 14 B 16 0 12 18 22 C 2 -12 0 4 6 D -18 -18 -4 0 -4 E -14 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 18 14 B 16 0 12 18 22 C 2 -12 0 4 6 D -18 -18 -4 0 -4 E -14 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 18 14 B 16 0 12 18 22 C 2 -12 0 4 6 D -18 -18 -4 0 -4 E -14 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6744: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) B D E C A (11) E C A B D (10) C A E D B (6) B D C A E (6) A C E D B (6) B D E A C (4) A C D E B (4) E B C A D (3) E A C D B (3) D B E A C (3) D B C A E (3) D A B C E (3) B E D C A (3) E D B A C (2) E B D C A (2) E B D A C (2) D B A E C (2) C A E B D (2) A C D B E (2) E C B A D (1) E C A D B (1) E B C D A (1) E A C B D (1) D A C B E (1) C E A D B (1) C B A D E (1) C A D B E (1) C A B E D (1) C A B D E (1) B D A C E (1) A E C D B (1) Total count = 100 A B C D E A 0 -12 -8 -10 4 B 12 0 14 0 8 C 8 -14 0 -8 0 D 10 0 8 0 8 E -4 -8 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.213770 C: 0.000000 D: 0.786230 E: 0.000000 Sum of squares = 0.663855713626 Cumulative probabilities = A: 0.000000 B: 0.213770 C: 0.213770 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -10 4 B 12 0 14 0 8 C 8 -14 0 -8 0 D 10 0 8 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=25 D=23 C=13 A=13 so C is eliminated. Round 2 votes counts: E=27 B=26 A=24 D=23 so D is eliminated. Round 3 votes counts: B=45 A=28 E=27 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:213 C:193 E:190 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -10 4 B 12 0 14 0 8 C 8 -14 0 -8 0 D 10 0 8 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -10 4 B 12 0 14 0 8 C 8 -14 0 -8 0 D 10 0 8 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -10 4 B 12 0 14 0 8 C 8 -14 0 -8 0 D 10 0 8 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6745: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (6) C A D E B (5) B E D C A (5) A D E B C (5) A C D E B (5) C D E B A (4) C D A E B (4) A D E C B (4) A B C D E (4) E B D A C (3) D E A C B (3) D A E C B (3) C B E D A (3) B E D A C (3) B E C D A (3) B E A D C (3) B A E D C (3) E D A B C (2) E C D B A (2) D E C A B (2) D C E A B (2) C E D B A (2) C D E A B (2) C B A D E (2) C A D B E (2) B C A E D (2) B A C E D (2) A D C E B (2) E C B D A (1) D E A B C (1) C A B D E (1) B E A C D (1) B C E A D (1) B A E C D (1) A E D B C (1) A D C B E (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 2 -2 2 B -6 0 -4 -6 -6 C -2 4 0 10 4 D 2 6 -10 0 10 E -2 6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040818 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -2 2 B -6 0 -4 -6 -6 C -2 4 0 10 4 D 2 6 -10 0 10 E -2 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408164 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=26 C=25 D=11 E=8 so E is eliminated. Round 2 votes counts: B=33 C=28 A=26 D=13 so D is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:208 A:204 D:204 E:195 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 -2 2 B -6 0 -4 -6 -6 C -2 4 0 10 4 D 2 6 -10 0 10 E -2 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408164 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -2 2 B -6 0 -4 -6 -6 C -2 4 0 10 4 D 2 6 -10 0 10 E -2 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408164 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -2 2 B -6 0 -4 -6 -6 C -2 4 0 10 4 D 2 6 -10 0 10 E -2 6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408164 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6746: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) C D B E A (10) E A C B D (7) C E D B A (6) A B E D C (6) D B C A E (5) D B A C E (5) C D E B A (5) B D A C E (5) E C A D B (4) E C A B D (4) B A D C E (4) A E B D C (4) A E B C D (4) E C D A B (3) D C B E A (3) E A C D B (2) C E D A B (2) C B D E A (2) B D C A E (2) E A B C D (1) D A B E C (1) B A D E C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 8 2 2 B -4 0 4 6 16 C -8 -4 0 2 -2 D -2 -6 -2 0 12 E -2 -16 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 2 2 B -4 0 4 6 16 C -8 -4 0 2 -2 D -2 -6 -2 0 12 E -2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997149 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=25 E=21 D=14 B=12 so B is eliminated. Round 2 votes counts: A=33 C=25 E=21 D=21 so E is eliminated. Round 3 votes counts: A=43 C=36 D=21 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:211 A:208 D:201 C:194 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 2 2 B -4 0 4 6 16 C -8 -4 0 2 -2 D -2 -6 -2 0 12 E -2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997149 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 2 2 B -4 0 4 6 16 C -8 -4 0 2 -2 D -2 -6 -2 0 12 E -2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997149 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 2 2 B -4 0 4 6 16 C -8 -4 0 2 -2 D -2 -6 -2 0 12 E -2 -16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997149 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6747: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (17) D C A E B (8) A E B D C (8) D C B A E (7) C D E A B (7) A E D C B (7) B E A C D (6) E A D C B (4) E A C D B (4) C D B A E (3) C B D E A (3) B A E D C (3) E A C B D (2) E A B C D (2) C D E B A (2) B C D E A (2) B C D A E (2) B A E C D (2) A E D B C (2) E C D A B (1) D C E A B (1) D B C A E (1) C E D A B (1) B E C A D (1) B D C A E (1) B D A E C (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -2 -14 -14 -6 B 2 0 -36 -32 -2 C 14 36 0 12 10 D 14 32 -12 0 12 E 6 2 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 -14 -6 B 2 0 -36 -32 -2 C 14 36 0 12 10 D 14 32 -12 0 12 E 6 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999139 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=19 B=18 D=17 E=13 so E is eliminated. Round 2 votes counts: C=34 A=31 B=18 D=17 so D is eliminated. Round 3 votes counts: C=50 A=31 B=19 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:236 D:223 E:193 A:182 B:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 -14 -6 B 2 0 -36 -32 -2 C 14 36 0 12 10 D 14 32 -12 0 12 E 6 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999139 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -14 -6 B 2 0 -36 -32 -2 C 14 36 0 12 10 D 14 32 -12 0 12 E 6 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999139 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -14 -6 B 2 0 -36 -32 -2 C 14 36 0 12 10 D 14 32 -12 0 12 E 6 2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999139 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6748: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) B D C A E (12) C E A B D (11) E A C D B (8) C B D E A (7) B D A E C (5) E C A D B (4) A E D B C (4) A E C D B (4) A E C B D (4) C E A D B (3) B D A C E (3) D B C E A (2) D B C A E (2) D B A C E (2) C B E D A (2) E C A B D (1) E A D B C (1) D B E A C (1) C D B E A (1) C B E A D (1) C A E B D (1) B D C E A (1) B C D E A (1) B C D A E (1) B A E D C (1) A E D C B (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 0 -4 12 B 8 0 4 6 12 C 0 -4 0 -2 0 D 4 -6 2 0 4 E -12 -12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -4 12 B 8 0 4 6 12 C 0 -4 0 -2 0 D 4 -6 2 0 4 E -12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 D=19 A=17 E=14 so E is eliminated. Round 2 votes counts: C=31 A=26 B=24 D=19 so D is eliminated. Round 3 votes counts: B=43 C=31 A=26 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:202 A:200 C:197 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -4 12 B 8 0 4 6 12 C 0 -4 0 -2 0 D 4 -6 2 0 4 E -12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -4 12 B 8 0 4 6 12 C 0 -4 0 -2 0 D 4 -6 2 0 4 E -12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -4 12 B 8 0 4 6 12 C 0 -4 0 -2 0 D 4 -6 2 0 4 E -12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6749: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (18) E C A D B (10) B D A C E (9) E A D B C (8) B D A E C (8) C B D A E (5) C E B D A (4) A D B E C (4) D A B E C (3) C B E D A (3) B D C A E (3) B C D A E (3) A D E B C (3) E A D C B (2) D B A E C (2) D B A C E (2) C E B A D (2) B E C D A (2) B E A D C (2) E A C D B (1) D A B C E (1) C E A B D (1) C D A E B (1) C D A B E (1) C B D E A (1) B D E C A (1) Total count = 100 A B C D E A 0 6 -10 2 -10 B -6 0 2 -12 0 C 10 -2 0 4 8 D -2 12 -4 0 -6 E 10 0 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.50617283965 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 2 -10 B -6 0 2 -12 0 C 10 -2 0 4 8 D -2 12 -4 0 -6 E 10 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839506 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=28 E=21 D=8 A=7 so A is eliminated. Round 2 votes counts: C=36 B=28 E=21 D=15 so D is eliminated. Round 3 votes counts: B=40 C=36 E=24 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:210 E:204 D:200 A:194 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 2 -10 B -6 0 2 -12 0 C 10 -2 0 4 8 D -2 12 -4 0 -6 E 10 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839506 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 2 -10 B -6 0 2 -12 0 C 10 -2 0 4 8 D -2 12 -4 0 -6 E 10 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839506 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 2 -10 B -6 0 2 -12 0 C 10 -2 0 4 8 D -2 12 -4 0 -6 E 10 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839506 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6750: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) E A D B C (8) D A E C B (7) C B D A E (7) B C E A D (7) D E A C B (5) B C D E A (5) B C A E D (5) E A B D C (4) D A C E B (3) C D A B E (3) B E A C D (3) E D A C B (2) E B D C A (2) E B A D C (2) E A D C B (2) B E C D A (2) B C E D A (2) B C A D E (2) A E D C B (2) A D C E B (2) E D A B C (1) E B D A C (1) E B A C D (1) D C E A B (1) D C A B E (1) C D B A E (1) C B D E A (1) C A D B E (1) C A B D E (1) B E C A D (1) B C D A E (1) B A E C D (1) A D E C B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 10 0 B 8 0 -2 18 12 C 6 2 0 12 10 D -10 -18 -12 0 6 E 0 -12 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 10 0 B 8 0 -2 18 12 C 6 2 0 12 10 D -10 -18 -12 0 6 E 0 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=24 E=23 D=17 A=7 so A is eliminated. Round 2 votes counts: B=30 E=25 C=25 D=20 so D is eliminated. Round 3 votes counts: E=38 C=32 B=30 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:218 C:215 A:198 E:186 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 10 0 B 8 0 -2 18 12 C 6 2 0 12 10 D -10 -18 -12 0 6 E 0 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 10 0 B 8 0 -2 18 12 C 6 2 0 12 10 D -10 -18 -12 0 6 E 0 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 10 0 B 8 0 -2 18 12 C 6 2 0 12 10 D -10 -18 -12 0 6 E 0 -12 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6751: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) B A D C E (7) E D C A B (6) E C D A B (6) E D C B A (5) E C D B A (5) E B A C D (4) A B D C E (4) D E C A B (3) D E B A C (3) D C A E B (3) D B A C E (3) D A B C E (3) C E D A B (3) A B C D E (3) E C B A D (2) E B C A D (2) D C A B E (2) D B E A C (2) D B A E C (2) C E A B D (2) C A B D E (2) B A D E C (2) B A C D E (2) E D B A C (1) E C B D A (1) E B D C A (1) D E B C A (1) D C E A B (1) D A E B C (1) D A C B E (1) C E B A D (1) C D E A B (1) C A E B D (1) C A D B E (1) C A B E D (1) B A E D C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -8 0 B 8 0 6 -8 -6 C 0 -6 0 -6 4 D 8 8 6 0 -4 E 0 6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 1.000000 A B C D E A 0 -8 0 -8 0 B 8 0 6 -8 -6 C 0 -6 0 -6 4 D 8 8 6 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=25 B=21 C=12 A=9 so A is eliminated. Round 2 votes counts: E=33 B=29 D=26 C=12 so C is eliminated. Round 3 votes counts: E=40 B=32 D=28 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:209 E:203 B:200 C:196 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -8 0 B 8 0 6 -8 -6 C 0 -6 0 -6 4 D 8 8 6 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -8 0 B 8 0 6 -8 -6 C 0 -6 0 -6 4 D 8 8 6 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -8 0 B 8 0 6 -8 -6 C 0 -6 0 -6 4 D 8 8 6 0 -4 E 0 6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.285714 E: 0.428571 Sum of squares = 0.346938775502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6752: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) A E B C D (11) B C A D E (9) E A D C B (7) A B C D E (7) D B C A E (6) A B C E D (5) E D A C B (4) D E C B A (4) B C D A E (4) E A C B D (3) B A C D E (3) A E B D C (3) E D C B A (2) C B D A E (2) A E D B C (2) E D C A B (1) E C D B A (1) E C B A D (1) E A D B C (1) E A B C D (1) D E A B C (1) D C E B A (1) C E D B A (1) C E B D A (1) C D B E A (1) C B D E A (1) C B A D E (1) B D C A E (1) A D B E C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 2 14 14 B 2 0 14 8 10 C -2 -14 0 6 12 D -14 -8 -6 0 10 E -14 -10 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 14 14 B 2 0 14 8 10 C -2 -14 0 6 12 D -14 -8 -6 0 10 E -14 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=24 E=21 B=17 C=7 so C is eliminated. Round 2 votes counts: A=31 D=25 E=23 B=21 so B is eliminated. Round 3 votes counts: A=44 D=33 E=23 so E is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:217 A:214 C:201 D:191 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 14 14 B 2 0 14 8 10 C -2 -14 0 6 12 D -14 -8 -6 0 10 E -14 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 14 14 B 2 0 14 8 10 C -2 -14 0 6 12 D -14 -8 -6 0 10 E -14 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 14 14 B 2 0 14 8 10 C -2 -14 0 6 12 D -14 -8 -6 0 10 E -14 -10 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6753: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (11) D E C A B (8) B C A E D (6) B A D E C (5) E D A C B (4) D E A C B (4) B D A E C (4) E C D A B (3) D E A B C (3) D B E C A (3) D B E A C (3) D B C E A (3) C E D A B (3) C E A D B (3) B D C E A (3) B A C D E (3) D C E A B (2) C E A B D (2) C B A E D (2) B D A C E (2) B A E D C (2) A E C D B (2) A E B C D (2) A B C E D (2) E A C D B (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E B A (1) D B A E C (1) C A E D B (1) B D C A E (1) B C D A E (1) B C A D E (1) A E D C B (1) A D E B C (1) A C E D B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 10 -4 2 B 10 0 20 -2 8 C -10 -20 0 -10 -2 D 4 2 10 0 4 E -2 -8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -4 2 B 10 0 20 -2 8 C -10 -20 0 -10 -2 D 4 2 10 0 4 E -2 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=31 C=11 A=11 E=8 so E is eliminated. Round 2 votes counts: B=39 D=35 C=14 A=12 so A is eliminated. Round 3 votes counts: B=44 D=37 C=19 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:210 A:199 E:194 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 10 -4 2 B 10 0 20 -2 8 C -10 -20 0 -10 -2 D 4 2 10 0 4 E -2 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -4 2 B 10 0 20 -2 8 C -10 -20 0 -10 -2 D 4 2 10 0 4 E -2 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -4 2 B 10 0 20 -2 8 C -10 -20 0 -10 -2 D 4 2 10 0 4 E -2 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6754: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) B E C D A (6) C B D A E (5) C A D B E (5) A E C D B (5) E B D C A (4) E B D A C (4) C D A B E (4) B D E C A (4) B D C E A (4) B C D E A (4) A D C B E (4) A C D B E (4) E B C D A (3) D C A B E (3) D A C B E (3) E C B A D (2) E B A D C (2) E A C B D (2) B E D C A (2) B C E D A (2) A D E C B (2) A D C E B (2) E B C A D (1) E B A C D (1) E A D B C (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C E A (1) C D B A E (1) C B D E A (1) C A B E D (1) B C D A E (1) A E D B C (1) A E C B D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -10 -8 8 B 0 0 -14 2 12 C 10 14 0 4 -4 D 8 -2 -4 0 2 E -8 -12 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.400000 D: 0.000000 E: 0.466667 Sum of squares = 0.395555555482 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.533333 D: 0.533333 E: 1.000000 A B C D E A 0 0 -10 -8 8 B 0 0 -14 2 12 C 10 14 0 4 -4 D 8 -2 -4 0 2 E -8 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.400000 D: 0.000000 E: 0.466667 Sum of squares = 0.395555555553 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.533333 D: 0.533333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=23 E=21 C=17 D=9 so D is eliminated. Round 2 votes counts: A=33 B=24 E=22 C=21 so C is eliminated. Round 3 votes counts: A=46 B=32 E=22 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 D:202 B:200 A:195 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -8 8 B 0 0 -14 2 12 C 10 14 0 4 -4 D 8 -2 -4 0 2 E -8 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.400000 D: 0.000000 E: 0.466667 Sum of squares = 0.395555555553 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.533333 D: 0.533333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -8 8 B 0 0 -14 2 12 C 10 14 0 4 -4 D 8 -2 -4 0 2 E -8 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.400000 D: 0.000000 E: 0.466667 Sum of squares = 0.395555555553 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.533333 D: 0.533333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -8 8 B 0 0 -14 2 12 C 10 14 0 4 -4 D 8 -2 -4 0 2 E -8 -12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.400000 D: 0.000000 E: 0.466667 Sum of squares = 0.395555555553 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.533333 D: 0.533333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6755: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (16) D C A E B (12) A E D C B (11) B C D E A (8) C D B A E (7) E B A D C (5) B E A D C (5) B E A C D (5) C D A E B (3) C B D A E (3) E A D C B (2) E A D B C (2) D C B A E (2) D C A B E (2) B E C A D (2) B C D A E (2) A D C E B (2) E A C B D (1) E A B C D (1) D A E B C (1) D A C E B (1) C D A B E (1) B E C D A (1) B D C E A (1) B D C A E (1) B C E D A (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 14 8 8 0 B -14 0 2 4 -18 C -8 -2 0 -28 -8 D -8 -4 28 0 -6 E 0 18 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.760352 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.239648 Sum of squares = 0.635565841535 Cumulative probabilities = A: 0.760352 B: 0.760352 C: 0.760352 D: 0.760352 E: 1.000000 A B C D E A 0 14 8 8 0 B -14 0 2 4 -18 C -8 -2 0 -28 -8 D -8 -4 28 0 -6 E 0 18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 D=18 A=15 C=14 so C is eliminated. Round 2 votes counts: D=29 B=29 E=27 A=15 so A is eliminated. Round 3 votes counts: E=39 D=32 B=29 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:215 D:205 B:187 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 8 0 B -14 0 2 4 -18 C -8 -2 0 -28 -8 D -8 -4 28 0 -6 E 0 18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 8 0 B -14 0 2 4 -18 C -8 -2 0 -28 -8 D -8 -4 28 0 -6 E 0 18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 8 0 B -14 0 2 4 -18 C -8 -2 0 -28 -8 D -8 -4 28 0 -6 E 0 18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6756: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) D B E C A (6) D E C B A (5) A E C D B (5) E A D C B (4) B C A D E (4) B A C D E (4) A E B C D (4) A B C E D (4) E D A C B (3) B D C A E (3) B D A E C (3) B C D E A (3) B C D A E (3) B A C E D (3) E C A D B (2) E A C D B (2) D E C A B (2) D E B A C (2) D E A B C (2) C A E B D (2) B D E C A (2) B D A C E (2) A C E D B (2) A C E B D (2) A C B E D (2) E D C A B (1) D E B C A (1) D C E B A (1) C E D B A (1) C E B D A (1) C E A D B (1) C D E B A (1) C B A E D (1) B C A E D (1) B A D C E (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -20 -6 -8 -4 B 20 0 22 14 8 C 6 -22 0 -4 8 D 8 -14 4 0 16 E 4 -8 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -8 -4 B 20 0 22 14 8 C 6 -22 0 -4 8 D 8 -14 4 0 16 E 4 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 A=21 D=19 E=12 C=7 so C is eliminated. Round 2 votes counts: B=42 A=23 D=20 E=15 so E is eliminated. Round 3 votes counts: B=43 A=32 D=25 so D is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:232 D:207 C:194 E:186 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -6 -8 -4 B 20 0 22 14 8 C 6 -22 0 -4 8 D 8 -14 4 0 16 E 4 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -8 -4 B 20 0 22 14 8 C 6 -22 0 -4 8 D 8 -14 4 0 16 E 4 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -8 -4 B 20 0 22 14 8 C 6 -22 0 -4 8 D 8 -14 4 0 16 E 4 -8 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6757: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) C D E B A (8) B A C E D (8) A E B D C (7) E D A C B (6) D E C A B (6) D C E A B (6) B A E D C (6) E D C A B (5) C D B E A (5) A B E D C (5) E A D B C (4) B A C D E (4) C B D E A (3) B C A D E (3) E A D C B (2) C D E A B (2) A E D B C (2) E A B D C (1) D E A C B (1) C D B A E (1) C D A E B (1) C B D A E (1) B E A C D (1) B C D E A (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 -2 14 6 -2 B 2 0 4 0 -4 C -14 -4 0 -4 -12 D -6 0 4 0 -14 E 2 4 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 14 6 -2 B 2 0 4 0 -4 C -14 -4 0 -4 -12 D -6 0 4 0 -14 E 2 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=21 E=18 A=15 D=13 so D is eliminated. Round 2 votes counts: B=33 C=27 E=25 A=15 so A is eliminated. Round 3 votes counts: B=38 E=35 C=27 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:208 B:201 D:192 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 14 6 -2 B 2 0 4 0 -4 C -14 -4 0 -4 -12 D -6 0 4 0 -14 E 2 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 6 -2 B 2 0 4 0 -4 C -14 -4 0 -4 -12 D -6 0 4 0 -14 E 2 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 6 -2 B 2 0 4 0 -4 C -14 -4 0 -4 -12 D -6 0 4 0 -14 E 2 4 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6758: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) A E B C D (9) A B E D C (8) C D E B A (7) C D A E B (6) D A C B E (5) A C D E B (4) E B C A D (3) D C A E B (3) C E B D A (3) B E C D A (3) B E A C D (3) E B C D A (2) E B A C D (2) D C B A E (2) D C A B E (2) D B E A C (2) B E D C A (2) B E D A C (2) B E A D C (2) A E C B D (2) A E B D C (2) A C E B D (2) E C B A D (1) D B C E A (1) D A B C E (1) C E D B A (1) C E B A D (1) C E A B D (1) C D E A B (1) C D B E A (1) C A E B D (1) B D E C A (1) A D C B E (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -4 -10 2 B -2 0 -8 4 -2 C 4 8 0 8 6 D 10 -4 -8 0 -2 E -2 2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -10 2 B -2 0 -8 4 -2 C 4 8 0 8 6 D 10 -4 -8 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=26 C=22 B=13 E=8 so E is eliminated. Round 2 votes counts: A=31 D=26 C=23 B=20 so B is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:198 E:198 B:196 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -10 2 B -2 0 -8 4 -2 C 4 8 0 8 6 D 10 -4 -8 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -10 2 B -2 0 -8 4 -2 C 4 8 0 8 6 D 10 -4 -8 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -10 2 B -2 0 -8 4 -2 C 4 8 0 8 6 D 10 -4 -8 0 -2 E -2 2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6759: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) C B A D E (8) E A B D C (7) E D A B C (6) A B E C D (5) E A B C D (4) D E C B A (4) D C E B A (4) C D A B E (4) B A E D C (4) A B C E D (4) C A B E D (3) C A B D E (3) B A C E D (3) E C D A B (2) D E B A C (2) D C B A E (2) D B A E C (2) D B A C E (2) B C A D E (2) B A C D E (2) A E B C D (2) E D C A B (1) E D B A C (1) E B A D C (1) E A C B D (1) D E C A B (1) D E A B C (1) D C E A B (1) D B E A C (1) C E A B D (1) C D E A B (1) B A D E C (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 2 6 22 B 4 0 2 6 20 C -2 -2 0 16 8 D -6 -6 -16 0 8 E -22 -20 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999291 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 6 22 B 4 0 2 6 20 C -2 -2 0 16 8 D -6 -6 -16 0 8 E -22 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=23 D=20 B=13 A=12 so A is eliminated. Round 2 votes counts: C=33 E=25 B=22 D=20 so D is eliminated. Round 3 votes counts: C=40 E=33 B=27 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:216 A:213 C:210 D:190 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 6 22 B 4 0 2 6 20 C -2 -2 0 16 8 D -6 -6 -16 0 8 E -22 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 22 B 4 0 2 6 20 C -2 -2 0 16 8 D -6 -6 -16 0 8 E -22 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 22 B 4 0 2 6 20 C -2 -2 0 16 8 D -6 -6 -16 0 8 E -22 -20 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6760: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) A B E C D (7) D C E B A (6) A B E D C (6) E B A C D (5) D C A E B (5) B A E D C (5) B E A D C (4) B A E C D (4) A B D E C (4) E D C B A (3) E C D B A (3) D C A B E (3) D A C B E (3) C D A E B (3) E B D A C (2) E B C D A (2) E B A D C (2) D C E A B (2) C E D B A (2) C E A D B (2) C D E A B (2) A D C B E (2) A C D B E (2) A B C D E (2) E C B D A (1) E B C A D (1) D E C B A (1) D A B C E (1) C A D B E (1) B E D A C (1) B E A C D (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 8 2 2 B 4 0 -2 0 -2 C -8 2 0 -2 -4 D -2 0 2 0 -4 E -2 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -4 8 2 2 B 4 0 -2 0 -2 C -8 2 0 -2 -4 D -2 0 2 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000038 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=21 E=19 C=19 B=15 so B is eliminated. Round 2 votes counts: A=35 E=25 D=21 C=19 so C is eliminated. Round 3 votes counts: A=36 D=35 E=29 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:204 E:204 B:200 D:198 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 2 2 B 4 0 -2 0 -2 C -8 2 0 -2 -4 D -2 0 2 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000038 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 2 2 B 4 0 -2 0 -2 C -8 2 0 -2 -4 D -2 0 2 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000038 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 2 2 B 4 0 -2 0 -2 C -8 2 0 -2 -4 D -2 0 2 0 -4 E -2 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000038 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6761: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) D E C A B (6) E D C B A (4) E D B C A (4) E C D A B (4) E C A D B (4) D B E A C (4) D A B C E (4) B A D C E (4) E D C A B (3) E B D C A (3) E B C D A (3) D E B C A (3) D B A C E (3) D A C E B (3) B A C D E (3) A C D B E (3) A B C D E (3) E C B A D (2) D B A E C (2) D A C B E (2) C E A D B (2) B D E A C (2) B D A E C (2) B D A C E (2) A C B E D (2) E B C A D (1) D E C B A (1) D C A E B (1) D A B E C (1) C E A B D (1) C D A E B (1) C B A E D (1) C A E D B (1) B E D C A (1) B E C A D (1) B C A E D (1) B A D E C (1) B A C E D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -12 -18 2 B -2 0 -2 -16 -12 C 12 2 0 -14 -4 D 18 16 14 0 4 E -2 12 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -18 2 B -2 0 -2 -16 -12 C 12 2 0 -14 -4 D 18 16 14 0 4 E -2 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=28 B=18 C=14 A=10 so A is eliminated. Round 2 votes counts: D=31 E=28 B=21 C=20 so C is eliminated. Round 3 votes counts: E=41 D=35 B=24 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:226 E:205 C:198 A:187 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -18 2 B -2 0 -2 -16 -12 C 12 2 0 -14 -4 D 18 16 14 0 4 E -2 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -18 2 B -2 0 -2 -16 -12 C 12 2 0 -14 -4 D 18 16 14 0 4 E -2 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -18 2 B -2 0 -2 -16 -12 C 12 2 0 -14 -4 D 18 16 14 0 4 E -2 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6762: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (14) D C E B A (7) A B E C D (7) B A D C E (5) B A C E D (5) E C A B D (4) D B C A E (4) D A B E C (4) E A C B D (3) D E A B C (3) B A C D E (3) E C D A B (2) E C A D B (2) E A B C D (2) D E C B A (2) D C B E A (2) D C B A E (2) D B A C E (2) C E D B A (2) C B A E D (2) B D A C E (2) B C A E D (2) A E B C D (2) A D B E C (2) A B D E C (2) A B C E D (2) E D C A B (1) E A D B C (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E A D (1) C B D E A (1) C B D A E (1) C B A D E (1) C A B E D (1) A E B D C (1) Total count = 100 A B C D E A 0 8 -6 -4 2 B -8 0 0 -6 4 C 6 0 0 -10 -6 D 4 6 10 0 20 E -2 -4 6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 -4 2 B -8 0 0 -6 4 C 6 0 0 -10 -6 D 4 6 10 0 20 E -2 -4 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 B=17 A=16 E=15 C=11 so C is eliminated. Round 2 votes counts: D=43 B=23 E=17 A=17 so E is eliminated. Round 3 votes counts: D=48 A=29 B=23 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:200 B:195 C:195 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -6 -4 2 B -8 0 0 -6 4 C 6 0 0 -10 -6 D 4 6 10 0 20 E -2 -4 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -4 2 B -8 0 0 -6 4 C 6 0 0 -10 -6 D 4 6 10 0 20 E -2 -4 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -4 2 B -8 0 0 -6 4 C 6 0 0 -10 -6 D 4 6 10 0 20 E -2 -4 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6763: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (11) D B A E C (8) B D A E C (7) D B A C E (4) C A E D B (4) B D E A C (4) A E C B D (4) A D C E B (4) A D B E C (4) E A C B D (3) D B C A E (3) D A B E C (3) C E D B A (3) A C E D B (3) E C A B D (2) D C B E A (2) D C B A E (2) C E B D A (2) C E B A D (2) C E A D B (2) C D E B A (2) C D E A B (2) C D A E B (2) A E B D C (2) A B E D C (2) E C B A D (1) E B A D C (1) D C A B E (1) C A E B D (1) C A D E B (1) B E C D A (1) B D E C A (1) A E B C D (1) A D E B C (1) A D C B E (1) A C E B D (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 14 10 6 22 B -14 0 -14 -6 -12 C -10 14 0 -2 8 D -6 6 2 0 8 E -22 12 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 6 22 B -14 0 -14 -6 -12 C -10 14 0 -2 8 D -6 6 2 0 8 E -22 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=25 D=23 B=13 E=7 so E is eliminated. Round 2 votes counts: C=35 A=28 D=23 B=14 so B is eliminated. Round 3 votes counts: C=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:226 C:205 D:205 E:187 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 6 22 B -14 0 -14 -6 -12 C -10 14 0 -2 8 D -6 6 2 0 8 E -22 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 6 22 B -14 0 -14 -6 -12 C -10 14 0 -2 8 D -6 6 2 0 8 E -22 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 6 22 B -14 0 -14 -6 -12 C -10 14 0 -2 8 D -6 6 2 0 8 E -22 12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6764: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) D B C A E (10) A C E D B (10) E A C B D (9) C A D B E (8) B D E C A (8) A C D E B (7) E B D A C (5) B D C E A (5) D C B A E (4) B D C A E (4) E A B C D (3) B E D C A (3) E B A D C (2) E A B D C (2) C D A B E (2) C A D E B (2) B E D A C (2) A E C D B (2) C D B A E (1) Total count = 100 A B C D E A 0 12 6 12 0 B -12 0 -12 -14 -6 C -6 12 0 10 6 D -12 14 -10 0 2 E 0 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.684282 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.315718 Sum of squares = 0.567919848986 Cumulative probabilities = A: 0.684282 B: 0.684282 C: 0.684282 D: 0.684282 E: 1.000000 A B C D E A 0 12 6 12 0 B -12 0 -12 -14 -6 C -6 12 0 10 6 D -12 14 -10 0 2 E 0 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500199 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499801 Sum of squares = 0.500000079236 Cumulative probabilities = A: 0.500199 B: 0.500199 C: 0.500199 D: 0.500199 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=22 A=19 D=14 C=13 so C is eliminated. Round 2 votes counts: E=32 A=29 B=22 D=17 so D is eliminated. Round 3 votes counts: B=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:215 C:211 E:199 D:197 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 12 0 B -12 0 -12 -14 -6 C -6 12 0 10 6 D -12 14 -10 0 2 E 0 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500199 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499801 Sum of squares = 0.500000079236 Cumulative probabilities = A: 0.500199 B: 0.500199 C: 0.500199 D: 0.500199 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 12 0 B -12 0 -12 -14 -6 C -6 12 0 10 6 D -12 14 -10 0 2 E 0 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500199 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499801 Sum of squares = 0.500000079236 Cumulative probabilities = A: 0.500199 B: 0.500199 C: 0.500199 D: 0.500199 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 12 0 B -12 0 -12 -14 -6 C -6 12 0 10 6 D -12 14 -10 0 2 E 0 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500199 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499801 Sum of squares = 0.500000079236 Cumulative probabilities = A: 0.500199 B: 0.500199 C: 0.500199 D: 0.500199 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6765: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (19) E B D A C (13) E D B A C (8) C A B D E (7) E B C A D (5) B E C A D (5) E B D C A (4) D A C B E (4) B E D A C (3) A C D B E (3) E C B A D (2) D A C E B (2) C B A E D (2) C A E D B (2) C A D E B (2) C A B E D (2) B D A C E (2) B C A D E (2) A D C B E (2) E D A C B (1) E C A D B (1) D E B A C (1) D E A C B (1) D E A B C (1) D A E C B (1) C A E B D (1) B E A C D (1) B D E A C (1) B C E A D (1) B A C D E (1) Total count = 100 A B C D E A 0 -2 -10 16 4 B 2 0 -4 4 10 C 10 4 0 12 4 D -16 -4 -12 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 16 4 B 2 0 -4 4 10 C 10 4 0 12 4 D -16 -4 -12 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=34 B=16 D=10 A=5 so A is eliminated. Round 2 votes counts: C=38 E=34 B=16 D=12 so D is eliminated. Round 3 votes counts: C=46 E=38 B=16 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:206 A:204 E:192 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 16 4 B 2 0 -4 4 10 C 10 4 0 12 4 D -16 -4 -12 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 16 4 B 2 0 -4 4 10 C 10 4 0 12 4 D -16 -4 -12 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 16 4 B 2 0 -4 4 10 C 10 4 0 12 4 D -16 -4 -12 0 -2 E -4 -10 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999547 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6766: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) C B D E A (8) A B D C E (8) A D E B C (6) A B C D E (5) C E B D A (4) B C A D E (4) E D C B A (3) E C D B A (3) E C B D A (3) E A D B C (3) B C D E A (3) A E D C B (3) E D C A B (2) E A D C B (2) D B A C E (2) D A E B C (2) C B E D A (2) C B D A E (2) C B A E D (2) C B A D E (2) B D C A E (2) B C D A E (2) A E C D B (2) A D B E C (2) E D B C A (1) E C A B D (1) E A C D B (1) D E C B A (1) D E B C A (1) D E A B C (1) D B E C A (1) D B C A E (1) C E D B A (1) C B E A D (1) B D C E A (1) B D A C E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -2 6 14 B 2 0 12 4 2 C 2 -12 0 -6 6 D -6 -4 6 0 12 E -14 -2 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998222 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 6 14 B 2 0 12 4 2 C 2 -12 0 -6 6 D -6 -4 6 0 12 E -14 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=22 E=19 B=13 D=9 so D is eliminated. Round 2 votes counts: A=39 E=22 C=22 B=17 so B is eliminated. Round 3 votes counts: A=42 C=35 E=23 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:210 A:208 D:204 C:195 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 6 14 B 2 0 12 4 2 C 2 -12 0 -6 6 D -6 -4 6 0 12 E -14 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 6 14 B 2 0 12 4 2 C 2 -12 0 -6 6 D -6 -4 6 0 12 E -14 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 6 14 B 2 0 12 4 2 C 2 -12 0 -6 6 D -6 -4 6 0 12 E -14 -2 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6767: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) E C B D A (8) D A C B E (8) B E A C D (7) A D B C E (7) A B D E C (7) C E D A B (6) B E A D C (6) E B C D A (4) D C A E B (4) D C E B A (3) B A D E C (3) E C D B A (2) E C B A D (2) E B C A D (2) D B E C A (2) A D B E C (2) D C A B E (1) D B A E C (1) D A C E B (1) D A B C E (1) C E B D A (1) C E A D B (1) C D E A B (1) B E D C A (1) B E C D A (1) B E C A D (1) B D E A C (1) B D A E C (1) B A E D C (1) A D C B E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -2 -16 -20 B 16 0 -2 -6 8 C 2 2 0 -4 -4 D 16 6 4 0 -8 E 20 -8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.363636 E: 0.272727 Sum of squares = 0.338842975199 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.727273 E: 1.000000 A B C D E A 0 -16 -2 -16 -20 B 16 0 -2 -6 8 C 2 2 0 -4 -4 D 16 6 4 0 -8 E 20 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.363636 E: 0.272727 Sum of squares = 0.338842975154 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 D=21 C=20 A=19 E=18 so E is eliminated. Round 2 votes counts: C=32 B=28 D=21 A=19 so A is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 D:209 B:208 C:198 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 -16 -20 B 16 0 -2 -6 8 C 2 2 0 -4 -4 D 16 6 4 0 -8 E 20 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.363636 E: 0.272727 Sum of squares = 0.338842975154 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.727273 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -16 -20 B 16 0 -2 -6 8 C 2 2 0 -4 -4 D 16 6 4 0 -8 E 20 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.363636 E: 0.272727 Sum of squares = 0.338842975154 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.727273 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -16 -20 B 16 0 -2 -6 8 C 2 2 0 -4 -4 D 16 6 4 0 -8 E 20 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.000000 D: 0.363636 E: 0.272727 Sum of squares = 0.338842975154 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.363636 D: 0.727273 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6768: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) B D A E C (8) A D B C E (7) C E B A D (6) D B A E C (5) B D A C E (5) C A E D B (4) E C D A B (3) E C B D A (3) E C A D B (3) D A B E C (3) A D E C B (3) A D C E B (3) A D C B E (3) E C B A D (2) D A E B C (2) C A D E B (2) C A D B E (2) B E C D A (2) B D E A C (2) B C E D A (2) B C D A E (2) B C A D E (2) A C D E B (2) E D A C B (1) E B C D A (1) E A C D B (1) D A E C B (1) C E A B D (1) C B E D A (1) C B E A D (1) B E D A C (1) B D C E A (1) B C D E A (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -2 12 12 B -10 0 -10 -18 -2 C 2 10 0 8 16 D -12 18 -8 0 12 E -12 2 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 12 12 B -10 0 -10 -18 -2 C 2 10 0 8 16 D -12 18 -8 0 12 E -12 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=26 A=20 E=14 D=11 so D is eliminated. Round 2 votes counts: B=31 C=29 A=26 E=14 so E is eliminated. Round 3 votes counts: C=40 B=32 A=28 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:216 D:205 E:181 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 12 12 B -10 0 -10 -18 -2 C 2 10 0 8 16 D -12 18 -8 0 12 E -12 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 12 12 B -10 0 -10 -18 -2 C 2 10 0 8 16 D -12 18 -8 0 12 E -12 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 12 12 B -10 0 -10 -18 -2 C 2 10 0 8 16 D -12 18 -8 0 12 E -12 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6769: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D C B A (9) D E C A B (9) C B A D E (8) D E C B A (7) A B C E D (7) E D A C B (6) B C A D E (4) E A B D C (3) D C E B A (3) C D B E A (3) A E D B C (3) E D A B C (2) E A D B C (2) D C E A B (2) D C B A E (2) D C A E B (2) C D B A E (2) C B D A E (2) C A B D E (2) A E B C D (2) A B E C D (2) E D C A B (1) E D B C A (1) D E A C B (1) D A C E B (1) B C E A D (1) B C A E D (1) B A C E D (1) A E B D C (1) Total count = 100 A B C D E A 0 12 -18 -6 0 B -12 0 -20 -12 -10 C 18 20 0 -10 2 D 6 12 10 0 16 E 0 10 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -18 -6 0 B -12 0 -20 -12 -10 C 18 20 0 -10 2 D 6 12 10 0 16 E 0 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 E=24 C=17 B=7 so B is eliminated. Round 2 votes counts: D=27 A=26 E=24 C=23 so C is eliminated. Round 3 votes counts: A=41 D=34 E=25 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:215 E:196 A:194 B:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -18 -6 0 B -12 0 -20 -12 -10 C 18 20 0 -10 2 D 6 12 10 0 16 E 0 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -18 -6 0 B -12 0 -20 -12 -10 C 18 20 0 -10 2 D 6 12 10 0 16 E 0 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -18 -6 0 B -12 0 -20 -12 -10 C 18 20 0 -10 2 D 6 12 10 0 16 E 0 10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6770: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (16) A C E D B (16) C A D E B (6) B D E A C (6) D E B C A (4) C A E D B (4) E D B A C (3) D E C B A (3) D B E C A (3) C A D B E (3) B A C D E (3) A C E B D (3) A C B E D (3) A C B D E (3) E D C A B (2) E D B C A (2) E B D A C (2) E A C D B (2) C D A E B (2) C A B D E (2) B D C E A (2) E D A C B (1) E C A D B (1) E B A D C (1) E A D C B (1) D B C E A (1) B E D A C (1) B C A D E (1) B A E C D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -4 4 -2 B -2 0 -6 -10 -8 C 4 6 0 4 0 D -4 10 -4 0 10 E 2 8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833677 D: 0.000000 E: 0.166323 Sum of squares = 0.72268020699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833677 D: 0.833677 E: 1.000000 A B C D E A 0 2 -4 4 -2 B -2 0 -6 -10 -8 C 4 6 0 4 0 D -4 10 -4 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.285714 Sum of squares = 0.591836735386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 C=17 E=15 D=11 so D is eliminated. Round 2 votes counts: B=34 A=27 E=22 C=17 so C is eliminated. Round 3 votes counts: A=44 B=34 E=22 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:207 D:206 A:200 E:200 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 4 -2 B -2 0 -6 -10 -8 C 4 6 0 4 0 D -4 10 -4 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.285714 Sum of squares = 0.591836735386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 -2 B -2 0 -6 -10 -8 C 4 6 0 4 0 D -4 10 -4 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.285714 Sum of squares = 0.591836735386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 -2 B -2 0 -6 -10 -8 C 4 6 0 4 0 D -4 10 -4 0 10 E 2 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.285714 Sum of squares = 0.591836735386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6771: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) C D A B E (9) E D C B A (7) A B E C D (6) E B A D C (5) D E C B A (5) D C E B A (5) D C E A B (5) A B C E D (5) A C B D E (4) E D B C A (3) E B D A C (3) B E A D C (3) B E A C D (3) B A C D E (3) A B C D E (3) E D C A B (2) C D A E B (2) B A E C D (2) A E B D C (2) E B D C A (1) D E C A B (1) D C B E A (1) D B C E A (1) C D B E A (1) C A D B E (1) C A B D E (1) B E D C A (1) B E D A C (1) B C D E A (1) B A E D C (1) A E D C B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 6 2 2 B -6 0 -12 -8 16 C -6 12 0 4 6 D -2 8 -4 0 6 E -2 -16 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999213 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 2 2 B -6 0 -12 -8 16 C -6 12 0 4 6 D -2 8 -4 0 6 E -2 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=21 D=18 B=15 C=14 so C is eliminated. Round 2 votes counts: A=34 D=30 E=21 B=15 so B is eliminated. Round 3 votes counts: A=40 D=31 E=29 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:208 D:204 B:195 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 2 2 B -6 0 -12 -8 16 C -6 12 0 4 6 D -2 8 -4 0 6 E -2 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 2 2 B -6 0 -12 -8 16 C -6 12 0 4 6 D -2 8 -4 0 6 E -2 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 2 2 B -6 0 -12 -8 16 C -6 12 0 4 6 D -2 8 -4 0 6 E -2 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6772: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) C E B D A (11) D B A C E (6) C E B A D (6) C E A B D (6) A E C B D (5) E C A B D (4) D A B E C (4) E C B A D (3) E A C B D (3) D B C E A (3) D B A E C (3) B D E C A (3) D C B E A (2) D B C A E (2) C B E D A (2) B D C E A (2) A E C D B (2) A E B C D (2) A D E C B (2) A D E B C (2) A D C E B (2) D A C B E (1) D A B C E (1) C E A D B (1) C D E B A (1) C D A E B (1) B E D A C (1) B C E D A (1) B C D E A (1) A E D B C (1) A E B D C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 10 0 B -6 0 -6 2 -8 C -2 6 0 0 2 D -10 -2 0 0 2 E 0 8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.740380 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.259620 Sum of squares = 0.615565192345 Cumulative probabilities = A: 0.740380 B: 0.740380 C: 0.740380 D: 0.740380 E: 1.000000 A B C D E A 0 6 2 10 0 B -6 0 -6 2 -8 C -2 6 0 0 2 D -10 -2 0 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500238 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499762 Sum of squares = 0.500000113393 Cumulative probabilities = A: 0.500238 B: 0.500238 C: 0.500238 D: 0.500238 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 D=22 E=10 B=8 so B is eliminated. Round 2 votes counts: A=32 C=30 D=27 E=11 so E is eliminated. Round 3 votes counts: C=37 A=35 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:203 E:202 D:195 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 10 0 B -6 0 -6 2 -8 C -2 6 0 0 2 D -10 -2 0 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500238 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499762 Sum of squares = 0.500000113393 Cumulative probabilities = A: 0.500238 B: 0.500238 C: 0.500238 D: 0.500238 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 10 0 B -6 0 -6 2 -8 C -2 6 0 0 2 D -10 -2 0 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500238 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499762 Sum of squares = 0.500000113393 Cumulative probabilities = A: 0.500238 B: 0.500238 C: 0.500238 D: 0.500238 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 10 0 B -6 0 -6 2 -8 C -2 6 0 0 2 D -10 -2 0 0 2 E 0 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500238 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499762 Sum of squares = 0.500000113393 Cumulative probabilities = A: 0.500238 B: 0.500238 C: 0.500238 D: 0.500238 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6773: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B C D A E (10) D C A B E (8) E B C D A (5) D C B A E (5) B E C D A (5) B C D E A (5) B C A D E (5) A E D C B (5) E B C A D (4) E D B C A (3) C B D A E (3) B D C E A (3) A C B D E (3) E A D B C (2) E A C B D (2) D A C B E (2) B C E D A (2) A D C B E (2) E D A C B (1) E B A D C (1) E B A C D (1) E A C D B (1) E A B C D (1) D E C B A (1) D E B C A (1) D B C A E (1) D A E C B (1) C B A D E (1) B E C A D (1) B C E A D (1) A E C D B (1) A E B C D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -28 -12 0 B 16 0 4 8 16 C 28 -4 0 6 6 D 12 -8 -6 0 6 E 0 -16 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -28 -12 0 B 16 0 4 8 16 C 28 -4 0 6 6 D 12 -8 -6 0 6 E 0 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=31 D=19 A=14 C=4 so C is eliminated. Round 2 votes counts: B=36 E=31 D=19 A=14 so A is eliminated. Round 3 votes counts: B=39 E=38 D=23 so D is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:218 D:202 E:186 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -28 -12 0 B 16 0 4 8 16 C 28 -4 0 6 6 D 12 -8 -6 0 6 E 0 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -28 -12 0 B 16 0 4 8 16 C 28 -4 0 6 6 D 12 -8 -6 0 6 E 0 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -28 -12 0 B 16 0 4 8 16 C 28 -4 0 6 6 D 12 -8 -6 0 6 E 0 -16 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6774: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) D C E A B (8) E A B C D (5) D C E B A (5) D B C A E (5) E C D A B (4) E C A D B (4) D B C E A (4) B E A D C (4) E A C B D (3) D C B A E (3) D B E A C (3) C D A E B (3) E A C D B (2) D E C A B (2) D C B E A (2) C D E A B (2) C A E D B (2) C A D E B (2) C A D B E (2) B D E A C (2) B D C A E (2) B D A E C (2) B D A C E (2) B A C E D (2) B A C D E (2) A E B C D (2) E D A B C (1) E A D C B (1) E A B D C (1) C A E B D (1) C A B E D (1) B E A C D (1) B A E D C (1) B A D E C (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 0 -8 B 2 0 4 -10 4 C 4 -4 0 2 0 D 0 10 -2 0 4 E 8 -4 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999984 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 0 -8 B 2 0 4 -10 4 C 4 -4 0 2 0 D 0 10 -2 0 4 E 8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 E=21 C=13 A=5 so A is eliminated. Round 2 votes counts: D=32 B=31 E=23 C=14 so C is eliminated. Round 3 votes counts: D=41 B=33 E=26 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:206 C:201 B:200 E:200 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 0 -8 B 2 0 4 -10 4 C 4 -4 0 2 0 D 0 10 -2 0 4 E 8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 0 -8 B 2 0 4 -10 4 C 4 -4 0 2 0 D 0 10 -2 0 4 E 8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 0 -8 B 2 0 4 -10 4 C 4 -4 0 2 0 D 0 10 -2 0 4 E 8 -4 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.250000 E: 0.000000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6775: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) A D C E B (9) D C E A B (7) B A E D C (6) C E D B A (5) C D E B A (5) B E C D A (5) A B E D C (5) A B D E C (5) A D B C E (4) A B D C E (4) E D C B A (3) E C D B A (3) E B C D A (2) D E C B A (2) B E C A D (2) B A E C D (2) A B C E D (2) A B C D E (2) E D B C A (1) E D B A C (1) E C B D A (1) E B D C A (1) D E C A B (1) D C E B A (1) D C A E B (1) C E B D A (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B E A C D (1) B C E A D (1) B C A E D (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -8 -2 -8 B -8 0 -6 -12 -10 C 8 6 0 -8 14 D 2 12 8 0 10 E 8 10 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -2 -8 B -8 0 -6 -12 -10 C 8 6 0 -8 14 D 2 12 8 0 10 E 8 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=22 B=20 E=12 D=12 so E is eliminated. Round 2 votes counts: A=34 C=26 B=23 D=17 so D is eliminated. Round 3 votes counts: C=41 A=34 B=25 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:216 C:210 E:197 A:195 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -2 -8 B -8 0 -6 -12 -10 C 8 6 0 -8 14 D 2 12 8 0 10 E 8 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -2 -8 B -8 0 -6 -12 -10 C 8 6 0 -8 14 D 2 12 8 0 10 E 8 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -2 -8 B -8 0 -6 -12 -10 C 8 6 0 -8 14 D 2 12 8 0 10 E 8 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6776: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (6) A B E D C (6) D C E A B (5) D E C B A (4) C B D E A (4) D E A B C (3) D A E C B (3) C B A D E (3) B E A D C (3) A D E C B (3) A D E B C (3) A C D B E (3) A C B D E (3) A B C E D (3) E D B A C (2) E B D C A (2) D E A C B (2) D A C E B (2) C D E B A (2) C D A E B (2) C D A B E (2) C B E D A (2) C B D A E (2) B E C D A (2) B C E A D (2) B A E C D (2) B A C E D (2) A E D B C (2) A E B D C (2) A D C E B (2) E D B C A (1) E D A B C (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E B A (1) D C A E B (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D C A (1) B E C A D (1) B C E D A (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 2 8 16 B -12 0 -4 2 4 C -2 4 0 -4 8 D -8 -2 4 0 12 E -16 -4 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 8 16 B -12 0 -4 2 4 C -2 4 0 -4 8 D -8 -2 4 0 12 E -16 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995277 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 C=21 B=20 E=8 so E is eliminated. Round 2 votes counts: A=31 D=26 B=22 C=21 so C is eliminated. Round 3 votes counts: B=34 A=34 D=32 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 C:203 D:203 B:195 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 8 16 B -12 0 -4 2 4 C -2 4 0 -4 8 D -8 -2 4 0 12 E -16 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995277 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 8 16 B -12 0 -4 2 4 C -2 4 0 -4 8 D -8 -2 4 0 12 E -16 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995277 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 8 16 B -12 0 -4 2 4 C -2 4 0 -4 8 D -8 -2 4 0 12 E -16 -4 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995277 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6777: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) E A B D C (7) D C B A E (7) D C E B A (6) A E B C D (6) E D C A B (5) E D A B C (5) E A D B C (5) D E C A B (5) D C B E A (5) C B A D E (5) E D A C B (4) D C E A B (4) C D B A E (4) C B D A E (4) B C A D E (4) A B E C D (3) B D C A E (2) B A E C D (2) E A D C B (1) D E B C A (1) D E B A C (1) D B C E A (1) B E A D C (1) B A C E D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 10 -6 -8 -20 B -10 0 -2 -8 -20 C 6 2 0 -20 -10 D 8 8 20 0 0 E 20 20 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.482229 E: 0.517771 Sum of squares = 0.500631599537 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.482229 E: 1.000000 A B C D E A 0 10 -6 -8 -20 B -10 0 -2 -8 -20 C 6 2 0 -20 -10 D 8 8 20 0 0 E 20 20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=30 C=13 B=11 A=10 so A is eliminated. Round 2 votes counts: E=42 D=30 C=14 B=14 so C is eliminated. Round 3 votes counts: E=43 D=34 B=23 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:225 D:218 C:189 A:188 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -6 -8 -20 B -10 0 -2 -8 -20 C 6 2 0 -20 -10 D 8 8 20 0 0 E 20 20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -8 -20 B -10 0 -2 -8 -20 C 6 2 0 -20 -10 D 8 8 20 0 0 E 20 20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -8 -20 B -10 0 -2 -8 -20 C 6 2 0 -20 -10 D 8 8 20 0 0 E 20 20 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6778: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (6) E B D A C (5) E A D C B (5) B C D A E (5) E A C D B (4) C B A D E (4) C A D E B (4) B E D C A (4) B C A D E (4) D B A E C (3) B D C A E (3) B C E D A (3) A D E C B (3) E D B A C (2) E D A C B (2) E D A B C (2) E C B A D (2) E C A D B (2) D E A B C (2) D B E A C (2) D B A C E (2) D A E C B (2) D A C E B (2) C A D B E (2) C A B D E (2) B E C D A (2) B E C A D (2) B D E A C (2) A C D E B (2) E B C A D (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A D B (1) C E A B D (1) C A E D B (1) B D E C A (1) B D A E C (1) B C E A D (1) B C D E A (1) B C A E D (1) A D C E B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 4 -8 -8 B 16 0 14 4 8 C -4 -14 0 -6 -12 D 8 -4 6 0 2 E 8 -8 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 -8 -8 B 16 0 14 4 8 C -4 -14 0 -6 -12 D 8 -4 6 0 2 E 8 -8 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=25 C=16 D=15 A=8 so A is eliminated. Round 2 votes counts: B=36 E=25 D=20 C=19 so C is eliminated. Round 3 votes counts: B=42 E=30 D=28 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:206 E:205 A:186 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 -8 -8 B 16 0 14 4 8 C -4 -14 0 -6 -12 D 8 -4 6 0 2 E 8 -8 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 -8 -8 B 16 0 14 4 8 C -4 -14 0 -6 -12 D 8 -4 6 0 2 E 8 -8 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 -8 -8 B 16 0 14 4 8 C -4 -14 0 -6 -12 D 8 -4 6 0 2 E 8 -8 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6779: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) A D C E B (6) D A C E B (5) C B E D A (5) E B C A D (4) E B A C D (4) E A B D C (4) D A C B E (4) C D B A E (4) B E A D C (4) B C D E A (4) A D E C B (4) D C A E B (3) C B D E A (3) B E C A D (3) A E D B C (3) D C A B E (2) C D E B A (2) C D A B E (2) B E C D A (2) B D C A E (2) A E B D C (2) A D E B C (2) E C A D B (1) E B A D C (1) E A D B C (1) E A C D B (1) D C B A E (1) C E B D A (1) C E B A D (1) C D B E A (1) C A E D B (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E D A (1) B A E D C (1) B A D E C (1) A E D C B (1) Total count = 100 A B C D E A 0 4 -6 -8 10 B -4 0 -16 -8 -14 C 6 16 0 2 16 D 8 8 -2 0 16 E -10 14 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -8 10 B -4 0 -16 -8 -14 C 6 16 0 2 16 D 8 8 -2 0 16 E -10 14 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=21 A=18 E=16 D=15 so D is eliminated. Round 2 votes counts: C=36 A=27 B=21 E=16 so E is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:215 A:200 E:186 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -8 10 B -4 0 -16 -8 -14 C 6 16 0 2 16 D 8 8 -2 0 16 E -10 14 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -8 10 B -4 0 -16 -8 -14 C 6 16 0 2 16 D 8 8 -2 0 16 E -10 14 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -8 10 B -4 0 -16 -8 -14 C 6 16 0 2 16 D 8 8 -2 0 16 E -10 14 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6780: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (15) E C A B D (8) E B A C D (7) D C A B E (5) E D C B A (4) E C A D B (4) E C D A B (3) D B E A C (3) C E D A B (3) C E A D B (3) B D A E C (3) B D A C E (3) B A D C E (3) D C B A E (2) C E A B D (2) C D A B E (2) C A D B E (2) B A E D C (2) B A E C D (2) B A C E D (2) E D B C A (1) E D B A C (1) E C B A D (1) E B D C A (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E A B (1) D B C A E (1) D B A E C (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B E A C D (1) B D E A C (1) B A D E C (1) B A C D E (1) A D B C E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 2 -4 6 B 14 0 6 -12 10 C -2 -6 0 -2 6 D 4 12 2 0 2 E -6 -10 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -4 6 B 14 0 6 -12 10 C -2 -6 0 -2 6 D 4 12 2 0 2 E -6 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=30 B=19 C=16 A=4 so A is eliminated. Round 2 votes counts: E=31 D=31 B=21 C=17 so C is eliminated. Round 3 votes counts: E=41 D=36 B=23 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:209 C:198 A:195 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 2 -4 6 B 14 0 6 -12 10 C -2 -6 0 -2 6 D 4 12 2 0 2 E -6 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -4 6 B 14 0 6 -12 10 C -2 -6 0 -2 6 D 4 12 2 0 2 E -6 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -4 6 B 14 0 6 -12 10 C -2 -6 0 -2 6 D 4 12 2 0 2 E -6 -10 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6781: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (14) A E D B C (12) A B E D C (11) D E C A B (10) E D A C B (7) C D E A B (7) B A E D C (7) C D E B A (6) B C A E D (6) B A C E D (4) D E A C B (3) B C A D E (3) D C E A B (2) B A E C D (2) D E A B C (1) C E D B A (1) C E A D B (1) C D B E A (1) C B D A E (1) B C D A E (1) Total count = 100 A B C D E A 0 8 -6 -8 -6 B -8 0 -6 -2 0 C 6 6 0 -6 -6 D 8 2 6 0 -2 E 6 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.284145 C: 0.000000 D: 0.000000 E: 0.715855 Sum of squares = 0.59318644413 Cumulative probabilities = A: 0.000000 B: 0.284145 C: 0.284145 D: 0.284145 E: 1.000000 A B C D E A 0 8 -6 -8 -6 B -8 0 -6 -2 0 C 6 6 0 -6 -6 D 8 2 6 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.51020408228 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=23 A=23 D=16 E=7 so E is eliminated. Round 2 votes counts: C=31 D=23 B=23 A=23 so D is eliminated. Round 3 votes counts: C=43 A=34 B=23 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:207 E:207 C:200 A:194 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 -8 -6 B -8 0 -6 -2 0 C 6 6 0 -6 -6 D 8 2 6 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.51020408228 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -8 -6 B -8 0 -6 -2 0 C 6 6 0 -6 -6 D 8 2 6 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.51020408228 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -8 -6 B -8 0 -6 -2 0 C 6 6 0 -6 -6 D 8 2 6 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.51020408228 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6782: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) D A E B C (8) A B D C E (7) D E C B A (6) C B E A D (6) D E C A B (5) B C A E D (5) A B C D E (5) E D C B A (4) A B C E D (4) E C D B A (3) D E A B C (3) D A B E C (3) C E B D A (3) A D E B C (3) E D C A B (2) E C B D A (2) D B A C E (2) C B E D A (2) B A C E D (2) B A C D E (2) A D B C E (2) E D A C B (1) E C D A B (1) E C B A D (1) E C A B D (1) D B C E A (1) D A B C E (1) C E B A D (1) C B D E A (1) C B A E D (1) B A D C E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 14 10 -14 -4 B -14 0 -2 -8 -8 C -10 2 0 -16 -4 D 14 8 16 0 18 E 4 8 4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 -14 -4 B -14 0 -2 -8 -8 C -10 2 0 -16 -4 D 14 8 16 0 18 E 4 8 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=23 E=15 C=14 B=10 so B is eliminated. Round 2 votes counts: D=38 A=28 C=19 E=15 so E is eliminated. Round 3 votes counts: D=45 A=28 C=27 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 A:203 E:199 C:186 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 10 -14 -4 B -14 0 -2 -8 -8 C -10 2 0 -16 -4 D 14 8 16 0 18 E 4 8 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -14 -4 B -14 0 -2 -8 -8 C -10 2 0 -16 -4 D 14 8 16 0 18 E 4 8 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -14 -4 B -14 0 -2 -8 -8 C -10 2 0 -16 -4 D 14 8 16 0 18 E 4 8 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6783: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (8) B D A E C (8) D B A E C (7) C E A D B (7) C A D E B (6) E C B A D (5) E C A B D (5) E B C D A (5) E B C A D (5) D A C B E (5) C A E D B (5) A D C B E (5) D B A C E (4) D A B C E (4) B E D C A (4) B E D A C (4) B E C A D (3) C E A B D (2) A D C E B (2) A D B C E (2) D C A E B (1) B E C D A (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 2 -2 2 B 8 0 12 2 12 C -2 -12 0 -10 -10 D 2 -2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -2 2 B 8 0 12 2 12 C -2 -12 0 -10 -10 D 2 -2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=21 E=20 C=20 A=11 so A is eliminated. Round 2 votes counts: D=30 B=29 C=21 E=20 so E is eliminated. Round 3 votes counts: B=39 C=31 D=30 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:209 A:197 E:194 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -2 2 B 8 0 12 2 12 C -2 -12 0 -10 -10 D 2 -2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 2 B 8 0 12 2 12 C -2 -12 0 -10 -10 D 2 -2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 2 B 8 0 12 2 12 C -2 -12 0 -10 -10 D 2 -2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6784: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) E C A B D (7) D C E B A (7) D B A E C (7) E A B C D (6) D C B A E (6) C E A B D (6) C D E A B (5) C A B E D (5) B A C E D (5) E D C A B (4) B A D C E (4) E C D A B (3) D E B A C (3) C B A D E (3) B A E C D (3) A B E C D (3) E A B D C (2) D B C A E (2) E D B A C (1) E D A B C (1) D E C B A (1) D E C A B (1) D C A E B (1) D B E A C (1) C E D A B (1) C D A B E (1) B A D E C (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -6 -6 2 B 6 0 -2 -6 0 C 6 2 0 0 10 D 6 6 0 0 4 E -2 0 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.357869 D: 0.642131 E: 0.000000 Sum of squares = 0.540402260563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.357869 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -6 2 B 6 0 -2 -6 0 C 6 2 0 0 10 D 6 6 0 0 4 E -2 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=24 C=21 B=14 A=4 so A is eliminated. Round 2 votes counts: D=37 E=25 C=21 B=17 so B is eliminated. Round 3 votes counts: D=42 E=31 C=27 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:209 D:208 B:199 A:192 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -6 2 B 6 0 -2 -6 0 C 6 2 0 0 10 D 6 6 0 0 4 E -2 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -6 2 B 6 0 -2 -6 0 C 6 2 0 0 10 D 6 6 0 0 4 E -2 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -6 2 B 6 0 -2 -6 0 C 6 2 0 0 10 D 6 6 0 0 4 E -2 0 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6785: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) A D C E B (8) B D C E A (5) D A C E B (4) C E A D B (4) E C B A D (3) D C A E B (3) D B A E C (3) D A C B E (3) D A B C E (3) C E B D A (3) B E C D A (3) B D E A C (3) B D A E C (3) A C E D B (3) E A C B D (2) D C B E A (2) D B C A E (2) D B A C E (2) D A B E C (2) C E B A D (2) C E A B D (2) C B E D A (2) B D E C A (2) B C E D A (2) E C A B D (1) E B C A D (1) E B A C D (1) E A B C D (1) D C B A E (1) D C A B E (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C A D (1) B E A D C (1) B E A C D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B C D (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -6 -20 -4 B 6 0 -2 -16 6 C 6 2 0 -24 28 D 20 16 24 0 20 E 4 -6 -28 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -20 -4 B 6 0 -2 -16 6 C 6 2 0 -24 28 D 20 16 24 0 20 E 4 -6 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=22 A=18 C=16 E=9 so E is eliminated. Round 2 votes counts: D=35 B=24 A=21 C=20 so C is eliminated. Round 3 votes counts: D=36 B=34 A=30 so A is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:240 C:206 B:197 A:182 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -20 -4 B 6 0 -2 -16 6 C 6 2 0 -24 28 D 20 16 24 0 20 E 4 -6 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -20 -4 B 6 0 -2 -16 6 C 6 2 0 -24 28 D 20 16 24 0 20 E 4 -6 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -20 -4 B 6 0 -2 -16 6 C 6 2 0 -24 28 D 20 16 24 0 20 E 4 -6 -28 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6786: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) B C A D E (9) C A E B D (6) B D C A E (6) A C E B D (6) D E A C B (5) C A B E D (5) B C A E D (5) E D A C B (4) D E B C A (4) D E B A C (4) D B E C A (4) B D C E A (4) A C B E D (4) B D E C A (3) E C A D B (2) E A C D B (2) B D E A C (2) A E C D B (2) E D C A B (1) D E C A B (1) D B E A C (1) C E A D B (1) C E A B D (1) C B A E D (1) B C D E A (1) B C D A E (1) B A C E D (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -10 18 20 B -8 0 -4 14 0 C 10 4 0 20 30 D -18 -14 -20 0 -6 E -20 0 -30 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 18 20 B -8 0 -4 14 0 C 10 4 0 20 30 D -18 -14 -20 0 -6 E -20 0 -30 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=26 D=19 C=14 E=9 so E is eliminated. Round 2 votes counts: B=32 A=28 D=24 C=16 so C is eliminated. Round 3 votes counts: A=43 B=33 D=24 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:232 A:218 B:201 E:178 D:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 18 20 B -8 0 -4 14 0 C 10 4 0 20 30 D -18 -14 -20 0 -6 E -20 0 -30 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 18 20 B -8 0 -4 14 0 C 10 4 0 20 30 D -18 -14 -20 0 -6 E -20 0 -30 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 18 20 B -8 0 -4 14 0 C 10 4 0 20 30 D -18 -14 -20 0 -6 E -20 0 -30 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6787: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) D A E C B (6) A D B E C (5) D A C E B (4) C E B A D (4) C D B E A (4) B C E D A (4) B C D E A (4) A E D C B (4) A E D B C (4) E C A B D (3) E A C B D (3) D C A E B (3) D C A B E (3) D B C A E (3) D B A C E (3) D A B E C (3) C B E D A (3) A E B C D (3) D A C B E (2) C B D E A (2) B D C E A (2) B D C A E (2) B A D E C (2) A E B D C (2) A D E B C (2) E B C A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D B A E C (1) C E D B A (1) C E B D A (1) C E A B D (1) B E C A D (1) B E A C D (1) B D A E C (1) B D A C E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 -8 10 B 0 0 -4 2 -2 C 0 4 0 -10 -4 D 8 -2 10 0 8 E -10 2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999795 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -8 10 B 0 0 -4 2 -2 C 0 4 0 -10 -4 D 8 -2 10 0 8 E -10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999279 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=21 B=18 E=16 C=16 so E is eliminated. Round 2 votes counts: D=29 C=26 A=25 B=20 so B is eliminated. Round 3 votes counts: C=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:201 B:198 C:195 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -8 10 B 0 0 -4 2 -2 C 0 4 0 -10 -4 D 8 -2 10 0 8 E -10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999279 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 10 B 0 0 -4 2 -2 C 0 4 0 -10 -4 D 8 -2 10 0 8 E -10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999279 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 10 B 0 0 -4 2 -2 C 0 4 0 -10 -4 D 8 -2 10 0 8 E -10 2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.125000 D: 0.250000 E: 0.000000 Sum of squares = 0.468749999279 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6788: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) D E C B A (7) C E B D A (7) E D C A B (6) E C D B A (6) A D E B C (6) B C A E D (5) C E D B A (4) B C A D E (4) B A C E D (4) A E D C B (4) E C D A B (3) D A E B C (3) C B E D A (3) B C D E A (3) A B D E C (3) A B D C E (3) E D C B A (2) D C E B A (2) B C E D A (2) A B C E D (2) E A D C B (1) D E C A B (1) D B E C A (1) D B C E A (1) C D E B A (1) C A B E D (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C D E (1) A D B E C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -24 -24 -20 B 12 0 -12 -14 -22 C 24 12 0 -2 -6 D 24 14 2 0 -4 E 20 22 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -24 -24 -20 B 12 0 -12 -14 -22 C 24 12 0 -2 -6 D 24 14 2 0 -4 E 20 22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 B=22 A=21 E=18 C=16 so C is eliminated. Round 2 votes counts: E=29 B=25 D=24 A=22 so A is eliminated. Round 3 votes counts: B=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=61 B=39 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:218 C:214 B:182 A:160 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -24 -24 -20 B 12 0 -12 -14 -22 C 24 12 0 -2 -6 D 24 14 2 0 -4 E 20 22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -24 -24 -20 B 12 0 -12 -14 -22 C 24 12 0 -2 -6 D 24 14 2 0 -4 E 20 22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -24 -24 -20 B 12 0 -12 -14 -22 C 24 12 0 -2 -6 D 24 14 2 0 -4 E 20 22 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6789: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (9) C E D A B (6) E D A C B (5) C D E A B (5) C B D E A (5) B C A D E (5) B A D E C (5) C B E A D (4) A E D B C (4) C E A D B (3) B D A E C (3) B A E D C (3) A B E D C (3) E A D C B (2) D E C A B (2) D C E A B (2) D B E A C (2) D B A E C (2) C B E D A (2) C B A E D (2) B C D A E (2) B A E C D (2) A E C B D (2) D B C E A (1) D A E B C (1) C D E B A (1) C D B E A (1) B D C A E (1) B C D E A (1) B A C E D (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -4 -4 -4 B 4 0 -4 2 12 C 4 4 0 2 2 D 4 -2 -2 0 4 E 4 -12 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -4 -4 B 4 0 -4 2 12 C 4 4 0 2 2 D 4 -2 -2 0 4 E 4 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=29 D=20 A=12 E=7 so E is eliminated. Round 2 votes counts: B=32 C=29 D=25 A=14 so A is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:206 D:202 E:193 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -4 -4 B 4 0 -4 2 12 C 4 4 0 2 2 D 4 -2 -2 0 4 E 4 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -4 -4 B 4 0 -4 2 12 C 4 4 0 2 2 D 4 -2 -2 0 4 E 4 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -4 -4 B 4 0 -4 2 12 C 4 4 0 2 2 D 4 -2 -2 0 4 E 4 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6790: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (8) C B D E A (7) B D C A E (6) A E D B C (6) E A D C B (5) A D B E C (5) B C D A E (4) A B E D C (4) E C A B D (3) E A C D B (3) E A C B D (3) D C B E A (3) C E D B A (3) B C D E A (3) B A D C E (3) E D C A B (2) E C A D B (2) D C E B A (2) D B C E A (2) D B C A E (2) D B A C E (2) D A B C E (2) C E B D A (2) C D B E A (2) C B E D A (2) E D A C B (1) E C D A B (1) E C B D A (1) D E C B A (1) D E A C B (1) C E B A D (1) B D A C E (1) B C A D E (1) A E D C B (1) A E C B D (1) A E B D C (1) A D E B C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 0 0 0 B -4 0 6 4 18 C 0 -6 0 -22 14 D 0 -4 22 0 16 E 0 -18 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.687830 B: 0.000000 C: 0.000000 D: 0.312170 E: 0.000000 Sum of squares = 0.570560163844 Cumulative probabilities = A: 0.687830 B: 0.687830 C: 0.687830 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 0 0 B -4 0 6 4 18 C 0 -6 0 -22 14 D 0 -4 22 0 16 E 0 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500308 B: 0.000000 C: 0.000000 D: 0.499692 E: 0.000000 Sum of squares = 0.500000189209 Cumulative probabilities = A: 0.500308 B: 0.500308 C: 0.500308 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=21 B=18 C=17 D=15 so D is eliminated. Round 2 votes counts: A=31 B=24 E=23 C=22 so C is eliminated. Round 3 votes counts: B=38 E=31 A=31 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:217 B:212 A:202 C:193 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 0 0 B -4 0 6 4 18 C 0 -6 0 -22 14 D 0 -4 22 0 16 E 0 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500308 B: 0.000000 C: 0.000000 D: 0.499692 E: 0.000000 Sum of squares = 0.500000189209 Cumulative probabilities = A: 0.500308 B: 0.500308 C: 0.500308 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 0 0 B -4 0 6 4 18 C 0 -6 0 -22 14 D 0 -4 22 0 16 E 0 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500308 B: 0.000000 C: 0.000000 D: 0.499692 E: 0.000000 Sum of squares = 0.500000189209 Cumulative probabilities = A: 0.500308 B: 0.500308 C: 0.500308 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 0 0 B -4 0 6 4 18 C 0 -6 0 -22 14 D 0 -4 22 0 16 E 0 -18 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500308 B: 0.000000 C: 0.000000 D: 0.499692 E: 0.000000 Sum of squares = 0.500000189209 Cumulative probabilities = A: 0.500308 B: 0.500308 C: 0.500308 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6791: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) D A C E B (8) B E C A D (8) C A D B E (6) C B E D A (5) B E C D A (5) A D E B C (5) E B C D A (4) D A E C B (4) E D B A C (3) C D A B E (3) C B A E D (3) B E A C D (3) B C E A D (3) B C A E D (3) A D C B E (3) E D C B A (2) C B E A D (2) A D C E B (2) A C D B E (2) E D A B C (1) E B A D C (1) E B A C D (1) D E A B C (1) D C E A B (1) D C A E B (1) D A E B C (1) C E B D A (1) C D E B A (1) C D B E A (1) C D B A E (1) C B D E A (1) C B A D E (1) B E A D C (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -4 -6 -8 B 18 0 4 6 8 C 4 -4 0 12 2 D 6 -6 -12 0 -12 E 8 -8 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -4 -6 -8 B 18 0 4 6 8 C 4 -4 0 12 2 D 6 -6 -12 0 -12 E 8 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 E=21 D=16 A=15 so A is eliminated. Round 2 votes counts: D=27 C=27 B=25 E=21 so E is eliminated. Round 3 votes counts: B=40 D=33 C=27 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:207 E:205 D:188 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -4 -6 -8 B 18 0 4 6 8 C 4 -4 0 12 2 D 6 -6 -12 0 -12 E 8 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -4 -6 -8 B 18 0 4 6 8 C 4 -4 0 12 2 D 6 -6 -12 0 -12 E 8 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -4 -6 -8 B 18 0 4 6 8 C 4 -4 0 12 2 D 6 -6 -12 0 -12 E 8 -8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6792: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) B C E A D (7) E D A B C (6) E B A D C (6) C D A E B (6) C B E D A (6) C B D E A (6) B E A D C (6) E B D A C (4) D A E C B (4) E A D B C (3) C D E A B (3) C B E A D (3) C B D A E (3) A E D B C (3) D C A E B (2) C D B A E (2) C D A B E (2) B E C A D (2) B A C E D (2) A E B D C (2) A D E C B (2) A D E B C (2) E D B A C (1) E A B D C (1) D E A B C (1) C E D B A (1) C B A D E (1) C A B D E (1) B C E D A (1) B A E D C (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 8 -10 -14 B 4 0 0 10 -8 C -8 0 0 -6 10 D 10 -10 6 0 -12 E 14 8 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.132653 B: 0.224490 C: 0.438776 D: 0.061224 E: 0.142857 Sum of squares = 0.28467305293 Cumulative probabilities = A: 0.132653 B: 0.357143 C: 0.795918 D: 0.857143 E: 1.000000 A B C D E A 0 -4 8 -10 -14 B 4 0 0 10 -8 C -8 0 0 -6 10 D 10 -10 6 0 -12 E 14 8 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.132653 B: 0.224490 C: 0.438776 D: 0.061224 E: 0.142857 Sum of squares = 0.284673052896 Cumulative probabilities = A: 0.132653 B: 0.357143 C: 0.795918 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=21 B=19 D=14 A=12 so A is eliminated. Round 2 votes counts: C=35 E=26 B=21 D=18 so D is eliminated. Round 3 votes counts: C=44 E=35 B=21 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:212 B:203 C:198 D:197 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 -10 -14 B 4 0 0 10 -8 C -8 0 0 -6 10 D 10 -10 6 0 -12 E 14 8 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.132653 B: 0.224490 C: 0.438776 D: 0.061224 E: 0.142857 Sum of squares = 0.284673052896 Cumulative probabilities = A: 0.132653 B: 0.357143 C: 0.795918 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -10 -14 B 4 0 0 10 -8 C -8 0 0 -6 10 D 10 -10 6 0 -12 E 14 8 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.132653 B: 0.224490 C: 0.438776 D: 0.061224 E: 0.142857 Sum of squares = 0.284673052896 Cumulative probabilities = A: 0.132653 B: 0.357143 C: 0.795918 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -10 -14 B 4 0 0 10 -8 C -8 0 0 -6 10 D 10 -10 6 0 -12 E 14 8 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.132653 B: 0.224490 C: 0.438776 D: 0.061224 E: 0.142857 Sum of squares = 0.284673052896 Cumulative probabilities = A: 0.132653 B: 0.357143 C: 0.795918 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6793: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (15) B D C E A (12) C E D B A (7) E C D B A (6) A E C B D (5) E A C D B (4) D B E C A (4) D B C E A (4) C B D E A (4) E D C B A (3) E C A D B (3) B D C A E (3) A E D C B (3) A C E B D (3) A B C D E (3) E A D B C (2) D B E A C (2) A E D B C (2) D E B C A (1) C E A D B (1) C D B E A (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A D E (1) B D A C E (1) B C D A E (1) B A D C E (1) B A C D E (1) A E B D C (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 -2 -12 B 10 0 -18 -18 -12 C 8 18 0 16 -6 D 2 18 -16 0 -16 E 12 12 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -8 -2 -12 B 10 0 -18 -18 -12 C 8 18 0 16 -6 D 2 18 -16 0 -16 E 12 12 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=19 E=18 C=17 D=11 so D is eliminated. Round 2 votes counts: A=35 B=29 E=19 C=17 so C is eliminated. Round 3 votes counts: B=38 A=35 E=27 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:223 C:218 D:194 A:184 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -8 -2 -12 B 10 0 -18 -18 -12 C 8 18 0 16 -6 D 2 18 -16 0 -16 E 12 12 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -2 -12 B 10 0 -18 -18 -12 C 8 18 0 16 -6 D 2 18 -16 0 -16 E 12 12 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -2 -12 B 10 0 -18 -18 -12 C 8 18 0 16 -6 D 2 18 -16 0 -16 E 12 12 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6794: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) C E D B A (5) A B D E C (5) D E A C B (4) C B E A D (4) B D A E C (4) B A C E D (4) E D C A B (3) E A C D B (3) D E C B A (3) D A B E C (3) B C A E D (3) B A D C E (3) A D E B C (3) A C E B D (3) A B D C E (3) D A E B C (2) C E D A B (2) C E B D A (2) C E B A D (2) C E A D B (2) C E A B D (2) B D C E A (2) B A D E C (2) B A C D E (2) A E C D B (2) A C E D B (2) A B C E D (2) E D A C B (1) E C D B A (1) E C A D B (1) E A D C B (1) D E C A B (1) D E B C A (1) D E B A C (1) D E A B C (1) D B A E C (1) C B E D A (1) C A E B D (1) B C E D A (1) B C D A E (1) B C A D E (1) A D E C B (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 12 10 8 0 B -12 0 -8 -2 -12 C -10 8 0 8 -2 D -8 2 -8 0 -10 E 0 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.433996 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.566004 Sum of squares = 0.508713005296 Cumulative probabilities = A: 0.433996 B: 0.433996 C: 0.433996 D: 0.433996 E: 1.000000 A B C D E A 0 12 10 8 0 B -12 0 -8 -2 -12 C -10 8 0 8 -2 D -8 2 -8 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=23 A=23 C=21 D=17 E=16 so E is eliminated. Round 2 votes counts: C=29 A=27 B=23 D=21 so D is eliminated. Round 3 votes counts: A=38 C=36 B=26 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:212 C:202 D:188 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 8 0 B -12 0 -8 -2 -12 C -10 8 0 8 -2 D -8 2 -8 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 8 0 B -12 0 -8 -2 -12 C -10 8 0 8 -2 D -8 2 -8 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 8 0 B -12 0 -8 -2 -12 C -10 8 0 8 -2 D -8 2 -8 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6795: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) A B C E D (8) A C B E D (7) D E B A C (6) C E D A B (6) E D C B A (5) C A E D B (5) B A E D C (5) D E C B A (4) B D E A C (4) A B C D E (4) E D B C A (3) C E A D B (3) C A E B D (3) C A B E D (3) B A D E C (3) D C E A B (2) D B E A C (2) C D E A B (2) C A B D E (2) E D C A B (1) E D B A C (1) E C D A B (1) E B D A C (1) E A B C D (1) D E C A B (1) D C E B A (1) C A D B E (1) B E D A C (1) B D A E C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -8 -6 -12 B -4 0 4 -10 -14 C 8 -4 0 -4 -4 D 6 10 4 0 -10 E 12 14 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -8 -6 -12 B -4 0 4 -10 -14 C 8 -4 0 -4 -4 D 6 10 4 0 -10 E 12 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 A=21 B=14 E=13 so E is eliminated. Round 2 votes counts: D=37 C=26 A=22 B=15 so B is eliminated. Round 3 votes counts: D=44 A=30 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:205 C:198 A:189 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 -6 -12 B -4 0 4 -10 -14 C 8 -4 0 -4 -4 D 6 10 4 0 -10 E 12 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -6 -12 B -4 0 4 -10 -14 C 8 -4 0 -4 -4 D 6 10 4 0 -10 E 12 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -6 -12 B -4 0 4 -10 -14 C 8 -4 0 -4 -4 D 6 10 4 0 -10 E 12 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6796: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) E C A B D (9) D B A C E (9) B D E C A (9) A D C E B (7) C E A D B (6) B D A E C (5) A C E D B (5) E C B A D (4) E A C B D (4) D B A E C (4) B E C A D (3) A C D E B (3) E B C A D (2) D C A E B (2) C E B A D (2) C A E D B (2) B E C D A (2) B A E C D (2) E C B D A (1) D A C B E (1) C E A B D (1) C D A E B (1) B E D C A (1) B A D E C (1) A E C D B (1) A E C B D (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 10 10 10 12 B -10 0 0 -6 -2 C -10 0 0 -2 0 D -10 6 2 0 8 E -12 2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 10 12 B -10 0 0 -6 -2 C -10 0 0 -2 0 D -10 6 2 0 8 E -12 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=23 E=20 A=19 C=12 so C is eliminated. Round 2 votes counts: E=29 D=27 B=23 A=21 so A is eliminated. Round 3 votes counts: D=39 E=38 B=23 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:221 D:203 C:194 B:191 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 10 12 B -10 0 0 -6 -2 C -10 0 0 -2 0 D -10 6 2 0 8 E -12 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 10 12 B -10 0 0 -6 -2 C -10 0 0 -2 0 D -10 6 2 0 8 E -12 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 10 12 B -10 0 0 -6 -2 C -10 0 0 -2 0 D -10 6 2 0 8 E -12 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6797: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (14) C E D B A (12) E C D B A (11) A B D C E (10) E C D A B (8) D E C B A (7) B A D C E (6) E C A D B (3) A E C B D (3) B D C A E (2) A E C D B (2) A D E B C (2) A B E D C (2) A B E C D (2) A B C E D (2) E D C B A (1) D C E B A (1) D B E C A (1) D B C E A (1) C E B A D (1) C E A B D (1) C D E B A (1) B D C E A (1) B D A C E (1) B C D E A (1) B A D E C (1) B A C E D (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -4 4 0 B -2 0 -4 -4 -8 C 4 4 0 -4 -18 D -4 4 4 0 0 E 0 8 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.570784 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.429216 Sum of squares = 0.510020642824 Cumulative probabilities = A: 0.570784 B: 0.570784 C: 0.570784 D: 0.570784 E: 1.000000 A B C D E A 0 2 -4 4 0 B -2 0 -4 -4 -8 C 4 4 0 -4 -18 D -4 4 4 0 0 E 0 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=23 C=15 B=13 D=10 so D is eliminated. Round 2 votes counts: A=39 E=30 C=16 B=15 so B is eliminated. Round 3 votes counts: A=48 E=31 C=21 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:213 D:202 A:201 C:193 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 4 0 B -2 0 -4 -4 -8 C 4 4 0 -4 -18 D -4 4 4 0 0 E 0 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 0 B -2 0 -4 -4 -8 C 4 4 0 -4 -18 D -4 4 4 0 0 E 0 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 0 B -2 0 -4 -4 -8 C 4 4 0 -4 -18 D -4 4 4 0 0 E 0 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6798: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) E C A B D (9) C E A B D (9) A C E D B (9) D B A C E (4) C E B D A (4) B E D C A (4) A E C B D (4) A D B C E (4) E C B D A (3) E B D C A (3) E B C D A (3) D B C E A (3) D A B C E (3) A C D E B (3) A C D B E (3) E A C B D (2) D B A E C (2) C E A D B (2) A D C B E (2) A D B E C (2) E C B A D (1) E B A D C (1) D C B A E (1) D B E A C (1) D B C A E (1) C E D B A (1) C A E D B (1) C A D B E (1) B E D A C (1) B D E A C (1) B C D E A (1) A E B C D (1) Total count = 100 A B C D E A 0 10 -14 8 -18 B -10 0 -10 14 -12 C 14 10 0 14 4 D -8 -14 -14 0 -16 E 18 12 -4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -14 8 -18 B -10 0 -10 14 -12 C 14 10 0 14 4 D -8 -14 -14 0 -16 E 18 12 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 C=18 B=17 D=15 so D is eliminated. Round 2 votes counts: A=31 B=28 E=22 C=19 so C is eliminated. Round 3 votes counts: E=38 A=33 B=29 so B is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:221 E:221 A:193 B:191 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -14 8 -18 B -10 0 -10 14 -12 C 14 10 0 14 4 D -8 -14 -14 0 -16 E 18 12 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 8 -18 B -10 0 -10 14 -12 C 14 10 0 14 4 D -8 -14 -14 0 -16 E 18 12 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 8 -18 B -10 0 -10 14 -12 C 14 10 0 14 4 D -8 -14 -14 0 -16 E 18 12 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6799: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) B E C D A (8) A D B C E (7) E C B D A (6) E B C D A (6) B C E A D (6) B C A D E (5) D A E B C (4) B D A E C (4) A D C E B (4) E D B A C (3) E D A C B (3) E C D A B (3) D A E C B (3) C E B A D (3) B A D C E (3) E C B A D (2) C A E D B (2) B C A E D (2) A D E C B (2) A D C B E (2) E C D B A (1) E B D C A (1) D A B C E (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D B E (1) B E D C A (1) B C E D A (1) B A C D E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -4 -8 8 B 10 0 26 6 8 C 4 -26 0 6 -12 D 8 -6 -6 0 -4 E -8 -8 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -8 8 B 10 0 26 6 8 C 4 -26 0 6 -12 D 8 -6 -6 0 -4 E -8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995699 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 D=17 A=17 C=10 so C is eliminated. Round 2 votes counts: B=33 E=29 A=21 D=17 so D is eliminated. Round 3 votes counts: A=38 B=33 E=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:200 D:196 A:193 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -8 8 B 10 0 26 6 8 C 4 -26 0 6 -12 D 8 -6 -6 0 -4 E -8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995699 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -8 8 B 10 0 26 6 8 C 4 -26 0 6 -12 D 8 -6 -6 0 -4 E -8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995699 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -8 8 B 10 0 26 6 8 C 4 -26 0 6 -12 D 8 -6 -6 0 -4 E -8 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995699 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6800: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) E C D A B (6) D B A E C (6) B C D A E (5) A E D B C (5) E C A D B (4) B A D C E (4) A D E B C (4) A B D E C (4) E D A B C (3) E A C D B (3) C E B D A (3) C E A B D (3) B D C A E (3) A E C B D (3) A D B E C (3) D E A B C (2) D A B E C (2) C E B A D (2) C B E D A (2) B D A C E (2) B C D E A (2) B A C D E (2) E D C A B (1) E C A B D (1) E A D C B (1) D E A C B (1) D C E B A (1) D A E B C (1) C E D B A (1) C E D A B (1) C D B E A (1) C B E A D (1) B D C E A (1) B C A D E (1) B A D E C (1) B A C E D (1) A E C D B (1) A E B D C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 2 -8 0 B -2 0 8 6 4 C -2 -8 0 8 -6 D 8 -6 -8 0 14 E 0 -4 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.406250000008 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -8 0 B -2 0 8 6 4 C -2 -8 0 8 -6 D 8 -6 -8 0 14 E 0 -4 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=22 A=22 E=19 D=13 so D is eliminated. Round 2 votes counts: B=28 C=25 A=25 E=22 so E is eliminated. Round 3 votes counts: C=37 A=35 B=28 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:208 D:204 A:198 C:196 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 2 -8 0 B -2 0 8 6 4 C -2 -8 0 8 -6 D 8 -6 -8 0 14 E 0 -4 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -8 0 B -2 0 8 6 4 C -2 -8 0 8 -6 D 8 -6 -8 0 14 E 0 -4 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -8 0 B -2 0 8 6 4 C -2 -8 0 8 -6 D 8 -6 -8 0 14 E 0 -4 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.500000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.40625 Cumulative probabilities = A: 0.375000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6801: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) D E B C A (10) C E D A B (10) C E D B A (9) B A D E C (8) A B D E C (6) A B C D E (6) A C E D B (5) D E C B A (4) C A E D B (4) C A B E D (4) B D E C A (3) B D E A C (3) A C B E D (3) E C D A B (2) D E B A C (2) C A E B D (2) E C D B A (1) C E A D B (1) C B E D A (1) B D A E C (1) B C D E A (1) B A C D E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -26 -16 -16 B 10 0 -16 -18 -24 C 26 16 0 4 -2 D 16 18 -4 0 -10 E 16 24 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -26 -16 -16 B 10 0 -16 -18 -24 C 26 16 0 4 -2 D 16 18 -4 0 -10 E 16 24 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999969231 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=22 B=17 D=16 E=14 so E is eliminated. Round 2 votes counts: C=34 D=27 A=22 B=17 so B is eliminated. Round 3 votes counts: C=35 D=34 A=31 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:226 C:222 D:210 B:176 A:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -26 -16 -16 B 10 0 -16 -18 -24 C 26 16 0 4 -2 D 16 18 -4 0 -10 E 16 24 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999969231 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -26 -16 -16 B 10 0 -16 -18 -24 C 26 16 0 4 -2 D 16 18 -4 0 -10 E 16 24 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999969231 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -26 -16 -16 B 10 0 -16 -18 -24 C 26 16 0 4 -2 D 16 18 -4 0 -10 E 16 24 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999969231 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6802: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (11) E C B A D (8) D A B C E (6) C A E D B (5) B E C D A (5) E C A D B (4) D A C B E (4) C E B A D (4) B E D C A (4) B D E A C (4) B D A C E (4) D A E B C (3) C A D E B (3) A D C B E (3) E B C D A (2) E B C A D (2) D B A E C (2) A D C E B (2) E D B A C (1) E C D A B (1) E C A B D (1) D E A B C (1) D B A C E (1) C E A B D (1) B E D A C (1) B E C A D (1) B C E A D (1) B C A E D (1) B C A D E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -10 0 -4 B 6 0 2 2 0 C 10 -2 0 6 -2 D 0 -2 -6 0 -6 E 4 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.582202 C: 0.000000 D: 0.000000 E: 0.417798 Sum of squares = 0.513514270146 Cumulative probabilities = A: 0.000000 B: 0.582202 C: 0.582202 D: 0.582202 E: 1.000000 A B C D E A 0 -6 -10 0 -4 B 6 0 2 2 0 C 10 -2 0 6 -2 D 0 -2 -6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=24 E=19 D=17 A=7 so A is eliminated. Round 2 votes counts: B=33 C=26 D=22 E=19 so E is eliminated. Round 3 votes counts: C=40 B=37 D=23 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:206 E:206 B:205 D:193 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 0 -4 B 6 0 2 2 0 C 10 -2 0 6 -2 D 0 -2 -6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 0 -4 B 6 0 2 2 0 C 10 -2 0 6 -2 D 0 -2 -6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 0 -4 B 6 0 2 2 0 C 10 -2 0 6 -2 D 0 -2 -6 0 -6 E 4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6803: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) C B D A E (7) D E C A B (6) D B C E A (6) A E C B D (5) E A C D B (4) C D B A E (4) A E B D C (4) E A D C B (3) C D E A B (3) C D B E A (3) B C D A E (3) B A E D C (3) E D C A B (2) E D A C B (2) E D A B C (2) D E C B A (2) C E D A B (2) C B A D E (2) C A E B D (2) B D C A E (2) A E B C D (2) A B E C D (2) E C D A B (1) E A B D C (1) D E B C A (1) D E B A C (1) D E A B C (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E A C (1) C E A D B (1) C D E B A (1) C A E D B (1) C A D E B (1) C A B E D (1) B D C E A (1) B D A E C (1) B C A D E (1) B A E C D (1) B A C D E (1) A E C D B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 14 -12 -10 -8 B -14 0 -16 -18 -18 C 12 16 0 0 -8 D 10 18 0 0 2 E 8 18 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.131205 D: 0.868795 E: 0.000000 Sum of squares = 0.772020021466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.131205 D: 1.000000 E: 1.000000 A B C D E A 0 14 -12 -10 -8 B -14 0 -16 -18 -18 C 12 16 0 0 -8 D 10 18 0 0 2 E 8 18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000152163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=22 D=21 A=16 B=13 so B is eliminated. Round 2 votes counts: C=32 D=25 E=22 A=21 so A is eliminated. Round 3 votes counts: E=41 C=34 D=25 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:215 C:210 A:192 B:167 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -12 -10 -8 B -14 0 -16 -18 -18 C 12 16 0 0 -8 D 10 18 0 0 2 E 8 18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000152163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -12 -10 -8 B -14 0 -16 -18 -18 C 12 16 0 0 -8 D 10 18 0 0 2 E 8 18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000152163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -12 -10 -8 B -14 0 -16 -18 -18 C 12 16 0 0 -8 D 10 18 0 0 2 E 8 18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000152163 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6804: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) C A D B E (6) A D B E C (6) E A D B C (5) B D C A E (5) A D E B C (5) E C B D A (4) C E B D A (4) A D C B E (4) A D B C E (4) E C A B D (3) D B A C E (3) C E A B D (3) A E D C B (3) E D B A C (2) E C B A D (2) E C A D B (2) E B D C A (2) E B D A C (2) D A B C E (2) C D B A E (2) C B D E A (2) C B D A E (2) C A E B D (2) B D C E A (2) A E D B C (2) A C D B E (2) E D A B C (1) E B C D A (1) E A C D B (1) D E A B C (1) D A B E C (1) C B E D A (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A E C (1) B C E D A (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 6 2 16 B -8 0 12 -18 8 C -6 -12 0 -20 -6 D -2 18 20 0 14 E -16 -8 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 2 16 B -8 0 12 -18 8 C -6 -12 0 -20 -6 D -2 18 20 0 14 E -16 -8 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 C=22 D=13 B=13 so D is eliminated. Round 2 votes counts: A=30 E=26 C=22 B=22 so C is eliminated. Round 3 votes counts: A=38 E=33 B=29 so B is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:225 A:216 B:197 E:184 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 2 16 B -8 0 12 -18 8 C -6 -12 0 -20 -6 D -2 18 20 0 14 E -16 -8 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 2 16 B -8 0 12 -18 8 C -6 -12 0 -20 -6 D -2 18 20 0 14 E -16 -8 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 2 16 B -8 0 12 -18 8 C -6 -12 0 -20 -6 D -2 18 20 0 14 E -16 -8 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6805: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) A C E B D (7) C A E B D (6) A E B D C (6) E B D A C (5) E A B D C (5) C D B A E (5) C A B D E (5) A E D B C (5) D B E C A (4) D B C E A (4) A E C B D (4) E D B A C (3) D B E A C (3) C A D B E (3) B D E C A (3) E B D C A (2) D C B A E (2) C D B E A (2) B E D C A (2) B D C E A (2) A E C D B (2) A C E D B (2) A C D E B (2) E B C D A (1) C B E D A (1) C B D A E (1) C A B E D (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 0 6 B 0 0 -8 22 0 C 8 8 0 6 8 D 0 -22 -6 0 -4 E -6 0 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 0 6 B 0 0 -8 22 0 C 8 8 0 6 8 D 0 -22 -6 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=30 E=16 D=13 B=7 so B is eliminated. Round 2 votes counts: C=34 A=30 E=18 D=18 so E is eliminated. Round 3 votes counts: C=35 A=35 D=30 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:207 A:199 E:195 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 0 6 B 0 0 -8 22 0 C 8 8 0 6 8 D 0 -22 -6 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 0 6 B 0 0 -8 22 0 C 8 8 0 6 8 D 0 -22 -6 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 0 6 B 0 0 -8 22 0 C 8 8 0 6 8 D 0 -22 -6 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6806: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) E C A B D (8) D B C A E (8) A B D E C (7) E C D A B (6) C E D B A (5) E A B C D (4) D C E B A (4) D C B E A (4) D C B A E (4) B D A C E (4) A E B C D (4) E C A D B (3) C E B A D (3) C D E B A (3) A E B D C (3) A B E C D (3) E A C D B (2) E A C B D (2) C D B E A (2) A B E D C (2) A B D C E (2) E D C A B (1) E A B D C (1) D E C A B (1) D E A C B (1) D A E B C (1) D A B C E (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 4 -4 -8 0 B -4 0 2 -10 -4 C 4 -2 0 -8 2 D 8 10 8 0 6 E 0 4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -8 0 B -4 0 2 -10 -4 C 4 -2 0 -8 2 D 8 10 8 0 6 E 0 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=27 A=21 C=13 B=6 so B is eliminated. Round 2 votes counts: D=37 E=27 A=23 C=13 so C is eliminated. Round 3 votes counts: D=42 E=35 A=23 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:198 E:198 A:196 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -8 0 B -4 0 2 -10 -4 C 4 -2 0 -8 2 D 8 10 8 0 6 E 0 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 0 B -4 0 2 -10 -4 C 4 -2 0 -8 2 D 8 10 8 0 6 E 0 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 0 B -4 0 2 -10 -4 C 4 -2 0 -8 2 D 8 10 8 0 6 E 0 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6807: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) A B E C D (8) D E B A C (7) C E B A D (6) D C E B A (5) E B A D C (4) D C A B E (4) C A B E D (4) E D B A C (3) E B A C D (3) D E C B A (3) D A B E C (3) C D E B A (3) C D A E B (3) C A D B E (3) A C B E D (3) A B E D C (3) A B D E C (3) E B D A C (2) D B A E C (2) D A C B E (2) C E B D A (2) B E A D C (2) B A E D C (2) E D B C A (1) E C B A D (1) D E B C A (1) D C A E B (1) D B E A C (1) C E D B A (1) C D E A B (1) C A E B D (1) C A D E B (1) C A B D E (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 -6 8 B -2 0 -6 -8 2 C 0 6 0 0 0 D 6 8 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.435053 D: 0.564947 E: 0.000000 Sum of squares = 0.508436173281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.435053 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -6 8 B -2 0 -6 -8 2 C 0 6 0 0 0 D 6 8 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=29 A=19 E=14 B=4 so B is eliminated. Round 2 votes counts: C=34 D=29 A=21 E=16 so E is eliminated. Round 3 votes counts: D=35 C=35 A=30 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:203 A:202 B:193 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -6 8 B -2 0 -6 -8 2 C 0 6 0 0 0 D 6 8 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -6 8 B -2 0 -6 -8 2 C 0 6 0 0 0 D 6 8 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -6 8 B -2 0 -6 -8 2 C 0 6 0 0 0 D 6 8 0 0 6 E -8 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6808: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (11) C A E B D (7) D A B E C (6) A C B D E (6) D E B C A (5) B E D A C (5) E B C D A (4) D B E A C (4) B E D C A (4) E B D C A (3) C A D E B (3) C A B E D (3) E D B C A (2) D E C B A (2) D E B A C (2) D A C E B (2) C E B A D (2) C D E A B (2) C D A E B (2) C A E D B (2) B E C A D (2) A D C B E (2) A C D B E (2) E C B D A (1) D C E B A (1) D B E C A (1) D A E B C (1) D A B C E (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A B D (1) C B A E D (1) B D A E C (1) B A E C D (1) A D B E C (1) A C E D B (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -2 -4 12 B -14 0 -10 10 6 C 2 10 0 12 10 D 4 -10 -12 0 -8 E -12 -6 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -4 12 B -14 0 -10 10 6 C 2 10 0 12 10 D 4 -10 -12 0 -8 E -12 -6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 D=25 B=13 E=10 so E is eliminated. Round 2 votes counts: D=27 C=27 A=26 B=20 so B is eliminated. Round 3 votes counts: D=40 C=33 A=27 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:210 B:196 E:190 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 -4 12 B -14 0 -10 10 6 C 2 10 0 12 10 D 4 -10 -12 0 -8 E -12 -6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -4 12 B -14 0 -10 10 6 C 2 10 0 12 10 D 4 -10 -12 0 -8 E -12 -6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -4 12 B -14 0 -10 10 6 C 2 10 0 12 10 D 4 -10 -12 0 -8 E -12 -6 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6809: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E C A B D (8) D B A C E (7) B D A C E (6) E C D A B (5) E D C B A (4) B A D C E (4) E D C A B (3) E C A D B (3) E B A C D (3) D C B A E (3) C E D A B (3) C A D B E (3) A C B E D (3) E D B C A (2) D E B C A (2) D C A B E (2) D B E A C (2) D B A E C (2) C E A D B (2) C D A B E (2) C A E B D (2) C A B E D (2) B A E C D (2) A B C D E (2) E D B A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D E B A C (1) D C E A B (1) C E A B D (1) C D E A B (1) C A D E B (1) C A B D E (1) B D E A C (1) B A D E C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 4 8 B 2 0 -4 0 6 C 2 4 0 14 14 D -4 0 -14 0 2 E -8 -6 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 4 8 B 2 0 -4 0 6 C 2 4 0 14 14 D -4 0 -14 0 2 E -8 -6 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=23 D=20 C=18 A=7 so A is eliminated. Round 2 votes counts: E=32 B=26 C=22 D=20 so D is eliminated. Round 3 votes counts: B=37 E=35 C=28 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:217 A:204 B:202 D:192 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 4 8 B 2 0 -4 0 6 C 2 4 0 14 14 D -4 0 -14 0 2 E -8 -6 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 8 B 2 0 -4 0 6 C 2 4 0 14 14 D -4 0 -14 0 2 E -8 -6 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 8 B 2 0 -4 0 6 C 2 4 0 14 14 D -4 0 -14 0 2 E -8 -6 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6810: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (31) E D C A B (29) E C D A B (6) B A D C E (4) E D C B A (3) D E C A B (3) C E D A B (3) B A E C D (2) B A D E C (2) B A C E D (2) A B C D E (2) E D B C A (1) E B D A C (1) D C E A B (1) C D E A B (1) C D A E B (1) C A E D B (1) C A B E D (1) C A B D E (1) B E A D C (1) B D A E C (1) B A E D C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 0 2 B -2 0 -2 2 0 C 2 2 0 4 0 D 0 -2 -4 0 -2 E -2 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.692691 D: 0.000000 E: 0.307308 Sum of squares = 0.574260019265 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.692692 D: 0.692692 E: 1.000000 A B C D E A 0 2 -2 0 2 B -2 0 -2 2 0 C 2 2 0 4 0 D 0 -2 -4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.000000 E: 0.499767 Sum of squares = 0.500000108748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.500233 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 E=40 C=8 D=4 A=4 so D is eliminated. Round 2 votes counts: B=44 E=43 C=9 A=4 so A is eliminated. Round 3 votes counts: B=47 E=43 C=10 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:204 A:201 E:200 B:199 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 0 2 B -2 0 -2 2 0 C 2 2 0 4 0 D 0 -2 -4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.000000 E: 0.499767 Sum of squares = 0.500000108748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.500233 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 2 B -2 0 -2 2 0 C 2 2 0 4 0 D 0 -2 -4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.000000 E: 0.499767 Sum of squares = 0.500000108748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.500233 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 2 B -2 0 -2 2 0 C 2 2 0 4 0 D 0 -2 -4 0 -2 E -2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.000000 E: 0.499767 Sum of squares = 0.500000108748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500233 D: 0.500233 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6811: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) B C E A D (6) E D A C B (5) D E C A B (4) D E A C B (4) C D E B A (4) B C D E A (4) A E B D C (4) A D E B C (4) E A D C B (3) D C B E A (3) C B D E A (3) B C D A E (3) B C A E D (3) B A E C D (3) B A C E D (3) A B E D C (3) E A C B D (2) D A E C B (2) C E D B A (2) C E D A B (2) C D B E A (2) B C A D E (2) B A D E C (2) A E D B C (2) E C A B D (1) D C E B A (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B C E (1) C D E A B (1) C B E A D (1) B E C A D (1) B D A C E (1) B C E D A (1) B A D C E (1) A E D C B (1) A E B C D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -8 -10 -14 B -4 0 -2 -8 -6 C 8 2 0 -8 10 D 10 8 8 0 10 E 14 6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -10 -14 B -4 0 -2 -8 -6 C 8 2 0 -8 10 D 10 8 8 0 10 E 14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=27 A=17 C=15 E=11 so E is eliminated. Round 2 votes counts: D=32 B=30 A=22 C=16 so C is eliminated. Round 3 votes counts: D=43 B=34 A=23 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:206 E:200 B:190 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -14 B -4 0 -2 -8 -6 C 8 2 0 -8 10 D 10 8 8 0 10 E 14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -14 B -4 0 -2 -8 -6 C 8 2 0 -8 10 D 10 8 8 0 10 E 14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -14 B -4 0 -2 -8 -6 C 8 2 0 -8 10 D 10 8 8 0 10 E 14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6812: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) B C A D E (9) B C E D A (6) B A C D E (6) A D E C B (6) A D E B C (6) E D A C B (5) D E A B C (4) C B E D A (4) C A E D B (4) B C D E A (4) A E D C B (4) E D B C A (3) C E D B A (3) C E D A B (3) B A D E C (3) E D C A B (2) B D E C A (2) B C A E D (2) A C D E B (2) A C B D E (2) A B D E C (2) E D C B A (1) D E B A C (1) C E B D A (1) C B A D E (1) C A B E D (1) C A B D E (1) B A D C E (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -12 22 22 B 12 0 2 10 10 C 12 -2 0 18 20 D -22 -10 -18 0 4 E -22 -10 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 22 22 B 12 0 2 10 10 C 12 -2 0 18 20 D -22 -10 -18 0 4 E -22 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=27 A=24 E=11 D=5 so D is eliminated. Round 2 votes counts: B=33 C=27 A=24 E=16 so E is eliminated. Round 3 votes counts: B=37 A=33 C=30 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:224 B:217 A:210 D:177 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 22 22 B 12 0 2 10 10 C 12 -2 0 18 20 D -22 -10 -18 0 4 E -22 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 22 22 B 12 0 2 10 10 C 12 -2 0 18 20 D -22 -10 -18 0 4 E -22 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 22 22 B 12 0 2 10 10 C 12 -2 0 18 20 D -22 -10 -18 0 4 E -22 -10 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6813: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (14) B C A E D (7) B C D E A (6) D A E C B (5) D A E B C (5) C B E A D (5) C B A E D (5) A E D C B (5) E D C A B (4) E D A C B (4) E A D C B (4) B C E A D (4) A D E C B (4) B C E D A (3) D E A B C (2) C B E D A (2) B D C E A (2) B D C A E (2) B C D A E (2) A D B E C (2) A C B E D (2) E D C B A (1) E C D B A (1) E C A D B (1) D E B C A (1) C E B D A (1) C A B E D (1) B C A D E (1) A D E B C (1) A C E B D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 2 -10 -10 B -14 0 -20 -8 -8 C -2 20 0 -14 -8 D 10 8 14 0 -4 E 10 8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 -10 -10 B -14 0 -20 -8 -8 C -2 20 0 -14 -8 D 10 8 14 0 -4 E 10 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 A=17 E=15 C=14 so C is eliminated. Round 2 votes counts: B=39 D=27 A=18 E=16 so E is eliminated. Round 3 votes counts: B=40 D=37 A=23 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:215 D:214 A:198 C:198 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 -10 -10 B -14 0 -20 -8 -8 C -2 20 0 -14 -8 D 10 8 14 0 -4 E 10 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 -10 -10 B -14 0 -20 -8 -8 C -2 20 0 -14 -8 D 10 8 14 0 -4 E 10 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 -10 -10 B -14 0 -20 -8 -8 C -2 20 0 -14 -8 D 10 8 14 0 -4 E 10 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6814: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) E B A D C (7) E B C A D (6) C D A B E (6) B E C D A (6) D A C B E (5) A D E C B (5) A D E B C (5) A D C B E (5) C B E D A (4) C A D E B (4) E A D B C (3) C E B D A (3) A D C E B (3) E C B D A (2) E C B A D (2) E C A D B (2) E A B D C (2) D A B E C (2) C E B A D (2) C D B A E (2) C B D A E (2) A D B E C (2) E B D A C (1) E B A C D (1) E A D C B (1) D A B C E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D A C E (1) B C E D A (1) B C D A E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 -6 6 -6 B 2 0 0 2 -18 C 6 0 0 6 -18 D -6 -2 -6 0 -10 E 6 18 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 6 -6 B 2 0 0 2 -18 C 6 0 0 6 -18 D -6 -2 -6 0 -10 E 6 18 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=24 A=22 B=11 D=8 so D is eliminated. Round 2 votes counts: E=35 A=30 C=24 B=11 so B is eliminated. Round 3 votes counts: E=43 A=31 C=26 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:197 A:196 B:193 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 6 -6 B 2 0 0 2 -18 C 6 0 0 6 -18 D -6 -2 -6 0 -10 E 6 18 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 6 -6 B 2 0 0 2 -18 C 6 0 0 6 -18 D -6 -2 -6 0 -10 E 6 18 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 6 -6 B 2 0 0 2 -18 C 6 0 0 6 -18 D -6 -2 -6 0 -10 E 6 18 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6815: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) C A B E D (7) B D C E A (7) B C D A E (7) A E C D B (7) E A D C B (6) C B A D E (5) C A B D E (5) B C D E A (5) B D C A E (4) A C E B D (4) E D B C A (3) E D B A C (3) D E B C A (3) D B E C A (3) A C B E D (3) A C B D E (3) E A C D B (2) D E B A C (2) E D A C B (1) E C B D A (1) E C B A D (1) E C A B D (1) D B E A C (1) D B A C E (1) D A E B C (1) C B D A E (1) C A E B D (1) B E D C A (1) B D E C A (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -12 -8 0 B -2 0 2 14 8 C 12 -2 0 8 8 D 8 -14 -8 0 -2 E 0 -8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593749999988 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -8 0 B -2 0 2 14 8 C 12 -2 0 8 8 D 8 -14 -8 0 -2 E 0 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593749999971 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 C=19 A=18 D=11 so D is eliminated. Round 2 votes counts: E=32 B=30 C=19 A=19 so C is eliminated. Round 3 votes counts: B=36 E=32 A=32 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 B:211 E:193 D:192 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -12 -8 0 B -2 0 2 14 8 C 12 -2 0 8 8 D 8 -14 -8 0 -2 E 0 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593749999971 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -8 0 B -2 0 2 14 8 C 12 -2 0 8 8 D 8 -14 -8 0 -2 E 0 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593749999971 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -8 0 B -2 0 2 14 8 C 12 -2 0 8 8 D 8 -14 -8 0 -2 E 0 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.750000 C: 0.125000 D: 0.000000 E: 0.000000 Sum of squares = 0.593749999971 Cumulative probabilities = A: 0.125000 B: 0.875000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6816: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) D B E A C (10) B D E C A (8) A C E D B (8) E D B A C (6) C A B D E (5) B D E A C (5) B D A E C (5) C A B E D (3) B D C E A (3) A E C D B (3) E D A B C (2) E A C D B (2) D E B C A (2) D E B A C (2) D B E C A (2) C A E B D (2) B D C A E (2) B D A C E (2) A E C B D (2) E D C A B (1) E D B C A (1) E D A C B (1) D B C E A (1) C E D A B (1) C E A D B (1) C B A D E (1) B C D A E (1) B A D E C (1) B A D C E (1) B A C D E (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 8 -10 4 B 8 0 14 -10 8 C -8 -14 0 -12 -8 D 10 10 12 0 6 E -4 -8 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -10 4 B 8 0 14 -10 8 C -8 -14 0 -12 -8 D 10 10 12 0 6 E -4 -8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=25 D=17 A=16 E=13 so E is eliminated. Round 2 votes counts: B=29 D=28 C=25 A=18 so A is eliminated. Round 3 votes counts: C=41 B=31 D=28 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:219 B:210 A:197 E:195 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -10 4 B 8 0 14 -10 8 C -8 -14 0 -12 -8 D 10 10 12 0 6 E -4 -8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -10 4 B 8 0 14 -10 8 C -8 -14 0 -12 -8 D 10 10 12 0 6 E -4 -8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -10 4 B 8 0 14 -10 8 C -8 -14 0 -12 -8 D 10 10 12 0 6 E -4 -8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6817: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) E B C A D (8) B C E D A (8) A D E C B (8) E B C D A (5) E A D C B (5) E A D B C (4) D A C B E (4) B C D A E (4) A E D C B (4) E A B C D (3) D B C A E (3) B C D E A (3) E D B A C (2) E A C D B (2) C E B A D (2) C B E D A (2) C B E A D (2) C B D A E (2) C B A D E (2) A D E B C (2) E D A B C (1) E A C B D (1) D E B C A (1) D E A B C (1) D C B A E (1) D A E B C (1) D A B E C (1) D A B C E (1) C B A E D (1) B E C D A (1) B C E A D (1) A E D B C (1) A E C D B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -12 18 -24 B 16 0 2 10 -24 C 12 -2 0 18 -24 D -18 -10 -18 0 -28 E 24 24 24 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -12 18 -24 B 16 0 2 10 -24 C 12 -2 0 18 -24 D -18 -10 -18 0 -28 E 24 24 24 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 A=18 B=17 D=13 C=11 so C is eliminated. Round 2 votes counts: E=43 B=26 A=18 D=13 so D is eliminated. Round 3 votes counts: E=45 B=30 A=25 so A is eliminated. Round 4 votes counts: E=62 B=38 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:250 B:202 C:202 A:183 D:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -12 18 -24 B 16 0 2 10 -24 C 12 -2 0 18 -24 D -18 -10 -18 0 -28 E 24 24 24 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 18 -24 B 16 0 2 10 -24 C 12 -2 0 18 -24 D -18 -10 -18 0 -28 E 24 24 24 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 18 -24 B 16 0 2 10 -24 C 12 -2 0 18 -24 D -18 -10 -18 0 -28 E 24 24 24 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6818: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) E B D C A (6) E A D B C (6) E B D A C (5) B E D C A (5) B D E C A (5) E A C B D (4) E A B C D (4) D B C A E (4) C D B A E (4) A C D E B (4) E B C A D (3) C D A B E (3) B D C E A (3) D C B A E (2) D B E C A (2) C B D E A (2) A E C D B (2) A C E D B (2) A C D B E (2) E C B A D (1) E C A B D (1) E B C D A (1) E B A C D (1) E A B D C (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C B E (1) C E B A D (1) C E A B D (1) C B E D A (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B E D (1) B E D A C (1) B E C D A (1) B D E A C (1) A E C B D (1) A D E B C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -20 -2 -12 B 4 0 4 6 4 C 20 -4 0 6 -6 D 2 -6 -6 0 -4 E 12 -4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -20 -2 -12 B 4 0 4 6 4 C 20 -4 0 6 -6 D 2 -6 -6 0 -4 E 12 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997454 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=25 B=16 A=14 D=12 so D is eliminated. Round 2 votes counts: E=33 C=28 B=24 A=15 so A is eliminated. Round 3 votes counts: C=39 E=37 B=24 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:209 E:209 C:208 D:193 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -20 -2 -12 B 4 0 4 6 4 C 20 -4 0 6 -6 D 2 -6 -6 0 -4 E 12 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997454 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -20 -2 -12 B 4 0 4 6 4 C 20 -4 0 6 -6 D 2 -6 -6 0 -4 E 12 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997454 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -20 -2 -12 B 4 0 4 6 4 C 20 -4 0 6 -6 D 2 -6 -6 0 -4 E 12 -4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997454 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6819: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) D B A C E (6) C D B E A (6) B D E C A (6) E C A B D (5) B D E A C (5) A E B D C (5) C E A D B (4) C A E D B (4) B D A E C (4) A E C B D (4) A C E B D (4) E A B D C (3) D B A E C (3) E A C B D (2) E A B C D (2) D B C A E (2) C E D B A (2) C E D A B (2) C E A B D (2) C D E B A (2) A D B C E (2) A C E D B (2) A B E D C (2) A B D E C (2) E C D B A (1) E C B A D (1) E B D C A (1) E B A D C (1) D C B E A (1) D B E C A (1) D B E A C (1) C D B A E (1) C D A B E (1) C A D B E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 0 -4 -10 B 2 0 6 -2 6 C 0 -6 0 -6 2 D 4 2 6 0 6 E 10 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -4 -10 B 2 0 6 -2 6 C 0 -6 0 -6 2 D 4 2 6 0 6 E 10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 D=21 E=16 B=15 so B is eliminated. Round 2 votes counts: D=36 C=25 A=23 E=16 so E is eliminated. Round 3 votes counts: D=37 C=32 A=31 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:206 E:198 C:195 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -4 -10 B 2 0 6 -2 6 C 0 -6 0 -6 2 D 4 2 6 0 6 E 10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -4 -10 B 2 0 6 -2 6 C 0 -6 0 -6 2 D 4 2 6 0 6 E 10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -4 -10 B 2 0 6 -2 6 C 0 -6 0 -6 2 D 4 2 6 0 6 E 10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6820: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) C E D B A (6) A B D E C (6) E C A B D (5) D B A C E (5) C E D A B (5) B A D E C (5) A B D C E (5) B D A E C (4) B A D C E (4) A B E D C (4) E D B C A (3) D E B C A (3) A B C E D (3) E D C B A (2) D E C B A (2) B E D A C (2) A E B C D (2) A C B D E (2) E C D A B (1) E C B D A (1) E B A D C (1) E B A C D (1) E A C B D (1) D C E B A (1) D C B E A (1) D C B A E (1) D B E C A (1) D B E A C (1) D B A E C (1) C E A D B (1) C E A B D (1) C D E B A (1) C D E A B (1) C D A B E (1) C A E D B (1) B D A C E (1) B A E D C (1) A D B C E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 4 -8 -2 B 16 0 12 2 2 C -4 -12 0 -10 -16 D 8 -2 10 0 -4 E 2 -2 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 -8 -2 B 16 0 12 2 2 C -4 -12 0 -10 -16 D 8 -2 10 0 -4 E 2 -2 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 C=17 B=17 D=16 so D is eliminated. Round 2 votes counts: E=30 B=25 A=25 C=20 so C is eliminated. Round 3 votes counts: E=46 B=27 A=27 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:216 E:210 D:206 A:189 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 -8 -2 B 16 0 12 2 2 C -4 -12 0 -10 -16 D 8 -2 10 0 -4 E 2 -2 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 -8 -2 B 16 0 12 2 2 C -4 -12 0 -10 -16 D 8 -2 10 0 -4 E 2 -2 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 -8 -2 B 16 0 12 2 2 C -4 -12 0 -10 -16 D 8 -2 10 0 -4 E 2 -2 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6821: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) A D E C B (7) E D B A C (6) C B E D A (6) C B A E D (6) C A B D E (6) C B E A D (5) C B A D E (5) B C A D E (5) E D A B C (4) D E A B C (4) A D E B C (4) A D C E B (4) A C B D E (4) D A E B C (3) B C A E D (3) E D C B A (2) D E A C B (2) D A E C B (2) B E C D A (2) A C D E B (2) E B D C A (1) D E B A C (1) C E B D A (1) C A D E B (1) C A B E D (1) B E D C A (1) B A D C E (1) B A C D E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -6 14 14 B 8 0 -12 12 12 C 6 12 0 14 22 D -14 -12 -14 0 8 E -14 -12 -22 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999786 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 14 14 B 8 0 -12 12 12 C 6 12 0 14 22 D -14 -12 -14 0 8 E -14 -12 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=23 B=21 E=13 D=12 so D is eliminated. Round 2 votes counts: C=31 A=28 B=21 E=20 so E is eliminated. Round 3 votes counts: A=38 C=33 B=29 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:210 A:207 D:184 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 14 14 B 8 0 -12 12 12 C 6 12 0 14 22 D -14 -12 -14 0 8 E -14 -12 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 14 14 B 8 0 -12 12 12 C 6 12 0 14 22 D -14 -12 -14 0 8 E -14 -12 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 14 14 B 8 0 -12 12 12 C 6 12 0 14 22 D -14 -12 -14 0 8 E -14 -12 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6822: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) D C B E A (5) B E A D C (5) A C E D B (5) A C D E B (5) E A B C D (4) D C B A E (4) D C A B E (4) B E D C A (4) A D C B E (4) E C B D A (3) E B C D A (3) C D E A B (3) A E B C D (3) A B E C D (3) E C A D B (2) E B A C D (2) D A C B E (2) C D E B A (2) C D A E B (2) B E D A C (2) B E A C D (2) B D E C A (2) B A E D C (2) B A E C D (2) A E C D B (2) A E C B D (2) A D C E B (2) E C B A D (1) E A C B D (1) D B C A E (1) D A C E B (1) C E D B A (1) C E A D B (1) C A D E B (1) B E C D A (1) B E C A D (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 4 6 10 B -14 0 -20 -10 -6 C -4 20 0 4 2 D -6 10 -4 0 -6 E -10 6 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 6 10 B -14 0 -20 -10 -6 C -4 20 0 4 2 D -6 10 -4 0 -6 E -10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=24 B=21 E=16 C=10 so C is eliminated. Round 2 votes counts: D=31 A=30 B=21 E=18 so E is eliminated. Round 3 votes counts: A=38 D=32 B=30 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:211 E:200 D:197 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 6 10 B -14 0 -20 -10 -6 C -4 20 0 4 2 D -6 10 -4 0 -6 E -10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 6 10 B -14 0 -20 -10 -6 C -4 20 0 4 2 D -6 10 -4 0 -6 E -10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 6 10 B -14 0 -20 -10 -6 C -4 20 0 4 2 D -6 10 -4 0 -6 E -10 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6823: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) A E C B D (8) C D B A E (6) E A B C D (5) A C E B D (5) E A C B D (4) C B D A E (4) B D C A E (4) E D B A C (3) E B A D C (3) E A C D B (3) E A B D C (3) D E C B A (3) D E B C A (3) D C B A E (3) D B C E A (3) C A B D E (3) B D E A C (3) E B D A C (2) E A D B C (2) D B E C A (2) C D A B E (2) A E B C D (2) E D C A B (1) E D B C A (1) E D A C B (1) E A D C B (1) C D E A B (1) C D A E B (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B E A D C (1) B C D A E (1) B A D C E (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 2 -6 8 B 4 0 0 8 -6 C -2 0 0 2 -6 D 6 -8 -2 0 2 E -8 6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691123 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 -4 2 -6 8 B 4 0 0 8 -6 C -2 0 0 2 -6 D 6 -8 -2 0 2 E -8 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691364 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=22 C=20 A=18 B=11 so B is eliminated. Round 2 votes counts: E=31 D=29 C=21 A=19 so A is eliminated. Round 3 votes counts: E=42 D=30 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:203 E:201 A:200 D:199 C:197 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 2 -6 8 B 4 0 0 8 -6 C -2 0 0 2 -6 D 6 -8 -2 0 2 E -8 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691364 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -6 8 B 4 0 0 8 -6 C -2 0 0 2 -6 D 6 -8 -2 0 2 E -8 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691364 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -6 8 B 4 0 0 8 -6 C -2 0 0 2 -6 D 6 -8 -2 0 2 E -8 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.444444 C: 0.000000 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691364 Cumulative probabilities = A: 0.333333 B: 0.777778 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6824: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) C D B E A (11) B E A C D (11) D C A E B (7) A E B D C (7) D C A B E (6) A E D B C (6) A E B C D (6) B E C A D (4) E B A C D (3) E A B C D (3) D C B A E (3) D A E C B (3) D A C E B (3) B C E A D (3) D B C A E (2) C B E D A (2) C B D E A (2) E C B A D (1) D B A E C (1) C D A E B (1) B C E D A (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -8 -6 -4 B 10 0 -4 -14 14 C 8 4 0 -2 4 D 6 14 2 0 4 E 4 -14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -6 -4 B 10 0 -4 -14 14 C 8 4 0 -2 4 D 6 14 2 0 4 E 4 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=22 B=19 C=16 E=7 so E is eliminated. Round 2 votes counts: D=36 A=25 B=22 C=17 so C is eliminated. Round 3 votes counts: D=48 B=27 A=25 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 C:207 B:203 E:191 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -8 -6 -4 B 10 0 -4 -14 14 C 8 4 0 -2 4 D 6 14 2 0 4 E 4 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -6 -4 B 10 0 -4 -14 14 C 8 4 0 -2 4 D 6 14 2 0 4 E 4 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -6 -4 B 10 0 -4 -14 14 C 8 4 0 -2 4 D 6 14 2 0 4 E 4 -14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6825: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) D E B C A (6) D E C B A (5) D B E C A (5) C E A D B (5) C D E A B (5) C A E B D (5) B A D E C (5) D C E B A (4) D B A E C (3) B E A C D (3) B A E C D (3) A C E B D (3) A B D C E (3) A B C E D (3) E C D B A (2) D C E A B (2) D B E A C (2) C A D E B (2) B E A D C (2) B D E A C (2) A C D B E (2) A B D E C (2) E D C B A (1) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B E A (1) D C A E B (1) D B A C E (1) D A C B E (1) C E D A B (1) C D A E B (1) C A E D B (1) B D A E C (1) B A E D C (1) A E B C D (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -2 10 -2 B 0 0 10 -4 0 C 2 -10 0 2 -14 D -10 4 -2 0 10 E 2 0 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.454545 Sum of squares = 0.42148760315 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 0.545455 E: 1.000000 A B C D E A 0 0 -2 10 -2 B 0 0 10 -4 0 C 2 -10 0 2 -14 D -10 4 -2 0 10 E 2 0 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.454545 Sum of squares = 0.42148760319 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 0.545455 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=24 C=20 B=17 E=8 so E is eliminated. Round 2 votes counts: D=32 A=25 C=24 B=19 so B is eliminated. Round 3 votes counts: A=39 D=35 C=26 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:203 B:203 E:203 D:201 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 10 -2 B 0 0 10 -4 0 C 2 -10 0 2 -14 D -10 4 -2 0 10 E 2 0 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.454545 Sum of squares = 0.42148760319 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 0.545455 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 10 -2 B 0 0 10 -4 0 C 2 -10 0 2 -14 D -10 4 -2 0 10 E 2 0 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.454545 Sum of squares = 0.42148760319 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 0.545455 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 10 -2 B 0 0 10 -4 0 C 2 -10 0 2 -14 D -10 4 -2 0 10 E 2 0 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.454545 Sum of squares = 0.42148760319 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 0.545455 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6826: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) D A E B C (10) D A B E C (7) A E D C B (7) B C E D A (6) A D E C B (6) E C A B D (5) E A C D B (5) D A E C B (5) C E B A D (5) C B E A D (5) B C D E A (5) E C A D B (4) C E A B D (4) D B C A E (3) B D C A E (3) B C E A D (2) E C B A D (1) E A C B D (1) D B A E C (1) C E A D B (1) C D E B A (1) B D C E A (1) B D A E C (1) A E C D B (1) Total count = 100 A B C D E A 0 12 8 -6 8 B -12 0 -2 -22 -12 C -8 2 0 -8 -8 D 6 22 8 0 6 E -8 12 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -6 8 B -12 0 -2 -22 -12 C -8 2 0 -8 -8 D 6 22 8 0 6 E -8 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=18 E=16 C=16 A=14 so A is eliminated. Round 2 votes counts: D=42 E=24 B=18 C=16 so C is eliminated. Round 3 votes counts: D=43 E=34 B=23 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:211 E:203 C:189 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -6 8 B -12 0 -2 -22 -12 C -8 2 0 -8 -8 D 6 22 8 0 6 E -8 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -6 8 B -12 0 -2 -22 -12 C -8 2 0 -8 -8 D 6 22 8 0 6 E -8 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -6 8 B -12 0 -2 -22 -12 C -8 2 0 -8 -8 D 6 22 8 0 6 E -8 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6827: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) C B E A D (11) E B C D A (6) A D E C B (6) A D C E B (6) A D C B E (6) E C B A D (5) C B A E D (5) B E C D A (5) C A E B D (4) D A B E C (3) C B A D E (3) B C E D A (3) E A D C B (2) D B A E C (2) C E B A D (2) A C D B E (2) E D B A C (1) E D A B C (1) E C B D A (1) E B D C A (1) E A C D B (1) D E B A C (1) D A B C E (1) C E B D A (1) C B E D A (1) C A B E D (1) C A B D E (1) B E D C A (1) B D E C A (1) B C D E A (1) A E D C B (1) A D E B C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -6 18 10 B 2 0 -20 8 -2 C 6 20 0 8 0 D -18 -8 -8 0 -6 E -10 2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.807254 D: 0.000000 E: 0.192746 Sum of squares = 0.688809600321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.807254 D: 0.807254 E: 1.000000 A B C D E A 0 -2 -6 18 10 B 2 0 -20 8 -2 C 6 20 0 8 0 D -18 -8 -8 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.375000 Sum of squares = 0.53125010717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=24 E=18 D=18 B=11 so B is eliminated. Round 2 votes counts: C=33 E=24 A=24 D=19 so D is eliminated. Round 3 votes counts: A=41 C=33 E=26 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:210 E:199 B:194 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 18 10 B 2 0 -20 8 -2 C 6 20 0 8 0 D -18 -8 -8 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.375000 Sum of squares = 0.53125010717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 18 10 B 2 0 -20 8 -2 C 6 20 0 8 0 D -18 -8 -8 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.375000 Sum of squares = 0.53125010717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 18 10 B 2 0 -20 8 -2 C 6 20 0 8 0 D -18 -8 -8 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.000000 E: 0.375000 Sum of squares = 0.53125010717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6828: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) C E B A D (6) A C B D E (6) D B E C A (5) B D C A E (5) A E D C B (5) A C E D B (5) C E A B D (4) A E C D B (4) E D B C A (3) B D C E A (3) A D B C E (3) A C B E D (3) E C B D A (2) E B D C A (2) D E B A C (2) D E A B C (2) D B E A C (2) C B E D A (2) C A B E D (2) B D E C A (2) A D E B C (2) A D B E C (2) E C D B A (1) E C D A B (1) E C A B D (1) E B C D A (1) E A D C B (1) E A C D B (1) D B A E C (1) C E B D A (1) C B D A E (1) C B A D E (1) C A E B D (1) C A B D E (1) B C E D A (1) B C D A E (1) A E C B D (1) A D E C B (1) A D C E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 16 6 24 14 B -16 0 -24 14 -14 C -6 24 0 16 16 D -24 -14 -16 0 -14 E -14 14 -16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 24 14 B -16 0 -24 14 -14 C -6 24 0 16 16 D -24 -14 -16 0 -14 E -14 14 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 C=19 E=13 D=12 B=12 so D is eliminated. Round 2 votes counts: A=44 B=20 C=19 E=17 so E is eliminated. Round 3 votes counts: A=48 B=28 C=24 so C is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:225 E:199 B:180 D:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 24 14 B -16 0 -24 14 -14 C -6 24 0 16 16 D -24 -14 -16 0 -14 E -14 14 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 24 14 B -16 0 -24 14 -14 C -6 24 0 16 16 D -24 -14 -16 0 -14 E -14 14 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 24 14 B -16 0 -24 14 -14 C -6 24 0 16 16 D -24 -14 -16 0 -14 E -14 14 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6829: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) D C E B A (8) A C B E D (8) D E B C A (7) C A D B E (7) A B E D C (6) A B E C D (6) C D E B A (5) C D E A B (5) C A D E B (5) A B C E D (5) D E C B A (4) B E D A C (4) B E A D C (4) E B D A C (3) C D A B E (3) E B A D C (2) A C B D E (2) E D C A B (1) E D B C A (1) E D B A C (1) C A B D E (1) B D E C A (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 18 -14 -4 8 B -18 0 -16 -12 -4 C 14 16 0 12 16 D 4 12 -12 0 14 E -8 4 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -14 -4 8 B -18 0 -16 -12 -4 C 14 16 0 12 16 D 4 12 -12 0 14 E -8 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=28 D=19 B=10 E=8 so E is eliminated. Round 2 votes counts: C=35 A=28 D=22 B=15 so B is eliminated. Round 3 votes counts: C=35 A=35 D=30 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:209 A:204 E:183 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -14 -4 8 B -18 0 -16 -12 -4 C 14 16 0 12 16 D 4 12 -12 0 14 E -8 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -14 -4 8 B -18 0 -16 -12 -4 C 14 16 0 12 16 D 4 12 -12 0 14 E -8 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -14 -4 8 B -18 0 -16 -12 -4 C 14 16 0 12 16 D 4 12 -12 0 14 E -8 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6830: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) E C A D B (11) E D C A B (10) D C A E B (7) A C D E B (7) B E A C D (6) B A C E D (6) B D A C E (4) D E C A B (3) D A C E B (3) D A C B E (3) C A D E B (3) B E C A D (3) B D E A C (3) A C D B E (3) E D B C A (2) E B D C A (2) B E D A C (2) B D E C A (2) E C D A B (1) E B C A D (1) D E B C A (1) C A E D B (1) B E D C A (1) A D C E B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 4 12 4 B -10 0 -10 -12 -8 C -4 10 0 12 4 D -12 12 -12 0 6 E -4 8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 12 4 B -10 0 -10 -12 -8 C -4 10 0 12 4 D -12 12 -12 0 6 E -4 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=27 D=17 A=13 C=4 so C is eliminated. Round 2 votes counts: B=39 E=27 D=17 A=17 so D is eliminated. Round 3 votes counts: B=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:215 C:211 D:197 E:197 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 12 4 B -10 0 -10 -12 -8 C -4 10 0 12 4 D -12 12 -12 0 6 E -4 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 12 4 B -10 0 -10 -12 -8 C -4 10 0 12 4 D -12 12 -12 0 6 E -4 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 12 4 B -10 0 -10 -12 -8 C -4 10 0 12 4 D -12 12 -12 0 6 E -4 8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6831: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) A E B D C (10) C D E A B (8) C D B E A (8) B C D E A (8) D C E A B (5) C D E B A (5) B C D A E (5) B A E D C (5) E A D C B (4) E D A C B (3) D E C A B (3) C B D E A (3) A E C D B (3) A B E D C (3) D C E B A (2) B D C E A (2) B A E C D (2) B A C E D (2) E D A B C (1) E C D A B (1) E A D B C (1) C E D A B (1) B D C A E (1) B D A C E (1) B C A D E (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 10 -6 -14 -10 B -10 0 -14 -12 -18 C 6 14 0 -4 4 D 14 12 4 0 4 E 10 18 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 -14 -10 B -10 0 -14 -12 -18 C 6 14 0 -4 4 D 14 12 4 0 4 E 10 18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 C=25 E=10 D=10 so E is eliminated. Round 2 votes counts: A=33 B=27 C=26 D=14 so D is eliminated. Round 3 votes counts: A=37 C=36 B=27 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:210 E:210 A:190 B:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -6 -14 -10 B -10 0 -14 -12 -18 C 6 14 0 -4 4 D 14 12 4 0 4 E 10 18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -14 -10 B -10 0 -14 -12 -18 C 6 14 0 -4 4 D 14 12 4 0 4 E 10 18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -14 -10 B -10 0 -14 -12 -18 C 6 14 0 -4 4 D 14 12 4 0 4 E 10 18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6832: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) D C B E A (8) D C E B A (6) B E A D C (6) A C E D B (6) E B A D C (5) E A B C D (5) C D A B E (5) A E B C D (5) B D E C A (4) A C D B E (4) E B A C D (3) C D E A B (3) A C D E B (3) D B E C A (2) B E D A C (2) A B E D C (2) E D B C A (1) E C A D B (1) E B D C A (1) E B D A C (1) E B C D A (1) E A C B D (1) D E C B A (1) D E B C A (1) D C A B E (1) D B C E A (1) C D E B A (1) C A D B E (1) B E D C A (1) B A E D C (1) A E C B D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 2 -10 B -8 0 -6 -8 -10 C -4 6 0 8 6 D -2 8 -8 0 8 E 10 10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000033 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 8 4 2 -10 B -8 0 -6 -8 -10 C -4 6 0 8 6 D -2 8 -8 0 8 E 10 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000003 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=20 E=19 C=19 B=14 so B is eliminated. Round 2 votes counts: A=29 E=28 D=24 C=19 so C is eliminated. Round 3 votes counts: D=42 A=30 E=28 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:208 D:203 E:203 A:202 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 4 2 -10 B -8 0 -6 -8 -10 C -4 6 0 8 6 D -2 8 -8 0 8 E 10 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000003 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 -10 B -8 0 -6 -8 -10 C -4 6 0 8 6 D -2 8 -8 0 8 E 10 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000003 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 -10 B -8 0 -6 -8 -10 C -4 6 0 8 6 D -2 8 -8 0 8 E 10 10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000003 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6833: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) A C E B D (7) A C B E D (7) B D E A C (6) D E B C A (5) D B E C A (5) E D B C A (4) C A E D B (4) E D C A B (3) E C D A B (3) E A C D B (3) D B E A C (3) C E A D B (3) C A B D E (3) B D A E C (3) A C E D B (3) A B C D E (3) E C A D B (2) D C B E A (2) C A E B D (2) C A D E B (2) B A C D E (2) E D C B A (1) E A B D C (1) D C E B A (1) C E D A B (1) C D E A B (1) C A D B E (1) B E D A C (1) B D E C A (1) B D C A E (1) B D A C E (1) B C D A E (1) B A E D C (1) B A D C E (1) A E C B D (1) A E B C D (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 8 0 -6 B -8 0 -2 -8 -10 C -8 2 0 6 -4 D 0 8 -6 0 -14 E 6 10 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 0 -6 B -8 0 -2 -8 -10 C -8 2 0 6 -4 D 0 8 -6 0 -14 E 6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 B=18 C=17 D=16 so D is eliminated. Round 2 votes counts: E=29 B=26 A=25 C=20 so C is eliminated. Round 3 votes counts: A=37 E=35 B=28 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:205 C:198 D:194 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 0 -6 B -8 0 -2 -8 -10 C -8 2 0 6 -4 D 0 8 -6 0 -14 E 6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 0 -6 B -8 0 -2 -8 -10 C -8 2 0 6 -4 D 0 8 -6 0 -14 E 6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 0 -6 B -8 0 -2 -8 -10 C -8 2 0 6 -4 D 0 8 -6 0 -14 E 6 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6834: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) B D A C E (8) E D C A B (7) E C A D B (7) D E B C A (6) E D C B A (5) D E C B A (4) D E C A B (4) C A E B D (4) A C B E D (4) E C D A B (3) C E A B D (3) B A C D E (3) A B C E D (3) A B C D E (3) D E B A C (2) D B A E C (2) D B A C E (2) C A B E D (2) B D E A C (2) B A D C E (2) A D C B E (2) E D B C A (1) E C A B D (1) E B C A D (1) D B E C A (1) D A C E B (1) D A C B E (1) D A B C E (1) C E A D B (1) C A E D B (1) B A C E D (1) A C E D B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -2 -18 -14 B -2 0 -6 -22 -6 C 2 6 0 -20 -10 D 18 22 20 0 8 E 14 6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -18 -14 B -2 0 -6 -22 -6 C 2 6 0 -20 -10 D 18 22 20 0 8 E 14 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=25 B=16 A=15 C=11 so C is eliminated. Round 2 votes counts: D=33 E=29 A=22 B=16 so B is eliminated. Round 3 votes counts: D=43 E=29 A=28 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:234 E:211 C:189 A:184 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -18 -14 B -2 0 -6 -22 -6 C 2 6 0 -20 -10 D 18 22 20 0 8 E 14 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -18 -14 B -2 0 -6 -22 -6 C 2 6 0 -20 -10 D 18 22 20 0 8 E 14 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -18 -14 B -2 0 -6 -22 -6 C 2 6 0 -20 -10 D 18 22 20 0 8 E 14 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6835: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) C B D A E (8) B C E D A (7) D A E B C (6) C B A D E (5) A D E C B (5) E D A B C (4) B C D A E (4) D E A B C (3) C B E D A (3) C A D B E (3) B E C D A (3) A D C E B (3) A D C B E (3) E C B A D (2) E C A B D (2) E B C D A (2) E B C A D (2) E A D B C (2) D B C A E (2) D A C B E (2) D A B C E (2) A E D C B (2) E D B A C (1) E B D C A (1) E B D A C (1) D B A C E (1) D A C E B (1) C E B A D (1) C E A B D (1) C B D E A (1) C B A E D (1) C A E B D (1) C A B E D (1) B D C A E (1) A E C B D (1) A D E B C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -20 -6 10 B 10 0 -14 14 14 C 20 14 0 18 24 D 6 -14 -18 0 6 E -10 -14 -24 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 -6 10 B 10 0 -14 14 14 C 20 14 0 18 24 D 6 -14 -18 0 6 E -10 -14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=17 D=17 A=17 B=15 so B is eliminated. Round 2 votes counts: C=45 E=20 D=18 A=17 so A is eliminated. Round 3 votes counts: C=47 D=30 E=23 so E is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:238 B:212 D:190 A:187 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -20 -6 10 B 10 0 -14 14 14 C 20 14 0 18 24 D 6 -14 -18 0 6 E -10 -14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 -6 10 B 10 0 -14 14 14 C 20 14 0 18 24 D 6 -14 -18 0 6 E -10 -14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 -6 10 B 10 0 -14 14 14 C 20 14 0 18 24 D 6 -14 -18 0 6 E -10 -14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6836: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (14) B D E A C (10) D B E A C (9) C A E B D (8) B C D E A (8) D E A B C (6) C B A E D (6) A E D C B (6) C B D A E (4) B D C E A (4) E A C D B (3) C A E D B (3) A C E D B (3) E D A B C (2) E A D B C (2) D B A E C (2) C B E A D (2) A E D B C (2) E A D C B (1) D A E B C (1) C B D E A (1) C A B E D (1) B D C A E (1) B C D A E (1) Total count = 100 A B C D E A 0 4 22 2 4 B -4 0 -4 -8 -2 C -22 4 0 8 -16 D -2 8 -8 0 -6 E -4 2 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 22 2 4 B -4 0 -4 -8 -2 C -22 4 0 8 -16 D -2 8 -8 0 -6 E -4 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=25 A=25 B=24 D=18 E=8 so E is eliminated. Round 2 votes counts: A=31 C=25 B=24 D=20 so D is eliminated. Round 3 votes counts: A=40 B=35 C=25 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:210 D:196 B:191 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 22 2 4 B -4 0 -4 -8 -2 C -22 4 0 8 -16 D -2 8 -8 0 -6 E -4 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 22 2 4 B -4 0 -4 -8 -2 C -22 4 0 8 -16 D -2 8 -8 0 -6 E -4 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 22 2 4 B -4 0 -4 -8 -2 C -22 4 0 8 -16 D -2 8 -8 0 -6 E -4 2 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6837: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) E C D B A (7) C E B A D (7) D A B E C (6) E C B A D (5) D A E B C (4) C E D B A (4) B A C D E (4) A D B E C (4) E C D A B (3) D E A B C (3) E D A C B (2) E C A B D (2) D E C A B (2) D E A C B (2) C E B D A (2) C D B E A (2) C B A E D (2) B C A E D (2) B C A D E (2) B A D C E (2) B A C E D (2) A D B C E (2) A B C E D (2) A B C D E (2) E D C B A (1) E D C A B (1) E C B D A (1) E A C D B (1) D E C B A (1) D C E B A (1) D B A C E (1) D A B C E (1) C D E B A (1) C B E A D (1) C B D E A (1) C B A D E (1) B C D A E (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 6 0 B 2 0 0 0 -2 C 0 0 0 12 6 D -6 0 -12 0 8 E 0 2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.611953 C: 0.388047 D: 0.000000 E: 0.000000 Sum of squares = 0.525066790671 Cumulative probabilities = A: 0.000000 B: 0.611953 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 6 0 B 2 0 0 0 -2 C 0 0 0 12 6 D -6 0 -12 0 8 E 0 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=22 D=21 C=21 B=13 so B is eliminated. Round 2 votes counts: A=30 C=26 E=23 D=21 so D is eliminated. Round 3 votes counts: A=42 E=31 C=27 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:209 A:202 B:200 D:195 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 6 0 B 2 0 0 0 -2 C 0 0 0 12 6 D -6 0 -12 0 8 E 0 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 6 0 B 2 0 0 0 -2 C 0 0 0 12 6 D -6 0 -12 0 8 E 0 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 6 0 B 2 0 0 0 -2 C 0 0 0 12 6 D -6 0 -12 0 8 E 0 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6838: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) B D E A C (8) B A C D E (7) D E A C B (6) A C B D E (6) B A D C E (5) B E D A C (4) E D C A B (3) E C D A B (3) D A E B C (3) C A E D B (3) C A D E B (3) C A B D E (3) B D A E C (3) E D C B A (2) E D B C A (2) E B D C A (2) D A E C B (2) C B A E D (2) C A B E D (2) B E D C A (2) B E C D A (2) B A D E C (2) A D C E B (2) A C D B E (2) E C B D A (1) E B D A C (1) D A C E B (1) D A B E C (1) C E D A B (1) C A D B E (1) B E A C D (1) B C E D A (1) A D C B E (1) A D B E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 34 -14 4 B 8 0 12 6 12 C -34 -12 0 -24 -16 D 14 -6 24 0 18 E -4 -12 16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 34 -14 4 B 8 0 12 6 12 C -34 -12 0 -24 -16 D 14 -6 24 0 18 E -4 -12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=23 C=15 A=14 D=13 so D is eliminated. Round 2 votes counts: B=35 E=29 A=21 C=15 so C is eliminated. Round 3 votes counts: B=37 A=33 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:225 B:219 A:208 E:191 C:157 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 34 -14 4 B 8 0 12 6 12 C -34 -12 0 -24 -16 D 14 -6 24 0 18 E -4 -12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 34 -14 4 B 8 0 12 6 12 C -34 -12 0 -24 -16 D 14 -6 24 0 18 E -4 -12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 34 -14 4 B 8 0 12 6 12 C -34 -12 0 -24 -16 D 14 -6 24 0 18 E -4 -12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998167 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6839: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D B C A E (7) D B A C E (7) E A C B D (6) D B C E A (6) B C D E A (5) A D E B C (5) E C B A D (4) E C A B D (4) D A B C E (4) A D B C E (4) E C B D A (3) E A D C B (3) C D B E A (3) C B D E A (3) B C D A E (3) A E B C D (3) E A C D B (2) D E C B A (2) B D C E A (2) A E D C B (2) A E D B C (2) A D B E C (2) E D C B A (1) D C B E A (1) C E B D A (1) C E B A D (1) C B E D A (1) A E C D B (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 2 4 B 0 0 6 -4 2 C -6 -6 0 2 0 D -2 4 -2 0 12 E -4 -2 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.845243 B: 0.154757 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.73838516795 Cumulative probabilities = A: 0.845243 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 2 4 B 0 0 6 -4 2 C -6 -6 0 2 0 D -2 4 -2 0 12 E -4 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564788 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=27 E=23 B=10 C=9 so C is eliminated. Round 2 votes counts: A=31 D=30 E=25 B=14 so B is eliminated. Round 3 votes counts: D=43 A=31 E=26 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:206 D:206 B:202 C:195 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 2 4 B 0 0 6 -4 2 C -6 -6 0 2 0 D -2 4 -2 0 12 E -4 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564788 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 2 4 B 0 0 6 -4 2 C -6 -6 0 2 0 D -2 4 -2 0 12 E -4 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564788 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 2 4 B 0 0 6 -4 2 C -6 -6 0 2 0 D -2 4 -2 0 12 E -4 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564788 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6840: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) E B A D C (6) B A E C D (5) E B D C A (4) D E A C B (4) C A B D E (4) B E A C D (4) A C B D E (4) E D B C A (3) E B D A C (3) D E C B A (3) D E C A B (3) D C E A B (3) D C A E B (3) C D A B E (3) C A D B E (3) B E C D A (3) B A C E D (3) A B C E D (3) E D B A C (2) C D A E B (2) C B A E D (2) C B A D E (2) B E C A D (2) B E A D C (2) A D E B C (2) E B C D A (1) E A B D C (1) D E A B C (1) D A E C B (1) D A C E B (1) C D E B A (1) C D B A E (1) C B E D A (1) C A B E D (1) B C A E D (1) A D C E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 14 6 B -2 0 -6 8 10 C -8 6 0 14 0 D -14 -8 -14 0 4 E -6 -10 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 14 6 B -2 0 -6 8 10 C -8 6 0 14 0 D -14 -8 -14 0 4 E -6 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=21 E=20 C=20 B=20 D=19 so D is eliminated. Round 2 votes counts: E=31 C=26 A=23 B=20 so B is eliminated. Round 3 votes counts: E=42 A=31 C=27 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:206 B:205 E:190 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 14 6 B -2 0 -6 8 10 C -8 6 0 14 0 D -14 -8 -14 0 4 E -6 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 14 6 B -2 0 -6 8 10 C -8 6 0 14 0 D -14 -8 -14 0 4 E -6 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 14 6 B -2 0 -6 8 10 C -8 6 0 14 0 D -14 -8 -14 0 4 E -6 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6841: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) B D E C A (6) D E C A B (5) D C E A B (5) C E A D B (5) A B D C E (5) D B E C A (4) D B A E C (4) B D A E C (4) A C E B D (4) A C B E D (4) D E C B A (3) C E A B D (3) C A E D B (3) B A E C D (3) B A C E D (3) A C D E B (3) A B C E D (3) E B C D A (2) D A B C E (2) C E D A B (2) B E D C A (2) B A C D E (2) E C D A B (1) E C B A D (1) E C A D B (1) E B C A D (1) D E B C A (1) D A C E B (1) D A B E C (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A E D (1) B A D E C (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 2 12 8 B -14 0 2 2 2 C -2 -2 0 8 14 D -12 -2 -8 0 -2 E -8 -2 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 12 8 B -14 0 2 2 2 C -2 -2 0 8 14 D -12 -2 -8 0 -2 E -8 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994819 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 D=26 C=13 E=6 so E is eliminated. Round 2 votes counts: B=30 A=28 D=26 C=16 so C is eliminated. Round 3 votes counts: A=40 B=31 D=29 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:209 B:196 E:189 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 12 8 B -14 0 2 2 2 C -2 -2 0 8 14 D -12 -2 -8 0 -2 E -8 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994819 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 12 8 B -14 0 2 2 2 C -2 -2 0 8 14 D -12 -2 -8 0 -2 E -8 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994819 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 12 8 B -14 0 2 2 2 C -2 -2 0 8 14 D -12 -2 -8 0 -2 E -8 -2 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994819 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6842: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (7) E C B A D (6) D A C E B (5) D A B C E (5) B E C D A (5) A E D C B (4) A E C D B (4) A D E C B (4) E C A B D (3) D B A C E (3) C E A D B (3) B E C A D (3) B D A E C (3) B C E D A (3) A D B E C (3) E C B D A (2) E B C A D (2) D B C A E (2) C E B D A (2) C D E A B (2) B D C E A (2) B D C A E (2) B C D E A (2) B A D E C (2) A E B D C (2) A B E D C (2) E C A D B (1) E B A C D (1) D C A E B (1) D B C E A (1) D A C B E (1) C E D B A (1) C E D A B (1) C B D E A (1) B E D C A (1) B E A D C (1) B E A C D (1) B D E C A (1) B D E A C (1) B A E D C (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 2 6 6 8 B -2 0 0 2 -8 C -6 0 0 -8 -10 D -6 -2 8 0 -4 E -8 8 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 6 8 B -2 0 0 2 -8 C -6 0 0 -8 -10 D -6 -2 8 0 -4 E -8 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 D=18 E=15 C=10 so C is eliminated. Round 2 votes counts: B=29 A=29 E=22 D=20 so D is eliminated. Round 3 votes counts: A=41 B=35 E=24 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:207 D:198 B:196 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 6 8 B -2 0 0 2 -8 C -6 0 0 -8 -10 D -6 -2 8 0 -4 E -8 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 6 8 B -2 0 0 2 -8 C -6 0 0 -8 -10 D -6 -2 8 0 -4 E -8 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 6 8 B -2 0 0 2 -8 C -6 0 0 -8 -10 D -6 -2 8 0 -4 E -8 8 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6843: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) A C E D B (11) D E C A B (8) C A E D B (7) B A E D C (5) B D E C A (4) D B E C A (3) B E A D C (3) B D A E C (3) B A C E D (3) A C E B D (3) E D A C B (2) D E B C A (2) D C E A B (2) D B C E A (2) C D E A B (2) B D E A C (2) B C A E D (2) B C A D E (2) B A E C D (2) A E C D B (2) A B E C D (2) E D C A B (1) E A D C B (1) E A D B C (1) E A C D B (1) D E C B A (1) D E B A C (1) D C E B A (1) D B E A C (1) C E A D B (1) C D A E B (1) C B D A E (1) C B A E D (1) C B A D E (1) B A D E C (1) A E D C B (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -6 4 0 B 4 0 4 -4 0 C 6 -4 0 -16 2 D -4 4 16 0 -2 E 0 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.095624 C: 0.071313 D: 0.071313 E: 0.761750 Sum of squares = 0.599578737453 Cumulative probabilities = A: 0.000000 B: 0.095624 C: 0.166937 D: 0.238250 E: 1.000000 A B C D E A 0 -4 -6 4 0 B 4 0 4 -4 0 C 6 -4 0 -16 2 D -4 4 16 0 -2 E 0 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.283581 C: 0.014926 D: 0.014926 E: 0.686568 Sum of squares = 0.552238805976 Cumulative probabilities = A: 0.000000 B: 0.283581 C: 0.298506 D: 0.313432 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=21 A=21 C=14 E=6 so E is eliminated. Round 2 votes counts: B=38 D=24 A=24 C=14 so C is eliminated. Round 3 votes counts: B=41 A=32 D=27 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:207 B:202 E:200 A:197 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 4 0 B 4 0 4 -4 0 C 6 -4 0 -16 2 D -4 4 16 0 -2 E 0 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.283581 C: 0.014926 D: 0.014926 E: 0.686568 Sum of squares = 0.552238805976 Cumulative probabilities = A: 0.000000 B: 0.283581 C: 0.298506 D: 0.313432 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 0 B 4 0 4 -4 0 C 6 -4 0 -16 2 D -4 4 16 0 -2 E 0 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.283581 C: 0.014926 D: 0.014926 E: 0.686568 Sum of squares = 0.552238805976 Cumulative probabilities = A: 0.000000 B: 0.283581 C: 0.298506 D: 0.313432 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 0 B 4 0 4 -4 0 C 6 -4 0 -16 2 D -4 4 16 0 -2 E 0 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.283581 C: 0.014926 D: 0.014926 E: 0.686568 Sum of squares = 0.552238805976 Cumulative probabilities = A: 0.000000 B: 0.283581 C: 0.298506 D: 0.313432 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6844: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D C A E B (9) E C B D A (8) D A C B E (8) A B D E C (8) C D E B A (6) E B A C D (5) D C E A B (5) B E A C D (5) E C D B A (4) A B E D C (4) B A E C D (3) D C A B E (2) C E D B A (2) A E B D C (2) A D C B E (2) A D B C E (2) A B E C D (2) A B D C E (2) E D C B A (1) E D C A B (1) E B C D A (1) E B A D C (1) D E C A B (1) D C E B A (1) D A C E B (1) C B E D A (1) B E C A D (1) B A C D E (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -4 -2 -4 B -2 0 -4 6 -16 C 4 4 0 -4 -16 D 2 -6 4 0 0 E 4 16 16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.528333 E: 0.471667 Sum of squares = 0.501605461231 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.528333 E: 1.000000 A B C D E A 0 2 -4 -2 -4 B -2 0 -4 6 -16 C 4 4 0 -4 -16 D 2 -6 4 0 0 E 4 16 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 A=24 B=10 C=9 so C is eliminated. Round 2 votes counts: D=33 E=32 A=24 B=11 so B is eliminated. Round 3 votes counts: E=39 D=33 A=28 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:218 D:200 A:196 C:194 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 -2 -4 B -2 0 -4 6 -16 C 4 4 0 -4 -16 D 2 -6 4 0 0 E 4 16 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 -4 B -2 0 -4 6 -16 C 4 4 0 -4 -16 D 2 -6 4 0 0 E 4 16 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 -4 B -2 0 -4 6 -16 C 4 4 0 -4 -16 D 2 -6 4 0 0 E 4 16 16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6845: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E A B C D (7) D C B A E (6) D B C A E (6) E A D C B (5) C B E A D (5) E A D B C (4) C B D E A (4) B C D A E (4) D C B E A (3) B C A E D (3) A E B C D (3) E D A C B (2) E C D B A (2) E C D A B (2) E C B A D (2) E C A B D (2) E A B D C (2) D E A B C (2) C E B A D (2) C D B E A (2) C B E D A (2) B D C A E (2) A E D B C (2) A B E C D (2) E C B D A (1) E C A D B (1) E A C D B (1) E A C B D (1) D C E B A (1) D A B E C (1) D A B C E (1) C E B D A (1) C D B A E (1) C B A E D (1) B C E A D (1) B C A D E (1) B A D C E (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -20 -28 -4 -10 B 20 0 -12 14 10 C 28 12 0 20 12 D 4 -14 -20 0 -10 E 10 -10 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -28 -4 -10 B 20 0 -12 14 10 C 28 12 0 20 12 D 4 -14 -20 0 -10 E 10 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=27 D=20 B=12 A=9 so A is eliminated. Round 2 votes counts: E=38 C=27 D=21 B=14 so B is eliminated. Round 3 votes counts: E=40 C=36 D=24 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:236 B:216 E:199 D:180 A:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -28 -4 -10 B 20 0 -12 14 10 C 28 12 0 20 12 D 4 -14 -20 0 -10 E 10 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -28 -4 -10 B 20 0 -12 14 10 C 28 12 0 20 12 D 4 -14 -20 0 -10 E 10 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -28 -4 -10 B 20 0 -12 14 10 C 28 12 0 20 12 D 4 -14 -20 0 -10 E 10 -10 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6846: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) D A E C B (9) A E D C B (9) C B D A E (6) D C A E B (5) C D B A E (5) B C D E A (5) B C E A D (4) B C A E D (4) E A B C D (3) D E A B C (3) B E A D C (3) B C D A E (3) A E C B D (3) E A B D C (2) D E B A C (2) D E A C B (2) C D A E B (2) C D A B E (2) B E A C D (2) B D E A C (2) E B A D C (1) E A D C B (1) D C B A E (1) D B C E A (1) D A C E B (1) C B A D E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C A D (1) B D C E A (1) A E C D B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 14 14 -2 12 B -14 0 -2 -12 -16 C -14 2 0 -8 -12 D 2 12 8 0 4 E -12 16 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 -2 12 B -14 0 -2 -12 -16 C -14 2 0 -8 -12 D 2 12 8 0 4 E -12 16 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=24 C=18 E=17 A=15 so A is eliminated. Round 2 votes counts: E=31 B=26 D=24 C=19 so C is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:219 D:213 E:206 C:184 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 14 -2 12 B -14 0 -2 -12 -16 C -14 2 0 -8 -12 D 2 12 8 0 4 E -12 16 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 -2 12 B -14 0 -2 -12 -16 C -14 2 0 -8 -12 D 2 12 8 0 4 E -12 16 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 -2 12 B -14 0 -2 -12 -16 C -14 2 0 -8 -12 D 2 12 8 0 4 E -12 16 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6847: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C B A D E (9) B C A D E (9) D E A C B (8) E D A C B (6) E B C D A (4) E D C A B (3) E B D A C (3) C D A B E (3) B C A E D (3) B A C E D (3) B A C D E (3) E D C B A (2) E D B C A (2) E D B A C (2) D A C E B (2) C D B A E (2) B E C A D (2) A D C B E (2) A C B D E (2) A B C D E (2) E C D B A (1) E B D C A (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) C A D B E (1) B C E A D (1) B A E C D (1) A D E B C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 6 -12 -4 B -2 0 8 -6 -8 C -6 -8 0 -8 -8 D 12 6 8 0 2 E 4 8 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -12 -4 B -2 0 8 -6 -8 C -6 -8 0 -8 -8 D 12 6 8 0 2 E 4 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=22 D=15 C=15 A=9 so A is eliminated. Round 2 votes counts: E=39 B=26 D=18 C=17 so C is eliminated. Round 3 votes counts: E=39 B=37 D=24 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:214 E:209 A:196 B:196 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -12 -4 B -2 0 8 -6 -8 C -6 -8 0 -8 -8 D 12 6 8 0 2 E 4 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -12 -4 B -2 0 8 -6 -8 C -6 -8 0 -8 -8 D 12 6 8 0 2 E 4 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -12 -4 B -2 0 8 -6 -8 C -6 -8 0 -8 -8 D 12 6 8 0 2 E 4 8 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6848: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) C A B E D (7) A C B E D (7) D E A B C (5) C B E D A (5) E D B C A (4) D E B A C (4) C D B E A (4) A E D B C (4) A E B C D (4) A C D B E (4) B E D C A (3) A D E B C (3) A C E B D (3) E B D C A (2) E B D A C (2) D C B E A (2) D B E C A (2) D A C E B (2) D A C B E (2) C D A B E (2) C B A E D (2) B E C D A (2) A E B D C (2) E B C A D (1) E B A D C (1) D E C B A (1) D A E C B (1) D A E B C (1) C B E A D (1) A E C B D (1) A D E C B (1) A D C E B (1) A D C B E (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 -10 0 B -6 0 4 -12 -10 C -2 -4 0 -10 -10 D 10 12 10 0 -4 E 0 10 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.167021 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.832979 Sum of squares = 0.721750116607 Cumulative probabilities = A: 0.167021 B: 0.167021 C: 0.167021 D: 0.167021 E: 1.000000 A B C D E A 0 6 2 -10 0 B -6 0 4 -12 -10 C -2 -4 0 -10 -10 D 10 12 10 0 -4 E 0 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836793544 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=31 C=21 E=10 B=5 so B is eliminated. Round 2 votes counts: A=33 D=31 C=21 E=15 so E is eliminated. Round 3 votes counts: D=42 A=34 C=24 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:212 A:199 B:188 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 -10 0 B -6 0 4 -12 -10 C -2 -4 0 -10 -10 D 10 12 10 0 -4 E 0 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836793544 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -10 0 B -6 0 4 -12 -10 C -2 -4 0 -10 -10 D 10 12 10 0 -4 E 0 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836793544 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -10 0 B -6 0 4 -12 -10 C -2 -4 0 -10 -10 D 10 12 10 0 -4 E 0 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836793544 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6849: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) D B E A C (7) E C D A B (4) E C A D B (4) C E A D B (4) C B A D E (4) C A E B D (4) B D A C E (4) A E D C B (4) E D C B A (3) E A D C B (3) D E B A C (3) D B A E C (3) C E A B D (3) C B D E A (3) C A B E D (3) B A D C E (3) A E D B C (3) C E D B A (2) C E B D A (2) C B D A E (2) C A B D E (2) B C D A E (2) B A C D E (2) A B D E C (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A B C (1) E A D B C (1) D E B C A (1) D A E B C (1) C E D A B (1) C B E D A (1) B D C E A (1) B C D E A (1) A D E B C (1) A C E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 2 -8 4 B 10 0 -4 2 2 C -2 4 0 -6 -6 D 8 -2 6 0 4 E -4 -2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -8 4 B 10 0 -4 2 2 C -2 4 0 -6 -6 D 8 -2 6 0 4 E -4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888536 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=22 E=19 D=15 A=13 so A is eliminated. Round 2 votes counts: C=33 E=26 B=25 D=16 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:208 B:205 E:198 C:195 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 -8 4 B 10 0 -4 2 2 C -2 4 0 -6 -6 D 8 -2 6 0 4 E -4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888536 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -8 4 B 10 0 -4 2 2 C -2 4 0 -6 -6 D 8 -2 6 0 4 E -4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888536 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -8 4 B 10 0 -4 2 2 C -2 4 0 -6 -6 D 8 -2 6 0 4 E -4 -2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888536 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6850: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) B C A D E (10) D E C B A (9) E D A B C (7) C B A D E (7) B A C D E (7) E D C A B (6) E D A C B (6) A E B D C (6) A E D B C (4) D E C A B (3) A B C E D (3) D E B A C (2) C D E B A (2) C D B E A (2) C B D A E (2) C B A E D (2) A C B E D (2) D E B C A (1) D C E B A (1) D C B E A (1) D B C E A (1) C B D E A (1) B D A C E (1) B C D A E (1) B A D C E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 2 8 -4 B 2 0 12 -10 -14 C -2 -12 0 -20 -12 D -8 10 20 0 4 E 4 14 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -2 2 8 -4 B 2 0 12 -10 -14 C -2 -12 0 -20 -12 D -8 10 20 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000425 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=20 D=18 A=17 C=16 so C is eliminated. Round 2 votes counts: B=32 E=29 D=22 A=17 so A is eliminated. Round 3 votes counts: E=39 B=39 D=22 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:213 A:202 B:195 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 8 -4 B 2 0 12 -10 -14 C -2 -12 0 -20 -12 D -8 10 20 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000425 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 8 -4 B 2 0 12 -10 -14 C -2 -12 0 -20 -12 D -8 10 20 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000425 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 8 -4 B 2 0 12 -10 -14 C -2 -12 0 -20 -12 D -8 10 20 0 4 E 4 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000425 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6851: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) D A E B C (7) D B C E A (6) A D E C B (5) A C E B D (5) E A B C D (4) D C B E A (4) A E B C D (4) D C B A E (3) D C A B E (3) D B E C A (3) D A C B E (3) C B E D A (3) C B D E A (3) A E D B C (3) A E C B D (3) E B D C A (2) D E B A C (2) C B E A D (2) C A E B D (2) B C E D A (2) A E D C B (2) A D E B C (2) A D C E B (2) A D C B E (2) E D B A C (1) E C B A D (1) E B C D A (1) E B A C D (1) E A C B D (1) D E B C A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E B A D (1) C A D B E (1) C A B E D (1) B D C E A (1) B C D E A (1) A E B D C (1) A C D B E (1) Total count = 100 A B C D E A 0 4 2 2 4 B -4 0 4 -8 -16 C -2 -4 0 -12 -6 D -2 8 12 0 6 E -4 16 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 4 B -4 0 4 -8 -16 C -2 -4 0 -12 -6 D -2 8 12 0 6 E -4 16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995413 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=30 E=18 C=13 B=4 so B is eliminated. Round 2 votes counts: D=36 A=30 E=18 C=16 so C is eliminated. Round 3 votes counts: D=40 A=34 E=26 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:206 E:206 B:188 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 4 B -4 0 4 -8 -16 C -2 -4 0 -12 -6 D -2 8 12 0 6 E -4 16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995413 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 4 B -4 0 4 -8 -16 C -2 -4 0 -12 -6 D -2 8 12 0 6 E -4 16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995413 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 4 B -4 0 4 -8 -16 C -2 -4 0 -12 -6 D -2 8 12 0 6 E -4 16 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995413 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6852: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) A E C B D (9) E C A B D (8) D B C E A (7) B D C E A (6) D B A E C (5) C E A B D (5) B C A E D (5) E A C B D (4) E C A D B (3) D B C A E (3) D B A C E (3) C B E A D (3) A E C D B (3) A C E B D (3) E C D A B (2) D E C A B (2) D E B C A (2) D A E C B (2) C A E B D (2) B D C A E (2) B C D A E (2) E D C A B (1) D E C B A (1) D E A B C (1) D B E C A (1) C E B A D (1) C B A E D (1) B D A C E (1) B C D E A (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -16 16 -14 B -12 0 -20 8 -18 C 16 20 0 24 -8 D -16 -8 -24 0 -22 E 14 18 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -16 16 -14 B -12 0 -20 8 -18 C 16 20 0 24 -8 D -16 -8 -24 0 -22 E 14 18 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=27 D=27 B=17 A=17 C=12 so C is eliminated. Round 2 votes counts: E=33 D=27 B=21 A=19 so A is eliminated. Round 3 votes counts: E=51 D=27 B=22 so B is eliminated. Round 4 votes counts: E=61 D=39 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:231 C:226 A:199 B:179 D:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -16 16 -14 B -12 0 -20 8 -18 C 16 20 0 24 -8 D -16 -8 -24 0 -22 E 14 18 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -16 16 -14 B -12 0 -20 8 -18 C 16 20 0 24 -8 D -16 -8 -24 0 -22 E 14 18 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -16 16 -14 B -12 0 -20 8 -18 C 16 20 0 24 -8 D -16 -8 -24 0 -22 E 14 18 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6853: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) C E D B A (7) E C A B D (6) E B A C D (6) E A B C D (6) C D A E B (6) D C A B E (5) C D E B A (5) C D E A B (5) B E A D C (5) C E D A B (4) B A E D C (4) B A D E C (4) A B E D C (4) D C B A E (3) E C B D A (2) E C B A D (2) A E B C D (2) A D C B E (2) A D B C E (2) A B D E C (2) E C D B A (1) E C D A B (1) E B C A D (1) E A C B D (1) D B E C A (1) D B A C E (1) D A B C E (1) B A D C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -14 0 -22 B 2 0 -18 -6 -12 C 14 18 0 12 4 D 0 6 -12 0 -4 E 22 12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 0 -22 B 2 0 -18 -6 -12 C 14 18 0 12 4 D 0 6 -12 0 -4 E 22 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=26 D=19 B=14 A=14 so B is eliminated. Round 2 votes counts: E=31 C=27 A=23 D=19 so D is eliminated. Round 3 votes counts: C=43 E=32 A=25 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:217 D:195 B:183 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 0 -22 B 2 0 -18 -6 -12 C 14 18 0 12 4 D 0 6 -12 0 -4 E 22 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 0 -22 B 2 0 -18 -6 -12 C 14 18 0 12 4 D 0 6 -12 0 -4 E 22 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 0 -22 B 2 0 -18 -6 -12 C 14 18 0 12 4 D 0 6 -12 0 -4 E 22 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6854: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (8) E B C D A (4) E B C A D (4) D C B E A (4) C D B E A (4) B E C D A (4) A D B E C (4) A C D E B (4) E C B D A (3) E C B A D (3) E C A B D (3) E B A C D (3) C D E B A (3) C D A B E (3) C A D E B (3) B E A D C (3) B D E C A (3) B D A E C (3) A D C E B (3) A C E D B (3) A B D E C (3) C E D B A (2) C D E A B (2) C D A E B (2) C A E D B (2) B E D C A (2) A E B C D (2) A D B C E (2) D C A B E (1) D B C E A (1) D B A E C (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) B E D A C (1) B D E A C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -10 4 -4 B 0 0 -12 -14 4 C 10 12 0 12 4 D -4 14 -12 0 16 E 4 -4 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 4 -4 B 0 0 -12 -14 4 C 10 12 0 12 4 D -4 14 -12 0 16 E 4 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=23 E=20 B=17 D=10 so D is eliminated. Round 2 votes counts: A=32 C=28 E=20 B=20 so E is eliminated. Round 3 votes counts: C=37 A=32 B=31 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:219 D:207 A:195 E:190 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 4 -4 B 0 0 -12 -14 4 C 10 12 0 12 4 D -4 14 -12 0 16 E 4 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 4 -4 B 0 0 -12 -14 4 C 10 12 0 12 4 D -4 14 -12 0 16 E 4 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 4 -4 B 0 0 -12 -14 4 C 10 12 0 12 4 D -4 14 -12 0 16 E 4 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999555 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6855: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) D A B C E (8) A B D E C (8) B A D C E (7) E C D A B (5) C E B D A (5) D A B E C (4) E B C A D (3) D C E A B (3) D A C B E (3) C E B A D (3) B A E C D (3) E C D B A (2) E B A C D (2) D C A B E (2) D B A C E (2) D A E B C (2) D A C E B (2) C E D A B (2) B E A C D (2) B C A E D (2) B A D E C (2) A D B E C (2) A B D C E (2) E C A D B (1) E C A B D (1) E A B D C (1) D C A E B (1) D B C A E (1) C D E B A (1) C D E A B (1) C D B A E (1) C B E D A (1) C B D E A (1) B C E A D (1) B A E D C (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 6 6 10 B 2 0 10 14 10 C -6 -10 0 -4 0 D -6 -14 4 0 6 E -10 -10 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 6 10 B 2 0 10 14 10 C -6 -10 0 -4 0 D -6 -14 4 0 6 E -10 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=25 B=19 C=15 A=13 so A is eliminated. Round 2 votes counts: D=30 B=30 E=25 C=15 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:218 A:210 D:195 C:190 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 6 10 B 2 0 10 14 10 C -6 -10 0 -4 0 D -6 -14 4 0 6 E -10 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 10 B 2 0 10 14 10 C -6 -10 0 -4 0 D -6 -14 4 0 6 E -10 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 10 B 2 0 10 14 10 C -6 -10 0 -4 0 D -6 -14 4 0 6 E -10 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6856: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) E D A C B (10) C A B D E (8) D E A B C (7) B E C D A (7) E D B A C (6) B C A E D (6) A C D E B (5) E B D C A (4) D E A C B (4) E D A B C (3) D A E C B (3) C A D B E (3) B E D C A (3) D A C E B (2) B C E A D (2) E C A B D (1) E A D C B (1) D E B A C (1) D B E A C (1) D B A E C (1) D B A C E (1) C B A E D (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D E A (1) B C D A E (1) A D C E B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -4 -14 -8 B 0 0 18 -2 2 C 4 -18 0 2 -8 D 14 2 -2 0 6 E 8 -2 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.090909 D: 0.818182 E: 0.000000 Sum of squares = 0.685950412306 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.181818 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -14 -8 B 0 0 18 -2 2 C 4 -18 0 2 -8 D 14 2 -2 0 6 E 8 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.090909 D: 0.818182 E: 0.000000 Sum of squares = 0.685950413148 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.181818 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=25 D=20 C=12 A=8 so A is eliminated. Round 2 votes counts: B=35 E=25 D=21 C=19 so C is eliminated. Round 3 votes counts: B=44 D=30 E=26 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 B:209 E:204 C:190 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -14 -8 B 0 0 18 -2 2 C 4 -18 0 2 -8 D 14 2 -2 0 6 E 8 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.090909 D: 0.818182 E: 0.000000 Sum of squares = 0.685950413148 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.181818 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -14 -8 B 0 0 18 -2 2 C 4 -18 0 2 -8 D 14 2 -2 0 6 E 8 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.090909 D: 0.818182 E: 0.000000 Sum of squares = 0.685950413148 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.181818 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -14 -8 B 0 0 18 -2 2 C 4 -18 0 2 -8 D 14 2 -2 0 6 E 8 -2 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.090909 D: 0.818182 E: 0.000000 Sum of squares = 0.685950413148 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.181818 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6857: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (6) E D B A C (5) C D B A E (5) B D C A E (5) E A C D B (4) E A B C D (4) B A C D E (4) E D C A B (3) E B D A C (3) D E C B A (3) D C E A B (3) D C B A E (3) D B E C A (3) D B C A E (3) A B C E D (3) E D C B A (2) E D B C A (2) E C D A B (2) E A D C B (2) D E B C A (2) C B A D E (2) C A D B E (2) B C A D E (2) B A C E D (2) A E C B D (2) A C E B D (2) A C B E D (2) E D A C B (1) E D A B C (1) E B A D C (1) E A B D C (1) D E C A B (1) D E B A C (1) D C E B A (1) D C B E A (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A B E (1) C A E D B (1) C A B E D (1) B E D A C (1) B D E A C (1) B D A C E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -14 -12 -2 B 6 0 -6 -10 -2 C 14 6 0 0 6 D 12 10 0 0 2 E 2 2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.767297 D: 0.232703 E: 0.000000 Sum of squares = 0.642895514762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.767297 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -12 -2 B 6 0 -6 -10 -2 C 14 6 0 0 6 D 12 10 0 0 2 E 2 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=21 C=21 B=16 A=11 so A is eliminated. Round 2 votes counts: E=34 C=25 D=21 B=20 so B is eliminated. Round 3 votes counts: E=36 C=36 D=28 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:212 E:198 B:194 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -12 -2 B 6 0 -6 -10 -2 C 14 6 0 0 6 D 12 10 0 0 2 E 2 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -12 -2 B 6 0 -6 -10 -2 C 14 6 0 0 6 D 12 10 0 0 2 E 2 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -12 -2 B 6 0 -6 -10 -2 C 14 6 0 0 6 D 12 10 0 0 2 E 2 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6858: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (11) D B C A E (9) E A C B D (4) E A B C D (4) C E A D B (4) B E A C D (4) A E B D C (4) A E B C D (4) D C A E B (3) D A C E B (3) C D B E A (3) C B D E A (3) B E C A D (3) B D C E A (3) B C E D A (3) A E D C B (3) A E D B C (3) A D E C B (3) D C B E A (2) D C B A E (2) D B A E C (2) D A B E C (2) C D E B A (2) B D C A E (2) D B A C E (1) D A E C B (1) C E D A B (1) C E B A D (1) C D E A B (1) C D A E B (1) B E A D C (1) B D E A C (1) B C E A D (1) B A D E C (1) A E C D B (1) A D E B C (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -10 -12 -4 B 10 0 22 2 10 C 10 -22 0 4 14 D 12 -2 -4 0 16 E 4 -10 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -12 -4 B 10 0 22 2 10 C 10 -22 0 4 14 D 12 -2 -4 0 16 E 4 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=25 A=21 C=16 E=8 so E is eliminated. Round 2 votes counts: B=30 A=29 D=25 C=16 so C is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:211 C:203 A:182 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 -12 -4 B 10 0 22 2 10 C 10 -22 0 4 14 D 12 -2 -4 0 16 E 4 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -12 -4 B 10 0 22 2 10 C 10 -22 0 4 14 D 12 -2 -4 0 16 E 4 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -12 -4 B 10 0 22 2 10 C 10 -22 0 4 14 D 12 -2 -4 0 16 E 4 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6859: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) D C E B A (11) C D E B A (11) A B E C D (9) C D E A B (8) D E C B A (6) A B C E D (6) B A E D C (5) B A D E C (4) D E B C A (3) E D C B A (2) E C D B A (2) C A D E B (2) A B D E C (2) E C B D A (1) E B D C A (1) D B E C A (1) C E D B A (1) C E D A B (1) C E A D B (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) A D B C E (1) A C E B D (1) A C B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 0 -4 B 4 0 0 -2 -4 C 4 0 0 -4 -6 D 0 2 4 0 6 E 4 4 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.247292 B: 0.000000 C: 0.000000 D: 0.752708 E: 0.000000 Sum of squares = 0.627723062751 Cumulative probabilities = A: 0.247292 B: 0.247292 C: 0.247292 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 0 -4 B 4 0 0 -2 -4 C 4 0 0 -4 -6 D 0 2 4 0 6 E 4 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=25 D=21 B=13 E=6 so E is eliminated. Round 2 votes counts: A=35 C=28 D=23 B=14 so B is eliminated. Round 3 votes counts: A=46 C=28 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:206 E:204 B:199 C:197 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 0 -4 B 4 0 0 -2 -4 C 4 0 0 -4 -6 D 0 2 4 0 6 E 4 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 -4 B 4 0 0 -2 -4 C 4 0 0 -4 -6 D 0 2 4 0 6 E 4 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 -4 B 4 0 0 -2 -4 C 4 0 0 -4 -6 D 0 2 4 0 6 E 4 4 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555555869 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6860: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) A B C D E (6) B D C A E (5) A C B E D (5) E A D C B (4) D E C B A (4) C E D B A (4) B C D A E (4) E D A C B (3) E A C D B (3) C B D E A (3) C B D A E (3) B D A C E (3) A C E B D (3) E D C B A (2) E C D A B (2) E A D B C (2) E A C B D (2) D E A B C (2) D C E B A (2) D B A E C (2) C D B E A (2) B A D C E (2) A E C B D (2) A E B D C (2) A E B C D (2) A B E D C (2) A B D C E (2) E D C A B (1) E C D B A (1) E C A D B (1) D E B C A (1) D C B E A (1) D B E C A (1) D B C E A (1) D B C A E (1) D B A C E (1) C B A E D (1) B D C E A (1) B C A D E (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 16 0 14 B -6 0 2 18 14 C -16 -2 0 8 20 D 0 -18 -8 0 0 E -14 -14 -20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.870148 B: 0.000000 C: 0.000000 D: 0.129852 E: 0.000000 Sum of squares = 0.774018968567 Cumulative probabilities = A: 0.870148 B: 0.870148 C: 0.870148 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 0 14 B -6 0 2 18 14 C -16 -2 0 8 20 D 0 -18 -8 0 0 E -14 -14 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000084012 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=21 B=17 D=16 C=13 so C is eliminated. Round 2 votes counts: A=33 E=25 B=24 D=18 so D is eliminated. Round 3 votes counts: E=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:214 C:205 D:187 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 0 14 B -6 0 2 18 14 C -16 -2 0 8 20 D 0 -18 -8 0 0 E -14 -14 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000084012 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 0 14 B -6 0 2 18 14 C -16 -2 0 8 20 D 0 -18 -8 0 0 E -14 -14 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000084012 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 0 14 B -6 0 2 18 14 C -16 -2 0 8 20 D 0 -18 -8 0 0 E -14 -14 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000084012 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6861: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) E C D B A (7) C E A D B (7) B D E C A (7) E C B D A (6) D B E C A (5) C E D A B (5) A C E D B (5) A B D C E (5) D B A E C (4) A D B C E (4) C E A B D (3) B E D C A (3) B E C D A (3) B A D E C (3) A C E B D (3) A B C E D (3) D E B C A (2) A D C E B (2) A C D E B (2) E C B A D (1) E B C D A (1) D E C B A (1) D C A E B (1) D A C E B (1) C E B A D (1) C A E D B (1) B D E A C (1) B A E D C (1) A D C B E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -8 -12 -6 B 10 0 4 2 2 C 8 -4 0 -2 -10 D 12 -2 2 0 0 E 6 -2 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -12 -6 B 10 0 4 2 2 C 8 -4 0 -2 -10 D 12 -2 2 0 0 E 6 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 C=17 E=15 D=14 so D is eliminated. Round 2 votes counts: B=36 A=28 E=18 C=18 so E is eliminated. Round 3 votes counts: B=39 C=33 A=28 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:207 D:206 C:196 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -12 -6 B 10 0 4 2 2 C 8 -4 0 -2 -10 D 12 -2 2 0 0 E 6 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -12 -6 B 10 0 4 2 2 C 8 -4 0 -2 -10 D 12 -2 2 0 0 E 6 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -12 -6 B 10 0 4 2 2 C 8 -4 0 -2 -10 D 12 -2 2 0 0 E 6 -2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6862: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) B D E A C (8) E D B C A (7) D B E A C (7) C A E B D (7) A C B D E (7) D E B C A (6) C A E D B (5) C A B D E (5) A C E B D (4) A C B E D (4) E C A D B (3) B D A C E (3) A B C D E (3) C E A D B (2) B A D C E (2) E D C B A (1) E D C A B (1) E C D A B (1) E C A B D (1) E B D A C (1) E B A D C (1) E A C B D (1) D C E A B (1) D B E C A (1) C E D A B (1) C D E A B (1) C D A B E (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B D C A E (1) B A C D E (1) Total count = 100 A B C D E A 0 2 4 -2 -8 B -2 0 2 2 -8 C -4 -2 0 0 2 D 2 -2 0 0 -2 E 8 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428302 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 2 4 -2 -8 B -2 0 2 2 -8 C -4 -2 0 0 2 D 2 -2 0 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428494 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 A=18 B=16 D=15 so D is eliminated. Round 2 votes counts: E=32 C=26 B=24 A=18 so A is eliminated. Round 3 votes counts: C=41 E=32 B=27 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:208 D:199 A:198 C:198 B:197 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -2 -8 B -2 0 2 2 -8 C -4 -2 0 0 2 D 2 -2 0 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428494 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 -8 B -2 0 2 2 -8 C -4 -2 0 0 2 D 2 -2 0 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428494 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 -8 B -2 0 2 2 -8 C -4 -2 0 0 2 D 2 -2 0 0 -2 E 8 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428494 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6863: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (15) C D A B E (8) B C E A D (7) B E C A D (6) E B C A D (5) D A C E B (5) E C B D A (3) E B C D A (3) D A E B C (3) C B A D E (3) A D B E C (3) A D B C E (3) E D A B C (2) E C B A D (2) E B A C D (2) D C A E B (2) D C A B E (2) D A E C B (2) C D E B A (2) C B E A D (2) B A E C D (2) A D C B E (2) E B D A C (1) D E C B A (1) D E A C B (1) D A B E C (1) C E B D A (1) C D E A B (1) C D B A E (1) C B D E A (1) C B D A E (1) C B A E D (1) C A B D E (1) B E A D C (1) A E B D C (1) A D E B C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -6 -12 18 B -10 0 -16 -12 24 C 6 16 0 6 20 D 12 12 -6 0 22 E -18 -24 -20 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 -12 18 B -10 0 -16 -12 24 C 6 16 0 6 20 D 12 12 -6 0 22 E -18 -24 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=22 E=18 B=16 A=12 so A is eliminated. Round 2 votes counts: D=41 C=23 E=19 B=17 so B is eliminated. Round 3 votes counts: D=42 C=30 E=28 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 D:220 A:205 B:193 E:158 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 -12 18 B -10 0 -16 -12 24 C 6 16 0 6 20 D 12 12 -6 0 22 E -18 -24 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -12 18 B -10 0 -16 -12 24 C 6 16 0 6 20 D 12 12 -6 0 22 E -18 -24 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -12 18 B -10 0 -16 -12 24 C 6 16 0 6 20 D 12 12 -6 0 22 E -18 -24 -20 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6864: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) C E B A D (6) C E B D A (5) A D B E C (5) E C B A D (4) D C A E B (4) C B E A D (4) A D B C E (4) E C B D A (3) D A E B C (3) D A C B E (3) C B A E D (3) B E A C D (3) E D B C A (2) E C D B A (2) E B C A D (2) D E B A C (2) D E A B C (2) D A E C B (2) D A B C E (2) C D E A B (2) B E C A D (2) B A C E D (2) E B D A C (1) E B C D A (1) D E C B A (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E A B (1) D B A E C (1) D A C E B (1) C E D B A (1) C E D A B (1) C D E B A (1) C A E D B (1) C A D B E (1) C A B E D (1) B E A D C (1) B D E A C (1) B A E D C (1) B A E C D (1) A D C B E (1) A C D B E (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 -6 -2 B 2 0 -4 -10 -2 C 0 4 0 -2 -4 D 6 10 2 0 4 E 2 2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -6 -2 B 2 0 -4 -10 -2 C 0 4 0 -2 -4 D 6 10 2 0 4 E 2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=26 E=15 A=15 B=11 so B is eliminated. Round 2 votes counts: D=34 C=26 E=21 A=19 so A is eliminated. Round 3 votes counts: D=46 C=31 E=23 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:202 C:199 A:195 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -6 -2 B 2 0 -4 -10 -2 C 0 4 0 -2 -4 D 6 10 2 0 4 E 2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 -2 B 2 0 -4 -10 -2 C 0 4 0 -2 -4 D 6 10 2 0 4 E 2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 -2 B 2 0 -4 -10 -2 C 0 4 0 -2 -4 D 6 10 2 0 4 E 2 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6865: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C A E (9) E C A B D (8) C E D A B (6) B D A E C (6) D B A C E (5) C E D B A (5) E C A D B (4) C E A D B (4) B D A C E (4) A B D E C (4) D B C E A (3) A E C B D (3) A E B C D (3) E A B C D (2) D C B A E (2) B A D E C (2) A E B D C (2) A D B C E (2) A C E D B (2) A C E B D (2) E D B C A (1) E C D B A (1) E B D C A (1) E A B D C (1) D B E C A (1) C D E B A (1) C D B E A (1) C D A E B (1) C D A B E (1) B A E D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 2 4 2 B -14 0 -2 2 -14 C -2 2 0 8 -2 D -4 -2 -8 0 -14 E -2 14 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 4 2 B -14 0 -2 2 -14 C -2 2 0 8 -2 D -4 -2 -8 0 -14 E -2 14 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=20 A=20 C=19 B=13 so B is eliminated. Round 2 votes counts: D=30 E=28 A=23 C=19 so C is eliminated. Round 3 votes counts: E=43 D=34 A=23 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:211 C:203 B:186 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 4 2 B -14 0 -2 2 -14 C -2 2 0 8 -2 D -4 -2 -8 0 -14 E -2 14 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 4 2 B -14 0 -2 2 -14 C -2 2 0 8 -2 D -4 -2 -8 0 -14 E -2 14 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 4 2 B -14 0 -2 2 -14 C -2 2 0 8 -2 D -4 -2 -8 0 -14 E -2 14 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6866: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (16) C A D B E (12) B C E A D (8) D A E C B (7) B E C D A (7) A D C E B (6) E B C D A (4) D A C E B (4) C B E A D (4) B E C A D (4) E D A B C (3) C B A D E (3) A D C B E (3) E D B A C (2) E A D B C (2) D E A B C (2) C D A B E (2) C A B D E (2) A D E C B (2) A C D B E (2) E B A D C (1) D A E B C (1) C B A E D (1) B C E D A (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 4 2 -8 B 2 0 4 2 -2 C -4 -4 0 0 -4 D -2 -2 0 0 -6 E 8 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 2 -8 B 2 0 4 2 -2 C -4 -4 0 0 -4 D -2 -2 0 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=24 B=20 D=14 A=14 so D is eliminated. Round 2 votes counts: E=30 A=26 C=24 B=20 so B is eliminated. Round 3 votes counts: E=41 C=33 A=26 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:203 A:198 D:195 C:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 2 -8 B 2 0 4 2 -2 C -4 -4 0 0 -4 D -2 -2 0 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 2 -8 B 2 0 4 2 -2 C -4 -4 0 0 -4 D -2 -2 0 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 2 -8 B 2 0 4 2 -2 C -4 -4 0 0 -4 D -2 -2 0 0 -6 E 8 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6867: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) B A D C E (7) D A E C B (6) C B E D A (6) D E A C B (4) D A B E C (4) B D A C E (4) B A C E D (4) A E C B D (4) C B E A D (3) B C E D A (3) B C E A D (3) B C D E A (3) A D E B C (3) A B D E C (3) E C A D B (2) D E C A B (2) D B C E A (2) D A E B C (2) C E B D A (2) C E A B D (2) B C A E D (2) B A D E C (2) A B E C D (2) E D C A B (1) E C A B D (1) E A D C B (1) E A C D B (1) D C E B A (1) C E D B A (1) C E D A B (1) C E B A D (1) C D E B A (1) C D B E A (1) C B D E A (1) B D A E C (1) B C D A E (1) B A C D E (1) A E D C B (1) A E B C D (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 0 6 -8 0 B 0 0 -2 14 8 C -6 2 0 8 0 D 8 -14 -8 0 2 E 0 -8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.499028 B: 0.500972 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500001876432 Cumulative probabilities = A: 0.499028 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -8 0 B 0 0 -2 14 8 C -6 2 0 8 0 D 8 -14 -8 0 2 E 0 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=21 C=19 A=16 E=13 so E is eliminated. Round 2 votes counts: B=31 C=29 D=22 A=18 so A is eliminated. Round 3 votes counts: B=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:210 C:202 A:199 E:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 6 -8 0 B 0 0 -2 14 8 C -6 2 0 8 0 D 8 -14 -8 0 2 E 0 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -8 0 B 0 0 -2 14 8 C -6 2 0 8 0 D 8 -14 -8 0 2 E 0 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -8 0 B 0 0 -2 14 8 C -6 2 0 8 0 D 8 -14 -8 0 2 E 0 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6868: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) A B E C D (8) A E C B D (6) E C A D B (5) D B C E A (5) D C B E A (4) B A D C E (4) B A C D E (4) E D C A B (3) E C D A B (3) D E C B A (3) D E C A B (3) C E D B A (3) C D E B A (3) B D A C E (3) B C D E A (3) B A C E D (3) A E B C D (3) A B D E C (3) D E B A C (2) C E B D A (2) C E A B D (2) E C A B D (1) E A D C B (1) E A C D B (1) E A C B D (1) D E A C B (1) D B E C A (1) D B C A E (1) D A E C B (1) C E B A D (1) B C D A E (1) B C A E D (1) A E C D B (1) A E B D C (1) A D E B C (1) A C E B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 0 -12 B 4 0 -8 0 -14 C 6 8 0 8 0 D 0 0 -8 0 2 E 12 14 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.609090 D: 0.000000 E: 0.390910 Sum of squares = 0.523801438272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.609090 D: 0.609090 E: 1.000000 A B C D E A 0 -4 -6 0 -12 B 4 0 -8 0 -14 C 6 8 0 8 0 D 0 0 -8 0 2 E 12 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=26 B=19 E=15 C=11 so C is eliminated. Round 2 votes counts: D=32 A=26 E=23 B=19 so B is eliminated. Round 3 votes counts: D=39 A=38 E=23 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:212 C:211 D:197 B:191 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 0 -12 B 4 0 -8 0 -14 C 6 8 0 8 0 D 0 0 -8 0 2 E 12 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 -12 B 4 0 -8 0 -14 C 6 8 0 8 0 D 0 0 -8 0 2 E 12 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 -12 B 4 0 -8 0 -14 C 6 8 0 8 0 D 0 0 -8 0 2 E 12 14 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6869: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) C B A D E (7) B E C A D (7) E D A B C (6) B C E A D (6) A D C B E (6) D A E C B (4) C A D B E (4) A B C D E (4) D C A B E (3) C D A B E (3) B C A E D (3) E B D A C (2) E B C D A (2) E B A D C (2) D C E A B (2) D C A E B (2) D A C B E (2) C E D B A (2) C D A E B (2) C B D A E (2) C A B D E (2) B E A C D (2) A C D B E (2) E D C A B (1) E D B C A (1) E D A C B (1) E A D B C (1) D E C A B (1) D E A C B (1) D E A B C (1) C B E A D (1) C B D E A (1) C B A E D (1) B A C E D (1) A D E B C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 20 -6 4 20 B -20 0 -20 -12 18 C 6 20 0 6 34 D -4 12 -6 0 22 E -20 -18 -34 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -6 4 20 B -20 0 -20 -12 18 C 6 20 0 6 34 D -4 12 -6 0 22 E -20 -18 -34 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 B=19 E=16 A=15 so A is eliminated. Round 2 votes counts: D=33 C=28 B=23 E=16 so E is eliminated. Round 3 votes counts: D=43 B=29 C=28 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:233 A:219 D:212 B:183 E:153 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -6 4 20 B -20 0 -20 -12 18 C 6 20 0 6 34 D -4 12 -6 0 22 E -20 -18 -34 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -6 4 20 B -20 0 -20 -12 18 C 6 20 0 6 34 D -4 12 -6 0 22 E -20 -18 -34 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -6 4 20 B -20 0 -20 -12 18 C 6 20 0 6 34 D -4 12 -6 0 22 E -20 -18 -34 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6870: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) D C E A B (7) B A E C D (7) E C D A B (5) D C E B A (5) C D E A B (5) B D A C E (5) B A E D C (5) D C A B E (4) B A D C E (4) D E C B A (3) A E C B D (3) E A C B D (2) E A B C D (2) D E B C A (2) D C B A E (2) D B C A E (2) C E D A B (2) C E A D B (2) C D A E B (2) B E D A C (2) B E A D C (2) B D E A C (2) B D A E C (2) B A C E D (2) A C E B D (2) A B E C D (2) E D B C A (1) E C A B D (1) E B D A C (1) E B C D A (1) E B A C D (1) D C B E A (1) D C A E B (1) D B C E A (1) D B A C E (1) Total count = 100 A B C D E A 0 -4 -10 -14 -12 B 4 0 -10 -8 -12 C 10 10 0 -6 -4 D 14 8 6 0 -2 E 12 12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 -14 -12 B 4 0 -10 -8 -12 C 10 10 0 -6 -4 D 14 8 6 0 -2 E 12 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 E=22 C=11 A=7 so A is eliminated. Round 2 votes counts: B=33 D=29 E=25 C=13 so C is eliminated. Round 3 votes counts: D=36 B=33 E=31 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:215 D:213 C:205 B:187 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 -14 -12 B 4 0 -10 -8 -12 C 10 10 0 -6 -4 D 14 8 6 0 -2 E 12 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -14 -12 B 4 0 -10 -8 -12 C 10 10 0 -6 -4 D 14 8 6 0 -2 E 12 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -14 -12 B 4 0 -10 -8 -12 C 10 10 0 -6 -4 D 14 8 6 0 -2 E 12 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6871: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) E B A D C (9) C D A B E (9) B E C D A (7) A D C E B (7) E A B D C (5) B C D E A (5) E B A C D (4) D B A C E (3) C D A E B (3) B E A D C (3) A D E B C (3) E B C A D (2) E A C D B (2) D C A B E (2) C B E D A (2) C B D E A (2) C B D A E (2) A E D C B (2) E C B A D (1) E C A D B (1) E A D B C (1) D B C A E (1) D A C E B (1) D A C B E (1) C E D A B (1) C E A D B (1) C D E A B (1) C A D E B (1) B C D A E (1) B A D E C (1) A E D B C (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -8 -6 6 B 10 0 -6 -12 6 C 8 6 0 14 12 D 6 12 -14 0 16 E -6 -6 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -6 6 B 10 0 -6 -12 6 C 8 6 0 14 12 D 6 12 -14 0 16 E -6 -6 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=25 B=17 A=16 D=8 so D is eliminated. Round 2 votes counts: C=36 E=25 B=21 A=18 so A is eliminated. Round 3 votes counts: C=46 E=32 B=22 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:210 B:199 A:191 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -6 6 B 10 0 -6 -12 6 C 8 6 0 14 12 D 6 12 -14 0 16 E -6 -6 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -6 6 B 10 0 -6 -12 6 C 8 6 0 14 12 D 6 12 -14 0 16 E -6 -6 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -6 6 B 10 0 -6 -12 6 C 8 6 0 14 12 D 6 12 -14 0 16 E -6 -6 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6872: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B D A C E (11) B C E D A (9) E C A D B (8) A D E C B (8) B E C A D (7) B D C A E (6) E C B A D (3) C E A D B (3) E B A D C (2) D A B C E (2) C E D A B (2) B D A E C (2) B C D E A (2) B A D E C (2) A E D C B (2) A D E B C (2) E C A B D (1) E B C A D (1) E A D C B (1) D C A E B (1) D B A C E (1) D A C B E (1) D A B E C (1) C E B A D (1) C D A E B (1) C B E D A (1) C B E A D (1) C B D A E (1) B E C D A (1) B C E A D (1) B C D A E (1) A E D B C (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -2 -8 12 B 6 0 6 8 2 C 2 -6 0 -12 14 D 8 -8 12 0 10 E -12 -2 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999239 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -8 12 B 6 0 6 8 2 C 2 -6 0 -12 14 D 8 -8 12 0 10 E -12 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 D=17 E=16 A=15 C=10 so C is eliminated. Round 2 votes counts: B=45 E=22 D=18 A=15 so A is eliminated. Round 3 votes counts: B=46 D=29 E=25 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:211 C:199 A:198 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -8 12 B 6 0 6 8 2 C 2 -6 0 -12 14 D 8 -8 12 0 10 E -12 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -8 12 B 6 0 6 8 2 C 2 -6 0 -12 14 D 8 -8 12 0 10 E -12 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -8 12 B 6 0 6 8 2 C 2 -6 0 -12 14 D 8 -8 12 0 10 E -12 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6873: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) A C D B E (9) E B D C A (7) E B A D C (6) B D C E A (6) A E C D B (6) A C D E B (6) D C B E A (5) B E D C A (5) E B D A C (4) E A C D B (4) B D C A E (4) E A B C D (3) D C B A E (3) A E B C D (3) A C E D B (3) E D C B A (2) E C A D B (2) B E D A C (2) B A D E C (2) D C E B A (1) D B C E A (1) C E D A B (1) C D A E B (1) C A D B E (1) B E A D C (1) B D E C A (1) B D A C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -8 2 C -2 6 0 -2 2 D 6 8 2 0 0 E 2 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.602070 E: 0.397930 Sum of squares = 0.520836598133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.602070 E: 1.000000 A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -8 2 C -2 6 0 -2 2 D 6 8 2 0 0 E 2 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 0.499652 Sum of squares = 0.500000241638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 B=22 C=12 D=10 so D is eliminated. Round 2 votes counts: E=28 A=28 B=23 C=21 so C is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:208 C:202 E:199 A:196 B:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -8 2 C -2 6 0 -2 2 D 6 8 2 0 0 E 2 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 0.499652 Sum of squares = 0.500000241638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -8 2 C -2 6 0 -2 2 D 6 8 2 0 0 E 2 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 0.499652 Sum of squares = 0.500000241638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -6 -2 B 2 0 -6 -8 2 C -2 6 0 -2 2 D 6 8 2 0 0 E 2 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 0.499652 Sum of squares = 0.500000241638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500348 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6874: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) E D C B A (8) D E C B A (6) B A C E D (6) A B E C D (6) A B C E D (6) E D B C A (5) D C E B A (4) C D E B A (4) B E A C D (4) C B D E A (3) B C A D E (3) A C D B E (3) A C B D E (3) A B E D C (3) E D B A C (2) D C E A B (2) C D B E A (2) C D B A E (2) C D A B E (2) C B A D E (2) A E D B C (2) E C B D A (1) E B D C A (1) E A D B C (1) D E C A B (1) D E A C B (1) D A C E B (1) C D E A B (1) C B E D A (1) B E C A D (1) B E A D C (1) B C E A D (1) B A C D E (1) A D E C B (1) Total count = 100 A B C D E A 0 -16 0 6 0 B 16 0 4 4 18 C 0 -4 0 22 12 D -6 -4 -22 0 2 E 0 -18 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 6 0 B 16 0 4 4 18 C 0 -4 0 22 12 D -6 -4 -22 0 2 E 0 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995647 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=18 C=17 B=17 D=15 so D is eliminated. Round 2 votes counts: A=34 E=26 C=23 B=17 so B is eliminated. Round 3 votes counts: A=41 E=32 C=27 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:221 C:215 A:195 D:185 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 6 0 B 16 0 4 4 18 C 0 -4 0 22 12 D -6 -4 -22 0 2 E 0 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995647 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 6 0 B 16 0 4 4 18 C 0 -4 0 22 12 D -6 -4 -22 0 2 E 0 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995647 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 6 0 B 16 0 4 4 18 C 0 -4 0 22 12 D -6 -4 -22 0 2 E 0 -18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995647 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6875: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) E C D A B (5) C D E B A (5) B D A E C (5) B A D E C (5) B A C D E (5) C D B E A (4) E A D C B (3) D E C A B (3) D B E A C (3) C E D A B (3) C A E B D (3) B C A D E (3) B A D C E (3) A E C B D (3) E D A C B (2) E D A B C (2) E C A D B (2) E A D B C (2) D B C E A (2) D B A E C (2) C A B E D (2) B A C E D (2) A B E D C (2) E D C A B (1) E A C D B (1) D E B C A (1) D E B A C (1) D E A B C (1) D C E B A (1) D C E A B (1) D B E C A (1) C E A D B (1) C E A B D (1) C D E A B (1) C B D A E (1) B D C A E (1) B D A C E (1) B C D A E (1) B A E D C (1) A E D B C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -6 4 6 B 16 0 -8 2 12 C 6 8 0 10 6 D -4 -2 -10 0 2 E -6 -12 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 4 6 B 16 0 -8 2 12 C 6 8 0 10 6 D -4 -2 -10 0 2 E -6 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=27 E=18 D=16 A=8 so A is eliminated. Round 2 votes counts: C=32 B=30 E=22 D=16 so D is eliminated. Round 3 votes counts: B=38 C=34 E=28 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:211 A:194 D:193 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -6 4 6 B 16 0 -8 2 12 C 6 8 0 10 6 D -4 -2 -10 0 2 E -6 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 4 6 B 16 0 -8 2 12 C 6 8 0 10 6 D -4 -2 -10 0 2 E -6 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 4 6 B 16 0 -8 2 12 C 6 8 0 10 6 D -4 -2 -10 0 2 E -6 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6876: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (16) E D C B A (11) C D E B A (9) A B C D E (8) D E C A B (6) B A C E D (6) A B D E C (5) E D C A B (4) A B E C D (4) D E C B A (3) C D E A B (3) E D B C A (2) E D B A C (2) D E A C B (2) D C E B A (2) D C E A B (2) C D B A E (2) B A E C D (2) E B A D C (1) C D B E A (1) C D A E B (1) C D A B E (1) C B A D E (1) C A D B E (1) B C A E D (1) B A C D E (1) A E D B C (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 0 -2 4 B -12 0 0 -8 0 C 0 0 0 -16 -20 D 2 8 16 0 -2 E -4 0 20 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000138 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 12 0 -2 4 B -12 0 0 -8 0 C 0 0 0 -16 -20 D 2 8 16 0 -2 E -4 0 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000123 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=20 C=19 D=15 B=10 so B is eliminated. Round 2 votes counts: A=45 E=20 C=20 D=15 so D is eliminated. Round 3 votes counts: A=45 E=31 C=24 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:212 E:209 A:207 B:190 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 -2 4 B -12 0 0 -8 0 C 0 0 0 -16 -20 D 2 8 16 0 -2 E -4 0 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000123 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -2 4 B -12 0 0 -8 0 C 0 0 0 -16 -20 D 2 8 16 0 -2 E -4 0 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000123 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -2 4 B -12 0 0 -8 0 C 0 0 0 -16 -20 D 2 8 16 0 -2 E -4 0 20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000123 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6877: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (13) D A B C E (13) E B C A D (6) B A D C E (6) E C D A B (5) C E B A D (5) C E A D B (5) D A C B E (4) B E A D C (4) C D A E B (3) A D B C E (3) E D C B A (2) E D C A B (2) D B A E C (2) C B A D E (2) C A E D B (2) B C A E D (2) E D B A C (1) E C D B A (1) E C B D A (1) E C A B D (1) E B C D A (1) E B A C D (1) D E C A B (1) D E A B C (1) D A E C B (1) D A E B C (1) C E D A B (1) C E A B D (1) C B E A D (1) C B A E D (1) C A B D E (1) B D A C E (1) B C A D E (1) B A E C D (1) B A C D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -16 18 -6 B 6 0 -6 4 -10 C 16 6 0 14 10 D -18 -4 -14 0 -14 E 6 10 -10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 18 -6 B 6 0 -6 4 -10 C 16 6 0 14 10 D -18 -4 -14 0 -14 E 6 10 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=23 C=22 B=16 A=5 so A is eliminated. Round 2 votes counts: E=34 D=26 C=22 B=18 so B is eliminated. Round 3 votes counts: E=39 D=34 C=27 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:223 E:210 B:197 A:195 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 18 -6 B 6 0 -6 4 -10 C 16 6 0 14 10 D -18 -4 -14 0 -14 E 6 10 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 18 -6 B 6 0 -6 4 -10 C 16 6 0 14 10 D -18 -4 -14 0 -14 E 6 10 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 18 -6 B 6 0 -6 4 -10 C 16 6 0 14 10 D -18 -4 -14 0 -14 E 6 10 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6878: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (16) C D A E B (15) A E B D C (11) E A B D C (6) C D B E A (5) D C B E A (4) A E B C D (4) D C E A B (3) C A E D B (3) C A D E B (3) B D E C A (3) A E C B D (3) D C A E B (2) D B C E A (2) B E D A C (2) B E A C D (2) B C E A D (2) B A E C D (2) A E C D B (2) A C E B D (2) D B E A C (1) C D B A E (1) C B D E A (1) C B D A E (1) B D C E A (1) A E D C B (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 14 8 18 4 B -14 0 8 14 -12 C -8 -8 0 -6 -10 D -18 -14 6 0 -16 E -4 12 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 18 4 B -14 0 8 14 -12 C -8 -8 0 -6 -10 D -18 -14 6 0 -16 E -4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998131 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 A=25 D=12 E=6 so E is eliminated. Round 2 votes counts: A=31 C=29 B=28 D=12 so D is eliminated. Round 3 votes counts: C=38 B=31 A=31 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:217 B:198 C:184 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 18 4 B -14 0 8 14 -12 C -8 -8 0 -6 -10 D -18 -14 6 0 -16 E -4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998131 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 18 4 B -14 0 8 14 -12 C -8 -8 0 -6 -10 D -18 -14 6 0 -16 E -4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998131 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 18 4 B -14 0 8 14 -12 C -8 -8 0 -6 -10 D -18 -14 6 0 -16 E -4 12 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998131 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6879: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (8) E B D A C (7) D E A B C (5) C B E A D (5) E D B A C (4) E D A B C (4) C B A E D (4) C B A D E (4) B E A D C (4) D E C A B (3) D E A C B (3) D A E B C (3) C D A E B (3) C D A B E (3) C A D B E (3) E B A D C (2) E A B D C (2) D C E A B (2) D C A E B (2) C D E B A (2) C D E A B (2) C B E D A (2) B E C A D (2) B C E A D (2) B A E D C (2) A E D B C (2) E D C B A (1) E A D B C (1) D E C B A (1) D A C E B (1) C D B A E (1) C B D E A (1) B E C D A (1) B E A C D (1) B C A E D (1) B A E C D (1) B A C E D (1) A D E C B (1) A D C E B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 10 -2 -14 B -2 0 10 -12 -20 C -10 -10 0 -20 -18 D 2 12 20 0 0 E 14 20 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.463281 E: 0.536719 Sum of squares = 0.50269651881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.463281 E: 1.000000 A B C D E A 0 2 10 -2 -14 B -2 0 10 -12 -20 C -10 -10 0 -20 -18 D 2 12 20 0 0 E 14 20 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=21 D=20 B=15 A=14 so A is eliminated. Round 2 votes counts: D=30 C=30 E=23 B=17 so B is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:217 A:198 B:188 C:171 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 10 -2 -14 B -2 0 10 -12 -20 C -10 -10 0 -20 -18 D 2 12 20 0 0 E 14 20 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -2 -14 B -2 0 10 -12 -20 C -10 -10 0 -20 -18 D 2 12 20 0 0 E 14 20 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -2 -14 B -2 0 10 -12 -20 C -10 -10 0 -20 -18 D 2 12 20 0 0 E 14 20 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6880: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) D A E B C (9) B C D A E (8) B C A D E (8) E C A D B (6) D B A E C (6) D A B E C (5) C E A D B (5) E D A C B (4) C E B A D (4) B C A E D (4) B A D C E (4) D E A B C (3) C E A B D (3) C B D A E (3) E A D C B (2) D E A C B (2) C B A E D (2) B D A E C (2) E D C A B (1) E D A B C (1) E A D B C (1) C E D A B (1) C B D E A (1) B D C A E (1) B D A C E (1) B A C D E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -14 4 12 B 10 0 12 4 14 C 14 -12 0 12 12 D -4 -4 -12 0 10 E -12 -14 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 4 12 B 10 0 12 4 14 C 14 -12 0 12 12 D -4 -4 -12 0 10 E -12 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=29 B=29 D=25 E=15 A=2 so A is eliminated. Round 2 votes counts: C=29 B=29 D=26 E=16 so E is eliminated. Round 3 votes counts: D=36 C=35 B=29 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:220 C:213 A:196 D:195 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 4 12 B 10 0 12 4 14 C 14 -12 0 12 12 D -4 -4 -12 0 10 E -12 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 4 12 B 10 0 12 4 14 C 14 -12 0 12 12 D -4 -4 -12 0 10 E -12 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 4 12 B 10 0 12 4 14 C 14 -12 0 12 12 D -4 -4 -12 0 10 E -12 -14 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6881: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) D B C E A (10) E B D A C (7) A E C B D (7) B D E C A (6) E A B D C (5) C D B A E (5) A E B D C (5) A D B E C (5) A C E D B (5) C A D B E (4) A C D B E (3) E C B D A (2) E B D C A (2) E B A D C (2) E A C B D (2) C E B D A (2) C A E D B (2) B D E A C (2) A E B C D (2) E C B A D (1) E C A B D (1) E A B C D (1) D B E C A (1) D B A E C (1) C B D E A (1) B E D A C (1) A E C D B (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 4 -2 -14 B 8 0 4 0 6 C -4 -4 0 0 -10 D 2 0 0 0 2 E 14 -6 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.506062 C: 0.000000 D: 0.493938 E: 0.000000 Sum of squares = 0.500073487476 Cumulative probabilities = A: 0.000000 B: 0.506062 C: 0.506062 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -2 -14 B 8 0 4 0 6 C -4 -4 0 0 -10 D 2 0 0 0 2 E 14 -6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=25 E=23 D=12 B=9 so B is eliminated. Round 2 votes counts: A=31 C=25 E=24 D=20 so D is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:209 E:208 D:202 C:191 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -2 -14 B 8 0 4 0 6 C -4 -4 0 0 -10 D 2 0 0 0 2 E 14 -6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -2 -14 B 8 0 4 0 6 C -4 -4 0 0 -10 D 2 0 0 0 2 E 14 -6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -2 -14 B 8 0 4 0 6 C -4 -4 0 0 -10 D 2 0 0 0 2 E 14 -6 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6882: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (15) E D B A C (10) D E C A B (9) D E B C A (6) A B C E D (6) E D A C B (5) D C E A B (5) B A C E D (5) A C B E D (4) A C B D E (3) E D A B C (2) E B A D C (2) E A D C B (2) D E C B A (2) D C A E B (2) C B A D E (2) C A D E B (2) B C D A E (2) B C A E D (2) E D B C A (1) E B D A C (1) E B A C D (1) D E B A C (1) D C B E A (1) D C B A E (1) C D A B E (1) C B D A E (1) C A D B E (1) B E D A C (1) B C A D E (1) A C E B D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 20 -8 -2 2 B -20 0 -16 -4 -6 C 8 16 0 -2 12 D 2 4 2 0 12 E -2 6 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -8 -2 2 B -20 0 -16 -4 -6 C 8 16 0 -2 12 D 2 4 2 0 12 E -2 6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 C=22 A=16 B=11 so B is eliminated. Round 2 votes counts: D=27 C=27 E=25 A=21 so A is eliminated. Round 3 votes counts: C=47 D=27 E=26 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:217 D:210 A:206 E:190 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 -8 -2 2 B -20 0 -16 -4 -6 C 8 16 0 -2 12 D 2 4 2 0 12 E -2 6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -8 -2 2 B -20 0 -16 -4 -6 C 8 16 0 -2 12 D 2 4 2 0 12 E -2 6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -8 -2 2 B -20 0 -16 -4 -6 C 8 16 0 -2 12 D 2 4 2 0 12 E -2 6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6883: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (17) C E D A B (10) A D C E B (10) E C D B A (7) B A E C D (5) B A D E C (5) A B D C E (5) A D B C E (4) D C A E B (3) D A C E B (3) C D E A B (3) A D C B E (3) E C B D A (2) D E C A B (2) D C E A B (2) B E C A D (2) B A E D C (2) B A D C E (2) A C D E B (2) A B D E C (2) E C D A B (1) E B C D A (1) D E A C B (1) C E D B A (1) C E B D A (1) B E D C A (1) B C E A D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -8 -10 -4 B -4 0 -2 -4 2 C 8 2 0 10 4 D 10 4 -10 0 -2 E 4 -2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -10 -4 B -4 0 -2 -4 2 C 8 2 0 10 4 D 10 4 -10 0 -2 E 4 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=27 C=15 E=11 D=11 so E is eliminated. Round 2 votes counts: B=37 A=27 C=25 D=11 so D is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:201 E:200 B:196 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -4 B -4 0 -2 -4 2 C 8 2 0 10 4 D 10 4 -10 0 -2 E 4 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -4 B -4 0 -2 -4 2 C 8 2 0 10 4 D 10 4 -10 0 -2 E 4 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -4 B -4 0 -2 -4 2 C 8 2 0 10 4 D 10 4 -10 0 -2 E 4 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6884: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (10) A E C D B (10) D B C E A (7) A E B C D (6) B A E D C (4) A C D E B (4) E B A D C (3) E A C B D (3) E A B C D (3) C D B E A (3) C D A B E (3) B E A D C (3) B D E A C (3) A C E D B (3) A B E D C (3) E B C D A (2) E A B D C (2) D C B E A (2) D C B A E (2) D C A B E (2) C A E D B (2) C A D E B (2) B D E C A (2) A E B D C (2) A C D B E (2) A B D E C (2) E B C A D (1) D B C A E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D A E B (1) C A D B E (1) B C D E A (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 8 10 16 4 B -8 0 10 4 4 C -10 -10 0 2 0 D -16 -4 -2 0 0 E -4 -4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 16 4 B -8 0 10 4 4 C -10 -10 0 2 0 D -16 -4 -2 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=23 C=15 E=14 D=14 so E is eliminated. Round 2 votes counts: A=42 B=29 C=15 D=14 so D is eliminated. Round 3 votes counts: A=42 B=37 C=21 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:205 E:196 C:191 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 16 4 B -8 0 10 4 4 C -10 -10 0 2 0 D -16 -4 -2 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 16 4 B -8 0 10 4 4 C -10 -10 0 2 0 D -16 -4 -2 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 16 4 B -8 0 10 4 4 C -10 -10 0 2 0 D -16 -4 -2 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6885: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (10) C B D E A (9) E A D C B (7) E A D B C (7) A D E B C (5) D B C A E (4) C E B D A (3) C B E D A (3) B A D C E (3) E C B A D (2) E A C D B (2) D E C B A (2) D C B A E (2) D B A C E (2) D A B C E (2) C E B A D (2) C D B E A (2) C B E A D (2) B D C A E (2) B C A D E (2) B A C D E (2) A E B D C (2) A B D C E (2) A B C E D (2) E D C B A (1) E D A C B (1) E C D A B (1) E C A B D (1) E A C B D (1) D E A B C (1) D C E B A (1) D A E B C (1) D A B E C (1) C E D B A (1) C D E B A (1) C B D A E (1) C B A E D (1) B D A C E (1) B C D A E (1) A E B C D (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 12 14 0 B 0 0 8 -12 -6 C -12 -8 0 -18 6 D -14 12 18 0 0 E 0 6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.540043 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.459957 Sum of squares = 0.503206935753 Cumulative probabilities = A: 0.540043 B: 0.540043 C: 0.540043 D: 0.540043 E: 1.000000 A B C D E A 0 0 12 14 0 B 0 0 8 -12 -6 C -12 -8 0 -18 6 D -14 12 18 0 0 E 0 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=25 A=25 E=23 D=16 B=11 so B is eliminated. Round 2 votes counts: A=30 C=28 E=23 D=19 so D is eliminated. Round 3 votes counts: C=37 A=37 E=26 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:208 E:200 B:195 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 14 0 B 0 0 8 -12 -6 C -12 -8 0 -18 6 D -14 12 18 0 0 E 0 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 14 0 B 0 0 8 -12 -6 C -12 -8 0 -18 6 D -14 12 18 0 0 E 0 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 14 0 B 0 0 8 -12 -6 C -12 -8 0 -18 6 D -14 12 18 0 0 E 0 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6886: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (16) D C B E A (9) C D B E A (7) E B C A D (5) E A B C D (5) A E B D C (5) D C B A E (4) D A C E B (4) D A C B E (4) A D E C B (4) B E A C D (3) A D C E B (3) A D C B E (3) A D B C E (3) E B C D A (2) E B A C D (2) D C A E B (2) D C A B E (2) C B E D A (2) B E C A D (2) B C D A E (2) A D E B C (2) E A D C B (1) D C E B A (1) C E B D A (1) C D E B A (1) B C E D A (1) A E D C B (1) A E D B C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 18 16 16 B -16 0 2 -4 -12 C -18 -2 0 0 -2 D -16 4 0 0 2 E -16 12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 16 16 B -16 0 2 -4 -12 C -18 -2 0 0 -2 D -16 4 0 0 2 E -16 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 D=26 E=15 C=11 B=8 so B is eliminated. Round 2 votes counts: A=40 D=26 E=20 C=14 so C is eliminated. Round 3 votes counts: A=40 D=36 E=24 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:233 E:198 D:195 C:189 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 16 16 B -16 0 2 -4 -12 C -18 -2 0 0 -2 D -16 4 0 0 2 E -16 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 16 16 B -16 0 2 -4 -12 C -18 -2 0 0 -2 D -16 4 0 0 2 E -16 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 16 16 B -16 0 2 -4 -12 C -18 -2 0 0 -2 D -16 4 0 0 2 E -16 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6887: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) A D C B E (9) E B C D A (7) A D E C B (6) E B C A D (5) E A C B D (5) B C D E A (5) B C D A E (5) E A D C B (4) E A D B C (4) C B D A E (4) B D C A E (4) B C E D A (4) A E D C B (4) E A B C D (3) E C B A D (2) D C B A E (2) A E D B C (2) A C D B E (2) E D A B C (1) E C A B D (1) E B A C D (1) E A C D B (1) D C A B E (1) D B E C A (1) D A B E C (1) D A B C E (1) C B E A D (1) C B A D E (1) C A B D E (1) A E C D B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 16 12 10 10 B -16 0 -12 -2 2 C -12 12 0 0 2 D -10 2 0 0 8 E -10 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 10 10 B -16 0 -12 -2 2 C -12 12 0 0 2 D -10 2 0 0 8 E -10 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=26 B=18 D=15 C=7 so C is eliminated. Round 2 votes counts: E=34 A=27 B=24 D=15 so D is eliminated. Round 3 votes counts: A=39 E=34 B=27 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:201 D:200 E:189 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 10 10 B -16 0 -12 -2 2 C -12 12 0 0 2 D -10 2 0 0 8 E -10 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 10 10 B -16 0 -12 -2 2 C -12 12 0 0 2 D -10 2 0 0 8 E -10 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 10 10 B -16 0 -12 -2 2 C -12 12 0 0 2 D -10 2 0 0 8 E -10 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6888: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) A D C E B (7) E C D A B (6) B C E A D (6) B A D C E (6) C A D E B (5) A D C B E (5) E D A C B (4) B E C D A (4) B C A D E (4) B A D E C (4) B A C D E (4) E C D B A (3) C D A E B (3) B C E D A (3) A D B E C (3) E D C A B (2) E B D A C (2) D E A C B (2) C B E A D (2) A D B C E (2) E D B A C (1) E D A B C (1) E B D C A (1) E B C D A (1) C E D B A (1) C E D A B (1) C B E D A (1) C B A D E (1) C A D B E (1) B E D C A (1) B E C A D (1) B E A D C (1) B C A E D (1) A D E B C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 12 12 B -4 0 -4 -12 4 C -4 4 0 -2 10 D -12 12 2 0 14 E -12 -4 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 12 12 B -4 0 -4 -12 4 C -4 4 0 -2 10 D -12 12 2 0 14 E -12 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998624 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=21 A=20 C=15 D=9 so D is eliminated. Round 2 votes counts: B=35 A=27 E=23 C=15 so C is eliminated. Round 3 votes counts: B=39 A=36 E=25 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:208 C:204 B:192 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 12 12 B -4 0 -4 -12 4 C -4 4 0 -2 10 D -12 12 2 0 14 E -12 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998624 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 12 12 B -4 0 -4 -12 4 C -4 4 0 -2 10 D -12 12 2 0 14 E -12 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998624 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 12 12 B -4 0 -4 -12 4 C -4 4 0 -2 10 D -12 12 2 0 14 E -12 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998624 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6889: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) D E C B A (6) E D B A C (5) A E B D C (5) A C B D E (5) A B C E D (5) C D E B A (4) A C D B E (4) A C B E D (4) A B E D C (4) D E A B C (3) D A E C B (3) C D E A B (3) C A B D E (3) B E A D C (3) A D C E B (3) E B D C A (2) E B C D A (2) E B A D C (2) D E B C A (2) D E B A C (2) C A B E D (2) B E C D A (2) A D E B C (2) A B E C D (2) D C E B A (1) D C E A B (1) D C A E B (1) C D A B E (1) C B E A D (1) C B A E D (1) C A D B E (1) B E D A C (1) B C E A D (1) B C A E D (1) B A E D C (1) B A E C D (1) B A C E D (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 8 16 8 4 B -8 0 8 -2 -10 C -16 -8 0 -12 -12 D -8 2 12 0 -6 E -4 10 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 8 4 B -8 0 8 -2 -10 C -16 -8 0 -12 -12 D -8 2 12 0 -6 E -4 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=19 E=18 C=16 B=11 so B is eliminated. Round 2 votes counts: A=39 E=24 D=19 C=18 so C is eliminated. Round 3 votes counts: A=47 D=27 E=26 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:212 D:200 B:194 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 8 4 B -8 0 8 -2 -10 C -16 -8 0 -12 -12 D -8 2 12 0 -6 E -4 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 8 4 B -8 0 8 -2 -10 C -16 -8 0 -12 -12 D -8 2 12 0 -6 E -4 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 8 4 B -8 0 8 -2 -10 C -16 -8 0 -12 -12 D -8 2 12 0 -6 E -4 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6890: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (18) D B E A C (11) D A C B E (10) D C A E B (6) C A E B D (5) C A D E B (5) B E D A C (5) A C D B E (5) A C B E D (5) E B D A C (3) C A D B E (3) E B D C A (2) D E B C A (2) D E B A C (2) D C A B E (2) C A B E D (2) B E A C D (2) A D C B E (2) E D B C A (1) E B A D C (1) E B A C D (1) D C E B A (1) D C E A B (1) D A B E C (1) C E A B D (1) B E A D C (1) B D E A C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 2 4 -6 B 2 0 4 -4 2 C -2 -4 0 -6 -2 D -4 4 6 0 6 E 6 -2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749999956 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 2 4 -6 B 2 0 4 -4 2 C -2 -4 0 -6 -2 D -4 4 6 0 6 E 6 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998954 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=26 C=16 A=13 B=9 so B is eliminated. Round 2 votes counts: D=37 E=34 C=16 A=13 so A is eliminated. Round 3 votes counts: D=40 E=34 C=26 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 B:202 E:200 A:199 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 4 -6 B 2 0 4 -4 2 C -2 -4 0 -6 -2 D -4 4 6 0 6 E 6 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998954 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -6 B 2 0 4 -4 2 C -2 -4 0 -6 -2 D -4 4 6 0 6 E 6 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998954 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -6 B 2 0 4 -4 2 C -2 -4 0 -6 -2 D -4 4 6 0 6 E 6 -2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.250000 Sum of squares = 0.343749998954 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6891: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) E C D A B (8) E C D B A (7) D B A E C (7) E D C B A (6) C A E B D (6) A C B E D (6) A B C D E (6) D B E A C (5) A B D C E (5) E C A D B (4) D E B C A (4) C E A D B (4) C E A B D (4) B A D C E (4) B D A E C (3) D B E C A (2) D E C B A (1) C E D B A (1) C E D A B (1) C B A E D (1) C A B E D (1) B C A D E (1) B A D E C (1) A D E B C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -2 -8 6 B 4 0 -4 -2 4 C 2 4 0 4 2 D 8 2 -4 0 0 E -6 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 6 B 4 0 -4 -2 4 C 2 4 0 4 2 D 8 2 -4 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=20 D=19 C=18 B=18 so C is eliminated. Round 2 votes counts: E=35 A=27 D=19 B=19 so D is eliminated. Round 3 votes counts: E=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:206 D:203 B:201 A:196 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 6 B 4 0 -4 -2 4 C 2 4 0 4 2 D 8 2 -4 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 6 B 4 0 -4 -2 4 C 2 4 0 4 2 D 8 2 -4 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 6 B 4 0 -4 -2 4 C 2 4 0 4 2 D 8 2 -4 0 0 E -6 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6892: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (7) D B C A E (6) E A B C D (5) D B A E C (5) C B E A D (5) C E A B D (4) B D E A C (4) A D E C B (4) E B A C D (3) E A C B D (3) D C B A E (3) D A E C B (3) D A C E B (3) A E D B C (3) E C A B D (2) D C A E B (2) D A C B E (2) C D B A E (2) C B D E A (2) C A E D B (2) C A D E B (2) B E A D C (2) B D C E A (2) B C E A D (2) A E C D B (2) A E C B D (2) E B A D C (1) E A B D C (1) D C A B E (1) D B C E A (1) D B A C E (1) D A E B C (1) D A B C E (1) C E B A D (1) C D B E A (1) C D A E B (1) B E D A C (1) B E C D A (1) B E A C D (1) B C E D A (1) B C D E A (1) A E D C B (1) A E B D C (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 2 10 -2 B 6 0 2 4 4 C -2 -2 0 0 -6 D -10 -4 0 0 -2 E 2 -4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 10 -2 B 6 0 2 4 4 C -2 -2 0 0 -6 D -10 -4 0 0 -2 E 2 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=22 C=20 E=15 A=14 so A is eliminated. Round 2 votes counts: D=34 E=24 B=22 C=20 so C is eliminated. Round 3 votes counts: D=40 E=31 B=29 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:208 E:203 A:202 C:195 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 10 -2 B 6 0 2 4 4 C -2 -2 0 0 -6 D -10 -4 0 0 -2 E 2 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 10 -2 B 6 0 2 4 4 C -2 -2 0 0 -6 D -10 -4 0 0 -2 E 2 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 10 -2 B 6 0 2 4 4 C -2 -2 0 0 -6 D -10 -4 0 0 -2 E 2 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6893: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) C B A D E (9) B C E D A (8) E D A B C (7) B E D C A (7) E D B A C (6) A C D E B (6) A C B E D (5) C B A E D (4) B C A E D (4) E D B C A (3) D E A C B (3) B C D E A (3) A E D C B (3) D E B C A (2) D E A B C (2) C A B D E (2) A C B D E (2) E B D C A (1) D E B A C (1) D C E A B (1) D A E C B (1) C B D E A (1) C B D A E (1) B E C D A (1) B C E A D (1) B C A D E (1) B A E C D (1) B A C E D (1) A E C D B (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 2 4 6 B 10 0 0 6 4 C -2 0 0 6 2 D -4 -6 -6 0 -10 E -6 -4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.379056 C: 0.620944 D: 0.000000 E: 0.000000 Sum of squares = 0.529255002121 Cumulative probabilities = A: 0.000000 B: 0.379056 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 4 6 B 10 0 0 6 4 C -2 0 0 6 2 D -4 -6 -6 0 -10 E -6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 E=17 C=17 D=10 so D is eliminated. Round 2 votes counts: A=30 B=27 E=25 C=18 so C is eliminated. Round 3 votes counts: B=42 A=32 E=26 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:203 A:201 E:199 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 4 6 B 10 0 0 6 4 C -2 0 0 6 2 D -4 -6 -6 0 -10 E -6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 4 6 B 10 0 0 6 4 C -2 0 0 6 2 D -4 -6 -6 0 -10 E -6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 4 6 B 10 0 0 6 4 C -2 0 0 6 2 D -4 -6 -6 0 -10 E -6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6894: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) C D E A B (6) A C E B D (6) D B E C A (5) C E D A B (5) C E A D B (4) B A E D C (4) E D B C A (3) E C D B A (3) E C D A B (3) D E C B A (3) D C E B A (3) C D A E B (3) C A E D B (3) B A D E C (3) A B D C E (3) E C A B D (2) E B D C A (2) E A C B D (2) D E B C A (2) D C B E A (2) D B A C E (2) B D E A C (2) A B C D E (2) E D C B A (1) E C B D A (1) E C A D B (1) E B C D A (1) E B A C D (1) D C E A B (1) D C A B E (1) D B C A E (1) D B A E C (1) D A B C E (1) C D E B A (1) C A D E B (1) B E D A C (1) B D A E C (1) B A D C E (1) A E C B D (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -16 -10 -10 B -12 0 -6 -12 -20 C 16 6 0 14 -6 D 10 12 -14 0 -10 E 10 20 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -16 -10 -10 B -12 0 -6 -12 -20 C 16 6 0 14 -6 D 10 12 -14 0 -10 E 10 20 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=23 A=23 D=22 E=20 B=12 so B is eliminated. Round 2 votes counts: A=31 D=25 C=23 E=21 so E is eliminated. Round 3 votes counts: C=34 A=34 D=32 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:215 D:199 A:188 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -16 -10 -10 B -12 0 -6 -12 -20 C 16 6 0 14 -6 D 10 12 -14 0 -10 E 10 20 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -16 -10 -10 B -12 0 -6 -12 -20 C 16 6 0 14 -6 D 10 12 -14 0 -10 E 10 20 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -16 -10 -10 B -12 0 -6 -12 -20 C 16 6 0 14 -6 D 10 12 -14 0 -10 E 10 20 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6895: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) E D A C B (6) C B E A D (6) D A E B C (5) C E D A B (5) B A D E C (5) B A C D E (5) A D B E C (5) E D C A B (4) C E B D A (4) B C A E D (4) B A D C E (4) A B D E C (4) D E A C B (3) D A B E C (3) C B E D A (3) B C E A D (3) B C A D E (3) A D E C B (3) D A E C B (2) A D E B C (2) A B D C E (2) E D C B A (1) E D B A C (1) E C D B A (1) D E C A B (1) C E D B A (1) C B A E D (1) C A D E B (1) C A B D E (1) B E D C A (1) A D C E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 14 6 4 6 B -14 0 -6 -6 4 C -6 6 0 -6 -8 D -4 6 6 0 4 E -6 -4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 4 6 B -14 0 -6 -6 4 C -6 6 0 -6 -8 D -4 6 6 0 4 E -6 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=22 E=20 A=19 D=14 so D is eliminated. Round 2 votes counts: A=29 B=25 E=24 C=22 so C is eliminated. Round 3 votes counts: B=35 E=34 A=31 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:215 D:206 E:197 C:193 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 4 6 B -14 0 -6 -6 4 C -6 6 0 -6 -8 D -4 6 6 0 4 E -6 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 4 6 B -14 0 -6 -6 4 C -6 6 0 -6 -8 D -4 6 6 0 4 E -6 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 4 6 B -14 0 -6 -6 4 C -6 6 0 -6 -8 D -4 6 6 0 4 E -6 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6896: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) C B A E D (6) D E A C B (5) B C A E D (4) A E D C B (4) E D A B C (3) E A D B C (3) D E A B C (3) D B C E A (3) C D A E B (3) C B D E A (3) C B D A E (3) C A B E D (3) B C D E A (3) A E C D B (3) A C E B D (3) D E C A B (2) D E B A C (2) D C E A B (2) D B E C A (2) C D B E A (2) C B A D E (2) B E D A C (2) B E A D C (2) B D E A C (2) A E B D C (2) A E B C D (2) A C E D B (2) A B E C D (2) E D B A C (1) D E B C A (1) D B E A C (1) C A E D B (1) C A D E B (1) C A D B E (1) B D E C A (1) B D C E A (1) B C E A D (1) B A E D C (1) B A E C D (1) A C D E B (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 12 10 10 10 B -12 0 4 -6 -2 C -10 -4 0 -2 -6 D -10 6 2 0 -12 E -10 2 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 10 10 B -12 0 4 -6 -2 C -10 -4 0 -2 -6 D -10 6 2 0 -12 E -10 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 D=21 B=18 E=7 so E is eliminated. Round 2 votes counts: A=32 D=25 C=25 B=18 so B is eliminated. Round 3 votes counts: A=36 C=33 D=31 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:205 D:193 B:192 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 10 10 B -12 0 4 -6 -2 C -10 -4 0 -2 -6 D -10 6 2 0 -12 E -10 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 10 10 B -12 0 4 -6 -2 C -10 -4 0 -2 -6 D -10 6 2 0 -12 E -10 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 10 10 B -12 0 4 -6 -2 C -10 -4 0 -2 -6 D -10 6 2 0 -12 E -10 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6897: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) D E B A C (6) D C A E B (5) C A B E D (5) B A E C D (5) E D B A C (4) E B A C D (4) D A B E C (4) C B A E D (4) C A B D E (4) A C D B E (4) A B C E D (4) D C E A B (3) C A D B E (3) B A C E D (3) A C B D E (3) E B D C A (2) E B D A C (2) E B C D A (2) D E C B A (2) D E B C A (2) D E A B C (2) D A C E B (2) D A C B E (2) C D E A B (2) B E A C D (2) E B C A D (1) D C E B A (1) C E B D A (1) C E B A D (1) C D A E B (1) C B E A D (1) B E C A D (1) B E A D C (1) B C A E D (1) B A E D C (1) A D B C E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 4 2 8 B 6 0 10 0 2 C -4 -10 0 6 4 D -2 0 -6 0 -4 E -8 -2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.749302 C: 0.000000 D: 0.250698 E: 0.000000 Sum of squares = 0.624302567032 Cumulative probabilities = A: 0.000000 B: 0.749302 C: 0.749302 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 2 8 B 6 0 10 0 2 C -4 -10 0 6 4 D -2 0 -6 0 -4 E -8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555615684 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=22 E=21 B=14 A=14 so B is eliminated. Round 2 votes counts: D=29 E=25 C=23 A=23 so C is eliminated. Round 3 votes counts: A=40 D=32 E=28 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:209 A:204 C:198 E:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 2 8 B 6 0 10 0 2 C -4 -10 0 6 4 D -2 0 -6 0 -4 E -8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555615684 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 2 8 B 6 0 10 0 2 C -4 -10 0 6 4 D -2 0 -6 0 -4 E -8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555615684 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 2 8 B 6 0 10 0 2 C -4 -10 0 6 4 D -2 0 -6 0 -4 E -8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555615684 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6898: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (8) D B A E C (7) D A B C E (7) D B E A C (6) C A E D B (5) E C A B D (4) D C A B E (4) D A C B E (4) C E A D B (4) C A D E B (4) B E D A C (4) E C B A D (3) E B C A D (3) D B E C A (3) C D E A B (3) B D A E C (3) A C D E B (3) D C A E B (2) D B A C E (2) B E A D C (2) B D E A C (2) A C E B D (2) A B D E C (2) E B D C A (1) E B C D A (1) D C B A E (1) C E D B A (1) C E D A B (1) C D A E B (1) C A E B D (1) B E A C D (1) B A E D C (1) B A D E C (1) A E B C D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 16 0 -6 6 B -16 0 -2 -18 4 C 0 2 0 -8 10 D 6 18 8 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 -6 6 B -16 0 -2 -18 4 C 0 2 0 -8 10 D 6 18 8 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=28 B=14 E=12 A=10 so A is eliminated. Round 2 votes counts: D=37 C=33 B=17 E=13 so E is eliminated. Round 3 votes counts: C=40 D=37 B=23 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:208 C:202 B:184 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 0 -6 6 B -16 0 -2 -18 4 C 0 2 0 -8 10 D 6 18 8 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 -6 6 B -16 0 -2 -18 4 C 0 2 0 -8 10 D 6 18 8 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 -6 6 B -16 0 -2 -18 4 C 0 2 0 -8 10 D 6 18 8 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6899: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) B E A D C (10) D C A E B (9) B C D E A (9) E B A D C (7) E A D C B (6) E A D B C (6) E A B D C (6) D A C E B (5) C D A B E (5) B E A C D (5) C D B A E (4) B C D A E (3) A E D C B (3) A D E C B (3) A D C E B (3) B E C A D (2) C B D A E (1) B E C D A (1) B A C E D (1) A C D E B (1) Total count = 100 A B C D E A 0 14 12 6 -4 B -14 0 0 -10 -18 C -12 0 0 -16 2 D -6 10 16 0 6 E 4 18 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.343749999999 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 A B C D E A 0 14 12 6 -4 B -14 0 0 -10 -18 C -12 0 0 -16 2 D -6 10 16 0 6 E 4 18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 C=20 D=14 A=10 so A is eliminated. Round 2 votes counts: B=31 E=28 C=21 D=20 so D is eliminated. Round 3 votes counts: C=38 E=31 B=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:214 D:213 E:207 C:187 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 12 6 -4 B -14 0 0 -10 -18 C -12 0 0 -16 2 D -6 10 16 0 6 E 4 18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 6 -4 B -14 0 0 -10 -18 C -12 0 0 -16 2 D -6 10 16 0 6 E 4 18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 6 -4 B -14 0 0 -10 -18 C -12 0 0 -16 2 D -6 10 16 0 6 E 4 18 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.375000 Sum of squares = 0.34375 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.375000 D: 0.625000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6900: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) E B D A C (7) C A D E B (6) C A D B E (6) A C D E B (6) E D B A C (5) E B D C A (5) B E C D A (5) B E A D C (4) A C D B E (4) D E A C B (3) C D A E B (3) C B A D E (3) B C A E D (3) A D E B C (3) C A B D E (2) B E D A C (2) B E C A D (2) B C E D A (2) B C E A D (2) B A E D C (2) A D E C B (2) A D C E B (2) A D C B E (2) E A D B C (1) D E C A B (1) D E A B C (1) D C E A B (1) D A E C B (1) C B E D A (1) C B A E D (1) B E D C A (1) B E A C D (1) B A C E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 12 10 -2 B -6 0 12 -8 -8 C -12 -12 0 -2 -8 D -10 8 2 0 -6 E 2 8 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 12 10 -2 B -6 0 12 -8 -8 C -12 -12 0 -2 -8 D -10 8 2 0 -6 E 2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=25 B=25 C=22 A=21 D=7 so D is eliminated. Round 2 votes counts: E=30 B=25 C=23 A=22 so A is eliminated. Round 3 votes counts: C=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:213 E:212 D:197 B:195 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 12 10 -2 B -6 0 12 -8 -8 C -12 -12 0 -2 -8 D -10 8 2 0 -6 E 2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 10 -2 B -6 0 12 -8 -8 C -12 -12 0 -2 -8 D -10 8 2 0 -6 E 2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 10 -2 B -6 0 12 -8 -8 C -12 -12 0 -2 -8 D -10 8 2 0 -6 E 2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6901: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) D A C E B (6) C E A D B (6) A E C D B (6) C E A B D (5) B E A C D (5) D A B E C (4) C E B A D (4) A E D C B (4) A E C B D (4) E C A B D (3) D C A E B (3) D B A C E (3) D A E C B (3) B C E A D (3) E A C B D (2) D B C E A (2) D B C A E (2) D B A E C (2) C B E A D (2) B D E A C (2) B D C E A (2) B A E D C (2) E C A D B (1) E B A C D (1) E A C D B (1) D C E A B (1) D C B E A (1) D C B A E (1) C E B D A (1) C D E B A (1) C D E A B (1) C B D E A (1) B E C A D (1) B D A C E (1) B A E C D (1) B A D E C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 18 8 8 B -4 0 -14 0 -8 C -18 14 0 -2 -8 D -8 0 2 0 -4 E -8 8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 8 8 B -4 0 -14 0 -8 C -18 14 0 -2 -8 D -8 0 2 0 -4 E -8 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=27 C=21 A=16 E=8 so E is eliminated. Round 2 votes counts: D=28 B=28 C=25 A=19 so A is eliminated. Round 3 votes counts: C=38 D=34 B=28 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:219 E:206 D:195 C:193 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 18 8 8 B -4 0 -14 0 -8 C -18 14 0 -2 -8 D -8 0 2 0 -4 E -8 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 8 8 B -4 0 -14 0 -8 C -18 14 0 -2 -8 D -8 0 2 0 -4 E -8 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 8 8 B -4 0 -14 0 -8 C -18 14 0 -2 -8 D -8 0 2 0 -4 E -8 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6902: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (13) D E C B A (8) D C E A B (8) A C D E B (7) D C E B A (6) B E D C A (5) D A C E B (3) B E C A D (3) B A E D C (3) A C E B D (3) A B E C D (3) E C B D A (2) E B D C A (2) E B C D A (2) D E B C A (2) D B A E C (2) C E D B A (2) C D E A B (2) C A D E B (2) B E A C D (2) A D B C E (2) A B D C E (2) D C A E B (1) D B E C A (1) D A C B E (1) D A B E C (1) D A B C E (1) C E D A B (1) C D A E B (1) B E D A C (1) B E C D A (1) B E A D C (1) B D A E C (1) A D C E B (1) A D C B E (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 2 -8 2 B 14 0 2 -6 -6 C -2 -2 0 -8 -8 D 8 6 8 0 10 E -2 6 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -8 2 B 14 0 2 -6 -6 C -2 -2 0 -8 -8 D 8 6 8 0 10 E -2 6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=30 A=22 C=8 E=6 so E is eliminated. Round 2 votes counts: D=34 B=34 A=22 C=10 so C is eliminated. Round 3 votes counts: D=40 B=36 A=24 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:202 E:201 A:191 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 2 -8 2 B 14 0 2 -6 -6 C -2 -2 0 -8 -8 D 8 6 8 0 10 E -2 6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -8 2 B 14 0 2 -6 -6 C -2 -2 0 -8 -8 D 8 6 8 0 10 E -2 6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -8 2 B 14 0 2 -6 -6 C -2 -2 0 -8 -8 D 8 6 8 0 10 E -2 6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) D E B C A (5) C A D B E (5) B E A D C (5) D E B A C (4) D C E A B (4) D C A E B (4) C D A E B (4) B A E C D (4) E D B A C (3) D A B E C (3) C E D B A (3) C E B A D (3) C A B D E (3) A D B C E (3) E B C A D (2) E B A D C (2) D E C B A (2) D A C B E (2) C D E B A (2) C D E A B (2) C B A E D (2) B E C A D (2) B E A C D (2) B C A E D (2) B A C E D (2) A C B D E (2) A B C E D (2) E D C B A (1) E D B C A (1) E C B D A (1) E B C D A (1) E B A C D (1) D E A B C (1) D C A B E (1) D A E B C (1) D A C E B (1) C E B D A (1) C D A B E (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -4 -10 -10 B 16 0 8 -10 -12 C 4 -8 0 -2 2 D 10 10 2 0 4 E 10 12 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -10 -10 B 16 0 8 -10 -12 C 4 -8 0 -2 2 D 10 10 2 0 4 E 10 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 E=19 B=17 A=10 so A is eliminated. Round 2 votes counts: D=32 C=29 B=20 E=19 so E is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:208 B:201 C:198 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -4 -10 -10 B 16 0 8 -10 -12 C 4 -8 0 -2 2 D 10 10 2 0 4 E 10 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -10 -10 B 16 0 8 -10 -12 C 4 -8 0 -2 2 D 10 10 2 0 4 E 10 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -10 -10 B 16 0 8 -10 -12 C 4 -8 0 -2 2 D 10 10 2 0 4 E 10 12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6904: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) E D A B C (7) B A C E D (7) A B E D C (7) C B A D E (6) D C E A B (5) C D E B A (5) B A E D C (5) E A B D C (4) D E C B A (4) D E C A B (4) C D B A E (4) C A B D E (4) B A E C D (4) C A B E D (3) E D B A C (2) D C E B A (2) C B D A E (2) B A C D E (2) A E B D C (2) A B E C D (2) A B C E D (2) E D A C B (1) E C D A B (1) E A D B C (1) D B C E A (1) B E A D C (1) B D A E C (1) B C A D E (1) B A D E C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 0 6 8 B -4 0 2 10 6 C 0 -2 0 4 4 D -6 -10 -4 0 0 E -8 -6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.612461 B: 0.000000 C: 0.387539 D: 0.000000 E: 0.000000 Sum of squares = 0.525294835695 Cumulative probabilities = A: 0.612461 B: 0.612461 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 6 8 B -4 0 2 10 6 C 0 -2 0 4 4 D -6 -10 -4 0 0 E -8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=22 E=16 D=16 A=14 so A is eliminated. Round 2 votes counts: B=33 C=32 E=19 D=16 so D is eliminated. Round 3 votes counts: C=39 B=34 E=27 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:207 C:203 E:191 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 6 8 B -4 0 2 10 6 C 0 -2 0 4 4 D -6 -10 -4 0 0 E -8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 8 B -4 0 2 10 6 C 0 -2 0 4 4 D -6 -10 -4 0 0 E -8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 8 B -4 0 2 10 6 C 0 -2 0 4 4 D -6 -10 -4 0 0 E -8 -6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6905: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (6) A D B C E (6) E C B D A (5) D E C A B (5) A B D C E (5) C E B A D (4) C B E A D (4) B C E A D (4) B A E C D (4) E B C A D (3) D E A C B (3) D A E B C (3) D A C E B (3) D A C B E (3) C E D B A (3) A B C D E (3) E C D B A (2) E C B A D (2) D C A B E (2) D A B C E (2) C E B D A (2) C B A E D (2) C A B D E (2) B E C A D (2) B E A C D (2) B C A E D (2) A D B E C (2) A B D E C (2) E D C B A (1) E D C A B (1) E B A D C (1) D E A B C (1) D C E A B (1) D C A E B (1) D A B E C (1) C D E B A (1) C B A D E (1) B A E D C (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 0 20 6 B 6 0 2 16 16 C 0 -2 0 12 18 D -20 -16 -12 0 -2 E -6 -16 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 20 6 B 6 0 2 16 16 C 0 -2 0 12 18 D -20 -16 -12 0 -2 E -6 -16 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=22 C=19 A=19 E=15 so E is eliminated. Round 2 votes counts: C=28 D=27 B=26 A=19 so A is eliminated. Round 3 votes counts: B=36 D=35 C=29 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:214 A:210 E:181 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 20 6 B 6 0 2 16 16 C 0 -2 0 12 18 D -20 -16 -12 0 -2 E -6 -16 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 20 6 B 6 0 2 16 16 C 0 -2 0 12 18 D -20 -16 -12 0 -2 E -6 -16 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 20 6 B 6 0 2 16 16 C 0 -2 0 12 18 D -20 -16 -12 0 -2 E -6 -16 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6906: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) D A C B E (6) C B E D A (6) A D E C B (6) D A C E B (5) C D B A E (5) C D A B E (5) B E C A D (5) C B D E A (4) C B D A E (4) B C E D A (4) A E D B C (4) A D B C E (4) E C B D A (3) E B A C D (3) A D C E B (3) A D B E C (3) E B A D C (2) E A D B C (2) D C A B E (2) B C D E A (2) A D C B E (2) E C D A B (1) E B C D A (1) E A B D C (1) D C B A E (1) D A E C B (1) C E D B A (1) C E B D A (1) C D E B A (1) C D E A B (1) B E C D A (1) B D C A E (1) B A E C D (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -10 -12 10 B 6 0 -16 -6 16 C 10 16 0 10 18 D 12 6 -10 0 16 E -10 -16 -18 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -12 10 B 6 0 -16 -6 16 C 10 16 0 10 18 D 12 6 -10 0 16 E -10 -16 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 E=19 D=15 B=15 so D is eliminated. Round 2 votes counts: A=35 C=31 E=19 B=15 so B is eliminated. Round 3 votes counts: C=38 A=37 E=25 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:212 B:200 A:191 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -12 10 B 6 0 -16 -6 16 C 10 16 0 10 18 D 12 6 -10 0 16 E -10 -16 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -12 10 B 6 0 -16 -6 16 C 10 16 0 10 18 D 12 6 -10 0 16 E -10 -16 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -12 10 B 6 0 -16 -6 16 C 10 16 0 10 18 D 12 6 -10 0 16 E -10 -16 -18 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6907: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) E C B D A (10) A D B C E (7) A C B D E (6) C A B D E (5) E C A B D (4) A B D C E (4) E D B A C (3) E A C D B (3) D B A E C (3) D B A C E (3) C A E B D (3) A D B E C (3) E D A B C (2) E B C D A (2) D B E C A (2) D B C A E (2) D A B E C (2) C E A B D (2) C B D E A (2) C B A D E (2) C A B E D (2) A E C D B (2) A C D B E (2) E C D B A (1) E B D C A (1) D E B C A (1) D E B A C (1) D A B C E (1) C E B A D (1) B E C D A (1) B D C E A (1) B D C A E (1) B A D C E (1) A E D B C (1) A D C B E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 2 6 B -4 0 6 -2 4 C 6 -6 0 0 -4 D -2 2 0 0 2 E -6 -4 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.220181 B: 0.220181 C: 0.250000 D: 0.309637 E: 0.000000 Sum of squares = 0.255334945331 Cumulative probabilities = A: 0.220181 B: 0.440363 C: 0.690363 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 2 6 B -4 0 6 -2 4 C 6 -6 0 0 -4 D -2 2 0 0 2 E -6 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=28 C=17 D=15 B=4 so B is eliminated. Round 2 votes counts: E=37 A=29 D=17 C=17 so D is eliminated. Round 3 votes counts: E=41 A=38 C=21 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:203 B:202 D:201 C:198 E:196 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -6 2 6 B -4 0 6 -2 4 C 6 -6 0 0 -4 D -2 2 0 0 2 E -6 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 2 6 B -4 0 6 -2 4 C 6 -6 0 0 -4 D -2 2 0 0 2 E -6 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 2 6 B -4 0 6 -2 4 C 6 -6 0 0 -4 D -2 2 0 0 2 E -6 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.25 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6908: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) D C E B A (8) D E C A B (7) B C A D E (5) B A C E D (5) A B C E D (5) E C D A B (4) D E C B A (4) B A C D E (4) A B E C D (4) E D A C B (3) D B E A C (3) C E D A B (3) C D E B A (3) A E B D C (3) A C E B D (3) A B E D C (3) E A D C B (2) D B E C A (2) B A D E C (2) A E D B C (2) A C B E D (2) D B C E A (1) C D E A B (1) C D B E A (1) C B D E A (1) C A B E D (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A C E (1) B C D A E (1) B C A E D (1) B A D C E (1) A E C D B (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -6 -8 -8 B -8 0 -4 -6 -6 C 6 4 0 -8 -4 D 8 6 8 0 -2 E 8 6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -6 -8 -8 B -8 0 -4 -6 -6 C 6 4 0 -8 -4 D 8 6 8 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 B=23 E=17 C=10 so C is eliminated. Round 2 votes counts: D=30 A=26 B=24 E=20 so E is eliminated. Round 3 votes counts: D=48 A=28 B=24 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:210 C:199 A:193 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 -8 -8 B -8 0 -4 -6 -6 C 6 4 0 -8 -4 D 8 6 8 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -8 -8 B -8 0 -4 -6 -6 C 6 4 0 -8 -4 D 8 6 8 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -8 -8 B -8 0 -4 -6 -6 C 6 4 0 -8 -4 D 8 6 8 0 -2 E 8 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6909: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) C A D E B (9) D C A B E (8) B E D A C (6) D B E A C (5) A C E B D (5) A B E C D (5) E B C A D (4) D C E B A (4) D B E C A (4) E B D C A (3) D A C B E (3) C A E D B (3) C A E B D (3) A C D E B (3) D C A E B (2) D A B C E (2) C E A B D (2) C D A E B (2) A C D B E (2) E D B C A (1) E C B D A (1) E C B A D (1) E A B C D (1) D C B E A (1) D B C E A (1) C E A D B (1) C D E A B (1) B E D C A (1) B E A D C (1) A E C B D (1) A E B C D (1) A D C B E (1) A D B C E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 16 -4 10 6 B -16 0 -10 -10 -16 C 4 10 0 12 12 D -10 10 -12 0 -2 E -6 16 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 10 6 B -16 0 -10 -10 -16 C 4 10 0 12 12 D -10 10 -12 0 -2 E -6 16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=21 A=21 E=20 B=8 so B is eliminated. Round 2 votes counts: D=30 E=28 C=21 A=21 so C is eliminated. Round 3 votes counts: A=36 D=33 E=31 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:219 A:214 E:200 D:193 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 10 6 B -16 0 -10 -10 -16 C 4 10 0 12 12 D -10 10 -12 0 -2 E -6 16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 10 6 B -16 0 -10 -10 -16 C 4 10 0 12 12 D -10 10 -12 0 -2 E -6 16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 10 6 B -16 0 -10 -10 -16 C 4 10 0 12 12 D -10 10 -12 0 -2 E -6 16 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6910: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) E C D B A (6) E C B D A (6) C E D B A (6) B A D C E (6) A B D C E (6) E C D A B (5) B D C A E (4) C E B D A (3) B D A C E (3) B C D E A (3) A D E B C (3) A D B C E (3) A B D E C (3) D C B A E (2) D B C A E (2) D B A C E (2) D A C B E (2) D A B C E (2) A E D C B (2) A E D B C (2) A D E C B (2) A D B E C (2) A B E D C (2) E C B A D (1) E A C D B (1) E A C B D (1) D C A B E (1) C E D A B (1) C D E B A (1) C D E A B (1) C D B E A (1) C B E D A (1) B E A C D (1) B A D E C (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -10 -4 2 B -2 0 -10 -18 -6 C 10 10 0 -2 2 D 4 18 2 0 2 E -2 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -4 2 B -2 0 -10 -18 -6 C 10 10 0 -2 2 D 4 18 2 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=26 B=18 C=14 D=11 so D is eliminated. Round 2 votes counts: E=31 A=30 B=22 C=17 so C is eliminated. Round 3 votes counts: E=43 A=31 B=26 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:213 C:210 E:200 A:195 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -10 -4 2 B -2 0 -10 -18 -6 C 10 10 0 -2 2 D 4 18 2 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -4 2 B -2 0 -10 -18 -6 C 10 10 0 -2 2 D 4 18 2 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -4 2 B -2 0 -10 -18 -6 C 10 10 0 -2 2 D 4 18 2 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6911: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (6) E D C A B (5) A D B E C (5) C E B D A (4) C E A D B (4) B D A E C (4) E D C B A (3) E C D A B (3) C B E D A (3) C A E D B (3) B E D C A (3) B D E A C (3) B A C D E (3) A B D C E (3) A B C D E (3) E D A C B (2) E C B D A (2) D E A B C (2) D A E B C (2) C E B A D (2) C B E A D (2) C A B E D (2) B E C D A (2) B C A E D (2) B C A D E (2) B A D C E (2) A D E C B (2) A C E D B (2) A C D E B (2) A C B D E (2) E D B C A (1) E D B A C (1) E C D B A (1) E B D C A (1) D E B A C (1) D E A C B (1) D B E A C (1) D A B E C (1) C A E B D (1) C A B D E (1) B D E C A (1) B D A C E (1) A D C E B (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 -8 -8 B 2 0 2 12 8 C 8 -2 0 6 6 D 8 -12 -6 0 -12 E 8 -8 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -8 -8 B 2 0 2 12 8 C 8 -2 0 6 6 D 8 -12 -6 0 -12 E 8 -8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995488 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 A=22 E=19 D=8 so D is eliminated. Round 2 votes counts: B=30 A=25 E=23 C=22 so C is eliminated. Round 3 votes counts: B=35 E=33 A=32 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:209 E:203 D:189 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 -8 -8 B 2 0 2 12 8 C 8 -2 0 6 6 D 8 -12 -6 0 -12 E 8 -8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995488 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -8 -8 B 2 0 2 12 8 C 8 -2 0 6 6 D 8 -12 -6 0 -12 E 8 -8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995488 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -8 -8 B 2 0 2 12 8 C 8 -2 0 6 6 D 8 -12 -6 0 -12 E 8 -8 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995488 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6912: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) A B E D C (9) A B C E D (8) D C E A B (7) D C E B A (5) A D B E C (4) D A E C B (3) C D E B A (3) C B E A D (3) B A E C D (3) A D E B C (3) A C B E D (3) A C B D E (3) E C D B A (2) E B C D A (2) D A E B C (2) D A C E B (2) C B E D A (2) A D C B E (2) A B D E C (2) A B D C E (2) E D B C A (1) E D B A C (1) E C B D A (1) E A B D C (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A B C (1) D C A E B (1) C E D B A (1) C E B D A (1) C D A E B (1) C D A B E (1) B E A C D (1) B C E A D (1) B A C E D (1) A E D B C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 38 32 20 28 B -38 0 14 8 16 C -32 -14 0 -2 -2 D -20 -8 2 0 -6 E -28 -16 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 38 32 20 28 B -38 0 14 8 16 C -32 -14 0 -2 -2 D -20 -8 2 0 -6 E -28 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=50 D=24 C=12 E=8 B=6 so B is eliminated. Round 2 votes counts: A=54 D=24 C=13 E=9 so E is eliminated. Round 3 votes counts: A=56 D=26 C=18 so C is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:259 B:200 D:184 E:182 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 38 32 20 28 B -38 0 14 8 16 C -32 -14 0 -2 -2 D -20 -8 2 0 -6 E -28 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 38 32 20 28 B -38 0 14 8 16 C -32 -14 0 -2 -2 D -20 -8 2 0 -6 E -28 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 38 32 20 28 B -38 0 14 8 16 C -32 -14 0 -2 -2 D -20 -8 2 0 -6 E -28 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6913: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) C B A D E (10) D C B A E (8) C D B A E (8) D C E A B (7) E A B C D (6) B A E C D (6) B A C E D (6) E D A B C (4) C A B E D (4) D E B A C (3) D E A B C (3) D C E B A (3) A B E C D (3) D E C A B (2) D E B C A (2) D C A B E (2) C B A E D (2) E B A D C (1) E B A C D (1) E A C B D (1) D E C B A (1) D C A E B (1) D B E A C (1) C D A B E (1) C A B D E (1) B A C D E (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -4 8 10 B 6 0 -4 8 8 C 4 4 0 4 10 D -8 -8 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 8 10 B 6 0 -4 8 8 C 4 4 0 4 10 D -8 -8 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=26 E=23 B=13 A=5 so A is eliminated. Round 2 votes counts: D=33 C=27 E=24 B=16 so B is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:209 A:204 D:194 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 8 10 B 6 0 -4 8 8 C 4 4 0 4 10 D -8 -8 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 8 10 B 6 0 -4 8 8 C 4 4 0 4 10 D -8 -8 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 8 10 B 6 0 -4 8 8 C 4 4 0 4 10 D -8 -8 -4 0 8 E -10 -8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6914: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) E B D A C (8) C A D B E (8) C A D E B (7) B D E A C (7) B E D A C (6) E A C D B (5) D B A C E (4) E C B A D (3) C B D A E (3) E A D B C (2) D B A E C (2) C D B A E (2) C B E D A (2) C A E B D (2) B D E C A (2) E D A B C (1) E C A B D (1) E B D C A (1) E A C B D (1) D E A B C (1) D B C A E (1) D A E B C (1) D A B E C (1) C E A D B (1) C E A B D (1) C D A B E (1) C B A D E (1) B E D C A (1) B E C D A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 0 10 B -2 0 -12 -10 -4 C 4 12 0 12 4 D 0 10 -12 0 0 E -10 4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999057 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 0 10 B -2 0 -12 -10 -4 C 4 12 0 12 4 D 0 10 -12 0 0 E -10 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=22 B=22 D=10 A=7 so A is eliminated. Round 2 votes counts: C=42 E=24 B=22 D=12 so D is eliminated. Round 3 votes counts: C=43 B=30 E=27 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:204 D:199 E:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 0 10 B -2 0 -12 -10 -4 C 4 12 0 12 4 D 0 10 -12 0 0 E -10 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 10 B -2 0 -12 -10 -4 C 4 12 0 12 4 D 0 10 -12 0 0 E -10 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 10 B -2 0 -12 -10 -4 C 4 12 0 12 4 D 0 10 -12 0 0 E -10 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6915: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (5) B C E D A (5) A D E C B (5) A D E B C (5) A E D B C (4) A B E D C (4) E C D B A (3) E B A D C (3) D A E C B (3) C E D B A (3) B C A D E (3) A E B D C (3) E D A C B (2) E C B D A (2) E B C D A (2) E A D B C (2) D E A C B (2) D C E A B (2) D A C E B (2) C D E A B (2) C D A B E (2) C B E D A (2) C B D E A (2) B E A D C (2) B E A C D (2) B C E A D (2) A D C E B (2) A D B E C (2) A B D E C (2) A B D C E (2) E D C A B (1) E D A B C (1) E A D C B (1) D C A B E (1) D A C B E (1) C E B D A (1) C D E B A (1) C D A E B (1) C B D A E (1) B E C D A (1) B E C A D (1) B C D A E (1) B C A E D (1) B A E D C (1) B A E C D (1) B A C E D (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 16 10 2 10 B -16 0 4 -8 -16 C -10 -4 0 -16 -10 D -2 8 16 0 -2 E -10 16 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999501 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 2 10 B -16 0 4 -8 -16 C -10 -4 0 -16 -10 D -2 8 16 0 -2 E -10 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998307 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=22 E=17 D=16 C=15 so C is eliminated. Round 2 votes counts: A=30 B=27 D=22 E=21 so E is eliminated. Round 3 votes counts: B=35 A=33 D=32 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:210 E:209 B:182 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 2 10 B -16 0 4 -8 -16 C -10 -4 0 -16 -10 D -2 8 16 0 -2 E -10 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998307 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 2 10 B -16 0 4 -8 -16 C -10 -4 0 -16 -10 D -2 8 16 0 -2 E -10 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998307 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 2 10 B -16 0 4 -8 -16 C -10 -4 0 -16 -10 D -2 8 16 0 -2 E -10 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998307 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6916: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) C A D B E (8) E B A C D (7) C D A B E (7) E B C A D (5) D C A B E (5) B E A D C (5) E D B A C (4) C D E A B (4) C D A E B (4) B E A C D (4) B A E D C (4) E D C B A (3) D E C A B (3) C E D B A (3) A B C D E (3) E C D B A (2) E B D C A (2) D C E A B (2) D C A E B (2) D A C B E (2) B A C E D (2) E D B C A (1) E B C D A (1) D E C B A (1) D A B C E (1) C E D A B (1) C E A D B (1) C A B D E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -12 4 -18 B 8 0 -2 -12 -12 C 12 2 0 8 -4 D -4 12 -8 0 -10 E 18 12 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 4 -18 B 8 0 -2 -12 -12 C 12 2 0 8 -4 D -4 12 -8 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=29 D=16 B=15 A=5 so A is eliminated. Round 2 votes counts: E=35 C=30 B=18 D=17 so D is eliminated. Round 3 votes counts: C=42 E=39 B=19 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:209 D:195 B:191 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 4 -18 B 8 0 -2 -12 -12 C 12 2 0 8 -4 D -4 12 -8 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 4 -18 B 8 0 -2 -12 -12 C 12 2 0 8 -4 D -4 12 -8 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 4 -18 B 8 0 -2 -12 -12 C 12 2 0 8 -4 D -4 12 -8 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6917: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) C B E D A (5) B C E A D (5) A D E B C (5) A D C E B (5) C D E B A (4) C D A E B (4) B C A E D (4) A D E C B (4) E B D A C (3) D E C A B (3) D E A C B (3) D A E C B (3) C B D E A (3) C B A D E (3) B E C D A (3) B E A D C (3) A E D B C (3) E D B C A (2) D E A B C (2) D C E A B (2) C D B E A (2) C D A B E (2) B E D A C (2) B A E D C (2) B A C E D (2) E D C B A (1) E D A C B (1) E B D C A (1) E A D B C (1) E A B D C (1) D A E B C (1) C D E A B (1) C A D E B (1) C A D B E (1) C A B D E (1) B E C A D (1) A E B D C (1) A D C B E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 8 2 -8 -10 B -8 0 -2 -18 -18 C -2 2 0 -14 -6 D 8 18 14 0 4 E 10 18 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -8 -10 B -8 0 -2 -18 -18 C -2 2 0 -14 -6 D 8 18 14 0 4 E 10 18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=22 A=21 E=16 D=14 so D is eliminated. Round 2 votes counts: C=29 A=25 E=24 B=22 so B is eliminated. Round 3 votes counts: C=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:222 E:215 A:196 C:190 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -8 -10 B -8 0 -2 -18 -18 C -2 2 0 -14 -6 D 8 18 14 0 4 E 10 18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -8 -10 B -8 0 -2 -18 -18 C -2 2 0 -14 -6 D 8 18 14 0 4 E 10 18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -8 -10 B -8 0 -2 -18 -18 C -2 2 0 -14 -6 D 8 18 14 0 4 E 10 18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6918: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (13) A D C B E (11) E A D C B (7) B C D A E (7) E A B D C (6) D C A B E (6) D A C B E (5) C D B A E (4) A D C E B (4) E B D C A (3) E A D B C (3) C D A B E (3) B C D E A (3) A E D C B (3) E B C A D (2) E B A C D (2) E A B C D (2) B E C D A (2) B D C A E (2) B C E D A (2) E B D A C (1) E B A D C (1) E A C B D (1) D C B A E (1) C B D A E (1) C A B D E (1) B E D C A (1) A E C D B (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -2 -8 2 B -10 0 0 0 0 C 2 0 0 -10 2 D 8 0 10 0 0 E -2 0 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.161199 C: 0.000000 D: 0.459106 E: 0.379695 Sum of squares = 0.380931560082 Cumulative probabilities = A: 0.000000 B: 0.161199 C: 0.161199 D: 0.620305 E: 1.000000 A B C D E A 0 10 -2 -8 2 B -10 0 0 0 0 C 2 0 0 -10 2 D 8 0 10 0 0 E -2 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.262294 C: 0.000000 D: 0.409838 E: 0.327869 Sum of squares = 0.344262712148 Cumulative probabilities = A: 0.000000 B: 0.262294 C: 0.262294 D: 0.672131 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 A=21 B=17 D=12 C=9 so C is eliminated. Round 2 votes counts: E=41 A=22 D=19 B=18 so B is eliminated. Round 3 votes counts: E=46 D=32 A=22 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:209 A:201 E:198 C:197 B:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -2 -8 2 B -10 0 0 0 0 C 2 0 0 -10 2 D 8 0 10 0 0 E -2 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.262294 C: 0.000000 D: 0.409838 E: 0.327869 Sum of squares = 0.344262712148 Cumulative probabilities = A: 0.000000 B: 0.262294 C: 0.262294 D: 0.672131 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -8 2 B -10 0 0 0 0 C 2 0 0 -10 2 D 8 0 10 0 0 E -2 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.262294 C: 0.000000 D: 0.409838 E: 0.327869 Sum of squares = 0.344262712148 Cumulative probabilities = A: 0.000000 B: 0.262294 C: 0.262294 D: 0.672131 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -8 2 B -10 0 0 0 0 C 2 0 0 -10 2 D 8 0 10 0 0 E -2 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.262294 C: 0.000000 D: 0.409838 E: 0.327869 Sum of squares = 0.344262712148 Cumulative probabilities = A: 0.000000 B: 0.262294 C: 0.262294 D: 0.672131 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6919: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) C B D A E (8) E D C B A (7) E C D B A (5) D C B E A (4) D B C A E (4) C B D E A (4) A B D C E (4) A B C D E (4) E D A B C (3) E A D C B (3) C E B D A (3) E C A D B (2) E C A B D (2) E A B C D (2) C E D B A (2) C D B E A (2) B D A C E (2) B C D A E (2) A E B D C (2) A D B E C (2) A C B E D (2) A B C E D (2) E D C A B (1) E D B C A (1) E D A C B (1) E C B A D (1) E A C B D (1) E A B D C (1) D E C B A (1) D C B A E (1) D B C E A (1) D B A E C (1) D B A C E (1) D A B C E (1) C D B A E (1) C B E D A (1) C B A E D (1) C B A D E (1) C A B E D (1) A E C B D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -12 -14 -12 B 8 0 -12 -4 6 C 12 12 0 -2 4 D 14 4 2 0 -10 E 12 -6 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.125000 Sum of squares = 0.468749999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.875000 E: 1.000000 A B C D E A 0 -8 -12 -14 -12 B 8 0 -12 -4 6 C 12 12 0 -2 4 D 14 4 2 0 -10 E 12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=24 A=20 D=14 B=4 so B is eliminated. Round 2 votes counts: E=38 C=26 A=20 D=16 so D is eliminated. Round 3 votes counts: E=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:206 D:205 B:199 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -14 -12 B 8 0 -12 -4 6 C 12 12 0 -2 4 D 14 4 2 0 -10 E 12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.875000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -14 -12 B 8 0 -12 -4 6 C 12 12 0 -2 4 D 14 4 2 0 -10 E 12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -14 -12 B 8 0 -12 -4 6 C 12 12 0 -2 4 D 14 4 2 0 -10 E 12 -6 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.250000 E: 0.125000 Sum of squares = 0.468750000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.625000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6920: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) E A B C D (7) C B E A D (7) D B C E A (5) D A C B E (5) C B D E A (5) D B E C A (4) D C B A E (3) D A C E B (3) B C E D A (3) B C D E A (3) A D C E B (3) E B C A D (2) E B A D C (2) E A C B D (2) D C B E A (2) D A E B C (2) C B E D A (2) B E C D A (2) B E C A D (2) B D C E A (2) A E D C B (2) A D E C B (2) A C E B D (2) E B A C D (1) E A B D C (1) D E B A C (1) D E A B C (1) D C A B E (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B E C (1) C D B E A (1) C D B A E (1) C A B E D (1) C A B D E (1) B E D C A (1) B C E A D (1) A E D B C (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 0 -2 -16 B 6 0 -8 16 12 C 0 8 0 10 6 D 2 -16 -10 0 -2 E 16 -12 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153662 B: 0.000000 C: 0.846338 D: 0.000000 E: 0.000000 Sum of squares = 0.739900240271 Cumulative probabilities = A: 0.153662 B: 0.153662 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 -16 B 6 0 -8 16 12 C 0 8 0 10 6 D 2 -16 -10 0 -2 E 16 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845369 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=22 C=18 E=15 B=14 so B is eliminated. Round 2 votes counts: D=33 C=25 A=22 E=20 so E is eliminated. Round 3 votes counts: A=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:213 C:212 E:200 A:188 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -2 -16 B 6 0 -8 16 12 C 0 8 0 10 6 D 2 -16 -10 0 -2 E 16 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845369 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 -16 B 6 0 -8 16 12 C 0 8 0 10 6 D 2 -16 -10 0 -2 E 16 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845369 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 -16 B 6 0 -8 16 12 C 0 8 0 10 6 D 2 -16 -10 0 -2 E 16 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.727273 D: 0.000000 E: 0.000000 Sum of squares = 0.603305845369 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6921: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) C A D B E (7) C A D E B (6) B E C D A (6) E B D A C (5) A D E B C (5) D A B E C (4) C A B D E (4) B E D C A (4) B E D A C (4) C E B D A (3) C E A B D (3) C B E A D (3) C A E D B (3) B D E A C (3) B C E D A (3) A D E C B (3) E C B D A (2) D E A B C (2) D A E B C (2) C B A E D (2) C B A D E (2) C A B E D (2) A D C B E (2) A C D E B (2) E C A D B (1) E B D C A (1) E A D C B (1) D B E A C (1) C E B A D (1) B D A C E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -26 -2 -6 B 2 0 -14 18 20 C 26 14 0 20 10 D 2 -18 -20 0 -8 E 6 -20 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -26 -2 -6 B 2 0 -14 18 20 C 26 14 0 20 10 D 2 -18 -20 0 -8 E 6 -20 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 B=21 A=14 E=10 D=9 so D is eliminated. Round 2 votes counts: C=46 B=22 A=20 E=12 so E is eliminated. Round 3 votes counts: C=49 B=28 A=23 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:235 B:213 E:192 A:182 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -26 -2 -6 B 2 0 -14 18 20 C 26 14 0 20 10 D 2 -18 -20 0 -8 E 6 -20 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -26 -2 -6 B 2 0 -14 18 20 C 26 14 0 20 10 D 2 -18 -20 0 -8 E 6 -20 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -26 -2 -6 B 2 0 -14 18 20 C 26 14 0 20 10 D 2 -18 -20 0 -8 E 6 -20 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6922: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) B C E D A (9) B C A D E (6) D E A C B (5) A D E B C (5) E C B D A (4) D A E B C (4) D A B E C (4) B C E A D (4) B C A E D (4) C E B D A (3) C B E A D (3) A E D C B (3) A D B C E (3) E D A C B (2) E C A D B (2) D E A B C (2) D B E C A (2) D A E C B (2) C B E D A (2) B A D C E (2) A D C B E (2) A B D C E (2) E D C A B (1) E D B C A (1) E C D A B (1) E B C D A (1) E A D C B (1) E A C D B (1) D E C A B (1) D B A E C (1) D B A C E (1) C E B A D (1) C A E B D (1) B D C E A (1) B D C A E (1) B C D A E (1) B A C D E (1) Total count = 100 A B C D E A 0 4 2 2 6 B -4 0 10 -8 -2 C -2 -10 0 -12 -6 D -2 8 12 0 12 E -6 2 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999599 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 6 B -4 0 10 -8 -2 C -2 -10 0 -12 -6 D -2 8 12 0 12 E -6 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=25 D=22 E=14 C=10 so C is eliminated. Round 2 votes counts: B=34 A=26 D=22 E=18 so E is eliminated. Round 3 votes counts: B=43 A=30 D=27 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:207 B:198 E:195 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 6 B -4 0 10 -8 -2 C -2 -10 0 -12 -6 D -2 8 12 0 12 E -6 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 6 B -4 0 10 -8 -2 C -2 -10 0 -12 -6 D -2 8 12 0 12 E -6 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 6 B -4 0 10 -8 -2 C -2 -10 0 -12 -6 D -2 8 12 0 12 E -6 2 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6923: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) B E D A C (8) A C B D E (8) C A E D B (7) C A D E B (7) E D C A B (6) B D E A C (5) B A C D E (5) E D B A C (4) C A B D E (4) A B C D E (4) E D C B A (3) D E C A B (3) B D A E C (3) A C B E D (3) E C D A B (2) E B D C A (2) D E B A C (2) C A E B D (2) C A B E D (2) B A E D C (2) D E C B A (1) C E A D B (1) C D A E B (1) B E A D C (1) B A E C D (1) B A D C E (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -2 0 4 B -2 0 -2 4 -2 C 2 2 0 -2 -6 D 0 -4 2 0 -10 E -4 2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888882 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 2 -2 0 4 B -2 0 -2 4 -2 C 2 2 0 -2 -6 D 0 -4 2 0 -10 E -4 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888848 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=27 B=27 C=24 A=16 D=6 so D is eliminated. Round 2 votes counts: E=33 B=27 C=24 A=16 so A is eliminated. Round 3 votes counts: C=36 E=33 B=31 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:202 B:199 C:198 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 0 4 B -2 0 -2 4 -2 C 2 2 0 -2 -6 D 0 -4 2 0 -10 E -4 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888848 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 4 B -2 0 -2 4 -2 C 2 2 0 -2 -6 D 0 -4 2 0 -10 E -4 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888848 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 4 B -2 0 -2 4 -2 C 2 2 0 -2 -6 D 0 -4 2 0 -10 E -4 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888848 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6924: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) E C A D B (6) C E A D B (6) A B C D E (5) E C D A B (4) C E D B A (4) C E D A B (4) B D A E C (4) B D A C E (4) B A D C E (4) C D E B A (3) C D B E A (3) B A D E C (3) A E B D C (3) A C B D E (3) A B D E C (3) E D C B A (2) E D B C A (2) E D B A C (2) E C D B A (2) D C B E A (2) D B E A C (2) A E C B D (2) E D C A B (1) E D A C B (1) E D A B C (1) E A D B C (1) E A C D B (1) E A C B D (1) D E B C A (1) D B C E A (1) D B C A E (1) D B A C E (1) C A E B D (1) C A B E D (1) C A B D E (1) B D C A E (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 16 8 8 0 B -16 0 0 -2 2 C -8 0 0 -2 14 D -8 2 2 0 4 E 0 -2 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.802300 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.197700 Sum of squares = 0.682771002062 Cumulative probabilities = A: 0.802300 B: 0.802300 C: 0.802300 D: 0.802300 E: 1.000000 A B C D E A 0 16 8 8 0 B -16 0 0 -2 2 C -8 0 0 -2 14 D -8 2 2 0 4 E 0 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190083594 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 C=23 B=16 D=8 so D is eliminated. Round 2 votes counts: A=29 E=25 C=25 B=21 so B is eliminated. Round 3 votes counts: A=45 C=28 E=27 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:202 D:200 B:192 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 8 0 B -16 0 0 -2 2 C -8 0 0 -2 14 D -8 2 2 0 4 E 0 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190083594 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 8 0 B -16 0 0 -2 2 C -8 0 0 -2 14 D -8 2 2 0 4 E 0 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190083594 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 8 0 B -16 0 0 -2 2 C -8 0 0 -2 14 D -8 2 2 0 4 E 0 -2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.636364 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.363636 Sum of squares = 0.537190083594 Cumulative probabilities = A: 0.636364 B: 0.636364 C: 0.636364 D: 0.636364 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6925: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) E D A B C (9) D E A B C (8) C D B A E (7) E A B D C (6) E A B C D (6) D C E B A (5) D C B A E (5) A B E C D (5) D E C A B (4) C B A D E (4) E D C A B (3) E C A B D (3) C E B A D (3) B A C E D (3) E C B A D (2) D B A C E (2) C B E A D (2) C B D A E (2) E C D A B (1) E A D B C (1) D E C B A (1) D C A E B (1) D A E B C (1) D A B E C (1) D A B C E (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -10 -2 -8 B -2 0 -10 0 -8 C 10 10 0 0 -4 D 2 0 0 0 -14 E 8 8 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -10 -2 -8 B -2 0 -10 0 -8 C 10 10 0 0 -4 D 2 0 0 0 -14 E 8 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 D=29 A=6 B=4 so B is eliminated. Round 2 votes counts: E=31 C=30 D=29 A=10 so A is eliminated. Round 3 votes counts: E=37 C=33 D=30 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:208 D:194 A:191 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -10 -2 -8 B -2 0 -10 0 -8 C 10 10 0 0 -4 D 2 0 0 0 -14 E 8 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -2 -8 B -2 0 -10 0 -8 C 10 10 0 0 -4 D 2 0 0 0 -14 E 8 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -2 -8 B -2 0 -10 0 -8 C 10 10 0 0 -4 D 2 0 0 0 -14 E 8 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6926: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) A D C E B (8) B E C D A (7) C A D B E (6) B C E D A (6) C B E D A (5) B E D C A (5) A D E B C (5) E B D A C (4) D E A B C (4) D A E B C (4) E D A B C (3) D A C E B (3) C A B E D (3) B E C A D (3) E D B A C (2) E A D B C (2) C D A B E (2) C B D E A (2) C B A D E (2) C A B D E (2) B C E A D (2) A C D E B (2) E B D C A (1) E B A D C (1) D E B A C (1) D B E C A (1) D A E C B (1) C B E A D (1) C B D A E (1) C B A E D (1) A E D C B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 0 -4 0 B -10 0 2 -8 0 C 0 -2 0 -8 -6 D 4 8 8 0 6 E 0 0 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -4 0 B -10 0 2 -8 0 C 0 -2 0 -8 -6 D 4 8 8 0 6 E 0 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=25 A=25 B=23 D=14 E=13 so E is eliminated. Round 2 votes counts: B=29 A=27 C=25 D=19 so D is eliminated. Round 3 votes counts: A=42 B=33 C=25 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:203 E:200 B:192 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -4 0 B -10 0 2 -8 0 C 0 -2 0 -8 -6 D 4 8 8 0 6 E 0 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -4 0 B -10 0 2 -8 0 C 0 -2 0 -8 -6 D 4 8 8 0 6 E 0 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -4 0 B -10 0 2 -8 0 C 0 -2 0 -8 -6 D 4 8 8 0 6 E 0 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6927: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) E A C D B (7) D B C E A (6) A E C D B (5) A D E B C (5) C B E D A (4) A E D C B (4) E C A D B (3) D B C A E (3) C E A D B (3) C E A B D (3) C B D E A (3) E C D A B (2) D B A C E (2) D A E C B (2) D A E B C (2) D A B E C (2) C E D A B (2) B D C A E (2) B D A C E (2) B C E A D (2) B A E C D (2) A E B C D (2) A D E C B (2) A B D E C (2) E A D C B (1) D C E B A (1) D C E A B (1) C E D B A (1) C E B A D (1) C D E B A (1) C D E A B (1) C D B E A (1) C B E A D (1) B D C E A (1) B D A E C (1) B C E D A (1) B C A E D (1) A E D B C (1) A E C B D (1) A E B D C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -8 -2 -12 B -8 0 0 -18 -4 C 8 0 0 16 6 D 2 18 -16 0 2 E 12 4 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300484 C: 0.699516 D: 0.000000 E: 0.000000 Sum of squares = 0.579613104554 Cumulative probabilities = A: 0.000000 B: 0.300484 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -2 -12 B -8 0 0 -18 -4 C 8 0 0 16 6 D 2 18 -16 0 2 E 12 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.470588 C: 0.529412 D: 0.000000 E: 0.000000 Sum of squares = 0.501730112387 Cumulative probabilities = A: 0.000000 B: 0.470588 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=22 C=21 D=19 E=13 so E is eliminated. Round 2 votes counts: A=33 C=26 B=22 D=19 so D is eliminated. Round 3 votes counts: A=39 B=33 C=28 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:215 E:204 D:203 A:193 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -2 -12 B -8 0 0 -18 -4 C 8 0 0 16 6 D 2 18 -16 0 2 E 12 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.470588 C: 0.529412 D: 0.000000 E: 0.000000 Sum of squares = 0.501730112387 Cumulative probabilities = A: 0.000000 B: 0.470588 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -2 -12 B -8 0 0 -18 -4 C 8 0 0 16 6 D 2 18 -16 0 2 E 12 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.470588 C: 0.529412 D: 0.000000 E: 0.000000 Sum of squares = 0.501730112387 Cumulative probabilities = A: 0.000000 B: 0.470588 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -2 -12 B -8 0 0 -18 -4 C 8 0 0 16 6 D 2 18 -16 0 2 E 12 4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.470588 C: 0.529412 D: 0.000000 E: 0.000000 Sum of squares = 0.501730112387 Cumulative probabilities = A: 0.000000 B: 0.470588 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6928: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) A D B E C (8) B E C D A (7) D B C E A (6) D A B C E (6) C E B D A (6) A E C D B (5) A E C B D (4) E C B D A (3) D B A C E (3) C E B A D (3) B D E C A (3) A E B D C (3) A D E B C (3) A D C E B (3) A D B C E (3) A C E D B (3) E C A B D (2) D B C A E (2) B C D E A (2) E A C B D (1) D C B A E (1) D B A E C (1) C D E B A (1) C B E D A (1) B E A D C (1) B D E A C (1) B D C E A (1) B C E D A (1) A E B C D (1) A D E C B (1) A C E B D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 0 10 0 B 8 0 6 6 -4 C 0 -6 0 6 -12 D -10 -6 -6 0 -8 E 0 4 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.185825 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.814175 Sum of squares = 0.69741147421 Cumulative probabilities = A: 0.185825 B: 0.185825 C: 0.185825 D: 0.185825 E: 1.000000 A B C D E A 0 -8 0 10 0 B 8 0 6 6 -4 C 0 -6 0 6 -12 D -10 -6 -6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556547 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=19 E=17 B=16 C=11 so C is eliminated. Round 2 votes counts: A=37 E=26 D=20 B=17 so B is eliminated. Round 3 votes counts: A=37 E=36 D=27 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:212 B:208 A:201 C:194 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 10 0 B 8 0 6 6 -4 C 0 -6 0 6 -12 D -10 -6 -6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556547 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 10 0 B 8 0 6 6 -4 C 0 -6 0 6 -12 D -10 -6 -6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556547 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 10 0 B 8 0 6 6 -4 C 0 -6 0 6 -12 D -10 -6 -6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556547 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6929: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) A C B E D (10) C A B E D (9) D C A B E (8) E D B A C (7) E B D A C (7) B E A C D (7) C A D B E (5) C A B D E (5) E B A C D (4) D C A E B (4) A C E B D (3) D E B C A (2) D B E A C (2) C A E D B (2) B A C E D (2) B A C D E (2) E C A D B (1) E B A D C (1) E A C D B (1) D E C A B (1) D C E A B (1) D B A E C (1) C D A B E (1) C A D E B (1) B D E A C (1) B A E C D (1) Total count = 100 A B C D E A 0 4 20 8 8 B -4 0 -4 4 8 C -20 4 0 8 6 D -8 -4 -8 0 -10 E -8 -8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 20 8 8 B -4 0 -4 4 8 C -20 4 0 8 6 D -8 -4 -8 0 -10 E -8 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=23 E=21 B=13 A=13 so B is eliminated. Round 2 votes counts: D=31 E=28 C=23 A=18 so A is eliminated. Round 3 votes counts: C=40 D=31 E=29 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:220 B:202 C:199 E:194 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 8 8 B -4 0 -4 4 8 C -20 4 0 8 6 D -8 -4 -8 0 -10 E -8 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 8 8 B -4 0 -4 4 8 C -20 4 0 8 6 D -8 -4 -8 0 -10 E -8 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 8 8 B -4 0 -4 4 8 C -20 4 0 8 6 D -8 -4 -8 0 -10 E -8 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6930: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) C D A E B (5) B C E D A (5) A D E C B (5) A D E B C (5) A D C E B (5) A E D B C (4) A C D E B (4) E B D C A (3) C D E B A (3) B A E D C (3) A E B D C (3) D E A C B (2) D C E A B (2) D C A E B (2) D A C E B (2) C B E D A (2) C A D E B (2) C A D B E (2) B E D A C (2) B E C A D (2) B E A D C (2) B C E A D (2) A D C B E (2) A B C D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D A C (1) E B A D C (1) E A B D C (1) D E C A B (1) D C E B A (1) C D E A B (1) C D B E A (1) C B D E A (1) C B D A E (1) C B A D E (1) B E D C A (1) B E A C D (1) B C A E D (1) B A E C D (1) A C D B E (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 2 6 8 B -8 0 4 -6 -12 C -2 -4 0 -4 -2 D -6 6 4 0 4 E -8 12 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 6 8 B -8 0 4 -6 -12 C -2 -4 0 -4 -2 D -6 6 4 0 4 E -8 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=28 C=19 D=10 E=9 so E is eliminated. Round 2 votes counts: A=35 B=33 C=19 D=13 so D is eliminated. Round 3 votes counts: A=39 B=35 C=26 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:204 E:201 C:194 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 6 8 B -8 0 4 -6 -12 C -2 -4 0 -4 -2 D -6 6 4 0 4 E -8 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 6 8 B -8 0 4 -6 -12 C -2 -4 0 -4 -2 D -6 6 4 0 4 E -8 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 6 8 B -8 0 4 -6 -12 C -2 -4 0 -4 -2 D -6 6 4 0 4 E -8 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6931: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) E D C A B (6) E C D B A (6) D C E B A (6) B C A D E (6) E D A C B (5) D E C A B (5) C E D B A (5) A B E D C (4) E D C B A (3) C D E B A (3) C D B E A (3) C B D E A (3) B C A E D (3) A E B D C (3) A D E B C (3) E A D B C (2) E A B C D (2) D E A C B (2) D C B E A (2) D C B A E (2) C B D A E (2) A B E C D (2) A B D E C (2) E B A C D (1) E A D C B (1) D B C A E (1) D A E B C (1) C E B D A (1) C B E D A (1) B C E A D (1) B A C D E (1) A E D B C (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -18 -14 -16 B 16 0 -12 -16 -12 C 18 12 0 -2 0 D 14 16 2 0 -12 E 16 12 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.546226 D: 0.000000 E: 0.453774 Sum of squares = 0.504273752702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.546226 D: 0.546226 E: 1.000000 A B C D E A 0 -16 -18 -14 -16 B 16 0 -12 -16 -12 C 18 12 0 -2 0 D 14 16 2 0 -12 E 16 12 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=19 B=19 C=18 A=18 so C is eliminated. Round 2 votes counts: E=32 D=25 B=25 A=18 so A is eliminated. Round 3 votes counts: E=36 B=35 D=29 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:214 D:210 B:188 A:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -18 -14 -16 B 16 0 -12 -16 -12 C 18 12 0 -2 0 D 14 16 2 0 -12 E 16 12 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 -14 -16 B 16 0 -12 -16 -12 C 18 12 0 -2 0 D 14 16 2 0 -12 E 16 12 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 -14 -16 B 16 0 -12 -16 -12 C 18 12 0 -2 0 D 14 16 2 0 -12 E 16 12 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6932: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) E A D B C (10) D E A C B (6) B C A E D (6) A B E C D (6) A E D B C (5) D E C B A (4) C B A D E (4) A E B D C (4) E D A B C (3) E A B D C (3) D C E B A (3) D C E A B (3) C B D A E (3) A B C E D (3) E D B C A (2) E B D C A (2) D A E C B (2) C D B A E (2) C B D E A (2) B A C E D (2) E D B A C (1) E B D A C (1) E B A D C (1) D E C A B (1) D E A B C (1) D C A E B (1) D A C E B (1) C D E B A (1) C D A E B (1) C B A E D (1) C A D B E (1) B A E C D (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 4 -2 -10 B -6 0 2 -20 -14 C -4 -2 0 -10 -8 D 2 20 10 0 -2 E 10 14 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 4 -2 -10 B -6 0 2 -20 -14 C -4 -2 0 -10 -8 D 2 20 10 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 D=22 A=20 B=9 so B is eliminated. Round 2 votes counts: C=32 E=23 A=23 D=22 so D is eliminated. Round 3 votes counts: C=39 E=35 A=26 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:215 A:199 C:188 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 -2 -10 B -6 0 2 -20 -14 C -4 -2 0 -10 -8 D 2 20 10 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -2 -10 B -6 0 2 -20 -14 C -4 -2 0 -10 -8 D 2 20 10 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -2 -10 B -6 0 2 -20 -14 C -4 -2 0 -10 -8 D 2 20 10 0 -2 E 10 14 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6933: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (13) E D C A B (9) E D B C A (5) E B D A C (5) C A D E B (4) B A D E C (4) E D C B A (3) D E A C B (3) C D E A B (3) B A E D C (3) B A C E D (3) B A C D E (3) A B D C E (3) A B C D E (3) E C D A B (2) E C B D A (2) D E C A B (2) D B A E C (2) C A E D B (2) B E D A C (2) B E A D C (2) B E A C D (2) B D A E C (2) B A D C E (2) A C B D E (2) E C D B A (1) E B D C A (1) D E B A C (1) D C E A B (1) D A E C B (1) D A E B C (1) C E B A D (1) C E A D B (1) C D A E B (1) C A D B E (1) C A B E D (1) B E C D A (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -10 -22 -20 B -8 0 -8 -14 -26 C 10 8 0 -4 -8 D 22 14 4 0 -20 E 20 26 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -10 -22 -20 B -8 0 -8 -14 -26 C 10 8 0 -4 -8 D 22 14 4 0 -20 E 20 26 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 B=25 D=11 A=9 so A is eliminated. Round 2 votes counts: B=31 C=30 E=28 D=11 so D is eliminated. Round 3 votes counts: E=36 B=33 C=31 so C is eliminated. Round 4 votes counts: E=63 B=37 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:237 D:210 C:203 A:178 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -10 -22 -20 B -8 0 -8 -14 -26 C 10 8 0 -4 -8 D 22 14 4 0 -20 E 20 26 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 -22 -20 B -8 0 -8 -14 -26 C 10 8 0 -4 -8 D 22 14 4 0 -20 E 20 26 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 -22 -20 B -8 0 -8 -14 -26 C 10 8 0 -4 -8 D 22 14 4 0 -20 E 20 26 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6934: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (11) D C B A E (11) B E A C D (6) D B C E A (5) E B A C D (4) E A D B C (4) C B A E D (4) E D A B C (3) D E A C B (3) D A C E B (3) B C E A D (3) B C D E A (3) A E C B D (3) A C E B D (3) D C B E A (2) D C A E B (2) D C A B E (2) D A E C B (2) C A B D E (2) B C A E D (2) A E D C B (2) A D E C B (2) A C E D B (2) A C D E B (2) E D A C B (1) E B A D C (1) E A D C B (1) D E B C A (1) D E B A C (1) D E A B C (1) C D A B E (1) C B D A E (1) C A B E D (1) B E C D A (1) B E C A D (1) B D C E A (1) B C D A E (1) A E C D B (1) Total count = 100 A B C D E A 0 4 12 10 -6 B -4 0 -2 -4 -6 C -12 2 0 4 2 D -10 4 -4 0 -8 E 6 6 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.460000000012 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 4 12 10 -6 B -4 0 -2 -4 -6 C -12 2 0 4 2 D -10 4 -4 0 -8 E 6 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.4599999999 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=25 B=18 A=15 C=9 so C is eliminated. Round 2 votes counts: D=34 E=25 B=23 A=18 so A is eliminated. Round 3 votes counts: D=38 E=36 B=26 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:210 E:209 C:198 B:192 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 12 10 -6 B -4 0 -2 -4 -6 C -12 2 0 4 2 D -10 4 -4 0 -8 E 6 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.4599999999 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 10 -6 B -4 0 -2 -4 -6 C -12 2 0 4 2 D -10 4 -4 0 -8 E 6 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.4599999999 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 10 -6 B -4 0 -2 -4 -6 C -12 2 0 4 2 D -10 4 -4 0 -8 E 6 6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.4599999999 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6935: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (12) B C D A E (6) E D B C A (5) D B E C A (5) D B E A C (5) C B D A E (5) C A B D E (5) B D C E A (4) A C B D E (4) E C D B A (3) E A C D B (3) B D A C E (3) A E C D B (3) A B C D E (3) E D C A B (2) E A D C B (2) D E B C A (2) C E A B D (2) C B E D A (2) C B D E A (2) C B A D E (2) A E D B C (2) A D B E C (2) E D C B A (1) E D B A C (1) E D A C B (1) E C A D B (1) E A D B C (1) D E B A C (1) D B A E C (1) C E B D A (1) C E B A D (1) C E A D B (1) C A B E D (1) B D C A E (1) B C D E A (1) B C A D E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -8 -4 6 B 6 0 -12 14 8 C 8 12 0 22 18 D 4 -14 -22 0 6 E -6 -8 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -4 6 B 6 0 -12 14 8 C 8 12 0 22 18 D 4 -14 -22 0 6 E -6 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 E=20 B=16 D=14 so D is eliminated. Round 2 votes counts: A=28 B=27 E=23 C=22 so C is eliminated. Round 3 votes counts: B=38 A=34 E=28 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:230 B:208 A:194 D:187 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -4 6 B 6 0 -12 14 8 C 8 12 0 22 18 D 4 -14 -22 0 6 E -6 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -4 6 B 6 0 -12 14 8 C 8 12 0 22 18 D 4 -14 -22 0 6 E -6 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -4 6 B 6 0 -12 14 8 C 8 12 0 22 18 D 4 -14 -22 0 6 E -6 -8 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6936: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (14) C D E B A (9) B E A C D (7) A D B E C (7) C E B D A (6) E B C A D (5) D A C B E (5) C D A E B (5) B E A D C (5) E B C D A (4) D C A E B (4) E C B A D (3) E B A C D (3) D C A B E (3) D A B C E (3) D C E B A (2) D C B E A (2) B A E D C (2) A C E B D (2) E C B D A (1) E A B C D (1) D B E C A (1) D A B E C (1) C D E A B (1) B E D C A (1) B E D A C (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 6 2 -4 B 4 0 12 12 4 C -6 -12 0 -2 -14 D -2 -12 2 0 -14 E 4 -4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 -4 B 4 0 12 12 4 C -6 -12 0 -2 -14 D -2 -12 2 0 -14 E 4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=21 C=21 E=17 B=16 so B is eliminated. Round 2 votes counts: E=31 A=27 D=21 C=21 so D is eliminated. Round 3 votes counts: A=36 E=32 C=32 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 E:214 A:200 D:187 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 -4 B 4 0 12 12 4 C -6 -12 0 -2 -14 D -2 -12 2 0 -14 E 4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 -4 B 4 0 12 12 4 C -6 -12 0 -2 -14 D -2 -12 2 0 -14 E 4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 -4 B 4 0 12 12 4 C -6 -12 0 -2 -14 D -2 -12 2 0 -14 E 4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998648 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6937: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) A E D B C (8) C B D E A (7) A E D C B (7) E D A B C (6) D E B C A (5) C B E D A (5) C B D A E (5) A D E B C (5) E D B C A (4) D B C E A (4) C B A D E (4) B C D E A (4) A D B C E (4) E B C D A (3) D E A B C (3) E B D C A (2) A C B D E (2) E A C B D (1) D E B A C (1) D B C A E (1) D A E B C (1) C B A E D (1) C A B E D (1) C A B D E (1) B C E D A (1) A E C D B (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 6 4 -4 -14 B -6 0 28 -24 -18 C -4 -28 0 -26 -20 D 4 24 26 0 -4 E 14 18 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 4 -4 -14 B -6 0 28 -24 -18 C -4 -28 0 -26 -20 D 4 24 26 0 -4 E 14 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 C=24 D=15 B=5 so B is eliminated. Round 2 votes counts: C=29 A=29 E=27 D=15 so D is eliminated. Round 3 votes counts: E=36 C=34 A=30 so A is eliminated. Round 4 votes counts: E=60 C=40 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:228 D:225 A:196 B:190 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 -4 -14 B -6 0 28 -24 -18 C -4 -28 0 -26 -20 D 4 24 26 0 -4 E 14 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 -14 B -6 0 28 -24 -18 C -4 -28 0 -26 -20 D 4 24 26 0 -4 E 14 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 -14 B -6 0 28 -24 -18 C -4 -28 0 -26 -20 D 4 24 26 0 -4 E 14 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6938: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E B A C D (7) D A E C B (7) D A C B E (7) E B C A D (6) B C A D E (6) A E D B C (5) C E B D A (4) C B E D A (4) B E C A D (4) E A B D C (3) C D B A E (3) C B D A E (3) B E A C D (3) B C E A D (3) A D B E C (3) E D A C B (2) E A D B C (2) D C A B E (2) C D E B A (2) C B D E A (2) B A E C D (2) A D E B C (2) A B D C E (2) D E C A B (1) D E A C B (1) D C A E B (1) D A C E B (1) C E D B A (1) B C A E D (1) B A C D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 0 4 -4 B 18 0 4 20 -4 C 0 -4 0 18 -14 D -4 -20 -18 0 -10 E 4 4 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 0 4 -4 B 18 0 4 20 -4 C 0 -4 0 18 -14 D -4 -20 -18 0 -10 E 4 4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=20 B=20 C=19 A=14 so A is eliminated. Round 2 votes counts: E=32 D=25 B=24 C=19 so C is eliminated. Round 3 votes counts: E=37 B=33 D=30 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:216 C:200 A:191 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 0 4 -4 B 18 0 4 20 -4 C 0 -4 0 18 -14 D -4 -20 -18 0 -10 E 4 4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 4 -4 B 18 0 4 20 -4 C 0 -4 0 18 -14 D -4 -20 -18 0 -10 E 4 4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 4 -4 B 18 0 4 20 -4 C 0 -4 0 18 -14 D -4 -20 -18 0 -10 E 4 4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6939: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) B A E C D (7) A E B D C (7) E A D B C (4) C D E A B (4) B E A C D (4) E D C A B (3) E B A D C (3) D C E A B (3) C D A B E (3) B C D A E (3) B C A D E (3) B A E D C (3) E D A C B (2) E B D A C (2) E A B D C (2) D E A C B (2) D C E B A (2) D C A E B (2) C D B E A (2) C B D E A (2) C B D A E (2) B E C A D (2) B E A D C (2) B A C E D (2) B A C D E (2) A E D B C (2) A C B D E (2) A B E D C (2) A B C D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D C A (1) D A E C B (1) C D B A E (1) B E C D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 10 4 -6 B 12 0 16 12 -6 C -10 -16 0 4 -12 D -4 -12 -4 0 -8 E 6 6 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 10 4 -6 B 12 0 16 12 -6 C -10 -16 0 4 -12 D -4 -12 -4 0 -8 E 6 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=22 E=20 A=18 D=10 so D is eliminated. Round 2 votes counts: B=30 C=29 E=22 A=19 so A is eliminated. Round 3 votes counts: B=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:216 A:198 D:186 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 10 4 -6 B 12 0 16 12 -6 C -10 -16 0 4 -12 D -4 -12 -4 0 -8 E 6 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 4 -6 B 12 0 16 12 -6 C -10 -16 0 4 -12 D -4 -12 -4 0 -8 E 6 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 4 -6 B 12 0 16 12 -6 C -10 -16 0 4 -12 D -4 -12 -4 0 -8 E 6 6 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6940: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (6) A D E B C (6) C E B A D (5) B C E D A (5) D B A C E (4) A E D C B (4) E D A B C (3) E C A D B (3) D A E B C (3) C B E A D (3) C A D B E (3) A D C B E (3) E D B A C (2) E C B A D (2) E C A B D (2) E B D C A (2) E B D A C (2) E A D B C (2) D E A B C (2) D B E A C (2) D A B E C (2) D A B C E (2) C E A B D (2) C A E D B (2) B D E A C (2) A C D E B (2) E B C D A (1) E A C D B (1) D E B A C (1) D B A E C (1) C B E D A (1) C B D A E (1) C A E B D (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D E A (1) B C D A E (1) A D E C B (1) A D C E B (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 2 10 4 B -2 0 6 -8 -4 C -2 -6 0 -6 2 D -10 8 6 0 6 E -4 4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 10 4 B -2 0 6 -8 -4 C -2 -6 0 -6 2 D -10 8 6 0 6 E -4 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990045 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=20 A=20 D=17 B=16 so B is eliminated. Round 2 votes counts: C=34 D=24 E=22 A=20 so A is eliminated. Round 3 votes counts: D=37 C=37 E=26 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:209 D:205 B:196 E:196 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 10 4 B -2 0 6 -8 -4 C -2 -6 0 -6 2 D -10 8 6 0 6 E -4 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990045 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 10 4 B -2 0 6 -8 -4 C -2 -6 0 -6 2 D -10 8 6 0 6 E -4 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990045 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 10 4 B -2 0 6 -8 -4 C -2 -6 0 -6 2 D -10 8 6 0 6 E -4 4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990045 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6941: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) C A B D E (7) D E C A B (6) D E B C A (6) C A D E B (5) B C A E D (5) B A C E D (5) D E A C B (4) D C A B E (4) A C B E D (4) E D A C B (3) E B D A C (3) C B A D E (3) C A D B E (3) B E A C D (3) B D E C A (3) B D C A E (3) E D B A C (2) E A D C B (2) D C B A E (2) D C A E B (2) D B E C A (2) D B C A E (2) C D A B E (2) A C E D B (2) A C E B D (2) E B A C D (1) E A C D B (1) D E C B A (1) D C E A B (1) B E A D C (1) B D C E A (1) Total count = 100 A B C D E A 0 -4 -34 6 20 B 4 0 -8 0 18 C 34 8 0 4 24 D -6 0 -4 0 32 E -20 -18 -24 -32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -34 6 20 B 4 0 -8 0 18 C 34 8 0 4 24 D -6 0 -4 0 32 E -20 -18 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=30 B=30 C=20 E=12 A=8 so A is eliminated. Round 2 votes counts: D=30 B=30 C=28 E=12 so E is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:235 D:211 B:207 A:194 E:153 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -34 6 20 B 4 0 -8 0 18 C 34 8 0 4 24 D -6 0 -4 0 32 E -20 -18 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -34 6 20 B 4 0 -8 0 18 C 34 8 0 4 24 D -6 0 -4 0 32 E -20 -18 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -34 6 20 B 4 0 -8 0 18 C 34 8 0 4 24 D -6 0 -4 0 32 E -20 -18 -24 -32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6942: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (13) D B E C A (10) A C D B E (7) B D E A C (6) C A D E B (5) C A E B D (4) C A D B E (4) B D E C A (4) E B D C A (3) D B A E C (3) C A E D B (3) E C A B D (2) E B A C D (2) E A C B D (2) E A B D C (2) D B E A C (2) C D E B A (2) B E D C A (2) B E D A C (2) A C E D B (2) A C D E B (2) E C B D A (1) E C B A D (1) E B D A C (1) E B A D C (1) D C E B A (1) D C B A E (1) D B C E A (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D B A (1) C E A D B (1) C D B E A (1) C D A B E (1) B A E D C (1) A E C B D (1) A E B C D (1) A D C B E (1) Total count = 100 A B C D E A 0 4 2 10 4 B -4 0 -14 -2 -2 C -2 14 0 12 6 D -10 2 -12 0 8 E -4 2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 10 4 B -4 0 -14 -2 -2 C -2 14 0 12 6 D -10 2 -12 0 8 E -4 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999124 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 D=21 E=15 B=15 so E is eliminated. Round 2 votes counts: A=31 C=26 B=22 D=21 so D is eliminated. Round 3 votes counts: B=40 A=32 C=28 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:210 D:194 E:192 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 10 4 B -4 0 -14 -2 -2 C -2 14 0 12 6 D -10 2 -12 0 8 E -4 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999124 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 10 4 B -4 0 -14 -2 -2 C -2 14 0 12 6 D -10 2 -12 0 8 E -4 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999124 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 10 4 B -4 0 -14 -2 -2 C -2 14 0 12 6 D -10 2 -12 0 8 E -4 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999124 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6943: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (6) D A C E B (6) A D C E B (6) E C B D A (5) D C E A B (5) B E C A D (5) B A C E D (5) E C D B A (4) E B C D A (4) C E D B A (4) A D B C E (4) D E A C B (3) B E C D A (3) B C E A D (3) B A E C D (3) A D C B E (3) A B C D E (3) E B D C A (2) D E B C A (2) D A B E C (2) C E B D A (2) B A E D C (2) B A D C E (2) A B D C E (2) E D C A B (1) D E C B A (1) D E C A B (1) C E A D B (1) C D E A B (1) C B E A D (1) C A B E D (1) B C E D A (1) B C A E D (1) B A D E C (1) A D B E C (1) A C D B E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 4 -6 2 B 2 0 -4 -4 -8 C -4 4 0 -2 6 D 6 4 2 0 2 E -2 8 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -6 2 B 2 0 -4 -4 -8 C -4 4 0 -2 6 D 6 4 2 0 2 E -2 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 A=22 E=16 C=10 so C is eliminated. Round 2 votes counts: D=27 B=27 E=23 A=23 so E is eliminated. Round 3 votes counts: B=40 D=36 A=24 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:207 C:202 A:199 E:199 B:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -6 2 B 2 0 -4 -4 -8 C -4 4 0 -2 6 D 6 4 2 0 2 E -2 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -6 2 B 2 0 -4 -4 -8 C -4 4 0 -2 6 D 6 4 2 0 2 E -2 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -6 2 B 2 0 -4 -4 -8 C -4 4 0 -2 6 D 6 4 2 0 2 E -2 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6944: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) B D E C A (7) A E C B D (6) E A C B D (5) D B C E A (5) C E A D B (5) C A E D B (5) D B E C A (4) D B C A E (4) E A B C D (3) D C B E A (3) D C B A E (3) C D A E B (3) B E A D C (3) A E B C D (3) A B E C D (3) E C A D B (2) E B A D C (2) C D E B A (2) B D C E A (2) B D A E C (2) A C B D E (2) E D C B A (1) E B C D A (1) E B C A D (1) E B A C D (1) E A C D B (1) D C E B A (1) D C A B E (1) D A B C E (1) C D E A B (1) C D A B E (1) B E D C A (1) B E D A C (1) B A E D C (1) B A D E C (1) A E C D B (1) A C E B D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -6 12 -4 B -8 0 -6 -6 -6 C 6 6 0 12 -2 D -12 6 -12 0 -12 E 4 6 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -6 12 -4 B -8 0 -6 -6 -6 C 6 6 0 12 -2 D -12 6 -12 0 -12 E 4 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=22 B=18 E=17 C=17 so E is eliminated. Round 2 votes counts: A=35 D=23 B=23 C=19 so C is eliminated. Round 3 votes counts: A=47 D=30 B=23 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:212 C:211 A:205 B:187 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 12 -4 B -8 0 -6 -6 -6 C 6 6 0 12 -2 D -12 6 -12 0 -12 E 4 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 12 -4 B -8 0 -6 -6 -6 C 6 6 0 12 -2 D -12 6 -12 0 -12 E 4 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 12 -4 B -8 0 -6 -6 -6 C 6 6 0 12 -2 D -12 6 -12 0 -12 E 4 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6945: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) E C D B A (7) C E A B D (6) E C B A D (5) B A D E C (5) E B D C A (4) E B A C D (4) D C A B E (4) C E D A B (4) C E A D B (4) A B D C E (4) E C B D A (3) E C A B D (3) E B D A C (3) D A B C E (3) A D B C E (3) E D B C A (2) E B A D C (2) C D A E B (2) C A E B D (2) B A E C D (2) B A D C E (2) E C D A B (1) E C A D B (1) E B C D A (1) D E B A C (1) D C E A B (1) D B A C E (1) D A C B E (1) C D A B E (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D A C (1) B E A C D (1) B D A E C (1) B D A C E (1) B A C E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -24 16 -8 B -6 0 -14 10 -14 C 24 14 0 22 6 D -16 -10 -22 0 -16 E 8 14 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -24 16 -8 B -6 0 -14 10 -14 C 24 14 0 22 6 D -16 -10 -22 0 -16 E 8 14 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=30 B=14 D=11 A=9 so A is eliminated. Round 2 votes counts: E=36 C=31 B=19 D=14 so D is eliminated. Round 3 votes counts: E=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:233 E:216 A:195 B:188 D:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -24 16 -8 B -6 0 -14 10 -14 C 24 14 0 22 6 D -16 -10 -22 0 -16 E 8 14 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -24 16 -8 B -6 0 -14 10 -14 C 24 14 0 22 6 D -16 -10 -22 0 -16 E 8 14 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -24 16 -8 B -6 0 -14 10 -14 C 24 14 0 22 6 D -16 -10 -22 0 -16 E 8 14 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6946: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (19) D C A E B (9) D A C B E (7) E B C A D (5) C A D E B (5) E C B A D (4) C D A E B (4) B E D C A (4) B E D A C (4) B E A D C (4) E B C D A (3) D B E C A (3) A C B E D (3) A B E C D (3) E B A C D (2) D C E B A (2) D B E A C (2) C E A D B (2) A C D E B (2) D C E A B (1) D C A B E (1) D A B C E (1) C E A B D (1) C D E B A (1) C A E B D (1) B E C A D (1) B D E A C (1) B D A E C (1) A E C B D (1) A D C B E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 6 12 -18 B 12 0 8 16 12 C -6 -8 0 18 -14 D -12 -16 -18 0 -18 E 18 -12 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 12 -18 B 12 0 8 16 12 C -6 -8 0 18 -14 D -12 -16 -18 0 -18 E 18 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=26 E=14 C=14 A=12 so A is eliminated. Round 2 votes counts: B=38 D=27 C=20 E=15 so E is eliminated. Round 3 votes counts: B=48 D=27 C=25 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:219 C:195 A:194 D:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 12 -18 B 12 0 8 16 12 C -6 -8 0 18 -14 D -12 -16 -18 0 -18 E 18 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 12 -18 B 12 0 8 16 12 C -6 -8 0 18 -14 D -12 -16 -18 0 -18 E 18 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 12 -18 B 12 0 8 16 12 C -6 -8 0 18 -14 D -12 -16 -18 0 -18 E 18 -12 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6947: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) C E D A B (8) A E D B C (7) E C A D B (4) E A D C B (4) C D E B A (4) C B D E A (4) E C D A B (3) E A D B C (3) D A B E C (3) C D B E A (3) C B E A D (3) B D A E C (3) B C A D E (3) B A D C E (3) E D A C B (2) C E D B A (2) C E A D B (2) C E A B D (2) C D E A B (2) B D C A E (2) B C D A E (2) A E B D C (2) A D B E C (2) A B E D C (2) A B D E C (2) E D C A B (1) D E C A B (1) D B C A E (1) D B A E C (1) C E B A D (1) C B D A E (1) B D A C E (1) B C A E D (1) B A E C D (1) B A C E D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 0 12 2 B -4 0 4 -6 2 C 0 -4 0 -2 -8 D -12 6 2 0 -2 E -2 -2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.857599 B: 0.000000 C: 0.142401 D: 0.000000 E: 0.000000 Sum of squares = 0.755753519399 Cumulative probabilities = A: 0.857599 B: 0.857599 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 12 2 B -4 0 4 -6 2 C 0 -4 0 -2 -8 D -12 6 2 0 -2 E -2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000004633 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=28 E=17 A=17 D=6 so D is eliminated. Round 2 votes counts: C=32 B=30 A=20 E=18 so E is eliminated. Round 3 votes counts: C=41 B=30 A=29 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:209 E:203 B:198 D:197 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 12 2 B -4 0 4 -6 2 C 0 -4 0 -2 -8 D -12 6 2 0 -2 E -2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000004633 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 12 2 B -4 0 4 -6 2 C 0 -4 0 -2 -8 D -12 6 2 0 -2 E -2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000004633 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 12 2 B -4 0 4 -6 2 C 0 -4 0 -2 -8 D -12 6 2 0 -2 E -2 -2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000004633 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6948: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (14) D E A C B (11) A D B E C (8) E D C A B (7) C E B D A (7) A B C D E (7) D A E B C (6) E D C B A (5) A D E B C (5) C B E D A (4) E C D B A (3) D E C A B (3) C B A E D (3) E C B D A (2) D E C B A (2) D E A B C (2) C B E A D (2) B A C E D (2) B A C D E (2) A B D C E (2) C E D B A (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -6 -6 2 B -6 0 -2 -8 -10 C 6 2 0 -6 -12 D 6 8 6 0 0 E -2 10 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.745924 E: 0.254076 Sum of squares = 0.620957595379 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.745924 E: 1.000000 A B C D E A 0 6 -6 -6 2 B -6 0 -2 -8 -10 C 6 2 0 -6 -12 D 6 8 6 0 0 E -2 10 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 B=18 E=17 C=17 so E is eliminated. Round 2 votes counts: D=36 A=24 C=22 B=18 so B is eliminated. Round 3 votes counts: D=36 C=36 A=28 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:210 A:198 C:195 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 -6 2 B -6 0 -2 -8 -10 C 6 2 0 -6 -12 D 6 8 6 0 0 E -2 10 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -6 2 B -6 0 -2 -8 -10 C 6 2 0 -6 -12 D 6 8 6 0 0 E -2 10 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -6 2 B -6 0 -2 -8 -10 C 6 2 0 -6 -12 D 6 8 6 0 0 E -2 10 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6949: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (13) A C E D B (10) B D E C A (9) D B E A C (8) B D C E A (5) A E C D B (5) A C E B D (5) C A B E D (4) E D B A C (3) C B D A E (3) C A D B E (3) E A B C D (2) D E B A C (2) D B C A E (2) C A E D B (2) C A B D E (2) B E D A C (2) B D E A C (2) B D C A E (2) A E D C B (2) E C B A D (1) E B C A D (1) E A D C B (1) E A D B C (1) E A C B D (1) D C B A E (1) D B E C A (1) D B C E A (1) D A E B C (1) C B E A D (1) C B A E D (1) B E D C A (1) B C D A E (1) A E C B D (1) Total count = 100 A B C D E A 0 6 -8 12 16 B -6 0 -12 14 -2 C 8 12 0 12 12 D -12 -14 -12 0 -14 E -16 2 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 12 16 B -6 0 -12 14 -2 C 8 12 0 12 12 D -12 -14 -12 0 -14 E -16 2 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=23 B=22 D=16 E=10 so E is eliminated. Round 2 votes counts: C=30 A=28 B=23 D=19 so D is eliminated. Round 3 votes counts: B=40 C=31 A=29 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:213 B:197 E:194 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 12 16 B -6 0 -12 14 -2 C 8 12 0 12 12 D -12 -14 -12 0 -14 E -16 2 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 12 16 B -6 0 -12 14 -2 C 8 12 0 12 12 D -12 -14 -12 0 -14 E -16 2 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 12 16 B -6 0 -12 14 -2 C 8 12 0 12 12 D -12 -14 -12 0 -14 E -16 2 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6950: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (18) C B A E D (16) E A B C D (9) D C B E A (8) E A B D C (6) A E B C D (5) C D B A E (4) C B D A E (4) D E A C B (3) D C E A B (3) B E A C D (3) B C A E D (3) E D A B C (2) E A D B C (2) D C B A E (2) C A B E D (2) E D B A C (1) E B A D C (1) E B A C D (1) D E B A C (1) D C A E B (1) D A E C B (1) D A E B C (1) C B E A D (1) B A E C D (1) A E B D C (1) Total count = 100 A B C D E A 0 8 12 2 -18 B -8 0 10 6 -12 C -12 -10 0 -2 -12 D -2 -6 2 0 -8 E 18 12 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 12 2 -18 B -8 0 10 6 -12 C -12 -10 0 -2 -12 D -2 -6 2 0 -8 E 18 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=27 E=22 B=7 A=6 so A is eliminated. Round 2 votes counts: D=38 E=28 C=27 B=7 so B is eliminated. Round 3 votes counts: D=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:202 B:198 D:193 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 12 2 -18 B -8 0 10 6 -12 C -12 -10 0 -2 -12 D -2 -6 2 0 -8 E 18 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 2 -18 B -8 0 10 6 -12 C -12 -10 0 -2 -12 D -2 -6 2 0 -8 E 18 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 2 -18 B -8 0 10 6 -12 C -12 -10 0 -2 -12 D -2 -6 2 0 -8 E 18 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6951: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (18) E C B D A (13) D A E B C (7) C B E A D (7) B C E A D (6) D A E C B (4) E D C B A (3) E D A C B (3) D A B C E (3) B C E D A (3) B C A E D (3) B C A D E (3) A D E C B (3) E D C A B (2) E C B A D (2) D E A C B (2) C E B A D (2) C B A E D (2) A B C D E (2) E D B C A (1) E C D B A (1) E B C D A (1) E A C B D (1) D E B C A (1) D E A B C (1) D B A C E (1) C E B D A (1) B A C D E (1) A E C B D (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -2 6 0 B 2 0 2 -2 2 C 2 -2 0 0 8 D -6 2 0 0 -4 E 0 -2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999469 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 6 0 B 2 0 2 -2 2 C 2 -2 0 0 8 D -6 2 0 0 -4 E 0 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 D=19 B=16 C=12 so C is eliminated. Round 2 votes counts: E=30 A=26 B=25 D=19 so D is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:204 B:202 A:201 E:197 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 6 0 B 2 0 2 -2 2 C 2 -2 0 0 8 D -6 2 0 0 -4 E 0 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 6 0 B 2 0 2 -2 2 C 2 -2 0 0 8 D -6 2 0 0 -4 E 0 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 6 0 B 2 0 2 -2 2 C 2 -2 0 0 8 D -6 2 0 0 -4 E 0 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6952: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) B E C A D (7) E C B A D (5) E A C B D (5) D C A E B (5) D B A C E (5) D A C E B (5) C D E B A (5) A B E D C (4) E B C A D (3) D C B E A (3) D B C A E (3) D A B C E (3) B E A C D (3) E A B C D (2) D A C B E (2) C E D B A (2) C E B D A (2) C B D E A (2) B C D E A (2) B A E D C (2) A E C D B (2) A D E B C (2) E C A D B (1) E B C D A (1) E B A C D (1) D C E B A (1) D C B A E (1) D C A B E (1) D B A E C (1) C E D A B (1) C D E A B (1) C D B E A (1) C B E D A (1) B D E C A (1) B D A E C (1) B C E D A (1) B A D E C (1) A E B C D (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -14 -16 -16 B 10 0 -6 -6 -4 C 14 6 0 0 8 D 16 6 0 0 10 E 16 4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.565713 D: 0.434287 E: 0.000000 Sum of squares = 0.5086363446 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.565713 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -16 -16 B 10 0 -6 -6 -4 C 14 6 0 0 8 D 16 6 0 0 10 E 16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=18 B=18 C=15 A=12 so A is eliminated. Round 2 votes counts: D=40 B=24 E=21 C=15 so C is eliminated. Round 3 votes counts: D=47 B=27 E=26 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:214 E:201 B:197 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -16 -16 B 10 0 -6 -6 -4 C 14 6 0 0 8 D 16 6 0 0 10 E 16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -16 -16 B 10 0 -6 -6 -4 C 14 6 0 0 8 D 16 6 0 0 10 E 16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -16 -16 B 10 0 -6 -6 -4 C 14 6 0 0 8 D 16 6 0 0 10 E 16 4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6953: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) C A E D B (8) B D E A C (6) D E B A C (5) E D A B C (4) D E B C A (4) B D E C A (4) A E B D C (4) E D C A B (3) E D B A C (3) C D B E A (3) B C A D E (3) E C D A B (2) E A D C B (2) D B E C A (2) C E D A B (2) C D E B A (2) C D E A B (2) C B A D E (2) B A D E C (2) B A D C E (2) B A C D E (2) A C E D B (2) A B E D C (2) A B E C D (2) E A D B C (1) D E C B A (1) D E C A B (1) D C E B A (1) D C B E A (1) D B C E A (1) C D B A E (1) C D A E B (1) C B D A E (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D E A (1) B C D A E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 2 -10 -4 B 2 0 22 -6 -2 C -2 -22 0 -4 -2 D 10 6 4 0 4 E 4 2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -10 -4 B 2 0 22 -6 -2 C -2 -22 0 -4 -2 D 10 6 4 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 C=22 D=16 E=15 so E is eliminated. Round 2 votes counts: D=26 A=26 C=24 B=24 so C is eliminated. Round 3 votes counts: D=39 A=34 B=27 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:208 E:202 A:193 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -10 -4 B 2 0 22 -6 -2 C -2 -22 0 -4 -2 D 10 6 4 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -10 -4 B 2 0 22 -6 -2 C -2 -22 0 -4 -2 D 10 6 4 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -10 -4 B 2 0 22 -6 -2 C -2 -22 0 -4 -2 D 10 6 4 0 4 E 4 2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6954: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (6) B E A D C (6) E C D B A (5) D C A E B (5) C D A E B (5) A D C B E (5) A B C D E (5) E D C B A (4) E B C D A (4) E D B C A (3) D E C A B (3) D C E A B (3) C E D B A (3) C B A D E (3) C A D B E (3) B E C A D (3) B A E C D (3) B A C E D (3) A D B C E (3) E D C A B (2) E B D C A (2) C B E D A (2) B E A C D (2) A C D B E (2) A B E D C (2) E D A B C (1) E B D A C (1) E B A D C (1) D E A B C (1) C E D A B (1) C E B D A (1) C D A B E (1) C A B D E (1) B C A E D (1) B A E D C (1) A D C E B (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -22 -6 -8 B -4 0 -14 -16 -4 C 22 14 0 10 10 D 6 16 -10 0 -2 E 8 4 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -22 -6 -8 B -4 0 -14 -16 -4 C 22 14 0 10 10 D 6 16 -10 0 -2 E 8 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 A=20 B=19 D=12 so D is eliminated. Round 2 votes counts: C=34 E=27 A=20 B=19 so B is eliminated. Round 3 votes counts: E=38 C=35 A=27 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 D:205 E:202 A:184 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -22 -6 -8 B -4 0 -14 -16 -4 C 22 14 0 10 10 D 6 16 -10 0 -2 E 8 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -22 -6 -8 B -4 0 -14 -16 -4 C 22 14 0 10 10 D 6 16 -10 0 -2 E 8 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -22 -6 -8 B -4 0 -14 -16 -4 C 22 14 0 10 10 D 6 16 -10 0 -2 E 8 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6955: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (21) B E C D A (10) D A C E B (9) E C B D A (5) E B C D A (5) B E C A D (5) A D B C E (5) E C D A B (4) D A C B E (4) E C A D B (3) B E A D C (3) D C A E B (2) C E A D B (2) C D A E B (2) B D A C E (2) B A E D C (2) A D C B E (2) E A B C D (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E A B (1) C A D E B (1) B E A C D (1) B A D E C (1) B A D C E (1) A E C D B (1) A D E C B (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 28 16 6 16 B -28 0 -20 -22 -20 C -16 20 0 -14 12 D -6 22 14 0 12 E -16 20 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 16 6 16 B -28 0 -20 -22 -20 C -16 20 0 -14 12 D -6 22 14 0 12 E -16 20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=25 E=18 D=16 C=8 so C is eliminated. Round 2 votes counts: A=34 B=25 E=22 D=19 so D is eliminated. Round 3 votes counts: A=52 B=25 E=23 so E is eliminated. Round 4 votes counts: A=64 B=36 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 D:221 C:201 E:190 B:155 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 16 6 16 B -28 0 -20 -22 -20 C -16 20 0 -14 12 D -6 22 14 0 12 E -16 20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 16 6 16 B -28 0 -20 -22 -20 C -16 20 0 -14 12 D -6 22 14 0 12 E -16 20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 16 6 16 B -28 0 -20 -22 -20 C -16 20 0 -14 12 D -6 22 14 0 12 E -16 20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6956: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) D C A B E (11) C D E A B (7) E C B A D (5) C D E B A (5) E B A D C (4) E A B C D (4) D A B E C (4) C D A E B (4) B E A D C (4) B A D E C (4) A B E D C (4) E B C A D (3) D B A C E (3) D A B C E (3) C E D B A (3) B A E D C (3) E C A B D (2) D C B A E (2) D A C B E (2) C E D A B (2) C E A D B (2) E A B D C (1) C E A B D (1) B E C D A (1) B D E A C (1) A E C B D (1) A E B D C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 2 4 4 -12 B -2 0 6 2 -14 C -4 -6 0 4 -10 D -4 -2 -4 0 -6 E 12 14 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999223 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 4 -12 B -2 0 6 2 -14 C -4 -6 0 4 -10 D -4 -2 -4 0 -6 E 12 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=25 C=24 B=13 A=8 so A is eliminated. Round 2 votes counts: E=33 D=26 C=24 B=17 so B is eliminated. Round 3 votes counts: E=45 D=31 C=24 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:199 B:196 C:192 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 4 -12 B -2 0 6 2 -14 C -4 -6 0 4 -10 D -4 -2 -4 0 -6 E 12 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 4 -12 B -2 0 6 2 -14 C -4 -6 0 4 -10 D -4 -2 -4 0 -6 E 12 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 4 -12 B -2 0 6 2 -14 C -4 -6 0 4 -10 D -4 -2 -4 0 -6 E 12 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6957: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) C B A E D (9) B C A E D (8) D E A B C (6) D C B E A (6) A B C E D (6) E D A B C (5) D E C B A (5) D E A C B (5) C B A D E (5) A E B C D (5) A B E C D (5) E A B D C (4) E A B C D (4) D C E B A (4) B A C E D (4) D E C A B (3) E D A C B (2) E A C B D (1) D C B A E (1) C D B A E (1) C A B E D (1) B C D A E (1) Total count = 100 A B C D E A 0 -6 -6 4 10 B 6 0 -4 24 12 C 6 4 0 18 10 D -4 -24 -18 0 -8 E -10 -12 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 4 10 B 6 0 -4 24 12 C 6 4 0 18 10 D -4 -24 -18 0 -8 E -10 -12 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=25 E=16 A=16 B=13 so B is eliminated. Round 2 votes counts: C=34 D=30 A=20 E=16 so E is eliminated. Round 3 votes counts: D=37 C=34 A=29 so A is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:219 C:219 A:201 E:188 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 4 10 B 6 0 -4 24 12 C 6 4 0 18 10 D -4 -24 -18 0 -8 E -10 -12 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 4 10 B 6 0 -4 24 12 C 6 4 0 18 10 D -4 -24 -18 0 -8 E -10 -12 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 4 10 B 6 0 -4 24 12 C 6 4 0 18 10 D -4 -24 -18 0 -8 E -10 -12 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6958: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (11) D C E A B (7) A E C B D (7) B D A C E (6) E C A D B (5) B A E C D (5) B A D E C (5) A C E B D (5) D B C E A (4) B D A E C (4) A B C E D (4) E C D A B (3) A C B E D (3) E A B C D (2) D E C B A (2) D E C A B (2) D C E B A (2) D C B E A (2) D B C A E (2) C E D A B (2) C E A D B (2) C A E D B (2) B D E A C (2) B A D C E (2) A B E C D (2) E D C A B (1) E B D C A (1) B E A D C (1) B D E C A (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 2 -4 0 B 4 0 10 6 14 C -2 -10 0 -10 -8 D 4 -6 10 0 8 E 0 -14 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -4 0 B 4 0 10 6 14 C -2 -10 0 -10 -8 D 4 -6 10 0 8 E 0 -14 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=22 E=12 C=6 so C is eliminated. Round 2 votes counts: D=32 B=28 A=24 E=16 so E is eliminated. Round 3 votes counts: D=38 A=33 B=29 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:217 D:208 A:197 E:193 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -4 0 B 4 0 10 6 14 C -2 -10 0 -10 -8 D 4 -6 10 0 8 E 0 -14 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -4 0 B 4 0 10 6 14 C -2 -10 0 -10 -8 D 4 -6 10 0 8 E 0 -14 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -4 0 B 4 0 10 6 14 C -2 -10 0 -10 -8 D 4 -6 10 0 8 E 0 -14 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6959: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D C A (8) E B C D A (6) D B E C A (5) C A D B E (5) E B D A C (4) E B A D C (4) D B E A C (4) A D C B E (4) D B C E A (3) C E B D A (3) C B E D A (3) B D E C A (3) E C B D A (2) E B A C D (2) E A B D C (2) C D B A E (2) B E D C A (2) A E D B C (2) A E C B D (2) A E B D C (2) A D B E C (2) A C E D B (2) A C D E B (2) E C B A D (1) E A C B D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B A C E (1) D A B C E (1) C D B E A (1) C A E B D (1) C A B E D (1) B E C D A (1) A D E B C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 2 -2 -12 B 12 0 10 0 6 C -2 -10 0 -2 -10 D 2 0 2 0 -2 E 12 -6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.512167 C: 0.000000 D: 0.487833 E: 0.000000 Sum of squares = 0.500296082788 Cumulative probabilities = A: 0.000000 B: 0.512167 C: 0.512167 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -2 -12 B 12 0 10 0 6 C -2 -10 0 -2 -10 D 2 0 2 0 -2 E 12 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=31 A=31 D=16 C=16 B=6 so B is eliminated. Round 2 votes counts: E=34 A=31 D=19 C=16 so C is eliminated. Round 3 votes counts: E=40 A=38 D=22 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:209 D:201 A:188 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 -2 -12 B 12 0 10 0 6 C -2 -10 0 -2 -10 D 2 0 2 0 -2 E 12 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -2 -12 B 12 0 10 0 6 C -2 -10 0 -2 -10 D 2 0 2 0 -2 E 12 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -2 -12 B 12 0 10 0 6 C -2 -10 0 -2 -10 D 2 0 2 0 -2 E 12 -6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6960: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) C A B D E (8) D E B C A (7) D E B A C (7) E D B A C (6) C A B E D (6) A C B E D (6) A C E D B (5) E D A C B (4) C B A D E (4) C A E D B (4) E D A B C (3) D B E C A (3) C A D E B (3) B C A D E (3) C B D E A (2) B E D A C (2) B D E A C (2) B C D E A (2) A B C E D (2) E A D B C (1) D E C A B (1) D E A C B (1) C D B E A (1) C A D B E (1) B C A E D (1) B A E D C (1) B A C E D (1) A E D C B (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -12 -2 -4 B 4 0 4 2 10 C 12 -4 0 0 0 D 2 -2 0 0 10 E 4 -10 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 -4 B 4 0 4 2 10 C 12 -4 0 0 0 D 2 -2 0 0 10 E 4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999525 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=22 D=19 A=16 E=14 so E is eliminated. Round 2 votes counts: D=32 C=29 B=22 A=17 so A is eliminated. Round 3 votes counts: C=41 D=35 B=24 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:210 D:205 C:204 E:192 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 -4 B 4 0 4 2 10 C 12 -4 0 0 0 D 2 -2 0 0 10 E 4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999525 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 -4 B 4 0 4 2 10 C 12 -4 0 0 0 D 2 -2 0 0 10 E 4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999525 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 -4 B 4 0 4 2 10 C 12 -4 0 0 0 D 2 -2 0 0 10 E 4 -10 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999525 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6961: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) E B C A D (7) A E D B C (7) A D C B E (7) E B D C A (5) D A C B E (5) B E C D A (5) A D E C B (5) E B A C D (3) E A B D C (3) C B A D E (3) C A B D E (3) E A D B C (2) D E B C A (2) D E B A C (2) D C B A E (2) D C A B E (2) D A E B C (2) B C E D A (2) A D E B C (2) A C D B E (2) E D A B C (1) E C B A D (1) E B D A C (1) E B A D C (1) E A C B D (1) E A B C D (1) D B E C A (1) D B C E A (1) C E B A D (1) C D B A E (1) C B E A D (1) C B D A E (1) C B A E D (1) B E D C A (1) A E D C B (1) A E C D B (1) A E B C D (1) A D C E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 2 14 -2 B 2 0 18 6 -20 C -2 -18 0 -8 -30 D -14 -6 8 0 -14 E 2 20 30 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 14 -2 B 2 0 18 6 -20 C -2 -18 0 -8 -30 D -14 -6 8 0 -14 E 2 20 30 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999915131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=29 D=17 C=11 B=8 so B is eliminated. Round 2 votes counts: E=41 A=29 D=17 C=13 so C is eliminated. Round 3 votes counts: E=45 A=36 D=19 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 A:206 B:203 D:187 C:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 14 -2 B 2 0 18 6 -20 C -2 -18 0 -8 -30 D -14 -6 8 0 -14 E 2 20 30 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999915131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 14 -2 B 2 0 18 6 -20 C -2 -18 0 -8 -30 D -14 -6 8 0 -14 E 2 20 30 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999915131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 14 -2 B 2 0 18 6 -20 C -2 -18 0 -8 -30 D -14 -6 8 0 -14 E 2 20 30 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999915131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6962: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) E A C D B (5) C E A D B (5) E A D B C (4) E A B D C (4) B D A E C (4) A B E C D (4) D E B A C (3) D C E B A (3) C D E B A (3) C D E A B (3) C D B E A (3) B A D E C (3) B A C D E (3) A B C E D (3) E D A B C (2) D B E C A (2) D B E A C (2) D B C E A (2) C E D A B (2) C B D A E (2) C A B D E (2) B E D A C (2) B D E A C (2) B A E D C (2) A B E D C (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A C B (1) E C D A B (1) E C A D B (1) D E C B A (1) D E B C A (1) C D B A E (1) C A E D B (1) C A D B E (1) C A B E D (1) B D E C A (1) B D A C E (1) B A D C E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 12 24 10 -12 B -12 0 16 -2 -6 C -24 -16 0 -6 -18 D -10 2 6 0 -6 E 12 6 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 24 10 -12 B -12 0 16 -2 -6 C -24 -16 0 -6 -18 D -10 2 6 0 -6 E 12 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=22 E=21 B=19 D=14 so D is eliminated. Round 2 votes counts: C=27 E=26 B=25 A=22 so A is eliminated. Round 3 votes counts: B=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:217 B:198 D:196 C:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 24 10 -12 B -12 0 16 -2 -6 C -24 -16 0 -6 -18 D -10 2 6 0 -6 E 12 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 24 10 -12 B -12 0 16 -2 -6 C -24 -16 0 -6 -18 D -10 2 6 0 -6 E 12 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 24 10 -12 B -12 0 16 -2 -6 C -24 -16 0 -6 -18 D -10 2 6 0 -6 E 12 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6963: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) B E C A D (8) A D C E B (7) B E D A C (6) B A D E C (6) D A C E B (5) C A D E B (5) A B D C E (5) D A E C B (4) B A D C E (4) A D C B E (4) E C D A B (3) E B C D A (3) B E A C D (3) A D B C E (3) A C D E B (3) A C D B E (3) E C D B A (2) E B D C A (2) C E D A B (2) B E D C A (2) B E A D C (2) E D C A B (1) E C B D A (1) D E C A B (1) C E B D A (1) C E B A D (1) C E A D B (1) B C E A D (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 16 18 4 B 4 0 12 12 16 C -16 -12 0 -4 -4 D -18 -12 4 0 4 E -4 -16 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 18 4 B 4 0 12 12 16 C -16 -12 0 -4 -4 D -18 -12 4 0 4 E -4 -16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=26 E=12 D=10 C=10 so D is eliminated. Round 2 votes counts: B=42 A=35 E=13 C=10 so C is eliminated. Round 3 votes counts: B=42 A=40 E=18 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:217 E:190 D:189 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 16 18 4 B 4 0 12 12 16 C -16 -12 0 -4 -4 D -18 -12 4 0 4 E -4 -16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 18 4 B 4 0 12 12 16 C -16 -12 0 -4 -4 D -18 -12 4 0 4 E -4 -16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 18 4 B 4 0 12 12 16 C -16 -12 0 -4 -4 D -18 -12 4 0 4 E -4 -16 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6964: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (12) C A B D E (9) D E B A C (8) B D E A C (7) E D B C A (6) E D A B C (5) C A E D B (5) C A B E D (5) E D C A B (4) D E B C A (4) D B E A C (4) A C B D E (4) A C B E D (3) E D C B A (2) C E A D B (2) C B A D E (2) B D A E C (2) B C A D E (2) B A D E C (2) A C E D B (2) A B C D E (2) D B E C A (1) C E D A B (1) C A E B D (1) B D C E A (1) B A D C E (1) B A C D E (1) A E D C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 10 -14 -14 B 10 0 18 -14 -6 C -10 -18 0 -22 -16 D 14 14 22 0 2 E 14 6 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -14 -14 B 10 0 18 -14 -6 C -10 -18 0 -22 -16 D 14 14 22 0 2 E 14 6 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=25 D=17 B=16 A=13 so A is eliminated. Round 2 votes counts: C=34 E=30 B=19 D=17 so D is eliminated. Round 3 votes counts: E=42 C=34 B=24 so B is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:226 E:217 B:204 A:186 C:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 10 -14 -14 B 10 0 18 -14 -6 C -10 -18 0 -22 -16 D 14 14 22 0 2 E 14 6 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -14 -14 B 10 0 18 -14 -6 C -10 -18 0 -22 -16 D 14 14 22 0 2 E 14 6 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -14 -14 B 10 0 18 -14 -6 C -10 -18 0 -22 -16 D 14 14 22 0 2 E 14 6 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6965: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) A E D C B (10) C B A E D (9) B C D E A (8) D E B A C (7) B C D A E (7) E D A B C (5) E A D C B (5) B C A D E (5) C B A D E (4) D B E C A (3) C A B E D (3) B D C E A (3) A C E B D (3) E D A C B (2) B D E C A (2) A E C D B (2) A D E C B (2) A C E D B (2) E D B A C (1) D E A C B (1) C B E A D (1) C A B D E (1) B E C D A (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 6 -4 -2 B 2 0 8 -4 -4 C -6 -8 0 -6 -6 D 4 4 6 0 10 E 2 4 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -4 -2 B 2 0 8 -4 -4 C -6 -8 0 -6 -6 D 4 4 6 0 10 E 2 4 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=23 A=20 C=18 E=13 so E is eliminated. Round 2 votes counts: D=31 B=26 A=25 C=18 so C is eliminated. Round 3 votes counts: B=40 D=31 A=29 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:201 E:201 A:199 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -4 -2 B 2 0 8 -4 -4 C -6 -8 0 -6 -6 D 4 4 6 0 10 E 2 4 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -4 -2 B 2 0 8 -4 -4 C -6 -8 0 -6 -6 D 4 4 6 0 10 E 2 4 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -4 -2 B 2 0 8 -4 -4 C -6 -8 0 -6 -6 D 4 4 6 0 10 E 2 4 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6966: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) E A C D B (8) B D C A E (6) D A C E B (5) C A E D B (5) D B A C E (4) C A D B E (4) B D E C A (4) B D A C E (4) A C E D B (4) E C A B D (3) D E B A C (3) D B E A C (3) D B C A E (3) D B A E C (3) C A D E B (3) B C D A E (3) E C A D B (2) D E A B C (2) C B A E D (2) C B A D E (2) C A B E D (2) B C A E D (2) B C A D E (2) E D A C B (1) E D A B C (1) E B C A D (1) E B A C D (1) D E A C B (1) D A C B E (1) C A E B D (1) C A B D E (1) B E D C A (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 6 -10 18 B 8 0 6 -10 14 C -6 -6 0 -6 10 D 10 10 6 0 30 E -18 -14 -10 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 -10 18 B 8 0 6 -10 14 C -6 -6 0 -6 10 D 10 10 6 0 30 E -18 -14 -10 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=25 C=20 E=17 A=6 so A is eliminated. Round 2 votes counts: B=32 D=26 C=24 E=18 so E is eliminated. Round 3 votes counts: C=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:228 B:209 A:203 C:196 E:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 6 -10 18 B 8 0 6 -10 14 C -6 -6 0 -6 10 D 10 10 6 0 30 E -18 -14 -10 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -10 18 B 8 0 6 -10 14 C -6 -6 0 -6 10 D 10 10 6 0 30 E -18 -14 -10 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -10 18 B 8 0 6 -10 14 C -6 -6 0 -6 10 D 10 10 6 0 30 E -18 -14 -10 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6967: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (21) C A B D E (16) A C D B E (10) E B D C A (8) B D E C A (6) A C E D B (6) B E D C A (4) A C D E B (4) C A E D B (3) C A B E D (3) B C A D E (3) E A C D B (2) C A D B E (2) E D B C A (1) E D A B C (1) E C A B D (1) E A D C B (1) D E B A C (1) D B E A C (1) C A E B D (1) C A D E B (1) B C D A E (1) A E C D B (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 0 12 6 B -8 0 -6 -12 -4 C 0 6 0 10 4 D -12 12 -10 0 -6 E -6 4 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.440113 B: 0.000000 C: 0.559887 D: 0.000000 E: 0.000000 Sum of squares = 0.507172891649 Cumulative probabilities = A: 0.440113 B: 0.440113 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 12 6 B -8 0 -6 -12 -4 C 0 6 0 10 4 D -12 12 -10 0 -6 E -6 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=26 A=23 B=14 D=2 so D is eliminated. Round 2 votes counts: E=36 C=26 A=23 B=15 so B is eliminated. Round 3 votes counts: E=47 C=30 A=23 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:210 E:200 D:192 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 12 6 B -8 0 -6 -12 -4 C 0 6 0 10 4 D -12 12 -10 0 -6 E -6 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 12 6 B -8 0 -6 -12 -4 C 0 6 0 10 4 D -12 12 -10 0 -6 E -6 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 12 6 B -8 0 -6 -12 -4 C 0 6 0 10 4 D -12 12 -10 0 -6 E -6 4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6968: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (10) E A C D B (9) D C B E A (7) E A B C D (6) B D C A E (6) C D B A E (5) B C D A E (5) D C E A B (4) B A C E D (4) D B C A E (3) C D E A B (3) C D B E A (3) B D E A C (3) B A E D C (3) B A E C D (3) E D C A B (2) E A D C B (2) E A B D C (2) D E C A B (2) D C E B A (2) D B C E A (2) C E A D B (2) B A D C E (2) A B E C D (2) E C A D B (1) E A D B C (1) E A C B D (1) D C B A E (1) C A E D B (1) B D A C E (1) B C A D E (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 0 2 -4 B 2 0 10 0 2 C 0 -10 0 12 4 D -2 0 -12 0 0 E 4 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.715617 C: 0.000000 D: 0.284383 E: 0.000000 Sum of squares = 0.592981109337 Cumulative probabilities = A: 0.000000 B: 0.715617 C: 0.715617 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 -4 B 2 0 10 0 2 C 0 -10 0 12 4 D -2 0 -12 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.50413223372 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 D=21 C=14 A=13 so A is eliminated. Round 2 votes counts: E=35 B=30 D=21 C=14 so C is eliminated. Round 3 votes counts: E=38 D=32 B=30 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:207 C:203 E:199 A:198 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 2 -4 B 2 0 10 0 2 C 0 -10 0 12 4 D -2 0 -12 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.50413223372 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 -4 B 2 0 10 0 2 C 0 -10 0 12 4 D -2 0 -12 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.50413223372 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 -4 B 2 0 10 0 2 C 0 -10 0 12 4 D -2 0 -12 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.50413223372 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6969: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (5) A E C B D (5) E A C B D (4) C A E D B (4) B A E D C (4) A C E D B (4) A B E C D (4) E C A D B (3) E A B C D (3) D C E B A (3) D C B E A (3) D B C A E (3) C E D B A (3) C E A D B (3) B A D E C (3) A E B C D (3) A B E D C (3) E C A B D (2) E B C D A (2) D C B A E (2) D B C E A (2) D A C B E (2) D A B C E (2) C D E B A (2) C D A E B (2) B E D A C (2) B E A D C (2) B D E C A (2) B D E A C (2) B D A C E (2) A B D E C (2) E D C B A (1) E C B D A (1) E B D C A (1) E B C A D (1) E B A C D (1) C E D A B (1) C D A B E (1) B E D C A (1) A D C B E (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 4 14 10 10 B -4 0 4 2 0 C -14 -4 0 0 -4 D -10 -2 0 0 -16 E -10 0 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 10 10 B -4 0 4 2 0 C -14 -4 0 0 -4 D -10 -2 0 0 -16 E -10 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=22 E=19 B=18 C=16 so C is eliminated. Round 2 votes counts: A=29 D=27 E=26 B=18 so B is eliminated. Round 3 votes counts: A=36 D=33 E=31 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:205 B:201 C:189 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 10 B -4 0 4 2 0 C -14 -4 0 0 -4 D -10 -2 0 0 -16 E -10 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 10 B -4 0 4 2 0 C -14 -4 0 0 -4 D -10 -2 0 0 -16 E -10 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 10 B -4 0 4 2 0 C -14 -4 0 0 -4 D -10 -2 0 0 -16 E -10 0 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6970: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) B D E C A (8) E A B D C (7) C D A B E (6) C A D E B (6) A E C D B (6) D B E A C (5) B D E A C (5) A E C B D (5) E A C B D (4) C D B A E (4) A C E D B (4) D C B A E (3) C D B E A (3) E B A D C (2) D B C A E (2) D A B E C (2) C A E D B (2) B E D A C (2) A E D B C (2) A E B D C (2) A E B C D (2) E A B C D (1) D B E C A (1) D B A E C (1) C E A B D (1) C D A E B (1) B D C E A (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 4 2 -10 -2 B -4 0 8 -18 8 C -2 -8 0 -8 -12 D 10 18 8 0 18 E 2 -8 12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -10 -2 B -4 0 8 -18 8 C -2 -8 0 -8 -12 D 10 18 8 0 18 E 2 -8 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 A=22 B=17 E=14 so E is eliminated. Round 2 votes counts: A=34 D=24 C=23 B=19 so B is eliminated. Round 3 votes counts: D=40 A=36 C=24 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:197 B:197 E:194 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -10 -2 B -4 0 8 -18 8 C -2 -8 0 -8 -12 D 10 18 8 0 18 E 2 -8 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -10 -2 B -4 0 8 -18 8 C -2 -8 0 -8 -12 D 10 18 8 0 18 E 2 -8 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -10 -2 B -4 0 8 -18 8 C -2 -8 0 -8 -12 D 10 18 8 0 18 E 2 -8 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6971: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (15) D E A C B (13) A E D B C (9) D E C B A (8) C B A E D (7) C B A D E (5) B C A E D (5) E D A C B (4) D E A B C (3) C D E B A (3) B C A D E (3) A E B D C (3) A D E B C (3) A B E D C (3) A B C E D (3) D E C A B (2) C B D A E (2) B A C E D (2) A B E C D (2) E D A B C (1) E A D B C (1) D C B E A (1) C B E D A (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -4 B 4 0 -22 4 0 C 4 22 0 -4 -4 D 6 -4 4 0 18 E 4 0 4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.133333 D: 0.733333 E: 0.000000 Sum of squares = 0.573333333393 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.266667 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -4 B 4 0 -22 4 0 C 4 22 0 -4 -4 D 6 -4 4 0 18 E 4 0 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.133333 D: 0.733333 E: 0.000000 Sum of squares = 0.573333333495 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.266667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=27 A=24 B=10 E=6 so E is eliminated. Round 2 votes counts: C=33 D=32 A=25 B=10 so B is eliminated. Round 3 votes counts: C=41 D=32 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:209 E:195 B:193 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -4 B 4 0 -22 4 0 C 4 22 0 -4 -4 D 6 -4 4 0 18 E 4 0 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.133333 D: 0.733333 E: 0.000000 Sum of squares = 0.573333333495 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.266667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -4 B 4 0 -22 4 0 C 4 22 0 -4 -4 D 6 -4 4 0 18 E 4 0 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.133333 D: 0.733333 E: 0.000000 Sum of squares = 0.573333333495 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.266667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -4 B 4 0 -22 4 0 C 4 22 0 -4 -4 D 6 -4 4 0 18 E 4 0 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.133333 C: 0.133333 D: 0.733333 E: 0.000000 Sum of squares = 0.573333333495 Cumulative probabilities = A: 0.000000 B: 0.133333 C: 0.266667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6972: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) E A B D C (5) B C A E D (5) A E B C D (5) E B A D C (4) D C A E B (4) A E C D B (4) E A D B C (3) D E A C B (3) D E A B C (3) D C B E A (3) D B C E A (3) C D B A E (3) C D A E B (3) C D A B E (3) C B A E D (3) B A E C D (3) A E C B D (3) D C E A B (2) D C B A E (2) C B D A E (2) C A B D E (2) B E D A C (2) B E A D C (2) B C D E A (2) A C E B D (2) E D B A C (1) E D A B C (1) E A B C D (1) D E C A B (1) D E B A C (1) C B D E A (1) C B A D E (1) C A D B E (1) C A B E D (1) B D E C A (1) B D C E A (1) B C D A E (1) B A C E D (1) A E D C B (1) A E D B C (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 2 10 14 6 B -2 0 6 10 0 C -10 -6 0 10 -6 D -14 -10 -10 0 -12 E -6 0 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 14 6 B -2 0 6 10 0 C -10 -6 0 10 -6 D -14 -10 -10 0 -12 E -6 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=22 C=20 A=18 E=15 so E is eliminated. Round 2 votes counts: B=29 A=27 D=24 C=20 so C is eliminated. Round 3 votes counts: B=36 D=33 A=31 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:207 E:206 C:194 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 14 6 B -2 0 6 10 0 C -10 -6 0 10 -6 D -14 -10 -10 0 -12 E -6 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 14 6 B -2 0 6 10 0 C -10 -6 0 10 -6 D -14 -10 -10 0 -12 E -6 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 14 6 B -2 0 6 10 0 C -10 -6 0 10 -6 D -14 -10 -10 0 -12 E -6 0 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999467 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6973: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) D A E C B (8) A C B D E (7) E D B C A (6) D E A B C (6) B C E A D (6) C B A E D (5) E D A B C (4) E B D C A (3) E B C D A (3) E B C A D (3) C B A D E (3) A D E C B (3) E C B A D (2) E A D C B (2) D A C B E (2) D A B C E (2) C B E A D (2) B C A D E (2) A E D C B (2) A D C B E (2) A C D E B (2) A C D B E (2) A C B E D (2) D E B C A (1) D E A C B (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A C E (1) D A E B C (1) C A B E D (1) C A B D E (1) B D C A E (1) B C E D A (1) B C D E A (1) B C D A E (1) Total count = 100 A B C D E A 0 12 12 -6 -2 B -12 0 -10 -12 -10 C -12 10 0 -12 -10 D 6 12 12 0 0 E 2 10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.362315 E: 0.637685 Sum of squares = 0.537914419233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.362315 E: 1.000000 A B C D E A 0 12 12 -6 -2 B -12 0 -10 -12 -10 C -12 10 0 -12 -10 D 6 12 12 0 0 E 2 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=25 A=20 C=12 B=12 so C is eliminated. Round 2 votes counts: E=31 D=25 B=22 A=22 so B is eliminated. Round 3 votes counts: E=40 A=32 D=28 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:211 A:208 C:188 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 12 -6 -2 B -12 0 -10 -12 -10 C -12 10 0 -12 -10 D 6 12 12 0 0 E 2 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 -6 -2 B -12 0 -10 -12 -10 C -12 10 0 -12 -10 D 6 12 12 0 0 E 2 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 -6 -2 B -12 0 -10 -12 -10 C -12 10 0 -12 -10 D 6 12 12 0 0 E 2 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6974: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (16) B E D C A (12) C A D E B (9) A C B D E (7) E D B C A (5) B E D A C (5) B A E D C (5) A B C E D (5) C D A E B (4) E B D C A (3) D E C B A (3) D E B C A (3) C D E B A (3) C D E A B (3) B E A D C (3) A B E D C (3) C A D B E (2) A C D B E (2) A B E C D (2) E B D A C (1) B C D E A (1) B A C E D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 4 14 16 B -10 0 -2 -2 -2 C -4 2 0 14 10 D -14 2 -14 0 6 E -16 2 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 14 16 B -10 0 -2 -2 -2 C -4 2 0 14 10 D -14 2 -14 0 6 E -16 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=27 C=21 E=9 D=6 so D is eliminated. Round 2 votes counts: A=37 B=27 C=21 E=15 so E is eliminated. Round 3 votes counts: B=39 A=37 C=24 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:211 B:192 D:190 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 14 16 B -10 0 -2 -2 -2 C -4 2 0 14 10 D -14 2 -14 0 6 E -16 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 14 16 B -10 0 -2 -2 -2 C -4 2 0 14 10 D -14 2 -14 0 6 E -16 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 14 16 B -10 0 -2 -2 -2 C -4 2 0 14 10 D -14 2 -14 0 6 E -16 2 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6975: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (12) D E A C B (9) A E C D B (6) D E C A B (5) B A C E D (5) C A E D B (4) B D E A C (4) B A D E C (4) D E C B A (3) D E A B C (3) D B E A C (3) B D C E A (3) E D A C B (2) D E B C A (2) D B E C A (2) C D E A B (2) B D E C A (2) B C D E A (2) B C A D E (2) B A C D E (2) A E D C B (2) A C B E D (2) E D C A B (1) E C D A B (1) E A D C B (1) E A C D B (1) D E B A C (1) D C B E A (1) C B A E D (1) C A E B D (1) C A B E D (1) B D A E C (1) B D A C E (1) B C D A E (1) B A E C D (1) B A D C E (1) A E D B C (1) A E B C D (1) A C E D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 8 2 4 B 8 0 10 -2 4 C -8 -10 0 -4 -12 D -2 2 4 0 8 E -4 -4 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000042 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 2 4 B 8 0 10 -2 4 C -8 -10 0 -4 -12 D -2 2 4 0 8 E -4 -4 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.49999999949 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=29 A=15 C=9 E=6 so E is eliminated. Round 2 votes counts: B=41 D=32 A=17 C=10 so C is eliminated. Round 3 votes counts: B=42 D=35 A=23 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:206 A:203 E:198 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 2 4 B 8 0 10 -2 4 C -8 -10 0 -4 -12 D -2 2 4 0 8 E -4 -4 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.49999999949 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 2 4 B 8 0 10 -2 4 C -8 -10 0 -4 -12 D -2 2 4 0 8 E -4 -4 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.49999999949 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 2 4 B 8 0 10 -2 4 C -8 -10 0 -4 -12 D -2 2 4 0 8 E -4 -4 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.49999999949 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6976: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) C D E A B (7) B E D A C (7) B C E D A (6) E D B A C (5) C D A E B (4) A D E C B (4) E D C A B (3) E D A B C (3) D E C A B (3) C E D B A (3) C E D A B (3) C E B D A (3) B E C D A (3) C B A D E (2) C A D B E (2) B E D C A (2) B C A E D (2) B A E D C (2) B A D E C (2) B A C E D (2) A D E B C (2) A D C E B (2) A B D E C (2) A B C D E (2) E C D B A (1) D E A C B (1) D E A B C (1) D A E C B (1) C B E A D (1) C B A E D (1) C A B D E (1) B E A D C (1) B C E A D (1) B A E C D (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -18 -12 -8 B -10 0 -8 -16 -16 C 18 8 0 14 12 D 12 16 -14 0 0 E 8 16 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -18 -12 -8 B -10 0 -8 -16 -16 C 18 8 0 14 12 D 12 16 -14 0 0 E 8 16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=29 A=15 E=12 D=6 so D is eliminated. Round 2 votes counts: C=38 B=29 E=17 A=16 so A is eliminated. Round 3 votes counts: C=41 B=35 E=24 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 D:207 E:206 A:186 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -18 -12 -8 B -10 0 -8 -16 -16 C 18 8 0 14 12 D 12 16 -14 0 0 E 8 16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -18 -12 -8 B -10 0 -8 -16 -16 C 18 8 0 14 12 D 12 16 -14 0 0 E 8 16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -18 -12 -8 B -10 0 -8 -16 -16 C 18 8 0 14 12 D 12 16 -14 0 0 E 8 16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6977: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) D E A B C (9) B C E A D (8) E D A B C (7) A D E C B (7) D E A C B (5) D A E C B (5) C B A D E (5) B E C D A (5) E B D C A (4) A C B D E (4) E A D B C (3) B C E D A (3) A C D B E (3) E D B A C (2) D E B A C (2) C A B D E (2) A E D B C (2) A D C E B (2) E D B C A (1) E B D A C (1) D E B C A (1) D C A E B (1) B E D C A (1) B E C A D (1) B C D E A (1) B C A E D (1) A E C D B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 10 4 -8 B -6 0 4 -4 -8 C -10 -4 0 -6 -14 D -4 4 6 0 -4 E 8 8 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 10 4 -8 B -6 0 4 -4 -8 C -10 -4 0 -6 -14 D -4 4 6 0 -4 E 8 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=21 B=20 E=18 C=18 so E is eliminated. Round 2 votes counts: D=33 B=25 A=24 C=18 so C is eliminated. Round 3 votes counts: B=41 D=33 A=26 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:217 A:206 D:201 B:193 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 10 4 -8 B -6 0 4 -4 -8 C -10 -4 0 -6 -14 D -4 4 6 0 -4 E 8 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 4 -8 B -6 0 4 -4 -8 C -10 -4 0 -6 -14 D -4 4 6 0 -4 E 8 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 4 -8 B -6 0 4 -4 -8 C -10 -4 0 -6 -14 D -4 4 6 0 -4 E 8 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6978: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (16) C B A E D (10) C B D A E (6) C B D E A (5) C B A D E (5) A E B C D (5) C D B A E (4) B C A E D (4) A E D B C (4) E D A B C (3) E A D B C (3) C A B E D (3) E A B D C (2) D E A C B (2) D C E B A (2) D C B E A (2) B E A C D (2) B C E A D (2) A E C B D (2) A C B E D (2) A B E C D (2) A B C E D (2) E D B A C (1) D E B C A (1) D E B A C (1) D C A E B (1) D C A B E (1) D A E C B (1) C B E A D (1) B E D C A (1) B C E D A (1) B A E C D (1) B A C E D (1) A E D C B (1) Total count = 100 A B C D E A 0 0 2 4 10 B 0 0 4 14 10 C -2 -4 0 16 4 D -4 -14 -16 0 -6 E -10 -10 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.528484 B: 0.471516 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501622691231 Cumulative probabilities = A: 0.528484 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 10 B 0 0 4 14 10 C -2 -4 0 16 4 D -4 -14 -16 0 -6 E -10 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=27 A=18 B=12 E=9 so E is eliminated. Round 2 votes counts: C=34 D=31 A=23 B=12 so B is eliminated. Round 3 votes counts: C=41 D=32 A=27 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:214 A:208 C:207 E:191 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 4 10 B 0 0 4 14 10 C -2 -4 0 16 4 D -4 -14 -16 0 -6 E -10 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 10 B 0 0 4 14 10 C -2 -4 0 16 4 D -4 -14 -16 0 -6 E -10 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 10 B 0 0 4 14 10 C -2 -4 0 16 4 D -4 -14 -16 0 -6 E -10 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6979: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) D C B E A (6) C D A E B (6) D C E B A (5) C D A B E (5) B A E C D (5) A B E C D (5) E B D A C (4) D E B C A (4) C A D E B (4) B E D A C (4) A B C E D (4) D C E A B (3) C A E B D (3) C A D B E (3) A C B E D (3) E B A C D (2) D C B A E (2) D B E C A (2) D B E A C (2) C A E D B (2) B E A D C (2) B A E D C (2) A E B C D (2) E D C B A (1) E D B A C (1) E A B C D (1) D E C B A (1) D B C E A (1) C E D A B (1) C D E A B (1) C A B E D (1) C A B D E (1) B E A C D (1) B D E A C (1) B A D C E (1) B A C D E (1) A E C B D (1) Total count = 100 A B C D E A 0 -8 -4 0 2 B 8 0 2 0 4 C 4 -2 0 4 6 D 0 0 -4 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.774569 C: 0.000000 D: 0.225431 E: 0.000000 Sum of squares = 0.650776770929 Cumulative probabilities = A: 0.000000 B: 0.774569 C: 0.774569 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 0 2 B 8 0 2 0 4 C 4 -2 0 4 6 D 0 0 -4 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560117 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 B=17 E=15 A=15 so E is eliminated. Round 2 votes counts: B=29 D=28 C=27 A=16 so A is eliminated. Round 3 votes counts: B=41 C=31 D=28 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:206 D:197 A:195 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 0 2 B 8 0 2 0 4 C 4 -2 0 4 6 D 0 0 -4 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560117 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 0 2 B 8 0 2 0 4 C 4 -2 0 4 6 D 0 0 -4 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560117 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 0 2 B 8 0 2 0 4 C 4 -2 0 4 6 D 0 0 -4 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560117 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6980: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) C D E B A (7) B E A C D (6) A D E B C (6) E B C D A (5) D A C E B (5) A B E D C (5) C B E D A (4) B E A D C (4) D A C B E (3) C D A E B (3) B A E C D (3) A D C B E (3) E D C B A (2) E B D C A (2) E B D A C (2) E B A D C (2) D A E C B (2) C D B A E (2) C D A B E (2) C B E A D (2) C B D E A (2) B E C A D (2) B C E A D (2) A D B E C (2) E A B D C (1) D E A C B (1) D C E B A (1) D C E A B (1) C E D B A (1) C E B D A (1) C D B E A (1) C A D B E (1) A E D B C (1) A E B D C (1) A D E C B (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 0 -12 2 B 2 0 -12 -10 -10 C 0 12 0 -10 4 D 12 10 10 0 8 E -2 10 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -12 2 B 2 0 -12 -10 -10 C 0 12 0 -10 4 D 12 10 10 0 8 E -2 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=22 A=21 B=17 E=14 so E is eliminated. Round 2 votes counts: B=28 C=26 D=24 A=22 so A is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:203 E:198 A:194 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -12 2 B 2 0 -12 -10 -10 C 0 12 0 -10 4 D 12 10 10 0 8 E -2 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -12 2 B 2 0 -12 -10 -10 C 0 12 0 -10 4 D 12 10 10 0 8 E -2 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -12 2 B 2 0 -12 -10 -10 C 0 12 0 -10 4 D 12 10 10 0 8 E -2 10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6981: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) B E A C D (6) C A D B E (5) B C A E D (5) D E C A B (4) B E C D A (4) B A E C D (4) E D B A C (3) E D A B C (3) D E A C B (3) B E D C A (3) B E A D C (3) B C E D A (3) B A E D C (3) B A C E D (3) A C D E B (3) E B D A C (2) E A B D C (2) D A C E B (2) C D E A B (2) C D A B E (2) C B D E A (2) C B D A E (2) C A B D E (2) B C E A D (2) A C D B E (2) A C B D E (2) E D C B A (1) E D B C A (1) E B D C A (1) E B A D C (1) E A D B C (1) D C A E B (1) C D E B A (1) C B A D E (1) B E D A C (1) A E D B C (1) A D E C B (1) A D C E B (1) A D B E C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 0 2 2 B 4 0 10 8 16 C 0 -10 0 20 0 D -2 -8 -20 0 -8 E -2 -16 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 2 2 B 4 0 10 8 16 C 0 -10 0 20 0 D -2 -8 -20 0 -8 E -2 -16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=25 E=15 A=13 D=10 so D is eliminated. Round 2 votes counts: B=37 C=26 E=22 A=15 so A is eliminated. Round 3 votes counts: B=40 C=36 E=24 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:205 A:200 E:195 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 2 2 B 4 0 10 8 16 C 0 -10 0 20 0 D -2 -8 -20 0 -8 E -2 -16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 2 B 4 0 10 8 16 C 0 -10 0 20 0 D -2 -8 -20 0 -8 E -2 -16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 2 B 4 0 10 8 16 C 0 -10 0 20 0 D -2 -8 -20 0 -8 E -2 -16 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6982: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) B C A E D (8) B A C D E (8) D E C A B (6) E C D A B (4) C E B D A (4) A B D E C (4) E D C A B (3) D A E B C (3) C E D B A (3) C E D A B (3) C B E A D (3) B A C E D (3) D E C B A (2) D E A B C (2) D C E B A (2) C E B A D (2) C B E D A (2) C B A E D (2) B A D C E (2) A D B E C (2) A C E D B (2) A B C E D (2) E A D C B (1) D E B C A (1) C D E B A (1) C D B E A (1) C B D E A (1) C A E D B (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D E A (1) B A D E C (1) A E C D B (1) A D E C B (1) A D E B C (1) A C E B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -4 -6 -12 B 0 0 -14 -2 -10 C 4 14 0 6 8 D 6 2 -6 0 8 E 12 10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -6 -12 B 0 0 -14 -2 -10 C 4 14 0 6 8 D 6 2 -6 0 8 E 12 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 C=23 A=16 E=8 so E is eliminated. Round 2 votes counts: D=30 C=27 B=26 A=17 so A is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:205 E:203 A:189 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -6 -12 B 0 0 -14 -2 -10 C 4 14 0 6 8 D 6 2 -6 0 8 E 12 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -6 -12 B 0 0 -14 -2 -10 C 4 14 0 6 8 D 6 2 -6 0 8 E 12 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -6 -12 B 0 0 -14 -2 -10 C 4 14 0 6 8 D 6 2 -6 0 8 E 12 10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6983: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) D E B C A (7) D A B E C (7) E C D B A (5) C A E B D (5) A B D C E (5) E C B D A (4) C E B A D (4) C E A B D (4) A D B C E (4) A C B E D (4) A C B D E (4) E D B C A (3) E B C D A (3) E B D C A (2) B D A E C (2) B A C E D (2) A C D E B (2) A B C D E (2) E C B A D (1) D E C B A (1) D E C A B (1) D B A E C (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A D B (1) C B E A D (1) C A B E D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B D E C A (1) B C A E D (1) B A D E C (1) B A C D E (1) A C E B D (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 -4 -6 B 8 0 16 10 10 C -2 -16 0 4 -6 D 4 -10 -4 0 2 E 6 -10 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -4 -6 B 8 0 16 10 10 C -2 -16 0 4 -6 D 4 -10 -4 0 2 E 6 -10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=24 E=18 C=17 B=13 so B is eliminated. Round 2 votes counts: D=31 A=28 E=23 C=18 so C is eliminated. Round 3 votes counts: A=35 E=34 D=31 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:200 D:196 A:192 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -4 -6 B 8 0 16 10 10 C -2 -16 0 4 -6 D 4 -10 -4 0 2 E 6 -10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -4 -6 B 8 0 16 10 10 C -2 -16 0 4 -6 D 4 -10 -4 0 2 E 6 -10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -4 -6 B 8 0 16 10 10 C -2 -16 0 4 -6 D 4 -10 -4 0 2 E 6 -10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6984: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) D C B E A (8) D B C A E (7) C B E D A (7) B C D E A (6) E C B D A (5) E C B A D (5) C B D E A (5) A E D C B (5) A E D B C (5) A E B D C (4) A E B C D (4) E A C D B (3) D C B A E (3) E B C A D (2) E A D C B (2) D A B C E (2) C D B E A (2) A D E C B (2) A D E B C (2) A D B E C (2) E B C D A (1) D C E B A (1) D B A C E (1) D A C B E (1) B D C A E (1) B C E D A (1) B C D A E (1) B A C D E (1) A D C B E (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -10 -4 -12 B 14 0 -16 4 2 C 10 16 0 4 -2 D 4 -4 -4 0 -6 E 12 -2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.000000 E: 0.800000 Sum of squares = 0.659999999857 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.200000 E: 1.000000 A B C D E A 0 -14 -10 -4 -12 B 14 0 -16 4 2 C 10 16 0 4 -2 D 4 -4 -4 0 -6 E 12 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.000000 E: 0.800000 Sum of squares = 0.66000000199 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=26 D=23 C=14 B=10 so B is eliminated. Round 2 votes counts: A=28 E=26 D=24 C=22 so C is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:209 B:202 D:195 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -10 -4 -12 B 14 0 -16 4 2 C 10 16 0 4 -2 D 4 -4 -4 0 -6 E 12 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.000000 E: 0.800000 Sum of squares = 0.66000000199 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -4 -12 B 14 0 -16 4 2 C 10 16 0 4 -2 D 4 -4 -4 0 -6 E 12 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.000000 E: 0.800000 Sum of squares = 0.66000000199 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -4 -12 B 14 0 -16 4 2 C 10 16 0 4 -2 D 4 -4 -4 0 -6 E 12 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.000000 E: 0.800000 Sum of squares = 0.66000000199 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6985: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (5) B C E A D (5) D B E A C (4) D A C E B (4) D A C B E (4) C A D E B (4) B E C A D (4) A D C E B (4) D C A B E (3) D A E B C (3) C E A B D (3) C A E D B (3) B D E A C (3) B D C A E (3) E A D B C (2) E A B D C (2) E A B C D (2) D B A C E (2) D A E C B (2) D A B C E (2) C D A B E (2) C B A E D (2) C B A D E (2) C A E B D (2) A D E C B (2) A C D E B (2) E C A B D (1) E B D A C (1) E B C A D (1) E B A D C (1) E A D C B (1) D E A B C (1) D C B A E (1) D C A E B (1) D B A E C (1) D A B E C (1) C E B A D (1) C D A E B (1) C B E A D (1) C B D A E (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 10 4 -4 12 B -10 0 0 -6 6 C -4 0 0 -14 16 D 4 6 14 0 14 E -12 -6 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 -4 12 B -10 0 0 -6 6 C -4 0 0 -14 16 D 4 6 14 0 14 E -12 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=27 C=22 E=11 A=11 so E is eliminated. Round 2 votes counts: B=30 D=29 C=23 A=18 so A is eliminated. Round 3 votes counts: D=39 B=34 C=27 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:211 C:199 B:195 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 4 -4 12 B -10 0 0 -6 6 C -4 0 0 -14 16 D 4 6 14 0 14 E -12 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 -4 12 B -10 0 0 -6 6 C -4 0 0 -14 16 D 4 6 14 0 14 E -12 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 -4 12 B -10 0 0 -6 6 C -4 0 0 -14 16 D 4 6 14 0 14 E -12 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6986: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (5) C B E A D (5) C B A E D (5) A B D C E (5) D E A B C (4) C B E D A (4) B A C E D (4) A D E B C (4) A D B E C (4) A C B D E (4) E D B C A (3) E C D B A (3) C D E A B (3) B E C D A (3) A D C E B (3) A B C D E (3) E D C B A (2) E B C D A (2) D E C A B (2) D A E B C (2) C E D B A (2) B C E A D (2) A D C B E (2) A C D B E (2) A B D E C (2) E D C A B (1) E D B A C (1) E B D C A (1) D E B A C (1) D E A C B (1) D B A E C (1) D A E C B (1) D A C E B (1) C E D A B (1) C D A E B (1) C A D E B (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D C A (1) B E A C D (1) B C A E D (1) B A E D C (1) B A E C D (1) A D B C E (1) Total count = 100 A B C D E A 0 2 -2 8 4 B -2 0 -4 4 10 C 2 4 0 12 16 D -8 -4 -12 0 0 E -4 -10 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 8 4 B -2 0 -4 4 10 C 2 4 0 12 16 D -8 -4 -12 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=30 A=30 B=14 E=13 D=13 so E is eliminated. Round 2 votes counts: C=33 A=30 D=20 B=17 so B is eliminated. Round 3 votes counts: C=41 A=37 D=22 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:206 B:204 D:188 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 8 4 B -2 0 -4 4 10 C 2 4 0 12 16 D -8 -4 -12 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 8 4 B -2 0 -4 4 10 C 2 4 0 12 16 D -8 -4 -12 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 8 4 B -2 0 -4 4 10 C 2 4 0 12 16 D -8 -4 -12 0 0 E -4 -10 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999413 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6987: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) D C E B A (8) A B C E D (6) D A E C B (5) C E B D A (5) A D C B E (5) A D B E C (5) A B E C D (5) D E C B A (4) C B E D A (4) C B E A D (4) E D B C A (3) D A E B C (3) E B C D A (2) D E B C A (2) D C A E B (2) D A C E B (2) C D E B A (2) B C E A D (2) B A E C D (2) B A C E D (2) A C B D E (2) E C B D A (1) E B A C D (1) D E C A B (1) D E A C B (1) D C E A B (1) C B D A E (1) C B A E D (1) B E C A D (1) B E A C D (1) B C A E D (1) A E D B C (1) A E B D C (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 10 6 14 B -6 0 0 -8 -8 C -10 0 0 -10 2 D -6 8 10 0 12 E -14 8 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 6 14 B -6 0 0 -8 -8 C -10 0 0 -10 2 D -6 8 10 0 12 E -14 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=29 C=17 B=9 E=7 so E is eliminated. Round 2 votes counts: A=38 D=32 C=18 B=12 so B is eliminated. Round 3 votes counts: A=44 D=32 C=24 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:212 C:191 E:190 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 6 14 B -6 0 0 -8 -8 C -10 0 0 -10 2 D -6 8 10 0 12 E -14 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 6 14 B -6 0 0 -8 -8 C -10 0 0 -10 2 D -6 8 10 0 12 E -14 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 6 14 B -6 0 0 -8 -8 C -10 0 0 -10 2 D -6 8 10 0 12 E -14 8 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6988: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) A C D B E (11) C A E D B (7) B D E A C (7) E B D C A (6) D B E A C (5) D B A C E (5) B D A C E (5) E C A D B (4) E D B C A (3) E D B A C (3) E B C D A (3) C A E B D (3) E B C A D (2) D A B C E (2) C A D E B (2) B E D A C (2) A C D E B (2) A B C D E (2) E D C A B (1) E C B D A (1) E B D A C (1) D A C B E (1) C E A B D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D C A (1) B D A E C (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 8 -2 6 -8 B -8 0 -2 4 -6 C 2 2 0 12 -8 D -6 -4 -12 0 -6 E 8 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -2 6 -8 B -8 0 -2 4 -6 C 2 2 0 12 -8 D -6 -4 -12 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=17 C=16 A=16 D=13 so D is eliminated. Round 2 votes counts: E=38 B=27 A=19 C=16 so C is eliminated. Round 3 votes counts: E=39 A=34 B=27 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:204 A:202 B:194 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -2 6 -8 B -8 0 -2 4 -6 C 2 2 0 12 -8 D -6 -4 -12 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 -8 B -8 0 -2 4 -6 C 2 2 0 12 -8 D -6 -4 -12 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 -8 B -8 0 -2 4 -6 C 2 2 0 12 -8 D -6 -4 -12 0 -6 E 8 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 6989: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (18) C D E B A (16) D C B A E (10) E A B C D (9) E B A D C (4) C D E A B (4) D C B E A (3) C E D A B (3) B A D E C (3) E C A D B (2) E A C D B (2) C D A B E (2) B E A D C (2) B A E D C (2) B A D C E (2) A E B D C (2) A B E C D (2) A B D C E (2) E B A C D (1) E A B D C (1) D C E B A (1) D C A B E (1) D B C E A (1) D B C A E (1) C D B A E (1) C D A E B (1) B D A E C (1) B D A C E (1) A E B C D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 8 8 2 B -2 0 6 2 6 C -8 -6 0 -12 0 D -8 -2 12 0 2 E -2 -6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 8 2 B -2 0 6 2 6 C -8 -6 0 -12 0 D -8 -2 12 0 2 E -2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 E=19 D=17 B=11 so B is eliminated. Round 2 votes counts: A=33 C=27 E=21 D=19 so D is eliminated. Round 3 votes counts: C=44 A=35 E=21 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:206 D:202 E:195 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 8 2 B -2 0 6 2 6 C -8 -6 0 -12 0 D -8 -2 12 0 2 E -2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 8 2 B -2 0 6 2 6 C -8 -6 0 -12 0 D -8 -2 12 0 2 E -2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 8 2 B -2 0 6 2 6 C -8 -6 0 -12 0 D -8 -2 12 0 2 E -2 -6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 6990: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) E B A D C (6) C A D E B (6) C A B E D (5) B E A D C (5) A E B C D (5) D C A B E (4) D A C E B (4) C B E A D (4) B E C D A (4) D C B A E (3) D A E C B (3) C D A B E (3) B E D A C (3) B C E A D (3) A C D E B (3) E A B C D (2) D C B E A (2) D A E B C (2) B E A C D (2) B C E D A (2) A D E C B (2) A C E D B (2) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) D B E C A (1) D B C E A (1) C B D E A (1) C B A E D (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B D E (1) B E C A D (1) A E D B C (1) A E C B D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 18 -6 16 18 B -18 0 -14 0 -6 C 6 14 0 0 14 D -16 0 0 0 -6 E -18 6 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.844266 D: 0.155734 E: 0.000000 Sum of squares = 0.737038513216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.844266 D: 1.000000 E: 1.000000 A B C D E A 0 18 -6 16 18 B -18 0 -14 0 -6 C 6 14 0 0 14 D -16 0 0 0 -6 E -18 6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.727273 D: 0.272727 E: 0.000000 Sum of squares = 0.603305855951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.727273 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 B=20 A=16 E=12 so E is eliminated. Round 2 votes counts: D=28 B=28 C=24 A=20 so A is eliminated. Round 3 votes counts: B=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:223 C:217 E:190 D:189 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -6 16 18 B -18 0 -14 0 -6 C 6 14 0 0 14 D -16 0 0 0 -6 E -18 6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.727273 D: 0.272727 E: 0.000000 Sum of squares = 0.603305855951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.727273 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -6 16 18 B -18 0 -14 0 -6 C 6 14 0 0 14 D -16 0 0 0 -6 E -18 6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.727273 D: 0.272727 E: 0.000000 Sum of squares = 0.603305855951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.727273 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -6 16 18 B -18 0 -14 0 -6 C 6 14 0 0 14 D -16 0 0 0 -6 E -18 6 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.727273 D: 0.272727 E: 0.000000 Sum of squares = 0.603305855951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.727273 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6991: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C E A B D (12) C D E A B (7) D C E A B (6) B A D E C (6) B D A E C (5) B A E D C (5) B A E C D (5) D A B E C (4) C E D A B (4) D C B A E (3) C E B A D (3) C E A D B (3) D C A E B (2) D B A C E (2) D A E C B (2) C B E A D (2) B C A E D (2) A B E D C (2) E C A B D (1) E A D B C (1) E A B C D (1) D E A C B (1) D B C A E (1) D A E B C (1) C B D A E (1) C B A E D (1) B C E A D (1) B A D C E (1) B A C E D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 2 -2 16 B 2 0 4 2 8 C -2 -4 0 -10 4 D 2 -2 10 0 8 E -16 -8 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 16 B 2 0 4 2 8 C -2 -4 0 -10 4 D 2 -2 10 0 8 E -16 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=33 B=26 A=4 E=3 so E is eliminated. Round 2 votes counts: D=34 C=34 B=26 A=6 so A is eliminated. Round 3 votes counts: D=35 C=34 B=31 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:208 A:207 C:194 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 16 B 2 0 4 2 8 C -2 -4 0 -10 4 D 2 -2 10 0 8 E -16 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 16 B 2 0 4 2 8 C -2 -4 0 -10 4 D 2 -2 10 0 8 E -16 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 16 B 2 0 4 2 8 C -2 -4 0 -10 4 D 2 -2 10 0 8 E -16 -8 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993523 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6992: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) D E B A C (6) D B E C A (6) E B D A C (5) C A D B E (5) B E D C A (5) B D C E A (5) A C D E B (5) A E C D B (4) A C E B D (4) C D A B E (3) C B D A E (3) A E C B D (3) E D B A C (2) E D A B C (2) E A D B C (2) E A B D C (2) D B C E A (2) C B D E A (2) C A D E B (2) C A B D E (2) B D E C A (2) B C D E A (2) A E D C B (2) A D E C B (2) A D C E B (2) E B A C D (1) D E A B C (1) D C A B E (1) D B E A C (1) D A E C B (1) C D B E A (1) C B A E D (1) C A E B D (1) B E C D A (1) B C E D A (1) B C E A D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 6 8 -4 0 B -6 0 10 -16 -12 C -8 -10 0 -12 -12 D 4 16 12 0 8 E 0 12 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999212 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -4 0 B -6 0 10 -16 -12 C -8 -10 0 -12 -12 D 4 16 12 0 8 E 0 12 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=20 D=18 B=17 E=14 so E is eliminated. Round 2 votes counts: A=35 B=23 D=22 C=20 so C is eliminated. Round 3 votes counts: A=45 B=29 D=26 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:220 E:208 A:205 B:188 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -4 0 B -6 0 10 -16 -12 C -8 -10 0 -12 -12 D 4 16 12 0 8 E 0 12 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -4 0 B -6 0 10 -16 -12 C -8 -10 0 -12 -12 D 4 16 12 0 8 E 0 12 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -4 0 B -6 0 10 -16 -12 C -8 -10 0 -12 -12 D 4 16 12 0 8 E 0 12 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6993: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (7) D E C B A (5) D E B A C (5) D E A C B (5) B C A E D (5) E D A C B (4) E A D B C (4) D E C A B (4) A E B D C (4) C D E A B (3) C D B E A (3) B D E A C (3) B A E C D (3) B A C E D (3) A B C E D (3) E D B A C (2) E D A B C (2) D E B C A (2) C B A D E (2) C A E D B (2) C A D E B (2) C A D B E (2) C A B E D (2) B D C E A (2) B C D A E (2) B A E D C (2) A E C D B (2) A C B E D (2) E A D C B (1) D E A B C (1) D C E B A (1) D B C E A (1) C D A E B (1) C B D E A (1) C A B D E (1) B E D A C (1) B C D E A (1) A E D C B (1) A E D B C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 2 -2 B 2 0 10 -10 -6 C -2 -10 0 -4 -6 D -2 10 4 0 8 E 2 6 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 -2 2 2 -2 B 2 0 10 -10 -6 C -2 -10 0 -4 -6 D -2 10 4 0 8 E 2 6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=24 C=19 A=15 E=13 so E is eliminated. Round 2 votes counts: D=32 B=29 A=20 C=19 so C is eliminated. Round 3 votes counts: D=39 B=32 A=29 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:203 A:200 B:198 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 -2 B 2 0 10 -10 -6 C -2 -10 0 -4 -6 D -2 10 4 0 8 E 2 6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 -2 B 2 0 10 -10 -6 C -2 -10 0 -4 -6 D -2 10 4 0 8 E 2 6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 -2 B 2 0 10 -10 -6 C -2 -10 0 -4 -6 D -2 10 4 0 8 E 2 6 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999797 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6994: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (16) D B A C E (14) A C E D B (10) D A B C E (8) A C D E B (6) B E C D A (4) E C B A D (3) E C A B D (3) E B C A D (3) C A E B D (3) B D C E A (3) E A C D B (2) E A C B D (2) D B E A C (2) D B C A E (2) C E A B D (2) A D C B E (2) E D A B C (1) E B D C A (1) E A D C B (1) D B E C A (1) D B A E C (1) D A C E B (1) D A C B E (1) C B E A D (1) C A B D E (1) B E D C A (1) B E D A C (1) B C E A D (1) A E C D B (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 10 -14 4 B 8 0 18 -10 18 C -10 -18 0 -16 12 D 14 10 16 0 20 E -4 -18 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 -14 4 B 8 0 18 -10 18 C -10 -18 0 -16 12 D 14 10 16 0 20 E -4 -18 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=26 A=21 E=16 C=7 so C is eliminated. Round 2 votes counts: D=30 B=27 A=25 E=18 so E is eliminated. Round 3 votes counts: A=35 B=34 D=31 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:230 B:217 A:196 C:184 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 10 -14 4 B 8 0 18 -10 18 C -10 -18 0 -16 12 D 14 10 16 0 20 E -4 -18 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 -14 4 B 8 0 18 -10 18 C -10 -18 0 -16 12 D 14 10 16 0 20 E -4 -18 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 -14 4 B 8 0 18 -10 18 C -10 -18 0 -16 12 D 14 10 16 0 20 E -4 -18 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 6995: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) E B A C D (9) D C A B E (9) D A B C E (8) C E B A D (8) C D E A B (7) C E D B A (6) C E B D A (6) C D E B A (6) E B C A D (3) D A C B E (3) B E A D C (3) B A E D C (3) E C B A D (2) C E A B D (2) B A D E C (2) E A B C D (1) D C E A B (1) D C B E A (1) D C A E B (1) D B A E C (1) C E D A B (1) C D A E B (1) B E A C D (1) A D B E C (1) A D B C E (1) A C E B D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -8 -22 -14 B 2 0 -10 -14 -10 C 8 10 0 8 24 D 22 14 -8 0 6 E 14 10 -24 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -22 -14 B 2 0 -10 -14 -10 C 8 10 0 8 24 D 22 14 -8 0 6 E 14 10 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=34 E=15 B=9 A=5 so A is eliminated. Round 2 votes counts: C=38 D=36 E=15 B=11 so B is eliminated. Round 3 votes counts: D=39 C=38 E=23 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 D:217 E:197 B:184 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -22 -14 B 2 0 -10 -14 -10 C 8 10 0 8 24 D 22 14 -8 0 6 E 14 10 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -22 -14 B 2 0 -10 -14 -10 C 8 10 0 8 24 D 22 14 -8 0 6 E 14 10 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -22 -14 B 2 0 -10 -14 -10 C 8 10 0 8 24 D 22 14 -8 0 6 E 14 10 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6996: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) C D B E A (8) C E A B D (7) C B D E A (7) B C D A E (7) C E A D B (6) B D C A E (6) A E D B C (6) D B A E C (4) B C A D E (4) A E C B D (4) A E B D C (4) E A C B D (3) D C B E A (3) D B C E A (3) B D A E C (3) D A E B C (2) C D E B A (2) B A E D C (2) A E B C D (2) E C A D B (1) E A D C B (1) D E A B C (1) D B E A C (1) C D E A B (1) C B E D A (1) C B E A D (1) B D C E A (1) B A D E C (1) Total count = 100 A B C D E A 0 -8 -16 0 -10 B 8 0 -6 6 4 C 16 6 0 24 14 D 0 -6 -24 0 8 E 10 -4 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 0 -10 B 8 0 -6 6 4 C 16 6 0 24 14 D 0 -6 -24 0 8 E 10 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=24 A=16 D=14 E=13 so E is eliminated. Round 2 votes counts: C=34 A=28 B=24 D=14 so D is eliminated. Round 3 votes counts: C=37 B=32 A=31 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:206 E:192 D:189 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -16 0 -10 B 8 0 -6 6 4 C 16 6 0 24 14 D 0 -6 -24 0 8 E 10 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 0 -10 B 8 0 -6 6 4 C 16 6 0 24 14 D 0 -6 -24 0 8 E 10 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 0 -10 B 8 0 -6 6 4 C 16 6 0 24 14 D 0 -6 -24 0 8 E 10 -4 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 6997: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) B E C A D (6) B A C E D (6) D A C E B (5) B E D C A (5) B D E A C (5) A C D E B (5) D E B C A (4) B E C D A (4) B C A E D (4) A C B E D (4) A C B D E (4) E B D C A (3) C E A D B (3) A C D B E (3) C A E D B (2) B E D A C (2) B C E A D (2) A D C E B (2) A B C D E (2) E D C B A (1) E C B D A (1) E C B A D (1) E B C D A (1) E B C A D (1) D E B A C (1) D E A C B (1) D E A B C (1) D C E A B (1) D B E C A (1) D B E A C (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B C E (1) C A E B D (1) C A B E D (1) B E A C D (1) B A E C D (1) B A D C E (1) A D C B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 4 6 -4 B 6 0 12 16 18 C -4 -12 0 8 2 D -6 -16 -8 0 -2 E 4 -18 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 6 -4 B 6 0 12 16 18 C -4 -12 0 8 2 D -6 -16 -8 0 -2 E 4 -18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=25 A=23 E=8 C=7 so C is eliminated. Round 2 votes counts: B=37 A=27 D=25 E=11 so E is eliminated. Round 3 votes counts: B=44 A=30 D=26 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:200 C:197 E:193 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 6 -4 B 6 0 12 16 18 C -4 -12 0 8 2 D -6 -16 -8 0 -2 E 4 -18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 6 -4 B 6 0 12 16 18 C -4 -12 0 8 2 D -6 -16 -8 0 -2 E 4 -18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 6 -4 B 6 0 12 16 18 C -4 -12 0 8 2 D -6 -16 -8 0 -2 E 4 -18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6998: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (12) B C D A E (11) A E B C D (9) A E D B C (8) E A D C B (7) C B D E A (7) B C A D E (6) D E A C B (5) A B E C D (5) E D A C B (3) D E C A B (3) B A C E D (3) A E B D C (3) E D C B A (2) D E C B A (2) B C A E D (2) A B C E D (2) E B C A D (1) E A C B D (1) D C E B A (1) D B C A E (1) D A B C E (1) C E B D A (1) C B E D A (1) C B E A D (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 -4 -4 -2 4 B 4 0 6 8 6 C 4 -6 0 2 0 D 2 -8 -2 0 0 E -4 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 4 B 4 0 6 8 6 C 4 -6 0 2 0 D 2 -8 -2 0 0 E -4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 B=23 E=14 C=10 so C is eliminated. Round 2 votes counts: B=32 A=28 D=25 E=15 so E is eliminated. Round 3 votes counts: A=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:200 A:197 D:196 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 4 B 4 0 6 8 6 C 4 -6 0 2 0 D 2 -8 -2 0 0 E -4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 4 B 4 0 6 8 6 C 4 -6 0 2 0 D 2 -8 -2 0 0 E -4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 4 B 4 0 6 8 6 C 4 -6 0 2 0 D 2 -8 -2 0 0 E -4 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 6999: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) E A B C D (7) D C B A E (7) B A D C E (6) A E B D C (5) A B E D C (5) E D C A B (4) D C E A B (4) B C D A E (4) A E D C B (4) E C D A B (3) E A D C B (3) C D B E A (3) B A C E D (3) B A C D E (3) E B A C D (2) E A C D B (2) D C E B A (2) C D E A B (2) B C A D E (2) A B E C D (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D B A (1) E A D B C (1) E A C B D (1) D C B E A (1) D C A E B (1) D C A B E (1) D A E C B (1) C D B A E (1) B E C A D (1) B C E D A (1) B A E C D (1) A E B C D (1) A D E C B (1) A D B C E (1) Total count = 100 A B C D E A 0 8 8 10 6 B -8 0 -2 -4 -8 C -8 2 0 -4 4 D -10 4 4 0 4 E -6 8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 10 6 B -8 0 -2 -4 -8 C -8 2 0 -4 4 D -10 4 4 0 4 E -6 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 B=21 D=17 C=14 so C is eliminated. Round 2 votes counts: D=31 E=25 A=23 B=21 so B is eliminated. Round 3 votes counts: A=38 D=35 E=27 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:201 C:197 E:197 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 10 6 B -8 0 -2 -4 -8 C -8 2 0 -4 4 D -10 4 4 0 4 E -6 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 10 6 B -8 0 -2 -4 -8 C -8 2 0 -4 4 D -10 4 4 0 4 E -6 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 10 6 B -8 0 -2 -4 -8 C -8 2 0 -4 4 D -10 4 4 0 4 E -6 8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7000: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (13) D B C E A (10) E A B D C (9) C B D A E (9) C D B A E (6) A E C B D (6) A E B D C (6) C D B E A (5) C A B D E (4) E A C D B (3) B D C E A (3) B D C A E (3) E D B C A (2) E D B A C (2) D C B E A (2) D C B A E (2) A C E B D (2) E D C B A (1) E C D B A (1) E A D C B (1) E A C B D (1) D B E C A (1) D B C A E (1) C E D A B (1) C A E D B (1) B D E C A (1) B D E A C (1) B D A E C (1) B C A D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 -4 -14 B 4 0 10 -4 2 C 8 -10 0 -18 2 D 4 4 18 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -4 -14 B 4 0 10 -4 2 C 8 -10 0 -18 2 D 4 4 18 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=26 D=16 A=15 B=10 so B is eliminated. Round 2 votes counts: E=33 C=27 D=25 A=15 so A is eliminated. Round 3 votes counts: E=45 C=30 D=25 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:214 B:206 E:204 C:191 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 -14 B 4 0 10 -4 2 C 8 -10 0 -18 2 D 4 4 18 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 -14 B 4 0 10 -4 2 C 8 -10 0 -18 2 D 4 4 18 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 -14 B 4 0 10 -4 2 C 8 -10 0 -18 2 D 4 4 18 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7001: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (11) C D A B E (10) B E C A D (10) C B E D A (8) E B A C D (7) D C A B E (7) D A C E B (7) C D B E A (6) E B A D C (4) E A B D C (4) D C A E B (4) B E A C D (4) A E B D C (3) D A E B C (2) C D B A E (2) C B E A D (2) A E D B C (2) A D C B E (2) D A C B E (1) B E C D A (1) B C E D A (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 8 -4 2 4 B -8 0 0 -10 10 C 4 0 0 4 2 D -2 10 -4 0 6 E -4 -10 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166898 C: 0.833102 D: 0.000000 E: 0.000000 Sum of squares = 0.721913944805 Cumulative probabilities = A: 0.000000 B: 0.166898 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 2 4 B -8 0 0 -10 10 C 4 0 0 4 2 D -2 10 -4 0 6 E -4 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836760162 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=21 A=19 B=17 E=15 so E is eliminated. Round 2 votes counts: C=28 B=28 A=23 D=21 so D is eliminated. Round 3 votes counts: C=39 A=33 B=28 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:205 C:205 D:205 B:196 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 2 4 B -8 0 0 -10 10 C 4 0 0 4 2 D -2 10 -4 0 6 E -4 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836760162 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 2 4 B -8 0 0 -10 10 C 4 0 0 4 2 D -2 10 -4 0 6 E -4 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836760162 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 2 4 B -8 0 0 -10 10 C 4 0 0 4 2 D -2 10 -4 0 6 E -4 -10 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836760162 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7002: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (14) D C E A B (9) A B D E C (9) E C D B A (7) C E D B A (6) B A E C D (6) B A D C E (5) E C B A D (4) D A B C E (4) D C A E B (3) A D B C E (3) E C A B D (2) D A C E B (2) C D E B A (2) B E A C D (2) B A E D C (2) A B E C D (2) E C D A B (1) E C B D A (1) E A C B D (1) D C E B A (1) D C B E A (1) D C B A E (1) D C A B E (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E A B (1) B E C A D (1) B D A C E (1) B C E D A (1) A E B C D (1) A D E C B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 16 14 10 18 B -16 0 8 6 14 C -14 -8 0 -22 16 D -10 -6 22 0 22 E -18 -14 -16 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 10 18 B -16 0 8 6 14 C -14 -8 0 -22 16 D -10 -6 22 0 22 E -18 -14 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=24 B=18 E=16 C=10 so C is eliminated. Round 2 votes counts: A=32 D=27 E=23 B=18 so B is eliminated. Round 3 votes counts: A=45 D=28 E=27 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:229 D:214 B:206 C:186 E:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 10 18 B -16 0 8 6 14 C -14 -8 0 -22 16 D -10 -6 22 0 22 E -18 -14 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 10 18 B -16 0 8 6 14 C -14 -8 0 -22 16 D -10 -6 22 0 22 E -18 -14 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 10 18 B -16 0 8 6 14 C -14 -8 0 -22 16 D -10 -6 22 0 22 E -18 -14 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7003: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) D A C E B (8) C D A B E (8) D C A E B (7) B E A C D (7) E B D A C (6) E B D C A (4) B E C A D (4) A D C E B (4) C D A E B (3) B E C D A (3) B C E A D (3) A D E C B (3) E B C D A (2) E B A D C (2) E A B D C (2) C A D B E (2) A E D B C (2) A D C B E (2) A C D B E (2) A B E D C (2) E D B A C (1) E C B D A (1) D E A C B (1) D C E A B (1) D A E C B (1) C D E B A (1) C D B E A (1) C B E A D (1) C B D E A (1) C B D A E (1) C B A D E (1) B E A D C (1) B C E D A (1) B A E C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -8 -20 -4 B 2 0 -16 4 0 C 8 16 0 4 12 D 20 -4 -4 0 -4 E 4 0 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -20 -4 B 2 0 -16 4 0 C 8 16 0 4 12 D 20 -4 -4 0 -4 E 4 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=20 E=18 D=18 A=16 so A is eliminated. Round 2 votes counts: C=30 D=28 B=22 E=20 so E is eliminated. Round 3 votes counts: B=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:204 E:198 B:195 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -20 -4 B 2 0 -16 4 0 C 8 16 0 4 12 D 20 -4 -4 0 -4 E 4 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -20 -4 B 2 0 -16 4 0 C 8 16 0 4 12 D 20 -4 -4 0 -4 E 4 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -20 -4 B 2 0 -16 4 0 C 8 16 0 4 12 D 20 -4 -4 0 -4 E 4 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7004: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B A C D E (6) C D B E A (5) B D A C E (5) A E C D B (5) A E C B D (5) D C E B A (4) B D C E A (4) B D C A E (4) B A D C E (4) A B C E D (4) E A D C B (3) D E C B A (3) C D E B A (3) A E B C D (3) A B D E C (3) E C D A B (2) E A C D B (2) D E A B C (2) D C B E A (2) D B C E A (2) C E D B A (2) A E D C B (2) A E D B C (2) A B E C D (2) A B C D E (2) E C D B A (1) D E C A B (1) D B A E C (1) D A E B C (1) C E D A B (1) C E B D A (1) B C D A E (1) B C A D E (1) B A C E D (1) A C E B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 12 -4 10 B 0 0 0 -2 -2 C -12 0 0 -4 8 D 4 2 4 0 10 E -10 2 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 -4 10 B 0 0 0 -2 -2 C -12 0 0 -4 8 D 4 2 4 0 10 E -10 2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=26 D=16 E=15 C=12 so C is eliminated. Round 2 votes counts: A=31 B=26 D=24 E=19 so E is eliminated. Round 3 votes counts: D=37 A=36 B=27 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:209 B:198 C:196 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 12 -4 10 B 0 0 0 -2 -2 C -12 0 0 -4 8 D 4 2 4 0 10 E -10 2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -4 10 B 0 0 0 -2 -2 C -12 0 0 -4 8 D 4 2 4 0 10 E -10 2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -4 10 B 0 0 0 -2 -2 C -12 0 0 -4 8 D 4 2 4 0 10 E -10 2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7005: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) A E C B D (9) D B C E A (8) D E A B C (7) C B A E D (7) A C E B D (7) E A C B D (6) D E A C B (6) B C D A E (6) B C A D E (5) E D A C B (4) E A D C B (4) E A C D B (4) D B E C A (3) D B C A E (3) E D A B C (2) D E B A C (2) C A B E D (2) D E B C A (1) C E A B D (1) B C A E D (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 8 6 -4 4 B -8 0 -4 10 -8 C -6 4 0 0 2 D 4 -10 0 0 2 E -4 8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900826443 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -4 4 B -8 0 -4 10 -8 C -6 4 0 0 2 D 4 -10 0 0 2 E -4 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900826346 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=22 E=20 A=18 C=10 so C is eliminated. Round 2 votes counts: D=30 B=29 E=21 A=20 so A is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:207 C:200 E:200 D:198 B:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 -4 4 B -8 0 -4 10 -8 C -6 4 0 0 2 D 4 -10 0 0 2 E -4 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900826346 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -4 4 B -8 0 -4 10 -8 C -6 4 0 0 2 D 4 -10 0 0 2 E -4 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900826346 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -4 4 B -8 0 -4 10 -8 C -6 4 0 0 2 D 4 -10 0 0 2 E -4 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.181818 C: 0.000000 D: 0.363636 E: 0.000000 Sum of squares = 0.371900826346 Cumulative probabilities = A: 0.454545 B: 0.636364 C: 0.636364 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7006: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (9) E D A B C (7) B C E A D (6) B C A E D (6) D E A B C (5) C B A E D (5) C B A D E (5) D E A C B (4) C A B D E (4) A D E C B (4) A D C B E (4) E D B A C (3) E B D C A (3) E B C D A (3) D A E C B (3) B C E D A (3) D E B C A (2) A C D B E (2) A C B E D (2) E A D B C (1) D A C E B (1) D A C B E (1) C D B E A (1) C D B A E (1) C B D A E (1) B E C D A (1) B A C E D (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -4 0 0 B 4 0 6 0 4 C 4 -6 0 2 4 D 0 0 -2 0 -6 E 0 -4 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.726661 C: 0.000000 D: 0.273339 E: 0.000000 Sum of squares = 0.602750347685 Cumulative probabilities = A: 0.000000 B: 0.726661 C: 0.726661 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 0 0 B 4 0 6 0 4 C 4 -6 0 2 4 D 0 0 -2 0 -6 E 0 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000001109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=22 C=17 B=17 D=16 so D is eliminated. Round 2 votes counts: E=39 A=27 C=17 B=17 so C is eliminated. Round 3 votes counts: E=39 A=31 B=30 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:207 C:202 E:199 A:196 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 0 0 B 4 0 6 0 4 C 4 -6 0 2 4 D 0 0 -2 0 -6 E 0 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000001109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 0 B 4 0 6 0 4 C 4 -6 0 2 4 D 0 0 -2 0 -6 E 0 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000001109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 0 B 4 0 6 0 4 C 4 -6 0 2 4 D 0 0 -2 0 -6 E 0 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000001109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7007: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C E B A D (6) E D B A C (5) D A C B E (5) C A E B D (5) D A B E C (4) A D C B E (4) E B C A D (3) E B A C D (3) D B E A C (3) C E A B D (3) C A D E B (3) B E A D C (3) A C D B E (3) E D B C A (2) E C B D A (2) E B D C A (2) E B C D A (2) D C A B E (2) D B A E C (2) C E D B A (2) C E D A B (2) C D E A B (2) C D A E B (2) C A D B E (2) C A B E D (2) C A B D E (2) B E A C D (2) A B C E D (2) E C D B A (1) E C B A D (1) D C A E B (1) C E A D B (1) B E D A C (1) B A E D C (1) B A D E C (1) A D B C E (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 4 2 -10 B 2 0 -6 6 -14 C -4 6 0 8 4 D -2 -6 -8 0 -24 E 10 14 -4 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407407505 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 -2 4 2 -10 B 2 0 -6 6 -14 C -4 6 0 8 4 D -2 -6 -8 0 -24 E 10 14 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407405709 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=30 D=17 A=13 B=8 so B is eliminated. Round 2 votes counts: E=36 C=32 D=17 A=15 so A is eliminated. Round 3 votes counts: C=39 E=38 D=23 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:222 C:207 A:197 B:194 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 4 2 -10 B 2 0 -6 6 -14 C -4 6 0 8 4 D -2 -6 -8 0 -24 E 10 14 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407405709 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 2 -10 B 2 0 -6 6 -14 C -4 6 0 8 4 D -2 -6 -8 0 -24 E 10 14 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407405709 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 2 -10 B 2 0 -6 6 -14 C -4 6 0 8 4 D -2 -6 -8 0 -24 E 10 14 -4 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.000000 E: 0.222222 Sum of squares = 0.407407405709 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7008: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (9) E A D C B (7) D E A C B (6) A E C D B (6) B C A D E (5) A E C B D (5) D C B E A (4) C B D A E (4) B C A E D (4) D E A B C (3) D B C E A (3) C B A D E (3) B D E A C (3) B D C E A (3) D E C A B (2) D C E B A (2) D B E C A (2) C D B A E (2) C D A E B (2) C B A E D (2) C A E D B (2) C A B E D (2) B D C A E (2) B A E C D (2) A E B C D (2) E D A C B (1) E D A B C (1) E A C D B (1) E A B D C (1) D C E A B (1) D B E A C (1) C E A D B (1) C D E A B (1) C A E B D (1) C A D E B (1) B A E D C (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -16 -4 14 B 6 0 -12 2 8 C 16 12 0 14 10 D 4 -2 -14 0 18 E -14 -8 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -4 14 B 6 0 -12 2 8 C 16 12 0 14 10 D 4 -2 -14 0 18 E -14 -8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=24 C=21 A=14 E=11 so E is eliminated. Round 2 votes counts: B=30 D=26 A=23 C=21 so C is eliminated. Round 3 votes counts: B=39 D=31 A=30 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:226 D:203 B:202 A:194 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -4 14 B 6 0 -12 2 8 C 16 12 0 14 10 D 4 -2 -14 0 18 E -14 -8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -4 14 B 6 0 -12 2 8 C 16 12 0 14 10 D 4 -2 -14 0 18 E -14 -8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -4 14 B 6 0 -12 2 8 C 16 12 0 14 10 D 4 -2 -14 0 18 E -14 -8 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7009: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) D B E A C (6) D C B A E (5) E A B C D (4) D C B E A (4) D C A B E (4) C D E A B (4) E C A B D (3) E B D A C (3) E B A C D (3) D B A E C (3) C E A B D (3) C D E B A (3) C A D B E (3) B E A D C (3) B A E D C (3) A B E C D (3) A B D E C (3) E D B C A (2) E A C B D (2) D C E B A (2) D B A C E (2) D A B C E (2) C D A B E (2) C A E D B (2) C A E B D (2) A C B D E (2) A B E D C (2) D E B C A (1) D E B A C (1) D B E C A (1) D A C B E (1) C E D A B (1) C D A E B (1) C A B E D (1) B E D A C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 12 2 -14 B 6 0 10 0 6 C -12 -10 0 -20 -8 D -2 0 20 0 4 E 14 -6 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.901491 C: 0.000000 D: 0.098509 E: 0.000000 Sum of squares = 0.822390130651 Cumulative probabilities = A: 0.000000 B: 0.901491 C: 0.901491 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 2 -14 B 6 0 10 0 6 C -12 -10 0 -20 -8 D -2 0 20 0 4 E 14 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=27 C=22 A=12 B=7 so B is eliminated. Round 2 votes counts: D=32 E=31 C=22 A=15 so A is eliminated. Round 3 votes counts: E=39 D=36 C=25 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:211 D:211 E:206 A:197 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 2 -14 B 6 0 10 0 6 C -12 -10 0 -20 -8 D -2 0 20 0 4 E 14 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 2 -14 B 6 0 10 0 6 C -12 -10 0 -20 -8 D -2 0 20 0 4 E 14 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 2 -14 B 6 0 10 0 6 C -12 -10 0 -20 -8 D -2 0 20 0 4 E 14 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7010: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) D B E C A (6) B D E C A (6) A C E D B (6) E A C B D (5) D B C E A (5) A E C B D (5) C A E D B (4) B D A C E (4) A C D B E (4) E C A D B (3) D C B A E (3) C A D B E (3) B E D A C (3) B D E A C (3) E B D A C (2) E A B C D (2) D C A B E (2) C E A D B (2) C D A B E (2) C A D E B (2) A B C D E (2) E D C B A (1) E B D C A (1) E B A D C (1) E B A C D (1) D E B C A (1) D C E B A (1) D A C B E (1) C D A E B (1) B E A D C (1) B D A E C (1) B A D E C (1) A E B C D (1) A C E B D (1) A C D E B (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -4 -4 12 B 0 0 2 -14 18 C 4 -2 0 -4 10 D 4 14 4 0 18 E -12 -18 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -4 12 B 0 0 2 -14 18 C 4 -2 0 -4 10 D 4 14 4 0 18 E -12 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=23 B=19 E=16 C=14 so C is eliminated. Round 2 votes counts: A=32 D=31 B=19 E=18 so E is eliminated. Round 3 votes counts: A=44 D=32 B=24 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:204 B:203 A:202 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -4 12 B 0 0 2 -14 18 C 4 -2 0 -4 10 D 4 14 4 0 18 E -12 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 12 B 0 0 2 -14 18 C 4 -2 0 -4 10 D 4 14 4 0 18 E -12 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 12 B 0 0 2 -14 18 C 4 -2 0 -4 10 D 4 14 4 0 18 E -12 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997276 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7011: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) D C E A B (12) C D E A B (9) B E A D C (7) B A E D C (5) B A E C D (5) E B A D C (4) D C E B A (3) C D E B A (3) C D B E A (3) C A D B E (3) B C A D E (3) A B E C D (3) E A B D C (2) D C A E B (2) C D B A E (2) C D A B E (2) B E A C D (2) B C D E A (2) E A D C B (1) E A D B C (1) D E C A B (1) C B D A E (1) C A D E B (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E A D (1) B A C E D (1) B A C D E (1) A E D C B (1) A E B D C (1) A D C E B (1) Total count = 100 A B C D E A 0 8 -30 -14 -8 B -8 0 -18 -18 -12 C 30 18 0 14 30 D 14 18 -14 0 28 E 8 12 -30 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -30 -14 -8 B -8 0 -18 -18 -12 C 30 18 0 14 30 D 14 18 -14 0 28 E 8 12 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=30 D=18 E=8 A=6 so A is eliminated. Round 2 votes counts: C=38 B=33 D=19 E=10 so E is eliminated. Round 3 votes counts: B=40 C=38 D=22 so D is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:246 D:223 E:181 A:178 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -30 -14 -8 B -8 0 -18 -18 -12 C 30 18 0 14 30 D 14 18 -14 0 28 E 8 12 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -30 -14 -8 B -8 0 -18 -18 -12 C 30 18 0 14 30 D 14 18 -14 0 28 E 8 12 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -30 -14 -8 B -8 0 -18 -18 -12 C 30 18 0 14 30 D 14 18 -14 0 28 E 8 12 -30 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7012: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) A C B E D (7) E B C D A (6) A D C B E (6) D A B E C (5) E C B D A (4) D B E C A (4) D A B C E (4) E C D B A (3) E C B A D (3) E B D C A (3) D B E A C (3) D B A E C (3) D A E B C (3) C A E B D (3) A D B C E (3) E D C B A (2) D A E C B (2) C E B A D (2) C E A B D (2) B E C D A (2) B D E A C (2) B C A E D (2) A C E D B (2) A B C E D (2) E D B C A (1) E C D A B (1) D A C E B (1) D A C B E (1) C E A D B (1) C B E A D (1) C A B E D (1) B E D C A (1) B E C A D (1) B D E C A (1) B C E A D (1) B A D C E (1) A C E B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -4 -18 -2 B 6 0 12 -6 6 C 4 -12 0 -8 -14 D 18 6 8 0 -4 E 2 -6 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 -4 -18 -2 B 6 0 12 -6 6 C 4 -12 0 -8 -14 D 18 6 8 0 -4 E 2 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343750000004 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=23 A=23 B=11 C=10 so C is eliminated. Round 2 votes counts: D=33 E=28 A=27 B=12 so B is eliminated. Round 3 votes counts: D=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:214 B:209 E:207 A:185 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -18 -2 B 6 0 12 -6 6 C 4 -12 0 -8 -14 D 18 6 8 0 -4 E 2 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343750000004 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -18 -2 B 6 0 12 -6 6 C 4 -12 0 -8 -14 D 18 6 8 0 -4 E 2 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343750000004 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -18 -2 B 6 0 12 -6 6 C 4 -12 0 -8 -14 D 18 6 8 0 -4 E 2 -6 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.375000 E: 0.375000 Sum of squares = 0.343750000004 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7013: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) A E D B C (8) E D C B A (7) B C E D A (6) D E A C B (5) D E C B A (4) C B D E A (4) A E B C D (4) A D E C B (4) A B E C D (4) D C E B A (3) D C B E A (3) C D B E A (3) B A C E D (3) A B C E D (3) D C A B E (2) D A E C B (2) B C E A D (2) B C D E A (2) B C A E D (2) A E D C B (2) A D E B C (2) A D B C E (2) A B C D E (2) E D A B C (1) E C D B A (1) E C B D A (1) E A D B C (1) D E C A B (1) D C E A B (1) B E C A D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E B D C (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 -8 -8 B 6 0 -4 -6 4 C 4 4 0 -2 -2 D 8 6 2 0 -14 E 8 -4 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000003 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -6 -4 -8 -8 B 6 0 -4 -6 4 C 4 4 0 -2 -2 D 8 6 2 0 -14 E 8 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000332 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=21 B=19 C=15 E=11 so E is eliminated. Round 2 votes counts: A=35 D=29 B=19 C=17 so C is eliminated. Round 3 votes counts: A=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:210 C:202 D:201 B:200 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 -8 B 6 0 -4 -6 4 C 4 4 0 -2 -2 D 8 6 2 0 -14 E 8 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000332 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 -8 B 6 0 -4 -6 4 C 4 4 0 -2 -2 D 8 6 2 0 -14 E 8 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000332 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 -8 B 6 0 -4 -6 4 C 4 4 0 -2 -2 D 8 6 2 0 -14 E 8 -4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000332 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7014: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) E B D A C (8) D A E B C (7) B E C A D (7) D A C E B (6) A D C B E (6) B E D A C (5) E D A B C (4) C B E A D (4) E D B A C (3) E B C D A (3) C E B D A (3) C B A E D (3) A D B E C (3) D A E C B (2) C A D E B (2) C A B D E (2) B E A D C (2) B A D E C (2) E C D B A (1) E C B D A (1) D E A C B (1) C E D A B (1) C D A E B (1) C B A D E (1) B E C D A (1) B E A C D (1) B C E A D (1) B C A D E (1) A D E B C (1) A D C E B (1) A D B C E (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 16 6 8 B -6 0 4 -4 8 C -16 -4 0 -8 -6 D -6 4 8 0 4 E -8 -8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 6 8 B -6 0 4 -4 8 C -16 -4 0 -8 -6 D -6 4 8 0 4 E -8 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=20 B=20 A=17 D=16 so D is eliminated. Round 2 votes counts: A=32 C=27 E=21 B=20 so B is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:205 B:201 E:193 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 6 8 B -6 0 4 -4 8 C -16 -4 0 -8 -6 D -6 4 8 0 4 E -8 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 6 8 B -6 0 4 -4 8 C -16 -4 0 -8 -6 D -6 4 8 0 4 E -8 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 6 8 B -6 0 4 -4 8 C -16 -4 0 -8 -6 D -6 4 8 0 4 E -8 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7015: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) A C D E B (10) D B E A C (9) B D E C A (7) B D E A C (7) E C A B D (5) E B C A D (5) C A E B D (5) B E D C A (5) A C E D B (5) D B A C E (4) E C B A D (3) D B A E C (3) D A C E B (3) E A C B D (2) C D A B E (2) A D C E B (2) E B D C A (1) E A D C B (1) D A E B C (1) D A B C E (1) C E A B D (1) C A E D B (1) B E D A C (1) B E C D A (1) B D C A E (1) A E C D B (1) Total count = 100 A B C D E A 0 6 26 -18 4 B -6 0 -8 -12 8 C -26 8 0 -18 -4 D 18 12 18 0 26 E -4 -8 4 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 26 -18 4 B -6 0 -8 -12 8 C -26 8 0 -18 -4 D 18 12 18 0 26 E -4 -8 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=22 A=18 E=17 C=9 so C is eliminated. Round 2 votes counts: D=36 A=24 B=22 E=18 so E is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:237 A:209 B:191 E:183 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 26 -18 4 B -6 0 -8 -12 8 C -26 8 0 -18 -4 D 18 12 18 0 26 E -4 -8 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 26 -18 4 B -6 0 -8 -12 8 C -26 8 0 -18 -4 D 18 12 18 0 26 E -4 -8 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 26 -18 4 B -6 0 -8 -12 8 C -26 8 0 -18 -4 D 18 12 18 0 26 E -4 -8 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7016: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) E B D A C (6) C A D B E (6) A C E B D (6) E B A C D (5) E A B C D (5) D C B A E (5) C A E D B (5) B E D A C (5) A C E D B (5) E A C B D (4) B E A D C (4) D C A B E (3) D B C E A (3) C A D E B (3) B D E C A (3) B D E A C (3) A E C B D (3) E B A D C (2) D E B C A (2) C D A B E (2) C A B D E (2) E D B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) C D B A E (1) C D A E B (1) B E A C D (1) B D C A E (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 4 8 -8 B 2 0 0 6 0 C -4 0 0 4 -2 D -8 -6 -4 0 -8 E 8 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.435649 C: 0.000000 D: 0.000000 E: 0.564351 Sum of squares = 0.508282144404 Cumulative probabilities = A: 0.000000 B: 0.435649 C: 0.435649 D: 0.435649 E: 1.000000 A B C D E A 0 -2 4 8 -8 B 2 0 0 6 0 C -4 0 0 4 -2 D -8 -6 -4 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=23 D=23 C=20 B=17 A=17 so B is eliminated. Round 2 votes counts: E=33 D=30 C=20 A=17 so A is eliminated. Round 3 votes counts: E=36 C=34 D=30 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:209 B:204 A:201 C:199 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 8 -8 B 2 0 0 6 0 C -4 0 0 4 -2 D -8 -6 -4 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 8 -8 B 2 0 0 6 0 C -4 0 0 4 -2 D -8 -6 -4 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 8 -8 B 2 0 0 6 0 C -4 0 0 4 -2 D -8 -6 -4 0 -8 E 8 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7017: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) E D C A B (5) D B A E C (5) C A B D E (5) E D C B A (4) E C D B A (4) E B A C D (4) E C D A B (3) E B A D C (3) D E C A B (3) D E B A C (3) D C A B E (3) D A C B E (3) C A B E D (3) B A D E C (3) E B C A D (2) D E A B C (2) D A B E C (2) C E D A B (2) C E B A D (2) C E A D B (2) C E A B D (2) C D A B E (2) C B A E D (2) B A E C D (2) B A C E D (2) B A C D E (2) A D B C E (2) E D B C A (1) E C B D A (1) E C B A D (1) E B D C A (1) E B D A C (1) D C A E B (1) D B E A C (1) D A B C E (1) C B E A D (1) C A E D B (1) C A E B D (1) C A D B E (1) B E D A C (1) B D E A C (1) B C A E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -8 -12 -12 B 8 0 -6 -14 -10 C 8 6 0 -6 -18 D 12 14 6 0 -16 E 12 10 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -8 -12 -12 B 8 0 -6 -14 -10 C 8 6 0 -6 -18 D 12 14 6 0 -16 E 12 10 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=24 C=24 B=12 A=4 so A is eliminated. Round 2 votes counts: E=36 D=26 C=25 B=13 so B is eliminated. Round 3 votes counts: E=39 D=31 C=30 so C is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:228 D:208 C:195 B:189 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -12 -12 B 8 0 -6 -14 -10 C 8 6 0 -6 -18 D 12 14 6 0 -16 E 12 10 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -12 -12 B 8 0 -6 -14 -10 C 8 6 0 -6 -18 D 12 14 6 0 -16 E 12 10 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -12 -12 B 8 0 -6 -14 -10 C 8 6 0 -6 -18 D 12 14 6 0 -16 E 12 10 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7018: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) E D A B C (9) A C B D E (8) E D B C A (7) A E B C D (6) D E A C B (5) E A D B C (4) C B A D E (4) A B E C D (4) D E C B A (3) D C B E A (3) A E D B C (3) E B D C A (2) B E C D A (2) B C E D A (2) A E D C B (2) A D E C B (2) A D C E B (2) A C D B E (2) A C B E D (2) E D B A C (1) E B D A C (1) E B C A D (1) E B A C D (1) E A B D C (1) E A B C D (1) D E C A B (1) D E B C A (1) D B C E A (1) D A E C B (1) D A C E B (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 30 34 12 2 B -30 0 18 -2 -12 C -34 -18 0 0 -18 D -12 2 0 0 -22 E -2 12 18 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 30 34 12 2 B -30 0 18 -2 -12 C -34 -18 0 0 -18 D -12 2 0 0 -22 E -2 12 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=28 D=16 C=8 B=6 so B is eliminated. Round 2 votes counts: A=42 E=31 D=16 C=11 so C is eliminated. Round 3 votes counts: A=47 E=33 D=20 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:239 E:225 B:187 D:184 C:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 30 34 12 2 B -30 0 18 -2 -12 C -34 -18 0 0 -18 D -12 2 0 0 -22 E -2 12 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 34 12 2 B -30 0 18 -2 -12 C -34 -18 0 0 -18 D -12 2 0 0 -22 E -2 12 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 34 12 2 B -30 0 18 -2 -12 C -34 -18 0 0 -18 D -12 2 0 0 -22 E -2 12 18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7019: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (6) C A B E D (6) D E A C B (5) D E B A C (4) D A E B C (4) C B E A D (4) C A E D B (4) B E D C A (4) E D C B A (3) E D B C A (3) B C E A D (3) B C A E D (3) B A C D E (3) A D B E C (3) A C B E D (3) A C B D E (3) A B C D E (3) E C B D A (2) D E B C A (2) D A B E C (2) C E D A B (2) C E B A D (2) C B A E D (2) B D A E C (2) B A D E C (2) A D C B E (2) A D B C E (2) E C D B A (1) E C D A B (1) E B D C A (1) E B C D A (1) D E C A B (1) D B A E C (1) C E A D B (1) C E A B D (1) B E C D A (1) B D E A C (1) B C E D A (1) B A D C E (1) B A C E D (1) A D E C B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 2 4 0 B -4 0 8 0 8 C -2 -8 0 0 -2 D -4 0 0 0 0 E 0 -8 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.772803 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.227197 Sum of squares = 0.648843117485 Cumulative probabilities = A: 0.772803 B: 0.772803 C: 0.772803 D: 0.772803 E: 1.000000 A B C D E A 0 4 2 4 0 B -4 0 8 0 8 C -2 -8 0 0 -2 D -4 0 0 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555661206 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=22 B=22 A=19 E=12 so E is eliminated. Round 2 votes counts: D=31 C=26 B=24 A=19 so A is eliminated. Round 3 votes counts: D=39 C=34 B=27 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:206 A:205 D:198 E:197 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 4 0 B -4 0 8 0 8 C -2 -8 0 0 -2 D -4 0 0 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555661206 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 4 0 B -4 0 8 0 8 C -2 -8 0 0 -2 D -4 0 0 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555661206 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 4 0 B -4 0 8 0 8 C -2 -8 0 0 -2 D -4 0 0 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555661206 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7020: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) D E A C B (10) E D B C A (7) D E B A C (6) A C D B E (5) A C B D E (5) E D C B A (4) B E D C A (4) B E C D A (4) A C D E B (4) A B C D E (4) E D C A B (3) D E B C A (3) D E A B C (3) D A E C B (3) C A B E D (3) B C E A D (3) A C B E D (3) D E C A B (2) D B E A C (2) B A C D E (2) A D C B E (2) E C B D A (1) E B D C A (1) D A C B E (1) C E B A D (1) C E A B D (1) B A C E D (1) A D C E B (1) Total count = 100 A B C D E A 0 0 4 -8 -10 B 0 0 2 -12 0 C -4 -2 0 -4 -6 D 8 12 4 0 6 E 10 0 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -8 -10 B 0 0 2 -12 0 C -4 -2 0 -4 -6 D 8 12 4 0 6 E 10 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=25 A=24 E=16 C=5 so C is eliminated. Round 2 votes counts: D=30 A=27 B=25 E=18 so E is eliminated. Round 3 votes counts: D=44 B=28 A=28 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:205 B:195 A:193 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 4 -8 -10 B 0 0 2 -12 0 C -4 -2 0 -4 -6 D 8 12 4 0 6 E 10 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -8 -10 B 0 0 2 -12 0 C -4 -2 0 -4 -6 D 8 12 4 0 6 E 10 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -8 -10 B 0 0 2 -12 0 C -4 -2 0 -4 -6 D 8 12 4 0 6 E 10 0 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7021: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) E D A C B (7) C B A D E (5) B D E A C (5) B C D A E (5) A E C D B (5) D B E C A (4) C A E D B (4) A C B E D (4) E D C A B (3) E A D B C (3) B A C D E (3) E C A D B (2) E A C D B (2) E A B D C (2) D E B C A (2) C E A D B (2) C B D A E (2) C A B E D (2) C A B D E (2) B D C E A (2) B C A D E (2) A E C B D (2) A C E B D (2) A B C D E (2) E D B A C (1) E D A B C (1) E A D C B (1) D E B A C (1) D C B E A (1) D B C E A (1) C D B E A (1) C D A B E (1) C A D B E (1) B D E C A (1) B D A E C (1) B D A C E (1) A E B D C (1) A E B C D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -4 2 16 B -6 0 2 14 16 C 4 -2 0 4 6 D -2 -14 -4 0 4 E -16 -16 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 2 16 B -6 0 2 14 16 C 4 -2 0 4 6 D -2 -14 -4 0 4 E -16 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=22 C=20 A=20 D=9 so D is eliminated. Round 2 votes counts: B=34 E=25 C=21 A=20 so A is eliminated. Round 3 votes counts: B=39 E=34 C=27 so C is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:210 C:206 D:192 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 2 16 B -6 0 2 14 16 C 4 -2 0 4 6 D -2 -14 -4 0 4 E -16 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 2 16 B -6 0 2 14 16 C 4 -2 0 4 6 D -2 -14 -4 0 4 E -16 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 2 16 B -6 0 2 14 16 C 4 -2 0 4 6 D -2 -14 -4 0 4 E -16 -16 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7022: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) A D E C B (11) B C D E A (8) A E D C B (5) D A E C B (4) B E C A D (4) A E D B C (4) A D B E C (4) D A C B E (3) C E B D A (3) B C E A D (3) B C A D E (3) A D E B C (3) E D A C B (2) E C B D A (2) E B C D A (2) E A D C B (2) D C B A E (2) C B E D A (2) A D C E B (2) A D B C E (2) E D C A B (1) E C D B A (1) E C B A D (1) E A B D C (1) D C E A B (1) D C A E B (1) D B C A E (1) D B A C E (1) D A C E B (1) C B D E A (1) B E C D A (1) B D C A E (1) B C D A E (1) B A D C E (1) B A C E D (1) B A C D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 0 8 B 2 0 10 -2 6 C 0 -10 0 -10 2 D 0 2 10 0 6 E -8 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.289262 B: 0.000000 C: 0.000000 D: 0.710738 E: 0.000000 Sum of squares = 0.588820891062 Cumulative probabilities = A: 0.289262 B: 0.289262 C: 0.289262 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 0 8 B 2 0 10 -2 6 C 0 -10 0 -10 2 D 0 2 10 0 6 E -8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.500258 E: 0.000000 Sum of squares = 0.500000133141 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=33 D=14 E=12 C=6 so C is eliminated. Round 2 votes counts: B=38 A=33 E=15 D=14 so D is eliminated. Round 3 votes counts: B=42 A=42 E=16 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:209 B:208 A:203 C:191 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 0 8 B 2 0 10 -2 6 C 0 -10 0 -10 2 D 0 2 10 0 6 E -8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.500258 E: 0.000000 Sum of squares = 0.500000133141 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 8 B 2 0 10 -2 6 C 0 -10 0 -10 2 D 0 2 10 0 6 E -8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.500258 E: 0.000000 Sum of squares = 0.500000133141 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 8 B 2 0 10 -2 6 C 0 -10 0 -10 2 D 0 2 10 0 6 E -8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499742 B: 0.000000 C: 0.000000 D: 0.500258 E: 0.000000 Sum of squares = 0.500000133141 Cumulative probabilities = A: 0.499742 B: 0.499742 C: 0.499742 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7023: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) C B A E D (8) A D E B C (6) D E A B C (5) C B E A D (5) A D B E C (5) D E A C B (4) C E B D A (4) B E D A C (4) D A E C B (3) D A E B C (3) C B E D A (3) B E C D A (3) B C A E D (3) A D E C B (3) A D C E B (3) A B C D E (3) E B D C A (2) E B C D A (2) C A B E D (2) B E D C A (2) B C E A D (2) B A E C D (2) A D B C E (2) A C D B E (2) E D B C A (1) E D B A C (1) E C B D A (1) D A C E B (1) C E D B A (1) C E A D B (1) C A B D E (1) B C E D A (1) B A E D C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 0 2 0 B 8 0 -2 0 2 C 0 2 0 -12 -14 D -2 0 12 0 -14 E 0 -2 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.777778 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629695 Cumulative probabilities = A: 0.000000 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 -8 0 2 0 B 8 0 -2 0 2 C 0 2 0 -12 -14 D -2 0 12 0 -14 E 0 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.777778 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629630428 Cumulative probabilities = A: 0.000000 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 B=18 D=16 E=15 so E is eliminated. Round 2 votes counts: D=26 C=26 A=26 B=22 so B is eliminated. Round 3 votes counts: C=37 D=34 A=29 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 B:204 D:198 A:197 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 2 0 B 8 0 -2 0 2 C 0 2 0 -12 -14 D -2 0 12 0 -14 E 0 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.777778 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629630428 Cumulative probabilities = A: 0.000000 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 2 0 B 8 0 -2 0 2 C 0 2 0 -12 -14 D -2 0 12 0 -14 E 0 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.777778 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629630428 Cumulative probabilities = A: 0.000000 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 2 0 B 8 0 -2 0 2 C 0 2 0 -12 -14 D -2 0 12 0 -14 E 0 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.777778 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629630428 Cumulative probabilities = A: 0.000000 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7024: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) E C D A B (7) A C E D B (5) B E D C A (4) B E A D C (4) B D A C E (4) B A D C E (4) B D E C A (3) B A E D C (3) A E C B D (3) A D C B E (3) A B E C D (3) A B C D E (3) E C D B A (2) E C A B D (2) E B D C A (2) E B C D A (2) D C E B A (2) D C A E B (2) D A C B E (2) C E D A B (2) C E A D B (2) C D E A B (2) B E D A C (2) A C D E B (2) E D C B A (1) E C B D A (1) E C A D B (1) E A B C D (1) D C E A B (1) D C B E A (1) D C A B E (1) D B E C A (1) D B C A E (1) D B A C E (1) C D E B A (1) C A D E B (1) B E C A D (1) B E A C D (1) B D A E C (1) B A E C D (1) A E C D B (1) A E B C D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 14 8 8 B -14 0 10 14 12 C -14 -10 0 -8 4 D -8 -14 8 0 -6 E -8 -12 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 8 8 B -14 0 10 14 12 C -14 -10 0 -8 4 D -8 -14 8 0 -6 E -8 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=28 E=19 D=12 C=8 so C is eliminated. Round 2 votes counts: A=34 B=28 E=23 D=15 so D is eliminated. Round 3 votes counts: A=39 B=32 E=29 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:211 E:191 D:190 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 8 8 B -14 0 10 14 12 C -14 -10 0 -8 4 D -8 -14 8 0 -6 E -8 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 8 8 B -14 0 10 14 12 C -14 -10 0 -8 4 D -8 -14 8 0 -6 E -8 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 8 8 B -14 0 10 14 12 C -14 -10 0 -8 4 D -8 -14 8 0 -6 E -8 -12 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7025: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) C A B D E (6) C A D E B (5) A D C B E (5) A D B E C (5) C E B D A (4) C D A E B (4) A B D E C (4) D E B A C (3) D C A E B (3) C D E A B (3) C A D B E (3) B E C D A (3) B A E D C (3) E D B C A (2) E C B D A (2) E B D A C (2) D E C A B (2) D C E A B (2) D A E B C (2) D A C E B (2) C E D B A (2) C B A E D (2) B E D A C (2) B E C A D (2) B E A D C (2) B E A C D (2) B A D E C (2) A D C E B (2) A C D E B (2) E B D C A (1) D E A C B (1) D B A E C (1) C E D A B (1) C B E A D (1) C A B E D (1) B A C E D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -14 0 10 B -10 0 -10 -2 -6 C 14 10 0 8 2 D 0 2 -8 0 18 E -10 6 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -14 0 10 B -10 0 -10 -2 -6 C 14 10 0 8 2 D 0 2 -8 0 18 E -10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=20 B=17 D=16 E=15 so E is eliminated. Round 2 votes counts: C=34 B=28 A=20 D=18 so D is eliminated. Round 3 votes counts: C=41 B=34 A=25 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:206 A:203 E:188 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -14 0 10 B -10 0 -10 -2 -6 C 14 10 0 8 2 D 0 2 -8 0 18 E -10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 0 10 B -10 0 -10 -2 -6 C 14 10 0 8 2 D 0 2 -8 0 18 E -10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 0 10 B -10 0 -10 -2 -6 C 14 10 0 8 2 D 0 2 -8 0 18 E -10 6 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7026: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) C D A E B (8) B E A D C (8) E B C A D (5) D A C B E (5) E B A C D (4) C E B A D (4) C E A D B (4) A B D E C (4) E C B A D (3) D C A B E (3) C D A B E (3) B E C D A (3) B E A C D (3) B A D E C (3) E C A B D (2) E A B D C (2) D B A E C (2) D A B C E (2) C E B D A (2) C D E A B (2) B A E D C (2) E B A D C (1) D C A E B (1) D B A C E (1) D A B E C (1) C E D B A (1) C E D A B (1) C E A B D (1) B E C A D (1) B D C E A (1) B D A E C (1) B D A C E (1) B C E D A (1) A D E C B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 6 8 22 2 B -6 0 14 4 12 C -8 -14 0 -2 -14 D -22 -4 2 0 2 E -2 -12 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998153 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 22 2 B -6 0 14 4 12 C -8 -14 0 -2 -14 D -22 -4 2 0 2 E -2 -12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 A=18 E=17 D=15 so D is eliminated. Round 2 votes counts: C=30 B=27 A=26 E=17 so E is eliminated. Round 3 votes counts: B=37 C=35 A=28 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:219 B:212 E:199 D:189 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 22 2 B -6 0 14 4 12 C -8 -14 0 -2 -14 D -22 -4 2 0 2 E -2 -12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 22 2 B -6 0 14 4 12 C -8 -14 0 -2 -14 D -22 -4 2 0 2 E -2 -12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 22 2 B -6 0 14 4 12 C -8 -14 0 -2 -14 D -22 -4 2 0 2 E -2 -12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7027: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E C B A D (9) C E B D A (7) E A B D C (5) C E D B A (5) A D B E C (5) C D B A E (4) A D E B C (4) E C D A B (3) B C A D E (3) B A D C E (3) A B E D C (3) E C B D A (2) E B C A D (2) E A D B C (2) C D E A B (2) C D B E A (2) C B D E A (2) A E B D C (2) A D B C E (2) A B D C E (2) E C A D B (1) E C A B D (1) E B A C D (1) E A C B D (1) E A B C D (1) D E A C B (1) D C B A E (1) D B C A E (1) D A C E B (1) D A C B E (1) D A B E C (1) C E B A D (1) C D E B A (1) C B E A D (1) B D C A E (1) B D A C E (1) B C A E D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 0 2 0 B -4 0 8 0 -4 C 0 -8 0 0 10 D -2 0 0 0 4 E 0 4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.751890 B: 0.000000 C: 0.248110 D: 0.000000 E: 0.000000 Sum of squares = 0.626896995406 Cumulative probabilities = A: 0.751890 B: 0.751890 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 2 0 B -4 0 8 0 -4 C 0 -8 0 0 10 D -2 0 0 0 4 E 0 4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555680845 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=25 D=19 A=19 B=9 so B is eliminated. Round 2 votes counts: C=29 E=28 A=22 D=21 so D is eliminated. Round 3 votes counts: A=39 C=32 E=29 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:203 C:201 D:201 B:200 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 0 B -4 0 8 0 -4 C 0 -8 0 0 10 D -2 0 0 0 4 E 0 4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555680845 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 0 B -4 0 8 0 -4 C 0 -8 0 0 10 D -2 0 0 0 4 E 0 4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555680845 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 0 B -4 0 8 0 -4 C 0 -8 0 0 10 D -2 0 0 0 4 E 0 4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555680845 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7028: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) B C A E D (7) B A C E D (7) E A D B C (6) C B D A E (6) D E C A B (4) A E B D C (4) A B E D C (4) A B E C D (4) E D A B C (3) D E A C B (3) E A B D C (2) D C E B A (2) D C E A B (2) D C B E A (2) D C B A E (2) D A B E C (2) C E B A D (2) C D B E A (2) C B D E A (2) C B A E D (2) C B A D E (2) B C A D E (2) B A E C D (2) A E B C D (2) E C A D B (1) E C A B D (1) E A D C B (1) D B A E C (1) D B A C E (1) D A E B C (1) C E D B A (1) C E A B D (1) C D B A E (1) B D A C E (1) B A D E C (1) B A D C E (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 16 8 10 B -6 0 26 8 6 C -16 -26 0 -10 -8 D -8 -8 10 0 -2 E -10 -6 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 8 10 B -6 0 26 8 6 C -16 -26 0 -10 -8 D -8 -8 10 0 -2 E -10 -6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=21 C=19 A=16 E=14 so E is eliminated. Round 2 votes counts: D=33 A=25 C=21 B=21 so C is eliminated. Round 3 votes counts: D=37 B=35 A=28 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:220 B:217 E:197 D:196 C:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 8 10 B -6 0 26 8 6 C -16 -26 0 -10 -8 D -8 -8 10 0 -2 E -10 -6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 8 10 B -6 0 26 8 6 C -16 -26 0 -10 -8 D -8 -8 10 0 -2 E -10 -6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 8 10 B -6 0 26 8 6 C -16 -26 0 -10 -8 D -8 -8 10 0 -2 E -10 -6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7029: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) A B D C E (11) E A C B D (5) B A D E C (5) A E B D C (5) D B C A E (4) C D E B A (4) B D A C E (4) A B C D E (4) E A B D C (3) C E D B A (3) B A D C E (3) A B D E C (3) E D B A C (2) E C D A B (2) D C E B A (2) D C B E A (2) D B C E A (2) C D B E A (2) C D B A E (2) B D A E C (2) A E C B D (2) A C B E D (2) E C A D B (1) E A D B C (1) E A C D B (1) E A B C D (1) D E C B A (1) D C B A E (1) D B E C A (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) C B D A E (1) A E B C D (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 14 2 6 B 4 0 6 8 0 C -14 -6 0 -6 4 D -2 -8 6 0 12 E -6 0 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.766154 C: 0.000000 D: 0.000000 E: 0.233846 Sum of squares = 0.641676292742 Cumulative probabilities = A: 0.000000 B: 0.766154 C: 0.766154 D: 0.766154 E: 1.000000 A B C D E A 0 -4 14 2 6 B 4 0 6 8 0 C -14 -6 0 -6 4 D -2 -8 6 0 12 E -6 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000032888 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=27 C=16 B=14 D=13 so D is eliminated. Round 2 votes counts: A=30 E=28 C=21 B=21 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:209 D:204 C:189 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 2 6 B 4 0 6 8 0 C -14 -6 0 -6 4 D -2 -8 6 0 12 E -6 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000032888 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 2 6 B 4 0 6 8 0 C -14 -6 0 -6 4 D -2 -8 6 0 12 E -6 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000032888 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 2 6 B 4 0 6 8 0 C -14 -6 0 -6 4 D -2 -8 6 0 12 E -6 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000032888 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7030: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) B C A E D (8) D E A C B (7) C B D A E (7) B C D E A (6) E A D B C (5) C B D E A (5) C A B E D (5) C A D E B (4) A E D B C (4) D E A B C (3) B A E C D (3) E A B D C (2) D C E B A (2) D C E A B (2) C D B E A (2) C B A E D (2) C B A D E (2) C A D B E (2) A D E C B (2) E D B A C (1) E B A D C (1) D E B A C (1) C D A E B (1) C D A B E (1) C A B D E (1) B E A C D (1) B D E C A (1) B D C E A (1) B C E D A (1) B A C E D (1) A E C D B (1) A C E D B (1) A C E B D (1) A C D E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -6 18 18 B -10 0 -18 0 2 C 6 18 0 14 14 D -18 0 -14 0 2 E -18 -2 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 18 18 B -10 0 -18 0 2 C 6 18 0 14 14 D -18 0 -14 0 2 E -18 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=22 A=22 D=15 E=9 so E is eliminated. Round 2 votes counts: C=32 A=29 B=23 D=16 so D is eliminated. Round 3 votes counts: A=39 C=36 B=25 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:220 B:187 D:185 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 18 18 B -10 0 -18 0 2 C 6 18 0 14 14 D -18 0 -14 0 2 E -18 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 18 18 B -10 0 -18 0 2 C 6 18 0 14 14 D -18 0 -14 0 2 E -18 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 18 18 B -10 0 -18 0 2 C 6 18 0 14 14 D -18 0 -14 0 2 E -18 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7031: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) D A B C E (7) C D E B A (7) E C B A D (6) A B E D C (6) D C B A E (5) D B A C E (5) A B D E C (5) A B D C E (5) D B C A E (4) B A D E C (4) E C D B A (3) C E D A B (3) E B A D C (2) D C B E A (2) C E D B A (2) C E A D B (2) C D E A B (2) C D B E A (2) C D A E B (2) B A E D C (2) A E B D C (2) A C E B D (2) E C A B D (1) E B A C D (1) D C E B A (1) D C A B E (1) D B E A C (1) D B C E A (1) C D A B E (1) B E A D C (1) B D A E C (1) B A D C E (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 10 0 12 B 2 0 14 -2 14 C -10 -14 0 -12 14 D 0 2 12 0 16 E -12 -14 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.391563 B: 0.000000 C: 0.000000 D: 0.608437 E: 0.000000 Sum of squares = 0.52351702734 Cumulative probabilities = A: 0.391563 B: 0.391563 C: 0.391563 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 0 12 B 2 0 14 -2 14 C -10 -14 0 -12 14 D 0 2 12 0 16 E -12 -14 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499558 B: 0.000000 C: 0.000000 D: 0.500442 E: 0.000000 Sum of squares = 0.500000389862 Cumulative probabilities = A: 0.499558 B: 0.499558 C: 0.499558 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 C=21 E=20 B=9 so B is eliminated. Round 2 votes counts: A=30 D=28 E=21 C=21 so E is eliminated. Round 3 votes counts: A=41 C=31 D=28 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 B:214 A:210 C:189 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 0 12 B 2 0 14 -2 14 C -10 -14 0 -12 14 D 0 2 12 0 16 E -12 -14 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499558 B: 0.000000 C: 0.000000 D: 0.500442 E: 0.000000 Sum of squares = 0.500000389862 Cumulative probabilities = A: 0.499558 B: 0.499558 C: 0.499558 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 0 12 B 2 0 14 -2 14 C -10 -14 0 -12 14 D 0 2 12 0 16 E -12 -14 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499558 B: 0.000000 C: 0.000000 D: 0.500442 E: 0.000000 Sum of squares = 0.500000389862 Cumulative probabilities = A: 0.499558 B: 0.499558 C: 0.499558 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 0 12 B 2 0 14 -2 14 C -10 -14 0 -12 14 D 0 2 12 0 16 E -12 -14 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499558 B: 0.000000 C: 0.000000 D: 0.500442 E: 0.000000 Sum of squares = 0.500000389862 Cumulative probabilities = A: 0.499558 B: 0.499558 C: 0.499558 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7032: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) C E B D A (6) B E A C D (6) A B E C D (6) D B C E A (5) C E D B A (5) A B D E C (5) B A E C D (4) A B E D C (4) E C B A D (3) D B A E C (3) D A B E C (3) B A E D C (3) A E B C D (3) A C E D B (3) D C B E A (2) D C A E B (2) C E B A D (2) C A E B D (2) B E C D A (2) B E C A D (2) B A D E C (2) A E C B D (2) A D B E C (2) A D B C E (2) E B C D A (1) E B C A D (1) E A C B D (1) D C E A B (1) D B A C E (1) D A C E B (1) D A C B E (1) C D E B A (1) B E D C A (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -20 8 10 2 B 20 0 12 12 10 C -8 -12 0 0 -8 D -10 -12 0 0 -14 E -2 -10 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999865 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 8 10 2 B 20 0 12 12 10 C -8 -12 0 0 -8 D -10 -12 0 0 -14 E -2 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 B=20 C=16 E=6 so E is eliminated. Round 2 votes counts: A=30 D=29 B=22 C=19 so C is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:205 A:200 C:186 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 8 10 2 B 20 0 12 12 10 C -8 -12 0 0 -8 D -10 -12 0 0 -14 E -2 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 8 10 2 B 20 0 12 12 10 C -8 -12 0 0 -8 D -10 -12 0 0 -14 E -2 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 8 10 2 B 20 0 12 12 10 C -8 -12 0 0 -8 D -10 -12 0 0 -14 E -2 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7033: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) E C B D A (7) A D B C E (7) D B E A C (6) E D A B C (5) E D B A C (4) C B A D E (4) B D E C A (4) A D B E C (4) A C B D E (4) E A D B C (3) D B A E C (3) C B D E A (3) C B D A E (3) C A E B D (3) E D B C A (2) E B D C A (2) E A D C B (2) E A C D B (2) D B A C E (2) C E B A D (2) B C D E A (2) E C A D B (1) D E B A C (1) D B E C A (1) D A E B C (1) D A B E C (1) C E A B D (1) C B E D A (1) C A B E D (1) B D C E A (1) B C E D A (1) B C D A E (1) A E D C B (1) A E C D B (1) A C E D B (1) A C E B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 -2 -2 B 0 0 4 2 20 C -2 -4 0 -2 -2 D 2 -2 2 0 18 E 2 -20 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.356404 B: 0.643596 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.541239347954 Cumulative probabilities = A: 0.356404 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -2 -2 B 0 0 4 2 20 C -2 -4 0 -2 -2 D 2 -2 2 0 18 E 2 -20 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499986 B: 0.500014 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000404 Cumulative probabilities = A: 0.499986 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 A=21 D=15 B=9 so B is eliminated. Round 2 votes counts: C=31 E=28 A=21 D=20 so D is eliminated. Round 3 votes counts: E=40 C=32 A=28 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:213 D:210 A:199 C:195 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -2 -2 B 0 0 4 2 20 C -2 -4 0 -2 -2 D 2 -2 2 0 18 E 2 -20 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499986 B: 0.500014 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000404 Cumulative probabilities = A: 0.499986 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -2 B 0 0 4 2 20 C -2 -4 0 -2 -2 D 2 -2 2 0 18 E 2 -20 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499986 B: 0.500014 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000404 Cumulative probabilities = A: 0.499986 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -2 B 0 0 4 2 20 C -2 -4 0 -2 -2 D 2 -2 2 0 18 E 2 -20 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499986 B: 0.500014 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000404 Cumulative probabilities = A: 0.499986 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7034: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (13) B D A E C (12) B D A C E (8) D B A E C (6) C E A D B (6) E C A D B (4) D E A B C (4) B D C A E (4) E A D C B (3) C B D E A (3) C B A E D (3) C A E B D (3) A E D C B (3) A E D B C (3) E A C D B (2) D E C B A (2) C E B D A (2) B C A D E (2) A D E B C (2) E D A C B (1) D B E A C (1) D B C E A (1) D A E B C (1) C E D A B (1) C B E A D (1) C B D A E (1) C A B E D (1) B D C E A (1) B C D E A (1) B A D E C (1) B A C E D (1) B A C D E (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 2 2 8 B 2 0 2 18 -2 C -2 -2 0 -8 6 D -2 -18 8 0 4 E -8 2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 -2 2 2 8 B 2 0 2 18 -2 C -2 -2 0 -8 6 D -2 -18 8 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=31 D=15 E=10 A=10 so E is eliminated. Round 2 votes counts: C=38 B=31 D=16 A=15 so A is eliminated. Round 3 votes counts: C=40 B=32 D=28 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:205 C:197 D:196 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 2 8 B 2 0 2 18 -2 C -2 -2 0 -8 6 D -2 -18 8 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 8 B 2 0 2 18 -2 C -2 -2 0 -8 6 D -2 -18 8 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 8 B 2 0 2 18 -2 C -2 -2 0 -8 6 D -2 -18 8 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.49999999988 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7035: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) A E D B C (10) D B C A E (9) D B A E C (9) D B A C E (7) C E B A D (7) E A C D B (5) D A B E C (5) C B D E A (5) E C A B D (4) E A C B D (4) C B E A D (3) B C D E A (3) A E D C B (3) A D E B C (3) B D C E A (2) A E C D B (2) E C B A D (1) E A D B C (1) E A B C D (1) D A E B C (1) D A C B E (1) C E A B D (1) B D C A E (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 8 -6 6 B 14 0 4 -16 10 C -8 -4 0 -6 2 D 6 16 6 0 -6 E -6 -10 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.000000 D: 0.312500 E: 0.500000 Sum of squares = 0.382812499909 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.187500 D: 0.500000 E: 1.000000 A B C D E A 0 -14 8 -6 6 B 14 0 4 -16 10 C -8 -4 0 -6 2 D 6 16 6 0 -6 E -6 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.000000 D: 0.312500 E: 0.500000 Sum of squares = 0.382812499985 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.187500 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=26 A=20 E=16 B=6 so B is eliminated. Round 2 votes counts: D=35 C=29 A=20 E=16 so E is eliminated. Round 3 votes counts: D=35 C=34 A=31 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:206 A:197 E:194 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 8 -6 6 B 14 0 4 -16 10 C -8 -4 0 -6 2 D 6 16 6 0 -6 E -6 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.000000 D: 0.312500 E: 0.500000 Sum of squares = 0.382812499985 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.187500 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -6 6 B 14 0 4 -16 10 C -8 -4 0 -6 2 D 6 16 6 0 -6 E -6 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.000000 D: 0.312500 E: 0.500000 Sum of squares = 0.382812499985 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.187500 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -6 6 B 14 0 4 -16 10 C -8 -4 0 -6 2 D 6 16 6 0 -6 E -6 -10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.187500 C: 0.000000 D: 0.312500 E: 0.500000 Sum of squares = 0.382812499985 Cumulative probabilities = A: 0.000000 B: 0.187500 C: 0.187500 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7036: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) A B E D C (7) B A D C E (6) A E B D C (5) E C D A B (4) E A D C B (4) E A B C D (4) C D E B A (4) D E A C B (3) D C B A E (3) C D B E A (3) C B D E A (3) B C D A E (3) B A C D E (3) A B D E C (3) E D C A B (2) E C B D A (2) E C A B D (2) D C B E A (2) D A B C E (2) B D A C E (2) A E D B C (2) A D B E C (2) A B E C D (2) E D A C B (1) E C D B A (1) E C B A D (1) E A D B C (1) E A C D B (1) E A C B D (1) D E C A B (1) D C E A B (1) D B C A E (1) D B A C E (1) D A E B C (1) C E B D A (1) C B E D A (1) C B D A E (1) B E A C D (1) B D C A E (1) B C A D E (1) A E B C D (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 10 -2 -2 B -4 0 2 4 -2 C -10 -2 0 -6 -6 D 2 -4 6 0 -2 E 2 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 -2 -2 B -4 0 2 4 -2 C -10 -2 0 -6 -6 D 2 -4 6 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 C=20 B=17 D=15 so D is eliminated. Round 2 votes counts: E=28 A=27 C=26 B=19 so B is eliminated. Round 3 votes counts: A=39 C=32 E=29 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:205 D:201 B:200 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 -2 -2 B -4 0 2 4 -2 C -10 -2 0 -6 -6 D 2 -4 6 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -2 -2 B -4 0 2 4 -2 C -10 -2 0 -6 -6 D 2 -4 6 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -2 -2 B -4 0 2 4 -2 C -10 -2 0 -6 -6 D 2 -4 6 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7037: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (7) A B E D C (6) D E C B A (5) C A B D E (4) A E C D B (4) E D B C A (3) E D B A C (3) E D A B C (3) D E B C A (3) D C E B A (3) C D E B A (3) B D E C A (3) B C D A E (3) A E D B C (3) A E B D C (3) A C E D B (3) A C B E D (3) A C B D E (3) E D C A B (2) C B A D E (2) C A E D B (2) B D E A C (2) B C D E A (2) B A D E C (2) A B E C D (2) E D C B A (1) E C A D B (1) E B D A C (1) E A D C B (1) E A D B C (1) D B E C A (1) C E A D B (1) C D E A B (1) C D B E A (1) C B D E A (1) C B D A E (1) C A D E B (1) B E D A C (1) B D C A E (1) B C A D E (1) B A E D C (1) B A D C E (1) B A C D E (1) A E D C B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 8 12 14 B -8 0 10 6 0 C -8 -10 0 -2 -6 D -12 -6 2 0 4 E -14 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999796 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 12 14 B -8 0 10 6 0 C -8 -10 0 -2 -6 D -12 -6 2 0 4 E -14 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=18 C=17 E=16 D=12 so D is eliminated. Round 2 votes counts: A=37 E=24 C=20 B=19 so B is eliminated. Round 3 votes counts: A=42 E=31 C=27 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:204 D:194 E:194 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 12 14 B -8 0 10 6 0 C -8 -10 0 -2 -6 D -12 -6 2 0 4 E -14 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 12 14 B -8 0 10 6 0 C -8 -10 0 -2 -6 D -12 -6 2 0 4 E -14 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 12 14 B -8 0 10 6 0 C -8 -10 0 -2 -6 D -12 -6 2 0 4 E -14 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7038: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) B A E D C (14) E A D C B (7) D C E A B (6) B A E C D (6) C D B E A (4) A B E D C (4) C B E A D (3) B C D A E (3) A E B D C (3) E A D B C (2) D C B A E (2) C E D A B (2) C D E B A (2) C D B A E (2) C B D E A (2) C B D A E (2) B C E A D (2) B C A E D (2) B A C E D (2) A E D B C (2) E D C A B (1) E D A C B (1) E C A D B (1) E C A B D (1) E B A C D (1) E A C D B (1) E A B D C (1) D E C A B (1) D E A C B (1) D A E B C (1) B E A C D (1) B A C D E (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 12 -8 B -2 0 -8 -2 2 C 0 8 0 4 -2 D -12 2 -4 0 -16 E 8 -2 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000074 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 2 0 12 -8 B -2 0 -8 -2 2 C 0 8 0 4 -2 D -12 2 -4 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=31 B=31 E=16 D=11 A=11 so D is eliminated. Round 2 votes counts: C=39 B=31 E=18 A=12 so A is eliminated. Round 3 votes counts: C=39 B=36 E=25 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:205 A:203 B:195 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 12 -8 B -2 0 -8 -2 2 C 0 8 0 4 -2 D -12 2 -4 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 12 -8 B -2 0 -8 -2 2 C 0 8 0 4 -2 D -12 2 -4 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 12 -8 B -2 0 -8 -2 2 C 0 8 0 4 -2 D -12 2 -4 0 -16 E 8 -2 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.666667 Sum of squares = 0.49999999995 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7039: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) C A B E D (7) E D B C A (6) D E B A C (6) D E A B C (6) B C A E D (6) E D C A B (5) A B C D E (5) E D C B A (4) A D E B C (4) C E D A B (3) B C E A D (3) B A C D E (3) D A E C B (2) C E D B A (2) C E B D A (2) B E D C A (2) B E C D A (2) A D E C B (2) A D B E C (2) A D B C E (2) A B D C E (2) D E A C B (1) D A B E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E D A (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B D E A C (1) B D A E C (1) B A D E C (1) B A D C E (1) A D C B E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -14 10 6 B -2 0 10 -2 0 C 14 -10 0 -2 6 D -10 2 2 0 -10 E -6 0 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.384615 B: 0.538462 C: 0.076923 D: 0.000000 E: 0.000000 Sum of squares = 0.443786982259 Cumulative probabilities = A: 0.384615 B: 0.923077 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 10 6 B -2 0 10 -2 0 C 14 -10 0 -2 6 D -10 2 2 0 -10 E -6 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.538462 C: 0.076923 D: 0.000000 E: 0.000000 Sum of squares = 0.443786982471 Cumulative probabilities = A: 0.384615 B: 0.923077 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=20 A=20 D=16 E=15 so E is eliminated. Round 2 votes counts: D=31 C=29 B=20 A=20 so B is eliminated. Round 3 votes counts: C=40 D=35 A=25 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:204 B:203 A:202 E:199 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -14 10 6 B -2 0 10 -2 0 C 14 -10 0 -2 6 D -10 2 2 0 -10 E -6 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.538462 C: 0.076923 D: 0.000000 E: 0.000000 Sum of squares = 0.443786982471 Cumulative probabilities = A: 0.384615 B: 0.923077 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 10 6 B -2 0 10 -2 0 C 14 -10 0 -2 6 D -10 2 2 0 -10 E -6 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.538462 C: 0.076923 D: 0.000000 E: 0.000000 Sum of squares = 0.443786982471 Cumulative probabilities = A: 0.384615 B: 0.923077 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 10 6 B -2 0 10 -2 0 C 14 -10 0 -2 6 D -10 2 2 0 -10 E -6 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.538462 C: 0.076923 D: 0.000000 E: 0.000000 Sum of squares = 0.443786982471 Cumulative probabilities = A: 0.384615 B: 0.923077 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7040: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (6) D B A C E (5) D A B C E (5) B D C A E (5) E C B A D (4) D A B E C (4) C E B A D (4) C B E A D (4) B C E D A (4) A E C D B (4) A C B D E (4) D E B A C (3) C B E D A (3) A E D C B (3) E D A B C (2) E C B D A (2) E B C D A (2) E A D C B (2) D B E C A (2) D B C A E (2) D A E B C (2) C A B E D (2) B D E C A (2) B C D E A (2) A E C B D (2) A D E C B (2) A C E B D (2) E A C B D (1) D E A B C (1) D B E A C (1) D B C E A (1) D B A E C (1) C E A B D (1) C B A E D (1) C B A D E (1) C A E B D (1) B C D A E (1) A D E B C (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 0 -12 6 B 12 0 8 10 20 C 0 -8 0 -4 16 D 12 -10 4 0 8 E -6 -20 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -12 6 B 12 0 8 10 20 C 0 -8 0 -4 16 D 12 -10 4 0 8 E -6 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 B=20 C=17 E=13 so E is eliminated. Round 2 votes counts: D=29 A=26 C=23 B=22 so B is eliminated. Round 3 votes counts: D=42 C=32 A=26 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:225 D:207 C:202 A:191 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -12 6 B 12 0 8 10 20 C 0 -8 0 -4 16 D 12 -10 4 0 8 E -6 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -12 6 B 12 0 8 10 20 C 0 -8 0 -4 16 D 12 -10 4 0 8 E -6 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -12 6 B 12 0 8 10 20 C 0 -8 0 -4 16 D 12 -10 4 0 8 E -6 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7041: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (6) E D C B A (5) D E C B A (5) D C B E A (4) D B C A E (4) B D C A E (4) B D A E C (4) B A D E C (4) A C E B D (4) E C D A B (3) E A C D B (3) D B E C A (3) C E A D B (3) C D E B A (3) C D B A E (3) A B C D E (3) E A C B D (2) E A B D C (2) D E B C A (2) D C E B A (2) C D E A B (2) C D B E A (2) C D A E B (2) B E A D C (2) B D A C E (2) B A D C E (2) B A C D E (2) A B E D C (2) E D C A B (1) E B A D C (1) E A D B C (1) E A B C D (1) D C B A E (1) D B C E A (1) C E D A B (1) C D A B E (1) C B D A E (1) C A E D B (1) C A B D E (1) B A E D C (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -10 -12 2 B 16 0 -4 -6 10 C 10 4 0 -6 2 D 12 6 6 0 16 E -2 -10 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -12 2 B 16 0 -4 -6 10 C 10 4 0 -6 2 D 12 6 6 0 16 E -2 -10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=22 B=21 C=20 E=19 A=18 so A is eliminated. Round 2 votes counts: B=33 C=25 D=22 E=20 so E is eliminated. Round 3 votes counts: B=37 C=34 D=29 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:220 B:208 C:205 E:185 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -10 -12 2 B 16 0 -4 -6 10 C 10 4 0 -6 2 D 12 6 6 0 16 E -2 -10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -12 2 B 16 0 -4 -6 10 C 10 4 0 -6 2 D 12 6 6 0 16 E -2 -10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -12 2 B 16 0 -4 -6 10 C 10 4 0 -6 2 D 12 6 6 0 16 E -2 -10 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7042: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D B C A E (7) E A D C B (5) D B C E A (5) E D A B C (4) C A B D E (4) E D B A C (3) D E A B C (3) D C B A E (3) D B E C A (3) C B D A E (3) C B A D E (3) C A D B E (3) B C E D A (3) E B D C A (2) E A D B C (2) E A B C D (2) D E B A C (2) D C A B E (2) C D B A E (2) B D E C A (2) B D C E A (2) B C D E A (2) A E C D B (2) E D A C B (1) E C A B D (1) E B D A C (1) E B C A D (1) E B A C D (1) E A B D C (1) D A E C B (1) D A B C E (1) C B A E D (1) B D C A E (1) B C E A D (1) B C D A E (1) A E C B D (1) A D C E B (1) A C E B D (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -4 -8 -20 B -2 0 0 -4 2 C 4 0 0 -4 -2 D 8 4 4 0 8 E 20 -2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -8 -20 B -2 0 0 -4 2 C 4 0 0 -4 -2 D 8 4 4 0 8 E 20 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=27 C=16 B=12 A=8 so A is eliminated. Round 2 votes counts: E=40 D=28 C=20 B=12 so B is eliminated. Round 3 votes counts: E=40 D=33 C=27 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:206 C:199 B:198 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -8 -20 B -2 0 0 -4 2 C 4 0 0 -4 -2 D 8 4 4 0 8 E 20 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -8 -20 B -2 0 0 -4 2 C 4 0 0 -4 -2 D 8 4 4 0 8 E 20 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -8 -20 B -2 0 0 -4 2 C 4 0 0 -4 -2 D 8 4 4 0 8 E 20 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7043: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (11) A C E B D (11) B E C D A (7) B D E C A (6) D A E C B (5) C E B A D (5) B C E A D (5) E C B D A (4) D B E C A (4) C E A B D (4) A D E C B (4) A C E D B (4) D B A E C (3) A D B C E (3) E C B A D (2) E C A D B (2) D B A C E (2) A D C B E (2) E C D B A (1) E C D A B (1) E B C D A (1) D E C B A (1) D E C A B (1) D E B C A (1) D E A C B (1) D A B E C (1) D A B C E (1) C A E B D (1) B E D C A (1) B D C E A (1) B D C A E (1) B C E D A (1) B A C D E (1) A C D E B (1) Total count = 100 A B C D E A 0 6 0 12 2 B -6 0 -22 2 -22 C 0 22 0 2 8 D -12 -2 -2 0 0 E -2 22 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.531443 B: 0.000000 C: 0.468557 D: 0.000000 E: 0.000000 Sum of squares = 0.501977309751 Cumulative probabilities = A: 0.531443 B: 0.531443 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 12 2 B -6 0 -22 2 -22 C 0 22 0 2 8 D -12 -2 -2 0 0 E -2 22 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=23 D=20 E=11 C=10 so C is eliminated. Round 2 votes counts: A=37 B=23 E=20 D=20 so E is eliminated. Round 3 votes counts: A=43 B=35 D=22 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:216 A:210 E:206 D:192 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 12 2 B -6 0 -22 2 -22 C 0 22 0 2 8 D -12 -2 -2 0 0 E -2 22 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 12 2 B -6 0 -22 2 -22 C 0 22 0 2 8 D -12 -2 -2 0 0 E -2 22 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 12 2 B -6 0 -22 2 -22 C 0 22 0 2 8 D -12 -2 -2 0 0 E -2 22 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999939 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7044: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (17) E A D B C (10) C E B D A (8) C B E D A (7) A D B E C (7) E C A D B (6) E C A B D (6) B D C A E (6) B D A C E (5) E A C D B (4) D A B E C (4) E A D C B (3) C E B A D (3) B C D A E (3) A E D B C (3) A D E B C (2) E C B A D (1) D B A C E (1) D A B C E (1) C E A B D (1) C B D E A (1) C A E D B (1) Total count = 100 A B C D E A 0 -4 -20 -6 0 B 4 0 -16 16 4 C 20 16 0 16 8 D 6 -16 -16 0 -6 E 0 -4 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -20 -6 0 B 4 0 -16 16 4 C 20 16 0 16 8 D 6 -16 -16 0 -6 E 0 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=30 B=14 A=12 D=6 so D is eliminated. Round 2 votes counts: C=38 E=30 A=17 B=15 so B is eliminated. Round 3 votes counts: C=47 E=30 A=23 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:204 E:197 A:185 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -20 -6 0 B 4 0 -16 16 4 C 20 16 0 16 8 D 6 -16 -16 0 -6 E 0 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -20 -6 0 B 4 0 -16 16 4 C 20 16 0 16 8 D 6 -16 -16 0 -6 E 0 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -20 -6 0 B 4 0 -16 16 4 C 20 16 0 16 8 D 6 -16 -16 0 -6 E 0 -4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7045: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (18) B A D C E (8) C E D B A (7) B A D E C (6) E D C A B (4) B A C D E (4) A B E D C (4) E C D B A (3) B C D A E (3) E D C B A (2) E D A C B (2) E C D A B (2) D E C B A (2) D C E B A (2) C E D A B (2) C E A D B (2) C E A B D (2) C B D E A (2) B D C A E (2) A E D B C (2) A B E C D (2) A B C D E (2) E C A D B (1) E A D C B (1) E A C D B (1) D E A B C (1) D B E C A (1) D B E A C (1) D A E B C (1) C E B D A (1) C D E B A (1) C B E D A (1) C B E A D (1) B D A C E (1) B C A E D (1) A E C B D (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 16 18 16 B -2 0 18 24 16 C -16 -18 0 -16 -12 D -18 -24 16 0 10 E -16 -16 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997811 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 18 16 B -2 0 18 24 16 C -16 -18 0 -16 -12 D -18 -24 16 0 10 E -16 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996168 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=25 C=19 E=16 D=8 so D is eliminated. Round 2 votes counts: A=33 B=27 C=21 E=19 so E is eliminated. Round 3 votes counts: A=38 C=35 B=27 so B is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:228 A:226 D:192 E:185 C:169 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 18 16 B -2 0 18 24 16 C -16 -18 0 -16 -12 D -18 -24 16 0 10 E -16 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996168 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 18 16 B -2 0 18 24 16 C -16 -18 0 -16 -12 D -18 -24 16 0 10 E -16 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996168 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 18 16 B -2 0 18 24 16 C -16 -18 0 -16 -12 D -18 -24 16 0 10 E -16 -16 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996168 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7046: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B D E C A (10) E A C B D (9) D B C A E (8) D C A B E (7) E B A C D (6) A C E D B (5) E A C D B (4) B E D A C (4) E B C A D (3) D A C B E (3) A C E B D (3) A C D E B (3) A C D B E (3) E D C A B (2) E B D C A (2) C D A B E (2) B E D C A (2) B D C A E (2) B D A C E (2) E C A D B (1) E C A B D (1) E B D A C (1) E A B C D (1) D B E C A (1) D B A C E (1) C A E D B (1) B E A C D (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 14 -4 6 4 B -14 0 -12 -4 -8 C 4 12 0 10 0 D -6 4 -10 0 4 E -4 8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.711794 D: 0.000000 E: 0.288206 Sum of squares = 0.589713666029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.711794 D: 0.711794 E: 1.000000 A B C D E A 0 14 -4 6 4 B -14 0 -12 -4 -8 C 4 12 0 10 0 D -6 4 -10 0 4 E -4 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.000000 E: 0.499410 Sum of squares = 0.500000697051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.500590 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=21 D=20 A=16 C=13 so C is eliminated. Round 2 votes counts: E=30 A=27 D=22 B=21 so B is eliminated. Round 3 votes counts: E=37 D=36 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:213 A:210 E:200 D:196 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 6 4 B -14 0 -12 -4 -8 C 4 12 0 10 0 D -6 4 -10 0 4 E -4 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.000000 E: 0.499410 Sum of squares = 0.500000697051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.500590 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 6 4 B -14 0 -12 -4 -8 C 4 12 0 10 0 D -6 4 -10 0 4 E -4 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.000000 E: 0.499410 Sum of squares = 0.500000697051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.500590 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 6 4 B -14 0 -12 -4 -8 C 4 12 0 10 0 D -6 4 -10 0 4 E -4 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.000000 E: 0.499410 Sum of squares = 0.500000697051 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500590 D: 0.500590 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7047: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) D E A B C (5) C A D B E (5) A C D B E (5) D B E C A (4) C B E A D (4) C A B D E (4) B E C D A (4) A D C E B (4) E B D A C (3) E B A C D (3) D C A B E (3) C D A B E (3) C B D A E (3) A C E B D (3) E D B A C (2) D A E C B (2) D A E B C (2) D A C E B (2) C A B E D (2) B E C A D (2) B C E A D (2) B C D E A (2) A E C B D (2) A D E C B (2) A D C B E (2) A C D E B (2) A C B E D (2) E D A B C (1) E B C D A (1) E B C A D (1) E B A D C (1) E A B C D (1) D E B A C (1) D C B E A (1) D B C E A (1) C B D E A (1) C B A E D (1) B E D C A (1) B C E D A (1) A E D B C (1) A E B C D (1) Total count = 100 A B C D E A 0 8 -6 0 2 B -8 0 -6 4 6 C 6 6 0 10 6 D 0 -4 -10 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 0 2 B -8 0 -6 4 6 C 6 6 0 10 6 D 0 -4 -10 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 D=21 E=20 B=12 so B is eliminated. Round 2 votes counts: C=28 E=27 A=24 D=21 so D is eliminated. Round 3 votes counts: E=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:202 B:198 D:197 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 0 2 B -8 0 -6 4 6 C 6 6 0 10 6 D 0 -4 -10 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 0 2 B -8 0 -6 4 6 C 6 6 0 10 6 D 0 -4 -10 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 0 2 B -8 0 -6 4 6 C 6 6 0 10 6 D 0 -4 -10 0 8 E -2 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7048: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (7) A E D C B (7) E D A B C (6) C B A E D (5) A C B E D (5) E D B A C (4) C B D A E (4) C B A D E (4) A E B C D (4) D E A B C (3) D C B E A (3) C A B E D (3) B E A C D (3) A E B D C (3) A C E B D (3) E B D A C (2) E A D B C (2) E A B D C (2) D E C A B (2) D E B A C (2) C D B E A (2) B A E C D (2) A B C E D (2) E B A D C (1) E A D C B (1) D E B C A (1) D C E B A (1) D C E A B (1) D A E C B (1) C D B A E (1) C D A E B (1) C A D E B (1) C A B D E (1) B E D C A (1) B E D A C (1) B D C E A (1) B C E D A (1) B C D E A (1) B C A E D (1) B A C E D (1) A E C D B (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 16 30 8 4 B -16 0 -10 4 -14 C -30 10 0 -6 -16 D -8 -4 6 0 -24 E -4 14 16 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 30 8 4 B -16 0 -10 4 -14 C -30 10 0 -6 -16 D -8 -4 6 0 -24 E -4 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 D=21 E=18 B=12 so B is eliminated. Round 2 votes counts: A=30 C=25 E=23 D=22 so D is eliminated. Round 3 votes counts: E=38 C=31 A=31 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:225 D:185 B:182 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 30 8 4 B -16 0 -10 4 -14 C -30 10 0 -6 -16 D -8 -4 6 0 -24 E -4 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 30 8 4 B -16 0 -10 4 -14 C -30 10 0 -6 -16 D -8 -4 6 0 -24 E -4 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 30 8 4 B -16 0 -10 4 -14 C -30 10 0 -6 -16 D -8 -4 6 0 -24 E -4 14 16 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7049: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) E C B A D (6) E B C D A (6) B D E C A (6) A C E D B (6) E C A B D (5) B E D C A (5) C E B D A (4) C E A B D (4) C A E D B (4) E B C A D (3) E B A C D (3) D B A C E (3) B E C D A (3) E C B D A (2) D C A B E (2) D B C E A (2) D B A E C (2) D A B C E (2) C E B A D (2) C B E D A (2) B D E A C (2) A E C B D (2) A D C B E (2) A D B E C (2) E B A D C (1) E A C B D (1) D C B E A (1) D A B E C (1) C E A D B (1) C A D E B (1) B E A D C (1) A E D C B (1) A E D B C (1) A D E B C (1) A D C E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -18 14 -20 B 8 0 -8 20 -24 C 18 8 0 24 -12 D -14 -20 -24 0 -28 E 20 24 12 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -18 14 -20 B 8 0 -8 20 -24 C 18 8 0 24 -12 D -14 -20 -24 0 -28 E 20 24 12 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=25 C=18 B=17 D=13 so D is eliminated. Round 2 votes counts: A=28 E=27 B=24 C=21 so C is eliminated. Round 3 votes counts: E=38 A=35 B=27 so B is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:242 C:219 B:198 A:184 D:157 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -18 14 -20 B 8 0 -8 20 -24 C 18 8 0 24 -12 D -14 -20 -24 0 -28 E 20 24 12 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 14 -20 B 8 0 -8 20 -24 C 18 8 0 24 -12 D -14 -20 -24 0 -28 E 20 24 12 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 14 -20 B 8 0 -8 20 -24 C 18 8 0 24 -12 D -14 -20 -24 0 -28 E 20 24 12 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7050: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (14) E B C A D (7) B E C A D (7) B A E D C (7) E C B A D (6) C D E A B (6) D C A E B (5) D A C E B (5) D A B C E (5) D A C B E (3) A D B E C (3) D C B E A (2) D C A B E (2) D B C E A (2) C E D B A (2) C D A E B (2) B D E C A (2) B D A E C (2) B A D E C (2) A B D E C (2) E B C D A (1) E A C B D (1) D C E A B (1) D A B E C (1) C E B A D (1) C E A B D (1) C D E B A (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E A C (1) A C D E B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 0 14 -12 B 16 0 18 14 16 C 0 -18 0 6 -18 D -14 -14 -6 0 -2 E 12 -16 18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 14 -12 B 16 0 18 14 16 C 0 -18 0 6 -18 D -14 -14 -6 0 -2 E 12 -16 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=26 E=15 C=15 A=8 so A is eliminated. Round 2 votes counts: B=40 D=29 C=16 E=15 so E is eliminated. Round 3 votes counts: B=48 D=29 C=23 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:232 E:208 A:193 C:185 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 14 -12 B 16 0 18 14 16 C 0 -18 0 6 -18 D -14 -14 -6 0 -2 E 12 -16 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 14 -12 B 16 0 18 14 16 C 0 -18 0 6 -18 D -14 -14 -6 0 -2 E 12 -16 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 14 -12 B 16 0 18 14 16 C 0 -18 0 6 -18 D -14 -14 -6 0 -2 E 12 -16 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999597 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7051: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) E B D C A (9) B E D C A (7) A D C B E (7) A C D E B (7) D B E A C (6) D A C B E (6) C A E D B (6) E C B A D (5) C E A B D (5) C A D E B (5) E B C D A (4) B D E A C (4) A C D B E (4) E B C A D (3) D A B C E (3) D B A E C (2) B E D A C (2) E C B D A (1) C E A D B (1) B D E C A (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 12 -14 10 4 B -12 0 -16 4 -14 C 14 16 0 4 12 D -10 -4 -4 0 -8 E -4 14 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -14 10 4 B -12 0 -16 4 -14 C 14 16 0 4 12 D -10 -4 -4 0 -8 E -4 14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 A=20 D=17 B=14 so B is eliminated. Round 2 votes counts: E=31 C=27 D=22 A=20 so A is eliminated. Round 3 votes counts: C=39 E=31 D=30 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:206 E:203 D:187 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -14 10 4 B -12 0 -16 4 -14 C 14 16 0 4 12 D -10 -4 -4 0 -8 E -4 14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -14 10 4 B -12 0 -16 4 -14 C 14 16 0 4 12 D -10 -4 -4 0 -8 E -4 14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -14 10 4 B -12 0 -16 4 -14 C 14 16 0 4 12 D -10 -4 -4 0 -8 E -4 14 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7052: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (19) C B D E A (14) A B E D C (7) A E D C B (6) D E A B C (5) D E A C B (4) C D E B A (4) B C D E A (4) E D A C B (3) D E B C A (3) E D A B C (2) D E C A B (2) D E B A C (2) C D B E A (2) C B A D E (2) C A E D B (2) A E B D C (2) A C E D B (2) E B D A C (1) D E C B A (1) D B E C A (1) C B D A E (1) C B A E D (1) C A B E D (1) B D E C A (1) B D C E A (1) B C D A E (1) B C A E D (1) B A E D C (1) B A C E D (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 16 16 -4 0 B -16 0 6 -16 -18 C -16 -6 0 -22 -20 D 4 16 22 0 -4 E 0 18 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.313309 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.686691 Sum of squares = 0.569707144192 Cumulative probabilities = A: 0.313309 B: 0.313309 C: 0.313309 D: 0.313309 E: 1.000000 A B C D E A 0 16 16 -4 0 B -16 0 6 -16 -18 C -16 -6 0 -22 -20 D 4 16 22 0 -4 E 0 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000124083 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=27 D=18 B=10 E=6 so E is eliminated. Round 2 votes counts: A=39 C=27 D=23 B=11 so B is eliminated. Round 3 votes counts: A=41 C=33 D=26 so D is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:221 D:219 A:214 B:178 C:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 16 -4 0 B -16 0 6 -16 -18 C -16 -6 0 -22 -20 D 4 16 22 0 -4 E 0 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000124083 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 -4 0 B -16 0 6 -16 -18 C -16 -6 0 -22 -20 D 4 16 22 0 -4 E 0 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000124083 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 -4 0 B -16 0 6 -16 -18 C -16 -6 0 -22 -20 D 4 16 22 0 -4 E 0 18 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000124083 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7053: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (14) D C E B A (7) B C A E D (7) D E A C B (6) C B D A E (6) E D A C B (5) E A D B C (5) B C D A E (5) A B E C D (5) A E D B C (4) A E B D C (4) E A D C B (3) E A B C D (3) D E C A B (3) D C B A E (3) C B D E A (3) E A B D C (2) D A E C B (2) C D B E A (2) C D B A E (2) B A C E D (2) A E B C D (2) D E C B A (1) D A E B C (1) B C E A D (1) B C A D E (1) B A E C D (1) Total count = 100 A B C D E A 0 -10 -10 -20 -10 B 10 0 -14 -16 4 C 10 14 0 -20 6 D 20 16 20 0 12 E 10 -4 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -20 -10 B 10 0 -14 -16 4 C 10 14 0 -20 6 D 20 16 20 0 12 E 10 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=18 B=17 A=15 C=13 so C is eliminated. Round 2 votes counts: D=41 B=26 E=18 A=15 so A is eliminated. Round 3 votes counts: D=41 B=31 E=28 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:234 C:205 E:194 B:192 A:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 -20 -10 B 10 0 -14 -16 4 C 10 14 0 -20 6 D 20 16 20 0 12 E 10 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -20 -10 B 10 0 -14 -16 4 C 10 14 0 -20 6 D 20 16 20 0 12 E 10 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -20 -10 B 10 0 -14 -16 4 C 10 14 0 -20 6 D 20 16 20 0 12 E 10 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7054: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) D E A B C (6) C A B D E (6) B C A E D (6) C B A E D (5) C B A D E (5) B C E D A (5) A D E C B (5) A C D E B (5) E D B C A (4) C B E D A (4) C A D B E (4) A D E B C (4) A D C E B (4) E D B A C (3) E B D A C (3) B E A D C (3) D A E C B (2) C A D E B (2) B E D C A (2) B E C D A (2) B A C E D (2) A E D B C (2) E D C B A (1) D E C B A (1) D E A C B (1) D C E A B (1) C B D E A (1) B E D A C (1) B C E A D (1) B A E D C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 0 14 10 B 0 0 6 -2 0 C 0 -6 0 -4 2 D -14 2 4 0 -4 E -10 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.381060 B: 0.618940 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.528293530206 Cumulative probabilities = A: 0.381060 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 14 10 B 0 0 6 -2 0 C 0 -6 0 -4 2 D -14 2 4 0 -4 E -10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=23 A=22 E=17 D=11 so D is eliminated. Round 2 votes counts: C=28 E=25 A=24 B=23 so B is eliminated. Round 3 votes counts: C=40 E=33 A=27 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:212 B:202 C:196 E:196 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 14 10 B 0 0 6 -2 0 C 0 -6 0 -4 2 D -14 2 4 0 -4 E -10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 14 10 B 0 0 6 -2 0 C 0 -6 0 -4 2 D -14 2 4 0 -4 E -10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 14 10 B 0 0 6 -2 0 C 0 -6 0 -4 2 D -14 2 4 0 -4 E -10 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7055: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) C B E A D (10) E B C A D (9) A D C B E (8) D A E B C (7) A D E C B (6) C B A E D (5) B C E D A (5) D E A B C (4) C B D A E (4) E B D C A (3) E A D B C (3) B E C D A (3) E D A B C (2) E C B A D (2) D A C B E (2) C B A D E (2) E D B C A (1) E D B A C (1) E A D C B (1) E A C B D (1) C A B E D (1) C A B D E (1) B E C A D (1) B D E C A (1) B C D E A (1) A E C D B (1) A D E B C (1) A D C E B (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -16 -18 12 -16 B 16 0 4 22 -8 C 18 -4 0 18 -14 D -12 -22 -18 0 -20 E 16 8 14 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -18 12 -16 B 16 0 4 22 -8 C 18 -4 0 18 -14 D -12 -22 -18 0 -20 E 16 8 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=23 A=20 D=13 B=11 so B is eliminated. Round 2 votes counts: E=37 C=29 A=20 D=14 so D is eliminated. Round 3 votes counts: E=42 C=29 A=29 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 B:217 C:209 A:181 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -18 12 -16 B 16 0 4 22 -8 C 18 -4 0 18 -14 D -12 -22 -18 0 -20 E 16 8 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 12 -16 B 16 0 4 22 -8 C 18 -4 0 18 -14 D -12 -22 -18 0 -20 E 16 8 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 12 -16 B 16 0 4 22 -8 C 18 -4 0 18 -14 D -12 -22 -18 0 -20 E 16 8 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7056: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (13) C B E A D (8) D A E B C (5) C B E D A (5) A D E C B (5) D E B A C (4) C B A E D (4) C A E D B (4) B C E D A (4) A C E D B (4) D E A B C (3) C A D B E (3) B E D C A (3) B D E C A (3) A D C E B (3) E D B A C (2) D B E A C (2) C E B A D (2) C A B D E (2) B E C D A (2) B D E A C (2) A E D B C (2) A E C D B (2) A C D E B (2) E C B D A (1) E B D C A (1) E B D A C (1) C B D E A (1) C B A D E (1) C A D E B (1) C A B E D (1) B D C E A (1) B C D A E (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 4 4 18 10 B -4 0 0 -14 -12 C -4 0 0 -4 -6 D -18 14 4 0 6 E -10 12 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 18 10 B -4 0 0 -14 -12 C -4 0 0 -4 -6 D -18 14 4 0 6 E -10 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=32 B=16 D=14 E=5 so E is eliminated. Round 2 votes counts: C=33 A=33 B=18 D=16 so D is eliminated. Round 3 votes counts: A=41 C=33 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:203 E:201 C:193 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 18 10 B -4 0 0 -14 -12 C -4 0 0 -4 -6 D -18 14 4 0 6 E -10 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 18 10 B -4 0 0 -14 -12 C -4 0 0 -4 -6 D -18 14 4 0 6 E -10 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 18 10 B -4 0 0 -14 -12 C -4 0 0 -4 -6 D -18 14 4 0 6 E -10 12 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999804 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7057: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) E A C D B (9) D E B C A (7) B C D A E (5) E D B C A (4) E D A C B (4) D B E C A (4) B D E C A (4) B D C E A (4) E B D A C (3) D B C E A (3) C A B D E (3) B C A D E (3) A C E D B (3) A C B D E (3) E D B A C (2) E A D C B (2) D C A B E (2) D B C A E (2) C A D E B (2) B A C D E (2) A E C D B (2) A E C B D (2) A C E B D (2) A C B E D (2) E D C A B (1) E D A B C (1) E B A C D (1) E A D B C (1) D C A E B (1) C D B A E (1) C A D B E (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -18 -20 0 B 16 0 18 -6 6 C 18 -18 0 -14 6 D 20 6 14 0 20 E 0 -6 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -18 -20 0 B 16 0 18 -6 6 C 18 -18 0 -14 6 D 20 6 14 0 20 E 0 -6 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=28 D=19 A=15 C=7 so C is eliminated. Round 2 votes counts: B=31 E=28 A=21 D=20 so D is eliminated. Round 3 votes counts: B=41 E=35 A=24 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:230 B:217 C:196 E:184 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -18 -20 0 B 16 0 18 -6 6 C 18 -18 0 -14 6 D 20 6 14 0 20 E 0 -6 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 -20 0 B 16 0 18 -6 6 C 18 -18 0 -14 6 D 20 6 14 0 20 E 0 -6 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 -20 0 B 16 0 18 -6 6 C 18 -18 0 -14 6 D 20 6 14 0 20 E 0 -6 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7058: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) E D C B A (8) A B D C E (8) B A D C E (7) D B E A C (6) E C D B A (5) C A B D E (5) A B C D E (5) E D B A C (4) B D A C E (4) E C D A B (3) E C A D B (3) D B A E C (3) C A E B D (3) B D A E C (3) E D B C A (2) E A D B C (2) D E B A C (2) D B E C A (2) C E A D B (2) C A B E D (2) E C A B D (1) E A B D C (1) D E B C A (1) D C E B A (1) D B C A E (1) D B A C E (1) C E D B A (1) C E D A B (1) B A D E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 18 4 10 B 4 0 10 4 20 C -18 -10 0 -18 2 D -4 -4 18 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 18 4 10 B 4 0 10 4 20 C -18 -10 0 -18 2 D -4 -4 18 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=25 D=17 B=15 C=14 so C is eliminated. Round 2 votes counts: A=35 E=33 D=17 B=15 so B is eliminated. Round 3 votes counts: A=43 E=33 D=24 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:219 D:216 A:214 C:178 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 18 4 10 B 4 0 10 4 20 C -18 -10 0 -18 2 D -4 -4 18 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 18 4 10 B 4 0 10 4 20 C -18 -10 0 -18 2 D -4 -4 18 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 18 4 10 B 4 0 10 4 20 C -18 -10 0 -18 2 D -4 -4 18 0 22 E -10 -20 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997526 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7059: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (17) D C E A B (11) E B A D C (9) A B E C D (8) B E A C D (6) B A E C D (6) A B C E D (6) E B A C D (4) C D B A E (4) D C A E B (3) D C A B E (3) E D B A C (2) E A B D C (2) D C E B A (2) C B A E D (2) E D A B C (1) E B D A C (1) D E C A B (1) D E A B C (1) D A C E B (1) C B D A E (1) C B A D E (1) C A B E D (1) B C A E D (1) B A C E D (1) A E B D C (1) A C B D E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 6 4 20 B -20 0 4 8 22 C -6 -4 0 20 14 D -4 -8 -20 0 -4 E -20 -22 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 4 20 B -20 0 4 8 22 C -6 -4 0 20 14 D -4 -8 -20 0 -4 E -20 -22 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=22 E=19 A=19 B=14 so B is eliminated. Round 2 votes counts: C=27 A=26 E=25 D=22 so D is eliminated. Round 3 votes counts: C=46 E=27 A=27 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:212 B:207 D:182 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 4 20 B -20 0 4 8 22 C -6 -4 0 20 14 D -4 -8 -20 0 -4 E -20 -22 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 4 20 B -20 0 4 8 22 C -6 -4 0 20 14 D -4 -8 -20 0 -4 E -20 -22 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 4 20 B -20 0 4 8 22 C -6 -4 0 20 14 D -4 -8 -20 0 -4 E -20 -22 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998762 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7060: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (10) A C D E B (7) D C A E B (5) B E D C A (5) E B D A C (4) C D A E B (4) B E D A C (4) A C B E D (4) C A D E B (3) B E A D C (3) B C D E A (3) E D B A C (2) E D A B C (2) E B A D C (2) E A B D C (2) D E C B A (2) D C E A B (2) D A E C B (2) C D B E A (2) C B A D E (2) C A B D E (2) B E A C D (2) A E D C B (2) A E B C D (2) A B E C D (2) A B C E D (2) E B D C A (1) E A D B C (1) D E C A B (1) D E B C A (1) D C B E A (1) D B C E A (1) C D B A E (1) C D A B E (1) C B D E A (1) C B D A E (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C E A (1) B A E C D (1) A E B D C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 14 -6 2 8 B -14 0 -10 -4 4 C 6 10 0 6 10 D -2 4 -6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 2 8 B -14 0 -10 -4 4 C 6 10 0 6 10 D -2 4 -6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=22 A=22 D=15 E=14 so E is eliminated. Round 2 votes counts: B=29 C=27 A=25 D=19 so D is eliminated. Round 3 votes counts: C=38 B=33 A=29 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:209 D:203 B:188 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 2 8 B -14 0 -10 -4 4 C 6 10 0 6 10 D -2 4 -6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 2 8 B -14 0 -10 -4 4 C 6 10 0 6 10 D -2 4 -6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 2 8 B -14 0 -10 -4 4 C 6 10 0 6 10 D -2 4 -6 0 10 E -8 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7061: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) B C E A D (9) C B E D A (8) A D B E C (6) E D A C B (5) E D C A B (4) E C B D A (4) B E C A D (4) A D E B C (4) C E B D A (3) C D B A E (3) A D B C E (3) E C D B A (2) D E A C B (2) D C A B E (2) D A C B E (2) C B D A E (2) B E A C D (2) B C A D E (2) B A C D E (2) E C D A B (1) E B C A D (1) E B A D C (1) E A D C B (1) E A B D C (1) D A C E B (1) D A B C E (1) C E D B A (1) C D E A B (1) C D A B E (1) C B E A D (1) C B D E A (1) B E A D C (1) B A E D C (1) B A D C E (1) B A C E D (1) A E B D C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 -10 -6 B 0 0 -12 -2 12 C 0 12 0 -2 -8 D 10 2 2 0 -6 E 6 -12 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 0 0 -10 -6 B 0 0 -12 -2 12 C 0 12 0 -2 -8 D 10 2 2 0 -6 E 6 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999932 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 C=21 E=20 D=19 A=17 so A is eliminated. Round 2 votes counts: D=32 B=26 E=21 C=21 so E is eliminated. Round 3 votes counts: D=42 B=30 C=28 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:204 E:204 C:201 B:199 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -10 -6 B 0 0 -12 -2 12 C 0 12 0 -2 -8 D 10 2 2 0 -6 E 6 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999932 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -10 -6 B 0 0 -12 -2 12 C 0 12 0 -2 -8 D 10 2 2 0 -6 E 6 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999932 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -10 -6 B 0 0 -12 -2 12 C 0 12 0 -2 -8 D 10 2 2 0 -6 E 6 -12 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999932 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7062: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E A B D C (6) D C B E A (6) A E B C D (6) A C B E D (6) E A D B C (5) A C E B D (5) E D A B C (4) D E C B A (4) D E B C A (4) C B A D E (4) B C D E A (4) C B D A E (3) A E D C B (3) A E D B C (3) D B E C A (2) C D A B E (2) A B E C D (2) E B A C D (1) E A D C B (1) D E C A B (1) D E B A C (1) D E A C B (1) D E A B C (1) D C E B A (1) D A E C B (1) C D B E A (1) C D A E B (1) C B A E D (1) C A D E B (1) C A B D E (1) B E D C A (1) B E D A C (1) B E A C D (1) B C E A D (1) B C D A E (1) B A C E D (1) A E C B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 4 2 6 B -6 0 -8 -6 -4 C -4 8 0 8 0 D -2 6 -8 0 0 E -6 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999305 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 6 B -6 0 -8 -6 -4 C -4 8 0 8 0 D -2 6 -8 0 0 E -6 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=23 D=22 E=17 B=10 so B is eliminated. Round 2 votes counts: C=29 A=29 D=22 E=20 so E is eliminated. Round 3 votes counts: A=43 C=29 D=28 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:206 E:199 D:198 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 6 B -6 0 -8 -6 -4 C -4 8 0 8 0 D -2 6 -8 0 0 E -6 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 6 B -6 0 -8 -6 -4 C -4 8 0 8 0 D -2 6 -8 0 0 E -6 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 6 B -6 0 -8 -6 -4 C -4 8 0 8 0 D -2 6 -8 0 0 E -6 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7063: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B E A (7) D C B E A (6) A E B C D (6) C D B A E (5) E A C D B (4) B A D E C (4) E A C B D (3) E A B C D (3) D C B A E (3) C D E B A (3) B A E D C (3) A E B D C (3) A B E D C (3) E B D A C (2) E B A D C (2) D B C E A (2) C D E A B (2) C D A B E (2) C A E D B (2) C A D E B (2) B D E A C (2) B D A C E (2) E D B C A (1) E C D B A (1) E C A D B (1) E A D C B (1) D E B C A (1) D C E B A (1) D B C A E (1) C E A D B (1) C D A E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E C A (1) B D A E C (1) B A D C E (1) A C E B D (1) A C D E B (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 12 10 -10 B 2 0 4 4 -2 C -12 -4 0 -4 -10 D -10 -4 4 0 2 E 10 2 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -2 12 10 -10 B 2 0 4 4 -2 C -12 -4 0 -4 -10 D -10 -4 4 0 2 E 10 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999988 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=26 A=17 B=16 D=14 so D is eliminated. Round 2 votes counts: C=36 E=28 B=19 A=17 so A is eliminated. Round 3 votes counts: C=38 E=37 B=25 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:205 B:204 D:196 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 12 10 -10 B 2 0 4 4 -2 C -12 -4 0 -4 -10 D -10 -4 4 0 2 E 10 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999988 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 12 10 -10 B 2 0 4 4 -2 C -12 -4 0 -4 -10 D -10 -4 4 0 2 E 10 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999988 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 12 10 -10 B 2 0 4 4 -2 C -12 -4 0 -4 -10 D -10 -4 4 0 2 E 10 2 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999988 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7064: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) C A E B D (6) D A B E C (5) A C D E B (5) E B C A D (4) E B A C D (4) D B E C A (4) D A C B E (4) C E A B D (4) B E D A C (4) D C A B E (3) D B E A C (3) C E B D A (3) C D A E B (3) B E A D C (3) A E B C D (3) A D C B E (3) E C B A D (2) E B C D A (2) D C B E A (2) D C B A E (2) C E B A D (2) B E D C A (2) B E C D A (2) A E B D C (2) A D C E B (2) A D B E C (2) A C E B D (2) A B E D C (2) D C A E B (1) C E D B A (1) C B E D A (1) B E A C D (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 16 -6 16 12 B -16 0 -12 -2 -12 C 6 12 0 10 10 D -16 2 -10 0 -2 E -12 12 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -6 16 12 B -16 0 -12 -2 -12 C 6 12 0 10 10 D -16 2 -10 0 -2 E -12 12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 A=23 E=12 B=12 so E is eliminated. Round 2 votes counts: C=31 D=24 A=23 B=22 so B is eliminated. Round 3 votes counts: C=39 A=31 D=30 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:219 E:196 D:187 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -6 16 12 B -16 0 -12 -2 -12 C 6 12 0 10 10 D -16 2 -10 0 -2 E -12 12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 16 12 B -16 0 -12 -2 -12 C 6 12 0 10 10 D -16 2 -10 0 -2 E -12 12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 16 12 B -16 0 -12 -2 -12 C 6 12 0 10 10 D -16 2 -10 0 -2 E -12 12 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7065: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) B E D C A (8) E D B C A (7) A C B E D (7) A D E C B (5) A B C E D (4) C B A E D (3) C A E D B (3) C A B E D (3) B D E A C (3) A D E B C (3) A D C E B (3) A D B E C (3) D E B A C (2) D B E C A (2) D A E C B (2) C E D B A (2) C E B D A (2) C B E D A (2) B C E D A (2) B C A E D (2) A C E D B (2) A B D E C (2) A B C D E (2) E D C B A (1) D E C A B (1) D B E A C (1) D A E B C (1) C E D A B (1) C E B A D (1) C E A D B (1) C B E A D (1) C A D E B (1) B E D A C (1) B E C D A (1) B D E C A (1) B A D E C (1) B A C E D (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 10 20 20 B -12 0 -4 -2 6 C -10 4 0 4 10 D -20 2 -4 0 -10 E -20 -6 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 20 20 B -12 0 -4 -2 6 C -10 4 0 4 10 D -20 2 -4 0 -10 E -20 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 C=20 B=20 D=9 E=8 so E is eliminated. Round 2 votes counts: A=43 C=20 B=20 D=17 so D is eliminated. Round 3 votes counts: A=46 B=32 C=22 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:231 C:204 B:194 E:187 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 20 20 B -12 0 -4 -2 6 C -10 4 0 4 10 D -20 2 -4 0 -10 E -20 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 20 20 B -12 0 -4 -2 6 C -10 4 0 4 10 D -20 2 -4 0 -10 E -20 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 20 20 B -12 0 -4 -2 6 C -10 4 0 4 10 D -20 2 -4 0 -10 E -20 -6 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7066: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) A B E C D (7) B E D C A (6) C D E A B (5) B E A D C (5) A C E B D (5) D E C B A (4) D E B C A (4) D C E B A (3) D A C B E (3) C A E D B (3) A C D B E (3) A C B E D (3) D C E A B (2) D C A E B (2) D B E C A (2) D B E A C (2) C A E B D (2) C A D E B (2) B E A C D (2) B D E C A (2) B D E A C (2) B A E D C (2) B A E C D (2) E D B C A (1) E C B D A (1) E B D C A (1) E B C D A (1) E B C A D (1) D B A C E (1) D A B C E (1) C E D A B (1) C E A D B (1) C E A B D (1) C D A E B (1) B E D A C (1) B E C D A (1) A D C B E (1) A D B C E (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 6 6 2 B -12 0 -2 0 4 C -6 2 0 4 4 D -6 0 -4 0 4 E -2 -4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 6 2 B -12 0 -2 0 4 C -6 2 0 4 4 D -6 0 -4 0 4 E -2 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=24 B=23 C=16 E=5 so E is eliminated. Round 2 votes counts: A=32 B=26 D=25 C=17 so C is eliminated. Round 3 votes counts: A=41 D=32 B=27 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:202 D:197 B:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 6 2 B -12 0 -2 0 4 C -6 2 0 4 4 D -6 0 -4 0 4 E -2 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 6 2 B -12 0 -2 0 4 C -6 2 0 4 4 D -6 0 -4 0 4 E -2 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 6 2 B -12 0 -2 0 4 C -6 2 0 4 4 D -6 0 -4 0 4 E -2 -4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7067: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) E C D A B (5) C E A B D (5) E C B A D (4) C E D A B (4) B D A E C (4) B A D E C (4) B A D C E (4) A B C D E (4) E B C A D (3) D B E A C (3) C A E B D (3) B A E C D (3) E D C B A (2) E D B C A (2) E C D B A (2) E C A D B (2) E B C D A (2) D E C B A (2) D C E A B (2) D C A E B (2) D A C E B (2) D A B C E (2) C D E A B (2) B E D A C (2) B A C E D (2) A D C B E (2) A B D C E (2) A B C E D (2) E D C A B (1) E C B D A (1) E C A B D (1) E B D C A (1) D E B C A (1) D E B A C (1) D B A C E (1) C E A D B (1) C A E D B (1) C A B E D (1) B D E A C (1) B D A C E (1) B A C D E (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 0 -4 0 B 6 0 4 6 -4 C 0 -4 0 4 -6 D 4 -6 -4 0 -2 E 0 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.184942 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.815058 Sum of squares = 0.698523671779 Cumulative probabilities = A: 0.184942 B: 0.184942 C: 0.184942 D: 0.184942 E: 1.000000 A B C D E A 0 -6 0 -4 0 B 6 0 4 6 -4 C 0 -4 0 4 -6 D 4 -6 -4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555568174 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=22 B=22 C=17 A=13 so A is eliminated. Round 2 votes counts: B=30 E=26 D=24 C=20 so C is eliminated. Round 3 votes counts: E=40 B=33 D=27 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:206 E:206 C:197 D:196 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 -4 0 B 6 0 4 6 -4 C 0 -4 0 4 -6 D 4 -6 -4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555568174 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -4 0 B 6 0 4 6 -4 C 0 -4 0 4 -6 D 4 -6 -4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555568174 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -4 0 B 6 0 4 6 -4 C 0 -4 0 4 -6 D 4 -6 -4 0 -2 E 0 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555568174 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7068: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) A C D E B (7) D E B C A (6) A C B D E (5) C A D B E (4) C A B D E (4) B E D C A (4) A B E C D (4) E B D A C (3) D E C B A (3) C D A E B (3) B E D A C (3) B E C D A (3) B E A D C (3) D E C A B (2) D E B A C (2) D E A B C (2) D A E C B (2) D A E B C (2) D A C E B (2) C D E B A (2) C A B E D (2) A E D B C (2) A E B D C (2) A C D B E (2) A B E D C (2) E D B A C (1) E B A D C (1) E A B D C (1) D C E B A (1) D C A E B (1) C D E A B (1) C D B E A (1) C D B A E (1) C B E A D (1) C B D E A (1) C A D E B (1) B E C A D (1) B C E A D (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 24 14 8 14 B -24 0 -12 -2 0 C -14 12 0 4 -2 D -8 2 -4 0 16 E -14 0 2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 14 8 14 B -24 0 -12 -2 0 C -14 12 0 4 -2 D -8 2 -4 0 16 E -14 0 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=23 C=21 B=15 E=6 so E is eliminated. Round 2 votes counts: A=36 D=24 C=21 B=19 so B is eliminated. Round 3 votes counts: A=40 D=34 C=26 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:230 D:203 C:200 E:186 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 14 8 14 B -24 0 -12 -2 0 C -14 12 0 4 -2 D -8 2 -4 0 16 E -14 0 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 14 8 14 B -24 0 -12 -2 0 C -14 12 0 4 -2 D -8 2 -4 0 16 E -14 0 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 14 8 14 B -24 0 -12 -2 0 C -14 12 0 4 -2 D -8 2 -4 0 16 E -14 0 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7069: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (7) A C B E D (7) C B D E A (6) D E B C A (4) D E A B C (4) C D B E A (4) B E C D A (4) A D E C B (4) C B A E D (3) C A D B E (3) B E D C A (3) B C E D A (3) E D B A C (2) E B D C A (2) E A B D C (2) D C E B A (2) C D A E B (2) C D A B E (2) C B E D A (2) C B A D E (2) C A D E B (2) C A B D E (2) B E A C D (2) A E D B C (2) A D C E B (2) A C D B E (2) A C B D E (2) A B E C D (2) A B C E D (2) E D B C A (1) E B D A C (1) E A D B C (1) D E C B A (1) D E A C B (1) D A E C B (1) D A C E B (1) C B E A D (1) C B D A E (1) B E C A D (1) B E A D C (1) A D E B C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -2 6 4 B -8 0 -24 0 10 C 2 24 0 28 20 D -6 0 -28 0 12 E -4 -10 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 6 4 B -8 0 -24 0 10 C 2 24 0 28 20 D -6 0 -28 0 12 E -4 -10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=30 D=14 B=14 E=9 so E is eliminated. Round 2 votes counts: A=36 C=30 D=17 B=17 so D is eliminated. Round 3 votes counts: A=43 C=33 B=24 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:237 A:208 B:189 D:189 E:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 6 4 B -8 0 -24 0 10 C 2 24 0 28 20 D -6 0 -28 0 12 E -4 -10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 4 B -8 0 -24 0 10 C 2 24 0 28 20 D -6 0 -28 0 12 E -4 -10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 4 B -8 0 -24 0 10 C 2 24 0 28 20 D -6 0 -28 0 12 E -4 -10 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999176 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7070: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A C B D E (7) C A B D E (5) E D B A C (4) E C D A B (4) E A C D B (4) E A B D C (4) B D A C E (4) A B D C E (4) D B E C A (3) D B C E A (3) D B C A E (3) C E A D B (3) C A E B D (3) A C E B D (3) A B C D E (3) E D C B A (2) D C B E A (2) D B E A C (2) C D E B A (2) C B D A E (2) B D C A E (2) B D A E C (2) B A D C E (2) A E B C D (2) A C B E D (2) A B D E C (2) E D A C B (1) E D A B C (1) E C D B A (1) E C A D B (1) E A D C B (1) E A D B C (1) E A C B D (1) D B A C E (1) C D B A E (1) C D A B E (1) C B A D E (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 10 6 2 4 B -10 0 4 2 4 C -6 -4 0 -6 8 D -2 -2 6 0 4 E -4 -4 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 2 4 B -10 0 4 2 4 C -6 -4 0 -6 8 D -2 -2 6 0 4 E -4 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=25 C=18 D=14 B=10 so B is eliminated. Round 2 votes counts: E=33 A=27 D=22 C=18 so C is eliminated. Round 3 votes counts: E=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:203 B:200 C:196 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 2 4 B -10 0 4 2 4 C -6 -4 0 -6 8 D -2 -2 6 0 4 E -4 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 4 B -10 0 4 2 4 C -6 -4 0 -6 8 D -2 -2 6 0 4 E -4 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 4 B -10 0 4 2 4 C -6 -4 0 -6 8 D -2 -2 6 0 4 E -4 -4 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7071: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) B C A E D (9) A D E B C (9) A B C E D (8) B A C E D (7) E D C B A (5) A B E D C (5) D E C B A (4) C E D B A (4) C B E D A (4) D E C A B (3) C D E B A (3) B C E A D (3) C B A E D (2) A B E C D (2) A B D C E (2) E D B C A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E A B C (1) D C E B A (1) D A E C B (1) D A E B C (1) C E B D A (1) C D A E B (1) C B E A D (1) C B D A E (1) B C E D A (1) B A E C D (1) A D C E B (1) A D B C E (1) A C D B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 6 8 8 B 2 0 10 4 0 C -6 -10 0 4 4 D -8 -4 -4 0 -14 E -8 0 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.902689 C: 0.000000 D: 0.000000 E: 0.097311 Sum of squares = 0.824316633539 Cumulative probabilities = A: 0.000000 B: 0.902689 C: 0.902689 D: 0.902689 E: 1.000000 A B C D E A 0 -2 6 8 8 B 2 0 10 4 0 C -6 -10 0 4 4 D -8 -4 -4 0 -14 E -8 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000108078 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=22 B=21 C=17 E=9 so E is eliminated. Round 2 votes counts: A=31 D=28 B=23 C=18 so C is eliminated. Round 3 votes counts: D=36 B=33 A=31 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:210 B:208 E:201 C:196 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 8 8 B 2 0 10 4 0 C -6 -10 0 4 4 D -8 -4 -4 0 -14 E -8 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000108078 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 8 8 B 2 0 10 4 0 C -6 -10 0 4 4 D -8 -4 -4 0 -14 E -8 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000108078 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 8 8 B 2 0 10 4 0 C -6 -10 0 4 4 D -8 -4 -4 0 -14 E -8 0 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000108078 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7072: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) D A B C E (6) E B C D A (5) D B A E C (5) C E A D B (5) C A E D B (5) B D E A C (5) A D B E C (5) A D B C E (5) E C B D A (4) C E A B D (4) B D A E C (4) E C A B D (3) B E D C A (3) B D E C A (3) A C E D B (3) A C D E B (3) A C D B E (3) E B D C A (2) D A C B E (2) C E B D A (2) C D B E A (2) C D B A E (2) A D C B E (2) E B C A D (1) E B A C D (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B E C (1) C E B A D (1) C D A B E (1) B E D A C (1) B E C D A (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -4 -4 -2 B 2 0 4 -6 10 C 4 -4 0 4 -4 D 4 6 -4 0 4 E 2 -10 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775458 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -2 B 2 0 4 -6 10 C 4 -4 0 4 -4 D 4 6 -4 0 4 E 2 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775501 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=22 C=22 A=22 D=17 B=17 so D is eliminated. Round 2 votes counts: A=31 B=25 E=22 C=22 so E is eliminated. Round 3 votes counts: C=35 B=34 A=31 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:205 D:205 C:200 E:196 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -2 B 2 0 4 -6 10 C 4 -4 0 4 -4 D 4 6 -4 0 4 E 2 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775501 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -2 B 2 0 4 -6 10 C 4 -4 0 4 -4 D 4 6 -4 0 4 E 2 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775501 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -2 B 2 0 4 -6 10 C 4 -4 0 4 -4 D 4 6 -4 0 4 E 2 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775501 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7073: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) B C A E D (9) B A C D E (8) C E D A B (6) B A D C E (6) E D C A B (5) E C D A B (5) B C E A D (5) A D C E B (4) D E A B C (3) C B E A D (3) B E C D A (3) B C E D A (3) A C D E B (3) E D B C A (2) C A E D B (2) C A D E B (2) B D A E C (2) B A D E C (2) A D E C B (2) A D C B E (2) A B D C E (2) E C D B A (1) E C B D A (1) E B C D A (1) D E C A B (1) D A E B C (1) D A B E C (1) C E A B D (1) C B A E D (1) C A B D E (1) B E D C A (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 10 18 B -4 0 0 2 2 C 6 0 0 14 20 D -10 -2 -14 0 2 E -18 -2 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.398530 C: 0.601470 D: 0.000000 E: 0.000000 Sum of squares = 0.520592125805 Cumulative probabilities = A: 0.000000 B: 0.398530 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 10 18 B -4 0 0 2 2 C 6 0 0 14 20 D -10 -2 -14 0 2 E -18 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=16 E=15 D=15 A=15 so E is eliminated. Round 2 votes counts: B=40 C=23 D=22 A=15 so A is eliminated. Round 3 votes counts: B=43 D=30 C=27 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:220 A:213 B:200 D:188 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 10 18 B -4 0 0 2 2 C 6 0 0 14 20 D -10 -2 -14 0 2 E -18 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 10 18 B -4 0 0 2 2 C 6 0 0 14 20 D -10 -2 -14 0 2 E -18 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 10 18 B -4 0 0 2 2 C 6 0 0 14 20 D -10 -2 -14 0 2 E -18 -2 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7074: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) D C A B E (6) B E C A D (6) D B C E A (4) D A C E B (4) C B E A D (4) C B D E A (4) A E B C D (4) E A B D C (3) D A E B C (3) C D A B E (3) C A D E B (3) B E D A C (3) A E C B D (3) A D C E B (3) A C D E B (3) E B A D C (2) D C B A E (2) D B C A E (2) C D B A E (2) C A E B D (2) C A B E D (2) B C E A D (2) B C D E A (2) A E D B C (2) A C E B D (2) E A D B C (1) E A B C D (1) D E A B C (1) D C A E B (1) D B E A C (1) C D B E A (1) C D A E B (1) C A D B E (1) B E D C A (1) B E C D A (1) B D E C A (1) B D E A C (1) B C E D A (1) A E D C B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 -20 -6 4 B -4 0 -12 -8 16 C 20 12 0 -4 26 D 6 8 4 0 16 E -4 -16 -26 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -20 -6 4 B -4 0 -12 -8 16 C 20 12 0 -4 26 D 6 8 4 0 16 E -4 -16 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=23 A=20 B=18 E=7 so E is eliminated. Round 2 votes counts: D=32 A=25 C=23 B=20 so B is eliminated. Round 3 votes counts: D=38 C=35 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:227 D:217 B:196 A:191 E:169 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -20 -6 4 B -4 0 -12 -8 16 C 20 12 0 -4 26 D 6 8 4 0 16 E -4 -16 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -6 4 B -4 0 -12 -8 16 C 20 12 0 -4 26 D 6 8 4 0 16 E -4 -16 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -6 4 B -4 0 -12 -8 16 C 20 12 0 -4 26 D 6 8 4 0 16 E -4 -16 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7075: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (18) B E C D A (13) A D C B E (13) E B C A D (7) D A C B E (6) A D C E B (6) A D B E C (6) E B A C D (4) A E D B C (3) A D E B C (3) E B A D C (2) A D E C B (2) E D A B C (1) E A B D C (1) D C A B E (1) D A C E B (1) C E B D A (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A C D (1) A E B D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 4 4 -6 B 6 0 26 6 -2 C -4 -26 0 4 -28 D -4 -6 -4 0 -10 E 6 2 28 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 4 -6 B 6 0 26 6 -2 C -4 -26 0 4 -28 D -4 -6 -4 0 -10 E 6 2 28 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999962022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=33 B=15 D=8 C=8 so D is eliminated. Round 2 votes counts: A=43 E=33 B=15 C=9 so C is eliminated. Round 3 votes counts: A=45 E=35 B=20 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:218 A:198 D:188 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 4 -6 B 6 0 26 6 -2 C -4 -26 0 4 -28 D -4 -6 -4 0 -10 E 6 2 28 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999962022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 4 -6 B 6 0 26 6 -2 C -4 -26 0 4 -28 D -4 -6 -4 0 -10 E 6 2 28 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999962022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 4 -6 B 6 0 26 6 -2 C -4 -26 0 4 -28 D -4 -6 -4 0 -10 E 6 2 28 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999962022 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7076: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) C B E A D (5) C A B E D (5) B E D C A (4) B E C D A (4) A D E C B (4) A C D E B (4) E B D C A (3) E B D A C (3) E B C A D (3) D B C E A (3) D A C B E (3) C B E D A (3) A D C E B (3) A C E D B (3) A C D B E (3) E B C D A (2) E B A D C (2) D E A B C (2) D A E C B (2) D A C E B (2) C B A E D (2) C B A D E (2) B D C E A (2) B C E D A (2) A E D B C (2) A D E B C (2) A C E B D (2) E D A B C (1) E A D B C (1) E A C B D (1) E A B D C (1) D E B A C (1) D C A B E (1) D B E C A (1) C E B A D (1) C D B A E (1) C D A B E (1) C B D E A (1) C B D A E (1) C A B D E (1) B C D E A (1) A E D C B (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 6 -10 12 6 B -6 0 -20 4 4 C 10 20 0 12 16 D -12 -4 -12 0 -6 E -6 -4 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 12 6 B -6 0 -20 4 4 C 10 20 0 12 16 D -12 -4 -12 0 -6 E -6 -4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 E=17 D=15 B=13 so B is eliminated. Round 2 votes counts: C=32 A=26 E=25 D=17 so D is eliminated. Round 3 votes counts: C=38 A=33 E=29 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 A:207 B:191 E:190 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 12 6 B -6 0 -20 4 4 C 10 20 0 12 16 D -12 -4 -12 0 -6 E -6 -4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 12 6 B -6 0 -20 4 4 C 10 20 0 12 16 D -12 -4 -12 0 -6 E -6 -4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 12 6 B -6 0 -20 4 4 C 10 20 0 12 16 D -12 -4 -12 0 -6 E -6 -4 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7077: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) E A B C D (7) A E B D C (7) C D B E A (6) D C B E A (5) C D E B A (5) B A E C D (5) A E B C D (5) A B E D C (5) D A B C E (4) E B A C D (3) C E D B A (3) B E C A D (3) E A C D B (2) E A C B D (2) D C E B A (2) D C E A B (2) D C A B E (2) A B E C D (2) A B D E C (2) E D C A B (1) E D A C B (1) E C D A B (1) E C B A D (1) E C A D B (1) E B C A D (1) E A D C B (1) D C A E B (1) D B A C E (1) D A C E B (1) D A C B E (1) C E D A B (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B A D C E (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 0 4 2 -2 B 0 0 6 0 2 C -4 -6 0 6 -2 D -2 0 -6 0 -12 E 2 -2 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307535 B: 0.692465 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.57408541007 Cumulative probabilities = A: 0.307535 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 2 -2 B 0 0 6 0 2 C -4 -6 0 6 -2 D -2 0 -6 0 -12 E 2 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499506 B: 0.500494 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000488236 Cumulative probabilities = A: 0.499506 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=22 E=21 C=15 B=15 so C is eliminated. Round 2 votes counts: D=38 E=25 A=22 B=15 so B is eliminated. Round 3 votes counts: D=41 E=30 A=29 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 B:204 A:202 C:197 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 2 -2 B 0 0 6 0 2 C -4 -6 0 6 -2 D -2 0 -6 0 -12 E 2 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499506 B: 0.500494 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000488236 Cumulative probabilities = A: 0.499506 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 2 -2 B 0 0 6 0 2 C -4 -6 0 6 -2 D -2 0 -6 0 -12 E 2 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499506 B: 0.500494 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000488236 Cumulative probabilities = A: 0.499506 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 2 -2 B 0 0 6 0 2 C -4 -6 0 6 -2 D -2 0 -6 0 -12 E 2 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499506 B: 0.500494 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000488236 Cumulative probabilities = A: 0.499506 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7078: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) E B C A D (5) D B A E C (5) D C A B E (4) D A C E B (4) C E B A D (4) C D A E B (4) B D E A C (4) E A C B D (3) D B A C E (3) D A E B C (3) C B E D A (3) C A E D B (3) C A D E B (3) B E A C D (3) B D C E A (3) A E D C B (3) A D C E B (3) E C A B D (2) E A B C D (2) D A E C B (2) D A B E C (2) C E A D B (2) C E A B D (2) C B E A D (2) C B D E A (2) B E C D A (2) B D A E C (2) B C E D A (2) A D E C B (2) D B C E A (1) D B C A E (1) D A C B E (1) C D B A E (1) B E A D C (1) B D E C A (1) B C D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 -12 -2 -6 B 8 0 0 4 4 C 12 0 0 8 -2 D 2 -4 -8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.702156 C: 0.297844 D: 0.000000 E: 0.000000 Sum of squares = 0.581734454185 Cumulative probabilities = A: 0.000000 B: 0.702156 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -2 -6 B 8 0 0 4 4 C 12 0 0 8 -2 D 2 -4 -8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 C=26 E=12 A=9 so A is eliminated. Round 2 votes counts: D=31 B=27 C=26 E=16 so E is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:208 E:200 D:197 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -12 -2 -6 B 8 0 0 4 4 C 12 0 0 8 -2 D 2 -4 -8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -2 -6 B 8 0 0 4 4 C 12 0 0 8 -2 D 2 -4 -8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -2 -6 B 8 0 0 4 4 C 12 0 0 8 -2 D 2 -4 -8 0 4 E 6 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7079: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (14) B A E D C (9) A D C B E (6) A B D C E (6) D C A E B (5) D A C E B (5) D C E A B (4) C D E A B (4) A D C E B (4) A B E D C (4) E D C A B (3) E B C D A (3) B E C D A (3) B E C A D (3) B E A C D (3) B A E C D (3) E C B D A (2) C D E B A (2) B C A D E (2) B A D C E (2) B A C D E (2) A D B C E (2) E D C B A (1) E D B C A (1) E B C A D (1) D E A C B (1) C E D B A (1) C B E D A (1) B A C E D (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 0 0 6 B 8 0 -8 -8 -4 C 0 8 0 -8 -4 D 0 8 8 0 -8 E -6 4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307692 B: 0.230769 C: 0.076923 D: 0.076923 E: 0.307692 Sum of squares = 0.254437869841 Cumulative probabilities = A: 0.307692 B: 0.538462 C: 0.615385 D: 0.692308 E: 1.000000 A B C D E A 0 -8 0 0 6 B 8 0 -8 -8 -4 C 0 8 0 -8 -4 D 0 8 8 0 -8 E -6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.230769 C: 0.076923 D: 0.076923 E: 0.307692 Sum of squares = 0.254437869824 Cumulative probabilities = A: 0.307692 B: 0.538462 C: 0.615385 D: 0.692308 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 A=24 D=15 C=8 so C is eliminated. Round 2 votes counts: B=29 E=26 A=24 D=21 so D is eliminated. Round 3 votes counts: E=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:205 D:204 A:199 C:198 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 0 6 B 8 0 -8 -8 -4 C 0 8 0 -8 -4 D 0 8 8 0 -8 E -6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.230769 C: 0.076923 D: 0.076923 E: 0.307692 Sum of squares = 0.254437869824 Cumulative probabilities = A: 0.307692 B: 0.538462 C: 0.615385 D: 0.692308 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 0 6 B 8 0 -8 -8 -4 C 0 8 0 -8 -4 D 0 8 8 0 -8 E -6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.230769 C: 0.076923 D: 0.076923 E: 0.307692 Sum of squares = 0.254437869824 Cumulative probabilities = A: 0.307692 B: 0.538462 C: 0.615385 D: 0.692308 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 0 6 B 8 0 -8 -8 -4 C 0 8 0 -8 -4 D 0 8 8 0 -8 E -6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.307692 B: 0.230769 C: 0.076923 D: 0.076923 E: 0.307692 Sum of squares = 0.254437869824 Cumulative probabilities = A: 0.307692 B: 0.538462 C: 0.615385 D: 0.692308 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7080: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) A D B C E (11) A D B E C (8) B D A E C (7) E C B D A (6) C E A D B (5) E C A B D (4) E B C D A (4) E A C D B (4) D B A C E (3) E C B A D (2) E B D C A (2) D A B C E (2) C E B A D (2) C D A B E (2) C B E D A (2) C B D E A (2) C A E D B (2) B D E A C (2) B D A C E (2) A E D C B (2) A C E D B (2) A C D B E (2) E B D A C (1) E A D B C (1) E A B D C (1) C E A B D (1) C B D A E (1) C A D B E (1) B E D C A (1) B D C E A (1) B D C A E (1) B C D E A (1) A D C B E (1) Total count = 100 A B C D E A 0 -2 -2 -2 -6 B 2 0 -4 8 0 C 2 4 0 8 10 D 2 -8 -8 0 -6 E 6 0 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -2 -6 B 2 0 -4 8 0 C 2 4 0 8 10 D 2 -8 -8 0 -6 E 6 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 E=25 B=15 D=5 so D is eliminated. Round 2 votes counts: C=29 A=28 E=25 B=18 so B is eliminated. Round 3 votes counts: A=40 C=32 E=28 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:203 E:201 A:194 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -2 -6 B 2 0 -4 8 0 C 2 4 0 8 10 D 2 -8 -8 0 -6 E 6 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -2 -6 B 2 0 -4 8 0 C 2 4 0 8 10 D 2 -8 -8 0 -6 E 6 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -2 -6 B 2 0 -4 8 0 C 2 4 0 8 10 D 2 -8 -8 0 -6 E 6 0 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7081: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) C E B A D (8) B A C E D (7) E C D B A (6) E C B D A (6) D A E C B (6) B C E A D (6) E D C B A (4) C E A D B (4) B E C D A (4) D E C A B (3) B A D C E (3) E D C A B (2) D A E B C (2) C E B D A (2) C B E A D (2) C A B E D (2) B D E C A (2) B D A E C (2) B A D E C (2) A B D C E (2) A B C E D (2) E D B C A (1) D E A C B (1) D B A E C (1) D A B E C (1) C E D A B (1) C A E D B (1) B E D C A (1) B D E A C (1) B C A E D (1) B A C D E (1) A D B E C (1) A D B C E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 -30 -10 -26 B 20 0 -16 10 -14 C 30 16 0 28 -10 D 10 -10 -28 0 -38 E 26 14 10 38 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -30 -10 -26 B 20 0 -16 10 -14 C 30 16 0 28 -10 D 10 -10 -28 0 -38 E 26 14 10 38 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 C=20 D=14 A=8 so A is eliminated. Round 2 votes counts: B=35 E=28 C=21 D=16 so D is eliminated. Round 3 votes counts: E=40 B=39 C=21 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:244 C:232 B:200 D:167 A:157 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -30 -10 -26 B 20 0 -16 10 -14 C 30 16 0 28 -10 D 10 -10 -28 0 -38 E 26 14 10 38 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -30 -10 -26 B 20 0 -16 10 -14 C 30 16 0 28 -10 D 10 -10 -28 0 -38 E 26 14 10 38 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -30 -10 -26 B 20 0 -16 10 -14 C 30 16 0 28 -10 D 10 -10 -28 0 -38 E 26 14 10 38 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999179 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7082: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) C B E D A (10) E B D C A (8) C B E A D (6) D A E B C (5) B C E D A (5) C A B D E (4) B E C D A (4) A D E C B (4) A C D B E (4) D E A B C (3) C B A E D (3) A E D B C (3) A D C E B (3) E D B A C (2) E D A B C (2) C B D E A (2) C B D A E (2) A C B E D (2) A C B D E (2) E B D A C (1) E B C A D (1) E A D B C (1) D E B A C (1) D B E C A (1) D A E C B (1) D A C E B (1) C B A D E (1) C A B E D (1) B E D C A (1) B E C A D (1) B E A C D (1) Total count = 100 A B C D E A 0 0 0 2 0 B 0 0 8 10 0 C 0 -8 0 -2 -8 D -2 -10 2 0 -4 E 0 0 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.517611 B: 0.176348 C: 0.000000 D: 0.000000 E: 0.306041 Sum of squares = 0.392680831516 Cumulative probabilities = A: 0.517611 B: 0.693959 C: 0.693959 D: 0.693959 E: 1.000000 A B C D E A 0 0 0 2 0 B 0 0 8 10 0 C 0 -8 0 -2 -8 D -2 -10 2 0 -4 E 0 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=29 E=15 D=12 B=12 so D is eliminated. Round 2 votes counts: A=39 C=29 E=19 B=13 so B is eliminated. Round 3 votes counts: A=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:209 E:206 A:201 D:193 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 2 0 B 0 0 8 10 0 C 0 -8 0 -2 -8 D -2 -10 2 0 -4 E 0 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 0 B 0 0 8 10 0 C 0 -8 0 -2 -8 D -2 -10 2 0 -4 E 0 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 0 B 0 0 8 10 0 C 0 -8 0 -2 -8 D -2 -10 2 0 -4 E 0 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7083: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) B C D A E (9) E A C D B (7) B C A D E (7) E B A C D (5) D E A C B (5) A E C D B (5) E D B C A (4) E D A C B (4) D A C E B (3) A C B D E (3) E A D C B (2) D E C A B (2) D E B C A (2) D C B A E (2) C B D A E (2) C A B D E (2) B E D C A (2) B D C A E (2) B A C E D (2) A C E D B (2) E D B A C (1) E B D C A (1) E B D A C (1) E A B C D (1) D E C B A (1) D C E B A (1) D C A E B (1) D B C E A (1) D B C A E (1) D A E C B (1) C B A D E (1) B E C A D (1) B D C E A (1) B C E A D (1) B C A E D (1) B A E C D (1) A E D C B (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 10 6 -4 B 0 0 -12 8 -20 C -10 12 0 22 -14 D -6 -8 -22 0 -6 E 4 20 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 10 6 -4 B 0 0 -12 8 -20 C -10 12 0 22 -14 D -6 -8 -22 0 -6 E 4 20 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=27 D=20 A=13 C=5 so C is eliminated. Round 2 votes counts: E=35 B=30 D=20 A=15 so A is eliminated. Round 3 votes counts: E=44 B=36 D=20 so D is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:206 C:205 B:188 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 6 -4 B 0 0 -12 8 -20 C -10 12 0 22 -14 D -6 -8 -22 0 -6 E 4 20 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 6 -4 B 0 0 -12 8 -20 C -10 12 0 22 -14 D -6 -8 -22 0 -6 E 4 20 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 6 -4 B 0 0 -12 8 -20 C -10 12 0 22 -14 D -6 -8 -22 0 -6 E 4 20 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7084: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) D A C E B (10) E C A D B (9) B E C A D (9) A D C E B (8) A C D E B (6) E C A B D (5) D A C B E (5) E D C A B (4) B D E A C (4) D A B C E (3) C A E D B (3) B E D C A (3) E C B A D (2) E B C A D (2) C E A D B (2) C A E B D (2) B D E C A (2) B D A E C (2) A C E D B (2) D B A E C (1) D A E C B (1) C E A B D (1) B E C D A (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 24 10 6 12 B -24 0 -22 -32 -14 C -10 22 0 -8 10 D -6 32 8 0 10 E -12 14 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 10 6 12 B -24 0 -22 -32 -14 C -10 22 0 -8 10 D -6 32 8 0 10 E -12 14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 B=22 A=17 C=8 so C is eliminated. Round 2 votes counts: D=31 E=25 B=22 A=22 so B is eliminated. Round 3 votes counts: D=39 E=38 A=23 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:226 D:222 C:207 E:191 B:154 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 10 6 12 B -24 0 -22 -32 -14 C -10 22 0 -8 10 D -6 32 8 0 10 E -12 14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 10 6 12 B -24 0 -22 -32 -14 C -10 22 0 -8 10 D -6 32 8 0 10 E -12 14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 10 6 12 B -24 0 -22 -32 -14 C -10 22 0 -8 10 D -6 32 8 0 10 E -12 14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7085: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D E C B A (6) D A C E B (6) D A C B E (5) B E C A D (4) A C B E D (4) E D B C A (3) E C B A D (3) D E C A B (3) D E B A C (3) D C E A B (3) D A E C B (3) C E D B A (3) C E B A D (3) B A C E D (3) A D C B E (3) A B C E D (3) E C B D A (2) E B C D A (2) D E B C A (2) D A B C E (2) C B A E D (2) C A E B D (2) B E A C D (2) A C D B E (2) A C B D E (2) A B D C E (2) E D C B A (1) E D B A C (1) E C D B A (1) E B D A C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E A B D (1) C A B E D (1) B E D A C (1) B C E A D (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -4 -2 -12 B 10 0 -12 -2 -18 C 4 12 0 0 -2 D 2 2 0 0 -8 E 12 18 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 -2 -12 B 10 0 -12 -2 -18 C 4 12 0 0 -2 D 2 2 0 0 -8 E 12 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=23 A=17 C=12 B=12 so C is eliminated. Round 2 votes counts: D=36 E=30 A=20 B=14 so B is eliminated. Round 3 votes counts: E=38 D=36 A=26 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:207 D:198 B:189 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 -2 -12 B 10 0 -12 -2 -18 C 4 12 0 0 -2 D 2 2 0 0 -8 E 12 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -2 -12 B 10 0 -12 -2 -18 C 4 12 0 0 -2 D 2 2 0 0 -8 E 12 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -2 -12 B 10 0 -12 -2 -18 C 4 12 0 0 -2 D 2 2 0 0 -8 E 12 18 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999982498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7086: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) C E B D A (6) A B D C E (6) E C D B A (5) C B E A D (5) B E C D A (5) A D B C E (5) A B C E D (5) D E C A B (4) D A E C B (4) C E B A D (4) B A C E D (4) A D B E C (4) A C B E D (3) A B C D E (3) E D C B A (2) E C B D A (2) E B C D A (2) D E A C B (2) C E A B D (2) C A B E D (2) B C E A D (2) E D B C A (1) D E C B A (1) D E B C A (1) D E B A C (1) D B E A C (1) D A B E C (1) C B E D A (1) C B A E D (1) C A E B D (1) B E D C A (1) B C E D A (1) B A D E C (1) A D C E B (1) A D C B E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 2 4 2 B -6 0 4 16 6 C -2 -4 0 10 8 D -4 -16 -10 0 -12 E -2 -6 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 4 2 B -6 0 4 16 6 C -2 -4 0 10 8 D -4 -16 -10 0 -12 E -2 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 C=22 B=14 E=12 so E is eliminated. Round 2 votes counts: A=30 C=29 D=25 B=16 so B is eliminated. Round 3 votes counts: C=39 A=35 D=26 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:210 A:207 C:206 E:198 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 4 2 B -6 0 4 16 6 C -2 -4 0 10 8 D -4 -16 -10 0 -12 E -2 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 4 2 B -6 0 4 16 6 C -2 -4 0 10 8 D -4 -16 -10 0 -12 E -2 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 4 2 B -6 0 4 16 6 C -2 -4 0 10 8 D -4 -16 -10 0 -12 E -2 -6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7087: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) B E C A D (10) E B C A D (9) D A C B E (9) E B A C D (5) E A B C D (5) E A B D C (4) D A C E B (4) C B D E A (4) B C E D A (4) A D E C B (4) C B E D A (3) A E D B C (3) E B A D C (2) D C B A E (2) C D B A E (2) A E D C B (2) E C B A D (1) E C A B D (1) E A D C B (1) E A C D B (1) D C A B E (1) D B C A E (1) D A B E C (1) C E B A D (1) C B E A D (1) C B D A E (1) B E C D A (1) B C D E A (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 14 32 -8 B -4 0 24 8 -16 C -14 -24 0 0 -32 D -32 -8 0 0 -10 E 8 16 32 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999343 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 14 32 -8 B -4 0 24 8 -16 C -14 -24 0 0 -32 D -32 -8 0 0 -10 E 8 16 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=25 D=18 B=16 C=12 so C is eliminated. Round 2 votes counts: E=30 B=25 A=25 D=20 so D is eliminated. Round 3 votes counts: A=40 E=30 B=30 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:233 A:221 B:206 D:175 C:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 14 32 -8 B -4 0 24 8 -16 C -14 -24 0 0 -32 D -32 -8 0 0 -10 E 8 16 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 32 -8 B -4 0 24 8 -16 C -14 -24 0 0 -32 D -32 -8 0 0 -10 E 8 16 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 32 -8 B -4 0 24 8 -16 C -14 -24 0 0 -32 D -32 -8 0 0 -10 E 8 16 32 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7088: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (13) D E C B A (10) C A B D E (10) A C B E D (8) D C E A B (5) A B E C D (5) D C A E B (4) B E A C D (4) A B C E D (4) D E C A B (3) D E B C A (3) C A D B E (3) C A B E D (3) A E B C D (3) E B A D C (2) D E A C B (2) D C E B A (2) C D B A E (2) C D A B E (2) C B A D E (2) E D B C A (1) E D A B C (1) D E B A C (1) B E D C A (1) B C E A D (1) B C A E D (1) B A C E D (1) A E B D C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -6 0 2 B -12 0 -16 -6 -4 C 6 16 0 2 0 D 0 6 -2 0 0 E -2 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.536623 D: 0.000000 E: 0.463377 Sum of squares = 0.502682443737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.536623 D: 0.536623 E: 1.000000 A B C D E A 0 12 -6 0 2 B -12 0 -16 -6 -4 C 6 16 0 2 0 D 0 6 -2 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 C=22 E=17 B=8 so B is eliminated. Round 2 votes counts: D=30 C=24 A=24 E=22 so E is eliminated. Round 3 votes counts: D=46 A=30 C=24 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:204 D:202 E:201 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 0 2 B -12 0 -16 -6 -4 C 6 16 0 2 0 D 0 6 -2 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 0 2 B -12 0 -16 -6 -4 C 6 16 0 2 0 D 0 6 -2 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 0 2 B -12 0 -16 -6 -4 C 6 16 0 2 0 D 0 6 -2 0 0 E -2 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999628 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7089: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (10) D A E B C (8) E C B D A (6) B C E D A (6) A D B E C (6) A D E C B (5) A B C D E (5) C B E A D (4) C B A E D (4) B C E A D (4) A D E B C (4) A D B C E (4) E D C B A (3) E C D B A (3) D E B C A (2) D B E C A (2) D A E C B (2) D A B E C (2) B E C D A (2) E D C A B (1) E C B A D (1) D A B C E (1) C E B A D (1) B D C A E (1) B C A D E (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -2 2 10 B 0 0 16 -6 6 C 2 -16 0 -2 -14 D -2 6 2 0 6 E -10 -6 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.757390 B: 0.242610 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.632499121909 Cumulative probabilities = A: 0.757390 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 2 10 B 0 0 16 -6 6 C 2 -16 0 -2 -14 D -2 6 2 0 6 E -10 -6 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000289403 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 B=24 E=14 C=9 so C is eliminated. Round 2 votes counts: B=32 D=27 A=26 E=15 so E is eliminated. Round 3 votes counts: B=40 D=34 A=26 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:208 D:206 A:205 E:196 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 2 10 B 0 0 16 -6 6 C 2 -16 0 -2 -14 D -2 6 2 0 6 E -10 -6 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000289403 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 10 B 0 0 16 -6 6 C 2 -16 0 -2 -14 D -2 6 2 0 6 E -10 -6 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000289403 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 10 B 0 0 16 -6 6 C 2 -16 0 -2 -14 D -2 6 2 0 6 E -10 -6 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000289403 Cumulative probabilities = A: 0.750000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7090: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) A B C D E (9) B A E D C (7) B A E C D (6) A B E C D (6) E C D A B (4) E B A C D (4) B D A E C (4) B A D C E (4) E C A D B (3) C D A E B (3) B A D E C (3) E B D A C (2) E B A D C (2) D C E A B (2) D C B A E (2) D C A B E (2) C E D A B (2) C D E A B (2) C D A B E (2) B E A D C (2) B D A C E (2) A B D C E (2) A B C E D (2) E C D B A (1) E C B A D (1) E C A B D (1) E A C B D (1) D C E B A (1) D C A E B (1) D B A C E (1) D A C B E (1) D A B C E (1) C E A D B (1) C A E D B (1) C A D B E (1) B E A C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 22 16 22 B 4 0 16 20 18 C -22 -16 0 4 -14 D -16 -20 -4 0 -14 E -22 -18 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 22 16 22 B 4 0 16 20 18 C -22 -16 0 4 -14 D -16 -20 -4 0 -14 E -22 -18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=28 A=20 C=12 D=11 so D is eliminated. Round 2 votes counts: B=30 E=28 A=22 C=20 so C is eliminated. Round 3 votes counts: E=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=61 E=39 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:229 A:228 E:194 C:176 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 22 16 22 B 4 0 16 20 18 C -22 -16 0 4 -14 D -16 -20 -4 0 -14 E -22 -18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 16 22 B 4 0 16 20 18 C -22 -16 0 4 -14 D -16 -20 -4 0 -14 E -22 -18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 16 22 B 4 0 16 20 18 C -22 -16 0 4 -14 D -16 -20 -4 0 -14 E -22 -18 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7091: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (10) A D C B E (9) E B C D A (6) A D B C E (6) A C D E B (6) A B D E C (6) E C B D A (4) B E D C A (4) A C E D B (4) A C E B D (4) D B E A C (3) C E A B D (3) B E D A C (3) B D E C A (3) A D B E C (3) D A B E C (2) C E D A B (2) C A E D B (2) C A E B D (2) A C B D E (2) A B D C E (2) E B D C A (1) E B C A D (1) D C E A B (1) D C B A E (1) D B E C A (1) D A C B E (1) D A B C E (1) C D A E B (1) B D E A C (1) B D A E C (1) B A E C D (1) A D C E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 20 16 8 14 B -20 0 -8 10 4 C -16 8 0 -2 18 D -8 -10 2 0 4 E -14 -4 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 16 8 14 B -20 0 -8 10 4 C -16 8 0 -2 18 D -8 -10 2 0 4 E -14 -4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 C=20 B=13 E=12 D=10 so D is eliminated. Round 2 votes counts: A=49 C=22 B=17 E=12 so E is eliminated. Round 3 votes counts: A=49 C=26 B=25 so B is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:204 D:194 B:193 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 16 8 14 B -20 0 -8 10 4 C -16 8 0 -2 18 D -8 -10 2 0 4 E -14 -4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 8 14 B -20 0 -8 10 4 C -16 8 0 -2 18 D -8 -10 2 0 4 E -14 -4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 8 14 B -20 0 -8 10 4 C -16 8 0 -2 18 D -8 -10 2 0 4 E -14 -4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7092: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (6) B D E C A (6) E C A D B (5) D B A C E (5) B C A E D (5) D B A E C (4) D A E C B (4) B E D C A (4) B D C E A (4) B D C A E (4) E C A B D (3) C A E B D (3) B D E A C (3) A D C E B (3) A C E D B (3) E D B A C (2) E D A B C (2) E C B A D (2) D E A B C (2) D B E A C (2) D A B C E (2) C E A B D (2) C B A E D (2) B C E D A (2) B C D A E (2) B C A D E (2) A D E C B (2) E D A C B (1) E B C D A (1) E A D C B (1) D E B A C (1) D A E B C (1) D A C E B (1) D A C B E (1) D A B E C (1) C E B A D (1) C B A D E (1) C A D B E (1) B E C A D (1) B D A C E (1) A E C D B (1) Total count = 100 A B C D E A 0 -10 -2 -12 -2 B 10 0 14 -2 6 C 2 -14 0 -14 -10 D 12 2 14 0 6 E 2 -6 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -12 -2 B 10 0 14 -2 6 C 2 -14 0 -14 -10 D 12 2 14 0 6 E 2 -6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=24 E=23 C=10 A=9 so A is eliminated. Round 2 votes counts: B=34 D=29 E=24 C=13 so C is eliminated. Round 3 votes counts: B=37 E=33 D=30 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:217 B:214 E:200 A:187 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 -12 -2 B 10 0 14 -2 6 C 2 -14 0 -14 -10 D 12 2 14 0 6 E 2 -6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -12 -2 B 10 0 14 -2 6 C 2 -14 0 -14 -10 D 12 2 14 0 6 E 2 -6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -12 -2 B 10 0 14 -2 6 C 2 -14 0 -14 -10 D 12 2 14 0 6 E 2 -6 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7093: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (5) D E B A C (4) D A C E B (4) D A C B E (4) C A B E D (4) C A B D E (4) A D C E B (4) E B D C A (3) E A D C B (3) D E B C A (3) C B A E D (3) C B A D E (3) A C E B D (3) A C B E D (3) E A C D B (2) E A C B D (2) E A B C D (2) D C B A E (2) D B E C A (2) D B C A E (2) D A E C B (2) C A D B E (2) B E D C A (2) B E C D A (2) B D E C A (2) B C D E A (2) B C D A E (2) A C E D B (2) A C B D E (2) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) E B D A C (1) E B C A D (1) E B A D C (1) E B A C D (1) E A D B C (1) D B C E A (1) B E C A D (1) B D C E A (1) B C E D A (1) B C E A D (1) B C A E D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 12 8 2 4 B -12 0 -10 -2 -4 C -8 10 0 -4 6 D -2 2 4 0 6 E -4 4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 2 4 B -12 0 -10 -2 -4 C -8 10 0 -4 6 D -2 2 4 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 A=19 C=16 B=15 so B is eliminated. Round 2 votes counts: D=32 E=26 C=23 A=19 so A is eliminated. Round 3 votes counts: D=36 C=35 E=29 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:213 D:205 C:202 E:194 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 2 4 B -12 0 -10 -2 -4 C -8 10 0 -4 6 D -2 2 4 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 2 4 B -12 0 -10 -2 -4 C -8 10 0 -4 6 D -2 2 4 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 2 4 B -12 0 -10 -2 -4 C -8 10 0 -4 6 D -2 2 4 0 6 E -4 4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7094: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (7) E A D C B (6) E A C D B (5) E A C B D (5) D B E C A (5) D B C A E (4) D A E B C (4) C B A D E (4) B D C E A (4) B C D A E (4) A C E B D (4) E D A B C (3) D E A B C (3) C B E A D (3) C A E B D (3) A E C D B (3) A E C B D (3) E D B C A (2) E D A C B (2) E C A B D (2) D E B C A (2) D B E A C (2) C A B E D (2) B C D E A (2) B C A D E (2) E D B A C (1) D E B A C (1) D B C E A (1) D B A E C (1) D A B E C (1) B E C D A (1) B C E D A (1) A E D C B (1) A D E C B (1) A D C B E (1) A D B C E (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 2 -2 -2 B -6 0 4 -2 -4 C -2 -4 0 -6 -8 D 2 2 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -2 -2 B -6 0 4 -2 -4 C -2 -4 0 -6 -8 D 2 2 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 B=21 A=17 C=12 so C is eliminated. Round 2 votes counts: B=28 E=26 D=24 A=22 so A is eliminated. Round 3 votes counts: E=41 B=32 D=27 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:206 E:206 A:202 B:196 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -2 -2 B -6 0 4 -2 -4 C -2 -4 0 -6 -8 D 2 2 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -2 -2 B -6 0 4 -2 -4 C -2 -4 0 -6 -8 D 2 2 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -2 -2 B -6 0 4 -2 -4 C -2 -4 0 -6 -8 D 2 2 6 0 2 E 2 4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7095: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) D B A C E (6) D B E A C (5) E C A B D (4) E A C D B (4) B D E C A (4) A D E C B (4) A D C E B (4) A C E D B (4) D A C B E (3) C E A B D (3) C B A D E (3) C A E B D (3) A E C D B (3) E B D A C (2) E B C A D (2) E A D B C (2) D B C A E (2) D A B E C (2) D A B C E (2) C B E D A (2) C B D A E (2) C A B D E (2) B E D C A (2) B E C D A (2) B D C E A (2) B C E D A (2) A C E B D (2) E D A B C (1) E C B A D (1) E B D C A (1) E B C D A (1) E B A D C (1) E A C B D (1) D E B A C (1) D B E C A (1) D A E B C (1) C B A E D (1) C A B E D (1) B D C A E (1) B C D E A (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -2 6 -4 B 6 0 -10 8 -4 C 2 10 0 4 10 D -6 -8 -4 0 -4 E 4 4 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999434 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 6 -4 B 6 0 -10 8 -4 C 2 10 0 4 10 D -6 -8 -4 0 -4 E 4 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999018 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 E=20 A=18 B=15 so B is eliminated. Round 2 votes counts: D=30 C=28 E=24 A=18 so A is eliminated. Round 3 votes counts: D=39 C=34 E=27 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:201 B:200 A:197 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 6 -4 B 6 0 -10 8 -4 C 2 10 0 4 10 D -6 -8 -4 0 -4 E 4 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999018 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 6 -4 B 6 0 -10 8 -4 C 2 10 0 4 10 D -6 -8 -4 0 -4 E 4 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999018 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 6 -4 B 6 0 -10 8 -4 C 2 10 0 4 10 D -6 -8 -4 0 -4 E 4 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999018 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7096: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) B A D C E (6) A B D C E (6) C E A B D (5) E C D A B (4) D E B C A (4) D B A E C (4) B C E D A (4) A E D C B (4) A D B E C (4) A C E B D (4) E C B D A (3) E C A D B (3) C E B D A (3) E D C B A (2) E D C A B (2) D B E A C (2) D B A C E (2) C E B A D (2) C B A E D (2) B D C E A (2) B D A E C (2) A C E D B (2) A C B E D (2) A B C D E (2) E C D B A (1) E C B A D (1) E A C D B (1) D B E C A (1) D A B E C (1) C B E D A (1) C B E A D (1) C A E D B (1) B E C D A (1) B D E C A (1) B D C A E (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 -18 4 -2 12 B 18 0 12 24 16 C -4 -12 0 -6 18 D 2 -24 6 0 0 E -12 -16 -18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 4 -2 12 B 18 0 12 24 16 C -4 -12 0 -6 18 D 2 -24 6 0 0 E -12 -16 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 E=17 C=15 D=14 so D is eliminated. Round 2 votes counts: B=39 A=25 E=21 C=15 so C is eliminated. Round 3 votes counts: B=43 E=31 A=26 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:235 A:198 C:198 D:192 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 4 -2 12 B 18 0 12 24 16 C -4 -12 0 -6 18 D 2 -24 6 0 0 E -12 -16 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 -2 12 B 18 0 12 24 16 C -4 -12 0 -6 18 D 2 -24 6 0 0 E -12 -16 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 -2 12 B 18 0 12 24 16 C -4 -12 0 -6 18 D 2 -24 6 0 0 E -12 -16 -18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7097: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (14) E D A B C (10) E D A C B (7) C B D A E (6) D E A C B (5) D A E C B (5) B E A C D (5) B C E A D (5) B C A D E (5) D A C E B (4) B C A E D (4) B E C A D (3) E D C B A (2) E B C D A (2) D C E A B (2) B A C D E (2) A D E C B (2) A D C E B (2) A D C B E (2) E D C A B (1) E D B C A (1) E C B D A (1) E B D A C (1) E B A D C (1) C D B A E (1) C D A B E (1) C B E D A (1) B C E D A (1) A E D B C (1) A E B D C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 0 -2 4 B 10 0 -12 6 2 C 0 12 0 2 0 D 2 -6 -2 0 4 E -4 -2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.383851 B: 0.000000 C: 0.616149 D: 0.000000 E: 0.000000 Sum of squares = 0.526981245959 Cumulative probabilities = A: 0.383851 B: 0.383851 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -2 4 B 10 0 -12 6 2 C 0 12 0 2 0 D 2 -6 -2 0 4 E -4 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499717 B: 0.000000 C: 0.500283 D: 0.000000 E: 0.000000 Sum of squares = 0.500000160028 Cumulative probabilities = A: 0.499717 B: 0.499717 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=25 C=23 D=16 A=10 so A is eliminated. Round 2 votes counts: E=28 B=26 D=23 C=23 so D is eliminated. Round 3 votes counts: E=41 C=33 B=26 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:203 D:199 A:196 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 -2 4 B 10 0 -12 6 2 C 0 12 0 2 0 D 2 -6 -2 0 4 E -4 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499717 B: 0.000000 C: 0.500283 D: 0.000000 E: 0.000000 Sum of squares = 0.500000160028 Cumulative probabilities = A: 0.499717 B: 0.499717 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -2 4 B 10 0 -12 6 2 C 0 12 0 2 0 D 2 -6 -2 0 4 E -4 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499717 B: 0.000000 C: 0.500283 D: 0.000000 E: 0.000000 Sum of squares = 0.500000160028 Cumulative probabilities = A: 0.499717 B: 0.499717 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -2 4 B 10 0 -12 6 2 C 0 12 0 2 0 D 2 -6 -2 0 4 E -4 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499717 B: 0.000000 C: 0.500283 D: 0.000000 E: 0.000000 Sum of squares = 0.500000160028 Cumulative probabilities = A: 0.499717 B: 0.499717 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7098: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (14) B D C A E (11) A E D B C (6) E C D A B (5) C D B E A (5) A E D C B (5) E C A D B (4) A E B D C (4) A B D E C (4) E A C B D (3) D C A B E (3) D B C A E (3) C D E B A (3) E A B C D (2) D C B A E (2) C E D A B (2) C B D E A (2) B D A C E (2) B C D E A (2) A D B C E (2) A B D C E (2) E C B D A (1) E C A B D (1) E B A D C (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E A B (1) C A E D B (1) B E A D C (1) B D C E A (1) B A E D C (1) B A D E C (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 24 4 8 2 B -24 0 -8 -16 -8 C -4 8 0 -6 -8 D -8 16 6 0 -6 E -2 8 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999116 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 4 8 2 B -24 0 -8 -16 -8 C -4 8 0 -6 -8 D -8 16 6 0 -6 E -2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=24 B=20 C=15 D=10 so D is eliminated. Round 2 votes counts: E=31 A=26 B=23 C=20 so C is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:219 E:210 D:204 C:195 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 4 8 2 B -24 0 -8 -16 -8 C -4 8 0 -6 -8 D -8 16 6 0 -6 E -2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 4 8 2 B -24 0 -8 -16 -8 C -4 8 0 -6 -8 D -8 16 6 0 -6 E -2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 4 8 2 B -24 0 -8 -16 -8 C -4 8 0 -6 -8 D -8 16 6 0 -6 E -2 8 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999425 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7099: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (17) D A C E B (14) D A C B E (10) B E C A D (9) A C D E B (5) C A E D B (4) E C A B D (3) D B A C E (3) B E D C A (3) E D C A B (2) E C B A D (2) E C A D B (2) E B C D A (2) D E B A C (2) D A B C E (2) C E A B D (2) C A E B D (2) A D C B E (2) A C D B E (2) A C B E D (2) E D C B A (1) D B E A C (1) D B A E C (1) C A D E B (1) B E C D A (1) B D E A C (1) B C A E D (1) B A C E D (1) B A C D E (1) A D C E B (1) Total count = 100 A B C D E A 0 8 -4 14 4 B -8 0 -10 -6 -20 C 4 10 0 14 6 D -14 6 -14 0 -8 E -4 20 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 14 4 B -8 0 -10 -6 -20 C 4 10 0 14 6 D -14 6 -14 0 -8 E -4 20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=29 B=17 A=12 C=9 so C is eliminated. Round 2 votes counts: D=33 E=31 A=19 B=17 so B is eliminated. Round 3 votes counts: E=44 D=34 A=22 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:217 A:211 E:209 D:185 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 14 4 B -8 0 -10 -6 -20 C 4 10 0 14 6 D -14 6 -14 0 -8 E -4 20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 14 4 B -8 0 -10 -6 -20 C 4 10 0 14 6 D -14 6 -14 0 -8 E -4 20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 14 4 B -8 0 -10 -6 -20 C 4 10 0 14 6 D -14 6 -14 0 -8 E -4 20 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7100: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) C B D A E (9) A E D C B (7) C D B E A (6) B C D A E (6) B C A E D (6) A E B C D (5) B C D E A (4) E A D C B (3) D E A C B (3) C D B A E (3) E D A B C (2) E B A D C (2) E A B D C (2) E A B C D (2) D E C A B (2) D C E B A (2) D C E A B (2) C B A E D (2) B D C E A (2) B A E C D (2) A E D B C (2) A E B D C (2) E D A C B (1) E B D A C (1) E A D B C (1) D C B A E (1) D C A E B (1) D B E C A (1) D A E C B (1) C B D E A (1) B E C D A (1) B E C A D (1) B C E A D (1) B A C E D (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -22 -20 -16 2 B 22 0 -12 6 14 C 20 12 0 6 16 D 16 -6 -6 0 6 E -2 -14 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -20 -16 2 B 22 0 -12 6 14 C 20 12 0 6 16 D 16 -6 -6 0 6 E -2 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 D=22 C=21 A=19 E=14 so E is eliminated. Round 2 votes counts: B=27 A=27 D=25 C=21 so C is eliminated. Round 3 votes counts: B=39 D=34 A=27 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:227 B:215 D:205 E:181 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -20 -16 2 B 22 0 -12 6 14 C 20 12 0 6 16 D 16 -6 -6 0 6 E -2 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -20 -16 2 B 22 0 -12 6 14 C 20 12 0 6 16 D 16 -6 -6 0 6 E -2 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -20 -16 2 B 22 0 -12 6 14 C 20 12 0 6 16 D 16 -6 -6 0 6 E -2 -14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7101: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) D E C B A (9) C E B D A (7) A B C E D (6) D E A C B (5) C E D B A (5) A B E C D (5) D E C A B (4) D A E B C (4) B A C E D (4) E D C B A (3) C B E A D (3) A D B E C (3) A B E D C (3) A B D C E (3) D E A B C (2) D C E B A (2) D A E C B (2) D A C B E (2) C B E D A (2) B C E A D (2) B C A E D (2) A B D E C (2) E D A B C (1) E C B D A (1) D A B C E (1) C D E B A (1) C D A B E (1) B E C A D (1) A E D B C (1) A E B D C (1) A D E B C (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -22 -14 B 2 0 -14 -14 -16 C 4 14 0 0 -14 D 22 14 0 0 -12 E 14 16 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -22 -14 B 2 0 -14 -14 -16 C 4 14 0 0 -14 D 22 14 0 0 -12 E 14 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=27 C=19 E=14 B=9 so B is eliminated. Round 2 votes counts: D=31 A=31 C=23 E=15 so E is eliminated. Round 3 votes counts: D=35 C=34 A=31 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:228 D:212 C:202 A:179 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -22 -14 B 2 0 -14 -14 -16 C 4 14 0 0 -14 D 22 14 0 0 -12 E 14 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -22 -14 B 2 0 -14 -14 -16 C 4 14 0 0 -14 D 22 14 0 0 -12 E 14 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -22 -14 B 2 0 -14 -14 -16 C 4 14 0 0 -14 D 22 14 0 0 -12 E 14 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7102: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (11) A D B E C (10) C E B D A (9) D B A C E (5) C B D E A (5) E C B A D (4) E C A B D (4) E A C D B (4) B D A C E (4) E C A D B (3) E A C B D (3) D B C A E (3) C E A D B (3) C D B A E (3) B C D E A (3) A E D C B (3) D A B E C (2) D A B C E (2) C E D B A (2) C B E D A (2) B C D A E (2) A E D B C (2) A D E B C (2) E C B D A (1) E A B C D (1) D A C E B (1) D A C B E (1) C E D A B (1) C E B A D (1) B E A D C (1) B D A E C (1) A D E C B (1) Total count = 100 A B C D E A 0 -14 -14 -16 6 B 14 0 -2 4 10 C 14 2 0 2 16 D 16 -4 -2 0 12 E -6 -10 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 -16 6 B 14 0 -2 4 10 C 14 2 0 2 16 D 16 -4 -2 0 12 E -6 -10 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=22 E=20 A=18 D=14 so D is eliminated. Round 2 votes counts: B=30 C=26 A=24 E=20 so E is eliminated. Round 3 votes counts: C=38 A=32 B=30 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:213 D:211 A:181 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -14 -16 6 B 14 0 -2 4 10 C 14 2 0 2 16 D 16 -4 -2 0 12 E -6 -10 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -16 6 B 14 0 -2 4 10 C 14 2 0 2 16 D 16 -4 -2 0 12 E -6 -10 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -16 6 B 14 0 -2 4 10 C 14 2 0 2 16 D 16 -4 -2 0 12 E -6 -10 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989585 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7103: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) A E C B D (11) B D C E A (8) C E B D A (7) E C A B D (6) D B A C E (6) D A B C E (6) C B E D A (6) A D B C E (5) E C B D A (3) B C D E A (3) A E C D B (3) A D B E C (3) E C B A D (2) D B C A E (2) C E A B D (2) A D E C B (2) A D E B C (2) A B D C E (2) E C D B A (1) E A C D B (1) E A C B D (1) D E B C A (1) D B E C A (1) D A B E C (1) A E D C B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -8 -14 -8 B 4 0 4 6 12 C 8 -4 0 -4 22 D 14 -6 4 0 10 E 8 -12 -22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -14 -8 B 4 0 4 6 12 C 8 -4 0 -4 22 D 14 -6 4 0 10 E 8 -12 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992187 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=29 C=15 E=14 B=11 so B is eliminated. Round 2 votes counts: D=37 A=31 C=18 E=14 so E is eliminated. Round 3 votes counts: D=37 A=33 C=30 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:213 C:211 D:211 A:183 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -14 -8 B 4 0 4 6 12 C 8 -4 0 -4 22 D 14 -6 4 0 10 E 8 -12 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992187 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -14 -8 B 4 0 4 6 12 C 8 -4 0 -4 22 D 14 -6 4 0 10 E 8 -12 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992187 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -14 -8 B 4 0 4 6 12 C 8 -4 0 -4 22 D 14 -6 4 0 10 E 8 -12 -22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992187 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7104: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) A B D C E (8) E D C A B (7) A D E B C (6) D A E C B (5) B C A E D (5) E C B D A (4) D E C A B (4) C B E D A (4) B A D C E (4) A D E C B (4) A B E D C (4) E D A C B (2) E B C D A (2) D E A C B (2) D C E A B (2) C D E B A (2) C D B E A (2) B C D E A (2) B C D A E (2) B C A D E (2) B A C D E (2) A E D C B (2) A D B E C (2) A D B C E (2) A B D E C (2) E C D A B (1) E A D C B (1) E A B C D (1) D C A E B (1) D C A B E (1) C E D B A (1) B E C A D (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 14 -2 -4 8 B -14 0 -6 -10 -10 C 2 6 0 -18 -18 D 4 10 18 0 10 E -8 10 18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -4 8 B -14 0 -6 -10 -10 C 2 6 0 -18 -18 D 4 10 18 0 10 E -8 10 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=26 B=20 D=15 C=9 so C is eliminated. Round 2 votes counts: A=30 E=27 B=24 D=19 so D is eliminated. Round 3 votes counts: E=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:208 E:205 C:186 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -4 8 B -14 0 -6 -10 -10 C 2 6 0 -18 -18 D 4 10 18 0 10 E -8 10 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -4 8 B -14 0 -6 -10 -10 C 2 6 0 -18 -18 D 4 10 18 0 10 E -8 10 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -4 8 B -14 0 -6 -10 -10 C 2 6 0 -18 -18 D 4 10 18 0 10 E -8 10 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7105: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) E A C B D (7) B D A E C (7) A E B D C (7) E A C D B (6) B D C E A (5) D B A C E (4) C D B A E (4) A E C D B (4) D B A E C (3) C E A D B (3) C D B E A (3) B D C A E (3) B C D E A (3) E C A B D (2) D C B A E (2) C E B A D (2) C E A B D (2) C B D E A (2) A E D C B (2) E C A D B (1) E B A D C (1) E B A C D (1) E A B D C (1) E A B C D (1) D A C E B (1) C E D A B (1) C D E A B (1) C A D E B (1) B E D C A (1) B E D A C (1) B D E C A (1) B D E A C (1) B A E D C (1) B A D E C (1) A E D B C (1) A E B C D (1) A D E B C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 8 -4 8 B 10 0 10 4 4 C -8 -10 0 -8 -6 D 4 -4 8 0 4 E -8 -4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -4 8 B 10 0 10 4 4 C -8 -10 0 -8 -6 D 4 -4 8 0 4 E -8 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=20 D=19 C=19 A=18 so A is eliminated. Round 2 votes counts: E=35 B=25 D=20 C=20 so D is eliminated. Round 3 votes counts: B=41 E=36 C=23 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:206 A:201 E:195 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 -4 8 B 10 0 10 4 4 C -8 -10 0 -8 -6 D 4 -4 8 0 4 E -8 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -4 8 B 10 0 10 4 4 C -8 -10 0 -8 -6 D 4 -4 8 0 4 E -8 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -4 8 B 10 0 10 4 4 C -8 -10 0 -8 -6 D 4 -4 8 0 4 E -8 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7106: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (17) D A E B C (16) C B E A D (13) C B E D A (11) B E C D A (3) A E D B C (3) A D C E B (3) E B D A C (2) E B A C D (2) E A B C D (2) D E B A C (2) D C B E A (2) C D B E A (2) C B D E A (2) C A E B D (2) C A B E D (2) E B C A D (1) E A B D C (1) D C B A E (1) D B E C A (1) D B E A C (1) D A B E C (1) C D A B E (1) C A D B E (1) B E D A C (1) B E C A D (1) B C E D A (1) A E D C B (1) A E B D C (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 12 6 4 B -8 0 14 -6 -12 C -12 -14 0 -6 -14 D -6 6 6 0 0 E -4 12 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 6 4 B -8 0 14 -6 -12 C -12 -14 0 -6 -14 D -6 6 6 0 0 E -4 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=28 D=24 E=8 B=6 so B is eliminated. Round 2 votes counts: C=35 A=28 D=24 E=13 so E is eliminated. Round 3 votes counts: C=40 A=33 D=27 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:211 D:203 B:194 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 6 4 B -8 0 14 -6 -12 C -12 -14 0 -6 -14 D -6 6 6 0 0 E -4 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 6 4 B -8 0 14 -6 -12 C -12 -14 0 -6 -14 D -6 6 6 0 0 E -4 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 6 4 B -8 0 14 -6 -12 C -12 -14 0 -6 -14 D -6 6 6 0 0 E -4 12 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7107: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) C E A D B (8) A B E D C (8) B A D E C (7) C D E B A (6) C A E D B (5) B D A E C (4) A B D E C (4) D E B A C (3) C E D B A (3) C A E B D (3) C A B E D (3) A C B E D (3) A B E C D (3) E C A D B (2) D E B C A (2) D C E B A (2) D B E C A (2) D B E A C (2) C D B E A (2) B D C E A (2) B C D A E (2) E D B A C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E C B A (1) D B C E A (1) C B D E A (1) B D A C E (1) B C D E A (1) B A D C E (1) A E D C B (1) A E D B C (1) A E B D C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 12 -10 10 -2 B -12 0 -4 -8 -6 C 10 4 0 8 8 D -10 8 -8 0 -12 E 2 6 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 10 -2 B -12 0 -4 -8 -6 C 10 4 0 8 8 D -10 8 -8 0 -12 E 2 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 A=23 B=18 D=13 E=6 so E is eliminated. Round 2 votes counts: C=42 A=26 B=18 D=14 so D is eliminated. Round 3 votes counts: C=45 B=29 A=26 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:206 A:205 D:189 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 10 -2 B -12 0 -4 -8 -6 C 10 4 0 8 8 D -10 8 -8 0 -12 E 2 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 10 -2 B -12 0 -4 -8 -6 C 10 4 0 8 8 D -10 8 -8 0 -12 E 2 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 10 -2 B -12 0 -4 -8 -6 C 10 4 0 8 8 D -10 8 -8 0 -12 E 2 6 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7108: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) B E C D A (12) A D E C B (6) A D C E B (6) A D C B E (6) E B C D A (5) B C E D A (5) A D E B C (5) C D B E A (4) C D B A E (4) E B D A C (3) C A D B E (3) A C D B E (3) E B D C A (2) D E A B C (2) D A E B C (2) C B D A E (2) C B A E D (2) A E D B C (2) E B A D C (1) E B A C D (1) E A B D C (1) D C B E A (1) D C A B E (1) D A C B E (1) C D A B E (1) C B E A D (1) C B D E A (1) B E D C A (1) B C E A D (1) A E B D C (1) Total count = 100 A B C D E A 0 -20 -20 -22 -10 B 20 0 -12 6 26 C 20 12 0 18 12 D 22 -6 -18 0 -4 E 10 -26 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999646 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -20 -22 -10 B 20 0 -12 6 26 C 20 12 0 18 12 D 22 -6 -18 0 -4 E 10 -26 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=29 B=19 E=13 D=7 so D is eliminated. Round 2 votes counts: C=34 A=32 B=19 E=15 so E is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:231 B:220 D:197 E:188 A:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -20 -22 -10 B 20 0 -12 6 26 C 20 12 0 18 12 D 22 -6 -18 0 -4 E 10 -26 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -20 -22 -10 B 20 0 -12 6 26 C 20 12 0 18 12 D 22 -6 -18 0 -4 E 10 -26 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -20 -22 -10 B 20 0 -12 6 26 C 20 12 0 18 12 D 22 -6 -18 0 -4 E 10 -26 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7109: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) D B C A E (8) D C B A E (7) C E A B D (7) E A B C D (6) B D A E C (6) E C A B D (5) D B E A C (5) B A E D C (5) D B A C E (4) E B A D C (3) E A C B D (3) D C B E A (3) C E A D B (3) C D E A B (3) B E A D C (3) D C E B A (2) C A E B D (2) A E B C D (2) E D C B A (1) E D B A C (1) E B A C D (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E A B (1) D B E C A (1) C D A E B (1) B D E A C (1) B A E C D (1) B A D E C (1) B A D C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -32 8 -12 -4 B 32 0 20 -4 10 C -8 -20 0 -30 -14 D 12 4 30 0 12 E 4 -10 14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -32 8 -12 -4 B 32 0 20 -4 10 C -8 -20 0 -30 -14 D 12 4 30 0 12 E 4 -10 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 E=20 B=18 C=16 A=3 so A is eliminated. Round 2 votes counts: D=43 E=22 B=18 C=17 so C is eliminated. Round 3 votes counts: D=47 E=35 B=18 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:229 D:229 E:198 A:180 C:164 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -32 8 -12 -4 B 32 0 20 -4 10 C -8 -20 0 -30 -14 D 12 4 30 0 12 E 4 -10 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -32 8 -12 -4 B 32 0 20 -4 10 C -8 -20 0 -30 -14 D 12 4 30 0 12 E 4 -10 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -32 8 -12 -4 B 32 0 20 -4 10 C -8 -20 0 -30 -14 D 12 4 30 0 12 E 4 -10 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999026 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7110: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) D B A E C (6) A C E D B (6) E C B D A (5) C E B A D (5) C A E B D (5) A D C B E (5) D A B E C (4) C E A B D (4) C B E D A (4) B D C E A (4) A D E B C (4) A D B E C (4) E C A B D (3) E B C D A (3) A E C D B (3) E B D C A (2) E A C B D (2) C A D B E (2) B E D C A (2) A D B C E (2) E D B A C (1) E C B A D (1) E B D A C (1) E B A D C (1) D B E C A (1) D B A C E (1) D A B C E (1) C B D E A (1) C B D A E (1) C A B E D (1) C A B D E (1) B D E C A (1) B C D E A (1) A E D B C (1) A D C E B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -10 6 0 B -2 0 -20 12 -14 C 10 20 0 16 10 D -6 -12 -16 0 -16 E 0 14 -10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 6 0 B -2 0 -20 12 -14 C 10 20 0 16 10 D -6 -12 -16 0 -16 E 0 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=28 E=19 D=13 B=8 so B is eliminated. Round 2 votes counts: C=33 A=28 E=21 D=18 so D is eliminated. Round 3 votes counts: A=40 C=37 E=23 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:210 A:199 B:188 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 6 0 B -2 0 -20 12 -14 C 10 20 0 16 10 D -6 -12 -16 0 -16 E 0 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 6 0 B -2 0 -20 12 -14 C 10 20 0 16 10 D -6 -12 -16 0 -16 E 0 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 6 0 B -2 0 -20 12 -14 C 10 20 0 16 10 D -6 -12 -16 0 -16 E 0 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7111: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) B A C D E (11) A B C E D (10) E D C A B (8) B C A D E (8) C B D E A (7) D E C A B (6) C B A D E (6) A B E D C (6) A E D B C (4) E D A C B (3) E D A B C (3) D C E B A (3) C D E B A (3) B A C E D (3) E D C B A (2) C D B E A (2) A B E C D (2) C E D B A (1) C B D A E (1) Total count = 100 A B C D E A 0 -16 -16 0 2 B 16 0 -6 8 12 C 16 6 0 8 10 D 0 -8 -8 0 16 E -2 -12 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 0 2 B 16 0 -6 8 12 C 16 6 0 8 10 D 0 -8 -8 0 16 E -2 -12 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=22 A=22 D=20 C=20 E=16 so E is eliminated. Round 2 votes counts: D=36 B=22 A=22 C=20 so C is eliminated. Round 3 votes counts: D=42 B=36 A=22 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:215 D:200 A:185 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -16 0 2 B 16 0 -6 8 12 C 16 6 0 8 10 D 0 -8 -8 0 16 E -2 -12 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 0 2 B 16 0 -6 8 12 C 16 6 0 8 10 D 0 -8 -8 0 16 E -2 -12 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 0 2 B 16 0 -6 8 12 C 16 6 0 8 10 D 0 -8 -8 0 16 E -2 -12 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7112: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) A E D B C (8) C B D E A (7) D B E A C (5) D A E B C (4) C B E A D (4) B D C E A (4) B C D E A (4) A D E B C (4) C B D A E (3) B D E C A (3) A E D C B (3) E B A D C (2) E B A C D (2) E A D B C (2) D E B A C (2) D E A B C (2) D C B A E (2) D B C E A (2) D B C A E (2) C A E B D (2) C A D E B (2) C A B E D (2) B E D C A (2) A E C D B (2) A E C B D (2) A C E B D (2) E A C B D (1) E A B C D (1) D A E C B (1) D A B E C (1) C D A E B (1) C B A E D (1) C A D B E (1) B E D A C (1) B E C A D (1) B C E D A (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -6 -12 8 B 14 0 6 -10 10 C 6 -6 0 -2 0 D 12 10 2 0 20 E -8 -10 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -12 8 B 14 0 6 -10 10 C 6 -6 0 -2 0 D 12 10 2 0 20 E -8 -10 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=23 D=21 B=16 E=8 so E is eliminated. Round 2 votes counts: C=32 A=27 D=21 B=20 so B is eliminated. Round 3 votes counts: C=38 D=31 A=31 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:222 B:210 C:199 A:188 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -12 8 B 14 0 6 -10 10 C 6 -6 0 -2 0 D 12 10 2 0 20 E -8 -10 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -12 8 B 14 0 6 -10 10 C 6 -6 0 -2 0 D 12 10 2 0 20 E -8 -10 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -12 8 B 14 0 6 -10 10 C 6 -6 0 -2 0 D 12 10 2 0 20 E -8 -10 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7113: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D A C B E (9) A C D B E (8) E B D C A (6) A C B D E (6) E D B C A (5) E D B A C (5) E B C D A (4) D E A B C (4) A D C B E (4) E D A C B (3) D E B C A (3) D B C A E (3) C A B E D (3) E A C B D (2) D B E C A (2) C A B D E (2) B E C D A (2) B E C A D (2) A C E B D (2) A C B E D (2) E C B A D (1) E B A C D (1) D E B A C (1) D E A C B (1) D C A B E (1) D A E C B (1) D A C E B (1) C B D A E (1) C B A D E (1) B D C E A (1) B C E A D (1) B C D E A (1) B C A E D (1) A D C E B (1) Total count = 100 A B C D E A 0 0 2 -8 -8 B 0 0 2 -4 0 C -2 -2 0 -2 -4 D 8 4 2 0 2 E 8 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -8 -8 B 0 0 2 -4 0 C -2 -2 0 -2 -4 D 8 4 2 0 2 E 8 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=26 A=23 B=8 C=7 so C is eliminated. Round 2 votes counts: E=36 A=28 D=26 B=10 so B is eliminated. Round 3 votes counts: E=41 A=30 D=29 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:208 E:205 B:199 C:195 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 2 -8 -8 B 0 0 2 -4 0 C -2 -2 0 -2 -4 D 8 4 2 0 2 E 8 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -8 -8 B 0 0 2 -4 0 C -2 -2 0 -2 -4 D 8 4 2 0 2 E 8 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -8 -8 B 0 0 2 -4 0 C -2 -2 0 -2 -4 D 8 4 2 0 2 E 8 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7114: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (12) D A B E C (10) B A D C E (9) D A E C B (8) E C B D A (7) A D B C E (7) C E A B D (6) E C D A B (4) B E C D A (4) D A E B C (3) B D A C E (3) C E A D B (2) C B E A D (2) B D A E C (2) B C E D A (2) B C E A D (2) B A C D E (2) A D C E B (2) E D C A B (1) E C D B A (1) D E A B C (1) D B A E C (1) C B A E D (1) B D E A C (1) B C A E D (1) B C A D E (1) A D E C B (1) A D B E C (1) A C D E B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 8 4 10 B 2 0 2 14 2 C -8 -2 0 -2 10 D -4 -14 2 0 10 E -10 -2 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 4 10 B 2 0 2 14 2 C -8 -2 0 -2 10 D -4 -14 2 0 10 E -10 -2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 C=23 A=14 E=13 so E is eliminated. Round 2 votes counts: C=35 B=27 D=24 A=14 so A is eliminated. Round 3 votes counts: C=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:210 B:210 C:199 D:197 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 4 10 B 2 0 2 14 2 C -8 -2 0 -2 10 D -4 -14 2 0 10 E -10 -2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 4 10 B 2 0 2 14 2 C -8 -2 0 -2 10 D -4 -14 2 0 10 E -10 -2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 4 10 B 2 0 2 14 2 C -8 -2 0 -2 10 D -4 -14 2 0 10 E -10 -2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7115: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) A C D B E (8) B A C D E (7) E D A C B (6) B A C E D (5) E D C A B (4) E A B C D (4) D C B A E (4) D C A E B (4) A E B C D (4) E B D C A (3) E A B D C (3) D C B E A (3) D C A B E (3) B E C D A (3) A B C E D (3) A B C D E (3) D C E A B (2) C D B A E (2) B D C E A (2) B C D A E (2) B C A D E (2) E D C B A (1) E D B C A (1) D E C B A (1) D C E B A (1) D A C E B (1) C D A B E (1) C B D A E (1) C A D B E (1) B E D C A (1) A E D C B (1) A E C B D (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 22 16 8 14 B -22 0 -14 -12 4 C -16 14 0 -2 18 D -8 12 2 0 2 E -14 -4 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 16 8 14 B -22 0 -14 -12 4 C -16 14 0 -2 18 D -8 12 2 0 2 E -14 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999427 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=24 B=22 D=19 C=5 so C is eliminated. Round 2 votes counts: E=30 A=25 B=23 D=22 so D is eliminated. Round 3 votes counts: E=34 A=34 B=32 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:207 D:204 E:181 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 16 8 14 B -22 0 -14 -12 4 C -16 14 0 -2 18 D -8 12 2 0 2 E -14 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999427 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 8 14 B -22 0 -14 -12 4 C -16 14 0 -2 18 D -8 12 2 0 2 E -14 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999427 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 8 14 B -22 0 -14 -12 4 C -16 14 0 -2 18 D -8 12 2 0 2 E -14 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999427 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7116: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B A C E D (6) D E A B C (5) C B A E D (5) B A D E C (5) E D C A B (4) C E D B A (4) C D E B A (4) A B C E D (4) E D A B C (3) E A D B C (3) D E C B A (3) D E B A C (3) C E A B D (3) C B D A E (3) B C A E D (3) B A C D E (3) E C D A B (2) E C A D B (2) C E D A B (2) C A B E D (2) B A D C E (2) A E B C D (2) A D B E C (2) A B E D C (2) A B E C D (2) A B D E C (2) E A C B D (1) D E B C A (1) D B A E C (1) D B A C E (1) D A B E C (1) C E B A D (1) C B E A D (1) B D A E C (1) B C A D E (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 2 8 0 B -4 0 8 2 -4 C -2 -8 0 4 -8 D -8 -2 -4 0 -8 E 0 4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.531458 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.468542 Sum of squares = 0.501979171138 Cumulative probabilities = A: 0.531458 B: 0.531458 C: 0.531458 D: 0.531458 E: 1.000000 A B C D E A 0 4 2 8 0 B -4 0 8 2 -4 C -2 -8 0 4 -8 D -8 -2 -4 0 -8 E 0 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=23 B=21 A=16 E=15 so E is eliminated. Round 2 votes counts: D=30 C=29 B=21 A=20 so A is eliminated. Round 3 votes counts: D=35 B=34 C=31 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:210 A:207 B:201 C:193 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 8 0 B -4 0 8 2 -4 C -2 -8 0 4 -8 D -8 -2 -4 0 -8 E 0 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 8 0 B -4 0 8 2 -4 C -2 -8 0 4 -8 D -8 -2 -4 0 -8 E 0 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 8 0 B -4 0 8 2 -4 C -2 -8 0 4 -8 D -8 -2 -4 0 -8 E 0 4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7117: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) E C D B A (6) C D E B A (5) B C D A E (5) E D C A B (4) D C E A B (4) C D B E A (4) B C E D A (4) A E B D C (4) E A D C B (3) D A E C B (3) C D B A E (3) C B E D A (3) A E D B C (3) A D B C E (3) A B D E C (3) E D A C B (2) D C A E B (2) C E D B A (2) B E C A D (2) B C D E A (2) B C A D E (2) B A E C D (2) B A C D E (2) A D E C B (2) A B D C E (2) E C D A B (1) E C B D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D A C E B (1) C E B D A (1) C B D E A (1) C B D A E (1) B E A C D (1) B C E A D (1) B A C E D (1) A E D C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -10 -14 -2 B -2 0 -6 -8 -2 C 10 6 0 0 0 D 14 8 0 0 -2 E 2 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.360144 D: 0.000000 E: 0.639856 Sum of squares = 0.539119106286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.360144 D: 0.360144 E: 1.000000 A B C D E A 0 2 -10 -14 -2 B -2 0 -6 -8 -2 C 10 6 0 0 0 D 14 8 0 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=22 C=20 E=19 D=12 so D is eliminated. Round 2 votes counts: A=31 C=26 B=22 E=21 so E is eliminated. Round 3 votes counts: C=39 A=39 B=22 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:208 E:203 B:191 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -14 -2 B -2 0 -6 -8 -2 C 10 6 0 0 0 D 14 8 0 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -14 -2 B -2 0 -6 -8 -2 C 10 6 0 0 0 D 14 8 0 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -14 -2 B -2 0 -6 -8 -2 C 10 6 0 0 0 D 14 8 0 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7118: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E C B D A (5) E C B A D (5) C E D A B (5) B A D E C (5) A D B C E (5) A C D B E (5) E C D B A (4) D A B C E (4) D C E A B (3) C E A D B (3) B E D A C (3) B D A E C (3) A D C B E (3) A B C D E (3) D E C B A (2) D E C A B (2) C E B A D (2) C E A B D (2) C A E D B (2) C A E B D (2) C A D E B (2) B E C D A (2) B E C A D (2) A C D E B (2) E D C B A (1) E C D A B (1) D E A C B (1) D C A E B (1) D B A E C (1) D A C B E (1) D A B E C (1) C D E A B (1) B E D C A (1) B D E A C (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C E D (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -10 0 -6 B -4 0 -14 0 -6 C 10 14 0 18 2 D 0 0 -18 0 -2 E 6 6 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 0 -6 B -4 0 -14 0 -6 C 10 14 0 18 2 D 0 0 -18 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 B=21 A=21 C=19 D=16 so D is eliminated. Round 2 votes counts: E=28 A=27 C=23 B=22 so B is eliminated. Round 3 votes counts: A=40 E=37 C=23 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:222 E:206 A:194 D:190 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 0 -6 B -4 0 -14 0 -6 C 10 14 0 18 2 D 0 0 -18 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 0 -6 B -4 0 -14 0 -6 C 10 14 0 18 2 D 0 0 -18 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 0 -6 B -4 0 -14 0 -6 C 10 14 0 18 2 D 0 0 -18 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7119: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (12) B C E D A (7) E D A B C (6) E A D C B (6) B C D E A (6) B C E A D (5) A E D C B (5) D E A B C (4) B C A E D (4) A E C B D (4) A D E C B (4) D E B C A (3) D A C B E (3) C B A E D (3) B C D A E (3) A C B E D (3) A C B D E (3) E A D B C (2) D E A C B (2) D A E C B (2) E D B A C (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B E C A (1) D A C E B (1) C B D A E (1) C A B D E (1) B C A D E (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -2 14 8 B 0 0 -8 14 10 C 2 8 0 14 14 D -14 -14 -14 0 0 E -8 -10 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 14 8 B 0 0 -8 14 10 C 2 8 0 14 14 D -14 -14 -14 0 0 E -8 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=21 E=18 D=18 C=17 so C is eliminated. Round 2 votes counts: B=42 A=22 E=18 D=18 so E is eliminated. Round 3 votes counts: B=44 A=31 D=25 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:219 A:210 B:208 E:184 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 14 8 B 0 0 -8 14 10 C 2 8 0 14 14 D -14 -14 -14 0 0 E -8 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 14 8 B 0 0 -8 14 10 C 2 8 0 14 14 D -14 -14 -14 0 0 E -8 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 14 8 B 0 0 -8 14 10 C 2 8 0 14 14 D -14 -14 -14 0 0 E -8 -10 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997102 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7120: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (6) A D C E B (6) E B A C D (5) C D A E B (5) A D C B E (5) A B D C E (5) E C D B A (4) E B C A D (4) C E D A B (4) C D E A B (4) B E A D C (4) B D A C E (4) A B E D C (4) E C D A B (3) D B C A E (3) C D E B A (3) A D B C E (3) E C B D A (2) E C A D B (2) E A C D B (2) D C A B E (2) C A D E B (2) B A D C E (2) A E B D C (2) E C A B D (1) E B A D C (1) D C B E A (1) D B A C E (1) D A B C E (1) C D B E A (1) B E D C A (1) B E C D A (1) B D E A C (1) B D C E A (1) B D A E C (1) B A E D C (1) A E D C B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 0 2 -2 B -6 0 4 -6 -14 C 0 -4 0 -2 6 D -2 6 2 0 4 E 2 14 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.596454 B: 0.000000 C: 0.403546 D: 0.000000 E: 0.000000 Sum of squares = 0.51860682305 Cumulative probabilities = A: 0.596454 B: 0.596454 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 2 -2 B -6 0 4 -6 -14 C 0 -4 0 -2 6 D -2 6 2 0 4 E 2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500489 B: 0.000000 C: 0.499511 D: 0.000000 E: 0.000000 Sum of squares = 0.50000047868 Cumulative probabilities = A: 0.500489 B: 0.500489 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=27 C=19 B=16 D=8 so D is eliminated. Round 2 votes counts: E=30 A=28 C=22 B=20 so B is eliminated. Round 3 votes counts: E=37 A=37 C=26 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:205 A:203 E:203 C:200 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 2 -2 B -6 0 4 -6 -14 C 0 -4 0 -2 6 D -2 6 2 0 4 E 2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500489 B: 0.000000 C: 0.499511 D: 0.000000 E: 0.000000 Sum of squares = 0.50000047868 Cumulative probabilities = A: 0.500489 B: 0.500489 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 2 -2 B -6 0 4 -6 -14 C 0 -4 0 -2 6 D -2 6 2 0 4 E 2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500489 B: 0.000000 C: 0.499511 D: 0.000000 E: 0.000000 Sum of squares = 0.50000047868 Cumulative probabilities = A: 0.500489 B: 0.500489 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 2 -2 B -6 0 4 -6 -14 C 0 -4 0 -2 6 D -2 6 2 0 4 E 2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500489 B: 0.000000 C: 0.499511 D: 0.000000 E: 0.000000 Sum of squares = 0.50000047868 Cumulative probabilities = A: 0.500489 B: 0.500489 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7121: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) E C A D B (8) B D E C A (6) B D A C E (6) E C A B D (5) B E C D A (5) A C E D B (5) A D C E B (4) E C B A D (3) E A C B D (3) D C A E B (3) D B C A E (3) D A C E B (3) B D C E A (3) A D B C E (3) E B C A D (2) E A C D B (2) D C E A B (2) D B C E A (2) D A C B E (2) C E D A B (2) A E B C D (2) E C D B A (1) E C B D A (1) D C B E A (1) D C A B E (1) D B A C E (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E B A (1) C A E D B (1) B E D C A (1) B E C A D (1) B E A C D (1) B D E A C (1) B D C A E (1) B D A E C (1) A C D E B (1) Total count = 100 A B C D E A 0 16 -10 2 -6 B -16 0 -20 -16 -20 C 10 20 0 10 -4 D -2 16 -10 0 -8 E 6 20 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -10 2 -6 B -16 0 -20 -16 -20 C 10 20 0 10 -4 D -2 16 -10 0 -8 E 6 20 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 A=24 D=19 C=6 so C is eliminated. Round 2 votes counts: E=29 B=26 A=25 D=20 so D is eliminated. Round 3 votes counts: A=35 B=33 E=32 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 C:218 A:201 D:198 B:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -10 2 -6 B -16 0 -20 -16 -20 C 10 20 0 10 -4 D -2 16 -10 0 -8 E 6 20 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -10 2 -6 B -16 0 -20 -16 -20 C 10 20 0 10 -4 D -2 16 -10 0 -8 E 6 20 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -10 2 -6 B -16 0 -20 -16 -20 C 10 20 0 10 -4 D -2 16 -10 0 -8 E 6 20 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7122: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (5) C B D E A (5) C B A D E (5) E A D C B (4) E A C D B (4) C E A B D (4) B D A C E (4) B C D A E (4) A E D B C (4) A E C B D (4) E D A B C (3) E C D A B (3) D B E A C (3) C E A D B (3) C B A E D (3) C A E B D (3) C A B E D (3) B D A E C (3) E D C A B (2) E D A C B (2) D E C B A (2) D B A E C (2) B D C A E (2) B A D E C (2) A C E B D (2) A B E D C (2) A B C E D (2) E A D B C (1) D C E B A (1) D B C E A (1) D A B E C (1) C D B E A (1) C B D A E (1) B C D E A (1) B C A D E (1) B A D C E (1) B A C D E (1) A E D C B (1) A E C D B (1) A E B D C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 14 0 18 10 B -14 0 -18 8 -2 C 0 18 0 12 -4 D -18 -8 -12 0 -14 E -10 2 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.697370 B: 0.000000 C: 0.302630 D: 0.000000 E: 0.000000 Sum of squares = 0.57790998614 Cumulative probabilities = A: 0.697370 B: 0.697370 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 18 10 B -14 0 -18 8 -2 C 0 18 0 12 -4 D -18 -8 -12 0 -14 E -10 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=24 B=19 A=19 D=10 so D is eliminated. Round 2 votes counts: C=29 E=26 B=25 A=20 so A is eliminated. Round 3 votes counts: E=38 C=31 B=31 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:221 C:213 E:205 B:187 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 18 10 B -14 0 -18 8 -2 C 0 18 0 12 -4 D -18 -8 -12 0 -14 E -10 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 18 10 B -14 0 -18 8 -2 C 0 18 0 12 -4 D -18 -8 -12 0 -14 E -10 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 18 10 B -14 0 -18 8 -2 C 0 18 0 12 -4 D -18 -8 -12 0 -14 E -10 2 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7123: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) C B A D E (7) E D A B C (6) D E A B C (5) C A D B E (5) B E C D A (5) C B A E D (4) A D C B E (4) E B C D A (3) D A E B C (3) C B E D A (3) C B E A D (3) B C E A D (3) A D E C B (3) A D C E B (3) E D C B A (2) E D B A C (2) E D A C B (2) E B D A C (2) C E D B A (2) C D E A B (2) C A D E B (2) B E D A C (2) B E C A D (2) B E A D C (2) B C A E D (2) A D B C E (2) E D C A B (1) D E A C B (1) D C A E B (1) D A C E B (1) C E B D A (1) C A B D E (1) B C A D E (1) B A D E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 0 -6 2 B -6 0 -14 -12 -2 C 0 14 0 -8 -4 D 6 12 8 0 6 E -2 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -6 2 B -6 0 -14 -12 -2 C 0 14 0 -8 -4 D 6 12 8 0 6 E -2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=20 E=18 B=18 A=14 so A is eliminated. Round 2 votes counts: D=32 C=30 B=20 E=18 so E is eliminated. Round 3 votes counts: D=45 C=30 B=25 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:201 C:201 E:199 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -6 2 B -6 0 -14 -12 -2 C 0 14 0 -8 -4 D 6 12 8 0 6 E -2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -6 2 B -6 0 -14 -12 -2 C 0 14 0 -8 -4 D 6 12 8 0 6 E -2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -6 2 B -6 0 -14 -12 -2 C 0 14 0 -8 -4 D 6 12 8 0 6 E -2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7124: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) D A B C E (7) A C D E B (7) C E A B D (6) B E C A D (6) E C B A D (5) D A C B E (5) D A C E B (4) C A E D B (4) B D E A C (4) E C A B D (3) E B C A D (3) D B E A C (3) D B A E C (3) C A E B D (3) B D A E C (3) E C A D B (2) E B D C A (2) E B C D A (2) B A D C E (2) B A C E D (2) A C B E D (2) A C B D E (2) E D C B A (1) E D B C A (1) D E C B A (1) D E B A C (1) D B A C E (1) C A D E B (1) B E D A C (1) B A E C D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 0 2 -2 B 4 0 4 14 6 C 0 -4 0 -2 -4 D -2 -14 2 0 -8 E 2 -6 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 2 -2 B 4 0 4 14 6 C 0 -4 0 -2 -4 D -2 -14 2 0 -8 E 2 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 E=19 C=14 A=13 so A is eliminated. Round 2 votes counts: B=29 D=27 C=25 E=19 so E is eliminated. Round 3 votes counts: B=36 C=35 D=29 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:204 A:198 C:195 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 2 -2 B 4 0 4 14 6 C 0 -4 0 -2 -4 D -2 -14 2 0 -8 E 2 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 -2 B 4 0 4 14 6 C 0 -4 0 -2 -4 D -2 -14 2 0 -8 E 2 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 -2 B 4 0 4 14 6 C 0 -4 0 -2 -4 D -2 -14 2 0 -8 E 2 -6 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7125: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (16) E B C D A (11) E B A D C (5) E B A C D (4) D A C E B (4) B E C A D (4) A D C E B (4) A B D C E (4) E C D B A (3) D A C B E (3) C D A B E (3) B C D A E (3) E D C A B (2) E C B D A (2) E B C A D (2) D C E A B (2) D C A E B (2) D C A B E (2) C B D E A (2) B E C D A (2) B C E D A (2) B A E D C (2) B A C D E (2) A D E C B (2) A B E D C (2) E D A C B (1) E A D C B (1) E A D B C (1) C B E D A (1) B E A D C (1) B E A C D (1) B A D C E (1) A E D B C (1) A E B D C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 14 10 6 B -4 0 0 4 4 C -14 0 0 -16 4 D -10 -4 16 0 2 E -6 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 10 6 B -4 0 0 4 4 C -14 0 0 -16 4 D -10 -4 16 0 2 E -6 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=31 B=18 D=13 C=6 so C is eliminated. Round 2 votes counts: E=32 A=31 B=21 D=16 so D is eliminated. Round 3 votes counts: A=45 E=34 B=21 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:202 D:202 E:192 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 10 6 B -4 0 0 4 4 C -14 0 0 -16 4 D -10 -4 16 0 2 E -6 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 10 6 B -4 0 0 4 4 C -14 0 0 -16 4 D -10 -4 16 0 2 E -6 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 10 6 B -4 0 0 4 4 C -14 0 0 -16 4 D -10 -4 16 0 2 E -6 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7126: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) A B C E D (8) D E A C B (7) D C B E A (6) A D E B C (6) A B C D E (6) B C A E D (5) A E B C D (5) C B D E A (4) A E D B C (4) A B E C D (4) E D C B A (3) E C B D A (3) D E C A B (3) D A E C B (3) C B E D A (3) C B E A D (3) A D B C E (3) D A C B E (2) B C A D E (2) B A C E D (2) E D A C B (1) E B C A D (1) E B A C D (1) E A D B C (1) D C E B A (1) D C B A E (1) D A E B C (1) B C E A D (1) B A C D E (1) Total count = 100 A B C D E A 0 8 10 6 6 B -8 0 2 -2 2 C -10 -2 0 -2 -4 D -6 2 2 0 10 E -6 -2 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 6 6 B -8 0 2 -2 2 C -10 -2 0 -2 -4 D -6 2 2 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=33 B=11 E=10 C=10 so E is eliminated. Round 2 votes counts: D=37 A=37 C=13 B=13 so C is eliminated. Round 3 votes counts: D=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:204 B:197 E:193 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 6 6 B -8 0 2 -2 2 C -10 -2 0 -2 -4 D -6 2 2 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 6 6 B -8 0 2 -2 2 C -10 -2 0 -2 -4 D -6 2 2 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 6 6 B -8 0 2 -2 2 C -10 -2 0 -2 -4 D -6 2 2 0 10 E -6 -2 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7127: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E B D A C (6) D E A B C (5) B E A C D (5) E B A D C (4) D C A E B (4) C A B E D (4) D E B A C (3) C D B E A (3) C D A E B (3) B D E C A (3) B C E A D (3) A E B C D (3) A C B E D (3) E D B A C (2) E A D B C (2) D C A B E (2) D A E C B (2) D A C E B (2) B E D C A (2) B E D A C (2) B E A D C (2) B A E C D (2) A E D C B (2) A E C B D (2) A E B D C (2) A C E D B (2) A C D E B (2) E D A B C (1) E A B C D (1) D C E A B (1) D C B E A (1) D B E C A (1) D A E B C (1) C D B A E (1) C B A E D (1) C A D B E (1) B E C D A (1) B E C A D (1) B C D E A (1) B C A E D (1) A D E C B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 10 18 -8 0 B -10 0 8 0 -6 C -18 -8 0 -4 -12 D 8 0 4 0 -10 E 0 6 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.283544 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.716456 Sum of squares = 0.593706433898 Cumulative probabilities = A: 0.283544 B: 0.283544 C: 0.283544 D: 0.283544 E: 1.000000 A B C D E A 0 10 18 -8 0 B -10 0 8 0 -6 C -18 -8 0 -4 -12 D 8 0 4 0 -10 E 0 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 D=22 C=20 A=19 E=16 so E is eliminated. Round 2 votes counts: B=33 D=25 A=22 C=20 so C is eliminated. Round 3 votes counts: D=39 B=34 A=27 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:214 A:210 D:201 B:196 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 18 -8 0 B -10 0 8 0 -6 C -18 -8 0 -4 -12 D 8 0 4 0 -10 E 0 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 -8 0 B -10 0 8 0 -6 C -18 -8 0 -4 -12 D 8 0 4 0 -10 E 0 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 -8 0 B -10 0 8 0 -6 C -18 -8 0 -4 -12 D 8 0 4 0 -10 E 0 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7128: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) E B C D A (6) B E D C A (6) A E C D B (6) E B A D C (4) C D A B E (4) A E B D C (4) E C B D A (3) E A C D B (3) D B C A E (3) C E D B A (3) C E A D B (3) C A D E B (3) B D C E A (3) A D B C E (3) A C D E B (3) E C B A D (2) E A C B D (2) E A B D C (2) D C B A E (2) D A C B E (2) C D A E B (2) B E D A C (2) B E C D A (2) B E A D C (2) B D A C E (2) B A D E C (2) E C D B A (1) E B D A C (1) D B A C E (1) C E D A B (1) C D B E A (1) C D B A E (1) C A E D B (1) B D A E C (1) A E D C B (1) A E B C D (1) A D C E B (1) A C E D B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 6 6 6 B -4 0 -6 -6 -8 C -6 6 0 -2 -4 D -6 6 2 0 -16 E -6 8 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 6 6 B -4 0 -6 -6 -8 C -6 6 0 -2 -4 D -6 6 2 0 -16 E -6 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 B=20 C=19 D=8 so D is eliminated. Round 2 votes counts: A=31 E=24 B=24 C=21 so C is eliminated. Round 3 votes counts: A=41 E=31 B=28 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:211 C:197 D:193 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 6 6 B -4 0 -6 -6 -8 C -6 6 0 -2 -4 D -6 6 2 0 -16 E -6 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 6 6 B -4 0 -6 -6 -8 C -6 6 0 -2 -4 D -6 6 2 0 -16 E -6 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 6 6 B -4 0 -6 -6 -8 C -6 6 0 -2 -4 D -6 6 2 0 -16 E -6 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7129: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (13) B E D C A (7) A E C D B (7) E B D C A (6) B A C D E (6) C A D E B (5) A C E D B (5) E D B C A (4) B A E C D (4) A B C E D (4) D C E A B (3) B D E C A (3) A C B D E (3) E D C B A (2) E D C A B (2) D C B E A (2) C D B E A (2) C D A E B (2) B E A D C (2) B D C E A (2) A B C D E (2) E D A C B (1) E B D A C (1) E A D C B (1) D E C B A (1) D B E C A (1) C D E A B (1) C B A D E (1) C A E D B (1) B C D A E (1) B A E D C (1) B A D E C (1) A E B D C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 8 18 18 B -6 0 -6 -6 -12 C -8 6 0 18 8 D -18 6 -18 0 -2 E -18 12 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 18 18 B -6 0 -6 -6 -12 C -8 6 0 18 8 D -18 6 -18 0 -2 E -18 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 B=27 E=17 C=12 D=7 so D is eliminated. Round 2 votes counts: A=37 B=28 E=18 C=17 so C is eliminated. Round 3 votes counts: A=45 B=33 E=22 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:212 E:194 B:185 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 18 18 B -6 0 -6 -6 -12 C -8 6 0 18 8 D -18 6 -18 0 -2 E -18 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 18 18 B -6 0 -6 -6 -12 C -8 6 0 18 8 D -18 6 -18 0 -2 E -18 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 18 18 B -6 0 -6 -6 -12 C -8 6 0 18 8 D -18 6 -18 0 -2 E -18 12 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999571 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7130: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (12) C D B A E (10) D C B E A (7) C B D A E (7) A E B C D (7) A E C B D (6) D B E A C (5) D B C E A (5) E A D B C (3) C A E D B (3) B D C A E (3) B D A E C (3) E D A B C (2) D C B A E (2) C D E A B (2) C A E B D (2) C A B E D (2) B D E A C (2) B C D A E (2) A E B D C (2) A C E B D (2) E D B A C (1) E D A C B (1) D E C A B (1) D E B A C (1) D B E C A (1) C E A D B (1) C D B E A (1) C A D E B (1) B A E D C (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 -6 0 -12 10 B 6 0 4 6 6 C 0 -4 0 -4 2 D 12 -6 4 0 6 E -10 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -12 10 B 6 0 4 6 6 C 0 -4 0 -4 2 D 12 -6 4 0 6 E -10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=22 E=19 A=17 B=13 so B is eliminated. Round 2 votes counts: C=31 D=30 A=20 E=19 so E is eliminated. Round 3 votes counts: A=35 D=34 C=31 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:211 D:208 C:197 A:196 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -12 10 B 6 0 4 6 6 C 0 -4 0 -4 2 D 12 -6 4 0 6 E -10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -12 10 B 6 0 4 6 6 C 0 -4 0 -4 2 D 12 -6 4 0 6 E -10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -12 10 B 6 0 4 6 6 C 0 -4 0 -4 2 D 12 -6 4 0 6 E -10 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7131: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (14) E D C B A (8) E C A D B (7) B A C D E (7) A C B E D (6) E C D A B (5) D B E A C (5) C A E B D (5) B D A C E (5) B A D C E (5) A B C D E (5) C E A D B (4) E D C A B (3) D B E C A (3) A C E B D (3) E A C D B (2) D E B A C (2) C A B D E (2) B D A E C (2) E D B C A (1) E D B A C (1) D E C B A (1) D B C A E (1) C E D A B (1) B A E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -10 -4 -14 B 12 0 6 -16 -14 C 10 -6 0 -4 -10 D 4 16 4 0 4 E 14 14 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -10 -4 -14 B 12 0 6 -16 -14 C 10 -6 0 -4 -10 D 4 16 4 0 4 E 14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 B=20 A=15 C=12 so C is eliminated. Round 2 votes counts: E=32 D=26 A=22 B=20 so B is eliminated. Round 3 votes counts: A=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:217 D:214 C:195 B:194 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -10 -4 -14 B 12 0 6 -16 -14 C 10 -6 0 -4 -10 D 4 16 4 0 4 E 14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -4 -14 B 12 0 6 -16 -14 C 10 -6 0 -4 -10 D 4 16 4 0 4 E 14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -4 -14 B 12 0 6 -16 -14 C 10 -6 0 -4 -10 D 4 16 4 0 4 E 14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7132: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) A D C B E (8) A C E B D (8) D B A E C (7) C A E B D (7) A C E D B (7) A C D E B (7) D B E A C (6) E C A B D (5) D A B C E (5) C E A B D (5) A D B C E (5) E B C D A (4) B D E C A (4) E C B D A (3) E C B A D (2) E B D C A (2) D B E C A (2) D B A C E (1) C A E D B (1) B E D C A (1) B D E A C (1) Total count = 100 A B C D E A 0 34 28 28 30 B -34 0 -24 -16 -2 C -28 24 0 16 26 D -28 16 -16 0 10 E -30 2 -26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 34 28 28 30 B -34 0 -24 -16 -2 C -28 24 0 16 26 D -28 16 -16 0 10 E -30 2 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 D=21 E=16 C=13 B=6 so B is eliminated. Round 2 votes counts: A=44 D=26 E=17 C=13 so C is eliminated. Round 3 votes counts: A=52 D=26 E=22 so E is eliminated. Round 4 votes counts: A=64 D=36 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:260 C:219 D:191 E:168 B:162 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 34 28 28 30 B -34 0 -24 -16 -2 C -28 24 0 16 26 D -28 16 -16 0 10 E -30 2 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 34 28 28 30 B -34 0 -24 -16 -2 C -28 24 0 16 26 D -28 16 -16 0 10 E -30 2 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 34 28 28 30 B -34 0 -24 -16 -2 C -28 24 0 16 26 D -28 16 -16 0 10 E -30 2 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7133: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) D A E C B (6) D E B A C (4) C A E B D (4) A D C E B (4) E D B A C (3) E B D C A (3) D B A E C (3) D A E B C (3) C E A D B (3) C A B E D (3) B D E A C (3) B D A C E (3) B C E D A (3) B A C D E (3) A C D E B (3) E D A B C (2) E C D A B (2) D B E A C (2) C E A B D (2) C B E A D (2) C B A E D (2) C A E D B (2) C A B D E (2) B E D C A (2) B E C D A (2) B A D C E (2) A D E C B (2) A C D B E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C B D A (1) E C A D B (1) D E A B C (1) D A B E C (1) C E B A D (1) B E D A C (1) B D A E C (1) B C E A D (1) B C A E D (1) B C A D E (1) A D C B E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 18 -12 2 B -8 0 -6 -12 -14 C -18 6 0 -16 -6 D 12 12 16 0 -2 E -2 14 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.593750000011 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 A B C D E A 0 8 18 -12 2 B -8 0 -6 -12 -14 C -18 6 0 -16 -6 D 12 12 16 0 -2 E -2 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.59375000004 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=22 C=21 D=20 A=14 so A is eliminated. Round 2 votes counts: D=28 C=26 B=24 E=22 so E is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:210 A:208 C:183 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 18 -12 2 B -8 0 -6 -12 -14 C -18 6 0 -16 -6 D 12 12 16 0 -2 E -2 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.59375000004 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 -12 2 B -8 0 -6 -12 -14 C -18 6 0 -16 -6 D 12 12 16 0 -2 E -2 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.59375000004 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 -12 2 B -8 0 -6 -12 -14 C -18 6 0 -16 -6 D 12 12 16 0 -2 E -2 14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.750000 Sum of squares = 0.59375000004 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7134: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) A B E D C (9) C D E B A (7) C D A E B (7) E C D B A (6) A B E C D (6) E B D C A (5) D C E B A (5) A C D B E (5) A B C D E (5) E D C B A (4) B E D C A (4) A C D E B (4) E B A C D (3) C D E A B (3) B E D A C (2) B E A D C (2) A B D E C (2) A B D C E (2) E B C D A (1) D B C E A (1) B D E C A (1) B D A E C (1) B A E C D (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 12 6 12 B 8 0 16 16 6 C -12 -16 0 2 -18 D -6 -16 -2 0 -14 E -12 -6 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 6 12 B 8 0 16 16 6 C -12 -16 0 2 -18 D -6 -16 -2 0 -14 E -12 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=22 E=19 C=17 D=6 so D is eliminated. Round 2 votes counts: A=36 B=23 C=22 E=19 so E is eliminated. Round 3 votes counts: A=36 C=32 B=32 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:211 E:207 D:181 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 6 12 B 8 0 16 16 6 C -12 -16 0 2 -18 D -6 -16 -2 0 -14 E -12 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 6 12 B 8 0 16 16 6 C -12 -16 0 2 -18 D -6 -16 -2 0 -14 E -12 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 6 12 B 8 0 16 16 6 C -12 -16 0 2 -18 D -6 -16 -2 0 -14 E -12 -6 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7135: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) E C D B A (6) A D C B E (6) E D C A B (4) E B C A D (4) D A E C B (4) C B D A E (4) A B D C E (4) E D A C B (3) E C B D A (3) E B A D C (3) D C A E B (3) D C A B E (3) D A C E B (3) D A C B E (3) B A C E D (3) E A D C B (2) E A B D C (2) D E A C B (2) C B E D A (2) B E A C D (2) B C E D A (2) B C E A D (2) A E D B C (2) A D E B C (2) A D B C E (2) E B A C D (1) D C E A B (1) C E D B A (1) C E B D A (1) C D E B A (1) C D B A E (1) C D A B E (1) C B D E A (1) B E C A D (1) B C D A E (1) B C A D E (1) B A E C D (1) B A C D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -4 -18 -6 B 2 0 -10 0 -14 C 4 10 0 0 -6 D 18 0 0 0 -12 E 6 14 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -18 -6 B 2 0 -10 0 -14 C 4 10 0 0 -6 D 18 0 0 0 -12 E 6 14 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=19 A=18 B=14 C=12 so C is eliminated. Round 2 votes counts: E=39 D=22 B=21 A=18 so A is eliminated. Round 3 votes counts: E=41 D=32 B=27 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:204 D:203 B:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -18 -6 B 2 0 -10 0 -14 C 4 10 0 0 -6 D 18 0 0 0 -12 E 6 14 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -18 -6 B 2 0 -10 0 -14 C 4 10 0 0 -6 D 18 0 0 0 -12 E 6 14 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -18 -6 B 2 0 -10 0 -14 C 4 10 0 0 -6 D 18 0 0 0 -12 E 6 14 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7136: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) E B A C D (6) C D E B A (6) E B C D A (5) D C E B A (5) A B E C D (5) E C D B A (4) D C E A B (4) D C A E B (4) B E A C D (4) E A B C D (3) D C B E A (3) D C A B E (3) D A C E B (3) B E C D A (3) B C E D A (3) B A E C D (3) A D B C E (3) E A C D B (2) C D B E A (2) A D C E B (2) A D C B E (2) E C D A B (1) E C B D A (1) E B C A D (1) E A C B D (1) D A C B E (1) C E D B A (1) C E B D A (1) B E C A D (1) B D C A E (1) B D A C E (1) B C D E A (1) B A D C E (1) B A C E D (1) A E D C B (1) A E B C D (1) A D E C B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 0 -6 -16 B 8 0 4 2 -6 C 0 -4 0 12 -2 D 6 -2 -12 0 -10 E 16 6 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 -6 -16 B 8 0 4 2 -6 C 0 -4 0 12 -2 D 6 -2 -12 0 -10 E 16 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 D=23 B=19 C=10 so C is eliminated. Round 2 votes counts: D=31 E=26 A=24 B=19 so B is eliminated. Round 3 votes counts: E=37 D=34 A=29 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:204 C:203 D:191 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -6 -16 B 8 0 4 2 -6 C 0 -4 0 12 -2 D 6 -2 -12 0 -10 E 16 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -6 -16 B 8 0 4 2 -6 C 0 -4 0 12 -2 D 6 -2 -12 0 -10 E 16 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -6 -16 B 8 0 4 2 -6 C 0 -4 0 12 -2 D 6 -2 -12 0 -10 E 16 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7137: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (15) B D E C A (9) B D C E A (8) E A C B D (7) E A B D C (6) D C B A E (6) C D B A E (6) D B C A E (5) B E D A C (5) A C E D B (5) E A B C D (4) C D A B E (4) C A D E B (4) E B D A C (3) E A C D B (3) B D C A E (3) E B A D C (2) C A D B E (2) E D B C A (1) D C B E A (1) B E D C A (1) Total count = 100 A B C D E A 0 0 0 -4 0 B 0 0 -6 -4 0 C 0 6 0 0 -12 D 4 4 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.161350 B: 0.258837 C: 0.000000 D: 0.000000 E: 0.579813 Sum of squares = 0.429213600198 Cumulative probabilities = A: 0.161350 B: 0.420187 C: 0.420187 D: 0.420187 E: 1.000000 A B C D E A 0 0 0 -4 0 B 0 0 -6 -4 0 C 0 6 0 0 -12 D 4 4 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000043195 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=26 B=26 A=20 C=16 D=12 so D is eliminated. Round 2 votes counts: B=31 E=26 C=23 A=20 so A is eliminated. Round 3 votes counts: E=41 B=31 C=28 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:208 D:202 A:198 C:197 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -4 0 B 0 0 -6 -4 0 C 0 6 0 0 -12 D 4 4 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000043195 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 0 B 0 0 -6 -4 0 C 0 6 0 0 -12 D 4 4 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000043195 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 0 B 0 0 -6 -4 0 C 0 6 0 0 -12 D 4 4 0 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000043195 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7138: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (13) A C E D B (10) B D E C A (8) A D E B C (6) B E D C A (5) D B E A C (4) C B D E A (4) A E C D B (4) D B A E C (3) C E B A D (3) C E A B D (3) A E D C B (3) E D B A C (2) E B D C A (2) E A C B D (2) D B C A E (2) C B E D A (2) C A E D B (2) C A B D E (2) B D C E A (2) A E D B C (2) A D B C E (2) E D A B C (1) E C B D A (1) E A D B C (1) D A B C E (1) C E B D A (1) C B D A E (1) B D C A E (1) B C D E A (1) A D E C B (1) A D C E B (1) A D C B E (1) A D B E C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 16 -6 18 16 B -16 0 -12 4 -20 C 6 12 0 2 8 D -18 -4 -2 0 -16 E -16 20 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -6 18 16 B -16 0 -12 4 -20 C 6 12 0 2 8 D -18 -4 -2 0 -16 E -16 20 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=31 B=17 D=10 E=9 so E is eliminated. Round 2 votes counts: A=36 C=32 B=19 D=13 so D is eliminated. Round 3 votes counts: A=38 C=32 B=30 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:222 C:214 E:206 D:180 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -6 18 16 B -16 0 -12 4 -20 C 6 12 0 2 8 D -18 -4 -2 0 -16 E -16 20 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 18 16 B -16 0 -12 4 -20 C 6 12 0 2 8 D -18 -4 -2 0 -16 E -16 20 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 18 16 B -16 0 -12 4 -20 C 6 12 0 2 8 D -18 -4 -2 0 -16 E -16 20 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7139: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) B A D C E (6) E C A D B (5) A B D E C (5) C E D A B (4) B A E C D (4) A B E D C (4) E C D A B (3) E C A B D (3) C E B D A (3) C E B A D (3) C E A B D (3) C D E B A (3) B D A C E (3) B C D E A (3) B A D E C (3) A E B C D (3) E A C B D (2) D E C A B (2) D E A C B (2) D C E B A (2) D C B E A (2) D B C A E (2) D A B E C (2) C B E D A (2) B D C A E (2) E A C D B (1) D C E A B (1) D B C E A (1) D B A E C (1) D B A C E (1) D A E C B (1) D A E B C (1) B C E A D (1) B C D A E (1) B C A D E (1) A E D C B (1) A E D B C (1) A E C B D (1) A D E C B (1) A D E B C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -8 2 -6 B 2 0 -4 8 -8 C 8 4 0 8 2 D -2 -8 -8 0 -4 E 6 8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 2 -6 B 2 0 -4 8 -8 C 8 4 0 8 2 D -2 -8 -8 0 -4 E 6 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=24 A=19 D=18 E=14 so E is eliminated. Round 2 votes counts: C=36 B=24 A=22 D=18 so D is eliminated. Round 3 votes counts: C=43 B=29 A=28 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:211 E:208 B:199 A:193 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 2 -6 B 2 0 -4 8 -8 C 8 4 0 8 2 D -2 -8 -8 0 -4 E 6 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 2 -6 B 2 0 -4 8 -8 C 8 4 0 8 2 D -2 -8 -8 0 -4 E 6 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 2 -6 B 2 0 -4 8 -8 C 8 4 0 8 2 D -2 -8 -8 0 -4 E 6 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7140: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) D B A C E (7) B D A E C (7) E C A B D (6) E A C B D (6) D B C A E (5) B D A C E (5) E C A D B (4) D B C E A (4) A B D C E (4) C D B E A (3) C D B A E (3) A B D E C (3) E C D B A (2) E A B D C (2) E A B C D (2) D C B E A (2) C E D B A (2) C E D A B (2) C A E D B (2) A E C B D (2) A E B D C (2) E D C B A (1) E B D A C (1) D B E C A (1) C E A B D (1) C A E B D (1) C A D B E (1) B E D A C (1) B A E D C (1) B A D E C (1) B A D C E (1) A E B C D (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 0 8 0 B -6 0 -2 0 4 C 0 2 0 2 12 D -8 0 -2 0 -2 E 0 -4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.541583 B: 0.000000 C: 0.458417 D: 0.000000 E: 0.000000 Sum of squares = 0.503458287712 Cumulative probabilities = A: 0.541583 B: 0.541583 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 8 0 B -6 0 -2 0 4 C 0 2 0 2 12 D -8 0 -2 0 -2 E 0 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 D=19 B=16 A=16 so B is eliminated. Round 2 votes counts: D=31 E=25 C=25 A=19 so A is eliminated. Round 3 votes counts: D=40 E=32 C=28 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:208 A:207 B:198 D:194 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 8 0 B -6 0 -2 0 4 C 0 2 0 2 12 D -8 0 -2 0 -2 E 0 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 8 0 B -6 0 -2 0 4 C 0 2 0 2 12 D -8 0 -2 0 -2 E 0 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 8 0 B -6 0 -2 0 4 C 0 2 0 2 12 D -8 0 -2 0 -2 E 0 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7141: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (6) A D C B E (6) E C D B A (5) E B A C D (4) E A C D B (4) C D A E B (4) B E A D C (4) A E C D B (4) A B E D C (4) A B D C E (4) E A B C D (3) D C A B E (3) D B C E A (3) B D C E A (3) B A E D C (3) E C D A B (2) E C A D B (2) E B D C A (2) E B C D A (2) D C B E A (2) C D E A B (2) C D B E A (2) C D A B E (2) B E D C A (2) B E D A C (2) B D C A E (2) A E B C D (2) A C E D B (2) A C D B E (2) E C B D A (1) E B C A D (1) E B A D C (1) E A C B D (1) D C B A E (1) D B C A E (1) C E D B A (1) C E D A B (1) B A D E C (1) B A D C E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 0 2 -12 B 0 0 -8 -12 -2 C 0 8 0 8 -2 D -2 12 -8 0 -6 E 12 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 2 -12 B 0 0 -8 -12 -2 C 0 8 0 8 -2 D -2 12 -8 0 -6 E 12 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 C=18 B=18 D=10 so D is eliminated. Round 2 votes counts: E=28 A=26 C=24 B=22 so B is eliminated. Round 3 votes counts: E=36 C=33 A=31 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:207 D:198 A:195 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 2 -12 B 0 0 -8 -12 -2 C 0 8 0 8 -2 D -2 12 -8 0 -6 E 12 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -12 B 0 0 -8 -12 -2 C 0 8 0 8 -2 D -2 12 -8 0 -6 E 12 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -12 B 0 0 -8 -12 -2 C 0 8 0 8 -2 D -2 12 -8 0 -6 E 12 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7142: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) E B A D C (6) C A D B E (6) A C D B E (6) E B D C A (5) B C A D E (5) A C D E B (5) D A E C B (4) B E C D A (4) E D A C B (3) E B D A C (3) D C A E B (3) D A C E B (3) C D A B E (3) B C D E A (3) E D B C A (2) D E C A B (2) C B A D E (2) B E D C A (2) B E C A D (2) B C E D A (2) E D B A C (1) E B A C D (1) E A D C B (1) E A B D C (1) D E A C B (1) D C A B E (1) C D A E B (1) C B D A E (1) C A B D E (1) B E A C D (1) B D C E A (1) B C E A D (1) B C D A E (1) B C A E D (1) B A E C D (1) A E D C B (1) A E C D B (1) A D E C B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 2 8 16 B -10 0 -12 -8 -8 C -2 12 0 0 14 D -8 8 0 0 22 E -16 8 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 8 16 B -10 0 -12 -8 -8 C -2 12 0 0 14 D -8 8 0 0 22 E -16 8 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=24 E=23 D=14 C=14 so D is eliminated. Round 2 votes counts: A=32 E=26 B=24 C=18 so C is eliminated. Round 3 votes counts: A=47 B=27 E=26 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:212 D:211 B:181 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 8 16 B -10 0 -12 -8 -8 C -2 12 0 0 14 D -8 8 0 0 22 E -16 8 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 8 16 B -10 0 -12 -8 -8 C -2 12 0 0 14 D -8 8 0 0 22 E -16 8 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 8 16 B -10 0 -12 -8 -8 C -2 12 0 0 14 D -8 8 0 0 22 E -16 8 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7143: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (12) D C E A B (7) B A D E C (6) D E C A B (5) D C E B A (5) D B A E C (4) C E D A B (4) C E B A D (4) B A C E D (4) D C B E A (3) D A B E C (3) B D C A E (3) B A E D C (3) A B E C D (3) E A C D B (2) C D E A B (2) C B E A D (2) B C A D E (2) A E C B D (2) A D E B C (2) A D B E C (2) E D C A B (1) E C A B D (1) E A D C B (1) D E A C B (1) D E A B C (1) D B C A E (1) D B A C E (1) D A E B C (1) C E D B A (1) C E B D A (1) C E A D B (1) C E A B D (1) C B A E D (1) B D A E C (1) B C A E D (1) B A D C E (1) B A C D E (1) A E B D C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 8 10 14 B 14 0 10 4 12 C -8 -10 0 -10 -8 D -10 -4 10 0 6 E -14 -12 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 10 14 B 14 0 10 4 12 C -8 -10 0 -10 -8 D -10 -4 10 0 6 E -14 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=32 C=17 A=12 E=5 so E is eliminated. Round 2 votes counts: B=34 D=33 C=18 A=15 so A is eliminated. Round 3 votes counts: B=40 D=38 C=22 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:209 D:201 E:188 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 10 14 B 14 0 10 4 12 C -8 -10 0 -10 -8 D -10 -4 10 0 6 E -14 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 10 14 B 14 0 10 4 12 C -8 -10 0 -10 -8 D -10 -4 10 0 6 E -14 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 10 14 B 14 0 10 4 12 C -8 -10 0 -10 -8 D -10 -4 10 0 6 E -14 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7144: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (6) A C B E D (5) D E A B C (4) D A E C B (4) D A B E C (4) C E D B A (4) C E B A D (4) C A E B D (4) A D B E C (4) A B C E D (4) E D C B A (3) D E C B A (3) D B A E C (3) C A B E D (3) B A C E D (3) A D B C E (3) D E C A B (2) D E B C A (2) D E B A C (2) D C E A B (2) D B E A C (2) C B E A D (2) C A E D B (2) B A E D C (2) A C D E B (2) A B D E C (2) A B C D E (2) E D B C A (1) E C D B A (1) E C B D A (1) E B D C A (1) E B C D A (1) D E A C B (1) D A E B C (1) C E D A B (1) C E B D A (1) C D E B A (1) C B A E D (1) B E D A C (1) B D A E C (1) B C E A D (1) B A E C D (1) B A D E C (1) A C D B E (1) Total count = 100 A B C D E A 0 14 18 6 18 B -14 0 -8 -6 4 C -18 8 0 2 4 D -6 6 -2 0 6 E -18 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 6 18 B -14 0 -8 -6 4 C -18 8 0 2 4 D -6 6 -2 0 6 E -18 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=29 C=23 B=10 E=8 so E is eliminated. Round 2 votes counts: D=34 A=29 C=25 B=12 so B is eliminated. Round 3 votes counts: D=37 A=36 C=27 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 D:202 C:198 B:188 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 6 18 B -14 0 -8 -6 4 C -18 8 0 2 4 D -6 6 -2 0 6 E -18 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 6 18 B -14 0 -8 -6 4 C -18 8 0 2 4 D -6 6 -2 0 6 E -18 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 6 18 B -14 0 -8 -6 4 C -18 8 0 2 4 D -6 6 -2 0 6 E -18 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7145: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) A B E D C (10) A B C E D (9) E D B A C (7) E B D A C (6) C D E B A (6) C D E A B (5) C D A E B (5) B A E D C (5) A B E C D (5) C A B D E (4) B E A D C (4) E D B C A (3) D C E B A (3) C A D B E (3) C D A B E (2) A C B D E (2) D E C A B (1) D E B C A (1) C D B E A (1) C B D E A (1) C B A E D (1) C B A D E (1) B E D C A (1) B E D A C (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 2 -8 -2 B 6 0 6 4 4 C -2 -6 0 -6 -10 D 8 -4 6 0 -8 E 2 -4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -8 -2 B 6 0 6 4 4 C -2 -6 0 -6 -10 D 8 -4 6 0 -8 E 2 -4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=27 E=16 D=16 B=12 so B is eliminated. Round 2 votes counts: A=33 C=29 E=22 D=16 so D is eliminated. Round 3 votes counts: E=35 A=33 C=32 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:208 D:201 A:193 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -8 -2 B 6 0 6 4 4 C -2 -6 0 -6 -10 D 8 -4 6 0 -8 E 2 -4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -8 -2 B 6 0 6 4 4 C -2 -6 0 -6 -10 D 8 -4 6 0 -8 E 2 -4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -8 -2 B 6 0 6 4 4 C -2 -6 0 -6 -10 D 8 -4 6 0 -8 E 2 -4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7146: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) D E C A B (9) E D B C A (8) B E D A C (7) C D A E B (6) C A D E B (5) E D C B A (4) B A C D E (4) A C B E D (4) A B C E D (4) B A E C D (3) A C D B E (3) A C B D E (3) A B C D E (3) D E B C A (2) D C A E B (2) C E D A B (2) C D E A B (2) B A E D C (2) B A D C E (2) E C B A D (1) E B D A C (1) D E C B A (1) D C E A B (1) D B E A C (1) C E A D B (1) C A E D B (1) C A D B E (1) B D E A C (1) B D A E C (1) B D A C E (1) B A D E C (1) Total count = 100 A B C D E A 0 -6 8 2 18 B 6 0 8 2 8 C -8 -8 0 12 16 D -2 -2 -12 0 -2 E -18 -8 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 2 18 B 6 0 8 2 8 C -8 -8 0 12 16 D -2 -2 -12 0 -2 E -18 -8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=18 A=17 D=16 E=14 so E is eliminated. Round 2 votes counts: B=36 D=28 C=19 A=17 so A is eliminated. Round 3 votes counts: B=43 C=29 D=28 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 A:211 C:206 D:191 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 2 18 B 6 0 8 2 8 C -8 -8 0 12 16 D -2 -2 -12 0 -2 E -18 -8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 2 18 B 6 0 8 2 8 C -8 -8 0 12 16 D -2 -2 -12 0 -2 E -18 -8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 2 18 B 6 0 8 2 8 C -8 -8 0 12 16 D -2 -2 -12 0 -2 E -18 -8 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7147: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) B C A E D (8) A D E B C (7) D E C A B (5) B C A D E (5) A E D B C (5) A B C D E (5) D C E B A (4) A B E C D (4) E D C B A (3) D E C B A (3) D C B E A (3) C B D E A (3) C B A D E (3) B C E A D (3) B A C E D (3) E D A B C (2) D A E C B (2) C D B E A (2) C B E D A (2) A B C E D (2) E D A C B (1) E A D B C (1) E A B C D (1) D E A B C (1) D C A B E (1) D A E B C (1) C E B D A (1) C D E B A (1) C D B A E (1) C B E A D (1) C B D A E (1) C B A E D (1) C A B D E (1) B E A C D (1) A E B D C (1) A E B C D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -4 10 8 B -2 0 6 -4 4 C 4 -6 0 0 4 D -10 4 0 0 16 E -8 -4 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 10 8 B -2 0 6 -4 4 C 4 -6 0 0 4 D -10 4 0 0 16 E -8 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888826 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=27 B=20 C=17 E=8 so E is eliminated. Round 2 votes counts: D=34 A=29 B=20 C=17 so C is eliminated. Round 3 votes counts: D=38 B=32 A=30 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:208 D:205 B:202 C:201 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -4 10 8 B -2 0 6 -4 4 C 4 -6 0 0 4 D -10 4 0 0 16 E -8 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888826 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 10 8 B -2 0 6 -4 4 C 4 -6 0 0 4 D -10 4 0 0 16 E -8 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888826 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 10 8 B -2 0 6 -4 4 C 4 -6 0 0 4 D -10 4 0 0 16 E -8 -4 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888826 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7148: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) C E D B A (10) B D A C E (8) B D C A E (7) B A D C E (7) E C D B A (5) E A C D B (5) C D B E A (5) A B D E C (5) E C D A B (4) A E D C B (4) A E C D B (4) D B C A E (3) C E B D A (3) A E D B C (3) D B C E A (2) A E C B D (2) A D B E C (2) A B E D C (2) A B D C E (2) E A C B D (1) D B A C E (1) C D E B A (1) C B D E A (1) B C D E A (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -4 -2 4 B 8 0 -10 -20 -6 C 4 10 0 6 4 D 2 20 -6 0 -8 E -4 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -2 4 B 8 0 -10 -20 -6 C 4 10 0 6 4 D 2 20 -6 0 -8 E -4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 B=23 C=20 D=6 so D is eliminated. Round 2 votes counts: B=29 A=26 E=25 C=20 so C is eliminated. Round 3 votes counts: E=39 B=35 A=26 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:212 D:204 E:203 A:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -4 -2 4 B 8 0 -10 -20 -6 C 4 10 0 6 4 D 2 20 -6 0 -8 E -4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -2 4 B 8 0 -10 -20 -6 C 4 10 0 6 4 D 2 20 -6 0 -8 E -4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -2 4 B 8 0 -10 -20 -6 C 4 10 0 6 4 D 2 20 -6 0 -8 E -4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7149: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) E C A B D (8) D B A C E (7) A C E B D (7) D A C E B (6) D B E A C (5) D E C A B (4) B A E C D (4) A D C E B (4) D E B C A (3) D B A E C (3) C E A D B (3) C E A B D (3) B E C A D (3) B D A C E (3) A C E D B (3) D E C B A (2) C A E D B (2) B D E A C (2) B D A E C (2) B A C E D (2) E C D A B (1) E C B A D (1) E C A D B (1) D C A E B (1) D B C E A (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E A B (1) C A E B D (1) B D E C A (1) A D C B E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 6 -10 0 B -2 0 -4 -24 -4 C -6 4 0 -16 0 D 10 24 16 0 18 E 0 4 0 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -10 0 B -2 0 -4 -24 -4 C -6 4 0 -16 0 D 10 24 16 0 18 E 0 4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=44 B=17 A=17 E=11 C=11 so E is eliminated. Round 2 votes counts: D=44 C=22 B=17 A=17 so B is eliminated. Round 3 votes counts: D=52 C=25 A=23 so A is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:234 A:199 E:193 C:191 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -10 0 B -2 0 -4 -24 -4 C -6 4 0 -16 0 D 10 24 16 0 18 E 0 4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -10 0 B -2 0 -4 -24 -4 C -6 4 0 -16 0 D 10 24 16 0 18 E 0 4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -10 0 B -2 0 -4 -24 -4 C -6 4 0 -16 0 D 10 24 16 0 18 E 0 4 0 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7150: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) B A D C E (9) C E D A B (8) E C D A B (7) D B E A C (7) D B A E C (7) C E A B D (7) D E C B A (6) B A D E C (5) E C A B D (4) A B C D E (4) E C D B A (3) D C E A B (2) D B A C E (2) C E A D B (2) C D E A B (2) C A B E D (2) A C B E D (2) A B D C E (2) E D C B A (1) D C E B A (1) D C A B E (1) D B E C A (1) D B C A E (1) C A E B D (1) B D A C E (1) B A E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 2 0 -2 B -10 0 2 -2 12 C -2 -2 0 6 14 D 0 2 -6 0 2 E 2 -12 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629628896 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 10 2 0 -2 B -10 0 2 -2 12 C -2 -2 0 6 14 D 0 2 -6 0 2 E 2 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629495 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=22 A=19 B=16 E=15 so E is eliminated. Round 2 votes counts: C=36 D=29 A=19 B=16 so B is eliminated. Round 3 votes counts: C=36 A=34 D=30 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:208 A:205 B:201 D:199 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 0 -2 B -10 0 2 -2 12 C -2 -2 0 6 14 D 0 2 -6 0 2 E 2 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629495 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 0 -2 B -10 0 2 -2 12 C -2 -2 0 6 14 D 0 2 -6 0 2 E 2 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629495 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 0 -2 B -10 0 2 -2 12 C -2 -2 0 6 14 D 0 2 -6 0 2 E 2 -12 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629495 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7151: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) B D E C A (10) E C A D B (8) E D C A B (7) A C E D B (7) B D E A C (6) B D A C E (6) B A C E D (6) B D A E C (5) B A D C E (5) E C D A B (4) D B E A C (4) A C E B D (4) D E A C B (3) B A C D E (3) A C B E D (3) C E A D B (2) C A E D B (2) D B E C A (1) B C E A D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 -14 -14 B -6 0 -2 2 4 C -8 2 0 -16 -18 D 14 -2 16 0 10 E 14 -4 18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.000000 Sum of squares = 0.48760330577 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 0.727273 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -14 -14 B -6 0 -2 2 4 C -8 2 0 -16 -18 D 14 -2 16 0 10 E 14 -4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.000000 Sum of squares = 0.487603305769 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 0.727273 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 E=19 D=19 A=16 C=4 so C is eliminated. Round 2 votes counts: B=42 E=21 D=19 A=18 so A is eliminated. Round 3 votes counts: B=47 E=34 D=19 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:219 E:209 B:199 A:193 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -14 -14 B -6 0 -2 2 4 C -8 2 0 -16 -18 D 14 -2 16 0 10 E 14 -4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.000000 Sum of squares = 0.487603305769 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 0.727273 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -14 -14 B -6 0 -2 2 4 C -8 2 0 -16 -18 D 14 -2 16 0 10 E 14 -4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.000000 Sum of squares = 0.487603305769 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 0.727273 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -14 -14 B -6 0 -2 2 4 C -8 2 0 -16 -18 D 14 -2 16 0 10 E 14 -4 18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.636364 C: 0.000000 D: 0.272727 E: 0.000000 Sum of squares = 0.487603305769 Cumulative probabilities = A: 0.090909 B: 0.727273 C: 0.727273 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7152: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) B A C E D (6) D E C A B (5) C E A B D (5) B D C E A (5) B A D C E (5) E D C A B (4) A C E B D (4) A B D E C (4) A B C E D (4) E C D A B (3) D E C B A (3) D B E C A (3) D B E A C (3) D B A E C (3) C E D A B (3) E A C D B (2) D E B C A (2) C E D B A (2) C A E B D (2) B D A C E (2) B C A E D (2) B A C D E (2) A C B E D (2) E D A C B (1) D E B A C (1) D E A C B (1) D C E B A (1) D B C E A (1) C E B A D (1) C B E A D (1) B D A E C (1) B C E A D (1) B C D E A (1) B C D A E (1) B A D E C (1) A E C D B (1) A E C B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -8 8 -14 B -4 0 0 6 0 C 8 0 0 6 6 D -8 -6 -6 0 -6 E 14 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.354038 C: 0.645962 D: 0.000000 E: 0.000000 Sum of squares = 0.542609895675 Cumulative probabilities = A: 0.000000 B: 0.354038 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 8 -14 B -4 0 0 6 0 C 8 0 0 6 6 D -8 -6 -6 0 -6 E 14 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=23 E=18 A=18 C=14 so C is eliminated. Round 2 votes counts: E=29 B=28 D=23 A=20 so A is eliminated. Round 3 votes counts: B=40 E=37 D=23 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:210 E:207 B:201 A:195 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 8 -14 B -4 0 0 6 0 C 8 0 0 6 6 D -8 -6 -6 0 -6 E 14 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 8 -14 B -4 0 0 6 0 C 8 0 0 6 6 D -8 -6 -6 0 -6 E 14 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 8 -14 B -4 0 0 6 0 C 8 0 0 6 6 D -8 -6 -6 0 -6 E 14 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7153: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) E A B D C (6) A D C B E (6) E B C D A (5) B E C D A (5) E B A C D (4) D A C E B (4) A D E C B (4) E C D B A (3) E B C A D (3) C D B E A (3) C D B A E (3) C B D A E (3) B A C D E (3) E B A D C (2) E A D B C (2) D C E A B (2) D C A E B (2) D C A B E (2) D A E C B (2) D A C B E (2) B E A C D (2) B C A E D (2) A E B D C (2) A D E B C (2) A D C E B (2) A D B C E (2) A B E D C (2) E D A C B (1) E C B D A (1) D E A C B (1) C E B D A (1) C D A B E (1) C B E D A (1) C B D E A (1) B E C A D (1) B C E A D (1) B C D E A (1) B C D A E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 0 -6 -8 B 8 0 10 12 2 C 0 -10 0 6 4 D 6 -12 -6 0 -4 E 8 -2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -6 -8 B 8 0 10 12 2 C 0 -10 0 6 4 D 6 -12 -6 0 -4 E 8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=24 A=21 D=15 C=13 so C is eliminated. Round 2 votes counts: B=29 E=28 D=22 A=21 so A is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:203 C:200 D:192 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -6 -8 B 8 0 10 12 2 C 0 -10 0 6 4 D 6 -12 -6 0 -4 E 8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -6 -8 B 8 0 10 12 2 C 0 -10 0 6 4 D 6 -12 -6 0 -4 E 8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -6 -8 B 8 0 10 12 2 C 0 -10 0 6 4 D 6 -12 -6 0 -4 E 8 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999668 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7154: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (11) A D B E C (11) C E B D A (10) B E C A D (10) D A C E B (9) E B C A D (7) A D E B C (7) A D B C E (7) E B C D A (4) B A D E C (3) A D C B E (3) E C B D A (2) C D A E B (2) B A E D C (2) D C A E B (1) C E D B A (1) C E D A B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) B E A D C (1) B E A C D (1) B C E D A (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 14 8 20 B -8 0 12 -12 12 C -14 -12 0 -14 2 D -8 12 14 0 18 E -20 -12 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 8 20 B -8 0 12 -12 12 C -14 -12 0 -14 2 D -8 12 14 0 18 E -20 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=21 B=19 C=18 E=13 so E is eliminated. Round 2 votes counts: B=30 A=29 D=21 C=20 so C is eliminated. Round 3 votes counts: B=43 A=29 D=28 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:218 B:202 C:181 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 8 20 B -8 0 12 -12 12 C -14 -12 0 -14 2 D -8 12 14 0 18 E -20 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 8 20 B -8 0 12 -12 12 C -14 -12 0 -14 2 D -8 12 14 0 18 E -20 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 8 20 B -8 0 12 -12 12 C -14 -12 0 -14 2 D -8 12 14 0 18 E -20 -12 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7155: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) A B E D C (11) D E C B A (9) E D C A B (7) A B C E D (7) C D E B A (6) B A D E C (6) E D A B C (5) B C A D E (5) C E D A B (4) D C E B A (3) C E D B A (3) C B A D E (3) A B D E C (3) C B A E D (2) E D C B A (1) E A D C B (1) E A D B C (1) C D B E A (1) C B D E A (1) C A B E D (1) B C D A E (1) B A D C E (1) A E D B C (1) A E B D C (1) A D E B C (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 6 18 16 B 8 0 14 14 14 C -6 -14 0 -2 4 D -18 -14 2 0 6 E -16 -14 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 18 16 B 8 0 14 14 14 C -6 -14 0 -2 4 D -18 -14 2 0 6 E -16 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=25 C=21 E=15 D=12 so D is eliminated. Round 2 votes counts: A=27 B=25 E=24 C=24 so E is eliminated. Round 3 votes counts: C=41 A=34 B=25 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:225 A:216 C:191 D:188 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 18 16 B 8 0 14 14 14 C -6 -14 0 -2 4 D -18 -14 2 0 6 E -16 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 18 16 B 8 0 14 14 14 C -6 -14 0 -2 4 D -18 -14 2 0 6 E -16 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 18 16 B 8 0 14 14 14 C -6 -14 0 -2 4 D -18 -14 2 0 6 E -16 -14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7156: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (15) D B C E A (11) A E B D C (10) E A B D C (7) C D B E A (7) A E C B D (7) B D E A C (5) C A E D B (4) B E D A C (4) A E B C D (4) C D A E B (3) E D B A C (2) D B E C A (2) C A D E B (2) B E A D C (2) E B D A C (1) E B A D C (1) E A D B C (1) E A B C D (1) D C B E A (1) D C B A E (1) D B E A C (1) C D E B A (1) C D E A B (1) C D A B E (1) C B D A E (1) C A B D E (1) B D C A E (1) B A E D C (1) A E C D B (1) Total count = 100 A B C D E A 0 -14 -4 -16 4 B 14 0 8 -8 8 C 4 -8 0 -2 0 D 16 8 2 0 8 E -4 -8 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -16 4 B 14 0 8 -8 8 C 4 -8 0 -2 0 D 16 8 2 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995458 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=22 D=16 E=13 B=13 so E is eliminated. Round 2 votes counts: C=36 A=31 D=18 B=15 so B is eliminated. Round 3 votes counts: C=36 A=35 D=29 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 B:211 C:197 E:190 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -4 -16 4 B 14 0 8 -8 8 C 4 -8 0 -2 0 D 16 8 2 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995458 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -16 4 B 14 0 8 -8 8 C 4 -8 0 -2 0 D 16 8 2 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995458 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -16 4 B 14 0 8 -8 8 C 4 -8 0 -2 0 D 16 8 2 0 8 E -4 -8 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995458 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7157: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) E A B C D (7) B A D C E (5) E C D A B (4) D B C A E (4) C E D B A (4) C D E B A (4) C D B E A (4) A B E D C (4) E C A B D (3) D C E A B (3) D B A C E (3) D A B E C (3) C D B A E (3) E C A D B (2) E A C D B (2) E A C B D (2) E A B D C (2) D E C A B (2) D C E B A (2) C E D A B (2) B C A E D (2) B C A D E (2) B A D E C (2) B A C D E (2) E D A C B (1) E D A B C (1) E C B A D (1) E A D C B (1) E A D B C (1) D E A B C (1) C E A D B (1) C B E A D (1) C B A D E (1) B D C A E (1) B D A C E (1) B C E A D (1) B A E D C (1) B A E C D (1) B A C E D (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -14 -6 -4 B 12 0 -6 -18 6 C 14 6 0 2 14 D 6 18 -2 0 8 E 4 -6 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999441 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -6 -4 B 12 0 -6 -18 6 C 14 6 0 2 14 D 6 18 -2 0 8 E 4 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 C=20 B=19 A=6 so A is eliminated. Round 2 votes counts: D=29 E=28 B=23 C=20 so C is eliminated. Round 3 votes counts: D=40 E=35 B=25 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:215 B:197 E:188 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -6 -4 B 12 0 -6 -18 6 C 14 6 0 2 14 D 6 18 -2 0 8 E 4 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -6 -4 B 12 0 -6 -18 6 C 14 6 0 2 14 D 6 18 -2 0 8 E 4 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -6 -4 B 12 0 -6 -18 6 C 14 6 0 2 14 D 6 18 -2 0 8 E 4 -6 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7158: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (13) E D C B A (7) A B C E D (7) C D E A B (6) B A C D E (6) E D C A B (5) B E D A C (5) B A E D C (5) C D A E B (4) D E C B A (3) D E B C A (3) B A D C E (3) A C E B D (3) A C B D E (3) E B D A C (2) D E C A B (2) D B C E A (2) C E D A B (2) C A E D B (2) C A D E B (2) C A D B E (2) E D B C A (1) E A C D B (1) D C E B A (1) C E A D B (1) C D E B A (1) B E A D C (1) B D E A C (1) B D C E A (1) B A D E C (1) B A C E D (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 12 10 8 10 B -12 0 6 8 8 C -10 -6 0 14 24 D -8 -8 -14 0 10 E -10 -8 -24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 8 10 B -12 0 6 8 8 C -10 -6 0 14 24 D -8 -8 -14 0 10 E -10 -8 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=24 C=20 E=16 D=11 so D is eliminated. Round 2 votes counts: A=29 B=26 E=24 C=21 so C is eliminated. Round 3 votes counts: A=39 E=35 B=26 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:211 B:205 D:190 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 8 10 B -12 0 6 8 8 C -10 -6 0 14 24 D -8 -8 -14 0 10 E -10 -8 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 8 10 B -12 0 6 8 8 C -10 -6 0 14 24 D -8 -8 -14 0 10 E -10 -8 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 8 10 B -12 0 6 8 8 C -10 -6 0 14 24 D -8 -8 -14 0 10 E -10 -8 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7159: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) A E D C B (6) D B C E A (5) E B D C A (4) E A D C B (4) E A B C D (4) D C B A E (4) D A C E B (4) B E C D A (4) B C E A D (4) A D C B E (4) E B C D A (3) E A D B C (3) C A B D E (3) B E C A D (3) A D E C B (3) E B C A D (2) E B A C D (2) D E C B A (2) D E A C B (2) D C A B E (2) C D B A E (2) C B D A E (2) B C E D A (2) B C D E A (2) A E C B D (2) A C D B E (2) E D B C A (1) E D A B C (1) E A B D C (1) D E B A C (1) D B E C A (1) D A E C B (1) C D A B E (1) C B A D E (1) B E A C D (1) B C D A E (1) B C A D E (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 0 -4 -4 B -4 0 -8 -14 6 C 0 8 0 -12 -4 D 4 14 12 0 2 E 4 -6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -4 -4 B -4 0 -8 -14 6 C 0 8 0 -12 -4 D 4 14 12 0 2 E 4 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 A=19 B=18 C=9 so C is eliminated. Round 2 votes counts: D=32 E=25 A=22 B=21 so B is eliminated. Round 3 votes counts: E=39 D=37 A=24 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:200 A:198 C:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -4 -4 B -4 0 -8 -14 6 C 0 8 0 -12 -4 D 4 14 12 0 2 E 4 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -4 -4 B -4 0 -8 -14 6 C 0 8 0 -12 -4 D 4 14 12 0 2 E 4 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -4 -4 B -4 0 -8 -14 6 C 0 8 0 -12 -4 D 4 14 12 0 2 E 4 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996004 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7160: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) A C D E B (6) D E B C A (5) C E D A B (5) B A E C D (5) B A D E C (5) B D E A C (4) A C B D E (4) A B D C E (4) A B C D E (4) D E C B A (3) D E C A B (3) C D E A B (3) B E D A C (3) A C E D B (3) A C D B E (3) E D C B A (2) E D B C A (2) E C D B A (2) C E D B A (2) B A E D C (2) A C B E D (2) E B D C A (1) D E A C B (1) D C E A B (1) D C A E B (1) D A E B C (1) D A C E B (1) C E B D A (1) C A E D B (1) B E C D A (1) B E C A D (1) B D E C A (1) B A C E D (1) A D C B E (1) A D B C E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 6 -10 -6 B 6 0 10 6 12 C -6 -10 0 -10 -10 D 10 -6 10 0 6 E 6 -12 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -10 -6 B 6 0 10 6 12 C -6 -10 0 -10 -10 D 10 -6 10 0 6 E 6 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=30 D=16 C=12 E=7 so E is eliminated. Round 2 votes counts: B=36 A=30 D=20 C=14 so C is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:210 E:199 A:192 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -10 -6 B 6 0 10 6 12 C -6 -10 0 -10 -10 D 10 -6 10 0 6 E 6 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -10 -6 B 6 0 10 6 12 C -6 -10 0 -10 -10 D 10 -6 10 0 6 E 6 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -10 -6 B 6 0 10 6 12 C -6 -10 0 -10 -10 D 10 -6 10 0 6 E 6 -12 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7161: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) A D C B E (7) A D B E C (7) E C A D B (5) E C A B D (5) C E B D A (5) A E C D B (5) D A B C E (4) B D C E A (4) B D A E C (4) E C B A D (3) E A C D B (3) B D C A E (3) A D B C E (3) A B D E C (3) E B C D A (2) C E A D B (2) C D B E A (2) C D B A E (2) C B E D A (2) B E D C A (2) B D E A C (2) A E D C B (2) E B C A D (1) E A C B D (1) E A B D C (1) D C B A E (1) D B C A E (1) D B A C E (1) D A B E C (1) C D A B E (1) C B D E A (1) C A E D B (1) C A D E B (1) B E D A C (1) B D A C E (1) A E D B C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 10 -2 6 2 B -10 0 -16 -4 6 C 2 16 0 -2 -14 D -6 4 2 0 2 E -2 -6 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629693 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 10 -2 6 2 B -10 0 -16 -4 6 C 2 16 0 -2 -14 D -6 4 2 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629744 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 C=17 B=17 D=8 so D is eliminated. Round 2 votes counts: A=35 E=28 B=19 C=18 so C is eliminated. Round 3 votes counts: A=38 E=35 B=27 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:202 C:201 D:201 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 6 2 B -10 0 -16 -4 6 C 2 16 0 -2 -14 D -6 4 2 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629744 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 6 2 B -10 0 -16 -4 6 C 2 16 0 -2 -14 D -6 4 2 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629744 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 6 2 B -10 0 -16 -4 6 C 2 16 0 -2 -14 D -6 4 2 0 2 E -2 -6 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.000000 C: 0.111111 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629744 Cumulative probabilities = A: 0.777778 B: 0.777778 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7162: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) E D A B C (8) E B C D A (8) D A E B C (8) C B E A D (6) A D C B E (6) E B D C A (5) B C E D A (5) A D E C B (5) E C B A D (4) D A B C E (4) B C A D E (4) E D A C B (3) B C D A E (3) A E D C B (3) E B D A C (2) D E A B C (2) B E C D A (2) B C E A D (2) A C D B E (2) E D B A C (1) E B C A D (1) E A D C B (1) D A E C B (1) D A B E C (1) C A D B E (1) C A B D E (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -4 -6 0 B 6 0 12 6 -6 C 4 -12 0 -2 -10 D 6 -6 2 0 -2 E 0 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.178120 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.821880 Sum of squares = 0.707213050835 Cumulative probabilities = A: 0.178120 B: 0.178120 C: 0.178120 D: 0.178120 E: 1.000000 A B C D E A 0 -6 -4 -6 0 B 6 0 12 6 -6 C 4 -12 0 -2 -10 D 6 -6 2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000237 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=18 A=17 D=16 B=16 so D is eliminated. Round 2 votes counts: E=35 A=31 C=18 B=16 so B is eliminated. Round 3 votes counts: E=37 C=32 A=31 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:209 E:209 D:200 A:192 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -6 0 B 6 0 12 6 -6 C 4 -12 0 -2 -10 D 6 -6 2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000237 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -6 0 B 6 0 12 6 -6 C 4 -12 0 -2 -10 D 6 -6 2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000237 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -6 0 B 6 0 12 6 -6 C 4 -12 0 -2 -10 D 6 -6 2 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000000237 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7163: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) C D B A E (9) A E C D B (8) C D A B E (6) B D C A E (6) C A D B E (5) E A C D B (4) B D A C E (4) E B D A C (3) E A B C D (3) A E B D C (3) A C E D B (3) A C D E B (3) A C D B E (3) A B D C E (3) E B A D C (2) D B C E A (2) C D B E A (2) B D C E A (2) A E C B D (2) E C D B A (1) E C A D B (1) E B D C A (1) E B C A D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B C A E (1) C D A E B (1) B E D C A (1) B E D A C (1) B D A E C (1) B A E D C (1) B A D E C (1) A E B C D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 18 18 14 28 B -18 0 -2 -4 4 C -18 2 0 8 6 D -14 4 -8 0 4 E -28 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 18 14 28 B -18 0 -2 -4 4 C -18 2 0 8 6 D -14 4 -8 0 4 E -28 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 C=23 B=17 D=5 so D is eliminated. Round 2 votes counts: A=28 E=27 C=25 B=20 so B is eliminated. Round 3 votes counts: C=36 A=35 E=29 so E is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:239 C:199 D:193 B:190 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 18 14 28 B -18 0 -2 -4 4 C -18 2 0 8 6 D -14 4 -8 0 4 E -28 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 14 28 B -18 0 -2 -4 4 C -18 2 0 8 6 D -14 4 -8 0 4 E -28 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 14 28 B -18 0 -2 -4 4 C -18 2 0 8 6 D -14 4 -8 0 4 E -28 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7164: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) C E D B A (6) A B E D C (6) D E C B A (5) D E B A C (5) C A B E D (5) A B C E D (5) D C E B A (4) D A B E C (4) C E B D A (4) A B E C D (4) E B C D A (3) C E B A D (3) C B A E D (3) C A D B E (3) A B D E C (3) E C B D A (2) E B C A D (2) E B A D C (2) D E B C A (2) D A E B C (2) C D E A B (2) B A E D C (2) A D B E C (2) E D B C A (1) E D B A C (1) D E A B C (1) D C E A B (1) D C A E B (1) C D A E B (1) C B E A D (1) B E A D C (1) B E A C D (1) B A E C D (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -12 -4 -8 B 12 0 -2 -2 -10 C 12 2 0 10 0 D 4 2 -10 0 -8 E 8 10 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.346957 D: 0.000000 E: 0.653043 Sum of squares = 0.54684448603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.346957 D: 0.346957 E: 1.000000 A B C D E A 0 -12 -12 -4 -8 B 12 0 -2 -2 -10 C 12 2 0 10 0 D 4 2 -10 0 -8 E 8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=25 A=24 E=11 B=5 so B is eliminated. Round 2 votes counts: C=35 A=27 D=25 E=13 so E is eliminated. Round 3 votes counts: C=42 A=31 D=27 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:212 B:199 D:194 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -12 -4 -8 B 12 0 -2 -2 -10 C 12 2 0 10 0 D 4 2 -10 0 -8 E 8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -4 -8 B 12 0 -2 -2 -10 C 12 2 0 10 0 D 4 2 -10 0 -8 E 8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -4 -8 B 12 0 -2 -2 -10 C 12 2 0 10 0 D 4 2 -10 0 -8 E 8 10 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7165: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) C A D B E (11) A C B D E (9) E D B C A (8) D E B C A (7) E B D A C (6) D E C A B (6) E D C A B (5) E D B A C (4) B E D A C (4) B A C D E (4) D C E A B (3) E B A C D (2) D C A E B (2) C A D E B (2) C A B D E (2) B E D C A (2) B E A C D (2) A C B E D (2) E B D C A (1) E A B C D (1) C D A B E (1) B C A D E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -2 2 -2 B 8 0 10 -2 2 C 2 -10 0 4 4 D -2 2 -4 0 -2 E 2 -2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333193 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 -8 -2 2 -2 B 8 0 10 -2 2 C 2 -10 0 4 4 D -2 2 -4 0 -2 E 2 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332869 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 D=18 C=16 A=13 so A is eliminated. Round 2 votes counts: C=29 E=27 B=26 D=18 so D is eliminated. Round 3 votes counts: E=40 C=34 B=26 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:209 C:200 E:199 D:197 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 2 -2 B 8 0 10 -2 2 C 2 -10 0 4 4 D -2 2 -4 0 -2 E 2 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332869 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 2 -2 B 8 0 10 -2 2 C 2 -10 0 4 4 D -2 2 -4 0 -2 E 2 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332869 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 2 -2 B 8 0 10 -2 2 C 2 -10 0 4 4 D -2 2 -4 0 -2 E 2 -2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333332869 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7166: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) E B D A C (6) C D A B E (6) C A D B E (6) B E A D C (6) C A B E D (5) C D E B A (4) C B E A D (4) C A B D E (4) A C B E D (4) C D A E B (3) A C D B E (3) E B A D C (2) E B A C D (2) D E B C A (2) D C E B A (2) D C A B E (2) D A E B C (2) D A C B E (2) D A B E C (2) C E B A D (2) C D E A B (2) A B E D C (2) A B C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E C B D A (1) E B D C A (1) E B C D A (1) E B C A D (1) D E C B A (1) D E B A C (1) D E A B C (1) D C E A B (1) D C A E B (1) C E D B A (1) C B A E D (1) B E C A D (1) B A E D C (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 10 -4 B 0 0 -14 8 20 C 8 14 0 22 12 D -10 -8 -22 0 -8 E 4 -20 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 10 -4 B 0 0 -14 8 20 C 8 14 0 22 12 D -10 -8 -22 0 -8 E 4 -20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=17 D=17 B=15 A=13 so A is eliminated. Round 2 votes counts: C=45 B=20 D=18 E=17 so E is eliminated. Round 3 votes counts: C=46 B=33 D=21 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:207 A:199 E:190 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 10 -4 B 0 0 -14 8 20 C 8 14 0 22 12 D -10 -8 -22 0 -8 E 4 -20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 10 -4 B 0 0 -14 8 20 C 8 14 0 22 12 D -10 -8 -22 0 -8 E 4 -20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 10 -4 B 0 0 -14 8 20 C 8 14 0 22 12 D -10 -8 -22 0 -8 E 4 -20 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7167: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E C B A D (10) A D E C B (7) C B E D A (6) D A B E C (5) C E B A D (5) B C E D A (5) E A C B D (4) D B C A E (4) D A E B C (4) B C E A D (4) A E D C B (4) A D E B C (4) D A E C B (3) C B E A D (3) B C D E A (3) B C D A E (3) E C A D B (2) E A C D B (2) B D C A E (2) E B C A D (1) E A D C B (1) D C A B E (1) D A C B E (1) B D A C E (1) B A E D C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 2 -2 8 B -4 0 2 -2 6 C -2 -2 0 -4 2 D 2 2 4 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -2 8 B -4 0 2 -2 6 C -2 -2 0 -4 2 D 2 2 4 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=20 B=19 A=16 C=14 so C is eliminated. Round 2 votes counts: D=31 B=28 E=25 A=16 so A is eliminated. Round 3 votes counts: D=42 E=29 B=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:206 D:205 B:201 C:197 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -2 8 B -4 0 2 -2 6 C -2 -2 0 -4 2 D 2 2 4 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -2 8 B -4 0 2 -2 6 C -2 -2 0 -4 2 D 2 2 4 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -2 8 B -4 0 2 -2 6 C -2 -2 0 -4 2 D 2 2 4 0 2 E -8 -6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7168: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E B C D A (6) A D C B E (6) C D B A E (5) C E A B D (4) B D E A C (4) E B D C A (3) E A B D C (3) D C B A E (3) D C A B E (3) D B A E C (3) D A B C E (3) C A D E B (3) A D B E C (3) A C D E B (3) E C B A D (2) E B A D C (2) E A B C D (2) D B A C E (2) D A C B E (2) C E B D A (2) C D E B A (2) C A D B E (2) B D E C A (2) A E C B D (2) A E B D C (2) A D C E B (2) E C A B D (1) E B D A C (1) E B A C D (1) D A B E C (1) C E B A D (1) C B E D A (1) C B D E A (1) C A E D B (1) B E D C A (1) B E D A C (1) B E C D A (1) B C D E A (1) A E D B C (1) A E C D B (1) A D E C B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -4 -10 16 B -10 0 -14 -12 6 C 4 14 0 2 12 D 10 12 -2 0 20 E -16 -6 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -10 16 B -10 0 -14 -12 6 C 4 14 0 2 12 D 10 12 -2 0 20 E -16 -6 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=23 E=21 D=17 B=10 so B is eliminated. Round 2 votes counts: C=30 E=24 D=23 A=23 so D is eliminated. Round 3 votes counts: C=36 A=34 E=30 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:216 A:206 B:185 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -10 16 B -10 0 -14 -12 6 C 4 14 0 2 12 D 10 12 -2 0 20 E -16 -6 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -10 16 B -10 0 -14 -12 6 C 4 14 0 2 12 D 10 12 -2 0 20 E -16 -6 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -10 16 B -10 0 -14 -12 6 C 4 14 0 2 12 D 10 12 -2 0 20 E -16 -6 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7169: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) A E D B C (8) B C E D A (7) B C E A D (7) B C A E D (7) D A E C B (5) A E D C B (5) B C D A E (4) A E B C D (4) A D E B C (4) B A C D E (3) E D A C B (2) E A D C B (2) E A B C D (2) D E C A B (2) D E A C B (2) D C E B A (2) D C B E A (2) D C B A E (2) B C D E A (2) A E B D C (2) A B E D C (2) A B E C D (2) E D C B A (1) D E C B A (1) D A C B E (1) C D B E A (1) C B E D A (1) C B D A E (1) B C A D E (1) A D E C B (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 4 8 B 8 0 14 16 14 C 6 -14 0 10 10 D -4 -16 -10 0 -6 E -8 -14 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 4 8 B 8 0 14 16 14 C 6 -14 0 10 10 D -4 -16 -10 0 -6 E -8 -14 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=30 D=17 C=15 E=7 so E is eliminated. Round 2 votes counts: A=34 B=31 D=20 C=15 so C is eliminated. Round 3 votes counts: B=45 A=34 D=21 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:206 A:199 E:187 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 4 8 B 8 0 14 16 14 C 6 -14 0 10 10 D -4 -16 -10 0 -6 E -8 -14 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 4 8 B 8 0 14 16 14 C 6 -14 0 10 10 D -4 -16 -10 0 -6 E -8 -14 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 4 8 B 8 0 14 16 14 C 6 -14 0 10 10 D -4 -16 -10 0 -6 E -8 -14 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7170: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) B C D A E (9) C B D E A (8) C E B D A (7) E A D C B (5) A D B E C (5) B D C A E (4) A E C D B (4) E C D A B (3) E C A D B (3) D A B E C (3) C E B A D (3) C E A B D (3) A E D B C (3) E D A C B (2) D E A B C (2) D B C E A (2) C B E D A (2) C B A E D (2) A C E B D (2) E A C D B (1) D B E C A (1) D B A E C (1) D A E B C (1) C E D A B (1) C E A D B (1) C B E A D (1) C A E B D (1) B D A E C (1) B C D E A (1) B A D C E (1) A E D C B (1) A E C B D (1) A D E B C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 -20 8 B 10 0 -2 20 10 C 4 2 0 8 22 D 20 -20 -8 0 6 E -8 -10 -22 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -20 8 B 10 0 -2 20 10 C 4 2 0 8 22 D 20 -20 -8 0 6 E -8 -10 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 A=19 E=14 D=10 so D is eliminated. Round 2 votes counts: B=32 C=29 A=23 E=16 so E is eliminated. Round 3 votes counts: C=35 A=33 B=32 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:219 C:218 D:199 A:187 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -20 8 B 10 0 -2 20 10 C 4 2 0 8 22 D 20 -20 -8 0 6 E -8 -10 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -20 8 B 10 0 -2 20 10 C 4 2 0 8 22 D 20 -20 -8 0 6 E -8 -10 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -20 8 B 10 0 -2 20 10 C 4 2 0 8 22 D 20 -20 -8 0 6 E -8 -10 -22 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7171: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (6) E D C B A (5) E D B C A (5) E A D C B (5) B D C E A (5) A C B D E (5) A B C E D (5) E D B A C (4) C A D E B (4) A C E D B (4) A C D E B (4) E B D C A (3) C D E A B (3) B E D C A (3) B C D E A (3) A E D B C (3) A E C D B (3) D E C B A (2) D B E C A (2) B C A D E (2) B A E D C (2) A E B D C (2) A B E D C (2) E D A C B (1) E B D A C (1) D E C A B (1) D C E B A (1) D C E A B (1) C D E B A (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A D C (1) B D E C A (1) B C D A E (1) B A C D E (1) A E D C B (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -6 0 -8 B -10 0 -4 -14 -20 C 6 4 0 -14 -8 D 0 14 14 0 -16 E 8 20 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -6 0 -8 B -10 0 -4 -14 -20 C 6 4 0 -14 -8 D 0 14 14 0 -16 E 8 20 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=30 B=19 C=12 D=7 so D is eliminated. Round 2 votes counts: E=33 A=32 B=21 C=14 so C is eliminated. Round 3 votes counts: E=39 A=38 B=23 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:206 A:198 C:194 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -6 0 -8 B -10 0 -4 -14 -20 C 6 4 0 -14 -8 D 0 14 14 0 -16 E 8 20 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 0 -8 B -10 0 -4 -14 -20 C 6 4 0 -14 -8 D 0 14 14 0 -16 E 8 20 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 0 -8 B -10 0 -4 -14 -20 C 6 4 0 -14 -8 D 0 14 14 0 -16 E 8 20 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7172: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (13) A D B E C (11) C E D A B (8) B A D E C (8) D A E C B (7) C B E A D (6) B C E A D (5) A B D E C (5) E C D A B (4) B C A E D (4) D A B E C (3) C E D B A (3) D A E B C (2) C A E D B (2) B A D C E (2) A D E C B (2) A D B C E (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A C B (1) E C D B A (1) E C B D A (1) E B C D A (1) C E A D B (1) C B E D A (1) B C E D A (1) B C A D E (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 2 2 B 0 0 -6 -2 2 C 8 6 0 8 2 D -2 2 -8 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998458 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 2 2 B 0 0 -6 -2 2 C 8 6 0 8 2 D -2 2 -8 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=22 A=21 D=12 E=11 so E is eliminated. Round 2 votes counts: C=40 B=23 A=21 D=16 so D is eliminated. Round 3 votes counts: C=42 A=34 B=24 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:202 A:198 B:197 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 2 2 B 0 0 -6 -2 2 C 8 6 0 8 2 D -2 2 -8 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 2 B 0 0 -6 -2 2 C 8 6 0 8 2 D -2 2 -8 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 2 B 0 0 -6 -2 2 C 8 6 0 8 2 D -2 2 -8 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7173: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (12) B E D A C (11) C B E D A (6) C B A D E (6) B C A E D (6) B A D E C (6) E D A C B (5) C D E A B (5) A D E B C (5) E D A B C (4) C A B D E (4) E D C A B (3) E D B A C (3) D E A C B (3) B C A D E (3) E D C B A (2) D E A B C (2) B A E D C (2) A C D E B (2) D E C A B (1) C E D B A (1) B E D C A (1) B E A D C (1) B C E D A (1) B A C D E (1) A D E C B (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 4 2 B 0 0 -4 0 2 C 2 4 0 -4 -2 D -4 0 4 0 8 E -2 -2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 4 2 B 0 0 -4 0 2 C 2 4 0 -4 -2 D -4 0 4 0 8 E -2 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=32 E=17 A=11 D=6 so D is eliminated. Round 2 votes counts: C=34 B=32 E=23 A=11 so A is eliminated. Round 3 votes counts: C=37 B=34 E=29 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:204 A:202 C:200 B:199 E:195 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 4 2 B 0 0 -4 0 2 C 2 4 0 -4 -2 D -4 0 4 0 8 E -2 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 4 2 B 0 0 -4 0 2 C 2 4 0 -4 -2 D -4 0 4 0 8 E -2 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 4 2 B 0 0 -4 0 2 C 2 4 0 -4 -2 D -4 0 4 0 8 E -2 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999994 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7174: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) C A D E B (7) C D A B E (6) C A E B D (6) B D E A C (6) E A B D C (5) C D A E B (5) C A E D B (5) A E B C D (4) D C A B E (3) C D B A E (3) B E D A C (3) B E A D C (3) A E B D C (3) E C B A D (2) E A B C D (2) D B E A C (2) D B C E A (2) D B A E C (2) D B A C E (2) C D B E A (2) A E C B D (2) E B C A D (1) E B A C D (1) E A C B D (1) D B E C A (1) D B C A E (1) D A B E C (1) C E A B D (1) C D E A B (1) B E C D A (1) B E C A D (1) B D C E A (1) B C E D A (1) A E D B C (1) A D E B C (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 0 14 10 B -14 0 6 4 -18 C 0 -6 0 8 -2 D -14 -4 -8 0 -4 E -10 18 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.562892 B: 0.000000 C: 0.437108 D: 0.000000 E: 0.000000 Sum of squares = 0.507910924994 Cumulative probabilities = A: 0.562892 B: 0.562892 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 14 10 B -14 0 6 4 -18 C 0 -6 0 8 -2 D -14 -4 -8 0 -4 E -10 18 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=20 B=16 D=14 A=14 so D is eliminated. Round 2 votes counts: C=39 B=26 E=20 A=15 so A is eliminated. Round 3 votes counts: C=42 E=31 B=27 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:219 E:207 C:200 B:189 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 14 10 B -14 0 6 4 -18 C 0 -6 0 8 -2 D -14 -4 -8 0 -4 E -10 18 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 14 10 B -14 0 6 4 -18 C 0 -6 0 8 -2 D -14 -4 -8 0 -4 E -10 18 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 14 10 B -14 0 6 4 -18 C 0 -6 0 8 -2 D -14 -4 -8 0 -4 E -10 18 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7175: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) D E B A C (7) A B D C E (5) E D C A B (4) C E D B A (4) C E B A D (4) C B A E D (4) C A B E D (4) B A C D E (4) A D B E C (4) E D B C A (3) E C D B A (3) D A E B C (3) D A B E C (3) B A D C E (3) A C B D E (3) E D B A C (2) E D A C B (2) E C D A B (2) D E A B C (2) D B A E C (2) C E B D A (2) C E A B D (2) B D A E C (2) B A D E C (2) A B D E C (2) A B C D E (2) E C B D A (1) D E B C A (1) C E D A B (1) C B A D E (1) C A E B D (1) C A B D E (1) B C E D A (1) B C D E A (1) B C A D E (1) A E D C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 2 -8 0 B 12 0 2 -8 -10 C -2 -2 0 -14 -10 D 8 8 14 0 0 E 0 10 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.389778 E: 0.610222 Sum of squares = 0.524297602836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.389778 E: 1.000000 A B C D E A 0 -12 2 -8 0 B 12 0 2 -8 -10 C -2 -2 0 -14 -10 D 8 8 14 0 0 E 0 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 A=19 D=18 B=14 so B is eliminated. Round 2 votes counts: A=28 C=27 E=25 D=20 so D is eliminated. Round 3 votes counts: A=38 E=35 C=27 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 E:210 B:198 A:191 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -8 0 B 12 0 2 -8 -10 C -2 -2 0 -14 -10 D 8 8 14 0 0 E 0 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -8 0 B 12 0 2 -8 -10 C -2 -2 0 -14 -10 D 8 8 14 0 0 E 0 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -8 0 B 12 0 2 -8 -10 C -2 -2 0 -14 -10 D 8 8 14 0 0 E 0 10 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7176: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) E A C B D (6) B D A E C (6) B D A C E (6) C A E B D (5) B A E D C (5) D B E A C (4) C E A D B (4) C A E D B (4) D C E B A (3) D C B E A (3) B D E A C (3) B D C A E (3) A B C D E (3) E D C B A (2) E D B A C (2) E B A D C (2) D E B C A (2) C E D A B (2) B A D E C (2) A E C B D (2) A C E B D (2) A B E C D (2) E D C A B (1) E C D A B (1) E C A D B (1) E A B C D (1) D E C B A (1) D C B A E (1) D B C E A (1) D B A E C (1) D B A C E (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A E B (1) C A D E B (1) C A B D E (1) B E D A C (1) B A D C E (1) B A C D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 6 -10 2 B 20 0 12 8 10 C -6 -12 0 -18 -6 D 10 -8 18 0 10 E -2 -10 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 -10 2 B 20 0 12 8 10 C -6 -12 0 -18 -6 D 10 -8 18 0 10 E -2 -10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=24 C=21 E=16 A=11 so A is eliminated. Round 2 votes counts: B=35 D=24 C=23 E=18 so E is eliminated. Round 3 votes counts: B=38 C=33 D=29 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:215 E:192 A:189 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 6 -10 2 B 20 0 12 8 10 C -6 -12 0 -18 -6 D 10 -8 18 0 10 E -2 -10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -10 2 B 20 0 12 8 10 C -6 -12 0 -18 -6 D 10 -8 18 0 10 E -2 -10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -10 2 B 20 0 12 8 10 C -6 -12 0 -18 -6 D 10 -8 18 0 10 E -2 -10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7177: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (6) C E B A D (6) E C A B D (5) D A B C E (4) C B E D A (4) B D E C A (4) B D A E C (4) A D C E B (4) A D B E C (4) E C B A D (3) D B A E C (3) D B A C E (3) D A B E C (3) C B D E A (3) B E C D A (3) B C E D A (3) A E C D B (3) E A C D B (2) D C B A E (2) C E A B D (2) C A E D B (2) B E D C A (2) A E D C B (2) A D E C B (2) A D E B C (2) A D C B E (2) A D B C E (2) A C D E B (2) E C B D A (1) E C A D B (1) E B C D A (1) E B C A D (1) E A B D C (1) C E A D B (1) C D A B E (1) C B D A E (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) Total count = 100 A B C D E A 0 -10 -12 -6 -2 B 10 0 -10 10 6 C 12 10 0 6 6 D 6 -10 -6 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -6 -2 B 10 0 -10 10 6 C 12 10 0 6 6 D 6 -10 -6 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 B=21 E=15 D=15 so E is eliminated. Round 2 votes counts: C=36 A=26 B=23 D=15 so D is eliminated. Round 3 votes counts: C=38 A=33 B=29 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:208 D:196 E:194 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -6 -2 B 10 0 -10 10 6 C 12 10 0 6 6 D 6 -10 -6 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -6 -2 B 10 0 -10 10 6 C 12 10 0 6 6 D 6 -10 -6 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -6 -2 B 10 0 -10 10 6 C 12 10 0 6 6 D 6 -10 -6 0 2 E 2 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7178: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) B E D C A (9) A C D E B (9) E B D C A (7) D C A B E (7) C D A B E (7) A C D B E (6) E A B C D (5) A C E D B (5) E B D A C (4) E B A D C (3) D C B A E (3) B E C D A (3) B D C E A (3) E A B D C (2) D A C E B (2) E A D B C (1) D E C B A (1) D C A E B (1) D B C E A (1) C D B A E (1) C B A D E (1) B E C A D (1) B E A C D (1) B C D E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E C B (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 8 0 -4 B -2 0 2 4 -10 C -8 -2 0 6 -2 D 0 -4 -6 0 -10 E 4 10 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 8 0 -4 B -2 0 2 4 -10 C -8 -2 0 6 -2 D 0 -4 -6 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=26 B=18 D=15 C=9 so C is eliminated. Round 2 votes counts: E=32 A=26 D=23 B=19 so B is eliminated. Round 3 votes counts: E=46 D=27 A=27 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:203 B:197 C:197 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 8 0 -4 B -2 0 2 4 -10 C -8 -2 0 6 -2 D 0 -4 -6 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 0 -4 B -2 0 2 4 -10 C -8 -2 0 6 -2 D 0 -4 -6 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 0 -4 B -2 0 2 4 -10 C -8 -2 0 6 -2 D 0 -4 -6 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7179: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) A E C B D (5) A D E B C (5) D E C B A (4) D E C A B (4) D B A C E (4) B C A E D (4) A E D C B (4) A B C E D (4) E C B A D (3) D C B E A (3) D A E B C (3) C B E D A (3) B D C E A (3) B C E D A (3) B C D E A (3) B A C D E (3) A E B C D (3) E C A B D (2) E A C D B (2) D A B E C (2) C E B D A (2) B C E A D (2) A E C D B (2) A D B C E (2) E D C B A (1) E C D B A (1) D E A C B (1) D C E B A (1) D B C A E (1) D A E C B (1) C E D B A (1) B D C A E (1) B D A C E (1) B C D A E (1) B C A D E (1) B A C E D (1) A D E C B (1) A D B E C (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 -4 6 B 10 0 18 -4 8 C 4 -18 0 -6 6 D 4 4 6 0 12 E -6 -8 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -4 6 B 10 0 18 -4 8 C 4 -18 0 -6 6 D 4 4 6 0 12 E -6 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=30 B=23 E=9 C=6 so C is eliminated. Round 2 votes counts: D=32 A=30 B=26 E=12 so E is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:213 A:194 C:193 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 -4 6 B 10 0 18 -4 8 C 4 -18 0 -6 6 D 4 4 6 0 12 E -6 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -4 6 B 10 0 18 -4 8 C 4 -18 0 -6 6 D 4 4 6 0 12 E -6 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -4 6 B 10 0 18 -4 8 C 4 -18 0 -6 6 D 4 4 6 0 12 E -6 -8 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7180: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (12) B E A C D (10) B E A D C (9) D C A E B (8) A E D C B (7) E A B D C (5) C D B A E (5) C B D A E (5) B C E A D (4) D A E C B (3) C E A B D (3) B D C E A (3) B C D E A (3) E A C B D (2) E A B C D (2) D C A B E (2) D A C E B (2) C A E D B (2) B C D A E (2) A D E C B (2) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) D C B A E (1) D B C A E (1) C B E A D (1) B D E A C (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -4 4 6 B -8 0 -16 2 -6 C 4 16 0 6 10 D -4 -2 -6 0 2 E -6 6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 4 6 B -8 0 -16 2 -6 C 4 16 0 6 10 D -4 -2 -6 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=28 D=17 E=13 A=10 so A is eliminated. Round 2 votes counts: B=32 C=29 E=20 D=19 so D is eliminated. Round 3 votes counts: C=42 B=33 E=25 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:207 D:195 E:194 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 4 6 B -8 0 -16 2 -6 C 4 16 0 6 10 D -4 -2 -6 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 4 6 B -8 0 -16 2 -6 C 4 16 0 6 10 D -4 -2 -6 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 4 6 B -8 0 -16 2 -6 C 4 16 0 6 10 D -4 -2 -6 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7181: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) B D A C E (8) D A C E B (7) B A D C E (7) E C D A B (6) E C A D B (6) D A B C E (5) E C A B D (4) D B A C E (4) A D C E B (4) A C E D B (4) E C B A D (3) D E C A B (3) D C E A B (3) C E D A B (3) C E A D B (3) B A C E D (3) E C B D A (2) D C A E B (2) B D E C A (2) B A E C D (2) A D C B E (2) A C D E B (2) E C D B A (1) D E C B A (1) C A E D B (1) B E D C A (1) B E C D A (1) B D A E C (1) A E B C D (1) Total count = 100 A B C D E A 0 12 0 0 6 B -12 0 -14 -14 -12 C 0 14 0 0 16 D 0 14 0 0 2 E -6 12 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.426252 B: 0.000000 C: 0.216628 D: 0.357120 E: 0.000000 Sum of squares = 0.356153288345 Cumulative probabilities = A: 0.426252 B: 0.426252 C: 0.642880 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 0 6 B -12 0 -14 -14 -12 C 0 14 0 0 16 D 0 14 0 0 2 E -6 12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=25 E=22 A=13 C=7 so C is eliminated. Round 2 votes counts: B=33 E=28 D=25 A=14 so A is eliminated. Round 3 votes counts: E=34 D=33 B=33 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:215 A:209 D:208 E:194 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 0 6 B -12 0 -14 -14 -12 C 0 14 0 0 16 D 0 14 0 0 2 E -6 12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 0 6 B -12 0 -14 -14 -12 C 0 14 0 0 16 D 0 14 0 0 2 E -6 12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 0 6 B -12 0 -14 -14 -12 C 0 14 0 0 16 D 0 14 0 0 2 E -6 12 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7182: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) E C A D B (5) D A B E C (5) C B A E D (5) B C A D E (5) E C B A D (4) D B A C E (4) D A B C E (4) C E B A D (4) E A C D B (3) D E B A C (3) D E A B C (3) A C D B E (3) E D C A B (2) E B D C A (2) E B C D A (2) E A D C B (2) D A E C B (2) C B E A D (2) C A E B D (2) C A B E D (2) B E D C A (2) B D E C A (2) B D C A E (2) B A D C E (2) A D C B E (2) A D B C E (2) A C B D E (2) A B D C E (2) E C D B A (1) E C D A B (1) E C A B D (1) D E A C B (1) D A E B C (1) C E A B D (1) C A B D E (1) B D A C E (1) B C D E A (1) B C D A E (1) B A C D E (1) A E D C B (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 12 4 8 4 B -12 0 -8 -4 4 C -4 8 0 -4 2 D -8 4 4 0 4 E -4 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 8 4 B -12 0 -8 -4 4 C -4 8 0 -4 2 D -8 4 4 0 4 E -4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=23 C=17 B=17 A=14 so A is eliminated. Round 2 votes counts: E=30 D=28 C=22 B=20 so B is eliminated. Round 3 votes counts: D=37 E=32 C=31 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:214 D:202 C:201 E:193 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 8 4 B -12 0 -8 -4 4 C -4 8 0 -4 2 D -8 4 4 0 4 E -4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 8 4 B -12 0 -8 -4 4 C -4 8 0 -4 2 D -8 4 4 0 4 E -4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 8 4 B -12 0 -8 -4 4 C -4 8 0 -4 2 D -8 4 4 0 4 E -4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7183: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (5) C B D E A (5) E C B A D (4) D C B A E (4) C D B E A (4) C D B A E (4) E A D B C (3) D E A C B (3) D C E B A (3) D C E A B (3) C E D B A (3) A D B E C (3) A B D E C (3) E D A C B (2) E C D B A (2) E C D A B (2) E C A B D (2) E B C A D (2) D E A B C (2) D C B E A (2) D B C A E (2) D A E B C (2) C B E A D (2) C B A D E (2) B C A E D (2) B A C E D (2) A D E B C (2) A B E D C (2) A B E C D (2) A B D C E (2) E D C B A (1) E D C A B (1) E C B D A (1) E C A D B (1) E B A C D (1) E A C B D (1) D C A E B (1) D C A B E (1) D B A C E (1) D A C B E (1) D A B C E (1) C B E D A (1) C B A E D (1) B E C A D (1) B C E A D (1) B C D A E (1) B A D C E (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 -18 -6 -16 B 8 0 -14 -10 4 C 18 14 0 6 2 D 6 10 -6 0 8 E 16 -4 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 -6 -16 B 8 0 -14 -10 4 C 18 14 0 6 2 D 6 10 -6 0 8 E 16 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 C=22 A=15 B=9 so B is eliminated. Round 2 votes counts: E=29 D=26 C=26 A=19 so A is eliminated. Round 3 votes counts: D=37 E=34 C=29 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:220 D:209 E:201 B:194 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -18 -6 -16 B 8 0 -14 -10 4 C 18 14 0 6 2 D 6 10 -6 0 8 E 16 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -6 -16 B 8 0 -14 -10 4 C 18 14 0 6 2 D 6 10 -6 0 8 E 16 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -6 -16 B 8 0 -14 -10 4 C 18 14 0 6 2 D 6 10 -6 0 8 E 16 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7184: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) B D A C E (7) D B E C A (5) D B E A C (5) D B A C E (5) B D C E A (4) B C A E D (4) E D A C B (3) E C A B D (3) E A C D B (3) D E B C A (3) B D E C A (3) B C E A D (3) A B C D E (3) D E A C B (2) D E A B C (2) D A B E C (2) D A B C E (2) C A E B D (2) B E C D A (2) B A C D E (2) A E D C B (2) A B D C E (2) E D C B A (1) E D C A B (1) E D B C A (1) E C B D A (1) E C A D B (1) E B C D A (1) E A D C B (1) D B C A E (1) D A E C B (1) D A E B C (1) C E B A D (1) C E A B D (1) C B E A D (1) C A B E D (1) B D C A E (1) B C E D A (1) B C D A E (1) B C A D E (1) A E C D B (1) A D B C E (1) A C E D B (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -22 12 -26 2 B 22 0 40 -6 32 C -12 -40 0 -26 -4 D 26 6 26 0 26 E -2 -32 4 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 12 -26 2 B 22 0 40 -6 32 C -12 -40 0 -26 -4 D 26 6 26 0 26 E -2 -32 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=29 E=16 A=13 C=6 so C is eliminated. Round 2 votes counts: D=36 B=30 E=18 A=16 so A is eliminated. Round 3 votes counts: D=38 B=38 E=24 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:244 D:242 A:183 E:172 C:159 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 12 -26 2 B 22 0 40 -6 32 C -12 -40 0 -26 -4 D 26 6 26 0 26 E -2 -32 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 12 -26 2 B 22 0 40 -6 32 C -12 -40 0 -26 -4 D 26 6 26 0 26 E -2 -32 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 12 -26 2 B 22 0 40 -6 32 C -12 -40 0 -26 -4 D 26 6 26 0 26 E -2 -32 4 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7185: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) A B D E C (11) E C A D B (7) D B A E C (7) C E A B D (6) B D A C E (6) A E B D C (6) C D B E A (5) D B A C E (4) A E C B D (4) D B C E A (3) D B C A E (3) C E A D B (3) A B D C E (3) E C A B D (2) E A C B D (2) C B D E A (2) B D A E C (2) A E D B C (2) E D B C A (1) E C D B A (1) E A C D B (1) C E B A D (1) C B D A E (1) B D C A E (1) B A D E C (1) A D B E C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 4 4 8 B 0 0 4 0 4 C -4 -4 0 -4 2 D -4 0 4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.534033 B: 0.465967 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.502316456246 Cumulative probabilities = A: 0.534033 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 4 8 B 0 0 4 0 4 C -4 -4 0 -4 2 D -4 0 4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=29 D=17 E=14 B=10 so B is eliminated. Round 2 votes counts: C=30 A=30 D=26 E=14 so E is eliminated. Round 3 votes counts: C=40 A=33 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:204 D:201 C:195 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 4 8 B 0 0 4 0 4 C -4 -4 0 -4 2 D -4 0 4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 8 B 0 0 4 0 4 C -4 -4 0 -4 2 D -4 0 4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 8 B 0 0 4 0 4 C -4 -4 0 -4 2 D -4 0 4 0 2 E -8 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7186: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A C D B E (7) E D B A C (5) E D A C B (5) D E A B C (4) E D A B C (3) E B D A C (3) E A D C B (3) D E B A C (3) D A B C E (3) C A B E D (3) B C A D E (3) A D C E B (3) A C E D B (3) A C D E B (3) E D B C A (2) E B D C A (2) E A C D B (2) D E A C B (2) D A E C B (2) C B E A D (2) B E C D A (2) B C E A D (2) A D C B E (2) E D C B A (1) E C B D A (1) E C A D B (1) E B C D A (1) D A C E B (1) D A B E C (1) C E B A D (1) C E A B D (1) C B A D E (1) C A E D B (1) C A E B D (1) C A D B E (1) B E D C A (1) B D E C A (1) B D C A E (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A E D (1) A D E C B (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 28 18 6 0 B -28 0 -14 -20 -10 C -18 14 0 0 8 D -6 20 0 0 4 E 0 10 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.609162 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.390837 Sum of squares = 0.523832887352 Cumulative probabilities = A: 0.609162 B: 0.609162 C: 0.609162 D: 0.609163 E: 1.000000 A B C D E A 0 28 18 6 0 B -28 0 -14 -20 -10 C -18 14 0 0 8 D -6 20 0 0 4 E 0 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 C=19 D=16 B=14 so B is eliminated. Round 2 votes counts: E=32 C=28 A=22 D=18 so D is eliminated. Round 3 votes counts: E=42 C=29 A=29 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:209 C:202 E:199 B:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 18 6 0 B -28 0 -14 -20 -10 C -18 14 0 0 8 D -6 20 0 0 4 E 0 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 18 6 0 B -28 0 -14 -20 -10 C -18 14 0 0 8 D -6 20 0 0 4 E 0 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 18 6 0 B -28 0 -14 -20 -10 C -18 14 0 0 8 D -6 20 0 0 4 E 0 10 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7187: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) E C B A D (7) D A B C E (7) C E A D B (5) E C B D A (4) D C A E B (4) D A C B E (4) B E D A C (4) E B D A C (3) E B C D A (3) D A C E B (3) C E A B D (3) C A D E B (3) A B D C E (3) A B C D E (3) E B D C A (2) D B A E C (2) C E D A B (2) C D E A B (2) C D A E B (2) C A D B E (2) C A B E D (2) A D C B E (2) A C D B E (2) A C B D E (2) E D C B A (1) E D B A C (1) E C D B A (1) E C A D B (1) E B C A D (1) D E C A B (1) D E B A C (1) D E A B C (1) D A E C B (1) C E B A D (1) C A E D B (1) B E A D C (1) B E A C D (1) B D A E C (1) B D A C E (1) B A C E D (1) Total count = 100 A B C D E A 0 12 4 -2 8 B -12 0 -12 2 -8 C -4 12 0 -2 26 D 2 -2 2 0 10 E -8 8 -26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749999999 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 -2 8 B -12 0 -12 2 -8 C -4 12 0 -2 26 D 2 -2 2 0 10 E -8 8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749999711 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 C=23 B=17 A=12 so A is eliminated. Round 2 votes counts: C=27 D=26 E=24 B=23 so B is eliminated. Round 3 votes counts: D=39 C=31 E=30 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:216 A:211 D:206 B:185 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 -2 8 B -12 0 -12 2 -8 C -4 12 0 -2 26 D 2 -2 2 0 10 E -8 8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749999711 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -2 8 B -12 0 -12 2 -8 C -4 12 0 -2 26 D 2 -2 2 0 10 E -8 8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749999711 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -2 8 B -12 0 -12 2 -8 C -4 12 0 -2 26 D 2 -2 2 0 10 E -8 8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749999711 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7188: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) C D E A B (7) E D C B A (6) E C D A B (5) E B A D C (5) E B A C D (5) D B A C E (5) E A B C D (4) D C A B E (4) A B C D E (4) E C D B A (3) E C A B D (3) D C E A B (3) C E D A B (3) C D A B E (3) E C A D B (2) D C B A E (2) D B C A E (2) C E A B D (2) C A B D E (2) A B D C E (2) A B C E D (2) E D B C A (1) E D B A C (1) D E C B A (1) D E C A B (1) D E B C A (1) D E B A C (1) D C E B A (1) D C B E A (1) C A D B E (1) B D A E C (1) B A E D C (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -8 -4 -12 B 2 0 -2 -8 -10 C 8 2 0 -4 16 D 4 8 4 0 10 E 12 10 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -4 -12 B 2 0 -2 -8 -10 C 8 2 0 -4 16 D 4 8 4 0 10 E 12 10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=22 C=18 B=16 A=9 so A is eliminated. Round 2 votes counts: E=35 B=24 D=22 C=19 so C is eliminated. Round 3 votes counts: E=40 D=33 B=27 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 C:211 E:198 B:191 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 -12 B 2 0 -2 -8 -10 C 8 2 0 -4 16 D 4 8 4 0 10 E 12 10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 -12 B 2 0 -2 -8 -10 C 8 2 0 -4 16 D 4 8 4 0 10 E 12 10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 -12 B 2 0 -2 -8 -10 C 8 2 0 -4 16 D 4 8 4 0 10 E 12 10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998505 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7189: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) C E A D B (10) A B D E C (10) E C A D B (7) A B D C E (7) E C A B D (5) E A C B D (5) D B C E A (5) C E D B A (5) B D A C E (5) D B A C E (3) C D B E A (3) A E B D C (3) D B E C A (2) D B C A E (2) C E A B D (2) C A B D E (2) B D A E C (2) A E C B D (2) A C E B D (2) E D B C A (1) D B E A C (1) C D E B A (1) B A D E C (1) B A D C E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 14 -12 18 -16 B -14 0 -12 -2 -8 C 12 12 0 12 -2 D -18 2 -12 0 -8 E 16 8 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -12 18 -16 B -14 0 -12 -2 -8 C 12 12 0 12 -2 D -18 2 -12 0 -8 E 16 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=26 C=23 D=13 B=9 so B is eliminated. Round 2 votes counts: E=29 A=28 C=23 D=20 so D is eliminated. Round 3 votes counts: A=38 E=32 C=30 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:217 E:217 A:202 B:182 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -12 18 -16 B -14 0 -12 -2 -8 C 12 12 0 12 -2 D -18 2 -12 0 -8 E 16 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -12 18 -16 B -14 0 -12 -2 -8 C 12 12 0 12 -2 D -18 2 -12 0 -8 E 16 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -12 18 -16 B -14 0 -12 -2 -8 C 12 12 0 12 -2 D -18 2 -12 0 -8 E 16 8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7190: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (13) B E A C D (12) D C B A E (11) A E B C D (9) D A E C B (7) D C A E B (5) A E D B C (5) E A B C D (4) D C A B E (4) D B C E A (4) B C E D A (4) C B D E A (3) B C E A D (3) A E C B D (3) A E B D C (3) C D B E A (2) B C D E A (2) A E D C B (2) D A C E B (1) C B E D A (1) C B E A D (1) A E C D B (1) Total count = 100 A B C D E A 0 -12 -6 -14 2 B 12 0 -8 -10 20 C 6 8 0 -10 8 D 14 10 10 0 4 E -2 -20 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -14 2 B 12 0 -8 -10 20 C 6 8 0 -10 8 D 14 10 10 0 4 E -2 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=45 A=23 B=21 C=7 E=4 so E is eliminated. Round 2 votes counts: D=45 A=27 B=21 C=7 so C is eliminated. Round 3 votes counts: D=47 A=27 B=26 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 B:207 C:206 A:185 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -14 2 B 12 0 -8 -10 20 C 6 8 0 -10 8 D 14 10 10 0 4 E -2 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -14 2 B 12 0 -8 -10 20 C 6 8 0 -10 8 D 14 10 10 0 4 E -2 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -14 2 B 12 0 -8 -10 20 C 6 8 0 -10 8 D 14 10 10 0 4 E -2 -20 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997377 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7191: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (13) B E C D A (11) D A C B E (7) C B E D A (7) E B C D A (6) A D C E B (6) D A E B C (4) C B D E A (4) E A B C D (3) D E A B C (3) D C B A E (3) B C E D A (3) A D E B C (3) A D C B E (3) E B A C D (2) D C A B E (2) C B E A D (2) A E D B C (2) A E B C D (2) A D E C B (2) E C B A D (1) E B D A C (1) E B A D C (1) D E B A C (1) D C B E A (1) D B E C A (1) D B C E A (1) C B A E D (1) B E D C A (1) B C E A D (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 -22 -16 -12 -26 B 22 0 20 20 -4 C 16 -20 0 14 -16 D 12 -20 -14 0 -16 E 26 4 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -22 -16 -12 -26 B 22 0 20 20 -4 C 16 -20 0 14 -16 D 12 -20 -14 0 -16 E 26 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 A=20 B=16 C=14 so C is eliminated. Round 2 votes counts: B=30 E=27 D=23 A=20 so A is eliminated. Round 3 votes counts: D=38 E=32 B=30 so B is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:229 C:197 D:181 A:162 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -22 -16 -12 -26 B 22 0 20 20 -4 C 16 -20 0 14 -16 D 12 -20 -14 0 -16 E 26 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -16 -12 -26 B 22 0 20 20 -4 C 16 -20 0 14 -16 D 12 -20 -14 0 -16 E 26 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -16 -12 -26 B 22 0 20 20 -4 C 16 -20 0 14 -16 D 12 -20 -14 0 -16 E 26 4 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7192: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D A B C E (7) B C D A E (7) E C A B D (6) E C B D A (5) E A D C B (5) C B E D A (5) A D C B E (5) B D A C E (4) A E D C B (4) E C A D B (3) E A C D B (3) B D C A E (3) B C E D A (3) A D E B C (3) A D B C E (3) E D A B C (2) E B D A C (2) E B C D A (2) D A E B C (2) C B D E A (2) B C D E A (2) A D E C B (2) E D B A C (1) E A D B C (1) D B A C E (1) C E B D A (1) C E B A D (1) C B E A D (1) C B A D E (1) C A B D E (1) B E C D A (1) B D C E A (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 12 -4 C 6 10 0 6 0 D 2 -12 -6 0 -8 E 10 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.594224 D: 0.000000 E: 0.405776 Sum of squares = 0.517756423636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.594224 D: 0.594224 E: 1.000000 A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 12 -4 C 6 10 0 6 0 D 2 -12 -6 0 -8 E 10 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=21 A=19 C=12 D=10 so D is eliminated. Round 2 votes counts: E=38 A=28 B=22 C=12 so C is eliminated. Round 3 votes counts: E=40 B=31 A=29 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:211 E:211 B:200 A:190 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 12 -4 C 6 10 0 6 0 D 2 -12 -6 0 -8 E 10 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 12 -4 C 6 10 0 6 0 D 2 -12 -6 0 -8 E 10 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -2 -10 B 2 0 -10 12 -4 C 6 10 0 6 0 D 2 -12 -6 0 -8 E 10 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7193: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (15) C E D A B (12) B A E C D (11) E C B D A (8) D A C E B (7) D A B C E (7) D C E A B (5) B E C A D (5) B A D C E (5) A B D C E (5) E C D B A (3) E C D A B (3) A D B C E (3) C E B A D (2) B A C E D (2) E B C A D (1) D E C A B (1) D C A E B (1) C E D B A (1) C E B D A (1) B E A C D (1) B A C D E (1) Total count = 100 A B C D E A 0 -12 14 2 14 B 12 0 12 14 10 C -14 -12 0 2 4 D -2 -14 -2 0 0 E -14 -10 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 14 2 14 B 12 0 12 14 10 C -14 -12 0 2 4 D -2 -14 -2 0 0 E -14 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 D=21 C=16 E=15 A=8 so A is eliminated. Round 2 votes counts: B=45 D=24 C=16 E=15 so E is eliminated. Round 3 votes counts: B=46 C=30 D=24 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:209 D:191 C:190 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 14 2 14 B 12 0 12 14 10 C -14 -12 0 2 4 D -2 -14 -2 0 0 E -14 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 14 2 14 B 12 0 12 14 10 C -14 -12 0 2 4 D -2 -14 -2 0 0 E -14 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 14 2 14 B 12 0 12 14 10 C -14 -12 0 2 4 D -2 -14 -2 0 0 E -14 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7194: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) E C A B D (7) C E B A D (7) C D B A E (6) E C D A B (5) D A B E C (5) D A B C E (5) E A D B C (4) D C B A E (4) C E B D A (4) E A C B D (3) E A B C D (3) B A D C E (3) A D B E C (3) E A C D B (2) C B E D A (2) C B E A D (2) C B D A E (2) A B D E C (2) E D A C B (1) E C A D B (1) E B A C D (1) E A D C B (1) E A B D C (1) D A E B C (1) D A C E B (1) D A C B E (1) C E D B A (1) C B A E D (1) C B A D E (1) B D C A E (1) B D A C E (1) B C E A D (1) B C A E D (1) B A E D C (1) B A E C D (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 8 2 8 B 2 0 -4 -2 10 C -8 4 0 6 10 D -2 2 -6 0 -8 E -8 -10 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.571429 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428558 Cumulative probabilities = A: 0.285714 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 2 8 B 2 0 -4 -2 10 C -8 4 0 6 10 D -2 2 -6 0 -8 E -8 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.571429 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571419334 Cumulative probabilities = A: 0.285714 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 C=26 B=11 A=8 so A is eliminated. Round 2 votes counts: E=31 D=29 C=26 B=14 so B is eliminated. Round 3 votes counts: D=36 E=34 C=30 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:208 C:206 B:203 D:193 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 2 8 B 2 0 -4 -2 10 C -8 4 0 6 10 D -2 2 -6 0 -8 E -8 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.571429 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571419334 Cumulative probabilities = A: 0.285714 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 2 8 B 2 0 -4 -2 10 C -8 4 0 6 10 D -2 2 -6 0 -8 E -8 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.571429 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571419334 Cumulative probabilities = A: 0.285714 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 2 8 B 2 0 -4 -2 10 C -8 4 0 6 10 D -2 2 -6 0 -8 E -8 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.571429 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571419334 Cumulative probabilities = A: 0.285714 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7195: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (15) A D E B C (14) C B E D A (13) E B C A D (12) A D C E B (7) B E C D A (6) E B A D C (5) C D A B E (3) C B D A E (3) A D C B E (3) E B A C D (2) E A D B C (2) D A E B C (2) D A C E B (2) B C E A D (2) A E D B C (2) A D E C B (2) E B D A C (1) C B E A D (1) C A E B D (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 8 16 8 10 B -8 0 -2 -6 -6 C -16 2 0 -12 2 D -8 6 12 0 2 E -10 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 8 10 B -8 0 -2 -6 -6 C -16 2 0 -12 2 D -8 6 12 0 2 E -10 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=22 C=21 D=19 B=9 so B is eliminated. Round 2 votes counts: A=29 E=28 C=24 D=19 so D is eliminated. Round 3 votes counts: A=48 E=28 C=24 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:206 E:196 B:189 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 8 10 B -8 0 -2 -6 -6 C -16 2 0 -12 2 D -8 6 12 0 2 E -10 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 8 10 B -8 0 -2 -6 -6 C -16 2 0 -12 2 D -8 6 12 0 2 E -10 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 8 10 B -8 0 -2 -6 -6 C -16 2 0 -12 2 D -8 6 12 0 2 E -10 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7196: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (11) A C E D B (8) B D C E A (6) A E C D B (5) A E B C D (5) E B D C A (4) E A B D C (4) C D B A E (4) A C D B E (4) E D C A B (3) E D B C A (3) E B A D C (3) B E D C A (3) B D C A E (3) B C D A E (3) A C D E B (3) A B C D E (3) E D C B A (2) C A D E B (2) B A C D E (2) A C B D E (2) E D B A C (1) E D A C B (1) E B D A C (1) E A C B D (1) E A B C D (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D A C (1) B D E A C (1) B D A E C (1) B A E D C (1) A E C B D (1) Total count = 100 A B C D E A 0 -8 -2 -6 2 B 8 0 18 16 2 C 2 -18 0 -6 -8 D 6 -16 6 0 4 E -2 -2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -6 2 B 8 0 18 16 2 C 2 -18 0 -6 -8 D 6 -16 6 0 4 E -2 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990049 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=31 E=24 C=9 D=4 so D is eliminated. Round 2 votes counts: B=34 A=31 E=24 C=11 so C is eliminated. Round 3 votes counts: B=39 A=36 E=25 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:200 E:200 A:193 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -6 2 B 8 0 18 16 2 C 2 -18 0 -6 -8 D 6 -16 6 0 4 E -2 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990049 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -6 2 B 8 0 18 16 2 C 2 -18 0 -6 -8 D 6 -16 6 0 4 E -2 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990049 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -6 2 B 8 0 18 16 2 C 2 -18 0 -6 -8 D 6 -16 6 0 4 E -2 -2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990049 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7197: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) C D E B A (8) A E B C D (8) B D A E C (7) D C B E A (6) C E A D B (5) B A E D C (5) B A D E C (5) A E C B D (5) A B E C D (5) B D A C E (4) D B C E A (3) D B C A E (3) C E D A B (3) A B E D C (3) A B D E C (3) E A B C D (2) C D E A B (2) C A D E B (2) A C D E B (2) E C D B A (1) E C B D A (1) E B A C D (1) D E C B A (1) D C E B A (1) C E D B A (1) B E A D C (1) B D E C A (1) B A E C D (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 8 16 10 B 0 0 4 4 -6 C -8 -4 0 14 -18 D -16 -4 -14 0 -4 E -10 6 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.518401 B: 0.481598 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500677192417 Cumulative probabilities = A: 0.518401 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 16 10 B 0 0 4 4 -6 C -8 -4 0 14 -18 D -16 -4 -14 0 -4 E -10 6 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500003 B: 0.499997 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.500003 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=24 C=21 D=14 E=13 so E is eliminated. Round 2 votes counts: C=31 A=30 B=25 D=14 so D is eliminated. Round 3 votes counts: C=39 B=31 A=30 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:217 E:209 B:201 C:192 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 16 10 B 0 0 4 4 -6 C -8 -4 0 14 -18 D -16 -4 -14 0 -4 E -10 6 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500003 B: 0.499997 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.500003 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 16 10 B 0 0 4 4 -6 C -8 -4 0 14 -18 D -16 -4 -14 0 -4 E -10 6 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500003 B: 0.499997 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.500003 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 16 10 B 0 0 4 4 -6 C -8 -4 0 14 -18 D -16 -4 -14 0 -4 E -10 6 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500003 B: 0.499997 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999839 Cumulative probabilities = A: 0.500003 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7198: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) D C E A B (9) E D C A B (7) B A E C D (7) B A C D E (7) D C E B A (5) E D C B A (4) E B A D C (4) E A B C D (4) D E C A B (4) C A B D E (4) B A C E D (4) A B C E D (4) E D B A C (3) E C A D B (3) D C B A E (3) D E C B A (2) D B A C E (2) C D A B E (2) A B E C D (2) E D A B C (1) E C D A B (1) E C A B D (1) E B A C D (1) E A C B D (1) D B C A E (1) C D E A B (1) B E A D C (1) B D A E C (1) B A D C E (1) Total count = 100 A B C D E A 0 8 6 8 -4 B -8 0 6 4 -2 C -6 -6 0 4 6 D -8 -4 -4 0 4 E 4 2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999941 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 8 6 8 -4 B -8 0 6 4 -2 C -6 -6 0 4 6 D -8 -4 -4 0 4 E 4 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 B=21 A=16 C=7 so C is eliminated. Round 2 votes counts: E=30 D=29 B=21 A=20 so A is eliminated. Round 3 votes counts: B=41 E=30 D=29 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:209 B:200 C:199 E:198 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 8 -4 B -8 0 6 4 -2 C -6 -6 0 4 6 D -8 -4 -4 0 4 E 4 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 8 -4 B -8 0 6 4 -2 C -6 -6 0 4 6 D -8 -4 -4 0 4 E 4 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 8 -4 B -8 0 6 4 -2 C -6 -6 0 4 6 D -8 -4 -4 0 4 E 4 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999996 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7199: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) C E B D A (9) E C B D A (8) E C A D B (7) D A B E C (7) B D A C E (7) A D B C E (7) D B A E C (4) C E A B D (4) B D A E C (4) A D B E C (4) C E B A D (3) B D C E A (3) A D E C B (3) A D C E B (3) E C B A D (2) D B A C E (2) B C E D A (2) A E C D B (2) A D C B E (2) C B E D A (1) C B E A D (1) C A E D B (1) B E D C A (1) B D E A C (1) B D C A E (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -6 0 -4 B -2 0 -14 -6 -6 C 6 14 0 2 14 D 0 6 -2 0 -2 E 4 6 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 0 -4 B -2 0 -14 -6 -6 C 6 14 0 2 14 D 0 6 -2 0 -2 E 4 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=22 B=19 E=17 D=13 so D is eliminated. Round 2 votes counts: C=29 A=29 B=25 E=17 so E is eliminated. Round 3 votes counts: C=46 A=29 B=25 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:201 E:199 A:196 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 0 -4 B -2 0 -14 -6 -6 C 6 14 0 2 14 D 0 6 -2 0 -2 E 4 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 -4 B -2 0 -14 -6 -6 C 6 14 0 2 14 D 0 6 -2 0 -2 E 4 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 -4 B -2 0 -14 -6 -6 C 6 14 0 2 14 D 0 6 -2 0 -2 E 4 6 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7200: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) C B E D A (8) E D C B A (6) B C E A D (6) A D E C B (6) D A E C B (5) A D E B C (5) E D A C B (4) C B D E A (4) A D C B E (4) E C B D A (3) E B D C A (3) E A B D C (3) D E A C B (3) E B C D A (2) E A D B C (2) D C A E B (2) D C A B E (2) B C A E D (2) A C B D E (2) E D B A C (1) E D A B C (1) E B C A D (1) E B A C D (1) D E C B A (1) D C E B A (1) D A C E B (1) C E B D A (1) C D B E A (1) C B D A E (1) B E A C D (1) B A E C D (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A C D B E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -6 -18 -24 B 10 0 -12 6 -8 C 6 12 0 -6 -4 D 18 -6 6 0 -20 E 24 8 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 -18 -24 B 10 0 -12 6 -8 C 6 12 0 -6 -4 D 18 -6 6 0 -20 E 24 8 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=22 B=21 D=15 C=15 so D is eliminated. Round 2 votes counts: E=31 A=28 B=21 C=20 so C is eliminated. Round 3 votes counts: B=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:204 D:199 B:198 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 -18 -24 B 10 0 -12 6 -8 C 6 12 0 -6 -4 D 18 -6 6 0 -20 E 24 8 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -18 -24 B 10 0 -12 6 -8 C 6 12 0 -6 -4 D 18 -6 6 0 -20 E 24 8 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -18 -24 B 10 0 -12 6 -8 C 6 12 0 -6 -4 D 18 -6 6 0 -20 E 24 8 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7201: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) D A E C B (6) A D E B C (5) C E B D A (4) B D A C E (4) A E D C B (4) E A C B D (3) D A E B C (3) C E B A D (3) C B E D A (3) B C E A D (3) B A D C E (3) B A C D E (3) A B D C E (3) E C D B A (2) E C B A D (2) E C A B D (2) E A D C B (2) D E A C B (2) D C B E A (2) D B A C E (2) D A B C E (2) B D C A E (2) B C A E D (2) A D E C B (2) A B E C D (2) A B D E C (2) E D A C B (1) E C D A B (1) E C B D A (1) E A C D B (1) D C E B A (1) D B C A E (1) D A B E C (1) C B E A D (1) B E C A D (1) B D C E A (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E C D B (1) A E B D C (1) A D B E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 14 2 10 B -8 0 -2 4 -8 C -14 2 0 -20 -8 D -2 -4 20 0 14 E -10 8 8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 2 10 B -8 0 -2 4 -8 C -14 2 0 -20 -8 D -2 -4 20 0 14 E -10 8 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=24 A=23 E=15 C=11 so C is eliminated. Round 2 votes counts: B=28 D=27 A=23 E=22 so E is eliminated. Round 3 votes counts: B=38 D=31 A=31 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:214 E:196 B:193 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 2 10 B -8 0 -2 4 -8 C -14 2 0 -20 -8 D -2 -4 20 0 14 E -10 8 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 2 10 B -8 0 -2 4 -8 C -14 2 0 -20 -8 D -2 -4 20 0 14 E -10 8 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 2 10 B -8 0 -2 4 -8 C -14 2 0 -20 -8 D -2 -4 20 0 14 E -10 8 8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7202: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) D E C A B (10) A B D E C (7) C E D B A (6) D C E B A (5) C D E B A (5) A B E C D (5) B A E C D (4) B C E A D (3) A D B E C (3) A B E D C (3) E A C D B (2) D C E A B (2) D B C E A (2) D A E C B (2) C B E D A (2) B C D E A (2) B C A E D (2) B A D C E (2) A E B C D (2) A D E C B (2) A D E B C (2) E D C A B (1) E C D A B (1) E A D C B (1) D A B E C (1) C E D A B (1) C E B D A (1) C E B A D (1) C D B E A (1) C B E A D (1) C B D E A (1) B C E D A (1) B C D A E (1) B A C D E (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 2 10 2 B 6 0 6 4 8 C -2 -6 0 14 4 D -10 -4 -14 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 10 2 B 6 0 6 4 8 C -2 -6 0 14 4 D -10 -4 -14 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=26 D=22 C=19 E=5 so E is eliminated. Round 2 votes counts: A=29 B=28 D=23 C=20 so C is eliminated. Round 3 votes counts: D=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:205 A:204 E:194 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 10 2 B 6 0 6 4 8 C -2 -6 0 14 4 D -10 -4 -14 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 10 2 B 6 0 6 4 8 C -2 -6 0 14 4 D -10 -4 -14 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 10 2 B 6 0 6 4 8 C -2 -6 0 14 4 D -10 -4 -14 0 -2 E -2 -8 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7203: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (12) A D C E B (9) D A E B C (8) B E C A D (8) A D B E C (7) E B C D A (6) C E B D A (5) A C B E D (5) B E A C D (4) B E D C A (3) B E D A C (3) A D C B E (3) D A C E B (2) C D A E B (2) C A D E B (2) C A B E D (2) A C D B E (2) E C B D A (1) E B D C A (1) D E B C A (1) D E B A C (1) D C E A B (1) C E D B A (1) C E B A D (1) C B E D A (1) C B E A D (1) C A E B D (1) B E A D C (1) A D E B C (1) A D B C E (1) A C D E B (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 4 -2 B 0 0 18 16 12 C -2 -18 0 14 -18 D -4 -16 -14 0 -16 E 2 -12 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.550103 B: 0.449897 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.505020695268 Cumulative probabilities = A: 0.550103 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 -2 B 0 0 18 16 12 C -2 -18 0 14 -18 D -4 -16 -14 0 -16 E 2 -12 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=31 C=16 D=13 E=8 so E is eliminated. Round 2 votes counts: B=38 A=32 C=17 D=13 so D is eliminated. Round 3 votes counts: A=42 B=40 C=18 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:223 E:212 A:202 C:188 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 4 -2 B 0 0 18 16 12 C -2 -18 0 14 -18 D -4 -16 -14 0 -16 E 2 -12 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 -2 B 0 0 18 16 12 C -2 -18 0 14 -18 D -4 -16 -14 0 -16 E 2 -12 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 -2 B 0 0 18 16 12 C -2 -18 0 14 -18 D -4 -16 -14 0 -16 E 2 -12 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7204: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) D E A B C (8) E D A B C (7) B C A E D (7) A E D C B (6) D E B C A (4) D A C E B (4) D A E C B (3) C B A E D (3) C B A D E (3) C A B D E (3) A C B E D (3) A C B D E (3) E D B A C (2) E A D C B (2) D B C E A (2) C B D A E (2) C A B E D (2) B C E D A (2) B C E A D (2) A D E C B (2) A D C E B (2) E D B C A (1) E B D C A (1) E B A C D (1) E A D B C (1) E A B C D (1) D E C B A (1) D E B A C (1) D C B E A (1) D C B A E (1) C D B E A (1) B E D C A (1) B E C A D (1) B D E C A (1) B A C E D (1) A E C B D (1) A E B C D (1) A C E B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 22 22 -4 0 B -22 0 -8 -18 -20 C -22 8 0 -20 -10 D 4 18 20 0 4 E 0 20 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 22 -4 0 B -22 0 -8 -18 -20 C -22 8 0 -20 -10 D 4 18 20 0 4 E 0 20 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=21 E=16 B=15 C=14 so C is eliminated. Round 2 votes counts: D=35 A=26 B=23 E=16 so E is eliminated. Round 3 votes counts: D=45 A=30 B=25 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:220 E:213 C:178 B:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 22 -4 0 B -22 0 -8 -18 -20 C -22 8 0 -20 -10 D 4 18 20 0 4 E 0 20 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 22 -4 0 B -22 0 -8 -18 -20 C -22 8 0 -20 -10 D 4 18 20 0 4 E 0 20 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 22 -4 0 B -22 0 -8 -18 -20 C -22 8 0 -20 -10 D 4 18 20 0 4 E 0 20 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7205: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) D A C B E (8) E C D A B (7) A D B C E (7) D A B E C (6) D A B C E (5) C E D A B (5) C E B A D (5) E D A B C (4) E C B A D (4) C B E A D (4) B A D C E (4) E B C A D (3) C D A E B (3) E D A C B (2) E B A D C (2) B E C A D (2) B E A D C (2) B A D E C (2) B A C D E (2) A D B E C (2) E D C A B (1) E C D B A (1) D A E C B (1) D A E B C (1) C E D B A (1) C D A B E (1) C B E D A (1) C B A E D (1) C B A D E (1) C A D B E (1) B C E A D (1) B C A D E (1) B A E D C (1) Total count = 100 A B C D E A 0 8 -2 -10 -6 B -8 0 -10 -12 4 C 2 10 0 4 2 D 10 12 -4 0 -10 E 6 -4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -10 -6 B -8 0 -10 -12 4 C 2 10 0 4 2 D 10 12 -4 0 -10 E 6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=23 D=21 B=15 A=9 so A is eliminated. Round 2 votes counts: E=32 D=30 C=23 B=15 so B is eliminated. Round 3 votes counts: E=37 D=36 C=27 so C is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:205 D:204 A:195 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 -10 -6 B -8 0 -10 -12 4 C 2 10 0 4 2 D 10 12 -4 0 -10 E 6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -10 -6 B -8 0 -10 -12 4 C 2 10 0 4 2 D 10 12 -4 0 -10 E 6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -10 -6 B -8 0 -10 -12 4 C 2 10 0 4 2 D 10 12 -4 0 -10 E 6 -4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7206: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) A D E C B (8) A B D E C (8) E D C A B (7) C B E D A (7) B C E D A (7) E C D B A (5) D E A C B (5) A B D C E (4) E D A C B (3) D E C A B (3) B C A E D (3) B C A D E (3) B A C E D (3) B A C D E (3) A D B E C (3) A D B C E (3) C E B D A (2) A E D B C (2) E C B D A (1) E B C A D (1) D C E A B (1) D A E C B (1) C D E A B (1) C D B E A (1) C B D E A (1) B A E D C (1) B A E C D (1) B A D C E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -4 -8 -8 B -2 0 -10 -6 0 C 4 10 0 -4 -2 D 8 6 4 0 -6 E 8 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.085139 C: 0.000000 D: 0.000000 E: 0.914861 Sum of squares = 0.844219009986 Cumulative probabilities = A: 0.000000 B: 0.085139 C: 0.085139 D: 0.085139 E: 1.000000 A B C D E A 0 2 -4 -8 -8 B -2 0 -10 -6 0 C 4 10 0 -4 -2 D 8 6 4 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222486 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=22 C=21 E=17 D=10 so D is eliminated. Round 2 votes counts: A=31 E=25 C=22 B=22 so C is eliminated. Round 3 votes counts: E=38 B=31 A=31 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 D:206 C:204 A:191 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 -8 -8 B -2 0 -10 -6 0 C 4 10 0 -4 -2 D 8 6 4 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222486 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -8 -8 B -2 0 -10 -6 0 C 4 10 0 -4 -2 D 8 6 4 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222486 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -8 -8 B -2 0 -10 -6 0 C 4 10 0 -4 -2 D 8 6 4 0 -6 E 8 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222222486 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7207: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D B C A E (12) E A C D B (8) B D C A E (8) E D B A C (6) B D A C E (5) E A D C B (4) D E B C A (4) B A C D E (4) A E C B D (4) A C E B D (4) E D B C A (3) E B A D C (2) D E C B A (2) D B E C A (2) D B C E A (2) C B D A E (2) C A E B D (2) E D A B C (1) E A D B C (1) E A B C D (1) D E B A C (1) D C E B A (1) D B E A C (1) C E A D B (1) C A E D B (1) C A B D E (1) B D E A C (1) B C D A E (1) B A E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 16 -4 -8 B 16 0 14 0 -18 C -16 -14 0 -14 -10 D 4 0 14 0 -4 E 8 18 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 16 -4 -8 B 16 0 14 0 -18 C -16 -14 0 -14 -10 D 4 0 14 0 -4 E 8 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=25 B=20 A=9 C=7 so C is eliminated. Round 2 votes counts: E=40 D=25 B=22 A=13 so A is eliminated. Round 3 votes counts: E=51 D=25 B=24 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:207 B:206 A:194 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 16 -4 -8 B 16 0 14 0 -18 C -16 -14 0 -14 -10 D 4 0 14 0 -4 E 8 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 16 -4 -8 B 16 0 14 0 -18 C -16 -14 0 -14 -10 D 4 0 14 0 -4 E 8 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 16 -4 -8 B 16 0 14 0 -18 C -16 -14 0 -14 -10 D 4 0 14 0 -4 E 8 18 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7208: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) C E A D B (6) C A E B D (5) C A D B E (5) B D A C E (5) B A D C E (5) E D C B A (4) E B D C A (4) C A E D B (4) B D E A C (4) E C D A B (3) E B D A C (3) D B E A C (3) C E A B D (3) B A C D E (3) A B C D E (3) E D C A B (2) E C D B A (2) E B C D A (2) E B C A D (2) D B A C E (2) D A B C E (2) C A D E B (2) C A B E D (2) B D A E C (2) A C B D E (2) A B D C E (2) E D B C A (1) E D B A C (1) E C B A D (1) E C A B D (1) D E B A C (1) D B A E C (1) C A B D E (1) B E D A C (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -16 14 -4 B -4 0 -4 4 -10 C 16 4 0 14 8 D -14 -4 -14 0 -12 E 4 10 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 14 -4 B -4 0 -4 4 -10 C 16 4 0 14 8 D -14 -4 -14 0 -12 E 4 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=28 B=21 D=9 A=8 so A is eliminated. Round 2 votes counts: E=34 C=31 B=26 D=9 so D is eliminated. Round 3 votes counts: E=35 B=34 C=31 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:221 E:209 A:199 B:193 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 14 -4 B -4 0 -4 4 -10 C 16 4 0 14 8 D -14 -4 -14 0 -12 E 4 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 14 -4 B -4 0 -4 4 -10 C 16 4 0 14 8 D -14 -4 -14 0 -12 E 4 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 14 -4 B -4 0 -4 4 -10 C 16 4 0 14 8 D -14 -4 -14 0 -12 E 4 10 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7209: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) C E B D A (9) D A B C E (6) B D A C E (6) E C B D A (5) E C B A D (5) A D B C E (5) E C A D B (4) C E D A B (4) A E D B C (4) E B A C D (3) E A C D B (3) C B D E A (3) B D C A E (3) B C E D A (3) B C D E A (3) B A D E C (3) A D E C B (3) C B E D A (2) E C A B D (1) E B C A D (1) E A C B D (1) D C B A E (1) D B C A E (1) D B A C E (1) D A C E B (1) C E D B A (1) C D E A B (1) C D B E A (1) C A D E B (1) B E C A D (1) B C D A E (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -2 -4 -2 B 6 0 8 2 4 C 2 -8 0 6 6 D 4 -2 -6 0 6 E 2 -4 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999244 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -4 -2 B 6 0 8 2 4 C 2 -8 0 6 6 D 4 -2 -6 0 6 E 2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 C=22 B=20 D=10 so D is eliminated. Round 2 votes counts: A=32 E=23 C=23 B=22 so B is eliminated. Round 3 votes counts: A=42 C=34 E=24 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:210 C:203 D:201 A:193 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 -2 B 6 0 8 2 4 C 2 -8 0 6 6 D 4 -2 -6 0 6 E 2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 -2 B 6 0 8 2 4 C 2 -8 0 6 6 D 4 -2 -6 0 6 E 2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 -2 B 6 0 8 2 4 C 2 -8 0 6 6 D 4 -2 -6 0 6 E 2 -4 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7210: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) C E D B A (8) E C A B D (6) E A B D C (6) D B A C E (5) C D B E A (5) A B D E C (5) E B A D C (4) D C B A E (4) D B A E C (4) D C B E A (3) C A E B D (3) E A C B D (2) D A B C E (2) C E A D B (2) C E A B D (2) C A E D B (2) B D A E C (2) A E B C D (2) A B E D C (2) E D C B A (1) E C B D A (1) E C B A D (1) E B D C A (1) E B D A C (1) E B C A D (1) D C E B A (1) D C A B E (1) D B C A E (1) D A C B E (1) C E D A B (1) C E B D A (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A B E (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E A C (1) B A E D C (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 2 -20 B 0 0 0 8 -16 C 0 0 0 0 -4 D -2 -8 0 0 -18 E 20 16 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 2 -20 B 0 0 0 8 -16 C 0 0 0 0 -4 D -2 -8 0 0 -18 E 20 16 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=29 D=22 A=11 B=6 so B is eliminated. Round 2 votes counts: E=34 C=29 D=25 A=12 so A is eliminated. Round 3 votes counts: E=40 D=31 C=29 so C is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:198 B:196 A:191 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 2 -20 B 0 0 0 8 -16 C 0 0 0 0 -4 D -2 -8 0 0 -18 E 20 16 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -20 B 0 0 0 8 -16 C 0 0 0 0 -4 D -2 -8 0 0 -18 E 20 16 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -20 B 0 0 0 8 -16 C 0 0 0 0 -4 D -2 -8 0 0 -18 E 20 16 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7211: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (10) E B D A C (6) C A E B D (6) E D B A C (4) C A D B E (4) B E D A C (4) E D C A B (3) D B A E C (3) D A B C E (3) C E A B D (3) B E A D C (3) B A C D E (3) A D B C E (3) A B C D E (3) E D C B A (2) E B D C A (2) E B C A D (2) D E C A B (2) D B E A C (2) D A C B E (2) D A B E C (2) C E D A B (2) C E A D B (2) B A D C E (2) B A C E D (2) A C B D E (2) E C D B A (1) E C D A B (1) E C B D A (1) E B C D A (1) E B A D C (1) D E B A C (1) D E A B C (1) D C A B E (1) D B A C E (1) D A E B C (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A C D (1) B D A E C (1) B C A E D (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 10 6 4 10 B -10 0 8 10 10 C -6 -8 0 -4 8 D -4 -10 4 0 2 E -10 -10 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 4 10 B -10 0 8 10 10 C -6 -8 0 -4 8 D -4 -10 4 0 2 E -10 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=24 D=19 B=18 A=9 so A is eliminated. Round 2 votes counts: C=32 E=24 D=23 B=21 so B is eliminated. Round 3 votes counts: C=41 E=32 D=27 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:215 B:209 D:196 C:195 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 4 10 B -10 0 8 10 10 C -6 -8 0 -4 8 D -4 -10 4 0 2 E -10 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 10 B -10 0 8 10 10 C -6 -8 0 -4 8 D -4 -10 4 0 2 E -10 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 10 B -10 0 8 10 10 C -6 -8 0 -4 8 D -4 -10 4 0 2 E -10 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7212: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (19) D E A C B (7) D E B C A (5) B A C D E (5) A C B D E (5) E D A C B (4) A C B E D (4) E D C A B (3) E C A D B (3) D B E C A (3) D B A C E (3) C A B E D (3) A C E B D (3) E D B C A (2) E C A B D (2) D E C A B (2) D E B A C (2) D B E A C (2) D A E C B (2) B E C D A (2) B C A D E (2) A E C D B (2) E D C B A (1) E C D A B (1) E C B A D (1) E B D C A (1) E A C D B (1) D E A B C (1) D A C B E (1) D A B C E (1) C A E B D (1) B D E C A (1) B D A C E (1) B C E A D (1) B A C E D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -6 10 10 B 4 0 4 4 10 C 6 -4 0 16 4 D -10 -4 -16 0 -12 E -10 -10 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 10 10 B 4 0 4 4 10 C 6 -4 0 16 4 D -10 -4 -16 0 -12 E -10 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=29 E=19 A=16 C=4 so C is eliminated. Round 2 votes counts: B=32 D=29 A=20 E=19 so E is eliminated. Round 3 votes counts: D=40 B=34 A=26 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:211 A:205 E:194 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 10 10 B 4 0 4 4 10 C 6 -4 0 16 4 D -10 -4 -16 0 -12 E -10 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 10 10 B 4 0 4 4 10 C 6 -4 0 16 4 D -10 -4 -16 0 -12 E -10 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 10 10 B 4 0 4 4 10 C 6 -4 0 16 4 D -10 -4 -16 0 -12 E -10 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999342 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7213: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (14) A D B E C (11) A C E B D (8) B D E C A (7) D B C E A (5) C E A B D (5) A C E D B (5) D B A E C (4) D A B E C (3) A E C B D (3) A D C E B (3) A C D E B (3) D B E C A (2) D B A C E (2) C A E B D (2) B E D C A (2) B E C D A (2) A D E B C (2) A D C B E (2) A D B C E (2) E C B D A (1) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E A B C D (1) D C B A E (1) D A B C E (1) C B E D A (1) C A E D B (1) B D C E A (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 10 4 6 10 B -10 0 -6 2 -8 C -4 6 0 2 12 D -6 -2 -2 0 -2 E -10 8 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 6 10 B -10 0 -6 2 -8 C -4 6 0 2 12 D -6 -2 -2 0 -2 E -10 8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=23 D=18 B=12 E=6 so E is eliminated. Round 2 votes counts: A=42 C=26 D=18 B=14 so B is eliminated. Round 3 votes counts: A=42 C=30 D=28 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:208 D:194 E:194 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 6 10 B -10 0 -6 2 -8 C -4 6 0 2 12 D -6 -2 -2 0 -2 E -10 8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 6 10 B -10 0 -6 2 -8 C -4 6 0 2 12 D -6 -2 -2 0 -2 E -10 8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 6 10 B -10 0 -6 2 -8 C -4 6 0 2 12 D -6 -2 -2 0 -2 E -10 8 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7214: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) A D C B E (9) E B C D A (5) D C A E B (5) C A D B E (5) B E D A C (5) E B D C A (4) C A B E D (4) B E C A D (4) E D B C A (3) E B D A C (3) D A E C B (3) C A D E B (3) B E A D C (3) B E A C D (3) A C D B E (3) A C B D E (3) D E C A B (2) D E B A C (2) B C E A D (2) E D B A C (1) E C B D A (1) D E B C A (1) D E A C B (1) D E A B C (1) D C E A B (1) D A E B C (1) C E B D A (1) C D A E B (1) C B E A D (1) C A E B D (1) B E D C A (1) B E C D A (1) B A E D C (1) B A E C D (1) B A C E D (1) A D B E C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 8 -6 8 B -12 0 -8 -8 0 C -8 8 0 -18 4 D 6 8 18 0 6 E -8 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -6 8 B -12 0 -8 -8 0 C -8 8 0 -18 4 D 6 8 18 0 6 E -8 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=22 A=18 E=17 C=16 so C is eliminated. Round 2 votes counts: A=31 D=28 B=23 E=18 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:211 C:193 E:191 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -6 8 B -12 0 -8 -8 0 C -8 8 0 -18 4 D 6 8 18 0 6 E -8 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -6 8 B -12 0 -8 -8 0 C -8 8 0 -18 4 D 6 8 18 0 6 E -8 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -6 8 B -12 0 -8 -8 0 C -8 8 0 -18 4 D 6 8 18 0 6 E -8 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7215: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) B E C D A (8) A D C E B (8) D A E C B (7) A D C B E (6) A D B C E (5) E C B D A (4) D E B C A (4) D A E B C (4) E B C D A (3) D E A B C (3) C E B D A (3) A C D E B (3) A C B E D (3) A C B D E (3) E D C B A (2) E D B C A (2) D A B E C (2) C B E D A (2) B C E A D (2) B C A E D (2) B A C E D (2) A D B E C (2) A C D B E (2) A B C E D (2) E B D C A (1) D E C B A (1) C E D A B (1) C E B A D (1) C A E D B (1) B E D C A (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 4 6 B -6 0 -12 -6 4 C -6 12 0 4 12 D -4 6 -4 0 2 E -6 -4 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 4 6 B -6 0 -12 -6 4 C -6 12 0 4 12 D -4 6 -4 0 2 E -6 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=21 C=17 B=15 E=12 so E is eliminated. Round 2 votes counts: A=35 D=25 C=21 B=19 so B is eliminated. Round 3 votes counts: A=37 C=36 D=27 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:211 D:200 B:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 4 6 B -6 0 -12 -6 4 C -6 12 0 4 12 D -4 6 -4 0 2 E -6 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 6 B -6 0 -12 -6 4 C -6 12 0 4 12 D -4 6 -4 0 2 E -6 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 6 B -6 0 -12 -6 4 C -6 12 0 4 12 D -4 6 -4 0 2 E -6 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7216: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (11) E B A C D (10) D C A B E (9) C D A B E (9) C A D E B (8) E B A D C (6) D B E A C (6) C A E B D (6) B E A D C (4) E B C A D (3) D B E C A (3) B E D A C (3) A C E B D (3) E B D A C (2) D C E B A (2) E A C B D (1) D E B C A (1) D C B E A (1) D C A E B (1) D B A E C (1) D A C B E (1) D A B C E (1) C D A E B (1) C A E D B (1) B E A C D (1) B D A E C (1) B A E D C (1) A D C B E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 -12 16 14 B -10 0 -12 -16 10 C 12 12 0 10 14 D -16 16 -10 0 16 E -14 -10 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 16 14 B -10 0 -12 -16 10 C 12 12 0 10 14 D -16 16 -10 0 16 E -14 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=26 E=22 B=10 A=6 so A is eliminated. Round 2 votes counts: C=40 D=28 E=22 B=10 so B is eliminated. Round 3 votes counts: C=40 E=31 D=29 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:214 D:203 B:186 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 16 14 B -10 0 -12 -16 10 C 12 12 0 10 14 D -16 16 -10 0 16 E -14 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 16 14 B -10 0 -12 -16 10 C 12 12 0 10 14 D -16 16 -10 0 16 E -14 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 16 14 B -10 0 -12 -16 10 C 12 12 0 10 14 D -16 16 -10 0 16 E -14 -10 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7217: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (7) B C D E A (6) D E A B C (5) C B A E D (5) B C D A E (5) E D C A B (4) E D A C B (4) D E B A C (4) C B D E A (4) E C D A B (3) E A D C B (3) D B E C A (3) D B A E C (3) A E D C B (3) A E D B C (3) E D C B A (2) E C D B A (2) E A C D B (2) D E B C A (2) C B E A D (2) C B A D E (2) C A B E D (2) B D C E A (2) B D C A E (2) A D E B C (2) A C E B D (2) E C A D B (1) D E C B A (1) D B E A C (1) D B C E A (1) D A E B C (1) D A B E C (1) C E D A B (1) C E A B D (1) B D A E C (1) B D A C E (1) B A D C E (1) B A C D E (1) A E C D B (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -16 -18 -8 B 16 0 10 -8 6 C 16 -10 0 -6 -8 D 18 8 6 0 18 E 8 -6 8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -18 -8 B 16 0 10 -8 6 C 16 -10 0 -6 -8 D 18 8 6 0 18 E 8 -6 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=22 E=21 C=17 A=14 so A is eliminated. Round 2 votes counts: E=28 B=28 D=25 C=19 so C is eliminated. Round 3 votes counts: B=43 E=32 D=25 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:225 B:212 C:196 E:196 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -16 -18 -8 B 16 0 10 -8 6 C 16 -10 0 -6 -8 D 18 8 6 0 18 E 8 -6 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -18 -8 B 16 0 10 -8 6 C 16 -10 0 -6 -8 D 18 8 6 0 18 E 8 -6 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -18 -8 B 16 0 10 -8 6 C 16 -10 0 -6 -8 D 18 8 6 0 18 E 8 -6 8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7218: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (15) E C D B A (8) A C D B E (8) E B D A C (5) C A D B E (5) E A C D B (4) E D B C A (3) E A B D C (3) D B C E A (3) C D B E A (3) A C E D B (3) A B E D C (3) E D C B A (2) E C A D B (2) C D B A E (2) C D A B E (2) B A D C E (2) A C B D E (2) A B C E D (2) E C D A B (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B C A E (1) C E A D B (1) C D E B A (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E C A (1) B D C A E (1) B D A E C (1) B D A C E (1) A E C B D (1) A E B D C (1) A E B C D (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -10 -8 -18 B 10 0 0 -6 -12 C 10 0 0 2 -14 D 8 6 -2 0 -22 E 18 12 14 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -10 -8 -18 B 10 0 0 -6 -12 C 10 0 0 2 -14 D 8 6 -2 0 -22 E 18 12 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=46 A=24 C=15 B=9 D=6 so D is eliminated. Round 2 votes counts: E=46 A=24 C=17 B=13 so B is eliminated. Round 3 votes counts: E=50 A=28 C=22 so C is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 C:199 B:196 D:195 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -10 -8 -18 B 10 0 0 -6 -12 C 10 0 0 2 -14 D 8 6 -2 0 -22 E 18 12 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -8 -18 B 10 0 0 -6 -12 C 10 0 0 2 -14 D 8 6 -2 0 -22 E 18 12 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -8 -18 B 10 0 0 -6 -12 C 10 0 0 2 -14 D 8 6 -2 0 -22 E 18 12 14 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7219: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) C A D B E (8) B E D C A (8) C B A E D (7) A C D E B (7) C A B D E (6) A D E C B (4) A D C E B (4) A C D B E (4) E B D C A (3) E B D A C (3) D E B A C (3) B E C D A (3) B D E A C (3) B C E A D (3) C B A D E (2) C A E B D (2) B E D A C (2) B D E C A (2) A C E D B (2) E D A C B (1) E D A B C (1) E C B A D (1) D E A B C (1) D B E A C (1) C E B A D (1) C B E A D (1) C A E D B (1) C A D E B (1) B E C A D (1) B D A C E (1) B C E D A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 -12 -6 14 4 B 12 0 -8 4 10 C 6 8 0 8 6 D -14 -4 -8 0 -2 E -4 -10 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 14 4 B 12 0 -8 4 10 C 6 8 0 8 6 D -14 -4 -8 0 -2 E -4 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=26 A=23 E=17 D=5 so D is eliminated. Round 2 votes counts: C=29 B=27 A=23 E=21 so E is eliminated. Round 3 votes counts: B=44 C=30 A=26 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:209 A:200 E:191 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -6 14 4 B 12 0 -8 4 10 C 6 8 0 8 6 D -14 -4 -8 0 -2 E -4 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 14 4 B 12 0 -8 4 10 C 6 8 0 8 6 D -14 -4 -8 0 -2 E -4 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 14 4 B 12 0 -8 4 10 C 6 8 0 8 6 D -14 -4 -8 0 -2 E -4 -10 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7220: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) B A D C E (10) A B C E D (8) E D C A B (5) E C D A B (5) C E D A B (5) E C A D B (4) D C E B A (3) C E A B D (3) B D A E C (3) B D A C E (3) B A C D E (3) A E D B C (3) A E C B D (3) A B E D C (3) A B E C D (3) E A C B D (2) D B A E C (2) C E D B A (2) C D E B A (2) B A D E C (2) A C E B D (2) E A D C B (1) E A C D B (1) D E C A B (1) D E B C A (1) D E A C B (1) D E A B C (1) D C B E A (1) D B C A E (1) C D B E A (1) C B A E D (1) C A B E D (1) B D C A E (1) B A C E D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 4 2 B -6 0 -8 0 -10 C -6 8 0 -6 -4 D -4 0 6 0 -6 E -2 10 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 4 2 B -6 0 -8 0 -10 C -6 8 0 -6 -4 D -4 0 6 0 -6 E -2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=23 A=23 D=21 E=18 C=15 so C is eliminated. Round 2 votes counts: E=28 D=24 B=24 A=24 so D is eliminated. Round 3 votes counts: E=47 B=29 A=24 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:209 E:209 D:198 C:196 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 4 2 B -6 0 -8 0 -10 C -6 8 0 -6 -4 D -4 0 6 0 -6 E -2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 2 B -6 0 -8 0 -10 C -6 8 0 -6 -4 D -4 0 6 0 -6 E -2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 2 B -6 0 -8 0 -10 C -6 8 0 -6 -4 D -4 0 6 0 -6 E -2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999421 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7221: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) D C E B A (8) E D A B C (7) B A C E D (7) D E C A B (6) C B A D E (6) A E B D C (6) E D A C B (4) E A D B C (4) C B D A E (4) B C A D E (4) A B E C D (4) A B C E D (4) E D C B A (3) E D C A B (3) D C E A B (3) C D E B A (3) B A C D E (3) E A B D C (2) C D B E A (2) C D B A E (2) E B A D C (1) D E A C B (1) C A B D E (1) B C D E A (1) A E D B C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 -10 -12 B 4 0 -8 -10 -20 C 8 8 0 -16 -2 D 10 10 16 0 6 E 12 20 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -10 -12 B 4 0 -8 -10 -20 C 8 8 0 -16 -2 D 10 10 16 0 6 E 12 20 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=24 C=18 A=17 B=15 so B is eliminated. Round 2 votes counts: A=27 D=26 E=24 C=23 so C is eliminated. Round 3 votes counts: D=38 A=38 E=24 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:214 C:199 A:183 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -10 -12 B 4 0 -8 -10 -20 C 8 8 0 -16 -2 D 10 10 16 0 6 E 12 20 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -10 -12 B 4 0 -8 -10 -20 C 8 8 0 -16 -2 D 10 10 16 0 6 E 12 20 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -10 -12 B 4 0 -8 -10 -20 C 8 8 0 -16 -2 D 10 10 16 0 6 E 12 20 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7222: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) C B E A D (9) C B A E D (8) E D A B C (7) D E A C B (7) D A E B C (6) D E A B C (5) E D A C B (4) D A E C B (3) B C A D E (3) E C D B A (2) E A B D C (2) D C A B E (2) D A C E B (2) C D E A B (2) C B E D A (2) C B D A E (2) B C E A D (2) B C A E D (2) B A D C E (2) A E D B C (2) A D E B C (2) E A D B C (1) D E C A B (1) D C E A B (1) D A B E C (1) D A B C E (1) C E B A D (1) C D E B A (1) C D B E A (1) C D A E B (1) C B D E A (1) B A D E C (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -4 -4 2 B -2 0 -22 -4 0 C 4 22 0 -2 10 D 4 4 2 0 12 E -2 0 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -4 2 B -2 0 -22 -4 0 C 4 22 0 -2 10 D 4 4 2 0 12 E -2 0 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=29 E=16 B=11 A=5 so A is eliminated. Round 2 votes counts: C=39 D=31 E=18 B=12 so B is eliminated. Round 3 votes counts: C=47 D=34 E=19 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:217 D:211 A:198 E:188 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -4 2 B -2 0 -22 -4 0 C 4 22 0 -2 10 D 4 4 2 0 12 E -2 0 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -4 2 B -2 0 -22 -4 0 C 4 22 0 -2 10 D 4 4 2 0 12 E -2 0 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -4 2 B -2 0 -22 -4 0 C 4 22 0 -2 10 D 4 4 2 0 12 E -2 0 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7223: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) B D E C A (8) D B E C A (7) D A C E B (6) B E D C A (5) A C D E B (5) D B A E C (4) C A E B D (4) A D C E B (4) A D C B E (4) A C E D B (4) D E B C A (3) D B E A C (3) C E A D B (3) E B D C A (2) C A E D B (2) B E C D A (2) B A C E D (2) A C B E D (2) E D C B A (1) E D C A B (1) E D B C A (1) E C D B A (1) E B C A D (1) D E C B A (1) D E C A B (1) D A E C B (1) D A B E C (1) D A B C E (1) C E D A B (1) C B E A D (1) C B A E D (1) B E C A D (1) B E A C D (1) B D E A C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 6 -2 10 B -8 0 -8 -10 -6 C -6 8 0 -12 8 D 2 10 12 0 4 E -10 6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -2 10 B -8 0 -8 -10 -6 C -6 8 0 -12 8 D 2 10 12 0 4 E -10 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=28 B=20 C=12 E=7 so E is eliminated. Round 2 votes counts: A=33 D=31 B=23 C=13 so C is eliminated. Round 3 votes counts: A=42 D=33 B=25 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:211 C:199 E:192 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -2 10 B -8 0 -8 -10 -6 C -6 8 0 -12 8 D 2 10 12 0 4 E -10 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -2 10 B -8 0 -8 -10 -6 C -6 8 0 -12 8 D 2 10 12 0 4 E -10 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -2 10 B -8 0 -8 -10 -6 C -6 8 0 -12 8 D 2 10 12 0 4 E -10 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7224: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) D E A C B (5) D B C E A (5) C B D A E (5) B C D A E (5) B C A E D (5) A E C D B (5) D E A B C (4) D C B A E (4) C B A E D (4) E A D B C (3) D E B A C (3) B D E C A (3) B D C E A (3) A E D C B (3) A C E D B (3) E D A B C (2) E B A C D (2) E A B D C (2) D C A B E (2) D B E A C (2) C B A D E (2) C A B E D (2) B C E A D (2) A E C B D (2) A C E B D (2) E D A C B (1) E B A D C (1) D B C A E (1) D A E C B (1) D A C E B (1) C D A B E (1) C A E B D (1) B E A C D (1) B D C A E (1) B C E D A (1) B C D E A (1) B C A D E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 2 -2 4 B 4 0 -4 -8 2 C -2 4 0 -8 6 D 2 8 8 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -2 4 B 4 0 -4 -8 2 C -2 4 0 -8 6 D 2 8 8 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=23 E=18 A=16 C=15 so C is eliminated. Round 2 votes counts: B=34 D=29 A=19 E=18 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 A:200 C:200 B:197 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -2 4 B 4 0 -4 -8 2 C -2 4 0 -8 6 D 2 8 8 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 4 B 4 0 -4 -8 2 C -2 4 0 -8 6 D 2 8 8 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 4 B 4 0 -4 -8 2 C -2 4 0 -8 6 D 2 8 8 0 2 E -4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7225: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (6) E D C A B (5) E C D B A (5) B C A D E (5) D A B C E (4) A D B E C (4) A D B C E (4) A B E C D (4) E C B D A (3) E C B A D (3) D C E B A (3) B A C E D (3) A E B D C (3) A D E B C (3) E D C B A (2) E B C A D (2) E B A C D (2) D E C A B (2) D E A C B (2) D A E B C (2) C D B E A (2) C B E D A (2) C B D E A (2) B C A E D (2) B A C D E (2) E D A C B (1) E A D C B (1) E A C B D (1) E A B D C (1) E A B C D (1) D E C B A (1) D C E A B (1) D C B E A (1) D C A E B (1) D A E C B (1) D A C E B (1) D A C B E (1) C E D B A (1) C D E B A (1) C B D A E (1) B E C A D (1) B D C A E (1) B C E A D (1) A E D B C (1) A E B C D (1) A D E C B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 8 6 B -8 0 10 -2 -4 C -4 -10 0 -8 -8 D -8 2 8 0 8 E -6 4 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 8 6 B -8 0 10 -2 -4 C -4 -10 0 -8 -8 D -8 2 8 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 D=20 B=15 C=9 so C is eliminated. Round 2 votes counts: A=29 E=28 D=23 B=20 so B is eliminated. Round 3 votes counts: A=41 E=32 D=27 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:205 E:199 B:198 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 8 6 B -8 0 10 -2 -4 C -4 -10 0 -8 -8 D -8 2 8 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 8 6 B -8 0 10 -2 -4 C -4 -10 0 -8 -8 D -8 2 8 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 8 6 B -8 0 10 -2 -4 C -4 -10 0 -8 -8 D -8 2 8 0 8 E -6 4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7226: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) B D C E A (9) C B D E A (8) D B C E A (5) D B E C A (4) C E B A D (4) C E A B D (4) B C D E A (4) A E D B C (4) A E C D B (4) E D A B C (3) D B C A E (3) C B E D A (3) C B E A D (3) E C B D A (2) E A D C B (2) D B A C E (2) C E B D A (2) B D C A E (2) A E D C B (2) A D E B C (2) A D B E C (2) A C E B D (2) E C D B A (1) E C B A D (1) E C A B D (1) E A D B C (1) D E B A C (1) D A B E C (1) C B D A E (1) C B A D E (1) C A E B D (1) A D E C B (1) A D B C E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -18 -2 -16 B 12 0 -10 22 2 C 18 10 0 8 14 D 2 -22 -8 0 -4 E 16 -2 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -18 -2 -16 B 12 0 -10 22 2 C 18 10 0 8 14 D 2 -22 -8 0 -4 E 16 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=27 D=16 B=15 E=11 so E is eliminated. Round 2 votes counts: A=34 C=32 D=19 B=15 so B is eliminated. Round 3 votes counts: C=36 A=34 D=30 so D is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:213 E:202 D:184 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -18 -2 -16 B 12 0 -10 22 2 C 18 10 0 8 14 D 2 -22 -8 0 -4 E 16 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -2 -16 B 12 0 -10 22 2 C 18 10 0 8 14 D 2 -22 -8 0 -4 E 16 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -2 -16 B 12 0 -10 22 2 C 18 10 0 8 14 D 2 -22 -8 0 -4 E 16 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7227: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) E B C D A (8) A B D E C (8) B E A D C (6) A D B E C (6) E C B D A (4) E B C A D (4) C E D B A (4) B E D C A (4) B A E D C (4) D A C B E (3) C D E B A (3) A D C E B (3) A D B C E (3) A C D E B (3) E B A C D (2) D B E C A (2) C E B D A (2) C E B A D (2) C D A E B (2) B E C D A (2) E C B A D (1) E C A B D (1) D C A E B (1) D B A E C (1) D A B E C (1) C E D A B (1) C D E A B (1) C A E B D (1) C A D E B (1) B E C A D (1) B E A C D (1) B D E C A (1) B D E A C (1) B D A E C (1) B A E C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 8 16 -2 B 10 0 14 10 10 C -8 -14 0 -8 -20 D -16 -10 8 0 0 E 2 -10 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 16 -2 B 10 0 14 10 10 C -8 -14 0 -8 -20 D -16 -10 8 0 0 E 2 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=22 E=20 C=17 D=8 so D is eliminated. Round 2 votes counts: A=37 B=25 E=20 C=18 so C is eliminated. Round 3 votes counts: A=42 E=33 B=25 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:222 A:206 E:206 D:191 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 16 -2 B 10 0 14 10 10 C -8 -14 0 -8 -20 D -16 -10 8 0 0 E 2 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 16 -2 B 10 0 14 10 10 C -8 -14 0 -8 -20 D -16 -10 8 0 0 E 2 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 16 -2 B 10 0 14 10 10 C -8 -14 0 -8 -20 D -16 -10 8 0 0 E 2 -10 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7228: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) B D C E A (8) E A C D B (7) B E A D C (7) A E C D B (7) C A D E B (6) B A E C D (4) E D C A B (3) E B A D C (3) D B C E A (3) C D A E B (3) B E D A C (3) B D E C A (3) A C E D B (3) E A D C B (2) E A C B D (2) D C E A B (2) D C A E B (2) C D A B E (2) B C A D E (2) A E B C D (2) A B E C D (2) E D B C A (1) E D A C B (1) E B D A C (1) E A B C D (1) D C E B A (1) C D B A E (1) C A D B E (1) B C D A E (1) B A E D C (1) B A C E D (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -2 6 4 B 6 0 12 10 6 C 2 -12 0 -6 0 D -6 -10 6 0 -4 E -4 -6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 6 4 B 6 0 12 10 6 C 2 -12 0 -6 0 D -6 -10 6 0 -4 E -4 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 E=21 A=15 C=13 D=8 so D is eliminated. Round 2 votes counts: B=46 E=21 C=18 A=15 so A is eliminated. Round 3 votes counts: B=48 E=30 C=22 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:201 E:197 D:193 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 6 4 B 6 0 12 10 6 C 2 -12 0 -6 0 D -6 -10 6 0 -4 E -4 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 6 4 B 6 0 12 10 6 C 2 -12 0 -6 0 D -6 -10 6 0 -4 E -4 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 6 4 B 6 0 12 10 6 C 2 -12 0 -6 0 D -6 -10 6 0 -4 E -4 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7229: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) B C D E A (8) D A E C B (6) D E A C B (5) B E A D C (5) A E D C B (5) E D A C B (4) E A D B C (4) C B D A E (4) B C E D A (4) E A B D C (3) C B A D E (3) B C D A E (3) B C A D E (3) A D E C B (3) E D A B C (2) E B A D C (2) D A C E B (2) C A D E B (2) C A D B E (2) B E C A D (2) B E A C D (2) B C A E D (2) A D C E B (2) A C D E B (2) D E B C A (1) D C E B A (1) D C E A B (1) D C A E B (1) C D E A B (1) C D B A E (1) B E D A C (1) B E C D A (1) B C E A D (1) B A E C D (1) A E D B C (1) Total count = 100 A B C D E A 0 10 0 -10 4 B -10 0 -8 -10 -14 C 0 8 0 2 4 D 10 10 -2 0 20 E -4 14 -4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.161386 B: 0.000000 C: 0.838614 D: 0.000000 E: 0.000000 Sum of squares = 0.729319524652 Cumulative probabilities = A: 0.161386 B: 0.161386 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -10 4 B -10 0 -8 -10 -14 C 0 8 0 2 4 D 10 10 -2 0 20 E -4 14 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222235831 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=22 D=17 E=15 A=13 so A is eliminated. Round 2 votes counts: B=33 C=24 D=22 E=21 so E is eliminated. Round 3 votes counts: D=38 B=38 C=24 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:207 A:202 E:193 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 0 -10 4 B -10 0 -8 -10 -14 C 0 8 0 2 4 D 10 10 -2 0 20 E -4 14 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222235831 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -10 4 B -10 0 -8 -10 -14 C 0 8 0 2 4 D 10 10 -2 0 20 E -4 14 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222235831 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -10 4 B -10 0 -8 -10 -14 C 0 8 0 2 4 D 10 10 -2 0 20 E -4 14 -4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.72222235831 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7230: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) E C B D A (8) E A D B C (6) D A E C B (6) B C A D E (6) A D B C E (6) E B C A D (5) A D E B C (5) C B D A E (4) B C E A D (4) B C A E D (4) A D B E C (4) E D C A B (3) E D A C B (3) B E C A D (3) E B C D A (2) D E A C B (2) D C A B E (2) D A C E B (2) B A C D E (2) A B E D C (2) E D A B C (1) E C D B A (1) E B A C D (1) D E C A B (1) D C E B A (1) D C B A E (1) D C A E B (1) D A C B E (1) A E D B C (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -12 2 -2 B 4 0 6 4 0 C 12 -6 0 2 -10 D -2 -4 -2 0 -8 E 2 0 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.733895 C: 0.000000 D: 0.000000 E: 0.266105 Sum of squares = 0.609413882198 Cumulative probabilities = A: 0.000000 B: 0.733895 C: 0.733895 D: 0.733895 E: 1.000000 A B C D E A 0 -4 -12 2 -2 B 4 0 6 4 0 C 12 -6 0 2 -10 D -2 -4 -2 0 -8 E 2 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=20 B=19 D=17 C=14 so C is eliminated. Round 2 votes counts: B=33 E=30 A=20 D=17 so D is eliminated. Round 3 votes counts: E=34 B=34 A=32 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:210 B:207 C:199 A:192 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -12 2 -2 B 4 0 6 4 0 C 12 -6 0 2 -10 D -2 -4 -2 0 -8 E 2 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 2 -2 B 4 0 6 4 0 C 12 -6 0 2 -10 D -2 -4 -2 0 -8 E 2 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 2 -2 B 4 0 6 4 0 C 12 -6 0 2 -10 D -2 -4 -2 0 -8 E 2 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7231: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E C A D B (9) D A C E B (9) D B A C E (6) B D E C A (6) E C A B D (5) D A B C E (4) C A E D B (4) B E C A D (4) B D A E C (4) A D C E B (4) A C E D B (4) E B C A D (3) B E D C A (3) B E C D A (3) E C B A D (2) E A C B D (2) C E A D B (2) B E D A C (2) B D C E A (2) A E C D B (2) A C D E B (2) E C B D A (1) E A C D B (1) D C A E B (1) D B C A E (1) B E A C D (1) B D C A E (1) A D B C E (1) Total count = 100 A B C D E A 0 0 6 -8 8 B 0 0 4 0 -2 C -6 -4 0 -10 4 D 8 0 10 0 4 E -8 2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.412356 C: 0.000000 D: 0.587644 E: 0.000000 Sum of squares = 0.515363043457 Cumulative probabilities = A: 0.000000 B: 0.412356 C: 0.412356 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 -8 8 B 0 0 4 0 -2 C -6 -4 0 -10 4 D 8 0 10 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=23 D=21 A=13 C=6 so C is eliminated. Round 2 votes counts: B=37 E=25 D=21 A=17 so A is eliminated. Round 3 votes counts: B=37 E=35 D=28 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:211 A:203 B:201 E:193 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 -8 8 B 0 0 4 0 -2 C -6 -4 0 -10 4 D 8 0 10 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -8 8 B 0 0 4 0 -2 C -6 -4 0 -10 4 D 8 0 10 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -8 8 B 0 0 4 0 -2 C -6 -4 0 -10 4 D 8 0 10 0 4 E -8 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999901 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7232: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) D A C B E (7) A D E C B (7) E B C D A (5) D E A B C (5) C B A E D (5) A D C B E (5) A C B E D (5) E D B C A (4) D E B C A (4) B E C D A (4) A C D B E (4) A C B D E (4) E A B C D (3) D A E C B (3) D A C E B (3) D C B A E (2) D B C E A (2) D A E B C (2) C B E A D (2) B C E D A (2) A C E B D (2) E B D C A (1) D B C A E (1) D A B E C (1) C B A D E (1) C A B D E (1) B C E A D (1) B C D E A (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 8 6 6 10 B -8 0 -4 -4 -4 C -6 4 0 2 -2 D -6 4 -2 0 10 E -10 4 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 6 10 B -8 0 -4 -4 -4 C -6 4 0 2 -2 D -6 4 -2 0 10 E -10 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=29 E=24 C=9 B=8 so B is eliminated. Round 2 votes counts: D=30 A=29 E=28 C=13 so C is eliminated. Round 3 votes counts: A=36 E=33 D=31 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:203 C:199 E:193 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 6 10 B -8 0 -4 -4 -4 C -6 4 0 2 -2 D -6 4 -2 0 10 E -10 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 6 10 B -8 0 -4 -4 -4 C -6 4 0 2 -2 D -6 4 -2 0 10 E -10 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 6 10 B -8 0 -4 -4 -4 C -6 4 0 2 -2 D -6 4 -2 0 10 E -10 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7233: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) E A B D C (7) C D B A E (7) C D E B A (6) B E A C D (5) E C B A D (4) D C A E B (4) D C A B E (4) D A E B C (4) D A B C E (4) A B D E C (4) D A C B E (3) C E D B A (3) A D E B C (3) E B A D C (2) E A D B C (2) D C E A B (2) D A E C B (2) D A B E C (2) C E B A D (2) C D E A B (2) C B E A D (2) B A E C D (2) B A C E D (2) B A C D E (2) A B E D C (2) E B C A D (1) D E C A B (1) D A C E B (1) C E B D A (1) C D B E A (1) C D A B E (1) C B A D E (1) B E A D C (1) A E B D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 16 4 2 B -2 0 6 -6 -10 C -16 -6 0 -2 -2 D -4 6 2 0 12 E -2 10 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 4 2 B -2 0 6 -6 -10 C -16 -6 0 -2 -2 D -4 6 2 0 12 E -2 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999554 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=26 E=23 B=12 A=12 so B is eliminated. Round 2 votes counts: E=29 D=27 C=26 A=18 so A is eliminated. Round 3 votes counts: D=36 E=34 C=30 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:212 D:208 E:199 B:194 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 4 2 B -2 0 6 -6 -10 C -16 -6 0 -2 -2 D -4 6 2 0 12 E -2 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999554 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 4 2 B -2 0 6 -6 -10 C -16 -6 0 -2 -2 D -4 6 2 0 12 E -2 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999554 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 4 2 B -2 0 6 -6 -10 C -16 -6 0 -2 -2 D -4 6 2 0 12 E -2 10 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999554 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7234: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) A B D E C (9) E C D A B (8) A B E D C (6) C E D B A (5) E C D B A (4) E A C B D (3) D E C B A (3) D B A E C (3) D B A C E (3) C E A D B (3) C D B A E (3) B D A C E (3) E D C B A (2) E C A D B (2) E A B C D (2) D C B E A (2) D B C A E (2) C D B E A (2) C A E B D (2) B A D E C (2) A E B D C (2) A E B C D (2) A B D C E (2) A B C D E (2) E D C A B (1) E D A B C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E A B C (1) D C E B A (1) D C B A E (1) D B E C A (1) D B E A C (1) C E A B D (1) C D E B A (1) C A B E D (1) B A C D E (1) Total count = 100 A B C D E A 0 2 10 4 6 B -2 0 6 -6 6 C -10 -6 0 -14 -12 D -4 6 14 0 4 E -6 -6 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 6 B -2 0 6 -6 6 C -10 -6 0 -14 -12 D -4 6 14 0 4 E -6 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=23 D=18 C=18 B=15 so B is eliminated. Round 2 votes counts: A=35 E=26 D=21 C=18 so C is eliminated. Round 3 votes counts: A=38 E=35 D=27 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:210 B:202 E:198 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 6 B -2 0 6 -6 6 C -10 -6 0 -14 -12 D -4 6 14 0 4 E -6 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 6 B -2 0 6 -6 6 C -10 -6 0 -14 -12 D -4 6 14 0 4 E -6 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 6 B -2 0 6 -6 6 C -10 -6 0 -14 -12 D -4 6 14 0 4 E -6 -6 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7235: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) D A C E B (8) B E C A D (7) D E C A B (6) E C D A B (5) E C B A D (5) C A E B D (4) E B C A D (3) D E B C A (3) B E D C A (3) B A C D E (3) A D C E B (3) E D C B A (2) E C D B A (2) E B C D A (2) D E B A C (2) D B E A C (2) D A E C B (2) D A B C E (2) C E A B D (2) C B E A D (2) C A E D B (2) B E C D A (2) B C E A D (2) B A D C E (2) A D B C E (2) A C D E B (2) E C B D A (1) D E C B A (1) D E A C B (1) D B E C A (1) C A B E D (1) B D E A C (1) B D A E C (1) B C A E D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -14 6 -10 B 16 0 0 6 -12 C 14 0 0 16 -4 D -6 -6 -16 0 -12 E 10 12 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -14 6 -10 B 16 0 0 6 -12 C 14 0 0 16 -4 D -6 -6 -16 0 -12 E 10 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=28 E=20 C=11 A=9 so A is eliminated. Round 2 votes counts: D=33 B=33 E=20 C=14 so C is eliminated. Round 3 votes counts: D=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:219 C:213 B:205 A:183 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -14 6 -10 B 16 0 0 6 -12 C 14 0 0 16 -4 D -6 -6 -16 0 -12 E 10 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 6 -10 B 16 0 0 6 -12 C 14 0 0 16 -4 D -6 -6 -16 0 -12 E 10 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 6 -10 B 16 0 0 6 -12 C 14 0 0 16 -4 D -6 -6 -16 0 -12 E 10 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7236: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (6) D A B E C (6) C E A D B (5) D C A B E (4) C D E A B (4) C D A E B (4) B D A E C (4) D C B A E (3) D B A E C (3) D B A C E (3) D A B C E (3) C E D A B (3) C D B A E (3) C B D E A (3) B A E D C (3) E C B A D (2) E C A B D (2) E A B D C (2) D A C B E (2) C E B D A (2) C E A B D (2) C D B E A (2) C B E D A (2) B E A D C (2) B C E D A (2) B C D E A (2) B A D E C (2) A E B D C (2) A D E B C (2) E C A D B (1) E B C A D (1) E A C D B (1) E A C B D (1) E A B C D (1) D B C A E (1) C E D B A (1) C E B A D (1) B D E C A (1) B D C E A (1) B D C A E (1) A E D C B (1) A E D B C (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -6 -20 0 B 2 0 0 -10 8 C 6 0 0 4 10 D 20 10 -4 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.158022 C: 0.841978 D: 0.000000 E: 0.000000 Sum of squares = 0.733897863538 Cumulative probabilities = A: 0.000000 B: 0.158022 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -20 0 B 2 0 0 -10 8 C 6 0 0 4 10 D 20 10 -4 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836817272 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=25 B=18 E=17 A=8 so A is eliminated. Round 2 votes counts: C=33 D=28 E=21 B=18 so B is eliminated. Round 3 votes counts: D=37 C=37 E=26 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:210 B:200 A:186 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -20 0 B 2 0 0 -10 8 C 6 0 0 4 10 D 20 10 -4 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836817272 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -20 0 B 2 0 0 -10 8 C 6 0 0 4 10 D 20 10 -4 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836817272 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -20 0 B 2 0 0 -10 8 C 6 0 0 4 10 D 20 10 -4 0 10 E 0 -8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836817272 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7237: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) E D B A C (7) C A B E D (7) D B E C A (5) C A B D E (5) A E B C D (5) A C B E D (5) C A E B D (4) E B D A C (3) D E B A C (3) D C B E A (3) B D C E A (3) A E B D C (3) D B E A C (2) C D E A B (2) C D A E B (2) C A E D B (2) B E A D C (2) B D E C A (2) E D A C B (1) E D A B C (1) E B A D C (1) E A D B C (1) E A C D B (1) E A B D C (1) D E C B A (1) D C E B A (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A B E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D A C E (1) B C D A E (1) B C A D E (1) B A E D C (1) B A C D E (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -8 -4 -4 B 2 0 12 2 -8 C 8 -12 0 -6 -8 D 4 -2 6 0 -4 E 4 8 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -8 -4 -4 B 2 0 12 2 -8 C 8 -12 0 -6 -8 D 4 -2 6 0 -4 E 4 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 A=19 E=16 B=13 so B is eliminated. Round 2 votes counts: D=30 C=30 A=21 E=19 so E is eliminated. Round 3 votes counts: D=43 C=30 A=27 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:212 B:204 D:202 A:191 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 -4 B 2 0 12 2 -8 C 8 -12 0 -6 -8 D 4 -2 6 0 -4 E 4 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 -4 B 2 0 12 2 -8 C 8 -12 0 -6 -8 D 4 -2 6 0 -4 E 4 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 -4 B 2 0 12 2 -8 C 8 -12 0 -6 -8 D 4 -2 6 0 -4 E 4 8 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7238: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) D E A B C (8) D A E B C (8) B C A E D (8) D E A C B (6) E A D B C (5) C B D E A (5) D C B A E (4) C B D A E (4) B A C E D (4) A E D B C (4) A B E C D (4) C B A E D (3) E D A C B (2) E A D C B (2) E A B C D (2) D E C A B (2) C B E D A (2) B C A D E (2) E D C A B (1) E D A B C (1) E C B A D (1) E A C D B (1) E A C B D (1) D C E B A (1) D C B E A (1) D A B E C (1) D A B C E (1) C D B E A (1) C D B A E (1) C B A D E (1) B C D A E (1) A E B D C (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 6 0 -2 B -4 0 4 0 6 C -6 -4 0 2 -2 D 0 0 -2 0 -4 E 2 -6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 6 0 -2 B -4 0 4 0 6 C -6 -4 0 2 -2 D 0 0 -2 0 -4 E 2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=26 E=16 B=15 A=11 so A is eliminated. Round 2 votes counts: D=32 C=26 E=22 B=20 so B is eliminated. Round 3 votes counts: C=41 D=33 E=26 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:204 B:203 E:201 D:197 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 0 -2 B -4 0 4 0 6 C -6 -4 0 2 -2 D 0 0 -2 0 -4 E 2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 0 -2 B -4 0 4 0 6 C -6 -4 0 2 -2 D 0 0 -2 0 -4 E 2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 0 -2 B -4 0 4 0 6 C -6 -4 0 2 -2 D 0 0 -2 0 -4 E 2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7239: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (12) C D B E A (12) A B C D E (9) E A D C B (8) B A C D E (8) D E C B A (7) B C D A E (5) B C A D E (5) E D A C B (4) D C B E A (3) A E D B C (3) A E B D C (3) E D C A B (2) E A C D B (2) C D E B A (2) A E B C D (2) A B E D C (2) A B E C D (2) E D A B C (1) E C D B A (1) E C D A B (1) E A C B D (1) D C E B A (1) D A B E C (1) C B D E A (1) B D C A E (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -6 -8 -16 B 16 0 -14 -20 0 C 6 14 0 2 -4 D 8 20 -2 0 12 E 16 0 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.111111 Sum of squares = 0.506172839495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.888889 E: 1.000000 A B C D E A 0 -16 -6 -8 -16 B 16 0 -14 -20 0 C 6 14 0 2 -4 D 8 20 -2 0 12 E 16 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.111111 Sum of squares = 0.506172839388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=22 B=19 C=15 D=12 so D is eliminated. Round 2 votes counts: E=39 A=23 C=19 B=19 so C is eliminated. Round 3 votes counts: E=42 B=35 A=23 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:219 C:209 E:204 B:191 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -6 -8 -16 B 16 0 -14 -20 0 C 6 14 0 2 -4 D 8 20 -2 0 12 E 16 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.111111 Sum of squares = 0.506172839388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.888889 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -8 -16 B 16 0 -14 -20 0 C 6 14 0 2 -4 D 8 20 -2 0 12 E 16 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.111111 Sum of squares = 0.506172839388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.888889 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -8 -16 B 16 0 -14 -20 0 C 6 14 0 2 -4 D 8 20 -2 0 12 E 16 0 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.222222 E: 0.111111 Sum of squares = 0.506172839388 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.888889 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7240: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) E A D C B (7) D E A C B (6) B C E A D (6) E D A B C (5) B C D A E (5) D A E C B (4) B D C E A (4) E D A C B (3) E A B C D (3) C B A D E (3) B C A D E (3) E D B A C (2) E A D B C (2) E A C B D (2) D E B A C (2) D E A B C (2) D C B A E (2) D C A E B (2) D B C E A (2) C D A B E (2) B E D C A (2) B C E D A (2) A E D C B (2) A E C D B (2) A C E D B (2) E B A C D (1) E A C D B (1) E A B D C (1) D B E C A (1) D B E A C (1) D A C E B (1) C B A E D (1) C A D B E (1) C A B D E (1) B E A C D (1) B D E C A (1) B D C A E (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 0 2 -2 -16 B 0 0 14 -6 -2 C -2 -14 0 -8 -4 D 2 6 8 0 -8 E 16 2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 -2 -16 B 0 0 14 -6 -2 C -2 -14 0 -8 -4 D 2 6 8 0 -8 E 16 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=27 D=23 C=8 A=7 so A is eliminated. Round 2 votes counts: B=35 E=31 D=24 C=10 so C is eliminated. Round 3 votes counts: B=40 E=33 D=27 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:204 B:203 A:192 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 -2 -16 B 0 0 14 -6 -2 C -2 -14 0 -8 -4 D 2 6 8 0 -8 E 16 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -16 B 0 0 14 -6 -2 C -2 -14 0 -8 -4 D 2 6 8 0 -8 E 16 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -16 B 0 0 14 -6 -2 C -2 -14 0 -8 -4 D 2 6 8 0 -8 E 16 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7241: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D A C E B (7) C B E D A (7) A D E B C (7) D C A B E (5) D A C B E (5) B E C A D (5) E A B D C (4) C D B A E (4) B C E D A (4) A D E C B (4) C B D E A (3) B E A C D (3) A E D B C (3) E B C A D (2) D A B E C (2) C E B D A (2) C D E B A (2) C D A E B (2) C D A B E (2) A E B D C (2) E C B A D (1) E B A D C (1) D C B A E (1) D C A E B (1) D B C A E (1) D B A C E (1) D A B C E (1) C D B E A (1) C B D A E (1) C A D E B (1) B E C D A (1) B E A D C (1) B D A E C (1) B C D E A (1) B A D E C (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 6 -10 8 B 4 0 0 -2 4 C -6 0 0 0 6 D 10 2 0 0 10 E -8 -4 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.421389 D: 0.578611 E: 0.000000 Sum of squares = 0.512359357866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.421389 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -10 8 B 4 0 0 -2 4 C -6 0 0 0 6 D 10 2 0 0 10 E -8 -4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 A=18 B=17 E=16 so E is eliminated. Round 2 votes counts: B=28 C=26 D=24 A=22 so A is eliminated. Round 3 votes counts: D=39 B=35 C=26 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:203 A:200 C:200 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -10 8 B 4 0 0 -2 4 C -6 0 0 0 6 D 10 2 0 0 10 E -8 -4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -10 8 B 4 0 0 -2 4 C -6 0 0 0 6 D 10 2 0 0 10 E -8 -4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -10 8 B 4 0 0 -2 4 C -6 0 0 0 6 D 10 2 0 0 10 E -8 -4 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7242: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) D B A E C (7) D B E C A (6) B D E A C (6) E C A B D (5) C E A D B (5) C E A B D (5) A E C B D (5) D B C A E (4) A C E B D (4) E C D B A (3) E B D A C (3) E A C B D (3) D B C E A (3) C A E D B (3) C A D E B (3) B D A E C (3) E B D C A (2) C D E B A (2) C D A B E (2) C A E B D (2) C A D B E (2) B E D A C (2) E C B D A (1) D C B A E (1) D C A B E (1) D B E A C (1) D A B C E (1) C D E A B (1) C D B E A (1) B A D E C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -4 -20 2 B 12 0 2 -12 4 C 4 -2 0 -4 0 D 20 12 4 0 12 E -2 -4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -20 2 B 12 0 2 -12 4 C 4 -2 0 -4 0 D 20 12 4 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=26 E=17 B=12 A=11 so A is eliminated. Round 2 votes counts: D=34 C=30 E=23 B=13 so B is eliminated. Round 3 votes counts: D=45 C=30 E=25 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:203 C:199 E:191 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -4 -20 2 B 12 0 2 -12 4 C 4 -2 0 -4 0 D 20 12 4 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -20 2 B 12 0 2 -12 4 C 4 -2 0 -4 0 D 20 12 4 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -20 2 B 12 0 2 -12 4 C 4 -2 0 -4 0 D 20 12 4 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994579 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7243: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (16) B E D A C (11) E C B A D (9) D A B C E (9) A D C E B (9) C E A D B (8) B E C D A (7) A D C B E (5) C A D E B (4) E B C D A (3) D A B E C (3) C E B A D (3) B D A C E (3) E C B D A (1) E C A D B (1) E B C A D (1) D B A E C (1) D A E C B (1) D A E B C (1) B E C A D (1) B D E A C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 24 -14 8 B 14 0 16 12 16 C -24 -16 0 -22 -14 D 14 -12 22 0 10 E -8 -16 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 24 -14 8 B 14 0 16 12 16 C -24 -16 0 -22 -14 D 14 -12 22 0 10 E -8 -16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 A=16 E=15 D=15 C=15 so E is eliminated. Round 2 votes counts: B=43 C=26 A=16 D=15 so D is eliminated. Round 3 votes counts: B=44 A=30 C=26 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:217 A:202 E:190 C:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 24 -14 8 B 14 0 16 12 16 C -24 -16 0 -22 -14 D 14 -12 22 0 10 E -8 -16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 24 -14 8 B 14 0 16 12 16 C -24 -16 0 -22 -14 D 14 -12 22 0 10 E -8 -16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 24 -14 8 B 14 0 16 12 16 C -24 -16 0 -22 -14 D 14 -12 22 0 10 E -8 -16 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7244: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) D C E A B (7) D A B E C (7) B A E C D (7) C E B A D (6) E A B D C (5) D A E B C (5) E B A C D (4) D E A B C (4) C E D B A (4) A B E D C (4) A B D E C (4) E C B A D (3) D C E B A (2) D C A B E (2) D A B C E (2) C E B D A (2) C D B A E (2) C B E A D (2) A D B E C (2) E D C B A (1) E B C A D (1) E B A D C (1) D E C B A (1) D E C A B (1) D E A C B (1) D A E C B (1) C B A D E (1) C A B D E (1) B E A C D (1) B A E D C (1) B A C E D (1) A D E B C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 6 -6 -14 B 2 0 4 -8 -22 C -6 -4 0 -6 -12 D 6 8 6 0 12 E 14 22 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -6 -14 B 2 0 4 -8 -22 C -6 -4 0 -6 -12 D 6 8 6 0 12 E 14 22 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=29 E=15 A=13 B=10 so B is eliminated. Round 2 votes counts: D=33 C=29 A=22 E=16 so E is eliminated. Round 3 votes counts: D=34 C=33 A=33 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:218 D:216 A:192 B:188 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -6 -14 B 2 0 4 -8 -22 C -6 -4 0 -6 -12 D 6 8 6 0 12 E 14 22 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -6 -14 B 2 0 4 -8 -22 C -6 -4 0 -6 -12 D 6 8 6 0 12 E 14 22 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -6 -14 B 2 0 4 -8 -22 C -6 -4 0 -6 -12 D 6 8 6 0 12 E 14 22 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7245: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) B A E D C (10) A E B C D (9) E A B C D (8) D C E B A (6) B A D E C (6) C E D A B (5) B D A C E (5) A B E C D (5) E C A D B (4) D B C A E (4) B A E C D (4) E A C B D (3) D C B E A (3) D C B A E (3) E C D A B (2) E A C D B (2) B D C A E (2) A E C B D (2) E D C A B (1) E B D A C (1) D C E A B (1) D B E C A (1) B D A E C (1) Total count = 100 A B C D E A 0 8 12 6 2 B -8 0 12 12 -12 C -12 -12 0 12 -18 D -6 -12 -12 0 -12 E -2 12 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 6 2 B -8 0 12 12 -12 C -12 -12 0 12 -18 D -6 -12 -12 0 -12 E -2 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996214 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=21 D=18 C=17 A=16 so A is eliminated. Round 2 votes counts: B=33 E=32 D=18 C=17 so C is eliminated. Round 3 votes counts: E=37 B=33 D=30 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:214 B:202 C:185 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 6 2 B -8 0 12 12 -12 C -12 -12 0 12 -18 D -6 -12 -12 0 -12 E -2 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996214 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 6 2 B -8 0 12 12 -12 C -12 -12 0 12 -18 D -6 -12 -12 0 -12 E -2 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996214 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 6 2 B -8 0 12 12 -12 C -12 -12 0 12 -18 D -6 -12 -12 0 -12 E -2 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996214 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7246: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) C D E A B (7) B A E D C (7) E C D B A (6) A B D E C (6) E C B D A (5) C D A E B (4) B A D E C (4) E D C B A (3) D B A E C (3) C E D A B (3) B E D A C (3) B E A C D (3) A D B E C (3) A D B C E (3) A B D C E (3) E B C A D (2) D C A E B (2) D A C B E (2) D A B E C (2) C E A D B (2) A D C B E (2) E C B A D (1) E B C D A (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E B A (1) D C E A B (1) C E B A D (1) C A E D B (1) C A D E B (1) C A D B E (1) B E A D C (1) B D E A C (1) A C D B E (1) A C B D E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 -8 -2 B 4 0 -6 -16 -2 C 2 6 0 -2 -10 D 8 16 2 0 4 E 2 2 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 -2 B 4 0 -6 -16 -2 C 2 6 0 -2 -10 D 8 16 2 0 4 E 2 2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=22 B=19 E=18 D=14 so D is eliminated. Round 2 votes counts: C=31 A=26 B=22 E=21 so E is eliminated. Round 3 votes counts: C=47 B=27 A=26 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:215 E:205 C:198 A:192 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 -2 B 4 0 -6 -16 -2 C 2 6 0 -2 -10 D 8 16 2 0 4 E 2 2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 -2 B 4 0 -6 -16 -2 C 2 6 0 -2 -10 D 8 16 2 0 4 E 2 2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 -2 B 4 0 -6 -16 -2 C 2 6 0 -2 -10 D 8 16 2 0 4 E 2 2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7247: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) C D A B E (8) B E A D C (7) D C B A E (6) D B C A E (6) C A D E B (6) E A C B D (5) E A B C D (5) E B A D C (4) D C A B E (4) A E C D B (4) E B A C D (3) B D C A E (3) A C E D B (3) E C A D B (2) E A C D B (2) C D B E A (2) C A E D B (2) B D C E A (2) A E B C D (2) A B D E C (2) E C A B D (1) E A B D C (1) C E D B A (1) C D B A E (1) B E C D A (1) B D E C A (1) B D E A C (1) B D A E C (1) B D A C E (1) B A E D C (1) A E D C B (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 18 -10 6 24 B -18 0 -16 -18 -6 C 10 16 0 16 10 D -6 18 -16 0 8 E -24 6 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -10 6 24 B -18 0 -16 -18 -6 C 10 16 0 16 10 D -6 18 -16 0 8 E -24 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=23 B=18 D=16 A=14 so A is eliminated. Round 2 votes counts: C=33 E=31 B=20 D=16 so D is eliminated. Round 3 votes counts: C=43 E=31 B=26 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:219 D:202 E:182 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -10 6 24 B -18 0 -16 -18 -6 C 10 16 0 16 10 D -6 18 -16 0 8 E -24 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -10 6 24 B -18 0 -16 -18 -6 C 10 16 0 16 10 D -6 18 -16 0 8 E -24 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -10 6 24 B -18 0 -16 -18 -6 C 10 16 0 16 10 D -6 18 -16 0 8 E -24 6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7248: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) C E A B D (10) D B A E C (9) C E B A D (6) B D E C A (4) A D E B C (4) C A E D B (3) C A D E B (3) B E D A C (3) E C B A D (2) E B C D A (2) D B A C E (2) C D A E B (2) C B D E A (2) C A E B D (2) B E C D A (2) B E A D C (2) B C E D A (2) B C D E A (2) A E C B D (2) A E B D C (2) A D B E C (2) A C D E B (2) E B A D C (1) E B A C D (1) E A C B D (1) E A B D C (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B C E (1) C E B D A (1) C D B E A (1) C B E D A (1) B D E A C (1) B D C E A (1) B D A E C (1) B A E D C (1) A E C D B (1) A D C B E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 2 -2 2 B 2 0 16 8 4 C -2 -16 0 0 -8 D 2 -8 0 0 4 E -2 -4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 2 B 2 0 16 8 4 C -2 -16 0 0 -8 D 2 -8 0 0 4 E -2 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993003 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=26 B=19 A=16 E=8 so E is eliminated. Round 2 votes counts: C=33 D=26 B=23 A=18 so A is eliminated. Round 3 votes counts: C=41 D=33 B=26 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:215 A:200 D:199 E:199 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 2 B 2 0 16 8 4 C -2 -16 0 0 -8 D 2 -8 0 0 4 E -2 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993003 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 2 B 2 0 16 8 4 C -2 -16 0 0 -8 D 2 -8 0 0 4 E -2 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993003 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 2 B 2 0 16 8 4 C -2 -16 0 0 -8 D 2 -8 0 0 4 E -2 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993003 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7249: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) C D B E A (8) B D E C A (7) D B C E A (5) C D A B E (5) C E B D A (4) A C D E B (4) E B A D C (3) D B E C A (3) D B E A C (3) A C E D B (3) E B C D A (2) E B A C D (2) E A B D C (2) E A B C D (2) C E D B A (2) C E A B D (2) C A E D B (2) C A E B D (2) A E C B D (2) A E B C D (2) A D C B E (2) A D B E C (2) A C D B E (2) E C B A D (1) E B D A C (1) E B C A D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B C A E (1) D B A E C (1) D A C B E (1) C D E B A (1) C D B A E (1) C B E D A (1) C B D E A (1) C A D B E (1) B E D A C (1) A E D C B (1) A E D B C (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -4 0 -8 B 2 0 0 -4 -6 C 4 0 0 6 2 D 0 4 -6 0 2 E 8 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.184670 C: 0.815330 D: 0.000000 E: 0.000000 Sum of squares = 0.698866411763 Cumulative probabilities = A: 0.000000 B: 0.184670 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 0 -8 B 2 0 0 -4 -6 C 4 0 0 6 2 D 0 4 -6 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000006087 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=30 D=16 E=15 B=8 so B is eliminated. Round 2 votes counts: A=31 C=30 D=23 E=16 so E is eliminated. Round 3 votes counts: A=41 C=34 D=25 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:206 E:205 D:200 B:196 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 0 -8 B 2 0 0 -4 -6 C 4 0 0 6 2 D 0 4 -6 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000006087 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 0 -8 B 2 0 0 -4 -6 C 4 0 0 6 2 D 0 4 -6 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000006087 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 0 -8 B 2 0 0 -4 -6 C 4 0 0 6 2 D 0 4 -6 0 2 E 8 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000006087 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7250: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) D E B C A (7) D A B E C (7) A B D C E (7) E C D B A (6) D E C B A (6) A B C E D (6) E C B D A (5) D B A E C (4) A C E B D (4) A C B E D (4) E D C B A (3) C E B A D (3) A D B C E (3) E C A D B (2) E C A B D (2) D E A C B (2) D B E A C (2) D A B C E (2) C E A B D (2) E D A C B (1) E C D A B (1) E C B A D (1) D E C A B (1) D B A C E (1) D A E B C (1) C A E B D (1) B D C E A (1) B C E D A (1) B A D C E (1) B A C E D (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -4 -22 -12 B 4 0 10 -22 2 C 4 -10 0 -20 -26 D 22 22 20 0 12 E 12 -2 26 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -22 -12 B 4 0 10 -22 2 C 4 -10 0 -20 -26 D 22 22 20 0 12 E 12 -2 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 A=26 E=21 C=6 B=4 so B is eliminated. Round 2 votes counts: D=44 A=28 E=21 C=7 so C is eliminated. Round 3 votes counts: D=44 A=29 E=27 so E is eliminated. Round 4 votes counts: D=61 A=39 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:238 E:212 B:197 A:179 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -22 -12 B 4 0 10 -22 2 C 4 -10 0 -20 -26 D 22 22 20 0 12 E 12 -2 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -22 -12 B 4 0 10 -22 2 C 4 -10 0 -20 -26 D 22 22 20 0 12 E 12 -2 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -22 -12 B 4 0 10 -22 2 C 4 -10 0 -20 -26 D 22 22 20 0 12 E 12 -2 26 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7251: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (12) C A E B D (9) B E C A D (9) D A E B C (6) B E D C A (6) B E C D A (5) D B E C A (4) D A E C B (4) C B E A D (4) B D E C A (4) A C E D B (4) A C D E B (4) D B A E C (3) D A B E C (3) D B E A C (2) C E B A D (2) C B D E A (2) B C E A D (2) A E C B D (2) A D C E B (2) A C E B D (2) E C B A D (1) E A C B D (1) E A B D C (1) E A B C D (1) C E A B D (1) C B A E D (1) C A D B E (1) B E D A C (1) B C E D A (1) Total count = 100 A B C D E A 0 6 -4 -6 6 B -6 0 -4 10 -4 C 4 4 0 4 -6 D 6 -10 -4 0 -6 E -6 4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999997 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 6 -4 -6 6 B -6 0 -4 10 -4 C 4 4 0 4 -6 D 6 -10 -4 0 -6 E -6 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=28 C=20 A=14 E=4 so E is eliminated. Round 2 votes counts: D=34 B=28 C=21 A=17 so A is eliminated. Round 3 votes counts: D=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:205 C:203 A:201 B:198 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -4 -6 6 B -6 0 -4 10 -4 C 4 4 0 4 -6 D 6 -10 -4 0 -6 E -6 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -6 6 B -6 0 -4 10 -4 C 4 4 0 4 -6 D 6 -10 -4 0 -6 E -6 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -6 6 B -6 0 -4 10 -4 C 4 4 0 4 -6 D 6 -10 -4 0 -6 E -6 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999911 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7252: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (12) C B A D E (8) D B E C A (7) D E B A C (6) D B C E A (6) C A B E D (6) B D C E A (5) E A D C B (4) C A B D E (4) B C D A E (4) A E C D B (4) B D E C A (3) B C A E D (3) E B D A C (2) D B C A E (2) C B D A E (2) A E C B D (2) E D B A C (1) E D A C B (1) E D A B C (1) E B A D C (1) E B A C D (1) E A C D B (1) E A C B D (1) E A B D C (1) D E A B C (1) D C A E B (1) C D A B E (1) C B A E D (1) C A E B D (1) C A D B E (1) B E D C A (1) B E A C D (1) B D C A E (1) B C D E A (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -16 8 10 B 12 0 -4 22 14 C 16 4 0 12 22 D -8 -22 -12 0 8 E -10 -14 -22 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 8 10 B 12 0 -4 22 14 C 16 4 0 12 22 D -8 -22 -12 0 8 E -10 -14 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 A=20 B=19 E=14 so E is eliminated. Round 2 votes counts: A=27 D=26 C=24 B=23 so B is eliminated. Round 3 votes counts: D=38 C=32 A=30 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:222 A:195 D:183 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -16 8 10 B 12 0 -4 22 14 C 16 4 0 12 22 D -8 -22 -12 0 8 E -10 -14 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 8 10 B 12 0 -4 22 14 C 16 4 0 12 22 D -8 -22 -12 0 8 E -10 -14 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 8 10 B 12 0 -4 22 14 C 16 4 0 12 22 D -8 -22 -12 0 8 E -10 -14 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7253: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) E D A C B (7) E B A D C (6) B C A D E (5) B A E C D (5) D E C A B (4) C D A E B (4) E B D A C (3) E A D C B (3) D C E A B (3) D C A E B (3) B E A D C (3) E A B C D (2) D E A C B (2) C D A B E (2) C A D B E (2) C A B D E (2) B E D C A (2) B E D A C (2) B E C A D (2) B D C E A (2) B C D A E (2) B A C E D (2) A C D E B (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A B C (1) E A D B C (1) E A C D B (1) D E C B A (1) C D B A E (1) C B A D E (1) C A D E B (1) B D C A E (1) B C A E D (1) B A C D E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 18 14 -14 B 2 0 6 8 -4 C -18 -6 0 0 -22 D -14 -8 0 0 -18 E 14 4 22 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 18 14 -14 B 2 0 6 8 -4 C -18 -6 0 0 -22 D -14 -8 0 0 -18 E 14 4 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=27 D=13 C=13 A=10 so A is eliminated. Round 2 votes counts: B=39 E=29 C=17 D=15 so D is eliminated. Round 3 votes counts: B=39 E=37 C=24 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:229 A:208 B:206 D:180 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 18 14 -14 B 2 0 6 8 -4 C -18 -6 0 0 -22 D -14 -8 0 0 -18 E 14 4 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 18 14 -14 B 2 0 6 8 -4 C -18 -6 0 0 -22 D -14 -8 0 0 -18 E 14 4 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 18 14 -14 B 2 0 6 8 -4 C -18 -6 0 0 -22 D -14 -8 0 0 -18 E 14 4 22 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7254: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (6) C E D A B (5) C A D E B (4) B E D A C (4) A B D C E (4) E D B C A (3) E C D A B (3) E B D C A (3) D E A C B (3) C E B A D (3) B C A E D (3) A C D B E (3) E D C A B (2) E C D B A (2) E B D A C (2) E B C D A (2) D E C A B (2) D E A B C (2) D B E A C (2) D B A E C (2) D A C E B (2) C D E A B (2) C A E D B (2) C A E B D (2) B D A E C (2) B A E D C (2) B A E C D (2) A D C B E (2) E D C B A (1) E D B A C (1) E C B D A (1) D E B A C (1) D C E A B (1) D C A E B (1) D A E C B (1) D A E B C (1) C E A D B (1) C E A B D (1) C D A E B (1) C B E A D (1) C B A E D (1) C A B E D (1) C A B D E (1) B E C A D (1) B D E A C (1) B C E A D (1) B A C D E (1) A D C E B (1) A D B E C (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 -6 -2 B -4 0 -2 -6 -12 C 2 2 0 -8 -6 D 6 6 8 0 0 E 2 12 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.699766 E: 0.300234 Sum of squares = 0.579812996497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.699766 E: 1.000000 A B C D E A 0 4 -2 -6 -2 B -4 0 -2 -6 -12 C 2 2 0 -8 -6 D 6 6 8 0 0 E 2 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 E=20 D=18 A=14 so A is eliminated. Round 2 votes counts: C=29 B=28 D=23 E=20 so E is eliminated. Round 3 votes counts: C=35 B=35 D=30 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:210 E:210 A:197 C:195 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -6 -2 B -4 0 -2 -6 -12 C 2 2 0 -8 -6 D 6 6 8 0 0 E 2 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -6 -2 B -4 0 -2 -6 -12 C 2 2 0 -8 -6 D 6 6 8 0 0 E 2 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -6 -2 B -4 0 -2 -6 -12 C 2 2 0 -8 -6 D 6 6 8 0 0 E 2 12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7255: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) B C A E D (11) B C D A E (8) C B A E D (7) B C A D E (7) E A C D B (6) D E A B C (6) D B E C A (6) E D A C B (5) D E B A C (4) C A B E D (4) A E C D B (4) A C E B D (4) E A D C B (3) D B C E A (3) C A E B D (3) B D C E A (2) B D C A E (2) D E B C A (1) C B A D E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -10 4 6 B 4 0 0 2 4 C 10 0 0 14 6 D -4 -2 -14 0 2 E -6 -4 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.643299 C: 0.356701 D: 0.000000 E: 0.000000 Sum of squares = 0.54106931218 Cumulative probabilities = A: 0.000000 B: 0.643299 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 4 6 B 4 0 0 2 4 C 10 0 0 14 6 D -4 -2 -14 0 2 E -6 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=30 C=15 E=14 A=10 so A is eliminated. Round 2 votes counts: D=31 B=30 C=20 E=19 so E is eliminated. Round 3 votes counts: D=39 C=31 B=30 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:205 A:198 D:191 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 4 6 B 4 0 0 2 4 C 10 0 0 14 6 D -4 -2 -14 0 2 E -6 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 4 6 B 4 0 0 2 4 C 10 0 0 14 6 D -4 -2 -14 0 2 E -6 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 4 6 B 4 0 0 2 4 C 10 0 0 14 6 D -4 -2 -14 0 2 E -6 -4 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7256: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) A B C D E (9) C A B E D (6) E C A B D (5) E A C B D (5) C A E B D (5) A C B E D (5) E C A D B (4) D B A C E (4) D E B A C (3) D B C E A (3) B D A C E (3) B A D C E (3) B A C D E (3) A B D C E (3) E D B A C (2) E C D B A (2) E C D A B (2) E A C D B (2) D E B C A (2) D B A E C (2) C E D B A (2) C D B E A (2) A D B E C (2) A C E B D (2) E A D B C (1) D B E C A (1) D B E A C (1) D B C A E (1) C E A B D (1) C B D A E (1) C B A D E (1) A E B D C (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 6 20 6 B -10 0 -8 10 2 C -6 8 0 18 10 D -20 -10 -18 0 -12 E -6 -2 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 20 6 B -10 0 -8 10 2 C -6 8 0 18 10 D -20 -10 -18 0 -12 E -6 -2 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=24 C=18 D=17 B=9 so B is eliminated. Round 2 votes counts: E=32 A=30 D=20 C=18 so C is eliminated. Round 3 votes counts: A=42 E=35 D=23 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:215 B:197 E:197 D:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 20 6 B -10 0 -8 10 2 C -6 8 0 18 10 D -20 -10 -18 0 -12 E -6 -2 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 20 6 B -10 0 -8 10 2 C -6 8 0 18 10 D -20 -10 -18 0 -12 E -6 -2 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 20 6 B -10 0 -8 10 2 C -6 8 0 18 10 D -20 -10 -18 0 -12 E -6 -2 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7257: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) A C B D E (7) E D B C A (5) A C E B D (5) E C D B A (4) A E C D B (4) E A D B C (3) D B E C A (3) D B C E A (3) B D C A E (3) A E B C D (3) A B D C E (3) A B C D E (3) E D C B A (2) E C D A B (2) E C A D B (2) E B D A C (2) E A C D B (2) D E C B A (2) C E D A B (2) C E A D B (2) C D E B A (2) C A E D B (2) C A B D E (2) B D C E A (2) B A D C E (2) A E B D C (2) A C E D B (2) E D B A C (1) E D A B C (1) E A B D C (1) D E B C A (1) D C E B A (1) D C B E A (1) C E D B A (1) C D E A B (1) C D B E A (1) C D B A E (1) C B D A E (1) C A D B E (1) B D E A C (1) B D A E C (1) B D A C E (1) A C B E D (1) Total count = 100 A B C D E A 0 18 6 10 4 B -18 0 -18 -4 -26 C -6 18 0 18 0 D -10 4 -18 0 -14 E -4 26 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 10 4 B -18 0 -18 -4 -26 C -6 18 0 18 0 D -10 4 -18 0 -14 E -4 26 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=25 C=16 D=11 B=10 so B is eliminated. Round 2 votes counts: A=40 E=25 D=19 C=16 so C is eliminated. Round 3 votes counts: A=45 E=30 D=25 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:218 C:215 D:181 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 10 4 B -18 0 -18 -4 -26 C -6 18 0 18 0 D -10 4 -18 0 -14 E -4 26 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 10 4 B -18 0 -18 -4 -26 C -6 18 0 18 0 D -10 4 -18 0 -14 E -4 26 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 10 4 B -18 0 -18 -4 -26 C -6 18 0 18 0 D -10 4 -18 0 -14 E -4 26 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7258: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) B E C D A (7) D A C B E (6) B E D A C (6) D C A B E (5) D A E B C (4) C E A B D (4) C D A B E (4) C A E B D (4) C A D E B (4) E B A C D (3) D E B A C (3) D B A E C (3) C B D E A (3) B D E A C (3) E B C A D (2) D B E A C (2) C B E A D (2) C A E D B (2) B E A D C (2) B C E D A (2) A E C D B (2) A C E D B (2) E C B A D (1) E B A D C (1) E A B D C (1) E A B C D (1) D C B A E (1) D C A E B (1) D A B E C (1) C D B A E (1) C D A E B (1) C A D B E (1) B E D C A (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 -8 -6 B 4 0 6 4 18 C 10 -6 0 12 -10 D 8 -4 -12 0 -6 E 6 -18 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -8 -6 B 4 0 6 4 18 C 10 -6 0 12 -10 D 8 -4 -12 0 -6 E 6 -18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=26 C=26 E=9 A=9 so E is eliminated. Round 2 votes counts: B=36 C=27 D=26 A=11 so A is eliminated. Round 3 votes counts: B=38 C=32 D=30 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:203 E:202 D:193 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -8 -6 B 4 0 6 4 18 C 10 -6 0 12 -10 D 8 -4 -12 0 -6 E 6 -18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -8 -6 B 4 0 6 4 18 C 10 -6 0 12 -10 D 8 -4 -12 0 -6 E 6 -18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -8 -6 B 4 0 6 4 18 C 10 -6 0 12 -10 D 8 -4 -12 0 -6 E 6 -18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7259: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) E A D C B (9) E D C B A (7) A E B C D (7) A B C D E (6) D E C B A (5) A B E C D (5) E A C B D (4) B A C D E (4) A E B D C (4) C D B E A (3) B C D A E (3) A E D B C (3) E D C A B (2) E A D B C (2) E A C D B (2) D C E B A (2) D A B C E (2) C D E B A (2) C B D E A (2) B D C A E (2) B C A D E (2) A B D C E (2) A B C E D (2) E A B C D (1) D E C A B (1) D E A B C (1) D C B A E (1) D B C A E (1) C E D B A (1) C E B D A (1) Total count = 100 A B C D E A 0 6 8 6 -12 B -6 0 -6 -10 -8 C -8 6 0 -10 -6 D -6 10 10 0 0 E 12 8 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.429766 E: 0.570234 Sum of squares = 0.509865745525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.429766 E: 1.000000 A B C D E A 0 6 8 6 -12 B -6 0 -6 -10 -8 C -8 6 0 -10 -6 D -6 10 10 0 0 E 12 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 D=24 B=11 C=9 so C is eliminated. Round 2 votes counts: E=29 D=29 A=29 B=13 so B is eliminated. Round 3 votes counts: D=36 A=35 E=29 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:213 D:207 A:204 C:191 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 6 -12 B -6 0 -6 -10 -8 C -8 6 0 -10 -6 D -6 10 10 0 0 E 12 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 6 -12 B -6 0 -6 -10 -8 C -8 6 0 -10 -6 D -6 10 10 0 0 E 12 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 6 -12 B -6 0 -6 -10 -8 C -8 6 0 -10 -6 D -6 10 10 0 0 E 12 8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7260: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (18) D E A B C (15) C B D A E (7) D E A C B (6) B C A E D (6) C D B A E (5) B A E C D (5) D E C A B (4) B A C E D (4) E A B D C (3) D A E B C (3) C D B E A (3) E D A B C (2) D C E A B (2) C E B A D (2) C D E B A (2) C B A D E (2) A E B D C (2) E C A B D (1) D C E B A (1) C E D B A (1) C B E D A (1) C B D E A (1) B D A C E (1) B C A D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -20 -14 -8 12 B 20 0 -12 12 12 C 14 12 0 20 14 D 8 -12 -20 0 6 E -12 -12 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -14 -8 12 B 20 0 -12 12 12 C 14 12 0 20 14 D 8 -12 -20 0 6 E -12 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 D=31 B=17 E=6 A=4 so A is eliminated. Round 2 votes counts: C=42 D=31 B=19 E=8 so E is eliminated. Round 3 votes counts: C=43 D=33 B=24 so B is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:216 D:191 A:185 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -14 -8 12 B 20 0 -12 12 12 C 14 12 0 20 14 D 8 -12 -20 0 6 E -12 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 -8 12 B 20 0 -12 12 12 C 14 12 0 20 14 D 8 -12 -20 0 6 E -12 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 -8 12 B 20 0 -12 12 12 C 14 12 0 20 14 D 8 -12 -20 0 6 E -12 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7261: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) B D C A E (7) E A C D B (6) E A D B C (5) A B E D C (5) D C B E A (4) D B C A E (4) C D E B A (4) C B D A E (4) A E C B D (4) E A D C B (3) D E C B A (3) D B C E A (3) C D B E A (3) B C D A E (3) E D C A B (2) E D A B C (2) E C D A B (2) E A B C D (2) D C E B A (2) C E D A B (2) C D E A B (2) B A E D C (2) A B E C D (2) A B C E D (2) E D C B A (1) E A C B D (1) E A B D C (1) D E C A B (1) D E B C A (1) C D B A E (1) C A E B D (1) C A B D E (1) B D A C E (1) B C A D E (1) B A D C E (1) B A C D E (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -4 -4 0 B -8 0 4 -2 -8 C 4 -4 0 2 -4 D 4 2 -2 0 -6 E 0 8 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.321923 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.678077 Sum of squares = 0.563423008487 Cumulative probabilities = A: 0.321923 B: 0.321923 C: 0.321923 D: 0.321923 E: 1.000000 A B C D E A 0 8 -4 -4 0 B -8 0 4 -2 -8 C 4 -4 0 2 -4 D 4 2 -2 0 -6 E 0 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499829 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500171 Sum of squares = 0.500000058247 Cumulative probabilities = A: 0.499829 B: 0.499829 C: 0.499829 D: 0.499829 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 D=18 C=18 B=16 so B is eliminated. Round 2 votes counts: A=27 D=26 E=25 C=22 so C is eliminated. Round 3 votes counts: D=43 A=30 E=27 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:209 A:200 C:199 D:199 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -4 -4 0 B -8 0 4 -2 -8 C 4 -4 0 2 -4 D 4 2 -2 0 -6 E 0 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499829 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500171 Sum of squares = 0.500000058247 Cumulative probabilities = A: 0.499829 B: 0.499829 C: 0.499829 D: 0.499829 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -4 0 B -8 0 4 -2 -8 C 4 -4 0 2 -4 D 4 2 -2 0 -6 E 0 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499829 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500171 Sum of squares = 0.500000058247 Cumulative probabilities = A: 0.499829 B: 0.499829 C: 0.499829 D: 0.499829 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -4 0 B -8 0 4 -2 -8 C 4 -4 0 2 -4 D 4 2 -2 0 -6 E 0 8 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499829 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500171 Sum of squares = 0.500000058247 Cumulative probabilities = A: 0.499829 B: 0.499829 C: 0.499829 D: 0.499829 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7262: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D C A E B (7) A D C E B (6) D C A B E (5) D A C E B (5) C E B D A (5) D A C B E (4) C D B E A (4) A E B D C (4) A E B C D (4) D C B E A (3) D C B A E (3) C D A E B (3) B E A C D (3) E C B A D (2) E A B C D (2) D A B E C (2) D A B C E (2) C D E B A (2) B E D A C (2) B E C D A (2) B E A D C (2) B C E D A (2) B A E D C (2) A E D B C (2) A D E B C (2) A C D E B (2) E B C A D (1) E B A C D (1) D B C E A (1) C E B A D (1) C E A B D (1) C B E D A (1) B E D C A (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -6 -8 10 B -6 0 -10 -10 -2 C 6 10 0 -10 14 D 8 10 10 0 6 E -10 2 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -8 10 B -6 0 -10 -10 -2 C 6 10 0 -10 14 D 8 10 10 0 6 E -10 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=23 A=22 C=17 E=6 so E is eliminated. Round 2 votes counts: D=32 B=25 A=24 C=19 so C is eliminated. Round 3 votes counts: D=41 B=34 A=25 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:210 A:201 B:186 E:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 -8 10 B -6 0 -10 -10 -2 C 6 10 0 -10 14 D 8 10 10 0 6 E -10 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -8 10 B -6 0 -10 -10 -2 C 6 10 0 -10 14 D 8 10 10 0 6 E -10 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -8 10 B -6 0 -10 -10 -2 C 6 10 0 -10 14 D 8 10 10 0 6 E -10 2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7263: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) D B A E C (8) C E B D A (8) B D C A E (8) E C A D B (7) E A C D B (7) D A B E C (6) A E D C B (6) C E A B D (5) C B E D A (5) A D E B C (5) C E B A D (4) E C A B D (3) E A D C B (3) B D C E A (3) B C D E A (3) D B A C E (2) D A E B C (2) A D B E C (2) E D A C B (1) C B D E A (1) B D A E C (1) B C D A E (1) Total count = 100 A B C D E A 0 -6 4 -16 0 B 6 0 0 2 -2 C -4 0 0 -12 -2 D 16 -2 12 0 2 E 0 2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333331 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 -6 4 -16 0 B 6 0 0 2 -2 C -4 0 0 -12 -2 D 16 -2 12 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 E=21 D=18 A=13 so A is eliminated. Round 2 votes counts: E=27 D=25 B=25 C=23 so C is eliminated. Round 3 votes counts: E=44 B=31 D=25 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:214 B:203 E:201 A:191 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -16 0 B 6 0 0 2 -2 C -4 0 0 -12 -2 D 16 -2 12 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -16 0 B 6 0 0 2 -2 C -4 0 0 -12 -2 D 16 -2 12 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -16 0 B 6 0 0 2 -2 C -4 0 0 -12 -2 D 16 -2 12 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7264: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) A E D C B (7) E C A B D (6) A D E C B (6) E B C D A (4) D A C B E (4) A D E B C (4) A D C B E (4) E A C B D (3) D B C A E (3) D A B C E (3) C B E D A (3) B C E D A (3) A D C E B (3) E C B A D (2) E B C A D (2) E A C D B (2) E A B D C (2) E A B C D (2) D C B A E (2) D B A C E (2) D A B E C (2) C E B D A (2) B E C D A (2) B D C A E (2) B D A C E (2) A D B E C (2) E C B D A (1) E C A D B (1) E B A C D (1) C D B A E (1) C D A E B (1) C B D E A (1) B E D C A (1) B E C A D (1) B E A D C (1) B D C E A (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 8 4 2 0 B -8 0 2 4 -2 C -4 -2 0 -6 -8 D -2 -4 6 0 4 E 0 2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.908377 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.091623 Sum of squares = 0.833544304583 Cumulative probabilities = A: 0.908377 B: 0.908377 C: 0.908377 D: 0.908377 E: 1.000000 A B C D E A 0 8 4 2 0 B -8 0 2 4 -2 C -4 -2 0 -6 -8 D -2 -4 6 0 4 E 0 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555649386 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=26 B=22 D=16 C=8 so C is eliminated. Round 2 votes counts: E=28 A=28 B=26 D=18 so D is eliminated. Round 3 votes counts: A=38 B=34 E=28 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:203 D:202 B:198 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 2 0 B -8 0 2 4 -2 C -4 -2 0 -6 -8 D -2 -4 6 0 4 E 0 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555649386 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 0 B -8 0 2 4 -2 C -4 -2 0 -6 -8 D -2 -4 6 0 4 E 0 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555649386 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 0 B -8 0 2 4 -2 C -4 -2 0 -6 -8 D -2 -4 6 0 4 E 0 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555649386 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7265: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) C A D B E (11) A C B D E (10) E B D C A (6) C A E D B (6) E C A D B (5) B D A E C (5) B D A C E (5) E C D B A (4) D B E C A (4) E B D A C (3) A E C B D (3) E A C B D (2) D B C A E (2) B D E C A (2) B A D E C (2) A C E B D (2) A B D C E (2) A B C D E (2) E D C B A (1) E C D A B (1) E B A D C (1) D C B A E (1) C E A D B (1) C D B A E (1) C A D E B (1) B E D A C (1) B D E A C (1) B D C A E (1) A E B C D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -16 2 14 B 2 0 0 0 4 C 16 0 0 2 -8 D -2 0 -2 0 2 E -14 -4 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.769767 C: 0.230233 D: 0.000000 E: 0.000000 Sum of squares = 0.645548407269 Cumulative probabilities = A: 0.000000 B: 0.769767 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 2 14 B 2 0 0 0 4 C 16 0 0 2 -8 D -2 0 -2 0 2 E -14 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555732856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=22 C=20 B=17 D=7 so D is eliminated. Round 2 votes counts: E=34 B=23 A=22 C=21 so C is eliminated. Round 3 votes counts: A=40 E=35 B=25 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:205 B:203 A:199 D:199 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -16 2 14 B 2 0 0 0 4 C 16 0 0 2 -8 D -2 0 -2 0 2 E -14 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555732856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 2 14 B 2 0 0 0 4 C 16 0 0 2 -8 D -2 0 -2 0 2 E -14 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555732856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 2 14 B 2 0 0 0 4 C 16 0 0 2 -8 D -2 0 -2 0 2 E -14 -4 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555732856 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7266: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (12) D C A B E (8) E B A C D (7) A C D E B (7) C D A B E (6) C A D B E (5) B E D C A (5) A C D B E (5) A C B E D (5) E B D C A (4) B E A C D (4) D E C B A (3) D C A E B (3) A B C E D (3) E D B C A (2) E B D A C (2) E B A D C (2) D C E B A (2) D C B E A (2) D B C E A (2) E A D C B (1) D C B A E (1) D B E C A (1) C D B A E (1) C A D E B (1) C A B D E (1) B E D A C (1) B D E C A (1) B C D A E (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 18 2 10 22 B -18 0 -6 -2 6 C -2 6 0 20 8 D -10 2 -20 0 2 E -22 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 10 22 B -18 0 -6 -2 6 C -2 6 0 20 8 D -10 2 -20 0 2 E -22 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982328 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=22 E=18 C=14 B=12 so B is eliminated. Round 2 votes counts: A=34 E=28 D=23 C=15 so C is eliminated. Round 3 votes counts: A=41 D=31 E=28 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 C:216 B:190 D:187 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 10 22 B -18 0 -6 -2 6 C -2 6 0 20 8 D -10 2 -20 0 2 E -22 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982328 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 10 22 B -18 0 -6 -2 6 C -2 6 0 20 8 D -10 2 -20 0 2 E -22 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982328 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 10 22 B -18 0 -6 -2 6 C -2 6 0 20 8 D -10 2 -20 0 2 E -22 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982328 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7267: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) A B C E D (9) D E C A B (7) D B A E C (6) A B D C E (6) D C E A B (5) C E D A B (5) B A C E D (5) B D A E C (4) B A E C D (4) B A D E C (4) A C E B D (4) E C D B A (3) D B E C A (3) C E B A D (3) C E A B D (3) E D C B A (2) E C B D A (2) D E B C A (2) D A B C E (2) B A D C E (2) E C B A D (1) E B C D A (1) C E A D B (1) C A E B D (1) B E C A D (1) B D E C A (1) B C E A D (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -4 0 B 8 0 6 8 2 C 2 -6 0 -8 0 D 4 -8 8 0 6 E 0 -2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -4 0 B 8 0 6 8 2 C 2 -6 0 -8 0 D 4 -8 8 0 6 E 0 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=22 A=22 C=13 E=9 so E is eliminated. Round 2 votes counts: D=36 B=23 A=22 C=19 so C is eliminated. Round 3 votes counts: D=44 B=29 A=27 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:205 E:196 C:194 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 0 B 8 0 6 8 2 C 2 -6 0 -8 0 D 4 -8 8 0 6 E 0 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 0 B 8 0 6 8 2 C 2 -6 0 -8 0 D 4 -8 8 0 6 E 0 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 0 B 8 0 6 8 2 C 2 -6 0 -8 0 D 4 -8 8 0 6 E 0 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7268: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) B C A D E (6) E A D B C (5) A E D C B (5) A E D B C (5) A D E C B (5) C D B E A (4) C B A D E (4) B E C D A (4) B C D E A (4) A D C E B (4) A C D B E (4) E D A C B (3) D E A C B (3) C B D E A (3) C B D A E (3) A B C D E (3) E D A B C (2) E A B D C (2) D A E C B (2) B C D A E (2) B C A E D (2) A B E C D (2) E D B A C (1) E B D C A (1) E B D A C (1) E A D C B (1) D E C A B (1) D C A E B (1) C D A B E (1) C A D B E (1) C A B D E (1) B E C A D (1) B C E A D (1) B A E C D (1) A E B D C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 4 12 10 B -8 0 6 4 14 C -4 -6 0 14 8 D -12 -4 -14 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 12 10 B -8 0 6 4 14 C -4 -6 0 14 8 D -12 -4 -14 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=29 C=17 E=16 D=7 so D is eliminated. Round 2 votes counts: A=33 B=29 E=20 C=18 so C is eliminated. Round 3 votes counts: B=43 A=37 E=20 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:208 C:206 D:188 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 12 10 B -8 0 6 4 14 C -4 -6 0 14 8 D -12 -4 -14 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 12 10 B -8 0 6 4 14 C -4 -6 0 14 8 D -12 -4 -14 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 12 10 B -8 0 6 4 14 C -4 -6 0 14 8 D -12 -4 -14 0 6 E -10 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7269: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (16) C A B E D (11) E D B A C (7) D E A B C (6) C A B D E (6) E B D A C (5) A B E D C (5) D E C B A (4) A B D E C (4) D A E B C (3) C E D B A (3) C D E A B (3) A C B E D (3) A B C E D (3) D E B C A (2) C B E A D (2) C B A E D (2) B A E C D (2) A D E B C (2) E D C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) C E B D A (1) C D E B A (1) C B E D A (1) B E C A D (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 18 -12 -12 B -2 0 18 -6 -14 C -18 -18 0 -20 -20 D 12 6 20 0 6 E 12 14 20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 -12 -12 B -2 0 18 -6 -14 C -18 -18 0 -20 -20 D 12 6 20 0 6 E 12 14 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=30 A=20 E=13 B=3 so B is eliminated. Round 2 votes counts: D=34 C=30 A=22 E=14 so E is eliminated. Round 3 votes counts: D=47 C=31 A=22 so A is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:220 A:198 B:198 C:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 18 -12 -12 B -2 0 18 -6 -14 C -18 -18 0 -20 -20 D 12 6 20 0 6 E 12 14 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 -12 -12 B -2 0 18 -6 -14 C -18 -18 0 -20 -20 D 12 6 20 0 6 E 12 14 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 -12 -12 B -2 0 18 -6 -14 C -18 -18 0 -20 -20 D 12 6 20 0 6 E 12 14 20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7270: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) E A C B D (10) B D A E C (10) C D B E A (7) B A D E C (7) E A B D C (6) E A B C D (6) C E A D B (6) B D A C E (5) A E B D C (5) D C B A E (4) C D E B A (3) C D E A B (3) E C A D B (2) D B C E A (2) B A E D C (2) A B E D C (2) D E C B A (1) D E B C A (1) D C B E A (1) D B A C E (1) C E D A B (1) C D B A E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -14 12 -4 2 B 14 0 20 10 8 C -12 -20 0 -18 -8 D 4 -10 18 0 16 E -2 -8 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 12 -4 2 B 14 0 20 10 8 C -12 -20 0 -18 -8 D 4 -10 18 0 16 E -2 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=24 B=24 D=22 C=21 A=9 so A is eliminated. Round 2 votes counts: E=31 B=26 D=22 C=21 so C is eliminated. Round 3 votes counts: E=38 D=36 B=26 so B is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:226 D:214 A:198 E:191 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 12 -4 2 B 14 0 20 10 8 C -12 -20 0 -18 -8 D 4 -10 18 0 16 E -2 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 12 -4 2 B 14 0 20 10 8 C -12 -20 0 -18 -8 D 4 -10 18 0 16 E -2 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 12 -4 2 B 14 0 20 10 8 C -12 -20 0 -18 -8 D 4 -10 18 0 16 E -2 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7271: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) C B A E D (7) E A D B C (6) D E A B C (5) B C A D E (5) B A C E D (5) D E A C B (4) A E B D C (4) E A D C B (3) D E C A B (3) C D B E A (3) C B D E A (3) B C D A E (3) A E C B D (3) A E B C D (3) E A C D B (2) D E B A C (2) D C B E A (2) D B C E A (2) D B A E C (2) C B A D E (2) C A E B D (2) C A B E D (2) B C A E D (2) B A D E C (2) A C B E D (2) A B E C D (2) E D A B C (1) D E C B A (1) D E B C A (1) D C E B A (1) C E D A B (1) C E A D B (1) C D E B A (1) C B D A E (1) B D C A E (1) B D A E C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 12 10 2 B -6 0 -2 4 -2 C -12 2 0 2 -6 D -10 -4 -2 0 -10 E -2 2 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 10 2 B -6 0 -2 4 -2 C -12 2 0 2 -6 D -10 -4 -2 0 -10 E -2 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=23 C=23 E=19 B=19 A=16 so A is eliminated. Round 2 votes counts: E=29 C=25 D=23 B=23 so D is eliminated. Round 3 votes counts: E=45 C=28 B=27 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:215 E:208 B:197 C:193 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 10 2 B -6 0 -2 4 -2 C -12 2 0 2 -6 D -10 -4 -2 0 -10 E -2 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 10 2 B -6 0 -2 4 -2 C -12 2 0 2 -6 D -10 -4 -2 0 -10 E -2 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 10 2 B -6 0 -2 4 -2 C -12 2 0 2 -6 D -10 -4 -2 0 -10 E -2 2 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7272: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) E C A D B (7) D B E C A (7) D B E A C (7) D B A C E (7) A C B D E (7) C A E B D (6) C A E D B (5) B D A C E (5) E B D A C (4) C A D B E (4) A C E B D (4) E D B C A (3) C E A B D (3) B E D A C (3) E A C B D (2) C E A D B (2) A C D B E (2) A B C D E (2) E C D B A (1) E C D A B (1) E C A B D (1) E B D C A (1) D E B C A (1) D B C A E (1) D A C B E (1) B D A E C (1) B A D C E (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 14 -6 -6 B 4 0 8 0 18 C -14 -8 0 -6 2 D 6 0 6 0 14 E 6 -18 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.104665 C: 0.000000 D: 0.895335 E: 0.000000 Sum of squares = 0.812579917092 Cumulative probabilities = A: 0.000000 B: 0.104665 C: 0.104665 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 -6 -6 B 4 0 8 0 18 C -14 -8 0 -6 2 D 6 0 6 0 14 E 6 -18 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=20 C=20 B=20 A=16 so A is eliminated. Round 2 votes counts: C=33 D=25 B=22 E=20 so E is eliminated. Round 3 votes counts: C=45 D=28 B=27 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:215 D:213 A:199 C:187 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 -6 -6 B 4 0 8 0 18 C -14 -8 0 -6 2 D 6 0 6 0 14 E 6 -18 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 -6 -6 B 4 0 8 0 18 C -14 -8 0 -6 2 D 6 0 6 0 14 E 6 -18 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 -6 -6 B 4 0 8 0 18 C -14 -8 0 -6 2 D 6 0 6 0 14 E 6 -18 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999925 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7273: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (13) B E C A D (12) A D E C B (8) C D A B E (7) C B D A E (7) E B A D C (6) E A D B C (4) D C A B E (4) B C D A E (4) D A C B E (3) C A D E B (3) B C E D A (3) A D C E B (3) E B C A D (2) E A D C B (2) D C A E B (2) B E C D A (2) B E A D C (2) B C E A D (2) E D B A C (1) E A C D B (1) D A E C B (1) C D A E B (1) C B E D A (1) C B E A D (1) B E D C A (1) B E D A C (1) B D C A E (1) B D A C E (1) A D E B C (1) Total count = 100 A B C D E A 0 6 -6 -6 18 B -6 0 -14 -8 4 C 6 14 0 -8 12 D 6 8 8 0 18 E -18 -4 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -6 18 B -6 0 -14 -8 4 C 6 14 0 -8 12 D 6 8 8 0 18 E -18 -4 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 C=20 E=16 A=12 so A is eliminated. Round 2 votes counts: D=35 B=29 C=20 E=16 so E is eliminated. Round 3 votes counts: D=42 B=37 C=21 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:212 A:206 B:188 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -6 -6 18 B -6 0 -14 -8 4 C 6 14 0 -8 12 D 6 8 8 0 18 E -18 -4 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -6 18 B -6 0 -14 -8 4 C 6 14 0 -8 12 D 6 8 8 0 18 E -18 -4 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -6 18 B -6 0 -14 -8 4 C 6 14 0 -8 12 D 6 8 8 0 18 E -18 -4 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7274: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) B E A C D (8) C D A E B (7) D A C B E (5) C D E A B (5) A D C B E (5) D C A B E (4) A B E C D (4) E A B C D (3) D C E B A (3) A E B C D (3) A D C E B (3) A C D E B (3) A B E D C (3) A B D C E (3) E C D B A (2) E B A C D (2) D C A E B (2) C E D B A (2) B E C D A (2) B E A D C (2) B D A C E (2) B A E D C (2) A D B C E (2) A B D E C (2) E C B D A (1) E B C A D (1) E A C B D (1) D C E A B (1) D C B E A (1) D C B A E (1) C E D A B (1) C D E B A (1) C A E D B (1) C A D E B (1) B E D A C (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 22 12 2 10 B -22 0 -4 -4 -6 C -12 4 0 16 8 D -2 4 -16 0 2 E -10 6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998391 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 2 10 B -22 0 -4 -4 -6 C -12 4 0 16 8 D -2 4 -16 0 2 E -10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981147 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=18 C=18 D=17 B=17 so D is eliminated. Round 2 votes counts: A=35 C=30 E=18 B=17 so B is eliminated. Round 3 votes counts: A=39 E=31 C=30 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:208 D:194 E:193 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 12 2 10 B -22 0 -4 -4 -6 C -12 4 0 16 8 D -2 4 -16 0 2 E -10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981147 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 2 10 B -22 0 -4 -4 -6 C -12 4 0 16 8 D -2 4 -16 0 2 E -10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981147 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 2 10 B -22 0 -4 -4 -6 C -12 4 0 16 8 D -2 4 -16 0 2 E -10 6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981147 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7275: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (15) D C B A E (14) E A B C D (12) C D E A B (7) C D B E A (7) E A C B D (5) D B A C E (5) C E D A B (5) E C A B D (4) A E B D C (3) A B E D C (3) D B C A E (2) C E A D B (2) C D E B A (2) B D A E C (2) B A D E C (2) E A C D B (1) D C E A B (1) D C B E A (1) D C A E B (1) D B A E C (1) C E B A D (1) C E A B D (1) C B D E A (1) B D A C E (1) B C D A E (1) Total count = 100 A B C D E A 0 -10 0 -2 0 B 10 0 -6 2 10 C 0 6 0 -2 4 D 2 -2 2 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -2 0 B 10 0 -6 2 10 C 0 6 0 -2 4 D 2 -2 2 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999961 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 E=22 B=21 A=6 so A is eliminated. Round 2 votes counts: C=26 E=25 D=25 B=24 so B is eliminated. Round 3 votes counts: E=43 D=30 C=27 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:208 C:204 D:199 E:195 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 -2 0 B 10 0 -6 2 10 C 0 6 0 -2 4 D 2 -2 2 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999961 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -2 0 B 10 0 -6 2 10 C 0 6 0 -2 4 D 2 -2 2 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999961 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -2 0 B 10 0 -6 2 10 C 0 6 0 -2 4 D 2 -2 2 0 -4 E 0 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999961 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7276: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) E D A B C (9) D E A C B (8) C B A D E (8) D A E C B (7) E B D A C (6) A C D B E (6) D A C E B (5) E D B A C (4) B C E A D (4) B C A E D (4) C A D B E (3) C A B D E (3) A D C B E (3) E B D C A (2) E B A D C (2) D A C B E (2) B E A C D (2) B C A D E (2) E B C A D (1) E B A C D (1) D E C A B (1) D C A B E (1) C D A B E (1) C B D A E (1) B E C D A (1) B C E D A (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 14 2 -4 B -2 0 -2 -2 6 C -14 2 0 -2 -8 D -2 2 2 0 6 E 4 -6 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888513 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 2 14 2 -4 B -2 0 -2 -2 6 C -14 2 0 -2 -8 D -2 2 2 0 6 E 4 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=24 B=24 C=16 A=11 so A is eliminated. Round 2 votes counts: D=28 E=25 B=24 C=23 so C is eliminated. Round 3 votes counts: D=38 B=37 E=25 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:204 B:200 E:200 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 14 2 -4 B -2 0 -2 -2 6 C -14 2 0 -2 -8 D -2 2 2 0 6 E 4 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 2 -4 B -2 0 -2 -2 6 C -14 2 0 -2 -8 D -2 2 2 0 6 E 4 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 2 -4 B -2 0 -2 -2 6 C -14 2 0 -2 -8 D -2 2 2 0 6 E 4 -6 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7277: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) B D E C A (5) A E D B C (5) A D B E C (5) C B D A E (4) C A E B D (4) B D E A C (4) A E D C B (4) A C E D B (4) A C B D E (4) E D A B C (3) C E D B A (3) C E B D A (3) C A E D B (3) B C D E A (3) A E C D B (3) A B D E C (3) E D B C A (2) E C D B A (2) D B E A C (2) C E B A D (2) C E A D B (2) C B E D A (2) C B A D E (2) C A B D E (2) B D C E A (2) B C D A E (2) A B C D E (2) E D B A C (1) E C D A B (1) E A D C B (1) E A C D B (1) D E B C A (1) D E B A C (1) D A B E C (1) B D A E C (1) B C A D E (1) B A D C E (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -4 2 6 B 0 0 -6 8 6 C 4 6 0 12 6 D -2 -8 -12 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 6 B 0 0 -6 8 6 C 4 6 0 12 6 D -2 -8 -12 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=32 B=19 E=11 D=5 so D is eliminated. Round 2 votes counts: C=33 A=33 B=21 E=13 so E is eliminated. Round 3 votes counts: A=38 C=36 B=26 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:204 A:202 D:193 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 2 6 B 0 0 -6 8 6 C 4 6 0 12 6 D -2 -8 -12 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 6 B 0 0 -6 8 6 C 4 6 0 12 6 D -2 -8 -12 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 6 B 0 0 -6 8 6 C 4 6 0 12 6 D -2 -8 -12 0 8 E -6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999591 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7278: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (13) B C A D E (11) E D B A C (7) B E C D A (5) A C B D E (5) E D B C A (4) D E A C B (4) D A E C B (4) B C E A D (4) B C A E D (4) A D C E B (4) E D A B C (3) C A B D E (3) B E A C D (3) A C D B E (3) E D C A B (2) E B D C A (2) E B D A C (2) D C A E B (2) D A C E B (2) B E D C A (2) B C D A E (2) B A C D E (2) E B A D C (1) C B A D E (1) C A D B E (1) B E C A D (1) B E A D C (1) B C E D A (1) B A C E D (1) Total count = 100 A B C D E A 0 -8 10 -10 -10 B 8 0 12 2 0 C -10 -12 0 -6 -8 D 10 -2 6 0 -12 E 10 0 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.682146 C: 0.000000 D: 0.000000 E: 0.317854 Sum of squares = 0.566354504368 Cumulative probabilities = A: 0.000000 B: 0.682146 C: 0.682146 D: 0.682146 E: 1.000000 A B C D E A 0 -8 10 -10 -10 B 8 0 12 2 0 C -10 -12 0 -6 -8 D 10 -2 6 0 -12 E 10 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=34 D=12 A=12 C=5 so C is eliminated. Round 2 votes counts: B=38 E=34 A=16 D=12 so D is eliminated. Round 3 votes counts: E=38 B=38 A=24 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:211 D:201 A:191 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 -10 -10 B 8 0 12 2 0 C -10 -12 0 -6 -8 D 10 -2 6 0 -12 E 10 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 -10 -10 B 8 0 12 2 0 C -10 -12 0 -6 -8 D 10 -2 6 0 -12 E 10 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 -10 -10 B 8 0 12 2 0 C -10 -12 0 -6 -8 D 10 -2 6 0 -12 E 10 0 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7279: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (10) A E D C B (7) C B E D A (6) B C D E A (6) B C A E D (5) A D E C B (5) D A E C B (4) D A E B C (4) A D B E C (4) A B D C E (4) A B C D E (4) C B E A D (3) B C E A D (3) A D E B C (3) E C D B A (2) E C B D A (2) E C A D B (2) C E B A D (2) B A C E D (2) A C E B D (2) A B C E D (2) E D C B A (1) E D A C B (1) E A D C B (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A C B (1) D B C E A (1) D B A C E (1) C E B D A (1) C A E B D (1) B D C E A (1) B D C A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 2 10 8 B 4 0 12 16 14 C -2 -12 0 12 18 D -10 -16 -12 0 -10 E -8 -14 -18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 10 8 B 4 0 12 16 14 C -2 -12 0 12 18 D -10 -16 -12 0 -10 E -8 -14 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=30 D=14 C=13 E=9 so E is eliminated. Round 2 votes counts: A=35 B=30 C=19 D=16 so D is eliminated. Round 3 votes counts: A=45 B=34 C=21 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:208 C:208 E:185 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 10 8 B 4 0 12 16 14 C -2 -12 0 12 18 D -10 -16 -12 0 -10 E -8 -14 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 10 8 B 4 0 12 16 14 C -2 -12 0 12 18 D -10 -16 -12 0 -10 E -8 -14 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 10 8 B 4 0 12 16 14 C -2 -12 0 12 18 D -10 -16 -12 0 -10 E -8 -14 -18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7280: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (11) E D B A C (10) C A D B E (8) E D B C A (7) D E C A B (7) E B D A C (5) D E B C A (5) B A C D E (5) E D C A B (4) B A C E D (4) D C A B E (3) C A B E D (3) A C B E D (3) A C B D E (3) E D C B A (2) D E C B A (2) D C E A B (2) C D A E B (2) B E D A C (2) B E A C D (2) E B A C D (1) E A B C D (1) D E B A C (1) D C B A E (1) C A E D B (1) C A D E B (1) B E A D C (1) B A D E C (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -18 -6 -4 B -2 0 -8 -12 -4 C 18 8 0 -6 -4 D 6 12 6 0 6 E 4 4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 -6 -4 B -2 0 -8 -12 -4 C 18 8 0 -6 -4 D 6 12 6 0 6 E 4 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 D=21 B=15 A=8 so A is eliminated. Round 2 votes counts: C=32 E=31 D=21 B=16 so B is eliminated. Round 3 votes counts: C=42 E=36 D=22 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:215 C:208 E:203 A:187 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -18 -6 -4 B -2 0 -8 -12 -4 C 18 8 0 -6 -4 D 6 12 6 0 6 E 4 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -6 -4 B -2 0 -8 -12 -4 C 18 8 0 -6 -4 D 6 12 6 0 6 E 4 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -6 -4 B -2 0 -8 -12 -4 C 18 8 0 -6 -4 D 6 12 6 0 6 E 4 4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7281: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (10) C A B D E (9) B C D E A (8) B D E C A (6) A E D C B (6) E D B A C (5) D E B A C (4) C B A E D (4) A D E C B (4) B E D C A (3) A C E D B (3) A C D E B (3) E B D C A (2) E A D B C (2) D E A B C (2) C A E D B (2) C A B E D (2) B D C E A (2) B C E D A (2) A D C E B (2) E C D A B (1) E A D C B (1) C B E A D (1) C A E B D (1) B D E A C (1) B C D A E (1) B A D C E (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 6 0 B 0 0 -2 6 2 C 8 2 0 -2 4 D -6 -6 2 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000009 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 6 0 B 0 0 -2 6 2 C 8 2 0 -2 4 D -6 -6 2 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000176 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=24 E=21 A=20 D=6 so D is eliminated. Round 2 votes counts: C=29 E=27 B=24 A=20 so A is eliminated. Round 3 votes counts: E=38 C=38 B=24 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:203 A:199 D:199 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 6 0 B 0 0 -2 6 2 C 8 2 0 -2 4 D -6 -6 2 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000176 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 6 0 B 0 0 -2 6 2 C 8 2 0 -2 4 D -6 -6 2 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000176 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 6 0 B 0 0 -2 6 2 C 8 2 0 -2 4 D -6 -6 2 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000176 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7282: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (11) B E C A D (8) D A C E B (7) E B A C D (6) C B D E A (6) A D E B C (6) C B E A D (5) D C B A E (4) A E D B C (4) A D E C B (4) E A B D C (3) B C E D A (3) E B A D C (2) E A B C D (2) D C A B E (2) D A E B C (2) D A C B E (2) C E B A D (2) C D B E A (2) C D B A E (2) C D A E B (2) B E D A C (2) B E C D A (2) B C E A D (2) B C D E A (2) D B E A C (1) D A E C B (1) D A B E C (1) C D A B E (1) B E D C A (1) B E A C D (1) A E B D C (1) Total count = 100 A B C D E A 0 -24 -10 -8 -22 B 24 0 -2 18 16 C 10 2 0 14 6 D 8 -18 -14 0 -10 E 22 -16 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -10 -8 -22 B 24 0 -2 18 16 C 10 2 0 14 6 D 8 -18 -14 0 -10 E 22 -16 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=21 D=20 A=15 E=13 so E is eliminated. Round 2 votes counts: C=31 B=29 D=20 A=20 so D is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:228 C:216 E:205 D:183 A:168 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -24 -10 -8 -22 B 24 0 -2 18 16 C 10 2 0 14 6 D 8 -18 -14 0 -10 E 22 -16 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -10 -8 -22 B 24 0 -2 18 16 C 10 2 0 14 6 D 8 -18 -14 0 -10 E 22 -16 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -10 -8 -22 B 24 0 -2 18 16 C 10 2 0 14 6 D 8 -18 -14 0 -10 E 22 -16 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7283: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) E D B C A (6) D B E C A (6) D B E A C (6) B D A C E (6) A C E B D (6) E C D B A (4) D B C E A (4) C E A B D (4) A E C B D (4) C E A D B (3) C A E B D (3) A C B E D (3) A B D C E (3) A B C D E (3) E C D A B (2) E C A B D (2) D B C A E (2) D B A E C (2) D B A C E (2) C E D B A (2) A E B D C (2) A E B C D (2) A C B D E (2) E D C B A (1) E A B D C (1) E A B C D (1) D E B A C (1) C D B A E (1) C B D A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B A D E C (1) Total count = 100 A B C D E A 0 8 -10 8 -8 B -8 0 -4 -6 -10 C 10 4 0 14 -4 D -8 6 -14 0 -16 E 8 10 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -10 8 -8 B -8 0 -4 -6 -10 C 10 4 0 14 -4 D -8 6 -14 0 -16 E 8 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 D=23 C=17 B=7 so B is eliminated. Round 2 votes counts: D=29 E=28 A=26 C=17 so C is eliminated. Round 3 votes counts: E=37 A=32 D=31 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:212 A:199 B:186 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -10 8 -8 B -8 0 -4 -6 -10 C 10 4 0 14 -4 D -8 6 -14 0 -16 E 8 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 8 -8 B -8 0 -4 -6 -10 C 10 4 0 14 -4 D -8 6 -14 0 -16 E 8 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 8 -8 B -8 0 -4 -6 -10 C 10 4 0 14 -4 D -8 6 -14 0 -16 E 8 10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7284: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) D E C B A (8) A E B C D (7) E D C A B (6) E D A C B (6) A B C E D (6) D B C E A (5) E D A B C (4) E A D B C (4) C B D A E (4) C B A D E (4) B C D A E (4) B C A D E (4) B A C D E (4) D E B A C (3) D C B E A (3) D B C A E (3) A C B E D (3) D C E B A (2) C A B D E (2) A B E C D (2) E D C B A (1) E D B A C (1) E A C B D (1) D E B C A (1) D C B A E (1) D B A E C (1) C D B A E (1) C A E B D (1) Total count = 100 A B C D E A 0 0 0 -8 -6 B 0 0 -2 -16 -6 C 0 2 0 -14 -6 D 8 16 14 0 0 E 6 6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.259933 E: 0.740067 Sum of squares = 0.6152641733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.259933 E: 1.000000 A B C D E A 0 0 0 -8 -6 B 0 0 -2 -16 -6 C 0 2 0 -14 -6 D 8 16 14 0 0 E 6 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=27 A=18 C=12 B=12 so C is eliminated. Round 2 votes counts: E=31 D=28 A=21 B=20 so B is eliminated. Round 3 votes counts: D=36 A=33 E=31 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:209 A:193 C:191 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -8 -6 B 0 0 -2 -16 -6 C 0 2 0 -14 -6 D 8 16 14 0 0 E 6 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 -6 B 0 0 -2 -16 -6 C 0 2 0 -14 -6 D 8 16 14 0 0 E 6 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 -6 B 0 0 -2 -16 -6 C 0 2 0 -14 -6 D 8 16 14 0 0 E 6 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7285: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (23) C D E A B (16) B A E D C (13) C D E B A (10) E D C A B (7) B A C D E (7) D C E A B (3) C D B E A (3) E D B A C (2) E D A B C (2) D E C A B (2) C B A D E (2) B A C E D (2) A E B D C (2) E A D C B (1) E A D B C (1) D E C B A (1) C D B A E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 18 10 6 4 B -18 0 6 0 4 C -10 -6 0 -16 -10 D -6 0 16 0 -10 E -4 -4 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 10 6 4 B -18 0 6 0 4 C -10 -6 0 -16 -10 D -6 0 16 0 -10 E -4 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=27 B=22 E=13 D=6 so D is eliminated. Round 2 votes counts: C=35 A=27 B=22 E=16 so E is eliminated. Round 3 votes counts: C=45 A=31 B=24 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:206 D:200 B:196 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 6 4 B -18 0 6 0 4 C -10 -6 0 -16 -10 D -6 0 16 0 -10 E -4 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 6 4 B -18 0 6 0 4 C -10 -6 0 -16 -10 D -6 0 16 0 -10 E -4 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 6 4 B -18 0 6 0 4 C -10 -6 0 -16 -10 D -6 0 16 0 -10 E -4 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7286: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) A D E B C (14) B C E D A (11) A D E C B (11) C B A D E (7) E C B D A (6) E D A C B (5) D A E B C (5) B C A D E (4) E D C B A (3) E D A B C (3) C E B D A (3) A D C B E (2) E D B A C (1) E C D A B (1) D B A E C (1) C E B A D (1) C B E A D (1) C B A E D (1) B C E A D (1) B C D E A (1) B C A E D (1) A E D C B (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -10 -8 -2 B 12 0 -14 2 -10 C 10 14 0 6 -4 D 8 -2 -6 0 -6 E 2 10 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -10 -8 -2 B 12 0 -14 2 -10 C 10 14 0 6 -4 D 8 -2 -6 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=27 E=19 B=18 D=6 so D is eliminated. Round 2 votes counts: A=35 C=27 E=19 B=19 so E is eliminated. Round 3 votes counts: A=43 C=37 B=20 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:211 D:197 B:195 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -10 -8 -2 B 12 0 -14 2 -10 C 10 14 0 6 -4 D 8 -2 -6 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -8 -2 B 12 0 -14 2 -10 C 10 14 0 6 -4 D 8 -2 -6 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -8 -2 B 12 0 -14 2 -10 C 10 14 0 6 -4 D 8 -2 -6 0 -6 E 2 10 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (10) B E A D C (8) C D A E B (7) C A D E B (7) D E C B A (6) C A D B E (6) A B C E D (6) B E D A C (5) E B D C A (4) E B D A C (4) B A E C D (4) B E A C D (3) A C D B E (3) D C E B A (2) D C E A B (2) B E D C A (2) A C D E B (2) E D B A C (1) D E C A B (1) D C A E B (1) C D B E A (1) C D B A E (1) A D E C B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -2 4 0 B 4 0 0 -4 2 C 2 0 0 2 -4 D -4 4 -2 0 0 E 0 -2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.222222 C: 0.111111 D: 0.277778 E: 0.222222 Sum of squares = 0.216049382714 Cumulative probabilities = A: 0.166667 B: 0.388889 C: 0.500000 D: 0.777778 E: 1.000000 A B C D E A 0 -4 -2 4 0 B 4 0 0 -4 2 C 2 0 0 2 -4 D -4 4 -2 0 0 E 0 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.222222 C: 0.111111 D: 0.277778 E: 0.222222 Sum of squares = 0.216049382717 Cumulative probabilities = A: 0.166667 B: 0.388889 C: 0.500000 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=22 B=22 E=20 D=12 so D is eliminated. Round 2 votes counts: E=27 C=27 A=24 B=22 so B is eliminated. Round 3 votes counts: E=45 A=28 C=27 so C is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:201 E:201 C:200 A:199 D:199 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D E , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 -2 4 0 B 4 0 0 -4 2 C 2 0 0 2 -4 D -4 4 -2 0 0 E 0 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.222222 C: 0.111111 D: 0.277778 E: 0.222222 Sum of squares = 0.216049382717 Cumulative probabilities = A: 0.166667 B: 0.388889 C: 0.500000 D: 0.777778 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 4 0 B 4 0 0 -4 2 C 2 0 0 2 -4 D -4 4 -2 0 0 E 0 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.222222 C: 0.111111 D: 0.277778 E: 0.222222 Sum of squares = 0.216049382717 Cumulative probabilities = A: 0.166667 B: 0.388889 C: 0.500000 D: 0.777778 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 4 0 B 4 0 0 -4 2 C 2 0 0 2 -4 D -4 4 -2 0 0 E 0 -2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.222222 C: 0.111111 D: 0.277778 E: 0.222222 Sum of squares = 0.216049382717 Cumulative probabilities = A: 0.166667 B: 0.388889 C: 0.500000 D: 0.777778 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7288: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (17) D E A C B (8) C B A E D (7) C B D A E (6) A E B C D (6) E A D B C (4) D E A B C (4) D B C E A (4) B A E C D (4) A E D C B (4) E A B D C (3) D C A E B (3) D A E C B (3) C D A E B (2) A E C D B (2) A B E C D (2) E D B A C (1) E D A B C (1) E A D C B (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B E A (1) D B E C A (1) D B E A C (1) D A C E B (1) C D B A E (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B E D (1) B E A D C (1) B D E C A (1) B C D E A (1) B C D A E (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -4 14 30 B -4 0 6 4 -2 C 4 -6 0 8 0 D -14 -4 -8 0 -16 E -30 2 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775512 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 14 30 B -4 0 6 4 -2 C 4 -6 0 8 0 D -14 -4 -8 0 -16 E -30 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775486 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=25 C=20 A=16 E=10 so E is eliminated. Round 2 votes counts: D=31 B=25 A=24 C=20 so C is eliminated. Round 3 votes counts: B=38 D=34 A=28 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:222 C:203 B:202 E:194 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 14 30 B -4 0 6 4 -2 C 4 -6 0 8 0 D -14 -4 -8 0 -16 E -30 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775486 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 14 30 B -4 0 6 4 -2 C 4 -6 0 8 0 D -14 -4 -8 0 -16 E -30 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775486 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 14 30 B -4 0 6 4 -2 C 4 -6 0 8 0 D -14 -4 -8 0 -16 E -30 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775486 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7289: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) A D E B C (11) E C B A D (9) C B E D A (8) D A E C B (6) D A C B E (6) E B C A D (4) C E B D A (4) C B D E A (4) A E D B C (4) E A D C B (3) B C D A E (3) E B A C D (2) E A B C D (2) D B C A E (2) D A B E C (2) C D B A E (2) B C E A D (2) B A C D E (2) A D B E C (2) E A D B C (1) D E C A B (1) D C A B E (1) D A E B C (1) C B E A D (1) C B D A E (1) B E C A D (1) B C D E A (1) A E B D C (1) Total count = 100 A B C D E A 0 8 12 -10 14 B -8 0 8 -10 2 C -12 -8 0 -8 0 D 10 10 8 0 16 E -14 -2 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -10 14 B -8 0 8 -10 2 C -12 -8 0 -8 0 D 10 10 8 0 16 E -14 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=21 C=20 A=18 B=9 so B is eliminated. Round 2 votes counts: D=32 C=26 E=22 A=20 so A is eliminated. Round 3 votes counts: D=45 C=28 E=27 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:212 B:196 C:186 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -10 14 B -8 0 8 -10 2 C -12 -8 0 -8 0 D 10 10 8 0 16 E -14 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -10 14 B -8 0 8 -10 2 C -12 -8 0 -8 0 D 10 10 8 0 16 E -14 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -10 14 B -8 0 8 -10 2 C -12 -8 0 -8 0 D 10 10 8 0 16 E -14 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7290: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) B E D C A (6) B E C D A (6) A C D E B (6) D B C A E (4) C B E A D (4) C B A E D (4) C B A D E (4) B D C E A (4) A C E D B (4) E B D C A (3) E B D A C (3) D A E B C (3) C A B D E (3) B D E C A (3) B C E A D (3) A D C E B (3) E D B A C (2) D E A B C (2) C B D A E (2) C A B E D (2) B C E D A (2) A E C D B (2) E D A B C (1) E C B A D (1) E C A B D (1) E B A D C (1) E A D C B (1) D E B A C (1) D B A E C (1) D B A C E (1) D A C B E (1) C D A B E (1) C A D B E (1) B D E A C (1) B D C A E (1) A E D C B (1) A D E C B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -30 -10 -12 -6 B 30 0 12 10 26 C 10 -12 0 -4 4 D 12 -10 4 0 4 E 6 -26 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -10 -12 -6 B 30 0 12 10 26 C 10 -12 0 -4 4 D 12 -10 4 0 4 E 6 -26 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=21 C=21 A=19 E=13 so E is eliminated. Round 2 votes counts: B=33 D=24 C=23 A=20 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:239 D:205 C:199 E:186 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -10 -12 -6 B 30 0 12 10 26 C 10 -12 0 -4 4 D 12 -10 4 0 4 E 6 -26 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -10 -12 -6 B 30 0 12 10 26 C 10 -12 0 -4 4 D 12 -10 4 0 4 E 6 -26 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -10 -12 -6 B 30 0 12 10 26 C 10 -12 0 -4 4 D 12 -10 4 0 4 E 6 -26 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7291: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) B E D A C (6) C E B A D (5) C A D E B (5) A C B D E (5) E D B C A (4) E C D A B (4) E B C D A (4) B A D E C (4) A C D B E (4) E D C B A (3) E B D C A (3) C E D A B (3) C A E D B (3) C A B E D (3) B D E A C (3) B D A E C (3) B A D C E (3) A D C B E (3) E C D B A (2) E C B D A (2) D B E A C (2) C D E A B (2) C A E B D (2) E D C A B (1) D E B A C (1) D B A E C (1) D A C B E (1) D A B C E (1) C E A D B (1) C E A B D (1) C D A E B (1) C B A E D (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -6 -8 -12 B 12 0 -8 14 -8 C 6 8 0 4 -4 D 8 -14 -4 0 -14 E 12 8 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -6 -8 -12 B 12 0 -8 14 -8 C 6 8 0 4 -4 D 8 -14 -4 0 -14 E 12 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=29 B=21 A=14 D=6 so D is eliminated. Round 2 votes counts: E=31 C=29 B=24 A=16 so A is eliminated. Round 3 votes counts: C=42 E=31 B=27 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:207 B:205 D:188 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -6 -8 -12 B 12 0 -8 14 -8 C 6 8 0 4 -4 D 8 -14 -4 0 -14 E 12 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -8 -12 B 12 0 -8 14 -8 C 6 8 0 4 -4 D 8 -14 -4 0 -14 E 12 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -8 -12 B 12 0 -8 14 -8 C 6 8 0 4 -4 D 8 -14 -4 0 -14 E 12 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7292: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) B E A C D (5) A C B D E (5) E B D A C (4) D E C B A (4) D E A C B (4) A C B E D (4) A B C E D (4) E A B D C (3) D E B C A (3) C D A B E (3) C B D A E (3) B E D C A (3) B C E A D (3) B C A E D (3) B A E C D (3) B A C E D (3) A C D B E (3) E D B A C (2) E B A D C (2) D C E A B (2) C B A D E (2) C A D B E (2) C A B D E (2) B C E D A (2) E B D C A (1) D E C A B (1) D C E B A (1) D C A E B (1) D C A B E (1) D A C E B (1) C D B E A (1) C D B A E (1) A E D C B (1) A E B D C (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 2 4 -2 B 12 0 6 10 8 C -2 -6 0 6 2 D -4 -10 -6 0 -14 E 2 -8 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 4 -2 B 12 0 6 10 8 C -2 -6 0 6 2 D -4 -10 -6 0 -14 E 2 -8 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=22 B=22 D=18 C=14 so C is eliminated. Round 2 votes counts: A=28 B=27 D=23 E=22 so E is eliminated. Round 3 votes counts: D=35 B=34 A=31 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:203 C:200 A:196 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 4 -2 B 12 0 6 10 8 C -2 -6 0 6 2 D -4 -10 -6 0 -14 E 2 -8 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 4 -2 B 12 0 6 10 8 C -2 -6 0 6 2 D -4 -10 -6 0 -14 E 2 -8 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 4 -2 B 12 0 6 10 8 C -2 -6 0 6 2 D -4 -10 -6 0 -14 E 2 -8 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7293: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) D C E B A (8) D C B A E (6) B A E C D (6) A B E C D (6) E D A B C (5) D E C A B (5) D C E A B (4) E A D B C (3) D E C B A (3) C B A D E (3) B E A C D (3) B C A D E (3) B A C E D (3) E D B A C (2) E A B D C (2) E A B C D (2) D E A C B (2) D C A E B (2) D C A B E (2) C D A B E (2) A C B D E (2) A B C E D (2) E D A C B (1) E B A D C (1) E B A C D (1) D C B E A (1) C D B E A (1) C B D A E (1) C A B D E (1) B E C D A (1) B C A E D (1) B A C D E (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -12 -16 10 B 14 0 -14 -20 14 C 12 14 0 6 12 D 16 20 -6 0 20 E -10 -14 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -16 10 B 14 0 -14 -20 14 C 12 14 0 6 12 D 16 20 -6 0 20 E -10 -14 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=20 B=18 E=17 A=12 so A is eliminated. Round 2 votes counts: D=33 B=26 C=23 E=18 so E is eliminated. Round 3 votes counts: D=44 B=33 C=23 so C is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 C:222 B:197 A:184 E:172 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -12 -16 10 B 14 0 -14 -20 14 C 12 14 0 6 12 D 16 20 -6 0 20 E -10 -14 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -16 10 B 14 0 -14 -20 14 C 12 14 0 6 12 D 16 20 -6 0 20 E -10 -14 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -16 10 B 14 0 -14 -20 14 C 12 14 0 6 12 D 16 20 -6 0 20 E -10 -14 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7294: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) E A C D B (9) D B A E C (9) C E A B D (8) E A C B D (6) D E A B C (6) E A D C B (5) D B C E A (5) A E C B D (5) B C D A E (4) A E D B C (4) C B E A D (3) A E D C B (3) E A D B C (2) D B C A E (2) D A E B C (2) C B A E D (2) B D A E C (2) B C A E D (2) E C A D B (1) D C B E A (1) D B E A C (1) C E B A D (1) C E A D B (1) C D B E A (1) C B E D A (1) C B D A E (1) C A E B D (1) B D C A E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 10 8 -22 B -8 0 -18 -4 -8 C -10 18 0 12 -12 D -8 4 -12 0 -10 E 22 8 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 10 8 -22 B -8 0 -18 -4 -8 C -10 18 0 12 -12 D -8 4 -12 0 -10 E 22 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 E=23 A=13 B=9 so B is eliminated. Round 2 votes counts: C=35 D=29 E=23 A=13 so A is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:204 A:202 D:187 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 8 -22 B -8 0 -18 -4 -8 C -10 18 0 12 -12 D -8 4 -12 0 -10 E 22 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 8 -22 B -8 0 -18 -4 -8 C -10 18 0 12 -12 D -8 4 -12 0 -10 E 22 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 8 -22 B -8 0 -18 -4 -8 C -10 18 0 12 -12 D -8 4 -12 0 -10 E 22 8 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7295: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) D A E C B (8) B C E A D (7) E D A C B (6) E B C A D (5) D A C E B (5) C B A D E (5) A D C B E (5) E D A B C (4) C A B D E (4) A C D B E (4) E D B A C (3) E B C D A (3) C B A E D (3) E B D C A (2) D E A C B (2) D E A B C (2) B E C A D (2) B C A E D (2) E D B C A (1) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) B E D C A (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A D E (1) A E D C B (1) A D C E B (1) A C B E D (1) Total count = 100 A B C D E A 0 18 18 -12 10 B -18 0 -20 -14 6 C -18 20 0 -18 12 D 12 14 18 0 10 E -10 -6 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 18 -12 10 B -18 0 -20 -14 6 C -18 20 0 -18 12 D 12 14 18 0 10 E -10 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=27 B=18 C=12 A=12 so C is eliminated. Round 2 votes counts: D=31 E=27 B=26 A=16 so A is eliminated. Round 3 votes counts: D=41 B=31 E=28 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:217 C:198 E:181 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 18 -12 10 B -18 0 -20 -14 6 C -18 20 0 -18 12 D 12 14 18 0 10 E -10 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 -12 10 B -18 0 -20 -14 6 C -18 20 0 -18 12 D 12 14 18 0 10 E -10 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 -12 10 B -18 0 -20 -14 6 C -18 20 0 -18 12 D 12 14 18 0 10 E -10 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7296: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A E C B D (8) E C A D B (6) C E A B D (6) B D C A E (6) B A C E D (6) D C E B A (5) D B C E A (5) D B C A E (4) B D A C E (4) B C D A E (4) E C A B D (3) E A C B D (3) E A C D B (2) D E C A B (2) D C E A B (2) C E D A B (2) B A E C D (2) A E B C D (2) A B E C D (2) A B C E D (2) E C D A B (1) E A D C B (1) D E A B C (1) D B E A C (1) D A E B C (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E B A (1) C B A E D (1) B D A E C (1) B A D E C (1) B A D C E (1) B A C D E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -18 2 -4 B 6 0 -8 12 0 C 18 8 0 10 22 D -2 -12 -10 0 -4 E 4 0 -22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 2 -4 B 6 0 -8 12 0 C 18 8 0 10 22 D -2 -12 -10 0 -4 E 4 0 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 E=16 A=16 C=13 so C is eliminated. Round 2 votes counts: D=30 E=27 B=27 A=16 so A is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:229 B:205 E:193 A:187 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -18 2 -4 B 6 0 -8 12 0 C 18 8 0 10 22 D -2 -12 -10 0 -4 E 4 0 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 2 -4 B 6 0 -8 12 0 C 18 8 0 10 22 D -2 -12 -10 0 -4 E 4 0 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 2 -4 B 6 0 -8 12 0 C 18 8 0 10 22 D -2 -12 -10 0 -4 E 4 0 -22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7297: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (5) A D C E B (5) E B D A C (4) C D E A B (4) B E A D C (4) B C E D A (4) C A D B E (3) B E D A C (3) B E C D A (3) B C E A D (3) B C A D E (3) B A E D C (3) A E D B C (3) A D E C B (3) E D A C B (2) E B A D C (2) E A D C B (2) E A D B C (2) E A B D C (2) D E C A B (2) D E A C B (2) D C E A B (2) C D E B A (2) C B A D E (2) B E D C A (2) B E C A D (2) B A C E D (2) B A C D E (2) E D C B A (1) E D A B C (1) E C D B A (1) E C D A B (1) E B C D A (1) D C A E B (1) D A E C B (1) C E D B A (1) C E D A B (1) C E B D A (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D E A (1) C A B D E (1) B E A C D (1) B C D E A (1) B C A E D (1) B A E C D (1) A E D C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 0 -20 B 6 0 4 2 -8 C 4 -4 0 0 -4 D 0 -2 0 0 -14 E 20 8 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 0 -20 B 6 0 4 2 -8 C 4 -4 0 0 -4 D 0 -2 0 0 -14 E 20 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=24 E=19 A=14 D=8 so D is eliminated. Round 2 votes counts: B=35 C=27 E=23 A=15 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:223 B:202 C:198 D:192 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 0 -20 B 6 0 4 2 -8 C 4 -4 0 0 -4 D 0 -2 0 0 -14 E 20 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 0 -20 B 6 0 4 2 -8 C 4 -4 0 0 -4 D 0 -2 0 0 -14 E 20 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 0 -20 B 6 0 4 2 -8 C 4 -4 0 0 -4 D 0 -2 0 0 -14 E 20 8 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7298: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (11) C D B E A (10) A E D B C (8) E A B D C (5) D C B A E (5) D C A B E (5) C B D E A (5) B C D E A (5) A E C D B (5) E B A C D (4) A E D C B (4) D B C A E (3) C D A B E (3) B E C D A (3) A E B D C (3) A E B C D (2) A D E C B (2) A C D E B (2) E C B A D (1) E B A D C (1) E A B C D (1) D C B E A (1) D B E A C (1) D A C B E (1) D A B C E (1) C E A B D (1) C B E D A (1) C A D E B (1) B E D C A (1) B E D A C (1) A E C B D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -12 -14 18 B 6 0 -20 -30 14 C 12 20 0 12 12 D 14 30 -12 0 16 E -18 -14 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999717 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 -14 18 B 6 0 -20 -30 14 C 12 20 0 12 12 D 14 30 -12 0 16 E -18 -14 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=29 D=17 E=12 B=10 so B is eliminated. Round 2 votes counts: C=37 A=29 E=17 D=17 so E is eliminated. Round 3 votes counts: C=41 A=40 D=19 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:228 D:224 A:193 B:185 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 -14 18 B 6 0 -20 -30 14 C 12 20 0 12 12 D 14 30 -12 0 16 E -18 -14 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 -14 18 B 6 0 -20 -30 14 C 12 20 0 12 12 D 14 30 -12 0 16 E -18 -14 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 -14 18 B 6 0 -20 -30 14 C 12 20 0 12 12 D 14 30 -12 0 16 E -18 -14 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7299: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (15) A C D E B (9) D E A B C (8) C A B D E (8) E D B A C (7) C A B E D (7) B E D C A (7) A D E C B (6) C B A E D (5) B C E D A (5) E D B C A (4) B D E A C (3) C A E D B (2) B E C D A (2) A D C E B (2) A D B E C (2) E D A C B (1) E B D C A (1) D E A C B (1) B D E C A (1) B A C D E (1) A D B C E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 16 -10 -10 B 2 0 14 -18 -12 C -16 -14 0 -18 -16 D 10 18 18 0 18 E 10 12 16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 -10 -10 B 2 0 14 -18 -12 C -16 -14 0 -18 -16 D 10 18 18 0 18 E 10 12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 A=22 B=19 E=13 so E is eliminated. Round 2 votes counts: D=36 C=22 A=22 B=20 so B is eliminated. Round 3 votes counts: D=48 C=29 A=23 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:232 E:210 A:197 B:193 C:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 16 -10 -10 B 2 0 14 -18 -12 C -16 -14 0 -18 -16 D 10 18 18 0 18 E 10 12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 -10 -10 B 2 0 14 -18 -12 C -16 -14 0 -18 -16 D 10 18 18 0 18 E 10 12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 -10 -10 B 2 0 14 -18 -12 C -16 -14 0 -18 -16 D 10 18 18 0 18 E 10 12 16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7300: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) B D E C A (8) E A D C B (6) C A B D E (6) A E C D B (6) A C E D B (6) E A C B D (5) B D C E A (5) B D C A E (5) E A C D B (4) C A D B E (4) B E D C A (4) D C A B E (3) C B D A E (3) E B A D C (2) E A D B C (2) E A B D C (2) C D B A E (2) C D A B E (2) C B A D E (2) C A D E B (2) E D B A C (1) E D A B C (1) E B A C D (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) C A B E D (1) B E D A C (1) B C D E A (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 0 -8 0 -8 B 0 0 -6 12 8 C 8 6 0 -6 -6 D 0 -12 6 0 0 E 8 -8 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.000000 E: 0.300000 Sum of squares = 0.34 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 0 -8 0 -8 B 0 0 -6 12 8 C 8 6 0 -6 -6 D 0 -12 6 0 0 E 8 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999718 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=26 C=22 A=12 D=8 so D is eliminated. Round 2 votes counts: E=32 B=30 C=26 A=12 so A is eliminated. Round 3 votes counts: E=38 C=32 B=30 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:207 E:203 C:201 D:197 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 0 -8 B 0 0 -6 12 8 C 8 6 0 -6 -6 D 0 -12 6 0 0 E 8 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999718 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 0 -8 B 0 0 -6 12 8 C 8 6 0 -6 -6 D 0 -12 6 0 0 E 8 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999718 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 0 -8 B 0 0 -6 12 8 C 8 6 0 -6 -6 D 0 -12 6 0 0 E 8 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.400000 D: 0.000000 E: 0.300000 Sum of squares = 0.339999999718 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7301: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) E A C B D (5) E A B D C (5) C E A D B (5) C D A E B (5) C D A B E (5) A E B D C (5) A B E D C (5) C D B A E (4) B E A D C (4) E C A D B (3) E B C D A (3) E A B C D (3) B D A E C (3) A B D E C (3) E C B A D (2) D C A B E (2) D B C A E (2) C E D A B (2) C D B E A (2) B D C E A (2) B D A C E (2) A C D B E (2) E C B D A (1) E C A B D (1) E A C D B (1) D C B E A (1) D B C E A (1) D B A C E (1) D A C B E (1) C E D B A (1) C D E A B (1) C A D E B (1) B D E C A (1) B D E A C (1) B D C A E (1) B A D E C (1) A E D B C (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 18 -8 0 10 B -18 0 -10 -4 6 C 8 10 0 -2 2 D 0 4 2 0 4 E -10 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.132967 B: 0.000000 C: 0.000000 D: 0.867033 E: 0.000000 Sum of squares = 0.769426951026 Cumulative probabilities = A: 0.132967 B: 0.132967 C: 0.132967 D: 1.000000 E: 1.000000 A B C D E A 0 18 -8 0 10 B -18 0 -10 -4 6 C 8 10 0 -2 2 D 0 4 2 0 4 E -10 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000066932 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 A=19 D=16 B=15 so B is eliminated. Round 2 votes counts: E=28 D=26 C=26 A=20 so A is eliminated. Round 3 votes counts: E=39 D=31 C=30 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:210 C:209 D:205 E:189 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -8 0 10 B -18 0 -10 -4 6 C 8 10 0 -2 2 D 0 4 2 0 4 E -10 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000066932 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 0 10 B -18 0 -10 -4 6 C 8 10 0 -2 2 D 0 4 2 0 4 E -10 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000066932 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 0 10 B -18 0 -10 -4 6 C 8 10 0 -2 2 D 0 4 2 0 4 E -10 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000066932 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7302: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) E C B A D (8) D A B C E (7) D A C B E (6) B E C D A (6) E C A B D (5) B C E D A (5) B C D E A (4) E A C D B (3) D C A B E (3) D B A C E (3) B D C A E (3) A D E C B (3) E A D B C (2) E A B D C (2) E A B C D (2) D A C E B (2) D A B E C (2) C E B A D (2) C B D E A (2) A D C E B (2) A D B E C (2) E B A C D (1) E A C B D (1) D C B A E (1) D B C A E (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D A E (1) B E C A D (1) B E A C D (1) B D E C A (1) B D C E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -14 -4 -14 B 6 0 12 14 6 C 14 -12 0 12 -4 D 4 -14 -12 0 -2 E 14 -6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -4 -14 B 6 0 12 14 6 C 14 -12 0 12 -4 D 4 -14 -12 0 -2 E 14 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=25 B=23 A=11 C=8 so C is eliminated. Round 2 votes counts: E=35 D=27 B=27 A=11 so A is eliminated. Round 3 votes counts: E=37 D=36 B=27 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:207 C:205 D:188 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 -4 -14 B 6 0 12 14 6 C 14 -12 0 12 -4 D 4 -14 -12 0 -2 E 14 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -4 -14 B 6 0 12 14 6 C 14 -12 0 12 -4 D 4 -14 -12 0 -2 E 14 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -4 -14 B 6 0 12 14 6 C 14 -12 0 12 -4 D 4 -14 -12 0 -2 E 14 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7303: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (16) B C E A D (9) C B D E A (8) C B E A D (7) A D E B C (6) D A E B C (5) E A D C B (4) C B E D A (4) B E C A D (4) C D B A E (3) A E D B C (3) A B E D C (3) E D A C B (2) E B C A D (2) E A B C D (2) D C A E B (2) D A B E C (2) D A B C E (2) C B D A E (2) B C A D E (2) B A E D C (2) E C A D B (1) D E C A B (1) D E A C B (1) D C E A B (1) D B A C E (1) C D E A B (1) B E A C D (1) B D C A E (1) B A E C D (1) B A D E C (1) Total count = 100 A B C D E A 0 4 4 -4 4 B -4 0 -6 -2 6 C -4 6 0 -6 -14 D 4 2 6 0 10 E -4 -6 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -4 4 B -4 0 -6 -2 6 C -4 6 0 -6 -14 D 4 2 6 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=25 B=21 A=12 E=11 so E is eliminated. Round 2 votes counts: D=33 C=26 B=23 A=18 so A is eliminated. Round 3 votes counts: D=46 B=28 C=26 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:204 B:197 E:197 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -4 4 B -4 0 -6 -2 6 C -4 6 0 -6 -14 D 4 2 6 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -4 4 B -4 0 -6 -2 6 C -4 6 0 -6 -14 D 4 2 6 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -4 4 B -4 0 -6 -2 6 C -4 6 0 -6 -14 D 4 2 6 0 10 E -4 -6 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7304: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) E A C D B (6) C D E A B (6) B A E C D (6) E C D A B (5) D C E A B (5) C E D A B (5) B D C A E (5) B A E D C (5) C D E B A (4) B D A C E (4) B A D E C (4) A E D C B (4) E C A D B (3) A B E D C (3) E A D C B (2) E A B C D (2) B E C A D (2) B A D C E (2) A E B D C (2) A D E C B (2) A D B E C (2) A B D E C (2) E C B D A (1) E C A B D (1) D C B E A (1) D C A E B (1) D C A B E (1) D A C E B (1) C E D B A (1) B C E D A (1) B C D E A (1) B C D A E (1) A E B C D (1) Total count = 100 A B C D E A 0 8 -4 -2 -8 B -8 0 -14 -14 -4 C 4 14 0 8 -6 D 2 14 -8 0 0 E 8 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.228961 E: 0.771039 Sum of squares = 0.646923989189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.228961 E: 1.000000 A B C D E A 0 8 -4 -2 -8 B -8 0 -14 -14 -4 C 4 14 0 8 -6 D 2 14 -8 0 0 E 8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.571429 Sum of squares = 0.510204133963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=24 E=20 A=16 D=9 so D is eliminated. Round 2 votes counts: C=32 B=31 E=20 A=17 so A is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 E:209 D:204 A:197 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -4 -2 -8 B -8 0 -14 -14 -4 C 4 14 0 8 -6 D 2 14 -8 0 0 E 8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.571429 Sum of squares = 0.510204133963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -2 -8 B -8 0 -14 -14 -4 C 4 14 0 8 -6 D 2 14 -8 0 0 E 8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.571429 Sum of squares = 0.510204133963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -2 -8 B -8 0 -14 -14 -4 C 4 14 0 8 -6 D 2 14 -8 0 0 E 8 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 0.571429 Sum of squares = 0.510204133963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.428571 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7305: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) B A D C E (7) E C A D B (6) B D C E A (6) E C D A B (5) B D A C E (5) C E D A B (4) A B C D E (4) D C E B A (3) D C E A B (3) D B C E A (3) B D E C A (3) B A E C D (3) A E C D B (3) E D C B A (2) E B A C D (2) E A C B D (2) E A B C D (2) D C B E A (2) C D E A B (2) C D A E B (2) B E D C A (2) B D E A C (2) A C D E B (2) A B E C D (2) E C D B A (1) E C B D A (1) E B C D A (1) D E C B A (1) D E C A B (1) D C A B E (1) D B C A E (1) C A E D B (1) B E A C D (1) B D C A E (1) B A D E C (1) B A C E D (1) A D C B E (1) A D B C E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -4 -4 -24 B -2 0 -2 -6 -4 C 4 2 0 8 4 D 4 6 -8 0 4 E 24 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999009 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -4 -24 B -2 0 -2 -6 -4 C 4 2 0 8 4 D 4 6 -8 0 4 E 24 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=29 D=15 A=15 C=9 so C is eliminated. Round 2 votes counts: E=33 B=32 D=19 A=16 so A is eliminated. Round 3 votes counts: B=39 E=38 D=23 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:209 D:203 B:193 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -4 -24 B -2 0 -2 -6 -4 C 4 2 0 8 4 D 4 6 -8 0 4 E 24 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -4 -24 B -2 0 -2 -6 -4 C 4 2 0 8 4 D 4 6 -8 0 4 E 24 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -4 -24 B -2 0 -2 -6 -4 C 4 2 0 8 4 D 4 6 -8 0 4 E 24 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7306: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) C D E A B (10) C D E B A (9) B A C E D (8) C E D B A (6) C B E D A (6) B A C D E (4) D E A C B (3) B A E C D (3) A D E B C (3) A B D E C (3) D E C A B (2) D C E A B (2) C D A E B (2) B E A D C (2) B C E A D (2) B C A E D (2) A E D B C (2) A B E D C (2) E D B A C (1) E D A B C (1) E C D B A (1) E B D A C (1) D E A B C (1) D A E B C (1) C E D A B (1) C E B D A (1) C D B E A (1) C B E A D (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B D E (1) B E C A D (1) B C E D A (1) B C A D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -28 -6 0 -6 B 28 0 2 6 6 C 6 -2 0 30 24 D 0 -6 -30 0 -8 E 6 -6 -24 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999472 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -6 0 -6 B 28 0 2 6 6 C 6 -2 0 30 24 D 0 -6 -30 0 -8 E 6 -6 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 B=35 A=11 D=9 E=4 so E is eliminated. Round 2 votes counts: C=42 B=36 D=11 A=11 so D is eliminated. Round 3 votes counts: C=46 B=37 A=17 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:229 B:221 E:192 A:180 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -6 0 -6 B 28 0 2 6 6 C 6 -2 0 30 24 D 0 -6 -30 0 -8 E 6 -6 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -6 0 -6 B 28 0 2 6 6 C 6 -2 0 30 24 D 0 -6 -30 0 -8 E 6 -6 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -6 0 -6 B 28 0 2 6 6 C 6 -2 0 30 24 D 0 -6 -30 0 -8 E 6 -6 -24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7307: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) C E B A D (6) E C B A D (5) B C E D A (5) D E C B A (4) C B E A D (4) B C E A D (4) E D C B A (3) E C D B A (3) E C D A B (3) D B C E A (3) D B A C E (3) D A E C B (3) D A E B C (3) D A B E C (3) B A C E D (3) A D E C B (3) A D B C E (3) A C B E D (3) D E B C A (2) D E A C B (2) B C D A E (2) B C A E D (2) B A D C E (2) A D E B C (2) A C E B D (2) A B D C E (2) A B C E D (2) E D A C B (1) E C A B D (1) D B A E C (1) D A B C E (1) C E B D A (1) C E A B D (1) C B A E D (1) B D C A E (1) B C A D E (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -26 -18 -2 -10 B 26 0 -8 14 -6 C 18 8 0 16 6 D 2 -14 -16 0 -16 E 10 6 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -18 -2 -10 B 26 0 -8 14 -6 C 18 8 0 16 6 D 2 -14 -16 0 -16 E 10 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=23 B=20 A=19 C=13 so C is eliminated. Round 2 votes counts: E=31 D=25 B=25 A=19 so A is eliminated. Round 3 votes counts: E=34 D=33 B=33 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:224 B:213 E:213 D:178 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -26 -18 -2 -10 B 26 0 -8 14 -6 C 18 8 0 16 6 D 2 -14 -16 0 -16 E 10 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -18 -2 -10 B 26 0 -8 14 -6 C 18 8 0 16 6 D 2 -14 -16 0 -16 E 10 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -18 -2 -10 B 26 0 -8 14 -6 C 18 8 0 16 6 D 2 -14 -16 0 -16 E 10 6 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7308: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (7) C A B E D (7) E D A C B (4) D C B A E (4) C B A D E (4) B D E A C (4) A E B C D (4) E A D C B (3) D E B C A (3) D C E A B (3) D B C E A (3) C D A E B (3) B C D A E (3) A E C B D (3) A C E B D (3) E D A B C (2) E A B C D (2) D E B A C (2) D E A B C (2) D B E C A (2) D B E A C (2) C D A B E (2) C B D A E (2) C A D E B (2) B E D A C (2) B E A C D (2) B D C E A (2) B C A E D (2) E D B A C (1) E B D A C (1) E B A D C (1) E A D B C (1) E A C D B (1) D C A E B (1) D C A B E (1) D B C A E (1) C D B A E (1) C A E D B (1) B D E C A (1) B D C A E (1) B C A D E (1) B A E C D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 8 0 -20 -2 B -8 0 -6 -6 -2 C 0 6 0 -8 -4 D 20 6 8 0 14 E 2 2 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -20 -2 B -8 0 -6 -6 -2 C 0 6 0 -8 -4 D 20 6 8 0 14 E 2 2 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=22 B=19 E=16 A=12 so A is eliminated. Round 2 votes counts: D=31 C=26 E=23 B=20 so B is eliminated. Round 3 votes counts: D=39 C=32 E=29 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:197 E:197 A:193 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 0 -20 -2 B -8 0 -6 -6 -2 C 0 6 0 -8 -4 D 20 6 8 0 14 E 2 2 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -20 -2 B -8 0 -6 -6 -2 C 0 6 0 -8 -4 D 20 6 8 0 14 E 2 2 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -20 -2 B -8 0 -6 -6 -2 C 0 6 0 -8 -4 D 20 6 8 0 14 E 2 2 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7309: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) A B C D E (8) A B E C D (6) E D C A B (4) C D B E A (4) B C E D A (4) B C D E A (4) A E D B C (4) A D E C B (4) E D C B A (3) D E C A B (3) C D E B A (3) C B D E A (3) B A C E D (3) A B E D C (3) A B C E D (3) E D A C B (2) E B C D A (2) E A D C B (2) E A B C D (2) D E C B A (2) D E A C B (2) D C B E A (2) B C E A D (2) B C A D E (2) A E B D C (2) A D E B C (2) A D C E B (2) E C D B A (1) D C A B E (1) D A E C B (1) C E D B A (1) C E B D A (1) C D B A E (1) C B E D A (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 6 -4 -6 -12 B -6 0 -4 -8 -2 C 4 4 0 2 10 D 6 8 -2 0 8 E 12 2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -6 -12 B -6 0 -4 -8 -2 C 4 4 0 2 10 D 6 8 -2 0 8 E 12 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=19 E=16 B=15 C=14 so C is eliminated. Round 2 votes counts: A=36 D=27 B=19 E=18 so E is eliminated. Round 3 votes counts: A=40 D=38 B=22 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:210 D:210 E:198 A:192 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -6 -12 B -6 0 -4 -8 -2 C 4 4 0 2 10 D 6 8 -2 0 8 E 12 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -6 -12 B -6 0 -4 -8 -2 C 4 4 0 2 10 D 6 8 -2 0 8 E 12 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -6 -12 B -6 0 -4 -8 -2 C 4 4 0 2 10 D 6 8 -2 0 8 E 12 2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7310: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) A B E C D (7) E C A B D (6) D E C A B (6) D B A C E (6) D C E B A (5) D A B E C (5) C E B D A (5) E C B A D (3) E A C B D (3) D B C A E (3) D A B C E (3) C B E A D (3) A D B E C (3) D A E B C (2) C E D B A (2) A E B C D (2) A D E B C (2) A B D E C (2) A B C E D (2) E D C B A (1) E D C A B (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D B C (1) E A B C D (1) D E C B A (1) D C B E A (1) D C B A E (1) C D E B A (1) C B E D A (1) C B A E D (1) B D A C E (1) B C A E D (1) B A C E D (1) B A C D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -12 6 -10 B 0 0 -10 6 -12 C 12 10 0 10 2 D -6 -6 -10 0 -10 E 10 12 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 6 -10 B 0 0 -10 6 -12 C 12 10 0 10 2 D -6 -6 -10 0 -10 E 10 12 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=24 A=20 E=19 B=4 so B is eliminated. Round 2 votes counts: D=34 C=25 A=22 E=19 so E is eliminated. Round 3 votes counts: C=37 D=36 A=27 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:215 A:192 B:192 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 6 -10 B 0 0 -10 6 -12 C 12 10 0 10 2 D -6 -6 -10 0 -10 E 10 12 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 6 -10 B 0 0 -10 6 -12 C 12 10 0 10 2 D -6 -6 -10 0 -10 E 10 12 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 6 -10 B 0 0 -10 6 -12 C 12 10 0 10 2 D -6 -6 -10 0 -10 E 10 12 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7311: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) B E C A D (9) E B C D A (8) D A E B C (8) C B E A D (8) C A D B E (8) E B D A C (6) C D A E B (6) E D A B C (5) D A C E B (3) A D C B E (3) C E D A B (2) C B A E D (2) B A D E C (2) E D B A C (1) E C D A B (1) E C B D A (1) E B D C A (1) D E A C B (1) D E A B C (1) D C A E B (1) D A E C B (1) D A C B E (1) C E B D A (1) C B A D E (1) B C E A D (1) B C A E D (1) B A D C E (1) A D B E C (1) A D B C E (1) A C B D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -2 4 -14 B 8 0 18 12 6 C 2 -18 0 0 -16 D -4 -12 0 0 -16 E 14 -6 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 4 -14 B 8 0 18 12 6 C 2 -18 0 0 -16 D -4 -12 0 0 -16 E 14 -6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 E=23 D=16 A=8 so A is eliminated. Round 2 votes counts: C=29 B=27 E=23 D=21 so D is eliminated. Round 3 votes counts: C=37 E=34 B=29 so B is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:220 A:190 C:184 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 4 -14 B 8 0 18 12 6 C 2 -18 0 0 -16 D -4 -12 0 0 -16 E 14 -6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 4 -14 B 8 0 18 12 6 C 2 -18 0 0 -16 D -4 -12 0 0 -16 E 14 -6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 4 -14 B 8 0 18 12 6 C 2 -18 0 0 -16 D -4 -12 0 0 -16 E 14 -6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7312: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) C A B E D (8) E A B C D (5) D C E A B (5) C A E B D (5) E D B A C (4) D C B A E (4) E D A C B (3) E A C B D (3) D E A C B (3) D E A B C (3) D B E A C (3) C B A D E (3) B A E C D (3) B A C E D (3) E D A B C (2) E B A D C (2) E A C D B (2) D E C A B (2) D C E B A (2) D C A E B (2) D B C A E (2) C D A B E (2) B E A D C (2) A C E B D (2) A C B E D (2) D B C E A (1) C E A D B (1) C D B A E (1) C B D A E (1) C B A E D (1) C A E D B (1) C A B D E (1) B E A C D (1) B D E A C (1) B D C A E (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 6 8 -2 -6 B -6 0 -8 -2 -12 C -8 8 0 0 4 D 2 2 0 0 -4 E 6 12 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024691107 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 A B C D E A 0 6 8 -2 -6 B -6 0 -8 -2 -12 C -8 8 0 0 4 D 2 2 0 0 -4 E 6 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024690767 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=24 E=21 B=15 A=5 so A is eliminated. Round 2 votes counts: D=35 C=28 E=22 B=15 so B is eliminated. Round 3 votes counts: D=37 C=35 E=28 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:209 A:203 C:202 D:200 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 -2 -6 B -6 0 -8 -2 -12 C -8 8 0 0 4 D 2 2 0 0 -4 E 6 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024690767 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -2 -6 B -6 0 -8 -2 -12 C -8 8 0 0 4 D 2 2 0 0 -4 E 6 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024690767 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -2 -6 B -6 0 -8 -2 -12 C -8 8 0 0 4 D 2 2 0 0 -4 E 6 12 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.444444 Sum of squares = 0.358024690767 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.555556 D: 0.555556 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7313: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) E C D B A (9) A B D C E (9) C E D B A (8) A B E D C (8) E C B D A (6) C D E B A (6) A D B C E (5) E B C D A (4) A E B C D (4) D B C E A (3) A B E C D (3) E C A B D (2) D C E B A (2) D C B E A (2) D C A B E (2) C E D A B (2) A E C D B (2) A C D E B (2) E C D A B (1) E C A D B (1) E B A C D (1) C D E A B (1) B E D C A (1) B D E C A (1) A E C B D (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 14 -2 4 0 B -14 0 0 6 -6 C 2 0 0 12 -10 D -4 -6 -12 0 -8 E 0 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.325729 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.674271 Sum of squares = 0.560740481534 Cumulative probabilities = A: 0.325729 B: 0.325729 C: 0.325729 D: 0.325729 E: 1.000000 A B C D E A 0 14 -2 4 0 B -14 0 0 6 -6 C 2 0 0 12 -10 D -4 -6 -12 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=48 E=24 C=17 D=9 B=2 so B is eliminated. Round 2 votes counts: A=48 E=25 C=17 D=10 so D is eliminated. Round 3 votes counts: A=48 E=26 C=26 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:212 A:208 C:202 B:193 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -2 4 0 B -14 0 0 6 -6 C 2 0 0 12 -10 D -4 -6 -12 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 4 0 B -14 0 0 6 -6 C 2 0 0 12 -10 D -4 -6 -12 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 4 0 B -14 0 0 6 -6 C 2 0 0 12 -10 D -4 -6 -12 0 -8 E 0 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7314: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) D B E C A (5) D B A C E (5) D A E C B (5) A C E B D (5) E A C D B (4) B D C E A (4) A E C D B (4) A D B C E (4) A B D C E (4) A B C E D (4) E C A B D (3) D B C E A (3) A E D C B (3) A E C B D (3) E C D B A (2) E C B D A (2) D E C B A (2) D B A E C (2) D A B E C (2) C B E A D (2) B D C A E (2) B C E D A (2) A D E C B (2) A D C B E (2) A C B E D (2) A B C D E (2) E D C A B (1) E C A D B (1) E A C B D (1) D E A C B (1) D B E A C (1) D A E B C (1) D A B C E (1) C E B A D (1) C E A B D (1) B D E C A (1) B D A C E (1) B C A E D (1) B A C E D (1) Total count = 100 A B C D E A 0 12 20 0 12 B -12 0 6 -16 2 C -20 -6 0 -18 -6 D 0 16 18 0 14 E -12 -2 6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.533852 B: 0.000000 C: 0.000000 D: 0.466148 E: 0.000000 Sum of squares = 0.502291861743 Cumulative probabilities = A: 0.533852 B: 0.533852 C: 0.533852 D: 1.000000 E: 1.000000 A B C D E A 0 12 20 0 12 B -12 0 6 -16 2 C -20 -6 0 -18 -6 D 0 16 18 0 14 E -12 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=35 A=35 E=14 B=12 C=4 so C is eliminated. Round 2 votes counts: D=35 A=35 E=16 B=14 so B is eliminated. Round 3 votes counts: D=43 A=37 E=20 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:224 A:222 B:190 E:189 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 20 0 12 B -12 0 6 -16 2 C -20 -6 0 -18 -6 D 0 16 18 0 14 E -12 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 20 0 12 B -12 0 6 -16 2 C -20 -6 0 -18 -6 D 0 16 18 0 14 E -12 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 20 0 12 B -12 0 6 -16 2 C -20 -6 0 -18 -6 D 0 16 18 0 14 E -12 -2 6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7315: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) A B D C E (8) E C D B A (5) D E C A B (5) D E A B C (5) C E B A D (4) A D B C E (4) E D C B A (3) E C D A B (3) D A C B E (3) D A B E C (3) C B A E D (3) A B C D E (3) E D C A B (2) E D B A C (2) E D A B C (2) E C B A D (2) D E A C B (2) D C E A B (2) D A B C E (2) C E D B A (2) C E D A B (2) C D A B E (2) C B E A D (2) B C A E D (2) B A D E C (2) B A C E D (2) E C B D A (1) E B D A C (1) D A E C B (1) C E B D A (1) C E A B D (1) C D E A B (1) B E A D C (1) B C A D E (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 12 0 2 B -4 0 6 -2 6 C -12 -6 0 0 18 D 0 2 0 0 18 E -2 -6 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.579555 B: 0.000000 C: 0.000000 D: 0.420445 E: 0.000000 Sum of squares = 0.51265794847 Cumulative probabilities = A: 0.579555 B: 0.579555 C: 0.579555 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 0 2 B -4 0 6 -2 6 C -12 -6 0 0 18 D 0 2 0 0 18 E -2 -6 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 B=22 E=21 C=18 A=16 so A is eliminated. Round 2 votes counts: B=34 D=27 E=21 C=18 so C is eliminated. Round 3 votes counts: B=39 E=31 D=30 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:209 B:203 C:200 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 0 2 B -4 0 6 -2 6 C -12 -6 0 0 18 D 0 2 0 0 18 E -2 -6 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 0 2 B -4 0 6 -2 6 C -12 -6 0 0 18 D 0 2 0 0 18 E -2 -6 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 0 2 B -4 0 6 -2 6 C -12 -6 0 0 18 D 0 2 0 0 18 E -2 -6 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7316: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) B C A E D (11) A C B D E (10) E D B C A (9) B C A D E (7) D A E C B (6) E D A C B (5) A D C B E (5) E D A B C (3) E B C A D (3) C B A D E (3) B C E A D (3) E B D C A (2) D E B C A (2) C A B D E (2) B C E D A (2) B C D A E (2) A C B E D (2) E D B A C (1) E B C D A (1) E A C D B (1) D C B A E (1) D A C B E (1) C A B E D (1) B E C D A (1) B E C A D (1) B D C A E (1) A E D C B (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -4 4 10 B 0 0 -2 4 8 C 4 2 0 2 4 D -4 -4 -2 0 6 E -10 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 4 10 B 0 0 -2 4 8 C 4 2 0 2 4 D -4 -4 -2 0 6 E -10 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 D=21 A=20 C=6 so C is eliminated. Round 2 votes counts: B=31 E=25 A=23 D=21 so D is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:206 A:205 B:205 D:198 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 4 10 B 0 0 -2 4 8 C 4 2 0 2 4 D -4 -4 -2 0 6 E -10 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 10 B 0 0 -2 4 8 C 4 2 0 2 4 D -4 -4 -2 0 6 E -10 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 10 B 0 0 -2 4 8 C 4 2 0 2 4 D -4 -4 -2 0 6 E -10 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7317: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B E D A C (8) B D E A C (6) E A D B C (5) B D C A E (5) D A C E B (4) C B E A D (4) C A E D B (4) A E D C B (4) D A E B C (3) B C D A E (3) E B C A D (2) E A B D C (2) D B A E C (2) C E B A D (2) C D A E B (2) C D A B E (2) C B D A E (2) C B A E D (2) C A E B D (2) B E A D C (2) B C E A D (2) A D E C B (2) E B A D C (1) E A C D B (1) D C A B E (1) D B E A C (1) D B A C E (1) D A E C B (1) D A B E C (1) C E A B D (1) C D B A E (1) C B D E A (1) C B A D E (1) B E D C A (1) B E C A D (1) B E A C D (1) B D C E A (1) B C E D A (1) B C D E A (1) A E C D B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -4 4 12 B 4 0 0 4 2 C 4 0 0 -2 10 D -4 -4 2 0 4 E -12 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.654926 C: 0.345074 D: 0.000000 E: 0.000000 Sum of squares = 0.548004281749 Cumulative probabilities = A: 0.000000 B: 0.654926 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 4 12 B 4 0 0 4 2 C 4 0 0 -2 10 D -4 -4 2 0 4 E -12 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=32 D=14 E=11 A=9 so A is eliminated. Round 2 votes counts: C=36 B=32 E=16 D=16 so E is eliminated. Round 3 votes counts: C=38 B=37 D=25 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:206 B:205 A:204 D:199 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 4 12 B 4 0 0 4 2 C 4 0 0 -2 10 D -4 -4 2 0 4 E -12 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 4 12 B 4 0 0 4 2 C 4 0 0 -2 10 D -4 -4 2 0 4 E -12 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 4 12 B 4 0 0 4 2 C 4 0 0 -2 10 D -4 -4 2 0 4 E -12 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7318: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) D B A C E (8) E A C B D (7) E C A B D (6) E B A C D (6) D C A B E (6) D C E A B (5) C E A D B (5) B D A E C (4) B A E C D (4) D C B A E (3) C D A E B (3) E A B C D (2) D E C A B (2) D C A E B (2) D B A E C (2) C E D A B (2) C D E A B (2) B E D A C (2) B E A D C (2) B E A C D (2) B D E A C (2) B D A C E (2) E C D A B (1) E C A D B (1) E B D A C (1) D B E A C (1) D B C E A (1) C A E B D (1) C A D E B (1) B A E D C (1) B A D C E (1) A E C B D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 -16 0 B 2 0 2 -8 2 C 0 -2 0 -8 4 D 16 8 8 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -16 0 B 2 0 2 -8 2 C 0 -2 0 -8 4 D 16 8 8 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=24 B=20 C=14 A=3 so A is eliminated. Round 2 votes counts: D=39 E=25 B=21 C=15 so C is eliminated. Round 3 votes counts: D=45 E=34 B=21 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:199 C:197 E:193 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -16 0 B 2 0 2 -8 2 C 0 -2 0 -8 4 D 16 8 8 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -16 0 B 2 0 2 -8 2 C 0 -2 0 -8 4 D 16 8 8 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -16 0 B 2 0 2 -8 2 C 0 -2 0 -8 4 D 16 8 8 0 8 E 0 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7319: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) D B C E A (5) B C D E A (5) E B D C A (4) D B E C A (4) C A B E D (4) C A B D E (4) A D C E B (4) E A B D C (3) D E B C A (3) C D B A E (3) B D E C A (3) A E D C B (3) A E C B D (3) A C D B E (3) A C B E D (3) E D B A C (2) E B D A C (2) E A D B C (2) E A B C D (2) D A E B C (2) D A C B E (2) C B D E A (2) B D C E A (2) A C E D B (2) E C B A D (1) E C A B D (1) E B C D A (1) E B C A D (1) E B A D C (1) D E B A C (1) D C B A E (1) D C A B E (1) D A E C B (1) C D A B E (1) C B D A E (1) C B A E D (1) B E D C A (1) B E C D A (1) B C E D A (1) A E D B C (1) A D C B E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -2 2 4 B -8 0 -6 14 0 C 2 6 0 2 14 D -2 -14 -2 0 2 E -4 0 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 2 4 B -8 0 -6 14 0 C 2 6 0 2 14 D -2 -14 -2 0 2 E -4 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=20 D=20 C=16 B=13 so B is eliminated. Round 2 votes counts: A=31 D=25 E=22 C=22 so E is eliminated. Round 3 votes counts: A=39 D=34 C=27 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:206 B:200 D:192 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 2 4 B -8 0 -6 14 0 C 2 6 0 2 14 D -2 -14 -2 0 2 E -4 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 2 4 B -8 0 -6 14 0 C 2 6 0 2 14 D -2 -14 -2 0 2 E -4 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 2 4 B -8 0 -6 14 0 C 2 6 0 2 14 D -2 -14 -2 0 2 E -4 0 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7320: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) C D E A B (6) C A D E B (6) A C B D E (6) B E D A C (5) A C B E D (5) D E B C A (4) C A D B E (4) D E C B A (3) D C B E A (3) C D E B A (3) B E D C A (3) B D E A C (3) A C D E B (3) A B E D C (3) E B D C A (2) E B A D C (2) D C E B A (2) D B E C A (2) C D B E A (2) B E A D C (2) B A E D C (2) A C E D B (2) A C D B E (2) A B C E D (2) E D C B A (1) E D B A C (1) E B D A C (1) E A D C B (1) E A D B C (1) E A B D C (1) C E D A B (1) C A B D E (1) B D E C A (1) B D C E A (1) B A D C E (1) A E C D B (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 -6 -20 B 6 0 -6 -14 -2 C 8 6 0 -8 4 D 6 14 8 0 8 E 20 2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -6 -20 B 6 0 -6 -14 -2 C 8 6 0 -8 4 D 6 14 8 0 8 E 20 2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=23 E=19 B=18 D=14 so D is eliminated. Round 2 votes counts: C=28 E=26 A=26 B=20 so B is eliminated. Round 3 votes counts: E=42 C=29 A=29 so C is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:218 C:205 E:205 B:192 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -8 -6 -20 B 6 0 -6 -14 -2 C 8 6 0 -8 4 D 6 14 8 0 8 E 20 2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -6 -20 B 6 0 -6 -14 -2 C 8 6 0 -8 4 D 6 14 8 0 8 E 20 2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -6 -20 B 6 0 -6 -14 -2 C 8 6 0 -8 4 D 6 14 8 0 8 E 20 2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7321: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B C E D A (9) A D E C B (7) E C A D B (6) C E B D A (5) C B E A D (5) A D B E C (5) D A B E C (4) B D A C E (4) D E A C B (3) D B A E C (3) A E D C B (3) E D A C B (2) D A E B C (2) C E B A D (2) C E A D B (2) C E A B D (2) C B E D A (2) B D C E A (2) B A C D E (2) A D E B C (2) A B D C E (2) A B C D E (2) E D C A B (1) E C D B A (1) E C B D A (1) E A C D B (1) D E C A B (1) D B E A C (1) D A E C B (1) C A E B D (1) B C E A D (1) B C D A E (1) B C A E D (1) A E C D B (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 20 -4 -4 -12 B -20 0 -18 -12 -10 C 4 18 0 14 -8 D 4 12 -14 0 -16 E 12 10 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 -4 -4 -12 B -20 0 -18 -12 -10 C 4 18 0 14 -8 D 4 12 -14 0 -16 E 12 10 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 B=20 C=19 D=15 so D is eliminated. Round 2 votes counts: A=32 E=25 B=24 C=19 so C is eliminated. Round 3 votes counts: E=36 A=33 B=31 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:214 A:200 D:193 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 -4 -4 -12 B -20 0 -18 -12 -10 C 4 18 0 14 -8 D 4 12 -14 0 -16 E 12 10 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -4 -4 -12 B -20 0 -18 -12 -10 C 4 18 0 14 -8 D 4 12 -14 0 -16 E 12 10 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -4 -4 -12 B -20 0 -18 -12 -10 C 4 18 0 14 -8 D 4 12 -14 0 -16 E 12 10 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7322: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B A D C E (9) E D C B A (6) A C E D B (6) A B C D E (6) D E C B A (5) B A D E C (5) E B D C A (4) C E D A B (4) E A C D B (3) D C E B A (3) B D E C A (3) A E C D B (3) A E C B D (3) E C A D B (2) D E B C A (2) D B E C A (2) D B C E A (2) C E A D B (2) C D E A B (2) B D A C E (2) A C E B D (2) A C B D E (2) A B E C D (2) C D E B A (1) C A D E B (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B A E D C (1) A C B E D (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 2 -4 B -2 0 -10 -6 -16 C 0 10 0 0 -6 D -2 6 0 0 0 E 4 16 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.413481 E: 0.586519 Sum of squares = 0.514971089497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.413481 E: 1.000000 A B C D E A 0 2 0 2 -4 B -2 0 -10 -6 -16 C 0 10 0 0 -6 D -2 6 0 0 0 E 4 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 B=24 D=14 C=10 so C is eliminated. Round 2 votes counts: E=30 A=29 B=24 D=17 so D is eliminated. Round 3 votes counts: E=43 A=29 B=28 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 C:202 D:202 A:200 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 2 -4 B -2 0 -10 -6 -16 C 0 10 0 0 -6 D -2 6 0 0 0 E 4 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 -4 B -2 0 -10 -6 -16 C 0 10 0 0 -6 D -2 6 0 0 0 E 4 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 -4 B -2 0 -10 -6 -16 C 0 10 0 0 -6 D -2 6 0 0 0 E 4 16 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7323: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) A E C D B (9) B C D E A (7) E A D C B (6) A E C B D (6) D E C A B (5) B C A E D (5) D B E C A (4) B D C E A (4) B D C A E (4) B C D A E (4) A C E B D (4) E A C D B (3) C B A E D (3) E C A D B (2) D A E B C (2) B D A C E (2) E D A C B (1) D E C B A (1) D E B C A (1) D E A C B (1) D E A B C (1) D C E B A (1) D C B E A (1) D B A E C (1) C E A D B (1) C B E D A (1) C A E B D (1) C A B E D (1) B D A E C (1) B C E D A (1) B C A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D C B (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -16 -6 -2 B 10 0 4 -2 8 C 16 -4 0 4 8 D 6 2 -4 0 6 E 2 -8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999998 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 -6 -2 B 10 0 4 -2 8 C 16 -4 0 4 8 D 6 2 -4 0 6 E 2 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=28 A=21 E=12 C=7 so C is eliminated. Round 2 votes counts: B=36 D=28 A=23 E=13 so E is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:212 B:210 D:205 E:190 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -16 -6 -2 B 10 0 4 -2 8 C 16 -4 0 4 8 D 6 2 -4 0 6 E 2 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 -6 -2 B 10 0 4 -2 8 C 16 -4 0 4 8 D 6 2 -4 0 6 E 2 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 -6 -2 B 10 0 4 -2 8 C 16 -4 0 4 8 D 6 2 -4 0 6 E 2 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7324: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (5) D C E A B (5) C E D A B (5) B D A C E (5) A E B C D (5) E C A D B (4) C B E A D (4) B A D E C (4) B A D C E (4) A E D B C (4) E C A B D (3) D C B E A (3) D B A C E (3) C E D B A (3) B D C A E (3) E A C D B (2) D E A C B (2) D C E B A (2) D A B E C (2) C D E B A (2) B D A E C (2) B C D A E (2) B A E C D (2) B A C E D (2) B A C D E (2) A E B D C (2) A D E B C (2) A B E C D (2) E D C A B (1) E D A C B (1) E C D A B (1) E A D C B (1) E A C B D (1) D B C E A (1) D B A E C (1) C E B D A (1) C E B A D (1) C B E D A (1) B C A E D (1) B C A D E (1) B A E D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 -2 2 B 2 0 4 0 -6 C -2 -4 0 -10 2 D 2 0 10 0 2 E -2 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.177258 C: 0.000000 D: 0.822742 E: 0.000000 Sum of squares = 0.70832461204 Cumulative probabilities = A: 0.000000 B: 0.177258 C: 0.177258 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 2 B 2 0 4 0 -6 C -2 -4 0 -10 2 D 2 0 10 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000143748 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=24 C=17 A=16 E=14 so E is eliminated. Round 2 votes counts: B=29 D=26 C=25 A=20 so A is eliminated. Round 3 votes counts: B=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:207 A:200 B:200 E:200 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -2 2 B 2 0 4 0 -6 C -2 -4 0 -10 2 D 2 0 10 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000143748 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 2 B 2 0 4 0 -6 C -2 -4 0 -10 2 D 2 0 10 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000143748 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 2 B 2 0 4 0 -6 C -2 -4 0 -10 2 D 2 0 10 0 2 E -2 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000143748 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7325: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) E C B D A (7) A D B C E (6) D B A E C (5) E C A D B (4) D A B E C (4) C B E D A (4) C A B D E (4) B D A C E (4) A D B E C (4) A C E D B (4) E D B A C (3) E A D B C (3) A D C B E (3) A C D E B (3) E B D C A (2) E A D C B (2) D E B A C (2) C B D E A (2) C A E D B (2) B D E C A (2) B D A E C (2) A C D B E (2) E D B C A (1) E D A B C (1) E B C D A (1) D E A B C (1) D B E A C (1) D A E B C (1) C E B A D (1) C B E A D (1) C B D A E (1) C A E B D (1) C A D B E (1) B D E A C (1) B C D A E (1) A E D C B (1) A D E C B (1) A D E B C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 14 -8 6 B -2 0 -6 -12 0 C -14 6 0 -2 0 D 8 12 2 0 8 E -6 0 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 -8 6 B -2 0 -6 -12 0 C -14 6 0 -2 0 D 8 12 2 0 8 E -6 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=25 E=24 D=14 B=10 so B is eliminated. Round 2 votes counts: A=27 C=26 E=24 D=23 so D is eliminated. Round 3 votes counts: A=43 E=31 C=26 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:207 C:195 E:193 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 14 -8 6 B -2 0 -6 -12 0 C -14 6 0 -2 0 D 8 12 2 0 8 E -6 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -8 6 B -2 0 -6 -12 0 C -14 6 0 -2 0 D 8 12 2 0 8 E -6 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -8 6 B -2 0 -6 -12 0 C -14 6 0 -2 0 D 8 12 2 0 8 E -6 0 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7326: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) D B E C A (9) A C E B D (7) C A E D B (6) A B E D C (5) D B C E A (4) D C E B A (3) D C B E A (3) D B A E C (3) D A B C E (3) B E D C A (3) B D E A C (3) B D A E C (3) D B E A C (2) D B A C E (2) C A E B D (2) B A E D C (2) A E B C D (2) A B E C D (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A C D (1) E A C B D (1) E A B C D (1) D C A E B (1) D C A B E (1) D B C A E (1) C E B D A (1) C E A D B (1) C E A B D (1) C D E B A (1) C D E A B (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) A D C B E (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 10 14 8 20 B -10 0 6 4 6 C -14 -6 0 -10 -12 D -8 -4 10 0 -4 E -20 -6 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 8 20 B -10 0 6 4 6 C -14 -6 0 -10 -12 D -8 -4 10 0 -4 E -20 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=32 C=15 B=13 E=6 so E is eliminated. Round 2 votes counts: A=36 D=32 C=17 B=15 so B is eliminated. Round 3 votes counts: D=42 A=40 C=18 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:203 D:197 E:195 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 8 20 B -10 0 6 4 6 C -14 -6 0 -10 -12 D -8 -4 10 0 -4 E -20 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 8 20 B -10 0 6 4 6 C -14 -6 0 -10 -12 D -8 -4 10 0 -4 E -20 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 8 20 B -10 0 6 4 6 C -14 -6 0 -10 -12 D -8 -4 10 0 -4 E -20 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7327: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) D A C B E (9) D A C E B (6) E B C A D (5) B E D A C (5) C A E B D (4) C A D B E (4) B E D C A (4) A D C E B (4) A C D E B (4) E D A C B (3) D A E C B (3) C A E D B (3) C A D E B (3) B E C D A (3) B D E A C (3) B C E A D (3) B C A E D (3) E B D A C (2) D E B A C (2) B D A C E (2) A D C B E (2) E D A B C (1) E C A D B (1) E B D C A (1) E A C D B (1) D B E A C (1) D A B C E (1) C E A D B (1) C B A E D (1) C A B E D (1) C A B D E (1) B D C A E (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 4 -2 4 8 B -4 0 -2 2 12 C 2 2 0 0 10 D -4 -2 0 0 -4 E -8 -12 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.738263 D: 0.261737 E: 0.000000 Sum of squares = 0.613538903559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.738263 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 4 8 B -4 0 -2 2 12 C 2 2 0 0 10 D -4 -2 0 0 -4 E -8 -12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555634272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=22 C=18 E=14 A=10 so A is eliminated. Round 2 votes counts: B=36 D=28 C=22 E=14 so E is eliminated. Round 3 votes counts: B=44 D=32 C=24 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:207 C:207 B:204 D:195 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 4 8 B -4 0 -2 2 12 C 2 2 0 0 10 D -4 -2 0 0 -4 E -8 -12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555634272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 4 8 B -4 0 -2 2 12 C 2 2 0 0 10 D -4 -2 0 0 -4 E -8 -12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555634272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 4 8 B -4 0 -2 2 12 C 2 2 0 0 10 D -4 -2 0 0 -4 E -8 -12 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555634272 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7328: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) A C E D B (9) B E D C A (6) B E D A C (6) E B D C A (4) C A D B E (4) E A C D B (3) E A C B D (3) D B E C A (3) C A D E B (3) A B C D E (3) E D C B A (2) E C A D B (2) E B A D C (2) E A B C D (2) D C E B A (2) D C B A E (2) D B C E A (2) D B C A E (2) C E A D B (2) C D E A B (2) B A E D C (2) A E C D B (2) A E C B D (2) A C E B D (2) A B E C D (2) E D C A B (1) E D B C A (1) E C D A B (1) E A D C B (1) D E C B A (1) C D B A E (1) C D A E B (1) C B D A E (1) B E A D C (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) A E B C D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -8 -4 -16 B 4 0 2 4 0 C 8 -2 0 -4 -18 D 4 -4 4 0 -16 E 16 0 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.750508 C: 0.000000 D: 0.000000 E: 0.249492 Sum of squares = 0.625508473662 Cumulative probabilities = A: 0.000000 B: 0.750508 C: 0.750508 D: 0.750508 E: 1.000000 A B C D E A 0 -4 -8 -4 -16 B 4 0 2 4 0 C 8 -2 0 -4 -18 D 4 -4 4 0 -16 E 16 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 E=22 C=14 D=12 so D is eliminated. Round 2 votes counts: B=36 E=23 A=23 C=18 so C is eliminated. Round 3 votes counts: B=40 A=31 E=29 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:225 B:205 D:194 C:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 -16 B 4 0 2 4 0 C 8 -2 0 -4 -18 D 4 -4 4 0 -16 E 16 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 -16 B 4 0 2 4 0 C 8 -2 0 -4 -18 D 4 -4 4 0 -16 E 16 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 -16 B 4 0 2 4 0 C 8 -2 0 -4 -18 D 4 -4 4 0 -16 E 16 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7329: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) D B A E C (7) C D A B E (7) C A D B E (6) E D B A C (4) E B A D C (4) A B D C E (4) E C B D A (3) E C B A D (3) D C A B E (3) C E D A B (3) C A B E D (3) B D A E C (3) A B D E C (3) E D B C A (2) E C D B A (2) E C A B D (2) E B D A C (2) E B A C D (2) C E A D B (2) B A E D C (2) A D B C E (2) A C D B E (2) D E B A C (1) D C E B A (1) D C B E A (1) D B E A C (1) D A B C E (1) C E D B A (1) C D B E A (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B A D E C (1) A E B C D (1) A D C B E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -8 12 2 B -14 0 -10 2 8 C 8 10 0 8 6 D -12 -2 -8 0 -4 E -2 -8 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 12 2 B -14 0 -10 2 8 C 8 10 0 8 6 D -12 -2 -8 0 -4 E -2 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=24 A=16 D=15 B=8 so B is eliminated. Round 2 votes counts: C=37 E=26 A=19 D=18 so D is eliminated. Round 3 votes counts: C=42 A=30 E=28 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:210 E:194 B:193 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -8 12 2 B -14 0 -10 2 8 C 8 10 0 8 6 D -12 -2 -8 0 -4 E -2 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 12 2 B -14 0 -10 2 8 C 8 10 0 8 6 D -12 -2 -8 0 -4 E -2 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 12 2 B -14 0 -10 2 8 C 8 10 0 8 6 D -12 -2 -8 0 -4 E -2 -8 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7330: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) D C E A B (7) D C B E A (6) E C D B A (5) E A D C B (5) E A C D B (4) D C B A E (4) A E B C D (4) E D C A B (3) E A B C D (3) D C E B A (3) B C D E A (3) A B E D C (3) A B D C E (3) E C D A B (2) E C B D A (2) C E D B A (2) C D B E A (2) B E C A D (2) B D C A E (2) B C A D E (2) B A E C D (2) B A D C E (2) B A C D E (2) A E D B C (2) A E B D C (2) E D A C B (1) E C A D B (1) E B C A D (1) D E C B A (1) D B A C E (1) D A E C B (1) D A B C E (1) C D E B A (1) C B D E A (1) B D A C E (1) B C D A E (1) B C A E D (1) A E D C B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 6 -4 0 -10 B -6 0 -6 -10 -6 C 4 6 0 -2 -10 D 0 10 2 0 -8 E 10 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 0 -10 B -6 0 -6 -10 -6 C 4 6 0 -2 -10 D 0 10 2 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=25 D=24 B=18 C=6 so C is eliminated. Round 2 votes counts: E=29 D=27 A=25 B=19 so B is eliminated. Round 3 votes counts: D=35 A=34 E=31 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:217 D:202 C:199 A:196 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 0 -10 B -6 0 -6 -10 -6 C 4 6 0 -2 -10 D 0 10 2 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 -10 B -6 0 -6 -10 -6 C 4 6 0 -2 -10 D 0 10 2 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 -10 B -6 0 -6 -10 -6 C 4 6 0 -2 -10 D 0 10 2 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7331: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (17) E A B C D (15) A E B C D (10) C D B A E (9) E A B D C (6) D C B E A (6) B E A D C (4) E A C D B (3) D C E A B (3) B A E D C (3) B A E C D (3) E A D C B (2) C D E A B (2) C D A E B (2) B D C A E (2) B C D A E (2) B C A E D (2) E B A C D (1) E A D B C (1) D C E B A (1) D C A E B (1) D B C A E (1) C A E D B (1) B D C E A (1) B A D E C (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 0 6 10 B 6 0 4 2 2 C 0 -4 0 2 0 D -6 -2 -2 0 -4 E -10 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 6 10 B 6 0 4 2 2 C 0 -4 0 2 0 D -6 -2 -2 0 -4 E -10 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 B=18 C=14 A=11 so A is eliminated. Round 2 votes counts: E=39 D=29 B=18 C=14 so C is eliminated. Round 3 votes counts: D=42 E=40 B=18 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:207 A:205 C:199 E:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 6 10 B 6 0 4 2 2 C 0 -4 0 2 0 D -6 -2 -2 0 -4 E -10 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 6 10 B 6 0 4 2 2 C 0 -4 0 2 0 D -6 -2 -2 0 -4 E -10 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 6 10 B 6 0 4 2 2 C 0 -4 0 2 0 D -6 -2 -2 0 -4 E -10 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7332: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) C A B E D (9) D E B A C (8) B C A D E (8) A C B E D (8) E D B C A (5) E B D C A (4) D E A C B (4) B D E C A (4) B C A E D (4) A C D E B (4) A C D B E (4) A C B D E (4) E D A C B (3) D B E C A (3) E B C A D (2) E A C D B (2) B D C A E (2) E A D C B (1) D B A C E (1) D A C B E (1) C B A E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B E C A D (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 -18 0 -2 B 16 0 14 2 8 C 18 -14 0 2 -2 D 0 -2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -18 0 -2 B 16 0 14 2 8 C 18 -14 0 2 -2 D 0 -2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=22 A=21 E=17 C=11 so C is eliminated. Round 2 votes counts: A=31 D=29 B=23 E=17 so E is eliminated. Round 3 votes counts: D=37 A=34 B=29 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:220 D:205 C:202 E:191 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -18 0 -2 B 16 0 14 2 8 C 18 -14 0 2 -2 D 0 -2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 0 -2 B 16 0 14 2 8 C 18 -14 0 2 -2 D 0 -2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 0 -2 B 16 0 14 2 8 C 18 -14 0 2 -2 D 0 -2 -2 0 14 E 2 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7333: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) D A B C E (6) A E B D C (5) B C A E D (4) B C A D E (4) A B E C D (4) A B D E C (4) A B D C E (4) E D C A B (3) E A C B D (3) D C B A E (3) C E D B A (3) C E B A D (3) C B E A D (3) B A C E D (3) A D B E C (3) D C E B A (2) D C E A B (2) D B A C E (2) D A E B C (2) C E B D A (2) C D B E A (2) C B E D A (2) C B D E A (2) B C D A E (2) B A D C E (2) B A C D E (2) A D E B C (2) E D A C B (1) E C D B A (1) E C D A B (1) E C A B D (1) E A B C D (1) D E A C B (1) D C B E A (1) D A E C B (1) C D E B A (1) C B D A E (1) B C E A D (1) A E D B C (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -2 18 14 B 6 0 8 22 12 C 2 -8 0 8 16 D -18 -22 -8 0 0 E -14 -12 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 18 14 B 6 0 8 22 12 C 2 -8 0 8 16 D -18 -22 -8 0 0 E -14 -12 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=20 C=19 E=18 B=18 so E is eliminated. Round 2 votes counts: C=29 A=29 D=24 B=18 so B is eliminated. Round 3 votes counts: C=40 A=36 D=24 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:224 A:212 C:209 E:179 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 18 14 B 6 0 8 22 12 C 2 -8 0 8 16 D -18 -22 -8 0 0 E -14 -12 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 18 14 B 6 0 8 22 12 C 2 -8 0 8 16 D -18 -22 -8 0 0 E -14 -12 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 18 14 B 6 0 8 22 12 C 2 -8 0 8 16 D -18 -22 -8 0 0 E -14 -12 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999081 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7334: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C A D (10) E B C D A (7) C B E A D (7) A D C B E (7) D A E B C (6) A D B E C (6) B E A C D (5) E B D A C (4) C E B D A (4) C A D B E (4) D A E C B (3) C B E D A (3) E B D C A (2) E B C A D (2) C D E B A (2) C D A E B (2) C A B D E (2) B A E D C (2) E B A D C (1) D E C B A (1) D E B A C (1) D E A B C (1) D A C B E (1) B E A D C (1) B C E A D (1) A D E B C (1) A D B C E (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 6 4 -4 B 6 0 4 4 4 C -6 -4 0 0 -8 D -4 -4 0 0 0 E 4 -4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 4 -4 B 6 0 4 4 4 C -6 -4 0 0 -8 D -4 -4 0 0 0 E 4 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 B=19 A=17 E=16 so E is eliminated. Round 2 votes counts: B=35 D=24 C=24 A=17 so A is eliminated. Round 3 votes counts: D=39 B=36 C=25 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:204 A:200 D:196 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 4 -4 B 6 0 4 4 4 C -6 -4 0 0 -8 D -4 -4 0 0 0 E 4 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 4 -4 B 6 0 4 4 4 C -6 -4 0 0 -8 D -4 -4 0 0 0 E 4 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 4 -4 B 6 0 4 4 4 C -6 -4 0 0 -8 D -4 -4 0 0 0 E 4 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7335: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) B C A D E (11) E D A B C (10) D A E C B (5) B C D A E (5) A C D B E (5) E B D C A (4) C B A D E (4) C A B D E (4) B C E D A (4) B C E A D (4) A D E C B (4) D E A B C (3) C A D B E (3) B C A E D (3) A D C E B (3) E D B A C (2) E A D C B (2) B E C D A (2) A E D C B (2) A C D E B (2) E B D A C (1) E B C D A (1) D E A C B (1) C B A E D (1) B E D C A (1) A D C B E (1) Total count = 100 A B C D E A 0 14 6 -2 6 B -14 0 2 -10 -4 C -6 -2 0 -2 0 D 2 10 2 0 2 E -6 4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -2 6 B -14 0 2 -10 -4 C -6 -2 0 -2 0 D 2 10 2 0 2 E -6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=30 A=17 C=12 D=9 so D is eliminated. Round 2 votes counts: E=36 B=30 A=22 C=12 so C is eliminated. Round 3 votes counts: E=36 B=35 A=29 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:212 D:208 E:198 C:195 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -2 6 B -14 0 2 -10 -4 C -6 -2 0 -2 0 D 2 10 2 0 2 E -6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -2 6 B -14 0 2 -10 -4 C -6 -2 0 -2 0 D 2 10 2 0 2 E -6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -2 6 B -14 0 2 -10 -4 C -6 -2 0 -2 0 D 2 10 2 0 2 E -6 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7336: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (21) B A C D E (18) D E B A C (14) C A B E D (9) A B C E D (5) E C D A B (4) D E C A B (4) C A E B D (4) D E C B A (3) B D A E C (3) E D C B A (2) D E B C A (2) C E A B D (2) B A D E C (2) A B C D E (2) D B E A C (1) D B A E C (1) C E D A B (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -4 -12 -8 B -6 0 -2 -6 -14 C 4 2 0 -6 -14 D 12 6 6 0 0 E 8 14 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666369 E: 0.333631 Sum of squares = 0.555356989139 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666369 E: 1.000000 A B C D E A 0 6 -4 -12 -8 B -6 0 -2 -6 -14 C 4 2 0 -6 -14 D 12 6 6 0 0 E 8 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 B=24 C=16 A=8 so A is eliminated. Round 2 votes counts: B=31 E=27 D=25 C=17 so C is eliminated. Round 3 votes counts: B=41 E=34 D=25 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:212 C:193 A:191 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -12 -8 B -6 0 -2 -6 -14 C 4 2 0 -6 -14 D 12 6 6 0 0 E 8 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -12 -8 B -6 0 -2 -6 -14 C 4 2 0 -6 -14 D 12 6 6 0 0 E 8 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -12 -8 B -6 0 -2 -6 -14 C 4 2 0 -6 -14 D 12 6 6 0 0 E 8 14 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7337: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (19) C B D A E (11) E A D B C (9) A E C B D (9) D B E C A (6) E D B C A (4) E A C B D (4) C A B D E (4) A E D B C (4) A C B E D (4) A C B D E (4) E A D C B (3) B D C E A (3) A E C D B (3) E D B A C (2) E A C D B (2) A C E B D (2) D B C A E (1) C B A D E (1) B D C A E (1) B C D A E (1) A E D C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -2 4 -4 B -2 0 2 -12 10 C 2 -2 0 -8 4 D -4 12 8 0 4 E 4 -10 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333233 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 2 -2 4 -4 B -2 0 2 -12 10 C 2 -2 0 -8 4 D -4 12 8 0 4 E 4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=26 E=24 C=16 B=5 so B is eliminated. Round 2 votes counts: D=30 A=29 E=24 C=17 so C is eliminated. Round 3 votes counts: D=42 A=34 E=24 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:200 B:199 C:198 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 4 -4 B -2 0 2 -12 10 C 2 -2 0 -8 4 D -4 12 8 0 4 E 4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 4 -4 B -2 0 2 -12 10 C 2 -2 0 -8 4 D -4 12 8 0 4 E 4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 4 -4 B -2 0 2 -12 10 C 2 -2 0 -8 4 D -4 12 8 0 4 E 4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7338: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (22) C B A D E (13) E B A D C (8) E D A C B (5) B E A D C (5) E A D B C (3) B A C E D (3) E D C A B (2) D E A C B (2) D C E A B (2) D A E B C (2) C E D A B (2) C D A B E (2) C B A E D (2) C A D B E (2) B E A C D (2) B C A D E (2) B A C D E (2) A B E D C (2) E C D B A (1) E C D A B (1) E A B D C (1) D E A B C (1) D C A E B (1) C E D B A (1) C D E A B (1) C B D A E (1) C A B D E (1) B C A E D (1) B A E C D (1) B A D E C (1) A D B E C (1) A D B C E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 30 8 -18 B -14 0 22 -4 -10 C -30 -22 0 -22 -22 D -8 4 22 0 -24 E 18 10 22 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 30 8 -18 B -14 0 22 -4 -10 C -30 -22 0 -22 -22 D -8 4 22 0 -24 E 18 10 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 C=25 B=17 D=8 A=7 so A is eliminated. Round 2 votes counts: E=43 C=25 B=22 D=10 so D is eliminated. Round 3 votes counts: E=48 C=28 B=24 so B is eliminated. Round 4 votes counts: E=61 C=39 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:237 A:217 B:197 D:197 C:152 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 30 8 -18 B -14 0 22 -4 -10 C -30 -22 0 -22 -22 D -8 4 22 0 -24 E 18 10 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 30 8 -18 B -14 0 22 -4 -10 C -30 -22 0 -22 -22 D -8 4 22 0 -24 E 18 10 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 30 8 -18 B -14 0 22 -4 -10 C -30 -22 0 -22 -22 D -8 4 22 0 -24 E 18 10 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7339: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) E B C A D (8) D A C B E (7) C E A B D (6) C A D E B (6) B D E A C (6) D C A E B (4) D B A E C (4) D A C E B (4) D A B C E (4) C A E D B (4) A C E D B (4) E B A C D (3) D B A C E (3) C D A E B (3) B E C D A (3) D C A B E (2) D B C E A (2) B E D C A (2) B E C A D (2) A D C E B (2) E C B A D (1) D A B E C (1) C E B D A (1) C E A D B (1) C A E B D (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E C A (1) B D A E C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 20 -12 2 4 B -20 0 -14 -4 -18 C 12 14 0 10 12 D -2 4 -10 0 2 E -4 18 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -12 2 4 B -20 0 -14 -4 -18 C 12 14 0 10 12 D -2 4 -10 0 2 E -4 18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=22 E=21 B=18 A=8 so A is eliminated. Round 2 votes counts: D=33 C=28 E=21 B=18 so B is eliminated. Round 3 votes counts: D=41 E=31 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:224 A:207 E:200 D:197 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -12 2 4 B -20 0 -14 -4 -18 C 12 14 0 10 12 D -2 4 -10 0 2 E -4 18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -12 2 4 B -20 0 -14 -4 -18 C 12 14 0 10 12 D -2 4 -10 0 2 E -4 18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -12 2 4 B -20 0 -14 -4 -18 C 12 14 0 10 12 D -2 4 -10 0 2 E -4 18 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7340: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) C A D B E (9) A C D B E (6) D B E A C (5) C E A B D (5) C A B E D (5) B D E A C (5) E D B C A (4) B D A E C (4) A C B D E (4) D E B A C (3) C A E D B (3) C A E B D (3) E D B A C (2) E B D C A (2) C D A B E (2) C A B D E (2) B E D A C (2) B A D E C (2) A C B E D (2) A B E C D (2) E C D B A (1) E C B A D (1) E B C A D (1) E B A D C (1) E B A C D (1) D E B C A (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B A D (1) C E A D B (1) C D E A B (1) C A D E B (1) B E A D C (1) B E A C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 10 8 -4 B 2 0 2 16 10 C -10 -2 0 10 -2 D -8 -16 -10 0 -6 E 4 -10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 8 -4 B 2 0 2 16 10 C -10 -2 0 10 -2 D -8 -16 -10 0 -6 E 4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=24 A=16 B=15 D=10 so D is eliminated. Round 2 votes counts: C=35 E=28 B=21 A=16 so A is eliminated. Round 3 votes counts: C=47 E=28 B=25 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 A:206 E:201 C:198 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 8 -4 B 2 0 2 16 10 C -10 -2 0 10 -2 D -8 -16 -10 0 -6 E 4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 8 -4 B 2 0 2 16 10 C -10 -2 0 10 -2 D -8 -16 -10 0 -6 E 4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 8 -4 B 2 0 2 16 10 C -10 -2 0 10 -2 D -8 -16 -10 0 -6 E 4 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984806 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7341: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) E D A B C (8) C B D A E (5) C A B D E (4) B D E A C (4) A E B D C (4) E D B C A (3) E A D B C (3) D E B A C (3) C E D B A (3) C E D A B (3) C D E B A (3) B A D E C (3) A C B E D (3) E D C B A (2) D B E C A (2) C E A D B (2) C D B E A (2) C A E B D (2) C A B E D (2) B D A E C (2) B D A C E (2) A C E B D (2) A B C E D (2) A B C D E (2) E D C A B (1) E A D C B (1) D E C B A (1) D E B C A (1) D B E A C (1) C B A D E (1) C A E D B (1) B D C E A (1) B D C A E (1) B C D A E (1) B C A D E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A C B D E (1) A B E D C (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 16 -16 -6 B 6 0 18 0 -8 C -16 -18 0 -12 -6 D 16 0 12 0 -10 E 6 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 16 -16 -6 B 6 0 18 0 -8 C -16 -18 0 -12 -6 D 16 0 12 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 A=20 B=17 D=8 so D is eliminated. Round 2 votes counts: E=32 C=28 B=20 A=20 so B is eliminated. Round 3 votes counts: E=39 C=32 A=29 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:209 B:208 A:194 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 16 -16 -6 B 6 0 18 0 -8 C -16 -18 0 -12 -6 D 16 0 12 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 16 -16 -6 B 6 0 18 0 -8 C -16 -18 0 -12 -6 D 16 0 12 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 16 -16 -6 B 6 0 18 0 -8 C -16 -18 0 -12 -6 D 16 0 12 0 -10 E 6 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7342: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) E B A D C (9) B A E C D (6) D C E A B (5) C E B A D (5) A B E D C (5) E D B A C (4) D C E B A (4) D C A B E (4) C D A B E (4) A B C E D (4) E D C B A (3) E B A C D (3) D E C B A (3) D E B A C (3) C A B E D (3) E C D B A (2) E B D A C (2) D E A B C (2) C E D B A (2) C D E A B (2) C B A E D (2) E D B C A (1) E D A B C (1) E C B D A (1) D E B C A (1) D A B E C (1) C A B D E (1) B C A E D (1) A D B C E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 -8 -10 -26 B 24 0 -6 -6 -26 C 8 6 0 0 2 D 10 6 0 0 -12 E 26 26 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.900035 D: 0.099965 E: 0.000000 Sum of squares = 0.82005593238 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.900035 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -8 -10 -26 B 24 0 -6 -6 -26 C 8 6 0 0 2 D 10 6 0 0 -12 E 26 26 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.857143 D: 0.142857 E: 0.000000 Sum of squares = 0.755102100909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=26 D=23 A=15 B=7 so B is eliminated. Round 2 votes counts: C=30 E=26 D=23 A=21 so A is eliminated. Round 3 votes counts: E=38 C=37 D=25 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:231 C:208 D:202 B:193 A:166 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -24 -8 -10 -26 B 24 0 -6 -6 -26 C 8 6 0 0 2 D 10 6 0 0 -12 E 26 26 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.857143 D: 0.142857 E: 0.000000 Sum of squares = 0.755102100909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -8 -10 -26 B 24 0 -6 -6 -26 C 8 6 0 0 2 D 10 6 0 0 -12 E 26 26 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.857143 D: 0.142857 E: 0.000000 Sum of squares = 0.755102100909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -8 -10 -26 B 24 0 -6 -6 -26 C 8 6 0 0 2 D 10 6 0 0 -12 E 26 26 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.857143 D: 0.142857 E: 0.000000 Sum of squares = 0.755102100909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7343: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B D A E (8) D E A C B (7) C B A D E (6) E D A B C (5) D E A B C (5) C B D E A (4) A D E B C (4) E B A D C (3) B E A C D (3) B C E D A (3) A E D B C (3) A D E C B (3) D C E A B (2) D C A E B (2) D A E C B (2) D A C E B (2) C D B E A (2) C D B A E (2) C B A E D (2) B C A E D (2) A D C E B (2) A C D B E (2) A B E C D (2) E B D C A (1) E B D A C (1) E A B D C (1) D E C A B (1) D C E B A (1) C D E B A (1) C D E A B (1) C D A B E (1) C B E D A (1) C A D B E (1) C A B D E (1) B E C D A (1) B E C A D (1) B C E A D (1) B A E C D (1) Total count = 100 A B C D E A 0 12 10 -6 -8 B -12 0 -8 -16 -12 C -10 8 0 -8 -6 D 6 16 8 0 20 E 8 12 6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 -6 -8 B -12 0 -8 -16 -12 C -10 8 0 -8 -6 D 6 16 8 0 20 E 8 12 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 E=20 A=16 B=12 so B is eliminated. Round 2 votes counts: C=36 E=25 D=22 A=17 so A is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:204 E:203 C:192 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 10 -6 -8 B -12 0 -8 -16 -12 C -10 8 0 -8 -6 D 6 16 8 0 20 E 8 12 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -6 -8 B -12 0 -8 -16 -12 C -10 8 0 -8 -6 D 6 16 8 0 20 E 8 12 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -6 -8 B -12 0 -8 -16 -12 C -10 8 0 -8 -6 D 6 16 8 0 20 E 8 12 6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7344: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) B D A E C (7) A B D E C (7) C E D B A (6) A E D B C (6) C B D E A (5) E C A D B (4) C E D A B (4) C E A D B (4) C D E B A (4) C D B E A (4) C A E D B (4) B A D E C (4) E A D B C (3) B D C E A (3) A E C D B (3) C B D A E (2) B C D A E (2) A B E D C (2) E D C B A (1) E D B A C (1) E D A B C (1) E A D C B (1) E A C D B (1) D E B A C (1) D B C E A (1) C B A D E (1) C A E B D (1) C A B E D (1) C A B D E (1) B D E A C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 4 14 10 B -14 0 4 2 -16 C -4 -4 0 -2 -10 D -14 -2 2 0 -14 E -10 16 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 14 10 B -14 0 4 2 -16 C -4 -4 0 -2 -10 D -14 -2 2 0 -14 E -10 16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=32 B=17 E=12 D=2 so D is eliminated. Round 2 votes counts: C=37 A=32 B=18 E=13 so E is eliminated. Round 3 votes counts: C=42 A=38 B=20 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:215 C:190 B:188 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 14 10 B -14 0 4 2 -16 C -4 -4 0 -2 -10 D -14 -2 2 0 -14 E -10 16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 14 10 B -14 0 4 2 -16 C -4 -4 0 -2 -10 D -14 -2 2 0 -14 E -10 16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 14 10 B -14 0 4 2 -16 C -4 -4 0 -2 -10 D -14 -2 2 0 -14 E -10 16 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7345: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) A D C E B (7) D E C A B (6) D A E C B (5) B E C D A (5) E D B C A (4) B E D C A (4) A D E C B (4) A D B E C (4) E C D B A (3) D E B A C (3) D E A C B (3) D A E B C (3) A C B D E (3) E D C A B (2) E B C D A (2) C E D B A (2) C E B A D (2) C A E B D (2) C A D E B (2) C A B E D (2) B C A E D (2) A D B C E (2) E D C B A (1) E C B D A (1) E B D C A (1) D E C B A (1) D E A B C (1) D C E A B (1) C E D A B (1) C E A B D (1) C B A E D (1) C A E D B (1) B E C A D (1) B E A C D (1) B D E A C (1) B C E D A (1) B C E A D (1) B A C E D (1) A C D E B (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -14 -16 -12 B -10 0 -20 -16 -34 C 14 20 0 -10 -14 D 16 16 10 0 0 E 12 34 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.635323 E: 0.364677 Sum of squares = 0.536624478499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.635323 E: 1.000000 A B C D E A 0 10 -14 -16 -12 B -10 0 -20 -16 -34 C 14 20 0 -10 -14 D 16 16 10 0 0 E 12 34 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 C=21 B=17 E=14 so E is eliminated. Round 2 votes counts: D=30 C=25 A=25 B=20 so B is eliminated. Round 3 votes counts: C=37 D=36 A=27 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:230 D:221 C:205 A:184 B:160 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -14 -16 -12 B -10 0 -20 -16 -34 C 14 20 0 -10 -14 D 16 16 10 0 0 E 12 34 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 -16 -12 B -10 0 -20 -16 -34 C 14 20 0 -10 -14 D 16 16 10 0 0 E 12 34 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 -16 -12 B -10 0 -20 -16 -34 C 14 20 0 -10 -14 D 16 16 10 0 0 E 12 34 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7346: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) A C B D E (11) A C D E B (10) E D B C A (8) D E B A C (7) B E D C A (7) D E B C A (4) C B A E D (4) C A E D B (4) B D E A C (4) C A E B D (3) A C E D B (3) E B D C A (2) D E A B C (2) D B E A C (2) B D E C A (2) B D A E C (2) A C B E D (2) E D C B A (1) D E A C B (1) D A E B C (1) D A B E C (1) C E A B D (1) C B E D A (1) B C E D A (1) A D E C B (1) A D C E B (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 10 0 8 14 B -10 0 -12 4 2 C 0 12 0 6 8 D -8 -4 -6 0 2 E -14 -2 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.304629 B: 0.000000 C: 0.695371 D: 0.000000 E: 0.000000 Sum of squares = 0.576339400026 Cumulative probabilities = A: 0.304629 B: 0.304629 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 8 14 B -10 0 -12 4 2 C 0 12 0 6 8 D -8 -4 -6 0 2 E -14 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=25 D=18 B=16 E=11 so E is eliminated. Round 2 votes counts: A=30 D=27 C=25 B=18 so B is eliminated. Round 3 votes counts: D=44 A=30 C=26 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:213 B:192 D:192 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 8 14 B -10 0 -12 4 2 C 0 12 0 6 8 D -8 -4 -6 0 2 E -14 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 8 14 B -10 0 -12 4 2 C 0 12 0 6 8 D -8 -4 -6 0 2 E -14 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 8 14 B -10 0 -12 4 2 C 0 12 0 6 8 D -8 -4 -6 0 2 E -14 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7347: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) D C A B E (8) C D B A E (5) C B D E A (5) B C E D A (5) D A C B E (4) C B D A E (4) A D B C E (4) E B C D A (3) E B A C D (3) E A D B C (3) E A B D C (3) C B E D A (3) B C D A E (3) A D E C B (3) E C B D A (2) D A C E B (2) B E C A D (2) B E A C D (2) B C E A D (2) B A D C E (2) A E D B C (2) A D E B C (2) A D C B E (2) E D A C B (1) E C D A B (1) E C B A D (1) E B C A D (1) E B A D C (1) D E C A B (1) D B C A E (1) D A E C B (1) C E D B A (1) C E B D A (1) C D A E B (1) C D A B E (1) B D C A E (1) B D A C E (1) B C D E A (1) B C A D E (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -8 -12 0 B 2 0 -12 -6 16 C 8 12 0 -4 16 D 12 6 4 0 8 E 0 -16 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -12 0 B 2 0 -12 -6 16 C 8 12 0 -4 16 D 12 6 4 0 8 E 0 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=21 B=20 D=17 A=15 so A is eliminated. Round 2 votes counts: E=30 D=29 C=21 B=20 so B is eliminated. Round 3 votes counts: E=34 D=33 C=33 so D is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:215 B:200 A:189 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -12 0 B 2 0 -12 -6 16 C 8 12 0 -4 16 D 12 6 4 0 8 E 0 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -12 0 B 2 0 -12 -6 16 C 8 12 0 -4 16 D 12 6 4 0 8 E 0 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -12 0 B 2 0 -12 -6 16 C 8 12 0 -4 16 D 12 6 4 0 8 E 0 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7348: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (7) E B A C D (6) D C A E B (6) E A B C D (5) D C A B E (5) B A E D C (5) E C B D A (4) E C B A D (4) D A C B E (4) C D E A B (4) B A D E C (4) A B E D C (4) D B A C E (3) C E D B A (3) C D E B A (3) B E A C D (3) B D A C E (3) A E B D C (3) A D B C E (3) E C A D B (2) E A B D C (2) A E C D B (2) A D B E C (2) E B C A D (1) E A C B D (1) D C B A E (1) D B C A E (1) C E D A B (1) C E B D A (1) C D B E A (1) C B E D A (1) B E A D C (1) B D C E A (1) B C E D A (1) B C D E A (1) A D C E B (1) Total count = 100 A B C D E A 0 4 18 12 8 B -4 0 12 16 2 C -18 -12 0 -12 -12 D -12 -16 12 0 0 E -8 -2 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 18 12 8 B -4 0 12 16 2 C -18 -12 0 -12 -12 D -12 -16 12 0 0 E -8 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=22 D=20 B=19 C=14 so C is eliminated. Round 2 votes counts: E=30 D=28 A=22 B=20 so B is eliminated. Round 3 votes counts: E=36 D=33 A=31 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:221 B:213 E:201 D:192 C:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 18 12 8 B -4 0 12 16 2 C -18 -12 0 -12 -12 D -12 -16 12 0 0 E -8 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 12 8 B -4 0 12 16 2 C -18 -12 0 -12 -12 D -12 -16 12 0 0 E -8 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 12 8 B -4 0 12 16 2 C -18 -12 0 -12 -12 D -12 -16 12 0 0 E -8 -2 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999361 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7349: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (7) C E D A B (5) C D A E B (5) A D B C E (5) C A D B E (4) A C D B E (4) E C D B A (3) E C B D A (3) E B C D A (3) D A C E B (3) D A B E C (3) C A B D E (3) B E D A C (3) E D C A B (2) E C D A B (2) E B D C A (2) E B D A C (2) E B C A D (2) D E C A B (2) D A B C E (2) B D E A C (2) B D A E C (2) B A E D C (2) B A D C E (2) A D C B E (2) A C B D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C B A D (1) D E A C B (1) D A C B E (1) C E B A D (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B E D (1) B E A D C (1) B E A C D (1) B C A E D (1) B A E C D (1) B A C E D (1) B A C D E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 2 18 B -8 0 -4 0 12 C -4 4 0 2 2 D -2 0 -2 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 2 18 B -8 0 -4 0 12 C -4 4 0 2 2 D -2 0 -2 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=24 E=23 A=16 D=12 so D is eliminated. Round 2 votes counts: E=26 C=25 A=25 B=24 so B is eliminated. Round 3 votes counts: A=41 E=33 C=26 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:203 C:202 B:200 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 2 18 B -8 0 -4 0 12 C -4 4 0 2 2 D -2 0 -2 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 18 B -8 0 -4 0 12 C -4 4 0 2 2 D -2 0 -2 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 18 B -8 0 -4 0 12 C -4 4 0 2 2 D -2 0 -2 0 10 E -18 -12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984048 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7350: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (6) B A D C E (6) E C D A B (5) E A D C B (5) C E B D A (5) B C D A E (5) A B D E C (5) E C D B A (4) C E D B A (4) B A C D E (4) A D B E C (4) A B D C E (4) E D A C B (3) B D C A E (3) B C A D E (3) A E D B C (3) A D E B C (3) A D B C E (3) E C B A D (2) E C A B D (2) D A E B C (2) D A B C E (2) C B D A E (2) B C A E D (2) A B E C D (2) E A C D B (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C A E (1) D A B E C (1) C D B E A (1) C D B A E (1) C B E A D (1) B C E A D (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 2 8 16 B 4 0 12 12 16 C -2 -12 0 6 10 D -8 -12 -6 0 2 E -16 -16 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 8 16 B 4 0 12 12 16 C -2 -12 0 6 10 D -8 -12 -6 0 2 E -16 -16 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 B=24 C=20 D=7 so D is eliminated. Round 2 votes counts: A=30 B=25 E=24 C=21 so C is eliminated. Round 3 votes counts: B=37 E=33 A=30 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:211 C:201 D:188 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 8 16 B 4 0 12 12 16 C -2 -12 0 6 10 D -8 -12 -6 0 2 E -16 -16 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 16 B 4 0 12 12 16 C -2 -12 0 6 10 D -8 -12 -6 0 2 E -16 -16 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 16 B 4 0 12 12 16 C -2 -12 0 6 10 D -8 -12 -6 0 2 E -16 -16 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7351: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) B D C A E (10) A E C B D (9) E A C D B (8) D B C E A (8) B A D E C (8) D C B E A (6) B D A C E (6) A E B C D (6) D C E B A (4) E C D A B (3) C D E B A (3) C D E A B (3) C E D A B (2) B D A E C (2) B A D C E (2) A E C D B (2) D B E A C (1) C E A D B (1) B D C E A (1) B A E D C (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -4 2 -2 B 4 0 -4 -4 -4 C 4 4 0 0 -6 D -2 4 0 0 8 E 2 4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.469593 D: 0.530407 E: 0.000000 Sum of squares = 0.501849218089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.469593 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 2 -2 B 4 0 -4 -4 -4 C 4 4 0 0 -6 D -2 4 0 0 8 E 2 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=22 A=20 D=19 C=9 so C is eliminated. Round 2 votes counts: B=30 E=25 D=25 A=20 so A is eliminated. Round 3 votes counts: E=42 B=33 D=25 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:205 E:202 C:201 A:196 B:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 2 -2 B 4 0 -4 -4 -4 C 4 4 0 0 -6 D -2 4 0 0 8 E 2 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 2 -2 B 4 0 -4 -4 -4 C 4 4 0 0 -6 D -2 4 0 0 8 E 2 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 2 -2 B 4 0 -4 -4 -4 C 4 4 0 0 -6 D -2 4 0 0 8 E 2 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7352: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) A D B E C (9) C E B D A (8) C E B A D (8) C B D A E (6) E C A D B (5) C E A B D (5) C B E D A (5) B D C A E (5) A E D C B (5) E A D B C (4) B D A C E (4) B C D A E (4) E A C D B (3) D A B E C (3) C B D E A (3) E C B D A (2) B D A E C (2) B C D E A (2) D B A E C (1) D A E B C (1) D A B C E (1) C E A D B (1) C A E D B (1) C A B D E (1) B C E D A (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 -14 4 6 B 2 0 -6 12 -6 C 14 6 0 10 10 D -4 -12 -10 0 2 E -6 6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -14 4 6 B 2 0 -6 12 -6 C 14 6 0 10 10 D -4 -12 -10 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=24 B=18 E=14 D=6 so D is eliminated. Round 2 votes counts: C=38 A=29 B=19 E=14 so E is eliminated. Round 3 votes counts: C=45 A=36 B=19 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:201 A:197 E:194 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -14 4 6 B 2 0 -6 12 -6 C 14 6 0 10 10 D -4 -12 -10 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 4 6 B 2 0 -6 12 -6 C 14 6 0 10 10 D -4 -12 -10 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 4 6 B 2 0 -6 12 -6 C 14 6 0 10 10 D -4 -12 -10 0 2 E -6 6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7353: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) A D B E C (7) C E B D A (6) B A D E C (6) A D C E B (6) B E C D A (4) B A E D C (4) A B D E C (4) E C B D A (3) D E C A B (3) D E A B C (3) C E D A B (3) C B E A D (3) B C E D A (3) E D B A C (2) D C E A B (2) D A E B C (2) C D E A B (2) C D A E B (2) C B E D A (2) C A D B E (2) B A C E D (2) A B D C E (2) E D C A B (1) E D B C A (1) E B D C A (1) E B C D A (1) D E B A C (1) D C A E B (1) D B E A C (1) C E A D B (1) C B A E D (1) C A E D B (1) B E D A C (1) B D E A C (1) B C E A D (1) A D E B C (1) A C D E B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -8 -12 -12 B 8 0 -2 -8 -8 C 8 2 0 0 6 D 12 8 0 0 -2 E 12 8 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.522967 D: 0.477033 E: 0.000000 Sum of squares = 0.501054986919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.522967 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -12 -12 B 8 0 -2 -8 -8 C 8 2 0 0 6 D 12 8 0 0 -2 E 12 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=23 B=22 D=13 E=9 so E is eliminated. Round 2 votes counts: C=36 B=24 A=23 D=17 so D is eliminated. Round 3 votes counts: C=43 B=29 A=28 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:209 C:208 E:208 B:195 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -12 -12 B 8 0 -2 -8 -8 C 8 2 0 0 6 D 12 8 0 0 -2 E 12 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -12 -12 B 8 0 -2 -8 -8 C 8 2 0 0 6 D 12 8 0 0 -2 E 12 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -12 -12 B 8 0 -2 -8 -8 C 8 2 0 0 6 D 12 8 0 0 -2 E 12 8 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7354: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) D A E C B (10) D C E B A (5) D C B E A (5) B A E C D (5) D A B E C (4) E C A D B (3) D E C A B (3) D B A C E (3) B D C E A (3) B C E A D (3) A E B C D (3) A B E D C (3) E C B A D (2) D B C E A (2) D B C A E (2) D A C E B (2) D A B C E (2) C E D B A (2) C E D A B (2) C D E B A (2) B D A E C (2) B C E D A (2) A D E C B (2) A B D E C (2) E C A B D (1) D B A E C (1) D A E B C (1) C E B D A (1) C E B A D (1) C E A B D (1) C D E A B (1) C B E A D (1) C B D E A (1) B E C A D (1) B A E D C (1) B A D C E (1) B A C E D (1) A E C B D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -8 -32 -4 B -6 0 -12 -26 -6 C 8 12 0 -30 6 D 32 26 30 0 30 E 4 6 -6 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -32 -4 B -6 0 -12 -26 -6 C 8 12 0 -30 6 D 32 26 30 0 30 E 4 6 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=50 B=19 A=13 C=12 E=6 so E is eliminated. Round 2 votes counts: D=50 B=19 C=18 A=13 so A is eliminated. Round 3 votes counts: D=53 B=28 C=19 so C is eliminated. Round 4 votes counts: D=63 B=37 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:259 C:198 E:187 A:181 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -32 -4 B -6 0 -12 -26 -6 C 8 12 0 -30 6 D 32 26 30 0 30 E 4 6 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -32 -4 B -6 0 -12 -26 -6 C 8 12 0 -30 6 D 32 26 30 0 30 E 4 6 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -32 -4 B -6 0 -12 -26 -6 C 8 12 0 -30 6 D 32 26 30 0 30 E 4 6 -6 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7355: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (17) D B E A C (10) E D A B C (6) C A B D E (6) C A E B D (5) C A B E D (5) D E B A C (4) C A E D B (4) E A D B C (3) C B D A E (3) B D E A C (3) E D A C B (2) E A C D B (2) C E A D B (2) C B D E A (2) B D E C A (2) B D A E C (2) B A D E C (2) B A D C E (2) A E C D B (2) A C E B D (2) A C B E D (2) E D B C A (1) E C D B A (1) E C D A B (1) E C A D B (1) C E D A B (1) C B A D E (1) B D C A E (1) B D A C E (1) A C B D E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 28 -14 -16 B 4 0 14 -14 -8 C -28 -14 0 -16 -20 D 14 14 16 0 -14 E 16 8 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 28 -14 -16 B 4 0 14 -14 -8 C -28 -14 0 -16 -20 D 14 14 16 0 -14 E 16 8 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=29 D=14 B=13 A=10 so A is eliminated. Round 2 votes counts: E=36 C=34 B=16 D=14 so D is eliminated. Round 3 votes counts: E=40 C=34 B=26 so B is eliminated. Round 4 votes counts: E=60 C=40 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:229 D:215 B:198 A:197 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 28 -14 -16 B 4 0 14 -14 -8 C -28 -14 0 -16 -20 D 14 14 16 0 -14 E 16 8 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 28 -14 -16 B 4 0 14 -14 -8 C -28 -14 0 -16 -20 D 14 14 16 0 -14 E 16 8 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 28 -14 -16 B 4 0 14 -14 -8 C -28 -14 0 -16 -20 D 14 14 16 0 -14 E 16 8 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999633 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7356: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (9) D E C B A (8) A B C E D (7) E A D B C (6) D E A B C (5) C B D A E (5) C B A D E (5) E D A B C (4) C B D E A (4) B C D A E (4) A E D B C (4) A B E C D (4) E D A C B (3) C B A E D (3) B C A D E (3) E D C A B (2) E A D C B (2) D C E B A (2) C D B E A (2) C A B E D (2) B A C D E (2) E C D B A (1) D E B A C (1) D E A C B (1) D C B E A (1) D B C A E (1) D A B E C (1) C E D B A (1) C E B D A (1) C D E B A (1) B D A C E (1) B A D C E (1) A E C B D (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 8 2 10 B -6 0 8 8 -4 C -8 -8 0 -6 -6 D -2 -8 6 0 -2 E -10 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 2 10 B -6 0 8 8 -4 C -8 -8 0 -6 -6 D -2 -8 6 0 -2 E -10 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=24 D=20 E=18 B=11 so B is eliminated. Round 2 votes counts: C=31 A=30 D=21 E=18 so E is eliminated. Round 3 votes counts: A=38 C=32 D=30 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:203 E:201 D:197 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 10 B -6 0 8 8 -4 C -8 -8 0 -6 -6 D -2 -8 6 0 -2 E -10 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 10 B -6 0 8 8 -4 C -8 -8 0 -6 -6 D -2 -8 6 0 -2 E -10 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 10 B -6 0 8 8 -4 C -8 -8 0 -6 -6 D -2 -8 6 0 -2 E -10 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7357: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) D C B A E (7) B D C A E (7) E A C B D (6) E A B C D (6) C A D B E (6) A C E D B (6) E B A C D (3) E A C D B (3) D B E C A (3) B D E C A (3) B C A D E (3) A C E B D (3) E B D C A (2) E B D A C (2) E A B D C (2) D E B C A (2) C A D E B (2) B E D C A (2) B E D A C (2) B E A C D (2) B D C E A (2) A E C B D (2) E D A C B (1) E B A D C (1) E A D B C (1) D C A E B (1) D C A B E (1) C D B A E (1) C A B D E (1) B E A D C (1) A E C D B (1) A E B C D (1) A C D E B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -8 6 12 B 8 0 12 6 8 C 8 -12 0 -2 8 D -6 -6 2 0 4 E -12 -8 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 6 12 B 8 0 12 6 8 C 8 -12 0 -2 8 D -6 -6 2 0 4 E -12 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 B=22 A=16 C=10 so C is eliminated. Round 2 votes counts: E=27 D=26 A=25 B=22 so B is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:217 A:201 C:201 D:197 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 6 12 B 8 0 12 6 8 C 8 -12 0 -2 8 D -6 -6 2 0 4 E -12 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 6 12 B 8 0 12 6 8 C 8 -12 0 -2 8 D -6 -6 2 0 4 E -12 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 6 12 B 8 0 12 6 8 C 8 -12 0 -2 8 D -6 -6 2 0 4 E -12 -8 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7358: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) D B C A E (6) C E B A D (6) C B D E A (6) E C B A D (5) E A B C D (5) D A E C B (5) D A B C E (5) A E D B C (5) A E B D C (5) C B E A D (4) B C D E A (4) A D E B C (4) E A C B D (3) D C B A E (3) D B A C E (3) C B E D A (3) A E D C B (3) D A B E C (2) C D E A B (2) E C A D B (1) E C A B D (1) E B C A D (1) E A C D B (1) D C B E A (1) D B A E C (1) D A C E B (1) D A C B E (1) C E D B A (1) C D B E A (1) B D C A E (1) B D A E C (1) B C E A D (1) B A C E D (1) Total count = 100 A B C D E A 0 2 6 -8 8 B -2 0 4 -6 -12 C -6 -4 0 -8 0 D 8 6 8 0 8 E -8 12 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -8 8 B -2 0 4 -6 -12 C -6 -4 0 -8 0 D 8 6 8 0 8 E -8 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=23 E=17 A=17 B=8 so B is eliminated. Round 2 votes counts: D=37 C=28 A=18 E=17 so E is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:204 E:198 B:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -8 8 B -2 0 4 -6 -12 C -6 -4 0 -8 0 D 8 6 8 0 8 E -8 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -8 8 B -2 0 4 -6 -12 C -6 -4 0 -8 0 D 8 6 8 0 8 E -8 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -8 8 B -2 0 4 -6 -12 C -6 -4 0 -8 0 D 8 6 8 0 8 E -8 12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7359: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (11) E A C B D (10) D B C A E (10) C D A B E (7) E B A D C (6) E A C D B (6) E C A D B (5) E A B C D (5) E A B D C (4) D C B A E (4) B D E C A (4) E B D A C (2) C A E D B (2) B D A E C (2) A E C B D (2) A C E D B (2) A C D B E (2) E C D A B (1) E B D C A (1) D C B E A (1) C E D A B (1) C E A D B (1) C D E A B (1) C D B E A (1) C D B A E (1) B E D C A (1) B E D A C (1) B D A C E (1) B A E D C (1) B A D C E (1) A E C D B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -4 0 -2 B -4 0 2 8 0 C 4 -2 0 -2 -6 D 0 -8 2 0 -6 E 2 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222262 C: 0.000000 D: 0.000000 E: 0.777738 Sum of squares = 0.654276247247 Cumulative probabilities = A: 0.000000 B: 0.222262 C: 0.222262 D: 0.222262 E: 1.000000 A B C D E A 0 4 -4 0 -2 B -4 0 2 8 0 C 4 -2 0 -2 -6 D 0 -8 2 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555710839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=22 D=15 C=14 A=9 so A is eliminated. Round 2 votes counts: E=43 B=23 C=19 D=15 so D is eliminated. Round 3 votes counts: E=43 B=33 C=24 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:207 B:203 A:199 C:197 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 0 -2 B -4 0 2 8 0 C 4 -2 0 -2 -6 D 0 -8 2 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555710839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 -2 B -4 0 2 8 0 C 4 -2 0 -2 -6 D 0 -8 2 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555710839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 -2 B -4 0 2 8 0 C 4 -2 0 -2 -6 D 0 -8 2 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555710839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7360: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) E B C D A (8) D A E B C (6) D A C B E (6) A E D C B (6) E B C A D (5) B C E D A (5) A D C B E (4) E C B A D (3) D C B A E (3) D B C A E (3) D A E C B (3) D A B C E (3) A C B D E (3) E D B C A (2) E A C B D (2) D B C E A (2) C B E A D (2) C B D A E (2) B E C D A (2) A E C B D (2) A D C E B (2) E D B A C (1) E D A B C (1) E B D C A (1) E A D C B (1) E A D B C (1) D E B C A (1) D E A B C (1) D C A B E (1) D B E C A (1) D A B E C (1) C E B A D (1) C B A E D (1) B E D C A (1) B D C E A (1) B C D A E (1) A E D B C (1) Total count = 100 A B C D E A 0 8 8 -12 16 B -8 0 -4 -20 -16 C -8 4 0 -26 -20 D 12 20 26 0 8 E -16 16 20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 -12 16 B -8 0 -4 -20 -16 C -8 4 0 -26 -20 D 12 20 26 0 8 E -16 16 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=28 E=25 B=10 C=6 so C is eliminated. Round 2 votes counts: D=31 A=28 E=26 B=15 so B is eliminated. Round 3 votes counts: E=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:233 A:210 E:206 B:176 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 8 -12 16 B -8 0 -4 -20 -16 C -8 4 0 -26 -20 D 12 20 26 0 8 E -16 16 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -12 16 B -8 0 -4 -20 -16 C -8 4 0 -26 -20 D 12 20 26 0 8 E -16 16 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -12 16 B -8 0 -4 -20 -16 C -8 4 0 -26 -20 D 12 20 26 0 8 E -16 16 20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7361: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) D A C B E (6) B C D E A (6) A E D C B (6) D C B A E (5) D B C E A (5) E B C A D (4) D B E C A (4) A C E B D (4) E A D B C (3) B C E D A (3) A E D B C (3) A E C D B (3) A E C B D (3) D E B A C (2) D E A B C (2) D A E B C (2) C B E A D (2) C B D A E (2) C B A E D (2) C A B E D (2) B E C A D (2) A C D B E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B A D (1) E B D C A (1) E B A C D (1) D A E C B (1) C D B A E (1) C D A B E (1) C B D E A (1) C B A D E (1) C A D B E (1) B E D C A (1) B E C D A (1) B D C E A (1) B C E A D (1) A D E B C (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 2 4 -2 B -2 0 6 -8 0 C -2 -6 0 6 -2 D -4 8 -6 0 -8 E 2 0 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.310204 C: 0.000000 D: 0.000000 E: 0.689796 Sum of squares = 0.57204471958 Cumulative probabilities = A: 0.000000 B: 0.310204 C: 0.310204 D: 0.310204 E: 1.000000 A B C D E A 0 2 2 4 -2 B -2 0 6 -8 0 C -2 -6 0 6 -2 D -4 8 -6 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499781 C: 0.000000 D: 0.000000 E: 0.500219 Sum of squares = 0.500000096187 Cumulative probabilities = A: 0.000000 B: 0.499781 C: 0.499781 D: 0.499781 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 E=20 B=15 C=13 so C is eliminated. Round 2 votes counts: D=29 A=28 B=23 E=20 so E is eliminated. Round 3 votes counts: A=38 D=32 B=30 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:203 B:198 C:198 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 4 -2 B -2 0 6 -8 0 C -2 -6 0 6 -2 D -4 8 -6 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499781 C: 0.000000 D: 0.000000 E: 0.500219 Sum of squares = 0.500000096187 Cumulative probabilities = A: 0.000000 B: 0.499781 C: 0.499781 D: 0.499781 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 -2 B -2 0 6 -8 0 C -2 -6 0 6 -2 D -4 8 -6 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499781 C: 0.000000 D: 0.000000 E: 0.500219 Sum of squares = 0.500000096187 Cumulative probabilities = A: 0.000000 B: 0.499781 C: 0.499781 D: 0.499781 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 -2 B -2 0 6 -8 0 C -2 -6 0 6 -2 D -4 8 -6 0 -8 E 2 0 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499781 C: 0.000000 D: 0.000000 E: 0.500219 Sum of squares = 0.500000096187 Cumulative probabilities = A: 0.000000 B: 0.499781 C: 0.499781 D: 0.499781 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7362: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) A B C E D (9) D E C B A (7) B A D C E (7) E D C A B (5) D E C A B (5) D C E A B (5) D B E A C (4) C E D A B (4) E D C B A (3) D E B C A (3) C E A B D (3) B A D E C (3) B A C D E (3) E C A B D (2) D B A E C (2) C E A D B (2) C D E A B (2) C A E B D (2) B D A E C (2) B A C E D (2) A C B E D (2) E D B C A (1) E C B A D (1) E C A D B (1) E B D A C (1) E B C A D (1) D C E B A (1) D C A E B (1) D C A B E (1) D B A C E (1) C A D B E (1) B E A D C (1) B A E D C (1) B A E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -20 -14 -22 B -10 0 -14 -16 -18 C 20 14 0 -10 -6 D 14 16 10 0 -2 E 22 18 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999642 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -20 -14 -22 B -10 0 -14 -16 -18 C 20 14 0 -10 -6 D 14 16 10 0 -2 E 22 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=24 B=20 C=14 A=12 so A is eliminated. Round 2 votes counts: D=30 B=30 E=24 C=16 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:219 C:209 A:177 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -20 -14 -22 B -10 0 -14 -16 -18 C 20 14 0 -10 -6 D 14 16 10 0 -2 E 22 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -20 -14 -22 B -10 0 -14 -16 -18 C 20 14 0 -10 -6 D 14 16 10 0 -2 E 22 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -20 -14 -22 B -10 0 -14 -16 -18 C 20 14 0 -10 -6 D 14 16 10 0 -2 E 22 18 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7363: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C A E (9) D B A C E (9) E C A B D (8) D B C E A (7) E D A C B (5) E A D C B (4) B A C D E (4) E A C D B (3) D E B C A (3) D E A B C (3) A C B E D (3) A B C E D (3) A B C D E (3) D E C B A (2) D B E C A (2) C B A E D (2) B D C A E (2) B C D A E (2) B C A D E (2) A E C B D (2) E D C A B (1) E C D B A (1) D B A E C (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C A B E D (1) B A D C E (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 4 12 0 -2 B -4 0 10 -8 6 C -12 -10 0 -6 4 D 0 8 6 0 8 E 2 -6 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.486890 B: 0.000000 C: 0.000000 D: 0.513110 E: 0.000000 Sum of squares = 0.500343711427 Cumulative probabilities = A: 0.486890 B: 0.486890 C: 0.486890 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 0 -2 B -4 0 10 -8 6 C -12 -10 0 -6 4 D 0 8 6 0 8 E 2 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=32 A=13 B=11 C=5 so C is eliminated. Round 2 votes counts: D=39 E=34 A=14 B=13 so B is eliminated. Round 3 votes counts: D=43 E=34 A=23 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:211 A:207 B:202 E:192 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 12 0 -2 B -4 0 10 -8 6 C -12 -10 0 -6 4 D 0 8 6 0 8 E 2 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 0 -2 B -4 0 10 -8 6 C -12 -10 0 -6 4 D 0 8 6 0 8 E 2 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 0 -2 B -4 0 10 -8 6 C -12 -10 0 -6 4 D 0 8 6 0 8 E 2 -6 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7364: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (10) E D B C A (6) E B C D A (6) C B E D A (6) A C B D E (6) D A B E C (5) C A B E D (5) E D B A C (4) A C D B E (4) E C B A D (3) D B E C A (3) D B A E C (3) C E B A D (3) C A E B D (3) A D B C E (3) D E B A C (2) D A E B C (2) C E B D A (2) C B D A E (2) B D E C A (2) B C E D A (2) A D E B C (2) A C B E D (2) E C B D A (1) D E B C A (1) D E A B C (1) D B E A C (1) D B C A E (1) D B A C E (1) D A B C E (1) C B E A D (1) C B A D E (1) C A B D E (1) B E D C A (1) A E C D B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -24 -20 -26 -10 B 24 0 14 16 4 C 20 -14 0 0 -8 D 26 -16 0 0 -14 E 10 -4 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -20 -26 -10 B 24 0 14 16 4 C 20 -14 0 0 -8 D 26 -16 0 0 -14 E 10 -4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=24 D=21 A=20 B=5 so B is eliminated. Round 2 votes counts: E=31 C=26 D=23 A=20 so A is eliminated. Round 3 votes counts: C=39 E=32 D=29 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:229 E:214 C:199 D:198 A:160 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -20 -26 -10 B 24 0 14 16 4 C 20 -14 0 0 -8 D 26 -16 0 0 -14 E 10 -4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -20 -26 -10 B 24 0 14 16 4 C 20 -14 0 0 -8 D 26 -16 0 0 -14 E 10 -4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -20 -26 -10 B 24 0 14 16 4 C 20 -14 0 0 -8 D 26 -16 0 0 -14 E 10 -4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992082 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7365: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) E A C D B (7) C B D E A (7) E A C B D (5) A E C B D (5) E A D C B (4) D A E B C (4) B D A C E (4) D B A E C (3) C E A B D (3) C D B E A (3) B D C E A (3) B D C A E (3) B C A E D (3) A E D C B (3) A E C D B (3) E C A D B (2) D E A C B (2) D B A C E (2) C A E B D (2) B D A E C (2) B C D A E (2) A E D B C (2) E D C A B (1) E C A B D (1) D C B E A (1) D B C A E (1) C E D B A (1) C E B A D (1) C B E D A (1) C B E A D (1) C B A E D (1) B C D E A (1) B A D E C (1) B A D C E (1) B A C E D (1) A E B D C (1) A E B C D (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 8 2 -4 B 2 0 -10 6 4 C -8 10 0 4 2 D -2 -6 -4 0 -2 E 4 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 A B C D E A 0 -2 8 2 -4 B 2 0 -10 6 4 C -8 10 0 4 2 D -2 -6 -4 0 -2 E 4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428561 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=21 B=21 E=20 C=20 A=18 so A is eliminated. Round 2 votes counts: E=35 B=23 D=21 C=21 so D is eliminated. Round 3 votes counts: E=41 B=37 C=22 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:204 A:202 B:201 E:200 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 2 -4 B 2 0 -10 6 4 C -8 10 0 4 2 D -2 -6 -4 0 -2 E 4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428561 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 2 -4 B 2 0 -10 6 4 C -8 10 0 4 2 D -2 -6 -4 0 -2 E 4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428561 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 2 -4 B 2 0 -10 6 4 C -8 10 0 4 2 D -2 -6 -4 0 -2 E 4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428561 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7366: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) A C B D E (10) B E C A D (9) B A C E D (8) D A C E B (7) D E A C B (6) E B D C A (5) B C A E D (5) D E C A B (4) B E A C D (4) D C A E B (3) A D C B E (3) A C D B E (3) D E A B C (2) C D E A B (2) C B A E D (2) C A D B E (2) C A B E D (2) E D C B A (1) E D B A C (1) E B D A C (1) E B C D A (1) D E C B A (1) D A C B E (1) C E A B D (1) C B E A D (1) B E D C A (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -4 4 -4 B 6 0 2 4 6 C 4 -2 0 4 2 D -4 -4 -4 0 -12 E 4 -6 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 4 -4 B 6 0 2 4 6 C 4 -2 0 4 2 D -4 -4 -4 0 -12 E 4 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=24 E=21 A=17 C=10 so C is eliminated. Round 2 votes counts: B=31 D=26 E=22 A=21 so A is eliminated. Round 3 votes counts: B=44 D=34 E=22 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 C:204 E:204 A:195 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 4 -4 B 6 0 2 4 6 C 4 -2 0 4 2 D -4 -4 -4 0 -12 E 4 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 4 -4 B 6 0 2 4 6 C 4 -2 0 4 2 D -4 -4 -4 0 -12 E 4 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 4 -4 B 6 0 2 4 6 C 4 -2 0 4 2 D -4 -4 -4 0 -12 E 4 -6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7367: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E A D (10) E B C D A (9) A D C B E (7) C B A E D (5) D A E C B (4) B C E A D (4) A D E B C (4) A B C E D (4) E C B D A (3) C E B D A (3) C B E D A (3) C A B D E (3) A D B E C (3) A D B C E (3) E D B C A (2) E B A D C (2) D E B A C (2) D A C E B (2) A D C E B (2) E D B A C (1) E D A B C (1) E B C A D (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E A B (1) C E D B A (1) C B D A E (1) C A B E D (1) B E C D A (1) B E C A D (1) B A E C D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -4 6 4 B 4 0 2 8 -2 C 4 -2 0 4 2 D -6 -8 -4 0 -8 E -4 2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333323 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -4 -4 6 4 B 4 0 2 8 -2 C 4 -2 0 4 2 D -6 -8 -4 0 -8 E -4 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=25 D=22 E=19 B=7 so B is eliminated. Round 2 votes counts: C=31 A=26 D=22 E=21 so E is eliminated. Round 3 votes counts: C=46 A=28 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:206 C:204 E:202 A:201 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 6 4 B 4 0 2 8 -2 C 4 -2 0 4 2 D -6 -8 -4 0 -8 E -4 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 6 4 B 4 0 2 8 -2 C 4 -2 0 4 2 D -6 -8 -4 0 -8 E -4 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 6 4 B 4 0 2 8 -2 C 4 -2 0 4 2 D -6 -8 -4 0 -8 E -4 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7368: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) E C D B A (7) E B C D A (6) C E D A B (6) C D A E B (6) A D C B E (6) B A E D C (5) A D B C E (5) D A C E B (4) C E D B A (4) C D E A B (4) C A D B E (4) B E C A D (4) B A D E C (4) A C D B E (4) A B D C E (4) D C A E B (3) C E B D A (3) E C D A B (1) D A C B E (1) D A B E C (1) C B A E D (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B E A C D (1) B D A E C (1) A D B E C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -18 -12 8 B -8 0 -28 -16 -6 C 18 28 0 24 16 D 12 16 -24 0 4 E -8 6 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 -12 8 B -8 0 -28 -16 -6 C 18 28 0 24 16 D 12 16 -24 0 4 E -8 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=22 A=22 B=17 D=9 so D is eliminated. Round 2 votes counts: C=33 A=28 E=22 B=17 so B is eliminated. Round 3 votes counts: A=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:243 D:204 A:193 E:189 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 -12 8 B -8 0 -28 -16 -6 C 18 28 0 24 16 D 12 16 -24 0 4 E -8 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -12 8 B -8 0 -28 -16 -6 C 18 28 0 24 16 D 12 16 -24 0 4 E -8 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -12 8 B -8 0 -28 -16 -6 C 18 28 0 24 16 D 12 16 -24 0 4 E -8 6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7369: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) B D E A C (6) C D A B E (5) E C B A D (4) E C A B D (4) C E B D A (4) C A E D B (4) B E A D C (4) E B C A D (3) D B E A C (3) D B A E C (3) C D B E A (3) B E D A C (3) B D A E C (3) A D B E C (3) E B C D A (2) E B A C D (2) C E D B A (2) C E B A D (2) C E A D B (2) C D E B A (2) C A D E B (2) A C D E B (2) A B E D C (2) E B D A C (1) E A C B D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B A C E (1) D A B E C (1) D A B C E (1) C D E A B (1) C D B A E (1) C D A E B (1) C A E B D (1) B D E C A (1) B C D E A (1) B A E D C (1) B A D E C (1) A E C B D (1) A D B C E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -14 6 -24 B 8 0 -10 18 -6 C 14 10 0 24 -2 D -6 -18 -24 0 -10 E 24 6 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -14 6 -24 B 8 0 -10 18 -6 C 14 10 0 24 -2 D -6 -18 -24 0 -10 E 24 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 B=20 E=18 D=11 A=11 so D is eliminated. Round 2 votes counts: C=42 B=27 E=18 A=13 so A is eliminated. Round 3 votes counts: C=45 B=36 E=19 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:223 E:221 B:205 A:180 D:171 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 6 -24 B 8 0 -10 18 -6 C 14 10 0 24 -2 D -6 -18 -24 0 -10 E 24 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 6 -24 B 8 0 -10 18 -6 C 14 10 0 24 -2 D -6 -18 -24 0 -10 E 24 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 6 -24 B 8 0 -10 18 -6 C 14 10 0 24 -2 D -6 -18 -24 0 -10 E 24 6 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7370: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) C E A D B (6) C D E A B (5) A E C B D (5) D B C E A (4) D B C A E (4) B E A D C (4) B D E C A (4) B D E A C (4) A D B E C (4) E A C B D (3) D A C B E (3) C E B D A (3) A E B C D (3) E C B A D (2) E C A B D (2) E B C A D (2) E B A C D (2) E A B C D (2) D C A B E (2) D B A E C (2) C E B A D (2) C D B E A (2) C D A E B (2) C A E D B (2) C A D E B (2) B E D C A (2) B A D E C (2) A E C D B (2) A D C E B (2) A C E D B (2) D B A C E (1) D A B C E (1) C E D A B (1) B E C D A (1) B D A E C (1) B C E D A (1) B A E D C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -10 2 -16 B 0 0 -10 -8 -2 C 10 10 0 6 4 D -2 8 -6 0 4 E 16 2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 2 -16 B 0 0 -10 -8 -2 C 10 10 0 6 4 D -2 8 -6 0 4 E 16 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=23 B=20 A=19 E=13 so E is eliminated. Round 2 votes counts: C=29 B=24 A=24 D=23 so D is eliminated. Round 3 votes counts: C=37 B=35 A=28 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:205 D:202 B:190 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 2 -16 B 0 0 -10 -8 -2 C 10 10 0 6 4 D -2 8 -6 0 4 E 16 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 2 -16 B 0 0 -10 -8 -2 C 10 10 0 6 4 D -2 8 -6 0 4 E 16 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 2 -16 B 0 0 -10 -8 -2 C 10 10 0 6 4 D -2 8 -6 0 4 E 16 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7371: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (16) E B D A C (14) E B D C A (6) E A B C D (6) A C E B D (6) A C D B E (6) C D B A E (5) A E C B D (5) D B C E A (4) D B C A E (4) E A B D C (3) D B E C A (3) D B E A C (3) C D B E A (3) C A E B D (3) B D E C A (3) B D E A C (2) E C B D A (1) E B C D A (1) D C B A E (1) C D A B E (1) C B D E A (1) A E B D C (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -4 -4 0 B 2 0 2 6 4 C 4 -2 0 12 2 D 4 -6 -12 0 4 E 0 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 0 B 2 0 2 6 4 C 4 -2 0 12 2 D 4 -6 -12 0 4 E 0 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=29 A=20 D=15 B=5 so B is eliminated. Round 2 votes counts: E=31 C=29 D=20 A=20 so D is eliminated. Round 3 votes counts: E=42 C=38 A=20 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:207 A:195 D:195 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 0 B 2 0 2 6 4 C 4 -2 0 12 2 D 4 -6 -12 0 4 E 0 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 0 B 2 0 2 6 4 C 4 -2 0 12 2 D 4 -6 -12 0 4 E 0 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 0 B 2 0 2 6 4 C 4 -2 0 12 2 D 4 -6 -12 0 4 E 0 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7372: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) B C A D E (5) A D E B C (5) C E B D A (4) B C E A D (4) B C D A E (4) E D A C B (3) E C B D A (3) E B C A D (3) D E A C B (3) D A C B E (3) C B D E A (3) C B D A E (3) A D B E C (3) A B E D C (3) E C B A D (2) E B A C D (2) E A D B C (2) E A B C D (2) D C E A B (2) D A B C E (2) C E D B A (2) C D B A E (2) B E C A D (2) B C A E D (2) B A C D E (2) A E D B C (2) A B E C D (2) A B D C E (2) E D C A B (1) E C D A B (1) D E C A B (1) D C B A E (1) D C A B E (1) D A E C B (1) C D E B A (1) C D B E A (1) B E A C D (1) B D C A E (1) B D A C E (1) B C E D A (1) B C D E A (1) B A D C E (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -18 -16 -6 -4 B 18 0 8 24 16 C 16 -8 0 20 12 D 6 -24 -20 0 0 E 4 -16 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 -6 -4 B 18 0 8 24 16 C 16 -8 0 20 12 D 6 -24 -20 0 0 E 4 -16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 E=19 A=19 D=14 so D is eliminated. Round 2 votes counts: C=27 B=25 A=25 E=23 so E is eliminated. Round 3 votes counts: C=35 A=35 B=30 so B is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:233 C:220 E:188 D:181 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 -6 -4 B 18 0 8 24 16 C 16 -8 0 20 12 D 6 -24 -20 0 0 E 4 -16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -6 -4 B 18 0 8 24 16 C 16 -8 0 20 12 D 6 -24 -20 0 0 E 4 -16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -6 -4 B 18 0 8 24 16 C 16 -8 0 20 12 D 6 -24 -20 0 0 E 4 -16 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7373: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) C B A D E (6) E D A B C (5) C D A B E (5) E C D B A (4) C E D B A (4) B A E D C (4) A B D C E (4) E C B D A (3) E B C A D (3) D A C B E (3) C D E A B (3) B E C A D (3) B A E C D (3) B A C D E (3) E D C A B (2) D E A C B (2) D C A B E (2) C D A E B (2) C B E D A (2) C B E A D (2) C A D B E (2) A D B C E (2) A B E D C (2) E D C B A (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D B C (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B E C (1) C B A E D (1) C A B D E (1) B E A D C (1) B E A C D (1) B C A E D (1) B A D C E (1) A E B D C (1) A D E B C (1) A D B E C (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -12 -2 18 B -8 0 -12 -4 8 C 12 12 0 6 10 D 2 4 -6 0 6 E -18 -8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 -2 18 B -8 0 -12 -4 8 C 12 12 0 6 10 D 2 4 -6 0 6 E -18 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=22 D=19 B=17 A=14 so A is eliminated. Round 2 votes counts: C=29 B=25 E=23 D=23 so E is eliminated. Round 3 votes counts: C=36 D=32 B=32 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:206 D:203 B:192 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 -2 18 B -8 0 -12 -4 8 C 12 12 0 6 10 D 2 4 -6 0 6 E -18 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 -2 18 B -8 0 -12 -4 8 C 12 12 0 6 10 D 2 4 -6 0 6 E -18 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 -2 18 B -8 0 -12 -4 8 C 12 12 0 6 10 D 2 4 -6 0 6 E -18 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7374: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A D C E B (9) D A B C E (8) E C B A D (7) D B A E C (6) D A B E C (5) C E A B D (5) B D A E C (5) E C A B D (4) C E A D B (4) B E C A D (4) D A C E B (3) B D E C A (3) E C B D A (2) D E A C B (2) B D E A C (2) B A D C E (2) B A C D E (2) A C D E B (2) E D C B A (1) E D C A B (1) E C D A B (1) E C A D B (1) D E B C A (1) D B A C E (1) D A E C B (1) C E B A D (1) C A E D B (1) B E D C A (1) B D A C E (1) B C E A D (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 2 -10 -4 B 2 0 8 2 8 C -2 -8 0 -8 -16 D 10 -2 8 0 10 E 4 -8 16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -10 -4 B 2 0 8 2 8 C -2 -8 0 -8 -16 D 10 -2 8 0 10 E 4 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=27 E=17 A=13 C=11 so C is eliminated. Round 2 votes counts: B=32 E=27 D=27 A=14 so A is eliminated. Round 3 votes counts: D=40 B=32 E=28 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:210 E:201 A:193 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -10 -4 B 2 0 8 2 8 C -2 -8 0 -8 -16 D 10 -2 8 0 10 E 4 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -10 -4 B 2 0 8 2 8 C -2 -8 0 -8 -16 D 10 -2 8 0 10 E 4 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -10 -4 B 2 0 8 2 8 C -2 -8 0 -8 -16 D 10 -2 8 0 10 E 4 -8 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7375: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) D B C E A (6) C D B E A (6) B D C A E (5) B C D A E (5) A E B D C (5) E D C A B (4) C D E A B (4) C D B A E (4) E C D A B (3) E A B D C (3) D C E B A (3) D C B E A (3) B A E D C (3) E A D C B (2) E A C D B (2) C D E B A (2) C B A D E (2) B D E A C (2) B D C E A (2) B D A C E (2) B A D C E (2) B A C D E (2) A E B C D (2) A B E C D (2) E D C B A (1) D E C B A (1) C E D A B (1) C A D E B (1) C A B D E (1) B E D A C (1) B D E C A (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -10 -12 6 B 6 0 8 8 24 C 10 -8 0 -12 12 D 12 -8 12 0 12 E -6 -24 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999515 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -12 6 B 6 0 8 8 24 C 10 -8 0 -12 12 D 12 -8 12 0 12 E -6 -24 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=25 C=21 E=15 D=13 so D is eliminated. Round 2 votes counts: B=31 C=27 A=26 E=16 so E is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:223 D:214 C:201 A:189 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 -12 6 B 6 0 8 8 24 C 10 -8 0 -12 12 D 12 -8 12 0 12 E -6 -24 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -12 6 B 6 0 8 8 24 C 10 -8 0 -12 12 D 12 -8 12 0 12 E -6 -24 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -12 6 B 6 0 8 8 24 C 10 -8 0 -12 12 D 12 -8 12 0 12 E -6 -24 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999409 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7376: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B A C D E (7) E D C A B (6) D A E C B (5) B A D C E (5) E D C B A (4) B C A E D (4) A D C E B (4) A B D C E (4) E D A C B (3) D A E B C (3) C E B D A (3) C B A E D (3) B E C D A (3) B C E A D (3) B A D E C (3) E D B C A (2) E C D A B (2) E C B D A (2) E B C D A (2) C E D A B (2) C B E A D (2) B C A D E (2) A B C D E (2) E C D B A (1) E B D C A (1) D E C A B (1) D A C E B (1) C E D B A (1) C B E D A (1) C B A D E (1) B E D A C (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 8 -8 0 B 2 0 -12 0 -12 C -8 12 0 -14 -4 D 8 0 14 0 8 E 0 12 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.246560 C: 0.000000 D: 0.753440 E: 0.000000 Sum of squares = 0.628463281301 Cumulative probabilities = A: 0.000000 B: 0.246560 C: 0.246560 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -8 0 B 2 0 -12 0 -12 C -8 12 0 -14 -4 D 8 0 14 0 8 E 0 12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000003355 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=23 D=20 A=16 C=13 so C is eliminated. Round 2 votes counts: B=35 E=29 D=20 A=16 so A is eliminated. Round 3 votes counts: B=42 E=29 D=29 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:215 E:204 A:199 C:193 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 -8 0 B 2 0 -12 0 -12 C -8 12 0 -14 -4 D 8 0 14 0 8 E 0 12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000003355 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -8 0 B 2 0 -12 0 -12 C -8 12 0 -14 -4 D 8 0 14 0 8 E 0 12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000003355 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -8 0 B 2 0 -12 0 -12 C -8 12 0 -14 -4 D 8 0 14 0 8 E 0 12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000003355 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7377: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) E C A B D (6) D B A C E (6) A B D C E (6) E D C B A (5) A E C B D (5) A D B E C (5) A B C D E (5) E C D B A (3) E C A D B (3) D B E C A (3) D B C E A (3) C A E B D (3) A E C D B (3) A C E B D (3) A C B E D (3) E C D A B (2) E C B D A (2) E A C D B (2) D B A E C (2) D A B E C (2) C E B D A (2) C E A B D (2) C B A E D (2) B D C A E (2) B C D A E (2) A C B D E (2) E D A C B (1) D E B C A (1) D B C A E (1) D A E B C (1) C B D A E (1) C B A D E (1) C A B E D (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 26 8 20 28 B -26 0 -4 -2 10 C -8 4 0 8 6 D -20 2 -8 0 4 E -28 -10 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 8 20 28 B -26 0 -4 -2 10 C -8 4 0 8 6 D -20 2 -8 0 4 E -28 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996161 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 E=24 D=19 C=12 B=5 so B is eliminated. Round 2 votes counts: A=40 E=24 D=21 C=15 so C is eliminated. Round 3 votes counts: A=47 E=28 D=25 so D is eliminated. Round 4 votes counts: A=64 E=36 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:241 C:205 B:189 D:189 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 8 20 28 B -26 0 -4 -2 10 C -8 4 0 8 6 D -20 2 -8 0 4 E -28 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996161 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 8 20 28 B -26 0 -4 -2 10 C -8 4 0 8 6 D -20 2 -8 0 4 E -28 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996161 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 8 20 28 B -26 0 -4 -2 10 C -8 4 0 8 6 D -20 2 -8 0 4 E -28 -10 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996161 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7378: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) C B A D E (11) B C D A E (10) E A D C B (6) C B E A D (5) E C B A D (4) D A E B C (4) B C E D A (4) C B D A E (3) B C D E A (3) A E D C B (3) A D C E B (3) E D B A C (2) E C A B D (2) E A C D B (2) D E A B C (2) D A B C E (2) C B E D A (2) C A B D E (2) A D E C B (2) A D C B E (2) A D B C E (2) A C D B E (2) E B D C A (1) E B C D A (1) E A D B C (1) D B A C E (1) D A B E C (1) C E A B D (1) C B A E D (1) C A E B D (1) B E C D A (1) B D C E A (1) B D C A E (1) Total count = 100 A B C D E A 0 -2 -8 0 2 B 2 0 -4 8 8 C 8 4 0 10 14 D 0 -8 -10 0 4 E -2 -8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999338 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 0 2 B 2 0 -4 8 8 C 8 4 0 10 14 D 0 -8 -10 0 4 E -2 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 B=20 A=14 D=10 so D is eliminated. Round 2 votes counts: E=32 C=26 B=21 A=21 so B is eliminated. Round 3 votes counts: C=45 E=33 A=22 so A is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:207 A:196 D:193 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 0 2 B 2 0 -4 8 8 C 8 4 0 10 14 D 0 -8 -10 0 4 E -2 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 0 2 B 2 0 -4 8 8 C 8 4 0 10 14 D 0 -8 -10 0 4 E -2 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 0 2 B 2 0 -4 8 8 C 8 4 0 10 14 D 0 -8 -10 0 4 E -2 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7379: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) E D C A B (9) A B C E D (8) B A C D E (6) D E C A B (5) C E D A B (5) B A C E D (5) A B E D C (5) E D A C B (4) E A D C B (4) B C A D E (4) E A D B C (3) D C E B A (3) C B A D E (3) C A B D E (3) C A E D B (2) B C D A E (2) B A E D C (2) A C B E D (2) E C A D B (1) D E B C A (1) D B C E A (1) C D E B A (1) C D E A B (1) C D B E A (1) C B D E A (1) C B D A E (1) B D E A C (1) B D A C E (1) B C D E A (1) A E D C B (1) A E C D B (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 12 -10 4 -4 B -12 0 -18 -6 -6 C 10 18 0 0 4 D -4 6 0 0 -8 E 4 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.822593 D: 0.177407 E: 0.000000 Sum of squares = 0.708133060856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.822593 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 4 -4 B -12 0 -18 -6 -6 C 10 18 0 0 4 D -4 6 0 0 -8 E 4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555624312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 E=21 D=20 A=19 C=18 so C is eliminated. Round 2 votes counts: B=27 E=26 A=24 D=23 so D is eliminated. Round 3 votes counts: E=47 B=29 A=24 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:216 E:207 A:201 D:197 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 4 -4 B -12 0 -18 -6 -6 C 10 18 0 0 4 D -4 6 0 0 -8 E 4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555624312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 4 -4 B -12 0 -18 -6 -6 C 10 18 0 0 4 D -4 6 0 0 -8 E 4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555624312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 4 -4 B -12 0 -18 -6 -6 C 10 18 0 0 4 D -4 6 0 0 -8 E 4 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555624312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7380: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) B E D C A (7) A E C D B (5) A C E D B (5) A C D E B (5) A C D B E (5) B D E C A (4) E B C D A (3) E B A C D (3) E A C D B (3) E A C B D (3) B D C A E (3) A D B C E (3) E C D B A (2) E C A D B (2) E A B C D (2) D C E B A (2) C D E A B (2) C A E D B (2) C A D B E (2) B D E A C (2) B D A C E (2) B A D C E (2) A B D C E (2) E D C B A (1) E C D A B (1) E C B D A (1) D C B A E (1) D C A B E (1) D A C B E (1) C E D A B (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D C E A (1) B A E D C (1) B A D E C (1) A E B C D (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 0 2 -8 B 0 0 2 6 -16 C 0 -2 0 2 -12 D -2 -6 -2 0 -14 E 8 16 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 0 2 -8 B 0 0 2 6 -16 C 0 -2 0 2 -12 D -2 -6 -2 0 -14 E 8 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=28 B=25 C=9 D=5 so D is eliminated. Round 2 votes counts: E=33 A=29 B=25 C=13 so C is eliminated. Round 3 votes counts: E=38 A=36 B=26 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:197 B:196 C:194 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 2 -8 B 0 0 2 6 -16 C 0 -2 0 2 -12 D -2 -6 -2 0 -14 E 8 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 2 -8 B 0 0 2 6 -16 C 0 -2 0 2 -12 D -2 -6 -2 0 -14 E 8 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 2 -8 B 0 0 2 6 -16 C 0 -2 0 2 -12 D -2 -6 -2 0 -14 E 8 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7381: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) E C D A B (8) E A D B C (6) E A B D C (6) C B D A E (6) B A E D C (6) B C D A E (5) C D E A B (4) A D B E C (4) E D A C B (3) E B C A D (3) E A D C B (3) D A B C E (3) C D B A E (3) B A D C E (3) E B A C D (2) C E D A B (2) C B D E A (2) B E A D C (2) B C A D E (2) A D E B C (2) E C B D A (1) E A C D B (1) D C E A B (1) D B C A E (1) D A C B E (1) C E B D A (1) C D B E A (1) C D A E B (1) C D A B E (1) C B E D A (1) B D C A E (1) B C E A D (1) A E D B C (1) A E B D C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 10 8 4 B 0 0 20 8 8 C -10 -20 0 -10 -20 D -8 -8 10 0 2 E -4 -8 20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.614312 B: 0.385688 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.526134293398 Cumulative probabilities = A: 0.614312 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 8 4 B 0 0 20 8 8 C -10 -20 0 -10 -20 D -8 -8 10 0 2 E -4 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=29 C=22 A=10 D=6 so D is eliminated. Round 2 votes counts: E=33 B=30 C=23 A=14 so A is eliminated. Round 3 votes counts: B=39 E=37 C=24 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:211 E:203 D:198 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 8 4 B 0 0 20 8 8 C -10 -20 0 -10 -20 D -8 -8 10 0 2 E -4 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 8 4 B 0 0 20 8 8 C -10 -20 0 -10 -20 D -8 -8 10 0 2 E -4 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 8 4 B 0 0 20 8 8 C -10 -20 0 -10 -20 D -8 -8 10 0 2 E -4 -8 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7382: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (14) E D C A B (13) A B E D C (11) C D E B A (7) C D B A E (6) D E C A B (5) D C E B A (5) C D B E A (5) B A E C D (5) B A C E D (5) D C E A B (4) C B D A E (3) B C A D E (3) A B E C D (3) E D A B C (2) B A E D C (2) A E B D C (2) E D A C B (1) E A D C B (1) E A B D C (1) C D E A B (1) B C D A E (1) Total count = 100 A B C D E A 0 -12 -6 -6 10 B 12 0 -2 0 16 C 6 2 0 6 8 D 6 0 -6 0 8 E -10 -16 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -6 10 B 12 0 -2 0 16 C 6 2 0 6 8 D 6 0 -6 0 8 E -10 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=22 E=18 A=16 D=14 so D is eliminated. Round 2 votes counts: C=31 B=30 E=23 A=16 so A is eliminated. Round 3 votes counts: B=44 C=31 E=25 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:213 C:211 D:204 A:193 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -6 -6 10 B 12 0 -2 0 16 C 6 2 0 6 8 D 6 0 -6 0 8 E -10 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -6 10 B 12 0 -2 0 16 C 6 2 0 6 8 D 6 0 -6 0 8 E -10 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -6 10 B 12 0 -2 0 16 C 6 2 0 6 8 D 6 0 -6 0 8 E -10 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7383: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) C D A E B (6) C A B E D (6) B A C E D (6) E B D A C (5) D E C A B (5) A B C D E (5) D E B A C (4) B A D C E (4) E D B A C (3) E B D C A (3) D E B C A (3) D A C B E (3) B A C D E (3) E D B C A (2) E B C A D (2) D C E A B (2) D C A E B (2) D B E A C (2) D B A E C (2) B E A C D (2) B A E C D (2) A C B D E (2) E D C A B (1) E C D B A (1) E C B A D (1) E C A B D (1) D E A C B (1) D E A B C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A D B (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B D E (1) B C A E D (1) B A D E C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 2 -12 4 B 12 0 6 -4 -8 C -2 -6 0 -10 0 D 12 4 10 0 6 E -4 8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -12 4 B 12 0 6 -4 -8 C -2 -6 0 -10 0 D 12 4 10 0 6 E -4 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=19 C=18 A=9 so A is eliminated. Round 2 votes counts: D=29 E=26 B=24 C=21 so C is eliminated. Round 3 votes counts: D=37 B=34 E=29 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:203 E:199 A:191 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 2 -12 4 B 12 0 6 -4 -8 C -2 -6 0 -10 0 D 12 4 10 0 6 E -4 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -12 4 B 12 0 6 -4 -8 C -2 -6 0 -10 0 D 12 4 10 0 6 E -4 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -12 4 B 12 0 6 -4 -8 C -2 -6 0 -10 0 D 12 4 10 0 6 E -4 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7384: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) D B E C A (7) A C B E D (7) A C B D E (7) A B C D E (5) D B E A C (4) A C D B E (4) A B C E D (4) E D B C A (3) C E D A B (3) C E A D B (3) B E D A C (3) B A D E C (3) E D C B A (2) E C D B A (2) E C A B D (2) E B D C A (2) D E C B A (2) D C A B E (2) D A B C E (2) C E A B D (2) C A E B D (2) B D E A C (2) B A E C D (2) B A D C E (2) A C E B D (2) A B D C E (2) E B D A C (1) E B C A D (1) D C A E B (1) D B A E C (1) D B A C E (1) C D A E B (1) B E A D C (1) B D A E C (1) B D A C E (1) B A E D C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 14 2 4 B 0 0 14 6 24 C -14 -14 0 -4 4 D -2 -6 4 0 14 E -4 -24 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.619656 B: 0.380344 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.528634945401 Cumulative probabilities = A: 0.619656 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 2 4 B 0 0 14 6 24 C -14 -14 0 -4 4 D -2 -6 4 0 14 E -4 -24 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=28 B=16 E=13 C=11 so C is eliminated. Round 2 votes counts: A=34 D=29 E=21 B=16 so B is eliminated. Round 3 votes counts: A=42 D=33 E=25 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:222 A:210 D:205 C:186 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 2 4 B 0 0 14 6 24 C -14 -14 0 -4 4 D -2 -6 4 0 14 E -4 -24 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 2 4 B 0 0 14 6 24 C -14 -14 0 -4 4 D -2 -6 4 0 14 E -4 -24 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 2 4 B 0 0 14 6 24 C -14 -14 0 -4 4 D -2 -6 4 0 14 E -4 -24 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7385: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (6) D B E C A (5) A B D E C (5) E C D A B (4) B D E A C (4) B D C E A (4) A C E B D (4) A C B E D (4) E D C A B (3) E D A C B (3) D E B C A (3) D E B A C (3) D B E A C (3) C E A D B (3) C B E D A (3) C A E D B (3) B D E C A (3) B D A E C (3) B A D E C (3) A C E D B (3) E C A D B (2) D E C B A (2) C B A E D (2) B D A C E (2) A E C D B (2) A D E B C (2) E D C B A (1) E A D C B (1) E A C D B (1) D B A E C (1) C E D A B (1) C E B D A (1) C E A B D (1) C A B E D (1) B D C A E (1) B C D E A (1) B C A D E (1) B A D C E (1) A E C B D (1) A D B E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -2 -2 -4 B -6 0 -4 6 0 C 2 4 0 -8 -12 D 2 -6 8 0 -2 E 4 0 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.268614 C: 0.000000 D: 0.000000 E: 0.731386 Sum of squares = 0.607078822953 Cumulative probabilities = A: 0.000000 B: 0.268614 C: 0.268614 D: 0.268614 E: 1.000000 A B C D E A 0 6 -2 -2 -4 B -6 0 -4 6 0 C 2 4 0 -8 -12 D 2 -6 8 0 -2 E 4 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000017718 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 C=21 D=17 E=15 so E is eliminated. Round 2 votes counts: C=27 A=26 D=24 B=23 so B is eliminated. Round 3 votes counts: D=41 A=30 C=29 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:209 D:201 A:199 B:198 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 -2 -4 B -6 0 -4 6 0 C 2 4 0 -8 -12 D 2 -6 8 0 -2 E 4 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000017718 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 -4 B -6 0 -4 6 0 C 2 4 0 -8 -12 D 2 -6 8 0 -2 E 4 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000017718 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 -4 B -6 0 -4 6 0 C 2 4 0 -8 -12 D 2 -6 8 0 -2 E 4 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000017718 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7386: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (15) A D C E B (11) B E C A D (10) B E C D A (7) E C B A D (5) E B C A D (5) E C A B D (4) D A E C B (4) A C D E B (4) D A C B E (3) C E A B D (3) B D E C A (3) B D C E A (3) D B A E C (2) D A B E C (2) D A B C E (2) C E B A D (2) C A E B D (2) B D C A E (2) A E C D B (2) E A C D B (1) E A C B D (1) D B A C E (1) C B E A D (1) C A B E D (1) B D E A C (1) B C E A D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 14 2 10 6 B -14 0 -22 2 -22 C -2 22 0 0 4 D -10 -2 0 0 8 E -6 22 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 10 6 B -14 0 -22 2 -22 C -2 22 0 0 4 D -10 -2 0 0 8 E -6 22 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=27 A=19 E=16 C=9 so C is eliminated. Round 2 votes counts: D=29 B=28 A=22 E=21 so E is eliminated. Round 3 votes counts: B=40 A=31 D=29 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 C:212 E:202 D:198 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 10 6 B -14 0 -22 2 -22 C -2 22 0 0 4 D -10 -2 0 0 8 E -6 22 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 10 6 B -14 0 -22 2 -22 C -2 22 0 0 4 D -10 -2 0 0 8 E -6 22 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 10 6 B -14 0 -22 2 -22 C -2 22 0 0 4 D -10 -2 0 0 8 E -6 22 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7387: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) A D B E C (8) B E C A D (7) A D C E B (7) D C A E B (5) B C E D A (5) D A C B E (4) A D E C B (4) D C E B A (3) D C B E A (3) C D E B A (3) B E C D A (3) A B E D C (3) E B C D A (2) D C E A B (2) D B C E A (2) C D B E A (2) C B E D A (2) B E A C D (2) B D A C E (2) B A E D C (2) A B E C D (2) E C B A D (1) D A B C E (1) C E B D A (1) C E A D B (1) C B D E A (1) B D C E A (1) B C D E A (1) B A E C D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A E B C D (1) A D E B C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 12 10 -8 16 B -12 0 -6 -22 8 C -10 6 0 -28 16 D 8 22 28 0 26 E -16 -8 -16 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 -8 16 B -12 0 -6 -22 8 C -10 6 0 -28 16 D 8 22 28 0 26 E -16 -8 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=31 B=24 C=10 E=3 so E is eliminated. Round 2 votes counts: A=32 D=31 B=26 C=11 so C is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:242 A:215 C:192 B:184 E:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 10 -8 16 B -12 0 -6 -22 8 C -10 6 0 -28 16 D 8 22 28 0 26 E -16 -8 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -8 16 B -12 0 -6 -22 8 C -10 6 0 -28 16 D 8 22 28 0 26 E -16 -8 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -8 16 B -12 0 -6 -22 8 C -10 6 0 -28 16 D 8 22 28 0 26 E -16 -8 -16 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7388: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (19) C D A B E (12) B E C D A (11) B E C A D (11) D A C E B (10) A D C E B (8) E A B D C (4) C D B E A (4) A D E B C (4) D C A E B (3) A D E C B (3) C D B A E (2) C B E D A (2) B C E D A (2) A E D B C (2) D A E B C (1) C B D E A (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 4 4 -8 B 4 0 10 2 -10 C -4 -10 0 -10 -12 D -4 -2 10 0 -4 E 8 10 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 4 -8 B 4 0 10 2 -10 C -4 -10 0 -10 -12 D -4 -2 10 0 -4 E 8 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=23 C=21 A=18 D=14 so D is eliminated. Round 2 votes counts: A=29 C=24 B=24 E=23 so E is eliminated. Round 3 votes counts: B=43 A=33 C=24 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:217 B:203 D:200 A:198 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 4 -8 B 4 0 10 2 -10 C -4 -10 0 -10 -12 D -4 -2 10 0 -4 E 8 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 -8 B 4 0 10 2 -10 C -4 -10 0 -10 -12 D -4 -2 10 0 -4 E 8 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 -8 B 4 0 10 2 -10 C -4 -10 0 -10 -12 D -4 -2 10 0 -4 E 8 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7389: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) B D E C A (8) A C E D B (8) A B C E D (6) D B E C A (5) C E A D B (5) B D A E C (5) A C B E D (5) C A E D B (4) E D C A B (3) E C D A B (3) E C A D B (3) D E B C A (3) B D E A C (3) B A D E C (3) B A C D E (3) A E C D B (3) D E C B A (2) D E C A B (2) C B D A E (2) B D C A E (2) B A C E D (2) E A D C B (1) E A D B C (1) D E B A C (1) D C E B A (1) D B E A C (1) C E D A B (1) C B D E A (1) B A E D C (1) B A D C E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 12 10 14 12 B -12 0 -8 6 -4 C -10 8 0 14 0 D -14 -6 -14 0 -14 E -12 4 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 14 12 B -12 0 -8 6 -4 C -10 8 0 14 0 D -14 -6 -14 0 -14 E -12 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=28 D=15 C=13 E=11 so E is eliminated. Round 2 votes counts: A=35 B=28 C=19 D=18 so D is eliminated. Round 3 votes counts: B=38 A=35 C=27 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:206 E:203 B:191 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 14 12 B -12 0 -8 6 -4 C -10 8 0 14 0 D -14 -6 -14 0 -14 E -12 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 14 12 B -12 0 -8 6 -4 C -10 8 0 14 0 D -14 -6 -14 0 -14 E -12 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 14 12 B -12 0 -8 6 -4 C -10 8 0 14 0 D -14 -6 -14 0 -14 E -12 4 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7390: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) C D B E A (7) A E B D C (7) D C A B E (6) B E C D A (6) B E A C D (6) E B A C D (5) C D E B A (5) C D A E B (5) A D C E B (5) C D E A B (4) A B E D C (4) D A C E B (3) B E A D C (3) E B C D A (2) E A B C D (2) B C E D A (2) A D E B C (2) A D C B E (2) A B D E C (2) E B C A D (1) E A B D C (1) D C B A E (1) C D A B E (1) C B E D A (1) C B D E A (1) B E C A D (1) B D C E A (1) B C D E A (1) B A E D C (1) B A D E C (1) A E D C B (1) A D E C B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 10 -4 -6 2 B -10 0 0 -4 -2 C 4 0 0 0 6 D 6 4 0 0 14 E -2 2 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.605382 D: 0.394618 E: 0.000000 Sum of squares = 0.522210607208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.605382 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -6 2 B -10 0 0 -4 -2 C 4 0 0 0 6 D 6 4 0 0 14 E -2 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 B=22 D=17 E=11 so E is eliminated. Round 2 votes counts: B=30 A=29 C=24 D=17 so D is eliminated. Round 3 votes counts: C=38 A=32 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:205 A:201 B:192 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -6 2 B -10 0 0 -4 -2 C 4 0 0 0 6 D 6 4 0 0 14 E -2 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -6 2 B -10 0 0 -4 -2 C 4 0 0 0 6 D 6 4 0 0 14 E -2 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -6 2 B -10 0 0 -4 -2 C 4 0 0 0 6 D 6 4 0 0 14 E -2 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7391: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (19) B E C A D (11) E B A D C (8) B C E D A (8) A E D B C (6) A D E B C (6) C B E D A (5) C B D E A (5) E B A C D (4) C D A E B (4) B E A C D (4) E A B D C (3) D C A E B (3) C D A B E (2) B E C D A (2) D B C A E (1) D A C B E (1) D A B E C (1) C D E A B (1) C D B E A (1) C D B A E (1) B E A D C (1) B C D E A (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 10 -10 -8 B 4 0 14 4 -10 C -10 -14 0 -2 6 D 10 -4 2 0 -4 E 8 10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.333333 D: 0.000000 E: 0.466667 Sum of squares = 0.368888888825 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.533333 D: 0.533333 E: 1.000000 A B C D E A 0 -4 10 -10 -8 B 4 0 14 4 -10 C -10 -14 0 -2 6 D 10 -4 2 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.333333 D: 0.000000 E: 0.466667 Sum of squares = 0.368888888887 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.533333 D: 0.533333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 C=19 E=15 A=14 so A is eliminated. Round 2 votes counts: D=33 B=27 E=21 C=19 so C is eliminated. Round 3 votes counts: D=42 B=37 E=21 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:208 B:206 D:202 A:194 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 10 -10 -8 B 4 0 14 4 -10 C -10 -14 0 -2 6 D 10 -4 2 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.333333 D: 0.000000 E: 0.466667 Sum of squares = 0.368888888887 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.533333 D: 0.533333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -10 -8 B 4 0 14 4 -10 C -10 -14 0 -2 6 D 10 -4 2 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.333333 D: 0.000000 E: 0.466667 Sum of squares = 0.368888888887 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.533333 D: 0.533333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -10 -8 B 4 0 14 4 -10 C -10 -14 0 -2 6 D 10 -4 2 0 -4 E 8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.333333 D: 0.000000 E: 0.466667 Sum of squares = 0.368888888887 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.533333 D: 0.533333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7392: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) B A C D E (8) E D C A B (7) E D A B C (7) E C D B A (7) D E C A B (6) C D B A E (5) A B E D C (5) A B D C E (5) C E D B A (4) A B C D E (4) C D E B A (3) B A C E D (3) E B A C D (2) C E B D A (2) C B D A E (2) B C A D E (2) A E B D C (2) A B D E C (2) E D A C B (1) E B D A C (1) E B C A D (1) D E A C B (1) D E A B C (1) D C E A B (1) D A E B C (1) D A C B E (1) D A B C E (1) C D B E A (1) C B A D E (1) B C A E D (1) B A E C D (1) A D E B C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 12 14 -4 -6 B -12 0 16 2 -12 C -14 -16 0 -4 -10 D 4 -2 4 0 -6 E 6 12 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 14 -4 -6 B -12 0 16 2 -12 C -14 -16 0 -4 -10 D 4 -2 4 0 -6 E 6 12 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=21 C=18 B=15 D=12 so D is eliminated. Round 2 votes counts: E=42 A=24 C=19 B=15 so B is eliminated. Round 3 votes counts: E=42 A=36 C=22 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:208 D:200 B:197 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 14 -4 -6 B -12 0 16 2 -12 C -14 -16 0 -4 -10 D 4 -2 4 0 -6 E 6 12 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 -4 -6 B -12 0 16 2 -12 C -14 -16 0 -4 -10 D 4 -2 4 0 -6 E 6 12 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 -4 -6 B -12 0 16 2 -12 C -14 -16 0 -4 -10 D 4 -2 4 0 -6 E 6 12 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7393: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) D C A B E (6) D A B C E (6) B E A C D (6) C E D B A (5) A B D E C (5) D A C E B (4) A B E D C (4) E B C A D (3) E A C B D (3) D B A C E (3) D A C B E (3) C D E B A (3) B A D E C (3) A E C D B (3) A E B C D (3) A D C E B (3) D C B E A (2) D C B A E (2) D C A E B (2) D B C E A (2) C E B D A (2) C D E A B (2) B A E D C (2) B A E C D (2) E A C D B (1) E A B C D (1) D C E A B (1) C E D A B (1) C D B E A (1) C B E D A (1) B E C A D (1) B D A C E (1) B C E D A (1) A E D B C (1) A E C B D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 20 30 4 28 B -20 0 6 -20 18 C -30 -6 0 -20 2 D -4 20 20 0 16 E -28 -18 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 30 4 28 B -20 0 6 -20 18 C -30 -6 0 -20 2 D -4 20 20 0 16 E -28 -18 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=30 B=16 C=15 E=8 so E is eliminated. Round 2 votes counts: A=35 D=31 B=19 C=15 so C is eliminated. Round 3 votes counts: D=43 A=35 B=22 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:241 D:226 B:192 C:173 E:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 30 4 28 B -20 0 6 -20 18 C -30 -6 0 -20 2 D -4 20 20 0 16 E -28 -18 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 30 4 28 B -20 0 6 -20 18 C -30 -6 0 -20 2 D -4 20 20 0 16 E -28 -18 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 30 4 28 B -20 0 6 -20 18 C -30 -6 0 -20 2 D -4 20 20 0 16 E -28 -18 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7394: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) A E B C D (10) D C B E A (7) D C E A B (6) C D A B E (6) E A B D C (5) C D B A E (5) E A D B C (3) D C E B A (3) D C A E B (3) C D A E B (3) B E D A C (3) B E A C D (3) A E C D B (3) E B A D C (2) D B E C A (2) B D E C A (2) B A E C D (2) A C E B D (2) A C B E D (2) A B E C D (2) E D A B C (1) E B D A C (1) D E C A B (1) D E B A C (1) D E A C B (1) C A D E B (1) C A B E D (1) B E D C A (1) B D C E A (1) B C A D E (1) B A C E D (1) A E D B C (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 6 14 6 -10 B -6 0 10 4 2 C -14 -10 0 -14 -14 D -6 -4 14 0 -14 E 10 -2 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.43209876536 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 6 14 6 -10 B -6 0 10 4 2 C -14 -10 0 -14 -14 D -6 -4 14 0 -14 E 10 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098764931 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=24 A=22 C=16 E=12 so E is eliminated. Round 2 votes counts: A=30 B=29 D=25 C=16 so C is eliminated. Round 3 votes counts: D=39 A=32 B=29 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:218 A:208 B:205 D:195 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 6 -10 B -6 0 10 4 2 C -14 -10 0 -14 -14 D -6 -4 14 0 -14 E 10 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098764931 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 6 -10 B -6 0 10 4 2 C -14 -10 0 -14 -14 D -6 -4 14 0 -14 E 10 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098764931 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 6 -10 B -6 0 10 4 2 C -14 -10 0 -14 -14 D -6 -4 14 0 -14 E 10 -2 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.555556 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.432098764931 Cumulative probabilities = A: 0.111111 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7395: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (11) E B C A D (8) D A B C E (7) B C A D E (7) E C B A D (5) D A B E C (5) B E C A D (5) E D A C B (4) E C D A B (3) C B E A D (3) E D C A B (2) E D B A C (2) E D A B C (2) E C D B A (2) E B D A C (2) E B C D A (2) D B A C E (2) C E B A D (2) C A B D E (2) B E A C D (2) B C E A D (2) A D C B E (2) E D B C A (1) D E A C B (1) D E A B C (1) D C E A B (1) D B A E C (1) D A E C B (1) D A E B C (1) D A C E B (1) C E D A B (1) C E A B D (1) C D A E B (1) C B A E D (1) C A D E B (1) B E A D C (1) B D A E C (1) B C A E D (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -2 -10 -6 B 2 0 10 -8 10 C 2 -10 0 0 -4 D 10 8 0 0 -4 E 6 -10 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826447 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.636364 E: 1.000000 A B C D E A 0 -2 -2 -10 -6 B 2 0 10 -8 10 C 2 -10 0 0 -4 D 10 8 0 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826441 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=32 B=20 C=12 A=3 so A is eliminated. Round 2 votes counts: D=35 E=33 B=20 C=12 so C is eliminated. Round 3 votes counts: E=37 D=37 B=26 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:207 D:207 E:202 C:194 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 -6 B 2 0 10 -8 10 C 2 -10 0 0 -4 D 10 8 0 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826441 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.636364 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 -6 B 2 0 10 -8 10 C 2 -10 0 0 -4 D 10 8 0 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826441 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.636364 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 -6 B 2 0 10 -8 10 C 2 -10 0 0 -4 D 10 8 0 0 -4 E 6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.454545 E: 0.363636 Sum of squares = 0.371900826441 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.636364 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7396: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (18) B A C D E (15) E D C B A (7) B C A D E (5) D E C A B (4) A C B D E (4) E B D C A (3) B E D C A (3) A D C E B (3) A B C D E (3) E D B C A (2) E D B A C (2) E D A C B (2) D E A C B (2) C D E A B (2) C D A E B (2) C A D E B (2) C A D B E (2) C A B D E (2) B E C D A (2) B E C A D (2) B C E A D (2) B C A E D (2) A C D E B (2) E B D A C (1) D C E A B (1) D C A E B (1) D A E C B (1) B E D A C (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -24 -8 -8 B -4 0 -12 -8 -10 C 24 12 0 -2 0 D 8 8 2 0 4 E 8 10 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -24 -8 -8 B -4 0 -12 -8 -10 C 24 12 0 -2 0 D 8 8 2 0 4 E 8 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=33 A=13 C=10 D=9 so D is eliminated. Round 2 votes counts: E=41 B=33 A=14 C=12 so C is eliminated. Round 3 votes counts: E=44 B=33 A=23 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:217 D:211 E:207 B:183 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -24 -8 -8 B -4 0 -12 -8 -10 C 24 12 0 -2 0 D 8 8 2 0 4 E 8 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -24 -8 -8 B -4 0 -12 -8 -10 C 24 12 0 -2 0 D 8 8 2 0 4 E 8 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -24 -8 -8 B -4 0 -12 -8 -10 C 24 12 0 -2 0 D 8 8 2 0 4 E 8 10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7397: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) B C E D A (11) A D E B C (8) C B A E D (7) D E B A C (6) A D E C B (5) E D B C A (4) A C E D B (4) A C B D E (4) E A C D B (3) D E A B C (3) C B E A D (3) A C D E B (3) C A E B D (2) C A B E D (2) B E D C A (2) B E C D A (2) B C D E A (2) A D C E B (2) A C D B E (2) E D B A C (1) E D A B C (1) E C D A B (1) E B D C A (1) E B C D A (1) D E B C A (1) D A E B C (1) C E B A D (1) C A B D E (1) B D A C E (1) B A D C E (1) A E C D B (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -6 0 -10 B 12 0 -8 6 2 C 6 8 0 24 18 D 0 -6 -24 0 -18 E 10 -2 -18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 0 -10 B 12 0 -8 6 2 C 6 8 0 24 18 D 0 -6 -24 0 -18 E 10 -2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=28 B=19 E=12 D=11 so D is eliminated. Round 2 votes counts: A=31 C=28 E=22 B=19 so B is eliminated. Round 3 votes counts: C=41 A=33 E=26 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:206 E:204 A:186 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -6 0 -10 B 12 0 -8 6 2 C 6 8 0 24 18 D 0 -6 -24 0 -18 E 10 -2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 0 -10 B 12 0 -8 6 2 C 6 8 0 24 18 D 0 -6 -24 0 -18 E 10 -2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 0 -10 B 12 0 -8 6 2 C 6 8 0 24 18 D 0 -6 -24 0 -18 E 10 -2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7398: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (17) C D B A E (10) B D C E A (10) E A B D C (8) C D B E A (5) C D A B E (5) A E B C D (5) B E D C A (4) B E A D C (4) A E B D C (4) B D E C A (3) B C D A E (3) A C E D B (3) D C B E A (2) C D A E B (2) B E D A C (2) A C D E B (2) E D C B A (1) E D B C A (1) E B D A C (1) E B A D C (1) E A D C B (1) E A C D B (1) C D E B A (1) C B D A E (1) C A D E B (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 2 -2 10 B -2 0 -8 -6 0 C -2 8 0 16 -8 D 2 6 -16 0 -8 E -10 0 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.100000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000223 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.900000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -2 10 B -2 0 -8 -6 0 C -2 8 0 16 -8 D 2 6 -16 0 -8 E -10 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.100000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000005 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.900000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=26 C=25 E=14 D=2 so D is eliminated. Round 2 votes counts: A=33 C=27 B=26 E=14 so E is eliminated. Round 3 votes counts: A=43 B=29 C=28 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:206 E:203 B:192 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 -2 10 B -2 0 -8 -6 0 C -2 8 0 16 -8 D 2 6 -16 0 -8 E -10 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.100000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000005 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.900000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -2 10 B -2 0 -8 -6 0 C -2 8 0 16 -8 D 2 6 -16 0 -8 E -10 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.100000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000005 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.900000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -2 10 B -2 0 -8 -6 0 C -2 8 0 16 -8 D 2 6 -16 0 -8 E -10 0 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.100000 D: 0.100000 E: 0.000000 Sum of squares = 0.660000000005 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 0.900000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7399: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E B D A C (6) A D C B E (6) E A D B C (5) A D E C B (5) D B A E C (4) C A D B E (4) E C B A D (3) E B C D A (3) B E D A C (3) A D E B C (3) E C A B D (2) E B A D C (2) E A C B D (2) E A B C D (2) D E B A C (2) D A B E C (2) C B E D A (2) C B E A D (2) C A D E B (2) B E D C A (2) B E C D A (2) B D E A C (2) B D C A E (2) B D A C E (2) B C E D A (2) A E D C B (2) A D C E B (2) A C D E B (2) E D B A C (1) E C B D A (1) E A D C B (1) E A B D C (1) D B A C E (1) D A E B C (1) C D B A E (1) C D A B E (1) C A E D B (1) C A B D E (1) B D E C A (1) B C D E A (1) B C D A E (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 18 0 4 B 6 0 2 4 0 C -18 -2 0 -12 -18 D 0 -4 12 0 8 E -4 0 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.793370 C: 0.000000 D: 0.000000 E: 0.206630 Sum of squares = 0.672131842773 Cumulative probabilities = A: 0.000000 B: 0.793370 C: 0.793370 D: 0.793370 E: 1.000000 A B C D E A 0 -6 18 0 4 B 6 0 2 4 0 C -18 -2 0 -12 -18 D 0 -4 12 0 8 E -4 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555633766 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 C=21 B=18 D=10 so D is eliminated. Round 2 votes counts: E=31 A=25 B=23 C=21 so C is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:208 D:208 B:206 E:203 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 18 0 4 B 6 0 2 4 0 C -18 -2 0 -12 -18 D 0 -4 12 0 8 E -4 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555633766 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 18 0 4 B 6 0 2 4 0 C -18 -2 0 -12 -18 D 0 -4 12 0 8 E -4 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555633766 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 18 0 4 B 6 0 2 4 0 C -18 -2 0 -12 -18 D 0 -4 12 0 8 E -4 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555633766 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7400: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) C E A B D (8) C D B A E (6) D B A C E (5) C E D A B (5) C D E B A (5) E C D A B (4) E C A B D (4) E A C B D (4) D B C A E (4) B A D E C (4) E A B D C (3) D C B A E (3) A E B D C (3) A B E D C (3) E A B C D (2) D B E A C (2) C E A D B (2) C D B E A (2) C A B E D (2) C A B D E (2) B D A E C (2) A B E C D (2) E C D B A (1) E A D B C (1) D C E B A (1) D B C E A (1) C E D B A (1) C B D A E (1) C B A D E (1) C A E B D (1) A E C B D (1) A E B C D (1) A C E B D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 -6 8 B -2 0 -10 -6 4 C 8 10 0 14 4 D 6 6 -14 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -6 8 B -2 0 -10 -6 4 C 8 10 0 14 4 D 6 6 -14 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=26 E=19 A=13 B=6 so B is eliminated. Round 2 votes counts: C=36 D=28 E=19 A=17 so A is eliminated. Round 3 votes counts: C=38 D=33 E=29 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:200 A:198 B:193 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -6 8 B -2 0 -10 -6 4 C 8 10 0 14 4 D 6 6 -14 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -6 8 B -2 0 -10 -6 4 C 8 10 0 14 4 D 6 6 -14 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -6 8 B -2 0 -10 -6 4 C 8 10 0 14 4 D 6 6 -14 0 2 E -8 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999512 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7401: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) C A B E D (7) D A B C E (6) E C A D B (5) E C A B D (5) D E B A C (5) D B A C E (5) B A D C E (4) A C B D E (4) E D C A B (3) E D B C A (3) E C D A B (3) E C B D A (3) D E A C B (3) C A E D B (3) B D E A C (3) B D A C E (3) E C D B A (2) E C B A D (2) E B C D A (2) D A B E C (2) C E A D B (2) C A E B D (2) A D C E B (2) A C D B E (2) E D C B A (1) E B D C A (1) D E A B C (1) D B A E C (1) C E B A D (1) C E A B D (1) C A D B E (1) B E C D A (1) B E C A D (1) B C E A D (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 6 0 -10 -12 B -6 0 -4 -16 0 C 0 4 0 -2 -8 D 10 16 2 0 2 E 12 0 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -10 -12 B -6 0 -4 -16 0 C 0 4 0 -2 -8 D 10 16 2 0 2 E 12 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=30 D=30 C=17 B=14 A=9 so A is eliminated. Round 2 votes counts: D=33 E=30 C=23 B=14 so B is eliminated. Round 3 votes counts: D=43 E=32 C=25 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:209 C:197 A:192 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -10 -12 B -6 0 -4 -16 0 C 0 4 0 -2 -8 D 10 16 2 0 2 E 12 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -10 -12 B -6 0 -4 -16 0 C 0 4 0 -2 -8 D 10 16 2 0 2 E 12 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -10 -12 B -6 0 -4 -16 0 C 0 4 0 -2 -8 D 10 16 2 0 2 E 12 0 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999803 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7402: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) A E B C D (8) E A B C D (6) C B D E A (6) A E D B C (6) D C B E A (5) D B C A E (5) D B A C E (5) C B E A D (5) E A C B D (4) C E B A D (4) C D B E A (4) B C D A E (3) E C B A D (2) D A E C B (2) D A E B C (2) D A C B E (2) D A B E C (2) D A B C E (2) C D E B A (2) C B E D A (2) B C E A D (2) A E B D C (2) A D E B C (2) E C A B D (1) E A D C B (1) E A C D B (1) D E C A B (1) B D C A E (1) B C E D A (1) B A C E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -4 -6 6 B 12 0 0 0 12 C 4 0 0 4 16 D 6 0 -4 0 6 E -6 -12 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.227940 C: 0.772060 D: 0.000000 E: 0.000000 Sum of squares = 0.64803373782 Cumulative probabilities = A: 0.000000 B: 0.227940 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -6 6 B 12 0 0 0 12 C 4 0 0 4 16 D 6 0 -4 0 6 E -6 -12 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999743 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=23 A=20 E=15 B=8 so B is eliminated. Round 2 votes counts: D=35 C=29 A=21 E=15 so E is eliminated. Round 3 votes counts: D=35 A=33 C=32 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:212 C:212 D:204 A:192 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 6 B 12 0 0 0 12 C 4 0 0 4 16 D 6 0 -4 0 6 E -6 -12 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999743 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 6 B 12 0 0 0 12 C 4 0 0 4 16 D 6 0 -4 0 6 E -6 -12 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999743 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 6 B 12 0 0 0 12 C 4 0 0 4 16 D 6 0 -4 0 6 E -6 -12 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999743 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7403: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) C D B E A (6) B C D A E (6) E D A C B (5) A B C E D (5) E A D C B (4) D E B C A (4) C B A D E (4) B D E A C (4) E A D B C (3) A E D C B (3) A C B E D (3) A B E D C (3) E D C A B (2) D C E B A (2) D C B E A (2) D B E C A (2) C D E B A (2) C A B E D (2) B C D E A (2) B C A D E (2) B A D C E (2) B A C D E (2) A E D B C (2) A E C B D (2) E D B A C (1) E D A B C (1) E C A D B (1) E B D A C (1) E A B D C (1) D E B A C (1) D B E A C (1) D B C E A (1) C E A D B (1) C D E A B (1) C B D E A (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B D E (1) A E C D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 8 2 B -6 0 6 6 2 C -10 -6 0 -6 -2 D -8 -6 6 0 -6 E -2 -2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998609 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 8 2 B -6 0 6 6 2 C -10 -6 0 -6 -2 D -8 -6 6 0 -6 E -2 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=21 E=19 B=18 D=13 so D is eliminated. Round 2 votes counts: A=29 C=25 E=24 B=22 so B is eliminated. Round 3 votes counts: C=36 A=33 E=31 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:204 E:202 D:193 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 8 2 B -6 0 6 6 2 C -10 -6 0 -6 -2 D -8 -6 6 0 -6 E -2 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 8 2 B -6 0 6 6 2 C -10 -6 0 -6 -2 D -8 -6 6 0 -6 E -2 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 8 2 B -6 0 6 6 2 C -10 -6 0 -6 -2 D -8 -6 6 0 -6 E -2 -2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999287 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7404: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) C E A B D (7) D B A C E (6) C A E B D (6) B D A E C (6) D B E A C (5) D B A E C (4) B A E C D (4) A C E B D (4) D E B C A (3) D C B A E (3) D B E C A (3) D B C A E (3) C E A D B (3) B D E A C (3) A C B E D (3) E D C A B (2) E D B C A (2) E C D A B (2) E C A D B (2) E A B C D (2) D C E A B (2) D C A B E (2) C E D A B (2) B D A C E (2) E A C B D (1) D E C A B (1) C D E A B (1) C D A E B (1) C D A B E (1) B E A D C (1) B A E D C (1) B A C E D (1) B A C D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -8 -8 0 B -4 0 -2 4 2 C 8 2 0 2 -2 D 8 -4 -2 0 -4 E 0 -2 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 -8 -8 0 B -4 0 -2 4 2 C 8 2 0 2 -2 D 8 -4 -2 0 -4 E 0 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=21 E=19 B=19 A=9 so A is eliminated. Round 2 votes counts: D=32 C=28 B=21 E=19 so E is eliminated. Round 3 votes counts: C=41 D=36 B=23 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:205 E:202 B:200 D:199 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -8 0 B -4 0 -2 4 2 C 8 2 0 2 -2 D 8 -4 -2 0 -4 E 0 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -8 0 B -4 0 -2 4 2 C 8 2 0 2 -2 D 8 -4 -2 0 -4 E 0 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -8 0 B -4 0 -2 4 2 C 8 2 0 2 -2 D 8 -4 -2 0 -4 E 0 -2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7405: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) C D E A B (9) B A E D C (7) A B E D C (7) E D B A C (6) C D A E B (5) B E A D C (4) A B C E D (4) E B D A C (3) D E C A B (3) D C E A B (3) B E D A C (3) E D B C A (2) D E C B A (2) D E A C B (2) D C E B A (2) C B A E D (2) C A D E B (2) B C A E D (2) B A E C D (2) A D E B C (2) A C B D E (2) E D A B C (1) E B D C A (1) D E B C A (1) D A E C B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A D B E (1) C A B D E (1) B E C A D (1) B C E D A (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -2 -16 -10 B 4 0 2 -12 -14 C 2 -2 0 -8 -4 D 16 12 8 0 0 E 10 14 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.521773 E: 0.478227 Sum of squares = 0.50094814521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.521773 E: 1.000000 A B C D E A 0 -4 -2 -16 -10 B 4 0 2 -12 -14 C 2 -2 0 -8 -4 D 16 12 8 0 0 E 10 14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=21 A=19 D=14 E=13 so E is eliminated. Round 2 votes counts: C=33 B=25 D=23 A=19 so A is eliminated. Round 3 votes counts: B=38 C=35 D=27 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:218 E:214 C:194 B:190 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -16 -10 B 4 0 2 -12 -14 C 2 -2 0 -8 -4 D 16 12 8 0 0 E 10 14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -16 -10 B 4 0 2 -12 -14 C 2 -2 0 -8 -4 D 16 12 8 0 0 E 10 14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -16 -10 B 4 0 2 -12 -14 C 2 -2 0 -8 -4 D 16 12 8 0 0 E 10 14 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7406: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) C A B E D (7) D E B A C (6) A C B E D (6) C E D B A (5) E B D A C (4) C A E B D (4) B E D A C (4) E D B A C (3) B D E A C (3) B A E D C (3) A B D C E (3) A B C E D (3) E D C B A (2) E D B C A (2) E B D C A (2) D A B E C (2) C D E A B (2) C D A E B (2) C A D E B (2) A C D B E (2) A B E C D (2) A B D E C (2) E C D B A (1) E C B D A (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A B E (1) D B E A C (1) D B A E C (1) D A B C E (1) C E B A D (1) C E A D B (1) C E A B D (1) C D E B A (1) C A E D B (1) B E A C D (1) B D A E C (1) B A E C D (1) A D C B E (1) A D B C E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 6 -12 -4 B 6 0 10 2 -4 C -6 -10 0 -10 -4 D 12 -2 10 0 -12 E 4 4 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 -12 -4 B 6 0 10 2 -4 C -6 -10 0 -10 -4 D 12 -2 10 0 -12 E 4 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=23 A=22 E=15 B=13 so B is eliminated. Round 2 votes counts: D=27 C=27 A=26 E=20 so E is eliminated. Round 3 votes counts: D=44 C=29 A=27 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:212 B:207 D:204 A:192 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 -12 -4 B 6 0 10 2 -4 C -6 -10 0 -10 -4 D 12 -2 10 0 -12 E 4 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -12 -4 B 6 0 10 2 -4 C -6 -10 0 -10 -4 D 12 -2 10 0 -12 E 4 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -12 -4 B 6 0 10 2 -4 C -6 -10 0 -10 -4 D 12 -2 10 0 -12 E 4 4 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998621 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7407: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) A C D E B (12) E A D C B (7) B C D A E (7) B C D E A (6) A C D B E (6) E D C A B (5) B A C D E (5) A E D C B (5) E D C B A (4) E D A C B (3) E B D C A (3) D C E A B (3) C D A E B (3) B E A D C (3) B C A D E (3) E D B C A (2) C D E A B (2) C A D B E (2) B A E D C (2) E B D A C (1) C A D E B (1) A E C D B (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 -2 -2 B -4 0 -10 -14 -6 C 6 10 0 0 4 D 2 14 0 0 2 E 2 6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.699849 D: 0.300151 E: 0.000000 Sum of squares = 0.57987903059 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.699849 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -2 -2 B -4 0 -10 -14 -6 C 6 10 0 0 4 D 2 14 0 0 2 E 2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=26 E=25 C=8 D=3 so D is eliminated. Round 2 votes counts: B=38 A=26 E=25 C=11 so C is eliminated. Round 3 votes counts: B=38 A=32 E=30 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:210 D:209 E:201 A:197 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -2 -2 B -4 0 -10 -14 -6 C 6 10 0 0 4 D 2 14 0 0 2 E 2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -2 -2 B -4 0 -10 -14 -6 C 6 10 0 0 4 D 2 14 0 0 2 E 2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -2 -2 B -4 0 -10 -14 -6 C 6 10 0 0 4 D 2 14 0 0 2 E 2 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7408: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) C A B E D (6) E A C D B (5) E A D C B (4) E A D B C (4) C E A B D (4) B D A E C (4) A E C B D (4) A B E D C (4) D C B E A (3) D B C E A (3) C E A D B (3) C A E B D (3) B D A C E (3) B A D C E (3) A B D E C (3) E D C A B (2) E D A B C (2) E C D A B (2) D B E A C (2) C E D A B (2) C D B E A (2) C B D A E (2) C B A E D (2) C B A D E (2) B A D E C (2) B A C D E (2) E D C B A (1) E C D B A (1) E C A D B (1) E C A B D (1) D E C B A (1) D E B A C (1) D B E C A (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B A D (1) B D C A E (1) B C A D E (1) A D B E C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 8 16 6 B -10 0 -8 0 10 C -8 8 0 -8 -8 D -16 0 8 0 -8 E -6 -10 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 16 6 B -10 0 -8 0 10 C -8 8 0 -8 -8 D -16 0 8 0 -8 E -6 -10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=23 D=19 B=16 A=14 so A is eliminated. Round 2 votes counts: C=29 E=27 B=24 D=20 so D is eliminated. Round 3 votes counts: B=38 C=32 E=30 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:220 E:200 B:196 C:192 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 16 6 B -10 0 -8 0 10 C -8 8 0 -8 -8 D -16 0 8 0 -8 E -6 -10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 16 6 B -10 0 -8 0 10 C -8 8 0 -8 -8 D -16 0 8 0 -8 E -6 -10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 16 6 B -10 0 -8 0 10 C -8 8 0 -8 -8 D -16 0 8 0 -8 E -6 -10 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999773 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7409: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) A E C B D (7) E C D A B (5) D B C E A (5) E D C A B (4) E C A D B (4) C B D E A (4) E D A B C (3) E A D B C (3) D B E C A (3) D B E A C (3) C E D B A (3) C E A B D (3) C D E B A (3) C A E B D (3) B D C A E (3) A B E D C (3) A B C D E (3) D C B E A (2) D B A E C (2) C E D A B (2) C D B E A (2) C A B E D (2) B C D A E (2) A E B D C (2) A B D E C (2) E D A C B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E B A C (1) C E A D B (1) C B D A E (1) C A B D E (1) B D C E A (1) B C D E A (1) B C A D E (1) B A D E C (1) B A D C E (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -14 -18 -14 B -6 0 -4 -4 2 C 14 4 0 0 2 D 18 4 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.493953 D: 0.506047 E: 0.000000 Sum of squares = 0.500073134466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.493953 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 -18 -14 B -6 0 -4 -4 2 C 14 4 0 0 2 D 18 4 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=22 A=19 D=17 B=17 so D is eliminated. Round 2 votes counts: B=30 C=27 E=24 A=19 so A is eliminated. Round 3 votes counts: B=39 E=34 C=27 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:212 C:210 E:204 B:194 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 -18 -14 B -6 0 -4 -4 2 C 14 4 0 0 2 D 18 4 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 -18 -14 B -6 0 -4 -4 2 C 14 4 0 0 2 D 18 4 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 -18 -14 B -6 0 -4 -4 2 C 14 4 0 0 2 D 18 4 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7410: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) E D C A B (10) B C D E A (6) A B C E D (6) A B C D E (6) D E C B A (5) B A D E C (5) A E C D B (5) E D C B A (4) C E D B A (4) C D E B A (4) E C D A B (3) B D E C A (3) B C A D E (3) A B D E C (3) E D A C B (2) C B D E A (2) B D C E A (2) A C E D B (2) A B E D C (2) D E B C A (1) D B C E A (1) C D B E A (1) C B A D E (1) C A E D B (1) C A B E D (1) B D C A E (1) A E D C B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -6 2 4 B 12 0 4 12 14 C 6 -4 0 20 12 D -2 -12 -20 0 14 E -4 -14 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 2 4 B 12 0 4 12 14 C 6 -4 0 20 12 D -2 -12 -20 0 14 E -4 -14 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=27 E=19 C=14 D=7 so D is eliminated. Round 2 votes counts: B=34 A=27 E=25 C=14 so C is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:217 A:194 D:190 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 2 4 B 12 0 4 12 14 C 6 -4 0 20 12 D -2 -12 -20 0 14 E -4 -14 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 2 4 B 12 0 4 12 14 C 6 -4 0 20 12 D -2 -12 -20 0 14 E -4 -14 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 2 4 B 12 0 4 12 14 C 6 -4 0 20 12 D -2 -12 -20 0 14 E -4 -14 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7411: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) C D B A E (9) E D C A B (6) E C D B A (6) C D E B A (6) A B D C E (6) B A D C E (5) E C D A B (4) E A B D C (4) B C A D E (4) D E C A B (3) D C B A E (3) D A B C E (3) C E D B A (3) B A C D E (3) A E B D C (3) E C B A D (2) D A B E C (2) C D B E A (2) C B D A E (2) C B A E D (2) C B A D E (2) A B D E C (2) E D A C B (1) E D A B C (1) E C A D B (1) E B A C D (1) E A D B C (1) D E A B C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -10 -4 -4 B 0 0 -4 -4 -6 C 10 4 0 16 0 D 4 4 -16 0 6 E 4 6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.528851 D: 0.000000 E: 0.471149 Sum of squares = 0.501664726106 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.528851 D: 0.528851 E: 1.000000 A B C D E A 0 0 -10 -4 -4 B 0 0 -4 -4 -6 C 10 4 0 16 0 D 4 4 -16 0 6 E 4 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=26 A=13 D=12 B=12 so D is eliminated. Round 2 votes counts: E=41 C=29 A=18 B=12 so B is eliminated. Round 3 votes counts: E=41 C=33 A=26 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:202 D:199 B:193 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -4 -4 B 0 0 -4 -4 -6 C 10 4 0 16 0 D 4 4 -16 0 6 E 4 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 -4 B 0 0 -4 -4 -6 C 10 4 0 16 0 D 4 4 -16 0 6 E 4 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 -4 B 0 0 -4 -4 -6 C 10 4 0 16 0 D 4 4 -16 0 6 E 4 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7412: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) C A E D B (8) E B D C A (7) D B A C E (6) C A D B E (6) C A E B D (5) B E D A C (5) E B D A C (4) D C A B E (4) A C E D B (4) E C A B D (3) E B C D A (3) E B C A D (3) E B A C D (3) D B E C A (3) D B E A C (3) D A C B E (3) B E D C A (3) A C E B D (3) E C B A D (2) E A C B D (2) C A D E B (2) E A B C D (1) D E B C A (1) D B C E A (1) D B C A E (1) C E A B D (1) C D E B A (1) B D E A C (1) B D A E C (1) A D C B E (1) Total count = 100 A B C D E A 0 4 -8 6 6 B -4 0 -8 -6 -6 C 8 8 0 12 10 D -6 6 -12 0 -14 E -6 6 -10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 6 6 B -4 0 -8 -6 -6 C 8 8 0 12 10 D -6 6 -12 0 -14 E -6 6 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 D=22 A=17 B=10 so B is eliminated. Round 2 votes counts: E=36 D=24 C=23 A=17 so A is eliminated. Round 3 votes counts: C=39 E=36 D=25 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:204 E:202 B:188 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 6 6 B -4 0 -8 -6 -6 C 8 8 0 12 10 D -6 6 -12 0 -14 E -6 6 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 6 6 B -4 0 -8 -6 -6 C 8 8 0 12 10 D -6 6 -12 0 -14 E -6 6 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 6 6 B -4 0 -8 -6 -6 C 8 8 0 12 10 D -6 6 -12 0 -14 E -6 6 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7413: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (16) B D C A E (10) E A D C B (8) E C D A B (7) E C A D B (6) B D A C E (6) B C D A E (6) E A B D C (5) E C D B A (4) C E D B A (4) B A D C E (4) A B D C E (4) A E D B C (3) A B D E C (3) C E B D A (2) C B D E A (2) C B D A E (2) A E B D C (2) E C B D A (1) C D B E A (1) C D B A E (1) C B E D A (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 8 2 2 -18 B -8 0 -10 0 -16 C -2 10 0 10 -10 D -2 0 -10 0 -20 E 18 16 10 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 2 -18 B -8 0 -10 0 -16 C -2 10 0 10 -10 D -2 0 -10 0 -20 E 18 16 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=47 B=28 C=13 A=12 so D is eliminated. Round 2 votes counts: E=47 B=28 C=13 A=12 so A is eliminated. Round 3 votes counts: E=52 B=35 C=13 so C is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:232 C:204 A:197 D:184 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 2 -18 B -8 0 -10 0 -16 C -2 10 0 10 -10 D -2 0 -10 0 -20 E 18 16 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 -18 B -8 0 -10 0 -16 C -2 10 0 10 -10 D -2 0 -10 0 -20 E 18 16 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 -18 B -8 0 -10 0 -16 C -2 10 0 10 -10 D -2 0 -10 0 -20 E 18 16 10 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7414: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) B E A D C (7) B E D A C (6) A D B E C (6) E B D A C (5) D E B A C (5) C A D E B (5) A C D B E (5) E B C D A (4) D A E B C (4) C E D B A (4) B E C D A (4) A D C B E (4) C A B E D (3) E B D C A (2) D A C E B (2) C D A E B (2) C B E D A (2) C B A E D (2) C A D B E (2) B E C A D (2) A D C E B (2) E C B D A (1) C E B A D (1) C D E B A (1) C B E A D (1) C A E D B (1) C A B D E (1) B E D C A (1) B E A C D (1) B C E A D (1) B A E C D (1) B A D E C (1) A D E B C (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 6 0 -10 B 18 0 8 12 6 C -6 -8 0 4 -8 D 0 -12 -4 0 -16 E 10 -6 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 0 -10 B 18 0 8 12 6 C -6 -8 0 4 -8 D 0 -12 -4 0 -16 E 10 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=24 A=21 E=12 D=11 so D is eliminated. Round 2 votes counts: C=32 A=27 B=24 E=17 so E is eliminated. Round 3 votes counts: B=40 C=33 A=27 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:214 C:191 A:189 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 0 -10 B 18 0 8 12 6 C -6 -8 0 4 -8 D 0 -12 -4 0 -16 E 10 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 0 -10 B 18 0 8 12 6 C -6 -8 0 4 -8 D 0 -12 -4 0 -16 E 10 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 0 -10 B 18 0 8 12 6 C -6 -8 0 4 -8 D 0 -12 -4 0 -16 E 10 -6 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7415: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) A D C E B (7) D A B E C (6) E C B D A (5) B E C D A (5) A D B E C (5) C E B D A (4) B A C E D (4) E C D B A (3) D A E C B (3) D A E B C (3) C E D A B (3) C E B A D (3) B C E A D (3) A B D C E (3) E D C B A (2) D E C B A (2) D B A E C (2) C A E B D (2) B D E C A (2) A D C B E (2) A D B C E (2) E D B C A (1) E C D A B (1) E B C D A (1) D E B C A (1) D A C E B (1) C E A D B (1) C E A B D (1) C B E A D (1) C B A E D (1) C A E D B (1) C A B E D (1) B E D C A (1) B E C A D (1) B C A E D (1) B A E C D (1) B A D E C (1) B A D C E (1) A C E D B (1) A C D E B (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -8 -6 4 B -8 0 -10 -12 -8 C 8 10 0 -4 -6 D 6 12 4 0 2 E -4 8 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -6 4 B -8 0 -10 -12 -8 C 8 10 0 -4 -6 D 6 12 4 0 2 E -4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=24 B=20 C=18 E=13 so E is eliminated. Round 2 votes counts: D=28 C=27 A=24 B=21 so B is eliminated. Round 3 votes counts: C=38 D=31 A=31 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:204 E:204 A:199 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -6 4 B -8 0 -10 -12 -8 C 8 10 0 -4 -6 D 6 12 4 0 2 E -4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -6 4 B -8 0 -10 -12 -8 C 8 10 0 -4 -6 D 6 12 4 0 2 E -4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -6 4 B -8 0 -10 -12 -8 C 8 10 0 -4 -6 D 6 12 4 0 2 E -4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7416: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B C D A E (7) E A D B C (5) C D B A E (5) A E D B C (5) D A E C B (4) B C E A D (4) B A E C D (4) E A B D C (3) D C E A B (3) C B D A E (3) B C D E A (3) B A C D E (3) A D E C B (3) E D A C B (2) D E A C B (2) D C A E B (2) D C A B E (2) C D E B A (2) C B D E A (2) B C E D A (2) B C A D E (2) B A C E D (2) A E D C B (2) A E B D C (2) A D E B C (2) A B D E C (2) E D C A B (1) E B A C D (1) D E C A B (1) D C B A E (1) C E B D A (1) C D E A B (1) C D B E A (1) B E C A D (1) B E A C D (1) B A E D C (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 6 12 10 8 B -6 0 4 -8 -4 C -12 -4 0 -10 -8 D -10 8 10 0 4 E -8 4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 10 8 B -6 0 4 -8 -4 C -12 -4 0 -10 -8 D -10 8 10 0 4 E -8 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=22 A=17 D=15 C=15 so D is eliminated. Round 2 votes counts: B=31 E=25 C=23 A=21 so A is eliminated. Round 3 votes counts: E=43 B=34 C=23 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:218 D:206 E:200 B:193 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 10 8 B -6 0 4 -8 -4 C -12 -4 0 -10 -8 D -10 8 10 0 4 E -8 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 10 8 B -6 0 4 -8 -4 C -12 -4 0 -10 -8 D -10 8 10 0 4 E -8 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 10 8 B -6 0 4 -8 -4 C -12 -4 0 -10 -8 D -10 8 10 0 4 E -8 4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7417: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) D E C B A (12) D E A C B (7) C E B D A (7) E D C B A (5) D E B C A (5) B C E A D (4) A D B E C (4) A C B E D (4) C E D B A (3) B A C E D (3) A B D C E (3) E C D B A (2) D E C A B (2) D B E C A (2) D A E B C (2) B E D C A (2) B C A E D (2) A D E C B (2) A D C E B (2) A B D E C (2) E C B D A (1) D C E A B (1) D B E A C (1) D A E C B (1) D A C E B (1) C E D A B (1) C B E A D (1) C B A E D (1) B D E C A (1) A D C B E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -12 -14 B 4 0 -10 -10 -8 C 4 10 0 -12 -2 D 12 10 12 0 2 E 14 8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -12 -14 B 4 0 -10 -10 -8 C 4 10 0 -12 -2 D 12 10 12 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=33 C=13 B=12 E=8 so E is eliminated. Round 2 votes counts: D=39 A=33 C=16 B=12 so B is eliminated. Round 3 votes counts: D=42 A=36 C=22 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:211 C:200 B:188 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -12 -14 B 4 0 -10 -10 -8 C 4 10 0 -12 -2 D 12 10 12 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -12 -14 B 4 0 -10 -10 -8 C 4 10 0 -12 -2 D 12 10 12 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -12 -14 B 4 0 -10 -10 -8 C 4 10 0 -12 -2 D 12 10 12 0 2 E 14 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7418: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) D E C A B (6) C E A B D (6) D B A E C (5) B A D C E (5) E D C B A (4) D B E A C (4) E D B C A (3) E C D B A (3) E C A D B (3) D E B A C (3) B E C A D (3) B D A C E (3) E D C A B (2) E C D A B (2) E C B D A (2) E C A B D (2) D E B C A (2) D E A C B (2) D A B C E (2) C E A D B (2) C A E D B (2) C A E B D (2) B A C E D (2) B A C D E (2) A D C B E (2) A C B E D (2) A C B D E (2) A B C D E (2) E B D C A (1) E B C A D (1) D E C B A (1) D E A B C (1) D C E A B (1) C A D E B (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E A D (1) B C A E D (1) B A E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -14 2 -26 B 12 0 -6 -2 -16 C 14 6 0 0 -20 D -2 2 0 0 -6 E 26 16 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -14 2 -26 B 12 0 -6 -2 -16 C 14 6 0 0 -20 D -2 2 0 0 -6 E 26 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=27 B=21 C=13 A=10 so A is eliminated. Round 2 votes counts: E=29 D=29 B=25 C=17 so C is eliminated. Round 3 votes counts: E=41 D=30 B=29 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:234 C:200 D:197 B:194 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -14 2 -26 B 12 0 -6 -2 -16 C 14 6 0 0 -20 D -2 2 0 0 -6 E 26 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 2 -26 B 12 0 -6 -2 -16 C 14 6 0 0 -20 D -2 2 0 0 -6 E 26 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 2 -26 B 12 0 -6 -2 -16 C 14 6 0 0 -20 D -2 2 0 0 -6 E 26 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7419: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) D C B E A (11) A B E D C (11) C A D E B (6) A D B E C (6) A E B C D (5) C D A E B (4) B E D A C (4) A C E B D (4) D B E A C (3) B E A D C (3) B A E D C (3) E B D C A (2) E B C D A (2) E B A C D (2) D C B A E (2) D B C E A (2) C E B D A (2) C D E A B (2) C D B E A (2) E D C B A (1) E B C A D (1) D E C B A (1) D C A B E (1) D B A E C (1) D A B E C (1) D A B C E (1) C A E B D (1) A E B D C (1) A C D B E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -4 -8 0 B 8 0 0 -14 8 C 4 0 0 -10 2 D 8 14 10 0 14 E 0 -8 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -8 0 B 8 0 0 -14 8 C 4 0 0 -10 2 D 8 14 10 0 14 E 0 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=29 D=23 B=10 E=8 so E is eliminated. Round 2 votes counts: A=30 C=29 D=24 B=17 so B is eliminated. Round 3 votes counts: A=38 C=32 D=30 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:223 B:201 C:198 A:190 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -4 -8 0 B 8 0 0 -14 8 C 4 0 0 -10 2 D 8 14 10 0 14 E 0 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -8 0 B 8 0 0 -14 8 C 4 0 0 -10 2 D 8 14 10 0 14 E 0 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -8 0 B 8 0 0 -14 8 C 4 0 0 -10 2 D 8 14 10 0 14 E 0 -8 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7420: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) B C D A E (7) E C B A D (6) D B C A E (5) A E D B C (5) C B D E A (4) A B D C E (4) E C D B A (3) E A D B C (3) D A E B C (3) D A B E C (3) C E B D A (3) C D B E A (3) B D A C E (3) A D E B C (3) D E A C B (2) D B A C E (2) C B E D A (2) C B D A E (2) B D C A E (2) B A D C E (2) A B E C D (2) E D C A B (1) E C D A B (1) E C B D A (1) E C A D B (1) E C A B D (1) E A C D B (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E B A (1) C E B A D (1) C D E B A (1) C D B A E (1) C B E A D (1) B C A E D (1) B A C E D (1) B A C D E (1) A E B D C (1) A E B C D (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 2 -2 4 B 8 0 6 0 -2 C -2 -6 0 -6 -6 D 2 0 6 0 4 E -4 2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.310041 C: 0.000000 D: 0.689959 E: 0.000000 Sum of squares = 0.572168710127 Cumulative probabilities = A: 0.000000 B: 0.310041 C: 0.310041 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -2 4 B 8 0 6 0 -2 C -2 -6 0 -6 -6 D 2 0 6 0 4 E -4 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=19 C=18 D=17 B=17 so D is eliminated. Round 2 votes counts: E=32 A=25 B=24 C=19 so C is eliminated. Round 3 votes counts: E=38 B=37 A=25 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:206 D:206 E:200 A:198 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -2 4 B 8 0 6 0 -2 C -2 -6 0 -6 -6 D 2 0 6 0 4 E -4 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -2 4 B 8 0 6 0 -2 C -2 -6 0 -6 -6 D 2 0 6 0 4 E -4 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -2 4 B 8 0 6 0 -2 C -2 -6 0 -6 -6 D 2 0 6 0 4 E -4 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7421: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (6) D B E C A (6) B D E C A (6) E C B D A (5) B D E A C (5) A C D B E (5) C E D B A (4) C E A D B (4) C A E D B (4) B E D A C (4) A D B C E (4) A B D E C (4) E C D B A (3) D B A C E (3) B D A E C (3) B A D E C (3) A C E D B (3) A C E B D (3) E D B C A (2) E C A B D (2) D B C E A (2) B E D C A (2) A E B D C (2) A C D E B (2) A C B D E (2) A B D C E (2) E B A D C (1) E A C B D (1) D C B E A (1) C E A B D (1) C D B E A (1) B A E D C (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 2 -6 -12 B 16 0 14 12 10 C -2 -14 0 -14 -16 D 6 -12 14 0 -2 E 12 -10 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 2 -6 -12 B 16 0 14 12 10 C -2 -14 0 -14 -16 D 6 -12 14 0 -2 E 12 -10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 E=20 C=14 D=12 so D is eliminated. Round 2 votes counts: B=35 A=30 E=20 C=15 so C is eliminated. Round 3 votes counts: B=37 A=34 E=29 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:210 D:203 A:184 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 2 -6 -12 B 16 0 14 12 10 C -2 -14 0 -14 -16 D 6 -12 14 0 -2 E 12 -10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 -6 -12 B 16 0 14 12 10 C -2 -14 0 -14 -16 D 6 -12 14 0 -2 E 12 -10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 -6 -12 B 16 0 14 12 10 C -2 -14 0 -14 -16 D 6 -12 14 0 -2 E 12 -10 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7422: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (6) D E B C A (5) D C E A B (5) D B A C E (5) B A E C D (5) A B C E D (5) C A D E B (4) B A C E D (4) E D B A C (3) E C A B D (3) D E C A B (3) D C A E B (3) D B E A C (3) D B A E C (3) C A D B E (3) B D A E C (3) A C B D E (3) E D C A B (2) E B A C D (2) D E B A C (2) C E A D B (2) C D A B E (2) C A E B D (2) C A B D E (2) B E A D C (2) A C E B D (2) A C B E D (2) E C A D B (1) E A B C D (1) D C E B A (1) D B E C A (1) D B C A E (1) C E A B D (1) C D E A B (1) C A E D B (1) C A B E D (1) B E A C D (1) B D A C E (1) B A D C E (1) B A C D E (1) A E C B D (1) Total count = 100 A B C D E A 0 12 0 0 22 B -12 0 -2 -14 10 C 0 2 0 0 18 D 0 14 0 0 18 E -22 -10 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.191645 B: 0.000000 C: 0.385974 D: 0.422381 E: 0.000000 Sum of squares = 0.364109534821 Cumulative probabilities = A: 0.191645 B: 0.191645 C: 0.577619 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 0 22 B -12 0 -2 -14 10 C 0 2 0 0 18 D 0 14 0 0 18 E -22 -10 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333334 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=19 B=18 A=13 E=12 so E is eliminated. Round 2 votes counts: D=43 C=23 B=20 A=14 so A is eliminated. Round 3 votes counts: D=43 C=31 B=26 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:217 D:216 C:210 B:191 E:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 0 22 B -12 0 -2 -14 10 C 0 2 0 0 18 D 0 14 0 0 18 E -22 -10 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333334 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 0 22 B -12 0 -2 -14 10 C 0 2 0 0 18 D 0 14 0 0 18 E -22 -10 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333334 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 0 22 B -12 0 -2 -14 10 C 0 2 0 0 18 D 0 14 0 0 18 E -22 -10 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.333334 E: 0.000000 Sum of squares = 0.333333333334 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666666 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7423: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (14) B D A C E (13) E C A D B (6) E A C B D (6) C E D A B (6) A B E D C (6) B A D C E (5) D C E B A (4) C D E B A (4) A B E C D (4) E A C D B (3) D B C E A (3) A E B C D (3) A B D C E (3) E C D A B (2) D C B E A (2) D B C A E (2) C E D B A (2) B D C A E (2) B A D E C (2) A B D E C (2) E C B D A (1) E C A B D (1) E B A C D (1) D B A C E (1) B D E C A (1) B D C E A (1) Total count = 100 A B C D E A 0 12 26 12 14 B -12 0 -2 30 -6 C -26 2 0 6 -4 D -12 -30 -6 0 -10 E -14 6 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 26 12 14 B -12 0 -2 30 -6 C -26 2 0 6 -4 D -12 -30 -6 0 -10 E -14 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=24 E=20 D=12 C=12 so D is eliminated. Round 2 votes counts: A=32 B=30 E=20 C=18 so C is eliminated. Round 3 votes counts: E=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:232 B:205 E:203 C:189 D:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 26 12 14 B -12 0 -2 30 -6 C -26 2 0 6 -4 D -12 -30 -6 0 -10 E -14 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 26 12 14 B -12 0 -2 30 -6 C -26 2 0 6 -4 D -12 -30 -6 0 -10 E -14 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 26 12 14 B -12 0 -2 30 -6 C -26 2 0 6 -4 D -12 -30 -6 0 -10 E -14 6 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7424: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) B C E A D (6) D A E C B (5) B E D C A (5) D E B A C (4) D B E C A (4) B E C D A (4) A C E D B (4) E B D C A (3) E B C A D (3) D E A B C (3) D B E A C (3) D A C B E (3) D A B C E (3) B E C A D (3) A C D B E (3) D E A C B (2) D B A C E (2) C B A E D (2) C A E B D (2) B C A D E (2) A E C D B (2) A C E B D (2) A C D E B (2) E D A C B (1) E C B A D (1) E C A B D (1) E B C D A (1) E A C D B (1) D E B C A (1) D A C E B (1) D A B E C (1) C E A B D (1) C B E A D (1) C A B E D (1) C A B D E (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D A E (1) B C A E D (1) A D C E B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 -10 -16 -16 B 16 0 22 6 18 C 10 -22 0 -6 -12 D 16 -6 6 0 6 E 16 -18 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -16 -16 B 16 0 22 6 18 C 10 -22 0 -6 -12 D 16 -6 6 0 6 E 16 -18 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=32 A=16 E=11 C=8 so C is eliminated. Round 2 votes counts: B=36 D=32 A=20 E=12 so E is eliminated. Round 3 votes counts: B=44 D=33 A=23 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 D:211 E:202 C:185 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -16 -16 B 16 0 22 6 18 C 10 -22 0 -6 -12 D 16 -6 6 0 6 E 16 -18 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -16 -16 B 16 0 22 6 18 C 10 -22 0 -6 -12 D 16 -6 6 0 6 E 16 -18 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -16 -16 B 16 0 22 6 18 C 10 -22 0 -6 -12 D 16 -6 6 0 6 E 16 -18 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7425: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (12) A D B E C (10) A E C D B (8) B D E C A (6) B D C E A (6) B D A E C (6) D B A E C (5) A C E D B (4) E C A D B (3) D A B E C (3) C E A D B (3) B C E D A (3) A D B C E (3) C E B A D (2) C E A B D (2) C B E D A (2) B E C D A (2) B D A C E (2) B C D E A (2) E C D B A (1) E C B D A (1) E B C D A (1) D B E A C (1) D B A C E (1) D A E B C (1) D A B C E (1) C B D E A (1) C A E D B (1) C A E B D (1) B C D A E (1) A E D C B (1) A D E C B (1) A D E B C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 0 -16 4 B 10 0 12 4 14 C 0 -12 0 2 -2 D 16 -4 -2 0 6 E -4 -14 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -16 4 B 10 0 12 4 14 C 0 -12 0 2 -2 D 16 -4 -2 0 6 E -4 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=28 C=24 D=12 E=6 so E is eliminated. Round 2 votes counts: A=30 C=29 B=29 D=12 so D is eliminated. Round 3 votes counts: B=36 A=35 C=29 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:208 C:194 A:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -16 4 B 10 0 12 4 14 C 0 -12 0 2 -2 D 16 -4 -2 0 6 E -4 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -16 4 B 10 0 12 4 14 C 0 -12 0 2 -2 D 16 -4 -2 0 6 E -4 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -16 4 B 10 0 12 4 14 C 0 -12 0 2 -2 D 16 -4 -2 0 6 E -4 -14 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7426: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B D A E C (5) B A D E C (5) B A C D E (5) C E D A B (4) C A D B E (4) E D C A B (3) E D B A C (3) E D A C B (3) E D A B C (3) E C B D A (3) E C B A D (3) E B D A C (3) D A B E C (3) C E B A D (3) C A B D E (3) B E C A D (3) B A D C E (3) E C D B A (2) E B C D A (2) E B C A D (2) C E A D B (2) C D A E B (2) C B E A D (2) C B A D E (2) C A E D B (2) C A D E B (2) A D B C E (2) E B D C A (1) D E A C B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E A B D (1) C D E A B (1) B E D A C (1) B E A D C (1) B C E A D (1) B C A D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -16 -4 -14 B 2 0 -8 2 -12 C 16 8 0 18 -12 D 4 -2 -18 0 -14 E 14 12 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -16 -4 -14 B 2 0 -8 2 -12 C 16 8 0 18 -12 D 4 -2 -18 0 -14 E 14 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=28 B=25 D=7 A=3 so A is eliminated. Round 2 votes counts: E=37 C=28 B=26 D=9 so D is eliminated. Round 3 votes counts: E=38 B=32 C=30 so C is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:215 B:192 D:185 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -16 -4 -14 B 2 0 -8 2 -12 C 16 8 0 18 -12 D 4 -2 -18 0 -14 E 14 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -4 -14 B 2 0 -8 2 -12 C 16 8 0 18 -12 D 4 -2 -18 0 -14 E 14 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -4 -14 B 2 0 -8 2 -12 C 16 8 0 18 -12 D 4 -2 -18 0 -14 E 14 12 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7427: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (6) C B E D A (5) B E C D A (5) A D E B C (5) E D B C A (4) E B D C A (4) D E B C A (4) C A B E D (4) B E D C A (4) B E D A C (4) A C D E B (4) D E A B C (3) C E B D A (3) B C E D A (3) A D C E B (3) A C B E D (3) A C B D E (3) A B D E C (3) D E C B A (2) D E B A C (2) D C E A B (2) C B E A D (2) C A E D B (2) B E A D C (2) B C E A D (2) B A E D C (2) A C D B E (2) E C D B A (1) E B C D A (1) D E A C B (1) D C A E B (1) D B E A C (1) C D E A B (1) C B A E D (1) B C A E D (1) B A C E D (1) A D B E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -16 -2 -12 B 8 0 8 10 2 C 16 -8 0 4 2 D 2 -10 -4 0 -10 E 12 -2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -2 -12 B 8 0 8 10 2 C 16 -8 0 4 2 D 2 -10 -4 0 -10 E 12 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 B=24 D=16 E=10 so E is eliminated. Round 2 votes counts: B=29 A=26 C=25 D=20 so D is eliminated. Round 3 votes counts: B=40 C=30 A=30 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:209 C:207 D:189 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -16 -2 -12 B 8 0 8 10 2 C 16 -8 0 4 2 D 2 -10 -4 0 -10 E 12 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -2 -12 B 8 0 8 10 2 C 16 -8 0 4 2 D 2 -10 -4 0 -10 E 12 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -2 -12 B 8 0 8 10 2 C 16 -8 0 4 2 D 2 -10 -4 0 -10 E 12 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7428: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (7) E D C A B (6) E D B C A (6) B A C D E (6) A B C D E (6) E D C B A (5) D E B C A (5) A C B D E (5) E C D A B (4) D E C B A (4) A B C E D (4) E C A D B (3) E A C D B (3) D B C E A (3) B D E C A (3) E D A B C (2) E A D C B (2) D C E B A (2) D C B E A (2) C D B A E (2) B A D E C (2) A E C D B (2) A C B E D (2) A B E C D (2) E D B A C (1) E B D A C (1) E A D B C (1) D B E C A (1) C E D A B (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E A C (1) B D A C E (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 2 -2 -14 B 6 0 8 -14 0 C -2 -8 0 -14 -12 D 2 14 14 0 2 E 14 0 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -2 -14 B 6 0 8 -14 0 C -2 -8 0 -14 -12 D 2 14 14 0 2 E 14 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=23 B=21 D=17 C=5 so C is eliminated. Round 2 votes counts: E=35 A=25 B=21 D=19 so D is eliminated. Round 3 votes counts: E=46 B=29 A=25 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:216 E:212 B:200 A:190 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -2 -14 B 6 0 8 -14 0 C -2 -8 0 -14 -12 D 2 14 14 0 2 E 14 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 -14 B 6 0 8 -14 0 C -2 -8 0 -14 -12 D 2 14 14 0 2 E 14 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 -14 B 6 0 8 -14 0 C -2 -8 0 -14 -12 D 2 14 14 0 2 E 14 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7429: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) B C D E A (7) E A D B C (6) B C E D A (6) E B C D A (5) D B C E A (5) C B D A E (5) A E D C B (5) A D C B E (5) A E C B D (4) A C D B E (4) E B D C A (3) D C B A E (3) E B A C D (2) D A C B E (2) C D B A E (2) B E C D A (2) B D C E A (2) A E C D B (2) A D C E B (2) A C E B D (2) A C B D E (2) E D B C A (1) E D B A C (1) E B C A D (1) E A C B D (1) D E B C A (1) D C A B E (1) C D A B E (1) C B A E D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D C A (1) B D E C A (1) B C E A D (1) A E D B C (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -4 2 -6 B 0 0 6 12 10 C 4 -6 0 18 10 D -2 -12 -18 0 -8 E 6 -10 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.328870 B: 0.671130 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.558570799219 Cumulative probabilities = A: 0.328870 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 2 -6 B 0 0 6 12 10 C 4 -6 0 18 10 D -2 -12 -18 0 -8 E 6 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 B=20 D=12 C=12 so D is eliminated. Round 2 votes counts: A=31 E=28 B=25 C=16 so C is eliminated. Round 3 votes counts: B=36 A=36 E=28 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:214 C:213 E:197 A:196 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 2 -6 B 0 0 6 12 10 C 4 -6 0 18 10 D -2 -12 -18 0 -8 E 6 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 2 -6 B 0 0 6 12 10 C 4 -6 0 18 10 D -2 -12 -18 0 -8 E 6 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 2 -6 B 0 0 6 12 10 C 4 -6 0 18 10 D -2 -12 -18 0 -8 E 6 -10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7430: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) A B E C D (11) E A B D C (10) C D B A E (9) D C E B A (8) E D C A B (7) D E C B A (6) D E C A B (5) C B A D E (5) B A C E D (4) B A C D E (4) C B D A E (3) D C E A B (2) C D B E A (2) A E B D C (2) E D C B A (1) E D A C B (1) E B A D C (1) E A D B C (1) D C B E A (1) D B C E A (1) C A B D E (1) B C A D E (1) B A E C D (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -4 -14 -14 B -6 0 -2 -10 -12 C 4 2 0 -14 -16 D 14 10 14 0 -4 E 14 12 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 -14 -14 B -6 0 -2 -10 -12 C 4 2 0 -14 -16 D 14 10 14 0 -4 E 14 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=23 C=20 A=15 B=10 so B is eliminated. Round 2 votes counts: E=32 A=24 D=23 C=21 so C is eliminated. Round 3 votes counts: D=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:217 C:188 A:187 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 -14 -14 B -6 0 -2 -10 -12 C 4 2 0 -14 -16 D 14 10 14 0 -4 E 14 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -14 -14 B -6 0 -2 -10 -12 C 4 2 0 -14 -16 D 14 10 14 0 -4 E 14 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -14 -14 B -6 0 -2 -10 -12 C 4 2 0 -14 -16 D 14 10 14 0 -4 E 14 12 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7431: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (14) C E A B D (9) C E B D A (7) A D B C E (6) E B D C A (5) D B A E C (5) C B D A E (5) A E D B C (5) C B E D A (4) E C B D A (3) C B D E A (3) C A B D E (3) B D C A E (3) A E C D B (3) E C A B D (2) E B C D A (2) B D E C A (2) B D C E A (2) E C A D B (1) D B E A C (1) D B A C E (1) D A B E C (1) C E B A D (1) C E A D B (1) C B A D E (1) C A E B D (1) C A D B E (1) B D A E C (1) B D A C E (1) B C D E A (1) A E D C B (1) A D E C B (1) A D E B C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -14 6 12 B -4 0 2 12 12 C 14 -2 0 0 4 D -6 -12 0 0 10 E -12 -12 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.540000000017 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 6 12 B -4 0 2 12 12 C 14 -2 0 0 4 D -6 -12 0 0 10 E -12 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.540000000703 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=33 E=13 B=10 D=8 so D is eliminated. Round 2 votes counts: C=36 A=34 B=17 E=13 so E is eliminated. Round 3 votes counts: C=42 A=34 B=24 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:208 A:204 D:196 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 6 12 B -4 0 2 12 12 C 14 -2 0 0 4 D -6 -12 0 0 10 E -12 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.540000000703 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 6 12 B -4 0 2 12 12 C 14 -2 0 0 4 D -6 -12 0 0 10 E -12 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.540000000703 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 6 12 B -4 0 2 12 12 C 14 -2 0 0 4 D -6 -12 0 0 10 E -12 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.700000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.540000000703 Cumulative probabilities = A: 0.100000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7432: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) D E A C B (8) E D A C B (7) B C A D E (7) E D B A C (6) C A B D E (6) B E D C A (6) A C D E B (6) D A C E B (5) E B D A C (3) A C B E D (3) D E C A B (2) B E C D A (2) B E C A D (2) A D C E B (2) A C D B E (2) E B A D C (1) E B A C D (1) E A D C B (1) D E A B C (1) D C A E B (1) D B C A E (1) C B A E D (1) C B A D E (1) C A D E B (1) B E D A C (1) B E A D C (1) B E A C D (1) B D E C A (1) B C E D A (1) B C D A E (1) B A C E D (1) A D E C B (1) A C E D B (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 8 8 10 B 2 0 0 10 4 C -8 0 0 4 10 D -8 -10 -4 0 -6 E -10 -4 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.877564 C: 0.122436 D: 0.000000 E: 0.000000 Sum of squares = 0.785108686749 Cumulative probabilities = A: 0.000000 B: 0.877564 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 8 10 B 2 0 0 10 4 C -8 0 0 4 10 D -8 -10 -4 0 -6 E -10 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000013866 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=19 D=18 A=17 C=9 so C is eliminated. Round 2 votes counts: B=39 A=24 E=19 D=18 so D is eliminated. Round 3 votes counts: B=40 E=30 A=30 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:212 B:208 C:203 E:191 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 8 10 B 2 0 0 10 4 C -8 0 0 4 10 D -8 -10 -4 0 -6 E -10 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000013866 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 8 10 B 2 0 0 10 4 C -8 0 0 4 10 D -8 -10 -4 0 -6 E -10 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000013866 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 8 10 B 2 0 0 10 4 C -8 0 0 4 10 D -8 -10 -4 0 -6 E -10 -4 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000013866 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7433: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) E D C A B (7) B A C D E (6) E C A D B (5) E C B A D (4) D E C A B (4) D E B C A (4) C E A B D (4) B D A C E (4) B C A E D (4) E C A B D (3) D B A C E (3) D A E C B (3) D A E B C (3) C A E B D (3) A C B E D (3) E C D A B (2) D E B A C (2) D B E A C (2) D A B C E (2) C E B A D (2) C B A E D (2) B D E C A (2) B C E A D (2) A D C B E (2) A B C D E (2) E C B D A (1) D E C B A (1) D B E C A (1) D B A E C (1) D A B E C (1) B E C A D (1) B D C A E (1) B A C E D (1) A C E B D (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -6 -4 -10 B -12 0 -12 -2 -14 C 6 12 0 -4 -10 D 4 2 4 0 8 E 10 14 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 -4 -10 B -12 0 -12 -2 -14 C 6 12 0 -4 -10 D 4 2 4 0 8 E 10 14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=22 B=21 C=11 A=11 so C is eliminated. Round 2 votes counts: D=35 E=28 B=23 A=14 so A is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:209 C:202 A:196 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -6 -4 -10 B -12 0 -12 -2 -14 C 6 12 0 -4 -10 D 4 2 4 0 8 E 10 14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -4 -10 B -12 0 -12 -2 -14 C 6 12 0 -4 -10 D 4 2 4 0 8 E 10 14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -4 -10 B -12 0 -12 -2 -14 C 6 12 0 -4 -10 D 4 2 4 0 8 E 10 14 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7434: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) A C D B E (7) B A E D C (6) E B D C A (4) D C E B A (4) C D A E B (4) B E A D C (4) A C B D E (4) A B C D E (4) D C B E A (3) B E D C A (3) B A D C E (3) A C E D B (3) A C B E D (3) A B E C D (3) E D B C A (2) E C D A B (2) E B A D C (2) E A C D B (2) C D E A B (2) C A D E B (2) B D E C A (2) B D E A C (2) B D A E C (2) B A D E C (2) A C E B D (2) E C D B A (1) E C A D B (1) D E B C A (1) D C B A E (1) D C A B E (1) C E D A B (1) C A E D B (1) C A D B E (1) B E D A C (1) B D A C E (1) B A E C D (1) B A C D E (1) A E C D B (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 12 10 10 B 8 0 -12 2 10 C -12 12 0 -4 -2 D -10 -2 4 0 -4 E -10 -10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999985 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 10 10 B 8 0 -12 2 10 C -12 12 0 -4 -2 D -10 -2 4 0 -4 E -10 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999987 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 E=22 C=11 D=10 so D is eliminated. Round 2 votes counts: A=29 B=28 E=23 C=20 so C is eliminated. Round 3 votes counts: A=38 B=32 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:212 B:204 C:197 D:194 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -8 12 10 10 B 8 0 -12 2 10 C -12 12 0 -4 -2 D -10 -2 4 0 -4 E -10 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999987 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 10 10 B 8 0 -12 2 10 C -12 12 0 -4 -2 D -10 -2 4 0 -4 E -10 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999987 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 10 10 B 8 0 -12 2 10 C -12 12 0 -4 -2 D -10 -2 4 0 -4 E -10 -10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.375000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.343749999987 Cumulative probabilities = A: 0.375000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7435: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (10) C E D B A (9) C E B D A (9) B E C A D (9) E C B D A (7) B A E D C (7) D A C E B (6) C D E A B (6) A D B C E (6) D C A E B (5) A D B E C (5) B A D E C (3) E C D B A (2) E B C A D (2) D C E A B (2) C E D A B (2) C D A E B (2) B E A C D (2) B A E C D (2) D E C A B (1) B E A D C (1) B A D C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -12 -2 -4 B 8 0 -2 8 -6 C 12 2 0 4 -2 D 2 -8 -4 0 -4 E 4 6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 -2 -4 B 8 0 -2 8 -6 C 12 2 0 4 -2 D 2 -8 -4 0 -4 E 4 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 A=22 D=14 E=11 so E is eliminated. Round 2 votes counts: C=37 B=27 A=22 D=14 so D is eliminated. Round 3 votes counts: C=45 A=28 B=27 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:208 E:208 B:204 D:193 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 -2 -4 B 8 0 -2 8 -6 C 12 2 0 4 -2 D 2 -8 -4 0 -4 E 4 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -2 -4 B 8 0 -2 8 -6 C 12 2 0 4 -2 D 2 -8 -4 0 -4 E 4 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -2 -4 B 8 0 -2 8 -6 C 12 2 0 4 -2 D 2 -8 -4 0 -4 E 4 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7436: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) E D A B C (6) E B D A C (6) A D C B E (6) E C D A B (4) D A E C B (4) C D A E B (4) C A D E B (4) C A B D E (4) B E D A C (4) B A D E C (4) E D C A B (3) C B E A D (3) B D A E C (3) A C D B E (3) E D B A C (2) E D A C B (2) E B D C A (2) E B C D A (2) D A E B C (2) D A C E B (2) C E D A B (2) C E B A D (2) B C E A D (2) B A D C E (2) E C D B A (1) D E C A B (1) D C A E B (1) C E B D A (1) C E A D B (1) C B A E D (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B C A D E (1) B A E D C (1) B A C D E (1) A D C E B (1) Total count = 100 A B C D E A 0 16 0 -8 4 B -16 0 -16 -14 -6 C 0 16 0 -8 -2 D 8 14 8 0 2 E -4 6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 -8 4 B -16 0 -16 -14 -6 C 0 16 0 -8 -2 D 8 14 8 0 2 E -4 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=28 B=22 D=10 A=10 so D is eliminated. Round 2 votes counts: C=31 E=29 B=22 A=18 so A is eliminated. Round 3 votes counts: C=43 E=35 B=22 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:216 A:206 C:203 E:201 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 0 -8 4 B -16 0 -16 -14 -6 C 0 16 0 -8 -2 D 8 14 8 0 2 E -4 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 -8 4 B -16 0 -16 -14 -6 C 0 16 0 -8 -2 D 8 14 8 0 2 E -4 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 -8 4 B -16 0 -16 -14 -6 C 0 16 0 -8 -2 D 8 14 8 0 2 E -4 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999219 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7437: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A C B E D (8) B C A E D (7) D E A C B (5) B C E D A (5) B A C D E (5) E D C A B (4) D E C B A (4) D E C A B (4) D E B A C (3) D B E A C (3) B D E C A (3) A B C E D (3) E D C B A (2) E D A C B (2) D A E C B (2) C E B A D (2) C B A E D (2) B A D C E (2) A E D C B (2) A D E C B (2) A B D C E (2) A B C D E (2) E D B C A (1) E C D A B (1) E C A D B (1) D E A B C (1) D B E C A (1) D A B E C (1) C E A B D (1) C B E A D (1) C A B E D (1) B D A E C (1) B D A C E (1) B C E A D (1) B C A D E (1) B A C E D (1) A D B C E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -2 -6 -8 B 10 0 8 0 4 C 2 -8 0 -12 -4 D 6 0 12 0 8 E 8 -4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.616027 C: 0.000000 D: 0.383973 E: 0.000000 Sum of squares = 0.526924322783 Cumulative probabilities = A: 0.000000 B: 0.616027 C: 0.616027 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -6 -8 B 10 0 8 0 4 C 2 -8 0 -12 -4 D 6 0 12 0 8 E 8 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=27 A=22 E=11 C=7 so C is eliminated. Round 2 votes counts: D=33 B=30 A=23 E=14 so E is eliminated. Round 3 votes counts: D=43 B=32 A=25 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:211 E:200 C:189 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -6 -8 B 10 0 8 0 4 C 2 -8 0 -12 -4 D 6 0 12 0 8 E 8 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -6 -8 B 10 0 8 0 4 C 2 -8 0 -12 -4 D 6 0 12 0 8 E 8 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -6 -8 B 10 0 8 0 4 C 2 -8 0 -12 -4 D 6 0 12 0 8 E 8 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7438: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) B A C D E (8) A B E C D (8) C E D A B (7) D C E B A (6) A B C E D (6) E C D A B (4) C D E B A (4) B A D E C (4) E D C B A (3) E C A D B (3) E A C B D (3) D B C A E (3) E D C A B (2) E D A B C (2) E A B C D (2) D E B C A (2) D C B E A (2) B C A D E (2) B A D C E (2) A E B C D (2) A B E D C (2) A B C D E (2) E D B A C (1) E A D B C (1) D E B A C (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E A B (1) C B D A E (1) C A B D E (1) B D C A E (1) A C E B D (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -2 0 -6 B 0 0 4 -4 -8 C 2 -4 0 14 0 D 0 4 -14 0 4 E 6 8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.443925 D: 0.000000 E: 0.556075 Sum of squares = 0.506288780812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.443925 D: 0.443925 E: 1.000000 A B C D E A 0 0 -2 0 -6 B 0 0 4 -4 -8 C 2 -4 0 14 0 D 0 4 -14 0 4 E 6 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=23 E=21 B=17 C=15 so C is eliminated. Round 2 votes counts: E=29 D=29 A=24 B=18 so B is eliminated. Round 3 votes counts: A=40 D=31 E=29 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:206 E:205 D:197 A:196 B:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 0 -6 B 0 0 4 -4 -8 C 2 -4 0 14 0 D 0 4 -14 0 4 E 6 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 0 -6 B 0 0 4 -4 -8 C 2 -4 0 14 0 D 0 4 -14 0 4 E 6 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 0 -6 B 0 0 4 -4 -8 C 2 -4 0 14 0 D 0 4 -14 0 4 E 6 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7439: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) C B D A E (7) A E D C B (7) D E A C B (6) C B D E A (6) B C D A E (6) A E D B C (6) B C A E D (5) B A C E D (4) E A D B C (3) C D B E A (3) B C A D E (3) B A E D C (3) B A E C D (3) A E B D C (3) A D C E B (3) E A B D C (2) D E C A B (2) D C E A B (2) C D E A B (2) C D B A E (2) C D A E B (2) B E A D C (2) B E A C D (2) B C D E A (2) A D E C B (2) A B E D C (2) E D C A B (1) C D E B A (1) C D A B E (1) Total count = 100 A B C D E A 0 2 10 0 18 B -2 0 -8 0 2 C -10 8 0 -2 -2 D 0 0 2 0 0 E -18 -2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.364065 B: 0.000000 C: 0.000000 D: 0.635935 E: 0.000000 Sum of squares = 0.536956764934 Cumulative probabilities = A: 0.364065 B: 0.364065 C: 0.364065 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 0 18 B -2 0 -8 0 2 C -10 8 0 -2 -2 D 0 0 2 0 0 E -18 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=24 A=23 E=13 D=10 so D is eliminated. Round 2 votes counts: B=30 C=26 A=23 E=21 so E is eliminated. Round 3 votes counts: A=41 B=30 C=29 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:201 C:197 B:196 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 0 18 B -2 0 -8 0 2 C -10 8 0 -2 -2 D 0 0 2 0 0 E -18 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 0 18 B -2 0 -8 0 2 C -10 8 0 -2 -2 D 0 0 2 0 0 E -18 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 0 18 B -2 0 -8 0 2 C -10 8 0 -2 -2 D 0 0 2 0 0 E -18 -2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999964 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7440: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (14) A E D C B (13) B C D E A (9) C A E D B (8) D E A C B (6) D E A B C (5) B C D A E (4) B C A E D (4) A E C D B (4) E A D B C (3) A E D B C (3) D B E C A (2) D B E A C (2) C D B E A (2) C B D A E (2) C B A E D (2) B E D A C (2) B D E C A (2) B D C E A (2) B C A D E (2) A C E D B (2) E A D C B (1) D E C A B (1) D E B A C (1) C B D E A (1) C A B E D (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 16 -10 -6 B 4 0 14 -6 4 C -16 -14 0 -18 -22 D 10 6 18 0 10 E 6 -4 22 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 -10 -6 B 4 0 14 -6 4 C -16 -14 0 -18 -22 D 10 6 18 0 10 E 6 -4 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=23 D=17 C=16 E=4 so E is eliminated. Round 2 votes counts: B=40 A=27 D=17 C=16 so C is eliminated. Round 3 votes counts: B=45 A=36 D=19 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:208 E:207 A:198 C:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 16 -10 -6 B 4 0 14 -6 4 C -16 -14 0 -18 -22 D 10 6 18 0 10 E 6 -4 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 -10 -6 B 4 0 14 -6 4 C -16 -14 0 -18 -22 D 10 6 18 0 10 E 6 -4 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 -10 -6 B 4 0 14 -6 4 C -16 -14 0 -18 -22 D 10 6 18 0 10 E 6 -4 22 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7441: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (13) E D B A C (6) D A C B E (5) D A B C E (5) E B C A D (4) E D A C B (3) C B E A D (3) B D A C E (3) B C A E D (3) B A D C E (3) A D C B E (3) E D C A B (2) E D A B C (2) E C D B A (2) E C D A B (2) E C B D A (2) E C A D B (2) E C A B D (2) E B C D A (2) D E A B C (2) D B A E C (2) D B A C E (2) D A B E C (2) C E B A D (2) C D A E B (2) C B A E D (2) C B A D E (2) C A B D E (2) B D A E C (2) A C D B E (2) E D B C A (1) E B A C D (1) D E A C B (1) D C A E B (1) C E A D B (1) C A E D B (1) B E A C D (1) B D E A C (1) B C A D E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -4 0 -10 B 16 0 -10 0 -8 C 4 10 0 4 -10 D 0 0 -4 0 -14 E 10 8 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -4 0 -10 B 16 0 -10 0 -8 C 4 10 0 4 -10 D 0 0 -4 0 -14 E 10 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 D=20 C=15 B=14 A=7 so A is eliminated. Round 2 votes counts: E=44 D=24 C=17 B=15 so B is eliminated. Round 3 votes counts: E=45 D=34 C=21 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:204 B:199 D:191 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -4 0 -10 B 16 0 -10 0 -8 C 4 10 0 4 -10 D 0 0 -4 0 -14 E 10 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 0 -10 B 16 0 -10 0 -8 C 4 10 0 4 -10 D 0 0 -4 0 -14 E 10 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 0 -10 B 16 0 -10 0 -8 C 4 10 0 4 -10 D 0 0 -4 0 -14 E 10 8 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7442: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (25) A E B D C (11) B C D E A (7) A E D C B (7) A E B C D (5) D C B E A (4) D C A E B (3) C D B A E (3) C D A B E (3) B E A C D (3) A E D B C (3) E B A D C (2) D C A B E (2) C B D E A (2) A D E C B (2) A B E C D (2) E B A C D (1) E A B D C (1) D E C A B (1) D C E B A (1) D C E A B (1) D A E C B (1) C D A E B (1) C B D A E (1) C B A D E (1) C A D B E (1) B E C D A (1) B E A D C (1) B A E C D (1) B A C D E (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -14 -12 0 B 8 0 -22 -20 16 C 14 22 0 18 16 D 12 20 -18 0 22 E 0 -16 -16 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -12 0 B 8 0 -22 -20 16 C 14 22 0 18 16 D 12 20 -18 0 22 E 0 -16 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=32 B=14 D=13 E=4 so E is eliminated. Round 2 votes counts: C=37 A=33 B=17 D=13 so D is eliminated. Round 3 votes counts: C=49 A=34 B=17 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:235 D:218 B:191 A:183 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -12 0 B 8 0 -22 -20 16 C 14 22 0 18 16 D 12 20 -18 0 22 E 0 -16 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -12 0 B 8 0 -22 -20 16 C 14 22 0 18 16 D 12 20 -18 0 22 E 0 -16 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -12 0 B 8 0 -22 -20 16 C 14 22 0 18 16 D 12 20 -18 0 22 E 0 -16 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7443: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) C B D A E (7) D B A C E (6) D A B E C (6) C E B A D (6) E D A B C (5) E C D A B (5) C B E D A (5) C B A D E (5) E C B A D (4) D A E B C (4) B D A C E (4) B C D A E (4) B C A D E (4) E D A C B (3) E C A D B (3) C E B D A (3) E A C B D (2) C B E A D (2) E C A B D (1) E A C D B (1) D E A B C (1) D C B A E (1) D A B C E (1) C B D E A (1) C B A E D (1) B D C A E (1) B A D C E (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -6 -14 2 B 10 0 0 8 4 C 6 0 0 8 8 D 14 -8 -8 0 0 E -2 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.778873 C: 0.221127 D: 0.000000 E: 0.000000 Sum of squares = 0.655540733773 Cumulative probabilities = A: 0.000000 B: 0.778873 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -14 2 B 10 0 0 8 4 C 6 0 0 8 8 D 14 -8 -8 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 D=19 B=14 A=6 so A is eliminated. Round 2 votes counts: E=33 C=30 D=21 B=16 so B is eliminated. Round 3 votes counts: C=38 E=33 D=29 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:211 D:199 E:193 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -14 2 B 10 0 0 8 4 C 6 0 0 8 8 D 14 -8 -8 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -14 2 B 10 0 0 8 4 C 6 0 0 8 8 D 14 -8 -8 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -14 2 B 10 0 0 8 4 C 6 0 0 8 8 D 14 -8 -8 0 0 E -2 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7444: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) B E D C A (7) B E C D A (6) C A E D B (5) E C B D A (4) D B A E C (4) C E A B D (4) B D A E C (4) A C D B E (4) E D B C A (3) E B D C A (3) D B E A C (3) C A E B D (3) A D C E B (3) A D C B E (3) A D B C E (3) E D C A B (2) E C D A B (2) E C A D B (2) E B C D A (2) D A B C E (2) C A B E D (2) B D E A C (2) B A C D E (2) A C D E B (2) A C B D E (2) E D C B A (1) E D B A C (1) E D A C B (1) E C B A D (1) D E B A C (1) C E A D B (1) C B E A D (1) C B A E D (1) C A B D E (1) B E D A C (1) B D A C E (1) B C A D E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -18 0 -10 B 12 0 -6 12 2 C 18 6 0 8 0 D 0 -12 -8 0 -22 E 10 -2 0 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.629811 D: 0.000000 E: 0.370189 Sum of squares = 0.533701798844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.629811 D: 0.629811 E: 1.000000 A B C D E A 0 -12 -18 0 -10 B 12 0 -6 12 2 C 18 6 0 8 0 D 0 -12 -8 0 -22 E 10 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=24 E=22 A=19 D=10 so D is eliminated. Round 2 votes counts: B=31 C=25 E=23 A=21 so A is eliminated. Round 3 votes counts: C=40 B=37 E=23 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:215 B:210 A:180 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -18 0 -10 B 12 0 -6 12 2 C 18 6 0 8 0 D 0 -12 -8 0 -22 E 10 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 0 -10 B 12 0 -6 12 2 C 18 6 0 8 0 D 0 -12 -8 0 -22 E 10 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 0 -10 B 12 0 -6 12 2 C 18 6 0 8 0 D 0 -12 -8 0 -22 E 10 -2 0 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7445: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (8) B A E D C (7) A B D C E (7) E D C B A (6) A B C D E (5) E B D C A (4) C E D B A (4) B A D E C (4) A D C B E (4) A B D E C (4) A B C E D (4) C E D A B (3) A D B C E (3) D E C B A (2) D E B A C (2) D C E A B (2) C E B A D (2) C D E A B (2) B E D A C (2) B E C A D (2) B A E C D (2) A C B E D (2) E D B C A (1) E C B D A (1) E B D A C (1) E B C D A (1) E B C A D (1) D A B E C (1) C E A D B (1) C E A B D (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D C A (1) B D E A C (1) B A C E D (1) A D B E C (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 8 14 4 B 6 0 10 10 6 C -8 -10 0 -6 -4 D -14 -10 6 0 -14 E -4 -6 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 14 4 B 6 0 10 10 6 C -8 -10 0 -6 -4 D -14 -10 6 0 -14 E -4 -6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=23 B=20 C=17 D=7 so D is eliminated. Round 2 votes counts: A=34 E=27 B=20 C=19 so C is eliminated. Round 3 votes counts: E=42 A=38 B=20 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:216 A:210 E:204 C:186 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 14 4 B 6 0 10 10 6 C -8 -10 0 -6 -4 D -14 -10 6 0 -14 E -4 -6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 14 4 B 6 0 10 10 6 C -8 -10 0 -6 -4 D -14 -10 6 0 -14 E -4 -6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 14 4 B 6 0 10 10 6 C -8 -10 0 -6 -4 D -14 -10 6 0 -14 E -4 -6 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7446: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) E A C D B (8) B D C A E (8) A D E B C (7) E C A B D (6) D A B C E (6) C B E D A (6) A E D C B (5) D B A E C (4) C E A B D (4) D A B E C (3) C D A B E (3) D B A C E (2) C B E A D (2) C B D E A (2) C B D A E (2) B E D A C (2) B D C E A (2) B C D A E (2) A E D B C (2) E C B A D (1) E C A D B (1) E B A D C (1) E A C B D (1) D B C A E (1) D A E B C (1) C E A D B (1) C A D E B (1) B E C D A (1) B D A C E (1) B C D E A (1) A E C D B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -8 6 2 B -4 0 -12 4 -4 C 8 12 0 8 10 D -6 -4 -8 0 -4 E -2 4 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 6 2 B -4 0 -12 4 -4 C 8 12 0 8 10 D -6 -4 -8 0 -4 E -2 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=18 D=17 B=17 A=17 so D is eliminated. Round 2 votes counts: C=31 A=27 B=24 E=18 so E is eliminated. Round 3 votes counts: C=39 A=36 B=25 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:202 E:198 B:192 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 6 2 B -4 0 -12 4 -4 C 8 12 0 8 10 D -6 -4 -8 0 -4 E -2 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 6 2 B -4 0 -12 4 -4 C 8 12 0 8 10 D -6 -4 -8 0 -4 E -2 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 6 2 B -4 0 -12 4 -4 C 8 12 0 8 10 D -6 -4 -8 0 -4 E -2 4 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7447: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (18) A B C D E (15) E B D C A (6) E D C A B (5) E A D C B (5) D C E B A (4) B D C E A (4) E D B C A (3) C D B A E (3) B A C D E (3) A C B D E (3) D C B E A (2) C D E B A (2) C A D B E (2) B D C A E (2) B C D A E (2) A E B D C (2) A E B C D (2) A C D B E (2) E C D A B (1) E B A D C (1) D E C B A (1) C D B E A (1) C D A B E (1) C B D A E (1) B E D C A (1) B E A D C (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -18 -14 -10 B 10 0 -12 -6 -10 C 18 12 0 -14 -2 D 14 6 14 0 -4 E 10 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -18 -14 -10 B 10 0 -12 -6 -10 C 18 12 0 -14 -2 D 14 6 14 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=31 B=13 C=10 D=7 so D is eliminated. Round 2 votes counts: E=40 A=31 C=16 B=13 so B is eliminated. Round 3 votes counts: E=42 A=34 C=24 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:213 C:207 B:191 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -18 -14 -10 B 10 0 -12 -6 -10 C 18 12 0 -14 -2 D 14 6 14 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -14 -10 B 10 0 -12 -6 -10 C 18 12 0 -14 -2 D 14 6 14 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -14 -10 B 10 0 -12 -6 -10 C 18 12 0 -14 -2 D 14 6 14 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7448: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) B C E D A (6) B C E A D (6) D E A B C (5) B C D E A (5) A E D C B (5) C A D B E (4) E D B A C (3) E B D A C (3) E B A D C (3) E A D B C (3) C D B A E (3) C B A D E (3) B E D C A (3) B D C E A (3) B C D A E (3) A E C B D (3) D E B A C (2) D C B A E (2) C D A B E (2) C B D A E (2) C B A E D (2) C A B E D (2) B E C D A (2) A D E C B (2) A D C E B (2) A C E D B (2) E A D C B (1) D C A B E (1) D B E C A (1) D B C E A (1) D A E C B (1) D A C E B (1) C A E B D (1) C A B D E (1) B E D A C (1) B E C A D (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -8 -14 -12 B 10 0 16 0 8 C 8 -16 0 0 6 D 14 0 0 0 -10 E 12 -8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.842164 C: 0.000000 D: 0.157836 E: 0.000000 Sum of squares = 0.734152946185 Cumulative probabilities = A: 0.000000 B: 0.842164 C: 0.842164 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -14 -12 B 10 0 16 0 8 C 8 -16 0 0 6 D 14 0 0 0 -10 E 12 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.506172885333 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=20 C=20 A=16 D=14 so D is eliminated. Round 2 votes counts: B=32 E=27 C=23 A=18 so A is eliminated. Round 3 votes counts: E=39 B=32 C=29 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:204 D:202 C:199 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -14 -12 B 10 0 16 0 8 C 8 -16 0 0 6 D 14 0 0 0 -10 E 12 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.506172885333 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -14 -12 B 10 0 16 0 8 C 8 -16 0 0 6 D 14 0 0 0 -10 E 12 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.506172885333 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -14 -12 B 10 0 16 0 8 C 8 -16 0 0 6 D 14 0 0 0 -10 E 12 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.444444 E: 0.000000 Sum of squares = 0.506172885333 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7449: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (17) B C E A D (13) A D E C B (11) D A B C E (10) E C B A D (9) C B E A D (7) A D B C E (6) D A E C B (4) A E C B D (4) E A C B D (2) D A E B C (2) B C D E A (2) A E D C B (2) E D A C B (1) E C A D B (1) E C A B D (1) E B C D A (1) E A D C B (1) D B C A E (1) D B A C E (1) D A B E C (1) B C D A E (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -6 18 -10 B 4 0 10 16 20 C 6 -10 0 20 18 D -18 -16 -20 0 -22 E 10 -20 -18 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 18 -10 B 4 0 10 16 20 C 6 -10 0 20 18 D -18 -16 -20 0 -22 E 10 -20 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999102 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=25 D=19 E=16 C=7 so C is eliminated. Round 2 votes counts: B=40 A=25 D=19 E=16 so E is eliminated. Round 3 votes counts: B=50 A=30 D=20 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:217 A:199 E:197 D:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 18 -10 B 4 0 10 16 20 C 6 -10 0 20 18 D -18 -16 -20 0 -22 E 10 -20 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999102 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 18 -10 B 4 0 10 16 20 C 6 -10 0 20 18 D -18 -16 -20 0 -22 E 10 -20 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999102 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 18 -10 B 4 0 10 16 20 C 6 -10 0 20 18 D -18 -16 -20 0 -22 E 10 -20 -18 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999102 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7450: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A D C B E (9) A D C E B (7) A B E D C (7) D C A E B (6) E C B D A (5) D A C E B (5) C D E B A (5) A D B C E (5) C E D B A (4) B E C D A (4) B E C A D (4) A B D E C (4) D C E B A (3) D A C B E (3) A B E C D (3) B E A C D (2) A C D E B (2) E C D B A (1) E B C A D (1) D C B A E (1) D B E C A (1) D B C E A (1) C E A D B (1) C D E A B (1) B E A D C (1) B D E C A (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 10 0 -4 8 B -10 0 -10 -12 -8 C 0 10 0 -8 10 D 4 12 8 0 8 E -8 8 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -4 8 B -10 0 -10 -12 -8 C 0 10 0 -8 10 D 4 12 8 0 8 E -8 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 D=20 E=18 B=12 C=11 so C is eliminated. Round 2 votes counts: A=39 D=26 E=23 B=12 so B is eliminated. Round 3 votes counts: A=39 E=34 D=27 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:207 C:206 E:191 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -4 8 B -10 0 -10 -12 -8 C 0 10 0 -8 10 D 4 12 8 0 8 E -8 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -4 8 B -10 0 -10 -12 -8 C 0 10 0 -8 10 D 4 12 8 0 8 E -8 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -4 8 B -10 0 -10 -12 -8 C 0 10 0 -8 10 D 4 12 8 0 8 E -8 8 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7451: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) E B C A D (7) E D B C A (6) E B D C A (6) E B D A C (6) B E C A D (6) B A C E D (5) A B C D E (5) E D C B A (3) E D C A B (3) D E C A B (3) D A C B E (3) B E A C D (3) B A C D E (3) A D C B E (3) A C B D E (3) D E A C B (2) D C A E B (2) C E D A B (2) C A D B E (2) B E D A C (2) B D A E C (2) A C D B E (2) E D B A C (1) E C D B A (1) E C D A B (1) E C B A D (1) D E A B C (1) D A E C B (1) D A B C E (1) C D E A B (1) C D A E B (1) C B A E D (1) C A D E B (1) B A E C D (1) Total count = 100 A B C D E A 0 -8 6 -14 -10 B 8 0 10 2 -16 C -6 -10 0 -8 -12 D 14 -2 8 0 -10 E 10 16 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 6 -14 -10 B 8 0 10 2 -16 C -6 -10 0 -8 -12 D 14 -2 8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=22 B=22 A=13 C=8 so C is eliminated. Round 2 votes counts: E=37 D=24 B=23 A=16 so A is eliminated. Round 3 votes counts: E=37 D=32 B=31 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:205 B:202 A:187 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 -14 -10 B 8 0 10 2 -16 C -6 -10 0 -8 -12 D 14 -2 8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -14 -10 B 8 0 10 2 -16 C -6 -10 0 -8 -12 D 14 -2 8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -14 -10 B 8 0 10 2 -16 C -6 -10 0 -8 -12 D 14 -2 8 0 -10 E 10 16 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7452: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (6) A E D C B (6) A B D E C (6) B C D E A (5) E A C D B (4) C E D B A (4) C B E D A (4) B D C A E (4) B D A C E (4) B C D A E (4) B A D C E (4) D B C E A (3) E D C A B (2) E C D A B (2) D B E A C (2) D B A E C (2) C E B D A (2) C D E B A (2) C D B E A (2) C B E A D (2) C B D E A (2) C B A E D (2) A C B E D (2) A B C E D (2) E D A C B (1) E C D B A (1) D E C B A (1) D E B C A (1) D C E B A (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) B D C E A (1) B D A E C (1) B C A D E (1) B A D E C (1) B A C D E (1) A E D B C (1) A E C D B (1) A E C B D (1) A D E B C (1) A D B E C (1) A C E B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -8 -6 0 B 16 0 -2 8 18 C 8 2 0 6 12 D 6 -8 -6 0 6 E 0 -18 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -6 0 B 16 0 -2 8 18 C 8 2 0 6 12 D 6 -8 -6 0 6 E 0 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 C=22 E=16 D=12 so D is eliminated. Round 2 votes counts: B=33 A=26 C=23 E=18 so E is eliminated. Round 3 votes counts: C=35 B=34 A=31 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:220 C:214 D:199 A:185 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -8 -6 0 B 16 0 -2 8 18 C 8 2 0 6 12 D 6 -8 -6 0 6 E 0 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -6 0 B 16 0 -2 8 18 C 8 2 0 6 12 D 6 -8 -6 0 6 E 0 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -6 0 B 16 0 -2 8 18 C 8 2 0 6 12 D 6 -8 -6 0 6 E 0 -18 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7453: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (10) D A E C B (8) B C E A D (6) B C D A E (6) E A B C D (5) D B C A E (5) A D E C B (5) E A D C B (4) E A C B D (4) E C B A D (3) D C A B E (3) D B A C E (3) D A B C E (3) C B E A D (3) C B D E A (3) E B C A D (2) D A E B C (2) D A C E B (2) B C E D A (2) A D E B C (2) E C A D B (1) E C A B D (1) D C B A E (1) C E B A D (1) C D B E A (1) C B E D A (1) B D C A E (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 0 -2 6 B -4 0 -2 0 -4 C 0 2 0 0 2 D 2 0 0 0 10 E -6 4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.663954 D: 0.336046 E: 0.000000 Sum of squares = 0.553761674173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.663954 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -2 6 B -4 0 -2 0 -4 C 0 2 0 0 2 D 2 0 0 0 10 E -6 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 E=20 A=19 C=9 so C is eliminated. Round 2 votes counts: B=32 D=28 E=21 A=19 so A is eliminated. Round 3 votes counts: D=35 E=33 B=32 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 A:204 C:202 B:195 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -2 6 B -4 0 -2 0 -4 C 0 2 0 0 2 D 2 0 0 0 10 E -6 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -2 6 B -4 0 -2 0 -4 C 0 2 0 0 2 D 2 0 0 0 10 E -6 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -2 6 B -4 0 -2 0 -4 C 0 2 0 0 2 D 2 0 0 0 10 E -6 4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7454: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (7) E B C A D (5) C A E D B (5) A C D E B (5) D B A E C (4) C E A D B (4) D C E B A (3) D C A E B (3) D B C E A (3) D A C B E (3) D A B C E (3) C E D B A (3) C D E A B (3) B E C A D (3) B D E C A (3) B D E A C (3) B A E D C (3) A D B C E (3) A C E B D (3) E B C D A (2) D C A B E (2) D B A C E (2) D A C E B (2) C E D A B (2) B E D C A (2) B A D E C (2) A E C B D (2) A D C B E (2) A C E D B (2) E D B C A (1) E C B A D (1) E B A C D (1) E A B C D (1) C A D E B (1) B A E C D (1) A E B C D (1) A D C E B (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 8 12 6 B -2 0 6 -16 -2 C -8 -6 0 6 10 D -12 16 -6 0 -2 E -6 2 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 12 6 B -2 0 6 -16 -2 C -8 -6 0 6 10 D -12 16 -6 0 -2 E -6 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997392 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=24 A=22 C=18 E=11 so E is eliminated. Round 2 votes counts: B=32 D=26 A=23 C=19 so C is eliminated. Round 3 votes counts: D=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:201 D:198 E:194 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 12 6 B -2 0 6 -16 -2 C -8 -6 0 6 10 D -12 16 -6 0 -2 E -6 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997392 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 12 6 B -2 0 6 -16 -2 C -8 -6 0 6 10 D -12 16 -6 0 -2 E -6 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997392 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 12 6 B -2 0 6 -16 -2 C -8 -6 0 6 10 D -12 16 -6 0 -2 E -6 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997392 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7455: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B A E D C (8) C D B E A (7) A D B E C (7) D A C E B (6) C D E B A (6) B E C A D (6) B E A C D (6) D C A B E (5) C D E A B (5) E C B A D (3) E A B C D (3) C B E D A (3) B C E D A (3) A B E D C (3) E B C A D (2) C E B D A (2) C B D E A (2) B C D E A (2) A E B D C (2) A D E C B (2) D C B A E (1) D B A C E (1) D A E C B (1) D A B E C (1) D A B C E (1) C E D B A (1) B E A D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -8 -14 -12 -4 B 8 0 -6 -8 14 C 14 6 0 2 8 D 12 8 -2 0 14 E 4 -14 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -12 -4 B 8 0 -6 -8 14 C 14 6 0 2 8 D 12 8 -2 0 14 E 4 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=26 B=26 D=25 A=15 E=8 so E is eliminated. Round 2 votes counts: C=29 B=28 D=25 A=18 so A is eliminated. Round 3 votes counts: B=36 D=35 C=29 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:215 B:204 E:184 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 -12 -4 B 8 0 -6 -8 14 C 14 6 0 2 8 D 12 8 -2 0 14 E 4 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -12 -4 B 8 0 -6 -8 14 C 14 6 0 2 8 D 12 8 -2 0 14 E 4 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -12 -4 B 8 0 -6 -8 14 C 14 6 0 2 8 D 12 8 -2 0 14 E 4 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7456: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (11) E B C A D (7) B E C A D (7) A D C E B (7) B C E A D (6) D A E C B (4) A C D E B (4) E D A B C (3) E B C D A (3) D A E B C (3) D A C E B (3) E D A C B (2) E C B A D (2) E C A D B (2) E B D C A (2) E B D A C (2) E A C D B (2) D A B C E (2) C B E A D (2) C B A E D (2) C B A D E (2) C A E D B (2) C A D B E (2) B E D C A (2) B E D A C (2) B C A D E (2) A C D B E (2) D E A C B (1) D E A B C (1) D A B E C (1) C A E B D (1) C A D E B (1) C A B D E (1) B E C D A (1) B D A C E (1) B C A E D (1) A D E C B (1) Total count = 100 A B C D E A 0 12 4 12 6 B -12 0 -8 -8 -6 C -4 8 0 4 4 D -12 8 -4 0 -2 E -6 6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 12 6 B -12 0 -8 -8 -6 C -4 8 0 4 4 D -12 8 -4 0 -2 E -6 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 B=22 A=14 C=13 so C is eliminated. Round 2 votes counts: B=28 D=26 E=25 A=21 so A is eliminated. Round 3 votes counts: D=43 B=29 E=28 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:217 C:206 E:199 D:195 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 12 6 B -12 0 -8 -8 -6 C -4 8 0 4 4 D -12 8 -4 0 -2 E -6 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 12 6 B -12 0 -8 -8 -6 C -4 8 0 4 4 D -12 8 -4 0 -2 E -6 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 12 6 B -12 0 -8 -8 -6 C -4 8 0 4 4 D -12 8 -4 0 -2 E -6 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7457: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) C E A D B (7) C A E D B (6) A D E C B (5) B E D A C (4) B D E A C (4) B D A E C (4) B C E D A (4) C E B D A (3) C E B A D (3) C B E A D (3) C B A D E (3) B E C D A (3) B D A C E (3) B C D A E (3) A D B C E (3) E D B A C (2) E C D B A (2) E C D A B (2) E C A D B (2) C B E D A (2) B A D C E (2) A D C B E (2) E D C A B (1) E D A B C (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) E A D C B (1) D B A E C (1) D B A C E (1) D A E B C (1) C E A B D (1) C A D E B (1) C A D B E (1) C A B D E (1) B D C A E (1) B C D E A (1) A D C E B (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -6 -12 0 B 6 0 2 -2 14 C 6 -2 0 2 6 D 12 2 -2 0 -2 E 0 -14 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333326 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -12 0 B 6 0 2 -2 14 C 6 -2 0 2 6 D 12 2 -2 0 -2 E 0 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=29 E=15 A=13 D=12 so D is eliminated. Round 2 votes counts: C=31 B=31 A=23 E=15 so E is eliminated. Round 3 votes counts: C=39 B=36 A=25 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:206 D:205 E:191 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -12 0 B 6 0 2 -2 14 C 6 -2 0 2 6 D 12 2 -2 0 -2 E 0 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -12 0 B 6 0 2 -2 14 C 6 -2 0 2 6 D 12 2 -2 0 -2 E 0 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -12 0 B 6 0 2 -2 14 C 6 -2 0 2 6 D 12 2 -2 0 -2 E 0 -14 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7458: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (13) C D A E B (8) B E A D C (8) E B A D C (6) D A C B E (6) B E A C D (6) A D C E B (6) E B C A D (4) D C A E B (4) B E C D A (4) B E C A D (4) D C A B E (3) A D C B E (3) E C B D A (2) E A B D C (2) C D B E A (2) C D B A E (2) B C E D A (2) A E B D C (2) A D E C B (2) E B A C D (1) E A C D B (1) D A C E B (1) C D E A B (1) C B E D A (1) B A E D C (1) B A D E C (1) A E D B C (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 0 2 12 B -12 0 -10 -8 18 C 0 10 0 2 6 D -2 8 -2 0 8 E -12 -18 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.402091 B: 0.000000 C: 0.597909 D: 0.000000 E: 0.000000 Sum of squares = 0.519172106247 Cumulative probabilities = A: 0.402091 B: 0.402091 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 2 12 B -12 0 -10 -8 18 C 0 10 0 2 6 D -2 8 -2 0 8 E -12 -18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=26 A=17 E=16 D=14 so D is eliminated. Round 2 votes counts: C=34 B=26 A=24 E=16 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:209 D:206 B:194 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 2 12 B -12 0 -10 -8 18 C 0 10 0 2 6 D -2 8 -2 0 8 E -12 -18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 2 12 B -12 0 -10 -8 18 C 0 10 0 2 6 D -2 8 -2 0 8 E -12 -18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 2 12 B -12 0 -10 -8 18 C 0 10 0 2 6 D -2 8 -2 0 8 E -12 -18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7459: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (12) E C A B D (9) B C E A D (6) C A E B D (5) B E C A D (5) B D E C A (5) A C D E B (5) A E C D B (4) E B C A D (3) D A C E B (3) B E D C A (3) E A C D B (2) D B A E C (2) D A B C E (2) C E A B D (2) B D C E A (2) B D A C E (2) A D C E B (2) A D B C E (2) A C E D B (2) E D B C A (1) E D A C B (1) E C D A B (1) E C B D A (1) E C B A D (1) E B D C A (1) D E A C B (1) D E A B C (1) D B E C A (1) D B E A C (1) D A E C B (1) D A B E C (1) C B A E D (1) B E C D A (1) B C A E D (1) B C A D E (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 14 4 B 0 0 8 4 4 C 0 -8 0 8 6 D -14 -4 -8 0 -4 E -4 -4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.335497 B: 0.664503 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.55412260175 Cumulative probabilities = A: 0.335497 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 14 4 B 0 0 8 4 4 C 0 -8 0 8 6 D -14 -4 -8 0 -4 E -4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 A=21 E=20 C=8 so C is eliminated. Round 2 votes counts: B=27 A=26 D=25 E=22 so E is eliminated. Round 3 votes counts: A=39 B=33 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:208 C:203 E:195 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 14 4 B 0 0 8 4 4 C 0 -8 0 8 6 D -14 -4 -8 0 -4 E -4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 14 4 B 0 0 8 4 4 C 0 -8 0 8 6 D -14 -4 -8 0 -4 E -4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 14 4 B 0 0 8 4 4 C 0 -8 0 8 6 D -14 -4 -8 0 -4 E -4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7460: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) E D B A C (8) C A B D E (6) B E C D A (6) D A C E B (5) A D C E B (5) E B D A C (4) D E C A B (4) D E A C B (4) A C D B E (4) D C A E B (3) D A E C B (3) A C B D E (3) E B D C A (2) D E A B C (2) C B E D A (2) C A D E B (2) C A D B E (2) B E D C A (2) B E C A D (2) B C A E D (2) A B E D C (2) E D B C A (1) E C D B A (1) E B A D C (1) D C E A B (1) C D A B E (1) C B E A D (1) C B A E D (1) C B A D E (1) B E A C D (1) B C E D A (1) B C E A D (1) B A C E D (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 24 16 -18 -6 B -24 0 -4 -20 -18 C -16 4 0 -20 -10 D 18 20 20 0 2 E 6 18 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999019 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 16 -18 -6 B -24 0 -4 -20 -18 C -16 4 0 -20 -10 D 18 20 20 0 2 E 6 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999957117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=22 A=20 C=16 B=16 so C is eliminated. Round 2 votes counts: A=30 E=26 D=23 B=21 so B is eliminated. Round 3 votes counts: E=42 A=35 D=23 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:230 E:216 A:208 C:179 B:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 16 -18 -6 B -24 0 -4 -20 -18 C -16 4 0 -20 -10 D 18 20 20 0 2 E 6 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999957117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 16 -18 -6 B -24 0 -4 -20 -18 C -16 4 0 -20 -10 D 18 20 20 0 2 E 6 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999957117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 16 -18 -6 B -24 0 -4 -20 -18 C -16 4 0 -20 -10 D 18 20 20 0 2 E 6 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999957117 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7461: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A D C B (6) B C D E A (6) A D E B C (6) A D B E C (6) C B E D A (5) C B D E A (4) D E A B C (3) D A E B C (3) B D C A E (3) B A D C E (3) A E D B C (3) E D A C B (2) E A C D B (2) D E C B A (2) D B A E C (2) C E D B A (2) C B E A D (2) B C A E D (2) B C A D E (2) A E D C B (2) A E C B D (2) A B D E C (2) A B C E D (2) A B C D E (2) E D C B A (1) E C D A B (1) E C A D B (1) D E C A B (1) D B E C A (1) D B C E A (1) C E B A D (1) C E A B D (1) C D E B A (1) C B A E D (1) B D C E A (1) B A C E D (1) B A C D E (1) A E C D B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 2 12 B 4 0 24 6 18 C -2 -24 0 2 4 D -2 -6 -2 0 22 E -12 -18 -4 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 2 12 B 4 0 24 6 18 C -2 -24 0 2 4 D -2 -6 -2 0 22 E -12 -18 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=28 C=17 E=13 D=13 so E is eliminated. Round 2 votes counts: A=36 B=29 C=19 D=16 so D is eliminated. Round 3 votes counts: A=44 B=33 C=23 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:206 D:206 C:190 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 2 12 B 4 0 24 6 18 C -2 -24 0 2 4 D -2 -6 -2 0 22 E -12 -18 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 2 12 B 4 0 24 6 18 C -2 -24 0 2 4 D -2 -6 -2 0 22 E -12 -18 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 2 12 B 4 0 24 6 18 C -2 -24 0 2 4 D -2 -6 -2 0 22 E -12 -18 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7462: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (18) C D E A B (17) C D E B A (8) A B E D C (8) B C A D E (7) E D C A B (5) E D A C B (4) D E C A B (4) C B D E A (4) C D B E A (3) B A C E D (3) E A D C B (2) C B D A E (2) B A D E C (2) B A C D E (2) A E D B C (2) A B E C D (2) E A D B C (1) D E C B A (1) D E B A C (1) C E D A B (1) C A E D B (1) B C D A E (1) B A E C D (1) Total count = 100 A B C D E A 0 -6 -8 -2 -2 B 6 0 -4 0 6 C 8 4 0 4 -2 D 2 0 -4 0 4 E 2 -6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888845 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -6 -8 -2 -2 B 6 0 -4 0 6 C 8 4 0 4 -2 D 2 0 -4 0 4 E 2 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=34 E=12 A=12 D=6 so D is eliminated. Round 2 votes counts: C=36 B=34 E=18 A=12 so A is eliminated. Round 3 votes counts: B=44 C=36 E=20 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:204 D:201 E:197 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -2 -2 B 6 0 -4 0 6 C 8 4 0 4 -2 D 2 0 -4 0 4 E 2 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -2 -2 B 6 0 -4 0 6 C 8 4 0 4 -2 D 2 0 -4 0 4 E 2 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -2 -2 B 6 0 -4 0 6 C 8 4 0 4 -2 D 2 0 -4 0 4 E 2 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.38888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7463: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) B C E D A (7) E D B C A (5) E D A B C (5) D E A C B (5) A D E C B (5) D E C B A (4) C A B D E (4) B E C D A (4) A D E B C (4) E B D C A (3) A E D B C (3) A D C E B (3) A B E D C (3) A B C D E (3) D A E B C (2) C D E A B (2) C B D E A (2) C B A E D (2) C B A D E (2) C A D B E (2) A C D B E (2) E D C B A (1) E D B A C (1) E C D B A (1) E B D A C (1) D E C A B (1) D E A B C (1) D A E C B (1) C D E B A (1) C D B E A (1) C D A E B (1) C D A B E (1) C B D A E (1) C A D E B (1) B E D C A (1) B E A C D (1) B C E A D (1) B C A E D (1) B A C E D (1) A E B D C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -12 -18 -10 B -4 0 -2 -6 -4 C 12 2 0 0 -6 D 18 6 0 0 0 E 10 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.498312 E: 0.501688 Sum of squares = 0.500005698961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.498312 E: 1.000000 A B C D E A 0 4 -12 -18 -10 B -4 0 -2 -6 -4 C 12 2 0 0 -6 D 18 6 0 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 E=17 B=16 D=14 so D is eliminated. Round 2 votes counts: A=29 E=28 C=27 B=16 so B is eliminated. Round 3 votes counts: C=36 E=34 A=30 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:210 C:204 B:192 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -12 -18 -10 B -4 0 -2 -6 -4 C 12 2 0 0 -6 D 18 6 0 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -18 -10 B -4 0 -2 -6 -4 C 12 2 0 0 -6 D 18 6 0 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -18 -10 B -4 0 -2 -6 -4 C 12 2 0 0 -6 D 18 6 0 0 0 E 10 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999264 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7464: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (17) D C B A E (8) D C E A B (6) C D B A E (6) D C E B A (5) A B E C D (5) E C A B D (4) D B A C E (4) C D E A B (4) B A E C D (4) B A D E C (4) B A C E D (4) D B A E C (3) B A D C E (3) E D C A B (2) C E D A B (2) C D B E A (2) E D A B C (1) E C A D B (1) E A C D B (1) E A B D C (1) D E C A B (1) D E A B C (1) D C B E A (1) D A E B C (1) C D E B A (1) C B E A D (1) C B A D E (1) B D A C E (1) B C A D E (1) B A C D E (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 8 2 -2 B 0 0 8 0 2 C -8 -8 0 12 2 D -2 0 -12 0 10 E 2 -2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.248369 B: 0.751631 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.626635865388 Cumulative probabilities = A: 0.248369 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 2 -2 B 0 0 8 0 2 C -8 -8 0 12 2 D -2 0 -12 0 10 E 2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000012 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 B=18 C=17 A=8 so A is eliminated. Round 2 votes counts: D=30 E=28 B=25 C=17 so C is eliminated. Round 3 votes counts: D=43 E=30 B=27 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:205 A:204 C:199 D:198 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 8 2 -2 B 0 0 8 0 2 C -8 -8 0 12 2 D -2 0 -12 0 10 E 2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000012 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 -2 B 0 0 8 0 2 C -8 -8 0 12 2 D -2 0 -12 0 10 E 2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000012 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 -2 B 0 0 8 0 2 C -8 -8 0 12 2 D -2 0 -12 0 10 E 2 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499998 B: 0.500002 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000012 Cumulative probabilities = A: 0.499998 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7465: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (18) E B D A C (12) C A D B E (11) A C E B D (7) E B D C A (6) C D A B E (6) E B A C D (5) D E B C A (4) C D A E B (3) B E A D C (3) A E B C D (3) A C B E D (3) A B E C D (3) E B A D C (2) D C E B A (2) B E D A C (2) E A B C D (1) D C B E A (1) D C A E B (1) D B E C A (1) D B C E A (1) C D E B A (1) C A D E B (1) B D E C A (1) B D E A C (1) A B E D C (1) Total count = 100 A B C D E A 0 16 22 16 14 B -16 0 -8 0 4 C -22 8 0 24 10 D -16 0 -24 0 4 E -14 -4 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 22 16 14 B -16 0 -8 0 4 C -22 8 0 24 10 D -16 0 -24 0 4 E -14 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=26 C=22 D=10 B=7 so B is eliminated. Round 2 votes counts: A=35 E=31 C=22 D=12 so D is eliminated. Round 3 votes counts: E=38 A=35 C=27 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:234 C:210 B:190 E:184 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 22 16 14 B -16 0 -8 0 4 C -22 8 0 24 10 D -16 0 -24 0 4 E -14 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 22 16 14 B -16 0 -8 0 4 C -22 8 0 24 10 D -16 0 -24 0 4 E -14 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 22 16 14 B -16 0 -8 0 4 C -22 8 0 24 10 D -16 0 -24 0 4 E -14 -4 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7466: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) D E A B C (10) D A E B C (10) C B A E D (9) A C B D E (6) C E B D A (4) C B E D A (4) E D C B A (3) C B E A D (3) C A B E D (3) A D C B E (3) A C B E D (3) A B D C E (3) E C B D A (2) E B C D A (2) D E B C A (2) D E B A C (2) D A C E B (2) B E C D A (2) B C E A D (2) A B C D E (2) E D B A C (1) E B D C A (1) D E A C B (1) D C E A B (1) D A E C B (1) C D A E B (1) C A D E B (1) C A B D E (1) B A C E D (1) A D E B C (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -2 -18 0 B -4 0 2 -2 -10 C 2 -2 0 -6 2 D 18 2 6 0 0 E 0 10 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.505562 E: 0.494438 Sum of squares = 0.500061803577 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.505562 E: 1.000000 A B C D E A 0 4 -2 -18 0 B -4 0 2 -2 -10 C 2 -2 0 -6 2 D 18 2 6 0 0 E 0 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 A=21 E=19 B=5 so B is eliminated. Round 2 votes counts: D=29 C=28 A=22 E=21 so E is eliminated. Round 3 votes counts: D=44 C=34 A=22 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:204 C:198 B:193 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -18 0 B -4 0 2 -2 -10 C 2 -2 0 -6 2 D 18 2 6 0 0 E 0 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -18 0 B -4 0 2 -2 -10 C 2 -2 0 -6 2 D 18 2 6 0 0 E 0 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -18 0 B -4 0 2 -2 -10 C 2 -2 0 -6 2 D 18 2 6 0 0 E 0 10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.499999 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500001 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7467: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (9) E D A C B (8) B E C D A (6) D C A E B (5) C B D A E (5) E B A D C (4) E A D B C (4) C D A B E (4) A B C D E (4) E D C B A (3) E D C A B (3) B E A C D (3) B A E C D (3) B A C E D (3) B A C D E (3) A C D B E (3) E B D C A (2) E A D C B (2) D E C A B (2) D E A C B (2) D C E A B (2) D A C E B (2) B E C A D (2) B C D A E (2) A D E C B (2) E D B C A (1) E D B A C (1) E D A B C (1) E B A C D (1) C D B A E (1) C B A D E (1) C A D B E (1) B C E D A (1) B C E A D (1) B C A E D (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 -2 2 B 6 0 6 4 8 C 4 -6 0 8 -2 D 2 -4 -8 0 -2 E -2 -8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -2 2 B 6 0 6 4 8 C 4 -6 0 8 -2 D 2 -4 -8 0 -2 E -2 -8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=30 D=13 C=12 A=11 so A is eliminated. Round 2 votes counts: B=39 E=30 D=16 C=15 so C is eliminated. Round 3 votes counts: B=45 E=30 D=25 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 C:202 E:197 A:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 2 B 6 0 6 4 8 C 4 -6 0 8 -2 D 2 -4 -8 0 -2 E -2 -8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 2 B 6 0 6 4 8 C 4 -6 0 8 -2 D 2 -4 -8 0 -2 E -2 -8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 2 B 6 0 6 4 8 C 4 -6 0 8 -2 D 2 -4 -8 0 -2 E -2 -8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7468: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (7) D E A B C (5) C E B D A (5) B A C E D (5) A D B E C (5) E D C B A (4) C B E D A (4) C B A E D (4) C A B E D (4) A B C E D (4) E D B C A (3) D E C B A (3) D A E B C (3) B C E D A (3) A C B D E (3) A B D E C (3) E B C D A (2) D E B C A (2) D E B A C (2) D E A C B (2) D A E C B (2) B A E D C (2) A D E C B (2) A D E B C (2) A D C E B (2) A C D E B (2) A B D C E (2) E C D B A (1) E C B D A (1) D E C A B (1) D C E A B (1) C E D B A (1) C D E A B (1) C B E A D (1) B D A E C (1) B A D E C (1) A D B C E (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 4 6 16 B 4 0 8 8 6 C -4 -8 0 2 6 D -6 -8 -2 0 -4 E -16 -6 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 6 16 B 4 0 8 8 6 C -4 -8 0 2 6 D -6 -8 -2 0 -4 E -16 -6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=21 C=20 B=19 E=11 so E is eliminated. Round 2 votes counts: A=29 D=28 C=22 B=21 so B is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:211 C:198 D:190 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 6 16 B 4 0 8 8 6 C -4 -8 0 2 6 D -6 -8 -2 0 -4 E -16 -6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 6 16 B 4 0 8 8 6 C -4 -8 0 2 6 D -6 -8 -2 0 -4 E -16 -6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 6 16 B 4 0 8 8 6 C -4 -8 0 2 6 D -6 -8 -2 0 -4 E -16 -6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7469: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (17) C E D A B (7) B A D E C (6) D C E B A (4) D B C E A (4) C E D B A (4) B A D C E (4) A E B D C (4) A B C E D (4) E C D A B (3) E A D C B (3) C D E B A (3) C D B E A (3) E D C B A (2) E D C A B (2) E D A B C (2) D E B C A (2) C B A D E (2) C A E D B (2) A B E C D (2) A B D E C (2) E D A C B (1) E C A D B (1) D E C B A (1) D B E C A (1) D B E A C (1) C E A D B (1) C B D E A (1) C A E B D (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D A E (1) B A C D E (1) A E B C D (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 6 8 6 B -12 0 14 4 10 C -6 -14 0 -20 -4 D -8 -4 20 0 -16 E -6 -10 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 8 6 B -12 0 14 4 10 C -6 -14 0 -20 -4 D -8 -4 20 0 -16 E -6 -10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 B=16 E=14 D=13 so D is eliminated. Round 2 votes counts: A=33 C=28 B=22 E=17 so E is eliminated. Round 3 votes counts: A=39 C=37 B=24 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:208 E:202 D:196 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 8 6 B -12 0 14 4 10 C -6 -14 0 -20 -4 D -8 -4 20 0 -16 E -6 -10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 8 6 B -12 0 14 4 10 C -6 -14 0 -20 -4 D -8 -4 20 0 -16 E -6 -10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 8 6 B -12 0 14 4 10 C -6 -14 0 -20 -4 D -8 -4 20 0 -16 E -6 -10 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7470: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) C B D A E (8) B A E C D (8) E A B D C (7) B C A E D (7) C D B A E (6) D C B A E (4) C D B E A (4) A B E C D (4) D E C A B (3) D C E B A (3) D C E A B (3) C B A D E (3) B A C E D (3) E A D B C (2) D C A E B (2) D C A B E (2) B E A C D (2) A E D B C (2) A D B C E (2) E D A C B (1) E D A B C (1) E B A C D (1) D E C B A (1) D E A C B (1) D C B E A (1) C D A B E (1) C B D E A (1) B C E A D (1) B C A D E (1) A E B D C (1) A E B C D (1) A D C B E (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 16 20 B 8 0 10 20 26 C 2 -10 0 22 12 D -16 -20 -22 0 0 E -20 -26 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 16 20 B 8 0 10 20 26 C 2 -10 0 22 12 D -16 -20 -22 0 0 E -20 -26 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 B=22 E=20 D=20 A=15 so A is eliminated. Round 2 votes counts: B=30 E=24 D=23 C=23 so D is eliminated. Round 3 votes counts: C=39 B=32 E=29 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 A:213 C:213 D:171 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 16 20 B 8 0 10 20 26 C 2 -10 0 22 12 D -16 -20 -22 0 0 E -20 -26 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 16 20 B 8 0 10 20 26 C 2 -10 0 22 12 D -16 -20 -22 0 0 E -20 -26 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 16 20 B 8 0 10 20 26 C 2 -10 0 22 12 D -16 -20 -22 0 0 E -20 -26 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999466 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7471: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) E D C A B (9) D C E A B (9) B A C E D (8) D E C A B (7) D C A B E (7) C A B D E (7) C B A D E (5) B A C D E (5) E B A D C (4) D E C B A (4) D C B A E (3) B A E C D (3) A B C E D (3) E B A C D (2) D C E B A (2) C D B A E (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E A B D C (1) E A B C D (1) D E B A C (1) D C B E A (1) A B E C D (1) Total count = 100 A B C D E A 0 16 -34 -16 12 B -16 0 -32 -16 14 C 34 32 0 -2 28 D 16 16 2 0 30 E -12 -14 -28 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -34 -16 12 B -16 0 -32 -16 14 C 34 32 0 -2 28 D 16 16 2 0 30 E -12 -14 -28 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=24 E=20 B=16 A=6 so A is eliminated. Round 2 votes counts: D=34 C=24 B=22 E=20 so E is eliminated. Round 3 votes counts: D=46 B=30 C=24 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:246 D:232 A:189 B:175 E:158 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -34 -16 12 B -16 0 -32 -16 14 C 34 32 0 -2 28 D 16 16 2 0 30 E -12 -14 -28 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -34 -16 12 B -16 0 -32 -16 14 C 34 32 0 -2 28 D 16 16 2 0 30 E -12 -14 -28 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -34 -16 12 B -16 0 -32 -16 14 C 34 32 0 -2 28 D 16 16 2 0 30 E -12 -14 -28 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7472: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (8) C B A D E (7) C B D E A (5) B C A E D (5) A E D B C (4) A E B D C (4) E D B A C (3) E A D B C (3) D E C A B (3) C D B E A (3) C D B A E (3) C D A B E (3) B C A D E (3) B A C E D (3) A B E D C (3) E D A B C (2) E B A D C (2) E A B D C (2) D C A E B (2) D A E C B (2) D A C E B (2) B C E D A (2) A D E C B (2) A D C B E (2) A B E C D (2) A B C E D (2) E D C B A (1) E D C A B (1) E D B C A (1) E D A C B (1) E B D C A (1) E B A C D (1) D E C B A (1) D C E B A (1) C D E B A (1) C B E D A (1) C B D A E (1) C A B D E (1) B E C D A (1) B E C A D (1) B A E D C (1) B A E C D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 4 2 10 B -2 0 -6 -2 2 C -4 6 0 -6 -2 D -2 2 6 0 4 E -10 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 2 10 B -2 0 -6 -2 2 C -4 6 0 -6 -2 D -2 2 6 0 4 E -10 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=21 D=19 E=18 B=17 so B is eliminated. Round 2 votes counts: C=35 A=26 E=20 D=19 so D is eliminated. Round 3 votes counts: C=38 E=32 A=30 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:209 D:205 C:197 B:196 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 2 10 B -2 0 -6 -2 2 C -4 6 0 -6 -2 D -2 2 6 0 4 E -10 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 10 B -2 0 -6 -2 2 C -4 6 0 -6 -2 D -2 2 6 0 4 E -10 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 10 B -2 0 -6 -2 2 C -4 6 0 -6 -2 D -2 2 6 0 4 E -10 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7473: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) A D E B C (10) C B E D A (8) D A B E C (6) B D E A C (6) D B A E C (5) C A E D B (5) B D E C A (5) A E D C B (5) D B E A C (3) C E A B D (3) B C E D A (3) A D E C B (3) E B D C A (2) D A E B C (2) B E C D A (2) B C D E A (2) B C D A E (2) A D B E C (2) A C E D B (2) A C D E B (2) E D B C A (1) E C B A D (1) D E B A C (1) D E A B C (1) C E B D A (1) C E A D B (1) C B A E D (1) C A D B E (1) B D C A E (1) B D A E C (1) A E D B C (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 2 -4 0 B 10 0 12 -4 -2 C -2 -12 0 -10 -14 D 4 4 10 0 8 E 0 2 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -4 0 B 10 0 12 -4 -2 C -2 -12 0 -10 -14 D 4 4 10 0 8 E 0 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=26 B=22 D=18 E=4 so E is eliminated. Round 2 votes counts: C=31 A=26 B=24 D=19 so D is eliminated. Round 3 votes counts: A=35 B=34 C=31 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:208 E:204 A:194 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 -4 0 B 10 0 12 -4 -2 C -2 -12 0 -10 -14 D 4 4 10 0 8 E 0 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -4 0 B 10 0 12 -4 -2 C -2 -12 0 -10 -14 D 4 4 10 0 8 E 0 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -4 0 B 10 0 12 -4 -2 C -2 -12 0 -10 -14 D 4 4 10 0 8 E 0 2 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7474: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) B D C E A (7) C E A B D (5) B A C D E (4) A E C D B (4) A B C E D (4) E A C D B (3) D E C B A (3) D E B C A (3) D E A C B (3) C E B D A (3) B A D C E (3) A E D B C (3) A D E B C (3) A C E B D (3) A B D C E (3) E D C A B (2) E C D B A (2) E A D C B (2) D B E C A (2) D A B E C (2) C D E B A (2) C B E D A (2) C A E B D (2) B D A E C (2) B C A D E (2) A D B E C (2) A C B E D (2) A B C D E (2) E D C B A (1) E C D A B (1) D E A B C (1) D C E B A (1) D B E A C (1) D B A E C (1) C E B A D (1) C E A D B (1) B C D A E (1) B A D E C (1) A E D C B (1) A E C B D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 4 6 -6 B -2 0 -4 -2 -16 C -4 4 0 4 10 D -6 2 -4 0 0 E 6 16 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.200000 Sum of squares = 0.379999999926 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 2 4 6 -6 B -2 0 -4 -2 -16 C -4 4 0 4 10 D -6 2 -4 0 0 E 6 16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000002 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=23 B=20 D=17 E=11 so E is eliminated. Round 2 votes counts: A=34 C=26 D=20 B=20 so D is eliminated. Round 3 votes counts: A=40 C=33 B=27 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:207 E:206 A:203 D:196 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 4 6 -6 B -2 0 -4 -2 -16 C -4 4 0 4 10 D -6 2 -4 0 0 E 6 16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000002 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 6 -6 B -2 0 -4 -2 -16 C -4 4 0 4 10 D -6 2 -4 0 0 E 6 16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000002 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 6 -6 B -2 0 -4 -2 -16 C -4 4 0 4 10 D -6 2 -4 0 0 E 6 16 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.200000 Sum of squares = 0.380000000002 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7475: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) C E A D B (6) B D E A C (5) E C A D B (4) E A C D B (4) D B E C A (4) D B C E A (4) B A E D C (4) D E C B A (3) C D E A B (3) B D C A E (3) A E C B D (3) A B E C D (3) A B C E D (3) E D B A C (2) D C E A B (2) C D A E B (2) C A E D B (2) C A D E B (2) C A D B E (2) B E A D C (2) B D A E C (2) B A D E C (2) A E B C D (2) A C E B D (2) A C B E D (2) E D C A B (1) E D A B C (1) E C D A B (1) E A B D C (1) E A B C D (1) D C E B A (1) D C B A E (1) D B E A C (1) C D E B A (1) C D B E A (1) C D A B E (1) C A B D E (1) B E D A C (1) B D A C E (1) B A D C E (1) B A C D E (1) A E C D B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -6 2 -14 B -4 0 -12 -18 8 C 6 12 0 0 4 D -2 18 0 0 6 E 14 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.476174 D: 0.523826 E: 0.000000 Sum of squares = 0.501135309396 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.476174 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 2 -14 B -4 0 -12 -18 8 C 6 12 0 0 4 D -2 18 0 0 6 E 14 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 C=21 A=18 E=15 so E is eliminated. Round 2 votes counts: D=28 C=26 A=24 B=22 so B is eliminated. Round 3 votes counts: D=40 A=34 C=26 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:211 E:198 A:193 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 2 -14 B -4 0 -12 -18 8 C 6 12 0 0 4 D -2 18 0 0 6 E 14 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 2 -14 B -4 0 -12 -18 8 C 6 12 0 0 4 D -2 18 0 0 6 E 14 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 2 -14 B -4 0 -12 -18 8 C 6 12 0 0 4 D -2 18 0 0 6 E 14 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7476: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) B A E D C (10) B D E C A (9) A C E D B (7) C D E A B (6) B E D A C (6) A C D E B (6) A E C D B (5) A B E D C (5) C D E B A (4) C D A E B (4) C D B E A (3) A E B C D (3) A B E C D (3) A E B D C (2) E D C A B (1) E D B C A (1) E C D A B (1) E A D B C (1) D E B C A (1) D C E B A (1) D C B E A (1) C D A B E (1) C A D B E (1) B E A D C (1) B D E A C (1) A C E B D (1) A C D B E (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 10 -2 4 B -2 0 10 10 12 C -10 -10 0 -2 -22 D 2 -10 2 0 -20 E -4 -12 22 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408196 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 -2 4 B -2 0 10 10 12 C -10 -10 0 -2 -22 D 2 -10 2 0 -20 E -4 -12 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408359 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=36 C=19 E=4 D=3 so D is eliminated. Round 2 votes counts: B=38 A=36 C=21 E=5 so E is eliminated. Round 3 votes counts: B=40 A=37 C=23 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:215 E:213 A:207 D:187 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 -2 4 B -2 0 10 10 12 C -10 -10 0 -2 -22 D 2 -10 2 0 -20 E -4 -12 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408359 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -2 4 B -2 0 10 10 12 C -10 -10 0 -2 -22 D 2 -10 2 0 -20 E -4 -12 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408359 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -2 4 B -2 0 10 10 12 C -10 -10 0 -2 -22 D 2 -10 2 0 -20 E -4 -12 22 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.142857 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408359 Cumulative probabilities = A: 0.714286 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7477: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (16) C E D B A (9) B C D E A (6) B C E D A (5) A B C E D (5) D E C A B (4) B A C E D (4) E D C A B (3) D E C B A (3) D A E C B (3) B A D C E (3) A E C D B (3) A C E B D (3) A B D E C (3) E C D A B (2) D B E C A (2) C B E D A (2) C B E A D (2) B D C E A (2) B C A E D (2) A D E B C (2) A D B E C (2) E D A C B (1) E C D B A (1) E A D C B (1) D E A C B (1) D C B E A (1) D B A E C (1) D A E B C (1) C E B A D (1) B C E A D (1) B A C D E (1) A E C B D (1) A C E D B (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 8 6 6 B -8 0 -16 -14 -12 C -8 16 0 -2 -2 D -6 14 2 0 4 E -6 12 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 6 6 B -8 0 -16 -14 -12 C -8 16 0 -2 -2 D -6 14 2 0 4 E -6 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=24 D=16 C=14 E=8 so E is eliminated. Round 2 votes counts: A=39 B=24 D=20 C=17 so C is eliminated. Round 3 votes counts: A=39 D=32 B=29 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:207 C:202 E:202 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 6 6 B -8 0 -16 -14 -12 C -8 16 0 -2 -2 D -6 14 2 0 4 E -6 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 6 B -8 0 -16 -14 -12 C -8 16 0 -2 -2 D -6 14 2 0 4 E -6 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 6 B -8 0 -16 -14 -12 C -8 16 0 -2 -2 D -6 14 2 0 4 E -6 12 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7478: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) B C A E D (9) D E C A B (7) E D C B A (6) D A E C B (6) C E B D A (6) A D B E C (6) A B C D E (6) D E A C B (5) C B A E D (4) A D E B C (4) D E A B C (3) D A E B C (3) B C E A D (3) E D B C A (2) C B E D A (2) B A C D E (2) A D E C B (2) A D B C E (2) E C D B A (1) E B D C A (1) E B C D A (1) D E B A C (1) C E D B A (1) C B E A D (1) B C E D A (1) B A C E D (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 10 8 18 B -16 0 16 0 2 C -10 -16 0 -24 2 D -8 0 24 0 22 E -18 -2 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 8 18 B -16 0 16 0 2 C -10 -16 0 -24 2 D -8 0 24 0 22 E -18 -2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=25 B=16 C=14 E=11 so E is eliminated. Round 2 votes counts: A=34 D=33 B=18 C=15 so C is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:219 B:201 E:178 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 8 18 B -16 0 16 0 2 C -10 -16 0 -24 2 D -8 0 24 0 22 E -18 -2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 8 18 B -16 0 16 0 2 C -10 -16 0 -24 2 D -8 0 24 0 22 E -18 -2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 8 18 B -16 0 16 0 2 C -10 -16 0 -24 2 D -8 0 24 0 22 E -18 -2 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7479: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) E B A D C (6) E A D B C (6) B E A C D (6) C D A B E (5) B E A D C (5) A E D B C (5) C D A E B (4) B E C A D (4) A E B D C (4) E A B D C (3) D E A C B (3) C D B E A (3) C B D E A (3) B E C D A (3) B C E D A (3) B A C E D (3) E D A B C (2) E B D A C (2) D A E C B (2) C D B A E (2) C B D A E (2) B C E A D (2) A B E D C (2) E D B C A (1) E D B A C (1) D C A E B (1) D A C E B (1) C D E B A (1) C B A D E (1) C A D B E (1) B C D E A (1) B C A E D (1) B A E C D (1) A E D C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 24 20 -10 B 2 0 22 4 -4 C -24 -22 0 -6 -28 D -20 -4 6 0 -22 E 10 4 28 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 24 20 -10 B 2 0 22 4 -4 C -24 -22 0 -6 -28 D -20 -4 6 0 -22 E 10 4 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 E=21 A=21 D=7 so D is eliminated. Round 2 votes counts: B=29 E=24 A=24 C=23 so C is eliminated. Round 3 votes counts: B=40 A=35 E=25 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:232 A:216 B:212 D:180 C:160 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 24 20 -10 B 2 0 22 4 -4 C -24 -22 0 -6 -28 D -20 -4 6 0 -22 E 10 4 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 24 20 -10 B 2 0 22 4 -4 C -24 -22 0 -6 -28 D -20 -4 6 0 -22 E 10 4 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 24 20 -10 B 2 0 22 4 -4 C -24 -22 0 -6 -28 D -20 -4 6 0 -22 E 10 4 28 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999988838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7480: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) B C A D E (8) E D C A B (5) D E C A B (5) B A C E D (5) E D A B C (4) C D A E B (4) A C D B E (4) E D A C B (3) C D E B A (3) C B D A E (3) C A B D E (3) B E D C A (3) B E A D C (3) B C E D A (3) A C B D E (3) E D C B A (2) C A D E B (2) B E D A C (2) B C A E D (2) B A E C D (2) A D C E B (2) E D B C A (1) E D B A C (1) E B D C A (1) E B D A C (1) D E A C B (1) D C E B A (1) D C E A B (1) C D E A B (1) C D B E A (1) C B D E A (1) C B A D E (1) C A D B E (1) B E C A D (1) B A E D C (1) A E D C B (1) A E D B C (1) A D E C B (1) A C D E B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 6 12 B 10 0 0 8 16 C 6 0 0 20 22 D -6 -8 -20 0 14 E -12 -16 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.550707 C: 0.449293 D: 0.000000 E: 0.000000 Sum of squares = 0.505142401933 Cumulative probabilities = A: 0.000000 B: 0.550707 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 6 12 B 10 0 0 8 16 C 6 0 0 20 22 D -6 -8 -20 0 14 E -12 -16 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=20 E=18 A=15 D=8 so D is eliminated. Round 2 votes counts: B=39 E=24 C=22 A=15 so A is eliminated. Round 3 votes counts: B=41 C=32 E=27 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:224 B:217 A:201 D:190 E:168 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 6 12 B 10 0 0 8 16 C 6 0 0 20 22 D -6 -8 -20 0 14 E -12 -16 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 6 12 B 10 0 0 8 16 C 6 0 0 20 22 D -6 -8 -20 0 14 E -12 -16 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 6 12 B 10 0 0 8 16 C 6 0 0 20 22 D -6 -8 -20 0 14 E -12 -16 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7481: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (5) E A B D C (5) C D B A E (5) E C D A B (4) D C A B E (4) C D E B A (4) B A E D C (4) A B E D C (4) E C A D B (3) E B C A D (3) D C E A B (3) D C B A E (3) D A B C E (3) C D B E A (3) B C E A D (3) B A E C D (3) B A D E C (3) B A D C E (3) A B D E C (3) D B A C E (2) D A C B E (2) C E D B A (2) C D E A B (2) C B D A E (2) A E B D C (2) A D E B C (2) E D C A B (1) E D A C B (1) E D A B C (1) E C B A D (1) E A B C D (1) D C A E B (1) D A E B C (1) D A C E B (1) D A B E C (1) C E D A B (1) C E B D A (1) C B D E A (1) B E A C D (1) B C A D E (1) B A C E D (1) B A C D E (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 2 8 B -6 0 10 -12 8 C -4 -10 0 -14 0 D -2 12 14 0 4 E -8 -8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 8 B -6 0 10 -12 8 C -4 -10 0 -14 0 D -2 12 14 0 4 E -8 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=21 C=21 B=20 A=13 so A is eliminated. Round 2 votes counts: E=28 B=28 D=23 C=21 so C is eliminated. Round 3 votes counts: D=37 E=32 B=31 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:210 B:200 E:190 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 8 B -6 0 10 -12 8 C -4 -10 0 -14 0 D -2 12 14 0 4 E -8 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 8 B -6 0 10 -12 8 C -4 -10 0 -14 0 D -2 12 14 0 4 E -8 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 8 B -6 0 10 -12 8 C -4 -10 0 -14 0 D -2 12 14 0 4 E -8 -8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998438 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7482: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) C D E A B (7) C D E B A (6) A B E D C (6) D C A B E (5) B A E C D (5) D C E A B (4) D C A E B (4) C D B E A (4) A E B D C (4) E A B D C (3) C D B A E (3) B E A C D (3) E B A C D (2) D A E C B (2) D A C B E (2) D A B C E (2) C E D B A (2) C E B D A (2) C D A B E (2) C B D E A (2) B E C A D (2) B C A E D (2) A D E B C (2) A D B E C (2) E C D A B (1) E B C A D (1) E B A D C (1) E A D B C (1) D E A B C (1) D A E B C (1) D A C E B (1) C B E D A (1) C B E A D (1) C B D A E (1) B E A D C (1) B A D C E (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 -6 10 B -2 0 0 -4 10 C 0 0 0 -4 6 D 6 4 4 0 8 E -10 -10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -6 10 B -2 0 0 -4 10 C 0 0 0 -4 6 D 6 4 4 0 8 E -10 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=23 D=22 A=15 E=9 so E is eliminated. Round 2 votes counts: C=32 B=27 D=22 A=19 so A is eliminated. Round 3 votes counts: B=41 C=32 D=27 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:211 A:203 B:202 C:201 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -6 10 B -2 0 0 -4 10 C 0 0 0 -4 6 D 6 4 4 0 8 E -10 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -6 10 B -2 0 0 -4 10 C 0 0 0 -4 6 D 6 4 4 0 8 E -10 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -6 10 B -2 0 0 -4 10 C 0 0 0 -4 6 D 6 4 4 0 8 E -10 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7483: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (9) A E C D B (8) A E D C B (7) E D A B C (6) D E B A C (6) C B A E D (6) C A B E D (6) B C D E A (6) B C A D E (5) D B E C A (4) B C D A E (4) D E B C A (3) C B A D E (3) B D C E A (3) A C E B D (3) E D A C B (2) E A D C B (2) D B E A C (2) B A C E D (2) A C E D B (2) A C B E D (2) E A D B C (1) C E D B A (1) C E A D B (1) C A E D B (1) C A E B D (1) B D E C A (1) B D E A C (1) B D C A E (1) A E D B C (1) Total count = 100 A B C D E A 0 4 8 2 4 B -4 0 10 -12 -8 C -8 -10 0 2 -6 D -2 12 -2 0 -4 E -4 8 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 2 4 B -4 0 10 -12 -8 C -8 -10 0 2 -6 D -2 12 -2 0 -4 E -4 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=23 A=23 C=19 E=11 so E is eliminated. Round 2 votes counts: D=32 A=26 B=23 C=19 so C is eliminated. Round 3 votes counts: A=35 D=33 B=32 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:207 D:202 B:193 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 2 4 B -4 0 10 -12 -8 C -8 -10 0 2 -6 D -2 12 -2 0 -4 E -4 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 2 4 B -4 0 10 -12 -8 C -8 -10 0 2 -6 D -2 12 -2 0 -4 E -4 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 2 4 B -4 0 10 -12 -8 C -8 -10 0 2 -6 D -2 12 -2 0 -4 E -4 8 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7484: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (16) B E A C D (13) C D B E A (9) A E B D C (8) E A B D C (7) D C A B E (6) C D B A E (6) B C D E A (5) E B A D C (3) C D A B E (3) C B D E A (3) E B A C D (2) D C E B A (2) A E D C B (2) A E D B C (2) A D E C B (2) E A D B C (1) D C E A B (1) D C B A E (1) D A C E B (1) B E C A D (1) B C E D A (1) B A E C D (1) A E B C D (1) A D C E B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -8 -8 4 B -6 0 -8 -8 2 C 8 8 0 -6 12 D 8 8 6 0 14 E -4 -2 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -8 4 B -6 0 -8 -8 2 C 8 8 0 -6 12 D 8 8 6 0 14 E -4 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=21 B=21 A=18 E=13 so E is eliminated. Round 2 votes counts: D=27 B=26 A=26 C=21 so C is eliminated. Round 3 votes counts: D=45 B=29 A=26 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:211 A:197 B:190 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -8 -8 4 B -6 0 -8 -8 2 C 8 8 0 -6 12 D 8 8 6 0 14 E -4 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -8 4 B -6 0 -8 -8 2 C 8 8 0 -6 12 D 8 8 6 0 14 E -4 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -8 4 B -6 0 -8 -8 2 C 8 8 0 -6 12 D 8 8 6 0 14 E -4 -2 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7485: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) C B D E A (9) E C D A B (7) B D A C E (5) A B D E C (5) E A C D B (4) C D B E A (4) C B D A E (4) B D C A E (4) E A D C B (3) D B A C E (3) B A D C E (3) A D B E C (3) E C A D B (2) D B A E C (2) D A B E C (2) C E B D A (2) C D B A E (2) B C D A E (2) A E C B D (2) A B D C E (2) E D C A B (1) E D A B C (1) E C D B A (1) E C A B D (1) E A D B C (1) E A C B D (1) D B C A E (1) C E B A D (1) C E A B D (1) C D E B A (1) C B E D A (1) C B E A D (1) C B A D E (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B D C (1) A D E B C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -14 -24 -4 B 18 0 -18 0 16 C 14 18 0 20 20 D 24 0 -20 0 12 E 4 -16 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 -24 -4 B 18 0 -18 0 16 C 14 18 0 20 20 D 24 0 -20 0 12 E 4 -16 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=22 A=17 B=16 D=8 so D is eliminated. Round 2 votes counts: C=37 E=22 B=22 A=19 so A is eliminated. Round 3 votes counts: C=37 B=36 E=27 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:236 B:208 D:208 E:178 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -14 -24 -4 B 18 0 -18 0 16 C 14 18 0 20 20 D 24 0 -20 0 12 E 4 -16 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 -24 -4 B 18 0 -18 0 16 C 14 18 0 20 20 D 24 0 -20 0 12 E 4 -16 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 -24 -4 B 18 0 -18 0 16 C 14 18 0 20 20 D 24 0 -20 0 12 E 4 -16 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7486: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (12) D C E B A (7) D B A C E (7) E C D A B (6) B A D C E (6) D B A E C (5) C E A B D (4) E C A B D (3) E A C B D (3) D C E A B (3) D C B A E (3) C E D B A (3) C B A E D (3) B A E C D (3) B A D E C (3) B A C E D (3) B A C D E (3) E D A B C (2) D E C B A (2) D E A B C (2) C E D A B (2) A E B C D (2) A B E D C (2) A B D E C (2) E D C A B (1) E A B C D (1) D E C A B (1) D A B E C (1) C E B A D (1) C A B E D (1) B C A D E (1) B A E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 18 10 18 B 2 0 14 10 14 C -18 -14 0 4 -4 D -10 -10 -4 0 -8 E -18 -14 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 18 10 18 B 2 0 14 10 14 C -18 -14 0 4 -4 D -10 -10 -4 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=20 A=19 E=16 C=14 so C is eliminated. Round 2 votes counts: D=31 E=26 B=23 A=20 so A is eliminated. Round 3 votes counts: B=41 D=31 E=28 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:222 B:220 E:190 C:184 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 18 10 18 B 2 0 14 10 14 C -18 -14 0 4 -4 D -10 -10 -4 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 18 10 18 B 2 0 14 10 14 C -18 -14 0 4 -4 D -10 -10 -4 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 18 10 18 B 2 0 14 10 14 C -18 -14 0 4 -4 D -10 -10 -4 0 -8 E -18 -14 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7487: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (11) D B A C E (11) E B C A D (8) D A C B E (8) B D A C E (8) B D E C A (6) E C A B D (4) D A C E B (4) C A E D B (4) A C E D B (4) D A B C E (3) B E D C A (3) B D E A C (3) A C D E B (3) E C D A B (2) E C B A D (2) C A D E B (2) B E C A D (2) B D A E C (2) E D C B A (1) E B D C A (1) E B C D A (1) D B E C A (1) D B C A E (1) B E C D A (1) B E A C D (1) A D C E B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 0 -12 6 B 4 0 6 -14 4 C 0 -6 0 -8 2 D 12 14 8 0 10 E -6 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -12 6 B 4 0 6 -14 4 C 0 -6 0 -8 2 D 12 14 8 0 10 E -6 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=28 B=26 A=10 C=6 so C is eliminated. Round 2 votes counts: E=30 D=28 B=26 A=16 so A is eliminated. Round 3 votes counts: E=38 D=35 B=27 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:200 A:195 C:194 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -12 6 B 4 0 6 -14 4 C 0 -6 0 -8 2 D 12 14 8 0 10 E -6 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -12 6 B 4 0 6 -14 4 C 0 -6 0 -8 2 D 12 14 8 0 10 E -6 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -12 6 B 4 0 6 -14 4 C 0 -6 0 -8 2 D 12 14 8 0 10 E -6 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7488: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) A E D C B (6) E C A D B (5) B A D C E (5) E C D A B (4) C B E D A (4) B D C E A (4) B C E D A (4) B C D E A (4) E A C D B (3) D B A C E (3) C E D B A (3) C E D A B (3) B E C A D (3) B D C A E (3) A D B C E (3) D A B C E (2) C E B D A (2) C D E B A (2) C D E A B (2) B A E D C (2) B A D E C (2) A D E C B (2) A D E B C (2) A D B E C (2) A B D E C (2) E C D B A (1) E C B A D (1) E C A B D (1) D C E A B (1) D C A E B (1) D C A B E (1) D A C B E (1) C B D E A (1) B E A C D (1) B A E C D (1) A E C D B (1) A E B D C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 0 -10 2 B 10 0 10 4 18 C 0 -10 0 -8 18 D 10 -4 8 0 6 E -2 -18 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -10 2 B 10 0 10 4 18 C 0 -10 0 -8 18 D 10 -4 8 0 6 E -2 -18 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=21 C=17 E=15 D=9 so D is eliminated. Round 2 votes counts: B=41 A=24 C=20 E=15 so E is eliminated. Round 3 votes counts: B=41 C=32 A=27 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:210 C:200 A:191 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -10 2 B 10 0 10 4 18 C 0 -10 0 -8 18 D 10 -4 8 0 6 E -2 -18 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -10 2 B 10 0 10 4 18 C 0 -10 0 -8 18 D 10 -4 8 0 6 E -2 -18 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -10 2 B 10 0 10 4 18 C 0 -10 0 -8 18 D 10 -4 8 0 6 E -2 -18 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7489: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (12) A D C B E (9) C E D A B (6) B E C D A (5) B E C A D (5) E C B A D (4) D A B C E (4) C D E A B (4) B A E C D (4) A B D C E (4) E B C D A (3) B E D A C (3) B D A E C (3) B A D E C (3) E C D B A (2) E B C A D (2) D E B C A (2) D C A E B (2) D B E C A (2) D A C B E (2) C A E D B (2) C A D E B (2) B E A C D (2) B A E D C (2) A D C E B (2) A C D E B (2) E C B D A (1) D C E A B (1) C E A B D (1) B D E A C (1) B A D C E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 12 -6 12 B -10 0 -6 -10 4 C -12 6 0 -8 12 D 6 10 8 0 16 E -12 -4 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 -6 12 B -10 0 -6 -10 4 C -12 6 0 -8 12 D 6 10 8 0 16 E -12 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 A=19 C=15 E=12 so E is eliminated. Round 2 votes counts: B=34 D=25 C=22 A=19 so A is eliminated. Round 3 votes counts: B=38 D=37 C=25 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:214 C:199 B:189 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 12 -6 12 B -10 0 -6 -10 4 C -12 6 0 -8 12 D 6 10 8 0 16 E -12 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -6 12 B -10 0 -6 -10 4 C -12 6 0 -8 12 D 6 10 8 0 16 E -12 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -6 12 B -10 0 -6 -10 4 C -12 6 0 -8 12 D 6 10 8 0 16 E -12 -4 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7490: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) E B A D C (5) E A C B D (5) B E D A C (5) B D C A E (5) A E C D B (5) C D B E A (4) B D E C A (4) E C D B A (3) E A B D C (3) E A B C D (3) D C B A E (3) C D B A E (3) B E D C A (3) A E C B D (3) A C E D B (3) A C D B E (3) A B D C E (3) E C A D B (2) E B D A C (2) E A C D B (2) D B C E A (2) D B C A E (2) C D E B A (2) C D A E B (2) C D A B E (2) B E A D C (2) B D A C E (2) A C D E B (2) E C A B D (1) C E A D B (1) C A E D B (1) C A D E B (1) B D E A C (1) B A D E C (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 4 -4 -14 B 12 0 4 14 6 C -4 -4 0 -4 -4 D 4 -14 4 0 0 E 14 -6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -4 -14 B 12 0 4 14 6 C -4 -4 0 -4 -4 D 4 -14 4 0 0 E 14 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 A=21 C=16 D=7 so D is eliminated. Round 2 votes counts: B=34 E=26 A=21 C=19 so C is eliminated. Round 3 votes counts: B=44 E=29 A=27 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:206 D:197 C:192 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 -4 -14 B 12 0 4 14 6 C -4 -4 0 -4 -4 D 4 -14 4 0 0 E 14 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -4 -14 B 12 0 4 14 6 C -4 -4 0 -4 -4 D 4 -14 4 0 0 E 14 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -4 -14 B 12 0 4 14 6 C -4 -4 0 -4 -4 D 4 -14 4 0 0 E 14 -6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999705 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7491: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) A B E D C (8) D E B C A (6) E D B A C (5) C E D B A (5) B A D E C (5) A B C D E (5) D E B A C (4) C E D A B (4) C D B E A (4) D B E A C (3) C E A D B (3) C D E B A (3) C A E D B (3) C A B D E (3) B D E A C (3) A C B E D (3) A C B D E (3) A B D E C (3) E C D B A (2) E C D A B (2) C A B E D (2) B D A E C (2) A B C E D (2) E D B C A (1) E D A B C (1) E A D C B (1) E A C D B (1) C B A D E (1) B D A C E (1) B C D A E (1) A E C D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 4 -10 -12 B 8 0 2 -14 0 C -4 -2 0 -4 -12 D 10 14 4 0 -4 E 12 0 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.133921 C: 0.000000 D: 0.000000 E: 0.866079 Sum of squares = 0.768026972969 Cumulative probabilities = A: 0.000000 B: 0.133921 C: 0.133921 D: 0.133921 E: 1.000000 A B C D E A 0 -8 4 -10 -12 B 8 0 2 -14 0 C -4 -2 0 -4 -12 D 10 14 4 0 -4 E 12 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.777778 Sum of squares = 0.654321213678 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.222222 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=26 E=21 D=13 B=12 so B is eliminated. Round 2 votes counts: A=31 C=29 E=21 D=19 so D is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:212 B:198 C:189 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 -10 -12 B 8 0 2 -14 0 C -4 -2 0 -4 -12 D 10 14 4 0 -4 E 12 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.777778 Sum of squares = 0.654321213678 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.222222 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -10 -12 B 8 0 2 -14 0 C -4 -2 0 -4 -12 D 10 14 4 0 -4 E 12 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.777778 Sum of squares = 0.654321213678 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.222222 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -10 -12 B 8 0 2 -14 0 C -4 -2 0 -4 -12 D 10 14 4 0 -4 E 12 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.777778 Sum of squares = 0.654321213678 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.222222 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7492: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) B A D C E (8) E C B D A (7) C D A B E (7) E B A D C (6) C D E A B (5) B E A D C (5) B A D E C (5) A D B C E (5) C D A E B (4) B A C D E (4) E D A B C (3) E B C A D (3) D A C E B (3) C E D A B (3) C B A D E (3) B C A D E (3) E B C D A (2) D A E C B (2) D A C B E (2) C A D B E (2) B E C A D (2) B E A C D (2) E D C A B (1) C E D B A (1) C B D A E (1) C A B D E (1) B A E D C (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -6 2 4 B 6 0 -2 6 4 C 6 2 0 18 6 D -2 -6 -18 0 12 E -4 -4 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 2 4 B 6 0 -2 6 4 C 6 2 0 18 6 D -2 -6 -18 0 12 E -4 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=30 B=30 C=27 D=7 A=6 so A is eliminated. Round 2 votes counts: E=30 B=30 C=28 D=12 so D is eliminated. Round 3 votes counts: B=35 C=33 E=32 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:207 A:197 D:193 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 2 4 B 6 0 -2 6 4 C 6 2 0 18 6 D -2 -6 -18 0 12 E -4 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 2 4 B 6 0 -2 6 4 C 6 2 0 18 6 D -2 -6 -18 0 12 E -4 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 2 4 B 6 0 -2 6 4 C 6 2 0 18 6 D -2 -6 -18 0 12 E -4 -4 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7493: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (6) E C B A D (5) C E D A B (5) C E B D A (5) B A E C D (5) E C A D B (4) A B D E C (4) E C D A B (3) D C E A B (3) D A E C B (3) D A B E C (3) C E D B A (3) C D E B A (3) B D A C E (3) B C E D A (3) B A D E C (3) E C A B D (2) E A D C B (2) D A C E B (2) C D E A B (2) C B E D A (2) B C D E A (2) B C D A E (2) B A E D C (2) B A C E D (2) A D B E C (2) A B E D C (2) E B C A D (1) E A C B D (1) E A B C D (1) D E C A B (1) D B C A E (1) D A E B C (1) D A B C E (1) C E B A D (1) C D B E A (1) B E C A D (1) B E A C D (1) B D C A E (1) B A D C E (1) B A C D E (1) A E D B C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -10 -14 0 -16 B 10 0 2 14 -2 C 14 -2 0 24 0 D 0 -14 -24 0 -16 E 16 2 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.145602 D: 0.000000 E: 0.854398 Sum of squares = 0.751195867253 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.145602 D: 0.145602 E: 1.000000 A B C D E A 0 -10 -14 0 -16 B 10 0 2 14 -2 C 14 -2 0 24 0 D 0 -14 -24 0 -16 E 16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.000000 E: 0.500356 Sum of squares = 0.500000253855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.499644 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=22 E=19 D=15 A=11 so A is eliminated. Round 2 votes counts: B=39 C=22 E=20 D=19 so D is eliminated. Round 3 votes counts: B=46 E=27 C=27 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:218 E:217 B:212 A:180 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -14 0 -16 B 10 0 2 14 -2 C 14 -2 0 24 0 D 0 -14 -24 0 -16 E 16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.000000 E: 0.500356 Sum of squares = 0.500000253855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.499644 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 0 -16 B 10 0 2 14 -2 C 14 -2 0 24 0 D 0 -14 -24 0 -16 E 16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.000000 E: 0.500356 Sum of squares = 0.500000253855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.499644 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 0 -16 B 10 0 2 14 -2 C 14 -2 0 24 0 D 0 -14 -24 0 -16 E 16 2 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.000000 E: 0.500356 Sum of squares = 0.500000253855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499644 D: 0.499644 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7494: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) D B C A E (8) E A B C D (6) D A B C E (6) C B E D A (6) B D C E A (5) E C B A D (4) E C A B D (4) D C B A E (4) A E C D B (4) A D B E C (4) D B A C E (3) A E D B C (3) A D E B C (3) C E B D A (2) C B D E A (2) B D C A E (2) B C E D A (2) B C D E A (2) A D E C B (2) E B A D C (1) E B A C D (1) E A C D B (1) E A C B D (1) E A B D C (1) D C A B E (1) C E B A D (1) C E A B D (1) C D B E A (1) C A E D B (1) B E A D C (1) B D A E C (1) B D A C E (1) B A D E C (1) A E B D C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 8 8 16 B -4 0 4 -8 0 C -8 -4 0 -20 0 D -8 8 20 0 -4 E -16 0 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 16 B -4 0 4 -8 0 C -8 -4 0 -20 0 D -8 8 20 0 -4 E -16 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 E=19 B=15 C=14 so C is eliminated. Round 2 votes counts: A=31 E=23 D=23 B=23 so E is eliminated. Round 3 votes counts: A=45 B=32 D=23 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:208 B:196 E:194 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 16 B -4 0 4 -8 0 C -8 -4 0 -20 0 D -8 8 20 0 -4 E -16 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 16 B -4 0 4 -8 0 C -8 -4 0 -20 0 D -8 8 20 0 -4 E -16 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 16 B -4 0 4 -8 0 C -8 -4 0 -20 0 D -8 8 20 0 -4 E -16 0 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7495: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (12) B C D E A (9) D B E C A (7) C B A E D (6) E D A B C (5) C A B E D (5) B D C E A (5) D E B C A (4) D E B A C (4) A C E D B (4) E D B A C (3) E A D B C (3) D E A C B (3) C B D A E (3) B D E C A (3) A E D C B (3) A C B E D (3) E A D C B (2) D E A B C (2) D A E C B (2) C A B D E (2) A E C D B (2) E D A C B (1) E B A D C (1) B C E A D (1) A E D B C (1) A D E C B (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -14 -2 -6 B 16 0 -4 2 14 C 14 4 0 0 6 D 2 -2 0 0 18 E 6 -14 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.679323 D: 0.320677 E: 0.000000 Sum of squares = 0.564313643063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.679323 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -2 -6 B 16 0 -4 2 14 C 14 4 0 0 6 D 2 -2 0 0 18 E 6 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=22 B=18 A=17 E=15 so E is eliminated. Round 2 votes counts: D=31 C=28 A=22 B=19 so B is eliminated. Round 3 votes counts: D=39 C=38 A=23 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:212 D:209 E:184 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -14 -2 -6 B 16 0 -4 2 14 C 14 4 0 0 6 D 2 -2 0 0 18 E 6 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -2 -6 B 16 0 -4 2 14 C 14 4 0 0 6 D 2 -2 0 0 18 E 6 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -2 -6 B 16 0 -4 2 14 C 14 4 0 0 6 D 2 -2 0 0 18 E 6 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7496: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) A B D E C (6) D C E B A (5) B C D A E (5) A E D B C (5) E C A B D (4) C E D B A (4) C B E D A (4) C B D E A (4) A E D C B (4) A E B D C (4) E C D A B (3) E A C D B (3) D C B E A (3) D B C A E (3) E A C B D (2) D E A C B (2) D B A C E (2) B D C A E (2) B A D C E (2) B A C E D (2) A E B C D (2) A D E B C (2) A D B E C (2) A B E D C (2) E D C A B (1) E D A C B (1) E C D B A (1) E C A D B (1) E A B C D (1) D E C A B (1) D A E C B (1) C E B A D (1) C D B E A (1) B C A D E (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 18 12 14 -6 B -18 0 -14 -12 -18 C -12 14 0 -18 -20 D -14 12 18 0 -14 E 6 18 20 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 12 14 -6 B -18 0 -14 -12 -18 C -12 14 0 -18 -20 D -14 12 18 0 -14 E 6 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 D=17 C=14 B=13 so B is eliminated. Round 2 votes counts: A=33 E=28 C=20 D=19 so D is eliminated. Round 3 votes counts: A=36 C=33 E=31 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:229 A:219 D:201 C:182 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 12 14 -6 B -18 0 -14 -12 -18 C -12 14 0 -18 -20 D -14 12 18 0 -14 E 6 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 12 14 -6 B -18 0 -14 -12 -18 C -12 14 0 -18 -20 D -14 12 18 0 -14 E 6 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 12 14 -6 B -18 0 -14 -12 -18 C -12 14 0 -18 -20 D -14 12 18 0 -14 E 6 18 20 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7497: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) B A C D E (7) B A D E C (6) E D C A B (5) D E C B A (4) C B A E D (4) B C D E A (4) B C A D E (4) D E C A B (3) D E A C B (3) D E A B C (3) D B E A C (3) C E D A B (3) C A B E D (3) B C D A E (3) B A C E D (3) A E C D B (3) A B C E D (3) D C E B A (2) C E D B A (2) C E A D B (2) C B E A D (2) B D C E A (2) B D C A E (2) B D A C E (2) A B E D C (2) D E B C A (1) D B A E C (1) C E A B D (1) C D E B A (1) C A E D B (1) C A E B D (1) B D E C A (1) B C A E D (1) B A D C E (1) A E D C B (1) A D B E C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 -4 -4 2 B 12 0 0 6 10 C 4 0 0 0 12 D 4 -6 0 0 8 E -2 -10 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.571859 C: 0.428141 D: 0.000000 E: 0.000000 Sum of squares = 0.510327476785 Cumulative probabilities = A: 0.000000 B: 0.571859 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -4 2 B 12 0 0 6 10 C 4 0 0 0 12 D 4 -6 0 0 8 E -2 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=20 C=20 E=12 A=12 so E is eliminated. Round 2 votes counts: B=36 D=32 C=20 A=12 so A is eliminated. Round 3 votes counts: B=41 D=34 C=25 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:208 D:203 A:191 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -4 2 B 12 0 0 6 10 C 4 0 0 0 12 D 4 -6 0 0 8 E -2 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -4 2 B 12 0 0 6 10 C 4 0 0 0 12 D 4 -6 0 0 8 E -2 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -4 2 B 12 0 0 6 10 C 4 0 0 0 12 D 4 -6 0 0 8 E -2 -10 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7498: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (6) B E C D A (6) E B A D C (5) D A C B E (5) B C D E A (5) A D C E B (5) E B C A D (4) E A D B C (4) E A B C D (4) D C A B E (4) A D E C B (4) E B A C D (3) D B E A C (3) D A E B C (3) C B D A E (3) C A D B E (3) B E D C A (3) E A B D C (2) D C B A E (2) C A E B D (2) B C E D A (2) A E D C B (2) A C D E B (2) E C B A D (1) E B C D A (1) D E B A C (1) D B C E A (1) D A E C B (1) D A C E B (1) C E B A D (1) C D B A E (1) C D A B E (1) C B E A D (1) C A B D E (1) B D E C A (1) B D C E A (1) B C E A D (1) A E C D B (1) A E C B D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 10 -2 -12 B 4 0 12 10 -12 C -10 -12 0 -8 -12 D 2 -10 8 0 -4 E 12 12 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 10 -2 -12 B 4 0 12 10 -12 C -10 -12 0 -8 -12 D 2 -10 8 0 -4 E 12 12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=21 B=19 A=17 C=13 so C is eliminated. Round 2 votes counts: E=31 D=23 B=23 A=23 so D is eliminated. Round 3 votes counts: A=38 E=32 B=30 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:207 D:198 A:196 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 10 -2 -12 B 4 0 12 10 -12 C -10 -12 0 -8 -12 D 2 -10 8 0 -4 E 12 12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -2 -12 B 4 0 12 10 -12 C -10 -12 0 -8 -12 D 2 -10 8 0 -4 E 12 12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -2 -12 B 4 0 12 10 -12 C -10 -12 0 -8 -12 D 2 -10 8 0 -4 E 12 12 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7499: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (16) C E A B D (10) B D A E C (7) D B A E C (6) C E A D B (5) C A E B D (5) B A E C D (5) A C E B D (5) D B E C A (4) B D E A C (4) E A C B D (3) D B C E A (3) D B A C E (3) C A E D B (3) A E C B D (3) E A B C D (2) D E C B A (2) D C E A B (2) C D E A B (2) C A D B E (2) B A E D C (2) E C A B D (1) D C B A E (1) D C A E B (1) D B C A E (1) C A D E B (1) A E B C D (1) Total count = 100 A B C D E A 0 -8 14 -4 -8 B 8 0 8 -4 8 C -14 -8 0 -4 -12 D 4 4 4 0 10 E 8 -8 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 14 -4 -8 B 8 0 8 -4 8 C -14 -8 0 -4 -12 D 4 4 4 0 10 E 8 -8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=28 B=18 A=9 E=6 so E is eliminated. Round 2 votes counts: D=39 C=29 B=18 A=14 so A is eliminated. Round 3 votes counts: C=40 D=39 B=21 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 B:210 E:201 A:197 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 14 -4 -8 B 8 0 8 -4 8 C -14 -8 0 -4 -12 D 4 4 4 0 10 E 8 -8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 -4 -8 B 8 0 8 -4 8 C -14 -8 0 -4 -12 D 4 4 4 0 10 E 8 -8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 -4 -8 B 8 0 8 -4 8 C -14 -8 0 -4 -12 D 4 4 4 0 10 E 8 -8 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7500: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) A E D C B (11) E A D B C (9) C D B A E (9) E A B D C (8) D A C E B (5) C B D A E (5) B C E D A (5) B E A C D (4) A D E C B (4) E B A C D (3) B E C A D (3) D C B A E (2) D A E C B (2) C D A B E (2) E B A D C (1) E A C B D (1) E A B C D (1) D C A E B (1) D A E B C (1) C B E A D (1) C B D E A (1) C A D E B (1) B E C D A (1) B E A D C (1) B D E C A (1) B C E A D (1) A E D B C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 8 4 -8 B 2 0 6 0 -2 C -8 -6 0 4 -4 D -4 0 -4 0 -4 E 8 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 4 -8 B 2 0 6 0 -2 C -8 -6 0 4 -4 D -4 0 -4 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 C=19 A=18 D=11 so D is eliminated. Round 2 votes counts: B=29 A=26 E=23 C=22 so C is eliminated. Round 3 votes counts: B=47 A=30 E=23 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:209 B:203 A:201 D:194 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 4 -8 B 2 0 6 0 -2 C -8 -6 0 4 -4 D -4 0 -4 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 4 -8 B 2 0 6 0 -2 C -8 -6 0 4 -4 D -4 0 -4 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 4 -8 B 2 0 6 0 -2 C -8 -6 0 4 -4 D -4 0 -4 0 -4 E 8 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7501: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (7) D A E B C (6) C B A E D (6) A D C B E (6) A C D B E (6) E D B A C (5) C B A D E (5) E D A C B (4) E B C D A (4) B E C D A (4) E C B A D (3) E B D C A (3) D E A C B (3) C A B D E (3) B C E A D (3) B C A D E (3) A D C E B (3) E D A B C (2) D A E C B (2) D A C B E (2) B E D C A (2) B D E A C (2) B D A C E (2) B C A E D (2) E C B D A (1) E C A D B (1) D E A B C (1) D A B C E (1) C E B A D (1) C E A D B (1) C A E D B (1) C A D B E (1) C A B E D (1) A D E C B (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -4 12 6 B 6 0 -18 4 14 C 4 18 0 8 12 D -12 -4 -8 0 -2 E -6 -14 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 12 6 B 6 0 -18 4 14 C 4 18 0 8 12 D -12 -4 -8 0 -2 E -6 -14 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=18 A=18 D=15 so D is eliminated. Round 2 votes counts: A=29 E=27 C=26 B=18 so B is eliminated. Round 3 votes counts: E=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:221 A:204 B:203 D:187 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 12 6 B 6 0 -18 4 14 C 4 18 0 8 12 D -12 -4 -8 0 -2 E -6 -14 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 12 6 B 6 0 -18 4 14 C 4 18 0 8 12 D -12 -4 -8 0 -2 E -6 -14 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 12 6 B 6 0 -18 4 14 C 4 18 0 8 12 D -12 -4 -8 0 -2 E -6 -14 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7502: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B E A C D (8) D C B A E (6) B C D A E (6) B A E C D (6) E A B C D (5) E A C D B (4) D E C A B (4) C D B A E (4) B D C E A (4) A E B C D (4) E A D B C (3) E A B D C (3) D C E A B (3) D C B E A (3) C D A E B (3) B E D A C (3) A E C D B (3) E B A C D (2) D C A B E (2) E D B A C (1) E D A C B (1) E B A D C (1) E A D C B (1) D E A C B (1) D B E C A (1) D B C E A (1) C D A B E (1) C B D A E (1) C A B E D (1) B E A D C (1) B C A D E (1) B A C E D (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 0 0 -8 0 B 0 0 2 -4 0 C 0 -2 0 2 -8 D 8 4 -2 0 0 E 0 0 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500021 E: 0.499979 Sum of squares = 0.5000000009 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500021 E: 1.000000 A B C D E A 0 0 0 -8 0 B 0 0 2 -4 0 C 0 -2 0 2 -8 D 8 4 -2 0 0 E 0 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=30 B=30 E=21 C=10 A=9 so A is eliminated. Round 2 votes counts: E=30 D=30 B=30 C=10 so C is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:205 E:204 B:199 A:196 C:196 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 0 -8 0 B 0 0 2 -4 0 C 0 -2 0 2 -8 D 8 4 -2 0 0 E 0 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 0 B 0 0 2 -4 0 C 0 -2 0 2 -8 D 8 4 -2 0 0 E 0 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 0 B 0 0 2 -4 0 C 0 -2 0 2 -8 D 8 4 -2 0 0 E 0 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7503: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) B D E A C (8) B E D C A (7) A C E D B (6) E D B A C (5) D E A B C (4) C A E B D (4) E D A C B (3) E D A B C (3) E C A D B (3) B D E C A (3) B C A D E (3) E B D C A (2) D B E A C (2) C E A B D (2) C A B D E (2) B E C D A (2) B D C E A (2) B D A C E (2) B C E D A (2) B C D E A (2) A D C E B (2) A C D E B (2) A C D B E (2) A C B D E (2) E D C A B (1) E D B C A (1) E C D A B (1) D A E B C (1) C E B A D (1) C E A D B (1) C B E A D (1) C A B E D (1) B D C A E (1) B D A E C (1) B C E A D (1) A E C D B (1) A D E C B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -6 -6 -14 B -8 0 6 0 -8 C 6 -6 0 0 2 D 6 0 0 0 -16 E 14 8 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999988 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 8 -6 -6 -14 B -8 0 6 0 -8 C 6 -6 0 0 2 D 6 0 0 0 -16 E 14 8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999953 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=22 E=19 A=18 D=7 so D is eliminated. Round 2 votes counts: B=36 E=23 C=22 A=19 so A is eliminated. Round 3 votes counts: C=37 B=37 E=26 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:218 C:201 B:195 D:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 -6 -14 B -8 0 6 0 -8 C 6 -6 0 0 2 D 6 0 0 0 -16 E 14 8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999953 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -6 -14 B -8 0 6 0 -8 C 6 -6 0 0 2 D 6 0 0 0 -16 E 14 8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999953 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -6 -14 B -8 0 6 0 -8 C 6 -6 0 0 2 D 6 0 0 0 -16 E 14 8 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.500000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999953 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7504: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) C E B A D (9) C E D A B (7) B D A C E (7) E C A B D (5) D A E C B (4) D A C E B (4) B C E A D (4) E C B A D (3) E C A D B (3) E A D C B (3) D C A E B (3) C E B D A (3) B A D E C (3) E B C A D (2) D A E B C (2) C B E D A (2) B A E D C (2) A E D B C (2) A D E B C (2) A B D E C (2) E B A C D (1) E A B C D (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D B A (1) C E A D B (1) C E A B D (1) C D E A B (1) C D B E A (1) C B D E A (1) B E C A D (1) B E A C D (1) B D C A E (1) B D A E C (1) B C E D A (1) B C A E D (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -6 -4 -4 B -6 0 -8 4 -16 C 6 8 0 0 4 D 4 -4 0 0 -8 E 4 16 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.842258 D: 0.157742 E: 0.000000 Sum of squares = 0.734281206907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.842258 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -4 -4 B -6 0 -8 4 -16 C 6 8 0 0 4 D 4 -4 0 0 -8 E 4 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.55555563432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 B=22 E=18 A=7 so A is eliminated. Round 2 votes counts: D=29 C=27 B=24 E=20 so E is eliminated. Round 3 votes counts: C=38 D=34 B=28 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:209 A:196 D:196 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -4 -4 B -6 0 -8 4 -16 C 6 8 0 0 4 D 4 -4 0 0 -8 E 4 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.55555563432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -4 -4 B -6 0 -8 4 -16 C 6 8 0 0 4 D 4 -4 0 0 -8 E 4 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.55555563432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -4 -4 B -6 0 -8 4 -16 C 6 8 0 0 4 D 4 -4 0 0 -8 E 4 16 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.55555563432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7505: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) A C E B D (10) E D B A C (9) D E B C A (9) A C B E D (8) C B A D E (7) B C D E A (7) E A D C B (6) C A B D E (5) D B E C A (4) A E C D B (4) A E C B D (4) B D E C A (3) A E D C B (3) E D A C B (2) B D C E A (2) B C D A E (2) D E C B A (1) D E B A C (1) C B D A E (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 8 18 -2 -8 B -8 0 -4 0 -22 C -18 4 0 -2 -14 D 2 0 2 0 -16 E 8 22 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 18 -2 -8 B -8 0 -4 0 -22 C -18 4 0 -2 -14 D 2 0 2 0 -16 E 8 22 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=27 D=15 B=14 C=13 so C is eliminated. Round 2 votes counts: A=36 E=27 B=22 D=15 so D is eliminated. Round 3 votes counts: E=38 A=36 B=26 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:230 A:208 D:194 C:185 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 18 -2 -8 B -8 0 -4 0 -22 C -18 4 0 -2 -14 D 2 0 2 0 -16 E 8 22 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 -2 -8 B -8 0 -4 0 -22 C -18 4 0 -2 -14 D 2 0 2 0 -16 E 8 22 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 -2 -8 B -8 0 -4 0 -22 C -18 4 0 -2 -14 D 2 0 2 0 -16 E 8 22 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7506: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (6) B D C E A (5) A E C D B (5) E A D B C (4) D A B C E (4) C B D A E (4) C A E B D (4) B C D E A (4) A D C E B (4) D C B A E (3) D A C B E (3) C B A D E (3) C A B D E (3) A E D B C (3) A C E D B (3) E B C D A (2) E A C B D (2) D B E A C (2) D A E B C (2) D A B E C (2) C B A E D (2) B E D C A (2) B E C D A (2) A D E B C (2) A C D E B (2) E D B A C (1) E D A B C (1) E C B A D (1) E B D A C (1) E B C A D (1) E A D C B (1) E A C D B (1) E A B D C (1) D E A B C (1) D B E C A (1) D B C E A (1) D B C A E (1) C E B A D (1) C E A B D (1) C D B A E (1) C B E A D (1) C A D B E (1) C A B E D (1) B D E C A (1) B C E D A (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 18 6 10 22 B -18 0 -10 -12 -2 C -6 10 0 -6 10 D -10 12 6 0 2 E -22 2 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 10 22 B -18 0 -10 -12 -2 C -6 10 0 -6 10 D -10 12 6 0 2 E -22 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 D=20 E=16 B=15 so B is eliminated. Round 2 votes counts: C=27 A=27 D=26 E=20 so E is eliminated. Round 3 votes counts: A=36 C=33 D=31 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:228 D:205 C:204 E:184 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 10 22 B -18 0 -10 -12 -2 C -6 10 0 -6 10 D -10 12 6 0 2 E -22 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 10 22 B -18 0 -10 -12 -2 C -6 10 0 -6 10 D -10 12 6 0 2 E -22 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 10 22 B -18 0 -10 -12 -2 C -6 10 0 -6 10 D -10 12 6 0 2 E -22 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997793 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7507: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (6) E B A D C (4) D B E A C (4) D B A E C (4) B D E C A (4) A C D E B (4) A C D B E (4) E B D C A (3) E B C D A (3) E B A C D (3) E A B D C (3) D B E C A (3) D B C E A (3) C E B D A (3) C E A B D (3) C D B E A (3) C A D B E (3) A D B C E (3) E C A B D (2) E B D A C (2) E A B C D (2) D C B A E (2) D B C A E (2) D A B C E (2) C E B A D (2) C D A B E (2) C B D E A (2) B D E A C (2) A E C B D (2) A D C B E (2) A C E D B (2) E C B D A (1) C E A D B (1) C B E D A (1) B E D C A (1) B E D A C (1) B D C E A (1) B C E D A (1) A E D B C (1) A E B C D (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -4 0 -16 B 10 0 8 -6 2 C 4 -8 0 2 6 D 0 6 -2 0 16 E 16 -2 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000006 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 0 -16 B 10 0 8 -6 2 C 4 -8 0 2 6 D 0 6 -2 0 16 E 16 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000013 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 A=21 D=20 B=10 so B is eliminated. Round 2 votes counts: D=27 C=27 E=25 A=21 so A is eliminated. Round 3 votes counts: C=37 D=34 E=29 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:210 B:207 C:202 E:196 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 0 -16 B 10 0 8 -6 2 C 4 -8 0 2 6 D 0 6 -2 0 16 E 16 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000013 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 0 -16 B 10 0 8 -6 2 C 4 -8 0 2 6 D 0 6 -2 0 16 E 16 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000013 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 0 -16 B 10 0 8 -6 2 C 4 -8 0 2 6 D 0 6 -2 0 16 E 16 -2 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000013 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7508: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (14) D B C E A (7) A E C D B (7) D C B A E (6) D B C A E (6) B D C E A (6) C D B E A (5) B D E C A (5) B D E A C (5) C D B A E (4) C A E D B (4) A E B D C (4) C A D E B (3) A C E D B (3) E A B D C (2) E A B C D (2) D C B E A (2) D B A E C (2) C D A E B (2) A E D C B (2) E A C B D (1) D A C E B (1) B E D A C (1) B E C D A (1) B E A D C (1) B C E A D (1) A E D B C (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -4 -6 22 B 4 0 -8 -12 4 C 4 8 0 -4 0 D 6 12 4 0 10 E -22 -4 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 22 B 4 0 -8 -12 4 C 4 8 0 -4 0 D 6 12 4 0 10 E -22 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=24 B=20 C=18 E=5 so E is eliminated. Round 2 votes counts: A=38 D=24 B=20 C=18 so C is eliminated. Round 3 votes counts: A=45 D=35 B=20 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:204 C:204 B:194 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 22 B 4 0 -8 -12 4 C 4 8 0 -4 0 D 6 12 4 0 10 E -22 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 22 B 4 0 -8 -12 4 C 4 8 0 -4 0 D 6 12 4 0 10 E -22 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 22 B 4 0 -8 -12 4 C 4 8 0 -4 0 D 6 12 4 0 10 E -22 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7509: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B D A E (9) C D B A E (6) B D A C E (6) A E B D C (6) A B D E C (5) E C A B D (4) B A D C E (4) E C D A B (3) E C A D B (3) E A C D B (3) E A C B D (3) E A B D C (3) E A B C D (3) C E D B A (3) C D E B A (3) C D B E A (3) B D C A E (3) A B D C E (3) E C D B A (2) A E D B C (2) A B E D C (2) E D C A B (1) E D A C B (1) E A D C B (1) D C B A E (1) D B C A E (1) D B A E C (1) C B D E A (1) B C D A E (1) B C A D E (1) A E B C D (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 10 10 8 B -10 0 6 12 -2 C -10 -6 0 0 -8 D -10 -12 0 0 -2 E -8 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 10 8 B -10 0 6 12 -2 C -10 -6 0 0 -8 D -10 -12 0 0 -2 E -8 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=25 A=21 B=15 D=3 so D is eliminated. Round 2 votes counts: E=36 C=26 A=21 B=17 so B is eliminated. Round 3 votes counts: E=36 C=32 A=32 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:203 E:202 C:188 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 10 8 B -10 0 6 12 -2 C -10 -6 0 0 -8 D -10 -12 0 0 -2 E -8 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 10 8 B -10 0 6 12 -2 C -10 -6 0 0 -8 D -10 -12 0 0 -2 E -8 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 10 8 B -10 0 6 12 -2 C -10 -6 0 0 -8 D -10 -12 0 0 -2 E -8 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7510: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) B C D A E (7) E D A B C (6) E A D C B (6) B D E C A (5) A E C D B (5) E A D B C (4) A B E C D (4) D E C A B (3) D E B C A (3) C B A D E (3) B A C E D (3) A E B D C (3) A E B C D (3) A C E D B (3) A C E B D (3) A B C E D (3) E D A C B (2) D C E B A (2) D C B E A (2) D B E C A (2) C D E A B (2) C D B A E (2) C D A E B (2) C A B D E (2) B E D A C (2) E D C A B (1) E D B A C (1) E A B D C (1) C B D A E (1) C A E D B (1) C A D B E (1) B D C A E (1) B C A D E (1) B A E C D (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 14 4 -2 2 B -14 0 14 2 -6 C -4 -14 0 -4 -8 D 2 -2 4 0 -8 E -2 6 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.500000000041 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 14 4 -2 2 B -14 0 14 2 -6 C -4 -14 0 -4 -8 D 2 -2 4 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 E=21 C=14 D=12 so D is eliminated. Round 2 votes counts: B=29 E=27 A=26 C=18 so C is eliminated. Round 3 votes counts: B=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:209 B:198 D:198 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 -2 2 B -14 0 14 2 -6 C -4 -14 0 -4 -8 D 2 -2 4 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -2 2 B -14 0 14 2 -6 C -4 -14 0 -4 -8 D 2 -2 4 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -2 2 B -14 0 14 2 -6 C -4 -14 0 -4 -8 D 2 -2 4 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7511: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (15) B E D A C (15) C A D E B (11) A C B E D (7) D E B C A (6) C A B E D (5) E D B C A (3) B A E D C (3) A C B D E (3) A B C E D (3) D E C B A (2) D C E B A (2) D A E B C (2) C D E B A (2) C D E A B (2) C D A E B (2) B E D C A (2) A D E C B (2) A B E D C (2) E B D C A (1) D E A C B (1) D B E A C (1) D A E C B (1) C E B D A (1) C A D B E (1) B E A D C (1) B D E A C (1) A C D B E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 20 -18 -10 B 10 0 14 -8 -6 C -20 -14 0 -22 -20 D 18 8 22 0 12 E 10 6 20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 20 -18 -10 B 10 0 14 -8 -6 C -20 -14 0 -22 -20 D 18 8 22 0 12 E 10 6 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=24 B=22 A=20 E=4 so E is eliminated. Round 2 votes counts: D=33 C=24 B=23 A=20 so A is eliminated. Round 3 votes counts: D=35 C=35 B=30 so B is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 E:212 B:205 A:191 C:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 20 -18 -10 B 10 0 14 -8 -6 C -20 -14 0 -22 -20 D 18 8 22 0 12 E 10 6 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 20 -18 -10 B 10 0 14 -8 -6 C -20 -14 0 -22 -20 D 18 8 22 0 12 E 10 6 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 20 -18 -10 B 10 0 14 -8 -6 C -20 -14 0 -22 -20 D 18 8 22 0 12 E 10 6 20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7512: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) B C E D A (8) D B A E C (6) B D E C A (5) A E C D B (5) C E B A D (4) C B E A D (4) B C D E A (4) E B D C A (3) D A E B C (3) A E D C B (3) A C D B E (3) E C B A D (2) E C A B D (2) E B C D A (2) E B C A D (2) E A D B C (2) D B E A C (2) D A B E C (2) D A B C E (2) C E A B D (2) B E D C A (2) B E C D A (2) B C D A E (2) A D C E B (2) E B D A C (1) E B A C D (1) E A C D B (1) E A B C D (1) D E B A C (1) D C A B E (1) D B E C A (1) D B A C E (1) D A C B E (1) C B E D A (1) C B A D E (1) C A E B D (1) C A B D E (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 0 0 -6 B 10 0 6 4 0 C 0 -6 0 0 -20 D 0 -4 0 0 0 E 6 0 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.639667 C: 0.000000 D: 0.000000 E: 0.360333 Sum of squares = 0.53901386494 Cumulative probabilities = A: 0.000000 B: 0.639667 C: 0.639667 D: 0.639667 E: 1.000000 A B C D E A 0 -10 0 0 -6 B 10 0 6 4 0 C 0 -6 0 0 -20 D 0 -4 0 0 0 E 6 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=23 D=20 E=17 C=14 so C is eliminated. Round 2 votes counts: B=29 A=28 E=23 D=20 so D is eliminated. Round 3 votes counts: B=39 A=37 E=24 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:213 B:210 D:198 A:192 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 0 -6 B 10 0 6 4 0 C 0 -6 0 0 -20 D 0 -4 0 0 0 E 6 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 0 -6 B 10 0 6 4 0 C 0 -6 0 0 -20 D 0 -4 0 0 0 E 6 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 0 -6 B 10 0 6 4 0 C 0 -6 0 0 -20 D 0 -4 0 0 0 E 6 0 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7513: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A D E B (10) B D E A C (10) B A D C E (6) B A C D E (6) E D C A B (4) B E C D A (4) E D A C B (3) C A E D B (3) A C D E B (3) A C D B E (3) A C B D E (3) E D C B A (2) E D B A C (2) E C D A B (2) E B D A C (2) D E A C B (2) D B E A C (2) C E D A B (2) C E A D B (2) B E D C A (2) B D A E C (2) B A D E C (2) E D B C A (1) E C D B A (1) E B D C A (1) D E A B C (1) D A E B C (1) D A C E B (1) C A B E D (1) C A B D E (1) B A E C D (1) A D C B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 28 -12 -8 B 10 0 12 8 14 C -28 -12 0 -14 -12 D 12 -8 14 0 12 E 8 -14 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 28 -12 -8 B 10 0 12 8 14 C -28 -12 0 -14 -12 D 12 -8 14 0 12 E 8 -14 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 C=19 E=18 A=12 D=7 so D is eliminated. Round 2 votes counts: B=46 E=21 C=19 A=14 so A is eliminated. Round 3 votes counts: B=48 C=30 E=22 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:215 A:199 E:197 C:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 28 -12 -8 B 10 0 12 8 14 C -28 -12 0 -14 -12 D 12 -8 14 0 12 E 8 -14 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 28 -12 -8 B 10 0 12 8 14 C -28 -12 0 -14 -12 D 12 -8 14 0 12 E 8 -14 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 28 -12 -8 B 10 0 12 8 14 C -28 -12 0 -14 -12 D 12 -8 14 0 12 E 8 -14 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7514: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (15) E A B D C (9) C D A B E (6) B D C A E (6) E A B C D (5) D C B E A (5) D C B A E (5) B D C E A (5) A E C B D (5) B E D A C (4) C D B E A (3) A E B C D (3) E B A D C (2) E A C B D (2) D B C E A (2) D B C A E (2) C A D E B (2) B D E A C (2) B A E D C (2) E B D A C (1) E A C D B (1) D E C B A (1) C E D A B (1) C D E A B (1) C D A E B (1) C A E D B (1) C A D B E (1) B E A D C (1) B D E C A (1) B D A C E (1) A E C D B (1) A E B D C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -16 -24 8 B 16 0 -6 2 22 C 16 6 0 0 18 D 24 -2 0 0 18 E -8 -22 -18 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.737241 D: 0.262759 E: 0.000000 Sum of squares = 0.612566827727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.737241 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -24 8 B 16 0 -6 2 22 C 16 6 0 0 18 D 24 -2 0 0 18 E -8 -22 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=22 E=20 D=15 A=12 so A is eliminated. Round 2 votes counts: C=33 E=30 B=22 D=15 so D is eliminated. Round 3 votes counts: C=43 E=31 B=26 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:220 B:217 A:176 E:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -16 -24 8 B 16 0 -6 2 22 C 16 6 0 0 18 D 24 -2 0 0 18 E -8 -22 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -24 8 B 16 0 -6 2 22 C 16 6 0 0 18 D 24 -2 0 0 18 E -8 -22 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -24 8 B 16 0 -6 2 22 C 16 6 0 0 18 D 24 -2 0 0 18 E -8 -22 -18 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7515: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) B D C E A (9) C B D E A (7) A E C D B (7) E A C B D (6) D B C A E (6) A E C B D (6) E C A B D (5) C E B D A (5) E A C D B (4) D B A C E (4) C D B E A (4) A D B E C (4) E C A D B (3) C E A B D (3) A E D B C (3) D B A E C (2) B D A C E (2) A E B D C (2) D A B E C (1) C E D B A (1) C E A D B (1) B D C A E (1) B D A E C (1) B C D E A (1) A E D C B (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -10 -6 -16 B 4 0 -6 0 4 C 10 6 0 8 6 D 6 0 -8 0 4 E 16 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -16 B 4 0 -6 0 4 C 10 6 0 8 6 D 6 0 -8 0 4 E 16 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=22 C=21 E=18 B=14 so B is eliminated. Round 2 votes counts: D=35 A=25 C=22 E=18 so E is eliminated. Round 3 votes counts: D=35 A=35 C=30 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:215 B:201 D:201 E:201 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -16 B 4 0 -6 0 4 C 10 6 0 8 6 D 6 0 -8 0 4 E 16 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -16 B 4 0 -6 0 4 C 10 6 0 8 6 D 6 0 -8 0 4 E 16 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -16 B 4 0 -6 0 4 C 10 6 0 8 6 D 6 0 -8 0 4 E 16 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7516: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (14) D B E C A (13) A C E B D (10) C A E B D (7) A C E D B (6) E C B A D (5) B E D C A (5) B D E C A (5) C E A B D (4) E B C D A (3) E B C A D (3) D B E A C (3) D A C B E (3) D A B C E (3) A D C E B (3) E A B C D (2) D B C E A (2) A D C B E (2) E C A B D (1) E B D C A (1) D C B A E (1) C E B D A (1) C D B E A (1) B D E A C (1) A E C B D (1) Total count = 100 A B C D E A 0 12 -4 16 0 B -12 0 -18 -2 -22 C 4 18 0 16 14 D -16 2 -16 0 2 E 0 22 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 16 0 B -12 0 -18 -2 -22 C 4 18 0 16 14 D -16 2 -16 0 2 E 0 22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=25 E=15 C=13 B=11 so B is eliminated. Round 2 votes counts: A=36 D=31 E=20 C=13 so C is eliminated. Round 3 votes counts: A=43 D=32 E=25 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:226 A:212 E:203 D:186 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 16 0 B -12 0 -18 -2 -22 C 4 18 0 16 14 D -16 2 -16 0 2 E 0 22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 16 0 B -12 0 -18 -2 -22 C 4 18 0 16 14 D -16 2 -16 0 2 E 0 22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 16 0 B -12 0 -18 -2 -22 C 4 18 0 16 14 D -16 2 -16 0 2 E 0 22 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7517: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) C A B D E (8) D E C A B (7) E D A B C (6) D E A B C (5) C B A E D (5) E D B C A (4) E D B A C (4) E B D A C (4) D E C B A (4) D E B C A (4) D E A C B (4) C B A D E (4) C D E B A (3) B A C E D (3) D A E C B (2) B C A E D (2) A D E B C (2) A C B D E (2) E D C B A (1) E A D B C (1) D E B A C (1) D C E B A (1) C D E A B (1) C B E A D (1) C B D A E (1) C A D E B (1) C A B E D (1) B E A D C (1) B C E D A (1) B A E C D (1) A E B D C (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 2 -6 -6 B -10 0 6 -2 -12 C -2 -6 0 -6 -8 D 6 2 6 0 2 E 6 12 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 -6 -6 B -10 0 6 -2 -12 C -2 -6 0 -6 -8 D 6 2 6 0 2 E 6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 E=20 A=19 B=8 so B is eliminated. Round 2 votes counts: D=28 C=28 A=23 E=21 so E is eliminated. Round 3 votes counts: D=47 C=28 A=25 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:212 D:208 A:200 B:191 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 2 -6 -6 B -10 0 6 -2 -12 C -2 -6 0 -6 -8 D 6 2 6 0 2 E 6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -6 -6 B -10 0 6 -2 -12 C -2 -6 0 -6 -8 D 6 2 6 0 2 E 6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -6 -6 B -10 0 6 -2 -12 C -2 -6 0 -6 -8 D 6 2 6 0 2 E 6 12 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7518: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) D A E C B (9) A D E B C (8) E B C A D (6) D A C E B (6) E B C D A (4) D A E B C (4) D A C B E (4) B C E A D (4) A D C B E (4) E C B D A (3) E A B D C (3) D C A B E (3) C B D E A (3) E B A C D (2) E A D B C (2) D C E A B (2) D C A E B (2) C B E A D (2) B E C A D (2) E D C B A (1) E D B A C (1) E B D A C (1) D E A B C (1) C E B D A (1) C D B A E (1) C D A B E (1) B E C D A (1) B E A C D (1) B C E D A (1) A E D B C (1) A E B D C (1) A D E C B (1) A D B E C (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 -18 -2 B -12 0 -6 -4 -18 C -6 6 0 -12 -10 D 18 4 12 0 2 E 2 18 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -18 -2 B -12 0 -6 -4 -18 C -6 6 0 -12 -10 D 18 4 12 0 2 E 2 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=23 A=19 C=18 B=9 so B is eliminated. Round 2 votes counts: D=31 E=27 C=23 A=19 so A is eliminated. Round 3 votes counts: D=46 E=30 C=24 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:214 A:199 C:189 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -18 -2 B -12 0 -6 -4 -18 C -6 6 0 -12 -10 D 18 4 12 0 2 E 2 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -18 -2 B -12 0 -6 -4 -18 C -6 6 0 -12 -10 D 18 4 12 0 2 E 2 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -18 -2 B -12 0 -6 -4 -18 C -6 6 0 -12 -10 D 18 4 12 0 2 E 2 18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7519: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (16) D C B E A (8) D E B C A (5) D E B A C (5) A E B D C (5) A C B E D (5) A B E C D (5) C D B A E (4) C A B E D (4) B E A C D (4) D C E B A (3) C D B E A (3) C D A B E (3) E D B A C (2) E B D C A (2) E B A D C (2) E A B D C (2) D B E C A (2) C B E A D (2) C A D B E (2) B C E A D (2) E B A C D (1) D C A B E (1) D B C E A (1) D A C E B (1) C B E D A (1) C B D E A (1) C B A D E (1) C A B D E (1) B E C D A (1) B C E D A (1) A E D B C (1) A C E B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 4 12 4 B 2 0 16 16 6 C -4 -16 0 20 -6 D -12 -16 -20 0 -16 E -4 -6 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 12 4 B 2 0 16 16 6 C -4 -16 0 20 -6 D -12 -16 -20 0 -16 E -4 -6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990277 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=26 C=22 E=9 B=8 so B is eliminated. Round 2 votes counts: A=35 D=26 C=25 E=14 so E is eliminated. Round 3 votes counts: A=44 D=30 C=26 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:209 E:206 C:197 D:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 12 4 B 2 0 16 16 6 C -4 -16 0 20 -6 D -12 -16 -20 0 -16 E -4 -6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990277 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 12 4 B 2 0 16 16 6 C -4 -16 0 20 -6 D -12 -16 -20 0 -16 E -4 -6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990277 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 12 4 B 2 0 16 16 6 C -4 -16 0 20 -6 D -12 -16 -20 0 -16 E -4 -6 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990277 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7520: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (15) D A B C E (9) D E B C A (7) E D C B A (4) D E C A B (4) D B E A C (4) B E C A D (4) E B C D A (3) E B C A D (3) D A C B E (3) C A E B D (3) A C B E D (3) E C B D A (2) E C A B D (2) D E C B A (2) D B E C A (2) D B A C E (2) D A C E B (2) D A B E C (2) C A B E D (2) B A C E D (2) A C B D E (2) A B C D E (2) E C A D B (1) E B D C A (1) D E B A C (1) D C E A B (1) D A E C B (1) D A E B C (1) C E A B D (1) C B A E D (1) B E D C A (1) B D E C A (1) B A D C E (1) B A C D E (1) A D B C E (1) A C E B D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -20 -6 -18 B 14 0 -2 4 -12 C 20 2 0 0 -22 D 6 -4 0 0 0 E 18 12 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.347362 E: 0.652638 Sum of squares = 0.54659654568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.347362 E: 1.000000 A B C D E A 0 -14 -20 -6 -18 B 14 0 -2 4 -12 C 20 2 0 0 -22 D 6 -4 0 0 0 E 18 12 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=31 A=11 B=10 C=7 so C is eliminated. Round 2 votes counts: D=41 E=32 A=16 B=11 so B is eliminated. Round 3 votes counts: D=42 E=37 A=21 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:226 B:202 D:201 C:200 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -20 -6 -18 B 14 0 -2 4 -12 C 20 2 0 0 -22 D 6 -4 0 0 0 E 18 12 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -6 -18 B 14 0 -2 4 -12 C 20 2 0 0 -22 D 6 -4 0 0 0 E 18 12 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -6 -18 B 14 0 -2 4 -12 C 20 2 0 0 -22 D 6 -4 0 0 0 E 18 12 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7521: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (7) D E B C A (5) D E A C B (5) B C A E D (5) E D A C B (4) E C B D A (4) B C E D A (4) B C A D E (4) A E D C B (4) A B C D E (4) E D C B A (3) C B E D A (3) C B A E D (3) B C D E A (3) A C B E D (3) E C B A D (2) E C A B D (2) D E C B A (2) D B C E A (2) D B C A E (2) D A E B C (2) D A B C E (2) C E B D A (2) C E B A D (2) C B E A D (2) B C D A E (2) A D E C B (2) A D B C E (2) A B C E D (2) E D C A B (1) E A D C B (1) E A C B D (1) D E C A B (1) D E A B C (1) D A E C B (1) B D C E A (1) B D C A E (1) A D E B C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -12 -2 -2 B 4 0 -12 18 -8 C 12 12 0 12 2 D 2 -18 -12 0 -12 E 2 8 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 -2 B 4 0 -12 18 -8 C 12 12 0 12 2 D 2 -18 -12 0 -12 E 2 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 B=20 E=18 C=12 so C is eliminated. Round 2 votes counts: B=28 A=27 D=23 E=22 so E is eliminated. Round 3 votes counts: B=38 D=31 A=31 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:219 E:210 B:201 A:190 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 -2 B 4 0 -12 18 -8 C 12 12 0 12 2 D 2 -18 -12 0 -12 E 2 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 -2 B 4 0 -12 18 -8 C 12 12 0 12 2 D 2 -18 -12 0 -12 E 2 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 -2 B 4 0 -12 18 -8 C 12 12 0 12 2 D 2 -18 -12 0 -12 E 2 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979566 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7522: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) B A C D E (11) B D E C A (7) B A C E D (6) E D A C B (3) E A C D B (3) D E C B A (3) D E C A B (3) D C E B A (3) D B C E A (3) B C A D E (3) A C B D E (3) D E B C A (2) D C E A B (2) D B E C A (2) C A D E B (2) B E D C A (2) B D C A E (2) B C D A E (2) A E C D B (2) A C E D B (2) A C B E D (2) A B C E D (2) E D B C A (1) E C D A B (1) E C A D B (1) E A D C B (1) E A B D C (1) C E D A B (1) C D E A B (1) C D A E B (1) C D A B E (1) C B D A E (1) C A E D B (1) C A D B E (1) B E A D C (1) B D A C E (1) B A E C D (1) A C E B D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -16 -8 -6 B 2 0 -4 -6 6 C 16 4 0 2 6 D 8 6 -2 0 10 E 6 -6 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999581 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -8 -6 B 2 0 -4 -6 6 C 16 4 0 2 6 D 8 6 -2 0 10 E 6 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=23 D=18 A=14 C=9 so C is eliminated. Round 2 votes counts: B=37 E=24 D=21 A=18 so A is eliminated. Round 3 votes counts: B=45 E=30 D=25 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:214 D:211 B:199 E:192 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 -8 -6 B 2 0 -4 -6 6 C 16 4 0 2 6 D 8 6 -2 0 10 E 6 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -8 -6 B 2 0 -4 -6 6 C 16 4 0 2 6 D 8 6 -2 0 10 E 6 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -8 -6 B 2 0 -4 -6 6 C 16 4 0 2 6 D 8 6 -2 0 10 E 6 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7523: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (14) E C A D B (10) E A C D B (7) B A D C E (6) B D C A E (5) B C D E A (5) B A D E C (5) C E D B A (3) C E D A B (3) C E B D A (3) A D E C B (3) E C D A B (2) E C B D A (2) E A B C D (2) D C A B E (2) D B A C E (2) C B D E A (2) B C E D A (2) B C D A E (2) A E B D C (2) E C B A D (1) E B C A D (1) E B A C D (1) E A D C B (1) E A C B D (1) D B C A E (1) D A B C E (1) C D E A B (1) C D B E A (1) C B E D A (1) C B D A E (1) B E A D C (1) B D A E C (1) B A E D C (1) A E D C B (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 4 -8 0 B 22 0 10 20 10 C -4 -10 0 2 10 D 8 -20 -2 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 4 -8 0 B 22 0 10 20 10 C -4 -10 0 2 10 D 8 -20 -2 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 E=28 C=15 A=9 D=6 so D is eliminated. Round 2 votes counts: B=45 E=28 C=17 A=10 so A is eliminated. Round 3 votes counts: B=48 E=35 C=17 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:231 C:199 D:198 A:187 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 4 -8 0 B 22 0 10 20 10 C -4 -10 0 2 10 D 8 -20 -2 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 4 -8 0 B 22 0 10 20 10 C -4 -10 0 2 10 D 8 -20 -2 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 4 -8 0 B 22 0 10 20 10 C -4 -10 0 2 10 D 8 -20 -2 0 10 E 0 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999573 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7524: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (5) C B A E D (5) A D E C B (5) A D E B C (5) E A C B D (4) D A C B E (4) B C D E A (4) E B C D A (3) E B C A D (3) C B E A D (3) B C E D A (3) A E C B D (3) A D C E B (3) A C E B D (3) E C B A D (2) E A D B C (2) D A B E C (2) D A B C E (2) C E B A D (2) C B E D A (2) C B D E A (2) C A B D E (2) B E C D A (2) A E D C B (2) A E D B C (2) A E C D B (2) A C D E B (2) E D B A C (1) E D A B C (1) E C A B D (1) E B D C A (1) E B D A C (1) E B A C D (1) E A B D C (1) E A B C D (1) D E A B C (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A C E (1) C D B A E (1) C A E B D (1) C A B E D (1) B D E C A (1) B D C E A (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 14 14 16 10 B -14 0 -6 6 -16 C -14 6 0 8 -8 D -16 -6 -8 0 -6 E -10 16 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 16 10 B -14 0 -6 6 -16 C -14 6 0 8 -8 D -16 -6 -8 0 -6 E -10 16 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=22 D=19 C=19 B=11 so B is eliminated. Round 2 votes counts: A=29 C=26 E=24 D=21 so D is eliminated. Round 3 votes counts: A=43 C=29 E=28 so E is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:210 C:196 B:185 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 16 10 B -14 0 -6 6 -16 C -14 6 0 8 -8 D -16 -6 -8 0 -6 E -10 16 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 16 10 B -14 0 -6 6 -16 C -14 6 0 8 -8 D -16 -6 -8 0 -6 E -10 16 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 16 10 B -14 0 -6 6 -16 C -14 6 0 8 -8 D -16 -6 -8 0 -6 E -10 16 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7525: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A D C B E (8) A C D E B (7) B E D C A (6) C A D E B (5) A C D B E (5) E A C D B (4) E B C D A (3) E B A D C (3) E A C B D (3) D C B A E (3) C D A B E (3) B D C A E (3) B D A C E (3) A D B C E (3) E C B D A (2) E B D A C (2) E A B C D (2) B E D A C (2) B D E C A (2) B D C E A (2) B A D E C (2) A C E D B (2) E C D B A (1) E C D A B (1) E C A D B (1) E B C A D (1) E B A C D (1) D C A B E (1) C E A D B (1) C D E B A (1) C D B E A (1) C D A E B (1) C A D B E (1) B E A D C (1) B D A E C (1) B A E D C (1) A E B D C (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 6 8 4 B -2 0 -2 2 -2 C -6 2 0 -6 2 D -8 -2 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 8 4 B -2 0 -2 2 -2 C -6 2 0 -6 2 D -8 -2 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=28 B=23 C=13 D=4 so D is eliminated. Round 2 votes counts: E=32 A=28 B=23 C=17 so C is eliminated. Round 3 votes counts: A=39 E=34 B=27 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:201 B:198 C:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 8 4 B -2 0 -2 2 -2 C -6 2 0 -6 2 D -8 -2 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 8 4 B -2 0 -2 2 -2 C -6 2 0 -6 2 D -8 -2 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 8 4 B -2 0 -2 2 -2 C -6 2 0 -6 2 D -8 -2 6 0 6 E -4 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7526: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) E B A D C (7) D A B E C (7) C A B E D (7) A B E D C (7) A B D E C (7) D C E A B (4) D C A B E (4) D A B C E (4) C E B A D (4) C D E B A (4) B A E D C (4) D E B A C (3) C E D B A (3) C A B D E (3) A B C E D (3) E D B A C (2) E C B A D (2) E B A C D (2) C D E A B (2) B A E C D (2) E D C B A (1) D E C B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D A C B E (1) C B A E D (1) B A D E C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 24 8 6 24 B -24 0 6 6 24 C -8 -6 0 -14 2 D -6 -6 14 0 8 E -24 -24 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 8 6 24 B -24 0 6 6 24 C -8 -6 0 -14 2 D -6 -6 14 0 8 E -24 -24 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=27 A=20 E=14 B=7 so B is eliminated. Round 2 votes counts: C=32 D=27 A=27 E=14 so E is eliminated. Round 3 votes counts: A=36 C=34 D=30 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:231 B:206 D:205 C:187 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 8 6 24 B -24 0 6 6 24 C -8 -6 0 -14 2 D -6 -6 14 0 8 E -24 -24 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 8 6 24 B -24 0 6 6 24 C -8 -6 0 -14 2 D -6 -6 14 0 8 E -24 -24 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 8 6 24 B -24 0 6 6 24 C -8 -6 0 -14 2 D -6 -6 14 0 8 E -24 -24 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999257 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7527: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) D C B E A (6) C D B E A (6) E C D B A (5) E C A B D (4) E A C B D (4) E A B C D (4) C B D A E (4) E D C A B (3) E D A C B (3) E A D C B (3) D C E B A (3) D C B A E (3) B D C A E (3) A E D B C (3) A E B D C (3) A B E D C (3) E C A D B (2) E A B D C (2) D E C B A (2) D B C A E (2) D B A C E (2) C E D B A (2) B A D C E (2) E D C B A (1) E C B A D (1) E A C D B (1) D E C A B (1) D E A C B (1) D A B E C (1) C E B D A (1) C D E B A (1) C B A D E (1) B D A C E (1) B C A E D (1) B C A D E (1) B A C E D (1) B A C D E (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -6 -2 -12 B 0 0 -16 -2 -16 C 6 16 0 0 -14 D 2 2 0 0 -12 E 12 16 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -6 -2 -12 B 0 0 -16 -2 -16 C 6 16 0 0 -14 D 2 2 0 0 -12 E 12 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=21 A=21 C=15 B=10 so B is eliminated. Round 2 votes counts: E=33 D=25 A=25 C=17 so C is eliminated. Round 3 votes counts: E=36 D=36 A=28 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:227 C:204 D:196 A:190 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 -2 -12 B 0 0 -16 -2 -16 C 6 16 0 0 -14 D 2 2 0 0 -12 E 12 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -2 -12 B 0 0 -16 -2 -16 C 6 16 0 0 -14 D 2 2 0 0 -12 E 12 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -2 -12 B 0 0 -16 -2 -16 C 6 16 0 0 -14 D 2 2 0 0 -12 E 12 16 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7528: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) A E C D B (9) C D B E A (6) C D B A E (6) A E B C D (6) A C D E B (4) E B D C A (3) E A B D C (3) B E D A C (3) A E B D C (3) A C D B E (3) A C B D E (3) A B C D E (3) E D C B A (2) E D B C A (2) E A D C B (2) E A C D B (2) D C B E A (2) C D E B A (2) B E D C A (2) B C D A E (2) B A E D C (2) B A C D E (2) A C E D B (2) E D C A B (1) E C D A B (1) E B D A C (1) E B A D C (1) D C E B A (1) D B C E A (1) C D E A B (1) C D A E B (1) C D A B E (1) C B D A E (1) C A D E B (1) C A D B E (1) B E A D C (1) B D C A E (1) B A D E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 6 2 8 B 2 0 -4 -2 4 C -6 4 0 16 6 D -2 2 -16 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.38888888888 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 2 8 B 2 0 -4 -2 4 C -6 4 0 16 6 D -2 2 -16 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=23 C=20 E=18 D=4 so D is eliminated. Round 2 votes counts: A=35 B=24 C=23 E=18 so E is eliminated. Round 3 votes counts: A=42 B=31 C=27 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:210 A:207 B:200 D:194 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 2 8 B 2 0 -4 -2 4 C -6 4 0 16 6 D -2 2 -16 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 2 8 B 2 0 -4 -2 4 C -6 4 0 16 6 D -2 2 -16 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 2 8 B 2 0 -4 -2 4 C -6 4 0 16 6 D -2 2 -16 0 4 E -8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7529: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) E D C A B (8) D E B A C (8) D E A C B (8) C A E D B (7) E D A C B (5) B E C A D (5) B D A C E (5) C A E B D (4) B D E A C (4) E C A D B (3) C A B E D (3) B C A D E (3) A C D E B (3) A C D B E (3) E D B A C (2) E C A B D (2) D B A C E (2) D A C E B (2) C A B D E (2) B A C D E (2) A C B D E (2) E B D A C (1) D E A B C (1) D B E A C (1) D A C B E (1) B E D C A (1) B E D A C (1) A C E D B (1) Total count = 100 A B C D E A 0 10 4 0 0 B -10 0 -8 -10 -10 C -4 8 0 0 0 D 0 10 0 0 -6 E 0 10 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.564804 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.435196 Sum of squares = 0.508399178673 Cumulative probabilities = A: 0.564804 B: 0.564804 C: 0.564804 D: 0.564804 E: 1.000000 A B C D E A 0 10 4 0 0 B -10 0 -8 -10 -10 C -4 8 0 0 0 D 0 10 0 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=23 E=21 C=16 A=9 so A is eliminated. Round 2 votes counts: B=31 C=25 D=23 E=21 so E is eliminated. Round 3 votes counts: D=38 B=32 C=30 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:208 A:207 C:202 D:202 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 0 0 B -10 0 -8 -10 -10 C -4 8 0 0 0 D 0 10 0 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 0 0 B -10 0 -8 -10 -10 C -4 8 0 0 0 D 0 10 0 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 0 0 B -10 0 -8 -10 -10 C -4 8 0 0 0 D 0 10 0 0 -6 E 0 10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7530: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (6) E C B D A (5) E A C D B (5) A D B C E (5) C D B E A (4) B D A C E (4) A E D B C (4) A D C E B (4) A D C B E (4) E C A D B (3) C E D A B (3) B E A C D (3) B D C A E (3) B A E D C (3) A D E C B (3) E B C A D (2) E A B C D (2) D C A E B (2) C D A E B (2) B E C D A (2) B E A D C (2) B C D E A (2) B A D E C (2) B A D C E (2) A E D C B (2) A E C D B (2) A E B D C (2) E C D A B (1) E B C D A (1) E B A C D (1) E A B D C (1) D C A B E (1) D B C A E (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E B A (1) B E D A C (1) B E C A D (1) B D C E A (1) B D A E C (1) B C E D A (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 12 12 0 B -12 0 0 -12 -8 C -12 0 0 -2 0 D -12 12 2 0 2 E 0 8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.437751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.562249 Sum of squares = 0.507749956407 Cumulative probabilities = A: 0.437751 B: 0.437751 C: 0.437751 D: 0.437751 E: 1.000000 A B C D E A 0 12 12 12 0 B -12 0 0 -12 -8 C -12 0 0 -2 0 D -12 12 2 0 2 E 0 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=28 A=28 E=21 C=18 D=5 so D is eliminated. Round 2 votes counts: B=29 A=29 E=21 C=21 so E is eliminated. Round 3 votes counts: A=37 B=33 C=30 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:203 D:202 C:193 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 12 0 B -12 0 0 -12 -8 C -12 0 0 -2 0 D -12 12 2 0 2 E 0 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 12 0 B -12 0 0 -12 -8 C -12 0 0 -2 0 D -12 12 2 0 2 E 0 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 12 0 B -12 0 0 -12 -8 C -12 0 0 -2 0 D -12 12 2 0 2 E 0 8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7531: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (9) C E B D A (7) E C A D B (6) B D A C E (6) A C E D B (6) B D C E A (5) E B D C A (4) D B A E C (4) B D C A E (4) A D B C E (4) D A B E C (3) C E A B D (3) C B E D A (3) C B D E A (3) A E D B C (3) E D B A C (2) E C D B A (2) D B E A C (2) C B D A E (2) B D E A C (2) B D A E C (2) A E C D B (2) E C D A B (1) E C B D A (1) E C A B D (1) E A D C B (1) E A D B C (1) E A C D B (1) D B A C E (1) C E B A D (1) C E A D B (1) C B A D E (1) C A E D B (1) B C D E A (1) A E D C B (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 6 -10 4 B 6 0 8 -4 10 C -6 -8 0 -10 4 D 10 4 10 0 4 E -4 -10 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -10 4 B 6 0 8 -4 10 C -6 -8 0 -10 4 D 10 4 10 0 4 E -4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 E=20 B=20 D=10 so D is eliminated. Round 2 votes counts: A=31 B=27 C=22 E=20 so E is eliminated. Round 3 votes counts: A=34 C=33 B=33 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:214 B:210 A:197 C:190 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 6 -10 4 B 6 0 8 -4 10 C -6 -8 0 -10 4 D 10 4 10 0 4 E -4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -10 4 B 6 0 8 -4 10 C -6 -8 0 -10 4 D 10 4 10 0 4 E -4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -10 4 B 6 0 8 -4 10 C -6 -8 0 -10 4 D 10 4 10 0 4 E -4 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7532: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B E A C D (8) C D B E A (6) A E B D C (6) A B E D C (6) D A C E B (5) C D E B A (5) D C B A E (3) D C A B E (3) B E C A D (3) B A E D C (3) B A E C D (3) A E D B C (3) E B A C D (2) E A B C D (2) D C E B A (2) C B D E A (2) B D C A E (2) A E D C B (2) A D E C B (2) A D E B C (2) A D C E B (2) E C D A B (1) E C A D B (1) E C A B D (1) E A D C B (1) D C E A B (1) D B C A E (1) D A E C B (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E A B (1) C B E D A (1) B C E D A (1) B C D E A (1) B C D A E (1) A D B E C (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 8 2 20 B -10 0 -2 -10 0 C -8 2 0 -18 2 D -2 10 18 0 8 E -20 0 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 2 20 B -10 0 -2 -10 0 C -8 2 0 -18 2 D -2 10 18 0 8 E -20 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 B=22 C=16 E=8 so E is eliminated. Round 2 votes counts: A=30 D=27 B=24 C=19 so C is eliminated. Round 3 votes counts: D=40 A=32 B=28 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:217 B:189 C:189 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 2 20 B -10 0 -2 -10 0 C -8 2 0 -18 2 D -2 10 18 0 8 E -20 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 2 20 B -10 0 -2 -10 0 C -8 2 0 -18 2 D -2 10 18 0 8 E -20 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 2 20 B -10 0 -2 -10 0 C -8 2 0 -18 2 D -2 10 18 0 8 E -20 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7533: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) A C D E B (14) E B D A C (8) C A B D E (7) B E D A C (6) C A D B E (5) E D B A C (4) C B A E D (4) D A E B C (3) C A E B D (3) C A B E D (3) E D A B C (2) D E B A C (2) D E A B C (2) C A E D B (2) C A D E B (2) B E C D A (2) B D E A C (2) B C E D A (2) B C D E A (2) A D E C B (2) D B E A C (1) D B C A E (1) D A B C E (1) C B E A D (1) C B D A E (1) B E C A D (1) B D E C A (1) B D C E A (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -4 -10 -2 B 6 0 10 16 10 C 4 -10 0 0 0 D 10 -16 0 0 -6 E 2 -10 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -10 -2 B 6 0 10 16 10 C 4 -10 0 0 0 D 10 -16 0 0 -6 E 2 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=28 A=17 E=14 D=10 so D is eliminated. Round 2 votes counts: B=33 C=28 A=21 E=18 so E is eliminated. Round 3 votes counts: B=47 C=28 A=25 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:199 C:197 D:194 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 -2 B 6 0 10 16 10 C 4 -10 0 0 0 D 10 -16 0 0 -6 E 2 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 -2 B 6 0 10 16 10 C 4 -10 0 0 0 D 10 -16 0 0 -6 E 2 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 -2 B 6 0 10 16 10 C 4 -10 0 0 0 D 10 -16 0 0 -6 E 2 -10 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7534: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (14) B C D A E (12) A E C D B (10) E D A B C (4) E A D C B (4) E A D B C (4) D B C E A (4) C D B A E (4) C B D A E (4) B D E A C (4) D E B A C (3) D B E A C (3) E A B D C (2) D E A B C (2) D C B E A (2) D B E C A (2) C D A E B (2) C B A D E (2) C A E B D (2) C A B E D (2) B D E C A (2) D E A C B (1) C A E D B (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -22 -16 -30 -8 B 22 0 22 2 22 C 16 -22 0 -8 12 D 30 -2 8 0 30 E 8 -22 -12 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -16 -30 -8 B 22 0 22 2 22 C 16 -22 0 -8 12 D 30 -2 8 0 30 E 8 -22 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=19 D=17 E=14 A=13 so A is eliminated. Round 2 votes counts: B=37 E=25 C=21 D=17 so D is eliminated. Round 3 votes counts: B=46 E=31 C=23 so C is eliminated. Round 4 votes counts: B=61 E=39 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:234 D:233 C:199 E:172 A:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -16 -30 -8 B 22 0 22 2 22 C 16 -22 0 -8 12 D 30 -2 8 0 30 E 8 -22 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -16 -30 -8 B 22 0 22 2 22 C 16 -22 0 -8 12 D 30 -2 8 0 30 E 8 -22 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -16 -30 -8 B 22 0 22 2 22 C 16 -22 0 -8 12 D 30 -2 8 0 30 E 8 -22 -12 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999965141 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7535: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) C D A B E (8) E B A D C (7) C D E A B (7) C D A E B (7) A B D E C (6) E B A C D (5) E C B D A (4) D A B C E (4) B A D E C (4) E C A B D (3) E B C A D (3) E C D B A (2) E C B A D (2) E A B C D (2) D A C B E (2) C E A D B (2) B E A D C (2) B D A E C (2) A D B C E (2) A C D B E (2) A B E D C (2) A B D C E (2) E C D A B (1) E B D C A (1) E B D A C (1) D B A C E (1) C E D B A (1) C E D A B (1) C A E B D (1) C A D B E (1) B E D A C (1) B A E D C (1) A D C B E (1) Total count = 100 A B C D E A 0 26 -6 -4 10 B -26 0 -8 -2 0 C 6 8 0 4 2 D 4 2 -4 0 16 E -10 0 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 -6 -4 10 B -26 0 -8 -2 0 C 6 8 0 4 2 D 4 2 -4 0 16 E -10 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=28 D=16 A=15 B=10 so B is eliminated. Round 2 votes counts: E=34 C=28 A=20 D=18 so D is eliminated. Round 3 votes counts: C=37 E=34 A=29 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:210 D:209 E:186 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 26 -6 -4 10 B -26 0 -8 -2 0 C 6 8 0 4 2 D 4 2 -4 0 16 E -10 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 -6 -4 10 B -26 0 -8 -2 0 C 6 8 0 4 2 D 4 2 -4 0 16 E -10 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 -6 -4 10 B -26 0 -8 -2 0 C 6 8 0 4 2 D 4 2 -4 0 16 E -10 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7536: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B D A E (10) B D C E A (7) A C E D B (7) B C D E A (6) E D B A C (4) D B E A C (4) A E D C B (4) E A B C D (3) C A B E D (3) A E C D B (3) A D E C B (3) E D A B C (2) E B D A C (2) E B A C D (2) D E A B C (2) D B C E A (2) D A E B C (2) C D B A E (2) C B A E D (2) C A E B D (2) C A D B E (2) B D E C A (2) B C E A D (2) E B A D C (1) E A C B D (1) E A B D C (1) D E B A C (1) D C B A E (1) C B A D E (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E A C (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 10 0 -10 B 4 0 12 -4 -6 C -10 -12 0 -2 2 D 0 4 2 0 -4 E 10 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999997 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 -4 10 0 -10 B 4 0 12 -4 -6 C -10 -12 0 -2 2 D 0 4 2 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=24 B=20 A=18 D=12 so D is eliminated. Round 2 votes counts: E=29 B=26 C=25 A=20 so A is eliminated. Round 3 votes counts: E=41 C=33 B=26 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:209 B:203 D:201 A:198 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 10 0 -10 B 4 0 12 -4 -6 C -10 -12 0 -2 2 D 0 4 2 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 0 -10 B 4 0 12 -4 -6 C -10 -12 0 -2 2 D 0 4 2 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 0 -10 B 4 0 12 -4 -6 C -10 -12 0 -2 2 D 0 4 2 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7537: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) D B C A E (6) A E B D C (6) E A B D C (5) D C B A E (5) B D C E A (5) D C B E A (4) C E A D B (4) B E A D C (4) A E C D B (4) C E D A B (3) C D B A E (3) B D E A C (3) B D A E C (3) B A E D C (3) A E C B D (3) A E B C D (3) E A C D B (2) E A C B D (2) E A B C D (2) D B C E A (2) C D A E B (2) B E D C A (2) A B D E C (2) E C A D B (1) E C A B D (1) E B A D C (1) E B A C D (1) D A B E C (1) D A B C E (1) C D E B A (1) C D E A B (1) C B D E A (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E C A (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 -8 -4 -6 -10 B 8 0 6 0 10 C 4 -6 0 -12 -4 D 6 0 12 0 -2 E 10 -10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.656495 C: 0.000000 D: 0.343505 E: 0.000000 Sum of squares = 0.548981503941 Cumulative probabilities = A: 0.000000 B: 0.656495 C: 0.656495 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -6 -10 B 8 0 6 0 10 C 4 -6 0 -12 -4 D 6 0 12 0 -2 E 10 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=23 D=19 A=19 E=15 so E is eliminated. Round 2 votes counts: A=30 C=26 B=25 D=19 so D is eliminated. Round 3 votes counts: C=35 B=33 A=32 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:208 E:203 C:191 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -6 -10 B 8 0 6 0 10 C 4 -6 0 -12 -4 D 6 0 12 0 -2 E 10 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -6 -10 B 8 0 6 0 10 C 4 -6 0 -12 -4 D 6 0 12 0 -2 E 10 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -6 -10 B 8 0 6 0 10 C 4 -6 0 -12 -4 D 6 0 12 0 -2 E 10 -10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7538: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (15) C E B A D (12) A D C E B (11) B E C D A (8) C A D E B (7) B E D A C (6) C E B D A (5) E B C A D (3) D A C B E (3) A D B E C (3) E B C D A (2) D B A E C (2) D A C E B (2) C B E D A (2) B E D C A (2) B E C A D (2) A D C B E (2) E C B D A (1) E C B A D (1) E B A C D (1) E A B D C (1) C E A B D (1) C A E D B (1) B E A D C (1) B D E A C (1) B C E D A (1) A E C D B (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 4 0 0 B 0 0 -2 0 -4 C -4 2 0 0 -2 D 0 0 0 0 -4 E 0 4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.517493 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.482507 Sum of squares = 0.500612020105 Cumulative probabilities = A: 0.517493 B: 0.517493 C: 0.517493 D: 0.517493 E: 1.000000 A B C D E A 0 0 4 0 0 B 0 0 -2 0 -4 C -4 2 0 0 -2 D 0 0 0 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=22 B=21 A=20 E=9 so E is eliminated. Round 2 votes counts: C=30 B=27 D=22 A=21 so A is eliminated. Round 3 votes counts: D=39 C=33 B=28 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:205 A:202 C:198 D:198 B:197 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 0 B 0 0 -2 0 -4 C -4 2 0 0 -2 D 0 0 0 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 0 B 0 0 -2 0 -4 C -4 2 0 0 -2 D 0 0 0 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 0 B 0 0 -2 0 -4 C -4 2 0 0 -2 D 0 0 0 0 -4 E 0 4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7539: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) E B D C A (7) D B E A C (7) C A D B E (6) E B D A C (5) D B A C E (4) C D B A E (4) E C B A D (3) E B C D A (3) D B E C A (3) D B C A E (3) C E A B D (3) A C E D B (3) A C D B E (3) E C B D A (2) E C A B D (2) E A D B C (2) D B C E A (2) D B A E C (2) D A B E C (2) C E B A D (2) C B E D A (2) C A E B D (2) C A B D E (2) A D E B C (2) A C E B D (2) E A C B D (1) E A B C D (1) D C B A E (1) D A B C E (1) B E D A C (1) B C E D A (1) B C D E A (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B C D (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -24 -16 -20 -14 B 24 0 18 2 10 C 16 -18 0 -6 -12 D 20 -2 6 0 8 E 14 -10 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997388 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -16 -20 -14 B 24 0 18 2 10 C 16 -18 0 -6 -12 D 20 -2 6 0 8 E 14 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=25 C=21 A=16 B=12 so B is eliminated. Round 2 votes counts: D=34 E=27 C=23 A=16 so A is eliminated. Round 3 votes counts: D=37 C=32 E=31 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:227 D:216 E:204 C:190 A:163 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -16 -20 -14 B 24 0 18 2 10 C 16 -18 0 -6 -12 D 20 -2 6 0 8 E 14 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -16 -20 -14 B 24 0 18 2 10 C 16 -18 0 -6 -12 D 20 -2 6 0 8 E 14 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -16 -20 -14 B 24 0 18 2 10 C 16 -18 0 -6 -12 D 20 -2 6 0 8 E 14 -10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7540: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E B D C (7) B D C E A (6) C E D B A (5) E A D B C (4) D B C E A (4) A C E B D (4) E D B C A (3) E D B A C (3) E C D B A (3) E A C D B (3) C E D A B (3) C B D A E (3) C A B D E (3) B D E C A (3) A B D E C (3) A B D C E (3) C E A D B (2) C D E B A (2) B D A E C (2) B D A C E (2) B C D A E (2) A E C B D (2) A C E D B (2) A C B D E (2) E D C B A (1) E C A D B (1) E A B D C (1) D B E C A (1) C B D E A (1) C A D B E (1) B D E A C (1) B D C A E (1) B A D C E (1) A E D B C (1) A E C D B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -12 -14 -16 B 10 0 0 -2 4 C 12 0 0 4 18 D 14 2 -4 0 6 E 16 -4 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.278040 C: 0.721960 D: 0.000000 E: 0.000000 Sum of squares = 0.598532828506 Cumulative probabilities = A: 0.000000 B: 0.278040 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -14 -16 B 10 0 0 -2 4 C 12 0 0 4 18 D 14 2 -4 0 6 E 16 -4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999908 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=27 E=19 B=18 D=5 so D is eliminated. Round 2 votes counts: C=31 A=27 B=23 E=19 so E is eliminated. Round 3 votes counts: C=36 A=35 B=29 so B is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:209 B:206 E:194 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -14 -16 B 10 0 0 -2 4 C 12 0 0 4 18 D 14 2 -4 0 6 E 16 -4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999908 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -14 -16 B 10 0 0 -2 4 C 12 0 0 4 18 D 14 2 -4 0 6 E 16 -4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999908 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -14 -16 B 10 0 0 -2 4 C 12 0 0 4 18 D 14 2 -4 0 6 E 16 -4 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999908 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7541: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) D C A E B (11) E B A C D (9) C A D B E (6) D E A C B (5) C A B E D (5) C A B D E (5) E B D A C (4) D E C A B (4) D C A B E (4) B A C E D (4) D E B C A (3) D E B A C (3) D A C E B (3) B C A E D (3) A C B E D (3) C B A E D (2) C A D E B (2) E D B A C (1) E D A B C (1) D E C B A (1) B E D A C (1) B E C A D (1) B E A D C (1) B D C E A (1) B A E C D (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 4 16 6 B -6 0 -10 6 -2 C -4 10 0 12 6 D -16 -6 -12 0 0 E -6 2 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 16 6 B -6 0 -10 6 -2 C -4 10 0 12 6 D -16 -6 -12 0 0 E -6 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=24 C=20 E=15 A=7 so A is eliminated. Round 2 votes counts: D=35 C=26 B=24 E=15 so E is eliminated. Round 3 votes counts: D=37 B=37 C=26 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:216 C:212 E:195 B:194 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 16 6 B -6 0 -10 6 -2 C -4 10 0 12 6 D -16 -6 -12 0 0 E -6 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 16 6 B -6 0 -10 6 -2 C -4 10 0 12 6 D -16 -6 -12 0 0 E -6 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 16 6 B -6 0 -10 6 -2 C -4 10 0 12 6 D -16 -6 -12 0 0 E -6 2 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7542: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) C A D E B (6) E B C D A (5) D A C B E (5) C E B D A (5) A D C B E (5) E B A C D (4) D B E C A (4) C D E B A (4) A D B E C (4) A C D E B (4) E B A D C (3) D C B E A (3) C D A E B (3) A C E B D (3) A B E D C (3) E B D C A (2) E B D A C (2) E B C A D (2) D A B E C (2) C E B A D (2) C A E D B (2) C A E B D (2) B D E C A (2) A E B D C (2) D C B A E (1) D B E A C (1) D B C E A (1) D B A E C (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E A C (1) A E C B D (1) A E B C D (1) A D C E B (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -6 -4 -4 B 8 0 4 2 -8 C 6 -4 0 -8 -2 D 4 -2 8 0 0 E 4 8 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.443778 E: 0.556222 Sum of squares = 0.506321863901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.443778 E: 1.000000 A B C D E A 0 -8 -6 -4 -4 B 8 0 4 2 -8 C 6 -4 0 -8 -2 D 4 -2 8 0 0 E 4 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 E=18 D=18 B=14 so B is eliminated. Round 2 votes counts: E=29 A=26 C=24 D=21 so D is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:207 D:205 B:203 C:196 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -4 B 8 0 4 2 -8 C 6 -4 0 -8 -2 D 4 -2 8 0 0 E 4 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -4 B 8 0 4 2 -8 C 6 -4 0 -8 -2 D 4 -2 8 0 0 E 4 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -4 B 8 0 4 2 -8 C 6 -4 0 -8 -2 D 4 -2 8 0 0 E 4 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7543: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) E B C D A (6) A B D E C (6) C E D A B (5) E B A D C (4) C D E B A (4) C D A E B (4) A B D C E (4) E C D B A (3) E C A B D (3) E B C A D (3) D A C B E (3) C E A D B (3) C D E A B (3) B E A D C (3) A D B C E (3) A C D B E (3) A B E D C (3) E B A C D (2) C A D E B (2) B D E A C (2) B D A E C (2) B A D E C (2) A E B C D (2) E C B D A (1) E C B A D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B E C A (1) D B C A E (1) D A B C E (1) C E A B D (1) C D A B E (1) C A D B E (1) B E D C A (1) B E D A C (1) B A E D C (1) A D C B E (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 8 4 4 B -8 0 16 4 2 C -8 -16 0 4 2 D -4 -4 -4 0 6 E -4 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 4 4 B -8 0 16 4 2 C -8 -16 0 4 2 D -4 -4 -4 0 6 E -4 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=24 C=24 D=15 B=12 so B is eliminated. Round 2 votes counts: E=29 A=28 C=24 D=19 so D is eliminated. Round 3 votes counts: A=41 E=32 C=27 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:207 D:197 E:193 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 4 4 B -8 0 16 4 2 C -8 -16 0 4 2 D -4 -4 -4 0 6 E -4 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 4 4 B -8 0 16 4 2 C -8 -16 0 4 2 D -4 -4 -4 0 6 E -4 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 4 4 B -8 0 16 4 2 C -8 -16 0 4 2 D -4 -4 -4 0 6 E -4 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7544: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) C E D A B (9) E C D A B (6) B D A E C (6) E C A D B (4) E C A B D (4) E A C B D (4) A B C D E (4) D E C B A (3) C D E B A (3) C A E B D (3) C A B D E (3) B A D E C (3) B A D C E (3) E C D B A (2) E A B C D (2) D E B A C (2) D C B E A (2) D B C A E (2) D B A E C (2) C E D B A (2) C E A B D (2) B D A C E (2) B A E D C (2) A B E D C (2) A B E C D (2) A B C E D (2) E D B C A (1) D B E A C (1) D B C E A (1) C D E A B (1) C D B A E (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -20 -14 -20 B -2 0 -26 -6 -22 C 20 26 0 14 6 D 14 6 -14 0 0 E 20 22 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -20 -14 -20 B -2 0 -26 -6 -22 C 20 26 0 14 6 D 14 6 -14 0 0 E 20 22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 E=23 B=16 A=13 so A is eliminated. Round 2 votes counts: C=26 B=26 E=24 D=24 so E is eliminated. Round 3 votes counts: C=47 B=28 D=25 so D is eliminated. Round 4 votes counts: C=63 B=37 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:233 E:218 D:203 A:174 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -20 -14 -20 B -2 0 -26 -6 -22 C 20 26 0 14 6 D 14 6 -14 0 0 E 20 22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -20 -14 -20 B -2 0 -26 -6 -22 C 20 26 0 14 6 D 14 6 -14 0 0 E 20 22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -20 -14 -20 B -2 0 -26 -6 -22 C 20 26 0 14 6 D 14 6 -14 0 0 E 20 22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999415 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7545: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (12) E B D C A (7) D B E C A (7) E B D A C (6) C A D B E (6) E A B D C (5) C D B E A (5) B E D C A (5) A E B D C (4) E B A D C (3) C D A B E (3) C B D E A (3) A E D B C (3) A E B C D (3) A C E D B (3) A C E B D (3) D C B E A (2) D B C E A (2) B D C E A (2) E C B A D (1) E B C A D (1) E A B C D (1) D E B A C (1) D C B A E (1) D A C B E (1) C B E D A (1) C B A D E (1) C A E B D (1) B D E C A (1) B C D E A (1) A E D C B (1) A E C B D (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 0 0 4 -8 B 0 0 4 0 8 C 0 -4 0 -6 0 D -4 0 6 0 2 E 8 -8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.386457 B: 0.613543 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.525784058692 Cumulative probabilities = A: 0.386457 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 -8 B 0 0 4 0 8 C 0 -4 0 -6 0 D -4 0 6 0 2 E 8 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499920 B: 0.500080 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012692 Cumulative probabilities = A: 0.499920 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=24 C=20 D=14 B=9 so B is eliminated. Round 2 votes counts: A=33 E=29 C=21 D=17 so D is eliminated. Round 3 votes counts: E=38 A=34 C=28 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:206 D:202 E:199 A:198 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 4 -8 B 0 0 4 0 8 C 0 -4 0 -6 0 D -4 0 6 0 2 E 8 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499920 B: 0.500080 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012692 Cumulative probabilities = A: 0.499920 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 -8 B 0 0 4 0 8 C 0 -4 0 -6 0 D -4 0 6 0 2 E 8 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499920 B: 0.500080 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012692 Cumulative probabilities = A: 0.499920 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 -8 B 0 0 4 0 8 C 0 -4 0 -6 0 D -4 0 6 0 2 E 8 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499920 B: 0.500080 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012692 Cumulative probabilities = A: 0.499920 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7546: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) E A B C D (7) C D B E A (7) C D B A E (7) E A C B D (6) C A E D B (6) A E B D C (6) D B C E A (4) D B A E C (4) C D A E B (4) B D E A C (4) E A B D C (3) D C B A E (3) D B C A E (3) C E A B D (3) B E A D C (3) B D A E C (3) A E B C D (3) C E A D B (2) A E D B C (2) D C B E A (1) D C A E B (1) D B E A C (1) C D E A B (1) C D A B E (1) C B D E A (1) B D C E A (1) B C D E A (1) B A E D C (1) A E C D B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 12 8 6 10 B -12 0 -8 0 -10 C -8 8 0 18 -4 D -6 0 -18 0 -4 E -10 10 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 6 10 B -12 0 -8 0 -10 C -8 8 0 18 -4 D -6 0 -18 0 -4 E -10 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=22 D=17 E=16 B=13 so B is eliminated. Round 2 votes counts: C=33 D=25 A=23 E=19 so E is eliminated. Round 3 votes counts: A=42 C=33 D=25 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:207 E:204 D:186 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 6 10 B -12 0 -8 0 -10 C -8 8 0 18 -4 D -6 0 -18 0 -4 E -10 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 6 10 B -12 0 -8 0 -10 C -8 8 0 18 -4 D -6 0 -18 0 -4 E -10 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 6 10 B -12 0 -8 0 -10 C -8 8 0 18 -4 D -6 0 -18 0 -4 E -10 10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7547: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (6) A B C D E (6) B C A D E (5) E D A C B (4) C B E A D (4) E D C B A (3) E D C A B (3) E C A D B (3) D E A B C (3) D B E C A (3) D A B E C (3) C E B A D (3) C B E D A (3) C A E B D (3) A D B C E (3) A C E D B (3) A C E B D (3) E A D C B (2) D B E A C (2) C E B D A (2) C B A E D (2) C A B E D (2) B A C D E (2) A E D C B (2) A D E C B (2) A C B E D (2) E C D B A (1) E C D A B (1) E C B D A (1) E A C D B (1) E A C B D (1) D E B C A (1) D E B A C (1) D B C E A (1) D A E B C (1) D A B C E (1) C E A B D (1) C B A D E (1) B D C E A (1) B D A C E (1) B C E D A (1) B C E A D (1) B C A E D (1) B A D C E (1) A D E B C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -6 14 -6 B -6 0 -6 8 8 C 6 6 0 18 20 D -14 -8 -18 0 -6 E 6 -8 -20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 14 -6 B -6 0 -6 8 8 C 6 6 0 18 20 D -14 -8 -18 0 -6 E 6 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=21 E=20 B=19 D=16 so D is eliminated. Round 2 votes counts: A=29 E=25 B=25 C=21 so C is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:225 A:204 B:202 E:192 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 14 -6 B -6 0 -6 8 8 C 6 6 0 18 20 D -14 -8 -18 0 -6 E 6 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 14 -6 B -6 0 -6 8 8 C 6 6 0 18 20 D -14 -8 -18 0 -6 E 6 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 14 -6 B -6 0 -6 8 8 C 6 6 0 18 20 D -14 -8 -18 0 -6 E 6 -8 -20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7548: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) A B C E D (10) E D B C A (7) D E C B A (7) B A E D C (7) A C D E B (6) C A D E B (5) A C B D E (5) E D C B A (4) C D E B A (4) B E D A C (4) C E D B A (3) C D E A B (3) C A E D B (3) E B D C A (2) D E B C A (2) B E A D C (2) A D C E B (2) A D B E C (2) E B C D A (1) D E B A C (1) D C E A B (1) D B A E C (1) C D A E B (1) B E C A D (1) A D C B E (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -10 -4 -6 B 14 0 8 -8 -4 C 10 -8 0 -12 -6 D 4 8 12 0 -10 E 6 4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -10 -4 -6 B 14 0 8 -8 -4 C 10 -8 0 -12 -6 D 4 8 12 0 -10 E 6 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=25 C=19 E=14 D=12 so D is eliminated. Round 2 votes counts: A=30 B=26 E=24 C=20 so C is eliminated. Round 3 votes counts: A=39 E=35 B=26 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:207 B:205 C:192 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -10 -4 -6 B 14 0 8 -8 -4 C 10 -8 0 -12 -6 D 4 8 12 0 -10 E 6 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -4 -6 B 14 0 8 -8 -4 C 10 -8 0 -12 -6 D 4 8 12 0 -10 E 6 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -4 -6 B 14 0 8 -8 -4 C 10 -8 0 -12 -6 D 4 8 12 0 -10 E 6 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7549: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) B D A E C (12) E A C B D (5) D C B A E (5) D B A E C (4) D B A C E (4) C D A E B (4) B A E D C (4) E C A B D (3) D B C A E (3) D C A B E (2) C E D A B (2) C E A B D (2) C B D E A (2) C A E D B (2) B E A D C (2) B A D E C (2) E C B A D (1) E C A D B (1) E B A C D (1) E A C D B (1) E A B D C (1) E A B C D (1) D A C E B (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B D A (1) C E B A D (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C A D (1) B E A C D (1) B D E A C (1) B D C A E (1) B C E D A (1) B C D E A (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -4 -8 8 B 6 0 -8 -4 10 C 4 8 0 0 4 D 8 4 0 0 0 E -8 -10 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.508845 D: 0.491155 E: 0.000000 Sum of squares = 0.500156484935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.508845 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -8 8 B 6 0 -8 -4 10 C 4 8 0 0 4 D 8 4 0 0 0 E -8 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=28 D=22 E=14 A=4 so A is eliminated. Round 2 votes counts: C=32 B=28 D=22 E=18 so E is eliminated. Round 3 votes counts: C=44 B=32 D=24 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:208 D:206 B:202 A:195 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 8 B 6 0 -8 -4 10 C 4 8 0 0 4 D 8 4 0 0 0 E -8 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 8 B 6 0 -8 -4 10 C 4 8 0 0 4 D 8 4 0 0 0 E -8 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 8 B 6 0 -8 -4 10 C 4 8 0 0 4 D 8 4 0 0 0 E -8 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7550: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) B E D C A (7) B E C D A (7) A C E B D (6) C E B A D (5) A D C E B (5) E B C A D (4) D A C E B (4) C A E B D (4) D C A E B (3) D B E C A (3) C E A B D (3) B E A C D (3) A E B C D (3) E C B A D (2) E B C D A (2) E B A C D (2) D A B E C (2) C A D E B (2) B E C A D (2) B D E C A (2) B D E A C (2) A C E D B (2) E C A B D (1) E A C B D (1) E A B C D (1) D C E B A (1) D C B A E (1) D B E A C (1) D B C E A (1) D B A C E (1) D A B C E (1) C D E A B (1) C A E D B (1) B E D A C (1) B E A D C (1) B A E D C (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -4 2 -6 B -2 0 -6 22 -10 C 4 6 0 8 2 D -2 -22 -8 0 -20 E 6 10 -2 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 2 -6 B -2 0 -6 22 -10 C 4 6 0 8 2 D -2 -22 -8 0 -20 E 6 10 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 A=18 C=16 E=13 so E is eliminated. Round 2 votes counts: B=34 D=27 A=20 C=19 so C is eliminated. Round 3 votes counts: B=41 A=31 D=28 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:217 C:210 B:202 A:197 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 2 -6 B -2 0 -6 22 -10 C 4 6 0 8 2 D -2 -22 -8 0 -20 E 6 10 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 2 -6 B -2 0 -6 22 -10 C 4 6 0 8 2 D -2 -22 -8 0 -20 E 6 10 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 2 -6 B -2 0 -6 22 -10 C 4 6 0 8 2 D -2 -22 -8 0 -20 E 6 10 -2 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7551: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (9) A C B D E (8) D E B A C (7) C A D E B (5) A C D B E (5) E D B C A (4) E B D C A (4) C A B E D (4) C E B A D (3) B E C A D (3) B A C E D (3) E D B A C (2) D E A C B (2) D A C E B (2) C D E A B (2) C A E D B (2) C A B D E (2) B E D C A (2) B E C D A (2) B A E C D (2) A D C B E (2) A C B E D (2) E D C B A (1) E D C A B (1) E B D A C (1) D E C A B (1) D E B C A (1) D E A B C (1) D C E A B (1) D C A E B (1) D A E B C (1) D A B E C (1) C E D B A (1) C D A E B (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D B E (1) B E A C D (1) B A E D C (1) B A D E C (1) A D C E B (1) A D B C E (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 0 10 4 0 B 0 0 -4 4 6 C -10 4 0 4 4 D -4 -4 -4 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.481323 B: 0.518677 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500697666893 Cumulative probabilities = A: 0.481323 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 4 0 B 0 0 -4 4 6 C -10 4 0 4 4 D -4 -4 -4 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 A=21 D=18 E=13 so E is eliminated. Round 2 votes counts: B=29 D=26 C=24 A=21 so A is eliminated. Round 3 votes counts: C=40 D=30 B=30 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:207 B:203 C:201 E:197 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 4 0 B 0 0 -4 4 6 C -10 4 0 4 4 D -4 -4 -4 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 4 0 B 0 0 -4 4 6 C -10 4 0 4 4 D -4 -4 -4 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 4 0 B 0 0 -4 4 6 C -10 4 0 4 4 D -4 -4 -4 0 -4 E 0 -6 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7552: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) D B C E A (6) A E C B D (6) D C A B E (5) C A B E D (4) C A B D E (4) B E C D A (4) A E D B C (4) E A B C D (3) D C B E A (3) D B E C A (3) C D B E A (3) B C E D A (3) A D C E B (3) A C E B D (3) E B D C A (2) E A D B C (2) D E B A C (2) D C B A E (2) D A C B E (2) C D A B E (2) C B E A D (2) B D E C A (2) B C D E A (2) A E D C B (2) A E C D B (2) A E B D C (2) E D B A C (1) D E A B C (1) D A E C B (1) D A E B C (1) C D B A E (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D B E (1) B D C E A (1) B C E A D (1) A E B C D (1) A D E C B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -22 -12 2 B -4 0 -4 0 24 C 22 4 0 -6 6 D 12 0 6 0 2 E -2 -24 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.265147 C: 0.000000 D: 0.734853 E: 0.000000 Sum of squares = 0.610312277986 Cumulative probabilities = A: 0.000000 B: 0.265147 C: 0.265147 D: 1.000000 E: 1.000000 A B C D E A 0 4 -22 -12 2 B -4 0 -4 0 24 C 22 4 0 -6 6 D 12 0 6 0 2 E -2 -24 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 C=20 B=20 E=8 so E is eliminated. Round 2 votes counts: A=31 D=27 B=22 C=20 so C is eliminated. Round 3 votes counts: A=40 D=33 B=27 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:210 B:208 A:186 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -22 -12 2 B -4 0 -4 0 24 C 22 4 0 -6 6 D 12 0 6 0 2 E -2 -24 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -22 -12 2 B -4 0 -4 0 24 C 22 4 0 -6 6 D 12 0 6 0 2 E -2 -24 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -22 -12 2 B -4 0 -4 0 24 C 22 4 0 -6 6 D 12 0 6 0 2 E -2 -24 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7553: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) C D B E A (9) E A C D B (8) B D C A E (8) A E B D C (7) A B E D C (6) E C A D B (5) A B D C E (5) C D E B A (4) A B D E C (4) D C B E A (3) C E D B A (3) B D C E A (3) B A D C E (3) A E B C D (3) E A D C B (2) D B C A E (2) A E D C B (2) A E C D B (2) E D C A B (1) E C A B D (1) D B C E A (1) C E B D A (1) C B D E A (1) B D A C E (1) B C D E A (1) B A C E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -8 2 -8 B 6 0 -6 -6 0 C 8 6 0 4 -6 D -2 6 -4 0 -8 E 8 0 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.323163 C: 0.000000 D: 0.000000 E: 0.676837 Sum of squares = 0.56254297186 Cumulative probabilities = A: 0.000000 B: 0.323163 C: 0.323163 D: 0.323163 E: 1.000000 A B C D E A 0 -6 -8 2 -8 B 6 0 -6 -6 0 C 8 6 0 4 -6 D -2 6 -4 0 -8 E 8 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499902 C: 0.000000 D: 0.000000 E: 0.500098 Sum of squares = 0.500000019015 Cumulative probabilities = A: 0.000000 B: 0.499902 C: 0.499902 D: 0.499902 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 C=18 B=18 D=6 so D is eliminated. Round 2 votes counts: A=30 E=28 C=21 B=21 so C is eliminated. Round 3 votes counts: E=36 B=34 A=30 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:211 C:206 B:197 D:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 2 -8 B 6 0 -6 -6 0 C 8 6 0 4 -6 D -2 6 -4 0 -8 E 8 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499902 C: 0.000000 D: 0.000000 E: 0.500098 Sum of squares = 0.500000019015 Cumulative probabilities = A: 0.000000 B: 0.499902 C: 0.499902 D: 0.499902 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 2 -8 B 6 0 -6 -6 0 C 8 6 0 4 -6 D -2 6 -4 0 -8 E 8 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499902 C: 0.000000 D: 0.000000 E: 0.500098 Sum of squares = 0.500000019015 Cumulative probabilities = A: 0.000000 B: 0.499902 C: 0.499902 D: 0.499902 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 2 -8 B 6 0 -6 -6 0 C 8 6 0 4 -6 D -2 6 -4 0 -8 E 8 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499902 C: 0.000000 D: 0.000000 E: 0.500098 Sum of squares = 0.500000019015 Cumulative probabilities = A: 0.000000 B: 0.499902 C: 0.499902 D: 0.499902 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7554: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (11) E A B D C (10) B A E C D (10) D C E A B (8) C D A B E (8) D C E B A (7) D C A E B (4) B A C D E (4) E D C A B (3) E B A D C (3) D C A B E (3) A B E C D (3) E A D C B (2) D E C A B (2) D C B E A (2) B E C D A (2) B A C E D (2) A C D B E (2) E D C B A (1) E D B C A (1) E D A C B (1) E A B C D (1) D C B A E (1) C B D A E (1) C A D B E (1) B E A C D (1) B C D A E (1) A E B D C (1) A E B C D (1) A D C E B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -12 -12 12 B -6 0 -16 -16 8 C 12 16 0 0 16 D 12 16 0 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.458244 D: 0.541756 E: 0.000000 Sum of squares = 0.503487131306 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.458244 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -12 12 B -6 0 -16 -16 8 C 12 16 0 0 16 D 12 16 0 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=22 C=21 B=20 A=10 so A is eliminated. Round 2 votes counts: D=28 B=25 E=24 C=23 so C is eliminated. Round 3 votes counts: D=50 B=26 E=24 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:222 D:221 A:197 B:185 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 -12 12 B -6 0 -16 -16 8 C 12 16 0 0 16 D 12 16 0 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -12 12 B -6 0 -16 -16 8 C 12 16 0 0 16 D 12 16 0 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -12 12 B -6 0 -16 -16 8 C 12 16 0 0 16 D 12 16 0 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7555: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) B E A D C (7) B E A C D (6) B E D C A (5) E D B A C (4) D E A B C (4) C D A B E (4) D E C B A (3) D C E A B (3) D C A E B (3) C D A E B (3) C A D B E (3) B C A E D (3) B A E C D (3) E B D C A (2) E B D A C (2) E B A D C (2) D A C E B (2) C D E B A (2) C D B E A (2) C B E D A (2) C B D E A (2) C A B D E (2) B E C D A (2) B C E A D (2) A D E C B (2) A C B D E (2) E D B C A (1) E A D B C (1) E A B D C (1) D E B C A (1) D E A C B (1) D A E C B (1) C B A E D (1) C A D E B (1) B C E D A (1) A D E B C (1) A C D E B (1) A C D B E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 4 0 -12 B 6 0 -2 -6 0 C -4 2 0 -12 0 D 0 6 12 0 6 E 12 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.183305 B: 0.000000 C: 0.000000 D: 0.816695 E: 0.000000 Sum of squares = 0.700592078376 Cumulative probabilities = A: 0.183305 B: 0.183305 C: 0.183305 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 0 -12 B 6 0 -2 -6 0 C -4 2 0 -12 0 D 0 6 12 0 6 E 12 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555589676 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 D=18 A=18 E=13 so E is eliminated. Round 2 votes counts: B=35 D=23 C=22 A=20 so A is eliminated. Round 3 votes counts: B=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:203 B:199 A:193 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 4 0 -12 B 6 0 -2 -6 0 C -4 2 0 -12 0 D 0 6 12 0 6 E 12 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555589676 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 0 -12 B 6 0 -2 -6 0 C -4 2 0 -12 0 D 0 6 12 0 6 E 12 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555589676 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 0 -12 B 6 0 -2 -6 0 C -4 2 0 -12 0 D 0 6 12 0 6 E 12 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555589676 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7556: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) B D E C A (7) A C E B D (7) D B E C A (6) A C E D B (6) E C A B D (5) A C D E B (5) C A E D B (4) B D E A C (4) D C E B A (3) D B C E A (3) D A B C E (3) C E D B A (3) C A E B D (3) B E C D A (3) B E A C D (3) E C B D A (2) E C B A D (2) E B C D A (2) E B C A D (2) D C A E B (2) D B E A C (2) D A C E B (2) B E C A D (2) A D C B E (2) E A C B D (1) D B A E C (1) D B A C E (1) D A C B E (1) C D A E B (1) B E D A C (1) B D A E C (1) A D C E B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -14 4 -16 B -4 0 -14 8 -16 C 14 14 0 18 8 D -4 -8 -18 0 -10 E 16 16 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 4 -16 B -4 0 -14 8 -16 C 14 14 0 18 8 D -4 -8 -18 0 -10 E 16 16 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=23 B=21 C=18 E=14 so E is eliminated. Round 2 votes counts: C=27 B=25 D=24 A=24 so D is eliminated. Round 3 votes counts: B=38 C=32 A=30 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:217 A:189 B:187 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 4 -16 B -4 0 -14 8 -16 C 14 14 0 18 8 D -4 -8 -18 0 -10 E 16 16 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 4 -16 B -4 0 -14 8 -16 C 14 14 0 18 8 D -4 -8 -18 0 -10 E 16 16 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 4 -16 B -4 0 -14 8 -16 C 14 14 0 18 8 D -4 -8 -18 0 -10 E 16 16 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7557: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) E C B A D (6) D E A C B (5) D A E B C (5) D A B C E (5) C E B A D (5) E C B D A (4) B C A D E (4) B A C D E (4) E D C A B (3) E D A C B (3) E C A D B (3) D B C A E (3) B C A E D (3) A D E C B (3) A B C D E (3) E D C B A (2) E C A B D (2) D E B C A (2) D E A B C (2) D B A C E (2) C B E A D (2) B D C E A (2) B A D C E (2) A C B E D (2) A B D C E (2) E C D A B (1) E A D C B (1) E A C D B (1) C E A B D (1) C B E D A (1) C B A E D (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) A E C B D (1) A D B E C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 0 2 B -6 0 6 0 0 C -2 -6 0 -2 -4 D 0 0 2 0 10 E -2 0 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600582 B: 0.000000 C: 0.000000 D: 0.399418 E: 0.000000 Sum of squares = 0.520233403762 Cumulative probabilities = A: 0.600582 B: 0.600582 C: 0.600582 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 0 2 B -6 0 6 0 0 C -2 -6 0 -2 -4 D 0 0 2 0 10 E -2 0 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=26 B=19 A=14 C=10 so C is eliminated. Round 2 votes counts: E=32 D=31 B=23 A=14 so A is eliminated. Round 3 votes counts: D=36 E=33 B=31 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 A:205 B:200 E:196 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 0 2 B -6 0 6 0 0 C -2 -6 0 -2 -4 D 0 0 2 0 10 E -2 0 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 2 B -6 0 6 0 0 C -2 -6 0 -2 -4 D 0 0 2 0 10 E -2 0 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 2 B -6 0 6 0 0 C -2 -6 0 -2 -4 D 0 0 2 0 10 E -2 0 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7558: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) D B E C A (10) B E D A C (10) A C E B D (10) B E A D C (8) A E B C D (7) E B A C D (6) C D A E B (6) D C A B E (5) C A D E B (5) D C B A E (3) C D A B E (3) C A E D B (3) E B A D C (2) D B E A C (2) D B C E A (2) C A E B D (2) B D E A C (2) A E C B D (2) C B A E D (1) B E A C D (1) Total count = 100 A B C D E A 0 -14 0 -6 -6 B 14 0 0 2 14 C 0 0 0 -8 0 D 6 -2 8 0 -4 E 6 -14 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.882313 C: 0.117687 D: 0.000000 E: 0.000000 Sum of squares = 0.792326687708 Cumulative probabilities = A: 0.000000 B: 0.882313 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -6 -6 B 14 0 0 2 14 C 0 0 0 -8 0 D 6 -2 8 0 -4 E 6 -14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000006303 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=21 C=20 A=19 E=8 so E is eliminated. Round 2 votes counts: D=32 B=29 C=20 A=19 so A is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:204 E:198 C:196 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -6 -6 B 14 0 0 2 14 C 0 0 0 -8 0 D 6 -2 8 0 -4 E 6 -14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000006303 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -6 -6 B 14 0 0 2 14 C 0 0 0 -8 0 D 6 -2 8 0 -4 E 6 -14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000006303 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -6 -6 B 14 0 0 2 14 C 0 0 0 -8 0 D 6 -2 8 0 -4 E 6 -14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000006303 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7559: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) C B E D A (7) D A E B C (6) D A C B E (6) E B A C D (5) C B E A D (5) B C E A D (5) E B C A D (4) C A B D E (4) A D E B C (4) A D C B E (4) E D A B C (3) D C A B E (3) C B A D E (3) B E C A D (3) D E A C B (2) D E A B C (2) C D B E A (2) C B D A E (2) C B A E D (2) A B E C D (2) E D B C A (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B D C (1) D C E B A (1) D A E C B (1) D A C E B (1) C E D B A (1) C D A B E (1) C A D B E (1) B C A E D (1) A E D B C (1) A E B C D (1) A D B E C (1) A D B C E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 6 -4 B 2 0 6 14 12 C 6 -6 0 18 6 D -6 -14 -18 0 -6 E 4 -12 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 6 -4 B 2 0 6 14 12 C 6 -6 0 18 6 D -6 -14 -18 0 -6 E 4 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993006 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=24 D=22 A=17 B=9 so B is eliminated. Round 2 votes counts: C=34 E=27 D=22 A=17 so A is eliminated. Round 3 votes counts: C=37 D=32 E=31 so E is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:217 C:212 A:197 E:196 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 6 -4 B 2 0 6 14 12 C 6 -6 0 18 6 D -6 -14 -18 0 -6 E 4 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993006 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 6 -4 B 2 0 6 14 12 C 6 -6 0 18 6 D -6 -14 -18 0 -6 E 4 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993006 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 6 -4 B 2 0 6 14 12 C 6 -6 0 18 6 D -6 -14 -18 0 -6 E 4 -12 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993006 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7560: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B A C E D (8) D E B A C (7) B E C D A (6) E D B C A (5) B E D A C (5) C A B E D (4) A C B D E (4) E C D B A (3) D E A B C (3) D B E A C (3) D A E C B (3) C E B D A (3) B E D C A (3) B C A E D (3) A D C E B (3) A B C E D (3) A B C D E (3) E B C D A (2) C E D A B (2) E D C B A (1) D A C E B (1) C D A E B (1) C B E D A (1) C B A E D (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B E C A D (1) B E A D C (1) B D E A C (1) B C E A D (1) B A D E C (1) B A D C E (1) A D C B E (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 2 -16 -12 B 14 0 16 12 8 C -2 -16 0 4 -6 D 16 -12 -4 0 -12 E 12 -8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -16 -12 B 14 0 16 12 8 C -2 -16 0 4 -6 D 16 -12 -4 0 -12 E 12 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=25 A=17 C=16 E=11 so E is eliminated. Round 2 votes counts: B=33 D=31 C=19 A=17 so A is eliminated. Round 3 votes counts: B=40 D=35 C=25 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:211 D:194 C:190 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 -16 -12 B 14 0 16 12 8 C -2 -16 0 4 -6 D 16 -12 -4 0 -12 E 12 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -16 -12 B 14 0 16 12 8 C -2 -16 0 4 -6 D 16 -12 -4 0 -12 E 12 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -16 -12 B 14 0 16 12 8 C -2 -16 0 4 -6 D 16 -12 -4 0 -12 E 12 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7561: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) A D C B E (7) E C B D A (6) D B A E C (6) B D E A C (6) C E A B D (5) C A E B D (5) B E D C A (5) E B C D A (4) E B C A D (4) D A B E C (4) A D B E C (4) E B D C A (3) C E B A D (3) B E D A C (3) D A B C E (2) C E B D A (2) C A E D B (2) A C D E B (2) A C D B E (2) E D B C A (1) E C D B A (1) D E C B A (1) D E B C A (1) D C E B A (1) D B E C A (1) C E A D B (1) C A D E B (1) B E C D A (1) B D E C A (1) A D B C E (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 2 -16 -18 B 18 0 14 4 10 C -2 -14 0 -12 -22 D 16 -4 12 0 0 E 18 -10 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 2 -16 -18 B 18 0 14 4 10 C -2 -14 0 -12 -22 D 16 -4 12 0 0 E 18 -10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=21 E=19 C=19 B=16 so B is eliminated. Round 2 votes counts: D=32 E=28 A=21 C=19 so C is eliminated. Round 3 votes counts: E=39 D=32 A=29 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:223 E:215 D:212 A:175 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 2 -16 -18 B 18 0 14 4 10 C -2 -14 0 -12 -22 D 16 -4 12 0 0 E 18 -10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 2 -16 -18 B 18 0 14 4 10 C -2 -14 0 -12 -22 D 16 -4 12 0 0 E 18 -10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 2 -16 -18 B 18 0 14 4 10 C -2 -14 0 -12 -22 D 16 -4 12 0 0 E 18 -10 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997219 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7562: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (10) D A C E B (9) E B C A D (6) D C A E B (6) C D A E B (5) C B E D A (5) B E C A D (5) B E A D C (5) E B A C D (4) D A C B E (4) C E B D A (4) E C B A D (3) C D E A B (3) A D E C B (3) A D E B C (3) C E D B A (2) C D E B A (2) B E A C D (2) B C D A E (2) B A D E C (2) A E D B C (2) A B D E C (2) E C D A B (1) E C A B D (1) E A D C B (1) D C A B E (1) D A B C E (1) C E D A B (1) C D B A E (1) C B D E A (1) B C E D A (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 8 0 2 6 B -8 0 -6 -12 -12 C 0 6 0 -2 -2 D -2 12 2 0 12 E -6 12 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.756538 B: 0.000000 C: 0.243462 D: 0.000000 E: 0.000000 Sum of squares = 0.631623761241 Cumulative probabilities = A: 0.756538 B: 0.756538 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 2 6 B -8 0 -6 -12 -12 C 0 6 0 -2 -2 D -2 12 2 0 12 E -6 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500348 B: 0.000000 C: 0.499652 D: 0.000000 E: 0.000000 Sum of squares = 0.500000241587 Cumulative probabilities = A: 0.500348 B: 0.500348 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=21 A=21 B=18 E=16 so E is eliminated. Round 2 votes counts: C=29 B=28 A=22 D=21 so D is eliminated. Round 3 votes counts: C=36 A=36 B=28 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:208 C:201 E:198 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 2 6 B -8 0 -6 -12 -12 C 0 6 0 -2 -2 D -2 12 2 0 12 E -6 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500348 B: 0.000000 C: 0.499652 D: 0.000000 E: 0.000000 Sum of squares = 0.500000241587 Cumulative probabilities = A: 0.500348 B: 0.500348 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 2 6 B -8 0 -6 -12 -12 C 0 6 0 -2 -2 D -2 12 2 0 12 E -6 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500348 B: 0.000000 C: 0.499652 D: 0.000000 E: 0.000000 Sum of squares = 0.500000241587 Cumulative probabilities = A: 0.500348 B: 0.500348 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 2 6 B -8 0 -6 -12 -12 C 0 6 0 -2 -2 D -2 12 2 0 12 E -6 12 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500348 B: 0.000000 C: 0.499652 D: 0.000000 E: 0.000000 Sum of squares = 0.500000241587 Cumulative probabilities = A: 0.500348 B: 0.500348 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7563: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) B A E C D (8) D C A E B (6) D B A C E (6) B D A E C (6) C E A D B (5) A E C B D (5) D C B A E (4) B A D E C (4) E C D A B (3) E C A D B (3) D B C A E (3) C E D A B (3) C D E A B (3) A C E B D (3) E B A C D (2) E A C B D (2) D C E A B (2) B D A C E (2) B A E D C (2) A B E C D (2) E D C B A (1) E C B A D (1) D E B C A (1) D C E B A (1) D C B E A (1) D C A B E (1) D B E C A (1) D B C E A (1) D A C B E (1) D A B C E (1) C A E D B (1) C A D E B (1) B E D C A (1) B E A D C (1) B D E C A (1) B D E A C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -4 0 16 B -4 0 -12 2 -2 C 4 12 0 2 -6 D 0 -2 -2 0 -4 E -16 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.615385 D: 0.000000 E: 0.153846 Sum of squares = 0.455621301769 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.846154 D: 0.846154 E: 1.000000 A B C D E A 0 4 -4 0 16 B -4 0 -12 2 -2 C 4 12 0 2 -6 D 0 -2 -2 0 -4 E -16 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.615385 D: 0.000000 E: 0.153846 Sum of squares = 0.455621300951 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.846154 D: 0.846154 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 E=20 C=13 A=12 so A is eliminated. Round 2 votes counts: D=29 B=29 E=25 C=17 so C is eliminated. Round 3 votes counts: E=37 D=33 B=30 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:208 C:206 E:198 D:196 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 0 16 B -4 0 -12 2 -2 C 4 12 0 2 -6 D 0 -2 -2 0 -4 E -16 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.615385 D: 0.000000 E: 0.153846 Sum of squares = 0.455621300951 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.846154 D: 0.846154 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 16 B -4 0 -12 2 -2 C 4 12 0 2 -6 D 0 -2 -2 0 -4 E -16 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.615385 D: 0.000000 E: 0.153846 Sum of squares = 0.455621300951 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.846154 D: 0.846154 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 16 B -4 0 -12 2 -2 C 4 12 0 2 -6 D 0 -2 -2 0 -4 E -16 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.615385 D: 0.000000 E: 0.153846 Sum of squares = 0.455621300951 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.846154 D: 0.846154 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7564: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) A C D E B (8) D E C B A (4) B E D A C (4) B A C E D (4) A D C B E (4) A C D B E (4) E B D C A (3) D E B C A (3) C A E D B (3) C A D E B (3) A C B D E (3) E D B C A (2) E C D B A (2) E C D A B (2) D C E A B (2) D B E A C (2) C E A D B (2) C E A B D (2) B E C A D (2) B E A C D (2) A D C E B (2) A B C D E (2) E D C B A (1) E B C D A (1) D E C A B (1) D E B A C (1) D E A C B (1) D C A E B (1) D B E C A (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B C E (1) C E D A B (1) C A E B D (1) B E C D A (1) B D E A C (1) B D A E C (1) B C E A D (1) B C A E D (1) A D B C E (1) A C E B D (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 -2 -8 B 0 0 -2 -10 4 C 4 2 0 -4 2 D 2 10 4 0 0 E 8 -4 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.643784 E: 0.356216 Sum of squares = 0.541347728521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.643784 E: 1.000000 A B C D E A 0 0 -4 -2 -8 B 0 0 -2 -10 4 C 4 2 0 -4 2 D 2 10 4 0 0 E 8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=28 D=20 C=12 E=11 so E is eliminated. Round 2 votes counts: B=33 A=28 D=23 C=16 so C is eliminated. Round 3 votes counts: A=39 B=33 D=28 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:208 C:202 E:201 B:196 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 -2 -8 B 0 0 -2 -10 4 C 4 2 0 -4 2 D 2 10 4 0 0 E 8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -2 -8 B 0 0 -2 -10 4 C 4 2 0 -4 2 D 2 10 4 0 0 E 8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -2 -8 B 0 0 -2 -10 4 C 4 2 0 -4 2 D 2 10 4 0 0 E 8 -4 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7565: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (18) C D E A B (17) D C E B A (9) A B E D C (8) C D E B A (7) D E C B A (5) C E D A B (4) A B C E D (4) D C B E A (2) C D A E B (2) C A E D B (2) A E C B D (2) A B E C D (2) E D C A B (1) E D B C A (1) E C D A B (1) E C A D B (1) D E B C A (1) C D B A E (1) C A D B E (1) C A B E D (1) B E D A C (1) B E A D C (1) B D E C A (1) B D E A C (1) B D A E C (1) B A E C D (1) B A D E C (1) A E C D B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -14 -10 -6 B 2 0 -18 -14 -10 C 14 18 0 -2 4 D 10 14 2 0 0 E 6 10 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.798205 E: 0.201795 Sum of squares = 0.677852103077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.798205 E: 1.000000 A B C D E A 0 -2 -14 -10 -6 B 2 0 -18 -14 -10 C 14 18 0 -2 4 D 10 14 2 0 0 E 6 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 0.333331 Sum of squares = 0.555557081839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=25 A=19 D=17 E=4 so E is eliminated. Round 2 votes counts: C=37 B=25 D=19 A=19 so D is eliminated. Round 3 votes counts: C=54 B=27 A=19 so A is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:213 E:206 A:184 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -14 -10 -6 B 2 0 -18 -14 -10 C 14 18 0 -2 4 D 10 14 2 0 0 E 6 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 0.333331 Sum of squares = 0.555557081839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 -10 -6 B 2 0 -18 -14 -10 C 14 18 0 -2 4 D 10 14 2 0 0 E 6 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 0.333331 Sum of squares = 0.555557081839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 -10 -6 B 2 0 -18 -14 -10 C 14 18 0 -2 4 D 10 14 2 0 0 E 6 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 0.333331 Sum of squares = 0.555557081839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666669 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7566: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) E D B A C (8) A C B D E (7) A B E D C (7) E D A B C (6) C E D B A (6) C A B D E (5) B D E A C (5) D B E A C (4) C D B E A (3) C B A D E (3) C A E D B (3) B D A E C (3) A C B E D (3) A B C D E (3) E C D B A (2) C E D A B (2) C E A D B (2) C D E B A (2) B A D E C (2) A E B D C (2) A B D E C (2) E D B C A (1) D E B C A (1) D B E C A (1) C B D E A (1) C A E B D (1) B D C E A (1) B D A C E (1) B C D A E (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 10 -16 -10 B 10 0 -4 -4 4 C -10 4 0 -8 -10 D 16 4 8 0 -10 E 10 -4 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407334 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 A B C D E A 0 -10 10 -16 -10 B 10 0 -4 -4 4 C -10 4 0 -8 -10 D 16 4 8 0 -10 E 10 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407382 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 A=26 B=13 D=6 so D is eliminated. Round 2 votes counts: E=28 C=28 A=26 B=18 so B is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:209 B:203 C:188 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 -16 -10 B 10 0 -4 -4 4 C -10 4 0 -8 -10 D 16 4 8 0 -10 E 10 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407382 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -16 -10 B 10 0 -4 -4 4 C -10 4 0 -8 -10 D 16 4 8 0 -10 E 10 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407382 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -16 -10 B 10 0 -4 -4 4 C -10 4 0 -8 -10 D 16 4 8 0 -10 E 10 -4 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.555556 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407382 Cumulative probabilities = A: 0.000000 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7567: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (15) E A D C B (7) C D B E A (6) E A D B C (5) D C B A E (5) C D B A E (4) C B D A E (4) E C A D B (3) E B A C D (3) E A C D B (3) E A B C D (3) C E D A B (3) B D A C E (3) E C B A D (2) D B C A E (2) D A B C E (2) C D E B A (2) C B E D A (2) C B D E A (2) B C E D A (2) B C D A E (2) B A E D C (2) B A D E C (2) A E B D C (2) A B D E C (2) E C A B D (1) E B C A D (1) D C A B E (1) D A C B E (1) C E D B A (1) C E B D A (1) B E A C D (1) B D C A E (1) B C D E A (1) A E D C B (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 8 10 -28 B -2 0 0 4 -8 C -8 0 0 -6 -10 D -10 -4 6 0 -16 E 28 8 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 8 10 -28 B -2 0 0 4 -8 C -8 0 0 -6 -10 D -10 -4 6 0 -16 E 28 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 C=25 B=14 D=11 A=7 so A is eliminated. Round 2 votes counts: E=46 C=25 B=16 D=13 so D is eliminated. Round 3 votes counts: E=47 C=32 B=21 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 B:197 A:196 C:188 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 8 10 -28 B -2 0 0 4 -8 C -8 0 0 -6 -10 D -10 -4 6 0 -16 E 28 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 10 -28 B -2 0 0 4 -8 C -8 0 0 -6 -10 D -10 -4 6 0 -16 E 28 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 10 -28 B -2 0 0 4 -8 C -8 0 0 -6 -10 D -10 -4 6 0 -16 E 28 8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) B D A E C (7) C E A D B (6) E C A B D (5) E A C B D (5) D B A E C (5) D B C A E (4) C E A B D (4) A C E B D (4) E C B A D (3) E B D C A (3) D B E A C (3) C A E D B (3) B D E A C (3) A E C B D (3) E D B C A (2) E C A D B (2) E B D A C (2) E B A C D (2) D B E C A (2) C D A B E (2) C A E B D (2) B E D A C (2) A C D B E (2) E D C B A (1) E A B C D (1) D E C B A (1) D C E B A (1) D A B C E (1) C E D B A (1) C E D A B (1) C A D E B (1) C A D B E (1) B D A C E (1) B A D E C (1) A E B C D (1) A C B D E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 10 0 0 B 4 0 2 6 -8 C -10 -2 0 2 -10 D 0 -6 -2 0 -8 E 0 8 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.294231 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.705769 Sum of squares = 0.584681981596 Cumulative probabilities = A: 0.294231 B: 0.294231 C: 0.294231 D: 0.294231 E: 1.000000 A B C D E A 0 -4 10 0 0 B 4 0 2 6 -8 C -10 -2 0 2 -10 D 0 -6 -2 0 -8 E 0 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=25 C=21 B=14 A=14 so B is eliminated. Round 2 votes counts: D=36 E=28 C=21 A=15 so A is eliminated. Round 3 votes counts: D=38 E=33 C=29 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:203 B:202 D:192 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 10 0 0 B 4 0 2 6 -8 C -10 -2 0 2 -10 D 0 -6 -2 0 -8 E 0 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 0 0 B 4 0 2 6 -8 C -10 -2 0 2 -10 D 0 -6 -2 0 -8 E 0 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 0 0 B 4 0 2 6 -8 C -10 -2 0 2 -10 D 0 -6 -2 0 -8 E 0 8 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7569: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) A B C D E (6) E C D B A (4) E C D A B (4) C E D A B (4) C A E D B (4) A D B C E (4) D E A C B (3) C E A D B (3) B C E A D (3) B A D E C (3) B A C E D (3) B A C D E (3) A C B E D (3) A B D C E (3) A B C E D (3) E C B D A (2) D E C A B (2) D E B C A (2) D B E A C (2) D B A E C (2) C E B D A (2) C E B A D (2) C E A B D (2) B E D C A (2) B E C D A (2) B D E A C (2) B A D C E (2) E D C B A (1) E D C A B (1) E D B C A (1) D E A B C (1) D B E C A (1) C B E A D (1) C B A E D (1) C A E B D (1) B C E D A (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D E B (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 8 6 4 B -16 0 12 -2 18 C -8 -12 0 12 10 D -6 2 -12 0 0 E -4 -18 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999165 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 6 4 B -16 0 12 -2 18 C -8 -12 0 12 10 D -6 2 -12 0 0 E -4 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994263 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=21 B=21 C=20 E=13 so E is eliminated. Round 2 votes counts: C=30 A=25 D=24 B=21 so B is eliminated. Round 3 votes counts: C=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:206 C:201 D:192 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 8 6 4 B -16 0 12 -2 18 C -8 -12 0 12 10 D -6 2 -12 0 0 E -4 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994263 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 6 4 B -16 0 12 -2 18 C -8 -12 0 12 10 D -6 2 -12 0 0 E -4 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994263 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 6 4 B -16 0 12 -2 18 C -8 -12 0 12 10 D -6 2 -12 0 0 E -4 -18 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994263 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7570: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) E A B D C (7) C D A E B (7) B E D A C (6) D C E B A (5) C D A B E (5) C A D E B (5) A C B E D (5) D C E A B (4) D C B E A (4) B E A D C (4) A C E B D (4) E B D A C (3) C D B E A (3) A E B D C (3) A B E C D (3) D E C B A (2) D E B C A (2) D B E C A (2) C A D B E (2) A E B C D (2) A C E D B (2) E D B A C (1) E D A C B (1) E D A B C (1) E A D B C (1) D B C E A (1) C D E A B (1) C D B A E (1) B E D C A (1) B D E C A (1) B C A D E (1) B A E C D (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 10 6 -2 -14 B -10 0 -4 0 -18 C -6 4 0 -14 0 D 2 0 14 0 -8 E 14 18 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.244326 D: 0.000000 E: 0.755674 Sum of squares = 0.630738018966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.244326 D: 0.244326 E: 1.000000 A B C D E A 0 10 6 -2 -14 B -10 0 -4 0 -18 C -6 4 0 -14 0 D 2 0 14 0 -8 E 14 18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.636364 Sum of squares = 0.537190151532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.363636 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 E=21 A=21 D=20 B=14 so B is eliminated. Round 2 votes counts: E=32 C=25 A=22 D=21 so D is eliminated. Round 3 votes counts: E=39 C=39 A=22 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:220 D:204 A:200 C:192 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 -2 -14 B -10 0 -4 0 -18 C -6 4 0 -14 0 D 2 0 14 0 -8 E 14 18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.636364 Sum of squares = 0.537190151532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.363636 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -2 -14 B -10 0 -4 0 -18 C -6 4 0 -14 0 D 2 0 14 0 -8 E 14 18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.636364 Sum of squares = 0.537190151532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.363636 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -2 -14 B -10 0 -4 0 -18 C -6 4 0 -14 0 D 2 0 14 0 -8 E 14 18 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.636364 Sum of squares = 0.537190151532 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.363636 D: 0.363636 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7571: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) D A E C B (6) E A D C B (5) C E D A B (5) C D E A B (5) B A D E C (5) D C E A B (4) B E A C D (4) B D A E C (4) D C A E B (3) C E B A D (3) C E A D B (3) C B E D A (3) E C A D B (2) E C A B D (2) E B A C D (2) D C A B E (2) C E D B A (2) C D B A E (2) C B D E A (2) C B D A E (2) B D C A E (2) B D A C E (2) B C E A D (2) B A E D C (2) A E D B C (2) A E B D C (2) E A D B C (1) E A C B D (1) E A B D C (1) E A B C D (1) D C B A E (1) D B C A E (1) D B A C E (1) D A C E B (1) C D B E A (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A D E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 10 -2 -18 -2 B -10 0 -10 -10 -2 C 2 10 0 -8 2 D 18 10 8 0 10 E 2 2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -18 -2 B -10 0 -10 -10 -2 C 2 10 0 -8 2 D 18 10 8 0 10 E 2 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 B=25 E=15 A=6 so A is eliminated. Round 2 votes counts: C=28 D=27 B=26 E=19 so E is eliminated. Round 3 votes counts: D=35 C=33 B=32 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:203 E:196 A:194 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -2 -18 -2 B -10 0 -10 -10 -2 C 2 10 0 -8 2 D 18 10 8 0 10 E 2 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -18 -2 B -10 0 -10 -10 -2 C 2 10 0 -8 2 D 18 10 8 0 10 E 2 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -18 -2 B -10 0 -10 -10 -2 C 2 10 0 -8 2 D 18 10 8 0 10 E 2 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7572: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (11) E D A B C (7) D E B C A (7) C A B D E (6) E D B A C (5) D E A C B (5) C B A D E (5) B C A E D (4) E D A C B (3) D E C A B (3) D C E B A (3) C B A E D (3) B C D E A (3) A E D C B (3) A B E C D (3) E A D C B (2) D E B A C (2) D C B E A (2) A B C E D (2) E D B C A (1) E A D B C (1) E A B D C (1) D E C B A (1) D B E C A (1) D A C E B (1) C B D A E (1) B E D C A (1) B E C A D (1) B E A D C (1) B D E C A (1) B D C E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E D B C (1) A E C B D (1) A E B D C (1) A E B C D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 8 2 -4 B -8 0 -4 0 -2 C -8 4 0 -10 -6 D -2 0 10 0 -8 E 4 2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 2 -4 B -8 0 -4 0 -2 C -8 4 0 -10 -6 D -2 0 10 0 -8 E 4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 E=20 C=15 B=15 so C is eliminated. Round 2 votes counts: A=31 D=25 B=24 E=20 so E is eliminated. Round 3 votes counts: D=41 A=35 B=24 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:207 D:200 B:193 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 2 -4 B -8 0 -4 0 -2 C -8 4 0 -10 -6 D -2 0 10 0 -8 E 4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 2 -4 B -8 0 -4 0 -2 C -8 4 0 -10 -6 D -2 0 10 0 -8 E 4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 2 -4 B -8 0 -4 0 -2 C -8 4 0 -10 -6 D -2 0 10 0 -8 E 4 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7573: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (13) C B D A E (10) A E C D B (8) D B C E A (7) D B C A E (7) C A E D B (7) E A C D B (6) A E C B D (6) E A C B D (5) C A E B D (5) E A D B C (4) D B E C A (3) B D C E A (3) C D B A E (2) B D C A E (2) E B D A C (1) E B A D C (1) D C B A E (1) D B A E C (1) D A E B C (1) C A B E D (1) B D E A C (1) B C D E A (1) A E D C B (1) A E D B C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 2 -6 10 B 6 0 -8 -26 4 C -2 8 0 6 -4 D 6 26 -6 0 4 E -10 -4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102032 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -6 10 B 6 0 -8 -26 4 C -2 8 0 6 -4 D 6 26 -6 0 4 E -10 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=25 A=18 E=17 B=7 so B is eliminated. Round 2 votes counts: D=39 C=26 A=18 E=17 so E is eliminated. Round 3 votes counts: D=40 A=34 C=26 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:204 A:200 E:193 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 -6 10 B 6 0 -8 -26 4 C -2 8 0 6 -4 D 6 26 -6 0 4 E -10 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -6 10 B 6 0 -8 -26 4 C -2 8 0 6 -4 D 6 26 -6 0 4 E -10 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -6 10 B 6 0 -8 -26 4 C -2 8 0 6 -4 D 6 26 -6 0 4 E -10 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.142857 E: 0.000000 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7574: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) E C D B A (7) D E C B A (7) A B C E D (7) D B C E A (6) A E C B D (6) D B A C E (5) E C A D B (4) E C A B D (4) B D C E A (4) D A B C E (3) B D A C E (3) A E C D B (3) A B E C D (3) E D C B A (2) E D A C B (2) E C D A B (2) E A C B D (2) D C E B A (2) D A B E C (2) B D C A E (2) A E B C D (2) A C E B D (2) A B D E C (2) E A D C B (1) E A C D B (1) D B E C A (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B E A (1) C B E A D (1) B C D E A (1) Total count = 100 A B C D E A 0 10 6 -4 -2 B -10 0 0 0 0 C -6 0 0 -2 -2 D 4 0 2 0 -2 E 2 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.106977 C: 0.000000 D: 0.000000 E: 0.893023 Sum of squares = 0.808934733735 Cumulative probabilities = A: 0.000000 B: 0.106977 C: 0.106977 D: 0.106977 E: 1.000000 A B C D E A 0 10 6 -4 -2 B -10 0 0 0 0 C -6 0 0 -2 -2 D 4 0 2 0 -2 E 2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222247015 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=26 E=25 B=10 C=5 so C is eliminated. Round 2 votes counts: A=34 D=28 E=27 B=11 so B is eliminated. Round 3 votes counts: D=38 A=34 E=28 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:205 E:203 D:202 B:195 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 6 -4 -2 B -10 0 0 0 0 C -6 0 0 -2 -2 D 4 0 2 0 -2 E 2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222247015 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -4 -2 B -10 0 0 0 0 C -6 0 0 -2 -2 D 4 0 2 0 -2 E 2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222247015 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -4 -2 B -10 0 0 0 0 C -6 0 0 -2 -2 D 4 0 2 0 -2 E 2 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222247015 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7575: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (14) A E D B C (12) D B C E A (7) A E C B D (6) E A D C B (5) E D C B A (4) C B D A E (4) B D C E A (4) A E C D B (4) E C D B A (3) A C B D E (3) E D B C A (2) E C A B D (2) E A D B C (2) E A C D B (2) E A C B D (2) D B E C A (2) C E B D A (2) C B E D A (2) B D C A E (2) B C D A E (2) A E B D C (2) A D B C E (2) E C B D A (1) E C A D B (1) D B C A E (1) D A B E C (1) C E A B D (1) C D B E A (1) C B A E D (1) C A B D E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 -4 -14 B 4 0 -18 0 -2 C 14 18 0 4 -4 D 4 0 -4 0 -8 E 14 2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -14 -4 -14 B 4 0 -18 0 -2 C 14 18 0 4 -4 D 4 0 -4 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=26 E=24 D=11 B=8 so B is eliminated. Round 2 votes counts: A=31 C=28 E=24 D=17 so D is eliminated. Round 3 votes counts: C=42 A=32 E=26 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:214 D:196 B:192 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -14 -4 -14 B 4 0 -18 0 -2 C 14 18 0 4 -4 D 4 0 -4 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -4 -14 B 4 0 -18 0 -2 C 14 18 0 4 -4 D 4 0 -4 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -4 -14 B 4 0 -18 0 -2 C 14 18 0 4 -4 D 4 0 -4 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7576: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (11) A E B C D (6) D C B A E (5) A D E C B (5) E B A C D (4) E A B C D (4) D E C B A (4) B C E D A (4) A E D C B (4) D C A B E (3) C D B A E (3) C B D A E (3) A D C E B (3) E D B C A (2) E B D C A (2) E B C A D (2) E A D B C (2) D E A C B (2) D A C E B (2) D A C B E (2) C B A D E (2) C A B D E (2) B C E A D (2) B C A E D (2) A D C B E (2) A C D B E (2) A B C E D (2) E D A B C (1) E A B D C (1) D B C E A (1) D A E C B (1) C B D E A (1) C A D B E (1) B E C D A (1) B E C A D (1) B C D E A (1) A E D B C (1) A E B D C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -6 2 8 B 2 0 -18 -14 6 C 6 18 0 -10 10 D -2 14 10 0 14 E -8 -6 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.111111 D: 0.333333 E: 0.000000 Sum of squares = 0.432098765393 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 2 8 B 2 0 -18 -14 6 C 6 18 0 -10 10 D -2 14 10 0 14 E -8 -6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.111111 D: 0.333333 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=28 E=18 C=12 B=11 so B is eliminated. Round 2 votes counts: D=31 A=28 C=21 E=20 so E is eliminated. Round 3 votes counts: A=39 D=36 C=25 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:218 C:212 A:201 B:188 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 2 8 B 2 0 -18 -14 6 C 6 18 0 -10 10 D -2 14 10 0 14 E -8 -6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.111111 D: 0.333333 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 2 8 B 2 0 -18 -14 6 C 6 18 0 -10 10 D -2 14 10 0 14 E -8 -6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.111111 D: 0.333333 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 2 8 B 2 0 -18 -14 6 C 6 18 0 -10 10 D -2 14 10 0 14 E -8 -6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.111111 D: 0.333333 E: 0.000000 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7577: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (11) B D C A E (10) E A C D B (9) E A C B D (8) D E A B C (7) D E B A C (6) A C E B D (6) D B E C A (5) D B C A E (5) C B A E D (5) A E C B D (5) E A D C B (4) D B C E A (4) C B A D E (3) E D A C B (2) B C A E D (2) E D B A C (1) E D A B C (1) D E A C B (1) D B E A C (1) C A E B D (1) C A B E D (1) B C A D E (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 4 -8 2 B 8 0 8 6 -4 C -4 -8 0 6 -2 D 8 -6 -6 0 8 E -2 4 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.333333 Sum of squares = 0.358024691359 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.666667 E: 1.000000 A B C D E A 0 -8 4 -8 2 B 8 0 8 6 -4 C -4 -8 0 6 -2 D 8 -6 -6 0 8 E -2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.333333 Sum of squares = 0.358024691276 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 B=24 A=12 C=10 so C is eliminated. Round 2 votes counts: B=32 D=29 E=25 A=14 so A is eliminated. Round 3 votes counts: E=38 B=33 D=29 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:209 D:202 E:198 C:196 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 -8 2 B 8 0 8 6 -4 C -4 -8 0 6 -2 D 8 -6 -6 0 8 E -2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.333333 Sum of squares = 0.358024691276 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -8 2 B 8 0 8 6 -4 C -4 -8 0 6 -2 D 8 -6 -6 0 8 E -2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.333333 Sum of squares = 0.358024691276 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -8 2 B 8 0 8 6 -4 C -4 -8 0 6 -2 D 8 -6 -6 0 8 E -2 4 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.222222 E: 0.333333 Sum of squares = 0.358024691276 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7578: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (14) B C E D A (9) E D A B C (8) C A D E B (7) E B D A C (6) B E D A C (6) C B A E D (5) C B A D E (5) C A D B E (5) C B E A D (4) C B E D A (3) B D A E C (3) B C A D E (3) E D B A C (2) E D A C B (2) D E A B C (2) C A B D E (2) A D C E B (2) A D B E C (2) A C D E B (2) E B C D A (1) C E A D B (1) C A E D B (1) B E D C A (1) B E C D A (1) B C E A D (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 0 10 4 B 2 0 -6 2 2 C 0 6 0 2 2 D -10 -2 -2 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.370407 B: 0.000000 C: 0.629593 D: 0.000000 E: 0.000000 Sum of squares = 0.533588846582 Cumulative probabilities = A: 0.370407 B: 0.370407 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 10 4 B 2 0 -6 2 2 C 0 6 0 2 2 D -10 -2 -2 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=25 A=21 E=19 D=2 so D is eliminated. Round 2 votes counts: C=33 B=25 E=21 A=21 so E is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:206 C:205 B:200 E:197 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 10 4 B 2 0 -6 2 2 C 0 6 0 2 2 D -10 -2 -2 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 10 4 B 2 0 -6 2 2 C 0 6 0 2 2 D -10 -2 -2 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 10 4 B 2 0 -6 2 2 C 0 6 0 2 2 D -10 -2 -2 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999971 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7579: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) C A B D E (5) E B D A C (4) C B A D E (4) B D E C A (4) A D C B E (4) E A D B C (3) D B E A C (3) D A E B C (3) D A C B E (3) D A B C E (3) C B E A D (3) C A B E D (3) B E D C A (3) B C E D A (3) E C B A D (2) E A D C B (2) E A C B D (2) D B E C A (2) D B C A E (2) D B A E C (2) C A E B D (2) A E D C B (2) A D E C B (2) A D C E B (2) A C E D B (2) A C E B D (2) A C D B E (2) E D A B C (1) E B C D A (1) E B A C D (1) E A B C D (1) D E B A C (1) D E A B C (1) D C B A E (1) D B C E A (1) D B A C E (1) C E A B D (1) C B D A E (1) C B A E D (1) C A D B E (1) B D C E A (1) B D C A E (1) A E D B C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 0 0 4 B 0 0 2 6 8 C 0 -2 0 -22 0 D 0 -6 22 0 2 E -4 -8 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.433233 B: 0.566767 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.508915635847 Cumulative probabilities = A: 0.433233 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 0 4 B 0 0 2 6 8 C 0 -2 0 -22 0 D 0 -6 22 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 C=21 A=19 B=12 so B is eliminated. Round 2 votes counts: D=29 E=28 C=24 A=19 so A is eliminated. Round 3 votes counts: D=37 E=32 C=31 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:208 A:202 E:193 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 0 4 B 0 0 2 6 8 C 0 -2 0 -22 0 D 0 -6 22 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 4 B 0 0 2 6 8 C 0 -2 0 -22 0 D 0 -6 22 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 4 B 0 0 2 6 8 C 0 -2 0 -22 0 D 0 -6 22 0 2 E -4 -8 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7580: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (14) E A C D B (9) B D C E A (7) A E C D B (7) A E B D C (7) E A B D C (6) C D E B A (6) C D B E A (6) E C D A B (5) A B D C E (5) E C D B A (4) D C B E A (4) B A D C E (3) A B E D C (3) E D C B A (2) C D B A E (2) A E C B D (2) E B D C A (1) C E D B A (1) C D A E B (1) C A D B E (1) B E D C A (1) B D A C E (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 -10 -4 B 4 0 -2 2 -6 C 10 2 0 -8 4 D 10 -2 8 0 2 E 4 6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000025 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 A B C D E A 0 -4 -10 -10 -4 B 4 0 -2 2 -6 C 10 2 0 -8 4 D 10 -2 8 0 2 E 4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.439999999969 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 A=26 C=17 D=4 so D is eliminated. Round 2 votes counts: E=27 B=26 A=26 C=21 so C is eliminated. Round 3 votes counts: B=38 E=34 A=28 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:209 C:204 E:202 B:199 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -10 -4 B 4 0 -2 2 -6 C 10 2 0 -8 4 D 10 -2 8 0 2 E 4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.439999999969 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -10 -4 B 4 0 -2 2 -6 C 10 2 0 -8 4 D 10 -2 8 0 2 E 4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.439999999969 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -10 -4 B 4 0 -2 2 -6 C 10 2 0 -8 4 D 10 -2 8 0 2 E 4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.439999999969 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7581: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) E C B D A (6) E B C D A (6) D A B E C (6) A B C E D (6) D E B C A (5) E D C B A (4) E D B C A (4) D E A C B (4) B E C A D (4) A D C B E (4) D E C B A (3) D A C E B (3) C E B A D (3) C B E A D (3) A D B C E (3) A C B E D (3) A C B D E (3) D E A B C (2) C A B E D (2) B C A E D (2) B A C E D (2) A C D B E (2) E C D B A (1) E C D A B (1) E B D C A (1) D E B A C (1) D A E C B (1) D A E B C (1) D A B C E (1) C E D A B (1) C E A B D (1) C B A E D (1) A D C E B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -10 -2 -14 B 6 0 4 2 0 C 10 -4 0 12 0 D 2 -2 -12 0 -16 E 14 0 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.682181 C: 0.000000 D: 0.000000 E: 0.317819 Sum of squares = 0.566379938361 Cumulative probabilities = A: 0.000000 B: 0.682181 C: 0.682181 D: 0.682181 E: 1.000000 A B C D E A 0 -6 -10 -2 -14 B 6 0 4 2 0 C 10 -4 0 12 0 D 2 -2 -12 0 -16 E 14 0 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=24 E=23 B=15 C=11 so C is eliminated. Round 2 votes counts: E=28 D=27 A=26 B=19 so B is eliminated. Round 3 votes counts: E=42 A=31 D=27 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:209 B:206 D:186 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 -2 -14 B 6 0 4 2 0 C 10 -4 0 12 0 D 2 -2 -12 0 -16 E 14 0 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -2 -14 B 6 0 4 2 0 C 10 -4 0 12 0 D 2 -2 -12 0 -16 E 14 0 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -2 -14 B 6 0 4 2 0 C 10 -4 0 12 0 D 2 -2 -12 0 -16 E 14 0 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7582: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) D C E A B (6) C E D B A (6) C E B D A (6) B E C D A (5) D E C A B (4) D C A E B (4) B C E A D (4) B A E C D (4) A D B E C (4) A C D B E (4) A B D E C (4) B E C A D (3) B A C E D (3) A D C E B (3) A D B C E (3) A B D C E (3) E D C B A (2) E C D B A (2) E C B D A (2) D E A B C (2) D A C E B (2) B A E D C (2) A B C D E (2) E D B C A (1) E D B A C (1) D E A C B (1) C D E A B (1) C B E A D (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E A C D (1) B C A E D (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -2 6 0 B -6 0 -6 -10 12 C 2 6 0 -4 18 D -6 10 4 0 6 E 0 -12 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 6 0 B -6 0 -6 -10 12 C 2 6 0 -4 18 D -6 10 4 0 6 E 0 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=26 D=19 C=16 E=8 so E is eliminated. Round 2 votes counts: A=31 B=26 D=23 C=20 so C is eliminated. Round 3 votes counts: B=35 A=33 D=32 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:207 A:205 B:195 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 6 0 B -6 0 -6 -10 12 C 2 6 0 -4 18 D -6 10 4 0 6 E 0 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 6 0 B -6 0 -6 -10 12 C 2 6 0 -4 18 D -6 10 4 0 6 E 0 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 6 0 B -6 0 -6 -10 12 C 2 6 0 -4 18 D -6 10 4 0 6 E 0 -12 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.500000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7583: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (14) D C A E B (9) B E A C D (7) A D C E B (6) D C A B E (5) D A C E B (5) C D A E B (5) A E B D C (5) E B C D A (4) C D B E A (4) E B A C D (3) C D E B A (3) E C B D A (2) E B C A D (2) D C B A E (2) D A C B E (2) C E D B A (2) C D E A B (2) B E C A D (2) B D C A E (2) A D E C B (2) E C B A D (1) C E D A B (1) C D A B E (1) B E A D C (1) B D A E C (1) B D A C E (1) B C E D A (1) B A D E C (1) A E D B C (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -24 -32 2 B 6 0 -4 -2 -6 C 24 4 0 8 4 D 32 2 -8 0 6 E -2 6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -24 -32 2 B 6 0 -4 -2 -6 C 24 4 0 8 4 D 32 2 -8 0 6 E -2 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 C=18 A=17 E=12 so E is eliminated. Round 2 votes counts: B=39 D=23 C=21 A=17 so A is eliminated. Round 3 votes counts: B=46 D=33 C=21 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:220 D:216 B:197 E:197 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -24 -32 2 B 6 0 -4 -2 -6 C 24 4 0 8 4 D 32 2 -8 0 6 E -2 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -24 -32 2 B 6 0 -4 -2 -6 C 24 4 0 8 4 D 32 2 -8 0 6 E -2 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -24 -32 2 B 6 0 -4 -2 -6 C 24 4 0 8 4 D 32 2 -8 0 6 E -2 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7584: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (14) B E D C A (9) A C D E B (7) E B A D C (5) B D C E A (5) C D B E A (4) C D A E B (4) C A D E B (4) E B C D A (3) E B A C D (3) C D A B E (3) B E A D C (3) A E B D C (3) A D C B E (3) A C E D B (3) E C A D B (2) E B C A D (2) E A C D B (2) B D E C A (2) A E C D B (2) A B E D C (2) E C D A B (1) E C B D A (1) D C B E A (1) D C B A E (1) C D E A B (1) B E D A C (1) B E C D A (1) B D C A E (1) B D A E C (1) B A D C E (1) A E C B D (1) A D C E B (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -18 -6 8 B -12 0 -12 -10 10 C 18 12 0 -10 12 D 6 10 10 0 12 E -8 -10 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999311 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -18 -6 8 B -12 0 -12 -10 10 C 18 12 0 -10 12 D 6 10 10 0 12 E -8 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=24 E=19 D=16 C=16 so D is eliminated. Round 2 votes counts: C=32 A=25 B=24 E=19 so E is eliminated. Round 3 votes counts: B=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:216 A:198 B:188 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -18 -6 8 B -12 0 -12 -10 10 C 18 12 0 -10 12 D 6 10 10 0 12 E -8 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -18 -6 8 B -12 0 -12 -10 10 C 18 12 0 -10 12 D 6 10 10 0 12 E -8 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -18 -6 8 B -12 0 -12 -10 10 C 18 12 0 -10 12 D 6 10 10 0 12 E -8 -10 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7585: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (6) E D B C A (5) D C E B A (5) A B C E D (5) E D B A C (4) D E C B A (4) C A B D E (4) A D E B C (4) C B E A D (3) B E C A D (3) A E B C D (3) A C D B E (3) E B D A C (2) E B C D A (2) E B A D C (2) E B A C D (2) D E B C A (2) D E A B C (2) D C A E B (2) D A C B E (2) C D B E A (2) C D B A E (2) C D A B E (2) C B A E D (2) C A D B E (2) B C E A D (2) B C A E D (2) B A E C D (2) A E B D C (2) A D C B E (2) A C B E D (2) E D A B C (1) E B D C A (1) E B C A D (1) D E B A C (1) D E A C B (1) D C E A B (1) D C B E A (1) D A E B C (1) C B D E A (1) C B D A E (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -12 -2 4 B 4 0 2 -10 4 C 12 -2 0 -6 8 D 2 10 6 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 4 B 4 0 2 -10 4 C 12 -2 0 -6 8 D 2 10 6 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=24 E=20 C=19 B=9 so B is eliminated. Round 2 votes counts: D=28 A=26 E=23 C=23 so E is eliminated. Round 3 votes counts: D=41 A=30 C=29 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:206 B:200 A:193 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 4 B 4 0 2 -10 4 C 12 -2 0 -6 8 D 2 10 6 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 4 B 4 0 2 -10 4 C 12 -2 0 -6 8 D 2 10 6 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 4 B 4 0 2 -10 4 C 12 -2 0 -6 8 D 2 10 6 0 6 E -4 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7586: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B D A E (7) D A E C B (6) C D E A B (6) D A E B C (5) B A E D C (5) E A B D C (4) D E A C B (4) C D A B E (4) B C A E D (4) E B A D C (3) D C A E B (3) C B E A D (3) B E A D C (3) B C E A D (3) B A D E C (3) A D E B C (3) C E D A B (2) C D B E A (2) C D B A E (2) C D A E B (2) A D B E C (2) A B E D C (2) E D A C B (1) E D A B C (1) E A D C B (1) D E C A B (1) D E A B C (1) C E D B A (1) C E B A D (1) C B E D A (1) C B D E A (1) C B A D E (1) B E A C D (1) B C A D E (1) A E B D C (1) Total count = 100 A B C D E A 0 16 10 0 2 B -16 0 2 -12 -10 C -10 -2 0 -16 -12 D 0 12 16 0 8 E -2 10 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.447997 B: 0.000000 C: 0.000000 D: 0.552003 E: 0.000000 Sum of squares = 0.505408613153 Cumulative probabilities = A: 0.447997 B: 0.447997 C: 0.447997 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 0 2 B -16 0 2 -12 -10 C -10 -2 0 -16 -12 D 0 12 16 0 8 E -2 10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=20 B=20 E=19 A=8 so A is eliminated. Round 2 votes counts: C=33 D=25 B=22 E=20 so E is eliminated. Round 3 votes counts: D=37 C=33 B=30 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:214 E:206 B:182 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 0 2 B -16 0 2 -12 -10 C -10 -2 0 -16 -12 D 0 12 16 0 8 E -2 10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 0 2 B -16 0 2 -12 -10 C -10 -2 0 -16 -12 D 0 12 16 0 8 E -2 10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 0 2 B -16 0 2 -12 -10 C -10 -2 0 -16 -12 D 0 12 16 0 8 E -2 10 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7587: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) E C A D B (9) E D C A B (6) D B A E C (6) D E A C B (5) D E A B C (5) B A D C E (5) C B A E D (4) B D A C E (4) D B E A C (3) D A B E C (3) C E A D B (3) C B E A D (3) C A E B D (3) B D A E C (3) B C A D E (3) E D A C B (2) E C D A B (2) D A E B C (2) C E D B A (2) C E B A D (2) B C D A E (2) B C A E D (2) E A C D B (1) D E B C A (1) D B E C A (1) C A B E D (1) B A C D E (1) A E D B C (1) A E C B D (1) A C E B D (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 16 -8 6 -10 B -16 0 -12 -4 -12 C 8 12 0 2 -6 D -6 4 -2 0 -10 E 10 12 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -8 6 -10 B -16 0 -12 -4 -12 C 8 12 0 2 -6 D -6 4 -2 0 -10 E 10 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 E=20 B=20 A=6 so A is eliminated. Round 2 votes counts: C=30 D=26 E=22 B=22 so E is eliminated. Round 3 votes counts: C=43 D=35 B=22 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:219 C:208 A:202 D:193 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -8 6 -10 B -16 0 -12 -4 -12 C 8 12 0 2 -6 D -6 4 -2 0 -10 E 10 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 6 -10 B -16 0 -12 -4 -12 C 8 12 0 2 -6 D -6 4 -2 0 -10 E 10 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 6 -10 B -16 0 -12 -4 -12 C 8 12 0 2 -6 D -6 4 -2 0 -10 E 10 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7588: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (6) D A B E C (6) B C A E D (6) A D B E C (6) E D C A B (5) C E B D A (5) B A D C E (5) C E D B A (4) B C E A D (4) B C A D E (4) B A C D E (4) E C A D B (3) D E A C B (3) C E B A D (3) C B E A D (3) A B D E C (3) E C B A D (2) D E C A B (2) D C B A E (2) D B A C E (2) C B E D A (2) A E D B C (2) A D E B C (2) A B D C E (2) E D A C B (1) E C D B A (1) E A D C B (1) E A C B D (1) E A B D C (1) D A E C B (1) D A E B C (1) D A B C E (1) C D E B A (1) C D B E A (1) B C D A E (1) B A C E D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -10 10 2 B 2 0 6 -2 8 C 10 -6 0 -8 14 D -10 2 8 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040829 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 10 2 B 2 0 6 -2 8 C 10 -6 0 -8 14 D -10 2 8 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408555 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 C=19 A=17 E=15 so E is eliminated. Round 2 votes counts: D=30 C=25 B=25 A=20 so A is eliminated. Round 3 votes counts: D=41 B=33 C=26 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:207 C:205 D:203 A:200 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -10 10 2 B 2 0 6 -2 8 C 10 -6 0 -8 14 D -10 2 8 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408555 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 10 2 B 2 0 6 -2 8 C 10 -6 0 -8 14 D -10 2 8 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408555 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 10 2 B 2 0 6 -2 8 C 10 -6 0 -8 14 D -10 2 8 0 6 E -2 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408555 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7589: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) B E D C A (7) A E C B D (7) A C E D B (7) D C A E B (6) D C A B E (6) B E A C D (6) B D E C A (6) D B C E A (5) B D E A C (4) A C D E B (4) E B A C D (3) D B C A E (3) B E D A C (3) B E A D C (3) E A C B D (2) E A B C D (2) D C B E A (2) C A E D B (2) E C A D B (1) E C A B D (1) E B C A D (1) D A C E B (1) D A C B E (1) C E A D B (1) C D E A B (1) C D A E B (1) B E C D A (1) B E C A D (1) B D A C E (1) A E C D B (1) A E B C D (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -4 -10 0 B 6 0 -4 0 16 C 4 4 0 -12 -2 D 10 0 12 0 -2 E 0 -16 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.450636 C: 0.000000 D: 0.549364 E: 0.000000 Sum of squares = 0.504873557641 Cumulative probabilities = A: 0.000000 B: 0.450636 C: 0.450636 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -10 0 B 6 0 -4 0 16 C 4 4 0 -12 -2 D 10 0 12 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=31 A=22 E=10 C=5 so C is eliminated. Round 2 votes counts: D=33 B=32 A=24 E=11 so E is eliminated. Round 3 votes counts: B=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:210 B:209 C:197 E:194 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 0 B 6 0 -4 0 16 C 4 4 0 -12 -2 D 10 0 12 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 0 B 6 0 -4 0 16 C 4 4 0 -12 -2 D 10 0 12 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 0 B 6 0 -4 0 16 C 4 4 0 -12 -2 D 10 0 12 0 -2 E 0 -16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7590: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) E A B C D (8) D C B A E (7) B A C D E (7) D C E B A (6) D B C A E (5) E D C A B (4) A E B C D (4) A B E C D (4) E A C B D (3) C D B A E (3) C B A D E (3) E C A B D (2) E A D B C (2) D E C B A (2) D C B E A (2) D B A C E (2) C D E B A (2) B C A D E (2) B A C E D (2) E D C B A (1) E D A C B (1) E C D A B (1) E A C D B (1) E A B D C (1) D E C A B (1) D E B A C (1) D B E A C (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A B D (1) C D B E A (1) C B D A E (1) B D C A E (1) B D A C E (1) B C D A E (1) B A D E C (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 6 8 16 B 10 0 14 10 16 C -6 -14 0 14 18 D -8 -10 -14 0 6 E -16 -16 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 8 16 B 10 0 14 10 16 C -6 -14 0 14 18 D -8 -10 -14 0 6 E -16 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 A=19 B=16 C=12 so C is eliminated. Round 2 votes counts: D=35 E=26 B=20 A=19 so A is eliminated. Round 3 votes counts: D=35 B=35 E=30 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:210 C:206 D:187 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 8 16 B 10 0 14 10 16 C -6 -14 0 14 18 D -8 -10 -14 0 6 E -16 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 8 16 B 10 0 14 10 16 C -6 -14 0 14 18 D -8 -10 -14 0 6 E -16 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 8 16 B 10 0 14 10 16 C -6 -14 0 14 18 D -8 -10 -14 0 6 E -16 -16 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7591: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (6) B E D A C (6) A D C E B (5) E B D C A (4) D A B E C (4) B C E A D (4) A C D E B (4) A C D B E (4) A C B D E (4) E D C A B (3) E D B C A (3) E D B A C (3) C E D A B (3) C A D E B (3) B E C A D (3) E C B A D (2) E B C D A (2) D E A C B (2) D E A B C (2) D A E B C (2) C B E A D (2) C A D B E (2) C A B D E (2) B E D C A (2) B E C D A (2) B D A E C (2) B A D C E (2) A D C B E (2) A D B C E (2) E D C B A (1) E C D B A (1) E C B D A (1) E B C A D (1) D B E A C (1) D B A E C (1) D A B C E (1) C E B A D (1) C B A E D (1) C A E B D (1) C A B E D (1) B C A E D (1) B A C D E (1) Total count = 100 A B C D E A 0 6 8 -4 2 B -6 0 -2 -10 0 C -8 2 0 -8 4 D 4 10 8 0 4 E -2 0 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -4 2 B -6 0 -2 -10 0 C -8 2 0 -8 4 D 4 10 8 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=21 A=21 D=19 C=16 so C is eliminated. Round 2 votes counts: A=30 B=26 E=25 D=19 so D is eliminated. Round 3 votes counts: A=43 E=29 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:206 C:195 E:195 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -4 2 B -6 0 -2 -10 0 C -8 2 0 -8 4 D 4 10 8 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -4 2 B -6 0 -2 -10 0 C -8 2 0 -8 4 D 4 10 8 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -4 2 B -6 0 -2 -10 0 C -8 2 0 -8 4 D 4 10 8 0 4 E -2 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7592: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (15) D C B E A (7) D C B A E (7) C D B E A (7) A E B D C (6) D C A B E (5) B E A D C (5) C D E A B (4) C D A E B (4) C E A B D (3) C D E B A (3) A E C B D (3) E A B C D (2) D C A E B (2) D B C E A (2) D A C B E (2) D A B E C (2) C A D E B (2) B E D A C (2) B E C D A (2) E C A B D (1) E B A C D (1) E A C B D (1) D A C E B (1) C E D B A (1) C A E D B (1) B D E C A (1) B D C E A (1) B D A E C (1) B A E D C (1) A E C D B (1) A D E C B (1) A C E D B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 18 -6 -8 14 B -18 0 -16 -8 -8 C 6 16 0 6 8 D 8 8 -6 0 6 E -14 8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -6 -8 14 B -18 0 -16 -8 -8 C 6 16 0 6 8 D 8 8 -6 0 6 E -14 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=28 C=25 B=13 E=5 so E is eliminated. Round 2 votes counts: A=32 D=28 C=26 B=14 so B is eliminated. Round 3 votes counts: A=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:218 A:209 D:208 E:190 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -6 -8 14 B -18 0 -16 -8 -8 C 6 16 0 6 8 D 8 8 -6 0 6 E -14 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -6 -8 14 B -18 0 -16 -8 -8 C 6 16 0 6 8 D 8 8 -6 0 6 E -14 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -6 -8 14 B -18 0 -16 -8 -8 C 6 16 0 6 8 D 8 8 -6 0 6 E -14 8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7593: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) C D E B A (8) A B E C D (8) A B E D C (7) C A D E B (6) C D A B E (4) E B D A C (3) E B A D C (3) D E B C A (3) D C B E A (3) D B E A C (3) C D E A B (3) C D A E B (3) C A D B E (3) A C E B D (3) D B E C A (2) C E D A B (2) C A E B D (2) C A B E D (2) B E D A C (2) B D E A C (2) B A E D C (2) A C B E D (2) E D B A C (1) E C B A D (1) D E C B A (1) D E B A C (1) D C B A E (1) D B C E A (1) D A B C E (1) C E A B D (1) C A E D B (1) C A B D E (1) B E A D C (1) A E B C D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -14 -6 0 B -6 0 -14 -12 -4 C 14 14 0 8 18 D 6 12 -8 0 14 E 0 4 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 -6 0 B -6 0 -14 -12 -4 C 14 14 0 8 18 D 6 12 -8 0 14 E 0 4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=25 A=24 E=8 B=7 so B is eliminated. Round 2 votes counts: C=36 D=27 A=26 E=11 so E is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:212 A:193 E:186 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 -6 0 B -6 0 -14 -12 -4 C 14 14 0 8 18 D 6 12 -8 0 14 E 0 4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 -6 0 B -6 0 -14 -12 -4 C 14 14 0 8 18 D 6 12 -8 0 14 E 0 4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 -6 0 B -6 0 -14 -12 -4 C 14 14 0 8 18 D 6 12 -8 0 14 E 0 4 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7594: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (12) D B E C A (8) A C E B D (7) A C E D B (6) A B D C E (5) E C A D B (4) C E A D B (4) B D A C E (4) E D C A B (3) B D C E A (3) B D A E C (3) B A D C E (3) A E C D B (3) A C B E D (3) E D C B A (2) E C D A B (2) D E C B A (2) D B E A C (2) C E A B D (2) C A E D B (2) B D E A C (2) A D B E C (2) A B C D E (2) E C D B A (1) E A D C B (1) E A C D B (1) D E B A C (1) D B A E C (1) D A B E C (1) C E B D A (1) C B A E D (1) B C D E A (1) B A D E C (1) B A C D E (1) A D E C B (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 2 4 2 -4 B -2 0 8 6 14 C -4 -8 0 -16 -8 D -2 -6 16 0 12 E 4 -14 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.539999999985 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 A B C D E A 0 2 4 2 -4 B -2 0 8 6 14 C -4 -8 0 -16 -8 D -2 -6 16 0 12 E 4 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.539999999369 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=30 D=15 E=14 C=10 so C is eliminated. Round 2 votes counts: A=33 B=31 E=21 D=15 so D is eliminated. Round 3 votes counts: B=42 A=34 E=24 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:213 D:210 A:202 E:193 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 4 2 -4 B -2 0 8 6 14 C -4 -8 0 -16 -8 D -2 -6 16 0 12 E 4 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.539999999369 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 -4 B -2 0 8 6 14 C -4 -8 0 -16 -8 D -2 -6 16 0 12 E 4 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.539999999369 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 -4 B -2 0 8 6 14 C -4 -8 0 -16 -8 D -2 -6 16 0 12 E 4 -14 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.539999999369 Cumulative probabilities = A: 0.700000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7595: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) E D A C B (7) D A C E B (7) A D E B C (6) C B E D A (5) B E C A D (5) E A D B C (4) A D E C B (4) C E B D A (3) C B D A E (3) B A C D E (3) A D B E C (3) E D A B C (2) E B A D C (2) E A B D C (2) C D B A E (2) C B D E A (2) B C E A D (2) B C A E D (2) B C A D E (2) B A E D C (2) A E D B C (2) A B E D C (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B D A (1) E B C D A (1) E B C A D (1) E B A C D (1) D E C A B (1) D E A C B (1) D C A B E (1) C E D A B (1) C D A B E (1) B E A C D (1) B C D A E (1) B A E C D (1) A D C E B (1) A D C B E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 18 26 -2 10 B -18 0 -6 -14 -18 C -26 6 0 -20 -24 D 2 14 20 0 0 E -10 18 24 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.869333 E: 0.130667 Sum of squares = 0.772813986738 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.869333 E: 1.000000 A B C D E A 0 18 26 -2 10 B -18 0 -6 -14 -18 C -26 6 0 -20 -24 D 2 14 20 0 0 E -10 18 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222403157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=21 D=19 B=19 C=17 so C is eliminated. Round 2 votes counts: B=29 E=28 D=22 A=21 so A is eliminated. Round 3 votes counts: D=38 B=32 E=30 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:226 D:218 E:216 B:172 C:168 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 26 -2 10 B -18 0 -6 -14 -18 C -26 6 0 -20 -24 D 2 14 20 0 0 E -10 18 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222403157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 26 -2 10 B -18 0 -6 -14 -18 C -26 6 0 -20 -24 D 2 14 20 0 0 E -10 18 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222403157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 26 -2 10 B -18 0 -6 -14 -18 C -26 6 0 -20 -24 D 2 14 20 0 0 E -10 18 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.166667 Sum of squares = 0.722222403157 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.833333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7596: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (12) C A D B E (9) E B D A C (7) C A D E B (7) C A B D E (7) E D C A B (6) E C D A B (5) A C D B E (5) B E D A C (4) E B C A D (3) C A E B D (3) B D A E C (3) B A C D E (3) E D B C A (2) E C A B D (2) D E A C B (2) D A C B E (2) C A E D B (2) E D C B A (1) E D A C B (1) E C A D B (1) E B C D A (1) D E B A C (1) D C A E B (1) D B E A C (1) D A B E C (1) C D A E B (1) C A B E D (1) B E A D C (1) B D E A C (1) B D A C E (1) B C A E D (1) B C A D E (1) B A C E D (1) Total count = 100 A B C D E A 0 12 -8 -6 -2 B -12 0 -12 -20 -16 C 8 12 0 6 -10 D 6 20 -6 0 -8 E 2 16 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -8 -6 -2 B -12 0 -12 -20 -16 C 8 12 0 6 -10 D 6 20 -6 0 -8 E 2 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997249 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=30 B=16 D=8 A=5 so A is eliminated. Round 2 votes counts: E=41 C=35 B=16 D=8 so D is eliminated. Round 3 votes counts: E=44 C=38 B=18 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:208 D:206 A:198 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -8 -6 -2 B -12 0 -12 -20 -16 C 8 12 0 6 -10 D 6 20 -6 0 -8 E 2 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997249 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 -6 -2 B -12 0 -12 -20 -16 C 8 12 0 6 -10 D 6 20 -6 0 -8 E 2 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997249 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 -6 -2 B -12 0 -12 -20 -16 C 8 12 0 6 -10 D 6 20 -6 0 -8 E 2 16 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997249 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7597: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (12) E D A C B (11) B C A D E (11) B C A E D (9) B A C E D (8) D E A C B (6) E D A B C (5) B A E D C (4) C D A E B (3) C A B D E (3) E D B A C (2) B E D A C (2) B E A D C (2) A C D E B (2) E D C A B (1) E D B C A (1) E B D A C (1) D C E A B (1) D C A E B (1) C D E A B (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D C A (1) B D C E A (1) B C D E A (1) B C D A E (1) B A E C D (1) A C E D B (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -4 -6 2 B -4 0 2 2 2 C 4 -2 0 -2 2 D 6 -2 2 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -6 2 B -4 0 2 2 2 C 4 -2 0 -2 2 D 6 -2 2 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=21 D=20 C=12 A=6 so A is eliminated. Round 2 votes counts: B=42 E=21 D=20 C=17 so C is eliminated. Round 3 votes counts: B=50 D=28 E=22 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:202 B:201 C:201 A:198 E:198 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -6 2 B -4 0 2 2 2 C 4 -2 0 -2 2 D 6 -2 2 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 2 B -4 0 2 2 2 C 4 -2 0 -2 2 D 6 -2 2 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 2 B -4 0 2 2 2 C 4 -2 0 -2 2 D 6 -2 2 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7598: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (8) E A C D B (7) E A C B D (7) B D C A E (7) D B C A E (6) D B A C E (5) B C D E A (4) A E D B C (4) A E C D B (4) E A B D C (3) D C B A E (3) C B D E A (3) B D C E A (3) E C A B D (2) E A D C B (2) E A D B C (2) E A B C D (2) C D B E A (2) C D A E B (2) C D A B E (2) B C D A E (2) A E B D C (2) E C A D B (1) E B C D A (1) E B A C D (1) D C A B E (1) D A B C E (1) C E B D A (1) C E A D B (1) C D E B A (1) C D B A E (1) C B E D A (1) C B D A E (1) B D A C E (1) B C E D A (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 12 8 2 10 B -12 0 -4 -16 -6 C -8 4 0 -6 2 D -2 16 6 0 0 E -10 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 2 10 B -12 0 -4 -16 -6 C -8 4 0 -6 2 D -2 16 6 0 0 E -10 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999326 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=23 B=18 D=16 C=15 so C is eliminated. Round 2 votes counts: E=30 D=24 B=23 A=23 so B is eliminated. Round 3 votes counts: D=45 E=32 A=23 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:210 E:197 C:196 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 2 10 B -12 0 -4 -16 -6 C -8 4 0 -6 2 D -2 16 6 0 0 E -10 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999326 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 2 10 B -12 0 -4 -16 -6 C -8 4 0 -6 2 D -2 16 6 0 0 E -10 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999326 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 2 10 B -12 0 -4 -16 -6 C -8 4 0 -6 2 D -2 16 6 0 0 E -10 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999326 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7599: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) A E D C B (6) B E A D C (5) B C D A E (5) A C D E B (5) E D C B A (3) E D A C B (3) E B D C A (3) E B A D C (3) E A D C B (3) E A D B C (3) D C A E B (3) C D E A B (3) C D A E B (3) C D A B E (3) C A D E B (3) B E D C A (3) B A E D C (3) B A E C D (3) D E C A B (2) B E C D A (2) B A C D E (2) A E B D C (2) A C B D E (2) A B E C D (2) A B C D E (2) E D C A B (1) E A B D C (1) D C E B A (1) D A C E B (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) C A D B E (1) B C A D E (1) B A C E D (1) A E D B C (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 0 2 6 B -4 0 6 6 -2 C 0 -6 0 6 0 D -2 -6 -6 0 0 E -6 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.728742 B: 0.000000 C: 0.271258 D: 0.000000 E: 0.000000 Sum of squares = 0.604646236258 Cumulative probabilities = A: 0.728742 B: 0.728742 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 2 6 B -4 0 6 6 -2 C 0 -6 0 6 0 D -2 -6 -6 0 0 E -6 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000034536 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=22 E=20 C=17 D=7 so D is eliminated. Round 2 votes counts: B=34 A=23 E=22 C=21 so C is eliminated. Round 3 votes counts: B=38 A=36 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:203 C:200 E:198 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 6 B -4 0 6 6 -2 C 0 -6 0 6 0 D -2 -6 -6 0 0 E -6 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000034536 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 6 B -4 0 6 6 -2 C 0 -6 0 6 0 D -2 -6 -6 0 0 E -6 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000034536 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 6 B -4 0 6 6 -2 C 0 -6 0 6 0 D -2 -6 -6 0 0 E -6 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000034536 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7600: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (17) D B A C E (14) A C B D E (11) E D B C A (10) C A E B D (5) A B D C E (4) E D B A C (3) E C B D A (3) E C B A D (3) C E A B D (3) A C E B D (3) E D C B A (2) E C D B A (2) E C A D B (2) D B E C A (2) D B E A C (2) C A B D E (2) B D E C A (2) A D B C E (2) E A C D B (1) D E B C A (1) D E B A C (1) D B A E C (1) C B E A D (1) B D A C E (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 4 -10 12 -10 B -4 0 -12 10 -14 C 10 12 0 6 -6 D -12 -10 -6 0 -10 E 10 14 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -10 12 -10 B -4 0 -12 10 -14 C 10 12 0 6 -6 D -12 -10 -6 0 -10 E 10 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 A=22 D=21 C=11 B=3 so B is eliminated. Round 2 votes counts: E=43 D=24 A=22 C=11 so C is eliminated. Round 3 votes counts: E=47 A=29 D=24 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 C:211 A:198 B:190 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -10 12 -10 B -4 0 -12 10 -14 C 10 12 0 6 -6 D -12 -10 -6 0 -10 E 10 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 12 -10 B -4 0 -12 10 -14 C 10 12 0 6 -6 D -12 -10 -6 0 -10 E 10 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 12 -10 B -4 0 -12 10 -14 C 10 12 0 6 -6 D -12 -10 -6 0 -10 E 10 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7601: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (5) D C B A E (5) E A C B D (4) D B C A E (4) D B A E C (4) C E A D B (4) C E A B D (4) C D B A E (4) C A E D B (4) B E A D C (4) B D C E A (4) E A B C D (3) C B E D A (3) C B D E A (3) B D E A C (3) B D C A E (3) A E D B C (3) E B A C D (2) E A C D B (2) D B A C E (2) D A C B E (2) D A B E C (2) C E B A D (2) C D A B E (2) C A D E B (2) B D A E C (2) A E B D C (2) A D E B C (2) A C E D B (2) E C A D B (1) E B A D C (1) D C A B E (1) C D E A B (1) C D B E A (1) C D A E B (1) B E D A C (1) B E C D A (1) B D A C E (1) B A D E C (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 0 0 2 B 2 0 0 -2 6 C 0 0 0 -6 12 D 0 2 6 0 2 E -2 -6 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.309804 B: 0.000000 C: 0.000000 D: 0.690196 E: 0.000000 Sum of squares = 0.572348705237 Cumulative probabilities = A: 0.309804 B: 0.309804 C: 0.309804 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 0 2 B 2 0 0 -2 6 C 0 0 0 -6 12 D 0 2 6 0 2 E -2 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499603 B: 0.000000 C: 0.000000 D: 0.500397 E: 0.000000 Sum of squares = 0.500000315303 Cumulative probabilities = A: 0.499603 B: 0.499603 C: 0.499603 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=20 B=20 E=18 A=11 so A is eliminated. Round 2 votes counts: C=34 E=24 D=22 B=20 so B is eliminated. Round 3 votes counts: D=36 C=34 E=30 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:205 B:203 C:203 A:200 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 0 2 B 2 0 0 -2 6 C 0 0 0 -6 12 D 0 2 6 0 2 E -2 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499603 B: 0.000000 C: 0.000000 D: 0.500397 E: 0.000000 Sum of squares = 0.500000315303 Cumulative probabilities = A: 0.499603 B: 0.499603 C: 0.499603 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 2 B 2 0 0 -2 6 C 0 0 0 -6 12 D 0 2 6 0 2 E -2 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499603 B: 0.000000 C: 0.000000 D: 0.500397 E: 0.000000 Sum of squares = 0.500000315303 Cumulative probabilities = A: 0.499603 B: 0.499603 C: 0.499603 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 2 B 2 0 0 -2 6 C 0 0 0 -6 12 D 0 2 6 0 2 E -2 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499603 B: 0.000000 C: 0.000000 D: 0.500397 E: 0.000000 Sum of squares = 0.500000315303 Cumulative probabilities = A: 0.499603 B: 0.499603 C: 0.499603 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7602: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (17) E B A C D (11) B A C D E (11) D C A B E (6) E D B C A (5) A C B D E (5) B A E C D (4) E D C B A (3) E B D A C (3) E B A D C (3) C D A B E (3) C A D B E (3) B A C E D (3) E A B C D (2) D E C B A (2) D C B A E (2) D C A E B (2) C A B D E (2) B E A D C (2) B E A C D (2) E D B A C (1) E C D A B (1) D E C A B (1) D E B C A (1) D B C E A (1) D B C A E (1) C E A D B (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -2 2 -12 B 10 0 4 0 -8 C 2 -4 0 0 -18 D -2 0 0 0 -18 E 12 8 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -2 2 -12 B 10 0 4 0 -8 C 2 -4 0 0 -18 D -2 0 0 0 -18 E 12 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=46 B=22 D=16 C=9 A=7 so A is eliminated. Round 2 votes counts: E=47 B=23 D=16 C=14 so C is eliminated. Round 3 votes counts: E=48 B=30 D=22 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 B:203 C:190 D:190 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -2 2 -12 B 10 0 4 0 -8 C 2 -4 0 0 -18 D -2 0 0 0 -18 E 12 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 2 -12 B 10 0 4 0 -8 C 2 -4 0 0 -18 D -2 0 0 0 -18 E 12 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 2 -12 B 10 0 4 0 -8 C 2 -4 0 0 -18 D -2 0 0 0 -18 E 12 8 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999156 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7603: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (12) C A B E D (12) A C D B E (8) E B D C A (7) A C B D E (6) A C B E D (5) E D B C A (3) D E A B C (3) D A C E B (3) B A E D C (3) A D C E B (3) E D B A C (2) E B D A C (2) D E C B A (2) D E B C A (2) D C E A B (2) C A E B D (2) C A D E B (2) B E D A C (2) B C E A D (2) B A D E C (2) A B C E D (2) E D C B A (1) E B C D A (1) D E C A B (1) D E A C B (1) C E B A D (1) C D E A B (1) C A E D B (1) B E C D A (1) B E C A D (1) B E A C D (1) A D B E C (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 16 8 4 B -10 0 -4 2 -6 C -16 4 0 -4 4 D -8 -2 4 0 2 E -4 6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 8 4 B -10 0 -4 2 -6 C -16 4 0 -4 4 D -8 -2 4 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 C=19 E=16 B=12 so B is eliminated. Round 2 votes counts: A=32 D=26 E=21 C=21 so E is eliminated. Round 3 votes counts: D=43 A=33 C=24 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:198 E:198 C:194 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 8 4 B -10 0 -4 2 -6 C -16 4 0 -4 4 D -8 -2 4 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 8 4 B -10 0 -4 2 -6 C -16 4 0 -4 4 D -8 -2 4 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 8 4 B -10 0 -4 2 -6 C -16 4 0 -4 4 D -8 -2 4 0 2 E -4 6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999251 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7604: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) C D E B A (7) B D A E C (7) B A D E C (6) A E B D C (5) E A C D B (4) D B E A C (4) C E D A B (4) C D B E A (4) C A E B D (4) E D A C B (3) E A D B C (3) D C B E A (3) D B C E A (3) C B D A E (3) C A E D B (3) A E C B D (3) D B E C A (2) B D E A C (2) B A D C E (2) A C E B D (2) A B E D C (2) A B C D E (2) E C A D B (1) E A D C B (1) D E B C A (1) D C E B A (1) C E D B A (1) C A B D E (1) B D C E A (1) B D C A E (1) B D A C E (1) B C A D E (1) A E D B C (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 4 -6 B 0 0 -8 -8 -6 C -2 8 0 2 6 D -4 8 -2 0 4 E 6 6 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102011 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 0 2 4 -6 B 0 0 -8 -8 -6 C -2 8 0 2 6 D -4 8 -2 0 4 E 6 6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=21 A=18 D=14 E=12 so E is eliminated. Round 2 votes counts: C=36 A=26 B=21 D=17 so D is eliminated. Round 3 votes counts: C=40 B=31 A=29 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 D:203 E:201 A:200 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 2 4 -6 B 0 0 -8 -8 -6 C -2 8 0 2 6 D -4 8 -2 0 4 E 6 6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 -6 B 0 0 -8 -8 -6 C -2 8 0 2 6 D -4 8 -2 0 4 E 6 6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 -6 B 0 0 -8 -8 -6 C -2 8 0 2 6 D -4 8 -2 0 4 E 6 6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102047 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7605: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (6) D C B A E (6) A E B D C (6) A E B C D (6) E B A C D (5) D C B E A (5) C D E B A (4) C D B E A (4) A D B C E (4) E C B D A (3) D C A B E (3) D B C A E (3) D B A C E (3) D A C B E (3) C D E A B (3) B A E D C (3) B A D E C (3) A B E D C (3) D B C E A (2) C E D B A (2) C E D A B (2) B E A D C (2) A B D E C (2) E C A B D (1) E B C A D (1) E A C B D (1) C E A D B (1) C D A E B (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B D E A C (1) B D C E A (1) B D A E C (1) A E C D B (1) A E C B D (1) A D E B C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 10 0 6 B 4 0 14 -2 6 C -10 -14 0 -10 -2 D 0 2 10 0 6 E -6 -6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.234411 B: 0.000000 C: 0.000000 D: 0.765589 E: 0.000000 Sum of squares = 0.641075275613 Cumulative probabilities = A: 0.234411 B: 0.234411 C: 0.234411 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 0 6 B 4 0 14 -2 6 C -10 -14 0 -10 -2 D 0 2 10 0 6 E -6 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555600447 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 C=18 E=17 B=14 so B is eliminated. Round 2 votes counts: A=32 D=28 E=22 C=18 so C is eliminated. Round 3 votes counts: D=40 A=33 E=27 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:211 D:209 A:206 E:192 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 10 0 6 B 4 0 14 -2 6 C -10 -14 0 -10 -2 D 0 2 10 0 6 E -6 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555600447 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 0 6 B 4 0 14 -2 6 C -10 -14 0 -10 -2 D 0 2 10 0 6 E -6 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555600447 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 0 6 B 4 0 14 -2 6 C -10 -14 0 -10 -2 D 0 2 10 0 6 E -6 -6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555600447 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7606: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) A E D B C (8) A E B C D (8) E A D B C (7) A E D C B (6) B C D E A (5) D C B E A (4) C B A D E (4) D E A B C (3) D A E C B (3) B C A E D (3) A D C E B (3) E D A B C (2) E A B D C (2) D C B A E (2) D B E C A (2) C D B A E (2) C B A E D (2) C A B E D (2) B D C E A (2) B C E D A (2) A E C B D (2) E D B A C (1) D E A C B (1) D A C E B (1) C B D A E (1) C A D B E (1) B E A C D (1) B C E A D (1) B A E C D (1) A E C D B (1) A D E C B (1) A C E D B (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 14 10 B -10 0 -2 2 -2 C -10 2 0 4 0 D -14 -2 -4 0 -6 E -10 2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 14 10 B -10 0 -2 2 -2 C -10 2 0 4 0 D -14 -2 -4 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 D=16 B=15 E=12 so E is eliminated. Round 2 votes counts: A=42 C=24 D=19 B=15 so B is eliminated. Round 3 votes counts: A=44 C=35 D=21 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:199 C:198 B:194 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 14 10 B -10 0 -2 2 -2 C -10 2 0 4 0 D -14 -2 -4 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 14 10 B -10 0 -2 2 -2 C -10 2 0 4 0 D -14 -2 -4 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 14 10 B -10 0 -2 2 -2 C -10 2 0 4 0 D -14 -2 -4 0 -6 E -10 2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7607: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) E A B D C (6) C D B A E (6) C A B E D (6) E A D B C (5) E A C B D (5) B D C A E (5) A E C B D (5) D E B A C (4) D C B E A (4) D B C A E (4) A C E B D (4) E A C D B (3) B D E A C (3) B C D A E (3) E D A B C (2) E A D C B (2) E A B C D (2) D E C B A (2) D B E A C (2) B E D A C (2) B C A D E (2) A E B C D (2) A B E C D (2) E B A D C (1) D E B C A (1) D E A B C (1) D C B A E (1) C D A E B (1) C A E D B (1) C A E B D (1) B D C E A (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 6 -2 -10 B 2 0 16 4 4 C -6 -16 0 -10 0 D 2 -4 10 0 -2 E 10 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -2 -10 B 2 0 16 4 4 C -6 -16 0 -10 0 D 2 -4 10 0 -2 E 10 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998558 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 B=17 C=15 A=14 so A is eliminated. Round 2 votes counts: E=33 D=28 C=20 B=19 so B is eliminated. Round 3 votes counts: E=37 D=37 C=26 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:204 D:203 A:196 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 -2 -10 B 2 0 16 4 4 C -6 -16 0 -10 0 D 2 -4 10 0 -2 E 10 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998558 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 -10 B 2 0 16 4 4 C -6 -16 0 -10 0 D 2 -4 10 0 -2 E 10 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998558 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 -10 B 2 0 16 4 4 C -6 -16 0 -10 0 D 2 -4 10 0 -2 E 10 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998558 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7608: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) E D A B C (8) D E A C B (7) C B D E A (7) C B A D E (7) E D A C B (5) D E C A B (5) C B A E D (4) B C A D E (4) B A C E D (4) B A E D C (3) A C B E D (3) D E B C A (2) D E B A C (2) D E A B C (2) D B E C A (2) C D B E A (2) C A D E B (2) C A B E D (2) B C D E A (2) A E D C B (2) A E D B C (2) A E B D C (2) E D B A C (1) E A D B C (1) D C E A B (1) D C B E A (1) C D E B A (1) C B D A E (1) B D E C A (1) B D E A C (1) A E C D B (1) A E C B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 -2 -2 B 8 0 -4 6 10 C 6 4 0 2 2 D 2 -6 -2 0 0 E 2 -10 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -2 -2 B 8 0 -4 6 10 C 6 4 0 2 2 D 2 -6 -2 0 0 E 2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 D=22 E=15 A=13 so A is eliminated. Round 2 votes counts: C=29 B=26 E=23 D=22 so D is eliminated. Round 3 votes counts: E=41 C=31 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:210 C:207 D:197 E:195 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -2 -2 B 8 0 -4 6 10 C 6 4 0 2 2 D 2 -6 -2 0 0 E 2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -2 -2 B 8 0 -4 6 10 C 6 4 0 2 2 D 2 -6 -2 0 0 E 2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -2 -2 B 8 0 -4 6 10 C 6 4 0 2 2 D 2 -6 -2 0 0 E 2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7609: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (20) E D B A C (9) D E B A C (9) E D C A B (6) D E C A B (6) D B A C E (6) E C A B D (5) D B E A C (4) C A E B D (4) C A B D E (4) D E B C A (3) B A C E D (3) E C A D B (2) E B D A C (2) D E C B A (2) D C A B E (2) C E A B D (2) B E A C D (2) E B A C D (1) E A C B D (1) D C A E B (1) B E A D C (1) B D E A C (1) B D A C E (1) B A D E C (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -14 -4 -12 B -8 0 -10 0 -6 C 14 10 0 -10 -10 D 4 0 10 0 -16 E 12 6 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -14 -4 -12 B -8 0 -10 0 -6 C 14 10 0 -10 -10 D 4 0 10 0 -16 E 12 6 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=30 E=26 B=10 A=1 so A is eliminated. Round 2 votes counts: D=33 C=30 E=26 B=11 so B is eliminated. Round 3 votes counts: D=37 C=34 E=29 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:222 C:202 D:199 A:189 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -14 -4 -12 B -8 0 -10 0 -6 C 14 10 0 -10 -10 D 4 0 10 0 -16 E 12 6 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -14 -4 -12 B -8 0 -10 0 -6 C 14 10 0 -10 -10 D 4 0 10 0 -16 E 12 6 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -14 -4 -12 B -8 0 -10 0 -6 C 14 10 0 -10 -10 D 4 0 10 0 -16 E 12 6 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7610: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) C D B E A (9) A B E D C (9) A E B D C (7) E C A D B (6) D C B A E (6) D B C A E (6) C D E B A (6) E C A B D (5) E A C B D (5) B A D E C (4) E A C D B (3) C E D A B (3) D B A C E (2) C D E A B (2) B D C A E (2) A B D E C (2) E C D A B (1) E C B D A (1) E C B A D (1) E B C A D (1) D C A E B (1) D A B C E (1) C E D B A (1) C B D E A (1) B D A C E (1) B C D E A (1) A E D C B (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 16 -6 12 -12 B -16 0 -4 0 -10 C 6 4 0 12 -14 D -12 0 -12 0 -10 E 12 10 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -6 12 -12 B -16 0 -4 0 -10 C 6 4 0 12 -14 D -12 0 -12 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=22 A=21 D=16 B=8 so B is eliminated. Round 2 votes counts: E=33 A=25 C=23 D=19 so D is eliminated. Round 3 votes counts: C=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:205 C:204 B:185 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -6 12 -12 B -16 0 -4 0 -10 C 6 4 0 12 -14 D -12 0 -12 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 12 -12 B -16 0 -4 0 -10 C 6 4 0 12 -14 D -12 0 -12 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 12 -12 B -16 0 -4 0 -10 C 6 4 0 12 -14 D -12 0 -12 0 -10 E 12 10 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7611: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) E A C D B (5) E D C A B (4) D E B C A (4) D C B A E (4) D B E C A (4) D B C E A (4) D B C A E (4) B E A D C (4) B D A C E (4) B A C E D (4) D C E A B (3) D C A E B (3) C D A E B (3) C A D E B (3) A B C E D (3) E D B A C (2) E B D A C (2) E A C B D (2) D E C B A (2) C A E D B (2) B E A C D (2) B A E D C (2) A E C B D (2) A C B E D (2) E D B C A (1) E D A C B (1) E B A C D (1) E A B C D (1) D C E B A (1) D C A B E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D A C (1) B D E A C (1) B A E C D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -16 -22 -12 B 0 0 -8 -30 -8 C 16 8 0 -20 0 D 22 30 20 0 10 E 12 8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -22 -12 B 0 0 -8 -30 -8 C 16 8 0 -20 0 D 22 30 20 0 10 E 12 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=20 E=19 C=15 A=8 so A is eliminated. Round 2 votes counts: D=38 B=23 E=21 C=18 so C is eliminated. Round 3 votes counts: D=48 E=26 B=26 so E is eliminated. Round 4 votes counts: D=65 B=35 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:241 E:205 C:202 B:177 A:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -16 -22 -12 B 0 0 -8 -30 -8 C 16 8 0 -20 0 D 22 30 20 0 10 E 12 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -22 -12 B 0 0 -8 -30 -8 C 16 8 0 -20 0 D 22 30 20 0 10 E 12 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -22 -12 B 0 0 -8 -30 -8 C 16 8 0 -20 0 D 22 30 20 0 10 E 12 8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7612: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (17) E D A C B (15) D E B C A (10) B C D A E (6) E D B C A (5) A C E B D (5) A C B E D (5) E D B A C (4) E A D C B (4) A E D C B (4) E D A B C (3) D B E C A (3) A C E D B (3) B D E C A (2) B C D E A (2) A E C D B (2) A C B D E (2) A B C E D (2) E A D B C (1) D E C A B (1) D B C E A (1) C B A D E (1) C A B D E (1) B D C E A (1) Total count = 100 A B C D E A 0 -4 0 -6 -4 B 4 0 14 -12 -14 C 0 -14 0 -8 -8 D 6 12 8 0 -6 E 4 14 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 0 -6 -4 B 4 0 14 -12 -14 C 0 -14 0 -8 -8 D 6 12 8 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=28 A=23 D=15 C=2 so C is eliminated. Round 2 votes counts: E=32 B=29 A=24 D=15 so D is eliminated. Round 3 votes counts: E=43 B=33 A=24 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:210 B:196 A:193 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -6 -4 B 4 0 14 -12 -14 C 0 -14 0 -8 -8 D 6 12 8 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -6 -4 B 4 0 14 -12 -14 C 0 -14 0 -8 -8 D 6 12 8 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -6 -4 B 4 0 14 -12 -14 C 0 -14 0 -8 -8 D 6 12 8 0 -6 E 4 14 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999295 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7613: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) E C D A B (8) E D A B C (7) E D C A B (6) B A C D E (6) C D E B A (5) C B D A E (5) C E D B A (4) C B E A D (4) B C A D E (4) E D A C B (3) E A D B C (3) E C D B A (2) D E C A B (2) D E A C B (2) C D E A B (2) C D B E A (2) C D B A E (2) C B A E D (2) B A D E C (2) B A D C E (2) B A C E D (2) A E D B C (2) A E B D C (2) A D B E C (2) A B E D C (2) E B A C D (1) D C B A E (1) D A E B C (1) C E D A B (1) C E B A D (1) C B E D A (1) B C E A D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -6 -8 -10 B -6 0 -6 -10 -4 C 6 6 0 6 -8 D 8 10 -6 0 -4 E 10 4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -6 -8 -10 B -6 0 -6 -10 -4 C 6 6 0 6 -8 D 8 10 -6 0 -4 E 10 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=29 A=18 B=17 D=6 so D is eliminated. Round 2 votes counts: E=34 C=30 A=19 B=17 so B is eliminated. Round 3 votes counts: C=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 C:205 D:204 A:191 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 -8 -10 B -6 0 -6 -10 -4 C 6 6 0 6 -8 D 8 10 -6 0 -4 E 10 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -8 -10 B -6 0 -6 -10 -4 C 6 6 0 6 -8 D 8 10 -6 0 -4 E 10 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -8 -10 B -6 0 -6 -10 -4 C 6 6 0 6 -8 D 8 10 -6 0 -4 E 10 4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7614: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (15) E C B A D (13) B C E D A (10) D A B C E (9) E A D C B (7) C B E D A (4) C B E A D (4) A D E B C (4) D A E B C (3) D A B E C (3) C E B A D (3) B D A C E (3) B C D A E (3) E C A D B (2) E B C D A (2) C B A D E (2) E B D A C (1) E A C D B (1) D A C B E (1) C E A D B (1) C B D A E (1) C A D E B (1) B E C D A (1) B D C A E (1) B D A E C (1) B C D E A (1) A E D C B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 0 2 12 0 B 0 0 -16 0 -10 C -2 16 0 -2 -8 D -12 0 2 0 0 E 0 10 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.337638 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.662362 Sum of squares = 0.552723132491 Cumulative probabilities = A: 0.337638 B: 0.337638 C: 0.337638 D: 0.337638 E: 1.000000 A B C D E A 0 0 2 12 0 B 0 0 -16 0 -10 C -2 16 0 -2 -8 D -12 0 2 0 0 E 0 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 B=20 D=16 C=16 so D is eliminated. Round 2 votes counts: A=38 E=26 B=20 C=16 so C is eliminated. Round 3 votes counts: A=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:207 C:202 D:195 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 12 0 B 0 0 -16 0 -10 C -2 16 0 -2 -8 D -12 0 2 0 0 E 0 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 12 0 B 0 0 -16 0 -10 C -2 16 0 -2 -8 D -12 0 2 0 0 E 0 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 12 0 B 0 0 -16 0 -10 C -2 16 0 -2 -8 D -12 0 2 0 0 E 0 10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7615: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) A E D B C (7) C A D E B (6) B E D A C (6) A E B D C (5) A C D E B (5) C D E A B (4) B D E C A (4) B C D E A (4) E A D B C (3) D C E B A (3) D B C E A (3) C D B E A (3) C B D A E (3) C A B D E (3) B D C E A (3) E D B A C (2) D E B C A (2) C D A E B (2) C A D B E (2) B E D C A (2) A E C D B (2) A C E D B (2) E D A C B (1) E D A B C (1) E B D A C (1) E A D C B (1) E A B D C (1) C D E B A (1) C B A D E (1) C A B E D (1) B C D A E (1) B C A E D (1) B A E D C (1) A E D C B (1) A D E C B (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -14 -8 -6 B -2 0 -4 -4 -4 C 14 4 0 2 16 D 8 4 -2 0 18 E 6 4 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 -8 -6 B -2 0 -4 -4 -4 C 14 4 0 2 16 D 8 4 -2 0 18 E 6 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992425 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=26 B=22 E=10 D=8 so D is eliminated. Round 2 votes counts: C=37 A=26 B=25 E=12 so E is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:214 B:193 E:188 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 -8 -6 B -2 0 -4 -4 -4 C 14 4 0 2 16 D 8 4 -2 0 18 E 6 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992425 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -8 -6 B -2 0 -4 -4 -4 C 14 4 0 2 16 D 8 4 -2 0 18 E 6 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992425 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -8 -6 B -2 0 -4 -4 -4 C 14 4 0 2 16 D 8 4 -2 0 18 E 6 4 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992425 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7616: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) B A E D C (8) B C E D A (7) B A D C E (7) A D B E C (5) A D B C E (5) C E B D A (4) C D E A B (4) A D E C B (4) A D C E B (4) A D C B E (4) E B C D A (3) E A D C B (3) D A C E B (3) B E C D A (3) B A D E C (3) E C D B A (2) E B C A D (2) E B A C D (2) D A B C E (2) B E A C D (2) B C D E A (2) A E D C B (2) E C D A B (1) E C B D A (1) E C A D B (1) E A B D C (1) D B C A E (1) D A E C B (1) C E D B A (1) C E D A B (1) B D C A E (1) B C E A D (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 14 26 2 B 16 0 28 12 20 C -14 -28 0 -10 -6 D -26 -12 10 0 -6 E -2 -20 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 14 26 2 B 16 0 28 12 20 C -14 -28 0 -10 -6 D -26 -12 10 0 -6 E -2 -20 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 A=25 E=16 C=10 D=7 so D is eliminated. Round 2 votes counts: B=43 A=31 E=16 C=10 so C is eliminated. Round 3 votes counts: B=43 A=31 E=26 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:238 A:213 E:195 D:183 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 14 26 2 B 16 0 28 12 20 C -14 -28 0 -10 -6 D -26 -12 10 0 -6 E -2 -20 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 14 26 2 B 16 0 28 12 20 C -14 -28 0 -10 -6 D -26 -12 10 0 -6 E -2 -20 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 14 26 2 B 16 0 28 12 20 C -14 -28 0 -10 -6 D -26 -12 10 0 -6 E -2 -20 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7617: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (14) D B A E C (9) D B C E A (7) D A B E C (6) D C E B A (4) D B A C E (4) C B E D A (4) A D B E C (4) E C A B D (3) C E B D A (3) C E B A D (3) B D C E A (3) A E C B D (3) E C A D B (2) C E A D B (2) C B E A D (2) B D C A E (2) B A D E C (2) A E C D B (2) A E B C D (2) A D E B C (2) E D A C B (1) E A C D B (1) D C E A B (1) D C B E A (1) D B C A E (1) C E D B A (1) C E D A B (1) C D E B A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D E A (1) B A E C D (1) A E B D C (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -16 -6 -14 B 6 0 0 -2 4 C 16 0 0 -4 16 D 6 2 4 0 2 E 14 -4 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -6 -14 B 6 0 0 -2 4 C 16 0 0 -4 16 D 6 2 4 0 2 E 14 -4 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=31 A=16 B=13 E=7 so E is eliminated. Round 2 votes counts: C=36 D=34 A=17 B=13 so B is eliminated. Round 3 votes counts: D=41 C=39 A=20 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:207 B:204 E:196 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -16 -6 -14 B 6 0 0 -2 4 C 16 0 0 -4 16 D 6 2 4 0 2 E 14 -4 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -6 -14 B 6 0 0 -2 4 C 16 0 0 -4 16 D 6 2 4 0 2 E 14 -4 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -6 -14 B 6 0 0 -2 4 C 16 0 0 -4 16 D 6 2 4 0 2 E 14 -4 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7618: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (27) E B C D A (18) E A B C D (11) E B C A D (8) D C B A E (6) B C D E A (5) A E D C B (5) C D B A E (4) C B D A E (4) E B A C D (2) D A C B E (2) A D C E B (2) E A D B C (1) B C E D A (1) B C D A E (1) A E D B C (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 18 8 B -2 0 -2 2 2 C -6 2 0 10 6 D -18 -2 -10 0 6 E -8 -2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 18 8 B -2 0 -2 2 2 C -6 2 0 10 6 D -18 -2 -10 0 6 E -8 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979817 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=37 D=8 C=8 B=7 so B is eliminated. Round 2 votes counts: E=40 A=37 C=15 D=8 so D is eliminated. Round 3 votes counts: E=40 A=39 C=21 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:206 B:200 E:189 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 18 8 B -2 0 -2 2 2 C -6 2 0 10 6 D -18 -2 -10 0 6 E -8 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979817 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 18 8 B -2 0 -2 2 2 C -6 2 0 10 6 D -18 -2 -10 0 6 E -8 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979817 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 18 8 B -2 0 -2 2 2 C -6 2 0 10 6 D -18 -2 -10 0 6 E -8 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999979817 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7619: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) D E C B A (8) D E B A C (8) B A E D C (7) D E B C A (5) C E B D A (5) B A E C D (5) C D E A B (4) C A B E D (4) C E B A D (3) C A D B E (3) A B E D C (3) A B D E C (3) A B C D E (3) D A B E C (2) C A D E B (2) A D B E C (2) A C B E D (2) E D B A C (1) E C D B A (1) E C B D A (1) E C B A D (1) D C E B A (1) D C A E B (1) D B E A C (1) D A C E B (1) C E D B A (1) C E D A B (1) C D E B A (1) C D A E B (1) C B E A D (1) C A E B D (1) C A B D E (1) B E D A C (1) B E A C D (1) B D A E C (1) A D C B E (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 8 10 10 B 6 0 8 8 6 C -8 -8 0 4 -2 D -10 -8 -4 0 4 E -10 -6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 10 10 B 6 0 8 8 6 C -8 -8 0 4 -2 D -10 -8 -4 0 4 E -10 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=27 A=26 B=15 E=4 so E is eliminated. Round 2 votes counts: C=31 D=28 A=26 B=15 so B is eliminated. Round 3 votes counts: A=39 C=31 D=30 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:214 A:211 C:193 D:191 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 10 10 B 6 0 8 8 6 C -8 -8 0 4 -2 D -10 -8 -4 0 4 E -10 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 10 10 B 6 0 8 8 6 C -8 -8 0 4 -2 D -10 -8 -4 0 4 E -10 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 10 10 B 6 0 8 8 6 C -8 -8 0 4 -2 D -10 -8 -4 0 4 E -10 -6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7620: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) E C B D A (7) C E D B A (7) D A B C E (6) A D B C E (6) C E D A B (5) B E C A D (5) D C E B A (4) D C E A B (4) C D E A B (4) B E A C D (4) A B D C E (4) E C D B A (3) E C B A D (3) D C A E B (3) B E C D A (3) B A E C D (3) D A C E B (2) B A E D C (2) A B D E C (2) E C A D B (1) E C A B D (1) E B C A D (1) E A B C D (1) D B A C E (1) D A C B E (1) C D E B A (1) B E D C A (1) B E A D C (1) B D E A C (1) B D A E C (1) B D A C E (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 -6 -10 -14 B 16 0 4 0 2 C 6 -4 0 0 2 D 10 0 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.508777 C: 0.000000 D: 0.491223 E: 0.000000 Sum of squares = 0.500154067932 Cumulative probabilities = A: 0.000000 B: 0.508777 C: 0.508777 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -10 -14 B 16 0 4 0 2 C 6 -4 0 0 2 D 10 0 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=21 E=17 C=17 A=14 so A is eliminated. Round 2 votes counts: B=37 D=28 C=18 E=17 so E is eliminated. Round 3 votes counts: B=39 C=33 D=28 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:206 E:204 C:202 A:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -10 -14 B 16 0 4 0 2 C 6 -4 0 0 2 D 10 0 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -10 -14 B 16 0 4 0 2 C 6 -4 0 0 2 D 10 0 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -10 -14 B 16 0 4 0 2 C 6 -4 0 0 2 D 10 0 0 0 2 E 14 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7621: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) E B D C A (6) B C A D E (6) A C D B E (6) E D B C A (5) A D C E B (5) E B A C D (4) A B C D E (4) E D A C B (3) E A D B C (3) D E A C B (3) D C A B E (3) C D B A E (3) C B D E A (3) B C E D A (3) A E D B C (3) A C B D E (3) E D C A B (2) E A D C B (2) C B D A E (2) B C E A D (2) B C A E D (2) A E D C B (2) E B C D A (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E B A (1) D C B A E (1) D C A E B (1) D A C E B (1) C B A D E (1) C A D B E (1) B E C A D (1) B E A C D (1) B C D E A (1) B C D A E (1) B A C E D (1) A E B D C (1) A D E C B (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -8 4 0 B 4 0 -6 -10 -6 C 8 6 0 -4 4 D -4 10 4 0 0 E 0 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.375 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 4 0 B 4 0 -6 -10 -6 C 8 6 0 -4 4 D -4 10 4 0 0 E 0 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999834 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=27 B=18 D=12 C=10 so C is eliminated. Round 2 votes counts: E=33 A=28 B=24 D=15 so D is eliminated. Round 3 votes counts: E=39 A=33 B=28 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:207 D:205 E:201 A:196 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 4 0 B 4 0 -6 -10 -6 C 8 6 0 -4 4 D -4 10 4 0 0 E 0 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999834 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 4 0 B 4 0 -6 -10 -6 C 8 6 0 -4 4 D -4 10 4 0 0 E 0 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999834 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 4 0 B 4 0 -6 -10 -6 C 8 6 0 -4 4 D -4 10 4 0 0 E 0 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999834 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7622: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) B D A E C (8) C D E B A (7) E A C D B (6) C E D A B (5) E C A D B (4) C E B D A (4) B A D E C (4) D B A E C (3) D B A C E (3) B D C E A (3) B D C A E (3) A E D B C (3) A B D E C (3) E A C B D (2) D C E A B (2) D A E C B (2) C E D B A (2) C E A D B (2) C B E D A (2) A E D C B (2) A E C D B (2) A E C B D (2) A D B E C (2) E C A B D (1) D C E B A (1) D C B E A (1) D A B E C (1) C E B A D (1) C E A B D (1) C D B E A (1) B E C A D (1) B E A C D (1) B C E A D (1) B A E C D (1) B A D C E (1) A E B D C (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 -14 16 -14 4 B 14 0 -2 0 -4 C -16 2 0 -6 -2 D 14 0 6 0 10 E -4 4 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.420869 C: 0.000000 D: 0.579131 E: 0.000000 Sum of squares = 0.51252332205 Cumulative probabilities = A: 0.000000 B: 0.420869 C: 0.420869 D: 1.000000 E: 1.000000 A B C D E A 0 -14 16 -14 4 B 14 0 -2 0 -4 C -16 2 0 -6 -2 D 14 0 6 0 10 E -4 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=25 A=17 E=13 D=13 so E is eliminated. Round 2 votes counts: B=32 C=30 A=25 D=13 so D is eliminated. Round 3 votes counts: B=38 C=34 A=28 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:215 B:204 A:196 E:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 16 -14 4 B 14 0 -2 0 -4 C -16 2 0 -6 -2 D 14 0 6 0 10 E -4 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 16 -14 4 B 14 0 -2 0 -4 C -16 2 0 -6 -2 D 14 0 6 0 10 E -4 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 16 -14 4 B 14 0 -2 0 -4 C -16 2 0 -6 -2 D 14 0 6 0 10 E -4 4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7623: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) E C D B A (7) E C D A B (6) D C E B A (6) C E D B A (4) A B D C E (4) E A C D B (3) E A C B D (3) E C B D A (2) E C A D B (2) D E C A B (2) D C E A B (2) D C B E A (2) D C A B E (2) D A E C B (2) D A C B E (2) D A B C E (2) C E B D A (2) C D E B A (2) C B E D A (2) C B D E A (2) B D C A E (2) B D A C E (2) B C E D A (2) B A E C D (2) B A D C E (2) B A C D E (2) A D E C B (2) A D B E C (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C A D (1) E A B C D (1) D C A E B (1) D B C E A (1) D B C A E (1) D B A C E (1) C D B E A (1) B C D E A (1) B C A E D (1) B A C E D (1) A E D B C (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -12 -18 -12 B 0 0 -20 -8 -4 C 12 20 0 16 0 D 18 8 -16 0 -4 E 12 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.381085 D: 0.000000 E: 0.618915 Sum of squares = 0.528281455118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.381085 D: 0.381085 E: 1.000000 A B C D E A 0 0 -12 -18 -12 B 0 0 -20 -8 -4 C 12 20 0 16 0 D 18 8 -16 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 A=21 B=15 C=13 so C is eliminated. Round 2 votes counts: E=33 D=27 A=21 B=19 so B is eliminated. Round 3 votes counts: E=37 D=34 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:224 E:210 D:203 B:184 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 -18 -12 B 0 0 -20 -8 -4 C 12 20 0 16 0 D 18 8 -16 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -18 -12 B 0 0 -20 -8 -4 C 12 20 0 16 0 D 18 8 -16 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -18 -12 B 0 0 -20 -8 -4 C 12 20 0 16 0 D 18 8 -16 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7624: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) D E A C B (6) B C E A D (6) E D B C A (5) B C A E D (5) A C B D E (5) C A B D E (4) A D E C B (4) A C D B E (4) E D B A C (3) E B C D A (3) D E C B A (3) D A E C B (3) C B A E D (3) C B A D E (3) B A C E D (3) A B C E D (3) E A B D C (2) D E A B C (2) D C E B A (2) C D B A E (2) C D A B E (2) C B E D A (2) B C E D A (2) A D C B E (2) A B E C D (2) E D C B A (1) E B D C A (1) E B D A C (1) D E C A B (1) D C E A B (1) D C A B E (1) C D B E A (1) A E D B C (1) A D E B C (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 4 0 0 B -8 0 -2 -8 2 C -4 2 0 2 6 D 0 8 -2 0 -2 E 0 -2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.695050 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.304950 Sum of squares = 0.576089319901 Cumulative probabilities = A: 0.695050 B: 0.695050 C: 0.695050 D: 0.695050 E: 1.000000 A B C D E A 0 8 4 0 0 B -8 0 -2 -8 2 C -4 2 0 2 6 D 0 8 -2 0 -2 E 0 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000006734 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 D=19 C=17 B=16 so B is eliminated. Round 2 votes counts: C=30 A=27 E=24 D=19 so D is eliminated. Round 3 votes counts: E=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:206 C:203 D:202 E:197 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 0 0 B -8 0 -2 -8 2 C -4 2 0 2 6 D 0 8 -2 0 -2 E 0 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000006734 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 0 0 B -8 0 -2 -8 2 C -4 2 0 2 6 D 0 8 -2 0 -2 E 0 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000006734 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 0 0 B -8 0 -2 -8 2 C -4 2 0 2 6 D 0 8 -2 0 -2 E 0 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000006734 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7625: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (10) B A E D C (10) C D E A B (6) C E D B A (5) B E A C D (5) E B A C D (4) E A B C D (4) C E D A B (4) C D E B A (4) E C B D A (3) E C B A D (3) E C A B D (3) D B C A E (3) D B A C E (3) D A B C E (3) B A D E C (3) A D B C E (3) A B E D C (3) E A C B D (2) D C E B A (2) D C A E B (2) B D A C E (2) A D C B E (2) E C A D B (1) E B C A D (1) D C E A B (1) D C B E A (1) D C B A E (1) B E A D C (1) B A E C D (1) A E C B D (1) A E B C D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 0 0 0 B 4 0 -2 -2 6 C 0 2 0 -2 6 D 0 2 2 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.109938 B: 0.000000 C: 0.381455 D: 0.381455 E: 0.127152 Sum of squares = 0.319269779751 Cumulative probabilities = A: 0.109938 B: 0.109938 C: 0.491393 D: 0.872848 E: 1.000000 A B C D E A 0 -4 0 0 0 B 4 0 -2 -2 6 C 0 2 0 -2 6 D 0 2 2 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.176470 B: 0.000000 C: 0.352941 D: 0.352941 E: 0.117647 Sum of squares = 0.294117698369 Cumulative probabilities = A: 0.176470 B: 0.176470 C: 0.529412 D: 0.882353 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=22 E=21 C=19 A=12 so A is eliminated. Round 2 votes counts: D=31 B=27 E=23 C=19 so C is eliminated. Round 3 votes counts: D=41 E=32 B=27 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:203 C:203 D:199 A:198 E:197 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 0 0 B 4 0 -2 -2 6 C 0 2 0 -2 6 D 0 2 2 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.176470 B: 0.000000 C: 0.352941 D: 0.352941 E: 0.117647 Sum of squares = 0.294117698369 Cumulative probabilities = A: 0.176470 B: 0.176470 C: 0.529412 D: 0.882353 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 0 0 B 4 0 -2 -2 6 C 0 2 0 -2 6 D 0 2 2 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.176470 B: 0.000000 C: 0.352941 D: 0.352941 E: 0.117647 Sum of squares = 0.294117698369 Cumulative probabilities = A: 0.176470 B: 0.176470 C: 0.529412 D: 0.882353 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 0 0 B 4 0 -2 -2 6 C 0 2 0 -2 6 D 0 2 2 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.176470 B: 0.000000 C: 0.352941 D: 0.352941 E: 0.117647 Sum of squares = 0.294117698369 Cumulative probabilities = A: 0.176470 B: 0.176470 C: 0.529412 D: 0.882353 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7626: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) D C B E A (5) C D B E A (5) C D B A E (5) B A E D C (4) A E B D C (4) E A D C B (3) E A D B C (3) E A B D C (3) D B C E A (3) C E D A B (3) B A C D E (3) A E C B D (3) A E B C D (3) E C A D B (2) E A C D B (2) D B E C A (2) D B E A C (2) D B C A E (2) C D E B A (2) C D E A B (2) C A B E D (2) B D A E C (2) B D A C E (2) B C D A E (2) B A D E C (2) B A D C E (2) E D C A B (1) E C D A B (1) E B A D C (1) E A B C D (1) D E C A B (1) D E B C A (1) D C E B A (1) D C B A E (1) C A E D B (1) C A B D E (1) B E A D C (1) B C A D E (1) B A C E D (1) A E C D B (1) A C E B D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -6 -4 10 B 18 0 14 2 20 C 6 -14 0 -12 10 D 4 -2 12 0 12 E -10 -20 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -4 10 B 18 0 14 2 20 C 6 -14 0 -12 10 D 4 -2 12 0 12 E -10 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978608 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=21 D=18 E=17 A=15 so A is eliminated. Round 2 votes counts: B=32 E=28 C=22 D=18 so D is eliminated. Round 3 votes counts: B=41 E=30 C=29 so C is eliminated. Round 4 votes counts: B=60 E=40 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:227 D:213 C:195 A:191 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -6 -4 10 B 18 0 14 2 20 C 6 -14 0 -12 10 D 4 -2 12 0 12 E -10 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978608 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -4 10 B 18 0 14 2 20 C 6 -14 0 -12 10 D 4 -2 12 0 12 E -10 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978608 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -4 10 B 18 0 14 2 20 C 6 -14 0 -12 10 D 4 -2 12 0 12 E -10 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978608 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7627: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) E C D B A (7) A B D C E (7) E C D A B (6) E D C B A (5) A B D E C (5) A B C D E (5) E D C A B (4) D E A B C (3) C E A B D (3) B A D C E (3) B A C D E (3) E A D B C (2) D B A E C (2) D A B E C (2) C E B D A (2) C B D E A (2) C B A E D (2) B D C A E (2) A D E B C (2) A B E D C (2) A B C E D (2) E C A D B (1) E A C D B (1) D C E B A (1) D C B E A (1) D B C E A (1) D B C A E (1) C B E A D (1) C B D A E (1) C B A D E (1) C A B E D (1) B D A C E (1) B C D A E (1) B C A D E (1) A E D B C (1) A E C B D (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -12 -8 -4 B 0 0 -6 -6 -4 C 12 6 0 8 10 D 8 6 -8 0 -8 E 4 4 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999558 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -8 -4 B 0 0 -6 -6 -4 C 12 6 0 8 10 D 8 6 -8 0 -8 E 4 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=26 C=25 D=11 B=11 so D is eliminated. Round 2 votes counts: E=29 A=29 C=27 B=15 so B is eliminated. Round 3 votes counts: A=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:203 D:199 B:192 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 -8 -4 B 0 0 -6 -6 -4 C 12 6 0 8 10 D 8 6 -8 0 -8 E 4 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -8 -4 B 0 0 -6 -6 -4 C 12 6 0 8 10 D 8 6 -8 0 -8 E 4 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -8 -4 B 0 0 -6 -6 -4 C 12 6 0 8 10 D 8 6 -8 0 -8 E 4 4 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7628: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (15) E A D C B (11) C B E D A (10) A D E B C (10) E C B A D (7) C B E A D (6) C B D E A (5) E C A B D (4) D A E B C (4) E A C D B (3) D A B E C (3) D A B C E (3) B D C A E (3) A E D B C (3) E A C B D (2) D B C A E (2) D B A C E (2) A E D C B (2) E D A B C (1) D C B A E (1) C E B A D (1) C B D A E (1) B C D E A (1) Total count = 100 A B C D E A 0 -8 -12 -2 -2 B 8 0 -6 10 4 C 12 6 0 10 0 D 2 -10 -10 0 0 E 2 -4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.580833 D: 0.000000 E: 0.419167 Sum of squares = 0.513067803015 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.580833 D: 0.580833 E: 1.000000 A B C D E A 0 -8 -12 -2 -2 B 8 0 -6 10 4 C 12 6 0 10 0 D 2 -10 -10 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 B=19 D=15 A=15 so D is eliminated. Round 2 votes counts: E=28 A=25 C=24 B=23 so B is eliminated. Round 3 votes counts: C=45 E=28 A=27 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:208 E:199 D:191 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -2 -2 B 8 0 -6 10 4 C 12 6 0 10 0 D 2 -10 -10 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -2 -2 B 8 0 -6 10 4 C 12 6 0 10 0 D 2 -10 -10 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -2 -2 B 8 0 -6 10 4 C 12 6 0 10 0 D 2 -10 -10 0 0 E 2 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7629: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) E D B C A (6) B A C E D (6) A E D C B (6) E D A C B (5) B C A D E (5) A E D B C (5) C B A D E (4) B C D A E (4) A B C E D (4) E D A B C (3) E A D B C (3) D C E B A (3) B C D E A (3) A C B D E (3) A B C D E (3) D E C A B (2) D E A C B (2) D C E A B (2) D C B E A (2) D A E C B (2) C B D E A (2) B C E A D (2) A E B C D (2) A B E C D (2) E D C B A (1) E D B A C (1) E B D C A (1) E B C D A (1) E A D C B (1) C D B E A (1) C D A B E (1) C B D A E (1) B E D C A (1) B C A E D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 0 -2 2 B 4 0 6 -10 -10 C 0 -6 0 -10 -4 D 2 10 10 0 -2 E -2 10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 -4 0 -2 2 B 4 0 6 -10 -10 C 0 -6 0 -10 -4 D 2 10 10 0 -2 E -2 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=22 B=22 D=20 C=9 so C is eliminated. Round 2 votes counts: B=29 A=27 E=22 D=22 so E is eliminated. Round 3 votes counts: D=38 B=31 A=31 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:207 A:198 B:195 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -2 2 B 4 0 6 -10 -10 C 0 -6 0 -10 -4 D 2 10 10 0 -2 E -2 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -2 2 B 4 0 6 -10 -10 C 0 -6 0 -10 -4 D 2 10 10 0 -2 E -2 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -2 2 B 4 0 6 -10 -10 C 0 -6 0 -10 -4 D 2 10 10 0 -2 E -2 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7630: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (16) E D A B C (11) C B A E D (11) A B C D E (10) E D C B A (8) C B A D E (8) D A B E C (6) B A C D E (6) D A E B C (5) E C B A D (3) E D A C B (2) E C D B A (2) D A B C E (2) C B E A D (2) B C A D E (2) A B D C E (2) E D C A B (1) D E A C B (1) C E B A D (1) C A B D E (1) Total count = 100 A B C D E A 0 14 22 -8 6 B -14 0 20 -8 0 C -22 -20 0 -8 -10 D 8 8 8 0 18 E -6 0 10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 22 -8 6 B -14 0 20 -8 0 C -22 -20 0 -8 -10 D 8 8 8 0 18 E -6 0 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 C=23 A=12 B=8 so B is eliminated. Round 2 votes counts: D=30 E=27 C=25 A=18 so A is eliminated. Round 3 votes counts: C=41 D=32 E=27 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:217 B:199 E:193 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 22 -8 6 B -14 0 20 -8 0 C -22 -20 0 -8 -10 D 8 8 8 0 18 E -6 0 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 22 -8 6 B -14 0 20 -8 0 C -22 -20 0 -8 -10 D 8 8 8 0 18 E -6 0 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 22 -8 6 B -14 0 20 -8 0 C -22 -20 0 -8 -10 D 8 8 8 0 18 E -6 0 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7631: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) E D C B A (6) A B C D E (6) D E A C B (5) A D B C E (5) A C B D E (5) E B C D A (4) B C E A D (4) B A C E D (4) A C D B E (4) C A B D E (3) A D C B E (3) E C B D A (2) E B D C A (2) E B C A D (2) D E C A B (2) D C E A B (2) D A E C B (2) D A C B E (2) C E D B A (2) C E B A D (2) C A B E D (2) B E A D C (2) A B D C E (2) E D C A B (1) E D B A C (1) E C D B A (1) E B D A C (1) D E B A C (1) D E A B C (1) D A C E B (1) C E D A B (1) C E B D A (1) C B A E D (1) C A E B D (1) B E D A C (1) B E C A D (1) B C A E D (1) B A E C D (1) B A D E C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 0 4 -8 B 0 0 2 0 0 C 0 -2 0 -4 4 D -4 0 4 0 -6 E 8 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.758332 C: 0.000000 D: 0.000000 E: 0.241668 Sum of squares = 0.633470451195 Cumulative probabilities = A: 0.000000 B: 0.758332 C: 0.758332 D: 0.758332 E: 1.000000 A B C D E A 0 0 0 4 -8 B 0 0 2 0 0 C 0 -2 0 -4 4 D -4 0 4 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555566794 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 D=16 B=15 C=13 so C is eliminated. Round 2 votes counts: E=35 A=33 D=16 B=16 so D is eliminated. Round 3 votes counts: E=46 A=38 B=16 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:205 B:201 C:199 A:198 D:197 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 4 -8 B 0 0 2 0 0 C 0 -2 0 -4 4 D -4 0 4 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555566794 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 -8 B 0 0 2 0 0 C 0 -2 0 -4 4 D -4 0 4 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555566794 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 -8 B 0 0 2 0 0 C 0 -2 0 -4 4 D -4 0 4 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.55555566794 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7632: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) D E C A B (5) C D B A E (5) B E A D C (5) A B E C D (5) E A B D C (4) D E B A C (4) A E B C D (4) A C E B D (4) E D B A C (3) E A D B C (3) D E C B A (3) D E B C A (3) D C E B A (3) C D A B E (3) B C A D E (3) A E C D B (3) A C B E D (3) A B C E D (3) E D A C B (2) E A D C B (2) D B E C A (2) C A D B E (2) C A B D E (2) B C A E D (2) B A E C D (2) E D A B C (1) E A C D B (1) D C E A B (1) D C B E A (1) D B C E A (1) C B A D E (1) C A D E B (1) B D C E A (1) B D C A E (1) B C D A E (1) B A C E D (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 2 4 6 10 B -2 0 -2 0 2 C -4 2 0 10 -6 D -6 0 -10 0 0 E -10 -2 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 6 10 B -2 0 -2 0 2 C -4 2 0 10 -6 D -6 0 -10 0 0 E -10 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 C=21 E=16 B=16 so E is eliminated. Round 2 votes counts: A=34 D=29 C=21 B=16 so B is eliminated. Round 3 votes counts: A=42 D=31 C=27 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:201 B:199 E:197 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 6 10 B -2 0 -2 0 2 C -4 2 0 10 -6 D -6 0 -10 0 0 E -10 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 6 10 B -2 0 -2 0 2 C -4 2 0 10 -6 D -6 0 -10 0 0 E -10 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 6 10 B -2 0 -2 0 2 C -4 2 0 10 -6 D -6 0 -10 0 0 E -10 -2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7633: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (13) C A B E D (10) E D A C B (9) E A C D B (9) E A D C B (6) D B E C A (5) B D E C A (5) B C A E D (5) B D C A E (4) A E C D B (4) E D B A C (3) C A E B D (3) B D C E A (3) B C D A E (3) B C A D E (3) A C E B D (3) D E A C B (2) A C E D B (2) E B D A C (1) D E A B C (1) D B C A E (1) C B D A E (1) B E D C A (1) B E C A D (1) B D E A C (1) B C E A D (1) Total count = 100 A B C D E A 0 -2 8 -6 -22 B 2 0 2 -10 -12 C -8 -2 0 -10 -22 D 6 10 10 0 -16 E 22 12 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 -6 -22 B 2 0 2 -10 -12 C -8 -2 0 -10 -22 D 6 10 10 0 -16 E 22 12 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 D=22 C=14 A=9 so A is eliminated. Round 2 votes counts: E=32 B=27 D=22 C=19 so C is eliminated. Round 3 votes counts: E=40 B=38 D=22 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:236 D:205 B:191 A:189 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 -6 -22 B 2 0 2 -10 -12 C -8 -2 0 -10 -22 D 6 10 10 0 -16 E 22 12 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -6 -22 B 2 0 2 -10 -12 C -8 -2 0 -10 -22 D 6 10 10 0 -16 E 22 12 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -6 -22 B 2 0 2 -10 -12 C -8 -2 0 -10 -22 D 6 10 10 0 -16 E 22 12 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7634: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (14) B D A E C (10) E C A D B (9) C E A B D (7) D B A E C (6) B D A C E (6) C E B A D (5) C B D E A (4) A D B E C (4) D A B E C (3) B C D A E (3) E C B D A (2) E A C D B (2) D B A C E (2) C A D E B (2) B E D A C (2) B D C A E (2) B C D E A (2) A E D C B (2) E C A B D (1) E B D A C (1) E B C D A (1) E A D C B (1) E A D B C (1) D A B C E (1) C E B D A (1) C B E D A (1) C A E D B (1) B E C D A (1) A E D B C (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -12 4 -10 B -2 0 -6 -2 -6 C 12 6 0 14 4 D -4 2 -14 0 -8 E 10 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 4 -10 B -2 0 -6 -2 -6 C 12 6 0 14 4 D -4 2 -14 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=26 E=18 D=12 A=9 so A is eliminated. Round 2 votes counts: C=36 B=26 E=21 D=17 so D is eliminated. Round 3 votes counts: B=42 C=36 E=22 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:210 A:192 B:192 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 4 -10 B -2 0 -6 -2 -6 C 12 6 0 14 4 D -4 2 -14 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 4 -10 B -2 0 -6 -2 -6 C 12 6 0 14 4 D -4 2 -14 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 4 -10 B -2 0 -6 -2 -6 C 12 6 0 14 4 D -4 2 -14 0 -8 E 10 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7635: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) D B C E A (8) A E C B D (7) E B C A D (6) D C B E A (6) A D C E B (6) D A C B E (5) D C A B E (4) B E D C A (4) B E C D A (4) A C D E B (4) D C B A E (3) D A C E B (3) A E B C D (3) A C E D B (3) E B A C D (2) D B C A E (2) C E B A D (2) B D E C A (2) A E C D B (2) E B A D C (1) D C A E B (1) D B E A C (1) D B A E C (1) C E B D A (1) C D B A E (1) C A D E B (1) B E D A C (1) B E C A D (1) B E A D C (1) B E A C D (1) B C D E A (1) A E B D C (1) A D C B E (1) Total count = 100 A B C D E A 0 2 6 4 -4 B -2 0 0 -4 -6 C -6 0 0 -2 4 D -4 4 2 0 0 E 4 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.34693877514 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 2 6 4 -4 B -2 0 0 -4 -6 C -6 0 0 -2 4 D -4 4 2 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.34693877543 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=27 E=19 B=15 C=5 so C is eliminated. Round 2 votes counts: D=35 A=28 E=22 B=15 so B is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:204 E:203 D:201 C:198 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 -4 B -2 0 0 -4 -6 C -6 0 0 -2 4 D -4 4 2 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.34693877543 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 -4 B -2 0 0 -4 -6 C -6 0 0 -2 4 D -4 4 2 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.34693877543 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 -4 B -2 0 0 -4 -6 C -6 0 0 -2 4 D -4 4 2 0 0 E 4 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.428571 Sum of squares = 0.34693877543 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7636: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) A E B D C (8) B D A C E (6) E A C B D (5) E A B D C (5) E A B C D (5) D C B A E (5) E C A B D (4) C D E B A (4) C D E A B (4) B A E D C (4) A B D E C (4) D B A E C (3) B A D E C (3) E C A D B (2) D B C A E (2) D B A C E (2) D A E C B (2) C E D B A (2) C E D A B (2) C E B A D (2) C E A D B (2) C D B E A (2) C B D A E (2) B D C A E (2) E C B A D (1) E B A C D (1) E A D C B (1) E A C D B (1) D A E B C (1) C D A B E (1) B E A C D (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 8 2 12 B 2 0 -2 8 -6 C -8 2 0 0 -6 D -2 -8 0 0 4 E -12 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.459999999919 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 A B C D E A 0 -2 8 2 12 B 2 0 -2 8 -6 C -8 2 0 0 -6 D -2 -8 0 0 4 E -12 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.459999999988 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=25 B=16 D=15 A=14 so A is eliminated. Round 2 votes counts: E=34 C=30 B=21 D=15 so D is eliminated. Round 3 votes counts: E=37 C=35 B=28 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:210 B:201 E:198 D:197 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 2 12 B 2 0 -2 8 -6 C -8 2 0 0 -6 D -2 -8 0 0 4 E -12 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.459999999988 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 2 12 B 2 0 -2 8 -6 C -8 2 0 0 -6 D -2 -8 0 0 4 E -12 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.459999999988 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 2 12 B 2 0 -2 8 -6 C -8 2 0 0 -6 D -2 -8 0 0 4 E -12 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.100000 Sum of squares = 0.459999999988 Cumulative probabilities = A: 0.300000 B: 0.900000 C: 0.900000 D: 0.900000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7637: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (14) E B D A C (10) C D A E B (10) C B E A D (8) B E C A D (7) C A D B E (5) A D E B C (5) D A E B C (4) C E B D A (3) C D A B E (3) B E A C D (3) D E B A C (2) D C A E B (2) D A C E B (2) C D E B A (2) C B E D A (2) A D C B E (2) A D B E C (2) A C D B E (2) E D B A C (1) E B D C A (1) E B C D A (1) E B A D C (1) D E A B C (1) D A E C B (1) C A B E D (1) B E D A C (1) B E C D A (1) A D E C B (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 8 6 -16 B 14 0 10 8 4 C -8 -10 0 -4 -14 D -6 -8 4 0 -10 E 16 -4 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 6 -16 B 14 0 10 8 4 C -8 -10 0 -4 -14 D -6 -8 4 0 -10 E 16 -4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=26 E=14 A=14 D=12 so D is eliminated. Round 2 votes counts: C=36 B=26 A=21 E=17 so E is eliminated. Round 3 votes counts: B=42 C=36 A=22 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:218 A:192 D:190 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 6 -16 B 14 0 10 8 4 C -8 -10 0 -4 -14 D -6 -8 4 0 -10 E 16 -4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 6 -16 B 14 0 10 8 4 C -8 -10 0 -4 -14 D -6 -8 4 0 -10 E 16 -4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 6 -16 B 14 0 10 8 4 C -8 -10 0 -4 -14 D -6 -8 4 0 -10 E 16 -4 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7638: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (7) C A D E B (6) A E C B D (5) B E D A C (4) A C E D B (4) E B C D A (3) E B C A D (3) D C B E A (3) D C A E B (3) D C A B E (3) D B A C E (3) C A E D B (3) B D E C A (3) B D A C E (3) B A E D C (3) A B D C E (3) E C D B A (2) E C A D B (2) E C A B D (2) D C B A E (2) D B C A E (2) B E D C A (2) B E A D C (2) B E A C D (2) B D C E A (2) B A D C E (2) A C D E B (2) A B E C D (2) E D C B A (1) E C B A D (1) E B D C A (1) E B A C D (1) E A C D B (1) E A B C D (1) D C E B A (1) D B E C A (1) D A C B E (1) C E D B A (1) C E A D B (1) B E C D A (1) B D E A C (1) B A D E C (1) A E B C D (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 2 14 16 B 2 0 -4 -2 8 C -2 4 0 -12 8 D -14 2 12 0 2 E -16 -8 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000183 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 14 16 B 2 0 -4 -2 8 C -2 4 0 -12 8 D -14 2 12 0 2 E -16 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 D=19 E=18 C=11 so C is eliminated. Round 2 votes counts: A=35 B=26 E=20 D=19 so D is eliminated. Round 3 votes counts: A=42 B=37 E=21 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:202 D:201 C:199 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 14 16 B 2 0 -4 -2 8 C -2 4 0 -12 8 D -14 2 12 0 2 E -16 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 14 16 B 2 0 -4 -2 8 C -2 4 0 -12 8 D -14 2 12 0 2 E -16 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 14 16 B 2 0 -4 -2 8 C -2 4 0 -12 8 D -14 2 12 0 2 E -16 -8 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7639: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) C E B A D (7) B A D E C (7) D E B A C (6) D B A E C (6) D A B E C (6) A B D C E (6) E D B A C (5) C E D B A (5) C A B E D (5) E C D B A (4) E B A D C (3) C A B D E (3) A B D E C (3) E B A C D (2) C E D A B (2) C E B D A (2) C E A B D (2) C D A E B (2) C B A E D (2) B A E D C (2) E C B D A (1) D E A C B (1) D E A B C (1) D A C B E (1) D A B C E (1) C E A D B (1) C D A B E (1) C A E B D (1) B E A D C (1) B A C E D (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 24 12 14 B 8 0 18 14 10 C -24 -18 0 0 2 D -12 -14 0 0 8 E -14 -10 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999647 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 24 12 14 B 8 0 18 14 10 C -24 -18 0 0 2 D -12 -14 0 0 8 E -14 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 A=19 E=15 B=11 so B is eliminated. Round 2 votes counts: C=33 A=29 D=22 E=16 so E is eliminated. Round 3 votes counts: C=38 A=35 D=27 so D is eliminated. Round 4 votes counts: A=62 C=38 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:225 A:221 D:191 E:183 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 24 12 14 B 8 0 18 14 10 C -24 -18 0 0 2 D -12 -14 0 0 8 E -14 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 24 12 14 B 8 0 18 14 10 C -24 -18 0 0 2 D -12 -14 0 0 8 E -14 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 24 12 14 B 8 0 18 14 10 C -24 -18 0 0 2 D -12 -14 0 0 8 E -14 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7640: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) A D B C E (10) A D C B E (7) C E A D B (6) B A D C E (6) E D A B C (5) E C D A B (5) C E B A D (5) C B E A D (5) E B C D A (4) D A E B C (4) B E D A C (3) B D A E C (3) B C E A D (3) E C B D A (2) E C B A D (2) D E A B C (2) D A E C B (2) D A B C E (2) C A D E B (2) C A B D E (2) E D C A B (1) E D B A C (1) E D A C B (1) E B D A C (1) D E A C B (1) C B A D E (1) C A D B E (1) B C A D E (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 24 20 6 6 B -24 0 14 -20 12 C -20 -14 0 -20 6 D -6 20 20 0 12 E -6 -12 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 20 6 6 B -24 0 14 -20 12 C -20 -14 0 -20 6 D -6 20 20 0 12 E -6 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=22 C=22 D=21 A=18 B=17 so B is eliminated. Round 2 votes counts: C=26 E=25 A=25 D=24 so D is eliminated. Round 3 votes counts: A=46 E=28 C=26 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:228 D:223 B:191 E:182 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 20 6 6 B -24 0 14 -20 12 C -20 -14 0 -20 6 D -6 20 20 0 12 E -6 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 20 6 6 B -24 0 14 -20 12 C -20 -14 0 -20 6 D -6 20 20 0 12 E -6 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 20 6 6 B -24 0 14 -20 12 C -20 -14 0 -20 6 D -6 20 20 0 12 E -6 -12 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999536 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7641: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) E D C A B (6) D A B C E (5) C E D B A (5) C B E A D (5) E C D A B (4) C E B D A (4) B C E A D (4) A D B E C (4) A B D C E (4) E C B A D (3) D E A C B (3) D A E C B (3) D A B E C (3) A B D E C (3) D C E B A (2) D C E A B (2) D C B A E (2) C B D A E (2) B C A E D (2) A E B D C (2) A D B C E (2) A B E D C (2) E D A C B (1) E C B D A (1) E C A B D (1) E A B C D (1) D E C A B (1) D E A B C (1) D B A C E (1) C E B A D (1) C D E B A (1) C D B E A (1) C D B A E (1) C B E D A (1) B E A C D (1) B C D A E (1) B C A D E (1) B A D C E (1) B A C E D (1) B A C D E (1) A E D B C (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 16 -2 -16 2 B -16 0 0 -14 -2 C 2 0 0 -14 0 D 16 14 14 0 6 E -2 2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -16 2 B -16 0 0 -14 -2 C 2 0 0 -14 0 D 16 14 14 0 6 E -2 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=21 A=20 E=17 B=12 so B is eliminated. Round 2 votes counts: D=30 C=29 A=23 E=18 so E is eliminated. Round 3 votes counts: C=38 D=37 A=25 so A is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:200 E:197 C:194 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -2 -16 2 B -16 0 0 -14 -2 C 2 0 0 -14 0 D 16 14 14 0 6 E -2 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -16 2 B -16 0 0 -14 -2 C 2 0 0 -14 0 D 16 14 14 0 6 E -2 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -16 2 B -16 0 0 -14 -2 C 2 0 0 -14 0 D 16 14 14 0 6 E -2 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7642: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) E D B C A (7) E C B D A (6) D E B A C (5) C A E B D (5) C A B E D (5) E B D C A (4) D A B E C (4) C B E A D (4) A C B E D (4) E C D B A (3) D B E C A (3) D B E A C (3) C B A E D (3) A D B C E (3) E D C B A (2) D E B C A (2) D E A B C (2) C B E D A (2) B D E C A (2) B C E D A (2) A E D C B (2) A D E B C (2) A C D B E (2) E C A D B (1) E A C D B (1) D B A C E (1) D A B C E (1) C E B A D (1) B E C D A (1) B D A C E (1) B C D E A (1) A D E C B (1) A D C B E (1) A D B E C (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 -4 -4 B 6 0 -8 4 10 C 8 8 0 6 -4 D 4 -4 -6 0 -8 E 4 -10 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.454545 D: 0.000000 E: 0.363636 Sum of squares = 0.371900826446 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.636364 D: 0.636364 E: 1.000000 A B C D E A 0 -6 -8 -4 -4 B 6 0 -8 4 10 C 8 8 0 6 -4 D 4 -4 -6 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.454545 D: 0.000000 E: 0.363636 Sum of squares = 0.371900826489 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.636364 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 D=21 C=20 B=7 so B is eliminated. Round 2 votes counts: A=28 E=25 D=24 C=23 so C is eliminated. Round 3 votes counts: A=41 E=34 D=25 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:209 B:206 E:203 D:193 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -4 -4 B 6 0 -8 4 10 C 8 8 0 6 -4 D 4 -4 -6 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.454545 D: 0.000000 E: 0.363636 Sum of squares = 0.371900826489 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.636364 D: 0.636364 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -4 -4 B 6 0 -8 4 10 C 8 8 0 6 -4 D 4 -4 -6 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.454545 D: 0.000000 E: 0.363636 Sum of squares = 0.371900826489 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.636364 D: 0.636364 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -4 -4 B 6 0 -8 4 10 C 8 8 0 6 -4 D 4 -4 -6 0 -8 E 4 -10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.454545 D: 0.000000 E: 0.363636 Sum of squares = 0.371900826489 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.636364 D: 0.636364 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7643: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) B D C E A (7) A E C D B (7) C E B D A (5) A E D C B (5) A D E B C (5) C E A D B (4) C E A B D (4) B C D E A (4) A C E B D (4) E C A D B (3) D B E C A (3) D A B E C (3) E D A C B (2) E A C D B (2) D E C B A (2) D B E A C (2) D B A E C (2) D A E C B (2) D A E B C (2) C E D B A (2) C E B A D (2) C B A E D (2) B D C A E (2) B C E D A (2) A D E C B (2) E C D A B (1) D E A C B (1) C E D A B (1) C D E B A (1) C A E B D (1) B C E A D (1) B C A E D (1) B A D E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 4 -8 2 B -2 0 -8 -4 -16 C -4 8 0 -2 10 D 8 4 2 0 -2 E -2 16 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408245 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 A B C D E A 0 2 4 -8 2 B -2 0 -8 -4 -16 C -4 8 0 -2 10 D 8 4 2 0 -2 E -2 16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=25 C=22 D=17 E=8 so E is eliminated. Round 2 votes counts: B=28 A=27 C=26 D=19 so D is eliminated. Round 3 votes counts: A=37 B=35 C=28 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:206 D:206 E:203 A:200 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -8 2 B -2 0 -8 -4 -16 C -4 8 0 -2 10 D 8 4 2 0 -2 E -2 16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -8 2 B -2 0 -8 -4 -16 C -4 8 0 -2 10 D 8 4 2 0 -2 E -2 16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -8 2 B -2 0 -8 -4 -16 C -4 8 0 -2 10 D 8 4 2 0 -2 E -2 16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.714286 E: 0.142857 Sum of squares = 0.551020408174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7644: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) D B E A C (9) D A E B C (8) C B D E A (6) D A B E C (5) C A E B D (5) D C A B E (3) D A E C B (3) C B D A E (3) C A E D B (3) B E C A D (3) E A D B C (2) D E B A C (2) D E A B C (2) D C A E B (2) D B C E A (2) C A D E B (2) B D E A C (2) A E D C B (2) A E D B C (2) A D E C B (2) A D E B C (2) E C A B D (1) E B A C D (1) E A B D C (1) D C B A E (1) D B C A E (1) D B A E C (1) D A C E B (1) D A B C E (1) C A D B E (1) B E D C A (1) B C E D A (1) B C E A D (1) B C D E A (1) A E C D B (1) A E C B D (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 2 -10 6 B -6 0 -4 -20 8 C -2 4 0 -12 -2 D 10 20 12 0 22 E -6 -8 2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -10 6 B -6 0 -4 -20 8 C -2 4 0 -12 -2 D 10 20 12 0 22 E -6 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 C=32 A=13 B=9 E=5 so E is eliminated. Round 2 votes counts: D=41 C=33 A=16 B=10 so B is eliminated. Round 3 votes counts: D=44 C=39 A=17 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:232 A:202 C:194 B:189 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -10 6 B -6 0 -4 -20 8 C -2 4 0 -12 -2 D 10 20 12 0 22 E -6 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -10 6 B -6 0 -4 -20 8 C -2 4 0 -12 -2 D 10 20 12 0 22 E -6 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -10 6 B -6 0 -4 -20 8 C -2 4 0 -12 -2 D 10 20 12 0 22 E -6 -8 2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7645: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) D A B C E (6) B D A C E (6) E A D C B (5) A D E B C (5) E C B A D (4) E C A D B (4) E C A B D (4) E A B C D (4) D A E C B (4) C E B D A (4) C B E D A (4) E A D B C (3) C E B A D (3) A D B E C (3) E A C D B (2) E A B D C (2) D C E A B (2) D C B A E (2) D A E B C (2) C E D B A (2) B C D A E (2) A D B C E (2) E D C A B (1) E D A C B (1) E C D A B (1) E B C A D (1) E A C B D (1) D B C A E (1) D B A C E (1) D A C E B (1) D A C B E (1) C D E A B (1) C D B E A (1) C B D E A (1) B E C A D (1) B D C A E (1) B C D E A (1) B A D C E (1) B A C D E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 26 18 -4 -6 B -26 0 2 -14 -14 C -18 -2 0 -16 -10 D 4 14 16 0 6 E 6 14 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 18 -4 -6 B -26 0 2 -14 -14 C -18 -2 0 -16 -10 D 4 14 16 0 6 E 6 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=26 C=16 B=13 A=12 so A is eliminated. Round 2 votes counts: D=36 E=33 C=16 B=15 so B is eliminated. Round 3 votes counts: D=46 E=34 C=20 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:217 E:212 C:177 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 26 18 -4 -6 B -26 0 2 -14 -14 C -18 -2 0 -16 -10 D 4 14 16 0 6 E 6 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 18 -4 -6 B -26 0 2 -14 -14 C -18 -2 0 -16 -10 D 4 14 16 0 6 E 6 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 18 -4 -6 B -26 0 2 -14 -14 C -18 -2 0 -16 -10 D 4 14 16 0 6 E 6 14 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7646: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (6) B E C A D (6) D B C A E (5) D B A E C (5) C E A B D (5) A E C D B (5) E C A B D (4) D B A C E (4) C E A D B (4) B C D E A (4) A E D C B (4) E A C B D (3) D A B E C (3) C D A E B (3) C A E D B (3) B E C D A (3) B D C E A (3) A D E B C (3) E C B A D (2) E B A C D (2) D C A B E (2) D A C E B (2) C E B A D (2) C B E D A (2) B E D C A (2) E B C A D (1) E A C D B (1) D A E C B (1) D A B C E (1) C B D E A (1) B E A C D (1) B D E A C (1) B C E D A (1) B C E A D (1) A E B D C (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 -8 2 2 B -8 0 6 -10 -10 C 8 -6 0 12 -10 D -2 10 -12 0 -8 E -2 10 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.419999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 8 -8 2 2 B -8 0 6 -10 -10 C 8 -6 0 12 -10 D -2 10 -12 0 -8 E -2 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.420000000243 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=22 C=20 A=16 E=13 so E is eliminated. Round 2 votes counts: D=29 C=26 B=25 A=20 so A is eliminated. Round 3 votes counts: D=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:213 A:202 C:202 D:194 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -8 2 2 B -8 0 6 -10 -10 C 8 -6 0 12 -10 D -2 10 -12 0 -8 E -2 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.420000000243 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 2 2 B -8 0 6 -10 -10 C 8 -6 0 12 -10 D -2 10 -12 0 -8 E -2 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.420000000243 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 2 2 B -8 0 6 -10 -10 C 8 -6 0 12 -10 D -2 10 -12 0 -8 E -2 10 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.400000 Sum of squares = 0.420000000243 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7647: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) B E D C A (7) A C D B E (7) A C B D E (6) B E D A C (5) D E C B A (4) D E C A B (4) C D E A B (4) C D A E B (4) A E D B C (4) E D B C A (3) E B D C A (3) C A D E B (3) C A B D E (3) B C D E A (3) E D C B A (2) C D E B A (2) C B D E A (2) B E A D C (2) B A E D C (2) A B C E D (2) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) D E A C B (1) D C E B A (1) C D B E A (1) C B D A E (1) C B A D E (1) B E C D A (1) B E A C D (1) B C E D A (1) B C A D E (1) B A E C D (1) B A C E D (1) A D E C B (1) A D C E B (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 -4 -2 B -4 0 -14 -4 2 C 2 14 0 10 8 D 4 4 -10 0 18 E 2 -2 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -4 -2 B -4 0 -14 -4 2 C 2 14 0 10 8 D 4 4 -10 0 18 E 2 -2 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=25 C=21 E=12 D=10 so D is eliminated. Round 2 votes counts: A=32 B=25 C=22 E=21 so E is eliminated. Round 3 votes counts: B=34 A=34 C=32 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:217 D:208 A:198 B:190 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 -4 -2 B -4 0 -14 -4 2 C 2 14 0 10 8 D 4 4 -10 0 18 E 2 -2 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -4 -2 B -4 0 -14 -4 2 C 2 14 0 10 8 D 4 4 -10 0 18 E 2 -2 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -4 -2 B -4 0 -14 -4 2 C 2 14 0 10 8 D 4 4 -10 0 18 E 2 -2 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7648: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) C A D E B (8) E B D C A (7) D C E A B (7) B E D A C (6) A C B E D (5) E D B C A (4) D E B C A (4) C D A E B (4) B E D C A (4) A C E D B (4) A C D B E (4) D C A E B (3) D C A B E (3) B E A D C (3) B E A C D (3) A B C E D (3) E D C B A (2) D E C B A (2) B D E C A (2) B A C E D (2) A C B D E (2) E C B A D (1) D C B A E (1) D B E C A (1) D B A C E (1) C A E D B (1) B D E A C (1) B A E C D (1) A E B C D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -8 -4 6 B -10 0 -12 -16 -14 C 8 12 0 -2 16 D 4 16 2 0 6 E -6 14 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 -4 6 B -10 0 -12 -16 -14 C 8 12 0 -2 16 D 4 16 2 0 6 E -6 14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=22 B=22 E=14 C=13 so C is eliminated. Round 2 votes counts: A=38 D=26 B=22 E=14 so E is eliminated. Round 3 votes counts: A=38 D=32 B=30 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:217 D:214 A:202 E:193 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -8 -4 6 B -10 0 -12 -16 -14 C 8 12 0 -2 16 D 4 16 2 0 6 E -6 14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 -4 6 B -10 0 -12 -16 -14 C 8 12 0 -2 16 D 4 16 2 0 6 E -6 14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 -4 6 B -10 0 -12 -16 -14 C 8 12 0 -2 16 D 4 16 2 0 6 E -6 14 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7649: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) A D E C B (6) E D A B C (4) D C B A E (4) B E C D A (4) B C D E A (4) E B C A D (3) E A D B C (3) E A C B D (3) D A E C B (3) D A C E B (3) D A C B E (3) C B E A D (3) C B A D E (3) B D C E A (3) B C E D A (3) B C E A D (3) A D C E B (3) E A B C D (2) D E A B C (2) C D B A E (2) C B D A E (2) C B A E D (2) C A D B E (2) B E C A D (2) B C D A E (2) A E C D B (2) E C B A D (1) E C A B D (1) E B D A C (1) E B C D A (1) E B A D C (1) E B A C D (1) E A D C B (1) E A B D C (1) D C A B E (1) D B C A E (1) D A E B C (1) D A B C E (1) C D A B E (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 4 8 6 B -8 0 -14 -8 -6 C -4 14 0 -4 -4 D -8 8 4 0 -2 E -6 6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 8 6 B -8 0 -14 -8 -6 C -4 14 0 -4 -4 D -8 8 4 0 -2 E -6 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=22 B=21 D=19 C=15 so C is eliminated. Round 2 votes counts: B=31 A=24 E=23 D=22 so D is eliminated. Round 3 votes counts: B=38 A=37 E=25 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:203 C:201 D:201 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 8 6 B -8 0 -14 -8 -6 C -4 14 0 -4 -4 D -8 8 4 0 -2 E -6 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 8 6 B -8 0 -14 -8 -6 C -4 14 0 -4 -4 D -8 8 4 0 -2 E -6 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 8 6 B -8 0 -14 -8 -6 C -4 14 0 -4 -4 D -8 8 4 0 -2 E -6 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7650: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) C D B A E (9) E D A B C (8) E A B D C (8) D E A B C (7) C B A D E (7) D E C A B (6) D C E B A (6) D E A C B (5) B C A E D (5) C D B E A (4) B A E C D (4) E A D B C (3) D C E A B (3) B A C E D (3) D C B E A (2) A E B D C (2) A B E C D (2) E B A D C (1) D C A E B (1) C B D A E (1) B C E A D (1) A E B C D (1) Total count = 100 A B C D E A 0 -8 -12 -4 -8 B 8 0 -10 -8 -2 C 12 10 0 -4 6 D 4 8 4 0 2 E 8 2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -4 -8 B 8 0 -10 -8 -2 C 12 10 0 -4 6 D 4 8 4 0 2 E 8 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=30 E=20 B=13 A=5 so A is eliminated. Round 2 votes counts: C=32 D=30 E=23 B=15 so B is eliminated. Round 3 votes counts: C=41 D=30 E=29 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:212 D:209 E:201 B:194 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 -4 -8 B 8 0 -10 -8 -2 C 12 10 0 -4 6 D 4 8 4 0 2 E 8 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -4 -8 B 8 0 -10 -8 -2 C 12 10 0 -4 6 D 4 8 4 0 2 E 8 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -4 -8 B 8 0 -10 -8 -2 C 12 10 0 -4 6 D 4 8 4 0 2 E 8 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7651: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) A B D E C (7) E C B A D (6) B A E C D (6) D C E A B (5) D A B E C (5) E C D A B (4) E B A C D (4) D C A B E (4) C E D B A (4) C D E A B (4) D A B C E (3) C E B D A (3) C E B A D (3) B A C D E (3) A B D C E (3) E D C A B (2) E D A B C (2) D E C A B (2) C E D A B (2) C D E B A (2) C D B A E (2) A B E D C (2) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D A C E B (1) D A C B E (1) C B A E D (1) B E C A D (1) B C A E D (1) B C A D E (1) B A D E C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 4 8 4 B -2 0 8 10 2 C -4 -8 0 -4 4 D -8 -10 4 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 4 B -2 0 8 10 2 C -4 -8 0 -4 4 D -8 -10 4 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 D=21 C=21 A=14 so A is eliminated. Round 2 votes counts: B=34 E=23 D=22 C=21 so C is eliminated. Round 3 votes counts: E=35 B=35 D=30 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:209 B:209 D:197 C:194 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 4 B -2 0 8 10 2 C -4 -8 0 -4 4 D -8 -10 4 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 4 B -2 0 8 10 2 C -4 -8 0 -4 4 D -8 -10 4 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 4 B -2 0 8 10 2 C -4 -8 0 -4 4 D -8 -10 4 0 8 E -4 -2 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7652: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) D C E A B (9) B A E C D (9) E D C A B (6) D E C A B (4) B A C D E (4) A B E D C (4) C E D B A (3) B A D E C (3) B A D C E (3) B A C E D (3) E C D A B (2) E A D B C (2) E A B D C (2) D C E B A (2) D C B A E (2) D A B C E (2) C D E A B (2) C B E A D (2) C B A E D (2) B C A D E (2) B A E D C (2) A E B D C (2) A D B E C (2) E D A B C (1) E C A D B (1) E A D C B (1) E A B C D (1) D C A B E (1) D B C A E (1) D A E C B (1) C E D A B (1) C D B E A (1) C D B A E (1) C B D A E (1) B C A E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -12 -4 -4 B 8 0 -8 -14 -4 C 12 8 0 -2 10 D 4 14 2 0 8 E 4 4 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -4 -4 B 8 0 -8 -14 -4 C 12 8 0 -2 10 D 4 14 2 0 8 E 4 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=25 D=22 E=16 A=10 so A is eliminated. Round 2 votes counts: B=33 C=25 D=24 E=18 so E is eliminated. Round 3 votes counts: B=38 D=34 C=28 so C is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:214 E:195 B:191 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 -4 -4 B 8 0 -8 -14 -4 C 12 8 0 -2 10 D 4 14 2 0 8 E 4 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -4 -4 B 8 0 -8 -14 -4 C 12 8 0 -2 10 D 4 14 2 0 8 E 4 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -4 -4 B 8 0 -8 -14 -4 C 12 8 0 -2 10 D 4 14 2 0 8 E 4 4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7653: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) B D E A C (6) A C E B D (5) E C D A B (4) E C A D B (4) D B C E A (4) C A E D B (4) B D C A E (4) B D A E C (4) B D A C E (4) A E B C D (4) A B E D C (4) E A C D B (3) D C B E A (3) C E A D B (3) C D E B A (3) A B E C D (3) D E B C A (2) D B C A E (2) C E D A B (2) C A B D E (2) A E C B D (2) A E B D C (2) A B C D E (2) E D B C A (1) E D B A C (1) D E C B A (1) D C E B A (1) D B E A C (1) C D E A B (1) C D B A E (1) C D A B E (1) C A E B D (1) C A D E B (1) C A D B E (1) B A E D C (1) B A D E C (1) B A C D E (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -6 -6 6 B -4 0 12 -2 10 C 6 -12 0 0 -2 D 6 2 0 0 8 E -6 -10 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.067379 D: 0.932621 E: 0.000000 Sum of squares = 0.874321344923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.067379 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -6 6 B -4 0 12 -2 10 C 6 -12 0 0 -2 D 6 2 0 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102067278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=21 B=21 C=20 E=13 so E is eliminated. Round 2 votes counts: C=28 A=28 D=23 B=21 so B is eliminated. Round 3 votes counts: D=41 A=31 C=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:208 D:208 A:199 C:196 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 -6 6 B -4 0 12 -2 10 C 6 -12 0 0 -2 D 6 2 0 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102067278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 6 B -4 0 12 -2 10 C 6 -12 0 0 -2 D 6 2 0 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102067278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 6 B -4 0 12 -2 10 C 6 -12 0 0 -2 D 6 2 0 0 8 E -6 -10 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102067278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7654: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D B C E A (5) C A E B D (5) B D C A E (5) B C D A E (5) B A C E D (5) D E A B C (4) C B A E D (4) A E C B D (4) E D A C B (3) E A C B D (3) D E C B A (3) D E C A B (3) D B A E C (3) C E A B D (3) A E B C D (3) E A D C B (2) D E A C B (2) D C E B A (2) D B C A E (2) C E A D B (2) C D B E A (2) C B D E A (2) B C A E D (2) B A E D C (2) B A E C D (2) A E B D C (2) A B E C D (2) D E B C A (1) D C E A B (1) D B E A C (1) C D E B A (1) B D A C E (1) B C A D E (1) B A D E C (1) B A D C E (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 8 4 B 2 0 -2 10 -6 C -2 2 0 10 0 D -8 -10 -10 0 -8 E -4 6 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888521 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 -2 2 8 4 B 2 0 -2 10 -6 C -2 2 0 10 0 D -8 -10 -10 0 -8 E -4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.38888888882 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 C=19 E=16 A=13 so A is eliminated. Round 2 votes counts: B=28 D=27 E=25 C=20 so C is eliminated. Round 3 votes counts: E=36 B=34 D=30 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:206 C:205 E:205 B:202 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 8 4 B 2 0 -2 10 -6 C -2 2 0 10 0 D -8 -10 -10 0 -8 E -4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.38888888882 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 8 4 B 2 0 -2 10 -6 C -2 2 0 10 0 D -8 -10 -10 0 -8 E -4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.38888888882 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 8 4 B 2 0 -2 10 -6 C -2 2 0 10 0 D -8 -10 -10 0 -8 E -4 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.166667 Sum of squares = 0.38888888882 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7655: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (19) D C B E A (14) C D B A E (10) E A B D C (7) C D B E A (7) A E B D C (6) C A D B E (4) E B A D C (3) D B C E A (3) B E A C D (3) A C E B D (3) D C A E B (2) C D A B E (2) C B D E A (2) C A B D E (2) A E C D B (2) E B D A C (1) D E B C A (1) D C A B E (1) C B D A E (1) C A D E B (1) B E A D C (1) B A E C D (1) A E D B C (1) A E C B D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 0 12 16 B -6 0 -8 0 2 C 0 8 0 20 8 D -12 0 -20 0 2 E -16 -2 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.474084 B: 0.000000 C: 0.525916 D: 0.000000 E: 0.000000 Sum of squares = 0.501343273288 Cumulative probabilities = A: 0.474084 B: 0.474084 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 12 16 B -6 0 -8 0 2 C 0 8 0 20 8 D -12 0 -20 0 2 E -16 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=29 D=21 E=11 B=5 so B is eliminated. Round 2 votes counts: A=35 C=29 D=21 E=15 so E is eliminated. Round 3 votes counts: A=49 C=29 D=22 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:217 B:194 E:186 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 12 16 B -6 0 -8 0 2 C 0 8 0 20 8 D -12 0 -20 0 2 E -16 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 12 16 B -6 0 -8 0 2 C 0 8 0 20 8 D -12 0 -20 0 2 E -16 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 12 16 B -6 0 -8 0 2 C 0 8 0 20 8 D -12 0 -20 0 2 E -16 -2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7656: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) C E D B A (6) E A D C B (5) C E B D A (5) C B D E A (5) B A D C E (4) A E D B C (4) A D B E C (4) A B E D C (4) E C D B A (3) B C D A E (3) A E B D C (3) A D E B C (3) E D C A B (2) E D A C B (2) E C A D B (2) E A C B D (2) E A B D C (2) D E C A B (2) D A E B C (2) D A B E C (2) D A B C E (2) C D E B A (2) B D A C E (2) B C A D E (2) E C D A B (1) E C B A D (1) E A D B C (1) E A C D B (1) D E A C B (1) D C E B A (1) D C A E B (1) D C A B E (1) D B A C E (1) D A C B E (1) C D B A E (1) C B D A E (1) B D C A E (1) B A C E D (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 -16 -8 B -2 0 -12 -4 -4 C 0 12 0 -6 4 D 16 4 6 0 -12 E 8 4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.181818 E: 0.272727 Sum of squares = 0.404958677701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.727273 E: 1.000000 A B C D E A 0 2 0 -16 -8 B -2 0 -12 -4 -4 C 0 12 0 -6 4 D 16 4 6 0 -12 E 8 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.181818 E: 0.272727 Sum of squares = 0.404958677687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=22 A=21 D=14 B=13 so B is eliminated. Round 2 votes counts: C=35 A=26 E=22 D=17 so D is eliminated. Round 3 votes counts: C=39 A=36 E=25 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:210 D:207 C:205 A:189 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -16 -8 B -2 0 -12 -4 -4 C 0 12 0 -6 4 D 16 4 6 0 -12 E 8 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.181818 E: 0.272727 Sum of squares = 0.404958677687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.727273 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -16 -8 B -2 0 -12 -4 -4 C 0 12 0 -6 4 D 16 4 6 0 -12 E 8 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.181818 E: 0.272727 Sum of squares = 0.404958677687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.727273 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -16 -8 B -2 0 -12 -4 -4 C 0 12 0 -6 4 D 16 4 6 0 -12 E 8 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.181818 E: 0.272727 Sum of squares = 0.404958677687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.545455 D: 0.727273 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7657: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) E C B A D (8) C E A D B (8) A D B C E (7) E B C A D (6) D B A E C (6) C E A B D (6) B D A E C (6) E B D A C (3) D B A C E (3) C E D A B (3) C D A B E (3) B A E D C (3) E C B D A (2) E B C D A (2) C D E A B (2) C A D B E (2) B A D E C (2) A C D B E (2) E C D B A (1) E C A B D (1) E B D C A (1) E B A D C (1) E B A C D (1) D E B A C (1) D B E A C (1) D A C B E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E A C (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 0 6 4 0 B 0 0 16 -8 4 C -6 -16 0 -2 2 D -4 8 2 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.722338 B: 0.277662 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.598868094439 Cumulative probabilities = A: 0.722338 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 0 B 0 0 16 -8 4 C -6 -16 0 -2 2 D -4 8 2 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555783252 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=26 C=26 D=23 B=14 A=11 so A is eliminated. Round 2 votes counts: D=31 C=28 E=26 B=15 so B is eliminated. Round 3 votes counts: D=41 E=31 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:206 A:205 D:204 E:196 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 0 B 0 0 16 -8 4 C -6 -16 0 -2 2 D -4 8 2 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555783252 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 0 B 0 0 16 -8 4 C -6 -16 0 -2 2 D -4 8 2 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555783252 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 0 B 0 0 16 -8 4 C -6 -16 0 -2 2 D -4 8 2 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555783252 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7658: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) C E B D A (6) C B E D A (6) E C D B A (5) A D E B C (5) B C E A D (4) B C D E A (4) B A D C E (4) B A C E D (4) E A D C B (3) D B C E A (3) D A B E C (3) C E D B A (3) C B E A D (3) A E D C B (3) A D B E C (3) E C A D B (2) E C A B D (2) E A C D B (2) D E C A B (2) C D B E A (2) A D E C B (2) A B E C D (2) A B D E C (2) E D C A B (1) E D A C B (1) E C B D A (1) E C B A D (1) D E C B A (1) D E A C B (1) D C E B A (1) D C B E A (1) D B C A E (1) D A E C B (1) C E B A D (1) C B D E A (1) B D A C E (1) B C E D A (1) B C D A E (1) B C A D E (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -22 -8 -30 B 12 0 -18 -10 -4 C 22 18 0 18 -2 D 8 10 -18 0 -18 E 30 4 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -22 -8 -30 B 12 0 -18 -10 -4 C 22 18 0 18 -2 D 8 10 -18 0 -18 E 30 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998451 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=22 B=20 A=19 D=14 so D is eliminated. Round 2 votes counts: E=29 C=24 B=24 A=23 so A is eliminated. Round 3 votes counts: E=41 B=35 C=24 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:228 E:227 D:191 B:190 A:164 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -22 -8 -30 B 12 0 -18 -10 -4 C 22 18 0 18 -2 D 8 10 -18 0 -18 E 30 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998451 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -22 -8 -30 B 12 0 -18 -10 -4 C 22 18 0 18 -2 D 8 10 -18 0 -18 E 30 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998451 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -22 -8 -30 B 12 0 -18 -10 -4 C 22 18 0 18 -2 D 8 10 -18 0 -18 E 30 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998451 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7659: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A B E D (9) C A E D B (7) C A E B D (7) B D E A C (7) A C E D B (7) E D A C B (5) D B E A C (5) A E D C B (5) B D E C A (4) E D A B C (3) D E B C A (3) C B A D E (3) C A B D E (3) E A D C B (2) D E A B C (2) D B E C A (2) C B E D A (2) B C A D E (2) A E C D B (2) A C B E D (2) A C B D E (2) E D B A C (1) E C A D B (1) C B A E D (1) B D C E A (1) B C D E A (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 18 8 10 4 B -18 0 -16 -12 -10 C -8 16 0 -2 -6 D -10 12 2 0 -8 E -4 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 8 10 4 B -18 0 -16 -12 -10 C -8 16 0 -2 -6 D -10 12 2 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=21 A=20 B=15 E=12 so E is eliminated. Round 2 votes counts: C=33 D=30 A=22 B=15 so B is eliminated. Round 3 votes counts: D=42 C=36 A=22 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:220 E:210 C:200 D:198 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 8 10 4 B -18 0 -16 -12 -10 C -8 16 0 -2 -6 D -10 12 2 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 8 10 4 B -18 0 -16 -12 -10 C -8 16 0 -2 -6 D -10 12 2 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 8 10 4 B -18 0 -16 -12 -10 C -8 16 0 -2 -6 D -10 12 2 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7660: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) A C D B E (10) B E C D A (7) A C D E B (7) E B D C A (6) C A B D E (5) A E D B C (5) B E D C A (4) A D E C B (4) A C B E D (4) C B A E D (3) C A D B E (3) C A B E D (3) A D C E B (3) E D B A C (2) E D A B C (2) E B D A C (2) D E B A C (2) D A E B C (2) C B E D A (2) C B E A D (2) C B D E A (2) B C E D A (2) E A B D C (1) D E B C A (1) D E A B C (1) D A E C B (1) C B D A E (1) C B A D E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 26 16 26 28 B -26 0 -6 -6 -2 C -16 6 0 8 -2 D -26 6 -8 0 6 E -28 2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 16 26 28 B -26 0 -6 -6 -2 C -16 6 0 8 -2 D -26 6 -8 0 6 E -28 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 C=22 E=13 B=13 D=7 so D is eliminated. Round 2 votes counts: A=48 C=22 E=17 B=13 so B is eliminated. Round 3 votes counts: A=48 E=28 C=24 so C is eliminated. Round 4 votes counts: A=64 E=36 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:248 C:198 D:189 E:185 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 16 26 28 B -26 0 -6 -6 -2 C -16 6 0 8 -2 D -26 6 -8 0 6 E -28 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 16 26 28 B -26 0 -6 -6 -2 C -16 6 0 8 -2 D -26 6 -8 0 6 E -28 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 16 26 28 B -26 0 -6 -6 -2 C -16 6 0 8 -2 D -26 6 -8 0 6 E -28 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7661: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (14) A E B C D (8) D C B A E (7) B C D A E (7) E D A C B (6) D C B E A (6) E A D C B (5) D E C B A (5) C B D A E (4) B C A D E (4) E D A B C (3) D E A C B (3) D C E B A (3) E D B C A (2) E B A C D (2) E A D B C (2) D C A B E (2) C B A D E (2) B D C E A (2) A E D C B (2) A B C E D (2) E A B D C (1) D E C A B (1) D C A E B (1) D B C E A (1) C D B A E (1) B E A C D (1) B A E C D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 -8 -14 B -4 0 2 0 -18 C -4 -2 0 -4 -16 D 8 0 4 0 -2 E 14 18 16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 -8 -14 B -4 0 2 0 -18 C -4 -2 0 -4 -16 D 8 0 4 0 -2 E 14 18 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=29 B=15 A=14 C=7 so C is eliminated. Round 2 votes counts: E=35 D=30 B=21 A=14 so A is eliminated. Round 3 votes counts: E=46 D=30 B=24 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 D:205 A:193 B:190 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 -8 -14 B -4 0 2 0 -18 C -4 -2 0 -4 -16 D 8 0 4 0 -2 E 14 18 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -8 -14 B -4 0 2 0 -18 C -4 -2 0 -4 -16 D 8 0 4 0 -2 E 14 18 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -8 -14 B -4 0 2 0 -18 C -4 -2 0 -4 -16 D 8 0 4 0 -2 E 14 18 16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999948939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7662: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C E A D B (8) B D A E C (8) E C B D A (6) E B D C A (5) D B A E C (5) C E A B D (5) C A E D B (5) A B D C E (5) A D B C E (4) A C D B E (4) E C D B A (3) C A E B D (3) C A D E B (3) B E D A C (3) B D E A C (3) E D B C A (2) E B C D A (2) C A D B E (2) A C B D E (2) E D C B A (1) E C D A B (1) E C A D B (1) E C A B D (1) D E B A C (1) D B A C E (1) D A C E B (1) D A B C E (1) B D A C E (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 -4 0 B 6 0 8 14 -12 C -4 -8 0 -2 -2 D 4 -14 2 0 -10 E 0 12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.433205 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.566795 Sum of squares = 0.508923120618 Cumulative probabilities = A: 0.433205 B: 0.433205 C: 0.433205 D: 0.433205 E: 1.000000 A B C D E A 0 -6 4 -4 0 B 6 0 8 14 -12 C -4 -8 0 -2 -2 D 4 -14 2 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 B=19 A=16 D=9 so D is eliminated. Round 2 votes counts: E=31 C=26 B=25 A=18 so A is eliminated. Round 3 votes counts: B=36 C=33 E=31 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:212 B:208 A:197 C:192 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 -4 0 B 6 0 8 14 -12 C -4 -8 0 -2 -2 D 4 -14 2 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -4 0 B 6 0 8 14 -12 C -4 -8 0 -2 -2 D 4 -14 2 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -4 0 B 6 0 8 14 -12 C -4 -8 0 -2 -2 D 4 -14 2 0 -10 E 0 12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7663: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (11) D C E B A (8) B D C E A (7) A E C D B (7) A E B C D (7) B E C D A (5) A E C B D (4) E C D A B (3) D C B E A (3) D B C E A (3) C D E B A (3) B E A C D (3) B C D E A (3) B A E C D (3) A E D C B (3) A B E D C (3) A B D C E (3) E C D B A (2) E B C D A (2) C E D B A (2) B E C A D (2) A D C E B (2) A B D E C (2) E C B D A (1) E C B A D (1) E C A D B (1) E B C A D (1) E B A C D (1) D C E A B (1) D B C A E (1) D A C B E (1) B D A C E (1) Total count = 100 A B C D E A 0 -4 2 8 -4 B 4 0 16 20 2 C -2 -16 0 24 -24 D -8 -20 -24 0 -24 E 4 -2 24 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 8 -4 B 4 0 16 20 2 C -2 -16 0 24 -24 D -8 -20 -24 0 -24 E 4 -2 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 B=24 D=17 E=12 C=5 so C is eliminated. Round 2 votes counts: A=42 B=24 D=20 E=14 so E is eliminated. Round 3 votes counts: A=43 B=30 D=27 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:225 B:221 A:201 C:191 D:162 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 8 -4 B 4 0 16 20 2 C -2 -16 0 24 -24 D -8 -20 -24 0 -24 E 4 -2 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 8 -4 B 4 0 16 20 2 C -2 -16 0 24 -24 D -8 -20 -24 0 -24 E 4 -2 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 8 -4 B 4 0 16 20 2 C -2 -16 0 24 -24 D -8 -20 -24 0 -24 E 4 -2 24 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7664: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) C B D E A (6) E A D B C (5) E D B A C (4) C B D A E (4) B C D A E (4) A E D B C (4) A D C B E (4) E D B C A (3) C B E D A (3) C B A D E (3) C A E B D (3) C A B D E (3) B D C E A (3) B C D E A (3) A E C D B (3) E B D C A (2) E B C D A (2) E A D C B (2) D E A B C (2) C B A E D (2) B D E C A (2) B D C A E (2) A D E B C (2) E C A B D (1) D E B A C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) C B E A D (1) C A B E D (1) B C E D A (1) B A C D E (1) A E D C B (1) A D C E B (1) A D B C E (1) A C E D B (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -2 -14 -10 B 0 0 12 0 -4 C 2 -12 0 -6 4 D 14 0 6 0 -4 E 10 4 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999958 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 0 -2 -14 -10 B 0 0 12 0 -4 C 2 -12 0 -6 4 D 14 0 6 0 -4 E 10 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999991 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=28 A=20 B=16 D=6 so D is eliminated. Round 2 votes counts: E=33 C=28 A=22 B=17 so B is eliminated. Round 3 votes counts: C=41 E=36 A=23 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:208 E:207 B:204 C:194 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -14 -10 B 0 0 12 0 -4 C 2 -12 0 -6 4 D 14 0 6 0 -4 E 10 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999991 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -14 -10 B 0 0 12 0 -4 C 2 -12 0 -6 4 D 14 0 6 0 -4 E 10 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999991 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -14 -10 B 0 0 12 0 -4 C 2 -12 0 -6 4 D 14 0 6 0 -4 E 10 4 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999991 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7665: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) B D E C A (7) A C E D B (7) E C A D B (6) A B D C E (6) E C D B A (5) C E A D B (5) D C E B A (4) B D A E C (4) A E C B D (4) B D C A E (3) A E C D B (3) E C B D A (2) E C A B D (2) E A C D B (2) D B C A E (2) C E D A B (2) B D C E A (2) B A E C D (2) A E B C D (2) E C D A B (1) E B C D A (1) E A C B D (1) E A B C D (1) D C E A B (1) D C B E A (1) D B C E A (1) D B A C E (1) C E D B A (1) C D E B A (1) C D E A B (1) C A D E B (1) B E D C A (1) B E C D A (1) B E A C D (1) B D E A C (1) B A D E C (1) B A D C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -2 -6 -2 B 6 0 -2 8 -8 C 2 2 0 6 2 D 6 -8 -6 0 -2 E 2 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -6 -2 B 6 0 -2 8 -8 C 2 2 0 6 2 D 6 -8 -6 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=24 E=21 C=11 D=10 so D is eliminated. Round 2 votes counts: B=38 A=24 E=21 C=17 so C is eliminated. Round 3 votes counts: B=39 E=36 A=25 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:206 E:205 B:202 D:195 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -6 -2 B 6 0 -2 8 -8 C 2 2 0 6 2 D 6 -8 -6 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -6 -2 B 6 0 -2 8 -8 C 2 2 0 6 2 D 6 -8 -6 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -6 -2 B 6 0 -2 8 -8 C 2 2 0 6 2 D 6 -8 -6 0 -2 E 2 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7666: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) E B C A D (6) B E C A D (6) B D E C A (6) B D E A C (6) A C E D B (6) A C D E B (6) E C A B D (5) C A E D B (5) B E D A C (5) E A C B D (4) D B A C E (4) D A B C E (4) C A D E B (4) D A C B E (3) D B E A C (2) D A C E B (2) C E A B D (2) C A E B D (2) B C E A D (2) A D E C B (2) E C B A D (1) D C A B E (1) D A E C B (1) D A E B C (1) C B E A D (1) B E C D A (1) B E A D C (1) B D C E A (1) B C E D A (1) Total count = 100 A B C D E A 0 -4 -6 6 -18 B 4 0 10 18 6 C 6 -10 0 4 -12 D -6 -18 -4 0 -14 E 18 -6 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999193 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 6 -18 B 4 0 10 18 6 C 6 -10 0 4 -12 D -6 -18 -4 0 -14 E 18 -6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=18 E=16 C=14 A=14 so C is eliminated. Round 2 votes counts: B=39 A=25 E=18 D=18 so E is eliminated. Round 3 votes counts: B=46 A=36 D=18 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:219 C:194 A:189 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 6 -18 B 4 0 10 18 6 C 6 -10 0 4 -12 D -6 -18 -4 0 -14 E 18 -6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 -18 B 4 0 10 18 6 C 6 -10 0 4 -12 D -6 -18 -4 0 -14 E 18 -6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 -18 B 4 0 10 18 6 C 6 -10 0 4 -12 D -6 -18 -4 0 -14 E 18 -6 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7667: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (11) D E A B C (6) B C A E D (6) D E B A C (5) D E A C B (5) C B D A E (5) E D B A C (4) E D A B C (4) E A D B C (4) A E D C B (4) C B A E D (3) B E D C A (3) A D E C B (3) A C E D B (3) D C B E A (2) C B A D E (2) C A B E D (2) B E C A D (2) B D E C A (2) B C E A D (2) A C D E B (2) E B D A C (1) E A D C B (1) E A B D C (1) D E C A B (1) D C E B A (1) D B E C A (1) D B C E A (1) D A E C B (1) C D B A E (1) C D A E B (1) C B D E A (1) C A D E B (1) C A B D E (1) B E A C D (1) B D E A C (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 2 -14 -20 B 10 0 16 -6 -4 C -2 -16 0 -6 -10 D 14 6 6 0 8 E 20 4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -14 -20 B 10 0 16 -6 -4 C -2 -16 0 -6 -10 D 14 6 6 0 8 E 20 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 C=17 A=17 E=15 so E is eliminated. Round 2 votes counts: D=31 B=29 A=23 C=17 so C is eliminated. Round 3 votes counts: B=40 D=33 A=27 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 E:213 B:208 C:183 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 -14 -20 B 10 0 16 -6 -4 C -2 -16 0 -6 -10 D 14 6 6 0 8 E 20 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -14 -20 B 10 0 16 -6 -4 C -2 -16 0 -6 -10 D 14 6 6 0 8 E 20 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -14 -20 B 10 0 16 -6 -4 C -2 -16 0 -6 -10 D 14 6 6 0 8 E 20 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7668: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (16) B D E C A (12) E A C B D (8) A E C B D (8) D B C E A (6) D B C A E (6) B D E A C (5) E C A B D (4) D C A E B (3) D C A B E (3) D B A C E (3) D A C E B (3) C A D E B (3) D A C B E (2) C A E D B (2) B E D C A (2) B E A C D (2) E C B D A (1) E C B A D (1) E A B C D (1) D B A E C (1) C E A D B (1) B E D A C (1) B E C D A (1) B D C E A (1) B D A E C (1) A E C D B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 8 -2 8 B -14 0 -16 -4 -8 C -8 16 0 0 2 D 2 4 0 0 2 E -8 8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.135367 D: 0.864633 E: 0.000000 Sum of squares = 0.765915129383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.135367 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 -2 8 B -14 0 -16 -4 -8 C -8 16 0 0 2 D 2 4 0 0 2 E -8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000579401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 B=25 E=15 C=6 so C is eliminated. Round 2 votes counts: A=32 D=27 B=25 E=16 so E is eliminated. Round 3 votes counts: A=46 D=27 B=27 so D is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:205 D:204 E:198 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 8 -2 8 B -14 0 -16 -4 -8 C -8 16 0 0 2 D 2 4 0 0 2 E -8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000579401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -2 8 B -14 0 -16 -4 -8 C -8 16 0 0 2 D 2 4 0 0 2 E -8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000579401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -2 8 B -14 0 -16 -4 -8 C -8 16 0 0 2 D 2 4 0 0 2 E -8 8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.800000 E: 0.000000 Sum of squares = 0.680000579401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7669: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) A B E C D (6) D C E B A (5) D B A C E (5) C E A B D (5) B A D E C (5) D C A B E (4) C D E A B (4) B E A D C (4) E C D B A (3) D E C B A (3) D E B C A (3) D C E A B (3) C E D B A (3) B A E D C (3) A B C D E (3) E C A B D (2) E B D A C (2) D B C A E (2) D B A E C (2) D A B C E (2) C E A D B (2) A D B C E (2) A C B D E (2) A B D C E (2) E D C B A (1) E D B C A (1) E B D C A (1) E B A D C (1) E A C B D (1) E A B C D (1) D C A E B (1) C E D A B (1) C D E B A (1) C D A E B (1) B E A C D (1) B D A E C (1) B A E C D (1) A C E B D (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 8 2 -10 B 10 0 12 0 -6 C -8 -12 0 -8 0 D -2 0 8 0 4 E 10 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.321166 C: 0.000000 D: 0.678834 E: 0.000000 Sum of squares = 0.563963192842 Cumulative probabilities = A: 0.000000 B: 0.321166 C: 0.321166 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 2 -10 B 10 0 12 0 -6 C -8 -12 0 -8 0 D -2 0 8 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000133023 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=20 A=18 C=17 B=15 so B is eliminated. Round 2 votes counts: D=31 A=27 E=25 C=17 so C is eliminated. Round 3 votes counts: D=37 E=36 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:208 E:206 D:205 A:195 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 2 -10 B 10 0 12 0 -6 C -8 -12 0 -8 0 D -2 0 8 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000133023 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 2 -10 B 10 0 12 0 -6 C -8 -12 0 -8 0 D -2 0 8 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000133023 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 2 -10 B 10 0 12 0 -6 C -8 -12 0 -8 0 D -2 0 8 0 4 E 10 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.520000133023 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7670: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (18) D C E B A (12) D A B E C (7) A D B E C (7) C E B A D (6) A B E D C (6) C D E B A (5) D C A E B (4) D B E C A (4) C E B D A (3) A B D E C (3) E B C A D (2) D E B C A (2) D A C B E (2) C A E B D (2) B A E D C (2) A C B E D (2) E C B D A (1) E B D C A (1) E B C D A (1) D E C B A (1) D B E A C (1) D B A E C (1) C E D B A (1) C E A B D (1) C A D E B (1) B E A C D (1) B D A E C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 10 6 6 16 B -10 0 14 2 10 C -6 -14 0 -10 -18 D -6 -2 10 0 4 E -16 -10 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 6 16 B -10 0 14 2 10 C -6 -14 0 -10 -18 D -6 -2 10 0 4 E -16 -10 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=34 C=19 E=5 B=4 so B is eliminated. Round 2 votes counts: A=40 D=35 C=19 E=6 so E is eliminated. Round 3 votes counts: A=41 D=36 C=23 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:208 D:203 E:194 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 6 16 B -10 0 14 2 10 C -6 -14 0 -10 -18 D -6 -2 10 0 4 E -16 -10 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 6 16 B -10 0 14 2 10 C -6 -14 0 -10 -18 D -6 -2 10 0 4 E -16 -10 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 6 16 B -10 0 14 2 10 C -6 -14 0 -10 -18 D -6 -2 10 0 4 E -16 -10 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998559 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7671: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (11) B D A C E (9) E C A D B (7) B D C A E (7) D B C A E (6) E A B C D (5) E A C D B (4) B A D C E (3) A C E D B (3) A C E B D (3) E C D A B (2) D C E B A (2) D C B A E (2) D B C E A (2) C E A D B (2) C D E A B (2) B E D C A (2) B E D A C (2) B E A D C (2) B A E D C (2) A C D B E (2) E D C A B (1) E D B C A (1) E C B D A (1) E B D C A (1) E B A C D (1) D E C B A (1) D E B C A (1) D C B E A (1) D C A E B (1) C E D A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B D A E C (1) B A E C D (1) A E C B D (1) A E B C D (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 8 4 -4 B -4 0 -4 10 -10 C -8 4 0 6 4 D -4 -10 -6 0 -10 E 4 10 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 8 4 -4 B -4 0 -4 10 -10 C -8 4 0 6 4 D -4 -10 -6 0 -10 E 4 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=29 D=16 A=13 C=8 so C is eliminated. Round 2 votes counts: E=37 B=29 D=20 A=14 so A is eliminated. Round 3 votes counts: E=46 B=31 D=23 so D is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:210 A:206 C:203 B:196 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 -4 B -4 0 -4 10 -10 C -8 4 0 6 4 D -4 -10 -6 0 -10 E 4 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 -4 B -4 0 -4 10 -10 C -8 4 0 6 4 D -4 -10 -6 0 -10 E 4 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 -4 B -4 0 -4 10 -10 C -8 4 0 6 4 D -4 -10 -6 0 -10 E 4 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7672: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) D E C A B (7) A B C E D (6) A B E C D (5) D E A C B (4) C E B D A (4) B A C D E (4) A E C B D (4) E D C A B (3) E D A C B (3) E A C D B (3) D E A B C (3) D C E B A (3) B C A D E (3) B A D C E (3) A B D E C (3) E C D B A (2) D B C E A (2) C E D B A (2) C D B E A (2) C B A E D (2) B D A C E (2) A E B D C (2) A D B E C (2) E C D A B (1) E C A D B (1) E A D C B (1) E A C B D (1) D E C B A (1) D C B E A (1) D B E C A (1) C E A B D (1) C D E B A (1) C B E D A (1) C B E A D (1) C B D E A (1) B D C A E (1) B C D E A (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 16 8 -2 B -4 0 0 12 4 C -16 0 0 12 2 D -8 -12 -12 0 -8 E 2 -4 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 4 16 8 -2 B -4 0 0 12 4 C -16 0 0 12 2 D -8 -12 -12 0 -8 E 2 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 D=22 E=15 C=15 so E is eliminated. Round 2 votes counts: A=29 D=28 B=24 C=19 so C is eliminated. Round 3 votes counts: D=36 B=33 A=31 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:206 E:202 C:199 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 8 -2 B -4 0 0 12 4 C -16 0 0 12 2 D -8 -12 -12 0 -8 E 2 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 8 -2 B -4 0 0 12 4 C -16 0 0 12 2 D -8 -12 -12 0 -8 E 2 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 8 -2 B -4 0 0 12 4 C -16 0 0 12 2 D -8 -12 -12 0 -8 E 2 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.360000000002 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7673: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) A E D B C (6) E A B C D (5) D B C E A (5) C B E A D (5) D C A B E (4) D A E C B (4) D A E B C (4) B E C A D (4) A E C B D (4) E B A C D (3) D C B E A (3) C A E B D (3) A D E C B (3) A D E B C (3) D C B A E (2) D B E A C (2) C B E D A (2) B D C E A (2) B C D E A (2) A C E D B (2) E D B A C (1) E B D A C (1) E B C A D (1) E A D B C (1) E A B D C (1) D E B A C (1) D A C E B (1) D A C B E (1) C E B A D (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D E B (1) C A D B E (1) B E D C A (1) B E C D A (1) B C E D A (1) B C E A D (1) A E D C B (1) A E C D B (1) A E B D C (1) A E B C D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -2 4 -4 B -2 0 -4 2 -4 C 2 4 0 4 0 D -4 -2 -4 0 0 E 4 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.466250 D: 0.000000 E: 0.533750 Sum of squares = 0.502278133725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.466250 D: 0.466250 E: 1.000000 A B C D E A 0 2 -2 4 -4 B -2 0 -4 2 -4 C 2 4 0 4 0 D -4 -2 -4 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 A=24 E=13 B=12 so B is eliminated. Round 2 votes counts: D=29 C=28 A=24 E=19 so E is eliminated. Round 3 votes counts: C=34 A=34 D=32 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:205 E:204 A:200 B:196 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -2 4 -4 B -2 0 -4 2 -4 C 2 4 0 4 0 D -4 -2 -4 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 4 -4 B -2 0 -4 2 -4 C 2 4 0 4 0 D -4 -2 -4 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 4 -4 B -2 0 -4 2 -4 C 2 4 0 4 0 D -4 -2 -4 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7674: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) D A E B C (6) B E C A D (6) D B E A C (5) D A C E B (5) C A E B D (5) B E D A C (5) C A E D B (4) B D E A C (4) A C D E B (4) E A D B C (3) C B E A D (3) C A B E D (3) B C D E A (3) E B A C D (2) D B C A E (2) C D A E B (2) C A D E B (2) B E A D C (2) B E A C D (2) B C E D A (2) A E C D B (2) A C E D B (2) E C A B D (1) E A D C B (1) E A C B D (1) E A B D C (1) E A B C D (1) D E A B C (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C B E (1) C D B A E (1) C B D A E (1) C B A D E (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D A E (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 6 10 -4 B 4 0 16 8 8 C -6 -16 0 12 4 D -10 -8 -12 0 -8 E 4 -8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 10 -4 B 4 0 16 8 8 C -6 -16 0 12 4 D -10 -8 -12 0 -8 E 4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=23 C=22 E=10 A=10 so E is eliminated. Round 2 votes counts: B=37 D=23 C=23 A=17 so A is eliminated. Round 3 votes counts: B=39 C=32 D=29 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 A:204 E:200 C:197 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 10 -4 B 4 0 16 8 8 C -6 -16 0 12 4 D -10 -8 -12 0 -8 E 4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 10 -4 B 4 0 16 8 8 C -6 -16 0 12 4 D -10 -8 -12 0 -8 E 4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 10 -4 B 4 0 16 8 8 C -6 -16 0 12 4 D -10 -8 -12 0 -8 E 4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7675: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) B D C E A (8) A C E B D (8) C B D A E (7) A E D B C (7) A C E D B (7) C A B D E (6) E D B A C (4) C B D E A (4) B D E C A (4) A E D C B (4) E B D A C (3) E A D B C (3) D B E C A (3) D B E A C (3) C A E B D (3) B C D E A (3) A E C D B (3) E D A B C (2) C B E D A (2) A E C B D (2) E B A D C (1) D E B A C (1) D C B A E (1) C D A B E (1) C B A E D (1) C A E D B (1) Total count = 100 A B C D E A 0 10 -4 8 18 B -10 0 -16 20 2 C 4 16 0 12 20 D -8 -20 -12 0 -18 E -18 -2 -20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 8 18 B -10 0 -16 20 2 C 4 16 0 12 20 D -8 -20 -12 0 -18 E -18 -2 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=31 B=15 E=13 D=8 so D is eliminated. Round 2 votes counts: C=34 A=31 B=21 E=14 so E is eliminated. Round 3 votes counts: A=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:216 B:198 E:189 D:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 8 18 B -10 0 -16 20 2 C 4 16 0 12 20 D -8 -20 -12 0 -18 E -18 -2 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 8 18 B -10 0 -16 20 2 C 4 16 0 12 20 D -8 -20 -12 0 -18 E -18 -2 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 8 18 B -10 0 -16 20 2 C 4 16 0 12 20 D -8 -20 -12 0 -18 E -18 -2 -20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7676: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) A B E D C (10) D E C B A (9) C D E A B (5) A B C E D (5) E D B C A (4) B A C E D (4) A C B E D (4) E D B A C (3) C D A E B (3) C A B D E (3) A D E B C (3) A C B D E (3) D E A B C (2) D C E B A (2) D A E C B (2) C B D E A (2) B E D C A (2) B E A D C (2) B A E D C (2) A D E C B (2) A B E C D (2) E B D C A (1) D E C A B (1) D E B C A (1) D E B A C (1) D C E A B (1) D C A E B (1) C B E D A (1) C B A E D (1) C A D E B (1) B E D A C (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E C D (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 -4 -8 -2 B 2 0 -4 -6 -8 C 4 4 0 -2 0 D 8 6 2 0 6 E 2 8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -8 -2 B 2 0 -4 -6 -8 C 4 4 0 -2 0 D 8 6 2 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=27 D=20 B=15 E=8 so E is eliminated. Round 2 votes counts: A=30 D=27 C=27 B=16 so B is eliminated. Round 3 votes counts: A=39 D=31 C=30 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:203 E:202 A:192 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -8 -2 B 2 0 -4 -6 -8 C 4 4 0 -2 0 D 8 6 2 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -8 -2 B 2 0 -4 -6 -8 C 4 4 0 -2 0 D 8 6 2 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -8 -2 B 2 0 -4 -6 -8 C 4 4 0 -2 0 D 8 6 2 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7677: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) E C B A D (8) E B A D C (7) C B A D E (7) D A B C E (6) C E D A B (6) C D A B E (5) B A D E C (5) B A D C E (5) E D A B C (4) D A B E C (3) C E B A D (3) C B D A E (3) A D B E C (3) E A D B C (2) E A B D C (2) C D E A B (2) C D A E B (2) B E A D C (2) B C A D E (2) B A E D C (2) E C B D A (1) E B C A D (1) D C A E B (1) D A E C B (1) D A E B C (1) C B E D A (1) C B E A D (1) B E A C D (1) B D A C E (1) B A C D E (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -4 8 0 B 2 0 0 8 0 C 4 0 0 6 -6 D -8 -8 -6 0 0 E 0 0 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.544373 C: 0.000000 D: 0.000000 E: 0.455627 Sum of squares = 0.503937860557 Cumulative probabilities = A: 0.000000 B: 0.544373 C: 0.544373 D: 0.544373 E: 1.000000 A B C D E A 0 -2 -4 8 0 B 2 0 0 8 0 C 4 0 0 6 -6 D -8 -8 -6 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=30 B=19 D=12 A=5 so A is eliminated. Round 2 votes counts: E=34 C=30 B=20 D=16 so D is eliminated. Round 3 votes counts: E=36 B=33 C=31 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:205 E:203 C:202 A:201 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 8 0 B 2 0 0 8 0 C 4 0 0 6 -6 D -8 -8 -6 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 0 B 2 0 0 8 0 C 4 0 0 6 -6 D -8 -8 -6 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 0 B 2 0 0 8 0 C 4 0 0 6 -6 D -8 -8 -6 0 0 E 0 0 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7678: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) A B E C D (8) E C A B D (6) B A E C D (6) A B D E C (6) E A B C D (5) B A D C E (5) E A C B D (4) D E C A B (4) D A B E C (4) D C B A E (3) C E B A D (3) B A C E D (3) E C D A B (2) E C A D B (2) D C E B A (2) D C E A B (2) D C B E A (2) D B C A E (2) D A B C E (2) C E D B A (2) C E D A B (2) C D E B A (2) B D A C E (2) D A E B C (1) C D E A B (1) C D B E A (1) C B E D A (1) C B E A D (1) B A E D C (1) B A D E C (1) B A C D E (1) A E B D C (1) A E B C D (1) A D E B C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 24 14 16 B -8 0 20 12 18 C -24 -20 0 2 -10 D -14 -12 -2 0 2 E -16 -18 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 24 14 16 B -8 0 20 12 18 C -24 -20 0 2 -10 D -14 -12 -2 0 2 E -16 -18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=19 B=19 A=19 C=13 so C is eliminated. Round 2 votes counts: D=34 E=26 B=21 A=19 so A is eliminated. Round 3 votes counts: D=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:231 B:221 D:187 E:187 C:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 24 14 16 B -8 0 20 12 18 C -24 -20 0 2 -10 D -14 -12 -2 0 2 E -16 -18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 24 14 16 B -8 0 20 12 18 C -24 -20 0 2 -10 D -14 -12 -2 0 2 E -16 -18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 24 14 16 B -8 0 20 12 18 C -24 -20 0 2 -10 D -14 -12 -2 0 2 E -16 -18 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7679: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) E A D C B (6) D C A E B (6) B C A E D (6) C B A D E (5) B C E A D (5) A E D C B (5) D E B A C (4) C A D E B (4) C B A E D (3) B E D A C (3) B E C A D (3) A E C D B (3) A D E C B (3) E D A C B (2) E A D B C (2) D B E C A (2) D A C E B (2) C A E B D (2) B E A D C (2) B D E C A (2) B C D E A (2) B C D A E (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B D C (1) D E A B C (1) D B C E A (1) D A E C B (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D B E (1) C A B E D (1) B E C D A (1) B C E D A (1) B C A D E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 10 -2 14 0 B -10 0 -16 -14 -16 C 2 16 0 -10 -8 D -14 14 10 0 -4 E 0 16 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.339324 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.660676 Sum of squares = 0.551633833029 Cumulative probabilities = A: 0.339324 B: 0.339324 C: 0.339324 D: 0.339324 E: 1.000000 A B C D E A 0 10 -2 14 0 B -10 0 -16 -14 -16 C 2 16 0 -10 -8 D -14 14 10 0 -4 E 0 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=26 C=19 E=14 A=13 so A is eliminated. Round 2 votes counts: D=29 B=28 E=23 C=20 so C is eliminated. Round 3 votes counts: B=37 D=36 E=27 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:214 A:211 D:203 C:200 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -2 14 0 B -10 0 -16 -14 -16 C 2 16 0 -10 -8 D -14 14 10 0 -4 E 0 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 14 0 B -10 0 -16 -14 -16 C 2 16 0 -10 -8 D -14 14 10 0 -4 E 0 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 14 0 B -10 0 -16 -14 -16 C 2 16 0 -10 -8 D -14 14 10 0 -4 E 0 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999912 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7680: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) B C A E D (7) D B C E A (6) D A E C B (6) B C D E A (4) D E A C B (3) B D C E A (3) B D C A E (3) B A C E D (3) A E D C B (3) A E C D B (3) A D E C B (3) A B E C D (3) E D C A B (2) E C A B D (2) D C E B A (2) D B E C A (2) D B A E C (2) D A E B C (2) C E B A D (2) B C E A D (2) A C E B D (2) E D A C B (1) E C D A B (1) E C A D B (1) E A D C B (1) E A C D B (1) E A C B D (1) D E C B A (1) D E C A B (1) D E A B C (1) D C B E A (1) D B E A C (1) D B C A E (1) D A B E C (1) C E A B D (1) C D E B A (1) C B E D A (1) C B E A D (1) B D A C E (1) B C E D A (1) B C D A E (1) B C A D E (1) A E B C D (1) A D E B C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 4 2 12 B -6 0 -2 4 -6 C -4 2 0 2 -10 D -2 -4 -2 0 -2 E -12 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999236 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 12 B -6 0 -2 4 -6 C -4 2 0 2 -10 D -2 -4 -2 0 -2 E -12 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=28 B=26 E=10 C=6 so C is eliminated. Round 2 votes counts: D=31 B=28 A=28 E=13 so E is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:203 B:195 C:195 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 12 B -6 0 -2 4 -6 C -4 2 0 2 -10 D -2 -4 -2 0 -2 E -12 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 12 B -6 0 -2 4 -6 C -4 2 0 2 -10 D -2 -4 -2 0 -2 E -12 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 12 B -6 0 -2 4 -6 C -4 2 0 2 -10 D -2 -4 -2 0 -2 E -12 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7681: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) A E D C B (6) A D E C B (6) E A C B D (5) C B D E A (5) E A D C B (4) E A D B C (4) E A B D C (4) B C E D A (4) B C D A E (4) E C B A D (3) E A B C D (3) D C B A E (3) D A B C E (3) C E B A D (3) C D B A E (3) C B D A E (3) B E C D A (3) B D C A E (3) A E D B C (3) A D E B C (3) A D C E B (3) E A C D B (2) D B C A E (2) D A C B E (2) D A B E C (2) B C D E A (2) E B A C D (1) D C A B E (1) C D A B E (1) A D B E C (1) Total count = 100 A B C D E A 0 6 4 18 -2 B -6 0 0 2 -16 C -4 0 0 0 -16 D -18 -2 0 0 -6 E 2 16 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998766 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 4 18 -2 B -6 0 0 2 -16 C -4 0 0 0 -16 D -18 -2 0 0 -6 E 2 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=22 B=16 C=15 D=13 so D is eliminated. Round 2 votes counts: E=34 A=29 C=19 B=18 so B is eliminated. Round 3 votes counts: E=37 C=34 A=29 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:213 B:190 C:190 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 18 -2 B -6 0 0 2 -16 C -4 0 0 0 -16 D -18 -2 0 0 -6 E 2 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 18 -2 B -6 0 0 2 -16 C -4 0 0 0 -16 D -18 -2 0 0 -6 E 2 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 18 -2 B -6 0 0 2 -16 C -4 0 0 0 -16 D -18 -2 0 0 -6 E 2 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7682: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) A D C E B (10) C A D B E (7) C D A E B (6) B E A D C (6) C A D E B (5) B E D A C (5) E B D C A (4) D A E C B (4) B E C D A (4) E D C A B (3) B E C A D (3) E D A B C (2) D A C E B (2) C D A B E (2) C B A D E (2) B E D C A (2) B E A C D (2) B C E A D (2) A C D E B (2) A C D B E (2) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E B C D A (1) E B A D C (1) D E A C B (1) C D E A B (1) C B E D A (1) B C E D A (1) B A E D C (1) B A E C D (1) A D C B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 6 -6 -6 B 0 0 -4 -2 -12 C -6 4 0 -10 -10 D 6 2 10 0 -8 E 6 12 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 6 -6 -6 B 0 0 -4 -2 -12 C -6 4 0 -10 -10 D 6 2 10 0 -8 E 6 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 C=24 A=17 D=7 so D is eliminated. Round 2 votes counts: B=27 E=26 C=24 A=23 so A is eliminated. Round 3 votes counts: C=42 E=30 B=28 so B is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 D:205 A:197 B:191 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 -6 -6 B 0 0 -4 -2 -12 C -6 4 0 -10 -10 D 6 2 10 0 -8 E 6 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -6 -6 B 0 0 -4 -2 -12 C -6 4 0 -10 -10 D 6 2 10 0 -8 E 6 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -6 -6 B 0 0 -4 -2 -12 C -6 4 0 -10 -10 D 6 2 10 0 -8 E 6 12 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7683: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (17) D A B E C (15) E B A C D (8) C D B E A (8) C D A B E (6) E B C A D (5) D C A B E (4) A E B D C (4) A B E D C (4) C E B D A (3) C D E B A (3) D A E B C (2) C E A B D (2) C D E A B (2) C D A E B (2) B E A C D (2) E C B A D (1) E A B D C (1) E A B C D (1) D B A E C (1) D A C B E (1) D A B C E (1) C E D B A (1) C D B A E (1) B E A D C (1) B D A E C (1) B A E D C (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 -10 -2 -10 B 6 0 -2 2 -6 C 10 2 0 24 2 D 2 -2 -24 0 -4 E 10 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -2 -10 B 6 0 -2 2 -6 C 10 2 0 24 2 D 2 -2 -24 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=45 D=24 E=16 A=10 B=5 so B is eliminated. Round 2 votes counts: C=45 D=25 E=19 A=11 so A is eliminated. Round 3 votes counts: C=45 E=29 D=26 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:209 B:200 A:186 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -2 -10 B 6 0 -2 2 -6 C 10 2 0 24 2 D 2 -2 -24 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -2 -10 B 6 0 -2 2 -6 C 10 2 0 24 2 D 2 -2 -24 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -2 -10 B 6 0 -2 2 -6 C 10 2 0 24 2 D 2 -2 -24 0 -4 E 10 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7684: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) A B E D C (8) C B D E A (6) C B A E D (6) D E C B A (5) D C E A B (5) C B A D E (5) A E B D C (5) E D A B C (4) C D B E A (4) D E A B C (3) D C E B A (3) B A E C D (3) A E D B C (3) A B C E D (3) C B E D A (2) C B D A E (2) C A B D E (2) B C E D A (2) B A C E D (2) A C B E D (2) A B E C D (2) E D B A C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C A E B (1) D A E B C (1) C D A E B (1) B E C D A (1) B E A D C (1) B E A C D (1) B C A E D (1) B A E D C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -10 -2 0 B 8 0 -12 12 8 C 10 12 0 6 12 D 2 -12 -6 0 0 E 0 -8 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -2 0 B 8 0 -12 12 8 C 10 12 0 6 12 D 2 -12 -6 0 0 E 0 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=25 D=20 B=12 E=7 so E is eliminated. Round 2 votes counts: C=36 A=27 D=25 B=12 so B is eliminated. Round 3 votes counts: C=40 A=35 D=25 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:208 D:192 A:190 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 -2 0 B 8 0 -12 12 8 C 10 12 0 6 12 D 2 -12 -6 0 0 E 0 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -2 0 B 8 0 -12 12 8 C 10 12 0 6 12 D 2 -12 -6 0 0 E 0 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -2 0 B 8 0 -12 12 8 C 10 12 0 6 12 D 2 -12 -6 0 0 E 0 -8 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7685: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (8) D C A E B (7) C D E B A (7) A E B D C (7) D C E A B (5) C D B A E (5) B E A C D (5) B A E C D (5) E D C A B (4) E B A D C (4) A E D B C (4) C B D A E (3) E D C B A (2) E B A C D (2) C D B E A (2) B E C D A (2) B C D E A (2) A D C E B (2) A D C B E (2) A C D B E (2) E D A C B (1) E C D B A (1) E C B D A (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E B A (1) D C A B E (1) D A C B E (1) C D A B E (1) C A B D E (1) B E C A D (1) B C E D A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E D C (1) A D E C B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 2 12 B -4 0 -6 -4 -4 C 2 6 0 -10 -4 D -2 4 10 0 -4 E -12 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.551020408246 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 2 12 B -4 0 -6 -4 -4 C 2 6 0 -10 -4 D -2 4 10 0 -4 E -12 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040816 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=20 C=19 E=17 D=16 so D is eliminated. Round 2 votes counts: C=33 A=29 B=20 E=18 so E is eliminated. Round 3 votes counts: C=42 A=32 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:208 D:204 E:200 C:197 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 2 12 B -4 0 -6 -4 -4 C 2 6 0 -10 -4 D -2 4 10 0 -4 E -12 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040816 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 12 B -4 0 -6 -4 -4 C 2 6 0 -10 -4 D -2 4 10 0 -4 E -12 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040816 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 12 B -4 0 -6 -4 -4 C 2 6 0 -10 -4 D -2 4 10 0 -4 E -12 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.000000 Sum of squares = 0.55102040816 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7686: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (11) D B C A E (6) D B A C E (6) E A D B C (5) E A C B D (5) D E C B A (5) D E A B C (4) D C B A E (4) A B C E D (4) E C A B D (3) D C E B A (3) D C B E A (3) B C D A E (3) E D A C B (2) E C D B A (2) E A D C B (2) E A B D C (2) D A B E C (2) C B A E D (2) B A C D E (2) A E C B D (2) A B E C D (2) A B D C E (2) E D C B A (1) E D C A B (1) E D A B C (1) E C D A B (1) E C B A D (1) E C A D B (1) D B E A C (1) C E B A D (1) C D E B A (1) C D B A E (1) C B E A D (1) C B D E A (1) C B A D E (1) C A B E D (1) A E B C D (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 10 14 4 -16 B -10 0 10 -6 -12 C -14 -10 0 -6 -16 D -4 6 6 0 -6 E 16 12 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 4 -16 B -10 0 10 -6 -12 C -14 -10 0 -6 -16 D -4 6 6 0 -6 E 16 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=34 A=14 C=9 B=5 so B is eliminated. Round 2 votes counts: E=38 D=34 A=16 C=12 so C is eliminated. Round 3 votes counts: E=40 D=40 A=20 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:206 D:201 B:191 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 4 -16 B -10 0 10 -6 -12 C -14 -10 0 -6 -16 D -4 6 6 0 -6 E 16 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 4 -16 B -10 0 10 -6 -12 C -14 -10 0 -6 -16 D -4 6 6 0 -6 E 16 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 4 -16 B -10 0 10 -6 -12 C -14 -10 0 -6 -16 D -4 6 6 0 -6 E 16 12 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7687: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) E C B A D (7) C E A D B (7) E C A B D (6) B D C A E (6) E C A D B (4) E B C D A (4) E A C D B (4) D B A C E (4) B D E A C (4) A D B C E (4) D A B C E (3) B D A C E (3) B C D E A (3) E B D A C (2) E A B D C (2) B E C D A (2) B C D A E (2) A E D C B (2) A D C B E (2) A C E D B (2) E C B D A (1) E A D B C (1) D B C A E (1) D B A E C (1) D A C B E (1) C E B D A (1) C E B A D (1) C D B E A (1) C B E D A (1) C A D E B (1) C A D B E (1) B D C E A (1) B C E D A (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C E B (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -2 0 -6 B 8 0 10 10 0 C 2 -10 0 0 -4 D 0 -10 0 0 2 E 6 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.527314 C: 0.000000 D: 0.000000 E: 0.472686 Sum of squares = 0.501492136891 Cumulative probabilities = A: 0.000000 B: 0.527314 C: 0.527314 D: 0.527314 E: 1.000000 A B C D E A 0 -8 -2 0 -6 B 8 0 10 10 0 C 2 -10 0 0 -4 D 0 -10 0 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=30 A=16 C=13 D=10 so D is eliminated. Round 2 votes counts: B=36 E=31 A=20 C=13 so C is eliminated. Round 3 votes counts: E=40 B=38 A=22 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:204 D:196 C:194 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 0 -6 B 8 0 10 10 0 C 2 -10 0 0 -4 D 0 -10 0 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 -6 B 8 0 10 10 0 C 2 -10 0 0 -4 D 0 -10 0 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 -6 B 8 0 10 10 0 C 2 -10 0 0 -4 D 0 -10 0 0 2 E 6 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7688: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (13) D B E C A (11) C A E D B (8) B D E A C (8) B D A C E (8) A C E B D (6) B D E C A (5) C E A D B (4) B A D C E (4) A C B E D (4) B D C A E (3) B D A E C (3) B A C D E (3) E C D A B (2) D E B C A (2) D B E A C (2) A C E D B (2) A B C E D (2) E A C D B (1) D E C B A (1) D C E A B (1) D B C E A (1) C D A B E (1) C A B D E (1) B E A D C (1) B D C E A (1) B A C E D (1) A E C D B (1) Total count = 100 A B C D E A 0 -8 -8 2 -6 B 8 0 10 0 18 C 8 -10 0 -2 0 D -2 0 2 0 10 E 6 -18 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.623885 C: 0.000000 D: 0.376115 E: 0.000000 Sum of squares = 0.530694976809 Cumulative probabilities = A: 0.000000 B: 0.623885 C: 0.623885 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 2 -6 B 8 0 10 0 18 C 8 -10 0 -2 0 D -2 0 2 0 10 E 6 -18 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999843 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=18 E=16 A=15 C=14 so C is eliminated. Round 2 votes counts: B=37 A=24 E=20 D=19 so D is eliminated. Round 3 votes counts: B=51 A=25 E=24 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:205 C:198 A:190 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 2 -6 B 8 0 10 0 18 C 8 -10 0 -2 0 D -2 0 2 0 10 E 6 -18 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999843 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 2 -6 B 8 0 10 0 18 C 8 -10 0 -2 0 D -2 0 2 0 10 E 6 -18 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999843 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 2 -6 B 8 0 10 0 18 C 8 -10 0 -2 0 D -2 0 2 0 10 E 6 -18 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999843 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7689: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) B D A C E (8) B A E C D (8) C E D A B (6) B D C E A (6) B A D C E (6) A E C B D (6) E C A D B (4) D B C E A (4) B E C D A (4) E C D A B (3) D C E B A (3) D C E A B (3) A B E C D (3) E C D B A (2) D A C E B (2) B D C A E (2) B A E D C (2) A E C D B (2) A D C E B (2) E C A B D (1) E B C A D (1) E B A C D (1) E A C D B (1) D C B E A (1) D C A E B (1) D B A C E (1) D A C B E (1) C D E B A (1) B E D C A (1) B E C A D (1) B D E C A (1) B D E A C (1) Total count = 100 A B C D E A 0 -30 10 -2 10 B 30 0 22 26 22 C -10 -22 0 -12 -6 D 2 -26 12 0 8 E -10 -22 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 10 -2 10 B 30 0 22 26 22 C -10 -22 0 -12 -6 D 2 -26 12 0 8 E -10 -22 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=51 D=16 E=13 A=13 C=7 so C is eliminated. Round 2 votes counts: B=51 E=19 D=17 A=13 so A is eliminated. Round 3 votes counts: B=54 E=27 D=19 so D is eliminated. Round 4 votes counts: B=61 E=39 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:250 D:198 A:194 E:183 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 10 -2 10 B 30 0 22 26 22 C -10 -22 0 -12 -6 D 2 -26 12 0 8 E -10 -22 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 10 -2 10 B 30 0 22 26 22 C -10 -22 0 -12 -6 D 2 -26 12 0 8 E -10 -22 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 10 -2 10 B 30 0 22 26 22 C -10 -22 0 -12 -6 D 2 -26 12 0 8 E -10 -22 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7690: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) E A C D B (8) B D E C A (8) B D C A E (8) E B A C D (7) B E D C A (7) E A C B D (5) C A D B E (5) D C A B E (4) D B C A E (4) A C D E B (4) E B D A C (3) D A C B E (3) C A D E B (2) C A B E D (2) B D C E A (2) A E C D B (2) E C A B D (1) E B A D C (1) E A D C B (1) E A D B C (1) D E A B C (1) D B E A C (1) D B A C E (1) D A C E B (1) C B A E D (1) C A E D B (1) B E D A C (1) B E C D A (1) B D E A C (1) B C E A D (1) B C D E A (1) A D C E B (1) Total count = 100 A B C D E A 0 4 4 6 -2 B -4 0 -2 0 2 C -4 2 0 2 2 D -6 0 -2 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999832 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 4 4 6 -2 B -4 0 -2 0 2 C -4 2 0 2 2 D -6 0 -2 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999972 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 A=17 D=15 C=11 so C is eliminated. Round 2 votes counts: B=31 E=27 A=27 D=15 so D is eliminated. Round 3 votes counts: B=37 A=35 E=28 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:202 C:201 B:198 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 6 -2 B -4 0 -2 0 2 C -4 2 0 2 2 D -6 0 -2 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999972 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 6 -2 B -4 0 -2 0 2 C -4 2 0 2 2 D -6 0 -2 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999972 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 6 -2 B -4 0 -2 0 2 C -4 2 0 2 2 D -6 0 -2 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999972 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7691: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) D C A B E (11) E B A C D (10) B A D E C (8) E C B D A (7) A B D C E (7) D A C B E (5) B A E D C (5) E C D B A (4) D A B C E (4) C E D A B (4) E C B A D (3) E B C A D (3) C E D B A (2) B A E C D (2) A D B C E (2) A B D E C (2) D C E A B (1) D C A E B (1) C D A E B (1) C A D B E (1) B E A D C (1) B E A C D (1) B D E A C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 6 0 -6 2 B -6 0 -4 4 4 C 0 4 0 2 4 D 6 -4 -2 0 14 E -2 -4 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.108805 B: 0.000000 C: 0.891195 D: 0.000000 E: 0.000000 Sum of squares = 0.806066325529 Cumulative probabilities = A: 0.108805 B: 0.108805 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -6 2 B -6 0 -4 4 4 C 0 4 0 2 4 D 6 -4 -2 0 14 E -2 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000225797 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=22 C=20 B=18 A=13 so A is eliminated. Round 2 votes counts: B=29 E=27 D=24 C=20 so C is eliminated. Round 3 votes counts: D=38 E=33 B=29 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:207 C:205 A:201 B:199 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 -6 2 B -6 0 -4 4 4 C 0 4 0 2 4 D 6 -4 -2 0 14 E -2 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000225797 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -6 2 B -6 0 -4 4 4 C 0 4 0 2 4 D 6 -4 -2 0 14 E -2 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000225797 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -6 2 B -6 0 -4 4 4 C 0 4 0 2 4 D 6 -4 -2 0 14 E -2 -4 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000225797 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7692: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (18) B D E C A (14) A B E D C (8) B E D C A (6) A C D E B (5) E B D C A (4) A E C D B (4) A C D B E (4) C E A D B (3) E D C B A (2) E C D A B (2) E A B C D (2) D C E B A (2) D B E C A (2) B D C E A (2) B A D E C (2) E D B C A (1) E C B D A (1) E C A D B (1) E A C D B (1) D E B C A (1) D C B A E (1) D B C E A (1) C E D B A (1) C E D A B (1) C A E D B (1) C A D B E (1) B E A D C (1) B D E A C (1) A E C B D (1) A E B C D (1) A C E B D (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 6 16 2 B -16 0 -2 -4 -6 C -6 2 0 2 -10 D -16 4 -2 0 -22 E -2 6 10 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 16 2 B -16 0 -2 -4 -6 C -6 2 0 2 -10 D -16 4 -2 0 -22 E -2 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 B=26 E=14 D=7 C=7 so D is eliminated. Round 2 votes counts: A=46 B=29 E=15 C=10 so C is eliminated. Round 3 votes counts: A=48 B=30 E=22 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:218 C:194 B:186 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 16 2 B -16 0 -2 -4 -6 C -6 2 0 2 -10 D -16 4 -2 0 -22 E -2 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 16 2 B -16 0 -2 -4 -6 C -6 2 0 2 -10 D -16 4 -2 0 -22 E -2 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 16 2 B -16 0 -2 -4 -6 C -6 2 0 2 -10 D -16 4 -2 0 -22 E -2 6 10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996575 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7693: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) A E C D B (9) E C A B D (8) E C B D A (7) D B C E A (6) B E C D A (6) A D B E C (6) D B A C E (5) B D A C E (5) A E C B D (5) D A B C E (4) C E B D A (4) E C B A D (3) C E D B A (3) A D B C E (3) A B D E C (3) D C E B A (2) E C D B A (1) D B C A E (1) C D E B A (1) C B E D A (1) B E C A D (1) B D A E C (1) B C E D A (1) B C D E A (1) A E B C D (1) A D E C B (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -10 -16 -8 B 16 0 8 14 8 C 10 -8 0 6 -6 D 16 -14 -6 0 -2 E 8 -8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -16 -8 B 16 0 8 14 8 C 10 -8 0 6 -6 D 16 -14 -6 0 -2 E 8 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=24 E=19 D=18 C=9 so C is eliminated. Round 2 votes counts: A=30 E=26 B=25 D=19 so D is eliminated. Round 3 votes counts: B=37 A=34 E=29 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:204 C:201 D:197 A:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -16 -8 B 16 0 8 14 8 C 10 -8 0 6 -6 D 16 -14 -6 0 -2 E 8 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -16 -8 B 16 0 8 14 8 C 10 -8 0 6 -6 D 16 -14 -6 0 -2 E 8 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -16 -8 B 16 0 8 14 8 C 10 -8 0 6 -6 D 16 -14 -6 0 -2 E 8 -8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7694: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) C B A D E (7) E C B A D (6) D A E B C (6) B C A D E (6) E D A C B (5) E C D A B (5) C B A E D (5) D A B E C (4) C E D A B (4) C E B A D (3) C B D A E (3) B A D C E (3) A D B E C (3) E D C A B (2) E D A B C (2) E C A B D (2) E A D B C (2) D E A B C (2) D A B C E (2) B A D E C (2) A E D B C (2) E B A C D (1) E A D C B (1) E A B D C (1) D C B A E (1) C D B A E (1) C B E D A (1) C B D E A (1) B E A D C (1) B C E A D (1) B C A E D (1) B A C D E (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -20 22 -2 B 10 0 -16 14 10 C 20 16 0 18 2 D -22 -14 -18 0 -12 E 2 -10 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 22 -2 B 10 0 -16 14 10 C 20 16 0 18 2 D -22 -14 -18 0 -12 E 2 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=27 D=15 B=15 A=7 so A is eliminated. Round 2 votes counts: C=36 E=29 D=19 B=16 so B is eliminated. Round 3 votes counts: C=45 E=30 D=25 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:209 E:201 A:195 D:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -20 22 -2 B 10 0 -16 14 10 C 20 16 0 18 2 D -22 -14 -18 0 -12 E 2 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 22 -2 B 10 0 -16 14 10 C 20 16 0 18 2 D -22 -14 -18 0 -12 E 2 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 22 -2 B 10 0 -16 14 10 C 20 16 0 18 2 D -22 -14 -18 0 -12 E 2 -10 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7695: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) E D C A B (7) C B A E D (7) B C A D E (7) A B D E C (6) E D A C B (5) C E B D A (5) D E A C B (4) E C D B A (3) C B E A D (3) B A D C E (3) A E D B C (3) A D E B C (3) E D A B C (2) E C D A B (2) D E C A B (2) D E A B C (2) D A E B C (2) C E B A D (2) B C D A E (2) B C A E D (2) A D B E C (2) E D C B A (1) D E C B A (1) D C E B A (1) D A B E C (1) C E D A B (1) C D B E A (1) C B D E A (1) C B A D E (1) B D C E A (1) B A D E C (1) B A C D E (1) A E D C B (1) A E B C D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -22 -10 -10 B 8 0 -18 -10 -18 C 22 18 0 4 0 D 10 10 -4 0 -14 E 10 18 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.477390 D: 0.000000 E: 0.522610 Sum of squares = 0.501022456029 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.477390 D: 0.477390 E: 1.000000 A B C D E A 0 -8 -22 -10 -10 B 8 0 -18 -10 -18 C 22 18 0 4 0 D 10 10 -4 0 -14 E 10 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=20 A=18 B=17 D=13 so D is eliminated. Round 2 votes counts: C=33 E=29 A=21 B=17 so B is eliminated. Round 3 votes counts: C=45 E=29 A=26 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:221 D:201 B:181 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -22 -10 -10 B 8 0 -18 -10 -18 C 22 18 0 4 0 D 10 10 -4 0 -14 E 10 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -22 -10 -10 B 8 0 -18 -10 -18 C 22 18 0 4 0 D 10 10 -4 0 -14 E 10 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -22 -10 -10 B 8 0 -18 -10 -18 C 22 18 0 4 0 D 10 10 -4 0 -14 E 10 18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7696: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (18) E D B C A (7) A B C E D (7) A C D B E (5) E B D A C (4) D E C B A (4) C A D B E (4) A C D E B (4) E D C B A (3) E B A D C (3) D C E A B (3) A B E C D (3) E B D C A (2) E A B D C (2) D E B C A (2) C D B A E (2) C D A E B (2) C A B D E (2) B E D C A (2) B E A D C (2) B C A D E (2) B A E C D (2) B A C E D (2) A E B D C (2) A C E D B (2) E D A C B (1) E A D C B (1) E A D B C (1) D C E B A (1) C D A B E (1) C A D E B (1) B A C D E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 22 24 32 24 B -22 0 -10 12 10 C -24 10 0 20 18 D -32 -12 -20 0 6 E -24 -10 -18 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 24 32 24 B -22 0 -10 12 10 C -24 10 0 20 18 D -32 -12 -20 0 6 E -24 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 E=24 C=12 B=11 D=10 so D is eliminated. Round 2 votes counts: A=43 E=30 C=16 B=11 so B is eliminated. Round 3 votes counts: A=48 E=34 C=18 so C is eliminated. Round 4 votes counts: A=62 E=38 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:251 C:212 B:195 D:171 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 24 32 24 B -22 0 -10 12 10 C -24 10 0 20 18 D -32 -12 -20 0 6 E -24 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 24 32 24 B -22 0 -10 12 10 C -24 10 0 20 18 D -32 -12 -20 0 6 E -24 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 24 32 24 B -22 0 -10 12 10 C -24 10 0 20 18 D -32 -12 -20 0 6 E -24 -10 -18 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7697: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) C D E B A (6) E A B D C (5) B D A C E (5) B A D C E (5) A E B D C (5) A E B C D (5) A B E D C (5) D C B E A (4) C E D B A (4) A B D C E (4) E C D A B (3) E C A D B (3) E A C B D (3) C E D A B (3) C D B E A (3) B D C A E (3) B A D E C (3) A B D E C (3) E A C D B (2) D B C E A (2) C D B A E (2) B D A E C (2) A E C B D (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E B D A C (1) E A D B C (1) E A B C D (1) C B D A E (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 10 -2 10 B 6 0 22 10 2 C -10 -22 0 -18 4 D 2 -10 18 0 4 E -10 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999054 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 -2 10 B 6 0 22 10 2 C -10 -22 0 -18 4 D 2 -10 18 0 4 E -10 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 C=19 B=18 D=14 so D is eliminated. Round 2 votes counts: B=28 A=26 E=23 C=23 so E is eliminated. Round 3 votes counts: A=38 C=31 B=31 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:207 A:206 E:190 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 -2 10 B 6 0 22 10 2 C -10 -22 0 -18 4 D 2 -10 18 0 4 E -10 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 -2 10 B 6 0 22 10 2 C -10 -22 0 -18 4 D 2 -10 18 0 4 E -10 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 -2 10 B 6 0 22 10 2 C -10 -22 0 -18 4 D 2 -10 18 0 4 E -10 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993584 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7698: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) B C E A D (6) D A E C B (5) C B A E D (5) B E C A D (5) A D E B C (5) A D C B E (5) E B C D A (4) D A E B C (4) B E C D A (4) A D E C B (4) A C D B E (4) E B D C A (3) C D B E A (3) A D C E B (3) A C B E D (3) E D B A C (2) E B C A D (2) E B A D C (2) E A D B C (2) D E B C A (2) D C A B E (2) D A C B E (2) C B E A D (2) C B D E A (2) C A D B E (2) E B D A C (1) D E B A C (1) D E A B C (1) D C A E B (1) C D B A E (1) C D A B E (1) C B A D E (1) C A B D E (1) B E A C D (1) B C E D A (1) Total count = 100 A B C D E A 0 -10 -10 6 -2 B 10 0 -8 0 16 C 10 8 0 10 4 D -6 0 -10 0 0 E 2 -16 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 6 -2 B 10 0 -8 0 16 C 10 8 0 10 4 D -6 0 -10 0 0 E 2 -16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=24 D=18 B=17 E=16 so E is eliminated. Round 2 votes counts: B=29 A=26 C=25 D=20 so D is eliminated. Round 3 votes counts: A=38 B=34 C=28 so C is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:209 A:192 D:192 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 6 -2 B 10 0 -8 0 16 C 10 8 0 10 4 D -6 0 -10 0 0 E 2 -16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 6 -2 B 10 0 -8 0 16 C 10 8 0 10 4 D -6 0 -10 0 0 E 2 -16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 6 -2 B 10 0 -8 0 16 C 10 8 0 10 4 D -6 0 -10 0 0 E 2 -16 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7699: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) E C A D B (6) A D B C E (5) E B C D A (4) E B C A D (4) B E C D A (4) B E A D C (4) B D E C A (4) A C D E B (4) E C D A B (3) D A C B E (3) C E D A B (3) B E D C A (3) A D C B E (3) A B E C D (3) E C B A D (2) E B A C D (2) D C E B A (2) C E A D B (2) C D E A B (2) C D A E B (2) C A D E B (2) B E D A C (2) B E A C D (2) B A E D C (2) B A D E C (2) B A D C E (2) A D C E B (2) A C E D B (2) A B D C E (2) E C A B D (1) D C A E B (1) D C A B E (1) D B C E A (1) D A B C E (1) C E D B A (1) B D C E A (1) B D A E C (1) B D A C E (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 -10 8 -18 B 0 0 2 6 -4 C 10 -2 0 12 -14 D -8 -6 -12 0 -14 E 18 4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -10 8 -18 B 0 0 2 6 -4 C 10 -2 0 12 -14 D -8 -6 -12 0 -14 E 18 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=28 B=28 A=23 C=12 D=9 so D is eliminated. Round 2 votes counts: B=29 E=28 A=27 C=16 so C is eliminated. Round 3 votes counts: E=38 A=33 B=29 so B is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 C:203 B:202 A:190 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 8 -18 B 0 0 2 6 -4 C 10 -2 0 12 -14 D -8 -6 -12 0 -14 E 18 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 8 -18 B 0 0 2 6 -4 C 10 -2 0 12 -14 D -8 -6 -12 0 -14 E 18 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 8 -18 B 0 0 2 6 -4 C 10 -2 0 12 -14 D -8 -6 -12 0 -14 E 18 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7700: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (17) D C A E B (8) D A B E C (8) B E A C D (8) D B E A C (4) D B A E C (4) C E B A D (4) C E A B D (4) B D E A C (4) C D E B A (3) C A D E B (3) E B A C D (2) D C A B E (2) D B E C A (2) D A C B E (2) D A B C E (2) C D E A B (2) C A E B D (2) B E D A C (2) B E A D C (2) A C E B D (2) E C B A D (1) E C A B D (1) E B C A D (1) E A C B D (1) E A B C D (1) D C B E A (1) D B C E A (1) C E D B A (1) C E B D A (1) B E C D A (1) B A E D C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 14 -10 -30 6 B -14 0 -12 -20 -12 C 10 12 0 14 10 D 30 20 -14 0 26 E -6 12 -10 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 -30 6 B -14 0 -12 -20 -12 C 10 12 0 14 10 D 30 20 -14 0 26 E -6 12 -10 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=34 B=18 E=7 A=4 so A is eliminated. Round 2 votes counts: C=39 D=34 B=18 E=9 so E is eliminated. Round 3 votes counts: C=43 D=34 B=23 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:231 C:223 A:190 E:185 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 -30 6 B -14 0 -12 -20 -12 C 10 12 0 14 10 D 30 20 -14 0 26 E -6 12 -10 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 -30 6 B -14 0 -12 -20 -12 C 10 12 0 14 10 D 30 20 -14 0 26 E -6 12 -10 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 -30 6 B -14 0 -12 -20 -12 C 10 12 0 14 10 D 30 20 -14 0 26 E -6 12 -10 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999369 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7701: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) A C E B D (9) D B E C A (8) B D A C E (8) E C D A B (7) D B E A C (7) A C B D E (7) C E A D B (6) E D B C A (5) C A E B D (5) A C B E D (5) E D C B A (4) B D E C A (3) D E B C A (2) C A E D B (2) B D E A C (2) B D A E C (2) A B C D E (2) E D A C B (1) E A C D B (1) D B A E C (1) C E A B D (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -6 0 -14 B -16 0 -18 -10 -8 C 6 18 0 12 -6 D 0 10 -12 0 -14 E 14 8 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -6 0 -14 B -16 0 -18 -10 -8 C 6 18 0 12 -6 D 0 10 -12 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 D=18 B=15 C=14 so C is eliminated. Round 2 votes counts: E=35 A=32 D=18 B=15 so B is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:215 A:198 D:192 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -6 0 -14 B -16 0 -18 -10 -8 C 6 18 0 12 -6 D 0 10 -12 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -6 0 -14 B -16 0 -18 -10 -8 C 6 18 0 12 -6 D 0 10 -12 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -6 0 -14 B -16 0 -18 -10 -8 C 6 18 0 12 -6 D 0 10 -12 0 -14 E 14 8 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7702: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (11) B E A D C (7) A D C B E (7) C E D A B (6) E D A C B (4) B E C D A (4) A D E B C (4) A D C E B (4) A B D E C (4) E C D A B (3) E C B D A (3) E B C D A (3) D A C E B (3) C D A E B (3) C B A D E (3) B C E D A (3) E D C A B (2) E D B A C (2) D E A C B (2) C E B D A (2) B E D A C (2) B C A D E (2) A D B E C (2) A D B C E (2) A B D C E (2) E C D B A (1) E B D C A (1) D C A E B (1) D A E C B (1) C A D E B (1) B E C A D (1) B C A E D (1) B A E D C (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 20 8 8 B -4 0 6 2 8 C -20 -6 0 -24 -16 D -8 -2 24 0 8 E -8 -8 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 20 8 8 B -4 0 6 2 8 C -20 -6 0 -24 -16 D -8 -2 24 0 8 E -8 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=26 E=19 C=15 D=7 so D is eliminated. Round 2 votes counts: B=33 A=30 E=21 C=16 so C is eliminated. Round 3 votes counts: B=36 A=35 E=29 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:211 B:206 E:196 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 8 8 B -4 0 6 2 8 C -20 -6 0 -24 -16 D -8 -2 24 0 8 E -8 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 8 8 B -4 0 6 2 8 C -20 -6 0 -24 -16 D -8 -2 24 0 8 E -8 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 8 8 B -4 0 6 2 8 C -20 -6 0 -24 -16 D -8 -2 24 0 8 E -8 -8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7703: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (8) A E C B D (7) A B C E D (7) E C D B A (6) D B C E A (6) E D C B A (4) E A C D B (4) B D C A E (4) B A D C E (4) E C A D B (3) E C A B D (3) D E C B A (3) D B C A E (3) A C B E D (3) E D A C B (2) D C E B A (2) A E D B C (2) A E B C D (2) A D E B C (2) A B D E C (2) A B C D E (2) E D C A B (1) E C D A B (1) E C B D A (1) E A D C B (1) E A C B D (1) D E B C A (1) D B E C A (1) D B A E C (1) D B A C E (1) C E D B A (1) C B D E A (1) B D A C E (1) B C A D E (1) B A C D E (1) A E D C B (1) A E C D B (1) A E B D C (1) A D B E C (1) A D B C E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 16 16 20 16 B -16 0 6 2 -2 C -16 -6 0 -6 -6 D -20 -2 6 0 -8 E -16 2 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 20 16 B -16 0 6 2 -2 C -16 -6 0 -6 -6 D -20 -2 6 0 -8 E -16 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=27 D=18 B=11 C=2 so C is eliminated. Round 2 votes counts: A=42 E=28 D=18 B=12 so B is eliminated. Round 3 votes counts: A=48 E=28 D=24 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:200 B:195 D:188 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 20 16 B -16 0 6 2 -2 C -16 -6 0 -6 -6 D -20 -2 6 0 -8 E -16 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 20 16 B -16 0 6 2 -2 C -16 -6 0 -6 -6 D -20 -2 6 0 -8 E -16 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 20 16 B -16 0 6 2 -2 C -16 -6 0 -6 -6 D -20 -2 6 0 -8 E -16 2 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7704: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (7) A B E D C (7) E D C A B (6) E C D B A (6) C D B E A (6) A B E C D (6) A B D C E (6) D C E B A (5) E C B D A (4) D C B E A (4) B C D A E (4) A E B D C (4) E A D C B (3) D C E A B (3) B A D C E (3) E C D A B (2) E A C D B (2) E A B C D (2) B A C D E (2) E D C B A (1) E C A D B (1) E A D B C (1) E A B D C (1) D C A E B (1) D C A B E (1) D A C E B (1) C E D B A (1) C D E B A (1) C D B A E (1) C B D E A (1) B E A C D (1) B C D E A (1) B A C E D (1) A E D C B (1) A E B C D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -12 -12 -4 B -2 0 -16 -10 6 C 12 16 0 -14 0 D 12 10 14 0 -2 E 4 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.333333 E: 0.555556 Sum of squares = 0.432098765433 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.444444 E: 1.000000 A B C D E A 0 2 -12 -12 -4 B -2 0 -16 -10 6 C 12 16 0 -14 0 D 12 10 14 0 -2 E 4 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.333333 E: 0.555556 Sum of squares = 0.432098765354 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 D=22 B=12 C=10 so C is eliminated. Round 2 votes counts: E=30 D=30 A=27 B=13 so B is eliminated. Round 3 votes counts: D=36 A=33 E=31 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:207 E:200 B:189 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -12 -12 -4 B -2 0 -16 -10 6 C 12 16 0 -14 0 D 12 10 14 0 -2 E 4 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.333333 E: 0.555556 Sum of squares = 0.432098765354 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.444444 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -12 -4 B -2 0 -16 -10 6 C 12 16 0 -14 0 D 12 10 14 0 -2 E 4 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.333333 E: 0.555556 Sum of squares = 0.432098765354 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -12 -4 B -2 0 -16 -10 6 C 12 16 0 -14 0 D 12 10 14 0 -2 E 4 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.333333 E: 0.555556 Sum of squares = 0.432098765354 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.444444 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7705: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E C A D B (7) B D A E C (6) D A B E C (5) E D A B C (4) C E B A D (4) C E A B D (4) A D C B E (4) E D C A B (3) D E A B C (3) D A E B C (3) C B A D E (3) B C D A E (3) B C A D E (3) A D B C E (3) E D A C B (2) E C D A B (2) E C B A D (2) E B C D A (2) E A C D B (2) D B A E C (2) C E A D B (2) C B E D A (2) B E D C A (2) B D A C E (2) B C E D A (2) E B D C A (1) E A D C B (1) D B A C E (1) D A B C E (1) C B E A D (1) C B A E D (1) C A E D B (1) C A D E B (1) C A B D E (1) B D E A C (1) B A D C E (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -8 -8 -8 B -8 0 -8 -2 -10 C 8 8 0 4 -14 D 8 2 -4 0 -4 E 8 10 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -8 -8 -8 B -8 0 -8 -2 -10 C 8 8 0 4 -14 D 8 2 -4 0 -4 E 8 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=20 B=20 D=15 A=12 so A is eliminated. Round 2 votes counts: E=33 D=25 C=22 B=20 so B is eliminated. Round 3 votes counts: E=35 D=35 C=30 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:203 D:201 A:192 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -8 -8 -8 B -8 0 -8 -2 -10 C 8 8 0 4 -14 D 8 2 -4 0 -4 E 8 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -8 -8 B -8 0 -8 -2 -10 C 8 8 0 4 -14 D 8 2 -4 0 -4 E 8 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -8 -8 B -8 0 -8 -2 -10 C 8 8 0 4 -14 D 8 2 -4 0 -4 E 8 10 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7706: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) E C D B A (7) C E D B A (5) A D C E B (5) D A E C B (4) B C E A D (4) B A D E C (4) B A C E D (4) A D B E C (4) A C B E D (4) A B D E C (4) C E D A B (3) B C E D A (3) A C D E B (3) A B D C E (3) A B C E D (3) D E B C A (2) D E A C B (2) D B E C A (2) D A E B C (2) C E B A D (2) C B E A D (2) C A E D B (2) B E D C A (2) B C A E D (2) B A C D E (2) E D C B A (1) E C B D A (1) D E C A B (1) C E A B D (1) C B E D A (1) B E C D A (1) B D E C A (1) B D E A C (1) B A E C D (1) A D E C B (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -2 6 0 B 12 0 -8 8 -2 C 2 8 0 18 16 D -6 -8 -18 0 -14 E 0 2 -16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998353 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 6 0 B 12 0 -8 8 -2 C 2 8 0 18 16 D -6 -8 -18 0 -14 E 0 2 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=25 C=24 D=13 E=9 so E is eliminated. Round 2 votes counts: C=32 A=29 B=25 D=14 so D is eliminated. Round 3 votes counts: A=37 C=34 B=29 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:205 E:200 A:196 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 6 0 B 12 0 -8 8 -2 C 2 8 0 18 16 D -6 -8 -18 0 -14 E 0 2 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 6 0 B 12 0 -8 8 -2 C 2 8 0 18 16 D -6 -8 -18 0 -14 E 0 2 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 6 0 B 12 0 -8 8 -2 C 2 8 0 18 16 D -6 -8 -18 0 -14 E 0 2 -16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7707: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) C E D A B (10) A B D E C (9) D E C A B (8) B A C D E (7) C E B D A (5) B C A E D (5) E D C A B (4) C B E D A (4) A D B E C (4) E C D A B (3) B A D E C (3) D E A C B (2) D A E C B (2) C B E A D (2) C A E B D (2) B C E A D (2) B A D C E (2) B A C E D (2) A D E B C (2) E C D B A (1) D E A B C (1) B E C D A (1) B E C A D (1) B C E D A (1) B A E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -26 -12 -18 B 2 0 -14 -2 -8 C 26 14 0 22 16 D 12 2 -22 0 -16 E 18 8 -16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -26 -12 -18 B 2 0 -14 -2 -8 C 26 14 0 22 16 D 12 2 -22 0 -16 E 18 8 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=25 A=17 D=13 E=8 so E is eliminated. Round 2 votes counts: C=41 B=25 D=17 A=17 so D is eliminated. Round 3 votes counts: C=53 B=25 A=22 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:239 E:213 B:189 D:188 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -26 -12 -18 B 2 0 -14 -2 -8 C 26 14 0 22 16 D 12 2 -22 0 -16 E 18 8 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -26 -12 -18 B 2 0 -14 -2 -8 C 26 14 0 22 16 D 12 2 -22 0 -16 E 18 8 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -26 -12 -18 B 2 0 -14 -2 -8 C 26 14 0 22 16 D 12 2 -22 0 -16 E 18 8 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7708: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (14) B C A E D (14) E A C B D (9) C A E B D (9) D B E C A (6) E A C D B (5) B A C E D (5) D B C A E (4) E D A C B (3) D B E A C (3) C B A E D (3) D E C A B (2) D E B A C (2) D E A B C (2) B A E C D (2) A C E B D (2) E C A D B (1) E B D A C (1) E A D B C (1) E A B C D (1) D C A E B (1) D B C E A (1) C A D B E (1) C A B E D (1) B E D A C (1) B E A D C (1) B D E A C (1) B D C A E (1) B C D A E (1) B C A D E (1) A E C B D (1) Total count = 100 A B C D E A 0 6 8 14 -8 B -6 0 -4 8 -8 C -8 4 0 12 -12 D -14 -8 -12 0 -20 E 8 8 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 14 -8 B -6 0 -4 8 -8 C -8 4 0 12 -12 D -14 -8 -12 0 -20 E 8 8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=27 E=21 C=14 A=3 so A is eliminated. Round 2 votes counts: D=35 B=27 E=22 C=16 so C is eliminated. Round 3 votes counts: D=36 E=33 B=31 so B is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:210 C:198 B:195 D:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 14 -8 B -6 0 -4 8 -8 C -8 4 0 12 -12 D -14 -8 -12 0 -20 E 8 8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 14 -8 B -6 0 -4 8 -8 C -8 4 0 12 -12 D -14 -8 -12 0 -20 E 8 8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 14 -8 B -6 0 -4 8 -8 C -8 4 0 12 -12 D -14 -8 -12 0 -20 E 8 8 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7709: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A D C E B (8) E B C D A (7) A D B C E (7) B A D E C (6) E C A D B (5) C E B D A (5) A D C B E (5) C E D A B (4) C D A E B (4) B E A D C (4) E C D A B (3) E C B D A (3) D A C E B (3) D A C B E (3) E B C A D (2) E A C D B (2) B C E D A (2) A D E C B (2) A D B E C (2) E C B A D (1) D C A E B (1) D C A B E (1) C D E A B (1) C D B A E (1) C B D E A (1) B E C A D (1) B D A C E (1) B A E D C (1) B A D C E (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -6 -2 -4 B -6 0 -8 -8 -6 C 6 8 0 8 -2 D 2 8 -8 0 -2 E 4 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -6 -2 -4 B -6 0 -8 -8 -6 C 6 8 0 8 -2 D 2 8 -8 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=26 E=23 C=16 D=8 so D is eliminated. Round 2 votes counts: A=32 B=27 E=23 C=18 so C is eliminated. Round 3 votes counts: A=38 E=33 B=29 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:207 D:200 A:197 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 -2 -4 B -6 0 -8 -8 -6 C 6 8 0 8 -2 D 2 8 -8 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -2 -4 B -6 0 -8 -8 -6 C 6 8 0 8 -2 D 2 8 -8 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -2 -4 B -6 0 -8 -8 -6 C 6 8 0 8 -2 D 2 8 -8 0 -2 E 4 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7710: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (6) D A B C E (6) B E C A D (6) A C D E B (6) A C B E D (6) D E B C A (5) D A C B E (5) E B C D A (4) E B C A D (4) D B E A C (4) A C E B D (4) D B A E C (3) C E A B D (3) C A E B D (3) B E C D A (3) B C E A D (3) A D C E B (3) E C B A D (2) E B D C A (2) D E B A C (2) D B E C A (2) D A B E C (2) C E B A D (2) C A B E D (2) B E D C A (2) A D C B E (2) E D B C A (1) D A E C B (1) C E D A B (1) B D E C A (1) B A D E C (1) A D B C E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 0 6 B -6 0 6 -2 2 C -8 -6 0 0 10 D 0 2 0 0 4 E -6 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.345580 B: 0.000000 C: 0.000000 D: 0.654420 E: 0.000000 Sum of squares = 0.547691134186 Cumulative probabilities = A: 0.345580 B: 0.345580 C: 0.345580 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 0 6 B -6 0 6 -2 2 C -8 -6 0 0 10 D 0 2 0 0 4 E -6 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=24 B=16 E=13 C=11 so C is eliminated. Round 2 votes counts: D=36 A=29 E=19 B=16 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:203 B:200 C:198 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 0 6 B -6 0 6 -2 2 C -8 -6 0 0 10 D 0 2 0 0 4 E -6 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 0 6 B -6 0 6 -2 2 C -8 -6 0 0 10 D 0 2 0 0 4 E -6 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 0 6 B -6 0 6 -2 2 C -8 -6 0 0 10 D 0 2 0 0 4 E -6 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7711: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) C D B A E (7) E A C D B (6) D B C A E (6) A E B C D (6) D C B A E (4) B A C D E (4) A B E C D (4) E D C A B (3) E D A C B (3) E A C B D (3) D C E B A (3) C D E A B (3) C A B D E (3) B D C A E (3) D E C B A (2) D B C E A (2) C E D A B (2) C D A B E (2) B A E D C (2) B A E C D (2) B A D E C (2) A E C B D (2) E C D A B (1) E B D A C (1) E B A D C (1) E A B D C (1) D C B E A (1) D B E C A (1) C D B E A (1) C D A E B (1) C B D A E (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D A C (1) B D A C E (1) B A D C E (1) A C B D E (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 2 2 16 B -6 0 -4 0 8 C -2 4 0 22 2 D -2 0 -22 0 6 E -16 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 2 16 B -6 0 -4 0 8 C -2 4 0 22 2 D -2 0 -22 0 6 E -16 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=23 D=19 B=16 A=16 so B is eliminated. Round 2 votes counts: E=27 A=27 D=23 C=23 so D is eliminated. Round 3 votes counts: C=42 E=30 A=28 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:213 B:199 D:191 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 2 16 B -6 0 -4 0 8 C -2 4 0 22 2 D -2 0 -22 0 6 E -16 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 16 B -6 0 -4 0 8 C -2 4 0 22 2 D -2 0 -22 0 6 E -16 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 16 B -6 0 -4 0 8 C -2 4 0 22 2 D -2 0 -22 0 6 E -16 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7712: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (11) E A D C B (10) D A B E C (6) C B E A D (6) C B A D E (5) E C A D B (4) C B E D A (4) B D A E C (4) B D A C E (4) E A D B C (3) D A E B C (3) C E B D A (3) C E B A D (3) C A D B E (3) B E D A C (3) A D E B C (3) A D B E C (3) E D A B C (2) C E A D B (2) C B D A E (2) A D E C B (2) E D A C B (1) E C D A B (1) E C B D A (1) E B D C A (1) E B D A C (1) E A C D B (1) D E A B C (1) C B A E D (1) B E C D A (1) B D E C A (1) B D E A C (1) B A D C E (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 2 -2 0 B 6 0 0 6 14 C -2 0 0 -4 -6 D 2 -6 4 0 4 E 0 -14 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.581213 C: 0.418787 D: 0.000000 E: 0.000000 Sum of squares = 0.513191033857 Cumulative probabilities = A: 0.000000 B: 0.581213 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -2 0 B 6 0 0 6 14 C -2 0 0 -4 -6 D 2 -6 4 0 4 E 0 -14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=26 E=25 D=10 A=10 so D is eliminated. Round 2 votes counts: C=29 E=26 B=26 A=19 so A is eliminated. Round 3 votes counts: B=36 E=34 C=30 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:202 A:197 C:194 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -2 0 B 6 0 0 6 14 C -2 0 0 -4 -6 D 2 -6 4 0 4 E 0 -14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 0 B 6 0 0 6 14 C -2 0 0 -4 -6 D 2 -6 4 0 4 E 0 -14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 0 B 6 0 0 6 14 C -2 0 0 -4 -6 D 2 -6 4 0 4 E 0 -14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7713: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) D C E B A (9) D E B C A (8) C A D B E (8) A C B D E (8) D E C B A (7) A C B E D (7) C D A E B (5) A B E C D (5) C D E A B (4) B E D A C (4) C A E D B (3) B A E D C (3) E D B A C (2) E B A C D (2) D C B A E (2) C D A B E (2) C A D E B (2) B E A D C (2) E D B C A (1) E C D B A (1) E C D A B (1) E B A D C (1) C A E B D (1) C A B D E (1) B D E A C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 -12 -4 B 4 0 -22 -10 -12 C 10 22 0 2 6 D 12 10 -2 0 14 E 4 12 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -12 -4 B 4 0 -22 -10 -12 C 10 22 0 2 6 D 12 10 -2 0 14 E 4 12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 A=21 E=17 B=10 so B is eliminated. Round 2 votes counts: D=27 C=26 A=24 E=23 so E is eliminated. Round 3 votes counts: D=43 A=29 C=28 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:220 D:217 E:198 A:185 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -12 -4 B 4 0 -22 -10 -12 C 10 22 0 2 6 D 12 10 -2 0 14 E 4 12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -12 -4 B 4 0 -22 -10 -12 C 10 22 0 2 6 D 12 10 -2 0 14 E 4 12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -12 -4 B 4 0 -22 -10 -12 C 10 22 0 2 6 D 12 10 -2 0 14 E 4 12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7714: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (7) E C D A B (5) C A D E B (5) E D C A B (4) E C A D B (4) E B D C A (4) E B A C D (4) E A C D B (4) E A B C D (4) B D A C E (4) B A E C D (4) A C D B E (4) E D C B A (3) D C E A B (3) B D C A E (3) E A C B D (2) D C A B E (2) B E A D C (2) B E A C D (2) A E C D B (2) A C E D B (2) A C D E B (2) A C B D E (2) E D B C A (1) E C A B D (1) E B D A C (1) E B A D C (1) D E C B A (1) D C E B A (1) D C B E A (1) D C B A E (1) D C A E B (1) D B C A E (1) C D E A B (1) C D A E B (1) C D A B E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D C E A (1) B D A E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -8 0 -18 B -4 0 -8 -2 -14 C 8 8 0 8 -18 D 0 2 -8 0 -6 E 18 14 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -8 0 -18 B -4 0 -8 -2 -14 C 8 8 0 8 -18 D 0 2 -8 0 -6 E 18 14 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 B=29 A=13 D=11 C=9 so C is eliminated. Round 2 votes counts: E=38 B=29 A=19 D=14 so D is eliminated. Round 3 votes counts: E=44 B=32 A=24 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:203 D:194 A:189 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -8 0 -18 B -4 0 -8 -2 -14 C 8 8 0 8 -18 D 0 2 -8 0 -6 E 18 14 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 0 -18 B -4 0 -8 -2 -14 C 8 8 0 8 -18 D 0 2 -8 0 -6 E 18 14 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 0 -18 B -4 0 -8 -2 -14 C 8 8 0 8 -18 D 0 2 -8 0 -6 E 18 14 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999513 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7715: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) B D E A C (7) D B C A E (6) B E C A D (6) B D C A E (6) E A C D B (5) C A E D B (5) C A D E B (5) E C A D B (4) D C A B E (4) B E A C D (4) E C A B D (3) E B A C D (3) D A C E B (3) B D A C E (3) A C E D B (3) E A C B D (2) D C A E B (2) D A C B E (2) A C D E B (2) E D A C B (1) E B C A D (1) D C B A E (1) D B A C E (1) C E A D B (1) C E A B D (1) C D B A E (1) C D A B E (1) B E C D A (1) B D E C A (1) B D C E A (1) B D A E C (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 0 -8 -6 B 10 0 6 4 16 C 0 -6 0 -4 -2 D 8 -4 4 0 -4 E 6 -16 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 -8 -6 B 10 0 6 4 16 C 0 -6 0 -4 -2 D 8 -4 4 0 -4 E 6 -16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 E=19 D=19 C=14 A=6 so A is eliminated. Round 2 votes counts: B=42 D=20 E=19 C=19 so E is eliminated. Round 3 votes counts: B=46 C=33 D=21 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:202 E:198 C:194 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 -8 -6 B 10 0 6 4 16 C 0 -6 0 -4 -2 D 8 -4 4 0 -4 E 6 -16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 -8 -6 B 10 0 6 4 16 C 0 -6 0 -4 -2 D 8 -4 4 0 -4 E 6 -16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 -8 -6 B 10 0 6 4 16 C 0 -6 0 -4 -2 D 8 -4 4 0 -4 E 6 -16 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999736 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7716: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (14) E D C A B (14) A B C D E (14) B C A D E (9) B A C D E (7) E D A C B (6) D E C A B (5) C B A D E (3) B C E D A (3) B A C E D (3) A E D B C (3) B C A E D (2) A D E C B (2) A C B D E (2) A B D E C (2) A B C E D (2) D E C B A (1) D E A C B (1) D A C E B (1) C E D B A (1) C D A E B (1) C B E D A (1) B C D E A (1) A E D C B (1) A B E D C (1) Total count = 100 A B C D E A 0 10 -10 2 6 B -10 0 -6 0 0 C 10 6 0 -2 0 D -2 0 2 0 -2 E -6 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.459448 D: 0.000000 E: 0.540552 Sum of squares = 0.503288893462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.459448 D: 0.459448 E: 1.000000 A B C D E A 0 10 -10 2 6 B -10 0 -6 0 0 C 10 6 0 -2 0 D -2 0 2 0 -2 E -6 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=27 B=25 D=8 C=6 so C is eliminated. Round 2 votes counts: E=35 B=29 A=27 D=9 so D is eliminated. Round 3 votes counts: E=42 B=29 A=29 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:204 D:199 E:198 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 2 6 B -10 0 -6 0 0 C 10 6 0 -2 0 D -2 0 2 0 -2 E -6 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 2 6 B -10 0 -6 0 0 C 10 6 0 -2 0 D -2 0 2 0 -2 E -6 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 2 6 B -10 0 -6 0 0 C 10 6 0 -2 0 D -2 0 2 0 -2 E -6 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7717: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) E A C D B (7) A E D B C (7) E A D B C (6) C B D A E (6) D B C E A (5) B D C A E (5) D B E A C (4) D B A E C (4) C E A B D (4) E C A B D (3) E D A B C (2) E C D A B (2) D E B C A (2) D B C A E (2) D B A C E (2) C B E A D (2) B D C E A (2) B C D E A (2) B C D A E (2) A D B E C (2) E D B C A (1) E D A C B (1) E C A D B (1) E A C B D (1) D E B A C (1) D E A B C (1) D C B E A (1) D B E C A (1) D A E B C (1) D A B E C (1) C E D B A (1) C A E B D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -10 -22 -24 B 10 0 6 -14 8 C 10 -6 0 -4 -2 D 22 14 4 0 14 E 24 -8 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -22 -24 B 10 0 6 -14 8 C 10 -6 0 -4 -2 D 22 14 4 0 14 E 24 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 E=24 A=14 B=11 so B is eliminated. Round 2 votes counts: D=32 C=30 E=24 A=14 so A is eliminated. Round 3 votes counts: E=34 D=34 C=32 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:227 B:205 E:202 C:199 A:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -10 -22 -24 B 10 0 6 -14 8 C 10 -6 0 -4 -2 D 22 14 4 0 14 E 24 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -22 -24 B 10 0 6 -14 8 C 10 -6 0 -4 -2 D 22 14 4 0 14 E 24 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -22 -24 B 10 0 6 -14 8 C 10 -6 0 -4 -2 D 22 14 4 0 14 E 24 -8 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7718: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (12) A B E C D (10) E A D C B (8) D C E B A (8) B A C D E (7) A E B D C (6) E A D B C (5) C B D A E (5) D E C A B (4) B C A D E (4) B A C E D (4) E D A C B (3) D C B E A (3) C D E B A (3) E D C A B (2) C B A D E (2) B A E D C (2) A E B C D (2) A B E D C (2) E D A B C (1) E C D A B (1) E A C D B (1) E A B D C (1) D E C B A (1) D B C E A (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 8 12 -8 B 6 0 -6 -6 8 C -8 6 0 6 0 D -12 6 -6 0 0 E 8 -8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.400000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999977 Cumulative probabilities = A: 0.300000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 12 -8 B 6 0 -6 -6 8 C -8 6 0 6 0 D -12 6 -6 0 0 E 8 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.400000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999909 Cumulative probabilities = A: 0.300000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=22 C=22 A=21 B=18 D=17 so D is eliminated. Round 2 votes counts: C=33 E=27 A=21 B=19 so B is eliminated. Round 3 votes counts: C=38 A=35 E=27 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:203 C:202 B:201 E:200 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 12 -8 B 6 0 -6 -6 8 C -8 6 0 6 0 D -12 6 -6 0 0 E 8 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.400000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999909 Cumulative probabilities = A: 0.300000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 12 -8 B 6 0 -6 -6 8 C -8 6 0 6 0 D -12 6 -6 0 0 E 8 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.400000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999909 Cumulative probabilities = A: 0.300000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 12 -8 B 6 0 -6 -6 8 C -8 6 0 6 0 D -12 6 -6 0 0 E 8 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.400000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.339999999909 Cumulative probabilities = A: 0.300000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7719: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) D B A E C (5) B D E A C (5) D A B E C (4) C E A B D (4) B C E D A (4) B C D E A (4) A D E B C (4) E C A B D (3) E B D A C (3) E A C D B (3) D A E B C (3) C E A D B (3) C B E D A (3) C A E D B (3) B D C A E (3) B D A E C (3) B D A C E (3) A D E C B (3) A C E D B (3) E B C D A (2) D B A C E (2) D A B C E (2) C B A D E (2) C A D E B (2) B E C D A (2) B C D A E (2) A D C E B (2) E D A B C (1) E C B A D (1) E C A D B (1) E B D C A (1) E A D C B (1) D B E A C (1) C B E A D (1) C A D B E (1) B E D C A (1) B E D A C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 2 -10 -2 B 10 0 12 8 2 C -2 -12 0 2 4 D 10 -8 -2 0 6 E 2 -2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -10 -2 B 10 0 12 8 2 C -2 -12 0 2 4 D 10 -8 -2 0 6 E 2 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 D=17 E=16 A=14 so A is eliminated. Round 2 votes counts: C=29 B=28 D=27 E=16 so E is eliminated. Round 3 votes counts: C=37 B=34 D=29 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:203 C:196 E:195 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 -10 -2 B 10 0 12 8 2 C -2 -12 0 2 4 D 10 -8 -2 0 6 E 2 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -10 -2 B 10 0 12 8 2 C -2 -12 0 2 4 D 10 -8 -2 0 6 E 2 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -10 -2 B 10 0 12 8 2 C -2 -12 0 2 4 D 10 -8 -2 0 6 E 2 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7720: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) D A B C E (8) E C B A D (6) D A B E C (6) C A E D B (6) A D C B E (6) E B D C A (5) C A D B E (5) E C B D A (4) D B A E C (4) E B D A C (3) C E A B D (3) C A B D E (3) A C D B E (3) E C A D B (2) D B E A C (2) C E B A D (2) C A E B D (2) B E D A C (2) A D B C E (2) E D C B A (1) E D B C A (1) E D B A C (1) E D A C B (1) D E B A C (1) D A E C B (1) C E A D B (1) C B A D E (1) C A B E D (1) B E C D A (1) B D E A C (1) B D A C E (1) B C E A D (1) B C A D E (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -12 -6 4 B -4 0 0 -6 -4 C 12 0 0 6 -6 D 6 6 -6 0 -6 E -4 4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677682 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.454545 E: 1.000000 A B C D E A 0 4 -12 -6 4 B -4 0 0 -6 -4 C 12 0 0 6 -6 D 6 6 -6 0 -6 E -4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677688 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.454545 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=24 D=22 A=13 B=7 so B is eliminated. Round 2 votes counts: E=37 C=26 D=24 A=13 so A is eliminated. Round 3 votes counts: E=37 D=33 C=30 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:206 E:206 D:200 A:195 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -12 -6 4 B -4 0 0 -6 -4 C 12 0 0 6 -6 D 6 6 -6 0 -6 E -4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677688 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.454545 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -6 4 B -4 0 0 -6 -4 C 12 0 0 6 -6 D 6 6 -6 0 -6 E -4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677688 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.454545 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -6 4 B -4 0 0 -6 -4 C 12 0 0 6 -6 D 6 6 -6 0 -6 E -4 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.000000 E: 0.545455 Sum of squares = 0.404958677688 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 0.454545 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7721: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) C A E B D (7) C A E D B (6) B D E A C (6) D B E A C (5) C A D E B (5) E D B A C (4) D B A C E (4) C A B E D (4) B D E C A (4) B D C A E (4) E D A C B (3) C A D B E (3) A C E D B (3) A C D E B (3) E B D A C (2) D A E C B (2) D A C E B (2) D A C B E (2) C A B D E (2) B E D C A (2) B E D A C (2) B E C A D (2) B C A E D (2) B C A D E (2) E C A B D (1) E B C A D (1) E A C B D (1) D E B A C (1) D E A C B (1) D E A B C (1) D B C A E (1) D B A E C (1) B E C D A (1) B D C E A (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 6 0 0 8 B -6 0 -4 -8 0 C 0 4 0 4 6 D 0 8 -4 0 2 E -8 0 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.289166 B: 0.000000 C: 0.710834 D: 0.000000 E: 0.000000 Sum of squares = 0.588902337608 Cumulative probabilities = A: 0.289166 B: 0.289166 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 8 B -6 0 -4 -8 0 C 0 4 0 4 6 D 0 8 -4 0 2 E -8 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=27 D=20 E=19 A=6 so A is eliminated. Round 2 votes counts: C=33 B=28 D=20 E=19 so E is eliminated. Round 3 votes counts: C=42 B=31 D=27 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:207 C:207 D:203 E:192 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 0 8 B -6 0 -4 -8 0 C 0 4 0 4 6 D 0 8 -4 0 2 E -8 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 8 B -6 0 -4 -8 0 C 0 4 0 4 6 D 0 8 -4 0 2 E -8 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 8 B -6 0 -4 -8 0 C 0 4 0 4 6 D 0 8 -4 0 2 E -8 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7722: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) A B C E D (10) C E D B A (8) C D E A B (6) B A E D C (6) E D C B A (5) C E D A B (4) A B C D E (4) E D B C A (3) D E B C A (3) C A D E B (3) B A C E D (3) A D C E B (3) A B D E C (3) E C D B A (2) D E C B A (2) D C E A B (2) C E A D B (2) B A E C D (2) A C D E B (2) A C B E D (2) D E C A B (1) D C E B A (1) D B E A C (1) D A E B C (1) C D A E B (1) C B A E D (1) C A E B D (1) B E D A C (1) B E A D C (1) B D E A C (1) B D A E C (1) B A D E C (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -10 -6 -6 B 4 0 4 -2 0 C 10 -4 0 6 12 D 6 2 -6 0 -22 E 6 0 -12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.778694 C: 0.000000 D: 0.000000 E: 0.221306 Sum of squares = 0.655340404577 Cumulative probabilities = A: 0.000000 B: 0.778694 C: 0.778694 D: 0.778694 E: 1.000000 A B C D E A 0 -4 -10 -6 -6 B 4 0 4 -2 0 C 10 -4 0 6 12 D 6 2 -6 0 -22 E 6 0 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000224333 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 B=26 D=11 E=10 so E is eliminated. Round 2 votes counts: C=28 A=27 B=26 D=19 so D is eliminated. Round 3 votes counts: C=39 B=33 A=28 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:212 E:208 B:203 D:190 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -6 B 4 0 4 -2 0 C 10 -4 0 6 12 D 6 2 -6 0 -22 E 6 0 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000224333 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -6 B 4 0 4 -2 0 C 10 -4 0 6 12 D 6 2 -6 0 -22 E 6 0 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000224333 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -6 B 4 0 4 -2 0 C 10 -4 0 6 12 D 6 2 -6 0 -22 E 6 0 -12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000224333 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7723: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) D A B E C (7) B C A E D (6) C E D A B (5) C E B D A (4) C E B A D (4) B D A E C (4) B A D E C (4) E A C D B (3) D C E A B (3) D A E B C (3) C E A D B (3) C E A B D (3) B D C A E (3) B D A C E (3) B C D E A (3) B A D C E (3) C B E A D (2) C B D E A (2) A E D C B (2) A D E B C (2) A D B E C (2) A B D E C (2) E D C A B (1) E D A C B (1) E C A B D (1) E A D C B (1) E A C B D (1) D E C A B (1) D E A C B (1) D B C A E (1) D B A E C (1) D B A C E (1) D A E C B (1) C E D B A (1) C D E B A (1) C D E A B (1) C B E D A (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C E D (1) A E C D B (1) Total count = 100 A B C D E A 0 4 -12 -2 -4 B -4 0 -2 0 -2 C 12 2 0 4 8 D 2 0 -4 0 2 E 4 2 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 -2 -4 B -4 0 -2 0 -2 C 12 2 0 4 8 D 2 0 -4 0 2 E 4 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=27 D=19 E=15 A=9 so A is eliminated. Round 2 votes counts: B=32 C=27 D=23 E=18 so E is eliminated. Round 3 votes counts: C=40 B=32 D=28 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 D:200 E:198 B:196 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -2 -4 B -4 0 -2 0 -2 C 12 2 0 4 8 D 2 0 -4 0 2 E 4 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -2 -4 B -4 0 -2 0 -2 C 12 2 0 4 8 D 2 0 -4 0 2 E 4 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -2 -4 B -4 0 -2 0 -2 C 12 2 0 4 8 D 2 0 -4 0 2 E 4 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7724: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) C E B A D (8) D A E B C (7) C A D B E (6) E B D A C (5) E B C D A (4) D E B A C (4) C B E A D (4) C A E B D (4) B E D A C (4) E D B C A (3) C E B D A (3) B E C A D (3) A D C B E (3) A D B E C (3) A C D B E (3) E D B A C (2) C E D B A (2) C B A E D (2) C A D E B (2) B C E A D (2) E C B D A (1) E B D C A (1) D E A B C (1) D C E A B (1) D B E A C (1) D A E C B (1) D A C E B (1) C D E B A (1) C A B D E (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D E C (1) A D C E B (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -14 16 -4 B 8 0 -10 14 -6 C 14 10 0 18 10 D -16 -14 -18 0 -22 E 4 6 -10 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 16 -4 B 8 0 -10 14 -6 C 14 10 0 18 10 D -16 -14 -18 0 -22 E 4 6 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 E=16 D=16 B=13 A=13 so B is eliminated. Round 2 votes counts: C=44 E=25 D=16 A=15 so A is eliminated. Round 3 votes counts: C=49 E=27 D=24 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:211 B:203 A:195 D:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -14 16 -4 B 8 0 -10 14 -6 C 14 10 0 18 10 D -16 -14 -18 0 -22 E 4 6 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 16 -4 B 8 0 -10 14 -6 C 14 10 0 18 10 D -16 -14 -18 0 -22 E 4 6 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 16 -4 B 8 0 -10 14 -6 C 14 10 0 18 10 D -16 -14 -18 0 -22 E 4 6 -10 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7725: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (15) D A C E B (11) A C B E D (6) D E C A B (5) D B E C A (5) B D E C A (4) A D C E B (4) A C E D B (4) E C A B D (3) D B E A C (3) D B A C E (3) D A B C E (3) B D A C E (3) A C E B D (3) E C B A D (2) E C A D B (2) E B C A D (2) D E A C B (2) D B A E C (2) D A E C B (2) B E C D A (2) E D B C A (1) E B D C A (1) D E B C A (1) D E B A C (1) D A C B E (1) C E A D B (1) C B A E D (1) C A E D B (1) C A E B D (1) B C E A D (1) B C A E D (1) A C D E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 0 -2 B -4 0 -2 -6 4 C -2 2 0 -4 -6 D 0 6 4 0 4 E 2 -4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.459657 B: 0.000000 C: 0.000000 D: 0.540343 E: 0.000000 Sum of squares = 0.503255105684 Cumulative probabilities = A: 0.459657 B: 0.459657 C: 0.459657 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 0 -2 B -4 0 -2 -6 4 C -2 2 0 -4 -6 D 0 6 4 0 4 E 2 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=26 A=20 E=11 C=4 so C is eliminated. Round 2 votes counts: D=39 B=27 A=22 E=12 so E is eliminated. Round 3 votes counts: D=40 B=32 A=28 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:207 A:202 E:200 B:196 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 0 -2 B -4 0 -2 -6 4 C -2 2 0 -4 -6 D 0 6 4 0 4 E 2 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 0 -2 B -4 0 -2 -6 4 C -2 2 0 -4 -6 D 0 6 4 0 4 E 2 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 0 -2 B -4 0 -2 -6 4 C -2 2 0 -4 -6 D 0 6 4 0 4 E 2 -4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999992 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7726: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (8) D E B C A (7) E D B C A (6) D E A B C (5) D A E C B (5) A C B E D (5) D E B A C (4) D B C E A (4) C B A E D (4) A D C B E (4) A C E B D (4) E C B A D (3) D B E C A (3) D A E B C (3) A E C D B (3) E D B A C (2) E B D C A (2) C B A D E (2) B C E D A (2) A E D C B (2) A D C E B (2) E D A B C (1) E B C D A (1) E B C A D (1) E A D C B (1) E A C B D (1) D B C A E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A B C E (1) C B E A D (1) C A E B D (1) C A B E D (1) C A B D E (1) B E C A D (1) B C E A D (1) B C A D E (1) A E C B D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 4 14 0 8 B -4 0 -4 -18 -16 C -14 4 0 -14 -8 D 0 18 14 0 10 E -8 16 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.365840 B: 0.000000 C: 0.000000 D: 0.634160 E: 0.000000 Sum of squares = 0.535998033766 Cumulative probabilities = A: 0.365840 B: 0.365840 C: 0.365840 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 0 8 B -4 0 -4 -18 -16 C -14 4 0 -14 -8 D 0 18 14 0 10 E -8 16 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=31 E=18 C=10 B=5 so B is eliminated. Round 2 votes counts: D=36 A=31 E=19 C=14 so C is eliminated. Round 3 votes counts: A=41 D=36 E=23 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:213 E:203 C:184 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 0 8 B -4 0 -4 -18 -16 C -14 4 0 -14 -8 D 0 18 14 0 10 E -8 16 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 0 8 B -4 0 -4 -18 -16 C -14 4 0 -14 -8 D 0 18 14 0 10 E -8 16 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 0 8 B -4 0 -4 -18 -16 C -14 4 0 -14 -8 D 0 18 14 0 10 E -8 16 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7727: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (15) E C D B A (10) B A E D C (9) E C D A B (8) D C A E B (6) A D B C E (6) A B D C E (5) E B C A D (4) C E D A B (4) C D E A B (4) B E A C D (4) A D C B E (4) C D A E B (3) B A D E C (3) E C B D A (2) E B A C D (2) D A C E B (2) E B C D A (1) E B A D C (1) D C A B E (1) D A C B E (1) C E D B A (1) B E C A D (1) B E A D C (1) B C D A E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 8 12 14 B 10 0 8 0 4 C -8 -8 0 -10 6 D -12 0 10 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.717916 C: 0.000000 D: 0.282084 E: 0.000000 Sum of squares = 0.594974924804 Cumulative probabilities = A: 0.000000 B: 0.717916 C: 0.717916 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 12 14 B 10 0 8 0 4 C -8 -8 0 -10 6 D -12 0 10 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.504132269134 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=28 A=16 C=12 D=10 so D is eliminated. Round 2 votes counts: B=34 E=28 C=19 A=19 so C is eliminated. Round 3 votes counts: E=37 B=34 A=29 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:212 B:211 D:201 C:190 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 12 14 B 10 0 8 0 4 C -8 -8 0 -10 6 D -12 0 10 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.504132269134 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 12 14 B 10 0 8 0 4 C -8 -8 0 -10 6 D -12 0 10 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.504132269134 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 12 14 B 10 0 8 0 4 C -8 -8 0 -10 6 D -12 0 10 0 4 E -14 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.454545 E: 0.000000 Sum of squares = 0.504132269134 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7728: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (12) A E B D C (8) B D C E A (7) D C B A E (6) E A C B D (5) E A C D B (4) E A B D C (4) B D C A E (4) A E C D B (4) A D C B E (4) E C A D B (3) C D E B A (3) C D B A E (3) E C D A B (2) E B D C A (2) E B C D A (2) E B A D C (2) E A B C D (2) D B C A E (2) A B E D C (2) A B D E C (2) A B D C E (2) E C D B A (1) E C B D A (1) E C A B D (1) E B D A C (1) E B C A D (1) D C B E A (1) D A C B E (1) C E D B A (1) C D A E B (1) C A D E B (1) B D A E C (1) A E B C D (1) A D B E C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -8 -2 -10 B 0 0 -10 -4 -2 C 8 10 0 -2 0 D 2 4 2 0 6 E 10 2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -2 -10 B 0 0 -10 -4 -2 C 8 10 0 -2 0 D 2 4 2 0 6 E 10 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=26 C=21 B=12 D=10 so D is eliminated. Round 2 votes counts: E=31 C=28 A=27 B=14 so B is eliminated. Round 3 votes counts: C=41 E=31 A=28 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 D:207 E:203 B:192 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -2 -10 B 0 0 -10 -4 -2 C 8 10 0 -2 0 D 2 4 2 0 6 E 10 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 -10 B 0 0 -10 -4 -2 C 8 10 0 -2 0 D 2 4 2 0 6 E 10 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 -10 B 0 0 -10 -4 -2 C 8 10 0 -2 0 D 2 4 2 0 6 E 10 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7729: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) E D B C A (8) E D C A B (5) E D B A C (5) E D A C B (4) E D A B C (4) D C B A E (4) C A B D E (4) A B C D E (4) E A B D C (3) E A B C D (3) D E C B A (3) D E B C A (3) D C B E A (3) C B A D E (3) A E C B D (3) A C B D E (3) D C E B A (2) C D B A E (2) C A D B E (2) B C D A E (2) A E B C D (2) E D C B A (1) E B D C A (1) E B D A C (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D B C A E (1) C D A E B (1) C B D A E (1) B E D C A (1) B D E C A (1) B D C E A (1) B C A D E (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 0 -8 -8 B -10 0 8 -4 -8 C 0 -8 0 -8 -8 D 8 4 8 0 -16 E 8 8 8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 0 -8 -8 B -10 0 8 -4 -8 C 0 -8 0 -8 -8 D 8 4 8 0 -16 E 8 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=25 D=16 C=13 B=6 so B is eliminated. Round 2 votes counts: E=41 A=25 D=18 C=16 so C is eliminated. Round 3 votes counts: E=41 A=35 D=24 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:202 A:197 B:193 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 0 -8 -8 B -10 0 8 -4 -8 C 0 -8 0 -8 -8 D 8 4 8 0 -16 E 8 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -8 -8 B -10 0 8 -4 -8 C 0 -8 0 -8 -8 D 8 4 8 0 -16 E 8 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -8 -8 B -10 0 8 -4 -8 C 0 -8 0 -8 -8 D 8 4 8 0 -16 E 8 8 8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7730: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E A D B C (8) B C D A E (6) B C A E D (6) D C E A B (5) A B E D C (5) D E A C B (4) C D E B A (4) C D B E A (4) B A E C D (4) D E A B C (3) C E D A B (3) C B A E D (3) B A E D C (3) E D A C B (2) E A D C B (2) D E C A B (2) D B A E C (2) C B E A D (2) B A C E D (2) A E D B C (2) A E B D C (2) E D C A B (1) E C D A B (1) E A C D B (1) E A C B D (1) E A B C D (1) D C E B A (1) D C B E A (1) D A E B C (1) D A B E C (1) C E A D B (1) C E A B D (1) C D E A B (1) C B D E A (1) B D A C E (1) B C A D E (1) B A D E C (1) B A C D E (1) Total count = 100 A B C D E A 0 -4 -6 -6 0 B 4 0 0 0 6 C 6 0 0 6 6 D 6 0 -6 0 -2 E 0 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.453765 C: 0.546235 D: 0.000000 E: 0.000000 Sum of squares = 0.504275335512 Cumulative probabilities = A: 0.000000 B: 0.453765 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -6 0 B 4 0 0 0 6 C 6 0 0 6 6 D 6 0 -6 0 -2 E 0 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=25 D=20 E=17 A=9 so A is eliminated. Round 2 votes counts: B=30 C=29 E=21 D=20 so D is eliminated. Round 3 votes counts: C=36 B=33 E=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:209 B:205 D:199 E:195 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -6 0 B 4 0 0 0 6 C 6 0 0 6 6 D 6 0 -6 0 -2 E 0 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -6 0 B 4 0 0 0 6 C 6 0 0 6 6 D 6 0 -6 0 -2 E 0 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -6 0 B 4 0 0 0 6 C 6 0 0 6 6 D 6 0 -6 0 -2 E 0 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7731: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) E B A C D (7) C D B E A (7) C D A E B (6) A E B D C (6) E B A D C (5) C D A B E (5) B E A D C (5) D C A B E (4) E B C A D (3) D C B E A (3) D B E A C (3) C E B D A (3) C B E D A (3) C A D E B (3) A D C E B (3) E C B A D (2) C D E B A (2) B E D A C (2) A D B E C (2) A C D E B (2) E A C B D (1) E A B C D (1) D C B A E (1) D B E C A (1) D B A E C (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A B D (1) C D E A B (1) C D B A E (1) C A E B D (1) A E D B C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 0 4 10 -2 B 0 0 -4 4 -2 C -4 4 0 4 -4 D -10 -4 -4 0 -6 E 2 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 10 -2 B 0 0 -4 4 -2 C -4 4 0 4 -4 D -10 -4 -4 0 -6 E 2 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=25 E=19 D=15 B=7 so B is eliminated. Round 2 votes counts: C=34 E=26 A=25 D=15 so D is eliminated. Round 3 votes counts: C=42 E=30 A=28 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:206 C:200 B:199 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 10 -2 B 0 0 -4 4 -2 C -4 4 0 4 -4 D -10 -4 -4 0 -6 E 2 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 -2 B 0 0 -4 4 -2 C -4 4 0 4 -4 D -10 -4 -4 0 -6 E 2 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 -2 B 0 0 -4 4 -2 C -4 4 0 4 -4 D -10 -4 -4 0 -6 E 2 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7732: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) E C D B A (5) C E B A D (5) E C D A B (4) D E C B A (4) D B A E C (4) D A E B C (4) A D E B C (4) E A C D B (3) D B C E A (3) B D A C E (3) B C A E D (3) A B D E C (3) E D C A B (2) E C A D B (2) D B E C A (2) D B A C E (2) C E D B A (2) C E A D B (2) B D C E A (2) B D C A E (2) B C D E A (2) B A D C E (2) A E D C B (2) A E C B D (2) A D B E C (2) A C B E D (2) A B C E D (2) A B C D E (2) D E B A C (1) D E A C B (1) D E A B C (1) D C B E A (1) D A B E C (1) C E D A B (1) C E A B D (1) C B E D A (1) C B E A D (1) C A E B D (1) C A B E D (1) B C E A D (1) B C A D E (1) B A C D E (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 2 -10 -8 B -4 0 -4 -22 -12 C -2 4 0 -6 -10 D 10 22 6 0 -4 E 8 12 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 -10 -8 B -4 0 -4 -22 -12 C -2 4 0 -6 -10 D 10 22 6 0 -4 E 8 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=23 A=21 B=17 C=15 so C is eliminated. Round 2 votes counts: E=34 D=24 A=23 B=19 so B is eliminated. Round 3 votes counts: E=37 D=33 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:217 E:217 A:194 C:193 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 -10 -8 B -4 0 -4 -22 -12 C -2 4 0 -6 -10 D 10 22 6 0 -4 E 8 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -10 -8 B -4 0 -4 -22 -12 C -2 4 0 -6 -10 D 10 22 6 0 -4 E 8 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -10 -8 B -4 0 -4 -22 -12 C -2 4 0 -6 -10 D 10 22 6 0 -4 E 8 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7733: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (14) D A E B C (8) A D C E B (8) A D E B C (7) B E C D A (6) A C D E B (5) E B D C A (4) D E A B C (4) A D E C B (4) E D B A C (3) E B D A C (3) D E B A C (3) C B A E D (3) C A D E B (3) C A B E D (3) A C D B E (3) C B E D A (2) B E D C A (2) B E D A C (2) A D B E C (2) A C B D E (2) E D B C A (1) E C B D A (1) E B C D A (1) C E B D A (1) C A D B E (1) B C E D A (1) B C E A D (1) B A C E D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 12 16 2 B -2 0 -2 -6 -12 C -12 2 0 -4 -2 D -16 6 4 0 2 E -2 12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 16 2 B -2 0 -2 -6 -12 C -12 2 0 -4 -2 D -16 6 4 0 2 E -2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990444 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=27 D=15 E=13 B=13 so E is eliminated. Round 2 votes counts: A=32 C=28 B=21 D=19 so D is eliminated. Round 3 votes counts: A=44 C=28 B=28 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:205 D:198 C:192 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 16 2 B -2 0 -2 -6 -12 C -12 2 0 -4 -2 D -16 6 4 0 2 E -2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990444 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 16 2 B -2 0 -2 -6 -12 C -12 2 0 -4 -2 D -16 6 4 0 2 E -2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990444 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 16 2 B -2 0 -2 -6 -12 C -12 2 0 -4 -2 D -16 6 4 0 2 E -2 12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990444 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7734: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) A E D B C (7) D A C B E (6) C B E A D (6) A D E B C (6) E A B D C (5) D A E B C (4) C E B A D (4) C B D E A (4) E A B C D (3) D C B A E (3) D C A B E (3) D A B E C (3) B E C D A (3) B E C A D (3) A E C B D (3) D B E A C (2) D B A E C (2) C D B E A (2) C D A B E (2) B E D C A (2) B C E D A (2) A E B D C (2) A D E C B (2) A C D E B (2) E C B A D (1) E B C A D (1) E B A D C (1) E B A C D (1) D B E C A (1) D B C E A (1) D A B C E (1) C A E B D (1) B D C E A (1) B C D E A (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 4 4 0 -2 B -4 0 6 4 10 C -4 -6 0 -6 -6 D 0 -4 6 0 -6 E 2 -10 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000197 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 4 4 0 -2 B -4 0 6 4 10 C -4 -6 0 -6 -6 D 0 -4 6 0 -6 E 2 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000063 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 A=24 E=12 B=12 so E is eliminated. Round 2 votes counts: A=32 C=27 D=26 B=15 so B is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:208 A:203 E:202 D:198 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 0 -2 B -4 0 6 4 10 C -4 -6 0 -6 -6 D 0 -4 6 0 -6 E 2 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000063 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 0 -2 B -4 0 6 4 10 C -4 -6 0 -6 -6 D 0 -4 6 0 -6 E 2 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000063 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 0 -2 B -4 0 6 4 10 C -4 -6 0 -6 -6 D 0 -4 6 0 -6 E 2 -10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000063 Cumulative probabilities = A: 0.625000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7735: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) C D B A E (7) E A B D C (6) B D C E A (6) A E D B C (6) D B A C E (5) C B D E A (4) B D E C A (4) B D E A C (4) A E C D B (4) E B D A C (3) E A C B D (3) D B C A E (3) D B A E C (3) C E A B D (3) C B D A E (3) C A D B E (3) B E D A C (3) A C E D B (3) C A E B D (2) E D A B C (1) E C A B D (1) E B C D A (1) E B C A D (1) E A D B C (1) E A C D B (1) D C B A E (1) D B E A C (1) C E B D A (1) C D A B E (1) C B E D A (1) C A D E B (1) A E D C B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -6 -4 10 B 2 0 -2 -8 2 C 6 2 0 0 12 D 4 8 0 0 -4 E -10 -2 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.587562 D: 0.412438 E: 0.000000 Sum of squares = 0.515334171908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.587562 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -4 10 B 2 0 -2 -8 2 C 6 2 0 0 12 D 4 8 0 0 -4 E -10 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=18 B=17 A=16 D=13 so D is eliminated. Round 2 votes counts: C=37 B=29 E=18 A=16 so A is eliminated. Round 3 votes counts: C=41 B=30 E=29 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:204 A:199 B:197 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 10 B 2 0 -2 -8 2 C 6 2 0 0 12 D 4 8 0 0 -4 E -10 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 10 B 2 0 -2 -8 2 C 6 2 0 0 12 D 4 8 0 0 -4 E -10 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 10 B 2 0 -2 -8 2 C 6 2 0 0 12 D 4 8 0 0 -4 E -10 -2 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7736: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (6) D C E A B (6) E D B C A (5) E B D A C (5) D E C B A (5) B E A D C (5) A C B E D (5) A B C E D (5) D E B A C (4) D C A B E (4) A B D C E (4) A B C D E (4) E D C B A (3) D E B C A (3) C D A E B (3) C A B E D (3) E C D B A (2) E B C A D (2) D C A E B (2) C A E B D (2) C A D B E (2) B E A C D (2) B A E C D (2) E B C D A (1) E B A D C (1) D B E A C (1) D A C B E (1) C E D B A (1) C D E A B (1) C D A B E (1) C A D E B (1) B E D A C (1) B D E A C (1) B D A E C (1) B A E D C (1) B A D E C (1) A D B C E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 6 -2 -10 B 6 0 14 8 -6 C -6 -14 0 -10 -6 D 2 -8 10 0 -6 E 10 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 -2 -10 B 6 0 14 8 -6 C -6 -14 0 -10 -6 D 2 -8 10 0 -6 E 10 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 A=21 C=14 B=14 so C is eliminated. Round 2 votes counts: D=31 A=29 E=26 B=14 so B is eliminated. Round 3 votes counts: E=34 D=33 A=33 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:211 D:199 A:194 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 -2 -10 B 6 0 14 8 -6 C -6 -14 0 -10 -6 D 2 -8 10 0 -6 E 10 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -2 -10 B 6 0 14 8 -6 C -6 -14 0 -10 -6 D 2 -8 10 0 -6 E 10 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -2 -10 B 6 0 14 8 -6 C -6 -14 0 -10 -6 D 2 -8 10 0 -6 E 10 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7737: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) A B E C D (8) D C B E A (7) C D B A E (7) E A B D C (6) C A B E D (5) A E B D C (5) D E B A C (4) C D B E A (4) C B D A E (4) E D A B C (3) D C A E B (3) B A E C D (3) D E C A B (2) D E A C B (2) D E A B C (2) D B E C A (2) C D A B E (2) C B D E A (2) C B A E D (2) C A B D E (2) A C E B D (2) A C B E D (2) E A D B C (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A B E (1) D A C E B (1) C B A D E (1) B E A D C (1) B E A C D (1) B D E C A (1) B C E A D (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 16 4 2 16 B -16 0 -2 12 14 C -4 2 0 12 -2 D -2 -12 -12 0 -2 E -16 -14 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 4 2 16 B -16 0 -2 12 14 C -4 2 0 12 -2 D -2 -12 -12 0 -2 E -16 -14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983336 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 A=26 E=11 B=8 so B is eliminated. Round 2 votes counts: C=30 A=30 D=27 E=13 so E is eliminated. Round 3 votes counts: A=40 D=30 C=30 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:204 C:204 E:187 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 4 2 16 B -16 0 -2 12 14 C -4 2 0 12 -2 D -2 -12 -12 0 -2 E -16 -14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983336 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 2 16 B -16 0 -2 12 14 C -4 2 0 12 -2 D -2 -12 -12 0 -2 E -16 -14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983336 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 2 16 B -16 0 -2 12 14 C -4 2 0 12 -2 D -2 -12 -12 0 -2 E -16 -14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983336 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7738: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) B C E D A (9) E D A B C (8) E A D C B (6) A E D C B (5) E C A B D (4) B C D E A (4) B C D A E (4) C B E A D (3) B E D C A (3) B C A D E (3) A D E C B (3) A D C E B (3) E D B A C (2) E C B A D (2) E B D C A (2) D E B A C (2) D E A B C (2) D B A C E (2) D A E C B (2) C B A E D (2) B C E A D (2) A D C B E (2) A C D E B (2) E A C D B (1) D B E A C (1) D B A E C (1) D A E B C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B A D (1) C A E D B (1) C A D E B (1) B D E A C (1) B D C E A (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 -2 4 -8 B 8 0 0 0 0 C 2 0 0 0 4 D -4 0 0 0 -6 E 8 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.258353 C: 0.544482 D: 0.197165 E: 0.000000 Sum of squares = 0.402080671211 Cumulative probabilities = A: 0.000000 B: 0.258353 C: 0.802835 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 4 -8 B 8 0 0 0 0 C 2 0 0 0 4 D -4 0 0 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.342105 C: 0.394737 D: 0.263158 E: 0.000000 Sum of squares = 0.342105263528 Cumulative probabilities = A: 0.000000 B: 0.342105 C: 0.736842 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 C=17 A=17 D=14 so D is eliminated. Round 2 votes counts: B=31 E=29 A=23 C=17 so C is eliminated. Round 3 votes counts: B=45 E=30 A=25 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:205 B:204 C:203 D:195 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 4 -8 B 8 0 0 0 0 C 2 0 0 0 4 D -4 0 0 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.342105 C: 0.394737 D: 0.263158 E: 0.000000 Sum of squares = 0.342105263528 Cumulative probabilities = A: 0.000000 B: 0.342105 C: 0.736842 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 4 -8 B 8 0 0 0 0 C 2 0 0 0 4 D -4 0 0 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.342105 C: 0.394737 D: 0.263158 E: 0.000000 Sum of squares = 0.342105263528 Cumulative probabilities = A: 0.000000 B: 0.342105 C: 0.736842 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 4 -8 B 8 0 0 0 0 C 2 0 0 0 4 D -4 0 0 0 -6 E 8 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.342105 C: 0.394737 D: 0.263158 E: 0.000000 Sum of squares = 0.342105263528 Cumulative probabilities = A: 0.000000 B: 0.342105 C: 0.736842 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7739: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) A B E C D (9) C D E A B (7) C E D A B (5) B A E D C (5) A C B D E (5) B D A E C (4) B A E C D (4) B A D E C (4) E B A C D (3) D C E A B (3) D C A B E (3) C E A B D (3) B E A D C (3) A E B C D (3) E D C B A (2) D B A E C (2) D B A C E (2) C D E B A (2) C D A E B (2) C A E D B (2) B E D A C (2) A C E B D (2) E C D B A (1) E B A D C (1) E A C B D (1) D E C B A (1) D E B C A (1) D C B E A (1) D C A E B (1) D B E C A (1) D B C E A (1) D B C A E (1) C E A D B (1) C A E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 4 -2 4 B 0 0 -6 4 -2 C -4 6 0 6 6 D 2 -4 -6 0 2 E -4 2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.633758 B: 0.366242 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.535782607395 Cumulative probabilities = A: 0.633758 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 -2 4 B 0 0 -6 4 -2 C -4 6 0 6 6 D 2 -4 -6 0 2 E -4 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044038 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 B=22 A=21 E=8 so E is eliminated. Round 2 votes counts: D=28 B=26 C=24 A=22 so A is eliminated. Round 3 votes counts: B=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 A:203 B:198 D:197 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 -2 4 B 0 0 -6 4 -2 C -4 6 0 6 6 D 2 -4 -6 0 2 E -4 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044038 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -2 4 B 0 0 -6 4 -2 C -4 6 0 6 6 D 2 -4 -6 0 2 E -4 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044038 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -2 4 B 0 0 -6 4 -2 C -4 6 0 6 6 D 2 -4 -6 0 2 E -4 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000044038 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7740: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) E B A D C (6) B C A E D (6) D C A E B (5) A E B D C (5) E D A B C (4) A D E B C (4) E D C A B (3) D E C A B (3) C D E B A (3) E D B C A (2) E B C D A (2) E A D B C (2) E A B D C (2) D E A C B (2) C D B E A (2) C B E D A (2) C B D A E (2) C A D B E (2) B E C A D (2) B E A C D (2) B C E A D (2) A E D B C (2) A B C D E (2) E D A C B (1) E C D B A (1) E C B D A (1) E B D C A (1) E B A C D (1) D E A B C (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) C D E A B (1) C D A B E (1) C B E A D (1) C B D E A (1) C B A E D (1) C A B D E (1) B C E D A (1) B C A D E (1) B A E C D (1) B A C E D (1) B A C D E (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -14 14 0 B 4 0 4 10 -12 C 14 -4 0 4 0 D -14 -10 -4 0 -4 E 0 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.462928 D: 0.000000 E: 0.537072 Sum of squares = 0.502748605139 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.462928 D: 0.462928 E: 1.000000 A B C D E A 0 -4 -14 14 0 B 4 0 4 10 -12 C 14 -4 0 4 0 D -14 -10 -4 0 -4 E 0 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=26 C=26 B=17 A=16 D=15 so D is eliminated. Round 2 votes counts: E=32 C=32 A=19 B=17 so B is eliminated. Round 3 votes counts: C=42 E=36 A=22 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:208 C:207 B:203 A:198 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -14 14 0 B 4 0 4 10 -12 C 14 -4 0 4 0 D -14 -10 -4 0 -4 E 0 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 14 0 B 4 0 4 10 -12 C 14 -4 0 4 0 D -14 -10 -4 0 -4 E 0 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 14 0 B 4 0 4 10 -12 C 14 -4 0 4 0 D -14 -10 -4 0 -4 E 0 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7741: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) E A D C B (6) B C D E A (6) B C D A E (6) A E B C D (5) A D E B C (5) D E C A B (4) D B C A E (4) C B D E A (4) D E A C B (3) D A E B C (3) B D A C E (3) B C A E D (3) A E D B C (3) E A C D B (2) C E B A D (2) C D E B A (2) C B E D A (2) B D C A E (2) B A D E C (2) A E C B D (2) A E B D C (2) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) E A C B D (1) D C E B A (1) D C B E A (1) D B C E A (1) D A E C B (1) C D B E A (1) C B E A D (1) C B A E D (1) B C E D A (1) B C E A D (1) B C A D E (1) B A D C E (1) B A C E D (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 6 6 14 B -6 0 2 -2 -14 C -6 -2 0 -10 -10 D -6 2 10 0 4 E -14 14 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 14 B -6 0 2 -2 -14 C -6 -2 0 -10 -10 D -6 2 10 0 4 E -14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 D=18 E=13 C=13 so E is eliminated. Round 2 votes counts: A=38 B=27 D=19 C=16 so C is eliminated. Round 3 votes counts: A=40 B=37 D=23 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:205 E:203 B:190 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 14 B -6 0 2 -2 -14 C -6 -2 0 -10 -10 D -6 2 10 0 4 E -14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 14 B -6 0 2 -2 -14 C -6 -2 0 -10 -10 D -6 2 10 0 4 E -14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 14 B -6 0 2 -2 -14 C -6 -2 0 -10 -10 D -6 2 10 0 4 E -14 14 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999687 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7742: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) C E B A D (10) A D C E B (9) D A B E C (8) B E D C A (7) A C D E B (6) D A C E B (5) E C B A D (4) E B C A D (3) C A E B D (3) B D E C A (3) E B D A C (2) D B E C A (2) D B E A C (2) D A E B C (2) D A C B E (2) C A E D B (2) B E D A C (2) B C E D A (2) E B A C D (1) D C A B E (1) D B C A E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A B C E (1) C B E A D (1) B E C A D (1) B C E A D (1) B C D E A (1) A E C D B (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 -6 -10 -6 B 12 0 4 6 -4 C 6 -4 0 -2 -4 D 10 -6 2 0 -6 E 6 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -6 -10 -6 B 12 0 4 6 -4 C 6 -4 0 -2 -4 D 10 -6 2 0 -6 E 6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=27 A=19 C=16 E=10 so E is eliminated. Round 2 votes counts: B=34 D=27 C=20 A=19 so A is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:210 B:209 D:200 C:198 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -6 -10 -6 B 12 0 4 6 -4 C 6 -4 0 -2 -4 D 10 -6 2 0 -6 E 6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -10 -6 B 12 0 4 6 -4 C 6 -4 0 -2 -4 D 10 -6 2 0 -6 E 6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -10 -6 B 12 0 4 6 -4 C 6 -4 0 -2 -4 D 10 -6 2 0 -6 E 6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7743: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E A B D (8) B C A D E (6) C E B D A (5) C B A E D (5) B D A E C (5) C E A D B (4) B A D C E (4) E D A C B (3) E C D A B (3) E A C D B (3) D E C B A (3) D E B A C (3) D B E A C (3) B D A C E (3) B A C D E (3) A B C E D (3) E D C A B (2) E A D C B (2) D E A B C (2) D A E B C (2) D A B E C (2) C E B A D (2) B D C E A (2) A E D B C (2) A D E B C (2) E C D B A (1) D E A C B (1) D B E C A (1) C B E D A (1) C B E A D (1) C B D E A (1) B D C A E (1) B C A E D (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -18 6 -4 -2 B 18 0 8 2 2 C -6 -8 0 -6 0 D 4 -2 6 0 6 E 2 -2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998416 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 -4 -2 B 18 0 8 2 2 C -6 -8 0 -6 0 D 4 -2 6 0 6 E 2 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 B=25 E=14 A=9 so A is eliminated. Round 2 votes counts: D=28 B=28 C=27 E=17 so E is eliminated. Round 3 votes counts: D=38 C=34 B=28 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:215 D:207 E:197 A:191 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 -4 -2 B 18 0 8 2 2 C -6 -8 0 -6 0 D 4 -2 6 0 6 E 2 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 -4 -2 B 18 0 8 2 2 C -6 -8 0 -6 0 D 4 -2 6 0 6 E 2 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 -4 -2 B 18 0 8 2 2 C -6 -8 0 -6 0 D 4 -2 6 0 6 E 2 -2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998506 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7744: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) B D C A E (8) A E D C B (8) B C D E A (7) E A D C B (6) C D B E A (6) A B E D C (6) B A E D C (5) C D E A B (4) B E C D A (4) B A D C E (4) E A B C D (3) A E B D C (3) E C D B A (2) E A D B C (2) D C A E B (2) D C A B E (2) C D B A E (2) B C D A E (2) A D C E B (2) E C D A B (1) E C A D B (1) E C A B D (1) E B A C D (1) E A B D C (1) D C B A E (1) D A C E B (1) C E D B A (1) C D E B A (1) C B D E A (1) B E C A D (1) B E A C D (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 4 4 8 -6 B -4 0 -2 -2 2 C -4 2 0 -4 -10 D -8 2 4 0 -12 E 6 -2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 4 8 -6 B -4 0 -2 -2 2 C -4 2 0 -4 -10 D -8 2 4 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=26 A=20 C=15 D=6 so D is eliminated. Round 2 votes counts: B=33 E=26 A=21 C=20 so C is eliminated. Round 3 votes counts: B=43 E=32 A=25 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:213 A:205 B:197 D:193 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 8 -6 B -4 0 -2 -2 2 C -4 2 0 -4 -10 D -8 2 4 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 8 -6 B -4 0 -2 -2 2 C -4 2 0 -4 -10 D -8 2 4 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 8 -6 B -4 0 -2 -2 2 C -4 2 0 -4 -10 D -8 2 4 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7745: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) D A E B C (8) C B E A D (8) B C E A D (8) E C B D A (7) A D B E C (7) D A C B E (6) D A C E B (4) A D C B E (4) A D B C E (4) E C B A D (3) E B C D A (3) D A E C B (3) B E C A D (3) A D E B C (3) E D A C B (2) E C D A B (2) B C A D E (2) E D C A B (1) E D A B C (1) E B D A C (1) D A B E C (1) D A B C E (1) C D E A B (1) C D A E B (1) C B E D A (1) C B D A E (1) C B A E D (1) B E A D C (1) B E A C D (1) Total count = 100 A B C D E A 0 -2 -6 -10 -8 B 2 0 -12 2 -2 C 6 12 0 6 6 D 10 -2 -6 0 -8 E 8 2 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -10 -8 B 2 0 -12 2 -2 C 6 12 0 6 6 D 10 -2 -6 0 -8 E 8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 E=20 A=18 B=15 so B is eliminated. Round 2 votes counts: C=34 E=25 D=23 A=18 so A is eliminated. Round 3 votes counts: D=41 C=34 E=25 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:206 D:197 B:195 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -10 -8 B 2 0 -12 2 -2 C 6 12 0 6 6 D 10 -2 -6 0 -8 E 8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -10 -8 B 2 0 -12 2 -2 C 6 12 0 6 6 D 10 -2 -6 0 -8 E 8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -10 -8 B 2 0 -12 2 -2 C 6 12 0 6 6 D 10 -2 -6 0 -8 E 8 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7746: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (6) D A C B E (5) D A B C E (5) B E D C A (5) B E C A D (5) A D C E B (5) E C B A D (4) D B A C E (4) A E C D B (4) A C E D B (4) E B C A D (3) E A C B D (3) C B E A D (3) B C E D A (3) A E D C B (3) E B D A C (2) E A B C D (2) D A E B C (2) C E B A D (2) C A E D B (2) B E D A C (2) B E C D A (2) B D E C A (2) B C D E A (2) A E D B C (2) A D E C B (2) E C A B D (1) E B A C D (1) E A D B C (1) D C B A E (1) D C A B E (1) D B E C A (1) D B E A C (1) D B C A E (1) D A C E B (1) C E A B D (1) C D A B E (1) C B D A E (1) C A E B D (1) C A D B E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 2 4 -4 B 2 0 4 4 4 C -2 -4 0 -4 4 D -4 -4 4 0 -14 E 4 -4 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 4 -4 B 2 0 4 4 4 C -2 -4 0 -4 4 D -4 -4 4 0 -14 E 4 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 A=22 E=17 C=12 so C is eliminated. Round 2 votes counts: B=31 A=26 D=23 E=20 so E is eliminated. Round 3 votes counts: B=43 A=34 D=23 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:207 E:205 A:200 C:197 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 4 -4 B 2 0 4 4 4 C -2 -4 0 -4 4 D -4 -4 4 0 -14 E 4 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -4 B 2 0 4 4 4 C -2 -4 0 -4 4 D -4 -4 4 0 -14 E 4 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -4 B 2 0 4 4 4 C -2 -4 0 -4 4 D -4 -4 4 0 -14 E 4 -4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7747: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) B A D C E (9) B E D A C (6) D A C E B (5) D A B C E (5) E D C A B (4) E B C D A (4) B D E A C (4) B D A E C (4) A D C B E (4) E C B A D (3) C E A D B (3) B E C D A (3) E D B A C (2) E C D A B (2) E C B D A (2) D A C B E (2) C D A E B (2) C A D B E (2) C A B D E (2) B E C A D (2) B A C D E (2) A D B C E (2) E C D B A (1) E C A D B (1) E C A B D (1) E B D C A (1) E B D A C (1) D E C A B (1) D A E B C (1) D A B E C (1) C E D A B (1) C E B A D (1) C A E B D (1) B D A C E (1) B C E A D (1) B C A D E (1) B A E D C (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 2 4 -6 12 B -2 0 4 2 6 C -4 -4 0 -12 10 D 6 -2 12 0 18 E -12 -6 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000012 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -6 12 B -2 0 4 2 6 C -4 -4 0 -12 10 D 6 -2 12 0 18 E -12 -6 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=22 C=21 D=15 A=8 so A is eliminated. Round 2 votes counts: B=35 E=22 D=22 C=21 so C is eliminated. Round 3 votes counts: B=37 D=35 E=28 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:217 A:206 B:205 C:195 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 4 -6 12 B -2 0 4 2 6 C -4 -4 0 -12 10 D 6 -2 12 0 18 E -12 -6 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -6 12 B -2 0 4 2 6 C -4 -4 0 -12 10 D 6 -2 12 0 18 E -12 -6 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -6 12 B -2 0 4 2 6 C -4 -4 0 -12 10 D 6 -2 12 0 18 E -12 -6 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7748: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (12) B A D E C (10) C E D A B (8) E D A B C (6) C A D E B (6) B A E D C (6) E D B A C (5) B A C D E (5) E D C A B (4) B C A E D (4) C D E A B (3) E D A C B (2) D E A C B (2) D A C E B (2) C D A E B (2) C A B D E (2) B E D A C (2) B E C D A (2) B E A D C (2) A B D E C (2) E C D B A (1) E C D A B (1) E B C D A (1) D A E C B (1) C B E A D (1) C B A E D (1) B E D C A (1) B C E D A (1) B C A D E (1) B A E C D (1) A D E C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -2 12 16 B 12 0 2 8 8 C 2 -2 0 4 -4 D -12 -8 -4 0 2 E -16 -8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 12 16 B 12 0 2 8 8 C 2 -2 0 4 -4 D -12 -8 -4 0 2 E -16 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=35 B=35 E=20 D=5 A=5 so D is eliminated. Round 2 votes counts: C=35 B=35 E=22 A=8 so A is eliminated. Round 3 votes counts: B=38 C=37 E=25 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:207 C:200 D:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 12 16 B 12 0 2 8 8 C 2 -2 0 4 -4 D -12 -8 -4 0 2 E -16 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 12 16 B 12 0 2 8 8 C 2 -2 0 4 -4 D -12 -8 -4 0 2 E -16 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 12 16 B 12 0 2 8 8 C 2 -2 0 4 -4 D -12 -8 -4 0 2 E -16 -8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990794 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7749: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) E C B D A (8) E B D C A (8) C A E D B (8) A D B E C (7) A D C B E (6) A D B C E (6) E B C D A (5) C E A B D (5) A C D B E (5) D B E A C (4) B D E A C (4) A B D E C (3) D B A E C (2) D A B E C (2) C E B A D (2) C A D E B (2) E D B C A (1) E C D B A (1) D E C B A (1) D B E C A (1) C E A D B (1) C D E B A (1) C A E B D (1) B E D C A (1) B E D A C (1) B D A E C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -14 -4 -10 B 4 0 -6 4 -10 C 14 6 0 4 -2 D 4 -4 -4 0 -8 E 10 10 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -14 -4 -10 B 4 0 -6 4 -10 C 14 6 0 4 -2 D 4 -4 -4 0 -8 E 10 10 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=29 E=23 D=10 B=7 so B is eliminated. Round 2 votes counts: C=31 A=29 E=25 D=15 so D is eliminated. Round 3 votes counts: E=35 A=34 C=31 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:211 B:196 D:194 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -14 -4 -10 B 4 0 -6 4 -10 C 14 6 0 4 -2 D 4 -4 -4 0 -8 E 10 10 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -4 -10 B 4 0 -6 4 -10 C 14 6 0 4 -2 D 4 -4 -4 0 -8 E 10 10 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -4 -10 B 4 0 -6 4 -10 C 14 6 0 4 -2 D 4 -4 -4 0 -8 E 10 10 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999317 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7750: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (15) D C B A E (13) B A E D C (13) E A B C D (10) B A E C D (7) D C E A B (5) D B C A E (4) B A D E C (4) E A B D C (3) D C B E A (3) C E D A B (3) C E A B D (3) B E A D C (3) E C A B D (2) D B A E C (2) C D A B E (2) B D A E C (2) E A C B D (1) D C E B A (1) D B A C E (1) C D B A E (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -4 -4 2 B 10 0 2 0 14 C 4 -2 0 -8 4 D 4 0 8 0 8 E -2 -14 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.384996 C: 0.000000 D: 0.615004 E: 0.000000 Sum of squares = 0.526451635369 Cumulative probabilities = A: 0.000000 B: 0.384996 C: 0.384996 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -4 2 B 10 0 2 0 14 C 4 -2 0 -8 4 D 4 0 8 0 8 E -2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=29 C=24 E=16 A=1 so A is eliminated. Round 2 votes counts: B=31 D=29 C=24 E=16 so E is eliminated. Round 3 votes counts: B=44 D=29 C=27 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:210 C:199 A:192 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -4 2 B 10 0 2 0 14 C 4 -2 0 -8 4 D 4 0 8 0 8 E -2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -4 2 B 10 0 2 0 14 C 4 -2 0 -8 4 D 4 0 8 0 8 E -2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -4 2 B 10 0 2 0 14 C 4 -2 0 -8 4 D 4 0 8 0 8 E -2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7751: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (19) E D B C A (18) A E C B D (8) E A C B D (6) D B C E A (5) C B A D E (5) A C B E D (5) D E B C A (4) E D B A C (3) E D A B C (3) E A D C B (3) E A C D B (3) D B E C A (3) D B C A E (3) C B D A E (3) B D C A E (2) E C B A D (1) E A D B C (1) C A B D E (1) B D C E A (1) B C D E A (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 0 4 6 -4 B 0 0 -10 8 -2 C -4 10 0 8 -6 D -6 -8 -8 0 -4 E 4 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 6 -4 B 0 0 -10 8 -2 C -4 10 0 8 -6 D -6 -8 -8 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=33 D=15 C=9 B=5 so B is eliminated. Round 2 votes counts: E=38 A=33 D=18 C=11 so C is eliminated. Round 3 votes counts: A=39 E=38 D=23 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:208 C:204 A:203 B:198 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 6 -4 B 0 0 -10 8 -2 C -4 10 0 8 -6 D -6 -8 -8 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 -4 B 0 0 -10 8 -2 C -4 10 0 8 -6 D -6 -8 -8 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 -4 B 0 0 -10 8 -2 C -4 10 0 8 -6 D -6 -8 -8 0 -4 E 4 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7752: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) D B A C E (7) E D C A B (6) E C A B D (6) D E C A B (6) E C A D B (5) C A B D E (5) D C E A B (4) D C A B E (4) E D B A C (3) E C D A B (3) B A E C D (3) B A C E D (3) A C B E D (3) E B A C D (2) D B E C A (2) D B E A C (2) C A E B D (2) C A D B E (2) B D A E C (2) B D A C E (2) B A D C E (2) A B C E D (2) E D C B A (1) E D B C A (1) E B D A C (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E B A (1) D C A E B (1) D B A E C (1) C E A D B (1) C E A B D (1) C D A B E (1) C A B E D (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -10 -2 4 B -12 0 -12 -8 4 C 10 12 0 2 4 D 2 8 -2 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 -2 4 B -12 0 -12 -8 4 C 10 12 0 2 4 D 2 8 -2 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=28 B=20 C=13 A=8 so A is eliminated. Round 2 votes counts: D=31 E=28 B=23 C=18 so C is eliminated. Round 3 votes counts: D=34 E=33 B=33 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:208 A:202 E:190 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 -2 4 B -12 0 -12 -8 4 C 10 12 0 2 4 D 2 8 -2 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 -2 4 B -12 0 -12 -8 4 C 10 12 0 2 4 D 2 8 -2 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 -2 4 B -12 0 -12 -8 4 C 10 12 0 2 4 D 2 8 -2 0 8 E -4 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7753: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) A D B C E (8) E B C D A (6) A B D E C (6) C D E B A (5) B E C D A (5) A D C B E (5) A E C B D (4) E B C A D (3) D B C E A (3) C E D B A (3) C E B D A (3) A E B D C (3) A D C E B (3) E B A C D (2) D C B E A (2) D C B A E (2) D C A B E (2) B E D C A (2) B D E C A (2) B D C E A (2) B D C A E (2) A C E D B (2) A B E D C (2) E A C D B (1) D B C A E (1) D A C B E (1) D A B C E (1) C E A D B (1) B E A D C (1) B D E A C (1) B D A E C (1) B D A C E (1) B A D E C (1) A E D C B (1) A E D B C (1) A E C D B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -4 -6 0 B 12 0 12 12 6 C 4 -12 0 -12 -4 D 6 -12 12 0 2 E 0 -6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -6 0 B 12 0 12 12 6 C 4 -12 0 -12 -4 D 6 -12 12 0 2 E 0 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=20 B=18 D=12 C=12 so D is eliminated. Round 2 votes counts: A=40 B=22 E=20 C=18 so C is eliminated. Round 3 votes counts: A=42 E=32 B=26 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:221 D:204 E:198 A:189 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -6 0 B 12 0 12 12 6 C 4 -12 0 -12 -4 D 6 -12 12 0 2 E 0 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -6 0 B 12 0 12 12 6 C 4 -12 0 -12 -4 D 6 -12 12 0 2 E 0 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -6 0 B 12 0 12 12 6 C 4 -12 0 -12 -4 D 6 -12 12 0 2 E 0 -6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7754: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) B E C D A (5) B D A E C (5) B C E A D (5) E C B D A (4) D E B A C (4) C E B A D (4) E D B C A (3) D B A E C (3) C B E A D (3) C A E B D (3) C A B E D (3) B A D C E (3) A D C B E (3) A D B C E (3) A C D B E (3) E D C B A (2) E C D A B (2) E B C D A (2) D E A C B (2) D A E C B (2) D A C E B (2) D A B C E (2) C E A B D (2) B C A E D (2) B A C E D (2) A D C E B (2) A C D E B (2) A C B E D (2) A C B D E (2) E D C A B (1) E C D B A (1) E C B A D (1) E B D C A (1) D E C A B (1) D E B C A (1) D B E A C (1) D A E B C (1) C E A D B (1) C B A E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 -2 8 B 6 0 2 4 12 C -4 -2 0 2 4 D 2 -4 -2 0 0 E -8 -12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -2 8 B 6 0 2 4 12 C -4 -2 0 2 4 D 2 -4 -2 0 0 E -8 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=22 A=19 E=17 C=17 so E is eliminated. Round 2 votes counts: D=31 C=25 B=25 A=19 so A is eliminated. Round 3 votes counts: D=39 C=34 B=27 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:212 A:202 C:200 D:198 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -2 8 B 6 0 2 4 12 C -4 -2 0 2 4 D 2 -4 -2 0 0 E -8 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 8 B 6 0 2 4 12 C -4 -2 0 2 4 D 2 -4 -2 0 0 E -8 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 8 B 6 0 2 4 12 C -4 -2 0 2 4 D 2 -4 -2 0 0 E -8 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996229 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7755: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (17) A C E B D (17) C A E B D (11) D B E A C (8) C E B A D (5) A C D E B (5) C E B D A (4) E B C D A (3) D C B E A (3) A D C B E (3) D C E B A (2) D B A E C (2) D A C B E (2) C D E B A (2) B E D C A (2) B D E C A (2) A D B E C (2) E C B D A (1) D B C E A (1) D B C A E (1) D A B E C (1) D A B C E (1) C A D E B (1) B E C D A (1) B E A C D (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -12 -6 -4 B 10 0 -14 -2 -6 C 12 14 0 6 16 D 6 2 -6 0 6 E 4 6 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -6 -4 B 10 0 -14 -2 -6 C 12 14 0 6 16 D 6 2 -6 0 6 E 4 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=29 C=23 B=6 E=4 so E is eliminated. Round 2 votes counts: D=38 A=29 C=24 B=9 so B is eliminated. Round 3 votes counts: D=42 A=30 C=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:224 D:204 B:194 E:194 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -6 -4 B 10 0 -14 -2 -6 C 12 14 0 6 16 D 6 2 -6 0 6 E 4 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -6 -4 B 10 0 -14 -2 -6 C 12 14 0 6 16 D 6 2 -6 0 6 E 4 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -6 -4 B 10 0 -14 -2 -6 C 12 14 0 6 16 D 6 2 -6 0 6 E 4 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999186 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7756: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) A C E B D (6) E D C B A (5) E C B D A (5) E C A D B (5) D B E C A (5) D B A E C (5) B D E C A (4) A B D C E (4) E C D A B (3) C E A D B (3) B D A E C (3) E C D B A (2) E C A B D (2) D B E A C (2) D A E C B (2) D A E B C (2) D A B E C (2) D A B C E (2) C E B A D (2) B E C D A (2) B C A E D (2) B A D C E (2) A D B C E (2) A C E D B (2) A C D E B (2) A C B E D (2) E D C A B (1) E B D C A (1) D E C B A (1) D E C A B (1) D B A C E (1) C B E A D (1) C A E B D (1) B E D C A (1) B D E A C (1) B C E A D (1) A E C D B (1) A D C E B (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -10 -2 -10 B -8 0 -12 0 -10 C 10 12 0 4 -12 D 2 0 -4 0 -12 E 10 10 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -10 -2 -10 B -8 0 -12 0 -10 C 10 12 0 4 -12 D 2 0 -4 0 -12 E 10 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=23 A=23 B=16 C=14 so C is eliminated. Round 2 votes counts: E=36 A=24 D=23 B=17 so B is eliminated. Round 3 votes counts: E=41 D=31 A=28 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:207 A:193 D:193 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -10 -2 -10 B -8 0 -12 0 -10 C 10 12 0 4 -12 D 2 0 -4 0 -12 E 10 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 -2 -10 B -8 0 -12 0 -10 C 10 12 0 4 -12 D 2 0 -4 0 -12 E 10 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 -2 -10 B -8 0 -12 0 -10 C 10 12 0 4 -12 D 2 0 -4 0 -12 E 10 10 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7757: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (14) D C A E B (8) E C D B A (7) E B C D A (6) C D E A B (6) B E A D C (6) B E A C D (5) B A D C E (5) E C D A B (4) A D C E B (4) A B D C E (4) E D C A B (3) C E D B A (3) B E C A D (3) C D A E B (2) B A E C D (2) B A C D E (2) A D C B E (2) A D B C E (2) E B A D C (1) D E C A B (1) D C E A B (1) D A C E B (1) C D A B E (1) B E C D A (1) B C E D A (1) B C A D E (1) B A D E C (1) A E D B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 4 10 4 B 16 0 14 6 4 C -4 -14 0 -12 -14 D -10 -6 12 0 -14 E -4 -4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 10 4 B 16 0 14 6 4 C -4 -14 0 -12 -14 D -10 -6 12 0 -14 E -4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=21 A=15 C=12 D=11 so D is eliminated. Round 2 votes counts: B=41 E=22 C=21 A=16 so A is eliminated. Round 3 votes counts: B=49 C=28 E=23 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:210 A:201 D:191 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 10 4 B 16 0 14 6 4 C -4 -14 0 -12 -14 D -10 -6 12 0 -14 E -4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 10 4 B 16 0 14 6 4 C -4 -14 0 -12 -14 D -10 -6 12 0 -14 E -4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 10 4 B 16 0 14 6 4 C -4 -14 0 -12 -14 D -10 -6 12 0 -14 E -4 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7758: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (8) B C E D A (8) B C D E A (7) D E A C B (5) D A E C B (5) A D E C B (5) E D C B A (4) D E C B A (4) A E D C B (4) A D E B C (4) A D B C E (4) B C E A D (3) B C A E D (3) A E C B D (3) A B C E D (3) A B C D E (3) E D A C B (2) E C D B A (2) E C B D A (2) E C B A D (2) D C E B A (2) D A E B C (2) B C D A E (2) B C A D E (2) E A C B D (1) D E C A B (1) D C B E A (1) D B C E A (1) D A B C E (1) C E B A D (1) C D B E A (1) C B E A D (1) C B D E A (1) B D C A E (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -14 -20 -14 B 12 0 -10 2 2 C 14 10 0 6 6 D 20 -2 -6 0 6 E 14 -2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -20 -14 B 12 0 -10 2 2 C 14 10 0 6 6 D 20 -2 -6 0 6 E 14 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 D=22 E=13 C=12 so C is eliminated. Round 2 votes counts: B=36 A=27 D=23 E=14 so E is eliminated. Round 3 votes counts: B=41 D=31 A=28 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:218 D:209 B:203 E:200 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -20 -14 B 12 0 -10 2 2 C 14 10 0 6 6 D 20 -2 -6 0 6 E 14 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -20 -14 B 12 0 -10 2 2 C 14 10 0 6 6 D 20 -2 -6 0 6 E 14 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -20 -14 B 12 0 -10 2 2 C 14 10 0 6 6 D 20 -2 -6 0 6 E 14 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7759: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (13) E A D C B (12) C B D A E (8) E C B A D (6) B C D E A (6) B C D A E (6) D A B C E (5) E A C B D (4) C B E D A (4) A E D C B (3) E A D B C (2) D E A B C (2) D A E B C (2) D A B E C (2) C B E A D (2) B D C E A (2) A E D B C (2) A E C B D (2) A D C B E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C A B D (1) E B C D A (1) D B E A C (1) D B C E A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A B D E (1) B D C A E (1) A D E B C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -12 -16 4 B 12 0 2 -4 18 C 12 -2 0 -10 12 D 16 4 10 0 14 E -4 -18 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -16 4 B 12 0 2 -4 18 C 12 -2 0 -10 12 D 16 4 10 0 14 E -4 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 C=18 B=15 A=12 so A is eliminated. Round 2 votes counts: E=36 D=30 C=19 B=15 so B is eliminated. Round 3 votes counts: E=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:214 C:206 A:182 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -12 -16 4 B 12 0 2 -4 18 C 12 -2 0 -10 12 D 16 4 10 0 14 E -4 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -16 4 B 12 0 2 -4 18 C 12 -2 0 -10 12 D 16 4 10 0 14 E -4 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -16 4 B 12 0 2 -4 18 C 12 -2 0 -10 12 D 16 4 10 0 14 E -4 -18 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7760: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (11) A C E B D (11) D B E C A (9) E A C B D (8) D B C E A (8) E C A B D (3) D E A B C (3) A E D C B (3) A E C B D (3) E C B A D (2) E B C A D (2) E A C D B (2) D E B C A (2) D B A C E (2) D A B C E (2) B E C A D (2) B D C A E (2) A E C D B (2) E D B C A (1) E B C D A (1) E B A C D (1) E A D C B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A E B C (1) D A C B E (1) C E A B D (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B E D (1) C A B D E (1) B D C E A (1) B C E D A (1) B C E A D (1) B C D E A (1) A D E C B (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -6 2 -2 B 0 0 6 -10 -4 C 6 -6 0 -2 0 D -2 10 2 0 2 E 2 4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999961 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 2 -2 B 0 0 6 -10 -4 C 6 -6 0 -2 0 D -2 10 2 0 2 E 2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 A=23 E=21 B=8 C=6 so C is eliminated. Round 2 votes counts: D=42 A=27 E=22 B=9 so B is eliminated. Round 3 votes counts: D=46 A=28 E=26 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:206 E:202 C:199 A:197 B:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 2 -2 B 0 0 6 -10 -4 C 6 -6 0 -2 0 D -2 10 2 0 2 E 2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 -2 B 0 0 6 -10 -4 C 6 -6 0 -2 0 D -2 10 2 0 2 E 2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 -2 B 0 0 6 -10 -4 C 6 -6 0 -2 0 D -2 10 2 0 2 E 2 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.439999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7761: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) A B E C D (6) E B A D C (5) C D A E B (5) C A D E B (5) A C E B D (5) D C A B E (4) D B E A C (4) C A D B E (4) B A E D C (4) E B D A C (3) D C E B A (3) D C B E A (3) C A E B D (3) A B E D C (3) E C A B D (2) E B A C D (2) E A C B D (2) D E C B A (2) C E D B A (2) C D E B A (2) C D E A B (2) C D A B E (2) B E A D C (2) B A D E C (2) A E B C D (2) E B D C A (1) E B C A D (1) E A B C D (1) D C B A E (1) D B C E A (1) D B A E C (1) D A C B E (1) C E A B D (1) C A E D B (1) B E D A C (1) B D E A C (1) B D A E C (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -2 8 6 B -4 0 -4 2 -2 C 2 4 0 2 -6 D -8 -2 -2 0 0 E -6 2 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102041 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 4 -2 8 6 B -4 0 -4 2 -2 C 2 4 0 2 -6 D -8 -2 -2 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102033 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 A=19 E=17 B=11 so B is eliminated. Round 2 votes counts: D=28 C=27 A=25 E=20 so E is eliminated. Round 3 votes counts: A=37 D=33 C=30 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:201 E:201 B:196 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 8 6 B -4 0 -4 2 -2 C 2 4 0 2 -6 D -8 -2 -2 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102033 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 8 6 B -4 0 -4 2 -2 C 2 4 0 2 -6 D -8 -2 -2 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102033 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 8 6 B -4 0 -4 2 -2 C 2 4 0 2 -6 D -8 -2 -2 0 0 E -6 2 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102033 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7762: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) E A B D C (7) C D B A E (6) C B A D E (6) E C D A B (5) C E D B A (5) E A B C D (4) D E C B A (4) D E B A C (4) D B A E C (4) C E A B D (3) C D E B A (3) C A B E D (3) A B E C D (3) A B C D E (3) E D C A B (2) E A C B D (2) D B E A C (2) D B A C E (2) D A B E C (2) C E B A D (2) A B D E C (2) A B D C E (2) E D C B A (1) E D A C B (1) E C D B A (1) E C A B D (1) E A D B C (1) D C E B A (1) D C B A E (1) D B C A E (1) D A E B C (1) C B D A E (1) B A D C E (1) B A C D E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 8 -14 -18 B -8 0 4 -14 -18 C -8 -4 0 -2 -18 D 14 14 2 0 -6 E 18 18 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 -14 -18 B -8 0 4 -14 -18 C -8 -4 0 -2 -18 D 14 14 2 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=29 D=22 A=12 B=2 so B is eliminated. Round 2 votes counts: E=35 C=29 D=22 A=14 so A is eliminated. Round 3 votes counts: E=40 C=33 D=27 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:230 D:212 A:192 C:184 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 -14 -18 B -8 0 4 -14 -18 C -8 -4 0 -2 -18 D 14 14 2 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -14 -18 B -8 0 4 -14 -18 C -8 -4 0 -2 -18 D 14 14 2 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -14 -18 B -8 0 4 -14 -18 C -8 -4 0 -2 -18 D 14 14 2 0 -6 E 18 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7763: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) A E B C D (8) E A B D C (7) C D B E A (6) B A E D C (6) C D B A E (4) B D C A E (4) E C D A B (3) E C A D B (3) E A B C D (3) D C E A B (3) D C B E A (3) C D E B A (3) C D E A B (3) B A D C E (3) A E B D C (3) E D C A B (2) E A D B C (2) E A C B D (2) D E C A B (2) D C B A E (2) D B C A E (2) C E D A B (2) B C D A E (2) B A C E D (2) A B E D C (2) E D A C B (1) D C E B A (1) C B D A E (1) C A E B D (1) C A B E D (1) B D A E C (1) B D A C E (1) B A D E C (1) A E C B D (1) Total count = 100 A B C D E A 0 16 4 8 -10 B -16 0 -6 -2 -18 C -4 6 0 8 -12 D -8 2 -8 0 -16 E 10 18 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 4 8 -10 B -16 0 -6 -2 -18 C -4 6 0 8 -12 D -8 2 -8 0 -16 E 10 18 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=21 B=20 A=14 D=13 so D is eliminated. Round 2 votes counts: E=34 C=30 B=22 A=14 so A is eliminated. Round 3 votes counts: E=46 C=30 B=24 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:209 C:199 D:185 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 4 8 -10 B -16 0 -6 -2 -18 C -4 6 0 8 -12 D -8 2 -8 0 -16 E 10 18 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 8 -10 B -16 0 -6 -2 -18 C -4 6 0 8 -12 D -8 2 -8 0 -16 E 10 18 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 8 -10 B -16 0 -6 -2 -18 C -4 6 0 8 -12 D -8 2 -8 0 -16 E 10 18 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7764: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (16) D A B C E (11) D A B E C (10) E C B A D (8) D A E B C (5) C B E D A (5) A D E B C (5) D A E C B (3) B C E A D (3) E C A D B (2) E C A B D (2) E B C A D (2) E A C D B (2) D B A C E (2) D A C B E (2) C E D B A (2) C E A D B (2) C B E A D (2) B D A E C (2) A D B E C (2) E B A C D (1) E A C B D (1) D C A E B (1) D B A E C (1) D A C E B (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A C D (1) B A E D C (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 2 4 6 -2 B -2 0 -4 -6 -10 C -4 4 0 4 -2 D -6 6 -4 0 -6 E 2 10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 6 -2 B -2 0 -4 -6 -10 C -4 4 0 4 -2 D -6 6 -4 0 -6 E 2 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=29 E=18 A=9 B=8 so B is eliminated. Round 2 votes counts: D=38 C=32 E=20 A=10 so A is eliminated. Round 3 votes counts: D=45 C=32 E=23 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:210 A:205 C:201 D:195 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 6 -2 B -2 0 -4 -6 -10 C -4 4 0 4 -2 D -6 6 -4 0 -6 E 2 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 6 -2 B -2 0 -4 -6 -10 C -4 4 0 4 -2 D -6 6 -4 0 -6 E 2 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 6 -2 B -2 0 -4 -6 -10 C -4 4 0 4 -2 D -6 6 -4 0 -6 E 2 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7765: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) A D C E B (10) A C D E B (9) B E C D A (7) C D E B A (6) A D B E C (6) C E B D A (5) A D C B E (4) A B D E C (4) D C A E B (3) C D A E B (3) B A E D C (3) E B C D A (2) D B E C A (2) C E D B A (2) C D E A B (2) C A D E B (2) B E D C A (2) B E D A C (2) B E A D C (2) A D B C E (2) E D B C A (1) E C B D A (1) D C E B A (1) D A E B C (1) C A E D B (1) C A B E D (1) B E C A D (1) B A D E C (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 24 16 20 28 B -24 0 -2 -10 2 C -16 2 0 -10 6 D -20 10 10 0 14 E -28 -2 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 16 20 28 B -24 0 -2 -10 2 C -16 2 0 -10 6 D -20 10 10 0 14 E -28 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=49 C=22 B=18 D=7 E=4 so E is eliminated. Round 2 votes counts: A=49 C=23 B=20 D=8 so D is eliminated. Round 3 votes counts: A=50 C=27 B=23 so B is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:244 D:207 C:191 B:183 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 16 20 28 B -24 0 -2 -10 2 C -16 2 0 -10 6 D -20 10 10 0 14 E -28 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 16 20 28 B -24 0 -2 -10 2 C -16 2 0 -10 6 D -20 10 10 0 14 E -28 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 16 20 28 B -24 0 -2 -10 2 C -16 2 0 -10 6 D -20 10 10 0 14 E -28 -2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7766: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) C E B A D (9) B E D A C (6) C E B D A (5) C A D E B (5) E B A C D (4) D A C B E (4) C E A B D (4) B E D C A (4) E B C A D (3) D C A B E (3) D A B C E (3) C A E D B (3) B E A D C (3) B D E A C (3) E C B A D (2) E B C D A (2) D B E C A (2) B E C D A (2) A E B D C (2) A D C E B (2) A D C B E (2) A C D E B (2) E C A B D (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A E C (1) C E D A B (1) C E A D B (1) C D A E B (1) B E C A D (1) B E A C D (1) B C E D A (1) A D B E C (1) A C E D B (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -4 -2 -14 B 4 0 6 10 2 C 4 -6 0 0 0 D 2 -10 0 0 -14 E 14 -2 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 -14 B 4 0 6 10 2 C 4 -6 0 0 0 D 2 -10 0 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 B=21 E=12 A=12 so E is eliminated. Round 2 votes counts: C=32 B=30 D=26 A=12 so A is eliminated. Round 3 votes counts: C=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:213 B:211 C:199 D:189 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -14 B 4 0 6 10 2 C 4 -6 0 0 0 D 2 -10 0 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -14 B 4 0 6 10 2 C 4 -6 0 0 0 D 2 -10 0 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -14 B 4 0 6 10 2 C 4 -6 0 0 0 D 2 -10 0 0 -14 E 14 -2 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7767: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (8) E C B D A (7) B D A E C (6) B D A C E (6) A B D E C (6) E C A B D (5) D B C E A (5) C A E D B (4) A C D B E (4) E C A D B (3) C E D B A (3) E C D A B (2) E B D C A (2) D B C A E (2) B E D A C (2) B E A D C (2) B D E C A (2) B D E A C (2) A E C B D (2) A C E D B (2) E B C D A (1) E B C A D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B A C E (1) C E A D B (1) C D E A B (1) C A D E B (1) B A D E C (1) A E B C D (1) A D B C E (1) A C E B D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -4 -8 -2 B 10 0 2 14 2 C 4 -2 0 2 -14 D 8 -14 -2 0 -2 E 2 -2 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997902 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -8 -2 B 10 0 2 14 2 C 4 -2 0 2 -14 D 8 -14 -2 0 -2 E 2 -2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=27 B=21 D=10 C=10 so D is eliminated. Round 2 votes counts: E=32 B=29 A=27 C=12 so C is eliminated. Round 3 votes counts: E=37 A=32 B=31 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:208 C:195 D:195 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -4 -8 -2 B 10 0 2 14 2 C 4 -2 0 2 -14 D 8 -14 -2 0 -2 E 2 -2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -8 -2 B 10 0 2 14 2 C 4 -2 0 2 -14 D 8 -14 -2 0 -2 E 2 -2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -8 -2 B 10 0 2 14 2 C 4 -2 0 2 -14 D 8 -14 -2 0 -2 E 2 -2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7768: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) B C D E A (8) E C B D A (6) D A B C E (6) C E B D A (6) E C A B D (5) A E D C B (5) A E D B C (5) A D E B C (5) A D B C E (5) E C B A D (4) E A C B D (4) E A C D B (3) D B C E A (3) E A D C B (2) D B C A E (2) C B D E A (2) B D C A E (2) A E C D B (2) E C A D B (1) D E C B A (1) D E B A C (1) D B A C E (1) D A E B C (1) D A B E C (1) C B E A D (1) C A E B D (1) B D C E A (1) B D A C E (1) B C D A E (1) A D B E C (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 -6 -16 B 0 0 -6 10 -4 C 8 6 0 10 4 D 6 -10 -10 0 -12 E 16 4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -6 -16 B 0 0 -6 10 -4 C 8 6 0 10 4 D 6 -10 -10 0 -12 E 16 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 C=20 D=16 B=13 so B is eliminated. Round 2 votes counts: C=29 A=26 E=25 D=20 so D is eliminated. Round 3 votes counts: C=37 A=36 E=27 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:214 E:214 B:200 D:187 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -6 -16 B 0 0 -6 10 -4 C 8 6 0 10 4 D 6 -10 -10 0 -12 E 16 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -6 -16 B 0 0 -6 10 -4 C 8 6 0 10 4 D 6 -10 -10 0 -12 E 16 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -6 -16 B 0 0 -6 10 -4 C 8 6 0 10 4 D 6 -10 -10 0 -12 E 16 4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7769: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) B C A D E (9) E D A C B (8) C B A E D (8) D A E B C (6) E A D C B (5) C B E D A (5) E C A B D (4) D B C E A (4) C B E A D (3) C B A D E (3) A E D C B (3) A E D B C (3) A D E B C (3) A B C D E (3) C E B A D (2) C B D E A (2) B C D A E (2) A E C B D (2) E D C B A (1) E C D B A (1) E C B D A (1) E A C B D (1) D E C B A (1) D E A C B (1) D B A C E (1) D A B E C (1) D A B C E (1) B D C A E (1) B A D C E (1) B A C D E (1) A D B C E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 6 8 2 B -8 0 -4 0 -4 C -6 4 0 -4 -2 D -8 0 4 0 4 E -2 4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999373 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 8 2 B -8 0 -4 0 -4 C -6 4 0 -4 -2 D -8 0 4 0 4 E -2 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 E=21 A=17 B=14 so B is eliminated. Round 2 votes counts: C=34 D=26 E=21 A=19 so A is eliminated. Round 3 votes counts: C=39 D=32 E=29 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:212 D:200 E:200 C:196 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 8 2 B -8 0 -4 0 -4 C -6 4 0 -4 -2 D -8 0 4 0 4 E -2 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 8 2 B -8 0 -4 0 -4 C -6 4 0 -4 -2 D -8 0 4 0 4 E -2 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 8 2 B -8 0 -4 0 -4 C -6 4 0 -4 -2 D -8 0 4 0 4 E -2 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7770: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (8) C D E A B (7) E B D A C (6) C A D B E (6) D C E B A (5) B A E D C (5) A C B E D (5) A C B D E (5) E B A D C (4) C D A B E (4) C A E D B (4) A B C D E (4) E D B C A (3) E C D A B (3) D E B C A (3) B E A D C (3) E D C B A (2) D C B A E (2) B E D A C (2) A E B C D (2) A B E C D (2) E C A D B (1) E A C B D (1) E A B C D (1) D C B E A (1) D B E C A (1) D B C E A (1) D B C A E (1) C E D A B (1) C E A D B (1) C D A E B (1) C A D E B (1) C A B D E (1) B A D E C (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 -2 -8 B -4 0 -4 4 6 C -2 4 0 4 2 D 2 -4 -4 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000022 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 4 2 -2 -8 B -4 0 -4 4 6 C -2 4 0 4 2 D 2 -4 -4 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000013 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=21 A=20 B=19 D=14 so D is eliminated. Round 2 votes counts: C=34 E=24 B=22 A=20 so A is eliminated. Round 3 votes counts: C=44 B=29 E=27 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:204 B:201 D:199 A:198 E:198 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 2 -2 -8 B -4 0 -4 4 6 C -2 4 0 4 2 D 2 -4 -4 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000013 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -2 -8 B -4 0 -4 4 6 C -2 4 0 4 2 D 2 -4 -4 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000013 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -2 -8 B -4 0 -4 4 6 C -2 4 0 4 2 D 2 -4 -4 0 4 E 8 -6 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.166667 Sum of squares = 0.500000000013 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7771: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (13) D E A B C (7) D E A C B (6) C B A E D (5) A B D E C (5) D A E B C (4) C D E B A (4) C B E A D (4) B A C E D (4) E D A B C (3) E C D B A (3) D E C A B (3) C B E D A (3) A B E D C (3) E D B A C (2) E C B D A (2) E B A D C (2) C D A B E (2) C B D A E (2) C A B D E (2) B C A E D (2) A D E B C (2) A D B E C (2) E D C B A (1) E D B C A (1) E B A C D (1) D A C B E (1) D A B E C (1) C E D B A (1) C D B E A (1) C A D B E (1) B E A C D (1) B C E A D (1) B A C D E (1) A D B C E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 6 8 B 8 0 -8 8 16 C 2 8 0 10 0 D -6 -8 -10 0 20 E -8 -16 0 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.876217 D: 0.000000 E: 0.123783 Sum of squares = 0.783078591692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.876217 D: 0.876217 E: 1.000000 A B C D E A 0 -8 -2 6 8 B 8 0 -8 8 16 C 2 8 0 10 0 D -6 -8 -10 0 20 E -8 -16 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000001625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=22 A=16 E=15 B=9 so B is eliminated. Round 2 votes counts: C=41 D=22 A=21 E=16 so E is eliminated. Round 3 votes counts: C=46 D=29 A=25 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:212 C:210 A:202 D:198 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 6 8 B 8 0 -8 8 16 C 2 8 0 10 0 D -6 -8 -10 0 20 E -8 -16 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000001625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 6 8 B 8 0 -8 8 16 C 2 8 0 10 0 D -6 -8 -10 0 20 E -8 -16 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000001625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 6 8 B 8 0 -8 8 16 C 2 8 0 10 0 D -6 -8 -10 0 20 E -8 -16 0 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.200000 Sum of squares = 0.680000001625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7772: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) E D A B C (8) B C E A D (6) E A D C B (5) C B A E D (5) B C D E A (5) B D C E A (4) B C D A E (4) A E D C B (4) D E B A C (3) D E A B C (3) C B D A E (3) C B A D E (3) B D E C A (3) A E C D B (3) E D B A C (2) D C B A E (2) D A C E B (2) C A D E B (2) C A D B E (2) B C E D A (2) A E C B D (2) A C E B D (2) E A B D C (1) E A B C D (1) D C A B E (1) D B E A C (1) D B C E A (1) D B C A E (1) C D A B E (1) C B E A D (1) C A B E D (1) B E D A C (1) B E C D A (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 4 4 -16 B -4 0 14 -10 -4 C -4 -14 0 -8 2 D -4 10 8 0 -12 E 16 4 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.700000 Sum of squares = 0.540000000132 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.300000 E: 1.000000 A B C D E A 0 4 4 4 -16 B -4 0 14 -10 -4 C -4 -14 0 -8 2 D -4 10 8 0 -12 E 16 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.700000 Sum of squares = 0.540000000062 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.300000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 C=18 A=15 D=14 so D is eliminated. Round 2 votes counts: E=33 B=29 C=21 A=17 so A is eliminated. Round 3 votes counts: E=43 B=29 C=28 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:201 A:198 B:198 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 4 -16 B -4 0 14 -10 -4 C -4 -14 0 -8 2 D -4 10 8 0 -12 E 16 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.700000 Sum of squares = 0.540000000062 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.300000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 -16 B -4 0 14 -10 -4 C -4 -14 0 -8 2 D -4 10 8 0 -12 E 16 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.700000 Sum of squares = 0.540000000062 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.300000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 -16 B -4 0 14 -10 -4 C -4 -14 0 -8 2 D -4 10 8 0 -12 E 16 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.000000 E: 0.700000 Sum of squares = 0.540000000062 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 0.300000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7773: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) C B D A E (9) E A B D C (8) C D B A E (6) E B A C D (4) C B E D A (4) B A E C D (4) A E B D C (4) A B E D C (4) E D A C B (3) E A B C D (3) C E D B A (3) B C A E D (3) A E D B C (3) E C D A B (2) E C B A D (2) E A D C B (2) D C E A B (2) D A E B C (2) C D E B A (2) B C D A E (2) A D E B C (2) A B D E C (2) E D C A B (1) E C A D B (1) E C A B D (1) D E C A B (1) D E A C B (1) D C B A E (1) D C A E B (1) C E B D A (1) C B D E A (1) B E C A D (1) B C A D E (1) B A D C E (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 16 -6 B -10 0 12 12 -18 C -10 -12 0 2 -26 D -16 -12 -2 0 -30 E 6 18 26 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 10 16 -6 B -10 0 12 12 -18 C -10 -12 0 2 -26 D -16 -12 -2 0 -30 E 6 18 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=26 A=17 B=12 D=8 so D is eliminated. Round 2 votes counts: E=39 C=30 A=19 B=12 so B is eliminated. Round 3 votes counts: E=40 C=36 A=24 so A is eliminated. Round 4 votes counts: E=63 C=37 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:240 A:215 B:198 C:177 D:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 10 16 -6 B -10 0 12 12 -18 C -10 -12 0 2 -26 D -16 -12 -2 0 -30 E 6 18 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 16 -6 B -10 0 12 12 -18 C -10 -12 0 2 -26 D -16 -12 -2 0 -30 E 6 18 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 16 -6 B -10 0 12 12 -18 C -10 -12 0 2 -26 D -16 -12 -2 0 -30 E 6 18 26 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7774: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) D C E B A (5) D B C E A (5) B E A D C (5) B D E C A (5) C D A E B (4) B E D A C (4) B E A C D (4) A E C D B (4) A C D E B (4) E C A D B (3) D C B A E (3) B E D C A (3) A E C B D (3) A C E D B (3) A B E C D (3) E C D A B (2) E B A C D (2) E A C D B (2) E A B C D (2) D C E A B (2) D C A B E (2) D B E C A (2) C D E A B (2) C A D E B (2) B D E A C (2) B A E D C (2) B A D C E (2) E B D C A (1) E A C B D (1) D B C A E (1) C E A D B (1) B D A E C (1) B D A C E (1) B A E C D (1) A E B C D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -2 C 0 4 0 -6 -6 D 4 8 6 0 2 E 6 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -2 C 0 4 0 -6 -6 D 4 8 6 0 2 E 6 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=27 A=21 E=13 C=9 so C is eliminated. Round 2 votes counts: D=33 B=30 A=23 E=14 so E is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:206 A:196 C:196 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -2 C 0 4 0 -6 -6 D 4 8 6 0 2 E 6 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -2 C 0 4 0 -6 -6 D 4 8 6 0 2 E 6 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 -6 B -2 0 -4 -8 -2 C 0 4 0 -6 -6 D 4 8 6 0 2 E 6 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7775: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) E C B D A (5) E B C A D (5) E B A D C (5) B A C D E (4) D E C A B (3) D C A E B (3) D A C E B (3) C D A B E (3) C B A D E (3) B E C A D (3) B A E C D (3) E D C A B (2) E B D A C (2) E B C D A (2) E B A C D (2) E A B D C (2) D C E A B (2) C D E B A (2) C D B E A (2) C A B D E (2) B E A C D (2) B C E A D (2) B A E D C (2) A D C B E (2) A B C D E (2) E D C B A (1) E D B C A (1) E D A C B (1) E D A B C (1) E C D B A (1) E B D C A (1) E A D B C (1) D C A B E (1) C E D B A (1) C D E A B (1) C D B A E (1) C B D E A (1) C B D A E (1) C A D B E (1) B C A E D (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B C E (1) A C D B E (1) A B E D C (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -2 4 -2 B 8 0 0 12 0 C 2 0 0 6 0 D -4 -12 -6 0 -2 E 2 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.147266 C: 0.363007 D: 0.000000 E: 0.489727 Sum of squares = 0.393293695494 Cumulative probabilities = A: 0.000000 B: 0.147266 C: 0.510273 D: 0.510273 E: 1.000000 A B C D E A 0 -8 -2 4 -2 B 8 0 0 12 0 C 2 0 0 6 0 D -4 -12 -6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=19 C=18 B=18 A=13 so A is eliminated. Round 2 votes counts: E=34 B=24 D=23 C=19 so C is eliminated. Round 3 votes counts: E=35 D=34 B=31 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:210 C:204 E:202 A:196 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 4 -2 B 8 0 0 12 0 C 2 0 0 6 0 D -4 -12 -6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 4 -2 B 8 0 0 12 0 C 2 0 0 6 0 D -4 -12 -6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 4 -2 B 8 0 0 12 0 C 2 0 0 6 0 D -4 -12 -6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7776: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) B A C D E (9) E C A D B (6) D E B A C (6) D B A C E (5) E C A B D (4) D E B C A (4) E D C A B (3) D E C B A (3) D B A E C (3) C E A D B (3) C A E B D (3) C A B D E (3) B A D C E (3) E C D A B (2) D C E B A (2) D B C E A (2) B D A E C (2) B D A C E (2) B A C E D (2) A C B E D (2) A C B D E (2) A B C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E B A D C (1) E A B C D (1) D E C A B (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C A E (1) C E A B D (1) C D E A B (1) C B D A E (1) C A B E D (1) A C E B D (1) Total count = 100 A B C D E A 0 -28 8 -12 -14 B 28 0 16 -20 10 C -8 -16 0 -12 -4 D 12 20 12 0 30 E 14 -10 4 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 8 -12 -14 B 28 0 16 -20 10 C -8 -16 0 -12 -4 D 12 20 12 0 30 E 14 -10 4 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 E=20 B=18 C=13 A=7 so A is eliminated. Round 2 votes counts: D=42 E=20 B=20 C=18 so C is eliminated. Round 3 votes counts: D=43 B=29 E=28 so E is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:237 B:217 E:189 C:180 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -28 8 -12 -14 B 28 0 16 -20 10 C -8 -16 0 -12 -4 D 12 20 12 0 30 E 14 -10 4 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 8 -12 -14 B 28 0 16 -20 10 C -8 -16 0 -12 -4 D 12 20 12 0 30 E 14 -10 4 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 8 -12 -14 B 28 0 16 -20 10 C -8 -16 0 -12 -4 D 12 20 12 0 30 E 14 -10 4 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7777: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) C A D E B (7) A E C B D (7) E B A D C (5) E A C B D (5) D B C A E (5) E B D A C (4) D B E C A (4) D B C E A (4) B D E A C (4) C A D B E (3) B E D A C (3) B D C A E (3) A C E B D (3) A C B D E (3) E D C A B (2) E B D C A (2) E A C D B (2) D C B A E (2) C D A E B (2) C D A B E (2) B E A D C (2) E D B C A (1) E B A C D (1) E A B C D (1) D E C B A (1) D C A B E (1) C E D A B (1) C E A D B (1) B D C E A (1) B D A E C (1) B D A C E (1) B A E D C (1) B A E C D (1) B A D C E (1) B A C D E (1) A E C D B (1) A E B C D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 16 12 12 B -4 0 -4 4 -12 C -16 4 0 4 0 D -12 -4 -4 0 -6 E -12 12 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 12 12 B -4 0 -4 4 -12 C -16 4 0 4 0 D -12 -4 -4 0 -6 E -12 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=23 B=19 D=17 C=16 so C is eliminated. Round 2 votes counts: A=35 E=25 D=21 B=19 so B is eliminated. Round 3 votes counts: A=39 D=31 E=30 so E is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:203 C:196 B:192 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 12 12 B -4 0 -4 4 -12 C -16 4 0 4 0 D -12 -4 -4 0 -6 E -12 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 12 12 B -4 0 -4 4 -12 C -16 4 0 4 0 D -12 -4 -4 0 -6 E -12 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 12 12 B -4 0 -4 4 -12 C -16 4 0 4 0 D -12 -4 -4 0 -6 E -12 12 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7778: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) A E D C B (8) B C D E A (7) B D C A E (5) E A C D B (4) C E B A D (4) C E A D B (4) C D A E B (4) E A B D C (3) D B C A E (3) D A C E B (3) C B D E A (3) A E D B C (3) E C A B D (2) D C B A E (2) C E A B D (2) C D E B A (2) C B E D A (2) C A E D B (2) B E C A D (2) B E A D C (2) B D A E C (2) A E B D C (2) E B A D C (1) E B A C D (1) E A B C D (1) D C A E B (1) D C A B E (1) D B A E C (1) D B A C E (1) D A E B C (1) D A B E C (1) C E D A B (1) C D B A E (1) C D A B E (1) C A D E B (1) B E A C D (1) B D C E A (1) B D A C E (1) B C E D A (1) B A D E C (1) A E C D B (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 12 -4 10 -4 B -12 0 -16 4 -22 C 4 16 0 10 6 D -10 -4 -10 0 -10 E 4 22 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 10 -4 B -12 0 -16 4 -22 C 4 16 0 10 6 D -10 -4 -10 0 -10 E 4 22 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=23 E=20 A=16 D=14 so D is eliminated. Round 2 votes counts: C=31 B=28 A=21 E=20 so E is eliminated. Round 3 votes counts: A=37 C=33 B=30 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:215 A:207 D:183 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 10 -4 B -12 0 -16 4 -22 C 4 16 0 10 6 D -10 -4 -10 0 -10 E 4 22 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 10 -4 B -12 0 -16 4 -22 C 4 16 0 10 6 D -10 -4 -10 0 -10 E 4 22 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 10 -4 B -12 0 -16 4 -22 C 4 16 0 10 6 D -10 -4 -10 0 -10 E 4 22 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7779: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (7) C B A D E (6) E D B A C (5) B D E C A (5) A E C D B (5) A C E B D (5) E D A C B (4) E A D C B (4) C A B D E (4) B C D A E (4) A C D E B (4) E B D C A (3) E B D A C (3) E A D B C (3) D B C E A (3) B E D C A (3) B D C E A (3) D E B C A (2) D C A B E (2) D B E C A (2) B D C A E (2) A E D C B (2) A E C B D (2) A C B E D (2) E D A B C (1) E B A C D (1) E A C D B (1) D E A C B (1) D B C A E (1) C D B A E (1) C D A B E (1) C A D B E (1) B C D E A (1) B C A E D (1) B C A D E (1) B A C E D (1) A D E C B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 8 6 10 B -4 0 -8 -2 -8 C -8 8 0 0 4 D -6 2 0 0 -8 E -10 8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 6 10 B -4 0 -8 -2 -8 C -8 8 0 0 4 D -6 2 0 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 B=21 C=13 D=11 so D is eliminated. Round 2 votes counts: A=30 E=28 B=27 C=15 so C is eliminated. Round 3 votes counts: A=38 B=34 E=28 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 C:202 E:201 D:194 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 10 B -4 0 -8 -2 -8 C -8 8 0 0 4 D -6 2 0 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 10 B -4 0 -8 -2 -8 C -8 8 0 0 4 D -6 2 0 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 10 B -4 0 -8 -2 -8 C -8 8 0 0 4 D -6 2 0 0 -8 E -10 8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7780: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) C B E D A (6) C B D E A (6) C A D B E (5) E B A D C (4) C D B A E (4) B E D C A (4) A E D B C (4) A D E C B (4) E B D A C (3) E A D B C (3) E A B D C (3) D C A B E (3) D A C E B (3) C A D E B (3) B E D A C (3) A D E B C (3) A D C E B (3) A C D E B (3) D C B A E (2) C D B E A (2) B E C D A (2) B C E D A (2) A C E D B (2) E D A B C (1) E B C A D (1) E B A C D (1) D C A E B (1) D B E A C (1) D B C E A (1) D B A E C (1) D A E B C (1) C B E A D (1) C B D A E (1) C B A E D (1) C A E B D (1) B E C A D (1) B D E C A (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 4 -10 -10 8 B -4 0 -18 -16 8 C 10 18 0 -2 14 D 10 16 2 0 10 E -8 -8 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -10 8 B -4 0 -18 -16 8 C 10 18 0 -2 14 D 10 16 2 0 10 E -8 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=21 E=16 D=13 B=13 so D is eliminated. Round 2 votes counts: C=43 A=25 E=16 B=16 so E is eliminated. Round 3 votes counts: C=43 A=32 B=25 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:219 A:196 B:185 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -10 -10 8 B -4 0 -18 -16 8 C 10 18 0 -2 14 D 10 16 2 0 10 E -8 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -10 8 B -4 0 -18 -16 8 C 10 18 0 -2 14 D 10 16 2 0 10 E -8 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -10 8 B -4 0 -18 -16 8 C 10 18 0 -2 14 D 10 16 2 0 10 E -8 -8 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7781: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) A B D E C (7) A B D C E (7) B D E C A (6) A D B E C (6) D B E C A (5) A E C D B (5) A C E B D (4) A B C D E (4) C E D A B (3) C E A D B (3) C A E B D (3) B D C E A (3) A D E B C (3) A C E D B (3) E C D B A (2) E C A D B (2) C E A B D (2) B D A E C (2) B D A C E (2) A E D C B (2) A C B E D (2) E C D A B (1) D E B C A (1) D B A E C (1) D A B E C (1) C E B D A (1) C E B A D (1) C B E D A (1) C A E D B (1) B C D E A (1) B C A D E (1) Total count = 100 A B C D E A 0 18 -2 12 8 B -18 0 0 -6 -2 C 2 0 0 8 12 D -12 6 -8 0 0 E -8 2 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.054974 C: 0.945026 D: 0.000000 E: 0.000000 Sum of squares = 0.896095927373 Cumulative probabilities = A: 0.000000 B: 0.054974 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -2 12 8 B -18 0 0 -6 -2 C 2 0 0 8 12 D -12 6 -8 0 0 E -8 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000014441 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 C=29 B=15 D=8 E=5 so E is eliminated. Round 2 votes counts: A=43 C=34 B=15 D=8 so D is eliminated. Round 3 votes counts: A=44 C=34 B=22 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:218 C:211 D:193 E:191 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -2 12 8 B -18 0 0 -6 -2 C 2 0 0 8 12 D -12 6 -8 0 0 E -8 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000014441 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -2 12 8 B -18 0 0 -6 -2 C 2 0 0 8 12 D -12 6 -8 0 0 E -8 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000014441 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -2 12 8 B -18 0 0 -6 -2 C 2 0 0 8 12 D -12 6 -8 0 0 E -8 2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.900000 D: 0.000000 E: 0.000000 Sum of squares = 0.820000014441 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7782: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) C B A E D (6) D E B C A (5) A B C E D (5) A B C D E (5) E D A C B (4) E C D B A (4) D A E B C (4) C B E D A (4) B C A D E (4) B A C D E (4) C A E B D (3) C A B E D (3) A E D C B (3) E D C B A (2) E D C A B (2) E C D A B (2) D B A E C (2) B D A C E (2) B C D E A (2) A D E B C (2) A C B E D (2) A B D E C (2) A B D C E (2) E D A B C (1) D E C B A (1) D E B A C (1) D E A C B (1) C E B D A (1) C E B A D (1) C E A B D (1) C B E A D (1) B D C A E (1) B C A E D (1) A E C D B (1) A E C B D (1) A D B E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 16 12 0 12 B -16 0 12 6 -4 C -12 -12 0 6 -2 D 0 -6 -6 0 0 E -12 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.517098 B: 0.000000 C: 0.000000 D: 0.482902 E: 0.000000 Sum of squares = 0.500584674551 Cumulative probabilities = A: 0.517098 B: 0.517098 C: 0.517098 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 0 12 B -16 0 12 6 -4 C -12 -12 0 6 -2 D 0 -6 -6 0 0 E -12 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 C=20 E=15 B=14 so B is eliminated. Round 2 votes counts: A=30 D=28 C=27 E=15 so E is eliminated. Round 3 votes counts: D=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:220 B:199 E:197 D:194 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 0 12 B -16 0 12 6 -4 C -12 -12 0 6 -2 D 0 -6 -6 0 0 E -12 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 0 12 B -16 0 12 6 -4 C -12 -12 0 6 -2 D 0 -6 -6 0 0 E -12 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 0 12 B -16 0 12 6 -4 C -12 -12 0 6 -2 D 0 -6 -6 0 0 E -12 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7783: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) E D A B C (6) C B D E A (6) A E D B C (6) E C A B D (5) E A D B C (5) C B E D A (5) C B D A E (5) B C D E A (5) B C D A E (5) A D E B C (5) E C B A D (4) E A D C B (3) E A C D B (3) A E D C B (3) E C B D A (2) D B C A E (2) D A B E C (2) C E B D A (2) A D B C E (2) D B E C A (1) D B A C E (1) D A E B C (1) C E B A D (1) C B A E D (1) C B A D E (1) C A E B D (1) C A B D E (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -18 -10 -2 B 6 0 6 16 2 C 18 -6 0 4 4 D 10 -16 -4 0 0 E 2 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999111 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 -10 -2 B 6 0 6 16 2 C 18 -6 0 4 4 D 10 -16 -4 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=23 B=22 A=20 D=7 so D is eliminated. Round 2 votes counts: E=28 B=26 C=23 A=23 so C is eliminated. Round 3 votes counts: B=44 E=31 A=25 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:210 E:198 D:195 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -18 -10 -2 B 6 0 6 16 2 C 18 -6 0 4 4 D 10 -16 -4 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 -10 -2 B 6 0 6 16 2 C 18 -6 0 4 4 D 10 -16 -4 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 -10 -2 B 6 0 6 16 2 C 18 -6 0 4 4 D 10 -16 -4 0 0 E 2 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7784: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) A B C E D (7) E C B A D (6) C E B A D (5) C B A E D (5) A B D C E (5) D A E B C (4) D A B C E (4) E D C A B (3) E D A B C (3) D E C A B (3) D A B E C (3) C E B D A (3) B A C D E (3) A D B E C (3) E D C B A (2) E A B C D (2) D E C B A (2) D C E B A (2) D B C A E (2) D B A C E (2) C E D B A (2) C B E A D (2) C B D A E (2) B C A E D (2) E C B D A (1) E C A D B (1) E C A B D (1) D E A C B (1) D C B A E (1) C B E D A (1) C B A D E (1) B A D C E (1) A B E D C (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 6 -4 -2 B -10 0 12 2 -4 C -6 -12 0 -8 2 D 4 -2 8 0 4 E 2 4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999983 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -4 -2 B -10 0 12 2 -4 C -6 -12 0 -8 2 D 4 -2 8 0 4 E 2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000013 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=21 E=19 A=19 B=6 so B is eliminated. Round 2 votes counts: D=35 C=23 A=23 E=19 so E is eliminated. Round 3 votes counts: D=43 C=32 A=25 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:207 A:205 B:200 E:200 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 -4 -2 B -10 0 12 2 -4 C -6 -12 0 -8 2 D 4 -2 8 0 4 E 2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000013 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -4 -2 B -10 0 12 2 -4 C -6 -12 0 -8 2 D 4 -2 8 0 4 E 2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000013 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -4 -2 B -10 0 12 2 -4 C -6 -12 0 -8 2 D 4 -2 8 0 4 E 2 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.250000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000013 Cumulative probabilities = A: 0.125000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7785: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (14) E A B C D (13) D C A B E (11) D C B A E (7) E A D C B (6) B C D A E (5) B E A C D (4) E D A C B (3) D C E A B (3) B C E A D (3) B C A D E (3) B A C E D (3) D E C A B (2) D C A E B (2) D B C E A (2) B E C A D (2) B D C E A (2) B D C A E (2) E B A D C (1) E A D B C (1) E A C D B (1) E A B D C (1) D E A C B (1) D B C A E (1) C D B A E (1) C D A B E (1) C A B D E (1) B E D C A (1) B D E C A (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 0 10 -22 B 4 0 20 16 2 C 0 -20 0 6 -4 D -10 -16 -6 0 -10 E 22 -2 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 10 -22 B 4 0 20 16 2 C 0 -20 0 6 -4 D -10 -16 -6 0 -10 E 22 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=29 B=26 C=3 A=2 so A is eliminated. Round 2 votes counts: E=41 D=29 B=26 C=4 so C is eliminated. Round 3 votes counts: E=41 D=31 B=28 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:221 E:217 A:192 C:191 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 10 -22 B 4 0 20 16 2 C 0 -20 0 6 -4 D -10 -16 -6 0 -10 E 22 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 10 -22 B 4 0 20 16 2 C 0 -20 0 6 -4 D -10 -16 -6 0 -10 E 22 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 10 -22 B 4 0 20 16 2 C 0 -20 0 6 -4 D -10 -16 -6 0 -10 E 22 -2 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7786: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (12) B D A E C (10) C E A D B (9) E D B C A (8) C A E B D (8) D E B C A (7) D B E A C (7) B A D C E (7) A B C D E (7) E C D A B (4) D E B A C (3) C A B E D (3) A C B D E (3) A B D C E (3) C E D A B (2) C A E D B (2) B A D E C (2) E D B A C (1) E C A D B (1) B D E A C (1) Total count = 100 A B C D E A 0 -16 -12 -10 -10 B 16 0 12 -12 -14 C 12 -12 0 -22 -12 D 10 12 22 0 0 E 10 14 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.505781 E: 0.494219 Sum of squares = 0.500066819488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.505781 E: 1.000000 A B C D E A 0 -16 -12 -10 -10 B 16 0 12 -12 -14 C 12 -12 0 -22 -12 D 10 12 22 0 0 E 10 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=24 B=20 D=17 A=13 so A is eliminated. Round 2 votes counts: B=30 C=27 E=26 D=17 so D is eliminated. Round 3 votes counts: B=37 E=36 C=27 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:222 E:218 B:201 C:183 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -12 -10 -10 B 16 0 12 -12 -14 C 12 -12 0 -22 -12 D 10 12 22 0 0 E 10 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -10 -10 B 16 0 12 -12 -14 C 12 -12 0 -22 -12 D 10 12 22 0 0 E 10 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -10 -10 B 16 0 12 -12 -14 C 12 -12 0 -22 -12 D 10 12 22 0 0 E 10 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.49999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7787: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) A D C E B (8) D A C B E (7) E B C A D (6) B D C A E (6) E A D B C (5) E B A D C (4) B E C D A (4) E C A B D (3) E A C D B (3) D C A B E (3) C B D A E (3) B E D A C (3) B C D A E (3) E C A D B (2) C D B A E (2) C D A B E (2) B D A C E (2) B C E D A (2) B C D E A (2) A D E C B (2) A C D E B (2) E C B A D (1) E B C D A (1) E B A C D (1) E A C B D (1) E A B D C (1) E A B C D (1) D B A C E (1) D A B E C (1) C E A D B (1) C B E D A (1) C A E D B (1) C A D E B (1) B E D C A (1) A E D C B (1) A E D B C (1) A D E B C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 14 10 12 -2 B -14 0 -6 -8 -10 C -10 6 0 -14 -2 D -12 8 14 0 -4 E 2 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 10 12 -2 B -14 0 -6 -8 -10 C -10 6 0 -14 -2 D -12 8 14 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=23 A=17 D=12 C=11 so C is eliminated. Round 2 votes counts: E=38 B=27 A=19 D=16 so D is eliminated. Round 3 votes counts: E=38 A=32 B=30 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:217 E:209 D:203 C:190 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 10 12 -2 B -14 0 -6 -8 -10 C -10 6 0 -14 -2 D -12 8 14 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 12 -2 B -14 0 -6 -8 -10 C -10 6 0 -14 -2 D -12 8 14 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 12 -2 B -14 0 -6 -8 -10 C -10 6 0 -14 -2 D -12 8 14 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7788: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) E D A C B (8) B C A D E (8) C B D A E (7) D C A B E (6) E B A C D (5) D E A C B (4) B C E A D (4) D A E C B (3) B C A E D (3) B A C E D (3) E D C B A (2) E C D B A (2) D E C A B (2) C D B E A (2) C D B A E (2) C B A D E (2) B E A C D (2) A E B D C (2) A D E B C (2) E D C A B (1) E D A B C (1) E B C D A (1) E B C A D (1) E B A D C (1) E A B D C (1) D C E B A (1) D C E A B (1) D C A E B (1) D A C E B (1) D A C B E (1) C D E B A (1) C B D E A (1) B E C A D (1) B A E C D (1) B A C D E (1) A E D B C (1) A D E C B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 4 -6 B 2 0 2 -10 -8 C -2 -2 0 -4 -6 D -4 10 4 0 -2 E 6 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 4 -6 B 2 0 2 -10 -8 C -2 -2 0 -4 -6 D -4 10 4 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=23 D=20 C=15 A=8 so A is eliminated. Round 2 votes counts: E=37 D=24 B=24 C=15 so C is eliminated. Round 3 votes counts: E=37 B=34 D=29 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:211 D:204 A:199 B:193 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 4 -6 B 2 0 2 -10 -8 C -2 -2 0 -4 -6 D -4 10 4 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 4 -6 B 2 0 2 -10 -8 C -2 -2 0 -4 -6 D -4 10 4 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 4 -6 B 2 0 2 -10 -8 C -2 -2 0 -4 -6 D -4 10 4 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7789: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (5) A E C B D (5) A C E D B (5) D C A E B (4) D B A C E (4) B E C A D (4) B D E C A (4) B D A E C (4) A D C B E (4) E C B A D (3) E C A B D (3) E B C A D (3) D C E B A (3) D A C E B (3) D A C B E (3) C E D B A (3) C E A D B (3) B E C D A (3) A B E C D (3) E B C D A (2) D C E A B (2) D B A E C (2) C D E A B (2) B E D C A (2) B D E A C (2) B A E D C (2) A D C E B (2) E C B D A (1) E A B C D (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C E A (1) D B C A E (1) D A B C E (1) C D E B A (1) B E D A C (1) B E A D C (1) B E A C D (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 4 -4 4 B 12 0 2 4 8 C -4 -2 0 -4 -12 D 4 -4 4 0 -4 E -4 -8 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -4 4 B 12 0 2 4 8 C -4 -2 0 -4 -12 D 4 -4 4 0 -4 E -4 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 A=22 E=13 C=9 so C is eliminated. Round 2 votes counts: D=30 B=29 A=22 E=19 so E is eliminated. Round 3 votes counts: B=38 D=33 A=29 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:202 D:200 A:196 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 -4 4 B 12 0 2 4 8 C -4 -2 0 -4 -12 D 4 -4 4 0 -4 E -4 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -4 4 B 12 0 2 4 8 C -4 -2 0 -4 -12 D 4 -4 4 0 -4 E -4 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -4 4 B 12 0 2 4 8 C -4 -2 0 -4 -12 D 4 -4 4 0 -4 E -4 -8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998364 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7790: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) C D E B A (8) A B E D C (6) C E D B A (5) B D A E C (5) B A D C E (5) E A C B D (4) D B C A E (4) A B D C E (4) E C D A B (3) E C A D B (3) E A B D C (3) C E A D B (3) C D B A E (3) B D A C E (3) A E B D C (3) E C D B A (2) E A B C D (2) D B E A C (2) C E D A B (2) C D B E A (2) A E B C D (2) A B C D E (2) E B A D C (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C E A (1) C E A B D (1) C D A E B (1) C D A B E (1) C B D A E (1) C A E B D (1) C A B D E (1) B D E A C (1) B A D E C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -8 -12 -8 B 12 0 -6 -4 -6 C 8 6 0 -4 2 D 12 4 4 0 0 E 8 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.587940 E: 0.412060 Sum of squares = 0.515466840979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.587940 E: 1.000000 A B C D E A 0 -12 -8 -12 -8 B 12 0 -6 -4 -6 C 8 6 0 -4 2 D 12 4 4 0 0 E 8 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=27 A=19 B=15 D=10 so D is eliminated. Round 2 votes counts: C=31 E=27 B=23 A=19 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:206 E:206 B:198 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -8 -12 -8 B 12 0 -6 -4 -6 C 8 6 0 -4 2 D 12 4 4 0 0 E 8 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -12 -8 B 12 0 -6 -4 -6 C 8 6 0 -4 2 D 12 4 4 0 0 E 8 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -12 -8 B 12 0 -6 -4 -6 C 8 6 0 -4 2 D 12 4 4 0 0 E 8 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7791: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (12) D B C E A (10) C A B E D (5) A C E B D (5) D B E A C (4) D B C A E (4) B D E A C (4) A E C B D (4) E A C D B (3) E A C B D (3) E A B D C (3) E A B C D (3) D C B A E (3) C A E D B (3) C A B D E (3) E D B A C (2) D E B A C (2) C D B A E (2) C D A E B (2) C B A D E (2) C A E B D (2) B E D A C (2) B D C E A (2) B D C A E (2) A E C D B (2) E D A C B (1) E D A B C (1) E B A D C (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E B A (1) C D A B E (1) B D E C A (1) B C D A E (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -14 -16 -16 B 10 0 12 -12 16 C 14 -12 0 -16 -4 D 16 12 16 0 14 E 16 -16 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -16 -16 B 10 0 12 -12 16 C 14 -12 0 -16 -4 D 16 12 16 0 14 E 16 -16 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=20 E=19 B=12 A=12 so B is eliminated. Round 2 votes counts: D=46 E=21 C=21 A=12 so A is eliminated. Round 3 votes counts: D=46 E=28 C=26 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 B:213 E:195 C:191 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -14 -16 -16 B 10 0 12 -12 16 C 14 -12 0 -16 -4 D 16 12 16 0 14 E 16 -16 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -16 -16 B 10 0 12 -12 16 C 14 -12 0 -16 -4 D 16 12 16 0 14 E 16 -16 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -16 -16 B 10 0 12 -12 16 C 14 -12 0 -16 -4 D 16 12 16 0 14 E 16 -16 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7792: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) B C D E A (9) E B D A C (5) E A D B C (5) E A B D C (5) C B D A E (5) B E D C A (5) A E D C B (4) E D A B C (3) C D A B E (3) C B D E A (3) C A D E B (3) C A D B E (3) B D E C A (3) A C E D B (3) D C B E A (2) C D B E A (2) C D A E B (2) C A B D E (2) A E C D B (2) A E C B D (2) A E B D C (2) A E B C D (2) E B A D C (1) D E B A C (1) D C B A E (1) D B C E A (1) D A C E B (1) C A E B D (1) B E D A C (1) B E C A D (1) B D C E A (1) B C E D A (1) B C E A D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -22 -22 0 B 10 0 -6 2 12 C 22 6 0 16 16 D 22 -2 -16 0 10 E 0 -12 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -22 -22 0 B 10 0 -6 2 12 C 22 6 0 16 16 D 22 -2 -16 0 10 E 0 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=22 E=19 A=17 D=6 so D is eliminated. Round 2 votes counts: C=39 B=23 E=20 A=18 so A is eliminated. Round 3 votes counts: C=45 E=32 B=23 so B is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:209 D:207 E:181 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -22 -22 0 B 10 0 -6 2 12 C 22 6 0 16 16 D 22 -2 -16 0 10 E 0 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -22 -22 0 B 10 0 -6 2 12 C 22 6 0 16 16 D 22 -2 -16 0 10 E 0 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -22 -22 0 B 10 0 -6 2 12 C 22 6 0 16 16 D 22 -2 -16 0 10 E 0 -12 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7793: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) D E A C B (8) D E C A B (7) B C A E D (7) B A C E D (7) B A C D E (7) E D C A B (6) C E D B A (5) C E B A D (4) D C E B A (3) A B E C D (3) E D A C B (2) E C D A B (2) E A D C B (2) D A B E C (2) C E B D A (2) C D E B A (2) C B E A D (2) B A D C E (2) A D B E C (2) A B D E C (2) E C A B D (1) E A C D B (1) E A C B D (1) D E A B C (1) D B C A E (1) D B A C E (1) C B E D A (1) C B D E A (1) C B A E D (1) B C D A E (1) A E B D C (1) A E B C D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 -8 -2 B -6 0 -2 -8 -16 C -8 2 0 -2 -4 D 8 8 2 0 0 E 2 16 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.659891 E: 0.340109 Sum of squares = 0.551130117548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.659891 E: 1.000000 A B C D E A 0 6 8 -8 -2 B -6 0 -2 -8 -16 C -8 2 0 -2 -4 D 8 8 2 0 0 E 2 16 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999997559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=24 C=18 E=15 A=11 so A is eliminated. Round 2 votes counts: D=34 B=31 C=18 E=17 so E is eliminated. Round 3 votes counts: D=44 B=33 C=23 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:209 A:202 C:194 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -8 -2 B -6 0 -2 -8 -16 C -8 2 0 -2 -4 D 8 8 2 0 0 E 2 16 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999997559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 -2 B -6 0 -2 -8 -16 C -8 2 0 -2 -4 D 8 8 2 0 0 E 2 16 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999997559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 -2 B -6 0 -2 -8 -16 C -8 2 0 -2 -4 D 8 8 2 0 0 E 2 16 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999997559 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7794: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) C E D B A (9) C E D A B (9) C D E B A (6) B A D E C (6) A B D E C (6) B D E A C (5) A C B E D (5) A E D B C (4) A B C D E (4) D E B C A (3) C A E D B (3) C A B D E (3) E D B A C (2) D B E C A (2) C E A D B (2) C D B E A (2) B D E C A (2) A E D C B (2) E D C A B (1) E D B C A (1) E D A C B (1) E D A B C (1) D E C B A (1) D E B A C (1) C A E B D (1) B E D A C (1) B D C E A (1) B C A D E (1) B A D C E (1) B A C D E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 10 6 4 0 B -10 0 8 -2 4 C -6 -8 0 -4 0 D -4 2 4 0 -10 E 0 -4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.620304 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.379696 Sum of squares = 0.528946237685 Cumulative probabilities = A: 0.620304 B: 0.620304 C: 0.620304 D: 0.620304 E: 1.000000 A B C D E A 0 10 6 4 0 B -10 0 8 -2 4 C -6 -8 0 -4 0 D -4 2 4 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=34 B=18 D=7 E=6 so E is eliminated. Round 2 votes counts: C=35 A=34 B=18 D=13 so D is eliminated. Round 3 votes counts: C=37 A=36 B=27 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 E:203 B:200 D:196 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 4 0 B -10 0 8 -2 4 C -6 -8 0 -4 0 D -4 2 4 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 4 0 B -10 0 8 -2 4 C -6 -8 0 -4 0 D -4 2 4 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 4 0 B -10 0 8 -2 4 C -6 -8 0 -4 0 D -4 2 4 0 -10 E 0 -4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7795: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) E A D B C (7) D E C B A (7) A E B C D (6) E D A C B (5) E A C B D (5) D E A B C (5) D C B E A (5) C B A E D (5) D C B A E (4) C B A D E (4) E D A B C (3) C B D A E (3) B C D A E (3) A E B D C (3) A B C E D (3) E C D A B (2) E A D C B (2) D E B C A (2) C D B E A (2) B C A E D (2) A C B E D (2) E D C B A (1) E A B C D (1) D E B A C (1) D E A C B (1) D C E B A (1) D B E C A (1) D B C E A (1) D B A C E (1) C E B D A (1) A E D B C (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -4 2 -6 B 4 0 -2 -4 -10 C 4 2 0 -2 -10 D -2 4 2 0 -2 E 6 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 2 -6 B 4 0 -2 -4 -10 C 4 2 0 -2 -10 D -2 4 2 0 -2 E 6 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=26 A=17 C=15 B=13 so B is eliminated. Round 2 votes counts: D=29 C=28 E=26 A=17 so A is eliminated. Round 3 votes counts: E=38 C=33 D=29 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:201 C:197 A:194 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 2 -6 B 4 0 -2 -4 -10 C 4 2 0 -2 -10 D -2 4 2 0 -2 E 6 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 2 -6 B 4 0 -2 -4 -10 C 4 2 0 -2 -10 D -2 4 2 0 -2 E 6 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 2 -6 B 4 0 -2 -4 -10 C 4 2 0 -2 -10 D -2 4 2 0 -2 E 6 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7796: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D B C A E (13) A B C D E (6) E A B C D (5) E A C D B (4) D C B A E (4) A E B C D (4) A B C E D (4) E D A B C (3) E A D C B (3) D C B E A (3) C D B A E (3) B D C A E (3) E D C B A (2) E D B C A (2) E C D A B (2) E A B D C (2) D E C B A (2) D B C E A (2) C B D A E (2) C B A D E (2) B C D A E (2) A C B D E (2) E D C A B (1) E D A C B (1) E C A D B (1) E B D A C (1) E A D B C (1) D B E C A (1) C A B D E (1) B C A D E (1) A E C B D (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 6 6 2 B -14 0 2 4 2 C -6 -2 0 12 0 D -6 -4 -12 0 -6 E -2 -2 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 6 2 B -14 0 2 4 2 C -6 -2 0 12 0 D -6 -4 -12 0 -6 E -2 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 D=25 A=20 C=8 B=6 so B is eliminated. Round 2 votes counts: E=41 D=28 A=20 C=11 so C is eliminated. Round 3 votes counts: E=41 D=35 A=24 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:214 C:202 E:201 B:197 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 6 2 B -14 0 2 4 2 C -6 -2 0 12 0 D -6 -4 -12 0 -6 E -2 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 6 2 B -14 0 2 4 2 C -6 -2 0 12 0 D -6 -4 -12 0 -6 E -2 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 6 2 B -14 0 2 4 2 C -6 -2 0 12 0 D -6 -4 -12 0 -6 E -2 -2 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7797: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) E A C B D (8) B D C A E (8) E B D A C (5) D B E C A (5) D B C A E (4) A C E B D (4) E D B A C (3) D C B A E (3) C D A B E (3) B E D A C (3) B D E C A (3) B D E A C (3) A E C B D (3) E C A D B (2) E A B C D (2) D B C E A (2) C D A E B (2) C B D A E (2) C A E D B (2) C A B D E (2) A E B C D (2) A C E D B (2) A C B E D (2) E D B C A (1) E B A D C (1) E A B D C (1) D E B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) C D B A E (1) B E A D C (1) B C D A E (1) B C A D E (1) B A C D E (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 8 -6 -12 B 2 0 -4 6 -8 C -8 4 0 6 -14 D 6 -6 -6 0 -10 E 12 8 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 -6 -12 B 2 0 -4 6 -8 C -8 4 0 6 -14 D 6 -6 -6 0 -10 E 12 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=21 D=18 A=14 C=12 so C is eliminated. Round 2 votes counts: E=35 D=24 B=23 A=18 so A is eliminated. Round 3 votes counts: E=49 B=27 D=24 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:198 A:194 C:194 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 -6 -12 B 2 0 -4 6 -8 C -8 4 0 6 -14 D 6 -6 -6 0 -10 E 12 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -6 -12 B 2 0 -4 6 -8 C -8 4 0 6 -14 D 6 -6 -6 0 -10 E 12 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -6 -12 B 2 0 -4 6 -8 C -8 4 0 6 -14 D 6 -6 -6 0 -10 E 12 8 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7798: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) D B E A C (5) C A E D B (5) B E D C A (5) D B A E C (4) A C B E D (4) E C B A D (3) E B C D A (3) D E C A B (3) D E B C A (3) C E A D B (3) C E A B D (3) C A E B D (3) B A C D E (3) A D B C E (3) A B C D E (3) E D C B A (2) E D C A B (2) E C D B A (2) D E C B A (2) D B E C A (2) C E D A B (2) C A B E D (2) B E C D A (2) B D E C A (2) B D E A C (2) A C E B D (2) A C B D E (2) A B D C E (2) E D B C A (1) E B C A D (1) D E A C B (1) D A C E B (1) D A B E C (1) C E B A D (1) C B E A D (1) B E A C D (1) B D A E C (1) B A D C E (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -24 -10 -22 B -6 0 -8 -4 -6 C 24 8 0 12 -14 D 10 4 -12 0 -16 E 22 6 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -24 -10 -22 B -6 0 -8 -4 -6 C 24 8 0 12 -14 D 10 4 -12 0 -16 E 22 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 D=22 C=20 A=18 B=17 so B is eliminated. Round 2 votes counts: E=31 D=27 A=22 C=20 so C is eliminated. Round 3 votes counts: E=41 A=32 D=27 so D is eliminated. Round 4 votes counts: E=61 A=39 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:215 D:193 B:188 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -24 -10 -22 B -6 0 -8 -4 -6 C 24 8 0 12 -14 D 10 4 -12 0 -16 E 22 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -24 -10 -22 B -6 0 -8 -4 -6 C 24 8 0 12 -14 D 10 4 -12 0 -16 E 22 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -24 -10 -22 B -6 0 -8 -4 -6 C 24 8 0 12 -14 D 10 4 -12 0 -16 E 22 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7799: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) D B A C E (9) E C A D B (8) D A B E C (7) C E B D A (7) C B D E A (7) A E D B C (7) E A D C B (5) E C A B D (4) C B E D A (4) B D C A E (4) A D E B C (4) A D B E C (3) D B C A E (2) D A E B C (2) C E B A D (2) B C D A E (2) B A D E C (2) E C D A B (1) E A C B D (1) D E A C B (1) D C E B A (1) D A E C B (1) C E D A B (1) C D E B A (1) C B E A D (1) B D A C E (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 14 10 -2 -8 B -14 0 -12 -28 -16 C -10 12 0 -2 -16 D 2 28 2 0 -6 E 8 16 16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 10 -2 -8 B -14 0 -12 -28 -16 C -10 12 0 -2 -16 D 2 28 2 0 -6 E 8 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=23 C=23 A=16 B=9 so B is eliminated. Round 2 votes counts: E=29 D=28 C=25 A=18 so A is eliminated. Round 3 votes counts: E=38 D=37 C=25 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:213 A:207 C:192 B:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 10 -2 -8 B -14 0 -12 -28 -16 C -10 12 0 -2 -16 D 2 28 2 0 -6 E 8 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -2 -8 B -14 0 -12 -28 -16 C -10 12 0 -2 -16 D 2 28 2 0 -6 E 8 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -2 -8 B -14 0 -12 -28 -16 C -10 12 0 -2 -16 D 2 28 2 0 -6 E 8 16 16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7800: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (14) B D C A E (8) E A C B D (6) C A E D B (6) E A C D B (5) D B E A C (5) D B C A E (5) B C D A E (5) E A D C B (3) D E B A C (3) D E A B C (3) B D E C A (3) B C A E D (3) A E C D B (3) E A D B C (2) D C A E B (2) D B C E A (2) C B A D E (2) C A D E B (2) C A B E D (2) B E D A C (2) B D E A C (2) A C E D B (2) E D B A C (1) E D A C B (1) E B A D C (1) E A B C D (1) C D B A E (1) C D A B E (1) C A D B E (1) B E A C D (1) B D C E A (1) B C A D E (1) Total count = 100 A B C D E A 0 8 -18 10 16 B -8 0 -2 4 -10 C 18 2 0 12 16 D -10 -4 -12 0 -6 E -16 10 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999269 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 10 16 B -8 0 -2 4 -10 C 18 2 0 12 16 D -10 -4 -12 0 -6 E -16 10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=26 E=20 D=20 A=5 so A is eliminated. Round 2 votes counts: C=31 B=26 E=23 D=20 so D is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:208 B:192 E:192 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 10 16 B -8 0 -2 4 -10 C 18 2 0 12 16 D -10 -4 -12 0 -6 E -16 10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 10 16 B -8 0 -2 4 -10 C 18 2 0 12 16 D -10 -4 -12 0 -6 E -16 10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 10 16 B -8 0 -2 4 -10 C 18 2 0 12 16 D -10 -4 -12 0 -6 E -16 10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999977719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7801: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (17) E D A B C (11) D A E C B (6) C A B D E (6) E B C D A (5) D E A C B (5) B C A E D (5) A D C B E (5) E A D B C (3) D A C B E (3) C D B A E (3) C A D B E (3) B C E A D (3) B C A D E (3) E D C B A (2) E D B A C (2) D A C E B (2) C D E B A (2) B E C A D (2) A E D B C (2) E D C A B (1) E D A C B (1) E C D B A (1) E B D A C (1) E B A D C (1) E A B D C (1) D E A B C (1) C B E D A (1) C B D A E (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -10 4 14 B -2 0 -18 -8 6 C 10 18 0 4 8 D -4 8 -4 0 16 E -14 -6 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 4 14 B -2 0 -18 -8 6 C 10 18 0 4 8 D -4 8 -4 0 16 E -14 -6 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=29 D=17 B=13 A=8 so A is eliminated. Round 2 votes counts: C=33 E=31 D=23 B=13 so B is eliminated. Round 3 votes counts: C=44 E=33 D=23 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:208 A:205 B:189 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 4 14 B -2 0 -18 -8 6 C 10 18 0 4 8 D -4 8 -4 0 16 E -14 -6 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 4 14 B -2 0 -18 -8 6 C 10 18 0 4 8 D -4 8 -4 0 16 E -14 -6 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 4 14 B -2 0 -18 -8 6 C 10 18 0 4 8 D -4 8 -4 0 16 E -14 -6 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7802: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) A D C B E (8) E B C D A (6) D B C A E (5) E C B A D (4) E A D B C (4) D C B A E (4) C B E D A (4) A E D C B (4) A E D B C (4) E A C B D (3) E A B D C (3) C B D A E (3) B C D E A (3) A E C D B (3) A E C B D (3) E C A B D (2) E B D C A (2) D B E C A (2) D B C E A (2) D A B C E (2) C A B D E (2) B E D C A (2) B D C E A (2) A D E B C (2) A C D B E (2) E D B A C (1) E D A B C (1) E C B D A (1) E A B C D (1) C E B A D (1) C D B A E (1) C D A B E (1) B C E D A (1) A D E C B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -12 -2 -6 B 4 0 -10 2 8 C 12 10 0 -2 0 D 2 -2 2 0 0 E 6 -8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.55102040817 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 -6 B 4 0 -10 2 8 C 12 10 0 -2 0 D 2 -2 2 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408091 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=28 C=20 D=15 B=8 so B is eliminated. Round 2 votes counts: E=30 A=29 C=24 D=17 so D is eliminated. Round 3 votes counts: C=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 B:202 D:201 E:199 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 -6 B 4 0 -10 2 8 C 12 10 0 -2 0 D 2 -2 2 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408091 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 -6 B 4 0 -10 2 8 C 12 10 0 -2 0 D 2 -2 2 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408091 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 -6 B 4 0 -10 2 8 C 12 10 0 -2 0 D 2 -2 2 0 0 E 6 -8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.142857 D: 0.714286 E: 0.000000 Sum of squares = 0.551020408091 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7803: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (6) A E D C B (6) D E B A C (5) B E D A C (5) E D B A C (4) E D A B C (4) C B D A E (4) C A B E D (4) B E D C A (4) B C D E A (4) A C D E B (4) D B E A C (3) B D E C A (3) B D C E A (3) A E C D B (3) A D E C B (3) A C E D B (3) E B D A C (2) E A D B C (2) C D B A E (2) C A E D B (2) B D E A C (2) B C E A D (2) B C D A E (2) E A D C B (1) E A B D C (1) E A B C D (1) D E A C B (1) D B C E A (1) D A C E B (1) C D A B E (1) C B A D E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E C A D (1) B E A C D (1) B C A E D (1) B C A D E (1) B A E C D (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 22 -14 -12 B 4 0 20 -8 -4 C -22 -20 0 -14 -22 D 14 8 14 0 0 E 12 4 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.633532 E: 0.366468 Sum of squares = 0.535661766494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.633532 E: 1.000000 A B C D E A 0 -4 22 -14 -12 B 4 0 20 -8 -4 C -22 -20 0 -14 -22 D 14 8 14 0 0 E 12 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=21 D=17 C=17 E=15 so E is eliminated. Round 2 votes counts: B=32 A=26 D=25 C=17 so C is eliminated. Round 3 votes counts: B=37 A=35 D=28 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:219 D:218 B:206 A:196 C:161 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 22 -14 -12 B 4 0 20 -8 -4 C -22 -20 0 -14 -22 D 14 8 14 0 0 E 12 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 22 -14 -12 B 4 0 20 -8 -4 C -22 -20 0 -14 -22 D 14 8 14 0 0 E 12 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 22 -14 -12 B 4 0 20 -8 -4 C -22 -20 0 -14 -22 D 14 8 14 0 0 E 12 4 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7804: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (10) D A E C B (9) B E C D A (7) B C E A D (7) A C D E B (7) C A E D B (5) B E D C A (5) B E C A D (5) A C E D B (4) D B A E C (3) D A C E B (3) B A D C E (3) E C B D A (2) E C A D B (2) E B C D A (2) D E B C A (2) D E A C B (2) D A E B C (2) C E B A D (2) C E A B D (2) B D E C A (2) B D E A C (2) E D B C A (1) E C D B A (1) E C D A B (1) D E C A B (1) C E A D B (1) C B E A D (1) C A E B D (1) B D A E C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 8 4 B -8 0 -12 -14 -20 C 0 12 0 4 0 D -8 14 -4 0 2 E -4 20 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.577056 B: 0.000000 C: 0.422944 D: 0.000000 E: 0.000000 Sum of squares = 0.511875136034 Cumulative probabilities = A: 0.577056 B: 0.577056 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 8 4 B -8 0 -12 -14 -20 C 0 12 0 4 0 D -8 14 -4 0 2 E -4 20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999721 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=25 D=22 C=12 E=9 so E is eliminated. Round 2 votes counts: B=34 A=25 D=23 C=18 so C is eliminated. Round 3 votes counts: B=39 A=36 D=25 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:208 E:207 D:202 B:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 8 4 B -8 0 -12 -14 -20 C 0 12 0 4 0 D -8 14 -4 0 2 E -4 20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999721 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 8 4 B -8 0 -12 -14 -20 C 0 12 0 4 0 D -8 14 -4 0 2 E -4 20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999721 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 8 4 B -8 0 -12 -14 -20 C 0 12 0 4 0 D -8 14 -4 0 2 E -4 20 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999721 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7805: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) E A B D C (6) C D E B A (6) A B D C E (6) E C D B A (5) B A D C E (5) E B A C D (4) E A B C D (4) D C E A B (4) C D B A E (4) A E B D C (4) E B C A D (3) E A D C B (3) D C A B E (3) B A E D C (3) B A E C D (3) A D B C E (3) E D C A B (2) E C D A B (2) E B C D A (2) D E C A B (2) D C A E B (2) D A C B E (2) B C D A E (2) A E D C B (2) A E D B C (2) E C B D A (1) D B A C E (1) C D E A B (1) C D B E A (1) B A D E C (1) B A C E D (1) B A C D E (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 14 20 20 8 B -14 0 18 8 -12 C -20 -18 0 -20 -14 D -20 -8 20 0 -8 E -8 12 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 20 8 B -14 0 18 8 -12 C -20 -18 0 -20 -14 D -20 -8 20 0 -8 E -8 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=26 B=16 D=14 C=12 so C is eliminated. Round 2 votes counts: E=32 D=26 A=26 B=16 so B is eliminated. Round 3 votes counts: A=40 E=32 D=28 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 E:213 B:200 D:192 C:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 20 8 B -14 0 18 8 -12 C -20 -18 0 -20 -14 D -20 -8 20 0 -8 E -8 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 20 8 B -14 0 18 8 -12 C -20 -18 0 -20 -14 D -20 -8 20 0 -8 E -8 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 20 8 B -14 0 18 8 -12 C -20 -18 0 -20 -14 D -20 -8 20 0 -8 E -8 12 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7806: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (5) D E A C B (5) C A D B E (5) B C A E D (5) E D A C B (4) D E C A B (4) D C A E B (4) B C D A E (4) C D B A E (3) C A B D E (3) B E D C A (3) A E B C D (3) A C B E D (3) E D A B C (2) E A D B C (2) D E C B A (2) D E B C A (2) D C E A B (2) D C B E A (2) D A C E B (2) C D A B E (2) C B D A E (2) B E C D A (2) B D C E A (2) B C A D E (2) B A C E D (2) A B E C D (2) E D B A C (1) E B D C A (1) E B A C D (1) E A B D C (1) E A B C D (1) D E B A C (1) D B E C A (1) D B C E A (1) D A E C B (1) C B A E D (1) C A D E B (1) B E C A D (1) B E A D C (1) B E A C D (1) B C E D A (1) B C D E A (1) B A E C D (1) A E D C B (1) A E D B C (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -14 -10 0 B -12 0 -6 -10 4 C 14 6 0 -4 -2 D 10 10 4 0 6 E 0 -4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -14 -10 0 B -12 0 -6 -10 4 C 14 6 0 -4 -2 D 10 10 4 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 E=18 C=17 A=12 so A is eliminated. Round 2 votes counts: B=29 D=28 E=23 C=20 so C is eliminated. Round 3 votes counts: D=39 B=38 E=23 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:207 E:196 A:194 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -14 -10 0 B -12 0 -6 -10 4 C 14 6 0 -4 -2 D 10 10 4 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -14 -10 0 B -12 0 -6 -10 4 C 14 6 0 -4 -2 D 10 10 4 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -14 -10 0 B -12 0 -6 -10 4 C 14 6 0 -4 -2 D 10 10 4 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7807: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) D B E C A (7) C E A D B (7) B D E A C (6) D B A C E (5) C A E D B (5) D E C B A (4) B E A D C (4) B D A E C (4) B D A C E (3) A C E B D (3) E D C B A (2) E C D A B (2) E A C B D (2) D B C E A (2) D B C A E (2) C D A E B (2) C A E B D (2) A E C B D (2) E D B C A (1) E C D B A (1) E C A D B (1) E B C D A (1) E B C A D (1) E B A D C (1) E B A C D (1) D E B C A (1) D C E B A (1) D C E A B (1) D C A B E (1) D B E A C (1) C E D A B (1) C D E A B (1) B E D A C (1) B A E D C (1) B A E C D (1) B A D C E (1) A D C B E (1) A C D E B (1) A C D B E (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -10 0 -18 B 4 0 -4 -2 -6 C 10 4 0 -2 -10 D 0 2 2 0 -4 E 18 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 0 -18 B 4 0 -4 -2 -6 C 10 4 0 -2 -10 D 0 2 2 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=22 B=21 C=18 A=14 so A is eliminated. Round 2 votes counts: D=26 C=25 B=25 E=24 so E is eliminated. Round 3 votes counts: C=42 D=29 B=29 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:219 C:201 D:200 B:196 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 0 -18 B 4 0 -4 -2 -6 C 10 4 0 -2 -10 D 0 2 2 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 0 -18 B 4 0 -4 -2 -6 C 10 4 0 -2 -10 D 0 2 2 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 0 -18 B 4 0 -4 -2 -6 C 10 4 0 -2 -10 D 0 2 2 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996183 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7808: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) C E B A D (10) B D A E C (9) E B C D A (6) B E C D A (5) C E A D B (4) C E A B D (4) B E D C A (4) B D A C E (4) A D C B E (4) A D B C E (4) E C B D A (3) D B A E C (3) D A E B C (3) B D E A C (3) E C B A D (2) C A E D B (2) C A D E B (2) B E D A C (2) A C D E B (2) A C D B E (2) E C A D B (1) E B D A C (1) D A E C B (1) D A B C E (1) C E B D A (1) C B E A D (1) C B A E D (1) C A E B D (1) B C E D A (1) B C E A D (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 -16 0 -14 2 B 16 0 16 20 12 C 0 -16 0 0 -6 D 14 -20 0 0 0 E -2 -12 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -14 2 B 16 0 16 20 12 C 0 -16 0 0 -6 D 14 -20 0 0 0 E -2 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=26 D=18 E=13 A=13 so E is eliminated. Round 2 votes counts: B=37 C=32 D=18 A=13 so A is eliminated. Round 3 votes counts: B=37 C=36 D=27 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 D:197 E:196 C:189 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -14 2 B 16 0 16 20 12 C 0 -16 0 0 -6 D 14 -20 0 0 0 E -2 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -14 2 B 16 0 16 20 12 C 0 -16 0 0 -6 D 14 -20 0 0 0 E -2 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -14 2 B 16 0 16 20 12 C 0 -16 0 0 -6 D 14 -20 0 0 0 E -2 -12 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7809: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (13) D A C B E (10) C A E D B (8) B E D C A (6) B D E A C (6) E C A B D (5) D B A C E (5) B E D A C (5) D A C E B (4) A C D E B (4) D C A E B (3) C A E B D (3) A C D B E (3) E C B A D (2) E C A D B (2) D B E C A (2) D B E A C (2) D A B C E (2) B E C A D (2) E C D B A (1) E B C D A (1) D E B C A (1) D B A E C (1) C E A D B (1) C E A B D (1) B E C D A (1) B E A D C (1) B E A C D (1) B A D C E (1) B A C E D (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -4 0 -6 B 4 0 4 0 0 C 4 -4 0 0 -4 D 0 0 0 0 -10 E 6 0 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.661923 C: 0.000000 D: 0.000000 E: 0.338077 Sum of squares = 0.552438054423 Cumulative probabilities = A: 0.000000 B: 0.661923 C: 0.661923 D: 0.661923 E: 1.000000 A B C D E A 0 -4 -4 0 -6 B 4 0 4 0 0 C 4 -4 0 0 -4 D 0 0 0 0 -10 E 6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=24 B=24 C=13 A=9 so A is eliminated. Round 2 votes counts: D=31 B=25 E=24 C=20 so C is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:204 C:198 D:195 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 0 -6 B 4 0 4 0 0 C 4 -4 0 0 -4 D 0 0 0 0 -10 E 6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 -6 B 4 0 4 0 0 C 4 -4 0 0 -4 D 0 0 0 0 -10 E 6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 -6 B 4 0 4 0 0 C 4 -4 0 0 -4 D 0 0 0 0 -10 E 6 0 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7810: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (11) A B E D C (10) E D C A B (9) C D E B A (9) A E B D C (7) A B C D E (7) D C E B A (5) A B E C D (5) E D C B A (4) E D A C B (4) E A D C B (4) D E C B A (4) A E D C B (4) C B D E A (3) B C A D E (3) A B C E D (3) B C D E A (2) A E D B C (2) C E D B A (1) C D B E A (1) B C D A E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 16 14 8 B -12 0 4 6 -6 C -16 -4 0 -8 -6 D -14 -6 8 0 -6 E -8 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 16 14 8 B -12 0 4 6 -6 C -16 -4 0 -8 -6 D -14 -6 8 0 -6 E -8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=21 B=17 C=14 D=9 so D is eliminated. Round 2 votes counts: A=39 E=25 C=19 B=17 so B is eliminated. Round 3 votes counts: A=50 E=25 C=25 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:205 B:196 D:191 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 16 14 8 B -12 0 4 6 -6 C -16 -4 0 -8 -6 D -14 -6 8 0 -6 E -8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 16 14 8 B -12 0 4 6 -6 C -16 -4 0 -8 -6 D -14 -6 8 0 -6 E -8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 16 14 8 B -12 0 4 6 -6 C -16 -4 0 -8 -6 D -14 -6 8 0 -6 E -8 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7811: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) A C E B D (11) D B E C A (8) D E B C A (7) A C B E D (6) B D C E A (5) D E A C B (4) B D C A E (4) A C E D B (4) E D A C B (3) C B A E D (3) C A B E D (3) A E C D B (3) E D C A B (2) E A D C B (2) E A C D B (2) E A C B D (2) D B C A E (2) B A C D E (2) E C A B D (1) D E C A B (1) D E A B C (1) D B C E A (1) D B A C E (1) D A B C E (1) C E A B D (1) C A E B D (1) B D E C A (1) B C E D A (1) B C E A D (1) B C A D E (1) B A C E D (1) A E C B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 16 14 B -2 0 0 16 8 C 8 0 0 14 24 D -16 -16 -14 0 -20 E -14 -8 -24 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.371130 C: 0.628870 D: 0.000000 E: 0.000000 Sum of squares = 0.53321481643 Cumulative probabilities = A: 0.000000 B: 0.371130 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 16 14 B -2 0 0 16 8 C 8 0 0 14 24 D -16 -16 -14 0 -20 E -14 -8 -24 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 D=26 E=12 C=8 so C is eliminated. Round 2 votes counts: A=31 B=30 D=26 E=13 so E is eliminated. Round 3 votes counts: A=39 D=31 B=30 so B is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:223 A:212 B:211 E:187 D:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 16 14 B -2 0 0 16 8 C 8 0 0 14 24 D -16 -16 -14 0 -20 E -14 -8 -24 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 16 14 B -2 0 0 16 8 C 8 0 0 14 24 D -16 -16 -14 0 -20 E -14 -8 -24 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 16 14 B -2 0 0 16 8 C 8 0 0 14 24 D -16 -16 -14 0 -20 E -14 -8 -24 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7812: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) B A E D C (9) C D E A B (8) C E D B A (6) E C B D A (5) A D B C E (5) D C A E B (4) A B D C E (4) D A C E B (3) C E B D A (3) B E C A D (3) A D C E B (3) A D C B E (3) A D B E C (3) A B D E C (3) E B C D A (2) D E C A B (2) B E A D C (2) B C E D A (2) B C A D E (2) B A E C D (2) B A C D E (2) A E D B C (2) E D C A B (1) E C D B A (1) E C D A B (1) D A E C B (1) C E D A B (1) C D A B E (1) B C E A D (1) B A D E C (1) B A C E D (1) A D E C B (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -6 -2 4 B 4 0 10 4 10 C 6 -10 0 4 0 D 2 -4 -4 0 -4 E -4 -10 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -2 4 B 4 0 10 4 10 C 6 -10 0 4 0 D 2 -4 -4 0 -4 E -4 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=26 C=19 E=10 D=10 so E is eliminated. Round 2 votes counts: B=37 C=26 A=26 D=11 so D is eliminated. Round 3 votes counts: B=37 C=33 A=30 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:200 A:196 D:195 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 4 B 4 0 10 4 10 C 6 -10 0 4 0 D 2 -4 -4 0 -4 E -4 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 4 B 4 0 10 4 10 C 6 -10 0 4 0 D 2 -4 -4 0 -4 E -4 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 4 B 4 0 10 4 10 C 6 -10 0 4 0 D 2 -4 -4 0 -4 E -4 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7813: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (11) C E A D B (7) B D A E C (7) E A C D B (6) C D E A B (6) A E B C D (6) C E D A B (5) B D C E A (5) B A E D C (5) A E C D B (5) D B C E A (4) B D A C E (4) D C B E A (3) D B C A E (3) C D B E A (3) A E C B D (3) A E B D C (3) E A C B D (2) B A D E C (2) A B E D C (2) E C A D B (1) D C E B A (1) D B A C E (1) C E A B D (1) C D E B A (1) C B D E A (1) A E D B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -4 -10 8 B 2 0 10 6 4 C 4 -10 0 -4 12 D 10 -6 4 0 4 E -8 -4 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -10 8 B 2 0 10 6 4 C 4 -10 0 -4 12 D 10 -6 4 0 4 E -8 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=24 A=21 D=12 E=9 so E is eliminated. Round 2 votes counts: B=34 A=29 C=25 D=12 so D is eliminated. Round 3 votes counts: B=42 C=29 A=29 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 D:206 C:201 A:196 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -10 8 B 2 0 10 6 4 C 4 -10 0 -4 12 D 10 -6 4 0 4 E -8 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -10 8 B 2 0 10 6 4 C 4 -10 0 -4 12 D 10 -6 4 0 4 E -8 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -10 8 B 2 0 10 6 4 C 4 -10 0 -4 12 D 10 -6 4 0 4 E -8 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999377 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7814: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) D E A C B (12) E D A C B (7) C B A E D (6) E A D C B (5) D E A B C (5) B C D A E (5) D E B A C (4) A E C D B (4) D B E C A (3) B D C E A (3) A B C E D (3) D E C A B (2) D E B C A (2) D A E B C (2) C A E B D (2) C A B E D (2) A E D C B (2) A D E B C (2) A C B E D (2) E D C A B (1) D E C B A (1) D B E A C (1) D B A E C (1) D A E C B (1) C E A B D (1) C D B E A (1) C B E A D (1) B D C A E (1) B D A C E (1) B C E D A (1) B C A D E (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 6 8 -8 0 B -6 0 0 -12 -6 C -8 0 0 -12 -10 D 8 12 12 0 -2 E 0 6 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.118685 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.881315 Sum of squares = 0.790801832221 Cumulative probabilities = A: 0.118685 B: 0.118685 C: 0.118685 D: 0.118685 E: 1.000000 A B C D E A 0 6 8 -8 0 B -6 0 0 -12 -6 C -8 0 0 -12 -10 D 8 12 12 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000030193 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=27 E=13 C=13 A=13 so E is eliminated. Round 2 votes counts: D=42 B=27 A=18 C=13 so C is eliminated. Round 3 votes counts: D=43 B=34 A=23 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:209 A:203 B:188 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 -8 0 B -6 0 0 -12 -6 C -8 0 0 -12 -10 D 8 12 12 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000030193 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 0 B -6 0 0 -12 -6 C -8 0 0 -12 -10 D 8 12 12 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000030193 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 0 B -6 0 0 -12 -6 C -8 0 0 -12 -10 D 8 12 12 0 -2 E 0 6 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000030193 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7815: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E D C A B (9) D E C B A (8) D E B C A (7) B A C D E (7) A C B E D (7) E A C B D (5) B C A D E (4) A E B C D (4) A B C E D (4) E A C D B (3) D B C E A (3) D B C A E (3) B D C A E (3) A E C B D (3) E D C B A (2) D B E C A (2) A C E B D (2) A B C D E (2) E D B C A (1) E D A B C (1) E C D A B (1) E A D C B (1) E A B D C (1) D C B E A (1) C D A B E (1) C B A D E (1) C A E B D (1) C A B E D (1) B C D A E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 2 -6 -10 B -14 0 -12 -6 -18 C -2 12 0 -4 -18 D 6 6 4 0 -14 E 10 18 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 -6 -10 B -14 0 -12 -6 -18 C -2 12 0 -4 -18 D 6 6 4 0 -14 E 10 18 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=24 A=23 B=15 C=4 so C is eliminated. Round 2 votes counts: E=34 D=25 A=25 B=16 so B is eliminated. Round 3 votes counts: A=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:230 D:201 A:200 C:194 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 -6 -10 B -14 0 -12 -6 -18 C -2 12 0 -4 -18 D 6 6 4 0 -14 E 10 18 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 -6 -10 B -14 0 -12 -6 -18 C -2 12 0 -4 -18 D 6 6 4 0 -14 E 10 18 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 -6 -10 B -14 0 -12 -6 -18 C -2 12 0 -4 -18 D 6 6 4 0 -14 E 10 18 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7816: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) A D C E B (8) C D A B E (6) A E D C B (6) B C D E A (5) C D B A E (4) C B D A E (4) B E C D A (4) B C A D E (4) E D A C B (3) E A B C D (3) D E A C B (3) B E A C D (3) B C D A E (3) A C D E B (3) E B D A C (2) E B A D C (2) E A D B C (2) E A B D C (2) C A D B E (2) A C D B E (2) A B C E D (2) E D A B C (1) E B D C A (1) E B C A D (1) E B A C D (1) D E C B A (1) D C E A B (1) D C B E A (1) D C A B E (1) D B C E A (1) D A E C B (1) C B A D E (1) C A B D E (1) B E C A D (1) B A C E D (1) A E B C D (1) A C E D B (1) A C B E D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 20 16 16 8 B -20 0 -18 -10 -2 C -16 18 0 12 6 D -16 10 -12 0 6 E -8 2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 16 16 8 B -20 0 -18 -10 -2 C -16 18 0 12 6 D -16 10 -12 0 6 E -8 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 A=26 B=21 C=18 D=9 so D is eliminated. Round 2 votes counts: E=30 A=27 B=22 C=21 so C is eliminated. Round 3 votes counts: A=37 B=32 E=31 so E is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:210 D:194 E:191 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 16 16 8 B -20 0 -18 -10 -2 C -16 18 0 12 6 D -16 10 -12 0 6 E -8 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 16 8 B -20 0 -18 -10 -2 C -16 18 0 12 6 D -16 10 -12 0 6 E -8 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 16 8 B -20 0 -18 -10 -2 C -16 18 0 12 6 D -16 10 -12 0 6 E -8 2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999366 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7817: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) C A B E D (10) C B E A D (7) A C D B E (7) D E B C A (6) C B E D A (6) E B D C A (5) E B C D A (5) D E B A C (5) A D C B E (5) A C B E D (5) C B A E D (4) B E C A D (4) E D B C A (3) D A E C B (3) A C B D E (3) D E A C B (2) D E A B C (2) B C E A D (2) A D C E B (2) D A C E B (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -4 2 6 B -6 0 -10 4 10 C 4 10 0 8 6 D -2 -4 -8 0 -4 E -6 -10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 2 6 B -6 0 -10 4 10 C 4 10 0 8 6 D -2 -4 -8 0 -4 E -6 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=27 A=24 E=13 B=6 so B is eliminated. Round 2 votes counts: D=30 C=29 A=24 E=17 so E is eliminated. Round 3 votes counts: D=38 C=38 A=24 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:205 B:199 D:191 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 2 6 B -6 0 -10 4 10 C 4 10 0 8 6 D -2 -4 -8 0 -4 E -6 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 2 6 B -6 0 -10 4 10 C 4 10 0 8 6 D -2 -4 -8 0 -4 E -6 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 2 6 B -6 0 -10 4 10 C 4 10 0 8 6 D -2 -4 -8 0 -4 E -6 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7818: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) B E A C D (8) E A C D B (7) E A C B D (7) B E A D C (6) D B C A E (5) C A D E B (5) B D E C A (5) E C A D B (4) E B A C D (4) D B E C A (4) B D E A C (4) E A B C D (3) D C B A E (3) B E D C A (3) A C E D B (3) D C B E A (2) D C A E B (2) D C A B E (2) C A E D B (2) B D C E A (2) B D A C E (2) D E C B A (1) D E C A B (1) D B C E A (1) B D C A E (1) B A E C D (1) B A D C E (1) A E C B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -22 14 8 -40 B 22 0 16 12 18 C -14 -16 0 -6 -34 D -8 -12 6 0 -14 E 40 -18 34 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999551 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 14 8 -40 B 22 0 16 12 18 C -14 -16 0 -6 -34 D -8 -12 6 0 -14 E 40 -18 34 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998195 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=25 D=21 C=7 A=6 so A is eliminated. Round 2 votes counts: B=41 E=26 D=21 C=12 so C is eliminated. Round 3 votes counts: B=41 E=31 D=28 so D is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:235 B:234 D:186 A:180 C:165 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 14 8 -40 B 22 0 16 12 18 C -14 -16 0 -6 -34 D -8 -12 6 0 -14 E 40 -18 34 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998195 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 14 8 -40 B 22 0 16 12 18 C -14 -16 0 -6 -34 D -8 -12 6 0 -14 E 40 -18 34 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998195 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 14 8 -40 B 22 0 16 12 18 C -14 -16 0 -6 -34 D -8 -12 6 0 -14 E 40 -18 34 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998195 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7819: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (13) A B C E D (8) D E C B A (6) A C B E D (6) A B C D E (6) B A C D E (5) E D C A B (4) E D A B C (4) C B A D E (3) B A D C E (3) A E B C D (3) E D A C B (2) D E B C A (2) D E B A C (2) D B A E C (2) C D E B A (2) C A B E D (2) B C D A E (2) B C A D E (2) A B E D C (2) A B E C D (2) A B D E C (2) E D B C A (1) E C D A B (1) E C A B D (1) E A D C B (1) D E A B C (1) D C B E A (1) D B E A C (1) D B A C E (1) D A E B C (1) C E B D A (1) C B D A E (1) C B A E D (1) B D A C E (1) A E C B D (1) A E B D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 14 2 14 B 0 0 8 8 6 C -14 -8 0 -6 -8 D -2 -8 6 0 -8 E -14 -6 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.248929 B: 0.751071 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.626073026238 Cumulative probabilities = A: 0.248929 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 2 14 B 0 0 8 8 6 C -14 -8 0 -6 -8 D -2 -8 6 0 -8 E -14 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=27 D=17 B=13 C=10 so C is eliminated. Round 2 votes counts: A=35 E=28 D=19 B=18 so B is eliminated. Round 3 votes counts: A=49 E=28 D=23 so D is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 B:211 E:198 D:194 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 2 14 B 0 0 8 8 6 C -14 -8 0 -6 -8 D -2 -8 6 0 -8 E -14 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 2 14 B 0 0 8 8 6 C -14 -8 0 -6 -8 D -2 -8 6 0 -8 E -14 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 2 14 B 0 0 8 8 6 C -14 -8 0 -6 -8 D -2 -8 6 0 -8 E -14 -6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7820: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (15) B C D E A (12) C B D E A (8) A E D B C (8) A B E D C (8) C D E B A (6) B C A D E (5) E D A C B (4) E A D C B (4) B C A E D (4) B A C E D (4) D E C A B (3) D E A C B (3) B C D A E (3) B A E C D (3) A E B D C (3) B A E D C (2) D C E A B (1) C D B E A (1) B C E D A (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 2 12 16 14 B -2 0 8 8 4 C -12 -8 0 -6 -10 D -16 -8 6 0 -12 E -14 -4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 16 14 B -2 0 8 8 4 C -12 -8 0 -6 -10 D -16 -8 6 0 -12 E -14 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986511 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=34 C=15 E=8 D=7 so D is eliminated. Round 2 votes counts: A=36 B=34 C=16 E=14 so E is eliminated. Round 3 votes counts: A=47 B=34 C=19 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:209 E:202 D:185 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 16 14 B -2 0 8 8 4 C -12 -8 0 -6 -10 D -16 -8 6 0 -12 E -14 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986511 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 16 14 B -2 0 8 8 4 C -12 -8 0 -6 -10 D -16 -8 6 0 -12 E -14 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986511 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 16 14 B -2 0 8 8 4 C -12 -8 0 -6 -10 D -16 -8 6 0 -12 E -14 -4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986511 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7821: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) C B D A E (7) E A D B C (6) E A B C D (5) E D A C B (4) E B A C D (4) B C A D E (4) E B C D A (3) D C B A E (3) D C A B E (3) B E C A D (3) B E A C D (3) B C D A E (3) A E D B C (3) A E B D C (3) A D E C B (3) E A B D C (2) D E C B A (2) D C B E A (2) D C A E B (2) D A C E B (2) D A C B E (2) B C D E A (2) E D C B A (1) E B C A D (1) E A D C B (1) D E C A B (1) D E A C B (1) D C E B A (1) C D B E A (1) C B D E A (1) B E C D A (1) B C E D A (1) B C E A D (1) B C A E D (1) B A C E D (1) A E D C B (1) A E B C D (1) A D C B E (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -8 -6 6 B 12 0 0 0 6 C 8 0 0 12 0 D 6 0 -12 0 4 E -6 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.531888 C: 0.468112 D: 0.000000 E: 0.000000 Sum of squares = 0.502033723514 Cumulative probabilities = A: 0.000000 B: 0.531888 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -6 6 B 12 0 0 0 6 C 8 0 0 12 0 D 6 0 -12 0 4 E -6 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999799 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=20 D=19 C=19 A=15 so A is eliminated. Round 2 votes counts: E=35 D=23 B=22 C=20 so C is eliminated. Round 3 votes counts: E=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 B:209 D:199 E:192 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -6 6 B 12 0 0 0 6 C 8 0 0 12 0 D 6 0 -12 0 4 E -6 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999799 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -6 6 B 12 0 0 0 6 C 8 0 0 12 0 D 6 0 -12 0 4 E -6 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999799 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -6 6 B 12 0 0 0 6 C 8 0 0 12 0 D 6 0 -12 0 4 E -6 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999799 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7822: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) E C A B D (9) D B A C E (9) B D E C A (9) D A C E B (6) B E C A D (6) E C A D B (5) A C E B D (4) E C B A D (3) D A B C E (3) B E D C A (3) B E C D A (3) B A C E D (3) A D C E B (3) D E C A B (2) D B E C A (2) D A C B E (2) C E A D B (2) C A E D B (2) B E A C D (2) B D E A C (2) B D A E C (2) E A B C D (1) D E C B A (1) D C A E B (1) C A E B D (1) B D A C E (1) B A D C E (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 2 8 0 B -6 0 -4 2 -2 C -2 4 0 6 0 D -8 -2 -6 0 -10 E 0 2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.706751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.293249 Sum of squares = 0.585491916405 Cumulative probabilities = A: 0.706751 B: 0.706751 C: 0.706751 D: 0.706751 E: 1.000000 A B C D E A 0 6 2 8 0 B -6 0 -4 2 -2 C -2 4 0 6 0 D -8 -2 -6 0 -10 E 0 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=26 A=19 E=18 C=5 so C is eliminated. Round 2 votes counts: B=32 D=26 A=22 E=20 so E is eliminated. Round 3 votes counts: A=39 B=35 D=26 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:206 C:204 B:195 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 8 0 B -6 0 -4 2 -2 C -2 4 0 6 0 D -8 -2 -6 0 -10 E 0 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 8 0 B -6 0 -4 2 -2 C -2 4 0 6 0 D -8 -2 -6 0 -10 E 0 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 8 0 B -6 0 -4 2 -2 C -2 4 0 6 0 D -8 -2 -6 0 -10 E 0 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7823: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (10) E D B C A (9) E B C D A (9) A D C B E (7) D E A B C (6) B C E A D (6) A C B E D (5) A C B D E (5) E B D C A (4) E B C A D (4) D E B C A (4) D E A C B (4) B C A E D (4) D E B A C (3) D A C E B (3) C B A E D (3) C A B E D (3) B E C D A (2) A C D B E (2) E D B A C (1) E A D C B (1) D A C B E (1) B E C A D (1) B C A D E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 0 -12 -8 B 2 0 8 -4 -20 C 0 -8 0 -6 -16 D 12 4 6 0 -8 E 8 20 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 -12 -8 B 2 0 8 -4 -20 C 0 -8 0 -6 -16 D 12 4 6 0 -8 E 8 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=28 A=21 B=14 C=6 so C is eliminated. Round 2 votes counts: D=31 E=28 A=24 B=17 so B is eliminated. Round 3 votes counts: E=37 A=32 D=31 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:207 B:193 A:189 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 -12 -8 B 2 0 8 -4 -20 C 0 -8 0 -6 -16 D 12 4 6 0 -8 E 8 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -12 -8 B 2 0 8 -4 -20 C 0 -8 0 -6 -16 D 12 4 6 0 -8 E 8 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -12 -8 B 2 0 8 -4 -20 C 0 -8 0 -6 -16 D 12 4 6 0 -8 E 8 20 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7824: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A D E C B (9) B C E D A (8) D A E C B (7) A E D C B (6) D A B C E (5) B C E A D (5) E A C D B (4) C B E D A (4) A D E B C (4) E C B A D (2) E C A B D (2) D E C A B (2) D C B E A (2) D C B A E (2) D B C A E (2) D A E B C (2) C E B A D (2) A E C B D (2) E D A C B (1) E C B D A (1) E B A C D (1) E A D C B (1) E A C B D (1) D E A C B (1) D C E A B (1) D B C E A (1) D B A C E (1) C E D B A (1) C D B E A (1) C B E A D (1) C B D E A (1) B D C E A (1) B D C A E (1) B D A C E (1) A E C D B (1) A E B C D (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 0 -12 -8 B -4 0 -10 -10 -4 C 0 10 0 -2 0 D 12 10 2 0 10 E 8 4 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -12 -8 B -4 0 -10 -10 -4 C 0 10 0 -2 0 D 12 10 2 0 10 E 8 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 A=25 E=13 C=10 so C is eliminated. Round 2 votes counts: B=32 D=27 A=25 E=16 so E is eliminated. Round 3 votes counts: B=38 A=33 D=29 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:217 C:204 E:201 A:192 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 0 -12 -8 B -4 0 -10 -10 -4 C 0 10 0 -2 0 D 12 10 2 0 10 E 8 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -12 -8 B -4 0 -10 -10 -4 C 0 10 0 -2 0 D 12 10 2 0 10 E 8 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -12 -8 B -4 0 -10 -10 -4 C 0 10 0 -2 0 D 12 10 2 0 10 E 8 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7825: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) A C B D E (8) B D E A C (6) C A B E D (5) B D A E C (5) A B C D E (5) E C D B A (4) C A B D E (4) B A D E C (4) A B D C E (4) E D B C A (3) E B D C A (3) D E B A C (3) C A E D B (3) C A E B D (3) C A D E B (3) E C B D A (2) E B D A C (2) D B E A C (2) D A B E C (2) C E D A B (2) C E B D A (2) A D B C E (2) E D C B A (1) E D C A B (1) E C D A B (1) D B A E C (1) C E D B A (1) C E B A D (1) C E A D B (1) C E A B D (1) C B A E D (1) C A D B E (1) B A D C E (1) A D C E B (1) A D B E C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 0 14 2 12 B 0 0 6 16 6 C -14 -6 0 -2 0 D -2 -16 2 0 10 E -12 -6 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.432258 B: 0.567742 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.509178080531 Cumulative probabilities = A: 0.432258 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 2 12 B 0 0 6 16 6 C -14 -6 0 -2 0 D -2 -16 2 0 10 E -12 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=25 A=23 B=16 D=8 so D is eliminated. Round 2 votes counts: E=28 C=28 A=25 B=19 so B is eliminated. Round 3 votes counts: E=36 A=36 C=28 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:214 D:197 C:189 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 2 12 B 0 0 6 16 6 C -14 -6 0 -2 0 D -2 -16 2 0 10 E -12 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 2 12 B 0 0 6 16 6 C -14 -6 0 -2 0 D -2 -16 2 0 10 E -12 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 2 12 B 0 0 6 16 6 C -14 -6 0 -2 0 D -2 -16 2 0 10 E -12 -6 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7826: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) C E D A B (6) B D A E C (5) B A C D E (5) B D E C A (4) B D E A C (4) B A D E C (4) B A D C E (4) A E C D B (4) A B C E D (4) E D C A B (3) E C D A B (3) C E D B A (3) C B D E A (3) B D C E A (3) D E C B A (2) D E B C A (2) C E A D B (2) C D E B A (2) C D B E A (2) C B A E D (2) A E C B D (2) A C E B D (2) A C B E D (2) A B E D C (2) A B D E C (2) E D A C B (1) E A D C B (1) D C B E A (1) D B E A C (1) D B A E C (1) C E B D A (1) C E A B D (1) C B E A D (1) C B D A E (1) C A E D B (1) C A B E D (1) B D A C E (1) B C D E A (1) B C D A E (1) A C E D B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 -4 -14 -6 B 20 0 4 16 26 C 4 -4 0 4 6 D 14 -16 -4 0 14 E 6 -26 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -14 -6 B 20 0 4 16 26 C 4 -4 0 4 6 D 14 -16 -4 0 14 E 6 -26 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=26 A=21 D=13 E=8 so E is eliminated. Round 2 votes counts: B=32 C=29 A=22 D=17 so D is eliminated. Round 3 votes counts: B=42 C=35 A=23 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:233 C:205 D:204 E:180 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -4 -14 -6 B 20 0 4 16 26 C 4 -4 0 4 6 D 14 -16 -4 0 14 E 6 -26 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -14 -6 B 20 0 4 16 26 C 4 -4 0 4 6 D 14 -16 -4 0 14 E 6 -26 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -14 -6 B 20 0 4 16 26 C 4 -4 0 4 6 D 14 -16 -4 0 14 E 6 -26 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7827: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) D E B C A (6) D C A E B (6) E B D A C (5) C A D E B (4) C A D B E (4) C A B D E (4) B E D C A (4) B A C E D (4) D C E A B (3) D C A B E (3) B E A C D (3) B D E C A (3) B C A E D (3) A E C B D (3) A C E D B (3) A C D E B (3) A C B E D (3) E A B D C (2) D E C B A (2) D B C E A (2) C D A E B (2) C D A B E (2) A C D B E (2) E D B C A (1) E B D C A (1) E B A D C (1) E B A C D (1) E A D C B (1) D C E B A (1) D C B A E (1) D B E C A (1) D A C E B (1) C D B A E (1) B D C E A (1) B A E C D (1) A E B C D (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -10 -8 8 B 0 0 0 -14 -12 C 10 0 0 -6 10 D 8 14 6 0 4 E -8 12 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -8 8 B 0 0 0 -14 -12 C 10 0 0 -6 10 D 8 14 6 0 4 E -8 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=20 B=19 A=18 C=17 so C is eliminated. Round 2 votes counts: D=31 A=30 E=20 B=19 so B is eliminated. Round 3 votes counts: A=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 C:207 A:195 E:195 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -10 -8 8 B 0 0 0 -14 -12 C 10 0 0 -6 10 D 8 14 6 0 4 E -8 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -8 8 B 0 0 0 -14 -12 C 10 0 0 -6 10 D 8 14 6 0 4 E -8 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -8 8 B 0 0 0 -14 -12 C 10 0 0 -6 10 D 8 14 6 0 4 E -8 12 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7828: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) E B D C A (9) B C A D E (9) E D C B A (8) E D A C B (8) B A C D E (8) A B C D E (5) A D C E B (4) A C B D E (4) D E A C B (3) B E D C A (3) B C E D A (3) A D E C B (3) A C D E B (3) E D B C A (2) D A E C B (2) B E D A C (2) B E C D A (2) B C E A D (2) B C A E D (2) E C D B A (1) E B D A C (1) C A D E B (1) C A D B E (1) C A B D E (1) B E A D C (1) B A C E D (1) A E D B C (1) Total count = 100 A B C D E A 0 -8 -8 -8 -10 B 8 0 2 6 -12 C 8 -2 0 -14 -12 D 8 -6 14 0 -12 E 10 12 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -8 -8 -10 B 8 0 2 6 -12 C 8 -2 0 -14 -12 D 8 -6 14 0 -12 E 10 12 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 B=33 A=20 D=5 C=3 so C is eliminated. Round 2 votes counts: E=39 B=33 A=23 D=5 so D is eliminated. Round 3 votes counts: E=42 B=33 A=25 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:202 D:202 C:190 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -8 -10 B 8 0 2 6 -12 C 8 -2 0 -14 -12 D 8 -6 14 0 -12 E 10 12 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -8 -10 B 8 0 2 6 -12 C 8 -2 0 -14 -12 D 8 -6 14 0 -12 E 10 12 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -8 -10 B 8 0 2 6 -12 C 8 -2 0 -14 -12 D 8 -6 14 0 -12 E 10 12 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7829: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (19) D E A C B (10) C B A E D (7) D E A B C (5) B A E C D (5) D B E A C (4) A E C B D (4) A E B C D (4) E A C D B (3) D E C A B (3) B C D A E (3) E A D C B (2) D C E A B (2) D B C E A (2) C E A D B (2) C D B E A (2) C A E B D (2) B D C A E (2) B D A E C (2) B A E D C (2) B A C E D (2) A E D B C (2) A B E C D (2) E C A D B (1) D B C A E (1) C E A B D (1) C D E A B (1) C B E A D (1) C B D A E (1) C A B E D (1) B C A D E (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -4 24 22 B 8 0 12 20 14 C 4 -12 0 26 2 D -24 -20 -26 0 -22 E -22 -14 -2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 24 22 B 8 0 12 20 14 C 4 -12 0 26 2 D -24 -20 -26 0 -22 E -22 -14 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=27 C=18 A=13 E=6 so E is eliminated. Round 2 votes counts: B=36 D=27 C=19 A=18 so A is eliminated. Round 3 votes counts: B=42 D=31 C=27 so C is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 A:217 C:210 E:192 D:154 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 24 22 B 8 0 12 20 14 C 4 -12 0 26 2 D -24 -20 -26 0 -22 E -22 -14 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 24 22 B 8 0 12 20 14 C 4 -12 0 26 2 D -24 -20 -26 0 -22 E -22 -14 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 24 22 B 8 0 12 20 14 C 4 -12 0 26 2 D -24 -20 -26 0 -22 E -22 -14 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7830: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) B D C A E (8) E A C D B (7) D B E A C (7) D B C E A (7) A C E B D (7) C E A D B (6) C A E B D (6) E A D C B (5) B D A E C (5) D B E C A (4) B D C E A (3) E D C A B (2) E C A D B (2) D E C B A (2) D E A C B (2) D C E B A (2) C D B E A (2) B D A C E (2) D E B C A (1) D E A B C (1) C E D A B (1) C B A E D (1) C B A D E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E C D B (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 4 2 -8 B -4 0 -14 -4 -8 C -4 14 0 -6 0 D -2 4 6 0 0 E 8 8 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.597510 E: 0.402490 Sum of squares = 0.519016316653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.597510 E: 1.000000 A B C D E A 0 4 4 2 -8 B -4 0 -14 -4 -8 C -4 14 0 -6 0 D -2 4 6 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=21 A=20 C=17 E=16 so E is eliminated. Round 2 votes counts: A=32 D=28 B=21 C=19 so C is eliminated. Round 3 votes counts: A=46 D=31 B=23 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:208 D:204 C:202 A:201 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 2 -8 B -4 0 -14 -4 -8 C -4 14 0 -6 0 D -2 4 6 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 -8 B -4 0 -14 -4 -8 C -4 14 0 -6 0 D -2 4 6 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 -8 B -4 0 -14 -4 -8 C -4 14 0 -6 0 D -2 4 6 0 0 E 8 8 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7831: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (22) E B D A C (20) B E D A C (12) C A D E B (8) D A C B E (6) C A E D B (5) E B C A D (4) B D E A C (4) E B D C A (3) D B E A C (3) D A B C E (3) C E A B D (2) C A E B D (2) E C B A D (1) E B A C D (1) E A B C D (1) D C A B E (1) D B A E C (1) C D A B E (1) Total count = 100 A B C D E A 0 2 2 -8 -2 B -2 0 4 0 6 C -2 -4 0 -6 0 D 8 0 6 0 -2 E 2 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.553443 C: 0.000000 D: 0.446557 E: 0.000000 Sum of squares = 0.505712320083 Cumulative probabilities = A: 0.000000 B: 0.553443 C: 0.553443 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -8 -2 B -2 0 4 0 6 C -2 -4 0 -6 0 D 8 0 6 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=30 B=16 D=14 so A is eliminated. Round 2 votes counts: C=40 E=30 B=16 D=14 so D is eliminated. Round 3 votes counts: C=47 E=30 B=23 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:206 B:204 E:199 A:197 C:194 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 2 -8 -2 B -2 0 4 0 6 C -2 -4 0 -6 0 D 8 0 6 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -8 -2 B -2 0 4 0 6 C -2 -4 0 -6 0 D 8 0 6 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -8 -2 B -2 0 4 0 6 C -2 -4 0 -6 0 D 8 0 6 0 -2 E 2 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7832: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) A E D B C (6) D E B A C (5) E D A B C (4) D C B E A (4) B C E D A (4) A D E C B (4) A C E B D (4) E A B D C (3) D E A B C (3) C B A E D (3) C A D B E (3) C A B E D (3) A E B D C (3) E B D A C (2) E A D B C (2) D B E C A (2) D B C E A (2) C D A E B (2) C D A B E (2) C B D E A (2) C B D A E (2) C A D E B (2) B E A D C (2) B C D E A (2) A E C D B (2) A C E D B (2) A C D E B (2) A B E C D (2) E B A D C (1) D E B C A (1) D A C E B (1) C D B A E (1) C B E A D (1) C A E B D (1) C A B D E (1) B E D A C (1) B E C D A (1) B D E C A (1) B D E A C (1) B C E A D (1) B C A E D (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 8 16 2 -2 B -8 0 12 -14 -16 C -16 -12 0 -10 -8 D -2 14 10 0 -14 E 2 16 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 16 2 -2 B -8 0 12 -14 -16 C -16 -12 0 -10 -8 D -2 14 10 0 -14 E 2 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 E=18 D=18 B=14 so B is eliminated. Round 2 votes counts: C=31 A=27 E=22 D=20 so D is eliminated. Round 3 votes counts: C=37 E=35 A=28 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:212 D:204 B:187 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 2 -2 B -8 0 12 -14 -16 C -16 -12 0 -10 -8 D -2 14 10 0 -14 E 2 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 2 -2 B -8 0 12 -14 -16 C -16 -12 0 -10 -8 D -2 14 10 0 -14 E 2 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 2 -2 B -8 0 12 -14 -16 C -16 -12 0 -10 -8 D -2 14 10 0 -14 E 2 16 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7833: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) D E B A C (10) A D C B E (6) C E B A D (5) C B A E D (5) A C B D E (5) B D E A C (4) A C B E D (4) E D B C A (3) D E A C B (3) D A B E C (3) B D A E C (3) A C D B E (3) E D C B A (2) E D C A B (2) E C D A B (2) E C B D A (2) D E A B C (2) D B A E C (2) D A E C B (2) C E A D B (2) B E C D A (2) B C A E D (2) B A D C E (2) A D C E B (2) E C D B A (1) E B D C A (1) C E D A B (1) C E A B D (1) C B E A D (1) C A D E B (1) B A D E C (1) B A C E D (1) B A C D E (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 12 10 12 B -4 0 -22 4 16 C -12 22 0 0 10 D -10 -4 0 0 4 E -12 -16 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999735 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 10 12 B -4 0 -22 4 16 C -12 22 0 0 10 D -10 -4 0 0 4 E -12 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 A=22 B=16 E=13 so E is eliminated. Round 2 votes counts: C=32 D=29 A=22 B=17 so B is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:210 B:197 D:195 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 10 12 B -4 0 -22 4 16 C -12 22 0 0 10 D -10 -4 0 0 4 E -12 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 10 12 B -4 0 -22 4 16 C -12 22 0 0 10 D -10 -4 0 0 4 E -12 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 10 12 B -4 0 -22 4 16 C -12 22 0 0 10 D -10 -4 0 0 4 E -12 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999741 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7834: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (9) C E D B A (7) D E C A B (6) E C D A B (5) A B D E C (5) B A C E D (4) A D E B C (4) E D A C B (3) D E C B A (3) D E A B C (3) C B E D A (3) C B A E D (3) B A D E C (3) E D C A B (2) C E D A B (2) C E B D A (2) B D E C A (2) B C E D A (2) B A D C E (2) A D B E C (2) A B C E D (2) A B C D E (2) E C D B A (1) D E B C A (1) D E B A C (1) D B E A C (1) D A E B C (1) D A B E C (1) C E A D B (1) C B E A D (1) C A E D B (1) B D A E C (1) B C E A D (1) B C D E A (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -6 -16 -16 B -2 0 -2 -10 -6 C 6 2 0 -2 -10 D 16 10 2 0 0 E 16 6 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.536071 E: 0.463929 Sum of squares = 0.502602258866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.536071 E: 1.000000 A B C D E A 0 2 -6 -16 -16 B -2 0 -2 -10 -6 C 6 2 0 -2 -10 D 16 10 2 0 0 E 16 6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 C=20 A=16 E=11 so E is eliminated. Round 2 votes counts: D=32 C=26 B=26 A=16 so A is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:214 C:198 B:190 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -16 -16 B -2 0 -2 -10 -6 C 6 2 0 -2 -10 D 16 10 2 0 0 E 16 6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -16 -16 B -2 0 -2 -10 -6 C 6 2 0 -2 -10 D 16 10 2 0 0 E 16 6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -16 -16 B -2 0 -2 -10 -6 C 6 2 0 -2 -10 D 16 10 2 0 0 E 16 6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7835: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (6) C E B D A (6) B D C A E (6) A E D C B (6) E A C D B (5) C E B A D (5) B C D E A (5) A D E B C (5) E A C B D (4) D A B E C (4) D A B C E (4) C B E D A (4) C B D E A (4) B D A C E (4) E C A D B (3) E A D C B (3) A E D B C (3) A D B E C (3) E C A B D (2) B A D E C (2) A B E D C (2) D C E A B (1) D B C A E (1) D B A E C (1) D A E C B (1) D A E B C (1) C E A D B (1) C E A B D (1) C D E B A (1) B D A E C (1) B C E D A (1) B A E D C (1) B A E C D (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 18 -2 8 B -2 0 6 2 2 C -18 -6 0 -14 0 D 2 -2 14 0 2 E -8 -2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333317 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 18 -2 8 B -2 0 6 2 2 C -18 -6 0 -14 0 D 2 -2 14 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=22 B=21 A=21 D=19 E=17 so E is eliminated. Round 2 votes counts: A=33 C=27 B=21 D=19 so D is eliminated. Round 3 votes counts: A=43 B=29 C=28 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:208 B:204 E:194 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 18 -2 8 B -2 0 6 2 2 C -18 -6 0 -14 0 D 2 -2 14 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 -2 8 B -2 0 6 2 2 C -18 -6 0 -14 0 D 2 -2 14 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 -2 8 B -2 0 6 2 2 C -18 -6 0 -14 0 D 2 -2 14 0 2 E -8 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7836: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) D E C A B (7) B A E D C (6) A B D E C (6) E C D B A (5) B A E C D (5) B A C D E (5) C D E A B (4) A B D C E (4) D C E A B (3) B E A C D (3) B A D E C (3) E D C B A (2) D C A E B (2) C E D B A (2) C D A B E (2) C B A E D (2) B E A D C (2) B C A E D (2) B A C E D (2) A B E D C (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C D A B (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) D A C E B (1) C E D A B (1) C E B D A (1) C D A E B (1) C B E A D (1) C A D B E (1) C A B D E (1) B E C A D (1) B C A D E (1) B A D C E (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -8 4 0 B 0 0 6 8 6 C 8 -6 0 -10 -20 D -4 -8 10 0 -10 E 0 -6 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.212219 B: 0.787781 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.665635896673 Cumulative probabilities = A: 0.212219 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 4 0 B 0 0 6 8 6 C 8 -6 0 -10 -20 D -4 -8 10 0 -10 E 0 -6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204153122 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 C=16 A=15 D=13 so D is eliminated. Round 2 votes counts: E=32 B=31 C=21 A=16 so A is eliminated. Round 3 votes counts: B=46 E=32 C=22 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:212 B:210 A:198 D:194 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 4 0 B 0 0 6 8 6 C 8 -6 0 -10 -20 D -4 -8 10 0 -10 E 0 -6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204153122 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 4 0 B 0 0 6 8 6 C 8 -6 0 -10 -20 D -4 -8 10 0 -10 E 0 -6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204153122 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 4 0 B 0 0 6 8 6 C 8 -6 0 -10 -20 D -4 -8 10 0 -10 E 0 -6 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.571429 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510204153122 Cumulative probabilities = A: 0.428571 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7837: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (13) A E D B C (8) E D A C B (5) E A D C B (5) D E A B C (5) B A C D E (5) D E C B A (4) C B E D A (4) B C D A E (4) C E D B A (3) B C A D E (3) A B D E C (3) D B C E A (2) D A B E C (2) C E B A D (2) C A B E D (2) B D C A E (2) B D A C E (2) B C D E A (2) A E B D C (2) A E B C D (2) A B C E D (2) E D C A B (1) E D A B C (1) E C A D B (1) D E B A C (1) D C E B A (1) D C B E A (1) D B A E C (1) D A E B C (1) C D E B A (1) C B E A D (1) C B A E D (1) C B A D E (1) A D E B C (1) A D B E C (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 -12 -6 B 8 0 8 10 12 C -2 -8 0 -2 8 D 12 -10 2 0 16 E 6 -12 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -12 -6 B 8 0 8 10 12 C -2 -8 0 -2 8 D 12 -10 2 0 16 E 6 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 D=18 B=18 E=13 so E is eliminated. Round 2 votes counts: C=29 A=28 D=25 B=18 so B is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:219 D:210 C:198 A:188 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -12 -6 B 8 0 8 10 12 C -2 -8 0 -2 8 D 12 -10 2 0 16 E 6 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -12 -6 B 8 0 8 10 12 C -2 -8 0 -2 8 D 12 -10 2 0 16 E 6 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -12 -6 B 8 0 8 10 12 C -2 -8 0 -2 8 D 12 -10 2 0 16 E 6 -12 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7838: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (5) D A C E B (5) B C E D A (5) B C D E A (5) B A E C D (5) A D E C B (5) E A C B D (4) D A E C B (4) D A B C E (4) A E D C B (4) A B D E C (4) E C A B D (3) E B C A D (3) E A B C D (3) C D E B A (3) B D C A E (3) E C B A D (2) D C B E A (2) D C B A E (2) C E B D A (2) C B E D A (2) A E D B C (2) A E B D C (2) A D E B C (2) A D B E C (2) E C D A B (1) E B A C D (1) D C E A B (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) C E D B A (1) C D B E A (1) B E C A D (1) B D A C E (1) B C E A D (1) B A D E C (1) B A D C E (1) B A C E D (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 10 18 8 8 B -10 0 0 2 -12 C -18 0 0 2 -12 D -8 -2 -2 0 0 E -8 12 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 18 8 8 B -10 0 0 2 -12 C -18 0 0 2 -12 D -8 -2 -2 0 0 E -8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 E=22 D=22 C=9 so C is eliminated. Round 2 votes counts: D=26 B=26 E=25 A=23 so A is eliminated. Round 3 votes counts: E=35 D=35 B=30 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:222 E:208 D:194 B:190 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 18 8 8 B -10 0 0 2 -12 C -18 0 0 2 -12 D -8 -2 -2 0 0 E -8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 8 8 B -10 0 0 2 -12 C -18 0 0 2 -12 D -8 -2 -2 0 0 E -8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 8 8 B -10 0 0 2 -12 C -18 0 0 2 -12 D -8 -2 -2 0 0 E -8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7839: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) D A E B C (12) B C D A E (10) C B E A D (9) B D C A E (7) D B C A E (6) C E A B D (6) E A C D B (5) E A C B D (5) C B D E A (3) A E D B C (3) E C A B D (2) D B A E C (2) D B A C E (2) D A B E C (2) C E B A D (2) B C D E A (2) A E D C B (2) A D E B C (2) E C A D B (1) D E A C B (1) D B C E A (1) C E A D B (1) C B E D A (1) C B A E D (1) Total count = 100 A B C D E A 0 8 -4 2 -2 B -8 0 -2 -4 -8 C 4 2 0 -4 2 D -2 4 4 0 0 E 2 8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.418128 E: 0.581872 Sum of squares = 0.513405923641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.418128 E: 1.000000 A B C D E A 0 8 -4 2 -2 B -8 0 -2 -4 -8 C 4 2 0 -4 2 D -2 4 4 0 0 E 2 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 0.500095 Sum of squares = 0.500000018085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 C=23 B=19 A=7 so A is eliminated. Round 2 votes counts: E=30 D=28 C=23 B=19 so B is eliminated. Round 3 votes counts: D=35 C=35 E=30 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:204 D:203 A:202 C:202 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -4 2 -2 B -8 0 -2 -4 -8 C 4 2 0 -4 2 D -2 4 4 0 0 E 2 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 0.500095 Sum of squares = 0.500000018085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 2 -2 B -8 0 -2 -4 -8 C 4 2 0 -4 2 D -2 4 4 0 0 E 2 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 0.500095 Sum of squares = 0.500000018085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 2 -2 B -8 0 -2 -4 -8 C 4 2 0 -4 2 D -2 4 4 0 0 E 2 8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 0.500095 Sum of squares = 0.500000018085 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499905 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7840: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) B A E D C (7) E B A C D (6) A E B C D (6) E A C B D (5) E A B C D (5) D C B E A (5) D C B A E (5) D C A B E (4) D B C E A (4) D B C A E (4) C D E B A (4) C D E A B (4) B E A D C (4) B A D E C (4) C E D A B (3) B D A E C (3) C E A D B (2) B D A C E (2) A E C B D (2) A E B D C (2) A B E D C (2) E C D B A (1) E C B A D (1) E C A D B (1) D A C B E (1) C D B E A (1) C D B A E (1) B D E C A (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 2 -2 6 B 6 0 4 2 0 C -2 -4 0 0 0 D 2 -2 0 0 6 E -6 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.826265 C: 0.000000 D: 0.000000 E: 0.173735 Sum of squares = 0.712897200712 Cumulative probabilities = A: 0.000000 B: 0.826265 C: 0.826265 D: 0.826265 E: 1.000000 A B C D E A 0 -6 2 -2 6 B 6 0 4 2 0 C -2 -4 0 0 0 D 2 -2 0 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000579 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=23 C=23 B=21 E=19 A=14 so A is eliminated. Round 2 votes counts: E=29 D=24 B=24 C=23 so C is eliminated. Round 3 votes counts: D=42 E=34 B=24 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:206 D:203 A:200 C:197 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -2 6 B 6 0 4 2 0 C -2 -4 0 0 0 D 2 -2 0 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000579 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 6 B 6 0 4 2 0 C -2 -4 0 0 0 D 2 -2 0 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000579 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 6 B 6 0 4 2 0 C -2 -4 0 0 0 D 2 -2 0 0 6 E -6 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000579 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7841: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) D E A C B (10) C A B D E (8) B C A E D (7) E D B A C (6) D A C B E (6) A C D B E (6) D A C E B (5) E B C A D (4) D E A B C (4) A C B E D (4) A C B D E (4) E B D C A (3) D C A B E (3) E B D A C (2) D E B C A (2) C B A E D (2) C A B E D (2) B E C A D (2) B C E A D (2) E B C D A (1) E B A C D (1) D E C B A (1) D E C A B (1) D E B A C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -12 -2 B -8 0 -4 -12 -4 C -2 4 0 -10 2 D 12 12 10 0 2 E 2 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -12 -2 B -8 0 -4 -12 -4 C -2 4 0 -10 2 D 12 12 10 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=28 A=15 C=12 B=12 so C is eliminated. Round 2 votes counts: D=33 E=28 A=25 B=14 so B is eliminated. Round 3 votes counts: A=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:201 A:198 C:197 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -12 -2 B -8 0 -4 -12 -4 C -2 4 0 -10 2 D 12 12 10 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -12 -2 B -8 0 -4 -12 -4 C -2 4 0 -10 2 D 12 12 10 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -12 -2 B -8 0 -4 -12 -4 C -2 4 0 -10 2 D 12 12 10 0 2 E 2 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985123 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7842: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) E C B A D (9) D A C E B (7) B E C D A (7) E C A B D (6) E B C A D (6) A D C E B (6) C E A D B (4) B E C A D (4) B D E C A (4) B D A E C (4) D A C B E (3) C E A B D (3) B D A C E (3) A C D E B (3) E C A D B (2) D B A C E (2) C E B A D (2) B E D C A (2) B C E D A (2) A C E D B (2) E D C A B (1) E B C D A (1) D E B A C (1) D B A E C (1) D A E B C (1) D A B E C (1) C B E A D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -8 -2 -10 B -2 0 2 10 -8 C 8 -2 0 6 0 D 2 -10 -6 0 -4 E 10 8 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.395376 D: 0.000000 E: 0.604624 Sum of squares = 0.521892244131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.395376 D: 0.395376 E: 1.000000 A B C D E A 0 2 -8 -2 -10 B -2 0 2 10 -8 C 8 -2 0 6 0 D 2 -10 -6 0 -4 E 10 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 E=25 A=12 C=10 so C is eliminated. Round 2 votes counts: E=34 D=27 B=27 A=12 so A is eliminated. Round 3 votes counts: E=36 D=36 B=28 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:206 B:201 A:191 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 -2 -10 B -2 0 2 10 -8 C 8 -2 0 6 0 D 2 -10 -6 0 -4 E 10 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 -10 B -2 0 2 10 -8 C 8 -2 0 6 0 D 2 -10 -6 0 -4 E 10 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 -10 B -2 0 2 10 -8 C 8 -2 0 6 0 D 2 -10 -6 0 -4 E 10 8 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7843: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) D E B A C (8) B C A E D (8) D E A B C (7) D E B C A (5) C B A D E (5) C A B E D (5) D B C E A (4) E A B C D (3) D C B A E (3) C B A E D (3) B C A D E (3) A C B E D (3) E A B D C (2) D C B E A (2) D B E C A (2) A E C D B (2) A E C B D (2) E D B A C (1) E D A C B (1) E D A B C (1) E B D C A (1) E A C D B (1) E A C B D (1) D E C A B (1) D C E B A (1) D A C E B (1) C D B A E (1) C B D A E (1) C A B D E (1) B E D C A (1) B E C D A (1) B D C E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D A E (1) B A C E D (1) A E B C D (1) A C E D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -6 -10 -10 B 12 0 10 -2 0 C 6 -10 0 -4 0 D 10 2 4 0 14 E 10 0 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -10 -10 B 12 0 10 -2 0 C 6 -10 0 -4 0 D 10 2 4 0 14 E 10 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 B=19 C=16 E=11 A=11 so E is eliminated. Round 2 votes counts: D=46 B=20 A=18 C=16 so C is eliminated. Round 3 votes counts: D=47 B=29 A=24 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:210 E:198 C:196 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -10 -10 B 12 0 10 -2 0 C 6 -10 0 -4 0 D 10 2 4 0 14 E 10 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -10 -10 B 12 0 10 -2 0 C 6 -10 0 -4 0 D 10 2 4 0 14 E 10 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -10 -10 B 12 0 10 -2 0 C 6 -10 0 -4 0 D 10 2 4 0 14 E 10 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7844: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A C E B D (9) B C E D A (8) A E D C B (7) A C B E D (6) D A E B C (5) C B A E D (5) E B C D A (4) D B C E A (4) A D E C B (4) A D C B E (4) C B E A D (3) A C B D E (3) E D A B C (2) D E A B C (2) D B C A E (2) B E C D A (2) B C D E A (2) B C A D E (2) A E C B D (2) E D B C A (1) E D B A C (1) E B D C A (1) E A D C B (1) E A C B D (1) D E B A C (1) D B E C A (1) D A B E C (1) D A B C E (1) C A B E D (1) B D C E A (1) B C E A D (1) B C D A E (1) A E C D B (1) A D B C E (1) Total count = 100 A B C D E A 0 2 4 2 10 B -2 0 6 4 -2 C -4 -6 0 2 8 D -2 -4 -2 0 -12 E -10 2 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998886 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 2 10 B -2 0 6 4 -2 C -4 -6 0 2 8 D -2 -4 -2 0 -12 E -10 2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=26 B=17 E=11 C=9 so C is eliminated. Round 2 votes counts: A=38 D=26 B=25 E=11 so E is eliminated. Round 3 votes counts: A=40 D=30 B=30 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:203 C:200 E:198 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 2 10 B -2 0 6 4 -2 C -4 -6 0 2 8 D -2 -4 -2 0 -12 E -10 2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 10 B -2 0 6 4 -2 C -4 -6 0 2 8 D -2 -4 -2 0 -12 E -10 2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 10 B -2 0 6 4 -2 C -4 -6 0 2 8 D -2 -4 -2 0 -12 E -10 2 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7845: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (6) B C A E D (6) D C B E A (5) C B D A E (5) B E A C D (5) E A B D C (4) D E A C B (4) B A E C D (4) A C E D B (4) E B A D C (3) D C E B A (3) D C E A B (3) D C B A E (3) C D A B E (3) C A B D E (3) E D A B C (2) E A D C B (2) E A D B C (2) E A B C D (2) D E B C A (2) D C A E B (2) D C A B E (2) C B A D E (2) C A B E D (2) B C A D E (2) A E C B D (2) A E B C D (2) A D E C B (2) A C E B D (2) E D A C B (1) D E C B A (1) D E C A B (1) D E B A C (1) D A C E B (1) C D A E B (1) B E C D A (1) B E A D C (1) B D C E A (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -8 4 12 B 4 0 -22 -2 6 C 8 22 0 8 16 D -4 2 -8 0 6 E -12 -6 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 4 12 B 4 0 -22 -2 6 C 8 22 0 8 16 D -4 2 -8 0 6 E -12 -6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=22 B=21 E=16 A=13 so A is eliminated. Round 2 votes counts: D=30 C=29 B=21 E=20 so E is eliminated. Round 3 votes counts: D=37 B=32 C=31 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:227 A:202 D:198 B:193 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 4 12 B 4 0 -22 -2 6 C 8 22 0 8 16 D -4 2 -8 0 6 E -12 -6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 4 12 B 4 0 -22 -2 6 C 8 22 0 8 16 D -4 2 -8 0 6 E -12 -6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 4 12 B 4 0 -22 -2 6 C 8 22 0 8 16 D -4 2 -8 0 6 E -12 -6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7846: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) C D E B A (7) D C A B E (6) A D B E C (5) D A C B E (4) C E B D A (4) A E B C D (4) E C B D A (3) E C B A D (3) E B C D A (3) E B A C D (3) D C B E A (3) C D A E B (3) C A D E B (3) A E C B D (3) E B C A D (2) D C B A E (2) D B A C E (2) D A B C E (2) C E D B A (2) C E A B D (2) B E D C A (2) B E D A C (2) B A E D C (2) A D B C E (2) A C E B D (2) E C A B D (1) E A B C D (1) D C A E B (1) D B E C A (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B A D (1) C D E A B (1) B E A D C (1) B D E A C (1) B D A E C (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -2 -6 12 B -8 0 -6 4 0 C 2 6 0 0 -2 D 6 -4 0 0 -6 E -12 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.207720 B: 0.000000 C: 0.253683 D: 0.330878 E: 0.207720 Sum of squares = 0.260130127621 Cumulative probabilities = A: 0.207720 B: 0.207720 C: 0.461402 D: 0.792280 E: 1.000000 A B C D E A 0 8 -2 -6 12 B -8 0 -6 4 0 C 2 6 0 0 -2 D 6 -4 0 0 -6 E -12 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.203704 B: 0.000000 C: 0.277778 D: 0.314815 E: 0.203704 Sum of squares = 0.259259259259 Cumulative probabilities = A: 0.203704 B: 0.203704 C: 0.481481 D: 0.796296 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=24 D=23 E=16 B=9 so B is eliminated. Round 2 votes counts: A=30 D=25 C=24 E=21 so E is eliminated. Round 3 votes counts: C=36 A=35 D=29 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:206 C:203 D:198 E:198 B:195 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 -6 12 B -8 0 -6 4 0 C 2 6 0 0 -2 D 6 -4 0 0 -6 E -12 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.203704 B: 0.000000 C: 0.277778 D: 0.314815 E: 0.203704 Sum of squares = 0.259259259259 Cumulative probabilities = A: 0.203704 B: 0.203704 C: 0.481481 D: 0.796296 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -6 12 B -8 0 -6 4 0 C 2 6 0 0 -2 D 6 -4 0 0 -6 E -12 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.203704 B: 0.000000 C: 0.277778 D: 0.314815 E: 0.203704 Sum of squares = 0.259259259259 Cumulative probabilities = A: 0.203704 B: 0.203704 C: 0.481481 D: 0.796296 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -6 12 B -8 0 -6 4 0 C 2 6 0 0 -2 D 6 -4 0 0 -6 E -12 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.203704 B: 0.000000 C: 0.277778 D: 0.314815 E: 0.203704 Sum of squares = 0.259259259259 Cumulative probabilities = A: 0.203704 B: 0.203704 C: 0.481481 D: 0.796296 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7847: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (12) C E A B D (11) D E C A B (8) D B A E C (7) D B A C E (7) B A C E D (7) E C B A D (6) B A D C E (6) E C A B D (5) D E C B A (5) D A B C E (5) E C D B A (3) E C D A B (3) D B E A C (2) C A E B D (2) B D A E C (2) A C B E D (2) E C A D B (1) D E B C A (1) C A E D B (1) B D A C E (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 12 10 B -6 0 4 14 6 C -8 -4 0 10 14 D -12 -14 -10 0 -8 E -10 -6 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 12 10 B -6 0 4 14 6 C -8 -4 0 10 14 D -12 -14 -10 0 -8 E -10 -6 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=18 A=17 B=16 C=14 so C is eliminated. Round 2 votes counts: D=35 E=29 A=20 B=16 so B is eliminated. Round 3 votes counts: D=38 A=33 E=29 so E is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:209 C:206 E:189 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 12 10 B -6 0 4 14 6 C -8 -4 0 10 14 D -12 -14 -10 0 -8 E -10 -6 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 12 10 B -6 0 4 14 6 C -8 -4 0 10 14 D -12 -14 -10 0 -8 E -10 -6 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 12 10 B -6 0 4 14 6 C -8 -4 0 10 14 D -12 -14 -10 0 -8 E -10 -6 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7848: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) C D A E B (7) D C B E A (6) D C B A E (6) C A D E B (6) B E D C A (6) A C E D B (6) E A B C D (5) B E D A C (5) B E A D C (5) D C A E B (4) D C A B E (4) B D E C A (4) A E B C D (3) A C D E B (3) E B A D C (2) D B C E A (2) C D A B E (2) B E A C D (2) B D C A E (2) A E C B D (2) A C E B D (2) E B D C A (1) E B D A C (1) D E B C A (1) C E D A B (1) C A D B E (1) B A E C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -6 -4 2 B 4 0 -2 0 -4 C 6 2 0 2 6 D 4 0 -2 0 -2 E -2 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 2 B 4 0 -2 0 -4 C 6 2 0 2 6 D 4 0 -2 0 -2 E -2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 A=18 E=17 C=17 so E is eliminated. Round 2 votes counts: B=37 D=23 A=23 C=17 so C is eliminated. Round 3 votes counts: B=37 D=33 A=30 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 D:200 B:199 E:199 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 2 B 4 0 -2 0 -4 C 6 2 0 2 6 D 4 0 -2 0 -2 E -2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 2 B 4 0 -2 0 -4 C 6 2 0 2 6 D 4 0 -2 0 -2 E -2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 2 B 4 0 -2 0 -4 C 6 2 0 2 6 D 4 0 -2 0 -2 E -2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7849: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) B E C A D (7) A D C E B (7) D C B A E (5) C D A E B (5) B D C A E (5) A E D C B (5) D C A E B (4) D A C B E (4) B E A C D (4) E B C A D (3) E A C D B (3) D A C E B (3) B E C D A (3) B E A D C (3) B D C E A (3) E A B C D (2) D C A B E (2) D B C A E (2) C D B E A (2) C D A B E (2) C A D E B (2) B E D C A (2) B C E D A (2) A E C D B (2) E C A D B (1) E C A B D (1) E A C B D (1) E A B D C (1) C E D A B (1) C D B A E (1) B C D E A (1) B A E D C (1) B A D E C (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -8 6 6 B 4 0 -4 -6 2 C 8 4 0 2 6 D -6 6 -2 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 6 6 B 4 0 -4 -6 2 C 8 4 0 2 6 D -6 6 -2 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=20 E=19 A=16 C=13 so C is eliminated. Round 2 votes counts: B=32 D=30 E=20 A=18 so A is eliminated. Round 3 votes counts: D=41 B=32 E=27 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:210 A:200 D:200 B:198 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 6 6 B 4 0 -4 -6 2 C 8 4 0 2 6 D -6 6 -2 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 6 6 B 4 0 -4 -6 2 C 8 4 0 2 6 D -6 6 -2 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 6 6 B 4 0 -4 -6 2 C 8 4 0 2 6 D -6 6 -2 0 2 E -6 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7850: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E B D A C (6) C A D B E (5) A C D B E (5) E B D C A (4) D E B A C (4) C E D A B (4) C A D E B (4) B E A D C (4) D A B E C (3) D A B C E (3) C A E D B (3) B E D A C (3) B A E C D (3) A D C B E (3) A D B C E (3) E B C A D (2) D C E A B (2) D C A E B (2) D A C B E (2) C E D B A (2) C E B A D (2) C D E A B (2) C D A E B (2) B D E A C (2) B D A E C (2) A B D E C (2) E D B C A (1) E D B A C (1) E C D B A (1) E C B A D (1) D B E A C (1) D A E B C (1) C D A B E (1) C A E B D (1) B E C A D (1) B E A C D (1) B A E D C (1) B A D E C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 6 -12 -2 B 2 0 2 -10 -4 C -6 -2 0 -2 -4 D 12 10 2 0 0 E 2 4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.366013 E: 0.633987 Sum of squares = 0.535904963233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.366013 E: 1.000000 A B C D E A 0 -2 6 -12 -2 B 2 0 2 -10 -4 C -6 -2 0 -2 -4 D 12 10 2 0 0 E 2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=19 D=18 A=14 so A is eliminated. Round 2 votes counts: C=31 D=24 E=23 B=22 so B is eliminated. Round 3 votes counts: E=36 C=33 D=31 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:205 A:195 B:195 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -12 -2 B 2 0 2 -10 -4 C -6 -2 0 -2 -4 D 12 10 2 0 0 E 2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -12 -2 B 2 0 2 -10 -4 C -6 -2 0 -2 -4 D 12 10 2 0 0 E 2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -12 -2 B 2 0 2 -10 -4 C -6 -2 0 -2 -4 D 12 10 2 0 0 E 2 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7851: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C A D E B (9) C D A B E (6) E B A D C (5) C E A B D (5) C B D E A (5) B D E A C (5) D A B C E (4) E A B D C (3) D B A E C (3) D A C B E (3) D A B E C (3) C E A D B (3) C B E D A (3) C A E D B (3) E A D B C (2) E A C B D (2) D B C A E (2) D B A C E (2) C E B A D (2) B D C A E (2) A E D B C (2) A C E D B (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A C D (1) D C A B E (1) C D B A E (1) C B E A D (1) B E C D A (1) B E C A D (1) B D A E C (1) B D A C E (1) A E D C B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 4 -6 -4 B -4 0 -2 2 10 C -4 2 0 -2 12 D 6 -2 2 0 0 E 4 -10 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888619 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -6 -4 B -4 0 -2 2 10 C -4 2 0 -2 12 D 6 -2 2 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.3888888885 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=21 D=18 E=16 A=7 so A is eliminated. Round 2 votes counts: C=41 B=21 E=19 D=19 so E is eliminated. Round 3 votes counts: C=45 B=31 D=24 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:204 B:203 D:203 A:199 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -6 -4 B -4 0 -2 2 10 C -4 2 0 -2 12 D 6 -2 2 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.3888888885 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -6 -4 B -4 0 -2 2 10 C -4 2 0 -2 12 D 6 -2 2 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.3888888885 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -6 -4 B -4 0 -2 2 10 C -4 2 0 -2 12 D 6 -2 2 0 0 E 4 -10 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.3888888885 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7852: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (12) E B C A D (8) D A C E B (7) D A B E C (7) D A C B E (6) B E A C D (5) B D A E C (5) D A B C E (4) B E C D A (4) E B C D A (3) C A E B D (3) B E D C A (3) A D C E B (3) E C B D A (2) E C B A D (2) C E A B D (2) C A D E B (2) B E A D C (2) A B C E D (2) E D B C A (1) D C A E B (1) D B E C A (1) C E D B A (1) C E D A B (1) C E B A D (1) C E A D B (1) C D E A B (1) C A E D B (1) B E D A C (1) B D E A C (1) B A E C D (1) A D B E C (1) A D B C E (1) A C E D B (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 0 2 -4 B 6 0 26 18 16 C 0 -26 0 12 -20 D -2 -18 -12 0 -16 E 4 -16 20 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 2 -4 B 6 0 26 18 16 C 0 -26 0 12 -20 D -2 -18 -12 0 -16 E 4 -16 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=26 E=16 C=13 A=11 so A is eliminated. Round 2 votes counts: B=37 D=31 E=16 C=16 so E is eliminated. Round 3 votes counts: B=48 D=32 C=20 so C is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:212 A:196 C:183 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 2 -4 B 6 0 26 18 16 C 0 -26 0 12 -20 D -2 -18 -12 0 -16 E 4 -16 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 2 -4 B 6 0 26 18 16 C 0 -26 0 12 -20 D -2 -18 -12 0 -16 E 4 -16 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 2 -4 B 6 0 26 18 16 C 0 -26 0 12 -20 D -2 -18 -12 0 -16 E 4 -16 20 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993547 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7853: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) D E B C A (7) D E C B A (6) B E A D C (6) C D A E B (5) A C B E D (5) A B E C D (4) A B C E D (4) E D B A C (3) D C A B E (3) A B C D E (3) E A B C D (2) D B E A C (2) C E A D B (2) C E A B D (2) C D E A B (2) C A E B D (2) C A D B E (2) B E D A C (2) B A E D C (2) B A E C D (2) B A D E C (2) A C E B D (2) A C B D E (2) E D C B A (1) E D B C A (1) E B D C A (1) E B D A C (1) D C B E A (1) D B A E C (1) C E D B A (1) C D A B E (1) C A E D B (1) C A D E B (1) C A B E D (1) C A B D E (1) B D E A C (1) B D A E C (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -6 -4 -6 B 6 0 -8 -6 -6 C 6 8 0 -6 8 D 4 6 6 0 8 E 6 6 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -4 -6 B 6 0 -8 -6 -6 C 6 8 0 -6 8 D 4 6 6 0 8 E 6 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=22 C=21 B=16 E=9 so E is eliminated. Round 2 votes counts: D=37 A=24 C=21 B=18 so B is eliminated. Round 3 votes counts: D=43 A=36 C=21 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 C:208 E:198 B:193 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 -6 B 6 0 -8 -6 -6 C 6 8 0 -6 8 D 4 6 6 0 8 E 6 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 -6 B 6 0 -8 -6 -6 C 6 8 0 -6 8 D 4 6 6 0 8 E 6 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 -6 B 6 0 -8 -6 -6 C 6 8 0 -6 8 D 4 6 6 0 8 E 6 6 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7854: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) D C E B A (8) D B A E C (5) B D A E C (5) B A E D C (5) E C B A D (4) D C A E B (4) B E C A D (4) B A E C D (4) A D B E C (4) E B C A D (3) D B A C E (3) D A B C E (3) C E D A B (3) C E A D B (3) B E A C D (3) A B E C D (3) E B A C D (2) D C B E A (2) D B C E A (2) D A C E B (2) D A B E C (2) C E B D A (2) C E A B D (2) C D E A B (2) B A D E C (2) E C A B D (1) E A B C D (1) D C A B E (1) C D E B A (1) B C E D A (1) A C D E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -6 -12 -8 B 12 0 8 -12 2 C 6 -8 0 -20 0 D 12 12 20 0 16 E 8 -2 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -12 -8 B 12 0 8 -12 2 C 6 -8 0 -20 0 D 12 12 20 0 16 E 8 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 B=24 C=13 E=11 A=10 so A is eliminated. Round 2 votes counts: D=46 B=29 C=14 E=11 so E is eliminated. Round 3 votes counts: D=46 B=35 C=19 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:230 B:205 E:195 C:189 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -12 -8 B 12 0 8 -12 2 C 6 -8 0 -20 0 D 12 12 20 0 16 E 8 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -12 -8 B 12 0 8 -12 2 C 6 -8 0 -20 0 D 12 12 20 0 16 E 8 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -12 -8 B 12 0 8 -12 2 C 6 -8 0 -20 0 D 12 12 20 0 16 E 8 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7855: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) D B C E A (9) B D E C A (9) A E C D B (6) A B D C E (6) B D C E A (5) A C E D B (5) A B D E C (5) B E C D A (4) A D C E B (4) A D B C E (4) E C D B A (3) E C A D B (3) A C D E B (3) A B E C D (3) B E D C A (2) B A D E C (2) B A D C E (2) A E C B D (2) A E B C D (2) A D C B E (2) E C A B D (1) D C E B A (1) C E D B A (1) C D E B A (1) B D E A C (1) B D C A E (1) B D A E C (1) B D A C E (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 2 4 2 B 6 0 18 16 18 C -2 -18 0 -12 -10 D -4 -16 12 0 14 E -2 -18 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 4 2 B 6 0 18 16 18 C -2 -18 0 -12 -10 D -4 -16 12 0 14 E -2 -18 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 B=29 E=16 D=10 C=2 so C is eliminated. Round 2 votes counts: A=43 B=29 E=17 D=11 so D is eliminated. Round 3 votes counts: A=43 B=38 E=19 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:203 A:201 E:188 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 4 2 B 6 0 18 16 18 C -2 -18 0 -12 -10 D -4 -16 12 0 14 E -2 -18 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 4 2 B 6 0 18 16 18 C -2 -18 0 -12 -10 D -4 -16 12 0 14 E -2 -18 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 4 2 B 6 0 18 16 18 C -2 -18 0 -12 -10 D -4 -16 12 0 14 E -2 -18 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998748 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7856: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (6) B C E A D (5) E C D A B (4) E C B D A (4) C B E D A (4) B C E D A (4) E C B A D (3) E B C A D (3) D A E C B (3) C D E A B (3) C B D E A (3) B E A C D (3) B A D C E (3) B A C E D (3) A D E C B (3) A D E B C (3) A D B E C (3) A D B C E (3) A B D E C (3) E A B D C (2) D A C E B (2) D A B C E (2) C E D A B (2) C E B D A (2) B A E C D (2) A E D B C (2) E D C A B (1) E B A D C (1) E A D C B (1) E A D B C (1) D C E A B (1) D C A E B (1) D C A B E (1) D A C B E (1) C D B A E (1) C D A E B (1) C D A B E (1) C B D A E (1) B D A C E (1) B C D E A (1) B C D A E (1) B A E D C (1) B A D E C (1) A E D C B (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -6 10 -8 B 6 0 12 18 10 C 6 -12 0 14 -4 D -10 -18 -14 0 -12 E 8 -10 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 10 -8 B 6 0 12 18 10 C 6 -12 0 14 -4 D -10 -18 -14 0 -12 E 8 -10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=20 A=20 C=18 D=11 so D is eliminated. Round 2 votes counts: B=31 A=28 C=21 E=20 so E is eliminated. Round 3 votes counts: B=35 C=33 A=32 so A is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:207 C:202 A:195 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 10 -8 B 6 0 12 18 10 C 6 -12 0 14 -4 D -10 -18 -14 0 -12 E 8 -10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 10 -8 B 6 0 12 18 10 C 6 -12 0 14 -4 D -10 -18 -14 0 -12 E 8 -10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 10 -8 B 6 0 12 18 10 C 6 -12 0 14 -4 D -10 -18 -14 0 -12 E 8 -10 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7857: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) E D A B C (6) C B A D E (6) B C E A D (5) B C D A E (5) E B A C D (4) B D E C A (4) A C E B D (4) C B A E D (3) B C A E D (3) A E D C B (3) A E C D B (3) E D B A C (2) E A D C B (2) E A D B C (2) E A B C D (2) D E A C B (2) D E A B C (2) D C B A E (2) D B E C A (2) D B C E A (2) C B D A E (2) C A B D E (2) B E C D A (2) B C D E A (2) B C A D E (2) A C E D B (2) A C D E B (2) E D A C B (1) E B D A C (1) E A C D B (1) E A C B D (1) D C A B E (1) D B C A E (1) D A C E B (1) C D B A E (1) C A D B E (1) B E D C A (1) B D C E A (1) B C E D A (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -20 2 0 -8 B 20 0 16 4 -2 C -2 -16 0 10 -2 D 0 -4 -10 0 0 E 8 2 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.082605 E: 0.917395 Sum of squares = 0.848437857474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.082605 E: 1.000000 A B C D E A 0 -20 2 0 -8 B 20 0 16 4 -2 C -2 -16 0 10 -2 D 0 -4 -10 0 0 E 8 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222223205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 D=21 A=16 C=15 so C is eliminated. Round 2 votes counts: B=37 E=22 D=22 A=19 so A is eliminated. Round 3 votes counts: B=39 E=35 D=26 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:206 C:195 D:193 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 2 0 -8 B 20 0 16 4 -2 C -2 -16 0 10 -2 D 0 -4 -10 0 0 E 8 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222223205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 2 0 -8 B 20 0 16 4 -2 C -2 -16 0 10 -2 D 0 -4 -10 0 0 E 8 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222223205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 2 0 -8 B 20 0 16 4 -2 C -2 -16 0 10 -2 D 0 -4 -10 0 0 E 8 2 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.833333 Sum of squares = 0.722222223205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7858: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) E B C D A (8) D A E C B (6) B E C D A (6) E C B D A (5) A D C B E (5) D A C E B (4) C B E A D (4) C A E B D (4) B E C A D (4) A C D E B (4) E C B A D (3) D B E C A (3) D B E A C (3) D A E B C (3) D A B E C (3) C E B A D (3) B E D C A (3) B C E A D (3) D E B C A (2) D A C B E (2) B D E C A (2) A C D B E (2) E D B C A (1) D E C A B (1) D A B C E (1) C B A E D (1) B C A E D (1) A C E D B (1) A C E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 -6 -2 B 4 0 -12 0 -10 C 8 12 0 4 -6 D 6 0 -4 0 2 E 2 10 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 A B C D E A 0 -4 -8 -6 -2 B 4 0 -12 0 -10 C 8 12 0 4 -6 D 6 0 -4 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=24 B=19 E=17 C=12 so C is eliminated. Round 2 votes counts: D=28 A=28 B=24 E=20 so E is eliminated. Round 3 votes counts: B=43 D=29 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:209 E:208 D:202 B:191 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 -2 B 4 0 -12 0 -10 C 8 12 0 4 -6 D 6 0 -4 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 -2 B 4 0 -12 0 -10 C 8 12 0 4 -6 D 6 0 -4 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 -2 B 4 0 -12 0 -10 C 8 12 0 4 -6 D 6 0 -4 0 2 E 2 10 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7859: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) B E D A C (6) C D A B E (5) C A D E B (5) D C A B E (4) C B D A E (4) B E C D A (4) B C D E A (4) A E C D B (4) E A D B C (3) D A C B E (3) C A E D B (3) B E D C A (3) A E D C B (3) A D C E B (3) E B C A D (2) E B A C D (2) E A D C B (2) E A B D C (2) E A B C D (2) D A C E B (2) C E B A D (2) C E A B D (2) C D B A E (2) B D E C A (2) B D E A C (2) B D C A E (2) B C E D A (2) A C D E B (2) E B D A C (1) E A C D B (1) D C B A E (1) D A E C B (1) D A B C E (1) C E A D B (1) C D A E B (1) C A E B D (1) B E C A D (1) B C E A D (1) A D E C B (1) Total count = 100 A B C D E A 0 4 -4 0 -4 B -4 0 -6 4 -6 C 4 6 0 2 2 D 0 -4 -2 0 -10 E 4 6 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 0 -4 B -4 0 -6 4 -6 C 4 6 0 2 2 D 0 -4 -2 0 -10 E 4 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998178 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 E=22 A=13 D=12 so D is eliminated. Round 2 votes counts: C=31 B=27 E=22 A=20 so A is eliminated. Round 3 votes counts: C=41 E=31 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:207 A:198 B:194 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 0 -4 B -4 0 -6 4 -6 C 4 6 0 2 2 D 0 -4 -2 0 -10 E 4 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998178 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 0 -4 B -4 0 -6 4 -6 C 4 6 0 2 2 D 0 -4 -2 0 -10 E 4 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998178 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 0 -4 B -4 0 -6 4 -6 C 4 6 0 2 2 D 0 -4 -2 0 -10 E 4 6 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998178 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7860: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) D E C B A (9) C A B D E (7) B A E C D (7) A C B E D (7) E B A D C (5) D E C A B (5) D E B C A (4) C D A B E (4) E B D A C (3) D C E A B (3) C A D B E (3) E D A C B (2) E A B D C (2) D E B A C (2) D C E B A (2) D C A E B (2) D B E C A (2) C A B E D (2) B E D A C (2) B E A C D (2) B A C E D (2) A E B C D (2) A B E C D (2) E A C D B (1) D C B E A (1) D C B A E (1) C B D A E (1) C B A D E (1) C A D E B (1) B E A D C (1) B D E C A (1) B D C A E (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 0 -8 -12 B 12 0 -6 -2 -6 C 0 6 0 -14 -22 D 8 2 14 0 0 E 12 6 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.746731 E: 0.253269 Sum of squares = 0.621752642436 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.746731 E: 1.000000 A B C D E A 0 -12 0 -8 -12 B 12 0 -6 -2 -6 C 0 6 0 -14 -22 D 8 2 14 0 0 E 12 6 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 C=19 B=16 A=12 so A is eliminated. Round 2 votes counts: D=31 C=27 E=24 B=18 so B is eliminated. Round 3 votes counts: E=38 D=33 C=29 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:212 B:199 C:185 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 0 -8 -12 B 12 0 -6 -2 -6 C 0 6 0 -14 -22 D 8 2 14 0 0 E 12 6 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -8 -12 B 12 0 -6 -2 -6 C 0 6 0 -14 -22 D 8 2 14 0 0 E 12 6 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -8 -12 B 12 0 -6 -2 -6 C 0 6 0 -14 -22 D 8 2 14 0 0 E 12 6 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7861: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (7) A D E B C (7) D E A C B (5) D B A C E (5) C E B A D (5) B D C A E (5) D E C A B (4) D C B E A (4) C D E B A (4) A E B C D (4) D A E B C (3) B C D A E (3) A B D E C (3) E C D A B (2) D C E A B (2) D B C A E (2) D A E C B (2) C E D B A (2) C E B D A (2) C B E D A (2) B C D E A (2) A E D B C (2) A B E C D (2) E C B A D (1) E C A D B (1) E C A B D (1) E A D C B (1) E A C D B (1) D C B A E (1) D A B E C (1) D A B C E (1) C B E A D (1) C B D E A (1) B D A C E (1) B C E D A (1) B C E A D (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C E D (1) B A C D E (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 2 2 -10 0 B -2 0 0 -4 -14 C -2 0 0 -4 -2 D 10 4 4 0 20 E 0 14 2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -10 0 B -2 0 0 -4 -14 C -2 0 0 -4 -2 D 10 4 4 0 20 E 0 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=20 B=19 C=17 E=14 so E is eliminated. Round 2 votes counts: D=30 A=29 C=22 B=19 so B is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:198 A:197 C:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 2 -10 0 B -2 0 0 -4 -14 C -2 0 0 -4 -2 D 10 4 4 0 20 E 0 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -10 0 B -2 0 0 -4 -14 C -2 0 0 -4 -2 D 10 4 4 0 20 E 0 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -10 0 B -2 0 0 -4 -14 C -2 0 0 -4 -2 D 10 4 4 0 20 E 0 14 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7862: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) C D B A E (7) E B D A C (6) C B D E A (6) E B C D A (5) E B A D C (4) E A C B D (4) D B A C E (4) B C D E A (4) E B D C A (3) E A B D C (3) D C B A E (3) A E D B C (3) A E C D B (3) A D C B E (3) A C E D B (3) A C D B E (3) E C B D A (2) E A D B C (2) D B C A E (2) C D A B E (2) C B E D A (2) B D E C A (2) B D C E A (2) A C D E B (2) E B A C D (1) E A B C D (1) D E B A C (1) D B E A C (1) D B A E C (1) C E B D A (1) C B D A E (1) C A E B D (1) B E D C A (1) B E C D A (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -20 -6 -14 -4 B 20 0 -4 0 8 C 6 4 0 14 8 D 14 0 -14 0 6 E 4 -8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -6 -14 -4 B 20 0 -4 0 8 C 6 4 0 14 8 D 14 0 -14 0 6 E 4 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=28 A=19 D=12 B=10 so B is eliminated. Round 2 votes counts: E=33 C=32 A=19 D=16 so D is eliminated. Round 3 votes counts: C=39 E=37 A=24 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:212 D:203 E:191 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -6 -14 -4 B 20 0 -4 0 8 C 6 4 0 14 8 D 14 0 -14 0 6 E 4 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -14 -4 B 20 0 -4 0 8 C 6 4 0 14 8 D 14 0 -14 0 6 E 4 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -14 -4 B 20 0 -4 0 8 C 6 4 0 14 8 D 14 0 -14 0 6 E 4 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7863: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) A B D E C (9) E D C A B (6) B A D C E (6) D B A E C (5) C E D A B (5) C E A D B (5) C E D B A (4) C E A B D (4) C B A E D (4) E C D B A (3) C B E D A (3) C B E A D (3) B D A E C (3) B C A E D (3) B A C D E (3) E D A C B (2) E C D A B (2) D E C A B (2) D E A B C (2) C E B D A (2) A B D C E (2) E C A D B (1) E A C D B (1) D E C B A (1) D E B A C (1) D E A C B (1) C E B A D (1) C A B E D (1) B D E A C (1) B D C E A (1) B D A C E (1) B C A D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -4 10 -2 B 12 0 -2 18 14 C 4 2 0 -6 0 D -10 -18 6 0 -2 E 2 -14 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.230769 C: 0.692308 D: 0.076923 E: 0.000000 Sum of squares = 0.538461538456 Cumulative probabilities = A: 0.000000 B: 0.230769 C: 0.923077 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 10 -2 B 12 0 -2 18 14 C 4 2 0 -6 0 D -10 -18 6 0 -2 E 2 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.230769 C: 0.692308 D: 0.076923 E: 0.000000 Sum of squares = 0.538461535405 Cumulative probabilities = A: 0.000000 B: 0.230769 C: 0.923077 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=29 E=15 D=12 A=12 so D is eliminated. Round 2 votes counts: B=34 C=32 E=22 A=12 so A is eliminated. Round 3 votes counts: B=46 C=32 E=22 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:200 A:196 E:195 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 10 -2 B 12 0 -2 18 14 C 4 2 0 -6 0 D -10 -18 6 0 -2 E 2 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.230769 C: 0.692308 D: 0.076923 E: 0.000000 Sum of squares = 0.538461535405 Cumulative probabilities = A: 0.000000 B: 0.230769 C: 0.923077 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 10 -2 B 12 0 -2 18 14 C 4 2 0 -6 0 D -10 -18 6 0 -2 E 2 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.230769 C: 0.692308 D: 0.076923 E: 0.000000 Sum of squares = 0.538461535405 Cumulative probabilities = A: 0.000000 B: 0.230769 C: 0.923077 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 10 -2 B 12 0 -2 18 14 C 4 2 0 -6 0 D -10 -18 6 0 -2 E 2 -14 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.230769 C: 0.692308 D: 0.076923 E: 0.000000 Sum of squares = 0.538461535405 Cumulative probabilities = A: 0.000000 B: 0.230769 C: 0.923077 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7864: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D C B A E (8) B D E A C (7) E B A D C (6) E A B C D (6) D B C E A (5) E A C B D (4) D B C A E (4) C D A B E (4) C A E D B (4) C A D E B (4) A E C B D (4) D C A B E (3) C D A E B (3) B E D A C (3) A C E D B (3) E C A B D (2) D B E C A (2) C D E A B (2) B E A D C (2) B A D E C (2) A C E B D (2) E B D A C (1) E A B D C (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E A C (1) D B A C E (1) C E D A B (1) C E A D B (1) B E D C A (1) B D E C A (1) B D A E C (1) Total count = 100 A B C D E A 0 -12 4 -2 -14 B 12 0 6 2 -8 C -4 -6 0 -4 -6 D 2 -2 4 0 2 E 14 8 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 A B C D E A 0 -12 4 -2 -14 B 12 0 6 2 -8 C -4 -6 0 -4 -6 D 2 -2 4 0 2 E 14 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 C=19 B=17 A=9 so A is eliminated. Round 2 votes counts: E=32 D=27 C=24 B=17 so B is eliminated. Round 3 votes counts: E=38 D=38 C=24 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:213 B:206 D:203 C:190 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -2 -14 B 12 0 6 2 -8 C -4 -6 0 -4 -6 D 2 -2 4 0 2 E 14 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -2 -14 B 12 0 6 2 -8 C -4 -6 0 -4 -6 D 2 -2 4 0 2 E 14 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -2 -14 B 12 0 6 2 -8 C -4 -6 0 -4 -6 D 2 -2 4 0 2 E 14 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7865: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (13) C A B D E (10) E B D A C (9) D E B C A (8) A C B E D (7) C D A E B (5) B E D A C (5) B E A D C (5) D E C B A (4) C A D B E (4) B A E C D (4) C D E A B (3) B A C E D (3) A B C E D (3) E C D A B (2) D E B A C (2) C D A B E (2) C A B E D (2) A C B D E (2) A B E C D (2) E D C A B (1) D C E A B (1) D B C A E (1) B E A C D (1) B A E D C (1) Total count = 100 A B C D E A 0 -12 14 -12 -8 B 12 0 14 8 4 C -14 -14 0 0 -14 D 12 -8 0 0 -16 E 8 -4 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 14 -12 -8 B 12 0 14 8 4 C -14 -14 0 0 -14 D 12 -8 0 0 -16 E 8 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 B=19 D=16 A=14 so A is eliminated. Round 2 votes counts: C=35 E=25 B=24 D=16 so D is eliminated. Round 3 votes counts: E=39 C=36 B=25 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:217 D:194 A:191 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 14 -12 -8 B 12 0 14 8 4 C -14 -14 0 0 -14 D 12 -8 0 0 -16 E 8 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 14 -12 -8 B 12 0 14 8 4 C -14 -14 0 0 -14 D 12 -8 0 0 -16 E 8 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 14 -12 -8 B 12 0 14 8 4 C -14 -14 0 0 -14 D 12 -8 0 0 -16 E 8 -4 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999496 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7866: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (6) C D A B E (6) C A D B E (6) E A C D B (5) E B A C D (4) D C B A E (4) D C A E B (4) D C A B E (4) D B C A E (4) A C D B E (4) B E D C A (3) B D C A E (3) A E C B D (3) E D B A C (2) E B D C A (2) E A D C B (2) E A C B D (2) E A B C D (2) D E C B A (2) B E A C D (2) B D C E A (2) B C A D E (2) B A E C D (2) A C E D B (2) A C E B D (2) A C B E D (2) E D B C A (1) E D A C B (1) D E C A B (1) D E B C A (1) D C B E A (1) D B E C A (1) D B C E A (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D A C (1) B E C D A (1) B D E C A (1) B A C E D (1) B A C D E (1) A E B C D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -6 -4 12 B -2 0 -12 -8 10 C 6 12 0 6 8 D 4 8 -6 0 6 E -12 -10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -4 12 B -2 0 -12 -8 10 C 6 12 0 6 8 D 4 8 -6 0 6 E -12 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 B=19 A=16 C=15 so C is eliminated. Round 2 votes counts: D=29 E=27 A=24 B=20 so B is eliminated. Round 3 votes counts: D=35 E=34 A=31 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:206 A:202 B:194 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -4 12 B -2 0 -12 -8 10 C 6 12 0 6 8 D 4 8 -6 0 6 E -12 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 12 B -2 0 -12 -8 10 C 6 12 0 6 8 D 4 8 -6 0 6 E -12 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 12 B -2 0 -12 -8 10 C 6 12 0 6 8 D 4 8 -6 0 6 E -12 -10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7867: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (12) D B E A C (7) C A E B D (6) E C A B D (5) E B A D C (5) E D B C A (4) E B D C A (4) E B D A C (4) A B E D C (4) C A E D B (3) A B D E C (3) E C B D A (2) E C B A D (2) D E C B A (2) C E D B A (2) C E A B D (2) C D E B A (2) C D A E B (2) C D A B E (2) C A D E B (2) B D E A C (2) B D A E C (2) A D B C E (2) A C E B D (2) A C D B E (2) E D B A C (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) D B A E C (1) D B A C E (1) B E D A C (1) B E A D C (1) B A D E C (1) A E B C D (1) A D C B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -14 12 0 B -2 0 -6 -2 -6 C 14 6 0 -2 -8 D -12 2 2 0 0 E 0 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.251342 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.748658 Sum of squares = 0.623661597145 Cumulative probabilities = A: 0.251342 B: 0.251342 C: 0.251342 D: 0.251342 E: 1.000000 A B C D E A 0 2 -14 12 0 B -2 0 -6 -2 -6 C 14 6 0 -2 -8 D -12 2 2 0 0 E 0 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.537190231502 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=27 A=17 D=16 B=7 so B is eliminated. Round 2 votes counts: C=33 E=29 D=20 A=18 so A is eliminated. Round 3 votes counts: C=39 E=34 D=27 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:207 C:205 A:200 D:196 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -14 12 0 B -2 0 -6 -2 -6 C 14 6 0 -2 -8 D -12 2 2 0 0 E 0 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.537190231502 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 12 0 B -2 0 -6 -2 -6 C 14 6 0 -2 -8 D -12 2 2 0 0 E 0 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.537190231502 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 12 0 B -2 0 -6 -2 -6 C 14 6 0 -2 -8 D -12 2 2 0 0 E 0 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.537190231502 Cumulative probabilities = A: 0.363636 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7868: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (7) E A B C D (6) D C B E A (6) B D C E A (6) E C A B D (5) C D B E A (5) B D E C A (4) A E B C D (4) A C E D B (4) E A C B D (3) D C B A E (3) D B C E A (3) C D B A E (3) A E C B D (3) C E B D A (2) C E A D B (2) C A E D B (2) C A D E B (2) B E D C A (2) B D E A C (2) B D A E C (2) A B D E C (2) E C B A D (1) E B C D A (1) E B A C D (1) D C A B E (1) D B A C E (1) C E D B A (1) C E B A D (1) C E A B D (1) C D A E B (1) C D A B E (1) C B D E A (1) B E D A C (1) B E A D C (1) B C D E A (1) B A E D C (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E C B (1) A D C E B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -10 6 -12 B -2 0 -6 18 -6 C 10 6 0 6 -2 D -6 -18 -6 0 -6 E 12 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -10 6 -12 B -2 0 -6 18 -6 C 10 6 0 6 -2 D -6 -18 -6 0 -6 E 12 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 B=20 E=17 D=14 so D is eliminated. Round 2 votes counts: C=32 A=27 B=24 E=17 so E is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:210 B:202 A:193 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -10 6 -12 B -2 0 -6 18 -6 C 10 6 0 6 -2 D -6 -18 -6 0 -6 E 12 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 6 -12 B -2 0 -6 18 -6 C 10 6 0 6 -2 D -6 -18 -6 0 -6 E 12 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 6 -12 B -2 0 -6 18 -6 C 10 6 0 6 -2 D -6 -18 -6 0 -6 E 12 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998545 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7869: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (5) C B E A D (5) C A B D E (5) D A C B E (4) C E B D A (4) A D B C E (4) A B E D C (4) E B C D A (3) C D E B A (3) C A D B E (3) A D C B E (3) A B D E C (3) E C D B A (2) E B D A C (2) E B A D C (2) D E A C B (2) D C E A B (2) D A E C B (2) D A E B C (2) C E D B A (2) C D E A B (2) C D A E B (2) B E A C D (2) B C E A D (2) A C B D E (2) A B E C D (2) A B C E D (2) A B C D E (2) E D B C A (1) E D B A C (1) E C B A D (1) E B C A D (1) E B A C D (1) D E C B A (1) D E A B C (1) D A C E B (1) D A B E C (1) C D B E A (1) C D A B E (1) C B A E D (1) B E C A D (1) B C A E D (1) B A E D C (1) B A E C D (1) B A C E D (1) A E D B C (1) A E B D C (1) A D B E C (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 20 2 10 16 B -20 0 -12 2 10 C -2 12 0 10 20 D -10 -2 -10 0 8 E -16 -10 -20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999659 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 2 10 16 B -20 0 -12 2 10 C -2 12 0 10 20 D -10 -2 -10 0 8 E -16 -10 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=27 D=21 E=14 B=9 so B is eliminated. Round 2 votes counts: C=32 A=30 D=21 E=17 so E is eliminated. Round 3 votes counts: C=40 A=35 D=25 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:220 D:193 B:190 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 2 10 16 B -20 0 -12 2 10 C -2 12 0 10 20 D -10 -2 -10 0 8 E -16 -10 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 2 10 16 B -20 0 -12 2 10 C -2 12 0 10 20 D -10 -2 -10 0 8 E -16 -10 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 2 10 16 B -20 0 -12 2 10 C -2 12 0 10 20 D -10 -2 -10 0 8 E -16 -10 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992135 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7870: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) B C D A E (8) B C A D E (7) E A D C B (5) D B E A C (5) E A C D B (4) D E A C B (4) D B C E A (4) C A E B D (4) B D C E A (4) C B A D E (3) B A C E D (3) A E C D B (3) A E B C D (3) E D A C B (2) E C A D B (2) D E C A B (2) D C E B A (2) C E A D B (2) C A E D B (2) C A B E D (2) B D C A E (2) B C A E D (2) B A D E C (2) E D C A B (1) E A D B C (1) D E B A C (1) D C E A B (1) D B E C A (1) C E D A B (1) C B D A E (1) B D E C A (1) B D E A C (1) B D A E C (1) B D A C E (1) A E C B D (1) A E B D C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -20 14 12 B 14 0 -2 14 14 C 20 2 0 14 16 D -14 -14 -14 0 2 E -12 -14 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 14 12 B 14 0 -2 14 14 C 20 2 0 14 16 D -14 -14 -14 0 2 E -12 -14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999127 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=23 D=20 E=15 A=10 so A is eliminated. Round 2 votes counts: B=33 C=24 E=23 D=20 so D is eliminated. Round 3 votes counts: B=43 E=30 C=27 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:226 B:220 A:196 D:180 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -20 14 12 B 14 0 -2 14 14 C 20 2 0 14 16 D -14 -14 -14 0 2 E -12 -14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999127 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 14 12 B 14 0 -2 14 14 C 20 2 0 14 16 D -14 -14 -14 0 2 E -12 -14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999127 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 14 12 B 14 0 -2 14 14 C 20 2 0 14 16 D -14 -14 -14 0 2 E -12 -14 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999127 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7871: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) E D A C B (7) C B A E D (5) B C A D E (5) E D C A B (4) E D B C A (4) E D A B C (4) C E B D A (4) B A D C E (4) E D C B A (3) E D B A C (3) D E A B C (3) D B A E C (3) C B A D E (3) B D A E C (3) E C D A B (2) E C B D A (2) E C A D B (2) D E B A C (2) D B E A C (2) D A E B C (2) C E B A D (2) C B E A D (2) B A C D E (2) A D B E C (2) E C D B A (1) E A D C B (1) D A B E C (1) C E A D B (1) C E A B D (1) C B E D A (1) C A E D B (1) B D E A C (1) B D A C E (1) B C E D A (1) A D E C B (1) A D E B C (1) A D C E B (1) A C E D B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -6 -8 -6 B 8 0 -8 -4 -6 C 6 8 0 -8 -8 D 8 4 8 0 -4 E 6 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 -8 -6 B 8 0 -8 -4 -6 C 6 8 0 -8 -8 D 8 4 8 0 -4 E 6 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=29 B=17 D=13 A=8 so A is eliminated. Round 2 votes counts: E=33 C=30 B=19 D=18 so D is eliminated. Round 3 votes counts: E=42 C=31 B=27 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 D:208 C:199 B:195 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -8 -6 B 8 0 -8 -4 -6 C 6 8 0 -8 -8 D 8 4 8 0 -4 E 6 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -8 -6 B 8 0 -8 -4 -6 C 6 8 0 -8 -8 D 8 4 8 0 -4 E 6 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -8 -6 B 8 0 -8 -4 -6 C 6 8 0 -8 -8 D 8 4 8 0 -4 E 6 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7872: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (8) E A D B C (7) E B A D C (5) E B A C D (5) D E A B C (5) E A B C D (4) C B D A E (4) C B A E D (4) D C B A E (3) D C A B E (3) D A C E B (3) C B E A D (3) A E D C B (3) E A D C B (2) E A C B D (2) E A B D C (2) D E B A C (2) D A E C B (2) C D A B E (2) C A E B D (2) C A D E B (2) B E A C D (2) B C D E A (2) E B C A D (1) D C A E B (1) D B E A C (1) D B C A E (1) D A E B C (1) D A B E C (1) C E B A D (1) C D B A E (1) C B A D E (1) C A D B E (1) C A B E D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E C A (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D A E (1) A E C D B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 4 20 -18 B 6 0 12 12 -6 C -4 -12 0 2 -4 D -20 -12 -2 0 -18 E 18 6 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 4 20 -18 B 6 0 12 12 -6 C -4 -12 0 2 -4 D -20 -12 -2 0 -18 E 18 6 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 C=22 B=21 A=6 so A is eliminated. Round 2 votes counts: E=32 D=24 C=23 B=21 so B is eliminated. Round 3 votes counts: E=38 C=35 D=27 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:212 A:200 C:191 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 4 20 -18 B 6 0 12 12 -6 C -4 -12 0 2 -4 D -20 -12 -2 0 -18 E 18 6 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 20 -18 B 6 0 12 12 -6 C -4 -12 0 2 -4 D -20 -12 -2 0 -18 E 18 6 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 20 -18 B 6 0 12 12 -6 C -4 -12 0 2 -4 D -20 -12 -2 0 -18 E 18 6 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999606 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7873: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) E D B A C (6) E C D A B (6) B A D E C (6) E C B A D (5) D A B E C (5) D A B C E (5) C D E A B (5) E C D B A (4) B A C D E (4) A B D C E (4) E B A D C (3) E B A C D (3) D E C A B (3) D C A B E (3) C E B A D (3) C A B D E (3) B A D C E (3) E D A B C (2) D E A B C (2) C E D A B (2) C D A B E (2) C B A E D (2) C A B E D (2) B A E D C (2) E D C B A (1) C E A B D (1) C B E A D (1) B A E C D (1) B A C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -4 -10 -12 B -10 0 -4 -10 -10 C 4 4 0 -8 -16 D 10 10 8 0 -8 E 12 10 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -4 -10 -12 B -10 0 -4 -10 -10 C 4 4 0 -8 -16 D 10 10 8 0 -8 E 12 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=21 D=18 B=17 A=5 so A is eliminated. Round 2 votes counts: E=39 B=22 C=21 D=18 so D is eliminated. Round 3 votes counts: E=44 B=32 C=24 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:210 A:192 C:192 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -4 -10 -12 B -10 0 -4 -10 -10 C 4 4 0 -8 -16 D 10 10 8 0 -8 E 12 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -10 -12 B -10 0 -4 -10 -10 C 4 4 0 -8 -16 D 10 10 8 0 -8 E 12 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -10 -12 B -10 0 -4 -10 -10 C 4 4 0 -8 -16 D 10 10 8 0 -8 E 12 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7874: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (20) B E C D A (6) E C D B A (5) E C B D A (5) D A C E B (5) B E C A D (5) A B D C E (5) E B C D A (4) D C E A B (4) B E A C D (4) B A E C D (4) A B E C D (4) D C E B A (3) D C A E B (3) C E D B A (3) A D B C E (3) A B D E C (3) C D E B A (2) B D C E A (2) B A D E C (2) A E C D B (2) A D C B E (2) C E D A B (1) C D E A B (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 12 12 10 B -8 0 -12 -8 -16 C -12 12 0 -8 10 D -12 8 8 0 12 E -10 16 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 12 10 B -8 0 -12 -8 -16 C -12 12 0 -8 10 D -12 8 8 0 12 E -10 16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=24 D=15 E=14 C=7 so C is eliminated. Round 2 votes counts: A=40 B=24 E=18 D=18 so E is eliminated. Round 3 votes counts: A=40 B=33 D=27 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:208 C:201 E:192 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 12 10 B -8 0 -12 -8 -16 C -12 12 0 -8 10 D -12 8 8 0 12 E -10 16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 12 10 B -8 0 -12 -8 -16 C -12 12 0 -8 10 D -12 8 8 0 12 E -10 16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 12 10 B -8 0 -12 -8 -16 C -12 12 0 -8 10 D -12 8 8 0 12 E -10 16 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999794 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7875: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) D B A E C (8) A E C D B (8) E A C B D (7) D B C A E (7) C E A B D (7) A E C B D (7) C B E A D (5) D B C E A (4) E C A B D (3) C B D A E (3) B C D E A (3) D B E A C (2) D B A C E (2) D A E B C (2) D A B E C (2) C A E B D (2) B D E A C (2) A E D C B (2) A E D B C (2) E B A D C (1) E A B C D (1) D C B A E (1) D A C B E (1) C E B A D (1) C D B A E (1) B D E C A (1) B D C A E (1) A D E B C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 0 -2 4 B 6 0 0 12 8 C 0 0 0 -2 2 D 2 -12 2 0 6 E -4 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.443137 C: 0.556863 D: 0.000000 E: 0.000000 Sum of squares = 0.506466804898 Cumulative probabilities = A: 0.000000 B: 0.443137 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 4 B 6 0 0 12 8 C 0 0 0 -2 2 D 2 -12 2 0 6 E -4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=22 C=19 B=18 E=12 so E is eliminated. Round 2 votes counts: A=30 D=29 C=22 B=19 so B is eliminated. Round 3 votes counts: D=44 A=31 C=25 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:213 C:200 D:199 A:198 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 -2 4 B 6 0 0 12 8 C 0 0 0 -2 2 D 2 -12 2 0 6 E -4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 4 B 6 0 0 12 8 C 0 0 0 -2 2 D 2 -12 2 0 6 E -4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 4 B 6 0 0 12 8 C 0 0 0 -2 2 D 2 -12 2 0 6 E -4 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7876: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (12) E B C A D (9) B E A C D (9) C A E B D (7) E C B A D (6) D B A E C (5) D A B C E (4) D C A E B (3) C E A D B (3) C E A B D (3) C A D E B (3) E B A C D (2) D A B E C (2) C D A E B (2) C A E D B (2) B E A D C (2) A D C B E (2) A D B C E (2) A C E B D (2) E B D C A (1) E B C D A (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A B E (1) D B E A C (1) D B A C E (1) D A C E B (1) C E D A B (1) C E B A D (1) C D E B A (1) C D E A B (1) B E D C A (1) B E D A C (1) B E C A D (1) B D E A C (1) B A E C D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 8 0 14 4 B -8 0 -10 -2 -6 C 0 10 0 14 12 D -14 2 -14 0 -8 E -4 6 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.518987 B: 0.000000 C: 0.481013 D: 0.000000 E: 0.000000 Sum of squares = 0.50072103937 Cumulative probabilities = A: 0.518987 B: 0.518987 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 14 4 B -8 0 -10 -2 -6 C 0 10 0 14 12 D -14 2 -14 0 -8 E -4 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=24 E=19 B=16 A=8 so A is eliminated. Round 2 votes counts: D=37 C=27 E=19 B=17 so B is eliminated. Round 3 votes counts: D=38 E=34 C=28 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:218 A:213 E:199 B:187 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 14 4 B -8 0 -10 -2 -6 C 0 10 0 14 12 D -14 2 -14 0 -8 E -4 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 14 4 B -8 0 -10 -2 -6 C 0 10 0 14 12 D -14 2 -14 0 -8 E -4 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 14 4 B -8 0 -10 -2 -6 C 0 10 0 14 12 D -14 2 -14 0 -8 E -4 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7877: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) D C A E B (8) D C B E A (7) D C B A E (6) A E B C D (6) B E A C D (5) D C A B E (4) D B E A C (4) D B C E A (4) C D A E B (4) C D A B E (4) C A E D B (4) B E D A C (4) C A D E B (3) E A B D C (2) D C E A B (2) B E C A D (2) A E C D B (2) A D E C B (2) A C E B D (2) E B A D C (1) E A B C D (1) D E B A C (1) D E A B C (1) D A E C B (1) D A C E B (1) C D B E A (1) C D B A E (1) C B D A E (1) C A E B D (1) B E D C A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D E A (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 -12 -18 2 B 2 0 -10 -20 14 C 12 10 0 -22 10 D 18 20 22 0 18 E -2 -14 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -18 2 B 2 0 -10 -20 14 C 12 10 0 -22 10 D 18 20 22 0 18 E -2 -14 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=25 C=19 A=13 E=4 so E is eliminated. Round 2 votes counts: D=39 B=26 C=19 A=16 so A is eliminated. Round 3 votes counts: D=41 B=35 C=24 so C is eliminated. Round 4 votes counts: D=60 B=40 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:239 C:205 B:193 A:185 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 -18 2 B 2 0 -10 -20 14 C 12 10 0 -22 10 D 18 20 22 0 18 E -2 -14 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -18 2 B 2 0 -10 -20 14 C 12 10 0 -22 10 D 18 20 22 0 18 E -2 -14 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -18 2 B 2 0 -10 -20 14 C 12 10 0 -22 10 D 18 20 22 0 18 E -2 -14 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7878: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) D A B C E (6) E B C A D (5) E A B C D (5) D A C B E (5) E C B A D (4) D C A B E (4) C D B E A (4) B C E D A (4) A E D C B (4) E B A C D (3) E A D C B (3) D C B A E (3) B E C A D (3) B E A C D (3) B C D E A (3) A E B D C (3) A D E C B (3) E C B D A (2) C E B D A (2) C B E D A (2) C B D E A (2) A E D B C (2) A D B E C (2) A B E D C (2) A B D E C (2) E C D B A (1) D C A E B (1) D B A C E (1) D A C E B (1) C E D B A (1) C D E B A (1) C D E A B (1) B C E A D (1) B A E C D (1) B A D C E (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 12 12 0 B -6 0 12 0 0 C -12 -12 0 -2 -12 D -12 0 2 0 -4 E 0 0 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.492326 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.507674 Sum of squares = 0.500117795337 Cumulative probabilities = A: 0.492326 B: 0.492326 C: 0.492326 D: 0.492326 E: 1.000000 A B C D E A 0 6 12 12 0 B -6 0 12 0 0 C -12 -12 0 -2 -12 D -12 0 2 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=23 D=21 B=16 C=13 so C is eliminated. Round 2 votes counts: D=27 A=27 E=26 B=20 so B is eliminated. Round 3 votes counts: E=39 D=32 A=29 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:215 E:208 B:203 D:193 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 12 0 B -6 0 12 0 0 C -12 -12 0 -2 -12 D -12 0 2 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 12 0 B -6 0 12 0 0 C -12 -12 0 -2 -12 D -12 0 2 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 12 0 B -6 0 12 0 0 C -12 -12 0 -2 -12 D -12 0 2 0 -4 E 0 0 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7879: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (9) D C E A B (7) E D A C B (6) E A D B C (6) E A B C D (6) A B E C D (6) D C E B A (5) D E C A B (4) C B D A E (4) D C B E A (3) D C B A E (3) C D B E A (3) A E B D C (3) A E B C D (3) D E A C B (2) C D E B A (2) B C A E D (2) B C A D E (2) B A E C D (2) B A C E D (2) B A C D E (2) A B E D C (2) E C D B A (1) E A D C B (1) D C A B E (1) D A E B C (1) D A B C E (1) C B E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 16 4 -6 -12 B -16 0 -4 -10 -12 C -4 4 0 -8 -4 D 6 10 8 0 0 E 12 12 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.467063 E: 0.532937 Sum of squares = 0.502169663687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.467063 E: 1.000000 A B C D E A 0 16 4 -6 -12 B -16 0 -4 -10 -12 C -4 4 0 -8 -4 D 6 10 8 0 0 E 12 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=27 C=19 A=14 B=11 so B is eliminated. Round 2 votes counts: E=29 D=27 C=24 A=20 so A is eliminated. Round 3 votes counts: E=45 C=28 D=27 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:212 A:201 C:194 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 16 4 -6 -12 B -16 0 -4 -10 -12 C -4 4 0 -8 -4 D 6 10 8 0 0 E 12 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 -6 -12 B -16 0 -4 -10 -12 C -4 4 0 -8 -4 D 6 10 8 0 0 E 12 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 -6 -12 B -16 0 -4 -10 -12 C -4 4 0 -8 -4 D 6 10 8 0 0 E 12 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7880: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) B C D E A (7) E A C B D (6) C E B A D (6) C B E D A (6) D B A C E (5) C B D A E (5) A D E B C (4) E B C A D (3) E A D B C (3) E A C D B (3) D C B A E (3) D A C B E (3) C B D E A (3) A E D C B (3) A E D B C (3) E C B A D (2) E A D C B (2) D A B E C (2) D A B C E (2) C B E A D (2) B D C A E (2) B C D A E (2) A D E C B (2) E A B D C (1) E A B C D (1) D B A E C (1) B E C D A (1) B D C E A (1) B C E D A (1) A E C D B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -24 -12 -12 4 B 24 0 2 0 16 C 12 -2 0 0 24 D 12 0 0 0 10 E -4 -16 -24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.367764 C: 0.000000 D: 0.632236 E: 0.000000 Sum of squares = 0.534972804986 Cumulative probabilities = A: 0.000000 B: 0.367764 C: 0.367764 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -12 -12 4 B 24 0 2 0 16 C 12 -2 0 0 24 D 12 0 0 0 10 E -4 -16 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=22 E=21 A=15 B=14 so B is eliminated. Round 2 votes counts: C=32 D=31 E=22 A=15 so A is eliminated. Round 3 votes counts: D=38 C=33 E=29 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:217 D:211 A:178 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -12 -12 4 B 24 0 2 0 16 C 12 -2 0 0 24 D 12 0 0 0 10 E -4 -16 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -12 -12 4 B 24 0 2 0 16 C 12 -2 0 0 24 D 12 0 0 0 10 E -4 -16 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -12 -12 4 B 24 0 2 0 16 C 12 -2 0 0 24 D 12 0 0 0 10 E -4 -16 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7881: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (13) B D C E A (11) C E B A D (10) B C D E A (8) A E D C B (6) D B A E C (5) C B E D A (5) B D C A E (5) E C A B D (4) C E A B D (4) A D E B C (4) A C E B D (4) D B A C E (3) C E B D A (3) A E D B C (3) E A C D B (2) D B C A E (2) D A B E C (2) E D C B A (1) E C B D A (1) E A C B D (1) C A B E D (1) A E C B D (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -10 8 0 B 8 0 -12 16 -14 C 10 12 0 14 12 D -8 -16 -14 0 -18 E 0 14 -12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 8 0 B 8 0 -12 16 -14 C 10 12 0 14 12 D -8 -16 -14 0 -18 E 0 14 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=24 C=23 D=12 E=9 so E is eliminated. Round 2 votes counts: A=35 C=28 B=24 D=13 so D is eliminated. Round 3 votes counts: A=37 B=34 C=29 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:224 E:210 B:199 A:195 D:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 8 0 B 8 0 -12 16 -14 C 10 12 0 14 12 D -8 -16 -14 0 -18 E 0 14 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 8 0 B 8 0 -12 16 -14 C 10 12 0 14 12 D -8 -16 -14 0 -18 E 0 14 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 8 0 B 8 0 -12 16 -14 C 10 12 0 14 12 D -8 -16 -14 0 -18 E 0 14 -12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7882: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (12) C E A B D (9) E C A B D (6) A E B D C (6) A C E D B (5) C A E B D (4) A D B E C (4) A D B C E (4) D B A E C (3) D B A C E (3) B D E C A (3) B D E A C (3) A E C B D (3) D C B E A (2) C E B A D (2) C A E D B (2) B E D C A (2) B E D A C (2) A E D B C (2) A D C B E (2) E C B A D (1) E A B C D (1) D C B A E (1) D B E A C (1) D B C A E (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E B A (1) C D B A E (1) C D A E B (1) C B E D A (1) C A D B E (1) B E C D A (1) B D A E C (1) B A E D C (1) A E B C D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 14 -8 14 0 B -14 0 8 -2 4 C 8 -8 0 -12 14 D -14 2 12 0 -6 E 0 -4 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.352941 B: 0.000000 C: 0.411765 D: 0.235294 E: 0.000000 Sum of squares = 0.349480968845 Cumulative probabilities = A: 0.352941 B: 0.352941 C: 0.764706 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 14 0 B -14 0 8 -2 4 C 8 -8 0 -12 14 D -14 2 12 0 -6 E 0 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.352941 B: 0.000000 C: 0.411765 D: 0.235294 E: 0.000000 Sum of squares = 0.349480968865 Cumulative probabilities = A: 0.352941 B: 0.352941 C: 0.764706 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=25 C=25 B=13 E=8 so E is eliminated. Round 2 votes counts: C=32 A=30 D=25 B=13 so B is eliminated. Round 3 votes counts: D=36 C=33 A=31 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:210 C:201 B:198 D:197 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 -8 14 0 B -14 0 8 -2 4 C 8 -8 0 -12 14 D -14 2 12 0 -6 E 0 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.352941 B: 0.000000 C: 0.411765 D: 0.235294 E: 0.000000 Sum of squares = 0.349480968865 Cumulative probabilities = A: 0.352941 B: 0.352941 C: 0.764706 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 14 0 B -14 0 8 -2 4 C 8 -8 0 -12 14 D -14 2 12 0 -6 E 0 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.352941 B: 0.000000 C: 0.411765 D: 0.235294 E: 0.000000 Sum of squares = 0.349480968865 Cumulative probabilities = A: 0.352941 B: 0.352941 C: 0.764706 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 14 0 B -14 0 8 -2 4 C 8 -8 0 -12 14 D -14 2 12 0 -6 E 0 -4 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.352941 B: 0.000000 C: 0.411765 D: 0.235294 E: 0.000000 Sum of squares = 0.349480968865 Cumulative probabilities = A: 0.352941 B: 0.352941 C: 0.764706 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7883: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (13) E C B D A (10) D A B C E (7) D B A C E (6) A D B C E (6) C B E D A (5) A D B E C (5) B C E D A (4) A D E C B (4) E C A B D (3) D B C E A (3) D A E B C (3) A E D C B (3) E A C D B (2) D A B E C (2) C E B D A (2) B D C A E (2) B C D A E (2) B C A D E (2) A D E B C (2) E C D B A (1) E A D C B (1) E A C B D (1) D E A C B (1) D B E C A (1) D B C A E (1) D A E C B (1) C E B A D (1) B D C E A (1) A E C D B (1) A E C B D (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -4 2 B 8 0 -2 0 0 C 2 2 0 0 -10 D 4 0 0 0 4 E -2 0 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.184629 D: 0.815371 E: 0.000000 Sum of squares = 0.698918192008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.184629 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -4 2 B 8 0 -2 0 0 C 2 2 0 0 -10 D 4 0 0 0 4 E -2 0 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 0.000000 Sum of squares = 0.591836865623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=25 A=25 B=11 C=8 so C is eliminated. Round 2 votes counts: E=34 D=25 A=25 B=16 so B is eliminated. Round 3 votes counts: E=43 D=30 A=27 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:204 B:203 E:202 C:197 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 2 B 8 0 -2 0 0 C 2 2 0 0 -10 D 4 0 0 0 4 E -2 0 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 0.000000 Sum of squares = 0.591836865623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 2 B 8 0 -2 0 0 C 2 2 0 0 -10 D 4 0 0 0 4 E -2 0 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 0.000000 Sum of squares = 0.591836865623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 2 B 8 0 -2 0 0 C 2 2 0 0 -10 D 4 0 0 0 4 E -2 0 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 0.000000 Sum of squares = 0.591836865623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7884: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) E D B C A (9) E C B A D (7) C A B E D (4) B C E D A (4) A D B C E (4) E C B D A (3) D B C A E (3) C B E A D (3) C B A D E (3) B E C D A (3) B D C E A (3) A D E C B (3) A D C B E (3) A C E B D (3) E D A B C (2) E C A B D (2) E A D C B (2) E A C B D (2) D B C E A (2) D B A C E (2) C B A E D (2) B C D E A (2) A E C B D (2) A C D B E (2) A C B D E (2) E B D C A (1) E B C D A (1) D E A B C (1) D B E C A (1) D B E A C (1) D A B E C (1) C E B A D (1) C B E D A (1) C A E B D (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 -14 -2 -2 B 6 0 4 2 14 C 14 -4 0 0 12 D 2 -2 0 0 -10 E 2 -14 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -2 -2 B 6 0 4 2 14 C 14 -4 0 0 12 D 2 -2 0 0 -10 E 2 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993457 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 D=21 C=15 B=13 so B is eliminated. Round 2 votes counts: E=32 D=24 C=22 A=22 so C is eliminated. Round 3 votes counts: E=41 A=32 D=27 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:213 C:211 D:195 E:193 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 -2 -2 B 6 0 4 2 14 C 14 -4 0 0 12 D 2 -2 0 0 -10 E 2 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993457 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -2 -2 B 6 0 4 2 14 C 14 -4 0 0 12 D 2 -2 0 0 -10 E 2 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993457 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -2 -2 B 6 0 4 2 14 C 14 -4 0 0 12 D 2 -2 0 0 -10 E 2 -14 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993457 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7885: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (14) A E C B D (13) D B E C A (10) A C E B D (10) C A E B D (7) D B E A C (6) D B A E C (4) E C B A D (3) C B E D A (3) A D E B C (3) E A C B D (2) D C B E A (2) D B C A E (2) C B E A D (2) B D E C A (2) B C D E A (2) A E D B C (2) E A B C D (1) D B A C E (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) B D C E A (1) A E C D B (1) A E B D C (1) A E B C D (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 0 2 0 B 6 0 4 0 2 C 0 -4 0 -2 -2 D -2 0 2 0 2 E 0 -2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.478987 C: 0.000000 D: 0.521013 E: 0.000000 Sum of squares = 0.500883103966 Cumulative probabilities = A: 0.000000 B: 0.478987 C: 0.478987 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 2 0 B 6 0 4 0 2 C 0 -4 0 -2 -2 D -2 0 2 0 2 E 0 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=34 C=14 E=6 B=5 so B is eliminated. Round 2 votes counts: D=44 A=34 C=16 E=6 so E is eliminated. Round 3 votes counts: D=44 A=37 C=19 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:206 D:201 E:199 A:198 C:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 2 0 B 6 0 4 0 2 C 0 -4 0 -2 -2 D -2 0 2 0 2 E 0 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 2 0 B 6 0 4 0 2 C 0 -4 0 -2 -2 D -2 0 2 0 2 E 0 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 2 0 B 6 0 4 0 2 C 0 -4 0 -2 -2 D -2 0 2 0 2 E 0 -2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7886: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) B C E D A (8) C B A D E (7) B C A E D (6) A D E C B (6) A C B D E (6) D E C B A (5) A E D B C (5) D A E C B (4) B C E A D (4) E D B C A (3) B E C D A (3) A D C E B (3) E D C B A (2) C B D E A (2) C B D A E (2) C A D B E (2) B E A C D (2) E D A C B (1) E B D C A (1) E B C D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C B E A (1) D C B A E (1) D C A E B (1) D C A B E (1) C D B A E (1) B E C A D (1) B A E C D (1) B A C E D (1) A D E B C (1) A D C B E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 0 2 B 4 0 4 -4 4 C 6 -4 0 -2 -2 D 0 4 2 0 -6 E -2 -4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 A B C D E A 0 -4 -6 0 2 B 4 0 4 -4 4 C 6 -4 0 -2 -2 D 0 4 2 0 -6 E -2 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775457 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 E=21 D=15 C=14 so C is eliminated. Round 2 votes counts: B=37 A=26 E=21 D=16 so D is eliminated. Round 3 votes counts: B=40 A=32 E=28 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:204 E:201 D:200 C:199 A:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 0 2 B 4 0 4 -4 4 C 6 -4 0 -2 -2 D 0 4 2 0 -6 E -2 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775457 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 2 B 4 0 4 -4 4 C 6 -4 0 -2 -2 D 0 4 2 0 -6 E -2 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775457 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 2 B 4 0 4 -4 4 C 6 -4 0 -2 -2 D 0 4 2 0 -6 E -2 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775457 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7887: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) A C D E B (6) E B A D C (5) D A C E B (5) B E C D A (5) A D C E B (5) E B D A C (4) C D A B E (4) B E A C D (4) E B A C D (3) D E A C B (3) D C A E B (3) D C A B E (3) C D B A E (3) B C A E D (3) A E D C B (3) E D B A C (2) E A D B C (2) E A B D C (2) C A D B E (2) B E A D C (2) B D C E A (2) A C D B E (2) E D A B C (1) D E B C A (1) D C E B A (1) D C B A E (1) D B C E A (1) D A E C B (1) C D B E A (1) C B D E A (1) C B D A E (1) C A B E D (1) C A B D E (1) B E D A C (1) B E C A D (1) B C D E A (1) B C A D E (1) A E C D B (1) A E B C D (1) A D E C B (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 14 -2 2 B 0 0 -4 -6 -2 C -14 4 0 -10 2 D 2 6 10 0 2 E -2 2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 -2 2 B 0 0 -4 -6 -2 C -14 4 0 -10 2 D 2 6 10 0 2 E -2 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=22 E=19 D=19 C=14 so C is eliminated. Round 2 votes counts: B=28 D=27 A=26 E=19 so E is eliminated. Round 3 votes counts: B=40 D=30 A=30 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:207 E:198 B:194 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 14 -2 2 B 0 0 -4 -6 -2 C -14 4 0 -10 2 D 2 6 10 0 2 E -2 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 -2 2 B 0 0 -4 -6 -2 C -14 4 0 -10 2 D 2 6 10 0 2 E -2 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 -2 2 B 0 0 -4 -6 -2 C -14 4 0 -10 2 D 2 6 10 0 2 E -2 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7888: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) C B D E A (7) A D B E C (6) E A B D C (5) C E B D A (5) E C B A D (4) A D B C E (4) E C B D A (3) E C A B D (3) D B C A E (3) D A B C E (3) C E A B D (3) C D B E A (3) A D E B C (3) E B C D A (2) E A C D B (2) E A C B D (2) E A B C D (2) D C A B E (2) D B A C E (2) C B D A E (2) B D C A E (2) B D A E C (2) A B D E C (2) E B D A C (1) E B A D C (1) D C B A E (1) D B A E C (1) D A C B E (1) D A B E C (1) C E D A B (1) C E A D B (1) C D B A E (1) C A E D B (1) C A D E B (1) B E D A C (1) B D E A C (1) B D C E A (1) B C E D A (1) A E D C B (1) A D E C B (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 12 6 6 2 B -12 0 6 0 -4 C -6 -6 0 -12 -6 D -6 0 12 0 4 E -2 4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 6 2 B -12 0 6 0 -4 C -6 -6 0 -12 -6 D -6 0 12 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=25 C=25 D=14 B=8 so B is eliminated. Round 2 votes counts: A=28 E=26 C=26 D=20 so D is eliminated. Round 3 votes counts: A=38 C=35 E=27 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:205 E:202 B:195 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 6 2 B -12 0 6 0 -4 C -6 -6 0 -12 -6 D -6 0 12 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 6 2 B -12 0 6 0 -4 C -6 -6 0 -12 -6 D -6 0 12 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 6 2 B -12 0 6 0 -4 C -6 -6 0 -12 -6 D -6 0 12 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999688 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7889: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (14) D A C B E (9) C E A D B (9) B E D A C (9) B D A E C (9) D A B C E (7) C A D E B (7) E C B A D (6) E B C A D (6) B D E A C (6) A D C E B (4) D B A E C (3) B E D C A (3) C A E D B (2) B E C A D (2) E C B D A (1) E B C D A (1) C E A B D (1) A D C B E (1) Total count = 100 A B C D E A 0 -20 -4 -24 -16 B 20 0 20 16 26 C 4 -20 0 -2 -20 D 24 -16 2 0 -8 E 16 -26 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -24 -16 B 20 0 20 16 26 C 4 -20 0 -2 -20 D 24 -16 2 0 -8 E 16 -26 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 D=19 C=19 E=14 A=5 so A is eliminated. Round 2 votes counts: B=43 D=24 C=19 E=14 so E is eliminated. Round 3 votes counts: B=50 C=26 D=24 so D is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:241 E:209 D:201 C:181 A:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -4 -24 -16 B 20 0 20 16 26 C 4 -20 0 -2 -20 D 24 -16 2 0 -8 E 16 -26 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -24 -16 B 20 0 20 16 26 C 4 -20 0 -2 -20 D 24 -16 2 0 -8 E 16 -26 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -24 -16 B 20 0 20 16 26 C 4 -20 0 -2 -20 D 24 -16 2 0 -8 E 16 -26 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7890: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) E B A D C (6) E D B A C (5) E B C A D (5) E B D A C (4) D C A E B (4) C D E A B (4) B E A C D (4) A B D E C (4) E C B D A (3) E B D C A (3) D A B E C (3) C D A E B (3) B E A D C (3) A D B E C (3) E D C B A (2) E D B C A (2) E B A C D (2) D E C A B (2) D C A B E (2) C E B D A (2) C D A B E (2) C B E A D (2) C A B D E (2) B C E A D (2) B A E C D (2) A D B C E (2) E C B A D (1) E B C D A (1) D E A B C (1) D A E B C (1) D A C B E (1) D A B C E (1) C E D B A (1) C E B A D (1) C A D B E (1) B E C A D (1) B C A E D (1) B A C E D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -24 6 6 -14 B 24 0 32 18 -6 C -6 -32 0 -14 -32 D -6 -18 14 0 -24 E 14 6 32 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 6 6 -14 B 24 0 32 18 -6 C -6 -32 0 -14 -32 D -6 -18 14 0 -24 E 14 6 32 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=22 C=18 D=15 A=11 so A is eliminated. Round 2 votes counts: E=34 B=27 D=20 C=19 so C is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:238 B:234 A:187 D:183 C:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 6 6 -14 B 24 0 32 18 -6 C -6 -32 0 -14 -32 D -6 -18 14 0 -24 E 14 6 32 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 6 6 -14 B 24 0 32 18 -6 C -6 -32 0 -14 -32 D -6 -18 14 0 -24 E 14 6 32 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 6 6 -14 B 24 0 32 18 -6 C -6 -32 0 -14 -32 D -6 -18 14 0 -24 E 14 6 32 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7891: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (15) A E D B C (14) A E D C B (7) E D B A C (6) D E B C A (6) B C D E A (6) E D A B C (5) B D C E A (5) D E B A C (4) C B A D E (4) E A D B C (3) C A B E D (3) B D E C A (3) A C E B D (3) E D B C A (2) D B E C A (2) C B D A E (2) C A B D E (2) A E C D B (2) A C E D B (2) B C D A E (1) B C A D E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 -4 -14 -14 B 14 0 16 -6 -8 C 4 -16 0 -14 -8 D 14 6 14 0 4 E 14 8 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 -14 -14 B 14 0 16 -6 -8 C 4 -16 0 -14 -8 D 14 6 14 0 4 E 14 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 E=16 B=16 D=12 so D is eliminated. Round 2 votes counts: A=30 E=26 C=26 B=18 so B is eliminated. Round 3 votes counts: C=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:219 E:213 B:208 C:183 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -4 -14 -14 B 14 0 16 -6 -8 C 4 -16 0 -14 -8 D 14 6 14 0 4 E 14 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -14 -14 B 14 0 16 -6 -8 C 4 -16 0 -14 -8 D 14 6 14 0 4 E 14 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -14 -14 B 14 0 16 -6 -8 C 4 -16 0 -14 -8 D 14 6 14 0 4 E 14 8 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7892: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (20) A C D E B (12) D A C B E (11) D B E A C (8) C A E B D (7) E B C A D (5) B E C A D (5) D B E C A (4) D A C E B (4) A C E B D (4) D C A E B (2) C A E D B (2) B E A C D (2) B D E C A (2) E C A B D (1) E B C D A (1) E B A C D (1) D C E B A (1) D B A E C (1) D A B C E (1) C E B A D (1) C A D E B (1) B E D A C (1) B E C D A (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -6 -14 -6 B 6 0 4 2 14 C 6 -4 0 -14 -4 D 14 -2 14 0 -2 E 6 -14 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -14 -6 B 6 0 4 2 14 C 6 -4 0 -14 -4 D 14 -2 14 0 -2 E 6 -14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999449 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=31 A=18 C=11 E=8 so E is eliminated. Round 2 votes counts: B=38 D=32 A=18 C=12 so C is eliminated. Round 3 votes counts: B=39 D=32 A=29 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:212 E:199 C:192 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -14 -6 B 6 0 4 2 14 C 6 -4 0 -14 -4 D 14 -2 14 0 -2 E 6 -14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999449 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -14 -6 B 6 0 4 2 14 C 6 -4 0 -14 -4 D 14 -2 14 0 -2 E 6 -14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999449 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -14 -6 B 6 0 4 2 14 C 6 -4 0 -14 -4 D 14 -2 14 0 -2 E 6 -14 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999449 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7893: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (18) D C B A E (13) B C A E D (12) D E A C B (7) C B A E D (7) D E A B C (6) E A D C B (3) D E C A B (3) D B C A E (3) D E B C A (2) D C E B A (2) C B D A E (2) C B A D E (2) B C D A E (2) A E C B D (2) E D A C B (1) E D A B C (1) E B A C D (1) E A D B C (1) E A B D C (1) D E B A C (1) D C E A B (1) D C A B E (1) D B C E A (1) C A B E D (1) B E A C D (1) B C A D E (1) B A E C D (1) A E B C D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -6 8 0 B 2 0 10 8 -2 C 6 -10 0 6 -2 D -8 -8 -6 0 -6 E 0 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.096628 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.903372 Sum of squares = 0.825417816715 Cumulative probabilities = A: 0.096628 B: 0.096628 C: 0.096628 D: 0.096628 E: 1.000000 A B C D E A 0 -2 -6 8 0 B 2 0 10 8 -2 C 6 -10 0 6 -2 D -8 -8 -6 0 -6 E 0 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000011376 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=26 B=17 C=12 A=5 so A is eliminated. Round 2 votes counts: D=40 E=29 B=19 C=12 so C is eliminated. Round 3 votes counts: D=40 B=31 E=29 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 E:205 A:200 C:200 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 8 0 B 2 0 10 8 -2 C 6 -10 0 6 -2 D -8 -8 -6 0 -6 E 0 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000011376 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 8 0 B 2 0 10 8 -2 C 6 -10 0 6 -2 D -8 -8 -6 0 -6 E 0 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000011376 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 8 0 B 2 0 10 8 -2 C 6 -10 0 6 -2 D -8 -8 -6 0 -6 E 0 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000011376 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7894: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) B C E A D (11) D A E C B (10) B C E D A (10) A B C E D (10) A D E C B (8) E C B D A (6) A D B C E (5) B C A E D (4) B A C E D (4) E D C B A (3) A D C E B (3) D E A C B (2) D A E B C (2) C E B A D (2) A B D C E (2) E C D B A (1) E B C D A (1) D E C A B (1) C B E D A (1) A D E B C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 4 2 B 8 0 4 4 -2 C 2 -4 0 2 6 D -4 -4 -2 0 -6 E -2 2 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -8 -2 4 2 B 8 0 4 4 -2 C 2 -4 0 2 6 D -4 -4 -2 0 -6 E -2 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=29 D=26 E=11 C=3 so C is eliminated. Round 2 votes counts: A=31 B=30 D=26 E=13 so E is eliminated. Round 3 votes counts: B=39 A=31 D=30 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:207 C:203 E:200 A:198 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 4 2 B 8 0 4 4 -2 C 2 -4 0 2 6 D -4 -4 -2 0 -6 E -2 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 4 2 B 8 0 4 4 -2 C 2 -4 0 2 6 D -4 -4 -2 0 -6 E -2 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 4 2 B 8 0 4 4 -2 C 2 -4 0 2 6 D -4 -4 -2 0 -6 E -2 2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888871 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7895: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) C D E B A (9) B A D E C (9) C E D A B (7) D B A E C (6) D B E A C (4) E C D A B (3) E C A D B (3) D C E B A (3) D B C E A (3) A E C B D (3) A E B C D (3) E A B D C (2) D E C B A (2) C E A B D (2) C D B E A (2) C D B A E (2) C B D A E (2) C A E B D (2) B D A E C (2) B D A C E (2) B C D A E (2) A B E C D (2) E A D B C (1) E A C D B (1) E A C B D (1) D B C A E (1) C E A D B (1) C B A D E (1) C A B E D (1) B D C A E (1) B A E D C (1) B A C D E (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 6 -2 12 B 6 0 10 4 12 C -6 -10 0 -2 -14 D 2 -4 2 0 4 E -12 -12 14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 -2 12 B 6 0 10 4 12 C -6 -10 0 -2 -14 D 2 -4 2 0 4 E -12 -12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=23 D=19 B=18 E=11 so E is eliminated. Round 2 votes counts: C=35 A=28 D=19 B=18 so B is eliminated. Round 3 votes counts: A=39 C=37 D=24 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 A:205 D:202 E:193 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 -2 12 B 6 0 10 4 12 C -6 -10 0 -2 -14 D 2 -4 2 0 4 E -12 -12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -2 12 B 6 0 10 4 12 C -6 -10 0 -2 -14 D 2 -4 2 0 4 E -12 -12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -2 12 B 6 0 10 4 12 C -6 -10 0 -2 -14 D 2 -4 2 0 4 E -12 -12 14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7896: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) E B D A C (5) D E A C B (5) D A E C B (5) B C E A D (5) E A C B D (4) D A C E B (4) C B A D E (4) C A B D E (4) B E C D A (4) E D A B C (3) E B C A D (3) E B A C D (3) E A D C B (3) D A C B E (3) C A D B E (3) A D C E B (3) A C D E B (3) A C D B E (3) E D A C B (2) C B A E D (2) B E C A D (2) B C D E A (2) B C D A E (2) B C A E D (2) A C E D B (2) E B D C A (1) D E B A C (1) D E A B C (1) D C B A E (1) D C A B E (1) C B D A E (1) C A E B D (1) B D C A E (1) B C E D A (1) B C A D E (1) A E D C B (1) A E C D B (1) A D C B E (1) Total count = 100 A B C D E A 0 6 6 2 -2 B -6 0 -14 8 -2 C -6 14 0 6 0 D -2 -8 -6 0 -2 E 2 2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.208992 D: 0.000000 E: 0.791008 Sum of squares = 0.669371346667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.208992 D: 0.208992 E: 1.000000 A B C D E A 0 6 6 2 -2 B -6 0 -14 8 -2 C -6 14 0 6 0 D -2 -8 -6 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000075455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=24 D=21 C=15 A=14 so A is eliminated. Round 2 votes counts: E=26 B=26 D=25 C=23 so C is eliminated. Round 3 votes counts: B=37 D=34 E=29 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:207 A:206 E:203 B:193 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 2 -2 B -6 0 -14 8 -2 C -6 14 0 6 0 D -2 -8 -6 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000075455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 2 -2 B -6 0 -14 8 -2 C -6 14 0 6 0 D -2 -8 -6 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000075455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 2 -2 B -6 0 -14 8 -2 C -6 14 0 6 0 D -2 -8 -6 0 -2 E 2 2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000075455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7897: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) A E B C D (9) D C B E A (8) E B A D C (5) E A B D C (5) A E C D B (5) A B E C D (5) E D B C A (4) C D B E A (4) C D B A E (4) B E D C A (3) A C D E B (3) A C D B E (3) E B D C A (2) E B D A C (2) E A D C B (2) D B C E A (2) C B D A E (2) C A D B E (2) B E A D C (2) A E C B D (2) A E B D C (2) A C E D B (2) A B C E D (2) E D C A B (1) D E C B A (1) D C E B A (1) D B E C A (1) C A B D E (1) B D C E A (1) B A E D C (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 14 4 6 12 B -14 0 -6 -10 6 C -4 6 0 12 -6 D -6 10 -12 0 -10 E -12 -6 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 6 12 B -14 0 -6 -10 6 C -4 6 0 12 -6 D -6 10 -12 0 -10 E -12 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=24 E=21 D=13 B=7 so B is eliminated. Round 2 votes counts: A=36 E=26 C=24 D=14 so D is eliminated. Round 3 votes counts: C=36 A=36 E=28 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:204 E:199 D:191 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 6 12 B -14 0 -6 -10 6 C -4 6 0 12 -6 D -6 10 -12 0 -10 E -12 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 6 12 B -14 0 -6 -10 6 C -4 6 0 12 -6 D -6 10 -12 0 -10 E -12 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 6 12 B -14 0 -6 -10 6 C -4 6 0 12 -6 D -6 10 -12 0 -10 E -12 -6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999707 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7898: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) B D A E C (7) A E D B C (7) C A E B D (6) A E C D B (6) A B D E C (6) C A E D B (4) B D E A C (4) B D C A E (4) E D B C A (3) D B E A C (3) C B D E A (3) C B D A E (3) C A B E D (3) C A B D E (3) B D C E A (3) E D A B C (2) E C D B A (2) D E B A C (2) C E D B A (2) B D A C E (2) B C D A E (2) E D B A C (1) E C A D B (1) E A D C B (1) E A C D B (1) D B E C A (1) C B E D A (1) C B A E D (1) C B A D E (1) B A D E C (1) A D B E C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -8 10 18 B -8 0 0 4 2 C 8 0 0 2 2 D -10 -4 -2 0 -6 E -18 -2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.376172 C: 0.623828 D: 0.000000 E: 0.000000 Sum of squares = 0.530666659609 Cumulative probabilities = A: 0.000000 B: 0.376172 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 10 18 B -8 0 0 4 2 C 8 0 0 2 2 D -10 -4 -2 0 -6 E -18 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499500 C: 0.500500 D: 0.000000 E: 0.000000 Sum of squares = 0.500000499494 Cumulative probabilities = A: 0.000000 B: 0.499500 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=23 A=22 E=11 D=6 so D is eliminated. Round 2 votes counts: C=38 B=27 A=22 E=13 so E is eliminated. Round 3 votes counts: C=41 B=33 A=26 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:214 C:206 B:199 E:192 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 10 18 B -8 0 0 4 2 C 8 0 0 2 2 D -10 -4 -2 0 -6 E -18 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499500 C: 0.500500 D: 0.000000 E: 0.000000 Sum of squares = 0.500000499494 Cumulative probabilities = A: 0.000000 B: 0.499500 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 10 18 B -8 0 0 4 2 C 8 0 0 2 2 D -10 -4 -2 0 -6 E -18 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499500 C: 0.500500 D: 0.000000 E: 0.000000 Sum of squares = 0.500000499494 Cumulative probabilities = A: 0.000000 B: 0.499500 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 10 18 B -8 0 0 4 2 C 8 0 0 2 2 D -10 -4 -2 0 -6 E -18 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499500 C: 0.500500 D: 0.000000 E: 0.000000 Sum of squares = 0.500000499494 Cumulative probabilities = A: 0.000000 B: 0.499500 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7899: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (14) B C E D A (11) E C B D A (9) D A B C E (6) A D B C E (6) C B E A D (5) D A E C B (4) B D A C E (4) A D B E C (4) E C A D B (3) C E B D A (3) C E B A D (3) B C D E A (3) E C B A D (2) D B A C E (2) D A E B C (2) D A B E C (2) B C E A D (2) A D E B C (2) E D C B A (1) E D A C B (1) E C D B A (1) E C A B D (1) E A D C B (1) D B A E C (1) C B E D A (1) B D C A E (1) B C D A E (1) B C A D E (1) B A D C E (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 4 -6 6 B 4 0 -2 -2 2 C -4 2 0 -6 2 D 6 2 6 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -6 6 B 4 0 -2 -2 2 C -4 2 0 -6 2 D 6 2 6 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=24 E=19 D=17 C=12 so C is eliminated. Round 2 votes counts: B=30 A=28 E=25 D=17 so D is eliminated. Round 3 votes counts: A=42 B=33 E=25 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 B:201 A:200 C:197 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -6 6 B 4 0 -2 -2 2 C -4 2 0 -6 2 D 6 2 6 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -6 6 B 4 0 -2 -2 2 C -4 2 0 -6 2 D 6 2 6 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -6 6 B 4 0 -2 -2 2 C -4 2 0 -6 2 D 6 2 6 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7900: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) E C B D A (7) A B C E D (6) D E B C A (5) D E A B C (5) D A E B C (5) B E C D A (4) B C E A D (4) A C B D E (4) E D C B A (3) E D B C A (3) E B C D A (3) C B A E D (3) A D C B E (3) A D B C E (3) A B C D E (3) D E C B A (2) D A E C B (2) D A C E B (2) D A B E C (2) C E B A D (2) A C D B E (2) A C B E D (2) E B D C A (1) D E B A C (1) D E A C B (1) D C E B A (1) C E B D A (1) C A B E D (1) B E C A D (1) B C A E D (1) B A E C D (1) A D C E B (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 4 -12 B 10 0 2 16 10 C 8 -2 0 14 4 D -4 -16 -14 0 -12 E 12 -10 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999909 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 4 -12 B 10 0 2 16 10 C 8 -2 0 14 4 D -4 -16 -14 0 -12 E 12 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989491 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 C=19 E=17 B=11 so B is eliminated. Round 2 votes counts: A=28 D=26 C=24 E=22 so E is eliminated. Round 3 votes counts: C=39 D=33 A=28 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:219 C:212 E:205 A:187 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 4 -12 B 10 0 2 16 10 C 8 -2 0 14 4 D -4 -16 -14 0 -12 E 12 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989491 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 4 -12 B 10 0 2 16 10 C 8 -2 0 14 4 D -4 -16 -14 0 -12 E 12 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989491 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 4 -12 B 10 0 2 16 10 C 8 -2 0 14 4 D -4 -16 -14 0 -12 E 12 -10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989491 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7901: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (8) B D E A C (8) D B E C A (7) C A E B D (7) D B E A C (6) A C E B D (6) C A B E D (5) D E B A C (4) D C B A E (4) C A D E B (4) B E D A C (4) D B C E A (3) C D A B E (3) A E C B D (3) E B A C D (2) E A C B D (2) D C A E B (2) D C A B E (2) D B C A E (2) B E A D C (2) B D C A E (2) A E C D B (2) A C B E D (2) E B D A C (1) E A D B C (1) E A B C D (1) D E A B C (1) D C E A B (1) D C B E A (1) C D A E B (1) C B A D E (1) C A B D E (1) B C A E D (1) Total count = 100 A B C D E A 0 4 -10 -4 12 B -4 0 -10 -4 8 C 10 10 0 -2 12 D 4 4 2 0 6 E -12 -8 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -4 12 B -4 0 -10 -4 8 C 10 10 0 -2 12 D 4 4 2 0 6 E -12 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=30 B=17 A=13 E=7 so E is eliminated. Round 2 votes counts: D=33 C=30 B=20 A=17 so A is eliminated. Round 3 votes counts: C=45 D=34 B=21 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:208 A:201 B:195 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -10 -4 12 B -4 0 -10 -4 8 C 10 10 0 -2 12 D 4 4 2 0 6 E -12 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -4 12 B -4 0 -10 -4 8 C 10 10 0 -2 12 D 4 4 2 0 6 E -12 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -4 12 B -4 0 -10 -4 8 C 10 10 0 -2 12 D 4 4 2 0 6 E -12 -8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999453 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7902: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) C A E D B (8) D E A B C (6) C A E B D (6) E A D C B (5) D A E C B (5) C B A E D (5) E A C B D (4) D E A C B (4) D C A E B (4) D B E A C (4) B C E A D (4) A E C D B (4) D B C A E (2) D B A C E (2) C D A B E (2) C A B E D (2) B D E C A (2) B D C E A (2) B C D A E (2) B C A E D (2) E D A B C (1) E B A D C (1) E A D B C (1) E A B C D (1) D C B A E (1) D B A E C (1) C E A B D (1) C D B A E (1) C D A E B (1) C B E A D (1) B E D C A (1) B E C A D (1) B D C A E (1) B C E D A (1) B C D E A (1) A C E D B (1) Total count = 100 A B C D E A 0 12 -2 -6 0 B -12 0 -10 -6 -6 C 2 10 0 -4 0 D 6 6 4 0 0 E 0 6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.627781 E: 0.372219 Sum of squares = 0.532655930497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.627781 E: 1.000000 A B C D E A 0 12 -2 -6 0 B -12 0 -10 -6 -6 C 2 10 0 -4 0 D 6 6 4 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 B=26 E=13 A=5 so A is eliminated. Round 2 votes counts: D=29 C=28 B=26 E=17 so E is eliminated. Round 3 votes counts: D=36 C=36 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:208 C:204 E:203 A:202 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -2 -6 0 B -12 0 -10 -6 -6 C 2 10 0 -4 0 D 6 6 4 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -6 0 B -12 0 -10 -6 -6 C 2 10 0 -4 0 D 6 6 4 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -6 0 B -12 0 -10 -6 -6 C 2 10 0 -4 0 D 6 6 4 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) E B A D C (10) B D A E C (10) E C A D B (7) C A D B E (6) C E D B A (4) C D A B E (4) B D A C E (4) A B D C E (4) E B D C A (3) D B A C E (3) C E A D B (3) A C D B E (3) E C D A B (2) E C B A D (2) E C A B D (2) E A B D C (2) D A B C E (2) B A D E C (2) E D B C A (1) E A C B D (1) D E B C A (1) D C B A E (1) D B E C A (1) D B C A E (1) C E D A B (1) C D E B A (1) C D B A E (1) C A E D B (1) C A D E B (1) B D E A C (1) B A E D C (1) B A D C E (1) A E B C D (1) A D B C E (1) Total count = 100 A B C D E A 0 -18 14 -4 -6 B 18 0 20 10 -8 C -14 -20 0 -20 -16 D 4 -10 20 0 -4 E 6 8 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 14 -4 -6 B 18 0 20 10 -8 C -14 -20 0 -20 -16 D 4 -10 20 0 -4 E 6 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 C=22 B=19 D=9 A=9 so D is eliminated. Round 2 votes counts: E=42 B=24 C=23 A=11 so A is eliminated. Round 3 votes counts: E=43 B=31 C=26 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:217 D:205 A:193 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 14 -4 -6 B 18 0 20 10 -8 C -14 -20 0 -20 -16 D 4 -10 20 0 -4 E 6 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 14 -4 -6 B 18 0 20 10 -8 C -14 -20 0 -20 -16 D 4 -10 20 0 -4 E 6 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 14 -4 -6 B 18 0 20 10 -8 C -14 -20 0 -20 -16 D 4 -10 20 0 -4 E 6 8 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7904: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (11) E A B D C (8) E A C D B (7) E A B C D (7) D C B A E (7) C D A E B (7) B E A D C (7) B D A C E (6) B D A E C (5) A E C D B (5) E A C B D (4) D B C A E (4) B E D A C (2) B D E A C (2) B D C E A (2) E C A D B (1) E C A B D (1) E B C D A (1) D C B E A (1) D C A B E (1) D A C B E (1) C E D A B (1) C D E B A (1) C D B E A (1) C D A B E (1) C A D E B (1) B D E C A (1) B A D E C (1) A D C E B (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 16 -10 6 B 4 0 16 18 8 C -16 -16 0 -22 -6 D 10 -18 22 0 10 E -6 -8 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 -10 6 B 4 0 16 18 8 C -16 -16 0 -22 -6 D 10 -18 22 0 10 E -6 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999253 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 E=29 D=14 C=12 A=8 so A is eliminated. Round 2 votes counts: B=38 E=34 D=15 C=13 so C is eliminated. Round 3 votes counts: B=38 E=35 D=27 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:212 A:204 E:191 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 16 -10 6 B 4 0 16 18 8 C -16 -16 0 -22 -6 D 10 -18 22 0 10 E -6 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999253 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 -10 6 B 4 0 16 18 8 C -16 -16 0 -22 -6 D 10 -18 22 0 10 E -6 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999253 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 -10 6 B 4 0 16 18 8 C -16 -16 0 -22 -6 D 10 -18 22 0 10 E -6 -8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999253 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7905: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) D A C E B (5) C D A B E (5) C B E D A (5) B E C A D (5) E B A D C (4) D C A E B (4) C D B E A (4) C B E A D (4) A D E B C (4) D E B A C (3) B E A C D (3) A D C B E (3) A D B E C (3) A B E D C (3) A B E C D (3) A B C E D (3) E B C D A (2) D E A C B (2) D C E B A (2) D C E A B (2) C E B D A (2) C A B E D (2) A E D B C (2) A E B D C (2) E A B D C (1) D E C B A (1) D E A B C (1) D C A B E (1) D A E C B (1) D A E B C (1) D A C B E (1) C E D B A (1) C D B A E (1) C B D E A (1) C A D B E (1) C A B D E (1) B E C D A (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -8 -10 -6 B -4 0 -16 -14 2 C 8 16 0 6 14 D 10 14 -6 0 14 E 6 -2 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -10 -6 B -4 0 -16 -14 2 C 8 16 0 6 14 D 10 14 -6 0 14 E 6 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=24 A=24 B=9 E=7 so E is eliminated. Round 2 votes counts: C=36 A=25 D=24 B=15 so B is eliminated. Round 3 votes counts: C=44 A=32 D=24 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:216 A:190 E:188 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -6 B -4 0 -16 -14 2 C 8 16 0 6 14 D 10 14 -6 0 14 E 6 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -6 B -4 0 -16 -14 2 C 8 16 0 6 14 D 10 14 -6 0 14 E 6 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -6 B -4 0 -16 -14 2 C 8 16 0 6 14 D 10 14 -6 0 14 E 6 -2 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7906: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (8) D B A C E (7) E C A D B (6) B D E C A (6) E C D A B (5) E C A B D (5) A C E D B (5) E B C A D (4) D C E A B (4) B D A E C (4) A D C B E (4) D B E C A (3) C E A D B (3) C A E D B (3) A C D E B (3) D C A E B (2) D B C E A (2) B E A C D (2) B D E A C (2) A C E B D (2) A C D B E (2) E C D B A (1) E C B A D (1) E B C D A (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E B A (1) D C A B E (1) C D E A B (1) C A E B D (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B A E C D (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -10 -2 -6 B -6 0 -10 -10 -6 C 10 10 0 8 4 D 2 10 -8 0 4 E 6 6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -2 -6 B -6 0 -10 -10 -6 C 10 10 0 8 4 D 2 10 -8 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 D=21 A=19 C=9 so C is eliminated. Round 2 votes counts: E=28 B=26 A=24 D=22 so D is eliminated. Round 3 votes counts: B=38 E=35 A=27 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:216 D:204 E:202 A:194 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 -2 -6 B -6 0 -10 -10 -6 C 10 10 0 8 4 D 2 10 -8 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -2 -6 B -6 0 -10 -10 -6 C 10 10 0 8 4 D 2 10 -8 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -2 -6 B -6 0 -10 -10 -6 C 10 10 0 8 4 D 2 10 -8 0 4 E 6 6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7907: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) C E A D B (7) A B D E C (7) D B E A C (5) A E D B C (5) B D E A C (4) A C E D B (4) A C E B D (4) E D B A C (3) D B E C A (3) B D C E A (3) B D A E C (3) A B C D E (3) E D C B A (2) E D B C A (2) E C D B A (2) D E B C A (2) C D B E A (2) C B D E A (2) C A E D B (2) B D E C A (2) B A D C E (2) A E C D B (2) E D C A B (1) E D A B C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E C B A (1) D E B A C (1) D B C E A (1) C E D A B (1) C D E B A (1) C B A D E (1) C A B E D (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) B A D E C (1) A E D C B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 4 -10 -18 B 12 0 6 -22 -8 C -4 -6 0 -8 -2 D 10 22 8 0 -2 E 18 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 4 -10 -18 B 12 0 6 -22 -8 C -4 -6 0 -8 -2 D 10 22 8 0 -2 E 18 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=26 B=19 E=14 D=13 so D is eliminated. Round 2 votes counts: B=28 A=28 C=26 E=18 so E is eliminated. Round 3 votes counts: B=36 C=32 A=32 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:219 E:215 B:194 C:190 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 4 -10 -18 B 12 0 6 -22 -8 C -4 -6 0 -8 -2 D 10 22 8 0 -2 E 18 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -10 -18 B 12 0 6 -22 -8 C -4 -6 0 -8 -2 D 10 22 8 0 -2 E 18 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -10 -18 B 12 0 6 -22 -8 C -4 -6 0 -8 -2 D 10 22 8 0 -2 E 18 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995796 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7908: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (11) E B C A D (9) A E D B C (9) D A C B E (8) E A B C D (6) C B E D A (6) B C E D A (6) A E D C B (6) A D E C B (6) E B C D A (5) C B D E A (4) A E B C D (3) A D C B E (3) E C B D A (2) E B A C D (2) D C B E A (2) D C A B E (2) C D B E A (2) B E C D A (2) A D E B C (2) E C B A D (1) D A B C E (1) B C D E A (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -6 -4 4 B 6 0 -8 -6 -4 C 6 8 0 -2 -6 D 4 6 2 0 -14 E -4 4 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999971 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 -6 -4 4 B 6 0 -8 -6 -4 C 6 8 0 -2 -6 D 4 6 2 0 -14 E -4 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 D=24 C=12 B=9 so B is eliminated. Round 2 votes counts: A=30 E=27 D=24 C=19 so C is eliminated. Round 3 votes counts: E=39 D=31 A=30 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:210 C:203 D:199 A:194 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 4 B 6 0 -8 -6 -4 C 6 8 0 -2 -6 D 4 6 2 0 -14 E -4 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 4 B 6 0 -8 -6 -4 C 6 8 0 -2 -6 D 4 6 2 0 -14 E -4 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 4 B 6 0 -8 -6 -4 C 6 8 0 -2 -6 D 4 6 2 0 -14 E -4 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7909: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D E C A B (5) D E A C B (5) B C A E D (5) D C E A B (4) A B C E D (4) E D C B A (3) D A E C B (3) C D A B E (3) C B A D E (3) C A D B E (3) A D E B C (3) A B E D C (3) A B E C D (3) A B C D E (3) E B D C A (2) D E A B C (2) D C E B A (2) D C A E B (2) D A C E B (2) C D E B A (2) C B E D A (2) B E C A D (2) B E A C D (2) B A E C D (2) A E D B C (2) A D C E B (2) A C D B E (2) E D B C A (1) E D B A C (1) E B D A C (1) E B C D A (1) E B A D C (1) D E C B A (1) C E B D A (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A E B (1) C B E A D (1) C B D A E (1) B E C D A (1) B E A D C (1) B C E A D (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 22 0 -8 2 B -22 0 -6 -20 -8 C 0 6 0 -8 -4 D 8 20 8 0 10 E -2 8 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 0 -8 2 B -22 0 -6 -20 -8 C 0 6 0 -8 -4 D 8 20 8 0 10 E -2 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 C=20 E=15 B=14 so B is eliminated. Round 2 votes counts: A=27 D=26 C=26 E=21 so E is eliminated. Round 3 votes counts: D=39 A=31 C=30 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 A:208 E:200 C:197 B:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 0 -8 2 B -22 0 -6 -20 -8 C 0 6 0 -8 -4 D 8 20 8 0 10 E -2 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 0 -8 2 B -22 0 -6 -20 -8 C 0 6 0 -8 -4 D 8 20 8 0 10 E -2 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 0 -8 2 B -22 0 -6 -20 -8 C 0 6 0 -8 -4 D 8 20 8 0 10 E -2 8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7910: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) C B A E D (9) C A B E D (8) D E A B C (7) D B E A C (5) B C A E D (5) D E A C B (4) B E A D C (4) C A E D B (3) C A D E B (3) B E D A C (3) B C E A D (3) B A E C D (3) D B E C A (2) D A E C B (2) C D A E B (2) B E A C D (2) B D E C A (2) B C E D A (2) A E B D C (2) E A D B C (1) D E C A B (1) D E B C A (1) D C E A B (1) D C A E B (1) D B C E A (1) C D B E A (1) C D A B E (1) C B D E A (1) C A E B D (1) B E D C A (1) B E C A D (1) B D E A C (1) A E D B C (1) A E C B D (1) A E B C D (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 0 2 -8 B 14 0 18 2 12 C 0 -18 0 0 -12 D -2 -2 0 0 -6 E 8 -12 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 2 -8 B 14 0 18 2 12 C 0 -18 0 0 -12 D -2 -2 0 0 -6 E 8 -12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=29 B=27 A=8 E=1 so E is eliminated. Round 2 votes counts: D=35 C=29 B=27 A=9 so A is eliminated. Round 3 votes counts: D=37 C=32 B=31 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:223 E:207 D:195 A:190 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 2 -8 B 14 0 18 2 12 C 0 -18 0 0 -12 D -2 -2 0 0 -6 E 8 -12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 2 -8 B 14 0 18 2 12 C 0 -18 0 0 -12 D -2 -2 0 0 -6 E 8 -12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 2 -8 B 14 0 18 2 12 C 0 -18 0 0 -12 D -2 -2 0 0 -6 E 8 -12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995265 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7911: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) C A B E D (10) A C B E D (8) D E B A C (5) C D B E A (5) B E A D C (5) A E B D C (5) A D E B C (5) A B E D C (5) C D A E B (4) C B E D A (4) A B E C D (4) D E B C A (3) D C E B A (3) C B A E D (3) E B D A C (2) D A E B C (2) B E D A C (2) B C E D A (2) A C D E B (2) D C E A B (1) C D B A E (1) C B E A D (1) C A D E B (1) C A B D E (1) B E D C A (1) B C E A D (1) B A E D C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 8 8 B 2 0 -10 14 12 C 4 10 0 20 20 D -8 -14 -20 0 -12 E -8 -12 -20 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 8 8 B 2 0 -10 14 12 C 4 10 0 20 20 D -8 -14 -20 0 -12 E -8 -12 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=30 D=14 B=13 E=2 so E is eliminated. Round 2 votes counts: C=41 A=30 B=15 D=14 so D is eliminated. Round 3 votes counts: C=45 A=32 B=23 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:227 B:209 A:205 E:186 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 8 8 B 2 0 -10 14 12 C 4 10 0 20 20 D -8 -14 -20 0 -12 E -8 -12 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 8 B 2 0 -10 14 12 C 4 10 0 20 20 D -8 -14 -20 0 -12 E -8 -12 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 8 B 2 0 -10 14 12 C 4 10 0 20 20 D -8 -14 -20 0 -12 E -8 -12 -20 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992283 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7912: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) C A D E B (9) B E D C A (8) D E A C B (7) A C D E B (6) E D B A C (5) E D B C A (4) E B D A C (4) C A B D E (4) B E D A C (4) B A C E D (4) A C B D E (4) C B A D E (3) B C A E D (3) B C A D E (3) E D A C B (2) E D A B C (2) D C E A B (2) D A E C B (2) D A C E B (2) B C E D A (2) B A E C D (2) E B D C A (1) D E C B A (1) C D A E B (1) B E C D A (1) B C E A D (1) B C D E A (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -10 -20 -12 B -6 0 -8 -8 -18 C 10 8 0 -10 -10 D 20 8 10 0 12 E 12 18 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -20 -12 B -6 0 -8 -8 -18 C 10 8 0 -10 -10 D 20 8 10 0 12 E 12 18 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 E=18 C=17 A=11 so A is eliminated. Round 2 votes counts: B=30 C=27 D=25 E=18 so E is eliminated. Round 3 votes counts: D=38 B=35 C=27 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:214 C:199 A:182 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -10 -20 -12 B -6 0 -8 -8 -18 C 10 8 0 -10 -10 D 20 8 10 0 12 E 12 18 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -20 -12 B -6 0 -8 -8 -18 C 10 8 0 -10 -10 D 20 8 10 0 12 E 12 18 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -20 -12 B -6 0 -8 -8 -18 C 10 8 0 -10 -10 D 20 8 10 0 12 E 12 18 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7913: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) C D A E B (6) E B D C A (5) D C E A B (5) A B C E D (5) E B D A C (4) D C A E B (4) C D A B E (4) B E A C D (4) B A E D C (4) A D C B E (4) A C D B E (4) A C B D E (4) E B C A D (3) D E C B A (3) D A B E C (3) E D C B A (2) E D B C A (2) E B C D A (2) E B A D C (2) D C A B E (2) C D E B A (2) C A D B E (2) E C D B A (1) E C B D A (1) D E A C B (1) D C E B A (1) D A C B E (1) C E D B A (1) C D E A B (1) C B E A D (1) C B A E D (1) C A B D E (1) B E A D C (1) B A C E D (1) A D B C E (1) A B E D C (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 0 -2 16 B -4 0 -4 0 8 C 0 4 0 6 4 D 2 0 -6 0 2 E -16 -8 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.367768 B: 0.000000 C: 0.632232 D: 0.000000 E: 0.000000 Sum of squares = 0.534970807986 Cumulative probabilities = A: 0.367768 B: 0.367768 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -2 16 B -4 0 -4 0 8 C 0 4 0 6 4 D 2 0 -6 0 2 E -16 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=22 A=22 D=20 C=19 B=17 so B is eliminated. Round 2 votes counts: A=34 E=27 D=20 C=19 so C is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:209 C:207 B:200 D:199 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -2 16 B -4 0 -4 0 8 C 0 4 0 6 4 D 2 0 -6 0 2 E -16 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -2 16 B -4 0 -4 0 8 C 0 4 0 6 4 D 2 0 -6 0 2 E -16 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -2 16 B -4 0 -4 0 8 C 0 4 0 6 4 D 2 0 -6 0 2 E -16 -8 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7914: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) D A B E C (9) D B A C E (7) D E C A B (5) C E B A D (5) A B E C D (5) E C D A B (4) E A C B D (4) D C E B A (4) C E D B A (4) A E B C D (4) D B C A E (3) B A D C E (3) E C A D B (2) E A B C D (2) D C B E A (2) D B A E C (2) C E A B D (2) C B D A E (2) B D C A E (2) B D A C E (2) B A D E C (2) B A C E D (2) E D C A B (1) E A C D B (1) D E C B A (1) C D E B A (1) C B E D A (1) C B E A D (1) C B A E D (1) B C A D E (1) A E D B C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -8 0 -4 B -8 0 -6 4 -6 C 8 6 0 10 -12 D 0 -4 -10 0 -6 E 4 6 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -8 0 -4 B -8 0 -6 4 -6 C 8 6 0 10 -12 D 0 -4 -10 0 -6 E 4 6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=26 C=17 B=12 A=12 so B is eliminated. Round 2 votes counts: D=37 E=26 A=19 C=18 so C is eliminated. Round 3 votes counts: D=40 E=39 A=21 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:206 A:198 B:192 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -8 0 -4 B -8 0 -6 4 -6 C 8 6 0 10 -12 D 0 -4 -10 0 -6 E 4 6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 0 -4 B -8 0 -6 4 -6 C 8 6 0 10 -12 D 0 -4 -10 0 -6 E 4 6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 0 -4 B -8 0 -6 4 -6 C 8 6 0 10 -12 D 0 -4 -10 0 -6 E 4 6 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7915: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) C E A D B (7) D C E A B (5) C E A B D (5) C D E A B (5) A E B C D (5) D C B E A (4) D B C E A (4) D B A E C (4) D A C E B (4) C E B D A (4) A E C B D (4) D C A E B (3) C D E B A (3) B E A C D (3) B D E C A (3) B A E D C (3) B A E C D (3) E A C B D (2) D A C B E (2) C E B A D (2) B E C A D (2) A B E C D (2) D C A B E (1) D B E C A (1) D A B E C (1) C E D A B (1) C B D E A (1) B E A D C (1) B D A E C (1) A E D C B (1) A E C D B (1) A D B E C (1) A C E B D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -18 -10 -22 B -6 0 -28 -12 -22 C 18 28 0 2 20 D 10 12 -2 0 4 E 22 22 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999342 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -18 -10 -22 B -6 0 -28 -12 -22 C 18 28 0 2 20 D 10 12 -2 0 4 E 22 22 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999906432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=28 A=17 B=16 E=2 so E is eliminated. Round 2 votes counts: D=37 C=28 A=19 B=16 so B is eliminated. Round 3 votes counts: D=41 C=30 A=29 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:234 D:212 E:210 A:178 B:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -18 -10 -22 B -6 0 -28 -12 -22 C 18 28 0 2 20 D 10 12 -2 0 4 E 22 22 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999906432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 -10 -22 B -6 0 -28 -12 -22 C 18 28 0 2 20 D 10 12 -2 0 4 E 22 22 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999906432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 -10 -22 B -6 0 -28 -12 -22 C 18 28 0 2 20 D 10 12 -2 0 4 E 22 22 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999906432 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7916: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) B E D C A (6) B C A E D (6) E B D C A (5) B A C E D (5) A C D B E (5) C A D E B (4) E D B C A (3) D C E A B (3) C D E B A (3) B E C D A (3) B E A D C (3) B C E D A (3) B A E C D (3) A C B D E (3) E D B A C (2) E D A B C (2) D E A C B (2) C E D B A (2) C B D E A (2) B E A C D (2) A E B D C (2) A D C E B (2) A B E C D (2) A B C D E (2) E D C A B (1) E B C D A (1) D E C B A (1) D E C A B (1) C E B D A (1) C D A B E (1) C B E D A (1) C B D A E (1) C B A D E (1) C A B D E (1) B E C A D (1) B C E A D (1) B A E D C (1) A E D B C (1) A D E C B (1) A D E B C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -28 -18 -2 -12 B 28 0 14 14 10 C 18 -14 0 8 -4 D 2 -14 -8 0 -28 E 12 -10 4 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -18 -2 -12 B 28 0 14 14 10 C 18 -14 0 8 -4 D 2 -14 -8 0 -28 E 12 -10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=21 A=21 C=17 D=7 so D is eliminated. Round 2 votes counts: B=34 E=25 A=21 C=20 so C is eliminated. Round 3 votes counts: B=39 E=34 A=27 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:217 C:204 D:176 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -18 -2 -12 B 28 0 14 14 10 C 18 -14 0 8 -4 D 2 -14 -8 0 -28 E 12 -10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -18 -2 -12 B 28 0 14 14 10 C 18 -14 0 8 -4 D 2 -14 -8 0 -28 E 12 -10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -18 -2 -12 B 28 0 14 14 10 C 18 -14 0 8 -4 D 2 -14 -8 0 -28 E 12 -10 4 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7917: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C E B A D (8) B E C D A (7) A E C B D (7) D B C E A (6) A D C E B (5) A D B E C (5) B D E C A (4) E A C B D (3) C E A B D (3) B C E D A (3) A D E C B (3) E C B A D (2) D B A C E (2) D A B E C (2) C E D B A (2) C E B D A (2) C A E B D (2) B E A D C (2) B D C E A (2) A E C D B (2) A C E B D (2) E B C D A (1) E B C A D (1) E B A C D (1) D C A B E (1) D B C A E (1) D A B C E (1) C E A D B (1) C D E A B (1) C D B E A (1) B E D C A (1) B E D A C (1) B D E A C (1) B D A E C (1) A D E B C (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 2 2 -6 B 16 0 6 8 4 C -2 -6 0 0 -8 D -2 -8 0 0 -4 E 6 -4 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 2 2 -6 B 16 0 6 8 4 C -2 -6 0 0 -8 D -2 -8 0 0 -4 E 6 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=22 B=22 C=20 E=8 so E is eliminated. Round 2 votes counts: A=31 B=25 D=22 C=22 so D is eliminated. Round 3 votes counts: B=43 A=34 C=23 so C is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:207 D:193 C:192 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 2 2 -6 B 16 0 6 8 4 C -2 -6 0 0 -8 D -2 -8 0 0 -4 E 6 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 2 2 -6 B 16 0 6 8 4 C -2 -6 0 0 -8 D -2 -8 0 0 -4 E 6 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 2 2 -6 B 16 0 6 8 4 C -2 -6 0 0 -8 D -2 -8 0 0 -4 E 6 -4 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7918: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C D E A B (6) C D A E B (6) B E A D C (6) D C A E B (5) E C B D A (4) B A D E C (4) A B D C E (4) E C D A B (3) E B C D A (3) D C A B E (3) C E D A B (3) B A E D C (3) B A D C E (3) A D C B E (3) A D B C E (3) A B D E C (3) E C A D B (2) C E A D B (2) C D E B A (2) B E A C D (2) B A E C D (2) A D C E B (2) A C D E B (2) A B E C D (2) E C B A D (1) E B A C D (1) E A B C D (1) D C E B A (1) D C B A E (1) D B C E A (1) D B C A E (1) D B A C E (1) D A B C E (1) B E D C A (1) B E C A D (1) B D C E A (1) B D A C E (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -8 -2 4 B -6 0 -8 -12 -4 C 8 8 0 2 6 D 2 12 -2 0 8 E -4 4 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 -2 4 B -6 0 -8 -12 -4 C 8 8 0 2 6 D 2 12 -2 0 8 E -4 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=22 A=21 C=19 D=14 so D is eliminated. Round 2 votes counts: C=29 B=27 E=22 A=22 so E is eliminated. Round 3 votes counts: C=46 B=31 A=23 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:210 A:200 E:193 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 -2 4 B -6 0 -8 -12 -4 C 8 8 0 2 6 D 2 12 -2 0 8 E -4 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -2 4 B -6 0 -8 -12 -4 C 8 8 0 2 6 D 2 12 -2 0 8 E -4 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -2 4 B -6 0 -8 -12 -4 C 8 8 0 2 6 D 2 12 -2 0 8 E -4 4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7919: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (10) C E D B A (8) E C B D A (7) E B C A D (7) B A E D C (7) D A C B E (5) A D B C E (5) D C A E B (4) C D E A B (4) E C D B A (3) E B C D A (3) D A C E B (3) C E B D A (3) C D A E B (3) A D C B E (3) C D A B E (2) B E C A D (2) B E A D C (2) B C E A D (2) B A C D E (2) A D B E C (2) A B D E C (2) E D B A C (1) D A E C B (1) D A E B C (1) C D E B A (1) C B E A D (1) C B A E D (1) C A B D E (1) B A E C D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -4 2 -8 B 22 0 0 8 2 C 4 0 0 24 0 D -2 -8 -24 0 -18 E 8 -2 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.557294 C: 0.442706 D: 0.000000 E: 0.000000 Sum of squares = 0.506565131202 Cumulative probabilities = A: 0.000000 B: 0.557294 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -4 2 -8 B 22 0 0 8 2 C 4 0 0 24 0 D -2 -8 -24 0 -18 E 8 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997691 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 E=21 A=15 D=14 so D is eliminated. Round 2 votes counts: C=28 B=26 A=25 E=21 so E is eliminated. Round 3 votes counts: C=38 B=37 A=25 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:214 E:212 A:184 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -4 2 -8 B 22 0 0 8 2 C 4 0 0 24 0 D -2 -8 -24 0 -18 E 8 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997691 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -4 2 -8 B 22 0 0 8 2 C 4 0 0 24 0 D -2 -8 -24 0 -18 E 8 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997691 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -4 2 -8 B 22 0 0 8 2 C 4 0 0 24 0 D -2 -8 -24 0 -18 E 8 -2 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999997691 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7920: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) D B E C A (7) B E D C A (7) D B C E A (6) B D E C A (5) E B A C D (4) D E B A C (4) D C B A E (4) C A D B E (4) A C E B D (4) D C A B E (3) D A C E B (3) C B A D E (3) C A B E D (3) A C D E B (3) E D B A C (2) E B A D C (2) E A B C D (2) D E B C A (2) B E A C D (2) A E D C B (2) A E B C D (2) A D C E B (2) A C E D B (2) D E C A B (1) D A E C B (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D E B (1) B E D A C (1) B E C A D (1) B C E A D (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -28 -4 -18 -14 B 28 0 16 0 2 C 4 -16 0 -26 -14 D 18 0 26 0 6 E 14 -2 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.460966 C: 0.000000 D: 0.539034 E: 0.000000 Sum of squares = 0.503047317923 Cumulative probabilities = A: 0.000000 B: 0.460966 C: 0.460966 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -4 -18 -14 B 28 0 16 0 2 C 4 -16 0 -26 -14 D 18 0 26 0 6 E 14 -2 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999127 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=20 B=17 A=17 C=15 so C is eliminated. Round 2 votes counts: D=33 A=25 B=22 E=20 so E is eliminated. Round 3 votes counts: B=38 D=35 A=27 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:225 B:223 E:210 C:174 A:168 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -4 -18 -14 B 28 0 16 0 2 C 4 -16 0 -26 -14 D 18 0 26 0 6 E 14 -2 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999127 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -4 -18 -14 B 28 0 16 0 2 C 4 -16 0 -26 -14 D 18 0 26 0 6 E 14 -2 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999127 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -4 -18 -14 B 28 0 16 0 2 C 4 -16 0 -26 -14 D 18 0 26 0 6 E 14 -2 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999127 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7921: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) E B A D C (6) D C E A B (6) E A B D C (5) C A B E D (5) B E A D C (5) D E B A C (4) B A E D C (4) D E A C B (3) C D B A E (3) C B A D E (3) A B E C D (3) E D B A C (2) E A D C B (2) E A D B C (2) D C B E A (2) C D A E B (2) C B D A E (2) C A E B D (2) C A D E B (2) B C D A E (2) B C A E D (2) A E C B D (2) A E B D C (2) A E B C D (2) A C E B D (2) A B C E D (2) E D A B C (1) D E C A B (1) D E B C A (1) D E A B C (1) D B E C A (1) D B C E A (1) C D E A B (1) C D A B E (1) C B A E D (1) C A E D B (1) C A B D E (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) A E C D B (1) Total count = 100 A B C D E A 0 0 14 24 6 B 0 0 14 24 -2 C -14 -14 0 -6 -14 D -24 -24 6 0 -18 E -6 2 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.770005 B: 0.229995 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.645805527015 Cumulative probabilities = A: 0.770005 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 24 6 B 0 0 14 24 -2 C -14 -14 0 -6 -14 D -24 -24 6 0 -18 E -6 2 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 D=20 E=18 A=14 so A is eliminated. Round 2 votes counts: B=29 C=26 E=25 D=20 so D is eliminated. Round 3 votes counts: E=35 C=34 B=31 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:222 B:218 E:214 C:176 D:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 24 6 B 0 0 14 24 -2 C -14 -14 0 -6 -14 D -24 -24 6 0 -18 E -6 2 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 24 6 B 0 0 14 24 -2 C -14 -14 0 -6 -14 D -24 -24 6 0 -18 E -6 2 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 24 6 B 0 0 14 24 -2 C -14 -14 0 -6 -14 D -24 -24 6 0 -18 E -6 2 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999965 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7922: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (6) C D B E A (6) C B E D A (6) D C B E A (5) D B E C A (5) D C A B E (4) D A B E C (4) C A E B D (4) A E B D C (4) A E B C D (4) A D E B C (4) E B A D C (3) E A B C D (3) D B C E A (3) D A B C E (3) B E D C A (3) E C B A D (2) E B D A C (2) D B E A C (2) C E A B D (2) C B D E A (2) B C E D A (2) A E D B C (2) A D C E B (2) A C E B D (2) A C D E B (2) E D B A C (1) E C A B D (1) E B D C A (1) E B C D A (1) E B C A D (1) D C B A E (1) D B A E C (1) D A E B C (1) D A C B E (1) C B E A D (1) B E C D A (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -14 -10 -18 B 10 0 2 4 0 C 14 -2 0 -4 6 D 10 -4 4 0 -8 E 18 0 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.925426 C: 0.000000 D: 0.000000 E: 0.074574 Sum of squares = 0.861975007027 Cumulative probabilities = A: 0.000000 B: 0.925426 C: 0.925426 D: 0.925426 E: 1.000000 A B C D E A 0 -10 -14 -10 -18 B 10 0 2 4 0 C 14 -2 0 -4 6 D 10 -4 4 0 -8 E 18 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000006801 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=27 A=22 E=15 B=6 so B is eliminated. Round 2 votes counts: D=30 C=29 A=22 E=19 so E is eliminated. Round 3 votes counts: D=37 C=35 A=28 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:210 B:208 C:207 D:201 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 -10 -18 B 10 0 2 4 0 C 14 -2 0 -4 6 D 10 -4 4 0 -8 E 18 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000006801 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -10 -18 B 10 0 2 4 0 C 14 -2 0 -4 6 D 10 -4 4 0 -8 E 18 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000006801 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -10 -18 B 10 0 2 4 0 C 14 -2 0 -4 6 D 10 -4 4 0 -8 E 18 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000006801 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7923: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (8) D A C E B (7) E D B C A (6) D E C A B (6) C A B E D (6) B E C A D (5) B A C E D (5) D E A B C (4) B C A E D (4) A C D B E (4) E B D C A (3) D E B A C (3) D E A C B (3) D A E C B (3) B E A C D (3) A B D C E (3) D E C B A (2) C B A E D (2) C A D B E (2) A B C D E (2) E D C A B (1) E D B A C (1) E C D B A (1) E C B D A (1) E B D A C (1) E B C D A (1) D B E A C (1) D A C B E (1) C E B D A (1) C E B A D (1) C E A D B (1) C D E A B (1) C D A E B (1) C A D E B (1) B E D A C (1) B C E A D (1) B A E C D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 2 4 B -12 0 -6 0 2 C -6 6 0 6 6 D -2 0 -6 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 2 4 B -12 0 -6 0 2 C -6 6 0 6 6 D -2 0 -6 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=20 A=19 C=16 E=15 so E is eliminated. Round 2 votes counts: D=38 B=25 A=19 C=18 so C is eliminated. Round 3 votes counts: D=41 B=30 A=29 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:212 C:206 D:199 B:192 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 2 4 B -12 0 -6 0 2 C -6 6 0 6 6 D -2 0 -6 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 2 4 B -12 0 -6 0 2 C -6 6 0 6 6 D -2 0 -6 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 2 4 B -12 0 -6 0 2 C -6 6 0 6 6 D -2 0 -6 0 6 E -4 -2 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7924: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (20) C A D B E (18) E C B D A (7) B D E A C (7) A C D B E (6) C E A B D (5) E B D C A (4) C A E D B (4) E C A B D (3) C E B D A (3) A C D E B (3) C E A D B (2) C B D A E (2) A D C B E (2) A D B E C (2) E C B A D (1) E B C D A (1) E A C D B (1) E A B D C (1) D B E A C (1) D B A E C (1) D A B E C (1) C D B A E (1) C A B D E (1) B E D A C (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 2 -4 2 -14 B -2 0 -22 12 -14 C 4 22 0 18 -6 D -2 -12 -18 0 -10 E 14 14 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -4 2 -14 B -2 0 -22 12 -14 C 4 22 0 18 -6 D -2 -12 -18 0 -10 E 14 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=36 A=15 B=8 D=3 so D is eliminated. Round 2 votes counts: E=38 C=36 A=16 B=10 so B is eliminated. Round 3 votes counts: E=47 C=36 A=17 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:219 A:193 B:187 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 2 -14 B -2 0 -22 12 -14 C 4 22 0 18 -6 D -2 -12 -18 0 -10 E 14 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 2 -14 B -2 0 -22 12 -14 C 4 22 0 18 -6 D -2 -12 -18 0 -10 E 14 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 2 -14 B -2 0 -22 12 -14 C 4 22 0 18 -6 D -2 -12 -18 0 -10 E 14 14 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7925: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) E D C A B (6) E A D C B (5) E A B C D (5) D C E B A (5) C D B A E (5) D E C A B (4) D C E A B (4) C B D A E (4) C B A D E (4) B C A D E (4) A B E C D (4) E D A C B (3) E A B D C (3) D C B A E (3) B E D A C (3) B C D A E (3) B A C E D (3) E B D A C (2) D C B E A (2) C D A B E (2) B A C D E (2) A E C D B (2) A E B C D (2) E D B A C (1) E D A B C (1) E B A D C (1) E A D B C (1) E A C D B (1) D E C B A (1) D C A B E (1) C D A E B (1) C A D B E (1) B E A D C (1) B D E C A (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -2 -4 0 B 4 0 -10 2 4 C 2 10 0 4 -8 D 4 -2 -4 0 -6 E 0 -4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.000000 E: 0.454545 Sum of squares = 0.371900826444 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 0.545455 E: 1.000000 A B C D E A 0 -4 -2 -4 0 B 4 0 -10 2 4 C 2 10 0 4 -8 D 4 -2 -4 0 -6 E 0 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.000000 E: 0.454545 Sum of squares = 0.371900826374 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 0.545455 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 D=20 C=17 A=10 so A is eliminated. Round 2 votes counts: E=33 B=29 D=20 C=18 so C is eliminated. Round 3 votes counts: B=38 E=33 D=29 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:205 C:204 B:200 D:196 A:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -2 -4 0 B 4 0 -10 2 4 C 2 10 0 4 -8 D 4 -2 -4 0 -6 E 0 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.000000 E: 0.454545 Sum of squares = 0.371900826374 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 0.545455 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -4 0 B 4 0 -10 2 4 C 2 10 0 4 -8 D 4 -2 -4 0 -6 E 0 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.000000 E: 0.454545 Sum of squares = 0.371900826374 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 0.545455 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -4 0 B 4 0 -10 2 4 C 2 10 0 4 -8 D 4 -2 -4 0 -6 E 0 -4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.181818 D: 0.000000 E: 0.454545 Sum of squares = 0.371900826374 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.545455 D: 0.545455 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7926: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (14) B C E D A (7) E C B A D (6) E A D C B (6) C E B A D (6) D A B E C (5) C B E D A (5) B C D A E (5) A D E B C (5) A D E C B (4) A D B E C (4) E C A B D (3) D B A C E (3) D A E C B (3) C E B D A (3) B D A C E (3) E C A D B (2) B C E A D (2) E D A C B (1) E B A C D (1) E A D B C (1) D A C E B (1) D A C B E (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B D C A E (1) B C D E A (1) A E D C B (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 6 10 -12 8 B -6 0 10 -8 12 C -10 -10 0 -10 12 D 12 8 10 0 8 E -8 -12 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 -12 8 B -6 0 10 -8 12 C -10 -10 0 -10 12 D 12 8 10 0 8 E -8 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=20 B=20 C=17 A=16 so A is eliminated. Round 2 votes counts: D=41 E=22 B=20 C=17 so C is eliminated. Round 3 votes counts: D=42 E=31 B=27 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:206 B:204 C:191 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -12 8 B -6 0 10 -8 12 C -10 -10 0 -10 12 D 12 8 10 0 8 E -8 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -12 8 B -6 0 10 -8 12 C -10 -10 0 -10 12 D 12 8 10 0 8 E -8 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -12 8 B -6 0 10 -8 12 C -10 -10 0 -10 12 D 12 8 10 0 8 E -8 -12 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7927: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) C A B D E (8) E B C D A (7) A D C B E (6) E B C A D (5) D E A B C (5) B C E A D (5) E D A C B (4) E B D C A (4) D A E C B (4) B E C D A (4) A C D B E (4) E D A B C (3) E C B A D (2) D E B A C (2) D A B E C (2) C B A E D (2) C B A D E (2) B E D C A (2) B C E D A (2) B C A D E (2) E B D A C (1) E A C D B (1) D B E A C (1) D A E B C (1) D A C B E (1) D A B C E (1) C B E A D (1) C A E D B (1) C A B E D (1) B E C A D (1) B D E C A (1) B C D A E (1) B C A E D (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -4 -10 -20 B 10 0 20 6 -2 C 4 -20 0 4 -18 D 10 -6 -4 0 -12 E 20 2 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999479 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 -10 -20 B 10 0 20 6 -2 C 4 -20 0 4 -18 D 10 -6 -4 0 -12 E 20 2 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=19 D=17 C=15 A=13 so A is eliminated. Round 2 votes counts: E=36 D=24 C=21 B=19 so B is eliminated. Round 3 votes counts: E=43 C=32 D=25 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:217 D:194 C:185 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 -10 -20 B 10 0 20 6 -2 C 4 -20 0 4 -18 D 10 -6 -4 0 -12 E 20 2 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -10 -20 B 10 0 20 6 -2 C 4 -20 0 4 -18 D 10 -6 -4 0 -12 E 20 2 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -10 -20 B 10 0 20 6 -2 C 4 -20 0 4 -18 D 10 -6 -4 0 -12 E 20 2 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7928: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) C E B A D (6) B C E D A (6) A D E C B (5) A D C E B (5) E B C D A (4) B E C D A (4) A D E B C (4) A C E D B (4) A C E B D (4) D A B E C (3) A D C B E (3) A D B C E (3) E D B C A (2) E C B D A (2) E B D C A (2) E A C D B (2) D E A B C (2) D B E C A (2) D B A E C (2) D B A C E (2) C E B D A (2) C B E A D (2) C B A E D (2) A C B D E (2) E D A C B (1) E C B A D (1) E C A B D (1) E A C B D (1) D E B A C (1) D B E A C (1) C E A B D (1) C B E D A (1) C A E B D (1) C A B E D (1) B E D C A (1) B D E C A (1) B D C E A (1) B C E A D (1) B C D A E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 10 2 4 B -6 0 2 -4 -20 C -10 -2 0 2 0 D -2 4 -2 0 -4 E -4 20 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 2 4 B -6 0 2 -4 -20 C -10 -2 0 2 0 D -2 4 -2 0 -4 E -4 20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=21 E=16 C=16 B=15 so B is eliminated. Round 2 votes counts: A=32 C=24 D=23 E=21 so E is eliminated. Round 3 votes counts: C=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:210 D:198 C:195 B:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 2 4 B -6 0 2 -4 -20 C -10 -2 0 2 0 D -2 4 -2 0 -4 E -4 20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 2 4 B -6 0 2 -4 -20 C -10 -2 0 2 0 D -2 4 -2 0 -4 E -4 20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 2 4 B -6 0 2 -4 -20 C -10 -2 0 2 0 D -2 4 -2 0 -4 E -4 20 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996389 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7929: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) A C E B D (11) D B E C A (6) E C B D A (5) D E B C A (5) C E B A D (5) A D B C E (5) A B C E D (5) E D C B A (4) D A E C B (4) A D E C B (4) A C B E D (4) E C D B A (3) A B D C E (3) E C A B D (2) D E C A B (2) C B E A D (2) B D E C A (2) B D C E A (2) B C E A D (2) A B C D E (2) E C D A B (1) D E A C B (1) D E A B C (1) D B E A C (1) D B A E C (1) D A B E C (1) B D A C E (1) B A D C E (1) B A C E D (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -4 -2 -10 B 4 0 -20 -4 -20 C 4 20 0 -12 -10 D 2 4 12 0 8 E 10 20 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999246 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 -10 B 4 0 -20 -4 -20 C 4 20 0 -12 -10 D 2 4 12 0 8 E 10 20 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=33 E=15 B=9 C=7 so C is eliminated. Round 2 votes counts: A=36 D=33 E=20 B=11 so B is eliminated. Round 3 votes counts: D=38 A=38 E=24 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:213 C:201 A:190 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -10 B 4 0 -20 -4 -20 C 4 20 0 -12 -10 D 2 4 12 0 8 E 10 20 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -10 B 4 0 -20 -4 -20 C 4 20 0 -12 -10 D 2 4 12 0 8 E 10 20 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -10 B 4 0 -20 -4 -20 C 4 20 0 -12 -10 D 2 4 12 0 8 E 10 20 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7930: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) C B E D A (7) E A D C B (6) A D B E C (6) D A E B C (4) D A B C E (4) C E B D A (4) B C E A D (4) A D B C E (4) E D A C B (3) E C B D A (3) E C B A D (3) C B E A D (3) B C D E A (3) B A D C E (3) B A C D E (3) E D C A B (2) E C D B A (2) D B C A E (2) D A E C B (2) C E B A D (2) C B D E A (2) B C D A E (2) B C A E D (2) A E D B C (2) E D C B A (1) E C D A B (1) E C A D B (1) E C A B D (1) E A C D B (1) E A C B D (1) B D C A E (1) B D A C E (1) B C E D A (1) B C A D E (1) A E D C B (1) A B D C E (1) Total count = 100 A B C D E A 0 0 4 10 -2 B 0 0 8 -4 0 C -4 -8 0 -6 0 D -10 4 6 0 -2 E 2 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.190917 C: 0.000000 D: 0.000000 E: 0.809082 Sum of squares = 0.691063975461 Cumulative probabilities = A: 0.000000 B: 0.190917 C: 0.190917 D: 0.190918 E: 1.000000 A B C D E A 0 0 4 10 -2 B 0 0 8 -4 0 C -4 -8 0 -6 0 D -10 4 6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555705389 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=24 B=21 C=18 D=12 so D is eliminated. Round 2 votes counts: A=34 E=25 B=23 C=18 so C is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:202 E:202 D:199 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 10 -2 B 0 0 8 -4 0 C -4 -8 0 -6 0 D -10 4 6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555705389 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 -2 B 0 0 8 -4 0 C -4 -8 0 -6 0 D -10 4 6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555705389 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 -2 B 0 0 8 -4 0 C -4 -8 0 -6 0 D -10 4 6 0 -2 E 2 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555705389 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7931: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) B A C E D (9) A B E C D (7) D E C A B (6) D E A C B (6) C B D A E (6) D C B E A (5) B C A D E (4) A E B C D (4) E D A C B (3) E D A B C (3) E A D B C (3) D E A B C (3) C D B A E (3) C B A E D (3) A E B D C (3) E A D C B (2) E A B C D (2) D C B A E (2) C D B E A (2) B C D A E (2) B A E C D (2) E D C A B (1) E A B D C (1) D B C A E (1) C B E A D (1) C B A D E (1) B D C A E (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -2 -12 0 B 10 0 -6 -4 2 C 2 6 0 -6 6 D 12 4 6 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 -12 0 B 10 0 -6 -4 2 C 2 6 0 -6 6 D 12 4 6 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=19 C=16 E=15 A=15 so E is eliminated. Round 2 votes counts: D=42 A=23 B=19 C=16 so C is eliminated. Round 3 votes counts: D=47 B=30 A=23 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:204 B:201 E:192 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 -12 0 B 10 0 -6 -4 2 C 2 6 0 -6 6 D 12 4 6 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -12 0 B 10 0 -6 -4 2 C 2 6 0 -6 6 D 12 4 6 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -12 0 B 10 0 -6 -4 2 C 2 6 0 -6 6 D 12 4 6 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7932: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (7) C E B A D (6) E B D C A (5) A D B C E (5) E D C B A (4) E C B D A (4) B E C A D (4) E C D B A (3) D E C B A (3) D C E A B (3) D B E A C (3) D A E C B (3) D A B C E (3) B A D E C (3) E C B A D (2) E B C D A (2) D E C A B (2) D E B C A (2) D B A E C (2) D A C E B (2) D A B E C (2) C E D A B (2) C E A B D (2) C A B E D (2) B E D C A (2) A C B D E (2) A B D E C (2) D E B A C (1) D E A C B (1) D C A E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C E A D B (1) C D E A B (1) C D A E B (1) C B E A D (1) B D E A C (1) B D A E C (1) B A E C D (1) B A C E D (1) A D C B E (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -10 -14 -14 B 6 0 -6 0 -8 C 10 6 0 -20 -6 D 14 0 20 0 10 E 14 8 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.343830 C: 0.000000 D: 0.656170 E: 0.000000 Sum of squares = 0.548778034182 Cumulative probabilities = A: 0.000000 B: 0.343830 C: 0.343830 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -14 -14 B 6 0 -6 0 -8 C 10 6 0 -20 -6 D 14 0 20 0 10 E 14 8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=20 A=20 C=18 B=13 so B is eliminated. Round 2 votes counts: D=31 E=26 A=25 C=18 so C is eliminated. Round 3 votes counts: E=40 D=33 A=27 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:209 B:196 C:195 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -10 -14 -14 B 6 0 -6 0 -8 C 10 6 0 -20 -6 D 14 0 20 0 10 E 14 8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -14 -14 B 6 0 -6 0 -8 C 10 6 0 -20 -6 D 14 0 20 0 10 E 14 8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -14 -14 B 6 0 -6 0 -8 C 10 6 0 -20 -6 D 14 0 20 0 10 E 14 8 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7933: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) B D C E A (7) C A B D E (6) E D A B C (5) C B D A E (5) B E D A C (5) B D E C A (5) A E C D B (5) E D B A C (4) A C E B D (4) E D A C B (3) E A D B C (3) D E B C A (3) D B E C A (3) D B C E A (3) C B A D E (3) C A B E D (3) D E C A B (2) C A D B E (2) B D E A C (2) A C B E D (2) E B A D C (1) E A D C B (1) D E A C B (1) D E A B C (1) C D E A B (1) C B D E A (1) C A E D B (1) C A D E B (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E C D (1) A E D C B (1) A B E C D (1) Total count = 100 A B C D E A 0 6 0 -6 -4 B -6 0 -4 0 6 C 0 4 0 0 6 D 6 0 0 0 -2 E 4 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.493425 D: 0.506575 E: 0.000000 Sum of squares = 0.500086470893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.493425 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -6 -4 B -6 0 -4 0 6 C 0 4 0 0 6 D 6 0 0 0 -2 E 4 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 A=23 E=17 D=13 so D is eliminated. Round 2 votes counts: B=30 E=24 C=23 A=23 so C is eliminated. Round 3 votes counts: B=39 A=36 E=25 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:205 D:202 A:198 B:198 E:197 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 -6 -4 B -6 0 -4 0 6 C 0 4 0 0 6 D 6 0 0 0 -2 E 4 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -6 -4 B -6 0 -4 0 6 C 0 4 0 0 6 D 6 0 0 0 -2 E 4 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -6 -4 B -6 0 -4 0 6 C 0 4 0 0 6 D 6 0 0 0 -2 E 4 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7934: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) B E C D A (10) D E B C A (6) D E A B C (6) E B D C A (5) D E B A C (5) C B E A D (5) C A B E D (5) D A E B C (4) A D E C B (4) A D C E B (4) A C B D E (4) B C E A D (3) A C D B E (3) A C B E D (3) E B C A D (2) D B E C A (2) D A E C B (2) B C E D A (2) B C D E A (2) A E C B D (2) E D B C A (1) E D A B C (1) E B A C D (1) C B E D A (1) C B A D E (1) B E C A D (1) A E D B C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -18 -16 6 -6 B 18 0 4 20 8 C 16 -4 0 18 -6 D -6 -20 -18 0 -12 E 6 -8 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 6 -6 B 18 0 4 20 8 C 16 -4 0 18 -6 D -6 -20 -18 0 -12 E 6 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998626 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=24 A=23 B=18 E=10 so E is eliminated. Round 2 votes counts: D=27 B=26 C=24 A=23 so A is eliminated. Round 3 votes counts: C=38 D=36 B=26 so B is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:225 C:212 E:208 A:183 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 6 -6 B 18 0 4 20 8 C 16 -4 0 18 -6 D -6 -20 -18 0 -12 E 6 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998626 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 6 -6 B 18 0 4 20 8 C 16 -4 0 18 -6 D -6 -20 -18 0 -12 E 6 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998626 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 6 -6 B 18 0 4 20 8 C 16 -4 0 18 -6 D -6 -20 -18 0 -12 E 6 -8 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998626 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7935: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (14) C D B A E (12) B A E C D (12) E A B D C (10) C D E B A (7) D C A B E (6) D E A B C (5) C B A E D (5) A B E D C (5) B A C E D (4) E B A C D (3) D C E B A (3) E D A B C (2) C B A D E (2) E C B A D (1) E B A D C (1) E A D B C (1) D E C A B (1) D A C E B (1) C B E A D (1) C B D A E (1) A E B D C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 -4 2 B 4 0 -8 -4 0 C 6 8 0 -2 14 D 4 4 2 0 8 E -2 0 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 2 B 4 0 -8 -4 0 C 6 8 0 -2 14 D 4 4 2 0 8 E -2 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=28 E=18 B=16 A=8 so A is eliminated. Round 2 votes counts: D=30 C=28 B=23 E=19 so E is eliminated. Round 3 votes counts: B=38 D=33 C=29 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:209 B:196 A:194 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 2 B 4 0 -8 -4 0 C 6 8 0 -2 14 D 4 4 2 0 8 E -2 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 2 B 4 0 -8 -4 0 C 6 8 0 -2 14 D 4 4 2 0 8 E -2 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 2 B 4 0 -8 -4 0 C 6 8 0 -2 14 D 4 4 2 0 8 E -2 0 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999337 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7936: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C D E A B (9) B E A D C (7) A C D E B (7) C A D E B (6) B A E D C (5) E D B A C (4) D E C A B (4) D E A C B (4) C D A E B (4) B E D C A (4) A C B D E (4) D E A B C (3) A E D B C (3) E D B C A (2) E D A B C (2) C B A D E (2) C A B D E (2) B E C D A (2) B C E D A (2) A D E C B (2) E B D A C (1) E A D B C (1) D E C B A (1) D C E A B (1) D A E C B (1) C B E D A (1) C B D E A (1) C B D A E (1) B A C E D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 10 16 -16 -20 B -10 0 -2 -12 -14 C -16 2 0 -16 -16 D 16 12 16 0 8 E 20 14 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 -16 -20 B -10 0 -2 -12 -14 C -16 2 0 -16 -16 D 16 12 16 0 8 E 20 14 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=26 A=18 D=14 E=10 so E is eliminated. Round 2 votes counts: B=33 C=26 D=22 A=19 so A is eliminated. Round 3 votes counts: C=37 B=33 D=30 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:226 E:221 A:195 B:181 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 16 -16 -20 B -10 0 -2 -12 -14 C -16 2 0 -16 -16 D 16 12 16 0 8 E 20 14 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 -16 -20 B -10 0 -2 -12 -14 C -16 2 0 -16 -16 D 16 12 16 0 8 E 20 14 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 -16 -20 B -10 0 -2 -12 -14 C -16 2 0 -16 -16 D 16 12 16 0 8 E 20 14 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7937: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (18) B C A E D (16) C B A E D (14) D E A B C (7) A E C D B (5) E A D C B (4) A E C B D (4) E D A C B (3) D E B A C (3) D B E C A (3) C A B E D (3) B D C E A (3) B C D A E (3) B C A D E (3) C A E B D (2) B C D E A (2) A C E B D (2) D E C A B (1) D E B C A (1) D B E A C (1) D B C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 0 -4 8 6 B 0 0 -14 4 -2 C 4 14 0 8 -2 D -8 -4 -8 0 -8 E -6 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888868 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 0 -4 8 6 B 0 0 -14 4 -2 C 4 14 0 8 -2 D -8 -4 -8 0 -8 E -6 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=27 C=19 A=12 E=7 so E is eliminated. Round 2 votes counts: D=38 B=27 C=19 A=16 so A is eliminated. Round 3 votes counts: D=43 C=30 B=27 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:205 E:203 B:194 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 8 6 B 0 0 -14 4 -2 C 4 14 0 8 -2 D -8 -4 -8 0 -8 E -6 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 8 6 B 0 0 -14 4 -2 C 4 14 0 8 -2 D -8 -4 -8 0 -8 E -6 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 8 6 B 0 0 -14 4 -2 C 4 14 0 8 -2 D -8 -4 -8 0 -8 E -6 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7938: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) A D E C B (8) D A B C E (7) A E D C B (6) E C B A D (5) E B C A D (5) E C A D B (3) E A C B D (3) D B A C E (3) B D A C E (3) B C D E A (3) B A E D C (3) A E C D B (3) A D E B C (3) E C D A B (2) E C B D A (2) D A C B E (2) C E D A B (2) C B E D A (2) B A E C D (2) B A D E C (2) A D B C E (2) A B D E C (2) E C A B D (1) E B A C D (1) D C A E B (1) D A C E B (1) C E D B A (1) C E B D A (1) C D B E A (1) C B D E A (1) B E A C D (1) B D C A E (1) B A D C E (1) B A C E D (1) A E D B C (1) A E B D C (1) A E B C D (1) A D C E B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 20 16 14 B -4 0 8 4 -4 C -20 -8 0 0 -14 D -16 -4 0 0 -14 E -14 4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 20 16 14 B -4 0 8 4 -4 C -20 -8 0 0 -14 D -16 -4 0 0 -14 E -14 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996408 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=26 E=22 D=14 C=8 so C is eliminated. Round 2 votes counts: A=30 B=29 E=26 D=15 so D is eliminated. Round 3 votes counts: A=41 B=33 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:209 B:202 D:183 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 16 14 B -4 0 8 4 -4 C -20 -8 0 0 -14 D -16 -4 0 0 -14 E -14 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996408 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 16 14 B -4 0 8 4 -4 C -20 -8 0 0 -14 D -16 -4 0 0 -14 E -14 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996408 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 16 14 B -4 0 8 4 -4 C -20 -8 0 0 -14 D -16 -4 0 0 -14 E -14 4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996408 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7939: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (13) E B A C D (9) D C A B E (9) B E A C D (8) D C E A B (7) C D A E B (4) B A C E D (4) E D A C B (3) E A B C D (3) C A D B E (3) A C B E D (3) E D A B C (2) E B D A C (2) D E C A B (2) D E B C A (2) C A D E B (2) B E D A C (2) B C D A E (2) B C A D E (2) B A E C D (2) A C E B D (2) E D B A C (1) E B A D C (1) E A D C B (1) E A C D B (1) D E C B A (1) D C E B A (1) D C B E A (1) D C B A E (1) D B E C A (1) D B C E A (1) C B A D E (1) B D E C A (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 -8 -12 0 B -14 0 -12 -14 -18 C 8 12 0 -4 14 D 12 14 4 0 8 E 0 18 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 -12 0 B -14 0 -12 -14 -18 C 8 12 0 -4 14 D 12 14 4 0 8 E 0 18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=23 B=21 C=10 A=7 so A is eliminated. Round 2 votes counts: D=39 E=24 B=21 C=16 so C is eliminated. Round 3 votes counts: D=48 E=27 B=25 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:215 E:198 A:197 B:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -8 -12 0 B -14 0 -12 -14 -18 C 8 12 0 -4 14 D 12 14 4 0 8 E 0 18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 -12 0 B -14 0 -12 -14 -18 C 8 12 0 -4 14 D 12 14 4 0 8 E 0 18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 -12 0 B -14 0 -12 -14 -18 C 8 12 0 -4 14 D 12 14 4 0 8 E 0 18 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7940: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) C E A B D (8) A B E C D (7) E D A B C (6) C A B E D (6) E C A B D (5) D E B A C (4) D C E B A (4) D C B A E (4) C B A E D (4) C B A D E (4) D E A B C (3) E D C A B (2) E A B C D (2) D E C A B (2) D B A E C (2) D B A C E (2) C D E B A (2) A B C E D (2) E C D A B (1) E A C B D (1) E A B D C (1) D E C B A (1) D E A C B (1) D B E A C (1) D B C A E (1) C D B E A (1) C D B A E (1) C A E B D (1) B D A C E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A D E C (1) B A D C E (1) B A C D E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 0 2 22 10 B 0 0 2 24 8 C -2 -2 0 26 14 D -22 -24 -26 0 -24 E -10 -8 -14 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.528436 B: 0.471564 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501617160041 Cumulative probabilities = A: 0.528436 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 22 10 B 0 0 2 24 8 C -2 -2 0 26 14 D -22 -24 -26 0 -24 E -10 -8 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999201 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 B=19 E=18 A=11 so A is eliminated. Round 2 votes counts: B=28 C=27 D=25 E=20 so E is eliminated. Round 3 votes counts: C=35 D=33 B=32 so B is eliminated. Round 4 votes counts: C=63 D=37 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:217 B:217 E:196 D:152 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 22 10 B 0 0 2 24 8 C -2 -2 0 26 14 D -22 -24 -26 0 -24 E -10 -8 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999201 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 22 10 B 0 0 2 24 8 C -2 -2 0 26 14 D -22 -24 -26 0 -24 E -10 -8 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999201 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 22 10 B 0 0 2 24 8 C -2 -2 0 26 14 D -22 -24 -26 0 -24 E -10 -8 -14 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999201 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7941: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) E A B D C (5) D C B E A (5) A E B C D (5) E B A C D (4) D C E B A (4) C D A B E (4) C A D B E (4) E D B A C (3) E A B C D (3) D E B A C (3) D C B A E (3) D C A E B (3) D C A B E (3) C D B A E (3) D B E C A (2) D B C E A (2) C B D A E (2) C B A E D (2) C A B E D (2) C A B D E (2) B E C A D (2) B E A D C (2) B E A C D (2) B C A E D (2) B A E C D (2) A C E B D (2) A C B E D (2) A B E C D (2) E B D A C (1) E A D B C (1) D E B C A (1) D E A B C (1) D C E A B (1) C D A E B (1) C B A D E (1) B C E D A (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -14 0 14 -6 B 14 0 8 10 0 C 0 -8 0 0 -2 D -14 -10 0 0 -10 E 6 0 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.266282 C: 0.000000 D: 0.000000 E: 0.733718 Sum of squares = 0.609248238203 Cumulative probabilities = A: 0.000000 B: 0.266282 C: 0.266282 D: 0.266282 E: 1.000000 A B C D E A 0 -14 0 14 -6 B 14 0 8 10 0 C 0 -8 0 0 -2 D -14 -10 0 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 C=21 A=13 B=11 so B is eliminated. Round 2 votes counts: E=33 D=28 C=24 A=15 so A is eliminated. Round 3 votes counts: E=44 D=28 C=28 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:216 E:209 A:197 C:195 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 14 -6 B 14 0 8 10 0 C 0 -8 0 0 -2 D -14 -10 0 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 14 -6 B 14 0 8 10 0 C 0 -8 0 0 -2 D -14 -10 0 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 14 -6 B 14 0 8 10 0 C 0 -8 0 0 -2 D -14 -10 0 0 -10 E 6 0 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7942: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) E D C B A (7) C B E A D (6) D E C A B (5) D A E B C (4) B A C E D (4) E C B D A (3) D E A C B (3) C B A D E (3) A D B E C (3) E D B A C (2) E D A B C (2) E B C D A (2) E B C A D (2) E B A D C (2) D C E A B (2) C E D B A (2) C B E D A (2) C B A E D (2) B E A C D (2) B A E C D (2) B A C D E (2) A D B C E (2) A B E D C (2) A B E C D (2) A B C D E (2) E D B C A (1) E B D C A (1) E B A C D (1) E A D B C (1) D E C B A (1) D C A E B (1) D C A B E (1) D A C E B (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B A E (1) C A D B E (1) C A B D E (1) B E C A D (1) B C E A D (1) B C A E D (1) A D E B C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 0 -6 -24 B 8 0 6 -6 -14 C 0 -6 0 -4 -20 D 6 6 4 0 -6 E 24 14 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 0 -6 -24 B 8 0 6 -6 -14 C 0 -6 0 -4 -20 D 6 6 4 0 -6 E 24 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 C=21 A=14 B=13 so B is eliminated. Round 2 votes counts: D=28 E=27 C=23 A=22 so A is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:232 D:205 B:197 C:185 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 0 -6 -24 B 8 0 6 -6 -14 C 0 -6 0 -4 -20 D 6 6 4 0 -6 E 24 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -6 -24 B 8 0 6 -6 -14 C 0 -6 0 -4 -20 D 6 6 4 0 -6 E 24 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -6 -24 B 8 0 6 -6 -14 C 0 -6 0 -4 -20 D 6 6 4 0 -6 E 24 14 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7943: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) E B D A C (9) C A D B E (9) B C D A E (9) E A C D B (8) A C D E B (6) E A D C B (4) C A D E B (4) B E C A D (4) E B C A D (3) B D E C A (3) A E C D B (3) E D A C B (2) E A D B C (2) E A C B D (2) D C A B E (2) C D A B E (2) C B A D E (2) B E D A C (2) B C A D E (2) E D A B C (1) E B A D C (1) E B A C D (1) E A B D C (1) D B C A E (1) C B D A E (1) C A E D B (1) C A E B D (1) B C A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -14 12 14 B 2 0 4 8 0 C 14 -4 0 20 8 D -12 -8 -20 0 6 E -14 0 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.917997 C: 0.000000 D: 0.000000 E: 0.082003 Sum of squares = 0.849443271337 Cumulative probabilities = A: 0.000000 B: 0.917997 C: 0.917997 D: 0.917997 E: 1.000000 A B C D E A 0 -2 -14 12 14 B 2 0 4 8 0 C 14 -4 0 20 8 D -12 -8 -20 0 6 E -14 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250004104 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=33 C=20 A=10 D=3 so D is eliminated. Round 2 votes counts: E=34 B=34 C=22 A=10 so A is eliminated. Round 3 votes counts: E=37 B=34 C=29 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:219 B:207 A:205 E:186 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -14 12 14 B 2 0 4 8 0 C 14 -4 0 20 8 D -12 -8 -20 0 6 E -14 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250004104 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 12 14 B 2 0 4 8 0 C 14 -4 0 20 8 D -12 -8 -20 0 6 E -14 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250004104 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 12 14 B 2 0 4 8 0 C 14 -4 0 20 8 D -12 -8 -20 0 6 E -14 0 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.875000 C: 0.000000 D: 0.000000 E: 0.125000 Sum of squares = 0.781250004104 Cumulative probabilities = A: 0.000000 B: 0.875000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7944: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (12) D B C E A (11) C B D A E (11) D B C A E (9) E A C B D (6) B C D A E (6) E A D C B (4) E A C D B (4) A E C D B (4) E A D B C (3) D E A B C (3) E D A B C (2) D E B A C (2) D C B A E (2) D A E B C (2) C D B A E (2) C B A E D (2) B D C E A (2) B C D E A (2) A C E B D (2) E A B C D (1) D B A E C (1) D B A C E (1) C B A D E (1) C A E B D (1) C A B E D (1) B D C A E (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -2 -14 20 B 6 0 -8 -4 4 C 2 8 0 12 10 D 14 4 -12 0 14 E -20 -4 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -14 20 B 6 0 -8 -4 4 C 2 8 0 12 10 D 14 4 -12 0 14 E -20 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=20 A=20 C=18 B=11 so B is eliminated. Round 2 votes counts: D=34 C=26 E=20 A=20 so E is eliminated. Round 3 votes counts: A=38 D=36 C=26 so C is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:210 A:199 B:199 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -14 20 B 6 0 -8 -4 4 C 2 8 0 12 10 D 14 4 -12 0 14 E -20 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -14 20 B 6 0 -8 -4 4 C 2 8 0 12 10 D 14 4 -12 0 14 E -20 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -14 20 B 6 0 -8 -4 4 C 2 8 0 12 10 D 14 4 -12 0 14 E -20 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7945: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) B A D E C (7) B E D A C (6) E D C B A (5) C A D E B (5) A C B E D (5) A C B D E (5) A B C D E (4) E B D C A (3) D E B C A (3) D E B A C (3) C A E B D (3) B D E A C (3) B D A E C (3) E C D B A (2) E C B D A (2) D E C A B (2) C E D A B (2) C D E A B (2) C D A E B (2) C A E D B (2) B A E C D (2) B A D C E (2) B A C E D (2) A C D B E (2) A B C E D (2) E D B A C (1) E C D A B (1) E B C D A (1) D E C B A (1) D B E A C (1) D A C E B (1) D A B E C (1) C E B D A (1) C E A D B (1) C E A B D (1) C A B D E (1) B A E D C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 6 -6 4 B 12 0 6 10 -2 C -6 -6 0 -4 -10 D 6 -10 4 0 0 E -4 2 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839396 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 -12 6 -6 4 B 12 0 6 10 -2 C -6 -6 0 -4 -10 D 6 -10 4 0 0 E -4 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839474 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 C=20 A=20 D=12 so D is eliminated. Round 2 votes counts: E=31 B=27 A=22 C=20 so C is eliminated. Round 3 votes counts: E=38 A=35 B=27 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:213 E:204 D:200 A:196 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 6 -6 4 B 12 0 6 10 -2 C -6 -6 0 -4 -10 D 6 -10 4 0 0 E -4 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839474 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -6 4 B 12 0 6 10 -2 C -6 -6 0 -4 -10 D 6 -10 4 0 0 E -4 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839474 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -6 4 B 12 0 6 10 -2 C -6 -6 0 -4 -10 D 6 -10 4 0 0 E -4 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.506172839474 Cumulative probabilities = A: 0.111111 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7946: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (6) C B E D A (6) B E C D A (6) A E D B C (6) A D C E B (6) E B A C D (5) D A C B E (5) C B D E A (5) E A B C D (4) D C A B E (4) C D B E A (4) E C B A D (3) C E B A D (3) A E B D C (3) E C A B D (2) D C B E A (2) D B C E A (2) D A B E C (2) B E D C A (2) B C E D A (2) B C D E A (2) A E C B D (2) A D E B C (2) E B C A D (1) E B A D C (1) E A C B D (1) E A B D C (1) D B E C A (1) D B C A E (1) D B A C E (1) C E A B D (1) C D B A E (1) C D A B E (1) C A E B D (1) B E D A C (1) B D C E A (1) A E C D B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -14 -10 -12 B 12 0 -12 6 10 C 14 12 0 4 10 D 10 -6 -4 0 -4 E 12 -10 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -10 -12 B 12 0 -12 6 10 C 14 12 0 4 10 D 10 -6 -4 0 -4 E 12 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 A=22 E=18 B=14 so B is eliminated. Round 2 votes counts: E=27 C=26 D=25 A=22 so A is eliminated. Round 3 votes counts: E=39 D=34 C=27 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:220 B:208 D:198 E:198 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -10 -12 B 12 0 -12 6 10 C 14 12 0 4 10 D 10 -6 -4 0 -4 E 12 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -10 -12 B 12 0 -12 6 10 C 14 12 0 4 10 D 10 -6 -4 0 -4 E 12 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -10 -12 B 12 0 -12 6 10 C 14 12 0 4 10 D 10 -6 -4 0 -4 E 12 -10 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7947: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) B C A E D (7) D E A C B (6) B D C E A (6) D E A B C (4) C B A E D (4) A C E B D (4) E C A D B (3) D B E C A (3) B C D E A (3) B C A D E (3) A C B E D (3) E C A B D (2) D E C B A (2) D E B C A (2) D E B A C (2) D B E A C (2) D B C E A (2) D A E C B (2) D A E B C (2) C E B A D (2) C B E A D (2) B D C A E (2) B C E D A (2) A E D C B (2) A E C D B (2) A D E C B (2) E D C B A (1) E A D C B (1) E A C D B (1) D B C A E (1) D B A E C (1) D B A C E (1) D A B E C (1) C E A B D (1) C A B E D (1) B E C D A (1) B D A C E (1) B C E A D (1) B C D A E (1) B A C D E (1) A D C E B (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -4 -10 -12 B 6 0 2 -4 2 C 4 -2 0 -10 2 D 10 4 10 0 6 E 12 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -10 -12 B 6 0 2 -4 2 C 4 -2 0 -10 2 D 10 4 10 0 6 E 12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=28 A=16 E=15 C=10 so C is eliminated. Round 2 votes counts: B=34 D=31 E=18 A=17 so A is eliminated. Round 3 votes counts: B=39 D=35 E=26 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:203 E:201 C:197 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -4 -10 -12 B 6 0 2 -4 2 C 4 -2 0 -10 2 D 10 4 10 0 6 E 12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -10 -12 B 6 0 2 -4 2 C 4 -2 0 -10 2 D 10 4 10 0 6 E 12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -10 -12 B 6 0 2 -4 2 C 4 -2 0 -10 2 D 10 4 10 0 6 E 12 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7948: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) B A C D E (10) E D C A B (6) E C D A B (6) D E C B A (6) D E C A B (6) D E B C A (4) B A C E D (4) D E B A C (3) D E A C B (3) D E A B C (3) D A E B C (3) C B A E D (3) A C B E D (3) A B C E D (3) E C A D B (2) C B E A D (2) C A B E D (2) B D A C E (2) B C D A E (2) A E C D B (2) A B C D E (2) E D C B A (1) E D A C B (1) E C B D A (1) E C A B D (1) E A C D B (1) D B E C A (1) D A B E C (1) C E A B D (1) C A E B D (1) B A D C E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -12 2 4 B 2 0 4 -2 -2 C 12 -4 0 14 -4 D -2 2 -14 0 -2 E -4 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.223052 B: 0.230524 C: 0.107790 D: 0.000000 E: 0.438633 Sum of squares = 0.306911516566 Cumulative probabilities = A: 0.223052 B: 0.453577 C: 0.561367 D: 0.561367 E: 1.000000 A B C D E A 0 -2 -12 2 4 B 2 0 4 -2 -2 C 12 -4 0 14 -4 D -2 2 -14 0 -2 E -4 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.228916 B: 0.289156 C: 0.084338 D: 0.000000 E: 0.397591 Sum of squares = 0.301204819283 Cumulative probabilities = A: 0.228916 B: 0.518071 C: 0.602409 D: 0.602409 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=30 B=30 E=19 A=12 C=9 so C is eliminated. Round 2 votes counts: B=35 D=30 E=20 A=15 so A is eliminated. Round 3 votes counts: B=45 D=32 E=23 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:209 E:202 B:201 A:196 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -12 2 4 B 2 0 4 -2 -2 C 12 -4 0 14 -4 D -2 2 -14 0 -2 E -4 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.228916 B: 0.289156 C: 0.084338 D: 0.000000 E: 0.397591 Sum of squares = 0.301204819283 Cumulative probabilities = A: 0.228916 B: 0.518071 C: 0.602409 D: 0.602409 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 2 4 B 2 0 4 -2 -2 C 12 -4 0 14 -4 D -2 2 -14 0 -2 E -4 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.228916 B: 0.289156 C: 0.084338 D: 0.000000 E: 0.397591 Sum of squares = 0.301204819283 Cumulative probabilities = A: 0.228916 B: 0.518071 C: 0.602409 D: 0.602409 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 2 4 B 2 0 4 -2 -2 C 12 -4 0 14 -4 D -2 2 -14 0 -2 E -4 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.228916 B: 0.289156 C: 0.084338 D: 0.000000 E: 0.397591 Sum of squares = 0.301204819283 Cumulative probabilities = A: 0.228916 B: 0.518071 C: 0.602409 D: 0.602409 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7949: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (6) E D C A B (4) E C D A B (4) C E A D B (4) C A E D B (4) C A E B D (4) B D A C E (4) B A C E D (4) A C B E D (4) D B E C A (3) D B E A C (3) D B A C E (3) D A B C E (3) B E D C A (3) B D A E C (3) A D C B E (3) A C E D B (3) A C E B D (3) E D C B A (2) E B C D A (2) D E B C A (2) D E A C B (2) D A C E B (2) B E C A D (2) B D E A C (2) A C D E B (2) A C D B E (2) A B D C E (2) E D B C A (1) E C D B A (1) E C A D B (1) D E C B A (1) D E C A B (1) D E B A C (1) C E B A D (1) C E A B D (1) B A D C E (1) B A C D E (1) A D C E B (1) A D B C E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 6 -6 6 B -8 0 -2 -8 6 C -6 2 0 -8 12 D 6 8 8 0 2 E -6 -6 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -6 6 B -8 0 -2 -8 6 C -6 2 0 -8 12 D 6 8 8 0 2 E -6 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 D=21 E=15 C=14 so C is eliminated. Round 2 votes counts: A=32 B=26 E=21 D=21 so E is eliminated. Round 3 votes counts: A=38 D=33 B=29 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 C:200 B:194 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -6 6 B -8 0 -2 -8 6 C -6 2 0 -8 12 D 6 8 8 0 2 E -6 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -6 6 B -8 0 -2 -8 6 C -6 2 0 -8 12 D 6 8 8 0 2 E -6 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -6 6 B -8 0 -2 -8 6 C -6 2 0 -8 12 D 6 8 8 0 2 E -6 -6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7950: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) C E A B D (8) E C D B A (7) C E A D B (6) E C B A D (5) C A E B D (5) B D A E C (5) A D B C E (5) A B D C E (5) D B E A C (4) A C E B D (4) A B C D E (4) E D B C A (3) D B A C E (3) A C D E B (3) A C B E D (3) E C D A B (2) D B E C A (2) D A B C E (2) B D A C E (2) E D C B A (1) E D C A B (1) E C B D A (1) E B C D A (1) D E C B A (1) D E C A B (1) D E B C A (1) D E A C B (1) C B A E D (1) B E D C A (1) B A D C E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 6 4 8 B -4 0 -4 -8 -2 C -6 4 0 2 8 D -4 8 -2 0 2 E -8 2 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 4 8 B -4 0 -4 -8 -2 C -6 4 0 2 8 D -4 8 -2 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 E=21 C=20 B=9 so B is eliminated. Round 2 votes counts: D=31 A=27 E=22 C=20 so C is eliminated. Round 3 votes counts: E=36 A=33 D=31 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:204 D:202 E:192 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 4 8 B -4 0 -4 -8 -2 C -6 4 0 2 8 D -4 8 -2 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 4 8 B -4 0 -4 -8 -2 C -6 4 0 2 8 D -4 8 -2 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 4 8 B -4 0 -4 -8 -2 C -6 4 0 2 8 D -4 8 -2 0 2 E -8 2 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7951: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (14) C A D E B (13) B E D C A (5) B A C E D (5) A B C E D (5) E D B A C (4) C A D B E (4) A C D E B (4) E B D A C (3) D E A C B (3) C A B D E (3) B E A D C (3) B D E C A (3) A E D C B (3) E D B C A (2) E D A B C (2) C D A E B (2) C B A D E (2) B C E D A (2) A E B D C (2) A C E D B (2) E B D C A (1) E A D B C (1) D E C B A (1) D E C A B (1) D C E B A (1) D C E A B (1) C D B E A (1) B C E A D (1) B C D A E (1) B C A E D (1) B A E D C (1) A E D B C (1) A D E C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 10 6 2 B 2 0 14 6 4 C -10 -14 0 -6 -2 D -6 -6 6 0 -18 E -2 -4 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 6 2 B 2 0 14 6 4 C -10 -14 0 -6 -2 D -6 -6 6 0 -18 E -2 -4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=25 A=19 E=13 D=7 so D is eliminated. Round 2 votes counts: B=36 C=27 A=19 E=18 so E is eliminated. Round 3 votes counts: B=46 C=29 A=25 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:208 E:207 D:188 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 6 2 B 2 0 14 6 4 C -10 -14 0 -6 -2 D -6 -6 6 0 -18 E -2 -4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 6 2 B 2 0 14 6 4 C -10 -14 0 -6 -2 D -6 -6 6 0 -18 E -2 -4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 6 2 B 2 0 14 6 4 C -10 -14 0 -6 -2 D -6 -6 6 0 -18 E -2 -4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996172 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7952: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) A C B D E (8) A B C D E (6) A B D C E (5) A B C E D (5) E D B C A (4) C E D A B (3) B D E C A (3) A E C B D (3) A E B D C (3) D E C B A (2) D B E C A (2) C E A D B (2) C D E B A (2) C D E A B (2) C A E D B (2) C A B D E (2) B D C A E (2) B D A C E (2) B A D C E (2) A E C D B (2) A E B C D (2) A C E B D (2) A C B E D (2) A B E D C (2) A B D E C (2) E D C A B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C A D B (1) E B D A C (1) E A D C B (1) D C E B A (1) D C B E A (1) C D B E A (1) C B D E A (1) C B D A E (1) B D E A C (1) B D C E A (1) B C D A E (1) B C A D E (1) A E D C B (1) A E D B C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 24 8 14 18 B -24 0 -4 18 4 C -8 4 0 6 12 D -14 -18 -6 0 -2 E -18 -4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 8 14 18 B -24 0 -4 18 4 C -8 4 0 6 12 D -14 -18 -6 0 -2 E -18 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 E=19 C=16 B=13 D=6 so D is eliminated. Round 2 votes counts: A=46 E=21 C=18 B=15 so B is eliminated. Round 3 votes counts: A=50 E=27 C=23 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:232 C:207 B:197 E:184 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 24 8 14 18 B -24 0 -4 18 4 C -8 4 0 6 12 D -14 -18 -6 0 -2 E -18 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 8 14 18 B -24 0 -4 18 4 C -8 4 0 6 12 D -14 -18 -6 0 -2 E -18 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 8 14 18 B -24 0 -4 18 4 C -8 4 0 6 12 D -14 -18 -6 0 -2 E -18 -4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7953: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (8) E C D A B (5) C E D B A (5) E D C A B (4) D E C A B (4) B D A C E (4) B A D C E (4) A D B E C (4) A B E C D (4) D E A C B (3) D B A E C (3) D B A C E (3) D A E B C (3) C E A B D (3) B C D A E (3) A D E B C (3) E C A D B (2) E C A B D (2) D C E B A (2) C E B A D (2) C B E D A (2) B D C A E (2) B A C E D (2) A B E D C (2) A B D E C (2) E D A C B (1) E A D C B (1) E A C B D (1) D E C B A (1) D E A B C (1) D C B E A (1) D B C E A (1) D A B E C (1) C E D A B (1) C D B E A (1) C B E A D (1) B C A D E (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B D C (1) A E B C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -2 -2 12 B -6 0 14 -2 4 C 2 -14 0 -8 -4 D 2 2 8 0 -2 E -12 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.593749999976 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 A B C D E A 0 6 -2 -2 12 B -6 0 14 -2 4 C 2 -14 0 -8 -4 D 2 2 8 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.59375000002 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 A=21 E=16 C=15 so C is eliminated. Round 2 votes counts: B=28 E=27 D=24 A=21 so A is eliminated. Round 3 votes counts: B=38 E=31 D=31 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:207 B:205 D:205 E:195 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 -2 12 B -6 0 14 -2 4 C 2 -14 0 -8 -4 D 2 2 8 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.59375000002 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 12 B -6 0 14 -2 4 C 2 -14 0 -8 -4 D 2 2 8 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.59375000002 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 12 B -6 0 14 -2 4 C 2 -14 0 -8 -4 D 2 2 8 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.59375000002 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7954: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) E B A C D (8) E A B D C (8) C D E B A (8) A B D C E (7) E C D B A (5) C D B A E (5) B A E C D (5) D C A B E (4) E C B D A (3) E C B A D (3) E B A D C (3) C D A B E (3) E D C A B (2) D E A B C (2) D C A E B (2) D A E B C (2) D A B C E (2) C E D B A (2) B C A D E (2) B A C D E (2) A D B C E (2) E D A C B (1) E D A B C (1) C E B D A (1) C E B A D (1) C B A D E (1) B E A C D (1) B A E D C (1) B A C E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -4 -6 -18 B 4 0 -2 -2 -24 C 4 2 0 2 6 D 6 2 -2 0 6 E 18 24 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -18 B 4 0 -2 -2 -24 C 4 2 0 2 6 D 6 2 -2 0 6 E 18 24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=22 C=21 B=12 A=11 so A is eliminated. Round 2 votes counts: E=34 D=24 C=21 B=21 so C is eliminated. Round 3 votes counts: D=40 E=38 B=22 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:215 C:207 D:206 B:188 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -18 B 4 0 -2 -2 -24 C 4 2 0 2 6 D 6 2 -2 0 6 E 18 24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -18 B 4 0 -2 -2 -24 C 4 2 0 2 6 D 6 2 -2 0 6 E 18 24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -18 B 4 0 -2 -2 -24 C 4 2 0 2 6 D 6 2 -2 0 6 E 18 24 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7955: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) E D B A C (8) E C D A B (7) D B A E C (7) B A D C E (7) B A C D E (6) E C A D B (5) C A B D E (5) E C A B D (4) B D A C E (4) D E B A C (3) D B E A C (3) D B A C E (3) C E A B D (3) B C A D E (3) A B C D E (3) E D C B A (2) D A B C E (2) C B A E D (2) A C B D E (2) E D B C A (1) E D A B C (1) E C B A D (1) E B C A D (1) D A C B E (1) D A B E C (1) C A E B D (1) C A B E D (1) B D A E C (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 8 -8 2 B 6 0 12 -8 6 C -8 -12 0 -10 -10 D 8 8 10 0 4 E -2 -6 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 -8 2 B 6 0 12 -8 6 C -8 -12 0 -10 -10 D 8 8 10 0 4 E -2 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=22 D=20 C=12 A=6 so A is eliminated. Round 2 votes counts: E=40 B=26 D=20 C=14 so C is eliminated. Round 3 votes counts: E=44 B=36 D=20 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:215 B:208 E:199 A:198 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 8 -8 2 B 6 0 12 -8 6 C -8 -12 0 -10 -10 D 8 8 10 0 4 E -2 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 -8 2 B 6 0 12 -8 6 C -8 -12 0 -10 -10 D 8 8 10 0 4 E -2 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 -8 2 B 6 0 12 -8 6 C -8 -12 0 -10 -10 D 8 8 10 0 4 E -2 -6 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7956: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (6) B A D C E (6) B C A D E (5) E D C A B (4) E C D B A (4) E C D A B (4) E A D B C (4) D A B C E (4) B A C E D (4) A B D C E (4) E A B D C (3) C E D B A (3) C B D A E (3) B C A E D (3) A D E B C (3) D E A C B (2) D C E A B (2) D C B A E (2) D A E B C (2) C E B D A (2) C B E A D (2) B C E A D (2) B A E C D (2) B A C D E (2) A E B D C (2) A D B E C (2) E D A B C (1) E C B D A (1) E C A D B (1) E A D C B (1) E A C D B (1) E A C B D (1) E A B C D (1) D C A B E (1) D A E C B (1) D A B E C (1) C E B A D (1) C D E B A (1) C D B A E (1) C B E D A (1) C B D E A (1) C B A E D (1) B C D A E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 8 4 2 B -4 0 6 -4 -2 C -8 -6 0 -4 6 D -4 4 4 0 -10 E -2 2 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 4 2 B -4 0 6 -4 -2 C -8 -6 0 -4 6 D -4 4 4 0 -10 E -2 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=25 C=16 D=15 A=12 so A is eliminated. Round 2 votes counts: E=34 B=29 D=21 C=16 so C is eliminated. Round 3 votes counts: E=40 B=37 D=23 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:209 E:202 B:198 D:197 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 2 B -4 0 6 -4 -2 C -8 -6 0 -4 6 D -4 4 4 0 -10 E -2 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 2 B -4 0 6 -4 -2 C -8 -6 0 -4 6 D -4 4 4 0 -10 E -2 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 2 B -4 0 6 -4 -2 C -8 -6 0 -4 6 D -4 4 4 0 -10 E -2 2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7957: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (7) D A E B C (6) C B E A D (6) A D B E C (5) E C D B A (4) E C B D A (4) C E B D A (4) B E D A C (4) A D C E B (4) A D B C E (4) E B C D A (3) D A E C B (3) B A C D E (3) A B D C E (3) E D C B A (2) E D C A B (2) E B D C A (2) D E A B C (2) D A B E C (2) C B A E D (2) B E C A D (2) B C A E D (2) B A D E C (2) A D E C B (2) A D E B C (2) A D C B E (2) E D B C A (1) D E B A C (1) D E A C B (1) D A C E B (1) C E D A B (1) C E B A D (1) C E A D B (1) C E A B D (1) B E C D A (1) B E A D C (1) B D A E C (1) B A D C E (1) B A C E D (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 8 10 -2 B 10 0 12 4 2 C -8 -12 0 -8 -6 D -10 -4 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 10 -2 B 10 0 12 4 2 C -8 -12 0 -8 -6 D -10 -4 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=25 A=25 E=18 D=16 C=16 so D is eliminated. Round 2 votes counts: A=37 B=25 E=22 C=16 so C is eliminated. Round 3 votes counts: A=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:205 A:203 D:195 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 8 10 -2 B 10 0 12 4 2 C -8 -12 0 -8 -6 D -10 -4 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 10 -2 B 10 0 12 4 2 C -8 -12 0 -8 -6 D -10 -4 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 10 -2 B 10 0 12 4 2 C -8 -12 0 -8 -6 D -10 -4 8 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7958: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (11) C B E A D (10) D E A B C (8) E D C B A (7) D A B E C (7) A D B C E (7) A B C D E (6) B C A E D (5) A C B D E (4) E C D B A (3) D E A C B (3) D A E B C (3) C A B E D (3) E D B C A (2) E D B A C (2) E C B D A (2) E C B A D (2) D A E C B (2) C E B A D (2) E B D C A (1) D E C A B (1) D E B A C (1) D A B C E (1) C A B D E (1) B E C A D (1) B D E A C (1) B C E A D (1) B A D E C (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -4 12 6 B 6 0 -4 4 22 C 4 4 0 4 6 D -12 -4 -4 0 -4 E -6 -22 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 12 6 B 6 0 -4 4 22 C 4 4 0 4 6 D -12 -4 -4 0 -4 E -6 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 E=19 A=18 B=10 so B is eliminated. Round 2 votes counts: C=33 D=27 E=20 A=20 so E is eliminated. Round 3 votes counts: C=41 D=39 A=20 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:209 A:204 D:188 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 12 6 B 6 0 -4 4 22 C 4 4 0 4 6 D -12 -4 -4 0 -4 E -6 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 12 6 B 6 0 -4 4 22 C 4 4 0 4 6 D -12 -4 -4 0 -4 E -6 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 12 6 B 6 0 -4 4 22 C 4 4 0 4 6 D -12 -4 -4 0 -4 E -6 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999696 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7959: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) B A D C E (7) D A B E C (6) E C D A B (5) B C E A D (5) E C D B A (4) D E C A B (4) D A E C B (4) C E B A D (4) B D A E C (4) E D C A B (3) D E A C B (3) D A E B C (3) B E C D A (3) B A C E D (3) A B D C E (3) A B C D E (3) D B E A C (2) D B A E C (2) C E A B D (2) B D E C A (2) B C A E D (2) B A C D E (2) A D C E B (2) A D B C E (2) E C B A D (1) D E C B A (1) D E B C A (1) D E A B C (1) D B E C A (1) C A E D B (1) C A E B D (1) B E D C A (1) B D A C E (1) A D C B E (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -2 -2 B -8 0 10 -10 2 C -2 -10 0 -8 -2 D 2 10 8 0 10 E 2 -2 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -2 -2 B -8 0 10 -10 2 C -2 -10 0 -8 -2 D 2 10 8 0 10 E 2 -2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=28 C=16 E=13 A=13 so E is eliminated. Round 2 votes counts: D=31 B=30 C=26 A=13 so A is eliminated. Round 3 votes counts: B=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:203 B:197 E:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -2 -2 B -8 0 10 -10 2 C -2 -10 0 -8 -2 D 2 10 8 0 10 E 2 -2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 -2 B -8 0 10 -10 2 C -2 -10 0 -8 -2 D 2 10 8 0 10 E 2 -2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 -2 B -8 0 10 -10 2 C -2 -10 0 -8 -2 D 2 10 8 0 10 E 2 -2 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7960: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B D A C E (9) B A D C E (9) A B C E D (7) E C A B D (5) B A E C D (5) E C D B A (4) E C A D B (4) D C E A B (4) C E D A B (4) A E C B D (4) A B E C D (4) D B E C A (3) E C B D A (2) D C E B A (2) D C B E A (2) D A B C E (2) C E A D B (2) B D A E C (2) A C E B D (2) A B D C E (2) E D C B A (1) E B C D A (1) E A C B D (1) D B C E A (1) D B A C E (1) C D E A B (1) B A D E C (1) A E B C D (1) A D C E B (1) A D C B E (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 10 4 8 B -14 0 -2 10 0 C -10 2 0 16 6 D -4 -10 -16 0 -14 E -8 0 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998333 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 4 8 B -14 0 -2 10 0 C -10 2 0 16 6 D -4 -10 -16 0 -14 E -8 0 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995825 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=26 A=25 D=15 C=7 so C is eliminated. Round 2 votes counts: E=33 B=26 A=25 D=16 so D is eliminated. Round 3 votes counts: E=40 B=33 A=27 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:218 C:207 E:200 B:197 D:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 4 8 B -14 0 -2 10 0 C -10 2 0 16 6 D -4 -10 -16 0 -14 E -8 0 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995825 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 4 8 B -14 0 -2 10 0 C -10 2 0 16 6 D -4 -10 -16 0 -14 E -8 0 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995825 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 4 8 B -14 0 -2 10 0 C -10 2 0 16 6 D -4 -10 -16 0 -14 E -8 0 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995825 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7961: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) A D E C B (8) D A C E B (7) C B E D A (7) D A B C E (5) E A D C B (4) D A E C B (4) C B D E A (4) B E C A D (4) E C B A D (3) E B C A D (3) E B A D C (3) D A C B E (3) C B D A E (3) B D C A E (3) B C D A E (3) A D E B C (3) E A D B C (2) E A B C D (2) C E D A B (2) C D B A E (2) C D A E B (2) E C A D B (1) E B A C D (1) E A C D B (1) E A B D C (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) C E B A D (1) C D A B E (1) B E A D C (1) B E A C D (1) B D A C E (1) Total count = 100 A B C D E A 0 -4 -4 -22 0 B 4 0 -12 0 2 C 4 12 0 0 16 D 22 0 0 0 8 E 0 -2 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.592679 D: 0.407321 E: 0.000000 Sum of squares = 0.517178876664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.592679 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -22 0 B 4 0 -12 0 2 C 4 12 0 0 16 D 22 0 0 0 8 E 0 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 B=22 E=21 A=11 so A is eliminated. Round 2 votes counts: D=35 C=22 B=22 E=21 so E is eliminated. Round 3 votes counts: D=41 B=32 C=27 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:216 D:215 B:197 E:187 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -22 0 B 4 0 -12 0 2 C 4 12 0 0 16 D 22 0 0 0 8 E 0 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -22 0 B 4 0 -12 0 2 C 4 12 0 0 16 D 22 0 0 0 8 E 0 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -22 0 B 4 0 -12 0 2 C 4 12 0 0 16 D 22 0 0 0 8 E 0 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7962: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) B D E A C (7) A C E B D (7) E C D B A (5) C D B E A (4) C D B A E (4) A B D C E (4) E D C B A (3) E C A B D (3) E B D A C (3) E A C B D (3) E A B D C (3) D B E C A (3) D B A C E (3) C E D A B (3) C E A B D (3) B D A E C (3) A E C B D (3) A C B D E (3) A B D E C (3) E D B C A (2) D C B A E (2) C D A B E (2) C A D E B (2) E C A D B (1) D C B E A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E D B A (1) C E A D B (1) C D A E B (1) C A E D B (1) C A E B D (1) B E D A C (1) A D B C E (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -2 -2 4 B -10 0 -24 -2 8 C 2 24 0 14 8 D 2 2 -14 0 10 E -4 -8 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -2 4 B -10 0 -24 -2 8 C 2 24 0 14 8 D 2 2 -14 0 10 E -4 -8 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=24 E=23 D=12 B=11 so B is eliminated. Round 2 votes counts: C=30 E=24 A=24 D=22 so D is eliminated. Round 3 votes counts: E=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:205 D:200 B:186 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 -2 4 B -10 0 -24 -2 8 C 2 24 0 14 8 D 2 2 -14 0 10 E -4 -8 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -2 4 B -10 0 -24 -2 8 C 2 24 0 14 8 D 2 2 -14 0 10 E -4 -8 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -2 4 B -10 0 -24 -2 8 C 2 24 0 14 8 D 2 2 -14 0 10 E -4 -8 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7963: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) C B D E A (8) D E A B C (7) D E C A B (5) C E D A B (4) B C A D E (4) B A D E C (4) C D E B A (3) C B A D E (3) A E D B C (3) E D C A B (2) E A D C B (2) E A D B C (2) D E C B A (2) D E B A C (2) D E A C B (2) D B E C A (2) D B E A C (2) D B A E C (2) C D E A B (2) C B A E D (2) C A E B D (2) B D A E C (2) B C D A E (2) B C A E D (2) B A D C E (2) B A C E D (2) A E C B D (2) A B E D C (2) A B C E D (2) E D A B C (1) C E B D A (1) C E A D B (1) C D B E A (1) C B E A D (1) B D E A C (1) B C D E A (1) B A E C D (1) B A C D E (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 4 -20 -20 B 2 0 -4 -10 -4 C -4 4 0 -8 -12 D 20 10 8 0 18 E 20 4 12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -20 -20 B 2 0 -4 -10 -4 C -4 4 0 -8 -12 D 20 10 8 0 18 E 20 4 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 B=22 E=15 A=11 so A is eliminated. Round 2 votes counts: C=28 B=26 D=25 E=21 so E is eliminated. Round 3 votes counts: D=43 C=31 B=26 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:209 B:192 C:190 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -20 -20 B 2 0 -4 -10 -4 C -4 4 0 -8 -12 D 20 10 8 0 18 E 20 4 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -20 -20 B 2 0 -4 -10 -4 C -4 4 0 -8 -12 D 20 10 8 0 18 E 20 4 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -20 -20 B 2 0 -4 -10 -4 C -4 4 0 -8 -12 D 20 10 8 0 18 E 20 4 12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7964: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) B A E C D (8) B D A E C (7) A E C D B (7) D C E A B (6) B D C A E (6) E A C D B (5) D B C E A (5) C D E A B (5) A E C B D (4) A E B C D (4) C E D A B (3) A E D C B (3) C E A D B (2) C E A B D (2) C D B E A (2) C B E D A (2) B D A C E (2) B C E A D (2) B A E D C (2) A E D B C (2) E A C B D (1) D C B E A (1) D C A E B (1) C D E B A (1) B E A C D (1) B C D E A (1) A E B D C (1) A D E C B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 -6 0 B 2 0 8 12 4 C 0 -8 0 2 4 D 6 -12 -2 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -6 0 B 2 0 8 12 4 C 0 -8 0 2 4 D 6 -12 -2 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998615 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=24 C=17 D=13 E=6 so E is eliminated. Round 2 votes counts: B=40 A=30 C=17 D=13 so D is eliminated. Round 3 votes counts: B=45 A=30 C=25 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:199 E:197 A:196 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -6 0 B 2 0 8 12 4 C 0 -8 0 2 4 D 6 -12 -2 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998615 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 0 B 2 0 8 12 4 C 0 -8 0 2 4 D 6 -12 -2 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998615 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 0 B 2 0 8 12 4 C 0 -8 0 2 4 D 6 -12 -2 0 -2 E 0 -4 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998615 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7965: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (17) E D B A C (11) D E B A C (9) C A D B E (7) E B A C D (6) D E B C A (5) C A B D E (4) B E A C D (4) E B D A C (3) D E C B A (3) D E C A B (3) C D A B E (3) E B A D C (2) D C A B E (2) B A E C D (2) B A C E D (2) A C E B D (2) A C B E D (2) E C A B D (1) E A B C D (1) D C E A B (1) D C A E B (1) D B E A C (1) D B C A E (1) D B A C E (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E A D C (1) B D E A C (1) Total count = 100 A B C D E A 0 -4 -4 8 -4 B 4 0 0 -2 -4 C 4 0 0 10 -6 D -8 2 -10 0 -12 E 4 4 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 8 -4 B 4 0 0 -2 -4 C 4 0 0 10 -6 D -8 2 -10 0 -12 E 4 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=27 E=24 B=10 A=4 so A is eliminated. Round 2 votes counts: C=39 D=27 E=24 B=10 so B is eliminated. Round 3 votes counts: C=41 E=31 D=28 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 C:204 B:199 A:198 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 8 -4 B 4 0 0 -2 -4 C 4 0 0 10 -6 D -8 2 -10 0 -12 E 4 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 8 -4 B 4 0 0 -2 -4 C 4 0 0 10 -6 D -8 2 -10 0 -12 E 4 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 8 -4 B 4 0 0 -2 -4 C 4 0 0 10 -6 D -8 2 -10 0 -12 E 4 4 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7966: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) A B D E C (7) D A B C E (5) C E B D A (5) E C B A D (4) E B C A D (4) C E D B A (4) C D E B A (4) B C E D A (4) A D E B C (4) E C A B D (3) B A D C E (3) A E B D C (3) A D B C E (3) D B C A E (2) D B A C E (2) D A E C B (2) D A C E B (2) C E D A B (2) C B E D A (2) B E C A D (2) B D C A E (2) B C D E A (2) A E B C D (2) A D E C B (2) E C D A B (1) E C B D A (1) E B A C D (1) E A C D B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) D A C B E (1) C D B E A (1) B D A C E (1) B A E C D (1) A E D C B (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 6 6 10 B -6 0 18 0 -2 C -6 -18 0 -8 0 D -6 0 8 0 12 E -10 2 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 10 B -6 0 18 0 -2 C -6 -18 0 -8 0 D -6 0 8 0 12 E -10 2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=18 C=18 E=17 B=15 so B is eliminated. Round 2 votes counts: A=36 C=24 D=21 E=19 so E is eliminated. Round 3 votes counts: A=40 C=39 D=21 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:207 B:205 E:190 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 10 B -6 0 18 0 -2 C -6 -18 0 -8 0 D -6 0 8 0 12 E -10 2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 10 B -6 0 18 0 -2 C -6 -18 0 -8 0 D -6 0 8 0 12 E -10 2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 10 B -6 0 18 0 -2 C -6 -18 0 -8 0 D -6 0 8 0 12 E -10 2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7967: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) A B E C D (9) B A E D C (8) B D E A C (6) B D A E C (6) D B E C A (5) A E C B D (5) D C E B A (4) C A E D B (4) B E D A C (4) A C E D B (4) D B C E A (3) C E A D B (3) C D E A B (3) A E B C D (3) A B C D E (3) E A C B D (2) D C B E A (2) B D E C A (2) E D C B A (1) E C A D B (1) E A C D B (1) E A B C D (1) D E C B A (1) D B C A E (1) D B A C E (1) C E D A B (1) C A D E B (1) B E A D C (1) B D A C E (1) B A D E C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 36 18 18 B -6 0 12 28 10 C -36 -12 0 4 -16 D -18 -28 -4 0 -18 E -18 -10 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 36 18 18 B -6 0 12 28 10 C -36 -12 0 4 -16 D -18 -28 -4 0 -18 E -18 -10 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990184 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=29 D=17 C=12 E=6 so E is eliminated. Round 2 votes counts: A=40 B=29 D=18 C=13 so C is eliminated. Round 3 votes counts: A=49 B=29 D=22 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:239 B:222 E:203 C:170 D:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 36 18 18 B -6 0 12 28 10 C -36 -12 0 4 -16 D -18 -28 -4 0 -18 E -18 -10 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990184 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 36 18 18 B -6 0 12 28 10 C -36 -12 0 4 -16 D -18 -28 -4 0 -18 E -18 -10 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990184 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 36 18 18 B -6 0 12 28 10 C -36 -12 0 4 -16 D -18 -28 -4 0 -18 E -18 -10 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990184 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7968: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (12) B A C D E (12) E C D B A (9) C E B D A (8) E D A C B (7) B C A D E (6) A D B E C (5) E D C A B (4) C E D B A (4) C B E A D (4) C B A E D (4) B A D C E (4) E C D A B (3) B C A E D (3) D E C A B (2) D E A B C (2) A D E B C (2) A B D E C (2) E D A B C (1) E A D B C (1) D E A C B (1) C E B A D (1) C B E D A (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 2 -8 4 B 14 0 4 -6 -14 C -2 -4 0 12 -2 D 8 6 -12 0 -2 E -4 14 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.321918 B: 0.047945 C: 0.184932 D: 0.123288 E: 0.321918 Sum of squares = 0.258960405329 Cumulative probabilities = A: 0.321918 B: 0.369863 C: 0.554795 D: 0.678082 E: 1.000000 A B C D E A 0 -14 2 -8 4 B 14 0 4 -6 -14 C -2 -4 0 12 -2 D 8 6 -12 0 -2 E -4 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.321918 B: 0.047945 C: 0.184932 D: 0.123288 E: 0.321918 Sum of squares = 0.25896040533 Cumulative probabilities = A: 0.321918 B: 0.369863 C: 0.554795 D: 0.678082 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=25 C=22 D=17 A=10 so A is eliminated. Round 2 votes counts: B=29 E=25 D=24 C=22 so C is eliminated. Round 3 votes counts: E=38 B=38 D=24 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:207 C:202 D:200 B:199 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 2 -8 4 B 14 0 4 -6 -14 C -2 -4 0 12 -2 D 8 6 -12 0 -2 E -4 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.321918 B: 0.047945 C: 0.184932 D: 0.123288 E: 0.321918 Sum of squares = 0.25896040533 Cumulative probabilities = A: 0.321918 B: 0.369863 C: 0.554795 D: 0.678082 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -8 4 B 14 0 4 -6 -14 C -2 -4 0 12 -2 D 8 6 -12 0 -2 E -4 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.321918 B: 0.047945 C: 0.184932 D: 0.123288 E: 0.321918 Sum of squares = 0.25896040533 Cumulative probabilities = A: 0.321918 B: 0.369863 C: 0.554795 D: 0.678082 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -8 4 B 14 0 4 -6 -14 C -2 -4 0 12 -2 D 8 6 -12 0 -2 E -4 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.321918 B: 0.047945 C: 0.184932 D: 0.123288 E: 0.321918 Sum of squares = 0.25896040533 Cumulative probabilities = A: 0.321918 B: 0.369863 C: 0.554795 D: 0.678082 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7969: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (9) E D C A B (8) D E C A B (8) C E D B A (6) A B E D C (6) C D E B A (5) C B A D E (5) A B D E C (5) C D A B E (4) E D A B C (3) D E A C B (3) C D E A B (3) B C E A D (3) B C A E D (3) E D A C B (2) D C E A B (2) B A C E D (2) A E B D C (2) A D B E C (2) E D C B A (1) E C D B A (1) E C D A B (1) E B D A C (1) E B C A D (1) E A B D C (1) D A E B C (1) D A C E B (1) C E B D A (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) C B A E D (1) B C A D E (1) A E D B C (1) A D E B C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -16 -10 -4 B -12 0 -12 -10 -6 C 16 12 0 2 4 D 10 10 -2 0 12 E 4 6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -16 -10 -4 B -12 0 -12 -10 -6 C 16 12 0 2 4 D 10 10 -2 0 12 E 4 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=19 A=19 B=18 D=15 so D is eliminated. Round 2 votes counts: C=31 E=30 A=21 B=18 so B is eliminated. Round 3 votes counts: C=38 A=32 E=30 so E is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:215 E:197 A:191 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -16 -10 -4 B -12 0 -12 -10 -6 C 16 12 0 2 4 D 10 10 -2 0 12 E 4 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -16 -10 -4 B -12 0 -12 -10 -6 C 16 12 0 2 4 D 10 10 -2 0 12 E 4 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -16 -10 -4 B -12 0 -12 -10 -6 C 16 12 0 2 4 D 10 10 -2 0 12 E 4 6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7970: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) A C E B D (10) E C A D B (7) A C B D E (6) E D C B A (5) D B E C A (5) B D E A C (5) C A E B D (4) B D E C A (4) B D A E C (4) E D B C A (3) E C D A B (3) E A C D B (3) D E B C A (3) C A E D B (3) A C E D B (3) A B D C E (3) E C D B A (2) D B E A C (2) B D A C E (2) B A D E C (2) B A C D E (2) A B C D E (2) D B A E C (1) C B E D A (1) C B E A D (1) C B A D E (1) B D C E A (1) B A D C E (1) Total count = 100 A B C D E A 0 10 -8 18 -12 B -10 0 -20 -2 -14 C 8 20 0 18 2 D -18 2 -18 0 -12 E 12 14 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999326 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 18 -12 B -10 0 -20 -2 -14 C 8 20 0 18 2 D -18 2 -18 0 -12 E 12 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 C=21 B=21 D=11 so D is eliminated. Round 2 votes counts: B=29 E=26 A=24 C=21 so C is eliminated. Round 3 votes counts: E=37 B=32 A=31 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:224 E:218 A:204 B:177 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 18 -12 B -10 0 -20 -2 -14 C 8 20 0 18 2 D -18 2 -18 0 -12 E 12 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 18 -12 B -10 0 -20 -2 -14 C 8 20 0 18 2 D -18 2 -18 0 -12 E 12 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 18 -12 B -10 0 -20 -2 -14 C 8 20 0 18 2 D -18 2 -18 0 -12 E 12 14 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998389 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7971: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (13) D A C E B (11) B C E A D (7) C A E D B (6) E B C A D (4) A C D E B (4) E D A C B (3) E B D C A (3) D A B C E (3) B E D C A (3) B D E A C (3) E D B A C (2) E C B A D (2) D E A C B (2) D A B E C (2) C E A B D (2) C B A E D (2) C A E B D (2) C A B E D (2) B E C D A (2) B D A C E (2) A D C E B (2) A C E D B (2) E C A B D (1) E B C D A (1) E A C D B (1) D B E A C (1) D B A E C (1) D B A C E (1) D A E C B (1) D A E B C (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C A E (1) B C A D E (1) A E C D B (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -10 12 -2 B 0 0 6 8 -2 C 10 -6 0 14 4 D -12 -8 -14 0 -18 E 2 2 -4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888896 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 0 -10 12 -2 B 0 0 6 8 -2 C 10 -6 0 14 4 D -12 -8 -14 0 -18 E 2 2 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=23 E=17 C=16 A=11 so A is eliminated. Round 2 votes counts: B=34 D=25 C=23 E=18 so E is eliminated. Round 3 votes counts: B=42 D=30 C=28 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 E:209 B:206 A:200 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -10 12 -2 B 0 0 6 8 -2 C 10 -6 0 14 4 D -12 -8 -14 0 -18 E 2 2 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 12 -2 B 0 0 6 8 -2 C 10 -6 0 14 4 D -12 -8 -14 0 -18 E 2 2 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 12 -2 B 0 0 6 8 -2 C 10 -6 0 14 4 D -12 -8 -14 0 -18 E 2 2 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888924 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7972: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) A C D B E (7) B E D C A (6) B E A D C (6) E D B C A (5) C D A E B (5) E B A D C (4) D C B E A (4) D C B A E (4) A E B C D (4) A C E D B (4) E B D C A (3) C D A B E (3) C A D B E (3) B D E C A (3) A C D E B (3) A C B E D (3) A B C D E (3) D E C B A (2) D C E A B (2) B A E C D (2) E D C B A (1) E B D A C (1) E A C D B (1) E A B D C (1) E A B C D (1) D E C A B (1) D B E C A (1) C D B A E (1) C A B D E (1) B D C E A (1) B D C A E (1) B A C D E (1) A E C B D (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 6 -4 -2 B 8 0 6 6 22 C -6 -6 0 -8 -4 D 4 -6 8 0 -6 E 2 -22 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 -4 -2 B 8 0 6 6 22 C -6 -6 0 -8 -4 D 4 -6 8 0 -6 E 2 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=28 A=28 E=17 D=14 C=13 so C is eliminated. Round 2 votes counts: A=32 B=28 D=23 E=17 so E is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:200 A:196 E:195 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 -4 -2 B 8 0 6 6 22 C -6 -6 0 -8 -4 D 4 -6 8 0 -6 E 2 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -4 -2 B 8 0 6 6 22 C -6 -6 0 -8 -4 D 4 -6 8 0 -6 E 2 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -4 -2 B 8 0 6 6 22 C -6 -6 0 -8 -4 D 4 -6 8 0 -6 E 2 -22 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997494 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7973: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (17) B D E A C (10) A C B E D (9) C A E D B (8) B A D E C (8) E D C B A (5) A B C E D (5) B A C E D (4) A C D E B (4) A B C D E (4) E D B C A (3) D E C B A (3) D E B C A (3) C E D B A (3) B E D A C (3) A D E C B (2) D E C A B (1) D E B A C (1) D E A B C (1) C A D E B (1) B C E D A (1) B A E D C (1) B A C D E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 10 -2 -2 B -8 0 -8 -6 -6 C -10 8 0 16 16 D 2 6 -16 0 -20 E 2 6 -16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.071429 D: 0.000000 E: 0.357143 Sum of squares = 0.459183673483 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.642857 D: 0.642857 E: 1.000000 A B C D E A 0 8 10 -2 -2 B -8 0 -8 -6 -6 C -10 8 0 16 16 D 2 6 -16 0 -20 E 2 6 -16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.071429 D: 0.000000 E: 0.357143 Sum of squares = 0.459183674079 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.642857 D: 0.642857 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 A=26 D=9 E=8 so E is eliminated. Round 2 votes counts: C=29 B=28 A=26 D=17 so D is eliminated. Round 3 votes counts: C=38 B=35 A=27 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:207 E:206 B:186 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 -2 -2 B -8 0 -8 -6 -6 C -10 8 0 16 16 D 2 6 -16 0 -20 E 2 6 -16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.071429 D: 0.000000 E: 0.357143 Sum of squares = 0.459183674079 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.642857 D: 0.642857 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 -2 -2 B -8 0 -8 -6 -6 C -10 8 0 16 16 D 2 6 -16 0 -20 E 2 6 -16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.071429 D: 0.000000 E: 0.357143 Sum of squares = 0.459183674079 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.642857 D: 0.642857 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 -2 -2 B -8 0 -8 -6 -6 C -10 8 0 16 16 D 2 6 -16 0 -20 E 2 6 -16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.071429 D: 0.000000 E: 0.357143 Sum of squares = 0.459183674079 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.642857 D: 0.642857 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7974: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) A C B E D (7) D E C A B (6) D E B C A (6) D B E A C (5) C A B D E (5) B D E A C (5) C A E D B (4) E D B C A (3) D E C B A (3) D C E A B (3) C A D B E (3) B A E C D (3) B A D C E (3) A B C E D (3) E D C B A (2) E D B A C (2) C D A E B (2) C A B E D (2) B E D A C (2) B A E D C (2) B A D E C (2) B A C E D (2) A C E B D (2) A B C D E (2) E D A B C (1) E C D A B (1) E C A D B (1) E B D A C (1) E A C B D (1) C E D A B (1) C E A D B (1) C D A B E (1) C A D E B (1) B E A C D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 -8 -6 -8 B -14 0 -10 -10 0 C 8 10 0 -10 -14 D 6 10 10 0 -4 E 8 0 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.182864 C: 0.000000 D: 0.000000 E: 0.817136 Sum of squares = 0.701150383215 Cumulative probabilities = A: 0.000000 B: 0.182864 C: 0.182864 D: 0.182864 E: 1.000000 A B C D E A 0 14 -8 -6 -8 B -14 0 -10 -10 0 C 8 10 0 -10 -14 D 6 10 10 0 -4 E 8 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735024 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=21 B=21 C=20 A=15 so A is eliminated. Round 2 votes counts: C=29 B=27 D=23 E=21 so E is eliminated. Round 3 votes counts: D=40 C=32 B=28 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:211 C:197 A:196 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -8 -6 -8 B -14 0 -10 -10 0 C 8 10 0 -10 -14 D 6 10 10 0 -4 E 8 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735024 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 -6 -8 B -14 0 -10 -10 0 C 8 10 0 -10 -14 D 6 10 10 0 -4 E 8 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735024 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 -6 -8 B -14 0 -10 -10 0 C 8 10 0 -10 -14 D 6 10 10 0 -4 E 8 0 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836735024 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7975: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) D C B A E (9) E A D B C (8) D A B C E (8) C E B A D (5) C B E A D (5) D A C B E (4) A E D B C (4) A D E B C (4) E A B C D (3) D B C A E (3) C B E D A (3) C B D A E (3) A D B E C (3) E C B D A (2) E B C A D (2) D E A C B (2) D C E B A (2) C E B D A (2) C D E B A (2) A D B C E (2) A B E C D (2) E D C B A (1) E D C A B (1) D E C B A (1) D C B E A (1) D B A C E (1) D A E C B (1) D A E B C (1) C D B A E (1) C B A D E (1) B E C A D (1) B A C E D (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 4 0 B 10 0 -10 -18 -2 C 8 10 0 -16 8 D -4 18 16 0 0 E 0 2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428064 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 4 0 B 10 0 -10 -18 -2 C 8 10 0 -16 8 D -4 18 16 0 0 E 0 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428542 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=26 C=22 A=17 B=2 so B is eliminated. Round 2 votes counts: D=33 E=27 C=22 A=18 so A is eliminated. Round 3 votes counts: D=43 E=34 C=23 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 C:205 E:197 A:193 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -8 4 0 B 10 0 -10 -18 -2 C 8 10 0 -16 8 D -4 18 16 0 0 E 0 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428542 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 4 0 B 10 0 -10 -18 -2 C 8 10 0 -16 8 D -4 18 16 0 0 E 0 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428542 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 4 0 B 10 0 -10 -18 -2 C 8 10 0 -16 8 D -4 18 16 0 0 E 0 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.142857 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428542 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7976: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E A D B C (8) D B C E A (6) B D C E A (6) E A D C B (5) B D E C A (5) A E C B D (5) C D B A E (4) A E C D B (4) A E B D C (4) E A B D C (3) D E B C A (3) D B E C A (3) C D B E A (3) A C E B D (3) E D B A C (2) D C B E A (2) C B D E A (2) C B A D E (2) B C D A E (2) A C E D B (2) A C B E D (2) A C B D E (2) E D B C A (1) E B A D C (1) D E C B A (1) C E A D B (1) C D A E B (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D A C (1) B D C A E (1) A E B C D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -10 -4 -6 B 8 0 -2 2 6 C 10 2 0 -4 4 D 4 -2 4 0 10 E 6 -6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000056 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -4 -6 B 8 0 -2 2 6 C 10 2 0 -4 4 D 4 -2 4 0 10 E 6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=25 A=25 E=20 D=15 B=15 so D is eliminated. Round 2 votes counts: C=27 A=25 E=24 B=24 so E is eliminated. Round 3 votes counts: A=41 B=31 C=28 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:208 B:207 C:206 E:193 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -10 -4 -6 B 8 0 -2 2 6 C 10 2 0 -4 4 D 4 -2 4 0 10 E 6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -4 -6 B 8 0 -2 2 6 C 10 2 0 -4 4 D 4 -2 4 0 10 E 6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -4 -6 B 8 0 -2 2 6 C 10 2 0 -4 4 D 4 -2 4 0 10 E 6 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7977: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) A E C D B (7) E D B A C (5) E D A B C (5) D E B A C (5) D B E C A (5) C A B E D (5) B D E C A (5) E D B C A (4) E A D C B (4) C A B D E (4) B C D E A (4) A C E D B (4) C B A E D (3) C A E B D (3) B C A D E (3) E C A D B (2) D E B C A (2) D E A B C (2) C B D A E (2) B D C E A (2) B D C A E (2) B C D A E (2) A C E B D (2) E D A C B (1) E A D B C (1) E A C D B (1) C B D E A (1) A E D C B (1) A E D B C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -18 6 2 B 10 0 -2 0 0 C 18 2 0 10 -2 D -6 0 -10 0 2 E -2 0 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020406772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 A B C D E A 0 -10 -18 6 2 B 10 0 -2 0 0 C 18 2 0 10 -2 D -6 0 -10 0 2 E -2 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=23 B=18 A=17 D=14 so D is eliminated. Round 2 votes counts: E=32 C=28 B=23 A=17 so A is eliminated. Round 3 votes counts: E=41 C=35 B=24 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:214 B:204 E:199 D:193 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -18 6 2 B 10 0 -2 0 0 C 18 2 0 10 -2 D -6 0 -10 0 2 E -2 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 6 2 B 10 0 -2 0 0 C 18 2 0 10 -2 D -6 0 -10 0 2 E -2 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 6 2 B 10 0 -2 0 0 C 18 2 0 10 -2 D -6 0 -10 0 2 E -2 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020408048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7978: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (9) D E A B C (8) C B E A D (8) C B A D E (8) D C B A E (7) C B A E D (6) C B D A E (5) E D A B C (4) E B C A D (4) E A D B C (4) E A B D C (4) E A B C D (4) D A C B E (3) D E A C B (2) D A E B C (2) C D B A E (2) B E C A D (2) A E B D C (2) A E B C D (2) E D C B A (1) E C B A D (1) E B A C D (1) D E C A B (1) D C E B A (1) D A E C B (1) D A C E B (1) D A B C E (1) C B E D A (1) B C E A D (1) B C A D E (1) B A C E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -16 20 6 B 18 0 4 24 14 C 16 -4 0 16 12 D -20 -24 -16 0 -14 E -6 -14 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 20 6 B 18 0 4 24 14 C 16 -4 0 16 12 D -20 -24 -16 0 -14 E -6 -14 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=27 E=23 B=14 A=6 so A is eliminated. Round 2 votes counts: C=30 E=27 D=27 B=16 so B is eliminated. Round 3 votes counts: C=43 E=30 D=27 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:230 C:220 A:196 E:191 D:163 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 20 6 B 18 0 4 24 14 C 16 -4 0 16 12 D -20 -24 -16 0 -14 E -6 -14 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 20 6 B 18 0 4 24 14 C 16 -4 0 16 12 D -20 -24 -16 0 -14 E -6 -14 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 20 6 B 18 0 4 24 14 C 16 -4 0 16 12 D -20 -24 -16 0 -14 E -6 -14 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7979: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) A B D E C (8) B A D C E (7) E C A D B (6) E C D B A (5) A B D C E (5) C E D B A (4) C E A B D (4) B D C E A (4) A E D C B (4) B D A C E (3) A E C B D (3) A D B E C (3) D B C E A (2) D A B E C (2) C E B A D (2) C B E A D (2) B A C D E (2) A E C D B (2) E C A B D (1) E A C D B (1) D E C B A (1) D E C A B (1) D C B E A (1) D B E C A (1) D B C A E (1) D B A E C (1) D A E B C (1) C E A D B (1) C D E B A (1) C D B E A (1) C B E D A (1) C B D E A (1) C A E B D (1) C A B E D (1) B D C A E (1) B C D E A (1) B C A E D (1) A E D B C (1) A D E B C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -8 16 0 B -14 0 -8 0 0 C 8 8 0 4 -2 D -16 0 -4 0 -2 E 0 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.171345 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.828655 Sum of squares = 0.716027678099 Cumulative probabilities = A: 0.171345 B: 0.171345 C: 0.171345 D: 0.171345 E: 1.000000 A B C D E A 0 14 -8 16 0 B -14 0 -8 0 0 C 8 8 0 4 -2 D -16 0 -4 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000081639 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=22 C=19 B=19 D=11 so D is eliminated. Round 2 votes counts: A=32 E=24 B=24 C=20 so C is eliminated. Round 3 votes counts: E=36 A=34 B=30 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:209 E:202 B:189 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -8 16 0 B -14 0 -8 0 0 C 8 8 0 4 -2 D -16 0 -4 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000081639 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 16 0 B -14 0 -8 0 0 C 8 8 0 4 -2 D -16 0 -4 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000081639 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 16 0 B -14 0 -8 0 0 C 8 8 0 4 -2 D -16 0 -4 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000081639 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7980: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) B A E D C (8) C D E B A (7) C D E A B (7) C A D B E (7) E B D A C (6) C A D E B (5) E D B A C (4) D E B C A (4) B E D A C (4) B E A D C (4) A C B D E (4) E D B C A (3) D E C B A (3) D C E B A (3) C D A E B (3) A C B E D (3) C E D B A (2) A B E C D (2) A B C E D (2) E D C B A (1) E C B D A (1) D C B E A (1) D B E A C (1) C A B E D (1) C A B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 2 0 -2 B 4 0 2 -2 2 C -2 -2 0 -10 -8 D 0 2 10 0 -6 E 2 -2 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.440000000003 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 -4 2 0 -2 B 4 0 2 -2 2 C -2 -2 0 -10 -8 D 0 2 10 0 -6 E 2 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=24 B=16 E=15 D=12 so D is eliminated. Round 2 votes counts: C=37 A=24 E=22 B=17 so B is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:207 B:203 D:203 A:198 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 0 -2 B 4 0 2 -2 2 C -2 -2 0 -10 -8 D 0 2 10 0 -6 E 2 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 0 -2 B 4 0 2 -2 2 C -2 -2 0 -10 -8 D 0 2 10 0 -6 E 2 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 0 -2 B 4 0 2 -2 2 C -2 -2 0 -10 -8 D 0 2 10 0 -6 E 2 -2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7981: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (13) A B D C E (12) D B A E C (9) C A B D E (8) D E B A C (7) E D C B A (6) E D B A C (6) C E A B D (6) C A B E D (6) B A D E C (4) A B C D E (4) E D B C A (3) B D A E C (3) A C B D E (3) D B E A C (2) C E D B A (2) D C E A B (1) D A B E C (1) C E D A B (1) C E B A D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 6 -8 4 B 12 0 6 -4 8 C -6 -6 0 -12 -10 D 8 4 12 0 12 E -4 -8 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -8 4 B 12 0 6 -4 8 C -6 -6 0 -12 -10 D 8 4 12 0 12 E -4 -8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=24 A=21 D=20 B=7 so B is eliminated. Round 2 votes counts: E=28 A=25 C=24 D=23 so D is eliminated. Round 3 votes counts: A=38 E=37 C=25 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:218 B:211 A:195 E:193 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 6 -8 4 B 12 0 6 -4 8 C -6 -6 0 -12 -10 D 8 4 12 0 12 E -4 -8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -8 4 B 12 0 6 -4 8 C -6 -6 0 -12 -10 D 8 4 12 0 12 E -4 -8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -8 4 B 12 0 6 -4 8 C -6 -6 0 -12 -10 D 8 4 12 0 12 E -4 -8 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7982: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) B A E D C (9) A B D E C (8) C D A E B (7) A D B C E (7) E C D B A (6) E C B D A (6) C E D A B (6) C D E A B (6) E B A D C (5) E B C A D (4) D A C B E (4) C D A B E (3) A D C B E (3) E B C D A (2) D C A E B (2) B E A C D (2) A B D C E (2) E C B A D (1) E B D C A (1) E B D A C (1) D E A C B (1) D A E B C (1) C E D B A (1) C E B A D (1) B A E C D (1) Total count = 100 A B C D E A 0 0 8 6 -6 B 0 0 6 6 -2 C -8 -6 0 -8 -16 D -6 -6 8 0 -12 E 6 2 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 6 -6 B 0 0 6 6 -2 C -8 -6 0 -8 -16 D -6 -6 8 0 -12 E 6 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999970335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=24 B=22 A=20 D=8 so D is eliminated. Round 2 votes counts: E=27 C=26 A=25 B=22 so B is eliminated. Round 3 votes counts: E=39 A=35 C=26 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 B:205 A:204 D:192 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 6 -6 B 0 0 6 6 -2 C -8 -6 0 -8 -16 D -6 -6 8 0 -12 E 6 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999970335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 6 -6 B 0 0 6 6 -2 C -8 -6 0 -8 -16 D -6 -6 8 0 -12 E 6 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999970335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 6 -6 B 0 0 6 6 -2 C -8 -6 0 -8 -16 D -6 -6 8 0 -12 E 6 2 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999970335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7983: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) A E B C D (10) E C B D A (4) D B E C A (4) C E D B A (4) C E A B D (4) B D A E C (4) A E C B D (4) E C A B D (3) D C B E A (3) C E A D B (3) C D E B A (3) C A D E B (3) A C E B D (3) A B D E C (3) E B A C D (2) D B C E A (2) D B A E C (2) D B A C E (2) D A B C E (2) C D E A B (2) C D A B E (2) C A E D B (2) B E A D C (2) E C B A D (1) E B C D A (1) E B C A D (1) E A C B D (1) D C B A E (1) D C A B E (1) D B C A E (1) D A C B E (1) C D A E B (1) C A E B D (1) B E D A C (1) B A E D C (1) B A D E C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 20 6 18 18 B -20 0 6 20 -6 C -6 -6 0 12 -16 D -18 -20 -12 0 -22 E -18 6 16 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 18 18 B -20 0 6 20 -6 C -6 -6 0 12 -16 D -18 -20 -12 0 -22 E -18 6 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=25 D=19 E=13 B=9 so B is eliminated. Round 2 votes counts: A=36 C=25 D=23 E=16 so E is eliminated. Round 3 votes counts: A=41 C=35 D=24 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:231 E:213 B:200 C:192 D:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 18 18 B -20 0 6 20 -6 C -6 -6 0 12 -16 D -18 -20 -12 0 -22 E -18 6 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 18 18 B -20 0 6 20 -6 C -6 -6 0 12 -16 D -18 -20 -12 0 -22 E -18 6 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 18 18 B -20 0 6 20 -6 C -6 -6 0 12 -16 D -18 -20 -12 0 -22 E -18 6 16 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999436 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7984: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) E C D A B (9) B A D C E (7) A B D E C (7) D E A B C (6) E D C A B (5) E D A B C (5) C E D A B (5) C B A E D (5) B A C D E (5) E D A C B (4) C E D B A (4) C B A D E (4) B A D E C (4) D A B E C (3) B C A D E (3) D E C A B (2) D A E B C (2) E C B A D (1) E B A C D (1) E A B D C (1) D C E A B (1) D A B C E (1) C B E A D (1) C B D A E (1) B C E A D (1) B A E C D (1) Total count = 100 A B C D E A 0 2 -6 4 -14 B -2 0 -6 6 -14 C 6 6 0 4 -2 D -4 -6 -4 0 -8 E 14 14 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -6 4 -14 B -2 0 -6 6 -14 C 6 6 0 4 -2 D -4 -6 -4 0 -8 E 14 14 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 B=21 D=15 A=7 so A is eliminated. Round 2 votes counts: C=31 B=28 E=26 D=15 so D is eliminated. Round 3 votes counts: E=36 C=32 B=32 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:207 A:193 B:192 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 4 -14 B -2 0 -6 6 -14 C 6 6 0 4 -2 D -4 -6 -4 0 -8 E 14 14 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 4 -14 B -2 0 -6 6 -14 C 6 6 0 4 -2 D -4 -6 -4 0 -8 E 14 14 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 4 -14 B -2 0 -6 6 -14 C 6 6 0 4 -2 D -4 -6 -4 0 -8 E 14 14 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999981066 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7985: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) A B E D C (7) A B E C D (7) A E B D C (5) A E B C D (5) E C D A B (4) D C B E A (4) C D E B A (4) C D B A E (4) B A C D E (4) A B C E D (4) E A C D B (3) D E C B A (3) D B C E A (3) C D B E A (3) C B D A E (3) E A D C B (2) D E B C A (2) B D C E A (2) B A D E C (2) B A D C E (2) A C E B D (2) E D C A B (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B D C (1) E A B C D (1) D C B A E (1) C E D A B (1) C E A D B (1) C A E D B (1) B E A D C (1) B D C A E (1) B C D A E (1) B C A D E (1) B A E D C (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 -4 4 8 6 B 4 0 6 6 2 C -4 -6 0 2 0 D -8 -6 -2 0 -4 E -6 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 8 6 B 4 0 6 6 2 C -4 -6 0 2 0 D -8 -6 -2 0 -4 E -6 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=21 C=17 E=15 B=15 so E is eliminated. Round 2 votes counts: A=40 D=22 C=21 B=17 so B is eliminated. Round 3 votes counts: A=51 D=26 C=23 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:209 A:207 E:198 C:196 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 8 6 B 4 0 6 6 2 C -4 -6 0 2 0 D -8 -6 -2 0 -4 E -6 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 8 6 B 4 0 6 6 2 C -4 -6 0 2 0 D -8 -6 -2 0 -4 E -6 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 8 6 B 4 0 6 6 2 C -4 -6 0 2 0 D -8 -6 -2 0 -4 E -6 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7986: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) C B D E A (9) E C A B D (8) D A B E C (8) E A C D B (7) C E B A D (7) A E D B C (6) E A C B D (5) D B C A E (4) B C D A E (4) A D B E C (4) C E B D A (3) C B E D A (3) A D E B C (3) C B E A D (2) B D C A E (2) B D A C E (2) A E D C B (2) A B D E C (2) E C D A B (1) E C B A D (1) E C A D B (1) E A D C B (1) D A E B C (1) D A B C E (1) C E D B A (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 2 8 2 2 B -2 0 -2 0 4 C -8 2 0 6 -2 D -2 0 -6 0 2 E -2 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999821 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 2 2 B -2 0 -2 0 4 C -8 2 0 6 -2 D -2 0 -6 0 2 E -2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 D=24 A=18 B=9 so B is eliminated. Round 2 votes counts: C=29 D=28 E=24 A=19 so A is eliminated. Round 3 votes counts: D=38 E=33 C=29 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:207 B:200 C:199 D:197 E:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 2 2 B -2 0 -2 0 4 C -8 2 0 6 -2 D -2 0 -6 0 2 E -2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 2 2 B -2 0 -2 0 4 C -8 2 0 6 -2 D -2 0 -6 0 2 E -2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 2 2 B -2 0 -2 0 4 C -8 2 0 6 -2 D -2 0 -6 0 2 E -2 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7987: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (14) C D B E A (7) D E A C B (6) C B D E A (6) B C A E D (6) A E D B C (6) A E B D C (5) B C A D E (4) E D A C B (3) E A D C B (3) E A D B C (3) C B D A E (3) B A E C D (3) A B E C D (3) E C D A B (2) D C E A B (2) D C B A E (2) D B C A E (2) C D E B A (2) C D B A E (2) A E B C D (2) E C A D B (1) E C A B D (1) E A B C D (1) D E C A B (1) D C B E A (1) D B A C E (1) D A E C B (1) D A E B C (1) C E B D A (1) B A D E C (1) B A D C E (1) B A C E D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -14 -14 20 B 14 0 12 6 18 C 14 -12 0 18 10 D 14 -6 -18 0 16 E -20 -18 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 -14 20 B 14 0 12 6 18 C 14 -12 0 18 10 D 14 -6 -18 0 16 E -20 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=21 A=18 D=17 E=14 so E is eliminated. Round 2 votes counts: B=30 C=25 A=25 D=20 so D is eliminated. Round 3 votes counts: A=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:215 D:203 A:189 E:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -14 -14 20 B 14 0 12 6 18 C 14 -12 0 18 10 D 14 -6 -18 0 16 E -20 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -14 20 B 14 0 12 6 18 C 14 -12 0 18 10 D 14 -6 -18 0 16 E -20 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -14 20 B 14 0 12 6 18 C 14 -12 0 18 10 D 14 -6 -18 0 16 E -20 -18 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7988: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) A C D B E (8) C A E D B (6) B E D C A (6) B E D A C (5) E B C D A (4) D A E C B (4) B E C A D (4) A D C B E (4) B C E A D (3) A C B D E (3) E B D A C (2) E B C A D (2) D E B A C (2) D B E A C (2) D A C E B (2) D A C B E (2) C E A B D (2) C A D E B (2) B E C D A (2) B D E A C (2) B C A E D (2) A B D C E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E C B D A (1) E C B A D (1) E C A D B (1) D E C A B (1) D E A C B (1) D B A E C (1) D B A C E (1) D A E B C (1) D A B C E (1) C E B A D (1) C E A D B (1) C B E A D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E A D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -8 -4 -14 B 10 0 8 8 8 C 8 -8 0 -4 -10 D 4 -8 4 0 -16 E 14 -8 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -4 -14 B 10 0 8 8 8 C 8 -8 0 -4 -10 D 4 -8 4 0 -16 E 14 -8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 D=18 A=18 C=16 so C is eliminated. Round 2 votes counts: A=29 E=27 B=26 D=18 so D is eliminated. Round 3 votes counts: A=39 E=31 B=30 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:216 C:193 D:192 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -4 -14 B 10 0 8 8 8 C 8 -8 0 -4 -10 D 4 -8 4 0 -16 E 14 -8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -4 -14 B 10 0 8 8 8 C 8 -8 0 -4 -10 D 4 -8 4 0 -16 E 14 -8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -4 -14 B 10 0 8 8 8 C 8 -8 0 -4 -10 D 4 -8 4 0 -16 E 14 -8 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7989: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) C E A D B (7) D A C E B (6) C D A E B (6) D B A C E (5) B D A E C (5) A D C E B (5) D A C B E (4) B E C D A (4) A C D E B (4) E C B A D (3) E C A D B (3) D C A B E (3) C A D E B (3) B D E A C (3) C A E D B (2) B E D A C (2) B E C A D (2) B E A C D (2) B D C A E (2) E B C A D (1) E A C B D (1) E A B C D (1) D C B A E (1) C E B A D (1) C E A B D (1) C D B E A (1) C B E D A (1) B E D C A (1) B E A D C (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A E B D C (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 24 10 -16 24 B -24 0 -10 -30 2 C -10 10 0 -8 32 D 16 30 8 0 26 E -24 -2 -32 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 10 -16 24 B -24 0 -10 -30 2 C -10 10 0 -8 32 D 16 30 8 0 26 E -24 -2 -32 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=25 C=22 A=14 E=9 so E is eliminated. Round 2 votes counts: D=30 C=28 B=26 A=16 so A is eliminated. Round 3 votes counts: D=37 C=35 B=28 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:240 A:221 C:212 B:169 E:158 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 10 -16 24 B -24 0 -10 -30 2 C -10 10 0 -8 32 D 16 30 8 0 26 E -24 -2 -32 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 10 -16 24 B -24 0 -10 -30 2 C -10 10 0 -8 32 D 16 30 8 0 26 E -24 -2 -32 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 10 -16 24 B -24 0 -10 -30 2 C -10 10 0 -8 32 D 16 30 8 0 26 E -24 -2 -32 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7990: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) D C B E A (5) B A E C D (5) A B E D C (5) E A B C D (4) B A D C E (4) B A C E D (4) D E C A B (3) D C B A E (3) D B C A E (3) D B A C E (3) C E D B A (3) C D E B A (3) C D B E A (3) A B E C D (3) A B D E C (3) E C D A B (2) E A D C B (2) D A B E C (2) D A B C E (2) C E B A D (2) B E A C D (2) B A E D C (2) B A C D E (2) A E B D C (2) A D E B C (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A C D (1) D E A C B (1) D C E B A (1) D C A E B (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E A B (1) C B E A D (1) C B D E A (1) B D C A E (1) A E D B C (1) A E B C D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 10 2 6 B 4 0 10 -4 12 C -10 -10 0 -16 8 D -2 4 16 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999991 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 2 6 B 4 0 10 -4 12 C -10 -10 0 -16 8 D -2 4 16 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000001 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=20 A=19 C=15 E=12 so E is eliminated. Round 2 votes counts: D=34 A=25 B=22 C=19 so C is eliminated. Round 3 votes counts: D=46 B=28 A=26 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:211 A:207 C:186 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 10 2 6 B 4 0 10 -4 12 C -10 -10 0 -16 8 D -2 4 16 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000001 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 2 6 B 4 0 10 -4 12 C -10 -10 0 -16 8 D -2 4 16 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000001 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 2 6 B 4 0 10 -4 12 C -10 -10 0 -16 8 D -2 4 16 0 12 E -6 -12 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.36000000001 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7991: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) C E D B A (7) A B E D C (7) C D E B A (6) B E A C D (6) B A E C D (6) E B C A D (4) E B A C D (4) D C A B E (4) D A C B E (4) E C D B A (3) B A E D C (3) A E B D C (3) A B D E C (3) E C D A B (2) E C B A D (2) D C B A E (2) D A B C E (2) C E B D A (2) A D E C B (2) A D B C E (2) A B D C E (2) E C B D A (1) E C A B D (1) E A B C D (1) D C A E B (1) D A C E B (1) C D B E A (1) C B D E A (1) B E C A D (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D A E (1) A E D B C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 0 2 -6 B 6 0 2 4 0 C 0 -2 0 4 -4 D -2 -4 -4 0 -14 E 6 0 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.730707 C: 0.000000 D: 0.000000 E: 0.269293 Sum of squares = 0.606451341024 Cumulative probabilities = A: 0.000000 B: 0.730707 C: 0.730707 D: 0.730707 E: 1.000000 A B C D E A 0 -6 0 2 -6 B 6 0 2 4 0 C 0 -2 0 4 -4 D -2 -4 -4 0 -14 E 6 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 A=22 B=20 E=18 C=17 so C is eliminated. Round 2 votes counts: D=30 E=27 A=22 B=21 so B is eliminated. Round 3 votes counts: E=36 D=33 A=31 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 B:206 C:199 A:195 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 2 -6 B 6 0 2 4 0 C 0 -2 0 4 -4 D -2 -4 -4 0 -14 E 6 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 2 -6 B 6 0 2 4 0 C 0 -2 0 4 -4 D -2 -4 -4 0 -14 E 6 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 2 -6 B 6 0 2 4 0 C 0 -2 0 4 -4 D -2 -4 -4 0 -14 E 6 0 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7992: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (17) E B D C A (14) E B A D C (7) B E D C A (7) A B E C D (7) D C E B A (6) C D A B E (6) A C D E B (6) D C A E B (4) C D B A E (3) E B D A C (2) D C E A B (2) D C B E A (2) B A E C D (2) A E B D C (2) A B C D E (2) E D C B A (1) E A B D C (1) C B D A E (1) C A D B E (1) B E C D A (1) B E A D C (1) B E A C D (1) A E D C B (1) A E B C D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 4 4 2 10 B -4 0 -4 -2 2 C -4 4 0 -2 4 D -2 2 2 0 2 E -10 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 2 10 B -4 0 -4 -2 2 C -4 4 0 -2 4 D -2 2 2 0 2 E -10 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=25 D=14 B=12 C=11 so C is eliminated. Round 2 votes counts: A=39 E=25 D=23 B=13 so B is eliminated. Round 3 votes counts: A=41 E=35 D=24 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:202 C:201 B:196 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 10 B -4 0 -4 -2 2 C -4 4 0 -2 4 D -2 2 2 0 2 E -10 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 10 B -4 0 -4 -2 2 C -4 4 0 -2 4 D -2 2 2 0 2 E -10 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 10 B -4 0 -4 -2 2 C -4 4 0 -2 4 D -2 2 2 0 2 E -10 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7993: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) D A C B E (7) A D B E C (7) C B E A D (6) B C A D E (6) E B A D C (4) D A E C B (4) C E B D A (4) C D A B E (4) E C D A B (3) D A B E C (3) C E D A B (3) C B A D E (3) B C E A D (3) B A D E C (3) A B D E C (3) E C B D A (2) E B C A D (2) E A D B C (2) D A B C E (2) C E D B A (2) C E B A D (2) C B D A E (2) B A C D E (2) A D B C E (2) E D C A B (1) E D A C B (1) E D A B C (1) E C B A D (1) E B D A C (1) E A B D C (1) C D A E B (1) B E A D C (1) B E A C D (1) B C A E D (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 4 8 4 18 B -4 0 8 0 16 C -8 -8 0 -4 2 D -4 0 4 0 14 E -18 -16 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 4 18 B -4 0 8 0 16 C -8 -8 0 -4 2 D -4 0 4 0 14 E -18 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=23 E=19 B=19 A=12 so A is eliminated. Round 2 votes counts: D=32 C=27 B=22 E=19 so E is eliminated. Round 3 votes counts: D=37 C=33 B=30 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:217 B:210 D:207 C:191 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 18 B -4 0 8 0 16 C -8 -8 0 -4 2 D -4 0 4 0 14 E -18 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 18 B -4 0 8 0 16 C -8 -8 0 -4 2 D -4 0 4 0 14 E -18 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 18 B -4 0 8 0 16 C -8 -8 0 -4 2 D -4 0 4 0 14 E -18 -16 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 7994: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) D E A C B (9) E D B C A (8) B C A D E (7) E D B A C (6) E B D C A (5) D A E C B (5) B E C A D (5) A C B D E (5) A C D B E (4) E D C B A (3) E D C A B (3) E D A C B (3) E B C A D (3) D A C E B (3) D E B A C (2) D A C B E (2) C A B E D (2) C A B D E (2) B C E A D (2) A D C B E (2) A B C D E (2) E C A D B (1) E B C D A (1) E A C D B (1) D E A B C (1) D A B C E (1) B A C D E (1) Total count = 100 A B C D E A 0 -8 -6 -4 -6 B 8 0 10 -8 -8 C 6 -10 0 -6 -12 D 4 8 6 0 -8 E 6 8 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 -4 -6 B 8 0 10 -8 -8 C 6 -10 0 -6 -12 D 4 8 6 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=26 D=23 A=13 C=4 so C is eliminated. Round 2 votes counts: E=34 B=26 D=23 A=17 so A is eliminated. Round 3 votes counts: B=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:205 B:201 C:189 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -6 B 8 0 10 -8 -8 C 6 -10 0 -6 -12 D 4 8 6 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -6 B 8 0 10 -8 -8 C 6 -10 0 -6 -12 D 4 8 6 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -6 B 8 0 10 -8 -8 C 6 -10 0 -6 -12 D 4 8 6 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 7995: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E A B D C (7) D C E B A (7) B A D E C (7) C D E B A (6) B A E D C (5) E D C B A (4) D E C B A (4) C D E A B (4) C D A B E (4) A B C E D (4) C E A B D (3) D E B A C (2) D B A C E (2) C E D A B (2) B A D C E (2) A B E D C (2) A B E C D (2) A B C D E (2) E D B A C (1) E C D A B (1) E C A D B (1) E C A B D (1) E B A D C (1) E A C B D (1) E A B C D (1) D E B C A (1) D C B A E (1) D B C A E (1) C E A D B (1) C D B E A (1) C A E B D (1) C A D B E (1) C A B D E (1) B E A D C (1) B D A E C (1) B D A C E (1) B A C D E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -18 -10 -6 0 B 18 0 -12 -8 2 C 10 12 0 0 14 D 6 8 0 0 20 E 0 -2 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.894643 D: 0.105357 E: 0.000000 Sum of squares = 0.811486145984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.894643 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -6 0 B 18 0 -12 -8 2 C 10 12 0 0 14 D 6 8 0 0 20 E 0 -2 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=18 D=18 B=18 A=12 so A is eliminated. Round 2 votes counts: C=36 B=28 E=18 D=18 so E is eliminated. Round 3 votes counts: C=40 B=37 D=23 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:217 B:200 A:183 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -10 -6 0 B 18 0 -12 -8 2 C 10 12 0 0 14 D 6 8 0 0 20 E 0 -2 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -6 0 B 18 0 -12 -8 2 C 10 12 0 0 14 D 6 8 0 0 20 E 0 -2 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -6 0 B 18 0 -12 -8 2 C 10 12 0 0 14 D 6 8 0 0 20 E 0 -2 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 7996: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (13) D E A B C (11) E D A C B (7) A E D B C (6) D E C A B (4) D E A C B (4) C A E B D (4) A E D C B (4) C B D E A (3) B D C E A (3) B C A E D (3) B C A D E (3) A E C D B (3) E D A B C (2) E C D A B (2) D B E A C (2) C E A D B (2) C B A D E (2) C A E D B (2) C A B E D (2) B D A E C (2) B C D E A (2) B A D E C (2) E A D C B (1) D E B A C (1) D B A E C (1) C D E B A (1) C D E A B (1) C D B E A (1) C B D A E (1) B D E A C (1) B C D A E (1) B A C E D (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 14 0 0 4 B -14 0 -16 -10 -14 C 0 16 0 -2 -8 D 0 10 2 0 -8 E -4 14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.754271 B: 0.000000 C: 0.000000 D: 0.245729 E: 0.000000 Sum of squares = 0.629307858281 Cumulative probabilities = A: 0.754271 B: 0.754271 C: 0.754271 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 0 4 B -14 0 -16 -10 -14 C 0 16 0 -2 -8 D 0 10 2 0 -8 E -4 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555575651 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=23 B=18 A=15 E=12 so E is eliminated. Round 2 votes counts: C=34 D=32 B=18 A=16 so A is eliminated. Round 3 votes counts: D=43 C=38 B=19 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 A:209 C:203 D:202 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 0 4 B -14 0 -16 -10 -14 C 0 16 0 -2 -8 D 0 10 2 0 -8 E -4 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555575651 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 0 4 B -14 0 -16 -10 -14 C 0 16 0 -2 -8 D 0 10 2 0 -8 E -4 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555575651 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 0 4 B -14 0 -16 -10 -14 C 0 16 0 -2 -8 D 0 10 2 0 -8 E -4 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555575651 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7997: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (9) E D C B A (6) E D A B C (6) C E D B A (6) C B D A E (6) D E B A C (4) C B D E A (4) C A B E D (4) D E B C A (3) C E B D A (3) B D C A E (3) A E D B C (3) A D B E C (3) A B D E C (3) E D C A B (2) E D A C B (2) E C D B A (2) E C D A B (2) E C A D B (2) C D E B A (2) C B E A D (2) C B A D E (2) B C A D E (2) A B E D C (2) A B D C E (2) E D B C A (1) E A D C B (1) E A D B C (1) D E A B C (1) D B E C A (1) D B E A C (1) D B C E A (1) C E A B D (1) B D C E A (1) B C D A E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -14 -16 -10 B 8 0 4 0 2 C 14 -4 0 0 4 D 16 0 0 0 6 E 10 -2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.678907 C: 0.000000 D: 0.321093 E: 0.000000 Sum of squares = 0.564015095542 Cumulative probabilities = A: 0.000000 B: 0.678907 C: 0.678907 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -14 -16 -10 B 8 0 4 0 2 C 14 -4 0 0 4 D 16 0 0 0 6 E 10 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999858 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=25 A=24 D=11 B=10 so B is eliminated. Round 2 votes counts: C=33 A=27 E=25 D=15 so D is eliminated. Round 3 votes counts: C=38 E=35 A=27 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:211 B:207 C:207 E:199 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -14 -16 -10 B 8 0 4 0 2 C 14 -4 0 0 4 D 16 0 0 0 6 E 10 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999858 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 -16 -10 B 8 0 4 0 2 C 14 -4 0 0 4 D 16 0 0 0 6 E 10 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999858 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 -16 -10 B 8 0 4 0 2 C 14 -4 0 0 4 D 16 0 0 0 6 E 10 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999858 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 7998: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (11) E D A C B (10) B C D A E (6) E A D B C (5) D E C A B (5) E D B C A (4) A E D B C (4) E D A B C (3) D E A C B (3) C B D A E (3) B A C D E (3) A C D B E (3) A B C D E (3) E D C A B (2) E B D C A (2) E A D C B (2) E A B D C (2) D C E B A (2) B C E D A (2) B C D E A (2) B A C E D (2) A E D C B (2) A E B C D (2) E D C B A (1) D E C B A (1) D C E A B (1) D C B E A (1) D C A B E (1) C D B E A (1) C D B A E (1) C A B D E (1) B E C D A (1) B E C A D (1) B C E A D (1) A E B D C (1) A D E C B (1) A D C E B (1) A D C B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 10 0 -4 -4 B -10 0 12 -10 -8 C 0 -12 0 -10 -4 D 4 10 10 0 4 E 4 8 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -4 -4 B -10 0 12 -10 -8 C 0 -12 0 -10 -4 D 4 10 10 0 4 E 4 8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=29 A=20 D=14 C=6 so C is eliminated. Round 2 votes counts: B=32 E=31 A=21 D=16 so D is eliminated. Round 3 votes counts: E=43 B=35 A=22 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:214 E:206 A:201 B:192 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -4 -4 B -10 0 12 -10 -8 C 0 -12 0 -10 -4 D 4 10 10 0 4 E 4 8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -4 -4 B -10 0 12 -10 -8 C 0 -12 0 -10 -4 D 4 10 10 0 4 E 4 8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -4 -4 B -10 0 12 -10 -8 C 0 -12 0 -10 -4 D 4 10 10 0 4 E 4 8 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 7999: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) C D B E A (6) A E B C D (6) E A B D C (5) B E A C D (5) A E D C B (5) D C E A B (4) E D A C B (3) E B A D C (3) D C E B A (3) D C B E A (3) D C A E B (3) C D A B E (3) B E C D A (3) B C E D A (3) B C D E A (3) A E B D C (3) A D C E B (3) E B D C A (2) D E C B A (2) D E A C B (2) C A D B E (2) B A E C D (2) A C D E B (2) E D B A C (1) E D A B C (1) E B D A C (1) E A D B C (1) D E C A B (1) D A E C B (1) C B A D E (1) B E D C A (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A E D (1) A D E C B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -4 -10 -12 B 4 0 -10 -14 -6 C 4 10 0 0 -4 D 10 14 0 0 0 E 12 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.311019 E: 0.688981 Sum of squares = 0.571427574991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.311019 E: 1.000000 A B C D E A 0 -4 -4 -10 -12 B 4 0 -10 -14 -6 C 4 10 0 0 -4 D 10 14 0 0 0 E 12 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=22 C=21 B=21 D=19 E=17 so E is eliminated. Round 2 votes counts: A=28 B=27 D=24 C=21 so C is eliminated. Round 3 votes counts: D=42 A=30 B=28 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:211 C:205 B:187 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -10 -12 B 4 0 -10 -14 -6 C 4 10 0 0 -4 D 10 14 0 0 0 E 12 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -10 -12 B 4 0 -10 -14 -6 C 4 10 0 0 -4 D 10 14 0 0 0 E 12 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -10 -12 B 4 0 -10 -14 -6 C 4 10 0 0 -4 D 10 14 0 0 0 E 12 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8000: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) E B A C D (7) D E B A C (6) A C B E D (6) E B A D C (5) C A B D E (5) B E A D C (5) C D A B E (4) C A D B E (4) A B C E D (4) D E B C A (3) D C E B A (3) D B A E C (3) B A E C D (3) B A D E C (3) E D B A C (2) E B D A C (2) D E C B A (2) D C E A B (2) D C A B E (2) C E A D B (2) A B E C D (2) E D B C A (1) E C D B A (1) E C B A D (1) E B C A D (1) D C B A E (1) D B E A C (1) D B A C E (1) D A C B E (1) D A B C E (1) C E D A B (1) C E A B D (1) C D E A B (1) C A E B D (1) C A D E B (1) B E A C D (1) B A E D C (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 10 24 4 B 6 0 4 14 14 C -10 -4 0 10 0 D -24 -14 -10 0 -10 E -4 -14 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999381 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 24 4 B 6 0 4 14 14 C -10 -4 0 10 0 D -24 -14 -10 0 -10 E -4 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 E=20 B=13 A=13 so B is eliminated. Round 2 votes counts: C=28 E=26 D=26 A=20 so A is eliminated. Round 3 votes counts: C=39 E=32 D=29 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:219 A:216 C:198 E:196 D:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 24 4 B 6 0 4 14 14 C -10 -4 0 10 0 D -24 -14 -10 0 -10 E -4 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 24 4 B 6 0 4 14 14 C -10 -4 0 10 0 D -24 -14 -10 0 -10 E -4 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 24 4 B 6 0 4 14 14 C -10 -4 0 10 0 D -24 -14 -10 0 -10 E -4 -14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998785 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8001: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (19) B A E C D (5) D C B A E (4) A E C B D (4) D C A B E (3) D B E C A (3) C D A E B (3) C B A E D (3) C A E B D (3) E B A C D (2) E A B D C (2) D E C A B (2) D E B A C (2) D E A C B (2) D E A B C (2) D C A E B (2) D B C E A (2) D B C A E (2) C D B A E (2) C D A B E (2) C A E D B (2) C A D E B (2) C A B E D (2) B E A D C (2) B D E A C (2) B C D A E (2) E D B A C (1) E D A C B (1) E B D A C (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D C B E A (1) D B E A C (1) C B D A E (1) C B A D E (1) B E D A C (1) B E A C D (1) B D C A E (1) B C A E D (1) B C A D E (1) B A C E D (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 14 10 14 -2 B -14 0 12 14 -10 C -10 -12 0 22 -16 D -14 -14 -22 0 -14 E 2 10 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 10 14 -2 B -14 0 12 14 -10 C -10 -12 0 22 -16 D -14 -14 -22 0 -14 E 2 10 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 C=21 B=17 A=6 so A is eliminated. Round 2 votes counts: E=34 D=26 C=22 B=18 so B is eliminated. Round 3 votes counts: E=44 D=29 C=27 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:218 B:201 C:192 D:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 10 14 -2 B -14 0 12 14 -10 C -10 -12 0 22 -16 D -14 -14 -22 0 -14 E 2 10 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 14 -2 B -14 0 12 14 -10 C -10 -12 0 22 -16 D -14 -14 -22 0 -14 E 2 10 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 14 -2 B -14 0 12 14 -10 C -10 -12 0 22 -16 D -14 -14 -22 0 -14 E 2 10 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8002: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) E A D C B (5) D C B E A (5) C E D A B (5) C A E D B (4) B D E A C (4) B D C E A (4) A C E D B (4) E A C D B (3) C E A D B (3) B D E C A (3) B D C A E (3) B A E D C (3) B A C D E (3) A E C B D (3) A C E B D (3) D E C B A (2) D C E B A (2) D B E C A (2) C D B E A (2) C D B A E (2) C B D A E (2) C A B D E (2) B D A E C (2) B A D E C (2) A E C D B (2) E D C A B (1) E D A C B (1) E C D A B (1) E A D B C (1) E A B D C (1) D E B A C (1) C D E A B (1) C D A E B (1) C B D E A (1) C A D B E (1) C A B E D (1) B E D A C (1) B D A C E (1) B C D A E (1) A E B C D (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 4 2 B -8 0 -16 2 10 C -2 16 0 -2 6 D -4 -2 2 0 -4 E -2 -10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 4 2 B -8 0 -16 2 10 C -2 16 0 -2 6 D -4 -2 2 0 -4 E -2 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=25 A=23 E=13 D=12 so D is eliminated. Round 2 votes counts: C=32 B=29 A=23 E=16 so E is eliminated. Round 3 votes counts: C=36 A=34 B=30 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:209 A:208 D:196 B:194 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 4 2 B -8 0 -16 2 10 C -2 16 0 -2 6 D -4 -2 2 0 -4 E -2 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 4 2 B -8 0 -16 2 10 C -2 16 0 -2 6 D -4 -2 2 0 -4 E -2 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 4 2 B -8 0 -16 2 10 C -2 16 0 -2 6 D -4 -2 2 0 -4 E -2 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999801 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8003: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (9) D B E A C (9) C B A E D (9) A E C D B (9) B D C E A (8) B D C A E (8) C A E B D (6) B C D A E (6) E A D C B (5) B C D E A (5) E A C D B (3) D B A E C (3) C E A B D (3) C B E A D (3) B D A E C (3) D A E B C (2) A E D C B (2) D E A C B (1) D B C E A (1) C E A D B (1) C B D E A (1) B C A E D (1) B C A D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -6 -12 2 B 16 0 14 10 18 C 6 -14 0 -4 6 D 12 -10 4 0 14 E -2 -18 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -12 2 B 16 0 14 10 18 C 6 -14 0 -4 6 D 12 -10 4 0 14 E -2 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=25 C=23 A=12 E=8 so E is eliminated. Round 2 votes counts: B=32 D=25 C=23 A=20 so A is eliminated. Round 3 votes counts: C=35 B=33 D=32 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:210 C:197 A:184 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -12 2 B 16 0 14 10 18 C 6 -14 0 -4 6 D 12 -10 4 0 14 E -2 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -12 2 B 16 0 14 10 18 C 6 -14 0 -4 6 D 12 -10 4 0 14 E -2 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -12 2 B 16 0 14 10 18 C 6 -14 0 -4 6 D 12 -10 4 0 14 E -2 -18 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8004: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (14) C B E A D (14) D A E C B (10) B C E A D (10) A E D B C (10) E A B D C (5) E A B C D (5) E B A C D (4) E A D B C (4) C B D E A (4) D C A B E (3) C D B A E (3) D C B A E (2) B E C A D (2) B C E D A (2) E D A B C (1) E B C A D (1) D E A B C (1) D B C A E (1) D A C E B (1) C D B E A (1) C B D A E (1) A E B D C (1) Total count = 100 A B C D E A 0 10 12 12 -8 B -10 0 22 -2 -14 C -12 -22 0 -6 -16 D -12 2 6 0 -18 E 8 14 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 12 12 -8 B -10 0 22 -2 -14 C -12 -22 0 -6 -16 D -12 2 6 0 -18 E 8 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=23 E=20 B=14 A=11 so A is eliminated. Round 2 votes counts: D=32 E=31 C=23 B=14 so B is eliminated. Round 3 votes counts: C=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:213 B:198 D:189 C:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 12 12 -8 B -10 0 22 -2 -14 C -12 -22 0 -6 -16 D -12 2 6 0 -18 E 8 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 12 -8 B -10 0 22 -2 -14 C -12 -22 0 -6 -16 D -12 2 6 0 -18 E 8 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 12 -8 B -10 0 22 -2 -14 C -12 -22 0 -6 -16 D -12 2 6 0 -18 E 8 14 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8005: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) B C A E D (8) E D B A C (5) E D A B C (5) E B D C A (5) D E A C B (5) C A B D E (4) B E D C A (4) B E C D A (4) A C D E B (4) E D A C B (3) E B D A C (3) A E D C B (3) A D C E B (3) A C D B E (3) E A D C B (2) D E B C A (2) D A E C B (2) B C E D A (2) B C E A D (2) B C D E A (2) B C A D E (2) A D E C B (2) A C B D E (2) E B A C D (1) E A B D C (1) D C E B A (1) D C E A B (1) C D B A E (1) C D A B E (1) C B D E A (1) B E C A D (1) B E A C D (1) B C D A E (1) B A C E D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -4 4 -2 B 14 0 2 12 2 C 4 -2 0 6 2 D -4 -12 -6 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 4 -2 B 14 0 2 12 2 C 4 -2 0 6 2 D -4 -12 -6 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 A=19 C=17 D=11 so D is eliminated. Round 2 votes counts: E=32 B=28 A=21 C=19 so C is eliminated. Round 3 votes counts: B=40 E=34 A=26 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:205 E:202 A:192 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 4 -2 B 14 0 2 12 2 C 4 -2 0 6 2 D -4 -12 -6 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 4 -2 B 14 0 2 12 2 C 4 -2 0 6 2 D -4 -12 -6 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 4 -2 B 14 0 2 12 2 C 4 -2 0 6 2 D -4 -12 -6 0 -6 E 2 -2 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978895 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8006: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (12) E A C D B (11) B C D E A (7) C B E A D (6) E C A B D (5) B C E D A (5) B C E A D (5) A E D C B (5) E A D C B (4) D A B E C (4) C E B A D (4) C B E D A (4) B C D A E (4) D B A C E (3) D A E B C (3) D B C A E (2) D A E C B (2) D A B C E (2) B D C A E (2) E C B A D (1) E B C A D (1) E A C B D (1) D C B A E (1) C E A D B (1) C B D E A (1) B E C A D (1) B D A E C (1) B D A C E (1) A D E B C (1) Total count = 100 A B C D E A 0 2 0 16 -14 B -2 0 -16 -2 -2 C 0 16 0 14 -4 D -16 2 -14 0 -8 E 14 2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 16 -14 B -2 0 -16 -2 -2 C 0 16 0 14 -4 D -16 2 -14 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 A=18 D=17 C=16 so C is eliminated. Round 2 votes counts: B=37 E=28 A=18 D=17 so D is eliminated. Round 3 votes counts: B=43 A=29 E=28 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:214 C:213 A:202 B:189 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 16 -14 B -2 0 -16 -2 -2 C 0 16 0 14 -4 D -16 2 -14 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 16 -14 B -2 0 -16 -2 -2 C 0 16 0 14 -4 D -16 2 -14 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 16 -14 B -2 0 -16 -2 -2 C 0 16 0 14 -4 D -16 2 -14 0 -8 E 14 2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8007: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) B A D E C (7) A B E D C (7) E A C D B (5) C D E B A (5) A E C B D (5) E C D A B (4) D E C B A (4) B D C E A (4) A E C D B (4) E D C B A (3) E C A D B (3) D B C E A (3) C E D A B (3) A E B C D (3) A C E D B (3) A B C E D (3) D C E B A (2) C E A D B (2) B D E A C (2) B D C A E (2) B D A C E (2) A B E C D (2) A B C D E (2) E B D A C (1) D E B C A (1) C E D B A (1) C D B E A (1) C B A D E (1) C A E D B (1) C A B D E (1) B D E C A (1) B D A E C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 16 20 10 B 0 0 2 10 -2 C -16 -2 0 0 -6 D -20 -10 0 0 -2 E -10 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571588 B: 0.428412 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510249705262 Cumulative probabilities = A: 0.571588 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 20 10 B 0 0 2 10 -2 C -16 -2 0 0 -6 D -20 -10 0 0 -2 E -10 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=28 E=16 C=15 D=10 so D is eliminated. Round 2 votes counts: B=31 A=31 E=21 C=17 so C is eliminated. Round 3 votes counts: E=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 B:205 E:200 C:188 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 16 20 10 B 0 0 2 10 -2 C -16 -2 0 0 -6 D -20 -10 0 0 -2 E -10 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 20 10 B 0 0 2 10 -2 C -16 -2 0 0 -6 D -20 -10 0 0 -2 E -10 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 20 10 B 0 0 2 10 -2 C -16 -2 0 0 -6 D -20 -10 0 0 -2 E -10 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8008: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (13) C D B A E (7) E D B A C (6) C A B D E (6) E A B C D (5) C B A D E (5) C A B E D (5) A B E C D (5) D E B A C (4) A E B C D (4) A B C E D (4) D E C B A (3) D E C A B (3) B A E D C (3) A C B E D (3) E D C A B (2) E A D B C (2) E A B D C (2) D E B C A (2) D B E A C (2) E D A C B (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C A E (1) D B A E C (1) C D E A B (1) C A E B D (1) B E A D C (1) B D C A E (1) B D A E C (1) B D A C E (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 16 20 -4 2 B -16 0 20 -2 -2 C -20 -20 0 -4 -22 D 4 2 4 0 -16 E -2 2 22 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.181818 Sum of squares = 0.570247933895 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 0.727273 D: 0.818182 E: 1.000000 A B C D E A 0 16 20 -4 2 B -16 0 20 -2 -2 C -20 -20 0 -4 -22 D 4 2 4 0 -16 E -2 2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.181818 Sum of squares = 0.570247933581 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 0.727273 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=25 D=19 A=17 B=8 so B is eliminated. Round 2 votes counts: E=32 C=25 D=22 A=21 so A is eliminated. Round 3 votes counts: E=44 C=34 D=22 so D is eliminated. Round 4 votes counts: E=61 C=39 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 A:217 B:200 D:197 C:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 20 -4 2 B -16 0 20 -2 -2 C -20 -20 0 -4 -22 D 4 2 4 0 -16 E -2 2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.181818 Sum of squares = 0.570247933581 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 0.727273 D: 0.818182 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 20 -4 2 B -16 0 20 -2 -2 C -20 -20 0 -4 -22 D 4 2 4 0 -16 E -2 2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.181818 Sum of squares = 0.570247933581 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 0.727273 D: 0.818182 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 20 -4 2 B -16 0 20 -2 -2 C -20 -20 0 -4 -22 D 4 2 4 0 -16 E -2 2 22 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.727273 B: 0.000000 C: 0.000000 D: 0.090909 E: 0.181818 Sum of squares = 0.570247933581 Cumulative probabilities = A: 0.727273 B: 0.727273 C: 0.727273 D: 0.818182 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8009: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (10) D E A C B (9) D B E A C (8) B D C A E (8) D B C E A (7) D E A B C (6) E A C D B (5) D E B A C (4) B D C E A (4) A E C B D (4) E D A C B (3) E A D C B (3) B C D A E (3) E A C B D (2) E A B D C (2) C B D A E (2) C B A E D (2) C A E D B (2) B D E A C (2) B C A E D (2) B A E C D (2) A E C D B (2) E D A B C (1) E A D B C (1) E A B C D (1) D C E A B (1) D C B A E (1) C D B A E (1) C D A E B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 12 -22 -18 B -8 0 2 -10 -16 C -12 -2 0 -20 -10 D 22 10 20 0 14 E 18 16 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -22 -18 B -8 0 2 -10 -16 C -12 -2 0 -20 -10 D 22 10 20 0 14 E 18 16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=21 E=18 C=18 A=7 so A is eliminated. Round 2 votes counts: D=36 E=24 B=21 C=19 so C is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:233 E:215 A:190 B:184 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -22 -18 B -8 0 2 -10 -16 C -12 -2 0 -20 -10 D 22 10 20 0 14 E 18 16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -22 -18 B -8 0 2 -10 -16 C -12 -2 0 -20 -10 D 22 10 20 0 14 E 18 16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -22 -18 B -8 0 2 -10 -16 C -12 -2 0 -20 -10 D 22 10 20 0 14 E 18 16 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8010: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) D C B E A (9) E A C B D (8) A B E C D (6) D E C B A (5) D B C A E (5) D C E B A (4) D B C E A (3) D A C E B (3) B C E D A (3) B A E C D (3) A B C E D (3) E B A C D (2) D A E C B (2) D A B C E (2) B D C A E (2) B C D E A (2) A E D C B (2) A B C D E (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B D A (1) E C A B D (1) E B C A D (1) E A C D B (1) E A B C D (1) D E C A B (1) D E A C B (1) D A C B E (1) C E B D A (1) C D B E A (1) C B D E A (1) B E C A D (1) B D C E A (1) B A D C E (1) B A C E D (1) A E C B D (1) A D E C B (1) A D E B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 10 -2 -2 B -4 0 6 10 -2 C -10 -6 0 6 -8 D 2 -10 -6 0 -2 E 2 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 -2 -2 B -4 0 6 10 -2 C -10 -6 0 6 -8 D 2 -10 -6 0 -2 E 2 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=29 E=18 B=14 C=3 so C is eliminated. Round 2 votes counts: D=37 A=29 E=19 B=15 so B is eliminated. Round 3 votes counts: D=43 A=34 E=23 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:207 A:205 B:205 D:192 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 -2 -2 B -4 0 6 10 -2 C -10 -6 0 6 -8 D 2 -10 -6 0 -2 E 2 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -2 -2 B -4 0 6 10 -2 C -10 -6 0 6 -8 D 2 -10 -6 0 -2 E 2 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -2 -2 B -4 0 6 10 -2 C -10 -6 0 6 -8 D 2 -10 -6 0 -2 E 2 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995455 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8011: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) B A D C E (7) E A B D C (6) E A B C D (6) E C D A B (5) D A B C E (5) C E D B A (5) E C A B D (4) D E A B C (4) A B D E C (4) E D A B C (3) E C D B A (3) D A E B C (3) A B E D C (3) E D C A B (2) E C B A D (2) E A C B D (2) D C E A B (2) C E B D A (2) C D E B A (2) C D B A E (2) C B D A E (2) A B D C E (2) E C A D B (1) E B A C D (1) D E C A B (1) D C E B A (1) D C A B E (1) D B A C E (1) D A B E C (1) C E B A D (1) C B E D A (1) C B E A D (1) C B A D E (1) B D A C E (1) B C D A E (1) B C A D E (1) B A E C D (1) A D B C E (1) Total count = 100 A B C D E A 0 12 2 -12 -10 B -12 0 2 -2 -12 C -2 -2 0 -12 -4 D 12 2 12 0 2 E 10 12 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 -12 -10 B -12 0 2 -2 -12 C -2 -2 0 -12 -4 D 12 2 12 0 2 E 10 12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=27 C=17 B=11 A=10 so A is eliminated. Round 2 votes counts: E=35 D=28 B=20 C=17 so C is eliminated. Round 3 votes counts: E=43 D=32 B=25 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 E:212 A:196 C:190 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 2 -12 -10 B -12 0 2 -2 -12 C -2 -2 0 -12 -4 D 12 2 12 0 2 E 10 12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -12 -10 B -12 0 2 -2 -12 C -2 -2 0 -12 -4 D 12 2 12 0 2 E 10 12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -12 -10 B -12 0 2 -2 -12 C -2 -2 0 -12 -4 D 12 2 12 0 2 E 10 12 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8012: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) B E C D A (8) E C B A D (7) C E D A B (7) C E A D B (7) B D A E C (7) A D C E B (6) A D B C E (5) D A C E B (4) D A B E C (3) C E B A D (3) C A D E B (3) B D E A C (3) A B D E C (3) E B C A D (2) B E A C D (2) B A E C D (2) B A D E C (2) E C B D A (1) E B C D A (1) D C A E B (1) D B E A C (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E A B (1) C D A E B (1) C A E D B (1) B E D C A (1) A D C B E (1) A C E D B (1) A C E B D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -10 16 -10 B 2 0 6 12 4 C 10 -6 0 20 -8 D -16 -12 -20 0 -12 E 10 -4 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998383 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 16 -10 B 2 0 6 12 4 C 10 -6 0 20 -8 D -16 -12 -20 0 -12 E 10 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=24 A=19 E=11 D=11 so E is eliminated. Round 2 votes counts: B=38 C=32 A=19 D=11 so D is eliminated. Round 3 votes counts: B=39 C=33 A=28 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:213 B:212 C:208 A:197 D:170 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -10 16 -10 B 2 0 6 12 4 C 10 -6 0 20 -8 D -16 -12 -20 0 -12 E 10 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 16 -10 B 2 0 6 12 4 C 10 -6 0 20 -8 D -16 -12 -20 0 -12 E 10 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 16 -10 B 2 0 6 12 4 C 10 -6 0 20 -8 D -16 -12 -20 0 -12 E 10 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999373 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8013: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (7) A D E B C (7) C B E D A (6) C B E A D (6) E B D C A (5) B E A D C (5) A D C E B (5) C A D E B (4) D C A E B (3) D A E B C (3) C A D B E (3) B E C A D (3) B E A C D (3) A E B D C (3) A B E D C (3) E B D A C (2) D E B C A (2) D E B A C (2) D E A B C (2) D A E C B (2) C A B E D (2) A C D E B (2) A B E C D (2) E D B C A (1) E D B A C (1) D C E B A (1) D A C E B (1) C E B D A (1) C D A E B (1) C B D E A (1) C B A D E (1) C A B D E (1) B E D C A (1) B E D A C (1) B C E A D (1) B A E C D (1) A E D B C (1) A D E C B (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 0 14 -2 B 2 0 12 12 0 C 0 -12 0 -6 -16 D -14 -12 6 0 -12 E 2 0 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.526807 C: 0.000000 D: 0.000000 E: 0.473193 Sum of squares = 0.50143722045 Cumulative probabilities = A: 0.000000 B: 0.526807 C: 0.526807 D: 0.526807 E: 1.000000 A B C D E A 0 -2 0 14 -2 B 2 0 12 12 0 C 0 -12 0 -6 -16 D -14 -12 6 0 -12 E 2 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999676 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 B=22 D=16 E=9 so E is eliminated. Round 2 votes counts: B=29 A=27 C=26 D=18 so D is eliminated. Round 3 votes counts: B=35 A=35 C=30 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:213 A:205 D:184 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 14 -2 B 2 0 12 12 0 C 0 -12 0 -6 -16 D -14 -12 6 0 -12 E 2 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999676 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 14 -2 B 2 0 12 12 0 C 0 -12 0 -6 -16 D -14 -12 6 0 -12 E 2 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999676 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 14 -2 B 2 0 12 12 0 C 0 -12 0 -6 -16 D -14 -12 6 0 -12 E 2 0 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999676 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8014: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (13) D A B C E (6) C E A B D (6) B D E C A (6) A C E D B (6) D B A C E (5) D A B E C (5) E C B A D (4) D A C E B (4) B E C A D (4) E A C B D (3) C E B A D (3) C A E D B (3) B E C D A (3) B D E A C (3) E C A B D (2) E B A C D (2) B E D C A (2) B D C E A (2) B C E D A (2) A E C D B (2) E B C A D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B E A C (1) D B C E A (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C B E (1) C B E D A (1) C A E B D (1) B C D E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -12 10 -22 4 B 12 0 20 -6 18 C -10 -20 0 -10 -10 D 22 6 10 0 8 E -4 -18 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -22 4 B 12 0 20 -6 18 C -10 -20 0 -10 -10 D 22 6 10 0 8 E -4 -18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 B=23 C=14 E=13 A=9 so A is eliminated. Round 2 votes counts: D=42 B=23 C=20 E=15 so E is eliminated. Round 3 votes counts: D=42 C=31 B=27 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:222 A:190 E:190 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 10 -22 4 B 12 0 20 -6 18 C -10 -20 0 -10 -10 D 22 6 10 0 8 E -4 -18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -22 4 B 12 0 20 -6 18 C -10 -20 0 -10 -10 D 22 6 10 0 8 E -4 -18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -22 4 B 12 0 20 -6 18 C -10 -20 0 -10 -10 D 22 6 10 0 8 E -4 -18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8015: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) C D A B E (9) D A E B C (7) B A E D C (5) E B C A D (4) C E B A D (4) C D A E B (4) A D B E C (4) E D B A C (3) E B D A C (3) E B A C D (3) D A C B E (3) C B E A D (3) D A B E C (2) D A B C E (2) C E D B A (2) C E B D A (2) C D E A B (2) C B E D A (2) C B A D E (2) C A B D E (2) B E A C D (2) E D C A B (1) E D B C A (1) E D A C B (1) E C B D A (1) E A D B C (1) D E A B C (1) D C A B E (1) D A E C B (1) C B D A E (1) C A D B E (1) B E C A D (1) B E A D C (1) B C E A D (1) B C A E D (1) B A D E C (1) A D B C E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 10 2 0 B 8 0 16 6 -4 C -10 -16 0 -4 -10 D -2 -6 4 0 -6 E 0 4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.193176 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.806824 Sum of squares = 0.688281330194 Cumulative probabilities = A: 0.193176 B: 0.193176 C: 0.193176 D: 0.193176 E: 1.000000 A B C D E A 0 -8 10 2 0 B 8 0 16 6 -4 C -10 -16 0 -4 -10 D -2 -6 4 0 -6 E 0 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555650027 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=29 D=17 B=12 A=8 so A is eliminated. Round 2 votes counts: C=34 E=29 D=22 B=15 so B is eliminated. Round 3 votes counts: E=38 C=37 D=25 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:213 E:210 A:202 D:195 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 10 2 0 B 8 0 16 6 -4 C -10 -16 0 -4 -10 D -2 -6 4 0 -6 E 0 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555650027 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 2 0 B 8 0 16 6 -4 C -10 -16 0 -4 -10 D -2 -6 4 0 -6 E 0 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555650027 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 2 0 B 8 0 16 6 -4 C -10 -16 0 -4 -10 D -2 -6 4 0 -6 E 0 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555650027 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8016: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) A D B E C (9) D B E A C (8) B E D C A (6) D B E C A (4) C E B D A (4) A C D E B (4) A C D B E (4) A C B E D (4) A B D E C (4) B D E C A (3) E D B C A (2) E B C D A (2) D C E B A (2) C E B A D (2) C A E D B (2) A D C B E (2) A D B C E (2) A C E D B (2) A B E D C (2) E C B D A (1) E B D C A (1) D E A C B (1) D B A E C (1) D A C E B (1) D A B E C (1) C E D B A (1) C E D A B (1) C A E B D (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E A C (1) B A E C D (1) B A D E C (1) A D C E B (1) A C E B D (1) A C B D E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 12 -6 -6 B 8 0 30 -18 20 C -12 -30 0 -28 -26 D 6 18 28 0 24 E 6 -20 26 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 -6 -6 B 8 0 30 -18 20 C -12 -30 0 -28 -26 D 6 18 28 0 24 E 6 -20 26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 D=29 B=15 C=11 E=6 so E is eliminated. Round 2 votes counts: A=39 D=31 B=18 C=12 so C is eliminated. Round 3 votes counts: A=42 D=33 B=25 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:238 B:220 A:196 E:194 C:152 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 12 -6 -6 B 8 0 30 -18 20 C -12 -30 0 -28 -26 D 6 18 28 0 24 E 6 -20 26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 -6 -6 B 8 0 30 -18 20 C -12 -30 0 -28 -26 D 6 18 28 0 24 E 6 -20 26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 -6 -6 B 8 0 30 -18 20 C -12 -30 0 -28 -26 D 6 18 28 0 24 E 6 -20 26 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8017: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (9) D A B E C (8) D A B C E (6) C E D A B (5) C D E A B (5) C E B A D (4) E C A B D (3) E B A C D (3) D C B A E (3) D A C B E (3) C E D B A (3) C E B D A (3) B E C A D (3) B D A E C (3) A B D E C (3) D C E A B (2) D C A E B (2) D A C E B (2) C D E B A (2) B D A C E (2) B C E D A (2) B A E D C (2) B A E C D (2) A E D C B (2) A D E C B (2) A D B E C (2) A B E D C (2) E C B A D (1) E C A D B (1) D C A B E (1) D B A C E (1) D A E C B (1) C E A D B (1) C B E D A (1) C B D E A (1) B E A C D (1) A E D B C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 8 14 -12 18 B -8 0 4 -8 10 C -14 -4 0 -18 -2 D 12 8 18 0 18 E -18 -10 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 -12 18 B -8 0 4 -8 10 C -14 -4 0 -18 -2 D 12 8 18 0 18 E -18 -10 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 B=24 A=14 E=8 so E is eliminated. Round 2 votes counts: C=30 D=29 B=27 A=14 so A is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:228 A:214 B:199 C:181 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 14 -12 18 B -8 0 4 -8 10 C -14 -4 0 -18 -2 D 12 8 18 0 18 E -18 -10 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 -12 18 B -8 0 4 -8 10 C -14 -4 0 -18 -2 D 12 8 18 0 18 E -18 -10 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 -12 18 B -8 0 4 -8 10 C -14 -4 0 -18 -2 D 12 8 18 0 18 E -18 -10 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8018: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) E A C D B (9) D C B E A (7) B A C D E (5) A E B C D (5) E D C A B (3) E D A C B (3) D E C A B (3) D C E B A (3) B D C A E (3) B C D A E (3) B A E C D (3) A E C B D (3) A B C E D (3) E B A D C (2) E A B C D (2) D E C B A (2) D C E A B (2) C D B A E (2) B C A D E (2) B A C E D (2) E C D A B (1) E C A D B (1) E B D A C (1) E A D B C (1) E A C B D (1) E A B D C (1) D C B A E (1) D B E C A (1) D B C E A (1) D B C A E (1) C E D A B (1) C D E A B (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D A C (1) B D E C A (1) B D E A C (1) B A E D C (1) A E C D B (1) A C E D B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 14 14 -16 B -10 0 -18 -10 -12 C -14 18 0 4 -14 D -14 10 -4 0 -14 E 16 12 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 14 -16 B -10 0 -18 -10 -12 C -14 18 0 4 -14 D -14 10 -4 0 -14 E 16 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=22 D=21 A=15 C=8 so C is eliminated. Round 2 votes counts: E=35 D=24 B=24 A=17 so A is eliminated. Round 3 votes counts: E=45 B=30 D=25 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 A:211 C:197 D:189 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 14 -16 B -10 0 -18 -10 -12 C -14 18 0 4 -14 D -14 10 -4 0 -14 E 16 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 14 -16 B -10 0 -18 -10 -12 C -14 18 0 4 -14 D -14 10 -4 0 -14 E 16 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 14 -16 B -10 0 -18 -10 -12 C -14 18 0 4 -14 D -14 10 -4 0 -14 E 16 12 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8019: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (13) D C B E A (9) D B E A C (9) A E B C D (9) C D A B E (6) C D B A E (5) A C E B D (5) D E B A C (4) E A B D C (3) D B E C A (3) C D B E A (3) A B E D C (3) E D B A C (2) D C E B A (2) D B C E A (2) C D E A B (2) C D A E B (2) C A D E B (2) C A B D E (2) B E A D C (2) A E C B D (2) E B A D C (1) D E B C A (1) C E A D B (1) C D E B A (1) C B D A E (1) C A E D B (1) C A D B E (1) B E D A C (1) B D E A C (1) A E B D C (1) Total count = 100 A B C D E A 0 6 -14 -8 6 B -6 0 -16 -12 -4 C 14 16 0 12 16 D 8 12 -12 0 12 E -6 4 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 -8 6 B -6 0 -16 -12 -4 C 14 16 0 12 16 D 8 12 -12 0 12 E -6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 D=30 A=20 E=6 B=4 so B is eliminated. Round 2 votes counts: C=40 D=31 A=20 E=9 so E is eliminated. Round 3 votes counts: C=40 D=34 A=26 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:210 A:195 E:185 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 -8 6 B -6 0 -16 -12 -4 C 14 16 0 12 16 D 8 12 -12 0 12 E -6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 -8 6 B -6 0 -16 -12 -4 C 14 16 0 12 16 D 8 12 -12 0 12 E -6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 -8 6 B -6 0 -16 -12 -4 C 14 16 0 12 16 D 8 12 -12 0 12 E -6 4 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8020: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (7) D E C A B (6) C E A D B (6) D E B A C (5) C A B E D (5) E D C A B (4) E C A D B (4) D B C E A (4) B D A E C (4) D E C B A (3) D B A E C (3) C A E B D (3) B A E D C (3) E D A B C (2) E A C D B (2) E A C B D (2) E A B D C (2) D E A B C (2) D C E B A (2) D B E A C (2) C E D A B (2) C D E A B (2) C D A E B (2) C B A D E (2) B D E A C (2) B D C A E (2) B A D E C (2) A E C B D (2) A E B C D (2) A B C E D (2) D E B C A (1) D E A C B (1) D C E A B (1) D B E C A (1) C D B A E (1) C B A E D (1) B A E C D (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -4 -4 -12 B -6 0 -4 -12 -14 C 4 4 0 -4 -12 D 4 12 4 0 -2 E 12 14 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 -4 -12 B -6 0 -4 -12 -14 C 4 4 0 -4 -12 D 4 12 4 0 -2 E 12 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=24 B=22 E=16 A=7 so A is eliminated. Round 2 votes counts: D=31 C=25 B=24 E=20 so E is eliminated. Round 3 votes counts: D=37 C=35 B=28 so B is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:209 C:196 A:193 B:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 -4 -12 B -6 0 -4 -12 -14 C 4 4 0 -4 -12 D 4 12 4 0 -2 E 12 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 -12 B -6 0 -4 -12 -14 C 4 4 0 -4 -12 D 4 12 4 0 -2 E 12 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 -12 B -6 0 -4 -12 -14 C 4 4 0 -4 -12 D 4 12 4 0 -2 E 12 14 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8021: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D A E B C (6) C E A B D (6) B A D E C (5) D B A C E (4) C E D A B (4) E C A D B (3) D C E A B (3) C E A D B (3) C B E A D (3) B D A E C (3) E C A B D (2) E A C B D (2) D E C A B (2) D C B E A (2) D B C E A (2) D B A E C (2) C B D E A (2) B D C A E (2) B D A C E (2) B C D A E (2) B A E C D (2) B A D C E (2) A E B C D (2) A B E C D (2) E D C A B (1) E A D C B (1) D E A B C (1) D C E B A (1) D B C A E (1) D A B E C (1) C E D B A (1) C E B D A (1) C E B A D (1) C D E B A (1) C D E A B (1) C D B E A (1) C A E B D (1) B C A D E (1) B A E D C (1) B A C E D (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 6 8 -4 B -14 0 -4 -6 -12 C -6 4 0 4 -4 D -8 6 -4 0 0 E 4 12 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.127905 E: 0.872095 Sum of squares = 0.776909035835 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.127905 E: 1.000000 A B C D E A 0 14 6 8 -4 B -14 0 -4 -6 -12 C -6 4 0 4 -4 D -8 6 -4 0 0 E 4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 B=21 E=17 A=12 so A is eliminated. Round 2 votes counts: D=27 C=25 B=25 E=23 so E is eliminated. Round 3 votes counts: C=42 D=30 B=28 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:212 E:210 C:199 D:197 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 6 8 -4 B -14 0 -4 -6 -12 C -6 4 0 4 -4 D -8 6 -4 0 0 E 4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 8 -4 B -14 0 -4 -6 -12 C -6 4 0 4 -4 D -8 6 -4 0 0 E 4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 8 -4 B -14 0 -4 -6 -12 C -6 4 0 4 -4 D -8 6 -4 0 0 E 4 12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.666667 Sum of squares = 0.555555559081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8022: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) B E D C A (8) E B A D C (7) C A B E D (5) A C D E B (5) E B A C D (4) D E B A C (4) B E C D A (4) A C D B E (4) E D B A C (3) E B D C A (3) E B D A C (3) C D B A E (3) C B E A D (3) C A D B E (3) A D C E B (3) A C B E D (3) D E B C A (2) D C A E B (2) D B E C A (2) C D B E A (2) C D A B E (2) B C E D A (2) A E B D C (2) A C E D B (2) E A B D C (1) E A B C D (1) D A C E B (1) C B E D A (1) B E C A D (1) B D C E A (1) B C E A D (1) A E C B D (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -26 -8 -4 -24 B 26 0 2 6 10 C 8 -2 0 -2 6 D 4 -6 2 0 -14 E 24 -10 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999726 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -8 -4 -24 B 26 0 2 6 10 C 8 -2 0 -2 6 D 4 -6 2 0 -14 E 24 -10 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=22 A=22 D=20 C=19 B=17 so B is eliminated. Round 2 votes counts: E=35 C=22 A=22 D=21 so D is eliminated. Round 3 votes counts: E=43 C=34 A=23 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:222 E:211 C:205 D:193 A:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -8 -4 -24 B 26 0 2 6 10 C 8 -2 0 -2 6 D 4 -6 2 0 -14 E 24 -10 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -8 -4 -24 B 26 0 2 6 10 C 8 -2 0 -2 6 D 4 -6 2 0 -14 E 24 -10 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -8 -4 -24 B 26 0 2 6 10 C 8 -2 0 -2 6 D 4 -6 2 0 -14 E 24 -10 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8023: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) B D E C A (7) A C E B D (6) D E B C A (5) B C A E D (5) A D E C B (4) A C B E D (4) E D C A B (3) E D A C B (3) D E A C B (3) D B E A C (3) B C E D A (3) A E C D B (3) A C E D B (3) E D C B A (2) E C D A B (2) E A C D B (2) D E C A B (2) D B A E C (2) D A B E C (2) C E A B D (2) B D C E A (2) B D A C E (2) B A C D E (2) A D C E B (2) A B C D E (2) E D B C A (1) E C B D A (1) E C A D B (1) E C A B D (1) D E C B A (1) D E A B C (1) D A E C B (1) D A B C E (1) C B E A D (1) C A E B D (1) B D A E C (1) B C D E A (1) B A C E D (1) A D B C E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 -14 -10 B -4 0 2 -12 0 C -2 -2 0 -14 -18 D 14 12 14 0 8 E 10 0 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -14 -10 B -4 0 2 -12 0 C -2 -2 0 -14 -18 D 14 12 14 0 8 E 10 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 B=24 E=16 C=4 so C is eliminated. Round 2 votes counts: D=29 A=28 B=25 E=18 so E is eliminated. Round 3 votes counts: D=40 A=34 B=26 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:210 B:193 A:191 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -14 -10 B -4 0 2 -12 0 C -2 -2 0 -14 -18 D 14 12 14 0 8 E 10 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -14 -10 B -4 0 2 -12 0 C -2 -2 0 -14 -18 D 14 12 14 0 8 E 10 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -14 -10 B -4 0 2 -12 0 C -2 -2 0 -14 -18 D 14 12 14 0 8 E 10 0 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8024: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (15) A D B C E (12) D A E B C (10) C B E A D (8) E B C D A (6) A D C B E (5) A C B D E (5) D E A C B (4) D E A B C (4) A B C D E (4) D A B E C (3) C E B A D (3) C B A E D (3) D A E C B (2) A C D B E (2) A B D C E (2) E D C B A (1) E D B A C (1) E C D B A (1) E C B A D (1) D B A E C (1) C B E D A (1) C A B E D (1) B E C D A (1) B D C E A (1) B C E D A (1) B C A E D (1) A D B E C (1) Total count = 100 A B C D E A 0 10 12 -4 4 B -10 0 -4 6 4 C -12 4 0 6 -2 D 4 -6 -6 0 12 E -4 -4 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.545455 E: 0.000000 Sum of squares = 0.404958677242 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 -4 4 B -10 0 -4 6 4 C -12 4 0 6 -2 D 4 -6 -6 0 12 E -4 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.545455 E: 0.000000 Sum of squares = 0.40495867732 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=25 D=24 C=16 B=4 so B is eliminated. Round 2 votes counts: A=31 E=26 D=25 C=18 so C is eliminated. Round 3 votes counts: E=39 A=36 D=25 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:202 B:198 C:198 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 -4 4 B -10 0 -4 6 4 C -12 4 0 6 -2 D 4 -6 -6 0 12 E -4 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.545455 E: 0.000000 Sum of squares = 0.40495867732 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 -4 4 B -10 0 -4 6 4 C -12 4 0 6 -2 D 4 -6 -6 0 12 E -4 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.545455 E: 0.000000 Sum of squares = 0.40495867732 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 -4 4 B -10 0 -4 6 4 C -12 4 0 6 -2 D 4 -6 -6 0 12 E -4 -4 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.272727 B: 0.000000 C: 0.181818 D: 0.545455 E: 0.000000 Sum of squares = 0.40495867732 Cumulative probabilities = A: 0.272727 B: 0.272727 C: 0.454545 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8025: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (13) B C A D E (8) B C D A E (7) A E C B D (7) E D A C B (6) B D C E A (6) E A C D B (4) D B C E A (4) B D C A E (4) E D B A C (3) C B A D E (3) A E C D B (3) A E B C D (3) A C E B D (3) E D A B C (2) E A B C D (2) D E C A B (2) D C B E A (2) D B E C A (2) C A B D E (2) B A C E D (2) A C B E D (2) A B C E D (2) D E C B A (1) D E B C A (1) D C E A B (1) D C B A E (1) D C A E B (1) C A D B E (1) B A E C D (1) A C B D E (1) Total count = 100 A B C D E A 0 10 8 14 2 B -10 0 -6 6 -4 C -8 6 0 2 0 D -14 -6 -2 0 -6 E -2 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 14 2 B -10 0 -6 6 -4 C -8 6 0 2 0 D -14 -6 -2 0 -6 E -2 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=28 A=21 D=15 C=6 so C is eliminated. Round 2 votes counts: B=31 E=30 A=24 D=15 so D is eliminated. Round 3 votes counts: B=40 E=35 A=25 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:217 E:204 C:200 B:193 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 14 2 B -10 0 -6 6 -4 C -8 6 0 2 0 D -14 -6 -2 0 -6 E -2 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 14 2 B -10 0 -6 6 -4 C -8 6 0 2 0 D -14 -6 -2 0 -6 E -2 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 14 2 B -10 0 -6 6 -4 C -8 6 0 2 0 D -14 -6 -2 0 -6 E -2 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8026: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (15) D B E C A (11) D B E A C (6) C A D E B (6) B D E A C (5) E B D C A (4) C A E D B (4) C A E B D (4) E C B D A (3) E C A B D (3) B E D C A (3) A C D B E (3) A C B E D (3) E B D A C (2) E B A D C (2) D C E B A (2) C E D B A (2) C D E A B (2) B E D A C (2) B D E C A (2) A D C B E (2) A B D E C (2) E B A C D (1) D E B C A (1) D A B C E (1) C E A D B (1) C D B E A (1) C D A B E (1) C A D B E (1) B D A E C (1) A D B C E (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -2 2 -6 B -4 0 -12 6 -8 C 2 12 0 6 4 D -2 -6 -6 0 0 E 6 8 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 2 -6 B -4 0 -12 6 -8 C 2 12 0 6 4 D -2 -6 -6 0 0 E 6 8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989289 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=22 D=21 E=15 B=13 so B is eliminated. Round 2 votes counts: D=29 A=29 C=22 E=20 so E is eliminated. Round 3 votes counts: D=40 A=32 C=28 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:212 E:205 A:199 D:193 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 2 -6 B -4 0 -12 6 -8 C 2 12 0 6 4 D -2 -6 -6 0 0 E 6 8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989289 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 -6 B -4 0 -12 6 -8 C 2 12 0 6 4 D -2 -6 -6 0 0 E 6 8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989289 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 -6 B -4 0 -12 6 -8 C 2 12 0 6 4 D -2 -6 -6 0 0 E 6 8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989289 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8027: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (16) D C B E A (14) B C D A E (13) A B C D E (10) E A D C B (9) A E B C D (9) B A C D E (5) E D C A B (4) E D A C B (2) D C E B A (2) C D B E A (2) B C D E A (2) A B E C D (2) E B C D A (1) E A B D C (1) D E C B A (1) C D B A E (1) C B D A E (1) B A C E D (1) A E D C B (1) A E B D C (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -14 -18 -8 B 18 0 -8 -6 4 C 14 8 0 -4 6 D 18 6 4 0 4 E 8 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 -18 -8 B 18 0 -8 -6 4 C 14 8 0 -4 6 D 18 6 4 0 4 E 8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=25 B=21 D=17 C=4 so C is eliminated. Round 2 votes counts: E=33 A=25 B=22 D=20 so D is eliminated. Round 3 votes counts: B=39 E=36 A=25 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:216 C:212 B:204 E:197 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -14 -18 -8 B 18 0 -8 -6 4 C 14 8 0 -4 6 D 18 6 4 0 4 E 8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 -18 -8 B 18 0 -8 -6 4 C 14 8 0 -4 6 D 18 6 4 0 4 E 8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 -18 -8 B 18 0 -8 -6 4 C 14 8 0 -4 6 D 18 6 4 0 4 E 8 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999143 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8028: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) B A C E D (6) A D B E C (6) D E A C B (5) D A E B C (5) B C A D E (5) B A C D E (5) C B E A D (4) B C A E D (4) E D C A B (3) E D A B C (3) E C D B A (3) D E C A B (3) D A E C B (3) C E B D A (3) A D E B C (3) A B E C D (3) A B D E C (3) E C D A B (2) D A C B E (2) C D E B A (2) C D B A E (2) C B D E A (2) A B E D C (2) E D A C B (1) E C B A D (1) E B A C D (1) E A D C B (1) D C B A E (1) D A C E B (1) D A B C E (1) C D B E A (1) C B E D A (1) B C E A D (1) B A D C E (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 8 -4 10 B 2 0 2 -12 2 C -8 -2 0 8 2 D 4 12 -8 0 4 E -10 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000001 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -4 10 B 2 0 2 -12 2 C -8 -2 0 8 2 D 4 12 -8 0 4 E -10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000012 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 B=22 D=21 A=19 E=15 so E is eliminated. Round 2 votes counts: C=29 D=28 B=23 A=20 so A is eliminated. Round 3 votes counts: D=38 B=33 C=29 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:206 D:206 C:200 B:197 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 8 -4 10 B 2 0 2 -12 2 C -8 -2 0 8 2 D 4 12 -8 0 4 E -10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000012 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -4 10 B 2 0 2 -12 2 C -8 -2 0 8 2 D 4 12 -8 0 4 E -10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000012 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -4 10 B 2 0 2 -12 2 C -8 -2 0 8 2 D 4 12 -8 0 4 E -10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.360000000012 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8029: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E C B A D (6) D B A C E (5) B C E A D (5) A D C E B (5) D A E C B (4) A D B C E (4) A C B D E (4) D B A E C (3) E D C A B (2) E D A C B (2) E C B D A (2) E C A B D (2) E B D C A (2) E B C D A (2) E A D C B (2) C A B E D (2) B C E D A (2) B C A E D (2) B C A D E (2) A D C B E (2) A C E D B (2) A C E B D (2) A C D B E (2) E D C B A (1) E C D A B (1) E C A D B (1) E B C A D (1) D E A C B (1) D B C A E (1) D A E B C (1) D A C E B (1) D A B E C (1) C E B A D (1) C E A B D (1) C B A E D (1) C A E B D (1) B E D C A (1) B D C E A (1) B D A C E (1) B C D E A (1) B C D A E (1) A E C D B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 18 16 2 26 B -18 0 -2 -14 8 C -16 2 0 -8 26 D -2 14 8 0 10 E -26 -8 -26 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998803 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 2 26 B -18 0 -2 -14 8 C -16 2 0 -8 26 D -2 14 8 0 10 E -26 -8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999927858 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=24 A=24 B=16 C=6 so C is eliminated. Round 2 votes counts: D=30 A=27 E=26 B=17 so B is eliminated. Round 3 votes counts: E=34 D=34 A=32 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:231 D:215 C:202 B:187 E:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 2 26 B -18 0 -2 -14 8 C -16 2 0 -8 26 D -2 14 8 0 10 E -26 -8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999927858 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 2 26 B -18 0 -2 -14 8 C -16 2 0 -8 26 D -2 14 8 0 10 E -26 -8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999927858 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 2 26 B -18 0 -2 -14 8 C -16 2 0 -8 26 D -2 14 8 0 10 E -26 -8 -26 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999927858 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8030: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) B C D E A (6) D E A C B (5) D E A B C (5) C B A D E (5) E D B A C (4) E B D C A (4) B C D A E (4) E D A B C (3) E A D B C (3) D E B C A (3) C A D B E (3) A E D C B (3) A C D E B (3) A C D B E (3) E D B C A (2) E A D C B (2) D B E C A (2) B E C D A (2) B D E C A (2) B D C E A (2) B C E A D (2) B C A E D (2) A E C D B (2) A E C B D (2) A C E B D (2) E D A C B (1) E B D A C (1) E B A C D (1) E A B C D (1) D E B A C (1) D C A B E (1) D B C E A (1) D A E C B (1) C B D A E (1) C B A E D (1) C A B D E (1) A E B C D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 8 12 -2 -6 B -8 0 4 -2 -6 C -12 -4 0 4 -4 D 2 2 -4 0 4 E 6 6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.111111 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839477 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -2 -6 B -8 0 4 -2 -6 C -12 -4 0 4 -4 D 2 2 -4 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.111111 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839269 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 B=20 D=19 C=11 so C is eliminated. Round 2 votes counts: A=32 B=27 E=22 D=19 so D is eliminated. Round 3 votes counts: E=36 A=34 B=30 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:206 E:206 D:202 B:194 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -2 -6 B -8 0 4 -2 -6 C -12 -4 0 4 -4 D 2 2 -4 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.111111 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839269 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -2 -6 B -8 0 4 -2 -6 C -12 -4 0 4 -4 D 2 2 -4 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.111111 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839269 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -2 -6 B -8 0 4 -2 -6 C -12 -4 0 4 -4 D 2 2 -4 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.111111 D: 0.666667 E: 0.000000 Sum of squares = 0.506172839269 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8031: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) E D B C A (6) A E D C B (5) A E C D B (5) E D B A C (4) E A D C B (4) C A B E D (4) B C D E A (4) E D A B C (3) D E B A C (3) D E A B C (3) C E B D A (3) C B A E D (3) C A B D E (3) B D E C A (3) B C D A E (3) A C E D B (3) A C B D E (3) E D A C B (2) C B E D A (2) C B D E A (2) C B D A E (2) C A E D B (2) C A E B D (2) B E D C A (2) B D C A E (2) A C D B E (2) E C D A B (1) E B D C A (1) D E B C A (1) C E A B D (1) C B E A D (1) B D E A C (1) B D C E A (1) A E D B C (1) A D E B C (1) A D B E C (1) A D B C E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -12 2 4 B 2 0 -18 2 -6 C 12 18 0 10 6 D -2 -2 -10 0 -12 E -4 6 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 2 4 B 2 0 -18 2 -6 C 12 18 0 10 6 D -2 -2 -10 0 -12 E -4 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=24 E=21 B=16 D=7 so D is eliminated. Round 2 votes counts: C=32 E=28 A=24 B=16 so B is eliminated. Round 3 votes counts: C=42 E=34 A=24 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 E:204 A:196 B:190 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 2 4 B 2 0 -18 2 -6 C 12 18 0 10 6 D -2 -2 -10 0 -12 E -4 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 2 4 B 2 0 -18 2 -6 C 12 18 0 10 6 D -2 -2 -10 0 -12 E -4 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 2 4 B 2 0 -18 2 -6 C 12 18 0 10 6 D -2 -2 -10 0 -12 E -4 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8032: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) E D B C A (8) E C B D A (8) E C A B D (8) A C B D E (8) A C E B D (6) D B A C E (5) D B C E A (4) D B C A E (4) D A B C E (4) E A C B D (3) D B E C A (3) A E C B D (3) E B D C A (2) B D C E A (2) A E D C B (2) A B C D E (2) E D C B A (1) E D B A C (1) E C D B A (1) E C B A D (1) E A D C B (1) E A D B C (1) E A C D B (1) D E B C A (1) C B E D A (1) C A B E D (1) B E D C A (1) B D C A E (1) B C D E A (1) B C D A E (1) A E D B C (1) A E C D B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 8 2 2 2 B -8 0 4 0 0 C -2 -4 0 -6 4 D -2 0 6 0 -6 E -2 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 2 2 B -8 0 4 0 0 C -2 -4 0 -6 4 D -2 0 6 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=35 D=21 B=6 C=2 so C is eliminated. Round 2 votes counts: E=36 A=36 D=21 B=7 so B is eliminated. Round 3 votes counts: E=38 A=36 D=26 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:200 D:199 B:198 C:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 2 2 B -8 0 4 0 0 C -2 -4 0 -6 4 D -2 0 6 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 2 2 B -8 0 4 0 0 C -2 -4 0 -6 4 D -2 0 6 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 2 2 B -8 0 4 0 0 C -2 -4 0 -6 4 D -2 0 6 0 -6 E -2 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8033: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) E B C A D (6) E B C D A (5) D A E B C (5) B E C D A (5) A D C E B (5) A C D B E (5) D A E C B (4) A C B E D (4) E D B C A (3) E C B A D (3) D E B A C (3) D B E C A (3) C B E A D (3) B C E A D (3) A D C B E (3) A C E B D (3) D E B C A (2) D E A B C (2) D A B C E (2) C E B A D (2) C B A E D (2) B E C A D (2) B D E C A (2) B C E D A (2) A C D E B (2) E B D C A (1) E A D C B (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A C E (1) D A B E C (1) C B A D E (1) C A E B D (1) C A B E D (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 -4 2 -2 0 B 4 0 2 -6 0 C -2 -2 0 2 -2 D 2 6 -2 0 2 E 0 0 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.143047 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.190286 Sum of squares = 0.278893530975 Cumulative probabilities = A: 0.143047 B: 0.143047 C: 0.476381 D: 0.809714 E: 1.000000 A B C D E A 0 -4 2 -2 0 B 4 0 2 -6 0 C -2 -2 0 2 -2 D 2 6 -2 0 2 E 0 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.166667 Sum of squares = 0.277777777777 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=24 E=19 B=14 C=10 so C is eliminated. Round 2 votes counts: D=33 A=26 E=21 B=20 so B is eliminated. Round 3 votes counts: E=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:204 B:200 E:200 A:198 C:198 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 -2 0 B 4 0 2 -6 0 C -2 -2 0 2 -2 D 2 6 -2 0 2 E 0 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.166667 Sum of squares = 0.277777777777 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 0 B 4 0 2 -6 0 C -2 -2 0 2 -2 D 2 6 -2 0 2 E 0 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.166667 Sum of squares = 0.277777777777 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 0 B 4 0 2 -6 0 C -2 -2 0 2 -2 D 2 6 -2 0 2 E 0 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.166667 Sum of squares = 0.277777777777 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8034: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (9) D B E C A (6) B D E A C (6) A C E D B (6) B D C A E (4) A E B D C (4) C E A D B (3) C B A D E (3) C A E D B (3) B A C D E (3) A C B D E (3) A B C D E (3) E C A D B (2) E A C D B (2) D B C E A (2) C D E B A (2) B E D A C (2) B A D E C (2) B A D C E (2) A E C D B (2) A C E B D (2) A C B E D (2) A B D E C (2) A B C E D (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D A C (1) D E C B A (1) D E B C A (1) D E B A C (1) D C B E A (1) C E D B A (1) C E D A B (1) C D B E A (1) C D B A E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E A D C (1) B D C E A (1) B C D A E (1) B C A D E (1) B A E D C (1) A E C B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 2 10 6 B 14 0 18 18 26 C -2 -18 0 -2 2 D -10 -18 2 0 18 E -6 -26 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 10 6 B 14 0 18 18 26 C -2 -18 0 -2 2 D -10 -18 2 0 18 E -6 -26 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=29 C=18 D=12 E=8 so E is eliminated. Round 2 votes counts: B=34 A=31 C=20 D=15 so D is eliminated. Round 3 votes counts: B=46 A=31 C=23 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:238 A:202 D:196 C:190 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 10 6 B 14 0 18 18 26 C -2 -18 0 -2 2 D -10 -18 2 0 18 E -6 -26 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 10 6 B 14 0 18 18 26 C -2 -18 0 -2 2 D -10 -18 2 0 18 E -6 -26 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 10 6 B 14 0 18 18 26 C -2 -18 0 -2 2 D -10 -18 2 0 18 E -6 -26 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8035: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (16) E B D A C (8) B D E A C (6) C A D B E (5) B C E D A (5) E D B A C (4) C A D E B (4) C A B D E (4) B D A E C (4) B E D C A (3) B E C D A (3) A D E B C (3) A D C E B (3) A D B E C (3) E D A B C (2) D B E A C (2) C E A B D (2) C B A D E (2) C A E D B (2) A C D E B (2) E C D B A (1) E B D C A (1) E B C D A (1) D E B A C (1) D E A B C (1) C E B A D (1) C E A D B (1) C A B E D (1) B D A C E (1) B C D A E (1) B A D C E (1) A E D C B (1) A D E C B (1) A D C B E (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -22 26 -20 -16 B 22 0 36 22 22 C -26 -36 0 -28 -24 D 20 -22 28 0 -4 E 16 -22 24 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 26 -20 -16 B 22 0 36 22 22 C -26 -36 0 -28 -24 D 20 -22 28 0 -4 E 16 -22 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 C=22 E=17 A=17 D=4 so D is eliminated. Round 2 votes counts: B=42 C=22 E=19 A=17 so A is eliminated. Round 3 votes counts: B=47 C=29 E=24 so E is eliminated. Round 4 votes counts: B=68 C=32 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:251 D:211 E:211 A:184 C:143 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 26 -20 -16 B 22 0 36 22 22 C -26 -36 0 -28 -24 D 20 -22 28 0 -4 E 16 -22 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 26 -20 -16 B 22 0 36 22 22 C -26 -36 0 -28 -24 D 20 -22 28 0 -4 E 16 -22 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 26 -20 -16 B 22 0 36 22 22 C -26 -36 0 -28 -24 D 20 -22 28 0 -4 E 16 -22 24 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8036: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) C B E A D (9) A D C E B (9) D E B A C (6) D E A B C (6) B C E D A (5) A C B D E (5) E B D C A (4) C B A E D (4) B E C D A (4) E D B C A (3) C B E D A (3) A D C B E (3) A C D B E (3) D A E C B (2) B E D C A (2) B C E A D (2) A D E C B (2) A D E B C (2) A C D E B (2) E D C B A (1) E D B A C (1) E C D B A (1) E C B D A (1) E B C D A (1) D B E A C (1) D A B E C (1) C E B D A (1) C E A B D (1) C A B E D (1) C A B D E (1) B C A D E (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 0 10 -6 -4 B 0 0 0 -8 -8 C -10 0 0 -8 2 D 6 8 8 0 10 E 4 8 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 -6 -4 B 0 0 0 -8 -8 C -10 0 0 -8 2 D 6 8 8 0 10 E 4 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 C=20 B=14 E=12 so E is eliminated. Round 2 votes counts: D=31 A=28 C=22 B=19 so B is eliminated. Round 3 votes counts: D=37 C=35 A=28 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:200 E:200 B:192 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 10 -6 -4 B 0 0 0 -8 -8 C -10 0 0 -8 2 D 6 8 8 0 10 E 4 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -6 -4 B 0 0 0 -8 -8 C -10 0 0 -8 2 D 6 8 8 0 10 E 4 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -6 -4 B 0 0 0 -8 -8 C -10 0 0 -8 2 D 6 8 8 0 10 E 4 8 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8037: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (13) B D A E C (10) A E B C D (10) C E A D B (8) D B C E A (7) C D E A B (7) B A E C D (7) B A E D C (6) D C B E A (5) A E C B D (4) E C A D B (3) D C E B A (2) D A B E C (2) C E D A B (2) B D C A E (2) E A C D B (1) E A C B D (1) D B A E C (1) D B A C E (1) D A C E B (1) C E B A D (1) B E C A D (1) B E A C D (1) B D A C E (1) B A D E C (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 8 -2 -8 -4 B -8 0 0 -10 -10 C 2 0 0 -6 0 D 8 10 6 0 8 E 4 10 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -8 -4 B -8 0 0 -10 -10 C 2 0 0 -6 0 D 8 10 6 0 8 E 4 10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=29 C=18 A=16 E=5 so E is eliminated. Round 2 votes counts: D=32 B=29 C=21 A=18 so A is eliminated. Round 3 votes counts: B=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:203 C:198 A:197 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -2 -8 -4 B -8 0 0 -10 -10 C 2 0 0 -6 0 D 8 10 6 0 8 E 4 10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -8 -4 B -8 0 0 -10 -10 C 2 0 0 -6 0 D 8 10 6 0 8 E 4 10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -8 -4 B -8 0 0 -10 -10 C 2 0 0 -6 0 D 8 10 6 0 8 E 4 10 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8038: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) D E C B A (8) E D A B C (7) C B D E A (7) A B E D C (6) D E B C A (5) C D B E A (5) C B A D E (5) C A D E B (4) A E B D C (4) A B C E D (4) E D B A C (3) D C E B A (3) C A B D E (3) B E D A C (3) A E D B C (3) A C B E D (3) E D A C B (2) E B D A C (2) C D E A B (2) B C D E A (2) E A D B C (1) D E C A B (1) D B C E A (1) C D A E B (1) C D A B E (1) C B D A E (1) B D E C A (1) B D C E A (1) B A E D C (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -20 -30 -26 B 14 0 -12 -14 -12 C 20 12 0 -4 6 D 30 14 4 0 20 E 26 12 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999252 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -30 -26 B 14 0 -12 -14 -12 C 20 12 0 -4 6 D 30 14 4 0 20 E 26 12 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=21 D=18 E=15 B=8 so B is eliminated. Round 2 votes counts: C=40 A=22 D=20 E=18 so E is eliminated. Round 3 votes counts: C=40 D=37 A=23 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:234 C:217 E:206 B:188 A:155 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -20 -30 -26 B 14 0 -12 -14 -12 C 20 12 0 -4 6 D 30 14 4 0 20 E 26 12 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -30 -26 B 14 0 -12 -14 -12 C 20 12 0 -4 6 D 30 14 4 0 20 E 26 12 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -30 -26 B 14 0 -12 -14 -12 C 20 12 0 -4 6 D 30 14 4 0 20 E 26 12 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992519 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8039: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) C A B E D (8) B A D C E (7) A C B E D (7) D B E A C (6) C E A D B (6) B A D E C (6) C E A B D (5) D E B C A (4) B D A E C (4) E D C B A (3) E D B A C (3) C E D A B (3) C A E B D (3) A C B D E (3) E C D B A (2) E C D A B (2) D E B A C (2) E D C A B (1) E D B C A (1) E C A D B (1) E B D A C (1) E B A D C (1) D E C B A (1) D B E C A (1) D B C A E (1) C D E B A (1) C D E A B (1) C A B D E (1) B D A C E (1) B A C D E (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 10 24 10 B -8 0 4 22 18 C -10 -4 0 10 20 D -24 -22 -10 0 4 E -10 -18 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 24 10 B -8 0 4 22 18 C -10 -4 0 10 20 D -24 -22 -10 0 4 E -10 -18 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 B=19 E=15 D=15 so E is eliminated. Round 2 votes counts: C=33 D=23 A=23 B=21 so B is eliminated. Round 3 votes counts: A=38 C=33 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:218 C:208 D:174 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 24 10 B -8 0 4 22 18 C -10 -4 0 10 20 D -24 -22 -10 0 4 E -10 -18 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 24 10 B -8 0 4 22 18 C -10 -4 0 10 20 D -24 -22 -10 0 4 E -10 -18 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 24 10 B -8 0 4 22 18 C -10 -4 0 10 20 D -24 -22 -10 0 4 E -10 -18 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8040: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) A B E D C (7) A B D E C (7) A B D C E (6) E C D A B (5) E D A C B (4) B A D C E (4) A D B E C (4) A B E C D (4) E A B C D (3) D C E B A (3) D A B C E (3) C B D A E (3) B A C D E (3) E C D B A (2) E C A B D (2) E A B D C (2) D C B A E (2) C E B D A (2) C E B A D (2) C B E D A (2) B D A C E (2) B C A D E (2) A E D B C (2) A D E B C (2) E C B D A (1) E C B A D (1) E A D B C (1) D E C A B (1) D B A C E (1) D A B E C (1) C E D B A (1) C D E B A (1) C D B E A (1) C D B A E (1) C B E A D (1) C B D E A (1) C B A E D (1) B D C A E (1) B C A E D (1) Total count = 100 A B C D E A 0 22 12 10 14 B -22 0 12 16 16 C -12 -12 0 -20 -12 D -10 -16 20 0 -2 E -14 -16 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 10 14 B -22 0 12 16 16 C -12 -12 0 -20 -12 D -10 -16 20 0 -2 E -14 -16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=28 C=16 B=13 D=11 so D is eliminated. Round 2 votes counts: A=36 E=29 C=21 B=14 so B is eliminated. Round 3 votes counts: A=46 E=29 C=25 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:229 B:211 D:196 E:192 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 12 10 14 B -22 0 12 16 16 C -12 -12 0 -20 -12 D -10 -16 20 0 -2 E -14 -16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 10 14 B -22 0 12 16 16 C -12 -12 0 -20 -12 D -10 -16 20 0 -2 E -14 -16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 10 14 B -22 0 12 16 16 C -12 -12 0 -20 -12 D -10 -16 20 0 -2 E -14 -16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8041: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D C A E B (8) B D C E A (8) B C D E A (7) A E B C D (7) B E A C D (6) A E C D B (5) A D E C B (5) D B C E A (4) A E C B D (4) B A E C D (3) A E D C B (3) A E B D C (3) E A C B D (2) E A B C D (2) D C B A E (2) D B C A E (2) C D B E A (2) E B C A D (1) E B A C D (1) D C E A B (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E B A (1) C D E A B (1) B E C A D (1) B D C A E (1) B D A E C (1) B C E D A (1) B A E D C (1) B A D E C (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 0 -6 4 B 4 0 6 2 4 C 0 -6 0 -10 2 D 6 -2 10 0 14 E -4 -4 -2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -6 4 B 4 0 6 2 4 C 0 -6 0 -10 2 D 6 -2 10 0 14 E -4 -4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=30 B=30 A=29 E=6 C=5 so C is eliminated. Round 2 votes counts: D=34 B=30 A=29 E=7 so E is eliminated. Round 3 votes counts: D=35 A=33 B=32 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:208 A:197 C:193 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 -6 4 B 4 0 6 2 4 C 0 -6 0 -10 2 D 6 -2 10 0 14 E -4 -4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -6 4 B 4 0 6 2 4 C 0 -6 0 -10 2 D 6 -2 10 0 14 E -4 -4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -6 4 B 4 0 6 2 4 C 0 -6 0 -10 2 D 6 -2 10 0 14 E -4 -4 -2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8042: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) E D A B C (5) C B A E D (5) B A C E D (5) A B E D C (5) E A D B C (4) B C A D E (4) B A D E C (4) A B E C D (4) E C A D B (3) C D E B A (3) C D B E A (3) C B D E A (3) C B D A E (3) B C A E D (3) A E B C D (3) E A D C B (2) D E C B A (2) D E A B C (2) D C E B A (2) D B A E C (2) C E D B A (2) C E D A B (2) C E A D B (2) C A E B D (2) A E B D C (2) A B D E C (2) E D C A B (1) E D A C B (1) E C D A B (1) D E A C B (1) D B C E A (1) D B A C E (1) D A B E C (1) C E A B D (1) C A B E D (1) B D C A E (1) B D A E C (1) B A E C D (1) B A D C E (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 4 -2 12 6 B -4 0 8 4 4 C 2 -8 0 4 -8 D -12 -4 -4 0 -12 E -6 -4 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428551 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 12 6 B -4 0 8 4 4 C 2 -8 0 4 -8 D -12 -4 -4 0 -12 E -6 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.42857142857 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=21 D=18 E=17 A=17 so E is eliminated. Round 2 votes counts: C=31 D=25 A=23 B=21 so B is eliminated. Round 3 votes counts: C=38 A=35 D=27 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:210 B:206 E:205 C:195 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 12 6 B -4 0 8 4 4 C 2 -8 0 4 -8 D -12 -4 -4 0 -12 E -6 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.42857142857 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 12 6 B -4 0 8 4 4 C 2 -8 0 4 -8 D -12 -4 -4 0 -12 E -6 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.42857142857 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 12 6 B -4 0 8 4 4 C 2 -8 0 4 -8 D -12 -4 -4 0 -12 E -6 -4 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.142857 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.42857142857 Cumulative probabilities = A: 0.571429 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8043: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) A B C E D (8) E D C A B (7) C D E B A (6) B A C D E (6) C A B D E (5) B A E D C (5) E D A B C (4) D E C B A (4) C B A D E (4) B C A D E (4) D C E B A (3) C D E A B (3) C D B E A (3) A B E C D (3) E D B A C (2) E C A D B (2) D E C A B (2) D B C E A (2) C D B A E (2) C A B E D (2) B A C E D (2) E D A C B (1) E A D C B (1) E A D B C (1) D E B C A (1) D E B A C (1) C B D A E (1) B D C A E (1) B A D E C (1) A E B D C (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -4 14 14 B -4 0 4 10 20 C 4 -4 0 6 8 D -14 -10 -6 0 0 E -14 -20 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 14 14 B -4 0 4 10 20 C 4 -4 0 6 8 D -14 -10 -6 0 0 E -14 -20 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 B=19 E=18 D=13 so D is eliminated. Round 2 votes counts: C=29 E=26 A=24 B=21 so B is eliminated. Round 3 votes counts: A=38 C=36 E=26 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:215 A:214 C:207 D:185 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 14 14 B -4 0 4 10 20 C 4 -4 0 6 8 D -14 -10 -6 0 0 E -14 -20 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 14 14 B -4 0 4 10 20 C 4 -4 0 6 8 D -14 -10 -6 0 0 E -14 -20 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 14 14 B -4 0 4 10 20 C 4 -4 0 6 8 D -14 -10 -6 0 0 E -14 -20 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8044: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (14) B C D E A (9) C B A E D (7) E A D B C (6) D E A B C (6) D B E A C (6) C B D E A (5) C A E D B (5) B D E A C (4) A C E D B (4) D E A C B (3) C B D A E (3) B D C E A (3) A E D B C (3) A E C D B (3) D E B A C (2) C A E B D (2) C A D E B (2) C A B E D (2) B C A E D (2) E D A B C (1) C D A E B (1) C A D B E (1) B E D A C (1) B E A D C (1) B D E C A (1) B C E A D (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 10 12 10 2 B -10 0 -6 -16 -8 C -12 6 0 -6 -6 D -10 16 6 0 -6 E -2 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 10 2 B -10 0 -6 -16 -8 C -12 6 0 -6 -6 D -10 16 6 0 -6 E -2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=26 B=22 D=17 E=7 so E is eliminated. Round 2 votes counts: A=32 C=28 B=22 D=18 so D is eliminated. Round 3 votes counts: A=42 B=30 C=28 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:209 D:203 C:191 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 10 2 B -10 0 -6 -16 -8 C -12 6 0 -6 -6 D -10 16 6 0 -6 E -2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 10 2 B -10 0 -6 -16 -8 C -12 6 0 -6 -6 D -10 16 6 0 -6 E -2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 10 2 B -10 0 -6 -16 -8 C -12 6 0 -6 -6 D -10 16 6 0 -6 E -2 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8045: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) B E C D A (7) C E D A B (6) B A D E C (6) A D B C E (6) E C B D A (5) E B C D A (5) A D C E B (4) A C D E B (4) E C D B A (3) D A C E B (3) C D E A B (3) A D C B E (3) D C E A B (2) D B E A C (2) D A B E C (2) C E D B A (2) C E A D B (2) C A D E B (2) B E A D C (2) B D A E C (2) B A E C D (2) A C B E D (2) E B D C A (1) D E C B A (1) D E B C A (1) D C E B A (1) D C A E B (1) D B A E C (1) C E B D A (1) C E B A D (1) C A E B D (1) B E D A C (1) B E C A D (1) B D E C A (1) B A E D C (1) A C E D B (1) A C D B E (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -8 -16 -10 B 8 0 2 -2 0 C 8 -2 0 2 -6 D 16 2 -2 0 -6 E 10 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.406003 C: 0.000000 D: 0.000000 E: 0.593997 Sum of squares = 0.517671038515 Cumulative probabilities = A: 0.000000 B: 0.406003 C: 0.406003 D: 0.406003 E: 1.000000 A B C D E A 0 -8 -8 -16 -10 B 8 0 2 -2 0 C 8 -2 0 2 -6 D 16 2 -2 0 -6 E 10 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 C=18 E=14 D=14 so E is eliminated. Round 2 votes counts: B=36 C=26 A=24 D=14 so D is eliminated. Round 3 votes counts: B=40 C=31 A=29 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:211 D:205 B:204 C:201 A:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -16 -10 B 8 0 2 -2 0 C 8 -2 0 2 -6 D 16 2 -2 0 -6 E 10 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -16 -10 B 8 0 2 -2 0 C 8 -2 0 2 -6 D 16 2 -2 0 -6 E 10 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -16 -10 B 8 0 2 -2 0 C 8 -2 0 2 -6 D 16 2 -2 0 -6 E 10 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8046: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) C D E B A (8) E B A C D (7) D C A E B (5) D C A B E (5) E A C D B (4) C D E A B (4) C D B E A (4) A D C E B (4) E A B D C (3) C D B A E (3) B E A C D (3) B A E D C (3) B A D C E (3) E C D A B (2) E B C D A (2) E B A D C (2) E A B C D (2) D C B A E (2) D A C B E (2) C E D A B (2) B E C D A (2) A E B D C (2) A D C B E (2) E C D B A (1) E C B D A (1) E C A D B (1) E B C A D (1) D B C A E (1) C E D B A (1) C D A B E (1) C A E D B (1) B E A D C (1) B D C A E (1) B C D E A (1) B A D E C (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -18 -16 -4 B -4 0 -28 -30 -30 C 18 28 0 24 22 D 16 30 -24 0 16 E 4 30 -22 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -18 -16 -4 B -4 0 -28 -30 -30 C 18 28 0 24 22 D 16 30 -24 0 16 E 4 30 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=26 D=15 B=15 A=10 so A is eliminated. Round 2 votes counts: C=35 E=29 D=21 B=15 so B is eliminated. Round 3 votes counts: E=38 C=36 D=26 so D is eliminated. Round 4 votes counts: C=61 E=39 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:246 D:219 E:198 A:183 B:154 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -18 -16 -4 B -4 0 -28 -30 -30 C 18 28 0 24 22 D 16 30 -24 0 16 E 4 30 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -18 -16 -4 B -4 0 -28 -30 -30 C 18 28 0 24 22 D 16 30 -24 0 16 E 4 30 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -18 -16 -4 B -4 0 -28 -30 -30 C 18 28 0 24 22 D 16 30 -24 0 16 E 4 30 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8047: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) B A C E D (6) A B C D E (6) E C D A B (4) D E C A B (4) D A C E B (4) C A E D B (4) C A D E B (4) B E C A D (4) B A E C D (4) E D C A B (3) D C E A B (3) D C A E B (3) B E D C A (3) B E A C D (3) A D C E B (3) A D C B E (3) A C D E B (3) E D B C A (2) D A C B E (2) C D E A B (2) C D A E B (2) B E D A C (2) B E A D C (2) A D B C E (2) A B D C E (2) E C D B A (1) E C B D A (1) E C B A D (1) D B A E C (1) C E D A B (1) C A E B D (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C D E (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 18 0 10 14 B -18 0 -4 -4 -6 C 0 4 0 2 12 D -10 4 -2 0 -2 E -14 6 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.515366 B: 0.000000 C: 0.484634 D: 0.000000 E: 0.000000 Sum of squares = 0.500472218187 Cumulative probabilities = A: 0.515366 B: 0.515366 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 10 14 B -18 0 -4 -4 -6 C 0 4 0 2 12 D -10 4 -2 0 -2 E -14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=22 E=19 D=17 C=14 so C is eliminated. Round 2 votes counts: A=31 B=28 D=21 E=20 so E is eliminated. Round 3 votes counts: B=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:221 C:209 D:195 E:191 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 10 14 B -18 0 -4 -4 -6 C 0 4 0 2 12 D -10 4 -2 0 -2 E -14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 10 14 B -18 0 -4 -4 -6 C 0 4 0 2 12 D -10 4 -2 0 -2 E -14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 10 14 B -18 0 -4 -4 -6 C 0 4 0 2 12 D -10 4 -2 0 -2 E -14 6 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8048: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (6) E C D A B (5) D E B A C (5) D E A B C (5) E D C A B (4) C E A D B (4) C A B E D (4) E C A D B (3) D A E C B (3) C B A E D (3) B A C D E (3) A C E D B (3) A C D E B (3) E D C B A (2) E D A C B (2) D E B C A (2) D E A C B (2) D B A E C (2) D A E B C (2) C E B A D (2) C A E D B (2) C A E B D (2) B D E C A (2) B D A E C (2) B D A C E (2) B C E D A (2) B A D C E (2) A E C D B (2) A B D C E (2) E C D B A (1) E C B D A (1) D B E A C (1) D A B E C (1) D A B C E (1) C E A B D (1) C B E A D (1) B E C D A (1) B D E A C (1) B D C A E (1) B C E A D (1) B C A D E (1) A E D C B (1) A D E C B (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 12 -2 0 4 B -12 0 -8 -20 -18 C 2 8 0 2 -2 D 0 20 -2 0 -6 E -4 18 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 12 -2 0 4 B -12 0 -8 -20 -18 C 2 8 0 2 -2 D 0 20 -2 0 -6 E -4 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000043 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=24 B=24 C=19 E=18 A=15 so A is eliminated. Round 2 votes counts: D=28 B=26 C=25 E=21 so E is eliminated. Round 3 votes counts: D=37 C=37 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 A:207 D:206 C:205 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 -2 0 4 B -12 0 -8 -20 -18 C 2 8 0 2 -2 D 0 20 -2 0 -6 E -4 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000043 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 0 4 B -12 0 -8 -20 -18 C 2 8 0 2 -2 D 0 20 -2 0 -6 E -4 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000043 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 0 4 B -12 0 -8 -20 -18 C 2 8 0 2 -2 D 0 20 -2 0 -6 E -4 18 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000043 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8049: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (15) B E A C D (12) E A D C B (7) B E A D C (6) B C A E D (5) E B A D C (4) D E A C B (4) C D B A E (4) A E D C B (4) E A D B C (3) E A B D C (3) D C B A E (3) B C D A E (3) B A E C D (3) E A B C D (2) D B C E A (2) D A E C B (2) B E C A D (2) B C E A D (2) A E C D B (2) E B A C D (1) D E C B A (1) D C B E A (1) D B E A C (1) C D A E B (1) C D A B E (1) C B A E D (1) B D E C A (1) B C E D A (1) B C D E A (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 12 18 -8 B 8 0 4 -4 -2 C -12 -4 0 -16 -20 D -18 4 16 0 -18 E 8 2 20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999355 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 12 18 -8 B 8 0 4 -4 -2 C -12 -4 0 -16 -20 D -18 4 16 0 -18 E 8 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=29 E=20 A=8 C=7 so C is eliminated. Round 2 votes counts: B=37 D=35 E=20 A=8 so A is eliminated. Round 3 votes counts: B=37 D=36 E=27 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:224 A:207 B:203 D:192 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 12 18 -8 B 8 0 4 -4 -2 C -12 -4 0 -16 -20 D -18 4 16 0 -18 E 8 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 18 -8 B 8 0 4 -4 -2 C -12 -4 0 -16 -20 D -18 4 16 0 -18 E 8 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 18 -8 B 8 0 4 -4 -2 C -12 -4 0 -16 -20 D -18 4 16 0 -18 E 8 2 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998473 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8050: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (6) E A C D B (5) E D A C B (4) E A D C B (4) B D A C E (4) B C D E A (4) E C D B A (3) D E C B A (3) D A E B C (3) B C E D A (3) B C A D E (3) B A C D E (3) A D E B C (3) E C A D B (2) D E A C B (2) D E A B C (2) D C B E A (2) C E B D A (2) C B E A D (2) C B D E A (2) B D C E A (2) B C A E D (2) A E D C B (2) A E C B D (2) A D B E C (2) A C E B D (2) A C B E D (2) A B D E C (2) A B C D E (2) E D C A B (1) E C D A B (1) E C A B D (1) E A C B D (1) D E B C A (1) D E B A C (1) D B E C A (1) D B E A C (1) D B C E A (1) C E B A D (1) C E A B D (1) C B E D A (1) C B A E D (1) C A E B D (1) B D C A E (1) B A D C E (1) B A C E D (1) A E C D B (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 16 10 -8 B -8 0 2 10 0 C -16 -2 0 12 2 D -10 -10 -12 0 -6 E 8 0 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.052677 B: 0.157597 C: 0.210709 D: 0.000000 E: 0.579016 Sum of squares = 0.407269752173 Cumulative probabilities = A: 0.052677 B: 0.210275 C: 0.420984 D: 0.420984 E: 1.000000 A B C D E A 0 8 16 10 -8 B -8 0 2 10 0 C -16 -2 0 12 2 D -10 -10 -12 0 -6 E 8 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.040650 B: 0.235772 C: 0.162602 D: 0.000000 E: 0.560976 Sum of squares = 0.398373983756 Cumulative probabilities = A: 0.040650 B: 0.276423 C: 0.439024 D: 0.439024 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 B=24 E=22 D=17 C=11 so C is eliminated. Round 2 votes counts: B=30 A=27 E=26 D=17 so D is eliminated. Round 3 votes counts: E=35 B=35 A=30 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:213 E:206 B:202 C:198 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 10 -8 B -8 0 2 10 0 C -16 -2 0 12 2 D -10 -10 -12 0 -6 E 8 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.040650 B: 0.235772 C: 0.162602 D: 0.000000 E: 0.560976 Sum of squares = 0.398373983756 Cumulative probabilities = A: 0.040650 B: 0.276423 C: 0.439024 D: 0.439024 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 10 -8 B -8 0 2 10 0 C -16 -2 0 12 2 D -10 -10 -12 0 -6 E 8 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.040650 B: 0.235772 C: 0.162602 D: 0.000000 E: 0.560976 Sum of squares = 0.398373983756 Cumulative probabilities = A: 0.040650 B: 0.276423 C: 0.439024 D: 0.439024 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 10 -8 B -8 0 2 10 0 C -16 -2 0 12 2 D -10 -10 -12 0 -6 E 8 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.040650 B: 0.235772 C: 0.162602 D: 0.000000 E: 0.560976 Sum of squares = 0.398373983756 Cumulative probabilities = A: 0.040650 B: 0.276423 C: 0.439024 D: 0.439024 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8051: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (13) C E D A B (7) B A D E C (6) E A B C D (5) E A C B D (4) D A E B C (4) C D E A B (4) A B E D C (4) E A B D C (3) D C B A E (3) D A E C B (3) C D B A E (3) C B D E A (3) C B D A E (3) B C D A E (3) E A D B C (2) E A C D B (2) D C E A B (2) C E A D B (2) C E A B D (2) C D B E A (2) E C A B D (1) E B A C D (1) E A D C B (1) D E A C B (1) D C A E B (1) D C A B E (1) D B C A E (1) D A B C E (1) C E B A D (1) C B E D A (1) B E C A D (1) B E A D C (1) B E A C D (1) B D C A E (1) B D A E C (1) B D A C E (1) B C E A D (1) B A E C D (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 2 14 8 4 B -2 0 6 20 6 C -14 -6 0 -2 -12 D -8 -20 2 0 -10 E -4 -6 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999281 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 8 4 B -2 0 6 20 6 C -14 -6 0 -2 -12 D -8 -20 2 0 -10 E -4 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=28 E=19 D=17 A=5 so A is eliminated. Round 2 votes counts: B=35 C=28 E=20 D=17 so D is eliminated. Round 3 votes counts: B=37 C=35 E=28 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:214 E:206 C:183 D:182 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 8 4 B -2 0 6 20 6 C -14 -6 0 -2 -12 D -8 -20 2 0 -10 E -4 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 8 4 B -2 0 6 20 6 C -14 -6 0 -2 -12 D -8 -20 2 0 -10 E -4 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 8 4 B -2 0 6 20 6 C -14 -6 0 -2 -12 D -8 -20 2 0 -10 E -4 -6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8052: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (7) D B C E A (6) A C E B D (6) E A B C D (4) D E B C A (4) D B E C A (4) C D A B E (4) B C D A E (4) A E C B D (4) E D B A C (3) E D A B C (3) D E A C B (3) C B A D E (3) B E D C A (3) E B D A C (2) E B A D C (2) D C B A E (2) D C A E B (2) D C A B E (2) D A E C B (2) D A C E B (2) C A D B E (2) B E A C D (2) B D E C A (2) B C A E D (2) B A C E D (2) A B E C D (2) E B A C D (1) E A D C B (1) E A D B C (1) E A C B D (1) E A B D C (1) D E C A B (1) D E B A C (1) D E A B C (1) C B D A E (1) B E D A C (1) B E C A D (1) B C E A D (1) A E D C B (1) A D C E B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -2 -6 2 B -6 0 6 6 4 C 2 -6 0 -2 -2 D 6 -6 2 0 10 E -2 -4 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333332 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -6 2 B -6 0 6 6 4 C 2 -6 0 -2 -2 D 6 -6 2 0 10 E -2 -4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333231 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=19 B=18 C=17 A=16 so A is eliminated. Round 2 votes counts: D=31 C=25 E=24 B=20 so B is eliminated. Round 3 votes counts: C=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:206 B:205 A:200 C:196 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 -2 -6 2 B -6 0 6 6 4 C 2 -6 0 -2 -2 D 6 -6 2 0 10 E -2 -4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333231 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -6 2 B -6 0 6 6 4 C 2 -6 0 -2 -2 D 6 -6 2 0 10 E -2 -4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333231 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -6 2 B -6 0 6 6 4 C 2 -6 0 -2 -2 D 6 -6 2 0 10 E -2 -4 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333231 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8053: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) B D C A E (7) D B E C A (6) E A C D B (5) A C B E D (5) E C A D B (4) D E B C A (4) B C A D E (4) B A D C E (4) A C E B D (4) D E B A C (3) C E A D B (3) C A E B D (3) C A B E D (3) B D A E C (3) B A C E D (3) A E C B D (3) E D C A B (2) E D A C B (2) E D A B C (2) D C B E A (2) C D B A E (2) C A E D B (2) B D C E A (2) B C D A E (2) B A D E C (2) E D B A C (1) E A D B C (1) E A C B D (1) D E C B A (1) D E A C B (1) D C E B A (1) C D B E A (1) B D A C E (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -14 -2 0 B 16 0 10 -2 14 C 14 -10 0 -6 18 D 2 2 6 0 10 E 0 -14 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -14 -2 0 B 16 0 10 -2 14 C 14 -10 0 -6 18 D 2 2 6 0 10 E 0 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 E=18 C=14 A=13 so A is eliminated. Round 2 votes counts: B=30 D=26 C=23 E=21 so E is eliminated. Round 3 votes counts: C=36 D=34 B=30 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:219 D:210 C:208 A:184 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -14 -2 0 B 16 0 10 -2 14 C 14 -10 0 -6 18 D 2 2 6 0 10 E 0 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -2 0 B 16 0 10 -2 14 C 14 -10 0 -6 18 D 2 2 6 0 10 E 0 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -2 0 B 16 0 10 -2 14 C 14 -10 0 -6 18 D 2 2 6 0 10 E 0 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999584 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8054: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (8) E A D B C (6) D B C E A (5) C B D E A (5) A E C B D (5) A C D B E (5) E B D C A (4) E A C B D (4) D B E C A (4) C B D A E (4) E B C D A (3) C A E B D (3) B D E C A (3) A E D B C (3) A C E D B (3) E C B D A (2) D B C A E (2) C E B D A (2) C E A B D (2) C D B A E (2) C B E D A (2) A D E C B (2) A D C B E (2) A D B E C (2) E D B C A (1) E D A B C (1) E C B A D (1) E C A B D (1) E B D A C (1) D E A B C (1) D C B A E (1) D A B E C (1) C B A E D (1) C A B E D (1) C A B D E (1) B E C D A (1) B D C E A (1) B C E D A (1) A D B C E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -6 6 -2 B -8 0 -18 16 -6 C 6 18 0 18 8 D -6 -16 -18 0 -14 E 2 6 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 6 -2 B -8 0 -18 16 -6 C 6 18 0 18 8 D -6 -16 -18 0 -14 E 2 6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=24 C=23 D=14 B=6 so B is eliminated. Round 2 votes counts: A=33 E=25 C=24 D=18 so D is eliminated. Round 3 votes counts: A=34 E=33 C=33 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:207 A:203 B:192 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 6 -2 B -8 0 -18 16 -6 C 6 18 0 18 8 D -6 -16 -18 0 -14 E 2 6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 6 -2 B -8 0 -18 16 -6 C 6 18 0 18 8 D -6 -16 -18 0 -14 E 2 6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 6 -2 B -8 0 -18 16 -6 C 6 18 0 18 8 D -6 -16 -18 0 -14 E 2 6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998341 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8055: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) A E B D C (7) C D B A E (6) B C D A E (6) E D C A B (5) E A D C B (4) E A B D C (4) C D B E A (4) C D A B E (4) E D C B A (3) E B A D C (3) D E C A B (3) D C E A B (3) C B D A E (3) B A E C D (3) B A C D E (3) C D E A B (2) C A D B E (2) B E C D A (2) B E A C D (2) B C D E A (2) B A E D C (2) B A C E D (2) E B D A C (1) D A C E B (1) C D E B A (1) C D A E B (1) C B D E A (1) C B A D E (1) B E A D C (1) B C E D A (1) B C A E D (1) B C A D E (1) A E D C B (1) A D E C B (1) A D C E B (1) A D C B E (1) A C D B E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -4 -14 0 B -2 0 -14 -4 2 C 4 14 0 0 -2 D 14 4 0 0 -2 E 0 -2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629451 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 A B C D E A 0 2 -4 -14 0 B -2 0 -14 -4 2 C 4 14 0 0 -2 D 14 4 0 0 -2 E 0 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629463 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=26 C=25 A=14 D=7 so D is eliminated. Round 2 votes counts: E=31 C=28 B=26 A=15 so A is eliminated. Round 3 votes counts: E=40 C=32 B=28 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:208 D:208 E:201 A:192 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -14 0 B -2 0 -14 -4 2 C 4 14 0 0 -2 D 14 4 0 0 -2 E 0 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629463 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -14 0 B -2 0 -14 -4 2 C 4 14 0 0 -2 D 14 4 0 0 -2 E 0 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629463 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -14 0 B -2 0 -14 -4 2 C 4 14 0 0 -2 D 14 4 0 0 -2 E 0 -2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.000000 E: 0.777778 Sum of squares = 0.629629629463 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.222222 D: 0.222222 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8056: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (18) B D C E A (17) E C A D B (10) E C D A B (8) D C E B A (6) B A D C E (5) D B C E A (4) B D A C E (4) B C E D A (4) B A E C D (3) A E C B D (3) A B E C D (3) A B D C E (3) E C B D A (2) D C E A B (2) B E C D A (2) C E D B A (1) C D E B A (1) C D E A B (1) A D C E B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -16 -4 -16 B -2 0 -6 -6 -6 C 16 6 0 12 0 D 4 6 -12 0 -8 E 16 6 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.577476 D: 0.000000 E: 0.422524 Sum of squares = 0.512005022954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.577476 D: 0.577476 E: 1.000000 A B C D E A 0 2 -16 -4 -16 B -2 0 -6 -6 -6 C 16 6 0 12 0 D 4 6 -12 0 -8 E 16 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=30 E=20 D=12 C=3 so C is eliminated. Round 2 votes counts: B=35 A=30 E=21 D=14 so D is eliminated. Round 3 votes counts: B=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:217 E:215 D:195 B:190 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 -4 -16 B -2 0 -6 -6 -6 C 16 6 0 12 0 D 4 6 -12 0 -8 E 16 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -4 -16 B -2 0 -6 -6 -6 C 16 6 0 12 0 D 4 6 -12 0 -8 E 16 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -4 -16 B -2 0 -6 -6 -6 C 16 6 0 12 0 D 4 6 -12 0 -8 E 16 6 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8057: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (17) D A E B C (13) C B E A D (12) D A C E B (10) A D E B C (10) B E C A D (9) D A C B E (5) E B C A D (4) C D B E A (3) A E B D C (3) E B C D A (2) E B A C D (2) D C B E A (2) D C A B E (2) A E B C D (2) E B D C A (1) D A E C B (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -4 -12 -4 B 4 0 -6 6 4 C 4 6 0 4 4 D 12 -6 -4 0 -6 E 4 -4 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -12 -4 B 4 0 -6 6 4 C 4 6 0 4 4 D 12 -6 -4 0 -6 E 4 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=32 A=17 E=9 B=9 so E is eliminated. Round 2 votes counts: D=33 C=32 B=18 A=17 so A is eliminated. Round 3 votes counts: D=44 C=32 B=24 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:204 E:201 D:198 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -12 -4 B 4 0 -6 6 4 C 4 6 0 4 4 D 12 -6 -4 0 -6 E 4 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -12 -4 B 4 0 -6 6 4 C 4 6 0 4 4 D 12 -6 -4 0 -6 E 4 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -12 -4 B 4 0 -6 6 4 C 4 6 0 4 4 D 12 -6 -4 0 -6 E 4 -4 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8058: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (10) D E C A B (8) B A C E D (6) C B A E D (5) D C E B A (4) C B D A E (4) C B A D E (4) A B E C D (4) E D C B A (3) E D C A B (3) E A B D C (3) C E D B A (3) B A C D E (3) A D B E C (3) E D A B C (2) E C D B A (2) E C B D A (2) D C A B E (2) C E B D A (2) C D E B A (2) C B E A D (2) A E D B C (2) A E B D C (2) A B D E C (2) E D A C B (1) E C B A D (1) E B C A D (1) E A D B C (1) E A B C D (1) D E A C B (1) D E A B C (1) D C E A B (1) D C B A E (1) D A E B C (1) D A C B E (1) D A B C E (1) C E B A D (1) B C A E D (1) B C A D E (1) B A E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -6 10 10 B -2 0 -6 14 4 C 6 6 0 -6 -10 D -10 -14 6 0 -18 E -10 -4 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.230769 Sum of squares = 0.349112426033 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.769231 D: 0.769231 E: 1.000000 A B C D E A 0 2 -6 10 10 B -2 0 -6 14 4 C 6 6 0 -6 -10 D -10 -14 6 0 -18 E -10 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.230769 Sum of squares = 0.349112426016 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.769231 D: 0.769231 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 D=21 E=20 B=12 so B is eliminated. Round 2 votes counts: A=34 C=25 D=21 E=20 so E is eliminated. Round 3 votes counts: A=39 C=31 D=30 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:208 E:207 B:205 C:198 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -6 10 10 B -2 0 -6 14 4 C 6 6 0 -6 -10 D -10 -14 6 0 -18 E -10 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.230769 Sum of squares = 0.349112426016 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.769231 D: 0.769231 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 10 10 B -2 0 -6 14 4 C 6 6 0 -6 -10 D -10 -14 6 0 -18 E -10 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.230769 Sum of squares = 0.349112426016 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.769231 D: 0.769231 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 10 10 B -2 0 -6 14 4 C 6 6 0 -6 -10 D -10 -14 6 0 -18 E -10 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.230769 Sum of squares = 0.349112426016 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.769231 D: 0.769231 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8059: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (13) E C A D B (9) B D A C E (8) A C D E B (8) B D E C A (5) A C D B E (5) D B A C E (4) B E D C A (4) B D E A C (4) B D A E C (4) E C A B D (3) C A E D B (3) B D C E A (3) E C B D A (2) E B D C A (2) E A C B D (2) D B C A E (2) B D C A E (2) A C E B D (2) A B D C E (2) E D C B A (1) E A B C D (1) D C B E A (1) D B C E A (1) C E D A B (1) C E A D B (1) B E A D C (1) B A E D C (1) B A E C D (1) A D C B E (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 20 12 18 B -8 0 -6 0 4 C -20 6 0 8 18 D -12 0 -8 0 2 E -18 -4 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 20 12 18 B -8 0 -6 0 4 C -20 6 0 8 18 D -12 0 -8 0 2 E -18 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=33 E=20 D=8 C=5 so C is eliminated. Round 2 votes counts: A=37 B=33 E=22 D=8 so D is eliminated. Round 3 votes counts: B=41 A=37 E=22 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:206 B:195 D:191 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 20 12 18 B -8 0 -6 0 4 C -20 6 0 8 18 D -12 0 -8 0 2 E -18 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 20 12 18 B -8 0 -6 0 4 C -20 6 0 8 18 D -12 0 -8 0 2 E -18 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 20 12 18 B -8 0 -6 0 4 C -20 6 0 8 18 D -12 0 -8 0 2 E -18 -4 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8060: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) C D A B E (7) D C B A E (6) C A B D E (6) D C A B E (5) C A D B E (5) E B A C D (4) E A C B D (4) A C B E D (4) E D C A B (3) D E B A C (3) D C E A B (3) D B C A E (3) B A E C D (3) E D B A C (2) D E C A B (2) D C E B A (2) D B C E A (2) C A E B D (2) B E D A C (2) B D C A E (2) E D A C B (1) E D A B C (1) E C A D B (1) E B D A C (1) E B A D C (1) E A D B C (1) E A C D B (1) D C B E A (1) D C A E B (1) D B E A C (1) C E A D B (1) C D B A E (1) C D A E B (1) C A D E B (1) B E A D C (1) B E A C D (1) B D A C E (1) B A E D C (1) B A C D E (1) A E B C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 22 -10 -2 6 B -22 0 -18 -10 10 C 10 18 0 8 14 D 2 10 -8 0 10 E -6 -10 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -10 -2 6 B -22 0 -18 -10 10 C 10 18 0 8 14 D 2 10 -8 0 10 E -6 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 C=24 B=12 A=7 so A is eliminated. Round 2 votes counts: E=29 D=29 C=29 B=13 so B is eliminated. Round 3 votes counts: E=37 D=32 C=31 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:225 A:208 D:207 B:180 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -10 -2 6 B -22 0 -18 -10 10 C 10 18 0 8 14 D 2 10 -8 0 10 E -6 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -10 -2 6 B -22 0 -18 -10 10 C 10 18 0 8 14 D 2 10 -8 0 10 E -6 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -10 -2 6 B -22 0 -18 -10 10 C 10 18 0 8 14 D 2 10 -8 0 10 E -6 -10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8061: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) E C B A D (10) E B D C A (10) C A E D B (8) E B C D A (7) B D E A C (7) C E A B D (6) D B A E C (5) B E D C A (5) E B C A D (4) C A D E B (4) B E D A C (4) A C D B E (4) D A C B E (3) D A B C E (3) A C D E B (3) E C A B D (2) D B E A C (1) D B A C E (1) C A E B D (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -14 8 -12 B 8 0 -4 12 -10 C 14 4 0 -2 -10 D -8 -12 2 0 -14 E 12 10 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -14 8 -12 B 8 0 -4 12 -10 C 14 4 0 -2 -10 D -8 -12 2 0 -14 E 12 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=19 A=19 B=16 D=13 so D is eliminated. Round 2 votes counts: E=33 A=25 B=23 C=19 so C is eliminated. Round 3 votes counts: E=39 A=38 B=23 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 B:203 C:203 A:187 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -14 8 -12 B 8 0 -4 12 -10 C 14 4 0 -2 -10 D -8 -12 2 0 -14 E 12 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -14 8 -12 B 8 0 -4 12 -10 C 14 4 0 -2 -10 D -8 -12 2 0 -14 E 12 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -14 8 -12 B 8 0 -4 12 -10 C 14 4 0 -2 -10 D -8 -12 2 0 -14 E 12 10 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8062: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) E D C A B (9) E D C B A (7) E D B A C (6) C A B D E (5) B A E D C (5) A C B D E (4) A B C E D (4) A B C D E (4) C B D A E (3) E D B C A (2) E C D A B (2) E A D B C (2) E A B D C (2) D C E B A (2) C D E B A (2) C D E A B (2) C A E D B (2) B D E A C (2) B D C E A (2) B A D E C (2) B A D C E (2) A E C D B (2) A E B D C (2) A C E B D (2) A B E C D (2) E D A B C (1) E B D A C (1) D E B C A (1) D B C E A (1) C E D A B (1) C D A E B (1) C D A B E (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 14 8 10 B 0 0 8 12 0 C -14 -8 0 2 2 D -8 -12 -2 0 -8 E -10 0 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.608318 B: 0.391682 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.523465679454 Cumulative probabilities = A: 0.608318 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 8 10 B 0 0 8 12 0 C -14 -8 0 2 2 D -8 -12 -2 0 -8 E -10 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=25 A=22 C=17 D=4 so D is eliminated. Round 2 votes counts: E=33 B=26 A=22 C=19 so C is eliminated. Round 3 votes counts: E=40 A=31 B=29 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:210 E:198 C:191 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 8 10 B 0 0 8 12 0 C -14 -8 0 2 2 D -8 -12 -2 0 -8 E -10 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 8 10 B 0 0 8 12 0 C -14 -8 0 2 2 D -8 -12 -2 0 -8 E -10 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 8 10 B 0 0 8 12 0 C -14 -8 0 2 2 D -8 -12 -2 0 -8 E -10 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8063: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (13) D E A C B (10) D E B A C (6) B C A E D (6) C A B D E (5) E D B A C (4) D B E C A (4) A C E B D (4) E D A C B (3) D E B C A (3) B E D A C (3) B E C A D (3) B E A C D (3) A C B E D (3) E A C D B (2) D C A E B (2) D B C A E (2) C B A D E (2) B E D C A (2) A C E D B (2) A C D E B (2) E B D A C (1) E B A C D (1) D E C A B (1) D E A B C (1) D C E A B (1) D C A B E (1) D B C E A (1) D A E C B (1) C D A B E (1) C A D E B (1) C A D B E (1) B D E C A (1) B D C A E (1) B C E A D (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 8 -8 0 -2 B -8 0 -10 2 10 C 8 10 0 4 2 D 0 -2 -4 0 -2 E 2 -10 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 0 -2 B -8 0 -10 2 10 C 8 10 0 4 2 D 0 -2 -4 0 -2 E 2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 B=22 E=11 A=11 so E is eliminated. Round 2 votes counts: D=40 B=24 C=23 A=13 so A is eliminated. Round 3 votes counts: D=40 C=36 B=24 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:199 B:197 D:196 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 0 -2 B -8 0 -10 2 10 C 8 10 0 4 2 D 0 -2 -4 0 -2 E 2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 0 -2 B -8 0 -10 2 10 C 8 10 0 4 2 D 0 -2 -4 0 -2 E 2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 0 -2 B -8 0 -10 2 10 C 8 10 0 4 2 D 0 -2 -4 0 -2 E 2 -10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8064: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) E B A D C (7) C E D B A (7) D C E A B (6) D A B E C (5) C D A B E (5) D A B C E (4) C D E A B (4) A B D E C (4) E C D B A (3) D B A E C (3) D A C B E (3) E D C B A (2) E D B C A (2) E D B A C (2) E B C A D (2) E B A C D (2) D C A B E (2) C D A E B (2) C A D B E (2) C A B E D (2) B A E C D (2) A B D C E (2) E C B A D (1) E B D C A (1) E B D A C (1) D E C B A (1) D E B A C (1) D E A B C (1) D C A E B (1) C E B A D (1) C E A D B (1) C E A B D (1) C D E B A (1) C B A E D (1) C A B D E (1) B E A C D (1) B A D E C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 2 -14 4 B 4 0 6 -20 0 C -2 -6 0 -22 -6 D 14 20 22 0 2 E -4 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -14 4 B 4 0 6 -20 0 C -2 -6 0 -22 -6 D 14 20 22 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=27 E=23 B=14 A=8 so A is eliminated. Round 2 votes counts: D=29 C=28 E=23 B=20 so B is eliminated. Round 3 votes counts: E=36 D=36 C=28 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:200 B:195 A:194 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -14 4 B 4 0 6 -20 0 C -2 -6 0 -22 -6 D 14 20 22 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -14 4 B 4 0 6 -20 0 C -2 -6 0 -22 -6 D 14 20 22 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -14 4 B 4 0 6 -20 0 C -2 -6 0 -22 -6 D 14 20 22 0 2 E -4 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999953669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8065: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) C B E D A (7) D C A E B (6) C D B A E (6) E B A C D (4) D A C E B (4) C D E B A (4) E B C D A (3) E A B D C (3) D C E B A (3) D C A B E (3) B E A C D (3) B C A E D (3) B A E C D (3) A E B D C (3) E B C A D (2) E A D B C (2) D E C A B (2) D A E C B (2) D A C B E (2) C D B E A (2) C D A B E (2) C B D E A (2) C B D A E (2) B C E D A (2) B C A D E (2) A D E C B (2) A D E B C (2) A D C B E (2) E D C B A (1) E C D B A (1) E B A D C (1) E A B C D (1) D E A C B (1) B E C A D (1) B A C E D (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -22 -24 -4 B 6 0 -22 -8 -8 C 22 22 0 2 22 D 24 8 -2 0 16 E 4 8 -22 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -22 -24 -4 B 6 0 -22 -8 -8 C 22 22 0 2 22 D 24 8 -2 0 16 E 4 8 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=25 E=18 B=15 A=12 so A is eliminated. Round 2 votes counts: D=36 C=25 E=21 B=18 so B is eliminated. Round 3 votes counts: D=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:234 D:223 E:187 B:184 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -22 -24 -4 B 6 0 -22 -8 -8 C 22 22 0 2 22 D 24 8 -2 0 16 E 4 8 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -22 -24 -4 B 6 0 -22 -8 -8 C 22 22 0 2 22 D 24 8 -2 0 16 E 4 8 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -22 -24 -4 B 6 0 -22 -8 -8 C 22 22 0 2 22 D 24 8 -2 0 16 E 4 8 -22 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999954541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8066: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (6) B D A E C (6) B A D E C (6) C E A B D (5) B D C A E (5) D B A E C (4) C A B E D (4) B D A C E (4) E D A C B (3) E A C D B (3) D B C A E (3) C D B E A (3) C A E B D (3) E C D A B (2) E C A D B (2) E A D B C (2) E A C B D (2) E A B D C (2) D E B A C (2) D E A B C (2) D C E B A (2) D B E A C (2) D B C E A (2) D B A C E (2) C D E B A (2) C D B A E (2) C B A E D (2) B C D A E (2) B A D C E (2) E D C A B (1) E D A B C (1) E C A B D (1) D E C B A (1) D E C A B (1) D C B E A (1) D A B E C (1) C B A D E (1) A C E B D (1) A C B E D (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -2 -20 4 B 8 0 2 0 12 C 2 -2 0 -14 6 D 20 0 14 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.504974 C: 0.000000 D: 0.495026 E: 0.000000 Sum of squares = 0.500049484361 Cumulative probabilities = A: 0.000000 B: 0.504974 C: 0.504974 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -20 4 B 8 0 2 0 12 C 2 -2 0 -14 6 D 20 0 14 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 D=23 E=19 A=5 so A is eliminated. Round 2 votes counts: C=30 B=28 D=23 E=19 so E is eliminated. Round 3 votes counts: C=40 D=30 B=30 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:224 B:211 C:196 A:187 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -20 4 B 8 0 2 0 12 C 2 -2 0 -14 6 D 20 0 14 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -20 4 B 8 0 2 0 12 C 2 -2 0 -14 6 D 20 0 14 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -20 4 B 8 0 2 0 12 C 2 -2 0 -14 6 D 20 0 14 0 14 E -4 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8067: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) C D A B E (9) B E D C A (9) A C D E B (9) A E C D B (6) B D C E A (5) B E A C D (4) B C D E A (4) A C E D B (4) E B A D C (3) E B A C D (3) E A D B C (3) C A D B E (3) E A D C B (2) E A B C D (2) D C B E A (2) D C A B E (2) D B C E A (2) C D B A E (2) B E C D A (2) B C D A E (2) E B D A C (1) E A B D C (1) D C E A B (1) D C A E B (1) C D A E B (1) C B A D E (1) C A D E B (1) B D E C A (1) B C A E D (1) A E D C B (1) A E C B D (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -16 -6 10 B 2 0 -12 -16 18 C 16 12 0 14 20 D 6 16 -14 0 10 E -10 -18 -20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -6 10 B 2 0 -12 -16 18 C 16 12 0 14 20 D 6 16 -14 0 10 E -10 -18 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=23 D=17 C=17 E=15 so E is eliminated. Round 2 votes counts: B=35 A=31 D=17 C=17 so D is eliminated. Round 3 votes counts: B=37 C=32 A=31 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:231 D:209 B:196 A:193 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 -6 10 B 2 0 -12 -16 18 C 16 12 0 14 20 D 6 16 -14 0 10 E -10 -18 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -6 10 B 2 0 -12 -16 18 C 16 12 0 14 20 D 6 16 -14 0 10 E -10 -18 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -6 10 B 2 0 -12 -16 18 C 16 12 0 14 20 D 6 16 -14 0 10 E -10 -18 -20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8068: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (10) E A D C B (8) D B C E A (8) E A D B C (6) A E C B D (6) E D A B C (5) C B D E A (5) C B D A E (4) E A C D B (3) B C D E A (3) A E C D B (3) A C B E D (3) A B C E D (3) A B C D E (3) D E C B A (2) D E B C A (2) D B E C A (2) C B A D E (2) C A B D E (2) B D C E A (2) B C D A E (2) E D C B A (1) E D B A C (1) E D A C B (1) E C D A B (1) D C E B A (1) D C B E A (1) D B E A C (1) C E D B A (1) C D E B A (1) C A B E D (1) B D C A E (1) B D A C E (1) A E D B C (1) A C E B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -4 -12 -30 B 2 0 4 -18 -8 C 4 -4 0 -10 -6 D 12 18 10 0 -10 E 30 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -12 -30 B 2 0 4 -18 -8 C 4 -4 0 -10 -6 D 12 18 10 0 -10 E 30 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=22 D=17 C=16 B=9 so B is eliminated. Round 2 votes counts: E=36 A=22 D=21 C=21 so D is eliminated. Round 3 votes counts: E=43 C=34 A=23 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:215 C:192 B:190 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -12 -30 B 2 0 4 -18 -8 C 4 -4 0 -10 -6 D 12 18 10 0 -10 E 30 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -12 -30 B 2 0 4 -18 -8 C 4 -4 0 -10 -6 D 12 18 10 0 -10 E 30 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -12 -30 B 2 0 4 -18 -8 C 4 -4 0 -10 -6 D 12 18 10 0 -10 E 30 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8069: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) E B A D C (6) E B A C D (5) C D E B A (5) C D A E B (5) B E A D C (5) A B E D C (5) E A B C D (4) C A D E B (4) A E B C D (4) C D A B E (3) B D E C A (3) A D B C E (3) A B D E C (3) D C B E A (2) D C B A E (2) D B A E C (2) C E D B A (2) C D E A B (2) C D B E A (2) C A E D B (2) B E D C A (2) B D E A C (2) B A E D C (2) A E B D C (2) A C E B D (2) A C D B E (2) E C B D A (1) E B D C A (1) E B C A D (1) D B A C E (1) D A C B E (1) C E B D A (1) C E B A D (1) C E A B D (1) B D A E C (1) A D C B E (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 6 10 8 B -6 0 6 4 0 C -6 -6 0 -4 0 D -10 -4 4 0 6 E -8 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 10 8 B -6 0 6 4 0 C -6 -6 0 -4 0 D -10 -4 4 0 6 E -8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=24 E=18 D=15 B=15 so D is eliminated. Round 2 votes counts: C=39 A=25 E=18 B=18 so E is eliminated. Round 3 votes counts: C=40 B=31 A=29 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:202 D:198 E:193 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 10 8 B -6 0 6 4 0 C -6 -6 0 -4 0 D -10 -4 4 0 6 E -8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 10 8 B -6 0 6 4 0 C -6 -6 0 -4 0 D -10 -4 4 0 6 E -8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 10 8 B -6 0 6 4 0 C -6 -6 0 -4 0 D -10 -4 4 0 6 E -8 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8070: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) A E C B D (8) D B C A E (6) E D A B C (5) D E A C B (5) D B E C A (4) C B A E D (4) B C A E D (4) B C A D E (4) A C E B D (4) E A D B C (3) E A B C D (3) D E B C A (3) D E A B C (3) D C A E B (3) D B C E A (3) C D B A E (3) B D C A E (3) E D A C B (2) E A C D B (2) E A C B D (2) D C B A E (2) C B D A E (2) C A B E D (2) B C D A E (2) E A D C B (1) E A B D C (1) D E C A B (1) D C E A B (1) C A E D B (1) C A E B D (1) B D E A C (1) A E C D B (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 2 2 -14 4 B -2 0 10 -14 -18 C -2 -10 0 -10 -10 D 14 14 10 0 8 E -4 18 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -14 4 B -2 0 10 -14 -18 C -2 -10 0 -10 -10 D 14 14 10 0 8 E -4 18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=19 A=15 B=14 C=13 so C is eliminated. Round 2 votes counts: D=42 B=20 E=19 A=19 so E is eliminated. Round 3 votes counts: D=49 A=31 B=20 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:208 A:197 B:188 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 2 -14 4 B -2 0 10 -14 -18 C -2 -10 0 -10 -10 D 14 14 10 0 8 E -4 18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -14 4 B -2 0 10 -14 -18 C -2 -10 0 -10 -10 D 14 14 10 0 8 E -4 18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -14 4 B -2 0 10 -14 -18 C -2 -10 0 -10 -10 D 14 14 10 0 8 E -4 18 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8071: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (15) D E B A C (12) E D B A C (10) C A B E D (10) D C B A E (5) D C A B E (5) E B A C D (4) C D A B E (4) D E C A B (3) B A C D E (3) A B C E D (3) E B D A C (2) E B A D C (2) D E B C A (2) D C E B A (2) D C E A B (2) D B A C E (2) A B E C D (2) E D C A B (1) D E C B A (1) D C A E B (1) D B E A C (1) D B A E C (1) C E A B D (1) C A E D B (1) C A D B E (1) B D E A C (1) B A E C D (1) B A C E D (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -8 -10 12 B 0 0 -6 -8 12 C 8 6 0 -6 14 D 10 8 6 0 22 E -12 -12 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999817 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -10 12 B 0 0 -6 -8 12 C 8 6 0 -6 14 D 10 8 6 0 22 E -12 -12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=32 E=19 B=6 A=6 so B is eliminated. Round 2 votes counts: D=38 C=32 E=19 A=11 so A is eliminated. Round 3 votes counts: C=40 D=38 E=22 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:211 B:199 A:197 E:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -10 12 B 0 0 -6 -8 12 C 8 6 0 -6 14 D 10 8 6 0 22 E -12 -12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -10 12 B 0 0 -6 -8 12 C 8 6 0 -6 14 D 10 8 6 0 22 E -12 -12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -10 12 B 0 0 -6 -8 12 C 8 6 0 -6 14 D 10 8 6 0 22 E -12 -12 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8072: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) E A B D C (8) C B E A D (8) D A E C B (7) C D E A B (6) D A E B C (5) C B D A E (5) D C A E B (4) C E A D B (4) E A D C B (3) D E A C B (3) D C E A B (3) C B E D A (3) B C A E D (3) A E D B C (3) C D B A E (2) B E A C D (2) B C E A D (2) B C D A E (2) B A E C D (2) B A D E C (2) E D C A B (1) E C A D B (1) E A D B C (1) D B C A E (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B A D (1) C D B E A (1) C D A E B (1) C B D E A (1) B E C A D (1) B D A E C (1) Total count = 100 A B C D E A 0 4 -2 2 0 B -4 0 -10 2 -4 C 2 10 0 -8 -4 D -2 -2 8 0 -8 E 0 4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.458628 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.541372 Sum of squares = 0.503423342658 Cumulative probabilities = A: 0.458628 B: 0.458628 C: 0.458628 D: 0.458628 E: 1.000000 A B C D E A 0 4 -2 2 0 B -4 0 -10 2 -4 C 2 10 0 -8 -4 D -2 -2 8 0 -8 E 0 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=25 B=25 E=14 A=3 so A is eliminated. Round 2 votes counts: C=33 D=25 B=25 E=17 so E is eliminated. Round 3 votes counts: C=34 D=33 B=33 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:208 A:202 C:200 D:198 B:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 2 0 B -4 0 -10 2 -4 C 2 10 0 -8 -4 D -2 -2 8 0 -8 E 0 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 2 0 B -4 0 -10 2 -4 C 2 10 0 -8 -4 D -2 -2 8 0 -8 E 0 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 2 0 B -4 0 -10 2 -4 C 2 10 0 -8 -4 D -2 -2 8 0 -8 E 0 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8073: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (15) B C E D A (14) B C E A D (12) C E B D A (11) A D B E C (11) B A D C E (7) D A E C B (6) E C D A B (4) A D E B C (4) A B D C E (4) E C D B A (2) B A C E D (2) E D C A B (1) E C B D A (1) E C A D B (1) D E C A B (1) C E D B A (1) C B E D A (1) B C A E D (1) B A C D E (1) Total count = 100 A B C D E A 0 -6 0 16 2 B 6 0 12 8 6 C 0 -12 0 2 8 D -16 -8 -2 0 -2 E -2 -6 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 16 2 B 6 0 12 8 6 C 0 -12 0 2 8 D -16 -8 -2 0 -2 E -2 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=34 C=13 E=9 D=7 so D is eliminated. Round 2 votes counts: A=40 B=37 C=13 E=10 so E is eliminated. Round 3 votes counts: A=40 B=37 C=23 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:206 C:199 E:193 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 16 2 B 6 0 12 8 6 C 0 -12 0 2 8 D -16 -8 -2 0 -2 E -2 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 16 2 B 6 0 12 8 6 C 0 -12 0 2 8 D -16 -8 -2 0 -2 E -2 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 16 2 B 6 0 12 8 6 C 0 -12 0 2 8 D -16 -8 -2 0 -2 E -2 -6 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8074: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) A C E D B (8) D B C E A (6) C D B A E (6) B D E C A (6) E A B D C (5) E A B C D (5) B D C E A (5) A E C D B (5) C D A B E (4) A E D B C (4) E B A D C (3) E A C B D (3) C A D E B (3) B E D C A (3) D C B A E (2) D B C A E (2) C B D A E (2) B E D A C (2) A E D C B (2) A C D E B (2) E D A B C (1) E B D A C (1) E B C A D (1) E B A C D (1) E A D B C (1) C B E A D (1) C B D E A (1) B E C D A (1) B D E A C (1) B C E A D (1) B C D E A (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 8 10 12 2 B -8 0 2 6 -12 C -10 -2 0 10 -12 D -12 -6 -10 0 -16 E -2 12 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 12 2 B -8 0 2 6 -12 C -10 -2 0 10 -12 D -12 -6 -10 0 -16 E -2 12 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=21 B=20 C=17 D=10 so D is eliminated. Round 2 votes counts: A=32 B=28 E=21 C=19 so C is eliminated. Round 3 votes counts: B=40 A=39 E=21 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 A:216 B:194 C:193 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 12 2 B -8 0 2 6 -12 C -10 -2 0 10 -12 D -12 -6 -10 0 -16 E -2 12 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 12 2 B -8 0 2 6 -12 C -10 -2 0 10 -12 D -12 -6 -10 0 -16 E -2 12 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 12 2 B -8 0 2 6 -12 C -10 -2 0 10 -12 D -12 -6 -10 0 -16 E -2 12 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8075: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A D C B E (11) D A B E C (9) A C D E B (8) D B E C A (7) C E B A D (7) D B E A C (6) A C E B D (6) A D B E C (5) E B C A D (4) C E B D A (4) C A E B D (4) B E D C A (4) E B D C A (2) D C A B E (2) D C B E A (1) D A B C E (1) C D B E A (1) C D A B E (1) C A D E B (1) B E D A C (1) B E A D C (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 2 2 0 2 B -2 0 4 -8 0 C -2 -4 0 0 -4 D 0 8 0 0 6 E -2 0 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.655409 B: 0.000000 C: 0.000000 D: 0.344591 E: 0.000000 Sum of squares = 0.548303991813 Cumulative probabilities = A: 0.655409 B: 0.655409 C: 0.655409 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 2 B -2 0 4 -8 0 C -2 -4 0 0 -4 D 0 8 0 0 6 E -2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=26 C=18 E=17 B=6 so B is eliminated. Round 2 votes counts: A=33 D=26 E=23 C=18 so C is eliminated. Round 3 votes counts: A=38 E=34 D=28 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:207 A:203 E:198 B:197 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 2 B -2 0 4 -8 0 C -2 -4 0 0 -4 D 0 8 0 0 6 E -2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 2 B -2 0 4 -8 0 C -2 -4 0 0 -4 D 0 8 0 0 6 E -2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 2 B -2 0 4 -8 0 C -2 -4 0 0 -4 D 0 8 0 0 6 E -2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8076: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (13) E A C B D (9) E D B A C (7) A C E B D (6) E B D C A (4) D B E C A (4) C A D B E (4) B D E C A (4) E D A B C (3) E A D C B (3) C A B D E (3) B D C A E (3) A E C B D (3) A C D B E (3) E B D A C (2) E B A C D (2) E A B C D (2) D C A B E (2) D A C B E (2) B E D C A (2) B D C E A (2) B C D A E (2) A E C D B (2) A C E D B (2) E D A C B (1) E C A B D (1) E A C D B (1) D E B C A (1) D E A C B (1) D B E A C (1) D B C E A (1) D B A C E (1) C B A D E (1) C A B E D (1) A C D E B (1) Total count = 100 A B C D E A 0 0 4 -12 -2 B 0 0 8 -6 -2 C -4 -8 0 -14 -6 D 12 6 14 0 -2 E 2 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 4 -12 -2 B 0 0 8 -6 -2 C -4 -8 0 -14 -6 D 12 6 14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=26 A=17 B=13 C=9 so C is eliminated. Round 2 votes counts: E=35 D=26 A=25 B=14 so B is eliminated. Round 3 votes counts: E=37 D=37 A=26 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:206 B:200 A:195 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 -12 -2 B 0 0 8 -6 -2 C -4 -8 0 -14 -6 D 12 6 14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 -12 -2 B 0 0 8 -6 -2 C -4 -8 0 -14 -6 D 12 6 14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 -12 -2 B 0 0 8 -6 -2 C -4 -8 0 -14 -6 D 12 6 14 0 -2 E 2 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8077: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (5) D E B C A (5) B D E C A (5) A C B D E (5) E D C B A (4) D E B A C (4) A C B E D (4) E C D A B (3) D B E C A (3) C E D B A (3) C B D E A (3) B A D C E (3) A E C D B (3) A B D E C (3) A B C D E (3) E C D B A (2) D B E A C (2) C E D A B (2) C E A D B (2) C A E D B (2) C A E B D (2) C A B E D (2) C A B D E (2) B D C E A (2) B C D E A (2) B C D A E (2) B A D E C (2) B A C D E (2) A C E D B (2) A C E B D (2) A B D C E (2) E D C A B (1) E D B A C (1) E D A B C (1) E C A D B (1) C B E A D (1) C B D A E (1) C B A E D (1) B D C A E (1) B D A E C (1) A E D C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -14 -6 -4 B 10 0 2 2 6 C 14 -2 0 4 2 D 6 -2 -4 0 10 E 4 -6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -6 -4 B 10 0 2 2 6 C 14 -2 0 4 2 D 6 -2 -4 0 10 E 4 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=21 B=20 E=18 D=14 so D is eliminated. Round 2 votes counts: E=27 A=27 B=25 C=21 so C is eliminated. Round 3 votes counts: A=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:210 C:209 D:205 E:193 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -14 -6 -4 B 10 0 2 2 6 C 14 -2 0 4 2 D 6 -2 -4 0 10 E 4 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -6 -4 B 10 0 2 2 6 C 14 -2 0 4 2 D 6 -2 -4 0 10 E 4 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -6 -4 B 10 0 2 2 6 C 14 -2 0 4 2 D 6 -2 -4 0 10 E 4 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8078: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) E B C D A (7) E B A C D (7) B A E D C (7) A E B C D (7) D C A B E (6) A D C B E (6) C D E A B (5) A B E D C (5) D C B E A (4) C D A E B (4) B E D C A (4) C E D B A (3) A D B C E (3) E C A D B (2) C E D A B (2) B E A D C (2) B E A C D (2) B D A C E (2) A D C E B (2) A B D C E (2) E C B D A (1) E C B A D (1) E A C B D (1) D C B A E (1) D C A E B (1) D A C B E (1) C D B E A (1) B D E C A (1) B A D C E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 -2 -2 B 4 0 0 0 -4 C 2 0 0 4 6 D 2 0 -4 0 -2 E 2 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.394918 C: 0.605082 D: 0.000000 E: 0.000000 Sum of squares = 0.522084530561 Cumulative probabilities = A: 0.000000 B: 0.394918 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -2 -2 B 4 0 0 0 -4 C 2 0 0 4 6 D 2 0 -4 0 -2 E 2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=23 E=19 B=19 D=13 so D is eliminated. Round 2 votes counts: C=35 A=27 E=19 B=19 so E is eliminated. Round 3 votes counts: C=39 B=33 A=28 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:206 E:201 B:200 D:198 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 -2 B 4 0 0 0 -4 C 2 0 0 4 6 D 2 0 -4 0 -2 E 2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 -2 B 4 0 0 0 -4 C 2 0 0 4 6 D 2 0 -4 0 -2 E 2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 -2 B 4 0 0 0 -4 C 2 0 0 4 6 D 2 0 -4 0 -2 E 2 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8079: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E A D B C (6) B C D A E (5) B C A E D (5) B A E D C (5) A E D C B (5) E D A B C (4) A E C D B (4) A C E B D (4) E A D C B (3) D E A B C (3) D B C E A (3) C B A E D (3) C A E B D (3) B D C E A (3) B A C E D (3) A E B C D (3) E D A C B (2) D E B A C (2) D C E B A (2) D B E C A (2) C A E D B (2) B D E C A (2) A E D B C (2) E A B D C (1) C D E A B (1) C D B E A (1) C B D E A (1) C B A D E (1) C A D E B (1) C A B E D (1) B E A D C (1) B D E A C (1) B D C A E (1) B C D E A (1) B C A D E (1) B A E C D (1) A E C B D (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 6 16 22 B 4 0 14 14 0 C -6 -14 0 2 2 D -16 -14 -2 0 -22 E -22 0 -2 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.913694 C: 0.000000 D: 0.000000 E: 0.086306 Sum of squares = 0.842285984628 Cumulative probabilities = A: 0.000000 B: 0.913694 C: 0.913694 D: 0.913694 E: 1.000000 A B C D E A 0 -4 6 16 22 B 4 0 14 14 0 C -6 -14 0 2 2 D -16 -14 -2 0 -22 E -22 0 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.846154 C: 0.000000 D: 0.000000 E: 0.153846 Sum of squares = 0.739645001541 Cumulative probabilities = A: 0.000000 B: 0.846154 C: 0.846154 D: 0.846154 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=22 A=21 E=16 D=12 so D is eliminated. Round 2 votes counts: B=34 C=24 E=21 A=21 so E is eliminated. Round 3 votes counts: A=40 B=36 C=24 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:220 B:216 E:199 C:192 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 16 22 B 4 0 14 14 0 C -6 -14 0 2 2 D -16 -14 -2 0 -22 E -22 0 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.846154 C: 0.000000 D: 0.000000 E: 0.153846 Sum of squares = 0.739645001541 Cumulative probabilities = A: 0.000000 B: 0.846154 C: 0.846154 D: 0.846154 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 16 22 B 4 0 14 14 0 C -6 -14 0 2 2 D -16 -14 -2 0 -22 E -22 0 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.846154 C: 0.000000 D: 0.000000 E: 0.153846 Sum of squares = 0.739645001541 Cumulative probabilities = A: 0.000000 B: 0.846154 C: 0.846154 D: 0.846154 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 16 22 B 4 0 14 14 0 C -6 -14 0 2 2 D -16 -14 -2 0 -22 E -22 0 -2 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.846154 C: 0.000000 D: 0.000000 E: 0.153846 Sum of squares = 0.739645001541 Cumulative probabilities = A: 0.000000 B: 0.846154 C: 0.846154 D: 0.846154 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8080: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) E D C B A (5) C D E A B (5) A B D E C (5) E D A B C (4) D A E B C (4) C E D B A (4) C B A E D (4) B A C E D (4) A D B E C (4) D E C A B (3) D E A B C (3) C E B A D (3) C B A D E (3) E D C A B (2) E C B D A (2) E B D A C (2) E B A C D (2) D A E C B (2) C E B D A (2) C A B D E (2) B C A E D (2) A D C B E (2) A D B C E (2) A C B D E (2) A B D C E (2) E D B C A (1) E D B A C (1) E C D B A (1) E B C A D (1) E B A D C (1) D A B E C (1) D A B C E (1) C D E B A (1) C D B A E (1) B E C A D (1) B E A D C (1) B E A C D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -8 10 -14 B 10 0 -10 6 2 C 8 10 0 8 6 D -10 -6 -8 0 -10 E 14 -2 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 10 -14 B 10 0 -10 6 2 C 8 10 0 8 6 D -10 -6 -8 0 -10 E 14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=22 A=18 D=14 B=10 so B is eliminated. Round 2 votes counts: C=38 E=25 A=23 D=14 so D is eliminated. Round 3 votes counts: C=38 E=31 A=31 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:208 B:204 A:189 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 10 -14 B 10 0 -10 6 2 C 8 10 0 8 6 D -10 -6 -8 0 -10 E 14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 10 -14 B 10 0 -10 6 2 C 8 10 0 8 6 D -10 -6 -8 0 -10 E 14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 10 -14 B 10 0 -10 6 2 C 8 10 0 8 6 D -10 -6 -8 0 -10 E 14 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8081: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A C E (8) D A B E C (8) E C B A D (7) C E B D A (5) E C A D B (4) D C E A B (4) C E D B A (4) C E A D B (4) B E C A D (4) A B D E C (4) D C E B A (3) D A B C E (3) D B A E C (2) D A E C B (2) D A C E B (2) C E D A B (2) C E B A D (2) C E A B D (2) B D A E C (2) B D A C E (2) B C E A D (2) B A E C D (2) B A D E C (2) E B A C D (1) D C B E A (1) D C A E B (1) C D E B A (1) B E A C D (1) B C E D A (1) B A E D C (1) A E C D B (1) A E C B D (1) A E B C D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -10 -2 -12 B 2 0 -8 -4 -12 C 10 8 0 6 -6 D 2 4 -6 0 -6 E 12 12 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -10 -2 -12 B 2 0 -8 -4 -12 C 10 8 0 6 -6 D 2 4 -6 0 -6 E 12 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=20 C=20 B=17 A=9 so A is eliminated. Round 2 votes counts: D=36 E=23 B=21 C=20 so C is eliminated. Round 3 votes counts: E=42 D=37 B=21 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:209 D:197 B:189 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -10 -2 -12 B 2 0 -8 -4 -12 C 10 8 0 6 -6 D 2 4 -6 0 -6 E 12 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -2 -12 B 2 0 -8 -4 -12 C 10 8 0 6 -6 D 2 4 -6 0 -6 E 12 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -2 -12 B 2 0 -8 -4 -12 C 10 8 0 6 -6 D 2 4 -6 0 -6 E 12 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8082: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) A D B C E (7) E C B D A (6) C E B A D (6) B A D C E (5) E D A C B (4) D E B A C (4) E D C A B (3) E D B C A (3) E B C D A (3) D E A C B (3) D E A B C (3) D B A E C (3) C B E A D (3) E C D B A (2) E C D A B (2) D A E B C (2) C E A B D (2) C B A E D (2) C A E D B (2) B E C D A (2) B D A C E (2) A C D B E (2) A C B D E (2) A B D C E (2) E D B A C (1) E C B A D (1) E C A D B (1) E C A B D (1) D B E A C (1) D A E C B (1) D A B C E (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B D E (1) B D A E C (1) B C E A D (1) B C A E D (1) A D E C B (1) A D C E B (1) A B C D E (1) Total count = 100 A B C D E A 0 6 10 -10 -4 B -6 0 2 -14 -8 C -10 -2 0 -12 -12 D 10 14 12 0 6 E 4 8 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 -10 -4 B -6 0 2 -14 -8 C -10 -2 0 -12 -12 D 10 14 12 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 C=19 A=16 B=12 so B is eliminated. Round 2 votes counts: E=29 D=29 C=21 A=21 so C is eliminated. Round 3 votes counts: E=41 A=30 D=29 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:221 E:209 A:201 B:187 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 -10 -4 B -6 0 2 -14 -8 C -10 -2 0 -12 -12 D 10 14 12 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -10 -4 B -6 0 2 -14 -8 C -10 -2 0 -12 -12 D 10 14 12 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -10 -4 B -6 0 2 -14 -8 C -10 -2 0 -12 -12 D 10 14 12 0 6 E 4 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8083: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) B D A E C (7) B A D C E (7) A C E B D (7) E C D A B (6) D E C B A (6) A B C E D (6) E C A D B (5) C A E B D (5) B A C E D (5) B D A C E (4) A B E C D (4) D E B C A (3) C E A D B (3) B A D E C (3) A C B E D (3) D E C A B (2) D C E A B (2) D B E A C (2) C A D B E (2) E D C A B (1) D B C E A (1) D B A E C (1) C E D A B (1) C D E A B (1) C D B A E (1) B D E A C (1) B A E C D (1) B A C D E (1) Total count = 100 A B C D E A 0 -4 4 4 14 B 4 0 10 8 16 C -4 -10 0 2 -2 D -4 -8 -2 0 6 E -14 -16 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 4 14 B 4 0 10 8 16 C -4 -10 0 2 -2 D -4 -8 -2 0 6 E -14 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999766 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 A=20 C=13 E=12 so E is eliminated. Round 2 votes counts: B=29 D=27 C=24 A=20 so A is eliminated. Round 3 votes counts: B=39 C=34 D=27 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:209 D:196 C:193 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 4 14 B 4 0 10 8 16 C -4 -10 0 2 -2 D -4 -8 -2 0 6 E -14 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999766 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 14 B 4 0 10 8 16 C -4 -10 0 2 -2 D -4 -8 -2 0 6 E -14 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999766 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 14 B 4 0 10 8 16 C -4 -10 0 2 -2 D -4 -8 -2 0 6 E -14 -16 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999766 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8084: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) A B C D E (8) A C B D E (7) E D C B A (6) E D B C A (5) E D B A C (5) E B D A C (4) D E C B A (4) C D E B A (4) B D E A C (4) C A E D B (3) B A E D C (3) B A D C E (3) A C B E D (3) A B E D C (3) A B C E D (3) D E B C A (2) C D E A B (2) C A B D E (2) B E D A C (2) B A D E C (2) B A C D E (2) A E C B D (2) E D C A B (1) E C D B A (1) C E D B A (1) C E A D B (1) C A D E B (1) B E A D C (1) B D A C E (1) A E B D C (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 12 -6 -8 B 0 0 0 4 -12 C -12 0 0 6 6 D 6 -4 -6 0 -16 E 8 12 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.307692 D: 0.000000 E: 0.461538 Sum of squares = 0.360946745563 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.538462 D: 0.538462 E: 1.000000 A B C D E A 0 0 12 -6 -8 B 0 0 0 4 -12 C -12 0 0 6 6 D 6 -4 -6 0 -16 E 8 12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.307692 D: 0.000000 E: 0.461538 Sum of squares = 0.360946745716 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.538462 D: 0.538462 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 E=22 B=18 D=6 so D is eliminated. Round 2 votes counts: A=29 E=28 C=25 B=18 so B is eliminated. Round 3 votes counts: A=40 E=35 C=25 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:200 A:199 B:196 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 12 -6 -8 B 0 0 0 4 -12 C -12 0 0 6 6 D 6 -4 -6 0 -16 E 8 12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.307692 D: 0.000000 E: 0.461538 Sum of squares = 0.360946745716 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.538462 D: 0.538462 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -6 -8 B 0 0 0 4 -12 C -12 0 0 6 6 D 6 -4 -6 0 -16 E 8 12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.307692 D: 0.000000 E: 0.461538 Sum of squares = 0.360946745716 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.538462 D: 0.538462 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -6 -8 B 0 0 0 4 -12 C -12 0 0 6 6 D 6 -4 -6 0 -16 E 8 12 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.230769 B: 0.000000 C: 0.307692 D: 0.000000 E: 0.461538 Sum of squares = 0.360946745716 Cumulative probabilities = A: 0.230769 B: 0.230769 C: 0.538462 D: 0.538462 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8085: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) B A E C D (9) C D E B A (8) E C D A B (7) D C E B A (6) A B E D C (6) B A D C E (5) E D C A B (4) C E D B A (4) B A C E D (4) A B D E C (4) B A C D E (3) A B E C D (3) E C A D B (2) E A D C B (2) E A B D C (2) E A B C D (2) D C E A B (2) C D E A B (2) C B D A E (2) C B A D E (2) B A D E C (2) E D A C B (1) E C A B D (1) D E C A B (1) D C B E A (1) C D B A E (1) C B E A D (1) B D A C E (1) B C A D E (1) B A E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -8 2 8 B 20 0 -12 0 12 C 8 12 0 4 6 D -2 0 -4 0 2 E -8 -12 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 2 8 B 20 0 -12 0 12 C 8 12 0 4 6 D -2 0 -4 0 2 E -8 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=21 C=20 D=19 A=14 so A is eliminated. Round 2 votes counts: B=40 E=21 C=20 D=19 so D is eliminated. Round 3 votes counts: B=40 C=38 E=22 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:210 D:198 A:191 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -8 2 8 B 20 0 -12 0 12 C 8 12 0 4 6 D -2 0 -4 0 2 E -8 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 2 8 B 20 0 -12 0 12 C 8 12 0 4 6 D -2 0 -4 0 2 E -8 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 2 8 B 20 0 -12 0 12 C 8 12 0 4 6 D -2 0 -4 0 2 E -8 -12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999271 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8086: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (10) D E A B C (6) A B D E C (6) E D A C B (5) C B E A D (5) C B A E D (5) B C A D E (5) E D C B A (4) C E B D A (4) B A D C E (4) B A C D E (4) E D B A C (3) E C D B A (3) E D C A B (2) E D A B C (2) E C D A B (2) D E A C B (2) D A E B C (2) C E D B A (2) C B E D A (2) B C E D A (2) B C A E D (2) A D E C B (2) A D B E C (2) A D B C E (2) A C B D E (2) A B D C E (2) A B C D E (2) E D B C A (1) D B E A C (1) C E A D B (1) C B A D E (1) C A B D E (1) A C E D B (1) Total count = 100 A B C D E A 0 4 16 14 6 B -4 0 12 -6 -4 C -16 -12 0 -12 -6 D -14 6 12 0 8 E -6 4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 16 14 6 B -4 0 12 -6 -4 C -16 -12 0 -12 -6 D -14 6 12 0 8 E -6 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=22 C=21 B=17 D=11 so D is eliminated. Round 2 votes counts: A=31 E=30 C=21 B=18 so B is eliminated. Round 3 votes counts: A=39 E=31 C=30 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:206 B:199 E:198 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 16 14 6 B -4 0 12 -6 -4 C -16 -12 0 -12 -6 D -14 6 12 0 8 E -6 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 16 14 6 B -4 0 12 -6 -4 C -16 -12 0 -12 -6 D -14 6 12 0 8 E -6 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 16 14 6 B -4 0 12 -6 -4 C -16 -12 0 -12 -6 D -14 6 12 0 8 E -6 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999276 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8087: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (14) A C E D B (10) E C B D A (9) E C A B D (8) E B C D A (6) D B A C E (6) C E B D A (5) D B A E C (4) C E A B D (4) B E D C A (4) A D B E C (4) C E A D B (3) B D E C A (3) E A C B D (2) D B C A E (2) C A E D B (2) B D A E C (2) A E C D B (2) A C D B E (2) E C B A D (1) D A B C E (1) C E D B A (1) C E B A D (1) A E D C B (1) A D E B C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 12 2 14 6 B -12 0 -6 -10 -14 C -2 6 0 14 6 D -14 10 -14 0 -18 E -6 14 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 14 6 B -12 0 -6 -10 -14 C -2 6 0 14 6 D -14 10 -14 0 -18 E -6 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=26 C=16 D=13 B=9 so B is eliminated. Round 2 votes counts: A=36 E=30 D=18 C=16 so C is eliminated. Round 3 votes counts: E=44 A=38 D=18 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:212 E:210 D:182 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 14 6 B -12 0 -6 -10 -14 C -2 6 0 14 6 D -14 10 -14 0 -18 E -6 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 14 6 B -12 0 -6 -10 -14 C -2 6 0 14 6 D -14 10 -14 0 -18 E -6 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 14 6 B -12 0 -6 -10 -14 C -2 6 0 14 6 D -14 10 -14 0 -18 E -6 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994219 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8088: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) A C B E D (10) D E B A C (7) C A B E D (7) E D C A B (6) B A C D E (6) D E B C A (5) A C E D B (4) E D A C B (3) E D A B C (3) E A D C B (3) B D E A C (3) A B C D E (3) D E C B A (2) D B E C A (2) C B D E A (2) B C D E A (2) B C D A E (2) B A D E C (2) A C E B D (2) E D C B A (1) E C D A B (1) D E A B C (1) C E D A B (1) C D E B A (1) C B D A E (1) C B A D E (1) C A E D B (1) C A E B D (1) B D E C A (1) B D C E A (1) B A D C E (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 0 10 10 B 4 0 4 16 14 C 0 -4 0 18 18 D -10 -16 -18 0 10 E -10 -14 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 10 10 B 4 0 4 16 14 C 0 -4 0 18 18 D -10 -16 -18 0 10 E -10 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998518 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=21 E=17 D=17 C=15 so C is eliminated. Round 2 votes counts: B=34 A=30 E=18 D=18 so E is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:216 A:208 D:183 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 10 10 B 4 0 4 16 14 C 0 -4 0 18 18 D -10 -16 -18 0 10 E -10 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998518 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 10 10 B 4 0 4 16 14 C 0 -4 0 18 18 D -10 -16 -18 0 10 E -10 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998518 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 10 10 B 4 0 4 16 14 C 0 -4 0 18 18 D -10 -16 -18 0 10 E -10 -14 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998518 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8089: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) A E D C B (10) A E B C D (10) D C B E A (9) A B C D E (9) E B C D A (7) B C D E A (7) A D C B E (6) E A D C B (4) E A D B C (3) D C B A E (3) B C E D A (3) A D E C B (3) E A B C D (2) B C D A E (2) A E D B C (2) A B C E D (2) E D B C A (1) E B D C A (1) E B C A D (1) D E C B A (1) D A E C B (1) C B D E A (1) B E C D A (1) A B E C D (1) Total count = 100 A B C D E A 0 6 6 6 -2 B -6 0 4 -6 -12 C -6 -4 0 -8 -16 D -6 6 8 0 -16 E 2 12 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 6 -2 B -6 0 4 -6 -12 C -6 -4 0 -8 -16 D -6 6 8 0 -16 E 2 12 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999968497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 E=29 D=14 B=13 C=1 so C is eliminated. Round 2 votes counts: A=43 E=29 D=14 B=14 so D is eliminated. Round 3 votes counts: A=44 E=30 B=26 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:223 A:208 D:196 B:190 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 6 -2 B -6 0 4 -6 -12 C -6 -4 0 -8 -16 D -6 6 8 0 -16 E 2 12 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999968497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 -2 B -6 0 4 -6 -12 C -6 -4 0 -8 -16 D -6 6 8 0 -16 E 2 12 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999968497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 -2 B -6 0 4 -6 -12 C -6 -4 0 -8 -16 D -6 6 8 0 -16 E 2 12 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999968497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8090: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) D B A C E (9) C A B D E (8) E D C B A (5) E D B A C (5) D B E A C (5) C B D A E (5) D A B E C (4) E C D B A (3) E A C B D (3) D B C A E (3) A B C D E (3) E D B C A (2) E D A B C (2) E C D A B (2) E C A D B (2) E A C D B (2) D E B A C (2) D B E C A (2) D B A E C (2) C B D E A (2) C A E B D (2) E D C A B (1) E C B D A (1) E A D B C (1) D B C E A (1) C E B D A (1) C E A B D (1) C B A D E (1) C A B E D (1) B C D A E (1) B A C D E (1) A E C B D (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -16 -16 -14 B 2 0 -10 -8 0 C 16 10 0 8 -18 D 16 8 -8 0 2 E 14 0 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.642857 E: 0.285714 Sum of squares = 0.499999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.714286 E: 1.000000 A B C D E A 0 -2 -16 -16 -14 B 2 0 -10 -8 0 C 16 10 0 8 -18 D 16 8 -8 0 2 E 14 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.642857 E: 0.285714 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 D=28 C=21 A=6 B=2 so B is eliminated. Round 2 votes counts: E=43 D=28 C=22 A=7 so A is eliminated. Round 3 votes counts: E=44 D=30 C=26 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:215 D:209 C:208 B:192 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -16 -16 -14 B 2 0 -10 -8 0 C 16 10 0 8 -18 D 16 8 -8 0 2 E 14 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.642857 E: 0.285714 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -16 -14 B 2 0 -10 -8 0 C 16 10 0 8 -18 D 16 8 -8 0 2 E 14 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.642857 E: 0.285714 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -16 -14 B 2 0 -10 -8 0 C 16 10 0 8 -18 D 16 8 -8 0 2 E 14 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.642857 E: 0.285714 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.071429 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8091: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) C A D B E (7) E B D C A (6) E B D A C (5) C B D A E (5) B E D A C (5) A C D E B (5) E A C D B (4) C D A B E (4) E B C D A (3) B E C D A (3) A D C B E (3) A C D B E (3) E A D C B (2) E A B D C (2) D B A E C (2) D A B C E (2) B E D C A (2) B D C E A (2) B D C A E (2) B D A E C (2) B C E D A (2) B C D A E (2) A D E B C (2) E C A B D (1) E B C A D (1) E A D B C (1) E A C B D (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C B E (1) C E B A D (1) C D B A E (1) C B E D A (1) C B D E A (1) C A E D B (1) C A B D E (1) B D A C E (1) B C D E A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 -14 2 -12 -2 B 14 0 10 14 12 C -2 -10 0 -4 -2 D 12 -14 4 0 2 E 2 -12 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -12 -2 B 14 0 10 14 12 C -2 -10 0 -4 -2 D 12 -14 4 0 2 E 2 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=22 B=22 A=15 D=8 so D is eliminated. Round 2 votes counts: E=33 B=26 C=23 A=18 so A is eliminated. Round 3 votes counts: E=37 C=35 B=28 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:225 D:202 E:195 C:191 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 -12 -2 B 14 0 10 14 12 C -2 -10 0 -4 -2 D 12 -14 4 0 2 E 2 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -12 -2 B 14 0 10 14 12 C -2 -10 0 -4 -2 D 12 -14 4 0 2 E 2 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -12 -2 B 14 0 10 14 12 C -2 -10 0 -4 -2 D 12 -14 4 0 2 E 2 -12 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8092: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) B C A E D (10) C B A E D (7) D E B A C (5) C A B E D (5) B C A D E (5) D E A B C (4) D A E C B (4) B E C A D (4) A C E B D (4) E A D C B (3) B C E A D (3) E D B A C (2) E A C B D (2) D C B A E (2) D B E C A (2) D A C E B (2) C B D A E (2) B E D C A (2) A C D E B (2) A C B E D (2) E D A C B (1) E D A B C (1) E B D C A (1) E B D A C (1) E A C D B (1) D E B C A (1) D C A B E (1) D B C E A (1) D B C A E (1) C D B A E (1) C B A D E (1) B E C D A (1) B D C E A (1) B D C A E (1) B C D A E (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 -6 2 6 B 10 0 -6 6 6 C 6 6 0 2 6 D -2 -6 -2 0 -2 E -6 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 2 6 B 10 0 -6 6 6 C 6 6 0 2 6 D -2 -6 -2 0 -2 E -6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=28 C=16 E=12 A=10 so A is eliminated. Round 2 votes counts: D=35 B=28 C=24 E=13 so E is eliminated. Round 3 votes counts: D=43 B=30 C=27 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:210 B:208 A:196 D:194 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -6 2 6 B 10 0 -6 6 6 C 6 6 0 2 6 D -2 -6 -2 0 -2 E -6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 2 6 B 10 0 -6 6 6 C 6 6 0 2 6 D -2 -6 -2 0 -2 E -6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 2 6 B 10 0 -6 6 6 C 6 6 0 2 6 D -2 -6 -2 0 -2 E -6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8093: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) D C B A E (9) A E D C B (8) A D E C B (7) A E D B C (6) E B A C D (5) B C D E A (5) E A D C B (4) D A C E B (4) B C E D A (4) E B C A D (3) C B D E A (3) B C D A E (3) D C A E B (2) D C A B E (2) D B C A E (2) D A E C B (2) C D E B A (2) C D B A E (2) B D C A E (2) A E B D C (2) E C B A D (1) E C A D B (1) E C A B D (1) E A C B D (1) E A B C D (1) D B A C E (1) D A B C E (1) C E B D A (1) C D B E A (1) C B E D A (1) B E C A D (1) B C E A D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 6 6 6 10 B -6 0 -4 -12 -22 C -6 4 0 -26 -8 D -6 12 26 0 0 E -10 22 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 10 B -6 0 -4 -12 -22 C -6 4 0 -26 -8 D -6 12 26 0 0 E -10 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=25 D=23 B=16 C=10 so C is eliminated. Round 2 votes counts: D=28 E=27 A=25 B=20 so B is eliminated. Round 3 votes counts: D=41 E=34 A=25 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:214 E:210 C:182 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 10 B -6 0 -4 -12 -22 C -6 4 0 -26 -8 D -6 12 26 0 0 E -10 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 10 B -6 0 -4 -12 -22 C -6 4 0 -26 -8 D -6 12 26 0 0 E -10 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 10 B -6 0 -4 -12 -22 C -6 4 0 -26 -8 D -6 12 26 0 0 E -10 22 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8094: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) B D C A E (8) A C D E B (8) D C A E B (6) B E A D C (6) B A C D E (6) A C D B E (6) E D C A B (5) E B D C A (5) A E C D B (5) A B C D E (4) E B A C D (3) E A B C D (3) D C E A B (3) B E A C D (3) E A C D B (2) D C B E A (2) D C B A E (2) D C A B E (2) C D A E B (2) C D A B E (2) A B E C D (2) E D C B A (1) D E C B A (1) D C E B A (1) B D C E A (1) B A E C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 2 2 12 B -4 0 2 4 10 C -2 -2 0 -2 10 D -2 -4 2 0 10 E -12 -10 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998783 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 2 12 B -4 0 2 4 10 C -2 -2 0 -2 10 D -2 -4 2 0 10 E -12 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=27 E=19 D=17 C=4 so C is eliminated. Round 2 votes counts: B=33 A=27 D=21 E=19 so E is eliminated. Round 3 votes counts: B=41 A=32 D=27 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:206 D:203 C:202 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 2 12 B -4 0 2 4 10 C -2 -2 0 -2 10 D -2 -4 2 0 10 E -12 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 12 B -4 0 2 4 10 C -2 -2 0 -2 10 D -2 -4 2 0 10 E -12 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 12 B -4 0 2 4 10 C -2 -2 0 -2 10 D -2 -4 2 0 10 E -12 -10 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8095: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (11) E A C D B (8) D E A C B (8) D B E C A (8) C A B E D (7) B C A D E (7) A C E B D (7) E D A C B (6) D E A B C (5) B D C A E (5) A E C D B (4) D E B A C (3) D A C E B (2) B D C E A (2) B C D A E (2) A E C B D (2) E D C A B (1) E C A B D (1) E A D C B (1) E A C B D (1) D E B C A (1) D B A C E (1) D A E C B (1) C E A B D (1) B E D C A (1) B E C D A (1) B C E A D (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 14 2 6 2 B -14 0 -4 -2 -6 C -2 4 0 8 -4 D -6 2 -8 0 -6 E -2 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999655 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 6 2 B -14 0 -4 -2 -6 C -2 4 0 8 -4 D -6 2 -8 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=29 E=18 A=15 C=8 so C is eliminated. Round 2 votes counts: B=30 D=29 A=22 E=19 so E is eliminated. Round 3 votes counts: D=36 A=34 B=30 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:207 C:203 D:191 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 6 2 B -14 0 -4 -2 -6 C -2 4 0 8 -4 D -6 2 -8 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 6 2 B -14 0 -4 -2 -6 C -2 4 0 8 -4 D -6 2 -8 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 6 2 B -14 0 -4 -2 -6 C -2 4 0 8 -4 D -6 2 -8 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8096: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (9) E B C A D (8) E B C D A (6) E A B C D (6) D A C B E (6) E D B C A (5) A E C B D (5) A D C B E (5) A E D C B (4) A D C E B (4) D E B C A (3) D A E C B (3) B C D E A (3) D C B E A (2) D C A B E (2) C B A D E (2) B C E D A (2) B C A D E (2) A E B C D (2) A D E C B (2) A B C E D (2) E D A B C (1) E B D C A (1) E A D B C (1) D E C B A (1) D E A C B (1) D B C E A (1) C D B A E (1) C B D A E (1) C A B D E (1) B E A C D (1) B C E A D (1) B C D A E (1) B C A E D (1) B A C E D (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -6 2 14 B 4 0 -4 -4 -6 C 6 4 0 -2 0 D -2 4 2 0 4 E -14 6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.44000000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 14 B 4 0 -4 -4 -6 C 6 4 0 -2 0 D -2 4 2 0 4 E -14 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000592 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 A=27 B=12 C=5 so C is eliminated. Round 2 votes counts: D=29 E=28 A=28 B=15 so B is eliminated. Round 3 votes counts: D=34 A=34 E=32 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:204 D:204 A:203 B:195 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 2 14 B 4 0 -4 -4 -6 C 6 4 0 -2 0 D -2 4 2 0 4 E -14 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000592 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 14 B 4 0 -4 -4 -6 C 6 4 0 -2 0 D -2 4 2 0 4 E -14 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000592 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 14 B 4 0 -4 -4 -6 C 6 4 0 -2 0 D -2 4 2 0 4 E -14 6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.200000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000592 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8097: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) B A D C E (6) D E B A C (5) D B A E C (5) C E A B D (5) C A B D E (5) B A C D E (5) E D C B A (4) E D C A B (4) E D B A C (4) E C A B D (4) C E A D B (4) A C B D E (4) E D B C A (3) E C D A B (3) E B D A C (3) E B C A D (3) D C A E B (3) D B E A C (3) A B C D E (3) D E C A B (2) D A B C E (2) C A B E D (2) A B D C E (2) E C A D B (1) D E A C B (1) D E A B C (1) C B A E D (1) C A E B D (1) C A D E B (1) B C A E D (1) Total count = 100 A B C D E A 0 -4 6 -4 0 B 4 0 10 -10 -4 C -6 -10 0 -14 8 D 4 10 14 0 14 E 0 4 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -4 0 B 4 0 10 -10 -4 C -6 -10 0 -14 8 D 4 10 14 0 14 E 0 4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=29 C=19 B=12 A=9 so A is eliminated. Round 2 votes counts: D=31 E=29 C=23 B=17 so B is eliminated. Round 3 votes counts: D=39 C=32 E=29 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 B:200 A:199 E:191 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -4 0 B 4 0 10 -10 -4 C -6 -10 0 -14 8 D 4 10 14 0 14 E 0 4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 0 B 4 0 10 -10 -4 C -6 -10 0 -14 8 D 4 10 14 0 14 E 0 4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 0 B 4 0 10 -10 -4 C -6 -10 0 -14 8 D 4 10 14 0 14 E 0 4 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8098: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (14) C B A E D (10) C B D E A (8) D E A B C (6) C A B E D (6) D E B A C (5) E D A B C (3) C D A E B (3) C B E A D (3) A C E B D (3) D C B E A (2) D A E B C (2) C D B A E (2) C A D E B (2) B D E C A (2) B D E A C (2) B C E D A (2) B C E A D (2) A E C D B (2) A E B D C (2) E A D B C (1) D E C A B (1) D C E B A (1) D C A E B (1) D B C E A (1) D A E C B (1) D A C E B (1) C D B E A (1) C B D A E (1) C A E B D (1) B E A D C (1) B E A C D (1) B A E D C (1) B A E C D (1) A E D C B (1) A E B C D (1) A D E C B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 8 2 10 16 B -8 0 -6 -4 -6 C -2 6 0 2 4 D -10 4 -2 0 -14 E -16 6 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999183 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 10 16 B -8 0 -6 -4 -6 C -2 6 0 2 4 D -10 4 -2 0 -14 E -16 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=26 D=21 B=12 E=4 so E is eliminated. Round 2 votes counts: C=37 A=27 D=24 B=12 so B is eliminated. Round 3 votes counts: C=41 A=31 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:205 E:200 D:189 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 10 16 B -8 0 -6 -4 -6 C -2 6 0 2 4 D -10 4 -2 0 -14 E -16 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 10 16 B -8 0 -6 -4 -6 C -2 6 0 2 4 D -10 4 -2 0 -14 E -16 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 10 16 B -8 0 -6 -4 -6 C -2 6 0 2 4 D -10 4 -2 0 -14 E -16 6 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981487 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8099: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) A C B D E (8) E D A C B (7) A B C E D (7) E D A B C (6) B A C E D (5) C B A D E (4) A E D C B (4) A B C D E (4) E D C B A (3) E D B C A (3) D E A C B (3) B E C D A (3) E D B A C (2) E B D C A (2) E B D A C (2) D C E B A (2) D C E A B (2) C D B E A (2) C B D E A (2) B C A E D (2) B C A D E (2) B A C D E (2) A E D B C (2) E A B D C (1) D E C A B (1) D C A E B (1) C D B A E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E D C A (1) A E B D C (1) A D C E B (1) A C D E B (1) A C D B E (1) A B E D C (1) Total count = 100 A B C D E A 0 8 16 -4 0 B -8 0 -8 -4 -4 C -16 8 0 -6 0 D 4 4 6 0 -4 E 0 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.350335 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.649665 Sum of squares = 0.544799459681 Cumulative probabilities = A: 0.350335 B: 0.350335 C: 0.350335 D: 0.350335 E: 1.000000 A B C D E A 0 8 16 -4 0 B -8 0 -8 -4 -4 C -16 8 0 -6 0 D 4 4 6 0 -4 E 0 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499575 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500425 Sum of squares = 0.500000361229 Cumulative probabilities = A: 0.499575 B: 0.499575 C: 0.499575 D: 0.499575 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=26 D=17 B=15 C=12 so C is eliminated. Round 2 votes counts: A=32 E=26 D=21 B=21 so D is eliminated. Round 3 votes counts: E=42 A=34 B=24 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:205 E:204 C:193 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 -4 0 B -8 0 -8 -4 -4 C -16 8 0 -6 0 D 4 4 6 0 -4 E 0 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499575 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500425 Sum of squares = 0.500000361229 Cumulative probabilities = A: 0.499575 B: 0.499575 C: 0.499575 D: 0.499575 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -4 0 B -8 0 -8 -4 -4 C -16 8 0 -6 0 D 4 4 6 0 -4 E 0 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499575 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500425 Sum of squares = 0.500000361229 Cumulative probabilities = A: 0.499575 B: 0.499575 C: 0.499575 D: 0.499575 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -4 0 B -8 0 -8 -4 -4 C -16 8 0 -6 0 D 4 4 6 0 -4 E 0 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499575 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500425 Sum of squares = 0.500000361229 Cumulative probabilities = A: 0.499575 B: 0.499575 C: 0.499575 D: 0.499575 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8100: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (13) C A B D E (10) D B E A C (9) B D A E C (9) C B D A E (7) C A E B D (5) E D B A C (4) C E A D B (4) E D A B C (3) D E B A C (3) C E D B A (3) C B A D E (3) C A B E D (3) E D B C A (2) E C D A B (2) D E B C A (2) B D C A E (2) B C D A E (2) A B D C E (2) E D C B A (1) E C D B A (1) E C A D B (1) E A C D B (1) D B E C A (1) D B C E A (1) D A E B C (1) C E D A B (1) C E B D A (1) B A D C E (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -4 -10 -6 B 4 0 12 -6 2 C 4 -12 0 -10 -8 D 10 6 10 0 8 E 6 -2 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -10 -6 B 4 0 12 -6 2 C 4 -12 0 -10 -8 D 10 6 10 0 8 E 6 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=28 D=17 B=14 A=4 so A is eliminated. Round 2 votes counts: C=38 E=28 D=17 B=17 so D is eliminated. Round 3 votes counts: C=38 E=34 B=28 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:217 B:206 E:202 A:188 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -10 -6 B 4 0 12 -6 2 C 4 -12 0 -10 -8 D 10 6 10 0 8 E 6 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -10 -6 B 4 0 12 -6 2 C 4 -12 0 -10 -8 D 10 6 10 0 8 E 6 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -10 -6 B 4 0 12 -6 2 C 4 -12 0 -10 -8 D 10 6 10 0 8 E 6 -2 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8101: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) E B A D C (9) D A B C E (8) C E D A B (8) B A D E C (8) C D A E B (7) E B C A D (6) C D A B E (6) E C B D A (4) D A B E C (4) E C D A B (3) D A C B E (3) C D E A B (3) B A D C E (3) A D B C E (3) E B D A C (2) C E B D A (2) C A D B E (2) B E A D C (2) A B D C E (2) D C A B E (1) C E D B A (1) C E B A D (1) C B E A D (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -10 -4 -4 B -2 0 -4 0 -12 C 10 4 0 8 2 D 4 0 -8 0 2 E 4 12 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 -4 -4 B -2 0 -4 0 -12 C 10 4 0 8 2 D 4 0 -8 0 2 E 4 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=31 D=16 B=13 A=6 so A is eliminated. Round 2 votes counts: E=34 C=31 D=20 B=15 so B is eliminated. Round 3 votes counts: E=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:212 E:206 D:199 A:192 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -4 -4 B -2 0 -4 0 -12 C 10 4 0 8 2 D 4 0 -8 0 2 E 4 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -4 -4 B -2 0 -4 0 -12 C 10 4 0 8 2 D 4 0 -8 0 2 E 4 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -4 -4 B -2 0 -4 0 -12 C 10 4 0 8 2 D 4 0 -8 0 2 E 4 12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8102: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (15) E C B D A (8) B D C A E (5) A D B E C (5) E C D B A (4) E A C D B (4) D B A C E (4) C B E D A (4) D B A E C (3) C E B D A (3) B D A C E (3) A E C D B (3) E D B A C (2) E C B A D (2) E C A D B (2) E C A B D (2) E A D C B (2) C E A B D (2) C B D E A (2) B D C E A (2) B C D E A (2) B C D A E (2) A E D B C (2) A D E B C (2) A D C B E (2) E D A B C (1) E B D C A (1) D E B A C (1) D B E A C (1) D A B E C (1) C B A D E (1) C A E B D (1) B A D C E (1) A E D C B (1) A D E C B (1) A C E B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 14 2 10 B 2 0 8 -12 10 C -14 -8 0 -12 4 D -2 12 12 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999575 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 2 10 B 2 0 8 -12 10 C -14 -8 0 -12 4 D -2 12 12 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000042 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=28 B=15 C=13 D=10 so D is eliminated. Round 2 votes counts: A=35 E=29 B=23 C=13 so C is eliminated. Round 3 votes counts: A=36 E=34 B=30 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:216 A:212 B:204 C:185 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 14 2 10 B 2 0 8 -12 10 C -14 -8 0 -12 4 D -2 12 12 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000042 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 2 10 B 2 0 8 -12 10 C -14 -8 0 -12 4 D -2 12 12 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000042 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 2 10 B 2 0 8 -12 10 C -14 -8 0 -12 4 D -2 12 12 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593750000042 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8103: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) D C E B A (6) E D A C B (4) D C B E A (4) B C D E A (4) B C D A E (4) B A C E D (4) A E D C B (4) A E B D C (4) A C D B E (4) A B E C D (4) A B C E D (4) E A D B C (3) C D B E A (3) B C A D E (3) E D B C A (2) E B A C D (2) E A D C B (2) D C E A B (2) B E D C A (2) B C A E D (2) A E D B C (2) A D C E B (2) A C B D E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D A B C (1) E B D C A (1) E B A D C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E C A B (1) D E A C B (1) D C B A E (1) C D B A E (1) C B D E A (1) C B D A E (1) B E A C D (1) B C E D A (1) B A E C D (1) A D C B E (1) Total count = 100 A B C D E A 0 6 16 14 6 B -6 0 14 6 0 C -16 -14 0 4 4 D -14 -6 -4 0 -12 E -6 0 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 14 6 B -6 0 14 6 0 C -16 -14 0 4 4 D -14 -6 -4 0 -12 E -6 0 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=22 E=20 D=16 C=6 so C is eliminated. Round 2 votes counts: A=36 B=24 E=20 D=20 so E is eliminated. Round 3 votes counts: A=43 D=29 B=28 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:207 E:201 C:189 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 14 6 B -6 0 14 6 0 C -16 -14 0 4 4 D -14 -6 -4 0 -12 E -6 0 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 14 6 B -6 0 14 6 0 C -16 -14 0 4 4 D -14 -6 -4 0 -12 E -6 0 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 14 6 B -6 0 14 6 0 C -16 -14 0 4 4 D -14 -6 -4 0 -12 E -6 0 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8104: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) D C B A E (6) D C A B E (6) E A B C D (5) C A B D E (5) B A C E D (5) E B A D C (4) E B A C D (4) D E A B C (4) C B A E D (4) B C A E D (4) E A B D C (3) D E A C B (3) D C E A B (3) C A B E D (3) E D B A C (2) E D A B C (2) E A D C B (2) D E C B A (2) D E B A C (2) D C A E B (2) B C D A E (2) B A E C D (2) A C E B D (2) E A D B C (1) D E B C A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B C A E (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) B C A D E (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -12 0 2 B -8 0 -8 -2 -2 C 12 8 0 -10 6 D 0 2 10 0 8 E -2 2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.316756 B: 0.000000 C: 0.000000 D: 0.683244 E: 0.000000 Sum of squares = 0.567156941391 Cumulative probabilities = A: 0.316756 B: 0.316756 C: 0.316756 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 0 2 B -8 0 -8 -2 -2 C 12 8 0 -10 6 D 0 2 10 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.000000 Sum of squares = 0.504132252469 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 E=23 C=17 B=14 A=5 so A is eliminated. Round 2 votes counts: D=41 E=23 C=20 B=16 so B is eliminated. Round 3 votes counts: D=41 C=33 E=26 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:208 A:199 E:193 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -12 0 2 B -8 0 -8 -2 -2 C 12 8 0 -10 6 D 0 2 10 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.000000 Sum of squares = 0.504132252469 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 0 2 B -8 0 -8 -2 -2 C 12 8 0 -10 6 D 0 2 10 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.000000 Sum of squares = 0.504132252469 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 0 2 B -8 0 -8 -2 -2 C 12 8 0 -10 6 D 0 2 10 0 8 E -2 2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.454545 B: 0.000000 C: 0.000000 D: 0.545455 E: 0.000000 Sum of squares = 0.504132252469 Cumulative probabilities = A: 0.454545 B: 0.454545 C: 0.454545 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8105: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (7) D A B C E (7) D B E C A (5) D B A C E (5) E C B D A (4) C B E A D (4) B C E D A (4) D E B C A (3) C A B E D (3) B D C E A (3) B C D E A (3) A E C D B (3) A E C B D (3) A D E C B (3) E D A C B (2) E B C D A (2) E A C B D (2) D E B A C (2) D E A B C (2) D B E A C (2) D B A E C (2) D A B E C (2) C E B A D (2) B C D A E (2) B C A D E (2) A D B C E (2) A C E B D (2) A C B D E (2) E D C A B (1) E D B C A (1) E C B A D (1) E C A D B (1) D B C A E (1) D A E B C (1) C E A B D (1) C B E D A (1) C B A E D (1) C A E B D (1) B D C A E (1) A D C B E (1) A D B E C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -8 -12 -6 B 2 0 8 4 12 C 8 -8 0 4 0 D 12 -4 -4 0 8 E 6 -12 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999087 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -12 -6 B 2 0 8 4 12 C 8 -8 0 4 0 D 12 -4 -4 0 8 E 6 -12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=21 A=19 B=15 C=13 so C is eliminated. Round 2 votes counts: D=32 E=24 A=23 B=21 so B is eliminated. Round 3 votes counts: D=41 E=33 A=26 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:213 D:206 C:202 E:193 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 -12 -6 B 2 0 8 4 12 C 8 -8 0 4 0 D 12 -4 -4 0 8 E 6 -12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -12 -6 B 2 0 8 4 12 C 8 -8 0 4 0 D 12 -4 -4 0 8 E 6 -12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -12 -6 B 2 0 8 4 12 C 8 -8 0 4 0 D 12 -4 -4 0 8 E 6 -12 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8106: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (18) C B D E A (13) C B A E D (8) B C D E A (8) D E B A C (6) E A D C B (5) C A B E D (5) B D E C A (4) A E D C B (4) E D A B C (3) C A E D B (3) C E A D B (2) C B D A E (2) C B A D E (2) B D E A C (2) A C E D B (2) E A D B C (1) D E B C A (1) D E A C B (1) D E A B C (1) D C E A B (1) D B E A C (1) C E D A B (1) C A E B D (1) B D C E A (1) B D A C E (1) B A E D C (1) B A D E C (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 -4 8 -2 B 2 0 -2 -2 -2 C 4 2 0 -4 0 D -8 2 4 0 -10 E 2 2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.512017 D: 0.000000 E: 0.487983 Sum of squares = 0.500288804829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.512017 D: 0.512017 E: 1.000000 A B C D E A 0 -2 -4 8 -2 B 2 0 -2 -2 -2 C 4 2 0 -4 0 D -8 2 4 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 A=25 B=18 D=11 E=9 so E is eliminated. Round 2 votes counts: C=37 A=31 B=18 D=14 so D is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:207 C:201 A:200 B:198 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 8 -2 B 2 0 -2 -2 -2 C 4 2 0 -4 0 D -8 2 4 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 -2 B 2 0 -2 -2 -2 C 4 2 0 -4 0 D -8 2 4 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 -2 B 2 0 -2 -2 -2 C 4 2 0 -4 0 D -8 2 4 0 -10 E 2 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8107: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (18) D A E B C (13) A D E B C (9) D A E C B (7) B C E A D (7) E A D B C (6) C B E D A (6) C D B A E (5) B E C A D (4) A E D B C (4) E B A D C (3) E A B D C (2) D C A E B (2) D A C E B (2) C B D E A (2) B E A D C (2) D C A B E (1) D A B E C (1) D A B C E (1) C D A E B (1) C D A B E (1) C B D A E (1) B E A C D (1) A D E C B (1) Total count = 100 A B C D E A 0 2 4 14 -2 B -2 0 6 -8 0 C -4 -6 0 -8 -6 D -14 8 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.215996 C: 0.000000 D: 0.000000 E: 0.784004 Sum of squares = 0.66131614025 Cumulative probabilities = A: 0.000000 B: 0.215996 C: 0.215996 D: 0.215996 E: 1.000000 A B C D E A 0 2 4 14 -2 B -2 0 6 -8 0 C -4 -6 0 -8 -6 D -14 8 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204087922 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=27 B=14 A=14 E=11 so E is eliminated. Round 2 votes counts: C=34 D=27 A=22 B=17 so B is eliminated. Round 3 votes counts: C=45 A=28 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:207 B:198 D:198 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 14 -2 B -2 0 6 -8 0 C -4 -6 0 -8 -6 D -14 8 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204087922 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 14 -2 B -2 0 6 -8 0 C -4 -6 0 -8 -6 D -14 8 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204087922 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 14 -2 B -2 0 6 -8 0 C -4 -6 0 -8 -6 D -14 8 8 0 -6 E 2 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204087922 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8108: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) E B A C D (5) C E A B D (5) C B E A D (5) B D E C A (5) A E C B D (5) A C E D B (5) E A C B D (4) C E B A D (4) D C B A E (3) D B C A E (3) B E C A D (3) B D C E A (3) A C D E B (3) E A B D C (2) D B E A C (2) D A C E B (2) D A B E C (2) C A D E B (2) B E D A C (2) B E C D A (2) B C E D A (2) B C D E A (2) A E D B C (2) A D C E B (2) E C B A D (1) E C A B D (1) E B C A D (1) E A B C D (1) D C A B E (1) D B E C A (1) D B C E A (1) D A E C B (1) D A B C E (1) C D B A E (1) C B D E A (1) C A E D B (1) C A E B D (1) B D E A C (1) A E D C B (1) A E C D B (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 4 2 28 -8 B -4 0 -16 14 -18 C -2 16 0 14 -2 D -28 -14 -14 0 -10 E 8 18 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 28 -8 B -4 0 -16 14 -18 C -2 16 0 14 -2 D -28 -14 -14 0 -10 E 8 18 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=20 B=20 D=17 E=15 so E is eliminated. Round 2 votes counts: A=35 B=26 C=22 D=17 so D is eliminated. Round 3 votes counts: A=41 B=33 C=26 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:219 A:213 C:213 B:188 D:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 28 -8 B -4 0 -16 14 -18 C -2 16 0 14 -2 D -28 -14 -14 0 -10 E 8 18 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 28 -8 B -4 0 -16 14 -18 C -2 16 0 14 -2 D -28 -14 -14 0 -10 E 8 18 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 28 -8 B -4 0 -16 14 -18 C -2 16 0 14 -2 D -28 -14 -14 0 -10 E 8 18 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8109: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (13) B D A C E (10) C A E B D (7) E D C B A (6) E D B C A (6) D B A C E (6) D B E A C (5) D B A E C (5) A C B D E (5) A B D C E (5) E C A B D (4) C A B D E (3) E D C A B (2) E C B D A (2) E B D C A (2) D E B A C (2) C A E D B (2) C A B E D (2) A D B C E (2) E D B A C (1) E C D B A (1) E C D A B (1) E A C D B (1) D A B C E (1) C E B A D (1) C E A B D (1) B D E C A (1) B A D C E (1) B A C D E (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -8 -2 2 B 0 0 -4 -10 -6 C 8 4 0 -10 -4 D 2 10 10 0 -6 E -2 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 A B C D E A 0 0 -8 -2 2 B 0 0 -4 -10 -6 C 8 4 0 -10 -4 D 2 10 10 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=19 C=16 B=13 A=13 so B is eliminated. Round 2 votes counts: E=39 D=30 C=16 A=15 so A is eliminated. Round 3 votes counts: E=39 D=38 C=23 so C is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:208 E:207 C:199 A:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -8 -2 2 B 0 0 -4 -10 -6 C 8 4 0 -10 -4 D 2 10 10 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -2 2 B 0 0 -4 -10 -6 C 8 4 0 -10 -4 D 2 10 10 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -2 2 B 0 0 -4 -10 -6 C 8 4 0 -10 -4 D 2 10 10 0 -6 E -2 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.571429 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8110: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (14) D A B E C (8) B E C D A (7) C B E D A (6) D B A E C (5) C E B A D (5) C E A B D (5) B D E C A (5) C B E A D (4) D B E A C (3) C A E D B (3) E C A B D (2) E B C A D (2) E A C B D (2) D A E B C (2) D A C B E (2) B E D C A (2) B E D A C (2) B E C A D (2) B D E A C (2) B C E D A (2) A D E C B (2) A D E B C (2) E B D A C (1) E A B C D (1) D B C A E (1) D A B C E (1) C D B A E (1) C A D E B (1) C A D B E (1) B D C E A (1) A E D C B (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 0 -2 -8 B 2 0 -2 2 10 C 0 2 0 -8 -4 D 2 -2 8 0 4 E 8 -10 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.500000000012 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -2 -8 B 2 0 -2 2 10 C 0 2 0 -8 -4 D 2 -2 8 0 4 E 8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=23 D=22 A=21 E=8 so E is eliminated. Round 2 votes counts: C=28 B=26 A=24 D=22 so D is eliminated. Round 3 votes counts: A=37 B=35 C=28 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:206 D:206 E:199 C:195 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 -2 -8 B 2 0 -2 2 10 C 0 2 0 -8 -4 D 2 -2 8 0 4 E 8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -2 -8 B 2 0 -2 2 10 C 0 2 0 -8 -4 D 2 -2 8 0 4 E 8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -2 -8 B 2 0 -2 2 10 C 0 2 0 -8 -4 D 2 -2 8 0 4 E 8 -10 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8111: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) E C D B A (9) C A E B D (7) B D A E C (7) E D B C A (6) A C B D E (6) D B A E C (5) C E A B D (5) D E B A C (4) D B E A C (4) C E A D B (4) C A B D E (4) A B D C E (4) E D C B A (3) C E D A B (3) B A D C E (3) E C D A B (2) D E B C A (2) C A B E D (2) E C A B D (1) E B A D C (1) C E D B A (1) C D E A B (1) C D A B E (1) C A D E B (1) C A D B E (1) B A D E C (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 -14 -10 B 10 0 -2 -12 -18 C 6 2 0 0 -10 D 14 12 0 0 -8 E 10 18 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 -14 -10 B 10 0 -2 -12 -18 C 6 2 0 0 -10 D 14 12 0 0 -8 E 10 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 D=15 A=13 B=11 so B is eliminated. Round 2 votes counts: E=31 C=30 D=22 A=17 so A is eliminated. Round 3 votes counts: C=38 E=31 D=31 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:223 D:209 C:199 B:189 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 -14 -10 B 10 0 -2 -12 -18 C 6 2 0 0 -10 D 14 12 0 0 -8 E 10 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -14 -10 B 10 0 -2 -12 -18 C 6 2 0 0 -10 D 14 12 0 0 -8 E 10 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -14 -10 B 10 0 -2 -12 -18 C 6 2 0 0 -10 D 14 12 0 0 -8 E 10 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8112: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) E B D C A (9) C A D E B (9) B E D A C (9) E B C D A (7) B D E A C (6) D A C B E (5) C A E D B (5) A D C B E (5) E B D A C (3) E B C A D (3) D B E A C (3) D A B C E (3) C A B E D (3) E D B A C (2) D B A E C (2) D B A C E (2) D A B E C (2) C A E B D (2) A C D E B (2) E D B C A (1) E C B A D (1) D E B A C (1) C E B A D (1) C E A B D (1) C A D B E (1) B E D C A (1) B E C A D (1) Total count = 100 A B C D E A 0 -4 10 -12 2 B 4 0 10 -6 6 C -10 -10 0 -8 -2 D 12 6 8 0 2 E -2 -6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -12 2 B 4 0 10 -6 6 C -10 -10 0 -8 -2 D 12 6 8 0 2 E -2 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=22 D=18 B=17 A=17 so B is eliminated. Round 2 votes counts: E=37 D=24 C=22 A=17 so A is eliminated. Round 3 votes counts: E=37 C=34 D=29 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:214 B:207 A:198 E:196 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 10 -12 2 B 4 0 10 -6 6 C -10 -10 0 -8 -2 D 12 6 8 0 2 E -2 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -12 2 B 4 0 10 -6 6 C -10 -10 0 -8 -2 D 12 6 8 0 2 E -2 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -12 2 B 4 0 10 -6 6 C -10 -10 0 -8 -2 D 12 6 8 0 2 E -2 -6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8113: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (14) C B E A D (9) E B C A D (7) D A C B E (7) C B E D A (6) A D E C B (6) D A B E C (5) B C E D A (5) B E C A D (4) E A B D C (3) E A B C D (3) C E B A D (3) A E D B C (3) E B A D C (2) E B A C D (2) D C B A E (2) D A E B C (2) C B D E A (2) B D E C A (2) E C A B D (1) D C A B E (1) D B C E A (1) D B C A E (1) D A B C E (1) C D B A E (1) C D A B E (1) C B D A E (1) C A E D B (1) C A D E B (1) B E D C A (1) B E C D A (1) B C D E A (1) Total count = 100 A B C D E A 0 -2 -4 18 -6 B 2 0 16 6 4 C 4 -16 0 -2 -12 D -18 -6 2 0 -2 E 6 -4 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998001 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 18 -6 B 2 0 16 6 4 C 4 -16 0 -2 -12 D -18 -6 2 0 -2 E 6 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 D=20 E=18 B=14 so B is eliminated. Round 2 votes counts: C=31 E=24 A=23 D=22 so D is eliminated. Round 3 votes counts: A=38 C=36 E=26 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:214 E:208 A:203 D:188 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 18 -6 B 2 0 16 6 4 C 4 -16 0 -2 -12 D -18 -6 2 0 -2 E 6 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 18 -6 B 2 0 16 6 4 C 4 -16 0 -2 -12 D -18 -6 2 0 -2 E 6 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 18 -6 B 2 0 16 6 4 C 4 -16 0 -2 -12 D -18 -6 2 0 -2 E 6 -4 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999029 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8114: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) C D A B E (8) B C D A E (8) A D E C B (8) E A D B C (6) B C E D A (6) E B A D C (5) E A D C B (5) A D C B E (5) C B D A E (4) E C D A B (3) E C B D A (3) D A C E B (3) D A C B E (3) B C A D E (3) A D E B C (3) B E C D A (2) A D C E B (2) E B C A D (1) C E D A B (1) C E B D A (1) C D A E B (1) C B E D A (1) C A D B E (1) B E C A D (1) B E A D C (1) B C E A D (1) B A E D C (1) B A D E C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 -10 -8 6 B -2 0 2 -2 -4 C 10 -2 0 10 -2 D 8 2 -10 0 4 E -6 4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765156 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.444444 D: 0.444444 E: 1.000000 A B C D E A 0 2 -10 -8 6 B -2 0 2 -2 -4 C 10 -2 0 10 -2 D 8 2 -10 0 4 E -6 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.444444 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=24 A=20 C=17 D=6 so D is eliminated. Round 2 votes counts: E=33 A=26 B=24 C=17 so C is eliminated. Round 3 votes counts: A=36 E=35 B=29 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:208 D:202 E:198 B:197 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 -8 6 B -2 0 2 -2 -4 C 10 -2 0 10 -2 D 8 2 -10 0 4 E -6 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.444444 D: 0.444444 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 -8 6 B -2 0 2 -2 -4 C 10 -2 0 10 -2 D 8 2 -10 0 4 E -6 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.444444 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 -8 6 B -2 0 2 -2 -4 C 10 -2 0 10 -2 D 8 2 -10 0 4 E -6 4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765428 Cumulative probabilities = A: 0.111111 B: 0.111111 C: 0.444444 D: 0.444444 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8115: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (14) D A E C B (8) C B E D A (7) C B E A D (7) A D E C B (7) B C E A D (6) E A D B C (5) C D B A E (5) B E C A D (5) B C E D A (5) B C D A E (4) E A D C B (3) D A C E B (3) E C B A D (2) E B C A D (2) E B A D C (2) D A E B C (2) C B D E A (2) E B A C D (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B A D (1) C D A E B (1) C D A B E (1) B E C D A (1) B D A C E (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 -30 -26 -14 2 B 30 0 -24 20 22 C 26 24 0 28 18 D 14 -20 -28 0 4 E -2 -22 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -26 -14 2 B 30 0 -24 20 22 C 26 24 0 28 18 D 14 -20 -28 0 4 E -2 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=22 D=16 E=15 A=9 so A is eliminated. Round 2 votes counts: C=38 D=24 B=22 E=16 so E is eliminated. Round 3 votes counts: C=40 D=33 B=27 so B is eliminated. Round 4 votes counts: C=64 D=36 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:248 B:224 D:185 E:177 A:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -30 -26 -14 2 B 30 0 -24 20 22 C 26 24 0 28 18 D 14 -20 -28 0 4 E -2 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -26 -14 2 B 30 0 -24 20 22 C 26 24 0 28 18 D 14 -20 -28 0 4 E -2 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -26 -14 2 B 30 0 -24 20 22 C 26 24 0 28 18 D 14 -20 -28 0 4 E -2 -22 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8116: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) A D B C E (6) E D A C B (5) E D A B C (5) C E B A D (5) B C A D E (5) D A B C E (4) C B A D E (4) E C A B D (3) E A C D B (3) D B E C A (3) D B A C E (3) C E A B D (3) C B E D A (3) B D A C E (3) E C A D B (2) E A D C B (2) D E A B C (2) D B A E C (2) C E B D A (2) C B E A D (2) A E D C B (2) A D E B C (2) A D B E C (2) A C B D E (2) A B D C E (2) E D C A B (1) E D B C A (1) E C D B A (1) E C D A B (1) E C B D A (1) E A D B C (1) D B E A C (1) C B A E D (1) C A B D E (1) B D C E A (1) B C E D A (1) B C D E A (1) B A D C E (1) B A C D E (1) A E D B C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 16 6 0 B -16 0 10 -12 14 C -16 -10 0 -12 6 D -6 12 12 0 6 E 0 -14 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.642303 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.357697 Sum of squares = 0.540500332608 Cumulative probabilities = A: 0.642303 B: 0.642303 C: 0.642303 D: 0.642303 E: 1.000000 A B C D E A 0 16 16 6 0 B -16 0 10 -12 14 C -16 -10 0 -12 6 D -6 12 12 0 6 E 0 -14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500451 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499549 Sum of squares = 0.500000407681 Cumulative probabilities = A: 0.500451 B: 0.500451 C: 0.500451 D: 0.500451 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=21 C=21 A=19 B=13 so B is eliminated. Round 2 votes counts: C=28 E=26 D=25 A=21 so A is eliminated. Round 3 votes counts: D=38 C=33 E=29 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:219 D:212 B:198 E:187 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 6 0 B -16 0 10 -12 14 C -16 -10 0 -12 6 D -6 12 12 0 6 E 0 -14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500451 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499549 Sum of squares = 0.500000407681 Cumulative probabilities = A: 0.500451 B: 0.500451 C: 0.500451 D: 0.500451 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 6 0 B -16 0 10 -12 14 C -16 -10 0 -12 6 D -6 12 12 0 6 E 0 -14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500451 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499549 Sum of squares = 0.500000407681 Cumulative probabilities = A: 0.500451 B: 0.500451 C: 0.500451 D: 0.500451 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 6 0 B -16 0 10 -12 14 C -16 -10 0 -12 6 D -6 12 12 0 6 E 0 -14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500451 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499549 Sum of squares = 0.500000407681 Cumulative probabilities = A: 0.500451 B: 0.500451 C: 0.500451 D: 0.500451 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8117: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (15) E D B C A (11) D E B A C (7) C A B D E (7) B D E C A (5) A B C D E (5) C E D B A (4) C A B E D (4) A B D E C (4) E D C B A (3) E D B A C (3) E D A B C (3) D B E A C (3) C B E D A (3) E C D B A (2) C E A B D (2) B C A D E (2) A E D B C (2) A C B E D (2) E C B D A (1) D E B C A (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) B E D C A (1) B D C E A (1) B C D A E (1) B A D E C (1) A D E B C (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -4 2 -2 B 2 0 4 16 14 C 4 -4 0 6 4 D -2 -16 -6 0 8 E 2 -14 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 2 -2 B 2 0 4 16 14 C 4 -4 0 6 4 D -2 -16 -6 0 8 E 2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=24 E=23 D=11 B=11 so D is eliminated. Round 2 votes counts: E=31 A=31 C=24 B=14 so B is eliminated. Round 3 votes counts: E=40 A=32 C=28 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:218 C:205 A:197 D:192 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 2 -2 B 2 0 4 16 14 C 4 -4 0 6 4 D -2 -16 -6 0 8 E 2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 2 -2 B 2 0 4 16 14 C 4 -4 0 6 4 D -2 -16 -6 0 8 E 2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 2 -2 B 2 0 4 16 14 C 4 -4 0 6 4 D -2 -16 -6 0 8 E 2 -14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8118: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (13) A D B C E (13) C E B D A (8) E B C D A (7) E B D C A (6) E B D A C (5) A D B E C (5) C A E D B (4) A D C B E (4) C E A B D (3) C D B A E (3) A C D B E (3) E A B D C (2) D A B C E (2) C A D B E (2) A E C D B (2) E C B A D (1) E C A B D (1) E B A C D (1) E A C D B (1) E A C B D (1) D B C A E (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A D B (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C D A (1) B D E C A (1) A E D B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -12 -4 -12 B 4 0 -6 6 -20 C 12 6 0 14 0 D 4 -6 -14 0 -20 E 12 20 0 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.632530 D: 0.000000 E: 0.367470 Sum of squares = 0.535128645382 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.632530 D: 0.632530 E: 1.000000 A B C D E A 0 -4 -12 -4 -12 B 4 0 -6 6 -20 C 12 6 0 14 0 D 4 -6 -14 0 -20 E 12 20 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=30 C=25 D=5 B=2 so B is eliminated. Round 2 votes counts: E=39 A=30 C=25 D=6 so D is eliminated. Round 3 votes counts: E=40 A=34 C=26 so C is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:216 B:192 A:184 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -4 -12 B 4 0 -6 6 -20 C 12 6 0 14 0 D 4 -6 -14 0 -20 E 12 20 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -4 -12 B 4 0 -6 6 -20 C 12 6 0 14 0 D 4 -6 -14 0 -20 E 12 20 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -4 -12 B 4 0 -6 6 -20 C 12 6 0 14 0 D 4 -6 -14 0 -20 E 12 20 0 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8119: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) C A E D B (10) A C E B D (10) E B D C A (6) C A D B E (6) A C D B E (6) D B E C A (4) E C D B A (3) E A B C D (3) D B A C E (3) B E D A C (3) B D E C A (3) E C A B D (2) E B D A C (2) E A C B D (2) D C B A E (2) D B C E A (2) D B C A E (2) C A D E B (2) A C E D B (2) A C B D E (2) E D B C A (1) E C B D A (1) E C A D B (1) E B C D A (1) D C B E A (1) D C A B E (1) D A C B E (1) C E A B D (1) C D A B E (1) B D A C E (1) A E B C D (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 0 0 4 B -6 0 -8 2 4 C 0 8 0 8 8 D 0 -2 -8 0 0 E -4 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.494941 B: 0.000000 C: 0.505059 D: 0.000000 E: 0.000000 Sum of squares = 0.500051191697 Cumulative probabilities = A: 0.494941 B: 0.494941 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 4 B -6 0 -8 2 4 C 0 8 0 8 8 D 0 -2 -8 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=22 C=20 B=19 D=16 so D is eliminated. Round 2 votes counts: B=30 C=24 A=24 E=22 so E is eliminated. Round 3 votes counts: B=40 C=31 A=29 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:205 B:196 D:195 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 0 4 B -6 0 -8 2 4 C 0 8 0 8 8 D 0 -2 -8 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 4 B -6 0 -8 2 4 C 0 8 0 8 8 D 0 -2 -8 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 4 B -6 0 -8 2 4 C 0 8 0 8 8 D 0 -2 -8 0 0 E -4 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999974 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8120: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) D C A E B (6) D B C E A (6) A E C D B (6) B E D A C (5) B D C E A (5) B C D E A (5) A E B C D (5) C D B A E (4) C B D A E (4) A E C B D (4) C D A E B (3) C B A E D (3) C A B E D (3) B D E A C (3) E A D B C (2) E A B C D (2) D C B A E (2) C A E D B (2) C A D E B (2) B D E C A (2) A E D C B (2) A C E B D (2) E B A D C (1) E B A C D (1) D E A C B (1) D C B E A (1) D B E C A (1) D B E A C (1) C D A B E (1) C A E B D (1) B E A D C (1) B E A C D (1) B C D A E (1) A E B D C (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 -4 -2 8 B -6 0 2 16 -2 C 4 -2 0 2 4 D 2 -16 -2 0 0 E -8 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888889 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -2 8 B -6 0 2 16 -2 C 4 -2 0 2 4 D 2 -16 -2 0 0 E -8 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888957 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=23 B=23 A=22 D=18 E=14 so E is eliminated. Round 2 votes counts: A=34 B=25 C=23 D=18 so D is eliminated. Round 3 votes counts: A=35 B=33 C=32 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:205 A:204 C:204 E:195 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -2 8 B -6 0 2 16 -2 C 4 -2 0 2 4 D 2 -16 -2 0 0 E -8 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888957 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -2 8 B -6 0 2 16 -2 C 4 -2 0 2 4 D 2 -16 -2 0 0 E -8 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888957 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -2 8 B -6 0 2 16 -2 C 4 -2 0 2 4 D 2 -16 -2 0 0 E -8 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.333333 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888957 Cumulative probabilities = A: 0.166667 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8121: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) B E A C D (8) E B C D A (7) E B C A D (7) D A C B E (7) A D C B E (7) D C A E B (6) B E A D C (6) B A E D C (6) C E B A D (3) C D A E B (3) A B E D C (3) A B D E C (3) A B D C E (3) E C B D A (2) D A B E C (2) C E B D A (2) C D E A B (2) A D B C E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B A D (1) E B D A C (1) E B A C D (1) D C E A B (1) D A B C E (1) C B E A D (1) B E D A C (1) B E C A D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 6 8 -10 B 16 0 8 16 6 C -6 -8 0 -2 -6 D -8 -16 2 0 -24 E 10 -6 6 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 8 -10 B 16 0 8 16 6 C -6 -8 0 -2 -6 D -8 -16 2 0 -24 E 10 -6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=22 B=22 A=20 C=19 D=17 so D is eliminated. Round 2 votes counts: A=30 C=26 E=22 B=22 so E is eliminated. Round 3 votes counts: B=39 C=31 A=30 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:217 A:194 C:189 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 8 -10 B 16 0 8 16 6 C -6 -8 0 -2 -6 D -8 -16 2 0 -24 E 10 -6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 8 -10 B 16 0 8 16 6 C -6 -8 0 -2 -6 D -8 -16 2 0 -24 E 10 -6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 8 -10 B 16 0 8 16 6 C -6 -8 0 -2 -6 D -8 -16 2 0 -24 E 10 -6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8122: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) C A D B E (7) A C D B E (7) E B D A C (6) D B E C A (6) A C E D B (6) E B D C A (5) A C D E B (4) E A C B D (3) D E B A C (3) D B E A C (3) B E D C A (3) B D E C A (3) E D B A C (2) E B C D A (2) E A B C D (2) C E B A D (2) C B E D A (2) C A B E D (2) A D E C B (2) A C E B D (2) E D A B C (1) E C B D A (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A B C (1) D C A B E (1) D B C A E (1) D B A C E (1) D A B E C (1) C E A B D (1) C B E A D (1) C B D E A (1) C B D A E (1) C A B D E (1) A E C D B (1) Total count = 100 A B C D E A 0 10 -4 12 -4 B -10 0 -12 4 -18 C 4 12 0 16 2 D -12 -4 -16 0 -14 E 4 18 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999177 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 12 -4 B -10 0 -12 4 -18 C 4 12 0 16 2 D -12 -4 -16 0 -14 E 4 18 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=26 A=22 D=17 B=6 so B is eliminated. Round 2 votes counts: E=29 C=29 A=22 D=20 so D is eliminated. Round 3 votes counts: E=45 C=31 A=24 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:217 A:207 B:182 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 12 -4 B -10 0 -12 4 -18 C 4 12 0 16 2 D -12 -4 -16 0 -14 E 4 18 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 12 -4 B -10 0 -12 4 -18 C 4 12 0 16 2 D -12 -4 -16 0 -14 E 4 18 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 12 -4 B -10 0 -12 4 -18 C 4 12 0 16 2 D -12 -4 -16 0 -14 E 4 18 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8123: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) C B D A E (8) E A D B C (7) B A D E C (7) E C A D B (6) D A E B C (6) C E D A B (6) E D A B C (5) C E B A D (5) C E A D B (4) C B E A D (4) B D A C E (4) E C D A B (3) C B A E D (3) C D B E A (2) B C D A E (2) A E D B C (2) A B D E C (2) E D A C B (1) E C A B D (1) E A D C B (1) E A C B D (1) D E A B C (1) D A B E C (1) C E B D A (1) C E A B D (1) C B D E A (1) C B A D E (1) B C A D E (1) B A E D C (1) B A E C D (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 2 0 0 B 2 0 2 8 -2 C -2 -2 0 4 -12 D 0 -8 -4 0 -6 E 0 2 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.376669 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.623331 Sum of squares = 0.530421081735 Cumulative probabilities = A: 0.376669 B: 0.376669 C: 0.376669 D: 0.376669 E: 1.000000 A B C D E A 0 -2 2 0 0 B 2 0 2 8 -2 C -2 -2 0 4 -12 D 0 -8 -4 0 -6 E 0 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499707 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500293 Sum of squares = 0.500000172195 Cumulative probabilities = A: 0.499707 B: 0.499707 C: 0.499707 D: 0.499707 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=26 E=25 D=8 A=5 so A is eliminated. Round 2 votes counts: C=36 B=28 E=27 D=9 so D is eliminated. Round 3 votes counts: C=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:210 B:205 A:200 C:194 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 0 0 B 2 0 2 8 -2 C -2 -2 0 4 -12 D 0 -8 -4 0 -6 E 0 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499707 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500293 Sum of squares = 0.500000172195 Cumulative probabilities = A: 0.499707 B: 0.499707 C: 0.499707 D: 0.499707 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 0 B 2 0 2 8 -2 C -2 -2 0 4 -12 D 0 -8 -4 0 -6 E 0 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499707 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500293 Sum of squares = 0.500000172195 Cumulative probabilities = A: 0.499707 B: 0.499707 C: 0.499707 D: 0.499707 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 0 B 2 0 2 8 -2 C -2 -2 0 4 -12 D 0 -8 -4 0 -6 E 0 2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499707 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500293 Sum of squares = 0.500000172195 Cumulative probabilities = A: 0.499707 B: 0.499707 C: 0.499707 D: 0.499707 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8124: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (7) D E A C B (5) B C A E D (5) D E A B C (4) D B E A C (4) D B A E C (4) B D A C E (4) E D C A B (3) E D A C B (3) D E B A C (3) C B E A D (3) C B A E D (3) C A E B D (3) B C D A E (3) A C B D E (3) E C B A D (2) E C A D B (2) E A D C B (2) D B A C E (2) D A E C B (2) D A B E C (2) C E A B D (2) C A B E D (2) B E C D A (2) B D E C A (2) B C E D A (2) A D C B E (2) E D C B A (1) E D B A C (1) E D A B C (1) E C D A B (1) E C A B D (1) E B D C A (1) E B C D A (1) D A B C E (1) C E B A D (1) B E D C A (1) B E C A D (1) B D C A E (1) B C E A D (1) B A C D E (1) A D E C B (1) A C E B D (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -2 -8 0 B 12 0 10 10 18 C 2 -10 0 -2 0 D 8 -10 2 0 6 E 0 -18 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -8 0 B 12 0 10 10 18 C 2 -10 0 -2 0 D 8 -10 2 0 6 E 0 -18 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=27 E=19 C=14 A=10 so A is eliminated. Round 2 votes counts: B=31 D=30 C=20 E=19 so E is eliminated. Round 3 votes counts: D=41 B=33 C=26 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:203 C:195 A:189 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 -8 0 B 12 0 10 10 18 C 2 -10 0 -2 0 D 8 -10 2 0 6 E 0 -18 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -8 0 B 12 0 10 10 18 C 2 -10 0 -2 0 D 8 -10 2 0 6 E 0 -18 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -8 0 B 12 0 10 10 18 C 2 -10 0 -2 0 D 8 -10 2 0 6 E 0 -18 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8125: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E D B C A (7) D E C B A (7) D E A C B (7) E D C B A (6) A B C E D (6) B C E A D (5) B C A E D (5) D A E C B (4) E D A B C (3) C B E D A (3) C B A D E (3) B A C E D (3) A D E B C (3) E C B D A (2) D A E B C (2) C B A E D (2) A E B D C (2) A D C B E (2) A D B E C (2) A C D B E (2) E D B A C (1) E C D B A (1) E B D C A (1) E B C D A (1) D E C A B (1) D E A B C (1) D A C E B (1) C B D A E (1) C A B D E (1) B E A C D (1) B C E D A (1) A E D B C (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 6 0 4 B 0 0 10 -4 -4 C -6 -10 0 -4 -6 D 0 4 4 0 -2 E -4 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.489075 B: 0.000000 C: 0.000000 D: 0.510925 E: 0.000000 Sum of squares = 0.500238728902 Cumulative probabilities = A: 0.489075 B: 0.489075 C: 0.489075 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 0 4 B 0 0 10 -4 -4 C -6 -10 0 -4 -6 D 0 4 4 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=23 E=22 B=15 C=10 so C is eliminated. Round 2 votes counts: A=31 B=24 D=23 E=22 so E is eliminated. Round 3 votes counts: D=41 A=31 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:205 E:204 D:203 B:201 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 0 4 B 0 0 10 -4 -4 C -6 -10 0 -4 -6 D 0 4 4 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 4 B 0 0 10 -4 -4 C -6 -10 0 -4 -6 D 0 4 4 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 4 B 0 0 10 -4 -4 C -6 -10 0 -4 -6 D 0 4 4 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8126: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) B C E A D (9) D C B A E (7) A E B C D (7) E A B C D (6) A E D B C (6) D A E C B (5) B C A E D (5) A E B D C (5) D E A C B (3) D C B E A (3) B A E C D (3) E D A C B (2) E C B A D (2) E A C B D (2) D C A B E (2) D A E B C (2) C B E A D (2) C B D A E (2) B E A C D (2) B C A D E (2) E B A C D (1) E A D C B (1) E A B D C (1) D C E B A (1) D C E A B (1) D C A E B (1) D B C A E (1) D A C B E (1) C D E B A (1) C D B E A (1) B E C A D (1) B C D A E (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -2 14 4 B 6 0 8 22 4 C 2 -8 0 12 -2 D -14 -22 -12 0 -12 E -4 -4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999221 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 14 4 B 6 0 8 22 4 C 2 -8 0 12 -2 D -14 -22 -12 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=23 A=20 E=15 C=15 so E is eliminated. Round 2 votes counts: A=30 D=29 B=24 C=17 so C is eliminated. Round 3 votes counts: B=39 D=31 A=30 so A is eliminated. Round 4 votes counts: B=61 D=39 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:205 E:203 C:202 D:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 14 4 B 6 0 8 22 4 C 2 -8 0 12 -2 D -14 -22 -12 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 14 4 B 6 0 8 22 4 C 2 -8 0 12 -2 D -14 -22 -12 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 14 4 B 6 0 8 22 4 C 2 -8 0 12 -2 D -14 -22 -12 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998937 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8127: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) E B A D C (7) D C B A E (6) E A B D C (5) E A B C D (5) D B C A E (4) E C D B A (3) E C A B D (3) E B D A C (3) E A C B D (3) D C B E A (3) C E A D B (3) C D E A B (3) C D B A E (3) C A D B E (3) B E A D C (3) A E B C D (3) A C D B E (3) E C A D B (2) D B C E A (2) B D E A C (2) A E C B D (2) A C E B D (2) E D B C A (1) E C D A B (1) E B A C D (1) D C E B A (1) C E D B A (1) C E D A B (1) C D E B A (1) C A D E B (1) B D A E C (1) B A D E C (1) A E B D C (1) A D B C E (1) A C B D E (1) A B E D C (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -2 10 -8 B -14 0 -10 -4 -6 C 2 10 0 12 0 D -10 4 -12 0 -4 E 8 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.414098 D: 0.000000 E: 0.585902 Sum of squares = 0.514758301189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.414098 D: 0.414098 E: 1.000000 A B C D E A 0 14 -2 10 -8 B -14 0 -10 -4 -6 C 2 10 0 12 0 D -10 4 -12 0 -4 E 8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=25 A=18 D=16 B=7 so B is eliminated. Round 2 votes counts: E=37 C=25 D=19 A=19 so D is eliminated. Round 3 votes counts: C=41 E=39 A=20 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:209 A:207 D:189 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 10 -8 B -14 0 -10 -4 -6 C 2 10 0 12 0 D -10 4 -12 0 -4 E 8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 10 -8 B -14 0 -10 -4 -6 C 2 10 0 12 0 D -10 4 -12 0 -4 E 8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 10 -8 B -14 0 -10 -4 -6 C 2 10 0 12 0 D -10 4 -12 0 -4 E 8 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8128: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) D C A E B (8) C D A B E (7) B C D E A (7) A E D C B (7) E A D C B (5) E A D B C (5) B E A C D (5) A C D E B (5) B E C D A (4) E A B D C (3) B C E D A (3) A E B D C (3) A E B C D (3) E B D A C (2) E B A D C (2) D C B E A (2) C B D E A (2) E D A C B (1) E B A C D (1) D C B A E (1) C D B E A (1) C D A E B (1) C B D A E (1) B E D C A (1) B E A D C (1) B C D A E (1) B A C E D (1) B A C D E (1) A E D B C (1) A D E C B (1) A D C E B (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 -4 10 B -8 0 -10 -14 2 C -2 10 0 12 10 D 4 14 -12 0 4 E -10 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839527 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -4 10 B -8 0 -10 -14 2 C -2 10 0 12 10 D 4 14 -12 0 4 E -10 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.000000 Sum of squares = 0.506172838832 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 C=22 E=19 D=11 so D is eliminated. Round 2 votes counts: C=33 B=24 A=24 E=19 so E is eliminated. Round 3 votes counts: A=38 C=33 B=29 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:208 D:205 E:187 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 2 -4 10 B -8 0 -10 -14 2 C -2 10 0 12 10 D 4 14 -12 0 4 E -10 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.000000 Sum of squares = 0.506172838832 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -4 10 B -8 0 -10 -14 2 C -2 10 0 12 10 D 4 14 -12 0 4 E -10 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.000000 Sum of squares = 0.506172838832 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -4 10 B -8 0 -10 -14 2 C -2 10 0 12 10 D 4 14 -12 0 4 E -10 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.222222 D: 0.111111 E: 0.000000 Sum of squares = 0.506172838832 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8129: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) B C A E D (6) E D A B C (5) E B A C D (5) D A E C B (5) D C A B E (4) C A D B E (4) E D B A C (3) E A D C B (3) E A B C D (3) D E B C A (3) D C B A E (3) D B C A E (3) C A B D E (3) B E C A D (3) E D A C B (2) E B C A D (2) E A D B C (2) D E A C B (2) D E A B C (2) B E C D A (2) B D E C A (2) B C A D E (2) A D C E B (2) A C D B E (2) E B D A C (1) E A C B D (1) D C E B A (1) D B E C A (1) D B C E A (1) C D B A E (1) C B A E D (1) C B A D E (1) C A B E D (1) B E D C A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) A E C D B (1) A E C B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 6 0 -8 4 B -6 0 4 -18 -6 C 0 -4 0 -10 0 D 8 18 10 0 8 E -4 6 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 -8 4 B -6 0 4 -18 -6 C 0 -4 0 -10 0 D 8 18 10 0 8 E -4 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=27 B=21 C=11 A=8 so A is eliminated. Round 2 votes counts: D=35 E=29 B=21 C=15 so C is eliminated. Round 3 votes counts: D=43 E=29 B=28 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:201 E:197 C:193 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 0 -8 4 B -6 0 4 -18 -6 C 0 -4 0 -10 0 D 8 18 10 0 8 E -4 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -8 4 B -6 0 4 -18 -6 C 0 -4 0 -10 0 D 8 18 10 0 8 E -4 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -8 4 B -6 0 4 -18 -6 C 0 -4 0 -10 0 D 8 18 10 0 8 E -4 6 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8130: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (10) B A C E D (9) C E D B A (8) B A D E C (8) A B D E C (7) D E C A B (6) E D C A B (4) D E A C B (3) B C A E D (3) A B C E D (3) D E B C A (2) D E B A C (2) C E A D B (2) B D E A C (2) B C E D A (2) A D E B C (2) A C D E B (2) A C B E D (2) A B D C E (2) A B C D E (2) E C D B A (1) D E C B A (1) D E A B C (1) D B E C A (1) D B E A C (1) D A E C B (1) C E B D A (1) C E B A D (1) C E A B D (1) C B E D A (1) C A E B D (1) C A B E D (1) B E D C A (1) B E C D A (1) B D A E C (1) B C E A D (1) B A C D E (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 4 2 0 -6 B -4 0 6 2 0 C -2 -6 0 6 6 D 0 -2 -6 0 -6 E 6 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.557731 C: 0.000000 D: 0.000000 E: 0.442269 Sum of squares = 0.506665643256 Cumulative probabilities = A: 0.000000 B: 0.557731 C: 0.557731 D: 0.557731 E: 1.000000 A B C D E A 0 4 2 0 -6 B -4 0 6 2 0 C -2 -6 0 6 6 D 0 -2 -6 0 -6 E 6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.000000 E: 0.499681 Sum of squares = 0.500000202989 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 0.500319 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=26 A=22 D=18 E=5 so E is eliminated. Round 2 votes counts: B=29 C=27 D=22 A=22 so D is eliminated. Round 3 votes counts: C=38 B=35 A=27 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:203 B:202 C:202 A:200 D:193 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 2 0 -6 B -4 0 6 2 0 C -2 -6 0 6 6 D 0 -2 -6 0 -6 E 6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.000000 E: 0.499681 Sum of squares = 0.500000202989 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 0.500319 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 0 -6 B -4 0 6 2 0 C -2 -6 0 6 6 D 0 -2 -6 0 -6 E 6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.000000 E: 0.499681 Sum of squares = 0.500000202989 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 0.500319 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 0 -6 B -4 0 6 2 0 C -2 -6 0 6 6 D 0 -2 -6 0 -6 E 6 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.000000 E: 0.499681 Sum of squares = 0.500000202989 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 0.500319 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8131: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (8) E D A B C (7) D E B C A (7) C B A E D (7) D A E C B (6) A C B D E (6) C B A D E (5) E D B C A (4) B C E A D (4) A D E C B (4) A C B E D (4) E B C A D (3) D E A B C (3) A D C B E (3) E B C D A (2) D C B A E (2) D A C B E (2) B C E D A (2) B C D E A (2) A E B C D (2) E B D C A (1) E A B C D (1) D C B E A (1) D A C E B (1) B E C A D (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 2 0 4 B -2 0 -6 -2 -6 C -2 6 0 -4 -4 D 0 2 4 0 4 E -4 6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.546848 B: 0.000000 C: 0.000000 D: 0.453152 E: 0.000000 Sum of squares = 0.504389511977 Cumulative probabilities = A: 0.546848 B: 0.546848 C: 0.546848 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 4 B -2 0 -6 -2 -6 C -2 6 0 -4 -4 D 0 2 4 0 4 E -4 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999558 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=21 E=18 B=17 C=12 so C is eliminated. Round 2 votes counts: D=32 B=29 A=21 E=18 so E is eliminated. Round 3 votes counts: D=43 B=35 A=22 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:205 A:204 E:201 C:198 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 4 B -2 0 -6 -2 -6 C -2 6 0 -4 -4 D 0 2 4 0 4 E -4 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999558 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 4 B -2 0 -6 -2 -6 C -2 6 0 -4 -4 D 0 2 4 0 4 E -4 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999558 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 4 B -2 0 -6 -2 -6 C -2 6 0 -4 -4 D 0 2 4 0 4 E -4 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999558 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8132: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (13) A D C B E (13) D A C B E (9) E B D A C (6) E B C A D (6) B E D A C (6) C A D B E (4) E C B A D (3) E B D C A (3) D A B E C (3) D A B C E (3) A D C E B (3) E C A B D (2) D B A E C (2) C E A B D (2) B E D C A (2) B D A C E (2) E A D B C (1) D B A C E (1) D A E B C (1) C E B A D (1) C E A D B (1) C B E A D (1) C B A D E (1) C A E D B (1) C A D E B (1) C A B E D (1) B E C D A (1) B D E A C (1) B D C A E (1) B D A E C (1) B C D A E (1) A E C D B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 10 -12 2 B 4 0 8 8 6 C -10 -8 0 -18 -6 D 12 -8 18 0 -2 E -2 -6 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -12 2 B 4 0 8 8 6 C -10 -8 0 -18 -6 D 12 -8 18 0 -2 E -2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=19 A=19 B=15 C=13 so C is eliminated. Round 2 votes counts: E=38 A=26 D=19 B=17 so B is eliminated. Round 3 votes counts: E=48 A=27 D=25 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:213 D:210 E:200 A:198 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 -12 2 B 4 0 8 8 6 C -10 -8 0 -18 -6 D 12 -8 18 0 -2 E -2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -12 2 B 4 0 8 8 6 C -10 -8 0 -18 -6 D 12 -8 18 0 -2 E -2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -12 2 B 4 0 8 8 6 C -10 -8 0 -18 -6 D 12 -8 18 0 -2 E -2 -6 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8133: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (13) C B D A E (11) C D B A E (7) E A B D C (6) A E D C B (5) E B D A C (4) B D C A E (4) E B A D C (3) C E B A D (3) C B D E A (3) C A E D B (3) C A D E B (3) B D E C A (3) B D C E A (3) B D A E C (3) A E D B C (3) A E C D B (3) E C A B D (2) D B A E C (2) C D A B E (2) B E D A C (2) B D E A C (2) E A D C B (1) E A C D B (1) E A C B D (1) C E A B D (1) C B E D A (1) C A D B E (1) B E D C A (1) A D E B C (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 4 4 0 B 4 0 0 6 -8 C -4 0 0 -14 -12 D -4 -6 14 0 -8 E 0 8 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.514441 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.485559 Sum of squares = 0.500417056513 Cumulative probabilities = A: 0.514441 B: 0.514441 C: 0.514441 D: 0.514441 E: 1.000000 A B C D E A 0 -4 4 4 0 B 4 0 0 6 -8 C -4 0 0 -14 -12 D -4 -6 14 0 -8 E 0 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=31 B=18 A=14 D=2 so D is eliminated. Round 2 votes counts: C=35 E=31 B=20 A=14 so A is eliminated. Round 3 votes counts: E=43 C=37 B=20 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:202 B:201 D:198 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 4 0 B 4 0 0 6 -8 C -4 0 0 -14 -12 D -4 -6 14 0 -8 E 0 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 0 B 4 0 0 6 -8 C -4 0 0 -14 -12 D -4 -6 14 0 -8 E 0 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 0 B 4 0 0 6 -8 C -4 0 0 -14 -12 D -4 -6 14 0 -8 E 0 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999875 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8134: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) A B D E C (6) A B C D E (6) C E D B A (5) E D B C A (4) E C D B A (4) C A B E D (4) B A D E C (4) E D C A B (3) D E B A C (3) D B E A C (3) C E B D A (3) C E A D B (3) C A E D B (3) B C E D A (3) A D B E C (3) A B D C E (3) D E A C B (2) D E A B C (2) D A B E C (2) C E D A B (2) B E D C A (2) B D E C A (2) B A C D E (2) A C E D B (2) A C D E B (2) A C B D E (2) E C D A B (1) D E B C A (1) D B A E C (1) D A E B C (1) C E B A D (1) C B E D A (1) C B A E D (1) C A E B D (1) B D E A C (1) B C E A D (1) B A D C E (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 0 -4 -6 -8 B 0 0 4 -10 -2 C 4 -4 0 -4 -6 D 6 10 4 0 -4 E 8 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -4 -6 -8 B 0 0 4 -10 -2 C 4 -4 0 -4 -6 D 6 10 4 0 -4 E 8 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=24 E=19 B=16 D=15 so D is eliminated. Round 2 votes counts: A=29 E=27 C=24 B=20 so B is eliminated. Round 3 votes counts: A=37 E=35 C=28 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:210 D:208 B:196 C:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 -6 -8 B 0 0 4 -10 -2 C 4 -4 0 -4 -6 D 6 10 4 0 -4 E 8 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -6 -8 B 0 0 4 -10 -2 C 4 -4 0 -4 -6 D 6 10 4 0 -4 E 8 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -6 -8 B 0 0 4 -10 -2 C 4 -4 0 -4 -6 D 6 10 4 0 -4 E 8 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8135: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) B A C D E (9) E D C A B (8) B A E D C (7) A B C D E (7) E D C B A (6) A B E D C (6) E A B D C (4) C D B E A (4) A B C E D (4) E A D B C (3) D E C B A (3) D C E B A (3) B E A D C (3) A E B D C (3) E D A B C (2) E B D A C (2) E A D C B (2) C D E A B (2) C D A B E (2) B E D C A (2) A C B D E (2) A B E C D (2) E B D C A (1) D E C A B (1) C D B A E (1) C B D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 14 6 -10 B 2 0 10 6 2 C -14 -10 0 -12 -10 D -6 -6 12 0 -10 E 10 -2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999397 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 6 -10 B 2 0 10 6 2 C -14 -10 0 -12 -10 D -6 -6 12 0 -10 E 10 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=25 B=21 C=19 D=7 so D is eliminated. Round 2 votes counts: E=32 A=25 C=22 B=21 so B is eliminated. Round 3 votes counts: A=41 E=37 C=22 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 B:210 A:204 D:195 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 14 6 -10 B 2 0 10 6 2 C -14 -10 0 -12 -10 D -6 -6 12 0 -10 E 10 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 6 -10 B 2 0 10 6 2 C -14 -10 0 -12 -10 D -6 -6 12 0 -10 E 10 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 6 -10 B 2 0 10 6 2 C -14 -10 0 -12 -10 D -6 -6 12 0 -10 E 10 -2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997799 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8136: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) A E C D B (8) E A D B C (6) E A B D C (6) D B C E A (6) C D B A E (6) A C E D B (6) D C B E A (5) C D B E A (5) C B D A E (5) B C D A E (5) A E C B D (5) B D E C A (3) A E B C D (3) E D C A B (2) E D B A C (2) C A D E B (2) B E A D C (2) B D E A C (2) A E B D C (2) A B E C D (2) E B A D C (1) D E B C A (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 -8 -8 -2 B 4 0 0 -8 4 C 8 0 0 6 10 D 8 8 -6 0 6 E 2 -4 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.251137 C: 0.748863 D: 0.000000 E: 0.000000 Sum of squares = 0.623865169771 Cumulative probabilities = A: 0.000000 B: 0.251137 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -8 -2 B 4 0 0 -8 4 C 8 0 0 6 10 D 8 8 -6 0 6 E 2 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204083335 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 B=21 E=17 D=12 so D is eliminated. Round 2 votes counts: C=28 B=27 A=27 E=18 so E is eliminated. Round 3 votes counts: A=39 B=31 C=30 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:212 D:208 B:200 E:191 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -8 -2 B 4 0 0 -8 4 C 8 0 0 6 10 D 8 8 -6 0 6 E 2 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204083335 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -8 -2 B 4 0 0 -8 4 C 8 0 0 6 10 D 8 8 -6 0 6 E 2 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204083335 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -8 -2 B 4 0 0 -8 4 C 8 0 0 6 10 D 8 8 -6 0 6 E 2 -4 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204083335 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8137: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) A B E C D (10) B E A D C (8) E A B D C (6) E B A D C (4) D E C B A (4) C D A E B (4) C D A B E (4) A E B D C (4) A C D E B (4) C D B E A (3) B E D C A (3) B A E C D (3) B A C E D (3) A C B D E (3) C B D A E (2) C A D B E (2) C A B D E (2) B E A C D (2) A C D B E (2) A B E D C (2) A B C E D (2) E D C B A (1) E D B C A (1) E D A C B (1) E B D C A (1) D C B E A (1) D C A E B (1) C D B A E (1) B E D A C (1) B C D A E (1) B C A E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 14 20 6 B 4 0 8 20 16 C -14 -8 0 2 -2 D -20 -20 -2 0 -6 E -6 -16 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 20 6 B 4 0 8 20 16 C -14 -8 0 2 -2 D -20 -20 -2 0 -6 E -6 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=23 C=18 D=17 E=14 so E is eliminated. Round 2 votes counts: A=34 B=28 D=20 C=18 so C is eliminated. Round 3 votes counts: A=38 D=32 B=30 so B is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:224 A:218 E:193 C:189 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 20 6 B 4 0 8 20 16 C -14 -8 0 2 -2 D -20 -20 -2 0 -6 E -6 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 20 6 B 4 0 8 20 16 C -14 -8 0 2 -2 D -20 -20 -2 0 -6 E -6 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 20 6 B 4 0 8 20 16 C -14 -8 0 2 -2 D -20 -20 -2 0 -6 E -6 -16 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8138: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) E A C D B (10) E C D A B (7) A B C D E (7) B A D C E (6) E B A D C (5) C A D B E (5) B D A C E (5) E B D C A (4) E A B C D (4) D C B A E (4) D B C A E (4) E D C B A (3) B E A D C (3) E C A D B (2) E B A C D (2) E A C B D (2) C D A B E (2) E D C A B (1) E D B C A (1) E C D B A (1) E B D A C (1) D C A B E (1) D B C E A (1) C D E A B (1) C D A E B (1) B D A E C (1) B A E D C (1) B A C D E (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 0 0 4 B 10 0 16 10 8 C 0 -16 0 -6 2 D 0 -10 6 0 4 E -4 -8 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 0 4 B 10 0 16 10 8 C 0 -16 0 -6 2 D 0 -10 6 0 4 E -4 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 B=29 D=10 C=9 A=9 so C is eliminated. Round 2 votes counts: E=43 B=29 D=14 A=14 so D is eliminated. Round 3 votes counts: E=44 B=38 A=18 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:200 A:197 E:191 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 0 4 B 10 0 16 10 8 C 0 -16 0 -6 2 D 0 -10 6 0 4 E -4 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 0 4 B 10 0 16 10 8 C 0 -16 0 -6 2 D 0 -10 6 0 4 E -4 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 0 4 B 10 0 16 10 8 C 0 -16 0 -6 2 D 0 -10 6 0 4 E -4 -8 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8139: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (15) C D E B A (12) A B E D C (9) C D E A B (8) C D B E A (8) B A E D C (5) B C A D E (4) D E C B A (3) C B D E A (3) A E D B C (3) E D A C B (2) E A D C B (2) E A D B C (2) E A B D C (2) D C E B A (2) D C B E A (2) C D B A E (2) B C D A E (2) A E D C B (2) E D C A B (1) D C E A B (1) D B C E A (1) C A D E B (1) C A D B E (1) B E D A C (1) B D C E A (1) B D C A E (1) B A C D E (1) A C E D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -6 0 -2 B -4 0 -2 -8 -14 C 6 2 0 -10 4 D 0 8 10 0 6 E 2 14 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428574 B: 0.000000 C: 0.000000 D: 0.571426 E: 0.000000 Sum of squares = 0.510203486039 Cumulative probabilities = A: 0.428574 B: 0.428574 C: 0.428574 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 0 -2 B -4 0 -2 -8 -14 C 6 2 0 -10 4 D 0 8 10 0 6 E 2 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=32 B=15 E=9 D=9 so E is eliminated. Round 2 votes counts: A=38 C=35 B=15 D=12 so D is eliminated. Round 3 votes counts: C=44 A=40 B=16 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 E:203 C:201 A:198 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -6 0 -2 B -4 0 -2 -8 -14 C 6 2 0 -10 4 D 0 8 10 0 6 E 2 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 0 -2 B -4 0 -2 -8 -14 C 6 2 0 -10 4 D 0 8 10 0 6 E 2 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 0 -2 B -4 0 -2 -8 -14 C 6 2 0 -10 4 D 0 8 10 0 6 E 2 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8140: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (7) E A C B D (5) C A D B E (5) E C A D B (4) D E B C A (4) D B E C A (4) C D B A E (4) C A B D E (4) A C B E D (4) E A C D B (3) E A B D C (3) B D E A C (3) B D C A E (3) B A C D E (3) A C E B D (3) E D B C A (2) E C D A B (2) E B D A C (2) E B A D C (2) E A B C D (2) D E C B A (2) D C E B A (2) D B E A C (2) D B C A E (2) A E C B D (2) A B C E D (2) E D C B A (1) E D C A B (1) E D A B C (1) D C B E A (1) D B C E A (1) C E D A B (1) C D E B A (1) C D A E B (1) C D A B E (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B E D (1) B D C E A (1) B D A E C (1) B D A C E (1) B A D C E (1) A E B C D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -2 -4 -14 B 2 0 -4 -6 -8 C 2 4 0 6 -8 D 4 6 -6 0 -2 E 14 8 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -2 -4 -14 B 2 0 -4 -6 -8 C 2 4 0 6 -8 D 4 6 -6 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=21 D=18 B=13 A=13 so B is eliminated. Round 2 votes counts: E=35 D=27 C=21 A=17 so A is eliminated. Round 3 votes counts: E=38 C=34 D=28 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:202 D:201 B:192 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -2 -4 -14 B 2 0 -4 -6 -8 C 2 4 0 6 -8 D 4 6 -6 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -4 -14 B 2 0 -4 -6 -8 C 2 4 0 6 -8 D 4 6 -6 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -4 -14 B 2 0 -4 -6 -8 C 2 4 0 6 -8 D 4 6 -6 0 -2 E 14 8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8141: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (10) E C B D A (9) E B D A C (8) B D A E C (6) A D B C E (5) E B D C A (4) C E B D A (4) C D A E B (4) B E D C A (4) B D E A C (4) D A C B E (3) D A B C E (3) C E D A B (3) C A E D B (3) C A D E B (3) C A D B E (3) A D B E C (3) E C B A D (2) E B C D A (2) E B C A D (2) C E A D B (2) C D B A E (2) E B A D C (1) E A C D B (1) E A B C D (1) C E A B D (1) C D A B E (1) B D A C E (1) B C D E A (1) B A D E C (1) A C E D B (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 0 -18 2 B 2 0 -6 4 -2 C 0 6 0 -8 2 D 18 -4 8 0 4 E -2 2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024690685 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -18 2 B 2 0 -6 4 -2 C 0 6 0 -8 2 D 18 -4 8 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691251 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 A=21 B=17 D=6 so D is eliminated. Round 2 votes counts: E=30 A=27 C=26 B=17 so B is eliminated. Round 3 votes counts: E=38 A=35 C=27 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:213 C:200 B:199 E:197 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -18 2 B 2 0 -6 4 -2 C 0 6 0 -8 2 D 18 -4 8 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691251 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -18 2 B 2 0 -6 4 -2 C 0 6 0 -8 2 D 18 -4 8 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691251 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -18 2 B 2 0 -6 4 -2 C 0 6 0 -8 2 D 18 -4 8 0 4 E -2 2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.222222 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691251 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8142: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (15) C A D E B (11) B D E A C (10) C B A D E (6) B E D A C (5) E D A B C (3) E B D A C (3) E A D B C (3) B C D E A (3) E D B A C (2) D A E B C (2) C E A D B (2) C B E A D (2) C B D A E (2) C A E B D (2) C A B D E (2) B E D C A (2) B C E D A (2) B C D A E (2) A E D C B (2) A E C D B (2) D E B A C (1) D E A B C (1) D B A E C (1) D A B E C (1) C E B A D (1) C B E D A (1) C B A E D (1) C A D B E (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A E C (1) B D A C E (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -16 8 14 B -2 0 -6 -2 -6 C 16 6 0 14 16 D -8 2 -14 0 2 E -14 6 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 8 14 B -2 0 -6 -2 -6 C 16 6 0 14 16 D -8 2 -14 0 2 E -14 6 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=46 B=29 E=11 A=8 D=6 so D is eliminated. Round 2 votes counts: C=46 B=30 E=13 A=11 so A is eliminated. Round 3 votes counts: C=48 B=31 E=21 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:204 B:192 D:191 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 8 14 B -2 0 -6 -2 -6 C 16 6 0 14 16 D -8 2 -14 0 2 E -14 6 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 8 14 B -2 0 -6 -2 -6 C 16 6 0 14 16 D -8 2 -14 0 2 E -14 6 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 8 14 B -2 0 -6 -2 -6 C 16 6 0 14 16 D -8 2 -14 0 2 E -14 6 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8143: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (10) C E A D B (8) A C B E D (8) A C E B D (7) E D C B A (6) D B E C A (6) B D A E C (6) A C B D E (4) E D B C A (3) D B E A C (3) C A E B D (3) B D E A C (3) A B C E D (3) E C D B A (2) E C D A B (2) D B A E C (2) C E A B D (2) C A E D B (2) B E D C A (2) B D A C E (2) A C D B E (2) A B D C E (2) A B C D E (2) E C B D A (1) E C A D B (1) E C A B D (1) E B D C A (1) B E C A D (1) B A D C E (1) B A C E D (1) A D C B E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -2 2 -4 B 0 0 -4 0 -2 C 2 4 0 4 0 D -2 0 -4 0 -10 E 4 2 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.670970 D: 0.000000 E: 0.329030 Sum of squares = 0.558461730742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.670970 D: 0.670970 E: 1.000000 A B C D E A 0 0 -2 2 -4 B 0 0 -4 0 -2 C 2 4 0 4 0 D -2 0 -4 0 -10 E 4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=21 E=17 B=16 C=15 so C is eliminated. Round 2 votes counts: A=36 E=27 D=21 B=16 so B is eliminated. Round 3 votes counts: A=38 D=32 E=30 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:208 C:205 A:198 B:197 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 2 -4 B 0 0 -4 0 -2 C 2 4 0 4 0 D -2 0 -4 0 -10 E 4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 -4 B 0 0 -4 0 -2 C 2 4 0 4 0 D -2 0 -4 0 -10 E 4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 -4 B 0 0 -4 0 -2 C 2 4 0 4 0 D -2 0 -4 0 -10 E 4 2 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8144: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (10) C E A D B (8) D B A C E (6) D A B C E (6) B D A C E (5) D B A E C (4) B E A D C (4) B C E D A (4) A D B E C (4) E C A D B (3) E A C D B (3) D B C A E (3) C E A B D (3) C B E D A (3) B C D E A (3) E C A B D (2) D C A B E (2) C D A E B (2) C A E D B (2) E C B A D (1) E B C A D (1) E A B C D (1) D C A E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C E B A D (1) C D B E A (1) C A D E B (1) B E D A C (1) B E C D A (1) B D C E A (1) B D C A E (1) B A E D C (1) B A D E C (1) A E D C B (1) A E C D B (1) A E B D C (1) A D E C B (1) A D C E B (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 8 -14 14 B 8 0 18 -6 26 C -8 -18 0 -12 16 D 14 6 12 0 10 E -14 -26 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -14 14 B 8 0 18 -6 26 C -8 -18 0 -12 16 D 14 6 12 0 10 E -14 -26 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=23 C=23 E=11 A=11 so E is eliminated. Round 2 votes counts: B=33 C=29 D=23 A=15 so A is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:221 A:200 C:189 E:167 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -14 14 B 8 0 18 -6 26 C -8 -18 0 -12 16 D 14 6 12 0 10 E -14 -26 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -14 14 B 8 0 18 -6 26 C -8 -18 0 -12 16 D 14 6 12 0 10 E -14 -26 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -14 14 B 8 0 18 -6 26 C -8 -18 0 -12 16 D 14 6 12 0 10 E -14 -26 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999189 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8145: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) E C D A B (6) E C A D B (6) C E D B A (6) E A C B D (5) B D A C E (5) B A D C E (5) C D E B A (4) B C D E A (4) A D B E C (4) A B D E C (4) E C D B A (3) D C B A E (3) D B A C E (3) A E B D C (3) A B E D C (3) E C B D A (2) E A B C D (2) C E D A B (2) C B D E A (2) A E D C B (2) A E D B C (2) A E B C D (2) E C B A D (1) E A C D B (1) D C A B E (1) D B C A E (1) D A E C B (1) D A C B E (1) D A B C E (1) C E B D A (1) C D B E A (1) B C E D A (1) B A E D C (1) A D E C B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 -4 4 -10 B -14 0 -14 0 -16 C 4 14 0 14 -14 D -4 0 -14 0 -14 E 10 16 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 -4 4 -10 B -14 0 -14 0 -16 C 4 14 0 14 -14 D -4 0 -14 0 -14 E 10 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=23 C=16 B=16 D=11 so D is eliminated. Round 2 votes counts: E=34 A=26 C=20 B=20 so C is eliminated. Round 3 votes counts: E=47 A=27 B=26 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:227 C:209 A:202 D:184 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 4 -10 B -14 0 -14 0 -16 C 4 14 0 14 -14 D -4 0 -14 0 -14 E 10 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 4 -10 B -14 0 -14 0 -16 C 4 14 0 14 -14 D -4 0 -14 0 -14 E 10 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 4 -10 B -14 0 -14 0 -16 C 4 14 0 14 -14 D -4 0 -14 0 -14 E 10 16 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8146: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (14) C D B A E (9) E A B C D (8) E C A D B (7) D B C A E (6) C E D B A (6) E A C B D (5) C D E B A (4) A B D E C (3) D E C B A (2) D C B E A (2) D B C E A (2) C E A D B (2) C D A B E (2) C A B D E (2) B D A C E (2) B A D C E (2) A E C B D (2) A B D C E (2) E D B C A (1) E C A B D (1) E B D A C (1) E A C D B (1) D C E B A (1) D C B A E (1) D B E C A (1) C E D A B (1) C E A B D (1) C D B E A (1) B D C A E (1) B D A E C (1) B C D A E (1) B A D E C (1) A E B C D (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -8 10 -22 B -10 0 -2 2 -16 C 8 2 0 12 0 D -10 -2 -12 0 -6 E 22 16 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500765 D: 0.000000 E: 0.499235 Sum of squares = 0.50000116405 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500765 D: 0.500765 E: 1.000000 A B C D E A 0 10 -8 10 -22 B -10 0 -2 2 -16 C 8 2 0 12 0 D -10 -2 -12 0 -6 E 22 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=28 D=15 A=11 B=8 so B is eliminated. Round 2 votes counts: E=38 C=29 D=19 A=14 so A is eliminated. Round 3 votes counts: E=42 C=31 D=27 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:222 C:211 A:195 B:187 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 10 -22 B -10 0 -2 2 -16 C 8 2 0 12 0 D -10 -2 -12 0 -6 E 22 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 10 -22 B -10 0 -2 2 -16 C 8 2 0 12 0 D -10 -2 -12 0 -6 E 22 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 10 -22 B -10 0 -2 2 -16 C 8 2 0 12 0 D -10 -2 -12 0 -6 E 22 16 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8147: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (6) A B D C E (6) E D C B A (5) C B A D E (5) A B C D E (5) E A D B C (4) E A C B D (4) D C E B A (4) D B C A E (4) A C B E D (4) E A B C D (3) D E A B C (3) D C B A E (3) A B C E D (3) E D C A B (2) E C D B A (2) C A B E D (2) B D C A E (2) B C A D E (2) B A C D E (2) A E B C D (2) A B E C D (2) E D B C A (1) E D A C B (1) E D A B C (1) E C D A B (1) E C A D B (1) E C A B D (1) E A B D C (1) D E C B A (1) D C B E A (1) D B E A C (1) D B A E C (1) D A B E C (1) C E D B A (1) C E A B D (1) C D E B A (1) C D B E A (1) C D B A E (1) C B D A E (1) C B A E D (1) B D A C E (1) B C D A E (1) B A D C E (1) A C B D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 10 6 18 B -2 0 10 6 20 C -10 -10 0 -4 20 D -6 -6 4 0 12 E -18 -20 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 6 18 B -2 0 10 6 20 C -10 -10 0 -4 20 D -6 -6 4 0 12 E -18 -20 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991076 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=25 A=25 C=14 B=9 so B is eliminated. Round 2 votes counts: D=28 A=28 E=27 C=17 so C is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:217 D:202 C:198 E:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 6 18 B -2 0 10 6 20 C -10 -10 0 -4 20 D -6 -6 4 0 12 E -18 -20 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991076 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 6 18 B -2 0 10 6 20 C -10 -10 0 -4 20 D -6 -6 4 0 12 E -18 -20 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991076 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 6 18 B -2 0 10 6 20 C -10 -10 0 -4 20 D -6 -6 4 0 12 E -18 -20 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991076 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8148: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) C A E B D (10) D B C A E (9) D C A E B (4) B E C A D (4) B D E C A (4) D B C E A (3) D B A E C (3) C E A B D (3) C B A E D (3) B D E A C (3) B C E A D (3) A E C B D (3) A C E D B (3) E A C B D (2) E A B C D (2) D C B A E (2) C E B A D (2) C B E A D (2) C A E D B (2) C A D E B (2) B E D A C (2) B E A C D (2) A E C D B (2) A C E B D (2) E D A B C (1) E B A D C (1) E A D C B (1) E A D B C (1) D C A B E (1) D B E C A (1) D B A C E (1) D A B C E (1) B E D C A (1) B E C D A (1) B D C E A (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 -18 -18 2 -2 B 18 0 10 4 16 C 18 -10 0 -2 10 D -2 -4 2 0 -8 E 2 -16 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -18 2 -2 B 18 0 10 4 16 C 18 -10 0 -2 10 D -2 -4 2 0 -8 E 2 -16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998686 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=24 B=22 A=11 E=8 so E is eliminated. Round 2 votes counts: D=36 C=24 B=23 A=17 so A is eliminated. Round 3 votes counts: D=39 C=36 B=25 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:224 C:208 D:194 E:192 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -18 2 -2 B 18 0 10 4 16 C 18 -10 0 -2 10 D -2 -4 2 0 -8 E 2 -16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998686 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -18 2 -2 B 18 0 10 4 16 C 18 -10 0 -2 10 D -2 -4 2 0 -8 E 2 -16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998686 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -18 2 -2 B 18 0 10 4 16 C 18 -10 0 -2 10 D -2 -4 2 0 -8 E 2 -16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998686 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8149: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) B E A D C (10) D B E A C (9) B E D A C (8) D B E C A (5) C A E B D (5) C A D E B (5) B D E A C (5) C D A B E (4) C A E D B (4) C A B E D (4) B D E C A (4) A E B C D (3) D E B A C (2) D C E A B (2) D C B E A (2) D B C E A (2) C D B E A (2) C B A E D (2) A C E B D (2) E B A D C (1) E A D B C (1) E A B D C (1) D C E B A (1) C D B A E (1) C A D B E (1) B E C A D (1) B E A C D (1) B D C E A (1) A E B D C (1) Total count = 100 A B C D E A 0 -14 -12 -16 -16 B 14 0 10 -2 24 C 12 -10 0 -10 -4 D 16 2 10 0 12 E 16 -24 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -16 -16 B 14 0 10 -2 24 C 12 -10 0 -10 -4 D 16 2 10 0 12 E 16 -24 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=30 D=23 A=6 E=3 so E is eliminated. Round 2 votes counts: C=38 B=31 D=23 A=8 so A is eliminated. Round 3 votes counts: C=40 B=36 D=24 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:220 C:194 E:192 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -12 -16 -16 B 14 0 10 -2 24 C 12 -10 0 -10 -4 D 16 2 10 0 12 E 16 -24 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -16 -16 B 14 0 10 -2 24 C 12 -10 0 -10 -4 D 16 2 10 0 12 E 16 -24 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -16 -16 B 14 0 10 -2 24 C 12 -10 0 -10 -4 D 16 2 10 0 12 E 16 -24 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8150: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (10) A C B E D (9) D E B C A (7) C A D E B (7) E B D A C (6) E D B C A (5) C D A E B (5) B A E C D (5) D E C B A (4) B E A D C (4) E B D C A (3) D C E B A (3) D C E A B (3) A C B D E (3) A B C E D (3) E B A C D (2) C D E A B (2) C A E B D (2) B E D A C (2) B E A C D (2) B D E A C (2) B A E D C (2) E B A D C (1) D B A E C (1) C E D A B (1) C D E B A (1) C D A B E (1) C A E D B (1) C A D B E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 0 8 8 4 B 0 0 12 12 -6 C -8 -12 0 12 -12 D -8 -12 -12 0 -16 E -4 6 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.792951 B: 0.207049 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.67164091861 Cumulative probabilities = A: 0.792951 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 8 4 B 0 0 12 12 -6 C -8 -12 0 12 -12 D -8 -12 -12 0 -16 E -4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000102476 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=21 D=18 E=17 B=17 so E is eliminated. Round 2 votes counts: B=29 A=27 D=23 C=21 so C is eliminated. Round 3 votes counts: A=38 D=33 B=29 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:210 B:209 C:190 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 8 4 B 0 0 12 12 -6 C -8 -12 0 12 -12 D -8 -12 -12 0 -16 E -4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000102476 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 8 4 B 0 0 12 12 -6 C -8 -12 0 12 -12 D -8 -12 -12 0 -16 E -4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000102476 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 8 4 B 0 0 12 12 -6 C -8 -12 0 12 -12 D -8 -12 -12 0 -16 E -4 6 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.400000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000102476 Cumulative probabilities = A: 0.600000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8151: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (19) A B D E C (13) B A C D E (7) A D B E C (7) B A C E D (6) E C D B A (5) C E B D A (5) E D C A B (4) D E A C B (4) C E D B A (4) C E B A D (4) C B E A D (4) B C A E D (4) E C D A B (3) D A E B C (3) B A D C E (3) A B D C E (2) D A B E C (1) B C E A D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -6 4 -6 B -14 0 -4 0 -2 C 6 4 0 -12 -18 D -4 0 12 0 20 E 6 2 18 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.133333 Sum of squares = 0.502222222212 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.866667 E: 1.000000 A B C D E A 0 14 -6 4 -6 B -14 0 -4 0 -2 C 6 4 0 -12 -18 D -4 0 12 0 20 E 6 2 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.133333 Sum of squares = 0.502222221805 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.866667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 B=21 C=17 E=12 so E is eliminated. Round 2 votes counts: D=31 C=25 A=23 B=21 so B is eliminated. Round 3 votes counts: A=39 D=31 C=30 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:214 A:203 E:203 B:190 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -6 4 -6 B -14 0 -4 0 -2 C 6 4 0 -12 -18 D -4 0 12 0 20 E 6 2 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.133333 Sum of squares = 0.502222221805 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.866667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 4 -6 B -14 0 -4 0 -2 C 6 4 0 -12 -18 D -4 0 12 0 20 E 6 2 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.133333 Sum of squares = 0.502222221805 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.866667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 4 -6 B -14 0 -4 0 -2 C 6 4 0 -12 -18 D -4 0 12 0 20 E 6 2 18 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.133333 Sum of squares = 0.502222221805 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.866667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8152: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) E B A C D (7) D A C E B (7) D A E B C (6) C D A B E (6) D C A B E (5) A E D B C (5) A D E B C (5) C B E A D (4) C B D E A (4) E B A D C (3) A D C E B (3) E B D A C (2) E A B D C (2) D C B A E (2) D B E A C (2) D A E C B (2) D A B E C (2) C E B A D (2) C D B E A (2) C D B A E (2) C D A E B (2) B C E D A (2) E D B A C (1) E C A B D (1) E A B C D (1) D C A E B (1) D B E C A (1) D A C B E (1) C B E D A (1) C B A E D (1) C A E B D (1) C A D E B (1) C A D B E (1) B E D A C (1) B C D E A (1) A C E D B (1) Total count = 100 A B C D E A 0 6 2 -6 8 B -6 0 0 -16 -6 C -2 0 0 -2 0 D 6 16 2 0 12 E -8 6 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -6 8 B -6 0 0 -16 -6 C -2 0 0 -2 0 D 6 16 2 0 12 E -8 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 E=17 A=14 B=13 so B is eliminated. Round 2 votes counts: C=30 D=29 E=27 A=14 so A is eliminated. Round 3 votes counts: D=37 E=32 C=31 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:205 C:198 E:193 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -6 8 B -6 0 0 -16 -6 C -2 0 0 -2 0 D 6 16 2 0 12 E -8 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -6 8 B -6 0 0 -16 -6 C -2 0 0 -2 0 D 6 16 2 0 12 E -8 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -6 8 B -6 0 0 -16 -6 C -2 0 0 -2 0 D 6 16 2 0 12 E -8 6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8153: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) C A B D E (12) A C B E D (11) E D B A C (8) E D A C B (6) E D A B C (5) D B E C A (5) D B C A E (5) D E B A C (4) C B A D E (4) E D B C A (3) E A C D B (3) B C A D E (3) B A C D E (3) A E C B D (3) A B C D E (3) E D C A B (2) E A C B D (2) A C E B D (2) E D C B A (1) E A B D C (1) D C B A E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 4 -4 -4 B -2 0 6 -10 -4 C -4 -6 0 -6 -10 D 4 10 6 0 4 E 4 4 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -4 -4 B -2 0 6 -10 -4 C -4 -6 0 -6 -10 D 4 10 6 0 4 E 4 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=27 A=20 C=16 B=6 so B is eliminated. Round 2 votes counts: E=31 D=27 A=23 C=19 so C is eliminated. Round 3 votes counts: A=42 E=31 D=27 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:207 A:199 B:195 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -4 -4 B -2 0 6 -10 -4 C -4 -6 0 -6 -10 D 4 10 6 0 4 E 4 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -4 -4 B -2 0 6 -10 -4 C -4 -6 0 -6 -10 D 4 10 6 0 4 E 4 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -4 -4 B -2 0 6 -10 -4 C -4 -6 0 -6 -10 D 4 10 6 0 4 E 4 4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8154: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) B A D E C (9) C E D A B (6) E B C D A (5) D C E A B (5) D A B C E (4) C E B A D (4) E C B D A (3) D C A E B (3) C A D E B (3) B E D A C (3) B A C E D (3) A D B C E (3) A B D C E (3) E C D A B (2) E B C A D (2) D A C E B (2) C B E A D (2) B A E C D (2) A D C B E (2) A C D B E (2) E D C A B (1) E D B C A (1) E B D A C (1) D E C A B (1) D E B A C (1) D E A B C (1) D B A E C (1) D A E C B (1) D A C B E (1) C E D B A (1) C E A D B (1) C D A E B (1) C A D B E (1) B E C D A (1) B E C A D (1) B E A D C (1) B D A E C (1) B C A E D (1) B A E D C (1) B A D C E (1) B A C D E (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 -8 -10 -4 B 10 0 -4 -8 -8 C 8 4 0 4 2 D 10 8 -4 0 -2 E 4 8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999381 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -10 -4 B 10 0 -4 -8 -8 C 8 4 0 4 2 D 10 8 -4 0 -2 E 4 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 D=20 C=19 A=12 so A is eliminated. Round 2 votes counts: B=28 D=26 E=24 C=22 so C is eliminated. Round 3 votes counts: E=36 D=33 B=31 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 D:206 E:206 B:195 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -10 -4 B 10 0 -4 -8 -8 C 8 4 0 4 2 D 10 8 -4 0 -2 E 4 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -10 -4 B 10 0 -4 -8 -8 C 8 4 0 4 2 D 10 8 -4 0 -2 E 4 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -10 -4 B 10 0 -4 -8 -8 C 8 4 0 4 2 D 10 8 -4 0 -2 E 4 8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8155: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D B E A C (5) A C E D B (5) E D B C A (4) C B A D E (4) C A B E D (4) B C D E A (4) B C D A E (4) A D E B C (4) E D A C B (3) E C B D A (3) D E A B C (3) C B A E D (3) B D A E C (3) A E C D B (3) A B C D E (3) C B E D A (2) C B E A D (2) B D E C A (2) B D C E A (2) A E D C B (2) A D B E C (2) A C D B E (2) E D B A C (1) E C D A B (1) E B D C A (1) E A C D B (1) D E B C A (1) D E B A C (1) D B E C A (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C A E B D (1) C A B D E (1) B D E A C (1) B D C A E (1) B D A C E (1) B A D C E (1) A E D B C (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D E B (1) A C B E D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 12 -2 8 B -2 0 8 -2 8 C -12 -8 0 0 -2 D 2 2 0 0 6 E -8 -8 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.052139 D: 0.947861 E: 0.000000 Sum of squares = 0.901158383499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.052139 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 -2 8 B -2 0 8 -2 8 C -12 -8 0 0 -2 D 2 2 0 0 6 E -8 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102044011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=20 E=19 B=19 D=13 so D is eliminated. Round 2 votes counts: A=30 B=26 E=24 C=20 so C is eliminated. Round 3 votes counts: B=37 A=36 E=27 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:206 D:205 E:190 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 12 -2 8 B -2 0 8 -2 8 C -12 -8 0 0 -2 D 2 2 0 0 6 E -8 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102044011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 -2 8 B -2 0 8 -2 8 C -12 -8 0 0 -2 D 2 2 0 0 6 E -8 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102044011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 -2 8 B -2 0 8 -2 8 C -12 -8 0 0 -2 D 2 2 0 0 6 E -8 -8 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.857143 E: 0.000000 Sum of squares = 0.755102044011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8156: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) C D B E A (7) A E B D C (6) D C E A B (5) D A E B C (5) D A B E C (4) C E B A D (4) C D E B A (4) B E A C D (4) D C B A E (3) D C A E B (3) D C A B E (3) D A B C E (3) B C E A D (3) B A E D C (3) B A E C D (3) E B A C D (2) E A B C D (2) D A E C B (2) C E D A B (2) C E A B D (2) C D E A B (2) C D B A E (2) C B E D A (2) A B E D C (2) E C A B D (1) E B C A D (1) E B A D C (1) E A D B C (1) E A C D B (1) E A B D C (1) D B A E C (1) C E B D A (1) C E A D B (1) C B D E A (1) B A D E C (1) B A C E D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -10 0 -12 B 4 0 -8 0 6 C 10 8 0 8 14 D 0 0 -8 0 -4 E 12 -6 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 0 -12 B 4 0 -8 0 6 C 10 8 0 8 14 D 0 0 -8 0 -4 E 12 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=29 B=15 E=10 A=10 so E is eliminated. Round 2 votes counts: C=37 D=29 B=19 A=15 so A is eliminated. Round 3 votes counts: C=38 D=31 B=31 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:201 E:198 D:194 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 0 -12 B 4 0 -8 0 6 C 10 8 0 8 14 D 0 0 -8 0 -4 E 12 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 0 -12 B 4 0 -8 0 6 C 10 8 0 8 14 D 0 0 -8 0 -4 E 12 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 0 -12 B 4 0 -8 0 6 C 10 8 0 8 14 D 0 0 -8 0 -4 E 12 -6 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8157: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (6) B C D A E (6) E D A C B (5) E D A B C (5) D E B A C (5) C B A E D (5) C B A D E (5) B C A E D (5) A E D C B (5) A C E D B (5) D E A C B (4) D B E C A (4) B D E C A (4) B D C E A (4) D E A B C (3) B C A D E (3) A E C D B (3) E A D C B (2) E A D B C (2) D B C E A (2) C A E B D (2) D B E A C (1) C D A E B (1) C D A B E (1) C B D A E (1) C A E D B (1) C A D E B (1) C A D B E (1) B E D C A (1) B D E A C (1) B C E A D (1) B C D E A (1) A D E C B (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -10 2 10 B -2 0 -4 -6 6 C 10 4 0 2 8 D -2 6 -2 0 0 E -10 -6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -10 2 10 B -2 0 -4 -6 6 C 10 4 0 2 8 D -2 6 -2 0 0 E -10 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 D=19 A=17 E=14 so E is eliminated. Round 2 votes counts: D=29 B=26 C=24 A=21 so A is eliminated. Round 3 votes counts: D=39 C=35 B=26 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:202 D:201 B:197 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -10 2 10 B -2 0 -4 -6 6 C 10 4 0 2 8 D -2 6 -2 0 0 E -10 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 2 10 B -2 0 -4 -6 6 C 10 4 0 2 8 D -2 6 -2 0 0 E -10 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 2 10 B -2 0 -4 -6 6 C 10 4 0 2 8 D -2 6 -2 0 0 E -10 -6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8158: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (13) A C E B D (13) D B E A C (11) D B E C A (7) C A E D B (7) C A E B D (6) B D A E C (6) C E A D B (3) C D B A E (3) A E C B D (3) E B D A C (2) E A C B D (2) E A B C D (2) D B C E A (2) D B C A E (2) C A B D E (2) E D C B A (1) E D B C A (1) E D B A C (1) E C D B A (1) E C D A B (1) E C A B D (1) D E C B A (1) D E B C A (1) C E D B A (1) C D E B A (1) C D B E A (1) B D A C E (1) B A E D C (1) B A D E C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 16 -14 -6 B 16 0 6 10 4 C -16 -6 0 -4 -14 D 14 -10 4 0 6 E 6 -4 14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 16 -14 -6 B 16 0 6 10 4 C -16 -6 0 -4 -14 D 14 -10 4 0 6 E 6 -4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 B=22 A=18 E=12 so E is eliminated. Round 2 votes counts: D=27 C=27 B=24 A=22 so A is eliminated. Round 3 votes counts: C=45 B=28 D=27 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:207 E:205 A:190 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 16 -14 -6 B 16 0 6 10 4 C -16 -6 0 -4 -14 D 14 -10 4 0 6 E 6 -4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 16 -14 -6 B 16 0 6 10 4 C -16 -6 0 -4 -14 D 14 -10 4 0 6 E 6 -4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 16 -14 -6 B 16 0 6 10 4 C -16 -6 0 -4 -14 D 14 -10 4 0 6 E 6 -4 14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8159: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (8) B D A E C (8) E C A B D (7) E A B C D (6) D C B A E (5) C D B E A (5) E A B D C (4) D C A B E (4) D B A C E (4) C B D E A (4) A E B D C (4) E A C B D (3) D B C A E (3) C E D B A (3) C E A D B (3) E B A D C (2) E A C D B (2) C E B D A (2) C A D E B (2) B D C E A (2) B A E D C (2) A E C D B (2) E C B D A (1) E C B A D (1) E C A D B (1) D A C B E (1) D A B E C (1) C E D A B (1) C D E B A (1) C D A E B (1) B E D A C (1) B E C D A (1) B C D E A (1) B A D E C (1) A D C E B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -12 -14 -2 B 10 0 -16 2 6 C 12 16 0 10 2 D 14 -2 -10 0 6 E 2 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -14 -2 B 10 0 -16 2 6 C 12 16 0 10 2 D 14 -2 -10 0 6 E 2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=27 D=18 B=16 A=9 so A is eliminated. Round 2 votes counts: E=33 C=30 D=20 B=17 so B is eliminated. Round 3 votes counts: E=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 D:204 B:201 E:194 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -12 -14 -2 B 10 0 -16 2 6 C 12 16 0 10 2 D 14 -2 -10 0 6 E 2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -14 -2 B 10 0 -16 2 6 C 12 16 0 10 2 D 14 -2 -10 0 6 E 2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -14 -2 B 10 0 -16 2 6 C 12 16 0 10 2 D 14 -2 -10 0 6 E 2 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8160: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) B E C A D (7) D C E A B (6) E C B D A (5) E B C D A (5) D A C E B (5) B A E D C (5) A B D C E (5) E C D B A (4) D E C A B (4) D C A E B (4) B E A C D (4) C E D A B (3) B A E C D (3) B A D E C (3) A B C D E (3) E D C B A (2) D E C B A (2) D A E C B (2) D A E B C (2) B E D A C (2) B C E A D (2) A D B C E (2) D E A B C (1) C E B D A (1) C E A B D (1) C D E A B (1) C D A E B (1) C A B E D (1) B E C D A (1) A C D B E (1) A C B D E (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 0 -2 -2 B -8 0 -6 2 2 C 0 6 0 -10 -6 D 2 -2 10 0 6 E 2 -2 6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000097 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -2 -2 B -8 0 -6 2 2 C 0 6 0 -10 -6 D 2 -2 10 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000043 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 A=23 E=16 C=8 so C is eliminated. Round 2 votes counts: D=28 B=27 A=24 E=21 so E is eliminated. Round 3 votes counts: B=38 D=37 A=25 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:208 A:202 E:200 B:195 C:195 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 0 -2 -2 B -8 0 -6 2 2 C 0 6 0 -10 -6 D 2 -2 10 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000043 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -2 -2 B -8 0 -6 2 2 C 0 6 0 -10 -6 D 2 -2 10 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000043 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -2 -2 B -8 0 -6 2 2 C 0 6 0 -10 -6 D 2 -2 10 0 6 E 2 -2 6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.500000000043 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8161: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) A B C D E (7) E B C A D (5) B A E C D (4) A B D C E (4) E D C B A (3) E D C A B (3) E C B D A (3) D C E A B (3) C D E B A (3) B E A C D (3) A B E D C (3) A B D E C (3) E D A B C (2) E C D B A (2) E C B A D (2) E A B D C (2) D E C B A (2) D C A E B (2) D C A B E (2) D A E B C (2) D A C E B (2) C E D B A (2) C E B D A (2) C E B A D (2) C D B E A (2) A B E C D (2) E D B A C (1) E D A C B (1) E B D C A (1) E B D A C (1) E B A D C (1) D E A C B (1) D A E C B (1) D A C B E (1) C D B A E (1) C B E A D (1) C B A D E (1) C A D B E (1) B E C A D (1) B C A E D (1) B A C E D (1) B A C D E (1) A D E B C (1) A D B E C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -6 -2 -14 B -8 0 -2 4 -16 C 6 2 0 -4 -18 D 2 -4 4 0 0 E 14 16 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.577913 E: 0.422087 Sum of squares = 0.512140829386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.577913 E: 1.000000 A B C D E A 0 8 -6 -2 -14 B -8 0 -2 4 -16 C 6 2 0 -4 -18 D 2 -4 4 0 0 E 14 16 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=24 A=23 C=15 B=11 so B is eliminated. Round 2 votes counts: E=31 A=29 D=24 C=16 so C is eliminated. Round 3 votes counts: E=38 A=32 D=30 so D is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 D:201 A:193 C:193 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 -2 -14 B -8 0 -2 4 -16 C 6 2 0 -4 -18 D 2 -4 4 0 0 E 14 16 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 -2 -14 B -8 0 -2 4 -16 C 6 2 0 -4 -18 D 2 -4 4 0 0 E 14 16 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 -2 -14 B -8 0 -2 4 -16 C 6 2 0 -4 -18 D 2 -4 4 0 0 E 14 16 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8162: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D C A E B (7) D C A B E (6) A C E B D (6) D B E A C (5) C A E B D (5) E B C A D (4) D E C B A (4) D E B C A (4) B E D A C (4) B E A D C (4) B E A C D (4) E B D C A (3) D C E B A (3) D A B C E (3) C D A E B (3) C A D E B (3) D B E C A (2) C A E D B (2) B A E C D (2) A D C B E (2) A C D B E (2) E C B A D (1) E C A B D (1) E B C D A (1) E A C B D (1) D C B E A (1) D A C B E (1) C E D B A (1) C A D B E (1) B A E D C (1) A E C B D (1) A E B C D (1) A D B C E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -4 4 -2 B 4 0 -4 -4 -20 C 4 4 0 -4 -4 D -4 4 4 0 -2 E 2 20 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 4 -2 B 4 0 -4 -4 -20 C 4 4 0 -4 -4 D -4 4 4 0 -2 E 2 20 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=19 C=15 B=15 A=15 so C is eliminated. Round 2 votes counts: D=39 A=26 E=20 B=15 so B is eliminated. Round 3 votes counts: D=39 E=32 A=29 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:201 C:200 A:197 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 4 -2 B 4 0 -4 -4 -20 C 4 4 0 -4 -4 D -4 4 4 0 -2 E 2 20 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 4 -2 B 4 0 -4 -4 -20 C 4 4 0 -4 -4 D -4 4 4 0 -2 E 2 20 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 4 -2 B 4 0 -4 -4 -20 C 4 4 0 -4 -4 D -4 4 4 0 -2 E 2 20 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8163: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) C D E A B (7) D C B E A (6) A B D E C (6) C E D B A (5) A E B C D (5) E C A D B (4) B A D E C (4) D C E B A (3) D C B A E (3) D C A B E (3) C D E B A (3) B D A C E (3) B A E C D (3) A B E D C (3) E C B D A (2) E C B A D (2) D B C A E (2) D B A C E (2) D A B C E (2) C E D A B (2) C E A D B (2) B E C D A (2) B E C A D (2) B A E D C (2) A D B C E (2) E A C D B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A E B (1) D B C E A (1) D A C B E (1) B D A E C (1) A D C E B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -20 0 -8 B -8 0 -14 -4 0 C 20 14 0 4 2 D 0 4 -4 0 6 E 8 0 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -20 0 -8 B -8 0 -14 -4 0 C 20 14 0 4 2 D 0 4 -4 0 6 E 8 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978508 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=20 C=19 A=19 B=17 so B is eliminated. Round 2 votes counts: D=29 A=28 E=24 C=19 so C is eliminated. Round 3 votes counts: D=39 E=33 A=28 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:220 D:203 E:200 A:190 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -20 0 -8 B -8 0 -14 -4 0 C 20 14 0 4 2 D 0 4 -4 0 6 E 8 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978508 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -20 0 -8 B -8 0 -14 -4 0 C 20 14 0 4 2 D 0 4 -4 0 6 E 8 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978508 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -20 0 -8 B -8 0 -14 -4 0 C 20 14 0 4 2 D 0 4 -4 0 6 E 8 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978508 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8164: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (5) A E D B C (5) A B E C D (5) D E A B C (4) C E D A B (4) C E A B D (4) C B D E A (4) D C B E A (3) C E D B A (3) B D C E A (3) B C D A E (3) B A D E C (3) B A C D E (3) E D A C B (2) E C D A B (2) E C A D B (2) E A D C B (2) D E C A B (2) D E A C B (2) D B A E C (2) D A E B C (2) C E B D A (2) C D B E A (2) C A E B D (2) B D C A E (2) B C D E A (2) B C A D E (2) A E C B D (2) A D E B C (2) A B D E C (2) E D A B C (1) D E B C A (1) D B C E A (1) D B C A E (1) D A B E C (1) C E B A D (1) C D E B A (1) C B E D A (1) C B E A D (1) C B A E D (1) C A B E D (1) B D A E C (1) B D A C E (1) B C A E D (1) B A D C E (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -4 -6 -10 B -8 0 2 0 -4 C 4 -2 0 8 0 D 6 0 -8 0 2 E 10 4 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.483956 D: 0.000000 E: 0.516044 Sum of squares = 0.500514808801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.483956 D: 0.483956 E: 1.000000 A B C D E A 0 8 -4 -6 -10 B -8 0 2 0 -4 C 4 -2 0 8 0 D 6 0 -8 0 2 E 10 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=22 D=19 A=18 E=14 so E is eliminated. Round 2 votes counts: C=31 A=25 D=22 B=22 so D is eliminated. Round 3 votes counts: A=37 C=36 B=27 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:206 C:205 D:200 B:195 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 -6 -10 B -8 0 2 0 -4 C 4 -2 0 8 0 D 6 0 -8 0 2 E 10 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -6 -10 B -8 0 2 0 -4 C 4 -2 0 8 0 D 6 0 -8 0 2 E 10 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -6 -10 B -8 0 2 0 -4 C 4 -2 0 8 0 D 6 0 -8 0 2 E 10 4 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8165: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) A C D B E (10) E B D C A (9) E A C B D (7) A C B D E (5) E D B C A (4) E B C A D (4) B C D A E (4) E B C D A (3) E B A C D (3) E A C D B (3) E A B C D (3) C B A D E (3) A E D C B (3) D C B A E (2) D B C E A (2) D A C B E (2) A D C E B (2) A C E D B (2) A C E B D (2) E D B A C (1) E D A B C (1) E B A D C (1) E A D C B (1) D A E C B (1) D A C E B (1) C D B A E (1) C A B D E (1) B E C D A (1) B D E C A (1) B D C E A (1) B D C A E (1) B C D E A (1) B C A E D (1) A E C D B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 2 8 8 B 6 0 2 2 -6 C -2 -2 0 12 6 D -8 -2 -12 0 0 E -8 6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102099 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -6 2 8 8 B 6 0 2 2 -6 C -2 -2 0 12 6 D -8 -2 -12 0 0 E -8 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755101917 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=27 D=18 B=10 C=5 so C is eliminated. Round 2 votes counts: E=40 A=28 D=19 B=13 so B is eliminated. Round 3 votes counts: E=41 A=32 D=27 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:207 A:206 B:202 E:196 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 2 8 8 B 6 0 2 2 -6 C -2 -2 0 12 6 D -8 -2 -12 0 0 E -8 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755101917 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 8 8 B 6 0 2 2 -6 C -2 -2 0 12 6 D -8 -2 -12 0 0 E -8 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755101917 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 8 8 B 6 0 2 2 -6 C -2 -2 0 12 6 D -8 -2 -12 0 0 E -8 6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755101917 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8166: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) D A B C E (8) E C B D A (7) C D E A B (5) A D B C E (5) A B D E C (5) E B C A D (4) C E D B A (4) C E B D A (4) A E B D C (4) C D E B A (3) B E A C D (3) A D E C B (3) A B E D C (3) E A C B D (2) D C B A E (2) D C A E B (2) D C A B E (2) D A C E B (2) D A C B E (2) B E C A D (2) B D C A E (2) B A D E C (2) B A D C E (2) A D E B C (2) A D B E C (2) E C D A B (1) E B A C D (1) E A B C D (1) B D A C E (1) B C E A D (1) B C D E A (1) B A E D C (1) B A E C D (1) A E B C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 4 8 6 B -2 0 6 14 -8 C -4 -6 0 -2 -6 D -8 -14 2 0 4 E -6 8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 6 B -2 0 6 14 -8 C -4 -6 0 -2 -6 D -8 -14 2 0 4 E -6 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=24 D=18 C=16 B=16 so C is eliminated. Round 2 votes counts: E=32 D=26 A=26 B=16 so B is eliminated. Round 3 votes counts: E=38 A=32 D=30 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:205 E:202 D:192 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 6 B -2 0 6 14 -8 C -4 -6 0 -2 -6 D -8 -14 2 0 4 E -6 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 6 B -2 0 6 14 -8 C -4 -6 0 -2 -6 D -8 -14 2 0 4 E -6 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 6 B -2 0 6 14 -8 C -4 -6 0 -2 -6 D -8 -14 2 0 4 E -6 8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999835 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8167: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) E B A C D (7) D C A B E (7) C D B A E (7) B E A D C (6) B C D E A (6) A E B D C (6) C D A E B (5) A B E D C (5) E A B D C (4) D C B A E (4) B E A C D (4) B D C A E (4) A E D C B (4) E C D B A (3) E B C D A (2) E A B C D (2) D C A E B (2) C D E B A (2) B A D C E (2) A D C B E (2) E C D A B (1) E B A D C (1) C D E A B (1) B C D A E (1) B A E D C (1) B A D E C (1) A D E C B (1) Total count = 100 A B C D E A 0 -20 -8 -8 4 B 20 0 22 22 18 C 8 -22 0 0 -14 D 8 -22 0 0 -10 E -4 -18 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 -8 4 B 20 0 22 22 18 C 8 -22 0 0 -14 D 8 -22 0 0 -10 E -4 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=20 A=18 C=15 D=13 so D is eliminated. Round 2 votes counts: B=34 C=28 E=20 A=18 so A is eliminated. Round 3 votes counts: B=39 E=31 C=30 so C is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:241 E:201 D:188 C:186 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -8 -8 4 B 20 0 22 22 18 C 8 -22 0 0 -14 D 8 -22 0 0 -10 E -4 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -8 4 B 20 0 22 22 18 C 8 -22 0 0 -14 D 8 -22 0 0 -10 E -4 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -8 4 B 20 0 22 22 18 C 8 -22 0 0 -14 D 8 -22 0 0 -10 E -4 -18 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8168: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) D E B C A (6) B C A D E (6) A B C D E (6) E C A B D (5) E A C B D (5) D B A C E (5) E D A C B (4) E C D B A (4) E C B A D (4) D A B C E (4) C A B E D (4) A C B E D (4) E C B D A (3) D E A B C (3) C B A E D (3) E D A B C (2) D E B A C (2) C B E A D (2) A C B D E (2) E D B C A (1) E D B A C (1) D B E C A (1) D B E A C (1) D B C E A (1) D A E B C (1) C E B A D (1) C E A B D (1) C B E D A (1) C B A D E (1) B D C A E (1) B C D A E (1) B A C D E (1) A E D C B (1) A D E B C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -10 -2 -14 B 10 0 -10 4 -10 C 10 10 0 10 -8 D 2 -4 -10 0 -12 E 14 10 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -10 -2 -14 B 10 0 -10 4 -10 C 10 10 0 10 -8 D 2 -4 -10 0 -12 E 14 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=24 A=16 C=13 B=9 so B is eliminated. Round 2 votes counts: E=38 D=25 C=20 A=17 so A is eliminated. Round 3 votes counts: E=39 C=34 D=27 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:211 B:197 D:188 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -10 -2 -14 B 10 0 -10 4 -10 C 10 10 0 10 -8 D 2 -4 -10 0 -12 E 14 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -2 -14 B 10 0 -10 4 -10 C 10 10 0 10 -8 D 2 -4 -10 0 -12 E 14 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -2 -14 B 10 0 -10 4 -10 C 10 10 0 10 -8 D 2 -4 -10 0 -12 E 14 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8169: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (5) D A E B C (5) A E D C B (5) A E C D B (5) E A D C B (4) C B D A E (4) C B A E D (4) B D C E A (4) B C D E A (4) D B C A E (3) D A E C B (3) C E A B D (3) C A E D B (3) B E D C A (3) B E D A C (3) B D E C A (3) B D C A E (3) E C A B D (2) E A C B D (2) D C A B E (2) D B A E C (2) C A D E B (2) C A D B E (2) B E C A D (2) B E A D C (2) B D E A C (2) B C D A E (2) A C E D B (2) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D C A E B (1) D B E A C (1) C D A B E (1) C B E A D (1) C B A D E (1) C A E B D (1) B E A C D (1) B C E D A (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -6 2 4 B -2 0 2 4 4 C 6 -2 0 -10 -8 D -2 -4 10 0 -8 E -4 -4 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000011 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 2 4 B -2 0 2 4 4 C 6 -2 0 -10 -8 D -2 -4 10 0 -8 E -4 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=22 E=17 D=17 A=13 so A is eliminated. Round 2 votes counts: B=31 E=27 C=24 D=18 so D is eliminated. Round 3 votes counts: B=37 E=35 C=28 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:204 E:204 A:201 D:198 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -6 2 4 B -2 0 2 4 4 C 6 -2 0 -10 -8 D -2 -4 10 0 -8 E -4 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 2 4 B -2 0 2 4 4 C 6 -2 0 -10 -8 D -2 -4 10 0 -8 E -4 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 2 4 B -2 0 2 4 4 C 6 -2 0 -10 -8 D -2 -4 10 0 -8 E -4 -4 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.440000000015 Cumulative probabilities = A: 0.200000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8170: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (10) A C B D E (8) E C A D B (7) C A B E D (7) D B E A C (6) D E B A C (5) B D A C E (5) B A C D E (5) E C A B D (4) D A C E B (4) A C B E D (4) E D B C A (3) D E B C A (3) D B A E C (3) D B A C E (3) B C A E D (3) C E A B D (2) C A E B D (2) E D C B A (1) E D A C B (1) E C B A D (1) E B D C A (1) E B C D A (1) E B C A D (1) D E C A B (1) D E A B C (1) D A E C B (1) C E A D B (1) B E D C A (1) B A D C E (1) A D C B E (1) A D B C E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 14 2 0 0 B -14 0 -12 -6 -2 C -2 12 0 -4 -2 D 0 6 4 0 -2 E 0 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.305156 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.694844 Sum of squares = 0.57592851714 Cumulative probabilities = A: 0.305156 B: 0.305156 C: 0.305156 D: 0.305156 E: 1.000000 A B C D E A 0 14 2 0 0 B -14 0 -12 -6 -2 C -2 12 0 -4 -2 D 0 6 4 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=27 A=16 B=15 C=12 so C is eliminated. Round 2 votes counts: E=33 D=27 A=25 B=15 so B is eliminated. Round 3 votes counts: E=34 A=34 D=32 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 D:204 E:203 C:202 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 0 0 B -14 0 -12 -6 -2 C -2 12 0 -4 -2 D 0 6 4 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 0 0 B -14 0 -12 -6 -2 C -2 12 0 -4 -2 D 0 6 4 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 0 0 B -14 0 -12 -6 -2 C -2 12 0 -4 -2 D 0 6 4 0 -2 E 0 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8171: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) D A C E B (9) C D A B E (9) C B E D A (7) E B C D A (5) B C E D A (5) A D C B E (5) C B D A E (4) B E C D A (4) B E C A D (4) A D C E B (4) E B C A D (3) E A D B C (3) E C D A B (2) E B A C D (2) D A E C B (2) C B D E A (2) B E A C D (2) A D E C B (2) A D E B C (2) E D A C B (1) E C D B A (1) E C B D A (1) E A D C B (1) D C A E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C D E B A (1) C D B E A (1) C B A D E (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -14 -6 -16 -16 B 14 0 -14 6 -8 C 6 14 0 14 6 D 16 -6 -14 0 -10 E 16 8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -16 -16 B 14 0 -14 6 -8 C 6 14 0 14 6 D 16 -6 -14 0 -10 E 16 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=27 B=16 A=14 D=13 so D is eliminated. Round 2 votes counts: E=30 C=28 A=26 B=16 so B is eliminated. Round 3 votes counts: E=40 C=33 A=27 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:214 B:199 D:193 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -6 -16 -16 B 14 0 -14 6 -8 C 6 14 0 14 6 D 16 -6 -14 0 -10 E 16 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -16 -16 B 14 0 -14 6 -8 C 6 14 0 14 6 D 16 -6 -14 0 -10 E 16 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -16 -16 B 14 0 -14 6 -8 C 6 14 0 14 6 D 16 -6 -14 0 -10 E 16 8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8172: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) C D A E B (8) C D A B E (7) B C E D A (7) E B A D C (6) E A B D C (6) C D B A E (5) E A D C B (4) E A D B C (3) D A C E B (3) C B D A E (3) B C D E A (3) B C D A E (3) A E D C B (3) D C A B E (2) C B D E A (2) B E C D A (2) B E A C D (2) B D C A E (2) B A E D C (2) A D C E B (2) E C D A B (1) E C A B D (1) E B C A D (1) E B A C D (1) E A C D B (1) C D E A B (1) C D B E A (1) C B E D A (1) B E C A D (1) B D A C E (1) B A D E C (1) B A D C E (1) A E D B C (1) A D E C B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -2 -4 -6 B 8 0 6 10 14 C 2 -6 0 2 8 D 4 -10 -2 0 -4 E 6 -14 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -4 -6 B 8 0 6 10 14 C 2 -6 0 2 8 D 4 -10 -2 0 -4 E 6 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=28 E=24 A=9 D=5 so D is eliminated. Round 2 votes counts: B=34 C=30 E=24 A=12 so A is eliminated. Round 3 votes counts: C=36 B=35 E=29 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:203 D:194 E:194 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -4 -6 B 8 0 6 10 14 C 2 -6 0 2 8 D 4 -10 -2 0 -4 E 6 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -4 -6 B 8 0 6 10 14 C 2 -6 0 2 8 D 4 -10 -2 0 -4 E 6 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -4 -6 B 8 0 6 10 14 C 2 -6 0 2 8 D 4 -10 -2 0 -4 E 6 -14 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8173: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) C B A E D (7) D E C A B (6) D C E A B (6) E C D B A (5) D A B E C (5) C E D B A (5) E D C B A (4) E D B C A (4) C E B A D (4) B A E C D (4) D E A B C (3) B A E D C (3) A B D E C (3) A B C D E (3) E D B A C (2) E B D A C (2) E B A D C (2) D E C B A (2) C A B E D (2) C A B D E (2) A B D C E (2) E C B A D (1) E B D C A (1) E B A C D (1) D E A C B (1) D C A E B (1) D A E B C (1) D A C B E (1) D A B C E (1) C D A E B (1) C B E A D (1) B E A D C (1) B E A C D (1) B C A E D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -6 -2 -4 B 20 0 2 2 -4 C 6 -2 0 -4 -4 D 2 -2 4 0 -22 E 4 4 4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -6 -2 -4 B 20 0 2 2 -4 C 6 -2 0 -4 -4 D 2 -2 4 0 -22 E 4 4 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=22 C=22 B=19 A=10 so A is eliminated. Round 2 votes counts: D=28 B=28 E=22 C=22 so E is eliminated. Round 3 votes counts: D=38 B=34 C=28 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:217 B:210 C:198 D:191 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -6 -2 -4 B 20 0 2 2 -4 C 6 -2 0 -4 -4 D 2 -2 4 0 -22 E 4 4 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -6 -2 -4 B 20 0 2 2 -4 C 6 -2 0 -4 -4 D 2 -2 4 0 -22 E 4 4 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -6 -2 -4 B 20 0 2 2 -4 C 6 -2 0 -4 -4 D 2 -2 4 0 -22 E 4 4 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993124 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8174: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (19) E A D B C (11) E D A C B (9) E B C A D (7) D A C B E (6) E B A C D (4) E D C A B (3) E C B D A (3) E A B D C (3) D C B A E (3) D A B C E (3) C D B A E (3) A D B C E (3) E D C B A (2) E A B C D (2) B C A D E (2) B A C D E (2) A D E B C (2) A B D C E (2) E D A B C (1) E C B A D (1) D E A C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C B E D A (1) C B E A D (1) C B D E A (1) B C E A D (1) A E D B C (1) Total count = 100 A B C D E A 0 -4 2 -16 -6 B 4 0 -12 0 -6 C -2 12 0 -2 0 D 16 0 2 0 -2 E 6 6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.176706 D: 0.000000 E: 0.823294 Sum of squares = 0.709038019771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.176706 D: 0.176706 E: 1.000000 A B C D E A 0 -4 2 -16 -6 B 4 0 -12 0 -6 C -2 12 0 -2 0 D 16 0 2 0 -2 E 6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.000000 E: 0.500728 Sum of squares = 0.500001059962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.499272 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=46 C=27 D=14 A=8 B=5 so B is eliminated. Round 2 votes counts: E=46 C=30 D=14 A=10 so A is eliminated. Round 3 votes counts: E=47 C=32 D=21 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:208 E:207 C:204 B:193 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -16 -6 B 4 0 -12 0 -6 C -2 12 0 -2 0 D 16 0 2 0 -2 E 6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.000000 E: 0.500728 Sum of squares = 0.500001059962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.499272 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -16 -6 B 4 0 -12 0 -6 C -2 12 0 -2 0 D 16 0 2 0 -2 E 6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.000000 E: 0.500728 Sum of squares = 0.500001059962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.499272 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -16 -6 B 4 0 -12 0 -6 C -2 12 0 -2 0 D 16 0 2 0 -2 E 6 6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.000000 E: 0.500728 Sum of squares = 0.500001059962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499272 D: 0.499272 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8175: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) B D C A E (7) D B C A E (6) E B D C A (5) E A C D B (5) E A C B D (5) C B D A E (5) A E C D B (5) A C E D B (5) E A D B C (4) C D B A E (4) C A D B E (4) A C D B E (4) E A B D C (3) B E D C A (3) E B D A C (2) E B A D C (2) D C B A E (2) A C D E B (2) E D A B C (1) E B C D A (1) D B E C A (1) D B E A C (1) D B A C E (1) C E B A D (1) C E A B D (1) C A E B D (1) C A B D E (1) B D E C A (1) B D E A C (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -12 -8 2 B 12 0 4 6 10 C 12 -4 0 -6 16 D 8 -6 6 0 8 E -2 -10 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -8 2 B 12 0 4 6 10 C 12 -4 0 -6 16 D 8 -6 6 0 8 E -2 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998465 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=25 A=19 C=17 D=11 so D is eliminated. Round 2 votes counts: B=34 E=28 C=19 A=19 so C is eliminated. Round 3 votes counts: B=45 E=30 A=25 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:209 D:208 A:185 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -8 2 B 12 0 4 6 10 C 12 -4 0 -6 16 D 8 -6 6 0 8 E -2 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998465 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -8 2 B 12 0 4 6 10 C 12 -4 0 -6 16 D 8 -6 6 0 8 E -2 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998465 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -8 2 B 12 0 4 6 10 C 12 -4 0 -6 16 D 8 -6 6 0 8 E -2 -10 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998465 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8176: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) A B D C E (9) B A D E C (7) C E D A B (6) C E B A D (6) D E C A B (5) D A B E C (5) A B D E C (5) E C B D A (4) A D B C E (4) D E C B A (3) D C E A B (3) D A C E B (3) C E D B A (3) B A C E D (3) D B A E C (2) C E B D A (2) C E A D B (2) B A E C D (2) E D B C A (1) E C D A B (1) E C B A D (1) D E B C A (1) D E B A C (1) D C A E B (1) D A E C B (1) D A C B E (1) D A B C E (1) C E A B D (1) B E A C D (1) B D A E C (1) B C E A D (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -4 -12 -6 B 2 0 -8 -12 -12 C 4 8 0 -10 -6 D 12 12 10 0 10 E 6 12 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -12 -6 B 2 0 -8 -12 -12 C 4 8 0 -10 -6 D 12 12 10 0 10 E 6 12 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=20 A=19 E=18 B=16 so B is eliminated. Round 2 votes counts: A=32 D=28 C=21 E=19 so E is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:222 E:207 C:198 A:188 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -12 -6 B 2 0 -8 -12 -12 C 4 8 0 -10 -6 D 12 12 10 0 10 E 6 12 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -12 -6 B 2 0 -8 -12 -12 C 4 8 0 -10 -6 D 12 12 10 0 10 E 6 12 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -12 -6 B 2 0 -8 -12 -12 C 4 8 0 -10 -6 D 12 12 10 0 10 E 6 12 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8177: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C A B E D (5) B C E D A (5) B C E A D (5) B C D A E (5) D E A B C (4) C B A E D (4) A C E D B (4) E A D C B (3) C B E A D (3) C B A D E (3) C A D E B (3) B E C A D (3) B D E A C (3) A C D E B (3) E D B A C (2) E D A B C (2) E A D B C (2) D E B A C (2) D A E C B (2) C D B A E (2) C A E B D (2) C A B D E (2) B C D E A (2) A E D C B (2) A D E C B (2) A D C E B (2) E B D A C (1) E A C D B (1) D C B A E (1) D C A B E (1) D B E A C (1) D B A E C (1) D B A C E (1) D A E B C (1) C B D A E (1) C A E D B (1) B E D C A (1) B D C E A (1) A E C D B (1) Total count = 100 A B C D E A 0 -14 0 2 -2 B 14 0 4 12 20 C 0 -4 0 10 12 D -2 -12 -10 0 -14 E 2 -20 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 2 -2 B 14 0 4 12 20 C 0 -4 0 10 12 D -2 -12 -10 0 -14 E 2 -20 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996471 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=26 D=14 A=14 E=11 so E is eliminated. Round 2 votes counts: B=36 C=26 A=20 D=18 so D is eliminated. Round 3 votes counts: B=43 A=29 C=28 so C is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 C:209 A:193 E:192 D:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 2 -2 B 14 0 4 12 20 C 0 -4 0 10 12 D -2 -12 -10 0 -14 E 2 -20 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996471 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 2 -2 B 14 0 4 12 20 C 0 -4 0 10 12 D -2 -12 -10 0 -14 E 2 -20 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996471 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 2 -2 B 14 0 4 12 20 C 0 -4 0 10 12 D -2 -12 -10 0 -14 E 2 -20 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996471 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8178: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) B D E A C (11) D B C E A (10) D B E C A (9) B E D A C (8) A E C B D (7) C A E B D (6) C A D E B (6) D B E A C (5) C D A E B (4) E B A D C (3) C D A B E (3) A C E B D (3) A E B C D (2) E D B A C (1) E B D A C (1) E A B D C (1) E A B C D (1) D C E A B (1) D C B E A (1) D C B A E (1) C D B A E (1) C B D A E (1) B E A D C (1) B A E D C (1) Total count = 100 A B C D E A 0 -8 -10 -14 -6 B 8 0 8 -8 4 C 10 -8 0 -8 -2 D 14 8 8 0 6 E 6 -4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -14 -6 B 8 0 8 -8 4 C 10 -8 0 -8 -2 D 14 8 8 0 6 E 6 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=27 B=21 A=12 E=7 so E is eliminated. Round 2 votes counts: C=33 D=28 B=25 A=14 so A is eliminated. Round 3 votes counts: C=43 B=29 D=28 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:218 B:206 E:199 C:196 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -14 -6 B 8 0 8 -8 4 C 10 -8 0 -8 -2 D 14 8 8 0 6 E 6 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -14 -6 B 8 0 8 -8 4 C 10 -8 0 -8 -2 D 14 8 8 0 6 E 6 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -14 -6 B 8 0 8 -8 4 C 10 -8 0 -8 -2 D 14 8 8 0 6 E 6 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8179: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (7) D B A C E (6) D A B E C (6) B D E A C (5) B D A E C (5) A D B E C (5) E C B A D (4) E C A B D (4) E B C D A (3) E A C D B (3) C E B D A (3) C E B A D (3) C A D B E (3) A E D B C (3) A D E B C (3) E C A D B (2) E B D A C (2) C E A B D (2) C D A B E (2) C A E D B (2) B E D C A (2) B E D A C (2) B D E C A (2) A D C E B (2) A D C B E (2) A D B C E (2) A C E D B (2) E C B D A (1) E B A D C (1) E A D C B (1) E A D B C (1) D A B C E (1) C D B A E (1) C B E D A (1) C B D E A (1) C B D A E (1) B D C E A (1) B D C A E (1) B D A C E (1) A E C D B (1) Total count = 100 A B C D E A 0 8 8 6 -2 B -8 0 4 -10 0 C -8 -4 0 -8 -12 D -6 10 8 0 0 E 2 0 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.141615 E: 0.858385 Sum of squares = 0.756878929454 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.141615 E: 1.000000 A B C D E A 0 8 8 6 -2 B -8 0 4 -10 0 C -8 -4 0 -8 -12 D -6 10 8 0 0 E 2 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000130481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=22 A=20 B=19 D=13 so D is eliminated. Round 2 votes counts: A=27 C=26 B=25 E=22 so E is eliminated. Round 3 votes counts: C=37 A=32 B=31 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 E:207 D:206 B:193 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 6 -2 B -8 0 4 -10 0 C -8 -4 0 -8 -12 D -6 10 8 0 0 E 2 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000130481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 -2 B -8 0 4 -10 0 C -8 -4 0 -8 -12 D -6 10 8 0 0 E 2 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000130481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 -2 B -8 0 4 -10 0 C -8 -4 0 -8 -12 D -6 10 8 0 0 E 2 0 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000130481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8180: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) C A E D B (7) E D B C A (6) B D E C A (6) A C B D E (6) D E B C A (5) D B E C A (5) B D A C E (5) A C E D B (5) E C A D B (4) E A C B D (4) B D E A C (4) E B D A C (3) C E A D B (3) C A D B E (3) B D C A E (3) E D C B A (2) E D C A B (2) E C D A B (2) D B C E A (2) C A D E B (2) E D B A C (1) E A C D B (1) E A B C D (1) D C B A E (1) D B C A E (1) C E D A B (1) C D E A B (1) C D A B E (1) B D A E C (1) B A E D C (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -14 -4 -6 B -8 0 -12 -10 -20 C 14 12 0 4 4 D 4 10 -4 0 -8 E 6 20 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999312 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -14 -4 -6 B -8 0 -12 -10 -20 C 14 12 0 4 4 D 4 10 -4 0 -8 E 6 20 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=22 B=20 C=18 D=14 so D is eliminated. Round 2 votes counts: E=31 B=28 A=22 C=19 so C is eliminated. Round 3 votes counts: E=36 A=35 B=29 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:217 E:215 D:201 A:192 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -14 -4 -6 B -8 0 -12 -10 -20 C 14 12 0 4 4 D 4 10 -4 0 -8 E 6 20 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -14 -4 -6 B -8 0 -12 -10 -20 C 14 12 0 4 4 D 4 10 -4 0 -8 E 6 20 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -14 -4 -6 B -8 0 -12 -10 -20 C 14 12 0 4 4 D 4 10 -4 0 -8 E 6 20 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999187 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8181: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (6) E A C B D (5) A E C B D (5) A C E D B (5) E C A B D (4) D B C A E (4) D B A E C (4) D B A C E (4) B E A D C (4) B D A E C (4) A D B E C (4) D C B A E (3) C E B A D (3) C D B E A (3) B E D C A (3) B E C D A (3) E B C D A (2) E B C A D (2) D B C E A (2) D A C B E (2) B E D A C (2) B D E A C (2) B A D E C (2) A E C D B (2) A E B C D (2) A D C E B (2) A D C B E (2) E C B D A (1) E A B C D (1) D C A B E (1) C E D B A (1) C D A E B (1) C B E D A (1) C A D E B (1) B C D E A (1) B A E D C (1) A E B D C (1) A D E C B (1) A D B C E (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 16 12 20 B 4 0 0 0 8 C -16 0 0 0 -12 D -12 0 0 0 -8 E -20 -8 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.806872 C: 0.085678 D: 0.107450 E: 0.000000 Sum of squares = 0.669928626995 Cumulative probabilities = A: 0.000000 B: 0.806872 C: 0.892550 D: 1.000000 E: 1.000000 A B C D E A 0 -4 16 12 20 B 4 0 0 0 8 C -16 0 0 0 -12 D -12 0 0 0 -8 E -20 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.761905 C: 0.047619 D: 0.190476 E: 0.000000 Sum of squares = 0.619047648264 Cumulative probabilities = A: 0.000000 B: 0.761905 C: 0.809524 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=22 D=20 C=16 E=15 so E is eliminated. Round 2 votes counts: A=33 B=26 C=21 D=20 so D is eliminated. Round 3 votes counts: B=40 A=35 C=25 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:222 B:206 E:196 D:190 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 16 12 20 B 4 0 0 0 8 C -16 0 0 0 -12 D -12 0 0 0 -8 E -20 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.761905 C: 0.047619 D: 0.190476 E: 0.000000 Sum of squares = 0.619047648264 Cumulative probabilities = A: 0.000000 B: 0.761905 C: 0.809524 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 16 12 20 B 4 0 0 0 8 C -16 0 0 0 -12 D -12 0 0 0 -8 E -20 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.761905 C: 0.047619 D: 0.190476 E: 0.000000 Sum of squares = 0.619047648264 Cumulative probabilities = A: 0.000000 B: 0.761905 C: 0.809524 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 16 12 20 B 4 0 0 0 8 C -16 0 0 0 -12 D -12 0 0 0 -8 E -20 -8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.761905 C: 0.047619 D: 0.190476 E: 0.000000 Sum of squares = 0.619047648264 Cumulative probabilities = A: 0.000000 B: 0.761905 C: 0.809524 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8182: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) D E B A C (8) A C B E D (8) D E B C A (7) E D B C A (6) A B C E D (6) B A D E C (5) C E D B A (4) C E D A B (4) C A E D B (4) C A E B D (4) C A B E D (4) D B A E C (3) C E A B D (3) C A D E B (3) A B C D E (3) D B E A C (2) C E B A D (2) B A E C D (2) A C B D E (2) E D C B A (1) E B D C A (1) D E C A B (1) D A C E B (1) D A B E C (1) C E B D A (1) C D E A B (1) B E D A C (1) B D E A C (1) B A E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -8 -2 -2 B 6 0 -2 -8 -18 C 8 2 0 2 2 D 2 8 -2 0 -4 E 2 18 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -2 -2 B 6 0 -2 -8 -18 C 8 2 0 2 2 D 2 8 -2 0 -4 E 2 18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=30 A=21 B=10 E=8 so E is eliminated. Round 2 votes counts: D=38 C=30 A=21 B=11 so B is eliminated. Round 3 votes counts: D=41 C=30 A=29 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 C:207 D:202 A:191 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -2 -2 B 6 0 -2 -8 -18 C 8 2 0 2 2 D 2 8 -2 0 -4 E 2 18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -2 -2 B 6 0 -2 -8 -18 C 8 2 0 2 2 D 2 8 -2 0 -4 E 2 18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -2 -2 B 6 0 -2 -8 -18 C 8 2 0 2 2 D 2 8 -2 0 -4 E 2 18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8183: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) C D E B A (8) B A C E D (8) D C E A B (6) C D E A B (5) C D A B E (5) E B A D C (4) C B A E D (4) B A E C D (4) D E A B C (3) D A E B C (3) C D A E B (3) B E A D C (3) B C A E D (3) B A E D C (3) E D A B C (2) E A B D C (2) D E C A B (2) C E D B A (2) C E B A D (2) C A B D E (2) A B E D C (2) A B C D E (2) E C D B A (1) E A D B C (1) D C A E B (1) D C A B E (1) C D B A E (1) C B E D A (1) C B E A D (1) C B D E A (1) B E A C D (1) B A C D E (1) A E D B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -16 10 10 B 14 0 -10 8 8 C 16 10 0 28 32 D -10 -8 -28 0 10 E -10 -8 -32 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 10 10 B 14 0 -10 8 8 C 16 10 0 28 32 D -10 -8 -28 0 10 E -10 -8 -32 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 B=23 D=16 E=10 A=7 so A is eliminated. Round 2 votes counts: C=44 B=28 D=17 E=11 so E is eliminated. Round 3 votes counts: C=45 B=34 D=21 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:243 B:210 A:195 D:182 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -16 10 10 B 14 0 -10 8 8 C 16 10 0 28 32 D -10 -8 -28 0 10 E -10 -8 -32 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 10 10 B 14 0 -10 8 8 C 16 10 0 28 32 D -10 -8 -28 0 10 E -10 -8 -32 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 10 10 B 14 0 -10 8 8 C 16 10 0 28 32 D -10 -8 -28 0 10 E -10 -8 -32 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8184: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (17) A E B C D (12) E D C A B (10) D C B E A (8) E D C B A (7) E A D C B (5) B A C D E (5) E A B D C (4) C D B A E (4) A B E C D (4) D C E B A (3) B C D A E (3) A B C E D (3) E A B C D (2) D E C B A (2) D C B A E (2) B C A D E (2) A E B D C (2) E B A C D (1) C B D A E (1) B C D E A (1) B A C E D (1) A D C B E (1) Total count = 100 A B C D E A 0 20 14 18 14 B -20 0 14 16 4 C -14 -14 0 12 2 D -18 -16 -12 0 -2 E -14 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 14 18 14 B -20 0 14 16 4 C -14 -14 0 12 2 D -18 -16 -12 0 -2 E -14 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=29 D=15 B=12 C=5 so C is eliminated. Round 2 votes counts: A=39 E=29 D=19 B=13 so B is eliminated. Round 3 votes counts: A=47 E=29 D=24 so D is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:233 B:207 C:193 E:191 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 14 18 14 B -20 0 14 16 4 C -14 -14 0 12 2 D -18 -16 -12 0 -2 E -14 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 14 18 14 B -20 0 14 16 4 C -14 -14 0 12 2 D -18 -16 -12 0 -2 E -14 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 14 18 14 B -20 0 14 16 4 C -14 -14 0 12 2 D -18 -16 -12 0 -2 E -14 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8185: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) E D C B A (6) C D B E A (6) A B D C E (6) E A B D C (5) B D C A E (5) B A D C E (5) A E B C D (5) D C B A E (4) A E B D C (4) A B E D C (4) D B C E A (3) C D E B A (3) A C B D E (3) A B C D E (3) E C D B A (2) E C A D B (2) E A C B D (2) D C B E A (2) D B C A E (2) C E D A B (2) C D B A E (2) C A B D E (2) B D A E C (2) E D B C A (1) E A D C B (1) E A D B C (1) E A B C D (1) D E C B A (1) C E D B A (1) C A D B E (1) B D E A C (1) B D A C E (1) B A D E C (1) A E C B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 -2 6 B -4 0 4 6 10 C 4 -4 0 -12 2 D 2 -6 12 0 8 E -6 -10 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888898 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -2 6 B -4 0 4 6 10 C 4 -4 0 -12 2 D 2 -6 12 0 8 E -6 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888996 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 C=17 B=15 D=12 so D is eliminated. Round 2 votes counts: E=29 A=28 C=23 B=20 so B is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:208 D:208 A:202 C:195 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 -2 6 B -4 0 4 6 10 C 4 -4 0 -12 2 D 2 -6 12 0 8 E -6 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888996 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -2 6 B -4 0 4 6 10 C 4 -4 0 -12 2 D 2 -6 12 0 8 E -6 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888996 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -2 6 B -4 0 4 6 10 C 4 -4 0 -12 2 D 2 -6 12 0 8 E -6 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888996 Cumulative probabilities = A: 0.500000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8186: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) A D C B E (6) A B E C D (6) C D B E A (5) B E C A D (5) A D E C B (5) C B D E A (4) E B A C D (3) D E C A B (3) D A C B E (3) C D B A E (3) A E B D C (3) E D C B A (2) E D A C B (2) D C B E A (2) D A C E B (2) C B E D A (2) C B D A E (2) B E A C D (2) B C E D A (2) B C D E A (2) B C D A E (2) B C A E D (2) B A E C D (2) A E D B C (2) A B E D C (2) A B C E D (2) A B C D E (2) E D A B C (1) E B C D A (1) E B C A D (1) E A D B C (1) D E A C B (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E C B (1) C E D B A (1) C E B D A (1) B E C D A (1) B A C D E (1) A D E B C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 0 -4 2 B 4 0 -10 -4 20 C 0 10 0 4 10 D 4 4 -4 0 12 E -2 -20 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.290339 B: 0.000000 C: 0.709661 D: 0.000000 E: 0.000000 Sum of squares = 0.587915576665 Cumulative probabilities = A: 0.290339 B: 0.290339 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -4 2 B 4 0 -10 -4 20 C 0 10 0 4 10 D 4 4 -4 0 12 E -2 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499688 B: 0.000000 C: 0.500312 D: 0.000000 E: 0.000000 Sum of squares = 0.500000194543 Cumulative probabilities = A: 0.499688 B: 0.499688 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=21 B=19 C=18 E=11 so E is eliminated. Round 2 votes counts: A=32 D=26 B=24 C=18 so C is eliminated. Round 3 votes counts: D=35 B=33 A=32 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:212 D:208 B:205 A:197 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 -4 2 B 4 0 -10 -4 20 C 0 10 0 4 10 D 4 4 -4 0 12 E -2 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499688 B: 0.000000 C: 0.500312 D: 0.000000 E: 0.000000 Sum of squares = 0.500000194543 Cumulative probabilities = A: 0.499688 B: 0.499688 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -4 2 B 4 0 -10 -4 20 C 0 10 0 4 10 D 4 4 -4 0 12 E -2 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499688 B: 0.000000 C: 0.500312 D: 0.000000 E: 0.000000 Sum of squares = 0.500000194543 Cumulative probabilities = A: 0.499688 B: 0.499688 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -4 2 B 4 0 -10 -4 20 C 0 10 0 4 10 D 4 4 -4 0 12 E -2 -20 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499688 B: 0.000000 C: 0.500312 D: 0.000000 E: 0.000000 Sum of squares = 0.500000194543 Cumulative probabilities = A: 0.499688 B: 0.499688 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8187: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) C D A E B (5) E C A D B (4) E B A D C (4) E B A C D (4) D C E B A (4) D C E A B (4) C A D E B (4) B E A D C (4) B A E C D (4) A B C E D (4) E B D C A (3) B E A C D (3) B D E C A (3) B A D C E (3) A E B C D (3) E B D A C (2) E A C B D (2) E A B C D (2) D C B A E (2) D B C A E (2) C D E A B (2) B D A C E (2) E D C B A (1) E D B C A (1) E C D A B (1) E C B D A (1) D E C B A (1) D E C A B (1) D C A E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B D E A C (1) B A E D C (1) B A C E D (1) B A C D E (1) A E C B D (1) A C E D B (1) A C D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -6 0 0 B 0 0 2 6 -4 C 6 -2 0 -6 0 D 0 -6 6 0 0 E 0 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.162740 B: 0.000000 C: 0.000000 D: 0.273968 E: 0.563293 Sum of squares = 0.41884109362 Cumulative probabilities = A: 0.162740 B: 0.162740 C: 0.162740 D: 0.436707 E: 1.000000 A B C D E A 0 0 -6 0 0 B 0 0 2 6 -4 C 6 -2 0 -6 0 D 0 -6 6 0 0 E 0 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938779009 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=25 B=25 C=12 A=12 so C is eliminated. Round 2 votes counts: D=33 E=25 B=25 A=17 so A is eliminated. Round 3 votes counts: D=39 B=31 E=30 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:202 E:202 D:200 C:199 A:197 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 0 0 B 0 0 2 6 -4 C 6 -2 0 -6 0 D 0 -6 6 0 0 E 0 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938779009 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 0 0 B 0 0 2 6 -4 C 6 -2 0 -6 0 D 0 -6 6 0 0 E 0 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938779009 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 0 0 B 0 0 2 6 -4 C 6 -2 0 -6 0 D 0 -6 6 0 0 E 0 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.428571 Sum of squares = 0.346938779009 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8188: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) C E A D B (6) B D A E C (6) B C A E D (6) D E A C B (5) C E A B D (5) C B E A D (4) D C E A B (3) D B A E C (3) C E D A B (3) C D E A B (3) B D C A E (3) B A C E D (3) A E D C B (3) A E D B C (3) E C D A B (2) E C A D B (2) D E C A B (2) D C B E A (2) D B C E A (2) D B C A E (2) D A E B C (2) B D C E A (2) B C D E A (2) B C A D E (2) B A E C D (2) B A D E C (2) E D C A B (1) E A C D B (1) C E B D A (1) C E B A D (1) C D E B A (1) C B D E A (1) C B A E D (1) B C D A E (1) B A E D C (1) B A C D E (1) A E C B D (1) A D B E C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -16 6 -10 B -2 0 -10 -6 -4 C 16 10 0 0 12 D -6 6 0 0 -6 E 10 4 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.581274 D: 0.418726 E: 0.000000 Sum of squares = 0.513210788615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.581274 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 6 -10 B -2 0 -10 -6 -4 C 16 10 0 0 12 D -6 6 0 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=26 D=21 E=12 A=10 so A is eliminated. Round 2 votes counts: B=32 C=27 D=22 E=19 so E is eliminated. Round 3 votes counts: D=35 C=33 B=32 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:204 D:197 A:191 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 6 -10 B -2 0 -10 -6 -4 C 16 10 0 0 12 D -6 6 0 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 6 -10 B -2 0 -10 -6 -4 C 16 10 0 0 12 D -6 6 0 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 6 -10 B -2 0 -10 -6 -4 C 16 10 0 0 12 D -6 6 0 0 -6 E 10 4 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8189: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (10) C E A B D (10) A E C D B (9) B D C A E (8) B D A C E (8) A E D C B (8) E A C D B (7) C E B D A (6) C B E D A (6) E C A D B (5) D B A E C (4) D B A C E (4) B D C E A (4) E C A B D (2) B C D E A (2) A D E B C (2) A D B E C (2) C E B A D (1) C E A D B (1) A D E C B (1) Total count = 100 A B C D E A 0 14 10 -4 12 B -14 0 -12 -6 -4 C -10 12 0 -2 0 D 4 6 2 0 -10 E -12 4 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.000000 D: 0.461538 E: 0.153846 Sum of squares = 0.384615384615 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.384615 D: 0.846154 E: 1.000000 A B C D E A 0 14 10 -4 12 B -14 0 -12 -6 -4 C -10 12 0 -2 0 D 4 6 2 0 -10 E -12 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.000000 D: 0.461538 E: 0.153846 Sum of squares = 0.384615384609 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.384615 D: 0.846154 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=22 A=22 D=18 E=14 so E is eliminated. Round 2 votes counts: C=31 A=29 B=22 D=18 so D is eliminated. Round 3 votes counts: A=39 C=31 B=30 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:201 E:201 C:200 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 -4 12 B -14 0 -12 -6 -4 C -10 12 0 -2 0 D 4 6 2 0 -10 E -12 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.000000 D: 0.461538 E: 0.153846 Sum of squares = 0.384615384609 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.384615 D: 0.846154 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 -4 12 B -14 0 -12 -6 -4 C -10 12 0 -2 0 D 4 6 2 0 -10 E -12 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.000000 D: 0.461538 E: 0.153846 Sum of squares = 0.384615384609 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.384615 D: 0.846154 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 -4 12 B -14 0 -12 -6 -4 C -10 12 0 -2 0 D 4 6 2 0 -10 E -12 4 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.384615 B: 0.000000 C: 0.000000 D: 0.461538 E: 0.153846 Sum of squares = 0.384615384609 Cumulative probabilities = A: 0.384615 B: 0.384615 C: 0.384615 D: 0.846154 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8190: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) E D C A B (6) E A D C B (6) B C D E A (6) B A E C D (6) B A C D E (6) A E B D C (6) B C A D E (5) C D E A B (4) B A E D C (4) E D C B A (3) E A B D C (3) D C E B A (3) D C E A B (3) A E D C B (3) A C B D E (3) A B E C D (3) E D B C A (2) C D B E A (2) A C D B E (2) E B A D C (1) D E C B A (1) D E C A B (1) D B C E A (1) C D B A E (1) C B D E A (1) C B D A E (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E C A (1) B D C E A (1) B A C E D (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -2 6 -12 B 14 0 4 8 0 C 2 -4 0 2 0 D -6 -8 -2 0 2 E 12 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.638119 C: 0.000000 D: 0.000000 E: 0.361881 Sum of squares = 0.53815395306 Cumulative probabilities = A: 0.000000 B: 0.638119 C: 0.638119 D: 0.638119 E: 1.000000 A B C D E A 0 -14 -2 6 -12 B 14 0 4 8 0 C 2 -4 0 2 0 D -6 -8 -2 0 2 E 12 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=21 A=20 C=17 D=9 so D is eliminated. Round 2 votes counts: B=34 E=23 C=23 A=20 so A is eliminated. Round 3 votes counts: B=40 E=32 C=28 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 E:205 C:200 D:193 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 6 -12 B 14 0 4 8 0 C 2 -4 0 2 0 D -6 -8 -2 0 2 E 12 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 6 -12 B 14 0 4 8 0 C 2 -4 0 2 0 D -6 -8 -2 0 2 E 12 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 6 -12 B 14 0 4 8 0 C 2 -4 0 2 0 D -6 -8 -2 0 2 E 12 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8191: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) D C E B A (6) D A C E B (5) B A E C D (5) A D C E B (5) A B E C D (5) A B D C E (5) D C E A B (4) E C D B A (3) E C B D A (3) D C A E B (3) C E D B A (3) C E B D A (3) B E C D A (3) B C E D A (3) B A C E D (3) A B E D C (3) D E C A B (2) D C B E A (2) C D E B A (2) C B D E A (2) B C E A D (2) A D E C B (2) A B D E C (2) E C D A B (1) E C B A D (1) E C A B D (1) E B C A D (1) E A D C B (1) E A C B D (1) D C B A E (1) D C A B E (1) D B A C E (1) C B E D A (1) B E A C D (1) B D C A E (1) B C D E A (1) A D E B C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -16 -2 -10 B 12 0 -8 10 4 C 16 8 0 6 12 D 2 -10 -6 0 -4 E 10 -4 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -2 -10 B 12 0 -8 10 4 C 16 8 0 6 12 D 2 -10 -6 0 -4 E 10 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 A=25 E=12 C=11 so C is eliminated. Round 2 votes counts: B=30 D=27 A=25 E=18 so E is eliminated. Round 3 votes counts: B=38 D=34 A=28 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:221 B:209 E:199 D:191 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -16 -2 -10 B 12 0 -8 10 4 C 16 8 0 6 12 D 2 -10 -6 0 -4 E 10 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -2 -10 B 12 0 -8 10 4 C 16 8 0 6 12 D 2 -10 -6 0 -4 E 10 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -2 -10 B 12 0 -8 10 4 C 16 8 0 6 12 D 2 -10 -6 0 -4 E 10 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8192: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) C A D E B (7) D C A E B (5) C D A E B (5) B E D C A (5) E A B C D (4) D E A C B (4) C D A B E (4) B E A D C (4) E B D A C (3) E B A D C (3) E B A C D (3) B E D A C (3) D E A B C (2) D C B A E (2) D B C E A (2) C A B E D (2) C A B D E (2) B D E C A (2) B C D A E (2) B C A E D (2) A E C D B (2) A C E D B (2) E A D C B (1) E A D B C (1) E A B D C (1) D E C B A (1) D E B C A (1) D E B A C (1) D C E A B (1) D C B E A (1) D C A B E (1) D A E C B (1) D A C E B (1) C D B A E (1) C B D A E (1) C A E D B (1) C A E B D (1) B E C D A (1) B C E A D (1) B A E C D (1) A E B C D (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -2 0 -8 B -2 0 6 4 -6 C 2 -6 0 8 -10 D 0 -4 -8 0 -4 E 8 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 0 -8 B -2 0 6 4 -6 C 2 -6 0 8 -10 D 0 -4 -8 0 -4 E 8 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=24 D=23 E=16 A=7 so A is eliminated. Round 2 votes counts: B=31 C=27 D=23 E=19 so E is eliminated. Round 3 votes counts: B=46 C=29 D=25 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:201 C:197 A:196 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 0 -8 B -2 0 6 4 -6 C 2 -6 0 8 -10 D 0 -4 -8 0 -4 E 8 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 0 -8 B -2 0 6 4 -6 C 2 -6 0 8 -10 D 0 -4 -8 0 -4 E 8 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 0 -8 B -2 0 6 4 -6 C 2 -6 0 8 -10 D 0 -4 -8 0 -4 E 8 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8193: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) A D B C E (6) D E C B A (5) D E B C A (5) A B D C E (5) E D C B A (4) D E C A B (4) C B E A D (3) C A E B D (3) B E C D A (3) B A C E D (3) A D C B E (3) A C D E B (3) A C B E D (3) E C D B A (2) E C B D A (2) E B C D A (2) D E A C B (2) D B A E C (2) D A C E B (2) D A B E C (2) C E A B D (2) B D E A C (2) B C E A D (2) A D C E B (2) E D B C A (1) E C B A D (1) E B C A D (1) D E B A C (1) D E A B C (1) D C A E B (1) D B E C A (1) D A E C B (1) D A E B C (1) C E A D B (1) C B A E D (1) C A E D B (1) C A B E D (1) B E C A D (1) B D E C A (1) B D A E C (1) B C A E D (1) B A D E C (1) A C E B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -12 8 -8 B 6 0 -12 -2 -12 C 12 12 0 -8 6 D -8 2 8 0 8 E 8 12 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -12 8 -8 B 6 0 -12 -2 -12 C 12 12 0 -8 6 D -8 2 8 0 8 E 8 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=25 C=19 B=15 E=13 so E is eliminated. Round 2 votes counts: D=33 A=25 C=24 B=18 so B is eliminated. Round 3 votes counts: D=37 C=34 A=29 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:211 D:205 E:203 A:191 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -12 8 -8 B 6 0 -12 -2 -12 C 12 12 0 -8 6 D -8 2 8 0 8 E 8 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -12 8 -8 B 6 0 -12 -2 -12 C 12 12 0 -8 6 D -8 2 8 0 8 E 8 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -12 8 -8 B 6 0 -12 -2 -12 C 12 12 0 -8 6 D -8 2 8 0 8 E 8 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.000000 Sum of squares = 0.346938775506 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8194: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) C B E D A (5) A B D C E (5) C E B D A (4) B E C A D (4) B C E D A (4) B C A D E (4) E D A C B (3) E C B D A (3) D A E C B (3) C E D B A (3) C B D A E (3) B A C E D (3) B A C D E (3) A E D B C (3) A D E B C (3) A D B C E (3) E C D B A (2) E C D A B (2) D E C A B (2) D A C E B (2) C E D A B (2) C B D E A (2) B A E D C (2) E B C A D (1) E A D B C (1) D E A C B (1) D C E A B (1) C D E B A (1) C D E A B (1) C D B A E (1) C D A E B (1) B E C D A (1) B C E A D (1) B C D E A (1) B C A E D (1) B A E C D (1) B A D C E (1) A D E C B (1) A D C B E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -24 -20 -14 B 2 0 -12 0 -4 C 24 12 0 8 6 D 20 0 -8 0 -18 E 14 4 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -24 -20 -14 B 2 0 -12 0 -4 C 24 12 0 8 6 D 20 0 -8 0 -18 E 14 4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=24 C=23 A=18 D=9 so D is eliminated. Round 2 votes counts: E=27 B=26 C=24 A=23 so A is eliminated. Round 3 votes counts: E=37 B=36 C=27 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:225 E:215 D:197 B:193 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -24 -20 -14 B 2 0 -12 0 -4 C 24 12 0 8 6 D 20 0 -8 0 -18 E 14 4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -24 -20 -14 B 2 0 -12 0 -4 C 24 12 0 8 6 D 20 0 -8 0 -18 E 14 4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -24 -20 -14 B 2 0 -12 0 -4 C 24 12 0 8 6 D 20 0 -8 0 -18 E 14 4 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8195: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D A C B E (8) A D C E B (6) C A D E B (5) A D E C B (5) E B D A C (4) E A D B C (4) C E B A D (4) D A B C E (3) C E A D B (3) C B E D A (3) C A E D B (3) B D E A C (3) B C E D A (3) A D E B C (3) E C B A D (2) E B A D C (2) E A B D C (2) D C A B E (2) D A B E C (2) C D A B E (2) C A D B E (2) B E D A C (2) B E C D A (2) B D C A E (2) B D A E C (2) E D A B C (1) E C A B D (1) D E A B C (1) D B C A E (1) D B A C E (1) D A E B C (1) C D B A E (1) C B E A D (1) C B D A E (1) B D A C E (1) B C D A E (1) A E C D B (1) A C E D B (1) Total count = 100 A B C D E A 0 12 6 6 8 B -12 0 -2 -12 -14 C -6 2 0 -12 8 D -6 12 12 0 6 E -8 14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999614 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 6 8 B -12 0 -2 -12 -14 C -6 2 0 -12 8 D -6 12 12 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 D=19 B=16 A=16 so B is eliminated. Round 2 votes counts: C=29 E=28 D=27 A=16 so A is eliminated. Round 3 votes counts: D=41 C=30 E=29 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:212 C:196 E:196 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 6 8 B -12 0 -2 -12 -14 C -6 2 0 -12 8 D -6 12 12 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 6 8 B -12 0 -2 -12 -14 C -6 2 0 -12 8 D -6 12 12 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 6 8 B -12 0 -2 -12 -14 C -6 2 0 -12 8 D -6 12 12 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8196: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) B E D C A (7) A C D B E (7) A C B D E (6) C A B E D (5) B E D A C (5) E B D C A (4) D E B A C (4) D B E A C (4) C A E D B (4) D A C E B (3) C A E B D (3) B D E A C (3) E B C D A (2) D A E C B (2) B C E A D (2) B C A E D (2) A D C E B (2) A D C B E (2) A C D E B (2) E D C A B (1) E D B C A (1) E D B A C (1) E C D A B (1) E C B D A (1) E B D A C (1) D E C A B (1) D E A C B (1) D B A E C (1) D A E B C (1) C E B A D (1) C E A D B (1) C B E A D (1) C B A E D (1) C A B D E (1) B E C D A (1) B E C A D (1) B E A C D (1) B D A E C (1) B A C D E (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -2 8 10 B -8 0 -14 2 6 C 2 14 0 10 10 D -8 -2 -10 0 4 E -10 -6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 8 10 B -8 0 -14 2 6 C 2 14 0 10 10 D -8 -2 -10 0 4 E -10 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 A=20 D=17 E=12 so E is eliminated. Round 2 votes counts: B=31 C=29 D=20 A=20 so D is eliminated. Round 3 votes counts: B=42 C=31 A=27 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:212 B:193 D:192 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 8 10 B -8 0 -14 2 6 C 2 14 0 10 10 D -8 -2 -10 0 4 E -10 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 8 10 B -8 0 -14 2 6 C 2 14 0 10 10 D -8 -2 -10 0 4 E -10 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 8 10 B -8 0 -14 2 6 C 2 14 0 10 10 D -8 -2 -10 0 4 E -10 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8197: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (9) C E D B A (8) C E B A D (8) E C B D A (5) D A B E C (5) A D B E C (5) D E B C A (4) C E B D A (4) B A E D C (4) E B C D A (3) C E D A B (3) A B D E C (3) E D C B A (2) E C D B A (2) D B E A C (2) D A B C E (2) C D E B A (2) C A D E B (2) B E A C D (2) B A D E C (2) A C D B E (2) A B E C D (2) A B D C E (2) A B C D E (2) E D B C A (1) E C B A D (1) E B C A D (1) D E C B A (1) D E B A C (1) D E A C B (1) D C E B A (1) D B A E C (1) D A C E B (1) D A C B E (1) C E A B D (1) C A E B D (1) B A E C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 0 -6 B 12 0 6 -12 -6 C 0 -6 0 4 2 D 0 12 -4 0 -2 E 6 6 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.428571 Sum of squares = 0.38775510204 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 -12 0 0 -6 B 12 0 6 -12 -6 C 0 -6 0 4 2 D 0 12 -4 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101999 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=27 D=20 E=15 B=9 so B is eliminated. Round 2 votes counts: A=34 C=29 D=20 E=17 so E is eliminated. Round 3 votes counts: C=41 A=36 D=23 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:206 D:203 B:200 C:200 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 0 0 -6 B 12 0 6 -12 -6 C 0 -6 0 4 2 D 0 12 -4 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101999 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 0 -6 B 12 0 6 -12 -6 C 0 -6 0 4 2 D 0 12 -4 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101999 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 0 -6 B 12 0 6 -12 -6 C 0 -6 0 4 2 D 0 12 -4 0 -2 E 6 6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.428571 D: 0.000000 E: 0.428571 Sum of squares = 0.387755101999 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8198: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) A E D C B (13) A D C E B (7) E A B D C (6) B E C D A (5) B C E D A (5) B C D A E (4) D C A E B (3) B E A C D (3) B A E C D (3) A E B D C (3) E D A C B (2) E B A D C (2) E A D B C (2) C D B E A (2) C D B A E (2) B A C D E (2) A E D B C (2) A D E C B (2) A D C B E (2) A C D B E (2) A B C D E (2) E D C A B (1) E B D C A (1) E B C D A (1) D C E B A (1) D C E A B (1) D A E C B (1) C D A B E (1) C B D E A (1) C B D A E (1) C A B D E (1) B E C A D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 10 8 4 B -4 0 14 12 4 C -10 -14 0 2 2 D -8 -12 -2 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999205 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 10 8 4 B -4 0 14 12 4 C -10 -14 0 2 2 D -8 -12 -2 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=34 E=15 C=8 D=6 so D is eliminated. Round 2 votes counts: B=37 A=35 E=15 C=13 so C is eliminated. Round 3 votes counts: B=43 A=40 E=17 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 B:213 E:196 C:190 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 10 8 4 B -4 0 14 12 4 C -10 -14 0 2 2 D -8 -12 -2 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 8 4 B -4 0 14 12 4 C -10 -14 0 2 2 D -8 -12 -2 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 8 4 B -4 0 14 12 4 C -10 -14 0 2 2 D -8 -12 -2 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999266 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8199: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (7) E D A B C (6) E D B A C (5) B A C E D (5) A C B E D (5) E A C D B (4) D E C B A (4) D E C A B (4) D B C E A (4) A B C E D (4) D C E B A (3) D C B A E (3) B D C A E (3) B C D A E (3) B C A D E (3) B A E C D (3) E D A C B (2) D C E A B (2) D C B E A (2) C D B A E (2) C B D A E (2) C B A D E (2) A C E B D (2) E D C A B (1) E C A D B (1) E A D C B (1) E A D B C (1) E A B D C (1) D E B C A (1) C D E A B (1) C A E D B (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D A C (1) B D E C A (1) B D E A C (1) B D A C E (1) B A D E C (1) A E C D B (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 8 -4 8 B 14 0 4 -2 10 C -8 -4 0 4 18 D 4 2 -4 0 4 E -8 -10 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 -4 8 B 14 0 4 -2 10 C -8 -4 0 4 18 D 4 2 -4 0 4 E -8 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999993 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 E=22 A=15 C=11 so C is eliminated. Round 2 votes counts: B=33 D=26 E=22 A=19 so A is eliminated. Round 3 votes counts: B=44 E=29 D=27 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:205 D:203 A:199 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 -4 8 B 14 0 4 -2 10 C -8 -4 0 4 18 D 4 2 -4 0 4 E -8 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999993 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -4 8 B 14 0 4 -2 10 C -8 -4 0 4 18 D 4 2 -4 0 4 E -8 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999993 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -4 8 B 14 0 4 -2 10 C -8 -4 0 4 18 D 4 2 -4 0 4 E -8 -10 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.200000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999993 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8200: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (10) E A B D C (7) D C B A E (7) C D B A E (7) B A D E C (6) C E B A D (5) E C B A D (4) D B A C E (4) E A D C B (3) E A B C D (3) D C A B E (3) D A B E C (3) C D E A B (3) C D B E A (3) E A D B C (2) D A E B C (2) C E D A B (2) C E B D A (2) C E A B D (2) C B E D A (2) C B D E A (2) C B D A E (2) B D A C E (2) E B C A D (1) E B A D C (1) E B A C D (1) E A C B D (1) D C E A B (1) D A C B E (1) C D A B E (1) C B E A D (1) B C E A D (1) B C A D E (1) A E D B C (1) A E B D C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 0 -14 6 B 4 0 -4 -8 14 C 0 4 0 -12 24 D 14 8 12 0 20 E -6 -14 -24 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -14 6 B 4 0 -4 -8 14 C 0 4 0 -12 24 D 14 8 12 0 20 E -6 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=31 E=23 B=10 A=4 so A is eliminated. Round 2 votes counts: D=32 C=32 E=25 B=11 so B is eliminated. Round 3 votes counts: D=41 C=34 E=25 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 C:208 B:203 A:194 E:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -14 6 B 4 0 -4 -8 14 C 0 4 0 -12 24 D 14 8 12 0 20 E -6 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -14 6 B 4 0 -4 -8 14 C 0 4 0 -12 24 D 14 8 12 0 20 E -6 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -14 6 B 4 0 -4 -8 14 C 0 4 0 -12 24 D 14 8 12 0 20 E -6 -14 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8201: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) B A C E D (7) C E D B A (5) D A B C E (4) C B D A E (4) C B A E D (4) B A C D E (4) A E B D C (4) D E A C B (3) C B E A D (3) C B D E A (3) B A E C D (3) B A D C E (3) E D A C B (2) E C D B A (2) E C D A B (2) E C B A D (2) C E B D A (2) C E B A D (2) C D B A E (2) B C A E D (2) B C A D E (2) A E D B C (2) A D E B C (2) A B E D C (2) E D C A B (1) E D A B C (1) E C A D B (1) E A D B C (1) E A C B D (1) E A B C D (1) D E C A B (1) D E A B C (1) D C E B A (1) D C E A B (1) D C B A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E B A (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 14 -2 24 B 8 0 8 2 -2 C -14 -8 0 10 8 D 2 -2 -10 0 -4 E -24 2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.058824 B: 0.705882 C: 0.000000 D: 0.000000 E: 0.235294 Sum of squares = 0.557093425698 Cumulative probabilities = A: 0.058824 B: 0.764706 C: 0.764706 D: 0.764706 E: 1.000000 A B C D E A 0 -8 14 -2 24 B 8 0 8 2 -2 C -14 -8 0 10 8 D 2 -2 -10 0 -4 E -24 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.705882 C: 0.000000 D: 0.000000 E: 0.235294 Sum of squares = 0.557093425594 Cumulative probabilities = A: 0.058824 B: 0.764706 C: 0.764706 D: 0.764706 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 B=21 E=14 A=13 so A is eliminated. Round 2 votes counts: D=28 C=27 B=25 E=20 so E is eliminated. Round 3 votes counts: D=35 C=35 B=30 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:214 B:208 C:198 D:193 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 14 -2 24 B 8 0 8 2 -2 C -14 -8 0 10 8 D 2 -2 -10 0 -4 E -24 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.705882 C: 0.000000 D: 0.000000 E: 0.235294 Sum of squares = 0.557093425594 Cumulative probabilities = A: 0.058824 B: 0.764706 C: 0.764706 D: 0.764706 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 14 -2 24 B 8 0 8 2 -2 C -14 -8 0 10 8 D 2 -2 -10 0 -4 E -24 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.705882 C: 0.000000 D: 0.000000 E: 0.235294 Sum of squares = 0.557093425594 Cumulative probabilities = A: 0.058824 B: 0.764706 C: 0.764706 D: 0.764706 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 14 -2 24 B 8 0 8 2 -2 C -14 -8 0 10 8 D 2 -2 -10 0 -4 E -24 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.058824 B: 0.705882 C: 0.000000 D: 0.000000 E: 0.235294 Sum of squares = 0.557093425594 Cumulative probabilities = A: 0.058824 B: 0.764706 C: 0.764706 D: 0.764706 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8202: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (11) B D C E A (8) C A B E D (5) A C E B D (5) E D A C B (4) C E A B D (4) B D C A E (4) E C D A B (3) D E C B A (3) D E A B C (3) C B A E D (3) B D A E C (3) B A D C E (3) E C D B A (2) E C A D B (2) C A E B D (2) B D A C E (2) B C D A E (2) B C A D E (2) B A C E D (2) A E C D B (2) A B D E C (2) A B C E D (2) A B C D E (2) E D C B A (1) E A D C B (1) E A C D B (1) D E A C B (1) D B E A C (1) D B A E C (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) C E A D B (1) C B E D A (1) C B E A D (1) C B D E A (1) B C D E A (1) B A C D E (1) A E C B D (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -18 -10 -4 B 10 0 8 20 20 C 18 -8 0 -4 10 D 10 -20 4 0 10 E 4 -20 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -18 -10 -4 B 10 0 8 20 20 C 18 -8 0 -4 10 D 10 -20 4 0 10 E 4 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=22 C=20 A=16 E=14 so E is eliminated. Round 2 votes counts: B=28 D=27 C=27 A=18 so A is eliminated. Round 3 votes counts: C=36 B=35 D=29 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 C:208 D:202 E:182 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -18 -10 -4 B 10 0 8 20 20 C 18 -8 0 -4 10 D 10 -20 4 0 10 E 4 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -18 -10 -4 B 10 0 8 20 20 C 18 -8 0 -4 10 D 10 -20 4 0 10 E 4 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -18 -10 -4 B 10 0 8 20 20 C 18 -8 0 -4 10 D 10 -20 4 0 10 E 4 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8203: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) E B D A C (5) D C A B E (5) C A D B E (5) C A B D E (5) A C B E D (5) E D B A C (4) D E B C A (4) C D B A E (4) C B A E D (4) B E D C A (4) D E A B C (3) D C B E A (3) D C A E B (3) B E A C D (3) A D E C B (3) A C D B E (3) E B A C D (2) D A E C B (2) C B D E A (2) C B A D E (2) B C E D A (2) B C A E D (2) A D C E B (2) E B A D C (1) D E C B A (1) D E A C B (1) D C E A B (1) D C B A E (1) D B C E A (1) D A E B C (1) D A C E B (1) C D A B E (1) C A B E D (1) B E C A D (1) B D E C A (1) B A E C D (1) A E D B C (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -6 -14 8 B 10 0 -12 -16 12 C 6 12 0 -10 6 D 14 16 10 0 24 E -8 -12 -6 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -14 8 B 10 0 -12 -16 12 C 6 12 0 -10 6 D 14 16 10 0 24 E -8 -12 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=24 A=16 B=14 E=12 so E is eliminated. Round 2 votes counts: D=38 C=24 B=22 A=16 so A is eliminated. Round 3 votes counts: D=44 C=33 B=23 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:232 C:207 B:197 A:189 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -6 -14 8 B 10 0 -12 -16 12 C 6 12 0 -10 6 D 14 16 10 0 24 E -8 -12 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -14 8 B 10 0 -12 -16 12 C 6 12 0 -10 6 D 14 16 10 0 24 E -8 -12 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -14 8 B 10 0 -12 -16 12 C 6 12 0 -10 6 D 14 16 10 0 24 E -8 -12 -6 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8204: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) A B E D C (9) C D E B A (8) C D A E B (8) C A D B E (8) D C E B A (6) E B D A C (5) D E B A C (5) C A B E D (4) E B C A D (3) D A B E C (3) C D A B E (3) A C D B E (3) E D B C A (2) E D B A C (2) E B D C A (2) D E B C A (2) C E B A D (2) A B E C D (2) E B C D A (1) E B A D C (1) D E C A B (1) D C A E B (1) D A E B C (1) D A C E B (1) D A C B E (1) C E B D A (1) C D E A B (1) C B A E D (1) B E C A D (1) A D C B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -10 -8 -4 B 2 0 -2 -14 -6 C 10 2 0 -4 2 D 8 14 4 0 6 E 4 6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -8 -4 B 2 0 -2 -14 -6 C 10 2 0 -4 2 D 8 14 4 0 6 E 4 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=21 A=17 E=16 B=10 so B is eliminated. Round 2 votes counts: C=36 E=26 D=21 A=17 so A is eliminated. Round 3 votes counts: C=41 E=37 D=22 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:216 C:205 E:201 B:190 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -10 -8 -4 B 2 0 -2 -14 -6 C 10 2 0 -4 2 D 8 14 4 0 6 E 4 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -8 -4 B 2 0 -2 -14 -6 C 10 2 0 -4 2 D 8 14 4 0 6 E 4 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -8 -4 B 2 0 -2 -14 -6 C 10 2 0 -4 2 D 8 14 4 0 6 E 4 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8205: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (17) D A E C B (10) E A B D C (8) E A D B C (7) E B A D C (6) C B D A E (6) B C E A D (6) C D A B E (5) B E A D C (5) D C A E B (4) D A C E B (4) C D B A E (3) C B E A D (3) B E C A D (3) A E D B C (3) A D E B C (3) D E A B C (2) D A E B C (1) C B E D A (1) C B A E D (1) C B A D E (1) B E A C D (1) Total count = 100 A B C D E A 0 28 0 -6 16 B -28 0 -10 -18 -30 C 0 10 0 -6 2 D 6 18 6 0 12 E -16 30 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 0 -6 16 B -28 0 -10 -18 -30 C 0 10 0 -6 2 D 6 18 6 0 12 E -16 30 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=21 D=21 B=15 A=6 so A is eliminated. Round 2 votes counts: C=37 E=24 D=24 B=15 so B is eliminated. Round 3 votes counts: C=43 E=33 D=24 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:221 A:219 C:203 E:200 B:157 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 28 0 -6 16 B -28 0 -10 -18 -30 C 0 10 0 -6 2 D 6 18 6 0 12 E -16 30 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 0 -6 16 B -28 0 -10 -18 -30 C 0 10 0 -6 2 D 6 18 6 0 12 E -16 30 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 0 -6 16 B -28 0 -10 -18 -30 C 0 10 0 -6 2 D 6 18 6 0 12 E -16 30 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8206: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) B D C E A (8) A C E B D (8) A E C D B (6) B C D E A (5) D B A E C (4) C B E D A (4) B D E C A (4) D B E A C (3) C E A B D (3) C A E B D (3) C A B E D (3) B D C A E (3) A D B E C (3) A C E D B (3) E D A B C (2) D E B A C (2) D E A B C (2) D A B E C (2) C B E A D (2) C B A D E (2) A E D C B (2) A D E B C (2) A C B D E (2) E D B C A (1) E D B A C (1) E C B D A (1) E C A D B (1) E C A B D (1) E A D C B (1) E A D B C (1) D A E B C (1) C E B D A (1) C E B A D (1) C B A E D (1) B D A C E (1) B C D A E (1) Total count = 100 A B C D E A 0 -8 -8 -10 -6 B 8 0 10 8 14 C 8 -10 0 -4 2 D 10 -8 4 0 8 E 6 -14 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -10 -6 B 8 0 10 8 14 C 8 -10 0 -4 2 D 10 -8 4 0 8 E 6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 B=22 C=20 E=9 so E is eliminated. Round 2 votes counts: A=28 D=27 C=23 B=22 so B is eliminated. Round 3 votes counts: D=43 C=29 A=28 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:220 D:207 C:198 E:191 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -10 -6 B 8 0 10 8 14 C 8 -10 0 -4 2 D 10 -8 4 0 8 E 6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -10 -6 B 8 0 10 8 14 C 8 -10 0 -4 2 D 10 -8 4 0 8 E 6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -10 -6 B 8 0 10 8 14 C 8 -10 0 -4 2 D 10 -8 4 0 8 E 6 -14 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8207: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) D A C E B (7) E B A C D (6) C D A B E (6) E B A D C (5) C D A E B (5) B E A D C (5) B E A C D (5) B E C D A (4) A E B D C (4) C D B A E (3) C B D E A (3) B C E D A (3) A E D B C (3) E B C A D (2) D C A B E (2) C E D A B (2) B C D A E (2) A D E B C (2) E C D A B (1) E C A D B (1) E B C D A (1) E A D C B (1) E A B D C (1) D C B A E (1) D A C B E (1) C E B D A (1) C D E A B (1) C D B E A (1) C B E D A (1) C B D A E (1) B E C A D (1) B D A C E (1) B C D E A (1) B A E D C (1) B A D E C (1) A D C E B (1) A D C B E (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -4 -14 8 B -2 0 2 2 -6 C 4 -2 0 2 8 D 14 -2 -2 0 4 E -8 6 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.092593 B: 0.518519 C: 0.277778 D: 0.037037 E: 0.074074 Sum of squares = 0.361454046561 Cumulative probabilities = A: 0.092593 B: 0.611111 C: 0.888889 D: 0.925926 E: 1.000000 A B C D E A 0 2 -4 -14 8 B -2 0 2 2 -6 C 4 -2 0 2 8 D 14 -2 -2 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.092593 B: 0.518519 C: 0.277778 D: 0.037037 E: 0.074074 Sum of squares = 0.361454046636 Cumulative probabilities = A: 0.092593 B: 0.611111 C: 0.888889 D: 0.925926 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 D=20 E=18 A=14 so A is eliminated. Round 2 votes counts: B=26 E=25 D=25 C=24 so C is eliminated. Round 3 votes counts: D=41 B=31 E=28 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:207 C:206 B:198 A:196 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 -14 8 B -2 0 2 2 -6 C 4 -2 0 2 8 D 14 -2 -2 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.092593 B: 0.518519 C: 0.277778 D: 0.037037 E: 0.074074 Sum of squares = 0.361454046636 Cumulative probabilities = A: 0.092593 B: 0.611111 C: 0.888889 D: 0.925926 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -14 8 B -2 0 2 2 -6 C 4 -2 0 2 8 D 14 -2 -2 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.092593 B: 0.518519 C: 0.277778 D: 0.037037 E: 0.074074 Sum of squares = 0.361454046636 Cumulative probabilities = A: 0.092593 B: 0.611111 C: 0.888889 D: 0.925926 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -14 8 B -2 0 2 2 -6 C 4 -2 0 2 8 D 14 -2 -2 0 4 E -8 6 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.092593 B: 0.518519 C: 0.277778 D: 0.037037 E: 0.074074 Sum of squares = 0.361454046636 Cumulative probabilities = A: 0.092593 B: 0.611111 C: 0.888889 D: 0.925926 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8208: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) B A D C E (11) B A C D E (8) C D E B A (7) E A D C B (6) A B E D C (6) E A B C D (4) D C E A B (4) A E B D C (4) A B D E C (4) E D C A B (3) E B A C D (3) B C D A E (3) B A E C D (3) E A B D C (2) D C B A E (2) B A C E D (2) A E D C B (2) E C D B A (1) E C B D A (1) E A C D B (1) E A C B D (1) D E C A B (1) D A C E B (1) C D E A B (1) C D B A E (1) C B D E A (1) B C A D E (1) A D E C B (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 12 24 24 4 B -12 0 8 12 -10 C -24 -8 0 2 -12 D -24 -12 -2 0 -4 E -4 10 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 24 24 4 B -12 0 8 12 -10 C -24 -8 0 2 -12 D -24 -12 -2 0 -4 E -4 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=28 A=20 C=10 D=8 so D is eliminated. Round 2 votes counts: E=35 B=28 A=21 C=16 so C is eliminated. Round 3 votes counts: E=47 B=32 A=21 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:232 E:211 B:199 C:179 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 24 24 4 B -12 0 8 12 -10 C -24 -8 0 2 -12 D -24 -12 -2 0 -4 E -4 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 24 24 4 B -12 0 8 12 -10 C -24 -8 0 2 -12 D -24 -12 -2 0 -4 E -4 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 24 24 4 B -12 0 8 12 -10 C -24 -8 0 2 -12 D -24 -12 -2 0 -4 E -4 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8209: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) B C A D E (7) E D A C B (6) C A B D E (5) B C E D A (4) A B C D E (4) E D B C A (3) C B A E D (3) B D E A C (3) B D A E C (3) A C D E B (3) E D C B A (2) E D C A B (2) E D A B C (2) E C D B A (2) E C D A B (2) E B D C A (2) D E B A C (2) D B A E C (2) D A E B C (2) C E D A B (2) C E A D B (2) C B E A D (2) C A B E D (2) B E D C A (2) B C A E D (2) A C D B E (2) A C B D E (2) E A D C B (1) C E D B A (1) C E B D A (1) C E A B D (1) C B E D A (1) C B A D E (1) C A E D B (1) C A E B D (1) B D E C A (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C D E (1) A E D C B (1) A D E C B (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -10 -8 -6 B -4 0 6 6 6 C 10 -6 0 8 4 D 8 -6 -8 0 2 E 6 -6 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.500000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.379999999995 Cumulative probabilities = A: 0.300000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -8 -6 B -4 0 6 6 6 C 10 -6 0 8 4 D 8 -6 -8 0 2 E 6 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.500000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.379999999983 Cumulative probabilities = A: 0.300000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=23 E=22 A=16 D=13 so D is eliminated. Round 2 votes counts: E=31 B=28 C=23 A=18 so A is eliminated. Round 3 votes counts: E=35 B=35 C=30 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:207 D:198 E:197 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -10 -8 -6 B -4 0 6 6 6 C 10 -6 0 8 4 D 8 -6 -8 0 2 E 6 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.500000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.379999999983 Cumulative probabilities = A: 0.300000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -8 -6 B -4 0 6 6 6 C 10 -6 0 8 4 D 8 -6 -8 0 2 E 6 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.500000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.379999999983 Cumulative probabilities = A: 0.300000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -8 -6 B -4 0 6 6 6 C 10 -6 0 8 4 D 8 -6 -8 0 2 E 6 -6 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.500000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.379999999983 Cumulative probabilities = A: 0.300000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8210: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A B E D (8) E D A B C (6) D E A B C (5) C E A D B (5) E D C A B (4) C B A E D (4) B A C D E (4) E D C B A (3) E D A C B (3) D B E C A (3) C A E B D (3) B D C E A (3) B C A D E (3) B A D C E (3) A E D B C (3) E C D A B (2) E C A D B (2) E A D C B (2) C B A D E (2) A E D C B (2) A E C D B (2) A C E B D (2) E C D B A (1) E A D B C (1) E A C D B (1) D E B C A (1) D B E A C (1) D A B E C (1) C E A B D (1) C B E A D (1) C B D E A (1) C A B D E (1) B D E C A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D A E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -2 4 -12 B -12 0 0 -14 -16 C 2 0 0 -12 -12 D -4 14 12 0 -14 E 12 16 12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -2 4 -12 B -12 0 0 -14 -16 C 2 0 0 -12 -12 D -4 14 12 0 -14 E 12 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 D=20 B=18 A=11 so A is eliminated. Round 2 votes counts: E=32 C=28 D=20 B=20 so D is eliminated. Round 3 votes counts: E=47 C=28 B=25 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:227 D:204 A:201 C:189 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -2 4 -12 B -12 0 0 -14 -16 C 2 0 0 -12 -12 D -4 14 12 0 -14 E 12 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 4 -12 B -12 0 0 -14 -16 C 2 0 0 -12 -12 D -4 14 12 0 -14 E 12 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 4 -12 B -12 0 0 -14 -16 C 2 0 0 -12 -12 D -4 14 12 0 -14 E 12 16 12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8211: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (5) C D E A B (5) B A D C E (5) E D C A B (4) B D A E C (4) A E C B D (4) A B E C D (4) A B C D E (4) D C E B A (3) C E D A B (3) C D B E A (3) C A D E B (3) B D E A C (3) B D C A E (3) B A C D E (3) A B E D C (3) E D B A C (2) E C D A B (2) E A D B C (2) C D A E B (2) C A E D B (2) C A D B E (2) B D E C A (2) B A E D C (2) A C B D E (2) E D A B C (1) E C A D B (1) E B D A C (1) E A D C B (1) E A C D B (1) E A B D C (1) D E B C A (1) D C B E A (1) D B E C A (1) D B C E A (1) C D E B A (1) C D B A E (1) C D A B E (1) C B D A E (1) B C A D E (1) B A D E C (1) A E C D B (1) A E B D C (1) A E B C D (1) A C E D B (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 2 -2 10 B -10 0 -4 -2 0 C -2 4 0 4 2 D 2 2 -4 0 8 E -10 0 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000018 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 -2 10 B -10 0 -4 -2 0 C -2 4 0 4 2 D 2 2 -4 0 8 E -10 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000093 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 A=24 E=21 D=7 so D is eliminated. Round 2 votes counts: C=28 B=26 A=24 E=22 so E is eliminated. Round 3 votes counts: C=40 B=30 A=30 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:204 D:204 B:192 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 -2 10 B -10 0 -4 -2 0 C -2 4 0 4 2 D 2 2 -4 0 8 E -10 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000093 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 -2 10 B -10 0 -4 -2 0 C -2 4 0 4 2 D 2 2 -4 0 8 E -10 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000093 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 -2 10 B -10 0 -4 -2 0 C -2 4 0 4 2 D 2 2 -4 0 8 E -10 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000093 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8212: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (14) E C B D A (12) B D A E C (10) E B C D A (6) B D A C E (6) E C A D B (5) C E B D A (5) C E A D B (5) E A D B C (4) C A D B E (4) A D C B E (4) A D B E C (4) B E D C A (3) E B D A C (2) D A B C E (2) B E D A C (2) A C D E B (2) E C B A D (1) E B A D C (1) D A B E C (1) C B E D A (1) C A D E B (1) B D C A E (1) B C D A E (1) B A D E C (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 10 -4 6 B 4 0 18 4 8 C -10 -18 0 -14 -6 D 4 -4 14 0 4 E -6 -8 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999693 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 10 -4 6 B 4 0 18 4 8 C -10 -18 0 -14 -6 D 4 -4 14 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=26 B=24 C=16 D=3 so D is eliminated. Round 2 votes counts: E=31 A=29 B=24 C=16 so C is eliminated. Round 3 votes counts: E=41 A=34 B=25 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:217 D:209 A:204 E:194 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 10 -4 6 B 4 0 18 4 8 C -10 -18 0 -14 -6 D 4 -4 14 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -4 6 B 4 0 18 4 8 C -10 -18 0 -14 -6 D 4 -4 14 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -4 6 B 4 0 18 4 8 C -10 -18 0 -14 -6 D 4 -4 14 0 4 E -6 -8 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999579 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8213: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) B C E D A (7) A D E C B (7) A D C E B (6) E B C D A (5) C D A B E (5) B E C A D (5) A D C B E (5) E C B D A (4) E B A D C (4) E B A C D (4) C D B E A (3) C D B A E (3) C B D E A (3) B C D E A (3) E D A C B (2) E A B D C (2) D C A E B (2) D C A B E (2) C D E B A (2) C B E D A (2) A E D B C (2) D A C E B (1) D A C B E (1) C D E A B (1) B E A D C (1) B E A C D (1) B C A D E (1) B A E D C (1) B A E C D (1) B A C E D (1) A E D C B (1) A E B D C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -14 -10 -16 B 20 0 0 12 10 C 14 0 0 20 -2 D 10 -12 -20 0 -6 E 16 -10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.640576 C: 0.359424 D: 0.000000 E: 0.000000 Sum of squares = 0.539523457734 Cumulative probabilities = A: 0.000000 B: 0.640576 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -14 -10 -16 B 20 0 0 12 10 C 14 0 0 20 -2 D 10 -12 -20 0 -6 E 16 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 E=21 C=19 D=6 so D is eliminated. Round 2 votes counts: B=30 A=26 C=23 E=21 so E is eliminated. Round 3 votes counts: B=43 A=30 C=27 so C is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:216 E:207 D:186 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -14 -10 -16 B 20 0 0 12 10 C 14 0 0 20 -2 D 10 -12 -20 0 -6 E 16 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 -10 -16 B 20 0 0 12 10 C 14 0 0 20 -2 D 10 -12 -20 0 -6 E 16 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 -10 -16 B 20 0 0 12 10 C 14 0 0 20 -2 D 10 -12 -20 0 -6 E 16 -10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8214: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (13) C A E B D (7) A D C B E (6) E B C D A (5) D B E A C (5) D A C B E (5) D A B E C (5) B E D A C (5) B D E A C (5) E B D C A (4) A D B C E (4) A C D E B (4) E C B A D (3) E B C A D (3) C A D E B (3) C E A B D (2) C D A E B (2) B E D C A (2) B E C A D (2) A C D B E (2) E C B D A (1) D E C B A (1) D C E B A (1) D C A E B (1) D B A E C (1) D A C E B (1) D A B C E (1) C E B D A (1) C D E A B (1) B E C D A (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -6 4 -10 B 6 0 -8 12 -6 C 6 8 0 2 8 D -4 -12 -2 0 0 E 10 6 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 4 -10 B 6 0 -8 12 -6 C 6 8 0 2 8 D -4 -12 -2 0 0 E 10 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 A=19 E=16 B=15 so B is eliminated. Round 2 votes counts: C=29 E=26 D=26 A=19 so A is eliminated. Round 3 votes counts: D=38 C=35 E=27 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:204 B:202 A:191 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 4 -10 B 6 0 -8 12 -6 C 6 8 0 2 8 D -4 -12 -2 0 0 E 10 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 4 -10 B 6 0 -8 12 -6 C 6 8 0 2 8 D -4 -12 -2 0 0 E 10 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 4 -10 B 6 0 -8 12 -6 C 6 8 0 2 8 D -4 -12 -2 0 0 E 10 6 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8215: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) C D B E A (6) C B D E A (6) A E B C D (6) A E B D C (5) D E B A C (4) D B E C A (4) B E A D C (4) E D A B C (3) E A D B C (3) D C E A B (3) D C B E A (3) C A B E D (3) A B E C D (3) E D B A C (2) E B A D C (2) E A B D C (2) C D B A E (2) C B D A E (2) C A D E B (2) B E D C A (2) B E D A C (2) B C A E D (2) E B D A C (1) D E C B A (1) D E A B C (1) D C E B A (1) D C A E B (1) D B C E A (1) C D E B A (1) C D A B E (1) C B A D E (1) C A D B E (1) C A B D E (1) B D C E A (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D C B (1) A E C B D (1) A D E B C (1) A C E D B (1) A C D E B (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 6 0 -10 B 2 0 20 -2 0 C -6 -20 0 -14 -14 D 0 2 14 0 -8 E 10 0 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.378612 C: 0.000000 D: 0.000000 E: 0.621388 Sum of squares = 0.529470168958 Cumulative probabilities = A: 0.000000 B: 0.378612 C: 0.378612 D: 0.378612 E: 1.000000 A B C D E A 0 -2 6 0 -10 B 2 0 20 -2 0 C -6 -20 0 -14 -14 D 0 2 14 0 -8 E 10 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=26 D=19 B=14 E=13 so E is eliminated. Round 2 votes counts: A=33 C=26 D=24 B=17 so B is eliminated. Round 3 votes counts: A=40 D=30 C=30 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:216 B:210 D:204 A:197 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 0 -10 B 2 0 20 -2 0 C -6 -20 0 -14 -14 D 0 2 14 0 -8 E 10 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 -10 B 2 0 20 -2 0 C -6 -20 0 -14 -14 D 0 2 14 0 -8 E 10 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 -10 B 2 0 20 -2 0 C -6 -20 0 -14 -14 D 0 2 14 0 -8 E 10 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8216: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (12) C E A D B (8) B D C E A (8) B D A E C (8) D B C E A (5) C A E B D (5) A E D B C (5) C E A B D (4) C B D E A (4) E A D C B (3) D B A E C (3) C A E D B (3) E D A C B (2) E C A D B (2) D E C A B (2) D E A B C (2) C B E D A (2) C B A E D (2) B D E C A (2) B C D E A (2) B C D A E (2) A E B D C (2) E A C D B (1) D E C B A (1) D C E B A (1) C E B A D (1) C D B E A (1) B D E A C (1) B D A C E (1) B C A D E (1) A E D C B (1) A E C B D (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -12 6 -4 B -10 0 -14 -4 -14 C 12 14 0 4 2 D -6 4 -4 0 -10 E 4 14 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999291 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 6 -4 B -10 0 -14 -4 -14 C 12 14 0 4 2 D -6 4 -4 0 -10 E 4 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=25 A=23 D=14 E=8 so E is eliminated. Round 2 votes counts: C=32 A=27 B=25 D=16 so D is eliminated. Round 3 votes counts: C=36 B=33 A=31 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:216 E:213 A:200 D:192 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 6 -4 B -10 0 -14 -4 -14 C 12 14 0 4 2 D -6 4 -4 0 -10 E 4 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 6 -4 B -10 0 -14 -4 -14 C 12 14 0 4 2 D -6 4 -4 0 -10 E 4 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 6 -4 B -10 0 -14 -4 -14 C 12 14 0 4 2 D -6 4 -4 0 -10 E 4 14 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8217: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) E D C B A (8) C B E D A (6) A B C D E (6) D E C B A (5) D E A C B (5) D A E C B (5) B C E D A (5) E D C A B (4) E C B D A (4) B C E A D (4) A D E B C (4) A D B C E (4) A B D C E (4) A B C E D (4) E C D B A (3) B C A E D (3) B A C E D (3) D E C A B (2) D A B E C (2) C E B D A (2) D B C E A (1) D B C A E (1) C E D B A (1) B D C E A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C B D (1) Total count = 100 A B C D E A 0 2 -4 -14 -4 B -2 0 -12 -10 -8 C 4 12 0 -12 -6 D 14 10 12 0 2 E 4 8 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -14 -4 B -2 0 -12 -10 -8 C 4 12 0 -12 -6 D 14 10 12 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=21 E=19 B=18 C=9 so C is eliminated. Round 2 votes counts: A=33 B=24 E=22 D=21 so D is eliminated. Round 3 votes counts: A=40 E=34 B=26 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:219 E:208 C:199 A:190 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -14 -4 B -2 0 -12 -10 -8 C 4 12 0 -12 -6 D 14 10 12 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -14 -4 B -2 0 -12 -10 -8 C 4 12 0 -12 -6 D 14 10 12 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -14 -4 B -2 0 -12 -10 -8 C 4 12 0 -12 -6 D 14 10 12 0 2 E 4 8 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999992622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8218: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) A C B D E (8) E D B C A (6) C A B D E (6) B E D C A (6) B C A E D (6) A C D E B (6) E D B A C (5) E D A B C (5) D E A B C (5) A D E C B (5) C B A E D (4) C B A D E (4) B E C D A (4) B C E D A (4) E B D C A (3) D A E C B (2) B C E A D (2) A D C E B (2) A C B E D (2) D E B A C (1) C B D A E (1) C A B E D (1) B C A D E (1) A E D C B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 4 -2 0 B -6 0 -4 4 0 C -4 4 0 0 -4 D 2 -4 0 0 2 E 0 0 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888881 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -2 0 B -6 0 -4 4 0 C -4 4 0 0 -4 D 2 -4 0 0 2 E 0 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=23 E=19 D=17 C=16 so C is eliminated. Round 2 votes counts: B=32 A=32 E=19 D=17 so D is eliminated. Round 3 votes counts: E=34 A=34 B=32 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:204 E:201 D:200 C:198 B:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 -2 0 B -6 0 -4 4 0 C -4 4 0 0 -4 D 2 -4 0 0 2 E 0 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -2 0 B -6 0 -4 4 0 C -4 4 0 0 -4 D 2 -4 0 0 2 E 0 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -2 0 B -6 0 -4 4 0 C -4 4 0 0 -4 D 2 -4 0 0 2 E 0 0 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8219: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (7) E C D B A (6) A D E C B (6) D A E C B (5) A B D C E (5) A E D B C (4) A B D E C (4) E D C A B (3) E C B D A (3) C D E B A (3) C B E D A (3) B C A E D (3) A D E B C (3) A D B E C (3) A B E D C (3) D C E B A (2) D C E A B (2) D A C E B (2) C E D B A (2) C E B D A (2) C D B E A (2) B C E A D (2) B C A D E (2) B A D C E (2) B A C E D (2) B A C D E (2) A D B C E (2) E D A C B (1) E C D A B (1) E C B A D (1) E B A C D (1) E A D C B (1) E A B C D (1) D E C A B (1) D E A C B (1) B E C A D (1) B C D E A (1) B C D A E (1) B A E C D (1) A E B D C (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 4 4 6 B -2 0 4 -2 -6 C -4 -4 0 -4 -4 D -4 2 4 0 0 E -6 6 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 4 6 B -2 0 4 -2 -6 C -4 -4 0 -4 -4 D -4 2 4 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=24 E=18 D=13 C=12 so C is eliminated. Round 2 votes counts: A=33 B=27 E=22 D=18 so D is eliminated. Round 3 votes counts: A=40 E=31 B=29 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:202 D:201 B:197 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 4 6 B -2 0 4 -2 -6 C -4 -4 0 -4 -4 D -4 2 4 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 4 6 B -2 0 4 -2 -6 C -4 -4 0 -4 -4 D -4 2 4 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 4 6 B -2 0 4 -2 -6 C -4 -4 0 -4 -4 D -4 2 4 0 0 E -6 6 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8220: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A D C B (8) E D A C B (5) D C E A B (5) D C B E A (5) C D E A B (5) B A E D C (5) A E B D C (5) A B E C D (5) B A E C D (4) D C E B A (3) B A C E D (3) A E C D B (3) A E B C D (3) E A C D B (2) D E C A B (2) D B E C A (2) C D B E A (2) B D E A C (2) B C A E D (2) B C A D E (2) A E D C B (2) A E D B C (2) A B E D C (2) E D C A B (1) E D A B C (1) E A D B C (1) C B D A E (1) C A E D B (1) B E A D C (1) B D C E A (1) B D A C E (1) B C D E A (1) B A C D E (1) A E C B D (1) Total count = 100 A B C D E A 0 8 14 6 6 B -8 0 8 0 0 C -14 -8 0 -8 -14 D -6 0 8 0 -14 E -6 0 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 6 6 B -8 0 8 0 0 C -14 -8 0 -8 -14 D -6 0 8 0 -14 E -6 0 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=23 E=18 D=17 C=9 so C is eliminated. Round 2 votes counts: B=34 D=24 A=24 E=18 so E is eliminated. Round 3 votes counts: A=35 B=34 D=31 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:211 B:200 D:194 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 6 6 B -8 0 8 0 0 C -14 -8 0 -8 -14 D -6 0 8 0 -14 E -6 0 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 6 6 B -8 0 8 0 0 C -14 -8 0 -8 -14 D -6 0 8 0 -14 E -6 0 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 6 6 B -8 0 8 0 0 C -14 -8 0 -8 -14 D -6 0 8 0 -14 E -6 0 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8221: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) E C A D B (9) E A C D B (9) C E B D A (9) D A B E C (7) B D A C E (7) A E C D B (6) A D B E C (6) B D C A E (5) D B A E C (4) B D A E C (4) B C D E A (4) C E A D B (3) C B E D A (3) B D C E A (3) A E D C B (3) A D E B C (3) C E B A D (2) E A D C B (1) Total count = 100 A B C D E A 0 18 0 8 -10 B -18 0 -14 -2 -14 C 0 14 0 14 -4 D -8 2 -14 0 -14 E 10 14 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 0 8 -10 B -18 0 -14 -2 -14 C 0 14 0 14 -4 D -8 2 -14 0 -14 E 10 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=23 E=19 A=18 D=11 so D is eliminated. Round 2 votes counts: C=29 B=27 A=25 E=19 so E is eliminated. Round 3 votes counts: C=38 A=35 B=27 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:221 C:212 A:208 D:183 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 0 8 -10 B -18 0 -14 -2 -14 C 0 14 0 14 -4 D -8 2 -14 0 -14 E 10 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 8 -10 B -18 0 -14 -2 -14 C 0 14 0 14 -4 D -8 2 -14 0 -14 E 10 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 8 -10 B -18 0 -14 -2 -14 C 0 14 0 14 -4 D -8 2 -14 0 -14 E 10 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8222: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) C D A E B (8) A B E D C (7) B E A D C (6) C D E A B (5) A E B D C (5) A B E C D (5) E A B D C (4) D E B C A (4) B A E D C (4) B A E C D (4) E D B A C (3) D C E A B (3) C D E B A (3) C D B A E (3) C D A B E (3) B A C E D (3) E B A D C (2) C A D B E (2) A C B E D (2) A B C E D (2) E D A C B (1) E D A B C (1) E B D A C (1) D E C B A (1) D E C A B (1) C D B E A (1) C B D E A (1) C A B D E (1) B E D A C (1) A E D C B (1) A E C B D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 10 2 6 B -8 0 4 0 -10 C -10 -4 0 -8 -4 D -2 0 8 0 -10 E -6 10 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 2 6 B -8 0 4 0 -10 C -10 -4 0 -8 -4 D -2 0 8 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=25 D=18 B=18 E=12 so E is eliminated. Round 2 votes counts: A=29 C=27 D=23 B=21 so B is eliminated. Round 3 votes counts: A=48 C=27 D=25 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:209 D:198 B:193 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 2 6 B -8 0 4 0 -10 C -10 -4 0 -8 -4 D -2 0 8 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 2 6 B -8 0 4 0 -10 C -10 -4 0 -8 -4 D -2 0 8 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 2 6 B -8 0 4 0 -10 C -10 -4 0 -8 -4 D -2 0 8 0 -10 E -6 10 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8223: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (12) B D A C E (7) C E B A D (6) A D B C E (6) E C B D A (5) E C B A D (5) B C E D A (5) E C A B D (4) D A B C E (4) E C D A B (3) D B A E C (3) C E B D A (3) A D E C B (3) A C E D B (3) D A E B C (2) D A B E C (2) C A E B D (2) B E C D A (2) B D C E A (2) B C E A D (2) A E D C B (2) A D C E B (2) A D C B E (2) A D B E C (2) E C D B A (1) E A C D B (1) D A E C B (1) C E A B D (1) C B E A D (1) B D E C A (1) B C D E A (1) B C A D E (1) B A D C E (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -14 16 -10 B -8 0 -16 0 -16 C 14 16 0 18 0 D -16 0 -18 0 -18 E 10 16 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.349652 D: 0.000000 E: 0.650348 Sum of squares = 0.545209112437 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.349652 D: 0.349652 E: 1.000000 A B C D E A 0 8 -14 16 -10 B -8 0 -16 0 -16 C 14 16 0 18 0 D -16 0 -18 0 -18 E 10 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=22 A=22 C=13 D=12 so D is eliminated. Round 2 votes counts: E=31 A=31 B=25 C=13 so C is eliminated. Round 3 votes counts: E=41 A=33 B=26 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:224 E:222 A:200 B:180 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -14 16 -10 B -8 0 -16 0 -16 C 14 16 0 18 0 D -16 0 -18 0 -18 E 10 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -14 16 -10 B -8 0 -16 0 -16 C 14 16 0 18 0 D -16 0 -18 0 -18 E 10 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -14 16 -10 B -8 0 -16 0 -16 C 14 16 0 18 0 D -16 0 -18 0 -18 E 10 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8224: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (13) B E D C A (7) E B D C A (5) E B C D A (5) B E D A C (5) E C B D A (4) D A B C E (3) C E A D B (3) C A E D B (3) B D E C A (3) B A E D C (3) A C E D B (3) A C E B D (3) A C D B E (3) E B C A D (2) C A D E B (2) B E A D C (2) B D E A C (2) B D A E C (2) A D B C E (2) A B E D C (2) A B D C E (2) E C D B A (1) E C B A D (1) E C A B D (1) E A B C D (1) D E C B A (1) D C E B A (1) D C B E A (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E B A (1) C D A E B (1) B E A C D (1) A E C B D (1) A D C B E (1) A C B E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 8 4 4 B -2 0 2 10 -8 C -8 -2 0 4 -2 D -4 -10 -4 0 -12 E -4 8 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 4 4 B -2 0 2 10 -8 C -8 -2 0 4 -2 D -4 -10 -4 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=25 E=20 D=11 C=11 so D is eliminated. Round 2 votes counts: A=37 B=27 E=21 C=15 so C is eliminated. Round 3 votes counts: A=45 B=28 E=27 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:209 B:201 C:196 D:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 4 4 B -2 0 2 10 -8 C -8 -2 0 4 -2 D -4 -10 -4 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 4 4 B -2 0 2 10 -8 C -8 -2 0 4 -2 D -4 -10 -4 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 4 4 B -2 0 2 10 -8 C -8 -2 0 4 -2 D -4 -10 -4 0 -12 E -4 8 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8225: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (8) D E B A C (6) C D A E B (5) E B D A C (4) C E D B A (4) C B E A D (4) A D C B E (4) A B D E C (4) D A E B C (3) D A C B E (3) C E B D A (3) C A B E D (3) B E A C D (3) A D B E C (3) E C B D A (2) E B D C A (2) E B C A D (2) D A C E B (2) D A B E C (2) C D E A B (2) C B A E D (2) C A B D E (2) B A E D C (2) B A E C D (2) A D B C E (2) A C D B E (2) A B E D C (2) A B C D E (2) E D B C A (1) E D B A C (1) E B A D C (1) D E C B A (1) D E C A B (1) D C E B A (1) D C A E B (1) D C A B E (1) D A E C B (1) C E D A B (1) C E B A D (1) C A E B D (1) B E C A D (1) B E A D C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 2 6 16 B -12 0 -10 -10 8 C -2 10 0 0 10 D -6 10 0 0 14 E -16 -8 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999194 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 6 16 B -12 0 -10 -10 8 C -2 10 0 0 10 D -6 10 0 0 14 E -16 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=22 A=20 E=13 B=9 so B is eliminated. Round 2 votes counts: C=36 A=24 D=22 E=18 so E is eliminated. Round 3 votes counts: C=41 D=30 A=29 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:218 C:209 D:209 B:188 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 6 16 B -12 0 -10 -10 8 C -2 10 0 0 10 D -6 10 0 0 14 E -16 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 6 16 B -12 0 -10 -10 8 C -2 10 0 0 10 D -6 10 0 0 14 E -16 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 6 16 B -12 0 -10 -10 8 C -2 10 0 0 10 D -6 10 0 0 14 E -16 -8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982195 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8226: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A C B E (9) C B E D A (9) E B A C D (8) A E B D C (8) C D B E A (6) D C A B E (5) B C E D A (5) A E D B C (5) D C B E A (4) D A C E B (4) B E C D A (3) A E B C D (3) A D E B C (3) A D C E B (3) A D C B E (3) E B C D A (2) E A B C D (2) D C B A E (2) E D B C A (1) C B D E A (1) B E C A D (1) A D E C B (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 -2 -4 B 4 0 4 6 0 C -2 -4 0 4 4 D 2 -6 -4 0 -16 E 4 0 -4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.769956 C: 0.000000 D: 0.000000 E: 0.230044 Sum of squares = 0.645752371077 Cumulative probabilities = A: 0.000000 B: 0.769956 C: 0.769956 D: 0.769956 E: 1.000000 A B C D E A 0 -4 2 -2 -4 B 4 0 4 6 0 C -2 -4 0 4 4 D 2 -6 -4 0 -16 E 4 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500257 C: 0.000000 D: 0.000000 E: 0.499743 Sum of squares = 0.500000132454 Cumulative probabilities = A: 0.000000 B: 0.500257 C: 0.500257 D: 0.500257 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=24 E=23 C=16 B=9 so B is eliminated. Round 2 votes counts: A=28 E=27 D=24 C=21 so C is eliminated. Round 3 votes counts: E=41 D=31 A=28 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:208 B:207 C:201 A:196 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -2 -4 B 4 0 4 6 0 C -2 -4 0 4 4 D 2 -6 -4 0 -16 E 4 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500257 C: 0.000000 D: 0.000000 E: 0.499743 Sum of squares = 0.500000132454 Cumulative probabilities = A: 0.000000 B: 0.500257 C: 0.500257 D: 0.500257 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 -4 B 4 0 4 6 0 C -2 -4 0 4 4 D 2 -6 -4 0 -16 E 4 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500257 C: 0.000000 D: 0.000000 E: 0.499743 Sum of squares = 0.500000132454 Cumulative probabilities = A: 0.000000 B: 0.500257 C: 0.500257 D: 0.500257 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 -4 B 4 0 4 6 0 C -2 -4 0 4 4 D 2 -6 -4 0 -16 E 4 0 -4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500257 C: 0.000000 D: 0.000000 E: 0.499743 Sum of squares = 0.500000132454 Cumulative probabilities = A: 0.000000 B: 0.500257 C: 0.500257 D: 0.500257 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8227: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) B C D A E (7) E A C B D (6) D E A C B (6) C B E D A (6) A E D B C (6) A E B C D (5) E A D C B (4) B D C A E (4) B C A E D (4) D C B E A (3) D A E B C (3) C B D E A (3) E D A C B (2) E C D B A (2) E C A B D (2) D B A C E (2) C B E A D (2) B C E A D (2) B C D E A (2) A D B E C (2) A B E C D (2) A B C E D (2) E D C A B (1) E C B D A (1) E C B A D (1) E A C D B (1) C E B D A (1) C E B A D (1) B A C D E (1) A E D C B (1) A E C B D (1) A E B D C (1) A D E C B (1) A D E B C (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -2 8 B 0 0 10 12 6 C 0 -10 0 6 2 D 2 -12 -6 0 -8 E -8 -6 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.520394 B: 0.479606 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50083181115 Cumulative probabilities = A: 0.520394 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -2 8 B 0 0 10 12 6 C 0 -10 0 6 2 D 2 -12 -6 0 -8 E -8 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=22 E=20 B=20 C=13 so C is eliminated. Round 2 votes counts: B=31 A=25 E=22 D=22 so E is eliminated. Round 3 votes counts: A=38 B=35 D=27 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:214 A:203 C:199 E:196 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -2 8 B 0 0 10 12 6 C 0 -10 0 6 2 D 2 -12 -6 0 -8 E -8 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 8 B 0 0 10 12 6 C 0 -10 0 6 2 D 2 -12 -6 0 -8 E -8 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 8 B 0 0 10 12 6 C 0 -10 0 6 2 D 2 -12 -6 0 -8 E -8 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8228: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) D E B C A (5) D E A C B (5) B C A E D (5) A E D C B (5) E A D B C (4) D A E C B (4) B E D A C (4) B E A D C (4) B C A D E (4) E D A C B (3) E B D A C (3) C A D B E (3) B E D C A (3) E D B A C (2) E D A B C (2) E B A D C (2) E A D C B (2) D E C A B (2) C A D E B (2) B E C A D (2) B E A C D (2) B C D E A (2) A D E C B (2) A C D E B (2) D E C B A (1) D E A B C (1) D C A E B (1) D A C E B (1) C D A B E (1) C B D E A (1) C B D A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D A E (1) B A E C D (1) B A C E D (1) A C E D B (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 8 -4 -6 B -4 0 4 -10 -14 C -8 -4 0 -16 -24 D 4 10 16 0 0 E 6 14 24 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.810665 E: 0.189335 Sum of squares = 0.693025066822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.810665 E: 1.000000 A B C D E A 0 4 8 -4 -6 B -4 0 4 -10 -14 C -8 -4 0 -16 -24 D 4 10 16 0 0 E 6 14 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.49999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=20 E=18 C=17 A=13 so A is eliminated. Round 2 votes counts: B=33 E=23 D=22 C=22 so D is eliminated. Round 3 votes counts: E=43 B=33 C=24 so C is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:215 A:201 B:188 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -4 -6 B -4 0 4 -10 -14 C -8 -4 0 -16 -24 D 4 10 16 0 0 E 6 14 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.49999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -4 -6 B -4 0 4 -10 -14 C -8 -4 0 -16 -24 D 4 10 16 0 0 E 6 14 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.49999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -4 -6 B -4 0 4 -10 -14 C -8 -4 0 -16 -24 D 4 10 16 0 0 E 6 14 24 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.500001 Sum of squares = 0.49999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499999 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8229: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) B C A D E (6) D C B A E (5) B C D A E (5) E D A C B (4) E D A B C (4) C D B A E (4) B C A E D (4) B A E C D (4) E A C D B (3) D B C E A (3) A E C B D (3) E B A D C (2) E A D C B (2) E A D B C (2) E A B D C (2) D E C A B (2) D E B C A (2) D C E A B (2) D C B E A (2) C B A D E (2) C A B E D (2) B E A C D (2) B D C E A (2) B A C E D (2) E D B A C (1) E A C B D (1) E A B C D (1) D C E B A (1) D C A E B (1) D C A B E (1) C B D A E (1) C B A E D (1) C A B D E (1) B E D A C (1) B D E C A (1) B D C A E (1) A E C D B (1) A E B C D (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 2 -10 -4 B 4 0 -4 -4 4 C -2 4 0 -6 -4 D 10 4 6 0 8 E 4 -4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -10 -4 B 4 0 -4 -4 4 C -2 4 0 -6 -4 D 10 4 6 0 8 E 4 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=28 E=22 C=11 A=8 so A is eliminated. Round 2 votes counts: D=31 B=30 E=27 C=12 so C is eliminated. Round 3 votes counts: B=37 D=35 E=28 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:200 E:198 C:196 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -10 -4 B 4 0 -4 -4 4 C -2 4 0 -6 -4 D 10 4 6 0 8 E 4 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -10 -4 B 4 0 -4 -4 4 C -2 4 0 -6 -4 D 10 4 6 0 8 E 4 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -10 -4 B 4 0 -4 -4 4 C -2 4 0 -6 -4 D 10 4 6 0 8 E 4 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8230: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (14) B E C A D (9) B A E C D (7) D A C E B (6) A B D E C (6) C E D B A (5) C E B D A (5) A D C E B (5) D C E A B (4) A B E C D (4) E C B D A (3) D B E C A (3) A C E D B (3) D C E B A (2) C E D A B (2) C E A B D (2) A D C B E (2) A D B E C (2) A C D E B (2) E C D B A (1) E C B A D (1) D E C B A (1) D B E A C (1) D B A C E (1) D A B E C (1) D A B C E (1) B E D C A (1) B E A C D (1) B D A E C (1) B A D E C (1) A D B C E (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -6 -4 -10 B 14 0 10 14 14 C 6 -10 0 20 -16 D 4 -14 -20 0 -20 E 10 -14 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -4 -10 B 14 0 10 14 14 C 6 -10 0 20 -16 D 4 -14 -20 0 -20 E 10 -14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=27 D=20 C=14 E=5 so E is eliminated. Round 2 votes counts: B=34 A=27 D=20 C=19 so C is eliminated. Round 3 votes counts: B=43 A=29 D=28 so D is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 E:216 C:200 A:183 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 -10 B 14 0 10 14 14 C 6 -10 0 20 -16 D 4 -14 -20 0 -20 E 10 -14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 -10 B 14 0 10 14 14 C 6 -10 0 20 -16 D 4 -14 -20 0 -20 E 10 -14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 -10 B 14 0 10 14 14 C 6 -10 0 20 -16 D 4 -14 -20 0 -20 E 10 -14 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8231: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (10) D A E C B (7) C B D E A (5) B C E A D (5) E B A C D (4) D C B A E (4) C B E A D (4) A E D B C (4) D A E B C (3) D A C B E (3) D A B E C (3) D A B C E (3) C D B A E (3) C B E D A (3) B E C A D (3) B C E D A (3) A E D C B (3) E A D C B (2) E A C D B (2) E A B D C (2) E A B C D (2) C E B D A (2) B D C A E (2) B C D A E (2) A D E C B (2) A D E B C (2) E C A D B (1) E B C A D (1) E A C B D (1) D C A E B (1) D B C A E (1) D B A C E (1) C E D A B (1) C D E B A (1) C D B E A (1) C D A E B (1) C B D A E (1) B C D E A (1) Total count = 100 A B C D E A 0 6 8 -24 12 B -6 0 -16 -18 -4 C -8 16 0 -6 16 D 24 18 6 0 14 E -12 4 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -24 12 B -6 0 -16 -18 -4 C -8 16 0 -6 16 D 24 18 6 0 14 E -12 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=22 B=16 E=15 A=11 so A is eliminated. Round 2 votes counts: D=40 E=22 C=22 B=16 so B is eliminated. Round 3 votes counts: D=42 C=33 E=25 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:231 C:209 A:201 E:181 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -24 12 B -6 0 -16 -18 -4 C -8 16 0 -6 16 D 24 18 6 0 14 E -12 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -24 12 B -6 0 -16 -18 -4 C -8 16 0 -6 16 D 24 18 6 0 14 E -12 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -24 12 B -6 0 -16 -18 -4 C -8 16 0 -6 16 D 24 18 6 0 14 E -12 4 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8232: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) B D A C E (7) D E A B C (5) D B A E C (4) C B E D A (4) B D C A E (4) B C D A E (4) B A D C E (4) E D C A B (3) E D A C B (3) D A B E C (3) B D C E A (3) B C E D A (3) B C A D E (3) A C B E D (3) E C D A B (2) D B E A C (2) C E B A D (2) C E A B D (2) C B E A D (2) C A B E D (2) B D E C A (2) A D B C E (2) A B D C E (2) A B C D E (2) E C D B A (1) E B C D A (1) E A D C B (1) E A C D B (1) D E B C A (1) D E B A C (1) D E A C B (1) C A E B D (1) B C A E D (1) B A C D E (1) A E C D B (1) A D E C B (1) A D E B C (1) A D B E C (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 0 0 -8 2 B 0 0 14 10 20 C 0 -14 0 -2 12 D 8 -10 2 0 12 E -2 -20 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.288876 B: 0.711124 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.589146545571 Cumulative probabilities = A: 0.288876 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -8 2 B 0 0 14 10 20 C 0 -14 0 -2 12 D 8 -10 2 0 12 E -2 -20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000009 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=21 D=17 A=17 C=13 so C is eliminated. Round 2 votes counts: B=38 E=25 A=20 D=17 so D is eliminated. Round 3 votes counts: B=44 E=33 A=23 so A is eliminated. Round 4 votes counts: B=60 E=40 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:206 C:198 A:197 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -8 2 B 0 0 14 10 20 C 0 -14 0 -2 12 D 8 -10 2 0 12 E -2 -20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000009 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -8 2 B 0 0 14 10 20 C 0 -14 0 -2 12 D 8 -10 2 0 12 E -2 -20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000009 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -8 2 B 0 0 14 10 20 C 0 -14 0 -2 12 D 8 -10 2 0 12 E -2 -20 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500002 B: 0.499998 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000009 Cumulative probabilities = A: 0.500002 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8233: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) C E A B D (6) C A B D E (6) B A D C E (6) E D B A C (5) E C B A D (5) D B E A C (4) C E B A D (4) C E A D B (4) E C D A B (3) B E D A C (3) B D E A C (3) B D A E C (3) E D B C A (2) E C B D A (2) E B D C A (2) D E B A C (2) D E A B C (2) D B A C E (2) D A B E C (2) D A B C E (2) C A E D B (2) B C A E D (2) B A C D E (2) A C D B E (2) A B D C E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C A B D (1) E B D A C (1) E B C A D (1) D A E C B (1) D A C B E (1) C A D B E (1) C A B E D (1) B D A C E (1) A D C B E (1) A D B C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 10 -2 -6 B 18 0 12 6 8 C -10 -12 0 -10 -4 D 2 -6 10 0 6 E 6 -8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 10 -2 -6 B 18 0 12 6 8 C -10 -12 0 -10 -4 D 2 -6 10 0 6 E 6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 D=23 B=20 A=8 so A is eliminated. Round 2 votes counts: C=27 E=25 D=25 B=23 so B is eliminated. Round 3 votes counts: D=40 C=32 E=28 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:222 D:206 E:198 A:192 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 10 -2 -6 B 18 0 12 6 8 C -10 -12 0 -10 -4 D 2 -6 10 0 6 E 6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 10 -2 -6 B 18 0 12 6 8 C -10 -12 0 -10 -4 D 2 -6 10 0 6 E 6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 10 -2 -6 B 18 0 12 6 8 C -10 -12 0 -10 -4 D 2 -6 10 0 6 E 6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8234: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) C B D A E (8) C B E D A (7) C D A B E (6) E B A D C (4) E A D B C (4) D A C B E (4) C E B A D (4) E B C A D (3) D A C E B (3) D A B E C (3) D A B C E (3) B D A C E (3) E C B A D (2) E C A D B (2) E B A C D (2) E A B D C (2) B E D A C (2) B E C A D (2) B D A E C (2) B C E A D (2) B C D A E (2) A D E C B (2) A D C E B (2) E A D C B (1) D C A E B (1) D C A B E (1) C E B D A (1) C E A D B (1) C D E A B (1) C D B A E (1) C D A E B (1) C B E A D (1) C B D E A (1) B E C D A (1) B D C A E (1) B C E D A (1) B A D E C (1) A E D B C (1) A E C D B (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 0 -6 12 B 2 0 -2 4 6 C 0 2 0 0 10 D 6 -4 0 0 12 E -12 -6 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.796899 D: 0.203101 E: 0.000000 Sum of squares = 0.676298082748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.796899 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -6 12 B 2 0 -2 4 6 C 0 2 0 0 10 D 6 -4 0 0 12 E -12 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555607008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=20 B=17 A=16 D=15 so D is eliminated. Round 2 votes counts: C=34 A=29 E=20 B=17 so B is eliminated. Round 3 votes counts: C=40 A=35 E=25 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:207 C:206 B:205 A:202 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 -6 12 B 2 0 -2 4 6 C 0 2 0 0 10 D 6 -4 0 0 12 E -12 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555607008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 12 B 2 0 -2 4 6 C 0 2 0 0 10 D 6 -4 0 0 12 E -12 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555607008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 12 B 2 0 -2 4 6 C 0 2 0 0 10 D 6 -4 0 0 12 E -12 -6 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555607008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8235: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) E B A D C (11) D C E A B (9) C A D B E (6) A B C D E (6) E B A C D (4) C D E A B (4) B E A C D (4) B A E C D (4) E D C B A (3) E B D A C (3) D C E B A (3) C D E B A (3) E D B A C (2) E B C D A (2) D E C B A (2) C D A E B (2) C A B D E (2) B E A D C (2) B A C E D (2) A C B D E (2) A B E D C (2) A B D C E (2) E C B D A (1) D C A E B (1) D C A B E (1) C B A D E (1) A D C B E (1) A D B E C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -4 4 -6 B -6 0 -6 0 0 C 4 6 0 14 16 D -4 0 -14 0 16 E 6 0 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 4 -6 B -6 0 -6 0 0 C 4 6 0 14 16 D -4 0 -14 0 16 E 6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=26 D=16 A=16 B=12 so B is eliminated. Round 2 votes counts: E=32 C=30 A=22 D=16 so D is eliminated. Round 3 votes counts: C=44 E=34 A=22 so A is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:200 D:199 B:194 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 4 -6 B -6 0 -6 0 0 C 4 6 0 14 16 D -4 0 -14 0 16 E 6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 4 -6 B -6 0 -6 0 0 C 4 6 0 14 16 D -4 0 -14 0 16 E 6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 4 -6 B -6 0 -6 0 0 C 4 6 0 14 16 D -4 0 -14 0 16 E 6 0 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8236: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) C A B E D (8) B D A C E (8) D B E A C (7) C A E B D (7) B A C D E (7) E C A D B (6) E D C A B (5) C E A B D (5) D E B A C (4) B D C A E (4) E D A C B (3) E C A B D (3) D B E C A (3) E A C D B (2) D E C B A (2) D E A B C (2) D B A C E (2) B C A D E (2) A B C E D (2) E C D A B (1) E A D C B (1) D E B C A (1) D E A C B (1) D B C E A (1) C B A E D (1) B A D C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 2 -8 6 B 6 0 8 -2 12 C -2 -8 0 -10 -2 D 8 2 10 0 10 E -6 -12 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -8 6 B 6 0 8 -2 12 C -2 -8 0 -10 -2 D 8 2 10 0 10 E -6 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=22 E=21 C=21 A=3 so A is eliminated. Round 2 votes counts: D=33 B=24 C=22 E=21 so E is eliminated. Round 3 votes counts: D=42 C=34 B=24 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:212 A:197 C:189 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -8 6 B 6 0 8 -2 12 C -2 -8 0 -10 -2 D 8 2 10 0 10 E -6 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -8 6 B 6 0 8 -2 12 C -2 -8 0 -10 -2 D 8 2 10 0 10 E -6 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -8 6 B 6 0 8 -2 12 C -2 -8 0 -10 -2 D 8 2 10 0 10 E -6 -12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8237: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) D C B A E (6) C E D B A (5) A B E D C (5) E C A B D (4) E B A D C (4) C E D A B (4) C D E B A (4) B A D E C (4) A B D C E (4) E C B D A (3) E B D A C (3) E A C B D (3) D B A E C (3) D B A C E (3) C D E A B (3) A B C D E (3) E A B C D (2) D A C B E (2) D A B C E (2) C A D B E (2) A E B C D (2) A C B D E (2) E D C B A (1) E D B A C (1) E C D B A (1) D E B C A (1) D E B A C (1) D B E C A (1) D B E A C (1) D B C E A (1) C E A B D (1) C A B D E (1) B E D A C (1) A C B E D (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 12 0 -18 10 B -12 0 -10 -8 14 C 0 10 0 10 14 D 18 8 -10 0 14 E -10 -14 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.213669 B: 0.000000 C: 0.786331 D: 0.000000 E: 0.000000 Sum of squares = 0.663970998929 Cumulative probabilities = A: 0.213669 B: 0.213669 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 -18 10 B -12 0 -10 -8 14 C 0 10 0 10 14 D 18 8 -10 0 14 E -10 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357143 B: 0.000000 C: 0.642857 D: 0.000000 E: 0.000000 Sum of squares = 0.540816374451 Cumulative probabilities = A: 0.357143 B: 0.357143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=22 D=21 A=20 B=5 so B is eliminated. Round 2 votes counts: C=32 A=24 E=23 D=21 so D is eliminated. Round 3 votes counts: C=39 A=34 E=27 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:217 D:215 A:202 B:192 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 0 -18 10 B -12 0 -10 -8 14 C 0 10 0 10 14 D 18 8 -10 0 14 E -10 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357143 B: 0.000000 C: 0.642857 D: 0.000000 E: 0.000000 Sum of squares = 0.540816374451 Cumulative probabilities = A: 0.357143 B: 0.357143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 -18 10 B -12 0 -10 -8 14 C 0 10 0 10 14 D 18 8 -10 0 14 E -10 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357143 B: 0.000000 C: 0.642857 D: 0.000000 E: 0.000000 Sum of squares = 0.540816374451 Cumulative probabilities = A: 0.357143 B: 0.357143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 -18 10 B -12 0 -10 -8 14 C 0 10 0 10 14 D 18 8 -10 0 14 E -10 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.357143 B: 0.000000 C: 0.642857 D: 0.000000 E: 0.000000 Sum of squares = 0.540816374451 Cumulative probabilities = A: 0.357143 B: 0.357143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8238: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) D C A B E (7) D E B A C (6) E B A D C (5) C E B A D (5) E D B A C (4) E B A C D (4) D C E A B (4) C A B D E (4) B E A C D (4) B A E C D (4) E D C B A (3) E B D A C (3) C D A B E (3) C A B E D (3) B E A D C (3) B A E D C (3) C E D B A (2) C D E B A (2) A B D E C (2) E D B C A (1) E B C D A (1) E B C A D (1) D E C A B (1) D E B C A (1) D E A C B (1) D E A B C (1) D C E B A (1) D C A E B (1) D A E B C (1) D A C B E (1) D A B E C (1) C D E A B (1) C B A E D (1) C A D B E (1) B C E A D (1) A D B C E (1) A C D B E (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -24 -2 -6 -24 B 24 0 0 -4 -12 C 2 0 0 -18 -18 D 6 4 18 0 -4 E 24 12 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999239 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 -2 -6 -24 B 24 0 0 -4 -12 C 2 0 0 -18 -18 D 6 4 18 0 -4 E 24 12 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=22 C=22 B=15 A=8 so A is eliminated. Round 2 votes counts: D=34 C=24 E=22 B=20 so B is eliminated. Round 3 votes counts: E=38 D=36 C=26 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 D:212 B:204 C:183 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 -2 -6 -24 B 24 0 0 -4 -12 C 2 0 0 -18 -18 D 6 4 18 0 -4 E 24 12 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -2 -6 -24 B 24 0 0 -4 -12 C 2 0 0 -18 -18 D 6 4 18 0 -4 E 24 12 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -2 -6 -24 B 24 0 0 -4 -12 C 2 0 0 -18 -18 D 6 4 18 0 -4 E 24 12 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8239: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) C D A B E (7) C A B D E (7) D C A B E (6) B A E C D (6) C B A E D (4) B A C E D (4) E D B A C (3) E C B A D (3) E B D A C (3) E B A C D (3) D C A E B (3) C E D B A (3) A B D C E (3) E D C B A (2) E D B C A (2) E C B D A (2) E B A D C (2) D E B A C (2) D E A B C (2) D C E A B (2) D A E B C (2) D A B E C (2) D A B C E (2) C E B A D (2) C D E B A (2) C A B E D (2) A C B D E (2) A B C E D (2) A B C D E (2) E C D B A (1) C E B D A (1) C D E A B (1) C A D B E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -16 -10 14 B -10 0 -16 0 4 C 16 16 0 10 12 D 10 0 -10 0 8 E -14 -4 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -16 -10 14 B -10 0 -16 0 4 C 16 16 0 10 12 D 10 0 -10 0 8 E -14 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=28 E=21 A=11 B=10 so B is eliminated. Round 2 votes counts: C=30 D=28 E=21 A=21 so E is eliminated. Round 3 votes counts: D=38 C=36 A=26 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:204 A:199 B:189 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -16 -10 14 B -10 0 -16 0 4 C 16 16 0 10 12 D 10 0 -10 0 8 E -14 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -16 -10 14 B -10 0 -16 0 4 C 16 16 0 10 12 D 10 0 -10 0 8 E -14 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -16 -10 14 B -10 0 -16 0 4 C 16 16 0 10 12 D 10 0 -10 0 8 E -14 -4 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8240: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) B E D A C (6) B C D A E (6) B C A D E (5) D B A E C (4) D A E B C (4) D A B C E (4) B C E D A (4) A D C B E (4) E D B A C (3) E D A C B (3) E D A B C (3) C B E A D (3) C B A D E (3) B D E A C (3) A D E C B (3) A C D B E (3) E C A D B (2) E B C D A (2) E A D C B (2) C A D E B (2) C A D B E (2) C A B D E (2) B D A E C (2) B D A C E (2) E C D A B (1) E C B D A (1) E C B A D (1) E B D C A (1) E B D A C (1) D E A B C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B A E D (1) C A E D B (1) B D C A E (1) B C E A D (1) A D C E B (1) A D B C E (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 16 -16 18 B 2 0 20 -4 28 C -16 -20 0 -10 2 D 16 4 10 0 22 E -18 -28 -2 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 -16 18 B 2 0 20 -4 28 C -16 -20 0 -10 2 D 16 4 10 0 22 E -18 -28 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=20 D=19 C=17 A=14 so A is eliminated. Round 2 votes counts: B=30 D=28 C=22 E=20 so E is eliminated. Round 3 votes counts: D=39 B=34 C=27 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:223 A:208 C:178 E:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 16 -16 18 B 2 0 20 -4 28 C -16 -20 0 -10 2 D 16 4 10 0 22 E -18 -28 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 -16 18 B 2 0 20 -4 28 C -16 -20 0 -10 2 D 16 4 10 0 22 E -18 -28 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 -16 18 B 2 0 20 -4 28 C -16 -20 0 -10 2 D 16 4 10 0 22 E -18 -28 -2 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8241: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) A E B D C (6) A B E D C (6) A B E C D (6) A E D B C (5) D E C B A (4) D E A B C (4) C D B E A (4) C B D E A (4) E D A B C (3) C D E B A (3) C B E D A (3) C B A D E (3) B C E A D (3) B A C E D (3) A B C E D (3) D A E B C (2) C D B A E (2) C B D A E (2) C A B D E (2) B E A C D (2) B C A E D (2) A C B E D (2) E D B C A (1) E C D B A (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C A E B (1) D A E C B (1) D A C E B (1) C D A B E (1) C B A E D (1) B A E C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 2 -2 2 B 4 0 4 4 0 C -2 -4 0 -4 2 D 2 -4 4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.610645 C: 0.000000 D: 0.000000 E: 0.389355 Sum of squares = 0.524484541953 Cumulative probabilities = A: 0.000000 B: 0.610645 C: 0.610645 D: 0.610645 E: 1.000000 A B C D E A 0 -4 2 -2 2 B 4 0 4 4 0 C -2 -4 0 -4 2 D 2 -4 4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=26 C=25 B=11 E=9 so E is eliminated. Round 2 votes counts: A=31 D=30 C=26 B=13 so B is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:206 E:200 A:199 D:199 C:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -2 2 B 4 0 4 4 0 C -2 -4 0 -4 2 D 2 -4 4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 2 B 4 0 4 4 0 C -2 -4 0 -4 2 D 2 -4 4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 2 B 4 0 4 4 0 C -2 -4 0 -4 2 D 2 -4 4 0 -4 E -2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8242: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) B A C D E (6) E D A B C (5) D E A C B (5) C B A D E (5) B C A D E (5) A D B C E (5) E C B D A (4) D A B C E (4) E D C B A (3) E D A C B (3) E B C A D (3) D A B E C (3) C E B A D (3) C B E A D (3) C B A E D (3) A B D C E (3) E C D B A (2) E C D A B (2) E C B A D (2) D E A B C (2) D A E B C (2) C A B D E (2) B C E A D (2) B C A E D (2) B A D C E (2) A C B D E (2) A B C D E (2) E D B C A (1) E B D C A (1) D A C E B (1) D A C B E (1) B A D E C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -2 6 2 B -4 0 -2 2 6 C 2 2 0 -2 6 D -6 -2 2 0 6 E -2 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000033 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 6 2 B -4 0 -2 2 6 C 2 2 0 -2 6 D -6 -2 2 0 6 E -2 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=18 B=18 C=16 A=14 so A is eliminated. Round 2 votes counts: E=34 D=24 B=23 C=19 so C is eliminated. Round 3 votes counts: B=38 E=37 D=25 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:205 C:204 B:201 D:200 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 6 2 B -4 0 -2 2 6 C 2 2 0 -2 6 D -6 -2 2 0 6 E -2 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 2 B -4 0 -2 2 6 C 2 2 0 -2 6 D -6 -2 2 0 6 E -2 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 2 B -4 0 -2 2 6 C 2 2 0 -2 6 D -6 -2 2 0 6 E -2 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999992 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8243: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (26) C B A D E (14) B C A D E (9) E D A C B (6) E D C B A (5) C B E D A (5) C B A E D (5) D E A B C (4) A D E B C (4) E C D B A (3) A B D C E (3) A B C D E (3) C B E A D (2) B A C D E (2) E D C A B (1) E A D B C (1) D A E B C (1) C E B D A (1) C B D E A (1) C B D A E (1) A E D B C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 4 6 -8 -10 B -4 0 12 -6 -8 C -6 -12 0 -8 -8 D 8 6 8 0 -14 E 10 8 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999766 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 6 -8 -10 B -4 0 12 -6 -8 C -6 -12 0 -8 -8 D 8 6 8 0 -14 E 10 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=29 A=13 B=11 D=5 so D is eliminated. Round 2 votes counts: E=46 C=29 A=14 B=11 so B is eliminated. Round 3 votes counts: E=46 C=38 A=16 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:204 B:197 A:196 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 -8 -10 B -4 0 12 -6 -8 C -6 -12 0 -8 -8 D 8 6 8 0 -14 E 10 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -8 -10 B -4 0 12 -6 -8 C -6 -12 0 -8 -8 D 8 6 8 0 -14 E 10 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -8 -10 B -4 0 12 -6 -8 C -6 -12 0 -8 -8 D 8 6 8 0 -14 E 10 8 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8244: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) A E D B C (8) E A D C B (6) E A D B C (6) E A C D B (6) D B C E A (6) B C D A E (6) D E A B C (4) C B E A D (4) D B A E C (3) B D C A E (3) A B D E C (3) E D C A B (2) E D A C B (2) E C A D B (2) D E B A C (2) D B A C E (2) D A E B C (2) C B D A E (2) C B A E D (2) A E D C B (2) A E C B D (2) A E B D C (2) E D C B A (1) E D A B C (1) E C A B D (1) E A C B D (1) D E B C A (1) D B E C A (1) C E B D A (1) C E A B D (1) C A E B D (1) B D A C E (1) B A C D E (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 10 14 2 -14 B -10 0 8 -14 -12 C -14 -8 0 -16 -18 D -2 14 16 0 -8 E 14 12 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 14 2 -14 B -10 0 8 -14 -12 C -14 -8 0 -16 -18 D -2 14 16 0 -8 E 14 12 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=21 C=20 A=20 B=11 so B is eliminated. Round 2 votes counts: E=28 C=26 D=25 A=21 so A is eliminated. Round 3 votes counts: E=43 C=29 D=28 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:210 A:206 B:186 C:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 2 -14 B -10 0 8 -14 -12 C -14 -8 0 -16 -18 D -2 14 16 0 -8 E 14 12 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 2 -14 B -10 0 8 -14 -12 C -14 -8 0 -16 -18 D -2 14 16 0 -8 E 14 12 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 2 -14 B -10 0 8 -14 -12 C -14 -8 0 -16 -18 D -2 14 16 0 -8 E 14 12 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8245: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) D A B C E (10) A D C E B (8) E B C D A (6) B C D A E (6) B E C D A (5) E B C A D (4) C E B A D (4) B D A C E (4) A D C B E (4) E A D C B (3) D A C B E (3) D A B E C (3) E C A B D (2) D B A C E (2) D A E C B (2) C E A B D (2) C B E A D (2) B C E D A (2) E D A C B (1) E C A D B (1) E B D C A (1) E A C D B (1) D B A E C (1) D A E B C (1) C E A D B (1) C B A E D (1) C B A D E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E C A D (1) B D C A E (1) B C E A D (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -8 2 2 B 8 0 -2 10 -4 C 8 2 0 8 10 D -2 -10 -8 0 -2 E -2 4 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 2 2 B 8 0 -2 10 -4 C 8 2 0 8 10 D -2 -10 -8 0 -2 E -2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=22 B=21 A=14 C=13 so C is eliminated. Round 2 votes counts: E=37 B=25 D=22 A=16 so A is eliminated. Round 3 votes counts: E=38 D=37 B=25 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:214 B:206 E:197 A:194 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 2 2 B 8 0 -2 10 -4 C 8 2 0 8 10 D -2 -10 -8 0 -2 E -2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 2 2 B 8 0 -2 10 -4 C 8 2 0 8 10 D -2 -10 -8 0 -2 E -2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 2 2 B 8 0 -2 10 -4 C 8 2 0 8 10 D -2 -10 -8 0 -2 E -2 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999383 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8246: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (16) B A E C D (10) A B C E D (9) A D C E B (6) B E C D A (5) B A C E D (5) D E C A B (4) D E B C A (4) A C E B D (4) D A C E B (3) B E D C A (3) A C D E B (3) E C D B A (2) D C E A B (2) D A E C B (2) C E D B A (2) C E D A B (2) B A E D C (2) A C E D B (2) E D C B A (1) E C B D A (1) D C A E B (1) C E A D B (1) C D E A B (1) B E C A D (1) B E A C D (1) B D E C A (1) B D A E C (1) B C E A D (1) B C A E D (1) A D B E C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 2 -2 4 B 14 0 -6 -8 -14 C -2 6 0 4 -10 D 2 8 -4 0 -6 E -4 14 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.437500 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.437500 Sum of squares = 0.398437499856 Cumulative probabilities = A: 0.437500 B: 0.562500 C: 0.562500 D: 0.562500 E: 1.000000 A B C D E A 0 -14 2 -2 4 B 14 0 -6 -8 -14 C -2 6 0 4 -10 D 2 8 -4 0 -6 E -4 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.437500 Sum of squares = 0.398437498379 Cumulative probabilities = A: 0.437500 B: 0.562500 C: 0.562500 D: 0.562500 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=31 A=27 C=6 E=4 so E is eliminated. Round 2 votes counts: D=33 B=31 A=27 C=9 so C is eliminated. Round 3 votes counts: D=40 B=32 A=28 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:200 C:199 A:195 B:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 2 -2 4 B 14 0 -6 -8 -14 C -2 6 0 4 -10 D 2 8 -4 0 -6 E -4 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.437500 Sum of squares = 0.398437498379 Cumulative probabilities = A: 0.437500 B: 0.562500 C: 0.562500 D: 0.562500 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -2 4 B 14 0 -6 -8 -14 C -2 6 0 4 -10 D 2 8 -4 0 -6 E -4 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.437500 Sum of squares = 0.398437498379 Cumulative probabilities = A: 0.437500 B: 0.562500 C: 0.562500 D: 0.562500 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -2 4 B 14 0 -6 -8 -14 C -2 6 0 4 -10 D 2 8 -4 0 -6 E -4 14 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.437500 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.437500 Sum of squares = 0.398437498379 Cumulative probabilities = A: 0.437500 B: 0.562500 C: 0.562500 D: 0.562500 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8247: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) B C D E A (7) D C A E B (6) C D A E B (6) B E A C D (6) C B D E A (5) C D B A E (4) C D A B E (4) B D C A E (4) A E D C B (4) A E D B C (4) E C A B D (3) E B A D C (3) E B A C D (3) E A B D C (3) D C A B E (3) E A B C D (2) C D E B A (2) C D E A B (2) C D B E A (2) C A D E B (2) B E C A D (2) E A D C B (1) E A C D B (1) D C B A E (1) D A C E B (1) C E B A D (1) C B E D A (1) C A E D B (1) B E C D A (1) B D C E A (1) B C E D A (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -18 -2 -14 B 8 0 -6 6 4 C 18 6 0 14 12 D 2 -6 -14 0 6 E 14 -4 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -18 -2 -14 B 8 0 -6 6 4 C 18 6 0 14 12 D 2 -6 -14 0 6 E 14 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=30 E=16 D=11 A=11 so D is eliminated. Round 2 votes counts: C=40 B=32 E=16 A=12 so A is eliminated. Round 3 votes counts: C=43 B=32 E=25 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:206 E:196 D:194 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -18 -2 -14 B 8 0 -6 6 4 C 18 6 0 14 12 D 2 -6 -14 0 6 E 14 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -18 -2 -14 B 8 0 -6 6 4 C 18 6 0 14 12 D 2 -6 -14 0 6 E 14 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -18 -2 -14 B 8 0 -6 6 4 C 18 6 0 14 12 D 2 -6 -14 0 6 E 14 -4 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8248: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (12) E A C D B (10) C B D A E (9) C A E B D (8) C A B E D (6) E D B A C (5) E D A B C (5) D B E A C (5) E A D C B (4) B C D A E (4) A C E B D (4) E A D B C (3) D B E C A (3) C B A D E (3) C A B D E (3) A E C D B (3) E C A B D (1) E A C B D (1) D E B A C (1) D E A B C (1) D B C A E (1) C E B D A (1) C E A B D (1) C B D E A (1) B D E C A (1) B D C E A (1) B C D E A (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -12 -2 10 B -4 0 -14 16 0 C 12 14 0 16 12 D 2 -16 -16 0 -8 E -10 0 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 -2 10 B -4 0 -14 16 0 C 12 14 0 16 12 D 2 -16 -16 0 -8 E -10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=29 B=19 D=11 A=9 so A is eliminated. Round 2 votes counts: C=37 E=33 B=19 D=11 so D is eliminated. Round 3 votes counts: C=37 E=35 B=28 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 A:200 B:199 E:193 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -2 10 B -4 0 -14 16 0 C 12 14 0 16 12 D 2 -16 -16 0 -8 E -10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -2 10 B -4 0 -14 16 0 C 12 14 0 16 12 D 2 -16 -16 0 -8 E -10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -2 10 B -4 0 -14 16 0 C 12 14 0 16 12 D 2 -16 -16 0 -8 E -10 0 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8249: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (16) C A E B D (9) C D E B A (5) B D E A C (5) C E D A B (4) A B E D C (4) E C A D B (3) E A B D C (3) D E B A C (3) C A E D B (3) B D A E C (3) A C E B D (3) D E C B A (2) D C B E A (2) C E A D B (2) C E A B D (2) C D B E A (2) C A B D E (2) B A E D C (2) A E C B D (2) A C B E D (2) E D B A C (1) E D A B C (1) E A C B D (1) D E B C A (1) D C E B A (1) D B E C A (1) D B C E A (1) D B A E C (1) D B A C E (1) C D E A B (1) C D A E B (1) C D A B E (1) C B D A E (1) C B A D E (1) C A B E D (1) B E D A C (1) B D C A E (1) B C D A E (1) A E B C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 4 -14 -16 B 4 0 -2 -6 2 C -4 2 0 -2 -4 D 14 6 2 0 8 E 16 -2 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -14 -16 B 4 0 -2 -6 2 C -4 2 0 -2 -4 D 14 6 2 0 8 E 16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=29 A=14 B=13 E=9 so E is eliminated. Round 2 votes counts: C=38 D=31 A=18 B=13 so B is eliminated. Round 3 votes counts: D=41 C=39 A=20 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:205 B:199 C:196 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -14 -16 B 4 0 -2 -6 2 C -4 2 0 -2 -4 D 14 6 2 0 8 E 16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -14 -16 B 4 0 -2 -6 2 C -4 2 0 -2 -4 D 14 6 2 0 8 E 16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -14 -16 B 4 0 -2 -6 2 C -4 2 0 -2 -4 D 14 6 2 0 8 E 16 -2 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999998682 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8250: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A C E (8) C E A D B (8) E C A D B (7) C E D A B (5) C D E B A (5) B A D E C (5) E C D B A (4) E A C B D (4) D B A E C (4) B D A E C (4) A E B D C (4) E A B C D (3) A B D E C (3) E A B D C (2) D B E A C (2) D B C E A (2) D B C A E (2) C E A B D (2) C A E B D (2) A B D C E (2) E D B C A (1) E C D A B (1) E B D C A (1) D C B E A (1) D C B A E (1) D A B C E (1) C E D B A (1) C D E A B (1) C D B E A (1) B A E D C (1) B A D C E (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -6 10 -18 B -12 0 -4 -10 -18 C 6 4 0 8 -10 D -10 10 -8 0 -12 E 18 18 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -6 10 -18 B -12 0 -4 -10 -18 C 6 4 0 8 -10 D -10 10 -8 0 -12 E 18 18 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=25 D=21 A=12 B=11 so B is eliminated. Round 2 votes counts: E=31 D=25 C=25 A=19 so A is eliminated. Round 3 votes counts: E=37 D=36 C=27 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 C:204 A:199 D:190 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -6 10 -18 B -12 0 -4 -10 -18 C 6 4 0 8 -10 D -10 10 -8 0 -12 E 18 18 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 10 -18 B -12 0 -4 -10 -18 C 6 4 0 8 -10 D -10 10 -8 0 -12 E 18 18 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 10 -18 B -12 0 -4 -10 -18 C 6 4 0 8 -10 D -10 10 -8 0 -12 E 18 18 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8251: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) D E C A B (7) E D A B C (5) D E B A C (5) D B C E A (5) C B A D E (5) B C D A E (5) B A C E D (5) E A D C B (4) E A B D C (3) D C B A E (3) A C B E D (3) A B C E D (3) E D C A B (2) E D A C B (2) E A C D B (2) D E B C A (2) D C B E A (2) D B E C A (2) D B C A E (2) C A B E D (2) B D C A E (2) A E B C D (2) E B A D C (1) E A D B C (1) D E C B A (1) D E A C B (1) D C E B A (1) C D A E B (1) C B D A E (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E A C (1) B C A E D (1) B A E C D (1) A E C B D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -12 -2 2 B 12 0 14 0 14 C 12 -14 0 -8 8 D 2 0 8 0 14 E -2 -14 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.747028 C: 0.000000 D: 0.252972 E: 0.000000 Sum of squares = 0.622046043047 Cumulative probabilities = A: 0.000000 B: 0.747028 C: 0.747028 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -2 2 B 12 0 14 0 14 C 12 -14 0 -8 8 D 2 0 8 0 14 E -2 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=25 E=20 C=13 A=11 so A is eliminated. Round 2 votes counts: D=31 B=29 E=23 C=17 so C is eliminated. Round 3 votes counts: B=42 D=34 E=24 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:212 C:199 A:188 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -2 2 B 12 0 14 0 14 C 12 -14 0 -8 8 D 2 0 8 0 14 E -2 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -2 2 B 12 0 14 0 14 C 12 -14 0 -8 8 D 2 0 8 0 14 E -2 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -2 2 B 12 0 14 0 14 C 12 -14 0 -8 8 D 2 0 8 0 14 E -2 -14 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8252: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) A D B E C (11) C E B D A (9) D B A E C (8) E C B A D (4) C D A B E (4) D B C A E (3) D A B E C (3) C E B A D (3) C A D E B (3) E B C D A (2) D A B C E (2) C A E D B (2) B D E C A (2) B D E A C (2) B D C E A (2) A E D C B (2) A E D B C (2) A E C B D (2) A D E B C (2) A C E D B (2) A C D E B (2) E C B D A (1) E C A B D (1) E B D A C (1) E B A D C (1) E A C B D (1) D C B A E (1) C E A D B (1) C D B A E (1) C D A E B (1) C B E D A (1) C A D B E (1) B E D C A (1) B E D A C (1) B D A E C (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 10 -10 8 10 B -10 0 -8 -2 -10 C 10 8 0 6 0 D -8 2 -6 0 -2 E -10 10 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.692319 D: 0.000000 E: 0.307681 Sum of squares = 0.573973385725 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.692319 D: 0.692319 E: 1.000000 A B C D E A 0 10 -10 8 10 B -10 0 -8 -2 -10 C 10 8 0 6 0 D -8 2 -6 0 -2 E -10 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.000000 E: 0.499692 Sum of squares = 0.50000018929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.500308 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=24 D=17 E=11 B=10 so B is eliminated. Round 2 votes counts: C=38 A=25 D=24 E=13 so E is eliminated. Round 3 votes counts: C=46 D=27 A=27 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:209 E:201 D:193 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -10 8 10 B -10 0 -8 -2 -10 C 10 8 0 6 0 D -8 2 -6 0 -2 E -10 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.000000 E: 0.499692 Sum of squares = 0.50000018929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.500308 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 8 10 B -10 0 -8 -2 -10 C 10 8 0 6 0 D -8 2 -6 0 -2 E -10 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.000000 E: 0.499692 Sum of squares = 0.50000018929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.500308 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 8 10 B -10 0 -8 -2 -10 C 10 8 0 6 0 D -8 2 -6 0 -2 E -10 10 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.000000 E: 0.499692 Sum of squares = 0.50000018929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500308 D: 0.500308 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8253: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) E A B C D (7) E B A C D (5) E B C A D (4) D A C B E (4) B C E D A (4) B C D E A (4) A E D C B (4) A D E C B (4) E B D C A (3) E B C D A (3) E A D B C (3) E A C B D (3) D C A B E (3) D B C A E (3) D A E C B (3) A D C B E (3) E D B A C (2) E D A B C (2) E A B D C (2) D C B A E (2) C B D A E (2) C B A D E (2) C A B D E (2) B D C E A (2) B C E A D (2) A D C E B (2) A C D B E (2) E B A D C (1) D E B A C (1) D B C E A (1) C D A B E (1) B E C D A (1) B C D A E (1) A E C B D (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 14 20 16 -16 B -14 0 2 0 -22 C -20 -2 0 -6 -14 D -16 0 6 0 -14 E 16 22 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 20 16 -16 B -14 0 2 0 -22 C -20 -2 0 -6 -14 D -16 0 6 0 -14 E 16 22 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 A=19 D=17 B=14 C=7 so C is eliminated. Round 2 votes counts: E=43 A=21 D=18 B=18 so D is eliminated. Round 3 votes counts: E=44 A=32 B=24 so B is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 A:217 D:188 B:183 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 20 16 -16 B -14 0 2 0 -22 C -20 -2 0 -6 -14 D -16 0 6 0 -14 E 16 22 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 16 -16 B -14 0 2 0 -22 C -20 -2 0 -6 -14 D -16 0 6 0 -14 E 16 22 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 16 -16 B -14 0 2 0 -22 C -20 -2 0 -6 -14 D -16 0 6 0 -14 E 16 22 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8254: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (12) C A E B D (10) B E D C A (9) A C D B E (9) A C E D B (8) E B D C A (7) D E B A C (6) A C D E B (5) C A B D E (4) E B C A D (3) D A B E C (3) B D E C A (3) A D C E B (3) D B E C A (2) D A C B E (2) C A B E D (2) E D A B C (1) E C B A D (1) E C A B D (1) E B D A C (1) E B C D A (1) E A C B D (1) D B A E C (1) D B A C E (1) C A D B E (1) B D C E A (1) B C E A D (1) A D C B E (1) Total count = 100 A B C D E A 0 2 8 0 0 B -2 0 4 -10 4 C -8 -4 0 -6 -4 D 0 10 6 0 8 E 0 -4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.424046 B: 0.000000 C: 0.000000 D: 0.575954 E: 0.000000 Sum of squares = 0.511538088456 Cumulative probabilities = A: 0.424046 B: 0.424046 C: 0.424046 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 0 0 B -2 0 4 -10 4 C -8 -4 0 -6 -4 D 0 10 6 0 8 E 0 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 C=17 E=16 B=14 so B is eliminated. Round 2 votes counts: D=31 A=26 E=25 C=18 so C is eliminated. Round 3 votes counts: A=43 D=31 E=26 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:212 A:205 B:198 E:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 0 0 B -2 0 4 -10 4 C -8 -4 0 -6 -4 D 0 10 6 0 8 E 0 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 0 0 B -2 0 4 -10 4 C -8 -4 0 -6 -4 D 0 10 6 0 8 E 0 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 0 0 B -2 0 4 -10 4 C -8 -4 0 -6 -4 D 0 10 6 0 8 E 0 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8255: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (5) D A B C E (5) C E B D A (5) B E A C D (5) C E D B A (4) A D C E B (4) A B D E C (4) E B A C D (3) D C B E A (3) D B C E A (3) D B A C E (3) D A C E B (3) D A C B E (3) C E A D B (3) B E C D A (3) B D E C A (3) B D A E C (3) A D B E C (3) A C E D B (3) A B E C D (3) E B C A D (2) D B C A E (2) D A B E C (2) C D E B A (2) B E D C A (2) A E B C D (2) A D B C E (2) E C B D A (1) E B C D A (1) D C E A B (1) D C A E B (1) D B A E C (1) C D E A B (1) C D A E B (1) C A E D B (1) B D E A C (1) B D C E A (1) B C E D A (1) A E C B D (1) A D E B C (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 8 -12 0 B 8 0 14 -6 10 C -8 -14 0 -6 6 D 12 6 6 0 8 E 0 -10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -12 0 B 8 0 14 -6 10 C -8 -14 0 -6 6 D 12 6 6 0 8 E 0 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 B=19 C=17 E=12 so E is eliminated. Round 2 votes counts: D=27 B=25 A=25 C=23 so C is eliminated. Round 3 votes counts: B=36 D=35 A=29 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:213 A:194 C:189 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -12 0 B 8 0 14 -6 10 C -8 -14 0 -6 6 D 12 6 6 0 8 E 0 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -12 0 B 8 0 14 -6 10 C -8 -14 0 -6 6 D 12 6 6 0 8 E 0 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -12 0 B 8 0 14 -6 10 C -8 -14 0 -6 6 D 12 6 6 0 8 E 0 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8256: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (14) D B C A E (10) A E C D B (10) D C B A E (6) E B D C A (5) A C D B E (5) E A C B D (4) E A B D C (4) B E D C A (4) A D C B E (4) E A C D B (3) E A B C D (3) C D B A E (3) C D A B E (3) A E D C B (3) A C D E B (3) E B C D A (2) D C A B E (2) C B D A E (2) A E D B C (2) E C B D A (1) E B A D C (1) E B A C D (1) C B D E A (1) B D E A C (1) B D C A E (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 -6 -10 -12 10 B 6 0 0 -10 14 C 10 0 0 -16 10 D 12 10 16 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -12 10 B 6 0 0 -10 14 C 10 0 0 -16 10 D 12 10 16 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=24 B=21 D=18 C=9 so C is eliminated. Round 2 votes counts: A=28 E=24 D=24 B=24 so E is eliminated. Round 3 votes counts: A=42 B=34 D=24 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:226 B:205 C:202 A:191 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -10 -12 10 B 6 0 0 -10 14 C 10 0 0 -16 10 D 12 10 16 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -12 10 B 6 0 0 -10 14 C 10 0 0 -16 10 D 12 10 16 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -12 10 B 6 0 0 -10 14 C 10 0 0 -16 10 D 12 10 16 0 14 E -10 -14 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8257: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (5) A C B D E (5) E D B C A (4) D E B A C (4) D A E C B (4) C A B E D (4) B C A E D (4) E B C A D (3) D E A C B (3) D B A E C (3) B C A D E (3) B A C D E (3) E D A C B (2) E C B A D (2) E C A B D (2) D B E C A (2) D B E A C (2) D A C B E (2) C E A B D (2) C B A E D (2) B D E C A (2) B C E D A (2) B C E A D (2) B C D A E (2) A D C B E (2) A D B C E (2) A C E D B (2) A C E B D (2) E D C B A (1) E D B A C (1) E C B D A (1) E C A D B (1) E B C D A (1) E A D C B (1) D E B C A (1) D E A B C (1) D B A C E (1) D A E B C (1) C E B A D (1) C A E B D (1) B E D C A (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A C E (1) A E C D B (1) A D C E B (1) A C D B E (1) A C B E D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 6 0 12 B 4 0 12 4 14 C -6 -12 0 0 10 D 0 -4 0 0 10 E -12 -14 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 0 12 B 4 0 12 4 14 C -6 -12 0 0 10 D 0 -4 0 0 10 E -12 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999163 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=23 E=19 A=19 C=10 so C is eliminated. Round 2 votes counts: D=29 B=25 A=24 E=22 so E is eliminated. Round 3 votes counts: D=37 B=33 A=30 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:207 D:203 C:196 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 0 12 B 4 0 12 4 14 C -6 -12 0 0 10 D 0 -4 0 0 10 E -12 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999163 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 0 12 B 4 0 12 4 14 C -6 -12 0 0 10 D 0 -4 0 0 10 E -12 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999163 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 0 12 B 4 0 12 4 14 C -6 -12 0 0 10 D 0 -4 0 0 10 E -12 -14 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999163 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8258: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (15) C E A B D (15) A E C D B (14) B D C E A (8) C E B A D (6) B C E D A (6) E C A D B (5) D A B E C (5) B D C A E (5) B D A C E (5) A D E C B (5) E C A B D (3) C B E A D (2) B C D E A (2) A E D C B (2) D A E B C (1) B E C D A (1) Total count = 100 A B C D E A 0 0 -6 4 4 B 0 0 -4 6 -2 C 6 4 0 8 -2 D -4 -6 -8 0 -8 E -4 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 0 -6 4 4 B 0 0 -4 6 -2 C 6 4 0 8 -2 D -4 -6 -8 0 -8 E -4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=23 D=21 A=21 E=8 so E is eliminated. Round 2 votes counts: C=31 B=27 D=21 A=21 so D is eliminated. Round 3 votes counts: B=42 C=31 A=27 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:208 E:204 A:201 B:200 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 4 4 B 0 0 -4 6 -2 C 6 4 0 8 -2 D -4 -6 -8 0 -8 E -4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 4 4 B 0 0 -4 6 -2 C 6 4 0 8 -2 D -4 -6 -8 0 -8 E -4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 4 4 B 0 0 -4 6 -2 C 6 4 0 8 -2 D -4 -6 -8 0 -8 E -4 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8259: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) A B E C D (9) D C E B A (8) E D B A C (7) D E B C A (6) A B C E D (6) C D A E B (5) A C B E D (5) A B E D C (5) D E C B A (4) E B D A C (3) C D A B E (3) C A B D E (3) B A E D C (3) E D B C A (2) C D E B A (2) C B A D E (2) C A D E B (2) C A D B E (2) B E D C A (2) A E C B D (2) A E B D C (2) E A B D C (1) D C E A B (1) C D E A B (1) B E D A C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 14 10 2 B 2 0 16 10 4 C -14 -16 0 -12 -16 D -10 -10 12 0 -20 E -2 -4 16 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 10 2 B 2 0 16 10 4 C -14 -16 0 -12 -16 D -10 -10 12 0 -20 E -2 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=20 D=19 B=17 E=13 so E is eliminated. Round 2 votes counts: A=32 D=28 C=20 B=20 so C is eliminated. Round 3 votes counts: D=39 A=39 B=22 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:216 E:215 A:212 D:186 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 14 10 2 B 2 0 16 10 4 C -14 -16 0 -12 -16 D -10 -10 12 0 -20 E -2 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 10 2 B 2 0 16 10 4 C -14 -16 0 -12 -16 D -10 -10 12 0 -20 E -2 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 10 2 B 2 0 16 10 4 C -14 -16 0 -12 -16 D -10 -10 12 0 -20 E -2 -4 16 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998768 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8260: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) D A E B C (9) E C B D A (8) E D B A C (7) C A B D E (7) A D B C E (7) D A B E C (5) C E B D A (5) D E A B C (4) E D A B C (3) E B D C A (3) C B A D E (3) A D B E C (3) E D B C A (2) E B D A C (2) C B A E D (2) B C A D E (2) A D C B E (2) E D C B A (1) E C D B A (1) E C D A B (1) D A E C B (1) C E D A B (1) C E B A D (1) C B E D A (1) C A D E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B A D C E (1) B A C D E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -4 -12 -6 B 6 0 4 2 0 C 4 -4 0 -2 -4 D 12 -2 2 0 -2 E 6 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999990428 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -6 -4 -12 -6 B 6 0 4 2 0 C 4 -4 0 -2 -4 D 12 -2 2 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=28 D=19 A=14 B=6 so B is eliminated. Round 2 votes counts: C=35 E=30 D=19 A=16 so A is eliminated. Round 3 votes counts: C=38 D=32 E=30 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:206 E:206 D:205 C:197 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -12 -6 B 6 0 4 2 0 C 4 -4 0 -2 -4 D 12 -2 2 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -12 -6 B 6 0 4 2 0 C 4 -4 0 -2 -4 D 12 -2 2 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -12 -6 B 6 0 4 2 0 C 4 -4 0 -2 -4 D 12 -2 2 0 -2 E 6 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8261: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) A B E D C (8) C D E A B (6) C D B E A (6) C D E B A (5) B A E C D (5) D C E A B (4) A E D B C (4) E A D C B (3) D E C A B (3) D E A C B (3) C B D E A (3) B D C E A (3) B A D E C (3) B A C E D (3) A E D C B (3) E D A C B (2) D C B E A (2) C B E D A (2) B C A E D (2) B C A D E (2) B A E D C (2) A E C D B (2) A E B D C (2) A D E B C (2) A B E C D (2) E D C A B (1) D B C E A (1) D A E B C (1) D A B E C (1) C E D A B (1) C E A D B (1) C B E A D (1) B D C A E (1) B C D A E (1) A E C B D (1) Total count = 100 A B C D E A 0 0 -6 -8 -10 B 0 0 2 -2 12 C 6 -2 0 2 4 D 8 2 -2 0 10 E 10 -12 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -8 -10 B 0 0 2 -2 12 C 6 -2 0 2 4 D 8 2 -2 0 10 E 10 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=25 A=24 D=15 E=6 so E is eliminated. Round 2 votes counts: B=30 A=27 C=25 D=18 so D is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:209 B:206 C:205 E:192 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 -8 -10 B 0 0 2 -2 12 C 6 -2 0 2 4 D 8 2 -2 0 10 E 10 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -8 -10 B 0 0 2 -2 12 C 6 -2 0 2 4 D 8 2 -2 0 10 E 10 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -8 -10 B 0 0 2 -2 12 C 6 -2 0 2 4 D 8 2 -2 0 10 E 10 -12 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8262: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) A D C B E (10) A D E B C (9) A D E C B (8) D A E B C (6) C B E D A (6) E D B C A (3) E B D C A (3) D A C B E (3) B C E D A (3) A C D B E (3) A C B D E (3) E B C A D (2) E A B C D (2) D E B C A (2) D B E C A (2) C B E A D (2) C B D E A (2) C A B D E (2) A E C B D (2) E D A B C (1) E B D A C (1) E B A C D (1) E A D B C (1) D E B A C (1) D C B E A (1) D B C E A (1) C D B E A (1) C B D A E (1) C A D B E (1) C A B E D (1) B C D E A (1) A E D B C (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 12 10 2 6 B -12 0 2 -10 -12 C -10 -2 0 -6 -12 D -2 10 6 0 16 E -6 12 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 2 6 B -12 0 2 -10 -12 C -10 -2 0 -6 -12 D -2 10 6 0 16 E -6 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=25 D=16 C=16 B=4 so B is eliminated. Round 2 votes counts: A=39 E=25 C=20 D=16 so D is eliminated. Round 3 votes counts: A=48 E=30 C=22 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:215 E:201 C:185 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 2 6 B -12 0 2 -10 -12 C -10 -2 0 -6 -12 D -2 10 6 0 16 E -6 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 2 6 B -12 0 2 -10 -12 C -10 -2 0 -6 -12 D -2 10 6 0 16 E -6 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 2 6 B -12 0 2 -10 -12 C -10 -2 0 -6 -12 D -2 10 6 0 16 E -6 12 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8263: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (12) C E B D A (10) C E B A D (9) D B E A C (8) C A E B D (8) A D B E C (8) D A B E C (7) C A D E B (6) A C D B E (6) E B D A C (4) E B C D A (4) A D C B E (4) A D B C E (3) E C B D A (2) C A D B E (2) B E D A C (2) D E B A C (1) C E A B D (1) C A E D B (1) C A B E D (1) B E D C A (1) Total count = 100 A B C D E A 0 -6 -14 -2 -8 B 6 0 0 8 -16 C 14 0 0 0 2 D 2 -8 0 0 -10 E 8 16 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.091189 C: 0.908811 D: 0.000000 E: 0.000000 Sum of squares = 0.834252293779 Cumulative probabilities = A: 0.000000 B: 0.091189 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -2 -8 B 6 0 0 8 -16 C 14 0 0 0 2 D 2 -8 0 0 -10 E 8 16 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469174162 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=22 A=21 D=16 B=3 so B is eliminated. Round 2 votes counts: C=38 E=25 A=21 D=16 so D is eliminated. Round 3 votes counts: C=38 E=34 A=28 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:216 C:208 B:199 D:192 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -2 -8 B 6 0 0 8 -16 C 14 0 0 0 2 D 2 -8 0 0 -10 E 8 16 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469174162 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -2 -8 B 6 0 0 8 -16 C 14 0 0 0 2 D 2 -8 0 0 -10 E 8 16 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469174162 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -2 -8 B 6 0 0 8 -16 C 14 0 0 0 2 D 2 -8 0 0 -10 E 8 16 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.888889 D: 0.000000 E: 0.000000 Sum of squares = 0.802469174162 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8264: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) B C E D A (8) D E A B C (6) E B C D A (4) B A C D E (4) A C D B E (4) D A E C B (3) C E B D A (3) C B A E D (3) C A B D E (3) B E D C A (3) B E D A C (3) B C A E D (3) B C A D E (3) A D C E B (3) E D B A C (2) E C B D A (2) D E B A C (2) D E A C B (2) D A E B C (2) C E D A B (2) C B E A D (2) C A D E B (2) C A B E D (2) B A D E C (2) B A D C E (2) A D E C B (2) A C D E B (2) A C B D E (2) E D C B A (1) E D C A B (1) E D B C A (1) E D A B C (1) E B D C A (1) C B A D E (1) C A E B D (1) C A D B E (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -10 -10 -4 B 16 0 16 22 12 C 10 -16 0 20 6 D 10 -22 -20 0 -2 E 4 -12 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -10 -4 B 16 0 16 22 12 C 10 -16 0 20 6 D 10 -22 -20 0 -2 E 4 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=20 A=16 D=15 E=13 so E is eliminated. Round 2 votes counts: B=41 C=22 D=21 A=16 so A is eliminated. Round 3 votes counts: B=42 C=30 D=28 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:233 C:210 E:194 D:183 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -10 -4 B 16 0 16 22 12 C 10 -16 0 20 6 D 10 -22 -20 0 -2 E 4 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -10 -4 B 16 0 16 22 12 C 10 -16 0 20 6 D 10 -22 -20 0 -2 E 4 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -10 -4 B 16 0 16 22 12 C 10 -16 0 20 6 D 10 -22 -20 0 -2 E 4 -12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8265: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) E B C D A (6) D C B E A (6) D A C B E (6) B C E D A (6) D C A B E (5) A D C B E (5) E C B D A (4) E A C B D (3) E A B C D (3) D C B A E (3) A E D C B (3) A E C D B (3) A E B D C (3) A D E C B (3) E B A C D (2) D C A E B (2) B D C E A (2) B A E C D (2) A E B C D (2) A D B C E (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A B D (1) E B C A D (1) D C E A B (1) D B C A E (1) D A C E B (1) D A B C E (1) C E B D A (1) C D B E A (1) C B E D A (1) C B D E A (1) B E C D A (1) B E A C D (1) B C D E A (1) B A D E C (1) A E D B C (1) A E C B D (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 6 -4 10 B -14 0 -24 -10 -6 C -6 24 0 -12 10 D 4 10 12 0 4 E -10 6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 -4 10 B -14 0 -24 -10 -6 C -6 24 0 -12 10 D 4 10 12 0 4 E -10 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=26 E=23 B=14 C=4 so C is eliminated. Round 2 votes counts: A=33 D=27 E=24 B=16 so B is eliminated. Round 3 votes counts: A=36 E=33 D=31 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:213 C:208 E:191 B:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 6 -4 10 B -14 0 -24 -10 -6 C -6 24 0 -12 10 D 4 10 12 0 4 E -10 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 -4 10 B -14 0 -24 -10 -6 C -6 24 0 -12 10 D 4 10 12 0 4 E -10 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 -4 10 B -14 0 -24 -10 -6 C -6 24 0 -12 10 D 4 10 12 0 4 E -10 6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998515 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8266: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) D B A E C (10) E A D B C (8) C E A B D (7) E A C B D (6) D E A B C (6) D B C A E (3) C D B A E (3) C B A D E (3) C A E B D (3) E D A B C (2) D B E A C (2) D B C E A (2) D A E B C (2) C B A E D (2) B D C A E (2) B A E D C (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A B D (1) E A C D B (1) D E B A C (1) D C E B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D B A C E (1) D A B E C (1) C E A D B (1) C D E B A (1) C D B E A (1) C B E A D (1) B D A E C (1) B D A C E (1) B C A D E (1) B A D E C (1) A E D B C (1) A E B D C (1) A E B C D (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 2 -16 8 B 6 0 0 -8 2 C -2 0 0 -8 -2 D 16 8 8 0 18 E -8 -2 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -16 8 B 6 0 0 -8 2 C -2 0 0 -8 -2 D 16 8 8 0 18 E -8 -2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=32 E=21 B=8 A=5 so A is eliminated. Round 2 votes counts: C=35 D=33 E=24 B=8 so B is eliminated. Round 3 votes counts: D=38 C=36 E=26 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:225 B:200 A:194 C:194 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -16 8 B 6 0 0 -8 2 C -2 0 0 -8 -2 D 16 8 8 0 18 E -8 -2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -16 8 B 6 0 0 -8 2 C -2 0 0 -8 -2 D 16 8 8 0 18 E -8 -2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -16 8 B 6 0 0 -8 2 C -2 0 0 -8 -2 D 16 8 8 0 18 E -8 -2 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8267: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (11) D C E A B (9) C D E A B (7) C D E B A (6) D A B E C (5) C E B A D (5) A B E D C (5) E C B A D (3) D A E B C (3) C E D B A (3) B C A E D (3) B A D E C (3) E C A D B (2) E B A C D (2) D B A C E (2) C E B D A (2) C B E A D (2) B D A C E (2) B C E A D (2) A B D E C (2) E D A B C (1) E C D A B (1) E A D B C (1) E A C B D (1) E A B C D (1) D E C A B (1) D E A C B (1) D C E B A (1) D C B A E (1) D C A E B (1) D A E C B (1) D A C E B (1) D A B C E (1) C E D A B (1) C D B E A (1) C B D E A (1) C B A E D (1) B E A C D (1) A E D B C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -6 -4 -10 B 4 0 -4 -4 -12 C 6 4 0 12 4 D 4 4 -12 0 0 E 10 12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -4 -10 B 4 0 -4 -4 -12 C 6 4 0 12 4 D 4 4 -12 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=27 B=22 E=12 A=10 so A is eliminated. Round 2 votes counts: C=29 B=29 D=28 E=14 so E is eliminated. Round 3 votes counts: C=36 B=33 D=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:209 D:198 B:192 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -4 -10 B 4 0 -4 -4 -12 C 6 4 0 12 4 D 4 4 -12 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -4 -10 B 4 0 -4 -4 -12 C 6 4 0 12 4 D 4 4 -12 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -4 -10 B 4 0 -4 -4 -12 C 6 4 0 12 4 D 4 4 -12 0 0 E 10 12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8268: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) B D E A C (9) E C B D A (7) A C D E B (6) C E B A D (5) B E D C A (5) D B A E C (4) C A E D B (4) B D A E C (4) A D B C E (4) E B D A C (3) E B C D A (3) D A B E C (3) C E B D A (3) A D E C B (3) E D B A C (2) E B D C A (2) C B D A E (2) B D A C E (2) A D C B E (2) A D B E C (2) E C B A D (1) E C A D B (1) E C A B D (1) D B A C E (1) C B E A D (1) C B A D E (1) C A D B E (1) B D E C A (1) B D C E A (1) B D C A E (1) B C E D A (1) A D E B C (1) A D C E B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -18 -2 -8 -12 B 18 0 -2 26 -8 C 2 2 0 -2 -4 D 8 -26 2 0 0 E 12 8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.164213 E: 0.835787 Sum of squares = 0.725506472286 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.164213 E: 1.000000 A B C D E A 0 -18 -2 -8 -12 B 18 0 -2 26 -8 C 2 2 0 -2 -4 D 8 -26 2 0 0 E 12 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 0.764707 Sum of squares = 0.640139778582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 A=21 E=20 D=8 so D is eliminated. Round 2 votes counts: B=29 C=27 A=24 E=20 so E is eliminated. Round 3 votes counts: B=39 C=37 A=24 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:217 E:212 C:199 D:192 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -2 -8 -12 B 18 0 -2 26 -8 C 2 2 0 -2 -4 D 8 -26 2 0 0 E 12 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 0.764707 Sum of squares = 0.640139778582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -2 -8 -12 B 18 0 -2 26 -8 C 2 2 0 -2 -4 D 8 -26 2 0 0 E 12 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 0.764707 Sum of squares = 0.640139778582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -2 -8 -12 B 18 0 -2 26 -8 C 2 2 0 -2 -4 D 8 -26 2 0 0 E 12 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 0.764707 Sum of squares = 0.640139778582 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.235293 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8269: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) B A D C E (9) E C D A B (8) C A D B E (8) C E D A B (7) C D E A B (5) C D A E B (5) A B D C E (5) A D B C E (4) E B C D A (3) D A C E B (3) B E A D C (3) E D C A B (2) E C B D A (2) E B D A C (2) B C A D E (2) B A E D C (2) E D B A C (1) E C D B A (1) E B D C A (1) D E A B C (1) D C E A B (1) D C A E B (1) C E D B A (1) C B E A D (1) C A B E D (1) C A B D E (1) B E D A C (1) B E C A D (1) B E A C D (1) B A E C D (1) B A C E D (1) A D C B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -2 10 16 B -10 0 4 2 12 C 2 -4 0 0 14 D -10 -2 0 0 20 E -16 -12 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999563 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 10 16 B -10 0 4 2 12 C 2 -4 0 0 14 D -10 -2 0 0 20 E -16 -12 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999926 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=29 E=20 A=12 D=6 so D is eliminated. Round 2 votes counts: B=33 C=31 E=21 A=15 so A is eliminated. Round 3 votes counts: B=44 C=35 E=21 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:217 C:206 B:204 D:204 E:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 10 16 B -10 0 4 2 12 C 2 -4 0 0 14 D -10 -2 0 0 20 E -16 -12 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999926 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 10 16 B -10 0 4 2 12 C 2 -4 0 0 14 D -10 -2 0 0 20 E -16 -12 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999926 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 10 16 B -10 0 4 2 12 C 2 -4 0 0 14 D -10 -2 0 0 20 E -16 -12 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999926 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8270: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) D A E C B (7) D A E B C (6) B E C A D (6) C E B A D (5) B C E A D (5) A D C B E (5) D A B E C (4) C B E A D (4) A D B E C (4) E C D B A (3) E C B D A (3) E B C D A (3) C E D A B (3) B A D E C (3) A B D C E (3) D E A C B (2) C E B D A (2) C A D E B (2) B C A E D (2) E D B C A (1) E D A C B (1) E D A B C (1) E B D C A (1) E B D A C (1) D E C A B (1) D E A B C (1) D C A E B (1) D A C E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D C E (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 12 12 6 B -8 0 18 -8 10 C -12 -18 0 -16 -8 D -12 8 16 0 8 E -6 -10 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 12 6 B -8 0 18 -8 10 C -12 -18 0 -16 -8 D -12 8 16 0 8 E -6 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 A=23 C=17 E=14 so E is eliminated. Round 2 votes counts: B=28 D=26 C=23 A=23 so C is eliminated. Round 3 votes counts: B=42 D=32 A=26 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:219 D:210 B:206 E:192 C:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 12 6 B -8 0 18 -8 10 C -12 -18 0 -16 -8 D -12 8 16 0 8 E -6 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 12 6 B -8 0 18 -8 10 C -12 -18 0 -16 -8 D -12 8 16 0 8 E -6 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 12 6 B -8 0 18 -8 10 C -12 -18 0 -16 -8 D -12 8 16 0 8 E -6 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8271: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) E D B A C (9) E D C B A (8) C A B D E (7) D E C A B (6) C D E A B (6) D E A B C (5) D C E A B (5) B A C E D (5) A B C D E (5) E C D B A (4) C E D A B (3) C A D B E (3) E D C A B (2) E D B C A (2) E B A D C (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D A B (1) D E A C B (1) D A E B C (1) C E D B A (1) C E B A D (1) C D A E B (1) C B A D E (1) B E A D C (1) B E A C D (1) B A C D E (1) A D B E C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -2 -12 -18 B -8 0 -4 -22 -18 C 2 4 0 -18 -14 D 12 22 18 0 -2 E 18 18 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -2 -12 -18 B -8 0 -4 -22 -18 C 2 4 0 -18 -14 D 12 22 18 0 -2 E 18 18 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=23 D=18 B=18 A=12 so A is eliminated. Round 2 votes counts: E=29 B=27 C=24 D=20 so D is eliminated. Round 3 votes counts: E=42 C=29 B=29 so C is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:225 A:188 C:187 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -2 -12 -18 B -8 0 -4 -22 -18 C 2 4 0 -18 -14 D 12 22 18 0 -2 E 18 18 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -12 -18 B -8 0 -4 -22 -18 C 2 4 0 -18 -14 D 12 22 18 0 -2 E 18 18 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -12 -18 B -8 0 -4 -22 -18 C 2 4 0 -18 -14 D 12 22 18 0 -2 E 18 18 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999994911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8272: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (16) B C E A D (10) D E A C B (8) A D B C E (6) E C B D A (5) B C A E D (5) E D C A B (4) B C E D A (4) B A C D E (4) A D E C B (4) E B C D A (3) D E C A B (3) A B D C E (3) E D C B A (2) E C D B A (2) C E B A D (2) A D E B C (2) A D B E C (2) A B C D E (2) E D B C A (1) D E C B A (1) D E B C A (1) D E A B C (1) D A B E C (1) C E B D A (1) C E A B D (1) C B E D A (1) C B E A D (1) B D A E C (1) B A C E D (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 10 6 -10 -2 B -10 0 -6 -12 -16 C -6 6 0 -14 -14 D 10 12 14 0 14 E 2 16 14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -10 -2 B -10 0 -6 -12 -16 C -6 6 0 -14 -14 D 10 12 14 0 14 E 2 16 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=25 A=21 E=17 C=6 so C is eliminated. Round 2 votes counts: D=31 B=27 E=21 A=21 so E is eliminated. Round 3 votes counts: D=40 B=38 A=22 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:209 A:202 C:186 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 -10 -2 B -10 0 -6 -12 -16 C -6 6 0 -14 -14 D 10 12 14 0 14 E 2 16 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -10 -2 B -10 0 -6 -12 -16 C -6 6 0 -14 -14 D 10 12 14 0 14 E 2 16 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -10 -2 B -10 0 -6 -12 -16 C -6 6 0 -14 -14 D 10 12 14 0 14 E 2 16 14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8273: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (6) C A D B E (5) B D C E A (5) E B A D C (4) C D A E B (4) B D C A E (4) B A E D C (4) E A B D C (3) D B C E A (3) C D B E A (3) C D B A E (3) B D E A C (3) A E C D B (3) A C E B D (3) E D B A C (2) E A C D B (2) D C E A B (2) D B E C A (2) C D E A B (2) C D A B E (2) C A E D B (2) C A D E B (2) B E A D C (2) B A C D E (2) A E C B D (2) A E B D C (2) A E B C D (2) A C E D B (2) A B E D C (2) A B C E D (2) E D A C B (1) E D A B C (1) E A D C B (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B E A (1) C B D A E (1) C A B D E (1) B D E C A (1) B C D A E (1) B A E C D (1) B A D E C (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 6 0 6 B 6 0 10 8 14 C -6 -10 0 -8 6 D 0 -8 8 0 4 E -6 -14 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 6 0 6 B 6 0 10 8 14 C -6 -10 0 -8 6 D 0 -8 8 0 4 E -6 -14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=25 A=19 E=14 D=11 so D is eliminated. Round 2 votes counts: B=36 C=29 A=19 E=16 so E is eliminated. Round 3 votes counts: B=43 C=30 A=27 so A is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 A:203 D:202 C:191 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 0 6 B 6 0 10 8 14 C -6 -10 0 -8 6 D 0 -8 8 0 4 E -6 -14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 0 6 B 6 0 10 8 14 C -6 -10 0 -8 6 D 0 -8 8 0 4 E -6 -14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 0 6 B 6 0 10 8 14 C -6 -10 0 -8 6 D 0 -8 8 0 4 E -6 -14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8274: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) C B A D E (7) A D C B E (7) E D A B C (5) E C B D A (5) D A E B C (5) E C B A D (4) D E A C B (4) C B A E D (4) B C E A D (4) E D C B A (3) E D A C B (3) D E A B C (3) D A B C E (3) C E B A D (3) B C A E D (3) E D C A B (2) E D B C A (2) E C A D B (2) E B D C A (2) C E A D B (2) C B E A D (2) C A B E D (2) C A B D E (2) B C A D E (2) A C B D E (2) E D B A C (1) E C D B A (1) E A D C B (1) D A C E B (1) B E D C A (1) B E C D A (1) B A D C E (1) A D E C B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -8 2 -2 B -8 0 -30 -8 -14 C 8 30 0 -8 -6 D -2 8 8 0 -6 E 2 14 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -8 2 -2 B -8 0 -30 -8 -14 C 8 30 0 -8 -6 D -2 8 8 0 -6 E 2 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=23 C=22 B=12 A=12 so B is eliminated. Round 2 votes counts: E=33 C=31 D=23 A=13 so A is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:214 C:212 D:204 A:200 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -8 2 -2 B -8 0 -30 -8 -14 C 8 30 0 -8 -6 D -2 8 8 0 -6 E 2 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 2 -2 B -8 0 -30 -8 -14 C 8 30 0 -8 -6 D -2 8 8 0 -6 E 2 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 2 -2 B -8 0 -30 -8 -14 C 8 30 0 -8 -6 D -2 8 8 0 -6 E 2 14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8275: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (14) A C B D E (11) C B A E D (9) E D B C A (7) D E B A C (6) A D E C B (6) D E A C B (5) B C E D A (5) B C E A D (4) B C A E D (4) A D E B C (4) E B D C A (3) A D C B E (3) A C D B E (3) E D C B A (2) D E B C A (2) C B E D A (2) A D B C E (2) A B C D E (2) E B C D A (1) D A E C B (1) D A E B C (1) C A B E D (1) B E D C A (1) A D B E C (1) Total count = 100 A B C D E A 0 8 18 0 -4 B -8 0 14 -14 -4 C -18 -14 0 -16 -8 D 0 14 16 0 22 E 4 4 8 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.516246 B: 0.000000 C: 0.000000 D: 0.483754 E: 0.000000 Sum of squares = 0.5005278496 Cumulative probabilities = A: 0.516246 B: 0.516246 C: 0.516246 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 0 -4 B -8 0 14 -14 -4 C -18 -14 0 -16 -8 D 0 14 16 0 22 E 4 4 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=29 B=14 E=13 C=12 so C is eliminated. Round 2 votes counts: A=33 D=29 B=25 E=13 so E is eliminated. Round 3 votes counts: D=38 A=33 B=29 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:226 A:211 E:197 B:194 C:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 18 0 -4 B -8 0 14 -14 -4 C -18 -14 0 -16 -8 D 0 14 16 0 22 E 4 4 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 0 -4 B -8 0 14 -14 -4 C -18 -14 0 -16 -8 D 0 14 16 0 22 E 4 4 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 0 -4 B -8 0 14 -14 -4 C -18 -14 0 -16 -8 D 0 14 16 0 22 E 4 4 8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8276: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (6) D E C A B (5) D C A E B (5) B E A C D (5) B A D C E (5) E B A C D (4) B A E C D (4) B A C E D (4) E C A B D (3) E B D C A (3) D C E A B (3) D B A C E (3) D A C B E (3) D A B C E (3) C E A D B (3) B A C D E (3) A C D B E (3) E D C B A (2) E C D A B (2) E C B A D (2) E B D A C (2) D E C B A (2) C D A E B (2) C A E D B (2) C A E B D (2) C A D B E (2) B E A D C (2) A C B E D (2) E D B C A (1) E B C D A (1) E B C A D (1) E B A D C (1) D E B C A (1) C A D E B (1) B D E A C (1) B A D E C (1) A D C B E (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 10 12 B -4 0 -4 0 4 C -2 4 0 -2 14 D -10 0 2 0 6 E -12 -4 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 10 12 B -4 0 -4 0 4 C -2 4 0 -2 14 D -10 0 2 0 6 E -12 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=25 E=22 C=12 A=10 so A is eliminated. Round 2 votes counts: D=32 B=28 E=22 C=18 so C is eliminated. Round 3 votes counts: D=40 B=31 E=29 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:214 C:207 D:199 B:198 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 10 12 B -4 0 -4 0 4 C -2 4 0 -2 14 D -10 0 2 0 6 E -12 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 10 12 B -4 0 -4 0 4 C -2 4 0 -2 14 D -10 0 2 0 6 E -12 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 10 12 B -4 0 -4 0 4 C -2 4 0 -2 14 D -10 0 2 0 6 E -12 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8277: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) D E B C A (7) D A E C B (7) C E B A D (5) A C B E D (5) E D C B A (4) E C B D A (4) A D C B E (4) A C B D E (4) E B C D A (3) D E A B C (3) B C E A D (3) E C B A D (2) E B D C A (2) D E C B A (2) D E A C B (2) D B A E C (2) D A B E C (2) C B E A D (2) B D E C A (2) B C A E D (2) A D C E B (2) A C E D B (2) A B C E D (2) E D C A B (1) E C D B A (1) E C D A B (1) D E C A B (1) D E B A C (1) D B E A C (1) D A E B C (1) D A C E B (1) D A B C E (1) C E A B D (1) C B A E D (1) C A E D B (1) C A B E D (1) B A C E D (1) A D B C E (1) A C E B D (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 -14 -12 B 6 0 -12 2 -12 C 8 12 0 4 -14 D 14 -2 -4 0 -6 E 12 12 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 -14 -12 B 6 0 -12 2 -12 C 8 12 0 4 -14 D 14 -2 -4 0 -6 E 12 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=24 E=18 B=16 C=11 so C is eliminated. Round 2 votes counts: D=31 A=26 E=24 B=19 so B is eliminated. Round 3 votes counts: E=37 D=33 A=30 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:205 D:201 B:192 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 -14 -12 B 6 0 -12 2 -12 C 8 12 0 4 -14 D 14 -2 -4 0 -6 E 12 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -14 -12 B 6 0 -12 2 -12 C 8 12 0 4 -14 D 14 -2 -4 0 -6 E 12 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -14 -12 B 6 0 -12 2 -12 C 8 12 0 4 -14 D 14 -2 -4 0 -6 E 12 12 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8278: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (13) E C D A B (7) E B D A C (6) E B C D A (6) C D A E B (6) B A D E C (4) A C D B E (4) E D C B A (3) E B D C A (3) B E D A C (3) B E A D C (3) E D C A B (2) E D B C A (2) E B C A D (2) D C A E B (2) D C A B E (2) D A C B E (2) C E D A B (2) C D E A B (2) B E A C D (2) B A E D C (2) B A D C E (2) B A C D E (2) A D C B E (2) A D B C E (2) A B D C E (2) A B C D E (2) E D B A C (1) E D A B C (1) E C D B A (1) E C B D A (1) E C B A D (1) D C E A B (1) D A B C E (1) C D A B E (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 8 -18 -10 0 B -8 0 -4 -14 2 C 18 4 0 8 0 D 10 14 -8 0 0 E 0 -2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.542297 D: 0.000000 E: 0.457703 Sum of squares = 0.503578091494 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.542297 D: 0.542297 E: 1.000000 A B C D E A 0 8 -18 -10 0 B -8 0 -4 -14 2 C 18 4 0 8 0 D 10 14 -8 0 0 E 0 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=24 B=20 A=12 D=8 so D is eliminated. Round 2 votes counts: E=36 C=29 B=20 A=15 so A is eliminated. Round 3 votes counts: C=37 E=36 B=27 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:208 E:199 A:190 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 -10 0 B -8 0 -4 -14 2 C 18 4 0 8 0 D 10 14 -8 0 0 E 0 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -10 0 B -8 0 -4 -14 2 C 18 4 0 8 0 D 10 14 -8 0 0 E 0 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -10 0 B -8 0 -4 -14 2 C 18 4 0 8 0 D 10 14 -8 0 0 E 0 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8279: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) B C D E A (8) A D E C B (7) B E D C A (6) A E D C B (6) A C D E B (6) E A D B C (5) C D B E A (5) A E D B C (5) B E D A C (4) B C E D A (4) A B E D C (4) C D A E B (3) B E A D C (3) E D A B C (2) E B A D C (2) C B D E A (2) B C E A D (2) A E B D C (2) E B D A C (1) E A D C B (1) D E C B A (1) D E B A C (1) C D E B A (1) C D E A B (1) C B D A E (1) C B A D E (1) C A D B E (1) C A B D E (1) B E C D A (1) B E A C D (1) B C A E D (1) B A E C D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 4 18 -2 B -8 0 8 -8 -6 C -4 -8 0 0 -8 D -18 8 0 0 -4 E 2 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 4 18 -2 B -8 0 8 -8 -6 C -4 -8 0 0 -8 D -18 8 0 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=31 A=31 C=25 E=11 D=2 so D is eliminated. Round 2 votes counts: B=31 A=31 C=25 E=13 so E is eliminated. Round 3 votes counts: A=39 B=35 C=26 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:210 B:193 D:193 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 4 18 -2 B -8 0 8 -8 -6 C -4 -8 0 0 -8 D -18 8 0 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 18 -2 B -8 0 8 -8 -6 C -4 -8 0 0 -8 D -18 8 0 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 18 -2 B -8 0 8 -8 -6 C -4 -8 0 0 -8 D -18 8 0 0 -4 E 2 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8280: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) D A E B C (6) C B A E D (6) E B D C A (4) B A D E C (4) B A C D E (4) E D C A B (3) D E A B C (3) D A E C B (3) C E A D B (3) C A B D E (3) A D C E B (3) A C D B E (3) A C B D E (3) E D C B A (2) E D A B C (2) E C D B A (2) E B C D A (2) C E D A B (2) C E B A D (2) C A E D B (2) C A D E B (2) C A D B E (2) B E D C A (2) B D E A C (2) B C A E D (2) A B D E C (2) A B C D E (2) E D B C A (1) E D A C B (1) D E A C B (1) D A B E C (1) C E D B A (1) C E B D A (1) C B E A D (1) C B A D E (1) B E D A C (1) B E C D A (1) B E C A D (1) B D A E C (1) B C E D A (1) B C E A D (1) B C A D E (1) A D B E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 2 8 B 0 0 6 -2 -4 C -2 -6 0 -2 -4 D -2 2 2 0 0 E -8 4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.678023 B: 0.321977 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.563384173822 Cumulative probabilities = A: 0.678023 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 2 8 B 0 0 6 -2 -4 C -2 -6 0 -2 -4 D -2 2 2 0 0 E -8 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500080 B: 0.499920 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012745 Cumulative probabilities = A: 0.500080 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=21 A=16 D=14 so D is eliminated. Round 2 votes counts: E=27 C=26 A=26 B=21 so B is eliminated. Round 3 votes counts: A=35 E=34 C=31 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 D:201 B:200 E:200 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 2 8 B 0 0 6 -2 -4 C -2 -6 0 -2 -4 D -2 2 2 0 0 E -8 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500080 B: 0.499920 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012745 Cumulative probabilities = A: 0.500080 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 8 B 0 0 6 -2 -4 C -2 -6 0 -2 -4 D -2 2 2 0 0 E -8 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500080 B: 0.499920 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012745 Cumulative probabilities = A: 0.500080 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 8 B 0 0 6 -2 -4 C -2 -6 0 -2 -4 D -2 2 2 0 0 E -8 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500080 B: 0.499920 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000012745 Cumulative probabilities = A: 0.500080 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8281: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (17) D E A B C (11) E D A C B (8) C B A E D (7) D E B C A (5) B C D A E (4) E A D C B (3) D B A C E (3) D A E B C (3) B C A E D (3) A E D C B (3) E D C B A (2) E A C D B (2) D E B A C (2) D E A C B (2) D B C E A (2) B C D E A (2) A E D B C (2) A E C B D (2) A D E B C (2) A B D C E (2) E D A B C (1) E C D B A (1) E C B D A (1) E C B A D (1) E C A B D (1) E A C B D (1) D E C B A (1) D B C A E (1) D B A E C (1) C B E A D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 2 0 6 B 8 0 26 -10 -8 C -2 -26 0 -8 -10 D 0 10 8 0 18 E -6 8 10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.343471 B: 0.000000 C: 0.000000 D: 0.656529 E: 0.000000 Sum of squares = 0.549002776312 Cumulative probabilities = A: 0.343471 B: 0.343471 C: 0.343471 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 0 6 B 8 0 26 -10 -8 C -2 -26 0 -8 -10 D 0 10 8 0 18 E -6 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 E=21 A=14 C=8 so C is eliminated. Round 2 votes counts: B=34 D=31 E=21 A=14 so A is eliminated. Round 3 votes counts: B=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:208 A:200 E:197 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 0 6 B 8 0 26 -10 -8 C -2 -26 0 -8 -10 D 0 10 8 0 18 E -6 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 0 6 B 8 0 26 -10 -8 C -2 -26 0 -8 -10 D 0 10 8 0 18 E -6 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 0 6 B 8 0 26 -10 -8 C -2 -26 0 -8 -10 D 0 10 8 0 18 E -6 8 10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8282: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) B D A E C (8) C E A D B (7) C E D A B (6) C A E B D (6) A C E B D (6) B A D E C (5) E C A D B (4) D B E C A (4) A E C B D (4) E D C A B (3) D E C B A (3) C A E D B (3) B D A C E (3) A C B E D (3) E D B A C (2) E C D A B (2) E A C B D (2) D B C E A (2) B D E A C (2) B A C D E (2) A B C E D (2) E A D C B (1) E A D B C (1) E A B D C (1) D E B C A (1) C D E B A (1) C D E A B (1) C D A B E (1) B A E D C (1) B A D C E (1) A E B D C (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 14 12 6 -2 B -14 0 -8 0 -10 C -12 8 0 4 -8 D -6 0 -4 0 -12 E 2 10 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 12 6 -2 B -14 0 -8 0 -10 C -12 8 0 4 -8 D -6 0 -4 0 -12 E 2 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=22 A=19 D=18 E=16 so E is eliminated. Round 2 votes counts: C=31 A=24 D=23 B=22 so B is eliminated. Round 3 votes counts: D=36 A=33 C=31 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:215 C:196 D:189 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 12 6 -2 B -14 0 -8 0 -10 C -12 8 0 4 -8 D -6 0 -4 0 -12 E 2 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 6 -2 B -14 0 -8 0 -10 C -12 8 0 4 -8 D -6 0 -4 0 -12 E 2 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 6 -2 B -14 0 -8 0 -10 C -12 8 0 4 -8 D -6 0 -4 0 -12 E 2 10 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8283: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (13) B A E C D (10) E A C D B (9) D C E B A (9) A E B C D (9) A B E C D (9) D C B E A (7) B D C A E (7) E C A D B (3) C E D A B (3) C D E A B (3) E C D A B (2) D B C A E (2) B D A C E (2) B A E D C (2) E D C A B (1) E A D C B (1) D E C A B (1) D C B A E (1) C D E B A (1) B C A D E (1) B A C E D (1) B A C D E (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 12 -8 -4 -6 B -12 0 -10 -12 -14 C 8 10 0 6 2 D 4 12 -6 0 -4 E 6 14 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -8 -4 -6 B -12 0 -10 -12 -14 C 8 10 0 6 2 D 4 12 -6 0 -4 E 6 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=24 A=20 E=16 C=7 so C is eliminated. Round 2 votes counts: D=37 B=24 A=20 E=19 so E is eliminated. Round 3 votes counts: D=43 A=33 B=24 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:213 E:211 D:203 A:197 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -8 -4 -6 B -12 0 -10 -12 -14 C 8 10 0 6 2 D 4 12 -6 0 -4 E 6 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 -4 -6 B -12 0 -10 -12 -14 C 8 10 0 6 2 D 4 12 -6 0 -4 E 6 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 -4 -6 B -12 0 -10 -12 -14 C 8 10 0 6 2 D 4 12 -6 0 -4 E 6 14 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8284: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (13) C D E B A (11) C D B E A (10) C D A E B (8) A B E D C (8) A E B D C (6) E B D A C (4) D E C B A (4) D C E B A (4) B E A D C (4) B E D C A (3) A C D B E (3) E D B C A (2) A E B C D (2) A C D E B (2) A C B D E (2) E B D C A (1) E B A D C (1) E A D C B (1) E A D B C (1) E A B D C (1) D E A C B (1) D A C E B (1) C D A B E (1) C A D B E (1) B E D A C (1) B E C D A (1) B C D A E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 6 -6 0 B -6 0 0 0 -2 C -6 0 0 14 -8 D 6 0 -14 0 -2 E 0 2 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142369 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.857631 Sum of squares = 0.755799550338 Cumulative probabilities = A: 0.142369 B: 0.142369 C: 0.142369 D: 0.142369 E: 1.000000 A B C D E A 0 6 6 -6 0 B -6 0 0 0 -2 C -6 0 0 14 -8 D 6 0 -14 0 -2 E 0 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000456056 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=31 E=11 D=10 B=10 so D is eliminated. Round 2 votes counts: A=39 C=35 E=16 B=10 so B is eliminated. Round 3 votes counts: A=39 C=36 E=25 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:203 C:200 B:196 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -6 0 B -6 0 0 0 -2 C -6 0 0 14 -8 D 6 0 -14 0 -2 E 0 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000456056 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -6 0 B -6 0 0 0 -2 C -6 0 0 14 -8 D 6 0 -14 0 -2 E 0 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000456056 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -6 0 B -6 0 0 0 -2 C -6 0 0 14 -8 D 6 0 -14 0 -2 E 0 2 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000456056 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8285: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) D E C B A (10) D E C A B (9) B A C E D (9) B C A E D (6) E D C B A (4) D C E A B (4) C A B D E (4) B A E C D (4) E D B C A (3) E B D A C (3) C D A E B (3) B E A D C (3) A C B D E (3) D E A C B (2) C B A E D (2) C A D B E (2) B E A C D (2) B A E D C (2) A B C E D (2) A B C D E (2) E D A B C (1) E B D C A (1) E B C D A (1) D E A B C (1) D C A E B (1) C E D B A (1) C D E B A (1) C B D A E (1) C B A D E (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -28 -8 -12 -12 B 28 0 0 -8 -12 C 8 0 0 -10 -12 D 12 8 10 0 -8 E 12 12 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -28 -8 -12 -12 B 28 0 0 -8 -12 C 8 0 0 -10 -12 D 12 8 10 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=26 E=23 C=15 A=9 so A is eliminated. Round 2 votes counts: B=30 D=28 E=23 C=19 so C is eliminated. Round 3 votes counts: B=41 D=35 E=24 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:222 D:211 B:204 C:193 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -28 -8 -12 -12 B 28 0 0 -8 -12 C 8 0 0 -10 -12 D 12 8 10 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -8 -12 -12 B 28 0 0 -8 -12 C 8 0 0 -10 -12 D 12 8 10 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -8 -12 -12 B 28 0 0 -8 -12 C 8 0 0 -10 -12 D 12 8 10 0 -8 E 12 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8286: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) A C B E D (8) D E B C A (7) C A E D B (6) C A E B D (6) B D E C A (5) B D E A C (5) B A D C E (5) A C E D B (5) E D C B A (4) E C D A B (3) D B E C A (3) B D A E C (3) A C D E B (3) A C B D E (3) E D B C A (2) E C B D A (2) D E C B A (2) D E C A B (2) D B E A C (2) C E B A D (2) C E A D B (2) B A C D E (2) E B D C A (1) D A B E C (1) D A B C E (1) B E D C A (1) B A C E D (1) A D C B E (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 4 12 14 B -6 0 -18 10 -14 C -4 18 0 8 14 D -12 -10 -8 0 -6 E -14 14 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 12 14 B -6 0 -18 10 -14 C -4 18 0 8 14 D -12 -10 -8 0 -6 E -14 14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=22 D=18 C=16 E=12 so E is eliminated. Round 2 votes counts: A=32 D=24 B=23 C=21 so C is eliminated. Round 3 votes counts: A=46 D=27 B=27 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:218 E:196 B:186 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 12 14 B -6 0 -18 10 -14 C -4 18 0 8 14 D -12 -10 -8 0 -6 E -14 14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 12 14 B -6 0 -18 10 -14 C -4 18 0 8 14 D -12 -10 -8 0 -6 E -14 14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 12 14 B -6 0 -18 10 -14 C -4 18 0 8 14 D -12 -10 -8 0 -6 E -14 14 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (12) D C B A E (12) E A C B D (7) D B C A E (7) D B A C E (7) C D E B A (6) B D A E C (6) C E D A B (5) C D B A E (5) B A D E C (5) A B E D C (4) C E A B D (3) B A E D C (3) E C A D B (2) D B A E C (2) B D A C E (2) E D C A B (1) E C D A B (1) E C A B D (1) E A D B C (1) E A B D C (1) C E A D B (1) C D B E A (1) B A C D E (1) A E D B C (1) A E B D C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 10 -10 16 B 14 0 10 -4 12 C -10 -10 0 -8 0 D 10 4 8 0 10 E -16 -12 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 -10 16 B 14 0 10 -4 12 C -10 -10 0 -8 0 D 10 4 8 0 10 E -16 -12 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=26 C=21 B=17 A=8 so A is eliminated. Round 2 votes counts: E=29 D=28 B=22 C=21 so C is eliminated. Round 3 votes counts: D=40 E=38 B=22 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:216 D:216 A:201 C:186 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 10 -10 16 B 14 0 10 -4 12 C -10 -10 0 -8 0 D 10 4 8 0 10 E -16 -12 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 -10 16 B 14 0 10 -4 12 C -10 -10 0 -8 0 D 10 4 8 0 10 E -16 -12 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 -10 16 B 14 0 10 -4 12 C -10 -10 0 -8 0 D 10 4 8 0 10 E -16 -12 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8288: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) A B E D C (7) A E B D C (6) C B D E A (5) B C D A E (5) B C A D E (5) A B C E D (5) E D A C B (4) E A D C B (4) C D B E A (4) B A C D E (4) E D C A B (3) E D B A C (3) E A D B C (3) D E C B A (3) D E B C A (3) C B D A E (3) A E D C B (3) B C D E A (2) B A C E D (2) A E C D B (2) A C B E D (2) E D A B C (1) D E C A B (1) D E B A C (1) D C E B A (1) C D E B A (1) C D E A B (1) C D B A E (1) C B A D E (1) C A E D B (1) C A B D E (1) B E A D C (1) B D C E A (1) B A D E C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 16 14 16 B -6 0 16 6 2 C -16 -16 0 -6 -8 D -14 -6 6 0 -12 E -16 -2 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 14 16 B -6 0 16 6 2 C -16 -16 0 -6 -8 D -14 -6 6 0 -12 E -16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=21 E=18 C=18 D=9 so D is eliminated. Round 2 votes counts: A=34 E=26 B=21 C=19 so C is eliminated. Round 3 votes counts: A=36 B=35 E=29 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:209 E:201 D:187 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 14 16 B -6 0 16 6 2 C -16 -16 0 -6 -8 D -14 -6 6 0 -12 E -16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 14 16 B -6 0 16 6 2 C -16 -16 0 -6 -8 D -14 -6 6 0 -12 E -16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 14 16 B -6 0 16 6 2 C -16 -16 0 -6 -8 D -14 -6 6 0 -12 E -16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8289: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (21) E A D B C (9) C B D A E (9) C B D E A (6) A D B C E (6) D B C A E (5) A E D B C (5) D A B C E (4) A D E B C (4) E C B A D (3) E C A B D (3) C E B D A (2) C B A D E (2) B D C A E (2) B C D A E (2) A B C D E (2) E D C B A (1) E D B C A (1) E D A B C (1) E C D B A (1) E A D C B (1) E A C D B (1) E A C B D (1) D B E C A (1) D B C E A (1) D A B E C (1) C B E A D (1) A E C B D (1) A D B E C (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -22 -16 -8 B 16 0 -8 14 -12 C 22 8 0 12 -12 D 16 -14 -12 0 -6 E 8 12 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -22 -16 -8 B 16 0 -8 14 -12 C 22 8 0 12 -12 D 16 -14 -12 0 -6 E 8 12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=43 A=21 C=20 D=12 B=4 so B is eliminated. Round 2 votes counts: E=43 C=22 A=21 D=14 so D is eliminated. Round 3 votes counts: E=44 C=30 A=26 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:215 B:205 D:192 A:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -22 -16 -8 B 16 0 -8 14 -12 C 22 8 0 12 -12 D 16 -14 -12 0 -6 E 8 12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -22 -16 -8 B 16 0 -8 14 -12 C 22 8 0 12 -12 D 16 -14 -12 0 -6 E 8 12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -22 -16 -8 B 16 0 -8 14 -12 C 22 8 0 12 -12 D 16 -14 -12 0 -6 E 8 12 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8290: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) C B A D E (9) E D A B C (7) D E C B A (7) E A B D C (6) D E A C B (6) D C E B A (6) E D A C B (5) C B D A E (5) A E B C D (5) E A D B C (4) C D B A E (3) B C A E D (3) B A C E D (3) B A C D E (3) A B C D E (3) D C B A E (2) B C A D E (2) E D C B A (1) E A B C D (1) D C E A B (1) D C B E A (1) D C A E B (1) D C A B E (1) D A E B C (1) C D B E A (1) A E D B C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 14 4 8 B -8 0 2 4 -4 C -14 -2 0 -2 8 D -4 -4 2 0 6 E -8 4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 4 8 B -8 0 2 4 -4 C -14 -2 0 -2 8 D -4 -4 2 0 6 E -8 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=24 A=21 C=18 B=11 so B is eliminated. Round 2 votes counts: A=27 D=26 E=24 C=23 so C is eliminated. Round 3 votes counts: A=41 D=35 E=24 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 D:200 B:197 C:195 E:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 4 8 B -8 0 2 4 -4 C -14 -2 0 -2 8 D -4 -4 2 0 6 E -8 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 4 8 B -8 0 2 4 -4 C -14 -2 0 -2 8 D -4 -4 2 0 6 E -8 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 4 8 B -8 0 2 4 -4 C -14 -2 0 -2 8 D -4 -4 2 0 6 E -8 4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998951 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8291: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (7) A C B E D (6) E D C B A (5) D E A B C (5) A D B E C (5) E D A C B (4) D E B A C (4) C E B D A (4) A E D C B (4) A C E B D (4) E D B C A (3) E C A B D (3) E A D C B (3) D B A C E (3) C B E D A (3) B C A D E (3) A E C D B (3) A C B D E (3) E C D B A (2) E C B D A (2) D B E C A (2) D A E B C (2) B C D A E (2) A D E B C (2) A B D C E (2) A B C D E (2) E A C D B (1) D E B C A (1) D B E A C (1) C E B A D (1) C B E A D (1) C B A D E (1) C A B E D (1) B D C E A (1) B D C A E (1) B A D C E (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 14 -4 -6 B -4 0 -2 -4 -8 C -14 2 0 -2 -6 D 4 4 2 0 -2 E 6 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 14 -4 -6 B -4 0 -2 -4 -8 C -14 2 0 -2 -6 D 4 4 2 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=23 D=18 B=15 C=11 so C is eliminated. Round 2 votes counts: A=34 E=28 B=20 D=18 so D is eliminated. Round 3 votes counts: E=38 A=36 B=26 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:204 D:204 B:191 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 14 -4 -6 B -4 0 -2 -4 -8 C -14 2 0 -2 -6 D 4 4 2 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -4 -6 B -4 0 -2 -4 -8 C -14 2 0 -2 -6 D 4 4 2 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -4 -6 B -4 0 -2 -4 -8 C -14 2 0 -2 -6 D 4 4 2 0 -2 E 6 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8292: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) E B C A D (7) D E B C A (6) D A C B E (5) A E C B D (5) A C B D E (5) E C B A D (4) E A D C B (4) D B C E A (4) B C E D A (4) A D C B E (4) E D A B C (3) E B C D A (3) D B C A E (3) B C D A E (3) A C B E D (3) E A C B D (2) D B E C A (2) D A E C B (2) D A B C E (2) C B A E D (2) C B A D E (2) B C E A D (2) B C A E D (2) B C A D E (2) A C D B E (2) E D A C B (1) D E A B C (1) D B A C E (1) D A B E C (1) C B E A D (1) C A B E D (1) B D C E A (1) A E D C B (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 -12 -12 2 -4 B 12 0 8 -2 4 C 12 -8 0 2 0 D -2 2 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 A B C D E A 0 -12 -12 2 -4 B 12 0 8 -2 4 C 12 -8 0 2 0 D -2 2 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=27 A=22 B=14 C=6 so C is eliminated. Round 2 votes counts: E=31 D=27 A=23 B=19 so B is eliminated. Round 3 votes counts: E=38 D=31 A=31 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:211 C:203 E:203 D:196 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 2 -4 B 12 0 8 -2 4 C 12 -8 0 2 0 D -2 2 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 2 -4 B 12 0 8 -2 4 C 12 -8 0 2 0 D -2 2 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 2 -4 B 12 0 8 -2 4 C 12 -8 0 2 0 D -2 2 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.333333 E: 0.166667 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8293: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) A B E D C (7) B E A D C (6) A B C E D (5) D E C B A (4) D E C A B (4) C B D E A (4) C A D E B (4) A E D B C (4) C D E A B (3) B C E D A (3) B A C E D (3) A E B D C (3) A C B E D (3) A C B D E (3) A B E C D (3) E D B C A (2) E D B A C (2) E D A B C (2) D E B C A (2) D A E C B (2) C B D A E (2) B E D C A (2) B C A E D (2) B A E C D (2) A D E C B (2) A C D B E (2) E A B D C (1) D E A C B (1) D C E B A (1) D C E A B (1) C D B E A (1) C A B D E (1) B E D A C (1) B E C D A (1) B A E D C (1) A D E B C (1) Total count = 100 A B C D E A 0 4 8 6 0 B -4 0 6 6 4 C -8 -6 0 2 -6 D -6 -6 -2 0 -6 E 0 -4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.733161 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.266839 Sum of squares = 0.60872853647 Cumulative probabilities = A: 0.733161 B: 0.733161 C: 0.733161 D: 0.733161 E: 1.000000 A B C D E A 0 4 8 6 0 B -4 0 6 6 4 C -8 -6 0 2 -6 D -6 -6 -2 0 -6 E 0 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500131 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499869 Sum of squares = 0.500000034118 Cumulative probabilities = A: 0.500131 B: 0.500131 C: 0.500131 D: 0.500131 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=24 B=21 D=15 E=7 so E is eliminated. Round 2 votes counts: A=34 C=24 D=21 B=21 so D is eliminated. Round 3 votes counts: A=39 C=34 B=27 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:206 E:204 C:191 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 0 B -4 0 6 6 4 C -8 -6 0 2 -6 D -6 -6 -2 0 -6 E 0 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500131 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499869 Sum of squares = 0.500000034118 Cumulative probabilities = A: 0.500131 B: 0.500131 C: 0.500131 D: 0.500131 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 0 B -4 0 6 6 4 C -8 -6 0 2 -6 D -6 -6 -2 0 -6 E 0 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500131 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499869 Sum of squares = 0.500000034118 Cumulative probabilities = A: 0.500131 B: 0.500131 C: 0.500131 D: 0.500131 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 0 B -4 0 6 6 4 C -8 -6 0 2 -6 D -6 -6 -2 0 -6 E 0 -4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500131 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499869 Sum of squares = 0.500000034118 Cumulative probabilities = A: 0.500131 B: 0.500131 C: 0.500131 D: 0.500131 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8294: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (5) B C A E D (5) B A C E D (5) A C E B D (5) E D B A C (4) E A C B D (4) C A E D B (4) C A E B D (4) B E D A C (4) B D E A C (4) E B D A C (3) D E C A B (3) D E A C B (3) C A B E D (3) C A B D E (3) E D A C B (2) E B A C D (2) E A C D B (2) D E B C A (2) D C B A E (2) D B E C A (2) D B E A C (2) C D A E B (2) C B A D E (2) B D C A E (2) B C A D E (2) B A E C D (2) A C B E D (2) E A B D C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) D B C A E (1) C D A B E (1) C B D A E (1) C A D E B (1) C A D B E (1) B E A C D (1) A E C D B (1) A E C B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 8 4 2 B 10 0 -2 14 -4 C -8 2 0 10 2 D -4 -14 -10 0 -12 E -2 4 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.468750000009 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 A B C D E A 0 -10 8 4 2 B 10 0 -2 14 -4 C -8 2 0 10 2 D -4 -14 -10 0 -12 E -2 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.46874999996 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=25 B=25 C=22 E=18 A=10 so A is eliminated. Round 2 votes counts: C=29 B=26 D=25 E=20 so E is eliminated. Round 3 votes counts: C=37 B=32 D=31 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:209 E:206 C:203 A:202 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 8 4 2 B 10 0 -2 14 -4 C -8 2 0 10 2 D -4 -14 -10 0 -12 E -2 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.46874999996 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 4 2 B 10 0 -2 14 -4 C -8 2 0 10 2 D -4 -14 -10 0 -12 E -2 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.46874999996 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 4 2 B 10 0 -2 14 -4 C -8 2 0 10 2 D -4 -14 -10 0 -12 E -2 4 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.000000 E: 0.625000 Sum of squares = 0.46874999996 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8295: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) C A E D B (8) D B A C E (6) D A B C E (6) B D E A C (6) E B C D A (5) B D A E C (5) E C B A D (4) E C A D B (4) C A D E B (4) A D C B E (4) A C D B E (4) E B C A D (3) D B E A C (3) B E C A D (3) E C A B D (2) E B D C A (2) C E A D B (2) B E D A C (2) B C A E D (2) A C D E B (2) E D C A B (1) E C D A B (1) E B D A C (1) D E A C B (1) D B A E C (1) D A C E B (1) D A C B E (1) C E A B D (1) C A E B D (1) B E D C A (1) B C E A D (1) B A D C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 10 -4 14 B 10 0 18 2 14 C -10 -18 0 -4 10 D 4 -2 4 0 12 E -14 -14 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999336 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -4 14 B 10 0 18 2 14 C -10 -18 0 -4 10 D 4 -2 4 0 12 E -14 -14 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=23 D=19 C=16 A=12 so A is eliminated. Round 2 votes counts: B=32 E=23 D=23 C=22 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:222 D:209 A:205 C:189 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 -4 14 B 10 0 18 2 14 C -10 -18 0 -4 10 D 4 -2 4 0 12 E -14 -14 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -4 14 B 10 0 18 2 14 C -10 -18 0 -4 10 D 4 -2 4 0 12 E -14 -14 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -4 14 B 10 0 18 2 14 C -10 -18 0 -4 10 D 4 -2 4 0 12 E -14 -14 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999964305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8296: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (12) A D E C B (12) D A C E B (7) C E B D A (5) B C E D A (4) B A E C D (4) E C D A B (3) D C A E B (3) D A B C E (3) C D B E A (3) A D B C E (3) E C D B A (2) D C B A E (2) D C A B E (2) C E D B A (2) C B E D A (2) B D C A E (2) B C A D E (2) B A E D C (2) B A D C E (2) A D C E B (2) A D B E C (2) A B E D C (2) A B D E C (2) E D C A B (1) E C B A D (1) E B C D A (1) E B A C D (1) E A B C D (1) C E D A B (1) C B D E A (1) B E C D A (1) B C E A D (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -6 6 14 B 4 0 0 -2 10 C 6 0 0 -2 0 D -6 2 2 0 4 E -14 -10 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.387755101865 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 6 14 B 4 0 0 -2 10 C 6 0 0 -2 0 D -6 2 2 0 4 E -14 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510153 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=27 D=17 C=14 E=10 so E is eliminated. Round 2 votes counts: B=34 A=28 C=20 D=18 so D is eliminated. Round 3 votes counts: A=38 B=34 C=28 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:206 A:205 C:202 D:201 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 6 14 B 4 0 0 -2 10 C 6 0 0 -2 0 D -6 2 2 0 4 E -14 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510153 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 6 14 B 4 0 0 -2 10 C 6 0 0 -2 0 D -6 2 2 0 4 E -14 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510153 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 6 14 B 4 0 0 -2 10 C 6 0 0 -2 0 D -6 2 2 0 4 E -14 -10 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.428571 D: 0.428571 E: 0.000000 Sum of squares = 0.38775510153 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.571429 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8297: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) E C D A B (7) A B C E D (7) B D C E A (6) B A D C E (6) A B E D C (6) D E C B A (5) B D E C A (5) A E C D B (5) A C E D B (5) E D C B A (4) C E D A B (4) A B E C D (4) E D C A B (3) C D E B A (3) B A D E C (3) C D B E A (2) D E B C A (1) D C E B A (1) C D E A B (1) C A D E B (1) B E D C A (1) B D E A C (1) B D C A E (1) B C D E A (1) B C D A E (1) B A E D C (1) A E D C B (1) A E C B D (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 16 6 6 10 B -16 0 8 12 12 C -6 -8 0 10 4 D -6 -12 -10 0 0 E -10 -12 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 6 10 B -16 0 8 12 12 C -6 -8 0 10 4 D -6 -12 -10 0 0 E -10 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 B=26 E=14 C=11 D=7 so D is eliminated. Round 2 votes counts: A=42 B=26 E=20 C=12 so C is eliminated. Round 3 votes counts: A=43 E=29 B=28 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:208 C:200 E:187 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 6 10 B -16 0 8 12 12 C -6 -8 0 10 4 D -6 -12 -10 0 0 E -10 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 6 10 B -16 0 8 12 12 C -6 -8 0 10 4 D -6 -12 -10 0 0 E -10 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 6 10 B -16 0 8 12 12 C -6 -8 0 10 4 D -6 -12 -10 0 0 E -10 -12 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8298: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (6) C A E D B (6) B D E A C (6) B A D E C (6) D B E A C (5) C D E A B (4) B A E D C (4) D E C A B (3) D B E C A (3) C D B E A (3) C B A D E (3) B D A E C (3) B C D A E (3) A E D B C (3) E D C A B (2) E A C D B (2) D E B C A (2) D E B A C (2) D C E A B (2) D C B E A (2) C D E B A (2) C A E B D (2) C A B E D (2) B D C E A (2) B A D C E (2) B A C E D (2) A E C D B (2) A E B D C (2) E D A C B (1) E D A B C (1) E A D C B (1) E A D B C (1) D B C E A (1) C E D A B (1) C B D A E (1) C B A E D (1) B D E C A (1) B D C A E (1) B D A C E (1) B A C D E (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -14 -6 -4 -6 B 14 0 6 -10 6 C 6 -6 0 -14 -4 D 4 10 14 0 18 E 6 -6 4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -4 -6 B 14 0 6 -10 6 C 6 -6 0 -14 -4 D 4 10 14 0 18 E 6 -6 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=31 D=20 A=9 E=8 so E is eliminated. Round 2 votes counts: B=32 C=31 D=24 A=13 so A is eliminated. Round 3 votes counts: C=36 B=35 D=29 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:223 B:208 E:193 C:191 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 -6 B 14 0 6 -10 6 C 6 -6 0 -14 -4 D 4 10 14 0 18 E 6 -6 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 -6 B 14 0 6 -10 6 C 6 -6 0 -14 -4 D 4 10 14 0 18 E 6 -6 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 -6 B 14 0 6 -10 6 C 6 -6 0 -14 -4 D 4 10 14 0 18 E 6 -6 4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8299: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (7) A B C E D (7) A B C D E (7) B A C D E (6) B D E C A (5) A E D C B (5) E D C A B (3) E C D A B (3) C D E B A (3) C B D E A (3) B D C E A (3) B A D C E (3) A E D B C (3) A E C D B (3) A C E D B (3) A B E D C (3) E D C B A (2) D E C B A (2) D E B C A (2) D C E B A (2) C E D B A (2) C E D A B (2) B D A E C (2) B C A D E (2) A C E B D (2) A C B E D (2) A B D E C (2) E A C D B (1) D B E C A (1) D B E A C (1) D B C E A (1) C A E D B (1) B C D A E (1) B C A E D (1) B A D E C (1) A E C B D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 0 8 10 14 B 0 0 20 20 16 C -8 -20 0 16 16 D -10 -20 -16 0 8 E -14 -16 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.617711 B: 0.382289 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.527711711839 Cumulative probabilities = A: 0.617711 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 10 14 B 0 0 20 20 16 C -8 -20 0 16 16 D -10 -20 -16 0 8 E -14 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=31 C=11 E=9 D=9 so E is eliminated. Round 2 votes counts: A=41 B=31 D=14 C=14 so D is eliminated. Round 3 votes counts: A=41 B=36 C=23 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:228 A:216 C:202 D:181 E:173 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 8 10 14 B 0 0 20 20 16 C -8 -20 0 16 16 D -10 -20 -16 0 8 E -14 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 10 14 B 0 0 20 20 16 C -8 -20 0 16 16 D -10 -20 -16 0 8 E -14 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 10 14 B 0 0 20 20 16 C -8 -20 0 16 16 D -10 -20 -16 0 8 E -14 -16 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8300: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) B E C D A (7) A D C E B (7) E C B D A (6) D C E A B (5) D A C E B (5) B E C A D (5) A D E C B (4) A D C B E (4) A B D E C (4) C E D B A (3) A B E D C (3) E C D A B (2) E C A D B (2) D C E B A (2) D C A E B (2) D A C B E (2) D A B C E (2) C D E B A (2) B A E C D (2) B A C E D (2) A B E C D (2) A B D C E (2) E D C A B (1) E C D B A (1) E B C D A (1) E A C D B (1) E A C B D (1) D C B E A (1) D C A B E (1) D B C A E (1) D B A C E (1) D A E C B (1) C E B D A (1) B D C E A (1) B C E D A (1) B C D A E (1) B C A E D (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 22 6 0 14 B -22 0 -10 -18 2 C -6 10 0 -16 10 D 0 18 16 0 12 E -14 -2 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.407138 B: 0.000000 C: 0.000000 D: 0.592862 E: 0.000000 Sum of squares = 0.517246857286 Cumulative probabilities = A: 0.407138 B: 0.407138 C: 0.407138 D: 1.000000 E: 1.000000 A B C D E A 0 22 6 0 14 B -22 0 -10 -18 2 C -6 10 0 -16 10 D 0 18 16 0 12 E -14 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=23 B=20 E=15 C=6 so C is eliminated. Round 2 votes counts: A=36 D=25 B=20 E=19 so E is eliminated. Round 3 votes counts: A=40 D=32 B=28 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:223 A:221 C:199 E:181 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 22 6 0 14 B -22 0 -10 -18 2 C -6 10 0 -16 10 D 0 18 16 0 12 E -14 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 6 0 14 B -22 0 -10 -18 2 C -6 10 0 -16 10 D 0 18 16 0 12 E -14 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 6 0 14 B -22 0 -10 -18 2 C -6 10 0 -16 10 D 0 18 16 0 12 E -14 -2 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8301: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) D E B A C (10) C A B E D (8) D B E A C (7) D C B A E (6) C D B A E (6) E B A D C (5) E A B D C (4) A E B C D (4) D B E C A (3) C A E B D (3) A B C E D (3) E D B A C (2) D E B C A (2) D C E A B (2) D B C E A (2) C D A E B (2) C D A B E (2) C B A D E (2) A E C B D (2) E D A B C (1) E B D A C (1) D E C A B (1) D E A B C (1) D C B E A (1) D B C A E (1) C E D A B (1) C E A D B (1) C A D B E (1) C A B D E (1) B E D A C (1) B E A D C (1) B C A D E (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 8 -4 -12 B 2 0 20 -4 -6 C -8 -20 0 -2 -12 D 4 4 2 0 2 E 12 6 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -4 -12 B 2 0 20 -4 -6 C -8 -20 0 -2 -12 D 4 4 2 0 2 E 12 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 C=27 E=23 A=11 B=3 so B is eliminated. Round 2 votes counts: D=36 C=28 E=25 A=11 so A is eliminated. Round 3 votes counts: D=36 E=32 C=32 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:214 B:206 D:206 A:195 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 -4 -12 B 2 0 20 -4 -6 C -8 -20 0 -2 -12 D 4 4 2 0 2 E 12 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -4 -12 B 2 0 20 -4 -6 C -8 -20 0 -2 -12 D 4 4 2 0 2 E 12 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -4 -12 B 2 0 20 -4 -6 C -8 -20 0 -2 -12 D 4 4 2 0 2 E 12 6 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999488 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8302: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) A D E B C (8) C D B E A (7) C B E D A (7) B C E D A (7) E B C A D (6) A D C E B (6) E B C D A (4) D C A B E (4) D A C E B (4) D A C B E (4) B E A C D (4) A E B D C (4) E B A C D (3) C B D E A (3) A D E C B (3) E B A D C (2) C D B A E (2) B E C D A (2) A D C B E (2) A D B E C (2) E A B D C (1) B C E A D (1) B C A E D (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -10 12 -16 B 20 0 16 14 16 C 10 -16 0 16 -4 D -12 -14 -16 0 -8 E 16 -16 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -10 12 -16 B 20 0 16 14 16 C 10 -16 0 16 -4 D -12 -14 -16 0 -8 E 16 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 C=19 E=16 D=12 so D is eliminated. Round 2 votes counts: A=35 B=26 C=23 E=16 so E is eliminated. Round 3 votes counts: B=41 A=36 C=23 so C is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:233 E:206 C:203 A:183 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -10 12 -16 B 20 0 16 14 16 C 10 -16 0 16 -4 D -12 -14 -16 0 -8 E 16 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -10 12 -16 B 20 0 16 14 16 C 10 -16 0 16 -4 D -12 -14 -16 0 -8 E 16 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -10 12 -16 B 20 0 16 14 16 C 10 -16 0 16 -4 D -12 -14 -16 0 -8 E 16 -16 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8303: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) E C A D B (10) E B C A D (8) D C A E B (8) B E A C D (8) E B A C D (6) D A C B E (6) B E D C A (5) D C A B E (4) C A D E B (4) B E D A C (4) D B A C E (3) B E C A D (3) B D E C A (3) E C A B D (2) A C E D B (2) A C D E B (2) E D C A B (1) E B D C A (1) E A C B D (1) E A B C D (1) D E B C A (1) D A C E B (1) C D A E B (1) B E A D C (1) B D C A E (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 -14 -4 0 -10 B 14 0 16 14 2 C 4 -16 0 -2 -10 D 0 -14 2 0 -6 E 10 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -4 0 -10 B 14 0 16 14 2 C 4 -16 0 -2 -10 D 0 -14 2 0 -6 E 10 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996086 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=30 D=23 C=5 A=4 so A is eliminated. Round 2 votes counts: B=38 E=30 D=23 C=9 so C is eliminated. Round 3 votes counts: B=38 E=32 D=30 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:212 D:191 C:188 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 0 -10 B 14 0 16 14 2 C 4 -16 0 -2 -10 D 0 -14 2 0 -6 E 10 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996086 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 0 -10 B 14 0 16 14 2 C 4 -16 0 -2 -10 D 0 -14 2 0 -6 E 10 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996086 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 0 -10 B 14 0 16 14 2 C 4 -16 0 -2 -10 D 0 -14 2 0 -6 E 10 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996086 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8304: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) A C B D E (10) E D B C A (8) A B C D E (6) E D A C B (4) B E D C A (4) A E D C B (4) A E D B C (4) A B E D C (4) A B C E D (4) D E C B A (3) C D E B A (3) C D B E A (3) C B D E A (3) B C A D E (3) A E B D C (3) C B D A E (2) B C D E A (2) A C D E B (2) A C D B E (2) E D A B C (1) E B D A C (1) E A D C B (1) E A D B C (1) E A B D C (1) D E C A B (1) D E B C A (1) C D B A E (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E C A (1) B C D A E (1) B A C D E (1) A C E B D (1) Total count = 100 A B C D E A 0 2 4 -2 0 B -2 0 -4 -2 -2 C -4 4 0 -8 -8 D 2 2 8 0 -6 E 0 2 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.517265 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.482735 Sum of squares = 0.500596166094 Cumulative probabilities = A: 0.517265 B: 0.517265 C: 0.517265 D: 0.517265 E: 1.000000 A B C D E A 0 2 4 -2 0 B -2 0 -4 -2 -2 C -4 4 0 -8 -8 D 2 2 8 0 -6 E 0 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 E=27 C=14 B=14 D=5 so D is eliminated. Round 2 votes counts: A=40 E=32 C=14 B=14 so C is eliminated. Round 3 votes counts: A=42 E=35 B=23 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:208 D:203 A:202 B:195 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 -2 0 B -2 0 -4 -2 -2 C -4 4 0 -8 -8 D 2 2 8 0 -6 E 0 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 0 B -2 0 -4 -2 -2 C -4 4 0 -8 -8 D 2 2 8 0 -6 E 0 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 0 B -2 0 -4 -2 -2 C -4 4 0 -8 -8 D 2 2 8 0 -6 E 0 2 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8305: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) D A E C B (10) C E D A B (8) A D E C B (8) E C D A B (6) C E B D A (6) B C E A D (6) A D B E C (6) A B D E C (5) D A C E B (4) B E C A D (4) B C E D A (3) B A E C D (3) D C E A B (2) D B A C E (2) C E D B A (2) B C D E A (2) B A D C E (2) E C B D A (1) E B C A D (1) E A C D B (1) D E C A B (1) D A B C E (1) C D E A B (1) C B E D A (1) B E A C D (1) B A C E D (1) Total count = 100 A B C D E A 0 6 12 0 8 B -6 0 -2 -4 -2 C -12 2 0 -6 -18 D 0 4 6 0 12 E -8 2 18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.379928 B: 0.000000 C: 0.000000 D: 0.620072 E: 0.000000 Sum of squares = 0.528834788979 Cumulative probabilities = A: 0.379928 B: 0.379928 C: 0.379928 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 0 8 B -6 0 -2 -4 -2 C -12 2 0 -6 -18 D 0 4 6 0 12 E -8 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999682 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=20 A=19 C=18 E=9 so E is eliminated. Round 2 votes counts: B=35 C=25 D=20 A=20 so D is eliminated. Round 3 votes counts: B=37 A=35 C=28 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:211 E:200 B:193 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 0 8 B -6 0 -2 -4 -2 C -12 2 0 -6 -18 D 0 4 6 0 12 E -8 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999682 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 0 8 B -6 0 -2 -4 -2 C -12 2 0 -6 -18 D 0 4 6 0 12 E -8 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999682 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 0 8 B -6 0 -2 -4 -2 C -12 2 0 -6 -18 D 0 4 6 0 12 E -8 2 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999682 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8306: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) C D B A E (8) A E C B D (8) E A D C B (7) B D C A E (7) E A D B C (6) C B D A E (4) D B E A C (3) B D E A C (3) B C D A E (3) A E B C D (3) E A C B D (2) E A B D C (2) E A B C D (2) D E C A B (2) D C E A B (2) D C B A E (2) D B C A E (2) C A E D B (2) C A E B D (2) C A B E D (2) B A E D C (2) E D A C B (1) E D A B C (1) D E A C B (1) D E A B C (1) D C B E A (1) D B E C A (1) D B C E A (1) C E A D B (1) C D A E B (1) C B A D E (1) B D E C A (1) B C A D E (1) B A E C D (1) B A C E D (1) B A C D E (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 12 10 8 B -14 0 -14 -2 -6 C -12 14 0 8 -10 D -10 2 -8 0 -8 E -8 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 12 10 8 B -14 0 -14 -2 -6 C -12 14 0 8 -10 D -10 2 -8 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=21 B=20 D=16 A=14 so A is eliminated. Round 2 votes counts: E=40 C=23 B=21 D=16 so D is eliminated. Round 3 votes counts: E=44 C=28 B=28 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:222 E:208 C:200 D:188 B:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 12 10 8 B -14 0 -14 -2 -6 C -12 14 0 8 -10 D -10 2 -8 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 10 8 B -14 0 -14 -2 -6 C -12 14 0 8 -10 D -10 2 -8 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 10 8 B -14 0 -14 -2 -6 C -12 14 0 8 -10 D -10 2 -8 0 -8 E -8 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8307: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (16) E A C B D (15) B D E A C (12) B D A C E (6) C E A D B (5) E A B C D (4) C A E D B (4) E C A D B (3) E A B D C (3) D B E C A (3) D B C E A (3) C D A B E (3) E D C B A (2) E B D A C (2) D B E A C (2) C A E B D (2) C A D E B (2) A C B D E (2) E C D A B (1) E C A B D (1) E B A D C (1) E A C D B (1) D C B E A (1) C D E A B (1) C A D B E (1) B E D A C (1) B D A E C (1) B A D C E (1) A E C B D (1) Total count = 100 A B C D E A 0 -2 4 -8 -22 B 2 0 10 4 4 C -4 -10 0 -8 -6 D 8 -4 8 0 8 E 22 -4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -8 -22 B 2 0 10 4 4 C -4 -10 0 -8 -6 D 8 -4 8 0 8 E 22 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=25 B=21 C=18 A=3 so A is eliminated. Round 2 votes counts: E=34 D=25 B=21 C=20 so C is eliminated. Round 3 votes counts: E=45 D=32 B=23 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:210 E:208 A:186 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 -8 -22 B 2 0 10 4 4 C -4 -10 0 -8 -6 D 8 -4 8 0 8 E 22 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -8 -22 B 2 0 10 4 4 C -4 -10 0 -8 -6 D 8 -4 8 0 8 E 22 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -8 -22 B 2 0 10 4 4 C -4 -10 0 -8 -6 D 8 -4 8 0 8 E 22 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999738 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8308: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E C A D B (6) E A D B C (5) C E D B A (5) C B D E A (5) A B D E C (5) C E B D A (4) C B A D E (4) B D A C E (4) E C D A B (3) E C A B D (3) E A C D B (3) C E A B D (3) B D C A E (3) A E B D C (3) E A D C B (2) D B E A C (2) C E D A B (2) C E A D B (2) C D B E A (2) C A E B D (2) B D A E C (2) A E D B C (2) A E C B D (2) A D B E C (2) E D C A B (1) E D B A C (1) E C D B A (1) E A C B D (1) D E B C A (1) D B C E A (1) D B A E C (1) C E B A D (1) C A B E D (1) B C D A E (1) B A D C E (1) A D E B C (1) A C E B D (1) A C B D E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -16 8 -8 B -8 0 -24 14 -10 C 16 24 0 22 4 D -8 -14 -22 0 -10 E 8 10 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -16 8 -8 B -8 0 -24 14 -10 C 16 24 0 22 4 D -8 -14 -22 0 -10 E 8 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 E=26 A=20 B=11 D=5 so D is eliminated. Round 2 votes counts: C=38 E=27 A=20 B=15 so B is eliminated. Round 3 votes counts: C=43 E=29 A=28 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:233 E:212 A:196 B:186 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -16 8 -8 B -8 0 -24 14 -10 C 16 24 0 22 4 D -8 -14 -22 0 -10 E 8 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -16 8 -8 B -8 0 -24 14 -10 C 16 24 0 22 4 D -8 -14 -22 0 -10 E 8 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -16 8 -8 B -8 0 -24 14 -10 C 16 24 0 22 4 D -8 -14 -22 0 -10 E 8 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985048 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8309: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (23) B A D E C (15) C E B A D (8) D A B E C (6) D A B C E (6) B A D C E (6) B D A C E (5) A D B E C (5) E C D A B (3) C E D B A (3) E B C A D (2) C D A E B (2) B E A C D (2) A D E B C (2) E C B A D (1) E C A B D (1) E B A D C (1) D C A B E (1) D A C E B (1) D A C B E (1) C E B D A (1) C B D A E (1) B E A D C (1) B D A E C (1) B C D A E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 6 -10 8 B -4 0 8 -6 4 C -6 -8 0 -4 18 D 10 6 4 0 8 E -8 -4 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -10 8 B -4 0 8 -6 4 C -6 -8 0 -4 18 D 10 6 4 0 8 E -8 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=31 D=15 E=8 A=8 so E is eliminated. Round 2 votes counts: C=43 B=34 D=15 A=8 so A is eliminated. Round 3 votes counts: C=43 B=35 D=22 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 A:204 B:201 C:200 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -10 8 B -4 0 8 -6 4 C -6 -8 0 -4 18 D 10 6 4 0 8 E -8 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -10 8 B -4 0 8 -6 4 C -6 -8 0 -4 18 D 10 6 4 0 8 E -8 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -10 8 B -4 0 8 -6 4 C -6 -8 0 -4 18 D 10 6 4 0 8 E -8 -4 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999681 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8310: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (15) D E C A B (10) D C E B A (10) B A C D E (9) E D C A B (7) D C B E A (6) A B E C D (6) B A E C D (3) E D A C B (2) D C B A E (2) D B C A E (2) C D B E A (2) B D A C E (2) B A D E C (2) A E B D C (2) E D A B C (1) E C D A B (1) E A D C B (1) E A D B C (1) E A B C D (1) D E A C B (1) D B A E C (1) D B A C E (1) C E D B A (1) C E D A B (1) C D E B A (1) C D E A B (1) C B D A E (1) C B A D E (1) B C D A E (1) B C A D E (1) B A D C E (1) A E C B D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -24 4 -8 6 B 24 0 2 -4 14 C -4 -2 0 -6 16 D 8 4 6 0 10 E -6 -14 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 4 -8 6 B 24 0 2 -4 14 C -4 -2 0 -6 16 D 8 4 6 0 10 E -6 -14 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=33 E=14 A=11 C=8 so C is eliminated. Round 2 votes counts: D=37 B=36 E=16 A=11 so A is eliminated. Round 3 votes counts: B=43 D=37 E=20 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:214 C:202 A:189 E:177 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 4 -8 6 B 24 0 2 -4 14 C -4 -2 0 -6 16 D 8 4 6 0 10 E -6 -14 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 4 -8 6 B 24 0 2 -4 14 C -4 -2 0 -6 16 D 8 4 6 0 10 E -6 -14 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 4 -8 6 B 24 0 2 -4 14 C -4 -2 0 -6 16 D 8 4 6 0 10 E -6 -14 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8311: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) C B E A D (6) B E D C A (5) A C D E B (5) E B D A C (4) C E A B D (4) A D E C B (4) C A E D B (3) C A E B D (3) C A D B E (3) B E C D A (3) B D E A C (3) A D C E B (3) A C E D B (3) E B C A D (2) E B A D C (2) E A C B D (2) E A B D C (2) D E B A C (2) D B C E A (2) D A C B E (2) D A B E C (2) C B E D A (2) B D C E A (2) B C D E A (2) A E C B D (2) A D E B C (2) E D B A C (1) E C A B D (1) E A D B C (1) D C B A E (1) D B E C A (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B C E (1) C E B A D (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D A C (1) B D E C A (1) A E D C B (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 6 8 -14 B 4 0 -4 2 -2 C -6 4 0 -4 0 D -8 -2 4 0 0 E 14 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.348889 E: 0.651111 Sum of squares = 0.545668977744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.348889 E: 1.000000 A B C D E A 0 -4 6 8 -14 B 4 0 -4 2 -2 C -6 4 0 -4 0 D -8 -2 4 0 0 E 14 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 0.500544 Sum of squares = 0.5000005923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=22 D=20 B=17 E=15 so E is eliminated. Round 2 votes counts: C=27 A=27 B=25 D=21 so D is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:208 B:200 A:198 C:197 D:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 8 -14 B 4 0 -4 2 -2 C -6 4 0 -4 0 D -8 -2 4 0 0 E 14 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 0.500544 Sum of squares = 0.5000005923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 8 -14 B 4 0 -4 2 -2 C -6 4 0 -4 0 D -8 -2 4 0 0 E 14 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 0.500544 Sum of squares = 0.5000005923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 8 -14 B 4 0 -4 2 -2 C -6 4 0 -4 0 D -8 -2 4 0 0 E 14 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 0.500544 Sum of squares = 0.5000005923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499456 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8312: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) B C D A E (7) D A B E C (6) C E B A D (6) C D B A E (5) C B D E A (5) C E A B D (4) C B E D A (4) C B D A E (4) E A D C B (3) E A C D B (3) E A C B D (3) D B C A E (3) C B E A D (3) E A D B C (2) E A B D C (2) E A B C D (2) D C B A E (2) D B A E C (2) D A E C B (2) D A B C E (2) C E D A B (2) B D C A E (2) B C E A D (2) B A E D C (2) A E D B C (2) A D B E C (2) E C B A D (1) E C A D B (1) E B A C D (1) D A C B E (1) C D E A B (1) B E C A D (1) B C E D A (1) B A D E C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -8 -14 6 B 4 0 0 4 10 C 8 0 0 12 8 D 14 -4 -12 0 8 E -6 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.455799 C: 0.544201 D: 0.000000 E: 0.000000 Sum of squares = 0.503907462579 Cumulative probabilities = A: 0.000000 B: 0.455799 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -14 6 B 4 0 0 4 10 C 8 0 0 12 8 D 14 -4 -12 0 8 E -6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=26 E=18 B=16 A=6 so A is eliminated. Round 2 votes counts: C=34 D=29 E=21 B=16 so B is eliminated. Round 3 votes counts: C=44 D=32 E=24 so E is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:209 D:203 A:190 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -14 6 B 4 0 0 4 10 C 8 0 0 12 8 D 14 -4 -12 0 8 E -6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -14 6 B 4 0 0 4 10 C 8 0 0 12 8 D 14 -4 -12 0 8 E -6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -14 6 B 4 0 0 4 10 C 8 0 0 12 8 D 14 -4 -12 0 8 E -6 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8313: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) C D A E B (8) B E A D C (8) D C E A B (6) C D B A E (5) B D E A C (4) B A E C D (4) A C E B D (4) E A C D B (3) E A B C D (3) D C B E A (3) C A E D B (3) E D A C B (2) E B A D C (2) C D E A B (2) C A E B D (2) B D C E A (2) B C D A E (2) B A E D C (2) A E B C D (2) E D A B C (1) E B A C D (1) E A D B C (1) E A C B D (1) E A B D C (1) D E C B A (1) D C E B A (1) D C A B E (1) D B C E A (1) D B C A E (1) C D A B E (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B E A C D (1) B D E C A (1) B D C A E (1) B D A E C (1) B C A E D (1) B A C D E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -10 -10 8 B 8 0 -16 -4 8 C 10 16 0 -2 18 D 10 4 2 0 8 E -8 -8 -18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -10 8 B 8 0 -16 -4 8 C 10 16 0 -2 18 D 10 4 2 0 8 E -8 -8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=24 C=24 E=15 A=8 so A is eliminated. Round 2 votes counts: C=29 B=29 D=24 E=18 so E is eliminated. Round 3 votes counts: B=38 C=34 D=28 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:212 B:198 A:190 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -10 8 B 8 0 -16 -4 8 C 10 16 0 -2 18 D 10 4 2 0 8 E -8 -8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -10 8 B 8 0 -16 -4 8 C 10 16 0 -2 18 D 10 4 2 0 8 E -8 -8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -10 8 B 8 0 -16 -4 8 C 10 16 0 -2 18 D 10 4 2 0 8 E -8 -8 -18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8314: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (15) A B E C D (15) E B A D C (14) C D A B E (6) E D B C A (5) D C A B E (5) A C B D E (4) A B C D E (4) E B A C D (3) C D E A B (3) E B D A C (2) D C E A B (2) D C A E B (2) C E D B A (2) C D E B A (2) C D A E B (2) C A D B E (2) B E A D C (2) A B C E D (2) E D C B A (1) E B D C A (1) B A E D C (1) B A E C D (1) A E B C D (1) A C B E D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 4 4 -4 B -2 0 6 6 -10 C -4 -6 0 -4 6 D -4 -6 4 0 -4 E 4 10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775508 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 2 4 4 -4 B -2 0 6 6 -10 C -4 -6 0 -4 6 D -4 -6 4 0 -4 E 4 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775537 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=26 D=24 C=17 B=4 so B is eliminated. Round 2 votes counts: A=31 E=28 D=24 C=17 so C is eliminated. Round 3 votes counts: D=37 A=33 E=30 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:206 A:203 B:200 C:196 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 4 -4 B -2 0 6 6 -10 C -4 -6 0 -4 6 D -4 -6 4 0 -4 E 4 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775537 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 4 -4 B -2 0 6 6 -10 C -4 -6 0 -4 6 D -4 -6 4 0 -4 E 4 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775537 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 4 -4 B -2 0 6 6 -10 C -4 -6 0 -4 6 D -4 -6 4 0 -4 E 4 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.285714 D: 0.000000 E: 0.285714 Sum of squares = 0.346938775537 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8315: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (8) D B A C E (8) B D C A E (8) E A C D B (7) B D A C E (7) D B A E C (5) C E A B D (5) E D B C A (4) E C B D A (4) E A D C B (4) D A B E C (4) E C B A D (3) B C D E A (3) B C D A E (3) A D B C E (3) E D B A C (2) D A B C E (2) C E B A D (2) C B A E D (2) A E D C B (2) E D A C B (1) E C A D B (1) D B E A C (1) D B C A E (1) C E B D A (1) C B E D A (1) C B D E A (1) A E D B C (1) A E C D B (1) A D E B C (1) A D B E C (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 6 -12 4 B 12 0 10 -2 4 C -6 -10 0 -12 0 D 12 2 12 0 0 E -4 -4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.686181 E: 0.313819 Sum of squares = 0.569326460217 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.686181 E: 1.000000 A B C D E A 0 -12 6 -12 4 B 12 0 10 -2 4 C -6 -10 0 -12 0 D 12 2 12 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555564557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=21 B=21 C=12 A=12 so C is eliminated. Round 2 votes counts: E=42 B=25 D=21 A=12 so A is eliminated. Round 3 votes counts: E=47 D=27 B=26 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:212 E:196 A:193 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 6 -12 4 B 12 0 10 -2 4 C -6 -10 0 -12 0 D 12 2 12 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555564557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -12 4 B 12 0 10 -2 4 C -6 -10 0 -12 0 D 12 2 12 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555564557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -12 4 B 12 0 10 -2 4 C -6 -10 0 -12 0 D 12 2 12 0 0 E -4 -4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555564557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8316: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (10) E D A B C (8) E A C D B (6) B C D A E (6) E D A C B (5) E A D C B (4) D C B E A (4) D B C E A (4) C B A D E (4) C A B E D (4) A E B C D (4) D E B C A (3) D E B A C (3) D B E C A (3) C A B D E (3) B D C E A (3) A E C B D (3) A C E B D (3) C D B A E (2) C B D A E (2) B D C A E (2) A E C D B (2) E A D B C (1) E A B D C (1) D E C B A (1) D E C A B (1) D C E B A (1) C D B E A (1) C A E D B (1) B D E A C (1) B A C E D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -10 0 0 B 2 0 4 0 6 C 10 -4 0 10 6 D 0 0 -10 0 8 E 0 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.855608 C: 0.000000 D: 0.144392 E: 0.000000 Sum of squares = 0.752914197062 Cumulative probabilities = A: 0.000000 B: 0.855608 C: 0.855608 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 0 0 B 2 0 4 0 6 C 10 -4 0 10 6 D 0 0 -10 0 8 E 0 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836856392 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=23 D=20 C=17 A=15 so A is eliminated. Round 2 votes counts: E=34 B=25 C=21 D=20 so D is eliminated. Round 3 votes counts: E=42 B=32 C=26 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:206 D:199 A:194 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -10 0 0 B 2 0 4 0 6 C 10 -4 0 10 6 D 0 0 -10 0 8 E 0 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836856392 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 0 0 B 2 0 4 0 6 C 10 -4 0 10 6 D 0 0 -10 0 8 E 0 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836856392 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 0 0 B 2 0 4 0 6 C 10 -4 0 10 6 D 0 0 -10 0 8 E 0 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.591836856392 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8317: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) D E B C A (6) E C B A D (5) E B C A D (5) D A C B E (5) D A B C E (5) C A D E B (5) D B E A C (4) D B A E C (4) A C D B E (4) E B D C A (3) E B C D A (3) C E A B D (3) A C E B D (3) D E C B A (2) D C A B E (2) C D A B E (2) C A D B E (2) B E D A C (2) B D E A C (2) A D C B E (2) A D B C E (2) E D C B A (1) E D B C A (1) E C A B D (1) E B D A C (1) E B A D C (1) E B A C D (1) D C E A B (1) D C B A E (1) D B E C A (1) D A B E C (1) C E D A B (1) C E B D A (1) C D E A B (1) C A E D B (1) B E A D C (1) B A E D C (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -18 0 6 B -6 0 -8 -8 -14 C 18 8 0 -2 4 D 0 8 2 0 8 E -6 14 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.071541 B: 0.000000 C: 0.000000 D: 0.928459 E: 0.000000 Sum of squares = 0.867154804721 Cumulative probabilities = A: 0.071541 B: 0.071541 C: 0.071541 D: 1.000000 E: 1.000000 A B C D E A 0 6 -18 0 6 B -6 0 -8 -8 -14 C 18 8 0 -2 4 D 0 8 2 0 8 E -6 14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012618 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=27 E=22 A=12 B=7 so B is eliminated. Round 2 votes counts: D=34 C=27 E=25 A=14 so A is eliminated. Round 3 votes counts: D=40 C=34 E=26 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:209 E:198 A:197 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -18 0 6 B -6 0 -8 -8 -14 C 18 8 0 -2 4 D 0 8 2 0 8 E -6 14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012618 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -18 0 6 B -6 0 -8 -8 -14 C 18 8 0 -2 4 D 0 8 2 0 8 E -6 14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012618 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -18 0 6 B -6 0 -8 -8 -14 C 18 8 0 -2 4 D 0 8 2 0 8 E -6 14 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.100000 B: 0.000000 C: 0.000000 D: 0.900000 E: 0.000000 Sum of squares = 0.820000012618 Cumulative probabilities = A: 0.100000 B: 0.100000 C: 0.100000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8318: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) C D E B A (9) C A D E B (7) A B E D C (7) A B E C D (7) D E C B A (5) D C E B A (5) C D E A B (5) A C E D B (5) C E D A B (3) B D E C A (3) D E B C A (2) C D A E B (2) C A E D B (2) C A D B E (2) B E D A C (2) B E A D C (2) B A D E C (2) B A D C E (2) A E B D C (2) A C B E D (2) A B C E D (2) A B C D E (2) E D C B A (1) E C D A B (1) D C B E A (1) C D A B E (1) B E D C A (1) B C D A E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -2 16 20 B -8 0 -10 -4 -4 C 2 10 0 10 6 D -16 4 -10 0 0 E -20 4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 16 20 B -8 0 -10 -4 -4 C 2 10 0 10 6 D -16 4 -10 0 0 E -20 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=31 A=31 B=23 D=13 E=2 so E is eliminated. Round 2 votes counts: C=32 A=31 B=23 D=14 so D is eliminated. Round 3 votes counts: C=44 A=31 B=25 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:221 C:214 D:189 E:189 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 16 20 B -8 0 -10 -4 -4 C 2 10 0 10 6 D -16 4 -10 0 0 E -20 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 16 20 B -8 0 -10 -4 -4 C 2 10 0 10 6 D -16 4 -10 0 0 E -20 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 16 20 B -8 0 -10 -4 -4 C 2 10 0 10 6 D -16 4 -10 0 0 E -20 4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8319: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) E A C D B (6) C E D B A (6) E C A B D (5) B D A C E (5) A D B E C (5) A B D E C (5) D C B E A (4) C E B D A (4) B D C A E (4) A E D C B (4) E C D A B (3) E C A D B (3) D B C E A (3) C D E B A (3) C B E D A (3) A D E B C (3) E C B A D (2) C B D E A (2) B C D E A (2) A E C D B (2) A E C B D (2) A E B C D (2) E A C B D (1) D C E B A (1) D C E A B (1) D B C A E (1) D A E C B (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) B D C E A (1) B C A D E (1) B A E C D (1) B A D E C (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 0 -8 -2 B 6 0 -8 -12 -4 C 0 8 0 -2 2 D 8 12 2 0 8 E 2 4 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -8 -2 B 6 0 -8 -12 -4 C 0 8 0 -2 2 D 8 12 2 0 8 E 2 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=21 E=20 C=19 B=16 so B is eliminated. Round 2 votes counts: D=31 A=27 C=22 E=20 so E is eliminated. Round 3 votes counts: C=35 A=34 D=31 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 C:204 E:198 A:192 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -8 -2 B 6 0 -8 -12 -4 C 0 8 0 -2 2 D 8 12 2 0 8 E 2 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -8 -2 B 6 0 -8 -12 -4 C 0 8 0 -2 2 D 8 12 2 0 8 E 2 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -8 -2 B 6 0 -8 -12 -4 C 0 8 0 -2 2 D 8 12 2 0 8 E 2 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8320: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (11) A C E D B (9) B D E C A (8) A E D C B (6) A E C D B (6) E D A C B (4) B C D E A (4) E D C A B (3) E A D C B (3) D E B C A (3) D B E C A (3) B C D A E (3) B C A D E (3) A E B D C (3) D E C B A (2) D C B E A (2) B A C D E (2) A E D B C (2) A C B D E (2) A B E D C (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D A C (1) E A C D B (1) D B C E A (1) C E D A B (1) C D E B A (1) C B D E A (1) C B A D E (1) C A E D B (1) C A D B E (1) C A B D E (1) B E D A C (1) B D E A C (1) B A E D C (1) A C E B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -8 B 4 0 6 -4 0 C 4 -6 0 -20 -6 D 6 4 20 0 2 E 8 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999624 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 -8 B 4 0 6 -4 0 C 4 -6 0 -20 -6 D 6 4 20 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=33 E=15 D=11 C=7 so C is eliminated. Round 2 votes counts: B=36 A=36 E=16 D=12 so D is eliminated. Round 3 votes counts: B=42 A=36 E=22 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:216 E:206 B:203 A:189 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -8 B 4 0 6 -4 0 C 4 -6 0 -20 -6 D 6 4 20 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -8 B 4 0 6 -4 0 C 4 -6 0 -20 -6 D 6 4 20 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -8 B 4 0 6 -4 0 C 4 -6 0 -20 -6 D 6 4 20 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984107 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8321: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) E C B D A (7) D E C B A (7) D E A C B (7) C E B A D (7) A D B C E (7) A B C E D (7) E C B A D (6) E D C B A (5) D A E B C (5) D A B E C (5) B A C E D (5) A B D C E (5) D A B C E (4) E C D B A (3) C B E A D (3) A B C D E (3) C E B D A (2) D A E C B (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 6 -2 -10 -14 B -6 0 -14 -6 -18 C 2 14 0 -10 -10 D 10 6 10 0 6 E 14 18 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -10 -14 B -6 0 -14 -6 -18 C 2 14 0 -10 -10 D 10 6 10 0 6 E 14 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=22 E=21 C=12 B=7 so B is eliminated. Round 2 votes counts: D=38 A=27 E=21 C=14 so C is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:218 D:216 C:198 A:190 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 -10 -14 B -6 0 -14 -6 -18 C 2 14 0 -10 -10 D 10 6 10 0 6 E 14 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -10 -14 B -6 0 -14 -6 -18 C 2 14 0 -10 -10 D 10 6 10 0 6 E 14 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -10 -14 B -6 0 -14 -6 -18 C 2 14 0 -10 -10 D 10 6 10 0 6 E 14 18 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999502 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8322: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) A C D E B (7) E D B A C (6) D C A E B (5) C A D E B (5) B E D C A (5) B E A C D (5) C D A B E (4) B A C E D (4) A C B E D (4) E B A D C (3) C D A E B (3) B A E C D (3) A D C E B (3) E B D A C (2) D E C A B (2) D E A C B (2) D A C E B (2) C A D B E (2) C A B D E (2) B C D A E (2) B C A E D (2) B C A D E (2) A C E D B (2) E D A C B (1) D E B C A (1) D E B A C (1) D C E A B (1) B E C D A (1) B E A D C (1) B D E C A (1) A E C B D (1) A E B C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 24 -2 12 B 2 0 6 4 4 C -24 -6 0 4 2 D 2 -4 -4 0 -8 E -12 -4 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999407 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 24 -2 12 B 2 0 6 4 4 C -24 -6 0 4 2 D 2 -4 -4 0 -8 E -12 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=20 C=16 D=14 E=12 so E is eliminated. Round 2 votes counts: B=43 D=21 A=20 C=16 so C is eliminated. Round 3 votes counts: B=43 A=29 D=28 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:208 E:195 D:193 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 24 -2 12 B 2 0 6 4 4 C -24 -6 0 4 2 D 2 -4 -4 0 -8 E -12 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 24 -2 12 B 2 0 6 4 4 C -24 -6 0 4 2 D 2 -4 -4 0 -8 E -12 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 24 -2 12 B 2 0 6 4 4 C -24 -6 0 4 2 D 2 -4 -4 0 -8 E -12 -4 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8323: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) E D A C B (6) C B D E A (6) A B E D C (5) E A D C B (4) D E C A B (4) C A B E D (4) A E D C B (4) A B E C D (4) D C E B A (3) C E D A B (3) C A E D B (3) B C A D E (3) A E D B C (3) D E B C A (2) D E B A C (2) C A E B D (2) B D C E A (2) B C D E A (2) B C D A E (2) B A D E C (2) B A C E D (2) B A C D E (2) A C E B D (2) A B C E D (2) E D C A B (1) E D A B C (1) D E C B A (1) D E A C B (1) D E A B C (1) D C E A B (1) D B E C A (1) D B E A C (1) D B C E A (1) C D E B A (1) C D E A B (1) C B A D E (1) B D E C A (1) B D E A C (1) B D C A E (1) B C A E D (1) A E C D B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 10 -8 8 6 B -10 0 -16 6 4 C 8 16 0 2 8 D -8 -6 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 8 6 B -10 0 -16 6 4 C 8 16 0 2 8 D -8 -6 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 B=19 D=18 E=12 so E is eliminated. Round 2 votes counts: C=28 A=27 D=26 B=19 so B is eliminated. Round 3 votes counts: C=36 A=33 D=31 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:217 A:208 E:198 B:192 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 8 6 B -10 0 -16 6 4 C 8 16 0 2 8 D -8 -6 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 8 6 B -10 0 -16 6 4 C 8 16 0 2 8 D -8 -6 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 8 6 B -10 0 -16 6 4 C 8 16 0 2 8 D -8 -6 -2 0 -14 E -6 -4 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8324: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) A D C B E (9) C A D B E (8) E B D A C (7) C E A D B (6) C A D E B (6) E C B A D (5) D A B C E (5) A D B C E (5) D B A C E (4) E B C D A (3) C D A B E (3) B D A E C (3) A C D E B (3) E B D C A (2) E B A D C (2) E A B D C (2) D B A E C (2) C A E D B (2) B D E A C (2) E C B D A (1) E C A D B (1) E B C A D (1) D A C B E (1) D A B E C (1) C E A B D (1) C D E A B (1) C D A E B (1) C B E D A (1) B D C A E (1) A C D B E (1) Total count = 100 A B C D E A 0 12 14 4 10 B -12 0 0 -18 12 C -14 0 0 -12 16 D -4 18 12 0 12 E -10 -12 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 4 10 B -12 0 0 -18 12 C -14 0 0 -12 16 D -4 18 12 0 12 E -10 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=24 A=18 B=16 D=13 so D is eliminated. Round 2 votes counts: C=29 A=25 E=24 B=22 so B is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:219 C:195 B:191 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 4 10 B -12 0 0 -18 12 C -14 0 0 -12 16 D -4 18 12 0 12 E -10 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 4 10 B -12 0 0 -18 12 C -14 0 0 -12 16 D -4 18 12 0 12 E -10 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 4 10 B -12 0 0 -18 12 C -14 0 0 -12 16 D -4 18 12 0 12 E -10 -12 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8325: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (6) D C B A E (6) D A B C E (6) E C B A D (5) C E D B A (5) C D B A E (5) E C A B D (3) E A C B D (3) D A E B C (3) C D B E A (3) C B E A D (3) C B D A E (3) B A D C E (3) E D C A B (2) E C D A B (2) E A D C B (2) E A B D C (2) C E B A D (2) C B D E A (2) B C A D E (2) B A E C D (2) B A C E D (2) A E B D C (2) A D E B C (2) A D B E C (2) A B E D C (2) E D A C B (1) E D A B C (1) E C D B A (1) E B A C D (1) E A D B C (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) D B C A E (1) D B A C E (1) D A C B E (1) C E B D A (1) C D E B A (1) C B A E D (1) B D C A E (1) B C D A E (1) B C A E D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 -2 0 B 8 0 -12 0 0 C 8 12 0 12 8 D 2 0 -12 0 -2 E 0 0 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -2 0 B 8 0 -12 0 0 C 8 12 0 12 8 D 2 0 -12 0 -2 E 0 0 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=26 D=22 B=13 A=9 so A is eliminated. Round 2 votes counts: E=32 D=26 C=26 B=16 so B is eliminated. Round 3 votes counts: E=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:198 E:197 D:194 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -2 0 B 8 0 -12 0 0 C 8 12 0 12 8 D 2 0 -12 0 -2 E 0 0 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -2 0 B 8 0 -12 0 0 C 8 12 0 12 8 D 2 0 -12 0 -2 E 0 0 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -2 0 B 8 0 -12 0 0 C 8 12 0 12 8 D 2 0 -12 0 -2 E 0 0 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8326: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) C E D A B (8) B A D C E (8) A B E C D (6) A B C E D (6) E C D A B (5) D E C B A (5) D B E C A (5) C E A B D (5) B D A E C (5) D B A E C (4) C E A D B (4) B D A C E (4) D C E B A (3) A C E B D (3) A C B E D (3) D B C E A (2) D B A C E (2) B A E D C (2) B A E C D (2) E C A D B (1) E C A B D (1) D C E A B (1) D C B E A (1) D B E A C (1) C A E B D (1) Total count = 100 A B C D E A 0 -12 16 8 16 B 12 0 18 16 26 C -16 -18 0 -10 2 D -8 -16 10 0 6 E -16 -26 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 16 8 16 B 12 0 18 16 26 C -16 -18 0 -10 2 D -8 -16 10 0 6 E -16 -26 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=24 C=18 A=18 E=7 so E is eliminated. Round 2 votes counts: B=33 C=25 D=24 A=18 so A is eliminated. Round 3 votes counts: B=45 C=31 D=24 so D is eliminated. Round 4 votes counts: B=59 C=41 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:236 A:214 D:196 C:179 E:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 16 8 16 B 12 0 18 16 26 C -16 -18 0 -10 2 D -8 -16 10 0 6 E -16 -26 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 16 8 16 B 12 0 18 16 26 C -16 -18 0 -10 2 D -8 -16 10 0 6 E -16 -26 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 16 8 16 B 12 0 18 16 26 C -16 -18 0 -10 2 D -8 -16 10 0 6 E -16 -26 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8327: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (8) C B A D E (8) A D E B C (7) E D A B C (6) D E A C B (6) C B E D A (6) B C E A D (6) E B C D A (4) C B D E A (4) E D A C B (3) E A D B C (3) C B E A D (3) C B D A E (3) C B A E D (3) B C A E D (3) D C A B E (2) B E C A D (2) A D C B E (2) A D B C E (2) E D C A B (1) E D B C A (1) E D B A C (1) E C B D A (1) E B A C D (1) D E A B C (1) D A E B C (1) D A C B E (1) C D E B A (1) C D A B E (1) B C A D E (1) B A C E D (1) A E D B C (1) A D E C B (1) A D B E C (1) A C D B E (1) A B E C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 -2 0 B -2 0 -10 -2 6 C 0 10 0 2 0 D 2 2 -2 0 4 E 0 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.214319 B: 0.000000 C: 0.595609 D: 0.000000 E: 0.190072 Sum of squares = 0.436810258363 Cumulative probabilities = A: 0.214319 B: 0.214319 C: 0.809928 D: 0.809928 E: 1.000000 A B C D E A 0 2 0 -2 0 B -2 0 -10 -2 6 C 0 10 0 2 0 D 2 2 -2 0 4 E 0 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571531635 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=21 D=19 A=18 B=13 so B is eliminated. Round 2 votes counts: C=39 E=23 D=19 A=19 so D is eliminated. Round 3 votes counts: C=41 E=30 A=29 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:206 D:203 A:200 B:196 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 -2 0 B -2 0 -10 -2 6 C 0 10 0 2 0 D 2 2 -2 0 4 E 0 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571531635 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 0 B -2 0 -10 -2 6 C 0 10 0 2 0 D 2 2 -2 0 4 E 0 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571531635 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 0 B -2 0 -10 -2 6 C 0 10 0 2 0 D 2 2 -2 0 4 E 0 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571531635 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8328: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) E C D A B (7) A D B E C (7) C B E D A (6) B A C D E (6) E D A C B (5) C E B D A (5) B A D C E (5) D A E B C (4) A D E B C (4) E D C A B (3) B C A E D (3) E A D C B (2) D E C A B (2) D E A C B (2) D B A E C (2) B C E D A (2) B C D E A (2) B C A D E (2) A D E C B (2) A B D C E (2) E C D B A (1) E C A D B (1) D E B C A (1) D B E C A (1) D B C E A (1) C E D B A (1) C E D A B (1) C E B A D (1) C E A D B (1) C D B E A (1) C B E A D (1) C B A E D (1) B C D A E (1) B A C E D (1) A E D C B (1) A E C D B (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 10 4 6 B -12 0 10 -2 10 C -10 -10 0 -6 -10 D -4 2 6 0 10 E -6 -10 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 4 6 B -12 0 10 -2 10 C -10 -10 0 -6 -10 D -4 2 6 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=22 E=19 C=18 D=13 so D is eliminated. Round 2 votes counts: A=32 B=26 E=24 C=18 so C is eliminated. Round 3 votes counts: B=35 E=33 A=32 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 D:207 B:203 E:192 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 4 6 B -12 0 10 -2 10 C -10 -10 0 -6 -10 D -4 2 6 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 4 6 B -12 0 10 -2 10 C -10 -10 0 -6 -10 D -4 2 6 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 4 6 B -12 0 10 -2 10 C -10 -10 0 -6 -10 D -4 2 6 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8329: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (15) C E B D A (9) B E C A D (8) A B E D C (8) D C A E B (5) D A C E B (4) B E A C D (4) A D B E C (4) A B E C D (4) D C E B A (3) D A E C B (3) C E D B A (3) E B C D A (2) D E A B C (2) D C E A B (2) C D E A B (2) B E A D C (2) A D C B E (2) A B D E C (2) A B C E D (2) E C B D A (1) E B D C A (1) D E C B A (1) D A C B E (1) C D A E B (1) C D A B E (1) C B E D A (1) C B E A D (1) C B A E D (1) B E C D A (1) A E B D C (1) A D E C B (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -16 -16 -16 B 6 0 -14 -4 -14 C 16 14 0 12 8 D 16 4 -12 0 2 E 16 14 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -16 -16 B 6 0 -14 -4 -14 C 16 14 0 12 8 D 16 4 -12 0 2 E 16 14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=26 D=21 B=15 E=4 so E is eliminated. Round 2 votes counts: C=35 A=26 D=21 B=18 so B is eliminated. Round 3 votes counts: C=46 A=32 D=22 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:210 D:205 B:187 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 -16 -16 B 6 0 -14 -4 -14 C 16 14 0 12 8 D 16 4 -12 0 2 E 16 14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -16 -16 B 6 0 -14 -4 -14 C 16 14 0 12 8 D 16 4 -12 0 2 E 16 14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -16 -16 B 6 0 -14 -4 -14 C 16 14 0 12 8 D 16 4 -12 0 2 E 16 14 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8330: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) A E C B D (10) C D B A E (8) A C E D B (8) E A B D C (7) D B C E A (6) C A E D B (6) E B A D C (5) B E D A C (5) B D E A C (4) E B D A C (3) C A D E B (3) A E B D C (3) E A B C D (2) D C B E A (2) D C B A E (2) C D B E A (2) C A D B E (2) A E C D B (2) E C B A D (1) E A C B D (1) D B E A C (1) D B C A E (1) C D A B E (1) B D C E A (1) A E B C D (1) A C E B D (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 10 8 0 B 2 0 -2 8 -8 C -10 2 0 0 -10 D -8 -8 0 0 -10 E 0 8 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.386109 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.613891 Sum of squares = 0.525942220003 Cumulative probabilities = A: 0.386109 B: 0.386109 C: 0.386109 D: 0.386109 E: 1.000000 A B C D E A 0 -2 10 8 0 B 2 0 -2 8 -8 C -10 2 0 0 -10 D -8 -8 0 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=22 B=20 E=19 D=12 so D is eliminated. Round 2 votes counts: B=28 A=27 C=26 E=19 so E is eliminated. Round 3 votes counts: A=37 B=36 C=27 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 A:208 B:200 C:191 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 10 8 0 B 2 0 -2 8 -8 C -10 2 0 0 -10 D -8 -8 0 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 8 0 B 2 0 -2 8 -8 C -10 2 0 0 -10 D -8 -8 0 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 8 0 B 2 0 -2 8 -8 C -10 2 0 0 -10 D -8 -8 0 0 -10 E 0 8 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999916 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8331: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) B E A C D (8) B D E C A (6) D B C E A (5) D A C B E (5) B E D C A (5) E B C D A (3) D C A B E (3) C A E D B (3) B A E D C (3) A E B C D (3) A D C E B (3) E B A C D (2) D C E B A (2) D C A E B (2) D B E C A (2) D A B C E (2) C D E A B (2) B E C A D (2) B A E C D (2) B A D E C (2) A E C B D (2) A D C B E (2) A B E C D (2) A B D E C (2) E C B A D (1) E B C A D (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D B A C E (1) D A C E B (1) C E A D B (1) C E A B D (1) C D E B A (1) C D A E B (1) B E C D A (1) B D E A C (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 12 10 6 B 0 0 8 0 12 C -12 -8 0 2 0 D -10 0 -2 0 12 E -6 -12 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.530791 B: 0.469209 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501896209643 Cumulative probabilities = A: 0.530791 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 10 6 B 0 0 8 0 12 C -12 -8 0 2 0 D -10 0 -2 0 12 E -6 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 D=25 E=9 C=9 so E is eliminated. Round 2 votes counts: B=36 A=29 D=25 C=10 so C is eliminated. Round 3 votes counts: B=37 A=34 D=29 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:210 D:200 C:191 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 10 6 B 0 0 8 0 12 C -12 -8 0 2 0 D -10 0 -2 0 12 E -6 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 10 6 B 0 0 8 0 12 C -12 -8 0 2 0 D -10 0 -2 0 12 E -6 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 10 6 B 0 0 8 0 12 C -12 -8 0 2 0 D -10 0 -2 0 12 E -6 -12 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8332: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (7) C A E B D (5) B C A E D (5) E A D C B (4) D E A B C (4) D C E B A (4) C B A E D (4) B C D A E (4) A E C D B (4) A E B C D (4) E D A B C (3) E A D B C (3) D E C A B (3) D E B A C (3) D C B E A (3) B D C E A (3) B A E D C (3) E D A C B (2) D C E A B (2) D B E A C (2) D B C E A (2) C D B E A (2) C A E D B (2) B E A D C (2) B D E A C (2) B C A D E (2) B A E C D (2) B A C E D (2) A E C B D (2) E A B D C (1) D E B C A (1) C D B A E (1) C B D A E (1) C B A D E (1) C A B E D (1) B A D E C (1) A E D B C (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 8 2 -6 B 0 0 2 -6 -12 C -8 -2 0 -12 -10 D -2 6 12 0 -4 E 6 12 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 2 -6 B 0 0 2 -6 -12 C -8 -2 0 -12 -10 D -2 6 12 0 -4 E 6 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 C=17 E=13 A=13 so E is eliminated. Round 2 votes counts: D=36 B=26 A=21 C=17 so C is eliminated. Round 3 votes counts: D=39 B=32 A=29 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:206 A:202 B:192 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 2 -6 B 0 0 2 -6 -12 C -8 -2 0 -12 -10 D -2 6 12 0 -4 E 6 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 -6 B 0 0 2 -6 -12 C -8 -2 0 -12 -10 D -2 6 12 0 -4 E 6 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 -6 B 0 0 2 -6 -12 C -8 -2 0 -12 -10 D -2 6 12 0 -4 E 6 12 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8333: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) B E D A C (8) D B A E C (5) C A D E B (5) E C B A D (4) C A E D B (4) A C D B E (4) E B C D A (3) E B C A D (3) D B E A C (3) D A C B E (3) C A E B D (3) B E D C A (3) B E C A D (3) A D C B E (3) E C A B D (2) D E B A C (2) D A E C B (2) D A B C E (2) C E A B D (2) B D E A C (2) B D A E C (2) A C B E D (2) E D B C A (1) E C D A B (1) E C B D A (1) E B D A C (1) D E C A B (1) D B A C E (1) D A E B C (1) D A C E B (1) C B A E D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C D A (1) B C E A D (1) B C A E D (1) B A D C E (1) B A C E D (1) A D C E B (1) A D B C E (1) A C D E B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -2 -4 0 B 12 0 10 14 6 C 2 -10 0 -4 -14 D 4 -14 4 0 -10 E 0 -6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -4 0 B 12 0 10 14 6 C 2 -10 0 -4 -14 D 4 -14 4 0 -10 E 0 -6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=23 D=21 C=18 A=14 so A is eliminated. Round 2 votes counts: D=26 C=26 E=24 B=24 so E is eliminated. Round 3 votes counts: B=39 C=34 D=27 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:209 D:192 A:191 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -2 -4 0 B 12 0 10 14 6 C 2 -10 0 -4 -14 D 4 -14 4 0 -10 E 0 -6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -4 0 B 12 0 10 14 6 C 2 -10 0 -4 -14 D 4 -14 4 0 -10 E 0 -6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -4 0 B 12 0 10 14 6 C 2 -10 0 -4 -14 D 4 -14 4 0 -10 E 0 -6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8334: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (6) B C A D E (6) B C E D A (5) A D E C B (5) A C D E B (5) C B E D A (4) B E D C A (4) B C A E D (4) E C D A B (3) D E A C B (3) C B E A D (3) C A E D B (3) C A D E B (3) B D E A C (3) A C D B E (3) A B C D E (3) E D C A B (2) E D A C B (2) E D A B C (2) D A E C B (2) C E D A B (2) C B A E D (2) C A B E D (2) B E D A C (2) B C E A D (2) B A D E C (2) B A C D E (2) A D C E B (2) A C B D E (2) E D B A C (1) E C D B A (1) E C B D A (1) E B D C A (1) D E B A C (1) D A E B C (1) C E A B D (1) C B A D E (1) B E C D A (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -2 6 0 B -8 0 -4 4 4 C 2 4 0 18 12 D -6 -4 -18 0 4 E 0 -4 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 6 0 B -8 0 -4 4 4 C 2 4 0 18 12 D -6 -4 -18 0 4 E 0 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=22 C=21 E=13 D=13 so E is eliminated. Round 2 votes counts: B=32 C=26 A=22 D=20 so D is eliminated. Round 3 votes counts: A=38 B=34 C=28 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:206 B:198 E:190 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 6 0 B -8 0 -4 4 4 C 2 4 0 18 12 D -6 -4 -18 0 4 E 0 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 0 B -8 0 -4 4 4 C 2 4 0 18 12 D -6 -4 -18 0 4 E 0 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 0 B -8 0 -4 4 4 C 2 4 0 18 12 D -6 -4 -18 0 4 E 0 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999982118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8335: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) A D E B C (6) D C A B E (5) A E D B C (5) D B C A E (4) B C D E A (4) A E C D B (4) A D E C B (4) E A D B C (3) D C B A E (3) D A E B C (3) D A C B E (3) C B E A D (3) C B D E A (3) C B D A E (3) A C D B E (3) E C B A D (2) E C A B D (2) E A C B D (2) D B C E A (2) C E A B D (2) C D B A E (2) C B E D A (2) C A E B D (2) B C E D A (2) A E C B D (2) E D B A C (1) E B D A C (1) E B A D C (1) E B A C D (1) E A B C D (1) D B E A C (1) D B A E C (1) D B A C E (1) D A B E C (1) D A B C E (1) C D A B E (1) C B A E D (1) C B A D E (1) C A B E D (1) B D C E A (1) A D C E B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -4 10 20 B -8 0 -8 -12 0 C 4 8 0 2 6 D -10 12 -2 0 10 E -20 0 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 10 20 B -8 0 -8 -12 0 C 4 8 0 2 6 D -10 12 -2 0 10 E -20 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=25 C=21 E=20 B=7 so B is eliminated. Round 2 votes counts: C=27 A=27 D=26 E=20 so E is eliminated. Round 3 votes counts: C=37 A=35 D=28 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:217 C:210 D:205 B:186 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 10 20 B -8 0 -8 -12 0 C 4 8 0 2 6 D -10 12 -2 0 10 E -20 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 10 20 B -8 0 -8 -12 0 C 4 8 0 2 6 D -10 12 -2 0 10 E -20 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 10 20 B -8 0 -8 -12 0 C 4 8 0 2 6 D -10 12 -2 0 10 E -20 0 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997578 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8336: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (13) E A B C D (12) B C D E A (11) A E D C B (8) D A E C B (7) C B D E A (5) B E A C D (5) E B A C D (4) E A B D C (4) B C E D A (4) D C A E B (3) B C E A D (3) A E D B C (3) E A D B C (2) C D B A E (2) C B D A E (2) C A D E B (2) A E C B D (2) A D E C B (2) D C B E A (1) D A E B C (1) D A C E B (1) C B E A D (1) B E C D A (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 4 -2 -6 B 4 0 2 10 -4 C -4 -2 0 10 -4 D 2 -10 -10 0 0 E 6 4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.161560 E: 0.838440 Sum of squares = 0.729082647987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.161560 E: 1.000000 A B C D E A 0 -4 4 -2 -6 B 4 0 2 10 -4 C -4 -2 0 10 -4 D 2 -10 -10 0 0 E 6 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591837161852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=24 E=22 A=16 C=12 so C is eliminated. Round 2 votes counts: B=32 D=28 E=22 A=18 so A is eliminated. Round 3 votes counts: E=36 D=32 B=32 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:207 B:206 C:200 A:196 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 -2 -6 B 4 0 2 10 -4 C -4 -2 0 10 -4 D 2 -10 -10 0 0 E 6 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591837161852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -2 -6 B 4 0 2 10 -4 C -4 -2 0 10 -4 D 2 -10 -10 0 0 E 6 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591837161852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -2 -6 B 4 0 2 10 -4 C -4 -2 0 10 -4 D 2 -10 -10 0 0 E 6 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.714286 Sum of squares = 0.591837161852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8337: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) E C B D A (10) A D B C E (8) E C D A B (5) E B C A D (5) C E D A B (5) A D B E C (5) D A E C B (4) C A D B E (4) E D A C B (3) E C B A D (3) E B C D A (3) C E B D A (3) C E B A D (3) B E C A D (3) A B D C E (3) D A E B C (2) D A B E C (2) D A B C E (2) C B E A D (2) B A D E C (2) E D B A C (1) E D A B C (1) E C D B A (1) E B D A C (1) D C A E B (1) D A C B E (1) C D A E B (1) C B A E D (1) B E A D C (1) B E A C D (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 16 -2 -14 -2 B -16 0 -16 -14 -26 C 2 16 0 2 -6 D 14 14 -2 0 -4 E 2 26 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -2 -14 -2 B -16 0 -16 -14 -26 C 2 16 0 2 -6 D 14 14 -2 0 -4 E 2 26 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=23 C=19 A=16 B=9 so B is eliminated. Round 2 votes counts: E=38 D=23 C=20 A=19 so A is eliminated. Round 3 votes counts: D=42 E=38 C=20 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:211 C:207 A:199 B:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -2 -14 -2 B -16 0 -16 -14 -26 C 2 16 0 2 -6 D 14 14 -2 0 -4 E 2 26 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -14 -2 B -16 0 -16 -14 -26 C 2 16 0 2 -6 D 14 14 -2 0 -4 E 2 26 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -14 -2 B -16 0 -16 -14 -26 C 2 16 0 2 -6 D 14 14 -2 0 -4 E 2 26 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8338: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) B A E C D (10) D C E A B (7) C D E A B (7) B A D E C (7) A B E C D (7) B D A C E (6) B A D C E (5) E C A D B (4) E C A B D (4) D E C B A (4) C E D A B (4) E C D A B (3) B A E D C (3) A E B C D (3) E B A C D (2) D B C E A (2) B E A C D (2) A B C E D (2) E D C B A (1) E A C B D (1) D C B E A (1) D B C A E (1) D B A C E (1) A E C B D (1) A C E B D (1) Total count = 100 A B C D E A 0 -12 2 4 -6 B 12 0 2 8 -6 C -2 -2 0 2 -4 D -4 -8 -2 0 4 E 6 6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.333333 E: 0.444444 Sum of squares = 0.358024691354 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 1.000000 A B C D E A 0 -12 2 4 -6 B 12 0 2 8 -6 C -2 -2 0 2 -4 D -4 -8 -2 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.333333 E: 0.444444 Sum of squares = 0.358024690629 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=27 E=15 A=14 C=11 so C is eliminated. Round 2 votes counts: D=34 B=33 E=19 A=14 so A is eliminated. Round 3 votes counts: B=42 D=34 E=24 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:206 C:197 D:195 A:194 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 2 4 -6 B 12 0 2 8 -6 C -2 -2 0 2 -4 D -4 -8 -2 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.333333 E: 0.444444 Sum of squares = 0.358024690629 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 4 -6 B 12 0 2 8 -6 C -2 -2 0 2 -4 D -4 -8 -2 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.333333 E: 0.444444 Sum of squares = 0.358024690629 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 4 -6 B 12 0 2 8 -6 C -2 -2 0 2 -4 D -4 -8 -2 0 4 E 6 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.222222 C: 0.000000 D: 0.333333 E: 0.444444 Sum of squares = 0.358024690629 Cumulative probabilities = A: 0.000000 B: 0.222222 C: 0.222222 D: 0.555556 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8339: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (15) D C E A B (10) B E A C D (6) A E B C D (6) D C E B A (5) D C B E A (5) D C A E B (5) C D A E B (5) C D E A B (4) A E C D B (4) D C B A E (3) B A E D C (3) A C E D B (3) E A C B D (2) E A B C D (2) C E A D B (2) B E A D C (2) A E C B D (2) E C A D B (1) E B D C A (1) E B A C D (1) D E B C A (1) D B C E A (1) D B C A E (1) C E D A B (1) C A E D B (1) C A D E B (1) B D E C A (1) B D C E A (1) B D A E C (1) B D A C E (1) B A D C E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 2 8 8 B -2 0 -10 -8 -14 C -2 10 0 14 4 D -8 8 -14 0 -4 E -8 14 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 8 8 B -2 0 -10 -8 -14 C -2 10 0 14 4 D -8 8 -14 0 -4 E -8 14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993714 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=31 B=31 A=17 C=14 E=7 so E is eliminated. Round 2 votes counts: B=33 D=31 A=21 C=15 so C is eliminated. Round 3 votes counts: D=41 B=33 A=26 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:213 A:210 E:203 D:191 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 8 8 B -2 0 -10 -8 -14 C -2 10 0 14 4 D -8 8 -14 0 -4 E -8 14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993714 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 8 8 B -2 0 -10 -8 -14 C -2 10 0 14 4 D -8 8 -14 0 -4 E -8 14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993714 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 8 8 B -2 0 -10 -8 -14 C -2 10 0 14 4 D -8 8 -14 0 -4 E -8 14 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993714 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8340: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) B D C E A (7) D A B C E (6) C B E D A (6) E C B A D (4) E B C A D (4) E A C D B (4) D A C B E (4) A D E C B (4) C E A B D (3) B D C A E (3) B D A E C (3) A D E B C (3) E C A B D (2) C E A D B (2) C D B A E (2) B E C D A (2) B C E D A (2) A E D B C (2) A D C B E (2) A D B E C (2) E C B D A (1) E C A D B (1) E B A C D (1) E A C B D (1) E A B C D (1) D C B A E (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B A D (1) C D E A B (1) C D B E A (1) C B D E A (1) B E D A C (1) B E A D C (1) B C D E A (1) B A D E C (1) A E D C B (1) A E C D B (1) A E B D C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -14 -12 -20 B 10 0 -14 16 -4 C 14 14 0 10 14 D 12 -16 -10 0 -8 E 20 4 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -12 -20 B 10 0 -14 16 -4 C 14 14 0 10 14 D 12 -16 -10 0 -8 E 20 4 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=21 E=19 A=18 D=13 so D is eliminated. Round 2 votes counts: C=30 A=29 B=22 E=19 so E is eliminated. Round 3 votes counts: C=38 A=35 B=27 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:209 B:204 D:189 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -12 -20 B 10 0 -14 16 -4 C 14 14 0 10 14 D 12 -16 -10 0 -8 E 20 4 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -12 -20 B 10 0 -14 16 -4 C 14 14 0 10 14 D 12 -16 -10 0 -8 E 20 4 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -12 -20 B 10 0 -14 16 -4 C 14 14 0 10 14 D 12 -16 -10 0 -8 E 20 4 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8341: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (8) E D B C A (6) E B A C D (6) E A B C D (6) E D C B A (5) D C B A E (5) E D C A B (4) E D A C B (4) B C A D E (4) D E C A B (3) C D A B E (3) C B A D E (3) A C D B E (3) A B C E D (3) A B C D E (3) E D B A C (2) E B D C A (2) E B A D C (2) E A D C B (2) E A C D B (2) D E C B A (2) B A C E D (2) B A C D E (2) A C B E D (2) E D A B C (1) E B D A C (1) E A C B D (1) D E B C A (1) D C E B A (1) D C E A B (1) D C A E B (1) D C A B E (1) D B C E A (1) C B D A E (1) C A D B E (1) C A B D E (1) B D C A E (1) A E C B D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 6 8 -6 B -6 0 -12 2 -10 C -6 12 0 8 -4 D -8 -2 -8 0 -8 E 6 10 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 8 -6 B -6 0 -12 2 -10 C -6 12 0 8 -4 D -8 -2 -8 0 -8 E 6 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 A=22 D=16 C=9 B=9 so C is eliminated. Round 2 votes counts: E=44 A=24 D=19 B=13 so B is eliminated. Round 3 votes counts: E=44 A=35 D=21 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:207 C:205 B:187 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 8 -6 B -6 0 -12 2 -10 C -6 12 0 8 -4 D -8 -2 -8 0 -8 E 6 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 -6 B -6 0 -12 2 -10 C -6 12 0 8 -4 D -8 -2 -8 0 -8 E 6 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 -6 B -6 0 -12 2 -10 C -6 12 0 8 -4 D -8 -2 -8 0 -8 E 6 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8342: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (13) B C E A D (8) D E A C B (7) A D B C E (5) E C B D A (4) B A C D E (4) A D E C B (4) A B D C E (4) E D A C B (3) D A E C B (3) D A E B C (3) C E B D A (3) C B E A D (3) B C E D A (3) E C D B A (2) D E C B A (2) D A B E C (2) C B A E D (2) B C A D E (2) A E D C B (2) A D E B C (2) A C B E D (2) A B C E D (2) E D C A B (1) E C A D B (1) E A D C B (1) D E B C A (1) D B E A C (1) D B A E C (1) C E B A D (1) C B E D A (1) C A E B D (1) B D E C A (1) B A D C E (1) A D C E B (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 2 24 14 B 6 0 8 14 14 C -2 -8 0 10 18 D -24 -14 -10 0 -8 E -14 -14 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 24 14 B 6 0 8 14 14 C -2 -8 0 10 18 D -24 -14 -10 0 -8 E -14 -14 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=25 D=20 E=12 C=11 so C is eliminated. Round 2 votes counts: B=38 A=26 D=20 E=16 so E is eliminated. Round 3 votes counts: B=46 A=28 D=26 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:217 C:209 E:181 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 24 14 B 6 0 8 14 14 C -2 -8 0 10 18 D -24 -14 -10 0 -8 E -14 -14 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 24 14 B 6 0 8 14 14 C -2 -8 0 10 18 D -24 -14 -10 0 -8 E -14 -14 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 24 14 B 6 0 8 14 14 C -2 -8 0 10 18 D -24 -14 -10 0 -8 E -14 -14 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8343: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) B D C A E (10) E C A D B (7) B E C D A (7) A E C D B (5) A C D E B (5) E B A C D (3) D C A E B (3) D A C B E (3) C D A E B (3) B D C E A (3) B D A C E (3) A D C E B (3) E B C A D (2) E A C B D (2) D C A B E (2) D B C A E (2) C D E A B (2) C A E D B (2) C A D E B (2) B E D A C (2) B E A D C (2) B D E C A (2) E C A B D (1) E B A D C (1) D C B A E (1) D A B C E (1) C E D A B (1) C E B D A (1) C B D E A (1) B E D C A (1) B D E A C (1) B A E D C (1) B A D C E (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -6 2 -2 B -12 0 -14 -10 -10 C 6 14 0 12 0 D -2 10 -12 0 0 E 2 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.455862 D: 0.000000 E: 0.544138 Sum of squares = 0.503896248354 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.455862 D: 0.455862 E: 1.000000 A B C D E A 0 12 -6 2 -2 B -12 0 -14 -10 -10 C 6 14 0 12 0 D -2 10 -12 0 0 E 2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=28 A=15 D=12 C=12 so D is eliminated. Round 2 votes counts: B=35 E=28 A=19 C=18 so C is eliminated. Round 3 votes counts: B=37 E=32 A=31 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:216 E:206 A:203 D:198 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 2 -2 B -12 0 -14 -10 -10 C 6 14 0 12 0 D -2 10 -12 0 0 E 2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 2 -2 B -12 0 -14 -10 -10 C 6 14 0 12 0 D -2 10 -12 0 0 E 2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 2 -2 B -12 0 -14 -10 -10 C 6 14 0 12 0 D -2 10 -12 0 0 E 2 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8344: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (12) B D E C A (9) C A E B D (8) D B C A E (6) A C E B D (6) D B E A C (5) E A C B D (4) E B D C A (3) E B D A C (3) D C B A E (3) D B E C A (3) E D B A C (2) E D A B C (2) E A C D B (2) D E B A C (2) C A D B E (2) B E D C A (2) B C D A E (2) A E C D B (2) A C D B E (2) E B C D A (1) E B C A D (1) E A D C B (1) E A B D C (1) E A B C D (1) D C A B E (1) D B A E C (1) D B A C E (1) D A E B C (1) C D A B E (1) C B D A E (1) C B A E D (1) C A E D B (1) C A B E D (1) C A B D E (1) B D E A C (1) B C E D A (1) B C D E A (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 2 -4 10 B -2 0 -2 -4 -10 C -2 2 0 4 4 D 4 4 -4 0 -12 E -10 10 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 -4 10 B -2 0 -2 -4 -10 C -2 2 0 4 4 D 4 4 -4 0 -12 E -10 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 E=21 C=16 B=16 so C is eliminated. Round 2 votes counts: A=37 D=24 E=21 B=18 so B is eliminated. Round 3 votes counts: D=38 A=38 E=24 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:205 C:204 E:204 D:196 B:191 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 2 -4 10 B -2 0 -2 -4 -10 C -2 2 0 4 4 D 4 4 -4 0 -12 E -10 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 -4 10 B -2 0 -2 -4 -10 C -2 2 0 4 4 D 4 4 -4 0 -12 E -10 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 -4 10 B -2 0 -2 -4 -10 C -2 2 0 4 4 D 4 4 -4 0 -12 E -10 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.400000 D: 0.200000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8345: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) C B D A E (7) A C E D B (7) E B D C A (6) B D E C A (6) E D B A C (5) E D A B C (5) C A E B D (5) B D C E A (5) A C D B E (5) E A D B C (4) D B E A C (4) E B D A C (3) D B A E C (3) E C B D A (2) D B C A E (2) D B A C E (2) A D C B E (2) E B C D A (1) E A C D B (1) E A C B D (1) D E A B C (1) D B E C A (1) C E B D A (1) C E A B D (1) C B A D E (1) C A D B E (1) C A B E D (1) B C D A E (1) A E C D B (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 0 -8 -10 6 B 0 0 0 8 8 C 8 0 0 0 10 D 10 -8 0 0 12 E -6 -8 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.394811 C: 0.605189 D: 0.000000 E: 0.000000 Sum of squares = 0.522129433997 Cumulative probabilities = A: 0.000000 B: 0.394811 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -10 6 B 0 0 0 8 8 C 8 0 0 0 10 D 10 -8 0 0 12 E -6 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=28 A=17 D=13 B=12 so B is eliminated. Round 2 votes counts: C=31 E=28 D=24 A=17 so A is eliminated. Round 3 votes counts: C=44 E=29 D=27 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:208 D:207 A:194 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -10 6 B 0 0 0 8 8 C 8 0 0 0 10 D 10 -8 0 0 12 E -6 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -10 6 B 0 0 0 8 8 C 8 0 0 0 10 D 10 -8 0 0 12 E -6 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -10 6 B 0 0 0 8 8 C 8 0 0 0 10 D 10 -8 0 0 12 E -6 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8346: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (16) C D E A B (10) E C B A D (9) D A B C E (9) C E D A B (8) E B A C D (7) D C A B E (6) D A B E C (5) E B A D C (4) C E B A D (4) B A D E C (4) D A C B E (3) C E D B A (3) B A E D C (3) C E B D A (2) C E A B D (2) E B C A D (1) C E A D B (1) C D A E B (1) B E A D C (1) A B D E C (1) Total count = 100 A B C D E A 0 24 -26 -26 -4 B -24 0 -30 -24 -4 C 26 30 0 28 30 D 26 24 -28 0 10 E 4 4 -30 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 -26 -26 -4 B -24 0 -30 -24 -4 C 26 30 0 28 30 D 26 24 -28 0 10 E 4 4 -30 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=47 D=23 E=21 B=8 A=1 so A is eliminated. Round 2 votes counts: C=47 D=23 E=21 B=9 so B is eliminated. Round 3 votes counts: C=47 D=28 E=25 so E is eliminated. Round 4 votes counts: C=64 D=36 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:257 D:216 A:184 E:184 B:159 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -26 -26 -4 B -24 0 -30 -24 -4 C 26 30 0 28 30 D 26 24 -28 0 10 E 4 4 -30 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -26 -26 -4 B -24 0 -30 -24 -4 C 26 30 0 28 30 D 26 24 -28 0 10 E 4 4 -30 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -26 -26 -4 B -24 0 -30 -24 -4 C 26 30 0 28 30 D 26 24 -28 0 10 E 4 4 -30 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8347: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) B A E D C (7) E D C B A (6) C D E A B (6) B E A D C (6) A B C D E (6) B E D C A (5) B E C D A (5) B A E C D (4) A C D E B (4) E B D C A (3) C D A E B (3) C A D E B (3) B A C E D (3) B A C D E (3) E D C A B (2) D C E B A (2) D C E A B (2) B E D A C (2) A D E C B (2) A D C B E (2) A C B D E (2) E D B C A (1) E D A C B (1) E B D A C (1) E B C D A (1) E B A D C (1) D E C A B (1) B C D E A (1) B C D A E (1) B C A D E (1) A E D C B (1) A E D B C (1) A D C E B (1) A C D B E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -20 0 0 -6 B 20 0 8 8 2 C 0 -8 0 4 -4 D 0 -8 -4 0 -4 E 6 -2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 0 0 -6 B 20 0 8 8 2 C 0 -8 0 4 -4 D 0 -8 -4 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=22 C=19 E=16 D=5 so D is eliminated. Round 2 votes counts: B=38 C=23 A=22 E=17 so E is eliminated. Round 3 votes counts: B=45 C=32 A=23 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:206 C:196 D:192 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 0 0 -6 B 20 0 8 8 2 C 0 -8 0 4 -4 D 0 -8 -4 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 0 0 -6 B 20 0 8 8 2 C 0 -8 0 4 -4 D 0 -8 -4 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 0 0 -6 B 20 0 8 8 2 C 0 -8 0 4 -4 D 0 -8 -4 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8348: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (12) E B A C D (6) D C B A E (6) C D B A E (6) E A B C D (5) C E A D B (5) E A C B D (4) C E B D A (4) A D B E C (4) E C B A D (3) E C A B D (3) E B A D C (3) C D A E B (3) B A E D C (3) A E B D C (3) E B C A D (2) E A D B C (2) D B C A E (2) D B A C E (2) D A B E C (2) C B E D A (2) C B D E A (2) A D E B C (2) A B E D C (2) A B D E C (2) E C A D B (1) E A D C B (1) C D E A B (1) C D B E A (1) B D A E C (1) B D A C E (1) B A D E C (1) A E D C B (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 10 18 34 -14 B -10 0 12 18 -26 C -18 -12 0 -4 -28 D -34 -18 4 0 -26 E 14 26 28 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 18 34 -14 B -10 0 12 18 -26 C -18 -12 0 -4 -28 D -34 -18 4 0 -26 E 14 26 28 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=24 A=16 D=12 B=6 so B is eliminated. Round 2 votes counts: E=42 C=24 A=20 D=14 so D is eliminated. Round 3 votes counts: E=42 C=32 A=26 so A is eliminated. Round 4 votes counts: E=64 C=36 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:247 A:224 B:197 C:169 D:163 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 18 34 -14 B -10 0 12 18 -26 C -18 -12 0 -4 -28 D -34 -18 4 0 -26 E 14 26 28 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 34 -14 B -10 0 12 18 -26 C -18 -12 0 -4 -28 D -34 -18 4 0 -26 E 14 26 28 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 34 -14 B -10 0 12 18 -26 C -18 -12 0 -4 -28 D -34 -18 4 0 -26 E 14 26 28 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8349: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) D E B C A (7) C E A D B (6) B D A E C (6) A C B E D (6) C A E D B (5) B A D E C (5) B A D C E (5) B A C D E (5) C E D A B (4) C A E B D (3) E D C B A (2) E D C A B (2) D E C B A (2) D E B A C (2) D B E A C (2) C E D B A (2) C D B E A (2) B D C E A (2) A C E D B (2) A C E B D (2) A B D E C (2) A B C E D (2) E D A C B (1) E C D A B (1) E C A D B (1) D E A B C (1) D B E C A (1) D B A E C (1) C B A E D (1) C A B E D (1) B D E C A (1) B D A C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 14 -4 -2 B 18 0 14 12 14 C -14 -14 0 -12 2 D 4 -12 12 0 18 E 2 -14 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 14 -4 -2 B 18 0 14 12 14 C -14 -14 0 -12 2 D 4 -12 12 0 18 E 2 -14 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=24 D=16 A=16 E=7 so E is eliminated. Round 2 votes counts: B=37 C=26 D=21 A=16 so A is eliminated. Round 3 votes counts: B=43 C=36 D=21 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:211 A:195 E:184 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 14 -4 -2 B 18 0 14 12 14 C -14 -14 0 -12 2 D 4 -12 12 0 18 E 2 -14 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 14 -4 -2 B 18 0 14 12 14 C -14 -14 0 -12 2 D 4 -12 12 0 18 E 2 -14 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 14 -4 -2 B 18 0 14 12 14 C -14 -14 0 -12 2 D 4 -12 12 0 18 E 2 -14 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8350: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (11) E B D A C (10) C A D B E (8) B E C D A (8) A C D E B (5) E B D C A (4) C A E B D (4) C A B E D (4) A D C E B (4) E C B A D (3) D E B A C (3) D B E A C (3) E D B A C (2) E A C B D (2) D A B E C (2) D A B C E (2) C E B A D (2) B E D C A (2) B E D A C (2) A D E C B (2) A C D B E (2) E B C A D (1) E A D B C (1) D B C E A (1) D B A E C (1) D A E B C (1) C E A B D (1) C D A B E (1) C B E D A (1) C B E A D (1) C B A D E (1) C A D E B (1) B D E A C (1) B C E D A (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -16 -10 -12 -20 B 16 0 14 18 -18 C 10 -14 0 14 -22 D 12 -18 -14 0 -22 E 20 18 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -10 -12 -20 B 16 0 14 18 -18 C 10 -14 0 14 -22 D 12 -18 -14 0 -22 E 20 18 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=24 A=15 B=14 D=13 so D is eliminated. Round 2 votes counts: E=37 C=24 A=20 B=19 so B is eliminated. Round 3 votes counts: E=53 C=26 A=21 so A is eliminated. Round 4 votes counts: E=61 C=39 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:241 B:215 C:194 D:179 A:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -10 -12 -20 B 16 0 14 18 -18 C 10 -14 0 14 -22 D 12 -18 -14 0 -22 E 20 18 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -12 -20 B 16 0 14 18 -18 C 10 -14 0 14 -22 D 12 -18 -14 0 -22 E 20 18 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -12 -20 B 16 0 14 18 -18 C 10 -14 0 14 -22 D 12 -18 -14 0 -22 E 20 18 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8351: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) C D E B A (8) A B E C D (7) E B C A D (6) A D B E C (6) D C A B E (5) D A B E C (5) C E B D A (5) C A E B D (4) A E B C D (4) E B A C D (3) D C E B A (3) D C B E A (3) C E B A D (3) B E A C D (3) D B E C A (2) D A C B E (2) C E D B A (2) C D E A B (2) C D A E B (2) E C B A D (1) D C A E B (1) D B E A C (1) C E A B D (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B A E D C (1) A C E B D (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 0 12 6 B -10 0 12 14 6 C 0 -12 0 8 -14 D -12 -14 -8 0 -16 E -6 -6 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.810224 B: 0.000000 C: 0.189776 D: 0.000000 E: 0.000000 Sum of squares = 0.692478058292 Cumulative probabilities = A: 0.810224 B: 0.810224 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 12 6 B -10 0 12 14 6 C 0 -12 0 8 -14 D -12 -14 -8 0 -16 E -6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000063945 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=27 D=22 E=10 B=8 so B is eliminated. Round 2 votes counts: A=34 C=27 D=22 E=17 so E is eliminated. Round 3 votes counts: A=41 C=35 D=24 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:211 E:209 C:191 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 12 6 B -10 0 12 14 6 C 0 -12 0 8 -14 D -12 -14 -8 0 -16 E -6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000063945 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 12 6 B -10 0 12 14 6 C 0 -12 0 8 -14 D -12 -14 -8 0 -16 E -6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000063945 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 12 6 B -10 0 12 14 6 C 0 -12 0 8 -14 D -12 -14 -8 0 -16 E -6 -6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.300000 D: 0.000000 E: 0.000000 Sum of squares = 0.580000063945 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8352: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) B A D C E (6) A B E C D (6) E C D A B (5) E C A D B (5) A E B C D (5) D C B E A (4) C E D B A (4) E C A B D (3) D B C A E (3) B A E C D (3) B A D E C (3) A E D C B (3) A E C D B (3) A D B E C (3) A B E D C (3) E C B A D (2) E A C D B (2) E A C B D (2) D C E A B (2) D C B A E (2) D B A C E (2) C D E B A (2) B D C E A (2) B D C A E (2) B D A C E (2) A B D E C (2) E C B D A (1) E B A C D (1) D C A E B (1) D C A B E (1) D B C E A (1) D A C E B (1) D A B C E (1) C E D A B (1) C D B E A (1) B E C A D (1) B E A C D (1) B C D E A (1) B A E D C (1) Total count = 100 A B C D E A 0 -2 0 10 6 B 2 0 -2 -6 2 C 0 2 0 -2 -10 D -10 6 2 0 -4 E -6 -2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.259259 B: 0.407407 C: 0.222222 D: 0.037037 E: 0.074074 Sum of squares = 0.289437585695 Cumulative probabilities = A: 0.259259 B: 0.666667 C: 0.888889 D: 0.925926 E: 1.000000 A B C D E A 0 -2 0 10 6 B 2 0 -2 -6 2 C 0 2 0 -2 -10 D -10 6 2 0 -4 E -6 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.259259 B: 0.407407 C: 0.222222 D: 0.037037 E: 0.074074 Sum of squares = 0.289437585732 Cumulative probabilities = A: 0.259259 B: 0.666667 C: 0.888889 D: 0.925926 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=24 B=22 E=21 C=8 so C is eliminated. Round 2 votes counts: D=27 E=26 A=25 B=22 so B is eliminated. Round 3 votes counts: A=38 D=34 E=28 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:203 B:198 D:197 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 10 6 B 2 0 -2 -6 2 C 0 2 0 -2 -10 D -10 6 2 0 -4 E -6 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.259259 B: 0.407407 C: 0.222222 D: 0.037037 E: 0.074074 Sum of squares = 0.289437585732 Cumulative probabilities = A: 0.259259 B: 0.666667 C: 0.888889 D: 0.925926 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 10 6 B 2 0 -2 -6 2 C 0 2 0 -2 -10 D -10 6 2 0 -4 E -6 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.259259 B: 0.407407 C: 0.222222 D: 0.037037 E: 0.074074 Sum of squares = 0.289437585732 Cumulative probabilities = A: 0.259259 B: 0.666667 C: 0.888889 D: 0.925926 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 10 6 B 2 0 -2 -6 2 C 0 2 0 -2 -10 D -10 6 2 0 -4 E -6 -2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.259259 B: 0.407407 C: 0.222222 D: 0.037037 E: 0.074074 Sum of squares = 0.289437585732 Cumulative probabilities = A: 0.259259 B: 0.666667 C: 0.888889 D: 0.925926 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8353: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (6) A B D E C (6) E C B A D (5) A B E D C (5) D C B E A (4) D C A E B (4) C E B A D (4) C E A D B (4) C D E B A (4) A E B C D (4) D A B E C (3) D A B C E (3) C E D B A (3) B E A C D (3) B C E D A (3) B A E D C (3) B A D E C (3) E C A B D (2) E A C B D (2) D C E A B (2) C D E A B (2) C D B E A (2) B E C A D (2) B D C E A (2) A D C E B (2) A B E C D (2) D C E B A (1) D C B A E (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C E B (1) D A C B E (1) C E D A B (1) C E B D A (1) C A D E B (1) B D A E C (1) B A E C D (1) A E C D B (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 10 2 16 6 B -10 0 0 0 8 C -2 0 0 -6 -4 D -16 0 6 0 6 E -6 -8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999115 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 16 6 B -10 0 0 0 8 C -2 0 0 -6 -4 D -16 0 6 0 6 E -6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=23 C=22 B=18 E=9 so E is eliminated. Round 2 votes counts: A=30 C=29 D=23 B=18 so B is eliminated. Round 3 votes counts: A=40 C=34 D=26 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:199 D:198 C:194 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 16 6 B -10 0 0 0 8 C -2 0 0 -6 -4 D -16 0 6 0 6 E -6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 16 6 B -10 0 0 0 8 C -2 0 0 -6 -4 D -16 0 6 0 6 E -6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 16 6 B -10 0 0 0 8 C -2 0 0 -6 -4 D -16 0 6 0 6 E -6 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8354: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) E B C D A (7) D A C B E (7) A E B C D (7) A D C B E (7) D C B E A (6) A D B C E (6) A D E C B (5) D C B A E (4) C B E D A (4) B C D E A (4) E C B D A (3) D C E B A (3) D C A B E (3) B C E D A (3) A D C E B (3) E C D B A (2) E A B C D (2) C D B E A (2) E C D A B (1) E A C B D (1) D B C E A (1) D A B C E (1) C E B D A (1) C B D E A (1) B E C D A (1) B D C E A (1) A E D C B (1) A E D B C (1) A D E B C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -12 -10 -4 B 4 0 -8 -8 6 C 12 8 0 -4 16 D 10 8 4 0 14 E 4 -6 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -10 -4 B 4 0 -8 -8 6 C 12 8 0 -4 16 D 10 8 4 0 14 E 4 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=25 D=25 B=9 C=8 so C is eliminated. Round 2 votes counts: A=33 D=27 E=26 B=14 so B is eliminated. Round 3 votes counts: E=34 D=33 A=33 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:218 C:216 B:197 A:185 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -12 -10 -4 B 4 0 -8 -8 6 C 12 8 0 -4 16 D 10 8 4 0 14 E 4 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -10 -4 B 4 0 -8 -8 6 C 12 8 0 -4 16 D 10 8 4 0 14 E 4 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -10 -4 B 4 0 -8 -8 6 C 12 8 0 -4 16 D 10 8 4 0 14 E 4 -6 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8355: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (13) A E C B D (8) E A C B D (6) A E C D B (6) E C A B D (4) D B C A E (4) C E B A D (4) C E A B D (4) C B E D A (4) B C D E A (4) A D B E C (4) D B E A C (3) D B C E A (3) A E D B C (3) A C E D B (3) D B A E C (2) D B A C E (2) C E B D A (2) C B D E A (2) A E D C B (2) A D E B C (2) A C E B D (2) E C B D A (1) E A D B C (1) D B E C A (1) D A B C E (1) C A E B D (1) C A B D E (1) B E D C A (1) B E C D A (1) B D E C A (1) B C E D A (1) A D E C B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -4 8 -12 B -2 0 -6 20 -2 C 4 6 0 10 6 D -8 -20 -10 0 -8 E 12 2 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 8 -12 B -2 0 -6 20 -2 C 4 6 0 10 6 D -8 -20 -10 0 -8 E 12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=21 C=18 D=16 E=12 so E is eliminated. Round 2 votes counts: A=40 C=23 B=21 D=16 so D is eliminated. Round 3 votes counts: A=41 B=36 C=23 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 E:208 B:205 A:197 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 8 -12 B -2 0 -6 20 -2 C 4 6 0 10 6 D -8 -20 -10 0 -8 E 12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 8 -12 B -2 0 -6 20 -2 C 4 6 0 10 6 D -8 -20 -10 0 -8 E 12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 8 -12 B -2 0 -6 20 -2 C 4 6 0 10 6 D -8 -20 -10 0 -8 E 12 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8356: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) E C D B A (6) D B E A C (6) A C E B D (6) A B D C E (6) E B D C A (5) C E A D B (5) B D A E C (5) E C A D B (4) C A E D B (4) B D E C A (4) B D E A C (4) B D A C E (4) A C B D E (4) E C B D A (3) C A E B D (3) E C D A B (2) D E B C A (2) D B A E C (2) B E D C A (2) A D B C E (2) A C E D B (2) E D C B A (1) E C B A D (1) E C A B D (1) E B C D A (1) D A B C E (1) C E A B D (1) B C E A D (1) B A D C E (1) B A C E D (1) A D C B E (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -4 -8 -10 B 10 0 6 8 -8 C 4 -6 0 -4 -10 D 8 -8 4 0 -12 E 10 8 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 -8 -10 B 10 0 6 8 -8 C 4 -6 0 -4 -10 D 8 -8 4 0 -12 E 10 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=24 B=22 C=13 D=11 so D is eliminated. Round 2 votes counts: E=32 B=30 A=25 C=13 so C is eliminated. Round 3 votes counts: E=38 A=32 B=30 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:208 D:196 C:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 -8 -10 B 10 0 6 8 -8 C 4 -6 0 -4 -10 D 8 -8 4 0 -12 E 10 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -8 -10 B 10 0 6 8 -8 C 4 -6 0 -4 -10 D 8 -8 4 0 -12 E 10 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -8 -10 B 10 0 6 8 -8 C 4 -6 0 -4 -10 D 8 -8 4 0 -12 E 10 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8357: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) D A C B E (7) D A B C E (6) C D A B E (6) B A D E C (6) E C D B A (5) D A B E C (5) B A E D C (5) E C D A B (4) D C A B E (4) A B D C E (4) C E D A B (3) A D B C E (3) E D A B C (2) E C B D A (2) E B A C D (2) D C A E B (2) C E B A D (2) C D E A B (2) B E A C D (2) B A C D E (2) A B C D E (2) E D C A B (1) E B C A D (1) E B A D C (1) D E C A B (1) D E B A C (1) D E A B C (1) D B A E C (1) C E D B A (1) C D A E B (1) C B A E D (1) C B A D E (1) B E A D C (1) B A D C E (1) B A C E D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 8 -10 18 B -10 0 -4 -12 18 C -8 4 0 -6 -2 D 10 12 6 0 14 E -18 -18 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 -10 18 B -10 0 -4 -12 18 C -8 4 0 -6 -2 D 10 12 6 0 14 E -18 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=27 B=18 C=17 A=10 so A is eliminated. Round 2 votes counts: D=31 E=27 B=25 C=17 so C is eliminated. Round 3 votes counts: D=40 E=33 B=27 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:213 B:196 C:194 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 8 -10 18 B -10 0 -4 -12 18 C -8 4 0 -6 -2 D 10 12 6 0 14 E -18 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 -10 18 B -10 0 -4 -12 18 C -8 4 0 -6 -2 D 10 12 6 0 14 E -18 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 -10 18 B -10 0 -4 -12 18 C -8 4 0 -6 -2 D 10 12 6 0 14 E -18 -18 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8358: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) D C A E B (8) C A D B E (8) B E A C D (7) D E C A B (6) B A C E D (6) B E D A C (5) A C B D E (5) A C D B E (4) E D A C B (3) D C E A B (3) B E A D C (3) E D C A B (2) E D B A C (2) E B D C A (2) D E B C A (2) C D A E B (2) C A B D E (2) B D E C A (2) B C D A E (2) B C A D E (2) B A E C D (2) B A C D E (2) A C D E B (2) A B C E D (2) E D B C A (1) D E A C B (1) D C A B E (1) D A C E B (1) B E D C A (1) A B C D E (1) Total count = 100 A B C D E A 0 2 12 -8 0 B -2 0 4 8 10 C -12 -4 0 -6 2 D 8 -8 6 0 8 E 0 -10 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.407407407407 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 -8 0 B -2 0 4 8 10 C -12 -4 0 -6 2 D 8 -8 6 0 8 E 0 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.407407407417 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=22 E=20 A=14 C=12 so C is eliminated. Round 2 votes counts: B=32 D=24 A=24 E=20 so E is eliminated. Round 3 votes counts: B=44 D=32 A=24 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:207 A:203 C:190 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 12 -8 0 B -2 0 4 8 10 C -12 -4 0 -6 2 D 8 -8 6 0 8 E 0 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.407407407417 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 -8 0 B -2 0 4 8 10 C -12 -4 0 -6 2 D 8 -8 6 0 8 E 0 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.407407407417 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 -8 0 B -2 0 4 8 10 C -12 -4 0 -6 2 D 8 -8 6 0 8 E 0 -10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.407407407417 Cumulative probabilities = A: 0.444444 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8359: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E C D A B (8) A B D E C (8) E A D B C (7) A B E D C (7) C E D B A (5) C D E B A (5) E D C A B (4) C B D A E (4) C B A D E (4) E D A B C (3) E C D B A (3) E A B D C (3) C D B E A (3) B D A C E (3) B A D C E (3) D C B A E (2) B C A D E (2) A E B D C (2) A B D C E (2) E D A C B (1) E C A B D (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E B A (1) D B C A E (1) D A E B C (1) D A B E C (1) C E B A D (1) B D C A E (1) B A C D E (1) Total count = 100 A B C D E A 0 0 -12 -14 4 B 0 0 -8 -12 4 C 12 8 0 -2 -4 D 14 12 2 0 6 E -4 -4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -14 4 B 0 0 -8 -12 4 C 12 8 0 -2 -4 D 14 12 2 0 6 E -4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=32 C=32 A=19 B=10 D=7 so D is eliminated. Round 2 votes counts: C=35 E=33 A=21 B=11 so B is eliminated. Round 3 votes counts: C=39 E=33 A=28 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:217 C:207 E:195 B:192 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -12 -14 4 B 0 0 -8 -12 4 C 12 8 0 -2 -4 D 14 12 2 0 6 E -4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -14 4 B 0 0 -8 -12 4 C 12 8 0 -2 -4 D 14 12 2 0 6 E -4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -14 4 B 0 0 -8 -12 4 C 12 8 0 -2 -4 D 14 12 2 0 6 E -4 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8360: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) E C A B D (9) B D A C E (6) C E A B D (5) E C B D A (4) E C A D B (4) E B D C A (4) D B E A C (4) C E A D B (4) A D B C E (4) A C B D E (4) E C B A D (3) D B E C A (3) D B A E C (3) C E B A D (3) B D E C A (3) E B C D A (2) C A E B D (2) B E D C A (2) B D A E C (2) B A D C E (2) A C E D B (2) E D B C A (1) E C D B A (1) E C D A B (1) E A C D B (1) D A B C E (1) B D E A C (1) B C E D A (1) A E C D B (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -4 0 -12 B 12 0 4 12 2 C 4 -4 0 0 0 D 0 -12 0 0 -2 E 12 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999565 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 0 -12 B 12 0 4 12 2 C 4 -4 0 0 0 D 0 -12 0 0 -2 E 12 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=22 B=17 A=17 C=14 so C is eliminated. Round 2 votes counts: E=42 D=22 A=19 B=17 so B is eliminated. Round 3 votes counts: E=45 D=34 A=21 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:206 C:200 D:193 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 0 -12 B 12 0 4 12 2 C 4 -4 0 0 0 D 0 -12 0 0 -2 E 12 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 0 -12 B 12 0 4 12 2 C 4 -4 0 0 0 D 0 -12 0 0 -2 E 12 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 0 -12 B 12 0 4 12 2 C 4 -4 0 0 0 D 0 -12 0 0 -2 E 12 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995751 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8361: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (16) C B A E D (8) D C E A B (7) C D B E A (7) B C A E D (7) D E C A B (5) C B D E A (5) B A E C D (5) D C E B A (4) E D A B C (3) D E A C B (3) C D E A B (3) C B A D E (3) A E D B C (3) A B E D C (3) E B A D C (2) C D E B A (2) B A C E D (2) A E B D C (2) A B E C D (2) E A D B C (1) E A B D C (1) C D B A E (1) C B D A E (1) C A B D E (1) B E C A D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -10 -14 -20 B -4 0 -2 -10 -4 C 10 2 0 0 6 D 14 10 0 0 16 E 20 4 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.752103 D: 0.247897 E: 0.000000 Sum of squares = 0.627112235744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.752103 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -14 -20 B -4 0 -2 -10 -4 C 10 2 0 0 6 D 14 10 0 0 16 E 20 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=31 B=15 A=12 E=7 so E is eliminated. Round 2 votes counts: D=38 C=31 B=17 A=14 so A is eliminated. Round 3 votes counts: D=42 C=32 B=26 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:209 E:201 B:190 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 -14 -20 B -4 0 -2 -10 -4 C 10 2 0 0 6 D 14 10 0 0 16 E 20 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -14 -20 B -4 0 -2 -10 -4 C 10 2 0 0 6 D 14 10 0 0 16 E 20 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -14 -20 B -4 0 -2 -10 -4 C 10 2 0 0 6 D 14 10 0 0 16 E 20 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8362: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (12) E A B C D (11) D C B A E (9) E C B A D (8) A B D C E (6) E C D B A (5) A B C D E (5) D E C B A (4) A B E C D (4) A B C E D (4) D B C A E (3) E D C B A (2) E D C A B (2) E C B D A (2) D C E B A (2) C D B A E (2) B A D C E (2) E B A C D (1) D E C A B (1) D E A B C (1) D C A B E (1) D B A C E (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A D E (1) B A C D E (1) A E B C D (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 6 -4 14 B -4 0 12 0 14 C -6 -12 0 0 8 D 4 0 0 0 16 E -14 -14 -8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.336464 C: 0.000000 D: 0.663536 E: 0.000000 Sum of squares = 0.553488096277 Cumulative probabilities = A: 0.000000 B: 0.336464 C: 0.336464 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 -4 14 B -4 0 12 0 14 C -6 -12 0 0 8 D 4 0 0 0 16 E -14 -14 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499828 C: 0.000000 D: 0.500172 E: 0.000000 Sum of squares = 0.500000058919 Cumulative probabilities = A: 0.000000 B: 0.499828 C: 0.499828 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=31 A=23 C=8 B=3 so B is eliminated. Round 2 votes counts: D=35 E=31 A=26 C=8 so C is eliminated. Round 3 votes counts: D=40 E=33 A=27 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:211 A:210 D:210 C:195 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 6 -4 14 B -4 0 12 0 14 C -6 -12 0 0 8 D 4 0 0 0 16 E -14 -14 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499828 C: 0.000000 D: 0.500172 E: 0.000000 Sum of squares = 0.500000058919 Cumulative probabilities = A: 0.000000 B: 0.499828 C: 0.499828 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 -4 14 B -4 0 12 0 14 C -6 -12 0 0 8 D 4 0 0 0 16 E -14 -14 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499828 C: 0.000000 D: 0.500172 E: 0.000000 Sum of squares = 0.500000058919 Cumulative probabilities = A: 0.000000 B: 0.499828 C: 0.499828 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 -4 14 B -4 0 12 0 14 C -6 -12 0 0 8 D 4 0 0 0 16 E -14 -14 -8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499828 C: 0.000000 D: 0.500172 E: 0.000000 Sum of squares = 0.500000058919 Cumulative probabilities = A: 0.000000 B: 0.499828 C: 0.499828 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8363: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (5) B C E D A (5) D C E A B (4) D C A E B (4) D A E C B (4) D A C E B (4) B E A C D (4) B C E A D (4) B C A D E (4) E B A C D (3) E A B D C (3) D C A B E (3) C D B A E (3) C B E D A (3) B E C A D (3) A D B C E (3) E D B A C (2) E D A C B (2) E B C D A (2) E B A D C (2) D A C B E (2) C D E B A (2) C B D E A (2) C B D A E (2) B E C D A (2) B C A E D (2) A E D B C (2) A E B D C (2) A D C B E (2) E D B C A (1) E B D C A (1) E B D A C (1) E B C A D (1) D E A C B (1) C E B D A (1) C D A B E (1) C B A D E (1) B A E D C (1) B A E C D (1) A E D C B (1) A D E C B (1) A D C E B (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -2 -4 -8 B 6 0 12 4 0 C 2 -12 0 -6 8 D 4 -4 6 0 -10 E 8 0 -8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.646857 C: 0.000000 D: 0.000000 E: 0.353143 Sum of squares = 0.543133978196 Cumulative probabilities = A: 0.000000 B: 0.646857 C: 0.646857 D: 0.646857 E: 1.000000 A B C D E A 0 -6 -2 -4 -8 B 6 0 12 4 0 C 2 -12 0 -6 8 D 4 -4 6 0 -10 E 8 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=23 D=22 C=15 A=14 so A is eliminated. Round 2 votes counts: D=29 E=28 B=28 C=15 so C is eliminated. Round 3 votes counts: B=36 D=35 E=29 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:205 D:198 C:196 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 -8 B 6 0 12 4 0 C 2 -12 0 -6 8 D 4 -4 6 0 -10 E 8 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 -8 B 6 0 12 4 0 C 2 -12 0 -6 8 D 4 -4 6 0 -10 E 8 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 -8 B 6 0 12 4 0 C 2 -12 0 -6 8 D 4 -4 6 0 -10 E 8 0 -8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8364: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) A C E D B (10) B D A E C (9) E C D B A (7) E C B D A (6) C E A B D (5) D B A E C (4) B D E C A (4) E C D A B (3) E B C D A (3) C A E D B (3) B E C D A (3) B D A C E (3) A D C B E (3) A D B C E (3) A C E B D (3) E C A D B (2) D B E A C (2) B E D C A (2) B D E A C (2) A B D C E (2) E D C B A (1) E C B A D (1) D A B E C (1) D A B C E (1) C E B A D (1) C B E A D (1) A D C E B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -8 -2 -10 B -2 0 -22 -8 -18 C 8 22 0 24 0 D 2 8 -24 0 -28 E 10 18 0 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.467227 D: 0.000000 E: 0.532773 Sum of squares = 0.502148080064 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.467227 D: 0.467227 E: 1.000000 A B C D E A 0 2 -8 -2 -10 B -2 0 -22 -8 -18 C 8 22 0 24 0 D 2 8 -24 0 -28 E 10 18 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 B=23 C=22 D=8 so D is eliminated. Round 2 votes counts: B=29 A=26 E=23 C=22 so C is eliminated. Round 3 votes counts: E=41 B=30 A=29 so A is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:227 A:191 D:179 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -2 -10 B -2 0 -22 -8 -18 C 8 22 0 24 0 D 2 8 -24 0 -28 E 10 18 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 -10 B -2 0 -22 -8 -18 C 8 22 0 24 0 D 2 8 -24 0 -28 E 10 18 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 -10 B -2 0 -22 -8 -18 C 8 22 0 24 0 D 2 8 -24 0 -28 E 10 18 0 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.49999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8365: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) D E C A B (6) A B D C E (6) B C A E D (5) D C E B A (4) C E B A D (4) B A C D E (4) A D B E C (4) E C A B D (3) D A B E C (3) C E B D A (3) C B E A D (3) B A C E D (3) E D C A B (2) E C D A B (2) D E C B A (2) D B C E A (2) D B A C E (2) D A E C B (2) D A B C E (2) C E D B A (2) B D A C E (2) B A D C E (2) A E C B D (2) A B D E C (2) E D C B A (1) E C A D B (1) E A D C B (1) E A C D B (1) D B C A E (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E D A (1) B C E D A (1) B C E A D (1) B C A D E (1) A E B C D (1) A D E B C (1) A D B C E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -14 -2 -4 B 12 0 -4 -6 0 C 14 4 0 6 8 D 2 6 -6 0 2 E 4 0 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -2 -4 B 12 0 -4 -6 0 C 14 4 0 6 8 D 2 6 -6 0 2 E 4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=21 A=20 B=19 C=15 so C is eliminated. Round 2 votes counts: E=30 D=27 B=23 A=20 so A is eliminated. Round 3 votes counts: B=34 E=33 D=33 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:202 B:201 E:197 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -2 -4 B 12 0 -4 -6 0 C 14 4 0 6 8 D 2 6 -6 0 2 E 4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -2 -4 B 12 0 -4 -6 0 C 14 4 0 6 8 D 2 6 -6 0 2 E 4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -2 -4 B 12 0 -4 -6 0 C 14 4 0 6 8 D 2 6 -6 0 2 E 4 0 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8366: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B A E C D (8) E C D A B (7) B A C E D (6) A B C E D (5) D E C A B (4) B A D E C (4) E C A D B (3) D C A B E (3) D B A C E (3) C E D A B (3) C A E B D (3) A C B E D (3) A C B D E (3) E D C B A (2) E D B C A (2) E B A C D (2) D E C B A (2) D C E B A (2) C E A D B (2) C E A B D (2) B E A C D (2) B A D C E (2) E D C A B (1) E C A B D (1) E B D A C (1) E B A D C (1) E A C B D (1) D E B C A (1) D C B E A (1) D C A E B (1) D B A E C (1) D A C B E (1) C D E A B (1) C D A E B (1) B E D A C (1) B D E A C (1) B A E D C (1) B A C D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -2 4 -4 B -12 0 -12 0 -4 C 2 12 0 10 6 D -4 0 -10 0 -16 E 4 4 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999449 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 4 -4 B -12 0 -12 0 -4 C 2 12 0 10 6 D -4 0 -10 0 -16 E 4 4 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=26 E=21 A=13 C=12 so C is eliminated. Round 2 votes counts: D=30 E=28 B=26 A=16 so A is eliminated. Round 3 votes counts: B=39 E=31 D=30 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:209 A:205 B:186 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 4 -4 B -12 0 -12 0 -4 C 2 12 0 10 6 D -4 0 -10 0 -16 E 4 4 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 4 -4 B -12 0 -12 0 -4 C 2 12 0 10 6 D -4 0 -10 0 -16 E 4 4 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 4 -4 B -12 0 -12 0 -4 C 2 12 0 10 6 D -4 0 -10 0 -16 E 4 4 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999776 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8367: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C E A B D (8) D A B E C (7) C A B E D (7) C A B D E (7) E C B A D (5) D E B A C (5) C D A B E (5) A B D E C (5) E D B A C (4) E B A D C (4) D A B C E (4) C E D B A (3) A B C E D (3) E B D A C (2) E B A C D (2) D B E A C (2) C D E A B (2) E C D B A (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) C E D A B (1) C E B A D (1) C D E B A (1) C A E B D (1) A B E D C (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 -4 10 B -10 0 12 0 12 C -10 -12 0 0 -8 D 4 0 0 0 10 E -10 -12 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.198999 C: 0.000000 D: 0.801001 E: 0.000000 Sum of squares = 0.681203788273 Cumulative probabilities = A: 0.000000 B: 0.198999 C: 0.198999 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -4 10 B -10 0 12 0 12 C -10 -12 0 0 -8 D 4 0 0 0 10 E -10 -12 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836744734 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=33 E=19 A=12 so B is eliminated. Round 2 votes counts: C=36 D=33 E=19 A=12 so A is eliminated. Round 3 votes counts: C=40 D=39 E=21 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:213 B:207 D:207 E:188 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -4 10 B -10 0 12 0 12 C -10 -12 0 0 -8 D 4 0 0 0 10 E -10 -12 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836744734 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -4 10 B -10 0 12 0 12 C -10 -12 0 0 -8 D 4 0 0 0 10 E -10 -12 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836744734 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -4 10 B -10 0 12 0 12 C -10 -12 0 0 -8 D 4 0 0 0 10 E -10 -12 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.714286 E: 0.000000 Sum of squares = 0.591836744734 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8368: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) E B D C A (9) C A D E B (7) B E D A C (7) A C B D E (6) E D B C A (5) D E B C A (5) B E C A D (5) B E A C D (5) A C D B E (5) D E C B A (4) D C A E B (4) A D C E B (4) D E C A B (3) A C D E B (3) E D B A C (2) E B D A C (2) B E C D A (2) D E B A C (1) D C E B A (1) D A E C B (1) D A C E B (1) C D A E B (1) C B A E D (1) C A B D E (1) B E A D C (1) B C A E D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -24 -20 -18 -26 B 24 0 14 6 -6 C 20 -14 0 -22 -26 D 18 -6 22 0 -6 E 26 6 26 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 -20 -18 -26 B 24 0 14 6 -6 C 20 -14 0 -22 -26 D 18 -6 22 0 -6 E 26 6 26 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=20 A=20 E=18 C=10 so C is eliminated. Round 2 votes counts: B=33 A=28 D=21 E=18 so E is eliminated. Round 3 votes counts: B=44 D=28 A=28 so D is eliminated. Round 4 votes counts: B=62 A=38 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:232 B:219 D:214 C:179 A:156 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 -20 -18 -26 B 24 0 14 6 -6 C 20 -14 0 -22 -26 D 18 -6 22 0 -6 E 26 6 26 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -20 -18 -26 B 24 0 14 6 -6 C 20 -14 0 -22 -26 D 18 -6 22 0 -6 E 26 6 26 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -20 -18 -26 B 24 0 14 6 -6 C 20 -14 0 -22 -26 D 18 -6 22 0 -6 E 26 6 26 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8369: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) C A E B D (9) A C D B E (8) E B D C A (7) D B E C A (7) B E D C A (7) A C D E B (7) E B C D A (4) B D E A C (4) E C B A D (3) D E B C A (3) D A B C E (3) A D C B E (3) A C E B D (3) D B A E C (2) D A C B E (2) C E A B D (2) C A E D B (2) E C D A B (1) E B C A D (1) D C A E B (1) D A C E B (1) D A B E C (1) C E B A D (1) C A D E B (1) B E C A D (1) B E A C D (1) B D E C A (1) B A E C D (1) A C E D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -2 -8 -6 B 6 0 8 -6 6 C 2 -8 0 -4 -8 D 8 6 4 0 8 E 6 -6 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -8 -6 B 6 0 8 -6 6 C 2 -8 0 -4 -8 D 8 6 4 0 8 E 6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=24 E=16 C=15 B=15 so C is eliminated. Round 2 votes counts: A=36 D=30 E=19 B=15 so B is eliminated. Round 3 votes counts: A=37 D=35 E=28 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:207 E:200 C:191 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 -8 -6 B 6 0 8 -6 6 C 2 -8 0 -4 -8 D 8 6 4 0 8 E 6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -8 -6 B 6 0 8 -6 6 C 2 -8 0 -4 -8 D 8 6 4 0 8 E 6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -8 -6 B 6 0 8 -6 6 C 2 -8 0 -4 -8 D 8 6 4 0 8 E 6 -6 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8370: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (12) D B A E C (11) B D A E C (7) E A B C D (6) E C A B D (5) D B A C E (5) C E A D B (5) C D E B A (5) B A E D C (4) B A D E C (4) E A C B D (3) C E D A B (3) C D E A B (3) D C B E A (2) D A B E C (2) C E B A D (2) A E C B D (2) D C B A E (1) D B C A E (1) D A C B E (1) C E B D A (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D C A (1) B E A D C (1) B E A C D (1) B A E C D (1) A E B C D (1) A D E B C (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 10 12 0 B -6 0 -4 6 -12 C -10 4 0 12 -4 D -12 -6 -12 0 -2 E 0 12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.558027 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.441973 Sum of squares = 0.506734376824 Cumulative probabilities = A: 0.558027 B: 0.558027 C: 0.558027 D: 0.558027 E: 1.000000 A B C D E A 0 6 10 12 0 B -6 0 -4 6 -12 C -10 4 0 12 -4 D -12 -6 -12 0 -2 E 0 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=23 B=19 E=14 A=9 so A is eliminated. Round 2 votes counts: C=37 D=26 B=20 E=17 so E is eliminated. Round 3 votes counts: C=47 B=27 D=26 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:214 E:209 C:201 B:192 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 12 0 B -6 0 -4 6 -12 C -10 4 0 12 -4 D -12 -6 -12 0 -2 E 0 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 12 0 B -6 0 -4 6 -12 C -10 4 0 12 -4 D -12 -6 -12 0 -2 E 0 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 12 0 B -6 0 -4 6 -12 C -10 4 0 12 -4 D -12 -6 -12 0 -2 E 0 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8371: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) D C A E B (7) C E B D A (6) B E C A D (6) E B D A C (5) D A E B C (5) D A C E B (5) E B C D A (4) C D E A B (4) C B E A D (4) A D C B E (4) A C D B E (4) E B D C A (3) D A E C B (3) B E A D C (3) E D B C A (2) E C B D A (2) E B C A D (2) C D A E B (2) C D A B E (2) A D B C E (2) A B D E C (2) D E B A C (1) D E A C B (1) D E A B C (1) D A C B E (1) C B A E D (1) C A D B E (1) C A B E D (1) B A E D C (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 20 6 -8 12 B -20 0 4 -16 -6 C -6 -4 0 -20 -10 D 8 16 20 0 16 E -12 6 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 -8 12 B -20 0 4 -16 -6 C -6 -4 0 -20 -10 D 8 16 20 0 16 E -12 6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=24 C=21 E=18 B=10 so B is eliminated. Round 2 votes counts: A=28 E=27 D=24 C=21 so C is eliminated. Round 3 votes counts: E=37 D=32 A=31 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:215 E:194 B:181 C:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 6 -8 12 B -20 0 4 -16 -6 C -6 -4 0 -20 -10 D 8 16 20 0 16 E -12 6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 -8 12 B -20 0 4 -16 -6 C -6 -4 0 -20 -10 D 8 16 20 0 16 E -12 6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 -8 12 B -20 0 4 -16 -6 C -6 -4 0 -20 -10 D 8 16 20 0 16 E -12 6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8372: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) C B A E D (9) D E A B C (7) D A B C E (6) B A C D E (6) D A E B C (5) C E B A D (5) E D C A B (4) E D C B A (3) E D A B C (3) E C D B A (3) D E C A B (3) C E B D A (3) C D B A E (3) C B A D E (3) D E A C B (2) D C E A B (2) D C A B E (2) D A C B E (2) B C A E D (2) A D B E C (2) E D A C B (1) E C B A D (1) E B C A D (1) E A B D C (1) D C A E B (1) D A C E B (1) D A B E C (1) C E D B A (1) C D B E A (1) C B E A D (1) B C E A D (1) B C A D E (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 10 0 -8 14 B -10 0 -2 -8 6 C 0 2 0 -16 26 D 8 8 16 0 20 E -14 -6 -26 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 -8 14 B -10 0 -2 -8 6 C 0 2 0 -16 26 D 8 8 16 0 20 E -14 -6 -26 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=26 E=17 A=14 B=11 so B is eliminated. Round 2 votes counts: D=32 C=30 A=21 E=17 so E is eliminated. Round 3 votes counts: D=43 C=35 A=22 so A is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:226 A:208 C:206 B:193 E:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 0 -8 14 B -10 0 -2 -8 6 C 0 2 0 -16 26 D 8 8 16 0 20 E -14 -6 -26 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 -8 14 B -10 0 -2 -8 6 C 0 2 0 -16 26 D 8 8 16 0 20 E -14 -6 -26 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 -8 14 B -10 0 -2 -8 6 C 0 2 0 -16 26 D 8 8 16 0 20 E -14 -6 -26 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8373: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) E C D B A (9) A D B E C (8) C A B D E (6) E D A B C (5) A B D E C (5) A B D C E (5) C E B A D (4) C E A B D (4) E D C A B (3) E D B A C (3) E C D A B (3) D A B E C (3) C E D A B (3) E D C B A (2) E D B C A (2) D A E B C (2) C E B D A (2) C B E A D (2) C B A E D (2) C B A D E (2) B D A E C (2) B A D C E (2) B A C D E (2) D E A B C (1) D B E A C (1) D B A E C (1) C E A D B (1) C A B E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -16 -8 -14 B -4 0 -14 -18 -12 C 16 14 0 10 0 D 8 18 -10 0 -16 E 14 12 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.411129 D: 0.000000 E: 0.588871 Sum of squares = 0.51579627164 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.411129 D: 0.411129 E: 1.000000 A B C D E A 0 4 -16 -8 -14 B -4 0 -14 -18 -12 C 16 14 0 10 0 D 8 18 -10 0 -16 E 14 12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=27 A=20 D=8 B=6 so B is eliminated. Round 2 votes counts: C=39 E=27 A=24 D=10 so D is eliminated. Round 3 votes counts: C=39 A=32 E=29 so E is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:221 C:220 D:200 A:183 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 -8 -14 B -4 0 -14 -18 -12 C 16 14 0 10 0 D 8 18 -10 0 -16 E 14 12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 -8 -14 B -4 0 -14 -18 -12 C 16 14 0 10 0 D 8 18 -10 0 -16 E 14 12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 -8 -14 B -4 0 -14 -18 -12 C 16 14 0 10 0 D 8 18 -10 0 -16 E 14 12 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8374: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (9) C D B A E (9) E A C B D (8) C A D B E (7) E A B D C (6) E C D B A (5) A E C B D (5) D C B A E (4) B A D C E (4) A C E B D (4) E C A B D (3) E B D A C (3) A B D C E (3) E D B C A (2) E C D A B (2) E C A D B (2) E B A D C (2) E A B C D (2) C E A D B (2) C D B E A (2) C A B D E (2) B D A E C (2) B D A C E (2) A B C D E (2) E D C B A (1) C E D B A (1) C E D A B (1) C D E B A (1) C D A B E (1) C A E D B (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -10 10 14 B -4 0 -22 0 -2 C 10 22 0 20 10 D -10 0 -20 0 -2 E -14 2 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 10 14 B -4 0 -22 0 -2 C 10 22 0 20 10 D -10 0 -20 0 -2 E -14 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=27 A=15 D=13 B=9 so B is eliminated. Round 2 votes counts: E=36 C=27 A=20 D=17 so D is eliminated. Round 3 votes counts: C=40 E=36 A=24 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:231 A:209 E:190 B:186 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 10 14 B -4 0 -22 0 -2 C 10 22 0 20 10 D -10 0 -20 0 -2 E -14 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 10 14 B -4 0 -22 0 -2 C 10 22 0 20 10 D -10 0 -20 0 -2 E -14 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 10 14 B -4 0 -22 0 -2 C 10 22 0 20 10 D -10 0 -20 0 -2 E -14 2 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8375: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) D E A B C (9) E A B D C (8) D C E A B (8) A E B C D (7) A B E C D (7) B A C E D (6) D E A C B (5) D C E B A (5) E A D B C (3) C B A E D (3) E D A B C (2) D E C A B (2) D C B E A (2) D C A B E (2) C D B E A (2) C B D A E (2) B E A C D (2) B A E C D (2) A B C E D (2) D E B A C (1) D C B A E (1) D C A E B (1) D A E C B (1) C B A D E (1) B C A E D (1) A E D B C (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 20 16 -10 2 B -20 0 4 -14 -8 C -16 -4 0 -4 -2 D 10 14 4 0 10 E -2 8 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 16 -10 2 B -20 0 4 -14 -8 C -16 -4 0 -4 -2 D 10 14 4 0 10 E -2 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 C=20 A=19 E=13 B=11 so B is eliminated. Round 2 votes counts: D=37 A=27 C=21 E=15 so E is eliminated. Round 3 votes counts: A=40 D=39 C=21 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:214 E:199 C:187 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 16 -10 2 B -20 0 4 -14 -8 C -16 -4 0 -4 -2 D 10 14 4 0 10 E -2 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 -10 2 B -20 0 4 -14 -8 C -16 -4 0 -4 -2 D 10 14 4 0 10 E -2 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 -10 2 B -20 0 4 -14 -8 C -16 -4 0 -4 -2 D 10 14 4 0 10 E -2 8 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995625 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8376: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) D A E C B (8) A D C B E (8) B C E A D (6) E D B C A (5) C B E A D (5) C B A E D (5) E D A C B (4) E D A B C (4) C B A D E (4) B E C D A (4) D A E B C (3) B C E D A (3) A D E C B (3) A D C E B (3) E D B A C (2) E C B D A (2) E B C D A (2) C E B A D (2) C A B D E (2) B C A D E (2) A D B C E (2) E C B A D (1) E C A D B (1) E B D C A (1) D E B A C (1) D E A C B (1) D A B E C (1) D A B C E (1) C A E D B (1) C A E B D (1) C A B E D (1) B C A E D (1) B A C D E (1) A C D E B (1) Total count = 100 A B C D E A 0 6 2 0 -4 B -6 0 -6 -14 -8 C -2 6 0 -10 -2 D 0 14 10 0 -2 E 4 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999839 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 2 0 -4 B -6 0 -6 -14 -8 C -2 6 0 -10 -2 D 0 14 10 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 E=22 C=21 B=17 A=17 so B is eliminated. Round 2 votes counts: C=33 E=26 D=23 A=18 so A is eliminated. Round 3 votes counts: D=39 C=35 E=26 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:208 A:202 C:196 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 0 -4 B -6 0 -6 -14 -8 C -2 6 0 -10 -2 D 0 14 10 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 0 -4 B -6 0 -6 -14 -8 C -2 6 0 -10 -2 D 0 14 10 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 0 -4 B -6 0 -6 -14 -8 C -2 6 0 -10 -2 D 0 14 10 0 -2 E 4 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8377: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (11) B C D A E (10) E D A B C (9) C A B E D (9) D E B A C (8) D B E C A (6) B D C E A (6) E A D C B (5) C B A D E (5) A E C D B (5) E A C D B (3) D B E A C (3) A E D C B (3) A C E D B (3) D E A B C (2) C B A E D (2) C A E B D (2) A C B E D (2) E D A C B (1) D E B C A (1) B D E C A (1) B D C A E (1) B C D E A (1) B C A D E (1) Total count = 100 A B C D E A 0 10 10 2 8 B -10 0 -2 2 -6 C -10 2 0 8 6 D -2 -2 -8 0 -10 E -8 6 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999498 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 2 8 B -10 0 -2 2 -6 C -10 2 0 8 6 D -2 -2 -8 0 -10 E -8 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=20 B=20 E=18 C=18 so E is eliminated. Round 2 votes counts: A=32 D=30 B=20 C=18 so C is eliminated. Round 3 votes counts: A=43 D=30 B=27 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:203 E:201 B:192 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 2 8 B -10 0 -2 2 -6 C -10 2 0 8 6 D -2 -2 -8 0 -10 E -8 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 2 8 B -10 0 -2 2 -6 C -10 2 0 8 6 D -2 -2 -8 0 -10 E -8 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 2 8 B -10 0 -2 2 -6 C -10 2 0 8 6 D -2 -2 -8 0 -10 E -8 6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999931 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8378: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) E A B C D (6) E A C B D (5) C D A B E (5) A C D E B (5) E C A D B (4) C D B A E (4) B D A C E (4) E C B D A (3) E B D C A (3) E B C D A (3) E B A D C (3) D B C A E (3) D B A C E (3) B E D A C (3) B D C A E (3) E B D A C (2) E B A C D (2) E A C D B (2) C E D B A (2) B E D C A (2) B D E C A (2) A E C D B (2) A D C B E (2) A B D C E (2) E C A B D (1) D C B A E (1) D C A B E (1) C E A D B (1) C D E A B (1) C D B E A (1) C D A E B (1) C A E D B (1) B E C D A (1) B E A D C (1) B D E A C (1) B D C E A (1) B D A E C (1) B A D E C (1) A E D B C (1) A E B D C (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 12 -2 0 B 0 0 14 4 0 C -12 -14 0 2 -2 D 2 -4 -2 0 -2 E 0 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.302471 B: 0.158289 C: 0.000000 D: 0.000000 E: 0.539239 Sum of squares = 0.40732352412 Cumulative probabilities = A: 0.302471 B: 0.460761 C: 0.460761 D: 0.460761 E: 1.000000 A B C D E A 0 0 12 -2 0 B 0 0 14 4 0 C -12 -14 0 2 -2 D 2 -4 -2 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=22 B=20 C=16 D=8 so D is eliminated. Round 2 votes counts: E=34 B=26 A=22 C=18 so C is eliminated. Round 3 votes counts: E=38 B=32 A=30 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:205 E:202 D:197 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 12 -2 0 B 0 0 14 4 0 C -12 -14 0 2 -2 D 2 -4 -2 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 -2 0 B 0 0 14 4 0 C -12 -14 0 2 -2 D 2 -4 -2 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 -2 0 B 0 0 14 4 0 C -12 -14 0 2 -2 D 2 -4 -2 0 -2 E 0 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8379: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (10) A B E D C (10) E C D A B (9) E D C A B (8) C D E B A (7) D C E A B (6) E C D B A (5) B A E C D (5) E A B D C (4) D C B A E (4) B A E D C (4) A B D C E (4) C D B A E (3) B A C D E (3) E A D C B (2) D C A B E (2) C D B E A (2) A E B D C (2) E B A C D (1) E A B C D (1) D C E B A (1) D C B E A (1) D A C B E (1) C D E A B (1) B D C A E (1) B C D A E (1) B A C E D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -2 -4 4 B -2 0 -4 -4 6 C 2 4 0 -22 -4 D 4 4 22 0 -4 E -4 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.189411 B: 0.123362 C: 0.000000 D: 0.374454 E: 0.312773 Sum of squares = 0.289137469136 Cumulative probabilities = A: 0.189411 B: 0.312773 C: 0.312773 D: 0.687227 E: 1.000000 A B C D E A 0 2 -2 -4 4 B -2 0 -4 -4 6 C 2 4 0 -22 -4 D 4 4 22 0 -4 E -4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.177778 B: 0.133333 C: 0.000000 D: 0.377778 E: 0.311111 Sum of squares = 0.288888888885 Cumulative probabilities = A: 0.177778 B: 0.311111 C: 0.311111 D: 0.688889 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=25 A=17 D=15 C=13 so C is eliminated. Round 2 votes counts: E=30 D=28 B=25 A=17 so A is eliminated. Round 3 votes counts: B=40 E=32 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:213 A:200 E:199 B:198 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 -4 4 B -2 0 -4 -4 6 C 2 4 0 -22 -4 D 4 4 22 0 -4 E -4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.177778 B: 0.133333 C: 0.000000 D: 0.377778 E: 0.311111 Sum of squares = 0.288888888885 Cumulative probabilities = A: 0.177778 B: 0.311111 C: 0.311111 D: 0.688889 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 4 B -2 0 -4 -4 6 C 2 4 0 -22 -4 D 4 4 22 0 -4 E -4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.177778 B: 0.133333 C: 0.000000 D: 0.377778 E: 0.311111 Sum of squares = 0.288888888885 Cumulative probabilities = A: 0.177778 B: 0.311111 C: 0.311111 D: 0.688889 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 4 B -2 0 -4 -4 6 C 2 4 0 -22 -4 D 4 4 22 0 -4 E -4 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.177778 B: 0.133333 C: 0.000000 D: 0.377778 E: 0.311111 Sum of squares = 0.288888888885 Cumulative probabilities = A: 0.177778 B: 0.311111 C: 0.311111 D: 0.688889 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8380: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B D A E (9) C A D B E (9) A D C B E (7) E B C D A (6) E A B D C (5) E A D C B (4) C D A B E (4) E C B A D (3) E B A D C (3) C B D E A (3) B C D E A (3) A C E D B (3) E B D A C (2) E B A C D (2) D B C A E (2) D A B C E (2) B E C D A (2) B D E A C (2) B D C A E (2) A E D C B (2) A D E B C (2) A C D E B (2) E D B A C (1) E C A D B (1) E A C D B (1) D C B A E (1) D A C B E (1) C D B A E (1) B E D C A (1) B D A C E (1) B C E D A (1) B C D A E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 2 10 2 B -8 0 -6 -8 4 C -2 6 0 2 6 D -10 8 -2 0 8 E -2 -4 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 10 2 B -8 0 -6 -8 4 C -2 6 0 2 6 D -10 8 -2 0 8 E -2 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=26 A=18 B=13 D=6 so D is eliminated. Round 2 votes counts: E=37 C=27 A=21 B=15 so B is eliminated. Round 3 votes counts: E=42 C=36 A=22 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:206 D:202 B:191 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 10 2 B -8 0 -6 -8 4 C -2 6 0 2 6 D -10 8 -2 0 8 E -2 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 10 2 B -8 0 -6 -8 4 C -2 6 0 2 6 D -10 8 -2 0 8 E -2 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 10 2 B -8 0 -6 -8 4 C -2 6 0 2 6 D -10 8 -2 0 8 E -2 -4 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8381: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (9) A E C B D (9) D C E B A (7) A B E C D (6) A B D C E (6) E C A B D (5) D B C E A (5) B A D E C (5) A B D E C (5) C E D B A (4) C E D A B (4) C E A D B (4) D A B C E (3) B D A E C (3) E C B D A (2) E C B A D (2) E C A D B (2) D C E A B (2) B D E C A (2) A D C E B (2) A D B C E (2) E C D B A (1) E B C A D (1) D C B E A (1) D A C B E (1) C D E B A (1) B E D C A (1) B E C D A (1) B E A D C (1) B D E A C (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 6 10 4 6 B -6 0 2 0 6 C -10 -2 0 -14 6 D -4 0 14 0 12 E -6 -6 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 4 6 B -6 0 2 0 6 C -10 -2 0 -14 6 D -4 0 14 0 12 E -6 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=28 B=14 E=13 C=13 so E is eliminated. Round 2 votes counts: A=32 D=28 C=25 B=15 so B is eliminated. Round 3 votes counts: A=38 D=35 C=27 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 D:211 B:201 C:190 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 4 6 B -6 0 2 0 6 C -10 -2 0 -14 6 D -4 0 14 0 12 E -6 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 4 6 B -6 0 2 0 6 C -10 -2 0 -14 6 D -4 0 14 0 12 E -6 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 4 6 B -6 0 2 0 6 C -10 -2 0 -14 6 D -4 0 14 0 12 E -6 -6 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8382: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) A E D C B (8) E A C B D (7) E A B C D (6) D C B A E (6) B C D E A (6) D A C E B (5) A E D B C (4) B E C A D (3) B D C A E (3) A E C D B (3) E C A B D (2) E B A C D (2) E A C D B (2) E A B D C (2) D A E C B (2) C E A D B (2) C B E D A (2) C B D E A (2) B D C E A (2) B C E A D (2) A D E C B (2) E B C A D (1) E A D C B (1) E A D B C (1) D C A B E (1) D B A E C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B A D (1) C B E A D (1) C A D E B (1) B E A D C (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E D A (1) A D E B C (1) Total count = 100 A B C D E A 0 6 8 8 2 B -6 0 2 -6 -6 C -8 -2 0 -10 -6 D -8 6 10 0 -6 E -2 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 8 2 B -6 0 2 -6 -6 C -8 -2 0 -10 -6 D -8 6 10 0 -6 E -2 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=24 B=21 A=18 C=9 so C is eliminated. Round 2 votes counts: D=28 E=27 B=26 A=19 so A is eliminated. Round 3 votes counts: E=42 D=32 B=26 so B is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:208 D:201 B:192 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 8 2 B -6 0 2 -6 -6 C -8 -2 0 -10 -6 D -8 6 10 0 -6 E -2 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 8 2 B -6 0 2 -6 -6 C -8 -2 0 -10 -6 D -8 6 10 0 -6 E -2 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 8 2 B -6 0 2 -6 -6 C -8 -2 0 -10 -6 D -8 6 10 0 -6 E -2 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8383: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (13) E D B C A (6) D E A B C (6) E C B D A (5) E C B A D (5) C B A D E (5) E D A C B (4) D E B A C (4) C A B D E (4) A D B C E (4) E D A B C (3) E B C D A (3) C B A E D (3) B C A D E (3) E D B A C (2) D B A E C (2) D B A C E (2) D A B C E (2) C E B A D (2) C A B E D (2) B D C A E (2) A C D B E (2) A B C D E (2) E D C A B (1) E C A D B (1) E C A B D (1) E B D C A (1) D A E C B (1) D A E B C (1) D A B E C (1) B E D C A (1) B D A C E (1) B C E A D (1) A D C B E (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 8 4 8 B -4 0 -6 12 4 C -8 6 0 10 4 D -4 -12 -10 0 12 E -8 -4 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 4 8 B -4 0 -6 12 4 C -8 6 0 10 4 D -4 -12 -10 0 12 E -8 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=25 D=19 C=16 B=8 so B is eliminated. Round 2 votes counts: E=33 A=25 D=22 C=20 so C is eliminated. Round 3 votes counts: A=42 E=36 D=22 so D is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:206 B:203 D:193 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 8 B -4 0 -6 12 4 C -8 6 0 10 4 D -4 -12 -10 0 12 E -8 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 8 B -4 0 -6 12 4 C -8 6 0 10 4 D -4 -12 -10 0 12 E -8 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 8 B -4 0 -6 12 4 C -8 6 0 10 4 D -4 -12 -10 0 12 E -8 -4 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8384: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (7) E D A B C (6) D E C B A (6) D C E B A (4) C D B E A (4) C B D A E (4) C B A D E (4) C A B D E (4) B A C E D (4) A E B D C (4) E D B A C (3) E A D C B (3) C D E A B (3) B E D A C (3) B A E D C (3) A C E B D (3) A C B E D (3) E D A C B (2) E A D B C (2) D E C A B (2) D E B C A (2) C D E B A (2) C B D E A (2) B D E C A (2) B C A D E (2) B A C D E (2) A E D B C (2) A B E D C (2) E D C A B (1) D C B E A (1) D B E C A (1) D B C E A (1) C A E D B (1) B E A D C (1) B D C E A (1) B C D A E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 4 -2 -4 B 6 0 2 8 8 C -4 -2 0 -4 8 D 2 -8 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998886 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -2 -4 B 6 0 2 8 8 C -4 -2 0 -4 8 D 2 -8 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 A=23 B=19 E=17 D=17 so E is eliminated. Round 2 votes counts: D=29 A=28 C=24 B=19 so B is eliminated. Round 3 votes counts: A=38 D=35 C=27 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:212 C:199 D:198 A:196 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -2 -4 B 6 0 2 8 8 C -4 -2 0 -4 8 D 2 -8 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 -4 B 6 0 2 8 8 C -4 -2 0 -4 8 D 2 -8 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 -4 B 6 0 2 8 8 C -4 -2 0 -4 8 D 2 -8 4 0 -2 E 4 -8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8385: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) C A D B E (9) B E D A C (9) E B D A C (7) D E B C A (7) C A B D E (7) B E A D C (6) D E C B A (5) D C E A B (5) B A C E D (5) D E C A B (4) C A D E B (4) E D B C A (3) E D B A C (3) B E A C D (3) A C B D E (3) A B C E D (3) E B A D C (2) C D A E B (2) B A E C D (2) D C A E B (1) Total count = 100 A B C D E A 0 -4 6 8 -8 B 4 0 0 14 14 C -6 0 0 -4 -2 D -8 -14 4 0 -6 E 8 -14 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.692124 C: 0.307876 D: 0.000000 E: 0.000000 Sum of squares = 0.573823436593 Cumulative probabilities = A: 0.000000 B: 0.692124 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 8 -8 B 4 0 0 14 14 C -6 0 0 -4 -2 D -8 -14 4 0 -6 E 8 -14 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=22 C=22 A=16 E=15 so E is eliminated. Round 2 votes counts: B=34 D=28 C=22 A=16 so A is eliminated. Round 3 votes counts: B=37 C=35 D=28 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:201 E:201 C:194 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 8 -8 B 4 0 0 14 14 C -6 0 0 -4 -2 D -8 -14 4 0 -6 E 8 -14 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 8 -8 B 4 0 0 14 14 C -6 0 0 -4 -2 D -8 -14 4 0 -6 E 8 -14 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 8 -8 B 4 0 0 14 14 C -6 0 0 -4 -2 D -8 -14 4 0 -6 E 8 -14 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000011109 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8386: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) C B A D E (9) B C E A D (8) E D B C A (6) E B D C A (6) D A E C B (5) A D E C B (5) A C B D E (5) E D A B C (4) E B C D A (4) D E A C B (4) B C A E D (4) E D B A C (2) E B C A D (2) D A C B E (2) C D A B E (2) C B D A E (2) B E C D A (2) B C D E A (2) A D C E B (2) A D C B E (2) E B D A C (1) E B A C D (1) E A D B C (1) E A B D C (1) D E C B A (1) D A C E B (1) C D B E A (1) C A B D E (1) B C E D A (1) B C D A E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -4 6 6 B 6 0 -6 2 6 C 4 6 0 12 8 D -6 -2 -12 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 6 6 B 6 0 -6 2 6 C 4 6 0 12 8 D -6 -2 -12 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=26 B=18 C=15 D=13 so D is eliminated. Round 2 votes counts: A=34 E=33 B=18 C=15 so C is eliminated. Round 3 votes counts: A=37 E=33 B=30 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:215 B:204 A:201 D:195 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 6 6 B 6 0 -6 2 6 C 4 6 0 12 8 D -6 -2 -12 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 6 6 B 6 0 -6 2 6 C 4 6 0 12 8 D -6 -2 -12 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 6 6 B 6 0 -6 2 6 C 4 6 0 12 8 D -6 -2 -12 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8387: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) C D A E B (7) B E A D C (6) E B C A D (5) E C B D A (4) E B A D C (4) C E B D A (4) B E C A D (4) B C E D A (4) A D E B C (4) D A B C E (3) C E D A B (3) C B E D A (3) B E A C D (3) A D B E C (3) A D B C E (3) E C D A B (2) D C A E B (2) D A C E B (2) C D B A E (2) C B D A E (2) B E C D A (2) B C D A E (2) B A D C E (2) A D C E B (2) E C B A D (1) E C A D B (1) E B C D A (1) E B A C D (1) E A D C B (1) E A B D C (1) D C A B E (1) C D E A B (1) C D B E A (1) C D A B E (1) B D C A E (1) B A E D C (1) B A D E C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -8 -10 -4 B 8 0 4 6 4 C 8 -4 0 8 6 D 10 -6 -8 0 -4 E 4 -4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -10 -4 B 8 0 4 6 4 C 8 -4 0 8 6 D 10 -6 -8 0 -4 E 4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 E=21 D=15 A=14 so A is eliminated. Round 2 votes counts: D=28 B=26 C=24 E=22 so E is eliminated. Round 3 votes counts: B=39 C=32 D=29 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:209 E:199 D:196 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -10 -4 B 8 0 4 6 4 C 8 -4 0 8 6 D 10 -6 -8 0 -4 E 4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -10 -4 B 8 0 4 6 4 C 8 -4 0 8 6 D 10 -6 -8 0 -4 E 4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -10 -4 B 8 0 4 6 4 C 8 -4 0 8 6 D 10 -6 -8 0 -4 E 4 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8388: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (11) C A B D E (7) E D A B C (6) D E B A C (6) C A B E D (5) B D E C A (5) A C E D B (5) C B A D E (4) E D B C A (3) D E A B C (3) D B E A C (3) C A E B D (3) B E D C A (3) A E D C B (3) A C D B E (3) A C B D E (3) E B D C A (2) E A C D B (2) C A E D B (2) B D A C E (2) B C E D A (2) B C D A E (2) A E C D B (2) A C D E B (2) E D A C B (1) E C B D A (1) E C A D B (1) E A D C B (1) D B E C A (1) D A E B C (1) C B A E D (1) B C A D E (1) A D C B E (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 14 -4 -2 B -6 0 4 -14 -12 C -14 -4 0 -4 -10 D 4 14 4 0 -10 E 2 12 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 14 -4 -2 B -6 0 4 -14 -12 C -14 -4 0 -4 -10 D 4 14 4 0 -10 E 2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=22 A=21 B=15 D=14 so D is eliminated. Round 2 votes counts: E=37 C=22 A=22 B=19 so B is eliminated. Round 3 votes counts: E=49 C=27 A=24 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:207 D:206 B:186 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 -4 -2 B -6 0 4 -14 -12 C -14 -4 0 -4 -10 D 4 14 4 0 -10 E 2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -4 -2 B -6 0 4 -14 -12 C -14 -4 0 -4 -10 D 4 14 4 0 -10 E 2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -4 -2 B -6 0 4 -14 -12 C -14 -4 0 -4 -10 D 4 14 4 0 -10 E 2 12 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8389: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E A C D B (10) C A B E D (9) A C E B D (9) E A D C B (8) B C A D E (7) A E C B D (7) D E B A C (6) B C D A E (6) D E A B C (5) C B A D E (4) B D C A E (4) E A C B D (3) D B E C A (3) D B C E A (3) D E B C A (2) D B C A E (2) D B A C E (1) C B A E D (1) Total count = 100 A B C D E A 0 22 18 16 0 B -22 0 -22 0 -20 C -18 22 0 12 -8 D -16 0 -12 0 -14 E 0 20 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.255986 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.744014 Sum of squares = 0.619085953317 Cumulative probabilities = A: 0.255986 B: 0.255986 C: 0.255986 D: 0.255986 E: 1.000000 A B C D E A 0 22 18 16 0 B -22 0 -22 0 -20 C -18 22 0 12 -8 D -16 0 -12 0 -14 E 0 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=22 B=17 A=16 C=14 so C is eliminated. Round 2 votes counts: E=31 A=25 D=22 B=22 so D is eliminated. Round 3 votes counts: E=44 B=31 A=25 so A is eliminated. Round 4 votes counts: E=60 B=40 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:228 E:221 C:204 D:179 B:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 22 18 16 0 B -22 0 -22 0 -20 C -18 22 0 12 -8 D -16 0 -12 0 -14 E 0 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 16 0 B -22 0 -22 0 -20 C -18 22 0 12 -8 D -16 0 -12 0 -14 E 0 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 16 0 B -22 0 -22 0 -20 C -18 22 0 12 -8 D -16 0 -12 0 -14 E 0 20 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8390: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) D E B C A (5) D A C E B (5) A C B E D (5) B E D C A (4) B E C A D (4) E C B A D (3) E B C D A (3) D E C A B (3) D B E A C (3) B A C E D (3) A C B D E (3) E D C B A (2) E D B C A (2) E C B D A (2) E B D C A (2) D C E A B (2) D B E C A (2) D B A E C (2) D A B E C (2) C E A D B (2) C E A B D (2) C B A E D (2) C A E B D (2) B D A E C (2) B A D C E (2) A D C B E (2) A C D E B (2) E D C A B (1) E C D A B (1) E C A B D (1) D E B A C (1) D E A B C (1) D C A E B (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E A B (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A C D (1) B D E C A (1) B D E A C (1) B D A C E (1) A D C E B (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -2 -20 -4 B 0 0 0 -8 0 C 2 0 0 -14 0 D 20 8 14 0 6 E 4 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -20 -4 B 0 0 0 -8 0 C 2 0 0 -14 0 D 20 8 14 0 6 E 4 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=20 E=17 A=16 C=12 so C is eliminated. Round 2 votes counts: D=36 E=22 B=22 A=20 so A is eliminated. Round 3 votes counts: D=43 B=30 E=27 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:199 B:196 C:194 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -2 -20 -4 B 0 0 0 -8 0 C 2 0 0 -14 0 D 20 8 14 0 6 E 4 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -20 -4 B 0 0 0 -8 0 C 2 0 0 -14 0 D 20 8 14 0 6 E 4 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -20 -4 B 0 0 0 -8 0 C 2 0 0 -14 0 D 20 8 14 0 6 E 4 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8391: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) D A C B E (9) D E C A B (5) D A B C E (5) E D B C A (4) E C B A D (4) D E A C B (4) D E A B C (4) B A C E D (4) A C B D E (4) D C A E B (3) D A C E B (3) B A C D E (3) D A B E C (2) C D A E B (2) C A D B E (2) C A B E D (2) B E A C D (2) B C A E D (2) A D B C E (2) A B C D E (2) E D C B A (1) E D C A B (1) E D B A C (1) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) E B A D C (1) D B A E C (1) D B A C E (1) D A E C B (1) D A E B C (1) C E A B D (1) C D A B E (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) B E C A D (1) B A D C E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 16 8 -8 12 B -16 0 -2 -14 -4 C -8 2 0 -6 6 D 8 14 6 0 16 E -12 4 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 8 -8 12 B -16 0 -2 -14 -4 C -8 2 0 -6 6 D 8 14 6 0 16 E -12 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=26 B=13 C=12 A=10 so A is eliminated. Round 2 votes counts: D=42 E=26 C=17 B=15 so B is eliminated. Round 3 votes counts: D=43 E=29 C=28 so C is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:214 C:197 E:185 B:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 8 -8 12 B -16 0 -2 -14 -4 C -8 2 0 -6 6 D 8 14 6 0 16 E -12 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 -8 12 B -16 0 -2 -14 -4 C -8 2 0 -6 6 D 8 14 6 0 16 E -12 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 -8 12 B -16 0 -2 -14 -4 C -8 2 0 -6 6 D 8 14 6 0 16 E -12 4 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8392: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (13) C D E B A (9) C E B D A (8) C B E A D (7) D C A E B (6) A B E D C (5) C A B E D (4) E B C D A (3) D E A B C (3) A D B E C (3) E B A D C (2) D E C B A (2) D E B C A (2) D E B A C (2) D C E B A (2) C D A E B (2) C D A B E (2) C B E D A (2) C B A E D (2) B E A D C (2) B C E A D (2) A D C B E (2) A C D B E (2) A B D E C (2) E D B C A (1) E D B A C (1) E C B D A (1) E B D A C (1) D C E A B (1) C E D B A (1) C E B A D (1) C A D B E (1) C A B D E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -20 -24 -6 B 4 0 -12 -10 -22 C 20 12 0 0 12 D 24 10 0 0 10 E 6 22 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636774 D: 0.363226 E: 0.000000 Sum of squares = 0.537414292701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636774 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -20 -24 -6 B 4 0 -12 -10 -22 C 20 12 0 0 12 D 24 10 0 0 10 E 6 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 D=31 A=15 E=9 B=5 so B is eliminated. Round 2 votes counts: C=42 D=31 A=16 E=11 so E is eliminated. Round 3 votes counts: C=46 D=34 A=20 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:222 E:203 B:180 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -20 -24 -6 B 4 0 -12 -10 -22 C 20 12 0 0 12 D 24 10 0 0 10 E 6 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -20 -24 -6 B 4 0 -12 -10 -22 C 20 12 0 0 12 D 24 10 0 0 10 E 6 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -20 -24 -6 B 4 0 -12 -10 -22 C 20 12 0 0 12 D 24 10 0 0 10 E 6 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8393: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) C B D A E (9) B C D E A (7) E A B D C (6) D C A E B (6) C D A E B (6) E B A D C (5) C D B A E (5) D C B A E (4) B E A D C (4) A E C D B (4) E A D B C (3) C D A B E (3) B E C A D (3) A C E D B (3) E A B C D (2) D A E C B (2) C A D E B (2) B E D A C (2) B E A C D (2) B D C E A (2) B C E A D (2) B C D A E (2) E A D C B (1) D E A B C (1) D A C E B (1) B E C D A (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -6 -4 16 B -2 0 -14 -4 -6 C 6 14 0 4 6 D 4 4 -4 0 2 E -16 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -4 16 B -2 0 -14 -4 -6 C 6 14 0 4 6 D 4 4 -4 0 2 E -16 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 A=18 E=17 D=14 so D is eliminated. Round 2 votes counts: C=35 B=26 A=21 E=18 so E is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:204 D:203 E:191 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -4 16 B -2 0 -14 -4 -6 C 6 14 0 4 6 D 4 4 -4 0 2 E -16 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 16 B -2 0 -14 -4 -6 C 6 14 0 4 6 D 4 4 -4 0 2 E -16 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 16 B -2 0 -14 -4 -6 C 6 14 0 4 6 D 4 4 -4 0 2 E -16 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999118 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8394: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (7) D E A B C (6) D B E A C (6) C B A E D (6) B C A D E (4) E A D C B (3) E A C D B (3) D C B E A (3) D B C E A (3) C E A D B (3) C B D E A (3) C B D A E (3) C B A D E (3) C A E B D (3) B D C A E (3) B A C D E (3) A E D B C (3) A E C D B (3) A E C B D (3) A E B C D (3) E D A B C (2) D E B A C (2) D B E C A (2) C E A B D (2) C A B E D (2) B D C E A (2) B A C E D (2) A B C E D (2) E C A D B (1) E A D B C (1) D E B C A (1) D C E B A (1) D B A E C (1) D A B E C (1) C D E B A (1) B D A E C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -14 -6 4 10 B 14 0 14 6 16 C 6 -14 0 14 12 D -4 -6 -14 0 14 E -10 -16 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 4 10 B 14 0 14 6 16 C 6 -14 0 14 12 D -4 -6 -14 0 14 E -10 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=26 C=26 B=22 A=16 E=10 so E is eliminated. Round 2 votes counts: D=28 C=27 A=23 B=22 so B is eliminated. Round 3 votes counts: C=38 D=34 A=28 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:225 C:209 A:197 D:195 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 4 10 B 14 0 14 6 16 C 6 -14 0 14 12 D -4 -6 -14 0 14 E -10 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 4 10 B 14 0 14 6 16 C 6 -14 0 14 12 D -4 -6 -14 0 14 E -10 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 4 10 B 14 0 14 6 16 C 6 -14 0 14 12 D -4 -6 -14 0 14 E -10 -16 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8395: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (9) D B C E A (8) A E D B C (8) E C A B D (7) E A C B D (5) D C B E A (5) C E D B A (5) A E B C D (5) D B C A E (4) D B A C E (4) C E B A D (4) B D C A E (4) E C D A B (3) A B D C E (3) D C E B A (2) D A B E C (2) C B D E A (2) B C D A E (2) B C A D E (2) A B E C D (2) E D C A B (1) E C A D B (1) E A C D B (1) D E C B A (1) C D E B A (1) C D B E A (1) C B E D A (1) B D A C E (1) B A C D E (1) A E D C B (1) A E B D C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -8 6 4 B -4 0 0 2 -12 C 8 0 0 4 0 D -6 -2 -4 0 -10 E -4 12 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.605528 D: 0.000000 E: 0.394472 Sum of squares = 0.522272160314 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.605528 D: 0.605528 E: 1.000000 A B C D E A 0 4 -8 6 4 B -4 0 0 2 -12 C 8 0 0 4 0 D -6 -2 -4 0 -10 E -4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=26 E=18 C=14 B=10 so B is eliminated. Round 2 votes counts: A=33 D=31 E=18 C=18 so E is eliminated. Round 3 votes counts: A=39 D=32 C=29 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:209 C:206 A:203 B:193 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 6 4 B -4 0 0 2 -12 C 8 0 0 4 0 D -6 -2 -4 0 -10 E -4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 6 4 B -4 0 0 2 -12 C 8 0 0 4 0 D -6 -2 -4 0 -10 E -4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 6 4 B -4 0 0 2 -12 C 8 0 0 4 0 D -6 -2 -4 0 -10 E -4 12 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8396: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) D B C A E (8) E D A C B (6) D E B C A (6) B C A E D (5) E D B A C (4) E B D C A (4) D E A C B (4) D B E C A (4) B D C A E (4) A C E B D (4) A C B E D (4) E B A C D (3) E A C D B (3) D C A B E (3) C A D B E (3) C A B E D (3) B D E C A (3) D B C E A (2) C A B D E (2) B E C A D (2) B C D A E (2) B C A D E (2) A C E D B (2) A C D E B (2) E D B C A (1) D E C A B (1) C B A D E (1) B E D C A (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -14 -6 -6 B 4 0 4 0 2 C 14 -4 0 -2 -4 D 6 0 2 0 -4 E 6 -2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.757197 C: 0.000000 D: 0.242803 E: 0.000000 Sum of squares = 0.632301039383 Cumulative probabilities = A: 0.000000 B: 0.757197 C: 0.757197 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -6 -6 B 4 0 4 0 2 C 14 -4 0 -2 -4 D 6 0 2 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555556030777 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=28 B=19 A=14 C=9 so C is eliminated. Round 2 votes counts: E=30 D=28 A=22 B=20 so B is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:206 B:205 C:202 D:202 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -14 -6 -6 B 4 0 4 0 2 C 14 -4 0 -2 -4 D 6 0 2 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555556030777 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -6 -6 B 4 0 4 0 2 C 14 -4 0 -2 -4 D 6 0 2 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555556030777 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -6 -6 B 4 0 4 0 2 C 14 -4 0 -2 -4 D 6 0 2 0 -4 E 6 -2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555556030777 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8397: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (11) C E A D B (8) E A D B C (6) B D C A E (5) A D E B C (5) D A B E C (4) C B D E A (4) C B D A E (4) B D E A C (4) B D A E C (4) B C D E A (4) E C A D B (3) E A B D C (3) C E B A D (3) C A E D B (3) B C D A E (3) A E D B C (3) E C B A D (2) E A D C B (2) C A D E B (2) B D A C E (2) A E C D B (2) E C A B D (1) E A C B D (1) D C A B E (1) D B A C E (1) C E B D A (1) C E A B D (1) C D A B E (1) C B E D A (1) C A E B D (1) C A D B E (1) B E D C A (1) B E D A C (1) A E D C B (1) Total count = 100 A B C D E A 0 20 0 18 -14 B -20 0 -8 -8 -18 C 0 8 0 14 -8 D -18 8 -14 0 -10 E 14 18 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 0 18 -14 B -20 0 -8 -8 -18 C 0 8 0 14 -8 D -18 8 -14 0 -10 E 14 18 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=29 B=24 A=11 D=6 so D is eliminated. Round 2 votes counts: C=31 E=29 B=25 A=15 so A is eliminated. Round 3 votes counts: E=40 C=31 B=29 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:212 C:207 D:183 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 0 18 -14 B -20 0 -8 -8 -18 C 0 8 0 14 -8 D -18 8 -14 0 -10 E 14 18 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 18 -14 B -20 0 -8 -8 -18 C 0 8 0 14 -8 D -18 8 -14 0 -10 E 14 18 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 18 -14 B -20 0 -8 -8 -18 C 0 8 0 14 -8 D -18 8 -14 0 -10 E 14 18 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8398: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (11) D A E B C (10) E D A C B (9) C B E A D (9) E C B D A (7) C E B A D (6) C B A D E (4) B C E A D (4) D A B C E (3) B A D E C (3) B A D C E (3) A D B E C (3) E D C A B (2) E D A B C (2) E C B A D (2) C E D A B (2) C E B D A (2) C B A E D (2) B A C D E (2) A D B C E (2) E C D A B (1) E B C D A (1) D E A B C (1) D A E C B (1) D A C B E (1) C D A E B (1) C D A B E (1) B E C A D (1) B E A C D (1) B C A E D (1) B C A D E (1) A B D E C (1) Total count = 100 A B C D E A 0 2 6 -10 0 B -2 0 0 0 6 C -6 0 0 -4 -12 D 10 0 4 0 -4 E 0 -6 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.566681 C: 0.000000 D: 0.433319 E: 0.000000 Sum of squares = 0.508892617618 Cumulative probabilities = A: 0.000000 B: 0.566681 C: 0.566681 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -10 0 B -2 0 0 0 6 C -6 0 0 -4 -12 D 10 0 4 0 -4 E 0 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 E=24 B=16 A=6 so A is eliminated. Round 2 votes counts: D=32 C=27 E=24 B=17 so B is eliminated. Round 3 votes counts: D=39 C=35 E=26 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:205 E:205 B:202 A:199 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 6 -10 0 B -2 0 0 0 6 C -6 0 0 -4 -12 D 10 0 4 0 -4 E 0 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -10 0 B -2 0 0 0 6 C -6 0 0 -4 -12 D 10 0 4 0 -4 E 0 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -10 0 B -2 0 0 0 6 C -6 0 0 -4 -12 D 10 0 4 0 -4 E 0 -6 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8399: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) E C D B A (10) E C D A B (7) B D C A E (6) E D C B A (5) D B C E A (5) E D B C A (4) E A C D B (4) D B C A E (4) E C A D B (3) E A C B D (3) B A D C E (3) A B E D C (3) E B A D C (2) C D E B A (2) C D B A E (2) B D C E A (2) B D A C E (2) B A D E C (2) A E C B D (2) A E B C D (2) A B E C D (2) A B C D E (2) E B D A C (1) E A B D C (1) D C B E A (1) D C B A E (1) D B E C A (1) C E D B A (1) C E D A B (1) C D B E A (1) C D A B E (1) C A D B E (1) B E D C A (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 -18 -16 -10 B 12 0 8 -10 2 C 18 -8 0 -10 -6 D 16 10 10 0 -6 E 10 -2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.111111 E: 0.555556 Sum of squares = 0.432098765467 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.444444 E: 1.000000 A B C D E A 0 -12 -18 -16 -10 B 12 0 8 -10 2 C 18 -8 0 -10 -6 D 16 10 10 0 -6 E 10 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.111111 E: 0.555556 Sum of squares = 0.43209876553 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=23 B=16 D=12 C=9 so C is eliminated. Round 2 votes counts: E=42 A=24 D=18 B=16 so B is eliminated. Round 3 votes counts: E=43 A=29 D=28 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:210 B:206 C:197 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -18 -16 -10 B 12 0 8 -10 2 C 18 -8 0 -10 -6 D 16 10 10 0 -6 E 10 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.111111 E: 0.555556 Sum of squares = 0.43209876553 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.444444 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -18 -16 -10 B 12 0 8 -10 2 C 18 -8 0 -10 -6 D 16 10 10 0 -6 E 10 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.111111 E: 0.555556 Sum of squares = 0.43209876553 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -18 -16 -10 B 12 0 8 -10 2 C 18 -8 0 -10 -6 D 16 10 10 0 -6 E 10 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.111111 E: 0.555556 Sum of squares = 0.43209876553 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.444444 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8400: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (17) A D E B C (10) D B C A E (9) A E D B C (9) E A C B D (7) C B E D A (5) E C A B D (3) E A D B C (3) B C D E A (3) A E D C B (3) A D B C E (3) D E B A C (2) D A B C E (2) C E B D A (2) C B D A E (2) C A B D E (2) A E C D B (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) E C B A D (1) E B D C A (1) E B C D A (1) D C B A E (1) D B A E C (1) D B A C E (1) D A B E C (1) C E B A D (1) B D C E A (1) B D C A E (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -4 -8 -2 B 4 0 2 -2 0 C 4 -2 0 -4 2 D 8 2 4 0 16 E 2 0 -2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -8 -2 B 4 0 2 -2 0 C 4 -2 0 -4 2 D 8 2 4 0 16 E 2 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=29 A=29 E=20 D=17 B=5 so B is eliminated. Round 2 votes counts: C=32 A=29 E=20 D=19 so D is eliminated. Round 3 votes counts: C=44 A=34 E=22 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:215 B:202 C:200 E:192 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -8 -2 B 4 0 2 -2 0 C 4 -2 0 -4 2 D 8 2 4 0 16 E 2 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -8 -2 B 4 0 2 -2 0 C 4 -2 0 -4 2 D 8 2 4 0 16 E 2 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -8 -2 B 4 0 2 -2 0 C 4 -2 0 -4 2 D 8 2 4 0 16 E 2 0 -2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8401: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (20) D C E B A (10) C E D B A (9) A B E D C (9) B E C A D (6) A D B E C (6) B E C D A (4) A B D E C (4) E C B D A (3) D C A E B (3) D A C E B (3) E B C D A (2) C E B A D (2) B A E C D (2) A D C E B (2) A D B C E (2) D E B C A (1) D C E A B (1) D A C B E (1) C E B D A (1) C D E B A (1) C D A E B (1) B E A C D (1) B D E C A (1) B A E D C (1) A E B C D (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 12 10 18 16 B -12 0 20 16 20 C -10 -20 0 10 -22 D -18 -16 -10 0 -24 E -16 -20 22 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 18 16 B -12 0 20 16 20 C -10 -20 0 10 -22 D -18 -16 -10 0 -24 E -16 -20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=47 D=19 B=15 C=14 E=5 so E is eliminated. Round 2 votes counts: A=47 D=19 C=17 B=17 so C is eliminated. Round 3 votes counts: A=47 D=30 B=23 so B is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 B:222 E:205 C:179 D:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 18 16 B -12 0 20 16 20 C -10 -20 0 10 -22 D -18 -16 -10 0 -24 E -16 -20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 18 16 B -12 0 20 16 20 C -10 -20 0 10 -22 D -18 -16 -10 0 -24 E -16 -20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 18 16 B -12 0 20 16 20 C -10 -20 0 10 -22 D -18 -16 -10 0 -24 E -16 -20 22 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8402: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) B E D A C (5) B D E A C (5) B E C A D (4) A D B C E (4) E D B C A (3) E C B D A (3) E B C D A (3) C E A B D (3) C A E D B (3) C A E B D (3) B E C D A (3) A C D E B (3) A C D B E (3) E C B A D (2) E B D C A (2) D E C A B (2) D E B A C (2) D A B E C (2) D A B C E (2) C A D E B (2) C A B E D (2) B C A E D (2) B A D E C (2) A C B D E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C D A B (1) E B C A D (1) D E B C A (1) D E A C B (1) D C E A B (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) C E B A D (1) C E A D B (1) C B E A D (1) C B A E D (1) B E D C A (1) B E A D C (1) B D A E C (1) B C E A D (1) B A E C D (1) B A C E D (1) B A C D E (1) A D C E B (1) A D C B E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 -2 -14 B 12 0 14 8 10 C 0 -14 0 0 -14 D 2 -8 0 0 -6 E 14 -10 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -2 -14 B 12 0 14 8 10 C 0 -14 0 0 -14 D 2 -8 0 0 -6 E 14 -10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=21 E=18 C=17 A=16 so A is eliminated. Round 2 votes counts: B=30 D=27 C=25 E=18 so E is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:212 D:194 A:186 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -2 -14 B 12 0 14 8 10 C 0 -14 0 0 -14 D 2 -8 0 0 -6 E 14 -10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -2 -14 B 12 0 14 8 10 C 0 -14 0 0 -14 D 2 -8 0 0 -6 E 14 -10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -2 -14 B 12 0 14 8 10 C 0 -14 0 0 -14 D 2 -8 0 0 -6 E 14 -10 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8403: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) E A B C D (9) E A D C B (8) A B E C D (8) D C B E A (7) D C E B A (6) C D B A E (6) B A C D E (6) D C B A E (5) A E B C D (5) E A B D C (4) A B C D E (4) E D C A B (3) C B D A E (3) B C D A E (3) A B C E D (3) E D A C B (2) D C A B E (2) E A D B C (1) D C E A B (1) B E C D A (1) B C E D A (1) B C A D E (1) A E D B C (1) A D C B E (1) Total count = 100 A B C D E A 0 4 4 2 -4 B -4 0 -6 -4 2 C -4 6 0 0 -2 D -2 4 0 0 -10 E 4 -2 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000004 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 4 4 2 -4 B -4 0 -6 -4 2 C -4 6 0 0 -2 D -2 4 0 0 -10 E 4 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999981 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=22 D=21 B=12 C=9 so C is eliminated. Round 2 votes counts: E=36 D=27 A=22 B=15 so B is eliminated. Round 3 votes counts: E=38 D=33 A=29 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 A:203 C:200 D:196 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 2 -4 B -4 0 -6 -4 2 C -4 6 0 0 -2 D -2 4 0 0 -10 E 4 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999981 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 -4 B -4 0 -6 -4 2 C -4 6 0 0 -2 D -2 4 0 0 -10 E 4 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999981 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 -4 B -4 0 -6 -4 2 C -4 6 0 0 -2 D -2 4 0 0 -10 E 4 -2 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999981 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8404: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) E B A C D (7) D C E A B (6) D C A B E (6) C D A B E (5) E D C B A (4) D E C B A (4) A B C D E (4) E C D B A (3) E B A D C (3) D C A E B (3) D A C B E (3) D A B C E (3) C E B A D (3) C A B E D (3) C A B D E (3) A B D C E (3) E B C A D (2) D E B A C (2) D B A E C (2) C E D B A (2) C D E A B (2) B A E D C (2) B A C E D (2) E D B C A (1) E C B D A (1) E C B A D (1) D E C A B (1) D E B C A (1) D A B E C (1) C E D A B (1) C D A E B (1) C B A E D (1) B A D E C (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 -4 12 B 4 0 -8 -2 4 C 8 8 0 8 6 D 4 2 -8 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -4 12 B 4 0 -8 -2 4 C 8 8 0 8 6 D 4 2 -8 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=22 C=21 B=15 A=10 so A is eliminated. Round 2 votes counts: D=32 B=24 E=22 C=22 so E is eliminated. Round 3 votes counts: D=37 B=36 C=27 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:201 B:199 A:198 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 12 B 4 0 -8 -2 4 C 8 8 0 8 6 D 4 2 -8 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 12 B 4 0 -8 -2 4 C 8 8 0 8 6 D 4 2 -8 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 12 B 4 0 -8 -2 4 C 8 8 0 8 6 D 4 2 -8 0 4 E -12 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8405: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) B A C E D (8) D E B C A (6) C A E D B (6) E D A C B (5) D C E A B (5) C A B E D (5) D E C B A (4) B D E C A (4) E D C A B (3) D E B A C (3) C B A D E (3) B C A D E (3) A C B E D (3) A B C E D (3) E D B A C (2) C A D E B (2) B E D A C (2) B D E A C (2) B A C D E (2) A C E D B (2) E D A B C (1) E B D A C (1) D C E B A (1) D B E C A (1) C E A D B (1) C D E A B (1) C A E B D (1) C A D B E (1) C A B D E (1) B D C E A (1) B D C A E (1) B D A C E (1) B A E C D (1) A E C D B (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -22 -10 -8 B -8 0 -14 -12 -14 C 22 14 0 -8 4 D 10 12 8 0 6 E 8 14 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -22 -10 -8 B -8 0 -14 -12 -14 C 22 14 0 -8 4 D 10 12 8 0 6 E 8 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=25 C=21 E=12 A=11 so A is eliminated. Round 2 votes counts: D=31 B=29 C=27 E=13 so E is eliminated. Round 3 votes counts: D=42 B=30 C=28 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:216 E:206 A:184 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -22 -10 -8 B -8 0 -14 -12 -14 C 22 14 0 -8 4 D 10 12 8 0 6 E 8 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -22 -10 -8 B -8 0 -14 -12 -14 C 22 14 0 -8 4 D 10 12 8 0 6 E 8 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -22 -10 -8 B -8 0 -14 -12 -14 C 22 14 0 -8 4 D 10 12 8 0 6 E 8 14 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8406: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E C A B D (12) D B A C E (12) E C B A D (8) A B D C E (6) D A B E C (5) E C A D B (4) B A D C E (4) B A C E D (4) B A C D E (4) A B C E D (4) E C D B A (3) D A E B C (3) E C D A B (2) D E C A B (2) B C E D A (2) A C E B D (2) E D C A B (1) E C B D A (1) D E A C B (1) D B C A E (1) D B A E C (1) C B A E D (1) B D A C E (1) B C E A D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 24 4 26 B -14 0 24 4 22 C -24 -24 0 0 14 D -4 -4 0 0 8 E -26 -22 -14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 24 4 26 B -14 0 24 4 22 C -24 -24 0 0 14 D -4 -4 0 0 8 E -26 -22 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990495 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=31 B=16 A=14 C=1 so C is eliminated. Round 2 votes counts: D=38 E=31 B=17 A=14 so A is eliminated. Round 3 votes counts: D=38 E=33 B=29 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:234 B:218 D:200 C:183 E:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 24 4 26 B -14 0 24 4 22 C -24 -24 0 0 14 D -4 -4 0 0 8 E -26 -22 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990495 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 24 4 26 B -14 0 24 4 22 C -24 -24 0 0 14 D -4 -4 0 0 8 E -26 -22 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990495 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 24 4 26 B -14 0 24 4 22 C -24 -24 0 0 14 D -4 -4 0 0 8 E -26 -22 -14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990495 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8407: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) C E D B A (7) B A D E C (7) B A D C E (7) E C D A B (6) C E D A B (6) D C E A B (5) B A E C D (5) A B E C D (5) D A B C E (4) C E B A D (3) B D A C E (3) A B E D C (3) E C A D B (2) E A C D B (2) C D E A B (2) C B E D A (2) B A C D E (2) A D B E C (2) A B D E C (2) E C B A D (1) E C A B D (1) E A D C B (1) E A C B D (1) E A B C D (1) D E C A B (1) D C B E A (1) D B C A E (1) D A B E C (1) B D C A E (1) B C D E A (1) B C D A E (1) B C A D E (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 0 0 -2 B 2 0 0 0 -2 C 0 0 0 2 10 D 0 0 -2 0 2 E 2 2 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.587038 C: 0.412962 D: 0.000000 E: 0.000000 Sum of squares = 0.515151151602 Cumulative probabilities = A: 0.000000 B: 0.587038 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 0 -2 B 2 0 0 0 -2 C 0 0 0 2 10 D 0 0 -2 0 2 E 2 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=21 C=20 A=16 E=15 so E is eliminated. Round 2 votes counts: C=30 B=28 D=21 A=21 so D is eliminated. Round 3 votes counts: C=45 B=29 A=26 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:206 B:200 D:200 A:198 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 0 -2 B 2 0 0 0 -2 C 0 0 0 2 10 D 0 0 -2 0 2 E 2 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 -2 B 2 0 0 0 -2 C 0 0 0 2 10 D 0 0 -2 0 2 E 2 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 -2 B 2 0 0 0 -2 C 0 0 0 2 10 D 0 0 -2 0 2 E 2 2 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8408: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) D C E A B (5) D B E C A (5) C D A E B (5) C B D E A (4) B E D A C (4) B E A D C (4) E D A B C (3) E B A D C (3) C D B A E (3) C D A B E (3) B C D E A (3) A E D C B (3) E D B A C (2) E A D B C (2) D E B A C (2) D C E B A (2) D C B E A (2) C D E A B (2) C D B E A (2) C B A E D (2) C A D E B (2) B C A E D (2) B A E D C (2) A E D B C (2) A C E B D (2) A C B E D (2) A B E C D (2) E A B D C (1) D E B C A (1) D E A B C (1) C D E B A (1) C B A D E (1) C A B E D (1) B D E C A (1) B A E C D (1) A E B C D (1) A C E D B (1) A C D E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 6 6 -2 0 B -6 0 12 0 -8 C -6 -12 0 -16 -6 D 2 0 16 0 -8 E 0 8 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.308960 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.691040 Sum of squares = 0.572992815978 Cumulative probabilities = A: 0.308960 B: 0.308960 C: 0.308960 D: 0.308960 E: 1.000000 A B C D E A 0 6 6 -2 0 B -6 0 12 0 -8 C -6 -12 0 -16 -6 D 2 0 16 0 -8 E 0 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=26 D=18 B=17 E=11 so E is eliminated. Round 2 votes counts: A=31 C=26 D=23 B=20 so B is eliminated. Round 3 votes counts: A=41 C=31 D=28 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:211 A:205 D:205 B:199 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -2 0 B -6 0 12 0 -8 C -6 -12 0 -16 -6 D 2 0 16 0 -8 E 0 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -2 0 B -6 0 12 0 -8 C -6 -12 0 -16 -6 D 2 0 16 0 -8 E 0 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -2 0 B -6 0 12 0 -8 C -6 -12 0 -16 -6 D 2 0 16 0 -8 E 0 8 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8409: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) D C B A E (9) E A B C D (7) D B C A E (7) C D A B E (7) C A E D B (4) E C D A B (3) E C A D B (3) E B A C D (3) C E A D B (3) B D A E C (3) B D A C E (3) E B D A C (2) D C A B E (2) C D A E B (2) C A E B D (2) C A D E B (2) B A E C D (2) B A D C E (2) E D C B A (1) E D B C A (1) E C A B D (1) E B A D C (1) D C E B A (1) D C B E A (1) D B E C A (1) D B E A C (1) D B A C E (1) C E D A B (1) C D E A B (1) C A D B E (1) C A B E D (1) C A B D E (1) B E A D C (1) B E A C D (1) B D E A C (1) A E C B D (1) A E B C D (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 -10 4 14 B -16 0 -18 -4 0 C 10 18 0 24 12 D -4 4 -24 0 -4 E -14 0 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -10 4 14 B -16 0 -18 -4 0 C 10 18 0 24 12 D -4 4 -24 0 -4 E -14 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=25 D=23 B=13 A=8 so A is eliminated. Round 2 votes counts: E=33 C=28 D=23 B=16 so B is eliminated. Round 3 votes counts: E=38 D=32 C=30 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:232 A:212 E:189 D:186 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -10 4 14 B -16 0 -18 -4 0 C 10 18 0 24 12 D -4 4 -24 0 -4 E -14 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -10 4 14 B -16 0 -18 -4 0 C 10 18 0 24 12 D -4 4 -24 0 -4 E -14 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -10 4 14 B -16 0 -18 -4 0 C 10 18 0 24 12 D -4 4 -24 0 -4 E -14 0 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8410: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (9) B E D A C (8) E D B C A (7) C A D E B (6) A C B E D (6) D E B C A (5) C D E A B (5) E B D A C (4) D E C B A (4) C A D B E (4) B E A D C (4) B A E D C (4) D E C A B (3) C D A E B (3) B D E A C (3) A B C D E (3) E D C A B (2) D C E B A (2) C D A B E (2) B A E C D (2) B A C D E (2) A C D B E (2) E B A D C (1) D C E A B (1) D B E C A (1) C A E D B (1) B D A E C (1) B A D E C (1) A E B C D (1) A C E D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 8 -2 0 B -2 0 -4 2 6 C -8 4 0 -2 -2 D 2 -2 2 0 14 E 0 -6 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.33333333328 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -2 0 B -2 0 -4 2 6 C -8 4 0 -2 -2 D 2 -2 2 0 14 E 0 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=24 C=21 D=16 E=14 so E is eliminated. Round 2 votes counts: B=30 D=25 A=24 C=21 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:208 A:204 B:201 C:196 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 -2 0 B -2 0 -4 2 6 C -8 4 0 -2 -2 D 2 -2 2 0 14 E 0 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -2 0 B -2 0 -4 2 6 C -8 4 0 -2 -2 D 2 -2 2 0 14 E 0 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -2 0 B -2 0 -4 2 6 C -8 4 0 -2 -2 D 2 -2 2 0 14 E 0 -6 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8411: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) E D A C B (6) D E A B C (6) B D E A C (6) B C A D E (6) D A E C B (4) B E C D A (4) B A D C E (4) B A C D E (4) E D B A C (3) E B D C A (3) D E A C B (3) C A E D B (3) C A B D E (3) A B C D E (3) E D C A B (2) E C D A B (2) D A E B C (2) C E B A D (2) C B A D E (2) C A D E B (2) B E D C A (2) B D A C E (2) B C A E D (2) A D B C E (2) A C D E B (2) A B D C E (2) E D C B A (1) E D A B C (1) D B E A C (1) D B A E C (1) C E A D B (1) C B E A D (1) C A E B D (1) B E D A C (1) B D A E C (1) B C E D A (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 12 -16 -4 B 6 0 26 0 -2 C -12 -26 0 -20 -10 D 16 0 20 0 16 E 4 2 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.410240 C: 0.000000 D: 0.589760 E: 0.000000 Sum of squares = 0.516113672667 Cumulative probabilities = A: 0.000000 B: 0.410240 C: 0.410240 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 -16 -4 B 6 0 26 0 -2 C -12 -26 0 -20 -10 D 16 0 20 0 16 E 4 2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 E=24 D=17 C=15 A=11 so A is eliminated. Round 2 votes counts: B=38 E=24 D=20 C=18 so C is eliminated. Round 3 votes counts: B=44 E=31 D=25 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:226 B:215 E:200 A:193 C:166 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 12 -16 -4 B 6 0 26 0 -2 C -12 -26 0 -20 -10 D 16 0 20 0 16 E 4 2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 -16 -4 B 6 0 26 0 -2 C -12 -26 0 -20 -10 D 16 0 20 0 16 E 4 2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 -16 -4 B 6 0 26 0 -2 C -12 -26 0 -20 -10 D 16 0 20 0 16 E 4 2 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8412: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) D B C E A (7) B D C A E (7) D B E C A (6) D B E A C (6) B D E A C (6) C A E D B (5) C A E B D (5) A E C B D (5) C B D A E (4) C A B E D (4) C A B D E (4) E A D C B (3) E A C D B (3) E A B D C (3) D E B A C (3) E A C B D (2) C D B E A (2) E D A B C (1) E B D A C (1) E B A D C (1) E A D B C (1) E A B C D (1) D C B A E (1) C D B A E (1) C D A B E (1) C B A D E (1) C A D B E (1) B C D A E (1) B C A D E (1) B A D E C (1) A C E D B (1) A C E B D (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -2 -10 -8 B 14 0 10 0 12 C 2 -10 0 -10 -4 D 10 0 10 0 8 E 8 -12 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.407617 C: 0.000000 D: 0.592383 E: 0.000000 Sum of squares = 0.517069328391 Cumulative probabilities = A: 0.000000 B: 0.407617 C: 0.407617 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -10 -8 B 14 0 10 0 12 C 2 -10 0 -10 -4 D 10 0 10 0 8 E 8 -12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=24 D=23 B=16 A=9 so A is eliminated. Round 2 votes counts: C=31 E=29 D=23 B=17 so B is eliminated. Round 3 votes counts: D=37 C=33 E=30 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:218 D:214 E:196 C:189 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -10 -8 B 14 0 10 0 12 C 2 -10 0 -10 -4 D 10 0 10 0 8 E 8 -12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -10 -8 B 14 0 10 0 12 C 2 -10 0 -10 -4 D 10 0 10 0 8 E 8 -12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -10 -8 B 14 0 10 0 12 C 2 -10 0 -10 -4 D 10 0 10 0 8 E 8 -12 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8413: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (10) A E D C B (10) B C D E A (9) B E A D C (8) A E B D C (6) B A E D C (5) E D A C B (4) B E D A C (4) B C D A E (4) B A C E D (4) E A D C B (3) A D E C B (3) E B D A C (2) D C E A B (2) C D E A B (2) C D B A E (2) B C A D E (2) B A E C D (2) A C D E B (2) A B C E D (2) E A D B C (1) D E C A B (1) D E B C A (1) D A E C B (1) D A C E B (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D C A (1) B E A C D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 22 8 20 B -2 0 8 12 0 C -22 -8 0 -6 -6 D -8 -12 6 0 -6 E -20 0 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 22 8 20 B -2 0 8 12 0 C -22 -8 0 -6 -6 D -8 -12 6 0 -6 E -20 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 A=24 C=19 E=10 D=6 so D is eliminated. Round 2 votes counts: B=41 A=26 C=21 E=12 so E is eliminated. Round 3 votes counts: B=44 A=34 C=22 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:209 E:196 D:190 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 22 8 20 B -2 0 8 12 0 C -22 -8 0 -6 -6 D -8 -12 6 0 -6 E -20 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 22 8 20 B -2 0 8 12 0 C -22 -8 0 -6 -6 D -8 -12 6 0 -6 E -20 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 22 8 20 B -2 0 8 12 0 C -22 -8 0 -6 -6 D -8 -12 6 0 -6 E -20 0 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976759 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8414: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) E D A C B (5) C B D A E (5) B C E D A (5) B C E A D (5) B C A E D (5) E D A B C (4) D E A C B (4) D A E C B (4) A E D B C (4) E B D C A (3) D A C E B (3) B C D E A (3) B C A D E (3) A E D C B (3) E D B A C (2) E B D A C (2) D E C A B (2) D C E A B (2) C A D E B (2) C A B D E (2) B E D C A (2) B E C D A (2) A D E C B (2) A C D B E (2) E D B C A (1) D E C B A (1) D C A E B (1) C D B A E (1) C D A E B (1) C B D E A (1) C B A D E (1) C A D B E (1) B E A C D (1) B A C E D (1) A D C E B (1) A C D E B (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 2 -8 -4 B -12 0 8 -8 -10 C -2 -8 0 -8 -2 D 8 8 8 0 -10 E 4 10 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 -8 -4 B -12 0 8 -8 -10 C -2 -8 0 -8 -2 D 8 8 8 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=24 A=18 D=17 C=14 so C is eliminated. Round 2 votes counts: B=34 E=24 A=23 D=19 so D is eliminated. Round 3 votes counts: B=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:207 A:201 C:190 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 -8 -4 B -12 0 8 -8 -10 C -2 -8 0 -8 -2 D 8 8 8 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -8 -4 B -12 0 8 -8 -10 C -2 -8 0 -8 -2 D 8 8 8 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -8 -4 B -12 0 8 -8 -10 C -2 -8 0 -8 -2 D 8 8 8 0 -10 E 4 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8415: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) E A C B D (10) D B C A E (8) D B A C E (8) B D A C E (8) A B D E C (7) C D B A E (5) C E D A B (4) C E A D B (4) C D E B A (4) E C A D B (3) D C B A E (3) E C D B A (2) E C D A B (2) E A B D C (2) E A B C D (2) D B E A C (2) C E A B D (2) A B E D C (2) A B D C E (2) D C B E A (1) D B A E C (1) C E D B A (1) C D B E A (1) B D A E C (1) B A D C E (1) A E C B D (1) A E B C D (1) A C E B D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 0 -2 0 B -8 0 -8 2 2 C 0 8 0 8 6 D 2 -2 -8 0 4 E 0 -2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.330932 B: 0.000000 C: 0.669068 D: 0.000000 E: 0.000000 Sum of squares = 0.557167782122 Cumulative probabilities = A: 0.330932 B: 0.330932 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -2 0 B -8 0 -8 2 2 C 0 8 0 8 6 D 2 -2 -8 0 4 E 0 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=23 C=21 A=15 B=10 so B is eliminated. Round 2 votes counts: D=32 E=31 C=21 A=16 so A is eliminated. Round 3 votes counts: D=42 E=36 C=22 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:211 A:203 D:198 B:194 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 0 -2 0 B -8 0 -8 2 2 C 0 8 0 8 6 D 2 -2 -8 0 4 E 0 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -2 0 B -8 0 -8 2 2 C 0 8 0 8 6 D 2 -2 -8 0 4 E 0 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -2 0 B -8 0 -8 2 2 C 0 8 0 8 6 D 2 -2 -8 0 4 E 0 -2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8416: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (12) C E B A D (7) E B C A D (5) C A B E D (5) A C D B E (5) D E B C A (4) D E B A C (4) D A C B E (4) B E A C D (4) A D C B E (4) E B D C A (3) D C E B A (3) D C A E B (3) D B E A C (3) C E B D A (3) C A E B D (3) B E C A D (3) E B D A C (2) E B C D A (2) C A D E B (2) B E D A C (2) B E A D C (2) A D B E C (2) E C B D A (1) E C B A D (1) E B A C D (1) D B A E C (1) D A C E B (1) C E A B D (1) C D E B A (1) C D A E B (1) C B E A D (1) C A D B E (1) B D E A C (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 0 -2 -8 B 8 0 4 -2 4 C 0 -4 0 -4 -8 D 2 2 4 0 4 E 8 -4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -2 -8 B 8 0 4 -2 4 C 0 -4 0 -4 -8 D 2 2 4 0 4 E 8 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=25 E=15 A=13 B=12 so B is eliminated. Round 2 votes counts: D=36 E=26 C=25 A=13 so A is eliminated. Round 3 votes counts: D=42 C=31 E=27 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:207 D:206 E:204 C:192 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -2 -8 B 8 0 4 -2 4 C 0 -4 0 -4 -8 D 2 2 4 0 4 E 8 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -2 -8 B 8 0 4 -2 4 C 0 -4 0 -4 -8 D 2 2 4 0 4 E 8 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -2 -8 B 8 0 4 -2 4 C 0 -4 0 -4 -8 D 2 2 4 0 4 E 8 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8417: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) E A C D B (7) C D A B E (6) B C D A E (6) E B A D C (5) E A D C B (5) C D B A E (5) B C D E A (5) E A B D C (4) C B D A E (4) E B C D A (3) E A D B C (3) D A C B E (3) C A E D B (3) B E D A C (3) B C E D A (3) A E D C B (3) D C A B E (2) D A B C E (2) C A D E B (2) A E D B C (2) E C B D A (1) E C A D B (1) E C A B D (1) D B C A E (1) D B A C E (1) C D A E B (1) C B E D A (1) C B D E A (1) C A D B E (1) B E C D A (1) B D A E C (1) B D A C E (1) A D E B C (1) A D C E B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -12 -18 12 B 0 0 4 -4 14 C 12 -4 0 4 16 D 18 4 -4 0 8 E -12 -14 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 -18 12 B 0 0 4 -4 14 C 12 -4 0 4 16 D 18 4 -4 0 8 E -12 -14 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=28 C=24 D=9 A=9 so D is eliminated. Round 2 votes counts: E=30 B=30 C=26 A=14 so A is eliminated. Round 3 votes counts: E=36 B=34 C=30 so C is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:214 D:213 B:207 A:191 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -12 -18 12 B 0 0 4 -4 14 C 12 -4 0 4 16 D 18 4 -4 0 8 E -12 -14 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 -18 12 B 0 0 4 -4 14 C 12 -4 0 4 16 D 18 4 -4 0 8 E -12 -14 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 -18 12 B 0 0 4 -4 14 C 12 -4 0 4 16 D 18 4 -4 0 8 E -12 -14 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8418: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) B A E D C (6) A D B C E (6) D A B C E (5) C E D A B (5) D A C B E (4) B D A E C (4) A B D E C (4) E B A C D (3) D C E A B (3) C E D B A (3) C D A E B (3) A B E C D (3) E D C B A (2) E C B A D (2) E B C A D (2) D C E B A (2) D C A B E (2) D B A E C (2) C E A D B (2) C D E A B (2) C A D B E (2) B A E C D (2) A D C B E (2) A C B D E (2) A B D C E (2) A B C D E (2) E D B C A (1) E C D B A (1) E C B D A (1) E C A B D (1) E B D C A (1) E B C D A (1) E A B C D (1) D E C B A (1) D B E A C (1) D B A C E (1) C E A B D (1) B E A D C (1) B E A C D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 8 24 10 24 B -8 0 18 0 22 C -24 -18 0 -18 -2 D -10 0 18 0 16 E -24 -22 2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 24 10 24 B -8 0 18 0 22 C -24 -18 0 -18 -2 D -10 0 18 0 16 E -24 -22 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 B=22 D=21 C=18 E=16 so E is eliminated. Round 2 votes counts: B=29 D=24 A=24 C=23 so C is eliminated. Round 3 votes counts: D=38 B=32 A=30 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:233 B:216 D:212 E:170 C:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 24 10 24 B -8 0 18 0 22 C -24 -18 0 -18 -2 D -10 0 18 0 16 E -24 -22 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 24 10 24 B -8 0 18 0 22 C -24 -18 0 -18 -2 D -10 0 18 0 16 E -24 -22 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 24 10 24 B -8 0 18 0 22 C -24 -18 0 -18 -2 D -10 0 18 0 16 E -24 -22 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8419: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) D B C E A (7) A C E B D (7) B D C E A (6) E A C D B (5) A B E D C (5) E D A B C (4) E A D B C (4) C D E B A (4) A E C B D (4) D C B E A (3) B D C A E (3) D E C B A (2) C B D A E (2) C A B D E (2) B D E A C (2) B D A E C (2) B C D A E (2) B A D C E (2) A E B C D (2) E D C A B (1) E D B A C (1) E A D C B (1) E A B D C (1) D E B C A (1) D E B A C (1) D B E C A (1) D B E A C (1) C E A D B (1) C D B E A (1) C D B A E (1) C A E D B (1) C A E B D (1) B D A C E (1) B C A D E (1) B A D E C (1) B A C D E (1) A E D C B (1) A E B D C (1) A C E D B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 20 8 6 B -8 0 2 -4 -8 C -20 -2 0 -4 -4 D -8 4 4 0 -4 E -6 8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 20 8 6 B -8 0 2 -4 -8 C -20 -2 0 -4 -4 D -8 4 4 0 -4 E -6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=21 E=17 D=16 C=13 so C is eliminated. Round 2 votes counts: A=37 B=23 D=22 E=18 so E is eliminated. Round 3 votes counts: A=49 D=28 B=23 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:205 D:198 B:191 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 20 8 6 B -8 0 2 -4 -8 C -20 -2 0 -4 -4 D -8 4 4 0 -4 E -6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 20 8 6 B -8 0 2 -4 -8 C -20 -2 0 -4 -4 D -8 4 4 0 -4 E -6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 20 8 6 B -8 0 2 -4 -8 C -20 -2 0 -4 -4 D -8 4 4 0 -4 E -6 8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8420: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) D C E B A (7) E A D B C (6) C D A E B (6) B C D A E (6) B A C E D (6) D C E A B (5) B C A D E (5) B A E C D (5) A B E C D (5) E A D C B (3) D C B E A (3) C A D B E (3) E A B D C (2) D E C A B (2) C D E A B (2) C D B A E (2) C A E D B (2) B E A D C (2) B D C A E (2) A B C E D (2) E D B A C (1) E D A C B (1) E D A B C (1) E A C D B (1) D E C B A (1) D E B A C (1) D B C E A (1) C D A B E (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D A C (1) B D E A C (1) B A C D E (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -2 12 18 B -8 0 14 0 -2 C 2 -14 0 18 16 D -12 0 -18 0 4 E -18 2 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.583333 B: 0.083333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.458333333604 Cumulative probabilities = A: 0.583333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 12 18 B -8 0 14 0 -2 C 2 -14 0 18 16 D -12 0 -18 0 4 E -18 2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.083333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.458333333767 Cumulative probabilities = A: 0.583333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=20 C=19 A=17 E=15 so E is eliminated. Round 2 votes counts: B=29 A=29 D=23 C=19 so C is eliminated. Round 3 votes counts: A=36 D=34 B=30 so B is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:211 B:202 D:187 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 12 18 B -8 0 14 0 -2 C 2 -14 0 18 16 D -12 0 -18 0 4 E -18 2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.083333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.458333333767 Cumulative probabilities = A: 0.583333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 12 18 B -8 0 14 0 -2 C 2 -14 0 18 16 D -12 0 -18 0 4 E -18 2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.083333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.458333333767 Cumulative probabilities = A: 0.583333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 12 18 B -8 0 14 0 -2 C 2 -14 0 18 16 D -12 0 -18 0 4 E -18 2 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.583333 B: 0.083333 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.458333333767 Cumulative probabilities = A: 0.583333 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8421: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) D C B A E (9) D C A B E (5) C D B A E (5) B C D A E (5) A E D C B (5) E B A C D (4) E A B D C (4) E A B C D (4) D C A E B (4) B D C E A (4) D E C B A (3) A E C D B (3) E D A C B (2) E A D B C (2) D A C E B (2) D A C B E (2) B E C D A (2) B C E D A (2) B C D E A (2) A E C B D (2) A E B C D (2) A B C E D (2) E D A B C (1) E B A D C (1) D E C A B (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C A E (1) D A E C B (1) C D A B E (1) B C A D E (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 18 4 -10 8 B -18 0 -26 -30 -14 C -4 26 0 -24 4 D 10 30 24 0 4 E -8 14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998787 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 -10 8 B -18 0 -26 -30 -14 C -4 26 0 -24 4 D 10 30 24 0 4 E -8 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=29 A=18 B=16 C=6 so C is eliminated. Round 2 votes counts: D=37 E=29 A=18 B=16 so B is eliminated. Round 3 votes counts: D=48 E=33 A=19 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:234 A:210 C:201 E:199 B:156 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 4 -10 8 B -18 0 -26 -30 -14 C -4 26 0 -24 4 D 10 30 24 0 4 E -8 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 -10 8 B -18 0 -26 -30 -14 C -4 26 0 -24 4 D 10 30 24 0 4 E -8 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 -10 8 B -18 0 -26 -30 -14 C -4 26 0 -24 4 D 10 30 24 0 4 E -8 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8422: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) D E C B A (6) D E B C A (5) C E D A B (5) A B E C D (5) E C A B D (4) C D E B A (4) B A D C E (4) A B C D E (4) E D C B A (3) E C D A B (3) D B C E A (3) C D E A B (3) B A E D C (3) A C E B D (3) A B E D C (3) E C A D B (2) E A B C D (2) D C B A E (2) C E A D B (2) B D A E C (2) A E B C D (2) A C B E D (2) E D B C A (1) E D B A C (1) E C D B A (1) D C E B A (1) D C B E A (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A E C (1) C D A B E (1) C A E B D (1) C A D E B (1) C A B D E (1) B D E A C (1) B D C A E (1) B D A C E (1) A E B D C (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 2 0 B 4 0 6 2 -2 C 6 -6 0 -4 -12 D -2 -2 4 0 10 E 0 2 12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.55102040821 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 A B C D E A 0 -4 -6 2 0 B 4 0 6 2 -2 C 6 -6 0 -4 -12 D -2 -2 4 0 10 E 0 2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408177 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=22 B=20 C=18 E=17 so E is eliminated. Round 2 votes counts: C=28 D=27 A=25 B=20 so B is eliminated. Round 3 votes counts: A=40 D=32 C=28 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:205 D:205 E:202 A:196 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 2 0 B 4 0 6 2 -2 C 6 -6 0 -4 -12 D -2 -2 4 0 10 E 0 2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408177 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 0 B 4 0 6 2 -2 C 6 -6 0 -4 -12 D -2 -2 4 0 10 E 0 2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408177 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 0 B 4 0 6 2 -2 C 6 -6 0 -4 -12 D -2 -2 4 0 10 E 0 2 12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408177 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8423: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) A C E B D (8) A C D E B (7) E B D A C (6) D C B E A (6) B E D C A (6) B E D A C (6) C D A B E (5) E B A D C (4) E B A C D (4) D B E C A (4) C A D E B (4) A E B C D (4) D C B A E (3) D B C E A (3) C A D B E (3) B D E C A (2) A E C B D (2) A E B D C (2) E B C D A (1) E B C A D (1) D B E A C (1) D A B E C (1) C E B A D (1) C D B E A (1) C B E D A (1) C A E B D (1) B E C D A (1) A E D B C (1) A D E B C (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -2 -10 4 B 2 0 -4 0 4 C 2 4 0 -8 6 D 10 0 8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.458407 C: 0.000000 D: 0.541593 E: 0.000000 Sum of squares = 0.5034599675 Cumulative probabilities = A: 0.000000 B: 0.458407 C: 0.458407 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -10 4 B 2 0 -4 0 4 C 2 4 0 -8 6 D 10 0 8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 E=16 C=16 B=15 so B is eliminated. Round 2 votes counts: E=29 D=28 A=27 C=16 so C is eliminated. Round 3 votes counts: A=35 D=34 E=31 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:209 C:202 B:201 A:195 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -10 4 B 2 0 -4 0 4 C 2 4 0 -8 6 D 10 0 8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -10 4 B 2 0 -4 0 4 C 2 4 0 -8 6 D 10 0 8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -10 4 B 2 0 -4 0 4 C 2 4 0 -8 6 D 10 0 8 0 0 E -4 -4 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8424: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (19) B E A D C (15) D C E A B (11) B A E C D (9) A B E C D (5) D C B E A (4) B D C E A (4) A E B C D (4) C A D E B (3) B E A C D (3) B D E C A (3) B D E A C (3) A C E D B (3) A E C D B (2) A E C B D (2) E A B D C (1) D E C B A (1) D C B A E (1) D C A E B (1) D B E C A (1) D B C E A (1) C D E A B (1) B E D A C (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -2 -4 2 B -6 0 2 4 2 C 2 -2 0 6 0 D 4 -4 -6 0 8 E -2 -2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.44 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -4 2 B -6 0 2 4 2 C 2 -2 0 6 0 D 4 -4 -6 0 8 E -2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=23 D=20 A=17 E=1 so E is eliminated. Round 2 votes counts: B=39 C=23 D=20 A=18 so A is eliminated. Round 3 votes counts: B=49 C=31 D=20 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:203 A:201 B:201 D:201 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 -4 2 B -6 0 2 4 2 C 2 -2 0 6 0 D 4 -4 -6 0 8 E -2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -4 2 B -6 0 2 4 2 C 2 -2 0 6 0 D 4 -4 -6 0 8 E -2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -4 2 B -6 0 2 4 2 C 2 -2 0 6 0 D 4 -4 -6 0 8 E -2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999994 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8425: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (12) D B C E A (9) C E B A D (7) A D B E C (7) C E B D A (5) C E A B D (5) A D C E B (5) D B C A E (4) D A B C E (4) E C A B D (3) D A B E C (3) A C E D B (3) D B E C A (2) D B A C E (2) D A C B E (2) C B D E A (2) B D E C A (2) A E B C D (2) A C E B D (2) E C B A D (1) E B C A D (1) D C B E A (1) D B E A C (1) D B A E C (1) C E D A B (1) C D A E B (1) C B E D A (1) C A E D B (1) C A E B D (1) B E C D A (1) B E C A D (1) B D C E A (1) B C E D A (1) A E D C B (1) A E B D C (1) A D E B C (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 14 -2 12 10 B -14 0 -10 0 -8 C 2 10 0 4 18 D -12 0 -4 0 -2 E -10 8 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 12 10 B -14 0 -10 0 -8 C 2 10 0 4 18 D -12 0 -4 0 -2 E -10 8 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=29 C=24 B=6 E=5 so E is eliminated. Round 2 votes counts: A=36 D=29 C=28 B=7 so B is eliminated. Round 3 votes counts: A=36 D=32 C=32 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:217 C:217 D:191 E:191 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 12 10 B -14 0 -10 0 -8 C 2 10 0 4 18 D -12 0 -4 0 -2 E -10 8 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 12 10 B -14 0 -10 0 -8 C 2 10 0 4 18 D -12 0 -4 0 -2 E -10 8 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 12 10 B -14 0 -10 0 -8 C 2 10 0 4 18 D -12 0 -4 0 -2 E -10 8 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8426: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) E D A B C (4) E A C B D (4) E A B D C (4) D E C B A (4) D C E B A (4) A B E C D (4) E D B A C (3) E C A D B (3) E A B C D (3) C D E A B (3) B D A C E (3) B A D C E (3) A E B C D (3) A C E B D (3) A B C E D (3) E D C A B (2) E A C D B (2) D C B E A (2) D B E C A (2) C D E B A (2) C B A D E (2) C A B D E (2) B C A D E (2) B A E D C (2) B A C D E (2) A B E D C (2) A B C D E (2) E D C B A (1) E D A C B (1) E A D C B (1) E A D B C (1) D E B C A (1) D E B A C (1) D C B A E (1) D B E A C (1) D B C E A (1) D B C A E (1) C E A D B (1) C D B E A (1) C B D A E (1) B E A D C (1) B D A E C (1) B C D A E (1) B A D E C (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 12 14 6 -18 B -12 0 4 2 -16 C -14 -4 0 4 -18 D -6 -2 -4 0 -12 E 18 16 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 14 6 -18 B -12 0 4 2 -16 C -14 -4 0 4 -18 D -6 -2 -4 0 -12 E 18 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=19 D=18 B=16 C=12 so C is eliminated. Round 2 votes counts: E=36 D=24 A=21 B=19 so B is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:232 A:207 B:189 D:188 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 14 6 -18 B -12 0 4 2 -16 C -14 -4 0 4 -18 D -6 -2 -4 0 -12 E 18 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 6 -18 B -12 0 4 2 -16 C -14 -4 0 4 -18 D -6 -2 -4 0 -12 E 18 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 6 -18 B -12 0 4 2 -16 C -14 -4 0 4 -18 D -6 -2 -4 0 -12 E 18 16 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8427: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) D C E B A (8) C D B E A (8) A B E D C (8) A B E C D (8) C D E B A (6) A E B D C (6) C A B E D (4) E D B A C (3) D C E A B (3) C D A B E (3) C B D A E (3) E D A B C (2) D E C A B (2) D E B A C (2) C B A E D (2) C B A D E (2) C A B D E (2) B A E C D (2) B A C E D (2) E B A D C (1) E A D B C (1) D E C B A (1) D E B C A (1) D E A C B (1) D E A B C (1) D C A E B (1) C D B A E (1) C D A E B (1) C B D E A (1) B E A D C (1) B E A C D (1) B A E D C (1) A D C E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 4 -2 B -8 0 -2 8 2 C -2 2 0 -4 0 D -4 -8 4 0 -4 E 2 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090177 B: 0.090177 C: 0.229469 D: 0.000000 E: 0.590177 Sum of squares = 0.41722867359 Cumulative probabilities = A: 0.090177 B: 0.180354 C: 0.409823 D: 0.409823 E: 1.000000 A B C D E A 0 8 2 4 -2 B -8 0 -2 8 2 C -2 2 0 -4 0 D -4 -8 4 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.416666666668 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=25 D=20 E=15 B=7 so B is eliminated. Round 2 votes counts: C=33 A=30 D=20 E=17 so E is eliminated. Round 3 votes counts: A=42 C=33 D=25 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:202 B:200 C:198 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 4 -2 B -8 0 -2 8 2 C -2 2 0 -4 0 D -4 -8 4 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.416666666668 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 4 -2 B -8 0 -2 8 2 C -2 2 0 -4 0 D -4 -8 4 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.416666666668 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 4 -2 B -8 0 -2 8 2 C -2 2 0 -4 0 D -4 -8 4 0 -4 E 2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.083333 B: 0.083333 C: 0.250000 D: 0.000000 E: 0.583333 Sum of squares = 0.416666666668 Cumulative probabilities = A: 0.083333 B: 0.166667 C: 0.416667 D: 0.416667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8428: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) B D A E C (8) B D E A C (7) A D C E B (7) B E D C A (5) A C D E B (5) E C A D B (4) E B C D A (4) D A B C E (4) C E A D B (4) C A E D B (4) E C A B D (3) D A B E C (3) C E B A D (3) A D C B E (3) E C B D A (2) E C B A D (2) D A E C B (2) D A E B C (2) C E A B D (2) C B E A D (2) C A D B E (2) B E C A D (2) B D A C E (2) E D B A C (1) E D A C B (1) E C D B A (1) D B E A C (1) D B A C E (1) D A C B E (1) B D C E A (1) B C E A D (1) B C A D E (1) A D E C B (1) Total count = 100 A B C D E A 0 -4 -2 -8 -8 B 4 0 2 6 4 C 2 -2 0 0 -14 D 8 -6 0 0 2 E 8 -4 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999758 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -8 -8 B 4 0 2 6 4 C 2 -2 0 0 -14 D 8 -6 0 0 2 E 8 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=18 C=17 A=16 D=14 so D is eliminated. Round 2 votes counts: B=37 A=28 E=18 C=17 so C is eliminated. Round 3 votes counts: B=39 A=34 E=27 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:208 D:202 C:193 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -8 -8 B 4 0 2 6 4 C 2 -2 0 0 -14 D 8 -6 0 0 2 E 8 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -8 -8 B 4 0 2 6 4 C 2 -2 0 0 -14 D 8 -6 0 0 2 E 8 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -8 -8 B 4 0 2 6 4 C 2 -2 0 0 -14 D 8 -6 0 0 2 E 8 -4 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999425 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8429: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) E D A C B (9) A B D E C (9) E D C A B (6) A D E B C (5) E C D A B (4) D E C A B (4) B C A D E (4) B A D C E (4) B A C D E (4) A B E D C (4) D E A B C (3) D C E B A (3) C D E B A (3) C B E D A (3) B C A E D (3) D E A C B (2) D A E B C (2) D A B E C (2) C B A E D (2) B A D E C (2) B A C E D (2) A D B E C (2) E A D C B (1) D B A C E (1) C E B D A (1) C E B A D (1) C B E A D (1) A E D B C (1) Total count = 100 A B C D E A 0 8 6 -10 -6 B -8 0 -4 -20 -14 C -6 4 0 -20 -12 D 10 20 20 0 0 E 6 14 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.114749 E: 0.885251 Sum of squares = 0.796836941495 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.114749 E: 1.000000 A B C D E A 0 8 6 -10 -6 B -8 0 -4 -20 -14 C -6 4 0 -20 -12 D 10 20 20 0 0 E 6 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 A=21 E=20 B=19 D=17 so D is eliminated. Round 2 votes counts: E=29 C=26 A=25 B=20 so B is eliminated. Round 3 votes counts: A=38 C=33 E=29 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:225 E:216 A:199 C:183 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -10 -6 B -8 0 -4 -20 -14 C -6 4 0 -20 -12 D 10 20 20 0 0 E 6 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -10 -6 B -8 0 -4 -20 -14 C -6 4 0 -20 -12 D 10 20 20 0 0 E 6 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -10 -6 B -8 0 -4 -20 -14 C -6 4 0 -20 -12 D 10 20 20 0 0 E 6 14 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8430: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) A C B D E (8) E B D A C (5) C A D B E (5) A B E C D (5) A B C E D (5) B E D C A (4) A B C D E (4) E D C A B (3) D E C B A (3) D E C A B (3) C D A E B (3) C D A B E (3) C B A D E (3) B C D A E (3) B A E C D (3) E D C B A (2) E D A C B (2) E A B D C (2) D C E B A (2) D C E A B (2) D C A E B (2) C A B D E (2) B A C D E (2) A E B D C (2) A C D B E (2) E D B A C (1) E B D C A (1) D C B E A (1) C D B A E (1) C B D A E (1) C A D E B (1) B D E C A (1) B C D E A (1) B A C E D (1) A E D C B (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 -10 -4 18 B -14 0 -4 6 10 C 10 4 0 10 6 D 4 -6 -10 0 8 E -18 -10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -10 -4 18 B -14 0 -4 6 10 C 10 4 0 10 6 D 4 -6 -10 0 8 E -18 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 C=19 B=15 D=13 so D is eliminated. Round 2 votes counts: E=30 A=29 C=26 B=15 so B is eliminated. Round 3 votes counts: E=35 A=35 C=30 so C is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:209 B:199 D:198 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -10 -4 18 B -14 0 -4 6 10 C 10 4 0 10 6 D 4 -6 -10 0 8 E -18 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -10 -4 18 B -14 0 -4 6 10 C 10 4 0 10 6 D 4 -6 -10 0 8 E -18 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -10 -4 18 B -14 0 -4 6 10 C 10 4 0 10 6 D 4 -6 -10 0 8 E -18 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999741 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8431: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) B C A E D (7) D E A C B (6) D E A B C (6) C B A E D (5) C A B D E (5) A C D B E (5) E D B A C (4) D A E C B (4) B E D A C (4) B C E A D (4) E D B C A (3) C A D E B (3) B E C D A (3) A D C E B (3) A C D E B (3) E B C D A (2) D E B A C (2) C A E D B (2) A D C B E (2) A C B D E (2) E D C B A (1) E D C A B (1) E D A C B (1) E C D A B (1) E B D C A (1) E B D A C (1) D B A E C (1) D A E B C (1) D A C E B (1) D A B E C (1) C B A D E (1) C A E B D (1) C A D B E (1) C A B E D (1) B E C A D (1) B C E D A (1) B A C D E (1) Total count = 100 A B C D E A 0 0 -4 -6 0 B 0 0 2 -4 6 C 4 -2 0 -2 -4 D 6 4 2 0 -4 E 0 -6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775519 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 A B C D E A 0 0 -4 -6 0 B 0 0 2 -4 6 C 4 -2 0 -2 -4 D 6 4 2 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=22 C=19 E=15 A=15 so E is eliminated. Round 2 votes counts: B=33 D=32 C=20 A=15 so A is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:204 B:202 E:201 C:198 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -6 0 B 0 0 2 -4 6 C 4 -2 0 -2 -4 D 6 4 2 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -6 0 B 0 0 2 -4 6 C 4 -2 0 -2 -4 D 6 4 2 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -6 0 B 0 0 2 -4 6 C 4 -2 0 -2 -4 D 6 4 2 0 -4 E 0 -6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775511 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8432: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C D A E B (8) E B A C D (7) E A B C D (4) D C B A E (4) C E A D B (4) B D C E A (4) D A B E C (3) C E A B D (3) B D A E C (3) A B E D C (3) E B C A D (2) D C A B E (2) D B A E C (2) C D E B A (2) C D B E A (2) C D A B E (2) C A D E B (2) B E D A C (2) B D E A C (2) A D C E B (2) A D B E C (2) E C B A D (1) E C A B D (1) E B A D C (1) E A B D C (1) D C B E A (1) D B C E A (1) D B A C E (1) D A C E B (1) D A C B E (1) C B E D A (1) C B D E A (1) C A E D B (1) B E D C A (1) B E A C D (1) B D E C A (1) B C D E A (1) B A E D C (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E C B (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 10 8 -10 B 6 0 14 10 8 C -10 -14 0 -8 -10 D -8 -10 8 0 2 E 10 -8 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 8 -10 B 6 0 14 10 8 C -10 -14 0 -8 -10 D -8 -10 8 0 2 E 10 -8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=26 E=17 D=16 A=14 so A is eliminated. Round 2 votes counts: B=31 C=27 E=21 D=21 so E is eliminated. Round 3 votes counts: B=47 C=31 D=22 so D is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:205 A:201 D:196 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 8 -10 B 6 0 14 10 8 C -10 -14 0 -8 -10 D -8 -10 8 0 2 E 10 -8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 8 -10 B 6 0 14 10 8 C -10 -14 0 -8 -10 D -8 -10 8 0 2 E 10 -8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 8 -10 B 6 0 14 10 8 C -10 -14 0 -8 -10 D -8 -10 8 0 2 E 10 -8 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8433: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) D B E C A (6) A C B E D (6) E D B C A (5) B D C E A (5) B C D A E (5) C B A E D (4) B C D E A (4) A E C B D (4) A C B D E (4) E A D C B (3) D E B A C (3) C A B E D (3) A E C D B (3) A C E B D (3) D E A B C (2) D A B E C (2) C A E B D (2) B D C A E (2) A E D C B (2) A D E B C (2) A C E D B (2) E D C B A (1) E D A C B (1) E D A B C (1) E C D A B (1) E C B D A (1) E B C D A (1) E A C D B (1) D B E A C (1) D B C E A (1) D B A C E (1) D A E B C (1) C E B A D (1) C B E D A (1) C B A D E (1) B D A C E (1) B C E D A (1) A E D B C (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -8 -12 2 B 8 0 12 0 0 C 8 -12 0 -2 -4 D 12 0 2 0 4 E -2 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.236906 C: 0.000000 D: 0.763094 E: 0.000000 Sum of squares = 0.638436605617 Cumulative probabilities = A: 0.000000 B: 0.236906 C: 0.236906 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -12 2 B 8 0 12 0 0 C 8 -12 0 -2 -4 D 12 0 2 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999714 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=26 B=18 E=15 C=12 so C is eliminated. Round 2 votes counts: A=34 D=26 B=24 E=16 so E is eliminated. Round 3 votes counts: A=38 D=35 B=27 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:209 E:199 C:195 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -12 2 B 8 0 12 0 0 C 8 -12 0 -2 -4 D 12 0 2 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999714 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -12 2 B 8 0 12 0 0 C 8 -12 0 -2 -4 D 12 0 2 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999714 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -12 2 B 8 0 12 0 0 C 8 -12 0 -2 -4 D 12 0 2 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999714 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8434: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) E B D C A (6) B A E C D (6) A C B E D (6) A C B D E (6) D E B C A (5) E D B C A (4) E B D A C (4) E B A C D (4) D E C A B (4) D C E A B (4) B E D A C (4) E C A D B (3) D C A E B (3) C A E D B (3) B A C D E (3) E C A B D (2) E A C B D (2) D B E C A (2) D B A C E (2) C D A E B (2) C A D B E (2) B E A C D (2) E C D A B (1) E B C D A (1) E B C A D (1) D B E A C (1) D A C B E (1) D A B C E (1) C E D A B (1) C D E A B (1) B E A D C (1) B D A E C (1) B A D C E (1) B A C E D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 4 -6 B -2 0 0 2 -16 C 4 0 0 10 -8 D -4 -2 -10 0 -4 E 6 16 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -4 4 -6 B -2 0 0 2 -16 C 4 0 0 10 -8 D -4 -2 -10 0 -4 E 6 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 B=19 C=16 A=14 so A is eliminated. Round 2 votes counts: C=29 E=28 D=24 B=19 so B is eliminated. Round 3 votes counts: E=41 C=33 D=26 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:203 A:198 B:192 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 4 -6 B -2 0 0 2 -16 C 4 0 0 10 -8 D -4 -2 -10 0 -4 E 6 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 -6 B -2 0 0 2 -16 C 4 0 0 10 -8 D -4 -2 -10 0 -4 E 6 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 -6 B -2 0 0 2 -16 C 4 0 0 10 -8 D -4 -2 -10 0 -4 E 6 16 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999604 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8435: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) E B C D A (8) B E C D A (6) A D B C E (6) B A D C E (5) E C B D A (4) D C A B E (4) B C D A E (4) A B D C E (4) E C D B A (3) D C A E B (3) C D E B A (3) C D E A B (3) B A D E C (3) A D C B E (3) D A C B E (2) D A B C E (2) C D A E B (2) B E A D C (2) B D A C E (2) B A E D C (2) A B E D C (2) E C A D B (1) E B A C D (1) E A D C B (1) E A C D B (1) E A C B D (1) D A C E B (1) C E D A B (1) C E B D A (1) C D B E A (1) C D A B E (1) B E A C D (1) B D C A E (1) A E B D C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 6 -16 -28 0 B -6 0 0 -4 2 C 16 0 0 8 0 D 28 4 -8 0 4 E 0 -2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.425357 C: 0.574643 D: 0.000000 E: 0.000000 Sum of squares = 0.511143034317 Cumulative probabilities = A: 0.000000 B: 0.425357 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -16 -28 0 B -6 0 0 -4 2 C 16 0 0 8 0 D 28 4 -8 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=26 A=18 D=12 C=12 so D is eliminated. Round 2 votes counts: E=32 B=26 A=23 C=19 so C is eliminated. Round 3 votes counts: E=40 A=33 B=27 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:214 C:212 E:197 B:196 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -16 -28 0 B -6 0 0 -4 2 C 16 0 0 8 0 D 28 4 -8 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -16 -28 0 B -6 0 0 -4 2 C 16 0 0 8 0 D 28 4 -8 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -16 -28 0 B -6 0 0 -4 2 C 16 0 0 8 0 D 28 4 -8 0 4 E 0 -2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8436: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C A E (8) A C B D E (7) E D B C A (6) B D C A E (6) D E B C A (5) D B C E A (5) C A B D E (5) A E C B D (5) E D B A C (4) E A B D C (4) A C E B D (4) C D B A E (3) C B D A E (3) C A E D B (3) C A D B E (3) E C A D B (2) E A C D B (2) C A E B D (2) B D E A C (2) B D A C E (2) E D C B A (1) E B D A C (1) E A D B C (1) D C B E A (1) D B E C A (1) D B E A C (1) C E A D B (1) C D B E A (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -12 0 4 B 0 0 -6 4 -2 C 12 6 0 2 8 D 0 -4 -2 0 6 E -4 2 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 0 4 B 0 0 -6 4 -2 C 12 6 0 2 8 D 0 -4 -2 0 6 E -4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=21 C=21 A=17 B=10 so B is eliminated. Round 2 votes counts: E=31 D=31 C=21 A=17 so A is eliminated. Round 3 votes counts: E=37 C=32 D=31 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:200 B:198 A:196 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 0 4 B 0 0 -6 4 -2 C 12 6 0 2 8 D 0 -4 -2 0 6 E -4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 0 4 B 0 0 -6 4 -2 C 12 6 0 2 8 D 0 -4 -2 0 6 E -4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 0 4 B 0 0 -6 4 -2 C 12 6 0 2 8 D 0 -4 -2 0 6 E -4 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998823 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8437: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) A B D E C (6) D E A C B (4) B A E D C (4) A D E C B (4) A D E B C (4) E D C A B (3) E D A C B (3) E D A B C (3) D E A B C (3) C B A E D (3) C B A D E (3) C A B D E (3) B C E D A (3) B C E A D (3) B C A E D (3) A B D C E (3) E B C D A (2) D A E B C (2) C E D B A (2) C E D A B (2) C D E A B (2) C D A E B (2) C B E A D (2) B E C D A (2) A D B E C (2) A C D B E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B D A (1) E B D C A (1) E B D A C (1) D A E C B (1) C A D E B (1) C A D B E (1) B E D C A (1) B A D E C (1) B A C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -10 -6 -6 B -4 0 -6 12 12 C 10 6 0 4 -2 D 6 -12 -4 0 -10 E 6 -12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.000000 E: 0.300000 Sum of squares = 0.460000000009 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.700000 E: 1.000000 A B C D E A 0 4 -10 -6 -6 B -4 0 -6 12 12 C 10 6 0 4 -2 D 6 -12 -4 0 -10 E 6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.000000 E: 0.300000 Sum of squares = 0.460000000036 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.700000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=22 B=18 E=17 D=10 so D is eliminated. Round 2 votes counts: C=33 A=25 E=24 B=18 so B is eliminated. Round 3 votes counts: C=42 A=31 E=27 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:209 B:207 E:203 A:191 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 -6 -6 B -4 0 -6 12 12 C 10 6 0 4 -2 D 6 -12 -4 0 -10 E 6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.000000 E: 0.300000 Sum of squares = 0.460000000036 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.700000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -6 -6 B -4 0 -6 12 12 C 10 6 0 4 -2 D 6 -12 -4 0 -10 E 6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.000000 E: 0.300000 Sum of squares = 0.460000000036 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.700000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -6 -6 B -4 0 -6 12 12 C 10 6 0 4 -2 D 6 -12 -4 0 -10 E 6 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.000000 E: 0.300000 Sum of squares = 0.460000000036 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 0.700000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8438: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) B A C D E (5) C B A E D (4) C A B E D (4) B E C D A (4) E C B D A (3) E C A D B (3) D E A C B (3) D E A B C (3) D B E A C (3) D A E C B (3) C E A B D (3) C A E B D (3) B D A C E (3) A D C E B (3) A D C B E (3) A D B C E (3) E D B C A (2) E C B A D (2) D B A E C (2) D B A C E (2) D A B C E (2) C E B A D (2) B C E A D (2) B C D A E (2) B C A D E (2) A B C D E (2) E D C A B (1) E D A C B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C A B D (1) E B D C A (1) E A D C B (1) D E B A C (1) C B E A D (1) C A E D B (1) B E D C A (1) B D E C A (1) B D A E C (1) B C E D A (1) B C A E D (1) A D E C B (1) A C E D B (1) A C D E B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 6 2 14 B -6 0 4 0 14 C -6 -4 0 2 6 D -2 0 -2 0 8 E -14 -14 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 2 14 B -6 0 4 0 14 C -6 -4 0 2 6 D -2 0 -2 0 8 E -14 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 B=23 E=18 C=18 A=16 so A is eliminated. Round 2 votes counts: D=35 B=26 C=21 E=18 so E is eliminated. Round 3 votes counts: D=41 C=32 B=27 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:214 B:206 D:202 C:199 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 2 14 B -6 0 4 0 14 C -6 -4 0 2 6 D -2 0 -2 0 8 E -14 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 2 14 B -6 0 4 0 14 C -6 -4 0 2 6 D -2 0 -2 0 8 E -14 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 2 14 B -6 0 4 0 14 C -6 -4 0 2 6 D -2 0 -2 0 8 E -14 -14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998789 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8439: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (15) B E A D C (15) A D C B E (12) C D A E B (9) C D A B E (7) D C A B E (6) D A C B E (5) C D E B A (5) E C B D A (4) E B A C D (4) B A E D C (4) E B C A D (3) C E D B A (3) E B A D C (2) B E A C D (2) C E B D A (1) A D B E C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -6 -10 -8 B 16 0 -4 2 8 C 6 4 0 6 -2 D 10 -2 -6 0 -8 E 8 -8 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 A B C D E A 0 -16 -6 -10 -8 B 16 0 -4 2 8 C 6 4 0 6 -2 D 10 -2 -6 0 -8 E 8 -8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=25 B=21 A=15 D=11 so D is eliminated. Round 2 votes counts: C=31 E=28 B=21 A=20 so A is eliminated. Round 3 votes counts: C=48 E=28 B=24 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:211 C:207 E:205 D:197 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -6 -10 -8 B 16 0 -4 2 8 C 6 4 0 6 -2 D 10 -2 -6 0 -8 E 8 -8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -10 -8 B 16 0 -4 2 8 C 6 4 0 6 -2 D 10 -2 -6 0 -8 E 8 -8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -10 -8 B 16 0 -4 2 8 C 6 4 0 6 -2 D 10 -2 -6 0 -8 E 8 -8 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.142857 C: 0.571429 D: 0.000000 E: 0.285714 Sum of squares = 0.428571428557 Cumulative probabilities = A: 0.000000 B: 0.142857 C: 0.714286 D: 0.714286 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8440: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E A C B (9) B C D A E (9) E A D C B (8) D E A B C (8) C B A E D (8) B D C A E (8) E A C D B (6) D B E A C (6) A E C B D (5) D B C E A (4) C E A B D (3) D B C A E (2) C A E B D (2) A C E B D (2) E A C B D (1) D E C A B (1) D B E C A (1) D B A E C (1) D A E B C (1) C B E A D (1) C A B E D (1) B D C E A (1) B C A D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -4 -2 2 B 4 0 6 6 6 C 4 -6 0 0 4 D 2 -6 0 0 4 E -2 -6 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 2 B 4 0 6 6 6 C 4 -6 0 0 4 D 2 -6 0 0 4 E -2 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=29 E=15 C=15 A=8 so A is eliminated. Round 2 votes counts: D=33 B=29 E=21 C=17 so C is eliminated. Round 3 votes counts: B=39 D=33 E=28 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:201 D:200 A:196 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 2 B 4 0 6 6 6 C 4 -6 0 0 4 D 2 -6 0 0 4 E -2 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 2 B 4 0 6 6 6 C 4 -6 0 0 4 D 2 -6 0 0 4 E -2 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 2 B 4 0 6 6 6 C 4 -6 0 0 4 D 2 -6 0 0 4 E -2 -6 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8441: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) C A B E D (7) A C B E D (7) E D C A B (5) D E C A B (4) C A E D B (4) B A C E D (4) B A C D E (4) A B C E D (4) E C A D B (3) D C E A B (3) D C B E A (3) C D E A B (3) C A E B D (3) C A D E B (3) B D E A C (3) E D A C B (2) E C D A B (2) E A C B D (2) D E B A C (2) D B E C A (2) D B E A C (2) D B C E A (2) C E D A B (2) C E A D B (2) B D A C E (2) B A E D C (2) E B A D C (1) E A B C D (1) C D B A E (1) B E D A C (1) B E A D C (1) B D A E C (1) B C D A E (1) B C A D E (1) B A E C D (1) B A D C E (1) A C E B D (1) Total count = 100 A B C D E A 0 16 -16 4 -6 B -16 0 -14 -4 0 C 16 14 0 12 16 D -4 4 -12 0 -10 E 6 0 -16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -16 4 -6 B -16 0 -14 -4 0 C 16 14 0 12 16 D -4 4 -12 0 -10 E 6 0 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 B=22 E=16 A=12 so A is eliminated. Round 2 votes counts: C=33 B=26 D=25 E=16 so E is eliminated. Round 3 votes counts: C=40 D=32 B=28 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 E:200 A:199 D:189 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -16 4 -6 B -16 0 -14 -4 0 C 16 14 0 12 16 D -4 4 -12 0 -10 E 6 0 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -16 4 -6 B -16 0 -14 -4 0 C 16 14 0 12 16 D -4 4 -12 0 -10 E 6 0 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -16 4 -6 B -16 0 -14 -4 0 C 16 14 0 12 16 D -4 4 -12 0 -10 E 6 0 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8442: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) B D A E C (8) A D C E B (8) C D A E B (7) B E A D C (7) C A D E B (6) B E D A C (5) E B C A D (4) D A C E B (4) E C B A D (3) D A C B E (3) D A B C E (3) C E B A D (3) B D E A C (3) E C A B D (2) E A B C D (2) D B A C E (2) C D B A E (2) B E A C D (2) E B A D C (1) E A C B D (1) D C A B E (1) D B C A E (1) D B A E C (1) D A B E C (1) C E D A B (1) C E A B D (1) C D B E A (1) C B D E A (1) C A E D B (1) B E C D A (1) B D E C A (1) A E D C B (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 8 8 8 2 B -8 0 -12 -10 -12 C -8 12 0 -4 10 D -8 10 4 0 10 E -2 12 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 8 2 B -8 0 -12 -10 -12 C -8 12 0 -4 10 D -8 10 4 0 10 E -2 12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=27 D=16 E=13 A=11 so A is eliminated. Round 2 votes counts: C=33 B=27 D=26 E=14 so E is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 D:208 C:205 E:195 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 8 2 B -8 0 -12 -10 -12 C -8 12 0 -4 10 D -8 10 4 0 10 E -2 12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 8 2 B -8 0 -12 -10 -12 C -8 12 0 -4 10 D -8 10 4 0 10 E -2 12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 8 2 B -8 0 -12 -10 -12 C -8 12 0 -4 10 D -8 10 4 0 10 E -2 12 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996637 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8443: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) A C B E D (12) D C A E B (9) D B E A C (7) E B D C A (5) C D A E B (5) B E A D C (5) C A D E B (4) A B E C D (3) E B C A D (2) D E C B A (2) D C E B A (2) D C E A B (2) C D E A B (2) C A E B D (2) B A E D C (2) B A E C D (2) A D C B E (2) A C D B E (2) A B C E D (2) E D C B A (1) E D B C A (1) E C D B A (1) E C B D A (1) E B D A C (1) E B A C D (1) D B A C E (1) D A C B E (1) D A B E C (1) C E D A B (1) C A E D B (1) B E D A C (1) B E A C D (1) B D E A C (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -6 -16 2 B 0 0 -2 -14 -10 C 6 2 0 -14 -2 D 16 14 14 0 10 E -2 10 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -16 2 B 0 0 -2 -14 -10 C 6 2 0 -14 -2 D 16 14 14 0 10 E -2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=22 C=15 E=13 B=13 so E is eliminated. Round 2 votes counts: D=39 B=22 A=22 C=17 so C is eliminated. Round 3 votes counts: D=48 A=29 B=23 so B is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 E:200 C:196 A:190 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 -16 2 B 0 0 -2 -14 -10 C 6 2 0 -14 -2 D 16 14 14 0 10 E -2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -16 2 B 0 0 -2 -14 -10 C 6 2 0 -14 -2 D 16 14 14 0 10 E -2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -16 2 B 0 0 -2 -14 -10 C 6 2 0 -14 -2 D 16 14 14 0 10 E -2 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8444: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) D B C E A (6) C A D E B (6) A C D E B (6) B E D C A (5) D A B E C (4) A E B C D (4) A C E D B (4) D B E C A (3) D B A E C (3) C D A B E (3) C B E D A (3) C B D E A (3) C A E D B (3) B E D A C (3) A E C B D (3) E B C D A (2) E B C A D (2) D B E A C (2) D A B C E (2) C E A B D (2) C A E B D (2) B E C D A (2) B D E A C (2) A D C E B (2) E C B A D (1) E C A B D (1) E B D A C (1) E A B C D (1) D C B E A (1) D C B A E (1) D C A B E (1) D A C B E (1) C E B D A (1) C E B A D (1) C A D B E (1) B D C E A (1) A E D B C (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -18 -16 -2 B 2 0 4 -2 10 C 18 -4 0 6 2 D 16 2 -6 0 12 E 2 -10 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888894 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -18 -16 -2 B 2 0 4 -2 10 C 18 -4 0 6 2 D 16 2 -6 0 12 E 2 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 A=22 B=21 E=8 so E is eliminated. Round 2 votes counts: C=27 B=26 D=24 A=23 so A is eliminated. Round 3 votes counts: C=42 B=31 D=27 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:212 C:211 B:207 E:189 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -18 -16 -2 B 2 0 4 -2 10 C 18 -4 0 6 2 D 16 2 -6 0 12 E 2 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -18 -16 -2 B 2 0 4 -2 10 C 18 -4 0 6 2 D 16 2 -6 0 12 E 2 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -18 -16 -2 B 2 0 4 -2 10 C 18 -4 0 6 2 D 16 2 -6 0 12 E 2 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.166667 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888916 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8445: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) B A D E C (8) C D E B A (6) C D E A B (5) A B E D C (5) E A B D C (4) D E B A C (4) D C B A E (4) E A B C D (3) D C E B A (3) D B A E C (3) C E D A B (3) B D A E C (3) B A C E D (3) A B E C D (3) E D A B C (2) D B E A C (2) D B A C E (2) C E A D B (2) C D B E A (2) C D B A E (2) C A E B D (2) C A B E D (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C B D (1) D E A B C (1) D C B E A (1) C B D A E (1) C B A E D (1) B D C A E (1) B A E D C (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 2 2 4 -14 B -2 0 -2 4 -10 C -2 2 0 -2 2 D -4 -4 2 0 -2 E 14 10 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 2 2 4 -14 B -2 0 -2 4 -10 C -2 2 0 -2 2 D -4 -4 2 0 -2 E 14 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=20 E=18 B=17 A=9 so A is eliminated. Round 2 votes counts: C=36 B=25 D=20 E=19 so E is eliminated. Round 3 votes counts: C=40 B=34 D=26 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:200 A:197 D:196 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 2 4 -14 B -2 0 -2 4 -10 C -2 2 0 -2 2 D -4 -4 2 0 -2 E 14 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 -14 B -2 0 -2 4 -10 C -2 2 0 -2 2 D -4 -4 2 0 -2 E 14 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 -14 B -2 0 -2 4 -10 C -2 2 0 -2 2 D -4 -4 2 0 -2 E 14 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8446: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) D A E B C (8) D A C E B (7) E B C D A (6) C A D B E (5) C A B D E (4) B E A D C (4) B C E A D (4) A C D B E (4) E B D C A (3) D E A B C (3) D A E C B (3) C E B D A (3) C B A E D (3) C A D E B (3) B E C A D (3) A D C E B (3) C D E B A (2) C D E A B (2) C B E A D (2) C A B E D (2) B E D A C (2) B E A C D (2) E D B A C (1) E B D A C (1) D C A E B (1) D B E A C (1) D A B E C (1) C E D B A (1) C E B A D (1) C E A D B (1) B E C D A (1) B A D E C (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 18 6 8 14 B -18 0 -12 -14 0 C -6 12 0 -2 14 D -8 14 2 0 20 E -14 0 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 8 14 B -18 0 -12 -14 0 C -6 12 0 -2 14 D -8 14 2 0 20 E -14 0 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 A=19 B=17 E=11 so E is eliminated. Round 2 votes counts: C=29 B=27 D=25 A=19 so A is eliminated. Round 3 votes counts: D=39 C=33 B=28 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:223 D:214 C:209 B:178 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 8 14 B -18 0 -12 -14 0 C -6 12 0 -2 14 D -8 14 2 0 20 E -14 0 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 8 14 B -18 0 -12 -14 0 C -6 12 0 -2 14 D -8 14 2 0 20 E -14 0 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 8 14 B -18 0 -12 -14 0 C -6 12 0 -2 14 D -8 14 2 0 20 E -14 0 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8447: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (6) A B E C D (6) D C A B E (5) B A E D C (5) E C B A D (4) E B D C A (4) D B A E C (4) C A D B E (4) A D B C E (4) A C D B E (4) E B C D A (3) E B C A D (3) E B A C D (3) D E C B A (3) C D A E B (3) B A E C D (3) A C B D E (3) D C A E B (2) D A B C E (2) C E B A D (2) B E A C D (2) B D E A C (2) A C B E D (2) A B D E C (2) A B D C E (2) E D B C A (1) E C A B D (1) D E B C A (1) D E B A C (1) D C E B A (1) D C E A B (1) C E D B A (1) C D E A B (1) C A E B D (1) C A D E B (1) B E D A C (1) B E A D C (1) B A D E C (1) A E B C D (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 16 16 28 B -8 0 10 10 24 C -16 -10 0 0 -6 D -16 -10 0 0 8 E -28 -24 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 16 28 B -8 0 10 10 24 C -16 -10 0 0 -6 D -16 -10 0 0 8 E -28 -24 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 E=19 B=15 C=13 so C is eliminated. Round 2 votes counts: A=33 D=30 E=22 B=15 so B is eliminated. Round 3 votes counts: A=42 D=32 E=26 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:234 B:218 D:191 C:184 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 16 28 B -8 0 10 10 24 C -16 -10 0 0 -6 D -16 -10 0 0 8 E -28 -24 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 16 28 B -8 0 10 10 24 C -16 -10 0 0 -6 D -16 -10 0 0 8 E -28 -24 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 16 28 B -8 0 10 10 24 C -16 -10 0 0 -6 D -16 -10 0 0 8 E -28 -24 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8448: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) D A C E B (9) C D A B E (9) C B D A E (9) E A D C B (8) B E C A D (7) B C D A E (7) E B A D C (6) A D E C B (6) B C E D A (4) B C E A D (4) E B A C D (3) E B D A C (2) E A B D C (2) D A E C B (2) B C A E D (2) E B C A D (1) D C A B E (1) D A C B E (1) C D B A E (1) C D A E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 2 2 2 0 B -2 0 2 0 -2 C -2 -2 0 2 0 D -2 0 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.559456 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.440544 Sum of squares = 0.507069952846 Cumulative probabilities = A: 0.559456 B: 0.559456 C: 0.559456 D: 0.559456 E: 1.000000 A B C D E A 0 2 2 2 0 B -2 0 2 0 -2 C -2 -2 0 2 0 D -2 0 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=27 C=21 D=13 A=7 so A is eliminated. Round 2 votes counts: E=33 B=27 C=21 D=19 so D is eliminated. Round 3 votes counts: E=41 C=32 B=27 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:203 E:203 B:199 C:199 D:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 2 0 B -2 0 2 0 -2 C -2 -2 0 2 0 D -2 0 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 0 B -2 0 2 0 -2 C -2 -2 0 2 0 D -2 0 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 0 B -2 0 2 0 -2 C -2 -2 0 2 0 D -2 0 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8449: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) A D C E B (6) A D B E C (6) B D A E C (5) A E C D B (5) E C B A D (4) D B A E C (4) C E B D A (4) C E A D B (4) B E C D A (4) B C E D A (4) D A B E C (3) C A E D B (3) B E D C A (3) E B A C D (2) D C A B E (2) D B C A E (2) D B A C E (2) D A C B E (2) C E D A B (2) C E B A D (2) C D A E B (2) C B E D A (2) B D C E A (2) A D E C B (2) E C A D B (1) E C A B D (1) E B C A D (1) E A C D B (1) E A C B D (1) D C A E B (1) D A C E B (1) C D E B A (1) C D B E A (1) C A D E B (1) B E D A C (1) B E A D C (1) B A D E C (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 8 6 -10 16 B -8 0 -2 -24 8 C -6 2 0 -6 8 D 10 24 6 0 6 E -16 -8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -10 16 B -8 0 -2 -24 8 C -6 2 0 -6 8 D 10 24 6 0 6 E -16 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 A=22 B=21 E=11 so E is eliminated. Round 2 votes counts: C=28 D=24 B=24 A=24 so D is eliminated. Round 3 votes counts: A=37 B=32 C=31 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:223 A:210 C:199 B:187 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -10 16 B -8 0 -2 -24 8 C -6 2 0 -6 8 D 10 24 6 0 6 E -16 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -10 16 B -8 0 -2 -24 8 C -6 2 0 -6 8 D 10 24 6 0 6 E -16 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -10 16 B -8 0 -2 -24 8 C -6 2 0 -6 8 D 10 24 6 0 6 E -16 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8450: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) C D A B E (9) B E A D C (9) E B A C D (7) D C A B E (6) C D A E B (6) D A C B E (5) D A B C E (5) C A D E B (5) E B D C A (4) E B C D A (3) A D C B E (3) A D B C E (3) E C B D A (2) E C B A D (2) C E A D B (2) C A D B E (2) B E D A C (2) B D E A C (2) B A E D C (2) E C D B A (1) E C A B D (1) D E C B A (1) D C B A E (1) D B C E A (1) D B A C E (1) C E D A B (1) C A E D B (1) B E A C D (1) B D E C A (1) B D A E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -16 -4 2 B 0 0 4 -6 10 C 16 -4 0 4 4 D 4 6 -4 0 6 E -2 -10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.346938775512 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -4 2 B 0 0 4 -6 10 C 16 -4 0 4 4 D 4 6 -4 0 6 E -2 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877549 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 D=20 B=18 A=7 so A is eliminated. Round 2 votes counts: E=29 D=26 C=26 B=19 so B is eliminated. Round 3 votes counts: E=43 D=31 C=26 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:210 D:206 B:204 A:191 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -4 2 B 0 0 4 -6 10 C 16 -4 0 4 4 D 4 6 -4 0 6 E -2 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877549 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -4 2 B 0 0 4 -6 10 C 16 -4 0 4 4 D 4 6 -4 0 6 E -2 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877549 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -4 2 B 0 0 4 -6 10 C 16 -4 0 4 4 D 4 6 -4 0 6 E -2 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.428571 D: 0.285714 E: 0.000000 Sum of squares = 0.34693877549 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8451: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) B E A D C (7) E B A D C (6) D C E B A (6) A C D E B (6) C D A E B (5) A E B D C (5) B A E C D (4) D C B E A (3) C D A B E (3) B E D C A (3) B E A C D (3) B C D E A (3) A E D C B (3) A B E C D (3) E B D C A (2) E A B D C (2) D C A E B (2) C D E B A (2) C A D B E (2) B A E D C (2) A E C D B (2) A E B C D (2) A C E D B (2) A B E D C (2) E D C B A (1) E D B C A (1) E B D A C (1) D C E A B (1) D A C E B (1) C D B A E (1) C A D E B (1) B D E C A (1) B C A D E (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 8 12 0 B 10 0 -2 -2 -2 C -8 2 0 2 0 D -12 2 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.158837 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.841163 Sum of squares = 0.732784992332 Cumulative probabilities = A: 0.158837 B: 0.158837 C: 0.158837 D: 0.158837 E: 1.000000 A B C D E A 0 -10 8 12 0 B 10 0 -2 -2 -2 C -8 2 0 2 0 D -12 2 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222333457 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=24 C=22 E=13 D=13 so E is eliminated. Round 2 votes counts: B=33 A=30 C=22 D=15 so D is eliminated. Round 3 votes counts: C=35 B=34 A=31 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:205 E:203 B:202 C:198 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 8 12 0 B 10 0 -2 -2 -2 C -8 2 0 2 0 D -12 2 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222333457 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 12 0 B 10 0 -2 -2 -2 C -8 2 0 2 0 D -12 2 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222333457 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 12 0 B 10 0 -2 -2 -2 C -8 2 0 2 0 D -12 2 -2 0 -4 E 0 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.833333 Sum of squares = 0.722222333457 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.166667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8452: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (16) C B E A D (16) C B E D A (6) D B E A C (5) C B D E A (5) C A E B D (4) B C E D A (4) A E D C B (4) D B A E C (3) B D E C A (3) A E C D B (3) A E C B D (3) D B C E A (2) D A B C E (2) C E B A D (2) C E A B D (2) B C D E A (2) A D E C B (2) A D E B C (2) E C B A D (1) E B C A D (1) E A C B D (1) E A B D C (1) D C B A E (1) D B E C A (1) D B C A E (1) D A C B E (1) D A B E C (1) C D A B E (1) B E C D A (1) B D C E A (1) A E D B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 -8 -12 -8 B 10 0 -6 6 12 C 8 6 0 6 2 D 12 -6 -6 0 -2 E 8 -12 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -12 -8 B 10 0 -6 6 12 C 8 6 0 6 2 D 12 -6 -6 0 -2 E 8 -12 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=33 A=16 B=11 E=4 so E is eliminated. Round 2 votes counts: C=37 D=33 A=18 B=12 so B is eliminated. Round 3 votes counts: C=45 D=37 A=18 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:211 D:199 E:198 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -12 -8 B 10 0 -6 6 12 C 8 6 0 6 2 D 12 -6 -6 0 -2 E 8 -12 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -12 -8 B 10 0 -6 6 12 C 8 6 0 6 2 D 12 -6 -6 0 -2 E 8 -12 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -12 -8 B 10 0 -6 6 12 C 8 6 0 6 2 D 12 -6 -6 0 -2 E 8 -12 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8453: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) B E D C A (7) E C B D A (6) E B C D A (6) D A B E C (6) C A E D B (6) A D C B E (5) B E C A D (4) A D B E C (4) A D B C E (4) A C D E B (4) E B C A D (3) D A C E B (3) A D C E B (3) D A C B E (2) C E B D A (2) C E A B D (2) B E D A C (2) B E A C D (2) B A E C D (2) E B D C A (1) D E B C A (1) D C E B A (1) D B E C A (1) D B E A C (1) D A E B C (1) C E D B A (1) C B E A D (1) C A D E B (1) B E C D A (1) B D E A C (1) A C E B D (1) A C D B E (1) A C B E D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -8 14 -6 B 6 0 0 10 -4 C 8 0 0 10 0 D -14 -10 -10 0 -18 E 6 4 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.459364 D: 0.000000 E: 0.540636 Sum of squares = 0.503302640483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.459364 D: 0.459364 E: 1.000000 A B C D E A 0 -6 -8 14 -6 B 6 0 0 10 -4 C 8 0 0 10 0 D -14 -10 -10 0 -18 E 6 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=23 B=19 E=16 D=16 so E is eliminated. Round 2 votes counts: C=29 B=29 A=26 D=16 so D is eliminated. Round 3 votes counts: A=38 B=32 C=30 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:214 C:209 B:206 A:197 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 14 -6 B 6 0 0 10 -4 C 8 0 0 10 0 D -14 -10 -10 0 -18 E 6 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 14 -6 B 6 0 0 10 -4 C 8 0 0 10 0 D -14 -10 -10 0 -18 E 6 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 14 -6 B 6 0 0 10 -4 C 8 0 0 10 0 D -14 -10 -10 0 -18 E 6 4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8454: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) D B E C A (6) B D C E A (6) E A C B D (5) C B D A E (5) B C D A E (5) E A B C D (4) D C B A E (4) D B C E A (4) D B C A E (4) E A D C B (3) E A B D C (3) D E A B C (3) D C A B E (3) B C D E A (3) A C E D B (3) E A D B C (2) D E B A C (2) D B E A C (2) C D B A E (2) C B A E D (2) C A B E D (2) C A B D E (2) A E C D B (2) E D B A C (1) E D A B C (1) E B D A C (1) E B A C D (1) D A E C B (1) C D A B E (1) C B A D E (1) C A E B D (1) B E D C A (1) B E C A D (1) A D E C B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -6 -10 2 B 2 0 0 8 8 C 6 0 0 4 0 D 10 -8 -4 0 12 E -2 -8 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.476929 C: 0.523071 D: 0.000000 E: 0.000000 Sum of squares = 0.501064575893 Cumulative probabilities = A: 0.000000 B: 0.476929 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -10 2 B 2 0 0 8 8 C 6 0 0 4 0 D 10 -8 -4 0 12 E -2 -8 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 A=18 C=16 B=16 so C is eliminated. Round 2 votes counts: D=32 B=24 A=23 E=21 so E is eliminated. Round 3 votes counts: A=40 D=34 B=26 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:209 C:205 D:205 A:192 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 -10 2 B 2 0 0 8 8 C 6 0 0 4 0 D 10 -8 -4 0 12 E -2 -8 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -10 2 B 2 0 0 8 8 C 6 0 0 4 0 D 10 -8 -4 0 12 E -2 -8 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -10 2 B 2 0 0 8 8 C 6 0 0 4 0 D 10 -8 -4 0 12 E -2 -8 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8455: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (10) E A B D C (6) E A B C D (6) D B C A E (6) D B A E C (6) D E A B C (5) D C B A E (5) C D B A E (5) E C A B D (4) E A C B D (4) C D E A B (4) E A D B C (3) D B A C E (3) A B E D C (3) E D A B C (2) D C E A B (2) C D E B A (2) C D B E A (2) C B D A E (2) B A E C D (2) A E B C D (2) E D C A B (1) E D A C B (1) D E C A B (1) D B E A C (1) C E D B A (1) C E B A D (1) C B A E D (1) C B A D E (1) C A B E D (1) B D C A E (1) B D A C E (1) B A E D C (1) B A D C E (1) B A C D E (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 14 0 -2 -12 B -14 0 4 0 -12 C 0 -4 0 0 0 D 2 0 0 0 -2 E 12 12 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.533563 D: 0.000000 E: 0.466437 Sum of squares = 0.502252917017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.533563 D: 0.533563 E: 1.000000 A B C D E A 0 14 0 -2 -12 B -14 0 4 0 -12 C 0 -4 0 0 0 D 2 0 0 0 -2 E 12 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=29 E=27 B=7 A=7 so B is eliminated. Round 2 votes counts: D=31 C=30 E=27 A=12 so A is eliminated. Round 3 votes counts: E=37 D=32 C=31 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:200 D:200 C:198 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 -2 -12 B -14 0 4 0 -12 C 0 -4 0 0 0 D 2 0 0 0 -2 E 12 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 -2 -12 B -14 0 4 0 -12 C 0 -4 0 0 0 D 2 0 0 0 -2 E 12 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 -2 -12 B -14 0 4 0 -12 C 0 -4 0 0 0 D 2 0 0 0 -2 E 12 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8456: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (6) D E B A C (5) B D E C A (5) E A D B C (4) C B A D E (4) A E C B D (4) A C E D B (4) E A B D C (3) E A B C D (3) D E A C B (3) D B C E A (3) C B D A E (3) B E D A C (3) B C D E A (3) B C D A E (3) A D C E B (3) E D A B C (2) E A D C B (2) D E B C A (2) D B E C A (2) C D B A E (2) C B A E D (2) B E A C D (2) B C A D E (2) A D E C B (2) A C E B D (2) A C D E B (2) E D A C B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A C B D (1) D E A B C (1) D C B A E (1) D C A B E (1) D A C E B (1) C D A B E (1) C A D E B (1) C A B E D (1) B E D C A (1) B E A D C (1) B C E A D (1) B C A E D (1) B A C E D (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -12 10 0 -14 B 12 0 16 12 -2 C -10 -16 0 -8 -4 D 0 -12 8 0 12 E 14 2 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.000000 D: 0.076923 E: 0.461538 Sum of squares = 0.431952662714 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.461538 D: 0.538462 E: 1.000000 A B C D E A 0 -12 10 0 -14 B 12 0 16 12 -2 C -10 -16 0 -8 -4 D 0 -12 8 0 12 E 14 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.000000 D: 0.076923 E: 0.461538 Sum of squares = 0.431952662684 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.461538 D: 0.538462 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=19 D=19 A=19 C=14 so C is eliminated. Round 2 votes counts: B=38 D=22 A=21 E=19 so E is eliminated. Round 3 votes counts: B=41 A=34 D=25 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:204 E:204 A:192 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 0 -14 B 12 0 16 12 -2 C -10 -16 0 -8 -4 D 0 -12 8 0 12 E 14 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.000000 D: 0.076923 E: 0.461538 Sum of squares = 0.431952662684 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.461538 D: 0.538462 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 0 -14 B 12 0 16 12 -2 C -10 -16 0 -8 -4 D 0 -12 8 0 12 E 14 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.000000 D: 0.076923 E: 0.461538 Sum of squares = 0.431952662684 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.461538 D: 0.538462 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 0 -14 B 12 0 16 12 -2 C -10 -16 0 -8 -4 D 0 -12 8 0 12 E 14 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.461538 C: 0.000000 D: 0.076923 E: 0.461538 Sum of squares = 0.431952662684 Cumulative probabilities = A: 0.000000 B: 0.461538 C: 0.461538 D: 0.538462 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8457: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) C B D E A (6) B C E D A (6) E A B C D (5) C B E D A (5) B C D E A (5) A E D B C (5) D A C B E (4) A D E C B (4) D C B A E (3) D A E B C (3) C A B E D (3) A E B C D (3) A D C E B (3) E D A B C (2) E B C A D (2) D E A B C (2) D B E C A (2) D B C E A (2) C B A D E (2) A E D C B (2) E D B C A (1) E B D C A (1) E B C D A (1) E B A C D (1) D E B C A (1) D E B A C (1) D B C A E (1) D A E C B (1) D A C E B (1) C D A B E (1) C B D A E (1) C B A E D (1) B D C E A (1) B C E A D (1) A E C D B (1) A E C B D (1) A E B D C (1) A D E B C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -14 -2 -12 B 10 0 -4 16 12 C 14 4 0 16 18 D 2 -16 -16 0 -8 E 12 -12 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -2 -12 B 10 0 -4 16 12 C 14 4 0 16 18 D 2 -16 -16 0 -8 E 12 -12 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=23 D=21 E=13 B=13 so E is eliminated. Round 2 votes counts: C=30 A=28 D=24 B=18 so B is eliminated. Round 3 votes counts: C=45 A=29 D=26 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:217 E:195 A:181 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -2 -12 B 10 0 -4 16 12 C 14 4 0 16 18 D 2 -16 -16 0 -8 E 12 -12 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -2 -12 B 10 0 -4 16 12 C 14 4 0 16 18 D 2 -16 -16 0 -8 E 12 -12 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -2 -12 B 10 0 -4 16 12 C 14 4 0 16 18 D 2 -16 -16 0 -8 E 12 -12 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998304 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8458: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) B D E A C (8) A C D B E (7) C A E D B (6) B E D C A (6) C E A D B (5) D B E A C (4) D A B E C (4) C A D E B (4) A C D E B (4) E C A D B (3) E B D C A (3) E B C D A (3) D E B A C (3) C E B A D (3) C E A B D (3) E D B A C (2) E C B D A (2) C A B D E (2) A D E C B (2) E C D B A (1) E C D A B (1) E C B A D (1) D A E B C (1) C B E A D (1) C B A E D (1) C B A D E (1) C A E B D (1) C A D B E (1) C A B E D (1) B E C D A (1) B D E C A (1) B D A E C (1) B C E D A (1) B C A D E (1) A E D C B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -6 0 -4 B 4 0 -4 -18 2 C 6 4 0 8 -12 D 0 18 -8 0 8 E 4 -2 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775507 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 1.000000 A B C D E A 0 -4 -6 0 -4 B 4 0 -4 -18 2 C 6 4 0 8 -12 D 0 18 -8 0 8 E 4 -2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=20 B=19 E=16 A=16 so E is eliminated. Round 2 votes counts: C=37 B=25 D=22 A=16 so A is eliminated. Round 3 votes counts: C=48 D=27 B=25 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:209 C:203 E:203 A:193 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 0 -4 B 4 0 -4 -18 2 C 6 4 0 8 -12 D 0 18 -8 0 8 E 4 -2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 -4 B 4 0 -4 -18 2 C 6 4 0 8 -12 D 0 18 -8 0 8 E 4 -2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 -4 B 4 0 -4 -18 2 C 6 4 0 8 -12 D 0 18 -8 0 8 E 4 -2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 0.285714 Sum of squares = 0.346938775061 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8459: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) A C D B E (9) D A C B E (8) C A B E D (8) E D B C A (6) E B D C A (5) A D C B E (5) D E B A C (4) D E A C B (4) D A C E B (4) A C B D E (4) D E B C A (3) C A E B D (3) B C A E D (3) E C B D A (2) D A B C E (2) C B E A D (2) C B A E D (2) B E D C A (2) E D C B A (1) E C B A D (1) D B E A C (1) D B A E C (1) D B A C E (1) D A E C B (1) C E B A D (1) C E A B D (1) C A E D B (1) B E C D A (1) A D C E B (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -4 8 12 B -8 0 -16 -4 4 C 4 16 0 0 16 D -8 4 0 0 0 E -12 -4 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.833638 D: 0.166362 E: 0.000000 Sum of squares = 0.722629212261 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.833638 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 8 12 B -8 0 -16 -4 4 C 4 16 0 0 16 D -8 4 0 0 0 E -12 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=25 A=22 C=18 B=6 so B is eliminated. Round 2 votes counts: D=29 E=28 A=22 C=21 so C is eliminated. Round 3 votes counts: A=39 E=32 D=29 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:212 D:198 B:188 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 8 12 B -8 0 -16 -4 4 C 4 16 0 0 16 D -8 4 0 0 0 E -12 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 8 12 B -8 0 -16 -4 4 C 4 16 0 0 16 D -8 4 0 0 0 E -12 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 8 12 B -8 0 -16 -4 4 C 4 16 0 0 16 D -8 4 0 0 0 E -12 -4 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559865 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8460: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (6) B C D A E (6) E A B C D (5) E A D C B (4) B D E C A (4) B C A D E (4) A E D C B (4) A E C D B (4) E B A D C (3) E A B D C (3) D A C E B (3) C D B A E (3) B E D C A (3) B D C E A (3) B C A E D (3) A C E B D (3) A C D E B (3) E B A C D (2) E A C D B (2) D E A C B (2) D C B A E (2) D B C E A (2) D A E C B (2) C D A B E (2) C B A D E (2) C A B E D (2) B E A C D (2) A E B C D (2) E D B A C (1) E D A C B (1) E D A B C (1) E B D A C (1) D E B C A (1) D E B A C (1) D C A B E (1) D B E C A (1) D B C A E (1) C B D A E (1) C A D E B (1) C A B D E (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 8 2 4 16 B -8 0 -2 2 -14 C -2 2 0 0 0 D -4 -2 0 0 4 E -16 14 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 4 16 B -8 0 -2 2 -14 C -2 2 0 0 0 D -4 -2 0 0 4 E -16 14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=23 D=22 A=18 C=12 so C is eliminated. Round 2 votes counts: B=28 D=27 E=23 A=22 so A is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:215 C:200 D:199 E:197 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 4 16 B -8 0 -2 2 -14 C -2 2 0 0 0 D -4 -2 0 0 4 E -16 14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 4 16 B -8 0 -2 2 -14 C -2 2 0 0 0 D -4 -2 0 0 4 E -16 14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 4 16 B -8 0 -2 2 -14 C -2 2 0 0 0 D -4 -2 0 0 4 E -16 14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8461: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) C A B E D (10) B E A C D (9) E B D A C (8) B E D A C (7) D C A E B (6) E B A C D (5) D E A C B (4) A C B E D (4) D E C A B (3) D E B C A (3) C A D E B (3) E D B A C (2) E B A D C (2) E A B C D (2) D C B A E (2) D B E C A (2) C A D B E (2) C A B D E (2) B C A E D (2) A C E B D (2) E D A C B (1) E D A B C (1) E A C B D (1) D C E A B (1) C D A E B (1) C D A B E (1) B E A D C (1) B D E A C (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 -12 24 -6 -26 B 12 0 14 16 -10 C -24 -14 0 -8 -26 D 6 -16 8 0 -18 E 26 10 26 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 24 -6 -26 B 12 0 14 16 -10 C -24 -14 0 -8 -26 D 6 -16 8 0 -18 E 26 10 26 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=22 B=22 C=19 A=6 so A is eliminated. Round 2 votes counts: D=31 C=25 E=22 B=22 so E is eliminated. Round 3 votes counts: B=39 D=35 C=26 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:240 B:216 A:190 D:190 C:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 24 -6 -26 B 12 0 14 16 -10 C -24 -14 0 -8 -26 D 6 -16 8 0 -18 E 26 10 26 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 24 -6 -26 B 12 0 14 16 -10 C -24 -14 0 -8 -26 D 6 -16 8 0 -18 E 26 10 26 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 24 -6 -26 B 12 0 14 16 -10 C -24 -14 0 -8 -26 D 6 -16 8 0 -18 E 26 10 26 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8462: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (7) D B C A E (5) D B A E C (5) B A E D C (5) E A B C D (4) D C E A B (4) C E A D B (4) E C A B D (3) E A B D C (3) D C B A E (3) C E A B D (3) C D E A B (3) C B D A E (3) C A E B D (3) A E B C D (3) E D A B C (2) D E B A C (2) D C E B A (2) D C B E A (2) D B A C E (2) C E D A B (2) C D E B A (2) C D B E A (2) C D B A E (2) C B A E D (2) B D C A E (2) B D A C E (2) B A D E C (2) B A D C E (2) A E B D C (2) E A C B D (1) D E C A B (1) D E B C A (1) D E A B C (1) D B E A C (1) C D A B E (1) C B A D E (1) B A C E D (1) B A C D E (1) A E C B D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 -2 -14 14 B 14 0 10 6 6 C 2 -10 0 -14 8 D 14 -6 14 0 18 E -14 -6 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -14 14 B 14 0 10 6 6 C 2 -10 0 -14 8 D 14 -6 14 0 18 E -14 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 B=22 E=13 A=8 so A is eliminated. Round 2 votes counts: D=29 C=28 B=24 E=19 so E is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:218 C:193 A:192 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -14 14 B 14 0 10 6 6 C 2 -10 0 -14 8 D 14 -6 14 0 18 E -14 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -14 14 B 14 0 10 6 6 C 2 -10 0 -14 8 D 14 -6 14 0 18 E -14 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -14 14 B 14 0 10 6 6 C 2 -10 0 -14 8 D 14 -6 14 0 18 E -14 -6 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8463: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) E C D A B (7) E B A C D (5) A E D C B (5) E C B D A (4) C D B E A (4) B A E D C (4) E C D B A (3) E B C A D (3) D C A E B (3) C E D A B (3) C D E B A (3) B E A C D (3) B D A C E (3) B A D E C (3) A B D C E (3) E C A D B (2) D C B A E (2) C E D B A (2) B C D E A (2) A E B D C (2) A D C E B (2) A D B C E (2) A B D E C (2) E C B A D (1) E B C D A (1) E A C D B (1) E A C B D (1) E A B C D (1) D C A B E (1) D A C B E (1) C B E D A (1) C B D E A (1) B E C A D (1) B D C A E (1) B C E D A (1) B A D C E (1) A E C D B (1) A E B C D (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -14 -8 -22 B -2 0 -20 -8 -26 C 14 20 0 28 -4 D 8 8 -28 0 -8 E 22 26 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -14 -8 -22 B -2 0 -20 -8 -26 C 14 20 0 28 -4 D 8 8 -28 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=25 A=20 B=19 D=7 so D is eliminated. Round 2 votes counts: C=31 E=29 A=21 B=19 so B is eliminated. Round 3 votes counts: C=35 E=33 A=32 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:230 C:229 D:190 A:179 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -14 -8 -22 B -2 0 -20 -8 -26 C 14 20 0 28 -4 D 8 8 -28 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -8 -22 B -2 0 -20 -8 -26 C 14 20 0 28 -4 D 8 8 -28 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -8 -22 B -2 0 -20 -8 -26 C 14 20 0 28 -4 D 8 8 -28 0 -8 E 22 26 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8464: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) B A D E C (7) D B A E C (5) B E D A C (5) B D A E C (5) E C D B A (4) D E B A C (4) C A D E B (4) C A B E D (4) E D C B A (3) C E B A D (3) C E A B D (3) C A E D B (3) A D C B E (3) A C B D E (3) A B C D E (3) E B D C A (2) E B C D A (2) D E C A B (2) C E D A B (2) C E A D B (2) A D B C E (2) A C D B E (2) A B D C E (2) E D B C A (1) E D B A C (1) E C B D A (1) E B D A C (1) E B C A D (1) D B E A C (1) C E B D A (1) C D A E B (1) C B E A D (1) C A E B D (1) C A D B E (1) B E A D C (1) B A E D C (1) B A E C D (1) A D B E C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 16 2 18 B 2 0 12 0 16 C -16 -12 0 -12 -16 D -2 0 12 0 10 E -18 -16 16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.735096 C: 0.000000 D: 0.264904 E: 0.000000 Sum of squares = 0.6105399721 Cumulative probabilities = A: 0.000000 B: 0.735096 C: 0.735096 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 2 18 B 2 0 12 0 16 C -16 -12 0 -12 -16 D -2 0 12 0 10 E -18 -16 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.499681 E: 0.000000 Sum of squares = 0.500000203089 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=20 B=20 A=18 E=16 so E is eliminated. Round 2 votes counts: C=31 B=26 D=25 A=18 so A is eliminated. Round 3 votes counts: C=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:215 D:210 E:186 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 2 18 B 2 0 12 0 16 C -16 -12 0 -12 -16 D -2 0 12 0 10 E -18 -16 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.499681 E: 0.000000 Sum of squares = 0.500000203089 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 2 18 B 2 0 12 0 16 C -16 -12 0 -12 -16 D -2 0 12 0 10 E -18 -16 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.499681 E: 0.000000 Sum of squares = 0.500000203089 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 2 18 B 2 0 12 0 16 C -16 -12 0 -12 -16 D -2 0 12 0 10 E -18 -16 16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500319 C: 0.000000 D: 0.499681 E: 0.000000 Sum of squares = 0.500000203089 Cumulative probabilities = A: 0.000000 B: 0.500319 C: 0.500319 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8465: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) B C E D A (6) A E D C B (6) E D A B C (5) E A D B C (5) E B D C A (4) C B D A E (4) B C D A E (4) A C D B E (4) E B A C D (3) B D C E A (3) B C E A D (3) A C B D E (3) E A D C B (2) D E B A C (2) D C B A E (2) D B E C A (2) C B A E D (2) C A D B E (2) A D E C B (2) A D C E B (2) A C E B D (2) E D B A C (1) E A C B D (1) E A B C D (1) D E A B C (1) D B C A E (1) D A E B C (1) D A C B E (1) C D B A E (1) C D A B E (1) C B A D E (1) C A B E D (1) C A B D E (1) B E D C A (1) B E C D A (1) B E C A D (1) B C A E D (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 -14 -12 -10 -12 B 14 0 20 16 22 C 12 -20 0 16 20 D 10 -16 -16 0 6 E 12 -22 -20 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -10 -12 B 14 0 20 16 22 C 12 -20 0 16 20 D 10 -16 -16 0 6 E 12 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=22 A=21 C=13 D=10 so D is eliminated. Round 2 votes counts: B=37 E=25 A=23 C=15 so C is eliminated. Round 3 votes counts: B=47 A=28 E=25 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:236 C:214 D:192 E:182 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -10 -12 B 14 0 20 16 22 C 12 -20 0 16 20 D 10 -16 -16 0 6 E 12 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -10 -12 B 14 0 20 16 22 C 12 -20 0 16 20 D 10 -16 -16 0 6 E 12 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -10 -12 B 14 0 20 16 22 C 12 -20 0 16 20 D 10 -16 -16 0 6 E 12 -22 -20 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8466: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (7) E C D A B (5) E D B C A (4) E A B C D (4) D C E B A (4) D B C A E (4) C D A B E (4) B A D C E (4) A E B C D (4) E B A D C (3) E A C B D (3) D C B A E (3) C A E D B (3) C A D B E (3) B E A D C (3) B A E D C (3) A C E B D (3) E D C B A (2) E B D A C (2) D E C B A (2) D E B C A (2) D B C E A (2) C E D A B (2) C D E A B (2) B E D A C (2) B A C D E (2) A E C B D (2) A C B E D (2) E D C A B (1) E C A D B (1) D E B A C (1) D C B E A (1) D B E C A (1) D B E A C (1) D B A C E (1) C D B A E (1) B D E A C (1) B D A C E (1) B A E C D (1) B A D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 6 2 2 B 4 0 12 0 0 C -6 -12 0 2 -12 D -2 0 -2 0 -16 E -2 0 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.615875 C: 0.000000 D: 0.000000 E: 0.384125 Sum of squares = 0.526853882679 Cumulative probabilities = A: 0.000000 B: 0.615875 C: 0.615875 D: 0.615875 E: 1.000000 A B C D E A 0 -4 6 2 2 B 4 0 12 0 0 C -6 -12 0 2 -12 D -2 0 -2 0 -16 E -2 0 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 A=20 B=18 C=15 so C is eliminated. Round 2 votes counts: D=29 E=27 A=26 B=18 so B is eliminated. Round 3 votes counts: A=37 E=32 D=31 so D is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:213 B:208 A:203 D:190 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 2 B 4 0 12 0 0 C -6 -12 0 2 -12 D -2 0 -2 0 -16 E -2 0 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 2 B 4 0 12 0 0 C -6 -12 0 2 -12 D -2 0 -2 0 -16 E -2 0 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 2 B 4 0 12 0 0 C -6 -12 0 2 -12 D -2 0 -2 0 -16 E -2 0 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999915 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8467: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) C D A B E (7) A D C E B (6) E B A D C (5) D B E C A (5) D A C E B (5) C D B E A (5) B E C D A (5) A C D E B (5) C B E A D (4) A D E B C (4) D C B E A (3) D B E A C (3) C B E D A (3) B E C A D (3) E B A C D (2) D E B A C (2) D C A B E (2) D A E B C (2) C B D E A (2) C A E B D (2) B E D C A (2) A E D B C (2) E B D A C (1) E B C A D (1) E A B D C (1) D E A B C (1) D A C B E (1) D A B E C (1) C D B A E (1) B E D A C (1) B C E A D (1) A E C B D (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 8 -4 0 B -2 0 2 -12 -2 C -8 -2 0 -12 -4 D 4 12 12 0 12 E 0 2 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -4 0 B -2 0 2 -12 -2 C -8 -2 0 -12 -4 D 4 12 12 0 12 E 0 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=25 C=24 B=12 E=10 so E is eliminated. Round 2 votes counts: A=30 D=25 C=24 B=21 so B is eliminated. Round 3 votes counts: A=37 C=34 D=29 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:220 A:203 E:197 B:193 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -4 0 B -2 0 2 -12 -2 C -8 -2 0 -12 -4 D 4 12 12 0 12 E 0 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -4 0 B -2 0 2 -12 -2 C -8 -2 0 -12 -4 D 4 12 12 0 12 E 0 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -4 0 B -2 0 2 -12 -2 C -8 -2 0 -12 -4 D 4 12 12 0 12 E 0 2 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8468: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) C E B A D (10) C A E B D (8) E B C D A (7) D B E C A (7) B E D C A (6) D A B E C (5) A D C B E (5) A C D E B (5) A D C E B (4) A D B E C (4) A C B E D (4) E B D C A (3) B E C D A (3) E C B D A (2) A D B C E (2) A C E B D (2) A C D B E (2) A C B D E (2) D E B C A (1) D E B A C (1) D C E A B (1) D A E B C (1) C E B D A (1) C E A D B (1) B D E A C (1) B C E A D (1) A C E D B (1) Total count = 100 A B C D E A 0 -6 -2 2 -10 B 6 0 4 0 4 C 2 -4 0 -2 -2 D -2 0 2 0 2 E 10 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.530222 C: 0.000000 D: 0.469778 E: 0.000000 Sum of squares = 0.501826693515 Cumulative probabilities = A: 0.000000 B: 0.530222 C: 0.530222 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 2 -10 B 6 0 4 0 4 C 2 -4 0 -2 -2 D -2 0 2 0 2 E 10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=26 C=20 E=12 B=11 so B is eliminated. Round 2 votes counts: A=31 D=27 E=21 C=21 so E is eliminated. Round 3 votes counts: D=36 C=33 A=31 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:207 E:203 D:201 C:197 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 2 -10 B 6 0 4 0 4 C 2 -4 0 -2 -2 D -2 0 2 0 2 E 10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 2 -10 B 6 0 4 0 4 C 2 -4 0 -2 -2 D -2 0 2 0 2 E 10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 2 -10 B 6 0 4 0 4 C 2 -4 0 -2 -2 D -2 0 2 0 2 E 10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8469: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (13) C B D A E (10) D B E A C (7) B D C A E (7) A E C B D (7) A E B D C (7) C D B E A (6) A E B C D (6) E A D C B (4) D C B E A (4) D B C E A (4) C B A E D (3) E D A B C (2) E C A D B (2) E A C D B (2) C D E B A (2) C A E B D (2) B C D A E (2) E A C B D (1) E A B D C (1) D E A B C (1) D C E B A (1) D C E A B (1) D B C A E (1) C E A D B (1) C B A D E (1) C A B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 4 4 4 -4 B -4 0 4 -2 -6 C -4 -4 0 -6 -8 D -4 2 6 0 -6 E 4 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 4 -4 B -4 0 4 -2 -6 C -4 -4 0 -6 -8 D -4 2 6 0 -6 E 4 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 A=21 D=19 B=9 so B is eliminated. Round 2 votes counts: C=28 D=26 E=25 A=21 so A is eliminated. Round 3 votes counts: E=46 C=28 D=26 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:204 D:199 B:196 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 4 -4 B -4 0 4 -2 -6 C -4 -4 0 -6 -8 D -4 2 6 0 -6 E 4 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 -4 B -4 0 4 -2 -6 C -4 -4 0 -6 -8 D -4 2 6 0 -6 E 4 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 -4 B -4 0 4 -2 -6 C -4 -4 0 -6 -8 D -4 2 6 0 -6 E 4 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8470: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) B C A D E (6) D E A B C (5) D A B C E (5) A D E C B (5) A D B C E (5) E C B D A (4) E C B A D (4) E A D C B (4) D A B E C (4) C B E D A (4) B C E D A (4) E D A C B (3) C B E A D (3) C B A E D (3) E D B C A (2) E C A B D (2) B E C D A (2) B C D A E (2) A D E B C (2) E D C A B (1) E D A B C (1) E C A D B (1) E A C D B (1) D E A C B (1) D B A E C (1) D B A C E (1) C E B A D (1) C B A D E (1) C A B E D (1) B D E C A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B A C D E (1) A E C D B (1) A D B E C (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 10 -10 8 B -12 0 18 -8 2 C -10 -18 0 -12 -12 D 10 8 12 0 14 E -8 -2 12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 -10 8 B -12 0 18 -8 2 C -10 -18 0 -12 -12 D 10 8 12 0 14 E -8 -2 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=23 B=20 A=17 C=13 so C is eliminated. Round 2 votes counts: B=31 D=27 E=24 A=18 so A is eliminated. Round 3 votes counts: D=40 B=34 E=26 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:210 B:200 E:194 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 10 -10 8 B -12 0 18 -8 2 C -10 -18 0 -12 -12 D 10 8 12 0 14 E -8 -2 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -10 8 B -12 0 18 -8 2 C -10 -18 0 -12 -12 D 10 8 12 0 14 E -8 -2 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -10 8 B -12 0 18 -8 2 C -10 -18 0 -12 -12 D 10 8 12 0 14 E -8 -2 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8471: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (9) A D B C E (7) D B E C A (6) D A B E C (5) C E A B D (5) B D E A C (5) E B C D A (4) C A E B D (4) E C B D A (3) D B A E C (3) D A B C E (3) C E D B A (3) C E B A D (3) C E A D B (3) B E D C A (3) E B D C A (2) D B E A C (2) B E D A C (2) A D C B E (2) A D B E C (2) A C E D B (2) A C D B E (2) A B D E C (2) A B D C E (2) E D B C A (1) E C D B A (1) D E B C A (1) D C A E B (1) C E D A B (1) C E B D A (1) C D E B A (1) C A E D B (1) B E A C D (1) B D E C A (1) B D A E C (1) B A D E C (1) A C B E D (1) A C B D E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 10 10 0 2 B -10 0 12 6 8 C -10 -12 0 -6 4 D 0 -6 6 0 -4 E -2 -8 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.768574 B: 0.000000 C: 0.000000 D: 0.231426 E: 0.000000 Sum of squares = 0.644264380312 Cumulative probabilities = A: 0.768574 B: 0.768574 C: 0.768574 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 0 2 B -10 0 12 6 8 C -10 -12 0 -6 4 D 0 -6 6 0 -4 E -2 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559572 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=22 D=21 B=14 E=11 so E is eliminated. Round 2 votes counts: A=32 C=26 D=22 B=20 so B is eliminated. Round 3 votes counts: D=36 A=34 C=30 so C is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:208 D:198 E:195 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 0 2 B -10 0 12 6 8 C -10 -12 0 -6 4 D 0 -6 6 0 -4 E -2 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559572 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 0 2 B -10 0 12 6 8 C -10 -12 0 -6 4 D 0 -6 6 0 -4 E -2 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559572 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 0 2 B -10 0 12 6 8 C -10 -12 0 -6 4 D 0 -6 6 0 -4 E -2 -8 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555559572 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8472: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) A D B C E (7) E B C D A (6) C A E B D (6) C E B D A (5) A D C B E (4) A D B E C (4) A C D B E (4) E C B A D (3) E B D C A (3) D B A C E (3) D A B C E (3) C B D A E (3) A E D C B (3) A C E D B (3) E D B A C (2) E C A B D (2) E A C B D (2) D B E C A (2) C E B A D (2) C E A B D (2) B D E C A (2) B D C E A (2) B D C A E (2) A D E B C (2) A C E B D (2) E B A C D (1) D B E A C (1) D B C A E (1) D B A E C (1) C D B A E (1) C A D B E (1) C A B D E (1) B E D C A (1) B C E D A (1) B C D E A (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -12 2 6 B 4 0 -10 12 -10 C 12 10 0 14 10 D -2 -12 -14 0 -8 E -6 10 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 2 6 B 4 0 -10 12 -10 C 12 10 0 14 10 D -2 -12 -14 0 -8 E -6 10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=28 C=21 D=11 B=9 so B is eliminated. Round 2 votes counts: A=31 E=29 C=23 D=17 so D is eliminated. Round 3 votes counts: A=38 E=34 C=28 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:223 E:201 B:198 A:196 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 2 6 B 4 0 -10 12 -10 C 12 10 0 14 10 D -2 -12 -14 0 -8 E -6 10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 2 6 B 4 0 -10 12 -10 C 12 10 0 14 10 D -2 -12 -14 0 -8 E -6 10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 2 6 B 4 0 -10 12 -10 C 12 10 0 14 10 D -2 -12 -14 0 -8 E -6 10 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8473: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) A B C D E (8) C B D E A (5) B C D E A (5) A E D B C (5) E C D B A (4) E A D C B (4) C D B E A (4) B C E A D (4) B C D A E (4) E D C A B (3) D C B E A (3) C B E D A (3) B C E D A (3) B A C D E (3) A E D C B (3) A D E C B (3) A B E C D (3) E D C B A (2) E D A C B (2) D E C B A (2) D E A C B (2) D C E B A (2) D C B A E (2) B C A E D (2) A D B C E (2) E B C A D (1) E A D B C (1) D A C E B (1) C E B D A (1) C D B A E (1) B A C E D (1) A D E B C (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -20 -18 2 -2 B 20 0 4 2 26 C 18 -4 0 20 26 D -2 -2 -20 0 16 E 2 -26 -26 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999739 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -18 2 -2 B 20 0 4 2 26 C 18 -4 0 20 26 D -2 -2 -20 0 16 E 2 -26 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 E=17 C=14 D=12 so D is eliminated. Round 2 votes counts: B=30 A=28 E=21 C=21 so E is eliminated. Round 3 votes counts: A=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:230 B:226 D:196 A:181 E:167 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -18 2 -2 B 20 0 4 2 26 C 18 -4 0 20 26 D -2 -2 -20 0 16 E 2 -26 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -18 2 -2 B 20 0 4 2 26 C 18 -4 0 20 26 D -2 -2 -20 0 16 E 2 -26 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -18 2 -2 B 20 0 4 2 26 C 18 -4 0 20 26 D -2 -2 -20 0 16 E 2 -26 -26 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998843 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8474: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A D E B (8) A C D B E (8) E B D C A (7) E D B C A (6) A C B D E (6) D E B C A (5) B A C E D (5) D C A E B (4) A C B E D (4) E B D A C (3) E B C A D (3) C A D B E (3) E D C B A (2) E C A D B (2) D C A B E (2) D B A C E (2) D A C B E (2) C A E D B (2) B E A D C (2) E D C A B (1) E B A C D (1) D E C B A (1) D C E A B (1) D B E A C (1) C E D A B (1) C D E A B (1) C D A E B (1) C A E B D (1) B E A C D (1) B D E A C (1) B A E C D (1) B A D C E (1) Total count = 100 A B C D E A 0 -6 -2 -4 0 B 6 0 0 -6 0 C 2 0 0 -4 4 D 4 6 4 0 -6 E 0 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775466 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 A B C D E A 0 -6 -2 -4 0 B 6 0 0 -6 0 C 2 0 0 -4 4 D 4 6 4 0 -6 E 0 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=22 D=18 A=18 C=17 so C is eliminated. Round 2 votes counts: A=32 E=26 B=22 D=20 so D is eliminated. Round 3 votes counts: A=41 E=34 B=25 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:204 C:201 E:201 B:200 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -4 0 B 6 0 0 -6 0 C 2 0 0 -4 4 D 4 6 4 0 -6 E 0 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -4 0 B 6 0 0 -6 0 C 2 0 0 -4 4 D 4 6 4 0 -6 E 0 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -4 0 B 6 0 0 -6 0 C 2 0 0 -4 4 D 4 6 4 0 -6 E 0 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8475: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) E D B A C (7) D E A B C (6) C B E D A (4) C B A E D (4) C A D B E (4) B A E C D (4) B A C E D (4) E D A B C (3) E B D C A (3) D E B C A (3) D E A C B (3) D A C E B (3) C A B D E (3) A C D B E (3) A C B D E (3) E B D A C (2) D C E B A (2) B E C D A (2) B E C A D (2) A E D B C (2) A D C B E (2) A B E C D (2) E A D B C (1) E A B D C (1) D E C B A (1) D E C A B (1) D C E A B (1) D C A E B (1) D A E C B (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A E B (1) C B E A D (1) C B D E A (1) C B A D E (1) C A B E D (1) B E D C A (1) B E A C D (1) B C A E D (1) A D E C B (1) A D E B C (1) A D C E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 4 -12 -10 B 8 0 8 -16 -6 C -4 -8 0 -8 -10 D 12 16 8 0 -8 E 10 6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 4 -12 -10 B 8 0 8 -16 -6 C -4 -8 0 -8 -10 D 12 16 8 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=23 D=22 A=16 B=15 so B is eliminated. Round 2 votes counts: E=30 C=24 A=24 D=22 so D is eliminated. Round 3 votes counts: E=44 C=28 A=28 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:214 B:197 A:187 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 4 -12 -10 B 8 0 8 -16 -6 C -4 -8 0 -8 -10 D 12 16 8 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -12 -10 B 8 0 8 -16 -6 C -4 -8 0 -8 -10 D 12 16 8 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -12 -10 B 8 0 8 -16 -6 C -4 -8 0 -8 -10 D 12 16 8 0 -8 E 10 6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8476: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (13) D A B E C (5) C E B A D (5) C B E A D (5) B C E A D (5) E C B D A (4) E B C D A (4) A D C B E (4) D A E C B (3) D A E B C (3) D A B C E (3) C E A D B (3) B E C D A (3) B C A E D (3) A D B C E (3) A C D B E (3) E D C A B (2) E C D A B (2) E C A B D (2) D A C E B (2) C A E D B (2) B E C A D (2) A C B D E (2) E D B C A (1) E D A C B (1) E C B A D (1) E C A D B (1) E B C A D (1) D B A E C (1) C B A E D (1) C A E B D (1) C A B E D (1) B E D C A (1) B D E C A (1) B D E A C (1) B D A C E (1) B A D C E (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 18 -2 24 10 B -18 0 -22 -10 -8 C 2 22 0 6 20 D -24 10 -6 0 -4 E -10 8 -20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -2 24 10 B -18 0 -22 -10 -8 C 2 22 0 6 20 D -24 10 -6 0 -4 E -10 8 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=19 C=18 B=18 D=17 so D is eliminated. Round 2 votes counts: A=44 E=19 B=19 C=18 so C is eliminated. Round 3 votes counts: A=48 E=27 B=25 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:225 E:191 D:188 B:171 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -2 24 10 B -18 0 -22 -10 -8 C 2 22 0 6 20 D -24 10 -6 0 -4 E -10 8 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -2 24 10 B -18 0 -22 -10 -8 C 2 22 0 6 20 D -24 10 -6 0 -4 E -10 8 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -2 24 10 B -18 0 -22 -10 -8 C 2 22 0 6 20 D -24 10 -6 0 -4 E -10 8 -20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999598 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8477: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (16) B D A C E (11) A B D E C (9) E C A D B (8) E C A B D (8) E C D A B (5) C E B A D (4) B A D C E (4) D C B E A (3) C D E B A (3) A B E D C (3) A B E C D (3) D C E B A (2) D B C E A (2) D B A C E (2) C E D A B (2) B A D E C (2) A E C B D (2) A E B C D (2) E D A C B (1) E A C B D (1) D E C B A (1) D A B E C (1) C E B D A (1) C E A D B (1) B D A E C (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -12 -2 -16 B 4 0 -16 4 -16 C 12 16 0 12 2 D 2 -4 -12 0 -16 E 16 16 -2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 -16 B 4 0 -16 4 -16 C 12 16 0 12 2 D 2 -4 -12 0 -16 E 16 16 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=23 A=21 B=18 D=11 so D is eliminated. Round 2 votes counts: C=32 E=24 B=22 A=22 so B is eliminated. Round 3 votes counts: A=42 C=34 E=24 so E is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:221 B:188 D:185 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 -16 B 4 0 -16 4 -16 C 12 16 0 12 2 D 2 -4 -12 0 -16 E 16 16 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 -16 B 4 0 -16 4 -16 C 12 16 0 12 2 D 2 -4 -12 0 -16 E 16 16 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 -16 B 4 0 -16 4 -16 C 12 16 0 12 2 D 2 -4 -12 0 -16 E 16 16 -2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8478: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) B A C E D (10) B D E C A (8) D E C A B (7) B D C E A (5) D E B A C (4) D B E C A (4) D E A B C (3) C E A D B (3) B D C A E (3) B C A E D (3) B A C D E (3) A C E D B (3) A C E B D (3) D E A C B (2) D B E A C (2) C A E D B (2) C A E B D (2) B C A D E (2) A E C D B (2) A C B E D (2) E D A B C (1) E C A D B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E B C A (1) C E D A B (1) C B D E A (1) C B A E D (1) C A B E D (1) B D A E C (1) B D A C E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -24 4 -14 -14 B 24 0 28 20 20 C -4 -28 0 -12 -4 D 14 -20 12 0 20 E 14 -20 4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 4 -14 -14 B 24 0 28 20 20 C -4 -28 0 -12 -4 D 14 -20 12 0 20 E 14 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=48 D=24 A=13 C=11 E=4 so E is eliminated. Round 2 votes counts: B=48 D=25 A=15 C=12 so C is eliminated. Round 3 votes counts: B=50 D=26 A=24 so A is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:246 D:213 E:189 A:176 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 4 -14 -14 B 24 0 28 20 20 C -4 -28 0 -12 -4 D 14 -20 12 0 20 E 14 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 4 -14 -14 B 24 0 28 20 20 C -4 -28 0 -12 -4 D 14 -20 12 0 20 E 14 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 4 -14 -14 B 24 0 28 20 20 C -4 -28 0 -12 -4 D 14 -20 12 0 20 E 14 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8479: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) D B E A C (6) C B A E D (6) A C E B D (6) D E A C B (5) D B E C A (5) C B A D E (5) B D C E A (5) A E D C B (5) B D E C A (4) A E C D B (4) E A B D C (3) D E A B C (3) C A D B E (3) C A B E D (3) A E C B D (3) D C B A E (2) D A E C B (2) C D A B E (2) C A E B D (2) C A B D E (2) A C E D B (2) E D B A C (1) E D A B C (1) E A D B C (1) E A C D B (1) E A B C D (1) D C A E B (1) C B D A E (1) C A D E B (1) B E D A C (1) B E C D A (1) B E C A D (1) B E A C D (1) B D E A C (1) B C E A D (1) B C D A E (1) B C A D E (1) Total count = 100 A B C D E A 0 2 6 4 4 B -2 0 -12 -2 4 C -6 12 0 -4 -12 D -4 2 4 0 12 E -4 -4 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 4 4 B -2 0 -12 -2 4 C -6 12 0 -4 -12 D -4 2 4 0 12 E -4 -4 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=25 A=20 B=17 E=8 so E is eliminated. Round 2 votes counts: D=32 A=26 C=25 B=17 so B is eliminated. Round 3 votes counts: D=43 C=30 A=27 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:208 D:207 E:196 C:195 B:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 4 B -2 0 -12 -2 4 C -6 12 0 -4 -12 D -4 2 4 0 12 E -4 -4 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 4 B -2 0 -12 -2 4 C -6 12 0 -4 -12 D -4 2 4 0 12 E -4 -4 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 4 B -2 0 -12 -2 4 C -6 12 0 -4 -12 D -4 2 4 0 12 E -4 -4 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8480: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) E D C A B (8) B A C E D (8) C E A D B (6) C A E D B (6) B D E A C (6) E C D A B (5) D E B C A (5) D E C B A (4) D E C A B (4) B D E C A (4) B A D C E (4) A C E D B (4) A C D E B (3) A B C E D (3) D B E C A (2) C E D A B (2) C A B E D (2) B D A E C (2) B A C D E (2) A B C D E (2) E D C B A (1) E D B C A (1) E C A D B (1) E B D C A (1) D B E A C (1) B E D C A (1) B E C A D (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 14 -8 6 -6 B -14 0 -14 -6 -4 C 8 14 0 12 6 D -6 6 -12 0 -20 E 6 4 -6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 6 -6 B -14 0 -14 -6 -4 C 8 14 0 12 6 D -6 6 -12 0 -20 E 6 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=23 E=17 D=16 C=16 so D is eliminated. Round 2 votes counts: B=31 E=30 A=23 C=16 so C is eliminated. Round 3 votes counts: E=38 B=31 A=31 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:220 E:212 A:203 D:184 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -8 6 -6 B -14 0 -14 -6 -4 C 8 14 0 12 6 D -6 6 -12 0 -20 E 6 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 6 -6 B -14 0 -14 -6 -4 C 8 14 0 12 6 D -6 6 -12 0 -20 E 6 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 6 -6 B -14 0 -14 -6 -4 C 8 14 0 12 6 D -6 6 -12 0 -20 E 6 4 -6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8481: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) C D A B E (7) E B D A C (5) E D B A C (4) E A B C D (4) D C E B A (4) B E A D C (4) B A E C D (4) D E C B A (3) D E B A C (3) D C E A B (3) D C A B E (3) B A E D C (3) E D C B A (2) E D C A B (2) E B A C D (2) E A C B D (2) D E B C A (2) D C B E A (2) D C B A E (2) D B E A C (2) C A B D E (2) B D A C E (2) A E C B D (2) A B E C D (2) A B C E D (2) E D B C A (1) E C D A B (1) E C A B D (1) D E C A B (1) D B C A E (1) C E A D B (1) C E A B D (1) C D E A B (1) C A E D B (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B C A D E (1) B A D E C (1) B A C D E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -18 10 -4 -22 B 18 0 10 4 -16 C -10 -10 0 -20 -24 D 4 -4 20 0 -14 E 22 16 24 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 10 -4 -22 B 18 0 10 4 -16 C -10 -10 0 -20 -24 D 4 -4 20 0 -14 E 22 16 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=26 B=17 C=16 A=8 so A is eliminated. Round 2 votes counts: E=36 D=26 B=21 C=17 so C is eliminated. Round 3 votes counts: E=40 D=36 B=24 so B is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:238 B:208 D:203 A:183 C:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 10 -4 -22 B 18 0 10 4 -16 C -10 -10 0 -20 -24 D 4 -4 20 0 -14 E 22 16 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 10 -4 -22 B 18 0 10 4 -16 C -10 -10 0 -20 -24 D 4 -4 20 0 -14 E 22 16 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 10 -4 -22 B 18 0 10 4 -16 C -10 -10 0 -20 -24 D 4 -4 20 0 -14 E 22 16 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8482: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) D B C E A (7) B C D A E (7) E D A B C (6) C B D E A (6) B C D E A (6) A E D C B (6) E A C D B (5) C B A E D (4) C B A D E (4) A E D B C (4) D E A B C (3) D B C A E (3) B D C E A (3) D E B C A (2) D B E A C (2) C E A B D (2) C B D A E (2) B D C A E (2) B C A D E (2) A D E B C (2) E D C B A (1) E C D B A (1) E C B A D (1) E C A B D (1) E A C B D (1) D E B A C (1) D B E C A (1) D B A C E (1) C E B D A (1) C B E D A (1) C B E A D (1) A E C D B (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -18 -16 -12 -20 B 18 0 4 -8 6 C 16 -4 0 -4 6 D 12 8 4 0 10 E 20 -6 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 -12 -20 B 18 0 4 -8 6 C 16 -4 0 -4 6 D 12 8 4 0 10 E 20 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=21 D=20 B=20 A=15 so A is eliminated. Round 2 votes counts: E=36 D=22 C=22 B=20 so B is eliminated. Round 3 votes counts: C=37 E=36 D=27 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:217 B:210 C:207 E:199 A:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -16 -12 -20 B 18 0 4 -8 6 C 16 -4 0 -4 6 D 12 8 4 0 10 E 20 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -12 -20 B 18 0 4 -8 6 C 16 -4 0 -4 6 D 12 8 4 0 10 E 20 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -12 -20 B 18 0 4 -8 6 C 16 -4 0 -4 6 D 12 8 4 0 10 E 20 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999499 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8483: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) E D A B C (6) C A D B E (6) E B D C A (4) E B D A C (4) E B C D A (4) C B A D E (4) C A D E B (4) E D B A C (3) E D A C B (3) E C A B D (3) E A D C B (3) C E A D B (3) B E D C A (3) B E C D A (3) B D A E C (3) A D B C E (3) D E A B C (2) C E A B D (2) C B A E D (2) C A E B D (2) B D E A C (2) B D A C E (2) A D E C B (2) D B E A C (1) D A C B E (1) D A B C E (1) C E B A D (1) C B E A D (1) C A E D B (1) B E D A C (1) B D C E A (1) B D C A E (1) B C E A D (1) B C D A E (1) B C A D E (1) B A D C E (1) A E D C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -12 8 -18 B -12 0 -4 -6 -18 C 12 4 0 0 -12 D -8 6 0 0 -20 E 18 18 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -12 8 -18 B -12 0 -4 -6 -18 C 12 4 0 0 -12 D -8 6 0 0 -20 E 18 18 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 C=26 B=20 A=11 D=5 so D is eliminated. Round 2 votes counts: E=40 C=26 B=21 A=13 so A is eliminated. Round 3 votes counts: E=43 C=32 B=25 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:234 C:202 A:195 D:189 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -12 8 -18 B -12 0 -4 -6 -18 C 12 4 0 0 -12 D -8 6 0 0 -20 E 18 18 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 8 -18 B -12 0 -4 -6 -18 C 12 4 0 0 -12 D -8 6 0 0 -20 E 18 18 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 8 -18 B -12 0 -4 -6 -18 C 12 4 0 0 -12 D -8 6 0 0 -20 E 18 18 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8484: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (14) C E D A B (11) B D A E C (9) B A D E C (9) C E A D B (7) C A E B D (6) A B C E D (6) A C B E D (5) A C E B D (4) E D C A B (3) D E C A B (3) B A D C E (3) A B D E C (3) E C D A B (2) D B E C A (2) D B E A C (2) D E B C A (1) D C E B A (1) D B A E C (1) C E A B D (1) C B D E A (1) C A B E D (1) B D E C A (1) B A C E D (1) B A C D E (1) A E C D B (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -8 -2 2 B -8 0 -20 4 -8 C 8 20 0 -6 -2 D 2 -4 6 0 4 E -2 8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.133333 D: 0.666667 E: 0.000000 Sum of squares = 0.50222222219 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.333333 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -2 2 B -8 0 -20 4 -8 C 8 20 0 -6 -2 D 2 -4 6 0 4 E -2 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.133333 D: 0.666667 E: 0.000000 Sum of squares = 0.502222222072 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=24 B=24 A=20 E=5 so E is eliminated. Round 2 votes counts: C=29 D=27 B=24 A=20 so A is eliminated. Round 3 votes counts: C=39 B=34 D=27 so D is eliminated. Round 4 votes counts: C=60 B=40 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:204 E:202 A:200 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -8 -2 2 B -8 0 -20 4 -8 C 8 20 0 -6 -2 D 2 -4 6 0 4 E -2 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.133333 D: 0.666667 E: 0.000000 Sum of squares = 0.502222222072 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -2 2 B -8 0 -20 4 -8 C 8 20 0 -6 -2 D 2 -4 6 0 4 E -2 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.133333 D: 0.666667 E: 0.000000 Sum of squares = 0.502222222072 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -2 2 B -8 0 -20 4 -8 C 8 20 0 -6 -2 D 2 -4 6 0 4 E -2 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.133333 D: 0.666667 E: 0.000000 Sum of squares = 0.502222222072 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8485: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (9) D A B C E (9) D C E B A (7) D B A C E (7) E A C B D (5) D B C A E (5) E C B A D (4) D B C E A (4) D A E C B (3) D A B E C (3) C E B D A (3) C E B A D (3) B D A C E (3) A D E B C (3) E C A D B (2) E A C D B (2) D C B E A (2) C B E D A (2) B C D E A (2) B A D E C (2) E A B C D (1) D E C A B (1) D C E A B (1) D A E B C (1) C E D B A (1) C D B E A (1) C B D E A (1) B D C E A (1) B D C A E (1) B C E A D (1) B A D C E (1) A E D C B (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A E B C D (1) A D E C B (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -2 -16 -6 B 2 0 -2 -14 -6 C 2 2 0 -20 10 D 16 14 20 0 22 E 6 6 -10 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -16 -6 B 2 0 -2 -14 -6 C 2 2 0 -20 10 D 16 14 20 0 22 E 6 6 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 E=23 A=12 C=11 B=11 so C is eliminated. Round 2 votes counts: D=44 E=30 B=14 A=12 so A is eliminated. Round 3 votes counts: D=49 E=36 B=15 so B is eliminated. Round 4 votes counts: D=61 E=39 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:236 C:197 B:190 E:190 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -16 -6 B 2 0 -2 -14 -6 C 2 2 0 -20 10 D 16 14 20 0 22 E 6 6 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -16 -6 B 2 0 -2 -14 -6 C 2 2 0 -20 10 D 16 14 20 0 22 E 6 6 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -16 -6 B 2 0 -2 -14 -6 C 2 2 0 -20 10 D 16 14 20 0 22 E 6 6 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8486: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) D B C E A (7) B C D A E (7) E A D B C (6) D E B A C (6) E A D C B (5) D E A B C (5) C B A E D (5) A E C B D (5) C A E B D (4) A E B C D (4) E A C D B (3) D C B E A (3) D B E C A (3) D B E A C (3) C D E A B (3) E D A B C (2) D C E A B (2) C B A D E (2) C A B E D (2) B D C A E (2) A C E B D (2) E D A C B (1) E A C B D (1) E A B D C (1) C A E D B (1) B D E A C (1) B D A E C (1) B C A D E (1) B A E D C (1) B A E C D (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 -2 -10 -4 B 4 0 4 0 -4 C 2 -4 0 0 0 D 10 0 0 0 10 E 4 4 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.496736 C: 0.000000 D: 0.503264 E: 0.000000 Sum of squares = 0.500021302982 Cumulative probabilities = A: 0.000000 B: 0.496736 C: 0.496736 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -10 -4 B 4 0 4 0 -4 C 2 -4 0 0 0 D 10 0 0 0 10 E 4 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=26 E=19 B=14 A=12 so A is eliminated. Round 2 votes counts: E=29 D=29 C=28 B=14 so B is eliminated. Round 3 votes counts: C=36 D=33 E=31 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:210 B:202 C:199 E:199 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 -10 -4 B 4 0 4 0 -4 C 2 -4 0 0 0 D 10 0 0 0 10 E 4 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -10 -4 B 4 0 4 0 -4 C 2 -4 0 0 0 D 10 0 0 0 10 E 4 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -10 -4 B 4 0 4 0 -4 C 2 -4 0 0 0 D 10 0 0 0 10 E 4 4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8487: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) C D A E B (9) A C D E B (9) D C E B A (7) A B E C D (6) A E B C D (5) A C E D B (5) D C B E A (4) C D E A B (4) B D C E A (4) B A E D C (4) D C B A E (3) B E D C A (3) E D C B A (2) E C D A B (2) E A B C D (2) C D A B E (2) B D C A E (2) A E C D B (2) A C D B E (2) A B C D E (2) E D B C A (1) E B A D C (1) D E C B A (1) C E D A B (1) C A D E B (1) C A D B E (1) B E D A C (1) B D E C A (1) B A D C E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 10 4 6 12 B -10 0 -14 -12 -6 C -4 14 0 10 16 D -6 12 -10 0 6 E -12 6 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 6 12 B -10 0 -14 -12 -6 C -4 14 0 10 16 D -6 12 -10 0 6 E -12 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=26 C=18 D=15 E=8 so E is eliminated. Round 2 votes counts: A=35 B=27 C=20 D=18 so D is eliminated. Round 3 votes counts: C=37 A=35 B=28 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:216 D:201 E:186 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 6 12 B -10 0 -14 -12 -6 C -4 14 0 10 16 D -6 12 -10 0 6 E -12 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 6 12 B -10 0 -14 -12 -6 C -4 14 0 10 16 D -6 12 -10 0 6 E -12 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 6 12 B -10 0 -14 -12 -6 C -4 14 0 10 16 D -6 12 -10 0 6 E -12 6 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8488: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) A B E D C (8) C D A B E (6) E A B D C (5) D C B E A (5) D B A E C (5) A E B D C (5) A E B C D (5) E B A D C (4) E A B C D (4) D C B A E (4) B E A D C (4) D B C E A (3) D B C A E (3) C E A B D (3) C D B E A (3) C D B A E (3) E C A B D (2) E A C B D (2) D B E C A (2) B D A E C (2) B A E D C (2) E C B D A (1) E B D A C (1) D A C B E (1) D A B C E (1) C E D B A (1) C E A D B (1) C D E A B (1) C D A E B (1) B E D A C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 2 -8 -6 B 8 0 10 0 6 C -2 -10 0 -12 -6 D 8 0 12 0 0 E 6 -6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.609502 C: 0.000000 D: 0.390498 E: 0.000000 Sum of squares = 0.523981515281 Cumulative probabilities = A: 0.000000 B: 0.609502 C: 0.609502 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -8 -6 B 8 0 10 0 6 C -2 -10 0 -12 -6 D 8 0 12 0 0 E 6 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999922 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=24 E=19 A=19 B=9 so B is eliminated. Round 2 votes counts: C=29 D=26 E=24 A=21 so A is eliminated. Round 3 votes counts: E=44 C=30 D=26 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:212 D:210 E:203 A:190 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -8 -6 B 8 0 10 0 6 C -2 -10 0 -12 -6 D 8 0 12 0 0 E 6 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999922 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -8 -6 B 8 0 10 0 6 C -2 -10 0 -12 -6 D 8 0 12 0 0 E 6 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999922 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -8 -6 B 8 0 10 0 6 C -2 -10 0 -12 -6 D 8 0 12 0 0 E 6 -6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999922 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8489: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) E C A B D (5) D B A C E (5) C A E D B (5) C A D E B (5) C A D B E (5) E B D A C (4) D A B C E (4) E C B A D (3) E B C D A (3) D B A E C (3) D A C B E (3) B D E A C (3) B D A E C (3) A D C E B (3) A D C B E (3) E A D C B (2) D A E B C (2) D A B E C (2) C E B A D (2) C E A D B (2) C B A D E (2) B E D C A (2) B E C D A (2) B C D A E (2) E D B A C (1) E D A B C (1) E C A D B (1) E B A D C (1) E A D B C (1) D B E A C (1) D B C A E (1) D A C E B (1) C D A B E (1) C B E A D (1) B E D A C (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -6 2 8 B 4 0 8 -8 -2 C 6 -8 0 0 2 D -2 8 0 0 10 E -8 2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.396429 D: 0.603571 E: 0.000000 Sum of squares = 0.52145378522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.396429 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 8 B 4 0 8 -8 -2 C 6 -8 0 0 2 D -2 8 0 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=23 D=22 B=17 A=8 so A is eliminated. Round 2 votes counts: E=30 D=28 C=25 B=17 so B is eliminated. Round 3 votes counts: D=37 E=35 C=28 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 B:201 A:200 C:200 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 2 8 B 4 0 8 -8 -2 C 6 -8 0 0 2 D -2 8 0 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 8 B 4 0 8 -8 -2 C 6 -8 0 0 2 D -2 8 0 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 8 B 4 0 8 -8 -2 C 6 -8 0 0 2 D -2 8 0 0 10 E -8 2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8490: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (15) A D E B C (9) D A C B E (8) B E C A D (6) E B C A D (5) D A C E B (5) B C E A D (5) D A E C B (4) E B A C D (3) E A D B C (3) E A B D C (3) D C A B E (3) C E B D A (3) E B A D C (2) C D B E A (2) C D B A E (2) C D A B E (2) C B D E A (2) C B D A E (2) B E A C D (2) B A E D C (2) E D A C B (1) D C A E B (1) C D E B A (1) C D A E B (1) B C A E D (1) B C A D E (1) B A E C D (1) A E D B C (1) A E B D C (1) A D C E B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -4 -4 -6 B 10 0 -6 10 12 C 4 6 0 8 12 D 4 -10 -8 0 -8 E 6 -12 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -4 -6 B 10 0 -6 10 12 C 4 6 0 8 12 D 4 -10 -8 0 -8 E 6 -12 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=21 B=18 E=17 A=14 so A is eliminated. Round 2 votes counts: D=32 C=30 E=19 B=19 so E is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:213 E:195 D:189 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -4 -6 B 10 0 -6 10 12 C 4 6 0 8 12 D 4 -10 -8 0 -8 E 6 -12 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -4 -6 B 10 0 -6 10 12 C 4 6 0 8 12 D 4 -10 -8 0 -8 E 6 -12 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -4 -6 B 10 0 -6 10 12 C 4 6 0 8 12 D 4 -10 -8 0 -8 E 6 -12 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8491: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) A E C D B (9) E A D B C (6) C D B E A (6) A E D B C (6) E B D A C (5) B D C E A (5) E D B A C (4) C B D E A (4) B C D E A (4) E A B D C (3) D B C E A (3) C D B A E (3) B E D C A (3) B D E C A (3) D E B C A (2) D C B E A (2) C D A B E (2) A C B D E (2) E D C A B (1) E D A B C (1) E B D C A (1) E B A D C (1) E A D C B (1) D E C B A (1) D C E A B (1) D B E C A (1) C D E B A (1) C B D A E (1) C A D E B (1) B C D A E (1) A E D C B (1) A E B C D (1) A C E B D (1) A C D B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 8 -10 -18 B 2 0 24 -6 -14 C -8 -24 0 -22 -22 D 10 6 22 0 -12 E 18 14 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 -10 -18 B 2 0 24 -6 -14 C -8 -24 0 -22 -22 D 10 6 22 0 -12 E 18 14 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=23 C=18 B=16 D=10 so D is eliminated. Round 2 votes counts: A=33 E=26 C=21 B=20 so B is eliminated. Round 3 votes counts: C=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:233 D:213 B:203 A:189 C:162 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 -10 -18 B 2 0 24 -6 -14 C -8 -24 0 -22 -22 D 10 6 22 0 -12 E 18 14 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -10 -18 B 2 0 24 -6 -14 C -8 -24 0 -22 -22 D 10 6 22 0 -12 E 18 14 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -10 -18 B 2 0 24 -6 -14 C -8 -24 0 -22 -22 D 10 6 22 0 -12 E 18 14 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8492: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) C A D B E (8) A C B D E (8) D E B C A (7) A C B E D (7) E D B C A (6) C D A E B (5) B A C E D (5) E B D A C (4) D E C A B (4) D C E A B (4) D C A E B (3) D C A B E (3) C A D E B (3) C A B D E (3) B E A C D (3) B A E C D (3) E B D C A (2) D E B A C (2) B E D A C (2) D E C B A (1) C A E D B (1) C A E B D (1) C A B E D (1) B E C A D (1) B E A D C (1) B A D C E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 -6 -6 6 B -4 0 -6 -16 -6 C 6 6 0 0 8 D 6 16 0 0 6 E -6 6 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.681837 D: 0.318163 E: 0.000000 Sum of squares = 0.566129506603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.681837 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -6 6 B -4 0 -6 -16 -6 C 6 6 0 0 8 D 6 16 0 0 6 E -6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=22 C=22 B=16 A=16 so B is eliminated. Round 2 votes counts: E=29 A=25 D=24 C=22 so C is eliminated. Round 3 votes counts: A=42 E=29 D=29 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:210 A:199 E:193 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -6 6 B -4 0 -6 -16 -6 C 6 6 0 0 8 D 6 16 0 0 6 E -6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 6 B -4 0 -6 -16 -6 C 6 6 0 0 8 D 6 16 0 0 6 E -6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 6 B -4 0 -6 -16 -6 C 6 6 0 0 8 D 6 16 0 0 6 E -6 6 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8493: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (9) E A C D B (7) B D C A E (6) D B A E C (5) D A C E B (5) E B A C D (4) C E B A D (4) B C E D A (4) B C E A D (4) A D E C B (4) C E A D B (3) C D B A E (3) B E A D C (3) B C D E A (3) E C A B D (2) E A B C D (2) D C B A E (2) C D A E B (2) C B E A D (2) B E C A D (2) A E D C B (2) A E C D B (2) E C A D B (1) E B C A D (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C B D (1) E A B D C (1) D B C A E (1) D B A C E (1) D A E C B (1) C E B D A (1) C B D E A (1) B E C D A (1) B E A C D (1) B D C E A (1) B D A C E (1) B A E D C (1) A E D B C (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -24 12 6 -4 B 24 0 10 12 2 C -12 -10 0 2 -10 D -6 -12 -2 0 -6 E 4 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 12 6 -4 B 24 0 10 12 2 C -12 -10 0 2 -10 D -6 -12 -2 0 -6 E 4 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=22 C=16 D=15 A=11 so A is eliminated. Round 2 votes counts: B=36 E=27 D=21 C=16 so C is eliminated. Round 3 votes counts: B=39 E=35 D=26 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:224 E:209 A:195 D:187 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 12 6 -4 B 24 0 10 12 2 C -12 -10 0 2 -10 D -6 -12 -2 0 -6 E 4 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 12 6 -4 B 24 0 10 12 2 C -12 -10 0 2 -10 D -6 -12 -2 0 -6 E 4 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 12 6 -4 B 24 0 10 12 2 C -12 -10 0 2 -10 D -6 -12 -2 0 -6 E 4 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999996856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8494: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (5) D A C E B (5) B E C A D (5) E C D B A (4) E B C A D (4) D C A E B (4) D A C B E (4) B E A C D (4) B A C E D (4) E D B C A (3) E B C D A (3) D E B A C (3) D C E A B (3) D A B E C (3) D A B C E (3) C D A E B (3) A D C B E (3) A C B D E (3) E D C B A (2) D E C A B (2) D B A E C (2) C A B E D (2) B A E C D (2) B A D E C (2) A B C E D (2) A B C D E (2) E B D C A (1) D B E A C (1) D A E B C (1) C E D B A (1) C E D A B (1) C E B A D (1) C B E A D (1) C A D B E (1) B E C D A (1) B E A D C (1) B C A E D (1) B A E D C (1) A D B E C (1) A D B C E (1) A C D B E (1) A C B E D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 4 -10 8 B 4 0 6 -14 8 C -4 -6 0 -6 -4 D 10 14 6 0 10 E -8 -8 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 -10 8 B 4 0 6 -14 8 C -4 -6 0 -6 -4 D 10 14 6 0 10 E -8 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=21 E=17 A=16 C=10 so C is eliminated. Round 2 votes counts: D=39 B=22 E=20 A=19 so A is eliminated. Round 3 votes counts: D=46 B=34 E=20 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:202 A:199 C:190 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -10 8 B 4 0 6 -14 8 C -4 -6 0 -6 -4 D 10 14 6 0 10 E -8 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -10 8 B 4 0 6 -14 8 C -4 -6 0 -6 -4 D 10 14 6 0 10 E -8 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -10 8 B 4 0 6 -14 8 C -4 -6 0 -6 -4 D 10 14 6 0 10 E -8 -8 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8495: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) D A B E C (7) C E D B A (5) C E B A D (5) B A E C D (5) E B A C D (4) D E B A C (4) A C B D E (4) A B C E D (4) E D C B A (3) E B A D C (3) D E C B A (3) D C E B A (3) B E A C D (3) B A C E D (3) E C B A D (2) E B D A C (2) D A C B E (2) C E B D A (2) C A D B E (2) C A B E D (2) A D B C E (2) A C B E D (2) A B E D C (2) A B D E C (2) E D B C A (1) E C D B A (1) E B C A D (1) D E C A B (1) D C A E B (1) D C A B E (1) D B A E C (1) D A B C E (1) C D E A B (1) C D A E B (1) C A D E B (1) C A B D E (1) B D A E C (1) B A E D C (1) B A D E C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 12 4 -2 B 8 0 0 4 -2 C -12 0 0 2 2 D -4 -4 -2 0 -4 E 2 2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.750000 Sum of squares = 0.59374999998 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.250000 D: 0.250000 E: 1.000000 A B C D E A 0 -8 12 4 -2 B 8 0 0 4 -2 C -12 0 0 2 2 D -4 -4 -2 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.750000 Sum of squares = 0.593749999805 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=20 A=18 E=17 B=14 so B is eliminated. Round 2 votes counts: D=32 A=28 E=20 C=20 so E is eliminated. Round 3 votes counts: D=38 A=38 C=24 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:205 A:203 E:203 C:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 4 -2 B 8 0 0 4 -2 C -12 0 0 2 2 D -4 -4 -2 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.750000 Sum of squares = 0.593749999805 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 4 -2 B 8 0 0 4 -2 C -12 0 0 2 2 D -4 -4 -2 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.750000 Sum of squares = 0.593749999805 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 4 -2 B 8 0 0 4 -2 C -12 0 0 2 2 D -4 -4 -2 0 -4 E 2 2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.750000 Sum of squares = 0.593749999805 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8496: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (8) D B E A C (7) C E A B D (7) E A C D B (6) D B A E C (6) E A D B C (5) E B D C A (4) C A E D B (4) B D E A C (4) E C B D A (3) E C A D B (3) E C A B D (3) C B E D A (3) C A E B D (3) B D E C A (3) B D C E A (3) A E D B C (3) E C B A D (2) C E B A D (2) C B D A E (2) A E C D B (2) A D B C E (2) E D B A C (1) D E B A C (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B D A (1) C B A E D (1) C A B D E (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) A D E C B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 4 4 10 -18 B -4 0 -8 -8 -20 C -4 8 0 8 -12 D -10 8 -8 0 -22 E 18 20 12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 10 -18 B -4 0 -8 -8 -20 C -4 8 0 8 -12 D -10 8 -8 0 -22 E 18 20 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=24 A=18 D=17 B=14 so B is eliminated. Round 2 votes counts: D=29 E=27 C=26 A=18 so A is eliminated. Round 3 votes counts: C=35 D=33 E=32 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:236 A:200 C:200 D:184 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 10 -18 B -4 0 -8 -8 -20 C -4 8 0 8 -12 D -10 8 -8 0 -22 E 18 20 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 10 -18 B -4 0 -8 -8 -20 C -4 8 0 8 -12 D -10 8 -8 0 -22 E 18 20 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 10 -18 B -4 0 -8 -8 -20 C -4 8 0 8 -12 D -10 8 -8 0 -22 E 18 20 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8497: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (12) C D A E B (8) E B A C D (7) C D B A E (7) B A E D C (6) A E B D C (6) E A C D B (4) D C B A E (4) C D A B E (4) E C A D B (3) D C A B E (3) D B C A E (3) C D E B A (3) C D E A B (3) C D B E A (3) E A C B D (2) E A B C D (2) C E B D A (2) B D C A E (2) A E D C B (2) A E C D B (2) A D E C B (2) E B C D A (1) E B A D C (1) D C A E B (1) D A C E B (1) C E D A B (1) C E A D B (1) B E D C A (1) B E C D A (1) B D A E C (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -2 2 6 B 8 0 -14 -12 -6 C 2 14 0 8 -6 D -2 12 -8 0 -8 E -6 6 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102046 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -8 -2 2 6 B 8 0 -14 -12 -6 C 2 14 0 8 -6 D -2 12 -8 0 -8 E -6 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102058 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=23 E=20 A=13 D=12 so D is eliminated. Round 2 votes counts: C=40 B=26 E=20 A=14 so A is eliminated. Round 3 votes counts: C=42 E=32 B=26 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:207 A:199 D:197 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 2 6 B 8 0 -14 -12 -6 C 2 14 0 8 -6 D -2 12 -8 0 -8 E -6 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102058 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 2 6 B 8 0 -14 -12 -6 C 2 14 0 8 -6 D -2 12 -8 0 -8 E -6 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102058 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 2 6 B 8 0 -14 -12 -6 C 2 14 0 8 -6 D -2 12 -8 0 -8 E -6 6 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102058 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8498: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) B E C A D (8) C B E A D (6) E B A C D (5) D C B E A (5) D A C E B (5) A E B D C (5) A D E B C (5) D C A E B (4) D B C E A (4) C E B A D (4) C B E D A (4) D A B C E (3) C E A B D (3) A D E C B (3) E B C A D (2) D A E C B (2) D A B E C (2) B E A C D (2) A E D B C (2) A E C B D (2) E A C B D (1) E A B C D (1) D C E B A (1) D C A B E (1) D A E B C (1) C D B E A (1) C A E D B (1) B E D C A (1) B E D A C (1) B D E C A (1) B D E A C (1) B C E D A (1) B C E A D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 4 4 24 -6 B -4 0 10 18 -16 C -4 -10 0 4 -8 D -24 -18 -4 0 -20 E 6 16 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 24 -6 B -4 0 10 18 -16 C -4 -10 0 4 -8 D -24 -18 -4 0 -20 E 6 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=28 A=28 C=19 B=16 E=9 so E is eliminated. Round 2 votes counts: A=30 D=28 B=23 C=19 so C is eliminated. Round 3 votes counts: B=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:225 A:213 B:204 C:191 D:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 24 -6 B -4 0 10 18 -16 C -4 -10 0 4 -8 D -24 -18 -4 0 -20 E 6 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 24 -6 B -4 0 10 18 -16 C -4 -10 0 4 -8 D -24 -18 -4 0 -20 E 6 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 24 -6 B -4 0 10 18 -16 C -4 -10 0 4 -8 D -24 -18 -4 0 -20 E 6 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8499: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (10) B E C A D (9) E C B D A (6) E B C D A (6) D A E C B (5) E D C B A (4) D E C A B (4) D E A C B (4) B E C D A (4) B C E A D (4) B C A E D (4) A D B C E (4) A B D C E (4) A B C D E (4) D E C B A (3) A D C E B (3) E C D B A (2) D C E A B (2) D A C E B (2) C E B D A (2) B A E C D (2) B A C E D (2) A D B E C (2) E D B A C (1) E B A D C (1) D C E B A (1) D C A E B (1) B C E D A (1) B A E D C (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -6 4 -8 B 6 0 0 4 6 C 6 0 0 -4 -8 D -4 -4 4 0 0 E 8 -6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.719396 C: 0.280604 D: 0.000000 E: 0.000000 Sum of squares = 0.596268808598 Cumulative probabilities = A: 0.000000 B: 0.719396 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 4 -8 B 6 0 0 4 6 C 6 0 0 -4 -8 D -4 -4 4 0 0 E 8 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.510204085823 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 D=22 E=20 C=2 so C is eliminated. Round 2 votes counts: A=29 B=27 E=22 D=22 so E is eliminated. Round 3 votes counts: B=42 D=29 A=29 so D is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:205 D:198 C:197 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 4 -8 B 6 0 0 4 6 C 6 0 0 -4 -8 D -4 -4 4 0 0 E 8 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.510204085823 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 4 -8 B 6 0 0 4 6 C 6 0 0 -4 -8 D -4 -4 4 0 0 E 8 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.510204085823 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 4 -8 B 6 0 0 4 6 C 6 0 0 -4 -8 D -4 -4 4 0 0 E 8 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.428571 D: 0.000000 E: 0.000000 Sum of squares = 0.510204085823 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8500: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (13) A C E D B (12) D B E A C (8) A C D E B (8) B E D C A (5) D E A C B (4) D A C E B (4) D B A C E (3) C A E D B (3) C A E B D (3) B E C A D (3) A C E B D (3) E D C A B (2) E C A D B (2) E C A B D (2) D B E C A (2) D B A E C (2) D A E C B (2) B E C D A (2) B D A C E (2) B A D C E (2) A D C E B (2) E D B C A (1) E C B A D (1) D E A B C (1) C B A E D (1) C A B E D (1) B D C E A (1) B D C A E (1) B C A E D (1) B A C D E (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 12 -6 6 B -2 0 -4 -14 0 C -12 4 0 -12 0 D 6 14 12 0 14 E -6 0 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 -6 6 B -2 0 -4 -14 0 C -12 4 0 -12 0 D 6 14 12 0 14 E -6 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=27 D=26 E=8 C=8 so E is eliminated. Round 2 votes counts: B=31 D=29 A=27 C=13 so C is eliminated. Round 3 votes counts: A=38 B=33 D=29 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:223 A:207 B:190 C:190 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 12 -6 6 B -2 0 -4 -14 0 C -12 4 0 -12 0 D 6 14 12 0 14 E -6 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 -6 6 B -2 0 -4 -14 0 C -12 4 0 -12 0 D 6 14 12 0 14 E -6 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 -6 6 B -2 0 -4 -14 0 C -12 4 0 -12 0 D 6 14 12 0 14 E -6 0 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8501: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (12) C E B D A (11) D A B E C (8) D C E B A (6) C A E B D (5) D A C E B (4) A D B E C (4) D E B C A (3) D E B A C (3) D C A E B (3) D B E A C (3) A C B E D (3) A B E D C (3) A B E C D (3) E C B D A (2) D E C B A (2) C E D B A (2) C D A E B (2) A C D B E (2) A B D E C (2) E B C D A (1) E B C A D (1) D C E A B (1) D A E B C (1) D A C B E (1) C D E B A (1) C B E A D (1) C A D E B (1) B E C A D (1) B E A C D (1) B A E D C (1) B A E C D (1) B A C E D (1) A D C B E (1) A D B C E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -10 -8 -2 B 6 0 -22 2 -22 C 10 22 0 6 20 D 8 -2 -6 0 0 E 2 22 -20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -8 -2 B 6 0 -22 2 -22 C 10 22 0 6 20 D 8 -2 -6 0 0 E 2 22 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=35 C=35 A=21 B=5 E=4 so E is eliminated. Round 2 votes counts: C=37 D=35 A=21 B=7 so B is eliminated. Round 3 votes counts: C=40 D=35 A=25 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 E:202 D:200 A:187 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 -2 B 6 0 -22 2 -22 C 10 22 0 6 20 D 8 -2 -6 0 0 E 2 22 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 -2 B 6 0 -22 2 -22 C 10 22 0 6 20 D 8 -2 -6 0 0 E 2 22 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 -2 B 6 0 -22 2 -22 C 10 22 0 6 20 D 8 -2 -6 0 0 E 2 22 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8502: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) E C A B D (7) D E B C A (5) D B C A E (5) D B A C E (5) A B C D E (5) E D C B A (4) D B C E A (4) C B A D E (4) A E C B D (4) E D A C B (3) D B E C A (3) D A B E C (3) C E B A D (3) B C D A E (3) A C B E D (3) A C B D E (3) E C D B A (2) E C B A D (2) D A B C E (2) C B D E A (2) E D B C A (1) E D B A C (1) E C D A B (1) E A D C B (1) D E B A C (1) D C B E A (1) C E A B D (1) C B E D A (1) C B E A D (1) C A B E D (1) C A B D E (1) B D C A E (1) B D A C E (1) B A D C E (1) B A C D E (1) A E B C D (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -6 2 -8 B 4 0 -12 16 4 C 6 12 0 14 2 D -2 -16 -14 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999734 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 2 -8 B 4 0 -12 16 4 C 6 12 0 14 2 D -2 -16 -14 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991263 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=29 A=18 C=14 B=7 so B is eliminated. Round 2 votes counts: E=32 D=31 A=20 C=17 so C is eliminated. Round 3 votes counts: E=38 D=36 A=26 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:217 B:206 E:199 A:192 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 2 -8 B 4 0 -12 16 4 C 6 12 0 14 2 D -2 -16 -14 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991263 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 2 -8 B 4 0 -12 16 4 C 6 12 0 14 2 D -2 -16 -14 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991263 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 2 -8 B 4 0 -12 16 4 C 6 12 0 14 2 D -2 -16 -14 0 4 E 8 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991263 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8503: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (7) C B D A E (7) C B E A D (6) D A E B C (5) A E D C B (5) A E D B C (5) E B D A C (4) B C D E A (4) E D A B C (3) D A C E B (3) B E C D A (3) B D C A E (3) E A C D B (2) D C A B E (2) C D A B E (2) C A E D B (2) C A D B E (2) B E D A C (2) B D C E A (2) B C E A D (2) A D E C B (2) E D B A C (1) E C A B D (1) E B C A D (1) E B A D C (1) E A D C B (1) E A C B D (1) D E A B C (1) D B A E C (1) D A E C B (1) D A C B E (1) D A B E C (1) D A B C E (1) C E B A D (1) C B E D A (1) C B A E D (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D C A (1) B E C A D (1) B D E A C (1) B C E D A (1) B C D A E (1) A E C D B (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 6 -4 4 B -8 0 6 -6 -4 C -6 -6 0 -10 -6 D 4 6 10 0 -10 E -4 4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.40740740741 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 A B C D E A 0 8 6 -4 4 B -8 0 6 -6 -4 C -6 -6 0 -10 -6 D 4 6 10 0 -10 E -4 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407523 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=22 B=21 D=16 A=16 so D is eliminated. Round 2 votes counts: A=28 C=27 E=23 B=22 so B is eliminated. Round 3 votes counts: C=40 E=31 A=29 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:208 A:207 D:205 B:194 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 6 -4 4 B -8 0 6 -6 -4 C -6 -6 0 -10 -6 D 4 6 10 0 -10 E -4 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407523 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -4 4 B -8 0 6 -6 -4 C -6 -6 0 -10 -6 D 4 6 10 0 -10 E -4 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407523 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -4 4 B -8 0 6 -6 -4 C -6 -6 0 -10 -6 D 4 6 10 0 -10 E -4 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.555556 B: 0.000000 C: 0.000000 D: 0.222222 E: 0.222222 Sum of squares = 0.407407407523 Cumulative probabilities = A: 0.555556 B: 0.555556 C: 0.555556 D: 0.777778 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8504: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) D A B E C (7) B D A C E (7) A D B E C (6) D A E C B (5) C E B A D (5) C E A D B (5) A D E B C (5) E A C D B (4) D B A C E (4) B D C A E (4) A E D C B (4) B C D E A (3) B C D A E (3) E C A B D (2) E A D C B (2) C B E A D (2) C B D E A (2) B D A E C (2) B C E A D (2) B C A E D (2) B A D E C (2) A D E C B (2) E B C A D (1) D C E A B (1) D C B E A (1) D A E B C (1) D A B C E (1) C E D A B (1) C E A B D (1) C D B E A (1) C B E D A (1) B C E D A (1) B C A D E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 12 8 10 16 B -12 0 8 -16 4 C -8 -8 0 -12 -4 D -10 16 12 0 16 E -16 -4 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 10 16 B -12 0 8 -16 4 C -8 -8 0 -12 -4 D -10 16 12 0 16 E -16 -4 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=20 A=19 C=18 E=16 so E is eliminated. Round 2 votes counts: B=28 C=27 A=25 D=20 so D is eliminated. Round 3 votes counts: A=39 B=32 C=29 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:223 D:217 B:192 C:184 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 10 16 B -12 0 8 -16 4 C -8 -8 0 -12 -4 D -10 16 12 0 16 E -16 -4 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 10 16 B -12 0 8 -16 4 C -8 -8 0 -12 -4 D -10 16 12 0 16 E -16 -4 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 10 16 B -12 0 8 -16 4 C -8 -8 0 -12 -4 D -10 16 12 0 16 E -16 -4 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999721 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8505: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) A C D E B (10) E B D C A (8) C D A B E (8) E B A D C (7) D C B A E (6) B D C E A (6) B E D C A (5) D C B E A (4) A C D B E (4) E D C B A (3) E A C D B (3) E B D A C (2) E B A C D (2) D C E B A (2) C D A E B (2) B E A D C (2) B A E C D (2) A E C D B (2) A B E C D (2) A B C D E (2) E A D C B (1) E A B D C (1) D C E A B (1) D C A B E (1) C D E A B (1) B E A C D (1) B A C D E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 6 2 -18 B 2 0 2 4 -10 C -6 -2 0 2 -2 D -2 -4 -2 0 -2 E 18 10 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 2 -18 B 2 0 2 4 -10 C -6 -2 0 2 -2 D -2 -4 -2 0 -2 E 18 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=21 B=17 D=14 C=11 so C is eliminated. Round 2 votes counts: E=37 D=25 A=21 B=17 so B is eliminated. Round 3 votes counts: E=45 D=31 A=24 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:199 C:196 D:195 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 2 -18 B 2 0 2 4 -10 C -6 -2 0 2 -2 D -2 -4 -2 0 -2 E 18 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 2 -18 B 2 0 2 4 -10 C -6 -2 0 2 -2 D -2 -4 -2 0 -2 E 18 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 2 -18 B 2 0 2 4 -10 C -6 -2 0 2 -2 D -2 -4 -2 0 -2 E 18 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8506: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (16) D B A C E (12) E C A D B (11) B E C A D (10) B D A C E (9) E C B A D (7) E C A B D (7) B E C D A (4) B D E C A (4) A C E D B (4) B D A E C (3) E C D A B (2) E B C A D (2) B A C E D (2) D E C A B (1) D A C B E (1) D A B C E (1) C A E B D (1) B D E A C (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 2 -8 2 B 10 0 -2 4 -2 C -2 2 0 2 -4 D 8 -4 -2 0 -2 E -2 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408171 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 A B C D E A 0 -10 2 -8 2 B 10 0 -2 4 -2 C -2 2 0 2 -4 D 8 -4 -2 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408211 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=31 E=29 A=5 C=1 so C is eliminated. Round 2 votes counts: B=34 D=31 E=29 A=6 so A is eliminated. Round 3 votes counts: B=35 E=34 D=31 so D is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:205 E:203 D:200 C:199 A:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 -8 2 B 10 0 -2 4 -2 C -2 2 0 2 -4 D 8 -4 -2 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408211 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -8 2 B 10 0 -2 4 -2 C -2 2 0 2 -4 D 8 -4 -2 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408211 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -8 2 B 10 0 -2 4 -2 C -2 2 0 2 -4 D 8 -4 -2 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.142857 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.551020408211 Cumulative probabilities = A: 0.142857 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8507: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) D E C A B (12) D A C E B (10) B E C A D (7) A B C E D (6) D E B C A (5) B E D C A (5) A D C E B (4) A C D E B (4) B D E C A (3) B A C D E (3) A C E D B (3) A C E B D (3) E D C B A (2) D E C B A (2) D B E C A (2) B D A C E (2) A C B E D (2) A C B D E (2) E D B C A (1) D E B A C (1) D E A C B (1) D C E A B (1) D B A C E (1) D A E C B (1) C A E D B (1) B E D A C (1) B C E A D (1) A D B C E (1) Total count = 100 A B C D E A 0 2 16 0 12 B -2 0 4 -4 -2 C -16 -4 0 -10 14 D 0 4 10 0 10 E -12 2 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.565529 B: 0.000000 C: 0.000000 D: 0.434471 E: 0.000000 Sum of squares = 0.508588195296 Cumulative probabilities = A: 0.565529 B: 0.565529 C: 0.565529 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 0 12 B -2 0 4 -4 -2 C -16 -4 0 -10 14 D 0 4 10 0 10 E -12 2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999872 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=35 A=25 E=3 C=1 so C is eliminated. Round 2 votes counts: D=36 B=35 A=26 E=3 so E is eliminated. Round 3 votes counts: D=39 B=35 A=26 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:215 D:212 B:198 C:192 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 0 12 B -2 0 4 -4 -2 C -16 -4 0 -10 14 D 0 4 10 0 10 E -12 2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999872 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 0 12 B -2 0 4 -4 -2 C -16 -4 0 -10 14 D 0 4 10 0 10 E -12 2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999872 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 0 12 B -2 0 4 -4 -2 C -16 -4 0 -10 14 D 0 4 10 0 10 E -12 2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999872 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8508: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (7) D B A E C (6) D B A C E (6) C E D A B (6) C D E B A (6) B A D E C (6) E C A B D (5) E A B C D (4) D C E B A (4) C E A B D (4) D C B A E (3) B A C E D (3) E C D A B (2) E C A D B (2) E A D B C (2) E A C B D (2) D E C A B (2) D E A B C (2) D C E A B (2) C D B E A (2) C D B A E (2) B A D C E (2) B A C D E (2) A E B C D (2) A B E C D (2) A B D E C (2) E D C A B (1) D E A C B (1) D C B E A (1) D B C A E (1) D A E B C (1) C D E A B (1) C B A D E (1) B D A E C (1) B D A C E (1) B A E C D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -4 -2 2 B 10 0 -8 -6 0 C 4 8 0 8 8 D 2 6 -8 0 10 E -2 0 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -2 2 B 10 0 -8 -6 0 C 4 8 0 8 8 D 2 6 -8 0 10 E -2 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=29 C=29 E=18 B=16 A=8 so A is eliminated. Round 2 votes counts: D=29 C=29 E=21 B=21 so E is eliminated. Round 3 votes counts: C=40 D=32 B=28 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:205 B:198 A:193 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -4 -2 2 B 10 0 -8 -6 0 C 4 8 0 8 8 D 2 6 -8 0 10 E -2 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -2 2 B 10 0 -8 -6 0 C 4 8 0 8 8 D 2 6 -8 0 10 E -2 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -2 2 B 10 0 -8 -6 0 C 4 8 0 8 8 D 2 6 -8 0 10 E -2 0 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8509: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) A C E B D (6) C A B E D (5) D E B A C (4) D C B A E (4) D B C E A (4) D A C B E (4) E B D A C (3) D E A B C (3) D A C E B (3) C B E A D (3) B D E C A (3) A E D B C (3) A E C B D (3) A D E B C (3) E B C A D (2) E B A D C (2) E A B C D (2) D B E A C (2) D A E B C (2) C B E D A (2) C A E B D (2) B E C D A (2) B E C A D (2) A E D C B (2) A E B C D (2) A D E C B (2) A D C E B (2) E A D B C (1) E A C B D (1) E A B D C (1) D C B E A (1) D C A B E (1) D B C A E (1) D A E C B (1) D A B E C (1) C D B A E (1) C D A B E (1) C B A D E (1) B C E D A (1) B C E A D (1) B C D E A (1) Total count = 100 A B C D E A 0 2 6 -8 0 B -2 0 10 -10 0 C -6 -10 0 -24 -12 D 8 10 24 0 8 E 0 0 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -8 0 B -2 0 10 -10 0 C -6 -10 0 -24 -12 D 8 10 24 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 A=23 C=15 E=12 B=10 so B is eliminated. Round 2 votes counts: D=43 A=23 C=18 E=16 so E is eliminated. Round 3 votes counts: D=46 A=30 C=24 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:225 E:202 A:200 B:199 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -8 0 B -2 0 10 -10 0 C -6 -10 0 -24 -12 D 8 10 24 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -8 0 B -2 0 10 -10 0 C -6 -10 0 -24 -12 D 8 10 24 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -8 0 B -2 0 10 -10 0 C -6 -10 0 -24 -12 D 8 10 24 0 8 E 0 0 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8510: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A C B E (10) C E B D A (7) E B A C D (6) E B A D C (5) D C A E B (5) A D B E C (5) D C A B E (4) D A E B C (4) C D A B E (4) D E A B C (3) C B E A D (3) B E C A D (3) B E A C D (3) E C B D A (2) E B C D A (2) C D E B A (2) C D B A E (2) A B E D C (2) E C D B A (1) E C B A D (1) E A B D C (1) D C E A B (1) D A E C B (1) D A C E B (1) D A B C E (1) C E D B A (1) C E B A D (1) C B E D A (1) C B D A E (1) C B A E D (1) C A D B E (1) B C E A D (1) B A E D C (1) A D E B C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -8 -6 -8 B 8 0 0 4 -10 C 8 0 0 8 -2 D 6 -4 -8 0 -4 E 8 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998287 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -8 -6 -8 B 8 0 0 4 -10 C 8 0 0 8 -2 D 6 -4 -8 0 -4 E 8 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=28 C=24 A=10 B=8 so B is eliminated. Round 2 votes counts: E=34 D=30 C=25 A=11 so A is eliminated. Round 3 votes counts: E=37 D=37 C=26 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:207 B:201 D:195 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -6 -8 B 8 0 0 4 -10 C 8 0 0 8 -2 D 6 -4 -8 0 -4 E 8 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -6 -8 B 8 0 0 4 -10 C 8 0 0 8 -2 D 6 -4 -8 0 -4 E 8 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -6 -8 B 8 0 0 4 -10 C 8 0 0 8 -2 D 6 -4 -8 0 -4 E 8 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8511: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) A E B C D (7) E A B C D (6) E D A B C (5) D E A B C (5) D C B E A (5) D C A B E (4) C D B E A (4) C B A D E (4) A E B D C (4) D A E B C (3) D A C E B (3) D A C B E (3) C B E A D (3) B C E A D (3) E D B C A (2) E D B A C (2) D C E B A (2) D C B A E (2) D A E C B (2) C D B A E (2) C B D A E (2) B C A E D (2) A C B D E (2) A B C E D (2) E D C B A (1) E C B D A (1) E B C A D (1) E A D B C (1) E A B D C (1) D E A C B (1) D C E A B (1) C B D E A (1) B A E C D (1) B A C E D (1) A D E B C (1) A D B C E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 20 -2 -4 B -8 0 14 0 -12 C -20 -14 0 0 -6 D 2 0 0 0 -2 E 4 12 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 20 -2 -4 B -8 0 14 0 -12 C -20 -14 0 0 -6 D 2 0 0 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=27 A=19 C=16 B=7 so B is eliminated. Round 2 votes counts: D=31 E=27 C=21 A=21 so C is eliminated. Round 3 votes counts: D=40 E=33 A=27 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:211 D:200 B:197 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 20 -2 -4 B -8 0 14 0 -12 C -20 -14 0 0 -6 D 2 0 0 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 20 -2 -4 B -8 0 14 0 -12 C -20 -14 0 0 -6 D 2 0 0 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 20 -2 -4 B -8 0 14 0 -12 C -20 -14 0 0 -6 D 2 0 0 0 -2 E 4 12 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999992561 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8512: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (18) E A C D B (16) B E D C A (7) A C D E B (7) E B D C A (6) E B D A C (5) E B A D C (5) C D A B E (4) A E C D B (4) C A D B E (3) B D C E A (3) A D C B E (3) A C D B E (3) E A C B D (2) D C B A E (2) D C A B E (2) C D B A E (2) A D C E B (2) E B A C D (1) E A B C D (1) D B C A E (1) D B A C E (1) C A D E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 2 -2 8 B 2 0 -4 -4 -2 C -2 4 0 -10 6 D 2 4 10 0 4 E -8 2 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 8 B 2 0 -4 -4 -2 C -2 4 0 -10 6 D 2 4 10 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=28 A=20 C=10 D=6 so D is eliminated. Round 2 votes counts: E=36 B=30 A=20 C=14 so C is eliminated. Round 3 votes counts: E=36 B=34 A=30 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:210 A:203 C:199 B:196 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -2 8 B 2 0 -4 -4 -2 C -2 4 0 -10 6 D 2 4 10 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 8 B 2 0 -4 -4 -2 C -2 4 0 -10 6 D 2 4 10 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 8 B 2 0 -4 -4 -2 C -2 4 0 -10 6 D 2 4 10 0 4 E -8 2 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999491 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8513: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) A C B E D (8) E D B A C (6) B D E C A (6) A C E D B (6) A C E B D (6) D E B A C (5) C A B E D (5) B E D A C (5) E A B D C (4) C A D E B (4) C A B D E (4) D E B C A (3) D B E A C (3) C B A D E (3) C A D B E (3) E D A B C (2) E B D A C (2) D B C E A (2) B E D C A (2) B C A E D (2) E B A D C (1) E A D C B (1) D E C A B (1) C D A B E (1) C B D A E (1) C A E B D (1) B E A D C (1) B C D A E (1) A E D C B (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 6 4 -4 B 2 0 8 6 10 C -6 -8 0 -6 -6 D -4 -6 6 0 -10 E 4 -10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 4 -4 B 2 0 8 6 10 C -6 -8 0 -6 -6 D -4 -6 6 0 -10 E 4 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=22 C=22 B=17 E=16 so E is eliminated. Round 2 votes counts: D=30 A=28 C=22 B=20 so B is eliminated. Round 3 votes counts: D=45 A=30 C=25 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:213 E:205 A:202 D:193 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 4 -4 B 2 0 8 6 10 C -6 -8 0 -6 -6 D -4 -6 6 0 -10 E 4 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 4 -4 B 2 0 8 6 10 C -6 -8 0 -6 -6 D -4 -6 6 0 -10 E 4 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 4 -4 B 2 0 8 6 10 C -6 -8 0 -6 -6 D -4 -6 6 0 -10 E 4 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999319 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8514: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) E A C D B (10) A C E B D (7) D B E A C (6) B C A D E (5) D B E C A (4) C B A E D (4) B D C A E (4) A E C B D (4) E C A D B (3) D E A B C (3) D B C E A (3) C E A D B (3) C A E B D (3) B D A E C (3) B C D A E (3) B A C E D (3) E D A C B (2) E A D C B (2) C E A B D (2) C A B E D (2) B C A E D (2) B A C D E (2) E C D A B (1) D E C B A (1) D E B C A (1) C E D A B (1) C B D E A (1) A E C D B (1) A E B C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 16 -8 12 -8 B -16 0 -18 -4 -12 C 8 18 0 20 -6 D -12 4 -20 0 -6 E 8 12 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 -8 12 -8 B -16 0 -18 -4 -12 C 8 18 0 20 -6 D -12 4 -20 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=22 E=18 C=16 A=15 so A is eliminated. Round 2 votes counts: D=29 E=24 C=24 B=23 so B is eliminated. Round 3 votes counts: C=40 D=36 E=24 so E is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:216 A:206 D:183 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 -8 12 -8 B -16 0 -18 -4 -12 C 8 18 0 20 -6 D -12 4 -20 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -8 12 -8 B -16 0 -18 -4 -12 C 8 18 0 20 -6 D -12 4 -20 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -8 12 -8 B -16 0 -18 -4 -12 C 8 18 0 20 -6 D -12 4 -20 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8515: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (9) E B D C A (8) B E D A C (7) C A D E B (6) B D A E C (6) B D E A C (5) E C B D A (4) C A E D B (4) A C B E D (4) E C D B A (3) D E B C A (3) D B E A C (3) C A E B D (3) B A D E C (3) A C B D E (3) E C B A D (2) E B D A C (2) D E C B A (2) D C E A B (2) D B A E C (2) C E D A B (2) B E A D C (2) B A E C D (2) E D C B A (1) E D B C A (1) E C D A B (1) E B C A D (1) E B A D C (1) D C A B E (1) D A B C E (1) C E A B D (1) A D C B E (1) A D B C E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 10 -8 -2 B 16 0 2 14 6 C -10 -2 0 -6 -20 D 8 -14 6 0 0 E 2 -6 20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 10 -8 -2 B 16 0 2 14 6 C -10 -2 0 -6 -20 D 8 -14 6 0 0 E 2 -6 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 A=21 C=16 D=14 so D is eliminated. Round 2 votes counts: B=30 E=29 A=22 C=19 so C is eliminated. Round 3 votes counts: A=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:208 D:200 A:192 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 10 -8 -2 B 16 0 2 14 6 C -10 -2 0 -6 -20 D 8 -14 6 0 0 E 2 -6 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 -8 -2 B 16 0 2 14 6 C -10 -2 0 -6 -20 D 8 -14 6 0 0 E 2 -6 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 -8 -2 B 16 0 2 14 6 C -10 -2 0 -6 -20 D 8 -14 6 0 0 E 2 -6 20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8516: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) B D C E A (10) B C D E A (7) A E C D B (6) D B E C A (5) D B C E A (5) C A B E D (5) A E D C B (5) E D A B C (4) E A D B C (4) D B E A C (4) A E D B C (4) E A D C B (3) D E B A C (3) C B A D E (3) B C D A E (3) A C E B D (3) C A E B D (2) C A B D E (2) B D E A C (2) E D B A C (1) E D A C B (1) E C A D B (1) D E B C A (1) C E A D B (1) C D B E A (1) C B D E A (1) B C A D E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -16 -16 -8 B 14 0 10 2 22 C 16 -10 0 -6 10 D 16 -2 6 0 16 E 8 -22 -10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999946 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -16 -8 B 14 0 10 2 22 C 16 -10 0 -6 10 D 16 -2 6 0 16 E 8 -22 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 B=23 A=20 D=18 E=14 so E is eliminated. Round 2 votes counts: A=27 C=26 D=24 B=23 so B is eliminated. Round 3 votes counts: C=37 D=36 A=27 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:224 D:218 C:205 E:180 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -16 -8 B 14 0 10 2 22 C 16 -10 0 -6 10 D 16 -2 6 0 16 E 8 -22 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -16 -8 B 14 0 10 2 22 C 16 -10 0 -6 10 D 16 -2 6 0 16 E 8 -22 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -16 -8 B 14 0 10 2 22 C 16 -10 0 -6 10 D 16 -2 6 0 16 E 8 -22 -10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998422 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8517: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (11) E D C B A (9) E D C A B (7) A B D C E (7) A E B D C (6) E D A C B (5) D E C B A (5) B C D A E (4) B C A D E (4) A B C E D (4) E C D B A (3) C B D E A (3) A E D C B (3) A E B C D (3) A B E C D (3) E A D C B (2) D C E B A (2) C D B E A (2) B C D E A (2) A D E B C (2) E A C B D (1) D C B A E (1) D A E C B (1) C E D B A (1) C D E B A (1) B E C A D (1) B E A C D (1) B D A C E (1) B A C E D (1) B A C D E (1) A E D B C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 10 6 10 B -16 0 8 8 -4 C -10 -8 0 -8 -8 D -6 -8 8 0 -2 E -10 4 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 6 10 B -16 0 8 8 -4 C -10 -8 0 -8 -8 D -6 -8 8 0 -2 E -10 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=42 E=27 B=15 D=9 C=7 so C is eliminated. Round 2 votes counts: A=42 E=28 B=18 D=12 so D is eliminated. Round 3 votes counts: A=43 E=36 B=21 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 E:202 B:198 D:196 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 6 10 B -16 0 8 8 -4 C -10 -8 0 -8 -8 D -6 -8 8 0 -2 E -10 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 6 10 B -16 0 8 8 -4 C -10 -8 0 -8 -8 D -6 -8 8 0 -2 E -10 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 6 10 B -16 0 8 8 -4 C -10 -8 0 -8 -8 D -6 -8 8 0 -2 E -10 4 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8518: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) C B A E D (7) B C A E D (7) B C A D E (6) D E A B C (5) C D E B A (5) C B D E A (5) E D A C B (4) D E C B A (4) D E C A B (4) A B C E D (4) C E D A B (3) C A B E D (3) A E D B C (3) E A D C B (2) D C E B A (2) C E A D B (2) B A D E C (2) B A C E D (2) B A C D E (2) A D E B C (2) A C B E D (2) A B E D C (2) A B E C D (2) A B D E C (2) E A D B C (1) D E B C A (1) D A E B C (1) C E D B A (1) C B A D E (1) B D E A C (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 8 -2 8 0 B -8 0 -12 -2 -4 C 2 12 0 6 4 D -8 2 -6 0 6 E 0 4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 8 0 B -8 0 -12 -2 -4 C 2 12 0 6 4 D -8 2 -6 0 6 E 0 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 B=20 A=19 E=7 so E is eliminated. Round 2 votes counts: D=31 C=27 A=22 B=20 so B is eliminated. Round 3 votes counts: C=40 D=32 A=28 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:207 D:197 E:197 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 8 0 B -8 0 -12 -2 -4 C 2 12 0 6 4 D -8 2 -6 0 6 E 0 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 8 0 B -8 0 -12 -2 -4 C 2 12 0 6 4 D -8 2 -6 0 6 E 0 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 8 0 B -8 0 -12 -2 -4 C 2 12 0 6 4 D -8 2 -6 0 6 E 0 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8519: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) B C D A E (9) A E D C B (9) E A D C B (7) B A C D E (6) A D E C B (5) E D C A B (4) D C B A E (4) C D B E A (4) E C D B A (3) D C A B E (3) B E A C D (3) A D C E B (3) D C A E B (2) C B D E A (2) B C A D E (2) A E B D C (2) A E B C D (2) A D C B E (2) E D C B A (1) E D A C B (1) E B C D A (1) E A B D C (1) D C E B A (1) D C E A B (1) D C B E A (1) D A C E B (1) C D E B A (1) C D B A E (1) C B D A E (1) B E C D A (1) B E C A D (1) B C E D A (1) B A E C D (1) B A D C E (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -6 -4 14 B 8 0 -14 -8 10 C 6 14 0 0 14 D 4 8 0 0 22 E -14 -10 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.867554 D: 0.132446 E: 0.000000 Sum of squares = 0.770192616645 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.867554 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -4 14 B 8 0 -14 -8 10 C 6 14 0 0 14 D 4 8 0 0 22 E -14 -10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=26 E=18 D=13 C=9 so C is eliminated. Round 2 votes counts: B=37 A=26 D=19 E=18 so E is eliminated. Round 3 votes counts: B=38 A=34 D=28 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:217 D:217 A:198 B:198 E:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 14 B 8 0 -14 -8 10 C 6 14 0 0 14 D 4 8 0 0 22 E -14 -10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 14 B 8 0 -14 -8 10 C 6 14 0 0 14 D 4 8 0 0 22 E -14 -10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 14 B 8 0 -14 -8 10 C 6 14 0 0 14 D 4 8 0 0 22 E -14 -10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8520: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A D C B (8) C D B E A (8) B A E C D (8) C B D E A (7) A E D C B (7) D C E A B (6) A E B D C (6) D C E B A (4) B C D E A (4) B C A E D (4) B A C E D (4) D E C A B (3) B C A D E (3) E D A C B (2) D E A C B (2) D C B E A (2) B A E D C (2) A E C D B (2) A E B C D (2) E A D B C (1) D E A B C (1) C E A D B (1) C B A E D (1) B D E A C (1) B D C E A (1) Total count = 100 A B C D E A 0 -18 -8 -2 -2 B 18 0 -6 6 10 C 8 6 0 8 10 D 2 -6 -8 0 4 E 2 -10 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -8 -2 -2 B 18 0 -6 6 10 C 8 6 0 8 10 D 2 -6 -8 0 4 E 2 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=18 C=17 A=17 E=11 so E is eliminated. Round 2 votes counts: B=37 A=26 D=20 C=17 so C is eliminated. Round 3 votes counts: B=45 D=28 A=27 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:214 D:196 E:189 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -8 -2 -2 B 18 0 -6 6 10 C 8 6 0 8 10 D 2 -6 -8 0 4 E 2 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -8 -2 -2 B 18 0 -6 6 10 C 8 6 0 8 10 D 2 -6 -8 0 4 E 2 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -8 -2 -2 B 18 0 -6 6 10 C 8 6 0 8 10 D 2 -6 -8 0 4 E 2 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8521: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B A D C E (6) E D A C B (5) E C D A B (4) D E C B A (4) C B E D A (4) B C A D E (4) A B D E C (4) D C B E A (3) D A E B C (3) C E D B A (3) C E A B D (3) C B A E D (3) B D A C E (3) B A C D E (3) A D E B C (3) A B C E D (3) D E A B C (2) C E B A D (2) C B E A D (2) C A B E D (2) B D A E C (2) B A D E C (2) B A C E D (2) A D B E C (2) E C A D B (1) D E C A B (1) D E B A C (1) D C E B A (1) D B E C A (1) D B A E C (1) D A B E C (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E B A (1) C D B E A (1) C B A D E (1) C A E B D (1) B C D A E (1) A E D C B (1) A E D B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -6 -2 2 B 4 0 -6 2 6 C 6 6 0 -12 6 D 2 -2 12 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 -2 2 B 4 0 -6 2 6 C 6 6 0 -12 6 D 2 -2 12 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=23 D=18 E=17 A=16 so A is eliminated. Round 2 votes counts: B=32 C=26 D=23 E=19 so E is eliminated. Round 3 votes counts: D=37 B=32 C=31 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:208 B:203 C:203 A:195 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 2 B 4 0 -6 2 6 C 6 6 0 -12 6 D 2 -2 12 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 2 B 4 0 -6 2 6 C 6 6 0 -12 6 D 2 -2 12 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 2 B 4 0 -6 2 6 C 6 6 0 -12 6 D 2 -2 12 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.100000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999993 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8522: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) D B A C E (8) D B E C A (7) D B A E C (5) C A E B D (5) A C B E D (5) E B D A C (4) E A C B D (4) D B E A C (4) E C A B D (3) D E B C A (3) D E B A C (3) B D E A C (3) B D A E C (3) B D A C E (3) A C E B D (3) E D C A B (2) E D B A C (2) E C D A B (2) D C B A E (2) D B C E A (2) C E A B D (2) C A D E B (2) C A B E D (2) C A B D E (2) E C A D B (1) E B A D C (1) D E C B A (1) C E D A B (1) C D E A B (1) C A E D B (1) B A D C E (1) B A C D E (1) A E C B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -22 6 -28 8 B 22 0 18 -10 16 C -6 -18 0 -24 2 D 28 10 24 0 20 E -8 -16 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 6 -28 8 B 22 0 18 -10 16 C -6 -18 0 -24 2 D 28 10 24 0 20 E -8 -16 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 E=19 C=16 B=11 A=11 so B is eliminated. Round 2 votes counts: D=52 E=19 C=16 A=13 so A is eliminated. Round 3 votes counts: D=53 C=27 E=20 so E is eliminated. Round 4 votes counts: D=62 C=38 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:241 B:223 A:182 C:177 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -22 6 -28 8 B 22 0 18 -10 16 C -6 -18 0 -24 2 D 28 10 24 0 20 E -8 -16 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 6 -28 8 B 22 0 18 -10 16 C -6 -18 0 -24 2 D 28 10 24 0 20 E -8 -16 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 6 -28 8 B 22 0 18 -10 16 C -6 -18 0 -24 2 D 28 10 24 0 20 E -8 -16 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8523: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (5) B C A E D (5) E C B D A (4) D C B A E (4) D C A B E (4) A E B D C (4) A D B C E (4) D E C A B (3) D E A C B (3) D A E C B (3) B C A D E (3) A D E B C (3) A B E D C (3) A B E C D (3) E D C A B (2) E D A C B (2) E C D B A (2) E B A C D (2) D C E A B (2) D C B E A (2) C B E D A (2) C B D E A (2) B C E A D (2) B A E C D (2) B A C E D (2) B A C D E (2) A E D B C (2) A B D C E (2) A B C E D (2) A B C D E (2) E C B A D (1) E A D B C (1) E A B D C (1) E A B C D (1) D E C B A (1) D C E B A (1) D C A E B (1) D A C B E (1) D A B C E (1) C E B D A (1) C D E B A (1) C D B E A (1) C D B A E (1) C B A D E (1) B E C A D (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -4 2 24 B -4 0 0 8 18 C 4 0 0 -4 8 D -2 -8 4 0 10 E -24 -18 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.448039 C: 0.551961 D: 0.000000 E: 0.000000 Sum of squares = 0.505399915695 Cumulative probabilities = A: 0.000000 B: 0.448039 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 2 24 B -4 0 0 8 18 C 4 0 0 -4 8 D -2 -8 4 0 10 E -24 -18 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499460 C: 0.500540 D: 0.000000 E: 0.000000 Sum of squares = 0.500000582969 Cumulative probabilities = A: 0.000000 B: 0.499460 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=26 B=17 E=16 C=14 so C is eliminated. Round 2 votes counts: D=29 B=27 A=27 E=17 so E is eliminated. Round 3 votes counts: D=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:211 C:204 D:202 E:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 2 24 B -4 0 0 8 18 C 4 0 0 -4 8 D -2 -8 4 0 10 E -24 -18 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499460 C: 0.500540 D: 0.000000 E: 0.000000 Sum of squares = 0.500000582969 Cumulative probabilities = A: 0.000000 B: 0.499460 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 2 24 B -4 0 0 8 18 C 4 0 0 -4 8 D -2 -8 4 0 10 E -24 -18 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499460 C: 0.500540 D: 0.000000 E: 0.000000 Sum of squares = 0.500000582969 Cumulative probabilities = A: 0.000000 B: 0.499460 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 2 24 B -4 0 0 8 18 C 4 0 0 -4 8 D -2 -8 4 0 10 E -24 -18 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499460 C: 0.500540 D: 0.000000 E: 0.000000 Sum of squares = 0.500000582969 Cumulative probabilities = A: 0.000000 B: 0.499460 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8524: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (13) A C D B E (9) B E D A C (8) E B D C A (7) A D C B E (6) E C B A D (5) E B C D A (4) D A C B E (4) C A E D B (4) D A B E C (3) C E B A D (3) E B C A D (2) C E D A B (2) C E A D B (2) C E A B D (2) C A E B D (2) B E C A D (2) B E A D C (2) B D E A C (2) B D A E C (2) E D C B A (1) E D B C A (1) D E C A B (1) D E B A C (1) D C A E B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A C E B (1) D A B C E (1) C D E A B (1) C B E A D (1) B E D C A (1) B E C D A (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -12 10 0 B -10 0 -20 -10 -10 C 12 20 0 6 8 D -10 10 -6 0 0 E 0 10 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 10 0 B -10 0 -20 -10 -10 C 12 20 0 6 8 D -10 10 -6 0 0 E 0 10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=20 B=18 A=17 D=15 so D is eliminated. Round 2 votes counts: C=31 A=27 E=22 B=20 so B is eliminated. Round 3 votes counts: E=39 C=31 A=30 so A is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:204 E:201 D:197 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 10 0 B -10 0 -20 -10 -10 C 12 20 0 6 8 D -10 10 -6 0 0 E 0 10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 10 0 B -10 0 -20 -10 -10 C 12 20 0 6 8 D -10 10 -6 0 0 E 0 10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 10 0 B -10 0 -20 -10 -10 C 12 20 0 6 8 D -10 10 -6 0 0 E 0 10 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8525: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) B E A C D (7) B A C E D (6) D B E C A (5) A C B D E (5) E B D A C (4) D C A E B (4) C A D E B (4) C A D B E (4) B E D A C (4) B E A D C (4) E D B C A (3) D E C A B (3) D E B C A (3) C D A E B (3) B D E C A (3) A C E B D (3) C D A B E (2) C A E D B (2) B D C A E (2) B A E C D (2) A C D E B (2) A C D B E (2) E B D C A (1) E B A D C (1) E B A C D (1) E A C B D (1) E A B C D (1) D C E B A (1) D C E A B (1) D C A B E (1) B E D C A (1) B D E A C (1) B D A C E (1) B A D C E (1) A E C B D (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 14 14 10 B 2 0 4 18 18 C -14 -4 0 12 6 D -14 -18 -12 0 -4 E -10 -18 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999826 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 14 10 B 2 0 4 18 18 C -14 -4 0 12 6 D -14 -18 -12 0 -4 E -10 -18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 D=18 C=15 E=12 so E is eliminated. Round 2 votes counts: B=39 A=25 D=21 C=15 so C is eliminated. Round 3 votes counts: B=39 A=35 D=26 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:218 C:200 E:185 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 14 14 10 B 2 0 4 18 18 C -14 -4 0 12 6 D -14 -18 -12 0 -4 E -10 -18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 14 10 B 2 0 4 18 18 C -14 -4 0 12 6 D -14 -18 -12 0 -4 E -10 -18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 14 10 B 2 0 4 18 18 C -14 -4 0 12 6 D -14 -18 -12 0 -4 E -10 -18 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8526: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) C B E A D (8) A B D E C (6) B C A E D (5) D E A C B (4) D C E A B (4) C E D B A (4) C D E B A (4) B A E C D (4) D A E B C (3) A D E B C (3) E D C A B (2) E C B D A (2) E B C A D (2) E B A C D (2) D E A B C (2) D C B A E (2) D A C E B (2) D A B C E (2) C E B D A (2) C D B A E (2) C B A D E (2) B C A D E (2) B A C D E (2) A E D B C (2) A B D C E (2) E D A B C (1) E C D B A (1) E C B A D (1) E A B C D (1) D E C A B (1) D A E C B (1) D A C B E (1) C D B E A (1) C B E D A (1) C B D E A (1) C B A E D (1) B E C A D (1) B E A C D (1) B C E A D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -22 0 14 6 B 22 0 6 16 10 C 0 -6 0 20 16 D -14 -16 -20 0 -6 E -6 -10 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 0 14 6 B 22 0 6 16 10 C 0 -6 0 20 16 D -14 -16 -20 0 -6 E -6 -10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 D=22 A=15 E=12 so E is eliminated. Round 2 votes counts: C=30 B=29 D=25 A=16 so A is eliminated. Round 3 votes counts: B=40 D=30 C=30 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:215 A:199 E:187 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 0 14 6 B 22 0 6 16 10 C 0 -6 0 20 16 D -14 -16 -20 0 -6 E -6 -10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 0 14 6 B 22 0 6 16 10 C 0 -6 0 20 16 D -14 -16 -20 0 -6 E -6 -10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 0 14 6 B 22 0 6 16 10 C 0 -6 0 20 16 D -14 -16 -20 0 -6 E -6 -10 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8527: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (5) B A E D C (5) E C A B D (4) D B A C E (4) C D B E A (4) B E C D A (4) B A D E C (4) E B C A D (3) E A B C D (3) D B C A E (3) D A C E B (3) C E A D B (3) B E A D C (3) B D C A E (3) B D A E C (3) B D A C E (3) E A C D B (2) D A C B E (2) C E D A B (2) C D E A B (2) C D A E B (2) C B D E A (2) B E C A D (2) B E A C D (2) B D C E A (2) B C E D A (2) A E D C B (2) A E C D B (2) A D C E B (2) E B A C D (1) E A C B D (1) D C B E A (1) D C B A E (1) D C A E B (1) D C A B E (1) D A B C E (1) C E D B A (1) C E B D A (1) C E A B D (1) B E D A C (1) B D E C A (1) B C D E A (1) A E B D C (1) A D E C B (1) A D B E C (1) A C E D B (1) Total count = 100 A B C D E A 0 -24 -4 -2 -8 B 24 0 6 16 12 C 4 -6 0 2 -2 D 2 -16 -2 0 -4 E 8 -12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -4 -2 -8 B 24 0 6 16 12 C 4 -6 0 2 -2 D 2 -16 -2 0 -4 E 8 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=19 C=18 D=17 A=10 so A is eliminated. Round 2 votes counts: B=36 E=24 D=21 C=19 so C is eliminated. Round 3 votes counts: B=38 E=33 D=29 so D is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:229 E:201 C:199 D:190 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -4 -2 -8 B 24 0 6 16 12 C 4 -6 0 2 -2 D 2 -16 -2 0 -4 E 8 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -4 -2 -8 B 24 0 6 16 12 C 4 -6 0 2 -2 D 2 -16 -2 0 -4 E 8 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -4 -2 -8 B 24 0 6 16 12 C 4 -6 0 2 -2 D 2 -16 -2 0 -4 E 8 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8528: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) D C A B E (11) E B C A D (6) C D B E A (6) B E A D C (6) C E B A D (5) C D A E B (5) C E B D A (4) B E D A C (4) C D E B A (3) A E B C D (3) A B E D C (3) E A B C D (2) D C B E A (2) D C B A E (2) D B E C A (2) C A D E B (2) B E C D A (2) A D E B C (2) A D C E B (2) A D B E C (2) E B A D C (1) D B E A C (1) D A C B E (1) D A B E C (1) D A B C E (1) C E D B A (1) C E A B D (1) C D E A B (1) C D B A E (1) B E A C D (1) B D E A C (1) B C E D A (1) A E C B D (1) A E B D C (1) A C D E B (1) Total count = 100 A B C D E A 0 -20 -10 0 -22 B 20 0 2 6 -4 C 10 -2 0 14 0 D 0 -6 -14 0 -6 E 22 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.445320 D: 0.000000 E: 0.554680 Sum of squares = 0.505979771002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.445320 D: 0.445320 E: 1.000000 A B C D E A 0 -20 -10 0 -22 B 20 0 2 6 -4 C 10 -2 0 14 0 D 0 -6 -14 0 -6 E 22 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.500004 Sum of squares = 0.499999999063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.499996 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=21 E=20 B=15 A=15 so B is eliminated. Round 2 votes counts: E=33 C=30 D=22 A=15 so A is eliminated. Round 3 votes counts: E=41 C=31 D=28 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:216 B:212 C:211 D:187 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -10 0 -22 B 20 0 2 6 -4 C 10 -2 0 14 0 D 0 -6 -14 0 -6 E 22 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.500004 Sum of squares = 0.499999999063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.499996 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -10 0 -22 B 20 0 2 6 -4 C 10 -2 0 14 0 D 0 -6 -14 0 -6 E 22 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.500004 Sum of squares = 0.499999999063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.499996 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -10 0 -22 B 20 0 2 6 -4 C 10 -2 0 14 0 D 0 -6 -14 0 -6 E 22 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.000000 E: 0.500004 Sum of squares = 0.499999999063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499996 D: 0.499996 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8529: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (14) B E C A D (9) B C E A D (7) C A E D B (5) B D E A C (5) D B A E C (4) C A D E B (4) B E D A C (4) A C D E B (4) D A E C B (3) D A B C E (3) C E B A D (3) B E C D A (3) E D A B C (2) E C A B D (2) E B C A D (2) C E A B D (2) C B E A D (2) B E D C A (2) B D E C A (2) B D A E C (2) A D C E B (2) A C E D B (2) E D B A C (1) D E B A C (1) D A C B E (1) D A B E C (1) C E A D B (1) C A E B D (1) C A D B E (1) C A B E D (1) C A B D E (1) B D C E A (1) B D A C E (1) B C D A E (1) Total count = 100 A B C D E A 0 0 0 -2 2 B 0 0 2 2 2 C 0 -2 0 2 14 D 2 -2 -2 0 2 E -2 -2 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.304802 B: 0.695198 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.576204261585 Cumulative probabilities = A: 0.304802 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -2 2 B 0 0 2 2 2 C 0 -2 0 2 14 D 2 -2 -2 0 2 E -2 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499358 B: 0.500642 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000824771 Cumulative probabilities = A: 0.499358 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=27 C=21 A=8 E=7 so E is eliminated. Round 2 votes counts: B=39 D=30 C=23 A=8 so A is eliminated. Round 3 votes counts: B=39 D=32 C=29 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:207 B:203 A:200 D:200 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -2 2 B 0 0 2 2 2 C 0 -2 0 2 14 D 2 -2 -2 0 2 E -2 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499358 B: 0.500642 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000824771 Cumulative probabilities = A: 0.499358 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 2 B 0 0 2 2 2 C 0 -2 0 2 14 D 2 -2 -2 0 2 E -2 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499358 B: 0.500642 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000824771 Cumulative probabilities = A: 0.499358 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 2 B 0 0 2 2 2 C 0 -2 0 2 14 D 2 -2 -2 0 2 E -2 -2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499358 B: 0.500642 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000824771 Cumulative probabilities = A: 0.499358 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8530: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) E B D A C (8) C B E A D (7) E B C D A (6) B C E A D (6) E D B C A (5) C A B D E (5) B E C A D (5) A D C B E (5) E D B A C (4) E D A B C (4) C B A E D (4) C B A D E (4) D A E B C (3) C D A B E (3) C A D B E (3) E B D C A (2) E B A D C (2) D E A B C (2) D A E C B (2) D A C B E (2) B E A C D (2) E C B D A (1) E A D B C (1) C D B A E (1) B C E D A (1) A E D B C (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -6 -6 -12 B 16 0 6 6 -2 C 6 -6 0 -2 2 D 6 -6 2 0 -18 E 12 2 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.440000000257 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 -16 -6 -6 -12 B 16 0 6 6 -2 C 6 -6 0 -2 2 D 6 -6 2 0 -18 E 12 2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=27 D=18 B=14 A=8 so A is eliminated. Round 2 votes counts: E=34 C=28 D=24 B=14 so B is eliminated. Round 3 votes counts: E=41 C=35 D=24 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:215 B:213 C:200 D:192 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -6 -12 B 16 0 6 6 -2 C 6 -6 0 -2 2 D 6 -6 2 0 -18 E 12 2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -6 -12 B 16 0 6 6 -2 C 6 -6 0 -2 2 D 6 -6 2 0 -18 E 12 2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -6 -12 B 16 0 6 6 -2 C 6 -6 0 -2 2 D 6 -6 2 0 -18 E 12 2 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.600000 Sum of squares = 0.439999999971 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8531: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (11) D C E B A (8) A B D E C (7) B A D E C (6) A B E C D (6) B A E C D (5) B A E D C (4) E C A B D (3) D C A B E (3) D A B C E (3) C E A D B (3) C D E A B (3) E C B A D (2) D C B E A (2) D B A C E (2) C E D A B (2) C E A B D (2) B E A C D (2) B D A E C (2) A E B C D (2) A B E D C (2) E C B D A (1) E B C D A (1) E B A C D (1) D E C B A (1) D C B A E (1) D C A E B (1) D B E C A (1) D B C E A (1) D B C A E (1) C E D B A (1) C E B D A (1) C E B A D (1) C D E B A (1) C A E B D (1) B E C A D (1) A E C B D (1) A D C E B (1) A D C B E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -6 6 2 B -8 0 -6 4 0 C 6 6 0 -16 4 D -6 -4 16 0 12 E -2 0 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.214286 D: 0.214286 E: 0.000000 Sum of squares = 0.418367346942 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.785714 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 6 2 B -8 0 -6 4 0 C 6 6 0 -16 4 D -6 -4 16 0 12 E -2 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.214286 D: 0.214286 E: 0.000000 Sum of squares = 0.418367346947 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.785714 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=22 B=20 C=15 E=8 so E is eliminated. Round 2 votes counts: D=35 B=22 A=22 C=21 so C is eliminated. Round 3 votes counts: D=42 A=31 B=27 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:209 A:205 C:200 B:195 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -6 6 2 B -8 0 -6 4 0 C 6 6 0 -16 4 D -6 -4 16 0 12 E -2 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.214286 D: 0.214286 E: 0.000000 Sum of squares = 0.418367346947 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.785714 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 6 2 B -8 0 -6 4 0 C 6 6 0 -16 4 D -6 -4 16 0 12 E -2 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.214286 D: 0.214286 E: 0.000000 Sum of squares = 0.418367346947 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.785714 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 6 2 B -8 0 -6 4 0 C 6 6 0 -16 4 D -6 -4 16 0 12 E -2 0 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.214286 D: 0.214286 E: 0.000000 Sum of squares = 0.418367346947 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.785714 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8532: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (16) A B D E C (13) C B E D A (6) B A D E C (6) A D E B C (6) C E D A B (5) E D C B A (4) E D A C B (4) A D B E C (4) D E A B C (3) C E B D A (3) E D C A B (2) E D B A C (2) E C D A B (2) C E A D B (2) C B E A D (2) C A B D E (2) A D E C B (2) A B D C E (2) A B C D E (2) D E B A C (1) D B E A C (1) D A E B C (1) C A E D B (1) C A E B D (1) C A B E D (1) B D E A C (1) B C A D E (1) B A C D E (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 4 -2 -8 B -12 0 -12 -16 -12 C -4 12 0 -4 -4 D 2 16 4 0 -2 E 8 12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 4 -2 -8 B -12 0 -12 -16 -12 C -4 12 0 -4 -4 D 2 16 4 0 -2 E 8 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 A=32 E=14 B=9 D=6 so D is eliminated. Round 2 votes counts: C=39 A=33 E=18 B=10 so B is eliminated. Round 3 votes counts: C=40 A=40 E=20 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:213 D:210 A:203 C:200 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 4 -2 -8 B -12 0 -12 -16 -12 C -4 12 0 -4 -4 D 2 16 4 0 -2 E 8 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 -2 -8 B -12 0 -12 -16 -12 C -4 12 0 -4 -4 D 2 16 4 0 -2 E 8 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 -2 -8 B -12 0 -12 -16 -12 C -4 12 0 -4 -4 D 2 16 4 0 -2 E 8 12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8533: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) D B A C E (12) B D E C A (11) C E A B D (9) A C E D B (8) E C A D B (6) B D C E A (6) D B A E C (5) B D A C E (4) E C B A D (2) E A C D B (2) D A B E C (2) B D C A E (2) B D A E C (2) B C E D A (2) E C B D A (1) E B D C A (1) E B C D A (1) E A D C B (1) C E A D B (1) C A E D B (1) C A E B D (1) B D E A C (1) B C D E A (1) B C D A E (1) A E D C B (1) A E C D B (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -18 -4 -16 B 4 0 4 16 0 C 18 -4 0 0 -2 D 4 -16 0 0 -2 E 16 0 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.401604 C: 0.000000 D: 0.000000 E: 0.598396 Sum of squares = 0.519363659467 Cumulative probabilities = A: 0.000000 B: 0.401604 C: 0.401604 D: 0.401604 E: 1.000000 A B C D E A 0 -4 -18 -4 -16 B 4 0 4 16 0 C 18 -4 0 0 -2 D 4 -16 0 0 -2 E 16 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 D=19 C=12 A=12 so C is eliminated. Round 2 votes counts: E=37 B=30 D=19 A=14 so A is eliminated. Round 3 votes counts: E=49 B=30 D=21 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:210 C:206 D:193 A:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -18 -4 -16 B 4 0 4 16 0 C 18 -4 0 0 -2 D 4 -16 0 0 -2 E 16 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -4 -16 B 4 0 4 16 0 C 18 -4 0 0 -2 D 4 -16 0 0 -2 E 16 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -4 -16 B 4 0 4 16 0 C 18 -4 0 0 -2 D 4 -16 0 0 -2 E 16 0 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999936 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8534: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (14) E D C B A (10) E D C A B (8) E D B C A (8) B A C D E (7) D C E B A (4) B A E D C (4) D E C B A (3) C D E A B (3) B D E C A (3) B D C E A (3) A B C E D (3) E D A C B (2) E A D C B (2) C D B E A (2) C D B A E (2) A E C D B (2) A E B D C (2) A C B D E (2) E D B A C (1) E C D A B (1) E B A D C (1) E A B D C (1) D B C E A (1) C B D A E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D C A (1) B E D A C (1) B D E A C (1) B A D E C (1) A E C B D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -6 -10 -12 B 8 0 8 -2 -2 C 6 -8 0 -14 -10 D 10 2 14 0 0 E 12 2 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.407845 E: 0.592155 Sum of squares = 0.516985208266 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.407845 E: 1.000000 A B C D E A 0 -8 -6 -10 -12 B 8 0 8 -2 -2 C 6 -8 0 -14 -10 D 10 2 14 0 0 E 12 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=26 B=21 C=11 D=8 so D is eliminated. Round 2 votes counts: E=37 A=26 B=22 C=15 so C is eliminated. Round 3 votes counts: E=44 A=29 B=27 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:212 B:206 C:187 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 -12 B 8 0 8 -2 -2 C 6 -8 0 -14 -10 D 10 2 14 0 0 E 12 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 -12 B 8 0 8 -2 -2 C 6 -8 0 -14 -10 D 10 2 14 0 0 E 12 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 -12 B 8 0 8 -2 -2 C 6 -8 0 -14 -10 D 10 2 14 0 0 E 12 2 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8535: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) B A E D C (10) E A B C D (7) D B A E C (7) C E A B D (7) C E D A B (5) B D A E C (5) E A B D C (4) B A D E C (4) E A C B D (3) C E A D B (3) C D B A E (3) E C A B D (2) E B A C D (2) D B C A E (2) B E A C D (2) B C A E D (2) E B C A D (1) E A D B C (1) D C E A B (1) D C B A E (1) D C A E B (1) D A C B E (1) D A B E C (1) C D E B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C B E A D (1) C B D E A (1) B E C A D (1) B E A D C (1) B C D A E (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 0 6 -18 B -4 0 6 14 -6 C 0 -6 0 20 -6 D -6 -14 -20 0 -10 E 18 6 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 0 6 -18 B -4 0 6 14 -6 C 0 -6 0 20 -6 D -6 -14 -20 0 -10 E 18 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=27 E=20 D=14 A=1 so A is eliminated. Round 2 votes counts: C=38 B=28 E=20 D=14 so D is eliminated. Round 3 votes counts: C=42 B=38 E=20 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:220 B:205 C:204 A:196 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 6 -18 B -4 0 6 14 -6 C 0 -6 0 20 -6 D -6 -14 -20 0 -10 E 18 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 -18 B -4 0 6 14 -6 C 0 -6 0 20 -6 D -6 -14 -20 0 -10 E 18 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 -18 B -4 0 6 14 -6 C 0 -6 0 20 -6 D -6 -14 -20 0 -10 E 18 6 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8536: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) E C B D A (7) B D E C A (7) C E A D B (5) A B D E C (5) D C E B A (4) D B E C A (4) A E C B D (4) A D C B E (4) A D B C E (4) D B A E C (3) D A B C E (3) B D A E C (3) A C E D B (3) A C E B D (3) A B D C E (3) E C D B A (2) E B C D A (2) D C B E A (2) C E B D A (2) B E D C A (2) A C D E B (2) D C A E B (1) D C A B E (1) D B C E A (1) D B A C E (1) D A C B E (1) D A B E C (1) C E D A B (1) C E A B D (1) C D E B A (1) C A E D B (1) C A E B D (1) C A D E B (1) B D E A C (1) B A E D C (1) B A E C D (1) A E B C D (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -8 -16 0 B 4 0 -12 -12 -2 C 8 12 0 -8 10 D 16 12 8 0 10 E 0 2 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -16 0 B 4 0 -12 -12 -2 C 8 12 0 -8 10 D 16 12 8 0 10 E 0 2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=22 C=21 B=15 E=11 so E is eliminated. Round 2 votes counts: A=31 C=30 D=22 B=17 so B is eliminated. Round 3 votes counts: D=35 A=33 C=32 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:211 E:191 B:189 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -16 0 B 4 0 -12 -12 -2 C 8 12 0 -8 10 D 16 12 8 0 10 E 0 2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -16 0 B 4 0 -12 -12 -2 C 8 12 0 -8 10 D 16 12 8 0 10 E 0 2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -16 0 B 4 0 -12 -12 -2 C 8 12 0 -8 10 D 16 12 8 0 10 E 0 2 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8537: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) D C E B A (6) E D B A C (5) C B D A E (5) D C E A B (4) C A B D E (4) B C A E D (4) B A C E D (4) E D A C B (3) E B D A C (3) E A B D C (3) D E B C A (3) B E A D C (3) A C E D B (3) A C B E D (3) E B A D C (2) D E C B A (2) D B E C A (2) C D B A E (2) C B A D E (2) C A D E B (2) B D C E A (2) B A E C D (2) A E C B D (2) A E B D C (2) A B E C D (2) A B C E D (2) E A D C B (1) D E C A B (1) D E B A C (1) D E A C B (1) D C B E A (1) D A C E B (1) C D A E B (1) C D A B E (1) C B A E D (1) C A D B E (1) B D E C A (1) B C D A E (1) B C A D E (1) B A E D C (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 6 -6 -2 B 8 0 4 4 -10 C -6 -4 0 -10 4 D 6 -4 10 0 -10 E 2 10 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888876 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -8 6 -6 -2 B 8 0 4 4 -10 C -6 -4 0 -10 4 D 6 -4 10 0 -10 E 2 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 C=19 B=19 A=16 so A is eliminated. Round 2 votes counts: E=29 C=26 B=23 D=22 so D is eliminated. Round 3 votes counts: C=38 E=37 B=25 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:209 B:203 D:201 A:195 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 -6 -2 B 8 0 4 4 -10 C -6 -4 0 -10 4 D 6 -4 10 0 -10 E 2 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -6 -2 B 8 0 4 4 -10 C -6 -4 0 -10 4 D 6 -4 10 0 -10 E 2 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -6 -2 B 8 0 4 4 -10 C -6 -4 0 -10 4 D 6 -4 10 0 -10 E 2 10 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888918 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8538: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (13) D B E A C (12) D B E C A (7) C A D B E (6) A C E B D (6) E B D A C (5) E B A D C (5) D B C E A (5) C D B A E (4) C A E D B (4) B E D A C (4) A E C B D (4) E A B D C (3) A E B D C (3) E D B A C (2) D E B A C (2) C D A B E (2) A E D B C (2) E A B C D (1) D C B E A (1) D C A B E (1) C A D E B (1) C A B E D (1) C A B D E (1) B D E C A (1) B D E A C (1) A E D C B (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 6 6 2 B -2 0 8 -2 -8 C -6 -8 0 -10 -8 D -6 2 10 0 -10 E -2 8 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 6 2 B -2 0 8 -2 -8 C -6 -8 0 -10 -8 D -6 2 10 0 -10 E -2 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999107 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=28 A=18 E=16 B=6 so B is eliminated. Round 2 votes counts: C=32 D=30 E=20 A=18 so A is eliminated. Round 3 votes counts: C=39 E=31 D=30 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:208 B:198 D:198 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 6 2 B -2 0 8 -2 -8 C -6 -8 0 -10 -8 D -6 2 10 0 -10 E -2 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999107 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 6 2 B -2 0 8 -2 -8 C -6 -8 0 -10 -8 D -6 2 10 0 -10 E -2 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999107 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 6 2 B -2 0 8 -2 -8 C -6 -8 0 -10 -8 D -6 2 10 0 -10 E -2 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999107 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8539: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) D A B C E (9) E C B A D (7) E B C A D (7) D A B E C (5) B E A D C (5) B A D E C (5) E C B D A (4) C E D A B (4) C D A E B (3) B E D A C (3) B D A E C (3) E C D B A (2) E C D A B (2) E B C D A (2) D A C E B (2) C E B A D (2) C A D E B (2) B E C A D (2) B E A C D (2) A D C B E (2) A D B C E (2) E B D A C (1) D E A B C (1) D C A E B (1) D B A E C (1) C E D B A (1) C E A D B (1) C E A B D (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B E D (1) Total count = 100 A B C D E A 0 2 8 -14 4 B -2 0 -4 -4 12 C -8 4 0 -6 -4 D 14 4 6 0 0 E -4 -12 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.821629 E: 0.178371 Sum of squares = 0.706890552896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.821629 E: 1.000000 A B C D E A 0 2 8 -14 4 B -2 0 -4 -4 12 C -8 4 0 -6 -4 D 14 4 6 0 0 E -4 -12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000015841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=25 B=20 C=19 A=4 so A is eliminated. Round 2 votes counts: D=36 E=25 B=20 C=19 so C is eliminated. Round 3 votes counts: D=42 E=35 B=23 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 B:201 A:200 E:194 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -14 4 B -2 0 -4 -4 12 C -8 4 0 -6 -4 D 14 4 6 0 0 E -4 -12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000015841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -14 4 B -2 0 -4 -4 12 C -8 4 0 -6 -4 D 14 4 6 0 0 E -4 -12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000015841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -14 4 B -2 0 -4 -4 12 C -8 4 0 -6 -4 D 14 4 6 0 0 E -4 -12 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000015841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8540: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) D E A C B (6) C B A E D (6) A B C D E (6) E C B D A (5) D A E B C (5) D A C B E (5) C B E A D (5) B C A E D (4) E B C D A (3) E B C A D (3) C E B D A (3) B C E A D (3) A C B D E (3) A B C E D (3) E D A B C (2) E C D B A (2) D E C A B (2) C B D E A (2) B E C A D (2) A E B C D (2) A D B C E (2) E D B C A (1) E D B A C (1) E B D A C (1) D E A B C (1) D C B E A (1) D C B A E (1) D A E C B (1) D A B E C (1) C D B E A (1) C D B A E (1) C B E D A (1) C B A D E (1) C A B D E (1) B A E C D (1) A E D B C (1) A E B D C (1) A D C B E (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -8 -4 -2 B 8 0 -8 16 8 C 8 8 0 16 2 D 4 -16 -16 0 -2 E 2 -8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -4 -2 B 8 0 -8 16 8 C 8 8 0 16 2 D 4 -16 -16 0 -2 E 2 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=22 C=21 E=18 B=10 so B is eliminated. Round 2 votes counts: D=29 C=28 A=23 E=20 so E is eliminated. Round 3 votes counts: C=43 D=34 A=23 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:212 E:197 A:189 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -4 -2 B 8 0 -8 16 8 C 8 8 0 16 2 D 4 -16 -16 0 -2 E 2 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -4 -2 B 8 0 -8 16 8 C 8 8 0 16 2 D 4 -16 -16 0 -2 E 2 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -4 -2 B 8 0 -8 16 8 C 8 8 0 16 2 D 4 -16 -16 0 -2 E 2 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8541: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (8) E A B D C (7) E A D B C (6) D C A E B (6) A E D C B (6) B E C A D (5) B E A D C (5) A D E C B (5) E B A D C (4) D A C E B (4) C B D E A (4) D C A B E (3) C D A B E (3) C B D A E (3) B C E D A (3) A D C E B (3) E B A C D (2) E A D C B (2) D C B A E (2) B E C D A (2) B E A C D (2) B C D E A (2) A E D B C (2) E B C A D (1) E A C D B (1) E A B C D (1) C D A E B (1) B E D C A (1) B D C A E (1) B D A C E (1) B C D A E (1) A E B D C (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 6 8 10 4 B -6 0 -4 -8 -6 C -8 4 0 -22 -8 D -10 8 22 0 -2 E -4 6 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 10 4 B -6 0 -4 -8 -6 C -8 4 0 -22 -8 D -10 8 22 0 -2 E -4 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 B=23 C=19 A=19 D=15 so D is eliminated. Round 2 votes counts: C=30 E=24 B=23 A=23 so B is eliminated. Round 3 votes counts: E=39 C=37 A=24 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:214 D:209 E:206 B:188 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 10 4 B -6 0 -4 -8 -6 C -8 4 0 -22 -8 D -10 8 22 0 -2 E -4 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 10 4 B -6 0 -4 -8 -6 C -8 4 0 -22 -8 D -10 8 22 0 -2 E -4 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 10 4 B -6 0 -4 -8 -6 C -8 4 0 -22 -8 D -10 8 22 0 -2 E -4 6 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8542: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (15) A E C D B (10) A B E D C (10) D C B E A (6) D B C E A (6) A E B C D (6) C D E B A (5) A E C B D (4) E A C D B (3) B A D E C (3) E C D A B (2) E A B C D (2) C E D A B (2) C D A E B (2) B D E C A (2) B D A C E (2) B A D C E (2) A C E D B (2) A C D B E (2) A B D C E (2) E C A D B (1) E B D C A (1) D C E B A (1) C E D B A (1) C E A D B (1) C D E A B (1) B E D C A (1) B D C A E (1) B D A E C (1) B A E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 4 2 0 B -4 0 14 10 12 C -4 -14 0 -10 2 D -2 -10 10 0 4 E 0 -12 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.854420 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.145580 Sum of squares = 0.751227436024 Cumulative probabilities = A: 0.854420 B: 0.854420 C: 0.854420 D: 0.854420 E: 1.000000 A B C D E A 0 4 4 2 0 B -4 0 14 10 12 C -4 -14 0 -10 2 D -2 -10 10 0 4 E 0 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000048294 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=28 D=13 C=12 E=9 so E is eliminated. Round 2 votes counts: A=43 B=29 C=15 D=13 so D is eliminated. Round 3 votes counts: A=43 B=35 C=22 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:216 A:205 D:201 E:191 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 0 B -4 0 14 10 12 C -4 -14 0 -10 2 D -2 -10 10 0 4 E 0 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000048294 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 0 B -4 0 14 10 12 C -4 -14 0 -10 2 D -2 -10 10 0 4 E 0 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000048294 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 0 B -4 0 14 10 12 C -4 -14 0 -10 2 D -2 -10 10 0 4 E 0 -12 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000048294 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8543: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (7) B C D E A (7) E A C B D (6) D B C E A (5) D A E C B (5) B C E D A (5) D B C A E (4) D A E B C (4) B D C E A (4) B C E A D (4) A E D C B (4) A E C D B (4) A D E C B (4) A C E B D (4) E C B A D (3) A E C B D (3) E B A C D (2) D E B A C (2) D C B A E (2) D A C E B (2) D A C B E (2) B E C A D (2) A D C E B (2) E C A B D (1) E A D B C (1) D E A B C (1) D C A B E (1) D B E A C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C B A E D (1) B E C D A (1) B C D A E (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 0 2 -8 B 4 0 -8 6 -2 C 0 8 0 8 10 D -2 -6 -8 0 0 E 8 2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400452 B: 0.000000 C: 0.599548 D: 0.000000 E: 0.000000 Sum of squares = 0.51981972985 Cumulative probabilities = A: 0.400452 B: 0.400452 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 2 -8 B 4 0 -8 6 -2 C 0 8 0 8 10 D -2 -6 -8 0 0 E 8 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=24 A=22 E=13 C=10 so C is eliminated. Round 2 votes counts: B=32 D=31 A=22 E=15 so E is eliminated. Round 3 votes counts: B=38 D=31 A=31 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:200 E:200 A:195 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 2 -8 B 4 0 -8 6 -2 C 0 8 0 8 10 D -2 -6 -8 0 0 E 8 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 -8 B 4 0 -8 6 -2 C 0 8 0 8 10 D -2 -6 -8 0 0 E 8 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 -8 B 4 0 -8 6 -2 C 0 8 0 8 10 D -2 -6 -8 0 0 E 8 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8544: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) B D C E A (8) D B E A C (7) A E D C B (7) D B A C E (6) D B C E A (5) E A C D B (4) D B E C A (4) D B A E C (4) C E A B D (4) B D C A E (4) E A C B D (3) D A B E C (3) C B E A D (3) C B D E A (3) B C D A E (3) E C A B D (2) D E B A C (2) D E A B C (2) D B C A E (2) C A E B D (2) A E C B D (2) E D A B C (1) E C B D A (1) E B C D A (1) E A D C B (1) D A E B C (1) C E B A D (1) C B E D A (1) C B A E D (1) C B A D E (1) B C D E A (1) A D E B C (1) Total count = 100 A B C D E A 0 -16 6 -18 -8 B 16 0 10 -18 12 C -6 -10 0 -16 -10 D 18 18 16 0 14 E 8 -12 10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 -18 -8 B 16 0 10 -18 12 C -6 -10 0 -16 -10 D 18 18 16 0 14 E 8 -12 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=19 C=16 B=16 E=13 so E is eliminated. Round 2 votes counts: D=37 A=27 C=19 B=17 so B is eliminated. Round 3 votes counts: D=49 A=27 C=24 so C is eliminated. Round 4 votes counts: D=59 A=41 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:233 B:210 E:196 A:182 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 6 -18 -8 B 16 0 10 -18 12 C -6 -10 0 -16 -10 D 18 18 16 0 14 E 8 -12 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -18 -8 B 16 0 10 -18 12 C -6 -10 0 -16 -10 D 18 18 16 0 14 E 8 -12 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -18 -8 B 16 0 10 -18 12 C -6 -10 0 -16 -10 D 18 18 16 0 14 E 8 -12 10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8545: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (10) C A E D B (9) E C B A D (7) D B A E C (7) C E B A D (5) B E D C A (5) D B A C E (4) D A B E C (4) C E A B D (4) A D C B E (4) E B C D A (3) D A B C E (3) B E C D A (3) A D C E B (3) A C D E B (3) E C A B D (2) E B C A D (2) C D A B E (2) C A D E B (2) C A D B E (2) A D E C B (2) A C E D B (2) E B D A C (1) E A B D C (1) D C A B E (1) D B C A E (1) C E A D B (1) C B E A D (1) C A E B D (1) B E D A C (1) B D E C A (1) B D C A E (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 -6 -8 4 4 B 6 0 -4 -2 2 C 8 4 0 0 0 D -4 2 0 0 2 E -4 -2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.505841 D: 0.494159 E: 0.000000 Sum of squares = 0.500068222521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.505841 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 4 4 B 6 0 -4 -2 2 C 8 4 0 0 0 D -4 2 0 0 2 E -4 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=22 D=20 E=16 A=15 so A is eliminated. Round 2 votes counts: C=32 D=29 B=22 E=17 so E is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:201 D:200 A:197 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 4 4 B 6 0 -4 -2 2 C 8 4 0 0 0 D -4 2 0 0 2 E -4 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 4 4 B 6 0 -4 -2 2 C 8 4 0 0 0 D -4 2 0 0 2 E -4 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 4 4 B 6 0 -4 -2 2 C 8 4 0 0 0 D -4 2 0 0 2 E -4 -2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8546: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) D E B A C (7) D B E A C (7) B A D C E (6) D E B C A (5) C A E B D (5) C A B E D (4) A C B E D (4) A C B D E (4) A B C D E (4) E D A B C (3) E C D A B (3) E C A D B (3) D B A E C (3) C E A D B (3) C E A B D (3) B D A E C (3) B C A D E (3) E D C B A (2) E D C A B (2) E D B A C (2) C A B D E (2) B D A C E (2) B A C D E (2) A B D C E (2) E A D B C (1) E A C D B (1) D A B E C (1) C B E D A (1) C B D E A (1) C B A D E (1) B C D A E (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 6 -2 -4 B 8 0 20 -2 2 C -6 -20 0 -8 -2 D 2 2 8 0 8 E 4 -2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 -2 -4 B 8 0 20 -2 2 C -6 -20 0 -8 -2 D 2 2 8 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 C=20 B=17 A=15 so A is eliminated. Round 2 votes counts: C=29 E=25 D=23 B=23 so D is eliminated. Round 3 votes counts: E=37 B=34 C=29 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:214 D:210 E:198 A:196 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 6 -2 -4 B 8 0 20 -2 2 C -6 -20 0 -8 -2 D 2 2 8 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -2 -4 B 8 0 20 -2 2 C -6 -20 0 -8 -2 D 2 2 8 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -2 -4 B 8 0 20 -2 2 C -6 -20 0 -8 -2 D 2 2 8 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993679 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8547: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) C B A E D (8) E A C D B (7) D B A C E (7) B C D A E (6) A E D C B (6) D B A E C (5) C E A B D (5) B D C A E (5) D E A B C (4) B C D E A (4) E A D B C (3) C B E A D (3) C B D A E (3) A E C D B (3) D B E A C (2) C B E D A (2) C B A D E (2) A E D B C (2) E D A B C (1) E C B D A (1) E C A B D (1) D E B A C (1) D B E C A (1) D A E B C (1) D A B C E (1) C E B A D (1) C B D E A (1) C A B E D (1) C A B D E (1) B E C D A (1) B D C E A (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -8 6 6 6 B 8 0 -8 -8 8 C -6 8 0 2 4 D -6 8 -2 0 -8 E -6 -8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975208 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 6 6 B 8 0 -8 -8 8 C -6 8 0 2 4 D -6 8 -2 0 -8 E -6 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975067 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=22 E=21 B=17 A=13 so A is eliminated. Round 2 votes counts: E=32 C=28 D=23 B=17 so B is eliminated. Round 3 votes counts: C=38 E=33 D=29 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:205 C:204 B:200 D:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 6 6 6 B 8 0 -8 -8 8 C -6 8 0 2 4 D -6 8 -2 0 -8 E -6 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975067 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 6 6 B 8 0 -8 -8 8 C -6 8 0 2 4 D -6 8 -2 0 -8 E -6 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975067 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 6 6 B 8 0 -8 -8 8 C -6 8 0 2 4 D -6 8 -2 0 -8 E -6 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.363636 B: 0.272727 C: 0.363636 D: 0.000000 E: 0.000000 Sum of squares = 0.338842975067 Cumulative probabilities = A: 0.363636 B: 0.636364 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8548: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (21) D B C A E (11) A C E B D (8) D B E C A (6) C A B E D (6) D E B A C (5) E D B A C (4) C A E B D (4) E A C D B (3) D B E A C (3) C A B D E (3) B D C A E (3) B C D A E (3) D B C E A (2) B C A E D (2) B C A D E (2) A E C B D (2) E D A B C (1) E A D C B (1) E A B C D (1) D E A B C (1) D C B A E (1) D C A B E (1) C D A B E (1) C B D A E (1) C A E D B (1) B D C E A (1) B A E C D (1) A C E D B (1) Total count = 100 A B C D E A 0 10 4 12 2 B -10 0 -8 16 -6 C -4 8 0 20 2 D -12 -16 -20 0 -12 E -2 6 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 12 2 B -10 0 -8 16 -6 C -4 8 0 20 2 D -12 -16 -20 0 -12 E -2 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=30 C=16 B=12 A=11 so A is eliminated. Round 2 votes counts: E=33 D=30 C=25 B=12 so B is eliminated. Round 3 votes counts: E=34 D=34 C=32 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:214 C:213 E:207 B:196 D:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 12 2 B -10 0 -8 16 -6 C -4 8 0 20 2 D -12 -16 -20 0 -12 E -2 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 12 2 B -10 0 -8 16 -6 C -4 8 0 20 2 D -12 -16 -20 0 -12 E -2 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 12 2 B -10 0 -8 16 -6 C -4 8 0 20 2 D -12 -16 -20 0 -12 E -2 6 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995025 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8549: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) C E B A D (6) C B E A D (6) B D E C A (6) B D E A C (5) D B E A C (4) C A B E D (4) A C D E B (4) D B C A E (3) D B A E C (3) D A E B C (3) C A E B D (3) C A D E B (3) B E D C A (3) B E C D A (3) E B C A D (2) E B A D C (2) D E B A C (2) D A C E B (2) C A E D B (2) B C D E A (2) A E C D B (2) A C E D B (2) E D A B C (1) E C B A D (1) E C A B D (1) E B D A C (1) E B A C D (1) E A D B C (1) E A C D B (1) E A B D C (1) D C B A E (1) D C A B E (1) D B A C E (1) D A E C B (1) D A C B E (1) C E A B D (1) C D B A E (1) C D A B E (1) C A D B E (1) B E C A D (1) B D C A E (1) B C E A D (1) A D C E B (1) Total count = 100 A B C D E A 0 -26 -8 -6 -18 B 26 0 8 16 12 C 8 -8 0 -2 -4 D 6 -16 2 0 -6 E 18 -12 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -8 -6 -18 B 26 0 8 16 12 C 8 -8 0 -2 -4 D 6 -16 2 0 -6 E 18 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999571 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=28 D=22 E=12 A=9 so A is eliminated. Round 2 votes counts: C=34 B=29 D=23 E=14 so E is eliminated. Round 3 votes counts: C=39 B=36 D=25 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:208 C:197 D:193 A:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -8 -6 -18 B 26 0 8 16 12 C 8 -8 0 -2 -4 D 6 -16 2 0 -6 E 18 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999571 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -8 -6 -18 B 26 0 8 16 12 C 8 -8 0 -2 -4 D 6 -16 2 0 -6 E 18 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999571 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -8 -6 -18 B 26 0 8 16 12 C 8 -8 0 -2 -4 D 6 -16 2 0 -6 E 18 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999571 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8550: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (11) E B D C A (10) A C D B E (6) E D C B A (5) E C A B D (4) D C E A B (4) D C A B E (4) B E D A C (4) C A D E B (3) B E A D C (3) B A D C E (3) A B C E D (3) E D B C A (2) E C D A B (2) E B D A C (2) C E D A B (2) C D E A B (2) C D A B E (2) C A D B E (2) B E A C D (2) B A E D C (2) B A E C D (2) B A D E C (2) A C B D E (2) E C D B A (1) E C B A D (1) E B C A D (1) E B A D C (1) E B A C D (1) D E C A B (1) D C B E A (1) D B C A E (1) D B A C E (1) C E A B D (1) B D A C E (1) B A C D E (1) A D C B E (1) A C E D B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -20 -12 0 B -6 0 -14 -6 -12 C 20 14 0 4 8 D 12 6 -4 0 0 E 0 12 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -20 -12 0 B -6 0 -14 -6 -12 C 20 14 0 4 8 D 12 6 -4 0 0 E 0 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=23 B=20 A=15 D=12 so D is eliminated. Round 2 votes counts: C=32 E=31 B=22 A=15 so A is eliminated. Round 3 votes counts: C=43 E=31 B=26 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:207 E:202 A:187 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -20 -12 0 B -6 0 -14 -6 -12 C 20 14 0 4 8 D 12 6 -4 0 0 E 0 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -20 -12 0 B -6 0 -14 -6 -12 C 20 14 0 4 8 D 12 6 -4 0 0 E 0 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -20 -12 0 B -6 0 -14 -6 -12 C 20 14 0 4 8 D 12 6 -4 0 0 E 0 12 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997764 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8551: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (19) D B E C A (16) C E A B D (9) A D B C E (9) D B A E C (8) B D E C A (8) A C E D B (8) E C B A D (4) E C B D A (3) A C D E B (3) E B C D A (2) C E B A D (2) C A E B D (2) A D C B E (2) D B E A C (1) D B C A E (1) D A B C E (1) B E D C A (1) B E C D A (1) Total count = 100 A B C D E A 0 6 2 16 6 B -6 0 -4 2 -4 C -2 4 0 6 12 D -16 -2 -6 0 -2 E -6 4 -12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 16 6 B -6 0 -4 2 -4 C -2 4 0 6 12 D -16 -2 -6 0 -2 E -6 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 D=27 C=13 B=10 E=9 so E is eliminated. Round 2 votes counts: A=41 D=27 C=20 B=12 so B is eliminated. Round 3 votes counts: A=41 D=36 C=23 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:210 B:194 E:194 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 16 6 B -6 0 -4 2 -4 C -2 4 0 6 12 D -16 -2 -6 0 -2 E -6 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 16 6 B -6 0 -4 2 -4 C -2 4 0 6 12 D -16 -2 -6 0 -2 E -6 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 16 6 B -6 0 -4 2 -4 C -2 4 0 6 12 D -16 -2 -6 0 -2 E -6 4 -12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996442 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8552: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) D B A E C (5) E C D A B (4) D E C A B (4) C D B E A (4) A E B D C (4) E D A C B (3) E A D C B (3) D E A C B (3) D E A B C (3) D C B E A (3) C D E B A (3) C B D E A (3) C B D A E (3) B D A C E (3) B C A E D (3) E A C D B (2) D C E B A (2) D A E B C (2) C E D A B (2) C E A B D (2) B C A D E (2) B A C E D (2) A E D B C (2) A D E B C (2) A B E D C (2) A B C E D (2) E D C A B (1) E D A B C (1) E C A D B (1) D E C B A (1) D E B A C (1) D B C E A (1) C E D B A (1) C E B A D (1) C B E D A (1) C B E A D (1) B D C A E (1) B D A E C (1) B C D A E (1) B A D E C (1) B A D C E (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -8 -14 -2 B 8 0 -16 -8 2 C 8 16 0 -2 4 D 14 8 2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999336 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -14 -2 B 8 0 -16 -8 2 C 8 16 0 -2 4 D 14 8 2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 E=15 B=15 A=15 so E is eliminated. Round 2 votes counts: C=35 D=30 A=20 B=15 so B is eliminated. Round 3 votes counts: C=41 D=35 A=24 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:213 D:213 E:197 B:193 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -14 -2 B 8 0 -16 -8 2 C 8 16 0 -2 4 D 14 8 2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -14 -2 B 8 0 -16 -8 2 C 8 16 0 -2 4 D 14 8 2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -14 -2 B 8 0 -16 -8 2 C 8 16 0 -2 4 D 14 8 2 0 2 E 2 -2 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999842 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8553: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) A D E B C (11) D A B C E (10) B C E D A (8) A D E C B (7) A E D B C (6) E A B C D (5) C B D E A (5) E C B A D (4) E A B D C (4) E B C A D (3) D C A B E (3) E A C B D (2) D C B A E (2) D B A C E (2) D A C E B (2) D A C B E (2) B C D E A (2) E C D A B (1) E C A D B (1) E A D C B (1) E A C D B (1) C B D A E (1) B E C A D (1) B D A C E (1) B C D A E (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 16 12 -4 0 B -16 0 10 -2 2 C -12 -10 0 -6 2 D 4 2 6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.752720 E: 0.247280 Sum of squares = 0.62773519325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.752720 E: 1.000000 A B C D E A 0 16 12 -4 0 B -16 0 10 -2 2 C -12 -10 0 -6 2 D 4 2 6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 0.499688 Sum of squares = 0.500000194907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 D=21 C=18 B=13 so B is eliminated. Round 2 votes counts: C=29 A=26 E=23 D=22 so D is eliminated. Round 3 votes counts: A=43 C=34 E=23 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:206 E:198 B:197 C:187 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 12 -4 0 B -16 0 10 -2 2 C -12 -10 0 -6 2 D 4 2 6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 0.499688 Sum of squares = 0.500000194907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 -4 0 B -16 0 10 -2 2 C -12 -10 0 -6 2 D 4 2 6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 0.499688 Sum of squares = 0.500000194907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 -4 0 B -16 0 10 -2 2 C -12 -10 0 -6 2 D 4 2 6 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 0.499688 Sum of squares = 0.500000194907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500312 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8554: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (12) E B A D C (10) C D A B E (8) B E C D A (8) D A C E B (5) E B D C A (4) D C A E B (4) C D B A E (4) E A D C B (3) E A B D C (3) D C A B E (3) B C E D A (3) B C D A E (3) B C A D E (3) A E D C B (3) E B A C D (2) E A D B C (2) B E C A D (2) B C D E A (2) A E B D C (2) A D E C B (2) A B C D E (2) E D B C A (1) E D A C B (1) E B D A C (1) D E C A B (1) D C E B A (1) C D B E A (1) C B D A E (1) B E D C A (1) B C A E D (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -2 -4 8 B -4 0 0 -2 -14 C 2 0 0 -18 8 D 4 2 18 0 6 E -8 14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -4 8 B -4 0 0 -2 -14 C 2 0 0 -18 8 D 4 2 18 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 A=22 D=14 C=14 so D is eliminated. Round 2 votes counts: E=28 A=27 B=23 C=22 so C is eliminated. Round 3 votes counts: A=42 E=29 B=29 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:203 C:196 E:196 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -4 8 B -4 0 0 -2 -14 C 2 0 0 -18 8 D 4 2 18 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -4 8 B -4 0 0 -2 -14 C 2 0 0 -18 8 D 4 2 18 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -4 8 B -4 0 0 -2 -14 C 2 0 0 -18 8 D 4 2 18 0 6 E -8 14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8555: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (11) A B C D E (8) E D C A B (5) C D E A B (5) B E D C A (5) B A E D C (5) B A E C D (5) E D C B A (4) E B D C A (4) C A D E B (4) B E A D C (4) A B E C D (4) A B C E D (4) C D A E B (3) B E D A C (3) D E C B A (2) D C E B A (2) D C E A B (2) C D E B A (2) C D A B E (2) B A C D E (2) A B E D C (2) E B D A C (1) E B A D C (1) E A D B C (1) D C B E A (1) D C A E B (1) B D E C A (1) B D C E A (1) B C D E A (1) A E D C B (1) A E C D B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 12 10 10 10 B -12 0 4 2 -2 C -10 -4 0 8 2 D -10 -2 -8 0 -2 E -10 2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 10 10 B -12 0 4 2 -2 C -10 -4 0 8 2 D -10 -2 -8 0 -2 E -10 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=27 E=16 C=16 D=8 so D is eliminated. Round 2 votes counts: A=33 B=27 C=22 E=18 so E is eliminated. Round 3 votes counts: A=34 C=33 B=33 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:198 B:196 E:196 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 10 10 B -12 0 4 2 -2 C -10 -4 0 8 2 D -10 -2 -8 0 -2 E -10 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 10 10 B -12 0 4 2 -2 C -10 -4 0 8 2 D -10 -2 -8 0 -2 E -10 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 10 10 B -12 0 4 2 -2 C -10 -4 0 8 2 D -10 -2 -8 0 -2 E -10 2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8556: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D C A E (9) E B D A C (7) E A D C B (7) B C D A E (7) A E C D B (5) B D E C A (4) B D C E A (4) E D B A C (3) E A D B C (3) E A B C D (3) C A B D E (3) B E C D A (3) E D A B C (2) E B A D C (2) E B A C D (2) E A C B D (2) D C A B E (2) D B C A E (2) C D B A E (2) C D A B E (2) C A D B E (2) B E D C A (2) B C D E A (2) A C D E B (2) E C A B D (1) D C B A E (1) D B A C E (1) D A C E B (1) C B A D E (1) C A E D B (1) C A D E B (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 0 -10 -14 B 6 0 14 6 -4 C 0 -14 0 -2 -10 D 10 -6 2 0 -6 E 14 4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 0 -10 -14 B 6 0 14 6 -4 C 0 -14 0 -2 -10 D 10 -6 2 0 -6 E 14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=32 C=12 A=8 D=7 so D is eliminated. Round 2 votes counts: E=41 B=35 C=15 A=9 so A is eliminated. Round 3 votes counts: E=46 B=35 C=19 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:211 D:200 C:187 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 -10 -14 B 6 0 14 6 -4 C 0 -14 0 -2 -10 D 10 -6 2 0 -6 E 14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -10 -14 B 6 0 14 6 -4 C 0 -14 0 -2 -10 D 10 -6 2 0 -6 E 14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -10 -14 B 6 0 14 6 -4 C 0 -14 0 -2 -10 D 10 -6 2 0 -6 E 14 4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8557: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (6) B D E A C (6) D C E A B (5) D B E C A (5) D B E A C (5) C D A E B (5) D B C A E (4) C E A D B (4) E B A C D (3) D C B E A (3) D C B A E (3) D C A B E (3) D B C E A (3) C A E B D (3) B E D A C (3) B E A D C (3) B D A C E (3) A C E B D (3) E A B C D (2) D E C B A (2) D E C A B (2) D C E B A (2) D B A C E (2) B A E C D (2) A E C B D (2) A C B E D (2) E C A D B (1) E C A B D (1) E B D A C (1) E B A D C (1) E A C B D (1) D E B A C (1) D C A E B (1) C D E A B (1) C A D E B (1) C A D B E (1) B E A C D (1) B A D E C (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -12 -20 -12 B 8 0 -6 -20 4 C 12 6 0 -18 12 D 20 20 18 0 20 E 12 -4 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -20 -12 B 8 0 -6 -20 4 C 12 6 0 -18 12 D 20 20 18 0 20 E 12 -4 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 C=21 B=19 E=10 A=9 so A is eliminated. Round 2 votes counts: D=41 C=27 B=20 E=12 so E is eliminated. Round 3 votes counts: D=41 C=32 B=27 so B is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:239 C:206 B:193 E:188 A:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -12 -20 -12 B 8 0 -6 -20 4 C 12 6 0 -18 12 D 20 20 18 0 20 E 12 -4 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -20 -12 B 8 0 -6 -20 4 C 12 6 0 -18 12 D 20 20 18 0 20 E 12 -4 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -20 -12 B 8 0 -6 -20 4 C 12 6 0 -18 12 D 20 20 18 0 20 E 12 -4 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8558: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (10) D C B A E (8) D A E C B (8) B C A E D (8) D E A C B (7) C B D E A (6) E A D C B (5) C D B E A (5) D C B E A (4) A E D B C (4) E A B C D (3) D A E B C (3) E D A C B (2) D E C A B (2) C B E D A (2) C B E A D (2) C B D A E (2) A E B D C (2) A E B C D (2) A D E B C (2) E C A B D (1) E B A C D (1) E A D B C (1) D C E B A (1) D C E A B (1) D C A E B (1) D A C E B (1) D A C B E (1) D A B C E (1) C D B A E (1) B C A D E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -10 -12 -6 B 4 0 -20 -16 6 C 10 20 0 -8 12 D 12 16 8 0 10 E 6 -6 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -12 -6 B 4 0 -20 -16 6 C 10 20 0 -8 12 D 12 16 8 0 10 E 6 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=20 C=18 E=13 A=11 so A is eliminated. Round 2 votes counts: D=40 E=21 B=21 C=18 so C is eliminated. Round 3 votes counts: D=46 B=33 E=21 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:217 E:189 B:187 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -12 -6 B 4 0 -20 -16 6 C 10 20 0 -8 12 D 12 16 8 0 10 E 6 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -12 -6 B 4 0 -20 -16 6 C 10 20 0 -8 12 D 12 16 8 0 10 E 6 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -12 -6 B 4 0 -20 -16 6 C 10 20 0 -8 12 D 12 16 8 0 10 E 6 -6 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8559: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) C B D E A (8) C B D A E (7) C B A D E (7) E D A B C (6) C E D B A (6) E D B A C (4) C A E D B (4) B D E A C (4) A E D B C (4) E D B C A (3) E C D B A (3) C E A D B (3) C A B D E (3) E D C A B (2) D B E A C (2) C A B E D (2) B D A E C (2) B C D E A (2) B A D E C (2) A C E D B (2) A B D C E (2) A B C D E (2) E A D C B (1) D E A B C (1) D B A E C (1) C E D A B (1) C E B D A (1) B D E C A (1) B D C E A (1) B D C A E (1) A E D C B (1) A E C D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 -10 -12 2 B 10 0 -6 10 14 C 10 6 0 6 6 D 12 -10 -6 0 10 E -2 -14 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -12 2 B 10 0 -6 10 14 C 10 6 0 6 6 D 12 -10 -6 0 10 E -2 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=42 A=22 E=19 B=13 D=4 so D is eliminated. Round 2 votes counts: C=42 A=22 E=20 B=16 so B is eliminated. Round 3 votes counts: C=46 E=27 A=27 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:214 C:214 D:203 A:185 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 -12 2 B 10 0 -6 10 14 C 10 6 0 6 6 D 12 -10 -6 0 10 E -2 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -12 2 B 10 0 -6 10 14 C 10 6 0 6 6 D 12 -10 -6 0 10 E -2 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -12 2 B 10 0 -6 10 14 C 10 6 0 6 6 D 12 -10 -6 0 10 E -2 -14 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8560: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (6) D E A B C (6) C D B E A (6) A E D B C (6) D E B C A (5) D C E B A (4) C B E D A (4) C A B E D (4) A E B D C (4) A C B E D (4) E D B A C (3) E B A D C (3) C A B D E (3) B E A C D (3) B C A E D (3) A C D E B (3) A B E C D (3) D E C B A (2) C B D E A (2) C A D B E (2) B A E C D (2) B A C E D (2) E D B C A (1) E B D C A (1) E A D B C (1) D E C A B (1) D A E B C (1) D A C E B (1) C D E B A (1) C D E A B (1) C D A E B (1) C D A B E (1) C B E A D (1) C B A E D (1) B E D C A (1) B E C A D (1) B C E A D (1) A D E C B (1) A D E B C (1) A D C E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 6 4 -8 B 6 0 12 -10 -8 C -6 -12 0 0 -6 D -4 10 0 0 -2 E 8 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 4 -8 B 6 0 12 -10 -8 C -6 -12 0 0 -6 D -4 10 0 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 A=25 B=13 E=9 so E is eliminated. Round 2 votes counts: D=30 C=27 A=26 B=17 so B is eliminated. Round 3 votes counts: A=36 D=32 C=32 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:212 D:202 B:200 A:198 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 4 -8 B 6 0 12 -10 -8 C -6 -12 0 0 -6 D -4 10 0 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 4 -8 B 6 0 12 -10 -8 C -6 -12 0 0 -6 D -4 10 0 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 4 -8 B 6 0 12 -10 -8 C -6 -12 0 0 -6 D -4 10 0 0 -2 E 8 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999666 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8561: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (6) E D C A B (5) B C A D E (5) A D E B C (5) E D A C B (4) E C D A B (4) E A D B C (4) D A E C B (4) C B E D A (4) C B D A E (4) D E A C B (3) C E B D A (3) B A C D E (3) A B D E C (3) D A C B E (2) C E D A B (2) C D E A B (2) B E A C D (2) B C A E D (2) B A E C D (2) B A D C E (2) B A C E D (2) A E D B C (2) A E B D C (2) A D B E C (2) E D A B C (1) E C D B A (1) E C B D A (1) E C B A D (1) E B A C D (1) D C E A B (1) D C A E B (1) D A C E B (1) D A B E C (1) D A B C E (1) C E D B A (1) C E B A D (1) C D E B A (1) C D B A E (1) C B E A D (1) C B D E A (1) C B A D E (1) B C D A E (1) B A D E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 12 0 12 B -4 0 0 0 -2 C -12 0 0 -6 -10 D 0 0 6 0 -4 E -12 2 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.507553 B: 0.000000 C: 0.000000 D: 0.492447 E: 0.000000 Sum of squares = 0.500114086826 Cumulative probabilities = A: 0.507553 B: 0.507553 C: 0.507553 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 0 12 B -4 0 0 0 -2 C -12 0 0 -6 -10 D 0 0 6 0 -4 E -12 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 C=22 A=16 D=14 so D is eliminated. Round 2 votes counts: B=26 E=25 A=25 C=24 so C is eliminated. Round 3 votes counts: B=38 E=36 A=26 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:214 E:202 D:201 B:197 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 0 12 B -4 0 0 0 -2 C -12 0 0 -6 -10 D 0 0 6 0 -4 E -12 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 0 12 B -4 0 0 0 -2 C -12 0 0 -6 -10 D 0 0 6 0 -4 E -12 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 0 12 B -4 0 0 0 -2 C -12 0 0 -6 -10 D 0 0 6 0 -4 E -12 2 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8562: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (23) C B A D E (13) E D B C A (4) C B E D A (4) A D E B C (4) A C B D E (4) E D A C B (3) E A D B C (3) D E A B C (3) C B D A E (3) C A B E D (3) B C E D A (3) A E D B C (3) A D B C E (3) E D B A C (2) E A D C B (2) C B D E A (2) C A B D E (2) A D C B E (2) E C B D A (1) E B D C A (1) E A C D B (1) D B E C A (1) D A B E C (1) C E B A D (1) C B E A D (1) C B A E D (1) B E C D A (1) B D A C E (1) B C D A E (1) A D E C B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 20 16 -8 -12 B -20 0 10 -14 -4 C -16 -10 0 -16 -8 D 8 14 16 0 -14 E 12 4 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 16 -8 -12 B -20 0 10 -14 -4 C -16 -10 0 -16 -8 D 8 14 16 0 -14 E 12 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=30 A=19 B=6 D=5 so D is eliminated. Round 2 votes counts: E=43 C=30 A=20 B=7 so B is eliminated. Round 3 votes counts: E=45 C=34 A=21 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:212 A:208 B:186 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 16 -8 -12 B -20 0 10 -14 -4 C -16 -10 0 -16 -8 D 8 14 16 0 -14 E 12 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 -8 -12 B -20 0 10 -14 -4 C -16 -10 0 -16 -8 D 8 14 16 0 -14 E 12 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 -8 -12 B -20 0 10 -14 -4 C -16 -10 0 -16 -8 D 8 14 16 0 -14 E 12 4 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996778 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8563: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (10) E B A D C (9) C D B A E (8) A E D B C (6) A D C E B (4) A D B C E (4) E C B D A (3) E B C D A (3) E A B D C (3) B E D C A (3) A E B D C (3) A D E B C (3) E B A C D (2) D C B A E (2) D C A B E (2) C E D B A (2) C D E B A (2) C D A E B (2) C D A B E (2) C B E D A (2) B E C D A (2) B E A D C (2) B A D E C (2) A E C D B (2) A B D E C (2) E C A D B (1) E B C A D (1) E A B C D (1) D B C A E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E A B (1) C B D E A (1) B E D A C (1) B D E C A (1) B D A C E (1) B C D E A (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 14 16 12 B 0 0 6 -10 -2 C -14 -6 0 -22 -4 D -16 10 22 0 0 E -12 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.570092 B: 0.429908 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.509825899732 Cumulative probabilities = A: 0.570092 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 16 12 B 0 0 6 -10 -2 C -14 -6 0 -22 -4 D -16 10 22 0 0 E -12 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.499996 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.500004 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=23 C=23 B=13 D=5 so D is eliminated. Round 2 votes counts: A=36 C=27 E=23 B=14 so B is eliminated. Round 3 votes counts: A=39 E=32 C=29 so C is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:208 B:197 E:197 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 14 16 12 B 0 0 6 -10 -2 C -14 -6 0 -22 -4 D -16 10 22 0 0 E -12 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.499996 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.500004 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 16 12 B 0 0 6 -10 -2 C -14 -6 0 -22 -4 D -16 10 22 0 0 E -12 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.499996 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.500004 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 16 12 B 0 0 6 -10 -2 C -14 -6 0 -22 -4 D -16 10 22 0 0 E -12 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500004 B: 0.499996 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.500004 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8564: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (18) B C D A E (9) A E D C B (9) C D E B A (7) B C D E A (7) C D B E A (6) A E D B C (5) A E B D C (5) C D E A B (4) B A C D E (4) C B D E A (3) E D C A B (2) E D A C B (2) D C E A B (2) D C A E B (2) B A E D C (2) B A D E C (2) B A D C E (2) A D E C B (2) E C D B A (1) E C B D A (1) E A B D C (1) D E A C B (1) C B D A E (1) B C E D A (1) B A C E D (1) Total count = 100 A B C D E A 0 6 8 2 -12 B -6 0 -22 -22 -24 C -8 22 0 -10 -2 D -2 22 10 0 4 E 12 24 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839502 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 A B C D E A 0 6 8 2 -12 B -6 0 -22 -22 -24 C -8 22 0 -10 -2 D -2 22 10 0 4 E 12 24 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839432 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 C=21 A=21 D=5 so D is eliminated. Round 2 votes counts: B=28 E=26 C=25 A=21 so A is eliminated. Round 3 votes counts: E=47 B=28 C=25 so C is eliminated. Round 4 votes counts: E=62 B=38 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:217 E:217 A:202 C:201 B:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 2 -12 B -6 0 -22 -22 -24 C -8 22 0 -10 -2 D -2 22 10 0 4 E 12 24 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839432 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 -12 B -6 0 -22 -22 -24 C -8 22 0 -10 -2 D -2 22 10 0 4 E 12 24 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839432 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 -12 B -6 0 -22 -22 -24 C -8 22 0 -10 -2 D -2 22 10 0 4 E 12 24 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.111111 Sum of squares = 0.506172839432 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.222222 D: 0.888889 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8565: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) B E C D A (8) E B C D A (7) E B A D C (6) C D A E B (6) B E A D C (6) E B D A C (5) D A C E B (5) C D A B E (5) B E C A D (4) A D C B E (4) D A E C B (3) B C A D E (3) B A D C E (3) A D C E B (3) E D C A B (2) E D A C B (2) E C D A B (2) E A D B C (2) B E A C D (2) B C E D A (2) B A D E C (2) A D E C B (2) A D B C E (2) E D A B C (1) E B A C D (1) D C A E B (1) C E D A B (1) C A D B E (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 0 -16 -2 B 14 0 10 14 0 C 0 -10 0 0 -12 D 16 -14 0 0 -2 E 2 0 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.533274 C: 0.000000 D: 0.000000 E: 0.466726 Sum of squares = 0.502214354708 Cumulative probabilities = A: 0.000000 B: 0.533274 C: 0.533274 D: 0.533274 E: 1.000000 A B C D E A 0 -14 0 -16 -2 B 14 0 10 14 0 C 0 -10 0 0 -12 D 16 -14 0 0 -2 E 2 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 C=21 A=12 D=9 so D is eliminated. Round 2 votes counts: B=30 E=28 C=22 A=20 so A is eliminated. Round 3 votes counts: E=34 C=34 B=32 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:208 D:200 C:189 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -16 -2 B 14 0 10 14 0 C 0 -10 0 0 -12 D 16 -14 0 0 -2 E 2 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -16 -2 B 14 0 10 14 0 C 0 -10 0 0 -12 D 16 -14 0 0 -2 E 2 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -16 -2 B 14 0 10 14 0 C 0 -10 0 0 -12 D 16 -14 0 0 -2 E 2 0 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8566: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) C A B D E (6) B A D C E (6) E C D A B (5) B E D C A (5) A D C E B (5) C E A B D (4) B D E A C (4) E C A D B (3) E B D C A (3) E B D A C (3) D A E C B (3) C E A D B (3) B D A E C (3) A D B C E (3) A C D B E (3) E B C D A (2) D E B A C (2) D E A B C (2) D B E A C (2) C A E D B (2) C A D B E (2) B E C D A (2) E D C A B (1) E D B C A (1) E D A C B (1) E C B A D (1) D E A C B (1) D B A E C (1) D A E B C (1) D A B E C (1) C E B A D (1) C B E A D (1) C B A D E (1) C A D E B (1) C A B E D (1) B E D A C (1) B D A C E (1) B C A D E (1) B A C D E (1) A D E C B (1) A D C B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 8 -2 -8 B -4 0 4 -2 -4 C -8 -4 0 -16 -10 D 2 2 16 0 8 E 8 4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -2 -8 B -4 0 4 -2 -4 C -8 -4 0 -16 -10 D 2 2 16 0 8 E 8 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=24 C=22 A=15 D=13 so D is eliminated. Round 2 votes counts: E=31 B=27 C=22 A=20 so A is eliminated. Round 3 votes counts: E=36 C=32 B=32 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:214 E:207 A:201 B:197 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -2 -8 B -4 0 4 -2 -4 C -8 -4 0 -16 -10 D 2 2 16 0 8 E 8 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 -8 B -4 0 4 -2 -4 C -8 -4 0 -16 -10 D 2 2 16 0 8 E 8 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 -8 B -4 0 4 -2 -4 C -8 -4 0 -16 -10 D 2 2 16 0 8 E 8 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8567: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (13) E A C B D (7) E A C D B (6) D B C E A (6) E D B C A (5) E A B C D (5) D B E C A (5) D B C A E (4) B E A D C (4) A C B D E (4) E D C A B (3) E B D A C (3) C A B D E (3) A E C B D (3) A C E D B (3) A C E B D (3) E D B A C (2) E B A D C (2) D E B C A (2) D C B A E (2) C A D B E (2) B C D A E (2) E D C B A (1) E D A C B (1) E A D B C (1) D C E B A (1) C E D A B (1) C D A B E (1) C A D E B (1) B E D A C (1) B A C D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -4 -6 -12 B 8 0 16 6 0 C 4 -16 0 -12 -4 D 6 -6 12 0 -4 E 12 0 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.485041 C: 0.000000 D: 0.000000 E: 0.514959 Sum of squares = 0.500447511192 Cumulative probabilities = A: 0.000000 B: 0.485041 C: 0.485041 D: 0.485041 E: 1.000000 A B C D E A 0 -8 -4 -6 -12 B 8 0 16 6 0 C 4 -16 0 -12 -4 D 6 -6 12 0 -4 E 12 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=21 D=20 A=15 C=8 so C is eliminated. Round 2 votes counts: E=37 D=21 B=21 A=21 so D is eliminated. Round 3 votes counts: E=40 B=38 A=22 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:210 D:204 C:186 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -6 -12 B 8 0 16 6 0 C 4 -16 0 -12 -4 D 6 -6 12 0 -4 E 12 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -6 -12 B 8 0 16 6 0 C 4 -16 0 -12 -4 D 6 -6 12 0 -4 E 12 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -6 -12 B 8 0 16 6 0 C 4 -16 0 -12 -4 D 6 -6 12 0 -4 E 12 0 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999982 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) E B A D C (6) D E A C B (6) C D A E B (6) B C A E D (6) E D A B C (5) B C E A D (4) E A D B C (3) D C A E B (3) C A D B E (3) C A B D E (3) B E D A C (3) B E A D C (3) B E A C D (3) A E D C B (3) A E D B C (3) A D E C B (3) E D B A C (2) D E A B C (2) C D B E A (2) C B D E A (2) C B A D E (2) B E D C A (2) B E C A D (2) B C E D A (2) E D A C B (1) D E C B A (1) D E C A B (1) D E B C A (1) D C B E A (1) D A C E B (1) C D B A E (1) C D A B E (1) C B A E D (1) C A D E B (1) B E C D A (1) A E B D C (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 10 8 -2 -6 B -10 0 0 -16 -16 C -8 0 0 -16 -20 D 2 16 16 0 -4 E 6 16 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 8 -2 -6 B -10 0 0 -16 -16 C -8 0 0 -16 -20 D 2 16 16 0 -4 E 6 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=23 C=22 E=17 A=12 so A is eliminated. Round 2 votes counts: D=26 B=26 E=25 C=23 so C is eliminated. Round 3 votes counts: D=41 B=34 E=25 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:223 D:215 A:205 B:179 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 8 -2 -6 B -10 0 0 -16 -16 C -8 0 0 -16 -20 D 2 16 16 0 -4 E 6 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 -2 -6 B -10 0 0 -16 -16 C -8 0 0 -16 -20 D 2 16 16 0 -4 E 6 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 -2 -6 B -10 0 0 -16 -16 C -8 0 0 -16 -20 D 2 16 16 0 -4 E 6 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8569: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (13) D A C B E (9) B E C D A (7) B C D A E (7) E B D A C (6) E B C A D (6) E A D C B (6) C A D B E (5) E B A D C (4) C B D A E (4) D A C E B (3) C D A B E (3) E D A C B (2) E D A B C (2) E B C D A (2) C B A D E (2) B E D C A (2) B C E D A (2) B C E A D (2) B C A D E (2) A D C B E (2) E C B A D (1) E B D C A (1) E A D B C (1) E A C D B (1) D C A B E (1) C D B A E (1) B E C A D (1) B C D E A (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 0 -6 6 B 2 0 -8 0 2 C 0 8 0 -4 16 D 6 0 4 0 8 E -6 -2 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.167227 C: 0.000000 D: 0.832773 E: 0.000000 Sum of squares = 0.721476005984 Cumulative probabilities = A: 0.000000 B: 0.167227 C: 0.167227 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -6 6 B 2 0 -8 0 2 C 0 8 0 -4 16 D 6 0 4 0 8 E -6 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555563839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=24 A=16 C=15 D=13 so D is eliminated. Round 2 votes counts: E=32 A=28 B=24 C=16 so C is eliminated. Round 3 votes counts: A=37 E=32 B=31 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:210 D:209 A:199 B:198 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -6 6 B 2 0 -8 0 2 C 0 8 0 -4 16 D 6 0 4 0 8 E -6 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555563839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 6 B 2 0 -8 0 2 C 0 8 0 -4 16 D 6 0 4 0 8 E -6 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555563839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 6 B 2 0 -8 0 2 C 0 8 0 -4 16 D 6 0 4 0 8 E -6 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.55555563839 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8570: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) B C E A D (8) A E D C B (7) D A E C B (6) B C E D A (6) D B A E C (5) D B A C E (5) C B A E D (4) C A E B D (4) E C A B D (3) E A C D B (3) D E A B C (3) C E A B D (3) B D C E A (3) B D C A E (3) B C D E A (3) B C A E D (3) E C B A D (2) D A E B C (2) B C D A E (2) A E C D B (2) A D E C B (2) E D A B C (1) E B D C A (1) E A D C B (1) D B E A C (1) D A B E C (1) C E B A D (1) C A B E D (1) B D E C A (1) B C A D E (1) A D C E B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 -16 14 2 B 16 0 -2 16 14 C 16 2 0 12 18 D -14 -16 -12 0 -20 E -2 -14 -18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 14 2 B 16 0 -2 16 14 C 16 2 0 12 18 D -14 -16 -12 0 -20 E -2 -14 -18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 C=22 A=14 E=11 so E is eliminated. Round 2 votes counts: B=31 C=27 D=24 A=18 so A is eliminated. Round 3 votes counts: D=36 C=33 B=31 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:222 E:193 A:192 D:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -16 14 2 B 16 0 -2 16 14 C 16 2 0 12 18 D -14 -16 -12 0 -20 E -2 -14 -18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 14 2 B 16 0 -2 16 14 C 16 2 0 12 18 D -14 -16 -12 0 -20 E -2 -14 -18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 14 2 B 16 0 -2 16 14 C 16 2 0 12 18 D -14 -16 -12 0 -20 E -2 -14 -18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8571: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) A C B E D (6) A B C D E (6) A B D E C (5) D A B E C (4) E B D C A (3) D E B A C (3) D C A E B (3) D B E A C (3) C E A B D (3) C A E B D (3) C A B E D (3) B E D A C (3) B D E A C (3) A D B C E (3) A C D B E (3) E C B D A (2) E B C D A (2) D E B C A (2) D C E A B (2) D A E B C (2) C E B D A (2) C D E A B (2) C B E A D (2) C A E D B (2) C A D E B (2) B C A E D (2) A D B E C (2) A C B D E (2) E D C B A (1) E D B C A (1) D E C A B (1) D E A B C (1) C E D B A (1) C E D A B (1) C E B A D (1) B E C D A (1) B E C A D (1) B A E D C (1) B A E C D (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 18 2 2 6 B -18 0 2 8 6 C -2 -2 0 0 4 D -2 -8 0 0 14 E -6 -6 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 2 6 B -18 0 2 8 6 C -2 -2 0 0 4 D -2 -8 0 0 14 E -6 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=27 C=22 B=12 E=9 so E is eliminated. Round 2 votes counts: A=30 D=29 C=24 B=17 so B is eliminated. Round 3 votes counts: D=38 A=32 C=30 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:202 C:200 B:199 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 2 6 B -18 0 2 8 6 C -2 -2 0 0 4 D -2 -8 0 0 14 E -6 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 2 6 B -18 0 2 8 6 C -2 -2 0 0 4 D -2 -8 0 0 14 E -6 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 2 6 B -18 0 2 8 6 C -2 -2 0 0 4 D -2 -8 0 0 14 E -6 -6 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8572: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) D E A B C (6) C B D A E (6) A E D B C (5) E A D B C (4) C D B A E (4) E A B D C (3) D A E C B (3) C D A B E (3) C B A E D (3) C A D B E (3) C A B E D (3) C A B D E (3) B E A C D (3) A E D C B (3) A E B C D (3) D E A C B (2) D C E B A (2) D C B E A (2) D A E B C (2) B E C D A (2) B E C A D (2) B C E D A (2) B C A E D (2) E D B A C (1) E D A B C (1) E B A D C (1) E B A C D (1) E A B C D (1) D E C B A (1) D E C A B (1) D E B C A (1) D C E A B (1) D C B A E (1) C B E D A (1) C B E A D (1) C B A D E (1) B C D E A (1) A E B D C (1) A D E C B (1) A D C E B (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -10 -4 2 B -6 0 -16 0 8 C 10 16 0 14 4 D 4 0 -14 0 14 E -2 -8 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -4 2 B -6 0 -16 0 8 C 10 16 0 14 4 D 4 0 -14 0 14 E -2 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=22 A=17 E=12 B=12 so E is eliminated. Round 2 votes counts: C=37 A=25 D=24 B=14 so B is eliminated. Round 3 votes counts: C=46 A=30 D=24 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:202 A:197 B:193 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 -4 2 B -6 0 -16 0 8 C 10 16 0 14 4 D 4 0 -14 0 14 E -2 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -4 2 B -6 0 -16 0 8 C 10 16 0 14 4 D 4 0 -14 0 14 E -2 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -4 2 B -6 0 -16 0 8 C 10 16 0 14 4 D 4 0 -14 0 14 E -2 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8573: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) B C D E A (9) D E A B C (8) D B E A C (8) B D C E A (8) D E B A C (6) A E C D B (6) C B A D E (5) E A D B C (4) C B D A E (4) C A E B D (4) B D E C A (4) E A D C B (3) B C D A E (3) A E D B C (3) E D A B C (2) D B E C A (2) C B A E D (2) C A B E D (2) E D A C B (1) D B A E C (1) D A B E C (1) C B D E A (1) B A D C E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 12 -16 -12 B 8 0 20 -12 2 C -12 -20 0 -24 -18 D 16 12 24 0 22 E 12 -2 18 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 -16 -12 B 8 0 20 -12 2 C -12 -20 0 -24 -18 D 16 12 24 0 22 E 12 -2 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=25 A=21 C=18 E=10 so E is eliminated. Round 2 votes counts: D=29 A=28 B=25 C=18 so C is eliminated. Round 3 votes counts: B=37 A=34 D=29 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:237 B:209 E:203 A:188 C:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 12 -16 -12 B 8 0 20 -12 2 C -12 -20 0 -24 -18 D 16 12 24 0 22 E 12 -2 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 -16 -12 B 8 0 20 -12 2 C -12 -20 0 -24 -18 D 16 12 24 0 22 E 12 -2 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 -16 -12 B 8 0 20 -12 2 C -12 -20 0 -24 -18 D 16 12 24 0 22 E 12 -2 18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8574: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C A B E D (9) E D B C A (7) C E B D A (7) A C B D E (7) A C B E D (6) E B D C A (5) C B A E D (5) E B C D A (4) D E B C A (3) D A E B C (3) D A B E C (3) C B E A D (3) E D C B A (2) E C B D A (2) D E A B C (2) D B E A C (2) D B A E C (2) B E D C A (2) B D E A C (2) B C E A D (2) B A C E D (2) A D B C E (2) A B C D E (2) D E A C B (1) C E B A D (1) C E A D B (1) C B E D A (1) A D B E C (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -8 -14 -12 B 22 0 8 22 4 C 8 -8 0 6 0 D 14 -22 -6 0 -18 E 12 -4 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -8 -14 -12 B 22 0 8 22 4 C 8 -8 0 6 0 D 14 -22 -6 0 -18 E 12 -4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989233 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 E=20 A=20 B=8 so B is eliminated. Round 2 votes counts: C=29 D=27 E=22 A=22 so E is eliminated. Round 3 votes counts: D=43 C=35 A=22 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:228 E:213 C:203 D:184 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 -8 -14 -12 B 22 0 8 22 4 C 8 -8 0 6 0 D 14 -22 -6 0 -18 E 12 -4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989233 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -8 -14 -12 B 22 0 8 22 4 C 8 -8 0 6 0 D 14 -22 -6 0 -18 E 12 -4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989233 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -8 -14 -12 B 22 0 8 22 4 C 8 -8 0 6 0 D 14 -22 -6 0 -18 E 12 -4 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989233 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8575: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) C A E B D (8) D B A C E (5) B D E A C (5) E C A D B (4) A C B E D (4) E D C B A (3) E D C A B (3) E D B C A (3) D E C A B (3) D B E A C (3) C A E D B (3) B A C D E (3) A C B D E (3) A B C D E (3) E C D A B (2) E C A B D (2) E B D C A (2) D E C B A (2) D C E A B (2) D A C E B (2) D A B C E (2) C E A D B (2) B E D C A (2) B E D A C (2) B D A E C (2) B D A C E (2) A D C B E (2) E C D B A (1) E B C A D (1) D E B A C (1) D C A E B (1) D B A E C (1) C E A B D (1) C B A E D (1) B A D E C (1) B A D C E (1) B A C E D (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -8 -14 -4 B 0 0 -4 -10 -12 C 8 4 0 -16 -2 D 14 10 16 0 6 E 4 12 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -14 -4 B 0 0 -4 -10 -12 C 8 4 0 -16 -2 D 14 10 16 0 6 E 4 12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=21 B=19 C=15 A=15 so C is eliminated. Round 2 votes counts: D=30 A=26 E=24 B=20 so B is eliminated. Round 3 votes counts: D=39 A=33 E=28 so E is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:206 C:197 A:187 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -8 -14 -4 B 0 0 -4 -10 -12 C 8 4 0 -16 -2 D 14 10 16 0 6 E 4 12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -14 -4 B 0 0 -4 -10 -12 C 8 4 0 -16 -2 D 14 10 16 0 6 E 4 12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -14 -4 B 0 0 -4 -10 -12 C 8 4 0 -16 -2 D 14 10 16 0 6 E 4 12 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8576: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E A B D C (8) D B C E A (8) C A D B E (7) A E C B D (5) B D C A E (4) A C E B D (4) E D B C A (3) E A C D B (3) C A B D E (3) B D E C A (3) B D C E A (3) A C E D B (3) A C B D E (3) E D C A B (2) E D B A C (2) E C A D B (2) E A D B C (2) E A B C D (2) C D B E A (2) B D A C E (2) A E C D B (2) A E B D C (2) E D C B A (1) E C D A B (1) E B A D C (1) E A D C B (1) E A C B D (1) D E C B A (1) D B E C A (1) C E D A B (1) C D E B A (1) C D E A B (1) C A E D B (1) C A D E B (1) B E D A C (1) B C A D E (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 12 -12 8 -2 B -12 0 -12 -10 -2 C 12 12 0 8 12 D -8 10 -8 0 2 E 2 2 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 8 -2 B -12 0 -12 -10 -2 C 12 12 0 8 12 D -8 10 -8 0 2 E 2 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=26 A=20 B=15 D=10 so D is eliminated. Round 2 votes counts: E=30 C=26 B=24 A=20 so A is eliminated. Round 3 votes counts: E=39 C=37 B=24 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:203 D:198 E:195 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -12 8 -2 B -12 0 -12 -10 -2 C 12 12 0 8 12 D -8 10 -8 0 2 E 2 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 8 -2 B -12 0 -12 -10 -2 C 12 12 0 8 12 D -8 10 -8 0 2 E 2 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 8 -2 B -12 0 -12 -10 -2 C 12 12 0 8 12 D -8 10 -8 0 2 E 2 2 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8577: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) D B C E A (8) E A B C D (7) A C E D B (7) B D E C A (6) A E C B D (6) E B A D C (5) C D A B E (5) E B D A C (4) E A B D C (4) C A D E B (4) B D C E A (4) D C B A E (3) B E D C A (3) D C B E A (2) D B C A E (2) C A D B E (2) B E D A C (2) A E C D B (2) D B E A C (1) C E A B D (1) C B E A D (1) C B D E A (1) C A E D B (1) C A E B D (1) B D E A C (1) B C E D A (1) B C D E A (1) A E B D C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -16 -12 -4 B 14 0 0 0 10 C 16 0 0 8 16 D 12 0 -8 0 6 E 4 -10 -16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.446797 C: 0.553203 D: 0.000000 E: 0.000000 Sum of squares = 0.505661159465 Cumulative probabilities = A: 0.000000 B: 0.446797 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -12 -4 B 14 0 0 0 10 C 16 0 0 8 16 D 12 0 -8 0 6 E 4 -10 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=20 B=18 A=18 D=16 so D is eliminated. Round 2 votes counts: C=33 B=29 E=20 A=18 so A is eliminated. Round 3 votes counts: C=42 E=29 B=29 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:212 D:205 E:186 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -16 -12 -4 B 14 0 0 0 10 C 16 0 0 8 16 D 12 0 -8 0 6 E 4 -10 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -12 -4 B 14 0 0 0 10 C 16 0 0 8 16 D 12 0 -8 0 6 E 4 -10 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -12 -4 B 14 0 0 0 10 C 16 0 0 8 16 D 12 0 -8 0 6 E 4 -10 -16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999986 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8578: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (5) E D A C B (4) E A C B D (4) D E A B C (4) D B A C E (4) D A B E C (4) C B D A E (4) A E B C D (4) E D A B C (3) E C D B A (3) C E B A D (3) C D B A E (3) C B A E D (3) A B E C D (3) A B D C E (3) E C A B D (2) E A D B C (2) D E C B A (2) D E C A B (2) D C E B A (2) D B C A E (2) D A E B C (2) C E B D A (2) B C A E D (2) A E D B C (2) A C B E D (2) E D C A B (1) E C D A B (1) E C B A D (1) E C A D B (1) E A D C B (1) E A C D B (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C E A (1) D B A E C (1) C E D B A (1) C D B E A (1) C B E D A (1) C B E A D (1) C B D E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E B D C (1) A D B E C (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 4 -14 -4 B -4 0 -6 -14 -2 C -4 6 0 0 -4 D 14 14 0 0 -4 E 4 2 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 4 -14 -4 B -4 0 -6 -14 -2 C -4 6 0 0 -4 D 14 14 0 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=25 C=20 A=19 B=5 so B is eliminated. Round 2 votes counts: D=31 E=25 C=24 A=20 so A is eliminated. Round 3 votes counts: D=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:207 C:199 A:195 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 4 -14 -4 B -4 0 -6 -14 -2 C -4 6 0 0 -4 D 14 14 0 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -14 -4 B -4 0 -6 -14 -2 C -4 6 0 0 -4 D 14 14 0 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -14 -4 B -4 0 -6 -14 -2 C -4 6 0 0 -4 D 14 14 0 0 -4 E 4 2 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8579: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) C D A E B (8) B A E C D (7) A C D E B (7) D C A B E (5) E B D C A (4) E B A C D (4) E A B C D (4) D C E B A (4) E A C D B (3) B E D C A (3) B D C A E (3) A C D B E (3) A B E C D (3) E A C B D (2) D E C B A (2) D C B E A (2) D C B A E (2) C A D E B (2) B E D A C (2) B E A C D (2) A E B C D (2) A C E D B (2) E D B C A (1) E B A D C (1) D C E A B (1) D B C E A (1) D B C A E (1) C D E A B (1) C A E D B (1) C A D B E (1) B D E C A (1) B A D C E (1) B A C E D (1) B A C D E (1) A E C B D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 14 18 6 B 4 0 4 6 0 C -14 -4 0 14 -2 D -18 -6 -14 0 -4 E -6 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.738608 C: 0.000000 D: 0.000000 E: 0.261392 Sum of squares = 0.613867543876 Cumulative probabilities = A: 0.000000 B: 0.738608 C: 0.738608 D: 0.738608 E: 1.000000 A B C D E A 0 -4 14 18 6 B 4 0 4 6 0 C -14 -4 0 14 -2 D -18 -6 -14 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000390267 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=20 E=19 D=18 C=13 so C is eliminated. Round 2 votes counts: B=30 D=27 A=24 E=19 so E is eliminated. Round 3 votes counts: B=39 A=33 D=28 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:207 E:200 C:197 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 18 6 B 4 0 4 6 0 C -14 -4 0 14 -2 D -18 -6 -14 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000390267 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 18 6 B 4 0 4 6 0 C -14 -4 0 14 -2 D -18 -6 -14 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000390267 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 18 6 B 4 0 4 6 0 C -14 -4 0 14 -2 D -18 -6 -14 0 -4 E -6 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000390267 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8580: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (12) A B D E C (9) A B E D C (7) C E D B A (6) E D C A B (5) C D E B A (5) C E B A D (4) B A D E C (4) D B A E C (3) C E D A B (3) A B E C D (3) E D A C B (2) E C D A B (2) E A B D C (2) D E C A B (2) D E A B C (2) D C E B A (2) D A B E C (2) C B A E D (2) B D A E C (2) B A C D E (2) A B C E D (2) E D A B C (1) E C D B A (1) E C A D B (1) E C A B D (1) E A D C B (1) E A D B C (1) E A C B D (1) D E C B A (1) D B A C E (1) C D B E A (1) C D B A E (1) C B A D E (1) B D A C E (1) B C A D E (1) A E B C D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 0 22 14 12 B 0 0 14 14 12 C -22 -14 0 -22 -8 D -14 -14 22 0 6 E -12 -12 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545321 B: 0.454679 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.504108063851 Cumulative probabilities = A: 0.545321 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 22 14 12 B 0 0 14 14 12 C -22 -14 0 -22 -8 D -14 -14 22 0 6 E -12 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 C=23 B=22 E=18 D=13 so D is eliminated. Round 2 votes counts: B=26 A=26 C=25 E=23 so E is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: A=61 C=39 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:220 D:200 E:189 C:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 22 14 12 B 0 0 14 14 12 C -22 -14 0 -22 -8 D -14 -14 22 0 6 E -12 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 22 14 12 B 0 0 14 14 12 C -22 -14 0 -22 -8 D -14 -14 22 0 6 E -12 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 22 14 12 B 0 0 14 14 12 C -22 -14 0 -22 -8 D -14 -14 22 0 6 E -12 -12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8581: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (11) C B A D E (8) A B E D C (8) C D E B A (7) B C A E D (7) A B C E D (7) D E C B A (6) C B D E A (5) D E A C B (4) B A C E D (4) C B E D A (3) B C E D A (3) B A E D C (3) A C B D E (3) E D B C A (2) E D A B C (2) D E C A B (2) C D E A B (2) C D B E A (2) C A D E B (2) A C D E B (2) E D C B A (1) D E A B C (1) D C E A B (1) D A E C B (1) C B A E D (1) B E D A C (1) A D E C B (1) Total count = 100 A B C D E A 0 -6 -4 14 16 B 6 0 -2 6 10 C 4 2 0 12 14 D -14 -6 -12 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 14 16 B 6 0 -2 6 10 C 4 2 0 12 14 D -14 -6 -12 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=30 B=18 D=15 E=5 so E is eliminated. Round 2 votes counts: A=32 C=30 D=20 B=18 so B is eliminated. Round 3 votes counts: C=40 A=39 D=21 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:210 B:210 E:183 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 14 16 B 6 0 -2 6 10 C 4 2 0 12 14 D -14 -6 -12 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 14 16 B 6 0 -2 6 10 C 4 2 0 12 14 D -14 -6 -12 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 14 16 B 6 0 -2 6 10 C 4 2 0 12 14 D -14 -6 -12 0 -6 E -16 -10 -14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8582: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) B A E C D (11) B A C E D (8) D C E A B (7) E D C A B (6) A C E B D (6) E C A D B (5) D B E C A (5) B A C D E (5) D E C B A (4) B E A C D (4) B D A C E (4) D E B C A (2) D B A C E (2) B D A E C (2) A E C B D (2) A C B E D (2) E C A B D (1) E B A C D (1) E A C B D (1) D B C A E (1) C E D A B (1) C D A E B (1) C A E D B (1) C A E B D (1) B E D A C (1) B E A D C (1) B D E A C (1) B A D C E (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 8 4 -2 B 6 0 2 8 -2 C -8 -2 0 4 -18 D -4 -8 -4 0 -8 E 2 2 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 8 4 -2 B 6 0 2 8 -2 C -8 -2 0 4 -18 D -4 -8 -4 0 -8 E 2 2 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=32 E=14 A=12 C=4 so C is eliminated. Round 2 votes counts: B=38 D=33 E=15 A=14 so A is eliminated. Round 3 votes counts: B=41 D=33 E=26 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:207 A:202 C:188 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 8 4 -2 B 6 0 2 8 -2 C -8 -2 0 4 -18 D -4 -8 -4 0 -8 E 2 2 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 4 -2 B 6 0 2 8 -2 C -8 -2 0 4 -18 D -4 -8 -4 0 -8 E 2 2 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 4 -2 B 6 0 2 8 -2 C -8 -2 0 4 -18 D -4 -8 -4 0 -8 E 2 2 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8583: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (6) D C E B A (6) E C B A D (5) D C B E A (5) B C E D A (5) A D E C B (5) A D B C E (5) A B E C D (5) E C D B A (4) A B D C E (4) E A C B D (3) D C E A B (3) C E B D A (3) B A C E D (3) E D C A B (2) E C A B D (2) D A E C B (2) D A B C E (2) C D E B A (2) B D C A E (2) B A C D E (2) A E C D B (2) A D E B C (2) E C D A B (1) E B C A D (1) E A B C D (1) D E C A B (1) D C A E B (1) D B C A E (1) D B A C E (1) D A C E B (1) D A C B E (1) C E D B A (1) B E C D A (1) B D C E A (1) B D A C E (1) B C D E A (1) B C A E D (1) B A D C E (1) A E D C B (1) A E B C D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -10 -8 -8 B 6 0 -14 0 -12 C 10 14 0 0 6 D 8 0 0 0 2 E 8 12 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.464670 D: 0.535330 E: 0.000000 Sum of squares = 0.502496434916 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.464670 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -8 -8 B 6 0 -14 0 -12 C 10 14 0 0 6 D 8 0 0 0 2 E 8 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 D=24 B=18 C=6 so C is eliminated. Round 2 votes counts: E=29 A=27 D=26 B=18 so B is eliminated. Round 3 votes counts: E=35 A=34 D=31 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:206 D:205 B:190 A:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 -8 B 6 0 -14 0 -12 C 10 14 0 0 6 D 8 0 0 0 2 E 8 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 -8 B 6 0 -14 0 -12 C 10 14 0 0 6 D 8 0 0 0 2 E 8 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 -8 B 6 0 -14 0 -12 C 10 14 0 0 6 D 8 0 0 0 2 E 8 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8584: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) C E D A B (8) B D E C A (6) B A E D C (6) C A E D B (5) B D E A C (5) B A D E C (5) A C E D B (5) A B C E D (5) D B E C A (4) E A C D B (3) D E B C A (3) D C E B A (3) C E A D B (3) C D E B A (3) C D E A B (3) A E C D B (3) A B E C D (3) B A C D E (2) A C E B D (2) A B E D C (2) E D B A C (1) E C D A B (1) E C A D B (1) E B A D C (1) C B D E A (1) C B A D E (1) B D C E A (1) B D A E C (1) B D A C E (1) B C D A E (1) B C A D E (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 0 -10 B 10 0 0 -8 -6 C 6 0 0 4 -6 D 0 8 -4 0 2 E 10 6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 A B C D E A 0 -10 -6 0 -10 B 10 0 0 -8 -6 C 6 0 0 4 -6 D 0 8 -4 0 2 E 10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=24 A=21 D=18 E=7 so E is eliminated. Round 2 votes counts: B=31 C=26 A=24 D=19 so D is eliminated. Round 3 votes counts: B=39 C=37 A=24 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:210 D:203 C:202 B:198 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 0 -10 B 10 0 0 -8 -6 C 6 0 0 4 -6 D 0 8 -4 0 2 E 10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 0 -10 B 10 0 0 -8 -6 C 6 0 0 4 -6 D 0 8 -4 0 2 E 10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 0 -10 B 10 0 0 -8 -6 C 6 0 0 4 -6 D 0 8 -4 0 2 E 10 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.500000 E: 0.333333 Sum of squares = 0.388888888801 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.166667 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8585: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) D E B A C (7) B C A D E (7) B A C E D (6) E D A B C (5) E D A C B (4) D E B C A (4) C D E A B (4) C A E D B (4) B D E A C (4) E D B A C (3) C D B E A (3) B A E C D (3) D B E C A (2) C A D E B (2) C A B D E (2) B D E C A (2) B C A E D (2) A E C D B (2) A B C E D (2) E D C A B (1) E A D C B (1) E A C D B (1) D E C B A (1) D E A B C (1) D C E B A (1) D C B E A (1) D B E A C (1) D B C E A (1) C D E B A (1) C D A B E (1) C B D E A (1) C B A E D (1) C B A D E (1) C A E B D (1) C A B E D (1) B E D A C (1) B E A D C (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D A E (1) A E C B D (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 -8 -20 -18 B 16 0 12 -18 -4 C 8 -12 0 -2 -6 D 20 18 2 0 16 E 18 4 6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -8 -20 -18 B 16 0 12 -18 -4 C 8 -12 0 -2 -6 D 20 18 2 0 16 E 18 4 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=26 C=22 E=15 A=7 so A is eliminated. Round 2 votes counts: B=32 D=26 C=24 E=18 so E is eliminated. Round 3 votes counts: D=40 B=32 C=28 so C is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:206 B:203 C:194 A:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -8 -20 -18 B 16 0 12 -18 -4 C 8 -12 0 -2 -6 D 20 18 2 0 16 E 18 4 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -8 -20 -18 B 16 0 12 -18 -4 C 8 -12 0 -2 -6 D 20 18 2 0 16 E 18 4 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -8 -20 -18 B 16 0 12 -18 -4 C 8 -12 0 -2 -6 D 20 18 2 0 16 E 18 4 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8586: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) C A B E D (5) B A D E C (5) A C D B E (5) A B C D E (5) E B D C A (4) A C B D E (4) E D C B A (3) E D B C A (3) D E C B A (3) D E B A C (3) D C E A B (3) D A B E C (3) C D E A B (3) B A E D C (3) B A C E D (3) A C B E D (3) A B D E C (3) D E A B C (2) D B E A C (2) C E D B A (2) C E B A D (2) C D A E B (2) C A E B D (2) B E C A D (2) B D A E C (2) A D B C E (2) A B D C E (2) E D B A C (1) E C D B A (1) E B D A C (1) E B C A D (1) D C A E B (1) C E B D A (1) C A D B E (1) C A B D E (1) B E D A C (1) B D E A C (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 6 16 22 B -12 0 0 4 10 C -6 0 0 2 12 D -16 -4 -2 0 22 E -22 -10 -12 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 16 22 B -12 0 0 4 10 C -6 0 0 2 12 D -16 -4 -2 0 22 E -22 -10 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 D=17 B=17 E=14 so E is eliminated. Round 2 votes counts: C=27 A=26 D=24 B=23 so B is eliminated. Round 3 votes counts: A=37 D=33 C=30 so C is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 C:204 B:201 D:200 E:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 16 22 B -12 0 0 4 10 C -6 0 0 2 12 D -16 -4 -2 0 22 E -22 -10 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 16 22 B -12 0 0 4 10 C -6 0 0 2 12 D -16 -4 -2 0 22 E -22 -10 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 16 22 B -12 0 0 4 10 C -6 0 0 2 12 D -16 -4 -2 0 22 E -22 -10 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8587: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) D B E C A (8) A E D B C (8) E A D B C (7) A E C D B (5) A C E B D (5) C B D A E (4) C A E B D (4) B D C E A (4) B C D A E (4) E D B A C (3) D B C E A (3) C B A D E (3) B D A E C (3) E D A B C (2) E A D C B (2) D E B A C (2) D B E A C (2) C D B E A (2) C A B E D (2) A E D C B (2) A E C B D (2) E D B C A (1) E D A C B (1) E A C D B (1) D E B C A (1) D A B E C (1) C E D B A (1) C E A B D (1) C D E B A (1) C B E D A (1) C A B D E (1) B D C A E (1) A C B D E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -2 -8 -4 B 6 0 4 -6 2 C 2 -4 0 -6 -6 D 8 6 6 0 2 E 4 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 -8 -4 B 6 0 4 -6 2 C 2 -4 0 -6 -6 D 8 6 6 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=25 E=17 D=17 B=12 so B is eliminated. Round 2 votes counts: C=33 D=25 A=25 E=17 so E is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:211 B:203 E:203 C:193 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 -8 -4 B 6 0 4 -6 2 C 2 -4 0 -6 -6 D 8 6 6 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -8 -4 B 6 0 4 -6 2 C 2 -4 0 -6 -6 D 8 6 6 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -8 -4 B 6 0 4 -6 2 C 2 -4 0 -6 -6 D 8 6 6 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8588: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (18) B C A E D (8) A E D C B (8) E A D B C (5) C D E A B (5) C B D A E (5) D E A B C (3) D C E A B (3) B E A D C (3) B A E D C (3) B A E C D (3) B A C E D (3) A E D B C (3) A E B D C (3) E D A B C (2) D E C A B (2) C D A E B (2) C B D E A (2) C B A E D (2) C B A D E (2) A E C D B (2) A B E C D (2) E A B D C (1) D E B A C (1) D C A E B (1) C A D E B (1) C A B D E (1) B E D A C (1) B D E A C (1) B C E A D (1) B C D E A (1) B C D A E (1) A B E D C (1) Total count = 100 A B C D E A 0 26 26 4 2 B -26 0 -8 -12 -20 C -26 8 0 -18 -24 D -4 12 18 0 -2 E -2 20 24 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999676 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 26 4 2 B -26 0 -8 -12 -20 C -26 8 0 -18 -24 D -4 12 18 0 -2 E -2 20 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 C=20 A=19 E=8 so E is eliminated. Round 2 votes counts: D=30 B=25 A=25 C=20 so C is eliminated. Round 3 votes counts: D=37 B=36 A=27 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:229 E:222 D:212 C:170 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 26 4 2 B -26 0 -8 -12 -20 C -26 8 0 -18 -24 D -4 12 18 0 -2 E -2 20 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 26 4 2 B -26 0 -8 -12 -20 C -26 8 0 -18 -24 D -4 12 18 0 -2 E -2 20 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 26 4 2 B -26 0 -8 -12 -20 C -26 8 0 -18 -24 D -4 12 18 0 -2 E -2 20 24 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8589: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) E C D B A (7) C E A D B (6) B D A E C (6) B D A C E (6) D E B C A (5) B A D E C (5) A B D C E (5) A B C E D (5) E C D A B (4) C E D A B (4) C A E B D (4) B A D C E (4) A C B E D (4) E D C B A (3) D B A E C (3) A B C D E (3) D E B A C (2) D B E C A (2) C E D B A (2) E C A D B (1) D B C E A (1) D B C A E (1) C D E B A (1) C B D A E (1) B A C D E (1) A E B D C (1) A C E B D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -4 -16 4 B 20 0 4 -4 -2 C 4 -4 0 -10 -2 D 16 4 10 0 14 E -4 2 2 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -16 4 B 20 0 4 -4 -2 C 4 -4 0 -10 -2 D 16 4 10 0 14 E -4 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=22 A=21 C=18 E=15 so E is eliminated. Round 2 votes counts: C=30 D=27 B=22 A=21 so A is eliminated. Round 3 votes counts: B=38 C=35 D=27 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:222 B:209 C:194 E:193 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -4 -16 4 B 20 0 4 -4 -2 C 4 -4 0 -10 -2 D 16 4 10 0 14 E -4 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -16 4 B 20 0 4 -4 -2 C 4 -4 0 -10 -2 D 16 4 10 0 14 E -4 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -16 4 B 20 0 4 -4 -2 C 4 -4 0 -10 -2 D 16 4 10 0 14 E -4 2 2 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8590: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (14) A B E C D (10) C D B E A (9) C D B A E (8) D C E B A (5) E D A C B (4) E A B D C (4) C D E B A (4) C B D A E (4) A E B C D (4) E A D B C (3) E D B C A (2) E A D C B (2) D E C B A (2) C B A D E (2) B E A D C (2) B A E C D (2) B A C E D (2) A C B D E (2) A B E D C (2) D E C A B (1) D E B C A (1) D B C E A (1) C D E A B (1) C D A E B (1) C D A B E (1) C B D E A (1) C A D E B (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 4 8 8 16 B -4 0 2 6 0 C -8 -2 0 12 -8 D -8 -6 -12 0 -6 E -16 0 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 16 B -4 0 2 6 0 C -8 -2 0 12 -8 D -8 -6 -12 0 -6 E -16 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=32 E=15 D=10 B=9 so B is eliminated. Round 2 votes counts: A=39 C=34 E=17 D=10 so D is eliminated. Round 3 votes counts: C=40 A=39 E=21 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:202 E:199 C:197 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 16 B -4 0 2 6 0 C -8 -2 0 12 -8 D -8 -6 -12 0 -6 E -16 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 16 B -4 0 2 6 0 C -8 -2 0 12 -8 D -8 -6 -12 0 -6 E -16 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 16 B -4 0 2 6 0 C -8 -2 0 12 -8 D -8 -6 -12 0 -6 E -16 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8591: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (9) B A C E D (9) D E C A B (8) E D C B A (7) E D B A C (6) D C E A B (5) E D B C A (4) E B A D C (4) D E C B A (4) C A D B E (4) C A B D E (4) B A E D C (4) B A E C D (4) A B C D E (4) C D E A B (3) E D A B C (2) E B D A C (2) C D A E B (2) C D A B E (2) B C A D E (2) B A C D E (2) A B C E D (2) E D A C B (1) C B A D E (1) B E D C A (1) B C E A D (1) A C D B E (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -20 4 6 -14 B 20 0 14 2 4 C -4 -14 0 -16 -14 D -6 -2 16 0 -14 E 14 -4 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 4 6 -14 B 20 0 14 2 4 C -4 -14 0 -16 -14 D -6 -2 16 0 -14 E 14 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=26 D=17 C=16 A=9 so A is eliminated. Round 2 votes counts: B=39 E=26 C=18 D=17 so D is eliminated. Round 3 votes counts: B=39 E=38 C=23 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:219 D:197 A:188 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 4 6 -14 B 20 0 14 2 4 C -4 -14 0 -16 -14 D -6 -2 16 0 -14 E 14 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 4 6 -14 B 20 0 14 2 4 C -4 -14 0 -16 -14 D -6 -2 16 0 -14 E 14 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 4 6 -14 B 20 0 14 2 4 C -4 -14 0 -16 -14 D -6 -2 16 0 -14 E 14 -4 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999027 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8592: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (11) D C B E A (9) E A D B C (5) B D E A C (5) D B E C A (4) C A E B D (4) B D C E A (4) A E C B D (4) E A B D C (3) D B C E A (3) C D B A E (3) C D A E B (3) C A E D B (3) C A B E D (3) B E D A C (3) A C E D B (3) E B A D C (2) E A D C B (2) C D A B E (2) C B D A E (2) C B A E D (2) C A B D E (2) B A E D C (2) A C E B D (2) E B D A C (1) E A B C D (1) D C E B A (1) D C E A B (1) C D E A B (1) C D B E A (1) C B A D E (1) C A D E B (1) B E A D C (1) B D E C A (1) A E B D C (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -2 -10 -18 B 12 0 -2 -6 22 C 2 2 0 -18 4 D 10 6 18 0 10 E 18 -22 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -10 -18 B 12 0 -2 -6 22 C 2 2 0 -18 4 D 10 6 18 0 10 E 18 -22 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=28 B=16 E=14 A=13 so A is eliminated. Round 2 votes counts: C=34 D=29 E=20 B=17 so B is eliminated. Round 3 votes counts: D=39 C=34 E=27 so E is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:213 C:195 E:191 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -2 -10 -18 B 12 0 -2 -6 22 C 2 2 0 -18 4 D 10 6 18 0 10 E 18 -22 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -10 -18 B 12 0 -2 -6 22 C 2 2 0 -18 4 D 10 6 18 0 10 E 18 -22 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -10 -18 B 12 0 -2 -6 22 C 2 2 0 -18 4 D 10 6 18 0 10 E 18 -22 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8593: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (13) E A D B C (12) A E D B C (11) B D C E A (5) A E D C B (5) E A B D C (4) C B A E D (4) B C D E A (4) D E B A C (3) D B E C A (3) C B D A E (3) C B A D E (3) B E A C D (3) E D A B C (2) D E A B C (2) D B C E A (2) C D B E A (2) C A E D B (2) C A B E D (2) A E C D B (2) A E C B D (2) A C E D B (2) E D B A C (1) D C B E A (1) D A E C B (1) C D B A E (1) C D A E B (1) C D A B E (1) B E A D C (1) B D E A C (1) B C E A D (1) Total count = 100 A B C D E A 0 -2 4 8 -20 B 2 0 10 -8 0 C -4 -10 0 -8 -6 D -8 8 8 0 -8 E 20 0 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.207493 C: 0.000000 D: 0.000000 E: 0.792507 Sum of squares = 0.671121078163 Cumulative probabilities = A: 0.000000 B: 0.207493 C: 0.207493 D: 0.207493 E: 1.000000 A B C D E A 0 -2 4 8 -20 B 2 0 10 -8 0 C -4 -10 0 -8 -6 D -8 8 8 0 -8 E 20 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499639 C: 0.000000 D: 0.000000 E: 0.500361 Sum of squares = 0.500000260048 Cumulative probabilities = A: 0.000000 B: 0.499639 C: 0.499639 D: 0.499639 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=22 E=19 B=15 D=12 so D is eliminated. Round 2 votes counts: C=33 E=24 A=23 B=20 so B is eliminated. Round 3 votes counts: C=45 E=32 A=23 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:202 D:200 A:195 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 8 -20 B 2 0 10 -8 0 C -4 -10 0 -8 -6 D -8 8 8 0 -8 E 20 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499639 C: 0.000000 D: 0.000000 E: 0.500361 Sum of squares = 0.500000260048 Cumulative probabilities = A: 0.000000 B: 0.499639 C: 0.499639 D: 0.499639 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 8 -20 B 2 0 10 -8 0 C -4 -10 0 -8 -6 D -8 8 8 0 -8 E 20 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499639 C: 0.000000 D: 0.000000 E: 0.500361 Sum of squares = 0.500000260048 Cumulative probabilities = A: 0.000000 B: 0.499639 C: 0.499639 D: 0.499639 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 8 -20 B 2 0 10 -8 0 C -4 -10 0 -8 -6 D -8 8 8 0 -8 E 20 0 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499639 C: 0.000000 D: 0.000000 E: 0.500361 Sum of squares = 0.500000260048 Cumulative probabilities = A: 0.000000 B: 0.499639 C: 0.499639 D: 0.499639 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8594: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) D C E A B (7) D A B C E (7) D B A C E (6) E C B A D (5) E A B C D (5) C E D B A (5) C D E B A (4) A D B E C (4) E C D B A (3) E B A C D (3) D C B E A (3) B E A C D (3) A B D C E (3) E C B D A (2) E C A B D (2) E A D C B (2) C E B D A (2) B A E C D (2) A D B C E (2) E B C A D (1) E A C D B (1) D E C A B (1) D C B A E (1) D C A B E (1) D B C A E (1) D A E C B (1) D A C B E (1) C E B A D (1) C D B E A (1) B C E A D (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A E B D C (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 18 8 -4 B -4 0 14 -4 6 C -18 -14 0 -12 -2 D -8 4 12 0 18 E 4 -6 2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.133333 E: 0.266667 Sum of squares = 0.448888888883 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.733333 E: 1.000000 A B C D E A 0 4 18 8 -4 B -4 0 14 -4 6 C -18 -14 0 -12 -2 D -8 4 12 0 18 E 4 -6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.133333 E: 0.266667 Sum of squares = 0.448888886879 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.733333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=24 A=24 C=13 B=10 so B is eliminated. Round 2 votes counts: A=30 D=29 E=27 C=14 so C is eliminated. Round 3 votes counts: E=36 D=34 A=30 so A is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:213 D:213 B:206 E:191 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 18 8 -4 B -4 0 14 -4 6 C -18 -14 0 -12 -2 D -8 4 12 0 18 E 4 -6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.133333 E: 0.266667 Sum of squares = 0.448888886879 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.733333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 18 8 -4 B -4 0 14 -4 6 C -18 -14 0 -12 -2 D -8 4 12 0 18 E 4 -6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.133333 E: 0.266667 Sum of squares = 0.448888886879 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.733333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 18 8 -4 B -4 0 14 -4 6 C -18 -14 0 -12 -2 D -8 4 12 0 18 E 4 -6 2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.133333 E: 0.266667 Sum of squares = 0.448888886879 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.733333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8595: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (12) B C D E A (8) B D C E A (7) C B D E A (5) A E D B C (5) E D A B C (4) C B A D E (4) E A D B C (3) C D B E A (3) C B D A E (3) C B A E D (3) C A B E D (3) B D E C A (3) B C D A E (3) A E C B D (3) A E B C D (3) E A D C B (2) D E B A C (2) D B E C A (2) C A E D B (2) A E B D C (2) E D A C B (1) E B A D C (1) E A B D C (1) D E A B C (1) D C E B A (1) D C E A B (1) D B E A C (1) D B C E A (1) C D B A E (1) C A D E B (1) B E A D C (1) B D E A C (1) B A E C D (1) B A C E D (1) A E C D B (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -2 4 2 B 4 0 4 12 6 C 2 -4 0 -4 -2 D -4 -12 4 0 -4 E -2 -6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 4 2 B 4 0 4 12 6 C 2 -4 0 -4 -2 D -4 -12 4 0 -4 E -2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999075 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 B=25 E=12 D=9 so D is eliminated. Round 2 votes counts: B=29 A=29 C=27 E=15 so E is eliminated. Round 3 votes counts: A=41 B=32 C=27 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:200 E:199 C:196 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 4 2 B 4 0 4 12 6 C 2 -4 0 -4 -2 D -4 -12 4 0 -4 E -2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999075 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 4 2 B 4 0 4 12 6 C 2 -4 0 -4 -2 D -4 -12 4 0 -4 E -2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999075 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 4 2 B 4 0 4 12 6 C 2 -4 0 -4 -2 D -4 -12 4 0 -4 E -2 -6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999075 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8596: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) E D C A B (6) E A D B C (5) D E C B A (5) C B D E A (5) A E B C D (5) D C B E A (4) C B D A E (4) B A C D E (4) E A D C B (3) D A E B C (3) B C D A E (3) B C A D E (3) A B E C D (3) A B C E D (3) A B C D E (3) E D C B A (2) E D A B C (2) D A B E C (2) C B E A D (2) C B A E D (2) B C A E D (2) B A C E D (2) A D E B C (2) A D B C E (2) E C D B A (1) E C D A B (1) E A C B D (1) D E C A B (1) D E A C B (1) C E B D A (1) C B E D A (1) B D A C E (1) A E D B C (1) A D B E C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 16 14 -6 -2 B -16 0 0 -4 0 C -14 0 0 -8 -14 D 6 4 8 0 -8 E 2 0 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.063825 C: 0.000000 D: 0.000000 E: 0.936175 Sum of squares = 0.880496473669 Cumulative probabilities = A: 0.000000 B: 0.063825 C: 0.063825 D: 0.063825 E: 1.000000 A B C D E A 0 16 14 -6 -2 B -16 0 0 -4 0 C -14 0 0 -8 -14 D 6 4 8 0 -8 E 2 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469143905 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=23 D=16 C=15 B=15 so C is eliminated. Round 2 votes counts: E=32 B=29 A=23 D=16 so D is eliminated. Round 3 votes counts: E=39 B=33 A=28 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:212 A:211 D:205 B:190 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 14 -6 -2 B -16 0 0 -4 0 C -14 0 0 -8 -14 D 6 4 8 0 -8 E 2 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469143905 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 -6 -2 B -16 0 0 -4 0 C -14 0 0 -8 -14 D 6 4 8 0 -8 E 2 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469143905 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 -6 -2 B -16 0 0 -4 0 C -14 0 0 -8 -14 D 6 4 8 0 -8 E 2 0 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.000000 E: 0.888889 Sum of squares = 0.802469143905 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.111111 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8597: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) C D B A E (6) A E B C D (6) E A B D C (5) D C B E A (5) B A E C D (5) D E A B C (4) D C A E B (4) B E A C D (4) E B D A C (3) C D A B E (3) A C E D B (3) E D A B C (2) D E A C B (2) D B E C A (2) D B C E A (2) D A E C B (2) C D B E A (2) C B D A E (2) C B A E D (2) B C E A D (2) A E B D C (2) E D B A C (1) E B A C D (1) E A D B C (1) E A B C D (1) D E B C A (1) D C E B A (1) D C E A B (1) D C A B E (1) C D A E B (1) C B A D E (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D E A (1) B C A D E (1) B A C E D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E C B D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 12 -2 -4 B 8 0 14 0 -8 C -12 -14 0 -4 -14 D 2 0 4 0 -6 E 4 8 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 12 -2 -4 B 8 0 14 0 -8 C -12 -14 0 -4 -14 D 2 0 4 0 -6 E 4 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=20 C=19 B=19 A=17 so A is eliminated. Round 2 votes counts: E=32 D=26 C=23 B=19 so B is eliminated. Round 3 votes counts: E=44 D=28 C=28 so D is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:207 D:200 A:199 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 12 -2 -4 B 8 0 14 0 -8 C -12 -14 0 -4 -14 D 2 0 4 0 -6 E 4 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 -2 -4 B 8 0 14 0 -8 C -12 -14 0 -4 -14 D 2 0 4 0 -6 E 4 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 -2 -4 B 8 0 14 0 -8 C -12 -14 0 -4 -14 D 2 0 4 0 -6 E 4 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8598: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) B C A E D (8) A D E C B (7) A C B D E (7) E D A B C (6) D E A C B (6) A D C E B (6) E D B A C (5) E B D C A (5) C A B D E (5) C B A E D (3) B E D C A (3) B C E D A (3) A C D B E (3) A C B E D (3) D A E C B (2) D A C E B (2) C B A D E (2) B E C D A (2) B E C A D (2) E A D B C (1) D E C B A (1) D E B C A (1) C A D B E (1) B D E C A (1) B C E A D (1) B C A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 0 -2 2 2 B 0 0 2 -8 -10 C 2 -2 0 -18 -10 D -2 8 18 0 -10 E -2 10 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.55102040837 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 0 -2 2 2 B 0 0 2 -8 -10 C 2 -2 0 -18 -10 D -2 8 18 0 -10 E -2 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020409436 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=27 B=21 D=12 C=11 so C is eliminated. Round 2 votes counts: A=33 E=29 B=26 D=12 so D is eliminated. Round 3 votes counts: E=37 A=37 B=26 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:214 D:207 A:201 B:192 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 2 2 B 0 0 2 -8 -10 C 2 -2 0 -18 -10 D -2 8 18 0 -10 E -2 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020409436 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 2 2 B 0 0 2 -8 -10 C 2 -2 0 -18 -10 D -2 8 18 0 -10 E -2 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020409436 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 2 2 B 0 0 2 -8 -10 C 2 -2 0 -18 -10 D -2 8 18 0 -10 E -2 10 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020409436 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8599: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (15) A E B D C (11) E A C B D (9) E A B D C (8) C D B A E (7) B D C E A (5) E A C D B (3) E A B C D (3) B D C A E (3) D C B A E (2) D B C A E (2) C E A D B (2) C D E B A (2) C A D E B (2) B E A D C (2) B D E C A (2) B D E A C (2) B D A E C (2) B D A C E (2) A E C D B (2) A C E D B (2) A B E D C (2) E C A D B (1) E B D A C (1) D C A B E (1) D B C E A (1) C E D B A (1) C D E A B (1) C D A B E (1) C B D E A (1) C A E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 0 0 -2 -18 B 0 0 -8 6 0 C 0 8 0 8 4 D 2 -6 -8 0 4 E 18 0 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.118519 B: 0.000000 C: 0.881481 D: 0.000000 E: 0.000000 Sum of squares = 0.791055218916 Cumulative probabilities = A: 0.118519 B: 0.118519 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -2 -18 B 0 0 -8 6 0 C 0 8 0 8 4 D 2 -6 -8 0 4 E 18 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.000000 Sum of squares = 0.702479395721 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=25 B=18 A=18 D=6 so D is eliminated. Round 2 votes counts: C=36 E=25 B=21 A=18 so A is eliminated. Round 3 votes counts: C=39 E=38 B=23 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 E:205 B:199 D:196 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -2 -18 B 0 0 -8 6 0 C 0 8 0 8 4 D 2 -6 -8 0 4 E 18 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.000000 Sum of squares = 0.702479395721 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -18 B 0 0 -8 6 0 C 0 8 0 8 4 D 2 -6 -8 0 4 E 18 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.000000 Sum of squares = 0.702479395721 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -18 B 0 0 -8 6 0 C 0 8 0 8 4 D 2 -6 -8 0 4 E 18 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.000000 Sum of squares = 0.702479395721 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8600: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (11) C D E B A (10) C B D E A (7) A E D B C (7) C B E D A (5) E D B C A (4) A B C E D (4) B E C D A (3) A E B D C (3) A D E C B (3) A D E B C (3) A D C E B (3) A B E D C (3) E D B A C (2) E A D B C (2) C D E A B (2) C D B E A (2) C D A E B (2) C B D A E (2) C B A D E (2) B E D C A (2) B E D A C (2) B C A E D (2) B A E D C (2) E B D A C (1) D E C B A (1) D E C A B (1) D A E C B (1) C E B D A (1) C B A E D (1) B E A C D (1) B C E A D (1) A C D E B (1) A C D B E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -24 -18 -18 -16 B 24 0 8 10 6 C 18 -8 0 20 16 D 18 -10 -20 0 -18 E 16 -6 -16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -18 -18 -16 B 24 0 8 10 6 C 18 -8 0 20 16 D 18 -10 -20 0 -18 E 16 -6 -16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999504 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=30 B=24 E=9 D=3 so D is eliminated. Round 2 votes counts: C=34 A=31 B=24 E=11 so E is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:223 E:206 D:185 A:162 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 -18 -18 -16 B 24 0 8 10 6 C 18 -8 0 20 16 D 18 -10 -20 0 -18 E 16 -6 -16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999504 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -18 -18 -16 B 24 0 8 10 6 C 18 -8 0 20 16 D 18 -10 -20 0 -18 E 16 -6 -16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999504 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -18 -18 -16 B 24 0 8 10 6 C 18 -8 0 20 16 D 18 -10 -20 0 -18 E 16 -6 -16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999504 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8601: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) C A E D B (5) B D A E C (5) A E C D B (5) E A D C B (4) D B E A C (4) C B A E D (4) B C D A E (4) E D C A B (3) D E A B C (3) C A B E D (3) B D E A C (3) B D C A E (3) A E D C B (3) A C E D B (3) E A D B C (2) D E C B A (2) D E B A C (2) C D B E A (2) C A E B D (2) B D E C A (2) B C A E D (2) B C A D E (2) B A D C E (2) A E D B C (2) A E C B D (2) A C E B D (2) E C A D B (1) D E C A B (1) D E A C B (1) D B E C A (1) D B C E A (1) D A E B C (1) C E D A B (1) C E A D B (1) C D E A B (1) C B D E A (1) B C D E A (1) B A E C D (1) B A D E C (1) B A C E D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -2 0 10 B 4 0 6 2 4 C 2 -6 0 -12 -2 D 0 -2 12 0 2 E -10 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 0 10 B 4 0 6 2 4 C 2 -6 0 -12 -2 D 0 -2 12 0 2 E -10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=20 A=19 D=16 E=10 so E is eliminated. Round 2 votes counts: B=35 A=25 C=21 D=19 so D is eliminated. Round 3 votes counts: B=43 A=30 C=27 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:206 A:202 E:193 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 0 10 B 4 0 6 2 4 C 2 -6 0 -12 -2 D 0 -2 12 0 2 E -10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 0 10 B 4 0 6 2 4 C 2 -6 0 -12 -2 D 0 -2 12 0 2 E -10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 0 10 B 4 0 6 2 4 C 2 -6 0 -12 -2 D 0 -2 12 0 2 E -10 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8602: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) A E C D B (8) B D C E A (6) D C B A E (5) E A C B D (4) C D B A E (4) C B E A D (4) B E C A D (4) A E C B D (4) D C A B E (3) D B C A E (3) D B A C E (3) D A E B C (3) C B D A E (3) E B A D C (2) E B A C D (2) E A B D C (2) E A B C D (2) D B E A C (2) D B A E C (2) D A C B E (2) C A E B D (2) B D E C A (2) A E D C B (2) A E D B C (2) E C B A D (1) E C A B D (1) E B D A C (1) E B C A D (1) D E A B C (1) D A E C B (1) D A B E C (1) C E A B D (1) C B D E A (1) C A D E B (1) C A D B E (1) B E A D C (1) B C E D A (1) B C E A D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -6 -4 4 B 14 0 0 8 14 C 6 0 0 10 0 D 4 -8 -10 0 6 E -4 -14 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.578763 C: 0.421237 D: 0.000000 E: 0.000000 Sum of squares = 0.512407171441 Cumulative probabilities = A: 0.000000 B: 0.578763 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -4 4 B 14 0 0 8 14 C 6 0 0 10 0 D 4 -8 -10 0 6 E -4 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=23 A=18 C=17 E=16 so E is eliminated. Round 2 votes counts: B=29 D=26 A=26 C=19 so C is eliminated. Round 3 votes counts: B=38 A=32 D=30 so D is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:208 D:196 A:190 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -4 4 B 14 0 0 8 14 C 6 0 0 10 0 D 4 -8 -10 0 6 E -4 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -4 4 B 14 0 0 8 14 C 6 0 0 10 0 D 4 -8 -10 0 6 E -4 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -4 4 B 14 0 0 8 14 C 6 0 0 10 0 D 4 -8 -10 0 6 E -4 -14 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8603: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (15) A D C B E (12) E B A C D (7) E B A D C (6) C D A B E (6) A D C E B (6) E B C D A (5) B E D C A (5) B D C A E (5) D C A B E (4) A C D E B (4) E B D C A (3) E A D C B (3) A D E C B (3) E B C A D (2) B C D E A (2) A C D B E (2) E C A D B (1) E B D A C (1) E A C B D (1) E A B D C (1) E A B C D (1) D B C A E (1) C D B A E (1) C A D E B (1) C A D B E (1) B D C E A (1) Total count = 100 A B C D E A 0 -8 -6 2 -8 B 8 0 10 10 10 C 6 -10 0 -2 -8 D -2 -10 2 0 -2 E 8 -10 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 2 -8 B 8 0 10 10 10 C 6 -10 0 -2 -8 D -2 -10 2 0 -2 E 8 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=28 A=27 C=9 D=5 so D is eliminated. Round 2 votes counts: E=31 B=29 A=27 C=13 so C is eliminated. Round 3 votes counts: A=39 E=31 B=30 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:204 D:194 C:193 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 2 -8 B 8 0 10 10 10 C 6 -10 0 -2 -8 D -2 -10 2 0 -2 E 8 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 2 -8 B 8 0 10 10 10 C 6 -10 0 -2 -8 D -2 -10 2 0 -2 E 8 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 2 -8 B 8 0 10 10 10 C 6 -10 0 -2 -8 D -2 -10 2 0 -2 E 8 -10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8604: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) D E C A B (5) C D E A B (5) C B A E D (5) C A B D E (5) B A C D E (5) D E A B C (4) B C A E D (4) A C B D E (4) A B C D E (4) E D C B A (3) E D C A B (3) E D B A C (3) E C B D A (3) D A E B C (3) C D A E B (3) B A E D C (3) B A C E D (3) A B D E C (3) E D A B C (2) D E A C B (2) C E D A B (2) C B E A D (2) B A D E C (2) E D B C A (1) E C D B A (1) E B C D A (1) E B C A D (1) C E B D A (1) C B A D E (1) C A D B E (1) B E C A D (1) B E A C D (1) B C A D E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -18 -4 -2 B 4 0 -12 2 -6 C 18 12 0 28 14 D 4 -2 -28 0 0 E 2 6 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 -4 -2 B 4 0 -12 2 -6 C 18 12 0 28 14 D 4 -2 -28 0 0 E 2 6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=20 E=18 D=14 A=13 so A is eliminated. Round 2 votes counts: C=39 B=28 E=18 D=15 so D is eliminated. Round 3 votes counts: C=39 E=32 B=29 so B is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:236 E:197 B:194 D:187 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -18 -4 -2 B 4 0 -12 2 -6 C 18 12 0 28 14 D 4 -2 -28 0 0 E 2 6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -4 -2 B 4 0 -12 2 -6 C 18 12 0 28 14 D 4 -2 -28 0 0 E 2 6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -4 -2 B 4 0 -12 2 -6 C 18 12 0 28 14 D 4 -2 -28 0 0 E 2 6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999755 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8605: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) A B E C D (7) E C D B A (6) C E D B A (5) C D A E B (5) B A E D C (5) E C B D A (4) D C E A B (4) D C A E B (4) B E A D C (4) A B C D E (4) D E C B A (3) D C E B A (3) B E A C D (3) A D C B E (3) E B D A C (2) E B C A D (2) D C A B E (2) D B E A C (2) D B A E C (2) D A C B E (2) B A E C D (2) A B D E C (2) A B D C E (2) A B C E D (2) E D B C A (1) E B D C A (1) D E B C A (1) C E A B D (1) C D E B A (1) C D A B E (1) C A D B E (1) B E D A C (1) B A D E C (1) A C B E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -6 -16 -4 B -2 0 -10 -8 -2 C 6 10 0 8 0 D 16 8 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.521884 D: 0.000000 E: 0.478116 Sum of squares = 0.500957828538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.521884 D: 0.521884 E: 1.000000 A B C D E A 0 2 -6 -16 -4 B -2 0 -10 -8 -2 C 6 10 0 8 0 D 16 8 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 A=23 C=22 E=16 B=16 so E is eliminated. Round 2 votes counts: C=32 D=24 A=23 B=21 so B is eliminated. Round 3 votes counts: A=38 C=34 D=28 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:210 E:201 B:189 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -16 -4 B -2 0 -10 -8 -2 C 6 10 0 8 0 D 16 8 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -16 -4 B -2 0 -10 -8 -2 C 6 10 0 8 0 D 16 8 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -16 -4 B -2 0 -10 -8 -2 C 6 10 0 8 0 D 16 8 -8 0 4 E 4 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8606: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) A E B D C (8) A E D B C (7) C D B E A (6) C B E A D (6) D C A B E (5) C D E B A (5) A D E B C (5) D A C E B (4) A B E D C (4) E B A D C (3) E A B D C (3) C D B A E (3) C B E D A (3) B C E A D (3) E D B A C (2) E B A C D (2) D A C B E (2) B E C A D (2) B E A C D (2) A D B E C (2) E D A B C (1) E B C A D (1) E A D B C (1) D C E B A (1) D C E A B (1) D A E C B (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B D A (1) C B D E A (1) C B A E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 2 4 10 B -14 0 0 -16 -14 C -2 0 0 -22 6 D -4 16 22 0 -6 E -10 14 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 4 10 B -14 0 0 -16 -14 C -2 0 0 -22 6 D -4 16 22 0 -6 E -10 14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=27 D=25 E=13 B=7 so B is eliminated. Round 2 votes counts: C=30 A=28 D=25 E=17 so E is eliminated. Round 3 votes counts: A=39 C=33 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:214 E:202 C:191 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 4 10 B -14 0 0 -16 -14 C -2 0 0 -22 6 D -4 16 22 0 -6 E -10 14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 4 10 B -14 0 0 -16 -14 C -2 0 0 -22 6 D -4 16 22 0 -6 E -10 14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 4 10 B -14 0 0 -16 -14 C -2 0 0 -22 6 D -4 16 22 0 -6 E -10 14 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989652 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8607: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) B D E C A (7) B A D E C (7) B C D E A (6) C A E D B (5) B D C E A (5) B A C D E (5) C B D E A (4) A E C D B (4) E D C A B (3) D B E C A (3) C B A D E (3) B D E A C (3) B C D A E (3) B C A D E (3) A E D C B (3) A C B E D (3) A B C E D (3) E D A C B (2) D E C B A (2) B A E D C (2) A E D B C (2) A C E D B (2) A C E B D (2) E A D C B (1) D E B C A (1) D B C E A (1) C E A D B (1) C D E B A (1) C B D A E (1) A E C B D (1) A E B D C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -14 0 4 B 14 0 8 22 22 C 14 -8 0 12 14 D 0 -22 -12 0 12 E -4 -22 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 0 4 B 14 0 8 22 22 C 14 -8 0 12 14 D 0 -22 -12 0 12 E -4 -22 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=23 A=23 D=7 E=6 so E is eliminated. Round 2 votes counts: B=41 A=24 C=23 D=12 so D is eliminated. Round 3 votes counts: B=46 C=28 A=26 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:233 C:216 D:189 A:188 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -14 0 4 B 14 0 8 22 22 C 14 -8 0 12 14 D 0 -22 -12 0 12 E -4 -22 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 0 4 B 14 0 8 22 22 C 14 -8 0 12 14 D 0 -22 -12 0 12 E -4 -22 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 0 4 B 14 0 8 22 22 C 14 -8 0 12 14 D 0 -22 -12 0 12 E -4 -22 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999926 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8608: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) B E D C A (7) A C D E B (7) E D A B C (6) C A B D E (6) B C A E D (5) A C B E D (4) E B D A C (3) D E B A C (3) C B A D E (3) B D E C A (3) B C E D A (3) A E D C B (3) A C E D B (3) A C E B D (3) D E B C A (2) D E A C B (2) C A B E D (2) B E D A C (2) B D C E A (2) B C D E A (2) A D E C B (2) E D B C A (1) E B D C A (1) E A D B C (1) E A B D C (1) D E A B C (1) D B E C A (1) C D B A E (1) C D A B E (1) C B D A E (1) B E C D A (1) B E C A D (1) B C D A E (1) A E D B C (1) A E B C D (1) A D C E B (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 12 -6 -4 B 4 0 20 10 -2 C -12 -20 0 -4 -6 D 6 -10 4 0 -20 E 4 2 6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999099 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 12 -6 -4 B 4 0 20 10 -2 C -12 -20 0 -4 -6 D 6 -10 4 0 -20 E 4 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=27 E=22 C=14 D=9 so D is eliminated. Round 2 votes counts: E=30 B=28 A=28 C=14 so C is eliminated. Round 3 votes counts: A=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:216 E:216 A:199 D:190 C:179 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 12 -6 -4 B 4 0 20 10 -2 C -12 -20 0 -4 -6 D 6 -10 4 0 -20 E 4 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -6 -4 B 4 0 20 10 -2 C -12 -20 0 -4 -6 D 6 -10 4 0 -20 E 4 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -6 -4 B 4 0 20 10 -2 C -12 -20 0 -4 -6 D 6 -10 4 0 -20 E 4 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8609: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) C E D B A (9) C A B E D (8) D E B A C (7) A B E D C (5) A B C E D (5) D E C B A (4) C E B A D (4) C D E B A (4) A B C D E (4) E D B A C (3) D C E B A (3) D B A E C (3) C E A B D (3) C A B D E (3) B A E D C (3) E D C B A (2) E B A D C (2) C E D A B (2) B A D E C (2) A B E C D (2) E B D A C (1) E B A C D (1) D A B E C (1) C E B D A (1) C E A D B (1) C D E A B (1) C D A E B (1) C D A B E (1) B E A D C (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 6 14 2 B 2 0 4 16 2 C -6 -4 0 4 2 D -14 -16 -4 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998718 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 14 2 B 2 0 4 16 2 C -6 -4 0 4 2 D -14 -16 -4 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=28 D=18 E=9 B=7 so B is eliminated. Round 2 votes counts: C=38 A=34 D=18 E=10 so E is eliminated. Round 3 votes counts: C=38 A=38 D=24 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 A:210 E:202 C:198 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 14 2 B 2 0 4 16 2 C -6 -4 0 4 2 D -14 -16 -4 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 14 2 B 2 0 4 16 2 C -6 -4 0 4 2 D -14 -16 -4 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 14 2 B 2 0 4 16 2 C -6 -4 0 4 2 D -14 -16 -4 0 -10 E -2 -2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999812 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8610: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) C B A D E (7) B C A D E (7) A D B C E (6) E C D A B (5) C E D A B (5) B A D C E (5) A D B E C (5) E D A B C (4) D A E C B (4) C E B D A (4) C D A E B (4) C E D B A (3) B A D E C (3) A D E B C (3) E D C A B (2) D A E B C (2) C B E D A (2) C B D A E (2) B A C D E (2) E C B D A (1) E B D A C (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A C B (1) D A C B E (1) C E B A D (1) C D B A E (1) C D A B E (1) C B E A D (1) B C E A D (1) Total count = 100 A B C D E A 0 14 4 -10 6 B -14 0 -14 -20 -12 C -4 14 0 -4 6 D 10 20 4 0 8 E -6 12 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999797 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 -10 6 B -14 0 -14 -20 -12 C -4 14 0 -4 6 D 10 20 4 0 8 E -6 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=29 B=18 A=14 D=8 so D is eliminated. Round 2 votes counts: C=31 E=30 A=21 B=18 so B is eliminated. Round 3 votes counts: C=39 A=31 E=30 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:207 C:206 E:196 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 4 -10 6 B -14 0 -14 -20 -12 C -4 14 0 -4 6 D 10 20 4 0 8 E -6 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -10 6 B -14 0 -14 -20 -12 C -4 14 0 -4 6 D 10 20 4 0 8 E -6 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -10 6 B -14 0 -14 -20 -12 C -4 14 0 -4 6 D 10 20 4 0 8 E -6 12 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8611: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (5) C B D A E (5) C B A E D (5) C B A D E (5) A C D E B (5) C A B D E (4) B C D E A (4) E D A B C (3) E A D C B (3) D E B A C (3) D B E C A (3) D A E C B (3) C A B E D (3) B E C D A (3) B D E C A (3) A E D C B (3) A C B D E (3) E B D C A (2) E B D A C (2) E A D B C (2) D E A B C (2) D C B A E (2) D B C A E (2) C A D B E (2) B E D C A (2) A E C D B (2) A C D B E (2) A C B E D (2) E B A D C (1) E B A C D (1) E A B D C (1) E A B C D (1) D B C E A (1) D A C B E (1) C D A B E (1) B D C E A (1) B C E A D (1) B C D A E (1) A E C B D (1) A D E C B (1) A D C E B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 0 2 12 B 4 0 -12 4 12 C 0 12 0 6 6 D -2 -4 -6 0 10 E -12 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.481704 B: 0.000000 C: 0.518296 D: 0.000000 E: 0.000000 Sum of squares = 0.500669451463 Cumulative probabilities = A: 0.481704 B: 0.481704 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 2 12 B 4 0 -12 4 12 C 0 12 0 6 6 D -2 -4 -6 0 10 E -12 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=22 E=21 D=17 B=15 so B is eliminated. Round 2 votes counts: C=31 E=26 A=22 D=21 so D is eliminated. Round 3 votes counts: E=37 C=37 A=26 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:205 B:204 D:199 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 0 2 12 B 4 0 -12 4 12 C 0 12 0 6 6 D -2 -4 -6 0 10 E -12 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 2 12 B 4 0 -12 4 12 C 0 12 0 6 6 D -2 -4 -6 0 10 E -12 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 2 12 B 4 0 -12 4 12 C 0 12 0 6 6 D -2 -4 -6 0 10 E -12 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8612: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A C E B D (8) E C A D B (6) E D B C A (5) E D B A C (5) B D E A C (5) B D C A E (5) A C B D E (5) E A C D B (4) D B C A E (4) B D A C E (4) E C D A B (3) E B D A C (3) C A E D B (3) C A D B E (3) E D C B A (2) D E B C A (2) D B C E A (2) C D B A E (2) B D A E C (2) A B D C E (2) E B A D C (1) E A D B C (1) E A B D C (1) D C B E A (1) D B E C A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D A B E (1) C B D A E (1) B D C E A (1) B A D C E (1) A E C B D (1) A E B C D (1) A C B E D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 -2 8 B -4 0 -4 4 2 C 6 4 0 2 12 D 2 -4 -2 0 4 E -8 -2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -2 8 B -4 0 -4 4 2 C 6 4 0 2 12 D 2 -4 -2 0 4 E -8 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=21 A=20 B=18 D=10 so D is eliminated. Round 2 votes counts: E=33 B=25 C=22 A=20 so A is eliminated. Round 3 votes counts: C=36 E=35 B=29 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:202 D:200 B:199 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -2 8 B -4 0 -4 4 2 C 6 4 0 2 12 D 2 -4 -2 0 4 E -8 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -2 8 B -4 0 -4 4 2 C 6 4 0 2 12 D 2 -4 -2 0 4 E -8 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -2 8 B -4 0 -4 4 2 C 6 4 0 2 12 D 2 -4 -2 0 4 E -8 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8613: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) D B E A C (7) B A D E C (7) A B C E D (7) D E B C A (6) D E C B A (5) C E A D B (5) B D A E C (5) A C B E D (5) D C E B A (4) C A E B D (4) E D C A B (3) E C D A B (3) C D E A B (3) A B E C D (3) D E B A C (2) D B C E A (2) B D E A C (2) B A D C E (2) B A C D E (2) A C E B D (2) E D B A C (1) E D A C B (1) E C A D B (1) E A B D C (1) D C B E A (1) D C B A E (1) D B E C A (1) C E A B D (1) C D B A E (1) C A E D B (1) B D A C E (1) B A E D C (1) Total count = 100 A B C D E A 0 -2 -2 -16 -16 B 2 0 0 -14 -4 C 2 0 0 -6 2 D 16 14 6 0 4 E 16 4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -16 -16 B 2 0 0 -14 -4 C 2 0 0 -6 2 D 16 14 6 0 4 E 16 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=24 B=20 A=17 E=10 so E is eliminated. Round 2 votes counts: D=34 C=28 B=20 A=18 so A is eliminated. Round 3 votes counts: C=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:207 C:199 B:192 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -16 -16 B 2 0 0 -14 -4 C 2 0 0 -6 2 D 16 14 6 0 4 E 16 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -16 -16 B 2 0 0 -14 -4 C 2 0 0 -6 2 D 16 14 6 0 4 E 16 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -16 -16 B 2 0 0 -14 -4 C 2 0 0 -6 2 D 16 14 6 0 4 E 16 4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999728 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8614: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) D E C A B (7) D E B C A (5) A C D E B (5) E C D A B (4) B A C E D (4) D A B C E (3) C A E D B (3) C A B E D (3) B E D A C (3) A C B E D (3) E D C A B (2) E B D C A (2) E B C D A (2) D E A C B (2) D B E A C (2) B E C D A (2) B E C A D (2) B D E C A (2) A D C E B (2) A C D B E (2) A B D C E (2) E D B C A (1) E C D B A (1) E C B D A (1) E C A B D (1) E B C A D (1) D E C B A (1) D E A B C (1) D C A E B (1) D B A C E (1) C E A D B (1) C E A B D (1) C D A E B (1) C A E B D (1) B E D C A (1) B E A C D (1) B D E A C (1) B D A E C (1) B C E A D (1) B C A E D (1) B A E C D (1) B A C D E (1) A D C B E (1) A C E D B (1) A C E B D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 24 4 -16 4 B -24 0 -14 -16 -18 C -4 14 0 -4 6 D 16 16 4 0 8 E -4 18 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 24 4 -16 4 B -24 0 -14 -16 -18 C -4 14 0 -4 6 D 16 16 4 0 8 E -4 18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=21 A=20 E=15 C=10 so C is eliminated. Round 2 votes counts: D=35 A=27 B=21 E=17 so E is eliminated. Round 3 votes counts: D=43 A=30 B=27 so B is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:208 C:206 E:200 B:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 24 4 -16 4 B -24 0 -14 -16 -18 C -4 14 0 -4 6 D 16 16 4 0 8 E -4 18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 4 -16 4 B -24 0 -14 -16 -18 C -4 14 0 -4 6 D 16 16 4 0 8 E -4 18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 4 -16 4 B -24 0 -14 -16 -18 C -4 14 0 -4 6 D 16 16 4 0 8 E -4 18 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999503 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8615: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) C A E B D (9) E B C A D (7) C A B E D (7) E C B A D (6) D E B A C (5) D B A C E (5) E C A B D (4) D A C B E (4) A C D B E (4) D E B C A (3) D A B C E (3) A C B E D (3) A C B D E (3) E B D C A (2) E B C D A (2) D E C A B (2) D C A E B (2) D B A E C (2) C A E D B (2) B E A C D (2) B D A C E (2) B A C E D (2) E D C B A (1) E B D A C (1) D E C B A (1) D C E A B (1) D A E C B (1) C E A D B (1) C A D E B (1) B E C A D (1) B D E A C (1) A D C B E (1) Total count = 100 A B C D E A 0 -4 -4 6 2 B 4 0 -6 4 -2 C 4 6 0 8 0 D -6 -4 -8 0 0 E -2 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.522227 D: 0.000000 E: 0.477773 Sum of squares = 0.500988060041 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.522227 D: 0.522227 E: 1.000000 A B C D E A 0 -4 -4 6 2 B 4 0 -6 4 -2 C 4 6 0 8 0 D -6 -4 -8 0 0 E -2 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=23 C=20 A=11 B=8 so B is eliminated. Round 2 votes counts: D=41 E=26 C=20 A=13 so A is eliminated. Round 3 votes counts: D=42 C=32 E=26 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:209 A:200 B:200 E:200 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 6 2 B 4 0 -6 4 -2 C 4 6 0 8 0 D -6 -4 -8 0 0 E -2 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 6 2 B 4 0 -6 4 -2 C 4 6 0 8 0 D -6 -4 -8 0 0 E -2 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 6 2 B 4 0 -6 4 -2 C 4 6 0 8 0 D -6 -4 -8 0 0 E -2 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8616: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) E C D A B (8) A B C D E (8) B A D C E (7) E D B C A (6) E D B A C (6) E B D A C (5) E B A D C (5) B A D E C (5) E D C A B (4) C D A B E (4) C A D B E (4) C A B D E (4) A C B D E (4) E B A C D (2) C D A E B (2) A B D C E (2) E C B A D (1) E B D C A (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E A B (1) D C A E B (1) C E A D B (1) C A E D B (1) B E D A C (1) B E A D C (1) B A E D C (1) B A C D E (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 6 4 -6 B 2 0 8 6 -10 C -6 -8 0 -10 -16 D -4 -6 10 0 -12 E 6 10 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 6 4 -6 B 2 0 8 6 -10 C -6 -8 0 -10 -16 D -4 -6 10 0 -12 E 6 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=48 A=17 C=16 B=16 D=3 so D is eliminated. Round 2 votes counts: E=49 C=18 A=17 B=16 so B is eliminated. Round 3 votes counts: E=51 A=31 C=18 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:203 A:201 D:194 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 4 -6 B 2 0 8 6 -10 C -6 -8 0 -10 -16 D -4 -6 10 0 -12 E 6 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 4 -6 B 2 0 8 6 -10 C -6 -8 0 -10 -16 D -4 -6 10 0 -12 E 6 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 4 -6 B 2 0 8 6 -10 C -6 -8 0 -10 -16 D -4 -6 10 0 -12 E 6 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8617: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (12) B A D C E (11) A B D E C (8) A D B E C (7) C E B D A (4) A E C D B (4) A D E C B (4) E C D A B (3) E C A D B (3) D C E B A (3) B D C E A (3) B D A C E (3) B C E D A (3) A E C B D (3) D A E C B (2) C B E D A (2) A E D C B (2) A D E B C (2) A B E D C (2) E D C A B (1) E C D B A (1) E A C D B (1) D E C B A (1) D E B C A (1) D E A C B (1) D B C E A (1) D A B E C (1) C E D A B (1) C D E B A (1) C A B E D (1) B D C A E (1) B D A E C (1) B C E A D (1) B C D E A (1) B C A E D (1) B A D E C (1) B A C E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 10 6 12 B 6 0 0 -4 0 C -10 0 0 -14 2 D -6 4 14 0 8 E -12 0 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999993 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 6 12 B 6 0 0 -4 0 C -10 0 0 -14 2 D -6 4 14 0 8 E -12 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=27 C=21 D=10 E=9 so E is eliminated. Round 2 votes counts: A=34 C=28 B=27 D=11 so D is eliminated. Round 3 votes counts: A=38 C=33 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:211 D:210 B:201 C:189 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 6 12 B 6 0 0 -4 0 C -10 0 0 -14 2 D -6 4 14 0 8 E -12 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 6 12 B 6 0 0 -4 0 C -10 0 0 -14 2 D -6 4 14 0 8 E -12 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 6 12 B 6 0 0 -4 0 C -10 0 0 -14 2 D -6 4 14 0 8 E -12 0 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.375000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.343749999998 Cumulative probabilities = A: 0.250000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8618: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (6) A B E D C (6) E C A B D (5) E A B C D (5) B E C D A (5) D C B E A (4) D B C A E (4) C D E A B (4) C D A E B (4) A D B C E (4) A C D E B (4) E B A C D (3) D C B A E (3) D C A E B (3) B D A E C (3) A E B C D (3) A B D E C (3) E C B D A (2) E B C A D (2) E A C B D (2) D C A B E (2) C E D B A (2) C D E B A (2) B E A C D (2) B D C E A (2) B A E D C (2) A E C D B (2) D B A C E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D B E A (1) B E A D C (1) B A D E C (1) A E D C B (1) A E C B D (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -8 12 -2 B -6 0 -2 14 -10 C 8 2 0 16 -12 D -12 -14 -16 0 -8 E 2 10 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 12 -2 B -6 0 -2 14 -10 C 8 2 0 16 -12 D -12 -14 -16 0 -8 E 2 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 D=17 C=16 B=16 so C is eliminated. Round 2 votes counts: E=30 D=28 A=26 B=16 so B is eliminated. Round 3 votes counts: E=38 D=33 A=29 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:207 A:204 B:198 D:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 12 -2 B -6 0 -2 14 -10 C 8 2 0 16 -12 D -12 -14 -16 0 -8 E 2 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 12 -2 B -6 0 -2 14 -10 C 8 2 0 16 -12 D -12 -14 -16 0 -8 E 2 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 12 -2 B -6 0 -2 14 -10 C 8 2 0 16 -12 D -12 -14 -16 0 -8 E 2 10 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8619: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) B E D A C (6) B C E A D (5) E D A C B (4) E B C A D (4) E A D C B (4) C A D E B (4) B E C A D (4) D A C E B (3) B C D A E (3) B C A D E (3) A E D C B (3) A D E C B (3) A D C E B (3) A C D E B (3) E D B A C (2) E D A B C (2) E C A D B (2) E A C D B (2) D E A C B (2) D B A C E (2) D A E C B (2) D A B C E (2) C E A B D (2) C B A D E (2) C A E D B (2) B C E D A (2) B C A E D (2) E B D A C (1) E B C D A (1) E B A D C (1) E A C B D (1) D C A B E (1) D A C B E (1) D A B E C (1) C D A B E (1) C B E A D (1) C B D A E (1) C A E B D (1) C A B D E (1) B E D C A (1) B D C A E (1) B D A C E (1) A C D B E (1) Total count = 100 A B C D E A 0 14 0 20 6 B -14 0 -12 -12 -4 C 0 12 0 8 8 D -20 12 -8 0 -6 E -6 4 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.378907 B: 0.000000 C: 0.621093 D: 0.000000 E: 0.000000 Sum of squares = 0.529326984594 Cumulative probabilities = A: 0.378907 B: 0.378907 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 20 6 B -14 0 -12 -12 -4 C 0 12 0 8 8 D -20 12 -8 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 C=21 D=14 A=13 so A is eliminated. Round 2 votes counts: B=28 E=27 C=25 D=20 so D is eliminated. Round 3 votes counts: E=34 C=33 B=33 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:220 C:214 E:198 D:189 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 14 0 20 6 B -14 0 -12 -12 -4 C 0 12 0 8 8 D -20 12 -8 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 20 6 B -14 0 -12 -12 -4 C 0 12 0 8 8 D -20 12 -8 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 20 6 B -14 0 -12 -12 -4 C 0 12 0 8 8 D -20 12 -8 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8620: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) C B A E D (6) C A E D B (6) C A D E B (6) B C E A D (6) D E B A C (4) D B E A C (4) B E D A C (4) E A D B C (3) D C A E B (3) D A E C B (3) C B D E A (3) C A B E D (3) B C D E A (3) A E D C B (3) E B D A C (2) E B A D C (2) D A C E B (2) C D A E B (2) C B D A E (2) C B A D E (2) C A E B D (2) A C E D B (2) E D B A C (1) D C E A B (1) D C B E A (1) D B C E A (1) D A E B C (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C B E A D (1) C A D B E (1) B E D C A (1) B E C A D (1) B E A D C (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E D A (1) A E D B C (1) A E C B D (1) Total count = 100 A B C D E A 0 -4 -16 -6 -4 B 4 0 -8 -10 -4 C 16 8 0 4 18 D 6 10 -4 0 4 E 4 4 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -6 -4 B 4 0 -8 -10 -4 C 16 8 0 4 18 D 6 10 -4 0 4 E 4 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=27 B=20 E=8 A=7 so A is eliminated. Round 2 votes counts: C=40 D=27 B=20 E=13 so E is eliminated. Round 3 votes counts: C=41 D=35 B=24 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:208 E:193 B:191 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 -6 -4 B 4 0 -8 -10 -4 C 16 8 0 4 18 D 6 10 -4 0 4 E 4 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -6 -4 B 4 0 -8 -10 -4 C 16 8 0 4 18 D 6 10 -4 0 4 E 4 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -6 -4 B 4 0 -8 -10 -4 C 16 8 0 4 18 D 6 10 -4 0 4 E 4 4 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8621: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) A C E B D (11) C A D E B (9) D E B C A (7) C A E D B (7) B E D A C (7) A C B E D (6) D B E C A (5) D E C A B (4) B A C D E (4) C D A E B (3) E D C A B (2) D C E A B (2) C E D A B (2) C A E B D (2) C A D B E (2) C A B D E (2) B A D C E (2) A C E D B (2) E D B C A (1) E C D A B (1) E B D A C (1) D B E A C (1) C A B E D (1) B D C E A (1) B A C E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 16 -2 2 8 B -16 0 -14 4 -8 C 2 14 0 10 18 D -2 -4 -10 0 10 E -8 8 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 2 8 B -16 0 -14 4 -8 C 2 14 0 10 18 D -2 -4 -10 0 10 E -8 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=27 A=21 D=19 E=5 so E is eliminated. Round 2 votes counts: C=29 B=28 D=22 A=21 so A is eliminated. Round 3 votes counts: C=49 B=29 D=22 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:212 D:197 E:186 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -2 2 8 B -16 0 -14 4 -8 C 2 14 0 10 18 D -2 -4 -10 0 10 E -8 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 2 8 B -16 0 -14 4 -8 C 2 14 0 10 18 D -2 -4 -10 0 10 E -8 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 2 8 B -16 0 -14 4 -8 C 2 14 0 10 18 D -2 -4 -10 0 10 E -8 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8622: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) D B C A E (9) D B C E A (7) E D A B C (6) A E C B D (6) E D B C A (5) E A D C B (5) C B A D E (5) A C B E D (5) E C B A D (4) B C D E A (3) A E C D B (3) E B C D A (2) E A C D B (2) D E B A C (2) D E A B C (2) D A B C E (2) C B D A E (2) B C E D A (2) B C D A E (2) A D C E B (2) A D C B E (2) E C A B D (1) E A B C D (1) D E B C A (1) D A E B C (1) C B E A D (1) C B A E D (1) C A B E D (1) C A B D E (1) B D C E A (1) B C E A D (1) B C A E D (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 0 6 -10 B -2 0 -4 0 -8 C 0 4 0 8 -2 D -6 0 -8 0 -16 E 10 8 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 6 -10 B -2 0 -4 0 -8 C 0 4 0 8 -2 D -6 0 -8 0 -16 E 10 8 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=24 A=20 C=11 B=10 so B is eliminated. Round 2 votes counts: E=35 D=25 C=20 A=20 so C is eliminated. Round 3 votes counts: E=39 D=32 A=29 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:205 A:199 B:193 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 6 -10 B -2 0 -4 0 -8 C 0 4 0 8 -2 D -6 0 -8 0 -16 E 10 8 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 -10 B -2 0 -4 0 -8 C 0 4 0 8 -2 D -6 0 -8 0 -16 E 10 8 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 -10 B -2 0 -4 0 -8 C 0 4 0 8 -2 D -6 0 -8 0 -16 E 10 8 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8623: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (5) C D A B E (5) B D A C E (5) A B E C D (5) E D C B A (4) E B D C A (4) E B A D C (4) D C B A E (4) D B C A E (4) C A D B E (4) D C E B A (3) C D E A B (3) C D A E B (3) B E D A C (3) A C E B D (3) A C D B E (3) A B C E D (3) A B C D E (3) E A C B D (2) D E C B A (2) D C B E A (2) D C A B E (2) C A D E B (2) B E A D C (2) B D E A C (2) B A E D C (2) B A D E C (2) A E C B D (2) A E B C D (2) E D B C A (1) D B E C A (1) D B C E A (1) D B A C E (1) C E A D B (1) C D E B A (1) B D E C A (1) B D A E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 4 -16 16 B 10 0 6 6 16 C -4 -6 0 -14 10 D 16 -6 14 0 14 E -16 -16 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 4 -16 16 B 10 0 6 6 16 C -4 -6 0 -14 10 D 16 -6 14 0 14 E -16 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=20 D=20 C=19 B=18 so B is eliminated. Round 2 votes counts: D=29 A=27 E=25 C=19 so C is eliminated. Round 3 votes counts: D=41 A=33 E=26 so E is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:219 D:219 A:197 C:193 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 4 -16 16 B 10 0 6 6 16 C -4 -6 0 -14 10 D 16 -6 14 0 14 E -16 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 -16 16 B 10 0 6 6 16 C -4 -6 0 -14 10 D 16 -6 14 0 14 E -16 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 -16 16 B 10 0 6 6 16 C -4 -6 0 -14 10 D 16 -6 14 0 14 E -16 -16 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8624: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (16) D A C E B (15) E B C A D (7) A D C E B (7) B E D C A (5) B E C D A (5) D B A E C (4) C A E D B (4) B E D A C (4) B D A E C (4) A C D E B (4) E C B A D (3) C E A D B (3) E C A B D (2) D A C B E (2) D A B C E (2) B D E C A (2) D B E C A (1) D B A C E (1) C E D B A (1) C E D A B (1) C E A B D (1) C A D E B (1) B E A C D (1) B A E D C (1) A D B C E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -4 6 -4 B 10 0 8 4 -2 C 4 -8 0 2 -10 D -6 -4 -2 0 -12 E 4 2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -4 6 -4 B 10 0 8 4 -2 C 4 -8 0 2 -10 D -6 -4 -2 0 -12 E 4 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=25 A=14 E=12 C=11 so C is eliminated. Round 2 votes counts: B=38 D=25 A=19 E=18 so E is eliminated. Round 3 votes counts: B=48 D=27 A=25 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:214 B:210 A:194 C:194 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -4 6 -4 B 10 0 8 4 -2 C 4 -8 0 2 -10 D -6 -4 -2 0 -12 E 4 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 6 -4 B 10 0 8 4 -2 C 4 -8 0 2 -10 D -6 -4 -2 0 -12 E 4 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 6 -4 B 10 0 8 4 -2 C 4 -8 0 2 -10 D -6 -4 -2 0 -12 E 4 2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999673 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8625: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (12) B C A E D (11) D E C A B (10) B A E D C (9) E D A B C (6) C D E A B (6) C D E B A (5) C B D E A (4) C B A D E (4) B A C E D (4) D C E A B (3) E D A C B (2) E A D C B (2) E A D B C (2) C D B E A (2) B E A D C (2) B C D E A (2) B C A D E (2) A B E D C (2) E D B A C (1) C D B A E (1) C B D A E (1) C A B E D (1) B E D A C (1) B D E C A (1) B A E C D (1) A E D C B (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 -6 -14 -22 B 2 0 -8 -8 -4 C 6 8 0 -12 -8 D 14 8 12 0 6 E 22 4 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -14 -22 B 2 0 -8 -8 -4 C 6 8 0 -12 -8 D 14 8 12 0 6 E 22 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=25 C=24 E=13 A=5 so A is eliminated. Round 2 votes counts: B=35 D=25 C=24 E=16 so E is eliminated. Round 3 votes counts: D=40 B=36 C=24 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:214 C:197 B:191 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -14 -22 B 2 0 -8 -8 -4 C 6 8 0 -12 -8 D 14 8 12 0 6 E 22 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -14 -22 B 2 0 -8 -8 -4 C 6 8 0 -12 -8 D 14 8 12 0 6 E 22 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -14 -22 B 2 0 -8 -8 -4 C 6 8 0 -12 -8 D 14 8 12 0 6 E 22 4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8626: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (15) D A B E C (15) C E B A D (14) A B E C D (10) C E D B A (7) D B A E C (4) B A E C D (3) A E C B D (3) A E B C D (3) A D B E C (3) A B E D C (3) D C E A B (2) D A E C B (2) D A C E B (2) C D E B A (2) C B E A D (2) B E C A D (2) B C E A D (2) E C A B D (1) D B C A E (1) D A B C E (1) C E B D A (1) C E A B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 0 -4 2 B 6 0 -4 -8 -6 C 0 4 0 2 0 D 4 8 -2 0 -4 E -2 6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.211901 B: 0.000000 C: 0.788099 D: 0.000000 E: 0.000000 Sum of squares = 0.666001772474 Cumulative probabilities = A: 0.211901 B: 0.211901 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -4 2 B 6 0 -4 -8 -6 C 0 4 0 2 0 D 4 8 -2 0 -4 E -2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555555973 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 C=27 A=23 B=7 E=1 so E is eliminated. Round 2 votes counts: D=42 C=28 A=23 B=7 so B is eliminated. Round 3 votes counts: D=42 C=32 A=26 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:204 C:203 D:203 A:196 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -4 2 B 6 0 -4 -8 -6 C 0 4 0 2 0 D 4 8 -2 0 -4 E -2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555555973 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -4 2 B 6 0 -4 -8 -6 C 0 4 0 2 0 D 4 8 -2 0 -4 E -2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555555973 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -4 2 B 6 0 -4 -8 -6 C 0 4 0 2 0 D 4 8 -2 0 -4 E -2 6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555555973 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8627: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (18) D E A C B (8) E B A C D (7) B C A D E (7) D C A B E (6) E B C A D (4) B E C A D (4) A C B D E (4) E D B C A (3) D B C A E (3) D A C E B (3) D E C A B (2) D C A E B (2) D A C B E (2) C A D B E (2) C A B D E (2) B D E C A (2) B D C A E (2) B C D A E (2) A C B E D (2) E D B A C (1) E B D A C (1) E B A D C (1) E A D C B (1) E A C D B (1) E A C B D (1) D E B C A (1) D C B A E (1) D A E C B (1) C B A D E (1) B E D C A (1) B A C E D (1) A E C D B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 16 10 -18 -12 B -16 0 -20 -16 -14 C -10 20 0 -18 -16 D 18 16 18 0 4 E 12 14 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999937 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 -18 -12 B -16 0 -20 -16 -14 C -10 20 0 -18 -16 D 18 16 18 0 4 E 12 14 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=29 B=19 A=9 C=5 so C is eliminated. Round 2 votes counts: E=38 D=29 B=20 A=13 so A is eliminated. Round 3 votes counts: E=40 D=32 B=28 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:219 A:198 C:188 B:167 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 10 -18 -12 B -16 0 -20 -16 -14 C -10 20 0 -18 -16 D 18 16 18 0 4 E 12 14 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 -18 -12 B -16 0 -20 -16 -14 C -10 20 0 -18 -16 D 18 16 18 0 4 E 12 14 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 -18 -12 B -16 0 -20 -16 -14 C -10 20 0 -18 -16 D 18 16 18 0 4 E 12 14 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8628: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (14) A E C D B (9) E A C B D (8) E A B C D (6) D C B A E (5) C D B A E (5) D B C A E (4) C E A D B (4) C D A E B (4) B E A C D (4) A E D B C (4) E B A C D (3) B C D E A (3) A E D C B (3) E B C A D (2) D C A E B (2) C A E D B (2) B E C D A (2) B E C A D (2) A C D E B (2) D C A B E (1) D A E C B (1) D A C E B (1) D A B E C (1) D A B C E (1) C D B E A (1) C B D E A (1) B E D A C (1) B E A D C (1) B D E A C (1) B C E D A (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -6 2 -8 B 0 0 0 -2 -4 C 6 0 0 18 2 D -2 2 -18 0 -4 E 8 4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.305789 C: 0.694211 D: 0.000000 E: 0.000000 Sum of squares = 0.575436114443 Cumulative probabilities = A: 0.000000 B: 0.305789 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 2 -8 B 0 0 0 -2 -4 C 6 0 0 18 2 D -2 2 -18 0 -4 E 8 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555768949 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=19 A=19 C=17 D=16 so D is eliminated. Round 2 votes counts: B=33 C=25 A=23 E=19 so E is eliminated. Round 3 votes counts: B=38 A=37 C=25 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 E:207 B:197 A:194 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 2 -8 B 0 0 0 -2 -4 C 6 0 0 18 2 D -2 2 -18 0 -4 E 8 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555768949 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 -8 B 0 0 0 -2 -4 C 6 0 0 18 2 D -2 2 -18 0 -4 E 8 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555768949 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 -8 B 0 0 0 -2 -4 C 6 0 0 18 2 D -2 2 -18 0 -4 E 8 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555768949 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8629: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) B E C D A (7) D B A E C (5) D A B E C (5) C E B A D (5) A D C E B (5) D E C A B (4) D A C E B (4) A C E B D (4) E C B D A (3) D E B C A (3) D A E C B (3) C E D A B (3) B D E C A (3) B C E A D (3) A D C B E (3) A C E D B (3) A B C E D (3) E C D B A (2) D E C B A (2) D B E C A (2) C A E D B (2) A C D E B (2) A B D C E (2) E D B C A (1) E C B A D (1) D C E A B (1) D B E A C (1) D A B C E (1) C E A B D (1) B E D C A (1) B D E A C (1) B D A E C (1) B C A E D (1) B A D C E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -8 -6 -6 B 4 0 2 -6 2 C 8 -2 0 0 -8 D 6 6 0 0 0 E 6 -2 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.518297 E: 0.481703 Sum of squares = 0.500669536167 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.518297 E: 1.000000 A B C D E A 0 -4 -8 -6 -6 B 4 0 2 -6 2 C 8 -2 0 0 -8 D 6 6 0 0 0 E 6 -2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=27 A=24 C=11 E=7 so E is eliminated. Round 2 votes counts: D=32 B=27 A=24 C=17 so C is eliminated. Round 3 votes counts: D=37 B=36 A=27 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:206 E:206 B:201 C:199 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 -6 B 4 0 2 -6 2 C 8 -2 0 0 -8 D 6 6 0 0 0 E 6 -2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 -6 B 4 0 2 -6 2 C 8 -2 0 0 -8 D 6 6 0 0 0 E 6 -2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 -6 B 4 0 2 -6 2 C 8 -2 0 0 -8 D 6 6 0 0 0 E 6 -2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8630: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (10) B D E A C (9) B E D A C (8) E D B A C (7) E B C A D (7) D A C B E (7) C A D E B (7) C A D B E (6) E B D C A (5) E C A B D (4) D B E A C (4) C A E D B (4) A C D E B (4) A C D B E (4) D B A C E (3) C A B E D (2) E C B A D (1) E C A D B (1) D A C E B (1) D A B C E (1) C A B D E (1) B E C D A (1) B E C A D (1) B D E C A (1) B D C A E (1) Total count = 100 A B C D E A 0 -16 16 -16 -18 B 16 0 16 2 -2 C -16 -16 0 -14 -18 D 16 -2 14 0 -2 E 18 2 18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999445 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 16 -16 -18 B 16 0 16 2 -2 C -16 -16 0 -14 -18 D 16 -2 14 0 -2 E 18 2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=21 C=20 D=16 A=8 so A is eliminated. Round 2 votes counts: E=35 C=28 B=21 D=16 so D is eliminated. Round 3 votes counts: C=36 E=35 B=29 so B is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:216 D:213 A:183 C:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 16 -16 -18 B 16 0 16 2 -2 C -16 -16 0 -14 -18 D 16 -2 14 0 -2 E 18 2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 16 -16 -18 B 16 0 16 2 -2 C -16 -16 0 -14 -18 D 16 -2 14 0 -2 E 18 2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 16 -16 -18 B 16 0 16 2 -2 C -16 -16 0 -14 -18 D 16 -2 14 0 -2 E 18 2 18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8631: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) D E A B C (8) E A D C B (7) D E B A C (7) E D A C B (6) B C A D E (5) C A B E D (4) B D C E A (4) A E D C B (4) A E C D B (4) D E B C A (3) A C E D B (3) A C E B D (3) E D C B A (2) E C D B A (2) D B E A C (2) C E A D B (2) C A E B D (2) B D E A C (2) B C D E A (2) B C D A E (2) E D B C A (1) E C D A B (1) D B E C A (1) D A E C B (1) C E A B D (1) C B E D A (1) C B E A D (1) C A E D B (1) B D E C A (1) B D A E C (1) B C E D A (1) A D E C B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 2 4 -10 B 0 0 -18 -12 -18 C -2 18 0 -2 -8 D -4 12 2 0 -18 E 10 18 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 2 4 -10 B 0 0 -18 -12 -18 C -2 18 0 -2 -8 D -4 12 2 0 -18 E 10 18 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=22 E=19 B=18 A=17 so A is eliminated. Round 2 votes counts: C=31 E=27 D=23 B=19 so B is eliminated. Round 3 votes counts: C=42 D=31 E=27 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:227 C:203 A:198 D:196 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 4 -10 B 0 0 -18 -12 -18 C -2 18 0 -2 -8 D -4 12 2 0 -18 E 10 18 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 -10 B 0 0 -18 -12 -18 C -2 18 0 -2 -8 D -4 12 2 0 -18 E 10 18 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 -10 B 0 0 -18 -12 -18 C -2 18 0 -2 -8 D -4 12 2 0 -18 E 10 18 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8632: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (15) C A E D B (10) C E A D B (8) B D A E C (7) B C D A E (7) D B E A C (5) E A D C B (4) C B A D E (4) E A C D B (3) D B A E C (3) C E A B D (3) C B A E D (3) C A E B D (3) B D C A E (3) B D A C E (3) B C D E A (3) A E C D B (3) E D B A C (2) B D E C A (2) E D A C B (1) E C D B A (1) D E A B C (1) D A B E C (1) C B D E A (1) B E D A C (1) B D C E A (1) B C A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 -24 0 -14 -2 B 24 0 10 14 20 C 0 -10 0 0 0 D 14 -14 0 0 14 E 2 -20 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 0 -14 -2 B 24 0 10 14 20 C 0 -10 0 0 0 D 14 -14 0 0 14 E 2 -20 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=43 C=32 E=11 D=10 A=4 so A is eliminated. Round 2 votes counts: B=43 C=32 E=15 D=10 so D is eliminated. Round 3 votes counts: B=52 C=32 E=16 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:234 D:207 C:195 E:184 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 0 -14 -2 B 24 0 10 14 20 C 0 -10 0 0 0 D 14 -14 0 0 14 E 2 -20 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 0 -14 -2 B 24 0 10 14 20 C 0 -10 0 0 0 D 14 -14 0 0 14 E 2 -20 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 0 -14 -2 B 24 0 10 14 20 C 0 -10 0 0 0 D 14 -14 0 0 14 E 2 -20 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8633: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (6) A B D E C (6) E D A B C (4) D E A C B (4) C E D B A (4) C E B D A (4) C B E D A (4) B E A D C (4) B A C E D (4) E A D B C (3) D C E A B (3) C D A E B (3) B C E A D (3) B C A E D (3) B A E D C (3) B A E C D (3) A D E B C (3) E D C B A (2) E C D B A (2) E B C D A (2) B E A C D (2) A D C E B (2) A D B C E (2) A B C D E (2) E D B C A (1) E B D A C (1) E B A D C (1) D E C B A (1) D E A B C (1) D C A E B (1) D A E B C (1) D A C E B (1) C D E B A (1) C B E A D (1) C B A E D (1) C B A D E (1) C A D B E (1) B C E D A (1) B A C D E (1) A E B D C (1) A D E C B (1) A D B E C (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 10 6 -10 B 0 0 10 2 -6 C -10 -10 0 -10 -8 D -6 -2 10 0 -10 E 10 6 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 10 6 -10 B 0 0 10 2 -6 C -10 -10 0 -10 -8 D -6 -2 10 0 -10 E 10 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=22 C=20 D=18 E=16 so E is eliminated. Round 2 votes counts: B=28 D=25 A=25 C=22 so C is eliminated. Round 3 votes counts: B=39 D=35 A=26 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:217 A:203 B:203 D:196 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 10 6 -10 B 0 0 10 2 -6 C -10 -10 0 -10 -8 D -6 -2 10 0 -10 E 10 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 6 -10 B 0 0 10 2 -6 C -10 -10 0 -10 -8 D -6 -2 10 0 -10 E 10 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 6 -10 B 0 0 10 2 -6 C -10 -10 0 -10 -8 D -6 -2 10 0 -10 E 10 6 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8634: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) E A B C D (6) C D B E A (6) C D E B A (5) A E B D C (5) E B A C D (4) E A C D B (4) B E C D A (4) A E C D B (4) A D C B E (4) E C D B A (3) E B C D A (3) D C B A E (3) B C D E A (3) A E D C B (3) A E B C D (3) A D E C B (3) A D C E B (3) E C B D A (2) E B C A D (2) D C A E B (2) C E D B A (2) C D E A B (2) B E A C D (2) A B D C E (2) E C A D B (1) D B C E A (1) C E D A B (1) C D A E B (1) C B E D A (1) C B D E A (1) B E D C A (1) B E A D C (1) B D C E A (1) B A E D C (1) B A D C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -4 2 -12 B -6 0 -16 -10 -18 C 4 16 0 20 -8 D -2 10 -20 0 -8 E 12 18 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -4 2 -12 B -6 0 -16 -10 -18 C 4 16 0 20 -8 D -2 10 -20 0 -8 E 12 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=25 C=19 B=14 D=13 so D is eliminated. Round 2 votes counts: C=31 A=29 E=25 B=15 so B is eliminated. Round 3 votes counts: C=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:216 A:196 D:190 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -4 2 -12 B -6 0 -16 -10 -18 C 4 16 0 20 -8 D -2 10 -20 0 -8 E 12 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 2 -12 B -6 0 -16 -10 -18 C 4 16 0 20 -8 D -2 10 -20 0 -8 E 12 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 2 -12 B -6 0 -16 -10 -18 C 4 16 0 20 -8 D -2 10 -20 0 -8 E 12 18 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8635: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) C D A E B (11) B E A D C (7) A C B D E (7) D C E A B (5) B A E D C (5) E D B C A (4) E B D A C (4) D E C B A (4) C A D E B (4) A C D B E (4) D C E B A (3) D C A E B (3) B E D A C (3) A C D E B (3) A B E D C (3) A B C E D (3) C D E B A (2) C A D B E (2) B E C D A (2) B E A C D (2) B A E C D (2) A B E C D (2) D A C E B (1) C D E A B (1) B E D C A (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -6 -10 2 B 0 0 -2 4 -14 C 6 2 0 -10 -2 D 10 -4 10 0 2 E -2 14 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.700000 E: 0.200000 Sum of squares = 0.540000000024 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.800000 E: 1.000000 A B C D E A 0 0 -6 -10 2 B 0 0 -2 4 -14 C 6 2 0 -10 -2 D 10 -4 10 0 2 E -2 14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.700000 E: 0.200000 Sum of squares = 0.539999999975 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 B=22 C=20 E=19 D=16 so D is eliminated. Round 2 votes counts: C=31 A=24 E=23 B=22 so B is eliminated. Round 3 votes counts: E=38 C=31 A=31 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:209 E:206 C:198 B:194 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -6 -10 2 B 0 0 -2 4 -14 C 6 2 0 -10 -2 D 10 -4 10 0 2 E -2 14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.700000 E: 0.200000 Sum of squares = 0.539999999975 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -10 2 B 0 0 -2 4 -14 C 6 2 0 -10 -2 D 10 -4 10 0 2 E -2 14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.700000 E: 0.200000 Sum of squares = 0.539999999975 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -10 2 B 0 0 -2 4 -14 C 6 2 0 -10 -2 D 10 -4 10 0 2 E -2 14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.700000 E: 0.200000 Sum of squares = 0.539999999975 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8636: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) D C E A B (9) E B A D C (8) D C A B E (7) C D B A E (7) E A B D C (6) D C E B A (6) E B A C D (5) A B E C D (4) E A B C D (3) D E B A C (3) D C B E A (3) C D A B E (3) C B A E D (3) C B A D E (3) E D B A C (2) D E C B A (2) D E A B C (2) D C B A E (2) C A B D E (2) A E B C D (2) E D A B C (1) E A D B C (1) D E C A B (1) D E A C B (1) D C A E B (1) C A B E D (1) B C A E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -2 0 -6 B 10 0 -2 -2 -6 C 2 2 0 -10 -2 D 0 2 10 0 4 E 6 6 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.103719 B: 0.000000 C: 0.000000 D: 0.896281 E: 0.000000 Sum of squares = 0.814076976466 Cumulative probabilities = A: 0.103719 B: 0.103719 C: 0.103719 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 0 -6 B 10 0 -2 -2 -6 C 2 2 0 -10 -2 D 0 2 10 0 4 E 6 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222229284 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=26 C=19 B=11 A=7 so A is eliminated. Round 2 votes counts: D=37 E=28 C=19 B=16 so B is eliminated. Round 3 votes counts: E=42 D=37 C=21 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 E:205 B:200 C:196 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -2 0 -6 B 10 0 -2 -2 -6 C 2 2 0 -10 -2 D 0 2 10 0 4 E 6 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222229284 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 0 -6 B 10 0 -2 -2 -6 C 2 2 0 -10 -2 D 0 2 10 0 4 E 6 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222229284 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 0 -6 B 10 0 -2 -2 -6 C 2 2 0 -10 -2 D 0 2 10 0 4 E 6 6 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.833333 E: 0.000000 Sum of squares = 0.722222229284 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8637: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (10) D C A E B (9) C E D B A (8) C D E B A (8) E C B A D (7) E B C A D (7) D A C B E (7) A B D E C (7) D A B C E (6) D C E A B (5) B A E C D (5) A D B E C (5) B A E D C (4) E B A C D (2) C E B D A (2) C E B A D (2) C D E A B (2) A B E D C (2) D A C E B (1) D A B E C (1) Total count = 100 A B C D E A 0 -10 0 2 -6 B 10 0 -2 -4 -6 C 0 2 0 6 0 D -2 4 -6 0 2 E 6 6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.465888 D: 0.000000 E: 0.534112 Sum of squares = 0.502327236586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.465888 D: 0.465888 E: 1.000000 A B C D E A 0 -10 0 2 -6 B 10 0 -2 -4 -6 C 0 2 0 6 0 D -2 4 -6 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=22 B=19 E=16 A=14 so A is eliminated. Round 2 votes counts: D=34 B=28 C=22 E=16 so E is eliminated. Round 3 votes counts: B=37 D=34 C=29 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:205 C:204 B:199 D:199 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 2 -6 B 10 0 -2 -4 -6 C 0 2 0 6 0 D -2 4 -6 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 2 -6 B 10 0 -2 -4 -6 C 0 2 0 6 0 D -2 4 -6 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 2 -6 B 10 0 -2 -4 -6 C 0 2 0 6 0 D -2 4 -6 0 2 E 6 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8638: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) B A E D C (7) D E C A B (6) B A C E D (5) E D A B C (4) C A B D E (4) A B E D C (4) A B C D E (4) E D C B A (3) E B C D A (3) C D A E B (3) C B E D A (3) C B A D E (3) B E C D A (3) B C A E D (3) A C B D E (3) E C D B A (2) D E A C B (2) D C E A B (2) D A E C B (2) C D E A B (2) B E A D C (2) B A E C D (2) A D C E B (2) A C D B E (2) E D C A B (1) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) D E C B A (1) C E D B A (1) C D A B E (1) C B D E A (1) C A D B E (1) B E A C D (1) B A C D E (1) A E D B C (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -10 -6 0 B 8 0 -10 6 2 C 10 10 0 18 2 D 6 -6 -18 0 0 E 0 -2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -6 0 B 8 0 -10 6 2 C 10 10 0 18 2 D 6 -6 -18 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=24 A=18 E=17 D=13 so D is eliminated. Round 2 votes counts: C=30 E=26 B=24 A=20 so A is eliminated. Round 3 votes counts: C=37 B=33 E=30 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 B:203 E:198 D:191 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 -6 0 B 8 0 -10 6 2 C 10 10 0 18 2 D 6 -6 -18 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -6 0 B 8 0 -10 6 2 C 10 10 0 18 2 D 6 -6 -18 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -6 0 B 8 0 -10 6 2 C 10 10 0 18 2 D 6 -6 -18 0 0 E 0 -2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999986846 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8639: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (12) B C D A E (12) B D C E A (9) A B C D E (9) A E C D B (7) A E B C D (7) E D C B A (6) D B C E A (6) D C B E A (5) E A B D C (4) E B D C A (3) B C A D E (3) E D C A B (2) E D A C B (2) D C E B A (2) B A C D E (2) E B D A C (1) D E C B A (1) C D B A E (1) C B D A E (1) B E D C A (1) B D E C A (1) A E C B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 -6 -10 B 8 0 18 12 4 C 6 -18 0 -10 4 D 6 -12 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -6 -10 B 8 0 18 12 4 C 6 -18 0 -10 4 D 6 -12 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=28 A=26 D=14 C=2 so C is eliminated. Round 2 votes counts: E=30 B=29 A=26 D=15 so D is eliminated. Round 3 votes counts: B=41 E=33 A=26 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:221 D:205 E:198 C:191 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -6 -10 B 8 0 18 12 4 C 6 -18 0 -10 4 D 6 -12 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -6 -10 B 8 0 18 12 4 C 6 -18 0 -10 4 D 6 -12 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -6 -10 B 8 0 18 12 4 C 6 -18 0 -10 4 D 6 -12 10 0 6 E 10 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999267 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8640: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (15) D B E A C (10) C E A B D (8) C A E B D (8) D C B E A (7) B E A C D (7) C B E A D (6) D C E A B (5) D C A E B (5) A E C B D (5) B C E A D (4) D C B A E (2) D B C E A (2) D A E C B (2) B E A D C (2) B D A E C (2) B A E C D (2) A E B C D (2) E A B C D (1) D B C A E (1) C B E D A (1) B D E A C (1) B D C E A (1) B C D E A (1) Total count = 100 A B C D E A 0 -28 -2 -10 -12 B 28 0 2 2 28 C 2 -2 0 -10 2 D 10 -2 10 0 8 E 12 -28 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997437 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -2 -10 -12 B 28 0 2 2 28 C 2 -2 0 -10 2 D 10 -2 10 0 8 E 12 -28 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999575 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=49 C=23 B=20 A=7 E=1 so E is eliminated. Round 2 votes counts: D=49 C=23 B=20 A=8 so A is eliminated. Round 3 votes counts: D=49 C=28 B=23 so B is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:230 D:213 C:196 E:187 A:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -2 -10 -12 B 28 0 2 2 28 C 2 -2 0 -10 2 D 10 -2 10 0 8 E 12 -28 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999575 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -2 -10 -12 B 28 0 2 2 28 C 2 -2 0 -10 2 D 10 -2 10 0 8 E 12 -28 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999575 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -2 -10 -12 B 28 0 2 2 28 C 2 -2 0 -10 2 D 10 -2 10 0 8 E 12 -28 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999575 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8641: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (5) E D C B A (4) E D A C B (4) D B E A C (4) C B A D E (4) B A C D E (4) A C B E D (4) A B E D C (4) A B D E C (4) E D C A B (3) E C D A B (3) C E D A B (3) C E A D B (3) B D A C E (3) A B C E D (3) E C D B A (2) E C A D B (2) D E C B A (2) D E B A C (2) D C B E A (2) C E A B D (2) C D E B A (2) C D B E A (2) C A E B D (2) B D A E C (2) B A D C E (2) A E D B C (2) A E B D C (2) E D B A C (1) E D A B C (1) E A D C B (1) E A D B C (1) D E B C A (1) D C E B A (1) D B E C A (1) D B A E C (1) C B E D A (1) C B D E A (1) C B D A E (1) B D C A E (1) B C D E A (1) B C D A E (1) B A D E C (1) A D E B C (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 0 -10 -10 B 4 0 -10 -8 -2 C 0 10 0 -4 2 D 10 8 4 0 -8 E 10 2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.714286 E: 1.000000 A B C D E A 0 -4 0 -10 -10 B 4 0 -10 -8 -2 C 0 10 0 -4 2 D 10 8 4 0 -8 E 10 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 E=22 B=15 D=14 so D is eliminated. Round 2 votes counts: C=29 E=27 A=23 B=21 so B is eliminated. Round 3 votes counts: A=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:209 D:207 C:204 B:192 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -10 -10 B 4 0 -10 -8 -2 C 0 10 0 -4 2 D 10 8 4 0 -8 E 10 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -10 -10 B 4 0 -10 -8 -2 C 0 10 0 -4 2 D 10 8 4 0 -8 E 10 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -10 -10 B 4 0 -10 -8 -2 C 0 10 0 -4 2 D 10 8 4 0 -8 E 10 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.142857 E: 0.285714 Sum of squares = 0.428571428534 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.714286 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8642: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) A B C E D (8) C A E B D (7) B D A E C (7) E C D A B (5) D E B C A (5) D B E A C (5) B A D C E (5) B A C D E (5) D E B A C (4) C E D A B (4) C E A D B (4) E D C B A (3) D E C B A (3) C A B E D (3) B A D E C (3) A C E B D (3) E D C A B (2) D B A E C (2) A B E C D (2) E C A D B (1) D C E B A (1) D B E C A (1) C E A B D (1) C D E A B (1) C B D A E (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E A C (1) B D A C E (1) B A E D C (1) Total count = 100 A B C D E A 0 2 12 6 16 B -2 0 2 16 10 C -12 -2 0 10 8 D -6 -16 -10 0 -8 E -16 -10 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 6 16 B -2 0 2 16 10 C -12 -2 0 10 8 D -6 -16 -10 0 -8 E -16 -10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998659 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 D=21 A=21 E=11 so E is eliminated. Round 2 votes counts: C=29 D=26 B=24 A=21 so A is eliminated. Round 3 votes counts: C=40 B=34 D=26 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:218 B:213 C:202 E:187 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 12 6 16 B -2 0 2 16 10 C -12 -2 0 10 8 D -6 -16 -10 0 -8 E -16 -10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998659 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 6 16 B -2 0 2 16 10 C -12 -2 0 10 8 D -6 -16 -10 0 -8 E -16 -10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998659 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 6 16 B -2 0 2 16 10 C -12 -2 0 10 8 D -6 -16 -10 0 -8 E -16 -10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998659 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8643: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) D B C A E (7) B D A E C (7) B A E D C (7) D C A B E (6) B E A C D (6) E C A B D (5) D C A E B (5) D C B A E (4) C D A E B (4) B A E C D (4) A E C B D (4) E A C B D (3) C D E A B (3) D C B E A (2) D B C E A (2) D B A C E (2) D A B C E (2) C E A B D (2) C A E D B (2) C A D E B (2) B D E A C (2) E B C A D (1) E B A C D (1) D C E B A (1) D B E C A (1) C E D A B (1) C D E B A (1) B E A D C (1) B D E C A (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -18 -2 16 B 0 0 -8 -8 10 C 18 8 0 -2 10 D 2 8 2 0 4 E -16 -10 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 -2 16 B 0 0 -8 -8 10 C 18 8 0 -2 10 D 2 8 2 0 4 E -16 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 C=24 E=10 A=6 so A is eliminated. Round 2 votes counts: D=32 B=29 C=24 E=15 so E is eliminated. Round 3 votes counts: C=36 D=32 B=32 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:208 A:198 B:197 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -18 -2 16 B 0 0 -8 -8 10 C 18 8 0 -2 10 D 2 8 2 0 4 E -16 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 -2 16 B 0 0 -8 -8 10 C 18 8 0 -2 10 D 2 8 2 0 4 E -16 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 -2 16 B 0 0 -8 -8 10 C 18 8 0 -2 10 D 2 8 2 0 4 E -16 -10 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8644: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (14) C A E D B (14) B D E A C (11) C D A E B (7) D C B A E (6) D C A B E (6) D B E A C (5) C A D E B (5) E A B C D (4) B E A D C (4) D B C A E (3) A E C D B (3) E B A D C (2) D B E C A (2) C A E B D (2) B E D A C (2) B D E C A (2) B D C E A (2) A E C B D (2) E A D B C (1) C B A D E (1) C A D B E (1) B C E A D (1) Total count = 100 A B C D E A 0 18 -4 8 0 B -18 0 -22 -6 -8 C 4 22 0 8 -4 D -8 6 -8 0 2 E 0 8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.398900 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.601100 Sum of squares = 0.520442592406 Cumulative probabilities = A: 0.398900 B: 0.398900 C: 0.398900 D: 0.398900 E: 1.000000 A B C D E A 0 18 -4 8 0 B -18 0 -22 -6 -8 C 4 22 0 8 -4 D -8 6 -8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000123909 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=22 B=22 E=21 A=5 so A is eliminated. Round 2 votes counts: C=30 E=26 D=22 B=22 so D is eliminated. Round 3 votes counts: C=42 B=32 E=26 so E is eliminated. Round 4 votes counts: C=61 B=39 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:211 E:205 D:196 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -4 8 0 B -18 0 -22 -6 -8 C 4 22 0 8 -4 D -8 6 -8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000123909 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 8 0 B -18 0 -22 -6 -8 C 4 22 0 8 -4 D -8 6 -8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000123909 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 8 0 B -18 0 -22 -6 -8 C 4 22 0 8 -4 D -8 6 -8 0 2 E 0 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499751 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500249 Sum of squares = 0.500000123909 Cumulative probabilities = A: 0.499751 B: 0.499751 C: 0.499751 D: 0.499751 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8645: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (12) D A C B E (8) E C D B A (7) E C B D A (6) A D B C E (6) A B D E C (6) A B D C E (6) E B C A D (5) D C A E B (5) C E D B A (5) B E A C D (4) A D C B E (4) D C E A B (3) C E D A B (3) C D E A B (3) E B A C D (2) D C A B E (2) B E A D C (2) B A D E C (2) E D C B A (1) E C B A D (1) E B C D A (1) D A B E C (1) D A B C E (1) C D E B A (1) C D A B E (1) C A D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 10 4 12 B -2 0 -2 -4 14 C -10 2 0 -18 0 D -4 4 18 0 0 E -12 -14 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999129 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 12 B -2 0 -2 -4 14 C -10 2 0 -18 0 D -4 4 18 0 0 E -12 -14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999644 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 A=23 D=20 B=20 C=14 so C is eliminated. Round 2 votes counts: E=31 D=25 A=24 B=20 so B is eliminated. Round 3 votes counts: A=38 E=37 D=25 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:209 B:203 C:187 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 12 B -2 0 -2 -4 14 C -10 2 0 -18 0 D -4 4 18 0 0 E -12 -14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999644 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 12 B -2 0 -2 -4 14 C -10 2 0 -18 0 D -4 4 18 0 0 E -12 -14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999644 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 12 B -2 0 -2 -4 14 C -10 2 0 -18 0 D -4 4 18 0 0 E -12 -14 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999644 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8646: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) E D A C B (8) E C A D B (7) E D A B C (6) B C D A E (6) C A B D E (5) E D B A C (4) C A D B E (4) B C A D E (4) E A C D B (3) D B A E C (3) C B A E D (3) C A E D B (3) B D A E C (3) E C B A D (2) E B D C A (2) E B C A D (2) E A D C B (2) D E A B C (2) D B A C E (2) D A E C B (2) B D C A E (2) A D C B E (2) E C A B D (1) E B D A C (1) E B C D A (1) D A B E C (1) C E A B D (1) C B E A D (1) C A E B D (1) B E D C A (1) B D E A C (1) B D A C E (1) B A D C E (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -12 8 10 B 0 0 -14 -2 0 C 12 14 0 10 -4 D -8 2 -10 0 0 E -10 0 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384615 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.538462 D: 0.538462 E: 1.000000 A B C D E A 0 0 -12 8 10 B 0 0 -14 -2 0 C 12 14 0 10 -4 D -8 2 -10 0 0 E -10 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384603 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.538462 D: 0.538462 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=28 B=19 D=10 A=4 so A is eliminated. Round 2 votes counts: E=39 C=29 B=19 D=13 so D is eliminated. Round 3 votes counts: E=43 C=32 B=25 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:216 A:203 E:197 B:192 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -12 8 10 B 0 0 -14 -2 0 C 12 14 0 10 -4 D -8 2 -10 0 0 E -10 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384603 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.538462 D: 0.538462 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 8 10 B 0 0 -14 -2 0 C 12 14 0 10 -4 D -8 2 -10 0 0 E -10 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384603 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.538462 D: 0.538462 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 8 10 B 0 0 -14 -2 0 C 12 14 0 10 -4 D -8 2 -10 0 0 E -10 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.384615 D: 0.000000 E: 0.461538 Sum of squares = 0.384615384603 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.538462 D: 0.538462 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8647: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) D A B E C (10) C E B A D (8) A D C B E (8) E C B A D (4) E B C A D (4) D B E A C (4) D A C B E (4) C B E A D (4) B E D C A (4) B E C D A (4) A C D E B (4) D A E B C (3) D A C E B (3) D A B C E (3) A D C E B (3) D B A E C (2) C A E B D (2) C A B E D (2) B E C A D (2) E D C B A (1) E C D A B (1) E B D C A (1) D E B C A (1) D E A B C (1) C E A B D (1) C A E D B (1) B D E A C (1) B C E A D (1) B C A E D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -4 B 4 0 4 -2 2 C 4 -4 0 2 -6 D 6 2 -2 0 -4 E 4 -2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000026 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 -4 -6 -4 B 4 0 4 -2 2 C 4 -4 0 2 -6 D 6 2 -2 0 -4 E 4 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=21 C=18 A=17 B=13 so B is eliminated. Round 2 votes counts: D=32 E=31 C=20 A=17 so A is eliminated. Round 3 votes counts: D=43 E=31 C=26 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:206 B:204 D:201 C:198 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -4 B 4 0 4 -2 2 C 4 -4 0 2 -6 D 6 2 -2 0 -4 E 4 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -4 B 4 0 4 -2 2 C 4 -4 0 2 -6 D 6 2 -2 0 -4 E 4 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -4 B 4 0 4 -2 2 C 4 -4 0 2 -6 D 6 2 -2 0 -4 E 4 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000004 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8648: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) B C E A D (7) C A B E D (6) B E C D A (6) E B D C A (5) C B A E D (5) A D E C B (5) A C D E B (5) E B C D A (4) D E A B C (4) D A C B E (4) C B E A D (4) A C D B E (4) E B C A D (3) D E B A C (3) D A E C B (3) D A C E B (3) C B A D E (3) C A B D E (3) C B D A E (2) C A D B E (2) E D B A C (1) E D A B C (1) E A C B D (1) D A E B C (1) D A B E C (1) B E D C A (1) B E C A D (1) B C E D A (1) B C D E A (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -8 18 14 B -6 0 -20 8 2 C 8 20 0 20 20 D -18 -8 -20 0 4 E -14 -2 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 18 14 B -6 0 -20 8 2 C 8 20 0 20 20 D -18 -8 -20 0 4 E -14 -2 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=24 D=19 B=17 E=15 so E is eliminated. Round 2 votes counts: B=29 C=25 A=25 D=21 so D is eliminated. Round 3 votes counts: A=42 B=33 C=25 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:234 A:215 B:192 E:180 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 18 14 B -6 0 -20 8 2 C 8 20 0 20 20 D -18 -8 -20 0 4 E -14 -2 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 18 14 B -6 0 -20 8 2 C 8 20 0 20 20 D -18 -8 -20 0 4 E -14 -2 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 18 14 B -6 0 -20 8 2 C 8 20 0 20 20 D -18 -8 -20 0 4 E -14 -2 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8649: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) C A D B E (8) C D A E B (7) B E D A C (7) E D B C A (6) E C D B A (6) E D B A C (4) C A D E B (4) A B C E D (4) A B C D E (4) D E B C A (3) C A E B D (3) A C B D E (3) A B E D C (3) D B E A C (2) C E D B A (2) B D E A C (2) B A E D C (2) B A D E C (2) A B D E C (2) E B A D C (1) D E C B A (1) D E B A C (1) D C E B A (1) D B A E C (1) D B A C E (1) C E B A D (1) C E A B D (1) C A B E D (1) C A B D E (1) B E A D C (1) A D B C E (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 10 -10 0 B 10 0 20 2 -4 C -10 -20 0 -6 -12 D 10 -2 6 0 -8 E 0 4 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.170955 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.829045 Sum of squares = 0.716540945169 Cumulative probabilities = A: 0.170955 B: 0.170955 C: 0.170955 D: 0.170955 E: 1.000000 A B C D E A 0 -10 10 -10 0 B 10 0 20 2 -4 C -10 -20 0 -6 -12 D 10 -2 6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836789549 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 A=20 B=14 D=10 so D is eliminated. Round 2 votes counts: E=33 C=29 A=20 B=18 so B is eliminated. Round 3 votes counts: E=45 C=29 A=26 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:214 E:212 D:203 A:195 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 10 -10 0 B 10 0 20 2 -4 C -10 -20 0 -6 -12 D 10 -2 6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836789549 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -10 0 B 10 0 20 2 -4 C -10 -20 0 -6 -12 D 10 -2 6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836789549 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -10 0 B 10 0 20 2 -4 C -10 -20 0 -6 -12 D 10 -2 6 0 -8 E 0 4 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.714286 Sum of squares = 0.591836789549 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8650: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) E D A C B (6) C B A D E (6) C A B D E (5) B E C A D (5) B C A E D (5) E D A B C (4) E B D C A (4) D A C E B (4) B C E A D (4) A C D B E (4) C A E B D (3) C A B E D (3) B D C E A (3) B C A D E (3) A C D E B (3) E B C D A (2) E B C A D (2) D E A C B (2) D E A B C (2) D A E C B (2) D A C B E (2) A D E C B (2) A C E D B (2) E B D A C (1) E A C B D (1) D B E A C (1) D B A E C (1) D B A C E (1) C E A B D (1) C B A E D (1) C A D B E (1) B E C D A (1) B D E C A (1) B D C A E (1) B C D A E (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 -4 4 2 B 4 0 2 6 -2 C 4 -2 0 6 8 D -4 -6 -6 0 -8 E -2 2 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.000000 E: 0.166667 Sum of squares = 0.50000000005 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 -4 -4 4 2 B 4 0 2 6 -2 C 4 -2 0 6 8 D -4 -6 -6 0 -8 E -2 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.000000 E: 0.166667 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 C=20 D=15 A=12 so A is eliminated. Round 2 votes counts: E=29 C=29 B=24 D=18 so D is eliminated. Round 3 votes counts: E=37 C=36 B=27 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:205 E:200 A:199 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 4 2 B 4 0 2 6 -2 C 4 -2 0 6 8 D -4 -6 -6 0 -8 E -2 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.000000 E: 0.166667 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 4 2 B 4 0 2 6 -2 C 4 -2 0 6 8 D -4 -6 -6 0 -8 E -2 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.000000 E: 0.166667 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 4 2 B 4 0 2 6 -2 C 4 -2 0 6 8 D -4 -6 -6 0 -8 E -2 2 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.166667 D: 0.000000 E: 0.166667 Sum of squares = 0.50000000003 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8651: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E B C D A (7) B E C A D (6) B C A D E (6) D A C E B (5) C D A B E (5) B C E D A (5) B C E A D (5) E D A B C (4) E B A D C (4) C B E D A (4) E A D B C (3) A D C B E (3) E D A C B (2) E A B D C (2) D A E C B (2) D A C B E (2) C D B E A (2) C B A D E (2) B E A C D (2) A D E C B (2) A D E B C (2) A D C E B (2) A C D B E (2) E B D C A (1) E B C A D (1) D E C A B (1) C D E A B (1) C D A E B (1) C B D E A (1) C A D B E (1) B E C D A (1) B C A E D (1) B A E C D (1) B A C D E (1) A E D B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 -18 -4 -4 B 16 0 8 18 18 C 18 -8 0 28 16 D 4 -18 -28 0 0 E 4 -18 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -18 -4 -4 B 16 0 8 18 18 C 18 -8 0 28 16 D 4 -18 -28 0 0 E 4 -18 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=25 E=24 A=13 D=10 so D is eliminated. Round 2 votes counts: B=28 E=25 C=25 A=22 so A is eliminated. Round 3 votes counts: C=39 E=32 B=29 so B is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:230 C:227 E:185 A:179 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -18 -4 -4 B 16 0 8 18 18 C 18 -8 0 28 16 D 4 -18 -28 0 0 E 4 -18 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -18 -4 -4 B 16 0 8 18 18 C 18 -8 0 28 16 D 4 -18 -28 0 0 E 4 -18 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -18 -4 -4 B 16 0 8 18 18 C 18 -8 0 28 16 D 4 -18 -28 0 0 E 4 -18 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8652: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (8) E D B C A (7) D C A B E (7) E B D C A (6) D C B E A (6) A C B D E (5) A B E C D (5) E D B A C (4) E B D A C (4) E B A C D (3) D E B C A (3) D C A E B (3) B C E A D (3) A E B C D (3) A C B E D (3) A B C E D (3) E A B C D (2) D E C B A (2) B A C E D (2) E B C D A (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E B A C (1) D E A C B (1) D A E C B (1) C D A B E (1) C B D E A (1) C B D A E (1) C B A E D (1) C B A D E (1) B E C A D (1) B E A C D (1) B D C E A (1) B C E D A (1) B C D E A (1) B C A E D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 2 -4 -6 B 8 0 14 6 6 C -2 -14 0 -2 0 D 4 -6 2 0 -10 E 6 -6 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -4 -6 B 8 0 14 6 6 C -2 -14 0 -2 0 D 4 -6 2 0 -10 E 6 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=29 D=24 B=11 C=5 so C is eliminated. Round 2 votes counts: E=31 A=29 D=25 B=15 so B is eliminated. Round 3 votes counts: E=37 A=34 D=29 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:205 D:195 A:192 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -4 -6 B 8 0 14 6 6 C -2 -14 0 -2 0 D 4 -6 2 0 -10 E 6 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -4 -6 B 8 0 14 6 6 C -2 -14 0 -2 0 D 4 -6 2 0 -10 E 6 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -4 -6 B 8 0 14 6 6 C -2 -14 0 -2 0 D 4 -6 2 0 -10 E 6 -6 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8653: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) E B C D A (10) D C E B A (10) D C A E B (8) B E C A D (6) A B E C D (6) E B A D C (4) D E B C A (4) C D E B A (4) A D C B E (4) A C B E D (4) D A C E B (3) A D B E C (3) E B A C D (2) D E B A C (2) D C A B E (2) D A E B C (2) C E B D A (2) A D E B C (2) A B E D C (2) E C B D A (1) E B D C A (1) E B C A D (1) D E C B A (1) C D A B E (1) C B E A D (1) A E B D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -22 -4 -2 -22 B 22 0 18 8 -16 C 4 -18 0 0 -20 D 2 -8 0 0 -6 E 22 16 20 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -22 -4 -2 -22 B 22 0 18 8 -16 C 4 -18 0 0 -20 D 2 -8 0 0 -6 E 22 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=23 E=19 B=18 C=8 so C is eliminated. Round 2 votes counts: D=37 A=23 E=21 B=19 so B is eliminated. Round 3 votes counts: E=40 D=37 A=23 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:232 B:216 D:194 C:183 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -22 -4 -2 -22 B 22 0 18 8 -16 C 4 -18 0 0 -20 D 2 -8 0 0 -6 E 22 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -4 -2 -22 B 22 0 18 8 -16 C 4 -18 0 0 -20 D 2 -8 0 0 -6 E 22 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -4 -2 -22 B 22 0 18 8 -16 C 4 -18 0 0 -20 D 2 -8 0 0 -6 E 22 16 20 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8654: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) D C A B E (7) E B A C D (5) D A C E B (5) B E D C A (5) B E C D A (5) E A B C D (4) B E C A D (4) A E D B C (4) E B C A D (3) E A B D C (3) D C B A E (3) C D A B E (3) B E D A C (3) B D E C A (3) E B A D C (2) D A C B E (2) C D B E A (2) C D B A E (2) C D A E B (2) C B D E A (2) C A E B D (2) B D C E A (2) B C D E A (2) A D E C B (2) A D C E B (2) D C B E A (1) D B C E A (1) D A E C B (1) D A B C E (1) C B E A D (1) C B D A E (1) C A D E B (1) C A D B E (1) B E A D C (1) A E C D B (1) A E B D C (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -6 -6 2 B -4 0 0 -2 4 C 6 0 0 0 4 D 6 2 0 0 10 E -2 -4 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.590146 D: 0.409854 E: 0.000000 Sum of squares = 0.516252487713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.590146 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -6 2 B -4 0 0 -2 4 C 6 0 0 0 4 D 6 2 0 0 10 E -2 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=21 A=20 E=17 C=17 so E is eliminated. Round 2 votes counts: B=35 A=27 D=21 C=17 so C is eliminated. Round 3 votes counts: B=39 A=31 D=30 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:209 C:205 B:199 A:197 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -6 2 B -4 0 0 -2 4 C 6 0 0 0 4 D 6 2 0 0 10 E -2 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -6 2 B -4 0 0 -2 4 C 6 0 0 0 4 D 6 2 0 0 10 E -2 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -6 2 B -4 0 0 -2 4 C 6 0 0 0 4 D 6 2 0 0 10 E -2 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8655: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (8) C D E B A (5) C D B E A (4) C B D A E (4) C A B D E (4) B A E D C (4) B A C D E (4) E D C A B (3) E D A C B (3) E A D B C (3) D E C B A (3) D C E B A (3) C D E A B (3) C B A D E (3) B D E A C (3) B D C E A (3) A E C D B (3) A E B D C (3) E C D A B (2) E A D C B (2) D B C E A (2) B D A C E (2) A E C B D (2) A E B C D (2) A C B E D (2) A C B D E (2) A B C D E (2) E D B A C (1) E D A B C (1) E B D A C (1) E A C D B (1) D E B C A (1) D B E A C (1) C E D B A (1) C E D A B (1) C E A D B (1) C D B A E (1) C A D B E (1) C A B E D (1) B D A E C (1) B A D C E (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 8 2 4 B -4 0 -12 8 8 C -8 12 0 2 4 D -2 -8 -2 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999825 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 2 4 B -4 0 -12 8 8 C -8 12 0 2 4 D -2 -8 -2 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=26 B=18 E=17 D=10 so D is eliminated. Round 2 votes counts: C=32 A=26 E=21 B=21 so E is eliminated. Round 3 votes counts: C=40 A=36 B=24 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:205 B:200 D:197 E:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 2 4 B -4 0 -12 8 8 C -8 12 0 2 4 D -2 -8 -2 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 2 4 B -4 0 -12 8 8 C -8 12 0 2 4 D -2 -8 -2 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 2 4 B -4 0 -12 8 8 C -8 12 0 2 4 D -2 -8 -2 0 6 E -4 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8656: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (9) E C B A D (8) D B C E A (8) A E C B D (8) A D E B C (8) C B E D A (7) D B C A E (6) A D B C E (6) A D E C B (5) E C B D A (4) E C A B D (4) B C E D A (4) C E B A D (3) B C D E A (3) A E C D B (3) D B A C E (2) D A B E C (2) B D C E A (2) A E D C B (2) E D C B A (1) E A D C B (1) E A C B D (1) D B E C A (1) B C D A E (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -4 0 6 B 0 0 6 -10 4 C 4 -6 0 -8 2 D 0 10 8 0 8 E -6 -4 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.339212 B: 0.000000 C: 0.000000 D: 0.660788 E: 0.000000 Sum of squares = 0.551705289816 Cumulative probabilities = A: 0.339212 B: 0.339212 C: 0.339212 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 0 6 B 0 0 6 -10 4 C 4 -6 0 -8 2 D 0 10 8 0 8 E -6 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=28 E=19 C=10 B=10 so C is eliminated. Round 2 votes counts: A=33 D=28 E=22 B=17 so B is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:213 A:201 B:200 C:196 E:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -4 0 6 B 0 0 6 -10 4 C 4 -6 0 -8 2 D 0 10 8 0 8 E -6 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 0 6 B 0 0 6 -10 4 C 4 -6 0 -8 2 D 0 10 8 0 8 E -6 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 0 6 B 0 0 6 -10 4 C 4 -6 0 -8 2 D 0 10 8 0 8 E -6 -4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8657: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) C D E A B (7) B E A C D (7) C D A E B (6) C B E D A (6) D C E A B (5) E C B D A (4) C D E B A (4) C D A B E (4) A B E D C (4) E B C D A (3) D A C E B (3) C E B D A (3) A D B E C (3) D A E C B (2) D A C B E (2) C E D B A (2) B A E D C (2) B A C E D (2) A E B D C (2) A D C B E (2) A B D E C (2) E D C B A (1) E D A C B (1) E C D B A (1) E B A D C (1) E A B D C (1) D E C A B (1) C D B E A (1) C B D E A (1) C B A D E (1) B E C D A (1) B E A D C (1) B C E A D (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E D B C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -22 -30 -4 B -8 0 -28 -6 -10 C 22 28 0 18 22 D 30 6 -18 0 8 E 4 10 -22 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -22 -30 -4 B -8 0 -28 -6 -10 C 22 28 0 18 22 D 30 6 -18 0 8 E 4 10 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=20 B=18 A=15 E=12 so E is eliminated. Round 2 votes counts: C=40 D=22 B=22 A=16 so A is eliminated. Round 3 votes counts: C=40 B=32 D=28 so D is eliminated. Round 4 votes counts: C=64 B=36 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:245 D:213 E:192 A:176 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -22 -30 -4 B -8 0 -28 -6 -10 C 22 28 0 18 22 D 30 6 -18 0 8 E 4 10 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -22 -30 -4 B -8 0 -28 -6 -10 C 22 28 0 18 22 D 30 6 -18 0 8 E 4 10 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -22 -30 -4 B -8 0 -28 -6 -10 C 22 28 0 18 22 D 30 6 -18 0 8 E 4 10 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8658: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) B E C A D (7) A E D B C (7) E A B D C (6) D C A E B (5) C B E D A (5) B E A C D (5) E B A D C (4) D A E C B (4) D A C E B (4) C D B A E (3) C B D E A (3) B C E D A (3) A D E C B (3) A D E B C (3) E A D B C (2) C D B E A (2) C D A E B (2) C B D A E (2) B C E A D (2) A E B D C (2) A C D B E (2) E D B C A (1) E D A B C (1) E B D A C (1) E B C D A (1) D E C A B (1) D E A C B (1) D E A B C (1) C A D B E (1) C A B D E (1) B E C D A (1) B E A D C (1) B A E D C (1) B A E C D (1) B A C E D (1) A D C E B (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 12 6 4 4 B -12 0 2 -4 0 C -6 -2 0 0 -8 D -4 4 0 0 -4 E -4 0 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 4 4 B -12 0 2 -4 0 C -6 -2 0 0 -8 D -4 4 0 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=22 A=20 E=16 D=16 so E is eliminated. Round 2 votes counts: B=28 A=28 C=26 D=18 so D is eliminated. Round 3 votes counts: A=39 C=32 B=29 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 E:204 D:198 B:193 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 4 4 B -12 0 2 -4 0 C -6 -2 0 0 -8 D -4 4 0 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 4 4 B -12 0 2 -4 0 C -6 -2 0 0 -8 D -4 4 0 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 4 4 B -12 0 2 -4 0 C -6 -2 0 0 -8 D -4 4 0 0 -4 E -4 0 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8659: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (14) C A D B E (11) A B E C D (10) D E B C A (7) E B A D C (5) C D A B E (5) C D E B A (4) C D A E B (4) A C B E D (4) D E C B A (3) D C A E B (3) B E A C D (3) A C D B E (3) A B C E D (3) E D B A C (2) E B D A C (2) E B C D A (2) D E B A C (2) D E A B C (2) C A B E D (2) E D B C A (1) E C B D A (1) E B A C D (1) D C E A B (1) D A C B E (1) C E B D A (1) B A E D C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -18 -10 -2 B -2 0 -16 -26 -10 C 18 16 0 10 14 D 10 26 -10 0 22 E 2 10 -14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -18 -10 -2 B -2 0 -16 -26 -10 C 18 16 0 10 14 D 10 26 -10 0 22 E 2 10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=27 A=22 E=14 B=4 so B is eliminated. Round 2 votes counts: D=33 C=27 A=23 E=17 so E is eliminated. Round 3 votes counts: D=38 A=32 C=30 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:229 D:224 E:188 A:186 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -18 -10 -2 B -2 0 -16 -26 -10 C 18 16 0 10 14 D 10 26 -10 0 22 E 2 10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -18 -10 -2 B -2 0 -16 -26 -10 C 18 16 0 10 14 D 10 26 -10 0 22 E 2 10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -18 -10 -2 B -2 0 -16 -26 -10 C 18 16 0 10 14 D 10 26 -10 0 22 E 2 10 -14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8660: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (13) B D C A E (12) D C B A E (7) E A D C B (6) B A E C D (6) E D C A B (5) D C E A B (5) C D A E B (5) B A C D E (5) D C B E A (4) A E B C D (4) E A B C D (3) D C E B A (2) B D E C A (2) B C D A E (2) E D C B A (1) E C D A B (1) E B D A C (1) E A D B C (1) D C A E B (1) D C A B E (1) D B C A E (1) C E D A B (1) C D A B E (1) B E A D C (1) B E A C D (1) B D C E A (1) B A E D C (1) B A D C E (1) A E C D B (1) A E C B D (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -4 -6 4 B -4 0 -14 -16 -6 C 4 14 0 -6 2 D 6 16 6 0 2 E -4 6 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 -6 4 B -4 0 -14 -16 -6 C 4 14 0 -6 2 D 6 16 6 0 2 E -4 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=31 D=21 A=9 C=7 so C is eliminated. Round 2 votes counts: E=32 B=32 D=27 A=9 so A is eliminated. Round 3 votes counts: E=39 B=33 D=28 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:215 C:207 A:199 E:199 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -4 -6 4 B -4 0 -14 -16 -6 C 4 14 0 -6 2 D 6 16 6 0 2 E -4 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 4 B -4 0 -14 -16 -6 C 4 14 0 -6 2 D 6 16 6 0 2 E -4 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 4 B -4 0 -14 -16 -6 C 4 14 0 -6 2 D 6 16 6 0 2 E -4 6 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984626 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8661: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (5) D E B C A (4) D E A B C (4) D A E B C (4) B A D C E (4) A E D B C (4) A B D C E (4) A B C D E (4) E C D B A (3) E C D A B (3) C E A B D (3) B C A D E (3) B A C E D (3) B A C D E (3) A E B C D (3) E D C B A (2) E D A C B (2) E C A D B (2) D E C B A (2) D E A C B (2) D B C E A (2) D B A C E (2) D A B E C (2) C E B A D (2) C D B E A (2) C B E D A (2) C B E A D (2) C B D E A (2) B C D A E (2) A C E B D (2) E C A B D (1) E A C B D (1) D E C A B (1) D E B A C (1) D B C A E (1) D B A E C (1) D A B C E (1) C B A E D (1) B D C A E (1) B D A C E (1) B C A E D (1) A E C D B (1) A D E B C (1) A D B E C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 16 6 14 B -6 0 26 4 4 C -16 -26 0 6 8 D -6 -4 -6 0 10 E -14 -4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 6 14 B -6 0 26 4 4 C -16 -26 0 6 8 D -6 -4 -6 0 10 E -14 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 B=18 E=14 C=14 so E is eliminated. Round 2 votes counts: D=31 A=28 C=23 B=18 so B is eliminated. Round 3 votes counts: A=38 D=33 C=29 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:221 B:214 D:197 C:186 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 6 14 B -6 0 26 4 4 C -16 -26 0 6 8 D -6 -4 -6 0 10 E -14 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 6 14 B -6 0 26 4 4 C -16 -26 0 6 8 D -6 -4 -6 0 10 E -14 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 6 14 B -6 0 26 4 4 C -16 -26 0 6 8 D -6 -4 -6 0 10 E -14 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8662: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (11) D B A C E (9) C E A B D (7) A B D E C (6) D B A E C (5) C E D B A (5) C D E B A (5) C D B A E (5) E A B D C (4) E A B C D (4) B D A C E (4) D B C A E (3) C E B A D (3) C B D A E (3) A B E D C (3) E C A D B (2) E A C B D (2) D C B A E (2) C D B E A (2) A E B D C (2) E D C B A (1) E C D A B (1) D E A B C (1) D C B E A (1) D A B E C (1) C E A D B (1) C B A D E (1) B D C A E (1) B D A E C (1) B A D C E (1) A D B E C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -8 0 0 B 4 0 -4 10 2 C 8 4 0 6 8 D 0 -10 -6 0 6 E 0 -2 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 0 0 B 4 0 -4 10 2 C 8 4 0 6 8 D 0 -10 -6 0 6 E 0 -2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=25 D=22 A=14 B=7 so B is eliminated. Round 2 votes counts: C=32 D=28 E=25 A=15 so A is eliminated. Round 3 votes counts: D=37 C=32 E=31 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:206 D:195 A:194 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 0 0 B 4 0 -4 10 2 C 8 4 0 6 8 D 0 -10 -6 0 6 E 0 -2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 0 0 B 4 0 -4 10 2 C 8 4 0 6 8 D 0 -10 -6 0 6 E 0 -2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 0 0 B 4 0 -4 10 2 C 8 4 0 6 8 D 0 -10 -6 0 6 E 0 -2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8663: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) A B D C E (7) B A E D C (6) D B A C E (5) E B A D C (4) D C A B E (4) C D E A B (4) B A D E C (4) E C D A B (3) E C A B D (3) E B A C D (3) D C B A E (3) D A B C E (3) C E D A B (3) A D B C E (3) A B E D C (3) E B C A D (2) E A B C D (2) D B C A E (2) C E D B A (2) B E A D C (2) B A D C E (2) A D C B E (2) A C D B E (2) E D C B A (1) E D B C A (1) E C B A D (1) E C A D B (1) E A C B D (1) D C E B A (1) D C B E A (1) D B C E A (1) D B A E C (1) C E A D B (1) C D E B A (1) C D A B E (1) B D A E C (1) A E C B D (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 14 10 10 B 2 0 8 -6 14 C -14 -8 0 -16 2 D -10 6 16 0 0 E -10 -14 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.43209876523 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 -2 14 10 10 B 2 0 8 -6 14 C -14 -8 0 -16 2 D -10 6 16 0 0 E -10 -14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765379 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=23 D=21 B=15 C=12 so C is eliminated. Round 2 votes counts: E=35 D=27 A=23 B=15 so B is eliminated. Round 3 votes counts: E=37 A=35 D=28 so D is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:209 D:206 E:187 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 14 10 10 B 2 0 8 -6 14 C -14 -8 0 -16 2 D -10 6 16 0 0 E -10 -14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765379 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 14 10 10 B 2 0 8 -6 14 C -14 -8 0 -16 2 D -10 6 16 0 0 E -10 -14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765379 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 14 10 10 B 2 0 8 -6 14 C -14 -8 0 -16 2 D -10 6 16 0 0 E -10 -14 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.555556 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.432098765379 Cumulative probabilities = A: 0.333333 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8664: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (17) D B C A E (11) E A C D B (10) B D C A E (9) B D C E A (8) A E C D B (8) A C E D B (7) B D E C A (5) D C B A E (4) C D B A E (3) E B A D C (2) E A B C D (2) C D A B E (2) E B D C A (1) E B C D A (1) E B C A D (1) E A D B C (1) D B E A C (1) D B A C E (1) D A B C E (1) C E A B D (1) C B D A E (1) C A D B E (1) B E C D A (1) A C D E B (1) Total count = 100 A B C D E A 0 2 2 2 -2 B -2 0 -10 -2 -4 C -2 10 0 12 0 D -2 2 -12 0 -4 E 2 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.223262 D: 0.000000 E: 0.776738 Sum of squares = 0.65316825869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.223262 D: 0.223262 E: 1.000000 A B C D E A 0 2 2 2 -2 B -2 0 -10 -2 -4 C -2 10 0 12 0 D -2 2 -12 0 -4 E 2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.000000 E: 0.500183 Sum of squares = 0.500000066627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.499817 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=23 D=18 A=16 C=8 so C is eliminated. Round 2 votes counts: E=36 B=24 D=23 A=17 so A is eliminated. Round 3 votes counts: E=51 D=25 B=24 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:210 E:205 A:202 D:192 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 2 2 -2 B -2 0 -10 -2 -4 C -2 10 0 12 0 D -2 2 -12 0 -4 E 2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.000000 E: 0.500183 Sum of squares = 0.500000066627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.499817 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 -2 B -2 0 -10 -2 -4 C -2 10 0 12 0 D -2 2 -12 0 -4 E 2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.000000 E: 0.500183 Sum of squares = 0.500000066627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.499817 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 -2 B -2 0 -10 -2 -4 C -2 10 0 12 0 D -2 2 -12 0 -4 E 2 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.000000 E: 0.500183 Sum of squares = 0.500000066627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499817 D: 0.499817 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8665: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) D B E A C (6) B C E D A (6) A E C D B (6) D B E C A (5) C B A E D (5) A D E C B (5) A C E D B (5) E B D C A (4) C B E A D (4) D B A E C (3) C A B E D (3) B D E C A (3) A D C E B (3) A D C B E (3) A C E B D (3) E C B A D (2) E A C B D (2) C A B D E (2) B E D C A (2) B C D E A (2) A C D E B (2) A C D B E (2) E D B A C (1) E B C D A (1) E A D C B (1) E A C D B (1) D E B A C (1) D B C A E (1) D A E B C (1) D A B E C (1) C B A D E (1) B E C D A (1) B D C E A (1) B D C A E (1) B C E A D (1) A E D C B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 -4 20 12 B 2 0 -18 4 8 C 4 18 0 14 6 D -20 -4 -14 0 -14 E -12 -8 -6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 20 12 B 2 0 -18 4 8 C 4 18 0 14 6 D -20 -4 -14 0 -14 E -12 -8 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=22 D=18 B=17 E=12 so E is eliminated. Round 2 votes counts: A=35 C=24 B=22 D=19 so D is eliminated. Round 3 votes counts: B=39 A=37 C=24 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:221 A:213 B:198 E:194 D:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 20 12 B 2 0 -18 4 8 C 4 18 0 14 6 D -20 -4 -14 0 -14 E -12 -8 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 20 12 B 2 0 -18 4 8 C 4 18 0 14 6 D -20 -4 -14 0 -14 E -12 -8 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 20 12 B 2 0 -18 4 8 C 4 18 0 14 6 D -20 -4 -14 0 -14 E -12 -8 -6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8666: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) E C B A D (12) A B C D E (12) D E C B A (7) C E B A D (7) E C D B A (6) D E C A B (6) B A C E D (6) D A B E C (5) C A B E D (3) B A D C E (3) A B D C E (3) A B C E D (3) E D C B A (2) E C B D A (2) C B E A D (2) B C A E D (2) E B C A D (1) D E A B C (1) D A E B C (1) D A C E B (1) B D A C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 0 10 8 B 2 0 2 16 8 C 0 -2 0 14 14 D -10 -16 -14 0 6 E -8 -8 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999683 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 10 8 B 2 0 2 16 8 C 0 -2 0 14 14 D -10 -16 -14 0 6 E -8 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=23 A=19 C=12 B=12 so C is eliminated. Round 2 votes counts: D=34 E=30 A=22 B=14 so B is eliminated. Round 3 votes counts: D=35 A=33 E=32 so E is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:214 C:213 A:208 D:183 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 10 8 B 2 0 2 16 8 C 0 -2 0 14 14 D -10 -16 -14 0 6 E -8 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 10 8 B 2 0 2 16 8 C 0 -2 0 14 14 D -10 -16 -14 0 6 E -8 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 10 8 B 2 0 2 16 8 C 0 -2 0 14 14 D -10 -16 -14 0 6 E -8 -8 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991098 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8667: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) B D C A E (11) A C D B E (9) E A C D B (8) A B D C E (8) E C B D A (6) E C D B A (5) E C A D B (5) C D B E A (5) A E C D B (5) E B D C A (4) B D A C E (4) A D B C E (4) E A B D C (3) C D B A E (2) A E B D C (2) E B D A C (1) E B C D A (1) D C B A E (1) B A D E C (1) B A D C E (1) A E D B C (1) A D C B E (1) Total count = 100 A B C D E A 0 -8 -4 -4 0 B 8 0 6 8 18 C 4 -6 0 -8 16 D 4 -8 8 0 18 E 0 -18 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -4 -4 0 B 8 0 6 8 18 C 4 -6 0 -8 16 D 4 -8 8 0 18 E 0 -18 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=30 B=29 C=7 D=1 so D is eliminated. Round 2 votes counts: E=33 A=30 B=29 C=8 so C is eliminated. Round 3 votes counts: B=37 E=33 A=30 so A is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:211 C:203 A:192 E:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -4 -4 0 B 8 0 6 8 18 C 4 -6 0 -8 16 D 4 -8 8 0 18 E 0 -18 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -4 -4 0 B 8 0 6 8 18 C 4 -6 0 -8 16 D 4 -8 8 0 18 E 0 -18 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -4 -4 0 B 8 0 6 8 18 C 4 -6 0 -8 16 D 4 -8 8 0 18 E 0 -18 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8668: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (5) C D A E B (5) B E A C D (5) B D A E C (5) A C B E D (5) E B A C D (4) D C E A B (4) D C A E B (4) D C A B E (4) B A E C D (4) A B C D E (4) E B C A D (3) D A C B E (3) C A E B D (3) C A D E B (3) B E A D C (3) B A D E C (3) A B D C E (3) A B C E D (3) E D C B A (2) E B D C A (2) D E B C A (2) C D E A B (2) A D C B E (2) A C D B E (2) E C D A B (1) E C B D A (1) E C A B D (1) E B D A C (1) E B C D A (1) E B A D C (1) D E C B A (1) D B A C E (1) C E A D B (1) C E A B D (1) C D A B E (1) C A D B E (1) B D E A C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 4 8 16 B -10 0 4 10 4 C -4 -4 0 4 8 D -8 -10 -4 0 6 E -16 -4 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 8 16 B -10 0 4 10 4 C -4 -4 0 4 8 D -8 -10 -4 0 6 E -16 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=22 B=21 A=21 D=19 C=17 so C is eliminated. Round 2 votes counts: A=28 D=27 E=24 B=21 so B is eliminated. Round 3 votes counts: A=35 D=33 E=32 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:204 C:202 D:192 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 8 16 B -10 0 4 10 4 C -4 -4 0 4 8 D -8 -10 -4 0 6 E -16 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 8 16 B -10 0 4 10 4 C -4 -4 0 4 8 D -8 -10 -4 0 6 E -16 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 8 16 B -10 0 4 10 4 C -4 -4 0 4 8 D -8 -10 -4 0 6 E -16 -4 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999463 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8669: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (15) D A C E B (13) E B C D A (8) D A E C B (5) A D C B E (5) A B D E C (5) C E B D A (4) B E C D A (4) B A E D C (4) E C B D A (3) D A E B C (3) C B E A D (3) D E A B C (2) D C A E B (2) D A C B E (2) C A D E B (2) B C E A D (2) A D C E B (2) A D B E C (2) A D B C E (2) E D B C A (1) E D B A C (1) E B D C A (1) D A B E C (1) C E D B A (1) C D E A B (1) C D A E B (1) C B A E D (1) B E A D C (1) B A D E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 -6 6 B 0 0 8 8 0 C -2 -8 0 -8 -14 D 6 -8 8 0 2 E -6 0 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.421493 B: 0.578507 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.512326538027 Cumulative probabilities = A: 0.421493 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -6 6 B 0 0 8 8 0 C -2 -8 0 -8 -14 D 6 -8 8 0 2 E -6 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=27 A=18 E=14 C=13 so C is eliminated. Round 2 votes counts: B=31 D=30 A=20 E=19 so E is eliminated. Round 3 votes counts: B=47 D=33 A=20 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:204 E:203 A:201 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -6 6 B 0 0 8 8 0 C -2 -8 0 -8 -14 D 6 -8 8 0 2 E -6 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -6 6 B 0 0 8 8 0 C -2 -8 0 -8 -14 D 6 -8 8 0 2 E -6 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -6 6 B 0 0 8 8 0 C -2 -8 0 -8 -14 D 6 -8 8 0 2 E -6 0 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8670: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (12) C B E D A (11) E B A D C (9) C E B A D (9) E B C A D (6) A D C E B (6) D A B E C (5) A D C B E (5) D A C B E (4) C D A B E (4) C A D E B (4) C A D B E (4) B E C D A (4) E B D A C (2) D A E B C (2) B C E D A (2) E C B A D (1) E B C D A (1) E A D B C (1) D A B C E (1) C B E A D (1) C B D A E (1) B E D C A (1) B E D A C (1) B D C A E (1) B D A E C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 0 18 2 B 2 0 0 2 -6 C 0 0 0 -4 6 D -18 -2 4 0 2 E -2 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.046875 B: 0.453125 C: 0.421875 D: 0.046875 E: 0.031250 Sum of squares = 0.38867187518 Cumulative probabilities = A: 0.046875 B: 0.500000 C: 0.921875 D: 0.968750 E: 1.000000 A B C D E A 0 -2 0 18 2 B 2 0 0 2 -6 C 0 0 0 -4 6 D -18 -2 4 0 2 E -2 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.046875 B: 0.453125 C: 0.421875 D: 0.046875 E: 0.031250 Sum of squares = 0.388671874965 Cumulative probabilities = A: 0.046875 B: 0.500000 C: 0.921875 D: 0.968750 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=24 E=20 D=12 B=10 so B is eliminated. Round 2 votes counts: C=36 E=26 A=24 D=14 so D is eliminated. Round 3 votes counts: C=37 A=37 E=26 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 C:201 B:199 E:198 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 18 2 B 2 0 0 2 -6 C 0 0 0 -4 6 D -18 -2 4 0 2 E -2 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.046875 B: 0.453125 C: 0.421875 D: 0.046875 E: 0.031250 Sum of squares = 0.388671874965 Cumulative probabilities = A: 0.046875 B: 0.500000 C: 0.921875 D: 0.968750 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 18 2 B 2 0 0 2 -6 C 0 0 0 -4 6 D -18 -2 4 0 2 E -2 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.046875 B: 0.453125 C: 0.421875 D: 0.046875 E: 0.031250 Sum of squares = 0.388671874965 Cumulative probabilities = A: 0.046875 B: 0.500000 C: 0.921875 D: 0.968750 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 18 2 B 2 0 0 2 -6 C 0 0 0 -4 6 D -18 -2 4 0 2 E -2 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.046875 B: 0.453125 C: 0.421875 D: 0.046875 E: 0.031250 Sum of squares = 0.388671874965 Cumulative probabilities = A: 0.046875 B: 0.500000 C: 0.921875 D: 0.968750 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8671: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (5) E A D B C (5) D A E B C (5) C E B A D (5) C B D E A (5) E A B D C (4) D C B A E (4) C B E A D (4) B C E A D (4) B C D A E (4) E C D A B (3) D E A C B (3) D C A E B (3) C B E D A (3) B D A C E (3) B C A E D (3) E D A C B (2) E C A B D (2) D A B E C (2) C E D A B (2) C D E A B (2) C B D A E (2) B D C A E (2) B D A E C (2) E B C A D (1) E A D C B (1) E A C D B (1) E A C B D (1) E A B C D (1) D E C A B (1) D B A E C (1) D A E C B (1) D A B C E (1) C E D B A (1) C E A D B (1) C E A B D (1) B E A C D (1) B A E D C (1) B A D E C (1) A E D B C (1) A E B D C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -16 -4 -18 B -6 0 -6 6 -8 C 16 6 0 4 0 D 4 -6 -4 0 -10 E 18 8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.723859 D: 0.000000 E: 0.276141 Sum of squares = 0.600225986929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.723859 D: 0.723859 E: 1.000000 A B C D E A 0 6 -16 -4 -18 B -6 0 -6 6 -8 C 16 6 0 4 0 D 4 -6 -4 0 -10 E 18 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=26 C=26 D=21 B=21 A=6 so A is eliminated. Round 2 votes counts: E=28 C=26 D=23 B=23 so D is eliminated. Round 3 votes counts: E=39 C=33 B=28 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:218 C:213 B:193 D:192 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -16 -4 -18 B -6 0 -6 6 -8 C 16 6 0 4 0 D 4 -6 -4 0 -10 E 18 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -16 -4 -18 B -6 0 -6 6 -8 C 16 6 0 4 0 D 4 -6 -4 0 -10 E 18 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -16 -4 -18 B -6 0 -6 6 -8 C 16 6 0 4 0 D 4 -6 -4 0 -10 E 18 8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8672: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (11) D A E B C (7) B E A C D (7) A D E B C (5) D C E A B (4) D C B A E (4) D A B E C (4) C B E D A (4) C B E A D (4) C B D E A (4) B A E C D (4) A E B D C (4) A B E D C (4) A E D B C (3) E C B A D (2) E B C A D (2) E A B C D (2) D C A E B (2) C E B D A (2) C E B A D (2) C D B E A (2) A D B E C (2) E D C A B (1) E A B D C (1) D E A C B (1) D C B E A (1) D B A C E (1) D A C E B (1) D A B C E (1) C E D B A (1) C D E A B (1) C B D A E (1) B D C A E (1) B D A C E (1) B C A E D (1) B A C E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 0 12 -4 B 12 0 26 18 18 C 0 -26 0 2 0 D -12 -18 -2 0 -12 E 4 -18 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 12 -4 B 12 0 26 18 18 C 0 -26 0 2 0 D -12 -18 -2 0 -12 E 4 -18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=26 B=26 C=21 A=19 E=8 so E is eliminated. Round 2 votes counts: B=28 D=27 C=23 A=22 so A is eliminated. Round 3 votes counts: B=40 D=37 C=23 so C is eliminated. Round 4 votes counts: B=59 D=41 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:237 E:199 A:198 C:188 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 12 -4 B 12 0 26 18 18 C 0 -26 0 2 0 D -12 -18 -2 0 -12 E 4 -18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 12 -4 B 12 0 26 18 18 C 0 -26 0 2 0 D -12 -18 -2 0 -12 E 4 -18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 12 -4 B 12 0 26 18 18 C 0 -26 0 2 0 D -12 -18 -2 0 -12 E 4 -18 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8673: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) D E C A B (9) B A C E D (9) D C E A B (7) A B E C D (7) E C D A B (5) E D C A B (4) C D E B A (4) C E D B A (3) B A E C D (3) A B E D C (3) A B D E C (3) E C D B A (2) E A B C D (2) D C B A E (2) D B A C E (2) D A E B C (2) C E B A D (2) C B D E A (2) B A D C E (2) A E B D C (2) E C A B D (1) E B C A D (1) E A D C B (1) D C B E A (1) D A B C E (1) C E B D A (1) C D B A E (1) B D C A E (1) B C D A E (1) B C A D E (1) B A C D E (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -20 -20 -14 B 2 0 -14 -14 -20 C 20 14 0 -6 6 D 20 14 6 0 6 E 14 20 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -20 -20 -14 B 2 0 -14 -14 -20 C 20 14 0 -6 6 D 20 14 6 0 6 E 14 20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=18 A=17 E=16 C=13 so C is eliminated. Round 2 votes counts: D=41 E=22 B=20 A=17 so A is eliminated. Round 3 votes counts: D=42 B=33 E=25 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 C:217 E:211 B:177 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -20 -20 -14 B 2 0 -14 -14 -20 C 20 14 0 -6 6 D 20 14 6 0 6 E 14 20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -20 -20 -14 B 2 0 -14 -14 -20 C 20 14 0 -6 6 D 20 14 6 0 6 E 14 20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -20 -20 -14 B 2 0 -14 -14 -20 C 20 14 0 -6 6 D 20 14 6 0 6 E 14 20 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8674: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E B A D C (6) C E D A B (6) D A B C E (5) C E A D B (5) E C A D B (4) D B A C E (4) C E D B A (4) B E A D C (4) B A D E C (4) E C B D A (3) E C B A D (3) E C A B D (3) E B C A D (3) C D A E B (3) B E D A C (3) B D A C E (3) A B D E C (3) E B A C D (2) D A C B E (2) C D B E A (2) C B D E A (2) C A D E B (2) B D A E C (2) A D C B E (2) E B D C A (1) E B D A C (1) E B C D A (1) E A B C D (1) D C B A E (1) D C A B E (1) C D E A B (1) B E D C A (1) B A E D C (1) A E D B C (1) A E B D C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -2 -6 -6 -12 B 2 0 -2 -4 -2 C 6 2 0 4 2 D 6 4 -4 0 -8 E 12 2 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -6 -12 B 2 0 -2 -4 -2 C 6 2 0 4 2 D 6 4 -4 0 -8 E 12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=28 B=18 D=13 A=9 so A is eliminated. Round 2 votes counts: C=32 E=30 B=21 D=17 so D is eliminated. Round 3 votes counts: C=38 B=32 E=30 so E is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:210 C:207 D:199 B:197 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -12 B 2 0 -2 -4 -2 C 6 2 0 4 2 D 6 4 -4 0 -8 E 12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -12 B 2 0 -2 -4 -2 C 6 2 0 4 2 D 6 4 -4 0 -8 E 12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -12 B 2 0 -2 -4 -2 C 6 2 0 4 2 D 6 4 -4 0 -8 E 12 2 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8675: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) E A B D C (7) E A B C D (6) B D C A E (6) E A C D B (5) D C B E A (4) D B C E A (4) C D E B A (4) E C A D B (3) E B A D C (3) C D B E A (3) C D A E B (3) B E D C A (3) A E C B D (3) A B D C E (3) E D B C A (2) D C E B A (2) D C B A E (2) D B C A E (2) C D A B E (2) C A D E B (2) B D E A C (2) A E C D B (2) A E B C D (2) A C D B E (2) A B E D C (2) E D C B A (1) E C D B A (1) E C D A B (1) E A C B D (1) D B E C A (1) C E A D B (1) C A E D B (1) B E D A C (1) B D E C A (1) B D A C E (1) B A E D C (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 2 8 -12 B -10 0 14 4 -16 C -2 -14 0 -14 -12 D -8 -4 14 0 -8 E 12 16 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 2 8 -12 B -10 0 14 4 -16 C -2 -14 0 -14 -12 D -8 -4 14 0 -8 E 12 16 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=23 C=16 B=16 D=15 so D is eliminated. Round 2 votes counts: E=30 C=24 B=23 A=23 so B is eliminated. Round 3 votes counts: E=38 C=36 A=26 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:204 D:197 B:196 C:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 2 8 -12 B -10 0 14 4 -16 C -2 -14 0 -14 -12 D -8 -4 14 0 -8 E 12 16 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 8 -12 B -10 0 14 4 -16 C -2 -14 0 -14 -12 D -8 -4 14 0 -8 E 12 16 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 8 -12 B -10 0 14 4 -16 C -2 -14 0 -14 -12 D -8 -4 14 0 -8 E 12 16 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8676: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) A B D E C (8) E B C D A (6) A D C E B (6) E C B D A (5) D E C A B (4) A D C B E (4) D C E A B (3) D C A E B (3) D A C E B (3) C E D B A (3) C D E B A (3) C D E A B (3) B E A D C (3) B A E D C (3) A D E B C (3) A B D C E (3) A B C D E (3) E D C B A (2) B E C A D (2) B C E D A (2) B C A E D (2) B A E C D (2) A C D B E (2) A B E C D (2) E C D B A (1) D E C B A (1) D A E B C (1) C B E D A (1) B E A C D (1) B C E A D (1) B A C E D (1) A C D E B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -4 -2 -2 B -2 0 8 14 4 C 4 -8 0 4 -10 D 2 -14 -4 0 4 E 2 -4 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000014 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 2 -4 -2 -2 B -2 0 8 14 4 C 4 -8 0 4 -10 D 2 -14 -4 0 4 E 2 -4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999956 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=27 D=15 E=14 C=10 so C is eliminated. Round 2 votes counts: A=34 B=28 D=21 E=17 so E is eliminated. Round 3 votes counts: B=39 A=34 D=27 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:202 A:197 C:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -4 -2 -2 B -2 0 8 14 4 C 4 -8 0 4 -10 D 2 -14 -4 0 4 E 2 -4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999956 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 -2 B -2 0 8 14 4 C 4 -8 0 4 -10 D 2 -14 -4 0 4 E 2 -4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999956 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 -2 B -2 0 8 14 4 C 4 -8 0 4 -10 D 2 -14 -4 0 4 E 2 -4 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999956 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8677: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (7) C D A E B (7) D C A B E (5) C A D B E (5) E C D B A (4) E B C A D (4) C D E A B (4) B E A D C (4) B A E D C (4) B A E C D (4) A D B C E (4) E B A C D (3) C D A B E (3) A C D B E (3) E D C B A (2) E B D A C (2) E B C D A (2) E B A D C (2) D C E A B (2) D C A E B (2) D B A E C (2) D A B C E (2) C E B A D (2) C E A B D (2) C A D E B (2) B E A C D (2) E D B C A (1) E D B A C (1) E C B A D (1) E B D C A (1) D E B A C (1) D B E A C (1) C E A D B (1) C A E B D (1) B D E A C (1) B A D E C (1) A C B D E (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -16 0 0 B -10 0 -8 -18 -8 C 16 8 0 20 10 D 0 18 -20 0 -2 E 0 8 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -16 0 0 B -10 0 -8 -18 -8 C 16 8 0 20 10 D 0 18 -20 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=23 B=16 D=15 A=12 so A is eliminated. Round 2 votes counts: C=38 E=23 B=20 D=19 so D is eliminated. Round 3 votes counts: C=47 B=29 E=24 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:200 D:198 A:197 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -16 0 0 B -10 0 -8 -18 -8 C 16 8 0 20 10 D 0 18 -20 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -16 0 0 B -10 0 -8 -18 -8 C 16 8 0 20 10 D 0 18 -20 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -16 0 0 B -10 0 -8 -18 -8 C 16 8 0 20 10 D 0 18 -20 0 -2 E 0 8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997703 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8678: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) D A C B E (6) B C E D A (6) A D C B E (6) D A B C E (5) A D C E B (5) B E C D A (4) A D E B C (4) E C A B D (3) E B C A D (3) D B A C E (3) D A E B C (3) C B E D A (3) C B E A D (3) A D E C B (3) E C B A D (2) E B A D C (2) C E A B D (2) C D A B E (2) C B D E A (2) C A D E B (2) C A D B E (2) B E D C A (2) B D E A C (2) A E C D B (2) E B D A C (1) E B C D A (1) E A D C B (1) E A D B C (1) E A C D B (1) E A C B D (1) D C A B E (1) D B C A E (1) C E B A D (1) C E A D B (1) C D B A E (1) C B D A E (1) B D C A E (1) B D A E C (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 20 12 -6 16 B -20 0 -4 -18 18 C -12 4 0 -12 10 D 6 18 12 0 18 E -16 -18 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999793 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 12 -6 16 B -20 0 -4 -18 18 C -12 4 0 -12 10 D 6 18 12 0 18 E -16 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=22 C=20 E=16 B=16 so E is eliminated. Round 2 votes counts: D=26 A=26 C=25 B=23 so B is eliminated. Round 3 votes counts: C=39 D=33 A=28 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:221 C:195 B:188 E:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 12 -6 16 B -20 0 -4 -18 18 C -12 4 0 -12 10 D 6 18 12 0 18 E -16 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 12 -6 16 B -20 0 -4 -18 18 C -12 4 0 -12 10 D 6 18 12 0 18 E -16 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 12 -6 16 B -20 0 -4 -18 18 C -12 4 0 -12 10 D 6 18 12 0 18 E -16 -18 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8679: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (9) B A D C E (9) B E A D C (8) C D A B E (7) B A E D C (7) C D A E B (6) C E D A B (5) E B C D A (4) E B A D C (4) D A C B E (4) B A D E C (4) E D C A B (3) C D E A B (3) E D A C B (2) E B C A D (2) E B A C D (2) E A D B C (2) D A C E B (2) A D B C E (2) A B D E C (2) A B D C E (2) E C B D A (1) E A D C B (1) E A B D C (1) C E D B A (1) C E B D A (1) C D B A E (1) C B D A E (1) B A C D E (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 8 12 0 2 B -8 0 2 -2 0 C -12 -2 0 -12 -8 D 0 2 12 0 -6 E -2 0 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.887290 B: 0.000000 C: 0.000000 D: 0.112710 E: 0.000000 Sum of squares = 0.799987001627 Cumulative probabilities = A: 0.887290 B: 0.887290 C: 0.887290 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 0 2 B -8 0 2 -2 0 C -12 -2 0 -12 -8 D 0 2 12 0 -6 E -2 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000097786 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=29 C=25 A=9 D=6 so D is eliminated. Round 2 votes counts: E=31 B=29 C=25 A=15 so A is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 E:206 D:204 B:196 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 12 0 2 B -8 0 2 -2 0 C -12 -2 0 -12 -8 D 0 2 12 0 -6 E -2 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000097786 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 0 2 B -8 0 2 -2 0 C -12 -2 0 -12 -8 D 0 2 12 0 -6 E -2 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000097786 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 0 2 B -8 0 2 -2 0 C -12 -2 0 -12 -8 D 0 2 12 0 -6 E -2 0 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.625000097786 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8680: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) B E A C D (8) E A C B D (6) D B E C A (6) E C A D B (5) D C E A B (5) D B C A E (5) B D E A C (5) D C A E B (3) C A E D B (3) B D A E C (3) A C B E D (3) E D C A B (2) E B D A C (2) E A C D B (2) D E C B A (2) D E C A B (2) D C E B A (2) D B E A C (2) D B C E A (2) B D A C E (2) B A D C E (2) B A C E D (2) E D B C A (1) E C A B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D C A B E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A E B (1) C D A B E (1) B E A D C (1) B A E D C (1) B A C D E (1) A E C B D (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 6 -2 -18 B 16 0 8 0 12 C -6 -8 0 -2 -22 D 2 0 2 0 -4 E 18 -12 22 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.561284 C: 0.000000 D: 0.438716 E: 0.000000 Sum of squares = 0.507511454219 Cumulative probabilities = A: 0.000000 B: 0.561284 C: 0.561284 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 -2 -18 B 16 0 8 0 12 C -6 -8 0 -2 -22 D 2 0 2 0 -4 E 18 -12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=32 E=20 C=8 A=6 so A is eliminated. Round 2 votes counts: B=34 D=32 E=22 C=12 so C is eliminated. Round 3 votes counts: B=37 D=35 E=28 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:216 D:200 A:185 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 -2 -18 B 16 0 8 0 12 C -6 -8 0 -2 -22 D 2 0 2 0 -4 E 18 -12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -2 -18 B 16 0 8 0 12 C -6 -8 0 -2 -22 D 2 0 2 0 -4 E 18 -12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -2 -18 B 16 0 8 0 12 C -6 -8 0 -2 -22 D 2 0 2 0 -4 E 18 -12 22 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999989 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8681: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (14) C E D A B (13) B A D E C (9) E C B A D (7) D C E A B (7) C E D B A (7) B E C A D (7) B A E C D (7) A B D E C (6) E C B D A (4) E C A D B (3) D C A E B (2) C D E A B (2) B E A C D (2) A D B E C (2) E C A B D (1) E B C A D (1) D B C E A (1) D B A C E (1) C E A D B (1) B A E D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 -12 -2 -12 B -6 0 6 -8 4 C 12 -6 0 12 -2 D 2 8 -12 0 -10 E 12 -4 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.454545 C: 0.000000 D: 0.181818 E: 0.363636 Sum of squares = 0.37190082643 Cumulative probabilities = A: 0.000000 B: 0.454545 C: 0.454545 D: 0.636364 E: 1.000000 A B C D E A 0 6 -12 -2 -12 B -6 0 6 -8 4 C 12 -6 0 12 -2 D 2 8 -12 0 -10 E 12 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.454545 C: 0.000000 D: 0.181818 E: 0.363636 Sum of squares = 0.371900826403 Cumulative probabilities = A: 0.000000 B: 0.454545 C: 0.454545 D: 0.636364 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=25 C=23 E=16 A=10 so A is eliminated. Round 2 votes counts: B=33 D=28 C=23 E=16 so E is eliminated. Round 3 votes counts: C=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:210 C:208 B:198 D:194 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -12 -2 -12 B -6 0 6 -8 4 C 12 -6 0 12 -2 D 2 8 -12 0 -10 E 12 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.454545 C: 0.000000 D: 0.181818 E: 0.363636 Sum of squares = 0.371900826403 Cumulative probabilities = A: 0.000000 B: 0.454545 C: 0.454545 D: 0.636364 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -2 -12 B -6 0 6 -8 4 C 12 -6 0 12 -2 D 2 8 -12 0 -10 E 12 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.454545 C: 0.000000 D: 0.181818 E: 0.363636 Sum of squares = 0.371900826403 Cumulative probabilities = A: 0.000000 B: 0.454545 C: 0.454545 D: 0.636364 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -2 -12 B -6 0 6 -8 4 C 12 -6 0 12 -2 D 2 8 -12 0 -10 E 12 -4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.454545 C: 0.000000 D: 0.181818 E: 0.363636 Sum of squares = 0.371900826403 Cumulative probabilities = A: 0.000000 B: 0.454545 C: 0.454545 D: 0.636364 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8682: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (6) A C D B E (6) E D A B C (5) D A C B E (5) C B E A D (5) C A D B E (5) E B D C A (4) B E D C A (4) E D B A C (3) E C B A D (3) E B C D A (3) D B E A C (3) B D E A C (3) A E D C B (3) E C A B D (2) E B D A C (2) E A D C B (2) E A D B C (2) E A C B D (2) D A B E C (2) C B A D E (2) C A E D B (2) C A B D E (2) B E C D A (2) B D E C A (2) B D C A E (2) A D C B E (2) D E B A C (1) D B C A E (1) D B A C E (1) D A E B C (1) C E A B D (1) C B D A E (1) C B A E D (1) C A D E B (1) C A B E D (1) B E D A C (1) B C E D A (1) B C D E A (1) B C D A E (1) A D E C B (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 6 -10 -4 B -6 0 2 -8 20 C -6 -2 0 -14 -2 D 10 8 14 0 2 E 4 -20 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -10 -4 B -6 0 2 -8 20 C -6 -2 0 -14 -2 D 10 8 14 0 2 E 4 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=21 D=20 B=17 A=14 so A is eliminated. Round 2 votes counts: E=31 C=28 D=24 B=17 so B is eliminated. Round 3 votes counts: E=38 D=31 C=31 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:217 B:204 A:199 E:192 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -10 -4 B -6 0 2 -8 20 C -6 -2 0 -14 -2 D 10 8 14 0 2 E 4 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -10 -4 B -6 0 2 -8 20 C -6 -2 0 -14 -2 D 10 8 14 0 2 E 4 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -10 -4 B -6 0 2 -8 20 C -6 -2 0 -14 -2 D 10 8 14 0 2 E 4 -20 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8683: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (11) E B D A C (10) C A D B E (8) A C B E D (8) E D B C A (6) C D A B E (6) A C D B E (6) E B D C A (5) E D B A C (4) D C B E A (3) C A B D E (3) E B A D C (2) E A B C D (2) D E C B A (2) D C A E B (2) B E D A C (2) A E B C D (2) A C D E B (2) A C B D E (2) A B E C D (2) E B A C D (1) E A B D C (1) D C E B A (1) D C A B E (1) D B E C A (1) C D A E B (1) C A D E B (1) B E D C A (1) B C A E D (1) A D C E B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -6 -12 -4 B 0 0 4 -12 -10 C 6 -4 0 -6 -4 D 12 12 6 0 2 E 4 10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999759 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -12 -4 B 0 0 4 -12 -10 C 6 -4 0 -6 -4 D 12 12 6 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=25 D=21 C=19 B=4 so B is eliminated. Round 2 votes counts: E=34 A=25 D=21 C=20 so C is eliminated. Round 3 votes counts: A=38 E=34 D=28 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:208 C:196 B:191 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -6 -12 -4 B 0 0 4 -12 -10 C 6 -4 0 -6 -4 D 12 12 6 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -12 -4 B 0 0 4 -12 -10 C 6 -4 0 -6 -4 D 12 12 6 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -12 -4 B 0 0 4 -12 -10 C 6 -4 0 -6 -4 D 12 12 6 0 2 E 4 10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8684: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (6) C E A D B (6) B D A C E (6) D B C E A (5) C E D A B (5) D E C B A (4) B A D E C (4) A C E B D (4) A B E C D (4) E D C A B (3) E C A D B (3) D C E B A (3) C D B E A (3) B D C E A (3) B A D C E (3) A E C B D (3) A B E D C (3) E A C D B (2) D C B E A (2) D B E C A (2) C D E B A (2) C A E B D (2) A E C D B (2) A E B C D (2) A B D E C (2) A B C E D (2) E D A C B (1) E A D C B (1) D E B C A (1) D E A B C (1) D B E A C (1) C E D B A (1) C A B E D (1) B D E C A (1) B D C A E (1) B D A E C (1) B C D E A (1) B C D A E (1) B A C D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -12 -8 -14 B -8 0 -8 -8 -4 C 12 8 0 2 6 D 8 8 -2 0 -2 E 14 4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 -8 -14 B -8 0 -8 -8 -4 C 12 8 0 2 6 D 8 8 -2 0 -2 E 14 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 B=22 C=20 D=19 E=16 so E is eliminated. Round 2 votes counts: C=29 A=26 D=23 B=22 so B is eliminated. Round 3 votes counts: D=35 A=34 C=31 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:214 E:207 D:206 A:187 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -12 -8 -14 B -8 0 -8 -8 -4 C 12 8 0 2 6 D 8 8 -2 0 -2 E 14 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 -8 -14 B -8 0 -8 -8 -4 C 12 8 0 2 6 D 8 8 -2 0 -2 E 14 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 -8 -14 B -8 0 -8 -8 -4 C 12 8 0 2 6 D 8 8 -2 0 -2 E 14 4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999268 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8685: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) A D E C B (9) A B E D C (7) D E A C B (6) A E D B C (6) E D C B A (4) D E C A B (4) B A E D C (4) B A C E D (4) E B D A C (3) C B D E A (3) B C A E D (3) E D B C A (2) E D A B C (2) D C E A B (2) C D E A B (2) C D A E B (2) B E D A C (2) B C E D A (2) B C A D E (2) B A E C D (2) A D C E B (2) A C D E B (2) A C B D E (2) E D C A B (1) E D B A C (1) E D A C B (1) E A D B C (1) D E C B A (1) D A E C B (1) C B E D A (1) C B D A E (1) C B A D E (1) C A D E B (1) C A B D E (1) B E D C A (1) B C E A D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 12 -2 2 B -6 0 -12 -18 -24 C -12 12 0 -20 -16 D 2 18 20 0 2 E -2 24 16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 -2 2 B -6 0 -12 -18 -24 C -12 12 0 -20 -16 D 2 18 20 0 2 E -2 24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999994305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=21 B=21 E=15 D=14 so D is eliminated. Round 2 votes counts: A=30 E=26 C=23 B=21 so B is eliminated. Round 3 votes counts: A=40 C=31 E=29 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:221 E:218 A:209 C:182 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 12 -2 2 B -6 0 -12 -18 -24 C -12 12 0 -20 -16 D 2 18 20 0 2 E -2 24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999994305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 -2 2 B -6 0 -12 -18 -24 C -12 12 0 -20 -16 D 2 18 20 0 2 E -2 24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999994305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 -2 2 B -6 0 -12 -18 -24 C -12 12 0 -20 -16 D 2 18 20 0 2 E -2 24 16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999994305 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8686: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) E B C A D (7) E B A C D (5) D A B E C (5) C E D B A (5) A D B C E (5) E C D B A (4) D C A B E (4) D A B C E (4) C E B A D (4) A D B E C (4) A B D E C (4) C B E A D (3) B C E A D (3) B A E C D (3) E D A C B (2) D E C A B (2) D A E B C (2) D A C B E (2) C E B D A (2) C D E A B (2) B E C A D (2) B C A E D (2) B A C D E (2) A B D C E (2) E D C A B (1) E D B A C (1) E C D A B (1) E B A D C (1) D E A C B (1) D E A B C (1) D C E A B (1) D A C E B (1) C D E B A (1) C B D E A (1) C B A D E (1) B A D C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -6 14 -14 B 10 0 10 2 -2 C 6 -10 0 10 -8 D -14 -2 -10 0 -8 E 14 2 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 14 -14 B 10 0 10 2 -2 C 6 -10 0 10 -8 D -14 -2 -10 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=23 C=19 A=16 B=13 so B is eliminated. Round 2 votes counts: E=31 C=24 D=23 A=22 so A is eliminated. Round 3 votes counts: D=39 E=35 C=26 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:210 C:199 A:192 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 14 -14 B 10 0 10 2 -2 C 6 -10 0 10 -8 D -14 -2 -10 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 14 -14 B 10 0 10 2 -2 C 6 -10 0 10 -8 D -14 -2 -10 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 14 -14 B 10 0 10 2 -2 C 6 -10 0 10 -8 D -14 -2 -10 0 -8 E 14 2 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999695 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8687: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (10) B C A D E (9) A E D C B (8) E A D C B (6) A C B D E (6) B E D C A (5) B C D E A (5) A C D E B (4) A B C E D (4) E D B C A (3) E D A B C (3) E B D C A (3) D C E B A (3) C A B D E (3) D E C A B (2) D E B C A (2) C D A E B (2) C B A D E (2) B C D A E (2) A B C D E (2) E D C A B (1) E D B A C (1) E A D B C (1) D E A C B (1) D C E A B (1) D B E C A (1) C B D A E (1) C A D E B (1) B E C D A (1) B E A D C (1) B C E D A (1) B A C E D (1) A E C D B (1) A D E C B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 18 4 4 -2 B -18 0 -10 -6 -10 C -4 10 0 -6 -2 D -4 6 6 0 -4 E 2 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 4 4 -2 B -18 0 -10 -6 -10 C -4 10 0 -6 -2 D -4 6 6 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=28 A=28 B=25 D=10 C=9 so C is eliminated. Round 2 votes counts: A=32 E=28 B=28 D=12 so D is eliminated. Round 3 votes counts: E=37 A=34 B=29 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:212 E:209 D:202 C:199 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 4 4 -2 B -18 0 -10 -6 -10 C -4 10 0 -6 -2 D -4 6 6 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 4 -2 B -18 0 -10 -6 -10 C -4 10 0 -6 -2 D -4 6 6 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 4 -2 B -18 0 -10 -6 -10 C -4 10 0 -6 -2 D -4 6 6 0 -4 E 2 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8688: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (13) C D E B A (8) B D C E A (8) A B E D C (7) A E C D B (6) A E B D C (6) E C D A B (5) E A C D B (5) A E B C D (5) A B E C D (4) E D C B A (3) D C B E A (3) C D B A E (3) B D C A E (3) D C E B A (2) B A D C E (2) E D C A B (1) E C D B A (1) E C A D B (1) E A D B C (1) C E D A B (1) C D E A B (1) C B D E A (1) C B D A E (1) C A D B E (1) B D E C A (1) B C D A E (1) B A D E C (1) B A C D E (1) A E C B D (1) A C E D B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -16 -12 -10 B 4 0 -18 -12 4 C 16 18 0 22 2 D 12 12 -22 0 2 E 10 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -12 -10 B 4 0 -18 -12 4 C 16 18 0 22 2 D 12 12 -22 0 2 E 10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999995974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=29 E=17 B=17 D=5 so D is eliminated. Round 2 votes counts: C=34 A=32 E=17 B=17 so E is eliminated. Round 3 votes counts: C=45 A=38 B=17 so B is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:202 E:201 B:189 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 -12 -10 B 4 0 -18 -12 4 C 16 18 0 22 2 D 12 12 -22 0 2 E 10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999995974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -12 -10 B 4 0 -18 -12 4 C 16 18 0 22 2 D 12 12 -22 0 2 E 10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999995974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -12 -10 B 4 0 -18 -12 4 C 16 18 0 22 2 D 12 12 -22 0 2 E 10 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999995974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8689: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) E A B C D (6) D C B A E (6) D B C A E (5) E D B A C (4) D C E A B (4) D B A C E (4) C A B D E (4) B A C D E (4) A B C E D (4) E D C A B (3) E B A D C (3) D E B C A (3) D E B A C (3) C D A B E (3) B D A E C (3) B A C E D (3) E D A B C (2) E C D A B (2) D B E A C (2) C E A B D (2) C D E A B (2) C A E B D (2) B D C A E (2) E D A C B (1) E C A B D (1) E B D A C (1) D E C A B (1) D C B E A (1) C E D A B (1) C D B A E (1) C B A D E (1) C A E D B (1) C A D B E (1) C A B E D (1) B C A D E (1) B A E D C (1) B A D C E (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 4 -8 0 B -2 0 4 0 -2 C -4 -4 0 0 10 D 8 0 0 0 4 E 0 2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.430757 C: 0.000000 D: 0.569243 E: 0.000000 Sum of squares = 0.50958904977 Cumulative probabilities = A: 0.000000 B: 0.430757 C: 0.430757 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -8 0 B -2 0 4 0 -2 C -4 -4 0 0 10 D 8 0 0 0 4 E 0 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=29 C=19 B=15 A=6 so A is eliminated. Round 2 votes counts: E=32 D=29 C=20 B=19 so B is eliminated. Round 3 votes counts: D=35 E=33 C=32 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:206 C:201 B:200 A:199 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -8 0 B -2 0 4 0 -2 C -4 -4 0 0 10 D 8 0 0 0 4 E 0 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -8 0 B -2 0 4 0 -2 C -4 -4 0 0 10 D 8 0 0 0 4 E 0 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -8 0 B -2 0 4 0 -2 C -4 -4 0 0 10 D 8 0 0 0 4 E 0 2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8690: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A D E B C (7) D C E B A (6) D B C A E (6) E C B A D (5) A D B E C (5) E A C B D (4) C B E D A (4) A E D C B (4) A E D B C (4) A E B C D (4) D B C E A (3) B D C A E (3) A B E C D (3) E C D B A (2) E C A B D (2) E A D C B (2) D B A C E (2) D A E B C (2) D A B C E (2) C D B E A (2) E D C A B (1) E C B D A (1) E A C D B (1) D E C A B (1) D E A C B (1) D A E C B (1) C E D B A (1) C E B D A (1) C E B A D (1) C D E B A (1) C B E A D (1) C B D E A (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) B C A E D (1) B A C D E (1) A E C D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -10 -4 -4 B 8 0 -4 -26 -6 C 10 4 0 -18 -4 D 4 26 18 0 10 E 4 6 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 -4 -4 B 8 0 -4 -26 -6 C 10 4 0 -18 -4 D 4 26 18 0 10 E 4 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=29 E=18 C=12 B=9 so B is eliminated. Round 2 votes counts: D=35 A=30 E=18 C=17 so C is eliminated. Round 3 votes counts: D=41 A=31 E=28 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:202 C:196 A:187 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -10 -4 -4 B 8 0 -4 -26 -6 C 10 4 0 -18 -4 D 4 26 18 0 10 E 4 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -4 -4 B 8 0 -4 -26 -6 C 10 4 0 -18 -4 D 4 26 18 0 10 E 4 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -4 -4 B 8 0 -4 -26 -6 C 10 4 0 -18 -4 D 4 26 18 0 10 E 4 6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999971363 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8691: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) D C A E B (6) B E A C D (5) B A E D C (5) B E C A D (4) B A D E C (4) E C A D B (3) E B C A D (3) E A B C D (3) D C B E A (3) D C A B E (3) B D A C E (3) B A E C D (3) A D C B E (3) E C B A D (2) D C B A E (2) D A C B E (2) C E D B A (2) C E D A B (2) C D E A B (2) B D C E A (2) B D A E C (2) B A D C E (2) A B E D C (2) E C D B A (1) E B A C D (1) E A C D B (1) E A C B D (1) D C E B A (1) D C E A B (1) D B C A E (1) D A B C E (1) C E A D B (1) C D E B A (1) C D B E A (1) C D A E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E A D C (1) B D C A E (1) A E B D C (1) A E B C D (1) A D E C B (1) A D B C E (1) A C D E B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 10 18 12 B 6 0 2 2 12 C -10 -2 0 -18 2 D -18 -2 18 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 18 12 B 6 0 2 2 12 C -10 -2 0 -18 2 D -18 -2 18 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=20 A=20 E=15 C=10 so C is eliminated. Round 2 votes counts: B=35 D=25 E=20 A=20 so E is eliminated. Round 3 votes counts: B=41 D=30 A=29 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:217 B:211 D:203 C:186 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 18 12 B 6 0 2 2 12 C -10 -2 0 -18 2 D -18 -2 18 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 18 12 B 6 0 2 2 12 C -10 -2 0 -18 2 D -18 -2 18 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 18 12 B 6 0 2 2 12 C -10 -2 0 -18 2 D -18 -2 18 0 8 E -12 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999695 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8692: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) C A B E D (8) C A B D E (7) E D B A C (6) B C D E A (5) A E D C B (5) A E D B C (5) C B E D A (4) B D E C A (4) B D E A C (4) D B E A C (3) C B D E A (3) C A E D B (3) A C D E B (3) E D A B C (2) E B D C A (2) E A D C B (2) C B E A D (2) C B A D E (2) C A E B D (2) A E C D B (2) A C D B E (2) E D C B A (1) E D C A B (1) E A D B C (1) D E B A C (1) D E A B C (1) D A B E C (1) C E B A D (1) C B D A E (1) C A D E B (1) B D A C E (1) B A C D E (1) A D E B C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 18 6 20 14 B -18 0 -22 -4 0 C -6 22 0 16 14 D -20 4 -16 0 -14 E -14 0 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 6 20 14 B -18 0 -22 -4 0 C -6 22 0 16 14 D -20 4 -16 0 -14 E -14 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=30 E=15 B=15 D=6 so D is eliminated. Round 2 votes counts: C=34 A=31 B=18 E=17 so E is eliminated. Round 3 votes counts: A=37 C=36 B=27 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 C:223 E:193 B:178 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 6 20 14 B -18 0 -22 -4 0 C -6 22 0 16 14 D -20 4 -16 0 -14 E -14 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 6 20 14 B -18 0 -22 -4 0 C -6 22 0 16 14 D -20 4 -16 0 -14 E -14 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 6 20 14 B -18 0 -22 -4 0 C -6 22 0 16 14 D -20 4 -16 0 -14 E -14 0 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8693: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (13) B A C E D (8) E D C B A (7) E D B A C (7) A B C D E (7) E D B C A (6) C A B D E (6) D E C B A (5) A C B D E (5) E B D A C (4) D C E A B (4) B E A D C (4) B A E C D (4) A C D B E (3) C D A E B (2) C A D B E (2) A B C E D (2) E B D C A (1) E A D B C (1) D E A C B (1) D C A E B (1) C B A D E (1) C A D E B (1) B E D C A (1) B C A E D (1) B C A D E (1) B A E D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 -4 -4 -8 B 2 0 -2 -8 -8 C 4 2 0 -14 -12 D 4 8 14 0 6 E 8 8 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -8 B 2 0 -2 -8 -8 C 4 2 0 -14 -12 D 4 8 14 0 6 E 8 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=24 B=20 A=18 C=12 so C is eliminated. Round 2 votes counts: A=27 E=26 D=26 B=21 so B is eliminated. Round 3 votes counts: A=43 E=31 D=26 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:216 E:211 B:192 A:191 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -8 B 2 0 -2 -8 -8 C 4 2 0 -14 -12 D 4 8 14 0 6 E 8 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -8 B 2 0 -2 -8 -8 C 4 2 0 -14 -12 D 4 8 14 0 6 E 8 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -8 B 2 0 -2 -8 -8 C 4 2 0 -14 -12 D 4 8 14 0 6 E 8 8 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999751 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8694: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (11) B A D E C (11) A B C D E (11) D E C B A (9) E D C B A (6) E C D B A (6) A C B E D (5) A B C E D (5) D B E C A (3) C E A D B (3) C D E A B (3) B D A E C (3) E D B C A (2) B D E C A (2) B D E A C (2) A C E D B (2) A B D C E (2) E C B D A (1) E B D C A (1) D E C A B (1) D E B C A (1) D B E A C (1) D A C E B (1) C A E D B (1) C A E B D (1) B E D C A (1) B E A C D (1) B A E D C (1) B A E C D (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 -4 -8 -8 B 4 0 -4 -2 0 C 4 4 0 6 -6 D 8 2 -6 0 2 E 8 0 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 A B C D E A 0 -4 -4 -8 -8 B 4 0 -4 -2 0 C 4 4 0 6 -6 D 8 2 -6 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=22 C=19 E=16 D=16 so E is eliminated. Round 2 votes counts: A=27 C=26 D=24 B=23 so B is eliminated. Round 3 votes counts: A=41 D=33 C=26 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:206 C:204 D:203 B:199 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 -8 -8 B 4 0 -4 -2 0 C 4 4 0 6 -6 D 8 2 -6 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -8 -8 B 4 0 -4 -2 0 C 4 4 0 6 -6 D 8 2 -6 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -8 -8 B 4 0 -4 -2 0 C 4 4 0 6 -6 D 8 2 -6 0 2 E 8 0 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102037 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8695: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (10) D E C B A (9) A B C E D (9) B C E A D (7) A D C E B (7) A B D E C (7) C E B D A (6) B C E D A (6) A D B E C (6) A D E C B (5) D E C A B (4) D C E B A (3) D A E C B (3) B E C D A (3) B E C A D (3) D C E A B (2) C E D B A (2) A C B E D (2) E C B D A (1) D B E C A (1) D A B E C (1) B A E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 18 6 B 4 0 12 14 16 C -6 -12 0 0 10 D -18 -14 0 0 -2 E -6 -16 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 18 6 B 4 0 12 14 16 C -6 -12 0 0 10 D -18 -14 0 0 -2 E -6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=30 D=23 C=8 E=1 so E is eliminated. Round 2 votes counts: A=38 B=30 D=23 C=9 so C is eliminated. Round 3 votes counts: A=38 B=37 D=25 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 A:213 C:196 E:185 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 18 6 B 4 0 12 14 16 C -6 -12 0 0 10 D -18 -14 0 0 -2 E -6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 18 6 B 4 0 12 14 16 C -6 -12 0 0 10 D -18 -14 0 0 -2 E -6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 18 6 B 4 0 12 14 16 C -6 -12 0 0 10 D -18 -14 0 0 -2 E -6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8696: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (11) A D C B E (10) A D E C B (9) B C E D A (7) B C A E D (5) A E D B C (4) E D A B C (3) D E C B A (3) D E A C B (3) D C A B E (3) D A C E B (3) C B D A E (3) B C A D E (3) A D E B C (3) E D C B A (2) E D A C B (2) E A D B C (2) D C B A E (2) D A E C B (2) B E C A D (2) B C E A D (2) A B E C D (2) A B C D E (2) E D B C A (1) E B C D A (1) D C E B A (1) D C B E A (1) D A C B E (1) C D B E A (1) C D B A E (1) C B E D A (1) C B A D E (1) B E C D A (1) A E B C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -4 -6 12 B -2 0 -20 -14 20 C 4 20 0 -10 18 D 6 14 10 0 26 E -12 -20 -18 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 -6 12 B -2 0 -20 -14 20 C 4 20 0 -10 18 D 6 14 10 0 26 E -12 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=20 D=19 C=18 E=11 so E is eliminated. Round 2 votes counts: A=34 D=27 B=21 C=18 so C is eliminated. Round 3 votes counts: B=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:228 C:216 A:202 B:192 E:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -6 12 B -2 0 -20 -14 20 C 4 20 0 -10 18 D 6 14 10 0 26 E -12 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -6 12 B -2 0 -20 -14 20 C 4 20 0 -10 18 D 6 14 10 0 26 E -12 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -6 12 B -2 0 -20 -14 20 C 4 20 0 -10 18 D 6 14 10 0 26 E -12 -20 -18 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8697: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (14) D E A C B (11) E A C D B (9) B D C A E (9) D B E A C (8) B C A E D (8) A C E D B (7) E C A B D (5) C A B E D (5) B D E C A (5) E D A C B (4) D B A C E (3) C A E D B (3) B C A D E (3) B E D C A (1) B E C A D (1) B D E A C (1) B D A E C (1) B D A C E (1) A E C D B (1) Total count = 100 A B C D E A 0 18 -8 12 10 B -18 0 -18 8 -8 C 8 18 0 12 6 D -12 -8 -12 0 -16 E -10 8 -6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -8 12 10 B -18 0 -18 8 -8 C 8 18 0 12 6 D -12 -8 -12 0 -16 E -10 8 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=22 C=22 E=18 A=8 so A is eliminated. Round 2 votes counts: B=30 C=29 D=22 E=19 so E is eliminated. Round 3 votes counts: C=44 B=30 D=26 so D is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:216 E:204 B:182 D:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -8 12 10 B -18 0 -18 8 -8 C 8 18 0 12 6 D -12 -8 -12 0 -16 E -10 8 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 12 10 B -18 0 -18 8 -8 C 8 18 0 12 6 D -12 -8 -12 0 -16 E -10 8 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 12 10 B -18 0 -18 8 -8 C 8 18 0 12 6 D -12 -8 -12 0 -16 E -10 8 -6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8698: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) E A D B C (6) E A C D B (6) E A C B D (6) A E D C B (6) A E C D B (6) A E C B D (6) E A D C B (5) D B A E C (5) D B E C A (4) D B C E A (4) B C D A E (4) D B C A E (3) D B A C E (3) C B D E A (3) C B D A E (3) A D E B C (3) D E A B C (2) C E A B D (2) C A E B D (2) B D C E A (2) B C D E A (2) E C A D B (1) E C A B D (1) D E B C A (1) D E B A C (1) D B E A C (1) D A E B C (1) C E B D A (1) C B E D A (1) C B A D E (1) B D C A E (1) Total count = 100 A B C D E A 0 20 28 16 2 B -20 0 0 -30 -26 C -28 0 0 -10 -36 D -16 30 10 0 -12 E -2 26 36 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999587 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 28 16 2 B -20 0 0 -30 -26 C -28 0 0 -10 -36 D -16 30 10 0 -12 E -2 26 36 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=25 D=25 C=13 B=9 so B is eliminated. Round 2 votes counts: D=28 A=28 E=25 C=19 so C is eliminated. Round 3 votes counts: D=40 A=31 E=29 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:236 A:233 D:206 C:163 B:162 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 28 16 2 B -20 0 0 -30 -26 C -28 0 0 -10 -36 D -16 30 10 0 -12 E -2 26 36 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 28 16 2 B -20 0 0 -30 -26 C -28 0 0 -10 -36 D -16 30 10 0 -12 E -2 26 36 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 28 16 2 B -20 0 0 -30 -26 C -28 0 0 -10 -36 D -16 30 10 0 -12 E -2 26 36 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999976883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8699: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (15) A C E D B (15) A C E B D (10) D B A C E (9) D B E C A (7) E C A B D (6) D A C E B (5) C A E B D (4) B E D C A (4) B E C A D (4) A C D E B (4) E B C A D (3) D B E A C (3) E A C B D (2) D B C A E (2) D A C B E (2) C E A B D (2) D C A B E (1) D B C E A (1) B D E A C (1) Total count = 100 A B C D E A 0 2 2 0 4 B -2 0 -2 2 -2 C -2 2 0 0 10 D 0 -2 0 0 0 E -4 2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.677941 B: 0.000000 C: 0.000000 D: 0.322059 E: 0.000000 Sum of squares = 0.563325679727 Cumulative probabilities = A: 0.677941 B: 0.677941 C: 0.677941 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 4 B -2 0 -2 2 -2 C -2 2 0 0 10 D 0 -2 0 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500214 B: 0.000000 C: 0.000000 D: 0.499786 E: 0.000000 Sum of squares = 0.500000091314 Cumulative probabilities = A: 0.500214 B: 0.500214 C: 0.500214 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=29 B=24 E=11 C=6 so C is eliminated. Round 2 votes counts: A=33 D=30 B=24 E=13 so E is eliminated. Round 3 votes counts: A=43 D=30 B=27 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:205 A:204 D:199 B:198 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 4 B -2 0 -2 2 -2 C -2 2 0 0 10 D 0 -2 0 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500214 B: 0.000000 C: 0.000000 D: 0.499786 E: 0.000000 Sum of squares = 0.500000091314 Cumulative probabilities = A: 0.500214 B: 0.500214 C: 0.500214 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 4 B -2 0 -2 2 -2 C -2 2 0 0 10 D 0 -2 0 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500214 B: 0.000000 C: 0.000000 D: 0.499786 E: 0.000000 Sum of squares = 0.500000091314 Cumulative probabilities = A: 0.500214 B: 0.500214 C: 0.500214 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 4 B -2 0 -2 2 -2 C -2 2 0 0 10 D 0 -2 0 0 0 E -4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500214 B: 0.000000 C: 0.000000 D: 0.499786 E: 0.000000 Sum of squares = 0.500000091314 Cumulative probabilities = A: 0.500214 B: 0.500214 C: 0.500214 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8700: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (13) A C E D B (12) B E D C A (5) E B D C A (4) C D E B A (4) A C D E B (4) E D B C A (3) C D E A B (3) B D C E A (3) B A E D C (3) A E C D B (3) A C E B D (3) A B E D C (3) D E B C A (2) D C E B A (2) D B C E A (2) C A D E B (2) B E D A C (2) B A D E C (2) A E C B D (2) A C D B E (2) A B C E D (2) E D C B A (1) E C D A B (1) E A D C B (1) E A C D B (1) D E C B A (1) D C B E A (1) C E D A B (1) C E A D B (1) C D A E B (1) C D A B E (1) B D E A C (1) B D A C E (1) B C D A E (1) B A D C E (1) A E B D C (1) A C B E D (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -6 -4 B 4 0 2 2 -6 C 4 -2 0 -6 2 D 6 -2 6 0 0 E 4 6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501229 E: 0.498771 Sum of squares = 0.500003014775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.501229 E: 1.000000 A B C D E A 0 -4 -4 -6 -4 B 4 0 2 2 -6 C 4 -2 0 -6 2 D 6 -2 6 0 0 E 4 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=32 C=13 E=11 D=8 so D is eliminated. Round 2 votes counts: A=36 B=34 C=16 E=14 so E is eliminated. Round 3 votes counts: B=43 A=38 C=19 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:205 E:204 B:201 C:199 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -4 B 4 0 2 2 -6 C 4 -2 0 -6 2 D 6 -2 6 0 0 E 4 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -4 B 4 0 2 2 -6 C 4 -2 0 -6 2 D 6 -2 6 0 0 E 4 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -4 B 4 0 2 2 -6 C 4 -2 0 -6 2 D 6 -2 6 0 0 E 4 6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8701: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (10) C A E D B (9) A C E B D (9) D B C A E (8) D B E C A (7) E B A C D (6) E A C B D (5) E A B C D (5) B E D A C (5) A E C B D (5) D C A B E (4) D B C E A (4) C A E B D (4) B D E A C (4) C D A B E (3) E B A D C (2) B E D C A (2) E A B D C (1) D C B A E (1) D A C E B (1) D A C B E (1) C D B E A (1) C D A E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 20 -8 16 16 B -20 0 -12 -4 -20 C 8 12 0 20 16 D -16 4 -20 0 -8 E -16 20 -16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -8 16 16 B -20 0 -12 -4 -20 C 8 12 0 20 16 D -16 4 -20 0 -8 E -16 20 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=26 E=19 A=16 B=11 so B is eliminated. Round 2 votes counts: D=30 C=28 E=26 A=16 so A is eliminated. Round 3 votes counts: C=39 E=31 D=30 so D is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:222 E:198 D:180 B:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -8 16 16 B -20 0 -12 -4 -20 C 8 12 0 20 16 D -16 4 -20 0 -8 E -16 20 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -8 16 16 B -20 0 -12 -4 -20 C 8 12 0 20 16 D -16 4 -20 0 -8 E -16 20 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -8 16 16 B -20 0 -12 -4 -20 C 8 12 0 20 16 D -16 4 -20 0 -8 E -16 20 -16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8702: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) B A D C E (7) D C B E A (6) D C B A E (6) A E B C D (5) E C D A B (3) E B C A D (3) E A C D B (3) D C E B A (3) C D E B A (3) B D A C E (3) A B E D C (3) A B E C D (3) E C D B A (2) E A D C B (2) E A C B D (2) D C E A B (2) D C A B E (2) D A B C E (2) C D E A B (2) C D B E A (2) C B D E A (2) B D C A E (2) B C E A D (2) B A E D C (2) A E D C B (2) A E B D C (2) E C B A D (1) E C A B D (1) E A B C D (1) D B C A E (1) D A E C B (1) C E B D A (1) B E C A D (1) B C E D A (1) B A D E C (1) B A C E D (1) B A C D E (1) A D E C B (1) A D E B C (1) A D B C E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -18 8 12 14 B 18 0 6 10 18 C -8 -6 0 -4 2 D -12 -10 4 0 2 E -14 -18 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 8 12 14 B 18 0 6 10 18 C -8 -6 0 -4 2 D -12 -10 4 0 2 E -14 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999138 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 A=20 E=18 C=10 so C is eliminated. Round 2 votes counts: B=31 D=30 A=20 E=19 so E is eliminated. Round 3 votes counts: B=36 D=35 A=29 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:208 C:192 D:192 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 8 12 14 B 18 0 6 10 18 C -8 -6 0 -4 2 D -12 -10 4 0 2 E -14 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999138 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 8 12 14 B 18 0 6 10 18 C -8 -6 0 -4 2 D -12 -10 4 0 2 E -14 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999138 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 8 12 14 B 18 0 6 10 18 C -8 -6 0 -4 2 D -12 -10 4 0 2 E -14 -18 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999138 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8703: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) A D C B E (7) A D B E C (7) C A D E B (6) E C B A D (5) D A C B E (5) D C A B E (4) D A B E C (4) C E B D A (4) C E B A D (4) C D B E A (4) E B A D C (3) D A B C E (3) C E A B D (3) B E C D A (3) A C D E B (3) E C B D A (2) E B C D A (2) E B A C D (2) D B A E C (2) C B E D A (2) B E D A C (2) B D E A C (2) A D E B C (2) A D C E B (2) E B C A D (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) B E D C A (1) B E A D C (1) A E C B D (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 6 10 0 2 B -6 0 -14 -22 10 C -10 14 0 -6 4 D 0 22 6 0 22 E -2 -10 -4 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.661847 B: 0.000000 C: 0.000000 D: 0.338153 E: 0.000000 Sum of squares = 0.552388692518 Cumulative probabilities = A: 0.661847 B: 0.661847 C: 0.661847 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 0 2 B -6 0 -14 -22 10 C -10 14 0 -6 4 D 0 22 6 0 22 E -2 -10 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 A=24 E=15 B=9 so B is eliminated. Round 2 votes counts: D=27 C=27 A=24 E=22 so E is eliminated. Round 3 votes counts: C=40 D=30 A=30 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:225 A:209 C:201 B:184 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 0 2 B -6 0 -14 -22 10 C -10 14 0 -6 4 D 0 22 6 0 22 E -2 -10 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 0 2 B -6 0 -14 -22 10 C -10 14 0 -6 4 D 0 22 6 0 22 E -2 -10 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 0 2 B -6 0 -14 -22 10 C -10 14 0 -6 4 D 0 22 6 0 22 E -2 -10 -4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8704: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) E A B D C (8) E D A C B (6) B C A E D (5) E D A B C (4) C D B A E (4) A D E B C (4) E B A C D (3) E A B C D (3) D E C A B (3) C B D A E (3) B A C E D (3) E C B A D (2) E B C A D (2) D E A C B (2) D C E B A (2) D C E A B (2) D C B A E (2) D C A B E (2) D A E B C (2) D A B C E (2) C B D E A (2) B E A C D (2) B C A D E (2) A E B C D (2) E D C B A (1) E D C A B (1) E C D B A (1) D E C B A (1) D E A B C (1) D A C E B (1) D A C B E (1) C E D B A (1) C E B D A (1) C D E B A (1) C B A E D (1) B C E A D (1) B A E C D (1) A E D B C (1) A E B D C (1) A B E C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 18 20 8 -18 B -18 0 20 -8 -30 C -20 -20 0 -14 -22 D -8 8 14 0 -22 E 18 30 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 20 8 -18 B -18 0 20 -8 -30 C -20 -20 0 -14 -22 D -8 8 14 0 -22 E 18 30 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=21 B=14 C=13 A=12 so A is eliminated. Round 2 votes counts: E=44 D=25 B=18 C=13 so C is eliminated. Round 3 votes counts: E=46 D=30 B=24 so B is eliminated. Round 4 votes counts: E=61 D=39 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:246 A:214 D:196 B:182 C:162 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 20 8 -18 B -18 0 20 -8 -30 C -20 -20 0 -14 -22 D -8 8 14 0 -22 E 18 30 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 8 -18 B -18 0 20 -8 -30 C -20 -20 0 -14 -22 D -8 8 14 0 -22 E 18 30 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 8 -18 B -18 0 20 -8 -30 C -20 -20 0 -14 -22 D -8 8 14 0 -22 E 18 30 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8705: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (5) C A D E B (5) B E A D C (5) E D B A C (4) D E B A C (4) D B E C A (4) C A D B E (4) B C A E D (4) A C E B D (4) A C B E D (4) E B D A C (3) D E B C A (3) D C E A B (3) C D A B E (3) C A B D E (3) B E A C D (3) E B A D C (2) E B A C D (2) D E C A B (2) D C B E A (2) D C A E B (2) D C A B E (2) C B A D E (2) C A E B D (2) B E D A C (2) B A E C D (2) B A C E D (2) A E C B D (2) A C E D B (2) E A D B C (1) E A B D C (1) D E C B A (1) D E A C B (1) D C E B A (1) C D B A E (1) C B A E D (1) C A E D B (1) C A B E D (1) B D E C A (1) B C A D E (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 12 10 B 0 0 -8 -2 -4 C 8 8 0 12 10 D -12 2 -12 0 0 E -10 4 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 12 10 B 0 0 -8 -2 -4 C 8 8 0 12 10 D -12 2 -12 0 0 E -10 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=25 B=20 A=14 E=13 so E is eliminated. Round 2 votes counts: D=29 C=28 B=27 A=16 so A is eliminated. Round 3 votes counts: C=40 D=30 B=30 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:207 B:193 E:192 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 12 10 B 0 0 -8 -2 -4 C 8 8 0 12 10 D -12 2 -12 0 0 E -10 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 12 10 B 0 0 -8 -2 -4 C 8 8 0 12 10 D -12 2 -12 0 0 E -10 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 12 10 B 0 0 -8 -2 -4 C 8 8 0 12 10 D -12 2 -12 0 0 E -10 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8706: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) B E D A C (8) C B A D E (7) A C D E B (7) C D A E B (6) C A D E B (5) B E A D C (5) B C A D E (5) E D A B C (4) E A D B C (4) D C A E B (3) D A E C B (3) C A D B E (3) B E C D A (3) B E C A D (3) B E A C D (3) B C A E D (3) E A B D C (2) D E A C B (2) C A B D E (2) B C D E A (2) A E D C B (2) E D A C B (1) D E B A C (1) D C E A B (1) D A C E B (1) C D A B E (1) C B D E A (1) B E D C A (1) B C E D A (1) B A C E D (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 6 6 0 B 4 0 8 10 -2 C -6 -8 0 6 0 D -6 -10 -6 0 2 E 0 2 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.302709 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.697291 Sum of squares = 0.577847642199 Cumulative probabilities = A: 0.302709 B: 0.302709 C: 0.302709 D: 0.302709 E: 1.000000 A B C D E A 0 -4 6 6 0 B 4 0 8 10 -2 C -6 -8 0 6 0 D -6 -10 -6 0 2 E 0 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555576272 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=25 E=19 D=11 A=10 so A is eliminated. Round 2 votes counts: B=35 C=32 E=21 D=12 so D is eliminated. Round 3 votes counts: C=38 B=35 E=27 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:204 E:200 C:196 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 6 0 B 4 0 8 10 -2 C -6 -8 0 6 0 D -6 -10 -6 0 2 E 0 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555576272 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 6 0 B 4 0 8 10 -2 C -6 -8 0 6 0 D -6 -10 -6 0 2 E 0 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555576272 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 6 0 B 4 0 8 10 -2 C -6 -8 0 6 0 D -6 -10 -6 0 2 E 0 2 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555576272 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8707: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (12) B E A C D (8) D B E C A (7) E C A B D (6) E A C B D (6) E B A C D (5) D B C A E (5) B D A C E (5) E C A D B (4) C A E D B (4) B D E A C (4) B D A E C (4) A C E B D (4) E B C A D (3) D E C A B (3) D C E A B (3) D C A B E (3) D B A C E (3) C A D E B (2) B E D A C (2) B A C E D (2) E A B C D (1) C D E A B (1) C D A E B (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -8 -6 -6 B -2 0 0 4 -10 C 8 0 0 -2 -8 D 6 -4 2 0 6 E 6 10 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.500000 E: 0.200000 Sum of squares = 0.379999999941 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.800000 E: 1.000000 A B C D E A 0 2 -8 -6 -6 B -2 0 0 4 -10 C 8 0 0 -2 -8 D 6 -4 2 0 6 E 6 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.500000 E: 0.200000 Sum of squares = 0.379999999999 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=26 E=25 C=8 A=5 so A is eliminated. Round 2 votes counts: D=36 B=26 E=25 C=13 so C is eliminated. Round 3 votes counts: D=40 E=33 B=27 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:209 D:205 C:199 B:196 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -6 -6 B -2 0 0 4 -10 C 8 0 0 -2 -8 D 6 -4 2 0 6 E 6 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.500000 E: 0.200000 Sum of squares = 0.379999999999 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -6 -6 B -2 0 0 4 -10 C 8 0 0 -2 -8 D 6 -4 2 0 6 E 6 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.500000 E: 0.200000 Sum of squares = 0.379999999999 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -6 -6 B -2 0 0 4 -10 C 8 0 0 -2 -8 D 6 -4 2 0 6 E 6 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.300000 C: 0.000000 D: 0.500000 E: 0.200000 Sum of squares = 0.379999999999 Cumulative probabilities = A: 0.000000 B: 0.300000 C: 0.300000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8708: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (8) C D A E B (7) C D A B E (7) B E C A D (7) D C B A E (5) B C D E A (5) E A B D C (4) B E A D C (4) A E C D B (4) B E D A C (3) A E D C B (3) A E D B C (3) E A B C D (2) D C A B E (2) D A E C B (2) D A E B C (2) D A C E B (2) C B D E A (2) C B D A E (2) C A E D B (2) B E A C D (2) B D C E A (2) B C E A D (2) A D E C B (2) E A D B C (1) E A C B D (1) D C A E B (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A E C (1) C B E A D (1) C A D E B (1) B E D C A (1) B E C D A (1) B D E A C (1) B C E D A (1) A E C B D (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -18 -16 16 B 2 0 -10 -16 20 C 18 10 0 14 8 D 16 16 -14 0 12 E -16 -20 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -18 -16 16 B 2 0 -10 -16 20 C 18 10 0 14 8 D 16 16 -14 0 12 E -16 -20 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=29 D=18 A=15 E=8 so E is eliminated. Round 2 votes counts: C=30 B=29 A=23 D=18 so D is eliminated. Round 3 votes counts: C=38 B=33 A=29 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 D:215 B:198 A:190 E:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -18 -16 16 B 2 0 -10 -16 20 C 18 10 0 14 8 D 16 16 -14 0 12 E -16 -20 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -18 -16 16 B 2 0 -10 -16 20 C 18 10 0 14 8 D 16 16 -14 0 12 E -16 -20 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -18 -16 16 B 2 0 -10 -16 20 C 18 10 0 14 8 D 16 16 -14 0 12 E -16 -20 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8709: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) E D B A C (7) E D A B C (7) D B E C A (7) E A D B C (6) A E C B D (6) D B C E A (5) D B C A E (5) A C E B D (4) E A C D B (3) E A B D C (3) C B A D E (3) A C B E D (3) E D C B A (2) E C D A B (2) E A C B D (2) D E B C A (2) D C B A E (2) D B E A C (2) C B D A E (2) B D A E C (2) A C B D E (2) E D B C A (1) E D A C B (1) E C A D B (1) D E C B A (1) D E B A C (1) C D B A E (1) C A B E D (1) B D C A E (1) B D A C E (1) B C D A E (1) B A C D E (1) A E B C D (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 10 -6 -6 B -6 0 12 -12 0 C -10 -12 0 -12 -16 D 6 12 12 0 -4 E 6 0 16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.146779 C: 0.000000 D: 0.000000 E: 0.853221 Sum of squares = 0.749529501049 Cumulative probabilities = A: 0.000000 B: 0.146779 C: 0.146779 D: 0.146779 E: 1.000000 A B C D E A 0 6 10 -6 -6 B -6 0 12 -12 0 C -10 -12 0 -12 -16 D 6 12 12 0 -4 E 6 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000059545 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=25 A=19 C=15 B=6 so B is eliminated. Round 2 votes counts: E=35 D=29 A=20 C=16 so C is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:213 E:213 A:202 B:197 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 10 -6 -6 B -6 0 12 -12 0 C -10 -12 0 -12 -16 D 6 12 12 0 -4 E 6 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000059545 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 -6 -6 B -6 0 12 -12 0 C -10 -12 0 -12 -16 D 6 12 12 0 -4 E 6 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000059545 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 -6 -6 B -6 0 12 -12 0 C -10 -12 0 -12 -16 D 6 12 12 0 -4 E 6 0 16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000059545 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8710: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (16) E A D B C (13) C B D A E (11) A E B D C (9) C B D E A (6) A E C B D (6) D B E A C (4) A E C D B (4) D C B E A (3) A E B C D (3) E A D C B (2) D E B A C (2) D B C E A (2) C D B E A (2) B D C E A (2) A E D C B (2) A C E B D (2) E D A B C (1) E A B D C (1) D C E B A (1) D B E C A (1) C B A D E (1) C A E B D (1) C A B E D (1) C A B D E (1) B D C A E (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 26 32 28 20 B -26 0 14 -6 -26 C -32 -14 0 -22 -30 D -28 6 22 0 -26 E -20 26 30 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 32 28 20 B -26 0 14 -6 -26 C -32 -14 0 -22 -30 D -28 6 22 0 -26 E -20 26 30 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 C=23 E=17 D=13 B=4 so B is eliminated. Round 2 votes counts: A=43 C=24 E=17 D=16 so D is eliminated. Round 3 votes counts: A=43 C=33 E=24 so E is eliminated. Round 4 votes counts: A=66 C=34 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:253 E:231 D:187 B:178 C:151 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 32 28 20 B -26 0 14 -6 -26 C -32 -14 0 -22 -30 D -28 6 22 0 -26 E -20 26 30 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 32 28 20 B -26 0 14 -6 -26 C -32 -14 0 -22 -30 D -28 6 22 0 -26 E -20 26 30 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 32 28 20 B -26 0 14 -6 -26 C -32 -14 0 -22 -30 D -28 6 22 0 -26 E -20 26 30 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8711: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (15) C D B A E (8) E A D C B (6) D C A E B (6) C B D A E (6) E B A D C (5) B A C D E (5) D C A B E (4) B E A C D (4) B C A D E (4) B A E C D (4) C D E B A (3) C D A E B (3) C D A B E (3) E D C A B (2) E D A C B (2) E A D B C (2) D C E A B (2) B E C A D (2) B A E D C (2) A E B D C (2) E C D B A (1) E B C A D (1) E B A C D (1) D A C B E (1) C D E A B (1) B E C D A (1) B C D A E (1) B A D C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 4 12 4 B -2 0 4 12 -4 C -4 -4 0 -4 -2 D -12 -12 4 0 -2 E -4 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 12 4 B -2 0 4 12 -4 C -4 -4 0 -4 -2 D -12 -12 4 0 -2 E -4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997116 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=24 B=24 D=13 A=4 so A is eliminated. Round 2 votes counts: E=37 B=26 C=24 D=13 so D is eliminated. Round 3 votes counts: E=37 C=37 B=26 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:211 B:205 E:202 C:193 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 12 4 B -2 0 4 12 -4 C -4 -4 0 -4 -2 D -12 -12 4 0 -2 E -4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997116 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 12 4 B -2 0 4 12 -4 C -4 -4 0 -4 -2 D -12 -12 4 0 -2 E -4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997116 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 12 4 B -2 0 4 12 -4 C -4 -4 0 -4 -2 D -12 -12 4 0 -2 E -4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997116 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8712: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (13) A C B E D (12) E D B A C (8) C A B E D (8) C A B D E (8) D E B A C (6) A C E D B (5) E D A B C (3) D B E C A (3) C B A D E (3) A C E B D (3) B D C A E (2) B C D A E (2) B A C E D (2) A E C B D (2) A B E C D (2) E D A C B (1) E B D A C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E C B A (1) D C B A E (1) D C A E B (1) C D A B E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A C D (1) B D C E A (1) B C A E D (1) B C A D E (1) A E C D B (1) A E B C D (1) Total count = 100 A B C D E A 0 6 4 10 16 B -6 0 -2 2 -2 C -4 2 0 12 6 D -10 -2 -12 0 -12 E -16 2 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 10 16 B -6 0 -2 2 -2 C -4 2 0 12 6 D -10 -2 -12 0 -12 E -16 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 C=22 E=16 B=11 so B is eliminated. Round 2 votes counts: D=28 A=28 C=26 E=18 so E is eliminated. Round 3 votes counts: D=42 A=32 C=26 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:208 B:196 E:196 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 10 16 B -6 0 -2 2 -2 C -4 2 0 12 6 D -10 -2 -12 0 -12 E -16 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 10 16 B -6 0 -2 2 -2 C -4 2 0 12 6 D -10 -2 -12 0 -12 E -16 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 10 16 B -6 0 -2 2 -2 C -4 2 0 12 6 D -10 -2 -12 0 -12 E -16 2 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8713: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) C A B E D (11) E D C B A (7) C E B A D (5) B A D E C (5) A B D E C (5) E C D B A (4) D A B E C (4) C E B D A (4) C A B D E (4) A B D C E (4) A B C D E (4) E D B C A (3) C E A B D (3) C B A E D (3) D B E A C (2) D B A E C (2) C E D B A (2) C E D A B (2) C A E D B (2) C A D B E (2) A C B D E (2) E D C A B (1) E D B A C (1) D E A B C (1) B E C A D (1) B D A E C (1) B C E A D (1) B A E C D (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -10 10 4 B 8 0 -4 10 8 C 10 4 0 4 2 D -10 -10 -4 0 -2 E -4 -8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -10 10 4 B 8 0 -4 10 8 C 10 4 0 4 2 D -10 -10 -4 0 -2 E -4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=20 E=16 A=16 B=10 so B is eliminated. Round 2 votes counts: C=39 A=23 D=21 E=17 so E is eliminated. Round 3 votes counts: C=44 D=33 A=23 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:210 A:198 E:194 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -10 10 4 B 8 0 -4 10 8 C 10 4 0 4 2 D -10 -10 -4 0 -2 E -4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 10 4 B 8 0 -4 10 8 C 10 4 0 4 2 D -10 -10 -4 0 -2 E -4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 10 4 B 8 0 -4 10 8 C 10 4 0 4 2 D -10 -10 -4 0 -2 E -4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998824 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8714: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) B A D E C (7) E C D B A (5) C D E A B (5) E D C B A (4) C D A E B (4) B A E C D (4) B A C E D (4) E C D A B (3) E B C A D (3) D C E A B (3) C E D A B (3) B E A D C (3) B E A C D (3) A D C B E (3) E C B A D (2) E B A C D (2) D E C A B (2) D A C B E (2) D A B C E (2) C E B A D (2) B A E D C (2) A B C D E (2) E D C A B (1) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B C D A (1) E B A D C (1) D E B A C (1) D C A E B (1) D C A B E (1) D A B E C (1) C E A B D (1) C D A B E (1) C A D B E (1) C A B D E (1) B E C A D (1) B D A E C (1) B A D C E (1) Total count = 100 A B C D E A 0 -2 4 10 0 B 2 0 8 12 4 C -4 -8 0 -2 -2 D -10 -12 2 0 2 E 0 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 10 0 B 2 0 8 12 4 C -4 -8 0 -2 -2 D -10 -12 2 0 2 E 0 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996164 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=26 B=26 C=18 A=17 D=13 so D is eliminated. Round 2 votes counts: E=29 B=26 C=23 A=22 so A is eliminated. Round 3 votes counts: B=43 E=29 C=28 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:206 E:198 C:192 D:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 10 0 B 2 0 8 12 4 C -4 -8 0 -2 -2 D -10 -12 2 0 2 E 0 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996164 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 0 B 2 0 8 12 4 C -4 -8 0 -2 -2 D -10 -12 2 0 2 E 0 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996164 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 0 B 2 0 8 12 4 C -4 -8 0 -2 -2 D -10 -12 2 0 2 E 0 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996164 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8715: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (17) B C D A E (10) C D A E B (7) B E A D C (6) B A D C E (6) E A B D C (5) C D E A B (5) B A E D C (5) A E D C B (4) D C A E B (3) B C D E A (3) E C D A B (2) E B A D C (2) C E D B A (2) C D B A E (2) C D A B E (2) C B D A E (2) B E C D A (2) B E A C D (2) A B D C E (2) E C A D B (1) C B D E A (1) B E C A D (1) B D C A E (1) B C E D A (1) B C E A D (1) A E D B C (1) A E B D C (1) A D C E B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 8 14 -2 B -6 0 2 6 -2 C -8 -2 0 -12 -2 D -14 -6 12 0 -8 E 2 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 8 14 -2 B -6 0 2 6 -2 C -8 -2 0 -12 -2 D -14 -6 12 0 -8 E 2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=27 C=21 A=11 D=3 so D is eliminated. Round 2 votes counts: B=38 E=27 C=24 A=11 so A is eliminated. Round 3 votes counts: B=42 E=33 C=25 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:213 E:207 B:200 D:192 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 14 -2 B -6 0 2 6 -2 C -8 -2 0 -12 -2 D -14 -6 12 0 -8 E 2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 14 -2 B -6 0 2 6 -2 C -8 -2 0 -12 -2 D -14 -6 12 0 -8 E 2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 14 -2 B -6 0 2 6 -2 C -8 -2 0 -12 -2 D -14 -6 12 0 -8 E 2 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8716: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (8) A C E B D (8) E B D A C (5) C D A E B (5) B E A D C (5) D E B C A (4) D C E A B (4) D B C E A (4) C A D E B (4) B E D A C (4) A C B E D (4) E B A D C (3) D B E C A (3) C D A B E (3) C A D B E (3) C A B D E (3) B D E A C (3) E D B A C (2) E A B D C (2) D E C B A (2) B D C E A (2) A C E D B (2) A B C E D (2) E B A C D (1) E A B C D (1) D C B E A (1) D C B A E (1) D C A E B (1) D C A B E (1) C D B A E (1) C B A D E (1) C A E D B (1) C A B E D (1) B D C A E (1) B C D A E (1) B A E C D (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -10 -12 -8 B 6 0 6 16 6 C 10 -6 0 -12 8 D 12 -16 12 0 12 E 8 -6 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -12 -8 B 6 0 6 16 6 C 10 -6 0 -12 8 D 12 -16 12 0 12 E 8 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=22 D=21 A=18 E=14 so E is eliminated. Round 2 votes counts: B=34 D=23 C=22 A=21 so A is eliminated. Round 3 votes counts: B=40 C=37 D=23 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:210 C:200 E:191 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 -12 -8 B 6 0 6 16 6 C 10 -6 0 -12 8 D 12 -16 12 0 12 E 8 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -12 -8 B 6 0 6 16 6 C 10 -6 0 -12 8 D 12 -16 12 0 12 E 8 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -12 -8 B 6 0 6 16 6 C 10 -6 0 -12 8 D 12 -16 12 0 12 E 8 -6 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8717: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (11) C B D A E (8) C D B E A (6) B D A E C (5) E D A B C (4) E C A D B (4) D B E A C (4) D B C E A (4) B D C A E (4) E C D A B (3) E A C D B (3) C B A D E (3) C A B D E (3) A E B D C (3) E D C B A (2) D C B E A (2) C A B E D (2) A E C B D (2) A C E B D (2) A C B D E (2) A B E D C (2) A B D E C (2) E D B A C (1) E C D B A (1) E A D B C (1) E A B D C (1) D E B C A (1) D E B A C (1) D B E C A (1) D B C A E (1) D B A C E (1) C E D B A (1) C E A D B (1) C E A B D (1) C D B A E (1) B C A D E (1) B A D E C (1) B A D C E (1) A E B C D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -22 -2 16 B 0 0 -18 14 12 C 22 18 0 16 14 D 2 -14 -16 0 6 E -16 -12 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -22 -2 16 B 0 0 -18 14 12 C 22 18 0 16 14 D 2 -14 -16 0 6 E -16 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=20 A=16 D=15 B=12 so B is eliminated. Round 2 votes counts: C=38 D=24 E=20 A=18 so A is eliminated. Round 3 votes counts: C=44 E=28 D=28 so E is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:235 B:204 A:196 D:189 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -22 -2 16 B 0 0 -18 14 12 C 22 18 0 16 14 D 2 -14 -16 0 6 E -16 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -22 -2 16 B 0 0 -18 14 12 C 22 18 0 16 14 D 2 -14 -16 0 6 E -16 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -22 -2 16 B 0 0 -18 14 12 C 22 18 0 16 14 D 2 -14 -16 0 6 E -16 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8718: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) D C E B A (8) A B E C D (7) E B C A D (5) D C A B E (5) D A E C B (5) D A C B E (5) E B C D A (4) A B C D E (4) E D C B A (3) E D B C A (3) E C D B A (3) E C B D A (3) E A B C D (3) C B D E A (3) B A C E D (3) A D C B E (3) A B C E D (3) D C B A E (2) B E A C D (2) A C B D E (2) E D B A C (1) E D A B C (1) E A B D C (1) D E C B A (1) D E A C B (1) D C B E A (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) B C E A D (1) A E D B C (1) A E B C D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 8 -4 -6 B 10 0 2 8 -8 C -8 -2 0 14 -12 D 4 -8 -14 0 -8 E 6 8 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 8 -4 -6 B 10 0 2 8 -8 C -8 -2 0 14 -12 D 4 -8 -14 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=28 A=23 C=7 B=6 so B is eliminated. Round 2 votes counts: E=38 D=28 A=26 C=8 so C is eliminated. Round 3 votes counts: E=39 D=34 A=27 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:206 C:196 A:194 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 8 -4 -6 B 10 0 2 8 -8 C -8 -2 0 14 -12 D 4 -8 -14 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -4 -6 B 10 0 2 8 -8 C -8 -2 0 14 -12 D 4 -8 -14 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -4 -6 B 10 0 2 8 -8 C -8 -2 0 14 -12 D 4 -8 -14 0 -8 E 6 8 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8719: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) D E B A C (5) C B A D E (5) C A B E D (5) B C A E D (5) D E A C B (4) D C A B E (4) A E C B D (4) D C B A E (3) D B C E A (3) C D B A E (3) C B A E D (3) C A E B D (3) B E C A D (3) B E A C D (3) A C E B D (3) E A D C B (2) D B C A E (2) D A E C B (2) C B D A E (2) C A D B E (2) B E D A C (2) A E C D B (2) E D B A C (1) E D A C B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A D B C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E A B C (1) D C E B A (1) D C A E B (1) D B E A C (1) D A C E B (1) C D A E B (1) C D A B E (1) C A D E B (1) B D E C A (1) B D C E A (1) B C E A D (1) A E D C B (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -16 2 14 B 10 0 -18 -8 12 C 16 18 0 2 8 D -2 8 -2 0 8 E -14 -12 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -16 2 14 B 10 0 -18 -8 12 C 16 18 0 2 8 D -2 8 -2 0 8 E -14 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996185 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=26 B=16 A=13 E=10 so E is eliminated. Round 2 votes counts: D=37 C=26 B=19 A=18 so A is eliminated. Round 3 votes counts: D=42 C=37 B=21 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:206 B:198 A:195 E:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -16 2 14 B 10 0 -18 -8 12 C 16 18 0 2 8 D -2 8 -2 0 8 E -14 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996185 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -16 2 14 B 10 0 -18 -8 12 C 16 18 0 2 8 D -2 8 -2 0 8 E -14 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996185 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -16 2 14 B 10 0 -18 -8 12 C 16 18 0 2 8 D -2 8 -2 0 8 E -14 -12 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996185 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8720: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) D B E C A (7) A C E B D (7) A E C D B (6) A B E C D (6) A B C E D (6) C E A D B (5) B D C E A (5) E C A D B (4) B D A C E (4) B A C E D (4) D C E A B (3) D B C E A (3) D E C A B (2) D B E A C (2) C E D A B (2) C D B E A (2) B D C A E (2) B A E D C (2) B A E C D (2) A E C B D (2) A C E D B (2) E D C A B (1) E C D A B (1) E A C D B (1) D C E B A (1) C E A B D (1) C A E B D (1) C A B E D (1) B D E C A (1) B D E A C (1) B A D E C (1) B A D C E (1) B A C D E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 16 10 16 B -6 0 14 16 22 C -16 -14 0 10 4 D -10 -16 -10 0 -10 E -16 -22 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999227 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 16 10 16 B -6 0 14 16 22 C -16 -14 0 10 4 D -10 -16 -10 0 -10 E -16 -22 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=31 D=18 C=12 E=7 so E is eliminated. Round 2 votes counts: B=32 A=32 D=19 C=17 so C is eliminated. Round 3 votes counts: A=44 B=32 D=24 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 B:223 C:192 E:184 D:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 10 16 B -6 0 14 16 22 C -16 -14 0 10 4 D -10 -16 -10 0 -10 E -16 -22 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 10 16 B -6 0 14 16 22 C -16 -14 0 10 4 D -10 -16 -10 0 -10 E -16 -22 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 10 16 B -6 0 14 16 22 C -16 -14 0 10 4 D -10 -16 -10 0 -10 E -16 -22 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999577 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8721: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (5) B C E D A (5) A D E C B (5) A B C D E (5) E D A C B (4) B E C A D (4) A D C B E (4) A D B C E (4) E C D B A (3) E B C D A (3) E A D B C (3) D E C A B (3) C B E D A (3) B C E A D (3) B C A E D (3) B C A D E (3) B A C E D (3) B A C D E (3) A D E B C (3) E B A D C (2) D E A C B (2) D A E C B (2) D A C E B (2) B E C D A (2) B A E C D (2) A E D B C (2) A B E D C (2) A B D C E (2) E D C B A (1) E B D A C (1) E A D C B (1) D C E A B (1) D C A E B (1) C E D B A (1) C D B E A (1) C B A D E (1) B C D A E (1) A D C E B (1) A C B D E (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 12 18 4 B -10 0 16 2 8 C -12 -16 0 -4 -4 D -18 -2 4 0 -8 E -4 -8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 18 4 B -10 0 16 2 8 C -12 -16 0 -4 -4 D -18 -2 4 0 -8 E -4 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=29 E=23 D=11 C=6 so C is eliminated. Round 2 votes counts: B=33 A=31 E=24 D=12 so D is eliminated. Round 3 votes counts: A=36 B=34 E=30 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:208 E:200 D:188 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 18 4 B -10 0 16 2 8 C -12 -16 0 -4 -4 D -18 -2 4 0 -8 E -4 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 18 4 B -10 0 16 2 8 C -12 -16 0 -4 -4 D -18 -2 4 0 -8 E -4 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 18 4 B -10 0 16 2 8 C -12 -16 0 -4 -4 D -18 -2 4 0 -8 E -4 -8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8722: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D C A B E (7) B E A D C (7) B A E D C (7) C D B E A (6) C D A E B (6) E B C A D (4) E B A D C (4) D C A E B (4) C D E A B (4) A B E D C (4) C E D A B (3) C D E B A (3) C D B A E (3) B E A C D (3) E A B D C (2) D C B A E (2) D A C E B (2) D A B C E (2) C B E D A (2) C B D E A (2) B D A C E (2) E C B A D (1) E C A D B (1) E A C D B (1) D A C B E (1) C D A B E (1) B E C A D (1) B C E D A (1) B A D E C (1) A E D B C (1) A E B D C (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -2 -2 -6 B 14 0 0 2 6 C 2 0 0 0 2 D 2 -2 0 0 -2 E 6 -6 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.350611 C: 0.649389 D: 0.000000 E: 0.000000 Sum of squares = 0.544634335535 Cumulative probabilities = A: 0.000000 B: 0.350611 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -2 -6 B 14 0 0 2 6 C 2 0 0 0 2 D 2 -2 0 0 -2 E 6 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=22 E=21 D=18 A=9 so A is eliminated. Round 2 votes counts: C=30 B=27 E=23 D=20 so D is eliminated. Round 3 votes counts: C=46 B=29 E=25 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:202 E:200 D:199 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -2 -6 B 14 0 0 2 6 C 2 0 0 0 2 D 2 -2 0 0 -2 E 6 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -2 -6 B 14 0 0 2 6 C 2 0 0 0 2 D 2 -2 0 0 -2 E 6 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -2 -6 B 14 0 0 2 6 C 2 0 0 0 2 D 2 -2 0 0 -2 E 6 -6 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8723: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (15) A D C B E (9) A B C D E (7) E A B C D (6) E B A C D (5) A B E C D (5) E B C D A (4) E A B D C (4) D C B E A (4) D C B A E (4) D C A B E (4) D E C A B (3) D C E B A (3) D A C B E (3) A E B C D (3) E B C A D (2) E A D B C (2) D E C B A (2) A B D C E (2) E D C A B (1) C D B E A (1) C D B A E (1) C B D A E (1) B E C A D (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C D E (1) A E D B C (1) A D E C B (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 4 8 -6 B -6 0 -8 -10 -6 C -4 8 0 -18 -12 D -8 10 18 0 -2 E 6 6 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 4 8 -6 B -6 0 -8 -10 -6 C -4 8 0 -18 -12 D -8 10 18 0 -2 E 6 6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=30 D=23 B=5 C=3 so C is eliminated. Round 2 votes counts: E=39 A=30 D=25 B=6 so B is eliminated. Round 3 votes counts: E=40 A=34 D=26 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:209 A:206 C:187 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 8 -6 B -6 0 -8 -10 -6 C -4 8 0 -18 -12 D -8 10 18 0 -2 E 6 6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 8 -6 B -6 0 -8 -10 -6 C -4 8 0 -18 -12 D -8 10 18 0 -2 E 6 6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 8 -6 B -6 0 -8 -10 -6 C -4 8 0 -18 -12 D -8 10 18 0 -2 E 6 6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997669 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8724: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (10) A D B C E (7) A B C E D (6) E C D B A (4) D A B E C (4) C A B E D (4) B E C A D (4) A C B E D (4) A B C D E (4) E C B A D (3) D E C B A (3) D E A B C (3) D A C E B (3) D A B C E (3) A B D C E (3) E C B D A (2) E B C A D (2) D E C A B (2) D A E B C (2) D A C B E (2) C E B D A (2) C D E A B (2) B C E A D (2) B A E C D (2) A D C B E (2) E B C D A (1) D E B A C (1) D E A C B (1) D C E A B (1) D C A E B (1) D B E A C (1) D B A E C (1) D A E C B (1) C E D B A (1) B E D A C (1) B D E A C (1) B A D E C (1) A D B E C (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 16 12 14 6 B -16 0 2 8 10 C -12 -2 0 8 16 D -14 -8 -8 0 4 E -6 -10 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 14 6 B -16 0 2 8 10 C -12 -2 0 8 16 D -14 -8 -8 0 4 E -6 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=29 A=29 C=19 E=12 B=11 so B is eliminated. Round 2 votes counts: A=32 D=30 C=21 E=17 so E is eliminated. Round 3 votes counts: C=37 A=32 D=31 so D is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:205 B:202 D:187 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 14 6 B -16 0 2 8 10 C -12 -2 0 8 16 D -14 -8 -8 0 4 E -6 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 14 6 B -16 0 2 8 10 C -12 -2 0 8 16 D -14 -8 -8 0 4 E -6 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 14 6 B -16 0 2 8 10 C -12 -2 0 8 16 D -14 -8 -8 0 4 E -6 -10 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8725: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (9) D C E B A (6) C D B E A (5) B C E A D (4) A B C E D (4) E B A C D (3) D E A B C (3) D C A B E (3) C B D E A (3) C A B D E (3) B E C A D (3) A E B D C (3) A C B E D (3) E B C A D (2) E A B D C (2) D E C B A (2) D E B C A (2) D E B A C (2) D E A C B (2) D A E C B (2) D A C E B (2) C D B A E (2) C D A B E (2) C B A D E (2) B E A C D (2) B A E C D (2) A C B D E (2) E D B C A (1) E D B A C (1) E B D A C (1) E B C D A (1) E B A D C (1) E A D B C (1) D C B E A (1) D C A E B (1) D A E B C (1) C B E D A (1) C B E A D (1) C A B E D (1) B E C D A (1) B C A E D (1) B A C E D (1) A E D B C (1) A E B C D (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -2 4 10 -2 B 2 0 8 14 12 C -4 -8 0 18 -2 D -10 -14 -18 0 -2 E 2 -12 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999903 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 10 -2 B 2 0 8 14 12 C -4 -8 0 18 -2 D -10 -14 -18 0 -2 E 2 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 C=20 B=14 E=13 so E is eliminated. Round 2 votes counts: D=29 A=29 B=22 C=20 so C is eliminated. Round 3 votes counts: D=38 A=33 B=29 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:218 A:205 C:202 E:197 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 4 10 -2 B 2 0 8 14 12 C -4 -8 0 18 -2 D -10 -14 -18 0 -2 E 2 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 -2 B 2 0 8 14 12 C -4 -8 0 18 -2 D -10 -14 -18 0 -2 E 2 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 -2 B 2 0 8 14 12 C -4 -8 0 18 -2 D -10 -14 -18 0 -2 E 2 -12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993617 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8726: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (7) E B D A C (6) D A B E C (6) E B A D C (5) D A B C E (5) A E B D C (5) C E B D A (4) C E B A D (4) C A E D B (4) A D C B E (4) A C D E B (4) E B C D A (3) D B E A C (3) C E A B D (3) B E D C A (3) B E D A C (3) A C D B E (3) E B C A D (2) D C A B E (2) D B A E C (2) C A D B E (2) B D E A C (2) A E C B D (2) A D E B C (2) A D B E C (2) A C E D B (2) E C B D A (1) D C B A E (1) D B C E A (1) D A C B E (1) C D A B E (1) C B E D A (1) C A D E B (1) A E C D B (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 18 20 10 18 B -18 0 2 4 -16 C -20 -2 0 -8 2 D -10 -4 8 0 -14 E -18 16 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 20 10 18 B -18 0 2 4 -16 C -20 -2 0 -8 2 D -10 -4 8 0 -14 E -18 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=27 A=27 D=21 E=17 B=8 so B is eliminated. Round 2 votes counts: C=27 A=27 E=23 D=23 so E is eliminated. Round 3 votes counts: D=35 C=33 A=32 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:233 E:205 D:190 B:186 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 20 10 18 B -18 0 2 4 -16 C -20 -2 0 -8 2 D -10 -4 8 0 -14 E -18 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 20 10 18 B -18 0 2 4 -16 C -20 -2 0 -8 2 D -10 -4 8 0 -14 E -18 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 20 10 18 B -18 0 2 4 -16 C -20 -2 0 -8 2 D -10 -4 8 0 -14 E -18 16 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997532 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8727: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) A B E C D (9) D C A B E (7) D C E B A (5) D C E A B (4) D B E C A (4) B E A D C (4) D C B A E (3) C D E A B (3) B D A C E (3) B A E C D (3) E C A B D (2) E A B C D (2) D E B C A (2) D C A E B (2) D B C A E (2) C E A D B (2) C D A E B (2) C D A B E (2) C A D E B (2) C A D B E (2) B E D A C (2) B D E A C (2) B D A E C (2) A E B C D (2) A C E B D (2) E D B C A (1) E C A D B (1) E B D C A (1) E B D A C (1) E B A D C (1) E A C B D (1) D E C B A (1) D A B C E (1) C E D A B (1) C A E B D (1) B E A C D (1) B A E D C (1) B A D C E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 0 -2 0 B -2 0 12 6 4 C 0 -12 0 -2 -4 D 2 -6 2 0 6 E 0 -4 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000002 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 0 B -2 0 12 6 4 C 0 -12 0 -2 -4 D 2 -6 2 0 6 E 0 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999926 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=19 B=19 A=16 C=15 so C is eliminated. Round 2 votes counts: D=38 E=22 A=21 B=19 so B is eliminated. Round 3 votes counts: D=45 E=29 A=26 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:202 A:200 E:197 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 -2 0 B -2 0 12 6 4 C 0 -12 0 -2 -4 D 2 -6 2 0 6 E 0 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999926 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 0 B -2 0 12 6 4 C 0 -12 0 -2 -4 D 2 -6 2 0 6 E 0 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999926 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 0 B -2 0 12 6 4 C 0 -12 0 -2 -4 D 2 -6 2 0 6 E 0 -4 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999926 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8728: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (8) E D B A C (6) D B A E C (6) B A D C E (6) E C D A B (5) D E B A C (5) C A B E D (5) B D A C E (5) E D C A B (4) C E A D B (4) E D A B C (3) D B E A C (3) C E B A D (3) C E A B D (3) E D B C A (2) E D A C B (2) E C D B A (2) E C B D A (2) E C A B D (2) D A B E C (2) C A E B D (2) B C A D E (2) A C B D E (2) A B D C E (2) A B C D E (2) E C B A D (1) D E A C B (1) D A E B C (1) C B E A D (1) C B A E D (1) C B A D E (1) C A B D E (1) B E D C A (1) B D A E C (1) A D C B E (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 16 -2 0 B 12 0 12 2 4 C -16 -12 0 -4 2 D 2 -2 4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 16 -2 0 B 12 0 12 2 4 C -16 -12 0 -4 2 D 2 -2 4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=23 C=21 D=18 A=9 so A is eliminated. Round 2 votes counts: E=29 B=27 C=24 D=20 so D is eliminated. Round 3 votes counts: B=39 E=36 C=25 so C is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:203 A:201 E:196 C:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 16 -2 0 B 12 0 12 2 4 C -16 -12 0 -4 2 D 2 -2 4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 16 -2 0 B 12 0 12 2 4 C -16 -12 0 -4 2 D 2 -2 4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 16 -2 0 B 12 0 12 2 4 C -16 -12 0 -4 2 D 2 -2 4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999779 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8729: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) B E A D C (8) E B C D A (7) B E C A D (7) A D C E B (7) B E C D A (6) A D C B E (5) D C A E B (3) C D E B A (3) B A E C D (3) A C D B E (3) E C D B A (2) E B D C A (2) E B D A C (2) C E D B A (2) C D E A B (2) C D A E B (2) C A B D E (2) B E A C D (2) B C A E D (2) B A E D C (2) B A C D E (2) A B D E C (2) A B C D E (2) E D C A B (1) E D B C A (1) E D B A C (1) D E C A B (1) D C E A B (1) C E B D A (1) C B E D A (1) B E D C A (1) B C E A D (1) B A C E D (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 4 4 -4 B 14 0 10 8 4 C -4 -10 0 2 0 D -4 -8 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 4 -4 B 14 0 10 8 4 C -4 -10 0 2 0 D -4 -8 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=22 E=16 D=14 C=13 so C is eliminated. Round 2 votes counts: B=36 A=24 D=21 E=19 so E is eliminated. Round 3 votes counts: B=48 D=28 A=24 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:203 A:195 C:194 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 4 -4 B 14 0 10 8 4 C -4 -10 0 2 0 D -4 -8 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 4 -4 B 14 0 10 8 4 C -4 -10 0 2 0 D -4 -8 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 4 -4 B 14 0 10 8 4 C -4 -10 0 2 0 D -4 -8 -2 0 -6 E 4 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999209 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8730: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (12) A E D B C (6) E C A B D (5) E C B A D (4) E A D B C (4) D B A C E (4) C B E D A (4) B D C A E (4) A E D C B (4) A D E B C (4) E A C B D (3) C B D E A (3) B D C E A (3) B C D E A (3) B C D A E (3) A E C D B (3) A D B E C (3) E C B D A (2) D B E A C (2) D B C A E (2) D A B C E (2) C B D A E (2) A E C B D (2) E D B A C (1) E D A B C (1) E B D C A (1) E A D C B (1) D B E C A (1) D B C E A (1) D B A E C (1) D A E B C (1) C E B A D (1) C B A E D (1) C B A D E (1) B D E C A (1) B C E D A (1) A D B C E (1) A C E B D (1) A C B D E (1) Total count = 100 A B C D E A 0 8 14 14 -8 B -8 0 0 -8 -12 C -14 0 0 4 -24 D -14 8 -4 0 -14 E 8 12 24 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 14 14 -8 B -8 0 0 -8 -12 C -14 0 0 4 -24 D -14 8 -4 0 -14 E 8 12 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=25 B=15 D=14 C=12 so C is eliminated. Round 2 votes counts: E=35 B=26 A=25 D=14 so D is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:229 A:214 D:188 B:186 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 14 14 -8 B -8 0 0 -8 -12 C -14 0 0 4 -24 D -14 8 -4 0 -14 E 8 12 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 14 -8 B -8 0 0 -8 -12 C -14 0 0 4 -24 D -14 8 -4 0 -14 E 8 12 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 14 -8 B -8 0 0 -8 -12 C -14 0 0 4 -24 D -14 8 -4 0 -14 E 8 12 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8731: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) C D B A E (7) E C D A B (6) D B A C E (6) C E D A B (5) C E B A D (5) E D A B C (4) E C A B D (4) B A E D C (4) E C B A D (3) D B A E C (3) D A B E C (3) C E D B A (3) C D E A B (3) C B A D E (3) B A E C D (3) B A D E C (3) B A D C E (3) A B D E C (3) D E A B C (2) C D E B A (2) C B D A E (2) A B E D C (2) E A D C B (1) E A C B D (1) E A B C D (1) D C E A B (1) D C A E B (1) D C A B E (1) D A E B C (1) D A B C E (1) C E B D A (1) C B A E D (1) B D A C E (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 -4 2 -6 0 B 4 0 0 0 -4 C -2 0 0 4 -4 D 6 0 -4 0 -4 E 0 4 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.268753 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.731247 Sum of squares = 0.606950099596 Cumulative probabilities = A: 0.268753 B: 0.268753 C: 0.268753 D: 0.268753 E: 1.000000 A B C D E A 0 -4 2 -6 0 B 4 0 0 0 -4 C -2 0 0 4 -4 D 6 0 -4 0 -4 E 0 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000000791 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=28 D=19 B=16 A=5 so A is eliminated. Round 2 votes counts: C=32 E=28 B=21 D=19 so D is eliminated. Round 3 votes counts: C=35 B=34 E=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:206 B:200 C:199 D:199 A:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -6 0 B 4 0 0 0 -4 C -2 0 0 4 -4 D 6 0 -4 0 -4 E 0 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000000791 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -6 0 B 4 0 0 0 -4 C -2 0 0 4 -4 D 6 0 -4 0 -4 E 0 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000000791 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -6 0 B 4 0 0 0 -4 C -2 0 0 4 -4 D 6 0 -4 0 -4 E 0 4 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.520000000791 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8732: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (8) B D C A E (6) B D A C E (6) E A D C B (5) E A C D B (5) D E B C A (5) D B E C A (5) D B E A C (5) A C E B D (5) C A E B D (4) B D C E A (4) B C D A E (4) D E B A C (3) D B C E A (3) C B D E A (3) A D B E C (3) E D A C B (2) E D A B C (2) D B C A E (2) A D E B C (2) A C B E D (2) A B C D E (2) E D B C A (1) E C D B A (1) E C D A B (1) E A C B D (1) D B A E C (1) C E A B D (1) C B D A E (1) C B A E D (1) C B A D E (1) B C D E A (1) B A D C E (1) A E C D B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 12 -12 4 B 8 0 14 6 6 C -12 -14 0 -14 -2 D 12 -6 14 0 20 E -4 -6 2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 -12 4 B 8 0 14 6 6 C -12 -14 0 -14 -2 D 12 -6 14 0 20 E -4 -6 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=24 B=22 E=18 C=11 so C is eliminated. Round 2 votes counts: A=29 B=28 D=24 E=19 so E is eliminated. Round 3 votes counts: A=41 D=31 B=28 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:220 B:217 A:198 E:186 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 -12 4 B 8 0 14 6 6 C -12 -14 0 -14 -2 D 12 -6 14 0 20 E -4 -6 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 -12 4 B 8 0 14 6 6 C -12 -14 0 -14 -2 D 12 -6 14 0 20 E -4 -6 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 -12 4 B 8 0 14 6 6 C -12 -14 0 -14 -2 D 12 -6 14 0 20 E -4 -6 2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999218 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8733: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) B C E A D (10) E C B D A (7) C E B D A (7) D A E C B (6) D A B E C (6) A D B C E (6) D A C E B (5) B E C A D (5) A D B E C (5) E B C D A (4) A D C E B (4) E B D A C (3) E D A C B (2) C E D B A (2) C E D A B (2) C E B A D (2) B C A D E (2) A B D C E (2) E D C A B (1) E C D A B (1) D E A C B (1) C D A E B (1) C B A E D (1) C A D B E (1) B D A E C (1) B C A E D (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -14 2 -14 B 12 0 -8 12 4 C 14 8 0 12 14 D -2 -12 -12 0 -16 E 14 -4 -14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 2 -14 B 12 0 -8 12 4 C 14 8 0 12 14 D -2 -12 -12 0 -16 E 14 -4 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=20 E=18 D=18 A=18 so E is eliminated. Round 2 votes counts: C=34 B=27 D=21 A=18 so A is eliminated. Round 3 votes counts: D=37 C=34 B=29 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:210 E:206 A:181 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 2 -14 B 12 0 -8 12 4 C 14 8 0 12 14 D -2 -12 -12 0 -16 E 14 -4 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 2 -14 B 12 0 -8 12 4 C 14 8 0 12 14 D -2 -12 -12 0 -16 E 14 -4 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 2 -14 B 12 0 -8 12 4 C 14 8 0 12 14 D -2 -12 -12 0 -16 E 14 -4 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8734: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (7) B D C A E (5) E A C D B (4) D B A E C (4) B C D E A (4) A E C D B (4) A C E B D (4) E D C A B (3) E D A C B (3) D E A B C (3) C E A D B (3) B D C E A (3) B D A E C (3) B C D A E (3) A C B E D (3) A B E D C (3) A B D E C (3) E A D C B (2) D E C B A (2) D E C A B (2) D E B A C (2) D A B E C (2) C E D B A (2) C B E A D (2) C B A E D (2) C A E B D (2) B A D C E (2) A E C B D (2) A B C E D (2) E C A D B (1) D E B C A (1) D B E C A (1) D B C E A (1) C E D A B (1) C D E B A (1) C B D E A (1) C A B E D (1) B C A E D (1) B A D E C (1) B A C D E (1) A E D C B (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 16 -8 2 B -2 0 8 -2 12 C -16 -8 0 -10 -10 D 8 2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 -8 2 B -2 0 8 -2 12 C -16 -8 0 -10 -10 D 8 2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=24 B=23 C=15 E=13 so E is eliminated. Round 2 votes counts: D=31 A=30 B=23 C=16 so C is eliminated. Round 3 votes counts: A=37 D=35 B=28 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:208 A:206 E:194 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 16 -8 2 B -2 0 8 -2 12 C -16 -8 0 -10 -10 D 8 2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 -8 2 B -2 0 8 -2 12 C -16 -8 0 -10 -10 D 8 2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 -8 2 B -2 0 8 -2 12 C -16 -8 0 -10 -10 D 8 2 10 0 8 E -2 -12 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991575 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8735: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (12) E C A B D (7) D E B C A (7) B D A C E (7) E C A D B (6) C A E D B (6) D B E A C (5) E D C B A (4) E D B C A (4) C A E B D (4) A C E B D (4) A B C D E (4) B A C D E (3) E C D A B (2) D B A E C (2) D B A C E (2) C E A D B (2) B D E A C (2) B A D C E (2) B A C E D (2) A C B E D (2) E D C A B (1) E C D B A (1) E B A C D (1) D C E A B (1) C E D A B (1) C A D E B (1) B E D A C (1) B D A E C (1) B A E C D (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 6 18 10 B -10 0 -12 6 -4 C -6 12 0 20 10 D -18 -6 -20 0 2 E -10 4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 18 10 B -10 0 -12 6 -4 C -6 12 0 20 10 D -18 -6 -20 0 2 E -10 4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=24 B=19 D=17 C=14 so C is eliminated. Round 2 votes counts: A=35 E=29 B=19 D=17 so D is eliminated. Round 3 votes counts: E=37 A=35 B=28 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:218 E:191 B:190 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 18 10 B -10 0 -12 6 -4 C -6 12 0 20 10 D -18 -6 -20 0 2 E -10 4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 18 10 B -10 0 -12 6 -4 C -6 12 0 20 10 D -18 -6 -20 0 2 E -10 4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 18 10 B -10 0 -12 6 -4 C -6 12 0 20 10 D -18 -6 -20 0 2 E -10 4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8736: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (14) D C B E A (11) E A C B D (9) C E A D B (8) B A D E C (7) B D A E C (6) E C A D B (5) D C E B A (5) D B C A E (5) B A E D C (5) A E C B D (4) A E B C D (4) D C E A B (3) B D A C E (3) A B E C D (3) E C A B D (2) A B E D C (2) E A C D B (1) D B C E A (1) C E D A B (1) B A E C D (1) Total count = 100 A B C D E A 0 12 -10 2 -20 B -12 0 -26 -8 -12 C 10 26 0 4 2 D -2 8 -4 0 10 E 20 12 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -10 2 -20 B -12 0 -26 -8 -12 C 10 26 0 4 2 D -2 8 -4 0 10 E 20 12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 B=22 E=17 A=13 so A is eliminated. Round 2 votes counts: B=27 E=25 D=25 C=23 so C is eliminated. Round 3 votes counts: D=39 E=34 B=27 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:221 E:210 D:206 A:192 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -10 2 -20 B -12 0 -26 -8 -12 C 10 26 0 4 2 D -2 8 -4 0 10 E 20 12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -10 2 -20 B -12 0 -26 -8 -12 C 10 26 0 4 2 D -2 8 -4 0 10 E 20 12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -10 2 -20 B -12 0 -26 -8 -12 C 10 26 0 4 2 D -2 8 -4 0 10 E 20 12 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8737: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) C D A E B (7) D C A E B (4) B E A D C (4) B C D E A (4) A D E C B (4) A D C E B (4) E B D A C (3) E B A D C (3) E A D B C (3) E A B D C (3) D C E A B (3) C D A B E (3) C B D A E (3) B E C A D (3) B C A E D (3) A C D E B (3) A C D B E (3) A B C D E (3) E D A C B (2) E A D C B (2) C D B A E (2) C B A D E (2) B E C D A (2) B A E C D (2) B A C E D (2) A E D C B (2) E D C B A (1) E D B A C (1) D A E C B (1) D A C E B (1) C A D E B (1) C A B D E (1) B E A C D (1) B C E A D (1) B C A D E (1) B A E D C (1) A E D B C (1) A E B D C (1) A C B D E (1) Total count = 100 A B C D E A 0 6 2 10 12 B -6 0 0 4 0 C -2 0 0 12 20 D -10 -4 -12 0 2 E -12 0 -20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 10 12 B -6 0 0 4 0 C -2 0 0 12 20 D -10 -4 -12 0 2 E -12 0 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=22 C=19 E=18 D=9 so D is eliminated. Round 2 votes counts: B=32 C=26 A=24 E=18 so E is eliminated. Round 3 votes counts: B=39 A=34 C=27 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:215 B:199 D:188 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 10 12 B -6 0 0 4 0 C -2 0 0 12 20 D -10 -4 -12 0 2 E -12 0 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 10 12 B -6 0 0 4 0 C -2 0 0 12 20 D -10 -4 -12 0 2 E -12 0 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 10 12 B -6 0 0 4 0 C -2 0 0 12 20 D -10 -4 -12 0 2 E -12 0 -20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992412 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8738: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (14) B C E D A (12) B A C D E (6) A D E B C (6) D A E C B (5) B A E C D (5) E D C A B (4) D E A C B (4) C E D B A (4) B A E D C (4) C D E B A (3) B E C D A (3) A B D E C (3) D E C A B (2) D C E A B (2) D A C E B (2) C E B D A (2) C D E A B (2) B E C A D (2) B A C E D (2) A D C E B (2) E C D B A (1) E B C D A (1) C E D A B (1) C D B A E (1) C B A D E (1) B E A C D (1) B C D E A (1) B A D E C (1) A D B E C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 16 0 10 B 0 0 0 -8 -10 C -16 0 0 -4 -16 D 0 8 4 0 14 E -10 10 16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.448150 B: 0.000000 C: 0.000000 D: 0.551850 E: 0.000000 Sum of squares = 0.505376857631 Cumulative probabilities = A: 0.448150 B: 0.448150 C: 0.448150 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 0 10 B 0 0 0 -8 -10 C -16 0 0 -4 -16 D 0 8 4 0 14 E -10 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=28 D=15 C=14 E=6 so E is eliminated. Round 2 votes counts: B=38 A=28 D=19 C=15 so C is eliminated. Round 3 votes counts: B=41 D=31 A=28 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:213 D:213 E:201 B:191 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 0 16 0 10 B 0 0 0 -8 -10 C -16 0 0 -4 -16 D 0 8 4 0 14 E -10 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 0 10 B 0 0 0 -8 -10 C -16 0 0 -4 -16 D 0 8 4 0 14 E -10 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 0 10 B 0 0 0 -8 -10 C -16 0 0 -4 -16 D 0 8 4 0 14 E -10 10 16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8739: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) A D C B E (8) A B E C D (6) D A E C B (5) E B C D A (4) E B C A D (4) D A C B E (4) C D B E A (4) B E A C D (4) E B A D C (3) D C E B A (3) D A E B C (3) D A C E B (3) C D B A E (3) B E C A D (3) E B A C D (2) C B E A D (2) B C E A D (2) B A E C D (2) A E D B C (2) A E B D C (2) A D E B C (2) A D C E B (2) A B E D C (2) E D B C A (1) E D B A C (1) E D A B C (1) E C D B A (1) E B D C A (1) D E C A B (1) D E A B C (1) D C E A B (1) D C B E A (1) D C B A E (1) C D E B A (1) C D A B E (1) A D B E C (1) A D B C E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 14 0 0 B 4 0 -2 -6 12 C -14 2 0 -2 -6 D 0 6 2 0 -4 E 0 -12 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677615 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 A B C D E A 0 -4 14 0 0 B 4 0 -2 -6 12 C -14 2 0 -2 -6 D 0 6 2 0 -4 E 0 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677704 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=23 C=20 E=18 B=11 so B is eliminated. Round 2 votes counts: A=30 E=25 D=23 C=22 so C is eliminated. Round 3 votes counts: E=38 D=32 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:205 B:204 D:202 E:199 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 14 0 0 B 4 0 -2 -6 12 C -14 2 0 -2 -6 D 0 6 2 0 -4 E 0 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677704 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 0 0 B 4 0 -2 -6 12 C -14 2 0 -2 -6 D 0 6 2 0 -4 E 0 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677704 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 0 0 B 4 0 -2 -6 12 C -14 2 0 -2 -6 D 0 6 2 0 -4 E 0 -12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.000000 D: 0.545455 E: 0.272727 Sum of squares = 0.404958677704 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.181818 D: 0.727273 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8740: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) B A C E D (7) D E C A B (6) C A B E D (6) A C B E D (5) E C D A B (4) E C B A D (3) E B D A C (3) E B C A D (3) D B E A C (3) A D B C E (3) A C B D E (3) A B C D E (3) E D B C A (2) D C A E B (2) D B A E C (2) D A B C E (2) C E B A D (2) C E A B D (2) C B A E D (2) C A E B D (2) C A D B E (2) B A D E C (2) B A D C E (2) A C D B E (2) A B D C E (2) E D C B A (1) E C D B A (1) E B C D A (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E A B (1) D A C E B (1) D A B E C (1) C E A D B (1) C D A E B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D A E C (1) B C E A D (1) B C A E D (1) A D C B E (1) Total count = 100 A B C D E A 0 4 6 14 8 B -4 0 -2 6 6 C -6 2 0 6 10 D -14 -6 -6 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 14 8 B -4 0 -2 6 6 C -6 2 0 6 10 D -14 -6 -6 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=19 A=19 E=18 B=16 so B is eliminated. Round 2 votes counts: A=30 D=29 C=21 E=20 so E is eliminated. Round 3 votes counts: D=36 C=33 A=31 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:216 C:206 B:203 D:188 E:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 14 8 B -4 0 -2 6 6 C -6 2 0 6 10 D -14 -6 -6 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 14 8 B -4 0 -2 6 6 C -6 2 0 6 10 D -14 -6 -6 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 14 8 B -4 0 -2 6 6 C -6 2 0 6 10 D -14 -6 -6 0 2 E -8 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999418 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8741: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (13) E C D A B (11) C E B A D (9) B A C D E (7) C E B D A (6) E D A C B (5) C E D B A (5) C B E A D (5) B C A D E (5) A D B E C (5) D A E B C (4) D A B E C (4) C B A D E (3) E D C A B (2) E D A B C (2) D E A C B (2) A D E B C (2) A B D E C (2) D E A B C (1) D C E A B (1) D B A C E (1) D A B C E (1) C B D A E (1) C B A E D (1) B D A C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 2 6 2 B 14 0 -2 8 0 C -2 2 0 6 20 D -6 -8 -6 0 8 E -2 0 -20 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.111111 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.629629629579 Cumulative probabilities = A: 0.111111 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 6 2 B 14 0 -2 8 0 C -2 2 0 6 20 D -6 -8 -6 0 8 E -2 0 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.111111 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.629629632515 Cumulative probabilities = A: 0.111111 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=26 E=20 D=14 A=10 so A is eliminated. Round 2 votes counts: C=30 B=29 D=21 E=20 so E is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:210 A:198 D:194 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 2 6 2 B 14 0 -2 8 0 C -2 2 0 6 20 D -6 -8 -6 0 8 E -2 0 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.111111 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.629629632515 Cumulative probabilities = A: 0.111111 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 6 2 B 14 0 -2 8 0 C -2 2 0 6 20 D -6 -8 -6 0 8 E -2 0 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.111111 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.629629632515 Cumulative probabilities = A: 0.111111 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 6 2 B 14 0 -2 8 0 C -2 2 0 6 20 D -6 -8 -6 0 8 E -2 0 -20 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.111111 C: 0.777778 D: 0.000000 E: 0.000000 Sum of squares = 0.629629632515 Cumulative probabilities = A: 0.111111 B: 0.222222 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8742: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) E A C D B (9) B E C A D (6) C A D E B (5) E D A C B (4) D A C E B (4) C A E D B (4) C A B E D (4) B E D A C (4) B C D A E (4) E C A D B (3) D A E B C (3) B C E A D (3) B C A E D (3) E C A B D (2) E B C A D (2) E A B D C (2) D C A E B (2) D A E C B (2) C B D A E (2) B D C A E (2) A E D C B (2) E D B A C (1) E C B A D (1) E B D A C (1) E B A D C (1) E A C B D (1) D E A C B (1) D E A B C (1) D C A B E (1) D B A C E (1) C E B A D (1) C E A B D (1) C D B A E (1) C B A E D (1) C A D B E (1) C A B D E (1) B E D C A (1) B E C D A (1) B D C E A (1) A C D E B (1) Total count = 100 A B C D E A 0 26 -6 26 -12 B -26 0 -26 -10 -26 C 6 26 0 14 -14 D -26 10 -14 0 -34 E 12 26 14 34 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 26 -6 26 -12 B -26 0 -26 -10 -26 C 6 26 0 14 -14 D -26 10 -14 0 -34 E 12 26 14 34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=25 C=21 D=15 A=3 so A is eliminated. Round 2 votes counts: E=38 B=25 C=22 D=15 so D is eliminated. Round 3 votes counts: E=45 C=29 B=26 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:243 A:217 C:216 D:168 B:156 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 26 -6 26 -12 B -26 0 -26 -10 -26 C 6 26 0 14 -14 D -26 10 -14 0 -34 E 12 26 14 34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 -6 26 -12 B -26 0 -26 -10 -26 C 6 26 0 14 -14 D -26 10 -14 0 -34 E 12 26 14 34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 -6 26 -12 B -26 0 -26 -10 -26 C 6 26 0 14 -14 D -26 10 -14 0 -34 E 12 26 14 34 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8743: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) C D E B A (7) A B E D C (7) D C E B A (6) C D A B E (6) A B E C D (6) E B D C A (4) D C B E A (4) C D A E B (4) B A E D C (4) E B D A C (3) C A E D B (3) B E D A C (3) A C D B E (3) A C B E D (3) A B C D E (3) E C D B A (2) E A B C D (2) D E C B A (2) D E B C A (2) D B E C A (2) A C E B D (2) A C B D E (2) E D C B A (1) E D B C A (1) E C B A D (1) E B C A D (1) D B E A C (1) C E A B D (1) C D E A B (1) C A D E B (1) A E C B D (1) A E B C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 2 2 -4 B 4 0 0 8 -8 C -2 0 0 2 -4 D -2 -8 -2 0 -10 E 4 8 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 2 -4 B 4 0 0 8 -8 C -2 0 0 2 -4 D -2 -8 -2 0 -10 E 4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=23 C=23 D=17 B=7 so B is eliminated. Round 2 votes counts: A=34 E=26 C=23 D=17 so D is eliminated. Round 3 votes counts: A=34 E=33 C=33 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:213 B:202 A:198 C:198 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 2 -4 B 4 0 0 8 -8 C -2 0 0 2 -4 D -2 -8 -2 0 -10 E 4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 2 -4 B 4 0 0 8 -8 C -2 0 0 2 -4 D -2 -8 -2 0 -10 E 4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 2 -4 B 4 0 0 8 -8 C -2 0 0 2 -4 D -2 -8 -2 0 -10 E 4 8 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8744: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) C B A E D (10) D E A B C (7) B C A D E (7) C B A D E (6) E D A C B (5) D E B C A (5) D E B A C (5) A E D C B (4) A C E D B (4) A C B E D (4) C A B E D (3) B D E C A (3) B D C E A (3) B C D E A (3) A E C D B (3) D E A C B (2) D B E C A (2) B C A E D (2) A C E B D (2) E A D C B (1) D E C B A (1) D B E A C (1) C D E B A (1) C A E D B (1) B D E A C (1) B C D A E (1) B A C E D (1) Total count = 100 A B C D E A 0 -4 4 -4 -4 B 4 0 6 -8 -6 C -4 -6 0 -4 -4 D 4 8 4 0 -4 E 4 6 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 -4 -4 B 4 0 6 -8 -6 C -4 -6 0 -4 -4 D 4 8 4 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=21 B=21 E=18 A=17 so A is eliminated. Round 2 votes counts: C=31 E=25 D=23 B=21 so B is eliminated. Round 3 votes counts: C=45 D=30 E=25 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:209 D:206 B:198 A:196 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 -4 -4 B 4 0 6 -8 -6 C -4 -6 0 -4 -4 D 4 8 4 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -4 -4 B 4 0 6 -8 -6 C -4 -6 0 -4 -4 D 4 8 4 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -4 -4 B 4 0 6 -8 -6 C -4 -6 0 -4 -4 D 4 8 4 0 -4 E 4 6 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8745: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) C D B A E (7) A E D B C (6) D C A B E (5) E B C A D (4) E A B C D (4) B D C E A (4) B C E D A (4) A E D C B (4) E B A C D (3) E A B D C (3) D C B A E (3) C D A B E (3) B D E C A (3) A D E B C (3) A C D E B (3) E A C B D (2) D B C A E (2) D A E B C (2) C E A B D (2) C D B E A (2) B C D E A (2) A E C D B (2) A D C E B (2) A C E D B (2) E C B A D (1) D E A B C (1) D C B E A (1) D B E A C (1) D B C E A (1) D A B E C (1) C A E D B (1) C A E B D (1) C A D B E (1) C A B E D (1) B E D C A (1) B E D A C (1) B C E A D (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 2 -16 -4 2 B -2 0 -6 -8 4 C 16 6 0 10 12 D 4 8 -10 0 12 E -2 -4 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 -4 2 B -2 0 -6 -8 4 C 16 6 0 10 12 D 4 8 -10 0 12 E -2 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=24 E=17 D=17 B=16 so B is eliminated. Round 2 votes counts: C=33 D=24 A=24 E=19 so E is eliminated. Round 3 votes counts: C=38 A=36 D=26 so D is eliminated. Round 4 votes counts: C=58 A=42 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:207 B:194 A:192 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 -4 2 B -2 0 -6 -8 4 C 16 6 0 10 12 D 4 8 -10 0 12 E -2 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -4 2 B -2 0 -6 -8 4 C 16 6 0 10 12 D 4 8 -10 0 12 E -2 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -4 2 B -2 0 -6 -8 4 C 16 6 0 10 12 D 4 8 -10 0 12 E -2 -4 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8746: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (13) C A D E B (9) C B E D A (7) C B E A D (7) E B D A C (5) D A E B C (5) C A D B E (5) C A B E D (5) B E D C A (5) A C D E B (5) A D E B C (4) D E A B C (3) C A B D E (3) B E C D A (3) B C E A D (3) A D C E B (3) D A C E B (2) C B D E A (2) B C E D A (2) A E D B C (2) D E B A C (1) D E A C B (1) D B E C A (1) D A E C B (1) C D A E B (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 2 -6 -4 -6 B -2 0 -2 14 12 C 6 2 0 6 8 D 4 -14 -6 0 -8 E 6 -12 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -4 -6 B -2 0 -2 14 12 C 6 2 0 6 8 D 4 -14 -6 0 -8 E 6 -12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=26 A=16 D=14 E=5 so E is eliminated. Round 2 votes counts: C=39 B=31 A=16 D=14 so D is eliminated. Round 3 votes counts: C=39 B=33 A=28 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:211 E:197 A:193 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -4 -6 B -2 0 -2 14 12 C 6 2 0 6 8 D 4 -14 -6 0 -8 E 6 -12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -4 -6 B -2 0 -2 14 12 C 6 2 0 6 8 D 4 -14 -6 0 -8 E 6 -12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -4 -6 B -2 0 -2 14 12 C 6 2 0 6 8 D 4 -14 -6 0 -8 E 6 -12 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8747: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (7) B A D E C (7) A B C D E (7) D E C B A (6) C E D A B (6) A C B D E (5) A B D E C (5) D E C A B (4) D E B A C (4) A B D C E (4) E D C B A (3) C E D B A (3) C D E A B (3) B A C E D (3) A B C E D (3) E B C D A (2) D E A B C (2) D C E A B (2) D A C E B (2) D A B E C (2) C E A B D (2) A C B E D (2) E D B C A (1) E C D B A (1) E C B D A (1) D E B C A (1) D C E B A (1) D B A E C (1) C E B D A (1) C E B A D (1) C A E B D (1) C A D E B (1) B D E A C (1) B D A E C (1) B A E D C (1) B A E C D (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 20 6 4 10 B -20 0 -4 10 4 C -6 4 0 0 10 D -4 -10 0 0 22 E -10 -4 -10 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 6 4 10 B -20 0 -4 10 4 C -6 4 0 0 10 D -4 -10 0 0 22 E -10 -4 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=25 C=25 B=14 E=8 so E is eliminated. Round 2 votes counts: D=29 A=28 C=27 B=16 so B is eliminated. Round 3 votes counts: A=40 D=31 C=29 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:204 D:204 B:195 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 6 4 10 B -20 0 -4 10 4 C -6 4 0 0 10 D -4 -10 0 0 22 E -10 -4 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 4 10 B -20 0 -4 10 4 C -6 4 0 0 10 D -4 -10 0 0 22 E -10 -4 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 4 10 B -20 0 -4 10 4 C -6 4 0 0 10 D -4 -10 0 0 22 E -10 -4 -10 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8748: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) C A B D E (7) A C B E D (7) D E B C A (6) C B A E D (6) E B D C A (5) D A E C B (5) B C E A D (5) B C D E A (5) A E D C B (5) E B C A D (4) A C D B E (4) E B C D A (3) E A D B C (3) D A C E B (3) A C B D E (3) E D B C A (2) E D B A C (2) D A C B E (2) C B D A E (2) C B A D E (2) C A B E D (2) A E C B D (2) E D A B C (1) E A B C D (1) D E B A C (1) D B C E A (1) B D C E A (1) B C E D A (1) B C D A E (1) A D C B E (1) Total count = 100 A B C D E A 0 6 -6 4 4 B -6 0 -2 14 0 C 6 2 0 10 6 D -4 -14 -10 0 2 E -4 0 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 4 4 B -6 0 -2 14 0 C 6 2 0 10 6 D -4 -14 -10 0 2 E -4 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=22 E=21 C=19 B=13 so B is eliminated. Round 2 votes counts: C=31 D=26 A=22 E=21 so E is eliminated. Round 3 votes counts: C=38 D=36 A=26 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 A:204 B:203 E:194 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 4 4 B -6 0 -2 14 0 C 6 2 0 10 6 D -4 -14 -10 0 2 E -4 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 4 4 B -6 0 -2 14 0 C 6 2 0 10 6 D -4 -14 -10 0 2 E -4 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 4 4 B -6 0 -2 14 0 C 6 2 0 10 6 D -4 -14 -10 0 2 E -4 0 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8749: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (13) B E A C D (7) C D A E B (6) C A D B E (6) C A B E D (6) B A E C D (5) B E A D C (4) A C B E D (4) E B A D C (3) E A B C D (3) C A E B D (3) B C A E D (3) A B E C D (3) E B D A C (2) D E B C A (2) D E B A C (2) D E A C B (2) D C E A B (2) D C A B E (2) C D B A E (2) B E D A C (2) B A C E D (2) E D B A C (1) E D A B C (1) D E C B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D C B E A (1) D B E C A (1) C D A B E (1) C A E D B (1) C A D E B (1) C A B D E (1) A E C D B (1) A E C B D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 22 -8 12 26 B -22 0 -14 2 2 C 8 14 0 16 14 D -12 -2 -16 0 -8 E -26 -2 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -8 12 26 B -22 0 -14 2 2 C 8 14 0 16 14 D -12 -2 -16 0 -8 E -26 -2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 B=23 A=11 E=10 so E is eliminated. Round 2 votes counts: D=31 B=28 C=27 A=14 so A is eliminated. Round 3 votes counts: B=35 C=34 D=31 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:226 C:226 B:184 E:183 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -8 12 26 B -22 0 -14 2 2 C 8 14 0 16 14 D -12 -2 -16 0 -8 E -26 -2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -8 12 26 B -22 0 -14 2 2 C 8 14 0 16 14 D -12 -2 -16 0 -8 E -26 -2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -8 12 26 B -22 0 -14 2 2 C 8 14 0 16 14 D -12 -2 -16 0 -8 E -26 -2 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8750: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (5) E C A B D (4) E B A C D (4) C A E B D (4) B D E A C (4) B D A E C (4) A C E B D (4) E B D C A (3) D C E A B (3) D C A E B (3) D B E C A (3) C E A B D (3) B A D C E (3) A C B E D (3) A B C D E (3) E D B C A (2) E C D A B (2) E B C A D (2) D E C B A (2) D E C A B (2) D C A B E (2) D B E A C (2) C E A D B (2) C D A E B (2) C A E D B (2) C A D E B (2) B E D A C (2) B A E C D (2) A B C E D (2) E C D B A (1) E B C D A (1) E B A D C (1) D E B C A (1) D C B E A (1) D C B A E (1) D B C E A (1) D B A E C (1) D A C B E (1) C E D A B (1) C D E A B (1) C A D B E (1) B E A C D (1) B D A C E (1) B A E D C (1) B A D E C (1) A C D B E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -4 -4 2 B 0 0 2 10 -4 C 4 -2 0 0 6 D 4 -10 0 0 4 E -2 4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888838 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 A B C D E A 0 0 -4 -4 2 B 0 0 2 10 -4 C 4 -2 0 0 6 D 4 -10 0 0 4 E -2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888674 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 E=20 B=19 C=18 A=15 so A is eliminated. Round 2 votes counts: D=28 C=27 B=25 E=20 so E is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:204 C:204 D:199 A:197 E:196 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 -4 2 B 0 0 2 10 -4 C 4 -2 0 0 6 D 4 -10 0 0 4 E -2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888674 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 2 B 0 0 2 10 -4 C 4 -2 0 0 6 D 4 -10 0 0 4 E -2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888674 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 2 B 0 0 2 10 -4 C 4 -2 0 0 6 D 4 -10 0 0 4 E -2 4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.000000 E: 0.166667 Sum of squares = 0.388888888674 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 0.833333 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8751: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) A E D C B (9) D A B E C (8) C E B A D (8) C B E D A (7) D B C A E (6) C B D E A (6) B D C A E (6) B C D E A (6) A D E B C (6) E A C B D (4) A E D B C (4) E C A B D (3) D B A E C (3) D B A C E (3) C E A B D (3) E A D C B (2) E D A C B (1) D B C E A (1) C E B D A (1) C B E A D (1) B C D A E (1) B A D E C (1) A B D E C (1) Total count = 100 A B C D E A 0 0 2 2 -4 B 0 0 -8 -4 0 C -2 8 0 -2 -2 D -2 4 2 0 -4 E 4 0 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.138546 C: 0.000000 D: 0.000000 E: 0.861454 Sum of squares = 0.761297713818 Cumulative probabilities = A: 0.000000 B: 0.138546 C: 0.138546 D: 0.138546 E: 1.000000 A B C D E A 0 0 2 2 -4 B 0 0 -8 -4 0 C -2 8 0 -2 -2 D -2 4 2 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000039445 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=21 A=20 E=19 B=14 so B is eliminated. Round 2 votes counts: C=33 D=27 A=21 E=19 so E is eliminated. Round 3 votes counts: C=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:205 C:201 A:200 D:200 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 2 2 -4 B 0 0 -8 -4 0 C -2 8 0 -2 -2 D -2 4 2 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000039445 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 -4 B 0 0 -8 -4 0 C -2 8 0 -2 -2 D -2 4 2 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000039445 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 -4 B 0 0 -8 -4 0 C -2 8 0 -2 -2 D -2 4 2 0 -4 E 4 0 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.800000 Sum of squares = 0.680000039445 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.200000 D: 0.200000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8752: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C E A D B (7) B D A E C (5) A D C B E (5) C A D E B (4) B D A C E (4) E C A B D (3) E A D B C (3) D A C B E (3) D A B E C (3) B E D A C (3) B E C D A (3) B D E A C (3) B D C A E (3) B C D E A (3) E D A B C (2) E C B A D (2) E C A D B (2) E B C A D (2) E A D C B (2) E A C D B (2) D A B C E (2) C B E A D (2) C A E D B (2) A D E C B (2) A D C E B (2) A C D E B (2) E B A D C (1) D B A E C (1) D B A C E (1) D A E B C (1) C E B A D (1) C B E D A (1) C B D A E (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D C A (1) B C D A E (1) A E D C B (1) A D E B C (1) Total count = 100 A B C D E A 0 2 16 0 -4 B -2 0 4 2 -2 C -16 -4 0 -16 -4 D 0 -2 16 0 0 E 4 2 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.257188 E: 0.742812 Sum of squares = 0.617915351758 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.257188 E: 1.000000 A B C D E A 0 2 16 0 -4 B -2 0 4 2 -2 C -16 -4 0 -16 -4 D 0 -2 16 0 0 E 4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 0.500073 Sum of squares = 0.500000010716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=26 C=22 A=13 D=11 so D is eliminated. Round 2 votes counts: E=28 B=28 C=22 A=22 so C is eliminated. Round 3 votes counts: E=36 B=34 A=30 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:207 D:207 E:205 B:201 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 16 0 -4 B -2 0 4 2 -2 C -16 -4 0 -16 -4 D 0 -2 16 0 0 E 4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 0.500073 Sum of squares = 0.500000010716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 0 -4 B -2 0 4 2 -2 C -16 -4 0 -16 -4 D 0 -2 16 0 0 E 4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 0.500073 Sum of squares = 0.500000010716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 0 -4 B -2 0 4 2 -2 C -16 -4 0 -16 -4 D 0 -2 16 0 0 E 4 2 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 0.500073 Sum of squares = 0.500000010716 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499927 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8753: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (19) A D C E B (13) A D C B E (9) D A B C E (7) C E B D A (7) E B C D A (6) E C B D A (4) E B C A D (3) D A C E B (3) A D B E C (3) A D B C E (3) D A C B E (2) B D C E A (2) A D E C B (2) A B E C D (2) A B D E C (2) E C A B D (1) E B A C D (1) D A B E C (1) C D E B A (1) C D B E A (1) C B E D A (1) B E C A D (1) B A E C D (1) A E C D B (1) A E B D C (1) A E B C D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 6 8 -8 6 B -6 0 6 4 8 C -8 -6 0 4 2 D 8 -4 -4 0 0 E -6 -8 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.444444 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691404 Cumulative probabilities = A: 0.222222 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -8 6 B -6 0 6 4 8 C -8 -6 0 4 2 D 8 -4 -4 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.444444 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691362 Cumulative probabilities = A: 0.222222 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=23 E=15 D=13 C=10 so C is eliminated. Round 2 votes counts: A=39 B=24 E=22 D=15 so D is eliminated. Round 3 votes counts: A=52 B=25 E=23 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:206 D:200 C:196 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -8 6 B -6 0 6 4 8 C -8 -6 0 4 2 D 8 -4 -4 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.444444 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691362 Cumulative probabilities = A: 0.222222 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -8 6 B -6 0 6 4 8 C -8 -6 0 4 2 D 8 -4 -4 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.444444 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691362 Cumulative probabilities = A: 0.222222 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -8 6 B -6 0 6 4 8 C -8 -6 0 4 2 D 8 -4 -4 0 0 E -6 -8 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.444444 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.358024691362 Cumulative probabilities = A: 0.222222 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8754: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) E A D B C (7) C B E A D (7) E D C B A (6) A D E B C (6) D E A C B (5) D A E C B (5) C B E D A (5) B C E A D (5) B C A E D (5) C B D A E (4) C B A D E (4) E D C A B (3) E B C A D (3) C E B D A (3) C B D E A (3) E D A B C (2) E B A C D (2) C D B A E (2) C B A E D (2) B C A D E (2) E D A C B (1) D A C B E (1) D A B C E (1) C E D B A (1) C D B E A (1) B A C E D (1) A E D B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -12 -6 -8 B 12 0 -6 -8 -10 C 12 6 0 0 -4 D 6 8 0 0 -8 E 8 10 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999307 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -12 -6 -8 B 12 0 -6 -8 -10 C 12 6 0 0 -4 D 6 8 0 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=24 D=22 B=13 A=9 so A is eliminated. Round 2 votes counts: C=32 D=30 E=25 B=13 so B is eliminated. Round 3 votes counts: C=45 D=30 E=25 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:207 D:203 B:194 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -12 -6 -8 B 12 0 -6 -8 -10 C 12 6 0 0 -4 D 6 8 0 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -6 -8 B 12 0 -6 -8 -10 C 12 6 0 0 -4 D 6 8 0 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -6 -8 B 12 0 -6 -8 -10 C 12 6 0 0 -4 D 6 8 0 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8755: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (14) B A C E D (9) D E C B A (7) A B C E D (5) A B C D E (5) D E B C A (4) E D C B A (3) E D C A B (3) E C B A D (3) D E B A C (3) C E D A B (3) C A D B E (3) C A B E D (3) B A E C D (3) E B D C A (2) D C E A B (2) D C A E B (2) D B E A C (2) D B A E C (2) D A C B E (2) A C B E D (2) E D B C A (1) E C D B A (1) E C D A B (1) E B C D A (1) E B C A D (1) D A B C E (1) C E A D B (1) C E A B D (1) C D A E B (1) C B A E D (1) C A E B D (1) B E A C D (1) B D E A C (1) B A C D E (1) A D C B E (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -18 -12 -10 B -8 0 -14 -16 -10 C 18 14 0 -2 -6 D 12 16 2 0 8 E 10 10 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998819 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 -12 -10 B -8 0 -14 -16 -10 C 18 14 0 -2 -6 D 12 16 2 0 8 E 10 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=16 A=16 B=15 C=14 so C is eliminated. Round 2 votes counts: D=40 A=23 E=21 B=16 so B is eliminated. Round 3 votes counts: D=41 A=37 E=22 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:212 E:209 A:184 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -18 -12 -10 B -8 0 -14 -16 -10 C 18 14 0 -2 -6 D 12 16 2 0 8 E 10 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 -12 -10 B -8 0 -14 -16 -10 C 18 14 0 -2 -6 D 12 16 2 0 8 E 10 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 -12 -10 B -8 0 -14 -16 -10 C 18 14 0 -2 -6 D 12 16 2 0 8 E 10 10 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997376 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8756: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) D A E C B (7) A E B C D (6) D C A E B (4) D C A B E (4) B C D E A (4) E A B C D (3) D C B E A (3) D C B A E (3) D B C A E (3) C D B E A (3) B E C A D (3) B C E D A (3) B C E A D (3) A E D B C (3) A D E B C (3) A D B E C (3) E C B A D (2) E A C B D (2) D A C E B (2) C E B A D (2) C D E A B (2) C B E D A (2) B D C E A (2) A E D C B (2) A E B D C (2) A D E C B (2) E C A D B (1) E C A B D (1) E B C A D (1) E A C D B (1) D B C E A (1) D B A C E (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B D A (1) C D E B A (1) C B E A D (1) B D C A E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 -20 -16 -2 B -4 0 -10 -4 2 C 20 10 0 2 12 D 16 4 -2 0 18 E 2 -2 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -20 -16 -2 B -4 0 -10 -4 2 C 20 10 0 2 12 D 16 4 -2 0 18 E 2 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=22 C=21 B=16 E=11 so E is eliminated. Round 2 votes counts: D=30 A=28 C=25 B=17 so B is eliminated. Round 3 votes counts: C=39 D=33 A=28 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:218 B:192 E:185 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -20 -16 -2 B -4 0 -10 -4 2 C 20 10 0 2 12 D 16 4 -2 0 18 E 2 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -16 -2 B -4 0 -10 -4 2 C 20 10 0 2 12 D 16 4 -2 0 18 E 2 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -16 -2 B -4 0 -10 -4 2 C 20 10 0 2 12 D 16 4 -2 0 18 E 2 -2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994493 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8757: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (12) C A D B E (10) C A D E B (8) D A C B E (7) B E C A D (6) A D C B E (6) B E D A C (5) E D A B C (4) E D B A C (3) E D A C B (3) D A C E B (3) C B A D E (3) B C E A D (3) E C D A B (2) E C B A D (2) E B C A D (2) D A E B C (2) C E B A D (2) C D A E B (2) C B A E D (2) B E A D C (2) B D A E C (2) E B C D A (1) D B A E C (1) D A B E C (1) D A B C E (1) C E D A B (1) B C A E D (1) B A D C E (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 2 10 0 4 B -2 0 -2 -10 6 C -10 2 0 -8 4 D 0 10 8 0 -2 E -4 -6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.556638 B: 0.000000 C: 0.000000 D: 0.443362 E: 0.000000 Sum of squares = 0.506415688011 Cumulative probabilities = A: 0.556638 B: 0.556638 C: 0.556638 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 0 4 B -2 0 -2 -10 6 C -10 2 0 -8 4 D 0 10 8 0 -2 E -4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=28 B=21 D=15 A=7 so A is eliminated. Round 2 votes counts: E=29 C=28 D=22 B=21 so B is eliminated. Round 3 votes counts: E=42 C=33 D=25 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:208 D:208 B:196 C:194 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 0 4 B -2 0 -2 -10 6 C -10 2 0 -8 4 D 0 10 8 0 -2 E -4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 0 4 B -2 0 -2 -10 6 C -10 2 0 -8 4 D 0 10 8 0 -2 E -4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 0 4 B -2 0 -2 -10 6 C -10 2 0 -8 4 D 0 10 8 0 -2 E -4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8758: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) E B A C D (7) C D A E B (7) B E A D C (6) B D C E A (6) A E B C D (6) A C D E B (6) D C B A E (5) E A B C D (4) A E C B D (4) E C A D B (3) D C B E A (3) D B C E A (3) B E D C A (3) A E B D C (3) A C E D B (3) E B C D A (2) E A C D B (2) E A C B D (2) D C A B E (2) B E D A C (2) B D E C A (2) E C A B D (1) E B A D C (1) D A B C E (1) C D B E A (1) C D A B E (1) C A D E B (1) B D C A E (1) B D A E C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 14 18 20 4 B -14 0 -2 2 -22 C -18 2 0 18 -16 D -20 -2 -18 0 -16 E -4 22 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 20 4 B -14 0 -2 2 -22 C -18 2 0 18 -16 D -20 -2 -18 0 -16 E -4 22 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=22 B=21 D=14 C=10 so C is eliminated. Round 2 votes counts: A=34 D=23 E=22 B=21 so B is eliminated. Round 3 votes counts: A=34 E=33 D=33 so E is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 E:225 C:193 B:182 D:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 20 4 B -14 0 -2 2 -22 C -18 2 0 18 -16 D -20 -2 -18 0 -16 E -4 22 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 20 4 B -14 0 -2 2 -22 C -18 2 0 18 -16 D -20 -2 -18 0 -16 E -4 22 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 20 4 B -14 0 -2 2 -22 C -18 2 0 18 -16 D -20 -2 -18 0 -16 E -4 22 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8759: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) E B A C D (6) A C E B D (6) D C A B E (5) D B C A E (5) C A E D B (5) E A B C D (4) D C B A E (4) C D A E B (4) C A D E B (4) B E A D C (4) E D B C A (3) D C A E B (3) B D E C A (3) B A E D C (3) E B D A C (2) E A C B D (2) D C E B A (2) D B C E A (2) C D A B E (2) B A D E C (2) A E C B D (2) A C E D B (2) A C D E B (2) A C D B E (2) A B E C D (2) E D C B A (1) E C A D B (1) E C A B D (1) E B A D C (1) D E C B A (1) D B E C A (1) B D E A C (1) B D A E C (1) B D A C E (1) B A E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 6 4 16 B 2 0 2 2 -4 C -6 -2 0 -6 2 D -4 -2 6 0 -6 E -16 4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.727273 C: 0.000000 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933855 Cumulative probabilities = A: 0.181818 B: 0.909091 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 -2 6 4 16 B 2 0 2 2 -4 C -6 -2 0 -6 2 D -4 -2 6 0 -6 E -16 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.727273 C: 0.000000 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933801 Cumulative probabilities = A: 0.181818 B: 0.909091 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 E=21 A=18 C=15 so C is eliminated. Round 2 votes counts: D=29 A=27 B=23 E=21 so E is eliminated. Round 3 votes counts: A=35 D=33 B=32 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:201 D:197 E:196 C:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 4 16 B 2 0 2 2 -4 C -6 -2 0 -6 2 D -4 -2 6 0 -6 E -16 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.727273 C: 0.000000 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933801 Cumulative probabilities = A: 0.181818 B: 0.909091 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 4 16 B 2 0 2 2 -4 C -6 -2 0 -6 2 D -4 -2 6 0 -6 E -16 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.727273 C: 0.000000 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933801 Cumulative probabilities = A: 0.181818 B: 0.909091 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 4 16 B 2 0 2 2 -4 C -6 -2 0 -6 2 D -4 -2 6 0 -6 E -16 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.727273 C: 0.000000 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933801 Cumulative probabilities = A: 0.181818 B: 0.909091 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8760: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) A D E B C (9) B C A E D (8) C B E D A (6) E D A B C (5) A D E C B (5) A B E C D (5) C B D E A (4) D A E C B (3) D A E B C (3) B C E A D (3) A D C B E (3) A C D B E (3) E C B D A (2) E A D B C (2) D C A E B (2) D A C E B (2) C B A E D (2) B A C E D (2) A B E D C (2) A B D E C (2) E D C B A (1) E B A D C (1) E B A C D (1) E A B D C (1) D E C A B (1) D C E B A (1) D C E A B (1) C D E B A (1) C D B A E (1) C D A B E (1) C B E A D (1) C B A D E (1) C A B D E (1) B E C A D (1) A E D B C (1) A E B D C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 28 24 14 18 B -28 0 -2 -10 -4 C -24 2 0 -12 -10 D -14 10 12 0 8 E -18 4 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 28 24 14 18 B -28 0 -2 -10 -4 C -24 2 0 -12 -10 D -14 10 12 0 8 E -18 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=22 C=18 B=14 E=13 so E is eliminated. Round 2 votes counts: A=36 D=28 C=20 B=16 so B is eliminated. Round 3 votes counts: A=40 C=32 D=28 so D is eliminated. Round 4 votes counts: A=62 C=38 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:242 D:208 E:194 B:178 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 28 24 14 18 B -28 0 -2 -10 -4 C -24 2 0 -12 -10 D -14 10 12 0 8 E -18 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 28 24 14 18 B -28 0 -2 -10 -4 C -24 2 0 -12 -10 D -14 10 12 0 8 E -18 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 28 24 14 18 B -28 0 -2 -10 -4 C -24 2 0 -12 -10 D -14 10 12 0 8 E -18 4 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8761: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) B A E D C (10) B D E C A (8) A B E D C (6) A C D E B (5) A B E C D (5) D C E B A (4) C A D E B (4) A B C E D (4) C D B E A (3) B E D C A (3) B E D A C (3) B D C E A (3) A E C D B (3) A E B D C (3) E D A C B (2) D E C B A (2) C D E B A (2) C D A E B (2) A C E D B (2) A C B D E (2) E D C A B (1) E B D A C (1) E B A D C (1) E A D B C (1) D E B C A (1) D C B E A (1) C B D A E (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 10 4 0 4 B -10 0 4 6 4 C -4 -4 0 -4 -4 D 0 -6 4 0 4 E -4 -4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.550710 B: 0.000000 C: 0.000000 D: 0.449290 E: 0.000000 Sum of squares = 0.505143005921 Cumulative probabilities = A: 0.550710 B: 0.550710 C: 0.550710 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 0 4 B -10 0 4 6 4 C -4 -4 0 -4 -4 D 0 -6 4 0 4 E -4 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=29 C=23 D=8 E=6 so E is eliminated. Round 2 votes counts: A=35 B=31 C=23 D=11 so D is eliminated. Round 3 votes counts: A=37 B=32 C=31 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:209 B:202 D:201 E:196 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 0 4 B -10 0 4 6 4 C -4 -4 0 -4 -4 D 0 -6 4 0 4 E -4 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 0 4 B -10 0 4 6 4 C -4 -4 0 -4 -4 D 0 -6 4 0 4 E -4 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 0 4 B -10 0 4 6 4 C -4 -4 0 -4 -4 D 0 -6 4 0 4 E -4 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8762: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (14) B E D C A (11) A D C E B (6) C A D E B (5) A B C D E (5) C D E A B (4) B E D A C (4) E D C B A (3) E D B C A (3) D E C A B (3) C E D A B (3) B A E D C (3) B A C E D (3) D E A C B (2) D C A E B (2) C E D B A (2) C B A E D (2) B A E C D (2) A C B D E (2) A B D E C (2) E C D B A (1) D E B C A (1) D E A B C (1) D B A E C (1) D A E C B (1) D A C E B (1) C D A E B (1) C B E D A (1) C A E D B (1) C A B E D (1) B E A D C (1) B D E A C (1) B C E A D (1) B A D E C (1) A D E C B (1) A D B E C (1) A C D B E (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 18 10 8 16 B -18 0 -16 -16 -10 C -10 16 0 0 14 D -8 16 0 0 14 E -16 10 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 10 8 16 B -18 0 -16 -16 -10 C -10 16 0 0 14 D -8 16 0 0 14 E -16 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=27 C=20 D=12 E=7 so E is eliminated. Round 2 votes counts: A=34 B=27 C=21 D=18 so D is eliminated. Round 3 votes counts: A=39 B=32 C=29 so C is eliminated. Round 4 votes counts: A=59 B=41 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 D:211 C:210 E:183 B:170 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 8 16 B -18 0 -16 -16 -10 C -10 16 0 0 14 D -8 16 0 0 14 E -16 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 8 16 B -18 0 -16 -16 -10 C -10 16 0 0 14 D -8 16 0 0 14 E -16 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 8 16 B -18 0 -16 -16 -10 C -10 16 0 0 14 D -8 16 0 0 14 E -16 10 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8763: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (8) A E B D C (7) D A B C E (5) A D B C E (5) E A B D C (4) C B D A E (4) A E D B C (4) E D A C B (3) E C D B A (3) E C B D A (3) E A D B C (3) D C E B A (3) D C B A E (3) D C A B E (3) C E D B A (3) C D E B A (3) B A C D E (3) E D C A B (2) E A B C D (2) D A C B E (2) C B E A D (2) B A D C E (2) A D E B C (2) A D B E C (2) E C B A D (1) E C A B D (1) E A D C B (1) E A C B D (1) D C B E A (1) D B C A E (1) D B A C E (1) D A E C B (1) C E B D A (1) C E B A D (1) C D B E A (1) C D B A E (1) C B E D A (1) C B D E A (1) B C D A E (1) B C A D E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 18 18 6 18 B -18 0 8 -6 2 C -18 -8 0 -30 14 D -6 6 30 0 10 E -18 -2 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 18 6 18 B -18 0 8 -6 2 C -18 -8 0 -30 14 D -6 6 30 0 10 E -18 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=24 D=20 C=18 B=7 so B is eliminated. Round 2 votes counts: A=36 E=24 D=20 C=20 so D is eliminated. Round 3 votes counts: A=45 C=31 E=24 so E is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 D:220 B:193 C:179 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 18 6 18 B -18 0 8 -6 2 C -18 -8 0 -30 14 D -6 6 30 0 10 E -18 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 18 6 18 B -18 0 8 -6 2 C -18 -8 0 -30 14 D -6 6 30 0 10 E -18 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 18 6 18 B -18 0 8 -6 2 C -18 -8 0 -30 14 D -6 6 30 0 10 E -18 -2 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8764: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (10) D E C A B (8) B A C E D (8) C E A D B (6) C A E D B (6) A E C B D (5) E C A D B (4) C E D A B (4) D B E C A (3) B D A C E (3) A E B C D (3) E A C D B (2) D E A C B (2) D C E B A (2) D B E A C (2) D B C E A (2) C D E A B (2) B D A E C (2) B A D E C (2) E C D A B (1) E A D B C (1) E A C B D (1) D E C B A (1) C A E B D (1) C A B E D (1) B D E A C (1) B D C A E (1) B C D A E (1) B A E D C (1) A C E B D (1) A C B E D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 22 -6 10 -4 B -22 0 -16 -12 -20 C 6 16 0 18 0 D -10 12 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.487212 D: 0.000000 E: 0.512788 Sum of squares = 0.500327061607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.487212 D: 0.487212 E: 1.000000 A B C D E A 0 22 -6 10 -4 B -22 0 -16 -12 -20 C 6 16 0 18 0 D -10 12 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=29 C=20 A=12 E=9 so E is eliminated. Round 2 votes counts: D=30 B=29 C=25 A=16 so A is eliminated. Round 3 votes counts: C=35 B=34 D=31 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:220 E:220 A:211 D:184 B:165 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -6 10 -4 B -22 0 -16 -12 -20 C 6 16 0 18 0 D -10 12 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -6 10 -4 B -22 0 -16 -12 -20 C 6 16 0 18 0 D -10 12 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -6 10 -4 B -22 0 -16 -12 -20 C 6 16 0 18 0 D -10 12 -18 0 -16 E 4 20 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8765: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (9) B D A E C (9) A C B D E (9) C A E B D (8) A C E B D (8) D B E A C (6) E C D B A (5) A B D E C (5) D B E C A (4) A B C D E (4) E C A D B (3) C E D B A (3) C A E D B (3) B D E A C (3) A B D C E (3) D E B C A (2) C A B D E (2) A C B E D (2) E D B C A (1) E D B A C (1) E D A B C (1) E A D B C (1) D E B A C (1) C E D A B (1) C D B E A (1) C D B A E (1) C B A D E (1) B D C A E (1) B A D C E (1) A E C B D (1) Total count = 100 A B C D E A 0 20 10 20 16 B -20 0 -14 14 4 C -10 14 0 22 14 D -20 -14 -22 0 6 E -16 -4 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 10 20 16 B -20 0 -14 14 4 C -10 14 0 22 14 D -20 -14 -22 0 6 E -16 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=29 B=14 D=13 E=12 so E is eliminated. Round 2 votes counts: C=37 A=33 D=16 B=14 so B is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:233 C:220 B:192 E:180 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 10 20 16 B -20 0 -14 14 4 C -10 14 0 22 14 D -20 -14 -22 0 6 E -16 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 10 20 16 B -20 0 -14 14 4 C -10 14 0 22 14 D -20 -14 -22 0 6 E -16 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 10 20 16 B -20 0 -14 14 4 C -10 14 0 22 14 D -20 -14 -22 0 6 E -16 -4 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8766: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (20) E D B C A (16) A C B D E (16) C A E D B (14) A C B E D (5) C A B E D (4) E D C A B (3) D E B C A (3) D E B A C (3) B A D C E (3) B A C D E (3) A B C D E (3) D B E A C (2) E D C B A (1) E C D B A (1) E C D A B (1) C A B D E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 12 0 0 B 4 0 6 10 14 C -12 -6 0 -2 0 D 0 -10 2 0 8 E 0 -14 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 0 0 B 4 0 6 10 14 C -12 -6 0 -2 0 D 0 -10 2 0 8 E 0 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=25 E=22 C=19 D=8 so D is eliminated. Round 2 votes counts: E=28 B=28 A=25 C=19 so C is eliminated. Round 3 votes counts: A=44 E=28 B=28 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:204 D:200 C:190 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 0 0 B 4 0 6 10 14 C -12 -6 0 -2 0 D 0 -10 2 0 8 E 0 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 0 0 B 4 0 6 10 14 C -12 -6 0 -2 0 D 0 -10 2 0 8 E 0 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 0 0 B 4 0 6 10 14 C -12 -6 0 -2 0 D 0 -10 2 0 8 E 0 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8767: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) A E D B C (9) E A B D C (7) D C A B E (7) C B D E A (7) D A C E B (5) C D B E A (5) B E C A D (5) A D E C B (5) E B C A D (4) B E A C D (4) E B A C D (3) D C A E B (3) B C D E A (3) A D E B C (3) E D C A B (2) D A C B E (2) C E B D A (2) B E A D C (2) E A D B C (1) D C E A B (1) C D E A B (1) C D A B E (1) B E C D A (1) B C E D A (1) B C E A D (1) B C D A E (1) B A C D E (1) A E D C B (1) A E B D C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -8 -2 0 B -2 0 -2 -12 4 C 8 2 0 -2 2 D 2 12 2 0 10 E 0 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -2 0 B -2 0 -2 -12 4 C 8 2 0 -2 2 D 2 12 2 0 10 E 0 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=21 B=19 D=18 E=17 so E is eliminated. Round 2 votes counts: A=29 B=26 C=25 D=20 so D is eliminated. Round 3 votes counts: C=38 A=36 B=26 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:205 A:196 B:194 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 -2 0 B -2 0 -2 -12 4 C 8 2 0 -2 2 D 2 12 2 0 10 E 0 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -2 0 B -2 0 -2 -12 4 C 8 2 0 -2 2 D 2 12 2 0 10 E 0 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -2 0 B -2 0 -2 -12 4 C 8 2 0 -2 2 D 2 12 2 0 10 E 0 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998224 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8768: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (13) D B C E A (7) D B A C E (7) A E C B D (7) E C A D B (6) B D A C E (6) A D B E C (5) B D C A E (4) A E C D B (4) D B C A E (3) C E A B D (3) B C D E A (3) E C B D A (2) E A C D B (2) E A C B D (2) D B A E C (2) C E B D A (2) C B E D A (2) B A D C E (2) A B D C E (2) E C D B A (1) E C B A D (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E B A (1) C B D E A (1) C A E B D (1) B D C E A (1) B A C E D (1) A E D C B (1) A E D B C (1) A D E B C (1) A C E B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 10 0 B -4 0 -6 12 -4 C 6 6 0 8 -2 D -10 -12 -8 0 -2 E 0 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.172579 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.827421 Sum of squares = 0.714408699378 Cumulative probabilities = A: 0.172579 B: 0.172579 C: 0.172579 D: 0.172579 E: 1.000000 A B C D E A 0 4 -6 10 0 B -4 0 -6 12 -4 C 6 6 0 8 -2 D -10 -12 -8 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028541 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=24 D=23 B=17 C=9 so C is eliminated. Round 2 votes counts: E=32 A=25 D=23 B=20 so B is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 A:204 E:204 B:199 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -6 10 0 B -4 0 -6 12 -4 C 6 6 0 8 -2 D -10 -12 -8 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028541 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 10 0 B -4 0 -6 12 -4 C 6 6 0 8 -2 D -10 -12 -8 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028541 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 10 0 B -4 0 -6 12 -4 C 6 6 0 8 -2 D -10 -12 -8 0 -2 E 0 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000028541 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8769: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (13) A D B E C (11) A D B C E (11) C E B D A (9) E C A B D (6) E A C D B (6) D B A C E (6) D A B C E (5) B D C A E (5) B C D E A (5) A E D C B (5) C B D E A (4) A D E B C (4) E C B A D (3) A E D B C (2) A E C D B (2) E A D C B (1) C B E D A (1) B D A E C (1) Total count = 100 A B C D E A 0 6 8 2 4 B -6 0 0 -6 -2 C -8 0 0 -2 -8 D -2 6 2 0 4 E -4 2 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999455 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 2 4 B -6 0 0 -6 -2 C -8 0 0 -2 -8 D -2 6 2 0 4 E -4 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=29 C=14 D=11 B=11 so D is eliminated. Round 2 votes counts: A=40 E=29 B=17 C=14 so C is eliminated. Round 3 votes counts: A=40 E=38 B=22 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:205 E:201 B:193 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 4 B -6 0 0 -6 -2 C -8 0 0 -2 -8 D -2 6 2 0 4 E -4 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 4 B -6 0 0 -6 -2 C -8 0 0 -2 -8 D -2 6 2 0 4 E -4 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 4 B -6 0 0 -6 -2 C -8 0 0 -2 -8 D -2 6 2 0 4 E -4 2 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8770: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) D E C A B (6) A E B D C (6) E D A C B (5) B C E A D (5) E D A B C (4) D C E A B (4) C D B E A (4) C B D A E (4) B A E C D (4) A D E C B (4) A B E C D (4) E D C B A (3) E A D B C (3) E A B D C (3) C A D B E (3) B A C E D (3) E B A D C (2) C D E B A (2) C B A D E (2) B E C A D (2) A E D B C (2) A C D B E (2) E B D A C (1) E A D C B (1) D E C B A (1) D E A C B (1) D C A E B (1) C D B A E (1) C D A B E (1) C B E D A (1) C B D E A (1) B E A C D (1) B C E D A (1) B C D E A (1) B C D A E (1) B C A E D (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -6 0 -18 B -2 0 -10 -12 -14 C 6 10 0 -10 -6 D 0 12 10 0 -4 E 18 14 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -6 0 -18 B -2 0 -10 -12 -14 C 6 10 0 -10 -6 D 0 12 10 0 -4 E 18 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=22 D=20 B=20 C=19 A=19 so C is eliminated. Round 2 votes counts: D=28 B=28 E=22 A=22 so E is eliminated. Round 3 votes counts: D=40 B=31 A=29 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:221 D:209 C:200 A:189 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 0 -18 B -2 0 -10 -12 -14 C 6 10 0 -10 -6 D 0 12 10 0 -4 E 18 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 -18 B -2 0 -10 -12 -14 C 6 10 0 -10 -6 D 0 12 10 0 -4 E 18 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 -18 B -2 0 -10 -12 -14 C 6 10 0 -10 -6 D 0 12 10 0 -4 E 18 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8771: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) B E A D C (8) D C B E A (7) B E D C A (7) A E C D B (7) A E B C D (7) E B A D C (5) C D A B E (5) A C D E B (4) A C D B E (4) E A B D C (3) C D A E B (3) A B E D C (3) E D C B A (2) E B D C A (2) E A B C D (2) C D E A B (2) C D B A E (2) C A D E B (2) A E B D C (2) A C E D B (2) E C D A B (1) E A C D B (1) D C B A E (1) D B C E A (1) C E A D B (1) C D E B A (1) C D B E A (1) B E D A C (1) B D E C A (1) B D C A E (1) B D A C E (1) A E C B D (1) Total count = 100 A B C D E A 0 0 2 4 -10 B 0 0 6 6 4 C -2 -6 0 -8 -6 D -4 -6 8 0 -10 E 10 -4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.199980 B: 0.800020 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680023501318 Cumulative probabilities = A: 0.199980 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 -10 B 0 0 6 6 4 C -2 -6 0 -8 -6 D -4 -6 8 0 -10 E 10 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836765134 Cumulative probabilities = A: 0.285714 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=28 C=17 E=16 D=9 so D is eliminated. Round 2 votes counts: A=30 B=29 C=25 E=16 so E is eliminated. Round 3 votes counts: B=36 A=36 C=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:211 B:208 A:198 D:194 C:189 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 4 -10 B 0 0 6 6 4 C -2 -6 0 -8 -6 D -4 -6 8 0 -10 E 10 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836765134 Cumulative probabilities = A: 0.285714 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 -10 B 0 0 6 6 4 C -2 -6 0 -8 -6 D -4 -6 8 0 -10 E 10 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836765134 Cumulative probabilities = A: 0.285714 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 -10 B 0 0 6 6 4 C -2 -6 0 -8 -6 D -4 -6 8 0 -10 E 10 -4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.591836765134 Cumulative probabilities = A: 0.285714 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8772: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (17) B C A D E (13) E D A C B (12) D E B C A (9) E A D C B (6) D B C E A (6) C A B E D (5) E D A B C (4) B C D A E (4) A E C D B (4) D E B A C (3) D E A C B (3) E D B A C (2) A C E B D (2) E A C D B (1) D E C B A (1) D C B E A (1) D B E C A (1) C B A D E (1) B D C E A (1) B D C A E (1) B C A E D (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 12 12 4 0 B -12 0 -8 -6 4 C -12 8 0 0 4 D -4 6 0 0 -12 E 0 -4 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.578867 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.421133 Sum of squares = 0.512439878214 Cumulative probabilities = A: 0.578867 B: 0.578867 C: 0.578867 D: 0.578867 E: 1.000000 A B C D E A 0 12 12 4 0 B -12 0 -8 -6 4 C -12 8 0 0 4 D -4 6 0 0 -12 E 0 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 D=24 B=20 C=6 so C is eliminated. Round 2 votes counts: A=30 E=25 D=24 B=21 so B is eliminated. Round 3 votes counts: A=45 D=30 E=25 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:202 C:200 D:195 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 4 0 B -12 0 -8 -6 4 C -12 8 0 0 4 D -4 6 0 0 -12 E 0 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 4 0 B -12 0 -8 -6 4 C -12 8 0 0 4 D -4 6 0 0 -12 E 0 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 4 0 B -12 0 -8 -6 4 C -12 8 0 0 4 D -4 6 0 0 -12 E 0 -4 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8773: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) A B E C D (6) E C A D B (5) D C E B A (5) B A E D C (5) A B C D E (5) D C B E A (4) D B C A E (4) C D E B A (4) B A D E C (4) A B E D C (4) E C D A B (3) E A C B D (3) E A B D C (3) C E D A B (3) B A C D E (3) A E B C D (3) E D C B A (2) E B D A C (2) E A C D B (2) D B C E A (2) B A D C E (2) A E B D C (2) E D B C A (1) D E C B A (1) D B E C A (1) C E D B A (1) C D A B E (1) C A E D B (1) C A B D E (1) B D E A C (1) B D A E C (1) B D A C E (1) A C E B D (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 6 14 B 8 0 24 20 16 C -2 -24 0 -12 2 D -6 -20 12 0 2 E -14 -16 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 6 14 B 8 0 24 20 16 C -2 -24 0 -12 2 D -6 -20 12 0 2 E -14 -16 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 E=21 D=17 C=11 so C is eliminated. Round 2 votes counts: B=27 A=26 E=25 D=22 so D is eliminated. Round 3 votes counts: B=38 E=35 A=27 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:234 A:207 D:194 E:183 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 6 14 B 8 0 24 20 16 C -2 -24 0 -12 2 D -6 -20 12 0 2 E -14 -16 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 6 14 B 8 0 24 20 16 C -2 -24 0 -12 2 D -6 -20 12 0 2 E -14 -16 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 6 14 B 8 0 24 20 16 C -2 -24 0 -12 2 D -6 -20 12 0 2 E -14 -16 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999782 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8774: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) E C A D B (8) E A C B D (8) C D E B A (7) B D C A E (7) B D A E C (7) C E D A B (6) D B C A E (5) B A E D C (5) A E B C D (5) B A D E C (4) A E B D C (4) E C D A B (3) C D E A B (3) C D B E A (3) B D A C E (3) A B E D C (3) E A C D B (2) A B E C D (2) E A B C D (1) D C E B A (1) D B A C E (1) C E A D B (1) B A E C D (1) B A D C E (1) Total count = 100 A B C D E A 0 -8 -6 -10 -4 B 8 0 -2 2 2 C 6 2 0 0 -6 D 10 -2 0 0 2 E 4 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999975 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -8 -6 -10 -4 B 8 0 -2 2 2 C 6 2 0 0 -6 D 10 -2 0 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999653 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=22 C=20 D=16 A=14 so A is eliminated. Round 2 votes counts: B=33 E=31 C=20 D=16 so D is eliminated. Round 3 votes counts: B=39 E=31 C=30 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:205 D:205 E:203 C:201 A:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 -4 B 8 0 -2 2 2 C 6 2 0 0 -6 D 10 -2 0 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999653 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 -4 B 8 0 -2 2 2 C 6 2 0 0 -6 D 10 -2 0 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999653 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 -4 B 8 0 -2 2 2 C 6 2 0 0 -6 D 10 -2 0 0 2 E 4 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999653 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8775: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (8) C D E B A (7) E A C D B (6) C E D B A (6) C D B A E (6) E A B D C (5) C D B E A (5) A E B D C (5) E A C B D (4) E A B C D (4) C D E A B (4) B A D C E (4) E C D A B (3) D B C A E (3) A B E D C (3) A B D E C (3) E C A D B (2) D C B A E (2) C E D A B (2) B D A C E (2) B A E D C (2) E C B D A (1) E C B A D (1) E B C A D (1) E B A D C (1) E B A C D (1) D B A C E (1) D A C B E (1) B D C E A (1) B C D E A (1) B C D A E (1) A E D C B (1) A E D B C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -8 -8 -8 -10 B 8 0 -4 -4 -12 C 8 4 0 10 12 D 8 4 -10 0 2 E 10 12 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -8 -10 B 8 0 -4 -4 -12 C 8 4 0 10 12 D 8 4 -10 0 2 E 10 12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=29 B=19 A=15 D=7 so D is eliminated. Round 2 votes counts: C=32 E=29 B=23 A=16 so A is eliminated. Round 3 votes counts: E=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:204 D:202 B:194 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -8 -10 B 8 0 -4 -4 -12 C 8 4 0 10 12 D 8 4 -10 0 2 E 10 12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -8 -10 B 8 0 -4 -4 -12 C 8 4 0 10 12 D 8 4 -10 0 2 E 10 12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -8 -10 B 8 0 -4 -4 -12 C 8 4 0 10 12 D 8 4 -10 0 2 E 10 12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8776: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (12) C D A B E (8) E A B D C (7) D C A E B (6) B E A C D (6) B E A D C (5) E B A C D (4) B E D A C (4) B C E D A (4) B C E A D (4) E B A D C (3) D E A B C (3) C D B A E (3) E D A B C (2) D C B A E (2) D B C A E (2) D A E C B (2) C A D E B (2) A E D C B (2) A D E C B (2) E B D A C (1) E A D B C (1) E A C B D (1) D C A B E (1) D A B E C (1) C D A E B (1) C B A E D (1) C A E D B (1) C A D B E (1) B E D C A (1) B E C A D (1) B D E A C (1) B C D E A (1) A E D B C (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 14 22 -10 2 B -14 0 4 -10 -8 C -22 -4 0 -18 2 D 10 10 18 0 -4 E -2 8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000178 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 A B C D E A 0 14 22 -10 2 B -14 0 4 -10 -8 C -22 -4 0 -18 2 D 10 10 18 0 -4 E -2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000042 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=27 E=19 C=17 A=8 so A is eliminated. Round 2 votes counts: D=31 B=27 E=23 C=19 so C is eliminated. Round 3 votes counts: D=46 B=28 E=26 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:214 E:204 B:186 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 22 -10 2 B -14 0 4 -10 -8 C -22 -4 0 -18 2 D 10 10 18 0 -4 E -2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000042 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 22 -10 2 B -14 0 4 -10 -8 C -22 -4 0 -18 2 D 10 10 18 0 -4 E -2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000042 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 22 -10 2 B -14 0 4 -10 -8 C -22 -4 0 -18 2 D 10 10 18 0 -4 E -2 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000042 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8777: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (13) E C D B A (7) D C E A B (7) D A C E B (7) A B D E C (7) E C B D A (6) A D C E B (6) A B D C E (5) D C A E B (4) B E C D A (4) B E C A D (4) B A E C D (4) A D C B E (3) E B C D A (2) C E D A B (2) C D E B A (2) C D E A B (2) B D E C A (2) B A E D C (2) E B C A D (1) D E C B A (1) D B C E A (1) D A C B E (1) C E D B A (1) C E A D B (1) C A E D B (1) B E A C D (1) B D E A C (1) B D A E C (1) B A D E C (1) Total count = 100 A B C D E A 0 18 4 -2 10 B -18 0 -2 -18 0 C -4 2 0 -24 12 D 2 18 24 0 28 E -10 0 -12 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 4 -2 10 B -18 0 -2 -18 0 C -4 2 0 -24 12 D 2 18 24 0 28 E -10 0 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999916099 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=21 B=20 E=16 C=9 so C is eliminated. Round 2 votes counts: A=35 D=25 E=20 B=20 so E is eliminated. Round 3 votes counts: A=36 D=35 B=29 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:236 A:215 C:193 B:181 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 4 -2 10 B -18 0 -2 -18 0 C -4 2 0 -24 12 D 2 18 24 0 28 E -10 0 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999916099 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 4 -2 10 B -18 0 -2 -18 0 C -4 2 0 -24 12 D 2 18 24 0 28 E -10 0 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999916099 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 4 -2 10 B -18 0 -2 -18 0 C -4 2 0 -24 12 D 2 18 24 0 28 E -10 0 -12 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999916099 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8778: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) E A D C B (7) C B D E A (6) B C D A E (6) A E D B C (6) E A B D C (5) C B D A E (5) A D E C B (5) E A B C D (4) D A E C B (4) D A C E B (4) C D B A E (4) E A D B C (3) C D B E A (3) B C E A D (3) B C D E A (3) E D A C B (2) E B A C D (2) E A C B D (2) D E C A B (2) D A B C E (2) A E B D C (2) A D E B C (2) E C D A B (1) E C B A D (1) D C E A B (1) D C A B E (1) D B A C E (1) C D E B A (1) C B E D A (1) B E C A D (1) B C E D A (1) B C A D E (1) Total count = 100 A B C D E A 0 6 2 -12 2 B -6 0 -16 -14 -8 C -2 16 0 -10 2 D 12 14 10 0 18 E -2 8 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -12 2 B -6 0 -16 -14 -8 C -2 16 0 -10 2 D 12 14 10 0 18 E -2 8 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 C=20 B=15 A=15 so B is eliminated. Round 2 votes counts: C=34 E=28 D=23 A=15 so A is eliminated. Round 3 votes counts: E=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:227 C:203 A:199 E:193 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -12 2 B -6 0 -16 -14 -8 C -2 16 0 -10 2 D 12 14 10 0 18 E -2 8 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -12 2 B -6 0 -16 -14 -8 C -2 16 0 -10 2 D 12 14 10 0 18 E -2 8 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -12 2 B -6 0 -16 -14 -8 C -2 16 0 -10 2 D 12 14 10 0 18 E -2 8 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8779: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (9) E C A B D (8) D B A C E (8) B D C A E (8) E A D C B (7) A D E B C (7) C B E D A (6) E C A D B (5) B C D A E (5) A E D B C (5) D A B E C (4) E A C D B (3) D A E B C (3) C B D E A (3) B D A C E (3) A D B E C (3) E C B A D (2) E A C B D (2) D E C A B (2) E D A C B (1) D B C A E (1) C E B D A (1) B C A E D (1) B A E C D (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 4 -2 10 2 B -4 0 -2 -2 -14 C 2 2 0 -8 -10 D -10 2 8 0 -4 E -2 14 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408193 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 4 -2 10 2 B -4 0 -2 -2 -14 C 2 2 0 -8 -10 D -10 2 8 0 -4 E -2 14 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.55102040814 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=19 D=18 B=18 A=17 so A is eliminated. Round 2 votes counts: E=34 D=29 C=19 B=18 so B is eliminated. Round 3 votes counts: D=40 E=35 C=25 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:207 D:198 C:193 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 10 2 B -4 0 -2 -2 -14 C 2 2 0 -8 -10 D -10 2 8 0 -4 E -2 14 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.55102040814 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 10 2 B -4 0 -2 -2 -14 C 2 2 0 -8 -10 D -10 2 8 0 -4 E -2 14 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.55102040814 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 10 2 B -4 0 -2 -2 -14 C 2 2 0 -8 -10 D -10 2 8 0 -4 E -2 14 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.714286 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.142857 Sum of squares = 0.55102040814 Cumulative probabilities = A: 0.714286 B: 0.714286 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8780: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A C B E D (8) E D B A C (7) D E B C A (7) A B C E D (6) D E C B A (5) D B E C A (4) C D E A B (4) B E A D C (4) B A C E D (4) E A D C B (3) D E C A B (3) C A D E B (3) B E D A C (3) E D A C B (2) E A D B C (2) D C E B A (2) D C E A B (2) C D E B A (2) C D A E B (2) C B A D E (2) B A E D C (2) B A C D E (2) A C E B D (2) E B D A C (1) C D A B E (1) B D E C A (1) B D E A C (1) B D C E A (1) B C A D E (1) B A E C D (1) A E D B C (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 0 4 4 -8 B 0 0 -4 -4 -2 C -4 4 0 -2 2 D -4 4 2 0 2 E 8 2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428517 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 A B C D E A 0 0 4 4 -8 B 0 0 -4 -4 -2 C -4 4 0 -2 2 D -4 4 2 0 2 E 8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428589 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 B=20 A=20 E=15 so E is eliminated. Round 2 votes counts: D=32 A=25 C=22 B=21 so B is eliminated. Round 3 votes counts: D=39 A=38 C=23 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:203 D:202 A:200 C:200 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 4 4 -8 B 0 0 -4 -4 -2 C -4 4 0 -2 2 D -4 4 2 0 2 E 8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428589 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 -8 B 0 0 -4 -4 -2 C -4 4 0 -2 2 D -4 4 2 0 2 E 8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428589 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 -8 B 0 0 -4 -4 -2 C -4 4 0 -2 2 D -4 4 2 0 2 E 8 2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.285714 Sum of squares = 0.428571428589 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 0.142857 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8781: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) D E C A B (8) D A B E C (8) C E B A D (7) D A E B C (6) C B E A D (6) A B D C E (6) E D C A B (4) D A B C E (4) B C A E D (4) B A C D E (4) C B A E D (3) B A C E D (3) A D B E C (3) E D C B A (2) E C D A B (2) E C B D A (2) D E A C B (2) C E B D A (2) E C B A D (1) E C A D B (1) E C A B D (1) E A C B D (1) D E C B A (1) D E A B C (1) D C E B A (1) D B C A E (1) C B E D A (1) B A D C E (1) A E D B C (1) A D B C E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -14 -10 -6 B -2 0 -12 -12 -6 C 14 12 0 0 -8 D 10 12 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -14 -10 -6 B -2 0 -12 -12 -6 C 14 12 0 0 -8 D 10 12 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 E=24 C=19 A=13 B=12 so B is eliminated. Round 2 votes counts: D=32 E=24 C=23 A=21 so A is eliminated. Round 3 votes counts: D=43 C=32 E=25 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:209 D:208 A:186 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -14 -10 -6 B -2 0 -12 -12 -6 C 14 12 0 0 -8 D 10 12 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -10 -6 B -2 0 -12 -12 -6 C 14 12 0 0 -8 D 10 12 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -10 -6 B -2 0 -12 -12 -6 C 14 12 0 0 -8 D 10 12 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8782: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) C E B D A (9) E C D A B (6) C E D A B (6) E A D C B (5) C B D A E (5) E C B A D (4) E C A D B (4) C B E D A (4) B D A C E (4) E B C A D (3) E A D B C (3) D A C B E (3) D A B C E (3) C D A E B (3) B E C A D (3) A D E B C (3) A D B E C (3) B E A D C (2) B C E A D (2) B C D A E (2) B A D C E (2) E B A D C (1) D C A E B (1) D A E C B (1) D A C E B (1) D A B E C (1) C D A B E (1) C B E A D (1) B C E D A (1) B C A D E (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -12 -2 -8 B 8 0 -10 8 -4 C 12 10 0 10 -2 D 2 -8 -10 0 -10 E 8 4 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -12 -2 -8 B 8 0 -10 8 -4 C 12 10 0 10 -2 D 2 -8 -10 0 -10 E 8 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=27 E=26 D=10 A=8 so A is eliminated. Round 2 votes counts: C=29 E=27 B=27 D=17 so D is eliminated. Round 3 votes counts: C=34 B=34 E=32 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:212 B:201 D:187 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -12 -2 -8 B 8 0 -10 8 -4 C 12 10 0 10 -2 D 2 -8 -10 0 -10 E 8 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -2 -8 B 8 0 -10 8 -4 C 12 10 0 10 -2 D 2 -8 -10 0 -10 E 8 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -2 -8 B 8 0 -10 8 -4 C 12 10 0 10 -2 D 2 -8 -10 0 -10 E 8 4 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8783: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (13) B D E A C (10) C B D E A (8) B D E C A (7) C A E D B (6) B C D E A (5) A E C D B (5) A C E D B (5) D B E A C (4) C B A E D (4) C A B E D (4) B D C E A (4) A E D C B (4) E D A B C (3) C A E B D (3) D E B A C (2) D E A B C (2) C E A D B (2) C B E D A (2) E D A C B (1) E A D C B (1) E A D B C (1) C E D A B (1) B C A D E (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 6 2 -6 B -4 0 8 0 2 C -6 -8 0 -8 -10 D -2 0 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888895 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 4 6 2 -6 B -4 0 8 0 2 C -6 -8 0 -8 -10 D -2 0 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888602 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=28 A=28 D=8 E=6 so E is eliminated. Round 2 votes counts: C=30 A=30 B=28 D=12 so D is eliminated. Round 3 votes counts: A=36 B=34 C=30 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:213 A:203 B:203 D:197 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 2 -6 B -4 0 8 0 2 C -6 -8 0 -8 -10 D -2 0 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888602 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 2 -6 B -4 0 8 0 2 C -6 -8 0 -8 -10 D -2 0 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888602 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 2 -6 B -4 0 8 0 2 C -6 -8 0 -8 -10 D -2 0 8 0 -12 E 6 -2 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888602 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8784: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (18) D B A E C (12) D B E A C (5) D B A C E (5) D A B C E (5) E C B A D (4) C E D A B (4) A D B C E (4) E C B D A (3) A C B D E (3) E C D B A (2) E C A B D (2) E B A D C (2) D E B A C (2) C A E B D (2) C A D B E (2) B D A E C (2) B A D E C (2) A B D C E (2) E D C B A (1) E D B C A (1) E D B A C (1) E B D C A (1) E B D A C (1) E B C A D (1) E B A C D (1) D E C B A (1) D C B E A (1) D C A B E (1) D B C A E (1) D A C B E (1) C E D B A (1) C D A B E (1) C A D E B (1) C A B D E (1) B D E A C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 -6 -6 B 2 0 0 -4 2 C -2 0 0 -4 10 D 6 4 4 0 8 E 6 -2 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -6 -6 B 2 0 0 -4 2 C -2 0 0 -4 10 D 6 4 4 0 8 E 6 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=30 E=20 A=11 B=5 so B is eliminated. Round 2 votes counts: D=37 C=30 E=20 A=13 so A is eliminated. Round 3 votes counts: D=45 C=35 E=20 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:202 B:200 A:194 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -6 -6 B 2 0 0 -4 2 C -2 0 0 -4 10 D 6 4 4 0 8 E 6 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -6 -6 B 2 0 0 -4 2 C -2 0 0 -4 10 D 6 4 4 0 8 E 6 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -6 -6 B 2 0 0 -4 2 C -2 0 0 -4 10 D 6 4 4 0 8 E 6 -2 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8785: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (18) B A C E D (14) D E B A C (9) C A B E D (9) D E C B A (6) B A C D E (4) A B C E D (4) E C D A B (3) E C A B D (3) D C E A B (3) D B A C E (3) E D C A B (2) E D B A C (2) E C B A D (2) D B A E C (2) C E A B D (2) B A E C D (2) B A D C E (2) A C B E D (2) A B C D E (2) E D C B A (1) E B A D C (1) E B A C D (1) D C A B E (1) D A B C E (1) C A E B D (1) Total count = 100 A B C D E A 0 2 -2 -2 -6 B -2 0 -6 -2 -8 C 2 6 0 -2 -4 D 2 2 2 0 2 E 6 8 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -2 -6 B -2 0 -6 -2 -8 C 2 6 0 -2 -4 D 2 2 2 0 2 E 6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 B=22 E=15 C=12 A=8 so A is eliminated. Round 2 votes counts: D=43 B=28 E=15 C=14 so C is eliminated. Round 3 votes counts: D=43 B=39 E=18 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:208 D:204 C:201 A:196 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -2 -6 B -2 0 -6 -2 -8 C 2 6 0 -2 -4 D 2 2 2 0 2 E 6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -2 -6 B -2 0 -6 -2 -8 C 2 6 0 -2 -4 D 2 2 2 0 2 E 6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -2 -6 B -2 0 -6 -2 -8 C 2 6 0 -2 -4 D 2 2 2 0 2 E 6 8 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8786: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) D B E A C (6) C A E B D (6) B A C E D (6) D E B A C (5) D E A B C (5) C E D A B (4) B D A E C (4) E A D B C (3) D E C A B (3) C D B E A (3) C B A D E (3) C A B E D (3) B C D A E (3) A B E C D (3) E D A B C (2) D E C B A (2) D E A C B (2) D C B E A (2) D B C E A (2) C B D A E (2) B D A C E (2) B C A D E (2) B A D E C (2) E D C A B (1) E D A C B (1) E C D A B (1) E C A D B (1) E A D C B (1) D E B C A (1) D B E C A (1) C E A D B (1) C D E B A (1) C A E D B (1) B D C A E (1) B C A E D (1) B A D C E (1) A E B D C (1) A C E B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -6 -8 4 B 16 0 6 2 14 C 6 -6 0 2 8 D 8 -2 -2 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999305 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 -8 4 B 16 0 6 2 14 C 6 -6 0 2 8 D 8 -2 -2 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=29 B=22 E=10 A=7 so A is eliminated. Round 2 votes counts: C=33 D=29 B=27 E=11 so E is eliminated. Round 3 votes counts: D=37 C=35 B=28 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:219 C:205 D:205 A:187 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 -8 4 B 16 0 6 2 14 C 6 -6 0 2 8 D 8 -2 -2 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 -8 4 B 16 0 6 2 14 C 6 -6 0 2 8 D 8 -2 -2 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 -8 4 B 16 0 6 2 14 C 6 -6 0 2 8 D 8 -2 -2 0 6 E -4 -14 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8787: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (11) B D E A C (10) B E D C A (8) E D B C A (7) A C D E B (7) E D C B A (6) A C D B E (5) B D A E C (4) B A D C E (4) A B C D E (4) D E B C A (3) B D E C A (3) A C B E D (3) A C B D E (3) E D C A B (2) D E A B C (2) C B A E D (2) B A D E C (2) B A C D E (2) E C D B A (1) E B D C A (1) D E C A B (1) D E B A C (1) C E D A B (1) C E A D B (1) C A E B D (1) C A B E D (1) B E D A C (1) B A C E D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 2 -2 4 B 12 0 6 0 8 C -2 -6 0 -14 -4 D 2 0 14 0 6 E -4 -8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.525550 C: 0.000000 D: 0.474450 E: 0.000000 Sum of squares = 0.501305605867 Cumulative probabilities = A: 0.000000 B: 0.525550 C: 0.525550 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 -2 4 B 12 0 6 0 8 C -2 -6 0 -14 -4 D 2 0 14 0 6 E -4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=24 E=17 C=17 D=7 so D is eliminated. Round 2 votes counts: B=35 E=24 A=24 C=17 so C is eliminated. Round 3 votes counts: B=37 A=37 E=26 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:211 A:196 E:193 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 -2 4 B 12 0 6 0 8 C -2 -6 0 -14 -4 D 2 0 14 0 6 E -4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -2 4 B 12 0 6 0 8 C -2 -6 0 -14 -4 D 2 0 14 0 6 E -4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -2 4 B 12 0 6 0 8 C -2 -6 0 -14 -4 D 2 0 14 0 6 E -4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8788: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (5) D E A C B (5) D E A B C (5) C B A E D (5) C A B E D (5) A C E B D (5) D E B C A (4) D E B A C (4) D B E C A (4) B C A E D (4) A E C B D (4) A C B E D (4) E A D C B (3) A E C D B (3) E D A C B (2) E B A C D (2) E A B C D (2) D C B A E (2) D C A E B (2) D C A B E (2) C B A D E (2) B D C A E (2) B C E A D (2) B C D A E (2) B C A D E (2) A C E D B (2) E D B A C (1) E B A D C (1) E A D B C (1) E A C B D (1) D C B E A (1) D B C E A (1) D A E C B (1) C D B A E (1) C A D E B (1) C A D B E (1) C A B D E (1) B E C D A (1) B E A D C (1) B E A C D (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 14 10 10 6 B -14 0 -8 -4 -12 C -10 8 0 6 -4 D -10 4 -6 0 -12 E -6 12 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 10 6 B -14 0 -8 -4 -12 C -10 8 0 6 -4 D -10 4 -6 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=20 E=18 C=16 B=15 so B is eliminated. Round 2 votes counts: D=33 C=26 E=21 A=20 so A is eliminated. Round 3 votes counts: C=38 D=33 E=29 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:220 E:211 C:200 D:188 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 10 6 B -14 0 -8 -4 -12 C -10 8 0 6 -4 D -10 4 -6 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 10 6 B -14 0 -8 -4 -12 C -10 8 0 6 -4 D -10 4 -6 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 10 6 B -14 0 -8 -4 -12 C -10 8 0 6 -4 D -10 4 -6 0 -12 E -6 12 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8789: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) D E A C B (7) D B A E C (6) B D C A E (5) D B E A C (4) D A E C B (4) B C E A D (4) B C A D E (4) E C A D B (3) D B E C A (3) C E A B D (3) B D A C E (3) B C D E A (3) B C D A E (3) B A D C E (3) E D C B A (2) E D A C B (2) E C B D A (2) E C A B D (2) D E A B C (2) D A E B C (2) D A B E C (2) C E B A D (2) C A E B D (2) B D C E A (2) A E D C B (2) A E C D B (2) A D E C B (2) A C E B D (2) E D C A B (1) E A D C B (1) E A C D B (1) C B E A D (1) B D E C A (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -6 -8 8 B 18 0 18 8 12 C 6 -18 0 -8 -2 D 8 -8 8 0 16 E -8 -12 2 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -6 -8 8 B 18 0 18 8 12 C 6 -18 0 -8 -2 D 8 -8 8 0 16 E -8 -12 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 D=30 E=14 A=9 C=8 so C is eliminated. Round 2 votes counts: B=40 D=30 E=19 A=11 so A is eliminated. Round 3 votes counts: B=41 D=32 E=27 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:228 D:212 C:189 A:188 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -6 -8 8 B 18 0 18 8 12 C 6 -18 0 -8 -2 D 8 -8 8 0 16 E -8 -12 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -6 -8 8 B 18 0 18 8 12 C 6 -18 0 -8 -2 D 8 -8 8 0 16 E -8 -12 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -6 -8 8 B 18 0 18 8 12 C 6 -18 0 -8 -2 D 8 -8 8 0 16 E -8 -12 2 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8790: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) E A B D C (6) A E C D B (6) E A D C B (4) C D B E A (4) C D B A E (4) B C D E A (4) B C D A E (4) E D C B A (3) E D A C B (3) E A D B C (3) D E C B A (3) D C B E A (3) B D C E A (3) B A E D C (3) A B C D E (3) E D C A B (2) E B A D C (2) D C E A B (2) D B C E A (2) B E A D C (2) B A E C D (2) A E D C B (2) A E B C D (2) A C B D E (2) A B E C D (2) E D B C A (1) E B D C A (1) D C E B A (1) C D E B A (1) C D A E B (1) C D A B E (1) C B D E A (1) C B D A E (1) C A D E B (1) B E D A C (1) B A C D E (1) A E D B C (1) A E C B D (1) A C D E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 6 14 8 -4 B -6 0 4 0 -10 C -14 -4 0 -12 -16 D -8 0 12 0 -12 E 4 10 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 14 8 -4 B -6 0 4 0 -10 C -14 -4 0 -12 -16 D -8 0 12 0 -12 E 4 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 B=20 C=14 D=11 so D is eliminated. Round 2 votes counts: A=30 E=28 B=22 C=20 so C is eliminated. Round 3 votes counts: B=35 A=33 E=32 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:221 A:212 D:196 B:194 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 8 -4 B -6 0 4 0 -10 C -14 -4 0 -12 -16 D -8 0 12 0 -12 E 4 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 8 -4 B -6 0 4 0 -10 C -14 -4 0 -12 -16 D -8 0 12 0 -12 E 4 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 8 -4 B -6 0 4 0 -10 C -14 -4 0 -12 -16 D -8 0 12 0 -12 E 4 10 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8791: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (13) D A C B E (10) E B C A D (8) C E B D A (8) D C A B E (7) E C B A D (6) B E A C D (4) A D B E C (4) D C A E B (3) A B D E C (3) E B A C D (2) D A C E B (2) D A B E C (2) C D E A B (2) C B D E A (2) B E C A D (2) A E D B C (2) A E B D C (2) A D E B C (2) A B E D C (2) E C D A B (1) E C A B D (1) E A C B D (1) E A B C D (1) D A E C B (1) C E D B A (1) C E D A B (1) C D E B A (1) C D B A E (1) C D A B E (1) C B E A D (1) B D A C E (1) B A D E C (1) B A C E D (1) Total count = 100 A B C D E A 0 22 8 -14 16 B -22 0 0 -8 10 C -8 0 0 -10 10 D 14 8 10 0 12 E -16 -10 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 8 -14 16 B -22 0 0 -8 10 C -8 0 0 -10 10 D 14 8 10 0 12 E -16 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=20 C=18 A=15 B=9 so B is eliminated. Round 2 votes counts: D=39 E=26 C=18 A=17 so A is eliminated. Round 3 votes counts: D=49 E=32 C=19 so C is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 A:216 C:196 B:190 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 8 -14 16 B -22 0 0 -8 10 C -8 0 0 -10 10 D 14 8 10 0 12 E -16 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 8 -14 16 B -22 0 0 -8 10 C -8 0 0 -10 10 D 14 8 10 0 12 E -16 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 8 -14 16 B -22 0 0 -8 10 C -8 0 0 -10 10 D 14 8 10 0 12 E -16 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8792: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) E D A B C (7) D E A B C (7) E D B A C (5) D E C A B (5) C A B D E (5) C B E A D (4) B C A E D (4) B A E C D (4) E D B C A (3) D C E A B (3) D C A E B (3) C D A B E (3) C B A E D (3) B A C E D (3) A B D E C (3) E B C A D (2) E B A D C (2) D A E B C (2) C E B D A (2) B E C A D (2) A D B C E (2) A B E D C (2) A B C D E (2) E C B A D (1) E A D B C (1) D E A C B (1) D A C E B (1) D A C B E (1) D A B E C (1) C D E B A (1) C B A D E (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 2 -4 -6 B -12 0 12 -14 -8 C -2 -12 0 -20 -14 D 4 14 20 0 -6 E 6 8 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999617 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 2 -4 -6 B -12 0 12 -14 -8 C -2 -12 0 -20 -14 D 4 14 20 0 -6 E 6 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=24 C=19 A=16 B=13 so B is eliminated. Round 2 votes counts: E=30 D=24 C=23 A=23 so C is eliminated. Round 3 votes counts: E=36 A=36 D=28 so D is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:216 A:202 B:189 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 2 -4 -6 B -12 0 12 -14 -8 C -2 -12 0 -20 -14 D 4 14 20 0 -6 E 6 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 -4 -6 B -12 0 12 -14 -8 C -2 -12 0 -20 -14 D 4 14 20 0 -6 E 6 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 -4 -6 B -12 0 12 -14 -8 C -2 -12 0 -20 -14 D 4 14 20 0 -6 E 6 8 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999144 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8793: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (7) E A B D C (6) D B E A C (6) C D B E A (5) C A E D B (5) B E D A C (5) E B A D C (4) A E B D C (4) C E A B D (3) C D A E B (3) C A E B D (3) B D E A C (3) A D C E B (3) D C B A E (2) D B A E C (2) D A E B C (2) D A C B E (2) D A B C E (2) C E B A D (2) C D A B E (2) C B E D A (2) C B D E A (2) B D E C A (2) B D C E A (2) E C B A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A B E (1) D B C E A (1) D A C E B (1) D A B E C (1) C D E B A (1) C D B A E (1) C B E A D (1) C A D E B (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A D C (1) B C E D A (1) A E C B D (1) A E B C D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 10 -4 -8 B -2 0 12 -2 -4 C -10 -12 0 -20 -4 D 4 2 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 10 -4 -8 B -2 0 12 -2 -4 C -10 -12 0 -20 -4 D 4 2 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=21 A=18 B=17 E=13 so E is eliminated. Round 2 votes counts: C=32 A=25 B=22 D=21 so D is eliminated. Round 3 votes counts: C=36 A=33 B=31 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:211 D:210 B:202 A:200 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 10 -4 -8 B -2 0 12 -2 -4 C -10 -12 0 -20 -4 D 4 2 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 -4 -8 B -2 0 12 -2 -4 C -10 -12 0 -20 -4 D 4 2 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 -4 -8 B -2 0 12 -2 -4 C -10 -12 0 -20 -4 D 4 2 20 0 -6 E 8 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995596 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8794: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (16) D E A B C (10) E D A B C (8) C B A E D (8) D E C B A (6) A B C E D (6) D E A C B (5) D C E B A (5) C B D A E (5) B C A E D (5) E A D B C (4) B A C E D (3) A B E C D (3) E A B D C (2) D E C A B (2) A E B C D (2) A B C D E (2) E B C A D (1) D C B E A (1) D C B A E (1) D A C B E (1) C D B E A (1) C D B A E (1) C B E A D (1) A E B D C (1) Total count = 100 A B C D E A 0 -8 -6 8 8 B 8 0 -6 10 8 C 6 6 0 8 12 D -8 -10 -8 0 12 E -8 -8 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 8 8 B 8 0 -6 10 8 C 6 6 0 8 12 D -8 -10 -8 0 12 E -8 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=31 E=15 A=14 B=8 so B is eliminated. Round 2 votes counts: C=37 D=31 A=17 E=15 so E is eliminated. Round 3 votes counts: D=39 C=38 A=23 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 B:210 A:201 D:193 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 8 8 B 8 0 -6 10 8 C 6 6 0 8 12 D -8 -10 -8 0 12 E -8 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 8 8 B 8 0 -6 10 8 C 6 6 0 8 12 D -8 -10 -8 0 12 E -8 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 8 8 B 8 0 -6 10 8 C 6 6 0 8 12 D -8 -10 -8 0 12 E -8 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8795: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (9) E A D B C (8) C B D A E (7) B D E A C (6) B C E A D (6) D A E C B (5) C B E A D (5) A E C D B (5) C B A E D (4) E A D C B (3) C A E D B (3) C A E B D (3) E A C D B (2) D E A B C (2) D B C A E (2) C D A E B (2) B E D A C (2) B D E C A (2) B C E D A (2) E D A B C (1) E B A C D (1) E A B D C (1) D B A E C (1) D B A C E (1) D A E B C (1) D A C E B (1) C D B A E (1) C A D E B (1) B E A D C (1) B D C A E (1) B C D A E (1) Total count = 100 A B C D E A 0 -6 0 4 -4 B 6 0 -2 4 4 C 0 2 0 6 0 D -4 -4 -6 0 -12 E 4 -4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.744181 D: 0.000000 E: 0.255819 Sum of squares = 0.61924864975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.744181 D: 0.744181 E: 1.000000 A B C D E A 0 -6 0 4 -4 B 6 0 -2 4 4 C 0 2 0 6 0 D -4 -4 -6 0 -12 E 4 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.000000 E: 0.333332 Sum of squares = 0.555556223014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.666668 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=26 E=16 A=14 D=13 so D is eliminated. Round 2 votes counts: B=35 C=26 A=21 E=18 so E is eliminated. Round 3 votes counts: A=38 B=36 C=26 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:206 E:206 C:204 A:197 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 4 -4 B 6 0 -2 4 4 C 0 2 0 6 0 D -4 -4 -6 0 -12 E 4 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.000000 E: 0.333332 Sum of squares = 0.555556223014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.666668 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 4 -4 B 6 0 -2 4 4 C 0 2 0 6 0 D -4 -4 -6 0 -12 E 4 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.000000 E: 0.333332 Sum of squares = 0.555556223014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.666668 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 4 -4 B 6 0 -2 4 4 C 0 2 0 6 0 D -4 -4 -6 0 -12 E 4 -4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.000000 E: 0.333332 Sum of squares = 0.555556223014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666668 D: 0.666668 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8796: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) A D E B C (10) C B E D A (9) B A C D E (7) B C A D E (6) E D C B A (5) E D C A B (5) E D A C B (5) A B D E C (5) A B C D E (5) B C A E D (4) E C D B A (3) C B A E D (3) C B A D E (3) D A E C B (2) C B E A D (2) C B D E A (2) A D B E C (2) E B C D A (1) E B A D C (1) D E A B C (1) D A E B C (1) C E D B A (1) C E B D A (1) B C E D A (1) B C E A D (1) B A E C D (1) A E D B C (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 6 6 4 B 2 0 -4 6 4 C -6 4 0 0 -8 D -6 -6 0 0 12 E -4 -4 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888877 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 6 4 B 2 0 -4 6 4 C -6 4 0 0 -8 D -6 -6 0 0 12 E -4 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888984 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=21 E=20 B=20 D=14 so D is eliminated. Round 2 votes counts: E=31 A=28 C=21 B=20 so B is eliminated. Round 3 votes counts: A=36 C=33 E=31 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:207 B:204 D:200 C:195 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 6 4 B 2 0 -4 6 4 C -6 4 0 0 -8 D -6 -6 0 0 12 E -4 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888984 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 4 B 2 0 -4 6 4 C -6 4 0 0 -8 D -6 -6 0 0 12 E -4 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888984 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 4 B 2 0 -4 6 4 C -6 4 0 0 -8 D -6 -6 0 0 12 E -4 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.388888888984 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8797: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) B E C D A (9) E B C A D (8) D A C B E (7) D A B C E (6) E C B A D (5) B E C A D (5) C E A D B (4) A D E C B (4) A D C E B (4) B D A C E (3) B C E D A (3) A D B E C (3) E B A C D (2) E A C D B (2) C E D A B (2) C E B D A (2) B D C A E (2) A E D C B (2) A E C D B (2) E C A D B (1) E A C B D (1) D C A E B (1) D B C A E (1) D B A C E (1) D A B E C (1) C E D B A (1) C D B E A (1) C D A B E (1) C B D E A (1) B E A C D (1) B D A E C (1) B C D E A (1) A B D E C (1) Total count = 100 A B C D E A 0 6 4 -10 2 B -6 0 -4 -10 -4 C -4 4 0 4 4 D 10 10 -4 0 0 E -2 4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407412 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -10 2 B -6 0 -4 -10 -4 C -4 4 0 4 4 D 10 10 -4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407407 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 E=19 A=16 C=12 so C is eliminated. Round 2 votes counts: D=30 E=28 B=26 A=16 so A is eliminated. Round 3 votes counts: D=41 E=32 B=27 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 C:204 A:201 E:199 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 4 -10 2 B -6 0 -4 -10 -4 C -4 4 0 4 4 D 10 10 -4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407407 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -10 2 B -6 0 -4 -10 -4 C -4 4 0 4 4 D 10 10 -4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407407 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -10 2 B -6 0 -4 -10 -4 C -4 4 0 4 4 D 10 10 -4 0 0 E -2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.555556 D: 0.222222 E: 0.000000 Sum of squares = 0.407407407407 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.777778 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8798: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (6) B D E C A (6) A C D E B (6) C D A B E (5) D E B C A (4) C D B E A (4) C A D B E (4) A C E B D (4) E D B A C (3) E B D A C (3) E B A D C (3) D C B E A (3) C D B A E (3) C D A E B (3) B E D C A (3) B E D A C (3) B C D E A (3) A E D C B (3) A C B E D (3) E A D B C (2) D C A E B (2) D B E C A (2) C B A D E (2) B D C E A (2) A E B C D (2) E D B C A (1) E D A B C (1) E B D C A (1) E A B D C (1) D A C E B (1) C B D E A (1) C B D A E (1) C A D E B (1) B E A D C (1) B C E D A (1) A E D B C (1) A E C B D (1) A D E C B (1) A C E D B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -16 -12 4 B 0 0 -12 -2 10 C 16 12 0 6 16 D 12 2 -6 0 22 E -4 -10 -16 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -12 4 B 0 0 -12 -2 10 C 16 12 0 6 16 D 12 2 -6 0 22 E -4 -10 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=24 B=19 E=15 D=12 so D is eliminated. Round 2 votes counts: C=35 A=25 B=21 E=19 so E is eliminated. Round 3 votes counts: B=36 C=35 A=29 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 D:215 B:198 A:188 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -12 4 B 0 0 -12 -2 10 C 16 12 0 6 16 D 12 2 -6 0 22 E -4 -10 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -12 4 B 0 0 -12 -2 10 C 16 12 0 6 16 D 12 2 -6 0 22 E -4 -10 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -12 4 B 0 0 -12 -2 10 C 16 12 0 6 16 D 12 2 -6 0 22 E -4 -10 -16 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8799: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) D B E A C (8) B D E A C (7) A C E D B (5) E C A D B (4) C E A B D (4) B D E C A (4) B D A C E (4) B C D A E (4) A C D B E (4) E D B A C (3) B D A E C (3) A D B C E (3) A C D E B (3) E D A C B (2) E D A B C (2) E C B D A (2) E C A B D (2) E A D C B (2) D B A E C (2) C E B A D (2) C E A D B (2) C A E B D (2) C A B D E (2) E C D A B (1) E B D C A (1) E B C D A (1) E A C D B (1) D E B A C (1) D B A C E (1) C B A D E (1) C A D B E (1) B E C D A (1) B D C A E (1) B C A D E (1) B A D C E (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 8 4 0 B -4 0 -4 -12 0 C -8 4 0 8 6 D -4 12 -8 0 6 E 0 0 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.718113 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.281887 Sum of squares = 0.59514654012 Cumulative probabilities = A: 0.718113 B: 0.718113 C: 0.718113 D: 0.718113 E: 1.000000 A B C D E A 0 4 8 4 0 B -4 0 -4 -12 0 C -8 4 0 8 6 D -4 12 -8 0 6 E 0 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000000394 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=24 E=21 A=17 D=12 so D is eliminated. Round 2 votes counts: B=37 C=24 E=22 A=17 so A is eliminated. Round 3 votes counts: B=40 C=38 E=22 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:208 C:205 D:203 E:194 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 4 0 B -4 0 -4 -12 0 C -8 4 0 8 6 D -4 12 -8 0 6 E 0 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000000394 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 4 0 B -4 0 -4 -12 0 C -8 4 0 8 6 D -4 12 -8 0 6 E 0 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000000394 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 4 0 B -4 0 -4 -12 0 C -8 4 0 8 6 D -4 12 -8 0 6 E 0 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000000394 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8800: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (8) D C B A E (7) A C D E B (7) B E A C D (6) D C B E A (5) D C A E B (5) C D A B E (5) B E D C A (5) B E D A C (5) E B A D C (4) D E B A C (4) B E A D C (4) D C A B E (3) D B E C A (3) C D A E B (3) B D C E A (3) E B A C D (2) E A B D C (2) E A B C D (2) D B C E A (2) D A C E B (2) C A D B E (2) B D E C A (2) A E B C D (2) E B D A C (1) C B D E A (1) C A B E D (1) B E C A D (1) A E D B C (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -10 -12 -12 -4 B 10 0 -2 -16 10 C 12 2 0 -18 12 D 12 16 18 0 26 E 4 -10 -12 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999441 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -12 -4 B 10 0 -2 -16 10 C 12 2 0 -18 12 D 12 16 18 0 26 E 4 -10 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=26 C=20 A=12 E=11 so E is eliminated. Round 2 votes counts: B=33 D=31 C=20 A=16 so A is eliminated. Round 3 votes counts: B=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:236 C:204 B:201 A:181 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -12 -12 -4 B 10 0 -2 -16 10 C 12 2 0 -18 12 D 12 16 18 0 26 E 4 -10 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -12 -4 B 10 0 -2 -16 10 C 12 2 0 -18 12 D 12 16 18 0 26 E 4 -10 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -12 -4 B 10 0 -2 -16 10 C 12 2 0 -18 12 D 12 16 18 0 26 E 4 -10 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8801: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (16) E A C D B (15) D B A C E (10) C E A B D (6) B D C A E (6) B C A E D (6) D E A C B (4) D B E C A (4) D B A E C (4) D A E C B (4) A E C D B (4) B C E A D (3) E A D C B (2) C A E B D (2) B E C A D (2) E D C B A (1) E A C B D (1) D E B A C (1) D B C A E (1) D A B E C (1) B D A C E (1) B C E D A (1) B A C E D (1) B A C D E (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 16 4 24 -12 B -16 0 -16 -8 -16 C -4 16 0 18 -18 D -24 8 -18 0 -20 E 12 16 18 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999527 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 4 24 -12 B -16 0 -16 -8 -16 C -4 16 0 18 -18 D -24 8 -18 0 -20 E 12 16 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=29 B=21 C=8 A=7 so A is eliminated. Round 2 votes counts: E=39 D=31 B=21 C=9 so C is eliminated. Round 3 votes counts: E=47 D=32 B=21 so B is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:233 A:216 C:206 D:173 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 4 24 -12 B -16 0 -16 -8 -16 C -4 16 0 18 -18 D -24 8 -18 0 -20 E 12 16 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 4 24 -12 B -16 0 -16 -8 -16 C -4 16 0 18 -18 D -24 8 -18 0 -20 E 12 16 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 4 24 -12 B -16 0 -16 -8 -16 C -4 16 0 18 -18 D -24 8 -18 0 -20 E 12 16 18 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8802: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (17) D E A B C (16) D E A C B (13) D B E A C (6) C A E B D (6) B C A E D (4) E A D C B (3) D B C E A (3) E D A C B (2) E A C D B (2) D C B E A (2) C D B E A (2) B D C E A (2) B C D A E (2) B A C E D (2) A E C B D (2) D E C A B (1) D E B A C (1) D C E A B (1) C E A D B (1) C D E A B (1) C B D E A (1) C B D A E (1) C A E D B (1) C A B E D (1) B C A D E (1) B A E C D (1) A E D C B (1) A E D B C (1) A E C D B (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 10 6 -8 -14 B -10 0 -18 -16 -10 C -6 18 0 -6 -4 D 8 16 6 0 6 E 14 10 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 -8 -14 B -10 0 -18 -16 -10 C -6 18 0 -6 -4 D 8 16 6 0 6 E 14 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 C=31 B=12 E=7 A=7 so E is eliminated. Round 2 votes counts: D=45 C=31 B=12 A=12 so B is eliminated. Round 3 votes counts: D=47 C=38 A=15 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:211 C:201 A:197 B:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 -8 -14 B -10 0 -18 -16 -10 C -6 18 0 -6 -4 D 8 16 6 0 6 E 14 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 -8 -14 B -10 0 -18 -16 -10 C -6 18 0 -6 -4 D 8 16 6 0 6 E 14 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 -8 -14 B -10 0 -18 -16 -10 C -6 18 0 -6 -4 D 8 16 6 0 6 E 14 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8803: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (7) E B C A D (7) D A C B E (7) E C A B D (6) C A E D B (6) B D A E C (6) D B A C E (5) B D E A C (5) E C B A D (4) E B D C A (4) C E A B D (3) B E D C A (3) B E C D A (3) A D C E B (3) A C D B E (3) E D B A C (2) D B A E C (2) C E A D B (2) C A D E B (2) B E D A C (2) A C D E B (2) E B D A C (1) E B C D A (1) E A C D B (1) D E A B C (1) D B E A C (1) D A E C B (1) D A C E B (1) D A B C E (1) C E B A D (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B D E (1) B E C A D (1) B D C E A (1) A D C B E (1) Total count = 100 A B C D E A 0 0 -10 6 -12 B 0 0 -8 2 -12 C 10 8 0 6 -16 D -6 -2 -6 0 -12 E 12 12 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -10 6 -12 B 0 0 -8 2 -12 C 10 8 0 6 -16 D -6 -2 -6 0 -12 E 12 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=21 D=19 C=18 A=9 so A is eliminated. Round 2 votes counts: E=33 D=23 C=23 B=21 so B is eliminated. Round 3 votes counts: E=42 D=35 C=23 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:204 A:192 B:191 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 6 -12 B 0 0 -8 2 -12 C 10 8 0 6 -16 D -6 -2 -6 0 -12 E 12 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 6 -12 B 0 0 -8 2 -12 C 10 8 0 6 -16 D -6 -2 -6 0 -12 E 12 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 6 -12 B 0 0 -8 2 -12 C 10 8 0 6 -16 D -6 -2 -6 0 -12 E 12 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8804: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (6) C E A B D (6) E C B A D (5) A B D E C (5) C E B D A (4) C A E D B (4) B D E A C (4) A E C B D (4) A D B E C (4) A C E B D (4) D C B E A (3) D A B E C (3) C A E B D (3) A E B C D (3) A D B C E (3) A B E C D (3) E C A B D (2) D B E A C (2) D A B C E (2) C E D B A (2) C E B A D (2) C D A E B (2) B E A C D (2) B D A E C (2) A C D E B (2) E B C D A (1) E B C A D (1) E B A C D (1) E A B C D (1) D C E B A (1) D C E A B (1) D B C E A (1) D B A C E (1) D A C B E (1) C E A D B (1) C D E B A (1) C D E A B (1) B E D A C (1) B E A D C (1) B A D E C (1) A D C B E (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 16 18 22 12 B -16 0 -2 14 -6 C -18 2 0 12 -6 D -22 -14 -12 0 -6 E -12 6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 22 12 B -16 0 -2 14 -6 C -18 2 0 12 -6 D -22 -14 -12 0 -6 E -12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=26 D=21 E=11 B=11 so E is eliminated. Round 2 votes counts: C=33 A=32 D=21 B=14 so B is eliminated. Round 3 votes counts: A=37 C=35 D=28 so D is eliminated. Round 4 votes counts: A=59 C=41 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:203 B:195 C:195 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 18 22 12 B -16 0 -2 14 -6 C -18 2 0 12 -6 D -22 -14 -12 0 -6 E -12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 22 12 B -16 0 -2 14 -6 C -18 2 0 12 -6 D -22 -14 -12 0 -6 E -12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 22 12 B -16 0 -2 14 -6 C -18 2 0 12 -6 D -22 -14 -12 0 -6 E -12 6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8805: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (11) C D A B E (11) D C A B E (8) D A C B E (7) E B A C D (6) A B E D C (5) E B C A D (3) E A B D C (3) D C E A B (3) B E A C D (3) A E B D C (3) A D C B E (3) E D A B C (2) E B C D A (2) D E A C B (2) D C A E B (2) D A E C B (2) C D E B A (2) C D B A E (2) B C E A D (2) A B E C D (2) E D C B A (1) E D B C A (1) D E C A B (1) D A C E B (1) C E D B A (1) C E B D A (1) C B D E A (1) C B D A E (1) B E A D C (1) B A E C D (1) A E D B C (1) A D E B C (1) A D B E C (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 22 16 -2 8 B -22 0 0 -6 2 C -16 0 0 -22 -6 D 2 6 22 0 2 E -8 -2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 16 -2 8 B -22 0 0 -6 2 C -16 0 0 -22 -6 D 2 6 22 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 C=19 A=19 B=7 so B is eliminated. Round 2 votes counts: E=33 D=26 C=21 A=20 so A is eliminated. Round 3 votes counts: E=45 D=33 C=22 so C is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:222 D:216 E:197 B:187 C:178 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 22 16 -2 8 B -22 0 0 -6 2 C -16 0 0 -22 -6 D 2 6 22 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 16 -2 8 B -22 0 0 -6 2 C -16 0 0 -22 -6 D 2 6 22 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 16 -2 8 B -22 0 0 -6 2 C -16 0 0 -22 -6 D 2 6 22 0 2 E -8 -2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8806: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) D B E A C (8) A C B E D (7) D E B C A (6) E C D A B (5) B D E A C (5) B A C E D (5) E D C B A (4) E D B C A (4) B A D C E (4) B A C D E (4) A B C E D (4) E D C A B (3) D E C B A (3) A C B D E (3) E C D B A (2) D B A E C (2) C A E D B (2) B D A E C (2) B D A C E (2) B A E C D (2) A B D C E (2) E C B A D (1) E C A D B (1) D E C A B (1) D E B A C (1) D C A E B (1) D B E C A (1) D B A C E (1) C E D A B (1) C E A B D (1) B A E D C (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 10 -4 6 B 16 0 10 6 10 C -10 -10 0 -2 -4 D 4 -6 2 0 -4 E -6 -10 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 10 -4 6 B 16 0 10 6 10 C -10 -10 0 -2 -4 D 4 -6 2 0 -4 E -6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=24 E=20 A=18 C=13 so C is eliminated. Round 2 votes counts: A=29 B=25 D=24 E=22 so E is eliminated. Round 3 votes counts: D=43 A=31 B=26 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:221 A:198 D:198 E:196 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 10 -4 6 B 16 0 10 6 10 C -10 -10 0 -2 -4 D 4 -6 2 0 -4 E -6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 10 -4 6 B 16 0 10 6 10 C -10 -10 0 -2 -4 D 4 -6 2 0 -4 E -6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 10 -4 6 B 16 0 10 6 10 C -10 -10 0 -2 -4 D 4 -6 2 0 -4 E -6 -10 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8807: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (14) A B D E C (11) C E D B A (10) D A B C E (6) C D E A B (6) B A E D C (6) E C B D A (5) C E D A B (5) B A D E C (5) A D B C E (5) E B C A D (4) D A C B E (4) C E B D A (2) B A E C D (2) A D B E C (2) E C D A B (1) E C A D B (1) D E A C B (1) D C A B E (1) C D E B A (1) C B D E A (1) B E C A D (1) B E A C D (1) B C E A D (1) B A D C E (1) A D E B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -6 14 -8 B 8 0 -4 12 -2 C 6 4 0 10 -12 D -14 -12 -10 0 -8 E 8 2 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -6 14 -8 B 8 0 -4 12 -2 C 6 4 0 10 -12 D -14 -12 -10 0 -8 E 8 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=25 C=25 A=21 B=17 D=12 so D is eliminated. Round 2 votes counts: A=31 E=26 C=26 B=17 so B is eliminated. Round 3 votes counts: A=45 E=28 C=27 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:207 C:204 A:196 D:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -6 14 -8 B 8 0 -4 12 -2 C 6 4 0 10 -12 D -14 -12 -10 0 -8 E 8 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 14 -8 B 8 0 -4 12 -2 C 6 4 0 10 -12 D -14 -12 -10 0 -8 E 8 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 14 -8 B 8 0 -4 12 -2 C 6 4 0 10 -12 D -14 -12 -10 0 -8 E 8 2 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8808: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (14) B E D A C (10) C E B D A (9) C E D A B (8) E C B D A (5) E D A C B (3) D A B E C (3) C E B A D (3) C E A D B (3) C A D E B (3) B E C D A (3) B D A E C (3) D B A E C (2) D A E B C (2) C A D B E (2) A D E C B (2) A D C E B (2) A B D E C (2) E D C B A (1) E D A B C (1) E C D A B (1) E B D C A (1) E B D A C (1) E B C D A (1) D E A B C (1) D B E A C (1) D A C E B (1) C E D B A (1) C B E D A (1) C B E A D (1) C B A E D (1) C B A D E (1) C A E D B (1) B D E A C (1) B C A D E (1) A D C B E (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 6 6 -20 -12 B -6 0 -2 -10 -2 C -6 2 0 -8 -16 D 20 10 8 0 -10 E 12 2 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998777 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 6 -20 -12 B -6 0 -2 -10 -2 C -6 2 0 -8 -16 D 20 10 8 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=24 B=18 E=14 D=10 so D is eliminated. Round 2 votes counts: C=34 A=30 B=21 E=15 so E is eliminated. Round 3 votes counts: C=41 A=35 B=24 so B is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:220 D:214 A:190 B:190 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 -20 -12 B -6 0 -2 -10 -2 C -6 2 0 -8 -16 D 20 10 8 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -20 -12 B -6 0 -2 -10 -2 C -6 2 0 -8 -16 D 20 10 8 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -20 -12 B -6 0 -2 -10 -2 C -6 2 0 -8 -16 D 20 10 8 0 -10 E 12 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999583 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8809: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (6) E B D C A (5) E D C B A (4) E C A B D (4) D B A C E (4) C D E A B (4) A B C E D (4) E C D A B (3) D C E A B (3) D A B C E (3) C E A D B (3) C A E D B (3) C A D E B (3) B E D A C (3) B A D E C (3) A C E B D (3) A C D B E (3) E D B C A (2) E C A D B (2) D C B A E (2) D C A E B (2) D B E C A (2) D A C B E (2) B D E A C (2) B A E D C (2) E C D B A (1) E C B A D (1) E B C A D (1) E B A C D (1) D E C A B (1) D E B C A (1) D C A B E (1) D B C A E (1) C E D A B (1) C D A E B (1) B E A D C (1) B E A C D (1) B D E C A (1) B D A E C (1) B D A C E (1) B A E C D (1) A E C B D (1) A D C B E (1) A D B C E (1) A C E D B (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -4 -2 6 B -6 0 -2 -10 -2 C 4 2 0 -10 10 D 2 10 10 0 0 E -6 2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.827423 E: 0.172577 Sum of squares = 0.714412276237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.827423 E: 1.000000 A B C D E A 0 6 -4 -2 6 B -6 0 -2 -10 -2 C 4 2 0 -10 10 D 2 10 10 0 0 E -6 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000000368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 B=22 A=17 C=15 so C is eliminated. Round 2 votes counts: E=28 D=27 A=23 B=22 so B is eliminated. Round 3 votes counts: A=35 E=33 D=32 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:211 A:203 C:203 E:193 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -2 6 B -6 0 -2 -10 -2 C 4 2 0 -10 10 D 2 10 10 0 0 E -6 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000000368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -2 6 B -6 0 -2 -10 -2 C 4 2 0 -10 10 D 2 10 10 0 0 E -6 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000000368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -2 6 B -6 0 -2 -10 -2 C 4 2 0 -10 10 D 2 10 10 0 0 E -6 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.250000 Sum of squares = 0.625000000368 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.750000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8810: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) B C E D A (7) D A E C B (6) B C E A D (6) D B A E C (5) C B E A D (5) D B C A E (4) A E C D B (4) E A B C D (3) D C B A E (3) D A C E B (3) D A B E C (3) C D B A E (3) B E A C D (3) B D E A C (3) B D C E A (3) B C D E A (3) A E D C B (3) E B A C D (2) E A C D B (2) E A C B D (2) D B A C E (2) C B D E A (2) B E A D C (2) A E D B C (2) A D E C B (2) E C A B D (1) E A B D C (1) D C A E B (1) D A B C E (1) C E A B D (1) C D A B E (1) B E C A D (1) B D C A E (1) A E B D C (1) Total count = 100 A B C D E A 0 -10 16 -18 6 B 10 0 22 -6 16 C -16 -22 0 -8 -8 D 18 6 8 0 8 E -6 -16 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 16 -18 6 B 10 0 22 -6 16 C -16 -22 0 -8 -8 D 18 6 8 0 8 E -6 -16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=29 C=12 A=12 E=11 so E is eliminated. Round 2 votes counts: D=36 B=31 A=20 C=13 so C is eliminated. Round 3 votes counts: D=40 B=38 A=22 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:221 D:220 A:197 E:189 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 16 -18 6 B 10 0 22 -6 16 C -16 -22 0 -8 -8 D 18 6 8 0 8 E -6 -16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 -18 6 B 10 0 22 -6 16 C -16 -22 0 -8 -8 D 18 6 8 0 8 E -6 -16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 -18 6 B 10 0 22 -6 16 C -16 -22 0 -8 -8 D 18 6 8 0 8 E -6 -16 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8811: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) B C E D A (10) B C E A D (9) D A E C B (8) B A D C E (8) C E B D A (7) A D B E C (6) E C D A B (5) D E C A B (5) B A C E D (5) D A B E C (3) C E D B A (3) A D E B C (3) A B D E C (3) C E D A B (2) E D C A B (1) E C A D B (1) D E A C B (1) D A C E B (1) D A B C E (1) C E B A D (1) C B E D A (1) B E C A D (1) B D A C E (1) B A D E C (1) B A C D E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 8 2 6 B -4 0 6 -4 2 C -8 -6 0 -8 2 D -2 4 8 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 2 6 B -4 0 6 -4 2 C -8 -6 0 -8 2 D -2 4 8 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=24 D=19 C=14 E=7 so E is eliminated. Round 2 votes counts: B=36 A=24 D=20 C=20 so D is eliminated. Round 3 votes counts: A=38 B=36 C=26 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:209 B:200 E:191 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 2 6 B -4 0 6 -4 2 C -8 -6 0 -8 2 D -2 4 8 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 2 6 B -4 0 6 -4 2 C -8 -6 0 -8 2 D -2 4 8 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 2 6 B -4 0 6 -4 2 C -8 -6 0 -8 2 D -2 4 8 0 8 E -6 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8812: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) E B D A C (6) C B A D E (6) E B D C A (5) D E A C B (5) D E A B C (5) C B A E D (5) A C D B E (5) E D B C A (4) D A C E B (4) C A D B E (4) C A B D E (4) B C A E D (4) E D B A C (3) D A E B C (3) E D C B A (2) E D A B C (2) E B C D A (2) C B E A D (2) B E A D C (2) B C E A D (2) A B D E C (2) A B D C E (2) A B C D E (2) E C B D A (1) D C E A B (1) D C A E B (1) D A C B E (1) C D A B E (1) C B E D A (1) B E C A D (1) B E A C D (1) B A C E D (1) B A C D E (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 8 -8 10 B -2 0 -4 2 -2 C -8 4 0 -12 -2 D 8 -2 12 0 12 E -10 2 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -8 10 B -2 0 -4 2 -2 C -8 4 0 -12 -2 D 8 -2 12 0 12 E -10 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=25 C=23 A=13 B=12 so B is eliminated. Round 2 votes counts: E=29 C=29 D=27 A=15 so A is eliminated. Round 3 votes counts: C=39 D=32 E=29 so E is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:206 B:197 C:191 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -8 10 B -2 0 -4 2 -2 C -8 4 0 -12 -2 D 8 -2 12 0 12 E -10 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -8 10 B -2 0 -4 2 -2 C -8 4 0 -12 -2 D 8 -2 12 0 12 E -10 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -8 10 B -2 0 -4 2 -2 C -8 4 0 -12 -2 D 8 -2 12 0 12 E -10 2 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.166667 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8813: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (8) D B C A E (7) E A C B D (6) B D E C A (6) E B C A D (5) D C A B E (5) A C E D B (5) C E A B D (4) B E D A C (4) E C B A D (3) E B A C D (3) D B A C E (3) D A C B E (3) E D B A C (2) E B A D C (2) D B C E A (2) C D A B E (2) C A E B D (2) C A D B E (2) B E D C A (2) B E C D A (2) B C D E A (2) A E D C B (2) A E C B D (2) A D C B E (2) E C A B D (1) E B D C A (1) E B C D A (1) D C B A E (1) D B E C A (1) D B A E C (1) C D B A E (1) C B A D E (1) C A E D B (1) C A D E B (1) B D E A C (1) A E C D B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -6 6 4 B 2 0 -10 -2 -4 C 6 10 0 8 8 D -6 2 -8 0 0 E -4 4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 6 4 B 2 0 -10 -2 -4 C 6 10 0 8 8 D -6 2 -8 0 0 E -4 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=23 A=22 B=17 C=14 so C is eliminated. Round 2 votes counts: E=28 A=28 D=26 B=18 so B is eliminated. Round 3 votes counts: E=36 D=35 A=29 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:216 A:201 E:196 D:194 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 6 4 B 2 0 -10 -2 -4 C 6 10 0 8 8 D -6 2 -8 0 0 E -4 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 6 4 B 2 0 -10 -2 -4 C 6 10 0 8 8 D -6 2 -8 0 0 E -4 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 6 4 B 2 0 -10 -2 -4 C 6 10 0 8 8 D -6 2 -8 0 0 E -4 4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8814: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (12) B C E D A (11) D A C E B (5) C B E D A (5) B E C A D (5) B A D C E (5) D A E C B (4) C D E B A (4) B C E A D (4) B A E D C (4) A D E B C (4) C E D A B (3) B E A C D (3) B C D A E (3) C E D B A (2) B C D E A (2) B A D E C (2) A E D B C (2) E D C A B (1) E C D A B (1) E C B D A (1) E B C A D (1) E B A D C (1) E A D C B (1) D E C A B (1) D E A C B (1) D C E A B (1) D C A E B (1) D A C B E (1) C E B D A (1) C D E A B (1) C D B E A (1) C B D E A (1) B A C E D (1) B A C D E (1) A E B D C (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 0 -2 -4 B 16 0 4 6 2 C 0 -4 0 2 6 D 2 -6 -2 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -2 -4 B 16 0 4 6 2 C 0 -4 0 2 6 D 2 -6 -2 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 A=21 C=18 D=14 E=6 so E is eliminated. Round 2 votes counts: B=43 A=22 C=20 D=15 so D is eliminated. Round 3 votes counts: B=43 A=33 C=24 so C is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:202 D:198 E:197 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -2 -4 B 16 0 4 6 2 C 0 -4 0 2 6 D 2 -6 -2 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -2 -4 B 16 0 4 6 2 C 0 -4 0 2 6 D 2 -6 -2 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -2 -4 B 16 0 4 6 2 C 0 -4 0 2 6 D 2 -6 -2 0 2 E 4 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8815: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (12) D C B A E (10) E A C B D (7) E A B D C (4) D B C E A (4) C D B A E (4) E D C A B (3) E A C D B (3) C A E D B (3) B E D A C (3) B A E D C (3) A E B C D (3) A B E C D (3) E D B A C (2) E D A B C (2) D E B C A (2) D C B E A (2) D B C A E (2) C D A E B (2) B E A D C (2) B D E A C (2) B C D A E (2) B A E C D (2) A E C B D (2) E D A C B (1) E B D A C (1) E B A D C (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E A B (1) D B E C A (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C A E (1) B D A E C (1) B D A C E (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 2 18 2 -14 B -2 0 16 8 -4 C -18 -16 0 -6 -28 D -2 -8 6 0 -20 E 14 4 28 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999282 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 18 2 -14 B -2 0 16 8 -4 C -18 -16 0 -6 -28 D -2 -8 6 0 -20 E 14 4 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=23 B=20 C=11 A=8 so A is eliminated. Round 2 votes counts: E=43 D=23 B=23 C=11 so C is eliminated. Round 3 votes counts: E=46 D=30 B=24 so B is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:233 B:209 A:204 D:188 C:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 18 2 -14 B -2 0 16 8 -4 C -18 -16 0 -6 -28 D -2 -8 6 0 -20 E 14 4 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 2 -14 B -2 0 16 8 -4 C -18 -16 0 -6 -28 D -2 -8 6 0 -20 E 14 4 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 2 -14 B -2 0 16 8 -4 C -18 -16 0 -6 -28 D -2 -8 6 0 -20 E 14 4 28 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999989719 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8816: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) A B D E C (8) E C D B A (4) E C B D A (4) E A B C D (4) C D E B A (4) A D B E C (4) E A C D B (3) D C E B A (3) D A C E B (3) D A B C E (3) C E D B A (3) B A D C E (3) A E B C D (3) A B D C E (3) D B C A E (2) D B A C E (2) C E B D A (2) B D A C E (2) B C E A D (2) A E B D C (2) A D B C E (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) E B C A D (1) E B A C D (1) E A D C B (1) E A C B D (1) D E A C B (1) D C E A B (1) D C B E A (1) D C B A E (1) D A C B E (1) C D B E A (1) C B E D A (1) B D C A E (1) B C E D A (1) B C D E A (1) B C A D E (1) B A E D C (1) B A E C D (1) B A C D E (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 10 12 -4 B 4 0 4 12 -8 C -10 -4 0 4 -10 D -12 -12 -4 0 0 E 4 8 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.172173 E: 0.827827 Sum of squares = 0.714940433346 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.172173 E: 1.000000 A B C D E A 0 -4 10 12 -4 B 4 0 4 12 -8 C -10 -4 0 4 -10 D -12 -12 -4 0 0 E 4 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000021141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=26 D=18 B=14 C=11 so C is eliminated. Round 2 votes counts: E=36 A=26 D=23 B=15 so B is eliminated. Round 3 votes counts: E=40 A=33 D=27 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 A:207 B:206 C:190 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 10 12 -4 B 4 0 4 12 -8 C -10 -4 0 4 -10 D -12 -12 -4 0 0 E 4 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000021141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 12 -4 B 4 0 4 12 -8 C -10 -4 0 4 -10 D -12 -12 -4 0 0 E 4 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000021141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 12 -4 B 4 0 4 12 -8 C -10 -4 0 4 -10 D -12 -12 -4 0 0 E 4 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000021141 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8817: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (15) B D A C E (15) C E A D B (11) E C A D B (8) C E B D A (7) C E A B D (5) E A C D B (4) C B E D A (4) B C D A E (4) E A D C B (3) C E D A B (3) B C D E A (3) D B A E C (2) D A B E C (2) A E D C B (2) A D E B C (2) A D B E C (2) C E D B A (1) C E B A D (1) C B E A D (1) C B D E A (1) B D C A E (1) B C E D A (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -2 -18 -6 B 12 0 -4 18 4 C 2 4 0 8 16 D 18 -18 -8 0 -2 E 6 -4 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 -18 -6 B 12 0 -4 18 4 C 2 4 0 8 16 D 18 -18 -8 0 -2 E 6 -4 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 C=34 E=15 A=8 D=4 so D is eliminated. Round 2 votes counts: B=41 C=34 E=15 A=10 so A is eliminated. Round 3 votes counts: B=46 C=34 E=20 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:215 D:195 E:194 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 -18 -6 B 12 0 -4 18 4 C 2 4 0 8 16 D 18 -18 -8 0 -2 E 6 -4 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -18 -6 B 12 0 -4 18 4 C 2 4 0 8 16 D 18 -18 -8 0 -2 E 6 -4 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -18 -6 B 12 0 -4 18 4 C 2 4 0 8 16 D 18 -18 -8 0 -2 E 6 -4 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8818: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) C D E B A (9) D B C A E (7) C E D B A (7) B D A C E (7) E C A D B (6) D C B E A (5) B A D C E (5) A E B C D (5) E A C B D (4) D C B A E (4) C E D A B (4) A B E D C (4) E C D A B (3) E A B C D (3) A B D C E (3) E C B D A (2) E C D B A (1) E C A B D (1) E B A C D (1) D C A B E (1) D B A C E (1) C D B E A (1) B D C A E (1) B A D E C (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 -4 -4 -6 6 B 4 0 2 0 4 C 4 -2 0 -4 10 D 6 0 4 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.715242 C: 0.000000 D: 0.284758 E: 0.000000 Sum of squares = 0.592658374182 Cumulative probabilities = A: 0.000000 B: 0.715242 C: 0.715242 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -6 6 B 4 0 2 0 4 C 4 -2 0 -4 10 D 6 0 4 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=21 C=21 D=18 B=14 so B is eliminated. Round 2 votes counts: A=32 D=26 E=21 C=21 so E is eliminated. Round 3 votes counts: A=40 C=34 D=26 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 B:205 C:204 A:196 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 6 B 4 0 2 0 4 C 4 -2 0 -4 10 D 6 0 4 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 6 B 4 0 2 0 4 C 4 -2 0 -4 10 D 6 0 4 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 6 B 4 0 2 0 4 C 4 -2 0 -4 10 D 6 0 4 0 14 E -6 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999979 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8819: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) A D C E B (11) D A C E B (8) D B A E C (5) C E B A D (5) B E D C A (5) B E C A D (5) D A E C B (4) D A B E C (4) E C B D A (3) E B C D A (3) E C D B A (2) D A B C E (2) C E D A B (2) C E B D A (2) B D E C A (2) B D E A C (2) B D A E C (2) A D B C E (2) A C D E B (2) E D C B A (1) D E C B A (1) D E B C A (1) D C E A B (1) D C A E B (1) D B E C A (1) D B E A C (1) C E A D B (1) C E A B D (1) C A E B D (1) B E D A C (1) B A D E C (1) A D C B E (1) A D B E C (1) A C E D B (1) A C B E D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 2 -30 -2 B 8 0 2 -6 -2 C -2 -2 0 -20 -14 D 30 6 20 0 10 E 2 2 14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -30 -2 B 8 0 2 -6 -2 C -2 -2 0 -20 -14 D 30 6 20 0 10 E 2 2 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=29 B=29 A=21 C=12 E=9 so E is eliminated. Round 2 votes counts: B=32 D=30 A=21 C=17 so C is eliminated. Round 3 votes counts: B=42 D=34 A=24 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:233 E:204 B:201 A:181 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 2 -30 -2 B 8 0 2 -6 -2 C -2 -2 0 -20 -14 D 30 6 20 0 10 E 2 2 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -30 -2 B 8 0 2 -6 -2 C -2 -2 0 -20 -14 D 30 6 20 0 10 E 2 2 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -30 -2 B 8 0 2 -6 -2 C -2 -2 0 -20 -14 D 30 6 20 0 10 E 2 2 14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999981108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8820: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) E A C D B (9) B D A E C (9) D B C E A (7) D C B E A (5) B D C A E (5) A E B C D (5) C D B E A (4) C A E B D (4) E A D C B (3) E A D B C (3) C D E A B (3) B D E A C (3) A E C B D (3) E C A D B (2) D E B A C (2) D E A C B (2) C E D A B (2) B D C E A (2) B A E D C (2) A C E B D (2) E D C A B (1) D E C A B (1) D E A B C (1) D C E B A (1) D B E C A (1) C B D A E (1) C B A E D (1) C A E D B (1) B A E C D (1) B A D C E (1) B A C E D (1) A E C D B (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -2 0 -24 B -8 0 -12 -18 -12 C 2 12 0 0 0 D 0 18 0 0 -4 E 24 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636015 D: 0.000000 E: 0.363985 Sum of squares = 0.537000392137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636015 D: 0.636015 E: 1.000000 A B C D E A 0 8 -2 0 -24 B -8 0 -12 -18 -12 C 2 12 0 0 0 D 0 18 0 0 -4 E 24 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 D=20 E=18 A=12 so A is eliminated. Round 2 votes counts: C=28 E=27 B=25 D=20 so D is eliminated. Round 3 votes counts: C=34 E=33 B=33 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:220 C:207 D:207 A:191 B:175 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 0 -24 B -8 0 -12 -18 -12 C 2 12 0 0 0 D 0 18 0 0 -4 E 24 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 0 -24 B -8 0 -12 -18 -12 C 2 12 0 0 0 D 0 18 0 0 -4 E 24 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 0 -24 B -8 0 -12 -18 -12 C 2 12 0 0 0 D 0 18 0 0 -4 E 24 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8821: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) C A B E D (8) C E B A D (7) D B E A C (5) D A E B C (5) D A B E C (5) C A E B D (5) E B D C A (4) E B C D A (4) A D C E B (4) A C D E B (4) A C B E D (4) E D B C A (3) D E B C A (3) D E B A C (3) C B E A D (3) A D C B E (3) A C B D E (3) C A E D B (2) B E D C A (2) B C E A D (2) A D B C E (2) E D C B A (1) E C D B A (1) E C B D A (1) D B A E C (1) B E C D A (1) B C A E D (1) B A D E C (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 4 22 20 B -14 0 -14 -4 4 C -4 14 0 16 20 D -22 4 -16 0 0 E -20 -4 -20 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 22 20 B -14 0 -14 -4 4 C -4 14 0 16 20 D -22 4 -16 0 0 E -20 -4 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997288 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=25 D=22 E=14 B=7 so B is eliminated. Round 2 votes counts: A=33 C=28 D=22 E=17 so E is eliminated. Round 3 votes counts: C=35 A=33 D=32 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:230 C:223 B:186 D:183 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 22 20 B -14 0 -14 -4 4 C -4 14 0 16 20 D -22 4 -16 0 0 E -20 -4 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997288 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 22 20 B -14 0 -14 -4 4 C -4 14 0 16 20 D -22 4 -16 0 0 E -20 -4 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997288 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 22 20 B -14 0 -14 -4 4 C -4 14 0 16 20 D -22 4 -16 0 0 E -20 -4 -20 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997288 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8822: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) C D E A B (6) B E C D A (6) E C A D B (4) E B C D A (4) D C E A B (4) D C A E B (4) C D E B A (4) B A E D C (4) B A E C D (4) B A D E C (4) A D C E B (4) C E D A B (3) B E A C D (3) B A D C E (3) E C D B A (2) E B C A D (2) D C B E A (2) B E A D C (2) B D C E A (2) B D A C E (2) A E C D B (2) A D B C E (2) A B D E C (2) E C B D A (1) E C B A D (1) E B A C D (1) D C E B A (1) D C A B E (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D B A (1) B E D C A (1) B D C A E (1) B C E D A (1) B C D E A (1) A E D C B (1) A D C B E (1) A C D E B (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -18 -14 -18 B 6 0 -4 -6 -8 C 18 4 0 10 -4 D 14 6 -10 0 -2 E 18 8 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -18 -14 -18 B 6 0 -4 -6 -8 C 18 4 0 10 -4 D 14 6 -10 0 -2 E 18 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=22 D=15 A=15 C=14 so C is eliminated. Round 2 votes counts: B=34 E=26 D=25 A=15 so A is eliminated. Round 3 votes counts: B=38 D=33 E=29 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:216 C:214 D:204 B:194 A:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -18 -14 -18 B 6 0 -4 -6 -8 C 18 4 0 10 -4 D 14 6 -10 0 -2 E 18 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 -14 -18 B 6 0 -4 -6 -8 C 18 4 0 10 -4 D 14 6 -10 0 -2 E 18 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 -14 -18 B 6 0 -4 -6 -8 C 18 4 0 10 -4 D 14 6 -10 0 -2 E 18 8 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8823: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) A D E C B (8) D E A B C (6) C B A E D (6) C B A D E (6) E D B A C (5) E D A B C (5) D E A C B (5) D A E C B (5) C B E D A (5) C A B D E (5) C D E B A (3) C B E A D (3) C A D B E (3) B E C D A (3) B C E A D (3) B C A E D (3) E B D A C (2) C B D E A (2) A D E B C (2) A C D E B (2) A C D B E (2) E D B C A (1) E B A D C (1) E A D B C (1) D C E A B (1) B A C E D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -6 -4 -10 B 6 0 -14 2 6 C 6 14 0 14 10 D 4 -2 -14 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999406 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -4 -10 B 6 0 -14 2 6 C 6 14 0 14 10 D 4 -2 -14 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=19 D=17 A=16 E=15 so E is eliminated. Round 2 votes counts: C=33 D=28 B=22 A=17 so A is eliminated. Round 3 votes counts: D=39 C=38 B=23 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:200 E:196 D:195 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 -10 B 6 0 -14 2 6 C 6 14 0 14 10 D 4 -2 -14 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 -10 B 6 0 -14 2 6 C 6 14 0 14 10 D 4 -2 -14 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 -10 B 6 0 -14 2 6 C 6 14 0 14 10 D 4 -2 -14 0 2 E 10 -6 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8824: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (10) A C E D B (9) B C E D A (7) A D E B C (7) B D E C A (5) E D B C A (4) C E B D A (4) B D A E C (4) A D E C B (4) D E B A C (3) D B E A C (3) A D B E C (3) E D C A B (2) E D A C B (2) D A E B C (2) C B E A D (2) C B A E D (2) B E D C A (2) B E C D A (2) B D E A C (2) B C D E A (2) E D B A C (1) E D A B C (1) E C D B A (1) D E A B C (1) D B A E C (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E D B (1) B D C E A (1) B D C A E (1) B C D A E (1) B C A D E (1) A E D C B (1) A D B C E (1) A C D E B (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -20 0 -26 -14 B 20 0 18 0 8 C 0 -18 0 -8 -6 D 26 0 8 0 -6 E 14 -8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.751171 C: 0.000000 D: 0.248829 E: 0.000000 Sum of squares = 0.626173285809 Cumulative probabilities = A: 0.000000 B: 0.751171 C: 0.751171 D: 1.000000 E: 1.000000 A B C D E A 0 -20 0 -26 -14 B 20 0 18 0 8 C 0 -18 0 -8 -6 D 26 0 8 0 -6 E 14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 C=21 E=11 D=11 so E is eliminated. Round 2 votes counts: A=29 B=28 C=22 D=21 so D is eliminated. Round 3 votes counts: B=40 A=36 C=24 so C is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:214 E:209 C:184 A:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 0 -26 -14 B 20 0 18 0 8 C 0 -18 0 -8 -6 D 26 0 8 0 -6 E 14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 0 -26 -14 B 20 0 18 0 8 C 0 -18 0 -8 -6 D 26 0 8 0 -6 E 14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 0 -26 -14 B 20 0 18 0 8 C 0 -18 0 -8 -6 D 26 0 8 0 -6 E 14 -8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8825: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) C E B D A (10) C E B A D (8) D B A E C (7) A D C B E (7) A D B E C (7) A C D E B (6) D A B E C (4) A D C E B (4) A D B C E (4) E B C D A (3) B E C D A (3) B D E A C (3) C E A B D (2) C A D E B (2) B E D C A (2) B C E D A (2) A D E B C (2) E C B A D (1) E B D A C (1) E A C D B (1) D E A B C (1) C B E D A (1) C B A D E (1) C A E B D (1) C A B D E (1) B E D A C (1) B D E C A (1) B D A E C (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 2 -2 -2 B 12 0 -16 6 -8 C -2 16 0 8 0 D 2 -6 -8 0 6 E 2 8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.486168 D: 0.000000 E: 0.513832 Sum of squares = 0.500382642521 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.486168 D: 0.486168 E: 1.000000 A B C D E A 0 -12 2 -2 -2 B 12 0 -16 6 -8 C -2 16 0 8 0 D 2 -6 -8 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.000000 E: 0.500374 Sum of squares = 0.50000027946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.499626 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=26 E=17 B=13 D=12 so D is eliminated. Round 2 votes counts: A=36 C=26 B=20 E=18 so E is eliminated. Round 3 votes counts: C=38 A=38 B=24 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:211 E:202 B:197 D:197 A:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 2 -2 -2 B 12 0 -16 6 -8 C -2 16 0 8 0 D 2 -6 -8 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.000000 E: 0.500374 Sum of squares = 0.50000027946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.499626 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 -2 -2 B 12 0 -16 6 -8 C -2 16 0 8 0 D 2 -6 -8 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.000000 E: 0.500374 Sum of squares = 0.50000027946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.499626 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 -2 -2 B 12 0 -16 6 -8 C -2 16 0 8 0 D 2 -6 -8 0 6 E 2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.000000 E: 0.500374 Sum of squares = 0.50000027946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499626 D: 0.499626 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8826: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (6) D A C B E (5) A E D B C (5) E A D C B (4) D C B A E (4) D C A B E (4) B C D E A (4) A E B D C (4) A E B C D (4) E C B D A (3) E B C A D (3) D A C E B (3) C B D E A (3) B C E A D (3) A D E C B (3) A D E B C (3) E D C A B (2) E B C D A (2) E A D B C (2) E A B C D (2) D E C B A (2) C D B E A (2) C B E D A (2) B C A D E (2) A E D C B (2) A D B C E (2) E D C B A (1) E D A C B (1) E B A C D (1) E A C D B (1) D E C A B (1) D C E B A (1) D A E C B (1) D A B C E (1) C E B D A (1) C D B A E (1) C B D A E (1) B E C A D (1) B C D A E (1) B A E C D (1) B A D C E (1) B A C E D (1) A D C E B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 0 -4 4 B -6 0 0 -6 -6 C 0 0 0 -8 2 D 4 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 6 0 -4 4 B -6 0 0 -6 -6 C 0 0 0 -8 2 D 4 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 D=22 B=20 C=10 so C is eliminated. Round 2 votes counts: B=26 A=26 D=25 E=23 so E is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:207 A:203 E:202 C:197 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 -4 4 B -6 0 0 -6 -6 C 0 0 0 -8 2 D 4 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -4 4 B -6 0 0 -6 -6 C 0 0 0 -8 2 D 4 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -4 4 B -6 0 0 -6 -6 C 0 0 0 -8 2 D 4 6 8 0 -4 E -4 6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8827: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (7) E B D A C (6) A C D E B (6) D E C A B (5) B C A E D (5) D E C B A (4) C B D A E (4) C A B D E (4) B E A D C (4) A C B E D (4) E D A B C (3) C B A D E (3) B C D E A (3) A C D B E (3) E D B A C (2) E D A C B (2) E A D B C (2) C D B E A (2) C D A B E (2) C A D B E (2) B E C A D (2) B C E D A (2) B A E C D (2) A C B D E (2) E D B C A (1) E B D C A (1) D E B C A (1) D E A C B (1) D C E A B (1) D A E C B (1) C D B A E (1) C D A E B (1) B E D A C (1) B E A C D (1) B D E C A (1) B C E A D (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E C B (1) A D C E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -6 -4 -6 B 8 0 -2 12 16 C 6 2 0 2 -4 D 4 -12 -2 0 -2 E 6 -16 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933965 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 -8 -6 -4 -6 B 8 0 -2 12 16 C 6 2 0 2 -4 D 4 -12 -2 0 -2 E 6 -16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933843 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=22 C=19 E=17 D=13 so D is eliminated. Round 2 votes counts: B=29 E=28 A=23 C=20 so C is eliminated. Round 3 votes counts: B=39 A=32 E=29 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:203 E:198 D:194 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -6 B 8 0 -2 12 16 C 6 2 0 2 -4 D 4 -12 -2 0 -2 E 6 -16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933843 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -6 B 8 0 -2 12 16 C 6 2 0 2 -4 D 4 -12 -2 0 -2 E 6 -16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933843 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -6 B 8 0 -2 12 16 C 6 2 0 2 -4 D 4 -12 -2 0 -2 E 6 -16 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.181818 C: 0.727273 D: 0.000000 E: 0.090909 Sum of squares = 0.570247933843 Cumulative probabilities = A: 0.000000 B: 0.181818 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8828: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (14) B C A D E (7) D E A C B (5) D A E C B (5) C E A D B (5) B C A E D (5) E D C A B (4) B D E A C (4) E C D A B (3) C A E B D (3) B D A C E (3) B C E A D (3) A C E D B (3) A C D E B (3) E C A D B (2) E B C D A (2) D B E A C (2) C B E A D (2) C B A E D (2) C A E D B (2) C A B E D (2) B D A E C (2) B A D C E (2) B A C D E (2) A E C D B (2) A D C E B (2) E D A B C (1) E A C D B (1) D E A B C (1) D A B E C (1) C E B A D (1) C A D B E (1) B E D C A (1) B E C D A (1) B D C A E (1) Total count = 100 A B C D E A 0 20 6 0 -4 B -20 0 -24 -14 -18 C -6 24 0 4 -2 D 0 14 -4 0 -18 E 4 18 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 6 0 -4 B -20 0 -24 -14 -18 C -6 24 0 4 -2 D 0 14 -4 0 -18 E 4 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=27 C=18 D=14 A=10 so A is eliminated. Round 2 votes counts: B=31 E=29 C=24 D=16 so D is eliminated. Round 3 votes counts: E=40 B=34 C=26 so C is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:211 C:210 D:196 B:162 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 6 0 -4 B -20 0 -24 -14 -18 C -6 24 0 4 -2 D 0 14 -4 0 -18 E 4 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 6 0 -4 B -20 0 -24 -14 -18 C -6 24 0 4 -2 D 0 14 -4 0 -18 E 4 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 6 0 -4 B -20 0 -24 -14 -18 C -6 24 0 4 -2 D 0 14 -4 0 -18 E 4 18 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8829: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (15) C E B D A (11) A B D E C (7) E C D B A (6) E D B A C (5) E C B D A (5) C E A D B (4) D A B E C (3) C E B A D (3) C A E B D (3) C A D B E (3) C A B D E (3) B D A E C (3) A D B C E (3) A B D C E (3) C E D B A (2) C E A B D (2) C A B E D (2) B D A C E (2) A C D B E (2) E D B C A (1) E C A D B (1) E B D C A (1) E B D A C (1) E B C D A (1) E A D C B (1) C E D A B (1) C B E D A (1) C A D E B (1) B D E A C (1) B A D E C (1) B A D C E (1) A D E B C (1) Total count = 100 A B C D E A 0 10 -2 12 6 B -10 0 -2 2 0 C 2 2 0 2 -6 D -12 -2 -2 0 -2 E -6 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102041 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 10 -2 12 6 B -10 0 -2 2 0 C 2 2 0 2 -6 D -12 -2 -2 0 -2 E -6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=31 E=22 B=8 D=3 so D is eliminated. Round 2 votes counts: C=36 A=34 E=22 B=8 so B is eliminated. Round 3 votes counts: A=41 C=36 E=23 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:213 E:201 C:200 B:195 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 12 6 B -10 0 -2 2 0 C 2 2 0 2 -6 D -12 -2 -2 0 -2 E -6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 12 6 B -10 0 -2 2 0 C 2 2 0 2 -6 D -12 -2 -2 0 -2 E -6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 12 6 B -10 0 -2 2 0 C 2 2 0 2 -6 D -12 -2 -2 0 -2 E -6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102036 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8830: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (15) E B C D A (12) E B A C D (10) D A C B E (10) A D C B E (10) C D A E B (7) B E A C D (7) E B C A D (6) D C A E B (6) D C A B E (3) C D E B A (2) A B E D C (2) E B A D C (1) D C E A B (1) D A C E B (1) C E D B A (1) C E B D A (1) C D E A B (1) C D A B E (1) B E D A C (1) A D B C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -12 18 6 -16 B 12 0 10 10 2 C -18 -10 0 -2 -8 D -6 -10 2 0 -12 E 16 -2 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 18 6 -16 B 12 0 10 10 2 C -18 -10 0 -2 -8 D -6 -10 2 0 -12 E 16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=23 D=21 A=14 C=13 so C is eliminated. Round 2 votes counts: D=32 E=31 B=23 A=14 so A is eliminated. Round 3 votes counts: D=44 E=31 B=25 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:217 A:198 D:187 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 18 6 -16 B 12 0 10 10 2 C -18 -10 0 -2 -8 D -6 -10 2 0 -12 E 16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 18 6 -16 B 12 0 10 10 2 C -18 -10 0 -2 -8 D -6 -10 2 0 -12 E 16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 18 6 -16 B 12 0 10 10 2 C -18 -10 0 -2 -8 D -6 -10 2 0 -12 E 16 -2 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996808 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8831: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (12) B C E D A (9) B C A D E (6) A D E B C (6) C E D A B (5) C B E D A (5) B A C D E (5) A B C D E (4) E D C A B (3) D E A C B (3) C B A E D (3) C B A D E (3) C A B D E (3) B E D A C (3) B A D E C (3) E D A C B (2) C E B D A (2) C B E A D (2) B C E A D (2) B A D C E (2) A B D E C (2) E D B C A (1) E D B A C (1) E D A B C (1) E C D A B (1) E C B D A (1) E B D C A (1) E B D A C (1) D E A B C (1) D A E C B (1) D A E B C (1) C E D B A (1) C A D E B (1) C A D B E (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -2 14 10 B 4 0 0 16 8 C 2 0 0 12 12 D -14 -16 -12 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.673564 C: 0.326436 D: 0.000000 E: 0.000000 Sum of squares = 0.560248808414 Cumulative probabilities = A: 0.000000 B: 0.673564 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 14 10 B 4 0 0 16 8 C 2 0 0 12 12 D -14 -16 -12 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=26 A=25 E=12 D=6 so D is eliminated. Round 2 votes counts: B=31 A=27 C=26 E=16 so E is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:213 A:209 D:184 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 14 10 B 4 0 0 16 8 C 2 0 0 12 12 D -14 -16 -12 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 14 10 B 4 0 0 16 8 C 2 0 0 12 12 D -14 -16 -12 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 14 10 B 4 0 0 16 8 C 2 0 0 12 12 D -14 -16 -12 0 10 E -10 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8832: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) E B C D A (6) C A E D B (6) A D B C E (6) D B A E C (5) D A B E C (5) A C D E B (5) E C B D A (4) B D A E C (4) D A B C E (3) C E A D B (3) A D C E B (3) C E B D A (2) C B E D A (2) C B D A E (2) B E D C A (2) B E C D A (2) B D E A C (2) B D A C E (2) B C D A E (2) A D E B C (2) A D C B E (2) A D B E C (2) A C E D B (2) E C B A D (1) E C A D B (1) E B D C A (1) E B D A C (1) E A D C B (1) E A D B C (1) E A C D B (1) D E B A C (1) D B E A C (1) C B E A D (1) C B A E D (1) C A E B D (1) C A D B E (1) C A B D E (1) B D E C A (1) B D C E A (1) B C D E A (1) B A C D E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 2 0 14 B 6 0 4 -4 2 C -2 -4 0 8 12 D 0 4 -8 0 8 E -14 -2 -12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 0 14 B 6 0 4 -4 2 C -2 -4 0 8 12 D 0 4 -8 0 8 E -14 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37499999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=23 B=18 E=17 D=15 so D is eliminated. Round 2 votes counts: A=31 C=27 B=24 E=18 so E is eliminated. Round 3 votes counts: A=34 C=33 B=33 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. C:207 A:205 B:204 D:202 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 0 14 B 6 0 4 -4 2 C -2 -4 0 8 12 D 0 4 -8 0 8 E -14 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37499999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 0 14 B 6 0 4 -4 2 C -2 -4 0 8 12 D 0 4 -8 0 8 E -14 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37499999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 0 14 B 6 0 4 -4 2 C -2 -4 0 8 12 D 0 4 -8 0 8 E -14 -2 -12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.37499999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8833: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (7) B E C A D (7) D A C E B (6) C D A E B (6) A E B D C (6) D A C B E (5) C D B E A (4) E C A B D (3) C E B D A (3) B E A C D (3) B C E D A (3) A D E C B (3) E C B A D (2) E A B C D (2) D C A B E (2) D B C A E (2) D A B E C (2) C E A D B (2) C B E D A (2) B E A D C (2) B D C A E (2) B C E A D (2) B C D E A (2) B A E D C (2) A E D C B (2) A E D B C (2) A B E D C (2) E B A C D (1) E A C B D (1) D C B A E (1) C E D A B (1) C E A B D (1) C D B A E (1) C B D E A (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A E C (1) A D E B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 12 -12 -8 10 B -12 0 -8 2 0 C 12 8 0 -6 10 D 8 -2 6 0 2 E -10 0 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999996 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 -8 10 B -12 0 -8 2 0 C 12 8 0 -6 10 D 8 -2 6 0 2 E -10 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999993 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 C=21 A=18 E=9 so E is eliminated. Round 2 votes counts: B=28 C=26 D=25 A=21 so A is eliminated. Round 3 votes counts: B=38 D=35 C=27 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:212 D:207 A:201 B:191 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -12 -8 10 B -12 0 -8 2 0 C 12 8 0 -6 10 D 8 -2 6 0 2 E -10 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999993 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 -8 10 B -12 0 -8 2 0 C 12 8 0 -6 10 D 8 -2 6 0 2 E -10 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999993 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 -8 10 B -12 0 -8 2 0 C 12 8 0 -6 10 D 8 -2 6 0 2 E -10 0 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999993 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8834: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (13) C E A B D (10) D B A E C (8) C A B D E (7) E C A B D (6) C A E B D (6) D E B A C (5) A B D C E (5) E D C B A (3) E C D B A (3) D B E A C (3) C D E A B (3) B A D E C (3) E D B A C (2) E A B C D (2) D C B A E (2) C E D A B (2) B D E A C (2) B A D C E (2) A B E D C (2) A B C E D (2) A B C D E (2) E D B C A (1) E C D A B (1) E B D A C (1) C A B E D (1) B D A E C (1) B D A C E (1) B A E D C (1) Total count = 100 A B C D E A 0 -2 10 -2 12 B 2 0 12 8 10 C -10 -12 0 -10 12 D 2 -8 10 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -2 12 B 2 0 12 8 10 C -10 -12 0 -10 12 D 2 -8 10 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=29 E=19 A=11 B=10 so B is eliminated. Round 2 votes counts: D=35 C=29 E=19 A=17 so A is eliminated. Round 3 votes counts: D=45 C=33 E=22 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:216 A:209 D:209 C:190 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 -2 12 B 2 0 12 8 10 C -10 -12 0 -10 12 D 2 -8 10 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -2 12 B 2 0 12 8 10 C -10 -12 0 -10 12 D 2 -8 10 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -2 12 B 2 0 12 8 10 C -10 -12 0 -10 12 D 2 -8 10 0 14 E -12 -10 -12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8835: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) C A D E B (5) B A E D C (5) E B D C A (4) D E C A B (4) C E D A B (4) C D A E B (4) C B A E D (4) E C D A B (3) D E A C B (3) C B E A D (3) B E C D A (3) B A C D E (3) A B D E C (3) E D C B A (2) E D C A B (2) E B C D A (2) D E A B C (2) C A D B E (2) C A B D E (2) B E D A C (2) B C A E D (2) B A D E C (2) A C D B E (2) A C B D E (2) A B D C E (2) A B C D E (2) E D B C A (1) E D A C B (1) D C E A B (1) C E D B A (1) C E B D A (1) C D E A B (1) C B E D A (1) C B A D E (1) B E D C A (1) B E A D C (1) B C E D A (1) B C E A D (1) B A E C D (1) B A C E D (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -12 -6 -8 B 4 0 -2 0 -2 C 12 2 0 4 -6 D 6 0 -4 0 -12 E 8 2 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -12 -6 -8 B 4 0 -2 0 -2 C 12 2 0 4 -6 D 6 0 -4 0 -12 E 8 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=24 B=23 A=14 D=10 so D is eliminated. Round 2 votes counts: E=33 C=30 B=23 A=14 so A is eliminated. Round 3 votes counts: C=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:206 B:200 D:195 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -12 -6 -8 B 4 0 -2 0 -2 C 12 2 0 4 -6 D 6 0 -4 0 -12 E 8 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -6 -8 B 4 0 -2 0 -2 C 12 2 0 4 -6 D 6 0 -4 0 -12 E 8 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -6 -8 B 4 0 -2 0 -2 C 12 2 0 4 -6 D 6 0 -4 0 -12 E 8 2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8836: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (13) B C A E D (10) A D E C B (9) B C E D A (8) A C B D E (8) D E A B C (7) C B A E D (7) E B D C A (6) B C E A D (6) D E A C B (5) D A E C B (4) C A B D E (4) A C D B E (4) B E C D A (3) B E D C A (2) A D C E B (2) E D B A C (1) D E B A C (1) Total count = 100 A B C D E A 0 -14 -18 0 -4 B 14 0 14 8 4 C 18 -14 0 0 -2 D 0 -8 0 0 -12 E 4 -4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999679 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -18 0 -4 B 14 0 14 8 4 C 18 -14 0 0 -2 D 0 -8 0 0 -12 E 4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=23 E=20 D=17 C=11 so C is eliminated. Round 2 votes counts: B=36 A=27 E=20 D=17 so D is eliminated. Round 3 votes counts: B=36 E=33 A=31 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 E:207 C:201 D:190 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -18 0 -4 B 14 0 14 8 4 C 18 -14 0 0 -2 D 0 -8 0 0 -12 E 4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -18 0 -4 B 14 0 14 8 4 C 18 -14 0 0 -2 D 0 -8 0 0 -12 E 4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -18 0 -4 B 14 0 14 8 4 C 18 -14 0 0 -2 D 0 -8 0 0 -12 E 4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8837: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) D B C A E (7) D B C E A (6) A C B E D (6) E D B A C (5) D E B A C (5) D B E C A (5) A C E B D (5) A E C B D (4) E A D C B (3) E A C B D (3) D E A B C (3) C B A D E (3) B C D A E (3) A E C D B (3) E D A B C (2) E A B C D (2) D C B A E (2) D A C B E (2) C B D A E (2) C B A E D (2) C A B E D (2) B D C E A (2) B C D E A (2) A C B D E (2) E B D C A (1) E B D A C (1) D C A B E (1) D B E A C (1) D A E B C (1) C A B D E (1) B E D C A (1) B E C D A (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 18 -6 -4 B -2 0 14 -14 4 C -18 -14 0 -14 0 D 6 14 14 0 -2 E 4 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999999 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 A B C D E A 0 2 18 -6 -4 B -2 0 14 -14 4 C -18 -14 0 -14 0 D 6 14 14 0 -2 E 4 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=26 A=22 C=10 B=9 so B is eliminated. Round 2 votes counts: D=35 E=28 A=22 C=15 so C is eliminated. Round 3 votes counts: D=42 A=30 E=28 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:205 B:201 E:201 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 18 -6 -4 B -2 0 14 -14 4 C -18 -14 0 -14 0 D 6 14 14 0 -2 E 4 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 18 -6 -4 B -2 0 14 -14 4 C -18 -14 0 -14 0 D 6 14 14 0 -2 E 4 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 18 -6 -4 B -2 0 14 -14 4 C -18 -14 0 -14 0 D 6 14 14 0 -2 E 4 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.000000 D: 0.200000 E: 0.700000 Sum of squares = 0.539999999961 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.100000 D: 0.300000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8838: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) D C E A B (6) B E A C D (6) A E D C B (6) A E B D C (6) E A B C D (5) B A D C E (4) A E D B C (4) A B E D C (4) A B D C E (4) E C B D A (3) E B A C D (3) C D B E A (3) B C E D A (3) E C D B A (2) E C D A B (2) D C B A E (2) D C A E B (2) D A C E B (2) C D E B A (2) C B D E A (2) B C D E A (2) B A C D E (2) A D E C B (2) E B C D A (1) E A D C B (1) E A C B D (1) D E C A B (1) D C E B A (1) D C B E A (1) D A E C B (1) B E C D A (1) B E C A D (1) B D C A E (1) B C D A E (1) B C A D E (1) B A E D C (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 24 22 6 B 4 0 20 22 -2 C -24 -20 0 0 -18 D -22 -22 0 0 -18 E -6 2 18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888798 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -4 24 22 6 B 4 0 20 22 -2 C -24 -20 0 0 -18 D -22 -22 0 0 -18 E -6 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888891154 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=27 E=18 D=16 C=7 so C is eliminated. Round 2 votes counts: B=34 A=27 D=21 E=18 so E is eliminated. Round 3 votes counts: B=41 A=34 D=25 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:224 B:222 E:216 C:169 D:169 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 24 22 6 B 4 0 20 22 -2 C -24 -20 0 0 -18 D -22 -22 0 0 -18 E -6 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888891154 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 24 22 6 B 4 0 20 22 -2 C -24 -20 0 0 -18 D -22 -22 0 0 -18 E -6 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888891154 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 24 22 6 B 4 0 20 22 -2 C -24 -20 0 0 -18 D -22 -22 0 0 -18 E -6 2 18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.388888891154 Cumulative probabilities = A: 0.166667 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8839: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) B D C E A (7) A C D B E (7) D B C E A (5) A E C D B (5) A E B D C (5) E B D A C (4) A C E D B (4) A B E D C (4) A B D C E (4) E B A D C (3) E A C D B (3) D B C A E (3) B E D C A (3) B D A C E (3) A E B C D (3) A B D E C (3) E C D B A (2) E C D A B (2) E B D C A (2) E A B D C (2) C D B A E (2) B D E C A (2) A C D E B (2) E C B D A (1) E C A D B (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) C D A B E (1) B E D A C (1) B E A D C (1) B D E A C (1) B A D E C (1) A E C B D (1) A D C B E (1) Total count = 100 A B C D E A 0 0 20 4 0 B 0 0 16 6 16 C -20 -16 0 -14 -4 D -4 -6 14 0 2 E 0 -16 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.427435 B: 0.572565 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510531210655 Cumulative probabilities = A: 0.427435 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 20 4 0 B 0 0 16 6 16 C -20 -16 0 -14 -4 D -4 -6 14 0 2 E 0 -16 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=22 B=19 D=10 C=10 so D is eliminated. Round 2 votes counts: A=39 B=27 E=22 C=12 so C is eliminated. Round 3 votes counts: A=40 B=38 E=22 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:219 A:212 D:203 E:193 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 20 4 0 B 0 0 16 6 16 C -20 -16 0 -14 -4 D -4 -6 14 0 2 E 0 -16 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 20 4 0 B 0 0 16 6 16 C -20 -16 0 -14 -4 D -4 -6 14 0 2 E 0 -16 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 20 4 0 B 0 0 16 6 16 C -20 -16 0 -14 -4 D -4 -6 14 0 2 E 0 -16 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8840: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (10) B C A D E (8) A D E C B (8) E D A C B (7) A D E B C (7) C B E D A (6) B C E D A (6) B C D A E (5) B A D E C (5) E A D C B (4) D A E C B (4) B A D C E (4) C D A E B (3) B A C D E (3) E C D A B (2) E B A D C (2) C E D A B (2) C B D A E (2) A D B C E (2) E B C D A (1) E B C A D (1) E A D B C (1) D A C E B (1) C E B D A (1) C B D E A (1) B E C A D (1) B C A E D (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -16 0 18 10 B 16 0 18 16 12 C 0 -18 0 6 10 D -18 -16 -6 0 8 E -10 -12 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 18 10 B 16 0 18 16 12 C 0 -18 0 6 10 D -18 -16 -6 0 8 E -10 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 E=18 A=18 C=15 D=5 so D is eliminated. Round 2 votes counts: B=44 A=23 E=18 C=15 so C is eliminated. Round 3 votes counts: B=53 A=26 E=21 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:231 A:206 C:199 D:184 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 18 10 B 16 0 18 16 12 C 0 -18 0 6 10 D -18 -16 -6 0 8 E -10 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 18 10 B 16 0 18 16 12 C 0 -18 0 6 10 D -18 -16 -6 0 8 E -10 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 18 10 B 16 0 18 16 12 C 0 -18 0 6 10 D -18 -16 -6 0 8 E -10 -12 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8841: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (7) E A D C B (6) B C A D E (6) E C B D A (5) D E A C B (5) E D A C B (4) B C D A E (4) A E D B C (4) E A D B C (3) D C B E A (3) D C B A E (3) C B E D A (3) B C E A D (3) A E B C D (3) A D B E C (3) A B E C D (3) E D C B A (2) E A B C D (2) D A E C B (2) D A C B E (2) C D B E A (2) B E C A D (2) B A C E D (2) A E B D C (2) A D E C B (2) A B C D E (2) E C A D B (1) E A B D C (1) D E C B A (1) D E C A B (1) D C E B A (1) C E D B A (1) C D B A E (1) C B D E A (1) B E A C D (1) B C A E D (1) B A E C D (1) B A C D E (1) A D E B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 12 18 18 4 B -12 0 8 -10 6 C -18 -8 0 -10 -12 D -18 10 10 0 -2 E -4 -6 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 18 18 4 B -12 0 8 -10 6 C -18 -8 0 -10 -12 D -18 10 10 0 -2 E -4 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 B=21 D=18 C=8 so C is eliminated. Round 2 votes counts: A=29 E=25 B=25 D=21 so D is eliminated. Round 3 votes counts: B=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:202 D:200 B:196 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 18 18 4 B -12 0 8 -10 6 C -18 -8 0 -10 -12 D -18 10 10 0 -2 E -4 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 18 18 4 B -12 0 8 -10 6 C -18 -8 0 -10 -12 D -18 10 10 0 -2 E -4 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 18 18 4 B -12 0 8 -10 6 C -18 -8 0 -10 -12 D -18 10 10 0 -2 E -4 -6 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998593 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8842: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (14) B D C E A (10) D B A C E (8) E A C B D (7) C B D E A (7) A E D B C (6) D A B C E (4) E C B A D (3) E C A B D (3) E A C D B (3) D C B A E (3) A D B E C (3) D A B E C (2) C E B D A (2) C B E D A (2) B D E C A (2) B D C A E (2) B C E D A (2) B C D E A (2) A E D C B (2) E C B D A (1) E A D B C (1) E A B D C (1) D E A B C (1) D B E A C (1) D B A E C (1) C E B A D (1) C E A B D (1) C D B A E (1) C D A B E (1) C A D B E (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -24 -16 -32 0 B 24 0 20 -8 32 C 16 -20 0 -22 24 D 32 8 22 0 28 E 0 -32 -24 -28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -16 -32 0 B 24 0 20 -8 32 C 16 -20 0 -22 24 D 32 8 22 0 28 E 0 -32 -24 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=19 B=18 C=16 A=13 so A is eliminated. Round 2 votes counts: D=37 E=28 B=18 C=17 so C is eliminated. Round 3 votes counts: D=41 E=32 B=27 so B is eliminated. Round 4 votes counts: D=64 E=36 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:245 B:234 C:199 A:164 E:158 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 -16 -32 0 B 24 0 20 -8 32 C 16 -20 0 -22 24 D 32 8 22 0 28 E 0 -32 -24 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -16 -32 0 B 24 0 20 -8 32 C 16 -20 0 -22 24 D 32 8 22 0 28 E 0 -32 -24 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -16 -32 0 B 24 0 20 -8 32 C 16 -20 0 -22 24 D 32 8 22 0 28 E 0 -32 -24 -28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999785 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8843: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (9) B A C D E (9) E D C A B (5) D E C A B (5) C D B E A (5) B A E D C (5) A E B C D (5) D C E B A (4) A B E D C (4) C D E B A (3) C D E A B (3) B C D A E (3) B A C E D (3) A C B E D (3) A B E C D (3) E D A C B (2) E C D A B (2) C B D A E (2) B D E A C (2) B A D E C (2) A E C D B (2) A E B D C (2) A B C E D (2) E D A B C (1) D C B E A (1) D B E A C (1) D B C E A (1) C E A D B (1) C B A D E (1) C A E D B (1) C A B E D (1) B D C E A (1) B D C A E (1) B D A E C (1) B C A D E (1) B A E C D (1) B A D C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 0 -4 8 B 14 0 -2 6 8 C 0 2 0 4 -4 D 4 -6 -4 0 12 E -8 -8 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428573 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -14 0 -4 8 B 14 0 -2 6 8 C 0 2 0 4 -4 D 4 -6 -4 0 12 E -8 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428247 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=22 D=21 C=17 E=10 so E is eliminated. Round 2 votes counts: B=30 D=29 A=22 C=19 so C is eliminated. Round 3 votes counts: D=42 B=33 A=25 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:203 C:201 A:195 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -4 8 B 14 0 -2 6 8 C 0 2 0 4 -4 D 4 -6 -4 0 12 E -8 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428247 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -4 8 B 14 0 -2 6 8 C 0 2 0 4 -4 D 4 -6 -4 0 12 E -8 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428247 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -4 8 B 14 0 -2 6 8 C 0 2 0 4 -4 D 4 -6 -4 0 12 E -8 -8 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428247 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8844: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (12) E C A D B (11) D B A C E (10) E C B A D (7) E D A C B (6) B D A C E (5) B C A D E (5) C E A B D (4) D A B C E (3) B E C D A (3) B A C D E (3) E A D C B (2) D A E B C (2) B D C A E (2) B C A E D (2) A C D E B (2) E D B A C (1) E C D A B (1) E B D C A (1) E A C D B (1) D E A B C (1) D A E C B (1) C E B A D (1) C B A E D (1) C B A D E (1) C A E D B (1) C A E B D (1) C A B E D (1) B D E C A (1) B D E A C (1) B D A E C (1) B C E A D (1) B A D C E (1) A E D C B (1) A E C D B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -12 22 -8 B -6 0 -14 8 -16 C 12 14 0 22 -8 D -22 -8 -22 0 -20 E 8 16 8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -12 22 -8 B -6 0 -14 8 -16 C 12 14 0 22 -8 D -22 -8 -22 0 -20 E 8 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 B=25 D=17 C=10 A=6 so A is eliminated. Round 2 votes counts: E=44 B=25 D=17 C=14 so C is eliminated. Round 3 votes counts: E=52 B=28 D=20 so D is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:220 A:204 B:186 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -12 22 -8 B -6 0 -14 8 -16 C 12 14 0 22 -8 D -22 -8 -22 0 -20 E 8 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 22 -8 B -6 0 -14 8 -16 C 12 14 0 22 -8 D -22 -8 -22 0 -20 E 8 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 22 -8 B -6 0 -14 8 -16 C 12 14 0 22 -8 D -22 -8 -22 0 -20 E 8 16 8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8845: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) C B D A E (7) C B A E D (7) B A C E D (6) D E C A B (4) D E A B C (4) D C B A E (4) D A E B C (4) E D A B C (3) E A B D C (3) C E D B A (3) C D B E A (3) B C A E D (3) A B E C D (3) E C A B D (2) E A D B C (2) D C E B A (2) D C B E A (2) D B A C E (2) D A B C E (2) C E B A D (2) C D E B A (2) C B A D E (2) A E B D C (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D B A (1) E A D C B (1) D E A C B (1) D C E A B (1) D C A B E (1) D A B E C (1) C E B D A (1) C D B A E (1) C B E A D (1) B C A D E (1) B A C D E (1) A E B C D (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 0 0 8 B 2 0 2 8 6 C 0 -2 0 10 14 D 0 -8 -10 0 0 E -8 -6 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 0 8 B 2 0 2 8 6 C 0 -2 0 10 14 D 0 -8 -10 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=28 E=20 A=12 B=11 so B is eliminated. Round 2 votes counts: C=33 D=28 E=20 A=19 so A is eliminated. Round 3 votes counts: C=41 D=33 E=26 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:209 A:203 D:191 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 0 8 B 2 0 2 8 6 C 0 -2 0 10 14 D 0 -8 -10 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 0 8 B 2 0 2 8 6 C 0 -2 0 10 14 D 0 -8 -10 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 0 8 B 2 0 2 8 6 C 0 -2 0 10 14 D 0 -8 -10 0 0 E -8 -6 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8846: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (9) E A B D C (7) E B D A C (6) D B E C A (6) A E C B D (6) C A D B E (5) A C E B D (4) E D B A C (3) D B C E A (3) C D B E A (3) B D E A C (3) E D B C A (2) E C D A B (2) E C A D B (2) E A C B D (2) D E B C A (2) D C B A E (2) D B C A E (2) C E A D B (2) C A E D B (2) C A E B D (2) C A B D E (2) B D C A E (2) A E B D C (2) A C B D E (2) A B C E D (2) E D C B A (1) E D A B C (1) E B A D C (1) D B E A C (1) C E D B A (1) C D A B E (1) C B D A E (1) B E D A C (1) B C A D E (1) B A D C E (1) A E B C D (1) A C B E D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -6 -4 2 B 2 0 0 0 2 C 6 0 0 4 0 D 4 0 -4 0 -4 E -2 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.625894 C: 0.374106 D: 0.000000 E: 0.000000 Sum of squares = 0.531698724935 Cumulative probabilities = A: 0.000000 B: 0.625894 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -4 2 B 2 0 0 0 2 C 6 0 0 4 0 D 4 0 -4 0 -4 E -2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999845 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 A=21 D=16 B=8 so B is eliminated. Round 2 votes counts: C=29 E=28 A=22 D=21 so D is eliminated. Round 3 votes counts: E=40 C=38 A=22 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:205 B:202 E:200 D:198 A:195 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 2 B 2 0 0 0 2 C 6 0 0 4 0 D 4 0 -4 0 -4 E -2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999845 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 2 B 2 0 0 0 2 C 6 0 0 4 0 D 4 0 -4 0 -4 E -2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999845 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 2 B 2 0 0 0 2 C 6 0 0 4 0 D 4 0 -4 0 -4 E -2 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999845 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8847: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (13) A C E D B (12) E D A B C (11) D E B A C (10) C A B D E (7) C B A D E (5) C A B E D (5) B D C E A (4) B C D E A (4) E A D C B (3) D B E C A (3) A E D C B (3) E D B A C (2) E B D C A (2) E A C D B (2) D B E A C (2) A E C D B (2) A C B D E (2) E D B C A (1) E D A C B (1) D E A B C (1) C B E D A (1) C A E B D (1) B D C A E (1) B C A D E (1) A D E C B (1) Total count = 100 A B C D E A 0 2 4 -12 -20 B -2 0 10 -8 -4 C -4 -10 0 -16 -14 D 12 8 16 0 8 E 20 4 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -12 -20 B -2 0 10 -8 -4 C -4 -10 0 -16 -14 D 12 8 16 0 8 E 20 4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=22 A=20 C=19 D=16 so D is eliminated. Round 2 votes counts: E=33 B=28 A=20 C=19 so C is eliminated. Round 3 votes counts: B=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:222 E:215 B:198 A:187 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -12 -20 B -2 0 10 -8 -4 C -4 -10 0 -16 -14 D 12 8 16 0 8 E 20 4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -12 -20 B -2 0 10 -8 -4 C -4 -10 0 -16 -14 D 12 8 16 0 8 E 20 4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -12 -20 B -2 0 10 -8 -4 C -4 -10 0 -16 -14 D 12 8 16 0 8 E 20 4 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8848: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (5) D B E A C (5) B D C E A (5) B D A C E (5) E D A B C (4) E C A D B (4) E A D B C (4) D E B A C (4) E D C B A (3) E D B C A (3) D B C E A (3) C D B E A (3) B D A E C (3) B A D C E (3) E D A C B (2) E C D A B (2) E A C D B (2) C E D B A (2) C B D E A (2) C A B D E (2) B D E A C (2) B D C A E (2) B C D A E (2) A E C B D (2) A E B D C (2) A C E D B (2) A C E B D (2) A C B E D (2) A C B D E (2) A B D E C (2) E D B A C (1) E C D B A (1) E A D C B (1) D E B C A (1) C E A D B (1) C D E B A (1) C B D A E (1) C B A D E (1) C A E D B (1) C A E B D (1) A E D B C (1) A E C D B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 8 -24 -22 B 16 0 18 -14 4 C -8 -18 0 -24 -10 D 24 14 24 0 12 E 22 -4 10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 8 -24 -22 B 16 0 18 -14 4 C -8 -18 0 -24 -10 D 24 14 24 0 12 E 22 -4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=22 D=18 A=18 C=15 so C is eliminated. Round 2 votes counts: E=30 B=26 D=22 A=22 so D is eliminated. Round 3 votes counts: B=42 E=36 A=22 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:237 B:212 E:208 A:173 C:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 8 -24 -22 B 16 0 18 -14 4 C -8 -18 0 -24 -10 D 24 14 24 0 12 E 22 -4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 8 -24 -22 B 16 0 18 -14 4 C -8 -18 0 -24 -10 D 24 14 24 0 12 E 22 -4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 8 -24 -22 B 16 0 18 -14 4 C -8 -18 0 -24 -10 D 24 14 24 0 12 E 22 -4 10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8849: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (11) D B A C E (8) B E A C D (8) A E C B D (8) D B C A E (7) B D A E C (6) B D E C A (5) C D E A B (4) E C A B D (3) E A C B D (3) B D E A C (3) E B A C D (2) E A B C D (2) D C E A B (2) D C B E A (2) D C A E B (2) D C A B E (2) D A C B E (2) B E A D C (2) B D C E A (2) B A D E C (2) A C E D B (2) A B E C D (2) E B C A D (1) D C B A E (1) D A C E B (1) D A B C E (1) C E B D A (1) C A E D B (1) B E D C A (1) B A E C D (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -26 10 -22 -4 B 26 0 28 4 32 C -10 -28 0 -20 0 D 22 -4 20 0 24 E 4 -32 0 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 10 -22 -4 B 26 0 28 4 32 C -10 -28 0 -20 0 D 22 -4 20 0 24 E 4 -32 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=30 A=14 E=11 C=6 so C is eliminated. Round 2 votes counts: D=43 B=30 A=15 E=12 so E is eliminated. Round 3 votes counts: D=43 B=34 A=23 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:245 D:231 A:179 E:174 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 10 -22 -4 B 26 0 28 4 32 C -10 -28 0 -20 0 D 22 -4 20 0 24 E 4 -32 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 10 -22 -4 B 26 0 28 4 32 C -10 -28 0 -20 0 D 22 -4 20 0 24 E 4 -32 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 10 -22 -4 B 26 0 28 4 32 C -10 -28 0 -20 0 D 22 -4 20 0 24 E 4 -32 0 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8850: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) E C B A D (8) C D E A B (6) B A E D C (6) A B D E C (6) D A C B E (5) C E B D A (5) B E A C D (5) A D B C E (5) E B C A D (4) C E D B A (4) C E B A D (4) D C A E B (3) D A B E C (3) E B A C D (2) D C A B E (2) D A C E B (2) C D A E B (2) B E C A D (2) B A E C D (2) A D B E C (2) E D A B C (1) E B A D C (1) E A B D C (1) D C E A B (1) D A E C B (1) C E D A B (1) B E A D C (1) B A C E D (1) A E D B C (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 16 6 8 B -10 0 12 0 6 C -16 -12 0 -8 6 D -6 0 8 0 0 E -8 -6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 16 6 8 B -10 0 12 0 6 C -16 -12 0 -8 6 D -6 0 8 0 0 E -8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=22 E=17 B=17 A=16 so A is eliminated. Round 2 votes counts: D=35 B=25 C=22 E=18 so E is eliminated. Round 3 votes counts: D=37 B=33 C=30 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:220 B:204 D:201 E:190 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 16 6 8 B -10 0 12 0 6 C -16 -12 0 -8 6 D -6 0 8 0 0 E -8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 6 8 B -10 0 12 0 6 C -16 -12 0 -8 6 D -6 0 8 0 0 E -8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 6 8 B -10 0 12 0 6 C -16 -12 0 -8 6 D -6 0 8 0 0 E -8 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998827 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8851: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) E D C A B (8) B C A E D (8) D E A B C (6) B C A D E (6) D B A E C (4) C B E D A (4) A B D C E (4) E C D B A (3) D E A C B (3) D A E B C (3) C B A E D (3) B C D A E (3) B A D C E (3) B A C D E (3) A B C D E (3) C E B D A (2) C E B A D (2) C B E A D (2) A E D C B (2) A C E B D (2) A B D E C (2) E D C B A (1) E A D C B (1) D E C B A (1) D E C A B (1) D E B A C (1) D B E C A (1) D A B E C (1) C E D B A (1) C E A B D (1) C A B E D (1) B D A C E (1) B C D E A (1) B A C E D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 2 -8 4 B 2 0 6 4 4 C -2 -6 0 -8 2 D 8 -4 8 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999878 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -8 4 B 2 0 6 4 4 C -2 -6 0 -8 2 D 8 -4 8 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 D=21 C=16 A=15 so A is eliminated. Round 2 votes counts: B=35 E=24 D=23 C=18 so C is eliminated. Round 3 votes counts: B=45 E=32 D=23 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 D:205 A:198 E:196 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -8 4 B 2 0 6 4 4 C -2 -6 0 -8 2 D 8 -4 8 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -8 4 B 2 0 6 4 4 C -2 -6 0 -8 2 D 8 -4 8 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -8 4 B 2 0 6 4 4 C -2 -6 0 -8 2 D 8 -4 8 0 -2 E -4 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8852: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) E D B A C (8) B E D C A (8) B C A E D (8) D E B C A (7) C A B D E (7) A C B E D (7) A C B D E (5) E D B C A (4) D E B A C (4) A C D E B (4) D E C A B (3) C B A D E (3) B C D E A (3) B C A D E (3) E D A C B (2) B E D A C (2) B A C E D (2) A E D C B (2) E B D C A (1) D E C B A (1) D E A B C (1) B E C D A (1) B D E C A (1) B C E D A (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -2 -16 -16 B 14 0 8 6 2 C 2 -8 0 -10 -12 D 16 -6 10 0 6 E 16 -2 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998627 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 -16 -16 B 14 0 8 6 2 C 2 -8 0 -10 -12 D 16 -6 10 0 6 E 16 -2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=27 A=19 E=15 C=10 so C is eliminated. Round 2 votes counts: B=32 D=27 A=26 E=15 so E is eliminated. Round 3 votes counts: D=41 B=33 A=26 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:213 E:210 C:186 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 -16 -16 B 14 0 8 6 2 C 2 -8 0 -10 -12 D 16 -6 10 0 6 E 16 -2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 -16 -16 B 14 0 8 6 2 C 2 -8 0 -10 -12 D 16 -6 10 0 6 E 16 -2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 -16 -16 B 14 0 8 6 2 C 2 -8 0 -10 -12 D 16 -6 10 0 6 E 16 -2 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999241 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8853: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (10) E C B D A (8) A D B E C (8) C E B A D (5) E D A C B (4) E D A B C (4) C E B D A (4) B C A D E (4) E D C A B (3) D A B E C (3) C B E D A (3) C B A D E (3) B E C D A (3) B A D C E (3) A D B C E (3) E C D B A (2) E C D A B (2) D A E B C (2) C A B D E (2) B D A E C (2) B D A C E (2) B C E D A (2) B C D A E (2) A D E B C (2) A C D B E (2) E B C D A (1) E A D C B (1) D E A B C (1) C E A D B (1) C A D B E (1) B E D C A (1) B A C D E (1) A D E C B (1) A D C E B (1) A C D E B (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -14 2 -10 B 12 0 -10 16 14 C 14 10 0 16 4 D -2 -16 -16 0 -8 E 10 -14 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 2 -10 B 12 0 -10 16 14 C 14 10 0 16 4 D -2 -16 -16 0 -8 E 10 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=25 B=20 A=20 D=6 so D is eliminated. Round 2 votes counts: C=29 E=26 A=25 B=20 so B is eliminated. Round 3 votes counts: C=37 A=33 E=30 so E is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:216 E:200 A:183 D:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 2 -10 B 12 0 -10 16 14 C 14 10 0 16 4 D -2 -16 -16 0 -8 E 10 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 2 -10 B 12 0 -10 16 14 C 14 10 0 16 4 D -2 -16 -16 0 -8 E 10 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 2 -10 B 12 0 -10 16 14 C 14 10 0 16 4 D -2 -16 -16 0 -8 E 10 -14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8854: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) C E B A D (6) E D B A C (5) D A C B E (5) C B A E D (5) E C D B A (4) E C B D A (4) E B A D C (4) D E A B C (4) D A B E C (4) D A B C E (4) A B C E D (4) E C B A D (3) E B A C D (3) D E C B A (3) D C A B E (3) A C B D E (3) A B C D E (3) E D C B A (2) E D B C A (2) D E B A C (2) D C A E B (2) C E D B A (2) C A B E D (2) C A B D E (2) B A C E D (2) A D C B E (2) D E C A B (1) D E A C B (1) D A E B C (1) C E B D A (1) C A D B E (1) B C E A D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -2 0 -10 B 12 0 -4 4 -14 C 2 4 0 6 -4 D 0 -4 -6 0 -16 E 10 14 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999887 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -2 0 -10 B 12 0 -4 4 -14 C 2 4 0 6 -4 D 0 -4 -6 0 -16 E 10 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=30 C=19 A=14 B=3 so B is eliminated. Round 2 votes counts: E=34 D=30 C=20 A=16 so A is eliminated. Round 3 votes counts: E=35 D=33 C=32 so C is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:204 B:199 A:188 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -2 0 -10 B 12 0 -4 4 -14 C 2 4 0 6 -4 D 0 -4 -6 0 -16 E 10 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 0 -10 B 12 0 -4 4 -14 C 2 4 0 6 -4 D 0 -4 -6 0 -16 E 10 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 0 -10 B 12 0 -4 4 -14 C 2 4 0 6 -4 D 0 -4 -6 0 -16 E 10 14 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8855: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D A E B C (8) D B A C E (7) C E B A D (7) C B E A D (6) B C D E A (5) A E D C B (5) D A B E C (4) C B E D A (4) B D C A E (4) D B A E C (3) D A E C B (3) C E A B D (3) A D E B C (3) A D B E C (3) E C A D B (2) D C B E A (2) C D B E A (2) B D A C E (2) B C E A D (2) A E D B C (2) A D E C B (2) E C B A D (1) E C A B D (1) E A D C B (1) D C E B A (1) D B C A E (1) C E D B A (1) C E D A B (1) C E B D A (1) B D A E C (1) B C E D A (1) B C D A E (1) B A D E C (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 8 -4 2 B 6 0 -2 -18 -2 C -8 2 0 -6 2 D 4 18 6 0 6 E -2 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 -4 2 B 6 0 -2 -18 -2 C -8 2 0 -6 2 D 4 18 6 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=25 B=17 A=16 E=13 so E is eliminated. Round 2 votes counts: D=29 C=29 A=25 B=17 so B is eliminated. Round 3 votes counts: C=38 D=36 A=26 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:200 E:196 C:195 B:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 8 -4 2 B 6 0 -2 -18 -2 C -8 2 0 -6 2 D 4 18 6 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 -4 2 B 6 0 -2 -18 -2 C -8 2 0 -6 2 D 4 18 6 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 -4 2 B 6 0 -2 -18 -2 C -8 2 0 -6 2 D 4 18 6 0 6 E -2 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8856: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (16) C D A B E (10) A B E D C (6) E D C B A (5) C E D B A (5) A B D C E (5) E B A C D (4) D C A B E (4) C D E A B (4) B A E D C (4) A D B C E (4) E B C A D (3) D A C B E (3) E C D B A (2) D E C A B (2) D C A E B (2) C D E B A (2) C D A E B (2) B A E C D (2) A B D E C (2) A B C D E (2) E C B D A (1) E B D A C (1) E B C D A (1) D C E A B (1) D A B C E (1) C E B A D (1) C B E A D (1) C B A E D (1) B E A D C (1) B A C E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 6 8 0 B 2 0 8 6 -4 C -6 -8 0 -14 0 D -8 -6 14 0 -12 E 0 4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.427217 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.572783 Sum of squares = 0.510594755821 Cumulative probabilities = A: 0.427217 B: 0.427217 C: 0.427217 D: 0.427217 E: 1.000000 A B C D E A 0 -2 6 8 0 B 2 0 8 6 -4 C -6 -8 0 -14 0 D -8 -6 14 0 -12 E 0 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=26 A=20 D=13 B=8 so B is eliminated. Round 2 votes counts: E=34 A=27 C=26 D=13 so D is eliminated. Round 3 votes counts: E=36 C=33 A=31 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:208 A:206 B:206 D:194 C:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 6 8 0 B 2 0 8 6 -4 C -6 -8 0 -14 0 D -8 -6 14 0 -12 E 0 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 8 0 B 2 0 8 6 -4 C -6 -8 0 -14 0 D -8 -6 14 0 -12 E 0 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 8 0 B 2 0 8 6 -4 C -6 -8 0 -14 0 D -8 -6 14 0 -12 E 0 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999961 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8857: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (11) C A D E B (7) E D B A C (5) E D A B C (5) D E A C B (5) A C D E B (5) C A B D E (4) B C A E D (4) B C A D E (4) D A E C B (3) C D E A B (3) C B A D E (3) C A D B E (3) B E A D C (3) B C E D A (3) B C E A D (3) E D A C B (2) E B D A C (2) C E D A B (2) C B A E D (2) B E D C A (2) A D C E B (2) E D C B A (1) E D B C A (1) E B D C A (1) D E C A B (1) D C E A B (1) C E D B A (1) C D A E B (1) C B E D A (1) B E C D A (1) B E C A D (1) B A E D C (1) B A C E D (1) B A C D E (1) A D E C B (1) A D E B C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 0 -4 -10 B 4 0 2 -2 0 C 0 -2 0 2 4 D 4 2 -2 0 -6 E 10 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.671096 C: 0.000000 D: 0.000000 E: 0.328904 Sum of squares = 0.558547997352 Cumulative probabilities = A: 0.000000 B: 0.671096 C: 0.671096 D: 0.671096 E: 1.000000 A B C D E A 0 -4 0 -4 -10 B 4 0 2 -2 0 C 0 -2 0 2 4 D 4 2 -2 0 -6 E 10 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560852 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=27 E=17 A=11 D=10 so D is eliminated. Round 2 votes counts: B=35 C=28 E=23 A=14 so A is eliminated. Round 3 votes counts: C=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:206 B:202 C:202 D:199 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 -4 -10 B 4 0 2 -2 0 C 0 -2 0 2 4 D 4 2 -2 0 -6 E 10 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560852 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -4 -10 B 4 0 2 -2 0 C 0 -2 0 2 4 D 4 2 -2 0 -6 E 10 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560852 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -4 -10 B 4 0 2 -2 0 C 0 -2 0 2 4 D 4 2 -2 0 -6 E 10 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.555555560852 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8858: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (7) D C A B E (7) B E D C A (6) B D C E A (6) A E C D B (6) A C D E B (6) E B A C D (5) A C E D B (5) E A B C D (4) B D E C A (4) E B A D C (3) C D A E B (3) B E A D C (3) B E A C D (3) B D C A E (3) A E B C D (3) A C D B E (3) D C B A E (2) C D A B E (2) B E D A C (2) A B E C D (2) E D C A B (1) E C A D B (1) E A C D B (1) E A C B D (1) D C E B A (1) D C B E A (1) D C A E B (1) D B C E A (1) D B C A E (1) C A D E B (1) B A E D C (1) A E C B D (1) A C E B D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 4 4 0 B -2 0 12 14 -2 C -4 -12 0 0 -8 D -4 -14 0 0 -14 E 0 2 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.649406 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.350594 Sum of squares = 0.544644260216 Cumulative probabilities = A: 0.649406 B: 0.649406 C: 0.649406 D: 0.649406 E: 1.000000 A B C D E A 0 2 4 4 0 B -2 0 12 14 -2 C -4 -12 0 0 -8 D -4 -14 0 0 -14 E 0 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 E=23 D=14 C=6 so C is eliminated. Round 2 votes counts: A=30 B=28 E=23 D=19 so D is eliminated. Round 3 votes counts: A=43 B=33 E=24 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:212 B:211 A:205 C:188 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 4 0 B -2 0 12 14 -2 C -4 -12 0 0 -8 D -4 -14 0 0 -14 E 0 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 4 0 B -2 0 12 14 -2 C -4 -12 0 0 -8 D -4 -14 0 0 -14 E 0 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 4 0 B -2 0 12 14 -2 C -4 -12 0 0 -8 D -4 -14 0 0 -14 E 0 2 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999935 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8859: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) A E C B D (8) C B D A E (7) E A D B C (6) B C D E A (6) A C B E D (6) E A B C D (5) D B C E A (5) C B A D E (4) D E B C A (3) D E A C B (3) D C B E A (3) A E B C D (3) A D E C B (3) E D B C A (2) E A D C B (2) D C B A E (2) C A B D E (2) B E C D A (2) B C E A D (2) B C A E D (2) A E D C B (2) E D B A C (1) E B C D A (1) E B C A D (1) E B A C D (1) D B E C A (1) D A C B E (1) C D B A E (1) C D A B E (1) C B A E D (1) B D E C A (1) B D C E A (1) A E C D B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 6 4 2 -8 B -6 0 2 8 -2 C -4 -2 0 12 -8 D -2 -8 -12 0 -10 E 8 2 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 4 2 -8 B -6 0 2 8 -2 C -4 -2 0 12 -8 D -2 -8 -12 0 -10 E 8 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=25 D=18 C=16 B=14 so B is eliminated. Round 2 votes counts: E=29 C=26 A=25 D=20 so D is eliminated. Round 3 votes counts: E=37 C=37 A=26 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 A:202 B:201 C:199 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 2 -8 B -6 0 2 8 -2 C -4 -2 0 12 -8 D -2 -8 -12 0 -10 E 8 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 -8 B -6 0 2 8 -2 C -4 -2 0 12 -8 D -2 -8 -12 0 -10 E 8 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 -8 B -6 0 2 8 -2 C -4 -2 0 12 -8 D -2 -8 -12 0 -10 E 8 2 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8860: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) E D C A B (5) C D A B E (5) B E A D C (5) B A E D C (5) E B A D C (4) D C E A B (4) C B A D E (4) E D A C B (3) D E C A B (3) D C A E B (3) B E A C D (3) B A E C D (3) B A C D E (3) A D C B E (3) A B E D C (3) E D A B C (2) E C D B A (2) E B D C A (2) E B D A C (2) E A D B C (2) C D E A B (2) B E C A D (2) B C A E D (2) B C A D E (2) B A C E D (2) A E B D C (2) A C B D E (2) A B C D E (2) E D C B A (1) E B C D A (1) C E D B A (1) C D B E A (1) C B E D A (1) C B D A E (1) C A B D E (1) A E D B C (1) A D E C B (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 2 10 8 B -6 0 2 6 4 C -2 -2 0 -8 -6 D -10 -6 8 0 -8 E -8 -4 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 10 8 B -6 0 2 6 4 C -2 -2 0 -8 -6 D -10 -6 8 0 -8 E -8 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998434 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=24 C=22 A=17 D=10 so D is eliminated. Round 2 votes counts: C=29 E=27 B=27 A=17 so A is eliminated. Round 3 votes counts: C=34 B=34 E=32 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:203 E:201 D:192 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 10 8 B -6 0 2 6 4 C -2 -2 0 -8 -6 D -10 -6 8 0 -8 E -8 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998434 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 10 8 B -6 0 2 6 4 C -2 -2 0 -8 -6 D -10 -6 8 0 -8 E -8 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998434 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 10 8 B -6 0 2 6 4 C -2 -2 0 -8 -6 D -10 -6 8 0 -8 E -8 -4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998434 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8861: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (12) A B E C D (11) D C B E A (8) C E D A B (7) B A D E C (7) D C E B A (6) C D E A B (6) D B C A E (4) D B A E C (4) D B A C E (4) B D A E C (4) E A C B D (3) D B C E A (3) A E B C D (3) D E C B A (2) D B E C A (2) C E A D B (2) B A E C D (2) E C A D B (1) E C A B D (1) E A B C D (1) D E B C A (1) D C E A B (1) D C B A E (1) C D E B A (1) C A E D B (1) B E A D C (1) A E C B D (1) Total count = 100 A B C D E A 0 -24 6 -8 8 B 24 0 18 -8 26 C -6 -18 0 -20 -12 D 8 8 20 0 8 E -8 -26 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999744 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 6 -8 8 B 24 0 18 -8 26 C -6 -18 0 -20 -12 D 8 8 20 0 8 E -8 -26 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=26 C=17 A=15 E=6 so E is eliminated. Round 2 votes counts: D=36 B=26 C=19 A=19 so C is eliminated. Round 3 votes counts: D=50 B=26 A=24 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:230 D:222 A:191 E:185 C:172 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 6 -8 8 B 24 0 18 -8 26 C -6 -18 0 -20 -12 D 8 8 20 0 8 E -8 -26 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 6 -8 8 B 24 0 18 -8 26 C -6 -18 0 -20 -12 D 8 8 20 0 8 E -8 -26 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 6 -8 8 B 24 0 18 -8 26 C -6 -18 0 -20 -12 D 8 8 20 0 8 E -8 -26 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999375 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8862: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (6) E B C D A (5) E B C A D (4) A E B D C (4) A D E B C (4) E D C A B (3) E A B D C (3) D C E B A (3) D A E C B (3) C E D B A (3) B E C A D (3) B C E D A (3) B C A E D (3) B A C D E (3) A D C B E (3) E D A C B (2) E B A C D (2) D E C A B (2) D C A B E (2) D A C E B (2) C D B E A (2) C B D E A (2) C B D A E (2) B C E A D (2) B C D A E (2) A E D B C (2) A D C E B (2) A D B C E (2) A B E D C (2) A B C D E (2) E A B C D (1) D E C B A (1) D E A C B (1) D C E A B (1) D C A E B (1) D A C B E (1) C E B D A (1) C D B A E (1) C B E D A (1) B E C D A (1) B E A C D (1) B C D E A (1) B C A D E (1) B A E C D (1) B A C E D (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 8 4 B -2 0 10 4 -12 C 0 -10 0 -4 -4 D -8 -4 4 0 4 E -4 12 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.902025 B: 0.000000 C: 0.097975 D: 0.000000 E: 0.000000 Sum of squares = 0.823247879984 Cumulative probabilities = A: 0.902025 B: 0.902025 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 8 4 B -2 0 10 4 -12 C 0 -10 0 -4 -4 D -8 -4 4 0 4 E -4 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222223944 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=22 E=20 D=17 C=12 so C is eliminated. Round 2 votes counts: A=29 B=27 E=24 D=20 so D is eliminated. Round 3 votes counts: A=38 E=32 B=30 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:204 B:200 D:198 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 8 4 B -2 0 10 4 -12 C 0 -10 0 -4 -4 D -8 -4 4 0 4 E -4 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222223944 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 8 4 B -2 0 10 4 -12 C 0 -10 0 -4 -4 D -8 -4 4 0 4 E -4 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222223944 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 8 4 B -2 0 10 4 -12 C 0 -10 0 -4 -4 D -8 -4 4 0 4 E -4 12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.722222223944 Cumulative probabilities = A: 0.833333 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8863: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (8) A B C E D (7) C E A D B (6) B D E C A (6) B D A E C (6) B A D E C (6) A C E B D (6) D E C B A (5) E C D B A (4) E C A D B (4) D B E C A (4) B D E A C (4) B A D C E (4) A C E D B (4) D E B C A (3) B D A C E (3) A B E C D (3) E C D A B (2) A C B E D (2) A B D C E (2) E D C B A (1) E C B A D (1) E A C B D (1) D C E B A (1) D B C E A (1) D B A C E (1) C A D E B (1) B E A D C (1) A E B C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 6 2 -4 B 2 0 8 10 4 C -6 -8 0 2 -6 D -2 -10 -2 0 -2 E 4 -4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 2 -4 B 2 0 8 10 4 C -6 -8 0 2 -6 D -2 -10 -2 0 -2 E 4 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 D=15 C=15 E=13 so E is eliminated. Round 2 votes counts: B=30 A=28 C=26 D=16 so D is eliminated. Round 3 votes counts: B=39 C=33 A=28 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:212 E:204 A:201 D:192 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 2 -4 B 2 0 8 10 4 C -6 -8 0 2 -6 D -2 -10 -2 0 -2 E 4 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 2 -4 B 2 0 8 10 4 C -6 -8 0 2 -6 D -2 -10 -2 0 -2 E 4 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 2 -4 B 2 0 8 10 4 C -6 -8 0 2 -6 D -2 -10 -2 0 -2 E 4 -4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8864: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) E C D A B (7) C B E D A (6) B D A C E (6) B C D A E (5) A D B E C (5) E C A D B (4) C E B D A (4) B A C D E (4) A E D B C (4) A D E B C (4) E C A B D (3) E A D C B (3) D B A C E (3) A B D E C (3) E D C A B (2) E D A C B (2) D A E C B (2) C E D B A (2) B D C A E (2) B C D E A (2) A E D C B (2) E A C D B (1) E A C B D (1) E A B C D (1) D E C A B (1) D B C A E (1) D A E B C (1) D A B E C (1) C E B A D (1) C D E B A (1) C B E A D (1) C B D E A (1) B C E D A (1) B C E A D (1) B C A E D (1) B C A D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 6 0 10 B 2 0 12 8 8 C -6 -12 0 -4 4 D 0 -8 4 0 4 E -10 -8 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 0 10 B 2 0 12 8 8 C -6 -12 0 -4 4 D 0 -8 4 0 4 E -10 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=24 A=20 C=16 D=9 so D is eliminated. Round 2 votes counts: B=35 E=25 A=24 C=16 so C is eliminated. Round 3 votes counts: B=43 E=33 A=24 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:207 D:200 C:191 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 0 10 B 2 0 12 8 8 C -6 -12 0 -4 4 D 0 -8 4 0 4 E -10 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 10 B 2 0 12 8 8 C -6 -12 0 -4 4 D 0 -8 4 0 4 E -10 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 10 B 2 0 12 8 8 C -6 -12 0 -4 4 D 0 -8 4 0 4 E -10 -8 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999658 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8865: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) C B D A E (6) A C D B E (6) E D B C A (5) E A B D C (5) D C B E A (5) D C B A E (5) C D B A E (4) B C D E A (4) E B D C A (3) E A B C D (3) D E C B A (3) B D C E A (3) B C D A E (3) A E C D B (3) A E B C D (3) A D C E B (3) E A D C B (2) C D A B E (2) B C E D A (2) A E C B D (2) A E B D C (2) A D C B E (2) A C B D E (2) E D A C B (1) E B C D A (1) E B A D C (1) D C E B A (1) D C A B E (1) D B E C A (1) D B C E A (1) D A C B E (1) C D B E A (1) B E C D A (1) B C A E D (1) A E D C B (1) A E D B C (1) A C E D B (1) A C D E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -6 -8 2 B 2 0 -6 -14 4 C 6 6 0 -6 10 D 8 14 6 0 10 E -2 -4 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -8 2 B 2 0 -6 -14 4 C 6 6 0 -6 10 D 8 14 6 0 10 E -2 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 D=18 B=14 C=13 so C is eliminated. Round 2 votes counts: A=28 E=27 D=25 B=20 so B is eliminated. Round 3 votes counts: D=41 E=30 A=29 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:208 A:193 B:193 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 2 B 2 0 -6 -14 4 C 6 6 0 -6 10 D 8 14 6 0 10 E -2 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 2 B 2 0 -6 -14 4 C 6 6 0 -6 10 D 8 14 6 0 10 E -2 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 2 B 2 0 -6 -14 4 C 6 6 0 -6 10 D 8 14 6 0 10 E -2 -4 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8866: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) C B A D E (10) B A D C E (8) D A E B C (7) D A B E C (7) C B E A D (7) C E B A D (6) B C A D E (5) E D A C B (4) E C D A B (4) C B D A E (3) C B A E D (3) B A C D E (3) E C B D A (2) D E A B C (2) D A B C E (2) C E B D A (2) A D B E C (2) E C D B A (1) E C A D B (1) E B C A D (1) D E A C B (1) D A C B E (1) C D E B A (1) C D B A E (1) C D A B E (1) B D A C E (1) B C E A D (1) B C D A E (1) B A C E D (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 0 -2 14 B 14 0 4 8 16 C 0 -4 0 8 16 D 2 -8 -8 0 14 E -14 -16 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -2 14 B 14 0 4 8 16 C 0 -4 0 8 16 D 2 -8 -8 0 14 E -14 -16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=23 D=20 B=20 A=3 so A is eliminated. Round 2 votes counts: C=34 E=23 D=23 B=20 so B is eliminated. Round 3 votes counts: C=45 D=32 E=23 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:210 D:200 A:199 E:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 -2 14 B 14 0 4 8 16 C 0 -4 0 8 16 D 2 -8 -8 0 14 E -14 -16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -2 14 B 14 0 4 8 16 C 0 -4 0 8 16 D 2 -8 -8 0 14 E -14 -16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -2 14 B 14 0 4 8 16 C 0 -4 0 8 16 D 2 -8 -8 0 14 E -14 -16 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8867: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) A C B E D (6) D E C A B (5) B E C A D (5) B A E C D (5) D A C E B (4) D A B E C (4) D C E A B (3) B E A C D (3) B D E A C (3) A D C E B (3) A D B C E (3) A B C E D (3) E D C B A (2) E C D B A (2) E C B D A (2) E B C D A (2) D B E A C (2) D A E C B (2) C A D E B (2) B E D C A (2) B D A E C (2) B C E A D (2) B A C E D (2) A B D C E (2) A B C D E (2) E C B A D (1) E B D C A (1) D E B C A (1) D E A B C (1) D C A E B (1) D B E C A (1) D B A E C (1) D A E B C (1) D A B C E (1) C E D A B (1) C D A E B (1) C A E D B (1) C A E B D (1) B E D A C (1) B E C D A (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 6 -14 -2 B 4 0 4 -4 6 C -6 -4 0 -14 -22 D 14 4 14 0 14 E 2 -6 22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999316 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -14 -2 B 4 0 4 -4 6 C -6 -4 0 -14 -22 D 14 4 14 0 14 E 2 -6 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991392 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=27 A=20 E=10 C=6 so C is eliminated. Round 2 votes counts: D=38 B=27 A=24 E=11 so E is eliminated. Round 3 votes counts: D=43 B=33 A=24 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:205 E:202 A:193 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -14 -2 B 4 0 4 -4 6 C -6 -4 0 -14 -22 D 14 4 14 0 14 E 2 -6 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991392 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -14 -2 B 4 0 4 -4 6 C -6 -4 0 -14 -22 D 14 4 14 0 14 E 2 -6 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991392 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -14 -2 B 4 0 4 -4 6 C -6 -4 0 -14 -22 D 14 4 14 0 14 E 2 -6 22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991392 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8868: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (11) D A B C E (9) D C B E A (7) A E B C D (7) D A E B C (6) C B E A D (6) E C B A D (5) D A E C B (5) B C E A D (4) B C D E A (4) D E A C B (3) D B C A E (3) E B C A D (2) E A B C D (2) D C E B A (2) C E B A D (2) B C D A E (2) B A C E D (2) A E D C B (2) A D E B C (2) E C D B A (1) E C B D A (1) E C A B D (1) E B A C D (1) D E C B A (1) D A C E B (1) C E B D A (1) C B E D A (1) C B D E A (1) B C A E D (1) B A C D E (1) A E D B C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 4 10 4 -12 B -4 0 -2 12 -14 C -10 2 0 14 -2 D -4 -12 -14 0 -4 E 12 14 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 4 -12 B -4 0 -2 12 -14 C -10 2 0 14 -2 D -4 -12 -14 0 -4 E 12 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=24 B=14 A=14 C=11 so C is eliminated. Round 2 votes counts: D=37 E=27 B=22 A=14 so A is eliminated. Round 3 votes counts: D=40 E=37 B=23 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 A:203 C:202 B:196 D:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 4 -12 B -4 0 -2 12 -14 C -10 2 0 14 -2 D -4 -12 -14 0 -4 E 12 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 4 -12 B -4 0 -2 12 -14 C -10 2 0 14 -2 D -4 -12 -14 0 -4 E 12 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 4 -12 B -4 0 -2 12 -14 C -10 2 0 14 -2 D -4 -12 -14 0 -4 E 12 14 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8869: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (17) A E C B D (17) A D B C E (13) E C B D A (11) C E B D A (4) A E D C B (4) E C A B D (3) A D E B C (3) A C E B D (3) E C B A D (2) D B A C E (2) D A B C E (2) C B E A D (2) B D C E A (2) B C E D A (2) B C D E A (2) A D B E C (2) A C B E D (2) E D B C A (1) E B C D A (1) D E B C A (1) D B E A C (1) D B C A E (1) C B E D A (1) A E C D B (1) Total count = 100 A B C D E A 0 0 0 4 0 B 0 0 0 4 -2 C 0 0 0 2 6 D -4 -4 -2 0 -8 E 0 2 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.328854 B: 0.241326 C: 0.429821 D: 0.000000 E: 0.000000 Sum of squares = 0.351128670941 Cumulative probabilities = A: 0.328854 B: 0.570179 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 4 0 B 0 0 0 4 -2 C 0 0 0 2 6 D -4 -4 -2 0 -8 E 0 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 D=24 E=18 C=7 B=6 so B is eliminated. Round 2 votes counts: A=45 D=26 E=18 C=11 so C is eliminated. Round 3 votes counts: A=45 D=28 E=27 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:204 A:202 E:202 B:201 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 4 0 B 0 0 0 4 -2 C 0 0 0 2 6 D -4 -4 -2 0 -8 E 0 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 4 0 B 0 0 0 4 -2 C 0 0 0 2 6 D -4 -4 -2 0 -8 E 0 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 4 0 B 0 0 0 4 -2 C 0 0 0 2 6 D -4 -4 -2 0 -8 E 0 2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8870: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (11) E B C A D (8) C A D B E (8) E B D C A (6) E B D A C (6) B E C A D (5) D A B E C (4) D A B C E (4) C E B A D (4) E C B A D (3) E B C D A (3) D E B A C (3) D E A B C (3) A D C B E (3) A C D B E (3) E D B A C (2) D E C A B (2) D A C E B (2) C B E A D (2) B E A C D (2) E D C A B (1) E D B C A (1) E D A B C (1) E C D A B (1) E C A B D (1) D B E A C (1) C E D A B (1) C E A D B (1) C E A B D (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B D E (1) B E A D C (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 4 -4 -6 -18 B -4 0 2 -8 -6 C 4 -2 0 -4 -10 D 6 8 4 0 -4 E 18 6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -4 -6 -18 B -4 0 2 -8 -6 C 4 -2 0 -4 -10 D 6 8 4 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=30 C=21 B=9 A=7 so A is eliminated. Round 2 votes counts: E=33 D=33 C=25 B=9 so B is eliminated. Round 3 votes counts: E=41 D=34 C=25 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 D:207 C:194 B:192 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 -6 -18 B -4 0 2 -8 -6 C 4 -2 0 -4 -10 D 6 8 4 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -6 -18 B -4 0 2 -8 -6 C 4 -2 0 -4 -10 D 6 8 4 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -6 -18 B -4 0 2 -8 -6 C 4 -2 0 -4 -10 D 6 8 4 0 -4 E 18 6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8871: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) B C E A D (7) C B E D A (6) A E D C B (6) A D E B C (6) C E D A B (5) C E B A D (5) C E D B A (4) C E B D A (4) B A D C E (4) A D B E C (4) A B D E C (4) E D C A B (3) D E A C B (3) B C A E D (3) D A E B C (2) D A B E C (2) C B E A D (2) B D C A E (2) B C E D A (2) B C A D E (2) B A C D E (2) E C A D B (1) E A C D B (1) D E B A C (1) D B E C A (1) D B C E A (1) D A E C B (1) C E A D B (1) C B D E A (1) B D C E A (1) B D A C E (1) B C D E A (1) A E C D B (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -20 2 -16 B 0 0 -6 -4 -8 C 20 6 0 14 8 D -2 4 -14 0 -20 E 16 8 -8 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -20 2 -16 B 0 0 -6 -4 -8 C 20 6 0 14 8 D -2 4 -14 0 -20 E 16 8 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=25 A=23 E=13 D=11 so D is eliminated. Round 2 votes counts: C=28 A=28 B=27 E=17 so E is eliminated. Round 3 votes counts: C=40 A=32 B=28 so B is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:218 B:191 D:184 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -20 2 -16 B 0 0 -6 -4 -8 C 20 6 0 14 8 D -2 4 -14 0 -20 E 16 8 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 2 -16 B 0 0 -6 -4 -8 C 20 6 0 14 8 D -2 4 -14 0 -20 E 16 8 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 2 -16 B 0 0 -6 -4 -8 C 20 6 0 14 8 D -2 4 -14 0 -20 E 16 8 -8 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999863 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8872: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (13) C A D B E (12) D A C E B (8) E B D A C (7) A C D B E (7) E D B A C (6) E B D C A (6) C A B E D (5) C A B D E (5) C B A E D (4) D A E C B (3) B C E A D (3) D B E A C (2) C A D E B (2) A C D E B (2) E D A C B (1) E B C D A (1) E B C A D (1) D E B A C (1) D E A C B (1) D E A B C (1) D A C B E (1) C E B A D (1) C B A D E (1) C A E B D (1) B E C D A (1) B D E A C (1) B C A E D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 2 -14 20 8 B -2 0 -12 2 14 C 14 12 0 20 10 D -20 -2 -20 0 -2 E -8 -14 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 20 8 B -2 0 -12 2 14 C 14 12 0 20 10 D -20 -2 -20 0 -2 E -8 -14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=22 B=19 D=17 A=11 so A is eliminated. Round 2 votes counts: C=40 E=22 D=19 B=19 so D is eliminated. Round 3 votes counts: C=51 E=28 B=21 so B is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:208 B:201 E:185 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 20 8 B -2 0 -12 2 14 C 14 12 0 20 10 D -20 -2 -20 0 -2 E -8 -14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 20 8 B -2 0 -12 2 14 C 14 12 0 20 10 D -20 -2 -20 0 -2 E -8 -14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 20 8 B -2 0 -12 2 14 C 14 12 0 20 10 D -20 -2 -20 0 -2 E -8 -14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8873: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) A D C E B (6) D A C E B (5) B E C D A (5) B D E C A (5) A C E D B (5) A B D E C (5) A D B E C (4) A C E B D (4) D C E B A (3) D C E A B (3) D B E C A (3) D B A E C (3) B E D C A (3) B E C A D (3) B A E C D (3) B A D E C (3) A B E C D (3) E C B D A (2) D B C E A (2) D A B E C (2) C E D B A (2) C E A B D (2) C A E D B (2) B D E A C (2) E C B A D (1) E B C D A (1) E B C A D (1) D C A E B (1) D B C A E (1) C E D A B (1) C E B A D (1) C E A D B (1) C D E A B (1) C D A E B (1) C A E B D (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 12 8 -6 16 B -12 0 14 -10 10 C -8 -14 0 -18 2 D 6 10 18 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 -6 16 B -12 0 14 -10 10 C -8 -14 0 -18 2 D 6 10 18 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=29 B=24 C=12 E=5 so E is eliminated. Round 2 votes counts: D=30 A=29 B=26 C=15 so C is eliminated. Round 3 votes counts: D=35 A=35 B=30 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 A:215 B:201 C:181 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 8 -6 16 B -12 0 14 -10 10 C -8 -14 0 -18 2 D 6 10 18 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 -6 16 B -12 0 14 -10 10 C -8 -14 0 -18 2 D 6 10 18 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 -6 16 B -12 0 14 -10 10 C -8 -14 0 -18 2 D 6 10 18 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8874: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (10) A E D C B (9) B D A E C (8) C B E D A (6) A C E D B (6) D E A B C (5) D A E B C (5) C B A E D (5) E D A C B (4) A D E B C (4) D E B A C (3) C E A D B (3) C B E A D (3) C A B E D (3) B D E A C (3) E D A B C (2) B C A D E (2) E D C A B (1) E C D A B (1) E A D C B (1) E A C D B (1) D B E A C (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B A D (1) C B D A E (1) C B A D E (1) C A E D B (1) C A E B D (1) B D E C A (1) B D C E A (1) B A D E C (1) A E D B C (1) A E C D B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 14 -10 2 B -2 0 -2 -4 -4 C -14 2 0 -2 -6 D 10 4 2 0 -4 E -2 4 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468750000343 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 A B C D E A 0 2 14 -10 2 B -2 0 -2 -4 -4 C -14 2 0 -2 -6 D 10 4 2 0 -4 E -2 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=26 A=22 D=15 E=10 so E is eliminated. Round 2 votes counts: C=28 B=26 A=24 D=22 so D is eliminated. Round 3 votes counts: A=40 B=31 C=29 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:206 E:206 A:204 B:194 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 14 -10 2 B -2 0 -2 -4 -4 C -14 2 0 -2 -6 D 10 4 2 0 -4 E -2 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 -10 2 B -2 0 -2 -4 -4 C -14 2 0 -2 -6 D 10 4 2 0 -4 E -2 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 -10 2 B -2 0 -2 -4 -4 C -14 2 0 -2 -6 D 10 4 2 0 -4 E -2 4 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.125000 E: 0.625000 Sum of squares = 0.468749999997 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8875: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) D B A C E (7) A E D C B (6) E A D C B (5) D B C A E (5) D A E C B (5) B C D E A (5) A D E B C (5) E A C B D (4) D E A C B (4) D A E B C (4) D A B E C (4) C B E D A (4) B C E A D (4) D B C E A (3) B C A E D (3) E C B A D (2) B D C A E (2) B D A C E (2) B C E D A (2) B C D A E (2) B C A D E (2) A B D C E (2) E C A B D (1) E A C D B (1) D E C A B (1) D E A B C (1) C D B E A (1) C B D E A (1) B A C E D (1) B A C D E (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 8 -6 6 B 10 0 12 -8 20 C -8 -12 0 -16 10 D 6 8 16 0 18 E -6 -20 -10 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 8 -6 6 B 10 0 12 -8 20 C -8 -12 0 -16 10 D 6 8 16 0 18 E -6 -20 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=24 A=15 C=14 E=13 so E is eliminated. Round 2 votes counts: D=34 A=25 B=24 C=17 so C is eliminated. Round 3 votes counts: B=39 D=35 A=26 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:217 A:199 C:187 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 8 -6 6 B 10 0 12 -8 20 C -8 -12 0 -16 10 D 6 8 16 0 18 E -6 -20 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -6 6 B 10 0 12 -8 20 C -8 -12 0 -16 10 D 6 8 16 0 18 E -6 -20 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -6 6 B 10 0 12 -8 20 C -8 -12 0 -16 10 D 6 8 16 0 18 E -6 -20 -10 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8876: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (8) A B E C D (5) E D A B C (4) E A D B C (4) E A B D C (4) D E A C B (4) B A E C D (4) E A B C D (3) D E C B A (3) D C E B A (3) D C E A B (3) D C B E A (3) D C A E B (3) D C A B E (3) C D B A E (3) C D A B E (3) C A B D E (3) E B A C D (2) D E C A B (2) D B E C A (2) C B D A E (2) C A D B E (2) B D C E A (2) A E C B D (2) A C D E B (2) A C B E D (2) A B C E D (2) E D A C B (1) E B D A C (1) E B A D C (1) D E B C A (1) D C B A E (1) D B C E A (1) D A E C B (1) C D B E A (1) C B A E D (1) C B A D E (1) B E D A C (1) B E C A D (1) B E A C D (1) B C D E A (1) B C A E D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 26 8 2 2 B -26 0 -2 -2 -8 C -8 2 0 4 -10 D -2 2 -4 0 0 E -2 8 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 26 8 2 2 B -26 0 -2 -2 -8 C -8 2 0 4 -10 D -2 2 -4 0 0 E -2 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999275 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 E=20 C=16 B=11 so B is eliminated. Round 2 votes counts: D=32 A=27 E=23 C=18 so C is eliminated. Round 3 votes counts: D=42 A=35 E=23 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:208 D:198 C:194 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 26 8 2 2 B -26 0 -2 -2 -8 C -8 2 0 4 -10 D -2 2 -4 0 0 E -2 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999275 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 26 8 2 2 B -26 0 -2 -2 -8 C -8 2 0 4 -10 D -2 2 -4 0 0 E -2 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999275 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 26 8 2 2 B -26 0 -2 -2 -8 C -8 2 0 4 -10 D -2 2 -4 0 0 E -2 8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999275 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8877: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E B D C A (6) E B C A D (6) D B C E A (5) A E B C D (5) A D C B E (5) E A B C D (4) A E B D C (4) D C A B E (3) C B E D A (3) B D C E A (3) A D E B C (3) A D C E B (3) E B A C D (2) D B E C A (2) D B A C E (2) D A B C E (2) C E B A D (2) C D B E A (2) C B D E A (2) B C E D A (2) A E D B C (2) A E C B D (2) E C B A D (1) E A B D C (1) D C B E A (1) D C B A E (1) D B E A C (1) D B A E C (1) D A C B E (1) D A B E C (1) C E B D A (1) C D B A E (1) C A E B D (1) C A D B E (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E C A (1) B C D E A (1) A E D C B (1) A D E C B (1) A D B C E (1) A C E D B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -10 -2 -10 B 10 0 28 16 -8 C 10 -28 0 -4 -6 D 2 -16 4 0 -10 E 10 8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -10 -2 -10 B 10 0 28 16 -8 C 10 -28 0 -4 -6 D 2 -16 4 0 -10 E 10 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=27 D=20 C=14 B=9 so B is eliminated. Round 2 votes counts: A=30 E=29 D=24 C=17 so C is eliminated. Round 3 votes counts: E=37 A=33 D=30 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:223 E:217 D:190 C:186 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -10 -2 -10 B 10 0 28 16 -8 C 10 -28 0 -4 -6 D 2 -16 4 0 -10 E 10 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -2 -10 B 10 0 28 16 -8 C 10 -28 0 -4 -6 D 2 -16 4 0 -10 E 10 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -2 -10 B 10 0 28 16 -8 C 10 -28 0 -4 -6 D 2 -16 4 0 -10 E 10 8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999471 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8878: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) B A C E D (6) D C E A B (5) B C A D E (5) C E A B D (4) E D A C B (3) D C E B A (3) D C B E A (3) D B A C E (3) C B D E A (3) C B D A E (3) C B A D E (3) A E D B C (3) A E B D C (3) E C A B D (2) E A D C B (2) E A D B C (2) E A C B D (2) E A B D C (2) D E C A B (2) D E A C B (2) D C B A E (2) D B C A E (2) D A E B C (2) C E D A B (2) C B A E D (2) B C D A E (2) B C A E D (2) A E B C D (2) E D C A B (1) D E A B C (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E B A (1) C D E A B (1) C D B E A (1) C B E A D (1) B D C A E (1) B D A E C (1) B D A C E (1) B A E C D (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -6 6 10 B -4 0 0 12 6 C 6 0 0 -10 16 D -6 -12 10 0 -2 E -10 -6 -16 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.515243 C: 0.484757 D: 0.000000 E: 0.000000 Sum of squares = 0.500464675798 Cumulative probabilities = A: 0.000000 B: 0.515243 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 6 10 B -4 0 0 12 6 C 6 0 0 -10 16 D -6 -12 10 0 -2 E -10 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=23 B=19 A=18 E=14 so E is eliminated. Round 2 votes counts: D=30 A=26 C=25 B=19 so B is eliminated. Round 3 votes counts: C=34 D=33 A=33 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:207 B:207 C:206 D:195 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 4 -6 6 10 B -4 0 0 12 6 C 6 0 0 -10 16 D -6 -12 10 0 -2 E -10 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 6 10 B -4 0 0 12 6 C 6 0 0 -10 16 D -6 -12 10 0 -2 E -10 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 6 10 B -4 0 0 12 6 C 6 0 0 -10 16 D -6 -12 10 0 -2 E -10 -6 -16 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999993 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8879: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) E A D C B (6) D C B A E (6) C B D E A (6) A D E C B (6) E A D B C (4) E A B C D (4) D A C B E (4) A E D B C (4) E A B D C (3) C D B E A (3) B C E A D (3) B A D C E (3) A D B E C (3) E C B D A (2) E C B A D (2) D E C A B (2) C E D B A (2) C B D A E (2) B E C A D (2) B C E D A (2) B C A D E (2) B A C E D (2) A D E B C (2) A B D C E (2) E D C A B (1) E A C B D (1) D E A C B (1) D C E B A (1) D C E A B (1) D A E C B (1) D A C E B (1) C D B A E (1) C B E D A (1) B D A C E (1) B C D E A (1) B A E D C (1) B A E C D (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 4 6 4 B 6 0 0 2 10 C -4 0 0 -8 6 D -6 -2 8 0 16 E -4 -10 -6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.855893 C: 0.144107 D: 0.000000 E: 0.000000 Sum of squares = 0.753320091228 Cumulative probabilities = A: 0.000000 B: 0.855893 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 6 4 B 6 0 0 2 10 C -4 0 0 -8 6 D -6 -2 8 0 16 E -4 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000036634 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=23 A=18 D=17 C=15 so C is eliminated. Round 2 votes counts: B=36 E=25 D=21 A=18 so A is eliminated. Round 3 votes counts: B=38 D=32 E=30 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:209 D:208 A:204 C:197 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 6 4 B 6 0 0 2 10 C -4 0 0 -8 6 D -6 -2 8 0 16 E -4 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000036634 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 6 4 B 6 0 0 2 10 C -4 0 0 -8 6 D -6 -2 8 0 16 E -4 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000036634 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 6 4 B 6 0 0 2 10 C -4 0 0 -8 6 D -6 -2 8 0 16 E -4 -10 -6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.800000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000036634 Cumulative probabilities = A: 0.000000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8880: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) B D C E A (8) A C E B D (6) E C A D B (5) D B E C A (5) B D C A E (5) B D A E C (5) E A C D B (4) C E D B A (4) C B D E A (4) B D A C E (4) D B E A C (3) C E A D B (3) A E D B C (3) A E B D C (3) A B D E C (3) D B C E A (2) C E D A B (2) B D E C A (2) A E D C B (2) A E C B D (2) A C E D B (2) E D C B A (1) E D A B C (1) E C D A B (1) D E C B A (1) D C B E A (1) C D B E A (1) C A E B D (1) C A B E D (1) B A D C E (1) A C B E D (1) A B E C D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 6 0 4 B -6 0 -2 0 0 C -6 2 0 -2 -2 D 0 0 2 0 -6 E -4 0 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.740448 B: 0.000000 C: 0.000000 D: 0.259552 E: 0.000000 Sum of squares = 0.615630126731 Cumulative probabilities = A: 0.740448 B: 0.740448 C: 0.740448 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 0 4 B -6 0 -2 0 0 C -6 2 0 -2 -2 D 0 0 2 0 -6 E -4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028022 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=25 C=16 E=12 D=12 so E is eliminated. Round 2 votes counts: A=39 B=25 C=22 D=14 so D is eliminated. Round 3 votes counts: A=40 B=35 C=25 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:202 D:198 B:196 C:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 0 4 B -6 0 -2 0 0 C -6 2 0 -2 -2 D 0 0 2 0 -6 E -4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028022 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 0 4 B -6 0 -2 0 0 C -6 2 0 -2 -2 D 0 0 2 0 -6 E -4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028022 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 0 4 B -6 0 -2 0 0 C -6 2 0 -2 -2 D 0 0 2 0 -6 E -4 0 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000028022 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8881: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (12) D A E C B (9) C B E A D (8) C A E D B (7) B C E D A (7) E A C D B (5) C E A D B (5) B D E A C (5) D A E B C (4) C B A E D (4) B C E A D (4) A D E C B (4) E D A C B (2) D A B E C (2) B E C D A (2) B C A D E (2) A E D C B (2) E D B A C (1) E D A B C (1) E C A D B (1) E A D C B (1) D E A B C (1) D B E A C (1) D B A E C (1) C E B A D (1) C E A B D (1) C B A D E (1) C A D E B (1) C A B D E (1) B D E C A (1) B C D E A (1) B C A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 4 0 4 B 4 0 -8 2 6 C -4 8 0 6 -10 D 0 -2 -6 0 -8 E -4 -6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999992 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 0 4 B 4 0 -8 2 6 C -4 8 0 6 -10 D 0 -2 -6 0 -8 E -4 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999929 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=29 D=18 E=11 A=7 so A is eliminated. Round 2 votes counts: B=35 C=30 D=22 E=13 so E is eliminated. Round 3 votes counts: C=36 B=35 D=29 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:204 A:202 B:202 C:200 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 4 0 4 B 4 0 -8 2 6 C -4 8 0 6 -10 D 0 -2 -6 0 -8 E -4 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999929 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 0 4 B 4 0 -8 2 6 C -4 8 0 6 -10 D 0 -2 -6 0 -8 E -4 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999929 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 0 4 B 4 0 -8 2 6 C -4 8 0 6 -10 D 0 -2 -6 0 -8 E -4 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999929 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8882: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (14) D C A E B (13) A C D B E (10) E D B C A (9) B E A C D (8) E B D A C (7) E B D C A (4) D E C B A (4) D E C A B (4) D C A B E (4) A C B D E (4) E B A C D (3) C A D E B (3) D C E A B (2) B A E C D (2) B A C E D (2) A B C E D (2) E B A D C (1) C D A E B (1) C D A B E (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 20 -18 2 16 B -20 0 -24 -32 -4 C 18 24 0 4 16 D -2 32 -4 0 20 E -16 4 -16 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -18 2 16 B -20 0 -24 -32 -4 C 18 24 0 4 16 D -2 32 -4 0 20 E -16 4 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 C=19 A=18 B=12 so B is eliminated. Round 2 votes counts: E=32 D=27 A=22 C=19 so C is eliminated. Round 3 votes counts: A=39 E=32 D=29 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:231 D:223 A:210 E:176 B:160 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -18 2 16 B -20 0 -24 -32 -4 C 18 24 0 4 16 D -2 32 -4 0 20 E -16 4 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -18 2 16 B -20 0 -24 -32 -4 C 18 24 0 4 16 D -2 32 -4 0 20 E -16 4 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -18 2 16 B -20 0 -24 -32 -4 C 18 24 0 4 16 D -2 32 -4 0 20 E -16 4 -16 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999687 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8883: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (14) E D B C A (10) E D B A C (7) B C E D A (7) A C D E B (7) D E A C B (6) A C B D E (6) D A E C B (5) A D C E B (5) C A B D E (4) E B D C A (3) B E C D A (3) A D E C B (3) C B A E D (2) B C E A D (2) B C A E D (2) A C D B E (2) A B C E D (2) E D A B C (1) E B D A C (1) D E A B C (1) C D A E B (1) C B A D E (1) C A D B E (1) B E D A C (1) B E C A D (1) B E A D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -2 -20 -16 B 10 0 14 2 0 C 2 -14 0 -18 -16 D 20 -2 18 0 -16 E 16 0 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.453843 C: 0.000000 D: 0.000000 E: 0.546157 Sum of squares = 0.504261016164 Cumulative probabilities = A: 0.000000 B: 0.453843 C: 0.453843 D: 0.453843 E: 1.000000 A B C D E A 0 -10 -2 -20 -16 B 10 0 14 2 0 C 2 -14 0 -18 -16 D 20 -2 18 0 -16 E 16 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=26 E=22 D=12 C=9 so C is eliminated. Round 2 votes counts: B=34 A=31 E=22 D=13 so D is eliminated. Round 3 votes counts: A=37 B=34 E=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:224 B:213 D:210 C:177 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 -20 -16 B 10 0 14 2 0 C 2 -14 0 -18 -16 D 20 -2 18 0 -16 E 16 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 -20 -16 B 10 0 14 2 0 C 2 -14 0 -18 -16 D 20 -2 18 0 -16 E 16 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 -20 -16 B 10 0 14 2 0 C 2 -14 0 -18 -16 D 20 -2 18 0 -16 E 16 0 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8884: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (11) E B D C A (8) A C B D E (8) E D A C B (7) C A B D E (7) A C D E B (5) E D B C A (4) D C A B E (4) B E D C A (4) B E C D A (4) D A C E B (3) A D C E B (3) E B A D C (2) E B A C D (2) D E C A B (2) D E B C A (2) D C A E B (2) B E C A D (2) B E A C D (2) B C A E D (2) E D C A B (1) E D B A C (1) E B D A C (1) E B C D A (1) E B C A D (1) D E A C B (1) D A E C B (1) D A C B E (1) C D A B E (1) C A D B E (1) B D C E A (1) B C E A D (1) B C D A E (1) B C A D E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 20 0 0 6 B -20 0 -18 0 6 C 0 18 0 4 8 D 0 0 -4 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.189469 B: 0.000000 C: 0.810531 D: 0.000000 E: 0.000000 Sum of squares = 0.692858871656 Cumulative probabilities = A: 0.189469 B: 0.189469 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 0 0 6 B -20 0 -18 0 6 C 0 18 0 4 8 D 0 0 -4 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=28 B=18 D=16 C=9 so C is eliminated. Round 2 votes counts: A=37 E=28 B=18 D=17 so D is eliminated. Round 3 votes counts: A=49 E=33 B=18 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:215 A:213 D:203 E:185 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 20 0 0 6 B -20 0 -18 0 6 C 0 18 0 4 8 D 0 0 -4 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 0 6 B -20 0 -18 0 6 C 0 18 0 4 8 D 0 0 -4 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 0 6 B -20 0 -18 0 6 C 0 18 0 4 8 D 0 0 -4 0 10 E -6 -6 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999984 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8885: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) E D A B C (8) E A B C D (8) E C A B D (5) D B A C E (5) E D B A C (4) D C B A E (4) D B A E C (4) C A B D E (4) B A D C E (4) E D C B A (3) E A B D C (3) D E C B A (3) D E B C A (3) E D C A B (2) E C D A B (2) E A C B D (2) D E B A C (2) D C B E A (2) D B C A E (2) C D B A E (2) C A B E D (2) A B E D C (2) A B D E C (2) A B C D E (2) E C D B A (1) E C A D B (1) E A D B C (1) D A B E C (1) C E D B A (1) C E A B D (1) C D E B A (1) C B D A E (1) A E B C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 -2 -6 B 2 0 6 -4 -4 C -2 -6 0 -10 -16 D 2 4 10 0 4 E 6 4 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 -6 B 2 0 6 -4 -4 C -2 -6 0 -10 -16 D 2 4 10 0 4 E 6 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=26 C=21 A=9 B=4 so B is eliminated. Round 2 votes counts: E=40 D=26 C=21 A=13 so A is eliminated. Round 3 votes counts: E=43 D=32 C=25 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:210 B:200 A:196 C:183 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -2 -6 B 2 0 6 -4 -4 C -2 -6 0 -10 -16 D 2 4 10 0 4 E 6 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 -6 B 2 0 6 -4 -4 C -2 -6 0 -10 -16 D 2 4 10 0 4 E 6 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 -6 B 2 0 6 -4 -4 C -2 -6 0 -10 -16 D 2 4 10 0 4 E 6 4 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998588 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8886: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) C A D E B (9) A D C B E (8) B D E A C (7) B E D C A (6) B E D A C (6) E B D C A (5) D B E A C (5) E B C D A (4) D B A E C (4) C E B D A (4) C A E D B (4) C A E B D (4) A C D B E (4) E C B D A (3) D A B C E (3) C E A B D (3) D A B E C (2) A D B E C (2) E A B C D (1) D C A B E (1) C A D B E (1) A D B C E (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -10 0 -10 B 10 0 -6 12 2 C 10 6 0 -2 8 D 0 -12 2 0 -4 E 10 -2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.300000 E: 0.000000 Sum of squares = 0.459999999999 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 0 -10 B 10 0 -6 12 2 C 10 6 0 -2 8 D 0 -12 2 0 -4 E 10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=19 A=17 D=15 E=13 so E is eliminated. Round 2 votes counts: C=39 B=28 A=18 D=15 so D is eliminated. Round 3 votes counts: C=40 B=37 A=23 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:209 E:202 D:193 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 0 -10 B 10 0 -6 12 2 C 10 6 0 -2 8 D 0 -12 2 0 -4 E 10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 0 -10 B 10 0 -6 12 2 C 10 6 0 -2 8 D 0 -12 2 0 -4 E 10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 0 -10 B 10 0 -6 12 2 C 10 6 0 -2 8 D 0 -12 2 0 -4 E 10 -2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.600000 D: 0.300000 E: 0.000000 Sum of squares = 0.460000000006 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.700000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8887: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) A B E D C (11) C D E B A (10) E B A C D (5) D A C B E (5) C E B D A (5) B E A C D (5) E B C A D (4) D C A E B (4) D A C E B (4) A B E C D (4) D C A B E (3) C B E A D (3) A D E B C (3) D C E A B (2) D C B E A (2) C E D B A (2) C B E D A (2) B E C A D (2) B A E C D (2) E D A B C (1) E C B D A (1) E C B A D (1) D E C A B (1) D A B C E (1) C D B E A (1) A E B D C (1) A E B C D (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -10 -12 -18 B 14 0 -16 -4 -14 C 10 16 0 -4 14 D 12 4 4 0 0 E 18 14 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.868162 E: 0.131838 Sum of squares = 0.771086772174 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.868162 E: 1.000000 A B C D E A 0 -14 -10 -12 -18 B 14 0 -16 -4 -14 C 10 16 0 -4 14 D 12 4 4 0 0 E 18 14 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.222222 Sum of squares = 0.654321051808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=23 A=22 E=12 B=9 so B is eliminated. Round 2 votes counts: D=34 A=24 C=23 E=19 so E is eliminated. Round 3 votes counts: D=35 A=34 C=31 so C is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:210 E:209 B:190 A:173 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -10 -12 -18 B 14 0 -16 -4 -14 C 10 16 0 -4 14 D 12 4 4 0 0 E 18 14 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.222222 Sum of squares = 0.654321051808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -12 -18 B 14 0 -16 -4 -14 C 10 16 0 -4 14 D 12 4 4 0 0 E 18 14 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.222222 Sum of squares = 0.654321051808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -12 -18 B 14 0 -16 -4 -14 C 10 16 0 -4 14 D 12 4 4 0 0 E 18 14 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 0.222222 Sum of squares = 0.654321051808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.777778 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8888: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (14) E A C B D (9) D B A E C (9) C E A B D (8) B D A E C (8) D B A C E (7) C E A D B (7) E A B D C (6) D B C A E (5) C D B E A (4) A E B D C (4) E C D B A (2) E A B C D (2) D C B A E (2) C E D A B (2) C D E B A (2) C D B A E (2) D B E A C (1) D B C E A (1) C A E B D (1) B D E A C (1) A E C B D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 0 8 -18 B -12 0 -8 12 -16 C 0 8 0 8 -18 D -8 -12 -8 0 -14 E 18 16 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 0 8 -18 B -12 0 -8 12 -16 C 0 8 0 8 -18 D -8 -12 -8 0 -14 E 18 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=26 D=25 B=9 A=7 so A is eliminated. Round 2 votes counts: E=38 C=26 D=25 B=11 so B is eliminated. Round 3 votes counts: E=39 D=35 C=26 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:233 A:201 C:199 B:188 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 0 8 -18 B -12 0 -8 12 -16 C 0 8 0 8 -18 D -8 -12 -8 0 -14 E 18 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 8 -18 B -12 0 -8 12 -16 C 0 8 0 8 -18 D -8 -12 -8 0 -14 E 18 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 8 -18 B -12 0 -8 12 -16 C 0 8 0 8 -18 D -8 -12 -8 0 -14 E 18 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8889: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) E C A D B (5) E A C B D (5) E A B C D (5) C E A D B (4) C A E D B (4) A E B C D (4) A B E D C (4) E C D A B (3) D B C A E (3) C E D A B (3) B D A E C (3) B D A C E (3) E D C B A (2) D E C B A (2) D C E B A (2) D C B A E (2) D B E C A (2) D B A C E (2) C D E A B (2) C D B A E (2) C D A E B (2) C D A B E (2) B E A D C (2) B D E C A (2) A C B E D (2) A B E C D (2) A B D C E (2) A B C E D (2) E D B C A (1) E B D C A (1) E B D A C (1) E B A D C (1) E B A C D (1) D E B C A (1) D C B E A (1) C D E B A (1) C A D E B (1) B D E A C (1) B D C A E (1) B A D C E (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -14 -6 -12 B -8 0 6 -12 -6 C 14 -6 0 4 4 D 6 12 -4 0 -8 E 12 6 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999971 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 8 -14 -6 -12 B -8 0 6 -12 -6 C 14 -6 0 4 4 D 6 12 -4 0 -8 E 12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999884 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 C=21 A=18 B=13 so B is eliminated. Round 2 votes counts: D=33 E=27 C=21 A=19 so A is eliminated. Round 3 votes counts: E=37 D=36 C=27 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:208 D:203 B:190 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -14 -6 -12 B -8 0 6 -12 -6 C 14 -6 0 4 4 D 6 12 -4 0 -8 E 12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999884 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -14 -6 -12 B -8 0 6 -12 -6 C 14 -6 0 4 4 D 6 12 -4 0 -8 E 12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999884 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -14 -6 -12 B -8 0 6 -12 -6 C 14 -6 0 4 4 D 6 12 -4 0 -8 E 12 6 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.375000 D: 0.000000 E: 0.375000 Sum of squares = 0.343749999884 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8890: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) A D C E B (10) E B A D C (9) B E C D A (8) A C D B E (8) E B D A C (5) A D E C B (5) E B D C A (4) A E D C B (4) D C A E B (3) C B D A E (3) B C E D A (3) B C D A E (3) B C A D E (3) D A C E B (2) C D B E A (2) C D B A E (2) B C D E A (2) A E D B C (2) A E B D C (2) E A D C B (1) D C E B A (1) D A E C B (1) C A B D E (1) B E D C A (1) B E A C D (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 4 -4 26 B -6 0 -12 -8 -2 C -4 12 0 0 12 D 4 8 0 0 18 E -26 2 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.402576 D: 0.597424 E: 0.000000 Sum of squares = 0.51898282457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.402576 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 -4 26 B -6 0 -12 -8 -2 C -4 12 0 0 12 D 4 8 0 0 18 E -26 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499586 D: 0.500414 E: 0.000000 Sum of squares = 0.500000342329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499586 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=21 C=20 E=19 D=7 so D is eliminated. Round 2 votes counts: A=36 C=24 B=21 E=19 so E is eliminated. Round 3 votes counts: B=39 A=37 C=24 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:215 C:210 B:186 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 4 -4 26 B -6 0 -12 -8 -2 C -4 12 0 0 12 D 4 8 0 0 18 E -26 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499586 D: 0.500414 E: 0.000000 Sum of squares = 0.500000342329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499586 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 -4 26 B -6 0 -12 -8 -2 C -4 12 0 0 12 D 4 8 0 0 18 E -26 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499586 D: 0.500414 E: 0.000000 Sum of squares = 0.500000342329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499586 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 -4 26 B -6 0 -12 -8 -2 C -4 12 0 0 12 D 4 8 0 0 18 E -26 2 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499586 D: 0.500414 E: 0.000000 Sum of squares = 0.500000342329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499586 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8891: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (8) C D B E A (7) A E D C B (7) A E B D C (6) B C A D E (5) A E D B C (4) A E B C D (4) E D A C B (3) E D A B C (3) D E C A B (3) D C E B A (3) D C E A B (3) D C B E A (3) B A C D E (3) E D B C A (2) D E C B A (2) C B D E A (2) B C D E A (2) A E C D B (2) A B C E D (2) E D C B A (1) E D B A C (1) E B A D C (1) E A D C B (1) D C A E B (1) D A E C B (1) D A C E B (1) C D E A B (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) C A D B E (1) B E D C A (1) B E A D C (1) B D C E A (1) B C E A D (1) B C D A E (1) B A E C D (1) B A C E D (1) A E C B D (1) A C D E B (1) A C B D E (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 12 10 0 B -16 0 2 -22 -20 C -12 -2 0 -16 -10 D -10 22 16 0 -6 E 0 20 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.576646 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.423354 Sum of squares = 0.511749286535 Cumulative probabilities = A: 0.576646 B: 0.576646 C: 0.576646 D: 0.576646 E: 1.000000 A B C D E A 0 16 12 10 0 B -16 0 2 -22 -20 C -12 -2 0 -16 -10 D -10 22 16 0 -6 E 0 20 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=20 D=17 B=17 C=15 so C is eliminated. Round 2 votes counts: A=32 D=27 B=21 E=20 so E is eliminated. Round 3 votes counts: A=41 D=37 B=22 so B is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:218 D:211 C:180 B:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 10 0 B -16 0 2 -22 -20 C -12 -2 0 -16 -10 D -10 22 16 0 -6 E 0 20 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 10 0 B -16 0 2 -22 -20 C -12 -2 0 -16 -10 D -10 22 16 0 -6 E 0 20 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 10 0 B -16 0 2 -22 -20 C -12 -2 0 -16 -10 D -10 22 16 0 -6 E 0 20 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8892: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) A E C D B (8) C D B E A (5) E A B C D (4) A D C E B (4) A D C B E (4) A B D C E (4) E C A D B (3) E B C D A (3) E A C D B (3) D C B E A (3) C D E B A (3) B E A C D (3) A E B D C (3) A E B C D (3) A C D E B (3) E C D B A (2) D C A E B (2) B E D C A (2) B E C D A (2) B E A D C (2) B D C A E (2) B C D E A (2) B A E D C (2) A B D E C (2) E B C A D (1) E B A C D (1) E A C B D (1) D C B A E (1) D B C E A (1) D B C A E (1) D B A C E (1) C E D B A (1) C D B A E (1) C D A E B (1) B D E C A (1) B A D E C (1) B A D C E (1) A E D C B (1) A E D B C (1) A E C B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 8 14 -4 B 2 0 6 2 2 C -8 -6 0 2 -2 D -14 -2 -2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 14 -4 B 2 0 6 2 2 C -8 -6 0 2 -2 D -14 -2 -2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=27 E=18 C=11 D=9 so D is eliminated. Round 2 votes counts: A=35 B=30 E=18 C=17 so C is eliminated. Round 3 votes counts: B=40 A=38 E=22 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:208 B:206 E:200 C:193 D:193 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 14 -4 B 2 0 6 2 2 C -8 -6 0 2 -2 D -14 -2 -2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 14 -4 B 2 0 6 2 2 C -8 -6 0 2 -2 D -14 -2 -2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 14 -4 B 2 0 6 2 2 C -8 -6 0 2 -2 D -14 -2 -2 0 4 E 4 -2 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8893: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E B D C (10) E A B D C (8) A E C B D (7) D B C E A (5) C D B A E (5) C A E D B (5) B D E A C (5) E B D A C (4) D B E C A (3) D B E A C (3) D B A E C (3) C D A B E (3) C A E B D (3) C A D E B (3) C A D B E (3) A C E B D (3) E B A D C (2) D C B E A (2) C E A D B (2) C E A B D (2) A E B C D (2) E B D C A (1) D B C A E (1) C D E B A (1) B E D A C (1) B D A E C (1) A B E D C (1) Total count = 100 A B C D E A 0 4 0 2 0 B -4 0 0 0 -6 C 0 0 0 0 -2 D -2 0 0 0 -2 E 0 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.526280 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.473720 Sum of squares = 0.501381241483 Cumulative probabilities = A: 0.526280 B: 0.526280 C: 0.526280 D: 0.526280 E: 1.000000 A B C D E A 0 4 0 2 0 B -4 0 0 0 -6 C 0 0 0 0 -2 D -2 0 0 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=23 D=17 E=15 B=7 so B is eliminated. Round 2 votes counts: C=38 D=23 A=23 E=16 so E is eliminated. Round 3 votes counts: C=38 A=33 D=29 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:205 A:203 C:199 D:198 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 0 B -4 0 0 0 -6 C 0 0 0 0 -2 D -2 0 0 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 0 B -4 0 0 0 -6 C 0 0 0 0 -2 D -2 0 0 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 0 B -4 0 0 0 -6 C 0 0 0 0 -2 D -2 0 0 0 -2 E 0 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8894: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (17) A E C B D (12) E A B D C (10) E B D A C (4) D B C E A (4) A C E B D (4) D C B E A (3) C A E D B (3) C A E B D (3) B D E A C (3) B D C E A (3) E D B A C (2) D B E C A (2) C D B E A (2) C D A B E (2) C A D B E (2) C A B E D (2) C A B D E (2) B D E C A (2) B A E D C (2) A E C D B (2) A E B C D (2) E D C A B (1) E D A C B (1) E D A B C (1) E A D B C (1) D E B C A (1) D B E A C (1) D B C A E (1) C D E A B (1) C B D A E (1) C B A D E (1) B E D A C (1) A E B D C (1) Total count = 100 A B C D E A 0 0 -6 -6 14 B 0 0 -18 6 2 C 6 18 0 12 2 D 6 -6 -12 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 -6 14 B 0 0 -18 6 2 C 6 18 0 12 2 D 6 -6 -12 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=21 E=20 D=12 B=11 so B is eliminated. Round 2 votes counts: C=36 A=23 E=21 D=20 so D is eliminated. Round 3 votes counts: C=47 E=30 A=23 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:201 B:195 E:193 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 -6 14 B 0 0 -18 6 2 C 6 18 0 12 2 D 6 -6 -12 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -6 14 B 0 0 -18 6 2 C 6 18 0 12 2 D 6 -6 -12 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -6 14 B 0 0 -18 6 2 C 6 18 0 12 2 D 6 -6 -12 0 -4 E -14 -2 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983078 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8895: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (18) E C B D A (12) E C A D B (5) A D C B E (5) C E D B A (4) C E D A B (4) C E A D B (4) B E C D A (4) B D A C E (4) B A D E C (4) A C D E B (4) E C A B D (3) E B C D A (3) D B A C E (3) E C D B A (2) E C D A B (2) B D E C A (2) B D A E C (2) A D B E C (2) A C E D B (2) E A B C D (1) D C B A E (1) D A B C E (1) C D E A B (1) C A D E B (1) B E A D C (1) B D C E A (1) B D C A E (1) A E C D B (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 12 0 6 2 B -12 0 -4 -22 0 C 0 4 0 6 12 D -6 22 -6 0 4 E -2 0 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.656268 B: 0.000000 C: 0.343732 D: 0.000000 E: 0.000000 Sum of squares = 0.548839332964 Cumulative probabilities = A: 0.656268 B: 0.656268 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 6 2 B -12 0 -4 -22 0 C 0 4 0 6 12 D -6 22 -6 0 4 E -2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999833 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=28 B=19 C=14 D=5 so D is eliminated. Round 2 votes counts: A=35 E=28 B=22 C=15 so C is eliminated. Round 3 votes counts: E=41 A=36 B=23 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:210 D:207 E:191 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 6 2 B -12 0 -4 -22 0 C 0 4 0 6 12 D -6 22 -6 0 4 E -2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999833 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 6 2 B -12 0 -4 -22 0 C 0 4 0 6 12 D -6 22 -6 0 4 E -2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999833 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 6 2 B -12 0 -4 -22 0 C 0 4 0 6 12 D -6 22 -6 0 4 E -2 0 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999833 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8896: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (13) B D C E A (13) A E C D B (10) C D A E B (9) D C B E A (6) B E D A C (6) D C B A E (5) C D B A E (5) A E C B D (5) E A B C D (4) A E B D C (4) C A D E B (3) C D B E A (2) B D C A E (2) A E B C D (2) A C E D B (2) A C D E B (2) E B A C D (1) E A C D B (1) E A B D C (1) D B C E A (1) D B C A E (1) C D A B E (1) B D E A C (1) Total count = 100 A B C D E A 0 -12 4 -4 2 B 12 0 -2 4 12 C -4 2 0 -6 4 D 4 -4 6 0 2 E -2 -12 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888892 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -4 2 B 12 0 -2 4 12 C -4 2 0 -6 4 D 4 -4 6 0 2 E -2 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=25 C=20 D=13 E=7 so E is eliminated. Round 2 votes counts: B=36 A=31 C=20 D=13 so D is eliminated. Round 3 votes counts: B=38 C=31 A=31 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:204 C:198 A:195 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 -4 2 B 12 0 -2 4 12 C -4 2 0 -6 4 D 4 -4 6 0 2 E -2 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -4 2 B 12 0 -2 4 12 C -4 2 0 -6 4 D 4 -4 6 0 2 E -2 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -4 2 B 12 0 -2 4 12 C -4 2 0 -6 4 D 4 -4 6 0 2 E -2 -12 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888878 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8897: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (14) E A D C B (11) D C A E B (9) B C D A E (7) E B A D C (5) C D A E B (5) C D A B E (5) B E C A D (5) B C E D A (5) E A D B C (4) B E A D C (4) A E D C B (4) D A C E B (3) C D B A E (3) C B D A E (3) A D E C B (3) B E A C D (2) E B A C D (1) E A C D B (1) C D E A B (1) B D A C E (1) B C D E A (1) B A E D C (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 24 12 14 -8 B -24 0 4 0 -24 C -12 -4 0 -22 -14 D -14 0 22 0 -16 E 8 24 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 12 14 -8 B -24 0 4 0 -24 C -12 -4 0 -22 -14 D -14 0 22 0 -16 E 8 24 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 B=26 C=17 D=12 A=9 so A is eliminated. Round 2 votes counts: E=41 B=27 C=17 D=15 so D is eliminated. Round 3 votes counts: E=44 C=29 B=27 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:231 A:221 D:196 B:178 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 12 14 -8 B -24 0 4 0 -24 C -12 -4 0 -22 -14 D -14 0 22 0 -16 E 8 24 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 12 14 -8 B -24 0 4 0 -24 C -12 -4 0 -22 -14 D -14 0 22 0 -16 E 8 24 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 12 14 -8 B -24 0 4 0 -24 C -12 -4 0 -22 -14 D -14 0 22 0 -16 E 8 24 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8898: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) D E A C B (6) E D C A B (5) C A E B D (5) A D E B C (4) E C D A B (3) D E C A B (3) D B E A C (3) C E D B A (3) B D A C E (3) B C D E A (3) A E D C B (3) A E C D B (3) A C E D B (3) A C E B D (3) E D A C B (2) E C A D B (2) D E B C A (2) D B E C A (2) D A E C B (2) D A B E C (2) C E B A D (2) C E A D B (2) B D A E C (2) B A C D E (2) A B C E D (2) E A D C B (1) E A C D B (1) D E B A C (1) D E A B C (1) D C E B A (1) D B A E C (1) D A E B C (1) C E D A B (1) C E B D A (1) C E A B D (1) C B E D A (1) C B E A D (1) C B A E D (1) C A B E D (1) B D E C A (1) B D C E A (1) B C E A D (1) B C A E D (1) B A D C E (1) A D E C B (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 20 0 -14 -14 B -20 0 -28 -32 -38 C 0 28 0 -14 -18 D 14 32 14 0 2 E 14 38 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999474 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 0 -14 -14 B -20 0 -28 -32 -38 C 0 28 0 -14 -18 D 14 32 14 0 2 E 14 38 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=21 C=19 B=15 E=14 so E is eliminated. Round 2 votes counts: D=38 C=24 A=23 B=15 so B is eliminated. Round 3 votes counts: D=45 C=29 A=26 so A is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:234 D:231 C:198 A:196 B:141 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 20 0 -14 -14 B -20 0 -28 -32 -38 C 0 28 0 -14 -18 D 14 32 14 0 2 E 14 38 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 0 -14 -14 B -20 0 -28 -32 -38 C 0 28 0 -14 -18 D 14 32 14 0 2 E 14 38 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 0 -14 -14 B -20 0 -28 -32 -38 C 0 28 0 -14 -18 D 14 32 14 0 2 E 14 38 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999965113 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8899: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) E A C D B (8) A D B C E (7) A E D B C (6) A B D C E (6) C E B D A (5) E C D B A (4) E C A D B (4) D B C A E (4) E C B D A (3) C B D E A (3) C B D A E (3) C B A D E (3) A D B E C (3) E C D A B (2) E C B A D (2) D B A C E (2) C A E B D (2) C A B E D (2) B C D A E (2) A D E B C (2) A C B D E (2) E D B C A (1) E C A B D (1) E A D C B (1) E A C B D (1) D E C B A (1) C E D B A (1) C E B A D (1) C E A B D (1) C D B E A (1) B D C A E (1) B D A C E (1) B A D C E (1) A E B C D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 22 6 32 2 B -22 0 -4 -12 -12 C -6 4 0 10 2 D -32 12 -10 0 -14 E -2 12 -2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 6 32 2 B -22 0 -4 -12 -12 C -6 4 0 10 2 D -32 12 -10 0 -14 E -2 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999933988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=30 C=22 D=7 B=5 so B is eliminated. Round 2 votes counts: E=36 A=31 C=24 D=9 so D is eliminated. Round 3 votes counts: E=37 A=34 C=29 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:231 E:211 C:205 D:178 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 6 32 2 B -22 0 -4 -12 -12 C -6 4 0 10 2 D -32 12 -10 0 -14 E -2 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999933988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 6 32 2 B -22 0 -4 -12 -12 C -6 4 0 10 2 D -32 12 -10 0 -14 E -2 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999933988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 6 32 2 B -22 0 -4 -12 -12 C -6 4 0 10 2 D -32 12 -10 0 -14 E -2 12 -2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999933988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8900: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) A E D B C (7) A E C D B (6) A D B E C (6) A C B D E (6) A C E B D (5) E D B C A (4) D B E A C (4) D B A E C (4) C E A B D (4) B D E C A (4) B D C E A (4) A D B C E (4) E A C D B (3) C B D E A (3) C A B D E (3) A C E D B (3) E B D C A (2) E A D B C (2) D B A C E (2) D A B E C (2) C A E B D (2) B D C A E (2) E D B A C (1) E D A B C (1) E C D B A (1) E C B D A (1) C E B D A (1) C B D A E (1) C A B E D (1) B C D E A (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 16 8 12 B -14 0 20 -18 12 C -16 -20 0 -18 -14 D -8 18 18 0 12 E -12 -12 14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 16 8 12 B -14 0 20 -18 12 C -16 -20 0 -18 -14 D -8 18 18 0 12 E -12 -12 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 D=20 E=15 C=15 B=11 so B is eliminated. Round 2 votes counts: A=39 D=30 C=16 E=15 so E is eliminated. Round 3 votes counts: A=44 D=38 C=18 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:220 B:200 E:189 C:166 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 16 8 12 B -14 0 20 -18 12 C -16 -20 0 -18 -14 D -8 18 18 0 12 E -12 -12 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 8 12 B -14 0 20 -18 12 C -16 -20 0 -18 -14 D -8 18 18 0 12 E -12 -12 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 8 12 B -14 0 20 -18 12 C -16 -20 0 -18 -14 D -8 18 18 0 12 E -12 -12 14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8901: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (6) A C D E B (6) E B A C D (5) C A D E B (4) B E D A C (4) B D E C A (4) A D C B E (4) E C A D B (3) E C A B D (3) D B A C E (3) C E A D B (3) B D A C E (3) A C E D B (3) E C D B A (2) E B D C A (2) E B C A D (2) E A C B D (2) D C B E A (2) D B C A E (2) C D E A B (2) B E A D C (2) B D E A C (2) B A D E C (2) B A D C E (2) A D B C E (2) A C E B D (2) A B E C D (2) E C D A B (1) E C B D A (1) E B C D A (1) E A B C D (1) D C E A B (1) D C B A E (1) D C A E B (1) D C A B E (1) D B E C A (1) D B C E A (1) D A C B E (1) D A B C E (1) C D A E B (1) C A E D B (1) B E A C D (1) B D C A E (1) B A E D C (1) B A E C D (1) A E C D B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 6 10 -4 B 4 0 8 4 4 C -6 -8 0 -4 -2 D -10 -4 4 0 0 E 4 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999553 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 10 -4 B 4 0 8 4 4 C -6 -8 0 -4 -2 D -10 -4 4 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=23 A=22 D=15 C=11 so C is eliminated. Round 2 votes counts: B=29 A=27 E=26 D=18 so D is eliminated. Round 3 votes counts: B=39 A=32 E=29 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:204 E:201 D:195 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 10 -4 B 4 0 8 4 4 C -6 -8 0 -4 -2 D -10 -4 4 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 10 -4 B 4 0 8 4 4 C -6 -8 0 -4 -2 D -10 -4 4 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 10 -4 B 4 0 8 4 4 C -6 -8 0 -4 -2 D -10 -4 4 0 0 E 4 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8902: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (6) A C B E D (6) E A C B D (5) D E B C A (5) D B C A E (5) D A B C E (5) A E C B D (5) A C E B D (5) E C B A D (4) E B C D A (4) B C A D E (4) E D B C A (3) E A D C B (3) D E B A C (3) A C B D E (3) E D A C B (2) E C A B D (2) D E A B C (2) D B C E A (2) D B A C E (2) C B A D E (2) C A B E D (2) B C D A E (2) A C D B E (2) E B C A D (1) D B E C A (1) D A E C B (1) D A E B C (1) C E B A D (1) C B E A D (1) B D C E A (1) B D C A E (1) B C E D A (1) B C D E A (1) B A D C E (1) A E D C B (1) A D E C B (1) A D C B E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 16 16 B 2 0 -8 18 0 C -2 8 0 18 12 D -16 -18 -18 0 -6 E -16 0 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000011 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 16 16 B 2 0 -8 18 0 C -2 8 0 18 12 D -16 -18 -18 0 -6 E -16 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 E=24 C=12 B=11 so B is eliminated. Round 2 votes counts: D=29 A=27 E=24 C=20 so C is eliminated. Round 3 votes counts: A=41 D=32 E=27 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:216 B:206 E:189 D:171 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 16 16 B 2 0 -8 18 0 C -2 8 0 18 12 D -16 -18 -18 0 -6 E -16 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 16 16 B 2 0 -8 18 0 C -2 8 0 18 12 D -16 -18 -18 0 -6 E -16 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 16 16 B 2 0 -8 18 0 C -2 8 0 18 12 D -16 -18 -18 0 -6 E -16 0 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.49999999993 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) E A B D C (7) E D C A B (6) E B A D C (6) D C A E B (6) C D A B E (6) B A C D E (6) B A D C E (5) A B D C E (4) E D A C B (3) E C D A B (3) E B C D A (3) D C A B E (3) C D E A B (3) B A E D C (3) A D C B E (3) E C B D A (2) E A D B C (2) C D E B A (2) C D B E A (2) B A D E C (2) A B E D C (2) E A D C B (1) D C E A B (1) D A C B E (1) C E D B A (1) C B D A E (1) B E C A D (1) B E A D C (1) B E A C D (1) B C D A E (1) B C A D E (1) B A E C D (1) A E D B C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 8 2 -2 -4 B -8 0 -4 -6 -10 C -2 4 0 -18 -4 D 2 6 18 0 -2 E 4 10 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 -2 -4 B -8 0 -4 -6 -10 C -2 4 0 -18 -4 D 2 6 18 0 -2 E 4 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 B=22 C=15 A=12 D=11 so D is eliminated. Round 2 votes counts: E=40 C=25 B=22 A=13 so A is eliminated. Round 3 votes counts: E=41 C=30 B=29 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:210 A:202 C:190 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 -2 -4 B -8 0 -4 -6 -10 C -2 4 0 -18 -4 D 2 6 18 0 -2 E 4 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 -4 B -8 0 -4 -6 -10 C -2 4 0 -18 -4 D 2 6 18 0 -2 E 4 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 -4 B -8 0 -4 -6 -10 C -2 4 0 -18 -4 D 2 6 18 0 -2 E 4 10 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997357 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8904: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (6) C D A E B (6) E B A C D (5) D B C A E (5) B D E C A (5) A E C B D (5) A C D E B (5) E C A B D (4) E B C A D (4) C A D E B (4) A D C B E (4) E A B C D (3) C E A D B (3) B D A E C (3) D C B E A (2) D C A B E (2) D B C E A (2) C E B A D (2) C D B E A (2) C A E D B (2) B E D C A (2) B E D A C (2) B C E D A (2) A E B D C (2) A D C E B (2) E A C B D (1) D B A C E (1) C E D B A (1) C E D A B (1) C E B D A (1) C B E D A (1) B E C D A (1) B E A D C (1) B D E A C (1) B D C E A (1) A E D B C (1) A E B C D (1) A D E B C (1) A D B C E (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -6 18 -6 B 0 0 2 8 -22 C 6 -2 0 10 2 D -18 -8 -10 0 -6 E 6 22 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.846154 D: 0.000000 E: 0.076923 Sum of squares = 0.727810650946 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.923077 D: 0.923077 E: 1.000000 A B C D E A 0 0 -6 18 -6 B 0 0 2 8 -22 C 6 -2 0 10 2 D -18 -8 -10 0 -6 E 6 22 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.846154 D: 0.000000 E: 0.076923 Sum of squares = 0.727810650688 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.923077 D: 0.923077 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 C=23 B=18 D=12 so D is eliminated. Round 2 votes counts: C=27 B=26 A=24 E=23 so E is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:216 C:208 A:203 B:194 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 18 -6 B 0 0 2 8 -22 C 6 -2 0 10 2 D -18 -8 -10 0 -6 E 6 22 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.846154 D: 0.000000 E: 0.076923 Sum of squares = 0.727810650688 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.923077 D: 0.923077 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 18 -6 B 0 0 2 8 -22 C 6 -2 0 10 2 D -18 -8 -10 0 -6 E 6 22 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.846154 D: 0.000000 E: 0.076923 Sum of squares = 0.727810650688 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.923077 D: 0.923077 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 18 -6 B 0 0 2 8 -22 C 6 -2 0 10 2 D -18 -8 -10 0 -6 E 6 22 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.076923 C: 0.846154 D: 0.000000 E: 0.076923 Sum of squares = 0.727810650688 Cumulative probabilities = A: 0.000000 B: 0.076923 C: 0.923077 D: 0.923077 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8905: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) B A D E C (8) A B C D E (8) C E D A B (6) C E A B D (6) D E B C A (5) E D C B A (4) E C D B A (4) D E B A C (4) D A B E C (4) C E D B A (4) A B D E C (4) D B A E C (3) A C B E D (3) D E C B A (2) C E A D B (2) C A E B D (2) C A B E D (2) B E D A C (2) B D A E C (2) B A D C E (2) B A C D E (2) A B C E D (2) E D B C A (1) E C D A B (1) E C B D A (1) D E C A B (1) C E B A D (1) C B E A D (1) B E A D C (1) B D E A C (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 2 14 10 6 B -2 0 20 16 12 C -14 -20 0 -8 2 D -10 -16 8 0 12 E -6 -12 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 14 10 6 B -2 0 20 16 12 C -14 -20 0 -8 2 D -10 -16 8 0 12 E -6 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=24 D=19 B=19 E=11 so E is eliminated. Round 2 votes counts: C=30 A=27 D=24 B=19 so B is eliminated. Round 3 votes counts: A=41 C=30 D=29 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:223 A:216 D:197 E:184 C:180 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 14 10 6 B -2 0 20 16 12 C -14 -20 0 -8 2 D -10 -16 8 0 12 E -6 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 14 10 6 B -2 0 20 16 12 C -14 -20 0 -8 2 D -10 -16 8 0 12 E -6 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 14 10 6 B -2 0 20 16 12 C -14 -20 0 -8 2 D -10 -16 8 0 12 E -6 -12 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8906: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) D C E B A (6) B E A C D (6) A B E C D (6) B E A D C (5) D C E A B (4) D A C B E (4) A D B E C (4) A B E D C (4) E B C D A (3) D C B E A (3) D C A E B (3) A D C B E (3) A C E B D (3) E C B D A (2) E C B A D (2) D B E C A (2) D B A E C (2) C E D B A (2) C A E D B (2) B E D C A (2) B E D A C (2) B D E A C (2) A D B C E (2) E C D B A (1) E B C A D (1) E B A C D (1) D C A B E (1) D B E A C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A D B (1) C D E A B (1) C A D E B (1) B E C A D (1) B D A E C (1) B A E D C (1) A E B C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 4 -6 -16 B 12 0 2 -8 12 C -4 -2 0 -10 -4 D 6 8 10 0 4 E 16 -12 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -6 -16 B 12 0 2 -8 12 C -4 -2 0 -10 -4 D 6 8 10 0 4 E 16 -12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=25 B=20 C=17 E=10 so E is eliminated. Round 2 votes counts: D=28 B=25 A=25 C=22 so C is eliminated. Round 3 votes counts: D=41 B=30 A=29 so A is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 B:209 E:202 C:190 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -6 -16 B 12 0 2 -8 12 C -4 -2 0 -10 -4 D 6 8 10 0 4 E 16 -12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -6 -16 B 12 0 2 -8 12 C -4 -2 0 -10 -4 D 6 8 10 0 4 E 16 -12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -6 -16 B 12 0 2 -8 12 C -4 -2 0 -10 -4 D 6 8 10 0 4 E 16 -12 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999733 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8907: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) D E A B C (8) E C B A D (6) D E C B A (6) B A C E D (6) E B C A D (5) A B C D E (5) C B A E D (4) C B A D E (4) E D A B C (3) D C A B E (3) D A C B E (3) C B E A D (3) C A B D E (3) E D B C A (2) E D B A C (2) E C B D A (2) E B A C D (2) E A B D C (2) E A B C D (2) D E A C B (2) A B E D C (2) A B C E D (2) E D C B A (1) E B D A C (1) E B C D A (1) D E C A B (1) D C E B A (1) D A E B C (1) D A B E C (1) C D B A E (1) B E C A D (1) B C A E D (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 10 0 -2 B -2 0 18 6 4 C -10 -18 0 -4 -2 D 0 -6 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999918 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 2 10 0 -2 B -2 0 18 6 4 C -10 -18 0 -4 -2 D 0 -6 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999437 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=29 C=15 A=11 B=8 so B is eliminated. Round 2 votes counts: D=37 E=30 A=17 C=16 so C is eliminated. Round 3 votes counts: D=38 E=33 A=29 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:213 A:205 D:201 E:198 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 0 -2 B -2 0 18 6 4 C -10 -18 0 -4 -2 D 0 -6 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999437 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 0 -2 B -2 0 18 6 4 C -10 -18 0 -4 -2 D 0 -6 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999437 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 0 -2 B -2 0 18 6 4 C -10 -18 0 -4 -2 D 0 -6 4 0 4 E 2 -4 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.250000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.374999999437 Cumulative probabilities = A: 0.500000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8908: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (12) E B D A C (7) C A D B E (7) A D C B E (6) E B C D A (5) B E D A C (5) E B C A D (3) D E A B C (3) D A E C B (3) C A E D B (3) B D A E C (3) A C D B E (3) E D B A C (2) E C B A D (2) E C A D B (2) E B D C A (2) D A C E B (2) D A B E C (2) D A B C E (2) C E A D B (2) C B E A D (2) B E D C A (2) B E C D A (2) B D E A C (2) A D C E B (2) A C D E B (2) E D A C B (1) E D A B C (1) E C D A B (1) D E B A C (1) D B E A C (1) D A E B C (1) D A C B E (1) C B A E D (1) C A E B D (1) C A B D E (1) B D A C E (1) B C A D E (1) Total count = 100 A B C D E A 0 16 2 0 8 B -16 0 -8 -20 -16 C -2 8 0 0 -2 D 0 20 0 0 12 E -8 16 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.746751 B: 0.000000 C: 0.000000 D: 0.253249 E: 0.000000 Sum of squares = 0.621772574386 Cumulative probabilities = A: 0.746751 B: 0.746751 C: 0.746751 D: 1.000000 E: 1.000000 A B C D E A 0 16 2 0 8 B -16 0 -8 -20 -16 C -2 8 0 0 -2 D 0 20 0 0 12 E -8 16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=26 D=16 B=16 A=13 so A is eliminated. Round 2 votes counts: C=34 E=26 D=24 B=16 so B is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:216 A:213 C:202 E:199 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 2 0 8 B -16 0 -8 -20 -16 C -2 8 0 0 -2 D 0 20 0 0 12 E -8 16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 2 0 8 B -16 0 -8 -20 -16 C -2 8 0 0 -2 D 0 20 0 0 12 E -8 16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 2 0 8 B -16 0 -8 -20 -16 C -2 8 0 0 -2 D 0 20 0 0 12 E -8 16 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999963 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8909: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) A D B E C (8) B A D C E (7) D A E C B (5) A B D E C (5) C E D B A (4) C D E A B (4) C B E D A (4) E C D A B (3) E C B A D (3) D A B C E (3) C E D A B (3) B A E D C (3) B A C D E (3) E C A D B (2) D C E A B (2) B E A C D (2) B D A C E (2) B C E A D (2) B A E C D (2) E D C A B (1) E D A C B (1) E C D B A (1) E C B D A (1) E A D C B (1) D E A C B (1) D C B A E (1) D C A E B (1) D C A B E (1) D A E B C (1) D A C E B (1) C E B A D (1) C D B A E (1) C B D E A (1) C B D A E (1) B E C A D (1) B C E D A (1) B C A E D (1) B C A D E (1) B A D E C (1) A E D B C (1) A D E C B (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 2 -4 6 B 4 0 -6 2 6 C -2 6 0 0 10 D 4 -2 0 0 8 E -6 -6 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.478764 D: 0.521236 E: 0.000000 Sum of squares = 0.500901942983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.478764 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -4 6 B 4 0 -6 2 6 C -2 6 0 0 10 D 4 -2 0 0 8 E -6 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=26 A=18 D=16 E=13 so E is eliminated. Round 2 votes counts: C=37 B=26 A=19 D=18 so D is eliminated. Round 3 votes counts: C=43 A=31 B=26 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:207 D:205 B:203 A:200 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 -4 6 B 4 0 -6 2 6 C -2 6 0 0 10 D 4 -2 0 0 8 E -6 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -4 6 B 4 0 -6 2 6 C -2 6 0 0 10 D 4 -2 0 0 8 E -6 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -4 6 B 4 0 -6 2 6 C -2 6 0 0 10 D 4 -2 0 0 8 E -6 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8910: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) B A D E C (8) D E C B A (6) D C E B A (6) B A E D C (6) A B C D E (6) E D C B A (3) E D B A C (3) E C D B A (3) E C A B D (3) D E B A C (3) D B A C E (3) C E A D B (3) C E A B D (3) C D E B A (3) B A D C E (3) A B E D C (3) A B E C D (3) E C D A B (2) D B E A C (2) D B A E C (2) A C B E D (2) A B D C E (2) A B C E D (2) E C A D B (1) E A C B D (1) E A B C D (1) D C B E A (1) D B C A E (1) C E D B A (1) C D E A B (1) C A E B D (1) C A B E D (1) B D A E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -10 4 0 -10 B 10 0 -2 -6 -6 C -4 2 0 -6 -2 D 0 6 6 0 -2 E 10 6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 4 0 -10 B 10 0 -2 -6 -6 C -4 2 0 -6 -2 D 0 6 6 0 -2 E 10 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=22 A=19 B=18 E=17 so E is eliminated. Round 2 votes counts: C=31 D=30 A=21 B=18 so B is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:210 D:205 B:198 C:195 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 4 0 -10 B 10 0 -2 -6 -6 C -4 2 0 -6 -2 D 0 6 6 0 -2 E 10 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 4 0 -10 B 10 0 -2 -6 -6 C -4 2 0 -6 -2 D 0 6 6 0 -2 E 10 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 4 0 -10 B 10 0 -2 -6 -6 C -4 2 0 -6 -2 D 0 6 6 0 -2 E 10 6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8911: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (13) B E C A D (8) E B A D C (7) D A C E B (6) E A B D C (5) A D C E B (5) E B D A C (4) E A D B C (4) D C A E B (4) D A E C B (3) B E A C D (3) A C D B E (3) E B D C A (2) D C E B A (2) D C A B E (2) C D B E A (2) C D B A E (2) B E C D A (2) B C E D A (2) B C D A E (2) B C A D E (2) A D E C B (2) E D C B A (1) E D C A B (1) E D A C B (1) E D A B C (1) E B C D A (1) E A D C B (1) D E C B A (1) D E C A B (1) C A D B E (1) B C D E A (1) B C A E D (1) B A E C D (1) B A C E D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 10 -2 -8 0 B -10 0 -2 -14 -8 C 2 2 0 -10 -2 D 8 14 10 0 4 E 0 8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 -8 0 B -10 0 -2 -14 -8 C 2 2 0 -10 -2 D 8 14 10 0 4 E 0 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=23 D=19 C=18 A=12 so A is eliminated. Round 2 votes counts: E=30 D=26 B=23 C=21 so C is eliminated. Round 3 votes counts: D=47 E=30 B=23 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:203 A:200 C:196 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -2 -8 0 B -10 0 -2 -14 -8 C 2 2 0 -10 -2 D 8 14 10 0 4 E 0 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -8 0 B -10 0 -2 -14 -8 C 2 2 0 -10 -2 D 8 14 10 0 4 E 0 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -8 0 B -10 0 -2 -14 -8 C 2 2 0 -10 -2 D 8 14 10 0 4 E 0 8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8912: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (7) A E C D B (5) E C A D B (4) B D C E A (4) B D C A E (4) B D A E C (4) A D C E B (4) E A C D B (3) D C B E A (3) B A E C D (3) B A D E C (3) A E B C D (3) A C E D B (3) A B E D C (3) D C E A B (2) D C B A E (2) D C A E B (2) D B C A E (2) D A C E B (2) D A C B E (2) D A B C E (2) C E D A B (2) C E A D B (2) C D E B A (2) C D E A B (2) C D B E A (2) B E C D A (2) B E A C D (2) B D E A C (2) A E C B D (2) A D E C B (2) E C B A D (1) E C A B D (1) E A C B D (1) E A B C D (1) D C A B E (1) D B C E A (1) D B A C E (1) C E B D A (1) C D A E B (1) B E D A C (1) B E C A D (1) B D A C E (1) A D E B C (1) Total count = 100 A B C D E A 0 2 16 4 20 B -2 0 -4 -6 6 C -16 4 0 -12 -4 D -4 6 12 0 4 E -20 -6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997816 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 4 20 B -2 0 -4 -6 6 C -16 4 0 -12 -4 D -4 6 12 0 4 E -20 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992047 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=23 D=20 C=12 E=11 so E is eliminated. Round 2 votes counts: B=34 A=28 D=20 C=18 so C is eliminated. Round 3 votes counts: B=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:209 B:197 E:187 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 16 4 20 B -2 0 -4 -6 6 C -16 4 0 -12 -4 D -4 6 12 0 4 E -20 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992047 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 4 20 B -2 0 -4 -6 6 C -16 4 0 -12 -4 D -4 6 12 0 4 E -20 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992047 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 4 20 B -2 0 -4 -6 6 C -16 4 0 -12 -4 D -4 6 12 0 4 E -20 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992047 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8913: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (11) E C A B D (9) D C B A E (9) C A B D E (7) E A B C D (6) E D B A C (5) E B A D C (5) D E B A C (5) E A C B D (4) C E A B D (4) A B C E D (4) D B C A E (3) C A E B D (3) B A E C D (3) E D A B C (2) D E C A B (2) C D A B E (2) C B A D E (2) C A B E D (2) A C B E D (2) E C D A B (1) E A B D C (1) D E C B A (1) D B C E A (1) D B A E C (1) C D B A E (1) C A D B E (1) B D A E C (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 0 4 10 8 B 0 0 0 10 4 C -4 0 0 4 8 D -10 -10 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.555696 B: 0.444304 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.506204095577 Cumulative probabilities = A: 0.555696 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 10 8 B 0 0 0 10 4 C -4 0 0 4 8 D -10 -10 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=33 D=33 C=22 B=6 A=6 so B is eliminated. Round 2 votes counts: D=34 E=33 C=22 A=11 so A is eliminated. Round 3 votes counts: E=36 D=35 C=29 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:211 B:207 C:204 E:192 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 10 8 B 0 0 0 10 4 C -4 0 0 4 8 D -10 -10 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 8 B 0 0 0 10 4 C -4 0 0 4 8 D -10 -10 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 8 B 0 0 0 10 4 C -4 0 0 4 8 D -10 -10 -4 0 -4 E -8 -4 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8914: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) A D B E C (6) D B A E C (5) A C D E B (5) E C B A D (4) E B C D A (4) E B A C D (4) D A C B E (4) C E A B D (4) B E D C A (4) A E C B D (4) A C E D B (4) E B C A D (3) D C A B E (3) B E D A C (3) B D E C A (3) B D E A C (3) E A C B D (2) E A B D C (2) D B A C E (2) C E B A D (2) C D B E A (2) C A E B D (2) B E C D A (2) A E C D B (2) A E B D C (2) E C A B D (1) E B A D C (1) E A B C D (1) D C B A E (1) D A B E C (1) D A B C E (1) C E B D A (1) C E A D B (1) C B E D A (1) C A E D B (1) B D C E A (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 8 6 -10 B 4 0 8 10 -2 C -8 -8 0 2 -14 D -6 -10 -2 0 -12 E 10 2 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 8 6 -10 B 4 0 8 10 -2 C -8 -8 0 2 -14 D -6 -10 -2 0 -12 E 10 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 E=22 B=16 C=14 so C is eliminated. Round 2 votes counts: E=30 A=28 D=25 B=17 so B is eliminated. Round 3 votes counts: E=40 D=32 A=28 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:219 B:210 A:200 C:186 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 8 6 -10 B 4 0 8 10 -2 C -8 -8 0 2 -14 D -6 -10 -2 0 -12 E 10 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 6 -10 B 4 0 8 10 -2 C -8 -8 0 2 -14 D -6 -10 -2 0 -12 E 10 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 6 -10 B 4 0 8 10 -2 C -8 -8 0 2 -14 D -6 -10 -2 0 -12 E 10 2 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998667 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8915: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (14) C A B E D (12) E D A B C (8) B C A D E (8) B A C D E (7) E D C A B (5) E D B A C (5) E D A C B (5) D B E A C (4) C B A D E (4) C B A E D (3) C A B D E (3) C E A D B (2) C A E B D (2) B D A E C (2) A C B E D (2) A B C D E (2) E D C B A (1) E C D B A (1) E C D A B (1) D B A E C (1) C E A B D (1) C A E D B (1) B D E C A (1) B C D E A (1) B A D C E (1) A E D C B (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 8 2 2 B 6 0 8 0 4 C -8 -8 0 4 2 D -2 0 -4 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.562567 C: 0.000000 D: 0.437433 E: 0.000000 Sum of squares = 0.507829160085 Cumulative probabilities = A: 0.000000 B: 0.562567 C: 0.562567 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 2 2 B 6 0 8 0 4 C -8 -8 0 4 2 D -2 0 -4 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 B=20 D=19 A=7 so A is eliminated. Round 2 votes counts: C=32 E=27 B=22 D=19 so D is eliminated. Round 3 votes counts: E=41 C=32 B=27 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:209 A:203 E:197 D:196 C:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 2 2 B 6 0 8 0 4 C -8 -8 0 4 2 D -2 0 -4 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 2 2 B 6 0 8 0 4 C -8 -8 0 4 2 D -2 0 -4 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 2 2 B 6 0 8 0 4 C -8 -8 0 4 2 D -2 0 -4 0 -2 E -2 -4 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8916: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) A B C E D (7) E D C A B (5) D E C B A (5) B A C D E (5) E D A B C (4) D E B C A (4) D C B A E (4) C A B D E (4) E D B A C (3) E A C B D (3) D E C A B (3) D B E A C (3) C A B E D (3) B C A D E (3) E D A C B (2) E C D A B (2) E C A D B (2) E A B C D (2) D E B A C (2) D C E B A (2) D C E A B (2) D B A E C (2) C E A D B (2) C D B A E (2) C A E B D (2) B A D C E (2) A C B E D (2) E B A D C (1) D B A C E (1) C E A B D (1) C B A D E (1) B D C A E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -20 -18 4 B 6 0 6 -24 6 C 20 -6 0 -16 14 D 18 24 16 0 18 E -4 -6 -14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -18 4 B 6 0 6 -24 6 C 20 -6 0 -16 14 D 18 24 16 0 18 E -4 -6 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=24 C=15 B=11 A=10 so A is eliminated. Round 2 votes counts: D=40 E=24 B=19 C=17 so C is eliminated. Round 3 votes counts: D=42 E=29 B=29 so E is eliminated. Round 4 votes counts: D=62 B=38 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:238 C:206 B:197 A:180 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -20 -18 4 B 6 0 6 -24 6 C 20 -6 0 -16 14 D 18 24 16 0 18 E -4 -6 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -18 4 B 6 0 6 -24 6 C 20 -6 0 -16 14 D 18 24 16 0 18 E -4 -6 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -18 4 B 6 0 6 -24 6 C 20 -6 0 -16 14 D 18 24 16 0 18 E -4 -6 -14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8917: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (7) D A C B E (6) C D E A B (6) C D A E B (6) D A B E C (5) A D B E C (5) E B D A C (4) E B C A D (4) C A D B E (4) B E A C D (4) A D B C E (4) E B C D A (3) B A E D C (3) D E B A C (2) D E A B C (2) D C A E B (2) D A E B C (2) C E B D A (2) C A D E B (2) C A B E D (2) B E C A D (2) A D C B E (2) A B D E C (2) E C B D A (1) E C B A D (1) E B D C A (1) E B A C D (1) D E C B A (1) D E C A B (1) D C E A B (1) D B E A C (1) D A C E B (1) D A B C E (1) C E D B A (1) C E B A D (1) C A E D B (1) C A E B D (1) B E D A C (1) B A E C D (1) B A D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 14 0 6 B -16 0 16 -12 6 C -14 -16 0 -10 -10 D 0 12 10 0 16 E -6 -6 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.464153 B: 0.000000 C: 0.000000 D: 0.535847 E: 0.000000 Sum of squares = 0.502570071317 Cumulative probabilities = A: 0.464153 B: 0.464153 C: 0.464153 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 0 6 B -16 0 16 -12 6 C -14 -16 0 -10 -10 D 0 12 10 0 16 E -6 -6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 B=19 E=15 A=15 so E is eliminated. Round 2 votes counts: B=32 C=28 D=25 A=15 so A is eliminated. Round 3 votes counts: D=36 B=36 C=28 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:218 B:197 E:191 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 0 6 B -16 0 16 -12 6 C -14 -16 0 -10 -10 D 0 12 10 0 16 E -6 -6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 0 6 B -16 0 16 -12 6 C -14 -16 0 -10 -10 D 0 12 10 0 16 E -6 -6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 0 6 B -16 0 16 -12 6 C -14 -16 0 -10 -10 D 0 12 10 0 16 E -6 -6 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8918: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (18) C D A B E (13) E B A D C (11) A B E C D (10) C A D B E (6) A B C E D (5) A C B D E (4) D E C B A (3) D E B C A (3) C D A E B (3) B E A D C (3) A C B E D (3) E D C B A (2) E B D C A (2) E B D A C (2) A E B C D (2) A C D B E (2) E D B C A (1) E B A C D (1) D C E A B (1) D C B E A (1) D C A B E (1) D B E C A (1) B E D A C (1) B A E C D (1) Total count = 100 A B C D E A 0 0 -10 -4 0 B 0 0 -14 -10 2 C 10 14 0 0 14 D 4 10 0 0 12 E 0 -2 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.513911 D: 0.486089 E: 0.000000 Sum of squares = 0.500387010701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.513911 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -4 0 B 0 0 -14 -10 2 C 10 14 0 0 14 D 4 10 0 0 12 E 0 -2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=26 C=22 E=19 B=5 so B is eliminated. Round 2 votes counts: D=28 A=27 E=23 C=22 so C is eliminated. Round 3 votes counts: D=44 A=33 E=23 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:219 D:213 A:193 B:189 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -4 0 B 0 0 -14 -10 2 C 10 14 0 0 14 D 4 10 0 0 12 E 0 -2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -4 0 B 0 0 -14 -10 2 C 10 14 0 0 14 D 4 10 0 0 12 E 0 -2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -4 0 B 0 0 -14 -10 2 C 10 14 0 0 14 D 4 10 0 0 12 E 0 -2 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8919: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) A C B D E (8) E B C D A (5) D E A C B (5) A C D B E (5) E B C A D (4) D A E C B (4) C B A D E (4) A E D C B (4) E D B C A (3) E D A C B (3) B C A E D (3) A D C E B (3) A D C B E (3) E A D B C (2) E A B D C (2) E A B C D (2) D A C E B (2) C B D A E (2) B E C A D (2) B C E A D (2) B C D A E (2) B C A D E (2) E D B A C (1) E D A B C (1) E B D C A (1) E B A C D (1) E A D C B (1) D E C B A (1) D E B C A (1) D C B E A (1) D C B A E (1) D C A B E (1) D A C B E (1) C D A B E (1) C A D B E (1) C A B D E (1) B E C D A (1) B C E D A (1) B C D E A (1) A E B C D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 22 18 22 18 B -22 0 -20 -6 -10 C -18 20 0 2 -8 D -22 6 -2 0 16 E -18 10 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 18 22 18 B -22 0 -20 -6 -10 C -18 20 0 2 -8 D -22 6 -2 0 16 E -18 10 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=26 D=17 B=14 C=9 so C is eliminated. Round 2 votes counts: A=36 E=26 B=20 D=18 so D is eliminated. Round 3 votes counts: A=45 E=33 B=22 so B is eliminated. Round 4 votes counts: A=59 E=41 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:240 D:199 C:198 E:192 B:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 18 22 18 B -22 0 -20 -6 -10 C -18 20 0 2 -8 D -22 6 -2 0 16 E -18 10 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 18 22 18 B -22 0 -20 -6 -10 C -18 20 0 2 -8 D -22 6 -2 0 16 E -18 10 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 18 22 18 B -22 0 -20 -6 -10 C -18 20 0 2 -8 D -22 6 -2 0 16 E -18 10 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8920: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (18) E B A D C (15) D C E B A (14) D E B A C (9) A B E C D (8) C A B E D (6) E D B A C (4) E B A C D (4) E B D A C (3) D E C B A (3) C A B D E (2) B E A D C (2) A B E D C (2) D C A B E (1) D B A E C (1) D A B E C (1) D A B C E (1) C D E A B (1) C D A E B (1) C A D B E (1) B A E D C (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 6 -14 -10 B 14 0 4 -10 -8 C -6 -4 0 -14 -8 D 14 10 14 0 6 E 10 8 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 6 -14 -10 B 14 0 4 -10 -8 C -6 -4 0 -14 -8 D 14 10 14 0 6 E 10 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=29 E=26 A=11 B=4 so B is eliminated. Round 2 votes counts: D=30 C=29 E=28 A=13 so A is eliminated. Round 3 votes counts: E=40 D=30 C=30 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:222 E:210 B:200 A:184 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 6 -14 -10 B 14 0 4 -10 -8 C -6 -4 0 -14 -8 D 14 10 14 0 6 E 10 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 -14 -10 B 14 0 4 -10 -8 C -6 -4 0 -14 -8 D 14 10 14 0 6 E 10 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 -14 -10 B 14 0 4 -10 -8 C -6 -4 0 -14 -8 D 14 10 14 0 6 E 10 8 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8921: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (15) D B A C E (9) E C A D B (8) B D E C A (6) D A B C E (5) A C E D B (5) D A C E B (4) E C B A D (3) C E A D B (3) B E C D A (3) B D A C E (3) A C D E B (3) E D C A B (2) D E C A B (2) D B E C A (2) D A C B E (2) C E A B D (2) B D E A C (2) B A D C E (2) B A C E D (2) E C D A B (1) E C B D A (1) E B C D A (1) E B C A D (1) D E C B A (1) D C A E B (1) C A E B D (1) C A D E B (1) B E C A D (1) B D A E C (1) A D C E B (1) A D C B E (1) A D B C E (1) A C E B D (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 24 -10 8 -8 B -24 0 -18 -4 -14 C 10 18 0 10 0 D -8 4 -10 0 -4 E 8 14 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.476518 D: 0.000000 E: 0.523482 Sum of squares = 0.501102850319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.476518 D: 0.476518 E: 1.000000 A B C D E A 0 24 -10 8 -8 B -24 0 -18 -4 -14 C 10 18 0 10 0 D -8 4 -10 0 -4 E 8 14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=26 B=20 A=15 C=7 so C is eliminated. Round 2 votes counts: E=37 D=26 B=20 A=17 so A is eliminated. Round 3 votes counts: E=44 D=33 B=23 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:213 A:207 D:191 B:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 24 -10 8 -8 B -24 0 -18 -4 -14 C 10 18 0 10 0 D -8 4 -10 0 -4 E 8 14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 -10 8 -8 B -24 0 -18 -4 -14 C 10 18 0 10 0 D -8 4 -10 0 -4 E 8 14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 -10 8 -8 B -24 0 -18 -4 -14 C 10 18 0 10 0 D -8 4 -10 0 -4 E 8 14 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8922: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) D C A E B (5) C D E A B (5) E B A D C (4) E B A C D (4) E A B D C (4) C D B A E (4) B A E D C (4) B A E C D (4) E D C A B (3) E C D B A (3) D E C A B (3) C E D B A (3) B E A C D (3) B A C E D (3) E D A C B (2) E B C A D (2) E A D B C (2) D E A C B (2) D C E A B (2) D C A B E (2) D A C B E (2) C D A E B (2) C B A D E (2) B E A D C (2) A B D C E (2) E D A B C (1) E B D A C (1) C D E B A (1) C D B E A (1) C B D E A (1) C A D B E (1) B E C A D (1) B C E A D (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B D C (1) A D B C E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -8 -8 -2 B -8 0 -10 -14 -2 C 8 10 0 8 2 D 8 14 -8 0 0 E 2 2 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -8 -8 -2 B -8 0 -10 -14 -2 C 8 10 0 8 2 D 8 14 -8 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=26 B=20 D=16 A=7 so A is eliminated. Round 2 votes counts: C=31 E=28 B=24 D=17 so D is eliminated. Round 3 votes counts: C=42 E=33 B=25 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:207 E:201 A:195 B:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -8 -2 B -8 0 -10 -14 -2 C 8 10 0 8 2 D 8 14 -8 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -8 -2 B -8 0 -10 -14 -2 C 8 10 0 8 2 D 8 14 -8 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -8 -2 B -8 0 -10 -14 -2 C 8 10 0 8 2 D 8 14 -8 0 0 E 2 2 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997444 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8923: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) B E A C D (9) E B A C D (8) D C B A E (7) B E A D C (7) E A B C D (6) B E D C A (6) B D C E A (6) A C D E B (5) C D A E B (4) B D E C A (4) A E C D B (4) E A C D B (3) B E D A C (3) B D C A E (3) D C A E B (2) D B C A E (2) C A D E B (2) A E C B D (2) E D C A B (1) E D B C A (1) E B D A C (1) E B A D C (1) C D A B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 0 -4 -12 B 16 0 14 12 18 C 0 -14 0 -10 -12 D 4 -12 10 0 -6 E 12 -18 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -4 -12 B 16 0 14 12 18 C 0 -14 0 -10 -12 D 4 -12 10 0 -6 E 12 -18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=22 E=21 A=12 C=7 so C is eliminated. Round 2 votes counts: B=38 D=27 E=21 A=14 so A is eliminated. Round 3 votes counts: B=38 D=34 E=28 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:230 E:206 D:198 A:184 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -4 -12 B 16 0 14 12 18 C 0 -14 0 -10 -12 D 4 -12 10 0 -6 E 12 -18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -4 -12 B 16 0 14 12 18 C 0 -14 0 -10 -12 D 4 -12 10 0 -6 E 12 -18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -4 -12 B 16 0 14 12 18 C 0 -14 0 -10 -12 D 4 -12 10 0 -6 E 12 -18 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8924: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) E D A B C (8) A E D B C (7) E A D C B (5) C B E D A (5) C B D E A (5) C B A E D (5) A D E B C (5) E D C B A (4) B A D C E (4) C B E A D (3) C B D A E (3) B C D E A (3) B C D A E (3) B C A D E (3) A B D C E (3) C E D B A (2) C E B D A (2) B D C E A (2) A B C D E (2) E D C A B (1) E D A C B (1) E A D B C (1) D E B C A (1) D E A B C (1) D C E B A (1) D B E A C (1) D A E B C (1) D A B E C (1) C E B A D (1) C A B E D (1) B D C A E (1) B A C D E (1) A E C D B (1) A D B E C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -10 8 6 B 18 0 0 16 14 C 10 0 0 0 20 D -8 -16 0 0 4 E -6 -14 -20 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.623767 C: 0.376233 D: 0.000000 E: 0.000000 Sum of squares = 0.530636498744 Cumulative probabilities = A: 0.000000 B: 0.623767 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 8 6 B 18 0 0 16 14 C 10 0 0 0 20 D -8 -16 0 0 4 E -6 -14 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=21 E=20 B=17 D=6 so D is eliminated. Round 2 votes counts: C=37 A=23 E=22 B=18 so B is eliminated. Round 3 votes counts: C=49 A=28 E=23 so E is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:215 A:193 D:190 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 8 6 B 18 0 0 16 14 C 10 0 0 0 20 D -8 -16 0 0 4 E -6 -14 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 8 6 B 18 0 0 16 14 C 10 0 0 0 20 D -8 -16 0 0 4 E -6 -14 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 8 6 B 18 0 0 16 14 C 10 0 0 0 20 D -8 -16 0 0 4 E -6 -14 -20 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8925: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) C A D E B (7) E D A C B (6) C D A E B (6) C A D B E (5) B E C D A (5) A C D B E (5) E D C A B (4) D C A E B (4) D A C E B (4) B C A D E (4) E B D A C (3) C A B D E (3) B E D C A (3) B E C A D (3) B E A C D (3) E D B A C (2) E A D C B (2) D E A C B (2) B E A D C (2) B C E A D (2) B C A E D (2) B A C E D (2) A C D E B (2) A C B D E (2) E D B C A (1) E D A B C (1) E B A D C (1) D A E C B (1) C B A D E (1) B A E C D (1) B A C D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 0 0 4 B -12 0 -8 -4 8 C 0 8 0 10 2 D 0 4 -10 0 -4 E -4 -8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.581640 B: 0.000000 C: 0.418360 D: 0.000000 E: 0.000000 Sum of squares = 0.513330144259 Cumulative probabilities = A: 0.581640 B: 0.581640 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 0 4 B -12 0 -8 -4 8 C 0 8 0 10 2 D 0 4 -10 0 -4 E -4 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=22 E=20 D=11 A=11 so D is eliminated. Round 2 votes counts: B=36 C=26 E=22 A=16 so A is eliminated. Round 3 votes counts: C=39 B=38 E=23 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:210 A:208 D:195 E:195 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 0 4 B -12 0 -8 -4 8 C 0 8 0 10 2 D 0 4 -10 0 -4 E -4 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 0 4 B -12 0 -8 -4 8 C 0 8 0 10 2 D 0 4 -10 0 -4 E -4 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 0 4 B -12 0 -8 -4 8 C 0 8 0 10 2 D 0 4 -10 0 -4 E -4 -8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999976 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8926: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (10) B C D A E (9) E D A C B (8) E A D C B (5) C B E A D (5) C B D A E (5) A D E B C (5) E A D B C (4) E A C B D (4) D E A C B (3) D A B E C (3) D A B C E (3) C B E D A (3) C B D E A (3) B C A E D (3) B A D C E (3) E A C D B (2) E A B D C (2) D B C A E (2) C E B D A (2) B D C A E (2) B C E A D (2) A E D B C (2) E C B A D (1) E C A D B (1) E C A B D (1) D B A C E (1) D A C E B (1) C E D A B (1) C E B A D (1) B A C D E (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 14 18 -12 4 B -14 0 8 -4 -6 C -18 -8 0 -12 -6 D 12 4 12 0 4 E -4 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 -12 4 B -14 0 8 -4 -6 C -18 -8 0 -12 -6 D 12 4 12 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=23 C=20 B=20 A=9 so A is eliminated. Round 2 votes counts: E=30 D=29 B=21 C=20 so C is eliminated. Round 3 votes counts: B=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:216 A:212 E:202 B:192 C:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 18 -12 4 B -14 0 8 -4 -6 C -18 -8 0 -12 -6 D 12 4 12 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 -12 4 B -14 0 8 -4 -6 C -18 -8 0 -12 -6 D 12 4 12 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 -12 4 B -14 0 8 -4 -6 C -18 -8 0 -12 -6 D 12 4 12 0 4 E -4 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998745 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8927: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (9) C A B D E (8) E A D B C (5) D E B C A (5) B D E C A (5) B D C E A (5) E D C B A (4) E D B A C (4) A E C B D (4) A C B E D (4) D B E C A (3) A C E B D (3) E B D A C (2) E A D C B (2) C D B E A (2) C B D A E (2) C A D E B (2) B D A C E (2) B C D A E (2) A B D C E (2) A B C D E (2) E D A C B (1) E A C D B (1) E A B D C (1) D C B E A (1) C E D B A (1) C D E B A (1) C D A B E (1) B D E A C (1) B D C A E (1) A E C D B (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -8 -8 -10 B 4 0 4 8 2 C 8 -4 0 -12 -2 D 8 -8 12 0 8 E 10 -2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999359 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -8 -10 B 4 0 4 8 2 C 8 -4 0 -12 -2 D 8 -8 12 0 8 E 10 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=27 C=17 B=16 D=9 so D is eliminated. Round 2 votes counts: E=36 A=27 B=19 C=18 so C is eliminated. Round 3 votes counts: E=38 A=38 B=24 so B is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 B:209 E:201 C:195 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 -8 -10 B 4 0 4 8 2 C 8 -4 0 -12 -2 D 8 -8 12 0 8 E 10 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -8 -10 B 4 0 4 8 2 C 8 -4 0 -12 -2 D 8 -8 12 0 8 E 10 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -8 -10 B 4 0 4 8 2 C 8 -4 0 -12 -2 D 8 -8 12 0 8 E 10 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8928: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) D E B A C (7) C E A B D (6) D B A C E (5) D A B C E (5) E C D B A (4) E C B A D (4) E B A C D (4) D A C B E (4) E C A B D (3) D C E A B (3) D B E A C (3) C A D B E (3) A C B D E (3) A B D C E (3) E D C B A (2) E B D A C (2) E B A D C (2) D E C B A (2) D E C A B (2) D E B C A (2) C E A D B (2) C A B E D (2) C A B D E (2) B A D E C (2) B A D C E (2) A B C D E (2) E D B A C (1) D C A B E (1) C E D A B (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E B D (1) B E A C D (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 14 -12 -4 B 6 0 2 -18 0 C -14 -2 0 -14 -2 D 12 18 14 0 26 E 4 0 2 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 14 -12 -4 B 6 0 2 -18 0 C -14 -2 0 -14 -2 D 12 18 14 0 26 E 4 0 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 E=22 C=20 A=9 B=6 so B is eliminated. Round 2 votes counts: D=43 E=23 C=20 A=14 so A is eliminated. Round 3 votes counts: D=50 C=26 E=24 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:235 A:196 B:195 E:190 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 14 -12 -4 B 6 0 2 -18 0 C -14 -2 0 -14 -2 D 12 18 14 0 26 E 4 0 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 14 -12 -4 B 6 0 2 -18 0 C -14 -2 0 -14 -2 D 12 18 14 0 26 E 4 0 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 14 -12 -4 B 6 0 2 -18 0 C -14 -2 0 -14 -2 D 12 18 14 0 26 E 4 0 2 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8929: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) A B D E C (7) B A E C D (6) B A D C E (5) E A D B C (4) D C E A B (4) D C B A E (4) B C E A D (4) B A C D E (4) E C D B A (3) E C D A B (3) D C A E B (3) C D E B A (3) A E B D C (3) A D B E C (3) E C A B D (2) E A B D C (2) D E C A B (2) D C E B A (2) D A E C B (2) C E B D A (2) C D B E A (2) B E A C D (2) B C D A E (2) A B E D C (2) A B D C E (2) E D C A B (1) E D A C B (1) E C A D B (1) E B C A D (1) D C A B E (1) D B A C E (1) D A B C E (1) C E D B A (1) C B E D A (1) C B E A D (1) B D C A E (1) B C A E D (1) B A C E D (1) A E B C D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 20 16 B -6 0 22 2 4 C -10 -22 0 -16 -8 D -20 -2 16 0 12 E -16 -4 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 20 16 B -6 0 22 2 4 C -10 -22 0 -16 -8 D -20 -2 16 0 12 E -16 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 D=20 E=18 C=10 so C is eliminated. Round 2 votes counts: B=28 A=26 D=25 E=21 so E is eliminated. Round 3 votes counts: A=35 D=34 B=31 so B is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:211 D:203 E:188 C:172 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 20 16 B -6 0 22 2 4 C -10 -22 0 -16 -8 D -20 -2 16 0 12 E -16 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 20 16 B -6 0 22 2 4 C -10 -22 0 -16 -8 D -20 -2 16 0 12 E -16 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 20 16 B -6 0 22 2 4 C -10 -22 0 -16 -8 D -20 -2 16 0 12 E -16 -4 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8930: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (9) A C D B E (7) D C B E A (6) A E B C D (6) E B D C A (5) C D A B E (5) C D B E A (4) A C B D E (4) E D C B A (3) E A D C B (3) C D A E B (3) B D C E A (3) B C D A E (3) A B E C D (3) A B C D E (3) D C E B A (2) C A D B E (2) B E D C A (2) A E D C B (2) A E C D B (2) A E C B D (2) A C D E B (2) E D C A B (1) E D B C A (1) E D A B C (1) E B A D C (1) E A D B C (1) D E B C A (1) D B C E A (1) B C D E A (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 16 -4 -2 10 B -16 0 -16 -14 6 C 4 16 0 14 12 D 2 14 -14 0 12 E -10 -6 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 -2 10 B -16 0 -16 -14 6 C 4 16 0 14 12 D 2 14 -14 0 12 E -10 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=25 C=23 D=10 B=9 so B is eliminated. Round 2 votes counts: A=33 E=27 C=27 D=13 so D is eliminated. Round 3 votes counts: C=39 A=33 E=28 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:223 A:210 D:207 B:180 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 -2 10 B -16 0 -16 -14 6 C 4 16 0 14 12 D 2 14 -14 0 12 E -10 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 -2 10 B -16 0 -16 -14 6 C 4 16 0 14 12 D 2 14 -14 0 12 E -10 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 -2 10 B -16 0 -16 -14 6 C 4 16 0 14 12 D 2 14 -14 0 12 E -10 -6 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8931: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (7) D E C B A (6) B A E C D (6) A B D C E (6) D C E A B (5) A B C E D (5) E C D B A (4) E B A C D (4) C D E A B (4) A B E C D (4) C E A D B (3) C D A E B (3) B D A E C (3) E B C A D (2) D B E C A (2) C E D A B (2) C A E D B (2) C A E B D (2) C A D E B (2) B E D A C (2) B E A D C (2) A C E B D (2) A C D B E (2) A C B E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E C B D A (1) E C B A D (1) E C A B D (1) E B D C A (1) E B D A C (1) D E C A B (1) D E B C A (1) C E A B D (1) C A D B E (1) B E A C D (1) B D E A C (1) B A E D C (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 2 -8 8 -10 B -2 0 -8 4 -16 C 8 8 0 14 4 D -8 -4 -14 0 -4 E 10 16 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 8 -10 B -2 0 -8 4 -16 C 8 8 0 14 4 D -8 -4 -14 0 -4 E 10 16 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=22 C=20 E=17 B=17 so E is eliminated. Round 2 votes counts: C=27 B=25 D=24 A=24 so D is eliminated. Round 3 votes counts: C=47 B=29 A=24 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:217 E:213 A:196 B:189 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 8 -10 B -2 0 -8 4 -16 C 8 8 0 14 4 D -8 -4 -14 0 -4 E 10 16 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 8 -10 B -2 0 -8 4 -16 C 8 8 0 14 4 D -8 -4 -14 0 -4 E 10 16 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 8 -10 B -2 0 -8 4 -16 C 8 8 0 14 4 D -8 -4 -14 0 -4 E 10 16 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8932: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) B E D C A (6) A E C B D (5) A D C E B (5) E B C D A (4) C D E A B (4) B D E C A (4) A D B C E (4) A B E C D (4) E C B D A (3) D B C E A (3) C D E B A (3) B E C D A (3) B A E D C (3) A C E D B (3) A B D E C (3) E B C A D (2) D C E B A (2) D C B E A (2) D A B C E (2) C E D B A (2) C E D A B (2) C E A D B (2) C D A E B (2) B E C A D (2) B D E A C (2) B D A E C (2) A D C B E (2) A C D E B (2) E C A B D (1) D C A E B (1) D C A B E (1) D B C A E (1) D A C E B (1) D A C B E (1) C A E D B (1) C A D E B (1) B E A D C (1) B E A C D (1) Total count = 100 A B C D E A 0 8 -4 -2 2 B -8 0 8 6 -6 C 4 -8 0 8 -6 D 2 -6 -8 0 -4 E -2 6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888875 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 8 -4 -2 2 B -8 0 8 6 -6 C 4 -8 0 8 -6 D 2 -6 -8 0 -4 E -2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888811 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=24 C=17 D=14 E=10 so E is eliminated. Round 2 votes counts: A=35 B=30 C=21 D=14 so D is eliminated. Round 3 votes counts: A=39 B=34 C=27 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:207 A:202 B:200 C:199 D:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -4 -2 2 B -8 0 8 6 -6 C 4 -8 0 8 -6 D 2 -6 -8 0 -4 E -2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888811 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -2 2 B -8 0 8 6 -6 C 4 -8 0 8 -6 D 2 -6 -8 0 -4 E -2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888811 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -2 2 B -8 0 8 6 -6 C 4 -8 0 8 -6 D 2 -6 -8 0 -4 E -2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.166667 D: 0.000000 E: 0.333333 Sum of squares = 0.388888888811 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8933: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) E D B A C (11) E D A C B (7) B C A D E (7) A C D E B (7) B E D C A (6) C A B E D (5) B D E C A (5) E B D A C (4) D E B A C (4) A C E D B (4) E B D C A (3) E A C D B (3) B C A E D (3) D E A C B (2) D B C A E (2) C A E B D (2) C A D B E (2) B D C A E (2) A C D B E (2) E C A B D (1) D B E A C (1) D A C E B (1) C A E D B (1) B E D A C (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -4 2 4 B -2 0 -2 4 -2 C 4 2 0 0 4 D -2 -4 0 0 -2 E -4 2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.760923 D: 0.239077 E: 0.000000 Sum of squares = 0.636162094 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.760923 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 2 4 B -2 0 -2 4 -2 C 4 2 0 0 4 D -2 -4 0 0 -2 E -4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555580153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 C=23 A=14 D=10 so D is eliminated. Round 2 votes counts: E=35 B=27 C=23 A=15 so A is eliminated. Round 3 votes counts: C=38 E=35 B=27 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:205 A:202 B:199 E:198 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 2 4 B -2 0 -2 4 -2 C 4 2 0 0 4 D -2 -4 0 0 -2 E -4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555580153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 2 4 B -2 0 -2 4 -2 C 4 2 0 0 4 D -2 -4 0 0 -2 E -4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555580153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 2 4 B -2 0 -2 4 -2 C 4 2 0 0 4 D -2 -4 0 0 -2 E -4 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555580153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8934: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (12) D C A B E (11) C D B E A (9) C B D E A (9) D A C E B (7) B E C A D (7) A E D B C (6) A E B D C (6) B C E D A (5) E B A C D (4) D C B A E (4) E A B C D (3) D A C B E (3) E B C A D (2) E A B D C (2) B E C D A (2) D C B E A (1) D C A E B (1) C B E D A (1) C A B D E (1) B E A C D (1) A E B C D (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 10 -6 -6 8 B -10 0 -20 -12 8 C 6 20 0 -10 6 D 6 12 10 0 20 E -8 -8 -6 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 -6 8 B -10 0 -20 -12 8 C 6 20 0 -10 6 D 6 12 10 0 20 E -8 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 C=20 B=15 E=11 so E is eliminated. Round 2 votes counts: A=32 D=27 B=21 C=20 so C is eliminated. Round 3 votes counts: D=36 A=33 B=31 so B is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:211 A:203 B:183 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -6 -6 8 B -10 0 -20 -12 8 C 6 20 0 -10 6 D 6 12 10 0 20 E -8 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -6 8 B -10 0 -20 -12 8 C 6 20 0 -10 6 D 6 12 10 0 20 E -8 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -6 8 B -10 0 -20 -12 8 C 6 20 0 -10 6 D 6 12 10 0 20 E -8 -8 -6 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8935: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) B D C E A (6) D B A C E (5) C E B A D (5) A D B E C (5) E C A B D (4) C E B D A (4) B C E D A (4) B C D E A (4) A E C B D (4) A D E B C (4) D A E C B (3) D A B E C (3) C B E A D (3) A E D C B (3) E C D B A (2) E C B A D (2) E A C D B (2) E A C B D (2) D B C E A (2) C B E D A (2) B C E A D (2) A E C D B (2) A C B E D (2) A B C D E (2) E D C A B (1) E D A C B (1) E C A D B (1) E A D C B (1) D E C B A (1) D E B C A (1) D E A C B (1) D B C A E (1) D B A E C (1) D A E B C (1) C E A B D (1) C B A E D (1) B D A C E (1) B A D C E (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 4 6 12 -4 B -4 0 -12 4 -8 C -6 12 0 0 -4 D -12 -4 0 0 0 E 4 8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.173334 E: 0.826666 Sum of squares = 0.713420729329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.173334 E: 1.000000 A B C D E A 0 4 6 12 -4 B -4 0 -12 4 -8 C -6 12 0 0 -4 D -12 -4 0 0 0 E 4 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000012001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=19 B=18 E=16 C=16 so E is eliminated. Round 2 votes counts: A=36 C=25 D=21 B=18 so B is eliminated. Round 3 votes counts: A=37 C=35 D=28 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:209 E:208 C:201 D:192 B:190 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 6 12 -4 B -4 0 -12 4 -8 C -6 12 0 0 -4 D -12 -4 0 0 0 E 4 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000012001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 12 -4 B -4 0 -12 4 -8 C -6 12 0 0 -4 D -12 -4 0 0 0 E 4 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000012001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 12 -4 B -4 0 -12 4 -8 C -6 12 0 0 -4 D -12 -4 0 0 0 E 4 8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 0.750000 Sum of squares = 0.625000012001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8936: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (13) C B D E A (9) E C A B D (8) C E A B D (7) B D C A E (6) E A C B D (5) E A D B C (4) D B A C E (4) C E B D A (4) C B D A E (4) A D E B C (4) A D B E C (4) E A C D B (3) B D C E A (3) C B E D A (2) C A D B E (2) A E C D B (2) E C B D A (1) E B D A C (1) E A D C B (1) E A B D C (1) D B C A E (1) D B A E C (1) D A B E C (1) D A B C E (1) C A E D B (1) B C D E A (1) B C D A E (1) A E D C B (1) A D C B E (1) A D B C E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 24 0 20 0 B -24 0 -6 6 -14 C 0 6 0 4 0 D -20 -6 -4 0 -8 E 0 14 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.186209 B: 0.000000 C: 0.495408 D: 0.000000 E: 0.318383 Sum of squares = 0.381470772499 Cumulative probabilities = A: 0.186209 B: 0.186209 C: 0.681617 D: 0.681617 E: 1.000000 A B C D E A 0 24 0 20 0 B -24 0 -6 6 -14 C 0 6 0 4 0 D -20 -6 -4 0 -8 E 0 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=28 E=24 B=11 D=8 so D is eliminated. Round 2 votes counts: A=30 C=29 E=24 B=17 so B is eliminated. Round 3 votes counts: C=41 A=35 E=24 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:211 C:205 B:181 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 24 0 20 0 B -24 0 -6 6 -14 C 0 6 0 4 0 D -20 -6 -4 0 -8 E 0 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 0 20 0 B -24 0 -6 6 -14 C 0 6 0 4 0 D -20 -6 -4 0 -8 E 0 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 0 20 0 B -24 0 -6 6 -14 C 0 6 0 4 0 D -20 -6 -4 0 -8 E 0 14 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8937: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) D B A C E (7) D A B C E (6) A D B E C (6) D B A E C (5) B E C D A (5) B D C E A (5) A D C E B (5) A E C D B (4) E C B A D (3) E C A B D (3) D B C A E (3) C E B D A (3) C E A B D (3) A E D C B (3) A D E C B (3) A D E B C (3) E C B D A (2) E A C B D (2) E A B C D (2) C E B A D (2) C E A D B (2) B C E D A (2) A D C B E (2) E B A D C (1) E A B D C (1) D A C B E (1) C B E D A (1) C A E D B (1) B E D C A (1) B D E C A (1) B D A E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 16 26 -4 22 B -16 0 16 -24 10 C -26 -16 0 -26 -10 D 4 24 26 0 16 E -22 -10 10 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999854 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 26 -4 22 B -16 0 16 -24 10 C -26 -16 0 -26 -10 D 4 24 26 0 16 E -22 -10 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 A=28 B=15 E=14 C=12 so C is eliminated. Round 2 votes counts: D=31 A=29 E=24 B=16 so B is eliminated. Round 3 votes counts: D=38 E=33 A=29 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:235 A:230 B:193 E:181 C:161 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 26 -4 22 B -16 0 16 -24 10 C -26 -16 0 -26 -10 D 4 24 26 0 16 E -22 -10 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 26 -4 22 B -16 0 16 -24 10 C -26 -16 0 -26 -10 D 4 24 26 0 16 E -22 -10 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 26 -4 22 B -16 0 16 -24 10 C -26 -16 0 -26 -10 D 4 24 26 0 16 E -22 -10 10 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998362 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8938: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (9) C B D A E (8) D E A B C (7) E A D B C (6) C B D E A (6) A E B D C (6) C B A D E (5) A E D B C (4) D C B E A (3) C B A E D (3) B D C A E (3) B C D A E (3) A E B C D (3) E D A C B (2) E A C D B (2) E A C B D (2) D E A C B (2) D B A E C (2) C E A B D (2) B C A D E (2) B A C E D (2) E D A B C (1) E C A D B (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B A C (1) D A E B C (1) C D E B A (1) C D B E A (1) C B E A D (1) B D A E C (1) B C A E D (1) B A E D C (1) B A E C D (1) B A D C E (1) A C B E D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 16 12 0 B -6 0 -2 10 -6 C -16 2 0 -10 -14 D -12 -10 10 0 0 E 0 6 14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.642590 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.357410 Sum of squares = 0.540663832537 Cumulative probabilities = A: 0.642590 B: 0.642590 C: 0.642590 D: 0.642590 E: 1.000000 A B C D E A 0 6 16 12 0 B -6 0 -2 10 -6 C -16 2 0 -10 -14 D -12 -10 10 0 0 E 0 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=24 D=18 A=16 B=15 so B is eliminated. Round 2 votes counts: C=33 E=24 D=22 A=21 so A is eliminated. Round 3 votes counts: E=40 C=36 D=24 so D is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:217 E:210 B:198 D:194 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 16 12 0 B -6 0 -2 10 -6 C -16 2 0 -10 -14 D -12 -10 10 0 0 E 0 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 16 12 0 B -6 0 -2 10 -6 C -16 2 0 -10 -14 D -12 -10 10 0 0 E 0 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 16 12 0 B -6 0 -2 10 -6 C -16 2 0 -10 -14 D -12 -10 10 0 0 E 0 6 14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8939: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) B C D E A (10) B C D A E (9) A E B C D (6) E A B C D (5) E D C B A (4) C B D A E (4) B C A D E (4) E D A C B (3) E A D C B (3) E A D B C (3) C D B E A (3) A D E C B (3) A C D B E (3) A B E C D (3) E B D C A (2) E A B D C (2) D A C B E (2) C D B A E (2) A E D B C (2) A B C D E (2) E B C D A (1) E B A C D (1) D C E B A (1) D C B A E (1) D C A B E (1) D A C E B (1) C B D E A (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C E A D (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 6 6 12 B -4 0 10 12 2 C -6 -10 0 20 -4 D -6 -12 -20 0 -2 E -12 -2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 6 6 12 B -4 0 10 12 2 C -6 -10 0 20 -4 D -6 -12 -20 0 -2 E -12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=28 E=24 C=10 D=6 so D is eliminated. Round 2 votes counts: A=35 B=28 E=24 C=13 so C is eliminated. Round 3 votes counts: B=39 A=36 E=25 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:210 C:200 E:196 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 6 6 12 B -4 0 10 12 2 C -6 -10 0 20 -4 D -6 -12 -20 0 -2 E -12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 6 6 12 B -4 0 10 12 2 C -6 -10 0 20 -4 D -6 -12 -20 0 -2 E -12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 6 6 12 B -4 0 10 12 2 C -6 -10 0 20 -4 D -6 -12 -20 0 -2 E -12 -2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999378 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8940: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (8) E D A C B (7) C A B E D (7) B C D A E (7) E D A B C (6) A C E B D (5) E D B A C (4) D B E C A (4) C A E B D (4) B D E A C (4) B C A D E (4) A C E D B (4) D E B A C (3) C A B D E (3) B D E C A (3) A E C D B (3) E A D C B (2) E A D B C (2) E A B D C (2) D E C A B (2) D E B C A (2) C A E D B (2) B D C E A (2) B C D E A (2) D C E B A (1) C B D A E (1) C A D E B (1) B A E D C (1) A E D C B (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 -6 4 8 B -8 0 -6 12 -4 C 6 6 0 8 6 D -4 -12 -8 0 -6 E -8 4 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -6 4 8 B -8 0 -6 12 -4 C 6 6 0 8 6 D -4 -12 -8 0 -6 E -8 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=23 B=23 A=16 D=12 so D is eliminated. Round 2 votes counts: E=30 C=27 B=27 A=16 so A is eliminated. Round 3 votes counts: C=37 E=35 B=28 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 A:207 E:198 B:197 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -6 4 8 B -8 0 -6 12 -4 C 6 6 0 8 6 D -4 -12 -8 0 -6 E -8 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 4 8 B -8 0 -6 12 -4 C 6 6 0 8 6 D -4 -12 -8 0 -6 E -8 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 4 8 B -8 0 -6 12 -4 C 6 6 0 8 6 D -4 -12 -8 0 -6 E -8 4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8941: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) C D E A B (6) B D A E C (6) B A E D C (6) D C E A B (5) D B E A C (5) C B A E D (5) A E B C D (5) D C E B A (4) D B C E A (4) D C B E A (3) B C A E D (3) B A E C D (3) B A C E D (3) E C A D B (2) E A C D B (2) D E C A B (2) D E A C B (2) C B D A E (2) C A E D B (2) B D C A E (2) B D A C E (2) A B E C D (2) E A D B C (1) E A C B D (1) E A B D C (1) D E B C A (1) D C B A E (1) D B C A E (1) C E D A B (1) C E A D B (1) C E A B D (1) C B A D E (1) C A E B D (1) B C D A E (1) B A D E C (1) A E C B D (1) A E B D C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 2 -10 2 B 8 0 12 0 4 C -2 -12 0 -12 0 D 10 0 12 0 12 E -2 -4 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.416908 C: 0.000000 D: 0.583092 E: 0.000000 Sum of squares = 0.513808697046 Cumulative probabilities = A: 0.000000 B: 0.416908 C: 0.416908 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -10 2 B 8 0 12 0 4 C -2 -12 0 -12 0 D 10 0 12 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=27 C=20 A=11 E=7 so E is eliminated. Round 2 votes counts: D=35 B=27 C=22 A=16 so A is eliminated. Round 3 votes counts: B=37 D=36 C=27 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:217 B:212 A:193 E:191 C:187 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -10 2 B 8 0 12 0 4 C -2 -12 0 -12 0 D 10 0 12 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -10 2 B 8 0 12 0 4 C -2 -12 0 -12 0 D 10 0 12 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -10 2 B 8 0 12 0 4 C -2 -12 0 -12 0 D 10 0 12 0 12 E -2 -4 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8942: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (10) C E B D A (8) D E B A C (6) E D C B A (5) A B C D E (5) E C D B A (4) D E A B C (4) D A E B C (4) C E D B A (4) C E D A B (4) B A D E C (4) A B D C E (4) D E C A B (3) C B E A D (3) C A B E D (3) B C A E D (3) E B D C A (2) D E A C B (2) D B A E C (2) C B A E D (2) C A E D B (2) A D B E C (2) A C B E D (2) E D B C A (1) D E B C A (1) D A B E C (1) C E A D B (1) B E D A C (1) B E C D A (1) B E A D C (1) B D E A C (1) B C E A D (1) B A C E D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 4 -8 -6 B 2 0 10 6 -4 C -4 -10 0 -8 -10 D 8 -6 8 0 2 E 6 4 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888962 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 A B C D E A 0 -2 4 -8 -6 B 2 0 10 6 -4 C -4 -10 0 -8 -10 D 8 -6 8 0 2 E 6 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=25 D=23 B=13 E=12 so E is eliminated. Round 2 votes counts: C=31 D=29 A=25 B=15 so B is eliminated. Round 3 votes counts: C=36 D=33 A=31 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:209 B:207 D:206 A:194 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 -8 -6 B 2 0 10 6 -4 C -4 -10 0 -8 -10 D 8 -6 8 0 2 E 6 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -8 -6 B 2 0 10 6 -4 C -4 -10 0 -8 -10 D 8 -6 8 0 2 E 6 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -8 -6 B 2 0 10 6 -4 C -4 -10 0 -8 -10 D 8 -6 8 0 2 E 6 4 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.000000 D: 0.333333 E: 0.500000 Sum of squares = 0.388888888979 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.166667 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8943: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) C D B A E (9) A E D C B (9) D C B A E (8) B C D E A (8) E A B C D (5) D C A B E (5) A E D B C (4) A E C D B (4) A D C E B (4) C D A B E (3) C B D E A (3) B E A C D (3) B D C E A (3) C D A E B (2) B E A D C (2) B C E D A (2) E B C A D (1) E B A D C (1) D C A E B (1) D B C A E (1) D A C E B (1) C D B E A (1) C A D E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B E C A D (1) B D A E C (1) A E B D C (1) A C D E B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -2 -2 12 B -6 0 -4 -8 8 C 2 4 0 -10 6 D 2 8 10 0 6 E -12 -8 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999831 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 -2 12 B -6 0 -4 -8 8 C 2 4 0 -10 6 D 2 8 10 0 6 E -12 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 C=19 E=18 D=16 so D is eliminated. Round 2 votes counts: C=33 A=25 B=24 E=18 so E is eliminated. Round 3 votes counts: A=41 C=33 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:213 A:207 C:201 B:195 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -2 -2 12 B -6 0 -4 -8 8 C 2 4 0 -10 6 D 2 8 10 0 6 E -12 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 -2 12 B -6 0 -4 -8 8 C 2 4 0 -10 6 D 2 8 10 0 6 E -12 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 -2 12 B -6 0 -4 -8 8 C 2 4 0 -10 6 D 2 8 10 0 6 E -12 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8944: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (6) C E A D B (5) C E A B D (5) B D A E C (5) D B A E C (4) D A C B E (4) C D B E A (4) B E C D A (4) B E A D C (4) B A E D C (4) D C A E B (3) D C A B E (3) C E B A D (3) C D E B A (3) C D A E B (3) B E A C D (3) E C A B D (2) E B C A D (2) E A B C D (2) D B A C E (2) D A B C E (2) B E D A C (2) B E C A D (2) B D E A C (2) B D C E A (2) A D E B C (2) A D C E B (2) A C E D B (2) A C D E B (2) E B A D C (1) E B A C D (1) E A C B D (1) D A C E B (1) C B E D A (1) C A E D B (1) C A D E B (1) B E D C A (1) A E D C B (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 0 10 -4 0 B 0 0 4 -4 10 C -10 -4 0 -4 -2 D 4 4 4 0 2 E 0 -10 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 -4 0 B 0 0 4 -4 10 C -10 -4 0 -4 -2 D 4 4 4 0 2 E 0 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=26 D=25 A=11 E=9 so E is eliminated. Round 2 votes counts: B=33 C=28 D=25 A=14 so A is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:207 B:205 A:203 E:195 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 10 -4 0 B 0 0 4 -4 10 C -10 -4 0 -4 -2 D 4 4 4 0 2 E 0 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 -4 0 B 0 0 4 -4 10 C -10 -4 0 -4 -2 D 4 4 4 0 2 E 0 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 -4 0 B 0 0 4 -4 10 C -10 -4 0 -4 -2 D 4 4 4 0 2 E 0 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8945: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (12) B E D A C (6) C D B E A (5) A E B C D (5) D C B E A (4) C D A E B (4) C A D E B (4) A E C B D (4) A C E D B (4) A C E B D (4) A B E C D (4) E B D A C (3) E B A D C (3) D C E B A (3) D B E C A (3) D B C E A (3) D E C B A (2) C D E A B (2) C D B A E (2) C A D B E (2) C A B D E (2) B D E C A (2) B D C E A (2) A B E D C (2) E D B A C (1) E A B D C (1) D E B C A (1) C D A B E (1) C B D E A (1) C B A D E (1) C A E D B (1) B D E A C (1) B A E D C (1) A E D B C (1) A E C D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 12 10 8 14 B -12 0 4 12 -12 C -10 -4 0 -2 -6 D -8 -12 2 0 -10 E -14 12 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 8 14 B -12 0 4 12 -12 C -10 -4 0 -2 -6 D -8 -12 2 0 -10 E -14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=25 D=16 B=12 E=8 so E is eliminated. Round 2 votes counts: A=40 C=25 B=18 D=17 so D is eliminated. Round 3 votes counts: A=40 C=34 B=26 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:222 E:207 B:196 C:189 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 8 14 B -12 0 4 12 -12 C -10 -4 0 -2 -6 D -8 -12 2 0 -10 E -14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 8 14 B -12 0 4 12 -12 C -10 -4 0 -2 -6 D -8 -12 2 0 -10 E -14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 8 14 B -12 0 4 12 -12 C -10 -4 0 -2 -6 D -8 -12 2 0 -10 E -14 12 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8946: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (8) A B D C E (8) E D B C A (7) E D B A C (7) C A E D B (7) A C B D E (7) C E A D B (5) E B D C A (4) C E B D A (4) C A B D E (4) D B A E C (3) C A B E D (3) A C D B E (3) E C B D A (2) D B E A C (2) C E D B A (2) B D E A C (2) B A C D E (2) A D B C E (2) A B D E C (2) E D A B C (1) E C D B A (1) E B D A C (1) E A D C B (1) C B E D A (1) C B D E A (1) C B D A E (1) C B A D E (1) B E D A C (1) B D E C A (1) B A D C E (1) A E D C B (1) A E C D B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 12 2 14 B 4 0 6 8 8 C -12 -6 0 -6 8 D -2 -8 6 0 0 E -14 -8 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999702 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 2 14 B 4 0 6 8 8 C -12 -6 0 -6 8 D -2 -8 6 0 0 E -14 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=27 E=24 B=15 D=5 so D is eliminated. Round 2 votes counts: C=29 A=27 E=24 B=20 so B is eliminated. Round 3 votes counts: A=41 E=30 C=29 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:213 A:212 D:198 C:192 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 2 14 B 4 0 6 8 8 C -12 -6 0 -6 8 D -2 -8 6 0 0 E -14 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 2 14 B 4 0 6 8 8 C -12 -6 0 -6 8 D -2 -8 6 0 0 E -14 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 2 14 B 4 0 6 8 8 C -12 -6 0 -6 8 D -2 -8 6 0 0 E -14 -8 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8947: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (5) E A D C B (4) C D B E A (4) B D C E A (4) B A E D C (4) A E C D B (4) E D A C B (3) D E C A B (3) D C E B A (3) D B C E A (3) C B D A E (3) C B A D E (3) C A E D B (3) B D E C A (3) B D A E C (3) B A C D E (3) A E B D C (3) A C B E D (3) A B C E D (3) E D A B C (2) E A D B C (2) D E B C A (2) D B E C A (2) D B E A C (2) C D E A B (2) B D C A E (2) B D A C E (2) B C D A E (2) A E C B D (2) A B E C D (2) E D C A B (1) E B A D C (1) D E C B A (1) D C B E A (1) C E D A B (1) C D A E B (1) C B D E A (1) C A B E D (1) B D E A C (1) B C A E D (1) B C A D E (1) B A D C E (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 4 -4 8 B 16 0 10 12 22 C -4 -10 0 -8 8 D 4 -12 8 0 6 E -8 -22 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 -4 8 B 16 0 10 12 22 C -4 -10 0 -8 8 D 4 -12 8 0 6 E -8 -22 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=19 A=19 D=17 E=13 so E is eliminated. Round 2 votes counts: B=33 A=25 D=23 C=19 so C is eliminated. Round 3 votes counts: B=40 D=31 A=29 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:230 D:203 A:196 C:193 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 4 -4 8 B 16 0 10 12 22 C -4 -10 0 -8 8 D 4 -12 8 0 6 E -8 -22 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 -4 8 B 16 0 10 12 22 C -4 -10 0 -8 8 D 4 -12 8 0 6 E -8 -22 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 -4 8 B 16 0 10 12 22 C -4 -10 0 -8 8 D 4 -12 8 0 6 E -8 -22 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8948: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) C B D E A (8) A E B D C (8) A E D B C (7) E A B D C (6) D A C E B (5) B E C A D (5) A E B C D (5) D C A B E (4) D C B E A (3) D A E C B (3) D A E B C (3) C B E D A (3) B C E A D (3) A E D C B (3) A D E C B (3) E A D B C (2) D C B A E (2) D B C E A (2) C D B E A (2) C B E A D (2) A D E B C (2) A D C E B (2) E D B A C (1) E A B C D (1) D E B C A (1) D C A E B (1) C D B A E (1) C D A B E (1) B E C D A (1) B C E D A (1) B C D E A (1) Total count = 100 A B C D E A 0 12 18 14 0 B -12 0 14 4 -22 C -18 -14 0 -16 -18 D -14 -4 16 0 -12 E 0 22 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.729726 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.270274 Sum of squares = 0.605548399605 Cumulative probabilities = A: 0.729726 B: 0.729726 C: 0.729726 D: 0.729726 E: 1.000000 A B C D E A 0 12 18 14 0 B -12 0 14 4 -22 C -18 -14 0 -16 -18 D -14 -4 16 0 -12 E 0 22 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 E=18 C=17 B=11 so B is eliminated. Round 2 votes counts: A=30 E=24 D=24 C=22 so C is eliminated. Round 3 votes counts: D=37 E=33 A=30 so A is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 A:222 D:193 B:192 C:167 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 12 18 14 0 B -12 0 14 4 -22 C -18 -14 0 -16 -18 D -14 -4 16 0 -12 E 0 22 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 18 14 0 B -12 0 14 4 -22 C -18 -14 0 -16 -18 D -14 -4 16 0 -12 E 0 22 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 18 14 0 B -12 0 14 4 -22 C -18 -14 0 -16 -18 D -14 -4 16 0 -12 E 0 22 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8949: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (15) A E D B C (10) D A E B C (8) D A E C B (5) B C E A D (5) E A B C D (4) D B C A E (4) C B E A D (4) C B D A E (3) B E A C D (3) A E D C B (3) A D E C B (3) E B A C D (2) E A D C B (2) E A D B C (2) E A C B D (2) E A B D C (2) D C B A E (2) D C A E B (2) C E B A D (2) C E A B D (2) C D B A E (2) B D C A E (2) B C E D A (2) B C D E A (2) E C A B D (1) D C A B E (1) D B A E C (1) D A C E B (1) D A B E C (1) C D A E B (1) B C D A E (1) Total count = 100 A B C D E A 0 0 -2 -6 0 B 0 0 -2 4 -4 C 2 2 0 2 2 D 6 -4 -2 0 8 E 0 4 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -6 0 B 0 0 -2 4 -4 C 2 2 0 2 2 D 6 -4 -2 0 8 E 0 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 A=16 E=15 B=15 so E is eliminated. Round 2 votes counts: C=30 A=28 D=25 B=17 so B is eliminated. Round 3 votes counts: C=40 A=33 D=27 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:204 D:204 B:199 E:197 A:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 -6 0 B 0 0 -2 4 -4 C 2 2 0 2 2 D 6 -4 -2 0 8 E 0 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -6 0 B 0 0 -2 4 -4 C 2 2 0 2 2 D 6 -4 -2 0 8 E 0 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -6 0 B 0 0 -2 4 -4 C 2 2 0 2 2 D 6 -4 -2 0 8 E 0 4 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8950: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) C D A E B (6) B E C A D (6) E D B A C (5) D A C E B (5) D A E C B (4) B E A D C (4) A D E C B (4) D E A B C (3) C D A B E (3) C A D B E (3) C A B D E (3) B E A C D (3) B C E D A (3) B C E A D (3) A E D B C (3) A D C E B (3) E B D C A (2) E B A D C (2) D A E B C (2) C B D E A (2) C B D A E (2) A C D B E (2) E D B C A (1) E D A B C (1) E A D B C (1) D E C B A (1) D E B C A (1) D E A C B (1) D C E A B (1) C D B E A (1) C D B A E (1) C B E A D (1) C B A E D (1) C A D E B (1) B E D A C (1) B E C D A (1) B A E C D (1) A D E B C (1) A C D E B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 14 -10 -4 B 0 0 6 -10 -14 C -14 -6 0 -8 -14 D 10 10 8 0 2 E 4 14 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 -10 -4 B 0 0 6 -10 -14 C -14 -6 0 -8 -14 D 10 10 8 0 2 E 4 14 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=22 E=20 D=18 A=16 so A is eliminated. Round 2 votes counts: C=28 D=26 E=23 B=23 so E is eliminated. Round 3 votes counts: D=37 B=35 C=28 so C is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 E:215 A:200 B:191 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 14 -10 -4 B 0 0 6 -10 -14 C -14 -6 0 -8 -14 D 10 10 8 0 2 E 4 14 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 -10 -4 B 0 0 6 -10 -14 C -14 -6 0 -8 -14 D 10 10 8 0 2 E 4 14 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 -10 -4 B 0 0 6 -10 -14 C -14 -6 0 -8 -14 D 10 10 8 0 2 E 4 14 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999401 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8951: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) A B E C D (8) A E B D C (7) E D C A B (6) E A B D C (6) C D B E A (6) B A C D E (6) E D C B A (4) D C E A B (4) C D B A E (4) E B A D C (3) E A D C B (3) C D A B E (3) A C D B E (3) B E A D C (2) B C D E A (2) B C A D E (2) B A E C D (2) A E B C D (2) E B D C A (1) E B D A C (1) E A D B C (1) D E C B A (1) D C B E A (1) D C A E B (1) C D E B A (1) C D E A B (1) C D A E B (1) C A D E B (1) B D E C A (1) B D C E A (1) B C D A E (1) B A E D C (1) A E D B C (1) A E C D B (1) A C D E B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 4 0 4 -6 B -4 0 -2 -4 -10 C 0 2 0 -8 -4 D -4 4 8 0 0 E 6 10 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.322676 E: 0.677324 Sum of squares = 0.562887940568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.322676 E: 1.000000 A B C D E A 0 4 0 4 -6 B -4 0 -2 -4 -10 C 0 2 0 -8 -4 D -4 4 8 0 0 E 6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 B=18 C=17 D=15 so D is eliminated. Round 2 votes counts: C=31 E=26 A=25 B=18 so B is eliminated. Round 3 votes counts: C=37 A=34 E=29 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:210 D:204 A:201 C:195 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 0 4 -6 B -4 0 -2 -4 -10 C 0 2 0 -8 -4 D -4 4 8 0 0 E 6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 4 -6 B -4 0 -2 -4 -10 C 0 2 0 -8 -4 D -4 4 8 0 0 E 6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 4 -6 B -4 0 -2 -4 -10 C 0 2 0 -8 -4 D -4 4 8 0 0 E 6 10 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8952: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D B E A C (9) B E A D C (9) B D E A C (9) C D A E B (8) A E B C D (7) C A E D B (6) D B C E A (5) A E C B D (5) E A B D C (4) D C B A E (4) C A E B D (4) E A B C D (3) C D B A E (3) B E A C D (3) B A E C D (2) E A D C B (1) D C A B E (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D E B (1) B E D A C (1) B C A E D (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 10 -2 -6 B 14 0 6 2 18 C -10 -6 0 -4 -8 D 2 -2 4 0 2 E 6 -18 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 -2 -6 B 14 0 6 2 18 C -10 -6 0 -4 -8 D 2 -2 4 0 2 E 6 -18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989852 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 B=25 A=14 E=8 so E is eliminated. Round 2 votes counts: D=28 C=25 B=25 A=22 so A is eliminated. Round 3 votes counts: B=39 C=32 D=29 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:203 E:197 A:194 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 -2 -6 B 14 0 6 2 18 C -10 -6 0 -4 -8 D 2 -2 4 0 2 E 6 -18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989852 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 -2 -6 B 14 0 6 2 18 C -10 -6 0 -4 -8 D 2 -2 4 0 2 E 6 -18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989852 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 -2 -6 B 14 0 6 2 18 C -10 -6 0 -4 -8 D 2 -2 4 0 2 E 6 -18 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989852 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8953: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) A D E B C (6) E B D A C (5) C B E D A (5) A D C B E (5) D A C B E (4) D A B C E (4) A D C E B (4) D B E A C (3) C E A B D (3) C D B A E (3) B E D C A (3) B E C D A (3) A D E C B (3) A C D E B (3) E B C D A (2) E B C A D (2) D B C A E (2) D B A E C (2) C E B A D (2) C D A B E (2) C A D B E (2) C A B D E (2) B D E C A (2) A E D C B (2) E D B A C (1) E C B A D (1) E C A B D (1) E B A D C (1) E A C B D (1) E A B D C (1) E A B C D (1) D C B A E (1) C E B D A (1) C B D E A (1) C B D A E (1) C A E D B (1) C A E B D (1) B C E D A (1) B C D E A (1) A E D B C (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 14 14 -10 18 B -14 0 0 -18 14 C -14 0 0 -18 0 D 10 18 18 0 22 E -18 -14 0 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 -10 18 B -14 0 0 -18 14 C -14 0 0 -18 0 D 10 18 18 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 C=24 E=16 B=10 so B is eliminated. Round 2 votes counts: D=26 C=26 A=26 E=22 so E is eliminated. Round 3 votes counts: D=35 C=35 A=30 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:234 A:218 B:191 C:184 E:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 14 -10 18 B -14 0 0 -18 14 C -14 0 0 -18 0 D 10 18 18 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 -10 18 B -14 0 0 -18 14 C -14 0 0 -18 0 D 10 18 18 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 -10 18 B -14 0 0 -18 14 C -14 0 0 -18 0 D 10 18 18 0 22 E -18 -14 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8954: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (13) C B E D A (9) C B E A D (9) D A B E C (6) A D E B C (6) E D A C B (5) E A D C B (5) D A E B C (5) D E A C B (4) B D A C E (4) B C A D E (4) E C A D B (3) C E A D B (3) A E D B C (3) B C A E D (2) A D B E C (2) E A C D B (1) D B C A E (1) D B A E C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A B D (1) C B D E A (1) B D C A E (1) B C E D A (1) B C D E A (1) B A D E C (1) B A D C E (1) A E D C B (1) A D E C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -4 -10 8 B 2 0 8 2 18 C 4 -8 0 2 8 D 10 -2 -2 0 6 E -8 -18 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998552 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -10 8 B 2 0 8 2 18 C 4 -8 0 2 8 D 10 -2 -2 0 6 E -8 -18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998073 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=26 D=17 A=15 E=14 so E is eliminated. Round 2 votes counts: C=29 B=28 D=22 A=21 so A is eliminated. Round 3 votes counts: D=40 C=30 B=30 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:206 C:203 A:196 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -10 8 B 2 0 8 2 18 C 4 -8 0 2 8 D 10 -2 -2 0 6 E -8 -18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998073 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -10 8 B 2 0 8 2 18 C 4 -8 0 2 8 D 10 -2 -2 0 6 E -8 -18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998073 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -10 8 B 2 0 8 2 18 C 4 -8 0 2 8 D 10 -2 -2 0 6 E -8 -18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998073 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8955: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (15) D E A B C (11) D E A C B (8) D E C A B (7) C B A E D (6) B C D E A (6) A E D C B (5) E D A B C (4) C D E A B (4) D E C B A (3) C A B E D (3) B C A D E (3) B A C E D (3) A C B E D (3) D E B A C (2) C A E D B (2) B D E A C (2) E D A C B (1) E A D C B (1) E A D B C (1) D B E C A (1) C D A E B (1) C B D E A (1) C B A D E (1) C A D E B (1) C A B D E (1) B D E C A (1) B A E D C (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 10 -12 -4 -6 B -10 0 2 -6 -6 C 12 -2 0 2 2 D 4 6 -2 0 6 E 6 6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999993 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -12 -4 -6 B -10 0 2 -6 -6 C 12 -2 0 2 2 D 4 6 -2 0 6 E 6 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999977 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=31 C=20 A=10 E=7 so E is eliminated. Round 2 votes counts: D=37 B=31 C=20 A=12 so A is eliminated. Round 3 votes counts: D=45 B=31 C=24 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:207 D:207 E:202 A:194 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -12 -4 -6 B -10 0 2 -6 -6 C 12 -2 0 2 2 D 4 6 -2 0 6 E 6 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999977 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 -4 -6 B -10 0 2 -6 -6 C 12 -2 0 2 2 D 4 6 -2 0 6 E 6 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999977 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 -4 -6 B -10 0 2 -6 -6 C 12 -2 0 2 2 D 4 6 -2 0 6 E 6 6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999977 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8956: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) C E A D B (7) D E C A B (6) B A C E D (6) A B C E D (6) E C B D A (5) A C E B D (5) E C D B A (4) D E C B A (4) B C E A D (4) B A D C E (4) A B D C E (4) D E B C A (3) D B E A C (3) C E B A D (3) A C E D B (3) D E A C B (2) D B E C A (2) C E A B D (2) B E C D A (2) B C E D A (2) A D C B E (2) A D B C E (2) A C B E D (2) A B C D E (2) E D C B A (1) E C D A B (1) D B A E C (1) D A E C B (1) C E D A B (1) C E B D A (1) B D E C A (1) A B D E C (1) Total count = 100 A B C D E A 0 8 2 6 -8 B -8 0 0 0 2 C -2 0 0 12 12 D -6 0 -12 0 -10 E 8 -2 -12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.090909 Sum of squares = 0.438016528909 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 8 2 6 -8 B -8 0 0 0 2 C -2 0 0 12 12 D -6 0 -12 0 -10 E 8 -2 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.090909 Sum of squares = 0.438016528904 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 A=27 B=19 C=14 E=11 so E is eliminated. Round 2 votes counts: D=30 A=27 C=24 B=19 so B is eliminated. Round 3 votes counts: A=37 C=32 D=31 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:204 E:202 B:197 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 2 6 -8 B -8 0 0 0 2 C -2 0 0 12 12 D -6 0 -12 0 -10 E 8 -2 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.090909 Sum of squares = 0.438016528904 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 6 -8 B -8 0 0 0 2 C -2 0 0 12 12 D -6 0 -12 0 -10 E 8 -2 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.090909 Sum of squares = 0.438016528904 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 6 -8 B -8 0 0 0 2 C -2 0 0 12 12 D -6 0 -12 0 -10 E 8 -2 -12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.545455 B: 0.000000 C: 0.363636 D: 0.000000 E: 0.090909 Sum of squares = 0.438016528904 Cumulative probabilities = A: 0.545455 B: 0.545455 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8957: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) C E A D B (9) C E A B D (9) E A C B D (7) B D A E C (6) A E B D C (6) C E D A B (5) A E B C D (5) D C B E A (4) A B E D C (4) D B C E A (3) D B A E C (3) D B A C E (3) C D E B A (3) C D B E A (3) C B D E A (2) B D C A E (2) B A D E C (2) A E C B D (2) E C A B D (1) D E B A C (1) D E A B C (1) D A E B C (1) C E B D A (1) C D E A B (1) C B A E D (1) B D A C E (1) B C D A E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -14 -6 0 B -4 0 4 2 -4 C 14 -4 0 2 20 D 6 -2 -2 0 -2 E 0 4 -20 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380109 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 -6 0 B -4 0 4 2 -4 C 14 -4 0 2 20 D 6 -2 -2 0 -2 E 0 4 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380152 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=28 A=18 B=12 E=8 so E is eliminated. Round 2 votes counts: C=35 D=28 A=25 B=12 so B is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 D:200 B:199 E:193 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -14 -6 0 B -4 0 4 2 -4 C 14 -4 0 2 20 D 6 -2 -2 0 -2 E 0 4 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380152 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 -6 0 B -4 0 4 2 -4 C 14 -4 0 2 20 D 6 -2 -2 0 -2 E 0 4 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380152 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 -6 0 B -4 0 4 2 -4 C 14 -4 0 2 20 D 6 -2 -2 0 -2 E 0 4 -20 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.636364 C: 0.181818 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380152 Cumulative probabilities = A: 0.181818 B: 0.818182 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8958: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (15) E C A D B (11) D B C E A (10) B D A C E (10) C E D B A (9) B A D E C (8) D C B E A (6) A E B C D (5) A B E C D (5) D C E B A (4) C E D A B (4) A B D E C (4) E A C D B (2) C D E B A (2) A B E D C (2) E C D A B (1) D B C A E (1) B D A E C (1) Total count = 100 A B C D E A 0 -2 4 4 2 B 2 0 -8 0 -6 C -4 8 0 8 -8 D -4 0 -8 0 -8 E -2 6 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.44000000001 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 -2 4 4 2 B 2 0 -8 0 -6 C -4 8 0 8 -8 D -4 0 -8 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=21 B=19 C=15 E=14 so E is eliminated. Round 2 votes counts: A=33 C=27 D=21 B=19 so B is eliminated. Round 3 votes counts: A=41 D=32 C=27 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:210 A:204 C:202 B:194 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 4 2 B 2 0 -8 0 -6 C -4 8 0 8 -8 D -4 0 -8 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 4 2 B 2 0 -8 0 -6 C -4 8 0 8 -8 D -4 0 -8 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 4 2 B 2 0 -8 0 -6 C -4 8 0 8 -8 D -4 0 -8 0 -8 E -2 6 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.000000 E: 0.200000 Sum of squares = 0.439999999983 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8959: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (13) D C A B E (8) C D E B A (8) A D B E C (7) E B C A D (6) D A C B E (6) D C A E B (5) C E B D A (5) B E A C D (4) A D C B E (4) D A C E B (3) C E D B A (3) C D B E A (3) B E C A D (3) E B C D A (2) C B E D A (2) C B D E A (2) B E C D A (2) A D C E B (2) A D B C E (2) E C B D A (1) E C A B D (1) D C B A E (1) C D E A B (1) B E A D C (1) B A E D C (1) A E B D C (1) A D E C B (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 -6 -4 12 B -12 0 -12 -10 20 C 6 12 0 -14 10 D 4 10 14 0 10 E -12 -20 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 -4 12 B -12 0 -12 -10 20 C 6 12 0 -14 10 D 4 10 14 0 10 E -12 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=24 D=23 B=11 E=10 so E is eliminated. Round 2 votes counts: A=32 C=26 D=23 B=19 so B is eliminated. Round 3 votes counts: C=39 A=38 D=23 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:219 A:207 C:207 B:193 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -6 -4 12 B -12 0 -12 -10 20 C 6 12 0 -14 10 D 4 10 14 0 10 E -12 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -4 12 B -12 0 -12 -10 20 C 6 12 0 -14 10 D 4 10 14 0 10 E -12 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -4 12 B -12 0 -12 -10 20 C 6 12 0 -14 10 D 4 10 14 0 10 E -12 -20 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999644 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8960: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (7) A E C B D (7) E A B D C (6) E A B C D (5) A C E D B (5) E B A D C (4) D C B E A (4) B E A D C (4) A C B E D (4) E D B A C (3) E A C D B (3) D E B C A (3) C D A B E (3) C A D B E (3) B D C A E (3) E D A B C (2) E B D A C (2) E A D C B (2) E A D B C (2) D B C E A (2) B E D A C (2) B C D A E (2) B A E C D (2) A E B C D (2) A C E B D (2) E D B C A (1) E C D A B (1) E A C B D (1) D E C B A (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C A E (1) C D A E B (1) C A E D B (1) B D E A C (1) B D C E A (1) B C A D E (1) B A C E D (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 4 24 14 -4 B -4 0 2 2 -12 C -24 -2 0 6 -12 D -14 -2 -6 0 -26 E 4 12 12 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 24 14 -4 B -4 0 2 2 -12 C -24 -2 0 6 -12 D -14 -2 -6 0 -26 E 4 12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=22 B=17 C=15 D=14 so D is eliminated. Round 2 votes counts: E=36 A=22 C=21 B=21 so C is eliminated. Round 3 votes counts: E=37 B=33 A=30 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:227 A:219 B:194 C:184 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 24 14 -4 B -4 0 2 2 -12 C -24 -2 0 6 -12 D -14 -2 -6 0 -26 E 4 12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 24 14 -4 B -4 0 2 2 -12 C -24 -2 0 6 -12 D -14 -2 -6 0 -26 E 4 12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 24 14 -4 B -4 0 2 2 -12 C -24 -2 0 6 -12 D -14 -2 -6 0 -26 E 4 12 12 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998826 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8961: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (13) D B A C E (10) B D E C A (10) C A E B D (8) B D C A E (7) E C A B D (5) C E A B D (5) D B A E C (4) E C B A D (3) E A C D B (3) D B E A C (3) D A B C E (3) B D C E A (3) A E C D B (3) E B C D A (2) D A B E C (2) B C D A E (2) A C D E B (2) E A D C B (1) E A C B D (1) D E A B C (1) C E B A D (1) C B E A D (1) C B D A E (1) C A B E D (1) B E D C A (1) B E C D A (1) B D E A C (1) A E D C B (1) A D C E B (1) Total count = 100 A B C D E A 0 0 -2 -2 16 B 0 0 0 6 0 C 2 0 0 4 16 D 2 -6 -4 0 0 E -16 0 -16 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.479607 C: 0.520393 D: 0.000000 E: 0.000000 Sum of squares = 0.500831727495 Cumulative probabilities = A: 0.000000 B: 0.479607 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -2 16 B 0 0 0 6 0 C 2 0 0 4 16 D 2 -6 -4 0 0 E -16 0 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 D=23 A=20 C=17 E=15 so E is eliminated. Round 2 votes counts: B=27 C=25 A=25 D=23 so D is eliminated. Round 3 votes counts: B=44 A=31 C=25 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:206 B:203 D:196 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -2 16 B 0 0 0 6 0 C 2 0 0 4 16 D 2 -6 -4 0 0 E -16 0 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 16 B 0 0 0 6 0 C 2 0 0 4 16 D 2 -6 -4 0 0 E -16 0 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 16 B 0 0 0 6 0 C 2 0 0 4 16 D 2 -6 -4 0 0 E -16 0 -16 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999978 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8962: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (15) C A B E D (10) D E C B A (7) C D B A E (7) C B A D E (7) E D A B C (6) B A C E D (5) E D A C B (4) D E A B C (3) E B D A C (2) E A C B D (2) D B E A C (2) D B C E A (2) A C E B D (2) E D B A C (1) E C A D B (1) E C A B D (1) E B A D C (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) E A B C D (1) D E A C B (1) D C E B A (1) D C E A B (1) D B C A E (1) C E A D B (1) C D A B E (1) C B D A E (1) C A E D B (1) C A E B D (1) C A B D E (1) B E A D C (1) B D C A E (1) B D A E C (1) B A E D C (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 10 -14 -14 B 12 0 -6 -16 -14 C -10 6 0 -8 -12 D 14 16 8 0 4 E 14 14 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -14 -14 B 12 0 -6 -16 -14 C -10 6 0 -8 -12 D 14 16 8 0 4 E 14 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=30 E=23 B=9 A=5 so A is eliminated. Round 2 votes counts: D=33 C=33 E=24 B=10 so B is eliminated. Round 3 votes counts: C=38 D=35 E=27 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:218 B:188 C:188 A:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 10 -14 -14 B 12 0 -6 -16 -14 C -10 6 0 -8 -12 D 14 16 8 0 4 E 14 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -14 -14 B 12 0 -6 -16 -14 C -10 6 0 -8 -12 D 14 16 8 0 4 E 14 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -14 -14 B 12 0 -6 -16 -14 C -10 6 0 -8 -12 D 14 16 8 0 4 E 14 14 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8963: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (8) D E A B C (7) C B A E D (7) C B A D E (7) E D A B C (5) E B A D C (4) D A E B C (4) C B E A D (4) C A B D E (4) A D B E C (4) E D B C A (3) E D B A C (3) D E C A B (3) D E A C B (3) B A C E D (3) E B C A D (2) E B A C D (2) D A E C B (2) C D A B E (2) B C E A D (2) A B D C E (2) E D C B A (1) E C D B A (1) E B D C A (1) E B C D A (1) E A D B C (1) D C A E B (1) D C A B E (1) D A C B E (1) D A B E C (1) C E B D A (1) C B E D A (1) B E C A D (1) A D E B C (1) A D C B E (1) A D B C E (1) A C D B E (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -2 16 8 B 4 0 18 6 8 C 2 -18 0 -2 -4 D -16 -6 2 0 -4 E -8 -8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 16 8 B 4 0 18 6 8 C 2 -18 0 -2 -4 D -16 -6 2 0 -4 E -8 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 D=23 B=14 A=13 so A is eliminated. Round 2 votes counts: D=30 C=27 E=24 B=19 so B is eliminated. Round 3 votes counts: C=41 D=33 E=26 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:218 A:209 E:196 C:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 16 8 B 4 0 18 6 8 C 2 -18 0 -2 -4 D -16 -6 2 0 -4 E -8 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 16 8 B 4 0 18 6 8 C 2 -18 0 -2 -4 D -16 -6 2 0 -4 E -8 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 16 8 B 4 0 18 6 8 C 2 -18 0 -2 -4 D -16 -6 2 0 -4 E -8 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999709 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8964: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) B C A D E (5) A C B D E (5) E C A D B (4) D B E A C (4) D B A E C (4) C A B D E (4) A C D E B (4) E C B D A (3) E C B A D (3) E C A B D (3) E B C D A (3) D E B A C (3) D A B C E (3) C B E A D (3) C A E B D (3) B D A C E (3) E D A C B (2) D A E C B (2) C A B E D (2) B D E C A (2) A D E C B (2) A D C B E (2) A D B C E (2) A C E D B (2) A C D B E (2) E D C B A (1) E D B A C (1) E B D C A (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B A D (1) C E A B D (1) C B A D E (1) B E D C A (1) B E C D A (1) B C E D A (1) B C E A D (1) B C D E A (1) B C D A E (1) A E C D B (1) A D C E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 8 8 B 2 0 -6 2 4 C 4 6 0 14 2 D -8 -2 -14 0 10 E -8 -4 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998933 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 8 8 B 2 0 -6 2 4 C 4 6 0 14 2 D -8 -2 -14 0 10 E -8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=23 D=19 B=16 C=15 so C is eliminated. Round 2 votes counts: A=32 E=29 B=20 D=19 so D is eliminated. Round 3 votes counts: A=39 E=32 B=29 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:213 A:205 B:201 D:193 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 8 8 B 2 0 -6 2 4 C 4 6 0 14 2 D -8 -2 -14 0 10 E -8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 8 8 B 2 0 -6 2 4 C 4 6 0 14 2 D -8 -2 -14 0 10 E -8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 8 8 B 2 0 -6 2 4 C 4 6 0 14 2 D -8 -2 -14 0 10 E -8 -4 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8965: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) E C A D B (7) C D E B A (6) C E A D B (5) C E D B A (4) C D B E A (4) A E B C D (4) A B E D C (4) C E D A B (3) B D A E C (3) B D A C E (3) B A D E C (3) A B E C D (3) E C D B A (2) E C D A B (2) D E C B A (2) D C E B A (2) D C B E A (2) D B C A E (2) C E A B D (2) C B D A E (2) C A E B D (2) B D C A E (2) B C D A E (2) A E C B D (2) A B D E C (2) A B C D E (2) E D C B A (1) E D A C B (1) E A C D B (1) D E B C A (1) D E A B C (1) D B E A C (1) D B A E C (1) C A B E D (1) B A D C E (1) B A C D E (1) A E D B C (1) A E B D C (1) A C E B D (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 -14 -2 C 24 6 0 14 14 D 10 14 -14 0 4 E 10 2 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 -14 -2 C 24 6 0 14 14 D 10 14 -14 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=22 D=20 B=15 E=14 so E is eliminated. Round 2 votes counts: C=40 A=23 D=22 B=15 so B is eliminated. Round 3 votes counts: C=42 D=30 A=28 so A is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:229 D:207 E:197 B:192 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 -14 -2 C 24 6 0 14 14 D 10 14 -14 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 -14 -2 C 24 6 0 14 14 D 10 14 -14 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -24 -10 -10 B 6 0 -6 -14 -2 C 24 6 0 14 14 D 10 14 -14 0 4 E 10 2 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999469 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8966: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) C A E D B (7) B D E A C (7) C B D E A (6) B D C E A (6) E D A B C (5) C A E B D (5) B D E C A (5) A C E D B (5) C B D A E (4) C A B D E (4) B C D E A (4) A E C D B (4) E A D B C (3) C B A D E (3) A E D C B (3) D E B A C (2) D B E A C (2) C A B E D (2) A D E B C (2) A C B E D (2) E D A C B (1) E C D B A (1) E A D C B (1) C E B D A (1) C E A D B (1) B D C A E (1) B D A E C (1) B D A C E (1) B C D A E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -2 4 10 B -10 0 0 8 2 C 2 0 0 0 6 D -4 -8 0 0 2 E -10 -2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.107568 C: 0.892432 D: 0.000000 E: 0.000000 Sum of squares = 0.808006458209 Cumulative probabilities = A: 0.000000 B: 0.107568 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 4 10 B -10 0 0 8 2 C 2 0 0 0 6 D -4 -8 0 0 2 E -10 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222265135 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=26 A=26 E=11 D=4 so D is eliminated. Round 2 votes counts: C=33 B=28 A=26 E=13 so E is eliminated. Round 3 votes counts: A=36 C=34 B=30 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:204 B:200 D:195 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 4 10 B -10 0 0 8 2 C 2 0 0 0 6 D -4 -8 0 0 2 E -10 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222265135 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 4 10 B -10 0 0 8 2 C 2 0 0 0 6 D -4 -8 0 0 2 E -10 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222265135 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 4 10 B -10 0 0 8 2 C 2 0 0 0 6 D -4 -8 0 0 2 E -10 -2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222265135 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8967: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) E D A C B (5) E C D A B (5) E A D B C (5) D A B E C (5) C B E A D (5) E C A B D (4) D A E B C (4) B A D C E (4) D B A C E (3) C B A E D (3) B C A E D (3) A D B E C (3) E D A B C (2) E C B A D (2) D E A B C (2) D C E A B (2) D C B A E (2) C E B D A (2) C D E B A (2) C D B A E (2) C B D E A (2) C B D A E (2) B D A C E (2) B A E C D (2) B A D E C (2) A B E D C (2) E D C A B (1) E C A D B (1) D E A C B (1) D C A E B (1) D A B C E (1) C D B E A (1) C B E D A (1) C B A D E (1) B E C A D (1) B D C A E (1) B C E A D (1) B C D A E (1) B C A D E (1) B A E D C (1) B A C D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -10 -8 0 -4 B 10 0 -4 2 6 C 8 4 0 0 2 D 0 -2 0 0 -8 E 4 -6 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.889732 D: 0.110268 E: 0.000000 Sum of squares = 0.803782572586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.889732 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 0 -4 B 10 0 -4 2 6 C 8 4 0 0 2 D 0 -2 0 0 -8 E 4 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.68000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=25 D=21 B=20 A=6 so A is eliminated. Round 2 votes counts: C=28 E=26 D=24 B=22 so B is eliminated. Round 3 votes counts: C=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:207 C:207 E:202 D:195 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 0 -4 B 10 0 -4 2 6 C 8 4 0 0 2 D 0 -2 0 0 -8 E 4 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.68000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 0 -4 B 10 0 -4 2 6 C 8 4 0 0 2 D 0 -2 0 0 -8 E 4 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.68000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 0 -4 B 10 0 -4 2 6 C 8 4 0 0 2 D 0 -2 0 0 -8 E 4 -6 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.68000000053 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8968: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (8) A E D C B (7) A B E D C (7) B C D E A (6) B A E C D (6) D C E A B (5) C D B E A (5) D C A E B (4) C B D E A (4) A D E C B (4) A B E C D (4) D C E B A (3) B E A C D (3) B C E D A (3) B C D A E (3) D A E C B (2) C D E B A (2) B C E A D (2) B A C E D (2) A E D B C (2) E D A C B (1) E C A D B (1) E B C D A (1) E B C A D (1) E A D C B (1) E A B D C (1) D E C A B (1) D E A C B (1) D C B E A (1) D A C E B (1) C E B D A (1) B E C A D (1) B C A E D (1) B C A D E (1) B A C D E (1) A D C B E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 8 12 12 B -6 0 10 14 6 C -8 -10 0 -4 -6 D -12 -14 4 0 -6 E -12 -6 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 12 12 B -6 0 10 14 6 C -8 -10 0 -4 -6 D -12 -14 4 0 -6 E -12 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=29 D=18 C=12 E=6 so E is eliminated. Round 2 votes counts: A=37 B=31 D=19 C=13 so C is eliminated. Round 3 votes counts: A=38 B=36 D=26 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:212 E:197 C:186 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 12 12 B -6 0 10 14 6 C -8 -10 0 -4 -6 D -12 -14 4 0 -6 E -12 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 12 12 B -6 0 10 14 6 C -8 -10 0 -4 -6 D -12 -14 4 0 -6 E -12 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 12 12 B -6 0 10 14 6 C -8 -10 0 -4 -6 D -12 -14 4 0 -6 E -12 -6 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8969: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) C D A E B (8) A D B C E (8) E B C D A (6) C E D A B (6) C A D B E (5) E D A B C (4) E B D A C (4) C B E A D (4) D A C E B (3) C B A D E (3) B E A D C (3) B C E A D (3) B C A D E (3) B A D E C (3) E D A C B (2) E C D A B (2) E C B A D (2) E B C A D (2) D A E C B (2) C A B D E (2) A D C B E (2) E D B A C (1) E C B D A (1) D E A C B (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C D E A B (1) C A D E B (1) B E D A C (1) B D A E C (1) B C A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 0 -22 8 -8 B 0 0 8 2 4 C 22 -8 0 22 6 D -8 -2 -22 0 -6 E 8 -4 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.173654 B: 0.826346 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.713002898441 Cumulative probabilities = A: 0.173654 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -22 8 -8 B 0 0 8 2 4 C 22 -8 0 22 6 D -8 -2 -22 0 -6 E 8 -4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.733333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.608888889408 Cumulative probabilities = A: 0.266667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=26 E=24 A=10 D=9 so D is eliminated. Round 2 votes counts: C=31 B=26 E=25 A=18 so A is eliminated. Round 3 votes counts: C=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:221 B:207 E:202 A:189 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -22 8 -8 B 0 0 8 2 4 C 22 -8 0 22 6 D -8 -2 -22 0 -6 E 8 -4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.733333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.608888889408 Cumulative probabilities = A: 0.266667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -22 8 -8 B 0 0 8 2 4 C 22 -8 0 22 6 D -8 -2 -22 0 -6 E 8 -4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.733333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.608888889408 Cumulative probabilities = A: 0.266667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -22 8 -8 B 0 0 8 2 4 C 22 -8 0 22 6 D -8 -2 -22 0 -6 E 8 -4 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.266667 B: 0.733333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.608888889408 Cumulative probabilities = A: 0.266667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8970: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) B D A C E (6) B C D A E (6) C E A B D (5) E C A D B (4) E A D C B (4) D A B E C (4) B D C A E (4) D B C E A (3) C B E D A (3) B C D E A (3) B C A D E (3) A E D B C (3) A D B E C (3) E D A B C (2) E C D A B (2) D E A B C (2) D B A E C (2) C E B A D (2) C E A D B (2) C B E A D (2) C B D E A (2) C A E B D (2) B D A E C (2) A E C D B (2) E C D B A (1) E A C B D (1) D E C B A (1) D E B A C (1) D B C A E (1) D B A C E (1) C E D B A (1) C E D A B (1) C E B D A (1) C B D A E (1) B A D C E (1) A E D C B (1) A E C B D (1) A E B D C (1) A D E B C (1) A C E B D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 0 0 -6 B -6 0 0 -4 -4 C 0 0 0 12 2 D 0 4 -12 0 -4 E 6 4 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.218371 B: 0.000000 C: 0.781629 D: 0.000000 E: 0.000000 Sum of squares = 0.658630140475 Cumulative probabilities = A: 0.218371 B: 0.218371 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 0 -6 B -6 0 0 -4 -4 C 0 0 0 12 2 D 0 4 -12 0 -4 E 6 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000012937 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 C=22 D=15 A=14 so A is eliminated. Round 2 votes counts: E=32 B=26 C=23 D=19 so D is eliminated. Round 3 votes counts: B=40 E=37 C=23 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:207 E:206 A:200 D:194 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 0 0 -6 B -6 0 0 -4 -4 C 0 0 0 12 2 D 0 4 -12 0 -4 E 6 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000012937 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 0 -6 B -6 0 0 -4 -4 C 0 0 0 12 2 D 0 4 -12 0 -4 E 6 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000012937 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 0 -6 B -6 0 0 -4 -4 C 0 0 0 12 2 D 0 4 -12 0 -4 E 6 4 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000012937 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8971: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) D E C B A (8) D C A B E (7) E D A B C (6) E B A C D (5) D E A C B (5) D C B A E (5) B C A E D (5) C A B D E (4) A C B D E (4) A B C E D (4) E D B A C (3) D C E B A (3) C B A E D (3) C B A D E (3) A D B C E (3) E D C B A (2) E D B C A (2) E C B D A (2) E B C A D (2) D A E C B (2) B A C E D (2) A E B C D (2) E C B A D (1) E A B D C (1) D E A B C (1) C E B D A (1) C D B A E (1) C B E A D (1) C B D A E (1) B C E A D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 -16 12 B 2 0 -24 -14 8 C 4 24 0 -14 16 D 16 14 14 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -16 12 B 2 0 -24 -14 8 C 4 24 0 -14 16 D 16 14 14 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=24 C=14 A=14 B=8 so B is eliminated. Round 2 votes counts: D=40 E=24 C=20 A=16 so A is eliminated. Round 3 votes counts: D=43 C=31 E=26 so E is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:229 C:215 A:195 B:186 E:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -16 12 B 2 0 -24 -14 8 C 4 24 0 -14 16 D 16 14 14 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -16 12 B 2 0 -24 -14 8 C 4 24 0 -14 16 D 16 14 14 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -16 12 B 2 0 -24 -14 8 C 4 24 0 -14 16 D 16 14 14 0 14 E -12 -8 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8972: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (12) B C A D E (8) E D C B A (7) A C B E D (7) D E C B A (6) C B A E D (6) A B D C E (6) D E B C A (5) E D A C B (4) D E A B C (3) A B C D E (3) E D C A B (2) E A D C B (2) D E A C B (2) D B E C A (2) B C A E D (2) B A C D E (2) A E D C B (2) A E C B D (2) A D E B C (2) A C E B D (2) E C D B A (1) E C A D B (1) E C A B D (1) D E C A B (1) D B C E A (1) C E B D A (1) C B E A D (1) C B D E A (1) C A E B D (1) B D C E A (1) B D C A E (1) B C D E A (1) B C D A E (1) Total count = 100 A B C D E A 0 6 -2 20 14 B -6 0 0 18 10 C 2 0 0 6 14 D -20 -18 -6 0 -8 E -14 -10 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.173493 C: 0.826507 D: 0.000000 E: 0.000000 Sum of squares = 0.713213570225 Cumulative probabilities = A: 0.000000 B: 0.173493 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 20 14 B -6 0 0 18 10 C 2 0 0 6 14 D -20 -18 -6 0 -8 E -14 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249999 C: 0.750001 D: 0.000000 E: 0.000000 Sum of squares = 0.625000549345 Cumulative probabilities = A: 0.000000 B: 0.249999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=20 E=18 B=16 C=10 so C is eliminated. Round 2 votes counts: A=37 B=24 D=20 E=19 so E is eliminated. Round 3 votes counts: A=41 D=34 B=25 so B is eliminated. Round 4 votes counts: A=60 D=40 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 B:211 C:211 E:185 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 20 14 B -6 0 0 18 10 C 2 0 0 6 14 D -20 -18 -6 0 -8 E -14 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249999 C: 0.750001 D: 0.000000 E: 0.000000 Sum of squares = 0.625000549345 Cumulative probabilities = A: 0.000000 B: 0.249999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 20 14 B -6 0 0 18 10 C 2 0 0 6 14 D -20 -18 -6 0 -8 E -14 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249999 C: 0.750001 D: 0.000000 E: 0.000000 Sum of squares = 0.625000549345 Cumulative probabilities = A: 0.000000 B: 0.249999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 20 14 B -6 0 0 18 10 C 2 0 0 6 14 D -20 -18 -6 0 -8 E -14 -10 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.249999 C: 0.750001 D: 0.000000 E: 0.000000 Sum of squares = 0.625000549345 Cumulative probabilities = A: 0.000000 B: 0.249999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 8973: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) D B C E A (7) D C E B A (5) C E D A B (5) B A D E C (5) A B D E C (5) E B C A D (4) D C B E A (4) D B C A E (4) D B A C E (4) C E D B A (4) C E A D B (3) B D A E C (3) B A E D C (3) A E C B D (3) E C D B A (2) E C B D A (2) E C A B D (2) E A C B D (2) D A C E B (2) C D E A B (2) B E C D A (2) B D E A C (2) A D C E B (2) E C B A D (1) E B A C D (1) D C E A B (1) D C B A E (1) D C A E B (1) D A C B E (1) C D E B A (1) C D A E B (1) B E D C A (1) B E A C D (1) A D C B E (1) A D B C E (1) A C E B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -6 -10 -4 B 14 0 6 -4 10 C 6 -6 0 -10 4 D 10 4 10 0 8 E 4 -10 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -10 -4 B 14 0 6 -4 10 C 6 -6 0 -10 4 D 10 4 10 0 8 E 4 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=23 B=17 C=16 E=14 so E is eliminated. Round 2 votes counts: D=30 A=25 C=23 B=22 so B is eliminated. Round 3 votes counts: D=36 A=35 C=29 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:213 C:197 E:191 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -6 -10 -4 B 14 0 6 -4 10 C 6 -6 0 -10 4 D 10 4 10 0 8 E 4 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -10 -4 B 14 0 6 -4 10 C 6 -6 0 -10 4 D 10 4 10 0 8 E 4 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -10 -4 B 14 0 6 -4 10 C 6 -6 0 -10 4 D 10 4 10 0 8 E 4 -10 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8974: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) C D B A E (8) C B D A E (7) E A B D C (6) E A B C D (6) D C B E A (6) E D C B A (5) D C B A E (5) A B C D E (5) E A D C B (4) E A D B C (4) A B C E D (4) E D C A B (3) B A C D E (3) A B E C D (3) E A C B D (2) D C E B A (2) C B A D E (2) B C D A E (2) B C A D E (2) E D A C B (1) E C D B A (1) E C D A B (1) D E C B A (1) D B E C A (1) D B C A E (1) B D C A E (1) B A D C E (1) A E B D C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 4 10 14 B -4 0 4 14 8 C -4 -4 0 10 0 D -10 -14 -10 0 -2 E -14 -8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 10 14 B -4 0 4 14 8 C -4 -4 0 10 0 D -10 -14 -10 0 -2 E -14 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=25 C=17 D=16 B=9 so B is eliminated. Round 2 votes counts: E=33 A=29 C=21 D=17 so D is eliminated. Round 3 votes counts: C=36 E=35 A=29 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:216 B:211 C:201 E:190 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 10 14 B -4 0 4 14 8 C -4 -4 0 10 0 D -10 -14 -10 0 -2 E -14 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 10 14 B -4 0 4 14 8 C -4 -4 0 10 0 D -10 -14 -10 0 -2 E -14 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 10 14 B -4 0 4 14 8 C -4 -4 0 10 0 D -10 -14 -10 0 -2 E -14 -8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8975: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (15) E B A D C (9) B A D E C (9) E C D B A (8) B A D C E (7) E C A B D (6) E B A C D (6) D C A B E (4) D B A C E (4) C E D A B (4) C D E A B (3) B A E D C (3) E C D A B (2) E B D A C (2) D B A E C (2) D A B C E (2) C D A E B (2) C A E B D (2) A B D C E (2) E D B C A (1) E D B A C (1) E C A D B (1) E A B C D (1) D C B E A (1) C A D B E (1) C A B E D (1) A D C B E (1) Total count = 100 A B C D E A 0 -6 -2 -2 10 B 6 0 -2 -4 4 C 2 2 0 4 -2 D 2 4 -4 0 6 E -10 -4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.082914 B: 0.104900 C: 0.624372 D: 0.000000 E: 0.187814 Sum of squares = 0.442992915332 Cumulative probabilities = A: 0.082914 B: 0.187814 C: 0.812186 D: 0.812186 E: 1.000000 A B C D E A 0 -6 -2 -2 10 B 6 0 -2 -4 4 C 2 2 0 4 -2 D 2 4 -4 0 6 E -10 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.032258 B: 0.193548 C: 0.548387 D: 0.000000 E: 0.225806 Sum of squares = 0.390218640458 Cumulative probabilities = A: 0.032258 B: 0.225806 C: 0.774194 D: 0.774194 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=28 B=19 D=13 A=3 so A is eliminated. Round 2 votes counts: E=37 C=28 B=21 D=14 so D is eliminated. Round 3 votes counts: E=37 C=34 B=29 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:204 C:203 B:202 A:200 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 -2 10 B 6 0 -2 -4 4 C 2 2 0 4 -2 D 2 4 -4 0 6 E -10 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.032258 B: 0.193548 C: 0.548387 D: 0.000000 E: 0.225806 Sum of squares = 0.390218640458 Cumulative probabilities = A: 0.032258 B: 0.225806 C: 0.774194 D: 0.774194 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -2 10 B 6 0 -2 -4 4 C 2 2 0 4 -2 D 2 4 -4 0 6 E -10 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.032258 B: 0.193548 C: 0.548387 D: 0.000000 E: 0.225806 Sum of squares = 0.390218640458 Cumulative probabilities = A: 0.032258 B: 0.225806 C: 0.774194 D: 0.774194 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -2 10 B 6 0 -2 -4 4 C 2 2 0 4 -2 D 2 4 -4 0 6 E -10 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.032258 B: 0.193548 C: 0.548387 D: 0.000000 E: 0.225806 Sum of squares = 0.390218640458 Cumulative probabilities = A: 0.032258 B: 0.225806 C: 0.774194 D: 0.774194 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8976: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) D B A E C (7) B E D C A (7) A D B C E (7) E C B D A (6) B D E C A (6) E B D C A (5) C E A B D (5) C A E D B (5) E C B A D (4) B D E A C (4) E C A B D (3) D A B C E (3) A D C B E (3) A C D E B (3) D B A C E (2) C E B D A (2) C A D E B (2) B E D A C (2) A C E D B (2) A C D B E (2) E B D A C (1) E B A C D (1) E A C B D (1) E A B C D (1) D B E A C (1) D B C A E (1) D A B E C (1) C A D B E (1) B C E D A (1) A E D B C (1) A E C B D (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -14 -10 -12 -14 B 14 0 20 16 -2 C 10 -20 0 -6 -22 D 12 -16 6 0 -12 E 14 2 22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -10 -12 -14 B 14 0 20 16 -2 C 10 -20 0 -6 -22 D 12 -16 6 0 -12 E 14 2 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=21 B=20 D=15 C=15 so D is eliminated. Round 2 votes counts: B=31 E=29 A=25 C=15 so C is eliminated. Round 3 votes counts: E=36 A=33 B=31 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:224 D:195 C:181 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -10 -12 -14 B 14 0 20 16 -2 C 10 -20 0 -6 -22 D 12 -16 6 0 -12 E 14 2 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 -12 -14 B 14 0 20 16 -2 C 10 -20 0 -6 -22 D 12 -16 6 0 -12 E 14 2 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 -12 -14 B 14 0 20 16 -2 C 10 -20 0 -6 -22 D 12 -16 6 0 -12 E 14 2 22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8977: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (12) D C E A B (11) A B E D C (9) E C B A D (6) C D E B A (6) E B C A D (4) E B A C D (4) D C A E B (4) C E D B A (4) A D B E C (4) A B D E C (4) D C A B E (3) D A C B E (3) D A B E C (2) D A B C E (2) C D E A B (2) C B E A D (2) C B A E D (2) B A E D C (2) A D B C E (2) E B A D C (1) D E A B C (1) D A E C B (1) C E B D A (1) C E B A D (1) C D B E A (1) B E C A D (1) B E A D C (1) B E A C D (1) B A C E D (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 4 18 6 B 0 0 6 6 6 C -4 -6 0 -4 -8 D -18 -6 4 0 -4 E -6 -6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.357356 B: 0.642644 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.540694455529 Cumulative probabilities = A: 0.357356 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 18 6 B 0 0 6 6 6 C -4 -6 0 -4 -8 D -18 -6 4 0 -4 E -6 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=21 C=19 B=18 E=15 so E is eliminated. Round 2 votes counts: D=27 B=27 C=25 A=21 so A is eliminated. Round 3 votes counts: B=41 D=34 C=25 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:209 E:200 C:189 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 18 6 B 0 0 6 6 6 C -4 -6 0 -4 -8 D -18 -6 4 0 -4 E -6 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 18 6 B 0 0 6 6 6 C -4 -6 0 -4 -8 D -18 -6 4 0 -4 E -6 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 18 6 B 0 0 6 6 6 C -4 -6 0 -4 -8 D -18 -6 4 0 -4 E -6 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8978: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) D A E C B (10) D E C A B (7) B C E A D (7) E C B D A (6) A B C E D (6) E D C B A (4) B E C D A (4) A D C E B (4) A B D C E (4) D E A C B (3) D B E C A (3) B A C E D (3) A D E C B (3) A D B E C (3) D E B C A (2) C E B D A (2) C E B A D (2) B C E D A (2) B C A E D (2) A C B E D (2) A B C D E (2) E C D B A (1) C E A B D (1) B E D C A (1) B A D E C (1) B A D C E (1) A D B C E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 -12 -12 B 4 0 -16 -6 -16 C 10 16 0 -16 -18 D 12 6 16 0 12 E 12 16 18 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -12 -12 B 4 0 -16 -6 -16 C 10 16 0 -16 -18 D 12 6 16 0 12 E 12 16 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=27 B=21 E=11 C=5 so C is eliminated. Round 2 votes counts: D=36 A=27 B=21 E=16 so E is eliminated. Round 3 votes counts: D=41 B=31 A=28 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:217 C:196 B:183 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -12 -12 B 4 0 -16 -6 -16 C 10 16 0 -16 -18 D 12 6 16 0 12 E 12 16 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -12 -12 B 4 0 -16 -6 -16 C 10 16 0 -16 -18 D 12 6 16 0 12 E 12 16 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -12 -12 B 4 0 -16 -6 -16 C 10 16 0 -16 -18 D 12 6 16 0 12 E 12 16 18 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999399 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8979: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) C E D B A (9) A B D C E (8) E C D B A (6) C A E D B (6) B D E A C (6) A D B C E (6) A B D E C (6) A C D B E (5) E B D C A (4) B D A E C (4) A C E B D (4) E B D A C (2) C E D A B (2) C E A D B (2) B D E C A (2) B A D E C (2) E C A B D (1) E B C D A (1) D C B E A (1) D C A B E (1) D B E C A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E B A (1) C D B E A (1) C D A B E (1) C A E B D (1) C A D B E (1) A D C B E (1) A C E D B (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -2 -6 2 B 2 0 -12 6 0 C 2 12 0 8 10 D 6 -6 -8 0 0 E -2 0 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -6 2 B 2 0 -12 6 0 C 2 12 0 8 10 D 6 -6 -8 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=25 E=23 B=14 D=5 so D is eliminated. Round 2 votes counts: A=33 C=27 E=23 B=17 so B is eliminated. Round 3 votes counts: A=41 E=32 C=27 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:216 B:198 A:196 D:196 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 2 B 2 0 -12 6 0 C 2 12 0 8 10 D 6 -6 -8 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 2 B 2 0 -12 6 0 C 2 12 0 8 10 D 6 -6 -8 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 2 B 2 0 -12 6 0 C 2 12 0 8 10 D 6 -6 -8 0 0 E -2 0 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999818 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8980: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) C B A D E (6) E D B A C (5) D B E C A (5) C A B D E (5) D E B C A (4) C A E B D (4) C A B E D (4) B C D A E (4) A E C D B (4) A C B D E (4) E D B C A (3) D E B A C (3) B D E C A (3) A C E B D (3) A C B E D (3) E D C B A (2) E A D B C (2) E A C D B (2) D B E A C (2) B C D E A (2) B C A D E (2) A D B E C (2) A B C D E (2) E D A C B (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D C B (1) C E B D A (1) C E A D B (1) C E A B D (1) C B E D A (1) C B D A E (1) B D C E A (1) B D A E C (1) B A D C E (1) A E D B C (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -6 2 -2 B -4 0 4 0 0 C 6 -4 0 8 -4 D -2 0 -8 0 -2 E 2 0 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.193501 C: 0.000000 D: 0.000000 E: 0.806499 Sum of squares = 0.687883621019 Cumulative probabilities = A: 0.000000 B: 0.193501 C: 0.193501 D: 0.193501 E: 1.000000 A B C D E A 0 4 -6 2 -2 B -4 0 4 0 0 C 6 -4 0 8 -4 D -2 0 -8 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555604558 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=24 A=21 D=14 B=14 so D is eliminated. Round 2 votes counts: E=34 C=24 B=21 A=21 so B is eliminated. Round 3 votes counts: E=44 C=33 A=23 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:204 C:203 B:200 A:199 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -6 2 -2 B -4 0 4 0 0 C 6 -4 0 8 -4 D -2 0 -8 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555604558 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 2 -2 B -4 0 4 0 0 C 6 -4 0 8 -4 D -2 0 -8 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555604558 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 2 -2 B -4 0 4 0 0 C 6 -4 0 8 -4 D -2 0 -8 0 -2 E 2 0 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555604558 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8981: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (10) B A C E D (9) C E D B A (7) C D E A B (6) A B D E C (6) E D C B A (5) D E C A B (5) D C E A B (5) B E A D C (4) C B A E D (3) C A B D E (3) B A E C D (3) A C B D E (3) E C B D A (2) D E A B C (2) D A E B C (2) C E B D A (2) C D A E B (2) C A D B E (2) B A E D C (2) A C D B E (2) A B C E D (2) E D B C A (1) E B D C A (1) E B D A C (1) E B C A D (1) D E A C B (1) D C A E B (1) D A C E B (1) C B E A D (1) B E C A D (1) A D C B E (1) A D B E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 4 12 10 B -14 0 -4 10 10 C -4 4 0 18 24 D -12 -10 -18 0 10 E -10 -10 -24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 12 10 B -14 0 -4 10 10 C -4 4 0 18 24 D -12 -10 -18 0 10 E -10 -10 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=26 B=19 D=17 E=11 so E is eliminated. Round 2 votes counts: C=28 A=27 D=23 B=22 so B is eliminated. Round 3 votes counts: A=45 C=30 D=25 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:221 A:220 B:201 D:185 E:173 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 12 10 B -14 0 -4 10 10 C -4 4 0 18 24 D -12 -10 -18 0 10 E -10 -10 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 12 10 B -14 0 -4 10 10 C -4 4 0 18 24 D -12 -10 -18 0 10 E -10 -10 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 12 10 B -14 0 -4 10 10 C -4 4 0 18 24 D -12 -10 -18 0 10 E -10 -10 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999353 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8982: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (10) E D A B C (8) D A E C B (8) E B C D A (7) D A E B C (6) E B C A D (5) D A C B E (5) C B A E D (5) D E A C B (4) A D C B E (4) A D B C E (4) E B D C A (3) E B A D C (2) D E A B C (2) D A C E B (2) C E B D A (2) C B E A D (2) B E C A D (2) B C E A D (2) B C A D E (2) B A E D C (2) E D C B A (1) E D B A C (1) E D A C B (1) E C B D A (1) D A B C E (1) C E D B A (1) C A B D E (1) B C A E D (1) B A C E D (1) B A C D E (1) A D E B C (1) A D B E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 10 -6 12 B 2 0 4 0 -10 C -10 -4 0 -12 -10 D 6 0 12 0 6 E -12 10 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.244267 C: 0.000000 D: 0.755733 E: 0.000000 Sum of squares = 0.63079864926 Cumulative probabilities = A: 0.000000 B: 0.244267 C: 0.244267 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -6 12 B 2 0 4 0 -10 C -10 -4 0 -12 -10 D 6 0 12 0 6 E -12 10 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250001005 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=28 C=21 B=11 A=11 so B is eliminated. Round 2 votes counts: E=31 D=28 C=26 A=15 so A is eliminated. Round 3 votes counts: D=38 E=33 C=29 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 E:201 B:198 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -6 12 B 2 0 4 0 -10 C -10 -4 0 -12 -10 D 6 0 12 0 6 E -12 10 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250001005 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -6 12 B 2 0 4 0 -10 C -10 -4 0 -12 -10 D 6 0 12 0 6 E -12 10 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250001005 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -6 12 B 2 0 4 0 -10 C -10 -4 0 -12 -10 D 6 0 12 0 6 E -12 10 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.375000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.531250001005 Cumulative probabilities = A: 0.000000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8983: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) D B A E C (10) C E A B D (10) C A E D B (9) A D B C E (9) A C D E B (9) A C D B E (7) A D C B E (6) E C B A D (4) E B C D A (4) E C B D A (3) D A B C E (3) C A E B D (3) B D E C A (3) B D E A C (3) B E D C A (2) D E B A C (1) D A B E C (1) C E B A D (1) A C E D B (1) Total count = 100 A B C D E A 0 16 0 18 16 B -16 0 -6 -12 -12 C 0 6 0 2 16 D -18 12 -2 0 4 E -16 12 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.522906 B: 0.000000 C: 0.477094 D: 0.000000 E: 0.000000 Sum of squares = 0.501049342771 Cumulative probabilities = A: 0.522906 B: 0.522906 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 18 16 B -16 0 -6 -12 -12 C 0 6 0 2 16 D -18 12 -2 0 4 E -16 12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=23 E=22 D=15 B=8 so B is eliminated. Round 2 votes counts: A=32 E=24 C=23 D=21 so D is eliminated. Round 3 votes counts: A=46 E=31 C=23 so C is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:225 C:212 D:198 E:188 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 18 16 B -16 0 -6 -12 -12 C 0 6 0 2 16 D -18 12 -2 0 4 E -16 12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 18 16 B -16 0 -6 -12 -12 C 0 6 0 2 16 D -18 12 -2 0 4 E -16 12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 18 16 B -16 0 -6 -12 -12 C 0 6 0 2 16 D -18 12 -2 0 4 E -16 12 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8984: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (17) E D C B A (14) E B D C A (9) B A C D E (7) C D B A E (6) D C A E B (5) D C E A B (4) B C D A E (4) E D C A B (3) E A B D C (3) D C A B E (3) C D A B E (3) B A E C D (3) E B C D A (2) E B A D C (2) E A D C B (2) A C D B E (2) E D B C A (1) E B A C D (1) D C E B A (1) D C B E A (1) D C B A E (1) B E A C D (1) B C A D E (1) A E D C B (1) A E B C D (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -16 -14 12 B 8 0 6 4 0 C 16 -6 0 -2 12 D 14 -4 2 0 12 E -12 0 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.854645 C: 0.000000 D: 0.000000 E: 0.145355 Sum of squares = 0.751546494503 Cumulative probabilities = A: 0.000000 B: 0.854645 C: 0.854645 D: 0.854645 E: 1.000000 A B C D E A 0 -8 -16 -14 12 B 8 0 6 4 0 C 16 -6 0 -2 12 D 14 -4 2 0 12 E -12 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.6250000339 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 A=23 B=16 D=15 C=9 so C is eliminated. Round 2 votes counts: E=37 D=24 A=23 B=16 so B is eliminated. Round 3 votes counts: E=38 A=34 D=28 so D is eliminated. Round 4 votes counts: A=56 E=44 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:212 C:210 B:209 A:187 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -16 -14 12 B 8 0 6 4 0 C 16 -6 0 -2 12 D 14 -4 2 0 12 E -12 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.6250000339 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -14 12 B 8 0 6 4 0 C 16 -6 0 -2 12 D 14 -4 2 0 12 E -12 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.6250000339 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -14 12 B 8 0 6 4 0 C 16 -6 0 -2 12 D 14 -4 2 0 12 E -12 0 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.6250000339 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8985: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C E A D B (7) A B E C D (7) E C A B D (6) D B A E C (6) B A D E C (6) D B A C E (5) C E D B A (5) C E A B D (5) B D A E C (5) D C E B A (4) A B D C E (4) D E B C A (3) D C E A B (3) D B C E A (3) D A B C E (3) C E D A B (3) D C A E B (2) B A E C D (2) A C E B D (2) E D C B A (1) E C D A B (1) E C B D A (1) E B D C A (1) D B E C A (1) C D E B A (1) C D E A B (1) C A D E B (1) B A E D C (1) A E C B D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -12 -12 -6 B 4 0 -4 -14 -10 C 12 4 0 4 2 D 12 14 -4 0 -2 E 6 10 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -12 -6 B 4 0 -4 -14 -10 C 12 4 0 4 2 D 12 14 -4 0 -2 E 6 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=23 E=17 A=16 B=14 so B is eliminated. Round 2 votes counts: D=35 A=25 C=23 E=17 so E is eliminated. Round 3 votes counts: C=38 D=37 A=25 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:210 E:208 B:188 A:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -12 -6 B 4 0 -4 -14 -10 C 12 4 0 4 2 D 12 14 -4 0 -2 E 6 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -12 -6 B 4 0 -4 -14 -10 C 12 4 0 4 2 D 12 14 -4 0 -2 E 6 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -12 -6 B 4 0 -4 -14 -10 C 12 4 0 4 2 D 12 14 -4 0 -2 E 6 10 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999017 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8986: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (12) E A D C B (10) B D C E A (8) C D B A E (7) C D A E B (7) D C E B A (6) D C E A B (6) A E B C D (5) B C D A E (4) D C B E A (3) C D E A B (3) A B E C D (3) E B A D C (2) C A D E B (2) B E A D C (2) B D E C A (2) B A E D C (2) A C E D B (2) A C D E B (2) E D C B A (1) E D C A B (1) E D B C A (1) E A B D C (1) C E D A B (1) B A E C D (1) B A C E D (1) B A C D E (1) A E D C B (1) A E B D C (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 18 -4 0 6 B -18 0 -30 -32 -28 C 4 30 0 6 10 D 0 32 -6 0 4 E -6 28 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -4 0 6 B -18 0 -30 -32 -28 C 4 30 0 6 10 D 0 32 -6 0 4 E -6 28 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=21 C=20 E=16 D=15 so D is eliminated. Round 2 votes counts: C=35 A=28 B=21 E=16 so E is eliminated. Round 3 votes counts: A=39 C=37 B=24 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:225 D:215 A:210 E:204 B:146 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -4 0 6 B -18 0 -30 -32 -28 C 4 30 0 6 10 D 0 32 -6 0 4 E -6 28 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 0 6 B -18 0 -30 -32 -28 C 4 30 0 6 10 D 0 32 -6 0 4 E -6 28 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 0 6 B -18 0 -30 -32 -28 C 4 30 0 6 10 D 0 32 -6 0 4 E -6 28 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8987: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (14) E C D B A (11) A C E B D (9) D B A E C (7) A B C D E (7) B D A E C (6) C E A B D (5) B D A C E (5) A C B E D (5) D E B C A (4) D B E C A (4) E D B C A (3) E C D A B (3) E C A D B (3) A B D C E (3) E D C B A (2) D B E A C (2) C A E B D (2) B A D C E (2) A C B D E (2) C E D A B (1) Total count = 100 A B C D E A 0 8 -4 4 -4 B -8 0 -14 -8 -14 C 4 14 0 24 10 D -4 8 -24 0 -16 E 4 14 -10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999888 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 4 -4 B -8 0 -14 -8 -14 C 4 14 0 24 10 D -4 8 -24 0 -16 E 4 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 C=22 D=17 B=13 so B is eliminated. Round 2 votes counts: D=28 A=28 E=22 C=22 so E is eliminated. Round 3 votes counts: C=39 D=33 A=28 so A is eliminated. Round 4 votes counts: C=62 D=38 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:212 A:202 D:182 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 4 -4 B -8 0 -14 -8 -14 C 4 14 0 24 10 D -4 8 -24 0 -16 E 4 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 4 -4 B -8 0 -14 -8 -14 C 4 14 0 24 10 D -4 8 -24 0 -16 E 4 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 4 -4 B -8 0 -14 -8 -14 C 4 14 0 24 10 D -4 8 -24 0 -16 E 4 14 -10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998421 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8988: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) D A C E B (8) C B A E D (7) E D A B C (6) E B C A D (5) D C A B E (5) D A E C B (4) D E A C B (3) D E A B C (3) C B E A D (3) B E C A D (3) A B C E D (3) E D B C A (2) E A D B C (2) D E C B A (2) C D B E A (2) C D B A E (2) C B D E A (2) C A D B E (2) B C E A D (2) B C A E D (2) A E D B C (2) A D C B E (2) E D B A C (1) E B D C A (1) E B A D C (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A E B (1) D A C B E (1) C B E D A (1) C B D A E (1) C B A D E (1) C A B E D (1) C A B D E (1) B C E D A (1) A E B D C (1) A E B C D (1) A D E B C (1) A D B C E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 2 0 2 0 B -2 0 -8 -6 -8 C 0 8 0 0 8 D -2 6 0 0 -8 E 0 8 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.722197 B: 0.000000 C: 0.277803 D: 0.000000 E: 0.000000 Sum of squares = 0.598743222713 Cumulative probabilities = A: 0.722197 B: 0.722197 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 2 0 B -2 0 -8 -6 -8 C 0 8 0 0 8 D -2 6 0 0 -8 E 0 8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=26 C=23 A=13 B=8 so B is eliminated. Round 2 votes counts: D=30 E=29 C=28 A=13 so A is eliminated. Round 3 votes counts: D=34 E=33 C=33 so E is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:208 E:204 A:202 D:198 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 2 0 B -2 0 -8 -6 -8 C 0 8 0 0 8 D -2 6 0 0 -8 E 0 8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 2 0 B -2 0 -8 -6 -8 C 0 8 0 0 8 D -2 6 0 0 -8 E 0 8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 2 0 B -2 0 -8 -6 -8 C 0 8 0 0 8 D -2 6 0 0 -8 E 0 8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8989: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) E B A C D (6) E A B D C (5) D C A E B (5) C E B D A (5) C D E A B (5) C D B A E (5) D C A B E (4) C D A B E (4) E B A D C (3) D A E B C (3) D A C E B (3) D A C B E (3) A D B E C (3) A B E D C (3) E D C A B (2) E D A C B (2) E C D A B (2) D A B C E (2) C D E B A (2) B E A C D (2) B A D C E (2) A B D E C (2) E C B A D (1) E C A D B (1) D C E A B (1) D A E C B (1) C E D B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C B E A D (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A D C (1) B A E D C (1) B A E C D (1) B A C E D (1) B A C D E (1) A E D B C (1) A E B D C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -2 -10 0 B -12 0 -6 -6 -14 C 2 6 0 0 4 D 10 6 0 0 4 E 0 14 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.578842 D: 0.421158 E: 0.000000 Sum of squares = 0.512432164428 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.578842 D: 1.000000 E: 1.000000 A B C D E A 0 12 -2 -10 0 B -12 0 -6 -6 -14 C 2 6 0 0 4 D 10 6 0 0 4 E 0 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 D=22 A=12 B=10 so B is eliminated. Round 2 votes counts: E=32 C=28 D=22 A=18 so A is eliminated. Round 3 votes counts: E=39 D=31 C=30 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:206 E:203 A:200 B:181 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -2 -10 0 B -12 0 -6 -6 -14 C 2 6 0 0 4 D 10 6 0 0 4 E 0 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -2 -10 0 B -12 0 -6 -6 -14 C 2 6 0 0 4 D 10 6 0 0 4 E 0 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -2 -10 0 B -12 0 -6 -6 -14 C 2 6 0 0 4 D 10 6 0 0 4 E 0 14 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8990: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (10) D C A E B (7) B E C A D (7) E B A D C (6) E B A C D (4) C D B E A (3) B A E C D (3) A B E D C (3) A B D E C (3) E D C B A (2) E D A B C (2) E B C A D (2) E A B D C (2) D A E B C (2) D A C E B (2) C E D B A (2) C D E B A (2) C B E A D (2) C B D A E (2) B E A D C (2) A D B C E (2) A B D C E (2) A B C D E (2) E C D B A (1) E C B D A (1) E B D A C (1) E B C D A (1) E A D B C (1) D E C A B (1) D E A B C (1) D C E A B (1) D C A B E (1) D A C B E (1) C E B D A (1) C D E A B (1) C D B A E (1) C D A E B (1) C B E D A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B D E (1) B E A C D (1) B C A E D (1) B A E D C (1) A D C B E (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -12 2 2 B 0 0 0 4 12 C 12 0 0 10 4 D -2 -4 -10 0 4 E -2 -12 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.507303 C: 0.492697 D: 0.000000 E: 0.000000 Sum of squares = 0.500106653663 Cumulative probabilities = A: 0.000000 B: 0.507303 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -12 2 2 B 0 0 0 4 12 C 12 0 0 10 4 D -2 -4 -10 0 4 E -2 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 D=16 B=15 A=15 so B is eliminated. Round 2 votes counts: E=33 C=32 A=19 D=16 so D is eliminated. Round 3 votes counts: C=41 E=35 A=24 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:208 A:196 D:194 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -12 2 2 B 0 0 0 4 12 C 12 0 0 10 4 D -2 -4 -10 0 4 E -2 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -12 2 2 B 0 0 0 4 12 C 12 0 0 10 4 D -2 -4 -10 0 4 E -2 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -12 2 2 B 0 0 0 4 12 C 12 0 0 10 4 D -2 -4 -10 0 4 E -2 -12 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8991: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (22) B E C D A (10) D A C E B (8) E B C A D (7) D A B C E (7) D A C B E (6) B E C A D (5) E C B D A (4) C E A D B (3) B D E C A (3) B D C E A (3) E A C B D (2) D C B A E (2) B C E D A (2) A E C D B (2) A E C B D (2) E C B A D (1) E C A B D (1) D B C E A (1) C E B D A (1) C B E D A (1) B D A E C (1) B A E D C (1) A D E C B (1) A D C B E (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 16 12 2 12 B -16 0 -18 -12 -12 C -12 18 0 -14 20 D -2 12 14 0 14 E -12 12 -20 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 2 12 B -16 0 -18 -12 -12 C -12 18 0 -14 20 D -2 12 14 0 14 E -12 12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=25 D=24 E=15 C=5 so C is eliminated. Round 2 votes counts: A=31 B=26 D=24 E=19 so E is eliminated. Round 3 votes counts: B=39 A=37 D=24 so D is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 D:219 C:206 E:183 B:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 2 12 B -16 0 -18 -12 -12 C -12 18 0 -14 20 D -2 12 14 0 14 E -12 12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 2 12 B -16 0 -18 -12 -12 C -12 18 0 -14 20 D -2 12 14 0 14 E -12 12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 2 12 B -16 0 -18 -12 -12 C -12 18 0 -14 20 D -2 12 14 0 14 E -12 12 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992683 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8992: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) E B C A D (7) E C D A B (5) B C E A D (5) B A D C E (5) E C B D A (4) E C B A D (4) D A B C E (4) C E B A D (4) C B E A D (4) B A C D E (4) D A E C B (3) D A E B C (3) D A C B E (3) C E D B A (3) C E B D A (3) A B D C E (3) E D A B C (2) E C D B A (2) D A C E B (2) C B D A E (2) B E C A D (2) A D B E C (2) E B C D A (1) E B A D C (1) E B A C D (1) E A D B C (1) D C A E B (1) C E D A B (1) C D E A B (1) C D A E B (1) C D A B E (1) C B A D E (1) B E A D C (1) B C A D E (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -10 -6 16 -4 B 10 0 10 8 -2 C 6 -10 0 14 18 D -16 -8 -14 0 -2 E 4 2 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.475555555579 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -10 -6 16 -4 B 10 0 10 8 -2 C 6 -10 0 14 18 D -16 -8 -14 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.475555555556 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=21 B=18 A=17 D=16 so D is eliminated. Round 2 votes counts: A=32 E=28 C=22 B=18 so B is eliminated. Round 3 votes counts: A=41 E=31 C=28 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:214 B:213 A:198 E:195 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 16 -4 B 10 0 10 8 -2 C 6 -10 0 14 18 D -16 -8 -14 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.475555555556 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 16 -4 B 10 0 10 8 -2 C 6 -10 0 14 18 D -16 -8 -14 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.475555555556 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 16 -4 B 10 0 10 8 -2 C 6 -10 0 14 18 D -16 -8 -14 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.066667 D: 0.000000 E: 0.333333 Sum of squares = 0.475555555556 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8993: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) E D C A B (9) E C D A B (5) D E A B C (5) B A C E D (5) A B D C E (5) A B C E D (5) D A B E C (4) E D C B A (3) E C D B A (3) D E C B A (3) C B A E D (3) B C A E D (3) B A C D E (3) A C B E D (3) D E C A B (2) D B A E C (2) D A E B C (2) C E B D A (2) C E B A D (2) C E A B D (2) C B E A D (2) C A E B D (2) A D E B C (2) A D B E C (2) E D A C B (1) E C A D B (1) D E B C A (1) D B A C E (1) C E D B A (1) C B E D A (1) C A B E D (1) B C D E A (1) B C A D E (1) B A D C E (1) A B D E C (1) Total count = 100 A B C D E A 0 24 4 -12 -8 B -24 0 -12 -14 -12 C -4 12 0 -8 -12 D 12 14 8 0 -8 E 8 12 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 24 4 -12 -8 B -24 0 -12 -14 -12 C -4 12 0 -8 -12 D 12 14 8 0 -8 E 8 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=22 A=18 C=16 B=14 so B is eliminated. Round 2 votes counts: D=30 A=27 E=22 C=21 so C is eliminated. Round 3 votes counts: A=37 E=32 D=31 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:213 A:204 C:194 B:169 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 24 4 -12 -8 B -24 0 -12 -14 -12 C -4 12 0 -8 -12 D 12 14 8 0 -8 E 8 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 24 4 -12 -8 B -24 0 -12 -14 -12 C -4 12 0 -8 -12 D 12 14 8 0 -8 E 8 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 24 4 -12 -8 B -24 0 -12 -14 -12 C -4 12 0 -8 -12 D 12 14 8 0 -8 E 8 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8994: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (11) C B D A E (7) C E A D B (6) E A D B C (5) E C D B A (4) B D A C E (4) A B D C E (4) E C A D B (3) C E B D A (3) C B D E A (3) B D C A E (3) B C D A E (3) A E D B C (3) A D B E C (3) A B C D E (3) E D C B A (2) E D B C A (2) E D A B C (2) E C D A B (2) D B E A C (2) C A E B D (2) C A B D E (2) B D C E A (2) A E C D B (2) E D B A C (1) E A C D B (1) D E B C A (1) D E B A C (1) D E A B C (1) D B E C A (1) D B C E A (1) D B A E C (1) C E A B D (1) C B E A D (1) B D A E C (1) B A D C E (1) B A C D E (1) A E C B D (1) A D E B C (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -20 -16 -12 B 12 0 -4 -12 -10 C 20 4 0 14 18 D 16 12 -14 0 -6 E 12 10 -18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -20 -16 -12 B 12 0 -4 -12 -10 C 20 4 0 14 18 D 16 12 -14 0 -6 E 12 10 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=22 A=19 B=15 D=8 so D is eliminated. Round 2 votes counts: C=36 E=25 B=20 A=19 so A is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:205 D:204 B:193 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -20 -16 -12 B 12 0 -4 -12 -10 C 20 4 0 14 18 D 16 12 -14 0 -6 E 12 10 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -20 -16 -12 B 12 0 -4 -12 -10 C 20 4 0 14 18 D 16 12 -14 0 -6 E 12 10 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -20 -16 -12 B 12 0 -4 -12 -10 C 20 4 0 14 18 D 16 12 -14 0 -6 E 12 10 -18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999652 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 8995: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (14) D B C A E (11) A C E B D (8) E C A D B (6) B D A C E (6) C A D B E (5) D B C E A (4) C A E D B (4) B D E A C (4) E B D A C (3) C E A D B (3) B D A E C (3) A C B D E (3) E C A B D (2) E B D C A (2) E A B D C (2) D C B E A (2) D B E C A (2) D B A C E (2) C D B A E (2) A C E D B (2) E B A D C (1) E A B C D (1) D C B A E (1) C D A B E (1) C A D E B (1) B D E C A (1) B A D E C (1) A E C B D (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 6 12 6 B -10 0 -10 8 0 C -6 10 0 6 12 D -12 -8 -6 0 0 E -6 0 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999428 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 6 12 6 B -10 0 -10 8 0 C -6 10 0 6 12 D -12 -8 -6 0 0 E -6 0 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=22 C=16 A=16 B=15 so B is eliminated. Round 2 votes counts: D=36 E=31 A=17 C=16 so C is eliminated. Round 3 votes counts: D=39 E=34 A=27 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:217 C:211 B:194 E:191 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 6 12 6 B -10 0 -10 8 0 C -6 10 0 6 12 D -12 -8 -6 0 0 E -6 0 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 12 6 B -10 0 -10 8 0 C -6 10 0 6 12 D -12 -8 -6 0 0 E -6 0 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 12 6 B -10 0 -10 8 0 C -6 10 0 6 12 D -12 -8 -6 0 0 E -6 0 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 8996: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (13) E D C B A (7) B A C D E (7) D E C B A (5) C E D B A (4) C B D E A (4) B C A D E (4) A D E B C (4) A D B E C (4) A B D E C (4) A B C D E (4) E D C A B (3) E D A B C (3) D E A B C (3) C E B D A (3) C B A D E (3) A E D B C (3) E C D B A (2) D A E B C (2) C B E A D (2) A B C E D (2) E C A B D (1) E A D C B (1) E A D B C (1) E A C B D (1) D E C A B (1) D E A C B (1) D C E B A (1) D B C A E (1) C D B E A (1) B A C E D (1) A E B D C (1) A C E B D (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 10 16 -8 -14 B -10 0 -8 -20 -22 C -16 8 0 -18 -22 D 8 20 18 0 0 E 14 22 22 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.758249 E: 0.241751 Sum of squares = 0.633384957879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.758249 E: 1.000000 A B C D E A 0 10 16 -8 -14 B -10 0 -8 -20 -22 C -16 8 0 -18 -22 D 8 20 18 0 0 E 14 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=25 C=17 D=14 B=12 so B is eliminated. Round 2 votes counts: A=33 E=32 C=21 D=14 so D is eliminated. Round 3 votes counts: E=42 A=35 C=23 so C is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 D:223 A:202 C:176 B:170 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 10 16 -8 -14 B -10 0 -8 -20 -22 C -16 8 0 -18 -22 D 8 20 18 0 0 E 14 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 16 -8 -14 B -10 0 -8 -20 -22 C -16 8 0 -18 -22 D 8 20 18 0 0 E 14 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 16 -8 -14 B -10 0 -8 -20 -22 C -16 8 0 -18 -22 D 8 20 18 0 0 E 14 22 22 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8997: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C D E B A (7) E B D C A (5) D C B E A (5) A B D E C (5) D B C E A (4) B D E C A (4) A E C B D (4) A E B C D (4) A C E D B (4) A B D C E (4) D C E B A (3) B A E D C (3) A B E D C (3) A B E C D (3) E D C B A (2) E B C D A (2) D C B A E (2) C A D E B (2) B E D C A (2) B E D A C (2) B E A D C (2) A E C D B (2) A E B D C (2) E D B C A (1) E B A D C (1) E B A C D (1) E A C D B (1) E A C B D (1) D E C B A (1) D B C A E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D B E A (1) C D A E B (1) C D A B E (1) B D C E A (1) B D A E C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 -10 -10 -12 B 16 0 4 2 -10 C 10 -4 0 -10 -18 D 10 -2 10 0 -10 E 12 10 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -16 -10 -10 -12 B 16 0 4 2 -10 C 10 -4 0 -10 -18 D 10 -2 10 0 -10 E 12 10 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=21 D=16 C=15 B=15 so C is eliminated. Round 2 votes counts: A=35 D=27 E=23 B=15 so B is eliminated. Round 3 votes counts: A=38 D=33 E=29 so E is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:225 B:206 D:204 C:189 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -16 -10 -10 -12 B 16 0 4 2 -10 C 10 -4 0 -10 -18 D 10 -2 10 0 -10 E 12 10 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -10 -12 B 16 0 4 2 -10 C 10 -4 0 -10 -18 D 10 -2 10 0 -10 E 12 10 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -10 -12 B 16 0 4 2 -10 C 10 -4 0 -10 -18 D 10 -2 10 0 -10 E 12 10 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 8998: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) D C E A B (6) D E C A B (5) B E D C A (5) B C A D E (5) B A E C D (5) C D A E B (4) C B D A E (4) E D A C B (3) D E C B A (3) B E D A C (3) B A C E D (3) A C B D E (3) E D B A C (2) E D A B C (2) E B D A C (2) E A D C B (2) D E A C B (2) D C E B A (2) D C A E B (2) C A D E B (2) C A D B E (2) B E A C D (2) B C D E A (2) A E D B C (2) A C E B D (2) A C B E D (2) E D B C A (1) E A B D C (1) D A C E B (1) C D B E A (1) C D B A E (1) C B D E A (1) C B A D E (1) B D C E A (1) B C E D A (1) A E D C B (1) A E B D C (1) A C D E B (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 2 -8 -10 B 6 0 -2 10 6 C -2 2 0 -12 -2 D 8 -10 12 0 0 E 10 -6 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.416667 D: 0.083333 E: 0.000000 Sum of squares = 0.430555555551 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.916667 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -8 -10 B 6 0 -2 10 6 C -2 2 0 -12 -2 D 8 -10 12 0 0 E 10 -6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.416667 D: 0.083333 E: 0.000000 Sum of squares = 0.430555555553 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.916667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=21 C=16 A=15 E=13 so E is eliminated. Round 2 votes counts: B=37 D=29 A=18 C=16 so C is eliminated. Round 3 votes counts: B=43 D=35 A=22 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:205 E:203 C:193 A:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -8 -10 B 6 0 -2 10 6 C -2 2 0 -12 -2 D 8 -10 12 0 0 E 10 -6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.416667 D: 0.083333 E: 0.000000 Sum of squares = 0.430555555553 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.916667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -8 -10 B 6 0 -2 10 6 C -2 2 0 -12 -2 D 8 -10 12 0 0 E 10 -6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.416667 D: 0.083333 E: 0.000000 Sum of squares = 0.430555555553 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.916667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -8 -10 B 6 0 -2 10 6 C -2 2 0 -12 -2 D 8 -10 12 0 0 E 10 -6 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.416667 D: 0.083333 E: 0.000000 Sum of squares = 0.430555555553 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.916667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 8999: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (6) C B E A D (6) B E D C A (6) B A C D E (6) B A D E C (5) E D C A B (4) B E C D A (4) A D C E B (4) A C D E B (4) A C D B E (4) E B D C A (3) E B C D A (3) D E A C B (3) C B A E D (3) E C D B A (2) E C D A B (2) D E A B C (2) D A E C B (2) C E B D A (2) C D E A B (2) C A E D B (2) B E D A C (2) B D E A C (2) B C E A D (2) A D B E C (2) E D C B A (1) E C B D A (1) D A E B C (1) C E A D B (1) C E A B D (1) C D A E B (1) C B E D A (1) C A E B D (1) C A D E B (1) C A B E D (1) B D A E C (1) B C A E D (1) B A D C E (1) A D E C B (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 -2 -12 B 4 0 -14 8 0 C 14 14 0 16 6 D 2 -8 -16 0 -10 E 12 0 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -2 -12 B 4 0 -14 8 0 C 14 14 0 16 6 D 2 -8 -16 0 -10 E 12 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=28 A=18 E=16 D=8 so D is eliminated. Round 2 votes counts: B=30 C=28 E=21 A=21 so E is eliminated. Round 3 votes counts: C=38 B=36 A=26 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:208 B:199 A:184 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -2 -12 B 4 0 -14 8 0 C 14 14 0 16 6 D 2 -8 -16 0 -10 E 12 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -2 -12 B 4 0 -14 8 0 C 14 14 0 16 6 D 2 -8 -16 0 -10 E 12 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -2 -12 B 4 0 -14 8 0 C 14 14 0 16 6 D 2 -8 -16 0 -10 E 12 0 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9000: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) E C B D A (7) D A B C E (7) A C B D E (7) C B D A E (5) E A D C B (4) E A D B C (4) E A C B D (4) C B A D E (4) A D C B E (4) E C B A D (3) C B E D A (3) A D E B C (3) A D B C E (3) E D B C A (2) E C A B D (2) D E A B C (2) D B C A E (2) D B A C E (2) D A E B C (2) D A B E C (2) C B A E D (2) B C E D A (2) B C D E A (2) B C D A E (2) E D B A C (1) E B C D A (1) D E B C A (1) D E B A C (1) D B E A C (1) D B C E A (1) C E B A D (1) C B D E A (1) C A B D E (1) B D C A E (1) A E D B C (1) A C D B E (1) Total count = 100 A B C D E A 0 10 14 -12 -2 B -10 0 2 -4 6 C -14 -2 0 -4 2 D 12 4 4 0 10 E 2 -6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 -12 -2 B -10 0 2 -4 6 C -14 -2 0 -4 2 D 12 4 4 0 10 E 2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=21 A=19 C=17 B=7 so B is eliminated. Round 2 votes counts: E=36 C=23 D=22 A=19 so A is eliminated. Round 3 votes counts: E=37 D=32 C=31 so C is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:205 B:197 E:192 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 14 -12 -2 B -10 0 2 -4 6 C -14 -2 0 -4 2 D 12 4 4 0 10 E 2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 -12 -2 B -10 0 2 -4 6 C -14 -2 0 -4 2 D 12 4 4 0 10 E 2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 -12 -2 B -10 0 2 -4 6 C -14 -2 0 -4 2 D 12 4 4 0 10 E 2 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999193 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9001: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (14) B E C D A (11) E C D A B (9) B A D C E (9) A D C B E (8) E B C D A (6) E C B D A (4) C D A E B (4) B E D C A (4) B A E D C (4) C E D A B (3) B E A C D (3) A C D E B (3) A B D C E (3) E C D B A (2) D A C B E (2) B E A D C (2) E D C A B (1) D C A E B (1) D A B C E (1) C D E A B (1) C A D E B (1) B E C A D (1) B A E C D (1) B A D E C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 4 2 6 B -4 0 -6 -2 2 C -4 6 0 -2 2 D -2 2 2 0 -2 E -6 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 2 6 B -4 0 -6 -2 2 C -4 6 0 -2 2 D -2 2 2 0 -2 E -6 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=29 E=22 C=9 D=4 so D is eliminated. Round 2 votes counts: B=36 A=32 E=22 C=10 so C is eliminated. Round 3 votes counts: A=38 B=36 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 C:201 D:200 E:196 B:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 6 B -4 0 -6 -2 2 C -4 6 0 -2 2 D -2 2 2 0 -2 E -6 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 6 B -4 0 -6 -2 2 C -4 6 0 -2 2 D -2 2 2 0 -2 E -6 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 6 B -4 0 -6 -2 2 C -4 6 0 -2 2 D -2 2 2 0 -2 E -6 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999636 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9002: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (6) E D C A B (5) A E D B C (5) E C B A D (4) D A E C B (4) A E D C B (4) A D B C E (4) A B E C D (4) A B D C E (4) E A D C B (3) C B D E A (3) B D A C E (3) B C E D A (3) B C A D E (3) B A C E D (3) A E B C D (3) E D A C B (2) D C E B A (2) D C E A B (2) D A C E B (2) D A C B E (2) C E B D A (2) C D E B A (2) C D B E A (2) B E C A D (2) B A D C E (2) B A C D E (2) A D E C B (2) E D C B A (1) E C D A B (1) E C A B D (1) E B C A D (1) E A C B D (1) E A B C D (1) D E C A B (1) D C B E A (1) D A B C E (1) C B E D A (1) B C D E A (1) B C D A E (1) A D B E C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 4 4 B -10 0 -8 -6 -10 C -10 8 0 -4 -6 D -4 6 4 0 -8 E -4 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 4 4 B -10 0 -8 -6 -10 C -10 8 0 -4 -6 D -4 6 4 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=26 B=20 D=15 C=10 so C is eliminated. Round 2 votes counts: A=29 E=28 B=24 D=19 so D is eliminated. Round 3 votes counts: A=38 E=35 B=27 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:210 D:199 C:194 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 4 4 B -10 0 -8 -6 -10 C -10 8 0 -4 -6 D -4 6 4 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 4 4 B -10 0 -8 -6 -10 C -10 8 0 -4 -6 D -4 6 4 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 4 4 B -10 0 -8 -6 -10 C -10 8 0 -4 -6 D -4 6 4 0 -8 E -4 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9003: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (8) A D C E B (8) E A C D B (7) E C A D B (6) B E C D A (6) E B A C D (5) C D A E B (5) B D C A E (5) B E A C D (4) E B C A D (3) D C A B E (3) B E D C A (3) B E C A D (3) B E A D C (3) A E C D B (3) A C D E B (3) E B C D A (2) D C A E B (2) D A C E B (2) C D E A B (2) B A D C E (2) E C B D A (1) E A C B D (1) D A C B E (1) C E D B A (1) C E B D A (1) C E A D B (1) C D B E A (1) C D B A E (1) C D A B E (1) B E D A C (1) B D E C A (1) B D C E A (1) B A E D C (1) B A D E C (1) A D C B E (1) Total count = 100 A B C D E A 0 -8 2 4 -6 B 8 0 -2 4 -6 C -2 2 0 14 -2 D -4 -4 -14 0 -4 E 6 6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 2 4 -6 B 8 0 -2 4 -6 C -2 2 0 14 -2 D -4 -4 -14 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=25 A=15 C=13 D=8 so D is eliminated. Round 2 votes counts: B=39 E=25 C=18 A=18 so C is eliminated. Round 3 votes counts: B=41 E=30 A=29 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:206 B:202 A:196 D:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 2 4 -6 B 8 0 -2 4 -6 C -2 2 0 14 -2 D -4 -4 -14 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 4 -6 B 8 0 -2 4 -6 C -2 2 0 14 -2 D -4 -4 -14 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 4 -6 B 8 0 -2 4 -6 C -2 2 0 14 -2 D -4 -4 -14 0 -4 E 6 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9004: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) D E A C B (8) B C A E D (8) C B A E D (6) A E B D C (6) E D A B C (5) E A D B C (5) A E D B C (5) C D B E A (4) B A E C D (4) D E C A B (3) B C D E A (3) B A C E D (3) E D A C B (2) C D E A B (2) C B D E A (2) C B D A E (2) C B A D E (2) C A E D B (2) B C A D E (2) E D C A B (1) D E C B A (1) D E B A C (1) D C E B A (1) D B E A C (1) D B C E A (1) B D E C A (1) B D E A C (1) B D C E A (1) B C D A E (1) A E C D B (1) A E B C D (1) A C B E D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 14 18 14 B 10 0 24 14 12 C -14 -24 0 -8 -16 D -18 -14 8 0 -26 E -14 -12 16 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 14 18 14 B 10 0 24 14 12 C -14 -24 0 -8 -16 D -18 -14 8 0 -26 E -14 -12 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=20 A=17 D=16 E=13 so E is eliminated. Round 2 votes counts: B=34 D=24 A=22 C=20 so C is eliminated. Round 3 votes counts: B=46 D=30 A=24 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:230 A:218 E:208 D:175 C:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 14 18 14 B 10 0 24 14 12 C -14 -24 0 -8 -16 D -18 -14 8 0 -26 E -14 -12 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 18 14 B 10 0 24 14 12 C -14 -24 0 -8 -16 D -18 -14 8 0 -26 E -14 -12 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 18 14 B 10 0 24 14 12 C -14 -24 0 -8 -16 D -18 -14 8 0 -26 E -14 -12 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9005: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (7) B D E C A (6) D B C E A (5) C B D E A (5) C A D B E (5) C A E B D (4) B E D C A (4) A C D B E (4) E B A D C (3) E A B C D (3) D B C A E (3) C D B A E (3) B E C D A (3) A D C B E (3) A C D E B (3) E C A B D (2) E B C D A (2) E B A C D (2) E A C B D (2) E A B D C (2) D B E A C (2) C E A B D (2) C D A B E (2) A E C D B (2) A D E B C (2) E B D C A (1) E B D A C (1) E A D B C (1) D E B A C (1) D C B A E (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C B E (1) C D B E A (1) C B D A E (1) C A E D B (1) C A D E B (1) B D E A C (1) A E D B C (1) A E C B D (1) A E B D C (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 6 -6 8 2 B -6 0 -8 -6 6 C 6 8 0 14 10 D -8 6 -14 0 8 E -2 -6 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 8 2 B -6 0 -8 -6 6 C 6 8 0 14 10 D -8 6 -14 0 8 E -2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 C=25 E=19 D=16 B=14 so B is eliminated. Round 2 votes counts: E=26 A=26 C=25 D=23 so D is eliminated. Round 3 votes counts: E=36 C=35 A=29 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:205 D:196 B:193 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 8 2 B -6 0 -8 -6 6 C 6 8 0 14 10 D -8 6 -14 0 8 E -2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 8 2 B -6 0 -8 -6 6 C 6 8 0 14 10 D -8 6 -14 0 8 E -2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 8 2 B -6 0 -8 -6 6 C 6 8 0 14 10 D -8 6 -14 0 8 E -2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9006: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (12) E C A B D (6) C E A B D (6) B D C E A (6) A C E B D (6) E A C D B (5) D E B C A (5) E C A D B (4) D B E C A (4) C A E B D (4) A B D C E (4) D B E A C (3) D B A E C (3) B D C A E (3) A C B D E (3) D E B A C (2) A E C D B (2) A E C B D (2) A C B E D (2) E D C B A (1) E D B A C (1) E D A B C (1) E C D B A (1) E A D C B (1) D B C E A (1) D B C A E (1) D B A C E (1) C E B A D (1) C B D A E (1) C B A E D (1) C B A D E (1) C A B E D (1) B C D E A (1) B C A D E (1) B A D C E (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 2 6 2 B 2 0 2 26 2 C -2 -2 0 -2 16 D -6 -26 2 0 8 E -2 -2 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998793 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 6 2 B 2 0 2 26 2 C -2 -2 0 -2 16 D -6 -26 2 0 8 E -2 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=21 E=20 D=20 C=15 so C is eliminated. Round 2 votes counts: E=27 B=27 A=26 D=20 so D is eliminated. Round 3 votes counts: B=40 E=34 A=26 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 C:205 A:204 D:189 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 6 2 B 2 0 2 26 2 C -2 -2 0 -2 16 D -6 -26 2 0 8 E -2 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 6 2 B 2 0 2 26 2 C -2 -2 0 -2 16 D -6 -26 2 0 8 E -2 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 6 2 B 2 0 2 26 2 C -2 -2 0 -2 16 D -6 -26 2 0 8 E -2 -2 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9007: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) A D B C E (6) E A D C B (4) D B A E C (4) D A B E C (4) C E B A D (4) C B E D A (4) B E C D A (4) A D B E C (4) E A D B C (3) E A C D B (3) C B D A E (3) B D E A C (3) B D A C E (3) E D A B C (2) D E A B C (2) D B A C E (2) C E A D B (2) C E A B D (2) C B A D E (2) C A D E B (2) C A B D E (2) B D C A E (2) B D A E C (2) A E D C B (2) A D C E B (2) A D C B E (2) E D B A C (1) E C B A D (1) E C A D B (1) E B C D A (1) C E B D A (1) C B E A D (1) C B D E A (1) C B A E D (1) C A E D B (1) C A D B E (1) B E D C A (1) B D C E A (1) B C D E A (1) A E C D B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 4 -2 -4 B 4 0 -6 -2 8 C -4 6 0 -2 -6 D 2 2 2 0 0 E 4 -8 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.879838 E: 0.120162 Sum of squares = 0.788553818901 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.879838 E: 1.000000 A B C D E A 0 -4 4 -2 -4 B 4 0 -6 -2 8 C -4 6 0 -2 -6 D 2 2 2 0 0 E 4 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000061089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=25 A=19 B=17 D=12 so D is eliminated. Round 2 votes counts: E=27 C=27 B=23 A=23 so B is eliminated. Round 3 votes counts: E=35 A=34 C=31 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:203 B:202 E:201 A:197 C:197 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 4 -2 -4 B 4 0 -6 -2 8 C -4 6 0 -2 -6 D 2 2 2 0 0 E 4 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000061089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 -2 -4 B 4 0 -6 -2 8 C -4 6 0 -2 -6 D 2 2 2 0 0 E 4 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000061089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 -2 -4 B 4 0 -6 -2 8 C -4 6 0 -2 -6 D 2 2 2 0 0 E 4 -8 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 0.200000 Sum of squares = 0.680000061089 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9008: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) A D C E B (7) D B E A C (6) D E B C A (5) B E D C A (5) B E C D A (5) D A E B C (4) C B E A D (4) A C D E B (4) E B D C A (3) E B C D A (3) C E B A D (3) C A E D B (3) C A B E D (3) A C D B E (3) A C B E D (3) D E B A C (2) D E A B C (2) D A B E C (2) B D E A C (2) A D C B E (2) A D B E C (2) A C B D E (2) A B C E D (2) E D B C A (1) D B E C A (1) D A C E B (1) C E A B D (1) C D A E B (1) C A D E B (1) B E C A D (1) B E A C D (1) B A E D C (1) B A D E C (1) A D B C E (1) A C E B D (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 2 14 10 B -12 0 4 4 -2 C -2 -4 0 2 4 D -14 -4 -2 0 0 E -10 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999884 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 14 10 B -12 0 4 4 -2 C -2 -4 0 2 4 D -14 -4 -2 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=25 D=23 B=16 E=7 so E is eliminated. Round 2 votes counts: A=29 C=25 D=24 B=22 so B is eliminated. Round 3 votes counts: D=34 C=34 A=32 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:219 C:200 B:197 E:194 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 14 10 B -12 0 4 4 -2 C -2 -4 0 2 4 D -14 -4 -2 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 14 10 B -12 0 4 4 -2 C -2 -4 0 2 4 D -14 -4 -2 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 14 10 B -12 0 4 4 -2 C -2 -4 0 2 4 D -14 -4 -2 0 0 E -10 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997477 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9009: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (14) C E A D B (7) B D A C E (7) B D A E C (6) C A E D B (5) D E B A C (4) D B E A C (4) B D E A C (4) A C E D B (4) B A D C E (3) B A C D E (3) A C E B D (3) E D C A B (2) E D A C B (2) E D A B C (2) E C A D B (2) E A C D B (2) C B A D E (2) C A B E D (2) B D E C A (2) B D C A E (2) B C D A E (2) B C A D E (2) A C B E D (2) E C D A B (1) D E B C A (1) D B E C A (1) C E D A B (1) C E B D A (1) C B D E A (1) C A B D E (1) B A D E C (1) A E C D B (1) A E C B D (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 2 14 26 B -8 0 -8 20 -6 C -2 8 0 16 26 D -14 -20 -16 0 -4 E -26 6 -26 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 14 26 B -8 0 -8 20 -6 C -2 8 0 16 26 D -14 -20 -16 0 -4 E -26 6 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=32 A=13 E=11 D=10 so D is eliminated. Round 2 votes counts: B=37 C=34 E=16 A=13 so A is eliminated. Round 3 votes counts: C=43 B=39 E=18 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:225 C:224 B:199 E:179 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 14 26 B -8 0 -8 20 -6 C -2 8 0 16 26 D -14 -20 -16 0 -4 E -26 6 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 14 26 B -8 0 -8 20 -6 C -2 8 0 16 26 D -14 -20 -16 0 -4 E -26 6 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 14 26 B -8 0 -8 20 -6 C -2 8 0 16 26 D -14 -20 -16 0 -4 E -26 6 -26 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994799 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9010: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) D A B E C (9) A B D E C (8) A B E C D (7) A C E B D (6) D C E B A (5) A D B E C (5) D B A E C (4) C E B D A (4) C E A B D (4) A D B C E (4) D A C E B (3) D A B C E (3) C E D B A (3) E C B D A (2) D C E A B (2) D C B E A (2) C D E B A (2) E C B A D (1) E B C A D (1) D E C B A (1) D C B A E (1) D C A B E (1) D B E A C (1) D B C A E (1) D A C B E (1) C E D A B (1) B E D C A (1) B E C A D (1) B E A C D (1) A E C B D (1) A E B C D (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 16 12 6 14 B -16 0 -6 -2 2 C -12 6 0 -6 12 D -6 2 6 0 10 E -14 -2 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 6 14 B -16 0 -6 -2 2 C -12 6 0 -6 12 D -6 2 6 0 10 E -14 -2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=34 A=34 C=25 E=4 B=3 so B is eliminated. Round 2 votes counts: D=34 A=34 C=25 E=7 so E is eliminated. Round 3 votes counts: D=35 A=35 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:206 C:200 B:189 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 6 14 B -16 0 -6 -2 2 C -12 6 0 -6 12 D -6 2 6 0 10 E -14 -2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 6 14 B -16 0 -6 -2 2 C -12 6 0 -6 12 D -6 2 6 0 10 E -14 -2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 6 14 B -16 0 -6 -2 2 C -12 6 0 -6 12 D -6 2 6 0 10 E -14 -2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9011: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) E D B A C (7) D B C A E (6) E A C B D (5) A C B D E (5) E D C B A (4) E D B C A (4) B D A C E (4) A E C B D (4) A B C D E (4) E D C A B (3) E A C D B (3) C A E B D (3) C A B D E (3) B A C D E (3) E A B D C (2) E A B C D (2) D E B C A (2) D C B E A (2) D B C E A (2) B D C A E (2) B D A E C (2) A E B C D (2) A C E B D (2) A B D C E (2) E C D B A (1) E C A D B (1) E A D B C (1) D E C B A (1) D B E A C (1) C D E B A (1) C D B A E (1) C A D B E (1) C A B E D (1) B D E A C (1) B A E D C (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -10 6 -6 -4 B 10 0 16 0 4 C -6 -16 0 -14 -12 D 6 0 14 0 6 E 4 -4 12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.528960 C: 0.000000 D: 0.471040 E: 0.000000 Sum of squares = 0.50167738579 Cumulative probabilities = A: 0.000000 B: 0.528960 C: 0.528960 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -6 -4 B 10 0 16 0 4 C -6 -16 0 -14 -12 D 6 0 14 0 6 E 4 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=23 A=20 B=14 C=10 so C is eliminated. Round 2 votes counts: E=33 A=28 D=25 B=14 so B is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:215 D:213 E:203 A:193 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 -6 -4 B 10 0 16 0 4 C -6 -16 0 -14 -12 D 6 0 14 0 6 E 4 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -6 -4 B 10 0 16 0 4 C -6 -16 0 -14 -12 D 6 0 14 0 6 E 4 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -6 -4 B 10 0 16 0 4 C -6 -16 0 -14 -12 D 6 0 14 0 6 E 4 -4 12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9012: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) C B E D A (8) B C D A E (7) E C B A D (6) B C A D E (6) A E D B C (6) A D E B C (6) C B D E A (4) D C B A E (3) D A E C B (3) D A E B C (3) D A B C E (3) B C A E D (3) A D B C E (3) A B D C E (3) E D C B A (2) E D A C B (2) E C B D A (2) D E C B A (2) B D A C E (2) B C E A D (2) A E B D C (2) A B C E D (2) E C D B A (1) E B C A D (1) E A B C D (1) D E A C B (1) D C E B A (1) C E B A D (1) C B E A D (1) B C D E A (1) B A C E D (1) B A C D E (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -2 10 12 B 10 0 8 10 0 C 2 -8 0 -2 4 D -10 -10 2 0 -2 E -12 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.667110 C: 0.000000 D: 0.000000 E: 0.332890 Sum of squares = 0.555851775317 Cumulative probabilities = A: 0.000000 B: 0.667110 C: 0.667110 D: 0.667110 E: 1.000000 A B C D E A 0 -10 -2 10 12 B 10 0 8 10 0 C 2 -8 0 -2 4 D -10 -10 2 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.000000 E: 0.454545 Sum of squares = 0.504132241545 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.545455 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=23 B=23 D=16 C=14 so C is eliminated. Round 2 votes counts: B=36 E=24 A=24 D=16 so D is eliminated. Round 3 votes counts: B=39 A=33 E=28 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:205 C:198 E:193 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 10 12 B 10 0 8 10 0 C 2 -8 0 -2 4 D -10 -10 2 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.000000 E: 0.454545 Sum of squares = 0.504132241545 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.545455 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 10 12 B 10 0 8 10 0 C 2 -8 0 -2 4 D -10 -10 2 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.000000 E: 0.454545 Sum of squares = 0.504132241545 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.545455 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 10 12 B 10 0 8 10 0 C 2 -8 0 -2 4 D -10 -10 2 0 -2 E -12 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.545455 C: 0.000000 D: 0.000000 E: 0.454545 Sum of squares = 0.504132241545 Cumulative probabilities = A: 0.000000 B: 0.545455 C: 0.545455 D: 0.545455 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9013: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) C A E D B (9) E A D B C (6) B D E A C (6) A E D C B (6) A E C D B (6) C B A D E (5) B D E C A (5) C B E A D (4) C B D E A (4) B D C E A (4) B C D E A (4) C B A E D (3) C A B E D (3) A C E D B (3) E D A B C (2) E A C D B (2) C B E D A (2) C A E B D (2) B D A C E (2) B C D A E (2) A E D B C (2) E A D C B (1) D E B A C (1) D B A C E (1) D A E B C (1) D A B C E (1) C B D A E (1) C A B D E (1) B D C A E (1) Total count = 100 A B C D E A 0 -10 0 6 -2 B 10 0 -4 -2 18 C 0 4 0 2 4 D -6 2 -2 0 -2 E 2 -18 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.192483 B: 0.000000 C: 0.807517 D: 0.000000 E: 0.000000 Sum of squares = 0.689132802674 Cumulative probabilities = A: 0.192483 B: 0.192483 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 6 -2 B 10 0 -4 -2 18 C 0 4 0 2 4 D -6 2 -2 0 -2 E 2 -18 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836806219 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=24 A=17 D=14 E=11 so E is eliminated. Round 2 votes counts: C=34 A=26 B=24 D=16 so D is eliminated. Round 3 votes counts: B=36 C=34 A=30 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:211 C:205 A:197 D:196 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 0 6 -2 B 10 0 -4 -2 18 C 0 4 0 2 4 D -6 2 -2 0 -2 E 2 -18 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836806219 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 -2 B 10 0 -4 -2 18 C 0 4 0 2 4 D -6 2 -2 0 -2 E 2 -18 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836806219 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 -2 B 10 0 -4 -2 18 C 0 4 0 2 4 D -6 2 -2 0 -2 E 2 -18 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591836806219 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9014: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (12) D B E A C (11) C A E B D (7) E A D B C (6) B C D A E (6) E D A B C (5) C B A D E (5) B D C A E (5) C E A D B (4) B D C E A (4) E A C D B (3) D E A B C (3) D B E C A (3) D B C E A (3) C A B E D (3) A E D B C (3) E A D C B (2) D C B E A (2) C B D E A (2) B D A E C (2) B C A D E (2) E D A C B (1) E C A D B (1) C E A B D (1) A E C D B (1) A E C B D (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -20 -18 -2 B 14 0 8 4 20 C 20 -8 0 -2 14 D 18 -4 2 0 20 E 2 -20 -14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -18 -2 B 14 0 8 4 20 C 20 -8 0 -2 14 D 18 -4 2 0 20 E 2 -20 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998252 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=22 B=19 E=18 A=7 so A is eliminated. Round 2 votes counts: C=35 E=24 D=22 B=19 so B is eliminated. Round 3 votes counts: C=43 D=33 E=24 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:223 D:218 C:212 E:174 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -20 -18 -2 B 14 0 8 4 20 C 20 -8 0 -2 14 D 18 -4 2 0 20 E 2 -20 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998252 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -18 -2 B 14 0 8 4 20 C 20 -8 0 -2 14 D 18 -4 2 0 20 E 2 -20 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998252 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -18 -2 B 14 0 8 4 20 C 20 -8 0 -2 14 D 18 -4 2 0 20 E 2 -20 -14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998252 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9015: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) C B E D A (9) A D E B C (7) A B D E C (6) D E C B A (5) A D B E C (5) D E C A B (4) B C A E D (4) E D C B A (3) B A C E D (3) A D E C B (3) A B C D E (3) E C D B A (2) D E A B C (2) D A E C B (2) D A E B C (2) C E B D A (2) C D E A B (2) B E D C A (2) B E C D A (2) A C D E B (2) A C B E D (2) A B D C E (2) A B C E D (2) E B D C A (1) E B C D A (1) D E A C B (1) C D E B A (1) C B E A D (1) B E D A C (1) B E C A D (1) B C E A D (1) B A E C D (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 -10 -12 -10 B 8 0 -8 -12 -10 C 10 8 0 6 -2 D 12 12 -6 0 -4 E 10 10 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -10 -12 -10 B 8 0 -8 -12 -10 C 10 8 0 6 -2 D 12 12 -6 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=29 D=16 B=15 E=7 so E is eliminated. Round 2 votes counts: A=33 C=31 D=19 B=17 so B is eliminated. Round 3 votes counts: C=40 A=37 D=23 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:213 C:211 D:207 B:189 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -12 -10 B 8 0 -8 -12 -10 C 10 8 0 6 -2 D 12 12 -6 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -12 -10 B 8 0 -8 -12 -10 C 10 8 0 6 -2 D 12 12 -6 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -12 -10 B 8 0 -8 -12 -10 C 10 8 0 6 -2 D 12 12 -6 0 -4 E 10 10 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9016: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (12) D E A C B (10) B C A E D (8) D A E C B (7) A E D B C (7) E D A B C (5) C B D A E (5) B C E A D (5) E A D B C (4) D E A B C (4) C B A E D (4) C B A D E (4) C B E D A (3) B C E D A (3) E B A D C (2) D E C A B (2) D C A E B (2) A B E C D (2) E B D A C (1) D E C B A (1) D C E B A (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A E B (1) C A B D E (1) A D E C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -10 -20 -10 B 4 0 -14 2 0 C 10 14 0 2 4 D 20 -2 -2 0 10 E 10 0 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -20 -10 B 4 0 -14 2 0 C 10 14 0 2 4 D 20 -2 -2 0 10 E 10 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=27 B=16 E=12 A=12 so E is eliminated. Round 2 votes counts: C=33 D=32 B=19 A=16 so A is eliminated. Round 3 votes counts: D=45 C=33 B=22 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 D:213 E:198 B:196 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -10 -20 -10 B 4 0 -14 2 0 C 10 14 0 2 4 D 20 -2 -2 0 10 E 10 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -20 -10 B 4 0 -14 2 0 C 10 14 0 2 4 D 20 -2 -2 0 10 E 10 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -20 -10 B 4 0 -14 2 0 C 10 14 0 2 4 D 20 -2 -2 0 10 E 10 0 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9017: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (11) D C A E B (7) D C A B E (7) B D E C A (7) E B D A C (6) A C D E B (6) B E D C A (5) E D A C B (4) D C B A E (4) B E A C D (4) B D C A E (4) D B E C A (3) C A D B E (3) B E D A C (3) E D C A B (2) E B A D C (2) E A D C B (2) E A B C D (2) D B C A E (2) B D C E A (2) A C D B E (2) E D B A C (1) E A C D B (1) E A C B D (1) D E C A B (1) D E B C A (1) C D A B E (1) B E A D C (1) B C A D E (1) B A C E D (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -16 0 -20 -18 B 16 0 12 2 0 C 0 -12 0 -28 -18 D 20 -2 28 0 2 E 18 0 18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.674568 C: 0.000000 D: 0.000000 E: 0.325432 Sum of squares = 0.560948087687 Cumulative probabilities = A: 0.000000 B: 0.674568 C: 0.674568 D: 0.674568 E: 1.000000 A B C D E A 0 -16 0 -20 -18 B 16 0 12 2 0 C 0 -12 0 -28 -18 D 20 -2 28 0 2 E 18 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500374 C: 0.000000 D: 0.000000 E: 0.499626 Sum of squares = 0.50000027955 Cumulative probabilities = A: 0.000000 B: 0.500374 C: 0.500374 D: 0.500374 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=28 D=25 A=11 C=4 so C is eliminated. Round 2 votes counts: E=32 B=28 D=26 A=14 so A is eliminated. Round 3 votes counts: D=37 E=35 B=28 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:217 B:215 A:173 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 0 -20 -18 B 16 0 12 2 0 C 0 -12 0 -28 -18 D 20 -2 28 0 2 E 18 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500374 C: 0.000000 D: 0.000000 E: 0.499626 Sum of squares = 0.50000027955 Cumulative probabilities = A: 0.000000 B: 0.500374 C: 0.500374 D: 0.500374 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -20 -18 B 16 0 12 2 0 C 0 -12 0 -28 -18 D 20 -2 28 0 2 E 18 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500374 C: 0.000000 D: 0.000000 E: 0.499626 Sum of squares = 0.50000027955 Cumulative probabilities = A: 0.000000 B: 0.500374 C: 0.500374 D: 0.500374 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -20 -18 B 16 0 12 2 0 C 0 -12 0 -28 -18 D 20 -2 28 0 2 E 18 0 18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500374 C: 0.000000 D: 0.000000 E: 0.499626 Sum of squares = 0.50000027955 Cumulative probabilities = A: 0.000000 B: 0.500374 C: 0.500374 D: 0.500374 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9018: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (9) D C B A E (9) C D B A E (9) D C A B E (7) E B A C D (6) D C E B A (4) C B D A E (4) B A E C D (4) A B E C D (4) E A B C D (3) D C E A B (3) B C A D E (3) A E B C D (3) E A B D C (2) D E C A B (2) D C A E B (2) D A E C B (2) B C E D A (2) B C D E A (2) A B C E D (2) E D C B A (1) E D A C B (1) E B D C A (1) E B C D A (1) E B C A D (1) E B A D C (1) E A D C B (1) E A D B C (1) D E C B A (1) D E A C B (1) C D A B E (1) C B A D E (1) C A B D E (1) B E A C D (1) B A C E D (1) A E D C B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -22 -28 -24 12 B 22 0 -24 -12 24 C 28 24 0 2 24 D 24 12 -2 0 24 E -12 -24 -24 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -28 -24 12 B 22 0 -24 -12 24 C 28 24 0 2 24 D 24 12 -2 0 24 E -12 -24 -24 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=19 C=16 B=13 A=12 so A is eliminated. Round 2 votes counts: D=40 E=23 B=19 C=18 so C is eliminated. Round 3 votes counts: D=51 B=26 E=23 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:239 D:229 B:205 A:169 E:158 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -28 -24 12 B 22 0 -24 -12 24 C 28 24 0 2 24 D 24 12 -2 0 24 E -12 -24 -24 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -28 -24 12 B 22 0 -24 -12 24 C 28 24 0 2 24 D 24 12 -2 0 24 E -12 -24 -24 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -28 -24 12 B 22 0 -24 -12 24 C 28 24 0 2 24 D 24 12 -2 0 24 E -12 -24 -24 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999974011 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9019: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) B C D A E (9) D E A B C (8) C B A E D (7) B D C A E (7) E A C D B (6) D A E C B (5) C E A B D (3) B D C E A (3) B C E A D (3) A E C D B (3) E A B D C (2) D E A C B (2) D B A E C (2) C D A E B (2) C A E D B (2) C A E B D (2) B E A D C (2) B E A C D (2) B D E A C (2) B D A E C (2) B C D E A (2) B C A E D (2) E D A C B (1) E A D B C (1) D C A E B (1) D B E A C (1) D A C E B (1) C D A B E (1) C B E A D (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D A C (1) Total count = 100 A B C D E A 0 4 4 -2 -2 B -4 0 -2 4 -2 C -4 2 0 -4 -2 D 2 -4 4 0 2 E 2 2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999834 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 4 4 -2 -2 B -4 0 -2 4 -2 C -4 2 0 -4 -2 D 2 -4 4 0 2 E 2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=21 C=21 D=20 A=3 so A is eliminated. Round 2 votes counts: B=35 E=24 C=21 D=20 so D is eliminated. Round 3 votes counts: E=39 B=38 C=23 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:202 D:202 E:202 B:198 C:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 -2 -2 B -4 0 -2 4 -2 C -4 2 0 -4 -2 D 2 -4 4 0 2 E 2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -2 -2 B -4 0 -2 4 -2 C -4 2 0 -4 -2 D 2 -4 4 0 2 E 2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -2 -2 B -4 0 -2 4 -2 C -4 2 0 -4 -2 D 2 -4 4 0 2 E 2 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999994 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9020: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (9) D E C A B (6) B A E D C (6) E D C B A (5) E C D B A (5) C A D E B (5) A B D C E (5) A B D E C (4) D E A B C (3) D C E A B (3) D A E C B (3) C D A E B (3) B E D A C (3) B A E C D (3) B A C E D (3) A B C D E (3) E D B C A (2) E B D C A (2) C E B D A (2) C A B D E (2) B E C D A (2) B E A D C (2) A C B D E (2) A B C E D (2) E D B A C (1) E C B D A (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A C B (1) D A C E B (1) C E D B A (1) C B E A D (1) C A D B E (1) C A B E D (1) B C A E D (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 14 -8 -12 -4 B -14 0 -12 -10 -14 C 8 12 0 -4 -4 D 12 10 4 0 14 E 4 14 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -8 -12 -4 B -14 0 -12 -10 -14 C 8 12 0 -4 -4 D 12 10 4 0 14 E 4 14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=20 B=20 A=19 E=16 so E is eliminated. Round 2 votes counts: C=31 D=28 B=22 A=19 so A is eliminated. Round 3 votes counts: B=36 C=34 D=30 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:220 C:206 E:204 A:195 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -8 -12 -4 B -14 0 -12 -10 -14 C 8 12 0 -4 -4 D 12 10 4 0 14 E 4 14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -8 -12 -4 B -14 0 -12 -10 -14 C 8 12 0 -4 -4 D 12 10 4 0 14 E 4 14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -8 -12 -4 B -14 0 -12 -10 -14 C 8 12 0 -4 -4 D 12 10 4 0 14 E 4 14 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9021: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (12) E A C B D (10) A C E B D (8) C A E D B (7) C A D E B (7) B D E A C (7) C A E B D (6) D C A B E (5) D B E C A (5) A C E D B (5) E B A C D (3) D C B A E (3) D B E A C (3) C D A B E (3) E B D A C (2) D E B A C (2) D C A E B (2) D B C E A (2) B E D A C (2) E B A D C (1) E A C D B (1) E A B C D (1) D A C E B (1) C B A D E (1) A E C D B (1) Total count = 100 A B C D E A 0 14 -6 2 22 B -14 0 -20 -18 -14 C 6 20 0 6 24 D -2 18 -6 0 6 E -22 14 -24 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 2 22 B -14 0 -20 -18 -14 C 6 20 0 6 24 D -2 18 -6 0 6 E -22 14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 C=24 E=18 A=14 B=9 so B is eliminated. Round 2 votes counts: D=42 C=24 E=20 A=14 so A is eliminated. Round 3 votes counts: D=42 C=37 E=21 so E is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:216 D:208 E:181 B:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 2 22 B -14 0 -20 -18 -14 C 6 20 0 6 24 D -2 18 -6 0 6 E -22 14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 2 22 B -14 0 -20 -18 -14 C 6 20 0 6 24 D -2 18 -6 0 6 E -22 14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 2 22 B -14 0 -20 -18 -14 C 6 20 0 6 24 D -2 18 -6 0 6 E -22 14 -24 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999173 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9022: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (10) C E B D A (8) D A C E B (7) A D B E C (7) D A E C B (6) C E D B A (5) C B E D A (4) B E C A D (4) A D C B E (4) E D A B C (3) E C D B A (3) D C E A B (3) D C A E B (3) C D E A B (3) C B A D E (3) B A C E D (3) A D B C E (3) E D C A B (2) B A E D C (2) A D E B C (2) E D C B A (1) E D A C B (1) E C D A B (1) E C B D A (1) E B C D A (1) E B C A D (1) E B A C D (1) D E A C B (1) D A C B E (1) C D A E B (1) B C A E D (1) B A E C D (1) B A C D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -10 -10 -6 B 0 0 -16 -14 -8 C 10 16 0 6 22 D 10 14 -6 0 -6 E 6 8 -22 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -10 -6 B 0 0 -16 -14 -8 C 10 16 0 6 22 D 10 14 -6 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=22 D=21 A=18 E=15 so E is eliminated. Round 2 votes counts: C=29 D=28 B=25 A=18 so A is eliminated. Round 3 votes counts: D=44 C=29 B=27 so B is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:206 E:199 A:187 B:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -10 -6 B 0 0 -16 -14 -8 C 10 16 0 6 22 D 10 14 -6 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -10 -6 B 0 0 -16 -14 -8 C 10 16 0 6 22 D 10 14 -6 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -10 -6 B 0 0 -16 -14 -8 C 10 16 0 6 22 D 10 14 -6 0 -6 E 6 8 -22 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9023: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) D E B A C (5) E B D A C (4) C D A E B (4) C B A E D (4) B E C A D (4) A D C B E (4) E D B C A (3) E C B D A (3) E B D C A (3) D C E A B (3) D A C E B (3) C A D E B (3) C A B D E (3) B E D A C (3) B E A D C (3) B A E C D (3) E D C B A (2) D E B C A (2) C E B A D (2) C D E B A (2) C D E A B (2) B A E D C (2) A C D B E (2) A C B D E (2) A B E D C (2) A B D C E (2) A B C D E (2) E C D B A (1) E B C D A (1) D E C B A (1) D C A E B (1) D A E B C (1) C E D B A (1) C E D A B (1) C B E A D (1) C A E B D (1) C A D B E (1) B E C D A (1) B E A C D (1) A D B E C (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -16 6 2 B 4 0 -12 14 2 C 16 12 0 8 6 D -6 -14 -8 0 -10 E -2 -2 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 6 2 B 4 0 -12 14 2 C 16 12 0 8 6 D -6 -14 -8 0 -10 E -2 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=17 B=17 A=17 D=16 so D is eliminated. Round 2 votes counts: C=37 E=25 A=21 B=17 so B is eliminated. Round 3 votes counts: E=37 C=37 A=26 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:204 E:200 A:194 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -16 6 2 B 4 0 -12 14 2 C 16 12 0 8 6 D -6 -14 -8 0 -10 E -2 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 6 2 B 4 0 -12 14 2 C 16 12 0 8 6 D -6 -14 -8 0 -10 E -2 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 6 2 B 4 0 -12 14 2 C 16 12 0 8 6 D -6 -14 -8 0 -10 E -2 -2 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9024: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) B C D A E (7) E A D C B (6) C D A E B (5) B A D E C (5) E B A D C (4) E A D B C (4) D A C E B (4) C B D E A (4) C B D A E (4) E C A D B (3) E B C A D (3) E A B D C (3) D C A E B (3) D A E C B (3) C D B A E (3) B E C A D (3) E C D A B (2) E C B A D (2) D C A B E (2) D A C B E (2) B D A C E (2) B A E D C (2) A E D B C (2) A D E C B (2) A D B E C (2) E B A C D (1) D E A C B (1) D A B C E (1) C E D A B (1) B E A D C (1) B D C A E (1) B C E D A (1) B C E A D (1) B A D C E (1) A D E B C (1) Total count = 100 A B C D E A 0 10 -6 -8 20 B -10 0 -10 -10 0 C 6 10 0 -4 0 D 8 10 4 0 22 E -20 0 0 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999752 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 -8 20 B -10 0 -10 -10 0 C 6 10 0 -4 0 D 8 10 4 0 22 E -20 0 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=25 B=24 D=16 A=7 so A is eliminated. Round 2 votes counts: E=30 C=25 B=24 D=21 so D is eliminated. Round 3 votes counts: E=37 C=36 B=27 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:222 A:208 C:206 B:185 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -6 -8 20 B -10 0 -10 -10 0 C 6 10 0 -4 0 D 8 10 4 0 22 E -20 0 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 -8 20 B -10 0 -10 -10 0 C 6 10 0 -4 0 D 8 10 4 0 22 E -20 0 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 -8 20 B -10 0 -10 -10 0 C 6 10 0 -4 0 D 8 10 4 0 22 E -20 0 0 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999292 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9025: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) A B D E C (8) D E A C B (5) C D E B A (5) B C A E D (5) D A E B C (4) C E D B A (4) A B E D C (4) E D C A B (3) E D A B C (3) E B A D C (3) C B E A D (3) B A E D C (3) B A C E D (3) A D E B C (3) A D B E C (3) D E A B C (2) D C E A B (2) C E B D A (2) C B E D A (2) C B A E D (2) C B A D E (2) B C A D E (2) B A E C D (2) A D B C E (2) A B C D E (2) E D C B A (1) E C D B A (1) E C B D A (1) E A D B C (1) D A E C B (1) D A C B E (1) C E D A B (1) C D E A B (1) C D A E B (1) B E A C D (1) B A C D E (1) A E D B C (1) A B D C E (1) Total count = 100 A B C D E A 0 14 8 4 2 B -14 0 8 -6 -6 C -8 -8 0 -18 -16 D -4 6 18 0 8 E -2 6 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 4 2 B -14 0 8 -6 -6 C -8 -8 0 -18 -16 D -4 6 18 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 C=23 B=17 E=13 so E is eliminated. Round 2 votes counts: D=30 C=25 A=25 B=20 so B is eliminated. Round 3 votes counts: A=38 C=32 D=30 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 D:214 E:206 B:191 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 8 4 2 B -14 0 8 -6 -6 C -8 -8 0 -18 -16 D -4 6 18 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 4 2 B -14 0 8 -6 -6 C -8 -8 0 -18 -16 D -4 6 18 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 4 2 B -14 0 8 -6 -6 C -8 -8 0 -18 -16 D -4 6 18 0 8 E -2 6 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999292 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9026: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) D A E B C (7) D E A C B (5) D C E A B (4) B A D C E (4) B A C E D (4) A B D E C (4) E D C A B (3) E C D B A (3) E A B D C (3) D A B E C (3) C E D B A (3) C B E A D (3) B C A D E (3) A B E D C (3) E D A C B (2) E C D A B (2) E C B D A (2) E B A C D (2) E A D B C (2) D A C B E (2) D A B C E (2) C D E B A (2) C B D A E (2) B A E C D (2) B A C D E (2) A E D B C (2) A B D C E (2) E C A D B (1) E A B C D (1) D C A E B (1) D C A B E (1) C E B D A (1) C E B A D (1) C D B A E (1) C B E D A (1) B A E D C (1) A E B D C (1) A D E B C (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 8 14 6 0 B -8 0 2 2 -16 C -14 -2 0 -10 -20 D -6 -2 10 0 -4 E 0 16 20 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.636569 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.363431 Sum of squares = 0.537302199163 Cumulative probabilities = A: 0.636569 B: 0.636569 C: 0.636569 D: 0.636569 E: 1.000000 A B C D E A 0 8 14 6 0 B -8 0 2 2 -16 C -14 -2 0 -10 -20 D -6 -2 10 0 -4 E 0 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=25 B=16 A=15 C=14 so C is eliminated. Round 2 votes counts: E=35 D=28 B=22 A=15 so A is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:214 D:199 B:190 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 6 0 B -8 0 2 2 -16 C -14 -2 0 -10 -20 D -6 -2 10 0 -4 E 0 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 6 0 B -8 0 2 2 -16 C -14 -2 0 -10 -20 D -6 -2 10 0 -4 E 0 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 6 0 B -8 0 2 2 -16 C -14 -2 0 -10 -20 D -6 -2 10 0 -4 E 0 16 20 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9027: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) B D A E C (7) B A D C E (7) D B E C A (6) C A E D B (5) B D E A C (5) A C E B D (5) A C B E D (5) E C D A B (4) B D A C E (4) A C E D B (4) E D C B A (3) E D C A B (3) D E C B A (3) A C B D E (3) A B C E D (3) E D B C A (2) E C A B D (2) E B D C A (2) A B C D E (2) E B A C D (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A B E (1) D B A C E (1) C E A D B (1) C D A E B (1) B E D A C (1) B E A D C (1) B A E C D (1) B A D E C (1) B A C D E (1) A E C B D (1) A E B C D (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 12 -8 10 B 12 0 10 8 4 C -12 -10 0 -16 -6 D 8 -8 16 0 8 E -10 -4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 12 -8 10 B 12 0 10 8 4 C -12 -10 0 -16 -6 D 8 -8 16 0 8 E -10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=26 D=22 E=17 C=7 so C is eliminated. Round 2 votes counts: A=31 B=28 D=23 E=18 so E is eliminated. Round 3 votes counts: D=35 A=34 B=31 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:217 D:212 A:201 E:192 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 12 -8 10 B 12 0 10 8 4 C -12 -10 0 -16 -6 D 8 -8 16 0 8 E -10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 12 -8 10 B 12 0 10 8 4 C -12 -10 0 -16 -6 D 8 -8 16 0 8 E -10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 12 -8 10 B 12 0 10 8 4 C -12 -10 0 -16 -6 D 8 -8 16 0 8 E -10 -4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998452 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9028: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (18) B D C E A (8) B C E D A (7) B D E C A (6) A B D E C (6) C E D B A (5) B C D E A (5) A B E C D (5) D E C B A (4) C E D A B (4) A E D C B (4) D C E B A (3) B A D C E (3) A B C E D (3) E C D A B (2) C B E D A (2) B A D E C (2) A D E C B (2) A C B E D (2) E D C A B (1) D B E C A (1) C A E D B (1) B D A E C (1) B D A C E (1) B A C E D (1) A D B E C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 2 2 0 4 B -2 0 2 8 10 C -2 -2 0 12 -8 D 0 -8 -12 0 -14 E -4 -10 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.924367 B: 0.000000 C: 0.000000 D: 0.075633 E: 0.000000 Sum of squares = 0.860175496751 Cumulative probabilities = A: 0.924367 B: 0.924367 C: 0.924367 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 0 4 B -2 0 2 8 10 C -2 -2 0 12 -8 D 0 -8 -12 0 -14 E -4 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.7551020409 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 B=34 C=12 D=8 E=3 so E is eliminated. Round 2 votes counts: A=43 B=34 C=14 D=9 so D is eliminated. Round 3 votes counts: A=43 B=35 C=22 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:209 A:204 E:204 C:200 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 0 4 B -2 0 2 8 10 C -2 -2 0 12 -8 D 0 -8 -12 0 -14 E -4 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.7551020409 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 0 4 B -2 0 2 8 10 C -2 -2 0 12 -8 D 0 -8 -12 0 -14 E -4 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.7551020409 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 0 4 B -2 0 2 8 10 C -2 -2 0 12 -8 D 0 -8 -12 0 -14 E -4 -10 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.857143 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.000000 Sum of squares = 0.7551020409 Cumulative probabilities = A: 0.857143 B: 0.857143 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9029: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) D E C B A (7) D B E C A (5) C A E D B (5) B D E A C (5) A B C E D (5) B A D C E (4) A C B E D (4) E C D A B (3) D E C A B (3) D E B C A (3) D C E A B (3) B E D C A (3) B A E C D (3) E C A D B (2) D C A E B (2) D B A C E (2) C E D A B (2) B D A C E (2) B A C E D (2) B A C D E (2) A C E D B (2) A C D B E (2) A B C D E (2) E D C A B (1) E C A B D (1) E B D C A (1) E B A C D (1) E A C B D (1) E A B C D (1) D C E B A (1) D B C A E (1) C E A D B (1) C A E B D (1) C A D E B (1) B E D A C (1) B E A D C (1) B E A C D (1) B D A E C (1) B A E D C (1) B A D E C (1) Total count = 100 A B C D E A 0 4 8 8 6 B -4 0 -4 8 -4 C -8 4 0 4 8 D -8 -8 -4 0 -6 E -6 4 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 8 6 B -4 0 -4 8 -4 C -8 4 0 4 8 D -8 -8 -4 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 A=25 E=11 C=10 so C is eliminated. Round 2 votes counts: A=32 D=27 B=27 E=14 so E is eliminated. Round 3 votes counts: A=38 D=33 B=29 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:204 B:198 E:198 D:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 8 6 B -4 0 -4 8 -4 C -8 4 0 4 8 D -8 -8 -4 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 6 B -4 0 -4 8 -4 C -8 4 0 4 8 D -8 -8 -4 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 6 B -4 0 -4 8 -4 C -8 4 0 4 8 D -8 -8 -4 0 -6 E -6 4 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9030: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) A E D C B (9) C D B A E (7) B E A C D (7) C B D A E (6) E A B D C (4) E B A D C (3) E A D B C (3) E A B C D (3) D E A B C (3) D C A E B (3) B C E D A (3) A E C D B (3) E D A B C (2) D C B E A (2) D C B A E (2) D B C E A (2) D A E C B (2) C B A E D (2) B E C A D (2) B D C E A (2) B C E A D (2) A E C B D (2) A E B C D (2) A C E D B (2) D E A C B (1) D C A B E (1) C D B E A (1) C D A E B (1) C D A B E (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B D E A C (1) B C A E D (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -2 0 -2 B 6 0 2 6 8 C 2 -2 0 16 2 D 0 -6 -16 0 -8 E 2 -8 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 0 -2 B 6 0 2 6 8 C 2 -2 0 16 2 D 0 -6 -16 0 -8 E 2 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=22 A=19 D=16 E=15 so E is eliminated. Round 2 votes counts: B=31 A=29 C=22 D=18 so D is eliminated. Round 3 votes counts: A=37 B=33 C=30 so C is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 C:209 E:200 A:195 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 0 -2 B 6 0 2 6 8 C 2 -2 0 16 2 D 0 -6 -16 0 -8 E 2 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 0 -2 B 6 0 2 6 8 C 2 -2 0 16 2 D 0 -6 -16 0 -8 E 2 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 0 -2 B 6 0 2 6 8 C 2 -2 0 16 2 D 0 -6 -16 0 -8 E 2 -8 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9031: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (16) D A E C B (12) D E A B C (11) D A E B C (7) C B A E D (7) C B E A D (5) C B D A E (5) E A D B C (3) E A B D C (3) E A B C D (3) B E A C D (3) D C A E B (2) B C D E A (2) A E D C B (2) E D A B C (1) D C B E A (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C E A (1) D A C E B (1) C D A E B (1) C D A B E (1) C A E D B (1) C A E B D (1) C A D E B (1) C A B E D (1) B E A D C (1) B D E A C (1) A E D B C (1) A E B D C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 10 6 2 -6 B -10 0 14 -2 -6 C -6 -14 0 -6 -6 D -2 2 6 0 2 E 6 6 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.439999999998 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 A B C D E A 0 10 6 2 -6 B -10 0 14 -2 -6 C -6 -14 0 -6 -6 D -2 2 6 0 2 E 6 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000005 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=23 B=23 E=10 A=6 so A is eliminated. Round 2 votes counts: D=40 C=23 B=23 E=14 so E is eliminated. Round 3 votes counts: D=47 B=30 C=23 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:208 A:206 D:204 B:198 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 10 6 2 -6 B -10 0 14 -2 -6 C -6 -14 0 -6 -6 D -2 2 6 0 2 E 6 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000005 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 6 2 -6 B -10 0 14 -2 -6 C -6 -14 0 -6 -6 D -2 2 6 0 2 E 6 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000005 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 6 2 -6 B -10 0 14 -2 -6 C -6 -14 0 -6 -6 D -2 2 6 0 2 E 6 6 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.200000 Sum of squares = 0.440000000005 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.200000 D: 0.800000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9032: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (17) E D A B C (13) C B A D E (9) E C B A D (7) D E A B C (6) E C B D A (5) D A E B C (5) C B A E D (5) E D C B A (4) E D A C B (3) E C D B A (3) D A B E C (3) C E B A D (3) B C A D E (3) B A C D E (3) A D B C E (3) E D C A B (2) C B E A D (2) E B C A D (1) D C E A B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 10 -24 0 B -10 0 12 -20 -6 C -10 -12 0 -16 -4 D 24 20 16 0 4 E 0 6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -24 0 B -10 0 12 -20 -6 C -10 -12 0 -16 -4 D 24 20 16 0 4 E 0 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=32 C=19 B=6 A=5 so A is eliminated. Round 2 votes counts: E=38 D=35 C=19 B=8 so B is eliminated. Round 3 votes counts: E=38 D=36 C=26 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:232 E:203 A:198 B:188 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -24 0 B -10 0 12 -20 -6 C -10 -12 0 -16 -4 D 24 20 16 0 4 E 0 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -24 0 B -10 0 12 -20 -6 C -10 -12 0 -16 -4 D 24 20 16 0 4 E 0 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -24 0 B -10 0 12 -20 -6 C -10 -12 0 -16 -4 D 24 20 16 0 4 E 0 6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999989478 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9033: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (15) D B A C E (13) A E C D B (13) D B C E A (10) A E C B D (9) C E B D A (8) B D C E A (8) A D B E C (6) E C B D A (4) A D E C B (4) B C E D A (3) D A B C E (2) A D B C E (2) E C B A D (1) D B A E C (1) C E A B D (1) Total count = 100 A B C D E A 0 4 0 2 0 B -4 0 -10 -2 -10 C 0 10 0 8 -6 D -2 2 -8 0 -8 E 0 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.556333 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.443667 Sum of squares = 0.506346878957 Cumulative probabilities = A: 0.556333 B: 0.556333 C: 0.556333 D: 0.556333 E: 1.000000 A B C D E A 0 4 0 2 0 B -4 0 -10 -2 -10 C 0 10 0 8 -6 D -2 2 -8 0 -8 E 0 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 D=26 E=20 B=11 C=9 so C is eliminated. Round 2 votes counts: A=34 E=29 D=26 B=11 so B is eliminated. Round 3 votes counts: D=34 A=34 E=32 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:212 C:206 A:203 D:192 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 0 B -4 0 -10 -2 -10 C 0 10 0 8 -6 D -2 2 -8 0 -8 E 0 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 0 B -4 0 -10 -2 -10 C 0 10 0 8 -6 D -2 2 -8 0 -8 E 0 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 0 B -4 0 -10 -2 -10 C 0 10 0 8 -6 D -2 2 -8 0 -8 E 0 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9034: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) C D B E A (6) B D A C E (5) A E C B D (5) E C D B A (4) C E A D B (4) B A D E C (4) A B D E C (4) E A C B D (3) D B E A C (3) D B A E C (3) C E D A B (3) C E A B D (3) C D B A E (3) C A B D E (3) A E B D C (3) A C E B D (3) A B E D C (3) E C A B D (2) E A D B C (2) E A B D C (2) D E B C A (2) D B C E A (2) D B C A E (2) C A E B D (2) B D C A E (2) B D A E C (2) B A D C E (2) A B D C E (2) E D A C B (1) E C D A B (1) E C A D B (1) E A C D B (1) D E B A C (1) C D E B A (1) C B D A E (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 2 2 2 B 0 0 -10 6 -2 C -2 10 0 10 4 D -2 -6 -10 0 -4 E -2 2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.898740 B: 0.101260 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.817987686273 Cumulative probabilities = A: 0.898740 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 2 2 B 0 0 -10 6 -2 C -2 10 0 10 4 D -2 -6 -10 0 -4 E -2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222245496 Cumulative probabilities = A: 0.833333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=22 E=17 B=15 D=13 so D is eliminated. Round 2 votes counts: C=33 B=25 A=22 E=20 so E is eliminated. Round 3 votes counts: C=41 A=31 B=28 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:211 A:203 E:200 B:197 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 2 2 B 0 0 -10 6 -2 C -2 10 0 10 4 D -2 -6 -10 0 -4 E -2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222245496 Cumulative probabilities = A: 0.833333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 2 B 0 0 -10 6 -2 C -2 10 0 10 4 D -2 -6 -10 0 -4 E -2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222245496 Cumulative probabilities = A: 0.833333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 2 B 0 0 -10 6 -2 C -2 10 0 10 4 D -2 -6 -10 0 -4 E -2 2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.833333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.722222245496 Cumulative probabilities = A: 0.833333 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9035: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) C B E A D (7) B C A E D (7) D A E B C (6) C B D E A (6) D E A C B (5) A B E C D (5) E A D B C (3) D E C A B (3) D E A B C (3) C E D B A (3) C D E B A (3) C B D A E (3) B C A D E (3) B A E C D (3) B A C D E (3) A E D B C (3) A E B D C (3) E D A B C (2) E A B D C (2) D C E B A (2) D C B E A (2) B A C E D (2) A B E D C (2) A B D E C (2) E A B C D (1) D C B A E (1) D A E C B (1) D A C B E (1) C E B A D (1) C D B A E (1) C B A E D (1) B E A C D (1) B C D A E (1) A D E B C (1) Total count = 100 A B C D E A 0 -14 -2 0 -2 B 14 0 6 6 16 C 2 -6 0 16 8 D 0 -6 -16 0 8 E 2 -16 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 0 -2 B 14 0 6 6 16 C 2 -6 0 16 8 D 0 -6 -16 0 8 E 2 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 B=20 A=16 E=8 so E is eliminated. Round 2 votes counts: C=32 D=26 A=22 B=20 so B is eliminated. Round 3 votes counts: C=43 A=31 D=26 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:221 C:210 D:193 A:191 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 0 -2 B 14 0 6 6 16 C 2 -6 0 16 8 D 0 -6 -16 0 8 E 2 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 0 -2 B 14 0 6 6 16 C 2 -6 0 16 8 D 0 -6 -16 0 8 E 2 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 0 -2 B 14 0 6 6 16 C 2 -6 0 16 8 D 0 -6 -16 0 8 E 2 -16 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998773 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9036: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (15) A D B C E (14) A D E B C (10) C B E D A (9) E C B A D (7) E A D C B (6) D A B C E (5) C B D E A (5) B C D A E (5) E A D B C (3) B C A D E (3) A E D B C (3) E A C D B (2) D B C A E (2) A D B E C (2) E C D B A (1) E A C B D (1) D E A C B (1) D B A C E (1) C E B D A (1) C B D A E (1) B D C A E (1) B C E A D (1) B A C D E (1) Total count = 100 A B C D E A 0 -6 -2 6 -4 B 6 0 2 0 0 C 2 -2 0 4 -2 D -6 0 -4 0 2 E 4 0 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.743070 C: 0.000000 D: 0.256930 E: 0.000000 Sum of squares = 0.618165632906 Cumulative probabilities = A: 0.000000 B: 0.743070 C: 0.743070 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 6 -4 B 6 0 2 0 0 C 2 -2 0 4 -2 D -6 0 -4 0 2 E 4 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555563905 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=29 C=16 B=11 D=9 so D is eliminated. Round 2 votes counts: E=36 A=34 C=16 B=14 so B is eliminated. Round 3 votes counts: E=36 A=36 C=28 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:204 E:202 C:201 A:197 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -2 6 -4 B 6 0 2 0 0 C 2 -2 0 4 -2 D -6 0 -4 0 2 E 4 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555563905 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 6 -4 B 6 0 2 0 0 C 2 -2 0 4 -2 D -6 0 -4 0 2 E 4 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555563905 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 6 -4 B 6 0 2 0 0 C 2 -2 0 4 -2 D -6 0 -4 0 2 E 4 0 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.333333 E: 0.000000 Sum of squares = 0.555555563905 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9037: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) E C A D B (5) D A E B C (5) C E A D B (5) C D B E A (5) C B D E A (5) B A D E C (5) A E B D C (5) E A D C B (4) D B A E C (4) D B A C E (4) A E D B C (4) D B C A E (3) C E A B D (3) C D E B A (3) B D C A E (3) E D A C B (2) E C A B D (2) D A B E C (2) C E D A B (2) C B E A D (2) B D A E C (2) B C D A E (2) E A C B D (1) E A B C D (1) D C E A B (1) D C B E A (1) C E D B A (1) C E B D A (1) C E B A D (1) C D B A E (1) C B E D A (1) C B D A E (1) B C A E D (1) B A E C D (1) B A C E D (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 2 2 -8 B -4 0 -10 -20 -10 C -2 10 0 8 -6 D -2 20 -8 0 -6 E 8 10 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 2 2 -8 B -4 0 -10 -20 -10 C -2 10 0 8 -6 D -2 20 -8 0 -6 E 8 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=23 D=20 B=15 A=11 so A is eliminated. Round 2 votes counts: E=33 C=31 D=20 B=16 so B is eliminated. Round 3 votes counts: E=35 C=35 D=30 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 C:205 D:202 A:200 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 2 2 -8 B -4 0 -10 -20 -10 C -2 10 0 8 -6 D -2 20 -8 0 -6 E 8 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 2 -8 B -4 0 -10 -20 -10 C -2 10 0 8 -6 D -2 20 -8 0 -6 E 8 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 2 -8 B -4 0 -10 -20 -10 C -2 10 0 8 -6 D -2 20 -8 0 -6 E 8 10 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9038: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (17) A E C B D (17) D B A E C (9) C E A B D (8) B C E A D (8) D A E B C (7) C B E A D (6) A E D B C (5) B D C E A (3) A E D C B (3) D C E A B (2) D C B E A (2) D B C A E (2) D A E C B (2) A E C D B (2) A E B C D (2) D A B E C (1) C E B A D (1) C D B E A (1) B A E C D (1) A B E C D (1) Total count = 100 A B C D E A 0 0 0 8 4 B 0 0 12 -6 2 C 0 -12 0 -6 0 D -8 6 6 0 -8 E -4 -2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.685224 B: 0.314776 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.568616116574 Cumulative probabilities = A: 0.685224 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 8 4 B 0 0 12 -6 2 C 0 -12 0 -6 0 D -8 6 6 0 -8 E -4 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 A=30 C=16 B=12 so E is eliminated. Round 2 votes counts: D=42 A=30 C=16 B=12 so B is eliminated. Round 3 votes counts: D=45 A=31 C=24 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:204 E:201 D:198 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 8 4 B 0 0 12 -6 2 C 0 -12 0 -6 0 D -8 6 6 0 -8 E -4 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 8 4 B 0 0 12 -6 2 C 0 -12 0 -6 0 D -8 6 6 0 -8 E -4 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 8 4 B 0 0 12 -6 2 C 0 -12 0 -6 0 D -8 6 6 0 -8 E -4 -2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.499999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.500001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9039: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) E A C D B (5) D C E A B (5) A E B C D (5) D C B E A (4) C D A E B (4) B E A D C (4) B D C E A (4) A E C D B (4) E B A D C (3) D C E B A (3) C A D E B (3) B D E C A (3) A E C B D (3) A C E D B (3) E D C A B (2) E A D C B (2) D E C B A (2) D C B A E (2) C D E A B (2) C D B A E (2) B D C A E (2) B A E D C (2) B A C D E (2) A B E C D (2) E D B C A (1) E C D A B (1) E A D B C (1) D B C E A (1) D B C A E (1) C E D A B (1) C D A B E (1) C B D A E (1) C A D B E (1) B E D C A (1) B E D A C (1) B D A E C (1) B C D A E (1) B A D E C (1) B A D C E (1) B A C E D (1) A C E B D (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 4 8 8 B 2 0 -8 -4 -2 C -4 8 0 6 -2 D -8 4 -6 0 -2 E -8 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428586 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 8 8 B 2 0 -8 -4 -2 C -4 8 0 6 -2 D -8 4 -6 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428528 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=21 D=18 E=15 C=15 so E is eliminated. Round 2 votes counts: B=34 A=29 D=21 C=16 so C is eliminated. Round 3 votes counts: B=35 A=33 D=32 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:209 C:204 E:199 B:194 D:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 4 8 8 B 2 0 -8 -4 -2 C -4 8 0 6 -2 D -8 4 -6 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428528 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 8 8 B 2 0 -8 -4 -2 C -4 8 0 6 -2 D -8 4 -6 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428528 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 8 8 B 2 0 -8 -4 -2 C -4 8 0 6 -2 D -8 4 -6 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.285714 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.428571428528 Cumulative probabilities = A: 0.571429 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9040: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) D E A B C (6) E C D B A (5) B A C D E (5) A B E D C (5) E D A B C (4) D B A C E (4) B D A C E (4) B C A D E (4) B A D C E (4) E A D B C (3) D B C A E (3) C B A D E (3) B C D A E (3) A B D E C (3) A B D C E (3) E D C A B (2) E C D A B (2) E C A D B (2) E A D C B (2) D E C B A (2) D A E B C (2) C E B D A (2) C E B A D (2) C B A E D (2) B D C A E (2) E D A C B (1) E A C D B (1) E A B C D (1) D C E B A (1) D C B E A (1) D A B E C (1) C D E B A (1) C B E D A (1) C B E A D (1) C B D E A (1) C A B E D (1) A E D B C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 4 -10 4 B 16 0 20 -2 6 C -4 -20 0 -8 12 D 10 2 8 0 6 E -4 -6 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 4 -10 4 B 16 0 20 -2 6 C -4 -20 0 -8 12 D 10 2 8 0 6 E -4 -6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 B=22 C=21 D=20 A=14 so A is eliminated. Round 2 votes counts: B=35 E=24 C=21 D=20 so D is eliminated. Round 3 votes counts: B=43 E=34 C=23 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:220 D:213 A:191 C:190 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 4 -10 4 B 16 0 20 -2 6 C -4 -20 0 -8 12 D 10 2 8 0 6 E -4 -6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 -10 4 B 16 0 20 -2 6 C -4 -20 0 -8 12 D 10 2 8 0 6 E -4 -6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 -10 4 B 16 0 20 -2 6 C -4 -20 0 -8 12 D 10 2 8 0 6 E -4 -6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9041: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) D E A B C (6) E D A B C (5) E A D C B (5) C A E D B (5) B C A E D (5) A E C D B (5) C B D E A (4) C A B E D (4) B D E A C (4) A E D B C (4) C E A D B (3) B D C E A (3) B C D E A (3) B A E C D (3) D E C A B (2) D E B A C (2) C A E B D (2) B C A D E (2) B A E D C (2) B A D E C (2) B A C E D (2) A E D C B (2) A C E B D (2) A C B E D (2) E C A D B (1) E A C D B (1) D E C B A (1) D E A C B (1) D C E A B (1) D C B E A (1) D B E A C (1) C E D A B (1) B D A E C (1) A E C B D (1) A E B D C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 12 8 28 10 B -12 0 -6 4 -4 C -8 6 0 12 -2 D -28 -4 -12 0 -32 E -10 4 2 32 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 28 10 B -12 0 -6 4 -4 C -8 6 0 12 -2 D -28 -4 -12 0 -32 E -10 4 2 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999618 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=27 B=27 A=19 D=15 E=12 so E is eliminated. Round 2 votes counts: C=28 B=27 A=25 D=20 so D is eliminated. Round 3 votes counts: A=37 C=33 B=30 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:214 C:204 B:191 D:162 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 28 10 B -12 0 -6 4 -4 C -8 6 0 12 -2 D -28 -4 -12 0 -32 E -10 4 2 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999618 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 28 10 B -12 0 -6 4 -4 C -8 6 0 12 -2 D -28 -4 -12 0 -32 E -10 4 2 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999618 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 28 10 B -12 0 -6 4 -4 C -8 6 0 12 -2 D -28 -4 -12 0 -32 E -10 4 2 32 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999618 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9042: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) C A D B E (7) B E A C D (7) A E C B D (7) E D B A C (4) E B D A C (4) D E B C A (4) D C E B A (4) B E D C A (4) E A B C D (3) D C B E A (3) D C A B E (3) D B E C A (3) C D A B E (3) B E A D C (3) A E B C D (3) E B A D C (2) E B A C D (2) E A D B C (2) D E B A C (2) C A D E B (2) C A B D E (2) A C E B D (2) A C D E B (2) E B D C A (1) E A B D C (1) D E C B A (1) D C E A B (1) D C B A E (1) D A E C B (1) C D B A E (1) C B D A E (1) B E C D A (1) B D C E A (1) B A E C D (1) A C D B E (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 0 -2 -2 -6 B 0 0 -4 -6 -12 C 2 4 0 -6 -12 D 2 6 6 0 4 E 6 12 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999809 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -2 -6 B 0 0 -4 -6 -12 C 2 4 0 -6 -12 D 2 6 6 0 4 E 6 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=19 B=17 A=17 C=16 so C is eliminated. Round 2 votes counts: D=35 A=28 E=19 B=18 so B is eliminated. Round 3 votes counts: D=37 E=34 A=29 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:209 A:195 C:194 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -2 -2 -6 B 0 0 -4 -6 -12 C 2 4 0 -6 -12 D 2 6 6 0 4 E 6 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 -6 B 0 0 -4 -6 -12 C 2 4 0 -6 -12 D 2 6 6 0 4 E 6 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 -6 B 0 0 -4 -6 -12 C 2 4 0 -6 -12 D 2 6 6 0 4 E 6 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9043: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) C A D E B (7) D A E B C (6) E B D A C (5) C E D A B (5) B E D C A (4) E D B A C (3) E B D C A (3) C E B D A (3) C B A E D (3) C A D B E (3) C A B D E (3) A D B E C (3) E D A C B (2) E D A B C (2) E B C D A (2) C E B A D (2) C B A D E (2) C A B E D (2) B E A D C (2) B A D E C (2) B A D C E (2) A C D B E (2) E C D B A (1) E C D A B (1) D E A B C (1) D B E A C (1) D A E C B (1) D A B E C (1) C E D B A (1) C E A D B (1) C B E D A (1) C B E A D (1) C A E D B (1) C A E B D (1) B E C D A (1) B D E A C (1) B D A E C (1) B C A D E (1) B A C D E (1) A D C E B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 2 -16 -10 B 10 0 10 12 2 C -2 -10 0 -8 -10 D 16 -12 8 0 -18 E 10 -2 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -16 -10 B 10 0 10 12 2 C -2 -10 0 -8 -10 D 16 -12 8 0 -18 E 10 -2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996507 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=27 E=19 D=10 A=8 so A is eliminated. Round 2 votes counts: C=38 B=28 E=19 D=15 so D is eliminated. Round 3 votes counts: C=40 B=33 E=27 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:217 D:197 C:185 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 -16 -10 B 10 0 10 12 2 C -2 -10 0 -8 -10 D 16 -12 8 0 -18 E 10 -2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996507 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -16 -10 B 10 0 10 12 2 C -2 -10 0 -8 -10 D 16 -12 8 0 -18 E 10 -2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996507 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -16 -10 B 10 0 10 12 2 C -2 -10 0 -8 -10 D 16 -12 8 0 -18 E 10 -2 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996507 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9044: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) B E C A D (7) C A E D B (6) E A C B D (5) D C A E B (5) B E A D C (5) E B A C D (4) D A C E B (4) B E D C A (4) A C D E B (4) E A B C D (3) C D A E B (3) B E C D A (3) B C E D A (3) A E C B D (3) A C E D B (3) E C B A D (2) E B C A D (2) D B C E A (2) D B C A E (2) D A B E C (2) C A D E B (2) B E D A C (2) E A B D C (1) D C B E A (1) D C B A E (1) D C A B E (1) D B A E C (1) D A E C B (1) D A C B E (1) C E B A D (1) C E A B D (1) C D B E A (1) B C D E A (1) A E D C B (1) A E C D B (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 4 6 -14 B 2 0 2 14 -8 C -4 -2 0 12 -14 D -6 -14 -12 0 -16 E 14 8 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 6 -14 B 2 0 2 14 -8 C -4 -2 0 12 -14 D -6 -14 -12 0 -16 E 14 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=21 E=17 C=14 A=14 so C is eliminated. Round 2 votes counts: B=34 D=25 A=22 E=19 so E is eliminated. Round 3 votes counts: B=43 A=32 D=25 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:226 B:205 A:197 C:196 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 6 -14 B 2 0 2 14 -8 C -4 -2 0 12 -14 D -6 -14 -12 0 -16 E 14 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 6 -14 B 2 0 2 14 -8 C -4 -2 0 12 -14 D -6 -14 -12 0 -16 E 14 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 6 -14 B 2 0 2 14 -8 C -4 -2 0 12 -14 D -6 -14 -12 0 -16 E 14 8 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9045: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) B E A D C (7) E B C A D (6) C E B D A (6) C D A E B (6) C E B A D (5) C A D E B (5) A C D E B (5) D A B E C (4) A D C B E (4) E B C D A (3) D B A E C (3) C E A B D (3) A D C E B (3) A D B E C (3) D C B E A (2) D A B C E (2) C D B E A (2) B E D C A (2) B E D A C (2) B E C D A (2) A E B C D (2) A C E B D (2) E C B D A (1) E C B A D (1) E C A B D (1) E A C B D (1) E A B C D (1) D C A E B (1) D C A B E (1) C E D B A (1) C B E D A (1) C A E D B (1) B D A E C (1) B C E D A (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 8 -2 4 4 B -8 0 -22 -2 -12 C 2 22 0 16 18 D -4 2 -16 0 -2 E -4 12 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 4 4 B -8 0 -22 -2 -12 C 2 22 0 16 18 D -4 2 -16 0 -2 E -4 12 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=21 D=20 B=15 E=14 so E is eliminated. Round 2 votes counts: C=33 B=24 A=23 D=20 so D is eliminated. Round 3 votes counts: C=37 A=36 B=27 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:229 A:207 E:196 D:190 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 4 4 B -8 0 -22 -2 -12 C 2 22 0 16 18 D -4 2 -16 0 -2 E -4 12 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 4 4 B -8 0 -22 -2 -12 C 2 22 0 16 18 D -4 2 -16 0 -2 E -4 12 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 4 4 B -8 0 -22 -2 -12 C 2 22 0 16 18 D -4 2 -16 0 -2 E -4 12 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9046: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (14) B E D C A (10) D A C E B (8) B E C D A (6) C E A D B (5) B D E A C (4) E D C B A (3) E B C D A (3) D B A E C (3) C A E D B (3) A D B C E (3) A C D B E (3) E D B C A (2) E B D C A (2) D A E C B (2) B E D A C (2) B C E A D (2) B A D E C (2) B A C E D (2) A C D E B (2) E D C A B (1) E C D A B (1) E C B D A (1) D E C A B (1) D E A B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C B E A D (1) C A E B D (1) C A D E B (1) B E C A D (1) B A E C D (1) B A C D E (1) A D C B E (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 -4 4 B -6 0 -2 -12 -6 C -8 2 0 -24 6 D 4 12 24 0 2 E -4 6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999563 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 8 -4 4 B -6 0 -2 -12 -6 C -8 2 0 -24 6 D 4 12 24 0 2 E -4 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=26 D=17 E=13 C=13 so E is eliminated. Round 2 votes counts: B=36 A=26 D=23 C=15 so C is eliminated. Round 3 votes counts: B=39 A=37 D=24 so D is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:221 A:207 E:197 C:188 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 8 -4 4 B -6 0 -2 -12 -6 C -8 2 0 -24 6 D 4 12 24 0 2 E -4 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 -4 4 B -6 0 -2 -12 -6 C -8 2 0 -24 6 D 4 12 24 0 2 E -4 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 -4 4 B -6 0 -2 -12 -6 C -8 2 0 -24 6 D 4 12 24 0 2 E -4 6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999972594 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9047: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A E B C D (8) D B C E A (6) C E A D B (5) D C A E B (4) B A E C D (4) B A D E C (4) A B D E C (4) D B C A E (3) D B A C E (3) C E D A B (3) C E B A D (3) C D E B A (3) B E C A D (3) B E A C D (3) B D A E C (3) B A E D C (3) A E C B D (3) A B E C D (3) E C A B D (2) D C E A B (2) D B A E C (2) D A B E C (2) C E A B D (2) E B C A D (1) D C E B A (1) D A C E B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D E A B (1) B E A D C (1) B D E C A (1) B D C E A (1) B C E D A (1) A E C D B (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -4 4 2 B 12 0 16 2 14 C 4 -16 0 -4 0 D -4 -2 4 0 4 E -2 -14 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999704 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 4 2 B 12 0 16 2 14 C 4 -16 0 -4 0 D -4 -2 4 0 4 E -2 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=24 A=21 C=19 E=3 so E is eliminated. Round 2 votes counts: D=33 B=25 C=21 A=21 so C is eliminated. Round 3 votes counts: D=41 A=30 B=29 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:222 D:201 A:195 C:192 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 4 2 B 12 0 16 2 14 C 4 -16 0 -4 0 D -4 -2 4 0 4 E -2 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 4 2 B 12 0 16 2 14 C 4 -16 0 -4 0 D -4 -2 4 0 4 E -2 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 4 2 B 12 0 16 2 14 C 4 -16 0 -4 0 D -4 -2 4 0 4 E -2 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991881 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9048: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) A E D B C (6) E C D A B (5) D A E B C (5) C B D E A (5) C B D A E (5) E A B C D (4) D C B A E (4) D E A C B (3) D C E A B (3) C B E D A (3) C B E A D (3) B C D A E (3) B C A E D (3) B A D C E (3) A E B D C (3) A D B E C (3) A B E D C (3) A B E C D (3) E D A C B (2) D A E C B (2) C D B E A (2) B C A D E (2) A D E B C (2) E C B A D (1) E C A D B (1) E C A B D (1) E B C A D (1) E A C B D (1) D E C A B (1) D C B E A (1) D C A E B (1) D B C A E (1) D B A C E (1) C E B D A (1) C E B A D (1) B C E A D (1) B A C E D (1) B A C D E (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 14 2 4 8 B -14 0 8 2 -2 C -2 -8 0 -2 -10 D -4 -2 2 0 -2 E -8 2 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999682 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 4 8 B -14 0 8 2 -2 C -2 -8 0 -2 -10 D -4 -2 2 0 -2 E -8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989235 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=22 D=22 A=22 C=20 B=14 so B is eliminated. Round 2 votes counts: C=29 A=27 E=22 D=22 so E is eliminated. Round 3 votes counts: C=38 A=38 D=24 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:203 B:197 D:197 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 2 4 8 B -14 0 8 2 -2 C -2 -8 0 -2 -10 D -4 -2 2 0 -2 E -8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989235 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 4 8 B -14 0 8 2 -2 C -2 -8 0 -2 -10 D -4 -2 2 0 -2 E -8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989235 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 4 8 B -14 0 8 2 -2 C -2 -8 0 -2 -10 D -4 -2 2 0 -2 E -8 2 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989235 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9049: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) D C A E B (10) D C B A E (6) D B E A C (6) C D B A E (5) C A E B D (5) B E A D C (5) A E C B D (5) E B A C D (4) D C B E A (4) C D B E A (3) B C E A D (3) A E B C D (3) A C E B D (3) D C E A B (2) D B E C A (2) D A E B C (2) C B A E D (2) A E B D C (2) E A B D C (1) E A B C D (1) D E B A C (1) D E A B C (1) D C A B E (1) D B C E A (1) D A E C B (1) D A C E B (1) C D A E B (1) C D A B E (1) C B D E A (1) B D E C A (1) B D E A C (1) A E D B C (1) A E C D B (1) A B E C D (1) Total count = 100 A B C D E A 0 -14 4 -2 2 B 14 0 -4 0 10 C -4 4 0 2 -2 D 2 0 -2 0 2 E -2 -10 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.636364 D: 0.000000 E: 0.000000 Sum of squares = 0.471074379918 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 4 -2 2 B 14 0 -4 0 10 C -4 4 0 2 -2 D 2 0 -2 0 2 E -2 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.636364 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=22 C=18 A=16 E=6 so E is eliminated. Round 2 votes counts: D=38 B=26 C=18 A=18 so C is eliminated. Round 3 votes counts: D=48 B=29 A=23 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:201 C:200 A:195 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 4 -2 2 B 14 0 -4 0 10 C -4 4 0 2 -2 D 2 0 -2 0 2 E -2 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.636364 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 4 -2 2 B 14 0 -4 0 10 C -4 4 0 2 -2 D 2 0 -2 0 2 E -2 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.636364 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 4 -2 2 B 14 0 -4 0 10 C -4 4 0 2 -2 D 2 0 -2 0 2 E -2 -10 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.636364 D: 0.000000 E: 0.000000 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9050: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) E A D C B (7) E D A B C (6) C B A D E (6) B C D A E (6) D B E C A (5) B D C E A (5) E D B A C (4) C B A E D (4) B C D E A (4) A C B E D (4) C B E A D (3) A E D C B (3) A C E B D (3) E C B A D (2) E A C D B (2) D A E B C (2) B D C A E (2) A C E D B (2) A C D E B (2) A C B D E (2) E C A B D (1) E B D C A (1) E A D B C (1) E A C B D (1) D B C E A (1) D B A C E (1) D A B E C (1) D A B C E (1) C B D E A (1) C A B E D (1) C A B D E (1) B D E C A (1) B C E D A (1) A E C D B (1) A E C B D (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 10 -2 -10 B 12 0 6 0 2 C -10 -6 0 -4 2 D 2 0 4 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.442363 C: 0.000000 D: 0.557637 E: 0.000000 Sum of squares = 0.506643988899 Cumulative probabilities = A: 0.000000 B: 0.442363 C: 0.442363 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -2 -10 B 12 0 6 0 2 C -10 -6 0 -4 2 D 2 0 4 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=20 A=20 B=19 C=16 so C is eliminated. Round 2 votes counts: B=33 E=25 A=22 D=20 so D is eliminated. Round 3 votes counts: B=40 E=34 A=26 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:205 E:201 A:193 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 -2 -10 B 12 0 6 0 2 C -10 -6 0 -4 2 D 2 0 4 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -2 -10 B 12 0 6 0 2 C -10 -6 0 -4 2 D 2 0 4 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -2 -10 B 12 0 6 0 2 C -10 -6 0 -4 2 D 2 0 4 0 4 E 10 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9051: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (9) E C D B A (9) C B D E A (9) E D C B A (6) B C D E A (5) A B D C E (5) E D A C B (4) A E D C B (4) A E D B C (4) A D E B C (4) A B E C D (4) E A D C B (3) D E C B A (3) D C E B A (3) C E D B A (3) B A C D E (3) D B C A E (2) C E B D A (2) B C D A E (2) B A D C E (2) B A C E D (2) A B D E C (2) A B C D E (2) E C B D A (1) E A C B D (1) D E A C B (1) C D E B A (1) B D C A E (1) B C E A D (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 -14 -22 -22 B 10 0 -18 -12 -20 C 14 18 0 -8 -14 D 22 12 8 0 -10 E 22 20 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -14 -22 -22 B 10 0 -18 -12 -20 C 14 18 0 -8 -14 D 22 12 8 0 -10 E 22 20 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=27 B=16 C=15 D=9 so D is eliminated. Round 2 votes counts: E=37 A=27 C=18 B=18 so C is eliminated. Round 3 votes counts: E=46 B=27 A=27 so B is eliminated. Round 4 votes counts: E=61 A=39 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:233 D:216 C:205 B:180 A:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -14 -22 -22 B 10 0 -18 -12 -20 C 14 18 0 -8 -14 D 22 12 8 0 -10 E 22 20 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -22 -22 B 10 0 -18 -12 -20 C 14 18 0 -8 -14 D 22 12 8 0 -10 E 22 20 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -22 -22 B 10 0 -18 -12 -20 C 14 18 0 -8 -14 D 22 12 8 0 -10 E 22 20 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9052: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (9) D C E A B (8) C D E A B (6) B A E C D (6) E C D A B (5) B C D E A (5) A E D C B (4) E C A B D (3) D C E B A (3) C E D B A (3) C B E D A (3) B A E D C (3) B A D C E (3) E C A D B (2) E B C A D (2) E A B C D (2) D A C E B (2) D A C B E (2) C E D A B (2) C D E B A (2) C D B E A (2) B D A C E (2) B C E A D (2) B A D E C (2) A B E D C (2) E C B A D (1) E A C B D (1) D E C A B (1) D C B E A (1) D C A E B (1) D A E C B (1) D A B C E (1) B E A C D (1) B D C A E (1) B C E D A (1) B C A E D (1) A E B C D (1) A D E C B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -28 -4 -30 B 6 0 -8 6 -2 C 28 8 0 22 2 D 4 -6 -22 0 -10 E 30 2 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -28 -4 -30 B 6 0 -8 6 -2 C 28 8 0 22 2 D 4 -6 -22 0 -10 E 30 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=20 C=18 E=16 A=10 so A is eliminated. Round 2 votes counts: B=40 E=21 D=21 C=18 so C is eliminated. Round 3 votes counts: B=43 D=31 E=26 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:230 E:220 B:201 D:183 A:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -28 -4 -30 B 6 0 -8 6 -2 C 28 8 0 22 2 D 4 -6 -22 0 -10 E 30 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -28 -4 -30 B 6 0 -8 6 -2 C 28 8 0 22 2 D 4 -6 -22 0 -10 E 30 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -28 -4 -30 B 6 0 -8 6 -2 C 28 8 0 22 2 D 4 -6 -22 0 -10 E 30 2 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999940821 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9053: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (17) D B C A E (13) E A B C D (7) D C B A E (7) D B C E A (6) E C A D B (4) B D C A E (4) E D C A B (3) E A D B C (3) B A E D C (3) E D A C B (2) E D A B C (2) E A C D B (2) E A B D C (2) D C B E A (2) C D A E B (2) B D A E C (2) B D A C E (2) B A E C D (2) A E C B D (2) E C D A B (1) E A D C B (1) D E C B A (1) D B E A C (1) C E A D B (1) C D B A E (1) C A D B E (1) B C A D E (1) B A C D E (1) A E B C D (1) A C E B D (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 8 6 2 -10 B -8 0 4 -6 -4 C -6 -4 0 -8 -14 D -2 6 8 0 -12 E 10 4 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 2 -10 B -8 0 4 -6 -4 C -6 -4 0 -8 -14 D -2 6 8 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 D=30 B=15 A=6 C=5 so C is eliminated. Round 2 votes counts: E=45 D=33 B=15 A=7 so A is eliminated. Round 3 votes counts: E=49 D=34 B=17 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 A:203 D:200 B:193 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 2 -10 B -8 0 4 -6 -4 C -6 -4 0 -8 -14 D -2 6 8 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 2 -10 B -8 0 4 -6 -4 C -6 -4 0 -8 -14 D -2 6 8 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 2 -10 B -8 0 4 -6 -4 C -6 -4 0 -8 -14 D -2 6 8 0 -12 E 10 4 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9054: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (12) B E A C D (12) B A C D E (8) B A E C D (7) C A D B E (6) E D C A B (5) E B D C A (5) E B C D A (4) A C D B E (4) E D B C A (3) E B D A C (3) D E C A B (3) D A C B E (3) B E A D C (3) E D C B A (2) D A E C B (2) B E C A D (2) B A C E D (2) A D C B E (2) A C B D E (2) A B C D E (2) E C D B A (1) E C D A B (1) E B C A D (1) D E A C B (1) D C E A B (1) D C A B E (1) C D A E B (1) C B E A D (1) Total count = 100 A B C D E A 0 -8 2 4 4 B 8 0 4 4 10 C -2 -4 0 8 -10 D -4 -4 -8 0 -4 E -4 -10 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 4 4 B 8 0 4 4 10 C -2 -4 0 8 -10 D -4 -4 -8 0 -4 E -4 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=25 D=23 A=10 C=8 so C is eliminated. Round 2 votes counts: B=35 E=25 D=24 A=16 so A is eliminated. Round 3 votes counts: B=39 D=36 E=25 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:213 A:201 E:200 C:196 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 4 4 B 8 0 4 4 10 C -2 -4 0 8 -10 D -4 -4 -8 0 -4 E -4 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 4 4 B 8 0 4 4 10 C -2 -4 0 8 -10 D -4 -4 -8 0 -4 E -4 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 4 4 B 8 0 4 4 10 C -2 -4 0 8 -10 D -4 -4 -8 0 -4 E -4 -10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9055: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (6) B A C D E (6) E A D C B (5) C D E B A (5) A E B D C (5) E C D B A (4) E B C D A (4) D E C A B (4) D C E A B (4) E B A C D (3) D A E C B (3) C D B E A (3) C D B A E (3) B E C A D (3) A D E C B (3) E D C A B (2) E D A C B (2) E B A D C (2) D E A C B (2) C B D E A (2) B C A D E (2) B A E C D (2) A E D C B (2) A E D B C (2) A B D C E (2) A B C D E (2) E D C B A (1) E C B D A (1) D C A E B (1) D C A B E (1) D A C E B (1) C E D B A (1) B E C D A (1) B E A D C (1) B C D E A (1) B C D A E (1) B C A E D (1) B A C E D (1) A D C E B (1) A D C B E (1) A B E D C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -2 -6 -14 B 8 0 -4 -2 -16 C 2 4 0 6 -10 D 6 2 -6 0 -2 E 14 16 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -2 -6 -14 B 8 0 -4 -2 -16 C 2 4 0 6 -10 D 6 2 -6 0 -2 E 14 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 A=21 D=16 C=14 so C is eliminated. Round 2 votes counts: D=27 B=27 E=25 A=21 so A is eliminated. Round 3 votes counts: E=34 B=34 D=32 so D is eliminated. Round 4 votes counts: E=58 B=42 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:201 D:200 B:193 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 -6 -14 B 8 0 -4 -2 -16 C 2 4 0 6 -10 D 6 2 -6 0 -2 E 14 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -6 -14 B 8 0 -4 -2 -16 C 2 4 0 6 -10 D 6 2 -6 0 -2 E 14 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -6 -14 B 8 0 -4 -2 -16 C 2 4 0 6 -10 D 6 2 -6 0 -2 E 14 16 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998393 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9056: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) A E B D C (10) D C B E A (8) B E A C D (8) D C A E B (7) E B A D C (6) A E B C D (5) C D A B E (3) C A D B E (3) C A B E D (3) B E C D A (3) A E D B C (3) A D C E B (3) D E B C A (2) C D A E B (2) C B E D A (2) B E D C A (2) B E A D C (2) A D E B C (2) E A B D C (1) D E C B A (1) D C E B A (1) D A E B C (1) D A C E B (1) C B E A D (1) C B D E A (1) C A D E B (1) C A B D E (1) B E D A C (1) B C E D A (1) B C E A D (1) B A E C D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 6 -4 B 4 0 0 0 6 C 8 0 0 -2 4 D -6 0 2 0 -2 E 4 -6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.721937 C: 0.000000 D: 0.278063 E: 0.000000 Sum of squares = 0.598512396494 Cumulative probabilities = A: 0.000000 B: 0.721937 C: 0.721937 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 6 -4 B 4 0 0 0 6 C 8 0 0 -2 4 D -6 0 2 0 -2 E 4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.399999 E: 0.000000 Sum of squares = 0.520000228589 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=25 D=21 B=19 E=7 so E is eliminated. Round 2 votes counts: C=28 A=26 B=25 D=21 so D is eliminated. Round 3 votes counts: C=45 A=28 B=27 so B is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:205 C:205 E:198 D:197 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 6 -4 B 4 0 0 0 6 C 8 0 0 -2 4 D -6 0 2 0 -2 E 4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.399999 E: 0.000000 Sum of squares = 0.520000228589 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 6 -4 B 4 0 0 0 6 C 8 0 0 -2 4 D -6 0 2 0 -2 E 4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.399999 E: 0.000000 Sum of squares = 0.520000228589 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 6 -4 B 4 0 0 0 6 C 8 0 0 -2 4 D -6 0 2 0 -2 E 4 -6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600001 C: 0.000000 D: 0.399999 E: 0.000000 Sum of squares = 0.520000228589 Cumulative probabilities = A: 0.000000 B: 0.600001 C: 0.600001 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9057: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) A C B D E (9) E D B A C (8) C A D B E (7) E C A B D (5) C A B D E (5) B D A C E (5) D B A C E (4) E D B C A (3) D B E A C (3) D B A E C (3) B D E A C (3) E C A D B (2) E B A C D (2) D E B A C (2) C E A D B (2) C E A B D (2) C A D E B (2) C A B E D (2) B D A E C (2) B A C D E (2) A C D B E (2) A B C D E (2) E D C B A (1) E D C A B (1) E C D B A (1) E C D A B (1) E C B A D (1) E B D C A (1) E B C A D (1) E B A D C (1) C E D A B (1) C D A E B (1) C A E D B (1) C A E B D (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 -8 18 2 0 B 8 0 6 10 2 C -18 -6 0 4 -2 D -2 -10 -4 0 8 E 0 -2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 18 2 0 B 8 0 6 10 2 C -18 -6 0 4 -2 D -2 -10 -4 0 8 E 0 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=24 B=14 A=13 D=12 so D is eliminated. Round 2 votes counts: E=39 C=24 B=24 A=13 so A is eliminated. Round 3 votes counts: E=39 C=35 B=26 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:213 A:206 D:196 E:196 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 18 2 0 B 8 0 6 10 2 C -18 -6 0 4 -2 D -2 -10 -4 0 8 E 0 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 18 2 0 B 8 0 6 10 2 C -18 -6 0 4 -2 D -2 -10 -4 0 8 E 0 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 18 2 0 B 8 0 6 10 2 C -18 -6 0 4 -2 D -2 -10 -4 0 8 E 0 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999823 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9058: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) B E C D A (8) C A E B D (7) C E B A D (6) B E D C A (6) A C E D B (6) D A B C E (5) B D E C A (5) A D C B E (5) D B E A C (4) E C B A D (3) D B A E C (3) D A C B E (3) C E A B D (3) A C D E B (3) E B C D A (2) D A E C B (2) D A E B C (2) D A C E B (2) D A B E C (2) C B E A D (2) A D E C B (2) E C A D B (1) E B C A D (1) E A D C B (1) D E B A C (1) D E A B C (1) C A E D B (1) B D C E A (1) B C E D A (1) B C E A D (1) B C D A E (1) Total count = 100 A B C D E A 0 10 2 2 6 B -10 0 -12 -6 -6 C -2 12 0 -8 12 D -2 6 8 0 2 E -6 6 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 2 6 B -10 0 -12 -6 -6 C -2 12 0 -8 12 D -2 6 8 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=25 A=25 B=23 C=19 E=8 so E is eliminated. Round 2 votes counts: B=26 A=26 D=25 C=23 so C is eliminated. Round 3 votes counts: A=38 B=37 D=25 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:207 D:207 E:193 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 2 6 B -10 0 -12 -6 -6 C -2 12 0 -8 12 D -2 6 8 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 2 6 B -10 0 -12 -6 -6 C -2 12 0 -8 12 D -2 6 8 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 2 6 B -10 0 -12 -6 -6 C -2 12 0 -8 12 D -2 6 8 0 2 E -6 6 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999905 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9059: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (14) D B A C E (11) E C A B D (8) D A B C E (7) A D B E C (7) E C B A D (5) E A C B D (5) C E B D A (5) A E D C B (5) E C B D A (4) A E C B D (4) A D E C B (4) D B C E A (3) D B C A E (2) C E A B D (2) C B E D A (2) A E C D B (2) E A C D B (1) D A E B C (1) C B D E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C D E A (1) B C A E D (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -2 -6 -6 B 4 0 4 10 6 C 2 -4 0 -14 6 D 6 -10 14 0 8 E 6 -6 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -6 -6 B 4 0 4 10 6 C 2 -4 0 -14 6 D 6 -10 14 0 8 E 6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=24 A=24 E=23 B=19 C=10 so C is eliminated. Round 2 votes counts: E=30 D=24 A=24 B=22 so B is eliminated. Round 3 votes counts: D=41 E=34 A=25 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:209 C:195 E:193 A:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 -6 -6 B 4 0 4 10 6 C 2 -4 0 -14 6 D 6 -10 14 0 8 E 6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -6 -6 B 4 0 4 10 6 C 2 -4 0 -14 6 D 6 -10 14 0 8 E 6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -6 -6 B 4 0 4 10 6 C 2 -4 0 -14 6 D 6 -10 14 0 8 E 6 -6 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9060: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) A B C D E (6) E B C D A (5) E D C B A (4) E D B C A (4) B C E A D (4) B C A E D (4) E C B D A (3) D E A C B (3) C B E D A (3) C A B D E (3) B E C D A (3) B A C D E (3) A D E B C (3) A D C E B (3) A D C B E (3) A D B C E (3) A B D C E (3) E D A C B (2) E C D B A (2) C B A E D (2) C B A D E (2) B E C A D (2) B C E D A (2) B C A D E (2) A D E C B (2) A D B E C (2) A C B D E (2) E D C A B (1) E D A B C (1) E B D A C (1) D E C A B (1) D A E B C (1) C E D B A (1) C E B D A (1) C D A E B (1) C A D B E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -2 4 14 B -4 0 0 4 4 C 2 0 0 6 0 D -4 -4 -6 0 10 E -14 -4 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.212541 C: 0.787459 D: 0.000000 E: 0.000000 Sum of squares = 0.665265338422 Cumulative probabilities = A: 0.000000 B: 0.212541 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 4 14 B -4 0 0 4 4 C 2 0 0 6 0 D -4 -4 -6 0 10 E -14 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555831252 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=23 B=20 D=14 C=14 so D is eliminated. Round 2 votes counts: A=39 E=27 B=20 C=14 so C is eliminated. Round 3 votes counts: A=44 E=29 B=27 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:204 B:202 D:198 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 4 14 B -4 0 0 4 4 C 2 0 0 6 0 D -4 -4 -6 0 10 E -14 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555831252 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 4 14 B -4 0 0 4 4 C 2 0 0 6 0 D -4 -4 -6 0 10 E -14 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555831252 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 4 14 B -4 0 0 4 4 C 2 0 0 6 0 D -4 -4 -6 0 10 E -14 -4 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555831252 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9061: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) A C B D E (7) D B C E A (6) B D E C A (5) A C E B D (5) B D E A C (4) E D C B A (3) E D B C A (3) E C A D B (3) E A B D C (3) D C B E A (3) C D A B E (3) B D C A E (3) A E C B D (3) E B D C A (2) E B D A C (2) E B A D C (2) E A D B C (2) E A C B D (2) D C E B A (2) D B E C A (2) C E D A B (2) B D A E C (2) A E B C D (2) E D A C B (1) E A B C D (1) D E C B A (1) D E B C A (1) D B C A E (1) C E A D B (1) C D E B A (1) C D B A E (1) C D A E B (1) C A E D B (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A D C (1) B D A C E (1) B A E D C (1) B A D E C (1) A C E D B (1) A C B E D (1) A B E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -8 -4 -10 B 0 0 0 6 8 C 8 0 0 -10 0 D 4 -6 10 0 10 E 10 -8 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.713176 C: 0.286824 D: 0.000000 E: 0.000000 Sum of squares = 0.590888289873 Cumulative probabilities = A: 0.000000 B: 0.713176 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -4 -10 B 0 0 0 6 8 C 8 0 0 -10 0 D 4 -6 10 0 10 E 10 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250112617 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 A=22 B=20 C=18 D=16 so D is eliminated. Round 2 votes counts: B=29 E=26 C=23 A=22 so A is eliminated. Round 3 votes counts: C=37 B=32 E=31 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:209 B:207 C:199 E:196 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -4 -10 B 0 0 0 6 8 C 8 0 0 -10 0 D 4 -6 10 0 10 E 10 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250112617 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -4 -10 B 0 0 0 6 8 C 8 0 0 -10 0 D 4 -6 10 0 10 E 10 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250112617 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -4 -10 B 0 0 0 6 8 C 8 0 0 -10 0 D 4 -6 10 0 10 E 10 -8 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.625000 C: 0.375000 D: 0.000000 E: 0.000000 Sum of squares = 0.531250112617 Cumulative probabilities = A: 0.000000 B: 0.625000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9062: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (16) D A C B E (10) E B C A D (9) B E C D A (9) B E C A D (7) D A C E B (5) B E D C A (4) B E A D C (4) A D E C B (4) D A B C E (3) C E B D A (3) E C B D A (2) D A B E C (2) B E D A C (2) E C B A D (1) E B C D A (1) E B A D C (1) E A C D B (1) D C B A E (1) D C A E B (1) D C A B E (1) D B A E C (1) C E B A D (1) C E A D B (1) C D A E B (1) C A E D B (1) B D C E A (1) B C E D A (1) B A D E C (1) A E C D B (1) A E B D C (1) A D E B C (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 10 4 4 B -2 0 -2 -4 -4 C -10 2 0 -20 -6 D -4 4 20 0 0 E -4 4 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 10 4 4 B -2 0 -2 -4 -4 C -10 2 0 -20 -6 D -4 4 20 0 0 E -4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=25 D=24 E=15 C=7 so C is eliminated. Round 2 votes counts: B=29 A=26 D=25 E=20 so E is eliminated. Round 3 votes counts: B=47 A=28 D=25 so D is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:210 E:203 B:194 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 10 4 4 B -2 0 -2 -4 -4 C -10 2 0 -20 -6 D -4 4 20 0 0 E -4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 10 4 4 B -2 0 -2 -4 -4 C -10 2 0 -20 -6 D -4 4 20 0 0 E -4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 10 4 4 B -2 0 -2 -4 -4 C -10 2 0 -20 -6 D -4 4 20 0 0 E -4 4 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998846 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9063: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (9) E D C A B (7) D E B A C (7) B A C E D (7) B A C D E (6) C E A D B (5) B D A E C (5) D E C A B (4) D E A C B (4) C A B E D (4) A C B E D (4) D B E A C (3) C A E D B (3) C A E B D (3) E C D A B (2) D E B C A (2) D B A E C (2) D A B E C (2) B C A E D (2) B A D E C (2) A B D C E (2) A B C D E (2) E D C B A (1) E B C D A (1) D E A B C (1) C E D A B (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) B E C A D (1) A D C B E (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 20 16 16 B 2 0 8 6 12 C -20 -8 0 -4 12 D -16 -6 4 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 20 16 16 B 2 0 8 6 12 C -20 -8 0 -4 12 D -16 -6 4 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993183 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=25 C=20 A=12 E=11 so E is eliminated. Round 2 votes counts: D=33 B=33 C=22 A=12 so A is eliminated. Round 3 votes counts: B=37 D=34 C=29 so C is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:225 B:214 D:196 C:190 E:175 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 20 16 16 B 2 0 8 6 12 C -20 -8 0 -4 12 D -16 -6 4 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993183 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 20 16 16 B 2 0 8 6 12 C -20 -8 0 -4 12 D -16 -6 4 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993183 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 20 16 16 B 2 0 8 6 12 C -20 -8 0 -4 12 D -16 -6 4 0 10 E -16 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993183 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9064: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (6) D E A B C (6) A D E C B (6) D A E B C (5) C B A E D (5) B E C D A (5) B C E D A (5) E D B A C (4) E D A C B (4) B C A E D (4) E B D C A (3) D B E A C (3) C B E A D (3) C A B D E (3) B E D C A (3) B C A D E (3) A C B D E (3) E C B D A (2) E A D C B (2) E A C D B (2) C A B E D (2) A E D C B (2) A C D E B (2) A C D B E (2) E D B C A (1) E C D B A (1) D E A C B (1) C E A D B (1) C A E B D (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A E C (1) B C E A D (1) B C D E A (1) B C D A E (1) A D C B E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 -10 -14 B -2 0 12 0 0 C -6 -12 0 -2 -18 D 10 0 2 0 -16 E 14 0 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.564696 C: 0.000000 D: 0.000000 E: 0.435304 Sum of squares = 0.508371220167 Cumulative probabilities = A: 0.000000 B: 0.564696 C: 0.564696 D: 0.564696 E: 1.000000 A B C D E A 0 2 6 -10 -14 B -2 0 12 0 0 C -6 -12 0 -2 -18 D 10 0 2 0 -16 E 14 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 A=18 D=15 C=15 so D is eliminated. Round 2 votes counts: E=32 B=30 A=23 C=15 so C is eliminated. Round 3 votes counts: B=38 E=33 A=29 so A is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:224 B:205 D:198 A:192 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 -10 -14 B -2 0 12 0 0 C -6 -12 0 -2 -18 D 10 0 2 0 -16 E 14 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -10 -14 B -2 0 12 0 0 C -6 -12 0 -2 -18 D 10 0 2 0 -16 E 14 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -10 -14 B -2 0 12 0 0 C -6 -12 0 -2 -18 D 10 0 2 0 -16 E 14 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999987 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9065: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) D E B C A (9) A B D C E (8) E D C B A (6) E C D B A (5) C E A D B (5) B D A E C (5) A B D E C (5) C E D B A (4) D B E A C (3) C E D A B (3) C A E B D (3) A C B E D (3) E D B C A (2) D E C B A (2) D A B E C (2) C E B D A (2) C A E D B (2) B E D C A (2) A B C D E (2) E B D C A (1) E B C D A (1) D E C A B (1) D B E C A (1) D B A E C (1) C E B A D (1) C E A B D (1) C D E A B (1) C A D E B (1) B E C D A (1) B D E C A (1) B D E A C (1) B A E D C (1) A D B C E (1) A C E B D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -10 -8 -6 B 18 0 16 2 -2 C 10 -16 0 -24 -20 D 8 -2 24 0 10 E 6 2 20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408186 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 A B C D E A 0 -18 -10 -8 -6 B 18 0 16 2 -2 C 10 -16 0 -24 -20 D 8 -2 24 0 10 E 6 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408076 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 A=22 B=21 D=19 E=15 so E is eliminated. Round 2 votes counts: C=28 D=27 B=23 A=22 so A is eliminated. Round 3 votes counts: B=39 C=33 D=28 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:220 B:217 E:209 A:179 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 -8 -6 B 18 0 16 2 -2 C 10 -16 0 -24 -20 D 8 -2 24 0 10 E 6 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408076 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -8 -6 B 18 0 16 2 -2 C 10 -16 0 -24 -20 D 8 -2 24 0 10 E 6 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408076 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -8 -6 B 18 0 16 2 -2 C 10 -16 0 -24 -20 D 8 -2 24 0 10 E 6 2 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.000000 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408076 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 0.714286 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9066: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (13) E C B D A (11) E C D B A (8) E A D B C (5) C E B D A (5) A E D B C (4) A D B E C (4) E A C B D (3) D B A C E (3) C B D E A (3) C B D A E (3) A D B C E (3) E D A C B (2) E C D A B (2) E C B A D (2) E C A D B (2) E C A B D (2) E A D C B (2) D B C A E (2) C E D B A (2) C E B A D (2) C B E D A (2) B D C A E (2) B C D A E (2) E D C A B (1) D E C B A (1) D E B A C (1) D E A B C (1) D A B E C (1) C E A B D (1) C D B E A (1) B C A D E (1) B A C D E (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 -6 -18 B 4 0 -10 10 -16 C 10 10 0 8 -6 D 6 -10 -8 0 -16 E 18 16 6 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 -18 B 4 0 -10 10 -16 C 10 10 0 8 -6 D 6 -10 -8 0 -16 E 18 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=26 C=19 D=9 B=6 so B is eliminated. Round 2 votes counts: E=40 A=27 C=22 D=11 so D is eliminated. Round 3 votes counts: E=43 A=31 C=26 so C is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:228 C:211 B:194 D:186 A:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 -18 B 4 0 -10 10 -16 C 10 10 0 8 -6 D 6 -10 -8 0 -16 E 18 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 -18 B 4 0 -10 10 -16 C 10 10 0 8 -6 D 6 -10 -8 0 -16 E 18 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 -18 B 4 0 -10 10 -16 C 10 10 0 8 -6 D 6 -10 -8 0 -16 E 18 16 6 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9067: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (10) D A E C B (9) C B A D E (9) E D A C B (7) E A D B C (7) E B A D C (6) C D A E B (6) C B D A E (6) B E A D C (5) B C E D A (5) B C E A D (4) B C A D E (4) E B D A C (3) D A C E B (2) C B E D A (2) C A D B E (2) A D B E C (2) E D B A C (1) D E A C B (1) D A E B C (1) C D B A E (1) C D A B E (1) B E D A C (1) B E C A D (1) B A D E C (1) B A D C E (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 0 18 -12 -6 B 0 0 6 -4 -8 C -18 -6 0 -18 -12 D 12 4 18 0 -4 E 6 8 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 18 -12 -6 B 0 0 6 -4 -8 C -18 -6 0 -18 -12 D 12 4 18 0 -4 E 6 8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=27 B=22 D=13 A=4 so A is eliminated. Round 2 votes counts: E=34 C=27 B=22 D=17 so D is eliminated. Round 3 votes counts: E=46 C=30 B=24 so B is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:215 A:200 B:197 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 18 -12 -6 B 0 0 6 -4 -8 C -18 -6 0 -18 -12 D 12 4 18 0 -4 E 6 8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 18 -12 -6 B 0 0 6 -4 -8 C -18 -6 0 -18 -12 D 12 4 18 0 -4 E 6 8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 18 -12 -6 B 0 0 6 -4 -8 C -18 -6 0 -18 -12 D 12 4 18 0 -4 E 6 8 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9068: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) B C A E D (8) C D E A B (6) C B E A D (6) E D A C B (5) D E A C B (5) D A E B C (5) C E D A B (5) A E D B C (5) A D E B C (5) B C E A D (4) E A D B C (3) E A B D C (3) C B E D A (3) B C D A E (3) B C A D E (3) B A E D C (3) D A E C B (2) C B D E A (2) B A D C E (2) E B C A D (1) E A D C B (1) D E C A B (1) D C A E B (1) D B C A E (1) D B A C E (1) D A B E C (1) C E B D A (1) C E A B D (1) C D E B A (1) C D B A E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 -14 -6 4 B -2 0 0 0 -2 C 14 0 0 8 16 D 6 0 -8 0 2 E -4 2 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.571718 C: 0.428282 D: 0.000000 E: 0.000000 Sum of squares = 0.510286834083 Cumulative probabilities = A: 0.000000 B: 0.571718 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -14 -6 4 B -2 0 0 0 -2 C 14 0 0 8 16 D 6 0 -8 0 2 E -4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=23 D=17 E=13 A=12 so A is eliminated. Round 2 votes counts: C=35 B=24 D=23 E=18 so E is eliminated. Round 3 votes counts: D=37 C=35 B=28 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 D:200 B:198 A:193 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 -6 4 B -2 0 0 0 -2 C 14 0 0 8 16 D 6 0 -8 0 2 E -4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -6 4 B -2 0 0 0 -2 C 14 0 0 8 16 D 6 0 -8 0 2 E -4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -6 4 B -2 0 0 0 -2 C 14 0 0 8 16 D 6 0 -8 0 2 E -4 2 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9069: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (13) D E A B C (9) D A E B C (9) C B A E D (9) E D C A B (6) E D A C B (4) D A B E C (4) E C D B A (3) E C D A B (3) D E B A C (3) C B E D A (3) C B E A D (3) B C A D E (3) B A D C E (3) A B D E C (3) D B A C E (2) C E D B A (2) C E A B D (2) B D A C E (2) A E C B D (2) A D B E C (2) A B D C E (2) E A C B D (1) D E A C B (1) C E B D A (1) C E B A D (1) C B D A E (1) C A B E D (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 24 -6 16 B -2 0 12 4 6 C -24 -12 0 0 0 D 6 -4 0 0 16 E -16 -6 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888894 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 2 24 -6 16 B -2 0 12 4 6 C -24 -12 0 0 0 D 6 -4 0 0 16 E -16 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888890149 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=23 B=21 E=17 A=11 so A is eliminated. Round 2 votes counts: D=30 B=27 C=24 E=19 so E is eliminated. Round 3 votes counts: D=40 C=33 B=27 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:218 B:210 D:209 C:182 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 24 -6 16 B -2 0 12 4 6 C -24 -12 0 0 0 D 6 -4 0 0 16 E -16 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888890149 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 24 -6 16 B -2 0 12 4 6 C -24 -12 0 0 0 D 6 -4 0 0 16 E -16 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888890149 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 24 -6 16 B -2 0 12 4 6 C -24 -12 0 0 0 D 6 -4 0 0 16 E -16 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.500000 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888890149 Cumulative probabilities = A: 0.333333 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9070: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (14) D A C E B (11) D C E A B (10) B A E C D (10) C E B D A (9) A B D E C (8) B E C D A (7) A D B E C (6) E C B D A (5) B E C A D (5) A D C E B (3) A D C B E (2) A D B C E (2) E B C D A (1) D E C B A (1) D A E C B (1) D A E B C (1) C D E A B (1) B E D A C (1) B E A C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -6 -24 -10 B 8 0 -14 -4 -14 C 6 14 0 6 4 D 24 4 -6 0 -8 E 10 14 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -24 -10 B 8 0 -14 -4 -14 C 6 14 0 6 4 D 24 4 -6 0 -8 E 10 14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=24 C=24 B=24 A=22 E=6 so E is eliminated. Round 2 votes counts: C=29 B=25 D=24 A=22 so A is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 E:214 D:207 B:188 A:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -24 -10 B 8 0 -14 -4 -14 C 6 14 0 6 4 D 24 4 -6 0 -8 E 10 14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -24 -10 B 8 0 -14 -4 -14 C 6 14 0 6 4 D 24 4 -6 0 -8 E 10 14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -24 -10 B 8 0 -14 -4 -14 C 6 14 0 6 4 D 24 4 -6 0 -8 E 10 14 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995293 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9071: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) D B C E A (6) B C E D A (5) A E C B D (5) D B E C A (4) C D A B E (4) C A D E B (4) D E B A C (3) D C B A E (3) D B C A E (3) C D B A E (3) C A E D B (3) C A E B D (3) C A D B E (3) B D E C A (3) B D E A C (3) B C D E A (3) A E D B C (3) A C E D B (3) A C E B D (3) E B C A D (2) E A B D C (2) D C A B E (2) B E D C A (2) B E D A C (2) A D C E B (2) E B C D A (1) E B A D C (1) E A C B D (1) D B E A C (1) D A E C B (1) D A E B C (1) D A B E C (1) C D B E A (1) C B E D A (1) C B E A D (1) B E C D A (1) B D C E A (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -18 -24 -2 B 14 0 12 -6 6 C 18 -12 0 -4 10 D 24 6 4 0 8 E 2 -6 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999814 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -18 -24 -2 B 14 0 12 -6 6 C 18 -12 0 -4 10 D 24 6 4 0 8 E 2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 B=20 A=18 E=14 so E is eliminated. Round 2 votes counts: B=31 D=25 C=23 A=21 so A is eliminated. Round 3 votes counts: C=36 B=33 D=31 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:221 B:213 C:206 E:189 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 -18 -24 -2 B 14 0 12 -6 6 C 18 -12 0 -4 10 D 24 6 4 0 8 E 2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -18 -24 -2 B 14 0 12 -6 6 C 18 -12 0 -4 10 D 24 6 4 0 8 E 2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -18 -24 -2 B 14 0 12 -6 6 C 18 -12 0 -4 10 D 24 6 4 0 8 E 2 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9072: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (8) D A C B E (7) B C D E A (6) C B E D A (5) C B D E A (5) D B C A E (4) A E C D B (4) A D E B C (4) E A B D C (3) E A B C D (3) D A E B C (3) C D B A E (3) E B A D C (2) D C B A E (2) D C A B E (2) D B C E A (2) D A C E B (2) C E A B D (2) C B E A D (2) B E C D A (2) B D E C A (2) B D C E A (2) A E D C B (2) A D C E B (2) A C E B D (2) E D A B C (1) E C B A D (1) E A D B C (1) D B E C A (1) D B A E C (1) D B A C E (1) D A B C E (1) C E B A D (1) C D A B E (1) C B D A E (1) C A E B D (1) C A B D E (1) B E C A D (1) B D E A C (1) B C E D A (1) A E C B D (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 4 -12 12 B -8 0 0 -10 8 C -4 0 0 -10 16 D 12 10 10 0 12 E -12 -8 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -12 12 B -8 0 0 -10 8 C -4 0 0 -10 16 D 12 10 10 0 12 E -12 -8 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 C=22 B=15 E=11 so E is eliminated. Round 2 votes counts: A=33 D=27 C=23 B=17 so B is eliminated. Round 3 votes counts: A=35 C=33 D=32 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:222 A:206 C:201 B:195 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -12 12 B -8 0 0 -10 8 C -4 0 0 -10 16 D 12 10 10 0 12 E -12 -8 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -12 12 B -8 0 0 -10 8 C -4 0 0 -10 16 D 12 10 10 0 12 E -12 -8 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -12 12 B -8 0 0 -10 8 C -4 0 0 -10 16 D 12 10 10 0 12 E -12 -8 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9073: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (17) E C A B D (16) E B C A D (10) D A C B E (9) D A C E B (8) C A E D B (5) C A E B D (4) B E C A D (4) E C B A D (3) D B A E C (3) D A B C E (3) D B E C A (2) C E A B D (2) B D E A C (2) B D A E C (2) E C A D B (1) D E B C A (1) D C E A B (1) D C A E B (1) D B E A C (1) C A D E B (1) B E D C A (1) B E A C D (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -4 -2 10 B -6 0 -6 -8 -8 C 4 6 0 -2 6 D 2 8 2 0 4 E -10 8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -2 10 B -6 0 -6 -8 -8 C 4 6 0 -2 6 D 2 8 2 0 4 E -10 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=46 E=30 C=12 B=10 A=2 so A is eliminated. Round 2 votes counts: D=46 E=30 C=14 B=10 so B is eliminated. Round 3 votes counts: D=50 E=36 C=14 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:208 C:207 A:205 E:194 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -2 10 B -6 0 -6 -8 -8 C 4 6 0 -2 6 D 2 8 2 0 4 E -10 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -2 10 B -6 0 -6 -8 -8 C 4 6 0 -2 6 D 2 8 2 0 4 E -10 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -2 10 B -6 0 -6 -8 -8 C 4 6 0 -2 6 D 2 8 2 0 4 E -10 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999077 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9074: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (13) D A E C B (6) B D A E C (5) A C E D B (5) E A D C B (4) C A E D B (4) C A E B D (4) B D E C A (4) B D C E A (4) B C E D A (4) B C A D E (4) A D E C B (4) D E A C B (3) D B A E C (3) D A E B C (3) B D E A C (3) B C A E D (3) E D A C B (2) D B E A C (2) D A B E C (2) C E A B D (2) C B E A D (2) A E D C B (2) A E C D B (2) E D C A B (1) E A C D B (1) D E B C A (1) D E A B C (1) C E B A D (1) B D C A E (1) B D A C E (1) B C E A D (1) B C D E A (1) B A D C E (1) Total count = 100 A B C D E A 0 18 0 6 0 B -18 0 -12 -18 -18 C 0 12 0 -6 2 D -6 18 6 0 -2 E 0 18 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.654814 B: 0.000000 C: 0.345186 D: 0.000000 E: 0.000000 Sum of squares = 0.5479345316 Cumulative probabilities = A: 0.654814 B: 0.654814 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 6 0 B -18 0 -12 -18 -18 C 0 12 0 -6 2 D -6 18 6 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500005 B: 0.000000 C: 0.499995 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000041 Cumulative probabilities = A: 0.500005 B: 0.500005 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=26 D=21 A=13 E=8 so E is eliminated. Round 2 votes counts: B=32 C=26 D=24 A=18 so A is eliminated. Round 3 votes counts: D=34 C=34 B=32 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:212 E:209 D:208 C:204 B:167 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 6 0 B -18 0 -12 -18 -18 C 0 12 0 -6 2 D -6 18 6 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500005 B: 0.000000 C: 0.499995 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000041 Cumulative probabilities = A: 0.500005 B: 0.500005 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 6 0 B -18 0 -12 -18 -18 C 0 12 0 -6 2 D -6 18 6 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500005 B: 0.000000 C: 0.499995 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000041 Cumulative probabilities = A: 0.500005 B: 0.500005 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 6 0 B -18 0 -12 -18 -18 C 0 12 0 -6 2 D -6 18 6 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500005 B: 0.000000 C: 0.499995 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000041 Cumulative probabilities = A: 0.500005 B: 0.500005 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9075: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (7) C B D E A (7) A E D B C (6) B E D C A (5) A D E B C (5) C A E B D (4) E A B D C (3) C D B E A (3) C D A B E (3) C B E A D (3) C A D E B (3) C A D B E (3) B D E A C (3) A C E D B (3) A C D E B (3) E B A D C (2) D B C E A (2) C B E D A (2) C A E D B (2) B D E C A (2) B C D E A (2) A E C D B (2) A E B D C (2) E D B A C (1) E C B A D (1) E B C A D (1) E A D B C (1) E A B C D (1) D E B A C (1) D C A B E (1) D B E C A (1) D B E A C (1) D B C A E (1) D A E B C (1) D A C B E (1) D A B C E (1) C E B A D (1) C E A B D (1) C A B E D (1) B E D A C (1) B E C D A (1) B C E D A (1) B C E A D (1) A E D C B (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 0 -18 4 4 B 0 0 -8 -8 6 C 18 8 0 14 16 D -4 8 -14 0 4 E -4 -6 -16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -18 4 4 B 0 0 -8 -8 6 C 18 8 0 14 16 D -4 8 -14 0 4 E -4 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 A=24 B=16 E=10 D=10 so E is eliminated. Round 2 votes counts: C=41 A=29 B=19 D=11 so D is eliminated. Round 3 votes counts: C=42 A=32 B=26 so B is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:228 D:197 A:195 B:195 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -18 4 4 B 0 0 -8 -8 6 C 18 8 0 14 16 D -4 8 -14 0 4 E -4 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -18 4 4 B 0 0 -8 -8 6 C 18 8 0 14 16 D -4 8 -14 0 4 E -4 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -18 4 4 B 0 0 -8 -8 6 C 18 8 0 14 16 D -4 8 -14 0 4 E -4 -6 -16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9076: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) A E C D B (8) E A B D C (7) C D A E B (7) A C E D B (6) C D B A E (5) B D E C A (5) A E C B D (5) E A B C D (4) C D E A B (4) E B A D C (3) D C B A E (3) D B C E A (3) C A D E B (3) B E A D C (3) B D C A E (3) A E B C D (3) E A C D B (2) C D B E A (2) C D A B E (2) B E D A C (2) A E B D C (2) E D C B A (1) D C E B A (1) D C B E A (1) C D E B A (1) B D E A C (1) B A E D C (1) Total count = 100 A B C D E A 0 6 -6 -6 -4 B -6 0 -2 2 -14 C 6 2 0 4 6 D 6 -2 -4 0 6 E 4 14 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -6 -4 B -6 0 -2 2 -14 C 6 2 0 4 6 D 6 -2 -4 0 6 E 4 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 A=24 E=17 D=8 so D is eliminated. Round 2 votes counts: B=30 C=29 A=24 E=17 so E is eliminated. Round 3 votes counts: A=37 B=33 C=30 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:209 D:203 E:203 A:195 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -6 -4 B -6 0 -2 2 -14 C 6 2 0 4 6 D 6 -2 -4 0 6 E 4 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -6 -4 B -6 0 -2 2 -14 C 6 2 0 4 6 D 6 -2 -4 0 6 E 4 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -6 -4 B -6 0 -2 2 -14 C 6 2 0 4 6 D 6 -2 -4 0 6 E 4 14 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997296 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9077: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (7) E D B C A (5) E B D C A (5) D C E B A (5) B E D C A (5) C D A E B (4) C B E D A (4) C A D B E (4) B E A D C (4) A D E B C (4) D E C B A (3) D C E A B (3) B E D A C (3) B E C D A (3) B A E D C (3) A D C E B (3) A B E D C (3) A B E C D (3) A B C E D (3) E B D A C (2) C D E B A (2) C A B D E (2) B A E C D (2) A C D E B (2) A C D B E (2) E D B A C (1) D E B C A (1) D E A B C (1) D A C E B (1) C E B D A (1) C B E A D (1) C B D E A (1) C A D E B (1) C A B E D (1) B E C A D (1) A E B D C (1) A D E C B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -4 0 -2 B 4 0 2 14 8 C 4 -2 0 -10 -4 D 0 -14 10 0 -4 E 2 -8 4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 0 -2 B 4 0 2 14 8 C 4 -2 0 -10 -4 D 0 -14 10 0 -4 E 2 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=21 B=21 D=14 E=13 so E is eliminated. Round 2 votes counts: A=31 B=28 C=21 D=20 so D is eliminated. Round 3 votes counts: B=35 A=33 C=32 so C is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:201 D:196 A:195 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 0 -2 B 4 0 2 14 8 C 4 -2 0 -10 -4 D 0 -14 10 0 -4 E 2 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 0 -2 B 4 0 2 14 8 C 4 -2 0 -10 -4 D 0 -14 10 0 -4 E 2 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 0 -2 B 4 0 2 14 8 C 4 -2 0 -10 -4 D 0 -14 10 0 -4 E 2 -8 4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999928 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9078: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) C D B E A (5) B E C D A (5) A B D E C (5) E A C B D (4) D A B C E (4) A D B E C (4) E B A C D (3) D C A E B (3) D B A C E (3) C E A D B (3) B D C E A (3) B D A E C (3) A E C D B (3) A E B D C (3) A D C E B (3) E C B A D (2) E C A D B (2) E C A B D (2) D B C A E (2) C D E B A (2) C B E D A (2) C A D E B (2) B E D C A (2) B E A C D (2) A C E D B (2) E B C A D (1) E A B C D (1) D C A B E (1) D B C E A (1) D A C E B (1) C D A E B (1) B E D A C (1) B E A D C (1) B D E A C (1) B D A C E (1) B C D E A (1) B A D E C (1) A E B C D (1) A D E C B (1) A D C B E (1) A C D E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 -2 -6 B 2 0 0 10 0 C -2 0 0 8 2 D 2 -10 -8 0 0 E 6 0 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.749860 C: 0.250140 D: 0.000000 E: 0.000000 Sum of squares = 0.62486038728 Cumulative probabilities = A: 0.000000 B: 0.749860 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 -6 B 2 0 0 10 0 C -2 0 0 8 2 D 2 -10 -8 0 0 E 6 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500252 C: 0.499748 D: 0.000000 E: 0.000000 Sum of squares = 0.500000126852 Cumulative probabilities = A: 0.000000 B: 0.500252 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=24 B=21 E=15 D=15 so E is eliminated. Round 2 votes counts: C=30 A=30 B=25 D=15 so D is eliminated. Round 3 votes counts: A=35 C=34 B=31 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:206 C:204 E:202 A:196 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 -2 -6 B 2 0 0 10 0 C -2 0 0 8 2 D 2 -10 -8 0 0 E 6 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500252 C: 0.499748 D: 0.000000 E: 0.000000 Sum of squares = 0.500000126852 Cumulative probabilities = A: 0.000000 B: 0.500252 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 -6 B 2 0 0 10 0 C -2 0 0 8 2 D 2 -10 -8 0 0 E 6 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500252 C: 0.499748 D: 0.000000 E: 0.000000 Sum of squares = 0.500000126852 Cumulative probabilities = A: 0.000000 B: 0.500252 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 -6 B 2 0 0 10 0 C -2 0 0 8 2 D 2 -10 -8 0 0 E 6 0 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500252 C: 0.499748 D: 0.000000 E: 0.000000 Sum of squares = 0.500000126852 Cumulative probabilities = A: 0.000000 B: 0.500252 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9079: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) E D C B A (7) D C E A B (6) B A E D C (6) A B D C E (6) E C D B A (5) E B C A D (4) E B A D C (4) C E D A B (4) C D A E B (4) B E A D C (4) A B D E C (4) D C A E B (3) D C A B E (3) C A D B E (3) E C B D A (2) E B A C D (2) A C D B E (2) A B C D E (2) E D B A C (1) E C B A D (1) E B D A C (1) E B C D A (1) D E C B A (1) D E C A B (1) D C E B A (1) D A C B E (1) D A B E C (1) D A B C E (1) C E D B A (1) C D E B A (1) C D A B E (1) C A B D E (1) B E A C D (1) B A E C D (1) A D C B E (1) A B C E D (1) Total count = 100 A B C D E A 0 12 -22 -14 -18 B -12 0 -20 -18 -22 C 22 20 0 -4 6 D 14 18 4 0 8 E 18 22 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -22 -14 -18 B -12 0 -20 -18 -22 C 22 20 0 -4 6 D 14 18 4 0 8 E 18 22 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=26 D=18 A=16 B=12 so B is eliminated. Round 2 votes counts: E=33 C=26 A=23 D=18 so D is eliminated. Round 3 votes counts: C=39 E=35 A=26 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:222 E:213 A:179 B:164 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -22 -14 -18 B -12 0 -20 -18 -22 C 22 20 0 -4 6 D 14 18 4 0 8 E 18 22 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -22 -14 -18 B -12 0 -20 -18 -22 C 22 20 0 -4 6 D 14 18 4 0 8 E 18 22 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -22 -14 -18 B -12 0 -20 -18 -22 C 22 20 0 -4 6 D 14 18 4 0 8 E 18 22 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9080: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) E A C B D (8) D C A E B (7) B D E A C (7) A C E D B (7) D B C A E (6) C A D E B (4) B D C A E (4) E A B C D (3) D C B A E (3) C A E D B (3) B E A C D (3) B A E C D (3) A E C B D (3) E D C A B (2) E B A C D (2) E A C D B (2) D B C E A (2) B E D A C (2) B D A E C (2) B A D C E (2) A C D B E (2) D C E B A (1) D C B E A (1) D C A B E (1) D B E C A (1) C E A D B (1) C D A E B (1) B E A D C (1) B D E C A (1) B D A C E (1) B A E D C (1) B A D E C (1) A C E B D (1) A C D E B (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 8 16 10 B -12 0 -14 -6 -8 C -8 14 0 8 0 D -16 6 -8 0 -2 E -10 8 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 8 16 10 B -12 0 -14 -6 -8 C -8 14 0 8 0 D -16 6 -8 0 -2 E -10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 D=22 A=16 C=9 so C is eliminated. Round 2 votes counts: B=28 E=26 D=23 A=23 so D is eliminated. Round 3 votes counts: B=41 A=32 E=27 so E is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:207 E:200 D:190 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 8 16 10 B -12 0 -14 -6 -8 C -8 14 0 8 0 D -16 6 -8 0 -2 E -10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 8 16 10 B -12 0 -14 -6 -8 C -8 14 0 8 0 D -16 6 -8 0 -2 E -10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 8 16 10 B -12 0 -14 -6 -8 C -8 14 0 8 0 D -16 6 -8 0 -2 E -10 8 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9081: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (11) B D A E C (7) C E A D B (6) E C D A B (5) D E A C B (5) D B E C A (5) B A C E D (5) A C E D B (5) E C B D A (4) D E B C A (4) A C E B D (4) A B C E D (4) E D C A B (3) D E C A B (3) B E C D A (3) B D E C A (3) B C E A D (3) D B A E C (2) D A E C B (2) B E D C A (2) A D B C E (2) E C D B A (1) D B E A C (1) C E B A D (1) C E A B D (1) B D E A C (1) B C E D A (1) B A C D E (1) A D C E B (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 10 -4 -4 B 10 0 16 10 10 C -10 -16 0 -8 -4 D 4 -10 8 0 2 E 4 -10 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -4 -4 B 10 0 16 10 10 C -10 -16 0 -8 -4 D 4 -10 8 0 2 E 4 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=22 A=20 E=13 C=8 so C is eliminated. Round 2 votes counts: B=37 D=22 E=21 A=20 so A is eliminated. Round 3 votes counts: B=45 E=30 D=25 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:202 E:198 A:196 C:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 -4 -4 B 10 0 16 10 10 C -10 -16 0 -8 -4 D 4 -10 8 0 2 E 4 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -4 -4 B 10 0 16 10 10 C -10 -16 0 -8 -4 D 4 -10 8 0 2 E 4 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -4 -4 B 10 0 16 10 10 C -10 -16 0 -8 -4 D 4 -10 8 0 2 E 4 -10 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9082: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (13) B C E D A (10) D E C A B (7) B A C E D (7) B E C D A (5) B C E A D (5) B A E C D (5) A D C E B (5) A B D C E (5) D A E C B (4) A B D E C (4) C E D B A (3) B C A E D (3) A D E B C (3) D C E A B (2) C E B D A (2) C B E D A (2) A D B E C (2) A B C D E (2) E D C B A (1) E B C D A (1) D E C B A (1) D E A C B (1) D E A B C (1) D A C E B (1) B E A D C (1) A D B C E (1) A C D E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 16 18 16 B -8 0 12 8 8 C -16 -12 0 -4 2 D -18 -8 4 0 8 E -16 -8 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 18 16 B -8 0 12 8 8 C -16 -12 0 -4 2 D -18 -8 4 0 8 E -16 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=36 D=17 C=7 E=2 so E is eliminated. Round 2 votes counts: A=38 B=37 D=18 C=7 so C is eliminated. Round 3 votes counts: B=41 A=38 D=21 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:229 B:210 D:193 C:185 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 18 16 B -8 0 12 8 8 C -16 -12 0 -4 2 D -18 -8 4 0 8 E -16 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 18 16 B -8 0 12 8 8 C -16 -12 0 -4 2 D -18 -8 4 0 8 E -16 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 18 16 B -8 0 12 8 8 C -16 -12 0 -4 2 D -18 -8 4 0 8 E -16 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9083: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) B D E A C (8) B D C A E (6) D B E C A (5) E C D A B (4) E A C D B (4) C A E D B (4) C A E B D (4) A C B E D (4) E C A D B (3) D E B C A (3) B C D A E (3) E D B C A (2) E D A B C (2) E A D C B (2) D B E A C (2) D B C E A (2) C E A D B (2) C D B A E (2) B D C E A (2) B D A C E (2) B A D E C (2) B A C D E (2) A C B D E (2) A B C D E (2) E D C B A (1) E D B A C (1) E D A C B (1) E B A D C (1) E A D B C (1) D E C B A (1) C B A D E (1) C A B E D (1) C A B D E (1) B E D A C (1) B A E D C (1) B A D C E (1) A E C B D (1) A E B D C (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 6 4 4 B -2 0 2 14 2 C -6 -2 0 4 6 D -4 -14 -4 0 -6 E -4 -2 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 4 4 B -2 0 2 14 2 C -6 -2 0 4 6 D -4 -14 -4 0 -6 E -4 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=22 A=22 C=15 D=13 so D is eliminated. Round 2 votes counts: B=37 E=26 A=22 C=15 so C is eliminated. Round 3 votes counts: B=40 A=32 E=28 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:208 B:208 C:201 E:197 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 4 4 B -2 0 2 14 2 C -6 -2 0 4 6 D -4 -14 -4 0 -6 E -4 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 4 4 B -2 0 2 14 2 C -6 -2 0 4 6 D -4 -14 -4 0 -6 E -4 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 4 4 B -2 0 2 14 2 C -6 -2 0 4 6 D -4 -14 -4 0 -6 E -4 -2 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999393 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9084: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) E B A D C (8) D E C B A (8) D C A E B (8) E D B A C (7) C A B D E (6) E B D A C (5) D E B C A (5) D C E A B (5) A C B E D (5) B A E C D (4) A B C E D (4) D E C A B (3) D C A B E (3) C D A B E (3) C A D B E (3) E B A C D (2) C D A E B (2) E D C A B (1) E C D A B (1) E A C D B (1) E A B C D (1) D E B A C (1) C B A D E (1) C A D E B (1) B E A D C (1) B E A C D (1) B A C E D (1) Total count = 100 A B C D E A 0 12 -18 -4 0 B -12 0 -20 -4 -18 C 18 20 0 -10 2 D 4 4 10 0 -2 E 0 18 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.55102040838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 A B C D E A 0 12 -18 -4 0 B -12 0 -20 -4 -18 C 18 20 0 -10 2 D 4 4 10 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020410975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=26 C=25 A=9 B=7 so B is eliminated. Round 2 votes counts: D=33 E=28 C=25 A=14 so A is eliminated. Round 3 votes counts: C=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:215 E:209 D:208 A:195 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -18 -4 0 B -12 0 -20 -4 -18 C 18 20 0 -10 2 D 4 4 10 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020410975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -18 -4 0 B -12 0 -20 -4 -18 C 18 20 0 -10 2 D 4 4 10 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020410975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -18 -4 0 B -12 0 -20 -4 -18 C 18 20 0 -10 2 D 4 4 10 0 -2 E 0 18 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.142857 E: 0.714286 Sum of squares = 0.551020410975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.285714 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9085: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (13) E B D A C (11) A C B E D (10) B E A C D (9) C A D B E (6) E B D C A (5) A C D B E (5) D E B A C (4) D C E B A (4) A B E C D (4) C D A E B (3) C D A B E (3) B E D C A (3) E D B C A (2) D C B E A (2) C A B E D (2) B E A D C (2) A D C E B (2) A C D E B (2) A B C E D (2) D E C A B (1) D C A E B (1) C D B E A (1) B E D A C (1) B E C A D (1) A C E B D (1) Total count = 100 A B C D E A 0 -16 6 -8 -18 B 16 0 14 2 2 C -6 -14 0 -2 -12 D 8 -2 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999065 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 6 -8 -18 B 16 0 14 2 2 C -6 -14 0 -2 -12 D 8 -2 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 E=18 B=16 C=15 so C is eliminated. Round 2 votes counts: A=34 D=32 E=18 B=16 so B is eliminated. Round 3 votes counts: E=34 A=34 D=32 so D is eliminated. Round 4 votes counts: E=59 A=41 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:217 E:217 D:201 C:183 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 -8 -18 B 16 0 14 2 2 C -6 -14 0 -2 -12 D 8 -2 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -8 -18 B 16 0 14 2 2 C -6 -14 0 -2 -12 D 8 -2 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -8 -18 B 16 0 14 2 2 C -6 -14 0 -2 -12 D 8 -2 2 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999569 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9086: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) E A D C B (10) C B D A E (8) B E C A D (8) E A D B C (7) B C D A E (7) A E D C B (6) C D B A E (5) D C A B E (4) D A C E B (4) E B A D C (3) E B A C D (3) E A B D C (3) C D A B E (3) B C E D A (3) D A E C B (2) B C E A D (2) B C D E A (2) A D C E B (2) D C A E B (1) D A C B E (1) C B D E A (1) B E C D A (1) B E A C D (1) B A E C D (1) A C D B E (1) Total count = 100 A B C D E A 0 10 10 16 12 B -10 0 -18 -14 -4 C -10 18 0 -8 -12 D -16 14 8 0 4 E -12 4 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 16 12 B -10 0 -18 -14 -4 C -10 18 0 -8 -12 D -16 14 8 0 4 E -12 4 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=25 A=20 C=17 D=12 so D is eliminated. Round 2 votes counts: A=27 E=26 B=25 C=22 so C is eliminated. Round 3 votes counts: B=39 A=35 E=26 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:205 E:200 C:194 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 16 12 B -10 0 -18 -14 -4 C -10 18 0 -8 -12 D -16 14 8 0 4 E -12 4 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 16 12 B -10 0 -18 -14 -4 C -10 18 0 -8 -12 D -16 14 8 0 4 E -12 4 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 16 12 B -10 0 -18 -14 -4 C -10 18 0 -8 -12 D -16 14 8 0 4 E -12 4 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9087: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) B A D C E (6) A E B D C (6) E D C A B (5) D E C B A (5) D E B A C (5) E C D A B (4) D C E B A (4) C E D B A (4) C D B E A (4) C B D A E (4) B D C A E (4) B A C D E (4) E D B A C (2) E D A C B (2) E D A B C (2) E A D B C (2) D B E A C (2) D B C E A (2) C A B E D (2) B D A C E (2) B C A D E (2) A E C B D (2) A B E D C (2) E C D B A (1) C E A D B (1) C B A D E (1) C A E B D (1) B D A E C (1) A E D B C (1) A E B C D (1) A C E B D (1) A C B E D (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -26 -8 -26 -10 B 26 0 -4 -12 -18 C 8 4 0 -30 -8 D 26 12 30 0 -2 E 10 18 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -26 -8 -26 -10 B 26 0 -4 -12 -18 C 8 4 0 -30 -8 D 26 12 30 0 -2 E 10 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=19 D=18 A=18 C=17 so C is eliminated. Round 2 votes counts: E=33 B=24 D=22 A=21 so A is eliminated. Round 3 votes counts: E=45 B=33 D=22 so D is eliminated. Round 4 votes counts: E=59 B=41 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:233 E:219 B:196 C:187 A:165 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -26 -8 -26 -10 B 26 0 -4 -12 -18 C 8 4 0 -30 -8 D 26 12 30 0 -2 E 10 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -8 -26 -10 B 26 0 -4 -12 -18 C 8 4 0 -30 -8 D 26 12 30 0 -2 E 10 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -8 -26 -10 B 26 0 -4 -12 -18 C 8 4 0 -30 -8 D 26 12 30 0 -2 E 10 18 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999159 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9088: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (14) E D B A C (12) D E C B A (9) C A B D E (9) C D E B A (7) B A C E D (5) A C B E D (4) D E C A B (3) D E B C A (3) A B E D C (3) E D B C A (2) D A E C B (2) C D E A B (2) C B A D E (2) C A B E D (2) B E A D C (2) B A E D C (2) D E B A C (1) D E A C B (1) D E A B C (1) D C E B A (1) D C E A B (1) C D B E A (1) C B E D A (1) C B D E A (1) C B A E D (1) C A D E B (1) C A D B E (1) B E D C A (1) B E D A C (1) B E C D A (1) B C A E D (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -8 0 -2 -2 B 8 0 2 2 4 C 0 -2 0 6 8 D 2 -2 -6 0 -8 E 2 -4 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -2 -2 B 8 0 2 2 4 C 0 -2 0 6 8 D 2 -2 -6 0 -8 E 2 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=23 D=22 E=14 B=13 so B is eliminated. Round 2 votes counts: A=30 C=29 D=22 E=19 so E is eliminated. Round 3 votes counts: D=38 A=32 C=30 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:208 C:206 E:199 A:194 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -2 -2 B 8 0 2 2 4 C 0 -2 0 6 8 D 2 -2 -6 0 -8 E 2 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -2 -2 B 8 0 2 2 4 C 0 -2 0 6 8 D 2 -2 -6 0 -8 E 2 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -2 -2 B 8 0 2 2 4 C 0 -2 0 6 8 D 2 -2 -6 0 -8 E 2 -4 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9089: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) B A E C D (9) C D E B A (7) C D B E A (6) E A B C D (5) D C B A E (5) E A B D C (4) A B E D C (4) E D C A B (3) E C D A B (3) B A C E D (3) B A C D E (3) A E B D C (3) E C D B A (2) E C B D A (2) D E C A B (2) B C A D E (2) B A D C E (2) A B E C D (2) A B D E C (2) A B D C E (2) E D A C B (1) E D A B C (1) E C B A D (1) E B A C D (1) E A D C B (1) D E A C B (1) D C E B A (1) D C B E A (1) D C A E B (1) D A E C B (1) D A E B C (1) C E D B A (1) C E B D A (1) C D B A E (1) C B D A E (1) B D C A E (1) B D A C E (1) B C A E D (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -2 -6 -6 B 4 0 -2 2 -6 C 2 2 0 4 -2 D 6 -2 -4 0 2 E 6 6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.37500000008 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 A B C D E A 0 -4 -2 -6 -6 B 4 0 -2 2 -6 C 2 2 0 4 -2 D 6 -2 -4 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 D=22 B=22 C=17 A=15 so A is eliminated. Round 2 votes counts: B=32 E=28 D=23 C=17 so C is eliminated. Round 3 votes counts: D=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:206 C:203 D:201 B:199 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -6 -6 B 4 0 -2 2 -6 C 2 2 0 4 -2 D 6 -2 -4 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -6 -6 B 4 0 -2 2 -6 C 2 2 0 4 -2 D 6 -2 -4 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -6 -6 B 4 0 -2 2 -6 C 2 2 0 4 -2 D 6 -2 -4 0 2 E 6 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 0.500000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9090: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (6) C A B D E (5) B A C D E (5) B D C A E (4) E D B A C (3) E A D C B (3) D E C B A (3) C D B A E (3) C B A D E (3) C A E B D (3) B D E A C (3) A E C B D (3) A E B D C (3) A C E B D (3) A B E D C (3) E A D B C (2) E A C D B (2) E A B D C (2) D E B C A (2) D B E C A (2) D B E A C (2) D B C E A (2) C B D A E (2) C A D B E (2) C A B E D (2) B D A C E (2) B C D A E (2) B A E D C (2) B A D E C (2) A E B C D (2) A B C E D (2) E D A C B (1) E D A B C (1) E C D A B (1) E B D A C (1) E A C B D (1) D E B A C (1) D C E B A (1) D C B E A (1) D B C A E (1) C D E B A (1) C D A E B (1) C A D E B (1) B D A E C (1) B C A D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 14 18 30 B 12 0 14 28 18 C -14 -14 0 -8 8 D -18 -28 8 0 18 E -30 -18 -8 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 14 18 30 B 12 0 14 28 18 C -14 -14 0 -8 8 D -18 -28 8 0 18 E -30 -18 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 E=17 A=17 D=15 so D is eliminated. Round 2 votes counts: B=35 C=25 E=23 A=17 so A is eliminated. Round 3 votes counts: B=40 E=31 C=29 so C is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:236 A:225 D:190 C:186 E:163 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 14 18 30 B 12 0 14 28 18 C -14 -14 0 -8 8 D -18 -28 8 0 18 E -30 -18 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 14 18 30 B 12 0 14 28 18 C -14 -14 0 -8 8 D -18 -28 8 0 18 E -30 -18 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 14 18 30 B 12 0 14 28 18 C -14 -14 0 -8 8 D -18 -28 8 0 18 E -30 -18 -8 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9091: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (11) D A B C E (9) E C A B D (6) E A C D B (6) A D C B E (6) B C E D A (5) E A D C B (4) C E B A D (4) B D C A E (4) B C D A E (4) E C B A D (3) E C A D B (3) E B C D A (3) C A E D B (3) A D B C E (3) D B A E C (2) C E A D B (2) B D E C A (2) E D B A C (1) E C B D A (1) E B D A C (1) D B A C E (1) D A E B C (1) D A C B E (1) D A B E C (1) C E B D A (1) C B E A D (1) C B D A E (1) C B A D E (1) C A D B E (1) B E C D A (1) B D C E A (1) B C D E A (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 4 -4 8 B -2 0 2 2 14 C -4 -2 0 0 28 D 4 -2 0 0 8 E -8 -14 -28 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999997 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -4 8 B -2 0 2 2 14 C -4 -2 0 0 28 D 4 -2 0 0 8 E -8 -14 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000014 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=28 D=15 C=14 A=14 so C is eliminated. Round 2 votes counts: E=35 B=32 A=18 D=15 so D is eliminated. Round 3 votes counts: E=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=57 E=43 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:208 A:205 D:205 E:171 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 4 -4 8 B -2 0 2 2 14 C -4 -2 0 0 28 D 4 -2 0 0 8 E -8 -14 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000014 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -4 8 B -2 0 2 2 14 C -4 -2 0 0 28 D 4 -2 0 0 8 E -8 -14 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000014 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -4 8 B -2 0 2 2 14 C -4 -2 0 0 28 D 4 -2 0 0 8 E -8 -14 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.000000 D: 0.250000 E: 0.000000 Sum of squares = 0.37500000014 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9092: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (14) D E B C A (12) D A C B E (10) D E A C B (8) E D B C A (6) E B C A D (6) A D C B E (6) B C A E D (5) B E C A D (4) B C E A D (4) A C D B E (4) A C B D E (4) D A C E B (3) E B C D A (2) D E C A B (2) C B A E D (2) E B D C A (1) D E C B A (1) D E A B C (1) D A E C B (1) D A E B C (1) B A C E D (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 12 10 4 6 B -12 0 -12 -12 10 C -10 12 0 -4 10 D -4 12 4 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999047 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 10 4 6 B -12 0 -12 -12 10 C -10 12 0 -4 10 D -4 12 4 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=30 E=15 B=14 C=2 so C is eliminated. Round 2 votes counts: D=39 A=30 B=16 E=15 so E is eliminated. Round 3 votes counts: D=45 A=30 B=25 so B is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:216 D:210 C:204 B:187 E:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 10 4 6 B -12 0 -12 -12 10 C -10 12 0 -4 10 D -4 12 4 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 4 6 B -12 0 -12 -12 10 C -10 12 0 -4 10 D -4 12 4 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 4 6 B -12 0 -12 -12 10 C -10 12 0 -4 10 D -4 12 4 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9093: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (8) C B A E D (7) D E B A C (6) C A B E D (6) A D E B C (6) D E B C A (4) C D B E A (4) C B E D A (4) B E D C A (4) D A E B C (3) B E C D A (3) A E D B C (3) E D B A C (2) E B D A C (2) E B A D C (2) D E C B A (2) D A C E B (2) C B E A D (2) C A D B E (2) B E A D C (2) B D E C A (2) A E B D C (2) A D E C B (2) A C B E D (2) E D A B C (1) E A D B C (1) D C E A B (1) D B C E A (1) D A E C B (1) C D A E B (1) C D A B E (1) C B D E A (1) C B A D E (1) C A D E B (1) C A B D E (1) B E A C D (1) B C E D A (1) B C E A D (1) A E C B D (1) A E B C D (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 0 -8 -10 B 4 0 12 -8 -8 C 0 -12 0 -16 -18 D 8 8 16 0 4 E 10 8 18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -8 -10 B 4 0 12 -8 -8 C 0 -12 0 -16 -18 D 8 8 16 0 4 E 10 8 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=28 A=19 B=14 E=8 so E is eliminated. Round 2 votes counts: D=31 C=31 A=20 B=18 so B is eliminated. Round 3 votes counts: D=39 C=36 A=25 so A is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 E:216 B:200 A:189 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -8 -10 B 4 0 12 -8 -8 C 0 -12 0 -16 -18 D 8 8 16 0 4 E 10 8 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -8 -10 B 4 0 12 -8 -8 C 0 -12 0 -16 -18 D 8 8 16 0 4 E 10 8 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -8 -10 B 4 0 12 -8 -8 C 0 -12 0 -16 -18 D 8 8 16 0 4 E 10 8 18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9094: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (6) B D C E A (6) B D C A E (6) A E B C D (6) A B E C D (5) E A C D B (4) B D E C A (4) B A E D C (4) A E C B D (4) A C E B D (4) E B A D C (3) D E B C A (3) D C B E A (3) E D C A B (2) E A B D C (2) D C E B A (2) D B E C A (2) D B C E A (2) C D B A E (2) B D E A C (2) B D A E C (2) A C E D B (2) E D C B A (1) E C D A B (1) E C A D B (1) E A B C D (1) D E C B A (1) D E C A B (1) D C E A B (1) D C B A E (1) C E D A B (1) C E A D B (1) C D A E B (1) C A D E B (1) C A B D E (1) B E D A C (1) B D A C E (1) B C D A E (1) B A D E C (1) B A D C E (1) B A C E D (1) B A C D E (1) A E C D B (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 -6 -2 B 2 0 12 20 0 C 2 -12 0 -4 -4 D 6 -20 4 0 6 E 2 0 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.445004 C: 0.000000 D: 0.000000 E: 0.554996 Sum of squares = 0.506049161368 Cumulative probabilities = A: 0.000000 B: 0.445004 C: 0.445004 D: 0.445004 E: 1.000000 A B C D E A 0 -2 -2 -6 -2 B 2 0 12 20 0 C 2 -12 0 -4 -4 D 6 -20 4 0 6 E 2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=25 D=16 E=15 C=13 so C is eliminated. Round 2 votes counts: B=31 A=27 D=25 E=17 so E is eliminated. Round 3 votes counts: A=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 E:200 D:198 A:194 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 -6 -2 B 2 0 12 20 0 C 2 -12 0 -4 -4 D 6 -20 4 0 6 E 2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -6 -2 B 2 0 12 20 0 C 2 -12 0 -4 -4 D 6 -20 4 0 6 E 2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -6 -2 B 2 0 12 20 0 C 2 -12 0 -4 -4 D 6 -20 4 0 6 E 2 0 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9095: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) C E D A B (6) C D E A B (6) B E C A D (6) E C D B A (5) E C B D A (4) A B D C E (4) E B C D A (3) E B C A D (3) D E C A B (3) C E D B A (3) C E B A D (3) B E A C D (3) B A E D C (3) A C D B E (3) E D C B A (2) E D B C A (2) D A C E B (2) D A B E C (2) B A E C D (2) B A C E D (2) A D B E C (2) A D B C E (2) A B C E D (2) A B C D E (2) E C B A D (1) D E B A C (1) D E A C B (1) D C A E B (1) D A C B E (1) D A B C E (1) C E B D A (1) C D A E B (1) C A E B D (1) C A D B E (1) C A B E D (1) B E A D C (1) B D A E C (1) B C A E D (1) B A D E C (1) A D C B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -22 -6 -22 B -4 0 -10 -6 -12 C 22 10 0 22 6 D 6 6 -22 0 -10 E 22 12 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -22 -6 -22 B -4 0 -10 -6 -12 C 22 10 0 22 6 D 6 6 -22 0 -10 E 22 12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 E=20 B=20 D=19 A=18 so A is eliminated. Round 2 votes counts: B=29 C=27 D=24 E=20 so E is eliminated. Round 3 votes counts: C=37 B=35 D=28 so D is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:230 E:219 D:190 B:184 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -22 -6 -22 B -4 0 -10 -6 -12 C 22 10 0 22 6 D 6 6 -22 0 -10 E 22 12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -22 -6 -22 B -4 0 -10 -6 -12 C 22 10 0 22 6 D 6 6 -22 0 -10 E 22 12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -22 -6 -22 B -4 0 -10 -6 -12 C 22 10 0 22 6 D 6 6 -22 0 -10 E 22 12 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999737 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9096: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) B E D C A (7) D A E C B (6) B E C A D (6) C A D E B (5) C A B D E (5) B C E A D (5) B C A E D (5) A C D B E (5) E B D C A (4) B C A D E (4) E D B A C (3) E D A C B (3) E B D A C (3) D E A C B (3) D E A B C (3) B E D A C (3) E B C D A (2) C B A D E (2) A D C E B (2) A C D E B (2) E D C A B (1) E C B A D (1) D A C B E (1) D A B E C (1) C E B A D (1) C E A D B (1) C A E D B (1) C A E B D (1) C A D B E (1) B E C D A (1) B D A E C (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -6 -2 6 B -4 0 -2 4 -2 C 6 2 0 -2 4 D 2 -4 2 0 4 E -6 2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -2 6 B -4 0 -2 4 -2 C 6 2 0 -2 4 D 2 -4 2 0 4 E -6 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000033 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=23 E=17 C=17 A=11 so A is eliminated. Round 2 votes counts: B=33 D=26 C=24 E=17 so E is eliminated. Round 3 votes counts: B=42 D=33 C=25 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:205 D:202 A:201 B:198 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -2 6 B -4 0 -2 4 -2 C 6 2 0 -2 4 D 2 -4 2 0 4 E -6 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000033 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -2 6 B -4 0 -2 4 -2 C 6 2 0 -2 4 D 2 -4 2 0 4 E -6 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000033 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -2 6 B -4 0 -2 4 -2 C 6 2 0 -2 4 D 2 -4 2 0 4 E -6 2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000033 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9097: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E A D C B (6) B C D A E (6) B A C D E (6) E D A C B (5) E B D C A (5) A C D B E (5) E B A D C (4) D C E A B (4) D C A E B (4) B E A C D (4) A D C E B (4) A B C D E (4) B E C D A (3) B C A D E (3) B A E C D (3) E D C A B (2) E B C D A (2) E B A C D (2) E A B D C (2) D C A B E (2) D A C E B (2) C D B A E (2) A E D C B (2) A D C B E (2) A B E C D (2) D C E B A (1) C D B E A (1) C B D E A (1) C A D B E (1) B E C A D (1) B C D E A (1) A E B D C (1) Total count = 100 A B C D E A 0 10 8 4 12 B -10 0 -2 0 8 C -8 2 0 8 12 D -4 0 -8 0 12 E -12 -8 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 4 12 B -10 0 -2 0 8 C -8 2 0 8 12 D -4 0 -8 0 12 E -12 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 A=20 D=13 C=12 so C is eliminated. Round 2 votes counts: E=28 B=28 D=23 A=21 so A is eliminated. Round 3 votes counts: D=35 B=34 E=31 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:217 C:207 D:200 B:198 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 4 12 B -10 0 -2 0 8 C -8 2 0 8 12 D -4 0 -8 0 12 E -12 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 4 12 B -10 0 -2 0 8 C -8 2 0 8 12 D -4 0 -8 0 12 E -12 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 4 12 B -10 0 -2 0 8 C -8 2 0 8 12 D -4 0 -8 0 12 E -12 -8 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9098: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (13) D E C A B (11) D E B C A (8) D A C B E (6) B E A C D (6) D C A E B (5) E D B C A (4) E B C A D (4) C A B E D (4) B E D A C (4) A C B D E (4) E B D C A (3) A C B E D (3) E D C A B (2) D E B A C (2) D B E A C (2) C A E D B (2) C A D B E (2) B A C D E (2) A C D B E (2) E B D A C (1) E B C D A (1) D E C B A (1) D E A B C (1) D C A B E (1) D B A C E (1) D A B C E (1) C A D E B (1) B C A E D (1) B A E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 -8 0 B 8 0 12 -4 8 C 0 -12 0 -6 -2 D 8 4 6 0 0 E 0 -8 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.785679 E: 0.214321 Sum of squares = 0.663225222867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.785679 E: 1.000000 A B C D E A 0 -8 0 -8 0 B 8 0 12 -4 8 C 0 -12 0 -6 -2 D 8 4 6 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555556335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=27 E=15 A=10 C=9 so C is eliminated. Round 2 votes counts: D=39 B=27 A=19 E=15 so E is eliminated. Round 3 votes counts: D=45 B=36 A=19 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:209 E:197 A:192 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 0 -8 0 B 8 0 12 -4 8 C 0 -12 0 -6 -2 D 8 4 6 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555556335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -8 0 B 8 0 12 -4 8 C 0 -12 0 -6 -2 D 8 4 6 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555556335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -8 0 B 8 0 12 -4 8 C 0 -12 0 -6 -2 D 8 4 6 0 0 E 0 -8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.555555556335 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9099: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) B E A D C (7) D C B A E (6) D C A B E (6) D A E B C (5) C D B A E (5) C B D E A (5) B E C A D (5) D C A E B (4) C D B E A (4) E A B D C (3) C E B A D (3) C B E D A (3) C A E D B (3) A E D B C (3) A E B D C (3) E B A D C (2) E A B C D (2) D A C E B (2) C A D E B (2) B E A C D (2) B C E D A (2) B C E A D (2) A D E B C (2) E B A C D (1) E A C B D (1) D B C A E (1) D B A E C (1) D A E C B (1) C E A B D (1) B C D E A (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 -18 -24 8 -6 B 18 0 -16 4 18 C 24 16 0 4 18 D -8 -4 -4 0 -8 E 6 -18 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -24 8 -6 B 18 0 -16 4 18 C 24 16 0 4 18 D -8 -4 -4 0 -8 E 6 -18 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=26 B=19 A=11 E=9 so E is eliminated. Round 2 votes counts: C=35 D=26 B=22 A=17 so A is eliminated. Round 3 votes counts: C=37 D=33 B=30 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:231 B:212 E:189 D:188 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -24 8 -6 B 18 0 -16 4 18 C 24 16 0 4 18 D -8 -4 -4 0 -8 E 6 -18 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -24 8 -6 B 18 0 -16 4 18 C 24 16 0 4 18 D -8 -4 -4 0 -8 E 6 -18 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -24 8 -6 B 18 0 -16 4 18 C 24 16 0 4 18 D -8 -4 -4 0 -8 E 6 -18 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9100: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (11) E B A C D (7) A D C E B (6) C D A B E (5) B D E C A (4) D C B A E (3) D B E C A (3) D B E A C (3) D B C E A (3) C D B A E (3) C D A E B (3) C A E B D (3) B E C A D (3) A E C D B (3) A E C B D (3) A C D E B (3) E B D A C (2) E B A D C (2) E A B D C (2) E A B C D (2) D C A E B (2) D A E B C (2) D A C E B (2) C A D E B (2) B E D C A (2) B E A C D (2) A D E B C (2) D A C B E (1) C B D E A (1) B E D A C (1) B E C D A (1) B E A D C (1) B D C E A (1) B C E D A (1) B C D E A (1) A E D B C (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 12 -4 -10 16 B -12 0 -6 -18 0 C 4 6 0 -8 6 D 10 18 8 0 22 E -16 0 -6 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 -10 16 B -12 0 -6 -18 0 C 4 6 0 -8 6 D 10 18 8 0 22 E -16 0 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=21 C=17 B=17 E=15 so E is eliminated. Round 2 votes counts: D=30 B=28 A=25 C=17 so C is eliminated. Round 3 votes counts: D=41 A=30 B=29 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:229 A:207 C:204 B:182 E:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -4 -10 16 B -12 0 -6 -18 0 C 4 6 0 -8 6 D 10 18 8 0 22 E -16 0 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 -10 16 B -12 0 -6 -18 0 C 4 6 0 -8 6 D 10 18 8 0 22 E -16 0 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 -10 16 B -12 0 -6 -18 0 C 4 6 0 -8 6 D 10 18 8 0 22 E -16 0 -6 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9101: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (7) E A B C D (5) C D E A B (5) B A E C D (5) B A D C E (5) E D C A B (4) B E A D C (4) B A C D E (4) E C D A B (3) D C E B A (3) D C B E A (3) D B C E A (3) C D A E B (3) C D A B E (3) B D C A E (3) A E B C D (3) A B E C D (3) E C A D B (2) E B D A C (2) C E D A B (2) C A D E B (2) C A D B E (2) B D A C E (2) B A E D C (2) A E C D B (2) A E C B D (2) A B C D E (2) E D C B A (1) E D B C A (1) D E C B A (1) D E C A B (1) D C B A E (1) D C A B E (1) D B C A E (1) B E D A C (1) B D C E A (1) A C E D B (1) A C E B D (1) A C D E B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -6 -4 2 B -14 0 -4 -6 -4 C 6 4 0 6 16 D 4 6 -6 0 10 E -2 4 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 -4 2 B -14 0 -4 -6 -4 C 6 4 0 6 16 D 4 6 -6 0 10 E -2 4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=21 E=18 C=17 A=17 so C is eliminated. Round 2 votes counts: D=32 B=27 A=21 E=20 so E is eliminated. Round 3 votes counts: D=43 B=29 A=28 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:207 A:203 E:188 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 -4 2 B -14 0 -4 -6 -4 C 6 4 0 6 16 D 4 6 -6 0 10 E -2 4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 -4 2 B -14 0 -4 -6 -4 C 6 4 0 6 16 D 4 6 -6 0 10 E -2 4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 -4 2 B -14 0 -4 -6 -4 C 6 4 0 6 16 D 4 6 -6 0 10 E -2 4 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9102: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (7) D B A E C (6) C B A E D (6) B D C A E (5) B A C E D (5) E D C A B (4) B C A E D (4) E C D A B (3) E A D C B (3) D E C B A (3) D E A C B (3) D B E A C (3) D B C E A (3) B A C D E (3) A B D E C (3) E C A D B (2) D E C A B (2) D E B A C (2) D E A B C (2) D C E B A (2) D B E C A (2) C B E D A (2) C A E B D (2) B C D A E (2) A E C D B (2) A E C B D (2) E D A C B (1) E A C D B (1) D E B C A (1) C E B D A (1) C E B A D (1) C E A B D (1) C B E A D (1) C B D E A (1) B D C E A (1) B D A E C (1) B D A C E (1) B C D E A (1) B C A D E (1) A E D B C (1) A E B D C (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -16 -2 -4 8 B 16 0 12 8 18 C 2 -12 0 0 2 D 4 -8 0 0 -4 E -8 -18 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 -4 8 B 16 0 12 8 18 C 2 -12 0 0 2 D 4 -8 0 0 -4 E -8 -18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=24 A=18 C=15 E=14 so E is eliminated. Round 2 votes counts: D=34 B=24 A=22 C=20 so C is eliminated. Round 3 votes counts: D=37 B=36 A=27 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 C:196 D:196 A:193 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 -4 8 B 16 0 12 8 18 C 2 -12 0 0 2 D 4 -8 0 0 -4 E -8 -18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -4 8 B 16 0 12 8 18 C 2 -12 0 0 2 D 4 -8 0 0 -4 E -8 -18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -4 8 B 16 0 12 8 18 C 2 -12 0 0 2 D 4 -8 0 0 -4 E -8 -18 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999062 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9103: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (7) D C B E A (6) A D C B E (6) E B C A D (5) C E D A B (5) B E A D C (5) A D B C E (5) C D E A B (4) A C D E B (4) A B E D C (4) A B D E C (4) E C B D A (3) E B C D A (3) C D E B A (3) C D A E B (3) A D C E B (3) E B A C D (2) D C A B E (2) D B C E A (2) C A E D B (2) B D A E C (2) B A D E C (2) A E C B D (2) E A C B D (1) E A B C D (1) D C B A E (1) D C A E B (1) D B C A E (1) D A C B E (1) C E B D A (1) C E A D B (1) B E D C A (1) B E C D A (1) B D E C A (1) B D E A C (1) B A E D C (1) A E C D B (1) A E B C D (1) A C E D B (1) Total count = 100 A B C D E A 0 4 -6 2 -6 B -4 0 -16 -18 -8 C 6 16 0 2 18 D -2 18 -2 0 4 E 6 8 -18 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 2 -6 B -4 0 -16 -18 -8 C 6 16 0 2 18 D -2 18 -2 0 4 E 6 8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=26 E=15 D=14 B=14 so D is eliminated. Round 2 votes counts: C=36 A=32 B=17 E=15 so E is eliminated. Round 3 votes counts: C=39 A=34 B=27 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 D:209 A:197 E:196 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 2 -6 B -4 0 -16 -18 -8 C 6 16 0 2 18 D -2 18 -2 0 4 E 6 8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 2 -6 B -4 0 -16 -18 -8 C 6 16 0 2 18 D -2 18 -2 0 4 E 6 8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 2 -6 B -4 0 -16 -18 -8 C 6 16 0 2 18 D -2 18 -2 0 4 E 6 8 -18 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991562 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9104: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (10) A E C B D (10) E A C B D (9) B D E A C (7) C A E D B (6) B D C A E (6) A C E B D (5) E C A D B (4) E A C D B (4) D B E C A (4) B D A C E (4) D B C E A (3) C E A D B (3) B D A E C (3) A E B C D (3) E A B C D (2) C E D A B (2) D E C B A (1) D C E B A (1) D C B E A (1) D B E A C (1) C D E A B (1) C D A B E (1) C A D E B (1) C A D B E (1) B E A D C (1) B E A C D (1) B A E D C (1) B A D E C (1) B A D C E (1) B A C E D (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 10 10 B -6 0 0 12 -4 C -10 0 0 10 -6 D -10 -12 -10 0 -6 E -10 4 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 10 10 B -6 0 0 12 -4 C -10 0 0 10 -6 D -10 -12 -10 0 -6 E -10 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=21 E=19 A=19 C=15 so C is eliminated. Round 2 votes counts: A=27 B=26 E=24 D=23 so D is eliminated. Round 3 votes counts: B=45 A=28 E=27 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 E:203 B:201 C:197 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 10 10 B -6 0 0 12 -4 C -10 0 0 10 -6 D -10 -12 -10 0 -6 E -10 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 10 10 B -6 0 0 12 -4 C -10 0 0 10 -6 D -10 -12 -10 0 -6 E -10 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 10 10 B -6 0 0 12 -4 C -10 0 0 10 -6 D -10 -12 -10 0 -6 E -10 4 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9105: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) C E B A D (11) D E C A B (9) A B D C E (9) C E B D A (7) A B D E C (5) A B C E D (5) E C D B A (4) D A B E C (4) B C A E D (4) E D C A B (3) E C D A B (3) D E C B A (3) D E A C B (3) C E D B A (3) C B E A D (3) E C A B D (2) D B A C E (2) B A D C E (2) A D B E C (2) E C B A D (1) D E A B C (1) D A E B C (1) B C E A D (1) Total count = 100 A B C D E A 0 -6 -8 14 -8 B 6 0 -4 24 -2 C 8 4 0 12 18 D -14 -24 -12 0 -18 E 8 2 -18 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 14 -8 B 6 0 -4 24 -2 C 8 4 0 12 18 D -14 -24 -12 0 -18 E 8 2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 A=21 B=19 E=13 so E is eliminated. Round 2 votes counts: C=34 D=26 A=21 B=19 so B is eliminated. Round 3 votes counts: C=39 A=35 D=26 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:212 E:205 A:196 D:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 14 -8 B 6 0 -4 24 -2 C 8 4 0 12 18 D -14 -24 -12 0 -18 E 8 2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 14 -8 B 6 0 -4 24 -2 C 8 4 0 12 18 D -14 -24 -12 0 -18 E 8 2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 14 -8 B 6 0 -4 24 -2 C 8 4 0 12 18 D -14 -24 -12 0 -18 E 8 2 -18 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9106: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (9) D B A E C (8) C E A B D (8) B D E C A (8) A D C E B (6) A D B C E (6) E C B D A (5) A C E D B (5) E C B A D (4) B E C D A (4) A C E B D (4) C E B A D (3) A E C D B (3) D B A C E (2) D A E C B (2) D A B E C (2) D A B C E (2) B D A C E (2) A D E C B (2) E C D B A (1) E C D A B (1) E C A D B (1) E C A B D (1) D E C B A (1) D E C A B (1) D E B C A (1) C E A D B (1) B D C E A (1) B D A E C (1) B C E D A (1) B C E A D (1) B C A E D (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 -6 -4 -4 B 8 0 0 -8 0 C 6 0 0 -10 -10 D 4 8 10 0 10 E 4 0 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -4 -4 B 8 0 0 -8 0 C 6 0 0 -10 -10 D 4 8 10 0 10 E 4 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=27 B=20 E=13 C=12 so C is eliminated. Round 2 votes counts: D=28 A=27 E=25 B=20 so B is eliminated. Round 3 votes counts: D=40 E=31 A=29 so A is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:202 B:200 C:193 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -4 B 8 0 0 -8 0 C 6 0 0 -10 -10 D 4 8 10 0 10 E 4 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -4 B 8 0 0 -8 0 C 6 0 0 -10 -10 D 4 8 10 0 10 E 4 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -4 B 8 0 0 -8 0 C 6 0 0 -10 -10 D 4 8 10 0 10 E 4 0 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997774 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9107: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (16) B E D C A (13) A C D E B (10) E D B A C (9) A C D B E (7) C A B E D (6) E B D C A (5) D E A C B (5) B C A E D (5) A D C E B (4) E D B C A (2) E D A C B (2) D E B A C (2) D A C E B (2) B D E C A (2) B C E A D (2) A C B D E (2) E D A B C (1) D E A B C (1) D A B C E (1) B E C A D (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 18 -4 10 10 B -18 0 -12 4 12 C 4 12 0 0 14 D -10 -4 0 0 6 E -10 -12 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.759149 D: 0.240851 E: 0.000000 Sum of squares = 0.634316269524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.759149 D: 1.000000 E: 1.000000 A B C D E A 0 18 -4 10 10 B -18 0 -12 4 12 C 4 12 0 0 14 D -10 -4 0 0 6 E -10 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836803198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 B=23 C=22 E=19 D=11 so D is eliminated. Round 2 votes counts: A=28 E=27 B=23 C=22 so C is eliminated. Round 3 votes counts: A=50 E=27 B=23 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:215 D:196 B:193 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -4 10 10 B -18 0 -12 4 12 C 4 12 0 0 14 D -10 -4 0 0 6 E -10 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836803198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 10 10 B -18 0 -12 4 12 C 4 12 0 0 14 D -10 -4 0 0 6 E -10 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836803198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 10 10 B -18 0 -12 4 12 C 4 12 0 0 14 D -10 -4 0 0 6 E -10 -12 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.285714 E: 0.000000 Sum of squares = 0.591836803198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9108: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (8) B D A E C (8) B C E A D (8) D E A C B (5) C A E D B (5) B C D A E (5) E A D C B (4) D A E C B (4) C B A E D (4) C A B D E (4) B D C A E (4) B D A C E (4) C B E A D (3) B C A D E (3) A C D E B (3) D A B E C (2) B D E A C (2) B C A E D (2) E D A B C (1) E C A D B (1) E B D A C (1) D E B A C (1) D E A B C (1) D B E A C (1) D A E B C (1) D A C E B (1) D A C B E (1) C E A D B (1) C E A B D (1) C B A D E (1) C A E B D (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B E C A D (1) B D E C A (1) B D C E A (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 -4 0 -8 14 B 4 0 -2 14 18 C 0 2 0 -6 14 D 8 -14 6 0 12 E -14 -18 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.636364 D: 0.090909 E: 0.000000 Sum of squares = 0.487603305771 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.909091 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -8 14 B 4 0 -2 14 18 C 0 2 0 -6 14 D 8 -14 6 0 12 E -14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.636364 D: 0.090909 E: 0.000000 Sum of squares = 0.487603305787 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.909091 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=23 D=17 E=15 A=4 so A is eliminated. Round 2 votes counts: B=41 C=26 D=18 E=15 so E is eliminated. Round 3 votes counts: B=42 D=31 C=27 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:206 C:205 A:201 E:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 -8 14 B 4 0 -2 14 18 C 0 2 0 -6 14 D 8 -14 6 0 12 E -14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.636364 D: 0.090909 E: 0.000000 Sum of squares = 0.487603305787 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.909091 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -8 14 B 4 0 -2 14 18 C 0 2 0 -6 14 D 8 -14 6 0 12 E -14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.636364 D: 0.090909 E: 0.000000 Sum of squares = 0.487603305787 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.909091 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -8 14 B 4 0 -2 14 18 C 0 2 0 -6 14 D 8 -14 6 0 12 E -14 -18 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.272727 C: 0.636364 D: 0.090909 E: 0.000000 Sum of squares = 0.487603305787 Cumulative probabilities = A: 0.000000 B: 0.272727 C: 0.909091 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9109: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (10) E B C A D (9) D A C B E (8) E B A C D (7) D C A B E (7) A D B C E (7) E C B D A (6) C D E A B (5) E B A D C (4) C D E B A (4) E B C D A (3) C B E D A (3) B E A C D (3) D E C A B (2) C E D B A (2) C E B D A (2) C D A B E (2) B A E D C (2) A D C B E (2) E D C A B (1) E D A B C (1) E C D B A (1) E C D A B (1) D E A C B (1) D A E C B (1) B E A D C (1) B A E C D (1) B A C E D (1) B A C D E (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -16 -20 -12 B 0 0 -16 -12 -20 C 16 16 0 2 10 D 20 12 -2 0 4 E 12 20 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 -20 -12 B 0 0 -16 -12 -20 C 16 16 0 2 10 D 20 12 -2 0 4 E 12 20 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=29 C=18 A=11 B=9 so B is eliminated. Round 2 votes counts: E=37 D=29 C=18 A=16 so A is eliminated. Round 3 votes counts: E=40 D=40 C=20 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:222 D:217 E:209 A:176 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 -20 -12 B 0 0 -16 -12 -20 C 16 16 0 2 10 D 20 12 -2 0 4 E 12 20 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 -20 -12 B 0 0 -16 -12 -20 C 16 16 0 2 10 D 20 12 -2 0 4 E 12 20 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 -20 -12 B 0 0 -16 -12 -20 C 16 16 0 2 10 D 20 12 -2 0 4 E 12 20 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990014 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9110: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) D C B E A (7) C D B E A (6) C D B A E (6) B A E D C (5) A E B C D (5) E A B D C (4) D C E B A (4) C D E A B (4) B C D A E (4) B C A D E (4) E D C A B (3) E A D C B (3) B A C E D (3) A B E C D (3) E D A C B (2) E A C D B (2) C D E B A (2) C B D A E (2) C B A E D (2) C A E D B (2) A E B D C (2) A B E D C (2) E D B A C (1) E D A B C (1) D E C B A (1) D E C A B (1) D E B C A (1) C E A D B (1) C B A D E (1) B E A D C (1) B D C A E (1) B D A C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -20 -10 -4 B 6 0 -20 -12 2 C 20 20 0 4 22 D 10 12 -4 0 8 E 4 -2 -22 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 -10 -4 B 6 0 -20 -12 2 C 20 20 0 4 22 D 10 12 -4 0 8 E 4 -2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=22 B=20 E=16 A=16 so E is eliminated. Round 2 votes counts: D=29 C=26 A=25 B=20 so B is eliminated. Round 3 votes counts: A=35 C=34 D=31 so D is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:233 D:213 B:188 E:186 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -20 -10 -4 B 6 0 -20 -12 2 C 20 20 0 4 22 D 10 12 -4 0 8 E 4 -2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 -10 -4 B 6 0 -20 -12 2 C 20 20 0 4 22 D 10 12 -4 0 8 E 4 -2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 -10 -4 B 6 0 -20 -12 2 C 20 20 0 4 22 D 10 12 -4 0 8 E 4 -2 -22 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998812 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9111: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) B C D E A (8) B C D A E (7) A E D C B (6) D A E B C (5) C B E D A (5) C B E A D (5) A E C B D (4) E A D C B (3) E A C D B (3) D C B E A (3) D B A C E (3) D A B E C (3) B D C A E (3) A E D B C (3) E D A C B (2) E A C B D (2) D E A C B (2) D B C E A (2) D B C A E (2) D B A E C (2) B A C D E (2) E D C A B (1) E C A B D (1) D E C B A (1) D E C A B (1) D E A B C (1) D C E B A (1) D B E A C (1) C E B D A (1) C E A B D (1) C D E B A (1) C B A E D (1) C A E B D (1) B C A E D (1) B A D E C (1) A E C D B (1) A E B D C (1) Total count = 100 A B C D E A 0 -18 -10 -28 -8 B 18 0 -10 6 16 C 10 10 0 6 12 D 28 -6 -6 0 16 E 8 -16 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -28 -8 B 18 0 -10 6 16 C 10 10 0 6 12 D 28 -6 -6 0 16 E 8 -16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=24 B=22 A=15 E=12 so E is eliminated. Round 2 votes counts: D=30 C=25 A=23 B=22 so B is eliminated. Round 3 votes counts: C=41 D=33 A=26 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:219 D:216 B:215 E:182 A:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -10 -28 -8 B 18 0 -10 6 16 C 10 10 0 6 12 D 28 -6 -6 0 16 E 8 -16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -28 -8 B 18 0 -10 6 16 C 10 10 0 6 12 D 28 -6 -6 0 16 E 8 -16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -28 -8 B 18 0 -10 6 16 C 10 10 0 6 12 D 28 -6 -6 0 16 E 8 -16 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9112: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) C B E D A (7) E B C D A (6) E A B D C (6) E C B A D (5) D A B E C (5) C E B D A (5) C D A B E (5) A D B E C (5) C A D B E (4) B D E A C (4) C D B A E (3) E C B D A (2) E B A D C (2) D C A B E (2) D B A E C (2) C E B A D (2) C B D E A (2) B E D C A (2) A E D C B (2) A E D B C (2) A D E B C (2) A D C B E (2) E C A B D (1) E B D A C (1) E B C A D (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C E A (1) D B C A E (1) D B A C E (1) C A E D B (1) B E C D A (1) B D E C A (1) A E C D B (1) A D B C E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -20 -4 -16 B 0 0 -10 14 2 C 20 10 0 14 -6 D 4 -14 -14 0 -12 E 16 -2 6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765515 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 A B C D E A 0 0 -20 -4 -16 B 0 0 -10 14 2 C 20 10 0 14 -6 D 4 -14 -14 0 -12 E 16 -2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765536 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=26 A=17 D=13 B=8 so B is eliminated. Round 2 votes counts: C=36 E=29 D=18 A=17 so A is eliminated. Round 3 votes counts: C=38 E=34 D=28 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:219 E:216 B:203 D:182 A:180 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -20 -4 -16 B 0 0 -10 14 2 C 20 10 0 14 -6 D 4 -14 -14 0 -12 E 16 -2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765536 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 -4 -16 B 0 0 -10 14 2 C 20 10 0 14 -6 D 4 -14 -14 0 -12 E 16 -2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765536 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 -4 -16 B 0 0 -10 14 2 C 20 10 0 14 -6 D 4 -14 -14 0 -12 E 16 -2 6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.111111 D: 0.000000 E: 0.555556 Sum of squares = 0.432098765536 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.444444 D: 0.444444 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9113: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) E A B D C (6) D C A E B (6) B D C E A (6) C D B A E (5) A E D C B (5) E B A C D (4) D C B A E (4) D A C E B (4) C B D E A (4) B E A C D (4) A E D B C (4) A E C D B (4) E A C B D (3) E A B C D (3) D A E C B (3) C D A E B (3) B E A D C (3) C E A B D (2) C A D E B (2) B D E A C (2) A D E C B (2) E C A B D (1) D B E A C (1) D B C E A (1) D A E B C (1) D A B C E (1) C B E D A (1) C A E D B (1) B E C A D (1) B D E C A (1) B C E A D (1) B C D E A (1) A E C B D (1) A D E B C (1) Total count = 100 A B C D E A 0 6 4 12 -6 B -6 0 -18 4 -12 C -4 18 0 -2 0 D -12 -4 2 0 -4 E 6 12 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.226322 D: 0.000000 E: 0.773678 Sum of squares = 0.649799284757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.226322 D: 0.226322 E: 1.000000 A B C D E A 0 6 4 12 -6 B -6 0 -18 4 -12 C -4 18 0 -2 0 D -12 -4 2 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=21 B=19 E=17 A=17 so E is eliminated. Round 2 votes counts: A=29 C=27 B=23 D=21 so D is eliminated. Round 3 votes counts: A=38 C=37 B=25 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:211 A:208 C:206 D:191 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 4 12 -6 B -6 0 -18 4 -12 C -4 18 0 -2 0 D -12 -4 2 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 12 -6 B -6 0 -18 4 -12 C -4 18 0 -2 0 D -12 -4 2 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 12 -6 B -6 0 -18 4 -12 C -4 18 0 -2 0 D -12 -4 2 0 -4 E 6 12 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9114: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (17) E D B C A (7) A C D B E (7) B E D C A (6) E B D C A (5) E B D A C (5) A C D E B (5) E D A C B (4) C A B D E (3) B C A D E (3) E D B A C (2) E D A B C (2) D E C B A (2) D E C A B (2) D E B C A (2) D E A C B (2) C A D B E (2) B C E A D (2) A B C E D (2) E B A D C (1) E B A C D (1) E A D B C (1) E A B D C (1) D C B E A (1) D C B A E (1) D C A E B (1) D C A B E (1) D B E C A (1) D A C E B (1) B E A C D (1) B D E C A (1) B D C E A (1) B C E D A (1) B C D A E (1) A E D C B (1) A E C D B (1) A C E D B (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 12 14 2 -2 B -12 0 -8 6 4 C -14 8 0 -2 4 D -2 -6 2 0 8 E 2 -4 -4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.500000000062 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 12 14 2 -2 B -12 0 -8 6 4 C -14 8 0 -2 4 D -2 -6 2 0 8 E 2 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999811 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=29 B=16 D=14 C=5 so C is eliminated. Round 2 votes counts: A=41 E=29 B=16 D=14 so D is eliminated. Round 3 votes counts: A=44 E=37 B=19 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:213 D:201 C:198 B:195 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 2 -2 B -12 0 -8 6 4 C -14 8 0 -2 4 D -2 -6 2 0 8 E 2 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999811 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 2 -2 B -12 0 -8 6 4 C -14 8 0 -2 4 D -2 -6 2 0 8 E 2 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999811 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 2 -2 B -12 0 -8 6 4 C -14 8 0 -2 4 D -2 -6 2 0 8 E 2 -4 -4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999811 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9115: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (9) C D B E A (6) A E D B C (6) B C E D A (5) A B E C D (5) E D A C B (4) C B D E A (4) A B E D C (4) E D C B A (3) E D B C A (3) D E C B A (3) D E C A B (3) D C E B A (3) D C E A B (3) B C D E A (3) B C A D E (3) B A C E D (3) A E B D C (3) E A D B C (2) D C A E B (2) D A E C B (2) C D E B A (2) A E D C B (2) A C D B E (2) E D B A C (1) E D A B C (1) E B D A C (1) E B C D A (1) E B A D C (1) E A B D C (1) C D B A E (1) C D A E B (1) C A B D E (1) B E C D A (1) B E A C D (1) B C E A D (1) B C D A E (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 -8 -6 B 12 0 16 -2 4 C 0 -16 0 2 -16 D 8 2 -2 0 -18 E 6 -4 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.166667 E: 0.083333 Sum of squares = 0.597222222217 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.916667 E: 1.000000 A B C D E A 0 -12 0 -8 -6 B 12 0 16 -2 4 C 0 -16 0 2 -16 D 8 2 -2 0 -18 E 6 -4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.166667 E: 0.083333 Sum of squares = 0.597222222206 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.916667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=24 E=18 D=16 C=15 so C is eliminated. Round 2 votes counts: B=31 D=26 A=25 E=18 so E is eliminated. Round 3 votes counts: D=38 B=34 A=28 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:218 B:215 D:195 A:187 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -8 -6 B 12 0 16 -2 4 C 0 -16 0 2 -16 D 8 2 -2 0 -18 E 6 -4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.166667 E: 0.083333 Sum of squares = 0.597222222206 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.916667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -8 -6 B 12 0 16 -2 4 C 0 -16 0 2 -16 D 8 2 -2 0 -18 E 6 -4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.166667 E: 0.083333 Sum of squares = 0.597222222206 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.916667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -8 -6 B 12 0 16 -2 4 C 0 -16 0 2 -16 D 8 2 -2 0 -18 E 6 -4 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.166667 E: 0.083333 Sum of squares = 0.597222222206 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.916667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9116: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (13) B D A E C (10) E C B D A (9) A D B C E (7) A C E D B (7) A B D C E (7) C E A B D (6) B D E C A (6) E C D B A (5) D B A E C (5) C E D A B (4) D B E C A (2) A D C E B (2) A C E B D (2) A B D E C (2) E C B A D (1) D C E A B (1) D A C E B (1) C E D B A (1) C E B D A (1) C A E D B (1) C A E B D (1) B E C D A (1) B D E A C (1) B A E C D (1) B A D E C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 -4 6 -2 B -12 0 -10 0 -10 C 4 10 0 8 8 D -6 0 -8 0 -8 E 2 10 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999923 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -4 6 -2 B -12 0 -10 0 -10 C 4 10 0 8 8 D -6 0 -8 0 -8 E 2 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 B=20 E=15 D=9 so D is eliminated. Round 2 votes counts: A=30 C=28 B=27 E=15 so E is eliminated. Round 3 votes counts: C=43 A=30 B=27 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 A:206 E:206 D:189 B:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -4 6 -2 B -12 0 -10 0 -10 C 4 10 0 8 8 D -6 0 -8 0 -8 E 2 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -4 6 -2 B -12 0 -10 0 -10 C 4 10 0 8 8 D -6 0 -8 0 -8 E 2 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -4 6 -2 B -12 0 -10 0 -10 C 4 10 0 8 8 D -6 0 -8 0 -8 E 2 10 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9117: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) B A C E D (7) B A D E C (6) E C D B A (5) C E D B A (5) A D C E B (5) A B D C E (5) E D C B A (4) D A E C B (4) D A B E C (4) B E C A D (4) D E C B A (3) D C E A B (3) D A C E B (3) C E B A D (3) B C E A D (3) B A E C D (3) A B C E D (3) E C B D A (2) D A E B C (2) C A E D B (2) A D B E C (2) E B C D A (1) D E B C A (1) D B E A C (1) C E D A B (1) C D E A B (1) C A E B D (1) B D E C A (1) B A C D E (1) A D C B E (1) A D B C E (1) A C E B D (1) A C D E B (1) A C D B E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 4 10 B 0 0 -6 -14 -10 C -6 6 0 -8 -2 D -4 14 8 0 10 E -10 10 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.870935 B: 0.129065 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.775185759186 Cumulative probabilities = A: 0.870935 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 10 B 0 0 -6 -14 -10 C -6 6 0 -8 -2 D -4 14 8 0 10 E -10 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.654321015438 Cumulative probabilities = A: 0.777778 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 A=22 C=13 E=12 so E is eliminated. Round 2 votes counts: D=32 B=26 A=22 C=20 so C is eliminated. Round 3 votes counts: D=44 B=31 A=25 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:210 E:196 C:195 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 10 B 0 0 -6 -14 -10 C -6 6 0 -8 -2 D -4 14 8 0 10 E -10 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.654321015438 Cumulative probabilities = A: 0.777778 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 10 B 0 0 -6 -14 -10 C -6 6 0 -8 -2 D -4 14 8 0 10 E -10 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.654321015438 Cumulative probabilities = A: 0.777778 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 10 B 0 0 -6 -14 -10 C -6 6 0 -8 -2 D -4 14 8 0 10 E -10 10 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.777778 B: 0.222222 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.654321015438 Cumulative probabilities = A: 0.777778 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9118: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (13) B C E D A (7) E B C A D (5) D E B A C (5) D A E B C (5) A E C B D (5) A C E B D (5) D A E C B (4) D A C B E (4) B C E A D (4) A D E C B (4) B E D C A (3) D E A B C (2) D A C E B (2) D A B C E (2) C B D E A (2) C A E B D (2) B E C D A (2) B E C A D (2) B C D E A (2) A D C E B (2) E C B A D (1) E B D C A (1) E B A C D (1) E A C B D (1) D C A B E (1) D B E C A (1) D B C E A (1) C E B A D (1) C E A B D (1) C B A E D (1) C B A D E (1) C A D B E (1) C A B E D (1) B D E C A (1) B D C E A (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -10 -10 8 -14 B 10 0 -8 26 0 C 10 8 0 20 12 D -8 -26 -20 0 -14 E 14 0 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 8 -14 B 10 0 -8 26 0 C 10 8 0 20 12 D -8 -26 -20 0 -14 E 14 0 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=23 B=22 A=19 E=9 so E is eliminated. Round 2 votes counts: B=29 D=27 C=24 A=20 so A is eliminated. Round 3 votes counts: C=37 D=34 B=29 so B is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:225 B:214 E:208 A:187 D:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 8 -14 B 10 0 -8 26 0 C 10 8 0 20 12 D -8 -26 -20 0 -14 E 14 0 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 8 -14 B 10 0 -8 26 0 C 10 8 0 20 12 D -8 -26 -20 0 -14 E 14 0 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 8 -14 B 10 0 -8 26 0 C 10 8 0 20 12 D -8 -26 -20 0 -14 E 14 0 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999754 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9119: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (8) E B A D C (7) A C D E B (7) C A D E B (6) B E D C A (6) B E D A C (6) D A C B E (5) D A B E C (4) C A E B D (4) B E C D A (4) E B C A D (3) E B A C D (3) C E B A D (3) A D C E B (3) D C B E A (2) D C A B E (2) D B E C A (2) D B E A C (2) D B C E A (2) C B E D A (2) C B E A D (2) C A D B E (2) A E B D C (2) A D E B C (2) E B D A C (1) E B C D A (1) E A B C D (1) D A E B C (1) C D B E A (1) C B D E A (1) B E C A D (1) A E C B D (1) A E B C D (1) A D E C B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 -4 0 0 B -2 0 -2 0 4 C 4 2 0 4 2 D 0 0 -4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 0 0 B -2 0 -2 0 4 C 4 2 0 4 2 D 0 0 -4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=20 A=18 B=17 E=16 so E is eliminated. Round 2 votes counts: B=32 C=29 D=20 A=19 so A is eliminated. Round 3 votes counts: C=38 B=36 D=26 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:206 B:200 A:199 D:199 E:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 0 0 B -2 0 -2 0 4 C 4 2 0 4 2 D 0 0 -4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 0 0 B -2 0 -2 0 4 C 4 2 0 4 2 D 0 0 -4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 0 0 B -2 0 -2 0 4 C 4 2 0 4 2 D 0 0 -4 0 2 E 0 -4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998531 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9120: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (12) B A E C D (8) D C E B A (6) D B C E A (5) C E D A B (5) B A E D C (5) A B E C D (5) D C A E B (4) C E A D B (4) B E D C A (3) B D A E C (3) A B D C E (3) E C D B A (2) E C D A B (2) E C B A D (2) E B C D A (2) D B C A E (2) B D A C E (2) B A D C E (2) A E C B D (2) A E B C D (2) A D C E B (2) A C E B D (2) A B C E D (2) E D C B A (1) E C A D B (1) E C A B D (1) E B C A D (1) E A C B D (1) D E B C A (1) D C B E A (1) D A C B E (1) C D E A B (1) B E A C D (1) B D E C A (1) B A D E C (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -14 -8 -6 B -2 0 -2 -2 -10 C 14 2 0 -10 10 D 8 2 10 0 -6 E 6 10 -10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.384615 E: 0.384615 Sum of squares = 0.349112426036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.615385 E: 1.000000 A B C D E A 0 2 -14 -8 -6 B -2 0 -2 -2 -10 C 14 2 0 -10 10 D 8 2 10 0 -6 E 6 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.384615 E: 0.384615 Sum of squares = 0.349112426036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.615385 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=26 A=19 E=13 C=10 so C is eliminated. Round 2 votes counts: D=33 B=26 E=22 A=19 so A is eliminated. Round 3 votes counts: B=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:208 D:207 E:206 B:192 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -14 -8 -6 B -2 0 -2 -2 -10 C 14 2 0 -10 10 D 8 2 10 0 -6 E 6 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.384615 E: 0.384615 Sum of squares = 0.349112426036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.615385 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -8 -6 B -2 0 -2 -2 -10 C 14 2 0 -10 10 D 8 2 10 0 -6 E 6 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.384615 E: 0.384615 Sum of squares = 0.349112426036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.615385 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -8 -6 B -2 0 -2 -2 -10 C 14 2 0 -10 10 D 8 2 10 0 -6 E 6 10 -10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.384615 E: 0.384615 Sum of squares = 0.349112426036 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.230769 D: 0.615385 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9121: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (9) C E A D B (6) A C D E B (6) C A E D B (5) B A D E C (5) E C D B A (4) E C B D A (4) B E D C A (4) B D A E C (4) A D B C E (4) D E C B A (3) C E D B A (3) C D E A B (3) A D C B E (3) A B D E C (3) A B D C E (3) D B A E C (2) C E B A D (2) B E D A C (2) B D E A C (2) B A E C D (2) A C E D B (2) E D C B A (1) E D B C A (1) E C B A D (1) E B C D A (1) E B C A D (1) D C E A B (1) D C A E B (1) D A B E C (1) D A B C E (1) C E B D A (1) C D A E B (1) B E C D A (1) B E C A D (1) B D E C A (1) B A E D C (1) A D C E B (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -10 -2 -4 B -6 0 -20 -16 -16 C 10 20 0 12 10 D 2 16 -12 0 -8 E 4 16 -10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -2 -4 B -6 0 -20 -16 -16 C 10 20 0 12 10 D 2 16 -12 0 -8 E 4 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=25 B=23 E=13 D=9 so D is eliminated. Round 2 votes counts: C=32 A=27 B=25 E=16 so E is eliminated. Round 3 votes counts: C=45 B=28 A=27 so A is eliminated. Round 4 votes counts: C=60 B=40 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:226 E:209 D:199 A:195 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 -2 -4 B -6 0 -20 -16 -16 C 10 20 0 12 10 D 2 16 -12 0 -8 E 4 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -2 -4 B -6 0 -20 -16 -16 C 10 20 0 12 10 D 2 16 -12 0 -8 E 4 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -2 -4 B -6 0 -20 -16 -16 C 10 20 0 12 10 D 2 16 -12 0 -8 E 4 16 -10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9122: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (9) C D A E B (5) C A D E B (5) A C D B E (5) C A D B E (4) B E C D A (4) A D E C B (4) A D C E B (4) E D A B C (3) D E C B A (3) D C E A B (3) C D E B A (3) C B A E D (3) B C E D A (3) A E D B C (3) E D B A C (2) E B D C A (2) D E C A B (2) B E D A C (2) B E A D C (2) B C E A D (2) B C A E D (2) A E B D C (2) A C D E B (2) A B E C D (2) E D B C A (1) E B D A C (1) E B A D C (1) D E A C B (1) D C A E B (1) D A E C B (1) D A E B C (1) D A C E B (1) C D E A B (1) C D B A E (1) C D A B E (1) C B E D A (1) C B A D E (1) B A E D C (1) B A C E D (1) A C B D E (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 10 -14 -4 8 B -10 0 -6 -14 -4 C 14 6 0 -4 2 D 4 14 4 0 2 E -8 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999905 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -14 -4 8 B -10 0 -6 -14 -4 C 14 6 0 -4 2 D 4 14 4 0 2 E -8 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 C=25 D=13 E=10 so E is eliminated. Round 2 votes counts: B=30 A=26 C=25 D=19 so D is eliminated. Round 3 votes counts: C=34 B=33 A=33 so B is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:212 C:209 A:200 E:196 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -14 -4 8 B -10 0 -6 -14 -4 C 14 6 0 -4 2 D 4 14 4 0 2 E -8 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 -4 8 B -10 0 -6 -14 -4 C 14 6 0 -4 2 D 4 14 4 0 2 E -8 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 -4 8 B -10 0 -6 -14 -4 C 14 6 0 -4 2 D 4 14 4 0 2 E -8 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987228 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9123: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) C D E B A (8) A B E D C (8) A B D E C (6) D C A B E (5) B E A C D (5) B A E C D (5) A E B D C (5) E B C A D (4) D C A E B (4) D A C B E (4) C E D B A (4) E B A C D (3) C D A B E (3) E D C A B (2) E C D B A (2) E C B D A (2) E B A D C (2) C D E A B (2) B C E A D (2) B A E D C (2) B A C E D (2) A E D B C (2) A B D C E (2) E B C D A (1) D A E C B (1) D A C E B (1) C E B D A (1) C D B A E (1) B E C A D (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 10 0 2 6 B -10 0 4 4 -4 C 0 -4 0 -6 -4 D -2 -4 6 0 -6 E -6 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.830052 B: 0.000000 C: 0.169948 D: 0.000000 E: 0.000000 Sum of squares = 0.717868859704 Cumulative probabilities = A: 0.830052 B: 0.830052 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 0 2 6 B -10 0 4 4 -4 C 0 -4 0 -6 -4 D -2 -4 6 0 -6 E -6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001568 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 C=19 B=17 E=16 so E is eliminated. Round 2 votes counts: B=27 D=25 A=25 C=23 so C is eliminated. Round 3 votes counts: D=45 B=30 A=25 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:209 E:204 B:197 D:197 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 0 2 6 B -10 0 4 4 -4 C 0 -4 0 -6 -4 D -2 -4 6 0 -6 E -6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001568 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 0 2 6 B -10 0 4 4 -4 C 0 -4 0 -6 -4 D -2 -4 6 0 -6 E -6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001568 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 0 2 6 B -10 0 4 4 -4 C 0 -4 0 -6 -4 D -2 -4 6 0 -6 E -6 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000001568 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9124: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (12) B C E D A (10) C E B A D (8) A D E C B (8) E A D C B (7) B D A C E (7) D A E C B (5) D A E B C (5) D A B E C (5) C B E A D (5) B C D A E (5) D A B C E (4) E C A D B (3) B D C A E (3) E C B A D (2) E A C D B (2) D B A E C (2) C B E D A (2) E C A B D (1) C E A D B (1) C D A E B (1) B E A C D (1) A E D C B (1) Total count = 100 A B C D E A 0 -14 -6 2 -8 B 14 0 8 12 12 C 6 -8 0 6 16 D -2 -12 -6 0 -10 E 8 -12 -16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 2 -8 B 14 0 8 12 12 C 6 -8 0 6 16 D -2 -12 -6 0 -10 E 8 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 D=21 C=17 E=15 A=9 so A is eliminated. Round 2 votes counts: B=38 D=29 C=17 E=16 so E is eliminated. Round 3 votes counts: B=38 D=37 C=25 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:210 E:195 A:187 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 2 -8 B 14 0 8 12 12 C 6 -8 0 6 16 D -2 -12 -6 0 -10 E 8 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 2 -8 B 14 0 8 12 12 C 6 -8 0 6 16 D -2 -12 -6 0 -10 E 8 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 2 -8 B 14 0 8 12 12 C 6 -8 0 6 16 D -2 -12 -6 0 -10 E 8 -12 -16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9125: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) E A C D B (7) A E C D B (7) A E C B D (7) B D C A E (6) B D C E A (5) A E B C D (5) C E D A B (4) C D E B A (4) C D E A B (4) B D E C A (4) A B E D C (4) E C A D B (3) B D E A C (3) B A D E C (3) B A D C E (3) E C D A B (2) D C B E A (2) C D B E A (2) C A E D B (2) B D A C E (2) B A E D C (2) D C E B A (1) D C B A E (1) D B C A E (1) C E A D B (1) C A D E B (1) C A D B E (1) B D A E C (1) A E B D C (1) A C E D B (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -6 -2 -2 B -2 0 0 -6 0 C 6 0 0 4 0 D 2 6 -4 0 6 E 2 0 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.220153 C: 0.685206 D: 0.000000 E: 0.094641 Sum of squares = 0.526931226124 Cumulative probabilities = A: 0.000000 B: 0.220153 C: 0.905358 D: 0.905359 E: 1.000000 A B C D E A 0 2 -6 -2 -2 B -2 0 0 -6 0 C 6 0 0 4 0 D 2 6 -4 0 6 E 2 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000055922 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=26 C=19 D=14 E=12 so E is eliminated. Round 2 votes counts: A=33 B=29 C=24 D=14 so D is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:205 D:205 E:198 A:196 B:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -2 -2 B -2 0 0 -6 0 C 6 0 0 4 0 D 2 6 -4 0 6 E 2 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000055922 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -2 -2 B -2 0 0 -6 0 C 6 0 0 4 0 D 2 6 -4 0 6 E 2 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000055922 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -2 -2 B -2 0 0 -6 0 C 6 0 0 4 0 D 2 6 -4 0 6 E 2 0 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.440000055922 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9126: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) B C E D A (8) A D E B C (6) E D B A C (5) E D A B C (5) C B D E A (5) C A D B E (5) E B D C A (4) E B A D C (4) D E A B C (4) D A E C B (4) E B D A C (3) D C A E B (3) D A C E B (3) C B E D A (3) C B E A D (3) B E C A D (3) A D E C B (3) A D C E B (3) D A E B C (2) C B D A E (2) C A B D E (2) B E C D A (2) A E D B C (2) A C D B E (2) D E A C B (1) C D E B A (1) C D B E A (1) C D A B E (1) C B A D E (1) Total count = 100 A B C D E A 0 -8 -6 -14 -22 B 8 0 14 -2 -6 C 6 -14 0 -4 4 D 14 2 4 0 -2 E 22 6 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.359999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 A B C D E A 0 -8 -6 -14 -22 B 8 0 14 -2 -6 C 6 -14 0 -4 4 D 14 2 4 0 -2 E 22 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.36000000121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 B=22 E=21 D=17 A=16 so A is eliminated. Round 2 votes counts: D=29 C=26 E=23 B=22 so B is eliminated. Round 3 votes counts: C=43 D=29 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:209 B:207 C:196 A:175 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -14 -22 B 8 0 14 -2 -6 C 6 -14 0 -4 4 D 14 2 4 0 -2 E 22 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.36000000121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -14 -22 B 8 0 14 -2 -6 C 6 -14 0 -4 4 D 14 2 4 0 -2 E 22 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.36000000121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -14 -22 B 8 0 14 -2 -6 C 6 -14 0 -4 4 D 14 2 4 0 -2 E 22 6 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.400000 E: 0.400000 Sum of squares = 0.36000000121 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.200000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9127: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (12) E C B A D (6) C B A E D (6) E C D B A (5) D A B E C (5) A B D C E (4) E D A C B (3) E B C A D (3) D E A C B (3) D A C B E (3) C B E A D (3) C A B D E (3) B A C D E (3) E D C B A (2) E D C A B (2) E D A B C (2) E C B D A (2) E B D A C (2) D E A B C (2) D C A E B (2) D A E B C (2) C E B A D (2) B E C A D (2) B C A E D (2) B A C E D (2) A D B C E (2) A B C D E (2) E B C D A (1) C E A B D (1) C D A E B (1) C D A B E (1) C A D B E (1) B E A C D (1) B C E A D (1) B A E D C (1) B A D E C (1) B A D C E (1) A D C B E (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 8 8 0 14 B -8 0 4 -2 18 C -8 -4 0 -2 8 D 0 2 2 0 2 E -14 -18 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.463682 B: 0.000000 C: 0.000000 D: 0.536318 E: 0.000000 Sum of squares = 0.502637963684 Cumulative probabilities = A: 0.463682 B: 0.463682 C: 0.463682 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 0 14 B -8 0 4 -2 18 C -8 -4 0 -2 8 D 0 2 2 0 2 E -14 -18 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=28 C=18 B=14 A=11 so A is eliminated. Round 2 votes counts: D=33 E=28 B=20 C=19 so C is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:215 B:206 D:203 C:197 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 0 14 B -8 0 4 -2 18 C -8 -4 0 -2 8 D 0 2 2 0 2 E -14 -18 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 0 14 B -8 0 4 -2 18 C -8 -4 0 -2 8 D 0 2 2 0 2 E -14 -18 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 0 14 B -8 0 4 -2 18 C -8 -4 0 -2 8 D 0 2 2 0 2 E -14 -18 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9128: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (13) C E A B D (10) D B A E C (7) A C B E D (7) D E B C A (5) C A E B D (5) E C A D B (4) E C A B D (4) D B A C E (4) B D A C E (4) E D C B A (3) E C D A B (3) D E C B A (3) B D A E C (3) A C E B D (3) A C B D E (3) E D B C A (2) D B E C A (2) C E A D B (2) B D E A C (2) B A D C E (2) A B C D E (2) E A C B D (1) D C B A E (1) D C A B E (1) D B C A E (1) B E A C D (1) B A D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 8 -8 -10 B 8 0 0 -2 10 C -8 0 0 -8 -8 D 8 2 8 0 8 E 10 -10 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -8 -10 B 8 0 0 -2 10 C -8 0 0 -8 -8 D 8 2 8 0 8 E 10 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=17 C=17 A=16 B=13 so B is eliminated. Round 2 votes counts: D=46 A=19 E=18 C=17 so C is eliminated. Round 3 votes counts: D=46 E=30 A=24 so A is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:208 E:200 A:191 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 8 -8 -10 B 8 0 0 -2 10 C -8 0 0 -8 -8 D 8 2 8 0 8 E 10 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -8 -10 B 8 0 0 -2 10 C -8 0 0 -8 -8 D 8 2 8 0 8 E 10 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -8 -10 B 8 0 0 -2 10 C -8 0 0 -8 -8 D 8 2 8 0 8 E 10 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9129: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (6) B E A C D (6) B D A E C (5) E C A B D (4) D B A E C (4) A C B D E (4) A B D C E (4) E D C B A (3) E C D A B (3) E B D C A (3) E B C A D (3) D B A C E (3) C E D A B (3) B A D E C (3) E C D B A (2) E C B A D (2) D B E C A (2) D A C B E (2) D A B C E (2) C A E D B (2) C A D E B (2) B E D A C (2) B D E A C (2) B A E D C (2) B A D C E (2) A D B C E (2) A C E B D (2) A C D B E (2) E C B D A (1) D E C A B (1) D E B C A (1) D C E A B (1) D C A E B (1) C E A B D (1) C D E A B (1) C D A E B (1) C A E B D (1) B E A D C (1) B D A C E (1) B A E C D (1) B A C D E (1) A D C B E (1) A C D E B (1) A C B E D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 12 12 4 B 0 0 4 8 10 C -12 -4 0 4 -2 D -12 -8 -4 0 0 E -4 -10 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444042 B: 0.555958 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.50626256662 Cumulative probabilities = A: 0.444042 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 12 4 B 0 0 4 8 10 C -12 -4 0 4 -2 D -12 -8 -4 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=21 A=19 D=17 C=17 so D is eliminated. Round 2 votes counts: B=35 E=23 A=23 C=19 so C is eliminated. Round 3 votes counts: E=35 B=35 A=30 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:211 E:194 C:193 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 12 4 B 0 0 4 8 10 C -12 -4 0 4 -2 D -12 -8 -4 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 12 4 B 0 0 4 8 10 C -12 -4 0 4 -2 D -12 -8 -4 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 12 4 B 0 0 4 8 10 C -12 -4 0 4 -2 D -12 -8 -4 0 0 E -4 -10 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9130: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (4) B D E A C (4) A D B C E (4) A C D B E (4) A B D C E (4) E D C B A (3) D E C A B (3) D A E B C (3) D A C E B (3) C E A D B (3) C D E A B (3) C A E D B (3) C A E B D (3) C A D E B (3) B D A E C (3) B A E D C (3) B A C D E (3) A B C D E (3) E C D A B (2) E C B D A (2) E B D C A (2) E B C D A (2) D E B A C (2) D C A E B (2) D A B E C (2) C E B A D (2) B A D E C (2) A C D E B (2) E D C A B (1) E D B C A (1) E B C A D (1) D E A C B (1) D B E A C (1) D B A E C (1) D A E C B (1) D A C B E (1) C E D A B (1) C E A B D (1) C D A E B (1) C B E A D (1) C A D B E (1) C A B E D (1) B E D C A (1) B E A D C (1) B C A E D (1) B A D C E (1) B A C E D (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 16 12 0 16 B -16 0 0 -8 -4 C -12 0 0 -12 8 D 0 8 12 0 20 E -16 4 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.591364 B: 0.000000 C: 0.000000 D: 0.408636 E: 0.000000 Sum of squares = 0.516694644266 Cumulative probabilities = A: 0.591364 B: 0.591364 C: 0.591364 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 0 16 B -16 0 0 -8 -4 C -12 0 0 -12 8 D 0 8 12 0 20 E -16 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 C=23 D=20 A=19 E=14 so E is eliminated. Round 2 votes counts: B=29 C=27 D=25 A=19 so A is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:222 D:220 C:192 B:186 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 0 16 B -16 0 0 -8 -4 C -12 0 0 -12 8 D 0 8 12 0 20 E -16 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 0 16 B -16 0 0 -8 -4 C -12 0 0 -12 8 D 0 8 12 0 20 E -16 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 0 16 B -16 0 0 -8 -4 C -12 0 0 -12 8 D 0 8 12 0 20 E -16 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9131: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (8) E A D C B (5) E A B D C (5) B C D A E (5) B A C D E (4) A E C B D (4) A C B E D (4) A C B D E (4) A B E D C (4) E D C B A (3) E D A C B (3) D E C B A (3) D C E B A (3) D C B E A (3) C D B E A (3) A B C E D (3) E D B C A (2) E D A B C (2) E A D B C (2) C D E B A (2) C B D A E (2) C A B D E (2) B D C A E (2) A E B D C (2) A B C D E (2) E D C A B (1) E D B A C (1) E C D A B (1) E A C D B (1) D E B C A (1) D B E C A (1) D B C E A (1) C D E A B (1) C A D E B (1) B E D A C (1) B D E A C (1) B D A E C (1) B C A D E (1) B A E D C (1) A E C D B (1) A E B C D (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 18 24 14 8 B -18 0 2 16 6 C -24 -2 0 4 -10 D -14 -16 -4 0 -14 E -8 -6 10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 24 14 8 B -18 0 2 16 6 C -24 -2 0 4 -10 D -14 -16 -4 0 -14 E -8 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=26 B=16 D=12 C=11 so C is eliminated. Round 2 votes counts: A=38 E=26 D=18 B=18 so D is eliminated. Round 3 votes counts: A=38 E=36 B=26 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:232 E:205 B:203 C:184 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 24 14 8 B -18 0 2 16 6 C -24 -2 0 4 -10 D -14 -16 -4 0 -14 E -8 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 24 14 8 B -18 0 2 16 6 C -24 -2 0 4 -10 D -14 -16 -4 0 -14 E -8 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 24 14 8 B -18 0 2 16 6 C -24 -2 0 4 -10 D -14 -16 -4 0 -14 E -8 -6 10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9132: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (13) C D A E B (8) B E A C D (7) E A D C B (6) C D B A E (6) B E A D C (6) B C D A E (6) A E D C B (6) B C E A D (5) B E C A D (3) B D C A E (3) A E D B C (3) E C A D B (2) E A B C D (2) D A E C B (2) D A C E B (2) C D E A B (2) B C D E A (2) B A E D C (2) E A D B C (1) E A C D B (1) E A C B D (1) E A B D C (1) D C A B E (1) D B A E C (1) C E A D B (1) C D B E A (1) C B D A E (1) B D A C E (1) B C E D A (1) A E B D C (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 10 -10 0 16 B -10 0 -8 -16 -8 C 10 8 0 -2 6 D 0 16 2 0 2 E -16 8 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.074809 B: 0.000000 C: 0.000000 D: 0.925191 E: 0.000000 Sum of squares = 0.86157559006 Cumulative probabilities = A: 0.074809 B: 0.074809 C: 0.074809 D: 1.000000 E: 1.000000 A B C D E A 0 10 -10 0 16 B -10 0 -8 -16 -8 C 10 8 0 -2 6 D 0 16 2 0 2 E -16 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166666 B: 0.000000 C: 0.000000 D: 0.833334 E: 0.000000 Sum of squares = 0.722222480362 Cumulative probabilities = A: 0.166666 B: 0.166666 C: 0.166666 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=19 C=19 E=14 A=12 so A is eliminated. Round 2 votes counts: B=36 E=24 D=21 C=19 so C is eliminated. Round 3 votes counts: D=38 B=37 E=25 so E is eliminated. Round 4 votes counts: D=58 B=42 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:211 D:210 A:208 E:192 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -10 0 16 B -10 0 -8 -16 -8 C 10 8 0 -2 6 D 0 16 2 0 2 E -16 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166666 B: 0.000000 C: 0.000000 D: 0.833334 E: 0.000000 Sum of squares = 0.722222480362 Cumulative probabilities = A: 0.166666 B: 0.166666 C: 0.166666 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 0 16 B -10 0 -8 -16 -8 C 10 8 0 -2 6 D 0 16 2 0 2 E -16 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166666 B: 0.000000 C: 0.000000 D: 0.833334 E: 0.000000 Sum of squares = 0.722222480362 Cumulative probabilities = A: 0.166666 B: 0.166666 C: 0.166666 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 0 16 B -10 0 -8 -16 -8 C 10 8 0 -2 6 D 0 16 2 0 2 E -16 8 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166666 B: 0.000000 C: 0.000000 D: 0.833334 E: 0.000000 Sum of squares = 0.722222480362 Cumulative probabilities = A: 0.166666 B: 0.166666 C: 0.166666 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9133: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (10) B A C E D (7) A B C E D (7) E D C A B (5) A B D E C (5) A B C D E (5) E D C B A (4) E C D B A (4) D E B C A (4) D A E B C (4) B A C D E (4) D E C A B (3) D E A C B (3) C E B D A (3) C B A E D (3) A D E B C (3) A B D C E (3) C E D B A (2) C B E D A (2) C B E A D (2) B C E D A (2) B C D E A (2) B C A E D (2) E C D A B (1) D E A B C (1) D A B E C (1) C E A B D (1) C A B E D (1) A E D C B (1) A E C D B (1) A D B E C (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -2 -2 2 B 2 0 4 4 -2 C 2 -4 0 4 -4 D 2 -4 -4 0 -2 E -2 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 -2 -2 -2 2 B 2 0 4 4 -2 C 2 -4 0 4 -4 D 2 -4 -4 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333224 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=26 B=17 E=14 C=14 so E is eliminated. Round 2 votes counts: D=35 A=29 C=19 B=17 so B is eliminated. Round 3 votes counts: A=40 D=35 C=25 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:204 E:203 C:199 A:198 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B E , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 -2 -2 2 B 2 0 4 4 -2 C 2 -4 0 4 -4 D 2 -4 -4 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333224 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -2 2 B 2 0 4 4 -2 C 2 -4 0 4 -4 D 2 -4 -4 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333224 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -2 2 B 2 0 4 4 -2 C 2 -4 0 4 -4 D 2 -4 -4 0 -2 E -2 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333224 Cumulative probabilities = A: 0.333333 B: 0.666667 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9134: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) D A E B C (7) A D E B C (7) A D C E B (6) C B E A D (5) A D E C B (5) D E B C A (4) D C B E A (4) E B C D A (3) E B C A D (3) D A C B E (3) C B E D A (3) A D C B E (3) A C E B D (3) E B D C A (2) E A B D C (2) D E A B C (2) C B A E D (2) B E C D A (2) B E C A D (2) A E B C D (2) A C B E D (2) E D B C A (1) E D B A C (1) E B D A C (1) E B A C D (1) E A D B C (1) E A B C D (1) D C A B E (1) D A E C B (1) C D B A E (1) C D A B E (1) C B A D E (1) C A D B E (1) C A B E D (1) C A B D E (1) B E D C A (1) B C E A D (1) A E C D B (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 2 6 2 B -6 0 6 0 -10 C -2 -6 0 -4 -2 D -6 0 4 0 -2 E -2 10 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 6 2 B -6 0 6 0 -10 C -2 -6 0 -4 -2 D -6 0 4 0 -2 E -2 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=22 E=16 C=16 B=15 so B is eliminated. Round 2 votes counts: A=31 C=26 D=22 E=21 so E is eliminated. Round 3 votes counts: C=36 A=36 D=28 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:208 E:206 D:198 B:195 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 6 2 B -6 0 6 0 -10 C -2 -6 0 -4 -2 D -6 0 4 0 -2 E -2 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 6 2 B -6 0 6 0 -10 C -2 -6 0 -4 -2 D -6 0 4 0 -2 E -2 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 6 2 B -6 0 6 0 -10 C -2 -6 0 -4 -2 D -6 0 4 0 -2 E -2 10 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998563 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9135: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) D A C E B (10) C E B D A (10) B E C A D (9) D C E A B (7) B A E C D (6) A B E D C (6) A B D E C (6) D A E C B (5) C E D B A (4) C B E D A (4) D C A E B (3) C D E B A (3) B E A C D (3) B C E D A (2) B C E A D (2) A D E B C (2) E C B D A (1) E B C A D (1) D A C B E (1) B A C E D (1) A D E C B (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 8 8 0 8 B -8 0 0 2 4 C -8 0 0 -8 -2 D 0 -2 8 0 2 E -8 -4 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.589837 B: 0.000000 C: 0.000000 D: 0.410163 E: 0.000000 Sum of squares = 0.51614147614 Cumulative probabilities = A: 0.589837 B: 0.589837 C: 0.589837 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 0 8 B -8 0 0 2 4 C -8 0 0 -8 -2 D 0 -2 8 0 2 E -8 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 B=23 C=21 E=2 so E is eliminated. Round 2 votes counts: A=28 D=26 B=24 C=22 so C is eliminated. Round 3 votes counts: B=39 D=33 A=28 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:212 D:204 B:199 E:194 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 0 8 B -8 0 0 2 4 C -8 0 0 -8 -2 D 0 -2 8 0 2 E -8 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 0 8 B -8 0 0 2 4 C -8 0 0 -8 -2 D 0 -2 8 0 2 E -8 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 0 8 B -8 0 0 2 4 C -8 0 0 -8 -2 D 0 -2 8 0 2 E -8 -4 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9136: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) E A C B D (6) A E D C B (6) A E C B D (6) D C B A E (5) C B D E A (5) E B A C D (4) D B C A E (4) B C E D A (4) B C D E A (4) A D C E B (4) E C B A D (3) E A B C D (3) D C B E A (3) D A C B E (3) B D C E A (3) A E D B C (3) A D E C B (3) E C A B D (2) E B C A D (2) D A B C E (2) C E B A D (2) A E C D B (2) E A D B C (1) E A B D C (1) D C A B E (1) D A B E C (1) C D B E A (1) C D B A E (1) C B E D A (1) B E D C A (1) B E C D A (1) A E B C D (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -4 2 -12 B 6 0 -10 -2 -2 C 4 10 0 -2 6 D -2 2 2 0 2 E 12 2 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.593749998947 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 A B C D E A 0 -6 -4 2 -12 B 6 0 -10 -2 -2 C 4 10 0 -2 6 D -2 2 2 0 2 E 12 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.593750000001 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=27 E=22 B=13 C=10 so C is eliminated. Round 2 votes counts: D=30 A=27 E=24 B=19 so B is eliminated. Round 3 votes counts: D=42 E=31 A=27 so A is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:209 E:203 D:202 B:196 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 2 -12 B 6 0 -10 -2 -2 C 4 10 0 -2 6 D -2 2 2 0 2 E 12 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.593750000001 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 2 -12 B 6 0 -10 -2 -2 C 4 10 0 -2 6 D -2 2 2 0 2 E 12 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.593750000001 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 2 -12 B 6 0 -10 -2 -2 C 4 10 0 -2 6 D -2 2 2 0 2 E 12 2 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.125000 Sum of squares = 0.593750000001 Cumulative probabilities = A: 0.125000 B: 0.125000 C: 0.125000 D: 0.875000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9137: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) A E D B C (10) B C D A E (9) E A C B D (7) E A D B C (6) C B D E A (6) C D B E A (5) C B E D A (5) D B C A E (4) D C B E A (3) A B D C E (3) E C B A D (2) E A C D B (2) D A B E C (2) C B D A E (2) B D C A E (2) B C A D E (2) A E B D C (2) A E B C D (2) A D E B C (2) E C D B A (1) E C D A B (1) D E C B A (1) D B A C E (1) D A E B C (1) D A B C E (1) C E D B A (1) C E B D A (1) C B E A D (1) B A C D E (1) A D B E C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 8 8 -6 B -6 0 2 -8 0 C -8 -2 0 -2 -4 D -8 8 2 0 -6 E 6 0 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.273925 C: 0.000000 D: 0.000000 E: 0.726075 Sum of squares = 0.602220180563 Cumulative probabilities = A: 0.000000 B: 0.273925 C: 0.273925 D: 0.273925 E: 1.000000 A B C D E A 0 6 8 8 -6 B -6 0 2 -8 0 C -8 -2 0 -2 -4 D -8 8 2 0 -6 E 6 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204088987 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=22 C=21 B=14 D=13 so D is eliminated. Round 2 votes counts: E=31 A=26 C=24 B=19 so B is eliminated. Round 3 votes counts: C=41 E=31 A=28 so A is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:208 E:208 D:198 B:194 C:192 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 8 8 -6 B -6 0 2 -8 0 C -8 -2 0 -2 -4 D -8 8 2 0 -6 E 6 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204088987 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 8 -6 B -6 0 2 -8 0 C -8 -2 0 -2 -4 D -8 8 2 0 -6 E 6 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204088987 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 8 -6 B -6 0 2 -8 0 C -8 -2 0 -2 -4 D -8 8 2 0 -6 E 6 0 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.571429 Sum of squares = 0.510204088987 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9138: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) A E D B C (7) A E B D C (7) E A D B C (5) C B D E A (5) C B A E D (5) B D E A C (5) B C A E D (5) A E D C B (5) D E B A C (4) D E A B C (4) C A E D B (4) C A E B D (3) B D C E A (3) A C E D B (3) A B E D C (3) D C B E A (2) C D B E A (2) C D A E B (2) C B A D E (2) C A D E B (2) B A E C D (2) A E C D B (2) E D A C B (1) E A B D C (1) D E A C B (1) D B E A C (1) C E A D B (1) C D E A B (1) C D B A E (1) B E D A C (1) B E A D C (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 8 10 20 24 B -8 0 0 4 -8 C -10 0 0 -2 -4 D -20 -4 2 0 -16 E -24 8 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 20 24 B -8 0 0 4 -8 C -10 0 0 -2 -4 D -20 -4 2 0 -16 E -24 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=29 B=17 D=12 E=7 so E is eliminated. Round 2 votes counts: C=35 A=35 B=17 D=13 so D is eliminated. Round 3 votes counts: A=41 C=37 B=22 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:231 E:202 B:194 C:192 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 20 24 B -8 0 0 4 -8 C -10 0 0 -2 -4 D -20 -4 2 0 -16 E -24 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 20 24 B -8 0 0 4 -8 C -10 0 0 -2 -4 D -20 -4 2 0 -16 E -24 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 20 24 B -8 0 0 4 -8 C -10 0 0 -2 -4 D -20 -4 2 0 -16 E -24 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999874 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9139: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (7) D A C E B (6) B E C A D (6) D E A B C (5) B C E A D (5) B C D A E (5) A E C D B (5) D B C A E (4) E A D C B (3) D E A C B (3) D A C B E (3) B D C A E (3) B C D E A (3) A C E D B (3) E C A B D (2) E B D A C (2) D E B A C (2) D B E A C (2) D B C E A (2) D B A C E (2) C B A E D (2) C B A D E (2) C A E B D (2) B E D C A (2) B D E C A (2) B C A E D (2) E D A C B (1) E B C A D (1) E A D B C (1) E A C D B (1) E A C B D (1) D B E C A (1) D A E B C (1) D A B C E (1) B E D A C (1) B D E A C (1) B D C E A (1) B C E D A (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 8 -22 2 B 4 0 12 -10 2 C -8 -12 0 -14 0 D 22 10 14 0 16 E -2 -2 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 -22 2 B 4 0 12 -10 2 C -8 -12 0 -14 0 D 22 10 14 0 16 E -2 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 B=32 E=12 A=11 C=6 so C is eliminated. Round 2 votes counts: D=39 B=36 A=13 E=12 so E is eliminated. Round 3 votes counts: D=40 B=39 A=21 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:231 B:204 A:192 E:190 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 8 -22 2 B 4 0 12 -10 2 C -8 -12 0 -14 0 D 22 10 14 0 16 E -2 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 -22 2 B 4 0 12 -10 2 C -8 -12 0 -14 0 D 22 10 14 0 16 E -2 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 -22 2 B 4 0 12 -10 2 C -8 -12 0 -14 0 D 22 10 14 0 16 E -2 -2 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9140: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) E C A D B (9) C B D A E (7) D B A E C (6) E D A B C (5) E C B D A (4) E A D C B (4) C B A D E (4) B D A E C (4) B D A C E (4) B C D A E (4) D A B E C (3) C E B A D (3) C E A B D (3) A E D B C (3) A D E B C (3) E A D B C (2) C B E D A (2) C B E A D (2) C A D B E (2) B D C A E (2) B C E D A (2) B C D E A (2) A E D C B (2) E B C D A (1) E A C D B (1) D B E A C (1) D A E B C (1) C B D E A (1) C A E D B (1) B D C E A (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -20 0 -6 B 0 0 -10 -8 -4 C 20 10 0 16 0 D 0 8 -16 0 -8 E 6 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.202653 D: 0.000000 E: 0.797347 Sum of squares = 0.676830654394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.202653 D: 0.202653 E: 1.000000 A B C D E A 0 0 -20 0 -6 B 0 0 -10 -8 -4 C 20 10 0 16 0 D 0 8 -16 0 -8 E 6 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=26 B=19 D=11 A=9 so A is eliminated. Round 2 votes counts: C=35 E=31 B=19 D=15 so D is eliminated. Round 3 votes counts: E=35 C=35 B=30 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 E:209 D:192 B:189 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -20 0 -6 B 0 0 -10 -8 -4 C 20 10 0 16 0 D 0 8 -16 0 -8 E 6 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -20 0 -6 B 0 0 -10 -8 -4 C 20 10 0 16 0 D 0 8 -16 0 -8 E 6 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -20 0 -6 B 0 0 -10 -8 -4 C 20 10 0 16 0 D 0 8 -16 0 -8 E 6 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9141: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E C B A D (7) D C A E B (6) D A B C E (6) E C D B A (5) B A E C D (5) C E B A D (4) E B C D A (3) E B C A D (3) D C E A B (3) D B A E C (3) D A C B E (3) D A B E C (3) A B D C E (3) D A C E B (2) C E D B A (2) C D E A B (2) C D A E B (2) C A D E B (2) B E C A D (2) B E A C D (2) B A E D C (2) B A D E C (2) A D B C E (2) A C D B E (2) A B D E C (2) E D C B A (1) E B D C A (1) D E C B A (1) C E D A B (1) C E B D A (1) C E A D B (1) C E A B D (1) C B A E D (1) B E D A C (1) B E A D C (1) B D A E C (1) B A C E D (1) A D B E C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -12 -8 2 B 12 0 -10 4 -10 C 12 10 0 12 -6 D 8 -4 -12 0 -8 E -2 10 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.46 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 -12 -12 -8 2 B 12 0 -10 4 -10 C 12 10 0 12 -6 D 8 -4 -12 0 -8 E -2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999784 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=27 D=27 C=17 B=17 A=12 so A is eliminated. Round 2 votes counts: D=30 E=27 B=23 C=20 so C is eliminated. Round 3 votes counts: D=38 E=37 B=25 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:211 B:198 D:192 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -12 -8 2 B 12 0 -10 4 -10 C 12 10 0 12 -6 D 8 -4 -12 0 -8 E -2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999784 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -8 2 B 12 0 -10 4 -10 C 12 10 0 12 -6 D 8 -4 -12 0 -8 E -2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999784 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -8 2 B 12 0 -10 4 -10 C 12 10 0 12 -6 D 8 -4 -12 0 -8 E -2 10 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.000000 C: 0.100000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999784 Cumulative probabilities = A: 0.300000 B: 0.300000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9142: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (9) A B D E C (9) B A D E C (8) B C A E D (7) C E D A B (6) A D E B C (6) D E C A B (5) C B E D A (5) A D E C B (4) D E A C B (3) B C D E A (3) B C A D E (3) B A C D E (3) E D C A B (2) D E C B A (2) C B A E D (2) B C E D A (2) B A D C E (2) B A C E D (2) A D B E C (2) A B C E D (2) E D A C B (1) E C D A B (1) D E A B C (1) D A E B C (1) D A B E C (1) C E B D A (1) C E A D B (1) C A E D B (1) C A B E D (1) B D E C A (1) B C E A D (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 -2 -6 12 12 B 2 0 8 4 8 C 6 -8 0 2 2 D -12 -4 -2 0 8 E -12 -8 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 12 12 B 2 0 8 4 8 C 6 -8 0 2 2 D -12 -4 -2 0 8 E -12 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=26 A=25 D=13 E=4 so E is eliminated. Round 2 votes counts: B=32 C=27 A=25 D=16 so D is eliminated. Round 3 votes counts: C=36 B=32 A=32 so B is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:211 A:208 C:201 D:195 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 12 12 B 2 0 8 4 8 C 6 -8 0 2 2 D -12 -4 -2 0 8 E -12 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 12 12 B 2 0 8 4 8 C 6 -8 0 2 2 D -12 -4 -2 0 8 E -12 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 12 12 B 2 0 8 4 8 C 6 -8 0 2 2 D -12 -4 -2 0 8 E -12 -8 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9143: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) B E C D A (5) E B A C D (4) E A B C D (4) D A B C E (4) C D A E B (4) A E C D B (4) E B C A D (3) D B C A E (3) D A C E B (3) C E A D B (3) C D A B E (3) C A E D B (3) C A D E B (3) B E A D C (3) B D E A C (3) B C D E A (3) A D E C B (3) A D C E B (3) E C B A D (2) E A C B D (2) D A B E C (2) C E B A D (2) C D B E A (2) B E D A C (2) A E D C B (2) A C D E B (2) E A B D C (1) D B A E C (1) D B A C E (1) D A C B E (1) C D B A E (1) C B E D A (1) C B D E A (1) B E D C A (1) B E C A D (1) B D C A E (1) B D A E C (1) B C E D A (1) B A E D C (1) A E D B C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 0 0 -2 -2 B 0 0 10 0 -2 C 0 -10 0 8 4 D 2 0 -8 0 6 E 2 2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.535466 C: 0.000000 D: 0.464534 E: 0.000000 Sum of squares = 0.502515676498 Cumulative probabilities = A: 0.000000 B: 0.535466 C: 0.535466 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -2 -2 B 0 0 10 0 -2 C 0 -10 0 8 4 D 2 0 -8 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=23 A=17 E=16 D=15 so D is eliminated. Round 2 votes counts: B=34 A=27 C=23 E=16 so E is eliminated. Round 3 votes counts: B=41 A=34 C=25 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:204 C:201 D:200 A:198 E:197 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -2 -2 B 0 0 10 0 -2 C 0 -10 0 8 4 D 2 0 -8 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -2 -2 B 0 0 10 0 -2 C 0 -10 0 8 4 D 2 0 -8 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -2 -2 B 0 0 10 0 -2 C 0 -10 0 8 4 D 2 0 -8 0 6 E 2 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500002 C: 0.000000 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.500002 C: 0.500002 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9144: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) B C E D A (6) B C A E D (6) B A C D E (6) A D E C B (6) A B D C E (6) C B E D A (5) B A C E D (5) D E A C B (4) A D B E C (4) E D A C B (3) E C D A B (3) E C B D A (3) B C E A D (3) B A D C E (3) E C D B A (2) E A C D B (2) D E C A B (2) D C E B A (2) D A E C B (2) C E D B A (2) C E B D A (2) A B D E C (2) A B C D E (2) E D C B A (1) D C A B E (1) C D E B A (1) C B D E A (1) B C D E A (1) B C A D E (1) B A E C D (1) A E D B C (1) A E B C D (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 0 2 -2 B 2 0 0 10 8 C 0 0 0 8 6 D -2 -10 -8 0 -10 E 2 -8 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.310306 C: 0.689694 D: 0.000000 E: 0.000000 Sum of squares = 0.571967710598 Cumulative probabilities = A: 0.000000 B: 0.310306 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 -2 B 2 0 0 10 8 C 0 0 0 8 6 D -2 -10 -8 0 -10 E 2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=24 E=22 D=11 C=11 so D is eliminated. Round 2 votes counts: B=32 E=28 A=26 C=14 so C is eliminated. Round 3 votes counts: B=38 E=35 A=27 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 C:207 A:199 E:199 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 0 2 -2 B 2 0 0 10 8 C 0 0 0 8 6 D -2 -10 -8 0 -10 E 2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 -2 B 2 0 0 10 8 C 0 0 0 8 6 D -2 -10 -8 0 -10 E 2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 -2 B 2 0 0 10 8 C 0 0 0 8 6 D -2 -10 -8 0 -10 E 2 -8 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9145: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (10) E D B A C (7) A C D B E (6) E D B C A (5) D E A B C (5) B C E A D (5) A C B D E (5) D A E C B (4) C A B D E (4) A D C B E (4) E B D C A (3) E B C D A (3) D A E B C (3) B D E A C (3) E C B D A (2) E A C D B (2) D E B A C (2) D B E A C (2) C B E A D (2) C B A E D (2) B E D C A (2) B E C D A (2) B C A D E (2) A D C E B (2) A C E D B (2) E C A D B (1) D E A C B (1) D A B E C (1) D A B C E (1) C E B A D (1) C A E B D (1) B D E C A (1) B D A C E (1) B A C D E (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 8 8 4 2 B -8 0 0 0 10 C -8 0 0 4 2 D -4 0 -4 0 0 E -2 -10 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 4 2 B -8 0 0 0 10 C -8 0 0 4 2 D -4 0 -4 0 0 E -2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=23 A=21 C=20 D=19 B=17 so B is eliminated. Round 2 votes counts: E=27 C=27 D=24 A=22 so A is eliminated. Round 3 votes counts: C=42 D=31 E=27 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:211 B:201 C:199 D:196 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 4 2 B -8 0 0 0 10 C -8 0 0 4 2 D -4 0 -4 0 0 E -2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 4 2 B -8 0 0 0 10 C -8 0 0 4 2 D -4 0 -4 0 0 E -2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 4 2 B -8 0 0 0 10 C -8 0 0 4 2 D -4 0 -4 0 0 E -2 -10 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9146: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) D E C A B (7) B A C E D (7) A C B D E (7) E D C B A (6) E D C A B (5) D A C E B (5) B C A D E (5) B A C D E (5) E D B A C (4) D E A C B (4) C A B D E (4) A C D B E (4) E B D C A (3) D E C B A (3) D C A B E (3) E D A C B (2) E B A D C (2) E B A C D (2) D A E C B (2) B E A C D (2) B C A E D (2) D C E B A (1) D C E A B (1) D C A E B (1) B C E A D (1) A D E C B (1) A D C E B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 -10 -2 B 2 0 -16 -18 -18 C 0 16 0 -18 -2 D 10 18 18 0 12 E 2 18 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 -10 -2 B 2 0 -16 -18 -18 C 0 16 0 -18 -2 D 10 18 18 0 12 E 2 18 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 D=27 B=22 A=15 C=4 so C is eliminated. Round 2 votes counts: E=32 D=27 B=22 A=19 so A is eliminated. Round 3 votes counts: D=34 B=34 E=32 so E is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:205 C:198 A:193 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -10 -2 B 2 0 -16 -18 -18 C 0 16 0 -18 -2 D 10 18 18 0 12 E 2 18 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -10 -2 B 2 0 -16 -18 -18 C 0 16 0 -18 -2 D 10 18 18 0 12 E 2 18 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -10 -2 B 2 0 -16 -18 -18 C 0 16 0 -18 -2 D 10 18 18 0 12 E 2 18 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9147: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (12) D E A C B (9) B C A E D (9) D E C A B (6) C E B A D (6) B A C E D (5) C E D B A (4) D A E B C (3) C D E B A (3) B C A D E (3) B A E C D (3) A B E C D (3) D C E B A (2) D C E A B (2) D A B E C (2) C E B D A (2) C B E D A (2) B C D A E (2) B A D C E (2) A E B D C (2) A D E B C (2) A B E D C (2) A B D E C (2) E D A C B (1) E C A D B (1) E A D C B (1) D C B E A (1) D B C A E (1) D B A E C (1) D B A C E (1) C E A B D (1) C B D A E (1) B A D E C (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 -24 -16 14 -6 B 24 0 -8 20 8 C 16 8 0 18 20 D -14 -20 -18 0 -10 E 6 -8 -20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 -16 14 -6 B 24 0 -8 20 8 C 16 8 0 18 20 D -14 -20 -18 0 -10 E 6 -8 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=28 B=26 A=12 E=3 so E is eliminated. Round 2 votes counts: C=32 D=29 B=26 A=13 so A is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:231 B:222 E:194 A:184 D:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -24 -16 14 -6 B 24 0 -8 20 8 C 16 8 0 18 20 D -14 -20 -18 0 -10 E 6 -8 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 -16 14 -6 B 24 0 -8 20 8 C 16 8 0 18 20 D -14 -20 -18 0 -10 E 6 -8 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 -16 14 -6 B 24 0 -8 20 8 C 16 8 0 18 20 D -14 -20 -18 0 -10 E 6 -8 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9148: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (9) E A B C D (7) D C B A E (7) E A D B C (5) A E D B C (5) A E C B D (5) D E A B C (4) D B C E A (4) C B E A D (4) A E C D B (4) C B D E A (3) C B D A E (3) C B A E D (3) B C E D A (3) A D E C B (3) A C E B D (3) E B C A D (2) D E B A C (2) D B E C A (2) B C E A D (2) A E D C B (2) E D A B C (1) E C B A D (1) E B A C D (1) D E B C A (1) D C A B E (1) D B A C E (1) D A E C B (1) D A C E B (1) D A C B E (1) C D B A E (1) C B E D A (1) C A B D E (1) B E C D A (1) B C D E A (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 14 18 4 10 B -14 0 2 -16 -16 C -18 -2 0 -4 -12 D -4 16 4 0 -2 E -10 16 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999669 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 4 10 B -14 0 2 -16 -16 C -18 -2 0 -4 -12 D -4 16 4 0 -2 E -10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998833 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=26 E=17 C=16 B=7 so B is eliminated. Round 2 votes counts: D=34 A=26 C=22 E=18 so E is eliminated. Round 3 votes counts: A=39 D=35 C=26 so C is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:210 D:207 C:182 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 4 10 B -14 0 2 -16 -16 C -18 -2 0 -4 -12 D -4 16 4 0 -2 E -10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998833 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 4 10 B -14 0 2 -16 -16 C -18 -2 0 -4 -12 D -4 16 4 0 -2 E -10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998833 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 4 10 B -14 0 2 -16 -16 C -18 -2 0 -4 -12 D -4 16 4 0 -2 E -10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998833 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9149: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (20) E B A C D (10) D C A E B (9) B E A C D (6) B A C E D (6) D E C A B (5) B A C D E (5) E D B C A (4) A C B D E (4) E D C A B (3) C A D B E (3) B A E C D (3) E B D C A (2) E B D A C (2) D C E A B (2) C D A B E (2) A B C D E (2) E D C B A (1) D E C B A (1) D B A C E (1) D A C B E (1) C D E A B (1) C A E B D (1) C A D E B (1) C A B D E (1) B E D A C (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 16 -12 -10 24 B -16 0 -16 -12 16 C 12 16 0 -6 24 D 10 12 6 0 20 E -24 -16 -24 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -12 -10 24 B -16 0 -16 -12 16 C 12 16 0 -6 24 D 10 12 6 0 20 E -24 -16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 E=22 B=21 C=9 A=9 so C is eliminated. Round 2 votes counts: D=42 E=22 B=21 A=15 so A is eliminated. Round 3 votes counts: D=48 B=29 E=23 so E is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:223 A:209 B:186 E:158 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -12 -10 24 B -16 0 -16 -12 16 C 12 16 0 -6 24 D 10 12 6 0 20 E -24 -16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -12 -10 24 B -16 0 -16 -12 16 C 12 16 0 -6 24 D 10 12 6 0 20 E -24 -16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -12 -10 24 B -16 0 -16 -12 16 C 12 16 0 -6 24 D 10 12 6 0 20 E -24 -16 -24 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9150: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (18) B A D E C (11) E C B A D (9) B D A C E (8) E C D A B (6) C E D B A (5) E C A D B (4) D A B C E (4) E A D C B (3) E C A B D (2) E B A C D (2) E A D B C (2) C E B D A (2) C D E A B (2) B A D C E (2) E B C A D (1) E A B C D (1) D E A C B (1) D C A E B (1) D B A C E (1) D A E C B (1) D A C B E (1) C E B A D (1) C D B E A (1) C D B A E (1) C B E A D (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A D C (1) B D C A E (1) B C E A D (1) B A E D C (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -18 -10 -30 B 4 0 -22 -6 -24 C 18 22 0 20 4 D 10 6 -20 0 -22 E 30 24 -4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999638 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 -10 -30 B 4 0 -22 -6 -24 C 18 22 0 20 4 D 10 6 -20 0 -22 E 30 24 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=30 B=26 D=9 A=2 so A is eliminated. Round 2 votes counts: C=33 E=30 B=26 D=11 so D is eliminated. Round 3 votes counts: C=35 E=33 B=32 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:236 C:232 D:187 B:176 A:169 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -18 -10 -30 B 4 0 -22 -6 -24 C 18 22 0 20 4 D 10 6 -20 0 -22 E 30 24 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -10 -30 B 4 0 -22 -6 -24 C 18 22 0 20 4 D 10 6 -20 0 -22 E 30 24 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -10 -30 B 4 0 -22 -6 -24 C 18 22 0 20 4 D 10 6 -20 0 -22 E 30 24 -4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9151: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) A D C E B (7) E B D C A (6) B E D C A (6) A E D C B (5) B C A D E (4) A C B D E (4) E D B C A (3) E D A B C (3) E B A C D (3) D E C B A (3) C B A D E (3) B E C D A (3) A C B E D (3) E B D A C (2) E A D C B (2) E A D B C (2) D E B C A (2) D C A B E (2) C A B D E (2) B E C A D (2) B C D E A (2) B C D A E (2) A B E C D (2) E D B A C (1) E D A C B (1) E B A D C (1) E A B C D (1) D E A C B (1) D C E A B (1) D C B E A (1) D C A E B (1) D B E C A (1) D A C E B (1) C B D A E (1) C A D B E (1) B E A C D (1) B C E D A (1) B C A E D (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 4 12 2 B -2 0 -2 0 4 C -4 2 0 -4 -4 D -12 0 4 0 0 E -2 -4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 12 2 B -2 0 -2 0 4 C -4 2 0 -4 -4 D -12 0 4 0 0 E -2 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=25 B=22 D=13 C=7 so C is eliminated. Round 2 votes counts: A=36 B=26 E=25 D=13 so D is eliminated. Round 3 votes counts: A=40 E=32 B=28 so B is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:200 E:199 D:196 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 12 2 B -2 0 -2 0 4 C -4 2 0 -4 -4 D -12 0 4 0 0 E -2 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 12 2 B -2 0 -2 0 4 C -4 2 0 -4 -4 D -12 0 4 0 0 E -2 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 12 2 B -2 0 -2 0 4 C -4 2 0 -4 -4 D -12 0 4 0 0 E -2 -4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9152: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) E B A D C (5) E B A C D (5) A C D E B (5) C A E B D (4) B E D C A (4) B C E D A (4) E B C A D (3) E A B D C (3) D B E A C (3) C D A B E (3) C A D B E (3) B E C A D (3) B D C E A (3) A E D B C (3) E A D B C (2) E A C B D (2) D A E C B (2) D A C E B (2) D A C B E (2) C E A B D (2) C B E D A (2) C B E A D (2) C B D E A (2) C A D E B (2) B E C D A (2) B D E C A (2) B D E A C (2) B C E A D (2) A D C E B (2) A C E D B (2) E A B C D (1) D E A B C (1) D C B A E (1) D B A E C (1) D A B C E (1) C B D A E (1) B C D E A (1) A D E C B (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 4 8 8 -12 B -4 0 18 12 -12 C -8 -18 0 4 -6 D -8 -12 -4 0 -4 E 12 12 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 8 -12 B -4 0 18 12 -12 C -8 -18 0 4 -6 D -8 -12 -4 0 -4 E 12 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=21 C=21 D=20 A=15 so A is eliminated. Round 2 votes counts: C=29 E=24 D=24 B=23 so B is eliminated. Round 3 votes counts: C=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:217 B:207 A:204 C:186 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 8 -12 B -4 0 18 12 -12 C -8 -18 0 4 -6 D -8 -12 -4 0 -4 E 12 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 8 -12 B -4 0 18 12 -12 C -8 -18 0 4 -6 D -8 -12 -4 0 -4 E 12 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 8 -12 B -4 0 18 12 -12 C -8 -18 0 4 -6 D -8 -12 -4 0 -4 E 12 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986452 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9153: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) C A B E D (6) B C A D E (6) B A C D E (6) E D A C B (5) E D A B C (5) D E B C A (5) A E C B D (4) E D C A B (3) E A D C B (3) D E C B A (3) C B A D E (3) B D C A E (3) A B C E D (3) E D B A C (2) D E C A B (2) D E B A C (2) D B E A C (2) C D B A E (2) C A E B D (2) C A D E B (2) B D E C A (2) B A E D C (2) A C E B D (2) E D C B A (1) D C E B A (1) D C B E A (1) D B E C A (1) C D E A B (1) C A E D B (1) C A D B E (1) B E D A C (1) B D C E A (1) B D A C E (1) B A D C E (1) B A C E D (1) A E B C D (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -12 12 18 B -6 0 -6 14 10 C 12 6 0 6 10 D -12 -14 -6 0 12 E -18 -10 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 12 18 B -6 0 -6 14 10 C 12 6 0 6 10 D -12 -14 -6 0 12 E -18 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=24 E=19 D=17 A=13 so A is eliminated. Round 2 votes counts: C=30 B=29 E=24 D=17 so D is eliminated. Round 3 votes counts: E=36 C=32 B=32 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:217 A:212 B:206 D:190 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 12 18 B -6 0 -6 14 10 C 12 6 0 6 10 D -12 -14 -6 0 12 E -18 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 12 18 B -6 0 -6 14 10 C 12 6 0 6 10 D -12 -14 -6 0 12 E -18 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 12 18 B -6 0 -6 14 10 C 12 6 0 6 10 D -12 -14 -6 0 12 E -18 -10 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9154: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) B A E D C (8) A B C D E (7) C A D E B (6) A C D B E (5) C D A E B (4) B E D A C (4) A B D E C (4) A B D C E (4) E D C B A (3) D C E A B (3) B E D C A (3) B E A D C (3) B A E C D (3) E D B C A (2) E C D B A (2) E B C D A (2) D E C A B (2) D A C E B (2) C E D B A (2) A D B C E (2) A C B D E (2) E B D A C (1) D E C B A (1) D C A E B (1) C E B D A (1) C B A E D (1) C A D B E (1) C A B E D (1) B E C D A (1) B E A C D (1) B C E D A (1) B A C E D (1) A D E B C (1) A D C B E (1) A D B E C (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 20 4 8 14 B -20 0 0 -2 10 C -4 0 0 8 16 D -8 2 -8 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 4 8 14 B -20 0 0 -2 10 C -4 0 0 8 16 D -8 2 -8 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 B=25 E=10 D=9 so D is eliminated. Round 2 votes counts: C=31 A=31 B=25 E=13 so E is eliminated. Round 3 votes counts: C=39 A=31 B=30 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:210 D:202 B:194 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 4 8 14 B -20 0 0 -2 10 C -4 0 0 8 16 D -8 2 -8 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 4 8 14 B -20 0 0 -2 10 C -4 0 0 8 16 D -8 2 -8 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 4 8 14 B -20 0 0 -2 10 C -4 0 0 8 16 D -8 2 -8 0 18 E -14 -10 -16 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9155: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) C E A B D (8) C A E D B (7) B D E A C (7) E C B A D (6) E C A B D (6) A C E D B (6) D B E C A (5) A C D E B (5) E C B D A (4) D B E A C (4) C E A D B (4) B D E C A (4) E B D C A (3) E B C D A (3) D B A E C (3) D A B C E (3) B E D C A (3) B D A E C (2) A C E B D (2) E B D A C (1) D B C E A (1) D A C B E (1) C A E B D (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -10 -4 -18 B 8 0 -2 2 -12 C 10 2 0 4 -4 D 4 -2 -4 0 -10 E 18 12 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -10 -4 -18 B 8 0 -2 2 -12 C 10 2 0 4 -4 D 4 -2 -4 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=23 C=20 B=16 A=16 so B is eliminated. Round 2 votes counts: D=38 E=26 C=20 A=16 so A is eliminated. Round 3 votes counts: D=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:222 C:206 B:198 D:194 A:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -10 -4 -18 B 8 0 -2 2 -12 C 10 2 0 4 -4 D 4 -2 -4 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -10 -4 -18 B 8 0 -2 2 -12 C 10 2 0 4 -4 D 4 -2 -4 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -10 -4 -18 B 8 0 -2 2 -12 C 10 2 0 4 -4 D 4 -2 -4 0 -10 E 18 12 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9156: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) A E D B C (9) D E A B C (8) C B D E A (8) C B A E D (8) E A D B C (5) B C A E D (5) A E B D C (5) D C E A B (4) C B D A E (4) B A E D C (4) B A E C D (4) C D E A B (3) C D B E A (3) B A C E D (3) D E C A B (2) C B E D A (2) C B A D E (2) A B E D C (2) E A B D C (1) D E B A C (1) D C E B A (1) D C A E B (1) D A E C B (1) C D E B A (1) B E A D C (1) B C E A D (1) B C D E A (1) A E D C B (1) Total count = 100 A B C D E A 0 2 8 2 -2 B -2 0 0 2 -4 C -8 0 0 -10 -6 D -2 -2 10 0 -2 E 2 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 8 2 -2 B -2 0 0 2 -4 C -8 0 0 -10 -6 D -2 -2 10 0 -2 E 2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=27 B=19 A=17 E=6 so E is eliminated. Round 2 votes counts: C=31 D=27 A=23 B=19 so B is eliminated. Round 3 votes counts: C=38 A=35 D=27 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:207 A:205 D:202 B:198 C:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 8 2 -2 B -2 0 0 2 -4 C -8 0 0 -10 -6 D -2 -2 10 0 -2 E 2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 2 -2 B -2 0 0 2 -4 C -8 0 0 -10 -6 D -2 -2 10 0 -2 E 2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 2 -2 B -2 0 0 2 -4 C -8 0 0 -10 -6 D -2 -2 10 0 -2 E 2 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999861 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9157: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (12) B D A C E (7) B A D C E (7) B C D A E (6) A D C E B (5) E A D C B (4) B E C D A (4) B C E D A (4) A E D C B (4) A D C B E (4) E C D A B (3) E C B D A (3) E B C D A (3) D A B C E (3) E C A D B (2) E C A B D (2) E B A C D (2) E A C D B (2) C E D B A (2) C E B D A (2) B E A D C (2) A D E C B (2) E C D B A (1) E C B A D (1) E B A D C (1) E A D B C (1) E A B C D (1) C E D A B (1) C D A E B (1) C B E D A (1) C B D E A (1) A E D B C (1) A D E B C (1) A D B E C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 26 16 14 B -6 0 18 0 10 C -26 -18 0 -16 14 D -16 0 16 0 4 E -14 -10 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 26 16 14 B -6 0 18 0 10 C -26 -18 0 -16 14 D -16 0 16 0 4 E -14 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=30 E=26 C=8 D=3 so D is eliminated. Round 2 votes counts: A=36 B=30 E=26 C=8 so C is eliminated. Round 3 votes counts: A=37 B=32 E=31 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:231 B:211 D:202 E:179 C:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 26 16 14 B -6 0 18 0 10 C -26 -18 0 -16 14 D -16 0 16 0 4 E -14 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 26 16 14 B -6 0 18 0 10 C -26 -18 0 -16 14 D -16 0 16 0 4 E -14 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 26 16 14 B -6 0 18 0 10 C -26 -18 0 -16 14 D -16 0 16 0 4 E -14 -10 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994299 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9158: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) C A B D E (7) E C B D A (6) A D B C E (6) E D B A C (4) E B C A D (4) A B C D E (4) E D B C A (3) E C B A D (3) E B D A C (3) E B C D A (3) D E A B C (3) D A E B C (3) C E B A D (3) B A D E C (3) E D A B C (2) E C D B A (2) E C D A B (2) E B D C A (2) D E A C B (2) D A C B E (2) C D A E B (2) C B E A D (2) C B A E D (2) C A D B E (2) B E A D C (2) D C A E B (1) D B E A C (1) D B A E C (1) D A B C E (1) C E D A B (1) C E A B D (1) C B A D E (1) C A B E D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A C D E (1) A D C B E (1) A D B E C (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 -4 0 B -2 0 16 6 0 C 0 -16 0 0 -16 D 4 -6 0 0 4 E 0 0 16 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888844 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -4 0 B -2 0 16 6 0 C 0 -16 0 0 -16 D 4 -6 0 0 4 E 0 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=22 D=21 A=14 B=9 so B is eliminated. Round 2 votes counts: E=36 C=24 D=22 A=18 so A is eliminated. Round 3 votes counts: E=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 E:206 D:201 A:199 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 0 -4 0 B -2 0 16 6 0 C 0 -16 0 0 -16 D 4 -6 0 0 4 E 0 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -4 0 B -2 0 16 6 0 C 0 -16 0 0 -16 D 4 -6 0 0 4 E 0 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -4 0 B -2 0 16 6 0 C 0 -16 0 0 -16 D 4 -6 0 0 4 E 0 0 16 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.333333 C: 0.000000 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888869 Cumulative probabilities = A: 0.500000 B: 0.833333 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9159: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) C B A E D (7) B C E A D (7) B C D E A (6) D E A B C (5) D C B A E (5) E A D B C (4) D B C E A (4) A E C B D (4) E A D C B (3) E A C B D (3) C B D A E (3) C A E B D (3) E B A C D (2) E A B D C (2) D E B A C (2) D C A E B (2) D B E A C (2) D B C A E (2) C D B A E (2) B D E A C (2) B C D A E (2) A D E C B (2) E D A B C (1) E B D A C (1) E A B C D (1) D C A B E (1) D B E C A (1) D A C E B (1) C B A D E (1) C A B E D (1) B E D A C (1) B E C A D (1) B E A C D (1) B D E C A (1) B D C E A (1) A E D C B (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 0 -8 -2 B 8 0 -2 2 2 C 0 2 0 -6 0 D 8 -2 6 0 8 E 2 -2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999953 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -8 -2 B 8 0 -2 2 2 C 0 2 0 -6 0 D 8 -2 6 0 8 E 2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999979 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=22 E=17 C=17 A=10 so A is eliminated. Round 2 votes counts: D=36 E=23 B=22 C=19 so C is eliminated. Round 3 votes counts: D=38 B=34 E=28 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:210 B:205 C:198 E:196 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -8 -2 B 8 0 -2 2 2 C 0 2 0 -6 0 D 8 -2 6 0 8 E 2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999979 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -8 -2 B 8 0 -2 2 2 C 0 2 0 -6 0 D 8 -2 6 0 8 E 2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999979 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -8 -2 B 8 0 -2 2 2 C 0 2 0 -6 0 D 8 -2 6 0 8 E 2 -2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.200000 D: 0.200000 E: 0.000000 Sum of squares = 0.43999999979 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9160: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) C B D E A (7) E A C B D (6) A E D B C (5) A E B C D (5) E A C D B (4) D E C B A (4) B A C D E (4) E D A C B (3) E C B D A (3) D B A C E (3) B C A E D (3) B A C E D (3) A D B E C (3) A B D C E (3) E D C B A (2) E C A B D (2) D C E B A (2) D B C A E (2) D A E C B (2) D A B C E (2) B C E A D (2) B A D C E (2) A B E C D (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B A D (1) E A D C B (1) D E C A B (1) D C B E A (1) D C A E B (1) D A B E C (1) C E D B A (1) C E B D A (1) C E B A D (1) C B E D A (1) B C A D E (1) A E D C B (1) A E B D C (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 6 4 8 B 6 0 4 12 -2 C -6 -4 0 14 -2 D -4 -12 -14 0 -2 E -8 2 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999977 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 A B C D E A 0 -6 6 4 8 B 6 0 4 12 -2 C -6 -4 0 14 -2 D -4 -12 -14 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999954 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=23 A=22 D=19 C=11 so C is eliminated. Round 2 votes counts: B=31 E=28 A=22 D=19 so D is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:210 A:206 C:201 E:199 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 6 4 8 B 6 0 4 12 -2 C -6 -4 0 14 -2 D -4 -12 -14 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999954 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 4 8 B 6 0 4 12 -2 C -6 -4 0 14 -2 D -4 -12 -14 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999954 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 4 8 B 6 0 4 12 -2 C -6 -4 0 14 -2 D -4 -12 -14 0 -2 E -8 2 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.375000 Sum of squares = 0.406249999954 Cumulative probabilities = A: 0.125000 B: 0.625000 C: 0.625000 D: 0.625000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9161: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) C B E D A (11) A B C D E (11) A D E B C (9) B A C E D (7) A B D E C (7) A D E C B (6) D E C A B (5) D E A C B (5) C E D B A (5) E D B C A (3) B C A E D (3) B C E D A (2) A B E D C (2) A B C E D (2) E D B A C (1) D E A B C (1) D A E C B (1) C D E A B (1) C B A E D (1) B E D C A (1) B A E D C (1) A D C E B (1) A D B E C (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 8 14 6 8 B -8 0 4 -2 0 C -14 -4 0 -12 -8 D -6 2 12 0 0 E -8 0 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 6 8 B -8 0 4 -2 0 C -14 -4 0 -12 -8 D -6 2 12 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=18 E=15 B=14 D=12 so D is eliminated. Round 2 votes counts: A=42 E=26 C=18 B=14 so B is eliminated. Round 3 votes counts: A=50 E=27 C=23 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:204 E:200 B:197 C:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 14 6 8 B -8 0 4 -2 0 C -14 -4 0 -12 -8 D -6 2 12 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 6 8 B -8 0 4 -2 0 C -14 -4 0 -12 -8 D -6 2 12 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 6 8 B -8 0 4 -2 0 C -14 -4 0 -12 -8 D -6 2 12 0 0 E -8 0 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9162: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (7) A D C E B (6) A C D B E (6) B E D C A (5) E D C B A (4) E D C A B (4) E B D C A (4) B C D E A (4) A C D E B (4) A C B D E (4) E B D A C (3) A E B D C (3) E D B C A (2) E D A C B (2) E B A D C (2) E A D C B (2) E A B D C (2) D C E B A (2) D C E A B (2) C D A B E (2) C A D B E (2) B E D A C (2) B E C D A (2) B C E D A (2) B A C E D (2) B A C D E (2) A B E C D (2) A B C D E (2) E D B A C (1) D E C B A (1) D E C A B (1) D A C E B (1) C D E B A (1) C D B E A (1) C D A E B (1) C B D A E (1) C A B D E (1) B E A D C (1) B C A D E (1) B A E C D (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 12 14 4 0 B -12 0 -12 -8 -14 C -14 12 0 -18 -6 D -4 8 18 0 4 E 0 14 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.669523 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.330477 Sum of squares = 0.557476222561 Cumulative probabilities = A: 0.669523 B: 0.669523 C: 0.669523 D: 0.669523 E: 1.000000 A B C D E A 0 12 14 4 0 B -12 0 -12 -8 -14 C -14 12 0 -18 -6 D -4 8 18 0 4 E 0 14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500423 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499577 Sum of squares = 0.50000035851 Cumulative probabilities = A: 0.500423 B: 0.500423 C: 0.500423 D: 0.500423 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=26 B=22 C=9 D=7 so D is eliminated. Round 2 votes counts: A=37 E=28 B=22 C=13 so C is eliminated. Round 3 votes counts: A=43 E=33 B=24 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 D:213 E:208 C:187 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 4 0 B -12 0 -12 -8 -14 C -14 12 0 -18 -6 D -4 8 18 0 4 E 0 14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500423 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499577 Sum of squares = 0.50000035851 Cumulative probabilities = A: 0.500423 B: 0.500423 C: 0.500423 D: 0.500423 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 4 0 B -12 0 -12 -8 -14 C -14 12 0 -18 -6 D -4 8 18 0 4 E 0 14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500423 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499577 Sum of squares = 0.50000035851 Cumulative probabilities = A: 0.500423 B: 0.500423 C: 0.500423 D: 0.500423 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 4 0 B -12 0 -12 -8 -14 C -14 12 0 -18 -6 D -4 8 18 0 4 E 0 14 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500423 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499577 Sum of squares = 0.50000035851 Cumulative probabilities = A: 0.500423 B: 0.500423 C: 0.500423 D: 0.500423 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9163: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) C D A E B (8) B E A D C (8) E B C A D (6) E B A C D (6) C E B A D (6) D A C B E (5) A B E D C (4) C E D B A (3) C D E A B (3) A E B D C (3) E C B A D (2) D C B E A (2) D C A E B (2) D B E C A (2) D B E A C (2) C E D A B (2) C E B D A (2) C E A B D (2) C A D E B (2) B E D A C (2) B A E D C (2) A D B E C (2) A C E B D (2) D B A E C (1) D A B E C (1) D A B C E (1) C D E B A (1) C D B E A (1) C D A B E (1) C A E D B (1) C A E B D (1) B E C A D (1) B E A C D (1) A E C B D (1) A E B C D (1) A D C E B (1) Total count = 100 A B C D E A 0 4 -14 4 -4 B -4 0 -14 0 -10 C 14 14 0 6 10 D -4 0 -6 0 -12 E 4 10 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -14 4 -4 B -4 0 -14 0 -10 C 14 14 0 6 10 D -4 0 -6 0 -12 E 4 10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=25 E=14 B=14 A=14 so E is eliminated. Round 2 votes counts: C=35 B=26 D=25 A=14 so A is eliminated. Round 3 votes counts: C=38 B=34 D=28 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:208 A:195 D:189 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -14 4 -4 B -4 0 -14 0 -10 C 14 14 0 6 10 D -4 0 -6 0 -12 E 4 10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -14 4 -4 B -4 0 -14 0 -10 C 14 14 0 6 10 D -4 0 -6 0 -12 E 4 10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -14 4 -4 B -4 0 -14 0 -10 C 14 14 0 6 10 D -4 0 -6 0 -12 E 4 10 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9164: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) D C B E A (6) D A E B C (5) C B E D A (5) C B E A D (5) A E B C D (5) D A E C B (4) C B D E A (4) C B A E D (4) A E B D C (4) A D E B C (4) E B A C D (3) C D B A E (3) C B D A E (3) B E C D A (3) B E C A D (3) B C E A D (3) A C B E D (3) E A D B C (2) E A B C D (2) D E B C A (2) D E A B C (2) D C A B E (2) D A C B E (2) B C E D A (2) A E D B C (2) A D C E B (2) E D B A C (1) E B D A C (1) E B C D A (1) D E B A C (1) D C B A E (1) C A B D E (1) A D E C B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -8 4 -4 B 14 0 4 18 2 C 8 -4 0 14 -4 D -4 -18 -14 0 -12 E 4 -2 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999032 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -8 4 -4 B 14 0 4 18 2 C 8 -4 0 14 -4 D -4 -18 -14 0 -12 E 4 -2 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=25 C=25 A=23 E=16 B=11 so B is eliminated. Round 2 votes counts: C=30 D=25 A=23 E=22 so E is eliminated. Round 3 votes counts: C=43 A=30 D=27 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:219 E:209 C:207 A:189 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -8 4 -4 B 14 0 4 18 2 C 8 -4 0 14 -4 D -4 -18 -14 0 -12 E 4 -2 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -8 4 -4 B 14 0 4 18 2 C 8 -4 0 14 -4 D -4 -18 -14 0 -12 E 4 -2 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -8 4 -4 B 14 0 4 18 2 C 8 -4 0 14 -4 D -4 -18 -14 0 -12 E 4 -2 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999968392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9165: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) C D A E B (10) A E B C D (10) C A E B D (7) C D B E A (5) C D B A E (5) C D A B E (5) E B A D C (4) B D E A C (4) D B C E A (3) C A D E B (3) B E A D C (3) A E C B D (3) A E B D C (3) D C E B A (2) D B E A C (2) C D E A B (2) C B A E D (2) C A B E D (2) B A E C D (2) A C E B D (2) A B E C D (2) E A B D C (1) D E B C A (1) D E B A C (1) D E A C B (1) D B E C A (1) C B A D E (1) C A E D B (1) B E D A C (1) B A E D C (1) Total count = 100 A B C D E A 0 4 -20 -6 18 B -4 0 -22 -4 -2 C 20 22 0 24 20 D 6 4 -24 0 12 E -18 2 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -20 -6 18 B -4 0 -22 -4 -2 C 20 22 0 24 20 D 6 4 -24 0 12 E -18 2 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=43 D=21 A=20 B=11 E=5 so E is eliminated. Round 2 votes counts: C=43 D=21 A=21 B=15 so B is eliminated. Round 3 votes counts: C=43 A=31 D=26 so D is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:243 D:199 A:198 B:184 E:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -20 -6 18 B -4 0 -22 -4 -2 C 20 22 0 24 20 D 6 4 -24 0 12 E -18 2 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -20 -6 18 B -4 0 -22 -4 -2 C 20 22 0 24 20 D 6 4 -24 0 12 E -18 2 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -20 -6 18 B -4 0 -22 -4 -2 C 20 22 0 24 20 D 6 4 -24 0 12 E -18 2 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9166: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) C E D A B (6) B D A C E (6) E C B A D (5) D A B E C (5) C D B A E (5) E C A B D (4) C E D B A (4) C B D A E (4) E C D A B (3) E A D B C (3) D A B C E (3) C E B D A (3) C B E A D (3) B C D A E (3) A D B E C (3) E C A D B (2) E B A D C (2) D A E B C (2) C D A E B (2) C D A B E (2) B E A D C (2) B C A D E (2) E D A C B (1) E A D C B (1) E A B D C (1) D E A C B (1) D C A B E (1) D A C E B (1) D A C B E (1) C E B A D (1) C E A D B (1) C D E B A (1) C D E A B (1) C D B E A (1) B E C A D (1) B E A C D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A D C E (1) Total count = 100 A B C D E A 0 -12 -14 -14 2 B 12 0 -8 0 10 C 14 8 0 14 8 D 14 0 -14 0 8 E -2 -10 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -14 2 B 12 0 -8 0 10 C 14 8 0 14 8 D 14 0 -14 0 8 E -2 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=27 E=22 D=14 A=3 so A is eliminated. Round 2 votes counts: C=34 B=27 E=22 D=17 so D is eliminated. Round 3 votes counts: B=38 C=37 E=25 so E is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 B:207 D:204 E:186 A:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -14 2 B 12 0 -8 0 10 C 14 8 0 14 8 D 14 0 -14 0 8 E -2 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -14 2 B 12 0 -8 0 10 C 14 8 0 14 8 D 14 0 -14 0 8 E -2 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -14 2 B 12 0 -8 0 10 C 14 8 0 14 8 D 14 0 -14 0 8 E -2 -10 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9167: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (10) B C D A E (9) B E D C A (8) E D B C A (7) B D C E A (7) A B C D E (7) A E C D B (6) A C D B E (6) A C B D E (6) E D C B A (5) E D C A B (5) E B D C A (4) B D E C A (4) B A C D E (3) A C D E B (3) E D A C B (2) D C E B A (1) C D B A E (1) B C D E A (1) B C A D E (1) A E D C B (1) A C E D B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -6 -8 -8 B 2 0 4 4 10 C 6 -4 0 -8 -6 D 8 -4 8 0 -2 E 8 -10 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -8 -8 B 2 0 4 4 10 C 6 -4 0 -8 -6 D 8 -4 8 0 -2 E 8 -10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=33 B=33 A=32 D=1 C=1 so D is eliminated. Round 2 votes counts: E=33 B=33 A=32 C=2 so C is eliminated. Round 3 votes counts: E=34 B=34 A=32 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:205 E:203 C:194 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 -8 B 2 0 4 4 10 C 6 -4 0 -8 -6 D 8 -4 8 0 -2 E 8 -10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 -8 B 2 0 4 4 10 C 6 -4 0 -8 -6 D 8 -4 8 0 -2 E 8 -10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 -8 B 2 0 4 4 10 C 6 -4 0 -8 -6 D 8 -4 8 0 -2 E 8 -10 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999043 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9168: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (9) D A C E B (8) B E C A D (8) C A E D B (7) B E D A C (7) A D C E B (7) B E C D A (6) E B D A C (5) C A D B E (4) E D A C B (3) D E A B C (3) D A E C B (3) B E D C A (3) A C D E B (3) E D A B C (2) D A C B E (2) D A B C E (2) C E A D B (2) C A B D E (2) B C E A D (2) B C A E D (2) E C D A B (1) E C B D A (1) E B C D A (1) D A E B C (1) C B E A D (1) C B A D E (1) C A E B D (1) B D A E C (1) B C A D E (1) A D C B E (1) Total count = 100 A B C D E A 0 22 -4 2 10 B -22 0 -12 -16 -14 C 4 12 0 4 10 D -2 16 -4 0 -4 E -10 14 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 -4 2 10 B -22 0 -12 -16 -14 C 4 12 0 4 10 D -2 16 -4 0 -4 E -10 14 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=27 D=19 E=13 A=11 so A is eliminated. Round 2 votes counts: C=30 B=30 D=27 E=13 so E is eliminated. Round 3 votes counts: B=36 D=32 C=32 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:215 C:215 D:203 E:199 B:168 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 22 -4 2 10 B -22 0 -12 -16 -14 C 4 12 0 4 10 D -2 16 -4 0 -4 E -10 14 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 -4 2 10 B -22 0 -12 -16 -14 C 4 12 0 4 10 D -2 16 -4 0 -4 E -10 14 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 -4 2 10 B -22 0 -12 -16 -14 C 4 12 0 4 10 D -2 16 -4 0 -4 E -10 14 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9169: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) E B C D A (8) D A B C E (6) E B D C A (5) D A B E C (5) B D E A C (4) A D C B E (4) E C B D A (3) E C A B D (3) C B A D E (3) B E D C A (3) A D C E B (3) A D B C E (3) E D A C B (2) D B A C E (2) D A E B C (2) C E B A D (2) C E A D B (2) C E A B D (2) C B E A D (2) C A E B D (2) B E C D A (2) B D A C E (2) A C D B E (2) E D B C A (1) E D A B C (1) E C A D B (1) E B D A C (1) D E B A C (1) D E A B C (1) D B E A C (1) D B A E C (1) C B A E D (1) C A E D B (1) C A D B E (1) C A B E D (1) B E C A D (1) B C E A D (1) B C D A E (1) B C A E D (1) B C A D E (1) B A D C E (1) A D E C B (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -12 -4 -10 B 12 0 10 18 0 C 12 -10 0 0 -10 D 4 -18 0 0 -8 E 10 0 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.227869 C: 0.000000 D: 0.000000 E: 0.772131 Sum of squares = 0.648110536167 Cumulative probabilities = A: 0.000000 B: 0.227869 C: 0.227869 D: 0.227869 E: 1.000000 A B C D E A 0 -12 -12 -4 -10 B 12 0 10 18 0 C 12 -10 0 0 -10 D 4 -18 0 0 -8 E 10 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=19 C=17 B=17 A=14 so A is eliminated. Round 2 votes counts: E=33 D=30 C=20 B=17 so B is eliminated. Round 3 votes counts: E=39 D=37 C=24 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:220 E:214 C:196 D:189 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -12 -4 -10 B 12 0 10 18 0 C 12 -10 0 0 -10 D 4 -18 0 0 -8 E 10 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -4 -10 B 12 0 10 18 0 C 12 -10 0 0 -10 D 4 -18 0 0 -8 E 10 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -4 -10 B 12 0 10 18 0 C 12 -10 0 0 -10 D 4 -18 0 0 -8 E 10 0 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9170: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) B A C E D (9) D E C A B (8) E A D B C (7) C D B E A (7) B A E C D (6) D E A C B (5) D C E A B (5) B A E D C (4) A E D B C (4) A E B D C (4) E D A B C (3) C B D A E (3) B C A D E (3) A B E D C (3) E D A C B (2) C D E B A (2) C D E A B (2) C D B A E (2) C D A B E (2) C B A D E (2) A B E C D (2) E D C A B (1) C B D E A (1) C A D E B (1) C A B D E (1) B C E A D (1) Total count = 100 A B C D E A 0 0 -2 14 12 B 0 0 12 -2 12 C 2 -12 0 8 2 D -14 2 -8 0 -12 E -12 -12 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.474028 B: 0.525972 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.501349035705 Cumulative probabilities = A: 0.474028 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 14 12 B 0 0 12 -2 12 C 2 -12 0 8 2 D -14 2 -8 0 -12 E -12 -12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=23 D=18 E=13 A=13 so E is eliminated. Round 2 votes counts: B=33 D=24 C=23 A=20 so A is eliminated. Round 3 votes counts: B=42 D=35 C=23 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:212 B:211 C:200 E:193 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -2 14 12 B 0 0 12 -2 12 C 2 -12 0 8 2 D -14 2 -8 0 -12 E -12 -12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 14 12 B 0 0 12 -2 12 C 2 -12 0 8 2 D -14 2 -8 0 -12 E -12 -12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 14 12 B 0 0 12 -2 12 C 2 -12 0 8 2 D -14 2 -8 0 -12 E -12 -12 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999914 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9171: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) E A D B C (7) D A B C E (5) C B E D A (5) E C B A D (4) E A D C B (4) D A E B C (4) C E B D A (4) E D A C B (3) C D B A E (3) B C D A E (3) A E D B C (3) A D E B C (3) E C D B A (2) E C D A B (2) E C A B D (2) E B C A D (2) E A C D B (2) E A C B D (2) D C A B E (2) D B A C E (2) C D E B A (2) B D A C E (2) B C E A D (2) B C A D E (2) A D B E C (2) E D C A B (1) E C B D A (1) E B A C D (1) E A B D C (1) D E A C B (1) D A E C B (1) D A C B E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D E A B (1) C B D E A (1) C B A D E (1) B E C A D (1) B E A D C (1) B A E D C (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 -6 -14 -10 B 2 0 -14 -6 -14 C 6 14 0 10 -4 D 14 6 -10 0 -10 E 10 14 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -6 -14 -10 B 2 0 -14 -6 -14 C 6 14 0 10 -4 D 14 6 -10 0 -10 E 10 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=29 D=16 B=12 A=9 so A is eliminated. Round 2 votes counts: E=38 C=29 D=21 B=12 so B is eliminated. Round 3 votes counts: E=41 C=36 D=23 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:213 D:200 A:184 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 -14 -10 B 2 0 -14 -6 -14 C 6 14 0 10 -4 D 14 6 -10 0 -10 E 10 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -14 -10 B 2 0 -14 -6 -14 C 6 14 0 10 -4 D 14 6 -10 0 -10 E 10 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -14 -10 B 2 0 -14 -6 -14 C 6 14 0 10 -4 D 14 6 -10 0 -10 E 10 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9172: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (5) C E D A B (5) B D C A E (5) D E A C B (4) C B D E A (4) B C E A D (4) A E B C D (4) E C A D B (3) E A C D B (3) D C E A B (3) D B C A E (3) C E D B A (3) C B E D A (3) B A E C D (3) A B D E C (3) E A D C B (2) D C B E A (2) D B A C E (2) D A E C B (2) D A B E C (2) C D E A B (2) C D B E A (2) B E C A D (2) B D A C E (2) B C A D E (2) B A E D C (2) B A D E C (2) B A C E D (2) A D B E C (2) E D A C B (1) E C A B D (1) E A B C D (1) D E C A B (1) D B A E C (1) D A E B C (1) D A C E B (1) C E B A D (1) C D E B A (1) C B E A D (1) B C D A E (1) B A D C E (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 -12 -8 B 0 0 -2 -6 6 C 8 2 0 10 2 D 12 6 -10 0 0 E 8 -6 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -12 -8 B 0 0 -2 -6 6 C 8 2 0 10 2 D 12 6 -10 0 0 E 8 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 C=22 E=16 A=13 so A is eliminated. Round 2 votes counts: B=31 D=24 E=23 C=22 so C is eliminated. Round 3 votes counts: B=39 E=32 D=29 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 D:204 E:200 B:199 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 -12 -8 B 0 0 -2 -6 6 C 8 2 0 10 2 D 12 6 -10 0 0 E 8 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -12 -8 B 0 0 -2 -6 6 C 8 2 0 10 2 D 12 6 -10 0 0 E 8 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -12 -8 B 0 0 -2 -6 6 C 8 2 0 10 2 D 12 6 -10 0 0 E 8 -6 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998081 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9173: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (9) E A C D B (6) D C B A E (6) C B D E A (6) C D B E A (4) B D C A E (4) A E D B C (4) A E B D C (4) E A C B D (3) E A B D C (3) E A B C D (3) B E C A D (3) B D A C E (3) B C E D A (3) E C A D B (2) E C A B D (2) E A D C B (2) D B C A E (2) D A C B E (2) C E B D A (2) C B E D A (2) B E A C D (2) B C D A E (2) B A E C D (2) B A D C E (2) A D B E C (2) A B D E C (2) E B C A D (1) D E A C B (1) D C B E A (1) C E D B A (1) C D E A B (1) C D B A E (1) B C E A D (1) B A C E D (1) A E D C B (1) A E B C D (1) A D E C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -6 0 -16 B 16 0 12 24 22 C 6 -12 0 16 6 D 0 -24 -16 0 0 E 16 -22 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -6 0 -16 B 16 0 12 24 22 C 6 -12 0 16 6 D 0 -24 -16 0 0 E 16 -22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=22 C=17 A=17 D=12 so D is eliminated. Round 2 votes counts: B=34 C=24 E=23 A=19 so A is eliminated. Round 3 votes counts: B=39 E=35 C=26 so C is eliminated. Round 4 votes counts: B=61 E=39 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:237 C:208 E:194 A:181 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -6 0 -16 B 16 0 12 24 22 C 6 -12 0 16 6 D 0 -24 -16 0 0 E 16 -22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -6 0 -16 B 16 0 12 24 22 C 6 -12 0 16 6 D 0 -24 -16 0 0 E 16 -22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -6 0 -16 B 16 0 12 24 22 C 6 -12 0 16 6 D 0 -24 -16 0 0 E 16 -22 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9174: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (8) E A B C D (6) D B E A C (6) C A E D B (6) A C E B D (6) B D E A C (4) B A E C D (4) A E C B D (4) D E A C B (3) D E A B C (3) C A E B D (3) C A D E B (3) C A B E D (3) E D B A C (2) E D A B C (2) D E B A C (2) D C E A B (2) D C B A E (2) D C A E B (2) C D B A E (2) C D A B E (2) C B A E D (2) C B A D E (2) B D C A E (2) B C A D E (2) B A C E D (2) A B C E D (2) E B A D C (1) E A B D C (1) D C B E A (1) D B E C A (1) D B C E A (1) C D A E B (1) C B D A E (1) B E D A C (1) B E A D C (1) B E A C D (1) B D C E A (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 6 18 22 B -2 0 8 18 4 C -6 -8 0 24 14 D -18 -18 -24 0 -14 E -22 -4 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999298 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 18 22 B -2 0 8 18 4 C -6 -8 0 24 14 D -18 -18 -24 0 -14 E -22 -4 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 C=25 D=23 A=14 E=12 so E is eliminated. Round 2 votes counts: D=27 B=27 C=25 A=21 so A is eliminated. Round 3 votes counts: B=37 C=36 D=27 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:224 B:214 C:212 E:187 D:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 18 22 B -2 0 8 18 4 C -6 -8 0 24 14 D -18 -18 -24 0 -14 E -22 -4 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 18 22 B -2 0 8 18 4 C -6 -8 0 24 14 D -18 -18 -24 0 -14 E -22 -4 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 18 22 B -2 0 8 18 4 C -6 -8 0 24 14 D -18 -18 -24 0 -14 E -22 -4 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999980306 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9175: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (9) A E C D B (8) E C A D B (6) D B A E C (6) D A B E C (6) B D A C E (6) C E A B D (5) E C A B D (4) C E B A D (4) B D C A E (4) B C E D A (4) E A C D B (3) D B A C E (3) A D B E C (3) E C B D A (2) D A E B C (2) C B E A D (2) B C D E A (2) B C A D E (2) A E D C B (2) A D E C B (2) A D E B C (2) E C D B A (1) E A D C B (1) D E B C A (1) D B E C A (1) D B E A C (1) C E B D A (1) C B E D A (1) C A E B D (1) B D E C A (1) B C D A E (1) A C E D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -4 2 B 4 0 10 2 8 C 4 -10 0 0 -4 D 4 -2 0 0 8 E -2 -8 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999413 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -4 2 B 4 0 10 2 8 C 4 -10 0 0 -4 D 4 -2 0 0 8 E -2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=20 A=20 E=17 C=14 so C is eliminated. Round 2 votes counts: B=32 E=27 A=21 D=20 so D is eliminated. Round 3 votes counts: B=43 A=29 E=28 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:205 A:195 C:195 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -4 2 B 4 0 10 2 8 C 4 -10 0 0 -4 D 4 -2 0 0 8 E -2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -4 2 B 4 0 10 2 8 C 4 -10 0 0 -4 D 4 -2 0 0 8 E -2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -4 2 B 4 0 10 2 8 C 4 -10 0 0 -4 D 4 -2 0 0 8 E -2 -8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9176: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (11) D E C B A (7) B E C A D (7) A B C E D (7) A C B E D (5) D E B C A (4) D A E B C (4) B C E A D (4) A D C B E (4) E B C D A (3) D C A E B (3) A D B E C (3) A C B D E (3) A B E D C (3) E C B D A (2) E B D C A (2) D E A B C (2) C E B D A (2) C B E A D (2) C A B E D (2) A D C E B (2) A D B C E (2) A B E C D (2) E D C B A (1) E B D A C (1) D E C A B (1) D E B A C (1) D C E B A (1) D C E A B (1) D A E C B (1) C E D B A (1) C D E B A (1) C A D E B (1) B C A E D (1) A D E B C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 8 2 14 B -20 0 -2 -4 -6 C -8 2 0 -12 10 D -2 4 12 0 10 E -14 6 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999161 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 8 2 14 B -20 0 -2 -4 -6 C -8 2 0 -12 10 D -2 4 12 0 10 E -14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=34 B=12 E=9 C=9 so E is eliminated. Round 2 votes counts: D=37 A=34 B=18 C=11 so C is eliminated. Round 3 votes counts: D=39 A=37 B=24 so B is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:222 D:212 C:196 E:186 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 8 2 14 B -20 0 -2 -4 -6 C -8 2 0 -12 10 D -2 4 12 0 10 E -14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 8 2 14 B -20 0 -2 -4 -6 C -8 2 0 -12 10 D -2 4 12 0 10 E -14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 8 2 14 B -20 0 -2 -4 -6 C -8 2 0 -12 10 D -2 4 12 0 10 E -14 6 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996167 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9177: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (7) A E C D B (6) B C D A E (5) A E B D C (5) A B E D C (5) A B E C D (5) E A D C B (4) C D E A B (4) D C E B A (3) D C B E A (3) B A E D C (3) B A C D E (3) A E C B D (3) A E B C D (3) E C D A B (2) E A C D B (2) D E C A B (2) C E D A B (2) B D E A C (2) B D C A E (2) B C A D E (2) B A D C E (2) A B C E D (2) E D C A B (1) E D A C B (1) E D A B C (1) E C A D B (1) E A B D C (1) D E C B A (1) D C E A B (1) D B C E A (1) C D E B A (1) C D B E A (1) C D A E B (1) C B D E A (1) C B D A E (1) C B A D E (1) C A E D B (1) B D E C A (1) B D A E C (1) B D A C E (1) B C D E A (1) B A D E C (1) B A C E D (1) A E D B C (1) A C E B D (1) Total count = 100 A B C D E A 0 10 8 6 12 B -10 0 12 20 4 C -8 -12 0 0 -4 D -6 -20 0 0 -2 E -12 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 6 12 B -10 0 12 20 4 C -8 -12 0 0 -4 D -6 -20 0 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=31 E=13 C=13 D=11 so D is eliminated. Round 2 votes counts: B=33 A=31 C=20 E=16 so E is eliminated. Round 3 votes counts: A=40 B=33 C=27 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 B:213 E:195 C:188 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 6 12 B -10 0 12 20 4 C -8 -12 0 0 -4 D -6 -20 0 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 6 12 B -10 0 12 20 4 C -8 -12 0 0 -4 D -6 -20 0 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 6 12 B -10 0 12 20 4 C -8 -12 0 0 -4 D -6 -20 0 0 -2 E -12 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999535 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9178: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (7) E C A D B (5) D B E C A (5) C E D A B (5) B D A E C (5) B D A C E (5) A C E B D (5) E A C B D (4) D B C A E (4) C A E B D (4) A E C B D (4) A B E D C (4) A B E C D (4) E C D A B (3) D B C E A (3) A B D C E (3) D E B C A (2) D C E B A (2) C D B A E (2) C A E D B (2) B D E A C (2) B A D E C (2) A B D E C (2) A B C D E (2) E C D B A (1) E B D A C (1) E A C D B (1) D E C B A (1) D B E A C (1) D B A E C (1) C D E B A (1) C D B E A (1) C D A E B (1) C A B D E (1) B A E D C (1) A E B C D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 6 14 8 24 B -6 0 10 18 14 C -14 -10 0 -2 0 D -8 -18 2 0 8 E -24 -14 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 8 24 B -6 0 10 18 14 C -14 -10 0 -2 0 D -8 -18 2 0 8 E -24 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=22 D=19 C=17 E=15 so E is eliminated. Round 2 votes counts: A=32 C=26 B=23 D=19 so D is eliminated. Round 3 votes counts: B=39 A=32 C=29 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:226 B:218 D:192 C:187 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 8 24 B -6 0 10 18 14 C -14 -10 0 -2 0 D -8 -18 2 0 8 E -24 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 8 24 B -6 0 10 18 14 C -14 -10 0 -2 0 D -8 -18 2 0 8 E -24 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 8 24 B -6 0 10 18 14 C -14 -10 0 -2 0 D -8 -18 2 0 8 E -24 -14 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999344 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9179: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (16) B E C D A (13) B E C A D (8) D A C E B (7) D C A E B (5) B A C E D (5) E B C D A (4) B A E C D (4) D C E A B (3) C D A E B (3) E C D B A (2) E B D C A (2) D E C B A (2) D E C A B (2) D A E C B (2) B E A C D (2) A D C B E (2) A B C D E (2) E C B D A (1) D A E B C (1) C E D B A (1) C E D A B (1) C E B D A (1) C E B A D (1) C B E A D (1) C A E B D (1) B E D A C (1) B A E D C (1) B A D E C (1) A D B C E (1) A C D E B (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 -2 -2 10 B 0 0 -6 0 -12 C 2 6 0 4 6 D 2 0 -4 0 2 E -10 12 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -2 10 B 0 0 -6 0 -12 C 2 6 0 4 6 D 2 0 -4 0 2 E -10 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=25 D=22 E=9 C=9 so E is eliminated. Round 2 votes counts: B=41 A=25 D=22 C=12 so C is eliminated. Round 3 votes counts: B=45 D=29 A=26 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:209 A:203 D:200 E:197 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -2 -2 10 B 0 0 -6 0 -12 C 2 6 0 4 6 D 2 0 -4 0 2 E -10 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -2 10 B 0 0 -6 0 -12 C 2 6 0 4 6 D 2 0 -4 0 2 E -10 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -2 10 B 0 0 -6 0 -12 C 2 6 0 4 6 D 2 0 -4 0 2 E -10 12 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999442 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9180: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D C A E (6) B D C E A (5) B D A C E (5) B A E D C (5) A E C D B (5) A B E D C (5) E C D A B (4) E C A D B (4) E A B C D (4) C D E B A (4) B A D C E (4) E B C D A (3) D C B E A (3) C D E A B (3) C A D E B (3) B E A D C (3) A E B D C (3) D C B A E (2) C E D B A (2) B E C D A (2) E C B D A (1) E C B A D (1) E B C A D (1) E B A C D (1) D C A B E (1) D B C A E (1) C E D A B (1) C D A E B (1) B E D A C (1) B A D E C (1) A E B C D (1) A D C E B (1) A D C B E (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 4 10 -4 B 2 0 6 6 -6 C -4 -6 0 2 -8 D -10 -6 -2 0 -12 E 4 6 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 10 -4 B 2 0 6 6 -6 C -4 -6 0 2 -8 D -10 -6 -2 0 -12 E 4 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=28 A=19 C=14 D=7 so D is eliminated. Round 2 votes counts: B=33 E=28 C=20 A=19 so A is eliminated. Round 3 votes counts: B=40 E=37 C=23 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:204 B:204 C:192 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 10 -4 B 2 0 6 6 -6 C -4 -6 0 2 -8 D -10 -6 -2 0 -12 E 4 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 10 -4 B 2 0 6 6 -6 C -4 -6 0 2 -8 D -10 -6 -2 0 -12 E 4 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 10 -4 B 2 0 6 6 -6 C -4 -6 0 2 -8 D -10 -6 -2 0 -12 E 4 6 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9181: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (10) B C E D A (6) B C A D E (6) A D E B C (5) E D C A B (4) D A C E B (4) A D C E B (4) A D B C E (4) E D A C B (3) E A D C B (3) D A E C B (3) B E C A D (3) B A C D E (3) A D B E C (3) E C D A B (2) E C B D A (2) E B C D A (2) D E A C B (2) C E D B A (2) C D E A B (2) C B D E A (2) C B D A E (2) B E C D A (2) B E A D C (2) B A E C D (2) B A D C E (2) A E D B C (2) E A D B C (1) D C A E B (1) D A C B E (1) C E D A B (1) C E B D A (1) C D B E A (1) C D A E B (1) C B E D A (1) B C E A D (1) B C D E A (1) B A D E C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 16 14 8 12 B -16 0 -6 -20 -10 C -14 6 0 -14 -4 D -8 20 14 0 20 E -12 10 4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 8 12 B -16 0 -6 -20 -10 C -14 6 0 -14 -4 D -8 20 14 0 20 E -12 10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=29 E=17 C=13 D=11 so D is eliminated. Round 2 votes counts: A=38 B=29 E=19 C=14 so C is eliminated. Round 3 votes counts: A=40 B=35 E=25 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:223 E:191 C:187 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 8 12 B -16 0 -6 -20 -10 C -14 6 0 -14 -4 D -8 20 14 0 20 E -12 10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 8 12 B -16 0 -6 -20 -10 C -14 6 0 -14 -4 D -8 20 14 0 20 E -12 10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 8 12 B -16 0 -6 -20 -10 C -14 6 0 -14 -4 D -8 20 14 0 20 E -12 10 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9182: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (7) D B E C A (6) E C B A D (4) D E B C A (4) D B E A C (4) D B A E C (4) B A E C D (4) A C E B D (4) D C E A B (3) D B A C E (3) C E A B D (3) C A E D B (3) C A E B D (3) A D C B E (3) A C E D B (3) A C D E B (3) D E C B A (2) D E C A B (2) D C A E B (2) D A C E B (2) D A C B E (2) D A B C E (2) B E C A D (2) B D E C A (2) B D E A C (2) B A C E D (2) A B D C E (2) E D C A B (1) E D B C A (1) E C B D A (1) E C A B D (1) E B D C A (1) E B C D A (1) E B C A D (1) B E D C A (1) B E C D A (1) B E A D C (1) B E A C D (1) B A D E C (1) B A D C E (1) A D B C E (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -4 6 -4 B 0 0 -2 -16 -4 C 4 2 0 -6 4 D -6 16 6 0 4 E 4 4 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343749999988 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 6 -4 B 0 0 -2 -16 -4 C 4 2 0 -6 4 D -6 16 6 0 4 E 4 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 A=19 B=18 C=16 E=11 so E is eliminated. Round 2 votes counts: D=38 C=22 B=21 A=19 so A is eliminated. Round 3 votes counts: D=42 C=34 B=24 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:210 C:202 E:200 A:199 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -4 6 -4 B 0 0 -2 -16 -4 C 4 2 0 -6 4 D -6 16 6 0 4 E 4 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 6 -4 B 0 0 -2 -16 -4 C 4 2 0 -6 4 D -6 16 6 0 4 E 4 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 6 -4 B 0 0 -2 -16 -4 C 4 2 0 -6 4 D -6 16 6 0 4 E 4 4 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.250000 E: 0.000000 Sum of squares = 0.343750000001 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9183: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (7) E C B A D (6) C E B D A (6) E B C A D (5) D C A E B (5) D A B C E (5) B E D A C (4) B E C A D (4) B E A D C (4) B E A C D (4) A C E D B (4) C E B A D (3) C E A B D (3) C D E A B (3) C D A E B (3) C A D E B (3) E C A B D (2) C E A D B (2) B D E A C (2) B A E D C (2) B A D E C (2) A E C B D (2) A D C E B (2) A D B E C (2) E B C D A (1) E B A C D (1) E A B C D (1) D B C A E (1) D B A C E (1) D A C E B (1) C E D A B (1) C D B E A (1) C A E D B (1) B E C D A (1) B D E C A (1) B D A E C (1) A D C B E (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 0 -4 12 -10 B 0 0 -12 12 -10 C 4 12 0 14 8 D -12 -12 -14 0 -16 E 10 10 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 12 -10 B 0 0 -12 12 -10 C 4 12 0 14 8 D -12 -12 -14 0 -16 E 10 10 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 D=20 E=16 A=13 so A is eliminated. Round 2 votes counts: C=30 D=26 B=26 E=18 so E is eliminated. Round 3 votes counts: C=40 B=34 D=26 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:214 A:199 B:195 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 12 -10 B 0 0 -12 12 -10 C 4 12 0 14 8 D -12 -12 -14 0 -16 E 10 10 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 12 -10 B 0 0 -12 12 -10 C 4 12 0 14 8 D -12 -12 -14 0 -16 E 10 10 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 12 -10 B 0 0 -12 12 -10 C 4 12 0 14 8 D -12 -12 -14 0 -16 E 10 10 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9184: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) A E D B C (8) B D C A E (6) E C A D B (5) B C D A E (5) E A D C B (4) C E A D B (4) C E A B D (4) B C E A D (4) E C A B D (3) E A C B D (3) D B A E C (3) D A E C B (3) C B D E A (3) B D A C E (3) B A E D C (3) B A D E C (3) A D E B C (3) D B A C E (2) D A E B C (2) D A B E C (2) C D B E A (2) C B E A D (2) B A E C D (2) D E A C B (1) D C B A E (1) D C A B E (1) C E D B A (1) C E D A B (1) C E B A D (1) C D E B A (1) C B E D A (1) C B D A E (1) B D A E C (1) B A C E D (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 8 20 4 B -8 0 0 -4 -6 C -8 0 0 4 -12 D -20 4 -4 0 -12 E -4 6 12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 20 4 B -8 0 0 -4 -6 C -8 0 0 4 -12 D -20 4 -4 0 -12 E -4 6 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=23 C=21 D=15 A=13 so A is eliminated. Round 2 votes counts: E=32 B=29 C=21 D=18 so D is eliminated. Round 3 votes counts: E=41 B=36 C=23 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:220 E:213 C:192 B:191 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 20 4 B -8 0 0 -4 -6 C -8 0 0 4 -12 D -20 4 -4 0 -12 E -4 6 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 20 4 B -8 0 0 -4 -6 C -8 0 0 4 -12 D -20 4 -4 0 -12 E -4 6 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 20 4 B -8 0 0 -4 -6 C -8 0 0 4 -12 D -20 4 -4 0 -12 E -4 6 12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9185: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) E C D B A (6) D B E C A (6) B D A C E (6) A E C D B (6) A B D C E (6) C E A B D (5) B D C E A (5) E C A D B (4) E A C D B (4) A D B E C (4) D E C B A (3) B D C A E (3) A E D C B (3) A E C B D (3) A D E B C (3) A C E B D (3) A B C E D (3) E C A B D (2) D B C E A (2) D B A E C (2) D B A C E (2) D A B E C (2) C E B A D (2) E C D A B (1) C B E D A (1) C B E A D (1) B C D E A (1) B A D C E (1) A D E C B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 2 6 0 B -4 0 -6 2 -6 C -2 6 0 0 -2 D -6 -2 0 0 -4 E 0 6 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.526257 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.473743 Sum of squares = 0.501378890671 Cumulative probabilities = A: 0.526257 B: 0.526257 C: 0.526257 D: 0.526257 E: 1.000000 A B C D E A 0 4 2 6 0 B -4 0 -6 2 -6 C -2 6 0 0 -2 D -6 -2 0 0 -4 E 0 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 E=17 D=17 C=16 B=16 so C is eliminated. Round 2 votes counts: A=34 E=31 B=18 D=17 so D is eliminated. Round 3 votes counts: A=36 E=34 B=30 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:206 C:201 D:194 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 6 0 B -4 0 -6 2 -6 C -2 6 0 0 -2 D -6 -2 0 0 -4 E 0 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 6 0 B -4 0 -6 2 -6 C -2 6 0 0 -2 D -6 -2 0 0 -4 E 0 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 6 0 B -4 0 -6 2 -6 C -2 6 0 0 -2 D -6 -2 0 0 -4 E 0 6 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9186: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (8) E D C B A (7) D E C A B (7) D C E B A (7) B A C E D (6) A B E C D (6) E D A B C (5) C D E B A (5) A B C E D (5) A B E D C (4) A B C D E (4) E A B D C (3) C B A D E (3) B A E C D (3) A B D C E (3) E D B C A (2) D E A C B (2) C B D E A (2) B C A E D (2) A E B D C (2) E C B D A (1) E B C D A (1) E B A D C (1) E A D B C (1) D E A B C (1) D A E C B (1) C E D B A (1) C D B E A (1) C D B A E (1) C B E D A (1) B E C A D (1) B C A D E (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 -2 -6 -14 B 8 0 2 -2 -10 C 2 -2 0 -10 -12 D 6 2 10 0 -4 E 14 10 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999902 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -2 -6 -14 B 8 0 2 -2 -10 C 2 -2 0 -10 -12 D 6 2 10 0 -4 E 14 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 E=21 C=14 B=13 so B is eliminated. Round 2 votes counts: A=35 D=26 E=22 C=17 so C is eliminated. Round 3 votes counts: A=41 D=35 E=24 so E is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:207 B:199 C:189 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -2 -6 -14 B 8 0 2 -2 -10 C 2 -2 0 -10 -12 D 6 2 10 0 -4 E 14 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -6 -14 B 8 0 2 -2 -10 C 2 -2 0 -10 -12 D 6 2 10 0 -4 E 14 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -6 -14 B 8 0 2 -2 -10 C 2 -2 0 -10 -12 D 6 2 10 0 -4 E 14 10 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999233 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9187: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) D C A E B (6) B A D E C (6) C D E A B (5) B E C A D (5) B E A C D (5) E C A D B (4) C E D A B (4) C D A E B (4) B C E D A (4) E B C A D (3) D A C B E (3) B E A D C (3) A B E D C (3) E C B A D (2) E C A B D (2) D A C E B (2) C D B E A (2) C D B A E (2) C B E D A (2) B D A C E (2) B C D A E (2) E B A D C (1) E B A C D (1) E A D C B (1) E A B C D (1) D C A B E (1) D B A C E (1) C E D B A (1) C E B D A (1) C D E B A (1) C B D E A (1) B D C A E (1) B C E A D (1) B C D E A (1) B A D C E (1) A E D C B (1) A E B D C (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 -10 8 -2 B 18 0 10 20 16 C 10 -10 0 8 -4 D -8 -20 -8 0 -12 E 2 -16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 8 -2 B 18 0 10 20 16 C 10 -10 0 8 -4 D -8 -20 -8 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=23 E=15 D=13 A=8 so A is eliminated. Round 2 votes counts: B=45 C=23 E=17 D=15 so D is eliminated. Round 3 votes counts: B=47 C=35 E=18 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 C:202 E:201 A:189 D:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -10 8 -2 B 18 0 10 20 16 C 10 -10 0 8 -4 D -8 -20 -8 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 8 -2 B 18 0 10 20 16 C 10 -10 0 8 -4 D -8 -20 -8 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 8 -2 B 18 0 10 20 16 C 10 -10 0 8 -4 D -8 -20 -8 0 -12 E 2 -16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9188: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (13) B E D C A (9) E B C D A (6) C A D E B (5) E B D C A (4) C E A B D (4) A D C B E (4) A C E D B (4) A C D B E (4) E C B D A (3) D C B A E (3) D A C B E (3) B E D A C (3) B D E C A (3) B D E A C (3) A D B C E (3) E C A B D (2) E B C A D (2) E A C B D (2) D C A B E (2) D B A C E (2) C A E D B (2) B D A E C (2) E C B A D (1) E B A C D (1) D B E C A (1) D B C E A (1) D B C A E (1) D B A E C (1) D A B C E (1) C E A D B (1) C A D B E (1) B E A D C (1) B A E D C (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -2 4 6 B -4 0 -10 -4 -2 C 2 10 0 4 8 D -4 4 -4 0 6 E -6 2 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 4 6 B -4 0 -10 -4 -2 C 2 10 0 4 8 D -4 4 -4 0 6 E -6 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=22 E=21 D=15 C=13 so C is eliminated. Round 2 votes counts: A=37 E=26 B=22 D=15 so D is eliminated. Round 3 votes counts: A=43 B=31 E=26 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:212 A:206 D:201 E:191 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 4 6 B -4 0 -10 -4 -2 C 2 10 0 4 8 D -4 4 -4 0 6 E -6 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 4 6 B -4 0 -10 -4 -2 C 2 10 0 4 8 D -4 4 -4 0 6 E -6 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 4 6 B -4 0 -10 -4 -2 C 2 10 0 4 8 D -4 4 -4 0 6 E -6 2 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999348 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9189: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) E C B D A (6) E B D C A (6) C E A B D (6) B E D C A (6) C E A D B (5) B D A E C (5) A D B C E (5) D A B C E (4) A D C B E (4) A C B D E (4) E D B C A (3) E C D A B (3) D B E A C (3) C A E B D (3) B D E A C (3) E C D B A (2) E C B A D (2) D E B A C (2) D A B E C (2) C E B A D (2) B A D C E (2) A C D B E (2) E D C B A (1) E D C A B (1) E C A D B (1) E B C D A (1) D A C B E (1) C E D A B (1) C D E A B (1) C A B E D (1) B E C A D (1) A D C E B (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -4 -16 -12 B 4 0 6 2 6 C 4 -6 0 -14 -10 D 16 -2 14 0 -4 E 12 -6 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -16 -12 B 4 0 6 2 6 C 4 -6 0 -14 -10 D 16 -2 14 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 D=19 C=19 A=19 B=17 so B is eliminated. Round 2 votes counts: E=33 D=27 A=21 C=19 so C is eliminated. Round 3 votes counts: E=47 D=28 A=25 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:212 E:210 B:209 C:187 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -16 -12 B 4 0 6 2 6 C 4 -6 0 -14 -10 D 16 -2 14 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -16 -12 B 4 0 6 2 6 C 4 -6 0 -14 -10 D 16 -2 14 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -16 -12 B 4 0 6 2 6 C 4 -6 0 -14 -10 D 16 -2 14 0 -4 E 12 -6 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996107 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9190: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (15) C B A E D (14) B C D E A (13) D E A B C (12) D E B C A (8) E A D C B (5) B C A D E (4) E D A B C (3) B C D A E (3) A C B E D (3) E D B C A (2) D B C A E (2) D A E B C (2) B D C E A (2) A E C B D (2) A C E B D (2) D A B E C (1) C B E D A (1) C B E A D (1) C B A D E (1) C A B E D (1) B D C A E (1) A D E C B (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -6 0 6 B 4 0 8 -4 -6 C 6 -8 0 -10 -4 D 0 4 10 0 2 E -6 6 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.343323 B: 0.000000 C: 0.000000 D: 0.656677 E: 0.000000 Sum of squares = 0.549095647083 Cumulative probabilities = A: 0.343323 B: 0.343323 C: 0.343323 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 0 6 B 4 0 8 -4 -6 C 6 -8 0 -10 -4 D 0 4 10 0 2 E -6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.000000 D: 0.500003 E: 0.000000 Sum of squares = 0.500000000018 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 0.499997 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=24 B=23 C=18 E=10 so E is eliminated. Round 2 votes counts: D=30 A=29 B=23 C=18 so C is eliminated. Round 3 votes counts: B=40 D=30 A=30 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:208 B:201 E:201 A:198 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 0 6 B 4 0 8 -4 -6 C 6 -8 0 -10 -4 D 0 4 10 0 2 E -6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.000000 D: 0.500003 E: 0.000000 Sum of squares = 0.500000000018 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 0.499997 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 6 B 4 0 8 -4 -6 C 6 -8 0 -10 -4 D 0 4 10 0 2 E -6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.000000 D: 0.500003 E: 0.000000 Sum of squares = 0.500000000018 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 0.499997 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 6 B 4 0 8 -4 -6 C 6 -8 0 -10 -4 D 0 4 10 0 2 E -6 6 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499997 B: 0.000000 C: 0.000000 D: 0.500003 E: 0.000000 Sum of squares = 0.500000000018 Cumulative probabilities = A: 0.499997 B: 0.499997 C: 0.499997 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9191: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) E A B D C (6) D A E B C (6) E B A D C (5) C B D A E (5) B E C A D (5) B C E D A (5) E A D B C (4) D B A E C (4) D A E C B (4) D A C E B (4) C B E A D (4) B C E A D (4) A D E C B (4) A D E B C (4) C D B A E (3) C B A E D (3) C B D E A (2) B C D E A (2) A E D B C (2) A D C E B (2) E B C A D (1) E B A C D (1) E A C B D (1) E A B C D (1) D E A B C (1) D C A E B (1) C D A E B (1) C A D E B (1) C A B D E (1) B E C D A (1) B E A D C (1) B D E C A (1) B D C E A (1) A E D C B (1) A C E D B (1) Total count = 100 A B C D E A 0 4 4 4 8 B -4 0 10 0 -2 C -4 -10 0 -2 -6 D -4 0 2 0 8 E -8 2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 4 8 B -4 0 10 0 -2 C -4 -10 0 -2 -6 D -4 0 2 0 8 E -8 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=20 B=20 E=19 A=14 so A is eliminated. Round 2 votes counts: D=30 C=28 E=22 B=20 so B is eliminated. Round 3 votes counts: C=39 D=32 E=29 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:210 D:203 B:202 E:196 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 4 8 B -4 0 10 0 -2 C -4 -10 0 -2 -6 D -4 0 2 0 8 E -8 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 4 8 B -4 0 10 0 -2 C -4 -10 0 -2 -6 D -4 0 2 0 8 E -8 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 4 8 B -4 0 10 0 -2 C -4 -10 0 -2 -6 D -4 0 2 0 8 E -8 2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9192: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) A E B C D (9) B C D A E (8) E D C B A (7) E A B C D (7) A B C D E (7) E D C A B (6) C B D A E (6) B A C D E (6) E D A C B (5) E A D C B (4) D C B A E (4) B C A D E (4) A B C E D (3) E A B D C (2) D E C B A (2) C D B A E (2) E C D B A (1) D C E B A (1) D B C A E (1) C D B E A (1) C B D E A (1) A E D B C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -8 -10 6 B 8 0 0 8 10 C 8 0 0 12 10 D 10 -8 -12 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.603803 C: 0.396197 D: 0.000000 E: 0.000000 Sum of squares = 0.521550162964 Cumulative probabilities = A: 0.000000 B: 0.603803 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -10 6 B 8 0 0 8 10 C 8 0 0 12 10 D 10 -8 -12 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=22 D=18 B=18 C=10 so C is eliminated. Round 2 votes counts: E=32 B=25 A=22 D=21 so D is eliminated. Round 3 votes counts: B=43 E=35 A=22 so A is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:213 D:199 A:190 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -10 6 B 8 0 0 8 10 C 8 0 0 12 10 D 10 -8 -12 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -10 6 B 8 0 0 8 10 C 8 0 0 12 10 D 10 -8 -12 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -10 6 B 8 0 0 8 10 C 8 0 0 12 10 D 10 -8 -12 0 8 E -6 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9193: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (6) E C D B A (5) E C A D B (5) D C B E A (5) D A B C E (5) D C E B A (4) C E D B A (4) C E B A D (4) B A D C E (4) B A C E D (4) A E B C D (4) E C B D A (3) D E C A B (3) D E A C B (3) B A C D E (3) E C D A B (2) E C A B D (2) D C E A B (2) D B C A E (2) D A E C B (2) B C E A D (2) B C D A E (2) A E D C B (2) A D B E C (2) A B E C D (2) A B D E C (2) E D C B A (1) E D C A B (1) E A C D B (1) E A C B D (1) D B A C E (1) C E B D A (1) C D E B A (1) C B E D A (1) B D C E A (1) B D C A E (1) B C A D E (1) A E C D B (1) A D E C B (1) A D E B C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -6 0 -4 B 0 0 -10 -10 -8 C 6 10 0 0 12 D 0 10 0 0 6 E 4 8 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.480632 D: 0.519368 E: 0.000000 Sum of squares = 0.500750235662 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.480632 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 0 -4 B 0 0 -10 -10 -8 C 6 10 0 0 12 D 0 10 0 0 6 E 4 8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 E=21 B=18 C=11 so C is eliminated. Round 2 votes counts: E=30 D=28 A=23 B=19 so B is eliminated. Round 3 votes counts: A=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:214 D:208 E:197 A:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 0 -4 B 0 0 -10 -10 -8 C 6 10 0 0 12 D 0 10 0 0 6 E 4 8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 0 -4 B 0 0 -10 -10 -8 C 6 10 0 0 12 D 0 10 0 0 6 E 4 8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 0 -4 B 0 0 -10 -10 -8 C 6 10 0 0 12 D 0 10 0 0 6 E 4 8 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9194: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (17) E C A B D (13) E C A D B (9) B D A C E (7) D B C A E (6) C E A B D (5) E C D A B (4) E A C B D (4) D E B C A (4) E A B C D (2) D E C B A (2) D B E C A (2) D B C E A (2) D B A E C (2) C D B A E (2) C A E B D (2) B A C D E (2) A E C B D (2) A E B C D (2) A C E B D (2) A B C E D (2) E D C B A (1) E D C A B (1) E A B D C (1) D E B A C (1) D B E A C (1) B D A E C (1) B A D E C (1) Total count = 100 A B C D E A 0 -2 -6 -6 -4 B 2 0 6 -8 -10 C 6 -6 0 2 -6 D 6 8 -2 0 0 E 4 10 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.175631 E: 0.824369 Sum of squares = 0.710430172101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.175631 E: 1.000000 A B C D E A 0 -2 -6 -6 -4 B 2 0 6 -8 -10 C 6 -6 0 2 -6 D 6 8 -2 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 E=35 B=11 C=9 A=8 so A is eliminated. Round 2 votes counts: E=39 D=37 B=13 C=11 so C is eliminated. Round 3 votes counts: E=48 D=39 B=13 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:210 D:206 C:198 B:195 A:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -6 -6 -4 B 2 0 6 -8 -10 C 6 -6 0 2 -6 D 6 8 -2 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -6 -4 B 2 0 6 -8 -10 C 6 -6 0 2 -6 D 6 8 -2 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -6 -4 B 2 0 6 -8 -10 C 6 -6 0 2 -6 D 6 8 -2 0 0 E 4 10 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9195: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (9) B A D C E (7) D B A C E (5) A E D C B (5) A B D C E (5) E C D B A (4) E C A B D (4) A E C B D (4) E C A D B (3) D C E B A (3) D C B E A (3) B D C A E (3) B D A C E (3) A E C D B (3) A B E C D (3) A B D E C (3) A B C E D (3) E C D A B (2) E C B D A (2) D E C B A (2) D E C A B (2) D E A C B (2) D A B E C (2) B C E D A (2) A D B E C (2) E A C D B (1) D B A E C (1) D A E C B (1) D A E B C (1) C E D B A (1) C E B D A (1) C B E A D (1) B D C E A (1) B C E A D (1) B C D A E (1) B A C D E (1) A D E C B (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 10 -2 12 B 2 0 10 -8 14 C -10 -10 0 -24 2 D 2 8 24 0 20 E -12 -14 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997056 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -2 12 B 2 0 10 -8 14 C -10 -10 0 -24 2 D 2 8 24 0 20 E -12 -14 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990448 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=31 A=31 B=19 E=16 C=3 so C is eliminated. Round 2 votes counts: D=31 A=31 B=20 E=18 so E is eliminated. Round 3 votes counts: A=39 D=38 B=23 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:209 B:209 C:179 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -2 12 B 2 0 10 -8 14 C -10 -10 0 -24 2 D 2 8 24 0 20 E -12 -14 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990448 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -2 12 B 2 0 10 -8 14 C -10 -10 0 -24 2 D 2 8 24 0 20 E -12 -14 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990448 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -2 12 B 2 0 10 -8 14 C -10 -10 0 -24 2 D 2 8 24 0 20 E -12 -14 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990448 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9196: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (14) A C B D E (12) E D C B A (10) B D A C E (6) A B C D E (6) C A E D B (5) C A B D E (4) E C A D B (3) E B D A C (3) D E B C A (3) B D E A C (3) B D A E C (3) B A D C E (3) B A C D E (3) E D C A B (2) E D B A C (2) E B A D C (2) A C B E D (2) E C D A B (1) E A B C D (1) D E C B A (1) D E B A C (1) D B E A C (1) C D B A E (1) C A E B D (1) C A D E B (1) C A D B E (1) B E A D C (1) B A E D C (1) A E B C D (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 6 -2 4 B 16 0 10 8 -4 C -6 -10 0 -12 -6 D 2 -8 12 0 -2 E -4 4 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.500000000003 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 A B C D E A 0 -16 6 -2 4 B 16 0 10 8 -4 C -6 -10 0 -12 -6 D 2 -8 12 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 A=23 B=20 C=13 D=6 so D is eliminated. Round 2 votes counts: E=43 A=23 B=21 C=13 so C is eliminated. Round 3 votes counts: E=43 A=35 B=22 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:215 E:204 D:202 A:196 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 6 -2 4 B 16 0 10 8 -4 C -6 -10 0 -12 -6 D 2 -8 12 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 6 -2 4 B 16 0 10 8 -4 C -6 -10 0 -12 -6 D 2 -8 12 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 6 -2 4 B 16 0 10 8 -4 C -6 -10 0 -12 -6 D 2 -8 12 0 -2 E -4 4 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.499999999638 Cumulative probabilities = A: 0.166667 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9197: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (12) A B D E C (11) E D C A B (7) C E D B A (7) B A C E D (7) B C E D A (6) B A D C E (4) D E C A B (3) D C E A B (3) D A E C B (3) C E D A B (3) A B E D C (3) C E B D A (2) C B E D A (2) B C D A E (2) B A E D C (2) B A E C D (2) B A D E C (2) B A C D E (2) A E D B C (2) A D B E C (2) A B D C E (2) E D A C B (1) E C D B A (1) D E A C B (1) C B D E A (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D C B (1) A D E C B (1) A D E B C (1) A D C B E (1) Total count = 100 A B C D E A 0 14 -4 -10 0 B -14 0 2 2 4 C 4 -2 0 0 -10 D 10 -2 0 0 -18 E 0 -4 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.447487 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.552513 Sum of squares = 0.505515269083 Cumulative probabilities = A: 0.447487 B: 0.447487 C: 0.447487 D: 0.447487 E: 1.000000 A B C D E A 0 14 -4 -10 0 B -14 0 2 2 4 C 4 -2 0 0 -10 D 10 -2 0 0 -18 E 0 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=24 E=21 C=15 D=10 so D is eliminated. Round 2 votes counts: B=30 A=27 E=25 C=18 so C is eliminated. Round 3 votes counts: E=40 B=33 A=27 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:212 A:200 B:197 C:196 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 -10 0 B -14 0 2 2 4 C 4 -2 0 0 -10 D 10 -2 0 0 -18 E 0 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -10 0 B -14 0 2 2 4 C 4 -2 0 0 -10 D 10 -2 0 0 -18 E 0 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -10 0 B -14 0 2 2 4 C 4 -2 0 0 -10 D 10 -2 0 0 -18 E 0 -4 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9198: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) D C B A E (8) D C A E B (8) B C D A E (8) A E C D B (6) E A D C B (5) D E C A B (5) B E D C A (4) E D A C B (3) E A B D C (3) D C B E A (3) A C D E B (3) A C D B E (3) E B D C A (2) D C E A B (2) C D B A E (2) C D A B E (2) C A D B E (2) B C A D E (2) B A C D E (2) E D C A B (1) E B D A C (1) E B A D C (1) E B A C D (1) E A C D B (1) D C A B E (1) D B C A E (1) D A E C B (1) C D A E B (1) C A B D E (1) B E A C D (1) B D C E A (1) B D C A E (1) B C E A D (1) B A E C D (1) B A C E D (1) A E C B D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 18 -12 -10 14 B -18 0 -20 -18 -8 C 12 20 0 -2 10 D 10 18 2 0 14 E -14 8 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999742 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 -10 14 B -18 0 -20 -18 -8 C 12 20 0 -2 10 D 10 18 2 0 14 E -14 8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=26 B=22 A=15 C=8 so C is eliminated. Round 2 votes counts: D=34 E=26 B=22 A=18 so A is eliminated. Round 3 votes counts: D=42 E=34 B=24 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:220 A:205 E:185 B:168 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -12 -10 14 B -18 0 -20 -18 -8 C 12 20 0 -2 10 D 10 18 2 0 14 E -14 8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 -10 14 B -18 0 -20 -18 -8 C 12 20 0 -2 10 D 10 18 2 0 14 E -14 8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 -10 14 B -18 0 -20 -18 -8 C 12 20 0 -2 10 D 10 18 2 0 14 E -14 8 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996769 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9199: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (9) D B C E A (7) E C D B A (5) E A B D C (5) C E D B A (5) C E A D B (5) B D A E C (5) E C A B D (4) C D B E A (4) A E C B D (4) A B D E C (4) E C A D B (3) E A C B D (3) D E C B A (3) A C E B D (3) E D C B A (2) E D B C A (2) E B D A C (2) D B A C E (2) C E D A B (2) B D A C E (2) B A D E C (2) A E B D C (2) A B C D E (2) D C B A E (1) D B E C A (1) D B E A C (1) D B C A E (1) D B A E C (1) C E A B D (1) C D E B A (1) C D B A E (1) C A E D B (1) C A D B E (1) B A D C E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 0 4 -12 B -2 0 0 2 -6 C 0 0 0 -6 0 D -4 -2 6 0 0 E 12 6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.477641 E: 0.522359 Sum of squares = 0.500999795699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.477641 E: 1.000000 A B C D E A 0 2 0 4 -12 B -2 0 0 2 -6 C 0 0 0 -6 0 D -4 -2 6 0 0 E 12 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 A=26 C=21 D=17 B=10 so B is eliminated. Round 2 votes counts: A=29 E=26 D=24 C=21 so C is eliminated. Round 3 votes counts: E=39 A=31 D=30 so D is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:209 D:200 A:197 B:197 C:197 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 4 -12 B -2 0 0 2 -6 C 0 0 0 -6 0 D -4 -2 6 0 0 E 12 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 4 -12 B -2 0 0 2 -6 C 0 0 0 -6 0 D -4 -2 6 0 0 E 12 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 4 -12 B -2 0 0 2 -6 C 0 0 0 -6 0 D -4 -2 6 0 0 E 12 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9200: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (15) D E A B C (10) E D C A B (8) C B A E D (7) E C B A D (6) D E C B A (5) C A B E D (5) A B C D E (5) D E B A C (4) D B A E C (4) D B A C E (4) D A B C E (4) E D C B A (3) E C A B D (3) D E B C A (3) C E A B D (2) B A D C E (2) A B C E D (2) E D A C B (1) E C A D B (1) D E A C B (1) D B C A E (1) C B E A D (1) B D A C E (1) B C A D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 8 2 4 B 14 0 12 2 6 C -8 -12 0 -2 2 D -2 -2 2 0 20 E -4 -6 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 8 2 4 B 14 0 12 2 6 C -8 -12 0 -2 2 D -2 -2 2 0 20 E -4 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 E=22 B=19 C=15 A=8 so A is eliminated. Round 2 votes counts: D=36 B=26 E=22 C=16 so C is eliminated. Round 3 votes counts: B=40 D=36 E=24 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:209 A:200 C:190 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 8 2 4 B 14 0 12 2 6 C -8 -12 0 -2 2 D -2 -2 2 0 20 E -4 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 2 4 B 14 0 12 2 6 C -8 -12 0 -2 2 D -2 -2 2 0 20 E -4 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 2 4 B 14 0 12 2 6 C -8 -12 0 -2 2 D -2 -2 2 0 20 E -4 -6 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996487 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9201: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (11) D C E B A (6) B E A D C (6) B A E D C (6) B A E C D (5) E A C D B (4) B E D A C (4) A E B C D (4) D E C B A (3) D C E A B (3) C D A E B (3) C D A B E (3) C A D E B (3) B A C D E (3) A B E C D (3) E D B C A (2) E B A D C (2) E A B D C (2) D C B E A (2) D B C E A (2) B D E C A (2) B D C A E (2) B C D A E (2) E D C B A (1) E D C A B (1) E D B A C (1) E D A C B (1) E B D A C (1) D E C A B (1) C D B E A (1) C D B A E (1) C A D B E (1) C A B D E (1) B D C E A (1) B D A C E (1) B C A D E (1) B A C E D (1) A E C D B (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -6 -10 -14 B 12 0 2 -2 -2 C 6 -2 0 0 -2 D 10 2 0 0 6 E 14 2 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.252482 D: 0.747518 E: 0.000000 Sum of squares = 0.622529830897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.252482 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -10 -14 B 12 0 2 -2 -2 C 6 -2 0 0 -2 D 10 2 0 0 6 E 14 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499738 D: 0.500262 E: 0.000000 Sum of squares = 0.500000137267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499738 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=24 D=17 E=15 A=10 so A is eliminated. Round 2 votes counts: B=37 C=25 E=21 D=17 so D is eliminated. Round 3 votes counts: B=39 C=36 E=25 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:209 E:206 B:205 C:201 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -6 -10 -14 B 12 0 2 -2 -2 C 6 -2 0 0 -2 D 10 2 0 0 6 E 14 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499738 D: 0.500262 E: 0.000000 Sum of squares = 0.500000137267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499738 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -10 -14 B 12 0 2 -2 -2 C 6 -2 0 0 -2 D 10 2 0 0 6 E 14 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499738 D: 0.500262 E: 0.000000 Sum of squares = 0.500000137267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499738 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -10 -14 B 12 0 2 -2 -2 C 6 -2 0 0 -2 D 10 2 0 0 6 E 14 2 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499738 D: 0.500262 E: 0.000000 Sum of squares = 0.500000137267 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499738 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9202: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) C A E D B (7) E B A D C (6) E A B C D (6) A E C B D (6) E A C D B (5) C D A B E (5) D B C E A (4) C A D B E (4) B D E A C (4) D C B E A (3) D B E C A (3) C A D E B (3) B E A D C (3) A C E D B (3) A C B E D (3) E D C A B (2) E B D A C (2) D C B A E (2) D B C A E (2) C D B A E (2) B E D A C (2) B D C A E (2) B A E D C (2) E D B A C (1) E A C B D (1) E A B D C (1) D E C A B (1) D E B C A (1) D C E B A (1) C D E A B (1) B D E C A (1) B D C E A (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 14 -6 4 2 B -14 0 -16 -16 -12 C 6 16 0 10 4 D -4 16 -10 0 -2 E -2 12 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 4 2 B -14 0 -16 -16 -12 C 6 16 0 10 4 D -4 16 -10 0 -2 E -2 12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=24 D=17 B=16 A=13 so A is eliminated. Round 2 votes counts: C=37 E=30 D=17 B=16 so B is eliminated. Round 3 votes counts: E=37 C=37 D=26 so D is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:207 E:204 D:200 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 4 2 B -14 0 -16 -16 -12 C 6 16 0 10 4 D -4 16 -10 0 -2 E -2 12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 4 2 B -14 0 -16 -16 -12 C 6 16 0 10 4 D -4 16 -10 0 -2 E -2 12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 4 2 B -14 0 -16 -16 -12 C 6 16 0 10 4 D -4 16 -10 0 -2 E -2 12 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999772 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9203: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (11) C D B E A (8) E A B D C (6) E C D A B (5) E A C D B (5) C D E B A (5) C D B A E (5) B C D A E (5) B A D C E (5) E D C A B (4) E C A D B (3) E A D C B (3) E A B C D (3) D C B E A (3) D C B A E (3) C D E A B (3) B A E D C (3) D B C A E (2) C E D A B (2) B D C A E (2) A B E D C (2) E A C B D (1) D E C A B (1) D C E B A (1) D C E A B (1) D C A E B (1) D B A C E (1) C B D A E (1) B A E C D (1) B A D E C (1) B A C E D (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -10 -6 -8 B -4 0 -10 -12 -12 C 10 10 0 0 0 D 6 12 0 0 -2 E 8 12 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.376024 D: 0.000000 E: 0.623976 Sum of squares = 0.530740001237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.376024 D: 0.376024 E: 1.000000 A B C D E A 0 4 -10 -6 -8 B -4 0 -10 -12 -12 C 10 10 0 0 0 D 6 12 0 0 -2 E 8 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=24 B=19 A=14 D=13 so D is eliminated. Round 2 votes counts: C=33 E=31 B=22 A=14 so A is eliminated. Round 3 votes counts: E=43 C=33 B=24 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:211 C:210 D:208 A:190 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 -6 -8 B -4 0 -10 -12 -12 C 10 10 0 0 0 D 6 12 0 0 -2 E 8 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -6 -8 B -4 0 -10 -12 -12 C 10 10 0 0 0 D 6 12 0 0 -2 E 8 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -6 -8 B -4 0 -10 -12 -12 C 10 10 0 0 0 D 6 12 0 0 -2 E 8 12 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999704 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9204: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) A C E D B (7) D E B C A (5) B D A E C (5) B C E D A (5) D B E A C (4) E C D A B (3) E C A D B (3) C B E D A (3) C B E A D (3) B A D C E (3) E D C B A (2) E D C A B (2) D B A E C (2) D A E B C (2) C E D A B (2) C E A B D (2) C A E B D (2) B D E A C (2) B D C E A (2) B C E A D (2) A D E C B (2) A D E B C (2) E A D C B (1) D E C A B (1) D E B A C (1) D E A C B (1) D B E C A (1) D A E C B (1) C E B D A (1) C E B A D (1) C B A E D (1) C A E D B (1) C A B E D (1) B E D C A (1) B D E C A (1) B C A E D (1) B A C E D (1) B A C D E (1) A E C D B (1) A D C E B (1) A D B E C (1) A B D E C (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -16 6 -22 B -4 0 -8 -16 -12 C 16 8 0 10 10 D -6 16 -10 0 -18 E 22 12 -10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -16 6 -22 B -4 0 -8 -16 -12 C 16 8 0 10 10 D -6 16 -10 0 -18 E 22 12 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=24 D=18 A=18 E=11 so E is eliminated. Round 2 votes counts: C=35 B=24 D=22 A=19 so A is eliminated. Round 3 votes counts: C=43 D=29 B=28 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:221 D:191 A:186 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -16 6 -22 B -4 0 -8 -16 -12 C 16 8 0 10 10 D -6 16 -10 0 -18 E 22 12 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -16 6 -22 B -4 0 -8 -16 -12 C 16 8 0 10 10 D -6 16 -10 0 -18 E 22 12 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -16 6 -22 B -4 0 -8 -16 -12 C 16 8 0 10 10 D -6 16 -10 0 -18 E 22 12 -10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9205: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (6) D C B E A (5) C D B E A (5) A E C B D (5) A B E D C (5) D B C E A (4) C D E B A (4) B D A E C (4) A E B C D (4) A B E C D (4) E D C A B (3) E C D A B (3) E A C D B (3) C B D A E (3) B C D A E (3) A E D B C (3) E A D C B (2) D E C B A (2) D E C A B (2) D B E A C (2) C E D A B (2) C E A D B (2) C B D E A (2) C A E B D (2) C A B E D (2) B D A C E (2) B A D C E (2) A E B D C (2) A C B E D (2) A B C E D (2) E D A C B (1) E C A D B (1) D E B C A (1) D E B A C (1) D C E B A (1) B D C E A (1) B A C E D (1) A C E B D (1) Total count = 100 A B C D E A 0 2 4 -2 6 B -2 0 -6 6 10 C -4 6 0 2 -8 D 2 -6 -2 0 0 E -6 -10 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.440000000003 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -2 6 B -2 0 -6 6 10 C -4 6 0 2 -8 D 2 -6 -2 0 0 E -6 -10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999877 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=22 B=19 D=18 E=13 so E is eliminated. Round 2 votes counts: A=33 C=26 D=22 B=19 so B is eliminated. Round 3 votes counts: A=42 D=29 C=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:205 B:204 C:198 D:197 E:196 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 -2 6 B -2 0 -6 6 10 C -4 6 0 2 -8 D 2 -6 -2 0 0 E -6 -10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999877 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -2 6 B -2 0 -6 6 10 C -4 6 0 2 -8 D 2 -6 -2 0 0 E -6 -10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999877 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -2 6 B -2 0 -6 6 10 C -4 6 0 2 -8 D 2 -6 -2 0 0 E -6 -10 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.000000 D: 0.200000 E: 0.000000 Sum of squares = 0.439999999877 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9206: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (14) D E C A B (10) D E A C B (10) C B A E D (9) B C A E D (9) B A C E D (5) E D C A B (4) D A B E C (3) B C A D E (3) A B E C D (3) E C D A B (2) E A B C D (2) D E C B A (2) D C E B A (2) D A E B C (2) C B E A D (2) C B D A E (2) B A D C E (2) A E B C D (2) E C A D B (1) E A C D B (1) D C B A E (1) D B C E A (1) D B A C E (1) C E D B A (1) C E A B D (1) C D E B A (1) B D A C E (1) B A C D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -2 -14 -8 B -14 0 2 -12 -10 C 2 -2 0 -8 -12 D 14 12 8 0 14 E 8 10 12 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -14 -8 B -14 0 2 -12 -10 C 2 -2 0 -8 -12 D 14 12 8 0 14 E 8 10 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=46 B=21 C=16 E=10 A=7 so A is eliminated. Round 2 votes counts: D=46 B=26 C=16 E=12 so E is eliminated. Round 3 votes counts: D=50 B=30 C=20 so C is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:208 A:195 C:190 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -14 -8 B -14 0 2 -12 -10 C 2 -2 0 -8 -12 D 14 12 8 0 14 E 8 10 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -14 -8 B -14 0 2 -12 -10 C 2 -2 0 -8 -12 D 14 12 8 0 14 E 8 10 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -14 -8 B -14 0 2 -12 -10 C 2 -2 0 -8 -12 D 14 12 8 0 14 E 8 10 12 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9207: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (8) B A C E D (8) E D B A C (7) C D A E B (7) A B C E D (6) D E C A B (5) C D E A B (5) C A B D E (5) E B D A C (4) D E C B A (4) B A E D C (4) B E A D C (3) B E A C D (3) E D C B A (2) E D B C A (2) C D A B E (2) C B A D E (2) C A D B E (2) A C B D E (2) A B E C D (2) E B A D C (1) D C A E B (1) D C A B E (1) D A C E B (1) C D E B A (1) B C A E D (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 -4 6 B 0 0 0 0 0 C 0 0 0 8 8 D 4 0 -8 0 -4 E -6 0 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.228540 B: 0.484301 C: 0.287159 D: 0.000000 E: 0.000000 Sum of squares = 0.36923836444 Cumulative probabilities = A: 0.228540 B: 0.712841 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 6 B 0 0 0 0 0 C 0 0 0 8 8 D 4 0 -8 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=24 D=22 E=16 A=11 so A is eliminated. Round 2 votes counts: B=36 C=26 D=22 E=16 so E is eliminated. Round 3 votes counts: B=41 D=33 C=26 so C is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 A:201 B:200 D:196 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 0 -4 6 B 0 0 0 0 0 C 0 0 0 8 8 D 4 0 -8 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 6 B 0 0 0 0 0 C 0 0 0 8 8 D 4 0 -8 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 6 B 0 0 0 0 0 C 0 0 0 8 8 D 4 0 -8 0 -4 E -6 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.333333 C: 0.333334 D: 0.000000 E: 0.000000 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.666666 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9208: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) E D C A B (5) E D A C B (5) E A D B C (5) C D E A B (5) C B D A E (5) E C D A B (4) B E A C D (4) B A D C E (4) A B D E C (4) C E D B A (3) C E D A B (3) B A C D E (3) E A B D C (2) D C A E B (2) D A E B C (2) D A C B E (2) C E B D A (2) C D E B A (2) C D B A E (2) C B E D A (2) C B A D E (2) B C E A D (2) B C A E D (2) B A D E C (2) B A C E D (2) A D E B C (2) E C B D A (1) E B C A D (1) E B A D C (1) D A E C B (1) D A B C E (1) C D A B E (1) C B E A D (1) C B D E A (1) B E A D C (1) B A E D C (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -8 2 0 B 4 0 2 6 8 C 8 -2 0 12 14 D -2 -6 -12 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 2 0 B 4 0 2 6 8 C 8 -2 0 12 14 D -2 -6 -12 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=29 B=29 E=24 A=10 D=8 so D is eliminated. Round 2 votes counts: C=31 B=29 E=24 A=16 so A is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:210 A:195 D:192 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -8 2 0 B 4 0 2 6 8 C 8 -2 0 12 14 D -2 -6 -12 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 2 0 B 4 0 2 6 8 C 8 -2 0 12 14 D -2 -6 -12 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 2 0 B 4 0 2 6 8 C 8 -2 0 12 14 D -2 -6 -12 0 4 E 0 -8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999916 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9209: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (7) E B C A D (5) B E D A C (5) A D C B E (5) E B C D A (4) B E D C A (4) A D B C E (4) D C B A E (3) C A D E B (3) B D E A C (3) B A D E C (3) A B E D C (3) E B A C D (2) E A C B D (2) E A B C D (2) D C A B E (2) D B C A E (2) D A C B E (2) D A B C E (2) C E D B A (2) C D A E B (2) C A E D B (2) B E C D A (2) B E A D C (2) B D E C A (2) B D C E A (2) B D A E C (2) B D A C E (2) A E C D B (2) A E C B D (2) A C E D B (2) E C B A D (1) E A C D B (1) D C B E A (1) D B C E A (1) C E D A B (1) C E B D A (1) C E A D B (1) C D B A E (1) C D A B E (1) B D C A E (1) A E B C D (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -2 -10 -2 B 16 0 10 18 12 C 2 -10 0 -6 -12 D 10 -18 6 0 -8 E 2 -12 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 -10 -2 B 16 0 10 18 12 C 2 -10 0 -6 -12 D 10 -18 6 0 -8 E 2 -12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 A=21 C=14 D=13 so D is eliminated. Round 2 votes counts: B=31 A=25 E=24 C=20 so C is eliminated. Round 3 votes counts: B=36 A=35 E=29 so E is eliminated. Round 4 votes counts: B=58 A=42 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:228 E:205 D:195 C:187 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 -10 -2 B 16 0 10 18 12 C 2 -10 0 -6 -12 D 10 -18 6 0 -8 E 2 -12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -10 -2 B 16 0 10 18 12 C 2 -10 0 -6 -12 D 10 -18 6 0 -8 E 2 -12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -10 -2 B 16 0 10 18 12 C 2 -10 0 -6 -12 D 10 -18 6 0 -8 E 2 -12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9210: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E A D C B (6) C B E D A (6) B D C A E (6) C E A B D (5) C B D E A (5) C B A D E (5) B C D A E (5) A E D B C (5) A D E B C (4) E A D B C (3) D E A B C (3) D B A E C (3) D A E B C (3) D A B E C (3) C E B A D (3) C A E B D (3) E A C D B (2) D B E A C (2) C B A E D (2) A D B C E (2) A C B E D (2) E D B C A (1) E D A B C (1) E C D B A (1) E C B A D (1) E C A D B (1) D E B A C (1) D B C A E (1) D B A C E (1) C A B E D (1) B D C E A (1) B D A C E (1) B C D E A (1) A E D C B (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 -10 -4 14 B 6 0 -6 10 10 C 10 6 0 4 16 D 4 -10 -4 0 10 E -14 -10 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -4 14 B 6 0 -6 10 10 C 10 6 0 4 16 D 4 -10 -4 0 10 E -14 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=17 E=16 A=16 B=14 so B is eliminated. Round 2 votes counts: C=43 D=25 E=16 A=16 so E is eliminated. Round 3 votes counts: C=46 D=27 A=27 so D is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:210 D:200 A:197 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -4 14 B 6 0 -6 10 10 C 10 6 0 4 16 D 4 -10 -4 0 10 E -14 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -4 14 B 6 0 -6 10 10 C 10 6 0 4 16 D 4 -10 -4 0 10 E -14 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -4 14 B 6 0 -6 10 10 C 10 6 0 4 16 D 4 -10 -4 0 10 E -14 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9211: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) E B C A D (6) D C B E A (6) D A C B E (6) B E C A D (6) E B A C D (5) C D E B A (5) D C A E B (4) C E B D A (4) C E B A D (4) C B E D A (4) A D B E C (4) D C A B E (3) A E B D C (3) D A C E B (2) D A B C E (2) C E D B A (2) B E A C D (2) B C E D A (2) A E D B C (2) A E C B D (2) A E B C D (2) A D E B C (2) E C B A D (1) E C A B D (1) D C E B A (1) D C E A B (1) D B C E A (1) D A E C B (1) C E A B D (1) C D B E A (1) C B D E A (1) B E C D A (1) B C D E A (1) B A E C D (1) A D E C B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -12 -10 -12 B 8 0 -2 -2 0 C 12 2 0 6 2 D 10 2 -6 0 -2 E 12 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -10 -12 B 8 0 -2 -2 0 C 12 2 0 6 2 D 10 2 -6 0 -2 E 12 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=22 A=18 E=13 B=13 so E is eliminated. Round 2 votes counts: D=34 C=24 B=24 A=18 so A is eliminated. Round 3 votes counts: D=43 B=31 C=26 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:211 E:206 B:202 D:202 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -10 -12 B 8 0 -2 -2 0 C 12 2 0 6 2 D 10 2 -6 0 -2 E 12 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -10 -12 B 8 0 -2 -2 0 C 12 2 0 6 2 D 10 2 -6 0 -2 E 12 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -10 -12 B 8 0 -2 -2 0 C 12 2 0 6 2 D 10 2 -6 0 -2 E 12 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9212: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (8) E A B C D (7) D B C A E (6) C A E D B (6) C D A B E (5) D C B E A (4) B E D A C (4) B D E A C (4) A E B C D (4) E B A D C (3) D B C E A (3) C D A E B (3) C A D E B (3) B E A D C (3) A E C B D (3) D B E C A (2) D B A E C (2) C D E B A (2) C D B E A (2) C A E B D (2) B E D C A (2) B D E C A (2) B A E D C (2) A C E D B (2) A C D E B (2) A B D E C (2) E B D A C (1) E B A C D (1) E A C B D (1) E A B D C (1) D C A B E (1) D B A C E (1) D A B C E (1) C E A B D (1) C D B A E (1) B D A E C (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -6 -10 14 B 8 0 6 -8 14 C 6 -6 0 -8 8 D 10 8 8 0 10 E -14 -14 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -10 14 B 8 0 6 -8 14 C 6 -6 0 -8 8 D 10 8 8 0 10 E -14 -14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 B=18 A=15 E=14 so E is eliminated. Round 2 votes counts: D=28 C=25 A=24 B=23 so B is eliminated. Round 3 votes counts: D=42 A=33 C=25 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:210 C:200 A:195 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -10 14 B 8 0 6 -8 14 C 6 -6 0 -8 8 D 10 8 8 0 10 E -14 -14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -10 14 B 8 0 6 -8 14 C 6 -6 0 -8 8 D 10 8 8 0 10 E -14 -14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -10 14 B 8 0 6 -8 14 C 6 -6 0 -8 8 D 10 8 8 0 10 E -14 -14 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9213: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (7) E A D B C (6) A B D E C (6) A B D C E (6) E C D A B (5) C E D B A (5) C D E B A (4) C B A D E (4) B A C D E (4) E C D B A (3) E A B C D (3) D A B E C (3) C E B A D (3) C D B E A (3) C D B A E (3) A E B D C (3) E D A B C (2) E A C B D (2) E A B D C (2) D A B C E (2) C E B D A (2) C B A E D (2) A D B E C (2) E D C A B (1) E D A C B (1) E C B A D (1) E C A D B (1) E A D C B (1) D E C B A (1) D B C A E (1) D A E B C (1) C B E A D (1) C B D E A (1) B C A D E (1) B A D C E (1) B A C E D (1) A E D B C (1) A E B C D (1) A D E B C (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 4 2 10 4 B -4 0 -2 6 0 C -2 2 0 16 4 D -10 -6 -16 0 4 E -4 0 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 10 4 B -4 0 -2 6 0 C -2 2 0 16 4 D -10 -6 -16 0 4 E -4 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=28 A=22 D=8 B=7 so B is eliminated. Round 2 votes counts: C=36 E=28 A=28 D=8 so D is eliminated. Round 3 votes counts: C=37 A=34 E=29 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:210 B:200 E:194 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 10 4 B -4 0 -2 6 0 C -2 2 0 16 4 D -10 -6 -16 0 4 E -4 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 10 4 B -4 0 -2 6 0 C -2 2 0 16 4 D -10 -6 -16 0 4 E -4 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 10 4 B -4 0 -2 6 0 C -2 2 0 16 4 D -10 -6 -16 0 4 E -4 0 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997395 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9214: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) B D C A E (7) E A D C B (6) E D A C B (4) D B E C A (4) D B C E A (4) A C E B D (4) A C B E D (4) E D B A C (3) D E B C A (3) C A B D E (3) B C D A E (3) A E B C D (3) A C E D B (3) E D C A B (2) E A B C D (2) D E C B A (2) D C B E A (2) D B C A E (2) C D B A E (2) C B D A E (2) C A D E B (2) B E D A C (2) B D E A C (2) B A C D E (2) A E C B D (2) A B C E D (2) E D C B A (1) E D B C A (1) E D A B C (1) E B D A C (1) E B A D C (1) E A D B C (1) E A C B D (1) D C B A E (1) C D A B E (1) C B A D E (1) C A D B E (1) B D E C A (1) B A D C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 10 -2 -4 B -4 0 -6 -8 -2 C -10 6 0 -4 -2 D 2 8 4 0 -4 E 4 2 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 10 -2 -4 B -4 0 -6 -8 -2 C -10 6 0 -4 -2 D 2 8 4 0 -4 E 4 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=20 D=18 B=18 C=12 so C is eliminated. Round 2 votes counts: E=32 A=26 D=21 B=21 so D is eliminated. Round 3 votes counts: E=37 B=36 A=27 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:206 D:205 A:204 C:195 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 10 -2 -4 B -4 0 -6 -8 -2 C -10 6 0 -4 -2 D 2 8 4 0 -4 E 4 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 10 -2 -4 B -4 0 -6 -8 -2 C -10 6 0 -4 -2 D 2 8 4 0 -4 E 4 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 10 -2 -4 B -4 0 -6 -8 -2 C -10 6 0 -4 -2 D 2 8 4 0 -4 E 4 2 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9215: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) C D A E B (12) C D A B E (12) B E C A D (11) B E A D C (8) D A C E B (7) B E A C D (6) D A E C B (4) A D E B C (4) C B E D A (3) E A D B C (2) E A B D C (2) D C A E B (2) C D B A E (2) C B D A E (2) C A D B E (2) A E D B C (2) A D E C B (2) C B D E A (1) B E C D A (1) B C E D A (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 6 2 6 6 B -6 0 0 -6 2 C -2 0 0 6 -10 D -6 6 -6 0 4 E -6 -2 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 6 6 B -6 0 0 -6 2 C -2 0 0 6 -10 D -6 6 -6 0 4 E -6 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=27 E=16 D=13 A=10 so A is eliminated. Round 2 votes counts: C=34 B=27 D=21 E=18 so E is eliminated. Round 3 votes counts: B=41 C=34 D=25 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:210 D:199 E:199 C:197 B:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 2 6 6 B -6 0 0 -6 2 C -2 0 0 6 -10 D -6 6 -6 0 4 E -6 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 6 6 B -6 0 0 -6 2 C -2 0 0 6 -10 D -6 6 -6 0 4 E -6 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 6 6 B -6 0 0 -6 2 C -2 0 0 6 -10 D -6 6 -6 0 4 E -6 -2 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9216: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (12) A D E C B (11) B C E D A (10) D A E C B (9) A D B E C (7) B A D C E (5) E C A D B (4) C E B A D (4) B D C E A (3) A E C D B (3) A C E B D (3) E C D A B (2) D E A C B (2) D A E B C (2) C B E D A (2) B C D E A (2) A E D C B (2) A D E B C (2) A B D C E (2) E D C A B (1) E D A C B (1) E C D B A (1) E A C D B (1) D B A C E (1) C E B D A (1) C E A B D (1) B D C A E (1) B D A C E (1) B A C E D (1) B A C D E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 10 12 22 6 B -10 0 2 -2 -2 C -12 -2 0 -2 4 D -22 2 2 0 0 E -6 2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 12 22 6 B -10 0 2 -2 -2 C -12 -2 0 -2 4 D -22 2 2 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 A=32 D=14 E=10 C=8 so C is eliminated. Round 2 votes counts: B=38 A=32 E=16 D=14 so D is eliminated. Round 3 votes counts: A=43 B=39 E=18 so E is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:196 B:194 C:194 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 12 22 6 B -10 0 2 -2 -2 C -12 -2 0 -2 4 D -22 2 2 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 12 22 6 B -10 0 2 -2 -2 C -12 -2 0 -2 4 D -22 2 2 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 12 22 6 B -10 0 2 -2 -2 C -12 -2 0 -2 4 D -22 2 2 0 0 E -6 2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9217: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) D C E A B (7) A B C D E (7) D E C B A (6) C B A D E (6) D E C A B (5) B A E C D (5) E D C B A (4) D E A B C (4) D C A B E (4) A B E D C (4) C E D B A (3) C B A E D (3) C A B D E (3) B A C E D (3) E D B C A (2) E D A B C (2) E C D B A (2) D C E B A (2) C D A B E (2) A B D E C (2) A B C E D (2) E C B D A (1) E C B A D (1) E B A D C (1) E B A C D (1) D A B E C (1) C E B A D (1) C D E A B (1) C B E A D (1) B E A C D (1) B A E D C (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -26 -10 -8 B 6 0 -22 -10 -4 C 26 22 0 6 12 D 10 10 -6 0 22 E 8 4 -12 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -26 -10 -8 B 6 0 -22 -10 -4 C 26 22 0 6 12 D 10 10 -6 0 22 E 8 4 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=29 C=29 A=18 E=14 B=10 so B is eliminated. Round 2 votes counts: D=29 C=29 A=27 E=15 so E is eliminated. Round 3 votes counts: D=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:233 D:218 E:189 B:185 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -26 -10 -8 B 6 0 -22 -10 -4 C 26 22 0 6 12 D 10 10 -6 0 22 E 8 4 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -26 -10 -8 B 6 0 -22 -10 -4 C 26 22 0 6 12 D 10 10 -6 0 22 E 8 4 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -26 -10 -8 B 6 0 -22 -10 -4 C 26 22 0 6 12 D 10 10 -6 0 22 E 8 4 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9218: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (16) B A E D C (16) B A E C D (6) D C E A B (5) B D E A C (4) B D C E A (4) E A C D B (3) B C D E A (3) B A D E C (3) A E C D B (3) A C E D B (3) D E C A B (2) D C E B A (2) D C B E A (2) C E A D B (2) C A E D B (2) B E A D C (2) B C D A E (2) B C A E D (2) B A C E D (2) A E B C D (2) A B E C D (2) E D C A B (1) E C D A B (1) C E D A B (1) C D E B A (1) C D B E A (1) C B D E A (1) C B D A E (1) C B A D E (1) B C A D E (1) B A D C E (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 -10 -2 6 -2 B 10 0 2 10 8 C 2 -2 0 14 6 D -6 -10 -14 0 0 E 2 -8 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -2 6 -2 B 10 0 2 10 8 C 2 -2 0 14 6 D -6 -10 -14 0 0 E 2 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=46 C=26 A=12 D=11 E=5 so E is eliminated. Round 2 votes counts: B=46 C=27 A=15 D=12 so D is eliminated. Round 3 votes counts: B=46 C=39 A=15 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:210 A:196 E:194 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -2 6 -2 B 10 0 2 10 8 C 2 -2 0 14 6 D -6 -10 -14 0 0 E 2 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -2 6 -2 B 10 0 2 10 8 C 2 -2 0 14 6 D -6 -10 -14 0 0 E 2 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -2 6 -2 B 10 0 2 10 8 C 2 -2 0 14 6 D -6 -10 -14 0 0 E 2 -8 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999791 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9219: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) D C A E B (7) C D A B E (6) B E C D A (6) A D C E B (6) C D E B A (5) B E A C D (5) D A C E B (4) B A E C D (4) A D C B E (4) E B D A C (3) D C E B A (3) C B D E A (3) B A E D C (3) A E D B C (3) A D E C B (3) D A E C B (2) C D B E A (2) C D A E B (2) B A C E D (2) A B E D C (2) E D C B A (1) E D A B C (1) D E C B A (1) D E C A B (1) D E A C B (1) C E B D A (1) C B E D A (1) C A D B E (1) B E A D C (1) B C E D A (1) B C E A D (1) B C A E D (1) A E B D C (1) A D E B C (1) A C D B E (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 14 -2 10 B 4 0 -12 -10 -8 C -14 12 0 -12 6 D 2 10 12 0 8 E -10 8 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 -2 10 B 4 0 -12 -10 -8 C -14 12 0 -12 6 D 2 10 12 0 8 E -10 8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 A=23 C=21 D=19 E=13 so E is eliminated. Round 2 votes counts: B=35 A=23 D=21 C=21 so D is eliminated. Round 3 votes counts: B=35 C=34 A=31 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:216 A:209 C:196 E:192 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 14 -2 10 B 4 0 -12 -10 -8 C -14 12 0 -12 6 D 2 10 12 0 8 E -10 8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 -2 10 B 4 0 -12 -10 -8 C -14 12 0 -12 6 D 2 10 12 0 8 E -10 8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 -2 10 B 4 0 -12 -10 -8 C -14 12 0 -12 6 D 2 10 12 0 8 E -10 8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9220: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) E B C A D (8) E B A C D (8) E C B D A (7) B A E C D (7) D C E A B (5) C E D B A (5) A B E D C (5) D C A B E (4) D A B C E (4) A D B C E (4) A B D E C (4) D A C B E (3) B E A C D (3) A B D C E (3) E A B C D (2) B D A C E (2) B A C E D (2) B A C D E (2) A E B D C (2) A D B E C (2) E C D B A (1) E C B A D (1) D C B A E (1) D C A E B (1) D A C E B (1) C E B D A (1) C D B A E (1) C B D E A (1) B E C A D (1) Total count = 100 A B C D E A 0 -20 8 8 -4 B 20 0 18 18 -2 C -8 -18 0 18 -2 D -8 -18 -18 0 -6 E 4 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 8 8 -4 B 20 0 18 18 -2 C -8 -18 0 18 -2 D -8 -18 -18 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=20 D=19 C=17 B=17 so C is eliminated. Round 2 votes counts: E=33 D=29 A=20 B=18 so B is eliminated. Round 3 votes counts: E=37 D=32 A=31 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:227 E:207 A:196 C:195 D:175 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 8 8 -4 B 20 0 18 18 -2 C -8 -18 0 18 -2 D -8 -18 -18 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 8 8 -4 B 20 0 18 18 -2 C -8 -18 0 18 -2 D -8 -18 -18 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 8 8 -4 B 20 0 18 18 -2 C -8 -18 0 18 -2 D -8 -18 -18 0 -6 E 4 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.9999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9221: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) D A B C E (7) C E B D A (7) C E A D B (6) A D B E C (5) D B A E C (4) C E B A D (4) B D E A C (4) E B C A D (3) D C E B A (3) D A C E B (3) C E D B A (3) B E D C A (3) B A D E C (3) A D C B E (3) E C B D A (2) E C B A D (2) E B C D A (2) D C A E B (2) D B C E A (2) C E D A B (2) C E A B D (2) B E C A D (2) B D A E C (2) A E C B D (2) A D C E B (2) A D B C E (2) A B D E C (2) D E C B A (1) D C E A B (1) D B E C A (1) D B A C E (1) D A C B E (1) B E D A C (1) B E C D A (1) B A E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 -20 -4 B 4 0 8 -12 6 C -2 -8 0 -22 2 D 20 12 22 0 12 E 4 -6 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -20 -4 B 4 0 8 -12 6 C -2 -8 0 -22 2 D 20 12 22 0 12 E 4 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=24 B=17 A=17 E=9 so E is eliminated. Round 2 votes counts: D=33 C=28 B=22 A=17 so A is eliminated. Round 3 votes counts: D=45 C=30 B=25 so B is eliminated. Round 4 votes counts: D=61 C=39 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:233 B:203 E:192 A:187 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 2 -20 -4 B 4 0 8 -12 6 C -2 -8 0 -22 2 D 20 12 22 0 12 E 4 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -20 -4 B 4 0 8 -12 6 C -2 -8 0 -22 2 D 20 12 22 0 12 E 4 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -20 -4 B 4 0 8 -12 6 C -2 -8 0 -22 2 D 20 12 22 0 12 E 4 -6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9222: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (11) C A B E D (11) A C B E D (10) D E B C A (8) D E B A C (8) C A D B E (6) C D A E B (5) B E A D C (5) A B E C D (5) E D B A C (4) D C E B A (4) C A B D E (4) E B A D C (3) C D A B E (3) B E A C D (3) C A D E B (2) E B A C D (1) D E C B A (1) D C A E B (1) D C A B E (1) D A C B E (1) C E B A D (1) B E D A C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 6 4 0 B 0 0 0 12 2 C -6 0 0 2 -2 D -4 -12 -2 0 -12 E 0 -2 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.423759 B: 0.576241 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.511625321032 Cumulative probabilities = A: 0.423759 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 0 B 0 0 0 12 2 C -6 0 0 2 -2 D -4 -12 -2 0 -12 E 0 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 E=19 A=16 B=9 so B is eliminated. Round 2 votes counts: C=32 E=28 D=24 A=16 so A is eliminated. Round 3 votes counts: C=42 E=34 D=24 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:207 E:206 A:205 C:197 D:185 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 0 B 0 0 0 12 2 C -6 0 0 2 -2 D -4 -12 -2 0 -12 E 0 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 0 B 0 0 0 12 2 C -6 0 0 2 -2 D -4 -12 -2 0 -12 E 0 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 0 B 0 0 0 12 2 C -6 0 0 2 -2 D -4 -12 -2 0 -12 E 0 -2 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999636 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9223: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) B C E D A (7) D A C E B (6) B E C A D (6) E C D A B (5) E C B D A (5) D A C B E (5) A D B E C (5) A D B C E (5) B E A D C (4) A D E B C (4) C B E D A (3) B C E A D (3) B A D C E (3) A D C B E (3) E B C D A (2) E B C A D (2) E A D C B (2) E A D B C (2) C E D A B (2) C D A E B (2) B A E D C (2) B A D E C (2) A B D E C (2) E B A D C (1) D A B C E (1) C E D B A (1) C D A B E (1) C B D A E (1) B E C D A (1) B E A C D (1) B C A E D (1) B A C E D (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 0 0 -10 B 8 0 10 10 14 C 0 -10 0 4 6 D 0 -10 -4 0 -18 E 10 -14 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 0 -10 B 8 0 10 10 14 C 0 -10 0 4 6 D 0 -10 -4 0 -18 E 10 -14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=20 E=19 C=18 D=12 so D is eliminated. Round 2 votes counts: A=32 B=31 E=19 C=18 so C is eliminated. Round 3 votes counts: B=35 A=35 E=30 so E is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:204 C:200 A:191 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 0 -10 B 8 0 10 10 14 C 0 -10 0 4 6 D 0 -10 -4 0 -18 E 10 -14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 0 -10 B 8 0 10 10 14 C 0 -10 0 4 6 D 0 -10 -4 0 -18 E 10 -14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 0 -10 B 8 0 10 10 14 C 0 -10 0 4 6 D 0 -10 -4 0 -18 E 10 -14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999675 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9224: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (16) D B C A E (15) E B A C D (11) D C A B E (5) E C A D B (4) C A E D B (4) C A D E B (4) B D E A C (4) E B D A C (3) E B A D C (3) C A D B E (3) B E D A C (3) B D A C E (3) A C E B D (3) E A B C D (2) D B E C A (2) D B C E A (2) A E C B D (2) E D B C A (1) D E C B A (1) D E C A B (1) D E B C A (1) D C B A E (1) D B A C E (1) C E A D B (1) C D E A B (1) C D A E B (1) C D A B E (1) B E A D C (1) Total count = 100 A B C D E A 0 -4 4 8 -14 B 4 0 4 2 -18 C -4 -4 0 6 -10 D -8 -2 -6 0 -8 E 14 18 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 4 8 -14 B 4 0 4 2 -18 C -4 -4 0 6 -10 D -8 -2 -6 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 D=29 C=15 B=11 A=5 so A is eliminated. Round 2 votes counts: E=42 D=29 C=18 B=11 so B is eliminated. Round 3 votes counts: E=46 D=36 C=18 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:225 A:197 B:196 C:194 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 4 8 -14 B 4 0 4 2 -18 C -4 -4 0 6 -10 D -8 -2 -6 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 8 -14 B 4 0 4 2 -18 C -4 -4 0 6 -10 D -8 -2 -6 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 8 -14 B 4 0 4 2 -18 C -4 -4 0 6 -10 D -8 -2 -6 0 -8 E 14 18 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999367 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9225: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (6) D A E C B (5) C B D E A (5) C B D A E (5) E A D B C (4) D A C E B (4) E A B D C (3) D E B A C (3) D C B A E (3) C D B A E (3) C B E A D (3) C A E B D (3) B E C A D (3) B C E A D (3) A E D C B (3) A E D B C (3) A E C B D (3) A D E C B (3) E B A D C (2) D E A B C (2) C D A B E (2) C B E D A (2) C A B E D (2) B E D C A (2) B E D A C (2) B D C E A (2) B C D E A (2) E B A C D (1) D E A C B (1) D C A E B (1) D B E C A (1) D B E A C (1) D B C E A (1) C D B E A (1) C B A E D (1) C B A D E (1) C A D E B (1) B E A D C (1) B C E D A (1) A E C D B (1) A E B C D (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 2 4 -10 8 B -2 0 -12 -2 -6 C -4 12 0 -8 -2 D 10 2 8 0 10 E -8 6 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999289 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -10 8 B -2 0 -12 -2 -6 C -4 12 0 -8 -2 D 10 2 8 0 10 E -8 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=28 A=17 B=16 E=10 so E is eliminated. Round 2 votes counts: C=29 D=28 A=24 B=19 so B is eliminated. Round 3 votes counts: C=38 D=34 A=28 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 A:202 C:199 E:195 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 4 -10 8 B -2 0 -12 -2 -6 C -4 12 0 -8 -2 D 10 2 8 0 10 E -8 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -10 8 B -2 0 -12 -2 -6 C -4 12 0 -8 -2 D 10 2 8 0 10 E -8 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -10 8 B -2 0 -12 -2 -6 C -4 12 0 -8 -2 D 10 2 8 0 10 E -8 6 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9226: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) E C D B A (7) B A D C E (6) A B D C E (6) B A D E C (5) D C E A B (4) D C A B E (4) E C B A D (3) E C A B D (3) D A B C E (3) C D E A B (3) C D A E B (3) C A D E B (3) C A D B E (3) B A E D C (3) E D B C A (2) D E C B A (2) D B E A C (2) C E D A B (2) B E A D C (2) B D A E C (2) A B C D E (2) E D C B A (1) E D B A C (1) E C B D A (1) E B D C A (1) E B C A D (1) D E C A B (1) D E B C A (1) D E B A C (1) D C E B A (1) D C B A E (1) D C A E B (1) D A C B E (1) C E A D B (1) C E A B D (1) C D A B E (1) B E A C D (1) A D B C E (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -2 8 16 B 4 0 0 -2 10 C 2 0 0 -4 2 D -8 2 4 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571428565 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 8 16 B 4 0 0 -2 10 C 2 0 0 -4 2 D -8 2 4 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571418198 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=22 E=20 C=17 A=14 so A is eliminated. Round 2 votes counts: B=37 D=23 E=20 C=20 so E is eliminated. Round 3 votes counts: B=39 C=34 D=27 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:209 D:208 B:206 C:200 E:177 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -2 8 16 B 4 0 0 -2 10 C 2 0 0 -4 2 D -8 2 4 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571418198 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 8 16 B 4 0 0 -2 10 C 2 0 0 -4 2 D -8 2 4 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571418198 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 8 16 B 4 0 0 -2 10 C 2 0 0 -4 2 D -8 2 4 0 18 E -16 -10 -2 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.571429 C: 0.000000 D: 0.285714 E: 0.000000 Sum of squares = 0.428571418198 Cumulative probabilities = A: 0.142857 B: 0.714286 C: 0.714286 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9227: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (10) D B E A C (7) A C D B E (6) E B D A C (5) D B C A E (5) C A E D B (5) B E D A C (5) B D E A C (5) E A C B D (4) C A E B D (4) E B A D C (3) C E A B D (3) C A D E B (3) C A D B E (3) A E C B D (3) E B A C D (2) E A B C D (2) D C A B E (2) D B E C A (2) D B C E A (2) D B A C E (2) D A C B E (2) B E D C A (2) B D E C A (2) E D B C A (1) E C B D A (1) E C A B D (1) E B C A D (1) D C B E A (1) D C B A E (1) C D E A B (1) B E A D C (1) A E B C D (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 20 8 -2 B -4 0 -4 10 -4 C -20 4 0 4 4 D -8 -10 -4 0 -10 E 2 4 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.076923 D: 0.000000 E: 0.769231 Sum of squares = 0.621301775145 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.230769 D: 0.230769 E: 1.000000 A B C D E A 0 4 20 8 -2 B -4 0 -4 10 -4 C -20 4 0 4 4 D -8 -10 -4 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.076923 D: 0.000000 E: 0.769231 Sum of squares = 0.621301774667 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.230769 D: 0.230769 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 E=20 C=19 B=15 so B is eliminated. Round 2 votes counts: D=31 E=28 A=22 C=19 so C is eliminated. Round 3 votes counts: A=37 D=32 E=31 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:206 B:199 C:196 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 20 8 -2 B -4 0 -4 10 -4 C -20 4 0 4 4 D -8 -10 -4 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.076923 D: 0.000000 E: 0.769231 Sum of squares = 0.621301774667 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.230769 D: 0.230769 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 20 8 -2 B -4 0 -4 10 -4 C -20 4 0 4 4 D -8 -10 -4 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.076923 D: 0.000000 E: 0.769231 Sum of squares = 0.621301774667 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.230769 D: 0.230769 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 20 8 -2 B -4 0 -4 10 -4 C -20 4 0 4 4 D -8 -10 -4 0 -10 E 2 4 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.153846 B: 0.000000 C: 0.076923 D: 0.000000 E: 0.769231 Sum of squares = 0.621301774667 Cumulative probabilities = A: 0.153846 B: 0.153846 C: 0.230769 D: 0.230769 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9228: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) B D E C A (8) E B D C A (6) C B A E D (6) C A B D E (5) D E A B C (4) C A E D B (4) C A B E D (4) B C E D A (4) B C A D E (4) A C E D B (4) A C D B E (4) C B A D E (3) B D A C E (3) A C D E B (3) E D A C B (2) E D A B C (2) E C A D B (2) D E B A C (2) A E D C B (2) A D E C B (2) A D C B E (2) E D C A B (1) E D B C A (1) E C D A B (1) E B C D A (1) E A D C B (1) D E A C B (1) C E B D A (1) C E B A D (1) C B E D A (1) B D E A C (1) B D C A E (1) B D A E C (1) B C E A D (1) B C D E A (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -12 0 0 B 8 0 -4 6 0 C 12 4 0 2 8 D 0 -6 -2 0 -6 E 0 0 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 0 0 B 8 0 -4 6 0 C 12 4 0 2 8 D 0 -6 -2 0 -6 E 0 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=25 C=25 B=24 A=19 D=7 so D is eliminated. Round 2 votes counts: E=32 C=25 B=24 A=19 so A is eliminated. Round 3 votes counts: C=40 E=36 B=24 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:205 E:199 D:193 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 0 0 B 8 0 -4 6 0 C 12 4 0 2 8 D 0 -6 -2 0 -6 E 0 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 0 0 B 8 0 -4 6 0 C 12 4 0 2 8 D 0 -6 -2 0 -6 E 0 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 0 0 B 8 0 -4 6 0 C 12 4 0 2 8 D 0 -6 -2 0 -6 E 0 0 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9229: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (7) C E D A B (5) A B C D E (5) E D C B A (4) E C A D B (4) E B A C D (4) B E D C A (4) B A D C E (4) D C A E B (3) C E A D B (3) C A E D B (3) B E A D C (3) B A E C D (3) A D C B E (3) E C B D A (2) E B D C A (2) D E B C A (2) D C E A B (2) D C A B E (2) C A D E B (2) B E D A C (2) B E A C D (2) B A E D C (2) B A D E C (2) A C D E B (2) A C B D E (2) A B D C E (2) E C D B A (1) E C B A D (1) E C A B D (1) E B C D A (1) E A C B D (1) E A B C D (1) D E C B A (1) D B E C A (1) D B C E A (1) D A C B E (1) C D E A B (1) B D E C A (1) B D A E C (1) B D A C E (1) A E C B D (1) A E B C D (1) A D B C E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 -8 10 -14 B -10 0 -8 0 -10 C 8 8 0 10 -10 D -10 0 -10 0 -18 E 14 10 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -8 10 -14 B -10 0 -8 0 -10 C 8 8 0 10 -10 D -10 0 -10 0 -18 E 14 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=25 A=19 C=14 D=13 so D is eliminated. Round 2 votes counts: E=32 B=27 C=21 A=20 so A is eliminated. Round 3 votes counts: B=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:208 A:199 B:186 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -8 10 -14 B -10 0 -8 0 -10 C 8 8 0 10 -10 D -10 0 -10 0 -18 E 14 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 10 -14 B -10 0 -8 0 -10 C 8 8 0 10 -10 D -10 0 -10 0 -18 E 14 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 10 -14 B -10 0 -8 0 -10 C 8 8 0 10 -10 D -10 0 -10 0 -18 E 14 10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9230: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D A E B C (6) D A B E C (6) C D A E B (6) C E B D A (5) C B E A D (5) E C B A D (4) E B C A D (4) E A B D C (4) B E A D C (4) E B A C D (3) C D E A B (3) C D B A E (3) B E C A D (3) B C E A D (3) B A E D C (3) A D B E C (3) E B A D C (2) C E D B A (2) C B A E D (2) B A D C E (2) A B D E C (2) E A D B C (1) D C A E B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A B C E (1) C E D A B (1) C D E B A (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D A E (1) B C A E D (1) B A C E D (1) A E B D C (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -18 -14 16 -10 B 18 0 4 20 -10 C 14 -4 0 18 2 D -16 -20 -18 0 -18 E 10 10 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999815 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -18 -14 16 -10 B 18 0 4 20 -10 C 14 -4 0 18 2 D -16 -20 -18 0 -18 E 10 10 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.46874999969 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=18 D=17 B=17 A=8 so A is eliminated. Round 2 votes counts: C=40 D=21 B=20 E=19 so E is eliminated. Round 3 votes counts: C=44 B=34 D=22 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:218 B:216 C:215 A:187 D:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -14 16 -10 B 18 0 4 20 -10 C 14 -4 0 18 2 D -16 -20 -18 0 -18 E 10 10 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.46874999969 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 16 -10 B 18 0 4 20 -10 C 14 -4 0 18 2 D -16 -20 -18 0 -18 E 10 10 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.46874999969 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 16 -10 B 18 0 4 20 -10 C 14 -4 0 18 2 D -16 -20 -18 0 -18 E 10 10 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.46874999969 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9231: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) B D A E C (8) C D A B E (6) E B A D C (5) C A D E B (5) B E D A C (5) C E B A D (4) B D A C E (4) E A D B C (3) E A C D B (3) D B A C E (3) C D B A E (3) C D A E B (3) A D E B C (3) E C B A D (2) E C A D B (2) E C A B D (2) E A B D C (2) E A B C D (2) D C A B E (2) D A C B E (2) C E B D A (2) C E A B D (2) C B E D A (2) C A E D B (2) B D E A C (2) E B C A D (1) E B A C D (1) E A C B D (1) D B A E C (1) D A B E C (1) D A B C E (1) B E C D A (1) B E A D C (1) B D E C A (1) B C E D A (1) A E D C B (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 6 2 4 -6 B -6 0 -6 -2 -12 C -2 6 0 6 2 D -4 2 -6 0 -8 E 6 12 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.4400000001 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 A B C D E A 0 6 2 4 -6 B -6 0 -6 -2 -12 C -2 6 0 6 2 D -4 2 -6 0 -8 E 6 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.43999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=24 B=23 D=10 A=6 so A is eliminated. Round 2 votes counts: C=37 E=26 B=23 D=14 so D is eliminated. Round 3 votes counts: C=42 E=29 B=29 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 C:206 A:203 D:192 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 6 2 4 -6 B -6 0 -6 -2 -12 C -2 6 0 6 2 D -4 2 -6 0 -8 E 6 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.43999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 4 -6 B -6 0 -6 -2 -12 C -2 6 0 6 2 D -4 2 -6 0 -8 E 6 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.43999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 4 -6 B -6 0 -6 -2 -12 C -2 6 0 6 2 D -4 2 -6 0 -8 E 6 12 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.600000 D: 0.000000 E: 0.200000 Sum of squares = 0.43999999987 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.800000 D: 0.800000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9232: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) E C D B A (10) A D C E B (9) E B C D A (8) C D E B A (7) B E C D A (7) D C A E B (6) A B E D C (6) E C B D A (4) C D E A B (4) B E A C D (4) B A E D C (4) A E C D B (3) A D C B E (3) A B D E C (2) E C D A B (1) E B C A D (1) E A B C D (1) D C E A B (1) D C B A E (1) D A C B E (1) C E D A B (1) C B D E A (1) B E D A C (1) B A E C D (1) B A D C E (1) A D B C E (1) Total count = 100 A B C D E A 0 0 -4 -6 -2 B 0 0 -4 4 -12 C 4 4 0 6 -6 D 6 -4 -6 0 -4 E 2 12 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -4 -6 -2 B 0 0 -4 4 -12 C 4 4 0 6 -6 D 6 -4 -6 0 -4 E 2 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=25 B=18 C=13 D=9 so D is eliminated. Round 2 votes counts: A=36 E=25 C=21 B=18 so B is eliminated. Round 3 votes counts: A=42 E=37 C=21 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:204 D:196 A:194 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -4 -6 -2 B 0 0 -4 4 -12 C 4 4 0 6 -6 D 6 -4 -6 0 -4 E 2 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -6 -2 B 0 0 -4 4 -12 C 4 4 0 6 -6 D 6 -4 -6 0 -4 E 2 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -6 -2 B 0 0 -4 4 -12 C 4 4 0 6 -6 D 6 -4 -6 0 -4 E 2 12 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999071 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9233: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (23) B D C E A (15) B D C A E (7) E C A D B (5) D C E A B (5) D C B E A (5) B A E C D (5) A B E C D (5) B D A C E (3) B A D C E (3) E A C D B (2) E A C B D (2) D C A E B (2) D B C E A (2) C D E A B (2) B A E D C (2) A E C B D (2) E C D B A (1) E C D A B (1) D C B A E (1) D B A C E (1) B E D C A (1) B E A C D (1) B D E A C (1) B D A E C (1) B A D E C (1) A E B C D (1) Total count = 100 A B C D E A 0 0 6 4 14 B 0 0 -2 0 8 C -6 2 0 0 -8 D -4 0 0 0 -2 E -14 -8 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.515023 B: 0.484977 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500451400726 Cumulative probabilities = A: 0.515023 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 6 4 14 B 0 0 -2 0 8 C -6 2 0 0 -8 D -4 0 0 0 -2 E -14 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=31 D=16 E=11 C=2 so C is eliminated. Round 2 votes counts: B=40 A=31 D=18 E=11 so E is eliminated. Round 3 votes counts: B=40 A=40 D=20 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:212 B:203 D:197 C:194 E:194 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 6 4 14 B 0 0 -2 0 8 C -6 2 0 0 -8 D -4 0 0 0 -2 E -14 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 4 14 B 0 0 -2 0 8 C -6 2 0 0 -8 D -4 0 0 0 -2 E -14 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 4 14 B 0 0 -2 0 8 C -6 2 0 0 -8 D -4 0 0 0 -2 E -14 -8 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999988 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9234: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (14) B D E C A (12) B E D A C (9) A C E D B (6) C A B D E (5) B D C E A (5) B C D A E (5) A C B E D (4) E D A C B (3) A C E B D (3) E D B A C (2) E B D A C (2) E A D C B (2) D E B C A (2) D E B A C (2) C A D B E (2) B E A D C (2) B C A E D (2) B C A D E (2) B A E C D (2) B A C E D (2) E D A B C (1) D E C B A (1) D E C A B (1) D C B E A (1) D C A E B (1) C D B A E (1) C A E D B (1) A E D C B (1) A E C D B (1) A C D E B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -10 4 10 B 4 0 4 14 12 C 10 -4 0 6 12 D -4 -14 -6 0 10 E -10 -12 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 4 10 B 4 0 4 14 12 C 10 -4 0 6 12 D -4 -14 -6 0 10 E -10 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=23 A=18 E=10 D=8 so D is eliminated. Round 2 votes counts: B=41 C=25 A=18 E=16 so E is eliminated. Round 3 votes counts: B=49 C=27 A=24 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:212 A:200 D:193 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -10 4 10 B 4 0 4 14 12 C 10 -4 0 6 12 D -4 -14 -6 0 10 E -10 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 4 10 B 4 0 4 14 12 C 10 -4 0 6 12 D -4 -14 -6 0 10 E -10 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 4 10 B 4 0 4 14 12 C 10 -4 0 6 12 D -4 -14 -6 0 10 E -10 -12 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9235: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (13) D E B C A (7) D E B A C (6) D E A B C (6) D B E A C (6) B A C D E (5) C A E D B (4) C A B E D (4) B A D C E (4) A C B E D (4) C E D A B (3) C A E B D (3) B A D E C (3) B A C E D (3) E D C B A (2) E D B C A (2) E C D A B (2) C E A D B (2) B D A E C (2) A C E D B (2) A B D E C (2) A B D C E (2) A B C D E (2) D E C A B (1) D E A C B (1) D B A E C (1) C E D B A (1) C E B A D (1) C B A E D (1) B D E C A (1) B D E A C (1) B D C E A (1) B C E D A (1) B C A D E (1) Total count = 100 A B C D E A 0 2 0 -14 -14 B -2 0 12 -18 -12 C 0 -12 0 -22 -12 D 14 18 22 0 4 E 14 12 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -14 -14 B -2 0 12 -18 -12 C 0 -12 0 -22 -12 D 14 18 22 0 4 E 14 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=22 E=19 C=19 A=12 so A is eliminated. Round 2 votes counts: D=28 B=28 C=25 E=19 so E is eliminated. Round 3 votes counts: D=45 B=28 C=27 so C is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:229 E:217 B:190 A:187 C:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -14 -14 B -2 0 12 -18 -12 C 0 -12 0 -22 -12 D 14 18 22 0 4 E 14 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -14 -14 B -2 0 12 -18 -12 C 0 -12 0 -22 -12 D 14 18 22 0 4 E 14 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -14 -14 B -2 0 12 -18 -12 C 0 -12 0 -22 -12 D 14 18 22 0 4 E 14 12 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9236: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) C E D A B (10) C E A D B (8) E A D B C (6) D B A E C (5) C D B E A (5) B D C A E (5) B A D E C (5) A E B D C (5) E C A D B (4) C E A B D (3) C D E B A (3) B D A C E (3) E A C D B (2) E A C B D (2) D B C A E (2) C B D E A (2) C B D A E (2) A D B E C (2) E D A C B (1) E C A B D (1) E A B C D (1) D B E A C (1) D A E B C (1) C E D B A (1) C E B D A (1) C D B A E (1) C B E D A (1) C B E A D (1) B C D A E (1) B A D C E (1) A E B C D (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -2 -12 -6 B 2 0 4 -4 0 C 2 -4 0 0 0 D 12 4 0 0 2 E 6 0 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.384896 D: 0.615104 E: 0.000000 Sum of squares = 0.526497741033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.384896 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -12 -6 B 2 0 4 -4 0 C 2 -4 0 0 0 D 12 4 0 0 2 E 6 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499148 D: 0.500852 E: 0.000000 Sum of squares = 0.500001450206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499148 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=26 E=17 A=10 D=9 so D is eliminated. Round 2 votes counts: C=38 B=34 E=17 A=11 so A is eliminated. Round 3 votes counts: C=38 B=38 E=24 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:209 E:202 B:201 C:199 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -12 -6 B 2 0 4 -4 0 C 2 -4 0 0 0 D 12 4 0 0 2 E 6 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499148 D: 0.500852 E: 0.000000 Sum of squares = 0.500001450206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499148 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -12 -6 B 2 0 4 -4 0 C 2 -4 0 0 0 D 12 4 0 0 2 E 6 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499148 D: 0.500852 E: 0.000000 Sum of squares = 0.500001450206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499148 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -12 -6 B 2 0 4 -4 0 C 2 -4 0 0 0 D 12 4 0 0 2 E 6 0 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499148 D: 0.500852 E: 0.000000 Sum of squares = 0.500001450206 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499148 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9237: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (11) B A D E C (9) D E C B A (7) D B A E C (7) A B C E D (7) E C D B A (6) C E D B A (6) C E D A B (6) A B D E C (6) D E B C A (5) C E A D B (4) C E A B D (4) B D A E C (4) D B E A C (3) D E C A B (2) E D C B A (1) E D C A B (1) E C B D A (1) D E B A C (1) D B E C A (1) D A B E C (1) C E B D A (1) C E B A D (1) C A E B D (1) B C E A D (1) A D B E C (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -8 4 -6 -2 B 8 0 14 -4 4 C -4 -14 0 -20 -12 D 6 4 20 0 16 E 2 -4 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 -6 -2 B 8 0 14 -4 4 C -4 -14 0 -20 -12 D 6 4 20 0 16 E 2 -4 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 C=23 B=14 E=9 so E is eliminated. Round 2 votes counts: C=30 D=29 A=27 B=14 so B is eliminated. Round 3 votes counts: A=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:211 E:197 A:194 C:175 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 4 -6 -2 B 8 0 14 -4 4 C -4 -14 0 -20 -12 D 6 4 20 0 16 E 2 -4 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 -6 -2 B 8 0 14 -4 4 C -4 -14 0 -20 -12 D 6 4 20 0 16 E 2 -4 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 -6 -2 B 8 0 14 -4 4 C -4 -14 0 -20 -12 D 6 4 20 0 16 E 2 -4 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9238: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (8) A D C B E (7) A D B E C (7) E B C A D (6) A D C E B (6) E B C D A (4) E B A C D (4) D A C B E (4) C B E D A (4) B E A D C (4) A D E B C (4) D C A E B (3) B E C A D (3) B D A E C (3) B C E D A (3) A D B C E (3) E C B A D (2) D C A B E (2) D B A C E (2) D A C E B (2) D A B C E (2) C E B D A (2) B D C E A (2) E C A B D (1) E A C D B (1) E A B C D (1) D B C A E (1) C E D B A (1) C E D A B (1) C D B E A (1) C B D E A (1) B D E C A (1) A E B D C (1) A E B C D (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 8 6 0 B 4 0 22 4 18 C -8 -22 0 -12 -6 D -6 -4 12 0 4 E 0 -18 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 6 0 B 4 0 22 4 18 C -8 -22 0 -12 -6 D -6 -4 12 0 4 E 0 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=24 E=19 D=16 C=10 so C is eliminated. Round 2 votes counts: A=31 B=29 E=23 D=17 so D is eliminated. Round 3 votes counts: A=44 B=33 E=23 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:205 D:203 E:192 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 6 0 B 4 0 22 4 18 C -8 -22 0 -12 -6 D -6 -4 12 0 4 E 0 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 6 0 B 4 0 22 4 18 C -8 -22 0 -12 -6 D -6 -4 12 0 4 E 0 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 6 0 B 4 0 22 4 18 C -8 -22 0 -12 -6 D -6 -4 12 0 4 E 0 -18 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999983764 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9239: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) D A C E B (8) D C E B A (6) B E C A D (6) E B C D A (5) B E A C D (5) A B E C D (5) D E C B A (4) A D C B E (4) A B E D C (4) A B C E D (4) E B D C A (3) D A E B C (3) C D E B A (3) A D B E C (3) E B C A D (2) D E B A C (2) D A C B E (2) C B E A D (2) B A E C D (2) A D B C E (2) A C B D E (2) E B D A C (1) E B A C D (1) D E B C A (1) D E A B C (1) D C E A B (1) D C A B E (1) C D B E A (1) C D B A E (1) C D A B E (1) C A D B E (1) B E A D C (1) B C E A D (1) A E B D C (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 -4 8 B -6 0 8 -6 0 C -6 -8 0 -14 -2 D 4 6 14 0 12 E -8 0 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 -4 8 B -6 0 8 -6 0 C -6 -8 0 -14 -2 D 4 6 14 0 12 E -8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=27 B=15 E=12 C=9 so C is eliminated. Round 2 votes counts: D=43 A=28 B=17 E=12 so E is eliminated. Round 3 votes counts: D=43 B=29 A=28 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 A:208 B:198 E:191 C:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 6 -4 8 B -6 0 8 -6 0 C -6 -8 0 -14 -2 D 4 6 14 0 12 E -8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 -4 8 B -6 0 8 -6 0 C -6 -8 0 -14 -2 D 4 6 14 0 12 E -8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 -4 8 B -6 0 8 -6 0 C -6 -8 0 -14 -2 D 4 6 14 0 12 E -8 0 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9240: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B D A C E (7) E C A D B (5) D C A E B (5) A E B C D (4) E C D A B (3) E A C B D (3) D C B E A (3) C D E A B (3) C D A E B (3) B E C D A (3) B E A D C (3) B D C E A (3) B A E D C (3) A E C D B (3) E A B C D (2) D C A B E (2) D B C E A (2) D B C A E (2) C D E B A (2) B E D C A (2) B E A C D (2) B D E C A (2) B D C A E (2) B A D C E (2) A D C B E (2) A B E D C (2) A B D E C (2) E C A B D (1) E B C D A (1) D C B A E (1) D A C B E (1) C E D B A (1) B E D A C (1) B D E A C (1) B D A E C (1) B C E D A (1) B A D E C (1) A E B D C (1) A D B C E (1) A C D E B (1) A B D C E (1) Total count = 100 A B C D E A 0 8 6 -4 -6 B -8 0 4 2 6 C -6 -4 0 -6 -10 D 4 -2 6 0 0 E 6 -6 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428576 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -4 -6 B -8 0 4 2 6 C -6 -4 0 -6 -10 D 4 -2 6 0 0 E 6 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428645 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=24 A=17 D=16 C=9 so C is eliminated. Round 2 votes counts: B=34 E=25 D=24 A=17 so A is eliminated. Round 3 votes counts: B=39 E=33 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:205 D:204 A:202 B:202 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -4 -6 B -8 0 4 2 6 C -6 -4 0 -6 -10 D 4 -2 6 0 0 E 6 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428645 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -4 -6 B -8 0 4 2 6 C -6 -4 0 -6 -10 D 4 -2 6 0 0 E 6 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428645 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -4 -6 B -8 0 4 2 6 C -6 -4 0 -6 -10 D 4 -2 6 0 0 E 6 -6 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428645 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9241: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) D C E B A (10) C D E A B (8) A B E D C (8) A B E C D (8) D E B A C (6) C A B E D (6) D B A E C (5) B A E D C (5) C E D A B (4) A E B C D (4) D C B A E (3) C E A B D (3) E A B D C (2) D E C B A (2) D B E A C (2) C D B A E (2) C D A E B (2) E D C A B (1) E D B A C (1) E C A B D (1) E B D A C (1) E B A D C (1) E A B C D (1) D B A C E (1) C A E B D (1) C A B D E (1) Total count = 100 A B C D E A 0 0 -10 -18 -8 B 0 0 -10 -16 -18 C 10 10 0 4 4 D 18 16 -4 0 6 E 8 18 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999613 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -10 -18 -8 B 0 0 -10 -16 -18 C 10 10 0 4 4 D 18 16 -4 0 6 E 8 18 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 D=29 A=20 E=8 B=5 so B is eliminated. Round 2 votes counts: C=38 D=29 A=25 E=8 so E is eliminated. Round 3 votes counts: C=39 D=32 A=29 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:218 C:214 E:208 A:182 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -10 -18 -8 B 0 0 -10 -16 -18 C 10 10 0 4 4 D 18 16 -4 0 6 E 8 18 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 -18 -8 B 0 0 -10 -16 -18 C 10 10 0 4 4 D 18 16 -4 0 6 E 8 18 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 -18 -8 B 0 0 -10 -16 -18 C 10 10 0 4 4 D 18 16 -4 0 6 E 8 18 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9242: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (8) E D A C B (5) B C E D A (5) E D B A C (4) D E A B C (4) D A B E C (4) C B E A D (4) C B A E D (4) C B A D E (4) C A E D B (4) A D E C B (4) A C D E B (4) E D A B C (3) E C D A B (3) B D E A C (3) E D C A B (2) E D B C A (2) E C A D B (2) E B D C A (2) D E A C B (2) D A E C B (2) D A E B C (2) B E D A C (2) B E C D A (2) B D E C A (2) B A D C E (2) E C D B A (1) D E B A C (1) D B A E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E D A (1) B E D C A (1) B D A E C (1) B D A C E (1) B C A E D (1) A E D C B (1) A D E B C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -2 -12 -8 B 6 0 6 -8 -4 C 2 -6 0 -6 -14 D 12 8 6 0 -4 E 8 4 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -2 -12 -8 B 6 0 6 -8 -4 C 2 -6 0 -6 -14 D 12 8 6 0 -4 E 8 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 C=20 D=16 A=12 so A is eliminated. Round 2 votes counts: B=29 E=25 C=25 D=21 so D is eliminated. Round 3 votes counts: E=41 B=34 C=25 so C is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:211 B:200 C:188 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -2 -12 -8 B 6 0 6 -8 -4 C 2 -6 0 -6 -14 D 12 8 6 0 -4 E 8 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -12 -8 B 6 0 6 -8 -4 C 2 -6 0 -6 -14 D 12 8 6 0 -4 E 8 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -12 -8 B 6 0 6 -8 -4 C 2 -6 0 -6 -14 D 12 8 6 0 -4 E 8 4 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9243: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (7) B C E D A (7) A D E C B (7) C B D E A (5) C B A E D (4) A D C E B (4) A C D B E (4) D E A B C (3) D A E C B (3) C D B A E (3) C B E D A (3) B E D C A (3) B E C D A (3) A E D B C (3) A C B D E (3) E D B C A (2) D C A E B (2) C D B E A (2) C B E A D (2) C B A D E (2) A E B D C (2) A E B C D (2) A C B E D (2) E D A B C (1) E B D C A (1) E B A D C (1) E B A C D (1) E A D B C (1) D E C B A (1) D E A C B (1) D C E B A (1) D B E C A (1) C D A B E (1) C B D A E (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C A D (1) B E A C D (1) B C E A D (1) B C A E D (1) B A E C D (1) A E D C B (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -14 -2 2 B 10 0 -12 0 10 C 14 12 0 10 6 D 2 0 -10 0 8 E -2 -10 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -2 2 B 10 0 -12 0 10 C 14 12 0 10 6 D 2 0 -10 0 8 E -2 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 D=19 B=18 E=7 so E is eliminated. Round 2 votes counts: A=31 C=26 D=22 B=21 so B is eliminated. Round 3 votes counts: C=39 A=35 D=26 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 B:204 D:200 A:188 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -2 2 B 10 0 -12 0 10 C 14 12 0 10 6 D 2 0 -10 0 8 E -2 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -2 2 B 10 0 -12 0 10 C 14 12 0 10 6 D 2 0 -10 0 8 E -2 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -2 2 B 10 0 -12 0 10 C 14 12 0 10 6 D 2 0 -10 0 8 E -2 -10 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9244: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (9) D B A C E (9) C E A B D (9) A C E D B (8) B D E C A (7) E C B D A (5) D B E C A (5) E C A B D (4) B D C E A (4) A D C B E (4) E C B A D (3) E B D C A (3) E B C D A (3) D B E A C (3) C A E B D (3) A C E B D (3) A C D E B (3) D B C A E (2) D A B E C (2) D A B C E (2) E D B C A (1) E A C B D (1) D E B A C (1) D B C E A (1) C B D E A (1) B D C A E (1) A E C D B (1) A D E C B (1) A C D B E (1) Total count = 100 A B C D E A 0 -16 -4 -18 -2 B 16 0 6 -6 2 C 4 -6 0 -10 2 D 18 6 10 0 12 E 2 -2 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -18 -2 B 16 0 6 -6 2 C 4 -6 0 -10 2 D 18 6 10 0 12 E 2 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 A=21 E=20 C=13 B=12 so B is eliminated. Round 2 votes counts: D=46 A=21 E=20 C=13 so C is eliminated. Round 3 votes counts: D=47 E=29 A=24 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:223 B:209 C:195 E:193 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 -4 -18 -2 B 16 0 6 -6 2 C 4 -6 0 -10 2 D 18 6 10 0 12 E 2 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -18 -2 B 16 0 6 -6 2 C 4 -6 0 -10 2 D 18 6 10 0 12 E 2 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -18 -2 B 16 0 6 -6 2 C 4 -6 0 -10 2 D 18 6 10 0 12 E 2 -2 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9245: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) E B C A D (8) D A E C B (8) B C E D A (7) E A D C B (6) C B D A E (6) B C E A D (5) A D E C B (5) E B D A C (4) D A C B E (4) B E C A D (4) B C D A E (4) A D C E B (4) E B A D C (3) D A C E B (3) C D A B E (3) E D A B C (2) E A B D C (2) D A E B C (2) C D B A E (2) C B E A D (2) D A B E C (1) C B A E D (1) C B A D E (1) B E D C A (1) B E C D A (1) B C D E A (1) A D C B E (1) Total count = 100 A B C D E A 0 0 8 2 -10 B 0 0 8 0 -12 C -8 -8 0 -10 -12 D -2 0 10 0 -10 E 10 12 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 2 -10 B 0 0 8 0 -12 C -8 -8 0 -10 -12 D -2 0 10 0 -10 E 10 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 B=23 D=18 C=15 A=10 so A is eliminated. Round 2 votes counts: E=34 D=28 B=23 C=15 so C is eliminated. Round 3 votes counts: E=34 D=33 B=33 so D is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 A:200 D:199 B:198 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 2 -10 B 0 0 8 0 -12 C -8 -8 0 -10 -12 D -2 0 10 0 -10 E 10 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 2 -10 B 0 0 8 0 -12 C -8 -8 0 -10 -12 D -2 0 10 0 -10 E 10 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 2 -10 B 0 0 8 0 -12 C -8 -8 0 -10 -12 D -2 0 10 0 -10 E 10 12 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9246: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (9) A E B C D (6) A D C E B (6) A E C B D (5) E B C D A (4) E B C A D (4) D C B E A (4) D A B C E (4) A D C B E (4) A D B C E (4) E C B D A (3) A D B E C (3) A B E D C (3) A B D E C (3) E C B A D (2) C E D B A (2) C E B D A (2) C E A D B (2) C D E B A (2) C D E A B (2) C A D E B (2) A C D E B (2) E C A B D (1) E B A C D (1) D C B A E (1) D C A B E (1) D B E C A (1) D B C E A (1) D B A E C (1) D B A C E (1) D A C E B (1) C E D A B (1) C E A B D (1) C D A E B (1) C A E D B (1) B E D C A (1) B E C A D (1) B D E C A (1) B C D E A (1) A E C D B (1) A E B D C (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 16 -2 12 8 B -16 0 0 0 -12 C 2 0 0 16 -4 D -12 0 -16 0 -6 E -8 12 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428382 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 16 -2 12 8 B -16 0 0 0 -12 C 2 0 0 16 -4 D -12 0 -16 0 -6 E -8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428558 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 C=16 E=15 D=15 B=13 so B is eliminated. Round 2 votes counts: A=41 E=26 C=17 D=16 so D is eliminated. Round 3 votes counts: A=48 E=28 C=24 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:207 E:207 B:186 D:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 -2 12 8 B -16 0 0 0 -12 C 2 0 0 16 -4 D -12 0 -16 0 -6 E -8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428558 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 12 8 B -16 0 0 0 -12 C 2 0 0 16 -4 D -12 0 -16 0 -6 E -8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428558 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 12 8 B -16 0 0 0 -12 C 2 0 0 16 -4 D -12 0 -16 0 -6 E -8 12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.142857 Sum of squares = 0.428571428558 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9247: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) A C E D B (7) A C B D E (7) E A B D C (5) A E C B D (4) A E B C D (4) E D B C A (3) C D A B E (3) C B D A E (3) A E B D C (3) A C E B D (3) E B D C A (2) E B A D C (2) D C B E A (2) D B E C A (2) D B C E A (2) C D B E A (2) C D B A E (2) C B D E A (2) C A D E B (2) C A D B E (2) C A B D E (2) B E D C A (2) B E A D C (2) B D E C A (2) B A E D C (2) A E C D B (2) A C D B E (2) A B E C D (2) E D B A C (1) E D A C B (1) E A D C B (1) E A D B C (1) C D E A B (1) B E D A C (1) B D C E A (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E D C B (1) A E D B C (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 12 24 18 14 B -12 0 -2 22 -4 C -24 2 0 12 -2 D -18 -22 -12 0 -18 E -14 4 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 24 18 14 B -12 0 -2 22 -4 C -24 2 0 12 -2 D -18 -22 -12 0 -18 E -14 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=23 C=19 B=14 D=6 so D is eliminated. Round 2 votes counts: A=38 E=23 C=21 B=18 so B is eliminated. Round 3 votes counts: A=41 E=32 C=27 so C is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:205 B:202 C:194 D:165 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 24 18 14 B -12 0 -2 22 -4 C -24 2 0 12 -2 D -18 -22 -12 0 -18 E -14 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 24 18 14 B -12 0 -2 22 -4 C -24 2 0 12 -2 D -18 -22 -12 0 -18 E -14 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 24 18 14 B -12 0 -2 22 -4 C -24 2 0 12 -2 D -18 -22 -12 0 -18 E -14 4 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9248: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (11) A C B D E (8) E C A D B (5) E D B C A (4) E B C A D (4) D E B A C (4) A D C B E (4) E D C A B (3) E B D C A (3) D A B C E (3) B E D C A (3) B D A C E (3) B A C D E (3) A C D B E (3) E B C D A (2) D B A E C (2) D B A C E (2) D A E C B (2) D A C B E (2) C A B D E (2) B E C A D (2) B D E A C (2) B C E A D (2) B C A E D (2) E D C B A (1) E D B A C (1) E C D A B (1) E C B A D (1) D E A C B (1) D E A B C (1) D B E A C (1) D A C E B (1) D A B E C (1) C E B A D (1) C B A E D (1) C A E D B (1) C A E B D (1) B E C D A (1) B A D C E (1) A D C E B (1) A D B C E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -2 12 14 B -8 0 0 10 24 C 2 0 0 4 10 D -12 -10 -4 0 0 E -14 -24 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.139396 C: 0.860604 D: 0.000000 E: 0.000000 Sum of squares = 0.760069800501 Cumulative probabilities = A: 0.000000 B: 0.139396 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 12 14 B -8 0 0 10 24 C 2 0 0 4 10 D -12 -10 -4 0 0 E -14 -24 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000114159 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=20 B=19 A=19 C=17 so C is eliminated. Round 2 votes counts: A=34 E=26 D=20 B=20 so D is eliminated. Round 3 votes counts: A=43 E=32 B=25 so B is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:216 B:213 C:208 D:187 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 12 14 B -8 0 0 10 24 C 2 0 0 4 10 D -12 -10 -4 0 0 E -14 -24 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000114159 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 12 14 B -8 0 0 10 24 C 2 0 0 4 10 D -12 -10 -4 0 0 E -14 -24 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000114159 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 12 14 B -8 0 0 10 24 C 2 0 0 4 10 D -12 -10 -4 0 0 E -14 -24 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000114159 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9249: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (11) B A C E D (9) E D C A B (8) E D B C A (8) D C E A B (7) B E A C D (7) A C B D E (7) E B D A C (6) C D A B E (6) E B A D C (5) B A C D E (5) C A D B E (4) A B C D E (4) E D C B A (3) D C A E B (3) E B A C D (2) B A E C D (2) E D B A C (1) D C A B E (1) C A B D E (1) Total count = 100 A B C D E A 0 4 -4 -8 -16 B -4 0 -2 -4 -8 C 4 2 0 -6 -6 D 8 4 6 0 -2 E 16 8 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -4 -8 -16 B -4 0 -2 -4 -8 C 4 2 0 -6 -6 D 8 4 6 0 -2 E 16 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=23 D=22 C=11 A=11 so C is eliminated. Round 2 votes counts: E=33 D=28 B=23 A=16 so A is eliminated. Round 3 votes counts: B=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:208 C:197 B:191 A:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -4 -8 -16 B -4 0 -2 -4 -8 C 4 2 0 -6 -6 D 8 4 6 0 -2 E 16 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -8 -16 B -4 0 -2 -4 -8 C 4 2 0 -6 -6 D 8 4 6 0 -2 E 16 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -8 -16 B -4 0 -2 -4 -8 C 4 2 0 -6 -6 D 8 4 6 0 -2 E 16 8 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999998882 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9250: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E C D B A (8) C E D B A (8) C E A D B (7) C E D A B (6) B D A E C (6) D B E C A (5) A B D E C (4) D E C B A (3) C A E D B (3) B A D E C (3) A E C B D (3) A C E B D (3) E D C B A (2) E C D A B (2) E C A B D (2) D B A C E (2) B D A C E (2) B A D C E (2) A C B E D (2) E B D C A (1) E A C B D (1) E A B C D (1) D E B C A (1) D C E B A (1) D B C E A (1) D A B C E (1) B E D C A (1) B D E C A (1) B A E D C (1) A D B C E (1) A C E D B (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -4 -2 -2 B -4 0 -4 -4 -6 C 4 4 0 -2 8 D 2 4 2 0 -8 E 2 6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.444444 E: 0.111111 Sum of squares = 0.407407407392 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.888889 E: 1.000000 A B C D E A 0 4 -4 -2 -2 B -4 0 -4 -4 -6 C 4 4 0 -2 8 D 2 4 2 0 -8 E 2 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.444444 E: 0.111111 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=24 E=17 B=16 D=14 so D is eliminated. Round 2 votes counts: A=30 C=25 B=24 E=21 so E is eliminated. Round 3 votes counts: C=42 A=32 B=26 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 E:204 D:200 A:198 B:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -4 -2 -2 B -4 0 -4 -4 -6 C 4 4 0 -2 8 D 2 4 2 0 -8 E 2 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.444444 E: 0.111111 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.888889 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 -2 -2 B -4 0 -4 -4 -6 C 4 4 0 -2 8 D 2 4 2 0 -8 E 2 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.444444 E: 0.111111 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.888889 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 -2 -2 B -4 0 -4 -4 -6 C 4 4 0 -2 8 D 2 4 2 0 -8 E 2 6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.444444 E: 0.111111 Sum of squares = 0.407407407409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.444444 D: 0.888889 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9251: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) D E B C A (5) C A E B D (5) A C B E D (5) E D C B A (4) E C B A D (4) A D B C E (4) D E C B A (3) D E C A B (3) D B E A C (3) D A B C E (3) C E A D B (3) C E A B D (3) B E C A D (3) B D A E C (3) B A D C E (3) A D C B E (3) A B D C E (3) E C D B A (2) E C D A B (2) E C B D A (2) C D E A B (2) C A D E B (2) C A B E D (2) B E D C A (2) B D E A C (2) B A C E D (2) E D B C A (1) D E B A C (1) D E A B C (1) D A E C B (1) D A C E B (1) D A C B E (1) D A B E C (1) C E D A B (1) C B A E D (1) C A E D B (1) B E D A C (1) B A D E C (1) B A C D E (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -2 -4 4 B 2 0 -4 -12 6 C 2 4 0 -14 -4 D 4 12 14 0 10 E -4 -6 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -4 4 B 2 0 -4 -12 6 C 2 4 0 -14 -4 D 4 12 14 0 10 E -4 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=20 B=18 A=17 E=15 so E is eliminated. Round 2 votes counts: D=35 C=30 B=18 A=17 so A is eliminated. Round 3 votes counts: D=42 C=36 B=22 so B is eliminated. Round 4 votes counts: D=57 C=43 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:220 A:198 B:196 C:194 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -2 -4 4 B 2 0 -4 -12 6 C 2 4 0 -14 -4 D 4 12 14 0 10 E -4 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -4 4 B 2 0 -4 -12 6 C 2 4 0 -14 -4 D 4 12 14 0 10 E -4 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -4 4 B 2 0 -4 -12 6 C 2 4 0 -14 -4 D 4 12 14 0 10 E -4 -6 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999727 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9252: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (11) E A B C D (7) E C A B D (5) E A C D B (4) D C A E B (4) D B C A E (4) D A C E B (4) C D B E A (4) B A E D C (4) D B A E C (3) C B D E A (3) B D A E C (3) B C D E A (3) A E D C B (3) D C B A E (2) C E D A B (2) C E B A D (2) C E A B D (2) C B E D A (2) C B E A D (2) A E D B C (2) A E C D B (2) A E B D C (2) A D E B C (2) E C B A D (1) E C A D B (1) E A C B D (1) D C A B E (1) D B C E A (1) D B A C E (1) D A B E C (1) D A B C E (1) C D E A B (1) C D A E B (1) C A E D B (1) B E C A D (1) B E A D C (1) B E A C D (1) B D C A E (1) B C E A D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 20 -12 16 -12 B -20 0 -18 -14 -16 C 12 18 0 16 8 D -16 14 -16 0 -18 E 12 16 -8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -12 16 -12 B -20 0 -18 -14 -16 C 12 18 0 16 8 D -16 14 -16 0 -18 E 12 16 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=22 E=19 B=15 A=13 so A is eliminated. Round 2 votes counts: C=31 E=28 D=25 B=16 so B is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:219 A:206 D:182 B:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -12 16 -12 B -20 0 -18 -14 -16 C 12 18 0 16 8 D -16 14 -16 0 -18 E 12 16 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -12 16 -12 B -20 0 -18 -14 -16 C 12 18 0 16 8 D -16 14 -16 0 -18 E 12 16 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -12 16 -12 B -20 0 -18 -14 -16 C 12 18 0 16 8 D -16 14 -16 0 -18 E 12 16 -8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9253: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) E B C A D (10) D A C B E (8) E B C D A (7) E C B D A (6) E D B A C (5) D A E B C (5) A D C B E (5) D A B E C (4) B E C A D (4) A D B C E (4) E C B A D (3) E D C B A (2) C D E A B (2) C B E A D (2) C B A E D (2) A C D B E (2) E D C A B (1) E D B C A (1) E D A B C (1) E C D B A (1) E B D C A (1) E B D A C (1) D E C A B (1) D C A E B (1) D C A B E (1) D B A E C (1) D A E C B (1) D A B C E (1) C E D B A (1) C D A E B (1) C A D B E (1) C A B E D (1) B A E D C (1) A C B D E (1) Total count = 100 A B C D E A 0 -18 -20 -6 -20 B 18 0 -8 0 -24 C 20 8 0 10 -12 D 6 0 -10 0 -22 E 20 24 12 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -20 -6 -20 B 18 0 -8 0 -24 C 20 8 0 10 -12 D 6 0 -10 0 -22 E 20 24 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 D=23 C=21 A=12 B=5 so B is eliminated. Round 2 votes counts: E=43 D=23 C=21 A=13 so A is eliminated. Round 3 votes counts: E=44 D=32 C=24 so C is eliminated. Round 4 votes counts: E=61 D=39 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:239 C:213 B:193 D:187 A:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -20 -6 -20 B 18 0 -8 0 -24 C 20 8 0 10 -12 D 6 0 -10 0 -22 E 20 24 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -20 -6 -20 B 18 0 -8 0 -24 C 20 8 0 10 -12 D 6 0 -10 0 -22 E 20 24 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -20 -6 -20 B 18 0 -8 0 -24 C 20 8 0 10 -12 D 6 0 -10 0 -22 E 20 24 12 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9254: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (11) A B C D E (10) E D C B A (8) A C B E D (7) A B C E D (6) B C D A E (5) E A D C B (3) D B C E A (3) B C D E A (3) E D C A B (2) E D A C B (2) E A D B C (2) D E C B A (2) D C B E A (2) C D B E A (2) B C A D E (2) A E B C D (2) A B E C D (2) E D B A C (1) E D A B C (1) D B E C A (1) C B D E A (1) C B D A E (1) C A B D E (1) B E D A C (1) B D C E A (1) B A C D E (1) A E D C B (1) A E D B C (1) A E C D B (1) A E C B D (1) A E B D C (1) A C E D B (1) Total count = 100 A B C D E A 0 10 10 6 8 B -10 0 8 12 20 C -10 -8 0 14 14 D -6 -12 -14 0 -8 E -8 -20 -14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 6 8 B -10 0 8 12 20 C -10 -8 0 14 14 D -6 -12 -14 0 -8 E -8 -20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998119 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=44 E=30 B=13 D=8 C=5 so C is eliminated. Round 2 votes counts: A=45 E=30 B=15 D=10 so D is eliminated. Round 3 votes counts: A=45 E=32 B=23 so B is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:215 C:205 E:183 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 6 8 B -10 0 8 12 20 C -10 -8 0 14 14 D -6 -12 -14 0 -8 E -8 -20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998119 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 6 8 B -10 0 8 12 20 C -10 -8 0 14 14 D -6 -12 -14 0 -8 E -8 -20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998119 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 6 8 B -10 0 8 12 20 C -10 -8 0 14 14 D -6 -12 -14 0 -8 E -8 -20 -14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998119 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9255: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) C D E B A (6) C D B A E (6) E C D A B (5) E A C B D (4) D C E B A (4) D B A C E (4) B D A C E (4) E C A B D (3) E A D B C (3) E A B C D (3) D B C A E (3) C E D A B (3) C E A B D (3) C B D A E (3) C B A D E (3) A B E D C (3) E C A D B (2) D E A B C (2) D C B A E (2) D B A E C (2) D A B E C (2) C D B E A (2) C A E B D (2) B C A D E (2) B A C D E (2) A B E C D (2) E D A C B (1) E D A B C (1) E A D C B (1) D E C B A (1) D C B E A (1) C E D B A (1) C E B D A (1) C E B A D (1) C B D E A (1) C B A E D (1) B C D A E (1) B A E C D (1) B A D C E (1) A E B C D (1) Total count = 100 A B C D E A 0 -6 -14 -12 -10 B 6 0 -14 -4 -8 C 14 14 0 18 14 D 12 4 -18 0 4 E 10 8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 -12 -10 B 6 0 -14 -4 -8 C 14 14 0 18 14 D 12 4 -18 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=29 D=21 B=11 A=6 so A is eliminated. Round 2 votes counts: C=33 E=30 D=21 B=16 so B is eliminated. Round 3 votes counts: C=38 E=36 D=26 so D is eliminated. Round 4 votes counts: C=57 E=43 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:230 D:201 E:200 B:190 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -12 -10 B 6 0 -14 -4 -8 C 14 14 0 18 14 D 12 4 -18 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -12 -10 B 6 0 -14 -4 -8 C 14 14 0 18 14 D 12 4 -18 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -12 -10 B 6 0 -14 -4 -8 C 14 14 0 18 14 D 12 4 -18 0 4 E 10 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9256: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (6) D A C E B (6) D C A E B (5) C D A E B (5) B E A D C (5) B E A C D (5) B A E C D (5) E D C B A (4) E B D C A (4) E B C D A (4) A D C B E (4) A C D B E (4) A B E D C (4) A B D E C (4) E B D A C (3) D C E A B (3) C D E B A (3) A D C E B (3) A C B D E (3) C A D B E (2) A B C E D (2) E C B D A (1) E B A D C (1) D E C A B (1) D C E B A (1) D A E C B (1) C E D B A (1) C E B D A (1) C D E A B (1) C B E D A (1) C B E A D (1) C B A E D (1) B C E A D (1) B A E D C (1) A E D B C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 -2 6 B -2 0 -16 -2 -10 C -8 16 0 -14 -2 D 2 2 14 0 6 E -6 10 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998618 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -2 6 B -2 0 -16 -2 -10 C -8 16 0 -14 -2 D 2 2 14 0 6 E -6 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=23 E=17 B=17 C=16 so C is eliminated. Round 2 votes counts: D=32 A=29 B=20 E=19 so E is eliminated. Round 3 votes counts: D=37 B=34 A=29 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:212 A:207 E:200 C:196 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -2 6 B -2 0 -16 -2 -10 C -8 16 0 -14 -2 D 2 2 14 0 6 E -6 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -2 6 B -2 0 -16 -2 -10 C -8 16 0 -14 -2 D 2 2 14 0 6 E -6 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -2 6 B -2 0 -16 -2 -10 C -8 16 0 -14 -2 D 2 2 14 0 6 E -6 10 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9257: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) A B D E C (10) D C E B A (9) B A D C E (9) E C D A B (7) C E D B A (7) A B E C D (7) E C A D B (4) A B D C E (4) D E C B A (3) A E B C D (3) E C D B A (2) C E B A D (2) B D C A E (2) B D A C E (2) B A C E D (2) B A C D E (2) A B E D C (2) E D C A B (1) E C A B D (1) E A C D B (1) E A C B D (1) D B A E C (1) D B A C E (1) C E B D A (1) C D E B A (1) A E C D B (1) A D E C B (1) A D E B C (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 0 6 0 B 8 0 16 -2 8 C 0 -16 0 -14 6 D -6 2 14 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999997 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 6 0 B 8 0 16 -2 8 C 0 -16 0 -14 6 D -6 2 14 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000552 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=24 E=17 B=17 C=11 so C is eliminated. Round 2 votes counts: A=31 E=27 D=25 B=17 so B is eliminated. Round 3 votes counts: A=44 D=29 E=27 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:215 D:212 A:199 C:188 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 6 0 B 8 0 16 -2 8 C 0 -16 0 -14 6 D -6 2 14 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000552 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 6 0 B 8 0 16 -2 8 C 0 -16 0 -14 6 D -6 2 14 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000552 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 6 0 B 8 0 16 -2 8 C 0 -16 0 -14 6 D -6 2 14 0 14 E 0 -8 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.375000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000552 Cumulative probabilities = A: 0.125000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9258: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (10) B C E A D (8) D A E C B (7) D A C E B (7) D E B A C (6) E B D C A (5) D E A B C (5) C B A E D (5) B E C A D (5) C A D B E (4) B E C D A (4) C A B D E (3) A E C D B (3) A D C E B (3) A C D E B (3) A C D B E (3) E B C D A (2) E B C A D (2) D A C B E (2) B C E D A (2) A D C B E (2) E D A B C (1) D E A C B (1) D B E A C (1) B E D C A (1) B D E C A (1) B D C E A (1) B C D E A (1) B C A E D (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 10 -14 -12 B 10 0 12 -18 -12 C -10 -12 0 -6 -8 D 14 18 6 0 0 E 12 12 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.622230 E: 0.377770 Sum of squares = 0.529880305339 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.622230 E: 1.000000 A B C D E A 0 -10 10 -14 -12 B 10 0 12 -18 -12 C -10 -12 0 -6 -8 D 14 18 6 0 0 E 12 12 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=24 E=20 A=15 C=12 so C is eliminated. Round 2 votes counts: D=29 B=29 A=22 E=20 so E is eliminated. Round 3 votes counts: D=40 B=38 A=22 so A is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:219 E:216 B:196 A:187 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 10 -14 -12 B 10 0 12 -18 -12 C -10 -12 0 -6 -8 D 14 18 6 0 0 E 12 12 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -14 -12 B 10 0 12 -18 -12 C -10 -12 0 -6 -8 D 14 18 6 0 0 E 12 12 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -14 -12 B 10 0 12 -18 -12 C -10 -12 0 -6 -8 D 14 18 6 0 0 E 12 12 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9259: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (11) C E B D A (10) A D B E C (8) D E C B A (5) A B E C D (5) E C D B A (4) B E C A D (4) A B C E D (4) D C A B E (3) D A E B C (3) D A C B E (3) C B E D A (3) C B E A D (3) B A E C D (3) E D C B A (2) E C B D A (2) E B C A D (2) E B A C D (2) D C E A B (2) D A C E B (2) D A B E C (2) D A B C E (2) A D B C E (2) E D A B C (1) E B C D A (1) D E C A B (1) D E A B C (1) C E D B A (1) C B A E D (1) B E A C D (1) B C A E D (1) B A C E D (1) A D E B C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -12 -18 -12 B 14 0 -6 -8 -2 C 12 6 0 -4 0 D 18 8 4 0 -4 E 12 2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.346588 D: 0.000000 E: 0.653412 Sum of squares = 0.547070367911 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.346588 D: 0.346588 E: 1.000000 A B C D E A 0 -14 -12 -18 -12 B 14 0 -6 -8 -2 C 12 6 0 -4 0 D 18 8 4 0 -4 E 12 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.000000 E: 0.500379 Sum of squares = 0.500000287872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.499621 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=23 C=18 E=14 B=10 so B is eliminated. Round 2 votes counts: D=35 A=27 E=19 C=19 so E is eliminated. Round 3 votes counts: D=38 C=32 A=30 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:209 C:207 B:199 A:172 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -12 -18 -12 B 14 0 -6 -8 -2 C 12 6 0 -4 0 D 18 8 4 0 -4 E 12 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.000000 E: 0.500379 Sum of squares = 0.500000287872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.499621 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -18 -12 B 14 0 -6 -8 -2 C 12 6 0 -4 0 D 18 8 4 0 -4 E 12 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.000000 E: 0.500379 Sum of squares = 0.500000287872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.499621 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -18 -12 B 14 0 -6 -8 -2 C 12 6 0 -4 0 D 18 8 4 0 -4 E 12 2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.000000 E: 0.500379 Sum of squares = 0.500000287872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499621 D: 0.499621 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9260: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (17) D E A B C (11) D E A C B (9) E A D C B (7) A E C D B (6) C A E B D (5) D B E A C (4) B C D E A (4) B C A E D (4) B C D A E (3) A E D C B (3) E A C D B (2) D E B A C (2) C E A D B (2) C B E A D (2) C A E D B (2) B D A E C (2) A C B E D (2) E D A C B (1) D B A E C (1) C E D A B (1) C E A B D (1) B D E A C (1) B D C E A (1) B D C A E (1) B D A C E (1) B C A D E (1) A E D B C (1) A E B C D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 12 12 16 4 B -12 0 -20 -4 -8 C -12 20 0 8 -6 D -16 4 -8 0 -18 E -4 8 6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999533 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 16 4 B -12 0 -20 -4 -8 C -12 20 0 8 -6 D -16 4 -8 0 -18 E -4 8 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=27 B=18 A=15 E=10 so E is eliminated. Round 2 votes counts: C=30 D=28 A=24 B=18 so B is eliminated. Round 3 votes counts: C=42 D=34 A=24 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:222 E:214 C:205 D:181 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 16 4 B -12 0 -20 -4 -8 C -12 20 0 8 -6 D -16 4 -8 0 -18 E -4 8 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 16 4 B -12 0 -20 -4 -8 C -12 20 0 8 -6 D -16 4 -8 0 -18 E -4 8 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 16 4 B -12 0 -20 -4 -8 C -12 20 0 8 -6 D -16 4 -8 0 -18 E -4 8 6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999056 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9261: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) D C B E A (10) B E A D C (10) A E B C D (10) C D A E B (6) D C A E B (5) D C B A E (4) B E D A C (4) A C E D B (4) D B E A C (3) D B C E A (3) C D B E A (3) B D E C A (3) B D E A C (3) A E C B D (3) E A B C D (2) D C A B E (2) C A E D B (2) E B A C D (1) E A B D C (1) D A C E B (1) C D B A E (1) C A D E B (1) B E C A D (1) B D C E A (1) B C E D A (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -18 14 0 -14 B 18 0 12 4 20 C -14 -12 0 -8 -12 D 0 -4 8 0 -6 E 14 -20 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 14 0 -14 B 18 0 12 4 20 C -14 -12 0 -8 -12 D 0 -4 8 0 -6 E 14 -20 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=28 A=21 C=13 E=4 so E is eliminated. Round 2 votes counts: B=35 D=28 A=24 C=13 so C is eliminated. Round 3 votes counts: D=38 B=35 A=27 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:227 E:206 D:199 A:191 C:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 14 0 -14 B 18 0 12 4 20 C -14 -12 0 -8 -12 D 0 -4 8 0 -6 E 14 -20 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 14 0 -14 B 18 0 12 4 20 C -14 -12 0 -8 -12 D 0 -4 8 0 -6 E 14 -20 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 14 0 -14 B 18 0 12 4 20 C -14 -12 0 -8 -12 D 0 -4 8 0 -6 E 14 -20 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998625 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9262: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) D E C B A (7) A B C E D (6) A B E C D (5) E D C A B (4) E A B D C (4) C A B E D (4) E D B A C (3) E D A C B (3) D C B A E (3) C B A D E (3) B A E D C (3) B A C D E (3) A C B E D (3) E D A B C (2) E B A D C (2) E A B C D (2) D E C A B (2) D E B A C (2) D B A C E (2) C D B A E (2) C D A B E (2) C A E B D (2) C A B D E (2) B A D E C (2) B A C E D (2) E C D A B (1) E C A D B (1) E A C B D (1) D E B C A (1) D C E A B (1) D B E C A (1) D B E A C (1) D B C E A (1) C E D A B (1) C E A D B (1) C D E B A (1) C B D A E (1) B C D A E (1) B C A D E (1) B A E C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 -2 0 C 4 8 0 -8 2 D 2 2 8 0 -4 E 2 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428659 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 1.000000 A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 -2 0 C 4 8 0 -8 2 D 2 2 8 0 -4 E 2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=23 C=19 A=15 B=13 so B is eliminated. Round 2 votes counts: D=30 A=26 E=23 C=21 so C is eliminated. Round 3 votes counts: A=38 D=37 E=25 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:204 C:203 E:202 B:197 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 -2 0 C 4 8 0 -8 2 D 2 2 8 0 -4 E 2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 -2 0 C 4 8 0 -8 2 D 2 2 8 0 -4 E 2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 -2 B 4 0 -8 -2 0 C 4 8 0 -8 2 D 2 2 8 0 -4 E 2 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.142857 E: 0.571429 Sum of squares = 0.428571428599 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9263: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) C A D E B (7) D A C E B (6) C A E D B (6) C A E B D (5) E B A C D (4) D B C A E (4) B E C A D (4) E B A D C (3) E A D C B (3) C D A B E (3) C A B E D (3) B D E C A (3) B D E A C (3) E D A B C (2) E B D A C (2) E A D B C (2) D E A C B (2) D C A E B (2) D C A B E (2) D B E A C (2) A C E D B (2) E D B A C (1) E D A C B (1) E B C A D (1) E A C D B (1) E A C B D (1) D E B A C (1) D C B A E (1) D A E C B (1) C B A D E (1) C A D B E (1) B E A C D (1) B D C E A (1) B D C A E (1) B C E A D (1) B C D E A (1) B C D A E (1) A E C D B (1) A D E C B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 8 4 0 0 B -8 0 -4 -6 -14 C -4 4 0 -8 0 D 0 6 8 0 -10 E 0 14 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.594260 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.405740 Sum of squares = 0.517769946962 Cumulative probabilities = A: 0.594260 B: 0.594260 C: 0.594260 D: 0.594260 E: 1.000000 A B C D E A 0 8 4 0 0 B -8 0 -4 -6 -14 C -4 4 0 -8 0 D 0 6 8 0 -10 E 0 14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=26 B=26 E=21 D=21 A=6 so A is eliminated. Round 2 votes counts: C=30 B=26 E=22 D=22 so E is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:212 A:206 D:202 C:196 B:184 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 0 0 B -8 0 -4 -6 -14 C -4 4 0 -8 0 D 0 6 8 0 -10 E 0 14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 0 0 B -8 0 -4 -6 -14 C -4 4 0 -8 0 D 0 6 8 0 -10 E 0 14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 0 0 B -8 0 -4 -6 -14 C -4 4 0 -8 0 D 0 6 8 0 -10 E 0 14 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999869 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9264: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (6) C B D A E (6) B C D E A (6) A E C D B (6) E D B A C (5) D B E A C (5) C A B D E (5) B D E C A (5) E A C D B (4) E C A B D (3) D B C A E (3) C A E B D (3) A D E B C (3) A C E D B (3) A C D B E (3) E A D C B (2) E A D B C (2) E A C B D (2) D E B A C (2) D B A C E (2) D A E B C (2) C B E A D (2) B D C E A (2) E D A B C (1) E B D C A (1) E B D A C (1) D E A B C (1) D A B C E (1) C E B D A (1) C B D E A (1) C B A D E (1) C A D B E (1) C A B E D (1) B E D C A (1) B C E D A (1) B C D A E (1) A E D C B (1) A E D B C (1) A D C B E (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 0 -8 -8 B 4 0 4 -12 8 C 0 -4 0 4 -8 D 8 12 -4 0 16 E 8 -8 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.114379 C: 0.587768 D: 0.236695 E: 0.061158 Sum of squares = 0.418318972366 Cumulative probabilities = A: 0.000000 B: 0.114379 C: 0.702148 D: 0.938842 E: 1.000000 A B C D E A 0 -4 0 -8 -8 B 4 0 4 -12 8 C 0 -4 0 4 -8 D 8 12 -4 0 16 E 8 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.250000 E: 0.083334 Sum of squares = 0.416666666646 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666666 D: 0.916666 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=22 E=21 C=21 A=20 B=16 so B is eliminated. Round 2 votes counts: D=29 C=29 E=22 A=20 so A is eliminated. Round 3 votes counts: C=37 D=33 E=30 so E is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:216 B:202 C:196 E:196 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 0 -8 -8 B 4 0 4 -12 8 C 0 -4 0 4 -8 D 8 12 -4 0 16 E 8 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.250000 E: 0.083334 Sum of squares = 0.416666666646 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666666 D: 0.916666 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -8 -8 B 4 0 4 -12 8 C 0 -4 0 4 -8 D 8 12 -4 0 16 E 8 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.250000 E: 0.083334 Sum of squares = 0.416666666646 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666666 D: 0.916666 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -8 -8 B 4 0 4 -12 8 C 0 -4 0 4 -8 D 8 12 -4 0 16 E 8 -8 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.083333 C: 0.583333 D: 0.250000 E: 0.083334 Sum of squares = 0.416666666646 Cumulative probabilities = A: 0.000000 B: 0.083333 C: 0.666666 D: 0.916666 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9265: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (14) E A B D C (4) D E C B A (4) D C E B A (4) B D A E C (4) A B E D C (4) E C D A B (3) D E B A C (3) D B C A E (3) C E A D B (3) C D E B A (3) C D E A B (3) C A B D E (3) B D C A E (3) A C B E D (3) A B C E D (3) E C A D B (2) E A C D B (2) E A C B D (2) D E B C A (2) D B E A C (2) C D B A E (2) C A E B D (2) B A D E C (2) A B E C D (2) E D C B A (1) E D C A B (1) E C A B D (1) E A D B C (1) C E D A B (1) C D A E B (1) C B D A E (1) C A D B E (1) C A B E D (1) B E A D C (1) B D E A C (1) B D A C E (1) B A E D C (1) A E C B D (1) A E B D C (1) A E B C D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 10 -14 -16 B 4 0 8 -12 -20 C -10 -8 0 -14 -20 D 14 12 14 0 -10 E 16 20 20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 10 -14 -16 B 4 0 8 -12 -20 C -10 -8 0 -14 -20 D 14 12 14 0 -10 E 16 20 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=21 D=18 A=17 B=13 so B is eliminated. Round 2 votes counts: E=32 D=27 C=21 A=20 so A is eliminated. Round 3 votes counts: E=42 D=29 C=29 so D is eliminated. Round 4 votes counts: E=60 C=40 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:233 D:215 B:190 A:188 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 10 -14 -16 B 4 0 8 -12 -20 C -10 -8 0 -14 -20 D 14 12 14 0 -10 E 16 20 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 10 -14 -16 B 4 0 8 -12 -20 C -10 -8 0 -14 -20 D 14 12 14 0 -10 E 16 20 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 10 -14 -16 B 4 0 8 -12 -20 C -10 -8 0 -14 -20 D 14 12 14 0 -10 E 16 20 20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9266: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) B D E C A (7) B C D E A (7) D E A B C (6) C A B E D (6) A C B D E (6) E D A B C (5) D E B C A (4) A E D C B (4) A D E C B (4) E D B A C (3) E A D C B (3) C B A D E (3) C A B D E (3) A C E D B (3) E D B C A (2) E A D B C (2) D B E C A (2) B C D A E (2) A C D E B (2) A C B E D (2) E C B D A (1) E B D C A (1) D B E A C (1) D B A E C (1) D A E B C (1) C B E D A (1) C B A E D (1) C A E B D (1) B D C E A (1) B D C A E (1) B C E D A (1) A E D B C (1) A E C D B (1) A D E B C (1) A D C E B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 8 12 -8 -8 B -8 0 12 -10 -8 C -12 -12 0 -16 -14 D 8 10 16 0 22 E 8 8 14 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 12 -8 -8 B -8 0 12 -10 -8 C -12 -12 0 -16 -14 D 8 10 16 0 22 E 8 8 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 D=22 B=19 E=17 C=15 so C is eliminated. Round 2 votes counts: A=37 B=24 D=22 E=17 so E is eliminated. Round 3 votes counts: A=42 D=32 B=26 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 E:204 A:202 B:193 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 12 -8 -8 B -8 0 12 -10 -8 C -12 -12 0 -16 -14 D 8 10 16 0 22 E 8 8 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 12 -8 -8 B -8 0 12 -10 -8 C -12 -12 0 -16 -14 D 8 10 16 0 22 E 8 8 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 12 -8 -8 B -8 0 12 -10 -8 C -12 -12 0 -16 -14 D 8 10 16 0 22 E 8 8 14 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9267: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (7) D B C E A (6) B D C A E (6) E A C D B (5) E A C B D (5) D C B E A (5) C D A E B (5) D B C A E (4) C A E D B (4) B D C E A (4) E C A D B (3) E A B C D (3) C E A D B (3) B D A C E (3) A E C B D (3) D C E A B (2) D C B A E (2) C D A B E (2) B E A D C (2) B E A C D (2) B D E C A (2) A E B C D (2) A C E D B (2) A C E B D (2) E D C B A (1) E C D A B (1) E B A D C (1) E B A C D (1) E A D C B (1) E A D B C (1) D E C A B (1) D E B C A (1) D C E B A (1) D B E C A (1) C D E A B (1) B A E D C (1) B A E C D (1) B A D C E (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -10 -10 -20 B 4 0 0 -4 2 C 10 0 0 -6 8 D 10 4 6 0 10 E 20 -2 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -10 -20 B 4 0 0 -4 2 C 10 0 0 -6 8 D 10 4 6 0 10 E 20 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=23 E=22 C=15 A=11 so A is eliminated. Round 2 votes counts: B=30 E=27 D=23 C=20 so C is eliminated. Round 3 votes counts: E=38 D=31 B=31 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:215 C:206 B:201 E:200 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -10 -20 B 4 0 0 -4 2 C 10 0 0 -6 8 D 10 4 6 0 10 E 20 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -10 -20 B 4 0 0 -4 2 C 10 0 0 -6 8 D 10 4 6 0 10 E 20 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -10 -20 B 4 0 0 -4 2 C 10 0 0 -6 8 D 10 4 6 0 10 E 20 -2 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9268: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) B C D A E (7) D B A C E (5) E A D C B (4) D E B A C (4) D B A E C (4) D A C B E (4) C A B D E (4) B D C A E (4) E D A B C (3) E C A B D (3) C A D B E (3) B D E A C (3) A C D E B (3) A C D B E (3) E D B A C (2) E D A C B (2) E B D C A (2) C D B A E (2) C A E B D (2) C A B E D (2) B E D C A (2) B C E A D (2) B C D E A (2) A E C D B (2) A D C E B (2) A C E D B (2) E C B A D (1) E C A D B (1) E B D A C (1) E B C A D (1) D E A B C (1) D B C A E (1) C B A D E (1) B E C D A (1) B D C E A (1) B D A E C (1) B C A D E (1) A E D C B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 14 -4 10 B -4 0 -4 -18 8 C -14 4 0 4 4 D 4 18 -4 0 14 E -10 -8 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074380248 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 -4 10 B -4 0 -4 -18 8 C -14 4 0 4 4 D 4 18 -4 0 14 E -10 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074379987 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=24 D=19 C=14 A=14 so C is eliminated. Round 2 votes counts: E=29 B=25 A=25 D=21 so D is eliminated. Round 3 votes counts: B=37 E=34 A=29 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:216 A:212 C:199 B:191 E:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 -4 10 B -4 0 -4 -18 8 C -14 4 0 4 4 D 4 18 -4 0 14 E -10 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074379987 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -4 10 B -4 0 -4 -18 8 C -14 4 0 4 4 D 4 18 -4 0 14 E -10 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074379987 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -4 10 B -4 0 -4 -18 8 C -14 4 0 4 4 D 4 18 -4 0 14 E -10 -8 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.000000 C: 0.181818 D: 0.636364 E: 0.000000 Sum of squares = 0.471074379987 Cumulative probabilities = A: 0.181818 B: 0.181818 C: 0.363636 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9269: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (12) E C D A B (11) B A D C E (10) A C D E B (10) E B C D A (7) C A D E B (6) B D A E C (5) C E D A B (4) B E D C A (4) B A C D E (4) B E C D A (3) B D E A C (3) D E A C B (2) C E A D B (2) B E C A D (2) A D C E B (2) A D C B E (2) A C D B E (2) E C B D A (1) E C B A D (1) E B D C A (1) D A C E B (1) B E A C D (1) B D A C E (1) B C A E D (1) B A D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 14 -10 -8 B 14 0 12 16 4 C -14 -12 0 10 -8 D 10 -16 -10 0 0 E 8 -4 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 14 -10 -8 B 14 0 12 16 4 C -14 -12 0 10 -8 D 10 -16 -10 0 0 E 8 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=47 E=21 A=17 C=12 D=3 so D is eliminated. Round 2 votes counts: B=47 E=23 A=18 C=12 so C is eliminated. Round 3 votes counts: B=47 E=29 A=24 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:206 D:192 A:191 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 14 -10 -8 B 14 0 12 16 4 C -14 -12 0 10 -8 D 10 -16 -10 0 0 E 8 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 14 -10 -8 B 14 0 12 16 4 C -14 -12 0 10 -8 D 10 -16 -10 0 0 E 8 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 14 -10 -8 B 14 0 12 16 4 C -14 -12 0 10 -8 D 10 -16 -10 0 0 E 8 -4 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9270: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (10) A D E C B (8) C D A E B (6) B C E D A (6) B C D E A (6) E A D B C (5) A E D B C (5) C B D E A (4) E D A C B (3) C D B E A (3) C B D A E (3) B E A C D (3) B A E D C (3) E D C A B (2) E A B D C (2) D C A E B (2) D A C E B (2) C D E A B (2) B C D A E (2) B C A E D (2) A D C E B (2) A B E D C (2) E A D C B (1) D C E A B (1) D A E C B (1) C D E B A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C D A (1) B E C A D (1) B C E A D (1) B C A D E (1) B A C E D (1) A E D C B (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 2 2 -8 B 2 0 6 4 10 C -2 -6 0 -4 0 D -2 -4 4 0 -4 E 8 -10 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 2 -8 B 2 0 6 4 10 C -2 -6 0 -4 0 D -2 -4 4 0 -4 E 8 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 C=24 A=19 E=13 D=6 so D is eliminated. Round 2 votes counts: B=38 C=27 A=22 E=13 so E is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:211 E:201 A:197 D:197 C:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 2 -8 B 2 0 6 4 10 C -2 -6 0 -4 0 D -2 -4 4 0 -4 E 8 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 2 -8 B 2 0 6 4 10 C -2 -6 0 -4 0 D -2 -4 4 0 -4 E 8 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 2 -8 B 2 0 6 4 10 C -2 -6 0 -4 0 D -2 -4 4 0 -4 E 8 -10 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998489 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9271: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) C B A E D (10) D E A C B (6) A E D C B (6) A E D B C (6) D A E C B (5) E A D B C (4) C D B E A (4) B E A D C (4) B C A E D (4) E D A B C (3) E A B D C (3) C D A E B (3) C B D E A (3) C B D A E (3) B C E A D (3) A E B D C (3) C A E D B (2) D E B A C (1) D B E C A (1) D A C E B (1) C D B A E (1) C A D E B (1) C A D B E (1) C A B E D (1) B E D A C (1) B E C A D (1) B E A C D (1) B D E A C (1) B D C E A (1) B C D E A (1) A D E C B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 20 20 6 0 B -20 0 0 -20 -18 C -20 0 0 -20 -18 D -6 20 20 0 -6 E 0 18 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.448219 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.551781 Sum of squares = 0.505362504128 Cumulative probabilities = A: 0.448219 B: 0.448219 C: 0.448219 D: 0.448219 E: 1.000000 A B C D E A 0 20 20 6 0 B -20 0 0 -20 -18 C -20 0 0 -20 -18 D -6 20 20 0 -6 E 0 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 A=18 B=17 E=10 so E is eliminated. Round 2 votes counts: D=29 C=29 A=25 B=17 so B is eliminated. Round 3 votes counts: C=38 D=32 A=30 so A is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:223 E:221 D:214 B:171 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 20 20 6 0 B -20 0 0 -20 -18 C -20 0 0 -20 -18 D -6 20 20 0 -6 E 0 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 20 6 0 B -20 0 0 -20 -18 C -20 0 0 -20 -18 D -6 20 20 0 -6 E 0 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 20 6 0 B -20 0 0 -20 -18 C -20 0 0 -20 -18 D -6 20 20 0 -6 E 0 18 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999957 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9272: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) A C E B D (8) E B D A C (6) D B C E A (5) C A E D B (5) C A D B E (5) B D E C A (5) A E C B D (5) E A B D C (4) B E D A C (4) A E B D C (4) E A C B D (3) D C B E A (3) D B C A E (3) C D E B A (3) C D A B E (3) A C D B E (3) E C A D B (2) E B D C A (2) D E B C A (2) C D B E A (2) C A D E B (2) B A E D C (2) A B E D C (2) E C A B D (1) E B A D C (1) D C B A E (1) C E D A B (1) C D B A E (1) B D E A C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 -12 -4 -10 B 2 0 4 -2 0 C 12 -4 0 -10 -10 D 4 2 10 0 -2 E 10 0 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.482854 C: 0.000000 D: 0.000000 E: 0.517146 Sum of squares = 0.500587973663 Cumulative probabilities = A: 0.000000 B: 0.482854 C: 0.482854 D: 0.482854 E: 1.000000 A B C D E A 0 -2 -12 -4 -10 B 2 0 4 -2 0 C 12 -4 0 -10 -10 D 4 2 10 0 -2 E 10 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499734 C: 0.000000 D: 0.000000 E: 0.500266 Sum of squares = 0.500000141741 Cumulative probabilities = A: 0.000000 B: 0.499734 C: 0.499734 D: 0.499734 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=23 C=22 E=19 B=12 so B is eliminated. Round 2 votes counts: D=30 A=25 E=23 C=22 so C is eliminated. Round 3 votes counts: D=39 A=37 E=24 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:211 D:207 B:202 C:194 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -12 -4 -10 B 2 0 4 -2 0 C 12 -4 0 -10 -10 D 4 2 10 0 -2 E 10 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499734 C: 0.000000 D: 0.000000 E: 0.500266 Sum of squares = 0.500000141741 Cumulative probabilities = A: 0.000000 B: 0.499734 C: 0.499734 D: 0.499734 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -4 -10 B 2 0 4 -2 0 C 12 -4 0 -10 -10 D 4 2 10 0 -2 E 10 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499734 C: 0.000000 D: 0.000000 E: 0.500266 Sum of squares = 0.500000141741 Cumulative probabilities = A: 0.000000 B: 0.499734 C: 0.499734 D: 0.499734 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -4 -10 B 2 0 4 -2 0 C 12 -4 0 -10 -10 D 4 2 10 0 -2 E 10 0 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499734 C: 0.000000 D: 0.000000 E: 0.500266 Sum of squares = 0.500000141741 Cumulative probabilities = A: 0.000000 B: 0.499734 C: 0.499734 D: 0.499734 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9273: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (16) A E D C B (9) E A C B D (8) B D C A E (7) E A B D C (5) D B C A E (5) C B D E A (5) B D C E A (5) A E C D B (5) E A B C D (4) B C D E A (4) E A D C B (3) B C D A E (3) A C E D B (3) D C B A E (2) C B D A E (2) A D E B C (2) E C A D B (1) E B C A D (1) E B A D C (1) E A D B C (1) D B A E C (1) D A E B C (1) C E B A D (1) C E A B D (1) C D B A E (1) B E C A D (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 22 22 28 -14 B -22 0 -16 -4 -28 C -22 16 0 12 -22 D -28 4 -12 0 -22 E 14 28 22 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 22 22 28 -14 B -22 0 -16 -4 -28 C -22 16 0 12 -22 D -28 4 -12 0 -22 E 14 28 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 A=21 B=20 C=10 D=9 so D is eliminated. Round 2 votes counts: E=40 B=26 A=22 C=12 so C is eliminated. Round 3 votes counts: E=42 B=36 A=22 so A is eliminated. Round 4 votes counts: E=64 B=36 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:243 A:229 C:192 D:171 B:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 22 22 28 -14 B -22 0 -16 -4 -28 C -22 16 0 12 -22 D -28 4 -12 0 -22 E 14 28 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 22 28 -14 B -22 0 -16 -4 -28 C -22 16 0 12 -22 D -28 4 -12 0 -22 E 14 28 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 22 28 -14 B -22 0 -16 -4 -28 C -22 16 0 12 -22 D -28 4 -12 0 -22 E 14 28 22 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9274: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) E B D A C (7) C A B D E (7) A C D E B (6) C A B E D (5) E D B A C (4) D B E A C (4) E C B A D (3) C A D B E (3) B D E C A (3) B D E A C (3) E D A B C (2) E B D C A (2) E A C D B (2) D A E C B (2) D A C B E (2) C A E D B (2) B E C A D (2) B C A E D (2) B C A D E (2) A D C B E (2) A C D B E (2) E D A C B (1) E C A D B (1) E B C D A (1) E B C A D (1) E A D C B (1) D B A C E (1) D A E B C (1) D A C E B (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) C B E A D (1) C B A E D (1) C B A D E (1) C A D E B (1) B E C D A (1) B D C E A (1) B C E D A (1) B C E A D (1) B C D E A (1) A E C D B (1) A D E C B (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -6 4 -6 B 4 0 0 12 12 C 6 0 0 2 -4 D -4 -12 -2 0 -6 E 6 -12 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.464714 C: 0.535286 D: 0.000000 E: 0.000000 Sum of squares = 0.502490191728 Cumulative probabilities = A: 0.000000 B: 0.464714 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -6 4 -6 B 4 0 0 12 12 C 6 0 0 2 -4 D -4 -12 -2 0 -6 E 6 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=25 B=25 C=23 A=14 D=13 so D is eliminated. Round 2 votes counts: B=30 E=25 C=23 A=22 so A is eliminated. Round 3 votes counts: C=38 B=32 E=30 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 C:202 E:202 A:194 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -6 4 -6 B 4 0 0 12 12 C 6 0 0 2 -4 D -4 -12 -2 0 -6 E 6 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 4 -6 B 4 0 0 12 12 C 6 0 0 2 -4 D -4 -12 -2 0 -6 E 6 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 4 -6 B 4 0 0 12 12 C 6 0 0 2 -4 D -4 -12 -2 0 -6 E 6 -12 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9275: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (12) D A B C E (7) D A E C B (6) B C D E A (6) A D E C B (6) E C B A D (5) E A C B D (5) D B C E A (5) D B C A E (4) C B E D A (4) A E B C D (4) E B C A D (3) C B D E A (3) B C E D A (3) A D E B C (3) D C B E A (2) D C A E B (2) D B A C E (2) D A C B E (2) B C E A D (2) A E D C B (2) A E C D B (2) E B A C D (1) D C E A B (1) D A E B C (1) D A B E C (1) C E B A D (1) B E C A D (1) B D C E A (1) B A C E D (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 14 0 14 B -12 0 -6 6 -10 C -14 6 0 6 -8 D 0 -6 -6 0 6 E -14 10 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.618775 B: 0.000000 C: 0.000000 D: 0.381225 E: 0.000000 Sum of squares = 0.528215208458 Cumulative probabilities = A: 0.618775 B: 0.618775 C: 0.618775 D: 1.000000 E: 1.000000 A B C D E A 0 12 14 0 14 B -12 0 -6 6 -10 C -14 6 0 6 -8 D 0 -6 -6 0 6 E -14 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=31 E=14 B=14 C=8 so C is eliminated. Round 2 votes counts: D=33 A=31 B=21 E=15 so E is eliminated. Round 3 votes counts: A=36 D=33 B=31 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:199 D:197 C:195 B:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 14 0 14 B -12 0 -6 6 -10 C -14 6 0 6 -8 D 0 -6 -6 0 6 E -14 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 14 0 14 B -12 0 -6 6 -10 C -14 6 0 6 -8 D 0 -6 -6 0 6 E -14 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 14 0 14 B -12 0 -6 6 -10 C -14 6 0 6 -8 D 0 -6 -6 0 6 E -14 10 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9276: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (8) B A E C D (7) B A C D E (6) E A D B C (5) C D E A B (5) B C A D E (5) B A C E D (5) E D C A B (4) E B A D C (4) C D E B A (4) C D B E A (4) C D B A E (4) C B D A E (4) E D C B A (3) D E C A B (3) B E A C D (3) E B C D A (2) E A B D C (2) C D A B E (2) B C A E D (2) B A E D C (2) A D C B E (2) A B C D E (2) E D B C A (1) D E A C B (1) C E D B A (1) C B A D E (1) C A D B E (1) B E A D C (1) B C E D A (1) B C D A E (1) A E B D C (1) A D E C B (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -22 -12 4 -4 B 22 0 0 -2 8 C 12 0 0 22 20 D -4 2 -22 0 12 E 4 -8 -20 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.478057 C: 0.521943 D: 0.000000 E: 0.000000 Sum of squares = 0.500962959567 Cumulative probabilities = A: 0.000000 B: 0.478057 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -12 4 -4 B 22 0 0 -2 8 C 12 0 0 22 20 D -4 2 -22 0 12 E 4 -8 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=26 E=21 D=12 A=8 so A is eliminated. Round 2 votes counts: B=35 C=27 E=22 D=16 so D is eliminated. Round 3 votes counts: C=38 B=35 E=27 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:227 B:214 D:194 A:183 E:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -12 4 -4 B 22 0 0 -2 8 C 12 0 0 22 20 D -4 2 -22 0 12 E 4 -8 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -12 4 -4 B 22 0 0 -2 8 C 12 0 0 22 20 D -4 2 -22 0 12 E 4 -8 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -12 4 -4 B 22 0 0 -2 8 C 12 0 0 22 20 D -4 2 -22 0 12 E 4 -8 -20 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9277: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (15) D B A E C (8) E C A D B (6) E A D B C (6) D B E A C (6) E D B A C (4) E C A B D (4) E A C D B (3) C E A D B (3) C B A D E (3) C A B D E (3) B D C A E (3) E A C B D (2) D E B C A (2) D B E C A (2) D B C E A (2) C E D B A (2) C D B E A (2) C D B A E (2) C B D A E (2) C A E B D (2) B D A C E (2) A B D E C (2) A B D C E (2) E D B C A (1) E D A B C (1) E A D C B (1) D B C A E (1) D B A C E (1) C E B D A (1) C A B E D (1) B D A E C (1) B C D A E (1) B A D C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -16 12 -26 B -6 0 -6 -6 -6 C 16 6 0 8 2 D -12 6 -8 0 -6 E 26 6 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -16 12 -26 B -6 0 -6 -6 -6 C 16 6 0 8 2 D -12 6 -8 0 -6 E 26 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=28 D=22 B=8 A=6 so A is eliminated. Round 2 votes counts: C=37 E=28 D=22 B=13 so B is eliminated. Round 3 votes counts: C=39 D=33 E=28 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:218 C:216 D:190 A:188 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -16 12 -26 B -6 0 -6 -6 -6 C 16 6 0 8 2 D -12 6 -8 0 -6 E 26 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -16 12 -26 B -6 0 -6 -6 -6 C 16 6 0 8 2 D -12 6 -8 0 -6 E 26 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -16 12 -26 B -6 0 -6 -6 -6 C 16 6 0 8 2 D -12 6 -8 0 -6 E 26 6 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993093 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9278: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (13) B E C A D (12) E B C D A (7) E B C A D (7) D A C E B (7) A C D B E (6) D A E C B (5) D A C B E (5) C B A E D (4) C B A D E (4) B C E A D (4) E D B A C (3) E B D C A (3) D A E B C (3) A C B D E (3) E D A B C (2) D E A B C (2) C A D B E (2) E D B C A (1) E B D A C (1) D E A C B (1) D C A B E (1) C D A B E (1) C B E A D (1) C A B D E (1) B C A D E (1) Total count = 100 A B C D E A 0 4 2 16 12 B -4 0 -8 -4 16 C -2 8 0 6 6 D -16 4 -6 0 10 E -12 -16 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 16 12 B -4 0 -8 -4 16 C -2 8 0 6 6 D -16 4 -6 0 10 E -12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985579 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=24 D=24 A=22 B=17 C=13 so C is eliminated. Round 2 votes counts: B=26 D=25 A=25 E=24 so E is eliminated. Round 3 votes counts: B=44 D=31 A=25 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:217 C:209 B:200 D:196 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 2 16 12 B -4 0 -8 -4 16 C -2 8 0 6 6 D -16 4 -6 0 10 E -12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985579 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 16 12 B -4 0 -8 -4 16 C -2 8 0 6 6 D -16 4 -6 0 10 E -12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985579 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 16 12 B -4 0 -8 -4 16 C -2 8 0 6 6 D -16 4 -6 0 10 E -12 -16 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985579 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9279: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (5) A D E B C (5) D E A B C (4) D B E C A (4) D B C E A (4) C B E A D (4) C B A E D (4) A C B D E (4) E D B A C (3) E A B D C (3) E A B C D (3) D C B A E (3) D A E B C (3) A E D B C (3) A C D B E (3) E D B C A (2) E D A B C (2) E B D A C (2) E B C D A (2) D E B C A (2) D C B E A (2) D A C B E (2) C B A D E (2) C A B E D (2) B E D C A (2) B C E D A (2) A E C B D (2) A E B C D (2) A D C E B (2) E B D C A (1) E B A D C (1) E B A C D (1) D E B A C (1) D C A B E (1) D A C E B (1) C B D E A (1) C B D A E (1) C A B D E (1) B E C D A (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C B E (1) A C E B D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 0 8 8 -4 B 0 0 16 -4 -10 C -8 -16 0 -12 -14 D -8 4 12 0 -2 E 4 10 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 8 8 -4 B 0 0 16 -4 -10 C -8 -16 0 -12 -14 D -8 4 12 0 -2 E 4 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=27 E=25 C=15 B=5 so B is eliminated. Round 2 votes counts: E=28 A=28 D=27 C=17 so C is eliminated. Round 3 votes counts: A=37 E=34 D=29 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:206 D:203 B:201 C:175 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 8 8 -4 B 0 0 16 -4 -10 C -8 -16 0 -12 -14 D -8 4 12 0 -2 E 4 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 8 -4 B 0 0 16 -4 -10 C -8 -16 0 -12 -14 D -8 4 12 0 -2 E 4 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 8 -4 B 0 0 16 -4 -10 C -8 -16 0 -12 -14 D -8 4 12 0 -2 E 4 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9280: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) C B D A E (5) B A C E D (5) A B C E D (5) E A D C B (4) D C B E A (4) C D A B E (4) A E B C D (4) E D A B C (3) E A D B C (3) D E C B A (3) C D B A E (3) C A B D E (3) B C D E A (3) B C A D E (3) B A E C D (3) A C B D E (3) E D B C A (2) E D B A C (2) E D A C B (2) D E B C A (2) D C A E B (2) B E D C A (2) B E C D A (2) B C A E D (2) A B E C D (2) E D C A B (1) E B A D C (1) E B A C D (1) E A B D C (1) D C B A E (1) D C A B E (1) D B C E A (1) D A E C B (1) D A C E B (1) C B A D E (1) B E A C D (1) B C D A E (1) B A C D E (1) A E D C B (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -8 -8 8 B -2 0 0 0 14 C 8 0 0 8 0 D 8 0 -8 0 4 E -8 -14 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.331874 C: 0.668126 D: 0.000000 E: 0.000000 Sum of squares = 0.556532453996 Cumulative probabilities = A: 0.000000 B: 0.331874 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -8 -8 8 B -2 0 0 0 14 C 8 0 0 8 0 D 8 0 -8 0 4 E -8 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=23 E=20 A=17 C=16 so C is eliminated. Round 2 votes counts: D=31 B=29 E=20 A=20 so E is eliminated. Round 3 votes counts: D=41 B=31 A=28 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:206 D:202 A:197 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -8 -8 8 B -2 0 0 0 14 C 8 0 0 8 0 D 8 0 -8 0 4 E -8 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 -8 8 B -2 0 0 0 14 C 8 0 0 8 0 D 8 0 -8 0 4 E -8 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 -8 8 B -2 0 0 0 14 C 8 0 0 8 0 D 8 0 -8 0 4 E -8 -14 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9281: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) E B D A C (6) B E C A D (6) B E A C D (5) D A C E B (4) C D A E B (4) C D A B E (4) B E D A C (4) B C A D E (4) A D C E B (4) A C D B E (4) E D B C A (3) E D A C B (3) E D A B C (3) D A E C B (3) C A D B E (3) B E C D A (3) B A C D E (3) E B D C A (2) D E C A B (2) B E D C A (2) B C E D A (2) B C E A D (2) A C D E B (2) A C B D E (2) E D C A B (1) E D B A C (1) E B A D C (1) C B A D E (1) C A D E B (1) B C D E A (1) B C A E D (1) B A E C D (1) B A C E D (1) A D E C B (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -2 -14 6 B -2 0 4 -4 0 C 2 -4 0 2 6 D 14 4 -2 0 6 E -6 0 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999999 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 -14 6 B -2 0 4 -4 0 C 2 -4 0 2 6 D 14 4 -2 0 6 E -6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999992 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=20 D=18 A=14 C=13 so C is eliminated. Round 2 votes counts: B=36 D=26 E=20 A=18 so A is eliminated. Round 3 votes counts: D=41 B=39 E=20 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:211 C:203 B:199 A:196 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -2 -14 6 B -2 0 4 -4 0 C 2 -4 0 2 6 D 14 4 -2 0 6 E -6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999992 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -14 6 B -2 0 4 -4 0 C 2 -4 0 2 6 D 14 4 -2 0 6 E -6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999992 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -14 6 B -2 0 4 -4 0 C 2 -4 0 2 6 D 14 4 -2 0 6 E -6 0 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999992 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9282: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (8) C D E B A (7) A B E D C (6) E C D A B (5) B A D C E (5) E C A D B (4) D C B E A (4) C D B E A (4) A B D E C (4) E A C B D (3) E A B C D (3) D C B A E (3) C E D B A (3) A C D B E (3) E C A B D (2) E A C D B (2) D B A C E (2) B E D C A (2) B E D A C (2) B D A C E (2) E C B D A (1) E B C D A (1) E B A D C (1) D B C E A (1) D B C A E (1) C E D A B (1) C D A E B (1) C D A B E (1) B E A D C (1) B D E C A (1) B A E D C (1) A E C D B (1) A E B D C (1) A D C B E (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -4 -4 -16 B 4 0 -14 -12 6 C 4 14 0 6 -2 D 4 12 -6 0 -2 E 16 -6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.000000 E: 0.636364 Sum of squares = 0.487603305743 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 0.363636 E: 1.000000 A B C D E A 0 -4 -4 -4 -16 B 4 0 -14 -12 6 C 4 14 0 6 -2 D 4 12 -6 0 -2 E 16 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.000000 E: 0.636364 Sum of squares = 0.487603304501 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 0.363636 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=26 C=17 B=14 D=11 so D is eliminated. Round 2 votes counts: E=32 A=26 C=24 B=18 so B is eliminated. Round 3 votes counts: E=38 A=36 C=26 so C is eliminated. Round 4 votes counts: E=58 A=42 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:211 E:207 D:204 B:192 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -4 -4 -16 B 4 0 -14 -12 6 C 4 14 0 6 -2 D 4 12 -6 0 -2 E 16 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.000000 E: 0.636364 Sum of squares = 0.487603304501 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 0.363636 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -4 -16 B 4 0 -14 -12 6 C 4 14 0 6 -2 D 4 12 -6 0 -2 E 16 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.000000 E: 0.636364 Sum of squares = 0.487603304501 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 0.363636 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -4 -16 B 4 0 -14 -12 6 C 4 14 0 6 -2 D 4 12 -6 0 -2 E 16 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.090909 C: 0.272727 D: 0.000000 E: 0.636364 Sum of squares = 0.487603304501 Cumulative probabilities = A: 0.000000 B: 0.090909 C: 0.363636 D: 0.363636 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9283: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (7) C D E B A (6) A B D E C (6) E B D A C (5) E C D B A (4) E B D C A (4) D C B E A (4) C A D B E (4) A C B D E (4) A B D C E (4) E D B C A (3) E C B D A (3) E A B D C (3) C E D B A (3) C A E B D (3) E B A D C (2) D B A C E (2) C E D A B (2) C D B A E (2) A E B C D (2) A B E C D (2) A B C D E (2) E B A C D (1) E A C B D (1) D E C B A (1) D C B A E (1) D B E C A (1) D B E A C (1) C E A D B (1) C D E A B (1) C D B E A (1) C D A E B (1) C A E D B (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B D E A C (1) B D A E C (1) B A E D C (1) B A D E C (1) A E B D C (1) A C D B E (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 4 4 0 B 0 0 6 18 2 C -4 -6 0 -2 -6 D -4 -18 2 0 -6 E 0 -2 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.567636 B: 0.432364 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.509149178205 Cumulative probabilities = A: 0.567636 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 4 0 B 0 0 6 18 2 C -4 -6 0 -2 -6 D -4 -18 2 0 -6 E 0 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=27 E=26 D=10 B=6 so B is eliminated. Round 2 votes counts: A=33 E=28 C=27 D=12 so D is eliminated. Round 3 votes counts: A=36 E=32 C=32 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:213 E:205 A:204 C:191 D:187 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 4 0 B 0 0 6 18 2 C -4 -6 0 -2 -6 D -4 -18 2 0 -6 E 0 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 0 B 0 0 6 18 2 C -4 -6 0 -2 -6 D -4 -18 2 0 -6 E 0 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 0 B 0 0 6 18 2 C -4 -6 0 -2 -6 D -4 -18 2 0 -6 E 0 -2 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9284: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (12) B E D C A (11) A C D E B (9) A C B D E (6) E B D C A (5) B C A D E (5) E D C A B (4) E D B A C (4) D A E C B (4) B E C D A (4) B C A E D (4) E D A B C (3) E B D A C (3) C A B D E (3) B A C E D (3) D E A C B (2) C B A D E (2) C A D E B (2) A D C E B (2) E D C B A (1) E D B C A (1) C D A E B (1) C A D B E (1) B E D A C (1) B E C A D (1) B E A D C (1) B D E C A (1) B C D A E (1) B A C D E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 -16 -8 B -2 0 0 6 -6 C -6 0 0 -10 -16 D 16 -6 10 0 -16 E 8 6 16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 6 -16 -8 B -2 0 0 6 -6 C -6 0 0 -10 -16 D 16 -6 10 0 -16 E 8 6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=33 B=33 A=19 C=9 D=6 so D is eliminated. Round 2 votes counts: E=35 B=33 A=23 C=9 so C is eliminated. Round 3 votes counts: E=35 B=35 A=30 so A is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:202 B:199 A:192 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 6 -16 -8 B -2 0 0 6 -6 C -6 0 0 -10 -16 D 16 -6 10 0 -16 E 8 6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -16 -8 B -2 0 0 6 -6 C -6 0 0 -10 -16 D 16 -6 10 0 -16 E 8 6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -16 -8 B -2 0 0 6 -6 C -6 0 0 -10 -16 D 16 -6 10 0 -16 E 8 6 16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999602 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9285: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) C D B A E (7) A E C B D (7) C B D E A (5) C A E B D (5) B D C E A (4) A E C D B (4) A C E D B (4) A C E B D (4) E A B D C (3) D B C A E (3) C A E D B (3) B C D E A (3) A E D B C (3) A D E B C (3) C E B A D (2) C B E D A (2) C B E A D (2) C B D A E (2) B D E C A (2) B D E A C (2) A E B C D (2) E B A D C (1) E B A C D (1) E A C B D (1) D C B E A (1) D C B A E (1) D C A B E (1) D B E A C (1) D B A C E (1) D A E B C (1) C E A B D (1) C D B E A (1) C D A B E (1) C B A E D (1) C A D E B (1) C A B E D (1) B E C D A (1) B E C A D (1) B C E D A (1) B C E A D (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -20 6 14 B 6 0 -18 10 4 C 20 18 0 30 32 D -6 -10 -30 0 -4 E -14 -4 -32 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -20 6 14 B 6 0 -18 10 4 C 20 18 0 30 32 D -6 -10 -30 0 -4 E -14 -4 -32 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=29 D=16 B=15 E=6 so E is eliminated. Round 2 votes counts: C=34 A=33 B=17 D=16 so D is eliminated. Round 3 votes counts: C=37 A=34 B=29 so B is eliminated. Round 4 votes counts: C=60 A=40 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:250 B:201 A:197 E:177 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -20 6 14 B 6 0 -18 10 4 C 20 18 0 30 32 D -6 -10 -30 0 -4 E -14 -4 -32 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -20 6 14 B 6 0 -18 10 4 C 20 18 0 30 32 D -6 -10 -30 0 -4 E -14 -4 -32 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -20 6 14 B 6 0 -18 10 4 C 20 18 0 30 32 D -6 -10 -30 0 -4 E -14 -4 -32 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9286: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (14) B E C A D (7) A C D E B (7) E D B A C (6) E B D C A (6) D E A C B (6) C A B D E (6) D E B A C (5) A C D B E (5) D A C B E (4) B E D C A (4) B C A E D (4) D A E C B (3) C B A E D (3) E D B C A (2) D B E A C (2) C A B E D (2) B D A C E (2) E D A C B (1) E C B A D (1) E B D A C (1) E B C A D (1) D E A B C (1) D B A C E (1) D A B E C (1) C A E B D (1) C A D B E (1) B E D A C (1) B D E A C (1) B C A D E (1) Total count = 100 A B C D E A 0 4 22 -22 10 B -4 0 -8 -18 -10 C -22 8 0 -22 2 D 22 18 22 0 20 E -10 10 -2 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 22 -22 10 B -4 0 -8 -18 -10 C -22 8 0 -22 2 D 22 18 22 0 20 E -10 10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=20 E=18 C=13 A=12 so A is eliminated. Round 2 votes counts: D=37 C=25 B=20 E=18 so E is eliminated. Round 3 votes counts: D=46 B=28 C=26 so C is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:241 A:207 E:189 C:183 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 22 -22 10 B -4 0 -8 -18 -10 C -22 8 0 -22 2 D 22 18 22 0 20 E -10 10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 22 -22 10 B -4 0 -8 -18 -10 C -22 8 0 -22 2 D 22 18 22 0 20 E -10 10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 22 -22 10 B -4 0 -8 -18 -10 C -22 8 0 -22 2 D 22 18 22 0 20 E -10 10 -2 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9287: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) C D B E A (8) A E C D B (8) C D E B A (7) B D C E A (7) A E B C D (6) D C B E A (5) A B E D C (5) E A C B D (4) E A B C D (4) D C B A E (4) B D C A E (4) A E C B D (4) E C D B A (3) E C A D B (3) D B C E A (2) D B C A E (2) D B A C E (2) C E D A B (2) A E B D C (2) E B C D A (1) E B A C D (1) C E D B A (1) C D A E B (1) B D A C E (1) B A D C E (1) A C E D B (1) A C D E B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 2 0 0 -12 B -2 0 -20 -16 -14 C 0 20 0 26 0 D 0 16 -26 0 -6 E 12 14 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.334992 D: 0.000000 E: 0.665008 Sum of squares = 0.554455513664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.334992 D: 0.334992 E: 1.000000 A B C D E A 0 2 0 0 -12 B -2 0 -20 -16 -14 C 0 20 0 26 0 D 0 16 -26 0 -6 E 12 14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=24 C=19 D=15 B=13 so B is eliminated. Round 2 votes counts: A=30 D=27 E=24 C=19 so C is eliminated. Round 3 votes counts: D=43 A=30 E=27 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:223 E:216 A:195 D:192 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 0 0 -12 B -2 0 -20 -16 -14 C 0 20 0 26 0 D 0 16 -26 0 -6 E 12 14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 0 -12 B -2 0 -20 -16 -14 C 0 20 0 26 0 D 0 16 -26 0 -6 E 12 14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 0 -12 B -2 0 -20 -16 -14 C 0 20 0 26 0 D 0 16 -26 0 -6 E 12 14 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.500000000002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9288: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (21) D C B E A (16) E B C D A (8) D A C B E (6) A D E C B (5) D C B A E (4) B C E D A (4) A D C E B (4) A D C B E (4) E B C A D (3) A D E B C (3) E B A C D (2) E A B C D (2) C B D E A (2) B E A D C (2) A E C B D (2) A E B D C (2) A D B E C (2) E C B A D (1) C B E D A (1) B E C D A (1) B C D E A (1) A E D B C (1) A E C D B (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 10 18 14 14 B -10 0 8 6 -10 C -18 -8 0 -2 -14 D -14 -6 2 0 -2 E -14 10 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 18 14 14 B -10 0 8 6 -10 C -18 -8 0 -2 -14 D -14 -6 2 0 -2 E -14 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=47 D=26 E=16 B=8 C=3 so C is eliminated. Round 2 votes counts: A=47 D=26 E=16 B=11 so B is eliminated. Round 3 votes counts: A=47 D=29 E=24 so E is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:228 E:206 B:197 D:190 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 18 14 14 B -10 0 8 6 -10 C -18 -8 0 -2 -14 D -14 -6 2 0 -2 E -14 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 18 14 14 B -10 0 8 6 -10 C -18 -8 0 -2 -14 D -14 -6 2 0 -2 E -14 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 18 14 14 B -10 0 8 6 -10 C -18 -8 0 -2 -14 D -14 -6 2 0 -2 E -14 10 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9289: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (6) C B E A D (6) C B D A E (6) B C E A D (6) D C A B E (5) B E C A D (5) B C E D A (5) E D A B C (4) D E A B C (4) D A E B C (4) C B A E D (4) C B A D E (4) A D E C B (4) E A B C D (3) D A C E B (3) A D C E B (3) E B D C A (2) E B A C D (2) D C B A E (2) C B D E A (2) A E D B C (2) A E B D C (2) A E B C D (2) E B C A D (1) E A D B C (1) E A B D C (1) D E B C A (1) D E B A C (1) D C B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A D B E (1) B E C D A (1) B C D E A (1) A E C B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -12 -2 4 B 4 0 -4 12 6 C 12 4 0 8 6 D 2 -12 -8 0 0 E -4 -6 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999768 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -12 -2 4 B 4 0 -4 12 6 C 12 4 0 8 6 D 2 -12 -8 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=26 B=18 A=15 E=14 so E is eliminated. Round 2 votes counts: D=31 C=26 B=23 A=20 so A is eliminated. Round 3 votes counts: D=41 B=31 C=28 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:215 B:209 A:193 E:192 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 4 B 4 0 -4 12 6 C 12 4 0 8 6 D 2 -12 -8 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 4 B 4 0 -4 12 6 C 12 4 0 8 6 D 2 -12 -8 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 4 B 4 0 -4 12 6 C 12 4 0 8 6 D 2 -12 -8 0 0 E -4 -6 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9290: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (10) A D E B C (8) C D E B A (6) C B D E A (6) E B A D C (5) C B E D A (5) B E C D A (5) B E A C D (5) A D C E B (4) A B E D C (4) D C A E B (3) D A E B C (3) D A C E B (3) B E C A D (3) A E B D C (3) A B E C D (3) E A D B C (2) C D A E B (2) B E A D C (2) B A E C D (2) A E D B C (2) A D E C B (2) A C D B E (2) E B D C A (1) E B D A C (1) D E B C A (1) D C E B A (1) D A E C B (1) B C E D A (1) B C E A D (1) B A E D C (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 10 2 -10 B 12 0 10 -2 4 C -10 -10 0 4 -8 D -2 2 -4 0 6 E 10 -4 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468749999957 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 2 -10 B 12 0 10 -2 4 C -10 -10 0 4 -8 D -2 2 -4 0 6 E 10 -4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000002 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=29 B=20 D=12 E=9 so E is eliminated. Round 2 votes counts: A=32 C=29 B=27 D=12 so D is eliminated. Round 3 votes counts: A=39 C=33 B=28 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:212 E:204 D:201 A:195 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 2 -10 B 12 0 10 -2 4 C -10 -10 0 4 -8 D -2 2 -4 0 6 E 10 -4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000002 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 2 -10 B 12 0 10 -2 4 C -10 -10 0 4 -8 D -2 2 -4 0 6 E 10 -4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000002 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 2 -10 B 12 0 10 -2 4 C -10 -10 0 4 -8 D -2 2 -4 0 6 E 10 -4 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.125000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000002 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9291: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (8) D B E C A (6) D B A C E (5) C A D E B (5) B D E C A (5) A C E D B (5) D C A E B (4) B D A C E (4) E B A C D (3) D B C A E (3) C D E A B (3) B E A C D (3) B D A E C (3) E C D B A (2) E C D A B (2) E C A B D (2) C E A D B (2) C D A E B (2) C A E D B (2) B E D C A (2) B D E A C (2) A C E B D (2) A B C D E (2) E D C B A (1) E D B C A (1) E C B A D (1) E B C D A (1) E B C A D (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A B E (1) D B C E A (1) D A C B E (1) C E D A B (1) B E D A C (1) B E A D C (1) B A E D C (1) B A E C D (1) A E C B D (1) A C D B E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -22 -12 -10 B 4 0 0 -22 -8 C 22 0 0 4 -4 D 12 22 -4 0 8 E 10 8 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.374999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 -4 -22 -12 -10 B 4 0 0 -22 -8 C 22 0 0 4 -4 D 12 22 -4 0 8 E 10 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.37500000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 B=23 C=15 A=13 so A is eliminated. Round 2 votes counts: B=27 D=26 E=24 C=23 so C is eliminated. Round 3 votes counts: D=37 E=36 B=27 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 C:211 E:207 B:187 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -22 -12 -10 B 4 0 0 -22 -8 C 22 0 0 4 -4 D 12 22 -4 0 8 E 10 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.37500000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -22 -12 -10 B 4 0 0 -22 -8 C 22 0 0 4 -4 D 12 22 -4 0 8 E 10 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.37500000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -22 -12 -10 B 4 0 0 -22 -8 C 22 0 0 4 -4 D 12 22 -4 0 8 E 10 8 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.37500000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9292: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (7) B D E C A (6) E B C D A (5) D A B C E (5) A E C B D (5) A C E D B (5) C A E D B (4) E C B A D (3) E B D C A (3) E A C B D (3) D C B A E (3) D B C A E (3) C E A B D (3) C D B E A (3) A D B C E (3) A C D E B (3) E C B D A (2) E C A B D (2) E B C A D (2) D B C E A (2) D B A C E (2) C D A B E (2) B E D A C (2) A E C D B (2) A E B C D (2) A C D B E (2) E B D A C (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A B E (1) D B A E C (1) C E D B A (1) C E D A B (1) C D E B A (1) C D B A E (1) C A E B D (1) B E D C A (1) B D C E A (1) B D A E C (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 8 -18 2 10 B -8 0 -16 -8 -18 C 18 16 0 26 14 D -2 8 -26 0 -2 E -10 18 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -18 2 10 B -8 0 -16 -8 -18 C 18 16 0 26 14 D -2 8 -26 0 -2 E -10 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 E=23 D=18 B=11 so B is eliminated. Round 2 votes counts: E=26 D=26 C=24 A=24 so C is eliminated. Round 3 votes counts: A=36 D=33 E=31 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:237 A:201 E:198 D:189 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -18 2 10 B -8 0 -16 -8 -18 C 18 16 0 26 14 D -2 8 -26 0 -2 E -10 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -18 2 10 B -8 0 -16 -8 -18 C 18 16 0 26 14 D -2 8 -26 0 -2 E -10 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -18 2 10 B -8 0 -16 -8 -18 C 18 16 0 26 14 D -2 8 -26 0 -2 E -10 18 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9293: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (6) D A B E C (6) C B E A D (6) D A C B E (5) A D B E C (5) E C D B A (4) E C B D A (4) B A D E C (4) A D B C E (4) A B D C E (4) D A E B C (3) D A C E B (3) D A B C E (3) B E C A D (3) B A D C E (3) B A C D E (3) E C B A D (2) E B C A D (2) D C E A B (2) C E B D A (2) C B A D E (2) B E A D C (2) B C E A D (2) A D C B E (2) E D B A C (1) E D A B C (1) E C D A B (1) E B D A C (1) E B C D A (1) E B A D C (1) D E C A B (1) D E A C B (1) C E D B A (1) C E D A B (1) C E B A D (1) C D E A B (1) C B A E D (1) C A B D E (1) B E A C D (1) B A E C D (1) B A C E D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 24 4 16 B -2 0 6 -2 20 C -24 -6 0 -18 -4 D -4 2 18 0 20 E -16 -20 4 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998614 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 24 4 16 B -2 0 6 -2 20 C -24 -6 0 -18 -4 D -4 2 18 0 20 E -16 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997476 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=20 E=18 C=16 A=16 so C is eliminated. Round 2 votes counts: D=31 B=29 E=23 A=17 so A is eliminated. Round 3 votes counts: D=42 B=35 E=23 so E is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:223 D:218 B:211 C:174 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 24 4 16 B -2 0 6 -2 20 C -24 -6 0 -18 -4 D -4 2 18 0 20 E -16 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997476 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 24 4 16 B -2 0 6 -2 20 C -24 -6 0 -18 -4 D -4 2 18 0 20 E -16 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997476 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 24 4 16 B -2 0 6 -2 20 C -24 -6 0 -18 -4 D -4 2 18 0 20 E -16 -20 4 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999997476 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9294: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) E D C B A (6) C A E D B (5) A C B D E (5) A B E D C (5) C D E A B (4) B A E D C (4) A C B E D (4) E D C A B (3) E D B C A (3) E A D C B (3) C A D E B (3) B D E C A (3) B C D E A (3) A E B C D (3) A C E D B (3) A B C E D (3) E D B A C (2) E A B D C (2) D E B C A (2) C E D A B (2) C B A D E (2) A E C D B (2) A B E C D (2) A B C D E (2) E D A C B (1) E C D B A (1) E B D C A (1) E B D A C (1) D E C B A (1) D C E B A (1) C D B E A (1) C D B A E (1) C B D E A (1) B E D A C (1) B E A D C (1) B D A E C (1) B C D A E (1) B A D E C (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 6 -4 4 -2 B -6 0 -14 -4 -16 C 4 14 0 14 0 D -4 4 -14 0 -18 E 2 16 0 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.586829 D: 0.000000 E: 0.413171 Sum of squares = 0.515078584922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.586829 D: 0.586829 E: 1.000000 A B C D E A 0 6 -4 4 -2 B -6 0 -14 -4 -16 C 4 14 0 14 0 D -4 4 -14 0 -18 E 2 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=27 E=23 B=16 D=4 so D is eliminated. Round 2 votes counts: A=30 C=28 E=26 B=16 so B is eliminated. Round 3 votes counts: A=37 C=32 E=31 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:218 C:216 A:202 D:184 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 4 -2 B -6 0 -14 -4 -16 C 4 14 0 14 0 D -4 4 -14 0 -18 E 2 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 4 -2 B -6 0 -14 -4 -16 C 4 14 0 14 0 D -4 4 -14 0 -18 E 2 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 4 -2 B -6 0 -14 -4 -16 C 4 14 0 14 0 D -4 4 -14 0 -18 E 2 16 0 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9295: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) D E C B A (7) C E D A B (7) A C B E D (7) B A C D E (6) B D C E A (5) D E B C A (4) D B E C A (4) E D C A B (3) E D A C B (3) C D E B A (3) C A E D B (3) B D E A C (3) B C A D E (3) A E D C B (3) A B E D C (3) E D A B C (2) D C E B A (2) B D E C A (2) A E C D B (2) A E B C D (2) A C E D B (2) A C E B D (2) A B E C D (2) D E B A C (1) C E A D B (1) C D E A B (1) C B D E A (1) C B A D E (1) C A E B D (1) C A B E D (1) B D A E C (1) B C D E A (1) B C D A E (1) B A D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -2 -2 0 B -8 0 0 4 2 C 2 0 0 12 14 D 2 -4 -12 0 -4 E 0 -2 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.156791 C: 0.843208 D: 0.000000 E: 0.000000 Sum of squares = 0.735584143109 Cumulative probabilities = A: 0.000000 B: 0.156792 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 -2 0 B -8 0 0 4 2 C 2 0 0 12 14 D 2 -4 -12 0 -4 E 0 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000010206 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=23 C=19 D=18 E=8 so E is eliminated. Round 2 votes counts: A=32 D=26 B=23 C=19 so C is eliminated. Round 3 votes counts: A=38 D=37 B=25 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:214 A:202 B:199 E:194 D:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 -2 0 B -8 0 0 4 2 C 2 0 0 12 14 D 2 -4 -12 0 -4 E 0 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000010206 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -2 0 B -8 0 0 4 2 C 2 0 0 12 14 D 2 -4 -12 0 -4 E 0 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000010206 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -2 0 B -8 0 0 4 2 C 2 0 0 12 14 D 2 -4 -12 0 -4 E 0 -2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000010206 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9296: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) C A D B E (6) C D E A B (5) B A E D C (5) A D C B E (5) E B C D A (4) D C A E B (4) B E C A D (4) E D C B A (3) E B D C A (3) E B C A D (3) D A C B E (3) D A B E C (3) C D A B E (3) E C D B A (2) E C B D A (2) E B A C D (2) D C A B E (2) C E D B A (2) C E D A B (2) B E A D C (2) B E A C D (2) B A C E D (2) A C B D E (2) A B D E C (2) A B D C E (2) E D C A B (1) E D B C A (1) E D B A C (1) E D A C B (1) E D A B C (1) D C E A B (1) D A C E B (1) D A B C E (1) C E B D A (1) C D A E B (1) C B E A D (1) C A B D E (1) B C E A D (1) B A E C D (1) B A C D E (1) A D B C E (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -6 -10 -4 B 0 0 0 -2 4 C 6 0 0 0 0 D 10 2 0 0 -8 E 4 -4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.474280 C: 0.170010 D: 0.237140 E: 0.118570 Sum of squares = 0.324139093107 Cumulative probabilities = A: 0.000000 B: 0.474280 C: 0.644290 D: 0.881430 E: 1.000000 A B C D E A 0 0 -6 -10 -4 B 0 0 0 -2 4 C 6 0 0 0 0 D 10 2 0 0 -8 E 4 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.300000 D: 0.200000 E: 0.100000 Sum of squares = 0.300000000034 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.700000 D: 0.900000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 B=18 D=15 A=14 so A is eliminated. Round 2 votes counts: E=31 C=25 B=23 D=21 so D is eliminated. Round 3 votes counts: C=41 E=31 B=28 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. E:204 C:203 D:202 B:201 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 -10 -4 B 0 0 0 -2 4 C 6 0 0 0 0 D 10 2 0 0 -8 E 4 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.300000 D: 0.200000 E: 0.100000 Sum of squares = 0.300000000034 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.700000 D: 0.900000 E: 1.000000 GTS winners are ['B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -10 -4 B 0 0 0 -2 4 C 6 0 0 0 0 D 10 2 0 0 -8 E 4 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.300000 D: 0.200000 E: 0.100000 Sum of squares = 0.300000000034 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.700000 D: 0.900000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -10 -4 B 0 0 0 -2 4 C 6 0 0 0 0 D 10 2 0 0 -8 E 4 -4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.400000 C: 0.300000 D: 0.200000 E: 0.100000 Sum of squares = 0.300000000034 Cumulative probabilities = A: 0.000000 B: 0.400000 C: 0.700000 D: 0.900000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9297: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (12) B A E D C (10) D C E B A (8) A B E C D (8) B D E A C (5) B D A C E (4) A E C B D (4) A B C D E (4) D C B E A (3) D B C E A (3) C E A D B (3) C A E D B (3) B A D E C (3) A C B E D (3) D C E A B (2) D B E C A (2) C E D A B (2) B D E C A (2) B A D C E (2) A E B C D (2) A C E D B (2) A C E B D (2) A B C E D (2) E D B A C (1) E A C B D (1) D E C B A (1) C D E B A (1) C D A B E (1) B D C A E (1) B D A E C (1) A E C D B (1) A B E D C (1) Total count = 100 A B C D E A 0 6 12 2 8 B -6 0 2 10 10 C -12 -2 0 2 16 D -2 -10 -2 0 10 E -8 -10 -16 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 2 8 B -6 0 2 10 10 C -12 -2 0 2 16 D -2 -10 -2 0 10 E -8 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=28 C=22 D=19 E=2 so E is eliminated. Round 2 votes counts: A=30 B=28 C=22 D=20 so D is eliminated. Round 3 votes counts: C=36 B=34 A=30 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:214 B:208 C:202 D:198 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 2 8 B -6 0 2 10 10 C -12 -2 0 2 16 D -2 -10 -2 0 10 E -8 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 2 8 B -6 0 2 10 10 C -12 -2 0 2 16 D -2 -10 -2 0 10 E -8 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 2 8 B -6 0 2 10 10 C -12 -2 0 2 16 D -2 -10 -2 0 10 E -8 -10 -16 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9298: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) D E B A C (7) E A D B C (5) D E A C B (5) D C E A B (5) B E D A C (5) B D E A C (5) D E A B C (4) C A D E B (4) C B D A E (3) C A E D B (3) B D E C A (3) B C A E D (3) E D A B C (2) E A D C B (2) D B E A C (2) D B C E A (2) C B A D E (2) B E A D C (2) B C D E A (2) A C E D B (2) A C E B D (2) A B E C D (2) E A B D C (1) D E B C A (1) D B E C A (1) C D B A E (1) C D A E B (1) C B D E A (1) C B A E D (1) C A E B D (1) C A B D E (1) B E D C A (1) B C E D A (1) A E D C B (1) A E C D B (1) A E B D C (1) A E B C D (1) A D C E B (1) A C D E B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 14 8 -4 -14 B -14 0 4 -2 -2 C -8 -4 0 -12 -4 D 4 2 12 0 4 E 14 2 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 -4 -14 B -14 0 4 -2 -2 C -8 -4 0 -12 -4 D 4 2 12 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=27 C=27 B=22 A=14 E=10 so E is eliminated. Round 2 votes counts: D=29 C=27 B=22 A=22 so B is eliminated. Round 3 votes counts: D=43 C=33 A=24 so A is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:211 E:208 A:202 B:193 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 8 -4 -14 B -14 0 4 -2 -2 C -8 -4 0 -12 -4 D 4 2 12 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -4 -14 B -14 0 4 -2 -2 C -8 -4 0 -12 -4 D 4 2 12 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -4 -14 B -14 0 4 -2 -2 C -8 -4 0 -12 -4 D 4 2 12 0 4 E 14 2 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990205 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9299: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (9) C E D A B (7) E C D B A (6) B E A C D (6) B A D E C (6) E C B A D (5) E B C A D (5) D A C B E (5) B E A D C (5) D A B C E (4) A D C B E (4) D C A E B (3) C E A D B (3) A D B C E (3) A B D C E (3) E B C D A (2) C D E A B (2) C D A E B (2) C A D E B (2) B E D A C (2) B D E A C (2) B A E D C (2) B A E C D (2) A C D B E (2) E D C B A (1) E B D C A (1) D B A E C (1) D A C E B (1) C E A B D (1) B A C E D (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 2 4 -14 B 12 0 -8 6 0 C -2 8 0 12 -12 D -4 -6 -12 0 -18 E 14 0 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.350456 C: 0.000000 D: 0.000000 E: 0.649544 Sum of squares = 0.544726989693 Cumulative probabilities = A: 0.000000 B: 0.350456 C: 0.350456 D: 0.350456 E: 1.000000 A B C D E A 0 -12 2 4 -14 B 12 0 -8 6 0 C -2 8 0 12 -12 D -4 -6 -12 0 -18 E 14 0 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=26 C=17 D=14 A=14 so D is eliminated. Round 2 votes counts: E=29 B=27 A=24 C=20 so C is eliminated. Round 3 votes counts: E=42 A=31 B=27 so B is eliminated. Round 4 votes counts: E=57 A=43 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:205 C:203 A:190 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 2 4 -14 B 12 0 -8 6 0 C -2 8 0 12 -12 D -4 -6 -12 0 -18 E 14 0 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 4 -14 B 12 0 -8 6 0 C -2 8 0 12 -12 D -4 -6 -12 0 -18 E 14 0 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 4 -14 B 12 0 -8 6 0 C -2 8 0 12 -12 D -4 -6 -12 0 -18 E 14 0 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9300: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (16) B E D C A (7) C B E A D (6) A C D E B (5) D B E A C (4) B E C D A (4) E B C A D (3) D B A E C (3) C A E B D (3) C A D B E (3) B E D A C (3) A D E C B (3) A D C E B (3) E B D A C (2) E B A C D (2) D E A B C (2) D A C E B (2) D A B E C (2) C B E D A (2) C A E D B (2) C A B E D (2) A D E B C (2) A C E D B (2) E D B A C (1) E D A B C (1) E A B C D (1) D E B A C (1) D C B A E (1) D B C E A (1) D A E C B (1) D A C B E (1) C E B A D (1) C B A E D (1) C B A D E (1) C A D E B (1) B E C A D (1) B D E C A (1) B D E A C (1) B C E D A (1) A E C D B (1) Total count = 100 A B C D E A 0 6 18 -14 10 B -6 0 18 -16 -10 C -18 -18 0 -16 -24 D 14 16 16 0 8 E -10 10 24 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 18 -14 10 B -6 0 18 -16 -10 C -18 -18 0 -16 -24 D 14 16 16 0 8 E -10 10 24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=22 B=18 A=16 E=10 so E is eliminated. Round 2 votes counts: D=36 B=25 C=22 A=17 so A is eliminated. Round 3 votes counts: D=44 C=30 B=26 so B is eliminated. Round 4 votes counts: D=58 C=42 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:227 A:210 E:208 B:193 C:162 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 18 -14 10 B -6 0 18 -16 -10 C -18 -18 0 -16 -24 D 14 16 16 0 8 E -10 10 24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 -14 10 B -6 0 18 -16 -10 C -18 -18 0 -16 -24 D 14 16 16 0 8 E -10 10 24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 -14 10 B -6 0 18 -16 -10 C -18 -18 0 -16 -24 D 14 16 16 0 8 E -10 10 24 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9301: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (14) C A E D B (8) A E C B D (8) D B C E A (6) E A B C D (5) B A C E D (5) B D E A C (4) A C E B D (4) E A C B D (3) D B C A E (3) C D E A B (3) A E C D B (3) A E B C D (3) D E C A B (2) D B E A C (2) C D A B E (2) B D E C A (2) B D C A E (2) E D C A B (1) E A D B C (1) E A B D C (1) D E B A C (1) D C B E A (1) D C B A E (1) D B E C A (1) C E D A B (1) C D B A E (1) C D A E B (1) C B D A E (1) C A E B D (1) C A D E B (1) C A D B E (1) B E D A C (1) B E A D C (1) B D A E C (1) B A E D C (1) B A E C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 30 22 26 0 B -30 0 -16 -8 -22 C -22 16 0 36 -14 D -26 8 -36 0 -28 E 0 22 14 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.138181 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.861819 Sum of squares = 0.761826585866 Cumulative probabilities = A: 0.138181 B: 0.138181 C: 0.138181 D: 0.138181 E: 1.000000 A B C D E A 0 30 22 26 0 B -30 0 -16 -8 -22 C -22 16 0 36 -14 D -26 8 -36 0 -28 E 0 22 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=20 A=20 B=18 D=17 so D is eliminated. Round 2 votes counts: B=30 E=28 C=22 A=20 so A is eliminated. Round 3 votes counts: E=42 B=31 C=27 so C is eliminated. Round 4 votes counts: E=61 B=39 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:239 E:232 C:208 B:162 D:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 30 22 26 0 B -30 0 -16 -8 -22 C -22 16 0 36 -14 D -26 8 -36 0 -28 E 0 22 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 22 26 0 B -30 0 -16 -8 -22 C -22 16 0 36 -14 D -26 8 -36 0 -28 E 0 22 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 22 26 0 B -30 0 -16 -8 -22 C -22 16 0 36 -14 D -26 8 -36 0 -28 E 0 22 14 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9302: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (5) C B D A E (5) C A E B D (5) E C A D B (4) E A C B D (4) D E A B C (4) D C B E A (4) B A E D C (4) B A D E C (4) E A D B C (3) D B E A C (3) D B C A E (3) D B A E C (3) C E A D B (3) C E A B D (3) C D B E A (3) B D C A E (3) B A C D E (3) E A B D C (2) D E A C B (2) D B C E A (2) C D E B A (2) C D E A B (2) C D B A E (2) C B A E D (2) C B A D E (2) B C A D E (2) B A C E D (2) A B E C D (2) E D C A B (1) E D A C B (1) E A D C B (1) E A C D B (1) D C E B A (1) D C E A B (1) C A B E D (1) B D A E C (1) A E C B D (1) A E B D C (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 -12 4 -4 B 2 0 -12 -2 4 C 12 12 0 10 12 D -4 2 -10 0 4 E 4 -4 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 4 -4 B 2 0 -12 -2 4 C 12 12 0 10 12 D -4 2 -10 0 4 E 4 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=23 B=19 E=17 A=6 so A is eliminated. Round 2 votes counts: C=35 D=23 B=22 E=20 so E is eliminated. Round 3 votes counts: C=45 D=29 B=26 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:223 B:196 D:196 A:193 E:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 4 -4 B 2 0 -12 -2 4 C 12 12 0 10 12 D -4 2 -10 0 4 E 4 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 4 -4 B 2 0 -12 -2 4 C 12 12 0 10 12 D -4 2 -10 0 4 E 4 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 4 -4 B 2 0 -12 -2 4 C 12 12 0 10 12 D -4 2 -10 0 4 E 4 -4 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9303: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (7) B E D C A (7) A D B C E (7) C E A B D (6) E C B D A (5) D B A E C (5) C A E D B (4) B D E A C (4) E C A D B (3) E B C D A (3) C E B A D (3) C B A D E (3) B D A E C (3) A D C B E (3) A C D B E (3) A B D C E (3) E D B A C (2) E B D C A (2) E B D A C (2) C E A D B (2) C A B E D (2) B C E D A (2) A D E C B (2) E C D A B (1) E C B A D (1) D E A B C (1) C E B D A (1) C B E A D (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D A C (1) B E C D A (1) B D E C A (1) A D E B C (1) A D C E B (1) A D B E C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 4 -4 4 2 B -4 0 6 6 14 C 4 -6 0 -6 -6 D -4 -6 6 0 -2 E -2 -14 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775497 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -4 4 2 B -4 0 6 6 14 C 4 -6 0 -6 -6 D -4 -6 6 0 -2 E -2 -14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775423 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 E=19 B=19 D=13 so D is eliminated. Round 2 votes counts: A=30 C=26 B=24 E=20 so E is eliminated. Round 3 votes counts: C=36 B=33 A=31 so A is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:203 D:197 E:196 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -4 4 2 B -4 0 6 6 14 C 4 -6 0 -6 -6 D -4 -6 6 0 -2 E -2 -14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775423 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -4 4 2 B -4 0 6 6 14 C 4 -6 0 -6 -6 D -4 -6 6 0 -2 E -2 -14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775423 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -4 4 2 B -4 0 6 6 14 C 4 -6 0 -6 -6 D -4 -6 6 0 -2 E -2 -14 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.285714 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.346938775423 Cumulative probabilities = A: 0.428571 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9304: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (11) A B C D E (8) C E D A B (5) B A D E C (5) E D C B A (4) E D B C A (4) C E A D B (4) B A D C E (4) A B D C E (4) D E C B A (3) C A E D B (3) B A E D C (3) A C B D E (3) E C D B A (2) E C B A D (2) D E B A C (2) D C A E B (2) D B A E C (2) C E A B D (2) C D E A B (2) C A E B D (2) B E D A C (2) B D E A C (2) A C D B E (2) E D C A B (1) E C A D B (1) E B D C A (1) E B C A D (1) D E B C A (1) D B E A C (1) D A C B E (1) D A B C E (1) C D A E B (1) C A D B E (1) B E A D C (1) B E A C D (1) B D A E C (1) B A C E D (1) B A C D E (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 12 -6 2 -6 B -12 0 -6 -12 -8 C 6 6 0 6 -2 D -2 12 -6 0 -2 E 6 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -6 2 -6 B -12 0 -6 -12 -8 C 6 6 0 6 -2 D -2 12 -6 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=21 C=20 A=19 D=13 so D is eliminated. Round 2 votes counts: E=33 B=24 C=22 A=21 so A is eliminated. Round 3 votes counts: B=38 E=33 C=29 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:208 A:201 D:201 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -6 2 -6 B -12 0 -6 -12 -8 C 6 6 0 6 -2 D -2 12 -6 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 2 -6 B -12 0 -6 -12 -8 C 6 6 0 6 -2 D -2 12 -6 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 2 -6 B -12 0 -6 -12 -8 C 6 6 0 6 -2 D -2 12 -6 0 -2 E 6 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999603 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9305: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) A B C D E (7) B A E D C (6) A B E D C (6) E D B C A (5) C D E A B (5) E D C B A (4) E B A D C (4) D E B C A (4) A C E D B (4) A C B E D (4) A C B D E (4) A B C E D (4) E C D A B (3) D C E B A (3) A C D E B (3) E B D A C (2) C E D A B (2) C A D E B (2) C A D B E (2) B E D C A (2) B E A D C (2) A B E C D (2) E B D C A (1) E A C D B (1) D E C B A (1) D B C E A (1) C D A E B (1) C A E D B (1) B A D E C (1) B A D C E (1) A E C D B (1) A E C B D (1) A E B C D (1) A C D B E (1) Total count = 100 A B C D E A 0 10 26 16 4 B -10 0 14 12 2 C -26 -14 0 -2 -10 D -16 -12 2 0 -28 E -4 -2 10 28 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998757 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 26 16 4 B -10 0 14 12 2 C -26 -14 0 -2 -10 D -16 -12 2 0 -28 E -4 -2 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=20 B=20 C=13 D=9 so D is eliminated. Round 2 votes counts: A=38 E=25 B=21 C=16 so C is eliminated. Round 3 votes counts: A=44 E=35 B=21 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:228 E:216 B:209 C:174 D:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 26 16 4 B -10 0 14 12 2 C -26 -14 0 -2 -10 D -16 -12 2 0 -28 E -4 -2 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 26 16 4 B -10 0 14 12 2 C -26 -14 0 -2 -10 D -16 -12 2 0 -28 E -4 -2 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 26 16 4 B -10 0 14 12 2 C -26 -14 0 -2 -10 D -16 -12 2 0 -28 E -4 -2 10 28 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998913 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9306: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) B A E D C (7) A B E D C (5) C D B E A (4) C A D E B (4) B D E C A (4) A B C E D (4) E D A B C (3) D C E A B (3) C D A B E (3) B E A D C (3) B A C E D (3) A C E B D (3) E B D A C (2) E A D C B (2) E A D B C (2) E A B D C (2) D E C B A (2) D C B E A (2) D B E C A (2) C D E B A (2) C D B A E (2) C A E D B (2) C A D B E (2) B E D A C (2) B D E A C (2) A E C B D (2) A C E D B (2) E D B A C (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E B A (1) C D E A B (1) C B D E A (1) C B D A E (1) C A B E D (1) B D C E A (1) B C A D E (1) B A E C D (1) A E C D B (1) A E B D C (1) A E B C D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 10 4 2 10 B -10 0 -2 -2 6 C -4 2 0 0 2 D -2 2 0 0 -4 E -10 -6 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 2 10 B -10 0 -2 -2 6 C -4 2 0 0 2 D -2 2 0 0 -4 E -10 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=24 A=21 D=13 E=12 so E is eliminated. Round 2 votes counts: C=30 A=27 B=26 D=17 so D is eliminated. Round 3 votes counts: C=39 A=31 B=30 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:213 C:200 D:198 B:196 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 2 10 B -10 0 -2 -2 6 C -4 2 0 0 2 D -2 2 0 0 -4 E -10 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 2 10 B -10 0 -2 -2 6 C -4 2 0 0 2 D -2 2 0 0 -4 E -10 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 2 10 B -10 0 -2 -2 6 C -4 2 0 0 2 D -2 2 0 0 -4 E -10 -6 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995818 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9307: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (11) C A D E B (9) B E D A C (9) B E D C A (8) C B E D A (6) E B D A C (5) A D C E B (5) C D A E B (4) C D A B E (4) C A D B E (4) A D E B C (4) C B E A D (3) C A B D E (3) A E D B C (3) A C D E B (3) E D B A C (2) E B A D C (2) D E A B C (2) B E A D C (2) A D E C B (2) E B D C A (1) C D E B A (1) C D B E A (1) C B D E A (1) C B D A E (1) C A B E D (1) B E C A D (1) B C E D A (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -20 -14 -12 B 10 0 2 10 14 C 20 -2 0 10 -4 D 14 -10 -10 0 -10 E 12 -14 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 -14 -12 B 10 0 2 10 14 C 20 -2 0 10 -4 D 14 -10 -10 0 -10 E 12 -14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=32 A=18 E=10 D=2 so D is eliminated. Round 2 votes counts: C=38 B=32 A=18 E=12 so E is eliminated. Round 3 votes counts: B=42 C=38 A=20 so A is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:218 C:212 E:206 D:192 A:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -20 -14 -12 B 10 0 2 10 14 C 20 -2 0 10 -4 D 14 -10 -10 0 -10 E 12 -14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 -14 -12 B 10 0 2 10 14 C 20 -2 0 10 -4 D 14 -10 -10 0 -10 E 12 -14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 -14 -12 B 10 0 2 10 14 C 20 -2 0 10 -4 D 14 -10 -10 0 -10 E 12 -14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999249 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9308: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) C B A E D (10) D E A C B (8) A E B D C (6) E D A B C (5) B C A E D (5) B A E C D (5) C D E B A (4) E A D B C (3) D C A E B (3) C D B E A (3) C D B A E (3) C B E A D (3) C B D A E (3) C B A D E (3) A E D B C (3) D C E A B (2) D A E B C (2) C B E D A (2) B E A C D (2) B A C E D (2) E D B A C (1) E B A D C (1) D E C A B (1) D C E B A (1) D A E C B (1) C E D B A (1) C D E A B (1) C D A B E (1) C B D E A (1) C A B D E (1) B E C A D (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 -8 -2 B 2 0 -4 -8 -8 C -2 4 0 2 -2 D 8 8 -2 0 -2 E 2 8 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 -8 -2 B 2 0 -4 -8 -8 C -2 4 0 2 -2 D 8 8 -2 0 -2 E 2 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 D=29 B=15 E=10 A=10 so E is eliminated. Round 2 votes counts: C=36 D=35 B=16 A=13 so A is eliminated. Round 3 votes counts: D=41 C=36 B=23 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:207 D:206 C:201 A:195 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -8 -2 B 2 0 -4 -8 -8 C -2 4 0 2 -2 D 8 8 -2 0 -2 E 2 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -8 -2 B 2 0 -4 -8 -8 C -2 4 0 2 -2 D 8 8 -2 0 -2 E 2 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -8 -2 B 2 0 -4 -8 -8 C -2 4 0 2 -2 D 8 8 -2 0 -2 E 2 8 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9309: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (18) C D A B E (14) D C A B E (9) E B A C D (7) C D A E B (6) E C A B D (3) E B C A D (3) C D E B A (3) B E D A C (3) A B D E C (3) E C B D A (2) E B C D A (2) C E D B A (2) C D E A B (2) B A E D C (2) A E B C D (2) A D C B E (2) A C D B E (2) E C B A D (1) E B D C A (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) D A C B E (1) D A B C E (1) C E A D B (1) C A E D B (1) B E D C A (1) B E A D C (1) B D E A C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -6 0 -8 B -2 0 -4 6 -12 C 6 4 0 8 -6 D 0 -6 -8 0 -6 E 8 12 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -6 0 -8 B -2 0 -4 6 -12 C 6 4 0 8 -6 D 0 -6 -8 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 C=29 D=13 A=11 B=8 so B is eliminated. Round 2 votes counts: E=44 C=29 D=14 A=13 so A is eliminated. Round 3 votes counts: E=49 C=31 D=20 so D is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:206 A:194 B:194 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -6 0 -8 B -2 0 -4 6 -12 C 6 4 0 8 -6 D 0 -6 -8 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 0 -8 B -2 0 -4 6 -12 C 6 4 0 8 -6 D 0 -6 -8 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 0 -8 B -2 0 -4 6 -12 C 6 4 0 8 -6 D 0 -6 -8 0 -6 E 8 12 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999936 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9310: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (9) B D A C E (8) B A D E C (7) A D B E C (7) E B C A D (6) E C B A D (5) B E A D C (5) C E D A B (4) C B E D A (4) B E C A D (4) B D A E C (4) D A C B E (3) C D A E B (3) C D A B E (3) B C E D A (3) E B A D C (2) D B A C E (2) C E A D B (2) A D E C B (2) A D E B C (2) A D C E B (2) E C B D A (1) E C A D B (1) E C A B D (1) E B A C D (1) E A D C B (1) E A C D B (1) D C A B E (1) D A B E C (1) C B D A E (1) C A D E B (1) B E C D A (1) B D C A E (1) B C D A E (1) Total count = 100 A B C D E A 0 -30 -4 0 -2 B 30 0 10 28 12 C 4 -10 0 4 -4 D 0 -28 -4 0 -2 E 2 -12 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -30 -4 0 -2 B 30 0 10 28 12 C 4 -10 0 4 -4 D 0 -28 -4 0 -2 E 2 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 C=27 E=19 A=13 D=7 so D is eliminated. Round 2 votes counts: B=36 C=28 E=19 A=17 so A is eliminated. Round 3 votes counts: B=44 C=33 E=23 so E is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:240 E:198 C:197 D:183 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -30 -4 0 -2 B 30 0 10 28 12 C 4 -10 0 4 -4 D 0 -28 -4 0 -2 E 2 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -30 -4 0 -2 B 30 0 10 28 12 C 4 -10 0 4 -4 D 0 -28 -4 0 -2 E 2 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -30 -4 0 -2 B 30 0 10 28 12 C 4 -10 0 4 -4 D 0 -28 -4 0 -2 E 2 -12 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998592 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9311: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (9) E D B A C (7) A B C E D (6) D E B C A (4) C D A E B (4) C A B D E (4) E B D A C (3) D E C B A (3) D A B C E (3) C D A B E (3) C A D B E (3) A C B D E (3) A B E C D (3) E C D B A (2) E B C A D (2) E B A C D (2) D E B A C (2) D C E B A (2) D C E A B (2) D C A B E (2) D B E A C (2) C E D A B (2) C E A B D (2) C D E A B (2) C A E B D (2) B A E D C (2) A C B E D (2) A B D E C (2) A B C D E (2) E D C B A (1) E D B C A (1) E C B D A (1) E C B A D (1) D C A E B (1) D B A E C (1) D A B E C (1) C E D B A (1) C E B D A (1) C E B A D (1) B E A D C (1) B D A E C (1) A D B C E (1) Total count = 100 A B C D E A 0 18 -12 -4 10 B -18 0 -8 0 2 C 12 8 0 16 16 D 4 0 -16 0 -4 E -10 -2 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 -4 10 B -18 0 -8 0 2 C 12 8 0 16 16 D 4 0 -16 0 -4 E -10 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=23 E=20 A=19 B=4 so B is eliminated. Round 2 votes counts: C=34 D=24 E=21 A=21 so E is eliminated. Round 3 votes counts: C=40 D=36 A=24 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:206 D:192 B:188 E:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -12 -4 10 B -18 0 -8 0 2 C 12 8 0 16 16 D 4 0 -16 0 -4 E -10 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 -4 10 B -18 0 -8 0 2 C 12 8 0 16 16 D 4 0 -16 0 -4 E -10 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 -4 10 B -18 0 -8 0 2 C 12 8 0 16 16 D 4 0 -16 0 -4 E -10 -2 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9312: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) D B E A C (5) C A E D B (5) A C E B D (5) A B D C E (5) E D C B A (4) C E D A B (4) C A D B E (4) B D A E C (4) A C B E D (4) E C D B A (3) E A C B D (3) D B E C A (3) B A D E C (3) A B E D C (3) A B D E C (3) E C D A B (2) E C A D B (2) D C B E A (2) D B A C E (2) C A E B D (2) B D E A C (2) A E C B D (2) A B C E D (2) E D B C A (1) E C A B D (1) E B D A C (1) D C E B A (1) D B C E A (1) D B C A E (1) D B A E C (1) C E D B A (1) C E A B D (1) C D E B A (1) C D B A E (1) C A B D E (1) B D A C E (1) B A E D C (1) B A D C E (1) A E B C D (1) A C D B E (1) A B C D E (1) Total count = 100 A B C D E A 0 20 2 18 8 B -20 0 -16 -6 4 C -2 16 0 10 10 D -18 6 -10 0 -12 E -8 -4 -10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 2 18 8 B -20 0 -16 -6 4 C -2 16 0 10 10 D -18 6 -10 0 -12 E -8 -4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 E=17 D=16 B=12 so B is eliminated. Round 2 votes counts: A=32 C=28 D=23 E=17 so E is eliminated. Round 3 votes counts: C=36 A=35 D=29 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:217 E:195 D:183 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 2 18 8 B -20 0 -16 -6 4 C -2 16 0 10 10 D -18 6 -10 0 -12 E -8 -4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 2 18 8 B -20 0 -16 -6 4 C -2 16 0 10 10 D -18 6 -10 0 -12 E -8 -4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 2 18 8 B -20 0 -16 -6 4 C -2 16 0 10 10 D -18 6 -10 0 -12 E -8 -4 -10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996906 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9313: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (17) C A D B E (12) A C E D B (10) B D E C A (8) E A C B D (6) C D A B E (5) A C D E B (5) E A C D B (4) D B C A E (4) C D B A E (4) B D C E A (4) A E C D B (4) E B D C A (3) E B A D C (3) E A B D C (2) D C B A E (2) D B C E A (2) B D C A E (2) A C D B E (2) E A B C D (1) Total count = 100 A B C D E A 0 2 8 -2 0 B -2 0 -8 -8 -10 C -8 8 0 6 4 D 2 8 -6 0 0 E 0 10 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999979 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -2 0 B -2 0 -8 -8 -10 C -8 8 0 6 4 D 2 8 -6 0 0 E 0 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999366 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=21 A=21 B=14 D=8 so D is eliminated. Round 2 votes counts: E=36 C=23 A=21 B=20 so B is eliminated. Round 3 votes counts: E=44 C=35 A=21 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:205 A:204 E:203 D:202 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 -2 0 B -2 0 -8 -8 -10 C -8 8 0 6 4 D 2 8 -6 0 0 E 0 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999366 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -2 0 B -2 0 -8 -8 -10 C -8 8 0 6 4 D 2 8 -6 0 0 E 0 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999366 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -2 0 B -2 0 -8 -8 -10 C -8 8 0 6 4 D 2 8 -6 0 0 E 0 10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.125000 D: 0.500000 E: 0.000000 Sum of squares = 0.406249999366 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9314: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C B A (7) B D A E C (6) A B E C D (6) C E D A B (5) B A E C D (5) B A D E C (5) D B E C A (4) D B C E A (4) C D E A B (4) B D A C E (4) B A D C E (4) D E C A B (3) D C E B A (3) D B E A C (3) E D C A B (2) E C D A B (2) D C E A B (2) D C B E A (2) D B A E C (2) C E A D B (2) C A E B D (2) B A C E D (2) A E C B D (2) A C E B D (2) E A C B D (1) D E B C A (1) D B A C E (1) C A B E D (1) B D C A E (1) B A E D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 10 -12 10 B 10 0 20 6 22 C -10 -20 0 -10 -2 D 12 -6 10 0 12 E -10 -22 2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 10 -12 10 B 10 0 20 6 22 C -10 -20 0 -10 -2 D 12 -6 10 0 12 E -10 -22 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 B=28 A=21 C=14 E=5 so E is eliminated. Round 2 votes counts: D=34 B=28 A=22 C=16 so C is eliminated. Round 3 votes counts: D=45 B=28 A=27 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:214 A:199 C:179 E:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 10 -12 10 B 10 0 20 6 22 C -10 -20 0 -10 -2 D 12 -6 10 0 12 E -10 -22 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 10 -12 10 B 10 0 20 6 22 C -10 -20 0 -10 -2 D 12 -6 10 0 12 E -10 -22 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 10 -12 10 B 10 0 20 6 22 C -10 -20 0 -10 -2 D 12 -6 10 0 12 E -10 -22 2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9315: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (10) B C D A E (7) B C A D E (7) A E C D B (7) B D C A E (6) E D A C B (5) E A D C B (5) E A C D B (5) A E C B D (4) D E B C A (3) D B E C A (3) B D E C A (3) B C A E D (3) D E C A B (2) D C E A B (2) B D C E A (2) A E B C D (2) A C E B D (2) E D B A C (1) E D A B C (1) E B A C D (1) E A D B C (1) E A C B D (1) E A B C D (1) D E B A C (1) D C B E A (1) D C B A E (1) D C A E B (1) D B E A C (1) D B C A E (1) C E D A B (1) C D A E B (1) C B D A E (1) C B A D E (1) C A E D B (1) C A B E D (1) B A E C D (1) A C E D B (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -16 -8 0 B 8 0 12 -10 2 C 16 -12 0 0 2 D 8 10 0 0 8 E 0 -2 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.287593 D: 0.712407 E: 0.000000 Sum of squares = 0.590233675483 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.287593 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -16 -8 0 B 8 0 12 -10 2 C 16 -12 0 0 2 D 8 10 0 0 8 E 0 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454541 D: 0.545459 E: 0.000000 Sum of squares = 0.504133056321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454541 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 E=21 A=18 C=6 so C is eliminated. Round 2 votes counts: B=31 D=27 E=22 A=20 so A is eliminated. Round 3 votes counts: E=39 B=34 D=27 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:206 C:203 E:194 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 0 B 8 0 12 -10 2 C 16 -12 0 0 2 D 8 10 0 0 8 E 0 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454541 D: 0.545459 E: 0.000000 Sum of squares = 0.504133056321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454541 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 0 B 8 0 12 -10 2 C 16 -12 0 0 2 D 8 10 0 0 8 E 0 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454541 D: 0.545459 E: 0.000000 Sum of squares = 0.504133056321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454541 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 0 B 8 0 12 -10 2 C 16 -12 0 0 2 D 8 10 0 0 8 E 0 -2 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.454541 D: 0.545459 E: 0.000000 Sum of squares = 0.504133056321 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.454541 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9316: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (8) D E B C A (7) E D C B A (5) A B D C E (5) E D B C A (4) D E C B A (4) D E A C B (4) C B A E D (4) D E A B C (3) B C A E D (3) A C D E B (3) A B C E D (3) E C D B A (2) E C D A B (2) D A E C B (2) C E B D A (2) C E A B D (2) C A E B D (2) C A B E D (2) B E D C A (2) B E C D A (2) B C E A D (2) B A D E C (2) B A C E D (2) A D E C B (2) A C B D E (2) E B C D A (1) D E C A B (1) D E B A C (1) D B E A C (1) D A B E C (1) C E D A B (1) C E B A D (1) C E A D B (1) B D E C A (1) B D A E C (1) B A D C E (1) B A C D E (1) A D C E B (1) A D B E C (1) A D B C E (1) A C E D B (1) A C E B D (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -2 6 2 B -2 0 6 2 -6 C 2 -6 0 0 2 D -6 -2 0 0 8 E -2 6 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.44 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 6 2 B -2 0 6 2 -6 C 2 -6 0 0 2 D -6 -2 0 0 8 E -2 6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999988 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=24 B=17 C=15 E=14 so E is eliminated. Round 2 votes counts: D=33 A=30 C=19 B=18 so B is eliminated. Round 3 votes counts: D=37 A=36 C=27 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:204 B:200 D:200 C:199 E:197 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 6 2 B -2 0 6 2 -6 C 2 -6 0 0 2 D -6 -2 0 0 8 E -2 6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999988 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 6 2 B -2 0 6 2 -6 C 2 -6 0 0 2 D -6 -2 0 0 8 E -2 6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999988 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 6 2 B -2 0 6 2 -6 C 2 -6 0 0 2 D -6 -2 0 0 8 E -2 6 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.200000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.439999999988 Cumulative probabilities = A: 0.600000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9317: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (15) C B D E A (11) D B E C A (10) A E D B C (6) D C B E A (4) C A B D E (4) A E C B D (4) E D B A C (3) E A B D C (3) D B C E A (3) C D B E A (3) B E D C A (3) E B D A C (2) C B A E D (2) C A B E D (2) B D C E A (2) A E D C B (2) A E B D C (2) A C B E D (2) E D A B C (1) D E B C A (1) D E A B C (1) D A E B C (1) C D A B E (1) C B E D A (1) C B D A E (1) C B A D E (1) B D E C A (1) A E C D B (1) A E B C D (1) A D E C B (1) A D E B C (1) A D C E B (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 4 0 2 2 B -4 0 -18 16 6 C 0 18 0 4 12 D -2 -16 -4 0 0 E -2 -6 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.753395 B: 0.000000 C: 0.246605 D: 0.000000 E: 0.000000 Sum of squares = 0.628417822898 Cumulative probabilities = A: 0.753395 B: 0.753395 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 2 2 B -4 0 -18 16 6 C 0 18 0 4 12 D -2 -16 -4 0 0 E -2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 C=26 D=20 E=9 B=6 so B is eliminated. Round 2 votes counts: A=39 C=26 D=23 E=12 so E is eliminated. Round 3 votes counts: A=42 D=32 C=26 so C is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:217 A:204 B:200 E:190 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 2 2 B -4 0 -18 16 6 C 0 18 0 4 12 D -2 -16 -4 0 0 E -2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 2 B -4 0 -18 16 6 C 0 18 0 4 12 D -2 -16 -4 0 0 E -2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 2 B -4 0 -18 16 6 C 0 18 0 4 12 D -2 -16 -4 0 0 E -2 -6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9318: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (7) C A B D E (7) C A D E B (5) C A D B E (5) B C A E D (5) E D C A B (4) E D B A C (4) A C B D E (4) E D A B C (3) D E A B C (3) D A E B C (3) D A B E C (3) B E D A C (3) E D C B A (2) E C D A B (2) E B D C A (2) E B C D A (2) D E A C B (2) D A B C E (2) C E B A D (2) C E A B D (2) C B A D E (2) B D A E C (2) B C E A D (2) B A D E C (2) A D B C E (2) A C D B E (2) E D B C A (1) D B E A C (1) D A E C B (1) D A C E B (1) C E A D B (1) C B E A D (1) C B A E D (1) C A E D B (1) C A E B D (1) B E C A D (1) B E A D C (1) B D E A C (1) B C A D E (1) B A C D E (1) A D C E B (1) A D C B E (1) Total count = 100 A B C D E A 0 12 0 2 6 B -12 0 4 0 0 C 0 -4 0 -4 0 D -2 0 4 0 4 E -6 0 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.753269 B: 0.000000 C: 0.246731 D: 0.000000 E: 0.000000 Sum of squares = 0.628290817412 Cumulative probabilities = A: 0.753269 B: 0.753269 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 2 6 B -12 0 4 0 0 C 0 -4 0 -4 0 D -2 0 4 0 4 E -6 0 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556576 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=27 B=19 D=16 A=10 so A is eliminated. Round 2 votes counts: C=34 E=27 D=20 B=19 so B is eliminated. Round 3 votes counts: C=43 E=32 D=25 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:210 D:203 B:196 C:196 E:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 2 6 B -12 0 4 0 0 C 0 -4 0 -4 0 D -2 0 4 0 4 E -6 0 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556576 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 2 6 B -12 0 4 0 0 C 0 -4 0 -4 0 D -2 0 4 0 4 E -6 0 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556576 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 2 6 B -12 0 4 0 0 C 0 -4 0 -4 0 D -2 0 4 0 4 E -6 0 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.000000 Sum of squares = 0.555555556576 Cumulative probabilities = A: 0.666667 B: 0.666667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9319: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) E A D C B (8) D C B E A (7) A E C D B (7) A E B C D (7) C D A E B (6) B C D A E (6) E A B D C (4) E A C D B (3) B E A D C (3) B D C E A (3) B D C A E (3) B A E D C (3) E B A D C (2) E A D B C (2) C D E A B (2) C A D E B (2) B A C D E (2) A B E C D (2) E D C A B (1) E B D C A (1) D C E B A (1) D C E A B (1) C D B E A (1) C D A B E (1) C A E D B (1) C A B D E (1) B E D A C (1) B D E C A (1) B A E C D (1) B A C E D (1) A E C B D (1) A E B D C (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 4 0 6 18 B -4 0 -12 -12 -2 C 0 12 0 16 4 D -6 12 -16 0 -2 E -18 2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.332067 B: 0.000000 C: 0.667933 D: 0.000000 E: 0.000000 Sum of squares = 0.556402996656 Cumulative probabilities = A: 0.332067 B: 0.332067 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 6 18 B -4 0 -12 -12 -2 C 0 12 0 16 4 D -6 12 -16 0 -2 E -18 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 E=21 A=20 D=9 so D is eliminated. Round 2 votes counts: C=35 B=24 E=21 A=20 so A is eliminated. Round 3 votes counts: E=37 C=36 B=27 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:214 D:194 E:191 B:185 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 6 18 B -4 0 -12 -12 -2 C 0 12 0 16 4 D -6 12 -16 0 -2 E -18 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 6 18 B -4 0 -12 -12 -2 C 0 12 0 16 4 D -6 12 -16 0 -2 E -18 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 6 18 B -4 0 -12 -12 -2 C 0 12 0 16 4 D -6 12 -16 0 -2 E -18 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9320: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) C D E A B (6) B A D C E (6) A B C D E (6) E C D A B (5) B D C A E (5) B A E D C (5) E C A D B (4) D C E B A (4) B D A C E (4) A E C B D (4) A B E D C (4) D B C E A (3) C D E B A (3) B D C E A (3) A E B C D (3) A B C E D (3) E A C D B (2) C D B A E (2) B E A D C (2) B D E C A (2) A E C D B (2) A B E C D (2) E D C B A (1) E B D C A (1) E B D A C (1) E A C B D (1) E A B D C (1) E A B C D (1) D B C A E (1) C D A E B (1) B E D A C (1) B A D E C (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 2 -2 2 B 6 0 10 14 16 C -2 -10 0 -6 14 D 2 -14 6 0 12 E -2 -16 -14 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999347 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -2 2 B 6 0 10 14 16 C -2 -10 0 -6 14 D 2 -14 6 0 12 E -2 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=26 E=17 D=16 C=12 so C is eliminated. Round 2 votes counts: B=29 D=28 A=26 E=17 so E is eliminated. Round 3 votes counts: A=35 D=34 B=31 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:223 D:203 A:198 C:198 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 -2 2 B 6 0 10 14 16 C -2 -10 0 -6 14 D 2 -14 6 0 12 E -2 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -2 2 B 6 0 10 14 16 C -2 -10 0 -6 14 D 2 -14 6 0 12 E -2 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -2 2 B 6 0 10 14 16 C -2 -10 0 -6 14 D 2 -14 6 0 12 E -2 -16 -14 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999645 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9321: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) B E C A D (9) E C B A D (8) D A C E B (8) B D A E C (8) B A D E C (7) C A D E B (6) A D C E B (6) D A C B E (5) E B C A D (4) B E C D A (3) D A B E C (2) D A B C E (2) C E B A D (2) C D A E B (2) A D C B E (2) A D B C E (2) E C D B A (1) E C B D A (1) E B C D A (1) C E D B A (1) C E D A B (1) C E A B D (1) C D E A B (1) C A E D B (1) B E D C A (1) B E D A C (1) B E A D C (1) B A E D C (1) B A E C D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -6 24 8 B 0 0 -12 -2 -8 C 6 12 0 6 0 D -24 2 -6 0 4 E -8 8 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.763645 D: 0.000000 E: 0.236355 Sum of squares = 0.639017737058 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.763645 D: 0.763645 E: 1.000000 A B C D E A 0 0 -6 24 8 B 0 0 -12 -2 -8 C 6 12 0 6 0 D -24 2 -6 0 4 E -8 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204101318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=25 D=17 E=15 A=11 so A is eliminated. Round 2 votes counts: B=32 D=28 C=25 E=15 so E is eliminated. Round 3 votes counts: B=37 C=35 D=28 so D is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:212 E:198 B:189 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 24 8 B 0 0 -12 -2 -8 C 6 12 0 6 0 D -24 2 -6 0 4 E -8 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204101318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 24 8 B 0 0 -12 -2 -8 C 6 12 0 6 0 D -24 2 -6 0 4 E -8 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204101318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 24 8 B 0 0 -12 -2 -8 C 6 12 0 6 0 D -24 2 -6 0 4 E -8 8 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.000000 E: 0.428571 Sum of squares = 0.510204101318 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9322: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) E A D C B (8) B A E C D (8) C D B E A (6) D C E A B (5) A E D B C (5) E D A C B (4) D A E C B (4) B C E A D (4) B C D E A (4) B C D A E (4) A E D C B (4) D E A C B (3) B E A C D (3) B C A E D (3) E A D B C (2) D C A E B (2) C D B A E (2) C B D A E (2) B C E D A (2) A D E C B (2) E B A C D (1) E A B C D (1) D E C A B (1) D C B A E (1) D C A B E (1) C D E B A (1) C D E A B (1) C D A B E (1) B E C A D (1) B C A D E (1) B A E D C (1) B A C E D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -2 -6 -12 B 8 0 -14 -6 10 C 2 14 0 10 0 D 6 6 -10 0 0 E 12 -10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.632274 D: 0.000000 E: 0.367726 Sum of squares = 0.534992753068 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.632274 D: 0.632274 E: 1.000000 A B C D E A 0 -8 -2 -6 -12 B 8 0 -14 -6 10 C 2 14 0 10 0 D 6 6 -10 0 0 E 12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=22 D=17 E=16 A=13 so A is eliminated. Round 2 votes counts: B=33 E=26 C=22 D=19 so D is eliminated. Round 3 votes counts: E=36 B=33 C=31 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:213 D:201 E:201 B:199 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 -6 -12 B 8 0 -14 -6 10 C 2 14 0 10 0 D 6 6 -10 0 0 E 12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -6 -12 B 8 0 -14 -6 10 C 2 14 0 10 0 D 6 6 -10 0 0 E 12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -6 -12 B 8 0 -14 -6 10 C 2 14 0 10 0 D 6 6 -10 0 0 E 12 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9323: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A D C B (9) E D A C B (7) B C A D E (6) E D B C A (5) D B C E A (5) A E C B D (4) E A B C D (3) D E A C B (3) B A C E D (3) A E D C B (3) A E B C D (3) E A B D C (2) D E C B A (2) D E C A B (2) D E B C A (2) D C B E A (2) C D B A E (2) C A B D E (2) B E C D A (2) B E C A D (2) B C A E D (2) A E C D B (2) E B A D C (1) E A C B D (1) D C A E B (1) D B E C A (1) C B D A E (1) C B A D E (1) B E D C A (1) B D C E A (1) B D C A E (1) B C D E A (1) A C E D B (1) A C E B D (1) A C B E D (1) A C B D E (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -4 2 -4 B 2 0 8 6 -4 C 4 -8 0 4 -12 D -2 -6 -4 0 -10 E 4 4 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999576 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 2 -4 B 2 0 8 6 -4 C 4 -8 0 4 -12 D -2 -6 -4 0 -10 E 4 4 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=28 A=19 D=18 C=6 so C is eliminated. Round 2 votes counts: B=31 E=28 A=21 D=20 so D is eliminated. Round 3 votes counts: B=41 E=37 A=22 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 B:206 A:196 C:194 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 2 -4 B 2 0 8 6 -4 C 4 -8 0 4 -12 D -2 -6 -4 0 -10 E 4 4 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 2 -4 B 2 0 8 6 -4 C 4 -8 0 4 -12 D -2 -6 -4 0 -10 E 4 4 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 2 -4 B 2 0 8 6 -4 C 4 -8 0 4 -12 D -2 -6 -4 0 -10 E 4 4 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9324: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (8) E D C A B (7) B A C D E (6) B A D E C (5) B A C E D (5) D E C B A (4) D B E A C (4) C E D A B (4) C A E B D (4) C A B E D (4) C A B D E (4) A B C E D (4) D E C A B (3) B D A C E (3) B A E D C (3) B A D C E (3) E D A B C (2) E B A D C (2) D E B C A (2) C E A D B (2) C D E A B (2) B D A E C (2) A B E C D (2) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) E A C D B (1) E A B D C (1) E A B C D (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C A E (1) C B A D E (1) B D E A C (1) B C A D E (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 10 4 -2 B 6 0 12 8 2 C -10 -12 0 -10 -4 D -4 -8 10 0 6 E 2 -2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 10 4 -2 B 6 0 12 8 2 C -10 -12 0 -10 -4 D -4 -8 10 0 6 E 2 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=25 C=21 E=18 A=7 so A is eliminated. Round 2 votes counts: B=35 D=25 C=22 E=18 so E is eliminated. Round 3 votes counts: B=39 D=35 C=26 so C is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:203 D:202 E:199 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 10 4 -2 B 6 0 12 8 2 C -10 -12 0 -10 -4 D -4 -8 10 0 6 E 2 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 10 4 -2 B 6 0 12 8 2 C -10 -12 0 -10 -4 D -4 -8 10 0 6 E 2 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 10 4 -2 B 6 0 12 8 2 C -10 -12 0 -10 -4 D -4 -8 10 0 6 E 2 -2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998725 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9325: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (8) D C A B E (8) E C D B A (7) E B A C D (7) A B C D E (7) E D B A C (6) E B A D C (6) D C E A B (6) A B C E D (5) D E C A B (4) D E A B C (4) B A E C D (4) C D E B A (3) C D A B E (3) D E C B A (2) C D B A E (2) C B A E D (2) C A B D E (2) B A C E D (2) A B E C D (2) A B D E C (2) A B D C E (2) E D A B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C B A D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 2 -12 -10 B 2 0 2 -12 -10 C -2 -2 0 -4 -10 D 12 12 4 0 -4 E 10 10 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 2 -12 -10 B 2 0 2 -12 -10 C -2 -2 0 -4 -10 D 12 12 4 0 -4 E 10 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=26 A=19 C=14 B=6 so B is eliminated. Round 2 votes counts: E=35 D=26 A=25 C=14 so C is eliminated. Round 3 votes counts: E=36 D=34 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 D:212 B:191 C:191 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 2 -12 -10 B 2 0 2 -12 -10 C -2 -2 0 -4 -10 D 12 12 4 0 -4 E 10 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -12 -10 B 2 0 2 -12 -10 C -2 -2 0 -4 -10 D 12 12 4 0 -4 E 10 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -12 -10 B 2 0 2 -12 -10 C -2 -2 0 -4 -10 D 12 12 4 0 -4 E 10 10 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9326: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) C D B A E (7) C A D B E (7) D C B E A (6) E B A D C (5) E B D A C (4) A C E D B (4) A C B D E (4) E D B C A (3) D B C E A (3) C E A D B (3) C D A B E (3) C A D E B (3) A E B D C (3) A E B C D (3) E A B C D (2) D B E C A (2) C A E D B (2) B E A D C (2) B D E C A (2) B D A C E (2) A E C B D (2) A C E B D (2) A B E D C (2) A B E C D (2) E C D B A (1) E A C D B (1) D E B C A (1) D C B A E (1) D B C A E (1) C D B E A (1) C D A E B (1) B E D A C (1) B D C E A (1) B A D E C (1) B A D C E (1) A C D E B (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 4 20 10 B -10 0 0 -4 4 C -4 0 0 2 12 D -20 4 -2 0 2 E -10 -4 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999778 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 20 10 B -10 0 0 -4 4 C -4 0 0 2 12 D -20 4 -2 0 2 E -10 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998096 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=26 E=23 D=14 B=10 so B is eliminated. Round 2 votes counts: A=28 C=27 E=26 D=19 so D is eliminated. Round 3 votes counts: C=39 E=31 A=30 so A is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:222 C:205 B:195 D:192 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 20 10 B -10 0 0 -4 4 C -4 0 0 2 12 D -20 4 -2 0 2 E -10 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998096 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 20 10 B -10 0 0 -4 4 C -4 0 0 2 12 D -20 4 -2 0 2 E -10 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998096 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 20 10 B -10 0 0 -4 4 C -4 0 0 2 12 D -20 4 -2 0 2 E -10 -4 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998096 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9327: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) E C D B A (7) A D C E B (7) A B D C E (6) B E C D A (5) D A E C B (4) D A B E C (4) B A D E C (4) E C B D A (3) D B E A C (3) C E A B D (3) C A D E B (3) B E D A C (3) A C D B E (3) E D C A B (2) D E A C B (2) C E B D A (2) C E A D B (2) C A B E D (2) B D E A C (2) B D A E C (2) B A D C E (2) A D B E C (2) D E B A C (1) D E A B C (1) D A E B C (1) C E D B A (1) C E D A B (1) C E B A D (1) C B E A D (1) C A E D B (1) B C E A D (1) B C A E D (1) B A E C D (1) B A C D E (1) A D C B E (1) A C D E B (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 8 -8 -2 B 2 0 4 6 14 C -8 -4 0 -16 -16 D 8 -6 16 0 6 E 2 -14 16 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999639 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 -8 -2 B 2 0 4 6 14 C -8 -4 0 -16 -16 D 8 -6 16 0 6 E 2 -14 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998517 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=23 C=17 D=16 E=12 so E is eliminated. Round 2 votes counts: B=32 C=27 A=23 D=18 so D is eliminated. Round 3 votes counts: B=36 A=35 C=29 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:212 E:199 A:198 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 8 -8 -2 B 2 0 4 6 14 C -8 -4 0 -16 -16 D 8 -6 16 0 6 E 2 -14 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998517 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 -8 -2 B 2 0 4 6 14 C -8 -4 0 -16 -16 D 8 -6 16 0 6 E 2 -14 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998517 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 -8 -2 B 2 0 4 6 14 C -8 -4 0 -16 -16 D 8 -6 16 0 6 E 2 -14 16 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998517 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9328: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) E B A C D (6) D B E C A (6) C D A B E (6) B E D C A (6) B E C D A (6) D C A B E (5) A D C E B (5) A C E B D (5) D C B E A (4) A E B D C (4) E B C A D (3) E B A D C (3) C E B A D (3) C D A E B (3) C B E D A (3) C A D E B (3) A E B C D (3) D A B E C (2) C E B D A (2) B E D A C (2) A E C B D (2) E B C D A (1) D C B A E (1) D A C B E (1) C D E B A (1) C B D E A (1) C A E B D (1) B E C A D (1) B E A C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -18 -30 -18 -16 B 18 0 -10 6 10 C 30 10 0 20 6 D 18 -6 -20 0 -4 E 16 -10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999261 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -30 -18 -16 B 18 0 -10 6 10 C 30 10 0 20 6 D 18 -6 -20 0 -4 E 16 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=20 D=19 B=16 E=13 so E is eliminated. Round 2 votes counts: C=32 B=29 A=20 D=19 so D is eliminated. Round 3 votes counts: C=42 B=35 A=23 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:233 B:212 E:202 D:194 A:159 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -18 -30 -18 -16 B 18 0 -10 6 10 C 30 10 0 20 6 D 18 -6 -20 0 -4 E 16 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -30 -18 -16 B 18 0 -10 6 10 C 30 10 0 20 6 D 18 -6 -20 0 -4 E 16 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -30 -18 -16 B 18 0 -10 6 10 C 30 10 0 20 6 D 18 -6 -20 0 -4 E 16 -10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9329: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (10) E A D B C (6) C D B A E (6) E B C A D (4) E A B D C (4) D A B C E (4) C B D A E (4) A D B C E (4) E C B A D (3) D C A B E (3) D A C B E (3) C E D A B (3) B C E A D (3) A E D B C (3) E D C A B (2) E C D A B (2) E C B D A (2) E C A D B (2) E B A C D (2) E A D C B (2) E A B C D (2) D A C E B (2) C B E D A (2) C B D E A (2) B C A D E (2) A D E B C (2) A D B E C (2) A B E D C (2) E D A C B (1) E C D B A (1) E C A B D (1) E B A D C (1) D E A C B (1) D C A E B (1) D A B E C (1) C E B D A (1) B E C A D (1) B D A C E (1) B C E D A (1) B A D C E (1) Total count = 100 A B C D E A 0 6 -12 -6 2 B -6 0 12 -2 4 C 12 -12 0 8 6 D 6 2 -8 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528661 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 -6 2 B -6 0 12 -2 4 C 12 -12 0 8 6 D 6 2 -8 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528869 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=19 C=18 D=15 A=13 so A is eliminated. Round 2 votes counts: E=38 D=23 B=21 C=18 so C is eliminated. Round 3 votes counts: E=42 D=29 B=29 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:207 B:204 D:199 A:195 E:195 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 6 -12 -6 2 B -6 0 12 -2 4 C 12 -12 0 8 6 D 6 2 -8 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528869 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 -6 2 B -6 0 12 -2 4 C 12 -12 0 8 6 D 6 2 -8 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528869 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 -6 2 B -6 0 12 -2 4 C 12 -12 0 8 6 D 6 2 -8 0 -2 E -2 -4 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.363636 C: 0.090909 D: 0.545455 E: 0.000000 Sum of squares = 0.438016528869 Cumulative probabilities = A: 0.000000 B: 0.363636 C: 0.454545 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9330: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (13) C B E A D (9) D A B E C (7) C E B D A (7) C E B A D (6) C D E B A (6) A B E D C (6) E B C A D (5) D A E B C (5) D C A E B (4) D C A B E (4) D A C B E (4) C D A B E (4) B E A C D (3) E B A D C (2) D A C E B (2) B E C A D (2) A D E B C (2) A B E C D (2) E B A C D (1) D C E B A (1) C E D B A (1) C D B E A (1) C B E D A (1) B E A D C (1) B C E A D (1) Total count = 100 A B C D E A 0 6 -4 6 6 B -6 0 0 -8 16 C 4 0 0 -2 2 D -6 8 2 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888907 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 6 6 B -6 0 0 -8 16 C 4 0 0 -2 2 D -6 8 2 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=27 A=23 E=8 B=7 so B is eliminated. Round 2 votes counts: C=36 D=27 A=23 E=14 so E is eliminated. Round 3 votes counts: C=43 A=30 D=27 so D is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:207 D:205 C:202 B:201 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 6 6 B -6 0 0 -8 16 C 4 0 0 -2 2 D -6 8 2 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 6 6 B -6 0 0 -8 16 C 4 0 0 -2 2 D -6 8 2 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 6 6 B -6 0 0 -8 16 C 4 0 0 -2 2 D -6 8 2 0 6 E -6 -16 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888897 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9331: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (6) C A E B D (5) B D A E C (5) B A D C E (5) A C E D B (5) D E B C A (4) B E D C A (4) B D E C A (4) A C B D E (4) E D C B A (3) E D C A B (3) E B D C A (3) D E B A C (3) D E A B C (3) D B E A C (3) B D E A C (3) B A D E C (3) B A C D E (3) C B A E D (2) B E C D A (2) B C A E D (2) A D E C B (2) A D C E B (2) A C D E B (2) A C B E D (2) E D B C A (1) E C D B A (1) E C D A B (1) D B E C A (1) D B A E C (1) D A E B C (1) C E D B A (1) C E D A B (1) C E A D B (1) C B E D A (1) C A B E D (1) B C E D A (1) A D B C E (1) A C D B E (1) A B D E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 4 0 12 B 12 0 12 6 4 C -4 -12 0 -14 -4 D 0 -6 14 0 8 E -12 -4 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 0 12 B 12 0 12 6 4 C -4 -12 0 -14 -4 D 0 -6 14 0 8 E -12 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=22 C=18 D=16 E=12 so E is eliminated. Round 2 votes counts: B=35 D=23 A=22 C=20 so C is eliminated. Round 3 votes counts: B=38 A=35 D=27 so D is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:208 A:202 E:190 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 0 12 B 12 0 12 6 4 C -4 -12 0 -14 -4 D 0 -6 14 0 8 E -12 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 0 12 B 12 0 12 6 4 C -4 -12 0 -14 -4 D 0 -6 14 0 8 E -12 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 0 12 B 12 0 12 6 4 C -4 -12 0 -14 -4 D 0 -6 14 0 8 E -12 -4 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999897 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9332: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (8) A E C D B (7) D C E A B (5) B D C E A (5) B A E D C (5) A E D C B (5) A D E C B (5) E C A D B (4) C E A D B (4) C D E B A (4) B C E D A (4) B C E A D (4) E A C D B (3) C E D A B (3) C D E A B (3) C B D E A (3) B D A E C (3) D C B E A (2) D B C E A (2) D A C E B (2) C E B D A (2) B E C A D (2) B D A C E (2) B A D E C (2) E C A B D (1) E A D C B (1) D E A C B (1) D C B A E (1) D B C A E (1) D A B E C (1) C B E A D (1) B A E C D (1) A E D B C (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -4 -18 -4 -24 B 4 0 -16 -10 -6 C 18 16 0 12 12 D 4 10 -12 0 0 E 24 6 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -18 -4 -24 B 4 0 -16 -10 -6 C 18 16 0 12 12 D 4 10 -12 0 0 E 24 6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 C=20 A=20 D=15 E=9 so E is eliminated. Round 2 votes counts: B=36 C=25 A=24 D=15 so D is eliminated. Round 3 votes counts: B=39 C=33 A=28 so A is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 E:209 D:201 B:186 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -18 -4 -24 B 4 0 -16 -10 -6 C 18 16 0 12 12 D 4 10 -12 0 0 E 24 6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -18 -4 -24 B 4 0 -16 -10 -6 C 18 16 0 12 12 D 4 10 -12 0 0 E 24 6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -18 -4 -24 B 4 0 -16 -10 -6 C 18 16 0 12 12 D 4 10 -12 0 0 E 24 6 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9333: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (6) B A E C D (6) E D C A B (5) E D A C B (5) D C E B A (5) D C E A B (5) C D B E A (5) C D A B E (4) E B A D C (3) E A D C B (3) E A D B C (3) C D B A E (3) C D A E B (3) B A C D E (3) A E D C B (3) A C D B E (3) A B E C D (3) A B C D E (3) E D B C A (2) C D E B A (2) C A D B E (2) B C D E A (2) B C D A E (2) B C A D E (2) A B E D C (2) E D C B A (1) D E C B A (1) D C A E B (1) C B D E A (1) C B D A E (1) B E C D A (1) B E A D C (1) B E A C D (1) B C E D A (1) B A C E D (1) A E D B C (1) A E B D C (1) A D E C B (1) A D C E B (1) A C B D E (1) Total count = 100 A B C D E A 0 12 2 0 -6 B -12 0 -12 -18 -4 C -2 12 0 0 2 D 0 18 0 0 2 E 6 4 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.164720 B: 0.000000 C: 0.000000 D: 0.835280 E: 0.000000 Sum of squares = 0.724825402325 Cumulative probabilities = A: 0.164720 B: 0.164720 C: 0.164720 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 0 -6 B -12 0 -12 -18 -4 C -2 12 0 0 2 D 0 18 0 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000022013 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=21 B=20 A=19 D=12 so D is eliminated. Round 2 votes counts: C=32 E=29 B=20 A=19 so A is eliminated. Round 3 votes counts: C=37 E=35 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:206 A:204 E:203 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 2 0 -6 B -12 0 -12 -18 -4 C -2 12 0 0 2 D 0 18 0 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000022013 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 0 -6 B -12 0 -12 -18 -4 C -2 12 0 0 2 D 0 18 0 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000022013 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 0 -6 B -12 0 -12 -18 -4 C -2 12 0 0 2 D 0 18 0 0 2 E 6 4 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.625000022013 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9334: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) B C A E D (8) A E D B C (8) B C D E A (6) C B D E A (5) E D A C B (4) D E A B C (4) C D E B A (4) B A D E C (4) B A C D E (4) A D E B C (4) A B C E D (4) E D C A B (3) D E C B A (3) C E D B A (3) C B E D A (3) B C A D E (3) A B E D C (3) D E B C A (2) C E D A B (2) C B A E D (2) B A C E D (2) D E C A B (1) D E A C B (1) D C E B A (1) C A B E D (1) B D E A C (1) A C E D B (1) A C B E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 6 14 14 B 2 0 10 -2 -2 C -6 -10 0 0 0 D -14 2 0 0 -12 E -14 2 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.111111 B: 0.777778 C: 0.000000 D: 0.000000 E: 0.111111 Sum of squares = 0.62962962962 Cumulative probabilities = A: 0.111111 B: 0.888889 C: 0.888889 D: 0.888889 E: 1.000000 A B C D E A 0 -2 6 14 14 B 2 0 10 -2 -2 C -6 -10 0 0 0 D -14 2 0 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.777778 C: 0.000000 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629583 Cumulative probabilities = A: 0.111111 B: 0.888889 C: 0.888889 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=28 C=20 D=12 E=7 so E is eliminated. Round 2 votes counts: A=33 B=28 C=20 D=19 so D is eliminated. Round 3 votes counts: A=42 B=30 C=28 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:204 E:200 C:192 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 6 14 14 B 2 0 10 -2 -2 C -6 -10 0 0 0 D -14 2 0 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.777778 C: 0.000000 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629583 Cumulative probabilities = A: 0.111111 B: 0.888889 C: 0.888889 D: 0.888889 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 14 14 B 2 0 10 -2 -2 C -6 -10 0 0 0 D -14 2 0 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.777778 C: 0.000000 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629583 Cumulative probabilities = A: 0.111111 B: 0.888889 C: 0.888889 D: 0.888889 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 14 14 B 2 0 10 -2 -2 C -6 -10 0 0 0 D -14 2 0 0 -12 E -14 2 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.111111 B: 0.777778 C: 0.000000 D: 0.000000 E: 0.111111 Sum of squares = 0.629629629583 Cumulative probabilities = A: 0.111111 B: 0.888889 C: 0.888889 D: 0.888889 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9335: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) C D B A E (6) D A C B E (5) E B A D C (4) D C A B E (4) B E C A D (4) B A C D E (4) A B D C E (4) E B A C D (3) E A D B C (3) D C A E B (3) C D B E A (3) C B D A E (3) A D B C E (3) A B E D C (3) E D A B C (2) E C D B A (2) E C B D A (2) D E C A B (2) D E A C B (2) D C E A B (2) C D E B A (2) C D A B E (2) C B D E A (2) C B A D E (2) A E B D C (2) A B C D E (2) E D C A B (1) E D B C A (1) E B D A C (1) E B C D A (1) D A E C B (1) D A C E B (1) D A B E C (1) C E D B A (1) C B E D A (1) C B E A D (1) C B A E D (1) C A D B E (1) B E A C D (1) B C A E D (1) B C A D E (1) B A E C D (1) A E D B C (1) Total count = 100 A B C D E A 0 -10 -12 -2 4 B 10 0 0 2 12 C 12 0 0 8 10 D 2 -2 -8 0 12 E -4 -12 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.494811 C: 0.505189 D: 0.000000 E: 0.000000 Sum of squares = 0.500053856545 Cumulative probabilities = A: 0.000000 B: 0.494811 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -2 4 B 10 0 0 2 12 C 12 0 0 8 10 D 2 -2 -8 0 12 E -4 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 D=21 A=15 B=12 so B is eliminated. Round 2 votes counts: E=32 C=27 D=21 A=20 so A is eliminated. Round 3 votes counts: E=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:212 D:202 A:190 E:181 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -2 4 B 10 0 0 2 12 C 12 0 0 8 10 D 2 -2 -8 0 12 E -4 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -2 4 B 10 0 0 2 12 C 12 0 0 8 10 D 2 -2 -8 0 12 E -4 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -2 4 B 10 0 0 2 12 C 12 0 0 8 10 D 2 -2 -8 0 12 E -4 -12 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9336: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) B C D A E (9) E A C D B (8) D E A B C (7) E A D C B (6) E A C B D (6) D B E A C (6) B C A E D (6) D E A C B (4) D B C A E (4) C E A B D (4) C B A E D (4) C A E B D (4) A E C D B (4) B D C E A (3) D A E B C (2) B C E A D (2) E A B D C (1) E A B C D (1) D B A E C (1) D A E C B (1) C E B A D (1) C B A D E (1) C A D E B (1) B E A C D (1) B D E C A (1) B C D E A (1) A E D C B (1) A C E B D (1) Total count = 100 A B C D E A 0 2 0 4 -4 B -2 0 8 10 -4 C 0 -8 0 8 0 D -4 -10 -8 0 0 E 4 4 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.228125 D: 0.000000 E: 0.771875 Sum of squares = 0.647831514798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.228125 D: 0.228125 E: 1.000000 A B C D E A 0 2 0 4 -4 B -2 0 8 10 -4 C 0 -8 0 8 0 D -4 -10 -8 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555591792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=25 E=22 C=15 A=6 so A is eliminated. Round 2 votes counts: B=32 E=27 D=25 C=16 so C is eliminated. Round 3 votes counts: E=37 B=37 D=26 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:206 E:204 A:201 C:200 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 4 -4 B -2 0 8 10 -4 C 0 -8 0 8 0 D -4 -10 -8 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555591792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 4 -4 B -2 0 8 10 -4 C 0 -8 0 8 0 D -4 -10 -8 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555591792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 4 -4 B -2 0 8 10 -4 C 0 -8 0 8 0 D -4 -10 -8 0 0 E 4 4 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.666667 Sum of squares = 0.555555591792 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9337: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (10) C E B A D (6) A D C B E (6) C E B D A (5) A C D B E (5) E B D A C (4) D A B E C (4) C A E B D (4) C A D E B (4) B E D A C (4) A D B E C (4) E C B D A (3) E B D C A (3) C A E D B (3) B E D C A (3) B D E A C (3) A D C E B (3) A D B C E (3) E B A D C (2) D B A E C (2) D A C B E (2) C B E D A (2) C A D B E (2) A C D E B (2) E C B A D (1) E C A B D (1) E B C A D (1) E A B C D (1) D B E A C (1) C E A B D (1) C D A B E (1) B E C D A (1) A E D B C (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 -2 4 -4 B 2 0 -6 12 -14 C 2 6 0 10 0 D -4 -12 -10 0 -16 E 4 14 0 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.760117 D: 0.000000 E: 0.239883 Sum of squares = 0.635321569766 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.760117 D: 0.760117 E: 1.000000 A B C D E A 0 -2 -2 4 -4 B 2 0 -6 12 -14 C 2 6 0 10 0 D -4 -12 -10 0 -16 E 4 14 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=26 A=26 B=11 D=9 so D is eliminated. Round 2 votes counts: A=32 C=28 E=26 B=14 so B is eliminated. Round 3 votes counts: E=38 A=34 C=28 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:209 A:198 B:197 D:179 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 4 -4 B 2 0 -6 12 -14 C 2 6 0 10 0 D -4 -12 -10 0 -16 E 4 14 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 4 -4 B 2 0 -6 12 -14 C 2 6 0 10 0 D -4 -12 -10 0 -16 E 4 14 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 4 -4 B 2 0 -6 12 -14 C 2 6 0 10 0 D -4 -12 -10 0 -16 E 4 14 0 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9338: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) E C B D A (7) D B A E C (7) C E D A B (7) D A B C E (5) C A D E B (5) E D B C A (3) D E B A C (3) C D A B E (3) C A D B E (3) B A E D C (3) E D B A C (2) E C B A D (2) E B D A C (2) E B C A D (2) E B A D C (2) D C A B E (2) D A C B E (2) C E A D B (2) C D E A B (2) C A E B D (2) B A D E C (2) A D B C E (2) A B D C E (2) E D C B A (1) E C D B A (1) E B D C A (1) E B C D A (1) D E B C A (1) D C E A B (1) D A B E C (1) C E D B A (1) C E B A D (1) C D A E B (1) C A B D E (1) B E A D C (1) B D E A C (1) B A E C D (1) A D B E C (1) A C D B E (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -18 -10 -6 B -10 0 -10 -14 -18 C 18 10 0 10 10 D 10 14 -10 0 -4 E 6 18 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -18 -10 -6 B -10 0 -10 -14 -18 C 18 10 0 10 10 D 10 14 -10 0 -4 E 6 18 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=24 D=22 A=9 B=8 so B is eliminated. Round 2 votes counts: C=37 E=25 D=23 A=15 so A is eliminated. Round 3 votes counts: C=41 D=30 E=29 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 E:209 D:205 A:188 B:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -18 -10 -6 B -10 0 -10 -14 -18 C 18 10 0 10 10 D 10 14 -10 0 -4 E 6 18 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -18 -10 -6 B -10 0 -10 -14 -18 C 18 10 0 10 10 D 10 14 -10 0 -4 E 6 18 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -18 -10 -6 B -10 0 -10 -14 -18 C 18 10 0 10 10 D 10 14 -10 0 -4 E 6 18 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999684 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9339: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) D A B E C (6) C E B A D (5) E C B A D (4) D B A E C (4) D A C B E (4) B A D E C (4) A B D C E (4) E C D B A (3) E B C A D (3) E B A C D (3) D C E A B (3) C D E A B (3) C A B E D (3) B A C E D (3) A D B C E (3) E D B A C (2) E C B D A (2) E B A D C (2) D E B A C (2) D E A B C (2) D C A B E (2) C E D A B (2) C E B D A (2) C E A B D (2) C A E B D (2) B E A C D (2) E D C B A (1) E D C A B (1) E D B C A (1) E B D C A (1) E B D A C (1) D E C B A (1) D E C A B (1) D E A C B (1) C E D B A (1) C D A E B (1) B D A E C (1) B A E D C (1) A D C B E (1) A C B D E (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 12 -10 -2 B -2 0 8 -4 -4 C -12 -8 0 -14 0 D 10 4 14 0 6 E 2 4 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 12 -10 -2 B -2 0 8 -4 -4 C -12 -8 0 -14 0 D 10 4 14 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=24 C=21 B=11 A=11 so B is eliminated. Round 2 votes counts: D=34 E=26 C=21 A=19 so A is eliminated. Round 3 votes counts: D=47 E=27 C=26 so C is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 A:201 E:200 B:199 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 12 -10 -2 B -2 0 8 -4 -4 C -12 -8 0 -14 0 D 10 4 14 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 12 -10 -2 B -2 0 8 -4 -4 C -12 -8 0 -14 0 D 10 4 14 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 12 -10 -2 B -2 0 8 -4 -4 C -12 -8 0 -14 0 D 10 4 14 0 6 E 2 4 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999701 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9340: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (8) A C B E D (7) D B A E C (6) A C B D E (6) D B E A C (5) A B C D E (5) E D C B A (4) E D B C A (4) E C D A B (4) B A C D E (4) D E B A C (3) B D A E C (3) B A C E D (3) A C D B E (3) E C D B A (2) E C B A D (2) E B D C A (2) E B C D A (2) D E C A B (2) D E B C A (2) C E A B D (2) C A E B D (2) C A D E B (2) B E D C A (2) B C A E D (2) E D C A B (1) E B C A D (1) D E A B C (1) D A E C B (1) D A E B C (1) D A C E B (1) D A C B E (1) D A B C E (1) C B E A D (1) B E C D A (1) B D E A C (1) B C E A D (1) B A E C D (1) Total count = 100 A B C D E A 0 -4 6 0 14 B 4 0 2 12 22 C -6 -2 0 18 -2 D 0 -12 -18 0 -4 E -14 -22 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 0 14 B 4 0 2 12 22 C -6 -2 0 18 -2 D 0 -12 -18 0 -4 E -14 -22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 E=22 A=21 B=18 C=15 so C is eliminated. Round 2 votes counts: A=33 E=24 D=24 B=19 so B is eliminated. Round 3 votes counts: A=43 E=29 D=28 so D is eliminated. Round 4 votes counts: A=57 E=43 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:220 A:208 C:204 E:185 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 0 14 B 4 0 2 12 22 C -6 -2 0 18 -2 D 0 -12 -18 0 -4 E -14 -22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 0 14 B 4 0 2 12 22 C -6 -2 0 18 -2 D 0 -12 -18 0 -4 E -14 -22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 0 14 B 4 0 2 12 22 C -6 -2 0 18 -2 D 0 -12 -18 0 -4 E -14 -22 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9341: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) B C A D E (7) B A C D E (6) E D C A B (5) D E A B C (5) D A B E C (5) C E A D B (5) C B E A D (5) E D A C B (4) E C D A B (4) C B A E D (4) B A D C E (4) E D A B C (3) C E D A B (3) B D A E C (3) B A D E C (3) A D B E C (3) E D C B A (2) E D B A C (2) D A E B C (2) C A E D B (2) C A B D E (2) B C D A E (2) E D B C A (1) E C B D A (1) E A D C B (1) D B E A C (1) C B E D A (1) B E D C A (1) B D E C A (1) B D C E A (1) B C E D A (1) A D E B C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -10 4 6 B 6 0 8 2 18 C 10 -8 0 2 4 D -4 -2 -2 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 4 6 B 6 0 8 2 18 C 10 -8 0 2 4 D -4 -2 -2 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=29 B=29 E=23 D=13 A=6 so A is eliminated. Round 2 votes counts: B=31 C=29 E=23 D=17 so D is eliminated. Round 3 votes counts: B=40 E=31 C=29 so C is eliminated. Round 4 votes counts: B=59 E=41 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 C:204 D:201 A:197 E:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 4 6 B 6 0 8 2 18 C 10 -8 0 2 4 D -4 -2 -2 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 4 6 B 6 0 8 2 18 C 10 -8 0 2 4 D -4 -2 -2 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 4 6 B 6 0 8 2 18 C 10 -8 0 2 4 D -4 -2 -2 0 10 E -6 -18 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998551 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9342: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (10) D E B C A (9) B E D A C (6) B D E A C (6) C A B E D (5) B A E D C (5) A B C E D (5) E D B A C (4) C A B D E (4) B A D E C (4) B E A D C (3) A B E C D (3) E A B D C (2) D E C B A (2) D C E B A (2) C E D A B (2) C D E A B (2) C A E B D (2) C A D B E (2) B D A E C (2) B A C E D (2) A C B E D (2) A B C D E (2) E D C A B (1) E D B C A (1) E C A D B (1) E B D A C (1) E A C D B (1) D E C A B (1) D B E A C (1) D B C E A (1) C E A D B (1) C D B E A (1) C B D A E (1) C A D E B (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 2 14 4 B 2 0 16 14 10 C -2 -16 0 -4 -10 D -14 -14 4 0 -18 E -4 -10 10 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 14 4 B 2 0 16 14 10 C -2 -16 0 -4 -10 D -14 -14 4 0 -18 E -4 -10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=28 D=16 A=14 E=11 so E is eliminated. Round 2 votes counts: C=32 B=29 D=22 A=17 so A is eliminated. Round 3 votes counts: B=42 C=36 D=22 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:209 E:207 C:184 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 14 4 B 2 0 16 14 10 C -2 -16 0 -4 -10 D -14 -14 4 0 -18 E -4 -10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 14 4 B 2 0 16 14 10 C -2 -16 0 -4 -10 D -14 -14 4 0 -18 E -4 -10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 14 4 B 2 0 16 14 10 C -2 -16 0 -4 -10 D -14 -14 4 0 -18 E -4 -10 10 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998729 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9343: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (10) E D C B A (6) D E C B A (6) B A C D E (6) B C A D E (5) A B C E D (5) E D C A B (4) E D A C B (4) D C E B A (4) B A C E D (4) A B E D C (4) E D B A C (3) C D E A B (3) A E B D C (3) A D E C B (3) A B C D E (3) E D B C A (2) E A D B C (2) D C E A B (2) C D B E A (2) C B A D E (2) B C D E A (2) B A E C D (2) E B D C A (1) C D B A E (1) C B D E A (1) C A D B E (1) B E D C A (1) B E C D A (1) B E A D C (1) B C E A D (1) B C D A E (1) B A E D C (1) A E D B C (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -26 -12 -8 -12 B 26 0 0 -10 -10 C 12 0 0 2 8 D 8 10 -2 0 6 E 12 10 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.094811 C: 0.905189 D: 0.000000 E: 0.000000 Sum of squares = 0.828355592533 Cumulative probabilities = A: 0.000000 B: 0.094811 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -12 -8 -12 B 26 0 0 -10 -10 C 12 0 0 2 8 D 8 10 -2 0 6 E 12 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222371468 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=22 A=21 C=20 D=12 so D is eliminated. Round 2 votes counts: E=28 C=26 B=25 A=21 so A is eliminated. Round 3 votes counts: B=38 E=35 C=27 so C is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:211 D:211 E:204 B:203 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -26 -12 -8 -12 B 26 0 0 -10 -10 C 12 0 0 2 8 D 8 10 -2 0 6 E 12 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222371468 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -12 -8 -12 B 26 0 0 -10 -10 C 12 0 0 2 8 D 8 10 -2 0 6 E 12 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222371468 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -12 -8 -12 B 26 0 0 -10 -10 C 12 0 0 2 8 D 8 10 -2 0 6 E 12 10 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.833333 D: 0.000000 E: 0.000000 Sum of squares = 0.722222371468 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9344: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) C B D A E (9) B C E A D (8) E B A C D (7) D C A B E (7) C D B A E (7) E A D B C (6) E B C A D (5) A E D B C (5) A D E B C (5) C B E D A (4) E A B D C (3) E A B C D (3) B E C A D (3) D A C B E (2) C B D E A (2) A D E C B (2) E D A C B (1) D E A C B (1) D C A E B (1) D A B C E (1) C B E A D (1) B C E D A (1) B C D A E (1) B C A D E (1) B A E C D (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 -2 0 4 10 B 2 0 4 2 -2 C 0 -4 0 6 -8 D -4 -2 -6 0 2 E -10 2 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408182 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 -2 0 4 10 B 2 0 4 2 -2 C 0 -4 0 6 -8 D -4 -2 -6 0 2 E -10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408161 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 C=23 B=16 A=13 so A is eliminated. Round 2 votes counts: E=31 D=30 C=23 B=16 so B is eliminated. Round 3 votes counts: E=35 C=34 D=31 so D is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:206 B:203 E:199 C:197 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 4 10 B 2 0 4 2 -2 C 0 -4 0 6 -8 D -4 -2 -6 0 2 E -10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408161 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 10 B 2 0 4 2 -2 C 0 -4 0 6 -8 D -4 -2 -6 0 2 E -10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408161 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 10 B 2 0 4 2 -2 C 0 -4 0 6 -8 D -4 -2 -6 0 2 E -10 2 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.714286 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.551020408161 Cumulative probabilities = A: 0.142857 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9345: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) C A B D E (7) A B D E C (7) E D B C A (6) E D B A C (5) E C D B A (5) B D A E C (5) E D C B A (4) D B A E C (4) C E D A B (4) A C B D E (4) D E B A C (3) D B E A C (3) C E A B D (3) C A B E D (3) B E D A C (3) A B C D E (3) E B C A D (2) C E D B A (2) C A D E B (2) C A D B E (2) A B D C E (2) A B C E D (2) E C B A D (1) E B A C D (1) D E C B A (1) D C A B E (1) D A C B E (1) C E A D B (1) C D E B A (1) B D E A C (1) B A E D C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 10 2 8 B 12 0 14 6 18 C -10 -14 0 -14 -20 D -2 -6 14 0 14 E -8 -18 20 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 2 8 B 12 0 14 6 18 C -10 -14 0 -14 -20 D -2 -6 14 0 14 E -8 -18 20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=24 A=20 B=18 D=13 so D is eliminated. Round 2 votes counts: E=28 C=26 B=25 A=21 so A is eliminated. Round 3 votes counts: B=40 C=32 E=28 so E is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:210 A:204 E:190 C:171 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 10 2 8 B 12 0 14 6 18 C -10 -14 0 -14 -20 D -2 -6 14 0 14 E -8 -18 20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 2 8 B 12 0 14 6 18 C -10 -14 0 -14 -20 D -2 -6 14 0 14 E -8 -18 20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 2 8 B 12 0 14 6 18 C -10 -14 0 -14 -20 D -2 -6 14 0 14 E -8 -18 20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999963 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9346: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) C E B D A (5) D B C A E (4) D B A E C (4) C E A D B (4) E C B A D (3) E A C B D (3) D A C B E (3) D A B C E (3) C E B A D (3) C D B E A (3) C B E D A (3) A E B D C (3) A E B C D (3) E B C A D (2) E B A C D (2) D C B A E (2) D A C E B (2) C E A B D (2) C D E A B (2) C B D E A (2) C A E D B (2) B E D C A (2) B E C D A (2) B D E C A (2) A D E B C (2) E C A B D (1) E A B C D (1) D C B E A (1) D C A E B (1) D B E C A (1) D B E A C (1) D B C E A (1) D B A C E (1) D A B E C (1) C D A E B (1) C D A B E (1) C A D E B (1) B E C A D (1) B E A D C (1) B D E A C (1) B D C E A (1) B C E D A (1) B C D E A (1) B A E D C (1) A E D B C (1) A E C D B (1) A E C B D (1) A D E C B (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -10 -4 -4 B 2 0 2 -4 4 C 10 -2 0 2 2 D 4 4 -2 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.375000000021 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -4 -4 B 2 0 2 -4 4 C 10 -2 0 2 2 D 4 4 -2 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 A=21 B=13 E=12 so E is eliminated. Round 2 votes counts: C=33 D=25 A=25 B=17 so B is eliminated. Round 3 votes counts: C=40 D=31 A=29 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:206 D:205 B:202 E:197 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 -4 -4 B 2 0 2 -4 4 C 10 -2 0 2 2 D 4 4 -2 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -4 -4 B 2 0 2 -4 4 C 10 -2 0 2 2 D 4 4 -2 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -4 -4 B 2 0 2 -4 4 C 10 -2 0 2 2 D 4 4 -2 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999995 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9347: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (12) A B E C D (8) D C E B A (7) D A B E C (6) B A E C D (6) A B E D C (6) D A B C E (5) C E D B A (5) D C A E B (4) D A C B E (4) E C B A D (3) D C E A B (3) C E B D A (3) B E A C D (3) A E B D C (3) D E A C B (2) D C A B E (2) C E B A D (2) C B E A D (2) E D C B A (1) E C D B A (1) E C B D A (1) E B C A D (1) E B A C D (1) E A B D C (1) D E C A B (1) C D B E A (1) B E C A D (1) B C E A D (1) A D E B C (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -2 -16 -4 B 2 0 -8 -14 -4 C 2 8 0 2 4 D 16 14 -2 0 2 E 4 4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -16 -4 B 2 0 -8 -14 -4 C 2 8 0 2 4 D 16 14 -2 0 2 E 4 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=25 A=21 B=11 E=9 so E is eliminated. Round 2 votes counts: D=35 C=30 A=22 B=13 so B is eliminated. Round 3 votes counts: D=35 C=33 A=32 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:215 C:208 E:201 A:188 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -2 -16 -4 B 2 0 -8 -14 -4 C 2 8 0 2 4 D 16 14 -2 0 2 E 4 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -16 -4 B 2 0 -8 -14 -4 C 2 8 0 2 4 D 16 14 -2 0 2 E 4 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -16 -4 B 2 0 -8 -14 -4 C 2 8 0 2 4 D 16 14 -2 0 2 E 4 4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999799 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9348: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (6) B E C A D (6) E B D C A (5) E B A D C (4) D E A B C (4) D C A E B (4) C B A E D (4) D E C B A (3) D E C A B (3) D C E B A (3) C A B D E (3) B E A D C (3) A D E B C (3) A C D B E (3) A B E D C (3) E D C B A (2) D A E C B (2) C D B E A (2) C B D A E (2) C A D B E (2) B C E A D (2) B C A E D (2) B A C E D (2) A D E C B (2) A D C E B (2) A B C E D (2) E D B C A (1) E D B A C (1) E B D A C (1) E B C A D (1) E A B D C (1) D E B C A (1) D E B A C (1) D A E B C (1) D A C E B (1) C E B D A (1) C B E D A (1) C B E A D (1) C B A D E (1) C A B E D (1) B E A C D (1) B A E C D (1) A D B E C (1) A C B E D (1) A C B D E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -14 10 4 B 4 0 -2 4 6 C 14 2 0 -6 -4 D -10 -4 6 0 4 E -4 -6 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888888903 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 10 4 B 4 0 -2 4 6 C 14 2 0 -6 -4 D -10 -4 6 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888889079 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=24 D=23 A=20 B=17 E=16 so E is eliminated. Round 2 votes counts: B=28 D=27 C=24 A=21 so A is eliminated. Round 3 votes counts: B=36 D=35 C=29 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:206 C:203 A:198 D:198 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -14 10 4 B 4 0 -2 4 6 C 14 2 0 -6 -4 D -10 -4 6 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888889079 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 10 4 B 4 0 -2 4 6 C 14 2 0 -6 -4 D -10 -4 6 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888889079 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 10 4 B 4 0 -2 4 6 C 14 2 0 -6 -4 D -10 -4 6 0 4 E -4 -6 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.333333 D: 0.166667 E: 0.000000 Sum of squares = 0.388888889079 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.833333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9349: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (11) B D C E A (8) E A B D C (6) C D B A E (6) E A C B D (5) A E D C B (4) A E D B C (4) E B C A D (3) E A B C D (3) D C B A E (3) D B C E A (3) D B C A E (3) C B D E A (3) C A E D B (3) B E D C A (3) B C D E A (3) A E C B D (3) E B A D C (2) D B A C E (2) D A B C E (2) C D A B E (2) B D E C A (2) B D E A C (2) A D C B E (2) E C B A D (1) E C A B D (1) E B D C A (1) E B A C D (1) D C A B E (1) D B E A C (1) D A C B E (1) C E B A D (1) C A D B E (1) B E D A C (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 2 4 6 0 B -2 0 2 -2 -2 C -4 -2 0 -4 -10 D -6 2 4 0 -6 E 0 2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.722919 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.277081 Sum of squares = 0.599385794915 Cumulative probabilities = A: 0.722919 B: 0.722919 C: 0.722919 D: 0.722919 E: 1.000000 A B C D E A 0 2 4 6 0 B -2 0 2 -2 -2 C -4 -2 0 -4 -10 D -6 2 4 0 -6 E 0 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=23 B=19 D=16 C=16 so D is eliminated. Round 2 votes counts: A=29 B=28 E=23 C=20 so C is eliminated. Round 3 votes counts: B=40 A=36 E=24 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:206 B:198 D:197 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 6 0 B -2 0 2 -2 -2 C -4 -2 0 -4 -10 D -6 2 4 0 -6 E 0 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 6 0 B -2 0 2 -2 -2 C -4 -2 0 -4 -10 D -6 2 4 0 -6 E 0 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 6 0 B -2 0 2 -2 -2 C -4 -2 0 -4 -10 D -6 2 4 0 -6 E 0 2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999865 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9350: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (8) E A C B D (7) E A D B C (6) D B C A E (6) E A C D B (4) D A E B C (4) D A B C E (4) B D C A E (4) A D E B C (4) E A D C B (3) D B C E A (3) D B A C E (3) C B E D A (3) C B D E A (3) B D C E A (3) B C D E A (3) A E D B C (3) E C B A D (2) E B C D A (2) D B A E C (2) A E D C B (2) A D C B E (2) A D B C E (2) A C B D E (2) E D B A C (1) E D A B C (1) E C B D A (1) D E B A C (1) D B E C A (1) D B E A C (1) D A B E C (1) C B E A D (1) C B A D E (1) C A E B D (1) C A B E D (1) B E C D A (1) B C D A E (1) A E C D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 10 -14 6 B 2 0 14 -10 12 C -10 -14 0 -14 4 D 14 10 14 0 18 E -6 -12 -4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 -14 6 B 2 0 14 -10 12 C -10 -14 0 -14 4 D 14 10 14 0 18 E -6 -12 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=26 C=18 A=17 B=12 so B is eliminated. Round 2 votes counts: D=33 E=28 C=22 A=17 so A is eliminated. Round 3 votes counts: D=41 E=34 C=25 so C is eliminated. Round 4 votes counts: D=59 E=41 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:209 A:200 C:183 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 10 -14 6 B 2 0 14 -10 12 C -10 -14 0 -14 4 D 14 10 14 0 18 E -6 -12 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 -14 6 B 2 0 14 -10 12 C -10 -14 0 -14 4 D 14 10 14 0 18 E -6 -12 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 -14 6 B 2 0 14 -10 12 C -10 -14 0 -14 4 D 14 10 14 0 18 E -6 -12 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9351: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) E C B D A (7) E B C A D (7) B E D A C (7) E B C D A (6) A D C B E (6) D C A E B (4) D B A E C (4) C E D B A (4) C E A D B (4) C A E D B (4) C A D E B (4) B E A D C (4) A D B C E (4) E C B A D (3) B A E D C (3) E B A C D (2) C E D A B (2) C E B D A (2) B E C D A (2) A D B E C (2) E C D B A (1) D A B C E (1) C D E A B (1) C D A E B (1) B E D C A (1) B E A C D (1) B D E A C (1) B D A E C (1) B A E C D (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -6 -6 -10 B 14 0 -4 0 -6 C 6 4 0 8 -6 D 6 0 -8 0 -24 E 10 6 6 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999879 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -6 -6 -10 B 14 0 -4 0 -6 C 6 4 0 8 -6 D 6 0 -8 0 -24 E 10 6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=22 B=21 D=17 A=14 so A is eliminated. Round 2 votes counts: D=29 E=26 C=23 B=22 so B is eliminated. Round 3 votes counts: E=45 D=31 C=24 so C is eliminated. Round 4 votes counts: E=62 D=38 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:206 B:202 D:187 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -6 -6 -10 B 14 0 -4 0 -6 C 6 4 0 8 -6 D 6 0 -8 0 -24 E 10 6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -6 -10 B 14 0 -4 0 -6 C 6 4 0 8 -6 D 6 0 -8 0 -24 E 10 6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -6 -10 B 14 0 -4 0 -6 C 6 4 0 8 -6 D 6 0 -8 0 -24 E 10 6 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9352: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (12) D A C B E (8) E B D C A (7) D A E C B (7) B C E A D (7) E B C A D (6) D E B A C (5) D A C E B (5) C B A E D (5) D E A B C (4) C A B E D (4) E D B A C (3) D C A B E (3) C A B D E (3) A C E B D (3) E B A C D (2) E A B C D (2) B C D A E (2) B C A E D (2) A E C B D (2) E B A D C (1) E A D C B (1) D B E C A (1) B E D C A (1) B E C D A (1) A E C D B (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 -8 6 -6 B 10 0 12 20 0 C 8 -12 0 8 -12 D -6 -20 -8 0 -20 E 6 0 12 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.543631 C: 0.000000 D: 0.000000 E: 0.456369 Sum of squares = 0.503807274518 Cumulative probabilities = A: 0.000000 B: 0.543631 C: 0.543631 D: 0.543631 E: 1.000000 A B C D E A 0 -10 -8 6 -6 B 10 0 12 20 0 C 8 -12 0 8 -12 D -6 -20 -8 0 -20 E 6 0 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=25 E=22 C=12 A=8 so A is eliminated. Round 2 votes counts: D=33 E=25 B=25 C=17 so C is eliminated. Round 3 votes counts: B=37 D=35 E=28 so E is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:219 C:196 A:191 D:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 6 -6 B 10 0 12 20 0 C 8 -12 0 8 -12 D -6 -20 -8 0 -20 E 6 0 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 6 -6 B 10 0 12 20 0 C 8 -12 0 8 -12 D -6 -20 -8 0 -20 E 6 0 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 6 -6 B 10 0 12 20 0 C 8 -12 0 8 -12 D -6 -20 -8 0 -20 E 6 0 12 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999951 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9353: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (9) D A E C B (8) B E C A D (7) C B D E A (6) B C E A D (6) E A D B C (5) D C A B E (5) D E A C B (4) C D B A E (4) B C A E D (4) E B A D C (3) D C A E B (3) C B A D E (3) B C E D A (3) B A C E D (3) A E D B C (3) D E C A B (2) D C E A B (2) D A C E B (2) B E A C D (2) B A E C D (2) A D E C B (2) A D E B C (2) E D A B C (1) E B A C D (1) E A B D C (1) D E C B A (1) D E A B C (1) D C E B A (1) C D A B E (1) A E B D C (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -14 -6 8 B 10 0 -8 4 12 C 14 8 0 4 6 D 6 -4 -4 0 14 E -8 -12 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -14 -6 8 B 10 0 -8 4 12 C 14 8 0 4 6 D 6 -4 -4 0 14 E -8 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=27 C=23 E=11 A=10 so A is eliminated. Round 2 votes counts: D=34 B=28 C=23 E=15 so E is eliminated. Round 3 votes counts: D=43 B=34 C=23 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:209 D:206 A:189 E:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -14 -6 8 B 10 0 -8 4 12 C 14 8 0 4 6 D 6 -4 -4 0 14 E -8 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -14 -6 8 B 10 0 -8 4 12 C 14 8 0 4 6 D 6 -4 -4 0 14 E -8 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -14 -6 8 B 10 0 -8 4 12 C 14 8 0 4 6 D 6 -4 -4 0 14 E -8 -12 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999857 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9354: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (9) B A C E D (9) D E A C B (8) D E C A B (4) D C E B A (4) D C B E A (4) C D E B A (4) B C A E D (4) A E D B C (4) A E B D C (4) A B E D C (4) E D C A B (3) C D B E A (3) C B E A D (3) C B D E A (3) B D C A E (2) A E B C D (2) E A D C B (1) D B A C E (1) C E D A B (1) C E B D A (1) C E B A D (1) C E A B D (1) C D E A B (1) C B E D A (1) B C E A D (1) B C D A E (1) B C A D E (1) B A D E C (1) B A C D E (1) A D E B C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 -6 -2 -8 B 8 0 -6 2 0 C 6 6 0 -4 14 D 2 -2 4 0 0 E 8 0 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888887 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -2 -8 B 8 0 -6 2 0 C 6 6 0 -4 14 D 2 -2 4 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=29 C=19 A=17 E=4 so E is eliminated. Round 2 votes counts: D=34 B=29 C=19 A=18 so A is eliminated. Round 3 votes counts: B=41 D=40 C=19 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:202 D:202 E:197 A:188 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -2 -8 B 8 0 -6 2 0 C 6 6 0 -4 14 D 2 -2 4 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -2 -8 B 8 0 -6 2 0 C 6 6 0 -4 14 D 2 -2 4 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -2 -8 B 8 0 -6 2 0 C 6 6 0 -4 14 D 2 -2 4 0 0 E 8 0 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.166667 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888901 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9355: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (8) C D A E B (7) B E A D C (7) A C D B E (6) D E C B A (5) D E B C A (5) E B D C A (4) D C A E B (4) C A D B E (4) A D C E B (4) E B D A C (3) C A D E B (3) A C B E D (3) A C B D E (3) D E B A C (2) D C E B A (2) D C E A B (2) D A C E B (2) C D E B A (2) C A B E D (2) B E D C A (2) B E A C D (2) B A E C D (2) A C D E B (2) A B E D C (2) A B E C D (2) A B C E D (2) E D B C A (1) E D B A C (1) C B E D A (1) C B E A D (1) C A B D E (1) B E C A D (1) B C E A D (1) B C A E D (1) Total count = 100 A B C D E A 0 -2 2 -2 0 B 2 0 -8 -4 2 C -2 8 0 -8 6 D 2 4 8 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -2 0 B 2 0 -8 -4 2 C -2 8 0 -8 6 D 2 4 8 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=24 A=24 D=22 C=21 E=9 so E is eliminated. Round 2 votes counts: B=31 D=24 A=24 C=21 so C is eliminated. Round 3 votes counts: A=34 D=33 B=33 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 C:202 A:199 B:196 E:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -2 0 B 2 0 -8 -4 2 C -2 8 0 -8 6 D 2 4 8 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -2 0 B 2 0 -8 -4 2 C -2 8 0 -8 6 D 2 4 8 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -2 0 B 2 0 -8 -4 2 C -2 8 0 -8 6 D 2 4 8 0 8 E 0 -2 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999323 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9356: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) C E D B A (7) A B D E C (7) C E A B D (6) B A D C E (6) D B A C E (5) E C D B A (4) E C A B D (4) D B E A C (4) C E D A B (4) C E A D B (4) A B C D E (4) D C E B A (3) D B C A E (3) B D A E C (3) A E B C D (3) E C D A B (2) E A C B D (2) D C B E A (2) C D E B A (2) C A E B D (2) A E C B D (2) A E B D C (2) A B E C D (2) E D C B A (1) E C A D B (1) D E B C A (1) D B E C A (1) D B C E A (1) C A B E D (1) B D A C E (1) B A D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 2 -4 2 B 6 0 6 -6 0 C -2 -6 0 0 4 D 4 6 0 0 6 E -2 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.312676 D: 0.687324 E: 0.000000 Sum of squares = 0.570180569961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.312676 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 -4 2 B 6 0 6 -6 0 C -2 -6 0 0 4 D 4 6 0 0 6 E -2 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499912 D: 0.500088 E: 0.000000 Sum of squares = 0.500000015325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499912 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 A=21 E=14 B=11 so B is eliminated. Round 2 votes counts: D=32 A=28 C=26 E=14 so E is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:208 B:203 C:198 A:197 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 2 -4 2 B 6 0 6 -6 0 C -2 -6 0 0 4 D 4 6 0 0 6 E -2 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499912 D: 0.500088 E: 0.000000 Sum of squares = 0.500000015325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499912 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 -4 2 B 6 0 6 -6 0 C -2 -6 0 0 4 D 4 6 0 0 6 E -2 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499912 D: 0.500088 E: 0.000000 Sum of squares = 0.500000015325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499912 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 -4 2 B 6 0 6 -6 0 C -2 -6 0 0 4 D 4 6 0 0 6 E -2 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499912 D: 0.500088 E: 0.000000 Sum of squares = 0.500000015325 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499912 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9357: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) E B A D C (5) D A E B C (5) D A C B E (5) C A D E B (5) A D C E B (5) B E C D A (4) B E C A D (4) A C D E B (4) E B D A C (3) D C B A E (3) C A E B D (3) C A D B E (3) B E D C A (3) B C D E A (3) A C E D B (3) E A C D B (2) E A B C D (2) D C A B E (2) D B E A C (2) D B C A E (2) D A C E B (2) C B D A E (2) C B A E D (2) C A E D B (2) B E D A C (2) B C E D A (2) A E C D B (2) A D E C B (2) E C A B D (1) E B A C D (1) E A D C B (1) E A D B C (1) D A B C E (1) C D B A E (1) C B E D A (1) C B D E A (1) B D C E A (1) A E D C B (1) Total count = 100 A B C D E A 0 4 -2 10 10 B -4 0 -6 -8 -12 C 2 6 0 8 6 D -10 8 -8 0 -2 E -10 12 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 10 10 B -4 0 -6 -8 -12 C 2 6 0 8 6 D -10 8 -8 0 -2 E -10 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=22 D=22 C=20 B=19 A=17 so A is eliminated. Round 2 votes counts: D=29 C=27 E=25 B=19 so B is eliminated. Round 3 votes counts: E=38 C=32 D=30 so D is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:211 E:199 D:194 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 10 10 B -4 0 -6 -8 -12 C 2 6 0 8 6 D -10 8 -8 0 -2 E -10 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 10 10 B -4 0 -6 -8 -12 C 2 6 0 8 6 D -10 8 -8 0 -2 E -10 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 10 10 B -4 0 -6 -8 -12 C 2 6 0 8 6 D -10 8 -8 0 -2 E -10 12 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9358: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (14) C B D A E (14) A E C B D (13) E A D B C (10) D B C A E (8) D B C E A (6) C B D E A (4) A C E B D (4) E A D C B (3) B D C A E (3) D E B C A (2) D B E A C (2) B C D A E (2) A E D B C (2) E D B C A (1) E D A B C (1) D E A B C (1) D B E C A (1) D B A C E (1) D A B E C (1) C E B A D (1) C B A E D (1) C A E B D (1) C A B E D (1) C A B D E (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 8 8 6 8 B -8 0 -16 22 -8 C -8 16 0 16 -4 D -6 -22 -16 0 -8 E -8 8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 6 8 B -8 0 -16 22 -8 C -8 16 0 16 -4 D -6 -22 -16 0 -8 E -8 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=23 D=22 A=21 B=5 so B is eliminated. Round 2 votes counts: E=29 D=25 C=25 A=21 so A is eliminated. Round 3 votes counts: E=45 C=30 D=25 so D is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:215 C:210 E:206 B:195 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 6 8 B -8 0 -16 22 -8 C -8 16 0 16 -4 D -6 -22 -16 0 -8 E -8 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 6 8 B -8 0 -16 22 -8 C -8 16 0 16 -4 D -6 -22 -16 0 -8 E -8 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 6 8 B -8 0 -16 22 -8 C -8 16 0 16 -4 D -6 -22 -16 0 -8 E -8 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9359: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (9) C A D B E (7) A D C E B (7) B E C D A (6) A C D B E (6) D A E C B (5) B E D C A (5) B E C A D (5) B C E A D (5) E B D A C (4) C A B D E (4) C A D E B (3) B C E D A (3) A D E C B (3) E D B A C (2) E B D C A (2) D A C E B (2) C D A B E (2) C B A D E (2) B E A D C (2) A B E C D (2) E D A B C (1) E B A D C (1) E A D B C (1) D E C B A (1) D E B C A (1) D E A B C (1) D C E B A (1) C D E B A (1) C B E A D (1) C B D E A (1) B A C E D (1) A D E B C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 12 0 24 12 B -12 0 -12 -8 8 C 0 12 0 18 12 D -24 8 -18 0 16 E -12 -8 -12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.497274 B: 0.000000 C: 0.502726 D: 0.000000 E: 0.000000 Sum of squares = 0.500014864516 Cumulative probabilities = A: 0.497274 B: 0.497274 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 24 12 B -12 0 -12 -8 8 C 0 12 0 18 12 D -24 8 -18 0 16 E -12 -8 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 B=27 C=21 E=11 D=11 so E is eliminated. Round 2 votes counts: B=34 A=31 C=21 D=14 so D is eliminated. Round 3 votes counts: A=40 B=37 C=23 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:224 C:221 D:191 B:188 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 24 12 B -12 0 -12 -8 8 C 0 12 0 18 12 D -24 8 -18 0 16 E -12 -8 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 24 12 B -12 0 -12 -8 8 C 0 12 0 18 12 D -24 8 -18 0 16 E -12 -8 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 24 12 B -12 0 -12 -8 8 C 0 12 0 18 12 D -24 8 -18 0 16 E -12 -8 -12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9360: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) D E A B C (9) C B D E A (8) B C D E A (6) C B D A E (5) A E C B D (5) C A B E D (4) A E C D B (4) D E B A C (3) C A E B D (3) B D E C A (3) A E D B C (3) A E B D C (3) A C E B D (3) E D A B C (2) E A B D C (2) D C B E A (2) D B E A C (2) D B C E A (2) C B A E D (2) C B A D E (2) B E D A C (2) B E A D C (2) B D E A C (2) B D C E A (2) A C E D B (2) E B D A C (1) D E B C A (1) C D A E B (1) C D A B E (1) B A E C D (1) A E D C B (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 8 16 -4 -16 B -8 0 12 14 -6 C -16 -12 0 -4 -12 D 4 -14 4 0 0 E 16 6 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.079564 E: 0.920436 Sum of squares = 0.853533050182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.079564 E: 1.000000 A B C D E A 0 8 16 -4 -16 B -8 0 12 14 -6 C -16 -12 0 -4 -12 D 4 -14 4 0 0 E 16 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.700000 Sum of squares = 0.580000096148 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=23 D=19 B=18 E=14 so E is eliminated. Round 2 votes counts: A=34 C=26 D=21 B=19 so B is eliminated. Round 3 votes counts: A=37 C=32 D=31 so D is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:217 B:206 A:202 D:197 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 16 -4 -16 B -8 0 12 14 -6 C -16 -12 0 -4 -12 D 4 -14 4 0 0 E 16 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.700000 Sum of squares = 0.580000096148 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 -4 -16 B -8 0 12 14 -6 C -16 -12 0 -4 -12 D 4 -14 4 0 0 E 16 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.700000 Sum of squares = 0.580000096148 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 -4 -16 B -8 0 12 14 -6 C -16 -12 0 -4 -12 D 4 -14 4 0 0 E 16 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 0.700000 Sum of squares = 0.580000096148 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.300000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9361: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) A E D B C (9) D C A E B (8) E B A D C (6) E B A C D (6) D C A B E (6) C D B A E (5) E A B D C (4) D A C E B (4) C D B E A (4) C B E D A (4) B E C A D (4) C D A B E (3) C B D E A (3) A D E C B (3) A D C E B (3) B C E D A (2) B C E A D (2) B C D E A (2) A E D C B (2) E A B C D (1) D E A B C (1) D C B E A (1) D B E A C (1) D B C E A (1) D A E C B (1) D A E B C (1) C D A E B (1) B E A D C (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 8 4 -4 B 2 0 4 -10 -4 C -8 -4 0 -8 -2 D -4 10 8 0 -2 E 4 4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999753 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 8 4 -4 B 2 0 4 -10 -4 C -8 -4 0 -8 -2 D -4 10 8 0 -2 E 4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=20 B=20 A=19 E=17 so E is eliminated. Round 2 votes counts: B=32 D=24 A=24 C=20 so C is eliminated. Round 3 votes counts: B=39 D=37 A=24 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:206 E:206 A:203 B:196 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 8 4 -4 B 2 0 4 -10 -4 C -8 -4 0 -8 -2 D -4 10 8 0 -2 E 4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 4 -4 B 2 0 4 -10 -4 C -8 -4 0 -8 -2 D -4 10 8 0 -2 E 4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 4 -4 B 2 0 4 -10 -4 C -8 -4 0 -8 -2 D -4 10 8 0 -2 E 4 4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9362: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (11) B C E A D (10) A D E B C (9) D A E C B (8) E A D B C (6) C B D A E (6) E B A D C (5) D A C E B (4) B E C A D (4) C B A D E (3) B E A C D (3) E B C A D (2) E B A C D (2) D C A E B (2) C D B A E (2) C D A E B (2) C D A B E (2) C B D E A (2) B E A D C (2) B C A D E (2) A D E C B (2) E D A C B (1) E D A B C (1) E C D A B (1) E A B D C (1) D A E B C (1) D A C B E (1) B A D C E (1) B A C D E (1) A E D B C (1) A D B C E (1) A B E D C (1) Total count = 100 A B C D E A 0 -12 2 12 -2 B 12 0 6 12 4 C -2 -6 0 6 0 D -12 -12 -6 0 -2 E 2 -4 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999649 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 12 -2 B 12 0 6 12 4 C -2 -6 0 6 0 D -12 -12 -6 0 -2 E 2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=23 E=19 D=16 A=14 so A is eliminated. Round 2 votes counts: D=28 C=28 B=24 E=20 so E is eliminated. Round 3 votes counts: D=37 B=34 C=29 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:217 A:200 E:200 C:199 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 12 -2 B 12 0 6 12 4 C -2 -6 0 6 0 D -12 -12 -6 0 -2 E 2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 12 -2 B 12 0 6 12 4 C -2 -6 0 6 0 D -12 -12 -6 0 -2 E 2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 12 -2 B 12 0 6 12 4 C -2 -6 0 6 0 D -12 -12 -6 0 -2 E 2 -4 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999511 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9363: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (13) D C B E A (9) E A B C D (8) C D B E A (7) D B C A E (6) A E B C D (5) D C B A E (4) C D E B A (4) A E B D C (4) E A C D B (3) C E B D A (3) C B E D A (3) A E D B C (3) A D E B C (3) A D B E C (3) E C B A D (2) E C A B D (2) D B A C E (2) D A B C E (2) C D E A B (2) B C D E A (2) E C D A B (1) E B A C D (1) D A C E B (1) D A C B E (1) C E D B A (1) B D C E A (1) B A D C E (1) A E D C B (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 8 6 2 -24 B -8 0 -14 -6 -14 C -6 14 0 16 -2 D -2 6 -16 0 -2 E 24 14 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 2 -24 B -8 0 -14 -6 -14 C -6 14 0 16 -2 D -2 6 -16 0 -2 E 24 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=25 A=21 C=20 B=4 so B is eliminated. Round 2 votes counts: E=30 D=26 C=22 A=22 so C is eliminated. Round 3 votes counts: D=41 E=37 A=22 so A is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:211 A:196 D:193 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 2 -24 B -8 0 -14 -6 -14 C -6 14 0 16 -2 D -2 6 -16 0 -2 E 24 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 2 -24 B -8 0 -14 -6 -14 C -6 14 0 16 -2 D -2 6 -16 0 -2 E 24 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 2 -24 B -8 0 -14 -6 -14 C -6 14 0 16 -2 D -2 6 -16 0 -2 E 24 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999457 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9364: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (7) D E C A B (5) A D B C E (5) E B C A D (4) D B A E C (4) D A C E B (4) B C E A D (4) A B D C E (4) A B C E D (4) E C D A B (3) D E B C A (3) D A B E C (3) C E B A D (3) C A E D B (3) B A C E D (3) A C B E D (3) A B C D E (3) E C B D A (2) E B C D A (2) D C A E B (2) D A C B E (2) B E C A D (2) B C A E D (2) B A D C E (2) A C D B E (2) E D C A B (1) E C D B A (1) E C A B D (1) D E A C B (1) D B E A C (1) D A E C B (1) D A E B C (1) D A B C E (1) C E A B D (1) C D E A B (1) C D A E B (1) C B A E D (1) C A E B D (1) B A D E C (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 16 0 22 16 B -16 0 -2 2 -2 C 0 2 0 14 14 D -22 -2 -14 0 2 E -16 2 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.411027 B: 0.000000 C: 0.588973 D: 0.000000 E: 0.000000 Sum of squares = 0.515832396248 Cumulative probabilities = A: 0.411027 B: 0.411027 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 22 16 B -16 0 -2 2 -2 C 0 2 0 14 14 D -22 -2 -14 0 2 E -16 2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=26 E=21 B=14 C=11 so C is eliminated. Round 2 votes counts: D=30 A=30 E=25 B=15 so B is eliminated. Round 3 votes counts: A=39 E=31 D=30 so D is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 C:215 B:191 E:185 D:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 22 16 B -16 0 -2 2 -2 C 0 2 0 14 14 D -22 -2 -14 0 2 E -16 2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 22 16 B -16 0 -2 2 -2 C 0 2 0 14 14 D -22 -2 -14 0 2 E -16 2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 22 16 B -16 0 -2 2 -2 C 0 2 0 14 14 D -22 -2 -14 0 2 E -16 2 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9365: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (8) D A B E C (7) D A B C E (7) B C E D A (7) A D E C B (7) E C B A D (5) E C A B D (5) C E B A D (5) A D E B C (5) E C A D B (4) A E C D B (4) E B C A D (3) D B A C E (3) B D C A E (3) B D A E C (3) B C D E A (3) D A C E B (2) C E B D A (2) C B E D A (2) B E C D A (2) A E D C B (2) E A C D B (1) D B C A E (1) D B A E C (1) D A C B E (1) C E A D B (1) B E D A C (1) B E C A D (1) B D E C A (1) A E D B C (1) A D C E B (1) A D B E C (1) Total count = 100 A B C D E A 0 -2 -6 -8 -2 B 2 0 16 2 4 C 6 -16 0 -10 -8 D 8 -2 10 0 8 E 2 -4 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -8 -2 B 2 0 16 2 4 C 6 -16 0 -10 -8 D 8 -2 10 0 8 E 2 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994106 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=22 A=21 E=18 C=10 so C is eliminated. Round 2 votes counts: B=31 E=26 D=22 A=21 so A is eliminated. Round 3 votes counts: D=36 E=33 B=31 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:212 E:199 A:191 C:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 -2 B 2 0 16 2 4 C 6 -16 0 -10 -8 D 8 -2 10 0 8 E 2 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994106 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 -2 B 2 0 16 2 4 C 6 -16 0 -10 -8 D 8 -2 10 0 8 E 2 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994106 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 -2 B 2 0 16 2 4 C 6 -16 0 -10 -8 D 8 -2 10 0 8 E 2 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994106 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9366: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (7) C A B D E (7) C A E B D (6) E A C D B (5) E A C B D (5) D B E C A (5) D B E A C (5) D E B A C (4) A C E D B (4) A C E B D (4) E B A C D (3) E A D C B (3) A C D E B (3) D B C A E (2) C B A E D (2) C B A D E (2) C A B E D (2) B E D C A (2) B D E C A (2) B D C E A (2) B D C A E (2) A E C D B (2) A D C E B (2) E D B A C (1) E D A B C (1) E B D A C (1) E B C A D (1) E B A D C (1) E A B C D (1) D E A C B (1) D C B A E (1) D C A B E (1) D B C E A (1) C E A B D (1) B E C A D (1) B C E D A (1) B C E A D (1) B C D A E (1) B C A E D (1) B C A D E (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 14 -4 34 4 B -14 0 -20 2 -2 C 4 20 0 24 8 D -34 -2 -24 0 0 E -4 2 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 34 4 B -14 0 -20 2 -2 C 4 20 0 24 8 D -34 -2 -24 0 0 E -4 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 D=20 A=17 B=14 so B is eliminated. Round 2 votes counts: C=32 D=26 E=25 A=17 so A is eliminated. Round 3 votes counts: C=43 D=29 E=28 so E is eliminated. Round 4 votes counts: C=62 D=38 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:224 E:195 B:183 D:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -4 34 4 B -14 0 -20 2 -2 C 4 20 0 24 8 D -34 -2 -24 0 0 E -4 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 34 4 B -14 0 -20 2 -2 C 4 20 0 24 8 D -34 -2 -24 0 0 E -4 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 34 4 B -14 0 -20 2 -2 C 4 20 0 24 8 D -34 -2 -24 0 0 E -4 2 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999984016 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9367: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) A E B D C (7) E C B D A (5) E C A B D (5) B D A E C (5) B D C E A (4) B A D E C (4) A D B E C (4) E A B C D (3) C D B E A (3) C D A B E (3) B E D A C (3) A E D B C (3) E C B A D (2) E B C D A (2) E B C A D (2) E A C B D (2) E A B D C (2) D B C A E (2) D B A C E (2) D A B C E (2) C E B A D (2) C B D E A (2) B D E C A (2) B D E A C (2) A E C D B (2) A E C B D (2) E B A C D (1) D C B E A (1) D C A B E (1) D B A E C (1) D A C B E (1) C E D B A (1) C D B A E (1) C B E D A (1) C A E D B (1) C A D E B (1) B E A D C (1) B C D E A (1) A E D C B (1) A D E C B (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 2 -4 -8 B 14 0 8 36 -4 C -2 -8 0 -2 -26 D 4 -36 2 0 -12 E 8 4 26 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 2 -4 -8 B 14 0 8 36 -4 C -2 -8 0 -2 -26 D 4 -36 2 0 -12 E 8 4 26 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=24 C=22 B=22 A=22 D=10 so D is eliminated. Round 2 votes counts: B=27 A=25 E=24 C=24 so E is eliminated. Round 3 votes counts: C=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:227 E:225 A:188 C:181 D:179 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 2 -4 -8 B 14 0 8 36 -4 C -2 -8 0 -2 -26 D 4 -36 2 0 -12 E 8 4 26 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -4 -8 B 14 0 8 36 -4 C -2 -8 0 -2 -26 D 4 -36 2 0 -12 E 8 4 26 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -4 -8 B 14 0 8 36 -4 C -2 -8 0 -2 -26 D 4 -36 2 0 -12 E 8 4 26 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9368: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (26) D E B C A (18) D A C B E (9) E D B C A (8) E B C A D (4) D A E C B (4) D A C E B (4) A D C B E (4) A C D B E (4) D E B A C (3) D E A B C (3) E B C D A (2) C A B E D (2) B E C A D (2) B C E A D (2) E B D C A (1) E B D A C (1) C D A B E (1) C B A E D (1) A C B D E (1) Total count = 100 A B C D E A 0 16 18 -8 12 B -16 0 -12 -16 4 C -18 12 0 -10 8 D 8 16 10 0 2 E -12 -4 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 -8 12 B -16 0 -12 -16 4 C -18 12 0 -10 8 D 8 16 10 0 2 E -12 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=41 A=35 E=16 C=4 B=4 so C is eliminated. Round 2 votes counts: D=42 A=37 E=16 B=5 so B is eliminated. Round 3 votes counts: D=42 A=38 E=20 so E is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:219 D:218 C:196 E:187 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 18 -8 12 B -16 0 -12 -16 4 C -18 12 0 -10 8 D 8 16 10 0 2 E -12 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 -8 12 B -16 0 -12 -16 4 C -18 12 0 -10 8 D 8 16 10 0 2 E -12 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 -8 12 B -16 0 -12 -16 4 C -18 12 0 -10 8 D 8 16 10 0 2 E -12 -4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999985319 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9369: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (10) B C D E A (8) E D C A B (7) A B D E C (7) C B E D A (6) D E C B A (5) E D A C B (4) D E A C B (4) A D E B C (4) E D C B A (3) D A E B C (3) C B D E A (3) B C A D E (3) B A C D E (3) A E D B C (3) A B C E D (3) E C D B A (2) E C D A B (2) C E B D A (2) E C A D B (1) E A D C B (1) E A C D B (1) D E C A B (1) D B C E A (1) C E D B A (1) C E D A B (1) C D B E A (1) B D A E C (1) B D A C E (1) B C A E D (1) B A D C E (1) B A C E D (1) A E C D B (1) A E C B D (1) A E B D C (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 4 -12 -8 B -14 0 -16 -12 -16 C -4 16 0 -14 -26 D 12 12 14 0 -8 E 8 16 26 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 4 -12 -8 B -14 0 -16 -12 -16 C -4 16 0 -14 -26 D 12 12 14 0 -8 E 8 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=21 B=19 D=14 C=14 so D is eliminated. Round 2 votes counts: A=35 E=31 B=20 C=14 so C is eliminated. Round 3 votes counts: E=35 A=35 B=30 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 D:215 A:199 C:186 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 4 -12 -8 B -14 0 -16 -12 -16 C -4 16 0 -14 -26 D 12 12 14 0 -8 E 8 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 -12 -8 B -14 0 -16 -12 -16 C -4 16 0 -14 -26 D 12 12 14 0 -8 E 8 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 -12 -8 B -14 0 -16 -12 -16 C -4 16 0 -14 -26 D 12 12 14 0 -8 E 8 16 26 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9370: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (5) D C E A B (5) B E C A D (5) E B D C A (4) E B C D A (4) D A C E B (4) C D A E B (4) B E C D A (4) B E A C D (4) B A E C D (4) A D E B C (4) A D C B E (4) A C D B E (4) E B D A C (3) C D A B E (3) C A B D E (3) A D C E B (3) E D B C A (2) E B A D C (2) D C A E B (2) B C E A D (2) B A E D C (2) B A C E D (2) A B E D C (2) A B C D E (2) E D C A B (1) E D A C B (1) E D A B C (1) E C B D A (1) E A B D C (1) D E A C B (1) D A E C B (1) C E B D A (1) C D B A E (1) C B D E A (1) C B A D E (1) C A D B E (1) B E A D C (1) B C A E D (1) A D B C E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 -2 2 2 B -10 0 4 4 0 C 2 -4 0 -2 -6 D -2 -4 2 0 4 E -2 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.442175 B: 0.000000 C: 0.278913 D: 0.197282 E: 0.081631 Sum of squares = 0.318894416805 Cumulative probabilities = A: 0.442175 B: 0.442175 C: 0.721087 D: 0.918369 E: 1.000000 A B C D E A 0 10 -2 2 2 B -10 0 4 4 0 C 2 -4 0 -2 -6 D -2 -4 2 0 4 E -2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.407407 B: 0.000000 C: 0.296296 D: 0.240741 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.407407 B: 0.407407 C: 0.703704 D: 0.944444 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 A=22 E=20 D=18 C=15 so C is eliminated. Round 2 votes counts: B=27 D=26 A=26 E=21 so E is eliminated. Round 3 votes counts: B=42 D=31 A=27 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:206 D:200 E:200 B:199 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 -2 2 2 B -10 0 4 4 0 C 2 -4 0 -2 -6 D -2 -4 2 0 4 E -2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.407407 B: 0.000000 C: 0.296296 D: 0.240741 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.407407 B: 0.407407 C: 0.703704 D: 0.944444 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 2 2 B -10 0 4 4 0 C 2 -4 0 -2 -6 D -2 -4 2 0 4 E -2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.407407 B: 0.000000 C: 0.296296 D: 0.240741 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.407407 B: 0.407407 C: 0.703704 D: 0.944444 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 2 2 B -10 0 4 4 0 C 2 -4 0 -2 -6 D -2 -4 2 0 4 E -2 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.407407 B: 0.000000 C: 0.296296 D: 0.240741 E: 0.055556 Sum of squares = 0.314814814814 Cumulative probabilities = A: 0.407407 B: 0.407407 C: 0.703704 D: 0.944444 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9371: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) A B E D C (10) C E D A B (9) C D E B A (8) E C A B D (6) D C B E A (5) D B A C E (5) B D A E C (5) E A C B D (4) E A B C D (4) D B C A E (4) C D B E A (4) B D A C E (4) A E B D C (4) E C A D B (3) D C B A E (3) C E A D B (3) A E B C D (3) C E A B D (2) E A B D C (1) C D E A B (1) B A E D C (1) B A D C E (1) Total count = 100 A B C D E A 0 0 4 4 0 B 0 0 4 10 4 C -4 -4 0 -6 -2 D -4 -10 6 0 0 E 0 -4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.545943 B: 0.454057 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.504221536688 Cumulative probabilities = A: 0.545943 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 4 0 B 0 0 4 10 4 C -4 -4 0 -6 -2 D -4 -10 6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=21 E=18 D=17 A=17 so D is eliminated. Round 2 votes counts: C=35 B=30 E=18 A=17 so A is eliminated. Round 3 votes counts: B=40 C=35 E=25 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:204 E:199 D:196 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 4 0 B 0 0 4 10 4 C -4 -4 0 -6 -2 D -4 -10 6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 0 B 0 0 4 10 4 C -4 -4 0 -6 -2 D -4 -10 6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 0 B 0 0 4 10 4 C -4 -4 0 -6 -2 D -4 -10 6 0 0 E 0 -4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9372: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (14) C A D B E (14) E B D C A (8) D B A C E (7) B D E C A (7) B D E A C (6) E C A B D (5) A C D B E (5) E B D A C (4) D B E A C (4) C A E B D (4) E D A B C (2) E C B A D (2) D B C A E (2) D A B C E (2) C E A B D (2) C A D E B (2) B D C A E (2) E B C D A (1) E A C B D (1) D B A E C (1) C E B A D (1) C B D A E (1) C B A D E (1) C A B E D (1) B E D A C (1) Total count = 100 A B C D E A 0 4 -34 4 12 B -4 0 -6 -6 8 C 34 6 0 8 16 D -4 6 -8 0 8 E -12 -8 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -34 4 12 B -4 0 -6 -6 8 C 34 6 0 8 16 D -4 6 -8 0 8 E -12 -8 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 E=23 D=16 B=16 A=5 so A is eliminated. Round 2 votes counts: C=45 E=23 D=16 B=16 so D is eliminated. Round 3 votes counts: C=45 B=32 E=23 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 D:201 B:196 A:193 E:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -34 4 12 B -4 0 -6 -6 8 C 34 6 0 8 16 D -4 6 -8 0 8 E -12 -8 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -34 4 12 B -4 0 -6 -6 8 C 34 6 0 8 16 D -4 6 -8 0 8 E -12 -8 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -34 4 12 B -4 0 -6 -6 8 C 34 6 0 8 16 D -4 6 -8 0 8 E -12 -8 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9373: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) B E C D A (6) E A C B D (5) B D E C A (5) D C A B E (4) D B A E C (4) C A E B D (4) B E D A C (4) E A D B C (3) C E A B D (3) C D B A E (3) B E C A D (3) B C E D A (3) B C D E A (3) A C E D B (3) E B D A C (2) E B A C D (2) D B E A C (2) D B C A E (2) D A E B C (2) D A B E C (2) C E B A D (2) B D C E A (2) A E C B D (2) E D A B C (1) E C B A D (1) E C A B D (1) E B C A D (1) E A B D C (1) D E A B C (1) D A C E B (1) D A C B E (1) D A B C E (1) C B E D A (1) C B D E A (1) C A E D B (1) C A D E B (1) B E A C D (1) B D E A C (1) B C E A D (1) A E D C B (1) A E D B C (1) A E C D B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -16 -12 -4 -28 B 16 0 10 26 16 C 12 -10 0 14 -8 D 4 -26 -14 0 -24 E 28 -16 8 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 -4 -28 B 16 0 10 26 16 C 12 -10 0 14 -8 D 4 -26 -14 0 -24 E 28 -16 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 C=24 D=20 E=17 A=10 so A is eliminated. Round 2 votes counts: B=29 C=27 E=22 D=22 so E is eliminated. Round 3 votes counts: C=37 B=35 D=28 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:234 E:222 C:204 A:170 D:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 -4 -28 B 16 0 10 26 16 C 12 -10 0 14 -8 D 4 -26 -14 0 -24 E 28 -16 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 -4 -28 B 16 0 10 26 16 C 12 -10 0 14 -8 D 4 -26 -14 0 -24 E 28 -16 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 -4 -28 B 16 0 10 26 16 C 12 -10 0 14 -8 D 4 -26 -14 0 -24 E 28 -16 8 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999769 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9374: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (10) B C D A E (8) A E D C B (8) B C D E A (6) E A D C B (5) A D C B E (5) E B C A D (3) D A C B E (3) C D B E A (3) C B D E A (3) B E C D A (3) A E D B C (3) A D E C B (3) A D C E B (3) E C D B A (2) E B C D A (2) E B A C D (2) E A B D C (2) D C E A B (2) D C B E A (2) D C A E B (2) B C E D A (2) E C B D A (1) D C B A E (1) D C A B E (1) D A E C B (1) D A C E B (1) C E B D A (1) C D E B A (1) C D B A E (1) C B D A E (1) B E A C D (1) B D C A E (1) B C E A D (1) B A C E D (1) B A C D E (1) A E B D C (1) A E B C D (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 6 6 4 -4 B -6 0 0 2 -10 C -6 0 0 8 2 D -4 -2 -8 0 2 E 4 10 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888893 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 6 6 4 -4 B -6 0 0 2 -10 C -6 0 0 8 2 D -4 -2 -8 0 2 E 4 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888868 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 A=26 B=24 D=13 C=10 so C is eliminated. Round 2 votes counts: E=28 B=28 A=26 D=18 so D is eliminated. Round 3 votes counts: B=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:205 C:202 D:194 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 6 4 -4 B -6 0 0 2 -10 C -6 0 0 8 2 D -4 -2 -8 0 2 E 4 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888868 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 4 -4 B -6 0 0 2 -10 C -6 0 0 8 2 D -4 -2 -8 0 2 E 4 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888868 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 4 -4 B -6 0 0 2 -10 C -6 0 0 8 2 D -4 -2 -8 0 2 E 4 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888868 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9375: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (7) E C D B A (6) D C E A B (5) A B D E C (5) E D C A B (4) E B C A D (4) C E D A B (4) C A B D E (4) A D B C E (4) E C D A B (3) E B A C D (3) D C A B E (3) D A C B E (3) D A B E C (3) C E D B A (3) B A E C D (3) B A C E D (3) B A C D E (3) E C B D A (2) E C B A D (2) D C A E B (2) C D E A B (2) C D A E B (2) B A E D C (2) B A D E C (2) B A D C E (2) A B C D E (2) E D B A C (1) E D A B C (1) E B D C A (1) E B D A C (1) E B C D A (1) E B A D C (1) D E C A B (1) D A E B C (1) C E B D A (1) C D A B E (1) B E A C D (1) A D B E C (1) Total count = 100 A B C D E A 0 16 -2 -2 6 B -16 0 4 0 -2 C 2 -4 0 0 2 D 2 0 0 0 6 E -6 2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.081129 C: 0.000000 D: 0.918871 E: 0.000000 Sum of squares = 0.850906024427 Cumulative probabilities = A: 0.000000 B: 0.081129 C: 0.081129 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -2 6 B -16 0 4 0 -2 C 2 -4 0 0 2 D 2 0 0 0 6 E -6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.888889 E: 0.000000 Sum of squares = 0.802469137633 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=19 D=18 C=17 B=16 so B is eliminated. Round 2 votes counts: A=34 E=31 D=18 C=17 so C is eliminated. Round 3 votes counts: E=39 A=38 D=23 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:209 D:204 C:200 E:194 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -2 -2 6 B -16 0 4 0 -2 C 2 -4 0 0 2 D 2 0 0 0 6 E -6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.888889 E: 0.000000 Sum of squares = 0.802469137633 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -2 6 B -16 0 4 0 -2 C 2 -4 0 0 2 D 2 0 0 0 6 E -6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.888889 E: 0.000000 Sum of squares = 0.802469137633 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -2 6 B -16 0 4 0 -2 C 2 -4 0 0 2 D 2 0 0 0 6 E -6 2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.888889 E: 0.000000 Sum of squares = 0.802469137633 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9376: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (6) C D B E A (5) C B D E A (4) B C E D A (4) B C E A D (4) A E B D C (4) A D E C B (4) A B E C D (4) E D A B C (3) C D A E B (3) B E A C D (3) B A E C D (3) E B D C A (2) E B D A C (2) E B A D C (2) E A B D C (2) D E C B A (2) D E C A B (2) D C E A B (2) D C A E B (2) D A C E B (2) C B A D E (2) B E C A D (2) B C A E D (2) B C A D E (2) B A C E D (2) A C D B E (2) A C B D E (2) A B C E D (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A C B (1) D C E B A (1) D A E C B (1) C D E B A (1) C D A B E (1) C B D A E (1) C A D B E (1) B E D A C (1) B E C D A (1) B C D E A (1) B C D A E (1) B A E D C (1) A E D C B (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 6 10 6 B 2 0 18 14 8 C -6 -18 0 10 0 D -10 -14 -10 0 -10 E -6 -8 0 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 10 6 B 2 0 18 14 8 C -6 -18 0 10 0 D -10 -14 -10 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989146 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 C=18 E=14 D=12 so D is eliminated. Round 2 votes counts: A=32 B=27 C=23 E=18 so E is eliminated. Round 3 votes counts: A=38 B=35 C=27 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 A:210 E:198 C:193 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 10 6 B 2 0 18 14 8 C -6 -18 0 10 0 D -10 -14 -10 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989146 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 10 6 B 2 0 18 14 8 C -6 -18 0 10 0 D -10 -14 -10 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989146 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 10 6 B 2 0 18 14 8 C -6 -18 0 10 0 D -10 -14 -10 0 -10 E -6 -8 0 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989146 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9377: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (8) A E B D C (7) E A B D C (6) C D B E A (5) C B D E A (5) A E B C D (5) D C A B E (4) D B C E A (4) C A D B E (4) D C B E A (3) D A C E B (3) C D B A E (3) A E D B C (3) A D C E B (3) E B D C A (2) E B A C D (2) D E B C A (2) D C E B A (2) D C B A E (2) D A E B C (2) C D A B E (2) C B A D E (2) B C E D A (2) E D B C A (1) E D B A C (1) E D A B C (1) E B D A C (1) E B A D C (1) E A B C D (1) D C A E B (1) D A E C B (1) D A C B E (1) C B E D A (1) C B A E D (1) C A B E D (1) B E C D A (1) B C E A D (1) B C D E A (1) A D E B C (1) A C D E B (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 14 0 -2 14 B -14 0 -10 -18 -10 C 0 10 0 -20 8 D 2 18 20 0 22 E -14 10 -8 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998672 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 0 -2 14 B -14 0 -10 -18 -10 C 0 10 0 -20 8 D 2 18 20 0 22 E -14 10 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=25 C=24 E=16 B=5 so B is eliminated. Round 2 votes counts: A=30 C=28 D=25 E=17 so E is eliminated. Round 3 votes counts: A=40 D=31 C=29 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:231 A:213 C:199 E:183 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 0 -2 14 B -14 0 -10 -18 -10 C 0 10 0 -20 8 D 2 18 20 0 22 E -14 10 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 -2 14 B -14 0 -10 -18 -10 C 0 10 0 -20 8 D 2 18 20 0 22 E -14 10 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 -2 14 B -14 0 -10 -18 -10 C 0 10 0 -20 8 D 2 18 20 0 22 E -14 10 -8 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999977386 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9378: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (10) C E D A B (9) C E D B A (8) E D C B A (7) E D B A C (7) C A B D E (7) E C D B A (6) D E B A C (6) A B C D E (6) D B A E C (5) A B D E C (5) C A B E D (3) A B D C E (3) D A B E C (2) C E A B D (2) C D A B E (2) C A E B D (2) D E C A B (1) C E B D A (1) C E B A D (1) C D E A B (1) C B E A D (1) B D A E C (1) B A E D C (1) B A D C E (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -12 -2 -12 0 B 12 0 -4 -8 -2 C 2 4 0 2 -2 D 12 8 -2 0 2 E 0 2 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 -12 -2 -12 0 B 12 0 -4 -8 -2 C 2 4 0 2 -2 D 12 8 -2 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 E=20 A=15 D=14 B=14 so D is eliminated. Round 2 votes counts: C=37 E=27 B=19 A=17 so A is eliminated. Round 3 votes counts: C=38 B=35 E=27 so E is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:203 E:201 B:199 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D E , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 -12 0 B 12 0 -4 -8 -2 C 2 4 0 2 -2 D 12 8 -2 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 -12 0 B 12 0 -4 -8 -2 C 2 4 0 2 -2 D 12 8 -2 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 -12 0 B 12 0 -4 -8 -2 C 2 4 0 2 -2 D 12 8 -2 0 2 E 0 2 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9379: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) A E D C B (10) B D E C A (7) B C D E A (7) D B E C A (6) C B A D E (6) A E C D B (6) A C B E D (6) E D A B C (4) A C E D B (4) E A D B C (3) D E B C A (3) B D C E A (3) B D A C E (3) E A D C B (2) C B D A E (2) C A B E D (2) A C B D E (2) E D B C A (1) E D B A C (1) E C B D A (1) E C A D B (1) D E A B C (1) D B E A C (1) C E B D A (1) C A B D E (1) B C A D E (1) A D E B C (1) A C E B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 16 -4 -6 B 8 0 8 -12 -2 C -16 -8 0 -16 -18 D 4 12 16 0 14 E 6 2 18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 16 -4 -6 B 8 0 8 -12 -2 C -16 -8 0 -16 -18 D 4 12 16 0 14 E 6 2 18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=22 B=21 E=13 C=12 so C is eliminated. Round 2 votes counts: A=35 B=29 D=22 E=14 so E is eliminated. Round 3 votes counts: A=41 B=31 D=28 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:223 E:206 B:201 A:199 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 16 -4 -6 B 8 0 8 -12 -2 C -16 -8 0 -16 -18 D 4 12 16 0 14 E 6 2 18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 16 -4 -6 B 8 0 8 -12 -2 C -16 -8 0 -16 -18 D 4 12 16 0 14 E 6 2 18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 16 -4 -6 B 8 0 8 -12 -2 C -16 -8 0 -16 -18 D 4 12 16 0 14 E 6 2 18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9380: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (8) A D B C E (6) E C D A B (4) D E C A B (4) B E C A D (4) B C A E D (4) B A C E D (4) B A C D E (4) E B D C A (3) D A C E B (3) C A B E D (3) A D C B E (3) A C D B E (3) E D C A B (2) E C D B A (2) E C B D A (2) E C B A D (2) E C A D B (2) C D A E B (2) C A E B D (2) A C D E B (2) E D B C A (1) E B C A D (1) D E C B A (1) D E B A C (1) D E A C B (1) D C E A B (1) D C A E B (1) D B E A C (1) D B A C E (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B A D (1) C E A D B (1) C E A B D (1) C D E A B (1) C B E A D (1) C A E D B (1) C A D E B (1) B E D C A (1) B E D A C (1) B E A C D (1) B D E A C (1) B C E A D (1) B A E C D (1) B A D E C (1) A D C E B (1) A C B E D (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 4 -2 26 14 B -4 0 -2 2 10 C 2 2 0 10 24 D -26 -2 -10 0 4 E -14 -10 -24 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 26 14 B -4 0 -2 2 10 C 2 2 0 10 24 D -26 -2 -10 0 4 E -14 -10 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=19 A=19 D=16 C=15 so C is eliminated. Round 2 votes counts: B=32 A=26 E=23 D=19 so D is eliminated. Round 3 votes counts: B=34 A=34 E=32 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:219 B:203 D:183 E:174 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 26 14 B -4 0 -2 2 10 C 2 2 0 10 24 D -26 -2 -10 0 4 E -14 -10 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 26 14 B -4 0 -2 2 10 C 2 2 0 10 24 D -26 -2 -10 0 4 E -14 -10 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 26 14 B -4 0 -2 2 10 C 2 2 0 10 24 D -26 -2 -10 0 4 E -14 -10 -24 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9381: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (7) A E C D B (6) A C D E B (6) E B A C D (5) B E D C A (5) B D C E A (5) D C B A E (4) D C A E B (4) D C A B E (4) A C E D B (4) E B D A C (3) E B A D C (3) E A D B C (3) E A B C D (3) E D A B C (2) E A D C B (2) E A C B D (2) D E B A C (2) D A E C B (2) D A C E B (2) C D B A E (2) C A D B E (2) C A B D E (2) B E D A C (2) B D E C A (2) E D B A C (1) E A C D B (1) D E B C A (1) D E A C B (1) D B E C A (1) D B C E A (1) D B C A E (1) C A B E D (1) B E C A D (1) B C E A D (1) B C A E D (1) B C A D E (1) A E D C B (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 16 8 -4 6 B -16 0 -12 -20 -12 C -8 12 0 -4 0 D 4 20 4 0 0 E -6 12 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.764322 E: 0.235678 Sum of squares = 0.639732523185 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.764322 E: 1.000000 A B C D E A 0 16 8 -4 6 B -16 0 -12 -20 -12 C -8 12 0 -4 0 D 4 20 4 0 0 E -6 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000027083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=23 A=20 B=18 C=14 so C is eliminated. Round 2 votes counts: D=32 E=25 A=25 B=18 so B is eliminated. Round 3 votes counts: D=39 E=34 A=27 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:213 E:203 C:200 B:170 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 8 -4 6 B -16 0 -12 -20 -12 C -8 12 0 -4 0 D 4 20 4 0 0 E -6 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000027083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 8 -4 6 B -16 0 -12 -20 -12 C -8 12 0 -4 0 D 4 20 4 0 0 E -6 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000027083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 8 -4 6 B -16 0 -12 -20 -12 C -8 12 0 -4 0 D 4 20 4 0 0 E -6 12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 0.400000 Sum of squares = 0.520000027083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.600000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9382: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (10) E D A C B (9) D E B C A (6) D E A B C (5) E D C A B (4) E C B D A (4) C B E A D (4) C B A E D (4) B C A D E (4) A D B C E (4) A B C D E (4) E A D C B (3) D B E C A (3) B C D A E (3) B A C D E (3) A C B E D (3) D B E A C (2) C B E D A (2) B D C A E (2) B C A E D (2) A C B D E (2) E D B C A (1) E C D B A (1) E C D A B (1) E C A D B (1) E C A B D (1) E A C D B (1) D B C E A (1) D A B E C (1) C E B A D (1) C E A B D (1) B D A C E (1) B C E D A (1) B C D E A (1) A E C B D (1) A D E C B (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -16 -16 -26 B 12 0 -10 -8 -4 C 16 10 0 -8 -10 D 16 8 8 0 -12 E 26 4 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -16 -16 -26 B 12 0 -10 -8 -4 C 16 10 0 -8 -10 D 16 8 8 0 -12 E 26 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 D=18 B=17 A=17 C=12 so C is eliminated. Round 2 votes counts: E=38 B=27 D=18 A=17 so A is eliminated. Round 3 votes counts: E=40 B=37 D=23 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:226 D:210 C:204 B:195 A:165 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -16 -16 -26 B 12 0 -10 -8 -4 C 16 10 0 -8 -10 D 16 8 8 0 -12 E 26 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -16 -26 B 12 0 -10 -8 -4 C 16 10 0 -8 -10 D 16 8 8 0 -12 E 26 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -16 -26 B 12 0 -10 -8 -4 C 16 10 0 -8 -10 D 16 8 8 0 -12 E 26 4 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998033 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9383: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) B A C E D (7) A B C E D (7) D E C B A (6) D E C A B (6) C A B E D (6) A C B E D (6) D E B C A (5) E D B C A (3) C E B A D (3) A C B D E (3) E D C B A (2) E C B D A (2) D A E C B (2) D A C E B (2) D A C B E (2) C D E A B (2) C B A E D (2) C A D B E (2) B E C A D (2) B A E D C (2) A D C B E (2) A C D B E (2) E C B A D (1) E B D A C (1) D E A C B (1) D E A B C (1) D B A E C (1) D A E B C (1) D A B E C (1) C D A E B (1) C A D E B (1) C A B D E (1) B E D A C (1) B E A D C (1) B C E A D (1) B C A E D (1) B A E C D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 6 10 B -2 0 -10 0 6 C -6 10 0 4 6 D -6 0 -4 0 2 E -10 -6 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 6 10 B -2 0 -10 0 6 C -6 10 0 4 6 D -6 0 -4 0 2 E -10 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 A=22 C=18 B=16 E=9 so E is eliminated. Round 2 votes counts: D=40 A=22 C=21 B=17 so B is eliminated. Round 3 votes counts: D=42 A=33 C=25 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 C:207 B:197 D:196 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 6 10 B -2 0 -10 0 6 C -6 10 0 4 6 D -6 0 -4 0 2 E -10 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 6 10 B -2 0 -10 0 6 C -6 10 0 4 6 D -6 0 -4 0 2 E -10 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 6 10 B -2 0 -10 0 6 C -6 10 0 4 6 D -6 0 -4 0 2 E -10 -6 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999881 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9384: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) D B C E A (7) B D C E A (7) A C E D B (7) D C B A E (6) A E C B D (6) E A B C D (5) B E A D C (5) A E C D B (5) C A E D B (4) B D E C A (4) A E B D C (4) A E B C D (4) E B A C D (3) E A C B D (3) D C A B E (2) D B C A E (2) C E B A D (2) C D B E A (2) C D A B E (2) B E D A C (2) E C A B D (1) D A B C E (1) C D B A E (1) C B D E A (1) C A D E B (1) B E C A D (1) B E A C D (1) B D A E C (1) A E D B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -2 8 -4 B 6 0 -4 0 6 C 2 4 0 0 8 D -8 0 0 0 -8 E 4 -6 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.872536 D: 0.127464 E: 0.000000 Sum of squares = 0.777566064301 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.872536 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -2 8 -4 B 6 0 -4 0 6 C 2 4 0 0 8 D -8 0 0 0 -8 E 4 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000049978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=26 B=21 C=13 E=12 so E is eliminated. Round 2 votes counts: A=36 D=26 B=24 C=14 so C is eliminated. Round 3 votes counts: A=42 D=31 B=27 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:207 B:204 E:199 A:198 D:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -2 8 -4 B 6 0 -4 0 6 C 2 4 0 0 8 D -8 0 0 0 -8 E 4 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000049978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 8 -4 B 6 0 -4 0 6 C 2 4 0 0 8 D -8 0 0 0 -8 E 4 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000049978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 8 -4 B 6 0 -4 0 6 C 2 4 0 0 8 D -8 0 0 0 -8 E 4 -6 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.800000 D: 0.200000 E: 0.000000 Sum of squares = 0.680000049978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.800000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9385: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (18) D A E C B (8) E A B C D (6) C B A E D (6) B C E A D (6) D E A C B (5) D B C E A (5) D C B A E (4) B C A E D (4) A E C B D (4) E A D B C (3) C A B E D (3) E A B D C (2) D B E A C (2) D A C E B (2) B E A C D (2) B D C E A (2) B C D E A (2) A E D C B (2) E B A C D (1) E A C B D (1) D E B A C (1) D C A E B (1) D B E C A (1) C B D A E (1) C A E B D (1) B E C D A (1) B E C A D (1) B D E C A (1) B C E D A (1) B C D A E (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 16 18 -12 -22 B -16 0 20 -8 -14 C -18 -20 0 -16 -20 D 12 8 16 0 10 E 22 14 20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 18 -12 -22 B -16 0 20 -8 -14 C -18 -20 0 -16 -20 D 12 8 16 0 10 E 22 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=47 B=21 E=13 C=11 A=8 so A is eliminated. Round 2 votes counts: D=48 B=21 E=20 C=11 so C is eliminated. Round 3 votes counts: D=48 B=31 E=21 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:223 E:223 A:200 B:191 C:163 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 18 -12 -22 B -16 0 20 -8 -14 C -18 -20 0 -16 -20 D 12 8 16 0 10 E 22 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 18 -12 -22 B -16 0 20 -8 -14 C -18 -20 0 -16 -20 D 12 8 16 0 10 E 22 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 18 -12 -22 B -16 0 20 -8 -14 C -18 -20 0 -16 -20 D 12 8 16 0 10 E 22 14 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9386: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (6) C D E B A (6) A E B D C (5) E D A C B (4) E C A D B (4) D E C A B (4) D E A B C (4) C B A E D (4) B A D E C (4) E D C A B (3) E C D A B (3) C E D A B (3) C B D A E (3) B C D A E (3) B C A D E (3) A E D B C (3) E D A B C (2) D B C A E (2) C D E A B (2) C D B E A (2) B D A C E (2) B A E D C (2) B A D C E (2) A B E D C (2) E A C D B (1) E A C B D (1) D E A C B (1) D C E B A (1) D C B E A (1) D B A E C (1) D A E B C (1) C E D B A (1) C E B D A (1) C E B A D (1) C E A D B (1) C E A B D (1) C B E D A (1) C B D E A (1) C B A D E (1) C A E B D (1) B D C A E (1) B C A E D (1) B A C D E (1) A E B C D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -10 -6 -10 B -10 0 -4 -12 -22 C 10 4 0 -4 -8 D 6 12 4 0 -6 E 10 22 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 10 -10 -6 -10 B -10 0 -4 -12 -22 C 10 4 0 -4 -8 D 6 12 4 0 -6 E 10 22 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=24 B=19 D=15 A=13 so A is eliminated. Round 2 votes counts: E=33 C=29 B=23 D=15 so D is eliminated. Round 3 votes counts: E=43 C=31 B=26 so B is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 D:208 C:201 A:192 B:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 -10 -6 -10 B -10 0 -4 -12 -22 C 10 4 0 -4 -8 D 6 12 4 0 -6 E 10 22 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -10 -6 -10 B -10 0 -4 -12 -22 C 10 4 0 -4 -8 D 6 12 4 0 -6 E 10 22 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -10 -6 -10 B -10 0 -4 -12 -22 C 10 4 0 -4 -8 D 6 12 4 0 -6 E 10 22 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999443 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9387: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (14) E B C A D (12) A D C E B (11) B E C D A (9) A D E B C (9) C B E D A (8) E B A C D (7) E B A D C (6) C D B E A (6) D C A B E (4) C D A B E (4) B E C A D (3) A E B D C (3) E B C D A (1) C D B A E (1) B E A D C (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 4 6 -6 B 8 0 2 0 0 C -4 -2 0 2 -4 D -6 0 -2 0 0 E 6 0 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.330351 C: 0.000000 D: 0.358566 E: 0.311083 Sum of squares = 0.33447399425 Cumulative probabilities = A: 0.000000 B: 0.330351 C: 0.330351 D: 0.688917 E: 1.000000 A B C D E A 0 -8 4 6 -6 B 8 0 2 0 0 C -4 -2 0 2 -4 D -6 0 -2 0 0 E 6 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=24 C=19 D=18 B=13 so B is eliminated. Round 2 votes counts: E=39 A=24 C=19 D=18 so D is eliminated. Round 3 votes counts: E=39 A=38 C=23 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:205 E:205 A:198 C:196 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 6 -6 B 8 0 2 0 0 C -4 -2 0 2 -4 D -6 0 -2 0 0 E 6 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 6 -6 B 8 0 2 0 0 C -4 -2 0 2 -4 D -6 0 -2 0 0 E 6 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 6 -6 B 8 0 2 0 0 C -4 -2 0 2 -4 D -6 0 -2 0 0 E 6 0 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9388: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) A D B C E (7) E C B D A (5) E C B A D (5) C E B D A (5) C D E A B (5) B A E C D (5) D A C E B (4) C E D B A (4) C E D A B (4) B A D E C (4) A B D E C (4) E B C A D (3) D C A E B (3) D A C B E (3) B E C A D (3) B A E D C (3) D C E A B (2) D A B E C (2) D A B C E (2) C E A B D (2) C A B E D (2) B D A E C (2) B A C E D (2) A D C B E (2) A B D C E (2) E D C B A (1) E B C D A (1) D E C B A (1) C E B A D (1) C D A E B (1) C B E A D (1) B E D A C (1) A D B E C (1) Total count = 100 A B C D E A 0 -8 -12 -6 -2 B 8 0 -16 2 -8 C 12 16 0 12 4 D 6 -2 -12 0 -10 E 2 8 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998538 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -6 -2 B 8 0 -16 2 -8 C 12 16 0 12 4 D 6 -2 -12 0 -10 E 2 8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 E=22 B=20 D=17 A=16 so A is eliminated. Round 2 votes counts: D=27 B=26 C=25 E=22 so E is eliminated. Round 3 votes counts: C=42 B=30 D=28 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:222 E:208 B:193 D:191 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -6 -2 B 8 0 -16 2 -8 C 12 16 0 12 4 D 6 -2 -12 0 -10 E 2 8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -6 -2 B 8 0 -16 2 -8 C 12 16 0 12 4 D 6 -2 -12 0 -10 E 2 8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -6 -2 B 8 0 -16 2 -8 C 12 16 0 12 4 D 6 -2 -12 0 -10 E 2 8 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999333 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9389: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (10) D B A E C (9) B D A C E (9) E C A B D (6) C A E B D (6) A D B C E (6) B D E C A (5) B D A E C (5) E C A D B (4) C E A B D (4) C A E D B (4) E C B A D (3) D B E A C (2) D B A C E (2) D A B E C (2) D A B C E (2) C E A D B (2) B D E A C (2) B D C A E (2) A C E D B (2) E D B A C (1) E C D B A (1) E B D C A (1) E A C D B (1) D E A B C (1) C A B E D (1) B E D C A (1) B C D E A (1) A D C E B (1) A D C B E (1) A C E B D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -2 -12 10 B 8 0 4 18 4 C 2 -4 0 -6 -8 D 12 -18 6 0 4 E -10 -4 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998814 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -12 10 B 8 0 4 18 4 C 2 -4 0 -6 -8 D 12 -18 6 0 4 E -10 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=25 D=18 C=17 A=13 so A is eliminated. Round 2 votes counts: E=27 D=26 B=26 C=21 so C is eliminated. Round 3 votes counts: E=46 B=28 D=26 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:202 E:195 A:194 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -12 10 B 8 0 4 18 4 C 2 -4 0 -6 -8 D 12 -18 6 0 4 E -10 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -12 10 B 8 0 4 18 4 C 2 -4 0 -6 -8 D 12 -18 6 0 4 E -10 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -12 10 B 8 0 4 18 4 C 2 -4 0 -6 -8 D 12 -18 6 0 4 E -10 -4 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9390: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) B C D A E (7) C D A B E (6) B E A D C (6) B E A C D (6) E A D B C (4) E A B D C (4) D C A E B (4) D A C E B (4) B A E D C (4) C D B A E (3) C D A E B (3) B A E C D (3) A D E C B (3) A D C E B (3) D C E A B (2) C D E B A (2) C D E A B (2) C B D E A (2) B E C D A (2) B C D E A (2) B C A E D (2) B C A D E (2) A E B D C (2) A B E D C (2) E D C A B (1) E C D B A (1) E A D C B (1) C D B E A (1) C B D A E (1) B C E D A (1) B C E A D (1) B A C E D (1) B A C D E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -14 10 12 6 B 14 0 20 16 6 C -10 -20 0 -2 0 D -12 -16 2 0 -4 E -6 -6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 12 6 B 14 0 20 16 6 C -10 -20 0 -2 0 D -12 -16 2 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 E=20 C=20 A=12 D=10 so D is eliminated. Round 2 votes counts: B=38 C=26 E=20 A=16 so A is eliminated. Round 3 votes counts: B=40 C=33 E=27 so E is eliminated. Round 4 votes counts: B=60 C=40 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:207 E:196 D:185 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 12 6 B 14 0 20 16 6 C -10 -20 0 -2 0 D -12 -16 2 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 12 6 B 14 0 20 16 6 C -10 -20 0 -2 0 D -12 -16 2 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 12 6 B 14 0 20 16 6 C -10 -20 0 -2 0 D -12 -16 2 0 -4 E -6 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999546 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9391: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (10) A E C B D (7) A C E D B (7) D C B E A (6) C A E D B (6) A B D C E (5) B D C E A (4) A E B C D (4) A B D E C (4) E C A D B (3) D B C E A (3) C E A D B (3) C D E B A (3) A E C D B (3) A B E D C (3) E C D B A (2) E C B D A (2) E A B C D (2) C E D A B (2) B E D C A (2) B D E A C (2) A C D E B (2) E D C B A (1) E C A B D (1) E A C D B (1) E A C B D (1) E A B D C (1) D C B A E (1) D B A C E (1) D A B C E (1) C E D B A (1) C D E A B (1) C A D E B (1) B D A C E (1) B A E D C (1) B A D E C (1) A C D B E (1) Total count = 100 A B C D E A 0 18 -4 14 -2 B -18 0 -10 2 -8 C 4 10 0 6 -2 D -14 -2 -6 0 -6 E 2 8 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 -4 14 -2 B -18 0 -10 2 -8 C 4 10 0 6 -2 D -14 -2 -6 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=21 C=17 E=14 D=12 so D is eliminated. Round 2 votes counts: A=37 B=25 C=24 E=14 so E is eliminated. Round 3 votes counts: A=42 C=33 B=25 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:209 E:209 D:186 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -4 14 -2 B -18 0 -10 2 -8 C 4 10 0 6 -2 D -14 -2 -6 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 14 -2 B -18 0 -10 2 -8 C 4 10 0 6 -2 D -14 -2 -6 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 14 -2 B -18 0 -10 2 -8 C 4 10 0 6 -2 D -14 -2 -6 0 -6 E 2 8 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999487 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9392: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (12) B C A E D (11) D E A C B (10) C B A D E (10) C B D E A (8) A E D B C (7) D E C B A (6) C B D A E (6) B A C E D (4) A B E D C (4) A B C E D (4) E A D B C (3) A B E C D (3) D E C A B (2) C D E B A (2) C D B E A (2) A E B D C (2) D E A B C (1) D C E B A (1) B E A C D (1) B C E D A (1) Total count = 100 A B C D E A 0 -4 2 -2 2 B 4 0 6 8 8 C -2 -6 0 4 -2 D 2 -8 -4 0 -4 E -2 -8 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 -2 2 B 4 0 6 8 8 C -2 -6 0 4 -2 D 2 -8 -4 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=20 A=20 B=17 E=15 so E is eliminated. Round 2 votes counts: D=32 C=28 A=23 B=17 so B is eliminated. Round 3 votes counts: C=40 D=32 A=28 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:213 A:199 E:198 C:197 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 2 -2 2 B 4 0 6 8 8 C -2 -6 0 4 -2 D 2 -8 -4 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 2 B 4 0 6 8 8 C -2 -6 0 4 -2 D 2 -8 -4 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 2 B 4 0 6 8 8 C -2 -6 0 4 -2 D 2 -8 -4 0 -4 E -2 -8 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9393: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (5) A E B D C (5) E D A B C (4) E A D B C (4) D E B A C (4) C D B E A (4) B D A C E (4) E C D B A (3) D B C E A (3) D B C A E (3) C D E B A (3) C A E B D (3) B A D E C (3) A B E D C (3) E D C B A (2) E D C A B (2) E D A C B (2) E C D A B (2) E C A D B (2) E A C D B (2) E A B D C (2) D E C B A (2) D B A E C (2) C A B E D (2) B C D A E (2) B A C D E (2) A E C B D (2) A B E C D (2) A B C E D (2) A B C D E (2) E D B A C (1) E A D C B (1) E A C B D (1) D E B C A (1) D B E C A (1) D B E A C (1) C E D B A (1) C E D A B (1) C E A B D (1) C D B A E (1) C B D E A (1) C B A D E (1) B D C A E (1) B D A E C (1) B A D C E (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 6 -14 -12 B 6 0 10 -14 -14 C -6 -10 0 -16 -12 D 14 14 16 0 -4 E 12 14 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999054 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 6 -14 -12 B 6 0 10 -14 -14 C -6 -10 0 -16 -12 D 14 14 16 0 -4 E 12 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=22 C=18 A=18 B=14 so B is eliminated. Round 2 votes counts: E=28 D=28 A=24 C=20 so C is eliminated. Round 3 votes counts: D=39 E=31 A=30 so A is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 D:220 B:194 A:187 C:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 6 -14 -12 B 6 0 10 -14 -14 C -6 -10 0 -16 -12 D 14 14 16 0 -4 E 12 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 6 -14 -12 B 6 0 10 -14 -14 C -6 -10 0 -16 -12 D 14 14 16 0 -4 E 12 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 6 -14 -12 B 6 0 10 -14 -14 C -6 -10 0 -16 -12 D 14 14 16 0 -4 E 12 14 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9394: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (13) A C E D B (11) D B E C A (7) A C E B D (6) B D A E C (5) C A E D B (4) B D E A C (4) A C B E D (4) E B C A D (3) D B A C E (3) C E A D B (3) B E D C A (3) A C B D E (3) D E B C A (2) D C E A B (2) D A B C E (2) B D A C E (2) B A E C D (2) B A D C E (2) A B C E D (2) E D C A B (1) E C D B A (1) E C A D B (1) E B D C A (1) D E C B A (1) D B C E A (1) D B A E C (1) D A C E B (1) D A C B E (1) C E D A B (1) C D A E B (1) C A E B D (1) B E A C D (1) B A E D C (1) B A C E D (1) A E C B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 8 -6 10 B 8 0 12 10 18 C -8 -12 0 -6 4 D 6 -10 6 0 4 E -10 -18 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -6 10 B 8 0 12 10 18 C -8 -12 0 -6 4 D 6 -10 6 0 4 E -10 -18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=28 D=21 C=10 E=7 so E is eliminated. Round 2 votes counts: B=38 A=28 D=22 C=12 so C is eliminated. Round 3 votes counts: B=38 A=37 D=25 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:224 D:203 A:202 C:189 E:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -6 10 B 8 0 12 10 18 C -8 -12 0 -6 4 D 6 -10 6 0 4 E -10 -18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -6 10 B 8 0 12 10 18 C -8 -12 0 -6 4 D 6 -10 6 0 4 E -10 -18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -6 10 B 8 0 12 10 18 C -8 -12 0 -6 4 D 6 -10 6 0 4 E -10 -18 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9395: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (12) B A E D C (11) E A B C D (8) D C B A E (8) B A E C D (8) D C E B A (6) C D E A B (6) C D A E B (5) D C B E A (4) A E B C D (4) D B E A C (2) C D A B E (2) C A E B D (2) B D A E C (2) B A D E C (2) A B E C D (2) E D A C B (1) E B A D C (1) E A C B D (1) E A B D C (1) D E C B A (1) D C A B E (1) D B C A E (1) D B A C E (1) C E A D B (1) C E A B D (1) C A E D B (1) C A B D E (1) B D E A C (1) B C D A E (1) B A C E D (1) B A C D E (1) Total count = 100 A B C D E A 0 -2 -6 -8 8 B 2 0 -6 -4 -2 C 6 6 0 -10 10 D 8 4 10 0 14 E -8 2 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -8 8 B 2 0 -6 -4 -2 C 6 6 0 -10 10 D 8 4 10 0 14 E -8 2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=27 C=19 E=12 A=6 so A is eliminated. Round 2 votes counts: D=36 B=29 C=19 E=16 so E is eliminated. Round 3 votes counts: B=43 D=37 C=20 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:218 C:206 A:196 B:195 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -8 8 B 2 0 -6 -4 -2 C 6 6 0 -10 10 D 8 4 10 0 14 E -8 2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -8 8 B 2 0 -6 -4 -2 C 6 6 0 -10 10 D 8 4 10 0 14 E -8 2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -8 8 B 2 0 -6 -4 -2 C 6 6 0 -10 10 D 8 4 10 0 14 E -8 2 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9396: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (12) C A E D B (8) D B C E A (6) E A C D B (5) D C B A E (4) D B E C A (4) A E C B D (4) E B A D C (3) C A E B D (3) B D C E A (3) B D C A E (3) A C E D B (3) E D A C B (2) E A C B D (2) D E B A C (2) D C E A B (2) C A B D E (2) B E D A C (2) B D E C A (2) B C A D E (2) A C E B D (2) E D A B C (1) E B D A C (1) E A D C B (1) E A B D C (1) E A B C D (1) D E A B C (1) D C B E A (1) D C A B E (1) D B E A C (1) D B C A E (1) C D E A B (1) C D A E B (1) C B D A E (1) C A D E B (1) C A D B E (1) C A B E D (1) B E A D C (1) B E A C D (1) B D A C E (1) B C A E D (1) B A E D C (1) B A C E D (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 2 -6 -12 B 8 0 6 4 8 C -2 -6 0 -14 2 D 6 -4 14 0 6 E 12 -8 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999654 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -6 -12 B 8 0 6 4 8 C -2 -6 0 -14 2 D 6 -4 14 0 6 E 12 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 C=19 E=17 A=11 so A is eliminated. Round 2 votes counts: B=31 C=24 D=23 E=22 so E is eliminated. Round 3 votes counts: B=37 C=36 D=27 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:213 D:211 E:198 C:190 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -6 -12 B 8 0 6 4 8 C -2 -6 0 -14 2 D 6 -4 14 0 6 E 12 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -6 -12 B 8 0 6 4 8 C -2 -6 0 -14 2 D 6 -4 14 0 6 E 12 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -6 -12 B 8 0 6 4 8 C -2 -6 0 -14 2 D 6 -4 14 0 6 E 12 -8 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999485 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9397: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (13) C D E B A (10) D C E A B (9) A D B C E (7) A B E D C (7) A B D E C (7) E C B D A (6) B E A C D (6) D A C B E (5) C E D B A (5) B A E C D (5) E B C A D (4) B E C A D (4) A D B E C (3) A B E C D (3) E C B A D (1) E A B C D (1) D C A E B (1) D A C E B (1) D A B C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -10 -6 -2 -18 B 10 0 -2 -10 -2 C 6 2 0 -8 6 D 2 10 8 0 14 E 18 2 -6 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -2 -18 B 10 0 -2 -10 -2 C 6 2 0 -8 6 D 2 10 8 0 14 E 18 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=27 B=16 C=15 E=12 so E is eliminated. Round 2 votes counts: D=30 A=28 C=22 B=20 so B is eliminated. Round 3 votes counts: A=40 D=30 C=30 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:203 E:200 B:198 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -6 -2 -18 B 10 0 -2 -10 -2 C 6 2 0 -8 6 D 2 10 8 0 14 E 18 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -2 -18 B 10 0 -2 -10 -2 C 6 2 0 -8 6 D 2 10 8 0 14 E 18 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -2 -18 B 10 0 -2 -10 -2 C 6 2 0 -8 6 D 2 10 8 0 14 E 18 2 -6 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998549 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9398: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E B D A C (5) E D B A C (4) E A B D C (4) D B C A E (4) C E D A B (4) C A B D E (4) A C E B D (4) A C B D E (4) A B E D C (4) E C D B A (3) E C A D B (3) D B E C A (3) C D A B E (3) A B D C E (3) E D B C A (2) E A C B D (2) E A B C D (2) D E B C A (2) D B E A C (2) C D B E A (2) C A E B D (2) C A D E B (2) B D E A C (2) B D A C E (2) A E B D C (2) A B D E C (2) A B C D E (2) E D C B A (1) E C D A B (1) E C A B D (1) D E C B A (1) D B C E A (1) D B A E C (1) C E D B A (1) C E A B D (1) C D E B A (1) C A D B E (1) B E D A C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -6 -12 2 B -4 0 -2 -4 2 C 6 2 0 8 2 D 12 4 -8 0 4 E -2 -2 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999302 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -6 -12 2 B -4 0 -2 -4 2 C 6 2 0 8 2 D 12 4 -8 0 4 E -2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=28 A=22 D=14 B=5 so B is eliminated. Round 2 votes counts: C=31 E=29 A=22 D=18 so D is eliminated. Round 3 votes counts: E=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:209 D:206 B:196 E:195 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -6 -12 2 B -4 0 -2 -4 2 C 6 2 0 8 2 D 12 4 -8 0 4 E -2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -6 -12 2 B -4 0 -2 -4 2 C 6 2 0 8 2 D 12 4 -8 0 4 E -2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -6 -12 2 B -4 0 -2 -4 2 C 6 2 0 8 2 D 12 4 -8 0 4 E -2 -2 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9399: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) E C D B A (7) D B E A C (7) D B A E C (6) C E A B D (6) C A E B D (6) E C A B D (5) C E D A B (4) B A E D C (4) D C A B E (3) C A D B E (3) A C B E D (3) E A C B D (2) D E B C A (2) D B C A E (2) B D A E C (2) B D A C E (2) B A D E C (2) B A D C E (2) A C E B D (2) E D C B A (1) E D B C A (1) E B D A C (1) E B C A D (1) E B A C D (1) D E C B A (1) D C B E A (1) D B E C A (1) D A B C E (1) C E A D B (1) C D A B E (1) C A E D B (1) C A B E D (1) B E D A C (1) B E A D C (1) A E C B D (1) A C D B E (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 4 -8 12 B 12 0 0 -8 14 C -4 0 0 -4 2 D 8 8 4 0 -6 E -12 -14 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.000000 D: 0.500000 E: 0.285714 Sum of squares = 0.377551020384 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.214286 D: 0.714286 E: 1.000000 A B C D E A 0 -12 4 -8 12 B 12 0 0 -8 14 C -4 0 0 -4 2 D 8 8 4 0 -6 E -12 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.000000 D: 0.500000 E: 0.285714 Sum of squares = 0.377551019956 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.214286 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=23 E=19 B=14 A=10 so A is eliminated. Round 2 votes counts: D=34 C=29 E=20 B=17 so B is eliminated. Round 3 votes counts: D=42 C=30 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:209 D:207 A:198 C:197 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -8 12 B 12 0 0 -8 14 C -4 0 0 -4 2 D 8 8 4 0 -6 E -12 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.000000 D: 0.500000 E: 0.285714 Sum of squares = 0.377551019956 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.214286 D: 0.714286 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -8 12 B 12 0 0 -8 14 C -4 0 0 -4 2 D 8 8 4 0 -6 E -12 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.000000 D: 0.500000 E: 0.285714 Sum of squares = 0.377551019956 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.214286 D: 0.714286 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -8 12 B 12 0 0 -8 14 C -4 0 0 -4 2 D 8 8 4 0 -6 E -12 -14 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.214286 C: 0.000000 D: 0.500000 E: 0.285714 Sum of squares = 0.377551019956 Cumulative probabilities = A: 0.000000 B: 0.214286 C: 0.214286 D: 0.714286 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9400: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (8) C E B D A (7) E C B D A (6) E C B A D (5) D A C E B (5) C E A B D (5) A D B C E (5) B E C D A (4) B D E C A (4) B D A E C (4) A C D E B (4) D A C B E (3) B E C A D (3) A D B E C (3) A C E D B (3) D A B E C (2) D A B C E (2) C E B A D (2) B E D C A (2) B E A C D (2) B A E D C (2) E C D B A (1) E B C D A (1) E B C A D (1) D E C B A (1) D C E B A (1) D C A E B (1) D B E C A (1) D B E A C (1) C E D B A (1) C E D A B (1) C D E A B (1) C D A E B (1) C A E D B (1) B A D E C (1) A C E B D (1) A B E C D (1) A B D E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 0 0 0 B 0 0 -16 8 -14 C 0 16 0 4 8 D 0 -8 -4 0 0 E 0 14 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.615607 B: 0.000000 C: 0.384393 D: 0.000000 E: 0.000000 Sum of squares = 0.526730096396 Cumulative probabilities = A: 0.615607 B: 0.615607 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 0 0 B 0 0 -16 8 -14 C 0 16 0 4 8 D 0 -8 -4 0 0 E 0 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 B=22 C=19 D=17 E=14 so E is eliminated. Round 2 votes counts: C=31 A=28 B=24 D=17 so D is eliminated. Round 3 votes counts: A=40 C=34 B=26 so B is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:214 E:203 A:200 D:194 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 0 0 B 0 0 -16 8 -14 C 0 16 0 4 8 D 0 -8 -4 0 0 E 0 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 0 0 B 0 0 -16 8 -14 C 0 16 0 4 8 D 0 -8 -4 0 0 E 0 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 0 0 B 0 0 -16 8 -14 C 0 16 0 4 8 D 0 -8 -4 0 0 E 0 14 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9401: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) D E C A B (10) B A C E D (8) A B C D E (6) E D C B A (5) C A D E B (5) A C B D E (5) B E D C A (4) B A C D E (4) A C D E B (4) C A B D E (3) B E D A C (3) E D B A C (2) E B D C A (2) D E C B A (2) D C A E B (2) C B A D E (2) C A D B E (2) B E C D A (2) B E A D C (2) B C A E D (2) B C A D E (2) B A E C D (2) E D A C B (1) E D A B C (1) E B A D C (1) D C E A B (1) C D A E B (1) B C E A D (1) B A E D C (1) A D C E B (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -14 6 4 B 12 0 10 2 2 C 14 -10 0 2 2 D -6 -2 -2 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999844 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 6 4 B 12 0 10 2 2 C 14 -10 0 2 2 D -6 -2 -2 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=23 A=18 D=15 C=13 so C is eliminated. Round 2 votes counts: B=33 A=28 E=23 D=16 so D is eliminated. Round 3 votes counts: E=36 B=33 A=31 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:213 C:204 D:196 E:195 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -14 6 4 B 12 0 10 2 2 C 14 -10 0 2 2 D -6 -2 -2 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 6 4 B 12 0 10 2 2 C 14 -10 0 2 2 D -6 -2 -2 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 6 4 B 12 0 10 2 2 C 14 -10 0 2 2 D -6 -2 -2 0 2 E -4 -2 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9402: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (6) D E C A B (5) C D B A E (5) C B E A D (5) D E C B A (4) D E A C B (4) D C E B A (4) A B C E D (4) A B C D E (4) E D C B A (3) E D A B C (3) E B C A D (3) D A E B C (3) D A B C E (3) B C A E D (3) A D B C E (3) A B D C E (3) E D C A B (2) E A B C D (2) D C E A B (2) D C B A E (2) D C A B E (2) C E D B A (2) C B D A E (2) A B E D C (2) A B E C D (2) E C D B A (1) E B A C D (1) E A D B C (1) E A B D C (1) D E A B C (1) D C B E A (1) D A E C B (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B E A (1) C B E D A (1) C B A E D (1) B A C E D (1) A E B D C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -4 -20 0 B -14 0 -14 -20 4 C 4 14 0 -16 16 D 20 20 16 0 16 E 0 -4 -16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -4 -20 0 B -14 0 -14 -20 4 C 4 14 0 -16 16 D 20 20 16 0 16 E 0 -4 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=21 C=20 E=17 B=4 so B is eliminated. Round 2 votes counts: D=38 C=23 A=22 E=17 so E is eliminated. Round 3 votes counts: D=46 C=27 A=27 so C is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:236 C:209 A:195 E:182 B:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -4 -20 0 B -14 0 -14 -20 4 C 4 14 0 -16 16 D 20 20 16 0 16 E 0 -4 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 -20 0 B -14 0 -14 -20 4 C 4 14 0 -16 16 D 20 20 16 0 16 E 0 -4 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 -20 0 B -14 0 -14 -20 4 C 4 14 0 -16 16 D 20 20 16 0 16 E 0 -4 -16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9403: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (10) B E A D C (6) E B C A D (5) D A C E B (5) C E B D A (5) B E A C D (5) A D B E C (5) D C A E B (4) C E D A B (4) C E B A D (4) B E C A D (4) A D B C E (4) E B D A C (3) C D A E B (3) C A D B E (3) E C B A D (2) E B D C A (2) E B A D C (2) E B A C D (2) D A E C B (2) C D A B E (2) B C A E D (2) B A E D C (2) A B D E C (2) E D A B C (1) E C B D A (1) D E C A B (1) D E A B C (1) D C E A B (1) C E D B A (1) C D E A B (1) C B E A D (1) C B A E D (1) B C E A D (1) B A D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 4 6 -6 B 0 0 -2 4 0 C -4 2 0 -6 6 D -6 -4 6 0 -8 E 6 0 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999448 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 0 4 6 -6 B 0 0 -2 4 0 C -4 2 0 -6 6 D -6 -4 6 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999989 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=24 B=21 E=18 A=12 so A is eliminated. Round 2 votes counts: D=33 C=25 B=24 E=18 so E is eliminated. Round 3 votes counts: B=38 D=34 C=28 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:204 A:202 B:201 C:199 D:194 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 4 6 -6 B 0 0 -2 4 0 C -4 2 0 -6 6 D -6 -4 6 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999989 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 6 -6 B 0 0 -2 4 0 C -4 2 0 -6 6 D -6 -4 6 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999989 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 6 -6 B 0 0 -2 4 0 C -4 2 0 -6 6 D -6 -4 6 0 -8 E 6 0 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.250000 Sum of squares = 0.343749999989 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9404: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) E A B C D (10) D C B A E (8) E B C D A (6) A E D C B (6) A E B D C (6) C D B E A (5) B C D E A (5) E B A C D (4) C B D E A (4) B E C D A (4) E C B D A (3) D A C B E (3) A E D B C (3) A D E C B (3) E A C D B (2) C D B A E (2) B D C E A (2) A E C D B (2) E B C A D (1) E A C B D (1) E A B D C (1) D B C A E (1) D A C E B (1) C D E B A (1) B D C A E (1) A D C E B (1) A D B E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 6 14 8 2 B -6 0 -6 -2 -2 C -14 6 0 0 -6 D -8 2 0 0 2 E -2 2 6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 8 2 B -6 0 -6 -2 -2 C -14 6 0 0 -6 D -8 2 0 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=28 D=13 C=12 B=12 so C is eliminated. Round 2 votes counts: A=35 E=28 D=21 B=16 so B is eliminated. Round 3 votes counts: A=35 D=33 E=32 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:215 E:202 D:198 C:193 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 8 2 B -6 0 -6 -2 -2 C -14 6 0 0 -6 D -8 2 0 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 8 2 B -6 0 -6 -2 -2 C -14 6 0 0 -6 D -8 2 0 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 8 2 B -6 0 -6 -2 -2 C -14 6 0 0 -6 D -8 2 0 0 2 E -2 2 6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9405: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (11) B E A C D (8) A C E B D (7) C D A B E (5) C A B E D (5) E B A D C (4) D C A E B (4) C D B A E (4) C D A E B (4) C A D E B (4) E A B D C (3) D C B E A (3) D C B A E (3) B E D A C (3) A E C B D (3) E B A C D (2) D E B C A (2) D E B A C (2) D E A C B (2) D B E C A (2) D A C E B (2) C A E D B (2) B E A D C (2) E B D A C (1) D E A B C (1) D B E A C (1) D A E C B (1) C B A E D (1) C B A D E (1) C A B D E (1) B E C A D (1) B D C E A (1) B C E A D (1) A E D C B (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 16 12 18 22 B -16 0 -8 12 -14 C -12 8 0 22 -4 D -18 -12 -22 0 -12 E -22 14 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 18 22 B -16 0 -8 12 -14 C -12 8 0 22 -4 D -18 -12 -22 0 -12 E -22 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=24 D=23 B=16 E=10 so E is eliminated. Round 2 votes counts: C=27 A=27 D=23 B=23 so D is eliminated. Round 3 votes counts: C=37 A=33 B=30 so B is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:234 C:207 E:204 B:187 D:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 18 22 B -16 0 -8 12 -14 C -12 8 0 22 -4 D -18 -12 -22 0 -12 E -22 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 18 22 B -16 0 -8 12 -14 C -12 8 0 22 -4 D -18 -12 -22 0 -12 E -22 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 18 22 B -16 0 -8 12 -14 C -12 8 0 22 -4 D -18 -12 -22 0 -12 E -22 14 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9406: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (9) C B D A E (8) C A D B E (7) E D A B C (6) C B A D E (6) B E C A D (6) E B A D C (5) D E A B C (4) C D A B E (4) B C E D A (4) E B D A C (3) E B A C D (3) A D C E B (3) E D B A C (2) E B D C A (2) E A D B C (2) E A B D C (2) D A E C B (2) D A E B C (2) D A C E B (2) D A C B E (2) C D B A E (2) C B E A D (2) C B A E D (2) B E C D A (2) E B C D A (1) E B C A D (1) E A C B D (1) D B C A E (1) C A B D E (1) B E D C A (1) B D E C A (1) A E C D B (1) Total count = 100 A B C D E A 0 -4 -2 2 4 B 4 0 -4 2 -2 C 2 4 0 2 -12 D -2 -2 -2 0 8 E -4 2 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.136364 B: 0.181818 C: 0.227273 D: 0.318182 E: 0.136364 Sum of squares = 0.223140495867 Cumulative probabilities = A: 0.136364 B: 0.318182 C: 0.545455 D: 0.863636 E: 1.000000 A B C D E A 0 -4 -2 2 4 B 4 0 -4 2 -2 C 2 4 0 2 -12 D -2 -2 -2 0 8 E -4 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.136364 B: 0.181818 C: 0.227273 D: 0.318182 E: 0.136364 Sum of squares = 0.223140495869 Cumulative probabilities = A: 0.136364 B: 0.318182 C: 0.545455 D: 0.863636 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=28 B=14 D=13 A=13 so D is eliminated. Round 2 votes counts: E=32 C=32 A=21 B=15 so B is eliminated. Round 3 votes counts: E=42 C=37 A=21 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:201 E:201 A:200 B:200 C:198 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -2 2 4 B 4 0 -4 2 -2 C 2 4 0 2 -12 D -2 -2 -2 0 8 E -4 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.136364 B: 0.181818 C: 0.227273 D: 0.318182 E: 0.136364 Sum of squares = 0.223140495869 Cumulative probabilities = A: 0.136364 B: 0.318182 C: 0.545455 D: 0.863636 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 2 4 B 4 0 -4 2 -2 C 2 4 0 2 -12 D -2 -2 -2 0 8 E -4 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.136364 B: 0.181818 C: 0.227273 D: 0.318182 E: 0.136364 Sum of squares = 0.223140495869 Cumulative probabilities = A: 0.136364 B: 0.318182 C: 0.545455 D: 0.863636 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 2 4 B 4 0 -4 2 -2 C 2 4 0 2 -12 D -2 -2 -2 0 8 E -4 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.136364 B: 0.181818 C: 0.227273 D: 0.318182 E: 0.136364 Sum of squares = 0.223140495869 Cumulative probabilities = A: 0.136364 B: 0.318182 C: 0.545455 D: 0.863636 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9407: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (11) C B A D E (11) E D C B A (8) D B C A E (6) A B D C E (6) E C A B D (5) E C B A D (4) E A C B D (4) E A D B C (3) D E A B C (3) D B A C E (3) C B E A D (3) A B C D E (3) E C B D A (2) E A D C B (2) E A B C D (2) D E B C A (2) D C B A E (2) D A B C E (2) C D B A E (2) C B A E D (2) E D C A B (1) E D B A C (1) E C D B A (1) D E B A C (1) D B A E C (1) D A B E C (1) C E B D A (1) C B D E A (1) C A B E D (1) C A B D E (1) B C A D E (1) A E B C D (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -8 2 -10 B 4 0 -2 -4 -6 C 8 2 0 -10 -8 D -2 4 10 0 -4 E 10 6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 2 -10 B 4 0 -2 -4 -6 C 8 2 0 -10 -8 D -2 4 10 0 -4 E 10 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=44 C=22 D=21 A=12 B=1 so B is eliminated. Round 2 votes counts: E=44 C=23 D=21 A=12 so A is eliminated. Round 3 votes counts: E=45 D=29 C=26 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:204 B:196 C:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 2 -10 B 4 0 -2 -4 -6 C 8 2 0 -10 -8 D -2 4 10 0 -4 E 10 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 2 -10 B 4 0 -2 -4 -6 C 8 2 0 -10 -8 D -2 4 10 0 -4 E 10 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 2 -10 B 4 0 -2 -4 -6 C 8 2 0 -10 -8 D -2 4 10 0 -4 E 10 6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9408: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (6) E B A D C (5) C D A B E (5) B E A D C (5) D B C E A (4) B E D A C (4) B D E C A (4) A C D E B (4) E B D A C (3) C D E B A (3) C D A E B (3) B A E D C (3) A B E C D (3) A B C E D (3) E D A B C (2) D C E B A (2) D C E A B (2) D B E C A (2) C D E A B (2) C D B A E (2) C B D A E (2) C A D E B (2) B C A E D (2) B A E C D (2) B A C E D (2) A E D C B (2) A E B D C (2) A E B C D (2) A C E D B (2) E D B A C (1) D E B C A (1) D E A C B (1) D C B E A (1) C B A D E (1) C A D B E (1) C A B D E (1) B E D C A (1) B D C E A (1) B C D E A (1) B C A D E (1) A E C D B (1) A E C B D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 -18 0 -6 -2 B 18 0 8 2 16 C 0 -8 0 8 10 D 6 -2 -8 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 0 -6 -2 B 18 0 8 2 16 C 0 -8 0 8 10 D 6 -2 -8 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987038 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=26 A=22 D=13 E=11 so E is eliminated. Round 2 votes counts: B=34 C=28 A=22 D=16 so D is eliminated. Round 3 votes counts: B=42 C=33 A=25 so A is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:205 D:200 A:187 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 0 -6 -2 B 18 0 8 2 16 C 0 -8 0 8 10 D 6 -2 -8 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987038 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 0 -6 -2 B 18 0 8 2 16 C 0 -8 0 8 10 D 6 -2 -8 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987038 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 0 -6 -2 B 18 0 8 2 16 C 0 -8 0 8 10 D 6 -2 -8 0 4 E 2 -16 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987038 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9409: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) D E A B C (7) E D A C B (5) D E B A C (5) E D A B C (4) B D C A E (4) B C D A E (4) B C A D E (4) A E C B D (4) A C E B D (4) D B E C A (3) C B A D E (3) C A B E D (3) B A C D E (3) A C B E D (3) E C A D B (2) E A D C B (2) E A D B C (2) E A C D B (2) D E B C A (2) D C B E A (2) D B E A C (2) D B C E A (2) C E D B A (2) C B D A E (2) B A C E D (2) A B C E D (2) D E C B A (1) D E A C B (1) D B A E C (1) C E D A B (1) C B E D A (1) C B D E A (1) C A E B D (1) B D A C E (1) B C A E D (1) B A D C E (1) A E C D B (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 6 -2 6 B 10 0 2 6 8 C -6 -2 0 8 10 D 2 -6 -8 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 -2 6 B 10 0 2 6 8 C -6 -2 0 8 10 D 2 -6 -8 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=22 B=20 E=17 A=15 so A is eliminated. Round 2 votes counts: C=29 D=26 B=23 E=22 so E is eliminated. Round 3 votes counts: D=39 C=38 B=23 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:213 C:205 A:200 D:193 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 -2 6 B 10 0 2 6 8 C -6 -2 0 8 10 D 2 -6 -8 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -2 6 B 10 0 2 6 8 C -6 -2 0 8 10 D 2 -6 -8 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -2 6 B 10 0 2 6 8 C -6 -2 0 8 10 D 2 -6 -8 0 -2 E -6 -8 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9410: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) B C E D A (7) E B A D C (6) B E C A D (6) E A D B C (5) D A C E B (5) C B D A E (5) E C B A D (4) D A E C B (4) D A E B C (4) A D C E B (4) D A C B E (3) C B E D A (3) C A D B E (3) A D E C B (3) E B D A C (2) E B C A D (2) E A C D B (2) D C A B E (2) C A D E B (2) B E D A C (2) B E C D A (2) B C D A E (2) A D E B C (2) D A B C E (1) C E B A D (1) C D A B E (1) C B A E D (1) B E D C A (1) B D C A E (1) B C E A D (1) B C D E A (1) A E D B C (1) Total count = 100 A B C D E A 0 -16 -12 8 -12 B 16 0 -8 16 6 C 12 8 0 8 8 D -8 -16 -8 0 -14 E 12 -6 -8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 8 -12 B 16 0 -8 16 6 C 12 8 0 8 8 D -8 -16 -8 0 -14 E 12 -6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 B=23 E=21 D=19 A=10 so A is eliminated. Round 2 votes counts: D=28 C=27 B=23 E=22 so E is eliminated. Round 3 votes counts: D=34 C=33 B=33 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:218 B:215 E:206 A:184 D:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -12 8 -12 B 16 0 -8 16 6 C 12 8 0 8 8 D -8 -16 -8 0 -14 E 12 -6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 8 -12 B 16 0 -8 16 6 C 12 8 0 8 8 D -8 -16 -8 0 -14 E 12 -6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 8 -12 B 16 0 -8 16 6 C 12 8 0 8 8 D -8 -16 -8 0 -14 E 12 -6 -8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9411: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (12) E D C A B (9) C A B D E (9) B A C D E (7) E C A B D (5) D E B A C (5) E C D A B (4) D E C A B (4) D B A C E (4) C A B E D (4) E D B C A (3) E B A C D (3) A B C D E (3) E D C B A (2) E B A D C (2) D B A E C (2) C A D B E (2) B A E C D (2) B A C E D (2) A C B D E (2) E B D A C (1) E B C D A (1) D C A B E (1) D A C B E (1) D A B C E (1) C E A B D (1) C D E A B (1) C D A E B (1) C D A B E (1) B D A E C (1) B A D E C (1) B A D C E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 4 -8 -6 B -2 0 6 -6 -8 C -4 -6 0 -2 -14 D 8 6 2 0 -4 E 6 8 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 -8 -6 B -2 0 6 -6 -8 C -4 -6 0 -2 -14 D 8 6 2 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 C=19 D=18 B=14 A=7 so A is eliminated. Round 2 votes counts: E=42 C=21 B=19 D=18 so D is eliminated. Round 3 votes counts: E=51 B=26 C=23 so C is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:216 D:206 A:196 B:195 C:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 -8 -6 B -2 0 6 -6 -8 C -4 -6 0 -2 -14 D 8 6 2 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -8 -6 B -2 0 6 -6 -8 C -4 -6 0 -2 -14 D 8 6 2 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -8 -6 B -2 0 6 -6 -8 C -4 -6 0 -2 -14 D 8 6 2 0 -4 E 6 8 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999864 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9412: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (9) C D B E A (6) C B D E A (5) A D C E B (5) D E B C A (4) B C E D A (4) A D E C B (4) E D A B C (3) E B D C A (3) D E A B C (3) C A B D E (3) B C D E A (3) E B D A C (2) E A D B C (2) D B E C A (2) D A E C B (2) C B A E D (2) C A D B E (2) B E C A D (2) B C E A D (2) A E D C B (2) A E B D C (2) A E B C D (2) A C E B D (2) A C D E B (2) A C D B E (2) A B E C D (2) E D B A C (1) E B A D C (1) D E A C B (1) D C E B A (1) D C E A B (1) D C B E A (1) D C A E B (1) D C A B E (1) D A E B C (1) C D A B E (1) B E D C A (1) B E C D A (1) B E A C D (1) B D E C A (1) B C A E D (1) A E C B D (1) A D E B C (1) A B C E D (1) Total count = 100 A B C D E A 0 12 4 2 -2 B -12 0 10 -16 -12 C -4 -10 0 -10 -8 D -2 16 10 0 6 E 2 12 8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.44000000003 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 12 4 2 -2 B -12 0 10 -16 -12 C -4 -10 0 -10 -8 D -2 16 10 0 6 E 2 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 C=19 D=18 B=16 E=12 so E is eliminated. Round 2 votes counts: A=37 D=22 B=22 C=19 so C is eliminated. Round 3 votes counts: A=42 D=29 B=29 so D is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:208 E:208 B:185 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 2 -2 B -12 0 10 -16 -12 C -4 -10 0 -10 -8 D -2 16 10 0 6 E 2 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 2 -2 B -12 0 10 -16 -12 C -4 -10 0 -10 -8 D -2 16 10 0 6 E 2 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 2 -2 B -12 0 10 -16 -12 C -4 -10 0 -10 -8 D -2 16 10 0 6 E 2 12 8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600000 B: 0.000000 C: 0.000000 D: 0.200000 E: 0.200000 Sum of squares = 0.440000000028 Cumulative probabilities = A: 0.600000 B: 0.600000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9413: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (18) B C A E D (13) D E A C B (8) C E B D A (6) C B E A D (6) E D C B A (5) A D E B C (5) C B E D A (4) C B A E D (4) B C E A D (4) A B D C E (4) D E A B C (3) A D B C E (3) E D C A B (2) E D A B C (2) D E C A B (2) D A E B C (2) E C D B A (1) E C B D A (1) E B C D A (1) D E C B A (1) C D E B A (1) C B A D E (1) B C E D A (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 -6 20 4 B -2 0 16 26 18 C 6 -16 0 22 30 D -20 -26 -22 0 -2 E -4 -18 -30 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.083333 D: 0.000000 E: 0.000000 Sum of squares = 0.513888888888 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 20 4 B -2 0 16 26 18 C 6 -16 0 22 30 D -20 -26 -22 0 -2 E -4 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.083333 D: 0.000000 E: 0.000000 Sum of squares = 0.513888888341 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=22 B=18 D=16 E=12 so E is eliminated. Round 2 votes counts: A=32 D=25 C=24 B=19 so B is eliminated. Round 3 votes counts: C=43 A=32 D=25 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:229 C:221 A:210 E:175 D:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -6 20 4 B -2 0 16 26 18 C 6 -16 0 22 30 D -20 -26 -22 0 -2 E -4 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.083333 D: 0.000000 E: 0.000000 Sum of squares = 0.513888888341 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 20 4 B -2 0 16 26 18 C 6 -16 0 22 30 D -20 -26 -22 0 -2 E -4 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.083333 D: 0.000000 E: 0.000000 Sum of squares = 0.513888888341 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 20 4 B -2 0 16 26 18 C 6 -16 0 22 30 D -20 -26 -22 0 -2 E -4 -18 -30 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.250000 C: 0.083333 D: 0.000000 E: 0.000000 Sum of squares = 0.513888888341 Cumulative probabilities = A: 0.666667 B: 0.916667 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9414: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) B D A C E (7) D C B E A (6) C E A D B (5) A E B C D (5) D B C E A (4) A C E D B (4) E C D A B (3) E A C B D (3) B D E C A (3) B D C A E (3) A E C B D (3) A B E C D (3) A B D E C (3) E C D B A (2) E C B A D (2) E A B C D (2) D C B A E (2) C E D B A (2) C D E B A (2) C A E D B (2) B E D A C (2) B D A E C (2) E D C B A (1) E B D C A (1) E A C D B (1) D C E B A (1) D B E C A (1) D B C A E (1) D B A C E (1) C E D A B (1) C D B E A (1) C D A B E (1) B E A D C (1) B D C E A (1) B A E D C (1) B A D E C (1) B A D C E (1) A E C D B (1) A C E B D (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -10 4 -10 B -2 0 -10 -4 -4 C 10 10 0 12 -2 D -4 4 -12 0 -14 E 10 4 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -10 4 -10 B -2 0 -10 -4 -4 C 10 10 0 12 -2 D -4 4 -12 0 -14 E 10 4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 B=22 D=16 C=14 so C is eliminated. Round 2 votes counts: E=33 A=25 B=22 D=20 so D is eliminated. Round 3 votes counts: B=38 E=36 A=26 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:215 E:215 A:193 B:190 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -10 4 -10 B -2 0 -10 -4 -4 C 10 10 0 12 -2 D -4 4 -12 0 -14 E 10 4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -10 4 -10 B -2 0 -10 -4 -4 C 10 10 0 12 -2 D -4 4 -12 0 -14 E 10 4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -10 4 -10 B -2 0 -10 -4 -4 C 10 10 0 12 -2 D -4 4 -12 0 -14 E 10 4 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996525 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9415: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (12) B A C D E (12) D E C A B (9) C D E A B (7) A C B D E (6) C A B D E (5) B E D A C (5) A B C D E (5) E D C B A (4) C A D E B (4) B E A D C (4) C A D B E (3) B A C E D (3) E D B C A (2) E D A C B (2) E B D C A (2) E B D A C (2) D C E A B (2) C B A D E (2) B A E D C (2) E C D B A (1) D A C E B (1) C E D A B (1) C D A E B (1) B A D C E (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 20 -10 -2 -6 B -20 0 -24 -2 -4 C 10 24 0 2 8 D 2 2 -2 0 20 E 6 4 -8 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -10 -2 -6 B -20 0 -24 -2 -4 C 10 24 0 2 8 D 2 2 -2 0 20 E 6 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 C=23 A=13 D=12 so D is eliminated. Round 2 votes counts: E=34 B=27 C=25 A=14 so A is eliminated. Round 3 votes counts: E=35 C=33 B=32 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:211 A:201 E:191 B:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -10 -2 -6 B -20 0 -24 -2 -4 C 10 24 0 2 8 D 2 2 -2 0 20 E 6 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -10 -2 -6 B -20 0 -24 -2 -4 C 10 24 0 2 8 D 2 2 -2 0 20 E 6 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -10 -2 -6 B -20 0 -24 -2 -4 C 10 24 0 2 8 D 2 2 -2 0 20 E 6 4 -8 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993394 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9416: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) C B A D E (6) E D B C A (5) A C B D E (5) E D A B C (4) E A D B C (4) C B D A E (4) A E B D C (4) A C B E D (4) A B C D E (4) E D B A C (3) E A D C B (3) D E B C A (3) C E D B A (3) C A B D E (3) B C D A E (3) A B C E D (3) E D A C B (2) D B C E A (2) C D B E A (2) B A D C E (2) A E C D B (2) A E C B D (2) A C E B D (2) E D C B A (1) E C D B A (1) E C D A B (1) E A C D B (1) D E C B A (1) D B E C A (1) D B E A C (1) B D C E A (1) B D C A E (1) B D A E C (1) B C D E A (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B C D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 4 2 4 B 4 0 -2 18 12 C -4 2 0 18 12 D -2 -18 -18 0 4 E -4 -12 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.360000000008 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 2 4 B 4 0 -2 18 12 C -4 2 0 18 12 D -2 -18 -18 0 4 E -4 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=26 E=25 B=11 D=8 so D is eliminated. Round 2 votes counts: A=30 E=29 C=26 B=15 so B is eliminated. Round 3 votes counts: C=35 A=34 E=31 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:216 C:214 A:203 E:184 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 2 4 B 4 0 -2 18 12 C -4 2 0 18 12 D -2 -18 -18 0 4 E -4 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 2 4 B 4 0 -2 18 12 C -4 2 0 18 12 D -2 -18 -18 0 4 E -4 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 2 4 B 4 0 -2 18 12 C -4 2 0 18 12 D -2 -18 -18 0 4 E -4 -12 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.400000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999978 Cumulative probabilities = A: 0.200000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9417: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (8) A C D B E (6) E C B A D (5) D B E A C (5) D B A E C (5) B D A E C (5) D E B C A (4) B D E A C (4) A C D E B (4) E D C B A (3) E B C D A (3) C A E D B (3) B E D C A (3) B A D C E (3) A C B D E (3) E B C A D (2) D A B C E (2) C E B A D (2) C E A D B (2) A C B E D (2) E D C A B (1) E C D A B (1) E C B D A (1) E C A D B (1) E C A B D (1) E B D C A (1) D E C B A (1) D E B A C (1) D E A C B (1) D E A B C (1) D B E C A (1) D A E B C (1) D A C E B (1) D A B E C (1) C E A B D (1) C A D E B (1) C A B E D (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E C A (1) B C A E D (1) B A C E D (1) A D C E B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 0 4 2 B 10 0 -2 2 -2 C 0 2 0 2 -12 D -4 -2 -2 0 8 E -2 2 12 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999823 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 A B C D E A 0 -10 0 4 2 B 10 0 -2 2 -2 C 0 2 0 2 -12 D -4 -2 -2 0 8 E -2 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 B=21 E=19 C=18 A=18 so C is eliminated. Round 2 votes counts: A=31 E=24 D=24 B=21 so B is eliminated. Round 3 votes counts: A=36 D=34 E=30 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:204 E:202 D:200 A:198 C:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 4 2 B 10 0 -2 2 -2 C 0 2 0 2 -12 D -4 -2 -2 0 8 E -2 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 4 2 B 10 0 -2 2 -2 C 0 2 0 2 -12 D -4 -2 -2 0 8 E -2 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 4 2 B 10 0 -2 2 -2 C 0 2 0 2 -12 D -4 -2 -2 0 8 E -2 2 12 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.666667 C: 0.000000 D: 0.166667 E: 0.166667 Sum of squares = 0.499999999866 Cumulative probabilities = A: 0.000000 B: 0.666667 C: 0.666667 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9418: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (14) E B A D C (9) B E A C D (6) D C B A E (5) B E A D C (5) C D A B E (4) E A D B C (3) E A B C D (3) D A C E B (3) C D B A E (3) C A D E B (3) B D C E A (3) E D A B C (2) E A D C B (2) D E A C B (2) D C E A B (2) D C A B E (2) B E D C A (2) B E C A D (2) B C E D A (2) B C E A D (2) B C D E A (2) B C D A E (2) B C A E D (2) A E D C B (2) E B D A C (1) E A B D C (1) D E B C A (1) C D A E B (1) C B D A E (1) C B A D E (1) C A B D E (1) B E D A C (1) B E C D A (1) A E C B D (1) A E B C D (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -12 -8 -4 B 2 0 4 0 -4 C 12 -4 0 -22 8 D 8 0 22 0 2 E 4 4 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.256816 C: 0.000000 D: 0.743184 E: 0.000000 Sum of squares = 0.618277383356 Cumulative probabilities = A: 0.000000 B: 0.256816 C: 0.256816 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -8 -4 B 2 0 4 0 -4 C 12 -4 0 -22 8 D 8 0 22 0 2 E 4 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555721159 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=29 E=21 C=14 A=6 so A is eliminated. Round 2 votes counts: B=31 D=30 E=25 C=14 so C is eliminated. Round 3 votes counts: D=41 B=34 E=25 so E is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:201 E:199 C:197 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 -8 -4 B 2 0 4 0 -4 C 12 -4 0 -22 8 D 8 0 22 0 2 E 4 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555721159 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -8 -4 B 2 0 4 0 -4 C 12 -4 0 -22 8 D 8 0 22 0 2 E 4 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555721159 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -8 -4 B 2 0 4 0 -4 C 12 -4 0 -22 8 D 8 0 22 0 2 E 4 4 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.666667 E: 0.000000 Sum of squares = 0.555555721159 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9419: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (10) E A B D C (9) E A B C D (8) B D C A E (8) A B C D E (6) E D C B A (5) D C B A E (5) D B C E A (4) E B A D C (3) D C E B A (3) D C B E A (3) B D E C A (3) B A D C E (3) A E B C D (3) E D B C A (2) E A D B C (2) E A C D B (2) D B C A E (2) C D A B E (2) B D C E A (2) A E C D B (2) E D B A C (1) E B D C A (1) E B D A C (1) E A D C B (1) C D E B A (1) C B A D E (1) B C D A E (1) B C A D E (1) A E C B D (1) A E B D C (1) A C B D E (1) A B E D C (1) A B D C E (1) Total count = 100 A B C D E A 0 -20 -8 -8 -2 B 20 0 26 10 8 C 8 -26 0 -22 8 D 8 -10 22 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 -8 -2 B 20 0 26 10 8 C 8 -26 0 -22 8 D 8 -10 22 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=18 D=17 A=16 C=14 so C is eliminated. Round 2 votes counts: E=35 D=30 B=19 A=16 so A is eliminated. Round 3 votes counts: E=42 D=30 B=28 so B is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:232 D:217 E:186 C:184 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -8 -8 -2 B 20 0 26 10 8 C 8 -26 0 -22 8 D 8 -10 22 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -8 -2 B 20 0 26 10 8 C 8 -26 0 -22 8 D 8 -10 22 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -8 -2 B 20 0 26 10 8 C 8 -26 0 -22 8 D 8 -10 22 0 14 E 2 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999439 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9420: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (10) C A B D E (7) E D B A C (6) D E A B C (6) B C E D A (6) E D A B C (5) A C D E B (5) D A E B C (4) C B E A D (4) C B A E D (4) C A D B E (4) A D E C B (4) A C D B E (4) E B D C A (3) E B D A C (3) C B A D E (3) B C D E A (3) D E B A C (2) B E D C A (2) B E C D A (2) A E D C B (2) A D C B E (2) D A E C B (1) D A B E C (1) D A B C E (1) C A E D B (1) C A E B D (1) B D E C A (1) B D E A C (1) B D C A E (1) A E C D B (1) Total count = 100 A B C D E A 0 18 16 4 12 B -18 0 -6 -18 -8 C -16 6 0 -10 12 D -4 18 10 0 20 E -12 8 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999594 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 4 12 B -18 0 -6 -18 -8 C -16 6 0 -10 12 D -4 18 10 0 20 E -12 8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 C=24 E=17 B=16 D=15 so D is eliminated. Round 2 votes counts: A=35 E=25 C=24 B=16 so B is eliminated. Round 3 votes counts: A=35 C=34 E=31 so E is eliminated. Round 4 votes counts: A=58 C=42 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 D:222 C:196 E:182 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 4 12 B -18 0 -6 -18 -8 C -16 6 0 -10 12 D -4 18 10 0 20 E -12 8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 4 12 B -18 0 -6 -18 -8 C -16 6 0 -10 12 D -4 18 10 0 20 E -12 8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 4 12 B -18 0 -6 -18 -8 C -16 6 0 -10 12 D -4 18 10 0 20 E -12 8 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999663 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9421: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (6) D C B A E (5) B E D A C (5) A E B D C (5) D C B E A (4) D C A E B (4) C D A E B (4) B E A D C (4) A E B C D (4) E B A C D (3) E A B C D (3) D C A B E (3) C A E B D (3) C A D E B (3) B E C A D (3) D B C A E (2) D B A E C (2) D A E B C (2) D A B E C (2) C E B A D (2) C E A B D (2) C B D E A (2) B E D C A (2) B D E A C (2) B C E A D (2) A E D C B (2) A E C B D (2) A D C E B (2) A C E B D (2) D B E A C (1) D B C E A (1) D A E C B (1) D A C E B (1) D A C B E (1) C D B E A (1) C D B A E (1) C D A B E (1) C B E D A (1) C A E D B (1) B E C D A (1) B D E C A (1) A D E B C (1) Total count = 100 A B C D E A 0 -2 2 0 8 B 2 0 4 10 6 C -2 -4 0 -6 -4 D 0 -10 6 0 -6 E -8 -6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 0 8 B 2 0 4 10 6 C -2 -4 0 -6 -4 D 0 -10 6 0 -6 E -8 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=26 C=21 A=18 E=6 so E is eliminated. Round 2 votes counts: D=29 B=29 C=21 A=21 so C is eliminated. Round 3 votes counts: D=36 B=34 A=30 so A is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:211 A:204 E:198 D:195 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 2 0 8 B 2 0 4 10 6 C -2 -4 0 -6 -4 D 0 -10 6 0 -6 E -8 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 8 B 2 0 4 10 6 C -2 -4 0 -6 -4 D 0 -10 6 0 -6 E -8 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 8 B 2 0 4 10 6 C -2 -4 0 -6 -4 D 0 -10 6 0 -6 E -8 -6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9422: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B A C E D (8) D A C E B (7) D A E C B (5) D A B C E (5) B E C A D (5) A D C B E (5) E B C A D (4) B C A E D (4) A C B E D (4) A B C E D (4) E D C B A (3) E C B A D (3) D A C B E (3) A B C D E (3) E D C A B (2) E B C D A (2) D E C B A (2) D E B C A (2) D E B A C (2) C B A E D (2) B E C D A (2) B C E A D (2) B A C D E (2) E D B C A (1) D C A E B (1) D B E A C (1) D A E B C (1) D A B E C (1) C E A B D (1) C A E B D (1) C A B E D (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 8 4 20 B -10 0 0 0 10 C -8 0 0 0 12 D -4 0 0 0 2 E -20 -10 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 4 20 B -10 0 0 0 10 C -8 0 0 0 12 D -4 0 0 0 2 E -20 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 B=23 A=19 E=15 C=5 so C is eliminated. Round 2 votes counts: D=38 B=25 A=21 E=16 so E is eliminated. Round 3 votes counts: D=44 B=34 A=22 so A is eliminated. Round 4 votes counts: D=50 B=50 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:221 C:202 B:200 D:199 E:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 4 20 B -10 0 0 0 10 C -8 0 0 0 12 D -4 0 0 0 2 E -20 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 4 20 B -10 0 0 0 10 C -8 0 0 0 12 D -4 0 0 0 2 E -20 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 4 20 B -10 0 0 0 10 C -8 0 0 0 12 D -4 0 0 0 2 E -20 -10 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998489 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9423: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (8) D E A B C (6) D B A E C (6) C E A B D (6) C B A E D (6) C E D A B (5) C D B E A (5) A B E D C (5) B A C E D (4) A B E C D (4) E A B C D (3) D C E B A (3) C D E A B (3) B A D C E (3) E D C A B (2) E C D A B (2) D C E A B (2) C B A D E (2) B C A E D (2) A B D E C (2) E D A C B (1) E C A D B (1) E A C B D (1) E A B D C (1) D E C B A (1) D E A C B (1) D C B A E (1) D B C A E (1) D B A C E (1) D A E B C (1) D A B E C (1) C E D B A (1) C E B A D (1) C D E B A (1) C D B A E (1) C B D A E (1) B D A E C (1) B D A C E (1) B A E D C (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 -2 6 4 8 B 2 0 6 8 12 C -6 -6 0 0 0 D -4 -8 0 0 4 E -8 -12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 4 8 B 2 0 6 8 12 C -6 -6 0 0 0 D -4 -8 0 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 B=20 A=13 E=11 so E is eliminated. Round 2 votes counts: C=35 D=27 B=20 A=18 so A is eliminated. Round 3 votes counts: B=37 C=36 D=27 so D is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 A:208 D:196 C:194 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 4 8 B 2 0 6 8 12 C -6 -6 0 0 0 D -4 -8 0 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 4 8 B 2 0 6 8 12 C -6 -6 0 0 0 D -4 -8 0 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 4 8 B 2 0 6 8 12 C -6 -6 0 0 0 D -4 -8 0 0 4 E -8 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999478 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9424: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (9) E D C A B (8) B A C D E (8) E D C B A (6) C B A D E (6) D E C B A (5) A B C D E (5) D E C A B (4) D C E B A (4) A B E D C (4) A B C E D (4) E C D B A (3) C E D B A (3) A B E C D (3) A B D C E (3) E A B D C (2) D E A B C (2) D C B A E (2) C D E B A (2) C B A E D (2) B C A D E (2) B A C E D (2) E D A C B (1) E A D B C (1) D B A C E (1) D A B E C (1) D A B C E (1) C E B A D (1) C D B E A (1) C D B A E (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 -8 -4 B -2 0 2 -12 -4 C 0 -2 0 -14 -4 D 8 12 14 0 0 E 4 4 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.169940 E: 0.830060 Sum of squares = 0.717879636329 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.169940 E: 1.000000 A B C D E A 0 2 0 -8 -4 B -2 0 2 -12 -4 C 0 -2 0 -14 -4 D 8 12 14 0 0 E 4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=22 D=20 C=16 B=12 so B is eliminated. Round 2 votes counts: A=32 E=30 D=20 C=18 so C is eliminated. Round 3 votes counts: A=42 E=34 D=24 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:217 E:206 A:195 B:192 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 0 -8 -4 B -2 0 2 -12 -4 C 0 -2 0 -14 -4 D 8 12 14 0 0 E 4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -8 -4 B -2 0 2 -12 -4 C 0 -2 0 -14 -4 D 8 12 14 0 0 E 4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -8 -4 B -2 0 2 -12 -4 C 0 -2 0 -14 -4 D 8 12 14 0 0 E 4 4 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9425: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (18) D C A E B (15) B E A C D (9) E B D A C (6) C A D B E (6) D C A B E (5) E B D C A (4) A C B D E (4) E B A C D (3) D E C A B (3) A C B E D (3) A B C E D (3) E D C B A (2) E D B C A (2) E B C A D (2) C A D E B (2) B A C E D (2) D E C B A (1) D E A B C (1) D A C B E (1) C D A E B (1) C B A E D (1) C A B E D (1) B E A D C (1) B D E A C (1) B A C D E (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 30 10 16 30 B -30 0 -28 -16 16 C -10 28 0 14 30 D -16 16 -14 0 22 E -30 -16 -30 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 30 10 16 30 B -30 0 -28 -16 16 C -10 28 0 14 30 D -16 16 -14 0 22 E -30 -16 -30 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=26 E=19 B=14 C=11 so C is eliminated. Round 2 votes counts: A=39 D=27 E=19 B=15 so B is eliminated. Round 3 votes counts: A=43 E=29 D=28 so D is eliminated. Round 4 votes counts: A=65 E=35 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:243 C:231 D:204 B:171 E:151 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 30 10 16 30 B -30 0 -28 -16 16 C -10 28 0 14 30 D -16 16 -14 0 22 E -30 -16 -30 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 10 16 30 B -30 0 -28 -16 16 C -10 28 0 14 30 D -16 16 -14 0 22 E -30 -16 -30 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 10 16 30 B -30 0 -28 -16 16 C -10 28 0 14 30 D -16 16 -14 0 22 E -30 -16 -30 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9426: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (10) C A E B D (7) C A B E D (7) D B E C A (5) D B E A C (5) B D C E A (5) A E C D B (5) E A D C B (4) D E A B C (4) C A E D B (4) B D C A E (4) E D A C B (3) E C A D B (3) C B A E D (3) B C A D E (3) A C E D B (3) A C E B D (3) D E B A C (2) B D E C A (2) B D E A C (2) B D A E C (2) B D A C E (2) B C D A E (2) A B E C D (2) E D A B C (1) D C B E A (1) D B C E A (1) C E D A B (1) C B A D E (1) A E B C D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 20 2 16 2 B -20 0 -10 -4 -2 C -2 10 0 12 -4 D -16 4 -12 0 -18 E -2 2 4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 2 16 2 B -20 0 -10 -4 -2 C -2 10 0 12 -4 D -16 4 -12 0 -18 E -2 2 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 B=22 E=21 D=18 A=16 so A is eliminated. Round 2 votes counts: C=29 E=27 B=26 D=18 so D is eliminated. Round 3 votes counts: B=37 E=33 C=30 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:220 E:211 C:208 B:182 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 2 16 2 B -20 0 -10 -4 -2 C -2 10 0 12 -4 D -16 4 -12 0 -18 E -2 2 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 2 16 2 B -20 0 -10 -4 -2 C -2 10 0 12 -4 D -16 4 -12 0 -18 E -2 2 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 2 16 2 B -20 0 -10 -4 -2 C -2 10 0 12 -4 D -16 4 -12 0 -18 E -2 2 4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999708 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9427: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (10) D C E A B (7) B A C E D (7) D E C A B (6) C D E A B (5) B E D A C (5) B A E D C (5) E D A C B (4) E A C D B (4) C A E D B (3) B D E C A (3) B D C E A (3) B C D A E (3) A E C D B (3) A B C E D (3) D E C B A (2) D C E B A (2) D B E C A (2) C A D E B (2) B C A D E (2) A E B C D (2) A C B E D (2) E A D C B (1) E A D B C (1) D E A C B (1) D C B E A (1) C D B E A (1) C D A E B (1) C B A E D (1) C B A D E (1) C A B E D (1) C A B D E (1) B E A D C (1) B D E A C (1) B A C D E (1) A E C B D (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 6 6 0 B 2 0 0 8 10 C -6 0 0 10 -6 D -6 -8 -10 0 -10 E 0 -10 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.862805 C: 0.137195 D: 0.000000 E: 0.000000 Sum of squares = 0.76325560209 Cumulative probabilities = A: 0.000000 B: 0.862805 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 6 0 B 2 0 0 8 10 C -6 0 0 10 -6 D -6 -8 -10 0 -10 E 0 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000554 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=21 C=16 A=12 E=10 so E is eliminated. Round 2 votes counts: B=41 D=25 A=18 C=16 so C is eliminated. Round 3 votes counts: B=43 D=32 A=25 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:205 E:203 C:199 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 6 0 B 2 0 0 8 10 C -6 0 0 10 -6 D -6 -8 -10 0 -10 E 0 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000554 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 6 0 B 2 0 0 8 10 C -6 0 0 10 -6 D -6 -8 -10 0 -10 E 0 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000554 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 6 0 B 2 0 0 8 10 C -6 0 0 10 -6 D -6 -8 -10 0 -10 E 0 -10 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000000554 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9428: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) E C B A D (6) C B E A D (6) B C A D E (6) C B E D A (5) A E D B C (5) E D A C B (4) E C D B A (4) D E A B C (4) C E B A D (4) B C D A E (4) A D E B C (4) E C D A B (3) E C B D A (3) E C A B D (3) E A D B C (3) A D B E C (3) A D B C E (3) E D C A B (2) E A D C B (2) D A E B C (2) D A B E C (2) C B D E A (2) C B D A E (2) B C A E D (2) E D A B C (1) E C A D B (1) E A C B D (1) D E A C B (1) C E B D A (1) C B A E D (1) B D C A E (1) B A C D E (1) A E B D C (1) Total count = 100 A B C D E A 0 4 -12 4 -12 B -4 0 -2 -2 -10 C 12 2 0 10 -10 D -4 2 -10 0 -16 E 12 10 10 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -12 4 -12 B -4 0 -2 -2 -10 C 12 2 0 10 -10 D -4 2 -10 0 -16 E 12 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=21 D=16 A=16 B=14 so B is eliminated. Round 2 votes counts: E=33 C=33 D=17 A=17 so D is eliminated. Round 3 votes counts: E=38 C=34 A=28 so A is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:224 C:207 A:192 B:191 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -12 4 -12 B -4 0 -2 -2 -10 C 12 2 0 10 -10 D -4 2 -10 0 -16 E 12 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 4 -12 B -4 0 -2 -2 -10 C 12 2 0 10 -10 D -4 2 -10 0 -16 E 12 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 4 -12 B -4 0 -2 -2 -10 C 12 2 0 10 -10 D -4 2 -10 0 -16 E 12 10 10 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9429: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (11) C E A B D (9) B E D C A (8) C A E D B (7) E C B D A (6) A D B C E (6) C E B A D (5) B D E A C (5) A D C B E (5) A C D E B (5) C E B D A (4) B D E C A (4) A C E D B (4) E B D C A (3) E B C D A (3) D A B E C (3) A D B E C (3) A C D B E (3) C A E B D (2) E C A B D (1) D B E A C (1) D A B C E (1) C A B D E (1) Total count = 100 A B C D E A 0 0 -6 2 2 B 0 0 -4 2 2 C 6 4 0 0 4 D -2 -2 0 0 -4 E -2 -2 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.641177 D: 0.358823 E: 0.000000 Sum of squares = 0.539861724794 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.641177 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 2 2 B 0 0 -4 2 2 C 6 4 0 0 4 D -2 -2 0 0 -4 E -2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500018 D: 0.499982 E: 0.000000 Sum of squares = 0.500000000656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500018 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=26 B=17 D=16 E=13 so E is eliminated. Round 2 votes counts: C=35 A=26 B=23 D=16 so D is eliminated. Round 3 votes counts: C=35 B=35 A=30 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:207 B:200 A:199 E:198 D:196 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -6 2 2 B 0 0 -4 2 2 C 6 4 0 0 4 D -2 -2 0 0 -4 E -2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500018 D: 0.499982 E: 0.000000 Sum of squares = 0.500000000656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500018 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 2 2 B 0 0 -4 2 2 C 6 4 0 0 4 D -2 -2 0 0 -4 E -2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500018 D: 0.499982 E: 0.000000 Sum of squares = 0.500000000656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500018 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 2 2 B 0 0 -4 2 2 C 6 4 0 0 4 D -2 -2 0 0 -4 E -2 -2 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500018 D: 0.499982 E: 0.000000 Sum of squares = 0.500000000656 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500018 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9430: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C E A D B (9) B D C A E (9) B D A E C (7) C E A B D (4) B A D E C (4) A E B C D (4) E A C D B (3) D B C E A (3) C D B E A (3) C B A E D (3) B D A C E (3) B C D A E (3) A E D B C (3) A E B D C (3) E A D C B (2) E A C B D (2) D E A B C (2) D C B E A (2) C B D E A (2) B A E D C (2) A E D C B (2) A E C B D (2) E A D B C (1) D E A C B (1) D B C A E (1) C E D A B (1) C E B A D (1) C B E D A (1) C B E A D (1) B C D E A (1) B A C E D (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -18 12 -2 22 B 18 0 22 10 20 C -12 -22 0 -16 -4 D 2 -10 16 0 8 E -22 -20 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 12 -2 22 B 18 0 22 10 20 C -12 -22 0 -16 -4 D 2 -10 16 0 8 E -22 -20 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=25 D=21 A=16 E=8 so E is eliminated. Round 2 votes counts: B=30 C=25 A=24 D=21 so D is eliminated. Round 3 votes counts: B=46 C=27 A=27 so C is eliminated. Round 4 votes counts: B=59 A=41 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:235 D:208 A:207 E:177 C:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 12 -2 22 B 18 0 22 10 20 C -12 -22 0 -16 -4 D 2 -10 16 0 8 E -22 -20 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 12 -2 22 B 18 0 22 10 20 C -12 -22 0 -16 -4 D 2 -10 16 0 8 E -22 -20 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 12 -2 22 B 18 0 22 10 20 C -12 -22 0 -16 -4 D 2 -10 16 0 8 E -22 -20 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9431: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (13) C B E D A (9) A D E B C (8) A D B C E (8) A D C B E (7) E D B C A (4) C B D A E (4) A C B D E (4) E D B A C (3) E D A B C (3) D B A C E (3) D A B E C (3) C B D E A (3) C B A D E (3) E C A B D (2) E B C D A (2) E A D B C (2) B C D A E (2) E C B A D (1) E B D C A (1) E A D C B (1) D E B C A (1) D A E B C (1) D A B C E (1) C E B D A (1) C B A E D (1) C A B E D (1) B C E D A (1) B C D E A (1) A E D B C (1) A E C D B (1) A E C B D (1) A D E C B (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 0 -12 4 B 6 0 -8 0 6 C 0 8 0 2 0 D 12 0 -2 0 4 E -4 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.116221 B: 0.000000 C: 0.883779 D: 0.000000 E: 0.000000 Sum of squares = 0.794572924472 Cumulative probabilities = A: 0.116221 B: 0.116221 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -12 4 B 6 0 -8 0 6 C 0 8 0 2 0 D 12 0 -2 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102047071 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=32 C=22 D=9 B=4 so B is eliminated. Round 2 votes counts: A=33 E=32 C=26 D=9 so D is eliminated. Round 3 votes counts: A=41 E=33 C=26 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:207 C:205 B:202 A:193 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 0 -12 4 B 6 0 -8 0 6 C 0 8 0 2 0 D 12 0 -2 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102047071 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -12 4 B 6 0 -8 0 6 C 0 8 0 2 0 D 12 0 -2 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102047071 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -12 4 B 6 0 -8 0 6 C 0 8 0 2 0 D 12 0 -2 0 4 E -4 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.000000 C: 0.857143 D: 0.000000 E: 0.000000 Sum of squares = 0.755102047071 Cumulative probabilities = A: 0.142857 B: 0.142857 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9432: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) B C A E D (11) E D A C B (7) D E C A B (7) D E A C B (7) E A D C B (4) D C E A B (4) B A E C D (4) C D A E B (3) C B A E D (3) B C A D E (3) A C B E D (3) E D A B C (2) C D B A E (2) C B D A E (2) C B A D E (2) C A B E D (2) B D E A C (2) A E C D B (2) A B E C D (2) E B D A C (1) E A B D C (1) D E C B A (1) D E B C A (1) D E B A C (1) D E A B C (1) D B E C A (1) C A E D B (1) C A D E B (1) C A B D E (1) B E A D C (1) B D C E A (1) B D C A E (1) B C D A E (1) A E B C D (1) A C E B D (1) Total count = 100 A B C D E A 0 0 4 10 16 B 0 0 -6 10 8 C -4 6 0 14 8 D -10 -10 -14 0 -16 E -16 -8 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.849743 B: 0.150257 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.744640721546 Cumulative probabilities = A: 0.849743 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 10 16 B 0 0 -6 10 8 C -4 6 0 14 8 D -10 -10 -14 0 -16 E -16 -8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.399999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000547485 Cumulative probabilities = A: 0.600001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=23 C=17 E=15 A=9 so A is eliminated. Round 2 votes counts: B=38 D=23 C=21 E=18 so E is eliminated. Round 3 votes counts: B=41 D=36 C=23 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:215 C:212 B:206 E:192 D:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 10 16 B 0 0 -6 10 8 C -4 6 0 14 8 D -10 -10 -14 0 -16 E -16 -8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.399999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000547485 Cumulative probabilities = A: 0.600001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 10 16 B 0 0 -6 10 8 C -4 6 0 14 8 D -10 -10 -14 0 -16 E -16 -8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.399999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000547485 Cumulative probabilities = A: 0.600001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 10 16 B 0 0 -6 10 8 C -4 6 0 14 8 D -10 -10 -14 0 -16 E -16 -8 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.399999 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.520000547485 Cumulative probabilities = A: 0.600001 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9433: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (9) B E D A C (6) E B D C A (5) D C E A B (5) D C A B E (5) E B A C D (4) D E C B A (4) D E B C A (4) D B E C A (4) C D A E B (4) C A E D B (4) C A D E B (4) A C D B E (4) E D B C A (3) D C A E B (3) B D E A C (3) B A E C D (3) A B C E D (3) E B C D A (2) D B A C E (2) C E A D B (2) B D A E C (2) A C E B D (2) A C B D E (2) E C D A B (1) E B D A C (1) D B E A C (1) D A C B E (1) C D A B E (1) C A E B D (1) B E A D C (1) B D A C E (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -4 -16 -10 B 10 0 10 -6 0 C 4 -10 0 -4 -6 D 16 6 4 0 4 E 10 0 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999715 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -16 -10 B 10 0 10 -6 0 C 4 -10 0 -4 -6 D 16 6 4 0 4 E 10 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=25 E=16 C=16 A=14 so A is eliminated. Round 2 votes counts: D=29 B=29 C=26 E=16 so E is eliminated. Round 3 votes counts: B=41 D=32 C=27 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:207 E:206 C:192 A:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 -16 -10 B 10 0 10 -6 0 C 4 -10 0 -4 -6 D 16 6 4 0 4 E 10 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -16 -10 B 10 0 10 -6 0 C 4 -10 0 -4 -6 D 16 6 4 0 4 E 10 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -16 -10 B 10 0 10 -6 0 C 4 -10 0 -4 -6 D 16 6 4 0 4 E 10 0 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9434: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) D A E C B (6) C E B A D (5) C B E A D (5) B C E A D (5) A D E B C (5) E C B A D (4) E B C A D (4) E B A C D (4) D A E B C (4) C D B A E (4) C B D E A (4) C B D A E (4) D A B C E (3) E A D C B (2) E A D B C (2) E A B D C (2) D B A C E (2) D A C B E (2) C D E A B (2) B E C A D (2) B C D A E (2) A D E C B (2) E D A C B (1) E C A D B (1) E B A D C (1) E A B C D (1) D C B A E (1) D C A E B (1) D C A B E (1) D A C E B (1) C E D A B (1) C E B D A (1) C D B E A (1) C D A E B (1) B E A C D (1) B D C A E (1) B C D E A (1) B C A E D (1) B C A D E (1) A E D B C (1) Total count = 100 A B C D E A 0 -22 -20 -2 -14 B 22 0 -14 12 -4 C 20 14 0 24 14 D 2 -12 -24 0 -2 E 14 4 -14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -20 -2 -14 B 22 0 -14 12 -4 C 20 14 0 24 14 D 2 -12 -24 0 -2 E 14 4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 E=22 D=21 B=14 A=8 so A is eliminated. Round 2 votes counts: C=35 D=28 E=23 B=14 so B is eliminated. Round 3 votes counts: C=45 D=29 E=26 so E is eliminated. Round 4 votes counts: C=62 D=38 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:236 B:208 E:203 D:182 A:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -20 -2 -14 B 22 0 -14 12 -4 C 20 14 0 24 14 D 2 -12 -24 0 -2 E 14 4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -20 -2 -14 B 22 0 -14 12 -4 C 20 14 0 24 14 D 2 -12 -24 0 -2 E 14 4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -20 -2 -14 B 22 0 -14 12 -4 C 20 14 0 24 14 D 2 -12 -24 0 -2 E 14 4 -14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9435: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (12) B A D E C (10) A D B C E (7) E C D A B (6) E C B D A (6) B E C A D (6) B D A E C (5) E B C D A (4) A D C E B (4) A C D E B (4) E C B A D (3) C E D A B (3) C E A D B (3) C A E D B (3) B A D C E (3) E C D B A (2) D A C E B (2) C D E A B (2) B E A C D (2) E D C A B (1) D B E C A (1) D B A C E (1) D A C B E (1) D A B C E (1) C D A E B (1) C A D E B (1) B E D C A (1) B E D A C (1) B D A C E (1) A D C B E (1) A C E D B (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -10 -2 -6 B 16 0 12 10 8 C 10 -12 0 18 -20 D 2 -10 -18 0 -8 E 6 -8 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 -2 -6 B 16 0 12 10 8 C 10 -12 0 18 -20 D 2 -10 -18 0 -8 E 6 -8 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=22 A=18 C=13 D=6 so D is eliminated. Round 2 votes counts: B=43 E=22 A=22 C=13 so C is eliminated. Round 3 votes counts: B=43 E=30 A=27 so A is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:223 E:213 C:198 A:183 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -10 -2 -6 B 16 0 12 10 8 C 10 -12 0 18 -20 D 2 -10 -18 0 -8 E 6 -8 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 -2 -6 B 16 0 12 10 8 C 10 -12 0 18 -20 D 2 -10 -18 0 -8 E 6 -8 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 -2 -6 B 16 0 12 10 8 C 10 -12 0 18 -20 D 2 -10 -18 0 -8 E 6 -8 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9436: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (10) D B E A C (6) C A E B D (5) A E D B C (5) E D B A C (4) E C D B A (4) E A D B C (4) C E A D B (4) C B D A E (4) C A B D E (4) B D A C E (4) E D B C A (3) C B D E A (3) B D E C A (3) B D A E C (3) A E C D B (3) A B D E C (3) E C A D B (2) D B E C A (2) C E B D A (2) C E A B D (2) C A E D B (2) C A B E D (2) A C E D B (2) A C E B D (2) A B D C E (2) A B C D E (2) E D C B A (1) D E B A C (1) C E D B A (1) C B A D E (1) B D C E A (1) B D C A E (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 10 4 12 14 B -10 0 -10 10 4 C -4 10 0 12 10 D -12 -10 -12 0 2 E -14 -4 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 4 12 14 B -10 0 -10 10 4 C -4 10 0 12 10 D -12 -10 -12 0 2 E -14 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=30 A=30 E=18 B=13 D=9 so D is eliminated. Round 2 votes counts: C=30 A=30 B=21 E=19 so E is eliminated. Round 3 votes counts: C=37 A=34 B=29 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:214 B:197 E:185 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 4 12 14 B -10 0 -10 10 4 C -4 10 0 12 10 D -12 -10 -12 0 2 E -14 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 4 12 14 B -10 0 -10 10 4 C -4 10 0 12 10 D -12 -10 -12 0 2 E -14 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 4 12 14 B -10 0 -10 10 4 C -4 10 0 12 10 D -12 -10 -12 0 2 E -14 -4 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9437: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (9) E C A D B (8) B E D C A (8) A C D E B (6) D A C E B (5) E B D C A (4) E B C A D (4) E D B A C (3) E C A B D (3) D B E A C (3) D B A C E (3) C A E B D (3) B D A C E (3) B C A E D (3) A C E D B (3) E D C A B (2) E D A C B (2) D E A C B (2) D A C B E (2) C A E D B (2) C A B D E (2) B C A D E (2) A D C E B (2) A C D B E (2) E D B C A (1) E C B A D (1) D B A E C (1) D A B C E (1) C E A B D (1) C B A E D (1) C A D E B (1) C A D B E (1) C A B E D (1) B E C D A (1) B E C A D (1) B D E C A (1) B D A E C (1) B C E D A (1) Total count = 100 A B C D E A 0 -2 -4 -6 -10 B 2 0 0 0 -6 C 4 0 0 -6 -10 D 6 0 6 0 -6 E 10 6 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999855 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -4 -6 -10 B 2 0 0 0 -6 C 4 0 0 -6 -10 D 6 0 6 0 -6 E 10 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 D=17 A=13 C=12 so C is eliminated. Round 2 votes counts: B=31 E=29 A=23 D=17 so D is eliminated. Round 3 votes counts: B=38 E=31 A=31 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:216 D:203 B:198 C:194 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -4 -6 -10 B 2 0 0 0 -6 C 4 0 0 -6 -10 D 6 0 6 0 -6 E 10 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -6 -10 B 2 0 0 0 -6 C 4 0 0 -6 -10 D 6 0 6 0 -6 E 10 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -6 -10 B 2 0 0 0 -6 C 4 0 0 -6 -10 D 6 0 6 0 -6 E 10 6 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9438: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (12) E D C A B (10) C A E B D (9) B D A E C (9) B D E A C (7) C E A D B (6) B D A C E (6) B A D C E (5) B A C D E (5) E C A D B (3) D B E A C (3) A C E D B (3) E C D A B (2) D E A C B (2) D E A B C (2) B D E C A (2) A C E B D (2) E D A C B (1) D E C B A (1) D E C A B (1) D E B C A (1) D B E C A (1) C E A B D (1) B E C D A (1) B D C E A (1) B A C E D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -2 0 10 B -14 0 -10 4 -12 C 2 10 0 -4 8 D 0 -4 4 0 -6 E -10 12 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.519752 B: 0.000000 C: 0.000000 D: 0.480248 E: 0.000000 Sum of squares = 0.500780287591 Cumulative probabilities = A: 0.519752 B: 0.519752 C: 0.519752 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 0 10 B -14 0 -10 4 -12 C 2 10 0 -4 8 D 0 -4 4 0 -6 E -10 12 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 C=28 E=16 D=11 A=8 so A is eliminated. Round 2 votes counts: B=38 C=35 E=16 D=11 so D is eliminated. Round 3 votes counts: B=42 C=35 E=23 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:211 C:208 E:200 D:197 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 -2 0 10 B -14 0 -10 4 -12 C 2 10 0 -4 8 D 0 -4 4 0 -6 E -10 12 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 0 10 B -14 0 -10 4 -12 C 2 10 0 -4 8 D 0 -4 4 0 -6 E -10 12 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 0 10 B -14 0 -10 4 -12 C 2 10 0 -4 8 D 0 -4 4 0 -6 E -10 12 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.000000 D: 0.499999 E: 0.000000 Sum of squares = 0.499999999969 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 0.500001 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9439: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) B A C E D (6) E C B A D (5) C D E B A (4) C B E A D (4) B E A C D (4) B A E C D (4) E D C B A (3) E C D B A (3) E B C A D (3) D E C A B (3) D C E A B (3) D A E B C (3) D A B E C (3) C D B A E (3) C B E D A (3) C B D A E (3) A B D C E (3) E D A B C (2) D A B C E (2) B C A E D (2) A E B D C (2) A D B E C (2) A D B C E (2) A B E D C (2) A B C D E (2) E C B D A (1) E B A C D (1) E A D B C (1) E A B D C (1) D C E B A (1) D C B A E (1) D C A E B (1) D A C B E (1) C E D B A (1) C E B A D (1) C D B E A (1) C B A D E (1) B C A D E (1) B A C D E (1) A B E C D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 -10 2 10 B 14 0 0 6 22 C 10 0 0 12 10 D -2 -6 -12 0 -2 E -10 -22 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.473424 C: 0.526576 D: 0.000000 E: 0.000000 Sum of squares = 0.501412541318 Cumulative probabilities = A: 0.000000 B: 0.473424 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -10 2 10 B 14 0 0 6 22 C 10 0 0 12 10 D -2 -6 -12 0 -2 E -10 -22 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=21 E=20 B=18 A=16 so A is eliminated. Round 2 votes counts: D=29 B=28 E=22 C=21 so C is eliminated. Round 3 votes counts: B=39 D=37 E=24 so E is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 C:216 A:194 D:189 E:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -10 2 10 B 14 0 0 6 22 C 10 0 0 12 10 D -2 -6 -12 0 -2 E -10 -22 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -10 2 10 B 14 0 0 6 22 C 10 0 0 12 10 D -2 -6 -12 0 -2 E -10 -22 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -10 2 10 B 14 0 0 6 22 C 10 0 0 12 10 D -2 -6 -12 0 -2 E -10 -22 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9440: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (8) D B E A C (7) D A E B C (7) B C D E A (7) A E C D B (6) B D C E A (5) B C E A D (4) B C D A E (4) A E D C B (4) E A C B D (3) D B C A E (3) D A E C B (3) C A E B D (3) A D E C B (3) E A C D B (2) D E A B C (2) C E B A D (2) C E A B D (2) C D B A E (2) C B D A E (2) C A B E D (2) B D C A E (2) A C E B D (2) E D A B C (1) E C A B D (1) E B C D A (1) E B C A D (1) E A D B C (1) E A B C D (1) D E B A C (1) D B E C A (1) D B C E A (1) D A C B E (1) C B E A D (1) C B A E D (1) B D E C A (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 8 8 -2 -6 B -8 0 0 -8 -12 C -8 0 0 -2 -10 D 2 8 2 0 4 E 6 12 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999815 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 -2 -6 B -8 0 0 -8 -12 C -8 0 0 -2 -10 D 2 8 2 0 4 E 6 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=23 E=19 A=17 C=15 so C is eliminated. Round 2 votes counts: D=28 B=27 E=23 A=22 so A is eliminated. Round 3 votes counts: E=40 D=31 B=29 so B is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:212 D:208 A:204 C:190 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 8 -2 -6 B -8 0 0 -8 -12 C -8 0 0 -2 -10 D 2 8 2 0 4 E 6 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 -2 -6 B -8 0 0 -8 -12 C -8 0 0 -2 -10 D 2 8 2 0 4 E 6 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 -2 -6 B -8 0 0 -8 -12 C -8 0 0 -2 -10 D 2 8 2 0 4 E 6 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9441: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (7) D B A C E (6) A B D C E (5) D B E A C (4) D B C A E (4) E D C B A (3) D C B E A (3) D B C E A (3) C D B E A (3) C B D A E (3) B A D C E (3) A E B D C (3) A C B E D (3) A B C D E (3) E D B C A (2) E D A C B (2) E C B A D (2) E A D B C (2) E A B C D (2) D E C B A (2) D E B C A (2) C B A D E (2) C A B E D (2) B C D A E (2) A E B C D (2) A C B D E (2) E D B A C (1) E D A B C (1) E C D B A (1) E C D A B (1) E C A D B (1) E A D C B (1) E A C B D (1) E A B D C (1) D E B A C (1) D E A B C (1) D C E B A (1) D B A E C (1) C E D B A (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B A E (1) C B A E D (1) B A C D E (1) A E D B C (1) A E C B D (1) A D B E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 -2 -4 B 12 0 6 -2 8 C 0 -6 0 -8 6 D 2 2 8 0 10 E 4 -8 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -2 -4 B 12 0 6 -2 8 C 0 -6 0 -8 6 D 2 2 8 0 10 E 4 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=28 D=28 A=22 C=16 B=6 so B is eliminated. Round 2 votes counts: E=28 D=28 A=26 C=18 so C is eliminated. Round 3 votes counts: D=38 E=31 A=31 so E is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:211 C:196 A:191 E:190 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 0 -2 -4 B 12 0 6 -2 8 C 0 -6 0 -8 6 D 2 2 8 0 10 E 4 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -2 -4 B 12 0 6 -2 8 C 0 -6 0 -8 6 D 2 2 8 0 10 E 4 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -2 -4 B 12 0 6 -2 8 C 0 -6 0 -8 6 D 2 2 8 0 10 E 4 -8 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999996614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9442: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (12) E B C A D (8) C A D E B (8) D E B C A (7) C A E B D (7) D E C B A (6) D C A E B (5) D B E A C (5) B E A C D (4) B E D A C (3) A C D B E (3) E C A B D (2) E B D C A (2) D B E C A (2) D B A C E (2) D A C B E (2) C E A B D (2) C A E D B (2) B A E C D (2) A C E B D (2) A B C E D (2) E C D B A (1) E C D A B (1) D E C A B (1) D C E A B (1) D B A E C (1) D A B C E (1) B E A D C (1) B D A E C (1) B A D C E (1) B A C E D (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -10 18 8 B -6 0 -12 4 -10 C 10 12 0 18 6 D -18 -4 -18 0 -4 E -8 10 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 18 8 B -6 0 -12 4 -10 C 10 12 0 18 6 D -18 -4 -18 0 -4 E -8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=21 C=19 E=14 B=13 so B is eliminated. Round 2 votes counts: D=34 A=25 E=22 C=19 so C is eliminated. Round 3 votes counts: A=42 D=34 E=24 so E is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:223 A:211 E:200 B:188 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 18 8 B -6 0 -12 4 -10 C 10 12 0 18 6 D -18 -4 -18 0 -4 E -8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 18 8 B -6 0 -12 4 -10 C 10 12 0 18 6 D -18 -4 -18 0 -4 E -8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 18 8 B -6 0 -12 4 -10 C 10 12 0 18 6 D -18 -4 -18 0 -4 E -8 10 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9443: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) A C D B E (7) D C B E A (6) B E D C A (6) C D E B A (5) A E C B D (5) A D C B E (5) A C E D B (5) C D B E A (4) B E D A C (4) A C D E B (4) E B D C A (3) C D A B E (3) A B E D C (3) E B D A C (2) D C B A E (2) D A C B E (2) C E D B A (2) C E A D B (2) C D B A E (2) C A D E B (2) A E B D C (2) A E B C D (2) A C E B D (2) E C A B D (1) E B C A D (1) E A B D C (1) E A B C D (1) D C A B E (1) D B C E A (1) D B A E C (1) C D E A B (1) B D E C A (1) B D E A C (1) A D B E C (1) A D B C E (1) A B D E C (1) Total count = 100 A B C D E A 0 4 0 -8 2 B -4 0 -22 -14 4 C 0 22 0 12 14 D 8 14 -12 0 2 E -2 -4 -14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.430676 B: 0.000000 C: 0.569324 D: 0.000000 E: 0.000000 Sum of squares = 0.509611759376 Cumulative probabilities = A: 0.430676 B: 0.430676 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 -8 2 B -4 0 -22 -14 4 C 0 22 0 12 14 D 8 14 -12 0 2 E -2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 C=21 E=16 D=13 B=12 so B is eliminated. Round 2 votes counts: A=38 E=26 C=21 D=15 so D is eliminated. Round 3 votes counts: A=41 C=31 E=28 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:224 D:206 A:199 E:189 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 -8 2 B -4 0 -22 -14 4 C 0 22 0 12 14 D 8 14 -12 0 2 E -2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 -8 2 B -4 0 -22 -14 4 C 0 22 0 12 14 D 8 14 -12 0 2 E -2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 -8 2 B -4 0 -22 -14 4 C 0 22 0 12 14 D 8 14 -12 0 2 E -2 -4 -14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999983 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9444: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) C E A B D (7) B D C A E (6) A E C D B (6) A C E B D (5) E C A B D (4) D B A C E (4) C B A E D (4) C A E B D (4) B D C E A (4) A C B E D (4) E A C D B (3) A E D C B (3) E D C B A (2) E C D B A (2) E C D A B (2) E A D C B (2) D E B C A (2) D E A B C (2) D B E A C (2) B C D A E (2) A D E C B (2) A C B D E (2) E D B C A (1) E D A C B (1) E C A D B (1) D E B A C (1) D B C E A (1) D A E B C (1) C E B A D (1) C B E D A (1) C A B D E (1) B D E C A (1) B D A C E (1) B C D E A (1) B A C D E (1) A D B E C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 -14 6 -2 B -6 0 -14 2 -4 C 14 14 0 4 2 D -6 -2 -4 0 -6 E 2 4 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999489 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -14 6 -2 B -6 0 -14 2 -4 C 14 14 0 4 2 D -6 -2 -4 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=23 E=18 C=18 B=16 so B is eliminated. Round 2 votes counts: D=35 A=26 C=21 E=18 so E is eliminated. Round 3 votes counts: D=39 A=31 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:217 E:205 A:198 D:191 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -14 6 -2 B -6 0 -14 2 -4 C 14 14 0 4 2 D -6 -2 -4 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 6 -2 B -6 0 -14 2 -4 C 14 14 0 4 2 D -6 -2 -4 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 6 -2 B -6 0 -14 2 -4 C 14 14 0 4 2 D -6 -2 -4 0 -6 E 2 4 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992237 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9445: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (16) D C B E A (14) C E B D A (7) A B E C D (6) E C B A D (5) A B E D C (4) D C E B A (3) D B C A E (3) C E B A D (3) E B A C D (2) E A C D B (2) E A C B D (2) E A B C D (2) D C E A B (2) D A C E B (2) C E D B A (2) C D E B A (2) C D E A B (2) C B D E A (2) B E A C D (2) B C D E A (2) B A E C D (2) A D B E C (2) E C D A B (1) D C B A E (1) D B A C E (1) C B E D A (1) B D C A E (1) B D A C E (1) B A E D C (1) B A D E C (1) A E D B C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -2 6 -12 B 12 0 -2 22 -10 C 2 2 0 22 -2 D -6 -22 -22 0 -18 E 12 10 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999142 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -2 6 -12 B 12 0 -2 22 -10 C 2 2 0 22 -2 D -6 -22 -22 0 -18 E 12 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=26 C=19 E=14 B=10 so B is eliminated. Round 2 votes counts: A=35 D=28 C=21 E=16 so E is eliminated. Round 3 votes counts: A=45 D=28 C=27 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:221 C:212 B:211 A:190 D:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -2 6 -12 B 12 0 -2 22 -10 C 2 2 0 22 -2 D -6 -22 -22 0 -18 E 12 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 6 -12 B 12 0 -2 22 -10 C 2 2 0 22 -2 D -6 -22 -22 0 -18 E 12 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 6 -12 B 12 0 -2 22 -10 C 2 2 0 22 -2 D -6 -22 -22 0 -18 E 12 10 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999986976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9446: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) C A B D E (10) E D B A C (6) E A D B C (6) D B E A C (5) A E C D B (5) E D B C A (4) C A E D B (4) C A E B D (4) B D E A C (4) B D C E A (4) A B D C E (4) E A C D B (3) D B E C A (3) A C B D E (3) E D A B C (2) C B D E A (2) B D E C A (2) B C A D E (2) A E D B C (2) A C E D B (2) D E B C A (1) D E B A C (1) C E D B A (1) C B A D E (1) B D C A E (1) B D A E C (1) B C D A E (1) A E D C B (1) A E C B D (1) A E B D C (1) A C E B D (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 2 0 4 12 B -2 0 2 8 10 C 0 -2 0 2 2 D -4 -8 -2 0 12 E -12 -10 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.627232 B: 0.000000 C: 0.372768 D: 0.000000 E: 0.000000 Sum of squares = 0.532376194706 Cumulative probabilities = A: 0.627232 B: 0.627232 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 4 12 B -2 0 2 8 10 C 0 -2 0 2 2 D -4 -8 -2 0 12 E -12 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500360 B: 0.000000 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258865 Cumulative probabilities = A: 0.500360 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=22 E=21 B=15 D=10 so D is eliminated. Round 2 votes counts: C=32 E=23 B=23 A=22 so A is eliminated. Round 3 votes counts: C=39 E=33 B=28 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:209 B:209 C:201 D:199 E:182 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 4 12 B -2 0 2 8 10 C 0 -2 0 2 2 D -4 -8 -2 0 12 E -12 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500360 B: 0.000000 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258865 Cumulative probabilities = A: 0.500360 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 4 12 B -2 0 2 8 10 C 0 -2 0 2 2 D -4 -8 -2 0 12 E -12 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500360 B: 0.000000 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258865 Cumulative probabilities = A: 0.500360 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 4 12 B -2 0 2 8 10 C 0 -2 0 2 2 D -4 -8 -2 0 12 E -12 -10 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500360 B: 0.000000 C: 0.499640 D: 0.000000 E: 0.000000 Sum of squares = 0.500000258865 Cumulative probabilities = A: 0.500360 B: 0.500360 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9447: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (14) D C E B A (10) A B E C D (10) D E C A B (8) D E A B C (7) D A E B C (6) D A B E C (4) C E D B A (3) A B E D C (3) A B C E D (3) E C D B A (2) E C B A D (2) E B A C D (2) D E C B A (2) D B A C E (2) D A B C E (2) C B E A D (2) C B A E D (2) A D B E C (2) A B D E C (2) E D C B A (1) E D C A B (1) E D A C B (1) E C A B D (1) E A B C D (1) D C B A E (1) D B C A E (1) C E B D A (1) C E B A D (1) C D E B A (1) C D B E A (1) B A D C E (1) Total count = 100 A B C D E A 0 2 20 -8 6 B -2 0 20 -10 0 C -20 -20 0 -8 -10 D 8 10 8 0 0 E -6 0 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.734588 E: 0.265412 Sum of squares = 0.610063026097 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.734588 E: 1.000000 A B C D E A 0 2 20 -8 6 B -2 0 20 -10 0 C -20 -20 0 -8 -10 D 8 10 8 0 0 E -6 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 A=20 B=15 E=11 C=11 so E is eliminated. Round 2 votes counts: D=46 A=21 B=17 C=16 so C is eliminated. Round 3 votes counts: D=53 B=25 A=22 so A is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:213 A:210 B:204 E:202 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 20 -8 6 B -2 0 20 -10 0 C -20 -20 0 -8 -10 D 8 10 8 0 0 E -6 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 20 -8 6 B -2 0 20 -10 0 C -20 -20 0 -8 -10 D 8 10 8 0 0 E -6 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 20 -8 6 B -2 0 20 -10 0 C -20 -20 0 -8 -10 D 8 10 8 0 0 E -6 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9448: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (10) D C E A B (10) D B C E A (8) B A E C D (7) B D C E A (6) A E C D B (6) C E A D B (5) A E C B D (5) A C E D B (5) E B C A D (4) C E D A B (4) D C A E B (3) B D A E C (3) B D A C E (3) A D C E B (3) E A C B D (2) D C E B A (2) B E C D A (2) B E A C D (2) B D E C A (2) A E B C D (2) D A C E B (1) C D E A B (1) B D E A C (1) B A E D C (1) B A D E C (1) A C D E B (1) Total count = 100 A B C D E A 0 16 -14 8 -18 B -16 0 -16 2 -28 C 14 16 0 12 4 D -8 -2 -12 0 -10 E 18 28 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -14 8 -18 B -16 0 -16 2 -28 C 14 16 0 12 4 D -8 -2 -12 0 -10 E 18 28 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=24 A=22 E=16 C=10 so C is eliminated. Round 2 votes counts: B=28 E=25 D=25 A=22 so A is eliminated. Round 3 votes counts: E=43 D=29 B=28 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:226 C:223 A:196 D:184 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -14 8 -18 B -16 0 -16 2 -28 C 14 16 0 12 4 D -8 -2 -12 0 -10 E 18 28 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -14 8 -18 B -16 0 -16 2 -28 C 14 16 0 12 4 D -8 -2 -12 0 -10 E 18 28 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -14 8 -18 B -16 0 -16 2 -28 C 14 16 0 12 4 D -8 -2 -12 0 -10 E 18 28 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999544 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9449: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (8) A C D B E (8) E D B A C (6) A C B E D (6) E B D C A (5) D A C E B (4) C A B D E (4) B D E C A (4) A C E B D (4) E D B C A (3) B E C A D (3) A E C B D (3) E B A D C (2) E A D C B (2) E A B C D (2) D E B A C (2) D E A C B (2) D C B A E (2) D B C E A (2) C B A D E (2) A C E D B (2) A C D E B (2) A C B D E (2) E D A B C (1) E B C D A (1) E B A C D (1) E A D B C (1) E A C B D (1) D E A B C (1) D C A B E (1) D B E C A (1) D B C A E (1) D A C B E (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D B E (1) B E D C A (1) B C E A D (1) B C A D E (1) A E D C B (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 4 12 4 0 B -4 0 -6 -10 -10 C -12 6 0 -6 -4 D -4 10 6 0 6 E 0 10 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.857338 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.142662 Sum of squares = 0.755380229499 Cumulative probabilities = A: 0.857338 B: 0.857338 C: 0.857338 D: 0.857338 E: 1.000000 A B C D E A 0 4 12 4 0 B -4 0 -6 -10 -10 C -12 6 0 -6 -4 D -4 10 6 0 6 E 0 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000360802 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=25 D=25 C=10 B=10 so C is eliminated. Round 2 votes counts: A=35 D=26 E=25 B=14 so B is eliminated. Round 3 votes counts: A=39 D=31 E=30 so E is eliminated. Round 4 votes counts: A=52 D=48 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 D:209 E:204 C:192 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 4 0 B -4 0 -6 -10 -10 C -12 6 0 -6 -4 D -4 10 6 0 6 E 0 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000360802 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 4 0 B -4 0 -6 -10 -10 C -12 6 0 -6 -4 D -4 10 6 0 6 E 0 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000360802 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 4 0 B -4 0 -6 -10 -10 C -12 6 0 -6 -4 D -4 10 6 0 6 E 0 10 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.600001 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.399999 Sum of squares = 0.520000360802 Cumulative probabilities = A: 0.600001 B: 0.600001 C: 0.600001 D: 0.600001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9450: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (15) B D A E C (10) E C A D B (6) C B A D E (6) B D E A C (4) B D A C E (4) B C D A E (4) E A D C B (3) D B A E C (3) C A E D B (3) B E C D A (3) A D E C B (3) A D E B C (3) E C D A B (2) E C B D A (2) E A D B C (2) C E B D A (2) C E B A D (2) C B D A E (2) C A D E B (2) C A D B E (2) B E D A C (2) B C E D A (2) E D B A C (1) E B D A C (1) E B C D A (1) E A C D B (1) D A B E C (1) C B E A D (1) B D C A E (1) A E D B C (1) A D C B E (1) A D B E C (1) A D B C E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 -12 10 0 B 2 0 -10 -6 -2 C 12 10 0 16 0 D -10 6 -16 0 0 E 0 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.355952 D: 0.000000 E: 0.644048 Sum of squares = 0.54149965595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.355952 D: 0.355952 E: 1.000000 A B C D E A 0 -2 -12 10 0 B 2 0 -10 -6 -2 C 12 10 0 16 0 D -10 6 -16 0 0 E 0 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 B=30 E=19 A=12 D=4 so D is eliminated. Round 2 votes counts: C=35 B=33 E=19 A=13 so A is eliminated. Round 3 votes counts: C=38 B=36 E=26 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:219 E:201 A:198 B:192 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -12 10 0 B 2 0 -10 -6 -2 C 12 10 0 16 0 D -10 6 -16 0 0 E 0 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 10 0 B 2 0 -10 -6 -2 C 12 10 0 16 0 D -10 6 -16 0 0 E 0 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 10 0 B 2 0 -10 -6 -2 C 12 10 0 16 0 D -10 6 -16 0 0 E 0 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9451: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) E A D C B (7) E D A C B (6) B C A D E (6) C B A E D (5) B C D E A (5) A C B D E (5) D A E C B (4) B C D A E (4) A E D C B (4) B C A E D (3) A C B E D (3) E D C A B (2) E C B A D (2) E C A B D (2) D E A C B (2) D A E B C (2) C A B E D (2) B C E D A (2) A D E C B (2) A D E B C (2) E D B C A (1) E D A B C (1) E C B D A (1) E C A D B (1) E B C D A (1) E A C B D (1) D B E C A (1) D A B E C (1) D A B C E (1) C B E D A (1) C B A D E (1) C A E B D (1) B E D C A (1) B D E C A (1) A E C D B (1) A D C B E (1) A D B C E (1) A C E B D (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 30 14 6 4 B -30 0 -12 -2 -10 C -14 12 0 0 -12 D -6 2 0 0 2 E -4 10 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 30 14 6 4 B -30 0 -12 -2 -10 C -14 12 0 0 -12 D -6 2 0 0 2 E -4 10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999566 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=22 A=22 D=21 C=10 so C is eliminated. Round 2 votes counts: B=29 E=25 A=25 D=21 so D is eliminated. Round 3 votes counts: E=37 A=33 B=30 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:227 E:208 D:199 C:193 B:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 30 14 6 4 B -30 0 -12 -2 -10 C -14 12 0 0 -12 D -6 2 0 0 2 E -4 10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999566 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 30 14 6 4 B -30 0 -12 -2 -10 C -14 12 0 0 -12 D -6 2 0 0 2 E -4 10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999566 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 30 14 6 4 B -30 0 -12 -2 -10 C -14 12 0 0 -12 D -6 2 0 0 2 E -4 10 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999566 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9452: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (8) B D A E C (8) C E A D B (6) C A E D B (6) A D C B E (6) A D C E B (5) E B D C A (4) D A B E C (4) C E B A D (4) C E A B D (4) E C B A D (3) E C A D B (3) B D E C A (3) B C E D A (3) A C D E B (3) E C B D A (2) D B A E C (2) D B A C E (2) D A E B C (2) D A B C E (2) B E D C A (2) E D A C B (1) E B D A C (1) E B C D A (1) E A D C B (1) D B E A C (1) C B E A D (1) C A D E B (1) C A D B E (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) A E C D B (1) A D E C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -2 2 -4 -2 B 2 0 0 0 -2 C -2 0 0 -14 0 D 4 0 14 0 10 E 2 2 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.577328 C: 0.000000 D: 0.422672 E: 0.000000 Sum of squares = 0.511959386599 Cumulative probabilities = A: 0.000000 B: 0.577328 C: 0.577328 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -4 -2 B 2 0 0 0 -2 C -2 0 0 -14 0 D 4 0 14 0 10 E 2 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999904 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=23 A=18 E=16 D=13 so D is eliminated. Round 2 votes counts: B=35 A=26 C=23 E=16 so E is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:214 B:200 A:197 E:197 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 2 -4 -2 B 2 0 0 0 -2 C -2 0 0 -14 0 D 4 0 14 0 10 E 2 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999904 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -4 -2 B 2 0 0 0 -2 C -2 0 0 -14 0 D 4 0 14 0 10 E 2 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999904 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -4 -2 B 2 0 0 0 -2 C -2 0 0 -14 0 D 4 0 14 0 10 E 2 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999904 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9453: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (13) D C B A E (10) C D A E B (9) C A E D B (9) E A B C D (8) B E A D C (8) E B A C D (6) D B C E A (4) D B C A E (4) E A C B D (3) C A D E B (3) B E D A C (3) B E A C D (3) D C A B E (2) D B A C E (2) B D A E C (2) E C A B D (1) E B A D C (1) D C B E A (1) D C A E B (1) D A C B E (1) D A B E C (1) C E A B D (1) A E D C B (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 10 -6 -4 B 14 0 10 0 8 C -10 -10 0 -8 -4 D 6 0 8 0 6 E 4 -8 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.432376 C: 0.000000 D: 0.567624 E: 0.000000 Sum of squares = 0.509145908717 Cumulative probabilities = A: 0.000000 B: 0.432376 C: 0.432376 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 -6 -4 B 14 0 10 0 8 C -10 -10 0 -8 -4 D 6 0 8 0 6 E 4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 C=22 E=19 A=4 so A is eliminated. Round 2 votes counts: B=29 D=26 C=23 E=22 so E is eliminated. Round 3 votes counts: B=44 C=29 D=27 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:216 D:210 E:197 A:193 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 -6 -4 B 14 0 10 0 8 C -10 -10 0 -8 -4 D 6 0 8 0 6 E 4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 -6 -4 B 14 0 10 0 8 C -10 -10 0 -8 -4 D 6 0 8 0 6 E 4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 -6 -4 B 14 0 10 0 8 C -10 -10 0 -8 -4 D 6 0 8 0 6 E 4 -8 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9454: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) E A D C B (9) A E D B C (8) E A C D B (6) C B E D A (5) B D C A E (5) A D B E C (5) A B D E C (5) D B A C E (4) B D A C E (4) B C D A E (4) E C A D B (3) E C A B D (3) D C B E A (3) D A E B C (3) D B C A E (2) C E B A D (2) C D B E A (2) A E B D C (2) A E B C D (2) E D C B A (1) E D C A B (1) E C D B A (1) E A D B C (1) E A C B D (1) C E D B A (1) C B D A E (1) C B A E D (1) B A D C E (1) A E C B D (1) A D E B C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 6 2 B -6 0 -2 -2 8 C -10 2 0 -12 -10 D -6 2 12 0 0 E -2 -8 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 6 2 B -6 0 -2 -2 8 C -10 2 0 -12 -10 D -6 2 12 0 0 E -2 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 A=26 C=22 B=14 D=12 so D is eliminated. Round 2 votes counts: A=29 E=26 C=25 B=20 so B is eliminated. Round 3 votes counts: A=38 C=36 E=26 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:212 D:204 E:200 B:199 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 6 2 B -6 0 -2 -2 8 C -10 2 0 -12 -10 D -6 2 12 0 0 E -2 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 6 2 B -6 0 -2 -2 8 C -10 2 0 -12 -10 D -6 2 12 0 0 E -2 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 6 2 B -6 0 -2 -2 8 C -10 2 0 -12 -10 D -6 2 12 0 0 E -2 -8 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997712 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9455: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) B C E D A (10) D A E B C (4) C E B D A (4) B D A C E (4) A E C D B (4) A D E C B (4) E C D B A (3) C E A B D (3) B D C E A (3) B C A E D (3) A D E B C (3) A C E D B (3) E C D A B (2) D E C B A (2) D E A C B (2) D B E A C (2) D B A E C (2) C E B A D (2) C E A D B (2) B D A E C (2) B A C D E (2) A C E B D (2) A B D C E (2) E D C B A (1) E D A C B (1) E C A D B (1) E A C D B (1) D A B E C (1) C E D B A (1) C E D A B (1) C B E D A (1) C B E A D (1) C B A E D (1) B D E C A (1) B D C A E (1) B C A D E (1) B A D E C (1) B A D C E (1) B A C E D (1) A E D C B (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 12 4 12 B 0 0 12 -4 6 C -12 -12 0 0 0 D -4 4 0 0 0 E -12 -6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.776492 B: 0.223508 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.65289595252 Cumulative probabilities = A: 0.776492 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 12 4 12 B 0 0 12 -4 6 C -12 -12 0 0 0 D -4 4 0 0 0 E -12 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500369 B: 0.499631 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272195 Cumulative probabilities = A: 0.500369 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=30 C=16 D=13 E=9 so E is eliminated. Round 2 votes counts: A=33 B=30 C=22 D=15 so D is eliminated. Round 3 votes counts: A=41 B=34 C=25 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 B:207 D:200 E:191 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 12 4 12 B 0 0 12 -4 6 C -12 -12 0 0 0 D -4 4 0 0 0 E -12 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500369 B: 0.499631 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272195 Cumulative probabilities = A: 0.500369 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 12 4 12 B 0 0 12 -4 6 C -12 -12 0 0 0 D -4 4 0 0 0 E -12 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500369 B: 0.499631 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272195 Cumulative probabilities = A: 0.500369 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 12 4 12 B 0 0 12 -4 6 C -12 -12 0 0 0 D -4 4 0 0 0 E -12 -6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500369 B: 0.499631 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500000272195 Cumulative probabilities = A: 0.500369 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9456: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (15) D B A E C (11) A B D E C (9) C E D A B (6) C E D B A (5) D E C B A (4) B A D E C (4) E C A B D (3) C D E B A (3) A B E C D (3) D E B C A (2) D C B A E (2) D B A C E (2) C E A D B (2) C D A B E (2) B A D C E (2) A C B E D (2) A B E D C (2) A B D C E (2) E D C B A (1) E D B C A (1) E C D B A (1) E C D A B (1) E B D A C (1) E A C B D (1) D E B A C (1) D C E B A (1) D C B E A (1) D B E A C (1) D A C B E (1) C A B E D (1) B D A E C (1) B A E D C (1) A E B C D (1) A C E B D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 10 -2 4 0 B -10 0 -8 4 0 C 2 8 0 0 2 D -4 -4 0 0 2 E 0 0 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.744548 D: 0.255452 E: 0.000000 Sum of squares = 0.619607707504 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.744548 D: 1.000000 E: 1.000000 A B C D E A 0 10 -2 4 0 B -10 0 -8 4 0 C 2 8 0 0 2 D -4 -4 0 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=26 A=23 E=9 B=8 so B is eliminated. Round 2 votes counts: C=34 A=30 D=27 E=9 so E is eliminated. Round 3 votes counts: C=39 A=31 D=30 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:206 C:206 E:198 D:197 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 4 0 B -10 0 -8 4 0 C 2 8 0 0 2 D -4 -4 0 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 4 0 B -10 0 -8 4 0 C 2 8 0 0 2 D -4 -4 0 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 4 0 B -10 0 -8 4 0 C 2 8 0 0 2 D -4 -4 0 0 2 E 0 0 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.666667 D: 0.333333 E: 0.000000 Sum of squares = 0.555555560409 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9457: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (15) C E A B D (10) D B A E C (9) B D A C E (9) D A B E C (6) B C E A D (6) D A E C B (5) D B A C E (4) B D E C A (4) E C A B D (3) B A C E D (3) D E A C B (2) B C E D A (2) A E C D B (2) A C E B D (2) E D C B A (1) E D C A B (1) E C D A B (1) E C B A D (1) D E C A B (1) D B E C A (1) D B E A C (1) D A B C E (1) C E B A D (1) C E A D B (1) C A B E D (1) B D C E A (1) B D C A E (1) B D A E C (1) B A C D E (1) A D E C B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -2 -2 -4 B -8 0 0 -6 4 C 2 0 0 2 -10 D 2 6 -2 0 -4 E 4 -4 10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.097959 B: 0.271720 C: 0.000000 D: 0.173761 E: 0.456560 Sum of squares = 0.322067509182 Cumulative probabilities = A: 0.097959 B: 0.369679 C: 0.369679 D: 0.543440 E: 1.000000 A B C D E A 0 8 -2 -2 -4 B -8 0 0 -6 4 C 2 0 0 2 -10 D 2 6 -2 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.101696 B: 0.271186 C: 0.000000 D: 0.169491 E: 0.457627 Sum of squares = 0.322033898305 Cumulative probabilities = A: 0.101696 B: 0.372882 C: 0.372882 D: 0.542373 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=28 E=22 C=13 A=7 so A is eliminated. Round 2 votes counts: D=31 B=28 E=24 C=17 so C is eliminated. Round 3 votes counts: E=39 D=31 B=30 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:207 D:201 A:200 C:197 B:195 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D E , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 -2 -4 B -8 0 0 -6 4 C 2 0 0 2 -10 D 2 6 -2 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.101696 B: 0.271186 C: 0.000000 D: 0.169491 E: 0.457627 Sum of squares = 0.322033898305 Cumulative probabilities = A: 0.101696 B: 0.372882 C: 0.372882 D: 0.542373 E: 1.000000 GTS winners are ['A', 'B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 -2 -4 B -8 0 0 -6 4 C 2 0 0 2 -10 D 2 6 -2 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.101696 B: 0.271186 C: 0.000000 D: 0.169491 E: 0.457627 Sum of squares = 0.322033898305 Cumulative probabilities = A: 0.101696 B: 0.372882 C: 0.372882 D: 0.542373 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 -2 -4 B -8 0 0 -6 4 C 2 0 0 2 -10 D 2 6 -2 0 -4 E 4 -4 10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.101696 B: 0.271186 C: 0.000000 D: 0.169491 E: 0.457627 Sum of squares = 0.322033898305 Cumulative probabilities = A: 0.101696 B: 0.372882 C: 0.372882 D: 0.542373 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9458: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (7) E A D C B (6) A D C B E (6) E D B A C (5) E B C A D (5) E D A B C (4) E B D C A (4) E B C D A (4) D B A C E (4) D A B C E (4) C A D B E (3) B E D C A (3) B E C D A (3) B C D A E (3) A D C E B (3) E D A C B (2) E C B A D (2) E B D A C (2) D E A B C (2) D A E B C (2) D A C B E (2) C A E B D (2) C A B E D (2) B D C A E (2) B C E D A (2) B C D E A (2) A C D B E (2) E C A B D (1) E A D B C (1) D B A E C (1) D A E C B (1) C E B A D (1) C B A E D (1) C A E D B (1) C A B D E (1) B C E A D (1) A E D C B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 0 -4 0 B 4 0 8 -4 -2 C 0 -8 0 -12 0 D 4 4 12 0 -8 E 0 2 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.139563 B: 0.000000 C: 0.088467 D: 0.000000 E: 0.771970 Sum of squares = 0.623241933196 Cumulative probabilities = A: 0.139563 B: 0.139563 C: 0.228030 D: 0.228030 E: 1.000000 A B C D E A 0 -4 0 -4 0 B 4 0 8 -4 -2 C 0 -8 0 -12 0 D 4 4 12 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.289474 B: 0.000000 C: 0.026316 D: 0.000000 E: 0.684211 Sum of squares = 0.55263158376 Cumulative probabilities = A: 0.289474 B: 0.289474 C: 0.315789 D: 0.315789 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 C=18 D=16 B=16 A=14 so A is eliminated. Round 2 votes counts: E=37 D=26 C=21 B=16 so B is eliminated. Round 3 votes counts: E=43 C=29 D=28 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:206 E:205 B:203 A:196 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 -4 0 B 4 0 8 -4 -2 C 0 -8 0 -12 0 D 4 4 12 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.289474 B: 0.000000 C: 0.026316 D: 0.000000 E: 0.684211 Sum of squares = 0.55263158376 Cumulative probabilities = A: 0.289474 B: 0.289474 C: 0.315789 D: 0.315789 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -4 0 B 4 0 8 -4 -2 C 0 -8 0 -12 0 D 4 4 12 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.289474 B: 0.000000 C: 0.026316 D: 0.000000 E: 0.684211 Sum of squares = 0.55263158376 Cumulative probabilities = A: 0.289474 B: 0.289474 C: 0.315789 D: 0.315789 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -4 0 B 4 0 8 -4 -2 C 0 -8 0 -12 0 D 4 4 12 0 -8 E 0 2 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.289474 B: 0.000000 C: 0.026316 D: 0.000000 E: 0.684211 Sum of squares = 0.55263158376 Cumulative probabilities = A: 0.289474 B: 0.289474 C: 0.315789 D: 0.315789 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9459: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E C D (14) C B E D A (10) D C E B A (8) D C B E A (7) C D B E A (5) A D E B C (5) E B C D A (4) E B C A D (4) D C A B E (4) B E A C D (3) A B E D C (3) A B C E D (3) D A E C B (2) D A C B E (2) C B E A D (2) B E C A D (2) A D B E C (2) E D C B A (1) E D B C A (1) E C B D A (1) E B D C A (1) E B A C D (1) D E C B A (1) D E A B C (1) D C E A B (1) D C B A E (1) D C A E B (1) C B D E A (1) C B D A E (1) B C A E D (1) B A C E D (1) A E D B C (1) A E B D C (1) A E B C D (1) A D C E B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 -14 -6 -8 B 12 0 -2 10 28 C 14 2 0 10 2 D 6 -10 -10 0 -12 E 8 -28 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -14 -6 -8 B 12 0 -2 10 28 C 14 2 0 10 2 D 6 -10 -10 0 -12 E 8 -28 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=28 C=19 E=13 B=7 so B is eliminated. Round 2 votes counts: A=34 D=28 C=20 E=18 so E is eliminated. Round 3 votes counts: A=38 D=31 C=31 so D is eliminated. Round 4 votes counts: C=57 A=43 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:224 C:214 E:195 D:187 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -14 -6 -8 B 12 0 -2 10 28 C 14 2 0 10 2 D 6 -10 -10 0 -12 E 8 -28 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 -6 -8 B 12 0 -2 10 28 C 14 2 0 10 2 D 6 -10 -10 0 -12 E 8 -28 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 -6 -8 B 12 0 -2 10 28 C 14 2 0 10 2 D 6 -10 -10 0 -12 E 8 -28 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997853 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9460: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (9) E D B A C (6) E B D A C (6) D E A C B (6) B A C E D (6) A C B D E (6) D E C B A (5) D E B C A (5) D C A E B (5) C A B D E (5) E D B C A (4) C A B E D (4) B E C A D (4) A C D B E (4) E B D C A (3) D E C A B (3) D A C E B (3) B E A C D (3) B C E A D (2) B C A E D (2) D E B A C (1) D C E A B (1) D A E C B (1) C D B E A (1) C D A E B (1) A D C E B (1) A D C B E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -8 -2 0 B -4 0 -12 -14 -2 C 8 12 0 -2 4 D 2 14 2 0 16 E 0 2 -4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -2 0 B -4 0 -12 -14 -2 C 8 12 0 -2 4 D 2 14 2 0 16 E 0 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=20 E=19 B=17 A=14 so A is eliminated. Round 2 votes counts: D=32 C=30 E=19 B=19 so E is eliminated. Round 3 votes counts: D=42 C=30 B=28 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:217 C:211 A:197 E:191 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -2 0 B -4 0 -12 -14 -2 C 8 12 0 -2 4 D 2 14 2 0 16 E 0 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -2 0 B -4 0 -12 -14 -2 C 8 12 0 -2 4 D 2 14 2 0 16 E 0 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -2 0 B -4 0 -12 -14 -2 C 8 12 0 -2 4 D 2 14 2 0 16 E 0 2 -4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999988145 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9461: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) E C B D A (7) A D B C E (7) A D E C B (6) E C D B A (4) D A B C E (4) B C E D A (4) A B D C E (4) E D C B A (3) D E C A B (3) D A E B C (3) C E B A D (3) C B E A D (3) B A D C E (3) E D C A B (2) E C D A B (2) E C A D B (2) D E B C A (2) D E A C B (2) D B A C E (2) D A E C B (2) B C E A D (2) B C D E A (2) B A C E D (2) A D B E C (2) A C E B D (2) E C B A D (1) E C A B D (1) D E C B A (1) D E A B C (1) D B E C A (1) D B A E C (1) C B E D A (1) C A E B D (1) B D A C E (1) B C A E D (1) B A C D E (1) A D C E B (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 10 8 -12 6 B -10 0 4 -20 0 C -8 -4 0 -18 -8 D 12 20 18 0 16 E -6 0 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 -12 6 B -10 0 4 -20 0 C -8 -4 0 -18 -8 D 12 20 18 0 16 E -6 0 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=24 E=22 B=16 C=8 so C is eliminated. Round 2 votes counts: D=30 E=25 A=25 B=20 so B is eliminated. Round 3 votes counts: E=35 D=33 A=32 so A is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:233 A:206 E:193 B:187 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 8 -12 6 B -10 0 4 -20 0 C -8 -4 0 -18 -8 D 12 20 18 0 16 E -6 0 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 -12 6 B -10 0 4 -20 0 C -8 -4 0 -18 -8 D 12 20 18 0 16 E -6 0 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 -12 6 B -10 0 4 -20 0 C -8 -4 0 -18 -8 D 12 20 18 0 16 E -6 0 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9462: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) D C E B A (10) A C D E B (8) B E A C D (7) B E D A C (5) A B E D C (5) D E C B A (4) C D A E B (4) B E A D C (4) A C D B E (4) A C B E D (4) A B C E D (4) E B D C A (3) D E B C A (3) C D E B A (3) A D C B E (3) C A D E B (2) A D C E B (2) A C B D E (2) A B E C D (2) E D B C A (1) D C E A B (1) D C A E B (1) D A E B C (1) C D E A B (1) C B E A D (1) B E C D A (1) B A E C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 6 0 -12 B 10 0 0 2 12 C -6 0 0 -12 0 D 0 -2 12 0 0 E 12 -12 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.900963 C: 0.099037 D: 0.000000 E: 0.000000 Sum of squares = 0.821543322996 Cumulative probabilities = A: 0.000000 B: 0.900963 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 0 -12 B 10 0 0 2 12 C -6 0 0 -12 0 D 0 -2 12 0 0 E 12 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041351 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=30 D=20 C=11 E=4 so E is eliminated. Round 2 votes counts: A=35 B=33 D=21 C=11 so C is eliminated. Round 3 votes counts: A=37 B=34 D=29 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:205 E:200 A:192 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 0 -12 B 10 0 0 2 12 C -6 0 0 -12 0 D 0 -2 12 0 0 E 12 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041351 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 0 -12 B 10 0 0 2 12 C -6 0 0 -12 0 D 0 -2 12 0 0 E 12 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041351 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 0 -12 B 10 0 0 2 12 C -6 0 0 -12 0 D 0 -2 12 0 0 E 12 -12 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.857143 C: 0.142857 D: 0.000000 E: 0.000000 Sum of squares = 0.755102041351 Cumulative probabilities = A: 0.000000 B: 0.857143 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9463: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) A C B E D (8) D A B C E (7) E C B A D (6) E B D C A (6) D A B E C (6) C E B A D (6) A D C B E (6) A C E B D (5) C A E B D (4) E B C A D (3) D B E C A (3) D B A E C (3) B E D C A (3) B D E C A (3) D E B C A (2) D B E A C (2) D A C E B (2) C E A B D (2) A C D E B (2) A C B D E (2) E D B C A (1) E C D B A (1) E C D A B (1) D A C B E (1) C B E A D (1) C A B E D (1) B E C D A (1) B C E A D (1) A D B C E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 0 -6 0 0 B 0 0 0 20 0 C 6 0 0 8 2 D 0 -20 -8 0 -18 E 0 0 -2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.257644 C: 0.742356 D: 0.000000 E: 0.000000 Sum of squares = 0.617472519624 Cumulative probabilities = A: 0.000000 B: 0.257644 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -6 0 0 B 0 0 0 20 0 C 6 0 0 8 2 D 0 -20 -8 0 -18 E 0 0 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A D E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=26 D=26 A=26 C=14 B=8 so B is eliminated. Round 2 votes counts: E=30 D=29 A=26 C=15 so C is eliminated. Round 3 votes counts: E=40 A=31 D=29 so D is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:210 C:208 E:208 A:197 D:177 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 0 0 B 0 0 0 20 0 C 6 0 0 8 2 D 0 -20 -8 0 -18 E 0 0 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 0 0 B 0 0 0 20 0 C 6 0 0 8 2 D 0 -20 -8 0 -18 E 0 0 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 0 0 B 0 0 0 20 0 C 6 0 0 8 2 D 0 -20 -8 0 -18 E 0 0 -2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999975 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9464: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (11) C D A E B (7) B D A C E (6) E A B C D (5) E A C D B (4) D A C E B (4) A E D C B (4) E B C A D (3) C D B A E (3) C D A B E (3) B E A D C (3) B E A C D (3) A D C E B (3) E C A D B (2) E B A D C (2) E B A C D (2) E A D C B (2) E A B D C (2) D C A B E (2) D A B C E (2) C E D A B (2) C B E D A (2) B C E D A (2) B A D E C (2) A D E C B (2) E C B A D (1) E C A B D (1) E A D B C (1) D B A E C (1) D B A C E (1) C E D B A (1) C E B A D (1) C B D E A (1) C A D E B (1) B E C D A (1) B D C E A (1) B D C A E (1) B D A E C (1) A E D B C (1) A E C D B (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 8 18 -6 B -2 0 6 4 -6 C -8 -6 0 14 -14 D -18 -4 -14 0 -16 E 6 6 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 8 18 -6 B -2 0 6 4 -6 C -8 -6 0 14 -14 D -18 -4 -14 0 -16 E 6 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 C=21 A=13 D=10 so D is eliminated. Round 2 votes counts: B=33 E=25 C=23 A=19 so A is eliminated. Round 3 votes counts: B=36 E=34 C=30 so C is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 A:211 B:201 C:193 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 8 18 -6 B -2 0 6 4 -6 C -8 -6 0 14 -14 D -18 -4 -14 0 -16 E 6 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 18 -6 B -2 0 6 4 -6 C -8 -6 0 14 -14 D -18 -4 -14 0 -16 E 6 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 18 -6 B -2 0 6 4 -6 C -8 -6 0 14 -14 D -18 -4 -14 0 -16 E 6 6 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9465: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (9) E A C B D (6) B D C A E (6) E D B C A (5) E B D A C (5) E D C B A (4) E B D C A (4) E A C D B (4) D B C A E (4) A C B D E (4) E C D B A (3) E C A D B (3) D C B E A (3) A C E D B (3) E B A D C (2) D C B A E (2) D B C E A (2) C E D B A (2) C D A B E (2) C A D B E (2) B D A C E (2) E D B A C (1) E C D A B (1) E A B C D (1) C D E B A (1) C D A E B (1) C A D E B (1) C A B D E (1) B E D C A (1) B C D A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -18 -16 -18 -14 B 18 0 -8 -8 -16 C 16 8 0 -2 -2 D 18 8 2 0 -12 E 14 16 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -16 -18 -14 B 18 0 -8 -8 -16 C 16 8 0 -2 -2 D 18 8 2 0 -12 E 14 16 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=48 C=19 B=12 D=11 A=10 so A is eliminated. Round 2 votes counts: E=50 C=26 B=13 D=11 so D is eliminated. Round 3 votes counts: E=50 C=31 B=19 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:210 D:208 B:193 A:167 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -16 -18 -14 B 18 0 -8 -8 -16 C 16 8 0 -2 -2 D 18 8 2 0 -12 E 14 16 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -18 -14 B 18 0 -8 -8 -16 C 16 8 0 -2 -2 D 18 8 2 0 -12 E 14 16 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -18 -14 B 18 0 -8 -8 -16 C 16 8 0 -2 -2 D 18 8 2 0 -12 E 14 16 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9466: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) E C B D A (8) E C A D B (7) E C A B D (6) C B E D A (5) B D A C E (5) C E B A D (4) E D A B C (3) E A D B C (3) D A B E C (3) C E A B D (3) C B A D E (3) B D C A E (3) A D E B C (3) E D B A C (2) E C B A D (2) E B C D A (2) E A D C B (2) E A C D B (2) D B A C E (2) D A B C E (2) C E B D A (2) B C D A E (2) A C D B E (2) E C D B A (1) E B D C A (1) E B D A C (1) D A E B C (1) C B D A E (1) C A D B E (1) C A B D E (1) B D A E C (1) B C D E A (1) B A D C E (1) A E C D B (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 6 -6 8 -10 B -6 0 -6 4 -10 C 6 6 0 8 -2 D -8 -4 -8 0 -10 E 10 10 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -6 8 -10 B -6 0 -6 4 -10 C 6 6 0 8 -2 D -8 -4 -8 0 -10 E 10 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=20 A=19 B=13 D=8 so D is eliminated. Round 2 votes counts: E=40 A=25 C=20 B=15 so B is eliminated. Round 3 votes counts: E=40 A=34 C=26 so C is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:216 C:209 A:199 B:191 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -6 8 -10 B -6 0 -6 4 -10 C 6 6 0 8 -2 D -8 -4 -8 0 -10 E 10 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 8 -10 B -6 0 -6 4 -10 C 6 6 0 8 -2 D -8 -4 -8 0 -10 E 10 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 8 -10 B -6 0 -6 4 -10 C 6 6 0 8 -2 D -8 -4 -8 0 -10 E 10 10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9467: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (5) B E A D C (5) E D B A C (4) C A D E B (4) C A B E D (4) B C E D A (4) D E A C B (3) C D B A E (3) B E A C D (3) B C A E D (3) A E D C B (3) A E B D C (3) A E B C D (3) E B D A C (2) E A D B C (2) D E C B A (2) D E B A C (2) D E A B C (2) D C E B A (2) D C A E B (2) D B E C A (2) C B A D E (2) C A B D E (2) B E D A C (2) B C E A D (2) B A E C D (2) A E C B D (2) A B E C D (2) A B C E D (2) E B A D C (1) D E C A B (1) D E B C A (1) D C E A B (1) D C B E A (1) D A E C B (1) D A C E B (1) C D A B E (1) C B D A E (1) C B A E D (1) C A E D B (1) C A E B D (1) C A D B E (1) B E C A D (1) B D E C A (1) B C D E A (1) A E D B C (1) A D C E B (1) A C E D B (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 4 0 10 10 B -4 0 2 2 -6 C 0 -2 0 8 -2 D -10 -2 -8 0 -12 E -10 6 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.680861 B: 0.000000 C: 0.319139 D: 0.000000 E: 0.000000 Sum of squares = 0.565421093923 Cumulative probabilities = A: 0.680861 B: 0.680861 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 10 10 B -4 0 2 2 -6 C 0 -2 0 8 -2 D -10 -2 -8 0 -12 E -10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=24 D=21 A=20 E=9 so E is eliminated. Round 2 votes counts: B=27 C=26 D=25 A=22 so A is eliminated. Round 3 votes counts: B=37 D=32 C=31 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:212 E:205 C:202 B:197 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 10 10 B -4 0 2 2 -6 C 0 -2 0 8 -2 D -10 -2 -8 0 -12 E -10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 10 10 B -4 0 2 2 -6 C 0 -2 0 8 -2 D -10 -2 -8 0 -12 E -10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 10 10 B -4 0 2 2 -6 C 0 -2 0 8 -2 D -10 -2 -8 0 -12 E -10 6 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500001 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999862 Cumulative probabilities = A: 0.500001 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9468: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A E C (11) E C A D B (10) B D A C E (10) C E A D B (8) B D C E A (8) A E C D B (6) E C A B D (5) B C E D A (4) E C B A D (3) C E D B A (3) B C D E A (3) B A D E C (3) A E C B D (3) E A C D B (2) D B A C E (2) D A C E B (2) C E D A B (2) E A C B D (1) E A B C D (1) D B C A E (1) D B A E C (1) D A E C B (1) D A C B E (1) C E B D A (1) C E B A D (1) C D E B A (1) C B E D A (1) C B D E A (1) B E C D A (1) B E C A D (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -8 -8 -14 B 12 0 -6 16 -2 C 8 6 0 16 -2 D 8 -16 -16 0 -8 E 14 2 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999616 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -8 -8 -14 B 12 0 -6 16 -2 C 8 6 0 16 -2 D 8 -16 -16 0 -8 E 14 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 E=22 C=18 A=11 D=8 so D is eliminated. Round 2 votes counts: B=45 E=22 C=18 A=15 so A is eliminated. Round 3 votes counts: B=46 E=33 C=21 so C is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. C:214 E:213 B:210 D:184 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 -8 -14 B 12 0 -6 16 -2 C 8 6 0 16 -2 D 8 -16 -16 0 -8 E 14 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -8 -14 B 12 0 -6 16 -2 C 8 6 0 16 -2 D 8 -16 -16 0 -8 E 14 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -8 -14 B 12 0 -6 16 -2 C 8 6 0 16 -2 D 8 -16 -16 0 -8 E 14 2 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9469: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (14) D E A B C (10) B C D E A (10) E A D C B (9) A E D C B (6) B C D A E (5) B C A E D (5) B C A D E (5) D B E A C (4) B D C E A (4) D E A C B (3) A E C B D (3) E D A C B (2) D B E C A (2) D B C E A (2) C B D E A (2) C A E B D (2) C A B E D (2) A E C D B (2) E A C D B (1) D C B E A (1) D A E B C (1) C B E D A (1) C B E A D (1) B D C A E (1) A E D B C (1) A C B E D (1) Total count = 100 A B C D E A 0 -14 -14 4 -4 B 14 0 0 12 20 C 14 0 0 8 12 D -4 -12 -8 0 0 E 4 -20 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.138970 C: 0.861030 D: 0.000000 E: 0.000000 Sum of squares = 0.760684748772 Cumulative probabilities = A: 0.000000 B: 0.138970 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -14 4 -4 B 14 0 0 12 20 C 14 0 0 8 12 D -4 -12 -8 0 0 E 4 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=23 C=22 A=13 E=12 so E is eliminated. Round 2 votes counts: B=30 D=25 A=23 C=22 so C is eliminated. Round 3 votes counts: B=48 A=27 D=25 so D is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:217 D:188 A:186 E:186 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -14 4 -4 B 14 0 0 12 20 C 14 0 0 8 12 D -4 -12 -8 0 0 E 4 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 4 -4 B 14 0 0 12 20 C 14 0 0 8 12 D -4 -12 -8 0 0 E 4 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 4 -4 B 14 0 0 12 20 C 14 0 0 8 12 D -4 -12 -8 0 0 E 4 -20 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9470: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (6) C A E D B (6) B A E D C (6) D C B E A (5) D B E C A (5) C D E A B (5) A E B C D (5) D C E B A (4) D B C E A (4) C A E B D (4) C D B E A (3) C D B A E (3) C D A E B (3) C A D B E (3) A E B D C (3) A B E D C (3) E A C B D (2) C E A D B (2) C D A B E (2) C A D E B (2) B E D A C (2) B E A D C (2) B D E A C (2) A E C B D (2) A C E B D (2) A C B E D (2) E D C B A (1) E C D A B (1) E C A B D (1) E B D A C (1) E A D B C (1) D E B C A (1) D B C A E (1) C D E B A (1) C A B E D (1) B D A E C (1) B A D E C (1) A B E C D (1) Total count = 100 A B C D E A 0 2 -16 -2 2 B -2 0 -10 -18 6 C 16 10 0 2 6 D 2 18 -2 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -16 -2 2 B -2 0 -10 -18 6 C 16 10 0 2 6 D 2 18 -2 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=26 A=18 B=14 E=7 so E is eliminated. Round 2 votes counts: C=37 D=27 A=21 B=15 so B is eliminated. Round 3 votes counts: C=37 D=33 A=30 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:217 D:211 A:193 E:191 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -16 -2 2 B -2 0 -10 -18 6 C 16 10 0 2 6 D 2 18 -2 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -16 -2 2 B -2 0 -10 -18 6 C 16 10 0 2 6 D 2 18 -2 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -16 -2 2 B -2 0 -10 -18 6 C 16 10 0 2 6 D 2 18 -2 0 4 E -2 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999993209 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9471: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (7) A D B E C (7) C D A B E (6) A D E B C (6) C E B D A (5) C E B A D (5) E B C A D (4) E B A D C (4) C E A B D (4) E B C D A (3) D B A E C (3) C D A E B (3) E C B D A (2) E C A B D (2) E A B D C (2) D C A B E (2) D A C B E (2) D A B E C (2) C D B E A (2) C D B A E (2) C A E D B (2) C A D E B (2) B E D C A (2) B E C D A (2) A E D B C (2) A E B D C (2) A D C B E (2) A C D E B (2) E C B A D (1) E B D A C (1) E B A C D (1) E A B C D (1) D A B C E (1) C E A D B (1) C B D E A (1) B E A D C (1) B D E A C (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 6 -2 6 -4 B -6 0 6 2 -12 C 2 -6 0 2 -12 D -6 -2 -2 0 -8 E 4 12 12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999477 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -2 6 -4 B -6 0 6 2 -12 C 2 -6 0 2 -12 D -6 -2 -2 0 -8 E 4 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=23 E=21 B=13 D=10 so D is eliminated. Round 2 votes counts: C=35 A=28 E=21 B=16 so B is eliminated. Round 3 votes counts: C=35 E=34 A=31 so A is eliminated. Round 4 votes counts: E=56 C=44 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:218 A:203 B:195 C:193 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -2 6 -4 B -6 0 6 2 -12 C 2 -6 0 2 -12 D -6 -2 -2 0 -8 E 4 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 6 -4 B -6 0 6 2 -12 C 2 -6 0 2 -12 D -6 -2 -2 0 -8 E 4 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 6 -4 B -6 0 6 2 -12 C 2 -6 0 2 -12 D -6 -2 -2 0 -8 E 4 12 12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9472: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) A E B D C (7) C D B E A (6) C D B A E (6) C B D E A (6) E A B C D (5) C E B D A (5) A D B C E (5) E C B D A (4) D A B C E (4) A D B E C (4) E A B D C (3) B D C A E (3) B C D E A (3) E C B A D (2) E C A D B (2) E B A C D (2) E A C D B (2) E A C B D (2) C E D B A (2) C D E B A (2) A E D B C (2) A B D E C (2) E C D B A (1) E C A B D (1) E B C D A (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C E B (1) C B E D A (1) B D A C E (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 -16 -16 -20 0 B 16 0 -10 -2 10 C 16 10 0 8 18 D 20 2 -8 0 14 E 0 -10 -18 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -16 -20 0 B 16 0 -10 -2 10 C 16 10 0 8 18 D 20 2 -8 0 14 E 0 -10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 E=25 A=21 D=18 B=8 so B is eliminated. Round 2 votes counts: C=32 E=25 D=22 A=21 so A is eliminated. Round 3 votes counts: E=35 D=33 C=32 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:226 D:214 B:207 E:179 A:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -16 -20 0 B 16 0 -10 -2 10 C 16 10 0 8 18 D 20 2 -8 0 14 E 0 -10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -16 -20 0 B 16 0 -10 -2 10 C 16 10 0 8 18 D 20 2 -8 0 14 E 0 -10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -16 -20 0 B 16 0 -10 -2 10 C 16 10 0 8 18 D 20 2 -8 0 14 E 0 -10 -18 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9473: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) D C A E B (8) C D A E B (6) B E A D C (5) B E A C D (5) E B A D C (4) E A B C D (4) D C B E A (4) D B E C A (4) C A D E B (4) B E C A D (4) A E C B D (4) E B A C D (3) D B C E A (3) A C E B D (3) D A E B C (2) D A C E B (2) C D B E A (2) C D A B E (2) C A E B D (2) B E C D A (2) A C D E B (2) E B D A C (1) E A B D C (1) D C A B E (1) D B E A C (1) D A E C B (1) C D B A E (1) C B D E A (1) C B A E D (1) B E D C A (1) B E D A C (1) B D E C A (1) B D C E A (1) B C E D A (1) A E D B C (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 2 8 2 B -8 0 8 8 -18 C -2 -8 0 14 -10 D -8 -8 -14 0 -6 E -2 18 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998839 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 8 2 B -8 0 8 8 -18 C -2 -8 0 14 -10 D -8 -8 -14 0 -6 E -2 18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=21 A=21 C=19 E=13 so E is eliminated. Round 2 votes counts: B=29 D=26 A=26 C=19 so C is eliminated. Round 3 votes counts: D=37 A=32 B=31 so B is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:216 A:210 C:197 B:195 D:182 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 8 2 B -8 0 8 8 -18 C -2 -8 0 14 -10 D -8 -8 -14 0 -6 E -2 18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 8 2 B -8 0 8 8 -18 C -2 -8 0 14 -10 D -8 -8 -14 0 -6 E -2 18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 8 2 B -8 0 8 8 -18 C -2 -8 0 14 -10 D -8 -8 -14 0 -6 E -2 18 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999756 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9474: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (12) C E B A D (11) D A B E C (10) D E A B C (8) C B A E D (6) E C D B A (5) A B D C E (5) E D C B A (4) C A B E D (4) A D B C E (4) D E B A C (3) D A E B C (3) B A C E D (3) A B C E D (3) E C B A D (2) D E C A B (2) D C E A B (2) D A B C E (2) E C B D A (1) E B D A C (1) D E C B A (1) D A C E B (1) D A C B E (1) C E D A B (1) C B E A D (1) B D E A C (1) B A E D C (1) B A D C E (1) B A C D E (1) Total count = 100 A B C D E A 0 16 20 8 14 B -16 0 16 6 10 C -20 -16 0 0 16 D -8 -6 0 0 14 E -14 -10 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 20 8 14 B -16 0 16 6 10 C -20 -16 0 0 16 D -8 -6 0 0 14 E -14 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=24 C=23 E=13 B=7 so B is eliminated. Round 2 votes counts: D=34 A=30 C=23 E=13 so E is eliminated. Round 3 votes counts: D=39 C=31 A=30 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:229 B:208 D:200 C:190 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 20 8 14 B -16 0 16 6 10 C -20 -16 0 0 16 D -8 -6 0 0 14 E -14 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 20 8 14 B -16 0 16 6 10 C -20 -16 0 0 16 D -8 -6 0 0 14 E -14 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 20 8 14 B -16 0 16 6 10 C -20 -16 0 0 16 D -8 -6 0 0 14 E -14 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999935 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9475: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (6) E D C B A (5) C E D A B (5) B A D E C (5) A B D C E (5) D E C A B (4) A D B C E (4) E C B D A (3) C E B D A (3) C A D E B (3) B C E A D (3) B A E D C (3) B A C D E (3) A D C B E (3) A D B E C (3) A B D E C (3) E D B A C (2) E C D B A (2) D A E B C (2) C E D B A (2) C D E A B (2) C D A E B (2) B E D A C (2) B E A D C (2) B C A E D (2) B A E C D (2) A D C E B (2) A B C D E (2) E D B C A (1) D E B A C (1) D E A B C (1) D C E A B (1) D A E C B (1) C B E D A (1) C B E A D (1) C A E D B (1) B E C A D (1) B E A C D (1) B D E A C (1) B A D C E (1) B A C E D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 4 6 0 B 8 0 14 4 14 C -4 -14 0 -4 -2 D -6 -4 4 0 2 E 0 -14 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 6 0 B 8 0 14 4 14 C -4 -14 0 -4 -2 D -6 -4 4 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 A=24 C=20 E=13 D=10 so D is eliminated. Round 2 votes counts: B=33 A=27 C=21 E=19 so E is eliminated. Round 3 votes counts: B=37 C=35 A=28 so A is eliminated. Round 4 votes counts: B=57 C=43 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:220 A:201 D:198 E:193 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 6 0 B 8 0 14 4 14 C -4 -14 0 -4 -2 D -6 -4 4 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 6 0 B 8 0 14 4 14 C -4 -14 0 -4 -2 D -6 -4 4 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 6 0 B 8 0 14 4 14 C -4 -14 0 -4 -2 D -6 -4 4 0 2 E 0 -14 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9476: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (24) E A B D C (18) D C B A E (7) C D B E A (7) A E B D C (6) E A C B D (5) E A B C D (4) D B C A E (4) E C A D B (3) C D E B A (2) B D C A E (2) B A E D C (2) A E B C D (2) A B D E C (2) E D C B A (1) E D B A C (1) D E C B A (1) C E A D B (1) C D E A B (1) C D A B E (1) B D E A C (1) B D A C E (1) B A D E C (1) A B E D C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -8 -8 -6 10 B 8 0 -6 -6 10 C 8 6 0 2 4 D 6 6 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -6 10 B 8 0 -6 -6 10 C 8 6 0 2 4 D 6 6 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 E=32 A=13 D=12 B=7 so B is eliminated. Round 2 votes counts: C=36 E=32 D=16 A=16 so D is eliminated. Round 3 votes counts: C=49 E=34 A=17 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:210 B:203 A:194 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -8 -6 10 B 8 0 -6 -6 10 C 8 6 0 2 4 D 6 6 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -6 10 B 8 0 -6 -6 10 C 8 6 0 2 4 D 6 6 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -6 10 B 8 0 -6 -6 10 C 8 6 0 2 4 D 6 6 -2 0 10 E -10 -10 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999885 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9477: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) A C D E B (8) B C D E A (7) C B D A E (6) D C A E B (4) C D A B E (4) B E C D A (4) B C A E D (4) E D B A C (3) E B A D C (3) E B A C D (3) E A D B C (3) B E D C A (3) B E D A C (3) B E C A D (3) E D A B C (2) E B D A C (2) D C B E A (2) D B E C A (2) C D B A E (2) C B A D E (2) C A D E B (2) C A D B E (2) B E A D C (2) A E D C B (2) A E C D B (2) D E A C B (1) D E A B C (1) D C B A E (1) D A E C B (1) D A C E B (1) C A B D E (1) B D C E A (1) B C E D A (1) A E C B D (1) A D E C B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -2 -2 -8 B 8 0 10 10 0 C 2 -10 0 8 0 D 2 -10 -8 0 -2 E 8 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.472497 C: 0.000000 D: 0.000000 E: 0.527503 Sum of squares = 0.501512832887 Cumulative probabilities = A: 0.000000 B: 0.472497 C: 0.472497 D: 0.472497 E: 1.000000 A B C D E A 0 -8 -2 -2 -8 B 8 0 10 10 0 C 2 -10 0 8 0 D 2 -10 -8 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=24 C=19 A=16 D=13 so D is eliminated. Round 2 votes counts: B=30 E=26 C=26 A=18 so A is eliminated. Round 3 votes counts: C=37 E=33 B=30 so B is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. B:214 E:205 C:200 D:191 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -2 -8 B 8 0 10 10 0 C 2 -10 0 8 0 D 2 -10 -8 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -2 -8 B 8 0 10 10 0 C 2 -10 0 8 0 D 2 -10 -8 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -2 -8 B 8 0 10 10 0 C 2 -10 0 8 0 D 2 -10 -8 0 -2 E 8 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9478: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) B D E A C (9) C A E D B (8) D B E C A (6) B D A C E (6) A C B D E (6) A B D C E (6) E C A D B (5) E B D A C (5) C E A D B (4) C A D B E (4) B D A E C (4) A C E B D (4) E A C B D (3) E C D B A (2) E C D A B (2) C D B A E (2) C A B D E (2) E A B D C (1) E A B C D (1) D C B E A (1) D B E A C (1) D B C A E (1) C E D B A (1) C E D A B (1) C D A B E (1) B A D E C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -2 -6 -6 B 0 0 8 0 4 C 2 -8 0 -6 -4 D 6 0 6 0 4 E 6 -4 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.372675 C: 0.000000 D: 0.627325 E: 0.000000 Sum of squares = 0.532423261553 Cumulative probabilities = A: 0.000000 B: 0.372675 C: 0.372675 D: 1.000000 E: 1.000000 A B C D E A 0 0 -2 -6 -6 B 0 0 8 0 4 C 2 -8 0 -6 -4 D 6 0 6 0 4 E 6 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=23 B=20 A=18 D=9 so D is eliminated. Round 2 votes counts: E=30 B=28 C=24 A=18 so A is eliminated. Round 3 votes counts: B=36 C=34 E=30 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:208 B:206 E:201 A:193 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -2 -6 -6 B 0 0 8 0 4 C 2 -8 0 -6 -4 D 6 0 6 0 4 E 6 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -6 -6 B 0 0 8 0 4 C 2 -8 0 -6 -4 D 6 0 6 0 4 E 6 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -6 -6 B 0 0 8 0 4 C 2 -8 0 -6 -4 D 6 0 6 0 4 E 6 -4 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9479: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) C B A E D (8) D E A B C (7) D A E B C (6) B E C D A (5) A D E B C (5) C B E D A (4) C A D B E (4) B C E D A (4) A D E C B (4) A D C E B (4) E B C A D (3) D E B A C (3) D A E C B (3) D A C E B (3) E D B C A (2) E D B A C (2) E B D A C (2) D A C B E (2) C B A D E (2) B E C A D (2) A E D B C (2) A C B E D (2) E B D C A (1) E A B D C (1) C B D A E (1) C A B D E (1) B C E A D (1) A D C B E (1) A C E B D (1) A C D E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 2 10 4 B 2 0 -8 0 0 C -2 8 0 4 4 D -10 0 -4 0 -2 E -4 0 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000042 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 10 4 B 2 0 -8 0 0 C -2 8 0 4 4 D -10 0 -4 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000007 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=24 A=22 B=12 E=11 so E is eliminated. Round 2 votes counts: C=31 D=28 A=23 B=18 so B is eliminated. Round 3 votes counts: C=46 D=31 A=23 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:207 C:207 B:197 E:197 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 2 10 4 B 2 0 -8 0 0 C -2 8 0 4 4 D -10 0 -4 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000007 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 10 4 B 2 0 -8 0 0 C -2 8 0 4 4 D -10 0 -4 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000007 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 10 4 B 2 0 -8 0 0 C -2 8 0 4 4 D -10 0 -4 0 -2 E -4 0 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000007 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9480: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (11) E A B D C (7) E B C A D (6) E B A D C (6) C D A B E (6) D C A B E (5) C B D A E (5) E B C D A (4) B E C D A (4) A D E C B (4) C B D E A (3) A E D B C (3) A D E B C (3) E B A C D (2) E A C D B (2) E A C B D (2) D A C B E (2) C B E D A (2) B C D E A (2) A E D C B (2) A D C E B (2) A D C B E (2) A C D E B (2) E C B A D (1) E A D B C (1) C E A D B (1) C D E A B (1) C A D E B (1) B E D C A (1) B D E C A (1) B D A E C (1) B D A C E (1) B C E D A (1) B C D A E (1) A E C D B (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 -12 -2 6 B 4 0 -10 0 -2 C 12 10 0 16 -2 D 2 0 -16 0 8 E -6 2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.076923 E: 0.615385 Sum of squares = 0.479289940813 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.384615 E: 1.000000 A B C D E A 0 -4 -12 -2 6 B 4 0 -10 0 -2 C 12 10 0 16 -2 D 2 0 -16 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.076923 E: 0.615385 Sum of squares = 0.479289936195 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.384615 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=30 A=20 B=12 D=7 so D is eliminated. Round 2 votes counts: C=35 E=31 A=22 B=12 so B is eliminated. Round 3 votes counts: C=39 E=37 A=24 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:218 D:197 B:196 E:195 A:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -12 -2 6 B 4 0 -10 0 -2 C 12 10 0 16 -2 D 2 0 -16 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.076923 E: 0.615385 Sum of squares = 0.479289936195 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.384615 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -12 -2 6 B 4 0 -10 0 -2 C 12 10 0 16 -2 D 2 0 -16 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.076923 E: 0.615385 Sum of squares = 0.479289936195 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.384615 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -12 -2 6 B 4 0 -10 0 -2 C 12 10 0 16 -2 D 2 0 -16 0 8 E -6 2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.076923 E: 0.615385 Sum of squares = 0.479289936195 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.307692 D: 0.384615 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9481: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (9) B E C A D (8) A D C E B (7) D B A C E (5) C E A B D (5) B D E C A (5) A C E D B (5) A C D E B (5) E B C A D (4) C E A D B (4) C A E D B (4) B E D A C (4) B E C D A (4) E C B A D (3) D B A E C (3) D A B E C (3) B E D C A (3) B D E A C (3) E C A B D (2) E A C D B (2) B D A E C (2) E C A D B (1) D A B C E (1) C E B A D (1) C B E A D (1) C A D E B (1) B D C A E (1) B D A C E (1) B C E D A (1) A E D C B (1) A D E C B (1) Total count = 100 A B C D E A 0 2 4 10 -2 B -2 0 -4 -4 -10 C -4 4 0 2 2 D -10 4 -2 0 -6 E 2 10 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999999 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 2 4 10 -2 B -2 0 -4 -4 -10 C -4 4 0 2 2 D -10 4 -2 0 -6 E 2 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=21 A=19 C=16 E=12 so E is eliminated. Round 2 votes counts: B=36 C=22 D=21 A=21 so D is eliminated. Round 3 votes counts: B=44 A=34 C=22 so C is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:208 A:207 C:202 D:193 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 10 -2 B -2 0 -4 -4 -10 C -4 4 0 2 2 D -10 4 -2 0 -6 E 2 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 10 -2 B -2 0 -4 -4 -10 C -4 4 0 2 2 D -10 4 -2 0 -6 E 2 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 10 -2 B -2 0 -4 -4 -10 C -4 4 0 2 2 D -10 4 -2 0 -6 E 2 10 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999998 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9482: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (18) B D E A C (7) A D B E C (7) C A B D E (6) A D E B C (6) C B D E A (5) C E B D A (4) C A E D B (4) B E D C A (4) B D A E C (4) A E D B C (4) E D B C A (3) B D E C A (3) A C E D B (3) D B E A C (2) C B E D A (2) C B A D E (2) A C D B E (2) A C B D E (2) A B D C E (2) E D A B C (1) E B D C A (1) E B D A C (1) D A B E C (1) C E B A D (1) C A E B D (1) B D C A E (1) B C D E A (1) B C D A E (1) A E D C B (1) Total count = 100 A B C D E A 0 -20 22 -18 -6 B 20 0 34 -4 4 C -22 -34 0 -32 -26 D 18 4 32 0 4 E 6 -4 26 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 22 -18 -6 B 20 0 34 -4 4 C -22 -34 0 -32 -26 D 18 4 32 0 4 E 6 -4 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=25 E=24 B=21 D=3 so D is eliminated. Round 2 votes counts: A=28 C=25 E=24 B=23 so B is eliminated. Round 3 votes counts: E=40 A=32 C=28 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:229 B:227 E:212 A:189 C:143 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 22 -18 -6 B 20 0 34 -4 4 C -22 -34 0 -32 -26 D 18 4 32 0 4 E 6 -4 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 22 -18 -6 B 20 0 34 -4 4 C -22 -34 0 -32 -26 D 18 4 32 0 4 E 6 -4 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 22 -18 -6 B 20 0 34 -4 4 C -22 -34 0 -32 -26 D 18 4 32 0 4 E 6 -4 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999987398 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9483: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) B D E C A (7) C A E B D (6) D E A B C (5) C B A E D (5) D B C E A (4) B D C E A (4) B C D A E (4) E A D C B (3) C D A B E (3) C A D E B (3) C A B E D (3) B E A C D (3) B D E A C (3) A E C D B (3) A C E D B (3) E D A B C (2) E A D B C (2) D A E C B (2) C B D A E (2) C B A D E (2) B E D A C (2) B C E A D (2) E D A C B (1) E B D A C (1) E A C B D (1) E A B C D (1) D E B A C (1) D E A C B (1) D C A B E (1) D B C A E (1) C D A E B (1) C A E D B (1) C A B D E (1) B E C A D (1) B E A D C (1) B D C A E (1) B C D E A (1) B C A E D (1) A E C B D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -8 -10 -8 B 8 0 10 8 20 C 8 -10 0 -2 2 D 10 -8 2 0 12 E 8 -20 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999221 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -10 -8 B 8 0 10 8 20 C 8 -10 0 -2 2 D 10 -8 2 0 12 E 8 -20 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=27 D=23 E=11 A=9 so A is eliminated. Round 2 votes counts: C=31 B=30 D=24 E=15 so E is eliminated. Round 3 votes counts: C=36 D=32 B=32 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:223 D:208 C:199 E:187 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -8 -10 -8 B 8 0 10 8 20 C 8 -10 0 -2 2 D 10 -8 2 0 12 E 8 -20 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -10 -8 B 8 0 10 8 20 C 8 -10 0 -2 2 D 10 -8 2 0 12 E 8 -20 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -10 -8 B 8 0 10 8 20 C 8 -10 0 -2 2 D 10 -8 2 0 12 E 8 -20 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999962 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9484: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) B C E A D (7) C A B D E (6) D E A C B (5) D E A B C (5) D C A B E (5) D E B C A (4) B C A E D (4) A C D B E (4) E D A B C (3) E B D C A (3) D C B A E (3) D B C E A (3) D A E C B (3) D A C E B (3) E A D C B (2) E A B C D (2) D A C B E (2) C B A D E (2) B E C A D (2) A E C B D (2) A E B C D (2) A C B D E (2) E D B A C (1) E D A C B (1) E B D A C (1) E B C D A (1) E A D B C (1) E A B D C (1) D E B A C (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A E D (1) B E C D A (1) A E D C B (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 6 -12 4 -10 B -6 0 6 0 -6 C 12 -6 0 2 -6 D -4 0 -2 0 4 E 10 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.038961 B: 0.142857 C: 0.155844 D: 0.545455 E: 0.116883 Sum of squares = 0.357395850879 Cumulative probabilities = A: 0.038961 B: 0.181818 C: 0.337662 D: 0.883117 E: 1.000000 A B C D E A 0 6 -12 4 -10 B -6 0 6 0 -6 C 12 -6 0 2 -6 D -4 0 -2 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.038961 B: 0.142857 C: 0.155844 D: 0.545455 E: 0.116883 Sum of squares = 0.357395850885 Cumulative probabilities = A: 0.038961 B: 0.181818 C: 0.337662 D: 0.883117 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=27 B=14 A=13 C=12 so C is eliminated. Round 2 votes counts: D=36 E=27 A=19 B=18 so B is eliminated. Round 3 votes counts: E=37 D=37 A=26 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:209 C:201 D:199 B:197 A:194 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -12 4 -10 B -6 0 6 0 -6 C 12 -6 0 2 -6 D -4 0 -2 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.038961 B: 0.142857 C: 0.155844 D: 0.545455 E: 0.116883 Sum of squares = 0.357395850885 Cumulative probabilities = A: 0.038961 B: 0.181818 C: 0.337662 D: 0.883117 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 4 -10 B -6 0 6 0 -6 C 12 -6 0 2 -6 D -4 0 -2 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.038961 B: 0.142857 C: 0.155844 D: 0.545455 E: 0.116883 Sum of squares = 0.357395850885 Cumulative probabilities = A: 0.038961 B: 0.181818 C: 0.337662 D: 0.883117 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 4 -10 B -6 0 6 0 -6 C 12 -6 0 2 -6 D -4 0 -2 0 4 E 10 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.038961 B: 0.142857 C: 0.155844 D: 0.545455 E: 0.116883 Sum of squares = 0.357395850885 Cumulative probabilities = A: 0.038961 B: 0.181818 C: 0.337662 D: 0.883117 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9485: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B A C E D (10) B C A E D (9) E D C A B (7) B D A E C (7) D E C A B (6) C E A D B (6) B A D E C (5) C E D A B (4) B D E A C (4) D E A B C (3) C A E D B (3) B A C D E (3) E C D A B (2) E A D C B (2) C B A E D (2) A C E D B (2) A C B E D (2) E D A C B (1) D E B C A (1) D B E A C (1) D A E B C (1) C E B A D (1) C B E A D (1) C A E B D (1) B D E C A (1) B C D E A (1) B A D C E (1) A E D C B (1) A D E C B (1) A B C E D (1) Total count = 100 A B C D E A 0 6 10 2 -2 B -6 0 -4 -2 -4 C -10 4 0 -4 -6 D -2 2 4 0 -10 E 2 4 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 10 2 -2 B -6 0 -4 -2 -4 C -10 4 0 -4 -6 D -2 2 4 0 -10 E 2 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 D=22 C=18 E=12 A=7 so A is eliminated. Round 2 votes counts: B=42 D=23 C=22 E=13 so E is eliminated. Round 3 votes counts: B=42 D=34 C=24 so C is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:211 A:208 D:197 B:192 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 10 2 -2 B -6 0 -4 -2 -4 C -10 4 0 -4 -6 D -2 2 4 0 -10 E 2 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 2 -2 B -6 0 -4 -2 -4 C -10 4 0 -4 -6 D -2 2 4 0 -10 E 2 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 2 -2 B -6 0 -4 -2 -4 C -10 4 0 -4 -6 D -2 2 4 0 -10 E 2 4 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999746 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9486: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D B E (10) C E D B A (9) A B D E C (7) C A E D B (6) E C B D A (5) C A D B E (5) E B D A C (4) A B E D C (4) E B D C A (3) B E D A C (3) B E A D C (3) A E B C D (3) A C E B D (3) E C D B A (2) D B E A C (2) D B A E C (2) C E A D B (2) C D E B A (2) B D E A C (2) B D A E C (2) A E C B D (2) A D B C E (2) A B D C E (2) E C A B D (1) E B C D A (1) D E C B A (1) D E B C A (1) D B E C A (1) C E D A B (1) C E B D A (1) C E A B D (1) C D B A E (1) C D A E B (1) C A E B D (1) A E B D C (1) A D C B E (1) A D B E C (1) A C B E D (1) Total count = 100 A B C D E A 0 10 10 12 10 B -10 0 -12 0 -2 C -10 12 0 16 -2 D -12 0 -16 0 -14 E -10 2 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 12 10 B -10 0 -12 0 -2 C -10 12 0 16 -2 D -12 0 -16 0 -14 E -10 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=30 E=16 B=10 D=7 so D is eliminated. Round 2 votes counts: A=37 C=30 E=18 B=15 so B is eliminated. Round 3 votes counts: A=41 C=30 E=29 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:221 C:208 E:204 B:188 D:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 12 10 B -10 0 -12 0 -2 C -10 12 0 16 -2 D -12 0 -16 0 -14 E -10 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 12 10 B -10 0 -12 0 -2 C -10 12 0 16 -2 D -12 0 -16 0 -14 E -10 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 12 10 B -10 0 -12 0 -2 C -10 12 0 16 -2 D -12 0 -16 0 -14 E -10 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9487: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (23) A C E B D (12) B E D C A (7) A C D E B (6) C A E B D (5) A D C B E (5) A C D B E (5) D B E A C (4) D B A E C (4) B D E C A (4) E C B A D (3) E B D C A (3) C E A B D (3) A C E D B (3) D B A C E (2) A D B C E (2) E C B D A (1) E C A B D (1) E B C D A (1) D E B C A (1) D A C B E (1) C A E D B (1) B E D A C (1) A E C B D (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -6 -4 -4 B 8 0 4 -14 18 C 6 -4 0 -14 -8 D 4 14 14 0 16 E 4 -18 8 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999951 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -4 -4 B 8 0 4 -14 18 C 6 -4 0 -14 -8 D 4 14 14 0 16 E 4 -18 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=35 A=35 B=12 E=9 C=9 so E is eliminated. Round 2 votes counts: D=35 A=35 B=16 C=14 so C is eliminated. Round 3 votes counts: A=45 D=35 B=20 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:208 C:190 A:189 E:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -6 -4 -4 B 8 0 4 -14 18 C 6 -4 0 -14 -8 D 4 14 14 0 16 E 4 -18 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -4 -4 B 8 0 4 -14 18 C 6 -4 0 -14 -8 D 4 14 14 0 16 E 4 -18 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -4 -4 B 8 0 4 -14 18 C 6 -4 0 -14 -8 D 4 14 14 0 16 E 4 -18 8 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997947 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9488: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (8) A C E D B (6) A C D E B (6) E B C D A (5) E B A C D (5) D B C A E (5) A D C B E (5) E C B A D (4) E B C A D (4) B D E C A (4) B D C E A (4) B E C D A (3) A E C D B (3) A D C E B (3) A D B C E (3) E C A D B (2) E A C B D (2) D C B A E (2) D C A B E (2) D A B C E (2) C D A E B (2) B D C A E (2) B D A E C (2) E C A B D (1) E A B C D (1) D C B E A (1) D B C E A (1) D A C B E (1) C E A D B (1) C A E D B (1) C A D E B (1) B E A D C (1) B D E A C (1) B D A C E (1) A E C B D (1) A E B C D (1) A D E C B (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -6 8 4 B 6 0 10 2 0 C 6 -10 0 -2 -2 D -8 -2 2 0 0 E -4 0 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.558657 C: 0.000000 D: 0.000000 E: 0.441343 Sum of squares = 0.506881169705 Cumulative probabilities = A: 0.000000 B: 0.558657 C: 0.558657 D: 0.558657 E: 1.000000 A B C D E A 0 -6 -6 8 4 B 6 0 10 2 0 C 6 -10 0 -2 -2 D -8 -2 2 0 0 E -4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 B=26 E=24 D=14 C=5 so C is eliminated. Round 2 votes counts: A=33 B=26 E=25 D=16 so D is eliminated. Round 3 votes counts: A=40 B=35 E=25 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:200 E:199 C:196 D:196 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 8 4 B 6 0 10 2 0 C 6 -10 0 -2 -2 D -8 -2 2 0 0 E -4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 8 4 B 6 0 10 2 0 C 6 -10 0 -2 -2 D -8 -2 2 0 0 E -4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 8 4 B 6 0 10 2 0 C 6 -10 0 -2 -2 D -8 -2 2 0 0 E -4 0 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9489: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (7) B A E C D (7) E A B D C (5) E A B C D (5) C B D E A (5) C B D A E (5) D C B E A (4) B C E A D (4) B C A E D (4) A E B D C (4) D C A E B (3) D A E C B (3) C D B A E (3) B A C E D (3) A E D B C (3) E B A C D (2) E A D B C (2) D E C A B (2) D C E A B (2) D C B A E (2) D C A B E (2) D A E B C (2) C B E A D (2) B E A C D (2) B A C D E (2) A B D E C (2) E C B A D (1) E C A B D (1) E A D C B (1) D E A C B (1) D C E B A (1) C E D B A (1) C D E B A (1) B C A D E (1) B A D C E (1) A D E B C (1) A D B E C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -16 -2 12 2 B 16 0 6 16 18 C 2 -6 0 14 6 D -12 -16 -14 0 2 E -2 -18 -6 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999958 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 12 2 B 16 0 6 16 18 C 2 -6 0 14 6 D -12 -16 -14 0 2 E -2 -18 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=24 B=24 D=22 E=17 A=13 so A is eliminated. Round 2 votes counts: B=28 E=24 D=24 C=24 so E is eliminated. Round 3 votes counts: B=44 D=30 C=26 so C is eliminated. Round 4 votes counts: B=58 D=42 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:228 C:208 A:198 E:186 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 12 2 B 16 0 6 16 18 C 2 -6 0 14 6 D -12 -16 -14 0 2 E -2 -18 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 12 2 B 16 0 6 16 18 C 2 -6 0 14 6 D -12 -16 -14 0 2 E -2 -18 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 12 2 B 16 0 6 16 18 C 2 -6 0 14 6 D -12 -16 -14 0 2 E -2 -18 -6 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9490: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (9) C A E D B (8) B D E A C (8) A C E D B (8) E D A C B (4) D E B A C (4) C A B E D (4) B D E C A (4) B D C E A (4) B C D E A (4) A E D C B (4) A E C D B (4) A C E B D (4) E D C A B (3) B A C D E (3) E D A B C (2) D E C B A (2) D E C A B (2) B C D A E (2) B C A D E (2) B A C E D (2) A B C E D (2) E C D A B (1) E A D B C (1) D E A C B (1) D E A B C (1) D B E C A (1) D B E A C (1) C D E A B (1) B D A E C (1) B A E D C (1) B A D C E (1) A B E C D (1) Total count = 100 A B C D E A 0 20 24 8 12 B -20 0 -10 4 0 C -24 10 0 10 8 D -8 -4 -10 0 -16 E -12 0 -8 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 24 8 12 B -20 0 -10 4 0 C -24 10 0 10 8 D -8 -4 -10 0 -16 E -12 0 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=32 A=32 C=13 D=12 E=11 so E is eliminated. Round 2 votes counts: A=33 B=32 D=21 C=14 so C is eliminated. Round 3 votes counts: A=45 B=32 D=23 so D is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:232 C:202 E:198 B:187 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 24 8 12 B -20 0 -10 4 0 C -24 10 0 10 8 D -8 -4 -10 0 -16 E -12 0 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 24 8 12 B -20 0 -10 4 0 C -24 10 0 10 8 D -8 -4 -10 0 -16 E -12 0 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 24 8 12 B -20 0 -10 4 0 C -24 10 0 10 8 D -8 -4 -10 0 -16 E -12 0 -8 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9491: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (13) D A E C B (9) E B A D C (6) B E A D C (6) C B E D A (5) B E C A D (5) A D E C B (5) A D E B C (5) C B D A E (4) B C E D A (4) A D C B E (4) E B D A C (3) D A C E B (3) C B A D E (3) B E C D A (3) E D A B C (2) E B C D A (2) E A D B C (2) C D A E B (2) B C E A D (2) B C A D E (2) A D C E B (2) A D B E C (2) E D C A B (1) E D B A C (1) E C D B A (1) C D E A B (1) C A D B E (1) B E A C D (1) Total count = 100 A B C D E A 0 4 2 -8 10 B -4 0 -8 -8 10 C -2 8 0 -2 -8 D 8 8 2 0 12 E -10 -10 8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -8 10 B -4 0 -8 -8 10 C -2 8 0 -2 -8 D 8 8 2 0 12 E -10 -10 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=23 E=18 A=18 D=12 so D is eliminated. Round 2 votes counts: A=30 C=29 B=23 E=18 so E is eliminated. Round 3 votes counts: B=35 A=34 C=31 so C is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:204 C:198 B:195 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -8 10 B -4 0 -8 -8 10 C -2 8 0 -2 -8 D 8 8 2 0 12 E -10 -10 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -8 10 B -4 0 -8 -8 10 C -2 8 0 -2 -8 D 8 8 2 0 12 E -10 -10 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -8 10 B -4 0 -8 -8 10 C -2 8 0 -2 -8 D 8 8 2 0 12 E -10 -10 8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9492: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) E A B D C (8) C D A B E (8) B E C D A (8) E B A D C (6) D A C B E (6) B C E D A (6) D C A B E (5) A D C E B (5) E B A C D (4) D A C E B (3) C B D E A (3) C B D A E (3) B E C A D (3) A E D B C (3) E B C A D (2) E A D B C (2) C D B A E (2) A E D C B (2) A D E B C (2) E C A B D (1) E A C B D (1) D A E B C (1) C E A B D (1) C B E D A (1) C A E D B (1) B E D A C (1) A E C D B (1) Total count = 100 A B C D E A 0 22 12 6 6 B -22 0 -8 -4 -8 C -12 8 0 -10 -12 D -6 4 10 0 -2 E -6 8 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 22 12 6 6 B -22 0 -8 -4 -8 C -12 8 0 -10 -12 D -6 4 10 0 -2 E -6 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=24 A=24 C=19 B=18 D=15 so D is eliminated. Round 2 votes counts: A=34 E=24 C=24 B=18 so B is eliminated. Round 3 votes counts: E=36 A=34 C=30 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:223 E:208 D:203 C:187 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 22 12 6 6 B -22 0 -8 -4 -8 C -12 8 0 -10 -12 D -6 4 10 0 -2 E -6 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 22 12 6 6 B -22 0 -8 -4 -8 C -12 8 0 -10 -12 D -6 4 10 0 -2 E -6 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 22 12 6 6 B -22 0 -8 -4 -8 C -12 8 0 -10 -12 D -6 4 10 0 -2 E -6 8 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999876 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9493: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (11) D A C E B (7) E B C D A (5) C B E A D (5) E C B D A (4) E B D C A (4) B E C A D (4) D A E B C (3) C E B A D (3) B E A C D (3) A C D B E (3) A C B D E (3) E D C B A (2) D E C A B (2) D C A E B (2) D A E C B (2) D A B E C (2) C E D B A (2) C E B D A (2) C D A E B (2) C A D E B (2) C A B E D (2) B E D A C (2) B C E A D (2) B A C E D (2) A D B C E (2) E C D B A (1) E B D A C (1) D E B A C (1) D E A C B (1) D E A B C (1) C D E A B (1) C A B D E (1) B E D C A (1) B E C D A (1) B C A E D (1) B A E D C (1) B A E C D (1) B A D E C (1) A D C E B (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 2 2 4 B -2 0 -18 4 2 C -2 18 0 4 12 D -2 -4 -4 0 0 E -4 -2 -12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 2 4 B -2 0 -18 4 2 C -2 18 0 4 12 D -2 -4 -4 0 0 E -4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 D=21 C=20 B=19 E=17 so E is eliminated. Round 2 votes counts: B=29 C=25 D=23 A=23 so D is eliminated. Round 3 votes counts: A=39 C=31 B=30 so B is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:216 A:205 D:195 B:193 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 2 4 B -2 0 -18 4 2 C -2 18 0 4 12 D -2 -4 -4 0 0 E -4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 2 4 B -2 0 -18 4 2 C -2 18 0 4 12 D -2 -4 -4 0 0 E -4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 2 4 B -2 0 -18 4 2 C -2 18 0 4 12 D -2 -4 -4 0 0 E -4 -2 -12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9494: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (10) E D A B C (7) C B A D E (7) E D B A C (6) E B A D C (6) D E B C A (6) B E A C D (5) D E C A B (4) D E A C B (4) A B C E D (4) D C E B A (3) B C A E D (3) E B D A C (2) D C A E B (2) C A D B E (2) C A B E D (2) C A B D E (2) B D C E A (2) A E B D C (2) E A D C B (1) E A D B C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E A B C (1) D C B E A (1) C D B E A (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A E D (1) C A D E B (1) B E D A C (1) B E C A D (1) B A C E D (1) A E D C B (1) A E B C D (1) A D E C B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 16 12 -10 B -2 0 2 8 -4 C -16 -2 0 -6 -6 D -12 -8 6 0 -14 E 10 4 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 16 12 -10 B -2 0 2 8 -4 C -16 -2 0 -6 -6 D -12 -8 6 0 -14 E 10 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=22 A=21 C=19 B=13 so B is eliminated. Round 2 votes counts: E=32 D=24 C=22 A=22 so C is eliminated. Round 3 votes counts: A=40 E=32 D=28 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:210 B:202 D:186 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 16 12 -10 B -2 0 2 8 -4 C -16 -2 0 -6 -6 D -12 -8 6 0 -14 E 10 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 12 -10 B -2 0 2 8 -4 C -16 -2 0 -6 -6 D -12 -8 6 0 -14 E 10 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 12 -10 B -2 0 2 8 -4 C -16 -2 0 -6 -6 D -12 -8 6 0 -14 E 10 4 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999247 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9495: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (11) B D E C A (6) A E B D C (6) E B A D C (5) B E D A C (5) A E C B D (5) E A B D C (4) D B E C A (4) C A D E B (4) B D C E A (4) A E C D B (4) E D A B C (3) C D E A B (3) C D B A E (3) C D A E B (3) C A E D B (3) A E B C D (3) A C E D B (3) D E B A C (2) D B C E A (2) C D A B E (2) B D E A C (2) E D B A C (1) E B D A C (1) D E C A B (1) D C E B A (1) D C B E A (1) D B E A C (1) C D E B A (1) C B D E A (1) C A E B D (1) B C D A E (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 -4 -18 -18 B 4 0 4 -6 -8 C 4 -4 0 0 -10 D 18 6 0 0 6 E 18 8 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.148016 D: 0.851984 E: 0.000000 Sum of squares = 0.747785167647 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.148016 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -18 -18 B 4 0 4 -6 -8 C 4 -4 0 0 -10 D 18 6 0 0 6 E 18 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.374999 D: 0.625001 E: 0.000000 Sum of squares = 0.531250375564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.374999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=24 B=18 E=14 D=12 so D is eliminated. Round 2 votes counts: C=34 B=25 A=24 E=17 so E is eliminated. Round 3 votes counts: C=35 B=34 A=31 so A is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:215 E:215 B:197 C:195 A:178 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -4 -18 -18 B 4 0 4 -6 -8 C 4 -4 0 0 -10 D 18 6 0 0 6 E 18 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.374999 D: 0.625001 E: 0.000000 Sum of squares = 0.531250375564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.374999 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -18 -18 B 4 0 4 -6 -8 C 4 -4 0 0 -10 D 18 6 0 0 6 E 18 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.374999 D: 0.625001 E: 0.000000 Sum of squares = 0.531250375564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.374999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -18 -18 B 4 0 4 -6 -8 C 4 -4 0 0 -10 D 18 6 0 0 6 E 18 8 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.374999 D: 0.625001 E: 0.000000 Sum of squares = 0.531250375564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.374999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9496: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) E A B C D (9) E A C D B (6) E B A D C (5) E D C A B (4) D B C E A (4) B D C A E (4) A C B D E (4) E D C B A (3) E A B D C (3) D C B A E (3) D B C A E (3) C D B A E (3) C D A B E (3) A C E B D (3) E D B A C (2) E C D A B (2) E A C B D (2) D C B E A (2) D B E C A (2) B D C E A (2) B A D C E (2) A E B C D (2) A C E D B (2) A C D B E (2) E D A C B (1) E B D C A (1) E B D A C (1) C D E A B (1) C D A E B (1) C B D A E (1) C B A D E (1) C A D E B (1) B E D C A (1) B A E D C (1) B A C D E (1) A E C B D (1) Total count = 100 A B C D E A 0 -6 -6 -10 -24 B 6 0 8 -12 -22 C 6 -8 0 -10 -14 D 10 12 10 0 -20 E 24 22 14 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -6 -10 -24 B 6 0 8 -12 -22 C 6 -8 0 -10 -14 D 10 12 10 0 -20 E 24 22 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=50 D=14 A=14 C=11 B=11 so C is eliminated. Round 2 votes counts: E=50 D=22 A=15 B=13 so B is eliminated. Round 3 votes counts: E=51 D=29 A=20 so A is eliminated. Round 4 votes counts: E=60 D=40 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:240 D:206 B:190 C:187 A:177 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -6 -10 -24 B 6 0 8 -12 -22 C 6 -8 0 -10 -14 D 10 12 10 0 -20 E 24 22 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -10 -24 B 6 0 8 -12 -22 C 6 -8 0 -10 -14 D 10 12 10 0 -20 E 24 22 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -10 -24 B 6 0 8 -12 -22 C 6 -8 0 -10 -14 D 10 12 10 0 -20 E 24 22 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9497: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) E D B C A (6) E B D A C (6) E D B A C (5) C E D A B (5) E D C B A (4) C A D E B (4) B A D E C (4) D E B C A (3) C E D B A (3) C D E A B (3) C A E B D (3) B E A D C (3) A B D C E (3) A B C E D (3) A B C D E (3) E B A D C (2) D E C B A (2) C D A B E (2) C A E D B (2) C A D B E (2) C A B E D (2) B E D A C (2) A C B E D (2) A C B D E (2) A B E C D (2) E C D A B (1) E C A D B (1) D E B A C (1) D B E A C (1) D B A E C (1) C E A D B (1) C E A B D (1) C D E B A (1) C D B A E (1) B D E A C (1) B D A E C (1) B A E D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -12 2 -4 B -4 0 -2 2 -10 C 12 2 0 8 4 D -2 -2 -8 0 -12 E 4 10 -4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999153 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 2 -4 B -4 0 -2 2 -10 C 12 2 0 8 4 D -2 -2 -8 0 -12 E 4 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 E=25 A=16 B=12 D=8 so D is eliminated. Round 2 votes counts: C=39 E=31 A=16 B=14 so B is eliminated. Round 3 votes counts: C=39 E=38 A=23 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:213 E:211 A:195 B:193 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 2 -4 B -4 0 -2 2 -10 C 12 2 0 8 4 D -2 -2 -8 0 -12 E 4 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 2 -4 B -4 0 -2 2 -10 C 12 2 0 8 4 D -2 -2 -8 0 -12 E 4 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 2 -4 B -4 0 -2 2 -10 C 12 2 0 8 4 D -2 -2 -8 0 -12 E 4 10 -4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999101 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9498: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) C E B A D (7) D B A E C (6) B C D E A (5) A E D C B (5) E C B A D (4) E A C B D (4) D B C A E (4) D B A C E (4) A D E C B (4) A D E B C (4) D A B E C (3) B D E A C (3) B D C E A (3) B C E A D (3) E C A B D (2) E A D B C (2) E A C D B (2) D E A B C (2) C E A B D (2) C B E A D (2) C A E B D (2) B D C A E (2) A C D E B (2) E D A B C (1) E B D A C (1) D C B A E (1) D A E C B (1) D A C E B (1) C D B A E (1) C D A B E (1) C B E D A (1) C B D A E (1) B D E C A (1) B C E D A (1) A D C E B (1) Total count = 100 A B C D E A 0 0 14 -8 8 B 0 0 12 -12 -16 C -14 -12 0 -20 -12 D 8 12 20 0 22 E -8 16 12 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 14 -8 8 B 0 0 12 -12 -16 C -14 -12 0 -20 -12 D 8 12 20 0 22 E -8 16 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=18 C=17 E=16 A=16 so E is eliminated. Round 2 votes counts: D=34 A=24 C=23 B=19 so B is eliminated. Round 3 votes counts: D=44 C=32 A=24 so A is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:231 A:207 E:199 B:192 C:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 14 -8 8 B 0 0 12 -12 -16 C -14 -12 0 -20 -12 D 8 12 20 0 22 E -8 16 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 14 -8 8 B 0 0 12 -12 -16 C -14 -12 0 -20 -12 D 8 12 20 0 22 E -8 16 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 14 -8 8 B 0 0 12 -12 -16 C -14 -12 0 -20 -12 D 8 12 20 0 22 E -8 16 12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999828 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9499: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) E B C D A (6) C B A D E (5) B E D C A (5) B E C D A (5) D E A B C (4) C B E A D (4) B C E D A (4) A C D E B (4) A C D B E (4) E B D A C (3) D E B A C (3) D B E A C (3) C B A E D (3) A E D C B (3) E D B A C (2) E A C B D (2) D B C A E (2) D A E B C (2) C B D A E (2) C A D B E (2) C A B D E (2) B D E C A (2) B D C E A (2) A D E C B (2) A C E B D (2) E B D C A (1) E B A C D (1) E A D C B (1) E A D B C (1) E A C D B (1) E A B D C (1) D C B A E (1) D A C E B (1) D A C B E (1) D A B E C (1) B C D E A (1) B C D A E (1) A D C E B (1) A D C B E (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 4 -18 -18 B 12 0 14 4 2 C -4 -14 0 0 -12 D 18 -4 0 0 -6 E 18 -2 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -18 -18 B 12 0 14 4 2 C -4 -14 0 0 -12 D 18 -4 0 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 B=20 D=18 C=18 A=18 so D is eliminated. Round 2 votes counts: E=33 B=25 A=23 C=19 so C is eliminated. Round 3 votes counts: B=40 E=33 A=27 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:217 B:216 D:204 C:185 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 4 -18 -18 B 12 0 14 4 2 C -4 -14 0 0 -12 D 18 -4 0 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -18 -18 B 12 0 14 4 2 C -4 -14 0 0 -12 D 18 -4 0 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -18 -18 B 12 0 14 4 2 C -4 -14 0 0 -12 D 18 -4 0 0 -6 E 18 -2 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999703 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9500: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (12) E C D A B (9) A B D E C (7) B A D C E (6) C E D B A (5) A D B C E (4) E C B D A (3) E C B A D (3) E A D C B (3) B D C A E (3) B A E C D (3) B A C D E (3) A B E D C (3) E C D B A (2) D C E A B (2) D A E C B (2) D A B C E (2) C D E B A (2) B C D A E (2) B A C E D (2) A E D C B (2) A D E C B (2) A B E C D (2) E D C A B (1) E A C D B (1) D E C A B (1) D E A C B (1) D C E B A (1) D C B E A (1) D B C A E (1) D A C B E (1) C D B E A (1) C B E D A (1) B D A C E (1) B C E D A (1) B C E A D (1) B C D E A (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 14 18 12 20 B -14 0 12 10 16 C -18 -12 0 -16 6 D -12 -10 16 0 12 E -20 -16 -6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 18 12 20 B -14 0 12 10 16 C -18 -12 0 -16 6 D -12 -10 16 0 12 E -20 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=23 E=22 D=12 C=9 so C is eliminated. Round 2 votes counts: A=34 E=27 B=24 D=15 so D is eliminated. Round 3 votes counts: A=39 E=34 B=27 so B is eliminated. Round 4 votes counts: A=60 E=40 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:232 B:212 D:203 C:180 E:173 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 18 12 20 B -14 0 12 10 16 C -18 -12 0 -16 6 D -12 -10 16 0 12 E -20 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 18 12 20 B -14 0 12 10 16 C -18 -12 0 -16 6 D -12 -10 16 0 12 E -20 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 18 12 20 B -14 0 12 10 16 C -18 -12 0 -16 6 D -12 -10 16 0 12 E -20 -16 -6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9501: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) A D E B C (9) D A C B E (8) C B E D A (8) D A C E B (5) B C E A D (5) D A E B C (4) A E B C D (4) A D C B E (4) E B C D A (3) E B C A D (3) E B A C D (3) D A E C B (3) C B D E A (3) E D B A C (2) E A D B C (2) D C E A B (2) C D B E A (2) A D E C B (2) E D C B A (1) E D A B C (1) E B A D C (1) D E A C B (1) D E A B C (1) D C E B A (1) D C B E A (1) D C A B E (1) C D A B E (1) C B D A E (1) C A B D E (1) B E C D A (1) B C A E D (1) B A E C D (1) A E D B C (1) A E B D C (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 10 0 -2 B -6 0 10 -6 0 C -10 -10 0 -4 -12 D 0 6 4 0 2 E 2 0 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.135706 B: 0.000000 C: 0.000000 D: 0.864294 E: 0.000000 Sum of squares = 0.765420582321 Cumulative probabilities = A: 0.135706 B: 0.135706 C: 0.135706 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 0 -2 B -6 0 10 -6 0 C -10 -10 0 -4 -12 D 0 6 4 0 2 E 2 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499824 B: 0.000000 C: 0.000000 D: 0.500176 E: 0.000000 Sum of squares = 0.500000062234 Cumulative probabilities = A: 0.499824 B: 0.499824 C: 0.499824 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=23 B=18 E=16 C=16 so E is eliminated. Round 2 votes counts: D=31 B=28 A=25 C=16 so C is eliminated. Round 3 votes counts: B=40 D=34 A=26 so A is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:206 E:206 B:199 C:182 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 10 0 -2 B -6 0 10 -6 0 C -10 -10 0 -4 -12 D 0 6 4 0 2 E 2 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499824 B: 0.000000 C: 0.000000 D: 0.500176 E: 0.000000 Sum of squares = 0.500000062234 Cumulative probabilities = A: 0.499824 B: 0.499824 C: 0.499824 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 0 -2 B -6 0 10 -6 0 C -10 -10 0 -4 -12 D 0 6 4 0 2 E 2 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499824 B: 0.000000 C: 0.000000 D: 0.500176 E: 0.000000 Sum of squares = 0.500000062234 Cumulative probabilities = A: 0.499824 B: 0.499824 C: 0.499824 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 0 -2 B -6 0 10 -6 0 C -10 -10 0 -4 -12 D 0 6 4 0 2 E 2 0 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.499824 B: 0.000000 C: 0.000000 D: 0.500176 E: 0.000000 Sum of squares = 0.500000062234 Cumulative probabilities = A: 0.499824 B: 0.499824 C: 0.499824 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9502: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (9) B E A C D (9) D A C E B (6) D C B A E (5) C D A E B (5) B C E D A (5) B A E D C (5) C D E A B (4) B E C A D (4) A E D C B (4) A D E C B (4) E B C A D (3) E A C D B (3) D C A B E (3) C D E B A (3) C D B E A (3) B E A D C (3) B D C A E (3) A E B D C (3) D A C B E (2) B D A C E (2) B C D E A (2) E C B A D (1) E B A C D (1) D B C A E (1) D A B C E (1) C E B D A (1) C B E D A (1) C B D E A (1) B A D E C (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 -8 -14 12 B 8 0 -10 -8 4 C 8 10 0 -8 14 D 14 8 8 0 10 E -12 -4 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -8 -14 12 B 8 0 -10 -8 4 C 8 10 0 -8 14 D 14 8 8 0 10 E -12 -4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=27 C=18 A=13 E=8 so E is eliminated. Round 2 votes counts: B=38 D=27 C=19 A=16 so A is eliminated. Round 3 votes counts: B=42 D=36 C=22 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:212 B:197 A:191 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -8 -14 12 B 8 0 -10 -8 4 C 8 10 0 -8 14 D 14 8 8 0 10 E -12 -4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -14 12 B 8 0 -10 -8 4 C 8 10 0 -8 14 D 14 8 8 0 10 E -12 -4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -14 12 B 8 0 -10 -8 4 C 8 10 0 -8 14 D 14 8 8 0 10 E -12 -4 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9503: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) B E D A C (7) B A E D C (7) A C D E B (7) C D E B A (5) C D E A B (5) C D A E B (5) B E D C A (5) E B D C A (4) A C B D E (4) E D B C A (3) D E B C A (3) A B E D C (3) A B E C D (3) A B D E C (3) A B C E D (3) D E C B A (2) C E D B A (2) C E B D A (2) B E C D A (2) B D E A C (2) A C B E D (2) D E C A B (1) D E A B C (1) C A E D B (1) C A B E D (1) B E A D C (1) B A C E D (1) A D C E B (1) A D B E C (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 -4 2 10 B -8 0 0 2 -6 C 4 0 0 10 4 D -2 -2 -10 0 6 E -10 6 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.151164 C: 0.848836 D: 0.000000 E: 0.000000 Sum of squares = 0.743372556469 Cumulative probabilities = A: 0.000000 B: 0.151164 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 2 10 B -8 0 0 2 -6 C 4 0 0 10 4 D -2 -2 -10 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555567435 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=29 B=25 E=7 D=7 so E is eliminated. Round 2 votes counts: C=32 B=29 A=29 D=10 so D is eliminated. Round 3 votes counts: C=35 B=35 A=30 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:209 A:208 D:196 B:194 E:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -4 2 10 B -8 0 0 2 -6 C 4 0 0 10 4 D -2 -2 -10 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555567435 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 2 10 B -8 0 0 2 -6 C 4 0 0 10 4 D -2 -2 -10 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555567435 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 2 10 B -8 0 0 2 -6 C 4 0 0 10 4 D -2 -2 -10 0 6 E -10 6 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.666667 D: 0.000000 E: 0.000000 Sum of squares = 0.555555567435 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9504: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (10) B A E C D (9) C D E B A (8) C D E A B (8) A B E D C (7) E A B D C (6) C D B A E (6) C B A D E (6) D C E A B (5) E D A B C (4) B A C E D (4) E D C A B (3) D E C A B (3) D E A B C (3) C B D A E (3) E D A C B (2) B A D E C (2) A E B D C (2) E C D A B (1) E A D B C (1) E A B C D (1) D C E B A (1) D C B A E (1) C E D A B (1) C B A E D (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 6 2 6 B 4 0 2 6 2 C -6 -2 0 0 -10 D -2 -6 0 0 -6 E -6 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999968 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 2 6 B 4 0 2 6 2 C -6 -2 0 0 -10 D -2 -6 0 0 -6 E -6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=26 E=18 D=13 A=10 so A is eliminated. Round 2 votes counts: B=34 C=33 E=20 D=13 so D is eliminated. Round 3 votes counts: C=40 B=34 E=26 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:207 A:205 E:204 D:193 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 2 6 B 4 0 2 6 2 C -6 -2 0 0 -10 D -2 -6 0 0 -6 E -6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 6 B 4 0 2 6 2 C -6 -2 0 0 -10 D -2 -6 0 0 -6 E -6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 6 B 4 0 2 6 2 C -6 -2 0 0 -10 D -2 -6 0 0 -6 E -6 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9505: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (7) E C B D A (6) C B E D A (6) D A E B C (5) C D E A B (5) B A D C E (5) A D B E C (5) A B D E C (5) C D B A E (4) B C A D E (4) E C D B A (3) E B A D C (3) E A D B C (3) C D A B E (3) C B A D E (3) B C A E D (3) E D C A B (2) E D A B C (2) E C D A B (2) E A B D C (2) D E A C B (2) D C E A B (2) C E D B A (2) B E C A D (2) E D A C B (1) E B C A D (1) D E C A B (1) D E A B C (1) D A E C B (1) C D E B A (1) C D B E A (1) C B D A E (1) B E A C D (1) B C E A D (1) B A E D C (1) B A E C D (1) B A D E C (1) B A C D E (1) Total count = 100 A B C D E A 0 -2 -6 -2 0 B 2 0 8 -6 -4 C 6 -8 0 2 -16 D 2 6 -2 0 16 E 0 4 16 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000017 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -6 -2 0 B 2 0 8 -6 -4 C 6 -8 0 2 -16 D 2 6 -2 0 16 E 0 4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000329 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=25 B=20 A=17 D=12 so D is eliminated. Round 2 votes counts: E=29 C=28 A=23 B=20 so B is eliminated. Round 3 votes counts: C=36 E=32 A=32 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:211 E:202 B:200 A:195 C:192 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -6 -2 0 B 2 0 8 -6 -4 C 6 -8 0 2 -16 D 2 6 -2 0 16 E 0 4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000329 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -2 0 B 2 0 8 -6 -4 C 6 -8 0 2 -16 D 2 6 -2 0 16 E 0 4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000329 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -2 0 B 2 0 8 -6 -4 C 6 -8 0 2 -16 D 2 6 -2 0 16 E 0 4 16 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.375000 D: 0.500000 E: 0.000000 Sum of squares = 0.406250000329 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9506: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (15) D A E B C (9) A E C B D (7) D B E A C (6) C B D E A (5) A E D B C (5) C A E B D (4) D B C E A (3) C D B E A (3) C B E D A (3) B D E C A (3) B C E A D (3) A C E B D (3) E A B C D (2) D E B A C (2) D C B E A (2) D B E C A (2) C D B A E (2) A E B C D (2) A C D E B (2) E B A D C (1) E B A C D (1) E A B D C (1) D E A B C (1) D A E C B (1) D A C E B (1) D A B C E (1) C B D A E (1) C A B D E (1) B E D C A (1) B E C A D (1) B C E D A (1) B C D E A (1) A E D C B (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 -12 -2 4 -14 B 12 0 -4 14 8 C 2 4 0 16 2 D -4 -14 -16 0 -6 E 14 -8 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -2 4 -14 B 12 0 -4 14 8 C 2 4 0 16 2 D -4 -14 -16 0 -6 E 14 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=28 A=23 B=10 E=5 so E is eliminated. Round 2 votes counts: C=34 D=28 A=26 B=12 so B is eliminated. Round 3 votes counts: C=40 D=32 A=28 so A is eliminated. Round 4 votes counts: C=58 D=42 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:215 C:212 E:205 A:188 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -2 4 -14 B 12 0 -4 14 8 C 2 4 0 16 2 D -4 -14 -16 0 -6 E 14 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -2 4 -14 B 12 0 -4 14 8 C 2 4 0 16 2 D -4 -14 -16 0 -6 E 14 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -2 4 -14 B 12 0 -4 14 8 C 2 4 0 16 2 D -4 -14 -16 0 -6 E 14 -8 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9507: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) D C A B E (10) C A D E B (10) B E D A C (8) D C A E B (5) C A E D B (5) E C A B D (4) D B A C E (4) A C D B E (4) E B D C A (3) E A C B D (3) B D E A C (3) E C A D B (2) E B C A D (2) D B C A E (2) C A D B E (2) B D E C A (2) B D A C E (2) A C E D B (2) A C E B D (2) E D C A B (1) E C D A B (1) E C B A D (1) E B D A C (1) E B C D A (1) E A B C D (1) D E C A B (1) D C E A B (1) D C B E A (1) D B E C A (1) D B E A C (1) D A C B E (1) D A B C E (1) C D A E B (1) B E D C A (1) Total count = 100 A B C D E A 0 14 -14 -4 2 B -14 0 -14 -12 -14 C 14 14 0 2 6 D 4 12 -2 0 4 E -2 14 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999874 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -14 -4 2 B -14 0 -14 -12 -14 C 14 14 0 2 6 D 4 12 -2 0 4 E -2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=28 C=18 B=16 A=8 so A is eliminated. Round 2 votes counts: E=30 D=28 C=26 B=16 so B is eliminated. Round 3 votes counts: E=39 D=35 C=26 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:209 E:201 A:199 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -14 -4 2 B -14 0 -14 -12 -14 C 14 14 0 2 6 D 4 12 -2 0 4 E -2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -14 -4 2 B -14 0 -14 -12 -14 C 14 14 0 2 6 D 4 12 -2 0 4 E -2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -14 -4 2 B -14 0 -14 -12 -14 C 14 14 0 2 6 D 4 12 -2 0 4 E -2 14 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9508: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) B A E C D (8) A B D C E (8) D E C A B (7) D B A E C (6) E C B D A (5) D C E A B (4) B A C E D (4) A B C E D (4) E C D A B (3) D E C B A (3) D B E A C (3) D A B E C (3) D A B C E (3) B A D E C (3) A D B C E (3) E C B A D (2) D B E C A (2) C E B A D (2) B E C A D (2) B D A E C (2) A C E B D (2) D E B C A (1) D E A C B (1) D A C E B (1) D A C B E (1) C E D A B (1) C E A D B (1) C E A B D (1) B E D A C (1) B E C D A (1) B A D C E (1) A C B E D (1) Total count = 100 A B C D E A 0 -12 10 -16 0 B 12 0 10 -6 12 C -10 -10 0 -6 -26 D 16 6 6 0 4 E 0 -12 26 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 10 -16 0 B 12 0 10 -6 12 C -10 -10 0 -6 -26 D 16 6 6 0 4 E 0 -12 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=22 E=20 A=18 C=5 so C is eliminated. Round 2 votes counts: D=35 E=25 B=22 A=18 so A is eliminated. Round 3 votes counts: D=38 B=35 E=27 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:214 E:205 A:191 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 10 -16 0 B 12 0 10 -6 12 C -10 -10 0 -6 -26 D 16 6 6 0 4 E 0 -12 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 10 -16 0 B 12 0 10 -6 12 C -10 -10 0 -6 -26 D 16 6 6 0 4 E 0 -12 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 10 -16 0 B 12 0 10 -6 12 C -10 -10 0 -6 -26 D 16 6 6 0 4 E 0 -12 26 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995896 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9509: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (9) C B A E D (7) D E A B C (6) B C E A D (6) C B E A D (5) C B D E A (4) B C E D A (4) A E D B C (4) E A D B C (3) E A B D C (3) D C A E B (3) D C A B E (3) D A C E B (3) B E C A D (3) E B A D C (2) E A B C D (2) D C B E A (2) C B D A E (2) C B A D E (2) B C D E A (2) A E D C B (2) A E B C D (2) A D E C B (2) A C E B D (2) E D A B C (1) D E B A C (1) D C B A E (1) D B E C A (1) D A E B C (1) C D B E A (1) C D B A E (1) C D A E B (1) C D A B E (1) C A D E B (1) C A B E D (1) B E D A C (1) B E A C D (1) B D E C A (1) A E C D B (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 6 -4 2 2 B -6 0 -8 2 -2 C 4 8 0 -2 4 D -2 -2 2 0 -2 E -2 2 -4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 2 2 B -6 0 -8 2 -2 C 4 8 0 -2 4 D -2 -2 2 0 -2 E -2 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=26 B=18 A=15 E=11 so E is eliminated. Round 2 votes counts: D=31 C=26 A=23 B=20 so B is eliminated. Round 3 votes counts: C=41 D=33 A=26 so A is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. C:207 A:203 E:199 D:198 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 2 2 B -6 0 -8 2 -2 C 4 8 0 -2 4 D -2 -2 2 0 -2 E -2 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 2 2 B -6 0 -8 2 -2 C 4 8 0 -2 4 D -2 -2 2 0 -2 E -2 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 2 2 B -6 0 -8 2 -2 C 4 8 0 -2 4 D -2 -2 2 0 -2 E -2 2 -4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.500000 E: 0.000000 Sum of squares = 0.37499999994 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9510: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (16) C A E B D (11) A C D B E (11) D B E A C (9) D A C B E (8) B E D C A (7) E B D A C (6) D B E C A (6) A C D E B (6) E B C A D (4) C A D B E (4) E D B A C (2) D E B A C (2) A C E B D (2) D A B C E (1) C E A B D (1) C A D E B (1) C A B E D (1) C A B D E (1) B E D A C (1) Total count = 100 A B C D E A 0 -6 -4 -16 -8 B 6 0 8 0 -2 C 4 -8 0 -16 -6 D 16 0 16 0 -2 E 8 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -4 -16 -8 B 6 0 8 0 -2 C 4 -8 0 -16 -6 D 16 0 16 0 -2 E 8 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=26 C=19 A=19 B=8 so B is eliminated. Round 2 votes counts: E=36 D=26 C=19 A=19 so C is eliminated. Round 3 votes counts: E=37 A=37 D=26 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:209 B:206 C:187 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -4 -16 -8 B 6 0 8 0 -2 C 4 -8 0 -16 -6 D 16 0 16 0 -2 E 8 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -16 -8 B 6 0 8 0 -2 C 4 -8 0 -16 -6 D 16 0 16 0 -2 E 8 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -16 -8 B 6 0 8 0 -2 C 4 -8 0 -16 -6 D 16 0 16 0 -2 E 8 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9511: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) B E A C D (8) D E A C B (6) C D A B E (6) B A E C D (6) D C E A B (4) B C E D A (4) D C E B A (3) C B A E D (3) B E A D C (3) B C E A D (3) B A C E D (3) A D E C B (3) E B A D C (2) E A B D C (2) D E C A B (2) D A E C B (2) C D B E A (2) C D B A E (2) C A D E B (2) B E D C A (2) B E C D A (2) B E C A D (2) B C A E D (2) A E B D C (2) A B E C D (2) E D B A C (1) E D A B C (1) E A D B C (1) C D A E B (1) C B D E A (1) C B D A E (1) C B A D E (1) C A D B E (1) B D E C A (1) B C D E A (1) A E D C B (1) A E C D B (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 -8 0 -2 B 6 0 -4 4 12 C 8 4 0 12 -2 D 0 -4 -12 0 -4 E 2 -12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.000000 E: 0.222222 Sum of squares = 0.506172839697 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 -6 -8 0 -2 B 6 0 -4 4 12 C 8 4 0 12 -2 D 0 -4 -12 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.000000 E: 0.222222 Sum of squares = 0.506172839519 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=25 C=20 A=11 E=7 so E is eliminated. Round 2 votes counts: B=39 D=27 C=20 A=14 so A is eliminated. Round 3 votes counts: B=45 D=32 C=23 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:209 E:198 A:192 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 0 -2 B 6 0 -4 4 12 C 8 4 0 12 -2 D 0 -4 -12 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.000000 E: 0.222222 Sum of squares = 0.506172839519 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 0 -2 B 6 0 -4 4 12 C 8 4 0 12 -2 D 0 -4 -12 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.000000 E: 0.222222 Sum of squares = 0.506172839519 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 0 -2 B 6 0 -4 4 12 C 8 4 0 12 -2 D 0 -4 -12 0 -4 E 2 -12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.666667 D: 0.000000 E: 0.222222 Sum of squares = 0.506172839519 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9512: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) A B E C D (7) D B E C A (6) B E A D C (6) A C D E B (6) A B E D C (6) C A D E B (5) B E D A C (5) D C B E A (4) E B C D A (3) C E D B A (3) C D E A B (3) C D A E B (3) B E D C A (3) A D C B E (3) A D B E C (3) A C E B D (3) E B D C A (2) C E B D A (2) A E B C D (2) E C B D A (1) E B C A D (1) E B A C D (1) D C E B A (1) D C A E B (1) D C A B E (1) D B E A C (1) D A C B E (1) B D E C A (1) B D E A C (1) A D B C E (1) A C D B E (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 -4 -2 -6 -10 B 4 0 0 -8 4 C 2 0 0 6 0 D 6 8 -6 0 8 E 10 -4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.339247 C: 0.660753 D: 0.000000 E: 0.000000 Sum of squares = 0.551682811906 Cumulative probabilities = A: 0.000000 B: 0.339247 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -6 -10 B 4 0 0 -8 4 C 2 0 0 6 0 D 6 8 -6 0 8 E 10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204099944 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=27 B=16 D=15 E=8 so E is eliminated. Round 2 votes counts: A=34 C=28 B=23 D=15 so D is eliminated. Round 3 votes counts: C=35 A=35 B=30 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:208 C:204 B:200 E:199 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -6 -10 B 4 0 0 -8 4 C 2 0 0 6 0 D 6 8 -6 0 8 E 10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204099944 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -6 -10 B 4 0 0 -8 4 C 2 0 0 6 0 D 6 8 -6 0 8 E 10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204099944 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -6 -10 B 4 0 0 -8 4 C 2 0 0 6 0 D 6 8 -6 0 8 E 10 -4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.428571 C: 0.571429 D: 0.000000 E: 0.000000 Sum of squares = 0.510204099944 Cumulative probabilities = A: 0.000000 B: 0.428571 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9513: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (9) A B C D E (9) E C D B A (8) E D C B A (7) D E A B C (6) C B E A D (6) C E B A D (5) B A C E D (5) A D B E C (5) A B D C E (5) E B C D A (4) D E C B A (3) D A E C B (3) B C A E D (3) A C B E D (3) C E B D A (2) C B A E D (2) B C E A D (2) A D B C E (2) A B C E D (2) E D C A B (1) E C B D A (1) D E B C A (1) D A E B C (1) D A C E B (1) D A B E C (1) C E D B A (1) C A B E D (1) B E D C A (1) Total count = 100 A B C D E A 0 -2 -14 0 -14 B 2 0 -6 2 -6 C 14 6 0 8 -2 D 0 -2 -8 0 -8 E 14 6 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -14 0 -14 B 2 0 -6 2 -6 C 14 6 0 8 -2 D 0 -2 -8 0 -8 E 14 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 E=21 C=17 B=11 so B is eliminated. Round 2 votes counts: A=31 D=25 E=22 C=22 so E is eliminated. Round 3 votes counts: C=35 D=34 A=31 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:215 C:213 B:196 D:191 A:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -14 0 -14 B 2 0 -6 2 -6 C 14 6 0 8 -2 D 0 -2 -8 0 -8 E 14 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -14 0 -14 B 2 0 -6 2 -6 C 14 6 0 8 -2 D 0 -2 -8 0 -8 E 14 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -14 0 -14 B 2 0 -6 2 -6 C 14 6 0 8 -2 D 0 -2 -8 0 -8 E 14 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998481 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9514: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (11) B C A E D (7) E D A B C (6) E A B D C (5) E A B C D (5) D E C A B (5) C D B A E (5) B A C E D (5) E D C B A (4) D A C B E (4) C B A E D (4) E D A C B (3) D E A C B (3) D E A B C (3) D C B A E (3) C B D A E (3) E D C A B (2) E A D B C (2) D C E A B (2) D C A B E (2) D A E C B (2) C B E A D (2) E C B A D (1) D C E B A (1) D C B E A (1) D C A E B (1) C E B D A (1) C E B A D (1) C B D E A (1) B E A C D (1) B A E C D (1) B A C D E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 -6 -14 -4 2 B 6 0 -24 0 2 C 14 24 0 -2 10 D 4 0 2 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468750000013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 A B C D E A 0 -6 -14 -4 2 B 6 0 -24 0 2 C 14 24 0 -2 10 D 4 0 2 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=28 C=28 D=27 B=15 A=2 so A is eliminated. Round 2 votes counts: E=30 C=28 D=27 B=15 so B is eliminated. Round 3 votes counts: C=41 E=32 D=27 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:223 D:201 E:195 B:192 A:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -14 -4 2 B 6 0 -24 0 2 C 14 24 0 -2 10 D 4 0 2 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 -4 2 B 6 0 -24 0 2 C 14 24 0 -2 10 D 4 0 2 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 -4 2 B 6 0 -24 0 2 C 14 24 0 -2 10 D 4 0 2 0 -4 E -2 -2 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.625000 E: 0.125000 Sum of squares = 0.468749999062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.875000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9515: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (7) E A C B D (6) E A B C D (6) C D B A E (6) D B C A E (5) B D E A C (5) A E C B D (5) D B A E C (4) A E D C B (4) A E D B C (4) D B C E A (3) C A E D B (3) B D C E A (3) A E B D C (3) E C A B D (2) E B A D C (2) E A B D C (2) D C B A E (2) D A E B C (2) C E B A D (2) C B E D A (2) B C D E A (2) A D E B C (2) D C A E B (1) D C A B E (1) D B E A C (1) D B A C E (1) D A C E B (1) D A B C E (1) C E A B D (1) C D B E A (1) C D A E B (1) C D A B E (1) C B E A D (1) C B D E A (1) C A D E B (1) B E D C A (1) B E A D C (1) A D C E B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 20 10 16 B -14 0 -4 -10 -16 C -20 4 0 0 -14 D -10 10 0 0 -6 E -16 16 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999697 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 20 10 16 B -14 0 -4 -10 -16 C -20 4 0 0 -14 D -10 10 0 0 -6 E -16 16 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=22 C=20 E=18 B=12 so B is eliminated. Round 2 votes counts: D=30 A=28 C=22 E=20 so E is eliminated. Round 3 votes counts: A=45 D=31 C=24 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:230 E:210 D:197 C:185 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 20 10 16 B -14 0 -4 -10 -16 C -20 4 0 0 -14 D -10 10 0 0 -6 E -16 16 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 20 10 16 B -14 0 -4 -10 -16 C -20 4 0 0 -14 D -10 10 0 0 -6 E -16 16 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 20 10 16 B -14 0 -4 -10 -16 C -20 4 0 0 -14 D -10 10 0 0 -6 E -16 16 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9516: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (11) B E A C D (8) A B E D C (8) D C A E B (7) B A E C D (7) C D B A E (6) C D E A B (5) C E D B A (4) B A E D C (4) C D E B A (3) C B D E A (3) A E B D C (3) E A D B C (2) E A B D C (2) D C B A E (2) D C A B E (2) C E B A D (2) C D B E A (2) B C A E D (2) A B D E C (2) E D A C B (1) E B C A D (1) E B A D C (1) E A B C D (1) D E C A B (1) D E A C B (1) D A E C B (1) D A E B C (1) C E D A B (1) C E B D A (1) C B E D A (1) C B E A D (1) C B A E D (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 2 -12 -6 -4 B -2 0 -12 -4 -2 C 12 12 0 -2 8 D 6 4 2 0 -4 E 4 2 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428592 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.857143 E: 1.000000 A B C D E A 0 2 -12 -6 -4 B -2 0 -12 -4 -2 C 12 12 0 -2 8 D 6 4 2 0 -4 E 4 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=26 B=21 A=15 E=8 so E is eliminated. Round 2 votes counts: C=30 D=27 B=23 A=20 so A is eliminated. Round 3 votes counts: B=39 D=31 C=30 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:204 E:201 A:190 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -6 -4 B -2 0 -12 -4 -2 C 12 12 0 -2 8 D 6 4 2 0 -4 E 4 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.857143 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -6 -4 B -2 0 -12 -4 -2 C 12 12 0 -2 8 D 6 4 2 0 -4 E 4 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -6 -4 B -2 0 -12 -4 -2 C 12 12 0 -2 8 D 6 4 2 0 -4 E 4 2 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.285714 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9517: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) D B A E C (6) B A D C E (6) E D B A C (5) A B D C E (5) E D C A B (4) D B E A C (4) D A B E C (4) D A B C E (4) C E A B D (4) C A B D E (4) B A C D E (4) E C B A D (3) E B D A C (3) C B A E D (3) A B C D E (3) E D C B A (2) E D A B C (2) E B A D C (2) D E A B C (2) C D A E B (2) C D A B E (2) C A D B E (2) B D A E C (2) B A D E C (2) E D A C B (1) E C D B A (1) E C B D A (1) E B A C D (1) D E C A B (1) C E D A B (1) C E B A D (1) C D E A B (1) C A B E D (1) B C A D E (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 2 16 -12 6 B -2 0 16 -4 10 C -16 -16 0 -10 -10 D 12 4 10 0 10 E -6 -10 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 16 -12 6 B -2 0 16 -4 10 C -16 -16 0 -10 -10 D 12 4 10 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=21 C=21 B=17 A=8 so A is eliminated. Round 2 votes counts: E=33 B=25 D=21 C=21 so D is eliminated. Round 3 votes counts: B=43 E=36 C=21 so C is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:218 B:210 A:206 E:192 C:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 16 -12 6 B -2 0 16 -4 10 C -16 -16 0 -10 -10 D 12 4 10 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 16 -12 6 B -2 0 16 -4 10 C -16 -16 0 -10 -10 D 12 4 10 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 16 -12 6 B -2 0 16 -4 10 C -16 -16 0 -10 -10 D 12 4 10 0 10 E -6 -10 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9518: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (9) D E B C A (6) D B E C A (6) A C B E D (6) B D E A C (5) A C E B D (5) C A E D B (4) E D B C A (3) E D B A C (3) E A C D B (3) D B C E A (3) C D B A E (3) C A B D E (3) B D E C A (3) B D A E C (3) A B C D E (3) E C A D B (2) E A D B C (2) D C E B A (2) D C B E A (2) A E C B D (2) E D C B A (1) E D C A B (1) E D A C B (1) E C D A B (1) E B D A C (1) E A D C B (1) E A B D C (1) D B C A E (1) C E A D B (1) C D E B A (1) C D E A B (1) C D B E A (1) C B D A E (1) C B A D E (1) B D C E A (1) B D A C E (1) B C A D E (1) B A D E C (1) A E B C D (1) A C B D E (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -18 -16 -20 -4 B 18 0 12 2 14 C 16 -12 0 -16 4 D 20 -2 16 0 18 E 4 -14 -4 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -16 -20 -4 B 18 0 12 2 14 C 16 -12 0 -16 4 D 20 -2 16 0 18 E 4 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=20 D=20 A=20 C=16 so C is eliminated. Round 2 votes counts: A=27 D=26 B=26 E=21 so E is eliminated. Round 3 votes counts: A=37 D=36 B=27 so B is eliminated. Round 4 votes counts: D=60 A=40 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:223 C:196 E:184 A:171 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -16 -20 -4 B 18 0 12 2 14 C 16 -12 0 -16 4 D 20 -2 16 0 18 E 4 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -16 -20 -4 B 18 0 12 2 14 C 16 -12 0 -16 4 D 20 -2 16 0 18 E 4 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -16 -20 -4 B 18 0 12 2 14 C 16 -12 0 -16 4 D 20 -2 16 0 18 E 4 -14 -4 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998542 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9519: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (8) B D C E A (8) B E D C A (7) E B A D C (6) E A C D B (6) A E C D B (6) D C B A E (5) E A B C D (4) D C A B E (4) D B C A E (4) B E A C D (4) E A C B D (3) D C A E B (3) B E A D C (3) B D C A E (3) E B D C A (2) C D A E B (2) B D E C A (2) B C D A E (2) A C D E B (2) A C D B E (2) E B D A C (1) E A D C B (1) D E B C A (1) D C E B A (1) D C E A B (1) D C B E A (1) D B C E A (1) C D A B E (1) C A D E B (1) C A D B E (1) B A C E D (1) B A C D E (1) A E C B D (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 0 2 -20 B 22 0 18 14 2 C 0 -18 0 -8 -10 D -2 -14 8 0 -6 E 20 -2 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998972 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 0 2 -20 B 22 0 18 14 2 C 0 -18 0 -8 -10 D -2 -14 8 0 -6 E 20 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B E , winner is: B compare: Computing IRV winner. Round 1 votes counts: E=31 B=31 D=21 A=12 C=5 so C is eliminated. Round 2 votes counts: E=31 B=31 D=24 A=14 so A is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:228 E:217 D:193 C:182 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -22 0 2 -20 B 22 0 18 14 2 C 0 -18 0 -8 -10 D -2 -14 8 0 -6 E 20 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 0 2 -20 B 22 0 18 14 2 C 0 -18 0 -8 -10 D -2 -14 8 0 -6 E 20 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 0 2 -20 B 22 0 18 14 2 C 0 -18 0 -8 -10 D -2 -14 8 0 -6 E 20 -2 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999988655 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9520: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (5) E A C B D (5) D B C A E (5) E A B C D (4) C D B E A (4) C D A E B (4) A E C D B (4) E C A B D (3) D C B A E (3) D C A B E (3) C D B A E (3) C B D E A (3) B E D C A (3) B C D E A (3) A E C B D (3) A E B D C (3) A D E B C (3) A D C E B (3) E B C A D (2) E B A D C (2) D B C E A (2) D A B E C (2) C D A B E (2) C A D E B (2) B E A D C (2) B C E D A (2) A E D C B (2) A E D B C (2) A E B C D (2) E C B A D (1) D C A E B (1) D B A C E (1) D A C E B (1) D A B C E (1) C E B A D (1) C E A D B (1) C A E D B (1) B D E C A (1) B D E A C (1) B D C E A (1) B D A E C (1) A D E C B (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -2 6 8 B -8 0 -2 -2 -12 C 2 2 0 10 -6 D -6 2 -10 0 4 E -8 12 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406250000006 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 A B C D E A 0 8 -2 6 8 B -8 0 -2 -2 -12 C 2 2 0 10 -6 D -6 2 -10 0 4 E -8 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999956 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 E=22 C=21 D=19 B=14 so B is eliminated. Round 2 votes counts: E=27 C=26 A=24 D=23 so D is eliminated. Round 3 votes counts: C=41 A=30 E=29 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. A:210 C:204 E:203 D:195 B:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 -2 6 8 B -8 0 -2 -2 -12 C 2 2 0 10 -6 D -6 2 -10 0 4 E -8 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999956 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 8 B -8 0 -2 -2 -12 C 2 2 0 10 -6 D -6 2 -10 0 4 E -8 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999956 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 8 B -8 0 -2 -2 -12 C 2 2 0 10 -6 D -6 2 -10 0 4 E -8 12 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.375000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.125000 Sum of squares = 0.406249999956 Cumulative probabilities = A: 0.375000 B: 0.375000 C: 0.875000 D: 0.875000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9521: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) C E B A D (7) C B E D A (7) D A E B C (5) B D E A C (5) B D A E C (5) E C A D B (4) C E A B D (4) C A D E B (4) B D A C E (4) E C B A D (3) E A D B C (3) D B A E C (3) D A B E C (3) C B E A D (3) B E D A C (3) B D C A E (3) B C E D A (3) B C D A E (3) E B A D C (2) D A B C E (2) C B A D E (2) A D E B C (2) E A D C B (1) E A C D B (1) E A B D C (1) D A C B E (1) C B D A E (1) C A B D E (1) B E D C A (1) B D C E A (1) B C D E A (1) A E D C B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -14 -12 -2 -16 B 14 0 0 20 4 C 12 0 0 6 12 D 2 -20 -6 0 -4 E 16 -4 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.304883 C: 0.695117 D: 0.000000 E: 0.000000 Sum of squares = 0.57614109022 Cumulative probabilities = A: 0.000000 B: 0.304883 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -12 -2 -16 B 14 0 0 20 4 C 12 0 0 6 12 D 2 -20 -6 0 -4 E 16 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=29 E=15 D=14 A=5 so A is eliminated. Round 2 votes counts: C=38 B=29 D=17 E=16 so E is eliminated. Round 3 votes counts: C=46 B=32 D=22 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 C:215 E:202 D:186 A:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -12 -2 -16 B 14 0 0 20 4 C 12 0 0 6 12 D 2 -20 -6 0 -4 E 16 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 -2 -16 B 14 0 0 20 4 C 12 0 0 6 12 D 2 -20 -6 0 -4 E 16 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 -2 -16 B 14 0 0 20 4 C 12 0 0 6 12 D 2 -20 -6 0 -4 E 16 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500001 C: 0.499999 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.000000 B: 0.500001 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9522: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (8) C E B D A (7) B E C D A (5) D A E B C (4) A D B C E (4) E C D B A (3) C E D B A (3) C E D A B (3) C B E D A (3) C A B D E (3) B E D A C (3) B C E A D (3) B A D E C (3) A B D E C (3) A B D C E (3) E D C B A (2) E D C A B (2) E B D C A (2) D E A C B (2) D A E C B (2) D A B E C (2) C B A E D (2) C A D E B (2) B C A E D (2) A C D B E (2) A B C D E (2) E D B C A (1) E D B A C (1) E D A B C (1) E C B D A (1) D E A B C (1) C E A D B (1) C B E A D (1) C B A D E (1) C A E D B (1) C A D B E (1) C A B E D (1) B E A D C (1) B C E D A (1) B A E D C (1) B A D C E (1) B A C E D (1) B A C D E (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -2 2 6 B -2 0 8 4 18 C 2 -8 0 4 4 D -2 -4 -4 0 -4 E -6 -18 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -2 2 6 B -2 0 8 4 18 C 2 -8 0 4 4 D -2 -4 -4 0 -4 E -6 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999536 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=25 B=22 E=13 D=11 so D is eliminated. Round 2 votes counts: A=33 C=29 B=22 E=16 so E is eliminated. Round 3 votes counts: C=37 A=37 B=26 so B is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. B:214 A:204 C:201 D:193 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 2 6 B -2 0 8 4 18 C 2 -8 0 4 4 D -2 -4 -4 0 -4 E -6 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999536 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 6 B -2 0 8 4 18 C 2 -8 0 4 4 D -2 -4 -4 0 -4 E -6 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999536 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 6 B -2 0 8 4 18 C 2 -8 0 4 4 D -2 -4 -4 0 -4 E -6 -18 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.166667 C: 0.166667 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999536 Cumulative probabilities = A: 0.666667 B: 0.833333 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9523: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) D E C A B (6) B C A D E (6) E D A C B (5) E D A B C (5) C B A D E (5) B C D E A (5) D E C B A (4) C B D A E (4) C A B D E (4) B A C E D (4) E D B C A (3) A E D C B (3) A C B E D (3) E A D C B (2) D C E B A (2) C D B A E (2) C B D E A (2) B A E C D (2) A C D E B (2) A C B D E (2) E D B A C (1) E A D B C (1) D E B C A (1) D E A C B (1) D C E A B (1) D C B E A (1) D B E C A (1) C D B E A (1) C A D B E (1) B E D C A (1) B E A D C (1) B D E C A (1) B D C E A (1) B C D A E (1) B C A E D (1) A E D B C (1) A E C B D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -8 2 8 B 0 0 -4 12 20 C 8 4 0 14 16 D -2 -12 -14 0 8 E -8 -20 -16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 2 8 B 0 0 -4 12 20 C 8 4 0 14 16 D -2 -12 -14 0 8 E -8 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 B=23 C=19 E=17 D=17 so E is eliminated. Round 2 votes counts: D=31 A=27 B=23 C=19 so C is eliminated. Round 3 votes counts: D=34 B=34 A=32 so A is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:221 B:214 A:201 D:190 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -8 2 8 B 0 0 -4 12 20 C 8 4 0 14 16 D -2 -12 -14 0 8 E -8 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 2 8 B 0 0 -4 12 20 C 8 4 0 14 16 D -2 -12 -14 0 8 E -8 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 2 8 B 0 0 -4 12 20 C 8 4 0 14 16 D -2 -12 -14 0 8 E -8 -20 -16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9524: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) B A D C E (8) E B C A D (7) C E D A B (6) A D B C E (6) D A C E B (4) E B A D C (3) D A C B E (3) C D A E B (3) C B A D E (3) A B D C E (3) E D A C B (2) E D A B C (2) E C B D A (2) E C B A D (2) E B D A C (2) D A E C B (2) D A E B C (2) C E D B A (2) C E B A D (2) C D E A B (2) C B E A D (2) B E A D C (2) B C E A D (2) A D B E C (2) E D C B A (1) E C D B A (1) E B A C D (1) D E A C B (1) D C E A B (1) D C A E B (1) D A B E C (1) D A B C E (1) C B A E D (1) C A B D E (1) B C A D E (1) B A E D C (1) B A E C D (1) B A D E C (1) B A C E D (1) B A C D E (1) A D C B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 4 6 -2 B -6 0 -2 -4 -14 C -4 2 0 -2 10 D -6 4 2 0 -2 E 2 14 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000008 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 6 4 6 -2 B -6 0 -2 -4 -14 C -4 2 0 -2 10 D -6 4 2 0 -2 E 2 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000481 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=22 B=18 D=16 A=13 so A is eliminated. Round 2 votes counts: E=31 D=25 C=22 B=22 so C is eliminated. Round 3 votes counts: E=41 D=30 B=29 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:207 E:204 C:203 D:199 B:187 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 6 -2 B -6 0 -2 -4 -14 C -4 2 0 -2 10 D -6 4 2 0 -2 E 2 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000481 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 6 -2 B -6 0 -2 -4 -14 C -4 2 0 -2 10 D -6 4 2 0 -2 E 2 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000481 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 6 -2 B -6 0 -2 -4 -14 C -4 2 0 -2 10 D -6 4 2 0 -2 E 2 14 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.125000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000481 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9525: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (7) D B C A E (6) A E C D B (6) C B A E D (4) B C D A E (4) A D E C B (4) A D C B E (4) A C E B D (4) E C B A D (3) D E B A C (3) D B E A C (3) D B A C E (3) C A B E D (3) B D C E A (3) B C D E A (3) A E C B D (3) A D C E B (3) E D B C A (2) E D A B C (2) E C A B D (2) E B C D A (2) E A D C B (2) E A C D B (2) E A C B D (2) D B C E A (2) C B E A D (2) C B A D E (2) C A E B D (2) B E C D A (2) A C B E D (2) D E A B C (1) D B A E C (1) D A B C E (1) B E C A D (1) B D C A E (1) B C E D A (1) B C E A D (1) A E D C B (1) Total count = 100 A B C D E A 0 -12 -6 6 8 B 12 0 -2 -6 12 C 6 2 0 2 2 D -6 6 -2 0 2 E -8 -12 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999374 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 6 8 B 12 0 -2 -6 12 C 6 2 0 2 2 D -6 6 -2 0 2 E -8 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=27 A=27 E=17 B=16 C=13 so C is eliminated. Round 2 votes counts: A=32 D=27 B=24 E=17 so E is eliminated. Round 3 votes counts: A=40 D=31 B=29 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:208 C:206 D:200 A:198 E:188 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -6 6 8 B 12 0 -2 -6 12 C 6 2 0 2 2 D -6 6 -2 0 2 E -8 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 6 8 B 12 0 -2 -6 12 C 6 2 0 2 2 D -6 6 -2 0 2 E -8 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 6 8 B 12 0 -2 -6 12 C 6 2 0 2 2 D -6 6 -2 0 2 E -8 -12 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999957 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9526: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (7) E C B D A (6) E C A D B (6) C E B D A (6) E A C D B (5) C E B A D (5) C E A B D (5) D A B C E (4) A D E B C (4) A D B C E (4) D B A C E (3) C E A D B (3) B D A E C (3) A E D C B (3) A E C D B (3) A D E C B (3) E C B A D (2) E A D C B (2) D B A E C (2) C B D E A (2) C A E D B (2) B D C A E (2) B D A C E (2) B C D E A (2) E C A B D (1) E B D C A (1) E B C D A (1) E A D B C (1) D B E A C (1) D A E B C (1) D A B E C (1) C B E D A (1) C B D A E (1) B D E C A (1) B C E D A (1) B C D A E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 2 16 -4 B -14 0 -16 -14 -24 C -2 16 0 10 -8 D -16 14 -10 0 -10 E 4 24 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 2 16 -4 B -14 0 -16 -14 -24 C -2 16 0 10 -8 D -16 14 -10 0 -10 E 4 24 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=25 C=25 D=12 B=12 so D is eliminated. Round 2 votes counts: A=32 E=25 C=25 B=18 so B is eliminated. Round 3 votes counts: A=42 C=31 E=27 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. E:223 A:214 C:208 D:189 B:166 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 2 16 -4 B -14 0 -16 -14 -24 C -2 16 0 10 -8 D -16 14 -10 0 -10 E 4 24 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 16 -4 B -14 0 -16 -14 -24 C -2 16 0 10 -8 D -16 14 -10 0 -10 E 4 24 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 16 -4 B -14 0 -16 -14 -24 C -2 16 0 10 -8 D -16 14 -10 0 -10 E 4 24 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9527: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (12) B A D E C (7) B A D C E (7) B D A C E (6) A B E C D (6) D C B E A (4) C E D B A (4) C E D A B (4) C D E B A (4) E C D A B (3) D B C E A (3) C E A B D (3) B D C A E (3) A E D C B (3) A E C B D (3) A B D E C (3) D E C B A (2) D E C A B (2) D B A E C (2) A E B C D (2) E D C A B (1) E D A C B (1) E C A D B (1) E C A B D (1) E A C B D (1) D E B C A (1) D B E C A (1) D B A C E (1) D A B E C (1) C E B D A (1) C A B E D (1) B D C E A (1) B A C D E (1) A E C D B (1) A D E C B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -4 -14 0 B 20 0 -6 -4 -2 C 4 6 0 -26 12 D 14 4 26 0 24 E 0 2 -12 -24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999154 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -4 -14 0 B 20 0 -6 -4 -2 C 4 6 0 -26 12 D 14 4 26 0 24 E 0 2 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=25 A=21 C=17 E=8 so E is eliminated. Round 2 votes counts: D=31 B=25 C=22 A=22 so C is eliminated. Round 3 votes counts: D=46 A=28 B=26 so B is eliminated. Round 4 votes counts: D=57 A=43 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:234 B:204 C:198 E:183 A:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -4 -14 0 B 20 0 -6 -4 -2 C 4 6 0 -26 12 D 14 4 26 0 24 E 0 2 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -4 -14 0 B 20 0 -6 -4 -2 C 4 6 0 -26 12 D 14 4 26 0 24 E 0 2 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -4 -14 0 B 20 0 -6 -4 -2 C 4 6 0 -26 12 D 14 4 26 0 24 E 0 2 -12 -24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.9999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9528: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) D E A C B (7) E D A C B (6) B C A E D (6) C D E B A (5) C B D E A (5) A D E B C (5) E C D A B (4) D E C A B (4) C D B E A (4) C B E D A (4) B C E A D (4) B C D A E (4) B A D C E (4) E D C A B (3) B A C D E (3) A D B E C (3) A B E D C (3) D C E A B (2) D A B E C (2) C B E A D (2) A E D B C (2) A B D E C (2) E A D C B (1) D C B E A (1) D B C A E (1) D A E B C (1) C E D B A (1) C E B A D (1) B A E C D (1) A E B D C (1) Total count = 100 A B C D E A 0 -8 -2 -8 -8 B 8 0 0 -4 14 C 2 0 0 4 10 D 8 4 -4 0 6 E 8 -14 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.272543 C: 0.727457 D: 0.000000 E: 0.000000 Sum of squares = 0.603472950413 Cumulative probabilities = A: 0.000000 B: 0.272543 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -8 -8 B 8 0 0 -4 14 C 2 0 0 4 10 D 8 4 -4 0 6 E 8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499931 C: 0.500069 D: 0.000000 E: 0.000000 Sum of squares = 0.500000009495 Cumulative probabilities = A: 0.000000 B: 0.499931 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=22 D=18 A=16 E=14 so E is eliminated. Round 2 votes counts: B=30 D=27 C=26 A=17 so A is eliminated. Round 3 votes counts: D=38 B=36 C=26 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:209 C:208 D:207 E:189 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -2 -8 -8 B 8 0 0 -4 14 C 2 0 0 4 10 D 8 4 -4 0 6 E 8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499931 C: 0.500069 D: 0.000000 E: 0.000000 Sum of squares = 0.500000009495 Cumulative probabilities = A: 0.000000 B: 0.499931 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -8 -8 B 8 0 0 -4 14 C 2 0 0 4 10 D 8 4 -4 0 6 E 8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499931 C: 0.500069 D: 0.000000 E: 0.000000 Sum of squares = 0.500000009495 Cumulative probabilities = A: 0.000000 B: 0.499931 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -8 -8 B 8 0 0 -4 14 C 2 0 0 4 10 D 8 4 -4 0 6 E 8 -14 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499931 C: 0.500069 D: 0.000000 E: 0.000000 Sum of squares = 0.500000009495 Cumulative probabilities = A: 0.000000 B: 0.499931 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9529: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) D E C B A (8) C A B E D (8) D E B A C (7) A B E C D (7) D C E B A (6) C D E A B (6) B A E C D (5) C A D B E (4) C A B D E (4) A C B E D (4) E D B A C (3) D E B C A (3) A B C D E (3) E B D C A (2) E B A D C (2) C D E B A (2) C D A E B (2) C D A B E (2) E D C B A (1) E D B C A (1) E B D A C (1) E B A C D (1) D A E B C (1) D A B C E (1) C E D B A (1) B E A D C (1) B E A C D (1) B C E A D (1) A D C B E (1) A C B D E (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -2 6 6 B -8 0 0 2 6 C 2 0 0 22 10 D -6 -2 -22 0 4 E -6 -6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.153599 C: 0.846401 D: 0.000000 E: 0.000000 Sum of squares = 0.739987689887 Cumulative probabilities = A: 0.000000 B: 0.153599 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 6 6 B -8 0 0 2 6 C 2 0 0 22 10 D -6 -2 -22 0 4 E -6 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000217586 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=26 A=26 E=11 B=8 so B is eliminated. Round 2 votes counts: A=31 C=30 D=26 E=13 so E is eliminated. Round 3 votes counts: A=36 D=34 C=30 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:217 A:209 B:200 D:187 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 6 6 B -8 0 0 2 6 C 2 0 0 22 10 D -6 -2 -22 0 4 E -6 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000217586 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 6 B -8 0 0 2 6 C 2 0 0 22 10 D -6 -2 -22 0 4 E -6 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000217586 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 6 B -8 0 0 2 6 C 2 0 0 22 10 D -6 -2 -22 0 4 E -6 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000217586 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9530: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (12) C A B D E (11) E C A B D (8) C E A B D (7) D B A C E (6) E D B A C (5) C A B E D (5) E D A B C (4) E A B D C (4) E C D A B (3) E A C B D (3) D E B A C (3) C D B A E (3) A B D E C (3) E D C B A (2) E D C A B (2) D B E A C (2) C B D A E (2) B D A C E (2) B A D C E (2) E D B C A (1) E C D B A (1) E A B C D (1) D C B A E (1) C E D A B (1) C B A D E (1) B A C D E (1) A C B E D (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 4 0 6 B -12 0 -4 8 10 C -4 4 0 0 -10 D 0 -8 0 0 2 E -6 -10 10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.551519 B: 0.000000 C: 0.000000 D: 0.448481 E: 0.000000 Sum of squares = 0.505308397428 Cumulative probabilities = A: 0.551519 B: 0.551519 C: 0.551519 D: 1.000000 E: 1.000000 A B C D E A 0 12 4 0 6 B -12 0 -4 8 10 C -4 4 0 0 -10 D 0 -8 0 0 2 E -6 -10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=30 D=24 A=7 B=5 so B is eliminated. Round 2 votes counts: E=34 C=30 D=26 A=10 so A is eliminated. Round 3 votes counts: E=35 C=34 D=31 so D is eliminated. Round 4 votes counts: E=55 C=45 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:211 B:201 D:197 E:196 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 4 0 6 B -12 0 -4 8 10 C -4 4 0 0 -10 D 0 -8 0 0 2 E -6 -10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 4 0 6 B -12 0 -4 8 10 C -4 4 0 0 -10 D 0 -8 0 0 2 E -6 -10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 4 0 6 B -12 0 -4 8 10 C -4 4 0 0 -10 D 0 -8 0 0 2 E -6 -10 10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9531: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (9) A C B E D (8) C B E A D (4) B D E C A (4) B C E D A (4) B C E A D (4) B A C E D (4) A D E C B (4) A D C E B (4) E B D C A (3) D E C B A (3) D E B A C (3) D E A B C (3) C E B A D (3) B E C D A (3) B C A E D (3) E D C A B (2) E D B C A (2) E C B D A (2) D E B C A (2) D B E C A (2) D A B E C (2) B D A C E (2) A D C B E (2) A C E D B (2) E D C B A (1) E C D A B (1) E A C D B (1) D B E A C (1) D B A E C (1) D B A C E (1) D A B C E (1) C E A B D (1) C B A E D (1) C A E B D (1) B E D C A (1) A D B C E (1) A C E B D (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 6 -6 -18 B 8 0 0 4 4 C -6 0 0 -10 0 D 6 -4 10 0 -6 E 18 -4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.838488 C: 0.161512 D: 0.000000 E: 0.000000 Sum of squares = 0.729147970336 Cumulative probabilities = A: 0.000000 B: 0.838488 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 -6 -18 B 8 0 0 4 4 C -6 0 0 -10 0 D 6 -4 10 0 -6 E 18 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836800353 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=25 A=25 E=12 C=10 so C is eliminated. Round 2 votes counts: B=30 D=28 A=26 E=16 so E is eliminated. Round 3 votes counts: B=38 D=34 A=28 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:210 B:208 D:203 C:192 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 -6 -18 B 8 0 0 4 4 C -6 0 0 -10 0 D 6 -4 10 0 -6 E 18 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836800353 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -6 -18 B 8 0 0 4 4 C -6 0 0 -10 0 D 6 -4 10 0 -6 E 18 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836800353 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -6 -18 B 8 0 0 4 4 C -6 0 0 -10 0 D 6 -4 10 0 -6 E 18 -4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.714286 C: 0.285714 D: 0.000000 E: 0.000000 Sum of squares = 0.591836800353 Cumulative probabilities = A: 0.000000 B: 0.714286 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9532: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (12) C E D A B (7) B A C E D (7) E D C A B (5) A B D E C (5) D E A B C (4) D A E B C (4) C E D B A (4) C B E A D (4) B A D C E (4) B A C D E (4) C E B D A (3) B D A E C (3) A D B E C (3) E C D A B (2) E C A D B (2) E A C D B (2) D E B C A (2) D B A E C (2) D A B E C (2) C E A D B (2) C E A B D (2) A D E B C (2) A B E C D (2) E D A C B (1) E A D C B (1) D E A C B (1) C E B A D (1) C B E D A (1) C B A E D (1) B D A C E (1) B C D A E (1) B C A D E (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 24 14 12 B 2 0 22 6 8 C -24 -22 0 -8 -14 D -14 -6 8 0 2 E -12 -8 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 24 14 12 B 2 0 22 6 8 C -24 -22 0 -8 -14 D -14 -6 8 0 2 E -12 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=25 D=15 A=14 E=13 so E is eliminated. Round 2 votes counts: B=33 C=29 D=21 A=17 so A is eliminated. Round 3 votes counts: B=41 C=31 D=28 so D is eliminated. Round 4 votes counts: B=61 C=39 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:224 B:219 E:196 D:195 C:166 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 24 14 12 B 2 0 22 6 8 C -24 -22 0 -8 -14 D -14 -6 8 0 2 E -12 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 24 14 12 B 2 0 22 6 8 C -24 -22 0 -8 -14 D -14 -6 8 0 2 E -12 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 24 14 12 B 2 0 22 6 8 C -24 -22 0 -8 -14 D -14 -6 8 0 2 E -12 -8 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999956 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9533: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) E D B C A (8) D C E B A (8) E B D C A (7) A B E C D (7) C A D B E (6) C D A E B (5) E B A D C (4) D E C B A (4) A E B C D (4) A C D B E (4) A B C E D (4) E B D A C (3) C D A B E (3) B A E D C (3) E D C B A (2) C D E B A (2) C D E A B (2) C A D E B (2) A C B E D (2) E A B C D (1) D E B C A (1) D C B E A (1) D B E C A (1) B E D A C (1) A C E B D (1) A C B D E (1) A B E D C (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 -4 4 -12 B 12 0 14 2 -8 C 4 -14 0 -10 -16 D -4 -2 10 0 -18 E 12 8 16 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -4 4 -12 B 12 0 14 2 -8 C 4 -14 0 -10 -16 D -4 -2 10 0 -18 E 12 8 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=25 A=25 C=20 D=15 B=15 so D is eliminated. Round 2 votes counts: E=30 C=29 A=25 B=16 so B is eliminated. Round 3 votes counts: E=43 C=29 A=28 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:227 B:210 D:193 A:188 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -4 4 -12 B 12 0 14 2 -8 C 4 -14 0 -10 -16 D -4 -2 10 0 -18 E 12 8 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 4 -12 B 12 0 14 2 -8 C 4 -14 0 -10 -16 D -4 -2 10 0 -18 E 12 8 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 4 -12 B 12 0 14 2 -8 C 4 -14 0 -10 -16 D -4 -2 10 0 -18 E 12 8 16 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9534: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (6) C B A D E (6) A E D B C (5) A E C B D (5) C B D E A (4) A C B E D (4) D E B C A (3) D E A B C (3) D B E C A (3) C E D B A (3) C B A E D (3) B D A C E (3) B C D E A (3) B C D A E (3) A D B E C (3) A C E B D (3) A B C D E (3) E D C B A (2) E D A B C (2) E C A D B (2) E A D C B (2) D E C B A (2) D E B A C (2) D B E A C (2) C E B D A (2) C A B E D (2) A E C D B (2) A C B D E (2) E D C A B (1) E D A C B (1) E C D B A (1) E A C D B (1) D C E B A (1) D B A E C (1) D A E B C (1) C E B A D (1) C B E D A (1) C B E A D (1) B D C A E (1) B C A D E (1) B A C D E (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -4 -2 2 B 12 0 -4 0 8 C 4 4 0 8 10 D 2 0 -8 0 12 E -2 -8 -10 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -2 2 B 12 0 -4 0 8 C 4 4 0 8 10 D 2 0 -8 0 12 E -2 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=24 C=23 E=12 B=12 so E is eliminated. Round 2 votes counts: A=32 D=30 C=26 B=12 so B is eliminated. Round 3 votes counts: D=34 C=33 A=33 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:213 B:208 D:203 A:192 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -4 -2 2 B 12 0 -4 0 8 C 4 4 0 8 10 D 2 0 -8 0 12 E -2 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -2 2 B 12 0 -4 0 8 C 4 4 0 8 10 D 2 0 -8 0 12 E -2 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -2 2 B 12 0 -4 0 8 C 4 4 0 8 10 D 2 0 -8 0 12 E -2 -8 -10 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9535: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) B C D A E (8) E A D C B (7) D E A B C (6) E D A B C (5) D B E A C (5) D B C E A (5) B C D E A (5) A E C B D (5) A E D C B (4) A C E B D (4) E D A C B (3) D E B C A (3) D B C A E (3) C B E A D (3) C B A D E (3) A D E C B (3) E D B C A (2) E A C B D (2) D B E C A (2) D A E B C (2) B C A D E (2) A C B E D (2) E D C A B (1) E C B D A (1) E C B A D (1) E A C D B (1) C A B E D (1) B D C E A (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 0 -4 -6 B 6 0 -2 -4 0 C 0 2 0 -4 -6 D 4 4 4 0 -2 E 6 0 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.159275 C: 0.000000 D: 0.000000 E: 0.840725 Sum of squares = 0.732186549829 Cumulative probabilities = A: 0.000000 B: 0.159275 C: 0.159275 D: 0.159275 E: 1.000000 A B C D E A 0 -6 0 -4 -6 B 6 0 -2 -4 0 C 0 2 0 -4 -6 D 4 4 4 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556316 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 A=19 C=16 B=16 so C is eliminated. Round 2 votes counts: B=31 D=26 E=23 A=20 so A is eliminated. Round 3 votes counts: E=36 B=35 D=29 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:207 D:205 B:200 C:196 A:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 0 -4 -6 B 6 0 -2 -4 0 C 0 2 0 -4 -6 D 4 4 4 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556316 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -4 -6 B 6 0 -2 -4 0 C 0 2 0 -4 -6 D 4 4 4 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556316 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -4 -6 B 6 0 -2 -4 0 C 0 2 0 -4 -6 D 4 4 4 0 -2 E 6 0 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.666667 Sum of squares = 0.555555556316 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.333333 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9536: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (5) E A B C D (5) E D A C B (4) E B A D C (4) D B E C A (4) B D E C A (4) B C A D E (4) A C E D B (4) A C B E D (4) E D B C A (3) D E B C A (3) D C B E A (3) D B C E A (3) C B A D E (3) C A B D E (3) B D C A E (3) B C D A E (3) B A C D E (3) A C D E B (3) E D B A C (2) E A D B C (2) D E C A B (2) D C A B E (2) C A D E B (2) C A D B E (2) B D C E A (2) B A C E D (2) E D C A B (1) E A D C B (1) E A C D B (1) D E C B A (1) D C E B A (1) B E D C A (1) B E D A C (1) B E A C D (1) B A E C D (1) A E C D B (1) A E C B D (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 0 4 -8 B 14 0 16 8 6 C 0 -16 0 -4 2 D -4 -8 4 0 6 E 8 -6 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 4 -8 B 14 0 16 8 6 C 0 -16 0 -4 2 D -4 -8 4 0 6 E 8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=25 D=19 A=18 C=10 so C is eliminated. Round 2 votes counts: E=28 B=28 A=25 D=19 so D is eliminated. Round 3 votes counts: B=38 E=35 A=27 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:222 D:199 E:197 A:191 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 0 4 -8 B 14 0 16 8 6 C 0 -16 0 -4 2 D -4 -8 4 0 6 E 8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 4 -8 B 14 0 16 8 6 C 0 -16 0 -4 2 D -4 -8 4 0 6 E 8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 4 -8 B 14 0 16 8 6 C 0 -16 0 -4 2 D -4 -8 4 0 6 E 8 -6 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9537: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (13) C D B A E (10) A B E C D (8) E D C A B (6) D C E B A (6) A B C D E (6) B A C D E (5) A E B C D (5) E D C B A (4) E A B C D (3) D E C B A (3) D C B E A (3) C B D A E (3) A B C E D (3) E D A B C (2) E A D B C (2) D C E A B (2) C D A B E (2) B A E D C (2) E D B A C (1) E D A C B (1) E B D A C (1) E A C D B (1) D E C A B (1) D C B A E (1) C D A E B (1) B C D A E (1) B C A D E (1) B A E C D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 16 12 4 2 B -16 0 10 8 -4 C -12 -10 0 2 -10 D -4 -8 -2 0 -8 E -2 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999907 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 4 2 B -16 0 10 8 -4 C -12 -10 0 2 -10 D -4 -8 -2 0 -8 E -2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 A=24 D=16 C=16 B=10 so B is eliminated. Round 2 votes counts: E=34 A=32 C=18 D=16 so D is eliminated. Round 3 votes counts: E=38 A=32 C=30 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:210 B:199 D:189 C:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 4 2 B -16 0 10 8 -4 C -12 -10 0 2 -10 D -4 -8 -2 0 -8 E -2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 4 2 B -16 0 10 8 -4 C -12 -10 0 2 -10 D -4 -8 -2 0 -8 E -2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 4 2 B -16 0 10 8 -4 C -12 -10 0 2 -10 D -4 -8 -2 0 -8 E -2 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9538: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (11) E C A B D (8) C E A D B (5) B D E A C (5) B D A E C (5) D B C E A (4) D B A C E (4) D A B C E (4) E C A D B (3) B E A D C (3) B E A C D (3) B D A C E (3) E B C D A (2) E B A C D (2) D C B A E (2) D C A B E (2) D B C A E (2) C E D A B (2) C D A E B (2) C A E D B (2) C A D E B (2) B E D A C (2) B D E C A (2) B A D E C (2) A E C B D (2) A D C B E (2) E C D B A (1) E B D C A (1) D C B E A (1) C D E A B (1) B E C D A (1) A E B C D (1) A D B C E (1) A C E D B (1) A C E B D (1) A C D E B (1) A C D B E (1) A B E C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 10 14 8 -14 B -10 0 0 14 4 C -14 0 0 6 -12 D -8 -14 -6 0 -4 E 14 -4 12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.357143 Sum of squares = 0.397959183677 Cumulative probabilities = A: 0.142857 B: 0.642857 C: 0.642857 D: 0.642857 E: 1.000000 A B C D E A 0 10 14 8 -14 B -10 0 0 14 4 C -14 0 0 6 -12 D -8 -14 -6 0 -4 E 14 -4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.357143 Sum of squares = 0.397959183678 Cumulative probabilities = A: 0.142857 B: 0.642857 C: 0.642857 D: 0.642857 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=26 D=19 C=14 A=13 so A is eliminated. Round 2 votes counts: E=31 B=29 D=22 C=18 so C is eliminated. Round 3 votes counts: E=42 D=29 B=29 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:213 A:209 B:204 C:190 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 14 8 -14 B -10 0 0 14 4 C -14 0 0 6 -12 D -8 -14 -6 0 -4 E 14 -4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.357143 Sum of squares = 0.397959183678 Cumulative probabilities = A: 0.142857 B: 0.642857 C: 0.642857 D: 0.642857 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 8 -14 B -10 0 0 14 4 C -14 0 0 6 -12 D -8 -14 -6 0 -4 E 14 -4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.357143 Sum of squares = 0.397959183678 Cumulative probabilities = A: 0.142857 B: 0.642857 C: 0.642857 D: 0.642857 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 8 -14 B -10 0 0 14 4 C -14 0 0 6 -12 D -8 -14 -6 0 -4 E 14 -4 12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.357143 Sum of squares = 0.397959183678 Cumulative probabilities = A: 0.142857 B: 0.642857 C: 0.642857 D: 0.642857 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9539: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (16) D B C A E (12) B D C A E (9) E D C A B (8) B A C D E (5) A C B D E (5) E D A C B (4) E A C D B (3) D B E C A (3) C A D B E (3) C A B D E (3) B D A C E (3) E D B C A (2) E D B A C (2) E A B C D (2) D C B A E (2) B A E C D (2) A C E B D (2) A C B E D (2) A B C D E (2) E D C B A (1) E D A B C (1) E C A D B (1) E B A C D (1) D E B C A (1) D C E A B (1) C D A B E (1) C B A D E (1) B E D A C (1) B C A D E (1) Total count = 100 A B C D E A 0 8 2 -2 6 B -8 0 -6 10 10 C -2 6 0 0 4 D 2 -10 0 0 4 E -6 -10 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.500000 B: 0.100000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999999912 Cumulative probabilities = A: 0.500000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -2 6 B -8 0 -6 10 10 C -2 6 0 0 4 D 2 -10 0 0 4 E -6 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.100000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999995418 Cumulative probabilities = A: 0.500000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=41 B=21 D=19 A=11 C=8 so C is eliminated. Round 2 votes counts: E=41 B=22 D=20 A=17 so A is eliminated. Round 3 votes counts: E=43 B=34 D=23 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:207 C:204 B:203 D:198 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 8 2 -2 6 B -8 0 -6 10 10 C -2 6 0 0 4 D 2 -10 0 0 4 E -6 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.100000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999995418 Cumulative probabilities = A: 0.500000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -2 6 B -8 0 -6 10 10 C -2 6 0 0 4 D 2 -10 0 0 4 E -6 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.100000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999995418 Cumulative probabilities = A: 0.500000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -2 6 B -8 0 -6 10 10 C -2 6 0 0 4 D 2 -10 0 0 4 E -6 -10 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.100000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.419999995418 Cumulative probabilities = A: 0.500000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9540: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) A E C D B (8) D B A E C (7) B C E A D (7) C E A B D (6) B D C E A (6) A E D C B (6) E A C D B (5) E A C B D (5) C B E A D (5) D C A E B (4) D B C A E (4) D A E C B (3) D A E B C (3) C E A D B (3) C B D E A (2) B D A E C (2) E C A D B (1) E C A B D (1) D C B A E (1) D A C E B (1) C E D B A (1) C D B E A (1) C D A E B (1) C B E D A (1) C A E D B (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 0 -16 0 -14 B 0 0 -12 -2 -2 C 16 12 0 22 14 D 0 2 -22 0 -4 E 14 2 -14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999848 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -16 0 -14 B 0 0 -12 -2 -2 C 16 12 0 22 14 D 0 2 -22 0 -4 E 14 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 C=21 A=16 E=12 so E is eliminated. Round 2 votes counts: B=28 A=26 D=23 C=23 so D is eliminated. Round 3 votes counts: B=39 A=33 C=28 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:232 E:203 B:192 D:188 A:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -16 0 -14 B 0 0 -12 -2 -2 C 16 12 0 22 14 D 0 2 -22 0 -4 E 14 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -16 0 -14 B 0 0 -12 -2 -2 C 16 12 0 22 14 D 0 2 -22 0 -4 E 14 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -16 0 -14 B 0 0 -12 -2 -2 C 16 12 0 22 14 D 0 2 -22 0 -4 E 14 2 -14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9541: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (16) A C D E B (13) B E D C A (9) A B C D E (8) C D E A B (6) E D B C A (5) E B D C A (4) B E A D C (4) D C E B A (3) C D A E B (3) C A D E B (3) A C D B E (3) D C E A B (2) B E D A C (2) B A E D C (2) B A E C D (2) B A C D E (2) A E C D B (2) A C B D E (2) A B E C D (2) E D C A B (1) E B A D C (1) D C B E A (1) C D B A E (1) A E B C D (1) A C E D B (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 -6 -8 B 4 0 -14 -20 -22 C 8 14 0 0 -2 D 6 20 0 0 -6 E 8 22 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999906 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 -6 -8 B 4 0 -14 -20 -22 C 8 14 0 0 -2 D 6 20 0 0 -6 E 8 22 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=27 B=21 C=13 D=6 so D is eliminated. Round 2 votes counts: A=33 E=27 B=21 C=19 so C is eliminated. Round 3 votes counts: A=39 E=38 B=23 so B is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:219 C:210 D:210 A:187 B:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 -6 -8 B 4 0 -14 -20 -22 C 8 14 0 0 -2 D 6 20 0 0 -6 E 8 22 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -6 -8 B 4 0 -14 -20 -22 C 8 14 0 0 -2 D 6 20 0 0 -6 E 8 22 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -6 -8 B 4 0 -14 -20 -22 C 8 14 0 0 -2 D 6 20 0 0 -6 E 8 22 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999993657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9542: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (8) B A C E D (8) E D A C B (7) E A D C B (4) D C E A B (4) C A D E B (4) B E D A C (4) B C A E D (4) D E A C B (3) C A B E D (3) B C A D E (3) E D A B C (2) E A B D C (2) D E C B A (2) D E B A C (2) C D A E B (2) C A E D B (2) C A B D E (2) B C D A E (2) B A E C D (2) A E B C D (2) A C E D B (2) A C E B D (2) A C B E D (2) E B A D C (1) E A D B C (1) E A C B D (1) D C B A E (1) D C A E B (1) D B E C A (1) D B E A C (1) D B C E A (1) D B C A E (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A E D (1) C B A D E (1) C A E B D (1) C A D B E (1) B E A D C (1) B D E C A (1) B D E A C (1) B D C E A (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 18 0 4 4 B -18 0 -16 -6 -8 C 0 16 0 0 6 D -4 6 0 0 -8 E -4 8 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.441319 B: 0.000000 C: 0.558681 D: 0.000000 E: 0.000000 Sum of squares = 0.50688697431 Cumulative probabilities = A: 0.441319 B: 0.441319 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 0 4 4 B -18 0 -16 -6 -8 C 0 16 0 0 6 D -4 6 0 0 -8 E -4 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=25 C=20 E=18 A=10 so A is eliminated. Round 2 votes counts: B=28 C=26 D=25 E=21 so E is eliminated. Round 3 votes counts: D=39 B=33 C=28 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. A:213 C:211 E:203 D:197 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 18 0 4 4 B -18 0 -16 -6 -8 C 0 16 0 0 6 D -4 6 0 0 -8 E -4 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 0 4 4 B -18 0 -16 -6 -8 C 0 16 0 0 6 D -4 6 0 0 -8 E -4 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 0 4 4 B -18 0 -16 -6 -8 C 0 16 0 0 6 D -4 6 0 0 -8 E -4 8 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9543: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) E D C A B (5) E D C B A (4) E B D C A (4) C E A D B (4) B E C A D (4) B A D E C (4) B A D C E (4) A B D C E (4) E C B D A (3) D E C A B (3) D E B A C (3) D A B C E (3) A D C E B (3) E C D A B (2) D B E A C (2) D B A E C (2) C E A B D (2) C B A E D (2) C A E D B (2) B E D C A (2) B E D A C (2) B D E A C (2) B D A E C (2) A D C B E (2) A C D B E (2) E D B C A (1) E C D B A (1) E C A D B (1) E C A B D (1) E B C A D (1) D E B C A (1) D C E A B (1) D C A E B (1) D A E C B (1) D A C B E (1) C D A E B (1) C A E B D (1) C A B E D (1) B E A D C (1) B E A C D (1) B C A E D (1) B A C D E (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -12 2 6 -2 B 12 0 10 6 8 C -2 -10 0 -18 -6 D -6 -6 18 0 -10 E 2 -8 6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 2 6 -2 B 12 0 10 6 8 C -2 -10 0 -18 -6 D -6 -6 18 0 -10 E 2 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 E=23 D=18 A=14 C=13 so C is eliminated. Round 2 votes counts: B=34 E=29 D=19 A=18 so A is eliminated. Round 3 votes counts: B=41 E=32 D=27 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 E:205 D:198 A:197 C:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 2 6 -2 B 12 0 10 6 8 C -2 -10 0 -18 -6 D -6 -6 18 0 -10 E 2 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 2 6 -2 B 12 0 10 6 8 C -2 -10 0 -18 -6 D -6 -6 18 0 -10 E 2 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 2 6 -2 B 12 0 10 6 8 C -2 -10 0 -18 -6 D -6 -6 18 0 -10 E 2 -8 6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999833 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9544: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) A D E C B (10) E A D C B (9) B C E D A (9) B D A C E (7) C E B A D (6) C B E A D (6) B C D A E (6) D A B E C (5) A D B C E (5) D A B C E (4) A D E B C (3) E D A C B (2) E C B A D (2) D A E C B (2) C B E D A (2) B C E A D (2) E D A B C (1) E C D A B (1) E B D A C (1) D B A E C (1) C B A D E (1) C A D B E (1) B E C D A (1) B C A D E (1) A E D C B (1) Total count = 100 A B C D E A 0 10 24 -6 16 B -10 0 14 -12 2 C -24 -14 0 -24 0 D 6 12 24 0 14 E -16 -2 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 24 -6 16 B -10 0 14 -12 2 C -24 -14 0 -24 0 D 6 12 24 0 14 E -16 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 D=23 A=19 E=16 C=16 so E is eliminated. Round 2 votes counts: A=28 B=27 D=26 C=19 so C is eliminated. Round 3 votes counts: B=44 A=29 D=27 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:228 A:222 B:197 E:184 C:169 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 24 -6 16 B -10 0 14 -12 2 C -24 -14 0 -24 0 D 6 12 24 0 14 E -16 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 24 -6 16 B -10 0 14 -12 2 C -24 -14 0 -24 0 D 6 12 24 0 14 E -16 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 24 -6 16 B -10 0 14 -12 2 C -24 -14 0 -24 0 D 6 12 24 0 14 E -16 -2 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9545: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (13) A C E D B (11) B D E C A (10) D B E C A (5) C E A B D (5) D E B C A (4) E C A B D (3) C A E B D (3) B E C D A (3) E C D B A (2) E C B D A (2) E C A D B (2) E B C D A (2) D B E A C (2) D A E C B (2) D A B C E (2) B D A C E (2) B C A E D (2) A D B C E (2) A C B E D (2) E D C B A (1) E D C A B (1) E C D A B (1) E C B A D (1) D E C B A (1) D E A C B (1) D B A E C (1) D B A C E (1) D A C E B (1) C B E A D (1) B E C A D (1) B D C A E (1) B C E A D (1) B C A D E (1) B A C D E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 10 -6 10 2 B -10 0 -18 8 -20 C 6 18 0 18 4 D -10 -8 -18 0 -18 E -2 20 -4 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -6 10 2 B -10 0 -18 8 -20 C 6 18 0 18 4 D -10 -8 -18 0 -18 E -2 20 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 B=22 D=20 E=15 C=9 so C is eliminated. Round 2 votes counts: A=37 B=23 E=20 D=20 so E is eliminated. Round 3 votes counts: A=47 B=28 D=25 so D is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:223 E:216 A:208 B:180 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -6 10 2 B -10 0 -18 8 -20 C 6 18 0 18 4 D -10 -8 -18 0 -18 E -2 20 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -6 10 2 B -10 0 -18 8 -20 C 6 18 0 18 4 D -10 -8 -18 0 -18 E -2 20 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -6 10 2 B -10 0 -18 8 -20 C 6 18 0 18 4 D -10 -8 -18 0 -18 E -2 20 -4 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999456 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9546: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B A E (10) D A C B E (10) A D C B E (9) A E D B C (8) E B C D A (7) A D E C B (7) B E C D A (6) B C E D A (6) E B C A D (5) C B D E A (5) A D C E B (5) A E B D C (4) E B A C D (3) E A B C D (3) D C B E A (2) B C D E A (2) A E B C D (2) A D E B C (2) D C A B E (1) C D B E A (1) C D B A E (1) A E D C B (1) Total count = 100 A B C D E A 0 4 8 -2 20 B -4 0 -4 -14 6 C -8 4 0 -18 4 D 2 14 18 0 10 E -20 -6 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999414 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -2 20 B -4 0 -4 -14 6 C -8 4 0 -18 4 D 2 14 18 0 10 E -20 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 D=23 E=18 B=14 C=7 so C is eliminated. Round 2 votes counts: A=38 D=25 B=19 E=18 so E is eliminated. Round 3 votes counts: A=41 B=34 D=25 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:222 A:215 B:192 C:191 E:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -2 20 B -4 0 -4 -14 6 C -8 4 0 -18 4 D 2 14 18 0 10 E -20 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 20 B -4 0 -4 -14 6 C -8 4 0 -18 4 D 2 14 18 0 10 E -20 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 20 B -4 0 -4 -14 6 C -8 4 0 -18 4 D 2 14 18 0 10 E -20 -6 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994062 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9547: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A B D (6) D A B E C (6) E C A D B (5) C E D A B (5) B D A C E (5) B A D E C (5) D C E B A (4) D C E A B (4) A B D E C (4) E A C B D (3) D B A E C (3) D A E C B (3) C E B A D (3) C E A D B (3) B C D E A (3) A E C B D (3) D C A E B (2) D B A C E (2) D A B C E (2) C E D B A (2) B D C E A (2) B C E D A (2) B A D C E (2) A E B C D (2) A D B E C (2) A B E C D (2) E D A C B (1) E C B A D (1) E A C D B (1) D B C E A (1) D B C A E (1) C D E B A (1) B D A E C (1) B A E D C (1) B A E C D (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B D C (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 18 10 0 6 B -18 0 2 -2 -6 C -10 -2 0 -12 -10 D 0 2 12 0 10 E -6 6 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.466206 B: 0.000000 C: 0.000000 D: 0.533794 E: 0.000000 Sum of squares = 0.502284111372 Cumulative probabilities = A: 0.466206 B: 0.466206 C: 0.466206 D: 1.000000 E: 1.000000 A B C D E A 0 18 10 0 6 B -18 0 2 -2 -6 C -10 -2 0 -12 -10 D 0 2 12 0 10 E -6 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=23 A=18 E=17 C=14 so C is eliminated. Round 2 votes counts: E=30 D=29 B=23 A=18 so A is eliminated. Round 3 votes counts: E=38 D=32 B=30 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. A:217 D:212 E:200 B:188 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 0 6 B -18 0 2 -2 -6 C -10 -2 0 -12 -10 D 0 2 12 0 10 E -6 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 0 6 B -18 0 2 -2 -6 C -10 -2 0 -12 -10 D 0 2 12 0 10 E -6 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 0 6 B -18 0 2 -2 -6 C -10 -2 0 -12 -10 D 0 2 12 0 10 E -6 6 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9548: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (12) E C D A B (7) E D C A B (6) E D B C A (6) E D C B A (5) E D B A C (5) C D A B E (5) B A C D E (4) E C A D B (3) E B D A C (3) D C A B E (3) C D A E B (3) C A B D E (3) B A D C E (3) A C B D E (3) A B C E D (3) E B A D C (2) E A C B D (2) D E B C A (2) D B A C E (2) B D E A C (2) B A C E D (2) E B A C D (1) D E C B A (1) D C B A E (1) D C A E B (1) D B C E A (1) C E A D B (1) C A E D B (1) C A D E B (1) B E D A C (1) B D A C E (1) B A E C D (1) B A D E C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -24 -10 4 B -12 0 -18 -32 0 C 24 18 0 8 4 D 10 32 -8 0 0 E -4 0 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -24 -10 4 B -12 0 -18 -32 0 C 24 18 0 8 4 D 10 32 -8 0 0 E -4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=40 C=26 B=15 D=11 A=8 so A is eliminated. Round 2 votes counts: E=40 C=30 B=19 D=11 so D is eliminated. Round 3 votes counts: E=43 C=35 B=22 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:227 D:217 E:196 A:191 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -24 -10 4 B -12 0 -18 -32 0 C 24 18 0 8 4 D 10 32 -8 0 0 E -4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -24 -10 4 B -12 0 -18 -32 0 C 24 18 0 8 4 D 10 32 -8 0 0 E -4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -24 -10 4 B -12 0 -18 -32 0 C 24 18 0 8 4 D 10 32 -8 0 0 E -4 0 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9549: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) C E B D A (6) C E A B D (6) C B A E D (6) E C A D B (5) D E A B C (4) B D A C E (4) A B D C E (4) D E B C A (3) D A B E C (3) B D C A E (3) B C D A E (3) E D C B A (2) E D C A B (2) E C D A B (2) D B E A C (2) D B A E C (2) D A E B C (2) C E B A D (2) C B E A D (2) C A B E D (2) B C D E A (2) B C A D E (2) B A D C E (2) B A C D E (2) A E C D B (2) A D B E C (2) A D B C E (2) E D A B C (1) E C D B A (1) E C A B D (1) E A C D B (1) D E B A C (1) D B E C A (1) D B C E A (1) C B D A E (1) C A E B D (1) A E D B C (1) A D E B C (1) A C B E D (1) Total count = 100 A B C D E A 0 4 -8 -10 -8 B -4 0 -4 0 -6 C 8 4 0 -4 4 D 10 0 4 0 -6 E 8 6 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.34693877553 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 A B C D E A 0 4 -8 -10 -8 B -4 0 -4 0 -6 C 8 4 0 -4 4 D 10 0 4 0 -6 E 8 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=24 D=19 B=18 A=13 so A is eliminated. Round 2 votes counts: E=27 C=27 D=24 B=22 so B is eliminated. Round 3 votes counts: D=37 C=36 E=27 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:208 C:206 D:204 B:193 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 -10 -8 B -4 0 -4 0 -6 C 8 4 0 -4 4 D 10 0 4 0 -6 E 8 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -10 -8 B -4 0 -4 0 -6 C 8 4 0 -4 4 D 10 0 4 0 -6 E 8 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -10 -8 B -4 0 -4 0 -6 C 8 4 0 -4 4 D 10 0 4 0 -6 E 8 6 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.285714 E: 0.285714 Sum of squares = 0.346938775528 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.714286 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9550: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) A C B D E (10) E C B A D (8) A D C B E (7) E B C A D (5) E D B C A (4) E B C D A (4) C A B E D (4) B C E A D (4) E D C B A (3) D E B A C (3) D E A C B (3) D A B E C (3) B E C D A (3) A C B E D (3) D A E B C (2) C E B A D (2) B E C A D (2) A C D B E (2) E D A C B (1) E C D A B (1) E C B D A (1) E C A B D (1) E B D C A (1) D E A B C (1) D A E C B (1) D A C E B (1) D A C B E (1) C B E A D (1) C B A E D (1) B C A E D (1) B C A D E (1) A E D C B (1) A D C E B (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 12 6 12 4 B -12 0 -8 6 10 C -6 8 0 10 4 D -12 -6 -10 0 -4 E -4 -10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 12 4 B -12 0 -8 6 10 C -6 8 0 10 4 D -12 -6 -10 0 -4 E -4 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 A=26 B=11 C=8 so C is eliminated. Round 2 votes counts: E=31 A=30 D=26 B=13 so B is eliminated. Round 3 votes counts: E=41 A=33 D=26 so D is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:217 C:208 B:198 E:193 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 6 12 4 B -12 0 -8 6 10 C -6 8 0 10 4 D -12 -6 -10 0 -4 E -4 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 12 4 B -12 0 -8 6 10 C -6 8 0 10 4 D -12 -6 -10 0 -4 E -4 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 12 4 B -12 0 -8 6 10 C -6 8 0 10 4 D -12 -6 -10 0 -4 E -4 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9551: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D C A B E (7) E C B D A (6) E B A C D (5) B E C A D (5) D C A E B (4) D A C B E (4) C D E B A (4) B A C D E (4) E D A C B (3) D E A C B (3) C D B E A (3) B C D A E (3) A B E D C (3) A B D C E (3) D C E A B (2) D A E C B (2) C E D B A (2) C B E D A (2) B C E A D (2) A E B C D (2) A D B E C (2) A D B C E (2) E C D B A (1) E A D C B (1) E A D B C (1) E A B C D (1) D E C A B (1) D C E B A (1) D C B E A (1) D C B A E (1) D A C E B (1) C D B A E (1) C B D E A (1) C B D A E (1) B E A C D (1) A E D B C (1) A D E C B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -16 -8 -12 B 8 0 -6 2 -4 C 16 6 0 10 0 D 8 -2 -10 0 8 E 12 4 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.671743 D: 0.000000 E: 0.328257 Sum of squares = 0.558991624278 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.671743 D: 0.671743 E: 1.000000 A B C D E A 0 -8 -16 -8 -12 B 8 0 -6 2 -4 C 16 6 0 10 0 D 8 -2 -10 0 8 E 12 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=27 A=16 B=15 C=14 so C is eliminated. Round 2 votes counts: D=35 E=30 B=19 A=16 so A is eliminated. Round 3 votes counts: D=40 E=33 B=27 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:216 E:204 D:202 B:200 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 -12 B 8 0 -6 2 -4 C 16 6 0 10 0 D 8 -2 -10 0 8 E 12 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 -12 B 8 0 -6 2 -4 C 16 6 0 10 0 D 8 -2 -10 0 8 E 12 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 -12 B 8 0 -6 2 -4 C 16 6 0 10 0 D 8 -2 -10 0 8 E 12 4 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.499999 Sum of squares = 0.499999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500001 D: 0.500001 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9552: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (8) D E B A C (7) D C E A B (7) C E D B A (6) A B C E D (6) D E C B A (5) E B D C A (4) C A D E B (4) B A E D C (4) A B D E C (4) A B D C E (4) C D A E B (3) C A B E D (3) A C B D E (3) D B E A C (2) D A C E B (2) C D E B A (2) C D E A B (2) B E D A C (2) A D B E C (2) A B E C D (2) A B C D E (2) E D B A C (1) E C B D A (1) E B D A C (1) E B C A D (1) D C A E B (1) C E A D B (1) C B E A D (1) C A E D B (1) C A D B E (1) B E C A D (1) B E A C D (1) B C E A D (1) B A D E C (1) A D C B E (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 2 -6 -8 -8 B -2 0 8 -14 -14 C 6 -8 0 -12 6 D 8 14 12 0 8 E 8 14 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -8 -8 B -2 0 8 -14 -14 C 6 -8 0 -12 6 D 8 14 12 0 8 E 8 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=24 C=24 E=16 B=10 so B is eliminated. Round 2 votes counts: A=31 C=25 D=24 E=20 so E is eliminated. Round 3 votes counts: D=40 A=32 C=28 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:221 E:204 C:196 A:190 B:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -6 -8 -8 B -2 0 8 -14 -14 C 6 -8 0 -12 6 D 8 14 12 0 8 E 8 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -8 -8 B -2 0 8 -14 -14 C 6 -8 0 -12 6 D 8 14 12 0 8 E 8 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -8 -8 B -2 0 8 -14 -14 C 6 -8 0 -12 6 D 8 14 12 0 8 E 8 14 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999899 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9553: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (7) D E C A B (5) B C A E D (5) A D B E C (5) A B D C E (5) E C D B A (4) E C D A B (4) C E D B A (4) B C E D A (4) B C D E A (4) B A C E D (4) A D E C B (4) D E C B A (3) D E B C A (3) D A E C B (3) B D C E A (3) B C E A D (3) B C A D E (3) A E D C B (3) A E C B D (3) E D C A B (2) D B E C A (2) C E B A D (2) B D A C E (2) B A D C E (2) A C E B D (2) E C A D B (1) D E A C B (1) D A E B C (1) C B E A D (1) B D C A E (1) B A C D E (1) A C B E D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 -16 -22 -6 -6 B 16 0 0 10 -4 C 22 0 0 8 10 D 6 -10 -8 0 -2 E 6 4 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.320299 C: 0.679701 D: 0.000000 E: 0.000000 Sum of squares = 0.564584753026 Cumulative probabilities = A: 0.000000 B: 0.320299 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -22 -6 -6 B 16 0 0 10 -4 C 22 0 0 8 10 D 6 -10 -8 0 -2 E 6 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=25 D=18 C=14 E=11 so E is eliminated. Round 2 votes counts: B=32 A=25 C=23 D=20 so D is eliminated. Round 3 votes counts: B=37 C=33 A=30 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:220 B:211 E:201 D:193 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -22 -6 -6 B 16 0 0 10 -4 C 22 0 0 8 10 D 6 -10 -8 0 -2 E 6 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -22 -6 -6 B 16 0 0 10 -4 C 22 0 0 8 10 D 6 -10 -8 0 -2 E 6 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -22 -6 -6 B 16 0 0 10 -4 C 22 0 0 8 10 D 6 -10 -8 0 -2 E 6 4 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999959 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9554: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) E A B D C (7) C B D E A (6) E C A B D (5) E A C B D (5) C E A D B (5) E A C D B (3) C E A B D (3) C D A B E (3) C A E D B (3) B D E A C (3) B D A E C (3) A E D B C (3) D C B A E (2) D B A C E (2) D A B C E (2) C E D B A (2) C E B D A (2) C D B E A (2) B C D E A (2) A E D C B (2) A E B D C (2) A D E C B (2) A D B E C (2) E C B A D (1) E C A D B (1) E B A D C (1) E A B C D (1) D C A B E (1) D B A E C (1) D A B E C (1) C E B A D (1) C B D A E (1) C A D E B (1) B E D A C (1) B D C A E (1) B A E D C (1) B A D E C (1) A E C D B (1) A D C E B (1) A B D E C (1) Total count = 100 A B C D E A 0 10 -8 6 -2 B -10 0 -30 -4 -4 C 8 30 0 20 4 D -6 4 -20 0 0 E 2 4 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -8 6 -2 B -10 0 -30 -4 -4 C 8 30 0 20 4 D -6 4 -20 0 0 E 2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 E=24 A=14 B=12 D=9 so D is eliminated. Round 2 votes counts: C=44 E=24 A=17 B=15 so B is eliminated. Round 3 votes counts: C=47 E=28 A=25 so A is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:231 A:203 E:201 D:189 B:176 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -8 6 -2 B -10 0 -30 -4 -4 C 8 30 0 20 4 D -6 4 -20 0 0 E 2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -8 6 -2 B -10 0 -30 -4 -4 C 8 30 0 20 4 D -6 4 -20 0 0 E 2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -8 6 -2 B -10 0 -30 -4 -4 C 8 30 0 20 4 D -6 4 -20 0 0 E 2 4 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999981258 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9555: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (13) D C A E B (13) B E A C D (13) D E B C A (12) C A D B E (9) E B D C A (5) C A D E B (5) A C B E D (5) E D B C A (4) B A C E D (3) E D B A C (2) C A B E D (2) B E D C A (2) B E D A C (2) B A E C D (2) A C D B E (2) D E C B A (1) D E C A B (1) C D A E B (1) C D A B E (1) B E C A D (1) A C B D E (1) Total count = 100 A B C D E A 0 -20 -14 -14 -12 B 20 0 18 -2 -14 C 14 -18 0 -10 -16 D 14 2 10 0 -8 E 12 14 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999564 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -20 -14 -14 -12 B 20 0 18 -2 -14 C 14 -18 0 -10 -16 D 14 2 10 0 -8 E 12 14 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 E=24 B=23 C=18 A=8 so A is eliminated. Round 2 votes counts: D=27 C=26 E=24 B=23 so B is eliminated. Round 3 votes counts: E=44 C=29 D=27 so D is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:225 B:211 D:209 C:185 A:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -20 -14 -14 -12 B 20 0 18 -2 -14 C 14 -18 0 -10 -16 D 14 2 10 0 -8 E 12 14 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -14 -14 -12 B 20 0 18 -2 -14 C 14 -18 0 -10 -16 D 14 2 10 0 -8 E 12 14 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -14 -14 -12 B 20 0 18 -2 -14 C 14 -18 0 -10 -16 D 14 2 10 0 -8 E 12 14 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999783 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9556: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (10) E C A B D (9) C A E B D (9) C A E D B (7) B D E C A (7) D B A C E (6) B D E A C (5) A C E D B (5) B D C A E (4) D B A E C (3) C E A B D (3) B E D C A (3) E C B A D (2) D E B A C (2) C A B E D (2) B C E D A (2) A E C D B (2) A C D E B (2) E D B A C (1) E B D A C (1) E B C A D (1) E A D C B (1) E A C D B (1) E A C B D (1) D B C A E (1) D A E C B (1) D A C B E (1) D A B C E (1) C B A D E (1) C A D E B (1) A E D C B (1) A D E C B (1) A D C E B (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 4 2 B -2 0 -6 0 -4 C 4 6 0 -2 -4 D -4 0 2 0 -2 E -2 4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999989 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 2 -4 4 2 B -2 0 -6 0 -4 C 4 6 0 -2 -4 D -4 0 2 0 -2 E -2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999989 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 B=21 E=17 A=14 so A is eliminated. Round 2 votes counts: C=31 D=28 B=21 E=20 so E is eliminated. Round 3 votes counts: C=46 D=31 B=23 so B is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:204 A:202 C:202 D:198 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -4 4 2 B -2 0 -6 0 -4 C 4 6 0 -2 -4 D -4 0 2 0 -2 E -2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999989 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 2 B -2 0 -6 0 -4 C 4 6 0 -2 -4 D -4 0 2 0 -2 E -2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999989 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 2 B -2 0 -6 0 -4 C 4 6 0 -2 -4 D -4 0 2 0 -2 E -2 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999989 Cumulative probabilities = A: 0.400000 B: 0.400000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9557: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (14) A C E D B (14) D B E C A (8) A C E B D (8) A C B E D (7) C A E D B (6) B D A C E (6) E C A D B (4) C A E B D (4) E C D A B (3) D E B C A (3) C E A D B (2) C A B E D (2) E D C B A (1) E D C A B (1) E D A C B (1) E C D B A (1) E A D C B (1) D E B A C (1) D E A B C (1) D B E A C (1) C E B D A (1) C B E A D (1) B D E A C (1) B D C A E (1) B D A E C (1) B C A D E (1) B A D C E (1) B A C D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 14 -6 10 10 B -14 0 -14 4 -4 C 6 14 0 14 16 D -10 -4 -14 0 -16 E -10 4 -16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -6 10 10 B -14 0 -14 4 -4 C 6 14 0 14 16 D -10 -4 -14 0 -16 E -10 4 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=26 C=16 D=14 E=12 so E is eliminated. Round 2 votes counts: A=33 B=26 C=24 D=17 so D is eliminated. Round 3 votes counts: B=39 A=35 C=26 so C is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:225 A:214 E:197 B:186 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -6 10 10 B -14 0 -14 4 -4 C 6 14 0 14 16 D -10 -4 -14 0 -16 E -10 4 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -6 10 10 B -14 0 -14 4 -4 C 6 14 0 14 16 D -10 -4 -14 0 -16 E -10 4 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -6 10 10 B -14 0 -14 4 -4 C 6 14 0 14 16 D -10 -4 -14 0 -16 E -10 4 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9558: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (7) E C D B A (6) C D E A B (6) A B C E D (6) A B C D E (6) E D B C A (5) C E D A B (5) E D C B A (4) C E A D B (4) B D A E C (4) A B D C E (4) D E C B A (3) D E B C A (3) D B E A C (3) C E A B D (3) C A E D B (3) A C B E D (3) D C E B A (2) C D A E B (2) C A E B D (2) C A D E B (2) B D E A C (2) B A E D C (2) A C B D E (2) E D B A C (1) E C B A D (1) E B A C D (1) D E B A C (1) D A C B E (1) C D E B A (1) C A D B E (1) C A B E D (1) B A E C D (1) B A D C E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -8 2 -2 B -4 0 -4 -6 -10 C 8 4 0 14 10 D -2 6 -14 0 2 E 2 10 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 2 -2 B -4 0 -4 -6 -10 C 8 4 0 14 10 D -2 6 -14 0 2 E 2 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=22 E=18 B=17 D=13 so D is eliminated. Round 2 votes counts: C=32 E=25 A=23 B=20 so B is eliminated. Round 3 votes counts: A=38 C=32 E=30 so E is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 E:200 A:198 D:196 B:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -8 2 -2 B -4 0 -4 -6 -10 C 8 4 0 14 10 D -2 6 -14 0 2 E 2 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 2 -2 B -4 0 -4 -6 -10 C 8 4 0 14 10 D -2 6 -14 0 2 E 2 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 2 -2 B -4 0 -4 -6 -10 C 8 4 0 14 10 D -2 6 -14 0 2 E 2 10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998838 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9559: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) B A D C E (7) E C A D B (6) E D C B A (5) E C D B A (5) D B A C E (5) A B D C E (5) D B A E C (4) B D A E C (4) A E C B D (4) E C D A B (3) E A B D C (3) D B E C A (3) C A B D E (3) B A D E C (3) A B E D C (3) C E A D B (2) C E A B D (2) C A E B D (2) A C B E D (2) A B E C D (2) A B D E C (2) A B C D E (2) E D B C A (1) E C A B D (1) E A C B D (1) D E C B A (1) D E B A C (1) D C E B A (1) D B E A C (1) D B C A E (1) C E D A B (1) C D E B A (1) C D E A B (1) C D B A E (1) B D E A C (1) A E B C D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 20 6 20 B 4 0 14 14 16 C -20 -14 0 -16 -10 D -6 -14 16 0 10 E -20 -16 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 20 6 20 B 4 0 14 14 16 C -20 -14 0 -16 -10 D -6 -14 16 0 10 E -20 -16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=23 B=22 D=17 C=13 so C is eliminated. Round 2 votes counts: E=30 A=28 B=22 D=20 so D is eliminated. Round 3 votes counts: B=37 E=35 A=28 so A is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:224 A:221 D:203 E:182 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 20 6 20 B 4 0 14 14 16 C -20 -14 0 -16 -10 D -6 -14 16 0 10 E -20 -16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 20 6 20 B 4 0 14 14 16 C -20 -14 0 -16 -10 D -6 -14 16 0 10 E -20 -16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 20 6 20 B 4 0 14 14 16 C -20 -14 0 -16 -10 D -6 -14 16 0 10 E -20 -16 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9560: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) C E D A B (8) A D B C E (7) A B D C E (7) E D C B A (5) B D A E C (5) A C D E B (5) E C B D A (3) C E D B A (3) B D E C A (3) B D E A C (3) A C B E D (3) E C D A B (2) D E B A C (2) D B E C A (2) C E B A D (2) C E A B D (2) C A E D B (2) C A E B D (2) B A D E C (2) B A D C E (2) B A C E D (2) A D C B E (2) A D B E C (2) E D B C A (1) D E C A B (1) D E B C A (1) D B E A C (1) D B A E C (1) D A E C B (1) C E B D A (1) C E A D B (1) B C E D A (1) B C E A D (1) A C E D B (1) A C D B E (1) A B D E C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 0 -6 -4 B 0 0 -8 -16 -4 C 0 8 0 2 10 D 6 16 -2 0 0 E 4 4 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.191424 B: 0.000000 C: 0.808576 D: 0.000000 E: 0.000000 Sum of squares = 0.690438241915 Cumulative probabilities = A: 0.191424 B: 0.191424 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -6 -4 B 0 0 -8 -16 -4 C 0 8 0 2 10 D 6 16 -2 0 0 E 4 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000023707 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=21 E=20 B=19 D=9 so D is eliminated. Round 2 votes counts: A=32 E=24 B=23 C=21 so C is eliminated. Round 3 votes counts: E=41 A=36 B=23 so B is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. C:210 D:210 E:199 A:195 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 0 -6 -4 B 0 0 -8 -16 -4 C 0 8 0 2 10 D 6 16 -2 0 0 E 4 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000023707 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -6 -4 B 0 0 -8 -16 -4 C 0 8 0 2 10 D 6 16 -2 0 0 E 4 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000023707 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -6 -4 B 0 0 -8 -16 -4 C 0 8 0 2 10 D 6 16 -2 0 0 E 4 4 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.750000 D: 0.000000 E: 0.000000 Sum of squares = 0.625000023707 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9561: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) B E C A D (6) E B D A C (5) D A C E B (5) C A B D E (4) B E C D A (4) E D B A C (3) E B D C A (3) D E B A C (3) D C A E B (3) D A E C B (3) C B E D A (3) B C A E D (3) A E D B C (3) A C D B E (3) E B A D C (2) D E C B A (2) D E B C A (2) D E A C B (2) D E A B C (2) C B A E D (2) C B A D E (2) C A D B E (2) B C E D A (2) B C E A D (2) A E B D C (2) A D C E B (2) A C B E D (2) E A B D C (1) D C E B A (1) C D E A B (1) C D B E A (1) C B D E A (1) C A B E D (1) B E D C A (1) B E A C D (1) B A E C D (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -14 -16 4 B 0 0 -2 2 10 C 14 2 0 8 4 D 16 -2 -8 0 0 E -4 -10 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -16 4 B 0 0 -2 2 10 C 14 2 0 8 4 D 16 -2 -8 0 0 E -4 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=23 B=20 A=15 E=14 so E is eliminated. Round 2 votes counts: B=30 C=28 D=26 A=16 so A is eliminated. Round 3 votes counts: B=36 C=33 D=31 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:205 D:203 E:191 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -14 -16 4 B 0 0 -2 2 10 C 14 2 0 8 4 D 16 -2 -8 0 0 E -4 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -16 4 B 0 0 -2 2 10 C 14 2 0 8 4 D 16 -2 -8 0 0 E -4 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -16 4 B 0 0 -2 2 10 C 14 2 0 8 4 D 16 -2 -8 0 0 E -4 -10 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9562: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) E D A C B (7) D A E B C (6) E C B D A (5) E A D C B (4) D E B A C (4) C B E A D (4) B C D A E (4) A D B C E (4) A B C D E (4) E D C A B (3) D E B C A (3) D E A B C (3) B C A D E (3) A D E B C (3) E D C B A (2) E D A B C (2) D A B E C (2) C E B A D (2) C B E D A (2) B D C A E (2) B C E D A (2) B A C D E (2) A E D C B (2) A D B E C (2) A C B E D (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A B D (1) E A C D B (1) D E A C B (1) D B E C A (1) D B A C E (1) B D C E A (1) B D A C E (1) B C D E A (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 0 6 -10 -4 B 0 0 4 -8 -6 C -6 -4 0 -12 -12 D 10 8 12 0 0 E 4 6 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.475699 E: 0.524301 Sum of squares = 0.501181119103 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.475699 E: 1.000000 A B C D E A 0 0 6 -10 -4 B 0 0 4 -8 -6 C -6 -4 0 -12 -12 D 10 8 12 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=21 A=19 C=16 B=16 so C is eliminated. Round 2 votes counts: E=30 B=30 D=21 A=19 so A is eliminated. Round 3 votes counts: B=37 E=32 D=31 so D is eliminated. Round 4 votes counts: E=53 B=47 so B is eliminated. IRV winner is E compare: Computing Borda winner. D:215 E:211 A:196 B:195 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 -10 -4 B 0 0 4 -8 -6 C -6 -4 0 -12 -12 D 10 8 12 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -10 -4 B 0 0 4 -8 -6 C -6 -4 0 -12 -12 D 10 8 12 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -10 -4 B 0 0 4 -8 -6 C -6 -4 0 -12 -12 D 10 8 12 0 0 E 4 6 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9563: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (15) D B C E A (10) A E D B C (8) C B D E A (7) A E C B D (7) B D C E A (6) A D B C E (6) C E B D A (5) E C A B D (4) A D B E C (4) D B E C A (3) D B E A C (3) C E B A D (3) E A C D B (2) E A C B D (2) D B C A E (2) D B A C E (2) C E A B D (2) A D E B C (2) E D B C A (1) E C B D A (1) D B A E C (1) D A B C E (1) C B E A D (1) B C D E A (1) A C E B D (1) Total count = 100 A B C D E A 0 8 8 14 -2 B -8 0 0 -20 -6 C -8 0 0 2 -6 D -14 20 -2 0 -4 E 2 6 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 8 14 -2 B -8 0 0 -20 -6 C -8 0 0 2 -6 D -14 20 -2 0 -4 E 2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=43 D=22 C=18 E=10 B=7 so B is eliminated. Round 2 votes counts: A=43 D=28 C=19 E=10 so E is eliminated. Round 3 votes counts: A=47 D=29 C=24 so C is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:209 D:200 C:194 B:183 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 14 -2 B -8 0 0 -20 -6 C -8 0 0 2 -6 D -14 20 -2 0 -4 E 2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 14 -2 B -8 0 0 -20 -6 C -8 0 0 2 -6 D -14 20 -2 0 -4 E 2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 14 -2 B -8 0 0 -20 -6 C -8 0 0 2 -6 D -14 20 -2 0 -4 E 2 6 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9564: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) E C A B D (7) D B C E A (7) A E C B D (6) E A D C B (4) D B E C A (4) D B C A E (4) C B D E A (4) B D C A E (4) B C D A E (4) A E D B C (4) E D C B A (3) E A C D B (3) C B D A E (3) C B A D E (3) C A B E D (3) A C E B D (3) E A D B C (2) D E B C A (2) C B A E D (2) A B D C E (2) E D A B C (1) E C B A D (1) E C A D B (1) D E B A C (1) D E A B C (1) D C B E A (1) D B A C E (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C E A B D (1) B D C E A (1) B C A D E (1) A E C D B (1) A E B C D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 -14 12 -6 B -6 0 -12 14 -4 C 14 12 0 10 -4 D -12 -14 -10 0 -8 E 6 4 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999664 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -14 12 -6 B -6 0 -12 14 -4 C 14 12 0 10 -4 D -12 -14 -10 0 -8 E 6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=24 A=19 C=17 B=10 so B is eliminated. Round 2 votes counts: E=30 D=29 C=22 A=19 so A is eliminated. Round 3 votes counts: E=42 D=31 C=27 so C is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:216 E:211 A:199 B:196 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -14 12 -6 B -6 0 -12 14 -4 C 14 12 0 10 -4 D -12 -14 -10 0 -8 E 6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -14 12 -6 B -6 0 -12 14 -4 C 14 12 0 10 -4 D -12 -14 -10 0 -8 E 6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -14 12 -6 B -6 0 -12 14 -4 C 14 12 0 10 -4 D -12 -14 -10 0 -8 E 6 4 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9565: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) A B D C E (10) C E D A B (8) D A B E C (6) D E A B C (5) B A C D E (5) E D C B A (4) C E D B A (4) C E B A D (4) A D B E C (4) C E B D A (3) C E A D B (3) C B A E D (3) C A B E D (3) B A D E C (3) E D C A B (2) E C D A B (2) D E A C B (2) D A E B C (2) C E A B D (2) C B E A D (2) B A D C E (2) A D B C E (2) A B D E C (2) E D B C A (1) D E C A B (1) D B E A C (1) D B A E C (1) C A E B D (1) B D A E C (1) A D E B C (1) Total count = 100 A B C D E A 0 12 -6 -6 -8 B -12 0 -8 -18 -10 C 6 8 0 0 4 D 6 18 0 0 -4 E 8 10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.725449 D: 0.274551 E: 0.000000 Sum of squares = 0.601654107757 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.725449 D: 1.000000 E: 1.000000 A B C D E A 0 12 -6 -6 -8 B -12 0 -8 -18 -10 C 6 8 0 0 4 D 6 18 0 0 -4 E 8 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=19 A=19 D=18 B=11 so B is eliminated. Round 2 votes counts: C=33 A=29 E=19 D=19 so E is eliminated. Round 3 votes counts: C=45 A=29 D=26 so D is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:209 E:209 A:196 B:176 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -6 -6 -8 B -12 0 -8 -18 -10 C 6 8 0 0 4 D 6 18 0 0 -4 E 8 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -6 -8 B -12 0 -8 -18 -10 C 6 8 0 0 4 D 6 18 0 0 -4 E 8 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -6 -8 B -12 0 -8 -18 -10 C 6 8 0 0 4 D 6 18 0 0 -4 E 8 10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500002 D: 0.499998 E: 0.000000 Sum of squares = 0.500000000005 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500002 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9566: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) A E C D B (11) E A D B C (8) D B E A C (8) C A E B D (8) B D C E A (8) D E A B C (7) C B A E D (6) B C D E A (6) C B D A E (5) E A D C B (3) C B A D E (3) C A B E D (3) A C E B D (3) E D A B C (2) D E B A C (2) B D E C A (2) C D A B E (1) B D E A C (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 16 14 14 6 B -16 0 -10 -8 -12 C -14 10 0 -4 -10 D -14 8 4 0 -12 E -6 12 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 14 14 6 B -16 0 -10 -8 -12 C -14 10 0 -4 -10 D -14 8 4 0 -12 E -6 12 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=26 A=26 B=18 D=17 E=13 so E is eliminated. Round 2 votes counts: A=37 C=26 D=19 B=18 so B is eliminated. Round 3 votes counts: A=37 C=33 D=30 so D is eliminated. Round 4 votes counts: A=57 C=43 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:225 E:214 D:193 C:191 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 14 14 6 B -16 0 -10 -8 -12 C -14 10 0 -4 -10 D -14 8 4 0 -12 E -6 12 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 14 14 6 B -16 0 -10 -8 -12 C -14 10 0 -4 -10 D -14 8 4 0 -12 E -6 12 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 14 14 6 B -16 0 -10 -8 -12 C -14 10 0 -4 -10 D -14 8 4 0 -12 E -6 12 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9567: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) B E C A D (6) E B A C D (5) C B E D A (5) D C A B E (4) C E A B D (4) C D B A E (4) D A E B C (3) D A C B E (3) C E B A D (3) B E D A C (3) A D E C B (3) A D E B C (3) E C B A D (2) E A B D C (2) D C B A E (2) D B C A E (2) D B A E C (2) D B A C E (2) D A E C B (2) C B E A D (2) C B D E A (2) C A E D B (2) B E C D A (2) B D E C A (2) B C E D A (2) A E C D B (2) E B C A D (1) E B A D C (1) E A B C D (1) D C B E A (1) D B E A C (1) D B C E A (1) D A C E B (1) D A B E C (1) C D B E A (1) C D A B E (1) B E D C A (1) B E A D C (1) B D E A C (1) B C E A D (1) B C D E A (1) A E D B C (1) A E C B D (1) A E B D C (1) A D C E B (1) A C E D B (1) Total count = 100 A B C D E A 0 -14 -16 -12 -4 B 14 0 -6 0 8 C 16 6 0 10 4 D 12 0 -10 0 0 E 4 -8 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999918 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -16 -12 -4 B 14 0 -6 0 8 C 16 6 0 10 4 D 12 0 -10 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 D=25 B=20 A=13 E=12 so E is eliminated. Round 2 votes counts: C=32 B=27 D=25 A=16 so A is eliminated. Round 3 votes counts: C=36 D=33 B=31 so B is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:208 D:201 E:196 A:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -16 -12 -4 B 14 0 -6 0 8 C 16 6 0 10 4 D 12 0 -10 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -16 -12 -4 B 14 0 -6 0 8 C 16 6 0 10 4 D 12 0 -10 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -16 -12 -4 B 14 0 -6 0 8 C 16 6 0 10 4 D 12 0 -10 0 0 E 4 -8 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999108 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9568: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (12) C B D E A (12) C B A E D (12) D E B C A (7) A E D B C (7) A E B C D (5) A C B E D (5) E D A B C (4) B C D E A (4) A C E B D (3) E A D B C (2) D E C B A (2) D E A C B (2) D C E B A (2) D C B E A (2) D B C E A (2) D A E B C (2) B C E A D (2) A E C B D (2) A B C E D (2) E D B C A (1) E B C D A (1) C D B A E (1) C B E A D (1) C B A D E (1) C A B E D (1) A E D C B (1) A E C D B (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -2 -8 -12 B 0 0 2 2 -10 C 2 -2 0 6 0 D 8 -2 -6 0 0 E 12 10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.612276 D: 0.000000 E: 0.387724 Sum of squares = 0.525211745619 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.612276 D: 0.612276 E: 1.000000 A B C D E A 0 0 -2 -8 -12 B 0 0 2 2 -10 C 2 -2 0 6 0 D 8 -2 -6 0 0 E 12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 C=28 A=27 E=8 B=6 so B is eliminated. Round 2 votes counts: C=34 D=31 A=27 E=8 so E is eliminated. Round 3 votes counts: D=36 C=35 A=29 so A is eliminated. Round 4 votes counts: C=53 D=47 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:211 C:203 D:200 B:197 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 -8 -12 B 0 0 2 2 -10 C 2 -2 0 6 0 D 8 -2 -6 0 0 E 12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 -8 -12 B 0 0 2 2 -10 C 2 -2 0 6 0 D 8 -2 -6 0 0 E 12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 -8 -12 B 0 0 2 2 -10 C 2 -2 0 6 0 D 8 -2 -6 0 0 E 12 10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9569: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) B D C E A (7) D C A E B (6) A C E D B (6) E A B C D (5) D C B E A (5) B E D C A (5) A E C D B (5) A E C B D (5) A C D E B (5) E A C B D (4) B E A D C (4) A E B C D (4) D C A B E (3) D B C A E (3) E B A C D (2) D C B A E (2) D B C E A (2) D A C B E (2) B E C D A (2) B D E C A (2) B D C A E (2) B D A E C (2) E C D B A (1) E A C D B (1) D A C E B (1) C D E B A (1) C D E A B (1) C A D E B (1) B D A C E (1) B C D E A (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 16 -2 -12 14 B -16 0 -14 -6 -10 C 2 14 0 2 14 D 12 6 -2 0 10 E -14 10 -14 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -12 14 B -16 0 -14 -6 -10 C 2 14 0 2 14 D 12 6 -2 0 10 E -14 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 D=24 E=13 C=10 so C is eliminated. Round 2 votes counts: D=33 A=28 B=26 E=13 so E is eliminated. Round 3 votes counts: A=38 D=34 B=28 so B is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:216 D:213 A:208 E:186 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -2 -12 14 B -16 0 -14 -6 -10 C 2 14 0 2 14 D 12 6 -2 0 10 E -14 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -12 14 B -16 0 -14 -6 -10 C 2 14 0 2 14 D 12 6 -2 0 10 E -14 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -12 14 B -16 0 -14 -6 -10 C 2 14 0 2 14 D 12 6 -2 0 10 E -14 10 -14 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991373 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9570: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (10) D C E A B (7) A D B C E (7) D C A E B (6) B A E C D (6) C E D A B (5) E C B D A (4) B E C D A (4) A D C E B (4) D A C E B (3) D A C B E (3) C E D B A (3) A B E C D (3) E C B A D (2) E B C D A (2) D B A C E (2) D A B C E (2) C D E A B (2) C A E D B (2) B E D C A (2) B E A C D (2) B A D E C (2) A D C B E (2) A C D E B (2) A B D C E (2) E B C A D (1) E A C B D (1) D E C B A (1) D C E B A (1) D B C E A (1) C E A D B (1) C A D E B (1) B D E A C (1) B D A E C (1) B A E D C (1) A C E D B (1) Total count = 100 A B C D E A 0 8 -10 0 0 B -8 0 -2 -12 2 C 10 2 0 4 14 D 0 12 -4 0 0 E 0 -2 -14 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999748 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 0 0 B -8 0 -2 -12 2 C 10 2 0 4 14 D 0 12 -4 0 0 E 0 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=26 A=21 C=14 E=10 so E is eliminated. Round 2 votes counts: B=32 D=26 A=22 C=20 so C is eliminated. Round 3 votes counts: B=38 D=36 A=26 so A is eliminated. Round 4 votes counts: D=56 B=44 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 D:204 A:199 E:192 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 0 0 B -8 0 -2 -12 2 C 10 2 0 4 14 D 0 12 -4 0 0 E 0 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 0 0 B -8 0 -2 -12 2 C 10 2 0 4 14 D 0 12 -4 0 0 E 0 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 0 0 B -8 0 -2 -12 2 C 10 2 0 4 14 D 0 12 -4 0 0 E 0 -2 -14 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999196 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9571: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (13) D C A E B (8) C D B E A (8) B E A C D (7) D A C E B (6) B A E C D (5) A E D B C (5) D C E A B (4) B E C D A (4) D C E B A (3) C D A E B (3) C D A B E (3) C B D E A (3) A E B D C (3) A D E C B (3) E A D B C (2) D A E C B (2) B C E A D (2) E D B A C (1) E A B D C (1) D E C B A (1) D A E B C (1) C D B A E (1) C B E D A (1) C B E A D (1) C B D A E (1) C B A E D (1) C A D B E (1) B E A D C (1) B C E D A (1) B C A E D (1) A E B C D (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 6 -2 -6 B 10 0 -2 -6 -16 C -6 2 0 -10 -2 D 2 6 10 0 -2 E 6 16 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 6 -2 -6 B 10 0 -2 -6 -16 C -6 2 0 -10 -2 D 2 6 10 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 C=23 B=21 E=17 A=14 so A is eliminated. Round 2 votes counts: D=29 E=26 C=23 B=22 so B is eliminated. Round 3 votes counts: E=44 D=29 C=27 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:213 D:208 A:194 B:193 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 6 -2 -6 B 10 0 -2 -6 -16 C -6 2 0 -10 -2 D 2 6 10 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 -2 -6 B 10 0 -2 -6 -16 C -6 2 0 -10 -2 D 2 6 10 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 -2 -6 B 10 0 -2 -6 -16 C -6 2 0 -10 -2 D 2 6 10 0 -2 E 6 16 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999462 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9572: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (7) A E B C D (7) E C B D A (6) E A D C B (6) E D C A B (5) E D C B A (4) A D E C B (4) E C D B A (3) D C B E A (3) C B E D A (3) B C E D A (3) B A C D E (3) A D E B C (3) A D B C E (3) E D A C B (2) E A B C D (2) D C E B A (2) D C B A E (2) D A B C E (2) C B D E A (2) B C A E D (2) A E D B C (2) A D C B E (2) A D B E C (2) A B D C E (2) A B C E D (2) A B C D E (2) E C B A D (1) E B C D A (1) E A C D B (1) E A C B D (1) D E C A B (1) D B C A E (1) D A C E B (1) C E D B A (1) C D E B A (1) B D C A E (1) B C E A D (1) A E D C B (1) A E B D C (1) A B D E C (1) Total count = 100 A B C D E A 0 6 0 -2 -12 B -6 0 -4 -4 -12 C 0 4 0 -2 -8 D 2 4 2 0 -10 E 12 12 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 0 -2 -12 B -6 0 -4 -4 -12 C 0 4 0 -2 -8 D 2 4 2 0 -10 E 12 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=32 A=32 B=17 D=12 C=7 so C is eliminated. Round 2 votes counts: E=33 A=32 B=22 D=13 so D is eliminated. Round 3 votes counts: E=37 A=35 B=28 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:221 D:199 C:197 A:196 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 0 -2 -12 B -6 0 -4 -4 -12 C 0 4 0 -2 -8 D 2 4 2 0 -10 E 12 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 -2 -12 B -6 0 -4 -4 -12 C 0 4 0 -2 -8 D 2 4 2 0 -10 E 12 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 -2 -12 B -6 0 -4 -4 -12 C 0 4 0 -2 -8 D 2 4 2 0 -10 E 12 12 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9573: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (9) A C E D B (9) E D B A C (8) C A E B D (8) B D E C A (6) D B E A C (5) C B D E A (5) B D C E A (5) A E C D B (5) C A B E D (4) C B A D E (3) E D A B C (2) E B D C A (2) E A C D B (2) D B A E C (2) C B E D A (2) C B D A E (2) B C D E A (2) A E D C B (2) A D B E C (2) A D B C E (2) E C A D B (1) E C A B D (1) E B D A C (1) E A D B C (1) D E B A C (1) C E B A D (1) C E A B D (1) C A E D B (1) B D E A C (1) B D C A E (1) A E D B C (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -8 10 6 B -6 0 -14 8 4 C 8 14 0 14 12 D -10 -8 -14 0 -4 E -6 -4 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 10 6 B -6 0 -14 8 4 C 8 14 0 14 12 D -10 -8 -14 0 -4 E -6 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=23 E=18 B=15 D=8 so D is eliminated. Round 2 votes counts: C=36 A=23 B=22 E=19 so E is eliminated. Round 3 votes counts: C=38 B=34 A=28 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:224 A:207 B:196 E:191 D:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 10 6 B -6 0 -14 8 4 C 8 14 0 14 12 D -10 -8 -14 0 -4 E -6 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 10 6 B -6 0 -14 8 4 C 8 14 0 14 12 D -10 -8 -14 0 -4 E -6 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 10 6 B -6 0 -14 8 4 C 8 14 0 14 12 D -10 -8 -14 0 -4 E -6 -4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9574: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (13) D B A E C (8) B C E A D (7) D A E B C (6) C E B A D (6) E C A D B (5) C E A B D (5) B D A E C (5) C E A D B (4) C B E A D (4) B D C E A (4) A E C D B (4) D A B E C (3) B D A C E (3) B C D E A (3) A E D C B (3) E A C D B (2) D E A C B (2) D B C A E (2) B D C A E (2) A E B C D (2) E C D A B (1) D B A C E (1) B A E C D (1) B A D E C (1) B A C E D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 4 14 -6 14 B -4 0 0 -10 -8 C -14 0 0 -8 -16 D 6 10 8 0 8 E -14 8 16 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 -6 14 B -4 0 0 -10 -8 C -14 0 0 -8 -16 D 6 10 8 0 8 E -14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=27 C=19 A=11 E=8 so E is eliminated. Round 2 votes counts: D=35 B=27 C=25 A=13 so A is eliminated. Round 3 votes counts: D=39 C=31 B=30 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:216 A:213 E:201 B:189 C:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 14 -6 14 B -4 0 0 -10 -8 C -14 0 0 -8 -16 D 6 10 8 0 8 E -14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 -6 14 B -4 0 0 -10 -8 C -14 0 0 -8 -16 D 6 10 8 0 8 E -14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 -6 14 B -4 0 0 -10 -8 C -14 0 0 -8 -16 D 6 10 8 0 8 E -14 8 16 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9575: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (12) D C B E A (6) D C A B E (6) A D C E B (6) A E B C D (5) D C A E B (4) C D B A E (4) B E C D A (4) B E A C D (4) E B D A C (3) E A B D C (3) C D B E A (3) C A B D E (3) B C D E A (3) A E B D C (3) A C B E D (3) A B E C D (3) E B A C D (2) E A B C D (2) D E B C A (2) D C E B A (2) B C E D A (2) A E D C B (2) A C D E B (2) A B C E D (2) E B A D C (1) E A D B C (1) D E A B C (1) D C B A E (1) D B C E A (1) C B D E A (1) C B D A E (1) C A D B E (1) A E D B C (1) Total count = 100 A B C D E A 0 20 -12 -12 18 B -20 0 -14 -10 20 C 12 14 0 14 26 D 12 10 -14 0 18 E -18 -20 -26 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -12 -12 18 B -20 0 -14 -10 20 C 12 14 0 14 26 D 12 10 -14 0 18 E -18 -20 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=25 D=23 B=13 E=12 so E is eliminated. Round 2 votes counts: A=33 C=25 D=23 B=19 so B is eliminated. Round 3 votes counts: A=40 C=34 D=26 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:233 D:213 A:207 B:188 E:159 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -12 -12 18 B -20 0 -14 -10 20 C 12 14 0 14 26 D 12 10 -14 0 18 E -18 -20 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -12 -12 18 B -20 0 -14 -10 20 C 12 14 0 14 26 D 12 10 -14 0 18 E -18 -20 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -12 -12 18 B -20 0 -14 -10 20 C 12 14 0 14 26 D 12 10 -14 0 18 E -18 -20 -26 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9576: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) E A B C D (9) D C B A E (8) D E A C B (5) C B D A E (5) A E D B C (5) C D B A E (4) C E B D A (3) C B A D E (3) B C A D E (3) A E B D C (3) E D C A B (2) E D A B C (2) E C B A D (2) E B A C D (2) E A B D C (2) D C A B E (2) D A E B C (2) D A B C E (2) C D B E A (2) C B E A D (2) C B A E D (2) B C E A D (2) B C A E D (2) B A C E D (2) A B D E C (2) E C D A B (1) D E C A B (1) D E A B C (1) D C E B A (1) D C E A B (1) D C B E A (1) D A E C B (1) C B E D A (1) B E A C D (1) B D A C E (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 6 4 8 0 B -6 0 6 -2 -4 C -4 -6 0 -6 -2 D -8 2 6 0 -6 E 0 4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.375495 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.624505 Sum of squares = 0.531002824321 Cumulative probabilities = A: 0.375495 B: 0.375495 C: 0.375495 D: 0.375495 E: 1.000000 A B C D E A 0 6 4 8 0 B -6 0 6 -2 -4 C -4 -6 0 -6 -2 D -8 2 6 0 -6 E 0 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=25 C=22 A=13 B=11 so B is eliminated. Round 2 votes counts: E=30 C=29 D=26 A=15 so A is eliminated. Round 3 votes counts: E=39 C=31 D=30 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. A:209 E:206 B:197 D:197 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 8 0 B -6 0 6 -2 -4 C -4 -6 0 -6 -2 D -8 2 6 0 -6 E 0 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 8 0 B -6 0 6 -2 -4 C -4 -6 0 -6 -2 D -8 2 6 0 -6 E 0 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 8 0 B -6 0 6 -2 -4 C -4 -6 0 -6 -2 D -8 2 6 0 -6 E 0 4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999895 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9577: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) E B C D A (9) A D C B E (8) A B C D E (8) E B A C D (5) D A C B E (5) A C D B E (4) E B D C A (3) E B C A D (3) C A B D E (3) B E A C D (3) E D A C B (2) D C E B A (2) D C E A B (2) D C A B E (2) B E C D A (2) B E C A D (2) B A C E D (2) A C B D E (2) A B C E D (2) E D C A B (1) E D B C A (1) E C B D A (1) E B A D C (1) E A B D C (1) D E C B A (1) D E A C B (1) D C B A E (1) D C A E B (1) D A E C B (1) C D B A E (1) C D A B E (1) C B D A E (1) C B A D E (1) B C E D A (1) B C E A D (1) B A E C D (1) B A C D E (1) A E B D C (1) A D E C B (1) A D C E B (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 2 4 -2 B 4 0 -4 10 6 C -2 4 0 10 0 D -4 -10 -10 0 -4 E 2 -6 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999945 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 4 -2 B 4 0 -4 10 6 C -2 4 0 10 0 D -4 -10 -10 0 -4 E 2 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=36 A=28 D=16 B=13 C=7 so C is eliminated. Round 2 votes counts: E=36 A=31 D=18 B=15 so B is eliminated. Round 3 votes counts: E=45 A=36 D=19 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:208 C:206 A:200 E:200 D:186 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 2 4 -2 B 4 0 -4 10 6 C -2 4 0 10 0 D -4 -10 -10 0 -4 E 2 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 4 -2 B 4 0 -4 10 6 C -2 4 0 10 0 D -4 -10 -10 0 -4 E 2 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 4 -2 B 4 0 -4 10 6 C -2 4 0 10 0 D -4 -10 -10 0 -4 E 2 -6 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999996 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9578: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (6) A D E C B (6) E C B A D (5) D E B A C (5) D E A B C (4) C B E A D (4) C B A E D (4) A D C B E (4) E D B C A (3) E B C D A (3) E A C D B (3) D B E A C (3) D A E B C (3) D A B E C (3) C A B D E (3) B E D C A (3) A D B C E (3) A C E D B (3) A C D B E (3) E D A C B (2) E C B D A (2) D A B C E (2) C A B E D (2) B D E C A (2) B D C A E (2) B C D E A (2) A D C E B (2) E D B A C (1) E C A B D (1) E A D C B (1) D B A C E (1) C B A D E (1) B E C D A (1) B D A C E (1) B C D A E (1) B C A D E (1) A E D C B (1) A E C D B (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 8 0 -2 B 2 0 2 -8 8 C -8 -2 0 -4 -6 D 0 8 4 0 8 E 2 -8 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.434608 B: 0.000000 C: 0.000000 D: 0.565392 E: 0.000000 Sum of squares = 0.50855218482 Cumulative probabilities = A: 0.434608 B: 0.434608 C: 0.434608 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 0 -2 B 2 0 2 -8 8 C -8 -2 0 -4 -6 D 0 8 4 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 E=21 D=21 B=19 C=14 so C is eliminated. Round 2 votes counts: A=30 B=28 E=21 D=21 so E is eliminated. Round 3 votes counts: B=38 A=35 D=27 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:210 A:202 B:202 E:196 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 0 -2 B 2 0 2 -8 8 C -8 -2 0 -4 -6 D 0 8 4 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 0 -2 B 2 0 2 -8 8 C -8 -2 0 -4 -6 D 0 8 4 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 0 -2 B 2 0 2 -8 8 C -8 -2 0 -4 -6 D 0 8 4 0 8 E 2 -8 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999996 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9579: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (9) C D E B A (7) C D E A B (6) C D B E A (6) A E B D C (5) D C A B E (4) C E D B A (4) B A D C E (4) A D B C E (4) A B D E C (4) E B C A D (3) D C B A E (3) B D C A E (3) E C D A B (2) E C B D A (2) E C B A D (2) E B A C D (2) E A C B D (2) D B C A E (2) D B A C E (2) D A C B E (2) C E D A B (2) B D A C E (2) B A E D C (2) A E D C B (2) A B E D C (2) D C A E B (1) C B E D A (1) C B D E A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C D E A (1) B A D E C (1) A E D B C (1) A D E B C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -4 -4 -6 B 2 0 4 0 -2 C 4 -4 0 6 16 D 4 0 -6 0 12 E 6 2 -16 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.662373 C: 0.000000 D: 0.337627 E: 0.000000 Sum of squares = 0.552729693036 Cumulative probabilities = A: 0.000000 B: 0.662373 C: 0.662373 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -4 -6 B 2 0 4 0 -2 C 4 -4 0 6 16 D 4 0 -6 0 12 E 6 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000148825 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 A=21 B=16 D=14 so D is eliminated. Round 2 votes counts: C=35 A=23 E=22 B=20 so B is eliminated. Round 3 votes counts: C=42 A=34 E=24 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:211 D:205 B:202 A:192 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -4 -4 -6 B 2 0 4 0 -2 C 4 -4 0 6 16 D 4 0 -6 0 12 E 6 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000148825 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -4 -6 B 2 0 4 0 -2 C 4 -4 0 6 16 D 4 0 -6 0 12 E 6 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000148825 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -4 -6 B 2 0 4 0 -2 C 4 -4 0 6 16 D 4 0 -6 0 12 E 6 2 -16 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.400000 E: 0.000000 Sum of squares = 0.520000148825 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9580: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (14) E B A D C (10) D C A E B (8) B E A C D (8) C D B A E (4) D E A C B (3) D C A B E (3) C B D A E (3) B A C D E (3) E D C A B (2) E A D B C (2) D C E A B (2) C D E B A (2) C D B E A (2) C D A E B (2) C B D E A (2) C B A D E (2) B C A D E (2) B A E C D (2) A D C E B (2) A B C D E (2) E D B C A (1) E C B D A (1) E B D C A (1) E B D A C (1) E B A C D (1) E A B D C (1) D E C A B (1) C A D B E (1) B E C D A (1) B E C A D (1) B C E D A (1) B C E A D (1) B C A E D (1) B A E D C (1) A E B D C (1) A D E C B (1) A D C B E (1) A C D B E (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -2 -16 -8 12 B 2 0 -14 -4 16 C 16 14 0 16 18 D 8 4 -16 0 22 E -12 -16 -18 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999856 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -16 -8 12 B 2 0 -14 -4 16 C 16 14 0 16 18 D 8 4 -16 0 22 E -12 -16 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=21 E=20 D=17 A=10 so A is eliminated. Round 2 votes counts: C=33 B=25 E=21 D=21 so E is eliminated. Round 3 votes counts: B=40 C=34 D=26 so D is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:232 D:209 B:200 A:193 E:166 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -16 -8 12 B 2 0 -14 -4 16 C 16 14 0 16 18 D 8 4 -16 0 22 E -12 -16 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -16 -8 12 B 2 0 -14 -4 16 C 16 14 0 16 18 D 8 4 -16 0 22 E -12 -16 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -16 -8 12 B 2 0 -14 -4 16 C 16 14 0 16 18 D 8 4 -16 0 22 E -12 -16 -18 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9581: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (8) C E D A B (8) D B A E C (6) E D C B A (4) E C D B A (4) E C B A D (4) D E B A C (4) C E A D B (4) C E A B D (4) A B D C E (4) E C D A B (3) E B A C D (3) B A D E C (3) A B C D E (3) E D B A C (2) E C B D A (2) E B C A D (2) C A E D B (2) C A E B D (2) C A B E D (2) B D E A C (2) B A E C D (2) B A D C E (2) A C B D E (2) A B E C D (2) E D B C A (1) E C A B D (1) E B C D A (1) E B A D C (1) D E C B A (1) D C E B A (1) D C E A B (1) D C A B E (1) D B E A C (1) C D A E B (1) C A D B E (1) B E D A C (1) B E A C D (1) B D A E C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 0 -6 -12 B 14 0 2 -8 -14 C 0 -2 0 10 -4 D 6 8 -10 0 -14 E 12 14 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 0 -6 -12 B 14 0 2 -8 -14 C 0 -2 0 10 -4 D 6 8 -10 0 -14 E 12 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=24 D=23 A=13 B=12 so B is eliminated. Round 2 votes counts: E=30 D=26 C=24 A=20 so A is eliminated. Round 3 votes counts: D=36 E=34 C=30 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:222 C:202 B:197 D:195 A:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 0 -6 -12 B 14 0 2 -8 -14 C 0 -2 0 10 -4 D 6 8 -10 0 -14 E 12 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -6 -12 B 14 0 2 -8 -14 C 0 -2 0 10 -4 D 6 8 -10 0 -14 E 12 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -6 -12 B 14 0 2 -8 -14 C 0 -2 0 10 -4 D 6 8 -10 0 -14 E 12 14 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998557 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9582: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (15) D A E B C (13) B C D A E (11) C E B A D (10) C B E D A (7) B D A C E (7) D A B E C (5) C B E A D (5) B C D E A (5) A D E B C (5) E C A D B (4) A E D C B (2) A D E C B (2) E C D A B (1) E A C D B (1) D E A C B (1) D B A E C (1) D A E C B (1) C E A B D (1) B D C A E (1) B C A E D (1) A D B E C (1) Total count = 100 A B C D E A 0 4 8 -6 0 B -4 0 0 -4 -12 C -8 0 0 -8 -4 D 6 4 8 0 6 E 0 12 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 -6 0 B -4 0 0 -4 -12 C -8 0 0 -8 -4 D 6 4 8 0 6 E 0 12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 C=23 E=21 D=21 A=10 so A is eliminated. Round 2 votes counts: D=29 B=25 E=23 C=23 so E is eliminated. Round 3 votes counts: D=46 C=29 B=25 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:205 A:203 B:190 C:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 8 -6 0 B -4 0 0 -4 -12 C -8 0 0 -8 -4 D 6 4 8 0 6 E 0 12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -6 0 B -4 0 0 -4 -12 C -8 0 0 -8 -4 D 6 4 8 0 6 E 0 12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -6 0 B -4 0 0 -4 -12 C -8 0 0 -8 -4 D 6 4 8 0 6 E 0 12 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999698 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9583: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (11) A C B D E (8) E D B A C (5) C A B D E (5) A E C D B (5) E D A B C (4) D E B A C (4) B D C E A (4) B C D E A (4) A C E B D (4) E D C B A (3) C B A D E (3) A D B E C (3) A C B E D (3) A B D C E (3) A B C D E (3) D E B C A (2) D B E C A (2) D B E A C (2) C A B E D (2) A D B C E (2) A C E D B (2) E D A C B (1) E C A D B (1) E A D C B (1) D B A E C (1) C B E D A (1) C B D E A (1) C B D A E (1) B E D C A (1) B E C D A (1) B D E C A (1) B C E D A (1) B C D A E (1) B C A D E (1) A E D C B (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 0 8 -2 0 B 0 0 16 -4 8 C -8 -16 0 -6 -2 D 2 4 6 0 4 E 0 -8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -2 0 B 0 0 16 -4 8 C -8 -16 0 -6 -2 D 2 4 6 0 4 E 0 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=26 B=14 C=13 D=11 so D is eliminated. Round 2 votes counts: A=36 E=32 B=19 C=13 so C is eliminated. Round 3 votes counts: A=43 E=32 B=25 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:210 D:208 A:203 E:195 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 8 -2 0 B 0 0 16 -4 8 C -8 -16 0 -6 -2 D 2 4 6 0 4 E 0 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -2 0 B 0 0 16 -4 8 C -8 -16 0 -6 -2 D 2 4 6 0 4 E 0 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -2 0 B 0 0 16 -4 8 C -8 -16 0 -6 -2 D 2 4 6 0 4 E 0 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999411 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9584: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (10) D E A B C (8) E D C B A (7) E D A C B (7) A B C D E (7) B C A D E (5) D A E B C (4) C B E A D (4) C B A D E (4) A C B D E (4) E D A B C (3) E C D B A (3) E C B D A (3) D A B E C (3) A D B C E (3) E D B C A (2) E C A B D (2) C E B D A (2) C E B A D (2) C E A B D (2) B A C D E (2) A E C D B (2) A B D C E (2) E D C A B (1) E C B A D (1) D E B A C (1) D B E C A (1) D A B C E (1) B D E C A (1) B C E D A (1) B C A E D (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -4 4 -2 B 0 0 -8 6 0 C 4 8 0 10 0 D -4 -6 -10 0 -6 E 2 0 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.471556 D: 0.000000 E: 0.528444 Sum of squares = 0.501618125315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.471556 D: 0.471556 E: 1.000000 A B C D E A 0 0 -4 4 -2 B 0 0 -8 6 0 C 4 8 0 10 0 D -4 -6 -10 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=24 A=19 D=18 B=10 so B is eliminated. Round 2 votes counts: C=31 E=29 A=21 D=19 so D is eliminated. Round 3 votes counts: E=40 C=31 A=29 so A is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 E:204 A:199 B:199 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 0 -4 4 -2 B 0 0 -8 6 0 C 4 8 0 10 0 D -4 -6 -10 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 4 -2 B 0 0 -8 6 0 C 4 8 0 10 0 D -4 -6 -10 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 4 -2 B 0 0 -8 6 0 C 4 8 0 10 0 D -4 -6 -10 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9585: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (9) A D E B C (9) D E C A B (7) C B E D A (6) E D A C B (5) C B E A D (5) B A C D E (5) B C D A E (4) A B D E C (4) C E B D A (3) B C A D E (3) B A D E C (3) B A D C E (3) B A C E D (3) E D C A B (2) C E B A D (2) A E D C B (2) A B E D C (2) E C D A B (1) E A D C B (1) D E B C A (1) D C E B A (1) D B E A C (1) C E D B A (1) C D E B A (1) C D B E A (1) C B D E A (1) B D A E C (1) B C D E A (1) A E D B C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 2 6 0 B 10 0 2 12 6 C -2 -2 0 -6 0 D -6 -12 6 0 12 E 0 -6 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 6 0 B 10 0 2 12 6 C -2 -2 0 -6 0 D -6 -12 6 0 12 E 0 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=20 C=20 A=19 E=9 so E is eliminated. Round 2 votes counts: B=32 D=27 C=21 A=20 so A is eliminated. Round 3 votes counts: D=40 B=39 C=21 so C is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:200 A:199 C:195 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 2 6 0 B 10 0 2 12 6 C -2 -2 0 -6 0 D -6 -12 6 0 12 E 0 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 6 0 B 10 0 2 12 6 C -2 -2 0 -6 0 D -6 -12 6 0 12 E 0 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 6 0 B 10 0 2 12 6 C -2 -2 0 -6 0 D -6 -12 6 0 12 E 0 -6 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9586: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) A E C D B (6) E D C A B (5) D E B C A (5) B C D A E (5) E D A C B (4) B A D E C (4) D E C B A (3) D E A B C (3) D B E A C (3) B D C E A (3) A E B D C (3) A C E B D (3) E C D A B (2) E A C D B (2) D E B A C (2) D C E B A (2) D B E C A (2) C B D E A (2) B D A E C (2) B C D E A (2) B C A E D (2) B C A D E (2) B A D C E (2) B A C D E (2) A E D B C (2) A E C B D (2) A C B E D (2) A B E C D (2) A B D E C (2) E A D C B (1) D E C A B (1) D E A C B (1) D B C E A (1) C E D A B (1) C D E B A (1) C B A E D (1) C A E B D (1) A E D C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 18 0 8 B -8 0 18 2 -4 C -18 -18 0 -8 -20 D 0 -2 8 0 2 E -8 4 20 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.599345 B: 0.000000 C: 0.000000 D: 0.400655 E: 0.000000 Sum of squares = 0.519738875728 Cumulative probabilities = A: 0.599345 B: 0.599345 C: 0.599345 D: 1.000000 E: 1.000000 A B C D E A 0 8 18 0 8 B -8 0 18 2 -4 C -18 -18 0 -8 -20 D 0 -2 8 0 2 E -8 4 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 B=24 D=23 E=14 C=6 so C is eliminated. Round 2 votes counts: A=34 B=27 D=24 E=15 so E is eliminated. Round 3 votes counts: A=37 D=36 B=27 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:207 B:204 D:204 C:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 18 0 8 B -8 0 18 2 -4 C -18 -18 0 -8 -20 D 0 -2 8 0 2 E -8 4 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 18 0 8 B -8 0 18 2 -4 C -18 -18 0 -8 -20 D 0 -2 8 0 2 E -8 4 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 18 0 8 B -8 0 18 2 -4 C -18 -18 0 -8 -20 D 0 -2 8 0 2 E -8 4 20 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999898 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9587: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) B A D C E (7) E C B D A (6) D A C B E (6) B A D E C (6) A D B C E (6) D C A B E (5) C D E A B (5) E C D A B (4) E B C A D (4) E B A C D (4) C E D A B (4) B E A D C (4) B E A C D (4) B A E D C (4) E B C D A (3) E C D B A (2) E C B A D (2) C A D E B (2) B C D E A (2) E C A D B (1) E A C D B (1) D C B A E (1) D C A E B (1) D A C E B (1) D A B C E (1) B E C D A (1) B E C A D (1) B D A C E (1) A D C B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 -4 -2 4 B 4 0 0 2 4 C 4 0 0 8 4 D 2 -2 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.486034 C: 0.513966 D: 0.000000 E: 0.000000 Sum of squares = 0.500390107641 Cumulative probabilities = A: 0.000000 B: 0.486034 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -4 -2 4 B 4 0 0 2 4 C 4 0 0 8 4 D 2 -2 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=27 C=19 D=15 A=9 so A is eliminated. Round 2 votes counts: B=32 E=27 D=22 C=19 so C is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:205 D:201 A:197 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -2 4 B 4 0 0 2 4 C 4 0 0 8 4 D 2 -2 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -2 4 B 4 0 0 2 4 C 4 0 0 8 4 D 2 -2 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -2 4 B 4 0 0 2 4 C 4 0 0 8 4 D 2 -2 -8 0 10 E -4 -4 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9588: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (10) D B C A E (9) D C B A E (8) C D B A E (5) E C A D B (4) E A B C D (4) D C B E A (4) D B A C E (4) E A C D B (3) D C E B A (3) C D E A B (3) B D A C E (3) B A E D C (3) E D C A B (2) E C D A B (2) D E B C A (2) D B C E A (2) C A B E D (2) B A C D E (2) A E B C D (2) A B E C D (2) A B C E D (2) E D A B C (1) E C A B D (1) E A B D C (1) D E C B A (1) D E C A B (1) D E B A C (1) D B E C A (1) D B E A C (1) D B A E C (1) C E A B D (1) C B D A E (1) B E A D C (1) B D C A E (1) B A D E C (1) B A D C E (1) A E C B D (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 -6 -12 2 B 10 0 -8 -16 12 C 6 8 0 -6 6 D 12 16 6 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -12 2 B 10 0 -8 -16 12 C 6 8 0 -6 6 D 12 16 6 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 E=28 C=12 B=12 A=10 so A is eliminated. Round 2 votes counts: D=38 E=31 B=17 C=14 so C is eliminated. Round 3 votes counts: D=46 E=33 B=21 so B is eliminated. Round 4 votes counts: D=55 E=45 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:207 B:199 A:187 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -6 -12 2 B 10 0 -8 -16 12 C 6 8 0 -6 6 D 12 16 6 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -12 2 B 10 0 -8 -16 12 C 6 8 0 -6 6 D 12 16 6 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -12 2 B 10 0 -8 -16 12 C 6 8 0 -6 6 D 12 16 6 0 10 E -2 -12 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9589: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) D A E C B (7) D A C E B (7) C E B A D (7) B E C D A (6) E B C D A (5) D A B C E (5) A D C B E (5) E B C A D (4) C B E A D (4) B C E A D (4) E C B D A (3) D A E B C (3) B E C A D (3) B C A D E (3) A D B C E (3) D A B E C (2) C E A D B (2) C A B D E (2) B C A E D (2) B A D C E (2) E D A C B (1) E C D B A (1) E C D A B (1) E B D C A (1) E A D C B (1) D E A C B (1) D A C B E (1) C B A E D (1) B E D C A (1) B C D A E (1) A D C E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -14 -20 8 -8 B 14 0 -10 16 -8 C 20 10 0 18 2 D -8 -16 -18 0 -12 E 8 8 -2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999543 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 8 -8 B 14 0 -10 16 -8 C 20 10 0 18 2 D -8 -16 -18 0 -12 E 8 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=26 D=26 B=22 C=16 A=10 so A is eliminated. Round 2 votes counts: D=35 E=26 B=22 C=17 so C is eliminated. Round 3 votes counts: D=36 E=35 B=29 so B is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:225 E:213 B:206 A:183 D:173 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -20 8 -8 B 14 0 -10 16 -8 C 20 10 0 18 2 D -8 -16 -18 0 -12 E 8 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 8 -8 B 14 0 -10 16 -8 C 20 10 0 18 2 D -8 -16 -18 0 -12 E 8 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 8 -8 B 14 0 -10 16 -8 C 20 10 0 18 2 D -8 -16 -18 0 -12 E 8 8 -2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999972242 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9590: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (9) D B C A E (7) D E C A B (6) E A C B D (5) D B E A C (5) A C B E D (5) E C A B D (4) D E B A C (4) C A E B D (4) C A B E D (4) B E A C D (4) B A E C D (4) E B A C D (3) C A E D B (3) B D A C E (3) A B C E D (3) E D B A C (2) E B D A C (2) D E B C A (2) D C A E B (2) D B A E C (2) D B A C E (2) C A D B E (2) B D A E C (2) B C A D E (2) B A D C E (2) B A C D E (2) E D C A B (1) E C D A B (1) E C A D B (1) E A B C D (1) B E A D C (1) Total count = 100 A B C D E A 0 -16 22 18 16 B 16 0 24 20 18 C -22 -24 0 14 0 D -18 -20 -14 0 -14 E -16 -18 0 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 22 18 16 B 16 0 24 20 18 C -22 -24 0 14 0 D -18 -20 -14 0 -14 E -16 -18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 B=29 E=20 C=13 A=8 so A is eliminated. Round 2 votes counts: B=32 D=30 E=20 C=18 so C is eliminated. Round 3 votes counts: B=41 D=32 E=27 so E is eliminated. Round 4 votes counts: B=60 D=40 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:239 A:220 E:190 C:184 D:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 22 18 16 B 16 0 24 20 18 C -22 -24 0 14 0 D -18 -20 -14 0 -14 E -16 -18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 22 18 16 B 16 0 24 20 18 C -22 -24 0 14 0 D -18 -20 -14 0 -14 E -16 -18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 22 18 16 B 16 0 24 20 18 C -22 -24 0 14 0 D -18 -20 -14 0 -14 E -16 -18 0 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9591: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (11) C D B A E (5) B C D A E (5) E B A C D (4) E A D B C (4) C B D E A (4) E C B D A (3) E B C A D (3) E B A D C (3) D C A B E (3) D A C E B (3) C D E A B (3) B E C A D (3) B E A D C (3) B A D C E (3) A D B C E (3) E C D B A (2) E B C D A (2) D A C B E (2) C D E B A (2) C D B E A (2) C D A B E (2) C B D A E (2) B C E D A (2) B C A D E (2) A B E D C (2) E C D A B (1) E A D C B (1) D E A C B (1) C E D A B (1) C B E D A (1) B E C D A (1) B E A C D (1) B C E A D (1) B A C D E (1) A E D C B (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C B E (1) A D B E C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 0 6 -18 B 10 0 18 18 4 C 0 -18 0 6 -2 D -6 -18 -6 0 -2 E 18 -4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 0 6 -18 B 10 0 18 18 4 C 0 -18 0 6 -2 D -6 -18 -6 0 -2 E 18 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=22 B=22 A=13 D=9 so D is eliminated. Round 2 votes counts: E=35 C=25 B=22 A=18 so A is eliminated. Round 3 votes counts: E=39 C=31 B=30 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:225 E:209 C:193 A:189 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 0 6 -18 B 10 0 18 18 4 C 0 -18 0 6 -2 D -6 -18 -6 0 -2 E 18 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 0 6 -18 B 10 0 18 18 4 C 0 -18 0 6 -2 D -6 -18 -6 0 -2 E 18 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 0 6 -18 B 10 0 18 18 4 C 0 -18 0 6 -2 D -6 -18 -6 0 -2 E 18 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9592: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (9) C D A B E (7) C B D E A (5) B D A C E (5) C D A E B (4) A D E C B (4) E A D B C (3) C E D A B (3) B D A E C (3) B C D A E (3) B A D E C (3) A B E D C (3) E B C A D (2) E B A C D (2) E A C D B (2) D C A E B (2) D B A C E (2) D A C B E (2) B C E D A (2) A E D C B (2) A E D B C (2) A D B E C (2) E C B D A (1) E C B A D (1) E C A D B (1) E B A D C (1) E A C B D (1) E A B C D (1) D C B A E (1) D B C A E (1) C D E A B (1) C D B E A (1) C B E D A (1) C B D A E (1) B E C D A (1) B E A D C (1) B E A C D (1) B D C A E (1) A E B D C (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 4 4 -12 20 B -4 0 0 -2 14 C -4 0 0 0 6 D 12 2 0 0 18 E -20 -14 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.508060 D: 0.491940 E: 0.000000 Sum of squares = 0.500129921327 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.508060 D: 1.000000 E: 1.000000 A B C D E A 0 4 4 -12 20 B -4 0 0 -2 14 C -4 0 0 0 6 D 12 2 0 0 18 E -20 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=24 B=20 A=16 D=8 so D is eliminated. Round 2 votes counts: C=35 E=24 B=23 A=18 so A is eliminated. Round 3 votes counts: C=38 E=33 B=29 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:216 A:208 B:204 C:201 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 4 -12 20 B -4 0 0 -2 14 C -4 0 0 0 6 D 12 2 0 0 18 E -20 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 -12 20 B -4 0 0 -2 14 C -4 0 0 0 6 D 12 2 0 0 18 E -20 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 -12 20 B -4 0 0 -2 14 C -4 0 0 0 6 D 12 2 0 0 18 E -20 -14 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999834 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9593: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (15) D E A C B (9) E D A C B (7) D E B C A (7) D B C E A (6) B C A E D (6) E D A B C (5) B C D A E (5) A E C B D (5) E A D C B (4) E A C B D (4) B C A D E (4) E A B C D (3) C B A D E (3) E D B C A (2) D B C A E (2) E B A C D (1) E A D B C (1) D C E B A (1) D C B E A (1) D C B A E (1) D C A B E (1) D B E C A (1) C B D A E (1) B E C A D (1) B C E D A (1) A E D C B (1) A E C D B (1) A B C E D (1) Total count = 100 A B C D E A 0 14 14 0 -8 B -14 0 -8 0 -2 C -14 8 0 2 -4 D 0 0 -2 0 -16 E 8 2 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999288 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 14 0 -8 B -14 0 -8 0 -2 C -14 8 0 2 -4 D 0 0 -2 0 -16 E 8 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=27 A=23 B=17 C=4 so C is eliminated. Round 2 votes counts: D=29 E=27 A=23 B=21 so B is eliminated. Round 3 votes counts: A=36 D=35 E=29 so E is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:210 C:196 D:191 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 14 0 -8 B -14 0 -8 0 -2 C -14 8 0 2 -4 D 0 0 -2 0 -16 E 8 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 0 -8 B -14 0 -8 0 -2 C -14 8 0 2 -4 D 0 0 -2 0 -16 E 8 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 0 -8 B -14 0 -8 0 -2 C -14 8 0 2 -4 D 0 0 -2 0 -16 E 8 2 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997653 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9594: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (6) B C E D A (6) E D B A C (5) B C A D E (5) E D A C B (4) D A B E C (4) C E B D A (4) C E A D B (4) C A B D E (4) A D E C B (4) D E A B C (3) C B A E D (3) C B A D E (3) C A E D B (3) A D B C E (3) E C D A B (2) E C B D A (2) E B C D A (2) D A E B C (2) C E A B D (2) B E D A C (2) B E C D A (2) B D E A C (2) B A D C E (2) A D E B C (2) A D B E C (2) A C D B E (2) E D C B A (1) E D C A B (1) E D A B C (1) E A C D B (1) D B A E C (1) D A E C B (1) C E B A D (1) C A D B E (1) C A B E D (1) B E D C A (1) B D A E C (1) B D A C E (1) B C D A E (1) A D C E B (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -10 2 -4 B 2 0 -2 4 8 C 10 2 0 10 8 D -2 -4 -10 0 -8 E 4 -8 -8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998331 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 2 -4 B 2 0 -2 4 8 C 10 2 0 10 8 D -2 -4 -10 0 -8 E 4 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=23 E=19 A=15 D=11 so D is eliminated. Round 2 votes counts: C=32 B=24 E=22 A=22 so E is eliminated. Round 3 votes counts: C=38 B=31 A=31 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:206 E:198 A:193 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 2 -4 B 2 0 -2 4 8 C 10 2 0 10 8 D -2 -4 -10 0 -8 E 4 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 2 -4 B 2 0 -2 4 8 C 10 2 0 10 8 D -2 -4 -10 0 -8 E 4 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 2 -4 B 2 0 -2 4 8 C 10 2 0 10 8 D -2 -4 -10 0 -8 E 4 -8 -8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998779 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9595: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (10) E A C D B (10) B D C A E (9) E A C B D (7) D B C A E (7) A C E D B (6) D C A E B (4) D C A B E (4) C A D E B (4) B D E C A (4) D B E C A (3) B E D A C (3) B E A C D (3) B C A D E (3) B A C E D (3) B E D C A (2) A E C D B (2) A C B E D (2) E D A C B (1) E B D A C (1) E A B C D (1) D E C A B (1) D C B A E (1) D B C E A (1) C B A D E (1) C A D B E (1) C A B D E (1) B C A E D (1) A E C B D (1) A C E B D (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 6 18 6 B 4 0 2 8 0 C -6 -2 0 18 2 D -18 -8 -18 0 -8 E -6 0 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.770267 C: 0.000000 D: 0.000000 E: 0.229733 Sum of squares = 0.646088197252 Cumulative probabilities = A: 0.000000 B: 0.770267 C: 0.770267 D: 0.770267 E: 1.000000 A B C D E A 0 -4 6 18 6 B 4 0 2 8 0 C -6 -2 0 18 2 D -18 -8 -18 0 -8 E -6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000005145 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 B=28 D=21 A=14 C=7 so C is eliminated. Round 2 votes counts: E=30 B=29 D=21 A=20 so A is eliminated. Round 3 votes counts: E=40 B=33 D=27 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:207 C:206 E:200 D:174 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 18 6 B 4 0 2 8 0 C -6 -2 0 18 2 D -18 -8 -18 0 -8 E -6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000005145 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 18 6 B 4 0 2 8 0 C -6 -2 0 18 2 D -18 -8 -18 0 -8 E -6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000005145 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 18 6 B 4 0 2 8 0 C -6 -2 0 18 2 D -18 -8 -18 0 -8 E -6 0 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.600000 C: 0.000000 D: 0.000000 E: 0.400000 Sum of squares = 0.520000005145 Cumulative probabilities = A: 0.000000 B: 0.600000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9596: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (7) C B E A D (6) B C D E A (6) E C A D B (5) E A D C B (5) C E B A D (5) C B E D A (5) B C E D A (5) B C D A E (4) A D E B C (4) E C D B A (3) E A C D B (3) D A B E C (3) B C A D E (3) B A D C E (3) A E D C B (3) A B D C E (3) E D C A B (2) D B A C E (2) B C A E D (2) A D B E C (2) E D C B A (1) E D A C B (1) E C D A B (1) E C B D A (1) D E C B A (1) D E C A B (1) D E A C B (1) D B E C A (1) D A B C E (1) C E B D A (1) C E A B D (1) C B A E D (1) C A E B D (1) B D C A E (1) B D A C E (1) B A C D E (1) A E C D B (1) A E C B D (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 -14 2 -10 B 6 0 0 2 2 C 14 0 0 12 6 D -2 -2 -12 0 -8 E 10 -2 -6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.671514 C: 0.328486 D: 0.000000 E: 0.000000 Sum of squares = 0.558833926609 Cumulative probabilities = A: 0.000000 B: 0.671514 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -14 2 -10 B 6 0 0 2 2 C 14 0 0 12 6 D -2 -2 -12 0 -8 E 10 -2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 E=22 C=20 D=17 A=15 so A is eliminated. Round 2 votes counts: B=29 E=27 D=24 C=20 so C is eliminated. Round 3 votes counts: B=41 E=35 D=24 so D is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:216 B:205 E:205 D:188 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -14 2 -10 B 6 0 0 2 2 C 14 0 0 12 6 D -2 -2 -12 0 -8 E 10 -2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -14 2 -10 B 6 0 0 2 2 C 14 0 0 12 6 D -2 -2 -12 0 -8 E 10 -2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -14 2 -10 B 6 0 0 2 2 C 14 0 0 12 6 D -2 -2 -12 0 -8 E 10 -2 -6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999933 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9597: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (10) E A D C B (8) D B C A E (6) D A E B C (6) D B E A C (5) E A C D B (4) E A C B D (4) D E A B C (4) D B C E A (4) A E C D B (4) C E B A D (3) C B E A D (3) C B D A E (3) C A E B D (3) A E D C B (3) A E C B D (3) E C A B D (2) D B A C E (2) C E A B D (2) C B E D A (2) C B A E D (2) B D C E A (2) B C D A E (2) E D A B C (1) E B D C A (1) E A D B C (1) D E B A C (1) D B A E C (1) C B D E A (1) C B A D E (1) C A B E D (1) B C E D A (1) B C D E A (1) A E D B C (1) A D E B C (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 0 -6 0 B 2 0 0 -4 -6 C 0 0 0 -14 0 D 6 4 14 0 0 E 0 6 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.430539 E: 0.569461 Sum of squares = 0.509649578655 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.430539 E: 1.000000 A B C D E A 0 -2 0 -6 0 B 2 0 0 -4 -6 C 0 0 0 -14 0 D 6 4 14 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=21 C=21 B=16 A=13 so A is eliminated. Round 2 votes counts: E=32 D=30 C=22 B=16 so B is eliminated. Round 3 votes counts: D=42 E=32 C=26 so C is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:203 A:196 B:196 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -6 0 B 2 0 0 -4 -6 C 0 0 0 -14 0 D 6 4 14 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 0 B 2 0 0 -4 -6 C 0 0 0 -14 0 D 6 4 14 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 0 B 2 0 0 -4 -6 C 0 0 0 -14 0 D 6 4 14 0 0 E 0 6 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9598: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (11) C A E B D (10) C A E D B (9) D B E A C (8) A C E D B (8) D E B A C (7) B D E A C (7) E A D C B (6) E A C D B (5) B D E C A (4) B D C E A (4) B D C A E (4) C A B E D (3) B C A D E (3) C B A E D (2) B C D A E (2) D E A B C (1) C B A D E (1) C A D E B (1) B C A E D (1) A E D C B (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 12 4 -6 B -14 0 -18 -18 -22 C -12 18 0 -6 -2 D -4 18 6 0 -14 E 6 22 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 12 4 -6 B -14 0 -18 -18 -22 C -12 18 0 -6 -2 D -4 18 6 0 -14 E 6 22 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 B=25 E=22 D=16 A=11 so A is eliminated. Round 2 votes counts: C=35 B=25 E=24 D=16 so D is eliminated. Round 3 votes counts: C=35 B=33 E=32 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:222 A:212 D:203 C:199 B:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 12 4 -6 B -14 0 -18 -18 -22 C -12 18 0 -6 -2 D -4 18 6 0 -14 E 6 22 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 4 -6 B -14 0 -18 -18 -22 C -12 18 0 -6 -2 D -4 18 6 0 -14 E 6 22 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 4 -6 B -14 0 -18 -18 -22 C -12 18 0 -6 -2 D -4 18 6 0 -14 E 6 22 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996739 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9599: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (9) C D B A E (8) A E B D C (8) E B A D C (6) E A B D C (6) D C B A E (5) E A B C D (4) C D A B E (4) B D C E A (4) A E C D B (4) E B C D A (3) C D A E B (3) C A D E B (3) A D C B E (3) E A C B D (2) C D E B A (2) A E B C D (2) A D E C B (2) A D C E B (2) A C D E B (2) E C B D A (1) E A C D B (1) D B C A E (1) C E D B A (1) C E D A B (1) C E A D B (1) C D E A B (1) C A E D B (1) B E D C A (1) B E D A C (1) B E A D C (1) B D C A E (1) A E D C B (1) A E C B D (1) A D B C E (1) A C E D B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 12 0 8 10 B -12 0 -18 -14 -18 C 0 18 0 10 8 D -8 14 -10 0 4 E -10 18 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.704500 B: 0.000000 C: 0.295500 D: 0.000000 E: 0.000000 Sum of squares = 0.583640829823 Cumulative probabilities = A: 0.704500 B: 0.704500 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 0 8 10 B -12 0 -18 -14 -18 C 0 18 0 10 8 D -8 14 -10 0 4 E -10 18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=29 E=23 B=8 D=6 so D is eliminated. Round 2 votes counts: C=39 A=29 E=23 B=9 so B is eliminated. Round 3 votes counts: C=45 A=29 E=26 so E is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:218 A:215 D:200 E:198 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 12 0 8 10 B -12 0 -18 -14 -18 C 0 18 0 10 8 D -8 14 -10 0 4 E -10 18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 0 8 10 B -12 0 -18 -14 -18 C 0 18 0 10 8 D -8 14 -10 0 4 E -10 18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 0 8 10 B -12 0 -18 -14 -18 C 0 18 0 10 8 D -8 14 -10 0 4 E -10 18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9600: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (13) D E C B A (8) C E D A B (8) B A D E C (8) A C B E D (8) D B E C A (5) C D E A B (4) C A E B D (4) E D C B A (3) D C E A B (3) A B D E C (3) E C D B A (2) E C B A D (2) D E B C A (2) D C E B A (2) D C A E B (2) D B E A C (2) C E A B D (2) B D E A C (2) B A E D C (2) B A E C D (2) A B E C D (2) A B C D E (2) E D B C A (1) D B A E C (1) C E A D B (1) B E D A C (1) B E A C D (1) B D A E C (1) A C E B D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 10 2 6 2 B -10 0 -2 10 10 C -2 2 0 6 4 D -6 -10 -6 0 -6 E -2 -10 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 2 6 2 B -10 0 -2 10 10 C -2 2 0 6 4 D -6 -10 -6 0 -6 E -2 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 D=25 C=19 B=17 E=8 so E is eliminated. Round 2 votes counts: A=31 D=29 C=23 B=17 so B is eliminated. Round 3 votes counts: A=44 D=33 C=23 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:210 C:205 B:204 E:195 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 2 6 2 B -10 0 -2 10 10 C -2 2 0 6 4 D -6 -10 -6 0 -6 E -2 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 2 6 2 B -10 0 -2 10 10 C -2 2 0 6 4 D -6 -10 -6 0 -6 E -2 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 2 6 2 B -10 0 -2 10 10 C -2 2 0 6 4 D -6 -10 -6 0 -6 E -2 -10 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997201 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9601: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (15) D B C A E (9) E A C D B (8) E A B C D (7) D C B A E (7) E A C B D (4) B D E A C (4) B A E C D (4) D C B E A (3) A E B C D (3) E D A C B (2) E C D A B (2) E C A D B (2) D C E A B (2) A E C B D (2) A B C E D (2) E D C A B (1) E A D C B (1) E A D B C (1) D E C A B (1) D C E B A (1) D C A E B (1) D C A B E (1) D B C E A (1) C E A D B (1) C D E A B (1) C D A E B (1) C D A B E (1) C A E D B (1) C A D E B (1) B E A D C (1) B E A C D (1) B D E C A (1) B D C E A (1) B C D A E (1) B C A D E (1) B A C E D (1) A E C D B (1) A C E D B (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 -12 -12 8 B 2 0 6 -2 8 C 12 -6 0 -6 8 D 12 2 6 0 6 E -8 -8 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -12 -12 8 B 2 0 6 -2 8 C 12 -6 0 -6 8 D 12 2 6 0 6 E -8 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 D=26 A=10 C=6 so C is eliminated. Round 2 votes counts: B=30 E=29 D=29 A=12 so A is eliminated. Round 3 votes counts: E=38 B=32 D=30 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:213 B:207 C:204 A:191 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -12 -12 8 B 2 0 6 -2 8 C 12 -6 0 -6 8 D 12 2 6 0 6 E -8 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -12 -12 8 B 2 0 6 -2 8 C 12 -6 0 -6 8 D 12 2 6 0 6 E -8 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -12 -12 8 B 2 0 6 -2 8 C 12 -6 0 -6 8 D 12 2 6 0 6 E -8 -8 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999661 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9602: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (8) A D E B C (8) C B E D A (6) C B A D E (6) B C E A D (6) D E A C B (5) D A E C B (4) D A E B C (4) C B A E D (4) B C A E D (4) A D B C E (4) D E A B C (3) C B D E A (3) E D C B A (2) E D A B C (2) E B C D A (2) E B C A D (2) E A D B C (2) D C A E B (2) D C A B E (2) C E D B A (2) C D A B E (2) A C B D E (2) E D B A C (1) E C D B A (1) E B A C D (1) D E C B A (1) D E C A B (1) D A C E B (1) C E B D A (1) C D B E A (1) C D B A E (1) B E C A D (1) A E D B C (1) A E B D C (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -16 6 -2 B 6 0 -10 -2 6 C 16 10 0 8 14 D -6 2 -8 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 6 -2 B 6 0 -10 -2 6 C 16 10 0 8 14 D -6 2 -8 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 D=23 A=19 E=13 B=11 so B is eliminated. Round 2 votes counts: C=44 D=23 A=19 E=14 so E is eliminated. Round 3 votes counts: C=50 D=28 A=22 so A is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:200 D:196 A:191 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -16 6 -2 B 6 0 -10 -2 6 C 16 10 0 8 14 D -6 2 -8 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 6 -2 B 6 0 -10 -2 6 C 16 10 0 8 14 D -6 2 -8 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 6 -2 B 6 0 -10 -2 6 C 16 10 0 8 14 D -6 2 -8 0 4 E 2 -6 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9603: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) E C B A D (9) C B A D E (6) D A B E C (5) B C E D A (5) E A D C B (4) C B E D A (4) C B D A E (4) E B D A C (3) D A E B C (3) D A B C E (3) C B E A D (3) B E C D A (3) B C D A E (3) A D C E B (3) A D C B E (3) E D A B C (2) E B C D A (2) E A D B C (2) D B A E C (2) C E A D B (2) C A D B E (2) B D A C E (2) E C B D A (1) E B D C A (1) E A C D B (1) D E A B C (1) D B A C E (1) D A C B E (1) C E B A D (1) C A D E B (1) B E D A C (1) B D E A C (1) B D C A E (1) B D A E C (1) B C D E A (1) A C D B E (1) Total count = 100 A B C D E A 0 -10 2 -2 6 B 10 0 -14 4 6 C -2 14 0 -2 -6 D 2 -4 2 0 12 E -6 -6 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.700000 E: 0.000000 Sum of squares = 0.539999999872 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 2 -2 6 B 10 0 -14 4 6 C -2 14 0 -2 -6 D 2 -4 2 0 12 E -6 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.700000 E: 0.000000 Sum of squares = 0.540000000084 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=23 B=18 A=18 D=16 so D is eliminated. Round 2 votes counts: A=30 E=26 C=23 B=21 so B is eliminated. Round 3 votes counts: A=36 C=33 E=31 so E is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:206 B:203 C:202 A:198 E:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 2 -2 6 B 10 0 -14 4 6 C -2 14 0 -2 -6 D 2 -4 2 0 12 E -6 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.700000 E: 0.000000 Sum of squares = 0.540000000084 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 2 -2 6 B 10 0 -14 4 6 C -2 14 0 -2 -6 D 2 -4 2 0 12 E -6 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.700000 E: 0.000000 Sum of squares = 0.540000000084 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 2 -2 6 B 10 0 -14 4 6 C -2 14 0 -2 -6 D 2 -4 2 0 12 E -6 -6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.100000 C: 0.200000 D: 0.700000 E: 0.000000 Sum of squares = 0.540000000084 Cumulative probabilities = A: 0.000000 B: 0.100000 C: 0.300000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9604: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (11) A C E D B (7) D B C A E (6) B D E C A (5) B D C E A (5) A E C D B (5) A E B D C (5) D B C E A (4) D B A E C (4) C E A B D (4) C D B A E (4) E A B D C (3) E A B C D (3) C B D E A (3) C A E D B (3) E C A B D (2) E B D A C (2) E B A D C (2) C D B E A (2) C A D B E (2) B D E A C (2) E C B A D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B E C A (1) D B E A C (1) D B A C E (1) D A B E C (1) C E B D A (1) C A E B D (1) C A D E B (1) B E D A C (1) A E D B C (1) A D C B E (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 6 6 10 12 B -6 0 -4 6 -8 C -6 4 0 4 -4 D -10 -6 -4 0 -8 E -12 8 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 10 12 B -6 0 -4 6 -8 C -6 4 0 4 -4 D -10 -6 -4 0 -8 E -12 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=21 D=20 E=14 B=13 so B is eliminated. Round 2 votes counts: D=32 A=32 C=21 E=15 so E is eliminated. Round 3 votes counts: A=41 D=35 C=24 so C is eliminated. Round 4 votes counts: A=55 D=45 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:217 E:204 C:199 B:194 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 10 12 B -6 0 -4 6 -8 C -6 4 0 4 -4 D -10 -6 -4 0 -8 E -12 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 10 12 B -6 0 -4 6 -8 C -6 4 0 4 -4 D -10 -6 -4 0 -8 E -12 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 10 12 B -6 0 -4 6 -8 C -6 4 0 4 -4 D -10 -6 -4 0 -8 E -12 8 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9605: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (7) E A C D B (6) D A B E C (6) A D B E C (6) E C A D B (5) E C A B D (5) C E B D A (5) A D E C B (5) C E B A D (4) B D C E A (4) A D E B C (4) E A C B D (3) D A C B E (3) B D C A E (3) D B C A E (2) D A B C E (2) C E A D B (2) C D E B A (2) C D B E A (2) B D A E C (2) B D A C E (2) A E D B C (2) A E C D B (2) E C B A D (1) E B C A D (1) D C A E B (1) D C A B E (1) D B A E C (1) D A C E B (1) C E D B A (1) C E A B D (1) C B E D A (1) B C E D A (1) B C E A D (1) B C D E A (1) B A D E C (1) A E D C B (1) A E B C D (1) A B E D C (1) Total count = 100 A B C D E A 0 16 12 4 8 B -16 0 -4 -24 -6 C -12 4 0 -10 -6 D -4 24 10 0 12 E -8 6 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 12 4 8 B -16 0 -4 -24 -6 C -12 4 0 -10 -6 D -4 24 10 0 12 E -8 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 A=22 E=21 C=18 B=15 so B is eliminated. Round 2 votes counts: D=35 A=23 E=21 C=21 so E is eliminated. Round 3 votes counts: D=35 C=33 A=32 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:221 A:220 E:196 C:188 B:175 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 12 4 8 B -16 0 -4 -24 -6 C -12 4 0 -10 -6 D -4 24 10 0 12 E -8 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 12 4 8 B -16 0 -4 -24 -6 C -12 4 0 -10 -6 D -4 24 10 0 12 E -8 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 12 4 8 B -16 0 -4 -24 -6 C -12 4 0 -10 -6 D -4 24 10 0 12 E -8 6 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999894 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9606: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C A E (11) E D B A C (8) E A C D B (7) A C E B D (7) D B C E A (6) D B C A E (6) D B E A C (5) C B D A E (5) A E C B D (5) E A D B C (4) D B E C A (3) C E A D B (3) C D B A E (3) C A B D E (3) B D A C E (3) E C D A B (2) E A C B D (2) D E B C A (2) C D B E A (2) C A E B D (2) C A B E D (2) E D B C A (1) E D A C B (1) E C A D B (1) E B D A C (1) E A B D C (1) D E C B A (1) D C B E A (1) B C D A E (1) A E B C D (1) Total count = 100 A B C D E A 0 -18 -10 -24 -2 B 18 0 6 -12 2 C 10 -6 0 -8 10 D 24 12 8 0 4 E 2 -2 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999565 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -10 -24 -2 B 18 0 6 -12 2 C 10 -6 0 -8 10 D 24 12 8 0 4 E 2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=24 C=20 B=15 A=13 so A is eliminated. Round 2 votes counts: E=34 C=27 D=24 B=15 so B is eliminated. Round 3 votes counts: D=38 E=34 C=28 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:207 C:203 E:193 A:173 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 -10 -24 -2 B 18 0 6 -12 2 C 10 -6 0 -8 10 D 24 12 8 0 4 E 2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -24 -2 B 18 0 6 -12 2 C 10 -6 0 -8 10 D 24 12 8 0 4 E 2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -24 -2 B 18 0 6 -12 2 C 10 -6 0 -8 10 D 24 12 8 0 4 E 2 -2 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998309 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9607: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (7) D A C E B (6) C D B E A (6) E C D B A (4) D A C B E (4) C D E B A (4) B C E D A (4) E B C A D (3) D C E A B (3) C B D E A (3) B E A C D (3) A D E B C (3) A D B C E (3) A B E C D (3) A B D E C (3) A B D C E (3) E A D C B (2) E A B C D (2) D C E B A (2) D C B E A (2) D C B A E (2) D C A E B (2) D C A B E (2) D A E C B (2) C E D B A (2) C B E D A (2) B E C A D (2) A E B D C (2) A B E D C (2) E C D A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E B A C D (1) E A D B C (1) E A C D B (1) D E C A B (1) D B A C E (1) D A B C E (1) B E C D A (1) A E D B C (1) Total count = 100 A B C D E A 0 -2 2 -12 -6 B 2 0 -8 -14 8 C -2 8 0 4 4 D 12 14 -4 0 6 E 6 -8 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839534 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 -2 2 -12 -6 B 2 0 -8 -14 8 C -2 8 0 4 4 D 12 14 -4 0 6 E 6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839511 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 A=20 E=18 C=17 B=17 so C is eliminated. Round 2 votes counts: D=38 B=22 E=20 A=20 so E is eliminated. Round 3 votes counts: D=45 B=28 A=27 so A is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 C:207 B:194 E:194 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 2 -12 -6 B 2 0 -8 -14 8 C -2 8 0 4 4 D 12 14 -4 0 6 E 6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839511 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 -12 -6 B 2 0 -8 -14 8 C -2 8 0 4 4 D 12 14 -4 0 6 E 6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839511 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 -12 -6 B 2 0 -8 -14 8 C -2 8 0 4 4 D 12 14 -4 0 6 E 6 -8 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.000000 C: 0.666667 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839511 Cumulative probabilities = A: 0.222222 B: 0.222222 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9608: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (10) A D C E B (10) A E B C D (5) A C E B D (4) A C D E B (4) A B D E C (4) E B C D A (3) D C A E B (3) D B E C A (3) C E B D A (3) C D E B A (3) B E D C A (3) B E C D A (3) B D E C A (3) B A E C D (3) A D C B E (3) A D B E C (3) A C E D B (3) E C A B D (2) C E D B A (2) C E B A D (2) C A E B D (2) B E D A C (2) A D B C E (2) A B E D C (2) E A C B D (1) E A B C D (1) D B E A C (1) D B C E A (1) D A C E B (1) D A B C E (1) C D A E B (1) C A E D B (1) B E A D C (1) B E A C D (1) B D E A C (1) B A D E C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 10 12 8 B -8 0 -10 -4 -22 C -10 10 0 -10 12 D -12 4 10 0 10 E -8 22 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 12 8 B -8 0 -10 -4 -22 C -10 10 0 -10 12 D -12 4 10 0 10 E -8 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 D=20 B=18 C=14 E=7 so E is eliminated. Round 2 votes counts: A=43 B=21 D=20 C=16 so C is eliminated. Round 3 votes counts: A=48 D=26 B=26 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:219 D:206 C:201 E:196 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 12 8 B -8 0 -10 -4 -22 C -10 10 0 -10 12 D -12 4 10 0 10 E -8 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 12 8 B -8 0 -10 -4 -22 C -10 10 0 -10 12 D -12 4 10 0 10 E -8 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 12 8 B -8 0 -10 -4 -22 C -10 10 0 -10 12 D -12 4 10 0 10 E -8 22 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999622 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9609: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (15) E A C D B (10) D B C A E (10) B D C A E (8) E A B D C (7) E A C B D (4) D B C E A (3) B D C E A (3) B D A C E (3) A C D B E (3) A B D C E (3) E C D B A (2) E B A D C (2) E A B C D (2) D C B A E (2) C A E D B (2) C A D B E (2) B D E C A (2) B D A E C (2) A E B D C (2) A C E D B (2) E D B C A (1) E C D A B (1) D C B E A (1) D B E C A (1) C D A B E (1) C A D E B (1) C A B D E (1) B D E A C (1) A E C B D (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -12 -12 20 B 12 0 0 -16 22 C 12 0 0 -2 24 D 12 16 2 0 26 E -20 -22 -24 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999964 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -12 -12 20 B 12 0 0 -16 22 C 12 0 0 -2 24 D 12 16 2 0 26 E -20 -22 -24 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999964945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 C=22 B=19 D=17 A=13 so A is eliminated. Round 2 votes counts: E=32 C=29 B=22 D=17 so D is eliminated. Round 3 votes counts: B=36 E=32 C=32 so E is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:228 C:217 B:209 A:192 E:154 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 -12 -12 20 B 12 0 0 -16 22 C 12 0 0 -2 24 D 12 16 2 0 26 E -20 -22 -24 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999964945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -12 20 B 12 0 0 -16 22 C 12 0 0 -2 24 D 12 16 2 0 26 E -20 -22 -24 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999964945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -12 20 B 12 0 0 -16 22 C 12 0 0 -2 24 D 12 16 2 0 26 E -20 -22 -24 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999964945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9610: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (9) B C A D E (9) E D C B A (7) D E A C B (7) C A B E D (7) D E C B A (5) E A C B D (4) D B C A E (4) B D C A E (4) A C B E D (4) D E A B C (3) D B C E A (3) E D C A B (2) E D A C B (2) E C D B A (2) E A C D B (2) D E C A B (2) D E B C A (2) D A B E C (2) B C A E D (2) B A C D E (2) A C E B D (2) A B C E D (2) E C A B D (1) D E B A C (1) D B E C A (1) D B A C E (1) C E B A D (1) B D C E A (1) B C D A E (1) B A D C E (1) B A C E D (1) A E C B D (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -14 -26 0 8 B 14 0 -16 6 12 C 26 16 0 2 12 D 0 -6 -2 0 2 E -8 -12 -12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -26 0 8 B 14 0 -16 6 12 C 26 16 0 2 12 D 0 -6 -2 0 2 E -8 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=21 E=20 C=17 A=11 so A is eliminated. Round 2 votes counts: D=32 B=24 C=23 E=21 so E is eliminated. Round 3 votes counts: D=43 C=33 B=24 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:228 B:208 D:197 A:184 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -26 0 8 B 14 0 -16 6 12 C 26 16 0 2 12 D 0 -6 -2 0 2 E -8 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -26 0 8 B 14 0 -16 6 12 C 26 16 0 2 12 D 0 -6 -2 0 2 E -8 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -26 0 8 B 14 0 -16 6 12 C 26 16 0 2 12 D 0 -6 -2 0 2 E -8 -12 -12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992024 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9611: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (13) B C D E A (13) A E D B C (12) A E D C B (11) E D A B C (7) D E B A C (6) B D E C A (6) A C E D B (6) C B A D E (5) B D C E A (5) C A B E D (3) C B D A E (2) B D E A C (2) A E C D B (2) E A D B C (1) D E B C A (1) D E A B C (1) D B E A C (1) C A E B D (1) C A B D E (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 0 -14 -12 B 8 0 10 4 2 C 0 -10 0 -6 0 D 14 -4 6 0 12 E 12 -2 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999363 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 0 -14 -12 B 8 0 10 4 2 C 0 -10 0 -6 0 D 14 -4 6 0 12 E 12 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=26 C=25 D=9 E=8 so E is eliminated. Round 2 votes counts: A=33 B=26 C=25 D=16 so D is eliminated. Round 3 votes counts: A=41 B=34 C=25 so C is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:214 B:212 E:199 C:192 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 0 -14 -12 B 8 0 10 4 2 C 0 -10 0 -6 0 D 14 -4 6 0 12 E 12 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 0 -14 -12 B 8 0 10 4 2 C 0 -10 0 -6 0 D 14 -4 6 0 12 E 12 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 0 -14 -12 B 8 0 10 4 2 C 0 -10 0 -6 0 D 14 -4 6 0 12 E 12 -2 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999408 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9612: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (9) D E C A B (7) B A C E D (7) B A C D E (6) E D C A B (5) E D B C A (5) C B A E D (4) A B C D E (4) E D B A C (3) D E A B C (3) C E B A D (3) B E C A D (3) A D B C E (3) A B D C E (3) E C D B A (2) D E A C B (2) D C E A B (2) D A E C B (2) D A C E B (2) D A B E C (2) C B E A D (2) C A B D E (2) B C A E D (2) A D C B E (2) E C B D A (1) E C B A D (1) E B D A C (1) E B C D A (1) D C A E B (1) D A C B E (1) C E B D A (1) C D E A B (1) C A D B E (1) B E A D C (1) B C E A D (1) B A E C D (1) A D B E C (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -8 -2 -8 B 8 0 -6 -10 -4 C 8 6 0 -10 0 D 2 10 10 0 -6 E 8 4 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.252437 D: 0.000000 E: 0.747563 Sum of squares = 0.622574413281 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.252437 D: 0.252437 E: 1.000000 A B C D E A 0 -8 -8 -2 -8 B 8 0 -6 -10 -4 C 8 6 0 -10 0 D 2 10 10 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250026836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.375000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 D=22 B=21 A=15 C=14 so C is eliminated. Round 2 votes counts: E=32 B=27 D=23 A=18 so A is eliminated. Round 3 votes counts: B=37 E=32 D=31 so D is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:209 D:208 C:202 B:194 A:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -8 -2 -8 B 8 0 -6 -10 -4 C 8 6 0 -10 0 D 2 10 10 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250026836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.375000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -8 -2 -8 B 8 0 -6 -10 -4 C 8 6 0 -10 0 D 2 10 10 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250026836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.375000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -8 -2 -8 B 8 0 -6 -10 -4 C 8 6 0 -10 0 D 2 10 10 0 -6 E 8 4 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.000000 E: 0.625000 Sum of squares = 0.531250026836 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.375000 D: 0.375000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9613: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (13) C E A D B (6) C E A B D (6) E D B C A (5) D B E A C (5) D B A E C (5) E D C B A (4) B D A E C (4) A B D C E (4) E C B D A (3) E C B A D (3) E B D C A (3) C E D A B (3) C A B E D (3) B E D A C (3) A D B C E (3) E B D A C (2) D E B C A (2) D E B A C (2) C A E D B (2) C A E B D (2) B D E A C (2) B A D C E (2) A C B D E (2) E C D A B (1) E C A D B (1) D B A C E (1) D A B C E (1) C A B D E (1) B A D E C (1) B A C E D (1) A D C B E (1) A C D B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -22 -16 -18 -28 B 22 0 -6 -12 -16 C 16 6 0 0 -18 D 18 12 0 0 -24 E 28 16 18 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -22 -16 -18 -28 B 22 0 -6 -12 -16 C 16 6 0 0 -18 D 18 12 0 0 -24 E 28 16 18 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=23 D=16 B=13 A=13 so B is eliminated. Round 2 votes counts: E=38 C=23 D=22 A=17 so A is eliminated. Round 3 votes counts: E=38 D=33 C=29 so C is eliminated. Round 4 votes counts: E=62 D=38 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:243 D:203 C:202 B:194 A:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -22 -16 -18 -28 B 22 0 -6 -12 -16 C 16 6 0 0 -18 D 18 12 0 0 -24 E 28 16 18 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -16 -18 -28 B 22 0 -6 -12 -16 C 16 6 0 0 -18 D 18 12 0 0 -24 E 28 16 18 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -16 -18 -28 B 22 0 -6 -12 -16 C 16 6 0 0 -18 D 18 12 0 0 -24 E 28 16 18 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9614: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (6) E A B D C (5) C A D B E (5) A E B C D (5) E D C A B (4) E D B C A (4) D C E B A (4) D C B E A (4) C D A E B (3) A C E D B (3) A C E B D (3) E D B A C (2) E B A D C (2) E A D C B (2) E A C D B (2) D C E A B (2) D B E C A (2) D B C E A (2) C D E A B (2) C D A B E (2) C A E D B (2) C A B D E (2) B C D A E (2) B A E D C (2) A E C D B (2) A E C B D (2) A C B E D (2) A C B D E (2) A B E C D (2) A B C E D (2) E C D A B (1) E A B C D (1) D E C B A (1) D E B C A (1) C D E B A (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E A C (1) B D A E C (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C E D (1) B A C D E (1) A B C D E (1) Total count = 100 A B C D E A 0 16 -4 8 10 B -16 0 -16 -16 -10 C 4 16 0 16 8 D -8 16 -16 0 -4 E -10 10 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -4 8 10 B -16 0 -16 -16 -10 C 4 16 0 16 8 D -8 16 -16 0 -4 E -10 10 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=24 A=24 E=23 D=16 B=13 so B is eliminated. Round 2 votes counts: A=30 C=27 E=25 D=18 so D is eliminated. Round 3 votes counts: C=39 A=31 E=30 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:222 A:215 E:198 D:194 B:171 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -4 8 10 B -16 0 -16 -16 -10 C 4 16 0 16 8 D -8 16 -16 0 -4 E -10 10 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -4 8 10 B -16 0 -16 -16 -10 C 4 16 0 16 8 D -8 16 -16 0 -4 E -10 10 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -4 8 10 B -16 0 -16 -16 -10 C 4 16 0 16 8 D -8 16 -16 0 -4 E -10 10 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999954 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9615: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (16) C B D E A (14) B C D E A (8) B C A E D (8) D E A B C (7) C B A E D (5) A D E C B (5) D A E C B (4) C B D A E (4) A E D C B (4) D E A C B (3) E A D B C (2) C D B E A (2) A E B D C (2) A C D E B (2) A B C E D (2) E D A B C (1) D E B A C (1) D C E B A (1) C B A D E (1) C A D E B (1) B E D C A (1) B E D A C (1) B C E A D (1) B A E C D (1) A E C D B (1) A E B C D (1) A C B E D (1) Total count = 100 A B C D E A 0 4 8 6 16 B -4 0 4 0 -2 C -8 -4 0 4 0 D -6 0 -4 0 6 E -16 2 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 6 16 B -4 0 4 0 -2 C -8 -4 0 4 0 D -6 0 -4 0 6 E -16 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=27 B=20 D=16 E=3 so E is eliminated. Round 2 votes counts: A=36 C=27 B=20 D=17 so D is eliminated. Round 3 votes counts: A=51 C=28 B=21 so B is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:217 B:199 D:198 C:196 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 6 16 B -4 0 4 0 -2 C -8 -4 0 4 0 D -6 0 -4 0 6 E -16 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 6 16 B -4 0 4 0 -2 C -8 -4 0 4 0 D -6 0 -4 0 6 E -16 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 6 16 B -4 0 4 0 -2 C -8 -4 0 4 0 D -6 0 -4 0 6 E -16 2 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9616: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (8) B C E D A (8) E A D B C (6) C B A D E (6) A D E C B (6) C A B D E (5) B E C D A (5) E D B A C (4) D E A C B (4) C B D A E (4) A E D C B (4) A D C E B (4) D A E C B (3) C A D B E (3) B C D E A (3) E D A C B (2) E B D A C (2) D C A B E (2) D A C E B (2) B E C A D (2) B C E A D (2) B C D A E (2) A C D E B (2) A C D B E (2) E B D C A (1) E B A D C (1) E B A C D (1) D E A B C (1) C D B A E (1) C D A B E (1) C B A E D (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 14 8 -6 0 B -14 0 -8 -12 -6 C -8 8 0 0 -2 D 6 12 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.244942 D: 0.755058 E: 0.000000 Sum of squares = 0.630109665129 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.244942 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 -6 0 B -14 0 -8 -12 -6 C -8 8 0 0 -2 D 6 12 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.571429 E: 0.000000 Sum of squares = 0.510204082254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=22 C=21 A=20 D=12 so D is eliminated. Round 2 votes counts: E=30 A=25 C=23 B=22 so B is eliminated. Round 3 votes counts: C=38 E=37 A=25 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:210 A:208 E:203 C:199 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 8 -6 0 B -14 0 -8 -12 -6 C -8 8 0 0 -2 D 6 12 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.571429 E: 0.000000 Sum of squares = 0.510204082254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -6 0 B -14 0 -8 -12 -6 C -8 8 0 0 -2 D 6 12 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.571429 E: 0.000000 Sum of squares = 0.510204082254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -6 0 B -14 0 -8 -12 -6 C -8 8 0 0 -2 D 6 12 0 0 2 E 0 6 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.571429 E: 0.000000 Sum of squares = 0.510204082254 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9617: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (8) A E B D C (8) C D B E A (7) C D A E B (7) D C B E A (6) E B A C D (5) D B E A C (5) D C A B E (4) C A E B D (4) A E B C D (4) A B E D C (4) D B A E C (3) C D E B A (3) C D A B E (3) C A E D B (3) A E C B D (3) E B A D C (2) D C B A E (2) D B C E A (2) D A B E C (2) C E B A D (2) A C E B D (2) D B E C A (1) C E B D A (1) C A D E B (1) B E D A C (1) B E C A D (1) B E A C D (1) B D A E C (1) B C E D A (1) B A E D C (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 4 2 8 B 6 0 0 -2 6 C -4 0 0 -4 -2 D -2 2 4 0 -2 E -8 -6 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.440000000004 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 2 8 B 6 0 0 -2 6 C -4 0 0 -4 -2 D -2 2 4 0 -2 E -8 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.44 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 A=23 B=14 E=7 so E is eliminated. Round 2 votes counts: C=31 D=25 A=23 B=21 so B is eliminated. Round 3 votes counts: A=40 C=33 D=27 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:205 A:204 D:201 C:195 E:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 2 8 B 6 0 0 -2 6 C -4 0 0 -4 -2 D -2 2 4 0 -2 E -8 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.44 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 2 8 B 6 0 0 -2 6 C -4 0 0 -4 -2 D -2 2 4 0 -2 E -8 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.44 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 2 8 B 6 0 0 -2 6 C -4 0 0 -4 -2 D -2 2 4 0 -2 E -8 -6 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.200000 C: 0.000000 D: 0.600000 E: 0.000000 Sum of squares = 0.44 Cumulative probabilities = A: 0.200000 B: 0.400000 C: 0.400000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9618: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (13) B E C A D (13) A B E C D (6) D C E B A (5) D A B E C (5) E B C D A (4) D E C B A (4) C E B D A (4) C E B A D (4) B E C D A (4) E C B D A (3) D E B C A (3) A D C E B (3) A D B C E (3) E C B A D (2) E B C A D (2) D A E C B (2) C E D B A (2) C B E A D (2) B E A C D (2) A C B E D (2) A B D E C (2) A B C E D (2) D E A C B (1) D C A E B (1) D B E A C (1) D A C B E (1) C D A E B (1) B C E A D (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 -12 -10 -8 -14 B 12 0 -2 6 -8 C 10 2 0 8 -10 D 8 -6 -8 0 -6 E 14 8 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -10 -8 -14 B 12 0 -2 6 -8 C 10 2 0 8 -10 D 8 -6 -8 0 -6 E 14 8 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=20 A=20 C=13 E=11 so E is eliminated. Round 2 votes counts: D=36 B=26 A=20 C=18 so C is eliminated. Round 3 votes counts: B=41 D=39 A=20 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:219 C:205 B:204 D:194 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -10 -8 -14 B 12 0 -2 6 -8 C 10 2 0 8 -10 D 8 -6 -8 0 -6 E 14 8 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -10 -8 -14 B 12 0 -2 6 -8 C 10 2 0 8 -10 D 8 -6 -8 0 -6 E 14 8 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -10 -8 -14 B 12 0 -2 6 -8 C 10 2 0 8 -10 D 8 -6 -8 0 -6 E 14 8 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9619: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) B D A C E (10) B D E C A (8) E C B D A (7) A D B C E (7) B D E A C (6) C E A B D (5) A D B E C (5) B D A E C (4) D B A C E (3) C E B D A (3) C E A D B (3) C A E D B (3) E C A B D (2) E B D C A (2) E B C D A (2) D A B E C (2) A E C D B (2) A C E D B (2) E D B A C (1) E C D B A (1) E C B A D (1) E C A D B (1) D B E A C (1) D A B C E (1) C E B A D (1) C B E D A (1) C A E B D (1) C A B D E (1) B E D C A (1) B D C E A (1) B C E D A (1) A E D C B (1) Total count = 100 A B C D E A 0 -28 10 -30 4 B 28 0 30 14 24 C -10 -30 0 -26 -14 D 30 -14 26 0 18 E -4 -24 14 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 10 -30 4 B 28 0 30 14 24 C -10 -30 0 -26 -14 D 30 -14 26 0 18 E -4 -24 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=18 E=17 D=17 A=17 so E is eliminated. Round 2 votes counts: B=35 C=30 D=18 A=17 so A is eliminated. Round 3 votes counts: B=35 C=34 D=31 so D is eliminated. Round 4 votes counts: B=65 C=35 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:248 D:230 E:184 A:178 C:160 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 10 -30 4 B 28 0 30 14 24 C -10 -30 0 -26 -14 D 30 -14 26 0 18 E -4 -24 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 10 -30 4 B 28 0 30 14 24 C -10 -30 0 -26 -14 D 30 -14 26 0 18 E -4 -24 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 10 -30 4 B 28 0 30 14 24 C -10 -30 0 -26 -14 D 30 -14 26 0 18 E -4 -24 14 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9620: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (13) C E D A B (11) A D B C E (8) E C D B A (7) B A D E C (7) C D E A B (4) B E A C D (4) B A E D C (4) A B D C E (4) E C B A D (3) E B C D A (3) D A C E B (3) D A C B E (3) C D A E B (3) B E C A D (3) C E D B A (2) C E B A D (2) B E D A C (2) B E A D C (2) E B C A D (1) D C A E B (1) D B A E C (1) D A B C E (1) C E A D B (1) C E A B D (1) C A E D B (1) B A C E D (1) A D C E B (1) A D B E C (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -12 -8 -18 B 10 0 -12 4 -14 C 12 12 0 22 -4 D 8 -4 -22 0 -22 E 18 14 4 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 -18 B 10 0 -12 4 -14 C 12 12 0 22 -4 D 8 -4 -22 0 -22 E 18 14 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=25 B=23 A=16 D=9 so D is eliminated. Round 2 votes counts: E=27 C=26 B=24 A=23 so A is eliminated. Round 3 votes counts: B=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:229 C:221 B:194 D:180 A:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 -18 B 10 0 -12 4 -14 C 12 12 0 22 -4 D 8 -4 -22 0 -22 E 18 14 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 -18 B 10 0 -12 4 -14 C 12 12 0 22 -4 D 8 -4 -22 0 -22 E 18 14 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 -18 B 10 0 -12 4 -14 C 12 12 0 22 -4 D 8 -4 -22 0 -22 E 18 14 4 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997806 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9621: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (12) D C B A E (11) D A E C B (7) B C E D A (7) A E D C B (7) E A B C D (6) D C A B E (6) B E C A D (6) D A E B C (4) D A C E B (4) E A D B C (3) E A B D C (3) A D E C B (3) E B C A D (2) E B A C D (2) E A C B D (2) D A C B E (2) C D B A E (2) C B D E A (2) B C E A D (2) D B A C E (1) C E B A D (1) C B E A D (1) C B D A E (1) B D C E A (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -8 -20 0 B 2 0 -2 -4 8 C 8 2 0 -4 8 D 20 4 4 0 14 E 0 -8 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -20 0 B 2 0 -2 -4 8 C 8 2 0 -4 8 D 20 4 4 0 14 E 0 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=28 E=18 A=12 C=7 so C is eliminated. Round 2 votes counts: D=37 B=32 E=19 A=12 so A is eliminated. Round 3 votes counts: D=41 B=32 E=27 so E is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:207 B:202 A:185 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -8 -20 0 B 2 0 -2 -4 8 C 8 2 0 -4 8 D 20 4 4 0 14 E 0 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -20 0 B 2 0 -2 -4 8 C 8 2 0 -4 8 D 20 4 4 0 14 E 0 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -20 0 B 2 0 -2 -4 8 C 8 2 0 -4 8 D 20 4 4 0 14 E 0 -8 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999993897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9622: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) E A D C B (7) E A C D B (6) D E B A C (6) B D C E A (6) E D B A C (5) B D E C A (5) E D A C B (4) E D A B C (4) A E C D B (4) A C E D B (4) E B D A C (3) C A D B E (3) B C D A E (3) A C E B D (3) E B D C A (2) D B E C A (2) C B A D E (2) C A B E D (2) B E D C A (2) B D C A E (2) B C E D A (2) B C A D E (2) A C D E B (2) E A C B D (1) D B E A C (1) D B C A E (1) B C E A D (1) B C D E A (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 4 2 -16 B -8 0 -4 -2 -6 C -4 4 0 -4 -6 D -2 2 4 0 0 E 16 6 6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571977 E: 0.428023 Sum of squares = 0.510361355171 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.571977 E: 1.000000 A B C D E A 0 8 4 2 -16 B -8 0 -4 -2 -6 C -4 4 0 -4 -6 D -2 2 4 0 0 E 16 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 B=24 C=19 A=15 D=10 so D is eliminated. Round 2 votes counts: E=38 B=28 C=19 A=15 so A is eliminated. Round 3 votes counts: E=43 C=29 B=28 so B is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:214 D:202 A:199 C:195 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 4 2 -16 B -8 0 -4 -2 -6 C -4 4 0 -4 -6 D -2 2 4 0 0 E 16 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 2 -16 B -8 0 -4 -2 -6 C -4 4 0 -4 -6 D -2 2 4 0 0 E 16 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 2 -16 B -8 0 -4 -2 -6 C -4 4 0 -4 -6 D -2 2 4 0 0 E 16 6 6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9623: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (11) D A E B C (10) B D C A E (10) C B E A D (8) B C D A E (8) C B E D A (7) D B A E C (5) C E A B D (5) C B D E A (5) A E D B C (5) E C A B D (4) E A C D B (4) A D E B C (3) E A C B D (2) D A B E C (2) C E B A D (2) B C D E A (2) E C A D B (1) D E A B C (1) D B C A E (1) D B A C E (1) B D C E A (1) A E D C B (1) A D B E C (1) Total count = 100 A B C D E A 0 0 -8 -6 -6 B 0 0 0 8 2 C 8 0 0 -4 0 D 6 -8 4 0 0 E 6 -2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.593532 C: 0.406468 D: 0.000000 E: 0.000000 Sum of squares = 0.517496319739 Cumulative probabilities = A: 0.000000 B: 0.593532 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -8 -6 -6 B 0 0 0 8 2 C 8 0 0 -4 0 D 6 -8 4 0 0 E 6 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=22 B=21 D=20 A=10 so A is eliminated. Round 2 votes counts: E=28 C=27 D=24 B=21 so B is eliminated. Round 3 votes counts: C=37 D=35 E=28 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:205 C:202 E:202 D:201 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -8 -6 -6 B 0 0 0 8 2 C 8 0 0 -4 0 D 6 -8 4 0 0 E 6 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -8 -6 -6 B 0 0 0 8 2 C 8 0 0 -4 0 D 6 -8 4 0 0 E 6 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -8 -6 -6 B 0 0 0 8 2 C 8 0 0 -4 0 D 6 -8 4 0 0 E 6 -2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9624: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (10) A B C D E (9) D A B E C (8) A D B C E (7) E D B A C (5) E C D B A (5) D A E B C (5) C E B A D (5) C A B D E (5) D E A B C (4) D B A E C (4) E D C A B (3) E D B C A (3) D E B A C (3) D A B C E (3) C E A B D (3) C A B E D (3) E D A B C (2) E C B D A (2) C B A E D (2) C A D B E (2) C E D A B (1) C B A D E (1) C A E B D (1) B D A E C (1) B C A D E (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 32 26 2 28 B -32 0 34 -10 16 C -26 -34 0 -20 8 D -2 10 20 0 30 E -28 -16 -8 -30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999966 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 32 26 2 28 B -32 0 34 -10 16 C -26 -34 0 -20 8 D -2 10 20 0 30 E -28 -16 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999889733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=26 C=23 E=20 B=4 so B is eliminated. Round 2 votes counts: D=28 A=28 C=24 E=20 so E is eliminated. Round 3 votes counts: D=41 C=31 A=28 so A is eliminated. Round 4 votes counts: D=60 C=40 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:244 D:229 B:204 C:164 E:159 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 32 26 2 28 B -32 0 34 -10 16 C -26 -34 0 -20 8 D -2 10 20 0 30 E -28 -16 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999889733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 32 26 2 28 B -32 0 34 -10 16 C -26 -34 0 -20 8 D -2 10 20 0 30 E -28 -16 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999889733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 32 26 2 28 B -32 0 34 -10 16 C -26 -34 0 -20 8 D -2 10 20 0 30 E -28 -16 -8 -30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999889733 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9625: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A D C (11) C D A E B (10) A D E B C (10) C B E A D (5) B E C A D (5) E B A C D (4) C D A B E (4) D C A B E (3) D A C E B (3) D A B E C (3) C B D E A (3) A E B D C (3) E B A D C (2) E A B C D (2) D C A E B (2) D A E B C (2) D A C B E (2) C E B A D (2) C E A B D (2) C D E A B (2) C D B E A (2) B E A C D (2) A E D B C (2) A B E D C (2) E C B A D (1) E B C A D (1) E A B D C (1) C D E B A (1) C B E D A (1) B E D A C (1) B D E A C (1) B C E A D (1) B A E D C (1) B A D E C (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 10 10 20 0 B -10 0 12 4 -2 C -10 -12 0 -4 -14 D -20 -4 4 0 2 E 0 2 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.485366 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.514634 Sum of squares = 0.500428311008 Cumulative probabilities = A: 0.485366 B: 0.485366 C: 0.485366 D: 0.485366 E: 1.000000 A B C D E A 0 10 10 20 0 B -10 0 12 4 -2 C -10 -12 0 -4 -14 D -20 -4 4 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 B=23 A=19 D=15 E=11 so E is eliminated. Round 2 votes counts: C=33 B=30 A=22 D=15 so D is eliminated. Round 3 votes counts: C=38 A=32 B=30 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 E:207 B:202 D:191 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 10 20 0 B -10 0 12 4 -2 C -10 -12 0 -4 -14 D -20 -4 4 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 20 0 B -10 0 12 4 -2 C -10 -12 0 -4 -14 D -20 -4 4 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 20 0 B -10 0 12 4 -2 C -10 -12 0 -4 -14 D -20 -4 4 0 2 E 0 2 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9626: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (12) B D E C A (6) C A B E D (5) A E C D B (4) A C B D E (4) E D B A C (3) E D A B C (3) E C A D B (3) C B E D A (3) C A E D B (3) A B D E C (3) E C D B A (2) E C D A B (2) D E B A C (2) D B E A C (2) D A B E C (2) C E B D A (2) C E B A D (2) C B A D E (2) C A E B D (2) B D E A C (2) B D C E A (2) B D A E C (2) B D A C E (2) B C D E A (2) A D E B C (2) E B D C A (1) E A D B C (1) E A C D B (1) D E A B C (1) D A E B C (1) C E D A B (1) C E A B D (1) C B E A D (1) C B D E A (1) C B A E D (1) B E D C A (1) B C D A E (1) B C A D E (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E C B (1) A C E B D (1) A C D B E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 14 10 12 10 B -14 0 -12 4 -4 C -10 12 0 20 8 D -12 -4 -20 0 -12 E -10 4 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 12 10 B -14 0 -12 4 -4 C -10 12 0 20 8 D -12 -4 -20 0 -12 E -10 4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 C=24 B=21 E=16 D=8 so D is eliminated. Round 2 votes counts: A=34 C=24 B=23 E=19 so E is eliminated. Round 3 votes counts: A=40 C=31 B=29 so B is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:223 C:215 E:199 B:187 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 12 10 B -14 0 -12 4 -4 C -10 12 0 20 8 D -12 -4 -20 0 -12 E -10 4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 12 10 B -14 0 -12 4 -4 C -10 12 0 20 8 D -12 -4 -20 0 -12 E -10 4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 12 10 B -14 0 -12 4 -4 C -10 12 0 20 8 D -12 -4 -20 0 -12 E -10 4 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9627: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (11) E B D C A (5) A C B D E (5) D E B C A (4) C A B E D (4) C A B D E (4) B E D A C (4) B E C D A (4) A C B E D (4) E D B C A (3) E D B A C (3) D A E C B (3) D A E B C (3) C B A E D (3) B E D C A (3) A C D B E (3) E B D A C (2) D E A C B (2) D E A B C (2) D A C E B (2) C E D B A (2) C A D E B (2) B E C A D (2) B A C E D (2) A D C E B (2) E D C B A (1) E C B D A (1) E B C D A (1) D E C B A (1) D E C A B (1) C E B D A (1) C D A E B (1) C A D B E (1) B E A D C (1) B C E A D (1) B C A E D (1) B A E D C (1) A D C B E (1) A D B E C (1) A C D E B (1) A B C E D (1) Total count = 100 A B C D E A 0 -14 8 -20 -10 B 14 0 10 0 -8 C -8 -10 0 -12 -18 D 20 0 12 0 0 E 10 8 18 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.484002 E: 0.515998 Sum of squares = 0.500511882648 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.484002 E: 1.000000 A B C D E A 0 -14 8 -20 -10 B 14 0 10 0 -8 C -8 -10 0 -12 -18 D 20 0 12 0 0 E 10 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=19 C=18 A=18 E=16 so E is eliminated. Round 2 votes counts: D=36 B=27 C=19 A=18 so A is eliminated. Round 3 votes counts: D=40 C=32 B=28 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:218 D:216 B:208 A:182 C:176 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 8 -20 -10 B 14 0 10 0 -8 C -8 -10 0 -12 -18 D 20 0 12 0 0 E 10 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 8 -20 -10 B 14 0 10 0 -8 C -8 -10 0 -12 -18 D 20 0 12 0 0 E 10 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 8 -20 -10 B 14 0 10 0 -8 C -8 -10 0 -12 -18 D 20 0 12 0 0 E 10 8 18 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9628: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (6) B C D E A (6) A E D C B (6) A E C D B (5) E A C B D (4) D B E A C (4) B E C A D (4) E A D B C (3) D C B A E (3) D B C E A (3) D B C A E (3) C B D A E (3) C B A D E (3) C A E B D (3) E C B A D (2) E A D C B (2) E A B C D (2) D E B A C (2) D B E C A (2) D A E C B (2) D A E B C (2) D A C E B (2) C B E A D (2) C A E D B (2) B C E A D (2) A C E D B (2) E B D A C (1) E B A C D (1) E A C D B (1) D E A B C (1) D C A B E (1) D B A E C (1) D B A C E (1) D A C B E (1) C D B A E (1) C D A B E (1) C A D E B (1) B E D A C (1) B D E C A (1) B D C A E (1) B C D A E (1) A E C B D (1) A D E C B (1) A D C E B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 -2 0 0 B 8 0 -4 -10 2 C 2 4 0 -2 2 D 0 10 2 0 10 E 0 -2 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.376826 B: 0.000000 C: 0.000000 D: 0.623174 E: 0.000000 Sum of squares = 0.530343578072 Cumulative probabilities = A: 0.376826 B: 0.376826 C: 0.376826 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 0 0 B 8 0 -4 -10 2 C 2 4 0 -2 2 D 0 10 2 0 10 E 0 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499670 B: 0.000000 C: 0.000000 D: 0.500330 E: 0.000000 Sum of squares = 0.50000021746 Cumulative probabilities = A: 0.499670 B: 0.499670 C: 0.499670 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=22 A=18 E=16 C=16 so E is eliminated. Round 2 votes counts: A=30 D=28 B=24 C=18 so C is eliminated. Round 3 votes counts: A=36 B=34 D=30 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:211 C:203 B:198 A:195 E:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -8 -2 0 0 B 8 0 -4 -10 2 C 2 4 0 -2 2 D 0 10 2 0 10 E 0 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499670 B: 0.000000 C: 0.000000 D: 0.500330 E: 0.000000 Sum of squares = 0.50000021746 Cumulative probabilities = A: 0.499670 B: 0.499670 C: 0.499670 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 0 0 B 8 0 -4 -10 2 C 2 4 0 -2 2 D 0 10 2 0 10 E 0 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499670 B: 0.000000 C: 0.000000 D: 0.500330 E: 0.000000 Sum of squares = 0.50000021746 Cumulative probabilities = A: 0.499670 B: 0.499670 C: 0.499670 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 0 0 B 8 0 -4 -10 2 C 2 4 0 -2 2 D 0 10 2 0 10 E 0 -2 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499670 B: 0.000000 C: 0.000000 D: 0.500330 E: 0.000000 Sum of squares = 0.50000021746 Cumulative probabilities = A: 0.499670 B: 0.499670 C: 0.499670 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9629: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) C E B D A (6) C B E D A (6) E D A C B (5) D A E C B (5) A D E C B (5) E D C A B (4) E C D B A (4) B C E D A (4) B C D E A (4) A D E B C (4) E D C B A (3) D E A C B (3) D A B C E (3) C E D B A (3) B C A E D (3) A D B C E (3) E C B A D (2) E A D C B (2) D C B E A (2) D A C E B (2) B A C D E (2) A D B E C (2) A B E C D (2) E C D A B (1) E C B D A (1) E B C A D (1) E A C D B (1) E A C B D (1) D E C B A (1) D C E B A (1) B E A C D (1) B D C A E (1) B A D C E (1) B A C E D (1) A E B C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 6 -8 -32 -26 B -6 0 -30 -22 -26 C 8 30 0 -12 -14 D 32 22 12 0 -4 E 26 26 14 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 -32 -26 B -6 0 -30 -22 -26 C 8 30 0 -12 -14 D 32 22 12 0 -4 E 26 26 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 D=24 A=19 B=17 C=15 so C is eliminated. Round 2 votes counts: E=34 D=24 B=23 A=19 so A is eliminated. Round 3 votes counts: D=38 E=35 B=27 so B is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:235 D:231 C:206 A:170 B:158 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 -32 -26 B -6 0 -30 -22 -26 C 8 30 0 -12 -14 D 32 22 12 0 -4 E 26 26 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 -32 -26 B -6 0 -30 -22 -26 C 8 30 0 -12 -14 D 32 22 12 0 -4 E 26 26 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 -32 -26 B -6 0 -30 -22 -26 C 8 30 0 -12 -14 D 32 22 12 0 -4 E 26 26 14 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9630: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (7) C A E D B (7) B D E A C (7) B E D C A (6) E B D C A (5) D A B C E (4) B E D A C (4) A D C B E (4) A C D E B (4) E D B C A (3) E C B A D (3) E C A B D (3) D A C E B (3) C A D E B (3) A B C D E (3) E B C A D (2) D B E C A (2) D B E A C (2) D B A C E (2) C A E B D (2) B E C A D (2) B A C E D (2) A C D B E (2) A C B D E (2) E D C B A (1) E C D B A (1) E C D A B (1) E C B D A (1) E C A D B (1) E B C D A (1) D E B C A (1) D C E A B (1) D B A E C (1) D A C B E (1) C E A B D (1) B E A C D (1) B D A E C (1) B A E D C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -8 4 -12 B -2 0 2 -6 -4 C 8 -2 0 0 0 D -4 6 0 0 -12 E 12 4 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.323688 D: 0.000000 E: 0.676312 Sum of squares = 0.562171933147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.323688 D: 0.323688 E: 1.000000 A B C D E A 0 2 -8 4 -12 B -2 0 2 -6 -4 C 8 -2 0 0 0 D -4 6 0 0 -12 E 12 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=24 E=22 C=20 D=17 A=17 so D is eliminated. Round 2 votes counts: B=31 A=25 E=23 C=21 so C is eliminated. Round 3 votes counts: A=37 E=32 B=31 so B is eliminated. Round 4 votes counts: E=56 A=44 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:214 C:203 B:195 D:195 A:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -8 4 -12 B -2 0 2 -6 -4 C 8 -2 0 0 0 D -4 6 0 0 -12 E 12 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 4 -12 B -2 0 2 -6 -4 C 8 -2 0 0 0 D -4 6 0 0 -12 E 12 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 4 -12 B -2 0 2 -6 -4 C 8 -2 0 0 0 D -4 6 0 0 -12 E 12 4 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9631: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (12) B D A E C (7) A C B D E (7) D E B A C (6) D B E A C (5) C E A D B (4) C A B D E (4) E D A B C (3) C A E D B (3) C A E B D (3) C A B E D (3) B D E A C (3) A B D C E (3) E D C B A (2) E D C A B (2) C E B A D (2) C B A D E (2) B D A C E (2) A D B E C (2) A B C D E (2) E D A C B (1) E C D B A (1) E C A D B (1) D E A B C (1) D B A E C (1) D A B E C (1) C E D B A (1) C E D A B (1) C B E D A (1) C B E A D (1) C B D E A (1) B E C D A (1) B D E C A (1) B D C A E (1) B C D E A (1) B C D A E (1) B C A D E (1) B A D E C (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 0 -12 -2 B 12 0 16 0 8 C 0 -16 0 -14 -6 D 12 0 14 0 12 E 2 -8 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.507085 C: 0.000000 D: 0.492915 E: 0.000000 Sum of squares = 0.500100400796 Cumulative probabilities = A: 0.000000 B: 0.507085 C: 0.507085 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -12 -2 B 12 0 16 0 8 C 0 -16 0 -14 -6 D 12 0 14 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 E=22 B=21 A=17 D=14 so D is eliminated. Round 2 votes counts: E=29 B=27 C=26 A=18 so A is eliminated. Round 3 votes counts: B=35 C=34 E=31 so E is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:219 B:218 E:194 A:187 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -12 -2 B 12 0 16 0 8 C 0 -16 0 -14 -6 D 12 0 14 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -12 -2 B 12 0 16 0 8 C 0 -16 0 -14 -6 D 12 0 14 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -12 -2 B 12 0 16 0 8 C 0 -16 0 -14 -6 D 12 0 14 0 12 E 2 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9632: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (5) B C D E A (5) B C D A E (5) D A C E B (4) C D B E A (4) B E C D A (4) B C E D A (4) A E D C B (4) A E D B C (4) E D C B A (3) D C B E A (3) B E C A D (3) B C A D E (3) B A C D E (3) A E B D C (3) A D E C B (3) A B C D E (3) E D C A B (2) E D A C B (2) E C D B A (2) E B D C A (2) E A B D C (2) D E C A B (2) D C E A B (2) D C A E B (2) D C A B E (2) C D B A E (2) C B D E A (2) B A E C D (2) B A C E D (2) A D C E B (2) A B E C D (2) E B C D A (1) E A B C D (1) D E C B A (1) D E A C B (1) C D E B A (1) A D E B C (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 -20 -22 -4 B 14 0 0 -4 0 C 20 0 0 0 10 D 22 4 0 0 14 E 4 0 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.421994 D: 0.578006 E: 0.000000 Sum of squares = 0.512169793981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.421994 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -20 -22 -4 B 14 0 0 -4 0 C 20 0 0 0 10 D 22 4 0 0 14 E 4 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=23 D=22 E=15 C=9 so C is eliminated. Round 2 votes counts: B=33 D=29 A=23 E=15 so E is eliminated. Round 3 votes counts: D=38 B=36 A=26 so A is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:220 C:215 B:205 E:190 A:170 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 -20 -22 -4 B 14 0 0 -4 0 C 20 0 0 0 10 D 22 4 0 0 14 E 4 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -20 -22 -4 B 14 0 0 -4 0 C 20 0 0 0 10 D 22 4 0 0 14 E 4 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -20 -22 -4 B 14 0 0 -4 0 C 20 0 0 0 10 D 22 4 0 0 14 E 4 0 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9633: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (9) A E B C D (7) D C B E A (6) B C D E A (6) E A C B D (4) E A B C D (4) C D B E A (4) B A E C D (4) A D E C B (4) E C D A B (3) E A C D B (3) C E D B A (3) B C E D A (3) B C D A E (3) A B D C E (3) E C B A D (2) E A D C B (2) C B E D A (2) B C A E D (2) B A C E D (2) B A C D E (2) A E D B C (2) E D A C B (1) E B C A D (1) E B A C D (1) D C E B A (1) D C B A E (1) D B A C E (1) D A C E B (1) D A C B E (1) C E B D A (1) C D E B A (1) C B D E A (1) B E C D A (1) B E C A D (1) B D C E A (1) B D A C E (1) B A D C E (1) A E C D B (1) A E B D C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -4 14 16 -4 B 4 0 -2 12 -4 C -14 2 0 26 -6 D -16 -12 -26 0 -24 E 4 4 6 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 14 16 -4 B 4 0 -2 12 -4 C -14 2 0 26 -6 D -16 -12 -26 0 -24 E 4 4 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 B=27 E=21 C=12 D=11 so D is eliminated. Round 2 votes counts: A=31 B=28 E=21 C=20 so C is eliminated. Round 3 votes counts: B=42 A=31 E=27 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:219 A:211 B:205 C:204 D:161 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 14 16 -4 B 4 0 -2 12 -4 C -14 2 0 26 -6 D -16 -12 -26 0 -24 E 4 4 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 16 -4 B 4 0 -2 12 -4 C -14 2 0 26 -6 D -16 -12 -26 0 -24 E 4 4 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 16 -4 B 4 0 -2 12 -4 C -14 2 0 26 -6 D -16 -12 -26 0 -24 E 4 4 6 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999869 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9634: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) A C D B E (8) E B D C A (6) A D C B E (6) E C D B A (5) E B C D A (5) B E D C A (5) B E A D C (5) D C A B E (4) B A E D C (4) E B C A D (3) A B D C E (3) A B C D E (3) E C A D B (2) E B A C D (2) D C E A B (2) D A C B E (2) B E A C D (2) B A D C E (2) A C E D B (2) A C D E B (2) E D C B A (1) E D B C A (1) E C B D A (1) E A C D B (1) D C E B A (1) D C B A E (1) D C A E B (1) C D E A B (1) C D A B E (1) C A D E B (1) B E D A C (1) B D E C A (1) B A D E C (1) A E C D B (1) A E C B D (1) A D B C E (1) A B E D C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 6 0 6 10 B -6 0 -4 -4 6 C 0 4 0 2 0 D -6 4 -2 0 -2 E -10 -6 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.281690 B: 0.000000 C: 0.718310 D: 0.000000 E: 0.000000 Sum of squares = 0.595318923946 Cumulative probabilities = A: 0.281690 B: 0.281690 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 6 10 B -6 0 -4 -4 6 C 0 4 0 2 0 D -6 4 -2 0 -2 E -10 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=27 B=21 D=11 C=11 so D is eliminated. Round 2 votes counts: A=32 E=27 B=21 C=20 so C is eliminated. Round 3 votes counts: A=47 E=31 B=22 so B is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:203 D:197 B:196 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 6 10 B -6 0 -4 -4 6 C 0 4 0 2 0 D -6 4 -2 0 -2 E -10 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 6 10 B -6 0 -4 -4 6 C 0 4 0 2 0 D -6 4 -2 0 -2 E -10 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 6 10 B -6 0 -4 -4 6 C 0 4 0 2 0 D -6 4 -2 0 -2 E -10 -6 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9635: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (8) C E A B D (7) C D A B E (6) E B A C D (5) D C B E A (5) D B A E C (5) B E A D C (5) B A E D C (5) D B E A C (4) C D E B A (4) C A D E B (4) D C A B E (3) C E D B A (3) C E B A D (3) E A B C D (2) D A C B E (2) C E B D A (2) B A D E C (2) A E B C D (2) A B E D C (2) E C B D A (1) E C B A D (1) E B D C A (1) E B D A C (1) E B C A D (1) E A B D C (1) D E B C A (1) D C B A E (1) D B C E A (1) D B A C E (1) C D E A B (1) C D B E A (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D B E (1) B D E A C (1) A D C B E (1) A D B E C (1) A B E C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -24 0 10 -18 B 24 0 2 6 -2 C 0 -2 0 -4 -2 D -10 -6 4 0 -6 E 18 2 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999762 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -24 0 10 -18 B 24 0 2 6 -2 C 0 -2 0 -4 -2 D -10 -6 4 0 -6 E 18 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=23 E=21 B=13 A=8 so A is eliminated. Round 2 votes counts: C=35 D=25 E=23 B=17 so B is eliminated. Round 3 votes counts: E=36 C=35 D=29 so D is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:214 C:196 D:191 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -24 0 10 -18 B 24 0 2 6 -2 C 0 -2 0 -4 -2 D -10 -6 4 0 -6 E 18 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 0 10 -18 B 24 0 2 6 -2 C 0 -2 0 -4 -2 D -10 -6 4 0 -6 E 18 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 0 10 -18 B 24 0 2 6 -2 C 0 -2 0 -4 -2 D -10 -6 4 0 -6 E 18 2 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9636: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (9) B A C D E (8) E D C A B (7) C B A D E (7) C D E A B (6) D A B E C (4) C E D B A (4) B A E C D (4) B A C E D (4) A B D C E (4) E D A B C (3) D E C A B (3) D E A B C (3) D C E A B (3) C D A B E (3) A B D E C (3) E C D B A (2) E B A D C (2) C E D A B (2) A D B E C (2) E D A C B (1) E C D A B (1) E B A C D (1) E A B D C (1) D A E B C (1) C E B D A (1) C D E B A (1) C D A E B (1) C B E D A (1) C B E A D (1) C B A E D (1) C A B D E (1) B C A D E (1) B A D E C (1) B A D C E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 6 14 B -2 0 6 6 14 C -8 -6 0 4 4 D -6 -6 -4 0 10 E -14 -14 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999678 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 6 14 B -2 0 6 6 14 C -8 -6 0 4 4 D -6 -6 -4 0 10 E -14 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 B=28 E=18 D=14 A=11 so A is eliminated. Round 2 votes counts: B=36 C=30 E=18 D=16 so D is eliminated. Round 3 votes counts: B=42 C=33 E=25 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:212 C:197 D:197 E:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 8 6 14 B -2 0 6 6 14 C -8 -6 0 4 4 D -6 -6 -4 0 10 E -14 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 6 14 B -2 0 6 6 14 C -8 -6 0 4 4 D -6 -6 -4 0 10 E -14 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 6 14 B -2 0 6 6 14 C -8 -6 0 4 4 D -6 -6 -4 0 10 E -14 -14 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9637: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) B A D E C (7) A D E C B (6) A E C B D (5) C E D B A (4) B E C D A (4) E C B A D (3) E C A D B (3) E C A B D (3) D C B E A (3) B D C E A (3) B C E D A (3) B C D E A (3) A E B C D (3) A D B C E (3) E B C A D (2) E A C D B (2) D C E A B (2) D B C A E (2) D B A C E (2) B D C A E (2) B D A C E (2) B A E D C (2) A E D C B (2) A D B E C (2) A B D E C (2) E C B D A (1) E A C B D (1) D A C E B (1) D A C B E (1) D A B C E (1) C E D A B (1) C E B D A (1) C E A D B (1) C D E B A (1) B A D C E (1) A E B D C (1) A D E B C (1) A D C E B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 8 16 26 18 B -8 0 -4 2 -10 C -16 4 0 4 -24 D -26 -2 -4 0 -8 E -18 10 24 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 16 26 18 B -8 0 -4 2 -10 C -16 4 0 4 -24 D -26 -2 -4 0 -8 E -18 10 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=27 E=15 D=12 C=8 so C is eliminated. Round 2 votes counts: A=38 B=27 E=22 D=13 so D is eliminated. Round 3 votes counts: A=41 B=34 E=25 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:234 E:212 B:190 C:184 D:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 16 26 18 B -8 0 -4 2 -10 C -16 4 0 4 -24 D -26 -2 -4 0 -8 E -18 10 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 16 26 18 B -8 0 -4 2 -10 C -16 4 0 4 -24 D -26 -2 -4 0 -8 E -18 10 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 16 26 18 B -8 0 -4 2 -10 C -16 4 0 4 -24 D -26 -2 -4 0 -8 E -18 10 24 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999601 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9638: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (12) D E C A B (10) B A E C D (6) D C E A B (5) B A E D C (5) D C E B A (4) C A B E D (4) B A C D E (4) A B E C D (4) A B C E D (4) E D A C B (3) D E B A C (3) C D E A B (3) E D C A B (2) E A D B C (2) E A B D C (2) D E B C A (2) D B E C A (2) C E D A B (2) C D B A E (2) C B D A E (2) C B A D E (2) C A E D B (2) E D A B C (1) E C A D B (1) D E C B A (1) D B C E A (1) C D A B E (1) C A E B D (1) B D E A C (1) B D C A E (1) B C A D E (1) A E C D B (1) A E B C D (1) A C E B D (1) A C B E D (1) Total count = 100 A B C D E A 0 2 2 8 10 B -2 0 4 4 6 C -2 -4 0 10 6 D -8 -4 -10 0 -10 E -10 -6 -6 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 8 10 B -2 0 4 4 6 C -2 -4 0 10 6 D -8 -4 -10 0 -10 E -10 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 D=28 C=19 A=12 E=11 so E is eliminated. Round 2 votes counts: D=34 B=30 C=20 A=16 so A is eliminated. Round 3 votes counts: B=41 D=36 C=23 so C is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:211 B:206 C:205 E:194 D:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 8 10 B -2 0 4 4 6 C -2 -4 0 10 6 D -8 -4 -10 0 -10 E -10 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 8 10 B -2 0 4 4 6 C -2 -4 0 10 6 D -8 -4 -10 0 -10 E -10 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 8 10 B -2 0 4 4 6 C -2 -4 0 10 6 D -8 -4 -10 0 -10 E -10 -6 -6 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9639: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (12) C B E A D (10) B D A C E (7) A D E C B (6) D A E B C (5) D A B C E (5) D A B E C (4) A D C E B (4) E D B A C (3) E C A D B (3) E A D C B (3) C A E D B (3) B D E A C (3) B C D A E (3) E C A B D (2) E B D A C (2) E B C D A (2) D E A B C (2) C E B A D (2) C E A B D (2) B E C D A (2) B C D E A (2) B C A D E (2) A D C B E (2) E D A B C (1) D B A E C (1) C E A D B (1) C B A D E (1) C A B D E (1) B E D C A (1) B D A E C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 -8 2 -12 -6 B 8 0 16 10 16 C -2 -16 0 0 18 D 12 -10 0 0 2 E 6 -16 -18 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 2 -12 -6 B 8 0 16 10 16 C -2 -16 0 0 18 D 12 -10 0 0 2 E 6 -16 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=20 D=17 E=16 A=14 so A is eliminated. Round 2 votes counts: B=33 D=29 C=22 E=16 so E is eliminated. Round 3 votes counts: B=37 D=36 C=27 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:202 C:200 A:188 E:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 2 -12 -6 B 8 0 16 10 16 C -2 -16 0 0 18 D 12 -10 0 0 2 E 6 -16 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 2 -12 -6 B 8 0 16 10 16 C -2 -16 0 0 18 D 12 -10 0 0 2 E 6 -16 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 2 -12 -6 B 8 0 16 10 16 C -2 -16 0 0 18 D 12 -10 0 0 2 E 6 -16 -18 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9640: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (16) E D B C A (11) A C B D E (11) B D E C A (10) E D B A C (9) A C E D B (7) D E B C A (4) A C E B D (4) D B E C A (3) C A D E B (3) B D C E A (3) A E C D B (3) A E C B D (3) E D A B C (2) E B D A C (2) E B D C A (1) E A D B C (1) E A C D B (1) E A B D C (1) C D B E A (1) C B A D E (1) B A E D C (1) A E B D C (1) A C B E D (1) Total count = 100 A B C D E A 0 8 -6 8 2 B -8 0 -2 10 -6 C 6 2 0 2 -6 D -8 -10 -2 0 4 E -2 6 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755102058 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 A B C D E A 0 8 -6 8 2 B -8 0 -2 10 -6 C 6 2 0 2 -6 D -8 -10 -2 0 4 E -2 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 E=28 C=21 B=14 D=7 so D is eliminated. Round 2 votes counts: E=32 A=30 C=21 B=17 so B is eliminated. Round 3 votes counts: E=45 A=31 C=24 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 E:203 C:202 B:197 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -6 8 2 B -8 0 -2 10 -6 C 6 2 0 2 -6 D -8 -10 -2 0 4 E -2 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -6 8 2 B -8 0 -2 10 -6 C 6 2 0 2 -6 D -8 -10 -2 0 4 E -2 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -6 8 2 B -8 0 -2 10 -6 C 6 2 0 2 -6 D -8 -10 -2 0 4 E -2 6 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.000000 C: 0.142857 D: 0.000000 E: 0.428571 Sum of squares = 0.387755102064 Cumulative probabilities = A: 0.428571 B: 0.428571 C: 0.571429 D: 0.571429 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9641: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (9) E D C A B (8) C D E A B (7) A B E D C (7) C D E B A (5) B A E D C (5) B A E C D (5) B A C D E (5) A B D C E (5) A B C D E (5) E C D B A (3) D E C A B (3) E D A C B (2) E B A D C (2) E A D B C (2) C E D B A (2) C B E D A (2) B E C A D (2) B E A C D (2) B C E A D (2) A D C B E (2) E D B A C (1) E C D A B (1) E B D C A (1) E B D A C (1) E B C D A (1) D C A B E (1) C D B A E (1) C D A B E (1) C B A D E (1) C A B D E (1) B E A D C (1) B C A D E (1) A E D B C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 14 -4 2 -14 B -14 0 2 0 0 C 4 -2 0 -6 0 D -2 0 6 0 -2 E 14 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.419186 C: 0.000000 D: 0.000000 E: 0.580814 Sum of squares = 0.513061887205 Cumulative probabilities = A: 0.000000 B: 0.419186 C: 0.419186 D: 0.419186 E: 1.000000 A B C D E A 0 14 -4 2 -14 B -14 0 2 0 0 C 4 -2 0 -6 0 D -2 0 6 0 -2 E 14 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499209 C: 0.000000 D: 0.000000 E: 0.500791 Sum of squares = 0.50000125062 Cumulative probabilities = A: 0.000000 B: 0.499209 C: 0.499209 D: 0.499209 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=23 E=22 A=22 C=20 D=13 so D is eliminated. Round 2 votes counts: C=30 E=25 B=23 A=22 so A is eliminated. Round 3 votes counts: B=41 C=32 E=27 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:208 D:201 A:199 C:198 B:194 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 -4 2 -14 B -14 0 2 0 0 C 4 -2 0 -6 0 D -2 0 6 0 -2 E 14 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499209 C: 0.000000 D: 0.000000 E: 0.500791 Sum of squares = 0.50000125062 Cumulative probabilities = A: 0.000000 B: 0.499209 C: 0.499209 D: 0.499209 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -4 2 -14 B -14 0 2 0 0 C 4 -2 0 -6 0 D -2 0 6 0 -2 E 14 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499209 C: 0.000000 D: 0.000000 E: 0.500791 Sum of squares = 0.50000125062 Cumulative probabilities = A: 0.000000 B: 0.499209 C: 0.499209 D: 0.499209 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -4 2 -14 B -14 0 2 0 0 C 4 -2 0 -6 0 D -2 0 6 0 -2 E 14 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499209 C: 0.000000 D: 0.000000 E: 0.500791 Sum of squares = 0.50000125062 Cumulative probabilities = A: 0.000000 B: 0.499209 C: 0.499209 D: 0.499209 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9642: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (6) C D E A B (6) B A E D C (5) C D A E B (4) B E C A D (4) B E A C D (4) B C A D E (4) B A E C D (4) E B C D A (3) D C A B E (3) C D E B A (3) C D B A E (3) C D A B E (3) B E A D C (3) A D C B E (3) A B E D C (3) E C D A B (2) E B A C D (2) D C E A B (2) D A C B E (2) C B D E A (2) B C E D A (2) A E B D C (2) A D E B C (2) A B D E C (2) E D C A B (1) E C D B A (1) E C B D A (1) E B A D C (1) E A D C B (1) E A D B C (1) E A B D C (1) D A E C B (1) D A C E B (1) C E D B A (1) C E B D A (1) C D B E A (1) C B E D A (1) C B D A E (1) B E C D A (1) B C D A E (1) B C A E D (1) B A D C E (1) B A C D E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -4 -16 -6 10 B 4 0 0 2 12 C 16 0 0 14 6 D 6 -2 -14 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.639068 C: 0.360932 D: 0.000000 E: 0.000000 Sum of squares = 0.538679941891 Cumulative probabilities = A: 0.000000 B: 0.639068 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -16 -6 10 B 4 0 0 2 12 C 16 0 0 14 6 D 6 -2 -14 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=26 D=15 E=14 A=14 so E is eliminated. Round 2 votes counts: B=37 C=30 A=17 D=16 so D is eliminated. Round 3 votes counts: C=42 B=37 A=21 so A is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:218 B:209 D:198 A:192 E:183 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -16 -6 10 B 4 0 0 2 12 C 16 0 0 14 6 D 6 -2 -14 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -16 -6 10 B 4 0 0 2 12 C 16 0 0 14 6 D 6 -2 -14 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -16 -6 10 B 4 0 0 2 12 C 16 0 0 14 6 D 6 -2 -14 0 6 E -10 -12 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9643: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (20) B D A E C (20) B D C E A (6) C B D E A (5) A E D B C (5) A E C D B (5) D B A E C (3) C E A B D (3) B D C A E (3) A E D C B (3) E D A B C (2) C A E D B (2) B D E A C (2) B A D C E (2) A C E D B (2) E D A C B (1) E C A D B (1) E A D B C (1) E A C D B (1) D B E A C (1) C E D B A (1) C E B D A (1) C E B A D (1) C B A E D (1) C A E B D (1) C A B E D (1) B D E C A (1) A E C B D (1) A D B E C (1) A C E B D (1) A C B E D (1) A B D E C (1) Total count = 100 A B C D E A 0 6 6 8 6 B -6 0 -4 2 -4 C -6 4 0 -4 2 D -8 -2 4 0 -10 E -6 4 -2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 8 6 B -6 0 -4 2 -4 C -6 4 0 -4 2 D -8 -2 4 0 -10 E -6 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=34 A=20 E=6 D=4 so D is eliminated. Round 2 votes counts: B=38 C=36 A=20 E=6 so E is eliminated. Round 3 votes counts: B=38 C=37 A=25 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 E:203 C:198 B:194 D:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 8 6 B -6 0 -4 2 -4 C -6 4 0 -4 2 D -8 -2 4 0 -10 E -6 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 8 6 B -6 0 -4 2 -4 C -6 4 0 -4 2 D -8 -2 4 0 -10 E -6 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 8 6 B -6 0 -4 2 -4 C -6 4 0 -4 2 D -8 -2 4 0 -10 E -6 4 -2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9644: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (12) C A D B E (9) E B C A D (7) E B D A C (6) B E C D A (5) B C E D A (5) A D C E B (5) E B C D A (4) A D E B C (4) E B D C A (3) E C B A D (2) D B A E C (2) D A C B E (2) C B E D A (2) C B D A E (2) C B A E D (2) C A B D E (2) A D E C B (2) A C D E B (2) A C D B E (2) E D A B C (1) E B A D C (1) E A D B C (1) E A C D B (1) E A B C D (1) D E A B C (1) D C A B E (1) D B E A C (1) D A E B C (1) D A B E C (1) D A B C E (1) C E B A D (1) C D A B E (1) C B E A D (1) C B D E A (1) C A E B D (1) B E D C A (1) B C D A E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 6 -2 16 10 B -6 0 -4 -4 8 C 2 4 0 4 6 D -16 4 -4 0 6 E -10 -8 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999875 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -2 16 10 B -6 0 -4 -4 8 C 2 4 0 4 6 D -16 4 -4 0 6 E -10 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 E=27 C=22 B=12 D=10 so D is eliminated. Round 2 votes counts: A=34 E=28 C=23 B=15 so B is eliminated. Round 3 votes counts: A=36 E=35 C=29 so C is eliminated. Round 4 votes counts: A=55 E=45 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:208 B:197 D:195 E:185 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -2 16 10 B -6 0 -4 -4 8 C 2 4 0 4 6 D -16 4 -4 0 6 E -10 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -2 16 10 B -6 0 -4 -4 8 C 2 4 0 4 6 D -16 4 -4 0 6 E -10 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -2 16 10 B -6 0 -4 -4 8 C 2 4 0 4 6 D -16 4 -4 0 6 E -10 -8 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997165 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9645: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (11) C B E A D (8) C A E D B (6) B C E D A (5) E A C D B (4) D B A E C (4) C B A D E (4) B D E A C (4) D A B E C (3) C E A D B (3) C D A B E (3) C A D E B (3) B C E A D (3) A E D C B (3) D A E C B (2) C E B A D (2) C E A B D (2) C B D A E (2) C B A E D (2) C A E B D (2) B E D A C (2) B E A D C (2) B D C A E (2) B D A E C (2) B C D A E (2) A D E C B (2) E D A B C (1) E C A D B (1) E B A C D (1) E A D C B (1) E A C B D (1) D E B A C (1) C D A E B (1) B E D C A (1) B D C E A (1) B D A C E (1) A E C D B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 -6 4 14 B -2 0 -8 -2 2 C 6 8 0 12 6 D -4 2 -12 0 -2 E -14 -2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 4 14 B -2 0 -8 -2 2 C 6 8 0 12 6 D -4 2 -12 0 -2 E -14 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=25 D=21 E=9 A=7 so A is eliminated. Round 2 votes counts: C=38 B=25 D=24 E=13 so E is eliminated. Round 3 votes counts: C=45 D=29 B=26 so B is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:207 B:195 D:192 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 4 14 B -2 0 -8 -2 2 C 6 8 0 12 6 D -4 2 -12 0 -2 E -14 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 4 14 B -2 0 -8 -2 2 C 6 8 0 12 6 D -4 2 -12 0 -2 E -14 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 4 14 B -2 0 -8 -2 2 C 6 8 0 12 6 D -4 2 -12 0 -2 E -14 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9646: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (9) D E C A B (7) C E B A D (7) B A C E D (7) A B C E D (7) E C D B A (6) E C A B D (5) D E A C B (5) D A B E C (4) D A B C E (4) A B D C E (4) E C D A B (3) D E C B A (3) D A E C B (3) A E C B D (3) D C E B A (2) D B A C E (2) C E D B A (2) C E B D A (2) B D A C E (2) B C E A D (2) B A D C E (2) A D B E C (2) A B E C D (2) E C B D A (1) E C A D B (1) D B C E A (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 4 -2 6 -12 B -4 0 -18 10 -20 C 2 18 0 16 -12 D -6 -10 -16 0 -16 E 12 20 12 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999849 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 -2 6 -12 B -4 0 -18 10 -20 C 2 18 0 16 -12 D -6 -10 -16 0 -16 E 12 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=25 A=20 B=13 C=11 so C is eliminated. Round 2 votes counts: E=36 D=31 A=20 B=13 so B is eliminated. Round 3 votes counts: E=38 D=33 A=29 so A is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:230 C:212 A:198 B:184 D:176 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 -2 6 -12 B -4 0 -18 10 -20 C 2 18 0 16 -12 D -6 -10 -16 0 -16 E 12 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 -12 B -4 0 -18 10 -20 C 2 18 0 16 -12 D -6 -10 -16 0 -16 E 12 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 -12 B -4 0 -18 10 -20 C 2 18 0 16 -12 D -6 -10 -16 0 -16 E 12 20 12 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9647: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (8) C B E D A (6) E C B D A (5) D A B E C (5) C B E A D (5) B D E C A (5) D B A E C (4) B C E D A (4) A D C E B (4) A D B E C (4) D A E B C (3) C E B A D (3) C A E B D (3) A D E C B (3) A C E D B (3) D E C A B (2) D A E C B (2) C A B E D (2) B D A E C (2) B C A E D (2) A D E B C (2) A D C B E (2) A D B C E (2) A C D E B (2) E D C B A (1) E A C D B (1) D E C B A (1) D E B A C (1) D E A C B (1) D E A B C (1) D B E C A (1) C B A E D (1) B E D C A (1) B D C E A (1) B C E A D (1) B A C D E (1) A E C D B (1) A C B E D (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -4 -8 4 B 6 0 -16 8 6 C 4 16 0 2 8 D 8 -8 -2 0 2 E -4 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -8 4 B 6 0 -16 8 6 C 4 16 0 2 8 D 8 -8 -2 0 2 E -4 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=27 D=21 B=17 E=7 so E is eliminated. Round 2 votes counts: C=33 A=28 D=22 B=17 so B is eliminated. Round 3 votes counts: C=40 D=31 A=29 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:215 B:202 D:200 A:193 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -4 -8 4 B 6 0 -16 8 6 C 4 16 0 2 8 D 8 -8 -2 0 2 E -4 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -8 4 B 6 0 -16 8 6 C 4 16 0 2 8 D 8 -8 -2 0 2 E -4 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -8 4 B 6 0 -16 8 6 C 4 16 0 2 8 D 8 -8 -2 0 2 E -4 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998234 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9648: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) A B C D E (6) D E B C A (5) D E B A C (5) E D C A B (4) E C D A B (4) C A B E D (4) B D A C E (4) B C A D E (4) A C B E D (4) E D C B A (3) D E C B A (3) C B D E A (3) C B A D E (3) C A E B D (3) A B C E D (3) E D A C B (2) E D A B C (2) E A D B C (2) E A C D B (2) D E A B C (2) C E D B A (2) C B A E D (2) B A D C E (2) A C E B D (2) A C B D E (2) E D B A C (1) E C A D B (1) D B E C A (1) D B E A C (1) C E D A B (1) C E A D B (1) C E A B D (1) B D A E C (1) A E D B C (1) A E C D B (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 0 10 12 8 B 0 0 4 12 2 C -10 -4 0 20 14 D -12 -12 -20 0 6 E -8 -2 -14 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.515212 B: 0.484788 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.500462833368 Cumulative probabilities = A: 0.515212 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 10 12 8 B 0 0 4 12 2 C -10 -4 0 20 14 D -12 -12 -20 0 6 E -8 -2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=21 B=21 A=21 C=20 D=17 so D is eliminated. Round 2 votes counts: E=36 B=23 A=21 C=20 so C is eliminated. Round 3 votes counts: E=41 B=31 A=28 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:215 C:210 B:209 E:185 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 10 12 8 B 0 0 4 12 2 C -10 -4 0 20 14 D -12 -12 -20 0 6 E -8 -2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 10 12 8 B 0 0 4 12 2 C -10 -4 0 20 14 D -12 -12 -20 0 6 E -8 -2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 10 12 8 B 0 0 4 12 2 C -10 -4 0 20 14 D -12 -12 -20 0 6 E -8 -2 -14 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9649: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (14) B C A E D (11) E A D C B (8) D E A C B (8) A E D B C (8) C B D E A (6) B C D E A (6) D A E C B (4) A E B D C (4) E A D B C (3) D C E A B (3) C D B E A (3) B C E A D (3) B A E C D (3) B C E D A (2) B C A D E (2) A E D C B (2) E D A C B (1) E A B C D (1) D C E B A (1) D C B A E (1) C D E B A (1) C B D A E (1) B E A C D (1) A D E C B (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -10 -8 -2 6 B 10 0 20 12 10 C 8 -20 0 10 8 D 2 -12 -10 0 2 E -6 -10 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -2 6 B 10 0 20 12 10 C 8 -20 0 10 8 D 2 -12 -10 0 2 E -6 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=42 D=17 A=17 E=13 C=11 so C is eliminated. Round 2 votes counts: B=49 D=21 A=17 E=13 so E is eliminated. Round 3 votes counts: B=49 A=29 D=22 so D is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:203 A:193 D:191 E:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 -2 6 B 10 0 20 12 10 C 8 -20 0 10 8 D 2 -12 -10 0 2 E -6 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -2 6 B 10 0 20 12 10 C 8 -20 0 10 8 D 2 -12 -10 0 2 E -6 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -2 6 B 10 0 20 12 10 C 8 -20 0 10 8 D 2 -12 -10 0 2 E -6 -10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9650: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) A E B D C (8) A B E D C (8) D C A E B (7) C D B E A (7) D C E A B (6) C D E B A (4) B A C E D (4) C D B A E (3) C B D E A (3) B E A C D (3) B C E D A (3) B C D A E (3) B C A E D (3) E B C A D (2) D E C A B (2) D C E B A (2) D C A B E (2) B C E A D (2) A D B E C (2) A B D C E (2) E D C A B (1) E D A C B (1) E C D B A (1) E B C D A (1) E B A C D (1) E A D B C (1) E A B D C (1) C E B D A (1) C B E D A (1) B E C A D (1) B C A D E (1) B A C D E (1) A E D B C (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -10 -12 4 12 B 10 0 14 14 14 C 12 -14 0 6 12 D -4 -14 -6 0 -4 E -12 -14 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 4 12 B 10 0 14 14 14 C 12 -14 0 6 12 D -4 -14 -6 0 -4 E -12 -14 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 A=24 D=19 C=19 E=9 so E is eliminated. Round 2 votes counts: B=33 A=26 D=21 C=20 so C is eliminated. Round 3 votes counts: B=38 D=36 A=26 so A is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:208 A:197 D:186 E:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 4 12 B 10 0 14 14 14 C 12 -14 0 6 12 D -4 -14 -6 0 -4 E -12 -14 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 4 12 B 10 0 14 14 14 C 12 -14 0 6 12 D -4 -14 -6 0 -4 E -12 -14 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 4 12 B 10 0 14 14 14 C 12 -14 0 6 12 D -4 -14 -6 0 -4 E -12 -14 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9651: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (16) A E D C B (10) E A D C B (7) C D B A E (6) E A B D C (5) E B A C D (4) C D A B E (4) E B D C A (3) E A B C D (3) D C A B E (3) B C D E A (3) A D C E B (3) A B C D E (3) E D C A B (2) D C E A B (2) D C B E A (2) D C B A E (2) D C A E B (2) B E C D A (2) B E A C D (2) B A C D E (2) A B E C D (2) E D C B A (1) E D A C B (1) E B D A C (1) E A D B C (1) D E C A B (1) B C E D A (1) B A E C D (1) A E C D B (1) A E B C D (1) A D C B E (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 0 -4 18 B -8 0 0 -2 2 C 0 0 0 6 4 D 4 2 -6 0 4 E -18 -2 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.384266 B: 0.000000 C: 0.615734 D: 0.000000 E: 0.000000 Sum of squares = 0.526788597498 Cumulative probabilities = A: 0.384266 B: 0.384266 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 0 -4 18 B -8 0 0 -2 2 C 0 0 0 6 4 D 4 2 -6 0 4 E -18 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 A=23 D=12 C=10 so C is eliminated. Round 2 votes counts: E=28 B=27 A=23 D=22 so D is eliminated. Round 3 votes counts: B=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:211 C:205 D:202 B:196 E:186 Borda winner is A compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 0 -4 18 B -8 0 0 -2 2 C 0 0 0 6 4 D 4 2 -6 0 4 E -18 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -4 18 B -8 0 0 -2 2 C 0 0 0 6 4 D 4 2 -6 0 4 E -18 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -4 18 B -8 0 0 -2 2 C 0 0 0 6 4 D 4 2 -6 0 4 E -18 -2 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999991 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9652: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) D A B E C (6) E C B A D (5) D A B C E (5) C B E A D (5) B A D C E (5) E C A B D (4) C E B D A (4) E C D A B (3) E C A D B (3) D C E B A (3) D A E B C (3) C E D B A (3) A B E D C (3) A B D E C (3) E D A C B (2) E A B C D (2) D C B A E (2) D B C A E (2) B D A C E (2) B C D A E (2) B A C E D (2) B A C D E (2) A D B E C (2) E D C A B (1) E B A C D (1) E A D B C (1) E A C D B (1) E A C B D (1) D E C A B (1) D E A C B (1) D C E A B (1) D B A C E (1) C D E B A (1) C B E D A (1) C B D E A (1) B C E A D (1) B C A E D (1) B C A D E (1) A E D B C (1) A D B C E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -6 10 -8 B 6 0 -2 12 0 C 6 2 0 8 10 D -10 -12 -8 0 -10 E 8 0 -10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 10 -8 B 6 0 -2 12 0 C 6 2 0 8 10 D -10 -12 -8 0 -10 E 8 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 E=24 C=23 B=16 A=12 so A is eliminated. Round 2 votes counts: D=28 E=25 B=24 C=23 so C is eliminated. Round 3 votes counts: E=40 B=31 D=29 so D is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:213 B:208 E:204 A:195 D:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 10 -8 B 6 0 -2 12 0 C 6 2 0 8 10 D -10 -12 -8 0 -10 E 8 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 10 -8 B 6 0 -2 12 0 C 6 2 0 8 10 D -10 -12 -8 0 -10 E 8 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 10 -8 B 6 0 -2 12 0 C 6 2 0 8 10 D -10 -12 -8 0 -10 E 8 0 -10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9653: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (9) D E A C B (7) C B A E D (6) E A D C B (5) B C A E D (5) D E A B C (4) D A E C B (4) C B A D E (4) B C D A E (4) A E C D B (4) E D A C B (3) E D A B C (3) E A D B C (3) D A C E B (3) C D B A E (3) C B D A E (3) B C E D A (3) B C D E A (3) B C A D E (3) A D E C B (3) C A B D E (2) B E C A D (2) B D E C A (2) A E D C B (2) E D B A C (1) E B A C D (1) E A B D C (1) D E B A C (1) D C A B E (1) D B E C A (1) D B C E A (1) B E D C A (1) B E D A C (1) A C E B D (1) Total count = 100 A B C D E A 0 -8 -6 2 -4 B 8 0 -2 2 8 C 6 2 0 6 2 D -2 -2 -6 0 -2 E 4 -8 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 2 -4 B 8 0 -2 2 8 C 6 2 0 6 2 D -2 -2 -6 0 -2 E 4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=22 C=18 E=17 A=10 so A is eliminated. Round 2 votes counts: B=33 D=25 E=23 C=19 so C is eliminated. Round 3 votes counts: B=48 D=28 E=24 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:208 C:208 E:198 D:194 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 2 -4 B 8 0 -2 2 8 C 6 2 0 6 2 D -2 -2 -6 0 -2 E 4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 2 -4 B 8 0 -2 2 8 C 6 2 0 6 2 D -2 -2 -6 0 -2 E 4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 2 -4 B 8 0 -2 2 8 C 6 2 0 6 2 D -2 -2 -6 0 -2 E 4 -8 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9654: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) E D C B A (7) D E A C B (7) A B C D E (7) B C A E D (6) E D B C A (4) C A B D E (4) B C E A D (4) E C D B A (3) E C B D A (3) D E C A B (3) A D C B E (3) E B D C A (2) D E A B C (2) D A E C B (2) D A E B C (2) D A C E B (2) C E D B A (2) C D E A B (2) C B E D A (2) C B A D E (2) A D B E C (2) A C D B E (2) A C B D E (2) A B D E C (2) E B D A C (1) E B C D A (1) D C E A B (1) D C A E B (1) C E B D A (1) C D A E B (1) C B A E D (1) C A D E B (1) C A D B E (1) B E D A C (1) B E C D A (1) B C E D A (1) B A C D E (1) A D E B C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 -8 -4 4 B 2 0 -6 2 2 C 8 6 0 12 12 D 4 -2 -12 0 4 E -4 -2 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -4 4 B 2 0 -6 2 2 C 8 6 0 12 12 D 4 -2 -12 0 4 E -4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 E=21 D=20 A=20 C=17 so C is eliminated. Round 2 votes counts: B=27 A=26 E=24 D=23 so D is eliminated. Round 3 votes counts: E=39 A=34 B=27 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:219 B:200 D:197 A:195 E:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 -4 4 B 2 0 -6 2 2 C 8 6 0 12 12 D 4 -2 -12 0 4 E -4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -4 4 B 2 0 -6 2 2 C 8 6 0 12 12 D 4 -2 -12 0 4 E -4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -4 4 B 2 0 -6 2 2 C 8 6 0 12 12 D 4 -2 -12 0 4 E -4 -2 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9655: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (13) A D E C B (9) B C E A D (8) D A B C E (7) B C E D A (7) E C B A D (4) B C D E A (4) A D E B C (4) D E C B A (3) C E B D A (3) A E D C B (3) A E C B D (3) A E B C D (3) A D B C E (3) E C B D A (2) D C E B A (2) D B C A E (2) C B E D A (2) B D C E A (2) B A C E D (2) A B E C D (2) E C D B A (1) D E C A B (1) D E A C B (1) D C B E A (1) D B A C E (1) B D C A E (1) B A C D E (1) A E C D B (1) A E B D C (1) A D B E C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 8 14 -6 18 B -8 0 2 -6 -8 C -14 -2 0 -12 -4 D 6 6 12 0 14 E -18 8 4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 14 -6 18 B -8 0 2 -6 -8 C -14 -2 0 -12 -4 D 6 6 12 0 14 E -18 8 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=31 B=25 E=7 C=5 so C is eliminated. Round 2 votes counts: A=32 D=31 B=27 E=10 so E is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:219 A:217 B:190 E:190 C:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 14 -6 18 B -8 0 2 -6 -8 C -14 -2 0 -12 -4 D 6 6 12 0 14 E -18 8 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 14 -6 18 B -8 0 2 -6 -8 C -14 -2 0 -12 -4 D 6 6 12 0 14 E -18 8 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 14 -6 18 B -8 0 2 -6 -8 C -14 -2 0 -12 -4 D 6 6 12 0 14 E -18 8 4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9656: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (7) B C A D E (7) D A B C E (6) C E B D A (5) E C B A D (4) E B C A D (4) E A D B C (4) C B A D E (4) B C A E D (4) A B D C E (4) D C A B E (3) D A C B E (3) C E D B A (3) C D B E A (3) E C D B A (2) E C B D A (2) D C B A E (2) D C A E B (2) C E B A D (2) C D E B A (2) C B E D A (2) C B E A D (2) C B D A E (2) A E B D C (2) A D B E C (2) A B E D C (2) E D C A B (1) E D A C B (1) E D A B C (1) E B A C D (1) E A B C D (1) D E C A B (1) D A B E C (1) C D E A B (1) C D B A E (1) B E C A D (1) B C E A D (1) B A E D C (1) B A E C D (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -22 12 -2 B 12 0 4 18 6 C 22 -4 0 10 20 D -12 -18 -10 0 -10 E 2 -6 -20 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -22 12 -2 B 12 0 4 18 6 C 22 -4 0 10 20 D -12 -18 -10 0 -10 E 2 -6 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 C=27 D=18 B=15 A=12 so A is eliminated. Round 2 votes counts: E=31 C=27 D=21 B=21 so D is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: C=60 E=40 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:224 B:220 E:193 A:188 D:175 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -22 12 -2 B 12 0 4 18 6 C 22 -4 0 10 20 D -12 -18 -10 0 -10 E 2 -6 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -22 12 -2 B 12 0 4 18 6 C 22 -4 0 10 20 D -12 -18 -10 0 -10 E 2 -6 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -22 12 -2 B 12 0 4 18 6 C 22 -4 0 10 20 D -12 -18 -10 0 -10 E 2 -6 -20 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999567 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9657: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (9) D A B E C (8) A D B C E (8) D A E B C (7) C B E A D (7) E C B D A (5) B C A D E (5) E D A B C (4) B A D C E (4) A B D C E (4) E D C A B (3) E C D B A (3) E D B A C (2) E C D A B (2) E B D A C (2) E B C D A (2) C E B D A (2) C B A E D (2) C B A D E (2) B C E A D (2) A C E D B (2) A C B D E (2) E B D C A (1) D E A C B (1) D E A B C (1) D B A E C (1) C E B A D (1) C E A B D (1) C B E D A (1) B D E A C (1) B D A E C (1) B C E D A (1) A D C E B (1) A D C B E (1) A B C D E (1) Total count = 100 A B C D E A 0 10 20 -14 -2 B -10 0 10 -6 2 C -20 -10 0 -18 -6 D 14 6 18 0 -4 E 2 -2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.085468 B: 0.219376 C: 0.000000 D: 0.066954 E: 0.628202 Sum of squares = 0.454551243275 Cumulative probabilities = A: 0.085468 B: 0.304844 C: 0.304844 D: 0.371798 E: 1.000000 A B C D E A 0 10 20 -14 -2 B -10 0 10 -6 2 C -20 -10 0 -18 -6 D 14 6 18 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166666 E: 0.500000 Sum of squares = 0.388888932684 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=19 D=18 C=16 B=14 so B is eliminated. Round 2 votes counts: E=33 C=24 A=23 D=20 so D is eliminated. Round 3 votes counts: A=40 E=36 C=24 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:217 A:207 E:205 B:198 C:173 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 10 20 -14 -2 B -10 0 10 -6 2 C -20 -10 0 -18 -6 D 14 6 18 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166666 E: 0.500000 Sum of squares = 0.388888932684 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 20 -14 -2 B -10 0 10 -6 2 C -20 -10 0 -18 -6 D 14 6 18 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166666 E: 0.500000 Sum of squares = 0.388888932684 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 20 -14 -2 B -10 0 10 -6 2 C -20 -10 0 -18 -6 D 14 6 18 0 -4 E 2 -2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.166666 E: 0.500000 Sum of squares = 0.388888932684 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9658: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (6) C B D A E (6) E B A C D (5) E A B C D (5) B C D A E (5) D C A B E (4) B C A D E (4) A C D B E (4) D A C E B (3) B E C D A (3) B E C A D (3) A D C E B (3) E D A B C (2) E B D C A (2) E B D A C (2) E B A D C (2) E A D B C (2) E A B D C (2) D E C A B (2) D E A C B (2) D C A E B (2) D B C E A (2) D A E C B (2) C B A D E (2) C A D B E (2) B C D E A (2) A E D C B (2) A C E B D (2) A C B D E (2) E D B C A (1) E D B A C (1) E D A C B (1) D E B C A (1) D C E B A (1) D C B E A (1) C A B D E (1) B D C E A (1) B C E D A (1) B A E C D (1) B A C E D (1) A E C D B (1) A E C B D (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 6 8 6 0 B -6 0 -4 6 -10 C -8 4 0 4 0 D -6 -6 -4 0 6 E 0 10 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.654441 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.345559 Sum of squares = 0.547703925966 Cumulative probabilities = A: 0.654441 B: 0.654441 C: 0.654441 D: 0.654441 E: 1.000000 A B C D E A 0 6 8 6 0 B -6 0 -4 6 -10 C -8 4 0 4 0 D -6 -6 -4 0 6 E 0 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500129 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499871 Sum of squares = 0.500000033375 Cumulative probabilities = A: 0.500129 B: 0.500129 C: 0.500129 D: 0.500129 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=21 D=20 A=17 C=11 so C is eliminated. Round 2 votes counts: E=31 B=29 D=20 A=20 so D is eliminated. Round 3 votes counts: E=37 B=32 A=31 so A is eliminated. Round 4 votes counts: E=55 B=45 so B is eliminated. IRV winner is E compare: Computing Borda winner. A:210 E:202 C:200 D:195 B:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 6 0 B -6 0 -4 6 -10 C -8 4 0 4 0 D -6 -6 -4 0 6 E 0 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500129 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499871 Sum of squares = 0.500000033375 Cumulative probabilities = A: 0.500129 B: 0.500129 C: 0.500129 D: 0.500129 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 6 0 B -6 0 -4 6 -10 C -8 4 0 4 0 D -6 -6 -4 0 6 E 0 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500129 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499871 Sum of squares = 0.500000033375 Cumulative probabilities = A: 0.500129 B: 0.500129 C: 0.500129 D: 0.500129 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 6 0 B -6 0 -4 6 -10 C -8 4 0 4 0 D -6 -6 -4 0 6 E 0 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.500129 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.499871 Sum of squares = 0.500000033375 Cumulative probabilities = A: 0.500129 B: 0.500129 C: 0.500129 D: 0.500129 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9659: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (6) C A B E D (5) E C B D A (4) D B E C A (4) A D E C B (4) D E B A C (3) D B E A C (3) D A B E C (3) C E B A D (3) C B A E D (3) C A E B D (3) B D A C E (3) B C A E D (3) A C E D B (3) E D C B A (2) E D C A B (2) E D B C A (2) E D A C B (2) D E B C A (2) D E A B C (2) D B A E C (2) D A E C B (2) C B E A D (2) B E D C A (2) B E C D A (2) B D C A E (2) B C E D A (2) A D C E B (2) A D B C E (2) A B D C E (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A B D (1) E B C D A (1) D E A C B (1) D A E B C (1) C B E D A (1) C A E D B (1) B D E C A (1) B D C E A (1) B C E A D (1) B A D C E (1) B A C D E (1) A E C D B (1) A D B E C (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 -6 -4 6 B 6 0 -4 6 10 C 6 4 0 -4 -2 D 4 -6 4 0 -10 E -6 -10 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.468749999905 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -6 -6 -4 6 B 6 0 -4 6 10 C 6 4 0 -4 -2 D 4 -6 4 0 -10 E -6 -10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000118 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=23 A=23 B=19 C=18 E=17 so E is eliminated. Round 2 votes counts: D=31 C=26 A=23 B=20 so B is eliminated. Round 3 votes counts: D=40 C=35 A=25 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:209 C:202 E:198 D:196 A:195 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 6 B 6 0 -4 6 10 C 6 4 0 -4 -2 D 4 -6 4 0 -10 E -6 -10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000118 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 6 B 6 0 -4 6 10 C 6 4 0 -4 -2 D 4 -6 4 0 -10 E -6 -10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000118 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 6 B 6 0 -4 6 10 C 6 4 0 -4 -2 D 4 -6 4 0 -10 E -6 -10 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.125000 C: 0.625000 D: 0.000000 E: 0.250000 Sum of squares = 0.468750000118 Cumulative probabilities = A: 0.000000 B: 0.125000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9660: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (12) A D B C E (7) A B D E C (6) D A B E C (5) C E D B A (5) C E B D A (5) E C B D A (4) B A D E C (4) A C E B D (4) A B E C D (4) B E C D A (3) B E C A D (3) A D C E B (3) A C B E D (3) E B C D A (2) D E C B A (2) D A C E B (2) C E B A D (2) C A E D B (2) B E D C A (2) A D C B E (2) D E B C A (1) D B E C A (1) D B A E C (1) D A B C E (1) C E D A B (1) C E A D B (1) C E A B D (1) C A E B D (1) B E A C D (1) B D E C A (1) B D A E C (1) B A E D C (1) B A E C D (1) A C E D B (1) A C D E B (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 20 26 26 30 B -20 0 20 4 24 C -26 -20 0 -6 -12 D -26 -4 6 0 2 E -30 -24 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 26 26 30 B -20 0 20 4 24 C -26 -20 0 -6 -12 D -26 -4 6 0 2 E -30 -24 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=46 C=18 B=17 D=13 E=6 so E is eliminated. Round 2 votes counts: A=46 C=22 B=19 D=13 so D is eliminated. Round 3 votes counts: A=54 C=24 B=22 so B is eliminated. Round 4 votes counts: A=63 C=37 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:251 B:214 D:189 E:178 C:168 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 26 26 30 B -20 0 20 4 24 C -26 -20 0 -6 -12 D -26 -4 6 0 2 E -30 -24 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 26 26 30 B -20 0 20 4 24 C -26 -20 0 -6 -12 D -26 -4 6 0 2 E -30 -24 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 26 26 30 B -20 0 20 4 24 C -26 -20 0 -6 -12 D -26 -4 6 0 2 E -30 -24 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9661: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (12) D C A B E (11) A C D E B (11) E B A C D (7) B D E C A (6) C A D B E (5) E B C A D (4) D C B A E (4) A E C D B (4) E A C B D (3) D B C A E (3) C D A B E (3) B E C D A (3) A D C E B (3) E B D A C (2) E B A D C (2) C A E D B (2) E D A B C (1) E C B A D (1) E C A B D (1) E A C D B (1) E A B D C (1) E A B C D (1) D E B A C (1) D A C B E (1) C E A B D (1) C A D E B (1) B D E A C (1) B D C E A (1) B C D E A (1) B C D A E (1) A C E D B (1) Total count = 100 A B C D E A 0 2 -20 -2 0 B -2 0 -6 -4 4 C 20 6 0 2 -2 D 2 4 -2 0 6 E 0 -4 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 A B C D E A 0 2 -20 -2 0 B -2 0 -6 -4 4 C 20 6 0 2 -2 D 2 4 -2 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=25 E=24 D=20 A=19 C=12 so C is eliminated. Round 2 votes counts: A=27 E=25 B=25 D=23 so D is eliminated. Round 3 votes counts: A=42 B=32 E=26 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:213 D:205 B:196 E:196 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -20 -2 0 B -2 0 -6 -4 4 C 20 6 0 2 -2 D 2 4 -2 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -20 -2 0 B -2 0 -6 -4 4 C 20 6 0 2 -2 D 2 4 -2 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -20 -2 0 B -2 0 -6 -4 4 C 20 6 0 2 -2 D 2 4 -2 0 6 E 0 -4 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.200000 E: 0.200000 Sum of squares = 0.439999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.600000 D: 0.800000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9662: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (5) E B D C A (4) D B A E C (4) D A B E C (4) C E B A D (4) C E A D B (4) C E A B D (4) A D B C E (4) A C D B E (4) E C B D A (3) D A E B C (3) C E D A B (3) C A E D B (3) B E D C A (3) A C D E B (3) E C D B A (2) E C D A B (2) E B C D A (2) D A B C E (2) C E D B A (2) C A E B D (2) B E D A C (2) B E C A D (2) B D E A C (2) B D A E C (2) B A D E C (2) B A C E D (2) A D C E B (2) A C B D E (2) E D B C A (1) E D B A C (1) E C B A D (1) D E C A B (1) D E B A C (1) D C A E B (1) D B E A C (1) D A C E B (1) D A C B E (1) C D E A B (1) C B E A D (1) C B A E D (1) C A D E B (1) C A B E D (1) B E C D A (1) B C E A D (1) B A D C E (1) Total count = 100 A B C D E A 0 8 -2 0 2 B -8 0 -10 -14 -4 C 2 10 0 4 12 D 0 14 -4 0 -4 E -2 4 -12 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 0 2 B -8 0 -10 -14 -4 C 2 10 0 4 12 D 0 14 -4 0 -4 E -2 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 A=20 D=19 B=18 E=16 so E is eliminated. Round 2 votes counts: C=35 B=24 D=21 A=20 so A is eliminated. Round 3 votes counts: C=44 D=32 B=24 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 A:204 D:203 E:197 B:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 0 2 B -8 0 -10 -14 -4 C 2 10 0 4 12 D 0 14 -4 0 -4 E -2 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 0 2 B -8 0 -10 -14 -4 C 2 10 0 4 12 D 0 14 -4 0 -4 E -2 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 0 2 B -8 0 -10 -14 -4 C 2 10 0 4 12 D 0 14 -4 0 -4 E -2 4 -12 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999079 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9663: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (14) D B E A C (10) B D E A C (9) A C E D B (8) C A D E B (7) C A E D B (5) B E D C A (4) D A C B E (3) D A B E C (3) B E D A C (3) A C E B D (3) E C A B D (2) E B A D C (2) E A C B D (2) D E B A C (2) D B E C A (2) D A B C E (2) C E A B D (2) C D A B E (2) E C B A D (1) E B D C A (1) E B D A C (1) E B C D A (1) E B C A D (1) E A D B C (1) D B A E C (1) D A C E B (1) C E B A D (1) C B E A D (1) B E C D A (1) A D E C B (1) A D E B C (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 18 10 8 6 B -18 0 -10 -2 -16 C -10 10 0 4 2 D -8 2 -4 0 -8 E -6 16 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 10 8 6 B -18 0 -10 -2 -16 C -10 10 0 4 2 D -8 2 -4 0 -8 E -6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=24 B=17 A=15 E=12 so E is eliminated. Round 2 votes counts: C=35 D=24 B=23 A=18 so A is eliminated. Round 3 votes counts: C=49 D=28 B=23 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:221 E:208 C:203 D:191 B:177 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 10 8 6 B -18 0 -10 -2 -16 C -10 10 0 4 2 D -8 2 -4 0 -8 E -6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 8 6 B -18 0 -10 -2 -16 C -10 10 0 4 2 D -8 2 -4 0 -8 E -6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 8 6 B -18 0 -10 -2 -16 C -10 10 0 4 2 D -8 2 -4 0 -8 E -6 16 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9664: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (6) E D A C B (4) E C D A B (4) E A D B C (4) C D E B A (4) B A D C E (4) A B E C D (4) A B D C E (4) E D C A B (3) E C A B D (3) E A C B D (3) E A B C D (3) D A E B C (3) C D B E A (3) B D A C E (3) A B D E C (3) E C B A D (2) E A B D C (2) D C B E A (2) D C B A E (2) D B C A E (2) C E D B A (2) C E B A D (2) C B E A D (2) C B D A E (2) B C D A E (2) B A C E D (2) A D B E C (2) A B E D C (2) E D A B C (1) E C D B A (1) E C B D A (1) E C A D B (1) D E A B C (1) D C E B A (1) D B A C E (1) D A B E C (1) D A B C E (1) C D B A E (1) C B D E A (1) B E A C D (1) B C A D E (1) B A C D E (1) A E D B C (1) A E B C D (1) Total count = 100 A B C D E A 0 14 16 8 -2 B -14 0 12 10 -6 C -16 -12 0 -6 -14 D -8 -10 6 0 -10 E 2 6 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999151 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 16 8 -2 B -14 0 12 10 -6 C -16 -12 0 -6 -14 D -8 -10 6 0 -10 E 2 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=23 C=17 D=14 B=14 so D is eliminated. Round 2 votes counts: E=33 A=28 C=22 B=17 so B is eliminated. Round 3 votes counts: A=39 E=34 C=27 so C is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. A:218 E:216 B:201 D:189 C:176 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 16 8 -2 B -14 0 12 10 -6 C -16 -12 0 -6 -14 D -8 -10 6 0 -10 E 2 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 16 8 -2 B -14 0 12 10 -6 C -16 -12 0 -6 -14 D -8 -10 6 0 -10 E 2 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 16 8 -2 B -14 0 12 10 -6 C -16 -12 0 -6 -14 D -8 -10 6 0 -10 E 2 6 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999611 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9665: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B E D (12) B E D C A (9) A C D E B (9) E B D A C (6) D E B A C (6) A C B E D (6) D E B C A (5) C B A E D (4) C A D E B (4) C A B D E (4) D C E A B (3) D C A E B (3) C A D B E (3) B E D A C (3) D A C E B (2) C D A E B (2) B E A C D (2) B C A E D (2) E D B A C (1) E B D C A (1) D E C A B (1) D E A C B (1) D A E C B (1) D A E B C (1) B E C D A (1) B E C A D (1) B E A D C (1) B D E C A (1) B C E A D (1) B A E C D (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 10 -14 6 14 B -10 0 -16 14 6 C 14 16 0 10 16 D -6 -14 -10 0 -4 E -14 -6 -16 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -14 6 14 B -10 0 -16 14 6 C 14 16 0 10 16 D -6 -14 -10 0 -4 E -14 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=23 B=22 A=18 E=8 so E is eliminated. Round 2 votes counts: C=29 B=29 D=24 A=18 so A is eliminated. Round 3 votes counts: C=47 B=29 D=24 so D is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:228 A:208 B:197 E:184 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -14 6 14 B -10 0 -16 14 6 C 14 16 0 10 16 D -6 -14 -10 0 -4 E -14 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -14 6 14 B -10 0 -16 14 6 C 14 16 0 10 16 D -6 -14 -10 0 -4 E -14 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -14 6 14 B -10 0 -16 14 6 C 14 16 0 10 16 D -6 -14 -10 0 -4 E -14 -6 -16 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9666: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (9) A D E B C (9) E D B A C (8) E D A B C (8) D E A B C (6) C B A E D (5) C B E D A (4) C A B D E (4) B E D C A (4) C A B E D (3) B C E D A (3) B C A D E (3) A D E C B (3) E D B C A (2) E D A C B (2) E C D B A (2) E C D A B (2) E B D C A (2) B E D A C (2) B A D E C (2) A D B E C (2) A C D E B (2) A C B D E (2) E D C A B (1) D E B A C (1) D A E B C (1) C E D B A (1) C E D A B (1) C B E A D (1) C A E D B (1) B A C D E (1) A E D C B (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 4 0 0 B 0 0 12 -6 -6 C -4 -12 0 -12 -18 D 0 6 12 0 -6 E 0 6 18 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.548357 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.451643 Sum of squares = 0.504676770865 Cumulative probabilities = A: 0.548357 B: 0.548357 C: 0.548357 D: 0.548357 E: 1.000000 A B C D E A 0 0 4 0 0 B 0 0 12 -6 -6 C -4 -12 0 -12 -18 D 0 6 12 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 E=27 A=21 B=15 D=8 so D is eliminated. Round 2 votes counts: E=34 C=29 A=22 B=15 so B is eliminated. Round 3 votes counts: E=40 C=35 A=25 so A is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 D:206 A:202 B:200 C:177 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 0 0 B 0 0 12 -6 -6 C -4 -12 0 -12 -18 D 0 6 12 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 0 0 B 0 0 12 -6 -6 C -4 -12 0 -12 -18 D 0 6 12 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 0 0 B 0 0 12 -6 -6 C -4 -12 0 -12 -18 D 0 6 12 0 -6 E 0 6 18 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999856 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9667: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (23) A E D B C (11) E A C B D (10) D B C A E (10) E A C D B (7) D B A C E (6) E C A B D (4) A D E B C (4) C E B D A (3) C B E D A (3) B C D E A (3) A D B E C (3) C E B A D (2) C E A B D (2) B D C A E (2) A E D C B (2) D B A E C (1) D A B E C (1) B D C E A (1) B C D A E (1) A E C D B (1) Total count = 100 A B C D E A 0 -10 -8 -8 -16 B 10 0 -14 8 8 C 8 14 0 18 12 D 8 -8 -18 0 10 E 16 -8 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 -8 -16 B 10 0 -14 8 8 C 8 14 0 18 12 D 8 -8 -18 0 10 E 16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=21 A=21 D=18 B=7 so B is eliminated. Round 2 votes counts: C=37 E=21 D=21 A=21 so E is eliminated. Round 3 votes counts: C=41 A=38 D=21 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:226 B:206 D:196 E:193 A:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -8 -8 -16 B 10 0 -14 8 8 C 8 14 0 18 12 D 8 -8 -18 0 10 E 16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 -8 -16 B 10 0 -14 8 8 C 8 14 0 18 12 D 8 -8 -18 0 10 E 16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 -8 -16 B 10 0 -14 8 8 C 8 14 0 18 12 D 8 -8 -18 0 10 E 16 -8 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999568 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9668: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (8) C B E D A (6) E B C D A (4) D A C E B (4) B E C A D (4) A D C B E (4) E D C B A (3) E D A C B (3) C B D E A (3) B A C E D (3) A E D B C (3) A E B D C (3) A B E D C (3) E C D B A (2) E C B D A (2) E B A C D (2) E A B D C (2) D C E A B (2) D C A B E (2) D A E C B (2) C D E B A (2) C D B A E (2) B E C D A (2) B E A C D (2) B C E A D (2) B A E C D (2) A D E C B (2) A D C E B (2) A B C D E (2) E D C A B (1) E B C A D (1) E A D B C (1) D E C B A (1) D E C A B (1) D C A E B (1) C E B D A (1) C B D A E (1) C A D B E (1) B C D A E (1) B C A D E (1) B A C D E (1) A D B E C (1) A D B C E (1) A C D B E (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -12 -8 -8 -10 B 12 0 2 16 10 C 8 -2 0 12 4 D 8 -16 -12 0 -20 E 10 -10 -4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 -8 -10 B 12 0 2 16 10 C 8 -2 0 12 4 D 8 -16 -12 0 -20 E 10 -10 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=26 A=24 E=21 C=16 D=13 so D is eliminated. Round 2 votes counts: A=30 B=26 E=23 C=21 so C is eliminated. Round 3 votes counts: B=38 A=34 E=28 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:220 C:211 E:208 A:181 D:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 -8 -10 B 12 0 2 16 10 C 8 -2 0 12 4 D 8 -16 -12 0 -20 E 10 -10 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -8 -10 B 12 0 2 16 10 C 8 -2 0 12 4 D 8 -16 -12 0 -20 E 10 -10 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -8 -10 B 12 0 2 16 10 C 8 -2 0 12 4 D 8 -16 -12 0 -20 E 10 -10 -4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999978522 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9669: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (12) B D A E C (9) A E D B C (9) C B D E A (8) E A C D B (5) B D C A E (5) B D A C E (5) C E A B D (4) C B E D A (4) B C D E A (4) E A D B C (3) D A E B C (3) B C D A E (3) A E D C B (3) A E C D B (3) A D B E C (3) E C A D B (2) D B A E C (2) D A B E C (2) C B E A D (2) A D E B C (2) E A D C B (1) D B E A C (1) C E D B A (1) C E B D A (1) C E B A D (1) B D E C A (1) B A D E C (1) Total count = 100 A B C D E A 0 4 4 2 0 B -4 0 6 -4 0 C -4 -6 0 0 0 D -2 4 0 0 -2 E 0 0 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.467796 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.532204 Sum of squares = 0.502074132103 Cumulative probabilities = A: 0.467796 B: 0.467796 C: 0.467796 D: 0.467796 E: 1.000000 A B C D E A 0 4 4 2 0 B -4 0 6 -4 0 C -4 -6 0 0 0 D -2 4 0 0 -2 E 0 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=28 A=20 E=11 D=8 so D is eliminated. Round 2 votes counts: C=33 B=31 A=25 E=11 so E is eliminated. Round 3 votes counts: C=35 A=34 B=31 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:205 E:201 D:200 B:199 C:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 4 4 2 0 B -4 0 6 -4 0 C -4 -6 0 0 0 D -2 4 0 0 -2 E 0 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 4 2 0 B -4 0 6 -4 0 C -4 -6 0 0 0 D -2 4 0 0 -2 E 0 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 4 2 0 B -4 0 6 -4 0 C -4 -6 0 0 0 D -2 4 0 0 -2 E 0 0 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9670: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (8) B C D A E (8) B C A E D (8) D B C E A (7) D E A C B (6) E A C B D (5) D E A B C (5) C A B E D (5) E D A C B (4) B C E A D (4) B C D E A (4) A E C D B (4) E D A B C (3) E A D C B (3) D E B A C (3) D B E C A (3) D A E C B (2) B D C E A (2) B C A D E (2) A C E B D (2) E A B C D (1) D B E A C (1) D B C A E (1) D A E B C (1) C B A D E (1) B E D C A (1) B D C A E (1) B C E D A (1) A E D C B (1) A E C B D (1) A D E C B (1) A D C B E (1) Total count = 100 A B C D E A 0 -10 -12 -6 -6 B 10 0 12 8 16 C 12 -12 0 8 10 D 6 -8 -8 0 -2 E 6 -16 -10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999575 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 -6 -6 B 10 0 12 8 16 C 12 -12 0 8 10 D 6 -8 -8 0 -2 E 6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=29 E=16 C=14 A=10 so A is eliminated. Round 2 votes counts: D=31 B=31 E=22 C=16 so C is eliminated. Round 3 votes counts: B=45 D=31 E=24 so E is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:223 C:209 D:194 E:191 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 -6 -6 B 10 0 12 8 16 C 12 -12 0 8 10 D 6 -8 -8 0 -2 E 6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -6 -6 B 10 0 12 8 16 C 12 -12 0 8 10 D 6 -8 -8 0 -2 E 6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -6 -6 B 10 0 12 8 16 C 12 -12 0 8 10 D 6 -8 -8 0 -2 E 6 -16 -10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9671: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (14) C D E B A (8) D C A E B (7) C E B D A (7) A D B E C (7) E B A C D (5) D C E A B (5) C E B A D (5) B A E D C (5) E B C A D (4) D A B E C (4) D A B C E (4) E C B A D (3) D A C B E (3) B E A C D (3) A B D E C (3) D C A B E (2) C D E A B (2) E D A B C (1) D A C E B (1) C E D B A (1) C D B E A (1) C B E A D (1) B E A D C (1) B A E C D (1) A E B D C (1) A D E B C (1) Total count = 100 A B C D E A 0 10 8 8 6 B -10 0 8 6 -2 C -8 -8 0 -18 -6 D -8 -6 18 0 -4 E -6 2 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999851 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 8 8 6 B -10 0 8 6 -2 C -8 -8 0 -18 -6 D -8 -6 18 0 -4 E -6 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=26 A=26 C=25 E=13 B=10 so B is eliminated. Round 2 votes counts: A=32 D=26 C=25 E=17 so E is eliminated. Round 3 votes counts: A=41 C=32 D=27 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:216 E:203 B:201 D:200 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 8 6 B -10 0 8 6 -2 C -8 -8 0 -18 -6 D -8 -6 18 0 -4 E -6 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 8 6 B -10 0 8 6 -2 C -8 -8 0 -18 -6 D -8 -6 18 0 -4 E -6 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 8 6 B -10 0 8 6 -2 C -8 -8 0 -18 -6 D -8 -6 18 0 -4 E -6 2 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999747 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9672: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (6) C A B E D (6) A B D E C (5) E D A B C (4) D E B A C (4) C E D B A (4) C B E D A (4) B D A E C (4) A B D C E (4) E D B A C (3) E C D B A (3) E C A D B (3) C E B D A (3) C A B D E (3) A B C D E (3) E D C B A (2) E C D A B (2) E A D C B (2) D E B C A (2) D B E C A (2) C B A D E (2) B A C D E (2) A E D B C (2) A D E B C (2) A C E D B (2) A C B D E (2) E D B C A (1) E D A C B (1) D B E A C (1) C E B A D (1) C E A B D (1) C B E A D (1) C B D A E (1) C A E D B (1) C A E B D (1) B D E C A (1) B D E A C (1) B D C E A (1) B D C A E (1) B C D A E (1) B C A D E (1) A E D C B (1) A E C D B (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -8 -4 -6 B 4 0 -8 14 6 C 8 8 0 12 6 D 4 -14 -12 0 -2 E 6 -6 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 -4 -6 B 4 0 -8 14 6 C 8 8 0 12 6 D 4 -14 -12 0 -2 E 6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 A=24 E=21 B=12 D=9 so D is eliminated. Round 2 votes counts: C=34 E=27 A=24 B=15 so B is eliminated. Round 3 votes counts: C=38 E=32 A=30 so A is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:217 B:208 E:198 A:189 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 -4 -6 B 4 0 -8 14 6 C 8 8 0 12 6 D 4 -14 -12 0 -2 E 6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 -4 -6 B 4 0 -8 14 6 C 8 8 0 12 6 D 4 -14 -12 0 -2 E 6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 -4 -6 B 4 0 -8 14 6 C 8 8 0 12 6 D 4 -14 -12 0 -2 E 6 -6 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9673: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (11) B D E C A (8) D E B A C (6) C A E D B (6) C A B E D (6) E D B A C (5) D B E A C (5) B D E A C (5) A E D B C (5) E D A B C (4) A E C D B (4) E A D B C (3) C A B D E (3) B D A E C (3) B C D E A (3) A E D C B (3) E A D C B (2) D B E C A (2) C B D A E (2) C B A D E (2) A C B D E (2) E D C B A (1) E D B C A (1) E D A C B (1) E C A D B (1) C E D B A (1) C E A D B (1) C A E B D (1) B D C E A (1) B D C A E (1) B C D A E (1) Total count = 100 A B C D E A 0 6 18 0 0 B -6 0 6 -24 -12 C -18 -6 0 -12 -18 D 0 24 12 0 -12 E 0 12 18 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.465986 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.534014 Sum of squares = 0.502313893544 Cumulative probabilities = A: 0.465986 B: 0.465986 C: 0.465986 D: 0.465986 E: 1.000000 A B C D E A 0 6 18 0 0 B -6 0 6 -24 -12 C -18 -6 0 -12 -18 D 0 24 12 0 -12 E 0 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=22 B=22 E=18 D=13 so D is eliminated. Round 2 votes counts: B=29 A=25 E=24 C=22 so C is eliminated. Round 3 votes counts: A=41 B=33 E=26 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:221 A:212 D:212 B:182 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 6 18 0 0 B -6 0 6 -24 -12 C -18 -6 0 -12 -18 D 0 24 12 0 -12 E 0 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 18 0 0 B -6 0 6 -24 -12 C -18 -6 0 -12 -18 D 0 24 12 0 -12 E 0 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 18 0 0 B -6 0 6 -24 -12 C -18 -6 0 -12 -18 D 0 24 12 0 -12 E 0 12 18 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999903 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9674: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (9) B A D E C (9) B D A C E (8) A B D C E (7) E A C B D (6) D C B E A (5) C D E B A (5) A E C B D (4) A B E D C (4) E C D A B (3) E A C D B (3) B A D C E (3) E C D B A (2) E A B C D (2) D B C E A (2) C E D B A (2) C E A D B (2) C D E A B (2) B D E C A (2) B D C E A (2) B D C A E (2) B D A E C (2) A E B C D (2) D B C A E (1) C E D A B (1) C D A E B (1) C A E D B (1) C A D E B (1) B E A D C (1) B A E D C (1) A E C D B (1) A E B D C (1) A D C B E (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 6 14 20 2 B -6 0 0 14 2 C -14 0 0 -4 -6 D -20 -14 4 0 8 E -2 -2 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 20 2 B -6 0 0 14 2 C -14 0 0 -4 -6 D -20 -14 4 0 8 E -2 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=25 A=22 C=15 D=8 so D is eliminated. Round 2 votes counts: B=33 E=25 A=22 C=20 so C is eliminated. Round 3 votes counts: B=38 E=37 A=25 so A is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:221 B:205 E:197 D:189 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 14 20 2 B -6 0 0 14 2 C -14 0 0 -4 -6 D -20 -14 4 0 8 E -2 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 20 2 B -6 0 0 14 2 C -14 0 0 -4 -6 D -20 -14 4 0 8 E -2 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 20 2 B -6 0 0 14 2 C -14 0 0 -4 -6 D -20 -14 4 0 8 E -2 -2 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998507 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9675: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (11) B A E C D (8) D C E A B (7) C D E B A (5) B E A C D (5) E D C A B (4) E C D A B (4) C E D B A (4) B A E D C (4) B A C D E (4) E C D B A (3) E A B C D (3) B C D A E (3) B A D C E (3) A E B D C (3) A B D E C (3) D C B A E (2) D C A E B (2) C D B A E (2) B C D E A (2) B A C E D (2) E D A C B (1) E C B D A (1) E A D B C (1) E A B D C (1) D E C A B (1) D A C B E (1) C E D A B (1) C B E D A (1) C B D A E (1) B D A C E (1) B C E D A (1) B A D E C (1) A E D C B (1) A E D B C (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 12 6 10 B 6 0 16 18 12 C -12 -16 0 0 -14 D -6 -18 0 0 -20 E -10 -12 14 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 6 10 B 6 0 16 18 12 C -12 -16 0 0 -14 D -6 -18 0 0 -20 E -10 -12 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=21 E=18 C=14 D=13 so D is eliminated. Round 2 votes counts: B=34 C=25 A=22 E=19 so E is eliminated. Round 3 votes counts: C=38 B=34 A=28 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:211 E:206 C:179 D:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 6 10 B 6 0 16 18 12 C -12 -16 0 0 -14 D -6 -18 0 0 -20 E -10 -12 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 6 10 B 6 0 16 18 12 C -12 -16 0 0 -14 D -6 -18 0 0 -20 E -10 -12 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 6 10 B 6 0 16 18 12 C -12 -16 0 0 -14 D -6 -18 0 0 -20 E -10 -12 14 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9676: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E A D (31) D A E C B (18) E C B A D (6) D A C E B (4) B C E D A (4) E C A D B (3) D B A C E (3) D A B C E (3) E A C D B (2) D B A E C (2) D A E B C (2) B E C A D (2) B D C A E (2) E D C A B (1) E C B D A (1) E A D C B (1) D E A C B (1) C E B A D (1) C E A B D (1) C B E A D (1) B E C D A (1) B D C E A (1) B C A E D (1) B A D C E (1) B A C D E (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C E B (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 -16 -12 14 -14 B 16 0 12 10 10 C 12 -12 0 12 14 D -14 -10 -12 0 -16 E 14 -10 -14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -12 14 -14 B 16 0 12 10 10 C 12 -12 0 12 14 D -14 -10 -12 0 -16 E 14 -10 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 D=33 E=14 A=6 C=3 so C is eliminated. Round 2 votes counts: B=45 D=33 E=16 A=6 so A is eliminated. Round 3 votes counts: B=46 D=36 E=18 so E is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:224 C:213 E:203 A:186 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -12 14 -14 B 16 0 12 10 10 C 12 -12 0 12 14 D -14 -10 -12 0 -16 E 14 -10 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -12 14 -14 B 16 0 12 10 10 C 12 -12 0 12 14 D -14 -10 -12 0 -16 E 14 -10 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -12 14 -14 B 16 0 12 10 10 C 12 -12 0 12 14 D -14 -10 -12 0 -16 E 14 -10 -14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9677: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B D A (11) A D C B E (10) C D A E B (7) A C D E B (6) E B C D A (5) A D B C E (5) E C B D A (4) E B C A D (4) D A B C E (4) B A E D C (4) A D B E C (4) A B D E C (4) C E B A D (3) C A D E B (3) D C A E B (2) D A B E C (2) C E D B A (2) B E D A C (2) B E A D C (2) B D E A C (2) B D A E C (2) A D C E B (2) E C B A D (1) D B C E A (1) D B C A E (1) D A C B E (1) C E D A B (1) C E A D B (1) B E D C A (1) B E C D A (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 6 4 2 18 B -6 0 -10 -4 -6 C -4 10 0 0 22 D -2 4 0 0 14 E -18 6 -22 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 4 2 18 B -6 0 -10 -4 -6 C -4 10 0 0 22 D -2 4 0 0 14 E -18 6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 C=28 B=15 E=14 D=11 so D is eliminated. Round 2 votes counts: A=39 C=30 B=17 E=14 so E is eliminated. Round 3 votes counts: A=39 C=35 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:214 D:208 B:187 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 4 2 18 B -6 0 -10 -4 -6 C -4 10 0 0 22 D -2 4 0 0 14 E -18 6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 4 2 18 B -6 0 -10 -4 -6 C -4 10 0 0 22 D -2 4 0 0 14 E -18 6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 4 2 18 B -6 0 -10 -4 -6 C -4 10 0 0 22 D -2 4 0 0 14 E -18 6 -22 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999991822 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9678: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (8) B C A D E (7) E D C B A (6) D E A C B (6) A D E B C (5) E C B D A (4) D A E C B (4) B C E A D (4) B C A E D (4) B A C E D (4) E D C A B (3) D E C B A (3) A E D B C (3) A E B D C (3) E D A C B (2) E C D B A (2) D E C A B (2) D C B E A (2) D A C B E (2) C B D E A (2) A E B C D (2) A D B C E (2) A B E C D (2) A B C D E (2) E B C A D (1) E B A C D (1) E A D B C (1) E A B C D (1) D C E B A (1) D C B A E (1) D C A E B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D E B A (1) C B D A E (1) B E C A D (1) B E A C D (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 4 8 8 B -2 0 10 2 -8 C -4 -10 0 2 -8 D -8 -2 -2 0 -12 E -8 8 8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 8 8 B -2 0 10 2 -8 C -4 -10 0 2 -8 D -8 -2 -2 0 -12 E -8 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=23 B=22 E=21 C=6 so C is eliminated. Round 2 votes counts: A=28 B=25 D=24 E=23 so E is eliminated. Round 3 votes counts: D=38 B=32 A=30 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:211 E:210 B:201 C:190 D:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 4 8 8 B -2 0 10 2 -8 C -4 -10 0 2 -8 D -8 -2 -2 0 -12 E -8 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 8 8 B -2 0 10 2 -8 C -4 -10 0 2 -8 D -8 -2 -2 0 -12 E -8 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 8 8 B -2 0 10 2 -8 C -4 -10 0 2 -8 D -8 -2 -2 0 -12 E -8 8 8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999586 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9679: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (8) C D E A B (6) A B D E C (6) C B E D A (5) B E A C D (5) D C E A B (4) D A C E B (4) C E D B A (4) B C E A D (4) D C A E B (3) D A C B E (3) C D A B E (3) C B D A E (3) B E C A D (3) B A E D C (3) B A E C D (3) E C D B A (2) E B A D C (2) E A D B C (2) D C A B E (2) D A E C B (2) C E B D A (2) C D E B A (2) B E A D C (2) A D B E C (2) E D A B C (1) E C B D A (1) E B C A D (1) E A B D C (1) C E D A B (1) C D B E A (1) C D A E B (1) C B D E A (1) C B A D E (1) B C A E D (1) B C A D E (1) A E D B C (1) A D C B E (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 6 -4 -2 0 B -6 0 -6 8 20 C 4 6 0 4 10 D 2 -8 -4 0 -4 E 0 -20 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -2 0 B -6 0 -6 8 20 C 4 6 0 4 10 D 2 -8 -4 0 -4 E 0 -20 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=22 A=20 D=18 E=10 so E is eliminated. Round 2 votes counts: C=33 B=25 A=23 D=19 so D is eliminated. Round 3 votes counts: C=42 A=33 B=25 so B is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:212 B:208 A:200 D:193 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 -2 0 B -6 0 -6 8 20 C 4 6 0 4 10 D 2 -8 -4 0 -4 E 0 -20 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -2 0 B -6 0 -6 8 20 C 4 6 0 4 10 D 2 -8 -4 0 -4 E 0 -20 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -2 0 B -6 0 -6 8 20 C 4 6 0 4 10 D 2 -8 -4 0 -4 E 0 -20 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996904 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9680: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (19) B C E D A (13) A D C E B (7) C B D E A (6) E B D C A (5) C B A D E (5) A D E B C (5) E D A B C (4) A E D B C (4) E D B C A (3) C B E D A (3) B E C D A (3) A C B D E (3) D E A C B (2) D A E C B (2) C B A E D (2) A E B D C (2) A E B C D (2) E A D B C (1) D E C B A (1) D E B C A (1) D E A B C (1) D C E A B (1) C A B E D (1) C A B D E (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 16 10 10 12 B -16 0 -10 -6 -20 C -10 10 0 -18 -12 D -10 6 18 0 12 E -12 20 12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 10 12 B -16 0 -10 -6 -20 C -10 10 0 -18 -12 D -10 6 18 0 12 E -12 20 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=45 C=18 B=16 E=13 D=8 so D is eliminated. Round 2 votes counts: A=47 C=19 E=18 B=16 so B is eliminated. Round 3 votes counts: A=47 C=32 E=21 so E is eliminated. Round 4 votes counts: A=55 C=45 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:213 E:204 C:185 B:174 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 10 12 B -16 0 -10 -6 -20 C -10 10 0 -18 -12 D -10 6 18 0 12 E -12 20 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 10 12 B -16 0 -10 -6 -20 C -10 10 0 -18 -12 D -10 6 18 0 12 E -12 20 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 10 12 B -16 0 -10 -6 -20 C -10 10 0 -18 -12 D -10 6 18 0 12 E -12 20 12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9681: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (14) C B E A D (10) E B D C A (9) B E C D A (8) A C D B E (8) D E B A C (6) C A D B E (6) C B E D A (5) A D C E B (5) E B D A C (4) C A B D E (4) A D C B E (3) A C D E B (3) C E B A D (2) C B A E D (2) B E D C A (2) B C E D A (2) E D B A C (1) D E A B C (1) C B A D E (1) C A B E D (1) B D E A C (1) B D C E A (1) A D E C B (1) Total count = 100 A B C D E A 0 -8 -6 -8 -4 B 8 0 -2 4 8 C 6 2 0 4 6 D 8 -4 -4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999798 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -6 -8 -4 B 8 0 -2 4 8 C 6 2 0 4 6 D 8 -4 -4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=21 A=20 E=14 B=14 so E is eliminated. Round 2 votes counts: C=31 B=27 D=22 A=20 so A is eliminated. Round 3 votes counts: C=42 D=31 B=27 so B is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:209 C:209 D:204 E:191 A:187 Borda winner is B compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -6 -8 -4 B 8 0 -2 4 8 C 6 2 0 4 6 D 8 -4 -4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -6 -8 -4 B 8 0 -2 4 8 C 6 2 0 4 6 D 8 -4 -4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -6 -8 -4 B 8 0 -2 4 8 C 6 2 0 4 6 D 8 -4 -4 0 8 E 4 -8 -6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9682: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) A D E C B (9) E B C D A (5) D A E B C (5) D A B C E (5) C B E A D (5) E A C B D (4) D A C B E (4) C B D A E (4) C B A D E (4) B C E D A (4) B C D E A (4) E C B A D (3) A E D C B (3) A D C B E (3) E A D C B (2) D E A B C (2) D C B A E (2) D A E C B (2) C B E D A (2) C A B D E (2) B E C D A (2) A E C B D (2) A D E B C (2) E D B C A (1) E D A B C (1) E A C D B (1) E A B C D (1) D B C A E (1) B C E A D (1) B C D A E (1) A E D B C (1) A E C D B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -2 10 4 B -2 0 -8 8 -10 C 2 8 0 14 -14 D -10 -8 -14 0 2 E -4 10 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.100000 Sum of squares = 0.540000000005 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.900000 D: 0.900000 E: 1.000000 A B C D E A 0 2 -2 10 4 B -2 0 -8 8 -10 C 2 8 0 14 -14 D -10 -8 -14 0 2 E -4 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.100000 Sum of squares = 0.540000000231 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.900000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=22 D=21 C=17 B=12 so B is eliminated. Round 2 votes counts: E=30 C=27 A=22 D=21 so D is eliminated. Round 3 votes counts: A=38 E=32 C=30 so C is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. E:209 A:207 C:205 B:194 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 -2 10 4 B -2 0 -8 8 -10 C 2 8 0 14 -14 D -10 -8 -14 0 2 E -4 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.100000 Sum of squares = 0.540000000231 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.900000 D: 0.900000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 10 4 B -2 0 -8 8 -10 C 2 8 0 14 -14 D -10 -8 -14 0 2 E -4 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.100000 Sum of squares = 0.540000000231 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.900000 D: 0.900000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 10 4 B -2 0 -8 8 -10 C 2 8 0 14 -14 D -10 -8 -14 0 2 E -4 10 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.700000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.100000 Sum of squares = 0.540000000231 Cumulative probabilities = A: 0.700000 B: 0.700000 C: 0.900000 D: 0.900000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9683: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (9) D B C A E (9) D C A B E (7) B E D A C (6) B E A D C (6) B D C E A (6) E A B C D (5) D C A E B (4) C D A E B (4) E B A D C (3) E A C B D (3) C A D E B (3) B E C A D (3) B D E C A (3) A E C D B (3) C D A B E (2) C B D A E (2) B E A C D (2) B D E A C (2) B C D E A (2) A C E D B (2) A C D E B (2) E C A B D (1) E A D B C (1) E A B D C (1) D C B A E (1) D B A C E (1) D A C E B (1) D A C B E (1) C E A B D (1) C D B A E (1) C B D E A (1) B E D C A (1) A E D C B (1) Total count = 100 A B C D E A 0 -16 -2 -8 -12 B 16 0 20 14 12 C 2 -20 0 -8 0 D 8 -14 8 0 4 E 12 -12 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -2 -8 -12 B 16 0 20 14 12 C 2 -20 0 -8 0 D 8 -14 8 0 4 E 12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=24 E=23 C=14 A=8 so A is eliminated. Round 2 votes counts: B=31 E=27 D=24 C=18 so C is eliminated. Round 3 votes counts: D=36 B=34 E=30 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 D:203 E:198 C:187 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -2 -8 -12 B 16 0 20 14 12 C 2 -20 0 -8 0 D 8 -14 8 0 4 E 12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -2 -8 -12 B 16 0 20 14 12 C 2 -20 0 -8 0 D 8 -14 8 0 4 E 12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -2 -8 -12 B 16 0 20 14 12 C 2 -20 0 -8 0 D 8 -14 8 0 4 E 12 -12 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9684: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (7) B D A C E (7) B A C D E (7) B D C A E (6) E C A D B (5) D B E C A (5) B A C E D (5) A C E B D (5) E D B A C (4) E A C D B (4) E A C B D (4) D E C B A (4) A B C E D (4) D E C A B (3) D B E A C (3) A C B E D (3) D C E A B (2) D B C A E (2) B D E A C (2) B D A E C (2) B A E C D (2) B A D C E (2) A E C B D (2) E D A C B (1) E C D A B (1) E A B C D (1) D E B C A (1) D C B E A (1) D C B A E (1) D C A E B (1) D B C E A (1) C A E B D (1) B E D A C (1) Total count = 100 A B C D E A 0 -12 18 -10 0 B 12 0 10 8 8 C -18 -10 0 -12 -4 D 10 -8 12 0 0 E 0 -8 4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 18 -10 0 B 12 0 10 8 8 C -18 -10 0 -12 -4 D 10 -8 12 0 0 E 0 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 E=27 D=24 A=14 C=1 so C is eliminated. Round 2 votes counts: B=34 E=27 D=24 A=15 so A is eliminated. Round 3 votes counts: B=41 E=35 D=24 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:207 A:198 E:198 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 18 -10 0 B 12 0 10 8 8 C -18 -10 0 -12 -4 D 10 -8 12 0 0 E 0 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 18 -10 0 B 12 0 10 8 8 C -18 -10 0 -12 -4 D 10 -8 12 0 0 E 0 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 18 -10 0 B 12 0 10 8 8 C -18 -10 0 -12 -4 D 10 -8 12 0 0 E 0 -8 4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999845 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9685: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B A E (12) A E B D C (12) E A B D C (7) D B C A E (7) D B A C E (7) C E A D B (7) C D B E A (7) B D A E C (6) C E A B D (5) A B D E C (5) E A C B D (4) D B A E C (4) E C A B D (3) C E D B A (3) E B D A C (2) D C B A E (2) C D E B A (2) E C B A D (1) C E D A B (1) C E B D A (1) C A D B E (1) B D E A C (1) Total count = 100 A B C D E A 0 -10 -4 -10 12 B 10 0 2 -6 4 C 4 -2 0 -6 10 D 10 6 6 0 8 E -12 -4 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -4 -10 12 B 10 0 2 -6 4 C 4 -2 0 -6 10 D 10 6 6 0 8 E -12 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 D=20 E=17 A=17 B=7 so B is eliminated. Round 2 votes counts: C=39 D=27 E=17 A=17 so E is eliminated. Round 3 votes counts: C=43 D=29 A=28 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:215 B:205 C:203 A:194 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 -4 -10 12 B 10 0 2 -6 4 C 4 -2 0 -6 10 D 10 6 6 0 8 E -12 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -4 -10 12 B 10 0 2 -6 4 C 4 -2 0 -6 10 D 10 6 6 0 8 E -12 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -4 -10 12 B 10 0 2 -6 4 C 4 -2 0 -6 10 D 10 6 6 0 8 E -12 -4 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999898 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9686: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (9) C E D A B (9) C E A D B (8) B A D E C (7) A B C D E (7) E D C B A (6) D B E A C (5) B D A E C (5) E C D B A (4) E C D A B (4) C A E B D (4) A C B E D (4) A B C E D (4) B A D C E (3) A B D E C (3) D E C B A (2) D E B C A (2) C E D B A (2) B A C D E (2) A B D C E (2) E D A C B (1) E D A B C (1) E C A D B (1) D C B E A (1) D B E C A (1) D B C E A (1) C E A B D (1) C A B E D (1) Total count = 100 A B C D E A 0 0 6 -6 -16 B 0 0 4 -14 -8 C -6 -4 0 2 -2 D 6 14 -2 0 0 E 16 8 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.464248 E: 0.535752 Sum of squares = 0.502556394201 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.464248 E: 1.000000 A B C D E A 0 0 6 -6 -16 B 0 0 4 -14 -8 C -6 -4 0 2 -2 D 6 14 -2 0 0 E 16 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 0.500651 Sum of squares = 0.500000848614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 D=21 A=20 E=17 B=17 so E is eliminated. Round 2 votes counts: C=34 D=29 A=20 B=17 so B is eliminated. Round 3 votes counts: D=34 C=34 A=32 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. E:213 D:209 C:195 A:192 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 6 -6 -16 B 0 0 4 -14 -8 C -6 -4 0 2 -2 D 6 14 -2 0 0 E 16 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 0.500651 Sum of squares = 0.500000848614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 -6 -16 B 0 0 4 -14 -8 C -6 -4 0 2 -2 D 6 14 -2 0 0 E 16 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 0.500651 Sum of squares = 0.500000848614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 -6 -16 B 0 0 4 -14 -8 C -6 -4 0 2 -2 D 6 14 -2 0 0 E 16 8 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 0.500651 Sum of squares = 0.500000848614 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499349 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9687: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (9) D B C A E (8) A C E D B (8) E A B C D (7) B D E C A (6) A E C D B (6) D C A B E (5) B D C E A (4) E B A C D (3) C A D E B (3) A E C B D (3) E B D A C (2) E B A D C (2) D B E A C (2) D B C E A (2) C D A B E (2) C A E B D (2) B E D C A (2) B E D A C (2) B E C A D (2) B E A C D (2) B D E A C (2) B D C A E (2) A C D E B (2) E A C D B (1) E A B D C (1) D C B A E (1) D B E C A (1) D A E C B (1) C B A E D (1) C A E D B (1) C A D B E (1) B E C D A (1) B C D A E (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 8 10 12 -2 B -8 0 4 8 -6 C -10 -4 0 10 -12 D -12 -8 -10 0 -10 E 2 6 12 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 10 12 -2 B -8 0 4 8 -6 C -10 -4 0 10 -12 D -12 -8 -10 0 -10 E 2 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=24 A=21 D=20 C=10 so C is eliminated. Round 2 votes counts: A=28 E=25 B=25 D=22 so D is eliminated. Round 3 votes counts: B=39 A=36 E=25 so E is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:215 A:214 B:199 C:192 D:180 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 10 12 -2 B -8 0 4 8 -6 C -10 -4 0 10 -12 D -12 -8 -10 0 -10 E 2 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 12 -2 B -8 0 4 8 -6 C -10 -4 0 10 -12 D -12 -8 -10 0 -10 E 2 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 12 -2 B -8 0 4 8 -6 C -10 -4 0 10 -12 D -12 -8 -10 0 -10 E 2 6 12 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999723 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9688: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (15) B A D E C (12) C D E A B (8) B A C D E (8) A B D E C (8) C E D B A (7) E D C A B (6) B A E D C (5) A D E B C (5) D E C A B (4) C B E D A (4) D E A C B (3) C B A D E (2) E D B A C (1) E D A B C (1) E C D B A (1) E B D C A (1) D E A B C (1) D A E C B (1) C A D E B (1) B C E A D (1) B C A E D (1) B A E C D (1) B A C E D (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 10 -2 -6 -6 B -10 0 -6 -12 -12 C 2 6 0 2 -2 D 6 12 -2 0 10 E 6 12 2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.142857 Sum of squares = 0.551020408232 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.857143 E: 1.000000 A B C D E A 0 10 -2 -6 -6 B -10 0 -6 -12 -12 C 2 6 0 2 -2 D 6 12 -2 0 10 E 6 12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.142857 Sum of squares = 0.551020407876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=29 A=15 E=10 D=9 so D is eliminated. Round 2 votes counts: C=37 B=29 E=18 A=16 so A is eliminated. Round 3 votes counts: C=38 B=38 E=24 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:213 E:205 C:204 A:198 B:180 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -2 -6 -6 B -10 0 -6 -12 -12 C 2 6 0 2 -2 D 6 12 -2 0 10 E 6 12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.142857 Sum of squares = 0.551020407876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.857143 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -2 -6 -6 B -10 0 -6 -12 -12 C 2 6 0 2 -2 D 6 12 -2 0 10 E 6 12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.142857 Sum of squares = 0.551020407876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.857143 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -2 -6 -6 B -10 0 -6 -12 -12 C 2 6 0 2 -2 D 6 12 -2 0 10 E 6 12 2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.142857 E: 0.142857 Sum of squares = 0.551020407876 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.714286 D: 0.857143 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9689: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) D C A B E (6) D E C A B (5) D C E A B (5) C D A B E (5) B A E C D (5) E B A D C (4) D C B A E (4) C A D B E (4) B C A E D (4) A E B C D (4) E D A B C (3) E B D A C (3) E A B C D (3) A B E C D (3) A B C E D (3) E D B A C (2) E D A C B (2) E A D B C (2) E A B D C (2) D E B C A (2) D E A C B (2) D C A E B (2) C B A D E (2) C A B E D (2) B E A C D (2) E A D C B (1) D E C B A (1) D E B A C (1) D C E B A (1) D B C E A (1) D A C E B (1) C D B A E (1) C B D A E (1) C A B D E (1) B D C E A (1) B A C E D (1) A C E B D (1) Total count = 100 A B C D E A 0 14 4 2 0 B -14 0 6 -2 -8 C -4 -6 0 -2 -8 D -2 2 2 0 -8 E 0 8 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.514064 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.485936 Sum of squares = 0.500395611767 Cumulative probabilities = A: 0.514064 B: 0.514064 C: 0.514064 D: 0.514064 E: 1.000000 A B C D E A 0 14 4 2 0 B -14 0 6 -2 -8 C -4 -6 0 -2 -8 D -2 2 2 0 -8 E 0 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=29 C=16 B=13 A=11 so A is eliminated. Round 2 votes counts: E=33 D=31 B=19 C=17 so C is eliminated. Round 3 votes counts: D=41 E=34 B=25 so B is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:210 D:197 B:191 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 2 0 B -14 0 6 -2 -8 C -4 -6 0 -2 -8 D -2 2 2 0 -8 E 0 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 2 0 B -14 0 6 -2 -8 C -4 -6 0 -2 -8 D -2 2 2 0 -8 E 0 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 2 0 B -14 0 6 -2 -8 C -4 -6 0 -2 -8 D -2 2 2 0 -8 E 0 8 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999944 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9690: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (6) C B E A D (6) E B A C D (5) D B C E A (5) B C E D A (5) A E D C B (5) D A E B C (4) D A C B E (4) D A B E C (4) B C E A D (4) D B E C A (3) D B E A C (3) D A C E B (3) C E B A D (3) C A D B E (3) B E C A D (3) B C D E A (3) D E A B C (2) D C A B E (2) B E D C A (2) B E C D A (2) B D E C A (2) A E C B D (2) A D E C B (2) A C D E B (2) E C B A D (1) E C A B D (1) E B D A C (1) E A D B C (1) D E B A C (1) D C B A E (1) D A B C E (1) C B D A E (1) C B A E D (1) C B A D E (1) C A E B D (1) C A B D E (1) A D C E B (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -18 -14 2 -18 B 18 0 14 4 16 C 14 -14 0 6 0 D -2 -4 -6 0 0 E 18 -16 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 -14 2 -18 B 18 0 14 4 16 C 14 -14 0 6 0 D -2 -4 -6 0 0 E 18 -16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=21 C=17 E=15 A=14 so A is eliminated. Round 2 votes counts: D=36 E=22 C=21 B=21 so C is eliminated. Round 3 votes counts: D=42 B=31 E=27 so E is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 C:203 E:201 D:194 A:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 -14 2 -18 B 18 0 14 4 16 C 14 -14 0 6 0 D -2 -4 -6 0 0 E 18 -16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -14 2 -18 B 18 0 14 4 16 C 14 -14 0 6 0 D -2 -4 -6 0 0 E 18 -16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -14 2 -18 B 18 0 14 4 16 C 14 -14 0 6 0 D -2 -4 -6 0 0 E 18 -16 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999074 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9691: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (17) D B A E C (9) A E C D B (9) E C A D B (6) C E B A D (6) C E A B D (5) A E D C B (5) C B E D A (4) B C D E A (4) E A C D B (3) D A B E C (3) C E B D A (3) B D A C E (3) E C A B D (2) D B C E A (2) D A E B C (2) C E A D B (2) C B E A D (2) B D C A E (2) B C E D A (2) A D E C B (2) E D C A B (1) D B E C A (1) C E D A B (1) A E D B C (1) A E C B D (1) A D E B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -10 -20 -8 -22 B 10 0 -4 2 0 C 20 4 0 0 6 D 8 -2 0 0 -6 E 22 0 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.724215 D: 0.275785 E: 0.000000 Sum of squares = 0.6005448459 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.724215 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -20 -8 -22 B 10 0 -4 2 0 C 20 4 0 0 6 D 8 -2 0 0 -6 E 22 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500725 D: 0.499275 E: 0.000000 Sum of squares = 0.500001052191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500725 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=23 A=20 D=17 E=12 so E is eliminated. Round 2 votes counts: C=31 B=28 A=23 D=18 so D is eliminated. Round 3 votes counts: B=40 C=32 A=28 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:215 E:211 B:204 D:200 A:170 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -20 -8 -22 B 10 0 -4 2 0 C 20 4 0 0 6 D 8 -2 0 0 -6 E 22 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500725 D: 0.499275 E: 0.000000 Sum of squares = 0.500001052191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500725 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -20 -8 -22 B 10 0 -4 2 0 C 20 4 0 0 6 D 8 -2 0 0 -6 E 22 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500725 D: 0.499275 E: 0.000000 Sum of squares = 0.500001052191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500725 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -20 -8 -22 B 10 0 -4 2 0 C 20 4 0 0 6 D 8 -2 0 0 -6 E 22 0 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500725 D: 0.499275 E: 0.000000 Sum of squares = 0.500001052191 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500725 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9692: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (13) C D B A E (9) B D C A E (8) A E C D B (7) D C B E A (5) B D C E A (5) E A D C B (3) E A B D C (3) D B C E A (3) A E C B D (3) A E B C D (3) A C E D B (3) E C A D B (2) E A C B D (2) C D A E B (2) C D A B E (2) B D E C A (2) B D A C E (2) B A D E C (2) A C B D E (2) E D C A B (1) E B D A C (1) E B A D C (1) E A B C D (1) D C E B A (1) D C B A E (1) D B E C A (1) D B C A E (1) C D E A B (1) C B D A E (1) C A E D B (1) C A D E B (1) C A D B E (1) C A B D E (1) B E D C A (1) B D A E C (1) B A E D C (1) B A D C E (1) A B E C D (1) Total count = 100 A B C D E A 0 6 0 4 8 B -6 0 -24 -16 2 C 0 24 0 12 2 D -4 16 -12 0 6 E -8 -2 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.544678 B: 0.000000 C: 0.455322 D: 0.000000 E: 0.000000 Sum of squares = 0.503992288481 Cumulative probabilities = A: 0.544678 B: 0.544678 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 0 4 8 B -6 0 -24 -16 2 C 0 24 0 12 2 D -4 16 -12 0 6 E -8 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 C=19 A=19 D=12 so D is eliminated. Round 2 votes counts: B=28 E=27 C=26 A=19 so A is eliminated. Round 3 votes counts: E=40 C=31 B=29 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:219 A:209 D:203 E:191 B:178 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 6 0 4 8 B -6 0 -24 -16 2 C 0 24 0 12 2 D -4 16 -12 0 6 E -8 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 0 4 8 B -6 0 -24 -16 2 C 0 24 0 12 2 D -4 16 -12 0 6 E -8 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 0 4 8 B -6 0 -24 -16 2 C 0 24 0 12 2 D -4 16 -12 0 6 E -8 -2 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.499999 B: 0.000000 C: 0.500001 D: 0.000000 E: 0.000000 Sum of squares = 0.500000000001 Cumulative probabilities = A: 0.499999 B: 0.499999 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9693: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (14) B E A C D (8) B C E D A (8) C D A B E (7) A D C E B (7) D C A E B (6) E A B D C (5) D A C E B (5) B E C D A (5) A D E C B (4) C D B A E (3) A E D C B (3) E B D A C (2) E A D B C (2) C D B E A (2) C B D E A (2) C B D A E (2) B E C A D (2) B E A D C (2) B C E A D (2) E B D C A (1) C D A E B (1) C A D E B (1) B E D C A (1) B E D A C (1) B C D E A (1) B C A E D (1) A E D B C (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 10 6 -16 B 14 0 12 16 -4 C -10 -12 0 -8 -2 D -6 -16 8 0 -16 E 16 4 2 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 10 6 -16 B 14 0 12 16 -4 C -10 -12 0 -8 -2 D -6 -16 8 0 -16 E 16 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=24 C=18 A=16 D=11 so D is eliminated. Round 2 votes counts: B=31 E=24 C=24 A=21 so A is eliminated. Round 3 votes counts: C=37 E=32 B=31 so B is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:219 E:219 A:193 D:185 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 10 6 -16 B 14 0 12 16 -4 C -10 -12 0 -8 -2 D -6 -16 8 0 -16 E 16 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 6 -16 B 14 0 12 16 -4 C -10 -12 0 -8 -2 D -6 -16 8 0 -16 E 16 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 6 -16 B 14 0 12 16 -4 C -10 -12 0 -8 -2 D -6 -16 8 0 -16 E 16 4 2 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999983551 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9694: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (12) D B E C A (7) E B C A D (6) A E B D C (6) D A B E C (5) A D E B C (5) A C D B E (5) E B D C A (4) C D B E A (4) A D C B E (4) A C E B D (4) A C D E B (4) A E D B C (3) D C B E A (2) D A C B E (2) C E B A D (2) C B D E A (2) C A E B D (2) C A D B E (2) A E B C D (2) A D C E B (2) E C B D A (1) E C B A D (1) E B C D A (1) E B A D C (1) E B A C D (1) E A B C D (1) D C A B E (1) D B C E A (1) D B A E C (1) D A E B C (1) D A B C E (1) C E A B D (1) B E D C A (1) B E C D A (1) A E C B D (1) Total count = 100 A B C D E A 0 4 -2 6 2 B -4 0 -4 0 2 C 2 4 0 6 2 D -6 0 -6 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999641 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 6 2 B -4 0 -4 0 2 C 2 4 0 6 2 D -6 0 -6 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 C=25 D=21 E=16 B=2 so B is eliminated. Round 2 votes counts: A=36 C=25 D=21 E=18 so E is eliminated. Round 3 votes counts: A=39 C=35 D=26 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:207 A:205 E:198 B:197 D:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 6 2 B -4 0 -4 0 2 C 2 4 0 6 2 D -6 0 -6 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 6 2 B -4 0 -4 0 2 C 2 4 0 6 2 D -6 0 -6 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 6 2 B -4 0 -4 0 2 C 2 4 0 6 2 D -6 0 -6 0 -2 E -2 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999692 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9695: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (10) E C A D B (6) B D A E C (6) B D A C E (6) B A E D C (6) D A C E B (5) C E D A B (5) B D C A E (5) E C A B D (4) E A C D B (4) A E C D B (4) D C A E B (3) D A B E C (3) C D E A B (3) B A D E C (3) A E D C B (3) D C B E A (2) D B A C E (2) C E D B A (2) C E A B D (2) B E A C D (2) B C D E A (2) D C E A B (1) D B C A E (1) D A E C B (1) D A C B E (1) D A B C E (1) C E B D A (1) C B D E A (1) B E C A D (1) B A E C D (1) A E B C D (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 18 2 -2 8 B -18 0 -16 -16 -12 C -2 16 0 -2 6 D 2 16 2 0 -4 E -8 12 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428449 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 A B C D E A 0 18 2 -2 8 B -18 0 -16 -16 -12 C -2 16 0 -2 6 D 2 16 2 0 -4 E -8 12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=24 D=20 E=14 A=10 so A is eliminated. Round 2 votes counts: B=33 C=24 E=22 D=21 so D is eliminated. Round 3 votes counts: B=40 C=36 E=24 so E is eliminated. Round 4 votes counts: C=58 B=42 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:213 C:209 D:208 E:201 B:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 -2 8 B -18 0 -16 -16 -12 C -2 16 0 -2 6 D 2 16 2 0 -4 E -8 12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 -2 8 B -18 0 -16 -16 -12 C -2 16 0 -2 6 D 2 16 2 0 -4 E -8 12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 -2 8 B -18 0 -16 -16 -12 C -2 16 0 -2 6 D 2 16 2 0 -4 E -8 12 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.285714 B: 0.000000 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.428571428416 Cumulative probabilities = A: 0.285714 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9696: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (12) E B D A C (10) C A B D E (8) E D B C A (6) A C B E D (6) D E B C A (5) D C A B E (5) C A D B E (5) E A B C D (4) B A C E D (4) E A C D B (3) E A C B D (3) D B C A E (3) B C A D E (3) E B A D C (2) D B E C A (2) D B C E A (2) B A C D E (2) A C E B D (2) A C B D E (2) E D C A B (1) E B A C D (1) D E C B A (1) D E C A B (1) D C B A E (1) C D A B E (1) C A D E B (1) B D E C A (1) A E C D B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 8 -2 -8 B 10 0 18 0 -6 C -8 -18 0 -4 -8 D 2 0 4 0 -14 E 8 6 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 8 -2 -8 B 10 0 18 0 -6 C -8 -18 0 -4 -8 D 2 0 4 0 -14 E 8 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=42 D=20 C=15 A=13 B=10 so B is eliminated. Round 2 votes counts: E=42 D=21 A=19 C=18 so C is eliminated. Round 3 votes counts: E=42 A=36 D=22 so D is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 B:211 D:196 A:194 C:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 8 -2 -8 B 10 0 18 0 -6 C -8 -18 0 -4 -8 D 2 0 4 0 -14 E 8 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 8 -2 -8 B 10 0 18 0 -6 C -8 -18 0 -4 -8 D 2 0 4 0 -14 E 8 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 8 -2 -8 B 10 0 18 0 -6 C -8 -18 0 -4 -8 D 2 0 4 0 -14 E 8 6 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9697: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (10) B E A D C (9) C D E A B (6) E C B D A (5) C E D A B (5) C D A B E (5) B A D E C (5) A B D E C (5) A B D C E (5) E C D B A (4) E B A C D (3) C E D B A (3) C A D B E (3) B A E D C (3) E D C B A (2) E C B A D (2) D E B A C (2) D B A E C (2) D A C B E (2) C E B A D (2) A B C D E (2) E D B A C (1) E B D C A (1) E B D A C (1) E B C D A (1) D E C B A (1) D C A B E (1) D A B C E (1) C D A E B (1) C A B E D (1) C A B D E (1) B E A C D (1) B D A E C (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 -18 12 10 -18 B 18 0 10 16 0 C -12 -10 0 -10 -20 D -10 -16 10 0 -8 E 18 0 20 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.695612 C: 0.000000 D: 0.000000 E: 0.304388 Sum of squares = 0.576527765737 Cumulative probabilities = A: 0.000000 B: 0.695612 C: 0.695612 D: 0.695612 E: 1.000000 A B C D E A 0 -18 12 10 -18 B 18 0 10 16 0 C -12 -10 0 -10 -20 D -10 -16 10 0 -8 E 18 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=27 B=19 A=15 D=9 so D is eliminated. Round 2 votes counts: E=33 C=28 B=21 A=18 so A is eliminated. Round 3 votes counts: B=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=50 B=50 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:223 B:222 A:193 D:188 C:174 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 12 10 -18 B 18 0 10 16 0 C -12 -10 0 -10 -20 D -10 -16 10 0 -8 E 18 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 12 10 -18 B 18 0 10 16 0 C -12 -10 0 -10 -20 D -10 -16 10 0 -8 E 18 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 12 10 -18 B 18 0 10 16 0 C -12 -10 0 -10 -20 D -10 -16 10 0 -8 E 18 0 20 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9698: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (8) C A D E B (8) D B E A C (7) B E D A C (5) A C D B E (5) D C A E B (4) C A E B D (4) E C A B D (3) E B C A D (3) D E B C A (3) D B E C A (3) D B A C E (3) C A E D B (3) E D C B A (2) E C D A B (2) E B C D A (2) E B A C D (2) D C E A B (2) D C A B E (2) D B A E C (2) B E A D C (2) B E A C D (2) B A D C E (2) A C E B D (2) E D B C A (1) E C D B A (1) E C B D A (1) E A C B D (1) E A B C D (1) D E C B A (1) D E C A B (1) D A C B E (1) D A B C E (1) C E A D B (1) C D A E B (1) B E D C A (1) B A E C D (1) A E B C D (1) A D C B E (1) A C B E D (1) A C B D E (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -14 -8 -10 B 4 0 4 -10 -16 C 14 -4 0 -6 -12 D 8 10 6 0 -2 E 10 16 12 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -14 -8 -10 B 4 0 4 -10 -16 C 14 -4 0 -6 -12 D 8 10 6 0 -2 E 10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=27 C=17 B=13 A=13 so B is eliminated. Round 2 votes counts: E=37 D=30 C=17 A=16 so A is eliminated. Round 3 votes counts: E=39 D=34 C=27 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:211 C:196 B:191 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -14 -8 -10 B 4 0 4 -10 -16 C 14 -4 0 -6 -12 D 8 10 6 0 -2 E 10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -8 -10 B 4 0 4 -10 -16 C 14 -4 0 -6 -12 D 8 10 6 0 -2 E 10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -8 -10 B 4 0 4 -10 -16 C 14 -4 0 -6 -12 D 8 10 6 0 -2 E 10 16 12 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999985631 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9699: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (10) E A B C D (6) A B D C E (6) E D C A B (4) E C B A D (4) B C A D E (4) E D C B A (3) E C B D A (3) E A D C B (3) D A B C E (3) C D B A E (3) C B D A E (3) B C D A E (3) B A C D E (3) A D E B C (3) A D B E C (3) A B E C D (3) E D A C B (2) E A D B C (2) D C E B A (2) D C B A E (2) D A E C B (2) D A C B E (2) C B D E A (2) B C A E D (2) E C D A B (1) E A C D B (1) D C E A B (1) D C B E A (1) D C A E B (1) D C A B E (1) D B A C E (1) C E D B A (1) C D B E A (1) C B E D A (1) C B E A D (1) B C E A D (1) A E D B C (1) A E B D C (1) A D B C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -10 -6 0 B 2 0 -10 -10 -2 C 10 10 0 10 -4 D 6 10 -10 0 -2 E 0 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.136940 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.863060 Sum of squares = 0.763624572824 Cumulative probabilities = A: 0.136940 B: 0.136940 C: 0.136940 D: 0.136940 E: 1.000000 A B C D E A 0 -2 -10 -6 0 B 2 0 -10 -10 -2 C 10 10 0 10 -4 D 6 10 -10 0 -2 E 0 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.624999999941 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=20 D=16 B=13 C=12 so C is eliminated. Round 2 votes counts: E=40 D=20 B=20 A=20 so D is eliminated. Round 3 votes counts: E=43 A=29 B=28 so B is eliminated. Round 4 votes counts: E=50 A=50 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:213 E:204 D:202 A:191 B:190 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -10 -6 0 B 2 0 -10 -10 -2 C 10 10 0 10 -4 D 6 10 -10 0 -2 E 0 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.624999999941 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -6 0 B 2 0 -10 -10 -2 C 10 10 0 10 -4 D 6 10 -10 0 -2 E 0 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.624999999941 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -6 0 B 2 0 -10 -10 -2 C 10 10 0 10 -4 D 6 10 -10 0 -2 E 0 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.750000 Sum of squares = 0.624999999941 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9700: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) C A D E B (8) E C B A D (5) D B A E C (5) D A C B E (5) C A E D B (5) B E D A C (5) E C A B D (4) D C A B E (4) D A B C E (4) B E D C A (4) B D E A C (4) D B E A C (3) D A B E C (3) C A E B D (3) B D E C A (3) A C D E B (3) E A B C D (2) D B C A E (2) C E A B D (2) C D A E B (2) C D A B E (2) A E C B D (2) E B A C D (1) E A B D C (1) D C B A E (1) D B A C E (1) C E B A D (1) C D E B A (1) B E C D A (1) A E D B C (1) A E C D B (1) A D E C B (1) A D E B C (1) A D C E B (1) Total count = 100 A B C D E A 0 10 -12 0 10 B -10 0 -2 -8 -6 C 12 2 0 2 -10 D 0 8 -2 0 8 E -10 6 10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.42000000013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 A B C D E A 0 10 -12 0 10 B -10 0 -2 -8 -6 C 12 2 0 2 -10 D 0 8 -2 0 8 E -10 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 E=21 B=17 A=10 so A is eliminated. Round 2 votes counts: D=31 C=27 E=25 B=17 so B is eliminated. Round 3 votes counts: D=38 E=35 C=27 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:207 A:204 C:203 E:199 B:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -12 0 10 B -10 0 -2 -8 -6 C 12 2 0 2 -10 D 0 8 -2 0 8 E -10 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -12 0 10 B -10 0 -2 -8 -6 C 12 2 0 2 -10 D 0 8 -2 0 8 E -10 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -12 0 10 B -10 0 -2 -8 -6 C 12 2 0 2 -10 D 0 8 -2 0 8 E -10 6 10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.500000 E: 0.100000 Sum of squares = 0.420000000073 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.400000 D: 0.900000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9701: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (8) E C B D A (6) E B C D A (5) C D A B E (5) E C D B A (4) E B C A D (4) E B A C D (4) D C A B E (4) C D E B A (4) A B E D C (4) A B D E C (4) E B A D C (3) E A B C D (3) D A C B E (3) C D E A B (3) C D A E B (3) A D C B E (3) C E D B A (2) C D B E A (2) C D B A E (2) B E D C A (2) B A D E C (2) B A D C E (2) E C D A B (1) E C B A D (1) E A C B D (1) D C B A E (1) D B C E A (1) D B A C E (1) D A B C E (1) C E D A B (1) C E B D A (1) C E A D B (1) B D A E C (1) B A E D C (1) A E C D B (1) A E B D C (1) A D B E C (1) A C E D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -6 -6 2 B -2 0 -2 -8 0 C 6 2 0 12 2 D 6 8 -12 0 6 E -2 0 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -6 -6 2 B -2 0 -2 -8 0 C 6 2 0 12 2 D 6 8 -12 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 A=25 C=24 D=11 B=8 so B is eliminated. Round 2 votes counts: E=34 A=30 C=24 D=12 so D is eliminated. Round 3 votes counts: A=36 E=34 C=30 so C is eliminated. Round 4 votes counts: A=51 E=49 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:211 D:204 A:196 E:195 B:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -6 -6 2 B -2 0 -2 -8 0 C 6 2 0 12 2 D 6 8 -12 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -6 -6 2 B -2 0 -2 -8 0 C 6 2 0 12 2 D 6 8 -12 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -6 -6 2 B -2 0 -2 -8 0 C 6 2 0 12 2 D 6 8 -12 0 6 E -2 0 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999699 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9702: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (14) A B D E C (12) A B D C E (11) C E D B A (9) E C D A B (5) E C A D B (4) E C A B D (4) B A D C E (4) A B E D C (3) E A D B C (2) D C B E A (2) D B A E C (2) D B A C E (2) C E A B D (2) C D E B A (2) A E B D C (2) A B C E D (2) E D C B A (1) D E C B A (1) D E B C A (1) D C E B A (1) D B C E A (1) D B C A E (1) C E D A B (1) C E B A D (1) C E A D B (1) C D B A E (1) C B D A E (1) C A E B D (1) C A B E D (1) B D A C E (1) B A D E C (1) A D B E C (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 -10 8 -4 B -8 0 -6 -4 -4 C 10 6 0 2 -6 D -8 4 -2 0 -8 E 4 4 6 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 -10 8 -4 B -8 0 -6 -4 -4 C 10 6 0 2 -6 D -8 4 -2 0 -8 E 4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=30 C=20 D=11 B=6 so B is eliminated. Round 2 votes counts: A=38 E=30 C=20 D=12 so D is eliminated. Round 3 votes counts: A=43 E=32 C=25 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:211 C:206 A:201 D:193 B:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 -10 8 -4 B -8 0 -6 -4 -4 C 10 6 0 2 -6 D -8 4 -2 0 -8 E 4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 8 -4 B -8 0 -6 -4 -4 C 10 6 0 2 -6 D -8 4 -2 0 -8 E 4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 8 -4 B -8 0 -6 -4 -4 C 10 6 0 2 -6 D -8 4 -2 0 -8 E 4 4 6 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9703: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (13) D C B E A (10) D B C A E (10) B D A E C (10) B A E D C (7) A E B C D (6) A B E D C (6) E C A B D (5) E A C B D (5) B A D E C (5) C E A D B (4) C E D A B (3) C D E B A (3) C D E A B (2) A B E C D (2) E C A D B (1) E A B C D (1) D B A E C (1) D A B E C (1) C E A B D (1) B D C A E (1) A E B D C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 20 -8 30 B 20 0 32 2 36 C -20 -32 0 -34 -6 D 8 -2 34 0 16 E -30 -36 6 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 20 -8 30 B 20 0 32 2 36 C -20 -32 0 -34 -6 D 8 -2 34 0 16 E -30 -36 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999898616 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 B=23 A=17 C=13 E=12 so E is eliminated. Round 2 votes counts: D=35 B=23 A=23 C=19 so C is eliminated. Round 3 votes counts: D=43 A=34 B=23 so B is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. B:245 D:228 A:211 E:162 C:154 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 20 -8 30 B 20 0 32 2 36 C -20 -32 0 -34 -6 D 8 -2 34 0 16 E -30 -36 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999898616 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 20 -8 30 B 20 0 32 2 36 C -20 -32 0 -34 -6 D 8 -2 34 0 16 E -30 -36 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999898616 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 20 -8 30 B 20 0 32 2 36 C -20 -32 0 -34 -6 D 8 -2 34 0 16 E -30 -36 6 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999898616 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9704: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) E D A C B (7) D C B E A (7) B C A D E (7) D C E B A (6) B A C E D (6) E A D B C (5) D E C A B (5) C B D A E (5) E A B D C (4) B A E C D (4) E D A B C (3) D C B A E (3) C D B A E (3) B C D A E (3) B C A E D (3) A B E C D (3) D E B C A (2) B A E D C (2) E A D C B (1) E A C D B (1) E A C B D (1) E A B C D (1) D E C B A (1) D C E A B (1) D B E A C (1) D B C A E (1) C D B E A (1) C B A D E (1) C A B D E (1) A E B D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 0 2 6 B 12 0 12 4 4 C 0 -12 0 0 -2 D -2 -4 0 0 -4 E -6 -4 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 2 6 B 12 0 12 4 4 C 0 -12 0 0 -2 D -2 -4 0 0 -4 E -6 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 B=25 E=23 A=14 C=11 so C is eliminated. Round 2 votes counts: D=31 B=31 E=23 A=15 so A is eliminated. Round 3 votes counts: B=36 E=33 D=31 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:216 A:198 E:198 D:195 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 2 6 B 12 0 12 4 4 C 0 -12 0 0 -2 D -2 -4 0 0 -4 E -6 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 2 6 B 12 0 12 4 4 C 0 -12 0 0 -2 D -2 -4 0 0 -4 E -6 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 2 6 B 12 0 12 4 4 C 0 -12 0 0 -2 D -2 -4 0 0 -4 E -6 -4 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999392 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9705: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (8) A D E C B (8) E C D B A (7) B C E D A (7) C D E A B (6) A D C E B (5) C D A E B (4) B A E C D (4) D C A E B (3) B A C D E (3) E D C B A (2) E D C A B (2) E D A C B (2) D A C E B (2) C E B D A (2) C D E B A (2) B E C D A (2) B E A D C (2) B E A C D (2) B C A E D (2) B C A D E (2) A D E B C (2) A D C B E (2) E B D A C (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E A B (1) C E D B A (1) C E D A B (1) C D A B E (1) C B D A E (1) C A D B E (1) B E C A D (1) B C D E A (1) B A C E D (1) A E D B C (1) A E B D C (1) A D B E C (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -2 0 2 10 B 2 0 -8 -12 -12 C 0 8 0 2 -2 D -2 12 -2 0 -4 E -10 12 2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.432367 B: 0.000000 C: 0.567633 D: 0.000000 E: 0.000000 Sum of squares = 0.509148489872 Cumulative probabilities = A: 0.432367 B: 0.432367 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 0 2 10 B 2 0 -8 -12 -12 C 0 8 0 2 -2 D -2 12 -2 0 -4 E -10 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 A=23 C=19 E=16 D=7 so D is eliminated. Round 2 votes counts: B=35 A=25 C=23 E=17 so E is eliminated. Round 3 votes counts: B=36 C=35 A=29 so A is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:205 C:204 E:204 D:202 B:185 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 0 2 10 B 2 0 -8 -12 -12 C 0 8 0 2 -2 D -2 12 -2 0 -4 E -10 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 2 10 B 2 0 -8 -12 -12 C 0 8 0 2 -2 D -2 12 -2 0 -4 E -10 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 2 10 B 2 0 -8 -12 -12 C 0 8 0 2 -2 D -2 12 -2 0 -4 E -10 12 2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999897 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9706: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (25) E D B C A (23) E D B A C (6) B D E C A (6) C A B D E (5) D B E A C (4) C E B D A (4) A C E D B (4) A D B E C (3) D B E C A (2) C A E B D (2) A D B C E (2) A B D C E (2) E D C B A (1) E D A B C (1) E A D B C (1) D E B A C (1) C E A B D (1) C B D E A (1) C A E D B (1) B D C E A (1) B C D E A (1) B A D C E (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 4 -2 -6 B 2 0 8 -2 6 C -4 -8 0 -8 2 D 2 2 8 0 10 E 6 -6 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 4 -2 -6 B 2 0 8 -2 6 C -4 -8 0 -8 2 D 2 2 8 0 10 E 6 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=32 C=14 B=9 D=7 so D is eliminated. Round 2 votes counts: A=38 E=33 B=15 C=14 so C is eliminated. Round 3 votes counts: A=46 E=38 B=16 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:211 B:207 A:197 E:194 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 4 -2 -6 B 2 0 8 -2 6 C -4 -8 0 -8 2 D 2 2 8 0 10 E 6 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -2 -6 B 2 0 8 -2 6 C -4 -8 0 -8 2 D 2 2 8 0 10 E 6 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -2 -6 B 2 0 8 -2 6 C -4 -8 0 -8 2 D 2 2 8 0 10 E 6 -6 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9707: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (6) E B C D A (5) C D E B A (5) B C E D A (5) E B A D C (4) D C A E B (4) D C A B E (4) C D B A E (4) A E D B C (4) A B C D E (4) E A D B C (3) D A C E B (3) C D B E A (3) E D C B A (2) E C D B A (2) D E C B A (2) D E C A B (2) D C E B A (2) C D A B E (2) C B E D A (2) B E C D A (2) B E C A D (2) B E A C D (2) B C A D E (2) B A E C D (2) A E B D C (2) A D E C B (2) A D E B C (2) A D B C E (2) A B E D C (2) E C B D A (1) E B C A D (1) E A B D C (1) D E A C B (1) D C E A B (1) C E D B A (1) C B D E A (1) C B D A E (1) C A B D E (1) B C E A D (1) B C A E D (1) A D C E B (1) Total count = 100 A B C D E A 0 -6 -18 -10 -2 B 6 0 -6 -16 -2 C 18 6 0 0 12 D 10 16 0 0 10 E 2 2 -12 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.516478 D: 0.483522 E: 0.000000 Sum of squares = 0.500543037615 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.516478 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -18 -10 -2 B 6 0 -6 -16 -2 C 18 6 0 0 12 D 10 16 0 0 10 E 2 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=20 E=19 D=19 B=17 so B is eliminated. Round 2 votes counts: C=29 A=27 E=25 D=19 so D is eliminated. Round 3 votes counts: C=40 E=30 A=30 so E is eliminated. Round 4 votes counts: C=59 A=41 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:218 D:218 B:191 E:191 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -18 -10 -2 B 6 0 -6 -16 -2 C 18 6 0 0 12 D 10 16 0 0 10 E 2 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -18 -10 -2 B 6 0 -6 -16 -2 C 18 6 0 0 12 D 10 16 0 0 10 E 2 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -18 -10 -2 B 6 0 -6 -16 -2 C 18 6 0 0 12 D 10 16 0 0 10 E 2 2 -12 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9708: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (9) E B A C D (8) B E A C D (7) D C A E B (6) A C D B E (5) E B C D A (4) D C E B A (4) B E C A D (4) B A E C D (4) A B C D E (4) E D C B A (3) E D C A B (3) E B D C A (3) E B A D C (3) E A B D C (3) D C E A B (3) C D B E A (2) C D A B E (2) C B D A E (2) B E C D A (2) A D C E B (2) A D C B E (2) E B D A C (1) E A D C B (1) D E C B A (1) D E C A B (1) D A E C B (1) D A C E B (1) C D E B A (1) C B A D E (1) C A D B E (1) B C E D A (1) B A C E D (1) A E B C D (1) A D E C B (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 -6 0 -10 B 4 0 -4 2 -2 C 6 4 0 4 -4 D 0 -2 -4 0 -2 E 10 2 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999808 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -6 0 -10 B 4 0 -4 2 -2 C 6 4 0 4 -4 D 0 -2 -4 0 -2 E 10 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=26 B=19 A=17 C=9 so C is eliminated. Round 2 votes counts: D=31 E=29 B=22 A=18 so A is eliminated. Round 3 votes counts: D=42 E=30 B=28 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:209 C:205 B:200 D:196 A:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -6 0 -10 B 4 0 -4 2 -2 C 6 4 0 4 -4 D 0 -2 -4 0 -2 E 10 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 0 -10 B 4 0 -4 2 -2 C 6 4 0 4 -4 D 0 -2 -4 0 -2 E 10 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 0 -10 B 4 0 -4 2 -2 C 6 4 0 4 -4 D 0 -2 -4 0 -2 E 10 2 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999651 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9709: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A C B (7) C E B A D (5) C B D A E (5) E A D B C (4) C B E A D (4) C B D E A (4) E D C A B (3) E C A D B (3) E A C D B (3) D C E A B (3) D B C A E (3) C E D B A (3) C E D A B (3) C D B E A (3) C B A E D (3) A E D B C (3) A E B D C (3) E C A B D (2) E A D C B (2) E A B C D (2) D B A E C (2) B A D E C (2) A D B E C (2) A B D E C (2) E D A B C (1) E C D A B (1) E B A C D (1) E A C B D (1) E A B D C (1) D E A C B (1) D E A B C (1) D C B A E (1) D C A B E (1) D A E B C (1) C E B D A (1) C E A D B (1) C D E B A (1) C D E A B (1) B E A C D (1) B D C A E (1) B D A C E (1) B C A E D (1) B C A D E (1) B A E D C (1) B A D C E (1) B A C E D (1) B A C D E (1) A B E D C (1) Total count = 100 A B C D E A 0 6 -8 4 -26 B -6 0 -24 -8 -16 C 8 24 0 4 -2 D -4 8 -4 0 -24 E 26 16 2 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 -8 4 -26 B -6 0 -24 -8 -16 C 8 24 0 4 -2 D -4 8 -4 0 -24 E 26 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=31 D=13 B=11 A=11 so B is eliminated. Round 2 votes counts: C=36 E=32 A=17 D=15 so D is eliminated. Round 3 votes counts: C=45 E=34 A=21 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:234 C:217 A:188 D:188 B:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 -8 4 -26 B -6 0 -24 -8 -16 C 8 24 0 4 -2 D -4 8 -4 0 -24 E 26 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 4 -26 B -6 0 -24 -8 -16 C 8 24 0 4 -2 D -4 8 -4 0 -24 E 26 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 4 -26 B -6 0 -24 -8 -16 C 8 24 0 4 -2 D -4 8 -4 0 -24 E 26 16 2 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999964765 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9710: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (12) B E A C D (8) B E C A D (5) B A E D C (5) A E B C D (5) D C B E A (4) D C B A E (4) D B C E A (4) D B C A E (4) B E A D C (4) B C E A D (4) E A C B D (3) D A C E B (3) C E B A D (3) B C D E A (3) C B E D A (2) C B D E A (2) A E D B C (2) A E C D B (2) A E C B D (2) A E B D C (2) E C A B D (1) E A B C D (1) D C E A B (1) D B A E C (1) D A E C B (1) D A E B C (1) D A B E C (1) C E A D B (1) C E A B D (1) C D B E A (1) C A E D B (1) B D E A C (1) B D C E A (1) A E D C B (1) A D E C B (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -8 8 0 B 12 0 8 8 10 C 8 -8 0 -8 2 D -8 -8 8 0 -8 E 0 -10 -2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -8 8 0 B 12 0 8 8 10 C 8 -8 0 -8 2 D -8 -8 8 0 -8 E 0 -10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=36 B=31 A=17 C=11 E=5 so E is eliminated. Round 2 votes counts: D=36 B=31 A=21 C=12 so C is eliminated. Round 3 votes counts: B=38 D=37 A=25 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:219 E:198 C:197 A:194 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -8 8 0 B 12 0 8 8 10 C 8 -8 0 -8 2 D -8 -8 8 0 -8 E 0 -10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 8 0 B 12 0 8 8 10 C 8 -8 0 -8 2 D -8 -8 8 0 -8 E 0 -10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 8 0 B 12 0 8 8 10 C 8 -8 0 -8 2 D -8 -8 8 0 -8 E 0 -10 -2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9711: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (13) D E A B C (6) A C B E D (6) C B A E D (5) C A B E D (5) E D A B C (4) D E B A C (4) A E C B D (4) D B E C A (3) C B A D E (3) C A B D E (3) B C D E A (3) B C D A E (3) B C A E D (3) E B D C A (2) E A D C B (2) E A D B C (2) E A B D C (2) D E A C B (2) D B C E A (2) B D C E A (2) A C E B D (2) A C D E B (2) E A C D B (1) E A B C D (1) D A E C B (1) D A C B E (1) C B D A E (1) C A D B E (1) B E C D A (1) B E C A D (1) B D E C A (1) B C E D A (1) B C E A D (1) A E D C B (1) A D E C B (1) A D C E B (1) A C E D B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 2 -8 0 -8 B -2 0 10 2 -4 C 8 -10 0 0 -4 D 0 -2 0 0 10 E 8 4 4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.329268 B: 0.243902 C: 0.012195 D: 0.365854 E: 0.048780 Sum of squares = 0.304283164731 Cumulative probabilities = A: 0.329268 B: 0.573171 C: 0.585366 D: 0.951220 E: 1.000000 A B C D E A 0 2 -8 0 -8 B -2 0 10 2 -4 C 8 -10 0 0 -4 D 0 -2 0 0 10 E 8 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.329268 B: 0.243902 C: 0.012195 D: 0.365854 E: 0.048780 Sum of squares = 0.304283164783 Cumulative probabilities = A: 0.329268 B: 0.573171 C: 0.585366 D: 0.951220 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=20 C=18 B=16 E=14 so E is eliminated. Round 2 votes counts: D=36 A=28 C=18 B=18 so C is eliminated. Round 3 votes counts: A=37 D=36 B=27 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:204 B:203 E:203 C:197 A:193 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -8 0 -8 B -2 0 10 2 -4 C 8 -10 0 0 -4 D 0 -2 0 0 10 E 8 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.329268 B: 0.243902 C: 0.012195 D: 0.365854 E: 0.048780 Sum of squares = 0.304283164783 Cumulative probabilities = A: 0.329268 B: 0.573171 C: 0.585366 D: 0.951220 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -8 0 -8 B -2 0 10 2 -4 C 8 -10 0 0 -4 D 0 -2 0 0 10 E 8 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.329268 B: 0.243902 C: 0.012195 D: 0.365854 E: 0.048780 Sum of squares = 0.304283164783 Cumulative probabilities = A: 0.329268 B: 0.573171 C: 0.585366 D: 0.951220 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -8 0 -8 B -2 0 10 2 -4 C 8 -10 0 0 -4 D 0 -2 0 0 10 E 8 4 4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.329268 B: 0.243902 C: 0.012195 D: 0.365854 E: 0.048780 Sum of squares = 0.304283164783 Cumulative probabilities = A: 0.329268 B: 0.573171 C: 0.585366 D: 0.951220 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9712: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) B C E A D (8) A E D C B (8) E A D C B (7) B C A E D (7) D E A C B (6) A E D B C (6) D C B E A (4) B C D A E (4) B A E C D (4) D A E B C (3) B C D E A (3) A E B C D (3) E A C D B (2) D C E A B (2) D A B E C (2) C B E A D (2) B D A E C (2) B A D E C (2) A D E B C (2) E D A C B (1) E A C B D (1) D E C A B (1) D C E B A (1) D B C E A (1) D B A E C (1) D B A C E (1) D A E C B (1) C E A D B (1) C D E A B (1) C D B E A (1) C B E D A (1) B A E D C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 10 12 -2 B 2 0 4 -4 6 C -10 -4 0 -8 -10 D -12 4 8 0 -8 E 2 -6 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.222222 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.506172839506 Cumulative probabilities = A: 0.222222 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 A B C D E A 0 -2 10 12 -2 B 2 0 4 -4 6 C -10 -4 0 -8 -10 D -12 4 8 0 -8 E 2 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.50617283951 Cumulative probabilities = A: 0.222222 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 D=23 A=21 C=14 E=11 so E is eliminated. Round 2 votes counts: B=31 A=31 D=24 C=14 so C is eliminated. Round 3 votes counts: B=42 A=32 D=26 so D is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:209 E:207 B:204 D:196 C:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 -2 10 12 -2 B 2 0 4 -4 6 C -10 -4 0 -8 -10 D -12 4 8 0 -8 E 2 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.50617283951 Cumulative probabilities = A: 0.222222 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 12 -2 B 2 0 4 -4 6 C -10 -4 0 -8 -10 D -12 4 8 0 -8 E 2 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.50617283951 Cumulative probabilities = A: 0.222222 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 12 -2 B 2 0 4 -4 6 C -10 -4 0 -8 -10 D -12 4 8 0 -8 E 2 -6 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.222222 B: 0.666667 C: 0.000000 D: 0.111111 E: 0.000000 Sum of squares = 0.50617283951 Cumulative probabilities = A: 0.222222 B: 0.888889 C: 0.888889 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9713: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) B C D A E (7) E A D C B (6) C B D E A (6) A E B C D (6) C B E A D (5) B C A E D (5) D C B E A (4) D B C E A (4) C D B E A (4) E A C D B (3) D E A C B (3) B C D E A (3) A B E C D (3) D E C A B (2) D B C A E (2) D A E B C (2) C E A B D (2) C B A E D (2) B D C A E (2) A E D B C (2) A E C B D (2) E D A C B (1) E C A B D (1) D C E B A (1) D C E A B (1) D B A E C (1) D A B E C (1) C E B A D (1) C D E B A (1) B C A D E (1) B A E C D (1) B A D E C (1) B A C E D (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -8 10 4 B 4 0 -12 2 8 C 8 12 0 8 4 D -10 -2 -8 0 -8 E -4 -8 -4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -8 10 4 B 4 0 -12 2 8 C 8 12 0 8 4 D -10 -2 -8 0 -8 E -4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=21 C=21 B=21 E=11 so E is eliminated. Round 2 votes counts: A=35 D=22 C=22 B=21 so B is eliminated. Round 3 votes counts: C=38 A=38 D=24 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:216 A:201 B:201 E:196 D:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -8 10 4 B 4 0 -12 2 8 C 8 12 0 8 4 D -10 -2 -8 0 -8 E -4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 10 4 B 4 0 -12 2 8 C 8 12 0 8 4 D -10 -2 -8 0 -8 E -4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 10 4 B 4 0 -12 2 8 C 8 12 0 8 4 D -10 -2 -8 0 -8 E -4 -8 -4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9714: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (10) A C E B D (9) D B C E A (8) D A B E C (8) D B E C A (6) A D E C B (6) D A C E B (5) C E B A D (4) D A E C B (3) B E C A D (3) B D E C A (3) B C E D A (3) A E C D B (3) A D C E B (3) E C B A D (2) E B C A D (2) D B E A C (2) B E D C A (2) B C E A D (2) A C E D B (2) E C A B D (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A E C (1) D A E B C (1) C E A B D (1) C B E A D (1) C A E B D (1) B E C D A (1) B D C E A (1) A E D C B (1) A E B C D (1) A D E B C (1) Total count = 100 A B C D E A 0 14 12 6 16 B -14 0 -8 -6 -12 C -12 8 0 -8 -14 D -6 6 8 0 2 E -16 12 14 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 12 6 16 B -14 0 -8 -6 -12 C -12 8 0 -8 -14 D -6 6 8 0 2 E -16 12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 A=36 B=15 C=7 E=5 so E is eliminated. Round 2 votes counts: D=37 A=36 B=17 C=10 so C is eliminated. Round 3 votes counts: A=39 D=37 B=24 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:224 D:205 E:204 C:187 B:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 12 6 16 B -14 0 -8 -6 -12 C -12 8 0 -8 -14 D -6 6 8 0 2 E -16 12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 12 6 16 B -14 0 -8 -6 -12 C -12 8 0 -8 -14 D -6 6 8 0 2 E -16 12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 12 6 16 B -14 0 -8 -6 -12 C -12 8 0 -8 -14 D -6 6 8 0 2 E -16 12 14 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999841 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9715: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (11) A D B C E (7) D A E C B (6) C B E A D (6) E D A C B (5) E D A B C (5) B C E A D (5) D E A C B (4) D A E B C (4) C B A D E (4) B C A D E (4) E C D B A (3) D A C B E (3) B A C D E (3) E D C A B (2) D E A B C (2) D A C E B (2) D A B E C (2) C E B D A (2) B A D C E (2) A D E B C (2) A D C B E (2) A D B E C (2) E D C B A (1) E C B A D (1) E B A D C (1) D A B C E (1) C D E A B (1) C D B A E (1) C B A E D (1) B E C A D (1) B C A E D (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 6 12 -10 0 B -6 0 -10 -10 -4 C -12 10 0 -10 -6 D 10 10 10 0 8 E 0 4 6 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 -10 0 B -6 0 -10 -10 -4 C -12 10 0 -10 -6 D 10 10 10 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=24 B=16 A=16 C=15 so C is eliminated. Round 2 votes counts: E=31 B=27 D=26 A=16 so A is eliminated. Round 3 votes counts: D=39 E=31 B=30 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:204 E:201 C:191 B:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 12 -10 0 B -6 0 -10 -10 -4 C -12 10 0 -10 -6 D 10 10 10 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 -10 0 B -6 0 -10 -10 -4 C -12 10 0 -10 -6 D 10 10 10 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 -10 0 B -6 0 -10 -10 -4 C -12 10 0 -10 -6 D 10 10 10 0 8 E 0 4 6 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9716: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (9) C B D A E (7) A D E B C (7) E A D B C (6) C D B A E (4) B C E D A (4) B C D E A (4) A E D C B (4) E B D A C (3) E B A D C (3) C D A B E (3) C B A D E (3) C A D B E (3) B D E C A (3) A E C D B (3) E A B D C (2) D E A B C (2) D B E A C (2) D A E B C (2) D A B E C (2) B E C D A (2) B D E A C (2) A E D B C (2) A D E C B (2) E D B A C (1) E C B A D (1) E B C A D (1) E A D C B (1) E A C B D (1) D E B A C (1) D B A E C (1) D A C B E (1) C E A B D (1) C B E A D (1) C B D E A (1) B D C E A (1) B D C A E (1) B C D A E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 0 -14 -4 B 12 0 6 2 10 C 0 -6 0 2 -8 D 14 -2 -2 0 8 E 4 -10 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999689 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -14 -4 B 12 0 6 2 10 C 0 -6 0 2 -8 D 14 -2 -2 0 8 E 4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=20 E=19 B=18 D=11 so D is eliminated. Round 2 votes counts: C=32 A=25 E=22 B=21 so B is eliminated. Round 3 votes counts: C=43 E=31 A=26 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 D:209 E:197 C:194 A:185 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 0 -14 -4 B 12 0 6 2 10 C 0 -6 0 2 -8 D 14 -2 -2 0 8 E 4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -14 -4 B 12 0 6 2 10 C 0 -6 0 2 -8 D 14 -2 -2 0 8 E 4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -14 -4 B 12 0 6 2 10 C 0 -6 0 2 -8 D 14 -2 -2 0 8 E 4 -10 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999995706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9717: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (18) B C A D E (17) C E D A B (12) E D A C B (8) A D B E C (7) D A E B C (6) B A D C E (5) A D E B C (5) E C D A B (4) C B E D A (4) C B E A D (4) C E D B A (3) E D C A B (2) D E A B C (1) C E B D A (1) C B A E D (1) B A C D E (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 4 18 22 B 8 0 22 4 16 C -4 -22 0 -6 -4 D -18 -4 6 0 22 E -22 -16 4 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999099 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 4 18 22 B 8 0 22 4 16 C -4 -22 0 -6 -4 D -18 -4 6 0 22 E -22 -16 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996266 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=41 C=25 E=14 A=13 D=7 so D is eliminated. Round 2 votes counts: B=41 C=25 A=19 E=15 so E is eliminated. Round 3 votes counts: B=41 C=31 A=28 so A is eliminated. Round 4 votes counts: B=61 C=39 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 A:218 D:203 C:182 E:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 4 18 22 B 8 0 22 4 16 C -4 -22 0 -6 -4 D -18 -4 6 0 22 E -22 -16 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996266 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 4 18 22 B 8 0 22 4 16 C -4 -22 0 -6 -4 D -18 -4 6 0 22 E -22 -16 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996266 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 4 18 22 B 8 0 22 4 16 C -4 -22 0 -6 -4 D -18 -4 6 0 22 E -22 -16 4 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996266 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9718: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (11) E D B C A (10) A E D B C (10) A C B D E (10) E D B A C (9) B C D E A (5) A C B E D (5) D E B C A (4) C B A D E (3) C A B D E (3) B D E C A (3) E D A B C (2) D B E C A (2) C E D B A (2) C B D A E (2) A C E D B (2) A C E B D (2) A B C D E (2) E D C B A (1) E A D C B (1) D E C B A (1) C D E B A (1) C D B E A (1) C A E D B (1) C A B E D (1) B D C E A (1) B D A E C (1) B A C D E (1) A E D C B (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -16 -4 -12 -8 B 16 0 4 4 6 C 4 -4 0 4 6 D 12 -4 -4 0 4 E 8 -6 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -4 -12 -8 B 16 0 4 4 6 C 4 -4 0 4 6 D 12 -4 -4 0 4 E 8 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=34 C=25 E=23 B=11 D=7 so D is eliminated. Round 2 votes counts: A=34 E=28 C=25 B=13 so B is eliminated. Round 3 votes counts: A=36 E=33 C=31 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:215 C:205 D:204 E:196 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -4 -12 -8 B 16 0 4 4 6 C 4 -4 0 4 6 D 12 -4 -4 0 4 E 8 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -4 -12 -8 B 16 0 4 4 6 C 4 -4 0 4 6 D 12 -4 -4 0 4 E 8 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -4 -12 -8 B 16 0 4 4 6 C 4 -4 0 4 6 D 12 -4 -4 0 4 E 8 -6 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999832 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9719: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (15) E B A C D (13) D C A E B (10) B E D A C (6) C D A E B (5) B E A C D (5) D C E A B (4) E B C A D (3) D B C A E (3) C E A D B (3) B D A E C (3) A E C B D (3) E C A B D (2) E B D C A (2) D B A C E (2) B E A D C (2) B A E C D (2) E C D B A (1) E C B D A (1) E C B A D (1) E B D A C (1) E A C B D (1) D E C B A (1) D C B A E (1) D B E C A (1) D B E A C (1) D B C E A (1) D A C B E (1) C D A B E (1) C A E D B (1) C A D E B (1) B A D E C (1) B A D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -14 -20 2 B 4 0 -4 -4 -6 C 14 4 0 -12 0 D 20 4 12 0 6 E -2 6 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -20 2 B 4 0 -4 -4 -6 C 14 4 0 -12 0 D 20 4 12 0 6 E -2 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=40 E=25 B=20 C=11 A=4 so A is eliminated. Round 2 votes counts: D=40 E=28 B=21 C=11 so C is eliminated. Round 3 votes counts: D=47 E=32 B=21 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:221 C:203 E:199 B:195 A:182 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -14 -20 2 B 4 0 -4 -4 -6 C 14 4 0 -12 0 D 20 4 12 0 6 E -2 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -20 2 B 4 0 -4 -4 -6 C 14 4 0 -12 0 D 20 4 12 0 6 E -2 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -20 2 B 4 0 -4 -4 -6 C 14 4 0 -12 0 D 20 4 12 0 6 E -2 6 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991131 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9720: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (12) D A B C E (6) C E D B A (6) A D B E C (6) D A B E C (5) C E B D A (5) D C E A B (4) C D E B A (4) A B E D C (4) A B E C D (4) D E C A B (3) D A E B C (3) B E C A D (3) B A E C D (3) E B C A D (2) E B A C D (2) D C E B A (2) D C A B E (2) C E B A D (2) B A C E D (2) A B D E C (2) A B D C E (2) E C D B A (1) E A B D C (1) E A B C D (1) D E C B A (1) D E A C B (1) D E A B C (1) D C B E A (1) D A E C B (1) C D B A E (1) C B E A D (1) B E A C D (1) B C A E D (1) A E B D C (1) A E B C D (1) A D B C E (1) A B C E D (1) Total count = 100 A B C D E A 0 0 -2 6 -8 B 0 0 6 2 -8 C 2 -6 0 6 -18 D -6 -2 -6 0 -8 E 8 8 18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -2 6 -8 B 0 0 6 2 -8 C 2 -6 0 6 -18 D -6 -2 -6 0 -8 E 8 8 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 A=22 E=19 C=19 B=10 so B is eliminated. Round 2 votes counts: D=30 A=27 E=23 C=20 so C is eliminated. Round 3 votes counts: E=37 D=35 A=28 so A is eliminated. Round 4 votes counts: E=54 D=46 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:200 A:198 C:192 D:189 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -2 6 -8 B 0 0 6 2 -8 C 2 -6 0 6 -18 D -6 -2 -6 0 -8 E 8 8 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -2 6 -8 B 0 0 6 2 -8 C 2 -6 0 6 -18 D -6 -2 -6 0 -8 E 8 8 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -2 6 -8 B 0 0 6 2 -8 C 2 -6 0 6 -18 D -6 -2 -6 0 -8 E 8 8 18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999128 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9721: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D C E (12) C E D B A (8) E C D A B (7) C E B A D (7) B A D E C (7) B A C D E (5) A B D E C (5) D E C A B (4) D A B E C (4) C E D A B (4) A D B E C (4) E D C A B (3) E D A B C (3) D E A C B (3) C E B D A (3) C B A D E (3) E C B A D (2) D E A B C (2) D C E A B (2) B A C E D (2) E C B D A (1) E A B C D (1) D A E B C (1) D A B C E (1) C B E A D (1) B E A C D (1) B C A E D (1) B C A D E (1) B A E D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -10 6 8 -4 B 10 0 4 8 -2 C -6 -4 0 -6 0 D -8 -8 6 0 8 E 4 2 0 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407405 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 A B C D E A 0 -10 6 8 -4 B 10 0 4 8 -2 C -6 -4 0 -6 0 D -8 -8 6 0 8 E 4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407354 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 C=26 E=17 D=17 A=10 so A is eliminated. Round 2 votes counts: B=36 C=26 D=21 E=17 so E is eliminated. Round 3 votes counts: B=37 C=36 D=27 so D is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:200 D:199 E:199 C:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 6 8 -4 B 10 0 4 8 -2 C -6 -4 0 -6 0 D -8 -8 6 0 8 E 4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407354 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 8 -4 B 10 0 4 8 -2 C -6 -4 0 -6 0 D -8 -8 6 0 8 E 4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407354 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 8 -4 B 10 0 4 8 -2 C -6 -4 0 -6 0 D -8 -8 6 0 8 E 4 2 0 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.444444 C: 0.000000 D: 0.111111 E: 0.444444 Sum of squares = 0.407407407354 Cumulative probabilities = A: 0.000000 B: 0.444444 C: 0.444444 D: 0.555556 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9722: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (8) D B E C A (8) B D E C A (7) A C E D B (6) A C E B D (6) E D B C A (5) C A B D E (5) A C B D E (5) C A E B D (4) A E C D B (4) D B A E C (3) B D C E A (3) B D C A E (3) B D A C E (3) E D A C B (2) E D A B C (2) E A C D B (2) D B E A C (2) D B A C E (2) C B E A D (2) E D C B A (1) E C B D A (1) E C A B D (1) E B D C A (1) E B C D A (1) E A D C B (1) D E B A C (1) C E B A D (1) C B E D A (1) C B A E D (1) C B A D E (1) C A B E D (1) B C D E A (1) B C D A E (1) B C A D E (1) B A D C E (1) B A C D E (1) A D E B C (1) Total count = 100 A B C D E A 0 -20 0 -12 -2 B 20 0 10 4 4 C 0 -10 0 -8 -2 D 12 -4 8 0 -2 E 2 -4 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 0 -12 -2 B 20 0 10 4 4 C 0 -10 0 -8 -2 D 12 -4 8 0 -2 E 2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 A=22 B=21 D=16 C=16 so D is eliminated. Round 2 votes counts: B=36 E=26 A=22 C=16 so C is eliminated. Round 3 votes counts: B=41 A=32 E=27 so E is eliminated. Round 4 votes counts: B=60 A=40 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:207 E:201 C:190 A:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 0 -12 -2 B 20 0 10 4 4 C 0 -10 0 -8 -2 D 12 -4 8 0 -2 E 2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 0 -12 -2 B 20 0 10 4 4 C 0 -10 0 -8 -2 D 12 -4 8 0 -2 E 2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 0 -12 -2 B 20 0 10 4 4 C 0 -10 0 -8 -2 D 12 -4 8 0 -2 E 2 -4 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9723: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) C B A E D (10) B C A D E (7) E A D C B (6) D B C E A (5) E C A B D (4) D E C B A (4) D E B C A (4) C E A B D (4) A E D B C (4) D B C A E (3) A B C E D (3) E D C B A (2) E D A C B (2) E D A B C (2) E A C D B (2) E A C B D (2) D B A E C (2) C E D B A (2) C B E A D (2) C B A D E (2) E D C A B (1) E C A D B (1) E A D B C (1) D E B A C (1) D E A C B (1) D B E A C (1) C A E B D (1) B D C A E (1) B C D E A (1) B C D A E (1) B A C E D (1) B A C D E (1) A E B C D (1) A D E B C (1) A C E B D (1) A B E D C (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 -10 14 -16 B 0 0 6 -10 -14 C 10 -6 0 -4 -8 D -14 10 4 0 -8 E 16 14 8 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 0 -10 14 -16 B 0 0 6 -10 -14 C 10 -6 0 -4 -8 D -14 10 4 0 -8 E 16 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 E=23 C=21 A=13 B=12 so B is eliminated. Round 2 votes counts: D=32 C=30 E=23 A=15 so A is eliminated. Round 3 votes counts: C=37 D=33 E=30 so E is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:223 C:196 D:196 A:194 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 0 -10 14 -16 B 0 0 6 -10 -14 C 10 -6 0 -4 -8 D -14 10 4 0 -8 E 16 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -10 14 -16 B 0 0 6 -10 -14 C 10 -6 0 -4 -8 D -14 10 4 0 -8 E 16 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -10 14 -16 B 0 0 6 -10 -14 C 10 -6 0 -4 -8 D -14 10 4 0 -8 E 16 14 8 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9724: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E C D (8) A C B E D (8) D C E A B (6) D C A E B (5) B E D A C (5) B E A D C (5) B E A C D (5) E B D A C (4) C D A E B (4) C A D E B (4) C A D B E (4) E D B C A (3) E B A C D (3) D E C B A (3) D E B C A (3) A C B D E (3) A B C E D (3) E D C A B (2) E B D C A (2) D C E B A (2) C D A B E (2) C A E D B (2) B D E C A (2) E C D A B (1) E C A D B (1) E B A D C (1) D E C A B (1) D C B A E (1) D C A B E (1) D B E C A (1) B A E D C (1) B A C E D (1) B A C D E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 0 4 0 B 2 0 -4 4 4 C 0 4 0 4 -2 D -4 -4 -4 0 -12 E 0 -4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999917 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 A B C D E A 0 -2 0 4 0 B 2 0 -4 4 4 C 0 4 0 4 -2 D -4 -4 -4 0 -12 E 0 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999846 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=23 E=17 C=16 A=16 so C is eliminated. Round 2 votes counts: D=29 B=28 A=26 E=17 so E is eliminated. Round 3 votes counts: B=38 D=35 A=27 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:205 B:203 C:203 A:201 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 0 4 0 B 2 0 -4 4 4 C 0 4 0 4 -2 D -4 -4 -4 0 -12 E 0 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999846 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 4 0 B 2 0 -4 4 4 C 0 4 0 4 -2 D -4 -4 -4 0 -12 E 0 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999846 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 4 0 B 2 0 -4 4 4 C 0 4 0 4 -2 D -4 -4 -4 0 -12 E 0 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.400000 Sum of squares = 0.359999999846 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 0.600000 D: 0.600000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9725: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (24) D E C B A (16) E C B A D (7) D A B C E (7) C B E A D (7) D A B E C (5) C E B A D (4) B C A E D (4) E C B D A (3) D C E B A (2) D A E B C (2) C B A E D (2) A B C D E (2) E D C B A (1) E C D B A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C E A B (1) D C A B E (1) D A C B E (1) C E B D A (1) A E B C D (1) A D B E C (1) A D B C E (1) A B E D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -2 12 8 B -4 0 2 16 16 C 2 -2 0 12 16 D -12 -16 -12 0 -12 E -8 -16 -16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000074 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 12 8 B -4 0 2 16 16 C 2 -2 0 12 16 D -12 -16 -12 0 -12 E -8 -16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000073 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 A=32 C=14 E=12 B=4 so B is eliminated. Round 2 votes counts: D=38 A=32 C=18 E=12 so E is eliminated. Round 3 votes counts: D=39 A=32 C=29 so C is eliminated. Round 4 votes counts: A=56 D=44 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:215 C:214 A:211 E:186 D:174 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 -2 12 8 B -4 0 2 16 16 C 2 -2 0 12 16 D -12 -16 -12 0 -12 E -8 -16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000073 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 12 8 B -4 0 2 16 16 C 2 -2 0 12 16 D -12 -16 -12 0 -12 E -8 -16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000073 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 12 8 B -4 0 2 16 16 C 2 -2 0 12 16 D -12 -16 -12 0 -12 E -8 -16 -16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.375000000073 Cumulative probabilities = A: 0.250000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9726: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C E D A (9) D A E C B (8) A D E C B (8) A E D C B (7) D B C E A (6) A E C D B (5) A B E C D (5) D C E B A (4) D B C A E (4) D A B C E (4) B C D E A (4) E C A B D (3) E A C D B (3) B C E A D (3) E A C B D (2) B D C E A (2) B D C A E (2) B A D C E (2) A E C B D (2) A E B C D (2) A D E B C (2) E D C A B (1) E C D B A (1) E C B D A (1) E C A D B (1) E B C A D (1) D E C A B (1) D B A C E (1) D A E B C (1) D A B E C (1) C E B D A (1) C E B A D (1) B C A D E (1) A B D E C (1) Total count = 100 A B C D E A 0 14 8 -2 12 B -14 0 2 -16 -10 C -8 -2 0 -10 -12 D 2 16 10 0 4 E -12 10 12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999574 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 8 -2 12 B -14 0 2 -16 -10 C -8 -2 0 -10 -12 D 2 16 10 0 4 E -12 10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 D=30 B=23 E=13 C=2 so C is eliminated. Round 2 votes counts: A=32 D=30 B=23 E=15 so E is eliminated. Round 3 votes counts: A=41 D=32 B=27 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. A:216 D:216 E:203 C:184 B:181 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 8 -2 12 B -14 0 2 -16 -10 C -8 -2 0 -10 -12 D 2 16 10 0 4 E -12 10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -2 12 B -14 0 2 -16 -10 C -8 -2 0 -10 -12 D 2 16 10 0 4 E -12 10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -2 12 B -14 0 2 -16 -10 C -8 -2 0 -10 -12 D 2 16 10 0 4 E -12 10 12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999995858 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9727: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (7) D A C E B (6) B E A D C (6) D A B E C (5) C D A E B (5) B E C A D (5) D C A B E (4) D A C B E (4) C E B A D (4) B C E D A (4) E B A C D (3) D C A E B (3) C D B A E (3) C B D E A (3) E C B A D (2) E A B C D (2) D A E B C (2) C E A D B (2) B E D A C (2) B D A E C (2) B C D E A (2) A E D B C (2) A D E B C (2) E C A B D (1) E B C A D (1) E A C B D (1) E A B D C (1) D B C A E (1) D B A C E (1) C E D B A (1) C E B D A (1) C E A B D (1) C D E A B (1) C D B E A (1) C D A B E (1) C B E D A (1) C B D A E (1) B E C D A (1) B E A C D (1) B D E A C (1) A E D C B (1) A D E C B (1) A D B E C (1) Total count = 100 A B C D E A 0 6 2 -26 4 B -6 0 4 -8 14 C -2 -4 0 -4 14 D 26 8 4 0 14 E -4 -14 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999419 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 2 -26 4 B -6 0 4 -8 14 C -2 -4 0 -4 14 D 26 8 4 0 14 E -4 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=25 B=24 E=11 A=7 so A is eliminated. Round 2 votes counts: D=37 C=25 B=24 E=14 so E is eliminated. Round 3 votes counts: D=40 B=31 C=29 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:226 B:202 C:202 A:193 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -26 4 B -6 0 4 -8 14 C -2 -4 0 -4 14 D 26 8 4 0 14 E -4 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -26 4 B -6 0 4 -8 14 C -2 -4 0 -4 14 D 26 8 4 0 14 E -4 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -26 4 B -6 0 4 -8 14 C -2 -4 0 -4 14 D 26 8 4 0 14 E -4 -14 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994781 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9728: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (7) D B E C A (7) C A B E D (6) A C B D E (6) C B A E D (5) B C A D E (5) E D A C B (4) B D E C A (4) B C E D A (4) A C E B D (4) A C B E D (4) E D A B C (3) D E B A C (3) C B A D E (3) B C D A E (3) A C E D B (3) E D B A C (2) E C B D A (2) E A D C B (2) D E B C A (2) D E A B C (2) D B E A C (2) B C D E A (2) A C D B E (2) A B C D E (2) E C A D B (1) E C A B D (1) D B A E C (1) C B E A D (1) B E D C A (1) B D C E A (1) A E D C B (1) A E C D B (1) A D E C B (1) A D C B E (1) A D B C E (1) Total count = 100 A B C D E A 0 -10 -10 0 -2 B 10 0 4 8 22 C 10 -4 0 10 6 D 0 -8 -10 0 -4 E 2 -22 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999981 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 0 -2 B 10 0 4 8 22 C 10 -4 0 10 6 D 0 -8 -10 0 -4 E 2 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 E=22 B=20 D=17 C=15 so C is eliminated. Round 2 votes counts: A=32 B=29 E=22 D=17 so D is eliminated. Round 3 votes counts: B=39 A=32 E=29 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:211 A:189 D:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -10 0 -2 B 10 0 4 8 22 C 10 -4 0 10 6 D 0 -8 -10 0 -4 E 2 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 0 -2 B 10 0 4 8 22 C 10 -4 0 10 6 D 0 -8 -10 0 -4 E 2 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 0 -2 B 10 0 4 8 22 C 10 -4 0 10 6 D 0 -8 -10 0 -4 E 2 -22 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9729: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) B C D A E (7) E A C D B (6) A E D B C (6) E A D B C (5) E C B D A (4) B D C A E (4) E C B A D (3) E C A D B (3) C E B D A (3) C D B A E (3) C B D A E (3) B E C D A (3) B D A C E (3) A E D C B (3) A D E C B (3) A D B C E (3) E B C A D (2) E A B D C (2) D A C E B (2) C B D E A (2) B C E D A (2) A D C E B (2) A D C B E (2) A D B E C (2) E C D A B (1) E B C D A (1) E B A D C (1) D C B A E (1) D C A B E (1) D A C B E (1) D A B C E (1) C E D B A (1) C D A B E (1) C B E D A (1) B D A E C (1) B C D E A (1) B A E D C (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 6 6 6 4 B -6 0 -8 -12 -14 C -6 8 0 -4 -10 D -6 12 4 0 -10 E -4 14 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 6 6 4 B -6 0 -8 -12 -14 C -6 8 0 -4 -10 D -6 12 4 0 -10 E -4 14 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 A=23 B=22 C=14 D=6 so D is eliminated. Round 2 votes counts: E=35 A=27 B=22 C=16 so C is eliminated. Round 3 votes counts: E=39 B=32 A=29 so A is eliminated. Round 4 votes counts: E=57 B=43 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:211 D:200 C:194 B:180 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 6 6 4 B -6 0 -8 -12 -14 C -6 8 0 -4 -10 D -6 12 4 0 -10 E -4 14 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 6 6 4 B -6 0 -8 -12 -14 C -6 8 0 -4 -10 D -6 12 4 0 -10 E -4 14 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 6 6 4 B -6 0 -8 -12 -14 C -6 8 0 -4 -10 D -6 12 4 0 -10 E -4 14 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9730: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) E A D B C (8) C E A B D (8) D B A E C (7) B D E A C (7) E A C D B (4) C B D E A (4) C B D A E (4) A E C D B (4) E B D A C (3) C E A D B (3) C B E D A (3) B D C E A (3) B D A C E (3) A E D B C (3) A D E B C (3) D B E A C (2) C B A D E (2) B D C A E (2) B D A E C (2) B C D E A (2) A E D C B (2) A D B E C (2) A D B C E (2) E D B A C (1) E C A D B (1) D E B A C (1) C E B A D (1) C A E B D (1) C A B E D (1) B E D A C (1) A C E D B (1) Total count = 100 A B C D E A 0 4 12 10 -4 B -4 0 4 -6 -6 C -12 -4 0 -4 -2 D -10 6 4 0 -8 E 4 6 2 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 12 10 -4 B -4 0 4 -6 -6 C -12 -4 0 -4 -2 D -10 6 4 0 -8 E 4 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=20 E=17 A=17 D=10 so D is eliminated. Round 2 votes counts: C=36 B=29 E=18 A=17 so A is eliminated. Round 3 votes counts: C=37 B=33 E=30 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:211 E:210 D:196 B:194 C:189 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 12 10 -4 B -4 0 4 -6 -6 C -12 -4 0 -4 -2 D -10 6 4 0 -8 E 4 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 10 -4 B -4 0 4 -6 -6 C -12 -4 0 -4 -2 D -10 6 4 0 -8 E 4 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 10 -4 B -4 0 4 -6 -6 C -12 -4 0 -4 -2 D -10 6 4 0 -8 E 4 6 2 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999997589 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9731: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (8) A B D E C (7) C A E D B (6) D B E A C (5) C E A D B (5) C A E B D (5) E D C B A (4) E D B A C (4) D E B A C (4) B D A E C (4) A C B D E (4) A B C D E (4) E C D B A (3) C E D B A (3) C E D A B (3) C D B E A (3) A B E D C (3) E D B C A (2) D B E C A (2) C D E B A (2) C B D A E (2) C A B E D (2) B D E A C (2) B A D E C (2) A C E B D (2) A C B E D (2) E D A B C (1) E A C D B (1) C B A D E (1) A E C D B (1) A E B C D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 14 -2 12 12 B -14 0 -14 2 6 C 2 14 0 18 8 D -12 -2 -18 0 2 E -12 -6 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 12 12 B -14 0 -14 2 6 C 2 14 0 18 8 D -12 -2 -18 0 2 E -12 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=40 A=26 E=15 D=11 B=8 so B is eliminated. Round 2 votes counts: C=40 A=28 D=17 E=15 so E is eliminated. Round 3 votes counts: C=43 A=29 D=28 so D is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:221 A:218 B:190 E:186 D:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 12 12 B -14 0 -14 2 6 C 2 14 0 18 8 D -12 -2 -18 0 2 E -12 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 12 12 B -14 0 -14 2 6 C 2 14 0 18 8 D -12 -2 -18 0 2 E -12 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 12 12 B -14 0 -14 2 6 C 2 14 0 18 8 D -12 -2 -18 0 2 E -12 -6 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997063 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9732: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (9) C B E D A (6) A E D B C (6) C B D E A (5) A D E B C (5) D B C A E (4) D A C B E (4) B C D E A (4) E B C D A (3) E B C A D (3) E A C B D (3) D A E B C (3) D A B C E (3) C B D A E (3) B C D A E (3) A E D C B (3) A D C B E (3) E D A B C (2) E B D C A (2) C D B A E (2) C B E A D (2) C A D B E (2) C A B D E (2) B D C A E (2) A D E C B (2) E D B A C (1) E C B A D (1) E C A B D (1) E B A C D (1) E A D C B (1) E A C D B (1) E A B D C (1) E A B C D (1) D C B A E (1) D B C E A (1) D B A E C (1) B E D C A (1) A D B E C (1) A C D E B (1) Total count = 100 A B C D E A 0 8 4 -2 2 B -8 0 14 -12 0 C -4 -14 0 -12 -4 D 2 12 12 0 4 E -2 0 4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 -2 2 B -8 0 14 -12 0 C -4 -14 0 -12 -4 D 2 12 12 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 C=22 A=21 D=17 B=10 so B is eliminated. Round 2 votes counts: E=31 C=29 A=21 D=19 so D is eliminated. Round 3 votes counts: C=37 A=32 E=31 so E is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:206 E:199 B:197 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 4 -2 2 B -8 0 14 -12 0 C -4 -14 0 -12 -4 D 2 12 12 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 -2 2 B -8 0 14 -12 0 C -4 -14 0 -12 -4 D 2 12 12 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 -2 2 B -8 0 14 -12 0 C -4 -14 0 -12 -4 D 2 12 12 0 4 E -2 0 4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999990407 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9733: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A C D (7) A E C D B (7) B E A C D (6) C D B E A (5) B E C A D (5) B D C E A (5) E A B C D (4) D B C A E (4) C B E D A (4) B C E D A (4) E A C B D (3) D C A E B (3) C D B A E (3) C D A E B (3) C B D E A (3) B E A D C (3) B C D E A (3) E C B A D (2) D C B A E (2) D B A E C (2) D A B E C (2) B D E A C (2) A E D C B (2) A D E C B (2) E B C A D (1) E B A D C (1) E A C D B (1) E A B D C (1) D C A B E (1) D A E B C (1) D A C E B (1) D A C B E (1) C E B A D (1) C B E A D (1) C A E D B (1) A E D B C (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -28 -2 2 -24 B 28 0 4 14 12 C 2 -4 0 32 -8 D -2 -14 -32 0 -14 E 24 -12 8 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999123 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -28 -2 2 -24 B 28 0 4 14 12 C 2 -4 0 32 -8 D -2 -14 -32 0 -14 E 24 -12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=21 E=20 D=17 A=14 so A is eliminated. Round 2 votes counts: E=31 B=28 C=22 D=19 so D is eliminated. Round 3 votes counts: B=36 E=34 C=30 so C is eliminated. Round 4 votes counts: B=56 E=44 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:229 E:217 C:211 A:174 D:169 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -28 -2 2 -24 B 28 0 4 14 12 C 2 -4 0 32 -8 D -2 -14 -32 0 -14 E 24 -12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -28 -2 2 -24 B 28 0 4 14 12 C 2 -4 0 32 -8 D -2 -14 -32 0 -14 E 24 -12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -28 -2 2 -24 B 28 0 4 14 12 C 2 -4 0 32 -8 D -2 -14 -32 0 -14 E 24 -12 8 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996385 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9734: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (12) A E C D B (9) A E D C B (7) C E D B A (5) C E B D A (5) B D C E A (5) B C D E A (5) A D E B C (5) C B E D A (4) A B D C E (4) A B C E D (4) E A C D B (3) B C E D A (3) E C A D B (2) D E C B A (2) B C D A E (2) B A D C E (2) B A C D E (2) A B E C D (2) E D A C B (1) E C D B A (1) D E C A B (1) D E B A C (1) D C E B A (1) D B C E A (1) C D E B A (1) C D B E A (1) C A B E D (1) B D C A E (1) B C A E D (1) B C A D E (1) B A D E C (1) A E D B C (1) A E C B D (1) A E B D C (1) A C E B D (1) Total count = 100 A B C D E A 0 10 14 20 16 B -10 0 8 16 4 C -14 -8 0 8 0 D -20 -16 -8 0 -4 E -16 -4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 14 20 16 B -10 0 8 16 4 C -14 -8 0 8 0 D -20 -16 -8 0 -4 E -16 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=47 B=23 C=17 E=7 D=6 so D is eliminated. Round 2 votes counts: A=47 B=24 C=18 E=11 so E is eliminated. Round 3 votes counts: A=51 B=25 C=24 so C is eliminated. Round 4 votes counts: A=55 B=45 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:230 B:209 C:193 E:192 D:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 10 14 20 16 B -10 0 8 16 4 C -14 -8 0 8 0 D -20 -16 -8 0 -4 E -16 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 14 20 16 B -10 0 8 16 4 C -14 -8 0 8 0 D -20 -16 -8 0 -4 E -16 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 14 20 16 B -10 0 8 16 4 C -14 -8 0 8 0 D -20 -16 -8 0 -4 E -16 -4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9735: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (9) D B C E A (9) D C B E A (7) B C D A E (6) A E C B D (6) A E B C D (6) E A D C B (4) D E C B A (4) B D C A E (4) A B C E D (4) D E B C A (3) D B E C A (3) C D B E A (3) C B A D E (3) E D C A B (2) E D A C B (2) E D A B C (2) E C D A B (2) D E B A C (2) D C E B A (2) C D E B A (2) B C A D E (2) A E B D C (2) A C E B D (2) E C A D B (1) E A C B D (1) C E D A B (1) C B D A E (1) C A B E D (1) B D A E C (1) B A D E C (1) B A C D E (1) A B E C D (1) Total count = 100 A B C D E A 0 -8 -12 -12 -18 B 8 0 -6 -16 -6 C 12 6 0 4 -4 D 12 16 -4 0 8 E 18 6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000007 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 A B C D E A 0 -8 -12 -12 -18 B 8 0 -6 -16 -6 C 12 6 0 4 -4 D 12 16 -4 0 8 E 18 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 E=23 A=21 B=15 C=11 so C is eliminated. Round 2 votes counts: D=35 E=24 A=22 B=19 so B is eliminated. Round 3 votes counts: D=47 A=29 E=24 so E is eliminated. Round 4 votes counts: D=56 A=44 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:216 E:210 C:209 B:190 A:175 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -12 -18 B 8 0 -6 -16 -6 C 12 6 0 4 -4 D 12 16 -4 0 8 E 18 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -12 -18 B 8 0 -6 -16 -6 C 12 6 0 4 -4 D 12 16 -4 0 8 E 18 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -12 -18 B 8 0 -6 -16 -6 C 12 6 0 4 -4 D 12 16 -4 0 8 E 18 6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.250000 E: 0.250000 Sum of squares = 0.375000000001 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9736: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (5) B C A E D (5) A D B E C (5) E C A D B (4) C B E A D (4) E A D C B (3) C E A B D (3) C B E D A (3) B D C A E (3) A E D B C (3) A E B D C (3) A D E B C (3) E C A B D (2) E A C D B (2) D E C A B (2) D E A C B (2) D B C E A (2) D B C A E (2) D B A E C (2) D A B E C (2) C E D A B (2) C E B D A (2) C E B A D (2) C B D E A (2) B D A C E (2) B C D E A (2) B A C E D (2) A E D C B (2) A E C B D (2) A B E D C (2) A B D E C (2) E D C A B (1) E D A C B (1) E C D A B (1) D C B E A (1) D B E A C (1) D B A C E (1) D A E C B (1) C E D B A (1) C E A D B (1) C D E B A (1) B D A E C (1) B C E A D (1) B C D A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A E B C D (1) A D E C B (1) Total count = 100 A B C D E A 0 14 4 12 8 B -14 0 6 -6 -4 C -4 -6 0 -8 -10 D -12 6 8 0 -8 E -8 4 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 12 8 B -14 0 6 -6 -4 C -4 -6 0 -8 -10 D -12 6 8 0 -8 E -8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 D=21 C=21 B=19 E=14 so E is eliminated. Round 2 votes counts: A=30 C=28 D=23 B=19 so B is eliminated. Round 3 votes counts: C=37 A=34 D=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:207 D:197 B:191 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 12 8 B -14 0 6 -6 -4 C -4 -6 0 -8 -10 D -12 6 8 0 -8 E -8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 12 8 B -14 0 6 -6 -4 C -4 -6 0 -8 -10 D -12 6 8 0 -8 E -8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 12 8 B -14 0 6 -6 -4 C -4 -6 0 -8 -10 D -12 6 8 0 -8 E -8 4 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999279 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9737: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (12) A C B E D (9) E D A C B (8) D E A B C (8) D E B C A (7) B C D E A (7) B C A D E (6) A E D C B (5) A C E B D (5) E A D C B (3) B D C E A (3) A B C E D (3) D E B A C (2) D E A C B (2) D B E C A (2) C B E D A (2) C A B E D (2) A E D B C (2) A E C D B (2) A D E B C (2) E D C A B (1) D E C B A (1) C E D B A (1) C B E A D (1) B C D A E (1) B A C D E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 8 8 10 4 B -8 0 -10 6 0 C -8 10 0 8 10 D -10 -6 -8 0 -14 E -4 0 -10 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 8 10 4 B -8 0 -10 6 0 C -8 10 0 8 10 D -10 -6 -8 0 -14 E -4 0 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 C=18 B=18 E=12 so E is eliminated. Round 2 votes counts: A=33 D=31 C=18 B=18 so C is eliminated. Round 3 votes counts: A=35 B=33 D=32 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:215 C:210 E:200 B:194 D:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 8 10 4 B -8 0 -10 6 0 C -8 10 0 8 10 D -10 -6 -8 0 -14 E -4 0 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 10 4 B -8 0 -10 6 0 C -8 10 0 8 10 D -10 -6 -8 0 -14 E -4 0 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 10 4 B -8 0 -10 6 0 C -8 10 0 8 10 D -10 -6 -8 0 -14 E -4 0 -10 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9738: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (8) A B C D E (7) D C E B A (6) B E C D A (6) A D C E B (6) C D B E A (5) B A E C D (5) A B E C D (4) E D C B A (3) D C E A B (3) B C D E A (3) A E B D C (3) A D C B E (3) A B E D C (3) E D C A B (2) E D A C B (2) E A D C B (2) D E C A B (2) D A C E B (2) B C E D A (2) B A C D E (2) A E D C B (2) A B D C E (2) E D B C A (1) E C D B A (1) E B C D A (1) E B A D C (1) E A D B C (1) D E C B A (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A C D (1) B C E A D (1) B C A D E (1) A E D B C (1) A D E B C (1) A D B C E (1) A C D B E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 2 0 -8 B 2 0 -2 -8 2 C -2 2 0 4 12 D 0 8 -4 0 12 E 8 -2 -12 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.540541 B: 0.054054 C: 0.270270 D: 0.081081 E: 0.054054 Sum of squares = 0.377647918197 Cumulative probabilities = A: 0.540541 B: 0.594595 C: 0.864865 D: 0.945946 E: 1.000000 A B C D E A 0 -2 2 0 -8 B 2 0 -2 -8 2 C -2 2 0 4 12 D 0 8 -4 0 12 E 8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.540541 B: 0.054054 C: 0.270270 D: 0.081081 E: 0.054054 Sum of squares = 0.377647918171 Cumulative probabilities = A: 0.540541 B: 0.594595 C: 0.864865 D: 0.945946 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=22 C=15 E=14 D=14 so E is eliminated. Round 2 votes counts: A=38 B=24 D=22 C=16 so C is eliminated. Round 3 votes counts: A=38 D=36 B=26 so B is eliminated. Round 4 votes counts: D=50 A=50 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:208 D:208 B:197 A:196 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 2 0 -8 B 2 0 -2 -8 2 C -2 2 0 4 12 D 0 8 -4 0 12 E 8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.540541 B: 0.054054 C: 0.270270 D: 0.081081 E: 0.054054 Sum of squares = 0.377647918171 Cumulative probabilities = A: 0.540541 B: 0.594595 C: 0.864865 D: 0.945946 E: 1.000000 GTS winners are ['A', 'B', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 2 0 -8 B 2 0 -2 -8 2 C -2 2 0 4 12 D 0 8 -4 0 12 E 8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.540541 B: 0.054054 C: 0.270270 D: 0.081081 E: 0.054054 Sum of squares = 0.377647918171 Cumulative probabilities = A: 0.540541 B: 0.594595 C: 0.864865 D: 0.945946 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 2 0 -8 B 2 0 -2 -8 2 C -2 2 0 4 12 D 0 8 -4 0 12 E 8 -2 -12 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.540541 B: 0.054054 C: 0.270270 D: 0.081081 E: 0.054054 Sum of squares = 0.377647918171 Cumulative probabilities = A: 0.540541 B: 0.594595 C: 0.864865 D: 0.945946 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9739: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (12) C D A B E (9) B E A C D (8) D C A E B (7) B E C A D (6) D A C E B (4) C D B E A (4) A E B D C (4) D A E C B (3) C B E D A (3) A C B E D (3) A B E C D (3) E B D A C (2) D E B A C (2) D E A B C (2) D C E B A (2) D A E B C (2) C D B A E (2) C D A E B (2) C B E A D (2) C B A E D (2) A E D B C (2) E B D C A (1) D E C B A (1) D E A C B (1) C B D A E (1) C A D B E (1) C A B E D (1) B E D C A (1) B E C D A (1) B C E D A (1) B C E A D (1) B A E C D (1) A D C E B (1) A C D E B (1) A C B D E (1) Total count = 100 A B C D E A 0 -6 4 -2 0 B 6 0 -2 8 2 C -4 2 0 6 -4 D 2 -8 -6 0 -8 E 0 -2 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000003 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 A B C D E A 0 -6 4 -2 0 B 6 0 -2 8 2 C -4 2 0 6 -4 D 2 -8 -6 0 -8 E 0 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=24 B=19 E=15 A=15 so E is eliminated. Round 2 votes counts: B=34 C=27 D=24 A=15 so A is eliminated. Round 3 votes counts: B=41 C=32 D=27 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:207 E:205 C:200 A:198 D:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -2 0 B 6 0 -2 8 2 C -4 2 0 6 -4 D 2 -8 -6 0 -8 E 0 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -2 0 B 6 0 -2 8 2 C -4 2 0 6 -4 D 2 -8 -6 0 -8 E 0 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -2 0 B 6 0 -2 8 2 C -4 2 0 6 -4 D 2 -8 -6 0 -8 E 0 -2 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.250000 Sum of squares = 0.375000000002 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9740: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) A D C E B (6) C D E B A (5) B E A C D (5) B A E D C (5) A D C B E (5) E B C D A (4) D A C B E (4) B E C D A (4) A B E D C (4) A B D E C (4) D C B E A (3) D C A E B (3) D A C E B (3) B A D E C (3) B A D C E (3) A B D C E (3) E C D B A (2) E C B D A (2) E C A D B (2) E B A C D (2) D C B A E (2) C D E A B (2) B D C E A (2) B C D E A (2) A D B C E (2) E C D A B (1) E A C D B (1) E A B C D (1) C D A E B (1) B E C A D (1) B E A D C (1) B D C A E (1) A E C B D (1) A E B C D (1) A C D E B (1) Total count = 100 A B C D E A 0 6 10 2 20 B -6 0 -4 -2 24 C -10 4 0 -24 12 D -2 2 24 0 26 E -20 -24 -12 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999774 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 2 20 B -6 0 -4 -2 24 C -10 4 0 -24 12 D -2 2 24 0 26 E -20 -24 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=27 A=27 D=23 E=15 C=8 so C is eliminated. Round 2 votes counts: D=31 B=27 A=27 E=15 so E is eliminated. Round 3 votes counts: B=35 D=34 A=31 so A is eliminated. Round 4 votes counts: D=51 B=49 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:225 A:219 B:206 C:191 E:159 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 2 20 B -6 0 -4 -2 24 C -10 4 0 -24 12 D -2 2 24 0 26 E -20 -24 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 2 20 B -6 0 -4 -2 24 C -10 4 0 -24 12 D -2 2 24 0 26 E -20 -24 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 2 20 B -6 0 -4 -2 24 C -10 4 0 -24 12 D -2 2 24 0 26 E -20 -24 -12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999105 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9741: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) C E D A B (8) C B E A D (6) D C E A B (5) B A D E C (5) B A C E D (5) B C E A D (4) B C A E D (4) B A E C D (4) A D E B C (4) D E A C B (3) D A E C B (3) C E B D A (3) C E A B D (3) C D E A B (3) B D A C E (3) B A E D C (3) E C A D B (2) D E C A B (2) C D E B A (2) B D A E C (2) B C D A E (2) B A D C E (2) A B E D C (2) A B D E C (2) E C D A B (1) E B C A D (1) E A D C B (1) E A C D B (1) D C A E B (1) C E D B A (1) C E B A D (1) B C A D E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 2 0 6 6 B -2 0 8 6 -8 C 0 -8 0 4 8 D -6 -6 -4 0 -2 E -6 8 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.905235 B: 0.000000 C: 0.094765 D: 0.000000 E: 0.000000 Sum of squares = 0.828430738772 Cumulative probabilities = A: 0.905235 B: 0.905235 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 6 6 B -2 0 8 6 -8 C 0 -8 0 4 8 D -6 -6 -4 0 -2 E -6 8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015097 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 C=27 D=22 A=10 E=6 so E is eliminated. Round 2 votes counts: B=36 C=30 D=22 A=12 so A is eliminated. Round 3 votes counts: B=40 C=31 D=29 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. A:207 B:202 C:202 E:198 D:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 6 6 B -2 0 8 6 -8 C 0 -8 0 4 8 D -6 -6 -4 0 -2 E -6 8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015097 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 6 6 B -2 0 8 6 -8 C 0 -8 0 4 8 D -6 -6 -4 0 -2 E -6 8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015097 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 6 6 B -2 0 8 6 -8 C 0 -8 0 4 8 D -6 -6 -4 0 -2 E -6 8 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.800000 B: 0.000000 C: 0.200000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000015097 Cumulative probabilities = A: 0.800000 B: 0.800000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9742: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (7) D B E A C (7) E D C B A (5) E C D A B (5) E C A D B (5) C E A B D (5) B A D C E (5) A C B E D (5) A B C E D (5) D C E B A (4) B D A C E (4) A C E B D (4) A B C D E (4) E C D B A (3) D E B C A (3) C E A D B (3) A E C B D (3) E D C A B (2) D B E C A (2) C A E B D (2) B D A E C (2) B A D E C (2) E A D C B (1) D B C E A (1) D B A E C (1) C E D A B (1) C D E B A (1) C A B E D (1) B D C A E (1) B A C D E (1) A E C D B (1) A C B D E (1) A B E D C (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 2 -2 2 -10 B -2 0 -18 -4 -10 C 2 18 0 2 -2 D -2 4 -2 0 -6 E 10 10 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 2 -10 B -2 0 -18 -4 -10 C 2 18 0 2 -2 D -2 4 -2 0 -6 E 10 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=25 E=21 B=15 C=13 so C is eliminated. Round 2 votes counts: E=30 A=29 D=26 B=15 so B is eliminated. Round 3 votes counts: A=37 D=33 E=30 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:214 C:210 D:197 A:196 B:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 2 -10 B -2 0 -18 -4 -10 C 2 18 0 2 -2 D -2 4 -2 0 -6 E 10 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 2 -10 B -2 0 -18 -4 -10 C 2 18 0 2 -2 D -2 4 -2 0 -6 E 10 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 2 -10 B -2 0 -18 -4 -10 C 2 18 0 2 -2 D -2 4 -2 0 -6 E 10 10 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999039 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9743: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (10) D A B C E (7) C E A B D (6) A D E B C (6) A E D B C (5) D A B E C (4) C E B D A (4) A E D C B (4) A D B E C (4) E C A B D (3) C B D A E (3) C A E D B (3) C A D B E (3) B D C A E (3) B D A C E (3) E D A B C (2) E C B A D (2) E A C D B (2) C E A D B (2) C B E D A (2) B E D C A (2) B D E A C (2) B C D E A (2) B C D A E (2) E B D A C (1) E B C D A (1) D E B A C (1) C D A B E (1) C B D E A (1) C B A E D (1) B D C E A (1) B D A E C (1) A E C D B (1) A D E C B (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 16 16 -6 32 B -16 0 14 -20 8 C -16 -14 0 -18 -4 D 6 20 18 0 16 E -32 -8 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 -6 32 B -16 0 14 -20 8 C -16 -14 0 -18 -4 D 6 20 18 0 16 E -32 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 A=25 D=22 B=16 E=11 so E is eliminated. Round 2 votes counts: C=31 A=27 D=24 B=18 so B is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: D=59 C=41 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 A:229 B:193 C:174 E:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 16 -6 32 B -16 0 14 -20 8 C -16 -14 0 -18 -4 D 6 20 18 0 16 E -32 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 -6 32 B -16 0 14 -20 8 C -16 -14 0 -18 -4 D 6 20 18 0 16 E -32 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 -6 32 B -16 0 14 -20 8 C -16 -14 0 -18 -4 D 6 20 18 0 16 E -32 -8 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9744: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) C D E A B (6) D A C B E (5) B A E D C (5) E B C D A (4) E B A C D (4) C D A E B (4) A D C E B (4) A B D C E (4) E C B D A (3) E A B D C (3) D A C E B (3) C E B D A (3) B E C D A (3) A D B E C (3) A D B C E (3) E C D A B (2) C D B E A (2) C D B A E (2) C B E D A (2) B C E D A (2) A D E C B (2) A D C B E (2) E B C A D (1) E B A D C (1) E A D C B (1) E A B C D (1) D C A B E (1) C E D B A (1) C E D A B (1) C D E B A (1) C D A B E (1) C B D E A (1) C B D A E (1) B E C A D (1) B E A C D (1) B D C A E (1) B C D A E (1) B A D C E (1) A E D C B (1) A E B D C (1) A D E B C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 18 -4 -16 12 B -18 0 -14 -8 -12 C 4 14 0 -4 20 D 16 8 4 0 16 E -12 12 -20 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -4 -16 12 B -18 0 -14 -8 -12 C 4 14 0 -4 20 D 16 8 4 0 16 E -12 12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=25 A=23 E=20 D=17 B=15 so B is eliminated. Round 2 votes counts: A=29 C=28 E=25 D=18 so D is eliminated. Round 3 votes counts: C=38 A=37 E=25 so E is eliminated. Round 4 votes counts: C=52 A=48 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:222 C:217 A:205 E:182 B:174 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -4 -16 12 B -18 0 -14 -8 -12 C 4 14 0 -4 20 D 16 8 4 0 16 E -12 12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -4 -16 12 B -18 0 -14 -8 -12 C 4 14 0 -4 20 D 16 8 4 0 16 E -12 12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -4 -16 12 B -18 0 -14 -8 -12 C 4 14 0 -4 20 D 16 8 4 0 16 E -12 12 -20 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999593 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9745: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (11) B E C A D (7) D C A E B (5) A E C B D (5) A C E D B (5) A B E C D (5) D B C E A (4) D A C E B (4) B E D C A (4) B E C D A (4) A D C E B (4) E C A B D (3) A C D E B (3) E B C A D (2) E B A C D (2) D C E A B (2) D C B E A (2) D A B C E (2) B E A D C (2) B D E C A (2) B A D E C (2) A D B C E (2) A C E B D (2) E C B A D (1) E A C B D (1) E A B C D (1) D C B A E (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C B E (1) C E D A B (1) C D E A B (1) C A E D B (1) C A D E B (1) B D A E C (1) B A E D C (1) A D C B E (1) A C B E D (1) Total count = 100 A B C D E A 0 4 14 26 0 B -4 0 8 14 12 C -14 -8 0 14 -8 D -26 -14 -14 0 -18 E 0 -12 8 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.867984 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.132016 Sum of squares = 0.770823777044 Cumulative probabilities = A: 0.867984 B: 0.867984 C: 0.867984 D: 0.867984 E: 1.000000 A B C D E A 0 4 14 26 0 B -4 0 8 14 12 C -14 -8 0 14 -8 D -26 -14 -14 0 -18 E 0 -12 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000064048 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=28 D=24 E=10 C=4 so C is eliminated. Round 2 votes counts: B=34 A=30 D=25 E=11 so E is eliminated. Round 3 votes counts: B=39 A=35 D=26 so D is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:222 B:215 E:207 C:192 D:164 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 26 0 B -4 0 8 14 12 C -14 -8 0 14 -8 D -26 -14 -14 0 -18 E 0 -12 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000064048 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 26 0 B -4 0 8 14 12 C -14 -8 0 14 -8 D -26 -14 -14 0 -18 E 0 -12 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000064048 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 26 0 B -4 0 8 14 12 C -14 -8 0 14 -8 D -26 -14 -14 0 -18 E 0 -12 8 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000064048 Cumulative probabilities = A: 0.750000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9746: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E C A (11) A C E D B (9) C E D A B (6) C E A D B (6) A C E B D (6) D E B C A (5) C A E D B (5) B D E A C (5) E D C B A (4) D B E C A (4) A B C D E (4) D E C B A (3) B A D E C (3) B A D C E (3) E C B D A (2) D B E A C (2) B E D C A (2) B A C E D (2) A D E C B (2) A C B E D (2) E C D B A (1) D E C A B (1) D E A C B (1) C E A B D (1) C A E B D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A C D E (1) A D C E B (1) A D B C E (1) A C D E B (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -8 4 -10 B 2 0 -4 -4 -10 C 8 4 0 0 6 D -4 4 0 0 0 E 10 10 -6 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.576676 D: 0.423324 E: 0.000000 Sum of squares = 0.511758296464 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.576676 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 4 -10 B 2 0 -4 -4 -10 C 8 4 0 0 6 D -4 4 0 0 0 E 10 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=28 C=19 D=16 E=7 so E is eliminated. Round 2 votes counts: B=30 A=28 C=22 D=20 so D is eliminated. Round 3 votes counts: B=41 C=30 A=29 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:209 E:207 D:200 A:192 B:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -8 4 -10 B 2 0 -4 -4 -10 C 8 4 0 0 6 D -4 4 0 0 0 E 10 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 4 -10 B 2 0 -4 -4 -10 C 8 4 0 0 6 D -4 4 0 0 0 E 10 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 4 -10 B 2 0 -4 -4 -10 C 8 4 0 0 6 D -4 4 0 0 0 E 10 10 -6 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9747: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (10) D B A C E (8) E C D B A (5) D C B E A (5) A E B D C (5) B C D A E (4) E C B D A (3) E C A B D (3) E A D C B (3) E A B C D (3) D C B A E (3) D A B C E (3) C E B D A (3) A B D C E (3) E C D A B (2) E A C D B (2) E A C B D (2) C B E D A (2) C B D E A (2) B D C A E (2) B C A E D (2) B C A D E (2) A E B C D (2) A D E B C (2) A B E C D (2) A B D E C (2) E D C B A (1) E D A C B (1) E A D B C (1) E A B D C (1) D E C A B (1) D C E B A (1) D B C A E (1) D A E C B (1) C E B A D (1) C D B E A (1) C D B A E (1) B D A C E (1) B A C D E (1) A E D B C (1) A D B E C (1) Total count = 100 A B C D E A 0 -18 -10 -2 -6 B 18 0 -6 12 -8 C 10 6 0 6 -8 D 2 -12 -6 0 -10 E 6 8 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -18 -10 -2 -6 B 18 0 -6 12 -8 C 10 6 0 6 -8 D 2 -12 -6 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 D=23 A=18 B=12 C=10 so C is eliminated. Round 2 votes counts: E=41 D=25 A=18 B=16 so B is eliminated. Round 3 votes counts: E=43 D=34 A=23 so A is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:216 B:208 C:207 D:187 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -18 -10 -2 -6 B 18 0 -6 12 -8 C 10 6 0 6 -8 D 2 -12 -6 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 -10 -2 -6 B 18 0 -6 12 -8 C 10 6 0 6 -8 D 2 -12 -6 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 -10 -2 -6 B 18 0 -6 12 -8 C 10 6 0 6 -8 D 2 -12 -6 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999775 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9748: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (7) E A B D C (5) B A E C D (5) D C E B A (4) D C E A B (4) B E A D C (4) B D C E A (4) B A E D C (4) A B E C D (4) C D B A E (3) B C D A E (3) B A C E D (3) A E C D B (3) A E C B D (3) A E B D C (3) E D A C B (2) E D A B C (2) E B D A C (2) E A D C B (2) C D E A B (2) C D B E A (2) C D A B E (2) C B D A E (2) C B A D E (2) B C D E A (2) A C B E D (2) A B C E D (2) E D B A C (1) E A D B C (1) D E C B A (1) D B E C A (1) D B C E A (1) C D A E B (1) C B D E A (1) C A D B E (1) C A B D E (1) B E D A C (1) B A C D E (1) A E D C B (1) A E D B C (1) A E B C D (1) A C E D B (1) A C E B D (1) A B E D C (1) Total count = 100 A B C D E A 0 -8 12 4 2 B 8 0 4 14 18 C -12 -4 0 -4 4 D -4 -14 4 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 12 4 2 B 8 0 4 14 18 C -12 -4 0 -4 4 D -4 -14 4 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 A=23 D=18 C=17 E=15 so E is eliminated. Round 2 votes counts: A=31 B=29 D=23 C=17 so C is eliminated. Round 3 votes counts: B=34 D=33 A=33 so D is eliminated. Round 4 votes counts: B=54 A=46 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:222 A:205 E:193 C:192 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 12 4 2 B 8 0 4 14 18 C -12 -4 0 -4 4 D -4 -14 4 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 12 4 2 B 8 0 4 14 18 C -12 -4 0 -4 4 D -4 -14 4 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 12 4 2 B 8 0 4 14 18 C -12 -4 0 -4 4 D -4 -14 4 0 -10 E -2 -18 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998749 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9749: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (13) E B D A C (9) E A D C B (7) B C D A E (7) C A D E B (6) B E D A C (6) B D C A E (5) E A C D B (4) B E C D A (4) B D C E A (4) E D A B C (3) D A C E B (3) B E D C A (3) B D E A C (3) B C A D E (3) E D A C B (2) D E A C B (2) C A B D E (2) B E C A D (2) E B A D C (1) E A B D C (1) D E A B C (1) D A C B E (1) C D A B E (1) C B A D E (1) C A E D B (1) B D E C A (1) B D A C E (1) B C D E A (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -8 -14 -8 B 2 0 10 8 16 C 8 -10 0 -8 0 D 14 -8 8 0 12 E 8 -16 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999592 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -8 -14 -8 B 2 0 10 8 16 C 8 -10 0 -8 0 D 14 -8 8 0 12 E 8 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985715 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 E=27 C=24 D=7 A=2 so A is eliminated. Round 2 votes counts: B=40 E=28 C=25 D=7 so D is eliminated. Round 3 votes counts: B=40 E=31 C=29 so C is eliminated. Round 4 votes counts: B=58 E=42 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:218 D:213 C:195 E:190 A:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -8 -14 -8 B 2 0 10 8 16 C 8 -10 0 -8 0 D 14 -8 8 0 12 E 8 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985715 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 -14 -8 B 2 0 10 8 16 C 8 -10 0 -8 0 D 14 -8 8 0 12 E 8 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985715 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 -14 -8 B 2 0 10 8 16 C 8 -10 0 -8 0 D 14 -8 8 0 12 E 8 -16 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985715 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9750: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (10) E D A B C (6) B A C E D (6) D E C A B (5) E D B A C (4) D E C B A (4) D E B C A (4) D E B A C (4) D C E B A (4) E D C A B (3) D B E A C (3) C A B E D (3) B C D A E (3) B C A D E (3) E D A C B (2) E C A D B (2) E A C D B (2) D C E A B (2) D C B A E (2) D B E C A (2) C E A D B (2) C A B D E (2) B D C A E (2) A E C B D (2) A C B E D (2) A B C E D (2) E C D A B (1) E A C B D (1) E A B C D (1) D C B E A (1) C E D A B (1) C D A E B (1) C D A B E (1) C B D A E (1) C A E D B (1) B E A D C (1) B D E A C (1) B A E D C (1) B A D E C (1) B A D C E (1) Total count = 100 A B C D E A 0 -16 0 -14 -12 B 16 0 10 -14 -4 C 0 -10 0 -6 0 D 14 14 6 0 14 E 12 4 0 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 0 -14 -12 B 16 0 10 -14 -4 C 0 -10 0 -6 0 D 14 14 6 0 14 E 12 4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=31 B=29 E=22 C=12 A=6 so A is eliminated. Round 2 votes counts: D=31 B=31 E=24 C=14 so C is eliminated. Round 3 votes counts: B=39 D=33 E=28 so E is eliminated. Round 4 votes counts: D=57 B=43 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:224 B:204 E:201 C:192 A:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 0 -14 -12 B 16 0 10 -14 -4 C 0 -10 0 -6 0 D 14 14 6 0 14 E 12 4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 0 -14 -12 B 16 0 10 -14 -4 C 0 -10 0 -6 0 D 14 14 6 0 14 E 12 4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 0 -14 -12 B 16 0 10 -14 -4 C 0 -10 0 -6 0 D 14 14 6 0 14 E 12 4 0 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999859 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9751: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) C D E B A (7) A B E D C (7) D E A C B (6) C B A D E (6) A E D B C (6) E D B A C (4) D E C A B (4) B C E D A (4) B A C E D (4) A D E B C (4) C D B E A (3) B C A D E (3) A D E C B (3) E D C B A (2) D E C B A (2) C D E A B (2) B C E A D (2) B C A E D (2) B A E C D (2) A E B D C (2) A B C D E (2) E D A C B (1) E A D B C (1) E A B D C (1) D A E C B (1) D A C E B (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E D A (1) C B D E A (1) C B D A E (1) B E C A D (1) B A E D C (1) A D C E B (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 6 12 0 2 B -6 0 6 -18 -12 C -12 -6 0 -8 -10 D 0 18 8 0 4 E -2 12 10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.604241 B: 0.000000 C: 0.000000 D: 0.395759 E: 0.000000 Sum of squares = 0.521732475675 Cumulative probabilities = A: 0.604241 B: 0.604241 C: 0.604241 D: 1.000000 E: 1.000000 A B C D E A 0 6 12 0 2 B -6 0 6 -18 -12 C -12 -6 0 -8 -10 D 0 18 8 0 4 E -2 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=24 B=19 E=16 D=14 so D is eliminated. Round 2 votes counts: A=29 E=28 C=24 B=19 so B is eliminated. Round 3 votes counts: A=36 C=35 E=29 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 A:210 E:208 B:185 C:182 Borda winner is D compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 12 0 2 B -6 0 6 -18 -12 C -12 -6 0 -8 -10 D 0 18 8 0 4 E -2 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 12 0 2 B -6 0 6 -18 -12 C -12 -6 0 -8 -10 D 0 18 8 0 4 E -2 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 12 0 2 B -6 0 6 -18 -12 C -12 -6 0 -8 -10 D 0 18 8 0 4 E -2 12 10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.49999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9752: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (7) E D A B C (5) E A C D B (5) D A C E B (5) C B A D E (5) C A E D B (4) B C D A E (4) E B A D C (3) E B A C D (3) E A D B C (3) D A E B C (3) C D A B E (3) C B A E D (3) B D C A E (3) B C D E A (3) A E D C B (3) E B D A C (2) D B E A C (2) C A D E B (2) C A B D E (2) B E D A C (2) B E C D A (2) B D E C A (2) B D E A C (2) B D C E A (2) B C E A D (2) E A C B D (1) E A B D C (1) E A B C D (1) D C B A E (1) D C A B E (1) D A E C B (1) D A C B E (1) C E A D B (1) C E A B D (1) C D A E B (1) C B D A E (1) C A E B D (1) B E D C A (1) B E C A D (1) B E A C D (1) B C E D A (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 8 6 4 -8 B -8 0 -2 0 -10 C -6 2 0 -2 -4 D -4 0 2 0 -8 E 8 10 4 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 6 4 -8 B -8 0 -2 0 -10 C -6 2 0 -2 -4 D -4 0 2 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 B=26 C=24 D=14 A=5 so A is eliminated. Round 2 votes counts: E=34 B=26 C=25 D=15 so D is eliminated. Round 3 votes counts: E=39 C=33 B=28 so B is eliminated. Round 4 votes counts: E=52 C=48 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:215 A:205 C:195 D:195 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 6 4 -8 B -8 0 -2 0 -10 C -6 2 0 -2 -4 D -4 0 2 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 4 -8 B -8 0 -2 0 -10 C -6 2 0 -2 -4 D -4 0 2 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 4 -8 B -8 0 -2 0 -10 C -6 2 0 -2 -4 D -4 0 2 0 -8 E 8 10 4 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9753: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) B E C A D (7) D E A C B (5) D A C B E (5) D B C A E (4) C A B E D (4) C A B D E (4) B C E A D (4) E D B A C (3) E B D A C (3) E B A C D (3) E A C B D (3) D C A B E (3) D B E C A (3) C B A D E (3) C A D B E (3) A D C E B (3) A C D E B (3) E A D C B (2) D C B A E (2) B D E C A (2) B C D A E (2) B C A E D (2) A C E D B (2) A C E B D (2) E D A C B (1) E B D C A (1) E B C A D (1) E A C D B (1) D E B A C (1) D E A B C (1) D B C E A (1) D A E C B (1) C D A B E (1) C B D A E (1) B E D C A (1) B E C D A (1) B D C E A (1) B C A D E (1) A E C D B (1) A E C B D (1) Total count = 100 A B C D E A 0 6 -4 0 10 B -6 0 -16 -6 10 C 4 16 0 0 16 D 0 6 0 0 14 E -10 -10 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.626865 D: 0.373135 E: 0.000000 Sum of squares = 0.532189448468 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.626865 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 0 10 B -6 0 -16 -6 10 C 4 16 0 0 16 D 0 6 0 0 14 E -10 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999253 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=21 E=18 C=16 A=12 so A is eliminated. Round 2 votes counts: D=36 C=23 B=21 E=20 so E is eliminated. Round 3 votes counts: D=42 C=29 B=29 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:218 D:210 A:206 B:191 E:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -4 0 10 B -6 0 -16 -6 10 C 4 16 0 0 16 D 0 6 0 0 14 E -10 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999253 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 0 10 B -6 0 -16 -6 10 C 4 16 0 0 16 D 0 6 0 0 14 E -10 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999253 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 0 10 B -6 0 -16 -6 10 C 4 16 0 0 16 D 0 6 0 0 14 E -10 -10 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999253 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9754: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (11) C E A D B (7) B D A E C (7) B C A E D (6) B A C E D (5) D B A E C (4) B D E C A (4) B D C E A (4) B A D E C (4) D B E C A (3) D B E A C (3) C B E A D (3) C A E D B (3) C A E B D (3) B C E A D (3) A E C D B (3) A C E D B (3) D E C A B (2) B D E A C (2) B D A C E (2) B C D E A (2) B A E D C (2) B A E C D (2) A E D C B (2) E D C A B (1) E C D A B (1) E C A D B (1) C E D A B (1) C D B E A (1) C B A E D (1) C A B E D (1) B A C D E (1) A E C B D (1) A D E C B (1) Total count = 100 A B C D E A 0 -18 6 4 2 B 18 0 8 6 20 C -6 -8 0 -4 -8 D -4 -6 4 0 2 E -2 -20 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999984 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -18 6 4 2 B 18 0 8 6 20 C -6 -8 0 -4 -8 D -4 -6 4 0 2 E -2 -20 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=44 D=23 C=20 A=10 E=3 so E is eliminated. Round 2 votes counts: B=44 D=24 C=22 A=10 so A is eliminated. Round 3 votes counts: B=44 C=29 D=27 so D is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:226 D:198 A:197 E:192 C:187 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -18 6 4 2 B 18 0 8 6 20 C -6 -8 0 -4 -8 D -4 -6 4 0 2 E -2 -20 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 6 4 2 B 18 0 8 6 20 C -6 -8 0 -4 -8 D -4 -6 4 0 2 E -2 -20 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 6 4 2 B 18 0 8 6 20 C -6 -8 0 -4 -8 D -4 -6 4 0 2 E -2 -20 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999743 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9755: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (11) E B D C A (6) D B E C A (6) A C E B D (6) E C B D A (5) C E D B A (5) E D B C A (4) E C D B A (4) E B D A C (4) E A C B D (4) D B A C E (4) C D B E A (4) C A E D B (4) B D E A C (3) E C A B D (2) C E A D B (2) C D B A E (2) C A E B D (2) C A D B E (2) A E C B D (2) A C D B E (2) A B D C E (2) E A B D C (1) D C B A E (1) D B C E A (1) D B A E C (1) C E D A B (1) C D E B A (1) B D E C A (1) B D A E C (1) A E B D C (1) A E B C D (1) A D B C E (1) A C E D B (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -6 -8 -8 B 6 0 -2 8 -12 C 6 2 0 2 -16 D 8 -8 -2 0 -12 E 8 12 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999447 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -6 -8 -8 B 6 0 -2 8 -12 C 6 2 0 2 -16 D 8 -8 -2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=29 C=23 D=13 B=5 so B is eliminated. Round 2 votes counts: E=30 A=29 C=23 D=18 so D is eliminated. Round 3 votes counts: E=40 A=35 C=25 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:224 B:200 C:197 D:193 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -6 -8 -8 B 6 0 -2 8 -12 C 6 2 0 2 -16 D 8 -8 -2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -8 -8 B 6 0 -2 8 -12 C 6 2 0 2 -16 D 8 -8 -2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -8 -8 B 6 0 -2 8 -12 C 6 2 0 2 -16 D 8 -8 -2 0 -12 E 8 12 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9756: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (11) B C D E A (8) B C E D A (5) D B E A C (4) C B A E D (4) C A E D B (4) B C A E D (4) E D A C B (3) D E B C A (3) D E A C B (3) D E A B C (3) B D A E C (3) B C E A D (3) B C A D E (3) A E C D B (3) A D E C B (3) E A D C B (2) E A C D B (2) D A E C B (2) D A E B C (2) C E B D A (2) C E A D B (2) C E A B D (2) C B E D A (2) C B E A D (2) B D E C A (2) A D E B C (2) E D C A B (1) E C A D B (1) D E C B A (1) D B A E C (1) D A B E C (1) C E B A D (1) B D E A C (1) B D C A E (1) B C D A E (1) B A D C E (1) A C E D B (1) Total count = 100 A B C D E A 0 -4 -4 2 -6 B 4 0 -4 -10 -8 C 4 4 0 0 -8 D -2 10 0 0 -10 E 6 8 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -4 2 -6 B 4 0 -4 -10 -8 C 4 4 0 0 -8 D -2 10 0 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=20 A=20 C=19 E=9 so E is eliminated. Round 2 votes counts: B=32 D=24 A=24 C=20 so C is eliminated. Round 3 votes counts: B=43 A=33 D=24 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. E:216 C:200 D:199 A:194 B:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -4 2 -6 B 4 0 -4 -10 -8 C 4 4 0 0 -8 D -2 10 0 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 2 -6 B 4 0 -4 -10 -8 C 4 4 0 0 -8 D -2 10 0 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 2 -6 B 4 0 -4 -10 -8 C 4 4 0 0 -8 D -2 10 0 0 -10 E 6 8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9757: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (14) C E A D B (10) D B C E A (9) C A E D B (6) C E D A B (5) B D A E C (5) B D E A C (4) A B E C D (4) E C A D B (3) D C B E A (3) B D E C A (3) B D A C E (3) A C E D B (3) A C E B D (3) E C A B D (2) E A C B D (2) C D E A B (2) E C D B A (1) E C B A D (1) E A C D B (1) D E C B A (1) D C E B A (1) D C B A E (1) D C A E B (1) D B E C A (1) D B C A E (1) D B A C E (1) D A C B E (1) B E A D C (1) B E A C D (1) B D C A E (1) B A E C D (1) B A D E C (1) A E B C D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 20 -4 12 -2 B -20 0 -24 -4 -14 C 4 24 0 22 6 D -12 4 -22 0 -18 E 2 14 -6 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998315 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 -4 12 -2 B -20 0 -24 -4 -14 C 4 24 0 22 6 D -12 4 -22 0 -18 E 2 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 C=23 D=20 B=20 E=10 so E is eliminated. Round 2 votes counts: C=30 A=30 D=20 B=20 so D is eliminated. Round 3 votes counts: C=37 B=32 A=31 so A is eliminated. Round 4 votes counts: C=62 B=38 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:228 E:214 A:213 D:176 B:169 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 20 -4 12 -2 B -20 0 -24 -4 -14 C 4 24 0 22 6 D -12 4 -22 0 -18 E 2 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 -4 12 -2 B -20 0 -24 -4 -14 C 4 24 0 22 6 D -12 4 -22 0 -18 E 2 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 -4 12 -2 B -20 0 -24 -4 -14 C 4 24 0 22 6 D -12 4 -22 0 -18 E 2 14 -6 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997915 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9758: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) C E D A B (8) B D A C E (8) D B A C E (6) B A D E C (6) D B A E C (4) E C D A B (3) E C A D B (3) E A C B D (3) D C E A B (3) D C B E A (3) D B C A E (3) C E A D B (3) C E A B D (3) A B E C D (3) A B D E C (3) E C A B D (2) D C E B A (2) C B E A D (2) B A D C E (2) A E C B D (2) A E B C D (2) A B E D C (2) E A D C B (1) E A C D B (1) D E C A B (1) D E A C B (1) D B C E A (1) D A E C B (1) D A B E C (1) C E B D A (1) C E B A D (1) C D E B A (1) B D C E A (1) B D C A E (1) B D A E C (1) B C A E D (1) B A C E D (1) A E D C B (1) Total count = 100 A B C D E A 0 -6 -2 -16 -4 B 6 0 -8 -10 -2 C 2 8 0 -4 18 D 16 10 4 0 -2 E 4 2 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.750000 E: 0.166667 Sum of squares = 0.597222222135 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.833333 E: 1.000000 A B C D E A 0 -6 -2 -16 -4 B 6 0 -8 -10 -2 C 2 8 0 -4 18 D 16 10 4 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.750000 E: 0.166667 Sum of squares = 0.59722222221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=26 B=21 E=13 A=13 so E is eliminated. Round 2 votes counts: C=35 D=26 B=21 A=18 so A is eliminated. Round 3 votes counts: C=41 B=31 D=28 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:214 C:212 E:195 B:193 A:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -2 -16 -4 B 6 0 -8 -10 -2 C 2 8 0 -4 18 D 16 10 4 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.750000 E: 0.166667 Sum of squares = 0.59722222221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.833333 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -2 -16 -4 B 6 0 -8 -10 -2 C 2 8 0 -4 18 D 16 10 4 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.750000 E: 0.166667 Sum of squares = 0.59722222221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -2 -16 -4 B 6 0 -8 -10 -2 C 2 8 0 -4 18 D 16 10 4 0 -2 E 4 2 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.750000 E: 0.166667 Sum of squares = 0.59722222221 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.083333 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9759: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (9) C B A D E (8) A D E C B (7) A D C B E (7) E D A B C (6) E B D C A (6) B E C D A (6) E B D A C (5) C A D B E (5) C A B D E (5) A D C E B (5) A C D B E (4) E D B A C (3) D A E C B (3) D A E B C (3) D A C B E (2) B C E D A (2) E C B A D (1) E C A B D (1) E B C A D (1) E A D C B (1) D E B A C (1) D B E A C (1) D A C E B (1) C B E A D (1) C B A E D (1) B E D C A (1) B D C A E (1) B C D E A (1) A E D C B (1) A E C D B (1) Total count = 100 A B C D E A 0 4 2 -2 8 B -4 0 -8 -2 -10 C -2 8 0 -8 -14 D 2 2 8 0 8 E -8 10 14 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999634 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 2 -2 8 B -4 0 -8 -2 -10 C -2 8 0 -8 -14 D 2 2 8 0 8 E -8 10 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 A=25 C=20 D=11 B=11 so D is eliminated. Round 2 votes counts: E=34 A=34 C=20 B=12 so B is eliminated. Round 3 votes counts: E=42 A=34 C=24 so C is eliminated. Round 4 votes counts: A=54 E=46 so E is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:206 E:204 C:192 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 2 -2 8 B -4 0 -8 -2 -10 C -2 8 0 -8 -14 D 2 2 8 0 8 E -8 10 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 2 -2 8 B -4 0 -8 -2 -10 C -2 8 0 -8 -14 D 2 2 8 0 8 E -8 10 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 2 -2 8 B -4 0 -8 -2 -10 C -2 8 0 -8 -14 D 2 2 8 0 8 E -8 10 14 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999356 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9760: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (11) A E D C B (9) D A B C E (7) E B C D A (6) D B C A E (5) E B C A D (4) D B A C E (4) C B E D A (4) B C E D A (4) A D C E B (4) E C A B D (3) C B D A E (3) A E C D B (3) A D E C B (3) E B D A C (2) E A D B C (2) D A C B E (2) C E B A D (2) C B D E A (2) B E C D A (2) B D C A E (2) B C D E A (2) A D E B C (2) A D C B E (2) E A C D B (1) E A C B D (1) E A B D C (1) D A B E C (1) C B A E D (1) C A D B E (1) B D C E A (1) B C D A E (1) A E D B C (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 -8 4 4 B 12 0 -4 4 -10 C 8 4 0 2 -4 D -4 -4 -2 0 -14 E -4 10 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999992 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -12 -8 4 4 B 12 0 -4 4 -10 C 8 4 0 2 -4 D -4 -4 -2 0 -14 E -4 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000074 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 A=25 D=19 C=13 B=12 so B is eliminated. Round 2 votes counts: E=33 A=25 D=22 C=20 so C is eliminated. Round 3 votes counts: E=43 D=30 A=27 so A is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:205 B:201 A:194 D:188 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -8 4 4 B 12 0 -4 4 -10 C 8 4 0 2 -4 D -4 -4 -2 0 -14 E -4 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000074 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 4 4 B 12 0 -4 4 -10 C 8 4 0 2 -4 D -4 -4 -2 0 -14 E -4 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000074 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 4 4 B 12 0 -4 4 -10 C 8 4 0 2 -4 D -4 -4 -2 0 -14 E -4 10 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000074 Cumulative probabilities = A: 0.250000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9761: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (11) C A D E B (7) E B C D A (6) D B E A C (6) C A E B D (6) A C B E D (6) D A B E C (5) C E B A D (5) A C D B E (5) E B C A D (4) D E B C A (4) D C A E B (4) C E D B A (4) B E D A C (4) B E A C D (4) B E A D C (3) A D C B E (3) A D B E C (3) D C E B A (2) D A C B E (2) D E B A C (1) D C E A B (1) D A C E B (1) C D E A B (1) C D A E B (1) B D E A C (1) Total count = 100 A B C D E A 0 -10 -12 -8 -14 B 10 0 4 0 -16 C 12 -4 0 -2 -4 D 8 0 2 0 -6 E 14 16 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -12 -8 -14 B 10 0 4 0 -16 C 12 -4 0 -2 -4 D 8 0 2 0 -6 E 14 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=24 E=21 A=17 B=12 so B is eliminated. Round 2 votes counts: E=32 D=27 C=24 A=17 so A is eliminated. Round 3 votes counts: C=35 D=33 E=32 so E is eliminated. Round 4 votes counts: D=51 C=49 so C is eliminated. IRV winner is D compare: Computing Borda winner. E:220 D:202 C:201 B:199 A:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -12 -8 -14 B 10 0 4 0 -16 C 12 -4 0 -2 -4 D 8 0 2 0 -6 E 14 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 -8 -14 B 10 0 4 0 -16 C 12 -4 0 -2 -4 D 8 0 2 0 -6 E 14 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 -8 -14 B 10 0 4 0 -16 C 12 -4 0 -2 -4 D 8 0 2 0 -6 E 14 16 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999595 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9762: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (8) E B C D A (8) C A E B D (8) B E D C A (7) A C E D B (6) D B E A C (5) D A B C E (4) A D C E B (4) A C D E B (4) A C D B E (4) D C B A E (3) C A D B E (3) A D C B E (3) E B A D C (2) E A C B D (2) D B C A E (2) C D A B E (2) B E D A C (2) A C E B D (2) E D B A C (1) E C A B D (1) E B D C A (1) E B C A D (1) E B A C D (1) E A B D C (1) E A B C D (1) D B E C A (1) D B A E C (1) D A B E C (1) C E B A D (1) C B D E A (1) C A E D B (1) C A D E B (1) B D E C A (1) B D C E A (1) B C D E A (1) A E C D B (1) A D E C B (1) A D E B C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 12 0 6 B -4 0 4 0 -14 C -12 -4 0 -2 4 D 0 0 2 0 -10 E -6 14 -4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.711850 B: 0.000000 C: 0.000000 D: 0.288150 E: 0.000000 Sum of squares = 0.589760907928 Cumulative probabilities = A: 0.711850 B: 0.711850 C: 0.711850 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 0 6 B -4 0 4 0 -14 C -12 -4 0 -2 4 D 0 0 2 0 -10 E -6 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250009139 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=27 A=27 D=17 C=17 B=12 so B is eliminated. Round 2 votes counts: E=36 A=27 D=19 C=18 so C is eliminated. Round 3 votes counts: A=40 E=37 D=23 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 E:207 D:196 B:193 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 0 6 B -4 0 4 0 -14 C -12 -4 0 -2 4 D 0 0 2 0 -10 E -6 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250009139 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 0 6 B -4 0 4 0 -14 C -12 -4 0 -2 4 D 0 0 2 0 -10 E -6 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250009139 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 0 6 B -4 0 4 0 -14 C -12 -4 0 -2 4 D 0 0 2 0 -10 E -6 14 -4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.625000 B: 0.000000 C: 0.000000 D: 0.375000 E: 0.000000 Sum of squares = 0.531250009139 Cumulative probabilities = A: 0.625000 B: 0.625000 C: 0.625000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9763: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) D B E A C (7) D B C E A (7) D B C A E (6) A E C B D (5) E A D C B (4) C A B D E (4) B D C A E (4) B D A C E (4) E D A B C (3) E C A D B (3) E A C D B (3) D C B E A (3) C D B A E (3) C B A D E (3) E D C A B (2) E A D B C (2) E A B D C (2) D C B A E (2) C A E B D (2) B C D A E (2) A B C E D (2) E D B A C (1) E D A C B (1) E C D A B (1) E A B C D (1) D E C B A (1) D E B C A (1) D C E B A (1) D B E C A (1) C E A D B (1) C D E B A (1) C A B E D (1) B D A E C (1) B C A D E (1) B A C D E (1) A E B C D (1) A C E B D (1) A C B E D (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 0 -4 -8 B 0 0 -2 -8 10 C 0 2 0 -8 0 D 4 8 8 0 6 E 8 -10 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 -4 -8 B 0 0 -2 -8 10 C 0 2 0 -8 0 D 4 8 8 0 6 E 8 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=29 C=15 B=13 A=12 so A is eliminated. Round 2 votes counts: E=37 D=29 C=17 B=17 so C is eliminated. Round 3 votes counts: E=41 D=33 B=26 so B is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 B:200 C:197 E:196 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -4 -8 B 0 0 -2 -8 10 C 0 2 0 -8 0 D 4 8 8 0 6 E 8 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 -8 B 0 0 -2 -8 10 C 0 2 0 -8 0 D 4 8 8 0 6 E 8 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 -8 B 0 0 -2 -8 10 C 0 2 0 -8 0 D 4 8 8 0 6 E 8 -10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9764: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (16) A C E D B (10) B D E C A (6) E C A B D (5) D B E C A (4) B D A C E (4) D E C A B (3) B E D C A (3) B D E A C (3) B A C E D (3) E D C A B (2) E B C D A (2) D C E A B (2) D B A C E (2) D A C E B (2) D A B C E (2) C A E D B (2) B E C A D (2) B D A E C (2) B A D C E (2) B A C D E (2) A D C B E (2) A C E B D (2) A C D E B (2) A B C D E (2) E C D A B (1) E B D C A (1) E B C A D (1) D E B C A (1) D C A E B (1) D A C B E (1) C E A D B (1) B E D A C (1) B E C D A (1) B E A C D (1) A C B E D (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 18 -8 12 -12 B -18 0 -8 -8 -8 C 8 8 0 12 -10 D -12 8 -12 0 -12 E 12 8 10 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999983 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 -8 12 -12 B -18 0 -8 -8 -8 C 8 8 0 12 -10 D -12 8 -12 0 -12 E 12 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=28 A=21 D=18 C=3 so C is eliminated. Round 2 votes counts: B=30 E=29 A=23 D=18 so D is eliminated. Round 3 votes counts: B=36 E=35 A=29 so A is eliminated. Round 4 votes counts: E=54 B=46 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:221 C:209 A:205 D:186 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -8 12 -12 B -18 0 -8 -8 -8 C 8 8 0 12 -10 D -12 8 -12 0 -12 E 12 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 12 -12 B -18 0 -8 -8 -8 C 8 8 0 12 -10 D -12 8 -12 0 -12 E 12 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 12 -12 B -18 0 -8 -8 -8 C 8 8 0 12 -10 D -12 8 -12 0 -12 E 12 8 10 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999417 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9765: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (8) B E C D A (8) B E A C D (8) A D C E B (8) E C B D A (6) D C A E B (6) D C E B A (4) C E D B A (4) C D E B A (4) B E C A D (4) B A E C D (4) A B E C D (4) D A C B E (3) A D C B E (3) A B E D C (3) A B D E C (3) E C D B A (2) D C E A B (2) D A C E B (2) A D B E C (2) A C D E B (2) E C B A D (1) E B C A D (1) D B A E C (1) D A B C E (1) C E A D B (1) C D E A B (1) B E D C A (1) B A E D C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 -6 -6 -10 B 14 0 0 6 -6 C 6 0 0 18 -14 D 6 -6 -18 0 -14 E 10 6 14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -6 -6 -10 B 14 0 0 6 -6 C 6 0 0 18 -14 D 6 -6 -18 0 -14 E 10 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 B=26 D=19 E=18 C=10 so C is eliminated. Round 2 votes counts: A=27 B=26 D=24 E=23 so E is eliminated. Round 3 votes counts: B=42 D=30 A=28 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:222 B:207 C:205 D:184 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -6 -6 -10 B 14 0 0 6 -6 C 6 0 0 18 -14 D 6 -6 -18 0 -14 E 10 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -6 -10 B 14 0 0 6 -6 C 6 0 0 18 -14 D 6 -6 -18 0 -14 E 10 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -6 -10 B 14 0 0 6 -6 C 6 0 0 18 -14 D 6 -6 -18 0 -14 E 10 6 14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9766: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (13) C E A B D (8) D B A E C (6) B D E A C (6) E B D A C (5) A D C B E (5) A D B E C (4) A C E D B (4) A C D B E (4) E C A B D (3) C B D E A (3) B D E C A (3) E C B A D (2) E B D C A (2) E B C D A (2) D B E A C (2) D A B E C (2) D A B C E (2) C E B D A (2) C A E B D (2) C A D E B (2) B E D C A (2) A E C D B (2) A D B C E (2) E C B D A (1) D B C E A (1) D B A C E (1) C E B A D (1) C D A B E (1) B E D A C (1) B E C D A (1) B D C E A (1) A E C B D (1) A D E C B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 16 0 12 8 B -16 0 -14 -6 -6 C 0 14 0 8 8 D -12 6 -8 0 -6 E -8 6 -8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.472493 B: 0.000000 C: 0.527507 D: 0.000000 E: 0.000000 Sum of squares = 0.501513276312 Cumulative probabilities = A: 0.472493 B: 0.472493 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 0 12 8 B -16 0 -14 -6 -6 C 0 14 0 8 8 D -12 6 -8 0 -6 E -8 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=25 E=15 D=14 B=14 so D is eliminated. Round 2 votes counts: C=32 A=29 B=24 E=15 so E is eliminated. Round 3 votes counts: C=38 B=33 A=29 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:218 C:215 E:198 D:190 B:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 16 0 12 8 B -16 0 -14 -6 -6 C 0 14 0 8 8 D -12 6 -8 0 -6 E -8 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 0 12 8 B -16 0 -14 -6 -6 C 0 14 0 8 8 D -12 6 -8 0 -6 E -8 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 0 12 8 B -16 0 -14 -6 -6 C 0 14 0 8 8 D -12 6 -8 0 -6 E -8 6 -8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9767: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) D B A C E (7) E D C A B (6) B D A C E (6) E C A D B (5) E C A B D (5) E B D C A (5) D E B A C (5) D A C B E (5) B A C D E (5) D E A C B (4) E C B A D (3) E B C D A (3) D A C E B (3) C A B E D (3) B D E A C (3) B A D C E (3) E D B C A (2) D B E A C (2) C A E B D (2) A C D B E (2) A C B D E (2) E D C B A (1) E D B A C (1) D B A E C (1) D A E B C (1) D A B C E (1) C E B A D (1) C E A D B (1) C A E D B (1) B E D C A (1) B E C A D (1) B C A E D (1) A C D E B (1) Total count = 100 A B C D E A 0 -16 4 -14 -12 B 16 0 10 2 -14 C -4 -10 0 -14 -12 D 14 -2 14 0 2 E 12 14 12 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.777778 E: 0.111111 Sum of squares = 0.629629629655 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.888889 E: 1.000000 A B C D E A 0 -16 4 -14 -12 B 16 0 10 2 -14 C -4 -10 0 -14 -12 D 14 -2 14 0 2 E 12 14 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.777778 E: 0.111111 Sum of squares = 0.629629629678 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.888889 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=38 D=29 B=20 C=8 A=5 so A is eliminated. Round 2 votes counts: E=38 D=29 B=20 C=13 so C is eliminated. Round 3 votes counts: E=43 D=32 B=25 so B is eliminated. Round 4 votes counts: D=51 E=49 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:218 D:214 B:207 A:181 C:180 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -16 4 -14 -12 B 16 0 10 2 -14 C -4 -10 0 -14 -12 D 14 -2 14 0 2 E 12 14 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.777778 E: 0.111111 Sum of squares = 0.629629629678 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.888889 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 4 -14 -12 B 16 0 10 2 -14 C -4 -10 0 -14 -12 D 14 -2 14 0 2 E 12 14 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.777778 E: 0.111111 Sum of squares = 0.629629629678 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.888889 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 4 -14 -12 B 16 0 10 2 -14 C -4 -10 0 -14 -12 D 14 -2 14 0 2 E 12 14 12 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.111111 C: 0.000000 D: 0.777778 E: 0.111111 Sum of squares = 0.629629629678 Cumulative probabilities = A: 0.000000 B: 0.111111 C: 0.111111 D: 0.888889 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9768: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A C B (10) B C A E D (9) C B D E A (6) C B D A E (6) B C E D A (5) A E B C D (5) E D A B C (4) D C B E A (4) C B A D E (4) A E D B C (4) A D E C B (4) B C A D E (3) E D A C B (2) E B C A D (2) E A D B C (2) E A B C D (2) D E C B A (2) D E C A B (2) D C B A E (2) D A E C B (2) C B E D A (2) A E B D C (2) A C D B E (2) A B C E D (2) A B C D E (2) E D C B A (1) E D B C A (1) E B C D A (1) D C E B A (1) D C A E B (1) D C A B E (1) C D B A E (1) C D A B E (1) B A C E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 -10 -10 6 B 2 0 -10 6 4 C 10 10 0 10 8 D 10 -6 -10 0 10 E -6 -4 -8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -10 -10 6 B 2 0 -10 6 4 C 10 10 0 10 8 D 10 -6 -10 0 10 E -6 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=25 A=22 C=20 B=18 E=15 so E is eliminated. Round 2 votes counts: D=33 A=26 B=21 C=20 so C is eliminated. Round 3 votes counts: B=39 D=35 A=26 so A is eliminated. Round 4 votes counts: B=53 D=47 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:219 D:202 B:201 A:192 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -10 -10 6 B 2 0 -10 6 4 C 10 10 0 10 8 D 10 -6 -10 0 10 E -6 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -10 -10 6 B 2 0 -10 6 4 C 10 10 0 10 8 D 10 -6 -10 0 10 E -6 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -10 -10 6 B 2 0 -10 6 4 C 10 10 0 10 8 D 10 -6 -10 0 10 E -6 -4 -8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9769: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (16) C B D A E (8) A D E C B (8) B C E A D (7) A E D B C (7) B C E D A (6) E B C A D (5) D A C E B (5) C D B A E (5) C B E D A (5) B E C A D (5) D A E C B (3) D A C B E (3) C B D E A (3) E B A D C (2) E B A C D (2) E A B D C (2) D C A B E (2) B E C D A (2) C D A B E (1) C B E A D (1) A D E B C (1) A C D B E (1) Total count = 100 A B C D E A 0 -2 0 14 -12 B 2 0 10 -4 -2 C 0 -10 0 2 -6 D -14 4 -2 0 -20 E 12 2 6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999552 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 0 14 -12 B 2 0 10 -4 -2 C 0 -10 0 2 -6 D -14 4 -2 0 -20 E 12 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 C=23 B=20 A=17 D=13 so D is eliminated. Round 2 votes counts: A=28 E=27 C=25 B=20 so B is eliminated. Round 3 votes counts: C=38 E=34 A=28 so A is eliminated. Round 4 votes counts: E=53 C=47 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:220 B:203 A:200 C:193 D:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 0 14 -12 B 2 0 10 -4 -2 C 0 -10 0 2 -6 D -14 4 -2 0 -20 E 12 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 14 -12 B 2 0 10 -4 -2 C 0 -10 0 2 -6 D -14 4 -2 0 -20 E 12 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 14 -12 B 2 0 10 -4 -2 C 0 -10 0 2 -6 D -14 4 -2 0 -20 E 12 2 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999731 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9770: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (7) A B C E D (7) D E C B A (6) B D A C E (6) A E C B D (5) B A D C E (4) A D B E C (4) A C E B D (4) E D C B A (3) E C D B A (3) E C D A B (3) E C A D B (3) D B E C A (3) D B C E A (3) B D C E A (3) A C B E D (3) A B D C E (3) E C A B D (2) D E C A B (2) D E A C B (2) D B E A C (2) D A E B C (2) D A B E C (2) C E A B D (2) C B E D A (2) A B D E C (2) A B C D E (2) E D C A B (1) E D A C B (1) E A C D B (1) E A C B D (1) D E A B C (1) D B A E C (1) D B A C E (1) C E B D A (1) C A E B D (1) B D C A E (1) Total count = 100 A B C D E A 0 8 8 2 -4 B -8 0 -6 12 -2 C -8 6 0 -6 0 D -2 -12 6 0 0 E 4 2 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.098767 E: 0.901233 Sum of squares = 0.821976260427 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.098767 E: 1.000000 A B C D E A 0 8 8 2 -4 B -8 0 -6 12 -2 C -8 6 0 -6 0 D -2 -12 6 0 0 E 4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.75510204637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=25 E=18 B=14 C=13 so C is eliminated. Round 2 votes counts: A=31 E=28 D=25 B=16 so B is eliminated. Round 3 votes counts: D=35 A=35 E=30 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:207 E:203 B:198 C:196 D:196 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 8 2 -4 B -8 0 -6 12 -2 C -8 6 0 -6 0 D -2 -12 6 0 0 E 4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.75510204637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 8 2 -4 B -8 0 -6 12 -2 C -8 6 0 -6 0 D -2 -12 6 0 0 E 4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.75510204637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 8 2 -4 B -8 0 -6 12 -2 C -8 6 0 -6 0 D -2 -12 6 0 0 E 4 2 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 0.857143 Sum of squares = 0.75510204637 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.142857 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9771: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (5) B A D C E (5) E D A C B (4) C E D A B (4) C D E B A (4) C D B E A (4) E A C B D (3) D B A C E (3) C B E A D (3) C B A E D (3) B C A D E (3) B A D E C (3) E D A B C (2) E C D A B (2) E A D C B (2) E A D B C (2) D E C A B (2) D E A B C (2) D C E B A (2) D C E A B (2) D C B E A (2) D C B A E (2) D B C A E (2) D B A E C (2) C B D A E (2) C B A D E (2) B D C A E (2) B C D A E (2) B A E D C (2) B A E C D (2) A E B D C (2) A B E D C (2) E C A D B (1) E C A B D (1) E A C D B (1) E A B C D (1) D E A C B (1) D A B E C (1) C E B A D (1) C E A D B (1) C E A B D (1) C D E A B (1) C B E D A (1) C B D E A (1) B A C E D (1) B A C D E (1) A E D B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -2 -10 -2 B 20 0 -6 0 14 C 2 6 0 -8 20 D 10 0 8 0 14 E 2 -14 -20 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.322083 C: 0.000000 D: 0.677917 E: 0.000000 Sum of squares = 0.563308714927 Cumulative probabilities = A: 0.000000 B: 0.322083 C: 0.322083 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -2 -10 -2 B 20 0 -6 0 14 C 2 6 0 -8 20 D 10 0 8 0 14 E 2 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 B=26 D=21 E=19 A=6 so A is eliminated. Round 2 votes counts: B=29 C=28 E=22 D=21 so D is eliminated. Round 3 votes counts: B=37 C=36 E=27 so E is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:216 B:214 C:210 A:183 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -20 -2 -10 -2 B 20 0 -6 0 14 C 2 6 0 -8 20 D 10 0 8 0 14 E 2 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -2 -10 -2 B 20 0 -6 0 14 C 2 6 0 -8 20 D 10 0 8 0 14 E 2 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -2 -10 -2 B 20 0 -6 0 14 C 2 6 0 -8 20 D 10 0 8 0 14 E 2 -14 -20 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.499999 C: 0.000000 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.499999 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9772: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (7) C E B D A (6) A E D C B (6) E C A D B (5) D A E B C (5) C B D E A (5) A E D B C (5) A D B E C (5) E C B D A (4) E C B A D (4) E C A B D (4) B C D A E (4) E A D C B (3) D A B E C (3) A E C B D (3) E C D A B (2) E A C D B (2) D E A B C (2) D B C A E (2) D B A C E (2) D A B C E (2) B C D E A (2) A E C D B (2) A D B C E (2) A B D C E (2) E D C B A (1) D B C E A (1) D B A E C (1) C E B A D (1) C E A B D (1) C B A E D (1) B D C E A (1) B D C A E (1) B C A D E (1) A D E B C (1) A B C D E (1) Total count = 100 A B C D E A 0 12 -6 -2 -2 B -12 0 -14 -4 -14 C 6 14 0 10 -16 D 2 4 -10 0 -14 E 2 14 16 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 -6 -2 -2 B -12 0 -14 -4 -14 C 6 14 0 10 -16 D 2 4 -10 0 -14 E 2 14 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 C=21 D=18 B=9 so B is eliminated. Round 2 votes counts: C=28 A=27 E=25 D=20 so D is eliminated. Round 3 votes counts: A=40 C=33 E=27 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:223 C:207 A:201 D:191 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 -6 -2 -2 B -12 0 -14 -4 -14 C 6 14 0 10 -16 D 2 4 -10 0 -14 E 2 14 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -6 -2 -2 B -12 0 -14 -4 -14 C 6 14 0 10 -16 D 2 4 -10 0 -14 E 2 14 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -6 -2 -2 B -12 0 -14 -4 -14 C 6 14 0 10 -16 D 2 4 -10 0 -14 E 2 14 16 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999995981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9773: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (11) A B D E C (8) A E D B C (7) E D A B C (5) C B D E A (5) C E D B A (4) C E B D A (4) C B A D E (4) A B C D E (4) E D A C B (3) E C D B A (3) D E A B C (3) A D E B C (3) A D B E C (3) A B D C E (3) E C D A B (2) E A D C B (2) D E B C A (2) C B A E D (2) B D A E C (2) B C D A E (2) B A C D E (2) A C B E D (2) E D C A B (1) E C A D B (1) D E C B A (1) D B E A C (1) C E D A B (1) C E A D B (1) C D B E A (1) C B E D A (1) C B D A E (1) B D C E A (1) B C D E A (1) B C A D E (1) A E D C B (1) A B C E D (1) Total count = 100 A B C D E A 0 2 0 -10 -8 B -2 0 -2 -12 -10 C 0 2 0 -14 -18 D 10 12 14 0 -4 E 8 10 18 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999908 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 -10 -8 B -2 0 -2 -12 -10 C 0 2 0 -14 -18 D 10 12 14 0 -4 E 8 10 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 E=28 C=24 B=9 D=7 so D is eliminated. Round 2 votes counts: E=34 A=32 C=24 B=10 so B is eliminated. Round 3 votes counts: A=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:216 A:192 B:187 C:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -10 -8 B -2 0 -2 -12 -10 C 0 2 0 -14 -18 D 10 12 14 0 -4 E 8 10 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -10 -8 B -2 0 -2 -12 -10 C 0 2 0 -14 -18 D 10 12 14 0 -4 E 8 10 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -10 -8 B -2 0 -2 -12 -10 C 0 2 0 -14 -18 D 10 12 14 0 -4 E 8 10 18 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999934 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9774: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (8) E C D A B (7) C B A E D (7) B A D C E (6) B A C D E (6) E D C B A (4) E C D B A (4) C E A B D (4) B D A C E (4) A B D C E (4) E D C A B (3) E C B D A (3) E A D C B (3) C E B A D (3) E D B C A (2) E D A B C (2) D E A B C (2) D B E A C (2) D A E B C (2) C E A D B (2) C B E A D (2) C A B E D (2) B C A D E (2) A D E C B (2) A B C D E (2) E D A C B (1) E C A D B (1) E B C D A (1) E A C D B (1) D B A E C (1) C E B D A (1) C B A D E (1) B D A E C (1) A D E B C (1) A D B E C (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 0 2 4 4 B 0 0 -4 0 2 C -2 4 0 0 -4 D -4 0 0 0 -6 E -4 -2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.777253 B: 0.222747 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.653738538332 Cumulative probabilities = A: 0.777253 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 4 4 B 0 0 -4 0 2 C -2 4 0 0 -4 D -4 0 0 0 -6 E -4 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564367 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=32 C=22 B=19 D=15 A=12 so A is eliminated. Round 2 votes counts: E=32 B=25 C=23 D=20 so D is eliminated. Round 3 votes counts: E=39 B=38 C=23 so C is eliminated. Round 4 votes counts: B=51 E=49 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:205 E:202 B:199 C:199 D:195 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 2 4 4 B 0 0 -4 0 2 C -2 4 0 0 -4 D -4 0 0 0 -6 E -4 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564367 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 4 4 B 0 0 -4 0 2 C -2 4 0 0 -4 D -4 0 0 0 -6 E -4 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564367 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 4 4 B 0 0 -4 0 2 C -2 4 0 0 -4 D -4 0 0 0 -6 E -4 -2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.333333 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.555555564367 Cumulative probabilities = A: 0.666667 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9775: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) E D A C B (6) A B E D C (6) A B E C D (6) C D E B A (5) C B D E A (5) B C A E D (5) D E C B A (4) B A C E D (4) E D A B C (3) E A D B C (3) D E A C B (3) C B E D A (3) A E D B C (3) A D B E C (3) E D C B A (2) D C E B A (2) D C E A B (2) D A E C B (2) C D B E A (2) C D B A E (2) B C A D E (2) B A E C D (2) A E B D C (2) A D E B C (2) A B C D E (2) E D C A B (1) E D B A C (1) E C D B A (1) E C B D A (1) E B C D A (1) E A B D C (1) D C B E A (1) C E D B A (1) C B A D E (1) B C E A D (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -10 -12 B -8 0 -2 -12 -6 C -2 2 0 -10 -22 D 10 12 10 0 -8 E 12 6 22 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 2 -10 -12 B -8 0 -2 -12 -6 C -2 2 0 -10 -22 D 10 12 10 0 -8 E 12 6 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=21 E=20 C=19 B=14 so B is eliminated. Round 2 votes counts: A=32 C=27 D=21 E=20 so E is eliminated. Round 3 votes counts: A=36 D=34 C=30 so C is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:224 D:212 A:194 B:186 C:184 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 2 -10 -12 B -8 0 -2 -12 -6 C -2 2 0 -10 -22 D 10 12 10 0 -8 E 12 6 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -10 -12 B -8 0 -2 -12 -6 C -2 2 0 -10 -22 D 10 12 10 0 -8 E 12 6 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -10 -12 B -8 0 -2 -12 -6 C -2 2 0 -10 -22 D 10 12 10 0 -8 E 12 6 22 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999998907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9776: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) E B D A C (5) B A C D E (5) D C B A E (4) D B C E A (4) D B C A E (4) B D C A E (4) E D C A B (3) E D B C A (3) E A C D B (3) D C E B A (3) D C E A B (3) D C A B E (3) C A D B E (3) B C D A E (3) A E C D B (3) E D A B C (2) E B A D C (2) E B A C D (2) D E C B A (2) D E C A B (2) B A C E D (2) A C E D B (2) A C B E D (2) A B C D E (2) E D B A C (1) E C D A B (1) E C A D B (1) E B D C A (1) E A D B C (1) E A C B D (1) D E B C A (1) D C B E A (1) D C A E B (1) D B E C A (1) C D A B E (1) C A D E B (1) B E A C D (1) B D E C A (1) B C A D E (1) A E C B D (1) A E B C D (1) A C E B D (1) A C D B E (1) A C B D E (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 -4 -8 -4 B 2 0 12 -10 -8 C 4 -12 0 -4 6 D 8 10 4 0 4 E 4 8 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999963 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 -8 -4 B 2 0 12 -10 -8 C 4 -12 0 -4 6 D 8 10 4 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 D=29 B=17 A=16 C=5 so C is eliminated. Round 2 votes counts: E=33 D=30 A=20 B=17 so B is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:213 E:201 B:198 C:197 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 -4 -8 -4 B 2 0 12 -10 -8 C 4 -12 0 -4 6 D 8 10 4 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 -8 -4 B 2 0 12 -10 -8 C 4 -12 0 -4 6 D 8 10 4 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 -8 -4 B 2 0 12 -10 -8 C 4 -12 0 -4 6 D 8 10 4 0 4 E 4 8 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999632 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9777: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (10) B A E C D (10) B A C E D (9) E A B C D (6) E A D B C (4) D E A C B (4) D C B A E (4) B A E D C (4) D E C A B (3) D C E B A (3) C D B A E (3) C B D A E (3) C B A D E (3) B C A E D (3) A E B C D (3) E D A B C (2) E A C B D (2) E A B D C (2) D E A B C (2) D C B E A (2) D B C A E (2) C B A E D (2) B A C D E (2) A B E C D (2) E D A C B (1) E A C D B (1) C D E B A (1) C D B E A (1) B D C A E (1) B D A E C (1) B D A C E (1) B C A D E (1) B A D C E (1) A E B D C (1) Total count = 100 A B C D E A 0 -14 16 12 12 B 14 0 14 14 10 C -16 -14 0 4 4 D -12 -14 -4 0 -4 E -12 -10 -4 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 16 12 12 B 14 0 14 14 10 C -16 -14 0 4 4 D -12 -14 -4 0 -4 E -12 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=30 E=18 C=13 A=6 so A is eliminated. Round 2 votes counts: B=35 D=30 E=22 C=13 so C is eliminated. Round 3 votes counts: B=43 D=35 E=22 so E is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:226 A:213 C:189 E:189 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 16 12 12 B 14 0 14 14 10 C -16 -14 0 4 4 D -12 -14 -4 0 -4 E -12 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 16 12 12 B 14 0 14 14 10 C -16 -14 0 4 4 D -12 -14 -4 0 -4 E -12 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 16 12 12 B 14 0 14 14 10 C -16 -14 0 4 4 D -12 -14 -4 0 -4 E -12 -10 -4 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9778: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A D E (12) C B D E A (10) E D A C B (8) E A D B C (7) E A D C B (6) D E A C B (5) C D B E A (5) B C A E D (5) B C D A E (4) A E D B C (4) D C E B A (3) D A E C B (3) C B D A E (3) B C E D A (3) B C E A D (3) E D A B C (2) C B A D E (2) B A E C D (2) E B A D C (1) E A B D C (1) D E C A B (1) D C E A B (1) C D A B E (1) B E A C D (1) B A C E D (1) A E B D C (1) A D E C B (1) A D E B C (1) A D C E B (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -6 2 -14 B 10 0 0 2 8 C 6 0 0 6 10 D -2 -2 -6 0 6 E 14 -8 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.755780 C: 0.244220 D: 0.000000 E: 0.000000 Sum of squares = 0.630846579233 Cumulative probabilities = A: 0.000000 B: 0.755780 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 2 -14 B 10 0 0 2 8 C 6 0 0 6 10 D -2 -2 -6 0 6 E 14 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 C=21 D=13 A=10 so A is eliminated. Round 2 votes counts: B=33 E=30 C=21 D=16 so D is eliminated. Round 3 votes counts: E=41 B=33 C=26 so C is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:211 B:210 D:198 E:195 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 2 -14 B 10 0 0 2 8 C 6 0 0 6 10 D -2 -2 -6 0 6 E 14 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 2 -14 B 10 0 0 2 8 C 6 0 0 6 10 D -2 -2 -6 0 6 E 14 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 2 -14 B 10 0 0 2 8 C 6 0 0 6 10 D -2 -2 -6 0 6 E 14 -8 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999981 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9779: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) A C E D B (8) E D B A C (7) B D E C A (7) E B D C A (6) D B A C E (6) C E A B D (6) D B E A C (5) E D B C A (4) D B A E C (4) C A B D E (4) B D C E A (4) E C A D B (2) E A C D B (2) C A B E D (2) B D E A C (2) B C D E A (2) E C B D A (1) E C B A D (1) E C A B D (1) E B C D A (1) D E B A C (1) D A B C E (1) C E A D B (1) C B A D E (1) C A E D B (1) B E D C A (1) B E C D A (1) B D C A E (1) B D A C E (1) B C E D A (1) B A D C E (1) A E D C B (1) A D C E B (1) A D C B E (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -16 -14 -12 -12 B 16 0 12 8 -6 C 14 -12 0 -8 6 D 12 -8 8 0 -12 E 12 6 -6 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.375000000015 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -16 -14 -12 -12 B 16 0 12 8 -6 C 14 -12 0 -8 6 D 12 -8 8 0 -12 E 12 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=24 B=21 D=17 A=13 so A is eliminated. Round 2 votes counts: C=34 E=26 B=21 D=19 so D is eliminated. Round 3 votes counts: B=37 C=36 E=27 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 E:212 C:200 D:200 A:173 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 -14 -12 -12 B 16 0 12 8 -6 C 14 -12 0 -8 6 D 12 -8 8 0 -12 E 12 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -14 -12 -12 B 16 0 12 8 -6 C 14 -12 0 -8 6 D 12 -8 8 0 -12 E 12 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -14 -12 -12 B 16 0 12 8 -6 C 14 -12 0 -8 6 D 12 -8 8 0 -12 E 12 6 -6 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.000000 E: 0.500000 Sum of squares = 0.374999999941 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9780: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (7) D A C B E (7) C D A B E (7) C A D E B (7) B E D C A (6) B D E A C (6) A D C E B (6) E B D A C (4) E B C D A (3) E B A C D (3) D A B C E (3) C B D A E (3) B D A E C (3) E C B A D (2) E B A D C (2) E A C B D (2) D B A E C (2) C E A D B (2) C A D B E (2) B E D A C (2) B E C D A (2) B D C A E (2) A D C B E (2) A C E D B (2) E C A D B (1) E C A B D (1) D C A B E (1) D B A C E (1) C D B A E (1) C A E D B (1) B D E C A (1) B D C E A (1) B D A C E (1) A E D C B (1) A E D B C (1) A D E B C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 0 -12 10 B 4 0 2 2 6 C 0 -2 0 -6 0 D 12 -2 6 0 16 E -10 -6 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999587 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 0 -12 10 B 4 0 2 2 6 C 0 -2 0 -6 0 D 12 -2 6 0 16 E -10 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 B=24 C=23 D=14 A=14 so D is eliminated. Round 2 votes counts: B=27 E=25 C=24 A=24 so C is eliminated. Round 3 votes counts: A=42 B=31 E=27 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:216 B:207 A:197 C:196 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 0 -12 10 B 4 0 2 2 6 C 0 -2 0 -6 0 D 12 -2 6 0 16 E -10 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 -12 10 B 4 0 2 2 6 C 0 -2 0 -6 0 D 12 -2 6 0 16 E -10 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 -12 10 B 4 0 2 2 6 C 0 -2 0 -6 0 D 12 -2 6 0 16 E -10 -6 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999912 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9781: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D A E (10) B C E D A (7) E A D C B (6) E A D B C (6) D A C E B (5) B C E A D (5) E D A B C (4) B C D E A (4) B C D A E (4) E A B D C (3) D A E C B (3) C B A D E (3) B E C D A (3) B E C A D (3) A D E C B (3) A D C E B (3) E B D A C (2) E A B C D (2) C D B A E (2) C A B D E (2) B D C E A (2) A E D C B (2) E B C A D (1) E B A D C (1) E B A C D (1) D E B A C (1) D B E C A (1) D B C A E (1) D A C B E (1) C D A B E (1) C B A E D (1) C A E B D (1) C A D B E (1) B E D C A (1) A E C D B (1) A E C B D (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -6 -4 -6 B 6 0 4 14 6 C 6 -4 0 10 10 D 4 -14 -10 0 -4 E 6 -6 -10 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -4 -6 B 6 0 4 14 6 C 6 -4 0 10 10 D 4 -14 -10 0 -4 E 6 -6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 E=26 C=21 D=12 A=12 so D is eliminated. Round 2 votes counts: B=31 E=27 C=21 A=21 so C is eliminated. Round 3 votes counts: B=47 E=27 A=26 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:211 E:197 A:189 D:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 -6 B 6 0 4 14 6 C 6 -4 0 10 10 D 4 -14 -10 0 -4 E 6 -6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 -6 B 6 0 4 14 6 C 6 -4 0 10 10 D 4 -14 -10 0 -4 E 6 -6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 -6 B 6 0 4 14 6 C 6 -4 0 10 10 D 4 -14 -10 0 -4 E 6 -6 -10 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999868 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9782: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (10) A C D E B (10) B C A D E (9) E D A C B (8) B C A E D (8) E D B A C (6) D E B C A (5) D A C E B (5) E B D C A (4) E D B C A (3) D E A C B (3) C A D B E (3) A C B E D (3) E B D A C (2) C A B D E (2) B D E C A (2) B A C E D (2) A C E D B (2) A C D B E (2) E A C B D (1) D E C B A (1) D C A E B (1) C D A B E (1) B E C D A (1) B E C A D (1) B E A C D (1) B C D A E (1) B A E C D (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -14 -4 -6 4 B 14 0 12 -2 -4 C 4 -12 0 0 0 D 6 2 0 0 -8 E -4 4 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.471074380154 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 A B C D E A 0 -14 -4 -6 4 B 14 0 12 -2 -4 C 4 -12 0 0 0 D 6 2 0 0 -8 E -4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 E=24 A=19 D=15 C=6 so C is eliminated. Round 2 votes counts: B=36 E=24 A=24 D=16 so D is eliminated. Round 3 votes counts: B=36 E=33 A=31 so A is eliminated. Round 4 votes counts: E=52 B=48 so B is eliminated. IRV winner is E compare: Computing Borda winner. B:210 E:204 D:200 C:196 A:190 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -4 -6 4 B 14 0 12 -2 -4 C 4 -12 0 0 0 D 6 2 0 0 -8 E -4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -4 -6 4 B 14 0 12 -2 -4 C 4 -12 0 0 0 D 6 2 0 0 -8 E -4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -4 -6 4 B 14 0 12 -2 -4 C 4 -12 0 0 0 D 6 2 0 0 -8 E -4 4 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.181818 B: 0.181818 C: 0.000000 D: 0.000000 E: 0.636364 Sum of squares = 0.471074380148 Cumulative probabilities = A: 0.181818 B: 0.363636 C: 0.363636 D: 0.363636 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9783: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (9) E C B D A (8) D B C E A (8) C D B E A (8) A E C B D (6) A D B C E (6) A B D C E (6) E B C D A (5) A E B D C (5) D B C A E (4) A E C D B (4) B D C E A (3) E B D C A (2) D B A C E (2) B D A C E (2) A E B C D (2) A D C B E (2) A B E D C (2) E C D A B (1) E C A D B (1) E C A B D (1) E B A D C (1) E A C D B (1) E A C B D (1) E A B C D (1) D C B E A (1) D C B A E (1) C E D B A (1) C E A D B (1) C D E B A (1) C D A B E (1) B D E C A (1) A C E D B (1) A B D E C (1) Total count = 100 A B C D E A 0 -14 -14 -16 -10 B 14 0 2 -6 -4 C 14 -2 0 6 -4 D 16 6 -6 0 -6 E 10 4 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -14 -16 -10 B 14 0 2 -6 -4 C 14 -2 0 6 -4 D 16 6 -6 0 -6 E 10 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=31 D=16 C=12 B=6 so B is eliminated. Round 2 votes counts: A=35 E=31 D=22 C=12 so C is eliminated. Round 3 votes counts: A=35 E=33 D=32 so D is eliminated. Round 4 votes counts: E=55 A=45 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 C:207 D:205 B:203 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -14 -16 -10 B 14 0 2 -6 -4 C 14 -2 0 6 -4 D 16 6 -6 0 -6 E 10 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -14 -16 -10 B 14 0 2 -6 -4 C 14 -2 0 6 -4 D 16 6 -6 0 -6 E 10 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -14 -16 -10 B 14 0 2 -6 -4 C 14 -2 0 6 -4 D 16 6 -6 0 -6 E 10 4 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9784: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) A B D E C (8) C E D A B (5) C B D E A (5) E D C B A (4) E D A B C (4) A E D B C (4) E D A C B (3) D E B A C (3) C B A D E (3) B D E C A (3) B D E A C (3) A B C D E (3) E D B C A (2) E C D A B (2) D E B C A (2) C A E D B (2) C A B E D (2) C A B D E (2) B D A E C (2) B C A D E (2) B A D E C (2) B A D C E (2) A C E D B (2) A C B E D (2) E D C A B (1) E D B A C (1) E C D B A (1) E A D C B (1) E A D B C (1) C E A D B (1) C D E B A (1) C B E D A (1) C B D A E (1) B D C E A (1) B C D E A (1) B C D A E (1) B A C D E (1) A E C B D (1) A D B E C (1) A C E B D (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -6 -14 -12 B 4 0 -4 -2 -4 C 6 4 0 2 0 D 14 2 -2 0 -4 E 12 4 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.635926 D: 0.000000 E: 0.364073 Sum of squares = 0.536952011083 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.635926 D: 0.635927 E: 1.000000 A B C D E A 0 -4 -6 -14 -12 B 4 0 -4 -2 -4 C 6 4 0 2 0 D 14 2 -2 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 A=24 E=20 B=18 D=5 so D is eliminated. Round 2 votes counts: C=33 E=25 A=24 B=18 so B is eliminated. Round 3 votes counts: C=38 E=31 A=31 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:210 C:206 D:205 B:197 A:182 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -6 -14 -12 B 4 0 -4 -2 -4 C 6 4 0 2 0 D 14 2 -2 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -14 -12 B 4 0 -4 -2 -4 C 6 4 0 2 0 D 14 2 -2 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -14 -12 B 4 0 -4 -2 -4 C 6 4 0 2 0 D 14 2 -2 0 -4 E 12 4 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9785: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (16) C E B A D (13) D A C B E (12) E C B A D (7) C D A B E (7) D A B C E (6) A D B C E (5) E C B D A (3) E B D A C (3) E B C A D (3) E B A D C (2) E B A C D (2) C D E A B (2) C B A E D (2) C A D B E (2) C A B D E (2) B A D E C (2) E C D A B (1) E B C D A (1) C D A E B (1) C B A D E (1) C A D E B (1) B E A D C (1) B E A C D (1) B C A D E (1) B A E D C (1) B A C D E (1) A D B E C (1) Total count = 100 A B C D E A 0 12 6 -4 22 B -12 0 -8 -8 22 C -6 8 0 2 12 D 4 8 -2 0 20 E -22 -22 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888954 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 A B C D E A 0 12 6 -4 22 B -12 0 -8 -8 22 C -6 8 0 2 12 D 4 8 -2 0 20 E -22 -22 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888825 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 C=31 E=22 B=7 A=6 so A is eliminated. Round 2 votes counts: D=40 C=31 E=22 B=7 so B is eliminated. Round 3 votes counts: D=42 C=33 E=25 so E is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:218 D:215 C:208 B:197 E:162 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 6 -4 22 B -12 0 -8 -8 22 C -6 8 0 2 12 D 4 8 -2 0 20 E -22 -22 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888825 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 6 -4 22 B -12 0 -8 -8 22 C -6 8 0 2 12 D 4 8 -2 0 20 E -22 -22 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888825 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 6 -4 22 B -12 0 -8 -8 22 C -6 8 0 2 12 D 4 8 -2 0 20 E -22 -22 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.333333 D: 0.500000 E: 0.000000 Sum of squares = 0.388888888825 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9786: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C E B (9) B C E D A (8) C D A E B (6) C B A D E (5) B E C D A (5) A D E B C (5) E D A B C (4) E B D C A (4) D A E C B (3) C B E D A (3) C A D B E (3) B E D C A (3) A B E D C (3) E B D A C (2) D E A C B (2) C D E B A (2) C B D A E (2) C A D E B (2) B E A D C (2) B C E A D (2) B A E D C (2) A D E C B (2) A B D E C (2) E D C B A (1) E C D B A (1) D E C A B (1) D E A B C (1) C E B D A (1) C D E A B (1) C B E A D (1) C B D E A (1) C B A E D (1) C A B D E (1) B E D A C (1) B C A E D (1) A E B D C (1) A D C B E (1) A D B E C (1) A C D E B (1) A C D B E (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 -10 -4 8 B -4 0 -4 6 2 C 10 4 0 -2 8 D 4 -6 2 0 8 E -8 -2 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.38888888889 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 -4 8 B -4 0 -4 6 2 C 10 4 0 -2 8 D 4 -6 2 0 8 E -8 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888975 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 A=28 B=24 E=12 D=7 so D is eliminated. Round 2 votes counts: A=31 C=29 B=24 E=16 so E is eliminated. Round 3 votes counts: A=38 C=32 B=30 so B is eliminated. Round 4 votes counts: C=55 A=45 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:204 B:200 A:199 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 -4 8 B -4 0 -4 6 2 C 10 4 0 -2 8 D 4 -6 2 0 8 E -8 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888975 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 -4 8 B -4 0 -4 6 2 C 10 4 0 -2 8 D 4 -6 2 0 8 E -8 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888975 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 -4 8 B -4 0 -4 6 2 C 10 4 0 -2 8 D 4 -6 2 0 8 E -8 -2 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888975 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9787: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (10) C E D A B (7) B E D C A (7) D E C B A (6) D E B C A (5) C A E D B (5) A C E B D (5) A C B E D (5) D E C A B (4) B A D E C (4) A C E D B (4) A B C D E (4) E C D B A (3) B D A E C (3) E B C D A (2) C A E B D (2) B A E D C (2) B A E C D (2) B A C E D (2) A C D E B (2) A C B D E (2) E D C B A (1) E D B C A (1) E C D A B (1) D E A C B (1) D B E C A (1) D B E A C (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E A B (1) B E C D A (1) B E A C D (1) B D E A C (1) B C A E D (1) Total count = 100 A B C D E A 0 8 -2 6 6 B -8 0 -4 10 -6 C 2 4 0 26 6 D -6 -10 -26 0 -30 E -6 6 -6 30 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999075 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -2 6 6 B -8 0 -4 10 -6 C 2 4 0 26 6 D -6 -10 -26 0 -30 E -6 6 -6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=24 D=18 C=18 E=8 so E is eliminated. Round 2 votes counts: A=32 B=26 C=22 D=20 so D is eliminated. Round 3 votes counts: B=34 C=33 A=33 so C is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:219 E:212 A:209 B:196 D:164 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -2 6 6 B -8 0 -4 10 -6 C 2 4 0 26 6 D -6 -10 -26 0 -30 E -6 6 -6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -2 6 6 B -8 0 -4 10 -6 C 2 4 0 26 6 D -6 -10 -26 0 -30 E -6 6 -6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -2 6 6 B -8 0 -4 10 -6 C 2 4 0 26 6 D -6 -10 -26 0 -30 E -6 6 -6 30 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999208 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9788: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (12) C B E A D (10) A D E B C (6) E B D A C (5) D B E A C (5) C B E D A (5) C A E B D (5) A C D E B (5) B E D C A (4) D A B E C (3) C A B E D (3) E B C A D (2) D E A B C (2) D A E C B (2) D A C B E (2) C B A E D (2) C A B D E (2) B E C D A (2) B C E D A (2) A D C E B (2) A C E B D (2) E B A C D (1) E A B D C (1) D B C E A (1) D A C E B (1) C E B A D (1) C D B E A (1) C D B A E (1) C B D E A (1) C A D E B (1) C A D B E (1) B D E C A (1) B D C E A (1) B C D E A (1) A E D B C (1) A D E C B (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 8 6 -4 8 B -8 0 0 2 -2 C -6 0 0 0 4 D 4 -2 0 0 6 E -8 2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.428571428567 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 A B C D E A 0 8 6 -4 8 B -8 0 0 2 -2 C -6 0 0 0 4 D 4 -2 0 0 6 E -8 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142846 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=28 A=19 B=11 E=9 so E is eliminated. Round 2 votes counts: C=33 D=28 A=20 B=19 so B is eliminated. Round 3 votes counts: C=40 D=39 A=21 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. A:209 D:204 C:199 B:196 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 6 -4 8 B -8 0 0 2 -2 C -6 0 0 0 4 D 4 -2 0 0 6 E -8 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142846 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 6 -4 8 B -8 0 0 2 -2 C -6 0 0 0 4 D 4 -2 0 0 6 E -8 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142846 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 6 -4 8 B -8 0 0 2 -2 C -6 0 0 0 4 D 4 -2 0 0 6 E -8 2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.142857 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.000000 Sum of squares = 0.42857142846 Cumulative probabilities = A: 0.142857 B: 0.428571 C: 0.428571 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9789: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) D B C A E (7) D C E A B (6) D C B E A (6) D B A C E (6) B A E D C (6) A E B C D (5) E C A B D (4) D B A E C (4) C E D A B (4) B A E C D (4) E A C B D (3) C E A B D (3) B A D E C (3) B A C E D (3) A B E C D (3) E D A C B (2) D C E B A (2) C B A E D (2) B D C A E (2) B D A E C (2) B D A C E (2) E D C A B (1) E A D B C (1) E A C D B (1) E A B D C (1) D E C A B (1) D C B A E (1) C D E A B (1) C D B E A (1) C A B E D (1) B C A D E (1) A E C B D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -4 14 4 10 B 4 0 20 12 12 C -14 -20 0 -8 -2 D -4 -12 8 0 -10 E -10 -12 2 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999706 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 14 4 10 B 4 0 20 12 12 C -14 -20 0 -8 -2 D -4 -12 8 0 -10 E -10 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=23 E=21 C=12 A=11 so A is eliminated. Round 2 votes counts: D=33 B=28 E=27 C=12 so C is eliminated. Round 3 votes counts: D=35 E=34 B=31 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:224 A:212 E:195 D:191 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 14 4 10 B 4 0 20 12 12 C -14 -20 0 -8 -2 D -4 -12 8 0 -10 E -10 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 14 4 10 B 4 0 20 12 12 C -14 -20 0 -8 -2 D -4 -12 8 0 -10 E -10 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 14 4 10 B 4 0 20 12 12 C -14 -20 0 -8 -2 D -4 -12 8 0 -10 E -10 -12 2 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999216 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9790: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (15) E D C B A (9) D E C B A (9) D C E B A (8) A B C E D (8) A B E C D (6) D E C A B (4) E D C A B (3) D E A C B (3) C D E B A (3) B A C D E (3) E C D B A (2) E C B A D (2) E A B D C (2) E A B C D (2) C D B A E (2) C B E A D (2) B A C E D (2) A E B C D (2) A B D E C (2) E D A C B (1) D A B C E (1) C E D B A (1) C E B D A (1) C D B E A (1) C B D E A (1) C B A D E (1) B C A D E (1) A E B D C (1) A D B E C (1) A B D C E (1) Total count = 100 A B C D E A 0 4 0 2 -8 B -4 0 -6 4 -6 C 0 6 0 10 2 D -2 -4 -10 0 12 E 8 6 -2 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.130362 B: 0.000000 C: 0.869638 D: 0.000000 E: 0.000000 Sum of squares = 0.77326406738 Cumulative probabilities = A: 0.130362 B: 0.130362 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 2 -8 B -4 0 -6 4 -6 C 0 6 0 10 2 D -2 -4 -10 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005124 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=25 E=21 C=12 B=6 so B is eliminated. Round 2 votes counts: A=41 D=25 E=21 C=13 so C is eliminated. Round 3 votes counts: A=43 D=32 E=25 so E is eliminated. Round 4 votes counts: A=51 D=49 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:209 E:200 A:199 D:198 B:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 0 2 -8 B -4 0 -6 4 -6 C 0 6 0 10 2 D -2 -4 -10 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005124 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 2 -8 B -4 0 -6 4 -6 C 0 6 0 10 2 D -2 -4 -10 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005124 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 2 -8 B -4 0 -6 4 -6 C 0 6 0 10 2 D -2 -4 -10 0 12 E 8 6 -2 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000005124 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9791: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (9) D E B A C (9) C A B E D (7) C A B D E (7) A C B E D (7) D C E B A (6) A B C E D (5) E B D A C (4) E B A D C (4) D E C B A (4) D C E A B (4) C D A B E (4) C D A E B (3) B A E C D (3) D E B C A (2) C D E A B (2) C B A E D (2) C A D B E (2) B E A D C (2) B A E D C (2) A E B C D (2) A B E C D (2) D E C A B (1) D C B E A (1) D B C E A (1) C A D E B (1) B E D A C (1) B D E A C (1) B D C E A (1) B A C E D (1) Total count = 100 A B C D E A 0 -6 4 -6 -4 B 6 0 -2 2 -2 C -4 2 0 -4 8 D 6 -2 4 0 -2 E 4 2 -8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999879 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -6 -4 B 6 0 -2 2 -2 C -4 2 0 -4 8 D 6 -2 4 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999692 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 E=17 A=16 B=11 so B is eliminated. Round 2 votes counts: D=30 C=28 A=22 E=20 so E is eliminated. Round 3 votes counts: D=44 C=28 A=28 so C is eliminated. Round 4 votes counts: D=53 A=47 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:203 B:202 C:201 E:200 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -6 -4 B 6 0 -2 2 -2 C -4 2 0 -4 8 D 6 -2 4 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999692 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -6 -4 B 6 0 -2 2 -2 C -4 2 0 -4 8 D 6 -2 4 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999692 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -6 -4 B 6 0 -2 2 -2 C -4 2 0 -4 8 D 6 -2 4 0 -2 E 4 2 -8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.250000 D: 0.250000 E: 0.000000 Sum of squares = 0.374999999692 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.750000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9792: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) E B C A D (11) B E D A C (8) C A D E B (7) E B C D A (6) C A E B D (6) E B D C A (5) D C A E B (5) D A B E C (5) C E B A D (4) C A D B E (4) C E B D A (3) C D A E B (3) B E A C D (3) A C D B E (3) E B D A C (2) D B E A C (2) B E A D C (2) D C E A B (1) D B A E C (1) D A E B C (1) D A B C E (1) B E C A D (1) A D C B E (1) A D B E C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 -12 -12 4 B -4 0 0 4 -8 C 12 0 0 4 2 D 12 -4 -4 0 -4 E -4 8 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.150439 C: 0.849561 D: 0.000000 E: 0.000000 Sum of squares = 0.744385835711 Cumulative probabilities = A: 0.000000 B: 0.150439 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 -12 4 B -4 0 0 4 -8 C 12 0 0 4 2 D 12 -4 -4 0 -4 E -4 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000021891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 C=27 E=24 B=14 A=6 so A is eliminated. Round 2 votes counts: D=31 C=30 E=24 B=15 so B is eliminated. Round 3 votes counts: E=39 D=31 C=30 so C is eliminated. Round 4 votes counts: E=52 D=48 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:209 E:203 D:200 B:196 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 -12 4 B -4 0 0 4 -8 C 12 0 0 4 2 D 12 -4 -4 0 -4 E -4 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000021891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 -12 4 B -4 0 0 4 -8 C 12 0 0 4 2 D 12 -4 -4 0 -4 E -4 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000021891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 -12 4 B -4 0 0 4 -8 C 12 0 0 4 2 D 12 -4 -4 0 -4 E -4 8 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.200000 C: 0.800000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000021891 Cumulative probabilities = A: 0.000000 B: 0.200000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9793: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (9) C E B A D (8) B D A C E (8) E A D C B (7) D A B E C (7) E C A B D (5) D B A E C (5) D A E B C (4) B D C A E (4) B C D A E (4) A D E B C (4) E C A D B (3) E A C D B (3) D B A C E (3) C B E D A (3) E C D A B (2) D A E C B (2) B C E A D (2) B A D C E (2) A E D C B (2) A D E C B (2) E C B A D (1) E A C B D (1) D E A C B (1) D B C A E (1) C E D A B (1) C B D E A (1) B C D E A (1) B C A E D (1) B C A D E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -8 6 6 4 B 8 0 -2 4 8 C -6 2 0 -8 -2 D -6 -4 8 0 4 E -4 -8 2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428581 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 A B C D E A 0 -8 6 6 4 B 8 0 -2 4 8 C -6 2 0 -8 -2 D -6 -4 8 0 4 E -4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428533 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=23 B=23 E=22 C=22 A=10 so A is eliminated. Round 2 votes counts: D=30 E=24 B=24 C=22 so C is eliminated. Round 3 votes counts: B=37 E=33 D=30 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 A:204 D:201 C:193 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 6 6 4 B 8 0 -2 4 8 C -6 2 0 -8 -2 D -6 -4 8 0 4 E -4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428533 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 6 4 B 8 0 -2 4 8 C -6 2 0 -8 -2 D -6 -4 8 0 4 E -4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428533 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 6 4 B 8 0 -2 4 8 C -6 2 0 -8 -2 D -6 -4 8 0 4 E -4 -8 2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.285714 D: 0.142857 E: 0.000000 Sum of squares = 0.428571428533 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.857143 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9794: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (14) E B A D C (10) E A D C B (9) C D A B E (8) B E A D C (6) E A D B C (5) B C D A E (4) E A B D C (3) D C A E B (3) C D B A E (3) E C D A B (2) D C A B E (2) D A C B E (2) C B D A E (2) B E A C D (2) B C A D E (2) B A E D C (2) A D E B C (2) A B D C E (2) E D C A B (1) E B C D A (1) D A C E B (1) C E D A B (1) C B E D A (1) B E C D A (1) B E C A D (1) B C E A D (1) B C D E A (1) B C A E D (1) B A D E C (1) B A D C E (1) B A C D E (1) A E D C B (1) A D E C B (1) A D C E B (1) A B E D C (1) Total count = 100 A B C D E A 0 18 2 6 10 B -18 0 -4 -12 -10 C -2 4 0 -8 2 D -6 12 8 0 2 E -10 10 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998723 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 2 6 10 B -18 0 -4 -12 -10 C -2 4 0 -8 2 D -6 12 8 0 2 E -10 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994334 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=29 B=24 D=8 A=8 so D is eliminated. Round 2 votes counts: C=34 E=31 B=24 A=11 so A is eliminated. Round 3 votes counts: C=38 E=35 B=27 so B is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. A:218 D:208 C:198 E:198 B:178 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 2 6 10 B -18 0 -4 -12 -10 C -2 4 0 -8 2 D -6 12 8 0 2 E -10 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994334 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 2 6 10 B -18 0 -4 -12 -10 C -2 4 0 -8 2 D -6 12 8 0 2 E -10 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994334 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 2 6 10 B -18 0 -4 -12 -10 C -2 4 0 -8 2 D -6 12 8 0 2 E -10 10 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994334 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9795: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (11) E C B D A (9) C D E A B (6) E B A C D (4) D A C B E (4) A D B C E (4) A B D E C (4) E C D A B (3) E B A D C (3) D C A B E (3) D A B C E (3) C E B D A (3) B E A C D (3) B A E D C (3) B A D E C (3) A D B E C (3) E C D B A (2) E C B A D (2) C E D B A (2) C E D A B (2) C D E B A (2) B C E D A (2) A B D C E (2) E C A D B (1) E A C D B (1) E A B C D (1) D C A E B (1) D B A C E (1) C D B A E (1) C D A E B (1) C D A B E (1) C B E D A (1) B E C A D (1) B D C A E (1) B C E A D (1) B A D C E (1) B A C D E (1) A D E C B (1) A D E B C (1) A B E D C (1) Total count = 100 A B C D E A 0 -14 -12 4 -20 B 14 0 8 14 -12 C 12 -8 0 22 -14 D -4 -14 -22 0 -12 E 20 12 14 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -14 -12 4 -20 B 14 0 8 14 -12 C 12 -8 0 22 -14 D -4 -14 -22 0 -12 E 20 12 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 C=19 B=16 A=16 D=12 so D is eliminated. Round 2 votes counts: E=37 C=23 A=23 B=17 so B is eliminated. Round 3 votes counts: E=41 A=32 C=27 so C is eliminated. Round 4 votes counts: E=60 A=40 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:229 B:212 C:206 A:179 D:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 -12 4 -20 B 14 0 8 14 -12 C 12 -8 0 22 -14 D -4 -14 -22 0 -12 E 20 12 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -12 4 -20 B 14 0 8 14 -12 C 12 -8 0 22 -14 D -4 -14 -22 0 -12 E 20 12 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -12 4 -20 B 14 0 8 14 -12 C 12 -8 0 22 -14 D -4 -14 -22 0 -12 E 20 12 14 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999816 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9796: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (7) E C B D A (6) E C A D B (6) D A B E C (6) C E B A D (6) B D A E C (5) D B A E C (4) E A D C B (3) D A E B C (3) C A E B D (3) B D E A C (3) A D C E B (3) A D B C E (3) E C D B A (2) E B D C A (2) E A C D B (2) D A B C E (2) C E A D B (2) C B A D E (2) B E D C A (2) B E C D A (2) B A D C E (2) B A C D E (2) A D E C B (2) A D C B E (2) E D B A C (1) E C D A B (1) E C B A D (1) E B D A C (1) D E B A C (1) D B A C E (1) C E B D A (1) C E A B D (1) C B E D A (1) C B A E D (1) C A B E D (1) C A B D E (1) B D E C A (1) B D C A E (1) B C D A E (1) B C A D E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 10 -8 10 B 14 0 4 10 6 C -10 -4 0 -12 -8 D 8 -10 12 0 10 E -10 -6 8 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 10 -8 10 B 14 0 4 10 6 C -10 -4 0 -12 -8 D 8 -10 12 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 E=25 C=19 D=17 A=12 so A is eliminated. Round 2 votes counts: B=28 D=27 E=25 C=20 so C is eliminated. Round 3 votes counts: E=38 B=34 D=28 so D is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:217 D:210 A:199 E:191 C:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 10 -8 10 B 14 0 4 10 6 C -10 -4 0 -12 -8 D 8 -10 12 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 10 -8 10 B 14 0 4 10 6 C -10 -4 0 -12 -8 D 8 -10 12 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 10 -8 10 B 14 0 4 10 6 C -10 -4 0 -12 -8 D 8 -10 12 0 10 E -10 -6 8 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999998938 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9797: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) B E D C A (6) A E D C B (5) A D E C B (5) A D C E B (5) E B D C A (4) E A D B C (4) C B D A E (4) A E D B C (4) E B D A C (3) D E A B C (3) D C B E A (3) B E C D A (3) B C D E A (3) A E C D B (3) A C D E B (3) A C B E D (3) E D B A C (2) D E B C A (2) D E B A C (2) C D A B E (2) C B A D E (2) B E C A D (2) B D E C A (2) B C E D A (2) B C A E D (2) E D B C A (1) E B A C D (1) E A B D C (1) D C A B E (1) D B E C A (1) D A C E B (1) C D B A E (1) C B A E D (1) C A D B E (1) A E C B D (1) A C E D B (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 6 14 -6 -4 B -6 0 10 -16 -18 C -14 -10 0 -24 -24 D 6 16 24 0 -14 E 4 18 24 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 14 -6 -4 B -6 0 10 -16 -18 C -14 -10 0 -24 -24 D 6 16 24 0 -14 E 4 18 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 E=23 B=20 D=13 C=11 so C is eliminated. Round 2 votes counts: A=34 B=27 E=23 D=16 so D is eliminated. Round 3 votes counts: A=38 B=32 E=30 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:230 D:216 A:205 B:185 C:164 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 14 -6 -4 B -6 0 10 -16 -18 C -14 -10 0 -24 -24 D 6 16 24 0 -14 E 4 18 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -6 -4 B -6 0 10 -16 -18 C -14 -10 0 -24 -24 D 6 16 24 0 -14 E 4 18 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -6 -4 B -6 0 10 -16 -18 C -14 -10 0 -24 -24 D 6 16 24 0 -14 E 4 18 24 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999991729 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9798: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (7) C E B D A (7) C E D B A (6) E C D A B (5) B D A C E (5) B A D C E (5) A D B C E (5) E C B A D (4) E C A B D (4) C B E D A (4) B D C A E (4) A D E B C (4) E A C B D (3) C B D E A (3) A E D B C (3) A E B C D (3) A D E C B (3) A D B E C (3) E A C D B (2) D E A C B (2) D C E A B (2) A E D C B (2) A B D C E (2) E C B D A (1) E C A D B (1) E A B C D (1) D A C E B (1) D A B C E (1) C E B A D (1) C D E B A (1) B C E D A (1) B C D E A (1) B C D A E (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 -6 -4 -12 B 2 0 -18 4 -26 C 6 18 0 12 0 D 4 -4 -12 0 -12 E 12 26 0 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.770927 D: 0.000000 E: 0.229073 Sum of squares = 0.646802793872 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.770927 D: 0.770927 E: 1.000000 A B C D E A 0 -2 -6 -4 -12 B 2 0 -18 4 -26 C 6 18 0 12 0 D 4 -4 -12 0 -12 E 12 26 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 A=27 C=22 B=17 D=6 so D is eliminated. Round 2 votes counts: E=30 A=29 C=24 B=17 so B is eliminated. Round 3 votes counts: A=39 C=31 E=30 so E is eliminated. Round 4 votes counts: C=53 A=47 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:225 C:218 A:188 D:188 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -6 -4 -12 B 2 0 -18 4 -26 C 6 18 0 12 0 D 4 -4 -12 0 -12 E 12 26 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -6 -4 -12 B 2 0 -18 4 -26 C 6 18 0 12 0 D 4 -4 -12 0 -12 E 12 26 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -6 -4 -12 B 2 0 -18 4 -26 C 6 18 0 12 0 D 4 -4 -12 0 -12 E 12 26 0 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.000000 E: 0.500001 Sum of squares = 0.499999999867 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.499999 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9799: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (12) D B E C A (11) A C B E D (8) D E A B C (7) C B E A D (5) A D E C B (5) A C E B D (5) C B A E D (3) B E D C A (3) A D C B E (3) A C E D B (3) A C D E B (3) A C D B E (3) D E B A C (2) D A E C B (2) D A E B C (2) C A B E D (2) B E C D A (2) E D A C B (1) E D A B C (1) E C B A D (1) E B D C A (1) E B C D A (1) D B E A C (1) D B C E A (1) D B C A E (1) D B A E C (1) D B A C E (1) D A B E C (1) D A B C E (1) C A E B D (1) B D E C A (1) B C E A D (1) B C D E A (1) A E D C B (1) A E C D B (1) A C B D E (1) Total count = 100 A B C D E A 0 2 6 -8 -4 B -2 0 4 -28 2 C -6 -4 0 -18 -14 D 8 28 18 0 20 E 4 -2 14 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 6 -8 -4 B -2 0 4 -28 2 C -6 -4 0 -18 -14 D 8 28 18 0 20 E 4 -2 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=43 A=33 C=11 B=8 E=5 so E is eliminated. Round 2 votes counts: D=45 A=33 C=12 B=10 so B is eliminated. Round 3 votes counts: D=50 A=33 C=17 so C is eliminated. Round 4 votes counts: D=54 A=46 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:237 A:198 E:198 B:188 C:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -8 -4 B -2 0 4 -28 2 C -6 -4 0 -18 -14 D 8 28 18 0 20 E 4 -2 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -8 -4 B -2 0 4 -28 2 C -6 -4 0 -18 -14 D 8 28 18 0 20 E 4 -2 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -8 -4 B -2 0 4 -28 2 C -6 -4 0 -18 -14 D 8 28 18 0 20 E 4 -2 14 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999259 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9800: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (18) E B D C A (12) A C D B E (12) B E A D C (7) B A E C D (6) A B E C D (6) E D C B A (5) D C E A B (4) D C E B A (3) D C A E B (3) A B C D E (3) E D B C A (2) E B D A C (2) D E C B A (2) B E D C A (2) B E D A C (2) E B A D C (1) C D E A B (1) C A D E B (1) C A D B E (1) B E C D A (1) B E A C D (1) B C D A E (1) B C A D E (1) B A E D C (1) A C B D E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 -14 -16 10 B -2 0 -2 -4 -8 C 14 2 0 6 -2 D 16 4 -6 0 2 E -10 8 2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.058080 B: 0.000000 C: 0.339393 D: 0.048991 E: 0.553536 Sum of squares = 0.427362679494 Cumulative probabilities = A: 0.058080 B: 0.058080 C: 0.397473 D: 0.446464 E: 1.000000 A B C D E A 0 2 -14 -16 10 B -2 0 -2 -4 -8 C 14 2 0 6 -2 D 16 4 -6 0 2 E -10 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.036723 B: 0.000000 C: 0.288136 D: 0.104520 E: 0.570621 Sum of squares = 0.420903954796 Cumulative probabilities = A: 0.036723 B: 0.036723 C: 0.324859 D: 0.429379 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=23 E=22 B=22 C=21 D=12 so D is eliminated. Round 2 votes counts: C=31 E=24 A=23 B=22 so B is eliminated. Round 3 votes counts: E=37 C=33 A=30 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. C:210 D:208 E:199 B:192 A:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -14 -16 10 B -2 0 -2 -4 -8 C 14 2 0 6 -2 D 16 4 -6 0 2 E -10 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.036723 B: 0.000000 C: 0.288136 D: 0.104520 E: 0.570621 Sum of squares = 0.420903954796 Cumulative probabilities = A: 0.036723 B: 0.036723 C: 0.324859 D: 0.429379 E: 1.000000 GTS winners are ['A', 'C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -14 -16 10 B -2 0 -2 -4 -8 C 14 2 0 6 -2 D 16 4 -6 0 2 E -10 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.036723 B: 0.000000 C: 0.288136 D: 0.104520 E: 0.570621 Sum of squares = 0.420903954796 Cumulative probabilities = A: 0.036723 B: 0.036723 C: 0.324859 D: 0.429379 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -14 -16 10 B -2 0 -2 -4 -8 C 14 2 0 6 -2 D 16 4 -6 0 2 E -10 8 2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.036723 B: 0.000000 C: 0.288136 D: 0.104520 E: 0.570621 Sum of squares = 0.420903954796 Cumulative probabilities = A: 0.036723 B: 0.036723 C: 0.324859 D: 0.429379 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9801: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (8) C D B A E (8) E B A D C (6) D C E A B (6) E A B D C (4) D C B E A (4) C D E A B (4) C A B E D (4) E D A C B (3) D E C A B (3) D B E C A (3) C D A E B (3) C D A B E (3) C B D A E (3) B E A D C (3) A B E C D (3) E D A B C (2) D E B C A (2) D E A C B (2) C B A D E (2) B C A E D (2) B A E C D (2) A E B C D (2) A B C E D (2) E D B A C (1) E B D A C (1) D E C B A (1) D E B A C (1) D E A B C (1) D B E A C (1) C D E B A (1) C D B E A (1) C A E D B (1) C A B D E (1) B E D A C (1) B C A D E (1) A E C B D (1) A C E B D (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 -22 -26 -18 B 4 0 -22 -18 -8 C 22 22 0 -8 12 D 26 18 8 0 18 E 18 8 -12 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -22 -26 -18 B 4 0 -22 -18 -8 C 22 22 0 -8 12 D 26 18 8 0 18 E 18 8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 C=31 E=17 A=11 B=9 so B is eliminated. Round 2 votes counts: C=34 D=32 E=21 A=13 so A is eliminated. Round 3 votes counts: C=38 D=32 E=30 so E is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:235 C:224 E:198 B:178 A:165 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -22 -26 -18 B 4 0 -22 -18 -8 C 22 22 0 -8 12 D 26 18 8 0 18 E 18 8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -22 -26 -18 B 4 0 -22 -18 -8 C 22 22 0 -8 12 D 26 18 8 0 18 E 18 8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -22 -26 -18 B 4 0 -22 -18 -8 C 22 22 0 -8 12 D 26 18 8 0 18 E 18 8 -12 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999897 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9802: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (8) B A E D C (8) D C B A E (7) C E D A B (7) B D A C E (7) C D B A E (5) B A D E C (5) D B A C E (4) C D E A B (4) E C A D B (3) D C A B E (3) C D B E A (3) B E A D C (3) B C D A E (3) E B A C D (2) E A D C B (2) E A C B D (2) E A B C D (2) D B C A E (2) C D A B E (2) C B E D A (2) B A D C E (2) A E B D C (2) A B D E C (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A D C (1) D B A E C (1) D A E C B (1) D A B E C (1) C E B A D (1) C D E B A (1) C B D E A (1) Total count = 100 A B C D E A 0 -20 6 -8 10 B 20 0 8 8 22 C -6 -8 0 -18 8 D 8 -8 18 0 8 E -10 -22 -8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999685 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 6 -8 10 B 20 0 8 8 22 C -6 -8 0 -18 8 D 8 -8 18 0 8 E -10 -22 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=26 E=23 D=19 A=4 so A is eliminated. Round 2 votes counts: B=30 C=26 E=25 D=19 so D is eliminated. Round 3 votes counts: B=38 C=36 E=26 so E is eliminated. Round 4 votes counts: B=54 C=46 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:229 D:213 A:194 C:188 E:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 6 -8 10 B 20 0 8 8 22 C -6 -8 0 -18 8 D 8 -8 18 0 8 E -10 -22 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 6 -8 10 B 20 0 8 8 22 C -6 -8 0 -18 8 D 8 -8 18 0 8 E -10 -22 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 6 -8 10 B 20 0 8 8 22 C -6 -8 0 -18 8 D 8 -8 18 0 8 E -10 -22 -8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999537 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9803: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B E C (11) A D C B E (10) C E B D A (9) E B C D A (7) B E D A C (4) E C B D A (3) E C B A D (3) E B C A D (3) D A B E C (3) C E D B A (3) C E A B D (3) C A E D B (3) C A D E B (3) E B D A C (2) E B A D C (2) E A B D C (2) D B A E C (2) C E B A D (2) B D A E C (2) A D B C E (2) E C A B D (1) E B A C D (1) D C A B E (1) D B E C A (1) D B C E A (1) D A C B E (1) C E D A B (1) C E A D B (1) C D E B A (1) C D E A B (1) C D B E A (1) C D A E B (1) C D A B E (1) C A D B E (1) B E A D C (1) B A E D C (1) A E B D C (1) A E B C D (1) A D E B C (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 2 -2 10 -6 B -2 0 -4 -4 -14 C 2 4 0 4 -4 D -10 4 -4 0 -10 E 6 14 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 10 -6 B -2 0 -4 -4 -14 C 2 4 0 4 -4 D -10 4 -4 0 -10 E 6 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 A=28 E=24 D=9 B=8 so B is eliminated. Round 2 votes counts: C=31 E=29 A=29 D=11 so D is eliminated. Round 3 votes counts: A=37 C=33 E=30 so E is eliminated. Round 4 votes counts: C=51 A=49 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:217 C:203 A:202 D:190 B:188 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 10 -6 B -2 0 -4 -4 -14 C 2 4 0 4 -4 D -10 4 -4 0 -10 E 6 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 10 -6 B -2 0 -4 -4 -14 C 2 4 0 4 -4 D -10 4 -4 0 -10 E 6 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 10 -6 B -2 0 -4 -4 -14 C 2 4 0 4 -4 D -10 4 -4 0 -10 E 6 14 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9804: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (8) B E A D C (7) C D E A B (6) C B E D A (6) C D A E B (5) A D C E B (5) B A E D C (4) A D E C B (4) E A D B C (3) C D E B A (3) B C E D A (3) E D C A B (2) E D A C B (2) E C D B A (2) E B C D A (2) E B A D C (2) D C A E B (2) C E D B A (2) C B D A E (2) B E A C D (2) B A E C D (2) A E D B C (2) A D E B C (2) A D C B E (2) A D B E C (2) A B D E C (2) E D B A C (1) E C D A B (1) E B D C A (1) E A D C B (1) E A B D C (1) D E C A B (1) D A E C B (1) C E D A B (1) C E B D A (1) C D A B E (1) C B D E A (1) C B A D E (1) B E C D A (1) B C A E D (1) B C A D E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -8 6 -20 B 6 0 -4 -2 -6 C 8 4 0 4 -14 D -6 2 -4 0 -18 E 20 6 14 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -6 -8 6 -20 B 6 0 -4 -2 -6 C 8 4 0 4 -14 D -6 2 -4 0 -18 E 20 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B C , winner is: B compare: Computing IRV winner. Round 1 votes counts: C=29 B=29 A=20 E=18 D=4 so D is eliminated. Round 2 votes counts: C=31 B=29 A=21 E=19 so E is eliminated. Round 3 votes counts: C=37 B=35 A=28 so A is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:229 C:201 B:197 D:187 A:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -6 -8 6 -20 B 6 0 -4 -2 -6 C 8 4 0 4 -14 D -6 2 -4 0 -18 E 20 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 6 -20 B 6 0 -4 -2 -6 C 8 4 0 4 -14 D -6 2 -4 0 -18 E 20 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 6 -20 B 6 0 -4 -2 -6 C 8 4 0 4 -14 D -6 2 -4 0 -18 E 20 6 14 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9805: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (8) B E C A D (8) C D A B E (7) D A E C B (6) B E A C D (6) A E D B C (6) C D B A E (5) C B D A E (5) E A B D C (4) B E A D C (4) E B A D C (3) D A E B C (3) D A C E B (3) C B D E A (3) B C E A D (3) E A D B C (2) D E A B C (2) D C B E A (2) B E C D A (2) A D E B C (2) A D C E B (2) E D A B C (1) E B A C D (1) D E B A C (1) D C E A B (1) D B C E A (1) C D A E B (1) C B E A D (1) C B A E D (1) C A D B E (1) C A B D E (1) B E D A C (1) B C E D A (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 4 -2 -6 6 B -4 0 6 -8 4 C 2 -6 0 -6 -8 D 6 8 6 0 8 E -6 -4 8 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 -6 6 B -4 0 6 -8 4 C 2 -6 0 -6 -8 D 6 8 6 0 8 E -6 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 C=25 B=25 A=12 E=11 so E is eliminated. Round 2 votes counts: B=29 D=28 C=25 A=18 so A is eliminated. Round 3 votes counts: D=40 B=35 C=25 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:214 A:201 B:199 E:195 C:191 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -2 -6 6 B -4 0 6 -8 4 C 2 -6 0 -6 -8 D 6 8 6 0 8 E -6 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 -6 6 B -4 0 6 -8 4 C 2 -6 0 -6 -8 D 6 8 6 0 8 E -6 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 -6 6 B -4 0 6 -8 4 C 2 -6 0 -6 -8 D 6 8 6 0 8 E -6 -4 8 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9806: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B D A (8) D A C E B (6) A D B E C (6) D A C B E (5) A D E C B (5) D C A E B (4) C E D B A (4) B E C D A (4) E B C A D (3) C E B D A (3) B C D A E (3) A E D B C (3) A D E B C (3) A B D E C (3) E C D B A (2) E C D A B (2) D C E A B (2) D C A B E (2) C D E B A (2) C D E A B (2) C D B A E (2) C B D E A (2) B E A C D (2) B C E D A (2) B C E A D (2) A D C E B (2) A D B C E (2) A B E D C (2) A B D C E (2) E C B A D (1) E B A C D (1) E A C B D (1) D C B A E (1) D A B C E (1) C D B E A (1) B E C A D (1) B D C A E (1) B C A E D (1) A E B D C (1) Total count = 100 A B C D E A 0 8 -10 -18 10 B -8 0 -14 -14 -10 C 10 14 0 -2 4 D 18 14 2 0 14 E -10 10 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999843 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -10 -18 10 B -8 0 -14 -14 -10 C 10 14 0 -2 4 D 18 14 2 0 14 E -10 10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 D=21 E=18 C=16 B=16 so C is eliminated. Round 2 votes counts: A=29 D=28 E=25 B=18 so B is eliminated. Round 3 votes counts: E=36 D=34 A=30 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:224 C:213 A:195 E:191 B:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -10 -18 10 B -8 0 -14 -14 -10 C 10 14 0 -2 4 D 18 14 2 0 14 E -10 10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 -18 10 B -8 0 -14 -14 -10 C 10 14 0 -2 4 D 18 14 2 0 14 E -10 10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 -18 10 B -8 0 -14 -14 -10 C 10 14 0 -2 4 D 18 14 2 0 14 E -10 10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999984961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9807: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) E C A B D (8) C D B E A (7) A B D E C (7) D B A C E (6) D C B E A (5) D C B A E (5) B D A E C (5) E A C B D (4) C E A D B (4) A B E D C (4) D B C E A (3) A E C B D (3) A E B C D (3) A D B C E (3) D A B C E (2) C D E B A (2) C A E D B (2) B D E C A (2) A E B D C (2) E C B D A (1) E C B A D (1) E B D C A (1) E B C D A (1) C E D A B (1) C E A B D (1) C D E A B (1) C D A E B (1) C D A B E (1) B E D A C (1) B E A D C (1) B A D E C (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -14 -10 -8 B 4 0 -16 -8 6 C 14 16 0 4 10 D 10 8 -4 0 4 E 8 -6 -10 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -14 -10 -8 B 4 0 -16 -8 6 C 14 16 0 4 10 D 10 8 -4 0 4 E 8 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998034 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 A=23 D=21 E=16 B=10 so B is eliminated. Round 2 votes counts: C=30 D=28 A=24 E=18 so E is eliminated. Round 3 votes counts: C=41 D=30 A=29 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:222 D:209 E:194 B:193 A:182 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -14 -10 -8 B 4 0 -16 -8 6 C 14 16 0 4 10 D 10 8 -4 0 4 E 8 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998034 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -14 -10 -8 B 4 0 -16 -8 6 C 14 16 0 4 10 D 10 8 -4 0 4 E 8 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998034 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -14 -10 -8 B 4 0 -16 -8 6 C 14 16 0 4 10 D 10 8 -4 0 4 E 8 -6 -10 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998034 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9808: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (9) D E B A C (8) C A E B D (8) A B D E C (7) E B D C A (6) C E D B A (6) C A D E B (5) C A B E D (5) A C B D E (5) C D E B A (4) E C B D A (3) E B C D A (3) C E B D A (3) A D B E C (3) A C D B E (3) A C B E D (3) E D B C A (2) E B D A C (2) D C E B A (2) C A D B E (2) A D B C E (2) E C B A D (1) D B A E C (1) C E B A D (1) C D A E B (1) C A B D E (1) B E D A C (1) B D E A C (1) A B E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 -6 -6 -4 -4 B 6 0 -6 4 -10 C 6 6 0 10 2 D 4 -4 -10 0 8 E 4 10 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999921 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -4 -4 B 6 0 -6 4 -10 C 6 6 0 10 2 D 4 -4 -10 0 8 E 4 10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 A=25 D=20 E=17 B=2 so B is eliminated. Round 2 votes counts: C=36 A=25 D=21 E=18 so E is eliminated. Round 3 votes counts: C=43 D=32 A=25 so A is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:202 D:199 B:197 A:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -4 -4 B 6 0 -6 4 -10 C 6 6 0 10 2 D 4 -4 -10 0 8 E 4 10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -4 -4 B 6 0 -6 4 -10 C 6 6 0 10 2 D 4 -4 -10 0 8 E 4 10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -4 -4 B 6 0 -6 4 -10 C 6 6 0 10 2 D 4 -4 -10 0 8 E 4 10 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9809: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (7) E C A D B (6) B D C E A (6) D B A E C (5) D A B E C (5) B D C A E (5) B D A C E (5) E C A B D (4) D B A C E (4) E C D A B (3) D E B A C (3) A E D C B (3) E D B A C (2) E D A B C (2) E C D B A (2) E C B D A (2) E A D C B (2) E A C D B (2) D A E B C (2) C E B A D (2) C B A D E (2) C A E B D (2) C A B E D (2) B A D C E (2) B A C D E (2) A D B E C (2) A D B C E (2) E D A C B (1) E C B A D (1) D B E A C (1) C E B D A (1) C B E A D (1) C B D A E (1) C B A E D (1) B C D E A (1) B C D A E (1) B C A D E (1) A C E B D (1) A C B D E (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 -2 -4 2 B 2 0 6 6 4 C 2 -6 0 -6 4 D 4 -6 6 0 6 E -2 -4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -4 2 B 2 0 6 6 4 C 2 -6 0 -6 4 D 4 -6 6 0 6 E -2 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990019 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 D=20 C=19 A=11 so A is eliminated. Round 2 votes counts: E=30 B=25 D=24 C=21 so C is eliminated. Round 3 votes counts: E=43 B=33 D=24 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:209 D:205 A:197 C:197 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 -4 2 B 2 0 6 6 4 C 2 -6 0 -6 4 D 4 -6 6 0 6 E -2 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990019 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -4 2 B 2 0 6 6 4 C 2 -6 0 -6 4 D 4 -6 6 0 6 E -2 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990019 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -4 2 B 2 0 6 6 4 C 2 -6 0 -6 4 D 4 -6 6 0 6 E -2 -4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990019 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9810: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D A C (10) E B D A C (8) C E A D B (8) A D C B E (8) C A D E B (7) E B C D A (6) D A B C E (5) E C B A D (4) D A C B E (4) B D A E C (4) E B D C A (3) E B C A D (3) D A B E C (3) C A D B E (3) B E C A D (3) E C B D A (2) E C A D B (2) D A C E B (2) C E B A D (2) A D B C E (2) E D C A B (1) E D B A C (1) E D A B C (1) D C A E B (1) D B A E C (1) C E D A B (1) C E A B D (1) C A E D B (1) C A B E D (1) B C A D E (1) B A D C E (1) Total count = 100 A B C D E A 0 2 0 -6 -12 B -2 0 4 -2 -8 C 0 -4 0 -10 -4 D 6 2 10 0 -16 E 12 8 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 0 -6 -12 B -2 0 4 -2 -8 C 0 -4 0 -10 -4 D 6 2 10 0 -16 E 12 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 C=24 B=19 D=16 A=10 so A is eliminated. Round 2 votes counts: E=31 D=26 C=24 B=19 so B is eliminated. Round 3 votes counts: E=44 D=31 C=25 so C is eliminated. Round 4 votes counts: E=58 D=42 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:220 D:201 B:196 A:192 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 0 -6 -12 B -2 0 4 -2 -8 C 0 -4 0 -10 -4 D 6 2 10 0 -16 E 12 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -6 -12 B -2 0 4 -2 -8 C 0 -4 0 -10 -4 D 6 2 10 0 -16 E 12 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -6 -12 B -2 0 4 -2 -8 C 0 -4 0 -10 -4 D 6 2 10 0 -16 E 12 8 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999322 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9811: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (20) E B A D C (15) D C A B E (8) E C D B A (6) A B D C E (6) C D E A B (5) B A E D C (5) E B C D A (3) E B A C D (3) C E D A B (3) C D A E B (3) E C D A B (2) D C B A E (2) D B C A E (2) A E B D C (2) A D B C E (2) A B E D C (2) E C B D A (1) E B D C A (1) E B C A D (1) E A B D C (1) E A B C D (1) D A C B E (1) B D E C A (1) B A D C E (1) A D C B E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 18 -16 -16 14 B -18 0 -6 -12 6 C 16 6 0 -2 10 D 16 12 2 0 8 E -14 -6 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999948 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -16 -16 14 B -18 0 -6 -12 6 C 16 6 0 -2 10 D 16 12 2 0 8 E -14 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=31 A=15 D=13 B=7 so B is eliminated. Round 2 votes counts: E=34 C=31 A=21 D=14 so D is eliminated. Round 3 votes counts: C=43 E=35 A=22 so A is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. D:219 C:215 A:200 B:185 E:181 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 18 -16 -16 14 B -18 0 -6 -12 6 C 16 6 0 -2 10 D 16 12 2 0 8 E -14 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -16 -16 14 B -18 0 -6 -12 6 C 16 6 0 -2 10 D 16 12 2 0 8 E -14 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -16 -16 14 B -18 0 -6 -12 6 C 16 6 0 -2 10 D 16 12 2 0 8 E -14 -6 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9812: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (15) C D B A E (13) D E A B C (10) D C B A E (9) E A B D C (7) C B A D E (7) C E A B D (6) D B A E C (4) D C E A B (3) C B D A E (3) E D A B C (2) D E B A C (2) D B E A C (2) C D E A B (2) C A E B D (2) E D C A B (1) E C A B D (1) E A D B C (1) E A B C D (1) D E C A B (1) D E A C B (1) D B C A E (1) B A E D C (1) B A E C D (1) B A D E C (1) B A C E D (1) A E C B D (1) A E B C D (1) Total count = 100 A B C D E A 0 -20 -28 -8 20 B 20 0 -30 -4 16 C 28 30 0 8 24 D 8 4 -8 0 18 E -20 -16 -24 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -28 -8 20 B 20 0 -30 -4 16 C 28 30 0 8 24 D 8 4 -8 0 18 E -20 -16 -24 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=48 D=33 E=13 B=4 A=2 so A is eliminated. Round 2 votes counts: C=48 D=33 E=15 B=4 so B is eliminated. Round 3 votes counts: C=49 D=34 E=17 so E is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:245 D:211 B:201 A:182 E:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -28 -8 20 B 20 0 -30 -4 16 C 28 30 0 8 24 D 8 4 -8 0 18 E -20 -16 -24 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -28 -8 20 B 20 0 -30 -4 16 C 28 30 0 8 24 D 8 4 -8 0 18 E -20 -16 -24 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -28 -8 20 B 20 0 -30 -4 16 C 28 30 0 8 24 D 8 4 -8 0 18 E -20 -16 -24 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998942 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9813: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D C E (16) E C D B A (11) E B C D A (9) B A D C E (9) D C A E B (6) C E D A B (6) C D E A B (6) E C B D A (5) E C D A B (4) B A E C D (4) A D C B E (4) D C E A B (3) D A C E B (3) B A E D C (3) A D B C E (3) B E C A D (2) B A D E C (2) D A C B E (1) C D A E B (1) B E A C D (1) B A C D E (1) Total count = 100 A B C D E A 0 6 -6 -10 6 B -6 0 0 4 -8 C 6 0 0 0 18 D 10 -4 0 0 10 E -6 8 -18 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.345614 C: 0.654386 D: 0.000000 E: 0.000000 Sum of squares = 0.547669986256 Cumulative probabilities = A: 0.000000 B: 0.345614 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -10 6 B -6 0 0 4 -8 C 6 0 0 0 18 D 10 -4 0 0 10 E -6 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499223 C: 0.500777 D: 0.000000 E: 0.000000 Sum of squares = 0.500001205937 Cumulative probabilities = A: 0.000000 B: 0.499223 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=23 B=22 D=13 C=13 so D is eliminated. Round 2 votes counts: E=29 A=27 C=22 B=22 so C is eliminated. Round 3 votes counts: E=44 A=34 B=22 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:212 D:208 A:198 B:195 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -10 6 B -6 0 0 4 -8 C 6 0 0 0 18 D 10 -4 0 0 10 E -6 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499223 C: 0.500777 D: 0.000000 E: 0.000000 Sum of squares = 0.500001205937 Cumulative probabilities = A: 0.000000 B: 0.499223 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -10 6 B -6 0 0 4 -8 C 6 0 0 0 18 D 10 -4 0 0 10 E -6 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499223 C: 0.500777 D: 0.000000 E: 0.000000 Sum of squares = 0.500001205937 Cumulative probabilities = A: 0.000000 B: 0.499223 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -10 6 B -6 0 0 4 -8 C 6 0 0 0 18 D 10 -4 0 0 10 E -6 8 -18 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499223 C: 0.500777 D: 0.000000 E: 0.000000 Sum of squares = 0.500001205937 Cumulative probabilities = A: 0.000000 B: 0.499223 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9814: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (9) D C E B A (6) C D E A B (6) C B A D E (5) D C E A B (4) C B D A E (4) E D A C B (3) E A D B C (3) D E C B A (3) C E D A B (3) C D B E A (3) B C D A E (3) A E B C D (3) A B E D C (3) E D A B C (2) E A D C B (2) E A C D B (2) D E B C A (2) D C B E A (2) C D E B A (2) B D E A C (2) B D C A E (2) B C A D E (2) A B E C D (2) E C D A B (1) E C A D B (1) E A C B D (1) E A B C D (1) D E C A B (1) D E B A C (1) D B E C A (1) C D A E B (1) C A E B D (1) B A D E C (1) B A C E D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -24 -22 -22 B -2 0 -22 -16 -16 C 24 22 0 6 12 D 22 16 -6 0 20 E 22 16 -12 -20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -24 -22 -22 B -2 0 -22 -16 -16 C 24 22 0 6 12 D 22 16 -6 0 20 E 22 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 E=25 D=20 B=11 A=10 so A is eliminated. Round 2 votes counts: C=35 E=28 D=20 B=17 so B is eliminated. Round 3 votes counts: C=42 E=33 D=25 so D is eliminated. Round 4 votes counts: C=56 E=44 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:232 D:226 E:203 B:172 A:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -24 -22 -22 B -2 0 -22 -16 -16 C 24 22 0 6 12 D 22 16 -6 0 20 E 22 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -24 -22 -22 B -2 0 -22 -16 -16 C 24 22 0 6 12 D 22 16 -6 0 20 E 22 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -24 -22 -22 B -2 0 -22 -16 -16 C 24 22 0 6 12 D 22 16 -6 0 20 E 22 16 -12 -20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9815: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C A D B (10) E A C D B (10) E A C B D (6) B D A C E (6) A C E B D (6) E D B C A (4) D B E C A (4) D B C E A (4) C A E D B (4) B D C A E (3) A B C D E (3) E D C B A (2) E C D A B (2) C A D B E (2) C A B D E (2) B E D A C (2) B D A E C (2) B A E D C (2) A E C B D (2) A C E D B (2) A C B E D (2) E B D A C (1) E A D C B (1) E A B D C (1) D E C B A (1) D E B C A (1) D C E B A (1) D C B A E (1) D B C A E (1) C E A D B (1) C D B A E (1) C D A B E (1) C B D A E (1) B E A D C (1) B D E A C (1) B C D A E (1) B A D C E (1) B A C D E (1) A E C D B (1) A C B D E (1) A B C E D (1) Total count = 100 A B C D E A 0 16 6 20 -6 B -16 0 -20 -8 -12 C -6 20 0 20 -8 D -20 8 -20 0 -22 E 6 12 8 22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 16 6 20 -6 B -16 0 -20 -8 -12 C -6 20 0 20 -8 D -20 8 -20 0 -22 E 6 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=37 B=20 A=18 D=13 C=12 so C is eliminated. Round 2 votes counts: E=38 A=26 B=21 D=15 so D is eliminated. Round 3 votes counts: E=41 B=32 A=27 so A is eliminated. Round 4 votes counts: E=56 B=44 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:224 A:218 C:213 D:173 B:172 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 16 6 20 -6 B -16 0 -20 -8 -12 C -6 20 0 20 -8 D -20 8 -20 0 -22 E 6 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 20 -6 B -16 0 -20 -8 -12 C -6 20 0 20 -8 D -20 8 -20 0 -22 E 6 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 20 -6 B -16 0 -20 -8 -12 C -6 20 0 20 -8 D -20 8 -20 0 -22 E 6 12 8 22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999767 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9816: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C A E (8) B D E A C (8) C A E D B (7) D C B A E (6) B E A D C (6) A C E D B (6) E B A C D (5) B D E C A (5) A E C D B (5) E A C B D (4) B E C A D (4) D B A C E (3) C E A B D (3) C D A E B (3) B E D A C (3) E A B C D (2) D C A B E (2) D B A E C (2) B E A C D (2) B D C E A (2) E C B A D (1) E C A B D (1) E B C A D (1) E B A D C (1) D C A E B (1) D B C E A (1) C E B A D (1) C A D E B (1) B E D C A (1) B E C D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -24 0 6 -6 B 24 0 12 4 10 C 0 -12 0 -2 -8 D -6 -4 2 0 -10 E 6 -10 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999761 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 0 6 -6 B 24 0 12 4 10 C 0 -12 0 -2 -8 D -6 -4 2 0 -10 E 6 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 D=23 E=15 C=15 A=14 so A is eliminated. Round 2 votes counts: B=33 D=24 C=22 E=21 so E is eliminated. Round 3 votes counts: B=42 C=33 D=25 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:225 E:207 D:191 C:189 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -24 0 6 -6 B 24 0 12 4 10 C 0 -12 0 -2 -8 D -6 -4 2 0 -10 E 6 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 0 6 -6 B 24 0 12 4 10 C 0 -12 0 -2 -8 D -6 -4 2 0 -10 E 6 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 0 6 -6 B 24 0 12 4 10 C 0 -12 0 -2 -8 D -6 -4 2 0 -10 E 6 -10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997818 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9817: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (13) E C B A D (7) D A B C E (5) C B E A D (5) A D B E C (5) D A B E C (4) B D A C E (4) A D E B C (4) D B A E C (3) D A E B C (3) E C A B D (2) E A D C B (2) E A B D C (2) D C B A E (2) C E D B A (2) C E D A B (2) C E A D B (2) C D B E A (2) C B E D A (2) C B D E A (2) C B D A E (2) B E A C D (2) B D C A E (2) B C D A E (2) A E D B C (2) E C A D B (1) E B A C D (1) E A C D B (1) E A B C D (1) D C A B E (1) D B A C E (1) D A E C B (1) C E B D A (1) C D E A B (1) C D B A E (1) B E C A D (1) B C E A D (1) B C D E A (1) B A D E C (1) A E D C B (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -16 -10 12 -8 B 16 0 -8 6 0 C 10 8 0 10 8 D -12 -6 -10 0 -4 E 8 0 -8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 -10 12 -8 B 16 0 -8 6 0 C 10 8 0 10 8 D -12 -6 -10 0 -4 E 8 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 D=20 E=17 B=14 A=14 so B is eliminated. Round 2 votes counts: C=39 D=26 E=20 A=15 so A is eliminated. Round 3 votes counts: C=39 D=37 E=24 so E is eliminated. Round 4 votes counts: C=55 D=45 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:218 B:207 E:202 A:189 D:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -16 -10 12 -8 B 16 0 -8 6 0 C 10 8 0 10 8 D -12 -6 -10 0 -4 E 8 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 -10 12 -8 B 16 0 -8 6 0 C 10 8 0 10 8 D -12 -6 -10 0 -4 E 8 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 -10 12 -8 B 16 0 -8 6 0 C 10 8 0 10 8 D -12 -6 -10 0 -4 E 8 0 -8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9818: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (7) C D A B E (7) A E B D C (7) C A B E D (6) E B A D C (5) D C E B A (5) C D B E A (5) D E B C A (4) A B E C D (4) D E A C B (3) D E A B C (3) D B E C A (3) C D B A E (3) C D A E B (3) C B A E D (3) C A E B D (3) E D B A C (2) C A E D B (2) B E D A C (2) B E A D C (2) B C A E D (2) A C E B D (2) E B D A C (1) D E C B A (1) D C B E A (1) D B E A C (1) C D E B A (1) C D E A B (1) C B D E A (1) C A D E B (1) C A B D E (1) B D E A C (1) B C E A D (1) B A E D C (1) B A E C D (1) A E D B C (1) A E B C D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -6 -8 -10 0 B 6 0 -2 -10 -8 C 8 2 0 0 0 D 10 10 0 0 6 E 0 8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.728547 D: 0.271453 E: 0.000000 Sum of squares = 0.604467884262 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.728547 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -8 -10 0 B 6 0 -2 -10 -8 C 8 2 0 0 0 D 10 10 0 0 6 E 0 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=28 A=17 B=10 E=8 so E is eliminated. Round 2 votes counts: C=37 D=30 A=17 B=16 so B is eliminated. Round 3 votes counts: C=40 D=34 A=26 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:213 C:205 E:201 B:193 A:188 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -8 -10 0 B 6 0 -2 -10 -8 C 8 2 0 0 0 D 10 10 0 0 6 E 0 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -8 -10 0 B 6 0 -2 -10 -8 C 8 2 0 0 0 D 10 10 0 0 6 E 0 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -8 -10 0 B 6 0 -2 -10 -8 C 8 2 0 0 0 D 10 10 0 0 6 E 0 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999928 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9819: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A D E C (7) B E C D A (6) E C D B A (5) C A B E D (5) A C B E D (5) E D C B A (4) E B C D A (4) B C E A D (4) D E B A C (3) C E B D A (3) B C A E D (3) A C B D E (3) D E B C A (2) D B E A C (2) D A B E C (2) C E D A B (2) C D E A B (2) C A E B D (2) B E D C A (2) B E D A C (2) B E C A D (2) B D A E C (2) B A E C D (2) A D C E B (2) A D B C E (2) A C D E B (2) A B D C E (2) E D B C A (1) E B D C A (1) D E C A B (1) D E A C B (1) D E A B C (1) D C E A B (1) D C A E B (1) D B A E C (1) D A E C B (1) D A E B C (1) C E D B A (1) C B E A D (1) C A D E B (1) B D E A C (1) B A E D C (1) A D E C B (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -20 -8 -6 -4 B 20 0 12 18 14 C 8 -12 0 6 -16 D 6 -18 -6 0 -12 E 4 -14 16 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -8 -6 -4 B 20 0 12 18 14 C 8 -12 0 6 -16 D 6 -18 -6 0 -12 E 4 -14 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 A=19 D=17 C=17 E=15 so E is eliminated. Round 2 votes counts: B=37 D=22 C=22 A=19 so A is eliminated. Round 3 votes counts: B=40 C=32 D=28 so D is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:232 E:209 C:193 D:185 A:181 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -20 -8 -6 -4 B 20 0 12 18 14 C 8 -12 0 6 -16 D 6 -18 -6 0 -12 E 4 -14 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -8 -6 -4 B 20 0 12 18 14 C 8 -12 0 6 -16 D 6 -18 -6 0 -12 E 4 -14 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -8 -6 -4 B 20 0 12 18 14 C 8 -12 0 6 -16 D 6 -18 -6 0 -12 E 4 -14 16 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9820: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (7) E B C A D (5) C A D E B (5) B E D A C (5) B E A D C (5) E A C B D (4) B D E A C (4) D C B A E (3) D C A B E (3) C D A E B (3) B D C E A (3) A E D B C (3) A E B D C (3) D B C E A (2) D B C A E (2) D A C B E (2) C E A D B (2) C B E D A (2) C A E D B (2) B D E C A (2) B D A E C (2) A E B C D (2) A C E D B (2) A C D E B (2) E B A D C (1) D C B E A (1) D B A C E (1) D A C E B (1) D A B E C (1) D A B C E (1) C E D A B (1) C E B D A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A B E (1) B E D C A (1) B E C D A (1) B E C A D (1) B E A C D (1) B C E D A (1) B C D E A (1) B A D E C (1) A E C D B (1) A E C B D (1) A D E C B (1) A D C E B (1) Total count = 100 A B C D E A 0 2 4 2 -10 B -2 0 12 10 -2 C -4 -12 0 2 -4 D -2 -10 -2 0 -6 E 10 2 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999927 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 4 2 -10 B -2 0 12 10 -2 C -4 -12 0 2 -4 D -2 -10 -2 0 -6 E 10 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 C=22 E=17 D=17 A=16 so A is eliminated. Round 2 votes counts: B=28 E=27 C=26 D=19 so D is eliminated. Round 3 votes counts: C=37 B=35 E=28 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:211 B:209 A:199 C:191 D:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 4 2 -10 B -2 0 12 10 -2 C -4 -12 0 2 -4 D -2 -10 -2 0 -6 E 10 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 2 -10 B -2 0 12 10 -2 C -4 -12 0 2 -4 D -2 -10 -2 0 -6 E 10 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 2 -10 B -2 0 12 10 -2 C -4 -12 0 2 -4 D -2 -10 -2 0 -6 E 10 2 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999949 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9821: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (10) B D E C A (6) E C A D B (5) C A E D B (5) E D C B A (4) B C E A D (4) D B A C E (3) D A B C E (3) C E A B D (3) B E D C A (3) B D E A C (3) B A D C E (3) A D C B E (3) A C E D B (3) E C D B A (2) E C D A B (2) E C B A D (2) E C A B D (2) D E B C A (2) D B A E C (2) D A B E C (2) C E A D B (2) B E C A D (2) B D A E C (2) A D C E B (2) A C B D E (2) A B C D E (2) E D C A B (1) E B D C A (1) E B C D A (1) D E C B A (1) D B E C A (1) D B E A C (1) D A E C B (1) C B E A D (1) C B A E D (1) C A B E D (1) B C A E D (1) B A C D E (1) A C E B D (1) A C D E B (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 -6 -2 2 B 14 0 8 8 18 C 6 -8 0 -10 8 D 2 -8 10 0 4 E -2 -18 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -6 -2 2 B 14 0 8 8 18 C 6 -8 0 -10 8 D 2 -8 10 0 4 E -2 -18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 E=20 D=16 A=16 C=13 so C is eliminated. Round 2 votes counts: B=37 E=25 A=22 D=16 so D is eliminated. Round 3 votes counts: B=44 E=28 A=28 so E is eliminated. Round 4 votes counts: B=57 A=43 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:224 D:204 C:198 A:190 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -6 -2 2 B 14 0 8 8 18 C 6 -8 0 -10 8 D 2 -8 10 0 4 E -2 -18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -6 -2 2 B 14 0 8 8 18 C 6 -8 0 -10 8 D 2 -8 10 0 4 E -2 -18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -6 -2 2 B 14 0 8 8 18 C 6 -8 0 -10 8 D 2 -8 10 0 4 E -2 -18 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999976 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9822: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (10) A E C D B (9) B D C E A (8) C A B D E (7) C B D E A (6) C B D A E (5) C A E B D (5) C A B E D (5) B C D E A (5) A E D C B (5) E D A B C (4) B D E C A (4) A E D B C (4) E C A B D (2) E A C D B (2) D E B A C (2) D B E A C (2) C B A D E (2) B D C A E (2) E D B A C (1) E C B D A (1) E B C D A (1) D B E C A (1) D A E B C (1) C E A B D (1) C B E D A (1) A C E D B (1) A C E B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 18 -12 12 -2 B -18 0 -10 14 0 C 12 10 0 12 2 D -12 -14 -12 0 -6 E 2 0 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 -12 12 -2 B -18 0 -10 14 0 C 12 10 0 12 2 D -12 -14 -12 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 A=22 E=21 B=19 D=6 so D is eliminated. Round 2 votes counts: C=32 E=23 A=23 B=22 so B is eliminated. Round 3 votes counts: C=47 E=30 A=23 so A is eliminated. Round 4 votes counts: C=51 E=49 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:218 A:208 E:203 B:193 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 18 -12 12 -2 B -18 0 -10 14 0 C 12 10 0 12 2 D -12 -14 -12 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -12 12 -2 B -18 0 -10 14 0 C 12 10 0 12 2 D -12 -14 -12 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -12 12 -2 B -18 0 -10 14 0 C 12 10 0 12 2 D -12 -14 -12 0 -6 E 2 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992718 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9823: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) D B E A C (7) A C D E B (6) A C D B E (6) B E C D A (5) E D B A C (4) E B C D A (4) B C E A D (4) D E B A C (3) D A E C B (3) D A B E C (3) D A B C E (3) C E A B D (3) C A B D E (3) D E A B C (2) C A E D B (2) C A E B D (2) C A D E B (2) C A B E D (2) B E C A D (2) B D E C A (2) B D E A C (2) A D C B E (2) E C B A D (1) E C A D B (1) E C A B D (1) E B D C A (1) E B D A C (1) E B C A D (1) D B A C E (1) D A C E B (1) D A C B E (1) C E B A D (1) C E A D B (1) C B E A D (1) C B A D E (1) B E D A C (1) B D A C E (1) A C E D B (1) A C B D E (1) Total count = 100 A B C D E A 0 -8 -2 -12 -18 B 8 0 16 2 18 C 2 -16 0 2 -10 D 12 -2 -2 0 0 E 18 -18 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -2 -12 -18 B 8 0 16 2 18 C 2 -16 0 2 -10 D 12 -2 -2 0 0 E 18 -18 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 D=24 C=18 A=16 E=14 so E is eliminated. Round 2 votes counts: B=35 D=28 C=21 A=16 so A is eliminated. Round 3 votes counts: C=35 B=35 D=30 so D is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 E:205 D:204 C:189 A:180 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -12 -18 B 8 0 16 2 18 C 2 -16 0 2 -10 D 12 -2 -2 0 0 E 18 -18 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -12 -18 B 8 0 16 2 18 C 2 -16 0 2 -10 D 12 -2 -2 0 0 E 18 -18 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -12 -18 B 8 0 16 2 18 C 2 -16 0 2 -10 D 12 -2 -2 0 0 E 18 -18 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998796 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9824: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (7) B C E D A (6) A D E C B (6) E C D A B (5) B E C A D (5) B E A C D (5) B A E D C (5) C D E A B (4) C D B A E (4) C B D E A (4) B A D C E (4) E D A C B (3) B C D A E (3) A B E D C (3) E C A D B (2) E A D B C (2) D C A E B (2) D A C B E (2) C B E D A (2) B C E A D (2) B C D E A (2) B A E C D (2) B A C D E (2) A E D C B (2) A E D B C (2) E D C A B (1) E B C A D (1) E B A C D (1) E A D C B (1) D E A C B (1) D C A B E (1) D A E C B (1) C D B E A (1) B E C D A (1) B C A E D (1) A E B D C (1) A D E B C (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -2 6 0 2 B 2 0 2 2 14 C -6 -2 0 6 -6 D 0 -2 -6 0 -6 E -2 -14 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999947 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 0 2 B 2 0 2 2 14 C -6 -2 0 6 -6 D 0 -2 -6 0 -6 E -2 -14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=38 A=17 E=16 C=15 D=14 so D is eliminated. Round 2 votes counts: B=38 A=27 C=18 E=17 so E is eliminated. Round 3 votes counts: B=40 A=34 C=26 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:210 A:203 E:198 C:196 D:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 6 0 2 B 2 0 2 2 14 C -6 -2 0 6 -6 D 0 -2 -6 0 -6 E -2 -14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 0 2 B 2 0 2 2 14 C -6 -2 0 6 -6 D 0 -2 -6 0 -6 E -2 -14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 0 2 B 2 0 2 2 14 C -6 -2 0 6 -6 D 0 -2 -6 0 -6 E -2 -14 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999519 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9825: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C B A (11) A E D C B (9) B C D E A (8) D B C E A (5) A E C B D (5) A B C E D (5) A D E C B (4) A D E B C (4) E D C A B (3) E D A C B (3) E C D B A (3) C B E D A (3) A E C D B (3) C E D B A (2) C D B E A (2) C B E A D (2) B C A E D (2) B A C E D (2) B A C D E (2) A D B E C (2) E C A B D (1) E A C B D (1) D E C A B (1) D E B C A (1) D E A C B (1) D C E B A (1) D C B E A (1) D A E C B (1) C E B D A (1) C B D E A (1) B D C E A (1) B D C A E (1) B C E D A (1) B C E A D (1) B C A D E (1) A E D B C (1) A C B E D (1) A B E C D (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -6 -2 -8 B 4 0 -20 -16 -12 C 6 20 0 -2 -12 D 2 16 2 0 0 E 8 12 12 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.556735 E: 0.443265 Sum of squares = 0.506437767416 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.556735 E: 1.000000 A B C D E A 0 -4 -6 -2 -8 B 4 0 -20 -16 -12 C 6 20 0 -2 -12 D 2 16 2 0 0 E 8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 D=22 B=19 E=11 C=11 so E is eliminated. Round 2 votes counts: A=38 D=28 B=19 C=15 so C is eliminated. Round 3 votes counts: A=39 D=35 B=26 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:216 D:210 C:206 A:190 B:178 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -6 -2 -8 B 4 0 -20 -16 -12 C 6 20 0 -2 -12 D 2 16 2 0 0 E 8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -6 -2 -8 B 4 0 -20 -16 -12 C 6 20 0 -2 -12 D 2 16 2 0 0 E 8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -6 -2 -8 B 4 0 -20 -16 -12 C 6 20 0 -2 -12 D 2 16 2 0 0 E 8 12 12 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9826: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (13) C B A E D (8) D E C A B (6) E D B A C (5) E B A C D (5) B A C E D (5) E B A D C (4) D C A B E (4) D E A C B (3) D A C B E (3) C A D B E (3) B A E C D (3) E B C A D (2) D E A B C (2) D C A E B (2) D A C E B (2) C B A D E (2) C A B E D (2) B E A C D (2) B C E A D (2) B C A E D (2) A C B D E (2) A B C D E (2) E D C B A (1) E D B C A (1) E D A B C (1) E B D C A (1) E B C D A (1) D E C B A (1) D C E A B (1) D A E B C (1) C D A B E (1) C B E A D (1) C B D A E (1) B E C A D (1) A D C B E (1) A B E D C (1) A B D E C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 -12 26 20 B -4 0 -14 24 22 C 12 14 0 18 16 D -26 -24 -18 0 2 E -20 -22 -16 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999907 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -12 26 20 B -4 0 -14 24 22 C 12 14 0 18 16 D -26 -24 -18 0 2 E -20 -22 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 D=25 E=21 B=15 A=8 so A is eliminated. Round 2 votes counts: C=33 D=26 E=21 B=20 so B is eliminated. Round 3 votes counts: C=45 E=28 D=27 so D is eliminated. Round 4 votes counts: C=58 E=42 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:230 A:219 B:214 E:170 D:167 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -12 26 20 B -4 0 -14 24 22 C 12 14 0 18 16 D -26 -24 -18 0 2 E -20 -22 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -12 26 20 B -4 0 -14 24 22 C 12 14 0 18 16 D -26 -24 -18 0 2 E -20 -22 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -12 26 20 B -4 0 -14 24 22 C 12 14 0 18 16 D -26 -24 -18 0 2 E -20 -22 -16 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9827: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C E D (8) D E C A B (7) D E B A C (7) E C D A B (6) B A D C E (6) E D B C A (5) E C B A D (4) D C E A B (4) D A B C E (4) E D C B A (3) D B A E C (3) C E D A B (3) A C B D E (3) A B C E D (3) A B C D E (3) E C D B A (2) D B E A C (2) D A C B E (2) C E A D B (2) C E A B D (2) B D A E C (2) B A E C D (2) B A D E C (2) A C D B E (2) E C A B D (1) E B C A D (1) D E C B A (1) D E B C A (1) D C A E B (1) D B A C E (1) C D A E B (1) C A E D B (1) C A E B D (1) C A B E D (1) B E D A C (1) B A E D C (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 6 -12 -4 B 4 0 6 -18 -6 C -6 -6 0 -8 -2 D 12 18 8 0 6 E 4 6 2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -12 -4 B 4 0 6 -18 -6 C -6 -6 0 -8 -2 D 12 18 8 0 6 E 4 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 E=22 B=22 A=12 C=11 so C is eliminated. Round 2 votes counts: D=34 E=29 B=22 A=15 so A is eliminated. Round 3 votes counts: D=37 B=32 E=31 so E is eliminated. Round 4 votes counts: D=59 B=41 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 E:203 A:193 B:193 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -12 -4 B 4 0 6 -18 -6 C -6 -6 0 -8 -2 D 12 18 8 0 6 E 4 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -12 -4 B 4 0 6 -18 -6 C -6 -6 0 -8 -2 D 12 18 8 0 6 E 4 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -12 -4 B 4 0 6 -18 -6 C -6 -6 0 -8 -2 D 12 18 8 0 6 E 4 6 2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999002 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9828: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (9) A D B E C (9) C E D B A (7) D C A E B (6) A B D E C (6) E C B D A (4) D A B E C (4) C E B A D (4) B E A C D (4) E C B A D (3) E B C A D (3) D C E A B (3) D A B C E (3) C E B D A (3) C D E A B (3) B E C A D (3) B E A D C (3) B A E C D (3) A D B C E (3) D A C E B (2) C D E B A (2) C D A E B (2) B A E D C (2) E C D B A (1) D A E B C (1) C A D B E (1) C A B E D (1) C A B D E (1) A C B E D (1) A C B D E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 6 0 14 B -16 0 -8 -12 12 C -6 8 0 -2 8 D 0 12 2 0 14 E -14 -12 -8 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.371737 B: 0.000000 C: 0.000000 D: 0.628263 E: 0.000000 Sum of squares = 0.532902818886 Cumulative probabilities = A: 0.371737 B: 0.371737 C: 0.371737 D: 1.000000 E: 1.000000 A B C D E A 0 16 6 0 14 B -16 0 -8 -12 12 C -6 8 0 -2 8 D 0 12 2 0 14 E -14 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 A=22 B=15 E=11 so E is eliminated. Round 2 votes counts: C=32 D=28 A=22 B=18 so B is eliminated. Round 3 votes counts: C=38 A=34 D=28 so D is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 D:214 C:204 B:188 E:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 16 6 0 14 B -16 0 -8 -12 12 C -6 8 0 -2 8 D 0 12 2 0 14 E -14 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 6 0 14 B -16 0 -8 -12 12 C -6 8 0 -2 8 D 0 12 2 0 14 E -14 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 6 0 14 B -16 0 -8 -12 12 C -6 8 0 -2 8 D 0 12 2 0 14 E -14 -12 -8 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999924 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9829: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (16) D A B E C (15) D A B C E (9) D E C A B (6) E C A B D (5) E C D A B (4) E C B A D (4) D B A C E (4) C E D B A (4) B C E A D (4) A B D E C (4) E C A D B (3) D A E C B (3) C E B D A (3) B A D E C (3) B D A C E (2) B A D C E (2) A D B E C (2) D C E A B (1) C E D A B (1) B D C A E (1) B A E C D (1) B A C E D (1) A D E C B (1) A B E C D (1) Total count = 100 A B C D E A 0 10 -4 -6 -2 B -10 0 -2 -6 -2 C 4 2 0 -6 -4 D 6 6 6 0 6 E 2 2 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -6 -2 B -10 0 -2 -6 -2 C 4 2 0 -6 -4 D 6 6 6 0 6 E 2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=38 C=24 E=16 B=14 A=8 so A is eliminated. Round 2 votes counts: D=41 C=24 B=19 E=16 so E is eliminated. Round 3 votes counts: D=41 C=40 B=19 so B is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:212 E:201 A:199 C:198 B:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 -4 -6 -2 B -10 0 -2 -6 -2 C 4 2 0 -6 -4 D 6 6 6 0 6 E 2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -6 -2 B -10 0 -2 -6 -2 C 4 2 0 -6 -4 D 6 6 6 0 6 E 2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -6 -2 B -10 0 -2 -6 -2 C 4 2 0 -6 -4 D 6 6 6 0 6 E 2 2 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9830: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (8) E A B D C (7) C E A D B (6) E C A B D (5) D B A C E (5) C E D B A (5) B D A E C (5) C E D A B (4) C E B D A (4) A D B E C (4) E A C B D (3) D A C B E (3) C D A E B (3) C D A B E (3) B A D E C (3) A B D E C (3) E C A D B (2) E B A D C (2) D C A B E (2) D B A E C (2) D A B C E (2) C E B A D (2) A E B D C (2) A D B C E (2) A B E D C (2) E B D A C (1) E B C A D (1) D C B A E (1) D B C A E (1) D A B E C (1) C D B A E (1) C B D E A (1) C A E D B (1) B D A C E (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 14 0 10 -2 B -14 0 -12 0 -16 C 0 12 0 2 -4 D -10 0 -2 0 -12 E 2 16 4 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999099 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 0 10 -2 B -14 0 -12 0 -16 C 0 12 0 2 -4 D -10 0 -2 0 -12 E 2 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=29 D=17 A=15 B=9 so B is eliminated. Round 2 votes counts: C=30 E=29 D=23 A=18 so A is eliminated. Round 3 votes counts: D=35 E=34 C=31 so C is eliminated. Round 4 votes counts: E=56 D=44 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:217 A:211 C:205 D:188 B:179 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 0 10 -2 B -14 0 -12 0 -16 C 0 12 0 2 -4 D -10 0 -2 0 -12 E 2 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 0 10 -2 B -14 0 -12 0 -16 C 0 12 0 2 -4 D -10 0 -2 0 -12 E 2 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 0 10 -2 B -14 0 -12 0 -16 C 0 12 0 2 -4 D -10 0 -2 0 -12 E 2 16 4 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999979572 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9831: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) B A D C E (8) C A B D E (6) B A C D E (6) E D C A B (5) E B D A C (5) C E D A B (5) C E A B D (5) C A B E D (5) B A D E C (4) E D B C A (3) E D B A C (3) D E C A B (3) C D A B E (3) A C B D E (3) E D C B A (2) E B C D A (2) C E A D B (2) C D E A B (2) B E A D C (2) A B C D E (2) E C A D B (1) D E B A C (1) D C E A B (1) D B A C E (1) D A C E B (1) D A B C E (1) C D A E B (1) C B A E D (1) C A E D B (1) C A D B E (1) B E C A D (1) B C E A D (1) B A E D C (1) B A C E D (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 16 -18 6 -4 B -16 0 -14 8 -2 C 18 14 0 16 18 D -6 -8 -16 0 -8 E 4 2 -18 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -18 6 -4 B -16 0 -14 8 -2 C 18 14 0 16 18 D -6 -8 -16 0 -8 E 4 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 E=29 B=24 D=8 A=7 so A is eliminated. Round 2 votes counts: C=36 E=29 B=27 D=8 so D is eliminated. Round 3 votes counts: C=38 E=33 B=29 so B is eliminated. Round 4 votes counts: C=59 E=41 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:233 A:200 E:198 B:188 D:181 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 16 -18 6 -4 B -16 0 -14 8 -2 C 18 14 0 16 18 D -6 -8 -16 0 -8 E 4 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -18 6 -4 B -16 0 -14 8 -2 C 18 14 0 16 18 D -6 -8 -16 0 -8 E 4 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -18 6 -4 B -16 0 -14 8 -2 C 18 14 0 16 18 D -6 -8 -16 0 -8 E 4 2 -18 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9832: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) A D E B C (9) C B E D A (6) B C E A D (6) E A B D C (5) E B A C D (4) D C A B E (4) D A E C B (4) D A C E B (4) C B D A E (4) D A C B E (3) C D B A E (3) C B D E A (3) B E C A D (3) E A D B C (2) D C B A E (2) D A E B C (2) D A B E C (2) C B E A D (2) B E C D A (2) A D E C B (2) A D C E B (2) E D B A C (1) E B D A C (1) E B A D C (1) E A B C D (1) D B E A C (1) D B C A E (1) D B A C E (1) D A B C E (1) C D A B E (1) C B A D E (1) C A B D E (1) B C E D A (1) A E D B C (1) A E C D B (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -6 0 6 2 B 6 0 10 4 -4 C 0 -10 0 2 -6 D -6 -4 -2 0 4 E -2 4 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888507 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 A B C D E A 0 -6 0 6 2 B 6 0 10 4 -4 C 0 -10 0 2 -6 D -6 -4 -2 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D E , winner is: D compare: Computing IRV winner. Round 1 votes counts: E=25 D=25 C=21 A=17 B=12 so B is eliminated. Round 2 votes counts: E=30 C=28 D=25 A=17 so A is eliminated. Round 3 votes counts: D=38 E=33 C=29 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:208 E:202 A:201 D:196 C:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 0 6 2 B 6 0 10 4 -4 C 0 -10 0 2 -6 D -6 -4 -2 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 6 2 B 6 0 10 4 -4 C 0 -10 0 2 -6 D -6 -4 -2 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 6 2 B 6 0 10 4 -4 C 0 -10 0 2 -6 D -6 -4 -2 0 4 E -2 4 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.166667 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.388888888885 Cumulative probabilities = A: 0.333333 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9833: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) E A B D C (11) C D B A E (10) A E D C B (10) D C A B E (8) E B A C D (7) B E C D A (6) D C B A E (5) E A D C B (4) E A D B C (4) C B D A E (3) A D C E B (3) E A B C D (2) D A C E B (2) B E C A D (2) B C E D A (2) E A C D B (1) D C A E B (1) C D E A B (1) B D C A E (1) B C D A E (1) A E D B C (1) A E B D C (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -6 -6 -6 B 0 0 2 -2 2 C 6 -2 0 -4 0 D 6 2 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333272 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 A B C D E A 0 0 -6 -6 -6 B 0 0 2 -2 2 C 6 -2 0 -4 0 D 6 2 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=25 D=16 A=16 C=14 so C is eliminated. Round 2 votes counts: E=29 B=28 D=27 A=16 so A is eliminated. Round 3 votes counts: E=41 D=31 B=28 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:205 E:203 B:201 C:200 A:191 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D E , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -6 -6 -6 B 0 0 2 -2 2 C 6 -2 0 -4 0 D 6 2 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -6 -6 -6 B 0 0 2 -2 2 C 6 -2 0 -4 0 D 6 2 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -6 -6 -6 B 0 0 2 -2 2 C 6 -2 0 -4 0 D 6 2 4 0 -2 E 6 -2 0 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.333333 C: 0.000000 D: 0.333333 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.000000 B: 0.333333 C: 0.333333 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9834: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (11) E C A B D (6) D B A C E (6) A E C D B (6) C E B D A (5) A D E C B (5) E C B A D (4) B D C E A (4) B D C A E (4) A E D C B (4) A D B E C (4) E C A D B (3) D B C A E (3) D B A E C (3) B C D E A (3) E A C B D (2) D C A B E (2) D A B E C (2) C B E D A (2) B C E D A (2) A D B C E (2) E B C A D (1) E B A D C (1) E A C D B (1) E A B D C (1) C E B A D (1) C E A D B (1) C D B E A (1) C D A E B (1) B E D C A (1) B E D A C (1) B E C D A (1) B D A E C (1) B D A C E (1) A D E B C (1) A D C E B (1) A D C B E (1) A C D E B (1) Total count = 100 A B C D E A 0 10 10 -8 18 B -10 0 6 -18 10 C -10 -6 0 -18 4 D 8 18 18 0 14 E -18 -10 -4 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 10 -8 18 B -10 0 6 -18 10 C -10 -6 0 -18 4 D 8 18 18 0 14 E -18 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=25 E=19 B=18 C=11 so C is eliminated. Round 2 votes counts: D=29 E=26 A=25 B=20 so B is eliminated. Round 3 votes counts: D=42 E=33 A=25 so A is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:229 A:215 B:194 C:185 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 10 10 -8 18 B -10 0 6 -18 10 C -10 -6 0 -18 4 D 8 18 18 0 14 E -18 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 10 -8 18 B -10 0 6 -18 10 C -10 -6 0 -18 4 D 8 18 18 0 14 E -18 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 10 -8 18 B -10 0 6 -18 10 C -10 -6 0 -18 4 D 8 18 18 0 14 E -18 -10 -4 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9835: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (9) A B D C E (7) C A E D B (6) B A D E C (5) D E B C A (4) C E D B A (4) C E D A B (4) E D B C A (3) D E C B A (3) D B E C A (3) D B A E C (3) C E A B D (3) C D E B A (3) C A E B D (3) B D E A C (3) A C E B D (3) E D C B A (2) E B D A C (2) D C E B A (2) C E A D B (2) B D A E C (2) B A E D C (2) A C B D E (2) A B E D C (2) A B C E D (2) E C D B A (1) E C B A D (1) E C A B D (1) E B D C A (1) E B C D A (1) D C B E A (1) D B E A C (1) D B C E A (1) D B C A E (1) D A B C E (1) C D A E B (1) C A D B E (1) A E B D C (1) A D B C E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 2 -4 6 8 B -2 0 12 4 -2 C 4 -12 0 -20 0 D -6 -4 20 0 10 E -8 2 0 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.666667 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839522 Cumulative probabilities = A: 0.666667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 6 8 B -2 0 12 4 -2 C 4 -12 0 -20 0 D -6 -4 20 0 10 E -8 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839545 Cumulative probabilities = A: 0.666667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=29 C=27 D=20 E=12 B=12 so E is eliminated. Round 2 votes counts: C=30 A=29 D=25 B=16 so B is eliminated. Round 3 votes counts: A=36 D=33 C=31 so C is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. D:210 A:206 B:206 E:192 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 -4 6 8 B -2 0 12 4 -2 C 4 -12 0 -20 0 D -6 -4 20 0 10 E -8 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839545 Cumulative probabilities = A: 0.666667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 6 8 B -2 0 12 4 -2 C 4 -12 0 -20 0 D -6 -4 20 0 10 E -8 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839545 Cumulative probabilities = A: 0.666667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 6 8 B -2 0 12 4 -2 C 4 -12 0 -20 0 D -6 -4 20 0 10 E -8 2 0 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.666667 B: 0.222222 C: 0.111111 D: 0.000000 E: 0.000000 Sum of squares = 0.506172839545 Cumulative probabilities = A: 0.666667 B: 0.888889 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9836: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (8) A B C D E (8) E D C B A (7) B A C D E (7) A C B D E (6) E D C A B (4) E D B C A (4) E D A C B (4) D E C B A (4) D C E B A (4) C B D E A (4) A E D B C (4) A B C E D (4) C D E A B (3) A E D C B (3) A C D E B (3) E D A B C (2) C D E B A (2) C A D E B (2) C A B D E (2) B D E C A (2) B C D E A (2) E D B A C (1) E B D C A (1) E A D B C (1) D E C A B (1) C D B E A (1) C B A D E (1) B C A D E (1) B A E D C (1) A E C D B (1) A C E D B (1) A C B E D (1) Total count = 100 A B C D E A 0 16 10 8 6 B -16 0 -8 -4 -4 C -10 8 0 -2 4 D -8 4 2 0 6 E -6 4 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999978 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 10 8 6 B -16 0 -8 -4 -4 C -10 8 0 -2 4 D -8 4 2 0 6 E -6 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=24 C=15 B=13 D=9 so D is eliminated. Round 2 votes counts: A=39 E=29 C=19 B=13 so B is eliminated. Round 3 votes counts: A=47 E=31 C=22 so C is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:202 C:200 E:194 B:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 10 8 6 B -16 0 -8 -4 -4 C -10 8 0 -2 4 D -8 4 2 0 6 E -6 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 10 8 6 B -16 0 -8 -4 -4 C -10 8 0 -2 4 D -8 4 2 0 6 E -6 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 10 8 6 B -16 0 -8 -4 -4 C -10 8 0 -2 4 D -8 4 2 0 6 E -6 4 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999792 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9837: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E C B (11) B C E D A (9) A D B C E (8) C B E D A (6) E C D B A (4) E C B D A (4) A D B E C (4) A C B E D (4) E D C B A (3) D A B E C (3) C E B D A (3) B C E A D (3) D E B C A (2) D E A C B (2) D A E C B (2) D A E B C (2) C E B A D (2) C B E A D (2) B A D C E (2) A D E B C (2) A C B D E (2) E D C A B (1) E D B C A (1) E D A C B (1) E C A B D (1) E B D C A (1) E B C D A (1) D E C B A (1) D E C A B (1) D B A E C (1) B E D C A (1) B E C D A (1) B D E C A (1) B D C E A (1) B C D A E (1) B A C E D (1) A E D C B (1) A D C E B (1) A C E B D (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -2 0 -6 -4 B 2 0 -6 -2 4 C 0 6 0 -8 -4 D 6 2 8 0 -2 E 4 -4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000006 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 A B C D E A 0 -2 0 -6 -4 B 2 0 -6 -2 4 C 0 6 0 -8 -4 D 6 2 8 0 -2 E 4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 B=20 E=17 D=14 C=13 so C is eliminated. Round 2 votes counts: A=36 B=28 E=22 D=14 so D is eliminated. Round 3 votes counts: A=43 B=29 E=28 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:207 E:203 B:199 C:197 A:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -6 -4 B 2 0 -6 -2 4 C 0 6 0 -8 -4 D 6 2 8 0 -2 E 4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -6 -4 B 2 0 -6 -2 4 C 0 6 0 -8 -4 D 6 2 8 0 -2 E 4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -6 -4 B 2 0 -6 -2 4 C 0 6 0 -8 -4 D 6 2 8 0 -2 E 4 -4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.250000 C: 0.000000 D: 0.500000 E: 0.250000 Sum of squares = 0.375000000066 Cumulative probabilities = A: 0.000000 B: 0.250000 C: 0.250000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9838: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (9) C D B A E (6) E A C B D (4) E A B C D (4) D C B A E (4) D C A B E (4) C D A B E (4) B E D C A (4) B D C E A (4) A E C D B (4) E B A D C (3) E B A C D (3) D B A E C (3) C B D E A (3) B E C D A (3) B D E C A (3) A E D B C (3) E B C A D (2) D B C E A (2) D B C A E (2) D A B E C (2) C E B A D (2) C E A B D (2) C A E D B (2) B E D A C (2) B C D E A (2) A E C B D (2) A E B D C (2) A D E B C (2) A D C E B (2) D B E C A (1) C D B E A (1) B D E A C (1) A C E D B (1) A C E B D (1) A C D E B (1) Total count = 100 A B C D E A 0 -2 -2 -2 -10 B 2 0 14 12 2 C 2 -14 0 -6 -14 D 2 -12 6 0 -6 E 10 -2 14 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998262 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -2 -2 -10 B 2 0 14 12 2 C 2 -14 0 -6 -14 D 2 -12 6 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994479 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=20 B=19 D=18 A=18 so D is eliminated. Round 2 votes counts: C=28 B=27 E=25 A=20 so A is eliminated. Round 3 votes counts: E=38 C=33 B=29 so B is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:215 E:214 D:195 A:192 C:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 -2 -2 -10 B 2 0 14 12 2 C 2 -14 0 -6 -14 D 2 -12 6 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994479 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -2 -2 -10 B 2 0 14 12 2 C 2 -14 0 -6 -14 D 2 -12 6 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994479 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -2 -2 -10 B 2 0 14 12 2 C 2 -14 0 -6 -14 D 2 -12 6 0 -6 E 10 -2 14 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999994479 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9839: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (13) A E D B C (8) C B A D E (6) C A B E D (6) D E B C A (5) C B D E A (5) A C B E D (5) D E C B A (4) D E B A C (4) C B D A E (4) C D E B A (3) B C A D E (3) A B E D C (3) A B C E D (3) E D B A C (2) D C E B A (2) D B E C A (2) C A B D E (2) B D A E C (2) B A D E C (2) B A C D E (2) A E D C B (2) A E B D C (2) A C E B D (2) E D A C B (1) E A D B C (1) D C B E A (1) D B C E A (1) C E D A B (1) C A E D B (1) B C D E A (1) A E C D B (1) Total count = 100 A B C D E A 0 2 6 -2 8 B -2 0 8 -4 -4 C -6 -8 0 -10 -4 D 2 4 10 0 -2 E -8 4 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.500000000121 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 A B C D E A 0 2 6 -2 8 B -2 0 8 -4 -4 C -6 -8 0 -10 -4 D 2 4 10 0 -2 E -8 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999781 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 A=26 D=19 E=17 B=10 so B is eliminated. Round 2 votes counts: C=32 A=30 D=21 E=17 so E is eliminated. Round 3 votes counts: D=37 C=32 A=31 so A is eliminated. Round 4 votes counts: D=55 C=45 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:207 D:207 E:201 B:199 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 2 6 -2 8 B -2 0 8 -4 -4 C -6 -8 0 -10 -4 D 2 4 10 0 -2 E -8 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999781 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -2 8 B -2 0 8 -4 -4 C -6 -8 0 -10 -4 D 2 4 10 0 -2 E -8 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999781 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -2 8 B -2 0 8 -4 -4 C -6 -8 0 -10 -4 D 2 4 10 0 -2 E -8 4 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.166667 Sum of squares = 0.499999999781 Cumulative probabilities = A: 0.166667 B: 0.166667 C: 0.166667 D: 0.833333 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9840: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (13) D C B A E (10) D C B E A (7) E A D C B (6) E A B D C (5) E A B C D (5) E D A C B (4) E A D B C (4) C D B A E (4) B C D E A (4) D E C B A (3) B D C E A (3) B C D A E (3) B C A D E (3) E D C B A (2) E D A B C (2) D B E C A (2) B E A C D (2) B A C D E (2) A C B D E (2) E D B A C (1) E B A D C (1) D E C A B (1) D C A B E (1) D B C E A (1) C B D A E (1) C A D B E (1) B E D C A (1) A E D C B (1) A E C B D (1) A D C E B (1) A C B E D (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 6 0 -8 B 0 0 8 -2 0 C -6 -8 0 -12 -10 D 0 2 12 0 0 E 8 0 10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.533519 E: 0.466481 Sum of squares = 0.502247013285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.533519 E: 1.000000 A B C D E A 0 0 6 0 -8 B 0 0 8 -2 0 C -6 -8 0 -12 -10 D 0 2 12 0 0 E 8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=25 A=21 B=18 C=6 so C is eliminated. Round 2 votes counts: E=30 D=29 A=22 B=19 so B is eliminated. Round 3 votes counts: D=40 E=33 A=27 so A is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. E:209 D:207 B:203 A:199 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 0 6 0 -8 B 0 0 8 -2 0 C -6 -8 0 -12 -10 D 0 2 12 0 0 E 8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 6 0 -8 B 0 0 8 -2 0 C -6 -8 0 -12 -10 D 0 2 12 0 0 E 8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 6 0 -8 B 0 0 8 -2 0 C -6 -8 0 -12 -10 D 0 2 12 0 0 E 8 0 10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999998235 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9841: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (7) B A E C D (6) E B A C D (5) E C A B D (4) D E C A B (4) D C E A B (4) C D A B E (4) B E A C D (4) B A D E C (4) A B C E D (4) D C A B E (3) D B E A C (3) C A D B E (3) A C B E D (3) A B C D E (3) E C D A B (2) E B D A C (2) D E B C A (2) D E B A C (2) D C B A E (2) D C A E B (2) C D A E B (2) C A E D B (2) C A B E D (2) B A E D C (2) B A C D E (2) A C B D E (2) E D B A C (1) E B D C A (1) E B A D C (1) D E C B A (1) D C E B A (1) D B A E C (1) D B A C E (1) C E A B D (1) C D E A B (1) C A B D E (1) B E A D C (1) B D A E C (1) B A D C E (1) B A C E D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 2 6 6 B 4 0 -4 2 10 C -2 4 0 6 -8 D -6 -2 -6 0 2 E -6 -10 8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.36 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 2 6 6 B 4 0 -4 2 10 C -2 4 0 6 -8 D -6 -2 -6 0 2 E -6 -10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999986 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 E=23 B=22 C=16 A=13 so A is eliminated. Round 2 votes counts: B=29 D=26 E=23 C=22 so C is eliminated. Round 3 votes counts: D=37 B=37 E=26 so E is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:206 A:205 C:200 E:195 D:194 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -4 2 6 6 B 4 0 -4 2 10 C -2 4 0 6 -8 D -6 -2 -6 0 2 E -6 -10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999986 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 6 6 B 4 0 -4 2 10 C -2 4 0 6 -8 D -6 -2 -6 0 2 E -6 -10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999986 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 6 6 B 4 0 -4 2 10 C -2 4 0 6 -8 D -6 -2 -6 0 2 E -6 -10 8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.400000 B: 0.200000 C: 0.400000 D: 0.000000 E: 0.000000 Sum of squares = 0.359999999986 Cumulative probabilities = A: 0.400000 B: 0.600000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9842: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D A E (8) E A D C B (7) E D A C B (6) B C A D E (6) D A E C B (5) D A B E C (5) C B E D A (5) A D E B C (5) A E D B C (4) D E A C B (3) C E B A D (3) C B E A D (3) C B D E A (3) B C E A D (3) B A D C E (3) E C B A D (2) E C A D B (2) D C B A E (2) B C D E A (2) B A C D E (2) A D B E C (2) A D B C E (2) E D C A B (1) E C D B A (1) E C D A B (1) E C B D A (1) E A C B D (1) D B C A E (1) D B A C E (1) D A B C E (1) C E D B A (1) C E B D A (1) C D E B A (1) B D C A E (1) B C E D A (1) B C A E D (1) B A E C D (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 0 -2 4 B 6 0 0 -4 8 C 0 0 0 -2 4 D 2 4 2 0 10 E -4 -8 -4 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -2 4 B 6 0 0 -4 8 C 0 0 0 -2 4 D 2 4 2 0 10 E -4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=22 D=18 C=17 A=15 so A is eliminated. Round 2 votes counts: B=29 D=28 E=26 C=17 so C is eliminated. Round 3 votes counts: B=40 E=31 D=29 so D is eliminated. Round 4 votes counts: B=54 E=46 so E is eliminated. IRV winner is B compare: Computing Borda winner. D:209 B:205 C:201 A:198 E:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -2 4 B 6 0 0 -4 8 C 0 0 0 -2 4 D 2 4 2 0 10 E -4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -2 4 B 6 0 0 -4 8 C 0 0 0 -2 4 D 2 4 2 0 10 E -4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -2 4 B 6 0 0 -4 8 C 0 0 0 -2 4 D 2 4 2 0 10 E -4 -8 -4 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9843: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (12) E B C A D (6) D A C E B (6) D A C B E (6) C A D E B (6) B E D A C (5) B E A D C (5) E B A C D (4) D C A E B (4) C D A E B (4) B E D C A (4) B E C A D (4) C E B A D (3) C A E B D (3) D B E C A (2) D B E A C (2) D B A E C (2) D A B E C (2) D A B C E (2) B E C D A (2) B D E A C (2) A E C D B (2) A D C E B (2) A C D E B (2) E C A B D (1) E A B C D (1) D C A B E (1) B D E C A (1) B C E D A (1) A E C B D (1) A D E B C (1) A C E D B (1) Total count = 100 A B C D E A 0 -10 16 8 -10 B 10 0 16 10 6 C -16 -16 0 6 -18 D -8 -10 -6 0 -10 E 10 -6 18 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999871 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 16 8 -10 B 10 0 16 10 6 C -16 -16 0 6 -18 D -8 -10 -6 0 -10 E 10 -6 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=27 C=16 E=12 A=9 so A is eliminated. Round 2 votes counts: B=36 D=30 C=19 E=15 so E is eliminated. Round 3 votes counts: B=47 D=30 C=23 so C is eliminated. Round 4 votes counts: B=55 D=45 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:216 A:202 D:183 C:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 16 8 -10 B 10 0 16 10 6 C -16 -16 0 6 -18 D -8 -10 -6 0 -10 E 10 -6 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 16 8 -10 B 10 0 16 10 6 C -16 -16 0 6 -18 D -8 -10 -6 0 -10 E 10 -6 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 16 8 -10 B 10 0 16 10 6 C -16 -16 0 6 -18 D -8 -10 -6 0 -10 E 10 -6 18 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999953 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9844: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (9) A B E C D (7) E D C B A (6) D C E A B (6) A E B D C (5) A D E C B (5) D C E B A (4) C D E B A (4) C B D E A (4) B C E D A (4) B C D E A (4) B A C E D (4) A B C E D (4) E D A C B (3) D E A C B (3) A E D C B (3) A B C D E (3) B E A C D (2) B C A E D (2) B C A D E (2) A D C E B (2) E D C A B (1) E D B A C (1) E C D B A (1) E B C D A (1) E A D C B (1) D E C B A (1) D A E C B (1) C D A B E (1) C B E D A (1) B E C D A (1) B A E C D (1) A D E B C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 14 14 6 4 B -14 0 2 8 2 C -14 -2 0 -6 -6 D -6 -8 6 0 -14 E -4 -2 6 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999914 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 14 6 4 B -14 0 2 8 2 C -14 -2 0 -6 -6 D -6 -8 6 0 -14 E -4 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=41 B=20 D=15 E=14 C=10 so C is eliminated. Round 2 votes counts: A=41 B=25 D=20 E=14 so E is eliminated. Round 3 votes counts: A=42 D=32 B=26 so B is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:219 E:207 B:199 D:189 C:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 14 6 4 B -14 0 2 8 2 C -14 -2 0 -6 -6 D -6 -8 6 0 -14 E -4 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 14 6 4 B -14 0 2 8 2 C -14 -2 0 -6 -6 D -6 -8 6 0 -14 E -4 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 14 6 4 B -14 0 2 8 2 C -14 -2 0 -6 -6 D -6 -8 6 0 -14 E -4 -2 6 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999862 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9845: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (10) A B D E C (9) C D E B A (8) D B A C E (7) D C B A E (6) C D B E A (6) E C A B D (5) D B C A E (5) B A D E C (5) E A C B D (4) E A B D C (4) C E D A B (4) B D A E C (4) A E B D C (3) A E B C D (3) E A B C D (2) D B A E C (2) C D B A E (2) A B E D C (2) E C D B A (1) E B A C D (1) D A B E C (1) D A B C E (1) C E A B D (1) C D E A B (1) C D A B E (1) A E C B D (1) A D B C E (1) Total count = 100 A B C D E A 0 -14 0 -18 6 B 14 0 0 -12 4 C 0 0 0 0 6 D 18 12 0 0 18 E -6 -4 -6 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.649364 D: 0.350636 E: 0.000000 Sum of squares = 0.544618973133 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.649364 D: 1.000000 E: 1.000000 A B C D E A 0 -14 0 -18 6 B 14 0 0 -12 4 C 0 0 0 0 6 D 18 12 0 0 18 E -6 -4 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 D=22 A=19 E=17 B=9 so B is eliminated. Round 2 votes counts: C=33 D=26 A=24 E=17 so E is eliminated. Round 3 votes counts: C=39 A=35 D=26 so D is eliminated. Round 4 votes counts: C=50 A=50 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:224 B:203 C:203 A:187 E:183 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -14 0 -18 6 B 14 0 0 -12 4 C 0 0 0 0 6 D 18 12 0 0 18 E -6 -4 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 0 -18 6 B 14 0 0 -12 4 C 0 0 0 0 6 D 18 12 0 0 18 E -6 -4 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 0 -18 6 B 14 0 0 -12 4 C 0 0 0 0 6 D 18 12 0 0 18 E -6 -4 -6 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999871 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9846: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (10) D A C B E (10) C A E D B (8) A C D E B (7) B E C D A (6) D B A C E (4) B E D C A (4) B C E D A (4) A C D B E (4) E C B A D (3) E A C D B (3) A D C E B (3) A C E D B (3) E C A B D (2) E A D B C (2) E A C B D (2) D B A E C (2) C E B A D (2) C E A B D (2) C A E B D (2) B E D A C (2) B D E A C (2) A D E C B (2) A D C B E (2) D B E A C (1) D B C A E (1) D A E B C (1) D A C E B (1) D A B E C (1) C B E A D (1) C A D E B (1) C A D B E (1) B E C A D (1) Total count = 100 A B C D E A 0 14 4 22 6 B -14 0 -18 -14 -8 C -4 18 0 24 12 D -22 14 -24 0 -14 E -6 8 -12 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999929 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 22 6 B -14 0 -18 -14 -8 C -4 18 0 24 12 D -22 14 -24 0 -14 E -6 8 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=22 D=21 A=21 B=19 C=17 so C is eliminated. Round 2 votes counts: A=33 E=26 D=21 B=20 so B is eliminated. Round 3 votes counts: E=44 A=33 D=23 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. C:225 A:223 E:202 D:177 B:173 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 22 6 B -14 0 -18 -14 -8 C -4 18 0 24 12 D -22 14 -24 0 -14 E -6 8 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 22 6 B -14 0 -18 -14 -8 C -4 18 0 24 12 D -22 14 -24 0 -14 E -6 8 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 22 6 B -14 0 -18 -14 -8 C -4 18 0 24 12 D -22 14 -24 0 -14 E -6 8 -12 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999549 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9847: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (10) E B A D C (5) D A B E C (5) C A E D B (5) C A D E B (5) A D C B E (5) E C B A D (4) D B A E C (4) C E B A D (4) A C D E B (4) E B C D A (3) C E A B D (3) C D B A E (3) B E D A C (3) A D E B C (3) A D C E B (3) E B D A C (2) E B C A D (2) D B C A E (2) D A C B E (2) B E D C A (2) B C D E A (2) A E D B C (2) A E C B D (2) E C A B D (1) E A C B D (1) E A B C D (1) D C A B E (1) D A B C E (1) C E B D A (1) C D A B E (1) C A D B E (1) B D E A C (1) B D C E A (1) B C E D A (1) A E D C B (1) A E C D B (1) A D E C B (1) A C E D B (1) Total count = 100 A B C D E A 0 0 -4 10 6 B 0 0 0 -2 -10 C 4 0 0 12 -8 D -10 2 -12 0 -10 E -6 10 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691357 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.777778 D: 0.777778 E: 1.000000 A B C D E A 0 0 -4 10 6 B 0 0 0 -2 -10 C 4 0 0 12 -8 D -10 2 -12 0 -10 E -6 10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691348 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.777778 D: 0.777778 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C , winner is: A compare: Computing IRV winner. Round 1 votes counts: C=23 A=23 B=20 E=19 D=15 so D is eliminated. Round 2 votes counts: A=31 B=26 C=24 E=19 so E is eliminated. Round 3 votes counts: B=38 A=33 C=29 so C is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. E:211 A:206 C:204 B:194 D:185 Borda winner is E compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 0 -4 10 6 B 0 0 0 -2 -10 C 4 0 0 12 -8 D -10 2 -12 0 -10 E -6 10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691348 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.777778 D: 0.777778 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 10 6 B 0 0 0 -2 -10 C 4 0 0 12 -8 D -10 2 -12 0 -10 E -6 10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691348 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.777778 D: 0.777778 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 10 6 B 0 0 0 -2 -10 C 4 0 0 12 -8 D -10 2 -12 0 -10 E -6 10 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.444444 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.222222 Sum of squares = 0.358024691348 Cumulative probabilities = A: 0.444444 B: 0.444444 C: 0.777778 D: 0.777778 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9848: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (8) D C A E B (7) D B C A E (7) E A C B D (6) C A D E B (5) E C A D B (4) B E D C A (4) B E D A C (4) B D E C A (4) B D C E A (4) E B A C D (3) C A E D B (3) B E A D C (3) B E A C D (3) B D C A E (3) A C E D B (3) E A B C D (2) D C A B E (2) C E D A B (2) C D A E B (2) B D A C E (2) B A E C D (2) A C E B D (2) A C D E B (2) E D B C A (1) E C D A B (1) E B D C A (1) D E C B A (1) D C E A B (1) D C B E A (1) D C B A E (1) D B C E A (1) B A D E C (1) A E C D B (1) A E C B D (1) A C D B E (1) A C B D E (1) Total count = 100 A B C D E A 0 8 -10 2 -8 B -8 0 -10 -8 -12 C 10 10 0 4 0 D -2 8 -4 0 -8 E 8 12 0 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.379543 D: 0.000000 E: 0.620457 Sum of squares = 0.529019835686 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.379543 D: 0.379543 E: 1.000000 A B C D E A 0 8 -10 2 -8 B -8 0 -10 -8 -12 C 10 10 0 4 0 D -2 8 -4 0 -8 E 8 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=26 D=21 C=12 A=11 so A is eliminated. Round 2 votes counts: B=30 E=28 D=21 C=21 so D is eliminated. Round 3 votes counts: B=38 C=33 E=29 so E is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:214 C:212 D:197 A:196 B:181 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -10 2 -8 B -8 0 -10 -8 -12 C 10 10 0 4 0 D -2 8 -4 0 -8 E 8 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -10 2 -8 B -8 0 -10 -8 -12 C 10 10 0 4 0 D -2 8 -4 0 -8 E 8 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -10 2 -8 B -8 0 -10 -8 -12 C 10 10 0 4 0 D -2 8 -4 0 -8 E 8 12 0 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9849: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) D C A B E (6) A D B C E (6) E B C A D (5) E C B D A (4) D E C B A (4) D A C B E (4) B C E A D (4) E B A C D (3) D A E C B (3) D A E B C (3) C E B D A (3) C D B E A (3) C D B A E (3) B C A E D (3) A B E C D (3) E D A B C (2) E C B A D (2) E A B C D (2) D E A B C (2) D C B A E (2) C B D A E (2) A C B D E (2) A B D E C (2) A B C D E (2) E B D C A (1) E B C D A (1) E B A D C (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C A E B (1) D A B C E (1) C B E D A (1) C B A E D (1) C B A D E (1) C A B D E (1) B E A C D (1) B A E C D (1) B A C E D (1) A E D B C (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -10 -8 2 B 6 0 -2 0 4 C 10 2 0 0 10 D 8 0 0 0 14 E -2 -4 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.636143 D: 0.363857 E: 0.000000 Sum of squares = 0.537069615541 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.636143 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -8 2 B 6 0 -2 0 4 C 10 2 0 0 10 D 8 0 0 0 14 E -2 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=23 A=18 C=15 B=10 so B is eliminated. Round 2 votes counts: D=34 E=24 C=22 A=20 so A is eliminated. Round 3 votes counts: D=43 E=29 C=28 so C is eliminated. Round 4 votes counts: D=57 E=43 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:211 D:211 B:204 A:189 E:185 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C D , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -10 -8 2 B 6 0 -2 0 4 C 10 2 0 0 10 D 8 0 0 0 14 E -2 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -8 2 B 6 0 -2 0 4 C 10 2 0 0 10 D 8 0 0 0 14 E -2 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -8 2 B 6 0 -2 0 4 C 10 2 0 0 10 D 8 0 0 0 14 E -2 -4 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9850: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D B E A (8) A B D E C (8) E A B D C (7) C D B A E (7) C E A B D (6) A E B D C (6) E A C B D (5) C E A D B (5) A B E D C (5) C E D B A (4) B D A E C (4) D B C A E (3) D B A E C (3) C E D A B (3) C D E B A (3) B A D E C (3) E B A D C (2) D C B A E (2) C A E D B (2) C A D B E (2) A E B C D (2) E C A B D (1) E B D A C (1) E A B C D (1) D B E A C (1) D B A C E (1) D A B C E (1) C A E B D (1) B E D A C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 14 6 16 4 B -14 0 2 8 2 C -6 -2 0 0 0 D -16 -8 0 0 -4 E -4 -2 0 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 6 16 4 B -14 0 2 8 2 C -6 -2 0 0 0 D -16 -8 0 0 -4 E -4 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998018 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=41 A=23 E=17 D=11 B=8 so B is eliminated. Round 2 votes counts: C=41 A=26 E=18 D=15 so D is eliminated. Round 3 votes counts: C=46 A=35 E=19 so E is eliminated. Round 4 votes counts: A=53 C=47 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 B:199 E:199 C:196 D:186 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 6 16 4 B -14 0 2 8 2 C -6 -2 0 0 0 D -16 -8 0 0 -4 E -4 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998018 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 6 16 4 B -14 0 2 8 2 C -6 -2 0 0 0 D -16 -8 0 0 -4 E -4 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998018 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 6 16 4 B -14 0 2 8 2 C -6 -2 0 0 0 D -16 -8 0 0 -4 E -4 -2 0 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998018 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9851: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (6) D C E A B (5) C A B D E (5) B E A D C (5) A C D B E (5) E B D C A (4) E B D A C (4) D E B A C (4) D E A C B (4) C B E A D (4) C A D B E (4) C A B E D (4) B A C E D (4) E D B A C (3) D E A B C (3) B E C A D (3) A C B D E (3) D C A E B (2) C D A E B (2) C A D E B (2) B E A C D (2) B A E D C (2) A C B E D (2) A B D E C (2) E C D B A (1) D E C B A (1) D E B C A (1) D A E C B (1) D A E B C (1) D A C E B (1) D A B E C (1) C D E A B (1) C B A E D (1) B D E A C (1) B C E A D (1) B A E C D (1) B A D E C (1) A D C E B (1) A D B E C (1) A B C D E (1) Total count = 100 A B C D E A 0 2 6 8 -6 B -2 0 2 0 6 C -6 -2 0 -8 -4 D -8 0 8 0 6 E 6 -6 4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102069 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 A B C D E A 0 2 6 8 -6 B -2 0 2 0 6 C -6 -2 0 -8 -4 D -8 0 8 0 6 E 6 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=24 C=23 B=20 E=18 A=15 so A is eliminated. Round 2 votes counts: C=33 D=26 B=23 E=18 so E is eliminated. Round 3 votes counts: D=35 C=34 B=31 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. A:205 B:203 D:203 E:199 C:190 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 2 6 8 -6 B -2 0 2 0 6 C -6 -2 0 -8 -4 D -8 0 8 0 6 E 6 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 8 -6 B -2 0 2 0 6 C -6 -2 0 -8 -4 D -8 0 8 0 6 E 6 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 8 -6 B -2 0 2 0 6 C -6 -2 0 -8 -4 D -8 0 8 0 6 E 6 -6 4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.428571 B: 0.428571 C: 0.000000 D: 0.000000 E: 0.142857 Sum of squares = 0.387755102043 Cumulative probabilities = A: 0.428571 B: 0.857143 C: 0.857143 D: 0.857143 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9852: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (7) B E D C A (6) E B D A C (5) D C A E B (5) C D A B E (5) C A D E B (5) C A D B E (5) B E A D C (5) A C D E B (5) C D A E B (4) D E B C A (3) D A C E B (3) C A B D E (3) B E D A C (3) B E C D A (3) D E A B C (2) D A E C B (2) C D B A E (2) C B E D A (2) C B E A D (2) C B D E A (2) C A B E D (2) B E A C D (2) B C E D A (2) B C E A D (2) A D E B C (2) A D C E B (2) E A D B C (1) D E A C B (1) D C B E A (1) D A E B C (1) C B D A E (1) C B A D E (1) B E C A D (1) A E D B C (1) A D E C B (1) Total count = 100 A B C D E A 0 0 -14 -6 0 B 0 0 -8 -2 0 C 14 8 0 -2 8 D 6 2 2 0 12 E 0 0 -8 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 -14 -6 0 B 0 0 -8 -2 0 C 14 8 0 -2 8 D 6 2 2 0 12 E 0 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=34 B=24 D=18 E=13 A=11 so A is eliminated. Round 2 votes counts: C=39 B=24 D=23 E=14 so E is eliminated. Round 3 votes counts: C=39 B=36 D=25 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:214 D:211 B:195 A:190 E:190 Borda winner is C compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 -14 -6 0 B 0 0 -8 -2 0 C 14 8 0 -2 8 D 6 2 2 0 12 E 0 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -14 -6 0 B 0 0 -8 -2 0 C 14 8 0 -2 8 D 6 2 2 0 12 E 0 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -14 -6 0 B 0 0 -8 -2 0 C 14 8 0 -2 8 D 6 2 2 0 12 E 0 0 -8 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999991198 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9853: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (12) A B D E C (10) D C E A B (9) B A E C D (8) A B D C E (8) C E D B A (6) B A E D C (6) E C B A D (5) E B C A D (5) D A B C E (5) D C A E B (4) E C B D A (3) C D E A B (2) B E C A D (2) B E A C D (2) A D C B E (2) A D B C E (2) A B C E D (2) A B C D E (2) D A C E B (1) C E D A B (1) C A D B E (1) B A D E C (1) A B E D C (1) Total count = 100 A B C D E A 0 0 0 14 6 B 0 0 8 10 4 C 0 -8 0 2 -10 D -14 -10 -2 0 -6 E -6 -4 10 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.339109 B: 0.660891 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.551771608147 Cumulative probabilities = A: 0.339109 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 0 14 6 B 0 0 8 10 4 C 0 -8 0 2 -10 D -14 -10 -2 0 -6 E -6 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=25 D=19 B=19 C=10 so C is eliminated. Round 2 votes counts: E=32 A=28 D=21 B=19 so B is eliminated. Round 3 votes counts: A=43 E=36 D=21 so D is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. B:211 A:210 E:203 C:192 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 0 14 6 B 0 0 8 10 4 C 0 -8 0 2 -10 D -14 -10 -2 0 -6 E -6 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 14 6 B 0 0 8 10 4 C 0 -8 0 2 -10 D -14 -10 -2 0 -6 E -6 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 14 6 B 0 0 8 10 4 C 0 -8 0 2 -10 D -14 -10 -2 0 -6 E -6 -4 10 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999952 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9854: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (15) B D E A C (13) D E B A C (7) B D E C A (7) A C E D B (7) C A B E D (6) E D A B C (5) A E C D B (4) C B A D E (3) B D C E A (3) B C D E A (3) E D A C B (2) E A D C B (2) D E A C B (2) D B E A C (2) A E D C B (2) A E D B C (2) D E C A B (1) D E B C A (1) D B E C A (1) C D E A B (1) C D B E A (1) C D B A E (1) C B D A E (1) C A E B D (1) C A D E B (1) C A B D E (1) B E D A C (1) B C A E D (1) B C A D E (1) B A E D C (1) A B E C D (1) Total count = 100 A B C D E A 0 6 2 -4 -4 B -6 0 -2 -14 -6 C -2 2 0 -4 -8 D 4 14 4 0 0 E 4 6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.410363 E: 0.589637 Sum of squares = 0.516069480465 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.410363 E: 1.000000 A B C D E A 0 6 2 -4 -4 B -6 0 -2 -14 -6 C -2 2 0 -4 -8 D 4 14 4 0 0 E 4 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=30 A=16 D=14 E=9 so E is eliminated. Round 2 votes counts: C=31 B=30 D=21 A=18 so A is eliminated. Round 3 votes counts: C=42 B=31 D=27 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:211 E:209 A:200 C:194 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 6 2 -4 -4 B -6 0 -2 -14 -6 C -2 2 0 -4 -8 D 4 14 4 0 0 E 4 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 -4 -4 B -6 0 -2 -14 -6 C -2 2 0 -4 -8 D 4 14 4 0 0 E 4 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 -4 -4 B -6 0 -2 -14 -6 C -2 2 0 -4 -8 D 4 14 4 0 0 E 4 6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9855: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D B C E (10) A B D C E (10) E C D B A (9) E C B D A (8) A D E C B (4) E C B A D (3) E B C D A (3) B C E D A (3) B C D A E (3) A D C B E (3) A B C D E (3) E D C B A (2) E D C A B (2) E B C A D (2) D C E B A (2) D A C B E (2) B E C D A (2) B C D E A (2) B A D C E (2) A E B C D (2) A D C E B (2) E D A C B (1) E A D C B (1) E A C D B (1) E A C B D (1) E A B C D (1) D E C B A (1) D C B E A (1) D C B A E (1) D C A E B (1) D A E C B (1) D A B C E (1) C E D B A (1) C B E D A (1) C B D E A (1) B C A D E (1) B A C D E (1) A E D C B (1) A E B D C (1) A D B E C (1) A B E C D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 2 4 4 B -2 0 0 4 0 C -2 0 0 0 4 D -4 -4 0 0 6 E -4 0 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999837 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 2 4 4 B -2 0 0 4 0 C -2 0 0 0 4 D -4 -4 0 0 6 E -4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 E=34 B=14 D=10 C=3 so C is eliminated. Round 2 votes counts: A=39 E=35 B=16 D=10 so D is eliminated. Round 3 votes counts: A=44 E=38 B=18 so B is eliminated. Round 4 votes counts: A=52 E=48 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:206 B:201 C:201 D:199 E:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 2 2 4 4 B -2 0 0 4 0 C -2 0 0 0 4 D -4 -4 0 0 6 E -4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 2 4 4 B -2 0 0 4 0 C -2 0 0 0 4 D -4 -4 0 0 6 E -4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 2 4 4 B -2 0 0 4 0 C -2 0 0 0 4 D -4 -4 0 0 6 E -4 0 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9856: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (6) B E A D C (6) E B D C A (5) C D E A B (5) B A E D C (5) A B E C D (5) E B C D A (4) B E D A C (4) A B E D C (4) A B D E C (4) D C A B E (3) D A C B E (3) B E D C A (3) A E B C D (3) A D C B E (3) E C B D A (2) E B A C D (2) D B C E A (2) D B A C E (2) D A B C E (2) C E D B A (2) C A D E B (2) A D B C E (2) A C E D B (2) A C D B E (2) E C A B D (1) E B C A D (1) E A B C D (1) D C E A B (1) D C B E A (1) D C B A E (1) D C A E B (1) D B E A C (1) C E A D B (1) C E A B D (1) C D E B A (1) C D A E B (1) B E A C D (1) B D E A C (1) B D A E C (1) B A D E C (1) A C E B D (1) Total count = 100 A B C D E A 0 -4 12 -4 -4 B 4 0 20 12 14 C -12 -20 0 -24 -10 D 4 -12 24 0 -8 E 4 -14 10 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 12 -4 -4 B 4 0 20 12 14 C -12 -20 0 -24 -10 D 4 -12 24 0 -8 E 4 -14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998256 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 B=22 E=16 C=13 so C is eliminated. Round 2 votes counts: D=30 A=28 B=22 E=20 so E is eliminated. Round 3 votes counts: B=36 D=32 A=32 so D is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:225 D:204 E:204 A:200 C:167 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 12 -4 -4 B 4 0 20 12 14 C -12 -20 0 -24 -10 D 4 -12 24 0 -8 E 4 -14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998256 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 12 -4 -4 B 4 0 20 12 14 C -12 -20 0 -24 -10 D 4 -12 24 0 -8 E 4 -14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998256 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 12 -4 -4 B 4 0 20 12 14 C -12 -20 0 -24 -10 D 4 -12 24 0 -8 E 4 -14 10 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998256 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9857: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E B D (9) C A B D E (8) C A B E D (7) A C E D B (6) D E B A C (4) D E A C B (4) D B E C A (4) D B E A C (4) B D E C A (4) B D E A C (4) E D B A C (3) E D A B C (3) D E A B C (3) C B A E D (3) C A E D B (3) C A D B E (3) B C D E A (3) E A D C B (2) C D B A E (2) C B D A E (2) B C A E D (2) A E C D B (2) A C E B D (2) E B A C D (1) D C E A B (1) D C B A E (1) D B C A E (1) B E D C A (1) B E D A C (1) B E C A D (1) B E A D C (1) B E A C D (1) B D C E A (1) A E D C B (1) A E C B D (1) A C D E B (1) Total count = 100 A B C D E A 0 12 -12 8 8 B -12 0 -16 4 8 C 12 16 0 14 10 D -8 -4 -14 0 0 E -8 -8 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999959 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -12 8 8 B -12 0 -16 4 8 C 12 16 0 14 10 D -8 -4 -14 0 0 E -8 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 D=22 B=19 A=13 E=9 so E is eliminated. Round 2 votes counts: C=37 D=28 B=20 A=15 so A is eliminated. Round 3 votes counts: C=49 D=31 B=20 so B is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:208 B:192 D:187 E:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -12 8 8 B -12 0 -16 4 8 C 12 16 0 14 10 D -8 -4 -14 0 0 E -8 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -12 8 8 B -12 0 -16 4 8 C 12 16 0 14 10 D -8 -4 -14 0 0 E -8 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -12 8 8 B -12 0 -16 4 8 C 12 16 0 14 10 D -8 -4 -14 0 0 E -8 -8 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9858: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (8) C B E A D (6) B A E D C (6) A E B D C (6) D C E A B (4) D A E C B (3) D A E B C (3) C E B A D (3) C D B E A (3) C B D A E (3) B A E C D (3) E C B A D (2) E C A D B (2) E C A B D (2) E A D B C (2) E A B D C (2) D E A C B (2) D B A E C (2) D A B C E (2) C E D A B (2) B C E A D (2) B A D E C (2) A D E B C (2) A B D E C (2) E B A C D (1) E A D C B (1) E A C D B (1) E A B C D (1) D E C A B (1) D C B A E (1) D C A E B (1) D C A B E (1) D B C A E (1) D B A C E (1) D A C E B (1) D A C B E (1) D A B E C (1) C E D B A (1) C E A B D (1) C D B A E (1) C B E D A (1) B E C A D (1) B E A D C (1) B E A C D (1) B C D A E (1) B C A E D (1) B C A D E (1) B A C D E (1) A E D B C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 0 10 -2 B -8 0 -4 2 -6 C 0 4 0 -2 -4 D -10 -2 2 0 -2 E 2 6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 10 -2 B -8 0 -4 2 -6 C 0 4 0 -2 -4 D -10 -2 2 0 -2 E 2 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=29 D=25 B=20 E=14 A=12 so A is eliminated. Round 2 votes counts: C=29 D=27 B=23 E=21 so E is eliminated. Round 3 votes counts: C=36 B=33 D=31 so D is eliminated. Round 4 votes counts: C=52 B=48 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:208 E:207 C:199 D:194 B:192 Borda winner is A compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 10 -2 B -8 0 -4 2 -6 C 0 4 0 -2 -4 D -10 -2 2 0 -2 E 2 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 10 -2 B -8 0 -4 2 -6 C 0 4 0 -2 -4 D -10 -2 2 0 -2 E 2 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 10 -2 B -8 0 -4 2 -6 C 0 4 0 -2 -4 D -10 -2 2 0 -2 E 2 6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999654 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9859: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (16) D C A E B (13) B E D C A (8) A C D E B (8) E B D C A (7) B E D A C (5) A C B D E (5) E D B C A (3) E B D A C (3) D C E A B (3) C A D E B (3) A C D B E (3) E D A C B (2) D E C A B (2) C D A E B (2) C A D B E (2) B D E C A (2) B A C D E (2) E D C A B (1) E A C B D (1) D E C B A (1) D C E B A (1) B E A D C (1) B A E C D (1) B A C E D (1) A E C D B (1) A C E D B (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -2 4 -6 -12 B 2 0 0 8 -4 C -4 0 0 -4 -8 D 6 -8 4 0 -6 E 12 4 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 4 -6 -12 B 2 0 0 8 -4 C -4 0 0 -4 -8 D 6 -8 4 0 -6 E 12 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=36 D=20 A=20 E=17 C=7 so C is eliminated. Round 2 votes counts: B=36 A=25 D=22 E=17 so E is eliminated. Round 3 votes counts: B=46 D=28 A=26 so A is eliminated. Round 4 votes counts: B=54 D=46 so D is eliminated. IRV winner is B compare: Computing Borda winner. E:215 B:203 D:198 A:192 C:192 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 4 -6 -12 B 2 0 0 8 -4 C -4 0 0 -4 -8 D 6 -8 4 0 -6 E 12 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 4 -6 -12 B 2 0 0 8 -4 C -4 0 0 -4 -8 D 6 -8 4 0 -6 E 12 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 4 -6 -12 B 2 0 0 8 -4 C -4 0 0 -4 -8 D 6 -8 4 0 -6 E 12 4 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9860: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (13) A D E B C (10) C B E D A (9) A D B E C (7) A E C D B (6) A C E D B (5) E D A C B (4) B D C E A (4) A E D C B (3) A D E C B (3) E C A D B (2) E A C D B (2) D E A B C (2) D B E A C (2) C E B D A (2) C B E A D (2) C A E B D (2) B C E D A (2) B A D C E (2) A B D C E (2) E D C B A (1) E D B C A (1) E C D A B (1) D E B C A (1) D E B A C (1) D B E C A (1) D B A E C (1) D A E B C (1) C E B A D (1) C B A E D (1) C A B E D (1) B D E C A (1) B D A E C (1) B C D A E (1) A C E B D (1) A B C D E (1) Total count = 100 A B C D E A 0 6 8 2 -4 B -6 0 8 -8 2 C -8 -8 0 4 -2 D -2 8 -4 0 8 E 4 -2 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428585 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 A B C D E A 0 6 8 2 -4 B -6 0 8 -8 2 C -8 -8 0 4 -2 D -2 8 -4 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428575 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 B=24 C=18 E=11 D=9 so D is eliminated. Round 2 votes counts: A=39 B=28 C=18 E=15 so E is eliminated. Round 3 votes counts: A=47 B=31 C=22 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:206 D:205 B:198 E:198 C:193 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 6 8 2 -4 B -6 0 8 -8 2 C -8 -8 0 4 -2 D -2 8 -4 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428575 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GTS winners are ['A', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 8 2 -4 B -6 0 8 -8 2 C -8 -8 0 4 -2 D -2 8 -4 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428575 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 8 2 -4 B -6 0 8 -8 2 C -8 -8 0 4 -2 D -2 8 -4 0 8 E 4 -2 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.571429 B: 0.000000 C: 0.000000 D: 0.285714 E: 0.142857 Sum of squares = 0.428571428575 Cumulative probabilities = A: 0.571429 B: 0.571429 C: 0.571429 D: 0.857143 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9861: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A C D E (12) D E C B A (11) B A D E C (9) A B C E D (9) D E B C A (8) C E D A B (8) B A C E D (6) E D C A B (5) C A E D B (5) A C B E D (5) D E C A B (4) B D E C A (4) C A B E D (3) B D E A C (3) E D C B A (2) A C E D B (2) E D A C B (1) D E B A C (1) D E A B C (1) B C A E D (1) Total count = 100 A B C D E A 0 -14 -2 4 4 B 14 0 8 4 4 C 2 -8 0 2 2 D -4 -4 -2 0 6 E -4 -4 -2 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 -2 4 4 B 14 0 8 4 4 C 2 -8 0 2 2 D -4 -4 -2 0 6 E -4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=35 D=25 C=16 A=16 E=8 so E is eliminated. Round 2 votes counts: B=35 D=33 C=16 A=16 so C is eliminated. Round 3 votes counts: D=41 B=35 A=24 so A is eliminated. Round 4 votes counts: B=52 D=48 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:199 D:198 A:196 E:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -14 -2 4 4 B 14 0 8 4 4 C 2 -8 0 2 2 D -4 -4 -2 0 6 E -4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 -2 4 4 B 14 0 8 4 4 C 2 -8 0 2 2 D -4 -4 -2 0 6 E -4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 -2 4 4 B 14 0 8 4 4 C 2 -8 0 2 2 D -4 -4 -2 0 6 E -4 -4 -2 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998935 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9862: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (13) E D B C A (11) B D E A C (10) A C E D B (7) D B E A C (5) A C E B D (5) E C B D A (4) B D A E C (4) E C D B A (3) C E A B D (3) C A E D B (3) C A E B D (3) B D E C A (3) B D A C E (3) A C B D E (3) A B D C E (3) D B A E C (2) A B C D E (2) E D C B A (1) E D A C B (1) E C D A B (1) E C A D B (1) E C A B D (1) D B E C A (1) D A B E C (1) C A B E D (1) B D C E A (1) B C D A E (1) A E C D B (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 2 -2 -4 -18 B -2 0 -2 -6 -16 C 2 2 0 4 -2 D 4 6 -4 0 -18 E 18 16 2 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 2 -2 -4 -18 B -2 0 -2 -6 -16 C 2 2 0 4 -2 D 4 6 -4 0 -18 E 18 16 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A C E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 C=23 A=23 B=22 D=9 so D is eliminated. Round 2 votes counts: B=30 A=24 E=23 C=23 so E is eliminated. Round 3 votes counts: B=41 C=34 A=25 so A is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:227 C:203 D:194 A:189 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 2 -2 -4 -18 B -2 0 -2 -6 -16 C 2 2 0 4 -2 D 4 6 -4 0 -18 E 18 16 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -2 -4 -18 B -2 0 -2 -6 -16 C 2 2 0 4 -2 D 4 6 -4 0 -18 E 18 16 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -2 -4 -18 B -2 0 -2 -6 -16 C 2 2 0 4 -2 D 4 6 -4 0 -18 E 18 16 2 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999947087 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9863: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) C E B D A (7) A E B C D (7) D B C E A (5) A D E B C (5) E C B A D (4) D A C B E (4) E A C B D (3) D B C A E (3) D A B C E (3) C B E D A (3) B D C E A (3) A E D C B (3) A E B D C (3) A D B E C (3) E C A B D (2) D B A C E (2) B C E D A (2) B C D E A (2) A E D B C (2) A E C B D (2) A D B C E (2) A C E B D (2) A C D E B (2) E C B D A (1) E B C D A (1) E B A C D (1) E A B C D (1) D C A B E (1) D A B E C (1) C E D B A (1) C D B A E (1) C B D E A (1) B E C D A (1) B E A D C (1) B D E C A (1) B A E D C (1) A E C D B (1) A D E C B (1) A D C E B (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 2 6 -2 4 B -2 0 2 0 0 C -6 -2 0 -10 8 D 2 0 10 0 0 E -4 0 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.203681 C: 0.000000 D: 0.605642 E: 0.190677 Sum of squares = 0.444645828639 Cumulative probabilities = A: 0.000000 B: 0.203681 C: 0.203681 D: 0.809323 E: 1.000000 A B C D E A 0 2 6 -2 4 B -2 0 2 0 0 C -6 -2 0 -10 8 D 2 0 10 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.42857146671 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 D=27 E=13 C=13 B=11 so B is eliminated. Round 2 votes counts: A=37 D=31 C=17 E=15 so E is eliminated. Round 3 votes counts: A=43 D=31 C=26 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:206 A:205 B:200 C:195 E:194 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 6 -2 4 B -2 0 2 0 0 C -6 -2 0 -10 8 D 2 0 10 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.42857146671 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 6 -2 4 B -2 0 2 0 0 C -6 -2 0 -10 8 D 2 0 10 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.42857146671 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 6 -2 4 B -2 0 2 0 0 C -6 -2 0 -10 8 D 2 0 10 0 0 E -4 0 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.000000 D: 0.571429 E: 0.142857 Sum of squares = 0.42857146671 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 0.285714 D: 0.857143 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9864: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (7) A B E D C (7) C D B A E (6) A E B D C (6) D C B E A (5) C D B E A (5) A B D E C (5) A B D C E (5) E A C B D (4) C E D B A (4) B D A E C (4) B A D E C (4) A E B C D (4) E C D B A (3) E B D A C (3) D B E C A (3) B D A C E (3) E C A D B (2) E A B C D (2) D B C E A (2) D B A C E (2) C E D A B (2) C A E D B (2) A E C B D (2) E C A B D (1) E A B D C (1) D B C A E (1) C E A D B (1) C D A B E (1) C A D B E (1) B E A D C (1) A C E B D (1) Total count = 100 A B C D E A 0 -6 8 -2 8 B 6 0 6 6 10 C -8 -6 0 -4 -4 D 2 -6 4 0 8 E -8 -10 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 8 -2 8 B 6 0 6 6 10 C -8 -6 0 -4 -4 D 2 -6 4 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 C=29 E=16 D=13 B=12 so B is eliminated. Round 2 votes counts: A=34 C=29 D=20 E=17 so E is eliminated. Round 3 votes counts: A=42 C=35 D=23 so D is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. B:214 A:204 D:204 C:189 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 8 -2 8 B 6 0 6 6 10 C -8 -6 0 -4 -4 D 2 -6 4 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 8 -2 8 B 6 0 6 6 10 C -8 -6 0 -4 -4 D 2 -6 4 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 8 -2 8 B 6 0 6 6 10 C -8 -6 0 -4 -4 D 2 -6 4 0 8 E -8 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9865: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (6) C B A D E (6) E B A D C (4) B E A D C (4) B C E D A (4) B C A D E (4) E D C A B (3) D E A C B (3) C B E D A (3) C B D E A (3) B E C D A (3) B C E A D (3) B C A E D (3) B A E D C (3) A E D B C (3) E D A C B (2) E B D A C (2) E A D B C (2) D E C A B (2) D A E C B (2) D A C E B (2) C E D B A (2) C D A B E (2) C A D B E (2) B A E C D (2) B A C E D (2) A D E C B (2) A D E B C (2) A D B E C (2) A C B D E (2) A B D E C (2) E D B C A (1) E D A B C (1) E B D C A (1) D C A E B (1) C D E A B (1) C B D A E (1) C A D E B (1) B E D C A (1) B E C A D (1) B E A C D (1) B A C D E (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -8 8 6 B 10 0 6 14 14 C 8 -6 0 8 2 D -8 -14 -8 0 -2 E -6 -14 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999989 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -8 8 6 B 10 0 6 14 14 C 8 -6 0 8 2 D -8 -14 -8 0 -2 E -6 -14 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 C=27 E=16 A=15 D=10 so D is eliminated. Round 2 votes counts: B=32 C=28 E=21 A=19 so A is eliminated. Round 3 votes counts: B=37 C=33 E=30 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:222 C:206 A:198 E:190 D:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -8 8 6 B 10 0 6 14 14 C 8 -6 0 8 2 D -8 -14 -8 0 -2 E -6 -14 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -8 8 6 B 10 0 6 14 14 C 8 -6 0 8 2 D -8 -14 -8 0 -2 E -6 -14 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -8 8 6 B 10 0 6 14 14 C 8 -6 0 8 2 D -8 -14 -8 0 -2 E -6 -14 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9866: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B C E (14) D C A B E (12) C E B A D (10) D A B E C (9) C E D B A (9) E B A C D (7) D C E A B (5) C D E B A (5) B A E C D (5) E C B A D (4) C D E A B (4) B A E D C (4) A B E D C (4) A B D E C (4) E B C A D (1) D C A E B (1) D A C B E (1) C E B D A (1) Total count = 100 A B C D E A 0 8 -4 -22 8 B -8 0 -4 -20 6 C 4 4 0 -8 24 D 22 20 8 0 10 E -8 -6 -24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -4 -22 8 B -8 0 -4 -20 6 C 4 4 0 -8 24 D 22 20 8 0 10 E -8 -6 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=42 C=29 E=12 B=9 A=8 so A is eliminated. Round 2 votes counts: D=42 C=29 B=17 E=12 so E is eliminated. Round 3 votes counts: D=42 C=33 B=25 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:230 C:212 A:195 B:187 E:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -4 -22 8 B -8 0 -4 -20 6 C 4 4 0 -8 24 D 22 20 8 0 10 E -8 -6 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -4 -22 8 B -8 0 -4 -20 6 C 4 4 0 -8 24 D 22 20 8 0 10 E -8 -6 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -4 -22 8 B -8 0 -4 -20 6 C 4 4 0 -8 24 D 22 20 8 0 10 E -8 -6 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999674 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9867: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D E B C (9) E D B A C (7) C A D E B (7) C B E D A (5) B E D C A (5) D E A B C (4) C B A E D (4) C A B D E (4) B E D A C (4) B C E D A (4) A D E C B (4) E D A B C (3) C D E B A (3) B E C D A (3) B E A D C (3) B A C E D (3) E D B C A (2) E B D C A (2) D E C A B (2) D E A C B (2) B C A E D (2) B A E D C (2) A C D E B (2) A C B D E (2) A B C E D (2) E B D A C (1) D E C B A (1) D E B A C (1) D A E C B (1) D A E B C (1) C D E A B (1) C B D E A (1) B C E A D (1) A E D B C (1) A D C E B (1) Total count = 100 A B C D E A 0 -8 6 -6 -10 B 8 0 20 -4 -10 C -6 -20 0 -12 -16 D 6 4 12 0 -8 E 10 10 16 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999893 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 6 -6 -10 B 8 0 20 -4 -10 C -6 -20 0 -12 -16 D 6 4 12 0 -8 E 10 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 C=25 A=21 E=15 D=12 so D is eliminated. Round 2 votes counts: B=27 E=25 C=25 A=23 so A is eliminated. Round 3 votes counts: E=41 C=30 B=29 so B is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:222 B:207 D:207 A:191 C:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 6 -6 -10 B 8 0 20 -4 -10 C -6 -20 0 -12 -16 D 6 4 12 0 -8 E 10 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 6 -6 -10 B 8 0 20 -4 -10 C -6 -20 0 -12 -16 D 6 4 12 0 -8 E 10 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 6 -6 -10 B 8 0 20 -4 -10 C -6 -20 0 -12 -16 D 6 4 12 0 -8 E 10 10 16 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9868: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (14) C E B A D (13) D A C B E (7) D A B E C (7) D A B C E (5) E B C A D (4) B E A C D (4) A B D C E (4) C E D A B (3) A D B C E (3) E D C B A (2) E D C A B (2) E C B D A (2) E B D A C (2) D C A E B (2) C B A E D (2) B E C A D (2) B A E D C (2) A D C B E (2) A C D B E (2) A B D E C (2) E D B C A (1) E D B A C (1) E C D B A (1) D C A B E (1) D B A E C (1) D A E B C (1) C E D B A (1) C D E A B (1) C D A E B (1) C D A B E (1) B D A E C (1) B A E C D (1) B A D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -10 -6 14 -6 B 10 0 -14 10 -2 C 6 14 0 6 -2 D -14 -10 -6 0 -14 E 6 2 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999771 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 -6 14 -6 B 10 0 -14 10 -2 C 6 14 0 6 -2 D -14 -10 -6 0 -14 E 6 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 D=24 C=22 A=14 B=11 so B is eliminated. Round 2 votes counts: E=35 D=25 C=22 A=18 so A is eliminated. Round 3 votes counts: E=38 D=37 C=25 so C is eliminated. Round 4 votes counts: E=57 D=43 so D is eliminated. IRV winner is E compare: Computing Borda winner. C:212 E:212 B:202 A:196 D:178 Borda winner is C compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 -6 14 -6 B 10 0 -14 10 -2 C 6 14 0 6 -2 D -14 -10 -6 0 -14 E 6 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 14 -6 B 10 0 -14 10 -2 C 6 14 0 6 -2 D -14 -10 -6 0 -14 E 6 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 14 -6 B 10 0 -14 10 -2 C 6 14 0 6 -2 D -14 -10 -6 0 -14 E 6 2 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999990895 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9869: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D C B (13) E A D B C (10) C B E D A (9) A D E B C (8) B C D A E (7) C E B D A (5) D A B E C (4) B D A C E (4) A E D B C (4) E D A B C (3) E C A D B (3) C E A D B (3) E C D A B (2) E A C D B (2) D A E B C (2) C E B A D (2) C B E A D (2) C B D A E (2) E C B D A (1) C B D E A (1) C B A D E (1) B E C D A (1) B D E A C (1) B D C A E (1) B D A E C (1) B C E D A (1) B C D E A (1) A E D C B (1) A D E C B (1) A D C E B (1) A D B E C (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 20 16 8 -20 B -20 0 0 -20 -24 C -16 0 0 -12 -16 D -8 20 12 0 -26 E 20 24 16 26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999714 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 20 16 8 -20 B -20 0 0 -20 -24 C -16 0 0 -12 -16 D -8 20 12 0 -26 E 20 24 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=25 A=18 B=17 D=6 so D is eliminated. Round 2 votes counts: E=34 C=25 A=24 B=17 so B is eliminated. Round 3 votes counts: E=36 C=35 A=29 so A is eliminated. Round 4 votes counts: E=58 C=42 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:243 A:212 D:199 C:178 B:168 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 20 16 8 -20 B -20 0 0 -20 -24 C -16 0 0 -12 -16 D -8 20 12 0 -26 E 20 24 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 16 8 -20 B -20 0 0 -20 -24 C -16 0 0 -12 -16 D -8 20 12 0 -26 E 20 24 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 16 8 -20 B -20 0 0 -20 -24 C -16 0 0 -12 -16 D -8 20 12 0 -26 E 20 24 16 26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999782 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9870: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (16) A B D E C (14) D B A E C (12) D B A C E (8) C E D A B (6) C E A B D (6) B A D E C (5) E C A B D (4) E C D B A (3) D C E B A (3) C E A D B (3) D B E C A (2) B D A E C (2) A B E D C (2) A B E C D (2) A B D C E (2) A B C E D (2) E C B D A (1) E C B A D (1) E B A C D (1) E A C B D (1) D E C B A (1) D E B C A (1) C A E B D (1) A D B C E (1) Total count = 100 A B C D E A 0 -12 4 -10 2 B 12 0 8 -12 4 C -4 -8 0 -6 -4 D 10 12 6 0 2 E -2 -4 4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 4 -10 2 B 12 0 8 -12 4 C -4 -8 0 -6 -4 D 10 12 6 0 2 E -2 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=32 D=27 A=23 E=11 B=7 so B is eliminated. Round 2 votes counts: C=32 D=29 A=28 E=11 so E is eliminated. Round 3 votes counts: C=41 A=30 D=29 so D is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:215 B:206 E:198 A:192 C:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 4 -10 2 B 12 0 8 -12 4 C -4 -8 0 -6 -4 D 10 12 6 0 2 E -2 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 4 -10 2 B 12 0 8 -12 4 C -4 -8 0 -6 -4 D 10 12 6 0 2 E -2 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 4 -10 2 B 12 0 8 -12 4 C -4 -8 0 -6 -4 D 10 12 6 0 2 E -2 -4 4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994643 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9871: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (11) C B A E D (10) B C A D E (9) D A E B C (6) E D A C B (5) B D A E C (5) B C D A E (5) C B E A D (4) E C D A B (3) E A D C B (3) C E D A B (3) C E A D B (3) A E D B C (3) D E B A C (2) D B A E C (2) C E D B A (2) C A B E D (2) B A D C E (2) A D E B C (2) A B D E C (2) E D C A B (1) E D A B C (1) E C A D B (1) D B E A C (1) C E B A D (1) C E A B D (1) C D E B A (1) C B E D A (1) C B D E A (1) C B A D E (1) C A E B D (1) B D C A E (1) B D A C E (1) B C A E D (1) A E B D C (1) A B E C D (1) Total count = 100 A B C D E A 0 0 -4 -4 10 B 0 0 12 0 -2 C 4 -12 0 2 0 D 4 0 -2 0 4 E -10 2 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.323869 C: 0.000000 D: 0.676131 E: 0.000000 Sum of squares = 0.562044471383 Cumulative probabilities = A: 0.000000 B: 0.323869 C: 0.323869 D: 1.000000 E: 1.000000 A B C D E A 0 0 -4 -4 10 B 0 0 12 0 -2 C 4 -12 0 2 0 D 4 0 -2 0 4 E -10 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=24 D=22 E=14 A=9 so A is eliminated. Round 2 votes counts: C=31 B=27 D=24 E=18 so E is eliminated. Round 3 votes counts: D=37 C=35 B=28 so B is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:205 D:203 A:201 C:197 E:194 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 0 -4 -4 10 B 0 0 12 0 -2 C 4 -12 0 2 0 D 4 0 -2 0 4 E -10 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 -4 -4 10 B 0 0 12 0 -2 C 4 -12 0 2 0 D 4 0 -2 0 4 E -10 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 -4 -4 10 B 0 0 12 0 -2 C 4 -12 0 2 0 D 4 0 -2 0 4 E -10 2 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9872: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D E A C (5) B A D E C (5) A D B C E (5) E C A B D (4) D B A C E (4) D A B C E (4) C E A D B (4) B E A C D (4) B D A E C (4) A D C B E (4) A B E C D (4) E C D B A (3) C E A B D (3) B E D C A (3) B D E C A (3) E B C A D (2) D B E C A (2) D A C B E (2) C E D A B (2) C A E D B (2) B E A D C (2) A C E B D (2) A C D B E (2) A B D E C (2) E D B C A (1) E C D A B (1) E C B D A (1) E C B A D (1) E B D C A (1) E B A C D (1) D E C B A (1) D E B C A (1) D C E A B (1) D C A E B (1) D B E A C (1) D B C E A (1) D B A E C (1) C E D B A (1) C D E A B (1) C A E B D (1) B E C A D (1) B A E D C (1) A C B E D (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 0 16 10 -2 B 0 0 22 10 30 C -16 -22 0 -12 -10 D -10 -10 12 0 6 E 2 -30 10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.548361 B: 0.451639 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.504677538601 Cumulative probabilities = A: 0.548361 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 16 10 -2 B 0 0 22 10 30 C -16 -22 0 -12 -10 D -10 -10 12 0 6 E 2 -30 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=24 D=19 E=15 C=14 so C is eliminated. Round 2 votes counts: B=28 A=27 E=25 D=20 so D is eliminated. Round 3 votes counts: B=37 A=34 E=29 so E is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:231 A:212 D:199 E:188 C:170 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 16 10 -2 B 0 0 22 10 30 C -16 -22 0 -12 -10 D -10 -10 12 0 6 E 2 -30 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 16 10 -2 B 0 0 22 10 30 C -16 -22 0 -12 -10 D -10 -10 12 0 6 E 2 -30 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 16 10 -2 B 0 0 22 10 30 C -16 -22 0 -12 -10 D -10 -10 12 0 6 E 2 -30 10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9873: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (8) A B E C D (7) C D E A B (6) E A B D C (5) B A E D C (5) A B E D C (5) E D C A B (4) D B C E A (4) C D B A E (4) D C E B A (3) C D E B A (3) B A C D E (3) A E B C D (3) E D B A C (2) E A D B C (2) C B D A E (2) C A E D B (2) C A B D E (2) B E D A C (2) B E A D C (2) B D C A E (2) B D A C E (2) E D C B A (1) E D B C A (1) E D A B C (1) E C D A B (1) E B D A C (1) E A C D B (1) D E C B A (1) D E B C A (1) D C B A E (1) C E D A B (1) C E A D B (1) C D B E A (1) C A D B E (1) B D E C A (1) B D E A C (1) B D C E A (1) B C A D E (1) B A D E C (1) B A D C E (1) A E B D C (1) A C E B D (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -6 -10 -10 B 10 0 12 0 16 C 6 -12 0 -18 2 D 10 0 18 0 0 E 10 -16 -2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.724472 C: 0.000000 D: 0.275528 E: 0.000000 Sum of squares = 0.600775049969 Cumulative probabilities = A: 0.000000 B: 0.724472 C: 0.724472 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -6 -10 -10 B 10 0 12 0 16 C 6 -12 0 -18 2 D 10 0 18 0 0 E 10 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=23 B=22 E=19 D=18 A=18 so D is eliminated. Round 2 votes counts: C=35 B=26 E=21 A=18 so A is eliminated. Round 3 votes counts: B=39 C=36 E=25 so E is eliminated. Round 4 votes counts: B=56 C=44 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:219 D:214 E:196 C:189 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -6 -10 -10 B 10 0 12 0 16 C 6 -12 0 -18 2 D 10 0 18 0 0 E 10 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -6 -10 -10 B 10 0 12 0 16 C 6 -12 0 -18 2 D 10 0 18 0 0 E 10 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -6 -10 -10 B 10 0 12 0 16 C 6 -12 0 -18 2 D 10 0 18 0 0 E 10 -16 -2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9874: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) A E C B D (7) E A C B D (6) B A D C E (6) E C A D B (5) D B C E A (5) C E A D B (5) D B C A E (4) C A E D B (4) B D E A C (4) B D C A E (3) A C E D B (3) A B D C E (3) A B C D E (3) E D B C A (2) E C D A B (2) D C B A E (2) D B A C E (2) B D E C A (2) A E B C D (2) A B E C D (2) E D C B A (1) E C D B A (1) E C A B D (1) E B A D C (1) E A C D B (1) D E C B A (1) C D E B A (1) C D E A B (1) C D A E B (1) C A D E B (1) B E A D C (1) B D C E A (1) B D A E C (1) B A E D C (1) A E C D B (1) A C D E B (1) A C D B E (1) A C B E D (1) A B E D C (1) Total count = 100 A B C D E A 0 4 14 14 18 B -4 0 6 10 4 C -14 -6 0 0 14 D -14 -10 0 0 4 E -18 -4 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 14 14 18 B -4 0 6 10 4 C -14 -6 0 0 14 D -14 -10 0 0 4 E -18 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=25 E=20 D=14 C=13 so C is eliminated. Round 2 votes counts: A=30 B=28 E=25 D=17 so D is eliminated. Round 3 votes counts: B=41 A=31 E=28 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:225 B:208 C:197 D:190 E:180 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 14 14 18 B -4 0 6 10 4 C -14 -6 0 0 14 D -14 -10 0 0 4 E -18 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 14 14 18 B -4 0 6 10 4 C -14 -6 0 0 14 D -14 -10 0 0 4 E -18 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 14 14 18 B -4 0 6 10 4 C -14 -6 0 0 14 D -14 -10 0 0 4 E -18 -4 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999990071 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9875: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B D C (10) C D B E A (9) D B C A E (8) E A C B D (6) D B A C E (6) E C A B D (5) C D B A E (5) E A B C D (4) D C B A E (4) C D E B A (4) B D A E C (4) A E B D C (4) C E D B A (3) C E D A B (3) C A E D B (3) B D A C E (3) A B E D C (3) C E A D B (2) E C D B A (1) E C B D A (1) E C A D B (1) D C E B A (1) D B C E A (1) C D A E B (1) C A D B E (1) B E D A C (1) B A E D C (1) B A D E C (1) A E C B D (1) A E B C D (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 -6 -6 -10 -4 B 6 0 -2 -6 -2 C 6 2 0 2 10 D 10 6 -2 0 0 E 4 2 -10 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999713 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -10 -4 B 6 0 -2 -6 -2 C 6 2 0 2 10 D 10 6 -2 0 0 E 4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 E=28 D=20 A=11 B=10 so B is eliminated. Round 2 votes counts: C=31 E=29 D=27 A=13 so A is eliminated. Round 3 votes counts: E=39 C=31 D=30 so D is eliminated. Round 4 votes counts: C=55 E=45 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:210 D:207 B:198 E:198 A:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 -10 -4 B 6 0 -2 -6 -2 C 6 2 0 2 10 D 10 6 -2 0 0 E 4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -10 -4 B 6 0 -2 -6 -2 C 6 2 0 2 10 D 10 6 -2 0 0 E 4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -10 -4 B 6 0 -2 -6 -2 C 6 2 0 2 10 D 10 6 -2 0 0 E 4 2 -10 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999539 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9876: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (10) C D B A E (9) E B D A C (8) C A D B E (8) E A B D C (7) E A C B D (6) E B A D C (5) E A B C D (5) D B C A E (5) B D E C A (5) D B C E A (4) B E D C A (4) B D C E A (4) A C D B E (4) D C B A E (3) C D A B E (3) A E C D B (3) E B D C A (2) A E C B D (2) E A C D B (1) D C B E A (1) A C D E B (1) Total count = 100 A B C D E A 0 0 4 4 -4 B 0 0 -2 -4 0 C -4 2 0 4 4 D -4 4 -4 0 -6 E 4 0 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333333 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 A B C D E A 0 0 4 4 -4 B 0 0 -2 -4 0 C -4 2 0 4 4 D -4 4 -4 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333095 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 C=20 A=20 D=13 B=13 so D is eliminated. Round 2 votes counts: E=34 C=24 B=22 A=20 so A is eliminated. Round 3 votes counts: E=39 C=39 B=22 so B is eliminated. Round 4 votes counts: C=52 E=48 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:203 E:203 A:202 B:197 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C E , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 4 -4 B 0 0 -2 -4 0 C -4 2 0 4 4 D -4 4 -4 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333095 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 4 -4 B 0 0 -2 -4 0 C -4 2 0 4 4 D -4 4 -4 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333095 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 4 -4 B 0 0 -2 -4 0 C -4 2 0 4 4 D -4 4 -4 0 -6 E 4 0 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.333333 B: 0.000000 C: 0.333333 D: 0.000000 E: 0.333333 Sum of squares = 0.333333333095 Cumulative probabilities = A: 0.333333 B: 0.333333 C: 0.666667 D: 0.666667 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9877: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A E D (6) A C B E D (6) D E A B C (5) D B E C A (5) C A B E D (5) A E C B D (5) A D E C B (5) E D A B C (4) D E B C A (4) D B C E A (4) A C E B D (4) C B A D E (3) B C E D A (3) B C E A D (3) B C D E A (3) A E D C B (3) A C B D E (3) E A D B C (2) D C B A E (2) D A E C B (2) D A E B C (2) C A B D E (2) B D C E A (2) A D C E B (2) A D C B E (2) E D B C A (1) E D B A C (1) E B D C A (1) E B C D A (1) E B C A D (1) E A C B D (1) D E B A C (1) C B E A D (1) C B D A E (1) B D E C A (1) A E C D B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 12 2 14 12 B -12 0 -12 4 4 C -2 12 0 2 8 D -14 -4 -2 0 0 E -12 -4 -8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999802 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 14 12 B -12 0 -12 4 4 C -2 12 0 2 8 D -14 -4 -2 0 0 E -12 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 D=25 C=18 E=12 B=12 so E is eliminated. Round 2 votes counts: A=36 D=31 C=18 B=15 so B is eliminated. Round 3 votes counts: A=36 D=35 C=29 so C is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:220 C:210 B:192 D:190 E:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 14 12 B -12 0 -12 4 4 C -2 12 0 2 8 D -14 -4 -2 0 0 E -12 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 14 12 B -12 0 -12 4 4 C -2 12 0 2 8 D -14 -4 -2 0 0 E -12 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 14 12 B -12 0 -12 4 4 C -2 12 0 2 8 D -14 -4 -2 0 0 E -12 -4 -8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999694 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9878: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) B A D C E (8) A D B E C (7) C E D A B (5) B C E A D (5) E C D B A (4) E C B A D (4) C B D A E (4) A D B C E (4) D E A C B (3) D C E A B (3) D A B C E (3) C D E A B (3) A B D E C (3) E D C A B (2) E B C A D (2) E A B D C (2) C E B D A (2) C B E D A (2) B E C A D (2) B E A C D (2) B A C D E (2) A B D C E (2) E D A C B (1) E C B D A (1) E B A D C (1) E B A C D (1) E A D B C (1) D C B A E (1) D C A E B (1) D C A B E (1) D A E C B (1) D A C E B (1) D A C B E (1) C D E B A (1) C D B A E (1) C D A B E (1) B C D A E (1) B C A D E (1) B A E D C (1) B A E C D (1) Total count = 100 A B C D E A 0 6 -10 -2 -10 B -6 0 -2 -6 6 C 10 2 0 6 6 D 2 6 -6 0 6 E 10 -6 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -10 -2 -10 B -6 0 -2 -6 6 C 10 2 0 6 6 D 2 6 -6 0 6 E 10 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 B=23 C=19 A=16 D=15 so D is eliminated. Round 2 votes counts: E=30 C=25 B=23 A=22 so A is eliminated. Round 3 votes counts: B=42 E=31 C=27 so C is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:212 D:204 B:196 E:196 A:192 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -10 -2 -10 B -6 0 -2 -6 6 C 10 2 0 6 6 D 2 6 -6 0 6 E 10 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -10 -2 -10 B -6 0 -2 -6 6 C 10 2 0 6 6 D 2 6 -6 0 6 E 10 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -10 -2 -10 B -6 0 -2 -6 6 C 10 2 0 6 6 D 2 6 -6 0 6 E 10 -6 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999732 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9879: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D B C (11) C B D E A (9) C A E D B (9) A E D C B (8) B C D E A (7) C D E A B (5) B D E A C (5) A E B D C (4) A C E D B (4) C B A D E (3) A E C D B (3) E D A C B (2) D B E A C (2) C D B E A (2) C B D A E (2) C A E B D (2) B D E C A (2) B D C E A (2) E D A B C (1) E A D C B (1) D E B C A (1) D E B A C (1) D E A C B (1) D E A B C (1) C D E B A (1) C A D E B (1) C A B E D (1) B E D A C (1) B E A D C (1) B C D A E (1) B A E D C (1) B A D E C (1) B A C E D (1) A C E B D (1) A C B E D (1) A B E C D (1) Total count = 100 A B C D E A 0 14 4 8 10 B -14 0 -12 -8 -14 C -4 12 0 8 4 D -8 8 -8 0 -6 E -10 14 -4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 8 10 B -14 0 -12 -8 -14 C -4 12 0 8 4 D -8 8 -8 0 -6 E -10 14 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=35 A=33 B=22 D=6 E=4 so E is eliminated. Round 2 votes counts: C=35 A=34 B=22 D=9 so D is eliminated. Round 3 votes counts: A=39 C=35 B=26 so B is eliminated. Round 4 votes counts: A=52 C=48 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:210 E:203 D:193 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 8 10 B -14 0 -12 -8 -14 C -4 12 0 8 4 D -8 8 -8 0 -6 E -10 14 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 8 10 B -14 0 -12 -8 -14 C -4 12 0 8 4 D -8 8 -8 0 -6 E -10 14 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 8 10 B -14 0 -12 -8 -14 C -4 12 0 8 4 D -8 8 -8 0 -6 E -10 14 -4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9880: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (8) D E B C A (7) D E B A C (7) E D C B A (6) E D B C A (5) D E A C B (5) B A C E D (5) C A E B D (4) B C A E D (4) A C D E B (4) D B E A C (3) C A E D B (3) B E D C A (3) B A D C E (3) A C D B E (3) A C B D E (3) D E C B A (2) D E A B C (2) D B A E C (2) C E B A D (2) B C E A D (2) E C D B A (1) E C B D A (1) D E C A B (1) D B E C A (1) D A E C B (1) D A C E B (1) D A B C E (1) C E A D B (1) C E A B D (1) C B E A D (1) C A B E D (1) B E D A C (1) B E C D A (1) A D C B E (1) A C E D B (1) A C E B D (1) A B D C E (1) Total count = 100 A B C D E A 0 -14 6 -2 -6 B 14 0 -4 -16 -12 C -6 4 0 -6 2 D 2 16 6 0 -4 E 6 12 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888804 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 A B C D E A 0 -14 6 -2 -6 B 14 0 -4 -16 -12 C -6 4 0 -6 2 D 2 16 6 0 -4 E 6 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 A=22 B=19 E=13 C=13 so E is eliminated. Round 2 votes counts: D=44 A=22 B=19 C=15 so C is eliminated. Round 3 votes counts: D=45 A=32 B=23 so B is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:210 E:210 C:197 A:192 B:191 Borda winner is D compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -14 6 -2 -6 B 14 0 -4 -16 -12 C -6 4 0 -6 2 D 2 16 6 0 -4 E 6 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 6 -2 -6 B 14 0 -4 -16 -12 C -6 4 0 -6 2 D 2 16 6 0 -4 E 6 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 6 -2 -6 B 14 0 -4 -16 -12 C -6 4 0 -6 2 D 2 16 6 0 -4 E 6 12 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.166667 E: 0.500000 Sum of squares = 0.388888888586 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.333333 D: 0.500000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9881: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (8) E A C B D (5) D B A C E (5) D A B E C (5) E C B A D (4) D C E A B (4) D C B E A (4) C B E D A (4) E C D A B (3) E A B C D (3) D B C A E (3) D A E C B (3) C E D B A (3) C D E B A (3) B C A E D (3) A E B C D (3) E B C A D (2) D C E B A (2) D A C E B (2) D A B C E (2) C E B D A (2) C B E A D (2) B C E A D (2) A D B E C (2) A B E D C (2) E D A C B (1) E C A D B (1) E C A B D (1) E B A C D (1) D C B A E (1) D C A E B (1) D A E B C (1) C E D A B (1) B E C A D (1) B E A C D (1) B D C A E (1) B C D E A (1) B C A D E (1) B A D E C (1) B A C E D (1) A E C B D (1) A E B D C (1) A D E B C (1) A B D E C (1) Total count = 100 A B C D E A 0 -12 -16 -4 -18 B 12 0 -12 4 -14 C 16 12 0 14 12 D 4 -4 -14 0 -12 E 18 14 -12 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -16 -4 -18 B 12 0 -12 4 -14 C 16 12 0 14 12 D 4 -4 -14 0 -12 E 18 14 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=23 E=21 B=12 A=11 so A is eliminated. Round 2 votes counts: D=36 E=26 C=23 B=15 so B is eliminated. Round 3 votes counts: D=39 C=31 E=30 so E is eliminated. Round 4 votes counts: C=57 D=43 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:227 E:216 B:195 D:187 A:175 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -12 -16 -4 -18 B 12 0 -12 4 -14 C 16 12 0 14 12 D 4 -4 -14 0 -12 E 18 14 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -16 -4 -18 B 12 0 -12 4 -14 C 16 12 0 14 12 D 4 -4 -14 0 -12 E 18 14 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -16 -4 -18 B 12 0 -12 4 -14 C 16 12 0 14 12 D 4 -4 -14 0 -12 E 18 14 -12 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9882: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (10) D B A C E (9) C E A B D (9) B D E C A (8) D B A E C (7) B D E A C (6) B D C E A (6) E A C B D (4) C A E B D (4) D B E A C (3) C A E D B (3) B C D E A (3) E C A B D (2) E B A C D (2) D A C E B (2) B C E D A (2) A E C B D (2) A C E D B (2) E B C A D (1) D B E C A (1) D B C E A (1) D B C A E (1) D A E C B (1) D A E B C (1) D A C B E (1) D A B E C (1) C D A E B (1) C B E A D (1) C A D E B (1) B E D A C (1) B E C A D (1) B C E A D (1) A E D C B (1) A C D E B (1) Total count = 100 A B C D E A 0 -8 8 -10 -4 B 8 0 10 6 6 C -8 -10 0 0 -4 D 10 -6 0 0 8 E 4 -6 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999873 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 8 -10 -4 B 8 0 10 6 6 C -8 -10 0 0 -4 D 10 -6 0 0 8 E 4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=28 B=28 C=19 A=16 E=9 so E is eliminated. Round 2 votes counts: B=31 D=28 C=21 A=20 so A is eliminated. Round 3 votes counts: C=40 B=31 D=29 so D is eliminated. Round 4 votes counts: B=55 C=45 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 D:206 E:197 A:193 C:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 8 -10 -4 B 8 0 10 6 6 C -8 -10 0 0 -4 D 10 -6 0 0 8 E 4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 8 -10 -4 B 8 0 10 6 6 C -8 -10 0 0 -4 D 10 -6 0 0 8 E 4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 8 -10 -4 B 8 0 10 6 6 C -8 -10 0 0 -4 D 10 -6 0 0 8 E 4 -6 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9883: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) B E A D C (10) A D C B E (9) C D A E B (8) A D C E B (8) E B C D A (7) E B A D C (7) B E C D A (7) E C B D A (4) D A C B E (4) E B C A D (2) D C A B E (2) C E B D A (2) C A D E B (2) B A D E C (2) A D B E C (2) A D B C E (2) A B E D C (2) E C A D B (1) E A B D C (1) C D B A E (1) B E D A C (1) B D A E C (1) B D A C E (1) B C D A E (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 8 4 0 16 B -8 0 -6 -4 12 C -4 6 0 -8 2 D 0 4 8 0 8 E -16 -12 -2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.307492 B: 0.000000 C: 0.000000 D: 0.692508 E: 0.000000 Sum of squares = 0.57411883093 Cumulative probabilities = A: 0.307492 B: 0.307492 C: 0.307492 D: 1.000000 E: 1.000000 A B C D E A 0 8 4 0 16 B -8 0 -6 -4 12 C -4 6 0 -8 2 D 0 4 8 0 8 E -16 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=25 C=24 B=23 E=22 D=6 so D is eliminated. Round 2 votes counts: A=29 C=26 B=23 E=22 so E is eliminated. Round 3 votes counts: B=39 C=31 A=30 so A is eliminated. Round 4 votes counts: C=53 B=47 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:214 D:210 C:198 B:197 E:181 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 8 4 0 16 B -8 0 -6 -4 12 C -4 6 0 -8 2 D 0 4 8 0 8 E -16 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 4 0 16 B -8 0 -6 -4 12 C -4 6 0 -8 2 D 0 4 8 0 8 E -16 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 4 0 16 B -8 0 -6 -4 12 C -4 6 0 -8 2 D 0 4 8 0 8 E -16 -12 -2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9884: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C D E A (14) A E D C B (14) B E C D A (6) B C D A E (6) E A D C B (5) E A B D C (5) E D C B A (4) C D A B E (3) B A E C D (3) B A C D E (3) A E B C D (3) E B D C A (2) E B C D A (2) E B A D C (2) E B A C D (2) D C B E A (2) D C A B E (2) C D B A E (2) B C A D E (2) A E C D B (2) A D C E B (2) A B C D E (2) D E C A B (1) D C E B A (1) D C E A B (1) D C B A E (1) D C A E B (1) C D B E A (1) B E A C D (1) B D E C A (1) B D C E A (1) A D C B E (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 -12 -6 -2 -2 B 12 0 12 12 6 C 6 -12 0 8 -8 D 2 -12 -8 0 -4 E 2 -6 8 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999807 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 -2 -2 B 12 0 12 12 6 C 6 -12 0 8 -8 D 2 -12 -8 0 -4 E 2 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 A=26 E=22 D=9 C=6 so C is eliminated. Round 2 votes counts: B=37 A=26 E=22 D=15 so D is eliminated. Round 3 votes counts: B=43 A=32 E=25 so E is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:221 E:204 C:197 A:189 D:189 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 -2 -2 B 12 0 12 12 6 C 6 -12 0 8 -8 D 2 -12 -8 0 -4 E 2 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 -2 -2 B 12 0 12 12 6 C 6 -12 0 8 -8 D 2 -12 -8 0 -4 E 2 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 -2 -2 B 12 0 12 12 6 C 6 -12 0 8 -8 D 2 -12 -8 0 -4 E 2 -6 8 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9885: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (9) E B A C D (8) B E A C D (7) B A E D C (7) D C A E B (6) D C A B E (6) C D A E B (6) B E A D C (6) D C B A E (4) C D E A B (4) A B E D C (4) A B D E C (4) D A B C E (3) E A B C D (2) D B C A E (2) C D B E A (2) B D C E A (2) A E C D B (2) E C A B D (1) E B C A D (1) E A C B D (1) D C B E A (1) D B A C E (1) D A C E B (1) D A C B E (1) C E D B A (1) C E D A B (1) C E B D A (1) C E A D B (1) C D A B E (1) B D E C A (1) A E B D C (1) A E B C D (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 0 -6 2 B 6 0 2 -6 6 C 0 -2 0 -2 6 D 6 6 2 0 10 E -2 -6 -6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999932 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 0 -6 2 B 6 0 2 -6 6 C 0 -2 0 -2 6 D 6 6 2 0 10 E -2 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=26 D=25 B=23 E=13 A=13 so E is eliminated. Round 2 votes counts: B=32 C=27 D=25 A=16 so A is eliminated. Round 3 votes counts: B=44 C=30 D=26 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:212 B:204 C:201 A:195 E:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 0 -6 2 B 6 0 2 -6 6 C 0 -2 0 -2 6 D 6 6 2 0 10 E -2 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 0 -6 2 B 6 0 2 -6 6 C 0 -2 0 -2 6 D 6 6 2 0 10 E -2 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 0 -6 2 B 6 0 2 -6 6 C 0 -2 0 -2 6 D 6 6 2 0 10 E -2 -6 -6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999997749 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9886: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (9) A D C E B (8) A D C B E (8) C E B A D (7) B E C D A (7) B E D A C (6) E C B D A (5) E B C D A (5) C E B D A (5) D B E A C (4) D A E B C (3) B E D C A (3) A D B C E (3) A C D E B (3) A C D B E (3) D B A E C (2) D A C E B (2) C A E D B (2) A D B E C (2) E D B A C (1) E B D C A (1) D E A C B (1) C E A B D (1) C B E A D (1) C A D E B (1) C A B E D (1) C A B D E (1) B D E A C (1) B D A E C (1) B C E A D (1) B A D E C (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 18 -12 2 B 2 0 0 -4 10 C -18 0 0 -12 -4 D 12 4 12 0 8 E -2 -10 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 18 -12 2 B 2 0 0 -4 10 C -18 0 0 -12 -4 D 12 4 12 0 8 E -2 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999192 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=21 B=20 C=19 E=12 so E is eliminated. Round 2 votes counts: A=28 B=26 C=24 D=22 so D is eliminated. Round 3 votes counts: A=43 B=33 C=24 so C is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:218 B:204 A:203 E:192 C:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 18 -12 2 B 2 0 0 -4 10 C -18 0 0 -12 -4 D 12 4 12 0 8 E -2 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999192 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 18 -12 2 B 2 0 0 -4 10 C -18 0 0 -12 -4 D 12 4 12 0 8 E -2 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999192 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 18 -12 2 B 2 0 0 -4 10 C -18 0 0 -12 -4 D 12 4 12 0 8 E -2 -10 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999192 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9887: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (8) C A D B E (8) E B D A C (7) B E D C A (7) B D E C A (7) A C D E B (7) D C A B E (6) C A E B D (5) B E D A C (4) A C E D B (4) E B A D C (3) D E B A C (3) A C E B D (3) E D B A C (2) E B A C D (2) E A C B D (2) E A B C D (2) C A D E B (2) C A B D E (2) B D C A E (2) A E C B D (2) E D A C B (1) E A D C B (1) D C B A E (1) D B E A C (1) D B C A E (1) C D B A E (1) C B D A E (1) C B A D E (1) C A B E D (1) B E C A D (1) B C D A E (1) A E C D B (1) Total count = 100 A B C D E A 0 -6 -10 -6 -2 B 6 0 2 6 6 C 10 -2 0 -8 -8 D 6 -6 8 0 4 E 2 -6 8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -10 -6 -2 B 6 0 2 6 6 C 10 -2 0 -8 -8 D 6 -6 8 0 4 E 2 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=22 C=21 E=20 D=20 A=17 so A is eliminated. Round 2 votes counts: C=35 E=23 B=22 D=20 so D is eliminated. Round 3 votes counts: C=42 B=32 E=26 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:210 D:206 E:200 C:196 A:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -10 -6 -2 B 6 0 2 6 6 C 10 -2 0 -8 -8 D 6 -6 8 0 4 E 2 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -10 -6 -2 B 6 0 2 6 6 C 10 -2 0 -8 -8 D 6 -6 8 0 4 E 2 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -10 -6 -2 B 6 0 2 6 6 C 10 -2 0 -8 -8 D 6 -6 8 0 4 E 2 -6 8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999324 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9888: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) C D A E B (10) B E A D C (7) A E C D B (7) A E B C D (6) E A D C B (4) C D B A E (4) C D A B E (4) A C E D B (4) E B A D C (3) D B E C A (3) B E D A C (3) B E A C D (3) B C A D E (3) E B D A C (2) E A B D C (2) D C E B A (2) D C B E A (2) B E D C A (2) B D E C A (2) B C D A E (2) A E C B D (2) A B E C D (2) E D A C B (1) E A B C D (1) D C E A B (1) D C A E B (1) D B C E A (1) C B D A E (1) B D C E A (1) B C D E A (1) B A E C D (1) A C B E D (1) Total count = 100 A B C D E A 0 14 -2 14 18 B -14 0 -10 -10 -14 C 2 10 0 26 -2 D -14 10 -26 0 -2 E -18 14 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.090909 Sum of squares = 0.685950413232 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.909091 D: 0.909091 E: 1.000000 A B C D E A 0 14 -2 14 18 B -14 0 -10 -10 -14 C 2 10 0 26 -2 D -14 10 -26 0 -2 E -18 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.090909 Sum of squares = 0.685950406188 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.909091 D: 0.909091 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=25 A=22 E=13 D=10 so D is eliminated. Round 2 votes counts: C=36 B=29 A=22 E=13 so E is eliminated. Round 3 votes counts: C=36 B=34 A=30 so A is eliminated. Round 4 votes counts: C=55 B=45 so B is eliminated. IRV winner is C compare: Computing Borda winner. A:222 C:218 E:200 D:184 B:176 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 14 -2 14 18 B -14 0 -10 -10 -14 C 2 10 0 26 -2 D -14 10 -26 0 -2 E -18 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.090909 Sum of squares = 0.685950406188 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.909091 D: 0.909091 E: 1.000000 GTS winners are ['A', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 14 18 B -14 0 -10 -10 -14 C 2 10 0 26 -2 D -14 10 -26 0 -2 E -18 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.090909 Sum of squares = 0.685950406188 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.909091 D: 0.909091 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 14 18 B -14 0 -10 -10 -14 C 2 10 0 26 -2 D -14 10 -26 0 -2 E -18 14 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.090909 B: 0.000000 C: 0.818182 D: 0.000000 E: 0.090909 Sum of squares = 0.685950406188 Cumulative probabilities = A: 0.090909 B: 0.090909 C: 0.909091 D: 0.909091 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9889: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (16) A E C D B (13) E C D B A (6) A C E D B (6) B D E C A (5) B A D C E (5) A B E D C (5) E D C B A (4) A B D C E (4) E C D A B (3) C E D A B (3) B E D C A (3) A C D B E (3) E D B C A (2) D C E B A (2) C D E B A (2) A E B D C (2) A C B D E (2) A B C D E (2) D E C B A (1) D B C E A (1) C E D B A (1) C D E A B (1) C D B E A (1) C D A E B (1) B D A C E (1) B A E D C (1) B A D E C (1) A E B C D (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -4 -4 -6 -2 B 4 0 -2 -2 2 C 4 2 0 -6 6 D 6 2 6 0 -2 E 2 -2 -6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755101852 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 A B C D E A 0 -4 -4 -6 -2 B 4 0 -2 -2 2 C 4 2 0 -6 6 D 6 2 6 0 -2 E 2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102023 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 B=32 E=15 C=9 D=4 so D is eliminated. Round 2 votes counts: A=40 B=33 E=16 C=11 so C is eliminated. Round 3 votes counts: A=41 B=34 E=25 so E is eliminated. Round 4 votes counts: B=52 A=48 so A is eliminated. IRV winner is B compare: Computing Borda winner. D:206 C:203 B:201 E:198 A:192 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 -4 -6 -2 B 4 0 -2 -2 2 C 4 2 0 -6 6 D 6 2 6 0 -2 E 2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102023 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 GTS winners are ['C', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 -2 -2 2 C 4 2 0 -6 6 D 6 2 6 0 -2 E 2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102023 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -4 -6 -2 B 4 0 -2 -2 2 C 4 2 0 -6 6 D 6 2 6 0 -2 E 2 -2 -6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.428571 E: 0.428571 Sum of squares = 0.387755102023 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.142857 D: 0.571429 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9890: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E A B (10) E B A D C (9) E B A C D (8) B E A D C (7) B A E D C (7) E C D B A (6) D C A B E (6) A B D C E (6) C D A E B (3) A B E D C (3) A B D E C (3) D C E B A (2) D C E A B (2) C D A B E (2) B A E C D (2) A D C B E (2) A D B C E (2) A C B D E (2) A B E C D (2) A B C D E (2) E D C B A (1) E D B C A (1) E C A B D (1) E B D C A (1) E B D A C (1) E B C D A (1) D E C B A (1) D E B C A (1) D C A E B (1) D B A C E (1) D A C B E (1) C D E B A (1) B E A C D (1) B A D E C (1) Total count = 100 A B C D E A 0 -4 20 16 -8 B 4 0 18 14 0 C -20 -18 0 -18 -14 D -16 -14 18 0 -2 E 8 0 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.655237 C: 0.000000 D: 0.000000 E: 0.344763 Sum of squares = 0.548196760778 Cumulative probabilities = A: 0.000000 B: 0.655237 C: 0.655237 D: 0.655237 E: 1.000000 A B C D E A 0 -4 20 16 -8 B 4 0 18 14 0 C -20 -18 0 -18 -14 D -16 -14 18 0 -2 E 8 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 A=22 B=18 C=16 D=15 so D is eliminated. Round 2 votes counts: E=31 C=27 A=23 B=19 so B is eliminated. Round 3 votes counts: E=39 A=34 C=27 so C is eliminated. Round 4 votes counts: E=54 A=46 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:218 A:212 E:212 D:193 C:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 20 16 -8 B 4 0 18 14 0 C -20 -18 0 -18 -14 D -16 -14 18 0 -2 E 8 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 20 16 -8 B 4 0 18 14 0 C -20 -18 0 -18 -14 D -16 -14 18 0 -2 E 8 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 20 16 -8 B 4 0 18 14 0 C -20 -18 0 -18 -14 D -16 -14 18 0 -2 E 8 0 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9891: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (9) B E C A D (7) D B C A E (6) B C E D A (6) B D C E A (5) E C B A D (4) D A E B C (4) D A C B E (4) A D E C B (4) E A C B D (3) B E C D A (3) A E C D B (3) A D C E B (3) E B A C D (2) E A D B C (2) E A B D C (2) D C A B E (2) D B A C E (2) D A C E B (2) D A B C E (2) B E D A C (2) B D E C A (2) B C D E A (2) A C E D B (2) E C A B D (1) E B A D C (1) E A B C D (1) D E A B C (1) D C B A E (1) D A B E C (1) C E B A D (1) C E A B D (1) C D B A E (1) C D A B E (1) C B D A E (1) C A E D B (1) C A E B D (1) C A D E B (1) B E D C A (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -12 -14 2 -12 B 12 0 24 10 -2 C 14 -24 0 2 -10 D -2 -10 -2 0 -10 E 12 2 10 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -14 2 -12 B 12 0 24 10 -2 C 14 -24 0 2 -10 D -2 -10 -2 0 -10 E 12 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 E=25 D=25 A=14 C=8 so C is eliminated. Round 2 votes counts: B=29 E=27 D=27 A=17 so A is eliminated. Round 3 votes counts: E=36 D=35 B=29 so B is eliminated. Round 4 votes counts: E=55 D=45 so D is eliminated. IRV winner is E compare: Computing Borda winner. B:222 E:217 C:191 D:188 A:182 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -14 2 -12 B 12 0 24 10 -2 C 14 -24 0 2 -10 D -2 -10 -2 0 -10 E 12 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -14 2 -12 B 12 0 24 10 -2 C 14 -24 0 2 -10 D -2 -10 -2 0 -10 E 12 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -14 2 -12 B 12 0 24 10 -2 C 14 -24 0 2 -10 D -2 -10 -2 0 -10 E 12 2 10 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999514 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9892: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (8) A C E D B (8) D A B C E (6) A D C B E (6) E C B A D (5) E B C D A (5) D B E A C (5) E C A B D (4) E B D C A (4) C E B A D (4) C B E D A (4) C A E B D (4) B D E C A (4) A D B E C (4) C E A B D (3) A D B C E (3) E C B D A (2) E A C B D (2) A C D E B (2) E B D A C (1) E B A D C (1) D B C E A (1) D B C A E (1) D A B E C (1) C E B D A (1) C D B A E (1) C B D E A (1) C A D E B (1) B E D A C (1) B D E A C (1) B D C E A (1) B C E D A (1) A E D C B (1) A E D B C (1) A C E B D (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 6 2 -2 B 4 0 -2 0 0 C -6 2 0 0 0 D -2 0 0 0 -6 E 2 0 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.158788 D: 0.000000 E: 0.841212 Sum of squares = 0.732851119365 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.158788 D: 0.158788 E: 1.000000 A B C D E A 0 -4 6 2 -2 B 4 0 -2 0 0 C -6 2 0 0 0 D -2 0 0 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000019926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=27 E=24 D=22 C=19 B=8 so B is eliminated. Round 2 votes counts: D=28 A=27 E=25 C=20 so C is eliminated. Round 3 votes counts: E=38 A=32 D=30 so D is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:204 A:201 B:201 C:198 D:196 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 6 2 -2 B 4 0 -2 0 0 C -6 2 0 0 0 D -2 0 0 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000019926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 2 -2 B 4 0 -2 0 0 C -6 2 0 0 0 D -2 0 0 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000019926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 2 -2 B 4 0 -2 0 0 C -6 2 0 0 0 D -2 0 0 0 -6 E 2 0 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.000000 E: 0.750000 Sum of squares = 0.625000019926 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.250000 D: 0.250000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9893: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C A D (8) D C E A B (8) D C A E B (8) D A C B E (8) A D B C E (6) E B C D A (4) C E D B A (4) B E A C D (4) E C D B A (3) E B A C D (3) D A B C E (3) B A E D C (3) B A E C D (3) A B D C E (3) A B C E D (3) E D C B A (2) D C A B E (2) D A C E B (2) C D A E B (2) C A D B E (2) A B E D C (2) A B D E C (2) E C B D A (1) E C B A D (1) E B D A C (1) E B A D C (1) D E C B A (1) D E C A B (1) D C E B A (1) D A B E C (1) C E D A B (1) C D E A B (1) B E C A D (1) B E A D C (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 16 -2 -8 6 B -16 0 2 -12 -6 C 2 -2 0 -12 12 D 8 12 12 0 6 E -6 6 -12 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999605 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 -2 -8 6 B -16 0 2 -12 -6 C 2 -2 0 -12 12 D 8 12 12 0 6 E -6 6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=35 E=24 A=19 B=12 C=10 so C is eliminated. Round 2 votes counts: D=38 E=29 A=21 B=12 so B is eliminated. Round 3 votes counts: D=38 E=35 A=27 so A is eliminated. Round 4 votes counts: D=53 E=47 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:219 A:206 C:200 E:191 B:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 16 -2 -8 6 B -16 0 2 -12 -6 C 2 -2 0 -12 12 D 8 12 12 0 6 E -6 6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 -2 -8 6 B -16 0 2 -12 -6 C 2 -2 0 -12 12 D 8 12 12 0 6 E -6 6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 -2 -8 6 B -16 0 2 -12 -6 C 2 -2 0 -12 12 D 8 12 12 0 6 E -6 6 -12 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9894: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D C E A (12) D B A C E (11) A E C D B (9) A D E C B (9) D A B C E (8) E C B A D (7) E C A B D (6) E A C B D (6) D B C A E (5) B D C A E (4) B C E D A (4) C E B A D (3) C B E D A (3) E A C D B (2) D A B E C (2) A D E B C (2) A D B E C (2) E B C D A (1) C E B D A (1) B E C D A (1) A E D C B (1) A D B C E (1) Total count = 100 A B C D E A 0 -4 6 -4 8 B 4 0 6 -4 6 C -6 -6 0 -14 4 D 4 4 14 0 12 E -8 -6 -4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 -4 8 B 4 0 6 -4 6 C -6 -6 0 -14 4 D 4 4 14 0 12 E -8 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=24 E=22 B=21 C=7 so C is eliminated. Round 2 votes counts: E=26 D=26 B=24 A=24 so B is eliminated. Round 3 votes counts: D=42 E=34 A=24 so A is eliminated. Round 4 votes counts: D=56 E=44 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 B:206 A:203 C:189 E:185 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 6 -4 8 B 4 0 6 -4 6 C -6 -6 0 -14 4 D 4 4 14 0 12 E -8 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 8 B 4 0 6 -4 6 C -6 -6 0 -14 4 D 4 4 14 0 12 E -8 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 8 B 4 0 6 -4 6 C -6 -6 0 -14 4 D 4 4 14 0 12 E -8 -6 -4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999886 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9895: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B C A (9) A B C E D (7) E D C A B (6) D E B A C (5) C A B E D (5) B A C D E (5) E D A B C (4) E C D A B (4) D E C B A (4) C E A D B (4) A C E B D (4) D B E C A (3) A C B E D (3) A B C D E (3) E D C B A (2) D B E A C (2) B D C E A (2) B D C A E (2) B D A E C (2) A E C D B (2) A B E C D (2) E D B A C (1) E D A C B (1) E A D C B (1) E A C D B (1) C E D A B (1) C D B E A (1) C B D E A (1) C B A E D (1) C B A D E (1) C A E D B (1) B D E C A (1) B C D A E (1) B C A D E (1) B A D E C (1) B A D C E (1) A E D C B (1) A E D B C (1) A E B C D (1) A C B D E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 0 -4 -6 B -8 0 10 -8 -6 C 0 -10 0 0 -8 D 4 8 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 -4 -6 B -8 0 10 -8 -6 C 0 -10 0 0 -8 D 4 8 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=26 D=23 E=20 B=16 C=15 so C is eliminated. Round 2 votes counts: A=32 E=25 D=24 B=19 so B is eliminated. Round 3 votes counts: A=42 D=33 E=25 so E is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:213 D:203 A:199 B:194 C:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 -4 -6 B -8 0 10 -8 -6 C 0 -10 0 0 -8 D 4 8 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -4 -6 B -8 0 10 -8 -6 C 0 -10 0 0 -8 D 4 8 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -4 -6 B -8 0 10 -8 -6 C 0 -10 0 0 -8 D 4 8 0 0 -6 E 6 6 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9896: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C B D (8) D B C E A (7) D B A C E (7) B D C E A (7) A E C B D (7) B C D E A (5) E C A B D (4) E A C D B (4) A D B C E (4) E C D B A (3) C B E D A (3) C B D E A (3) A B C E D (3) E C D A B (2) E C B D A (2) E C B A D (2) D B C A E (2) D A B C E (2) C E B D A (2) A E D C B (2) A E C D B (2) A E B C D (2) A D E B C (2) A D B E C (2) E D A C B (1) E C A D B (1) D C E B A (1) D C B E A (1) C E D B A (1) C B E A D (1) B D C A E (1) B C D A E (1) B A D C E (1) A E D B C (1) A E B D C (1) A B D E C (1) A B D C E (1) Total count = 100 A B C D E A 0 0 2 -2 -16 B 0 0 0 10 4 C -2 0 0 12 6 D 2 -10 -12 0 -4 E 16 -4 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.120292 B: 0.879708 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.788355792746 Cumulative probabilities = A: 0.120292 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 2 -2 -16 B 0 0 0 10 4 C -2 0 0 12 6 D 2 -10 -12 0 -4 E 16 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000009902 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=27 D=20 B=15 C=10 so C is eliminated. Round 2 votes counts: E=30 A=28 B=22 D=20 so D is eliminated. Round 3 votes counts: B=39 E=31 A=30 so A is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. C:208 B:207 E:205 A:192 D:188 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 0 2 -2 -16 B 0 0 0 10 4 C -2 0 0 12 6 D 2 -10 -12 0 -4 E 16 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000009902 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 -2 -16 B 0 0 0 10 4 C -2 0 0 12 6 D 2 -10 -12 0 -4 E 16 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000009902 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 -2 -16 B 0 0 0 10 4 C -2 0 0 12 6 D 2 -10 -12 0 -4 E 16 -4 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.800000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.680000009902 Cumulative probabilities = A: 0.200000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9897: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (5) C E D A B (5) D A C E B (4) D A B E C (4) C D E A B (4) C A D E B (4) A D C B E (4) A B D C E (4) E D B C A (3) E C D B A (3) E B D C A (3) D C E A B (3) C E B A D (3) C E A B D (3) C A B E D (3) B D E A C (3) B A D E C (3) A C B D E (3) E D C B A (2) D C A E B (2) D B A E C (2) C E A D B (2) B E D A C (2) B E A C D (2) A D B C E (2) A B C D E (2) E C D A B (1) E C B D A (1) E B C D A (1) D E C A B (1) D E A C B (1) D E A B C (1) D B E A C (1) C E B D A (1) C B A E D (1) C A D B E (1) B E C A D (1) B E A D C (1) B D A E C (1) B C E A D (1) B A E D C (1) B A E C D (1) B A C E D (1) A D B E C (1) A C D B E (1) A C B E D (1) Total count = 100 A B C D E A 0 14 2 -8 -8 B -14 0 -8 -14 -6 C -2 8 0 -8 10 D 8 14 8 0 14 E 8 6 -10 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 2 -8 -8 B -14 0 -8 -14 -6 C -2 8 0 -8 10 D 8 14 8 0 14 E 8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=24 A=18 B=17 E=14 so E is eliminated. Round 2 votes counts: C=32 D=29 B=21 A=18 so A is eliminated. Round 3 votes counts: C=37 D=36 B=27 so B is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:222 C:204 A:200 E:195 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 2 -8 -8 B -14 0 -8 -14 -6 C -2 8 0 -8 10 D 8 14 8 0 14 E 8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 2 -8 -8 B -14 0 -8 -14 -6 C -2 8 0 -8 10 D 8 14 8 0 14 E 8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 2 -8 -8 B -14 0 -8 -14 -6 C -2 8 0 -8 10 D 8 14 8 0 14 E 8 6 -10 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9898: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (9) A D C B E (7) E B C D A (6) E B A D C (6) B E A D C (5) C D A E B (4) B E D C A (4) A D B C E (4) E A B C D (3) D C A B E (3) B E D A C (3) A E B D C (3) E C B D A (2) E C A B D (2) E B D C A (2) D A C B E (2) C D B E A (2) C A D B E (2) B E C D A (2) B D E A C (2) B A E D C (2) A C D E B (2) A B E D C (2) E C B A D (1) E B D A C (1) E B C A D (1) E B A C D (1) E A C B D (1) D C B A E (1) D B C A E (1) C E B D A (1) C E A D B (1) C D B A E (1) C B E D A (1) C A D E B (1) B D E C A (1) B D C E A (1) B D A E C (1) B C D E A (1) A E C D B (1) A E B C D (1) A D B E C (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 0 -2 0 B -2 0 8 12 18 C 0 -8 0 -4 -8 D 2 -12 4 0 -6 E 0 -18 8 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999608 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 A B C D E A 0 2 0 -2 0 B -2 0 8 12 18 C 0 -8 0 -4 -8 D 2 -12 4 0 -6 E 0 -18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999935 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 A=23 C=22 B=22 D=7 so D is eliminated. Round 2 votes counts: E=26 C=26 A=25 B=23 so B is eliminated. Round 3 votes counts: E=43 C=29 A=28 so A is eliminated. Round 4 votes counts: E=54 C=46 so C is eliminated. IRV winner is E compare: Computing Borda winner. B:218 A:200 E:198 D:194 C:190 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 2 0 -2 0 B -2 0 8 12 18 C 0 -8 0 -4 -8 D 2 -12 4 0 -6 E 0 -18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999935 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 0 -2 0 B -2 0 8 12 18 C 0 -8 0 -4 -8 D 2 -12 4 0 -6 E 0 -18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999935 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 0 -2 0 B -2 0 8 12 18 C 0 -8 0 -4 -8 D 2 -12 4 0 -6 E 0 -18 8 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.750000 B: 0.125000 C: 0.000000 D: 0.125000 E: 0.000000 Sum of squares = 0.593749999935 Cumulative probabilities = A: 0.750000 B: 0.875000 C: 0.875000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9899: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (10) D E A B C (9) E B D A C (8) B C A E D (7) E D A C B (6) C B A D E (6) C A D E B (4) B E D A C (4) E D B A C (3) C B E A D (3) B E C D A (3) B C E A D (3) B C A D E (3) A C D B E (3) E B C D A (2) D E A C B (2) D A E B C (2) C B A E D (2) C A D B E (2) A D C E B (2) A C B D E (2) E D C A B (1) E D B C A (1) E D A B C (1) E C D A B (1) D A E C B (1) D A C E B (1) C E D B A (1) C E D A B (1) C D E A B (1) B A E C D (1) B A C D E (1) A D E C B (1) A D C B E (1) A D B E C (1) Total count = 100 A B C D E A 0 4 -2 4 0 B -4 0 -2 10 4 C 2 2 0 12 6 D -4 -10 -12 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999941 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -2 4 0 B -4 0 -2 10 4 C 2 2 0 12 6 D -4 -10 -12 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 E=23 B=22 D=15 A=10 so A is eliminated. Round 2 votes counts: C=35 E=23 B=22 D=20 so D is eliminated. Round 3 votes counts: C=39 E=38 B=23 so B is eliminated. Round 4 votes counts: C=53 E=47 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:211 B:204 A:203 E:193 D:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -2 4 0 B -4 0 -2 10 4 C 2 2 0 12 6 D -4 -10 -12 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -2 4 0 B -4 0 -2 10 4 C 2 2 0 12 6 D -4 -10 -12 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -2 4 0 B -4 0 -2 10 4 C 2 2 0 12 6 D -4 -10 -12 0 4 E 0 -4 -6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.9999999943 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9900: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (9) D C E B A (6) A B E C D (5) E A B D C (4) D C B E A (4) A E B D C (4) A C D B E (4) E D B C A (3) A E D C B (3) A E D B C (3) A C B D E (3) E D A B C (2) E B D C A (2) E B D A C (2) E B A D C (2) E A D B C (2) E A B C D (2) D E C B A (2) D C A E B (2) D C A B E (2) D A C E B (2) C D A B E (2) B E A C D (2) B C A E D (2) B A C E D (2) A D E C B (2) A B C E D (2) E D C B A (1) E D B A C (1) E D A C B (1) E B A C D (1) D E C A B (1) D C B A E (1) D A C B E (1) C D B E A (1) C D B A E (1) C B D E A (1) C B D A E (1) C B A D E (1) C A B D E (1) B E C D A (1) B E C A D (1) B C D E A (1) B A E C D (1) A D C E B (1) A C D E B (1) A B C D E (1) Total count = 100 A B C D E A 0 20 26 18 14 B -20 0 10 2 -18 C -26 -10 0 -8 -14 D -18 -2 8 0 -16 E -14 18 14 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 26 18 14 B -20 0 10 2 -18 C -26 -10 0 -8 -14 D -18 -2 8 0 -16 E -14 18 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=38 E=23 D=21 B=10 C=8 so C is eliminated. Round 2 votes counts: A=39 D=25 E=23 B=13 so B is eliminated. Round 3 votes counts: A=45 D=28 E=27 so E is eliminated. Round 4 votes counts: A=59 D=41 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:239 E:217 B:187 D:186 C:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 26 18 14 B -20 0 10 2 -18 C -26 -10 0 -8 -14 D -18 -2 8 0 -16 E -14 18 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 26 18 14 B -20 0 10 2 -18 C -26 -10 0 -8 -14 D -18 -2 8 0 -16 E -14 18 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 26 18 14 B -20 0 10 2 -18 C -26 -10 0 -8 -14 D -18 -2 8 0 -16 E -14 18 14 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9901: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (29) C A D B E (15) E C A B D (9) D B A C E (5) B D E A C (4) B D A E C (4) A D B C E (3) A C D B E (3) E B D C A (2) D B E A C (2) C E A D B (2) C E A B D (2) C A E D B (2) C A D E B (2) B D A C E (2) E D B C A (1) E C D A B (1) E C B D A (1) E C B A D (1) E C A D B (1) E A B D C (1) D C B A E (1) D B C A E (1) D A B C E (1) C E D B A (1) C A E B D (1) B E D A C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 14 -12 -16 B 10 0 14 16 -12 C -14 -14 0 -18 -14 D 12 -16 18 0 -10 E 16 12 14 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999805 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -10 14 -12 -16 B 10 0 14 16 -12 C -14 -14 0 -18 -14 D 12 -16 18 0 -10 E 16 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=46 C=25 B=11 D=10 A=8 so A is eliminated. Round 2 votes counts: E=46 C=28 D=14 B=12 so B is eliminated. Round 3 votes counts: E=47 C=28 D=25 so D is eliminated. Round 4 votes counts: E=57 C=43 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:226 B:214 D:202 A:188 C:170 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -10 14 -12 -16 B 10 0 14 16 -12 C -14 -14 0 -18 -14 D 12 -16 18 0 -10 E 16 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 14 -12 -16 B 10 0 14 16 -12 C -14 -14 0 -18 -14 D 12 -16 18 0 -10 E 16 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 14 -12 -16 B 10 0 14 16 -12 C -14 -14 0 -18 -14 D 12 -16 18 0 -10 E 16 12 14 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999998 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9902: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (7) E A C B D (6) B C D E A (5) E B C A D (4) E A D B C (4) D B E C A (4) D A E B C (4) A D E C B (4) E D B C A (3) E D B A C (3) D E B C A (3) D A B C E (3) C B A D E (3) A C B D E (3) E D A B C (2) E B D C A (2) E B C D A (2) E A B D C (2) D E B A C (2) C B E A D (2) C B D E A (2) C B D A E (2) C B A E D (2) C A B E D (2) B C E D A (2) A E D B C (2) A C E B D (2) A C B E D (2) E B A C D (1) E A B C D (1) D E A B C (1) D B C A E (1) D A C B E (1) C B E D A (1) C A E B D (1) B E C D A (1) A E D C B (1) A E C D B (1) A E C B D (1) A D E B C (1) A D C E B (1) A D C B E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 -4 2 -2 -20 B 4 0 20 -2 -10 C -2 -20 0 -4 -10 D 2 2 4 0 -2 E 20 10 10 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 2 -2 -20 B 4 0 20 -2 -10 C -2 -20 0 -4 -10 D 2 2 4 0 -2 E 20 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 D=26 A=21 C=15 B=8 so B is eliminated. Round 2 votes counts: E=31 D=26 C=22 A=21 so A is eliminated. Round 3 votes counts: E=36 D=33 C=31 so C is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:221 B:206 D:203 A:188 C:182 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 2 -2 -20 B 4 0 20 -2 -10 C -2 -20 0 -4 -10 D 2 2 4 0 -2 E 20 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 2 -2 -20 B 4 0 20 -2 -10 C -2 -20 0 -4 -10 D 2 2 4 0 -2 E 20 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 2 -2 -20 B 4 0 20 -2 -10 C -2 -20 0 -4 -10 D 2 2 4 0 -2 E 20 10 10 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9903: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B C A (9) E C D B A (8) A B D E C (8) B D E A C (7) A C B D E (6) B D E C A (5) A C E B D (5) D B E C A (4) C E D B A (4) C A E D B (4) C A B D E (4) B D A E C (4) A B D C E (4) D B C E A (3) A E B D C (3) E D B A C (2) B A D E C (2) A B E D C (2) E D C B A (1) E D A C B (1) E B D A C (1) D E B C A (1) C E D A B (1) C E A D B (1) C D E B A (1) C D B E A (1) C A D E B (1) C A D B E (1) B D C E A (1) B D C A E (1) B C D A E (1) A E C D B (1) A E C B D (1) A C E D B (1) Total count = 100 A B C D E A 0 -12 -4 -12 -2 B 12 0 16 10 8 C 4 -16 0 -18 -20 D 12 -10 18 0 10 E 2 -8 20 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999694 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -4 -12 -2 B 12 0 16 10 8 C 4 -16 0 -18 -20 D 12 -10 18 0 10 E 2 -8 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=22 B=21 C=18 D=8 so D is eliminated. Round 2 votes counts: A=31 B=28 E=23 C=18 so C is eliminated. Round 3 votes counts: A=41 E=30 B=29 so B is eliminated. Round 4 votes counts: E=51 A=49 so A is eliminated. IRV winner is E compare: Computing Borda winner. B:223 D:215 E:202 A:185 C:175 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -4 -12 -2 B 12 0 16 10 8 C 4 -16 0 -18 -20 D 12 -10 18 0 10 E 2 -8 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -4 -12 -2 B 12 0 16 10 8 C 4 -16 0 -18 -20 D 12 -10 18 0 10 E 2 -8 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -4 -12 -2 B 12 0 16 10 8 C 4 -16 0 -18 -20 D 12 -10 18 0 10 E 2 -8 20 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999952 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9904: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) A C E D B (10) D B E C A (9) B D E C A (9) E B D C A (8) D B A E C (8) E B C D A (5) C A E B D (5) D B E A C (4) A D B C E (4) A C D B E (4) E C B D A (3) E C A B D (3) A C E B D (3) C E A B D (2) C B E D A (2) A D C B E (2) A C B D E (2) E D A B C (1) E C B A D (1) D A B C E (1) B E D C A (1) B D C E A (1) B D C A E (1) A C D E B (1) Total count = 100 A B C D E A 0 -24 0 -26 2 B 24 0 24 -8 16 C 0 -24 0 -18 -4 D 26 8 18 0 12 E -2 -16 4 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -24 0 -26 2 B 24 0 24 -8 16 C 0 -24 0 -18 -4 D 26 8 18 0 12 E -2 -16 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=32 A=26 E=21 B=12 C=9 so C is eliminated. Round 2 votes counts: D=32 A=31 E=23 B=14 so B is eliminated. Round 3 votes counts: D=43 A=31 E=26 so E is eliminated. Round 4 votes counts: D=63 A=37 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:232 B:228 E:187 C:177 A:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -24 0 -26 2 B 24 0 24 -8 16 C 0 -24 0 -18 -4 D 26 8 18 0 12 E -2 -16 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -24 0 -26 2 B 24 0 24 -8 16 C 0 -24 0 -18 -4 D 26 8 18 0 12 E -2 -16 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -24 0 -26 2 B 24 0 24 -8 16 C 0 -24 0 -18 -4 D 26 8 18 0 12 E -2 -16 4 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9905: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (11) C E A D B (11) B A E C D (10) B D A E C (8) D C E A B (7) B E A C D (7) D A C E B (6) B A D E C (5) E A C B D (4) D C A E B (4) E C A B D (3) D B C A E (3) D B A C E (3) C E A B D (2) B A E D C (2) A E C B D (2) A C E D B (2) E C B A D (1) E C A D B (1) D C B E A (1) D B A E C (1) D A C B E (1) C E D A B (1) B D E C A (1) B D E A C (1) B D C E A (1) A E B C D (1) Total count = 100 A B C D E A 0 -10 6 2 -4 B 10 0 8 -4 10 C -6 -8 0 -10 6 D -2 4 10 0 6 E 4 -10 -6 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000002 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 6 2 -4 B 10 0 8 -4 10 C -6 -8 0 -10 6 D -2 4 10 0 6 E 4 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000058 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=37 B=35 C=14 E=9 A=5 so A is eliminated. Round 2 votes counts: D=37 B=35 C=16 E=12 so E is eliminated. Round 3 votes counts: D=37 B=36 C=27 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:212 D:209 A:197 C:191 E:191 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -10 6 2 -4 B 10 0 8 -4 10 C -6 -8 0 -10 6 D -2 4 10 0 6 E 4 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000058 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 6 2 -4 B 10 0 8 -4 10 C -6 -8 0 -10 6 D -2 4 10 0 6 E 4 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000058 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 6 2 -4 B 10 0 8 -4 10 C -6 -8 0 -10 6 D -2 4 10 0 6 E 4 -10 -6 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.125000 C: 0.000000 D: 0.625000 E: 0.000000 Sum of squares = 0.468750000058 Cumulative probabilities = A: 0.250000 B: 0.375000 C: 0.375000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9906: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (8) C A E B D (8) E A C B D (7) C A E D B (7) A C E D B (6) D B C A E (4) A C D E B (4) C A D B E (3) B E D C A (3) B E C A D (3) B D E C A (3) B D E A C (3) A E C D B (3) E D B A C (2) E C A B D (2) E B C A D (2) D C B A E (2) D A C B E (2) D A B C E (2) C E A B D (2) B E C D A (2) B C D A E (2) E C B A D (1) E B A C D (1) D E B A C (1) D E A B C (1) D C A B E (1) D B A E C (1) D B A C E (1) D A E C B (1) D A C E B (1) D A B E C (1) C D A B E (1) C B D A E (1) C A D E B (1) C A B D E (1) B E D A C (1) B D C E A (1) B D C A E (1) A E C B D (1) A D C E B (1) A C E B D (1) Total count = 100 A B C D E A 0 14 -2 8 14 B -14 0 -14 -8 -6 C 2 14 0 18 6 D -8 8 -18 0 -4 E -14 6 -6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999404 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 8 14 B -14 0 -14 -8 -6 C 2 14 0 18 6 D -8 8 -18 0 -4 E -14 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=24 B=19 A=16 E=15 so E is eliminated. Round 2 votes counts: D=28 C=27 A=23 B=22 so B is eliminated. Round 3 votes counts: D=40 C=36 A=24 so A is eliminated. Round 4 votes counts: C=59 D=41 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:220 A:217 E:195 D:189 B:179 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 8 14 B -14 0 -14 -8 -6 C 2 14 0 18 6 D -8 8 -18 0 -4 E -14 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 8 14 B -14 0 -14 -8 -6 C 2 14 0 18 6 D -8 8 -18 0 -4 E -14 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 8 14 B -14 0 -14 -8 -6 C 2 14 0 18 6 D -8 8 -18 0 -4 E -14 6 -6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997912 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9907: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E C A (10) B E D A C (10) C A D E B (9) C D E A B (5) C A E D B (5) A C B E D (5) C A B E D (4) B E A D C (4) D E B C A (3) C D A E B (3) B D E A C (3) A B E D C (3) A B E C D (3) E D B A C (2) E B D A C (2) D E B A C (2) D B E A C (2) C B A D E (2) C A E B D (2) C A D B E (2) C A B D E (2) A E B D C (2) A C E D B (2) A C E B D (2) A B C E D (2) D E C B A (1) D C E B A (1) C D E B A (1) C D B E A (1) C D B A E (1) C D A B E (1) B D E C A (1) B C A D E (1) B A C D E (1) Total count = 100 A B C D E A 0 4 -10 2 4 B -4 0 2 -2 16 C 10 -2 0 8 4 D -2 2 -8 0 4 E -4 -16 -4 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999985 Cumulative probabilities = A: 0.125000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -10 2 4 B -4 0 2 -2 16 C 10 -2 0 8 4 D -2 2 -8 0 4 E -4 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999939 Cumulative probabilities = A: 0.125000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 B=20 D=19 A=19 E=4 so E is eliminated. Round 2 votes counts: C=38 B=22 D=21 A=19 so A is eliminated. Round 3 votes counts: C=47 B=32 D=21 so D is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:210 B:206 A:200 D:198 E:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 4 -10 2 4 B -4 0 2 -2 16 C 10 -2 0 8 4 D -2 2 -8 0 4 E -4 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999939 Cumulative probabilities = A: 0.125000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -10 2 4 B -4 0 2 -2 16 C 10 -2 0 8 4 D -2 2 -8 0 4 E -4 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999939 Cumulative probabilities = A: 0.125000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -10 2 4 B -4 0 2 -2 16 C 10 -2 0 8 4 D -2 2 -8 0 4 E -4 -16 -4 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.625000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.468749999939 Cumulative probabilities = A: 0.125000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9908: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (7) C D A E B (6) B A C E D (6) E B D C A (5) D E C B A (5) D E C A B (4) C A D E B (4) A B C D E (4) E D B C A (3) D E B A C (3) D C E A B (3) B E D A C (3) B D A E C (3) A C B D E (3) A B D C E (3) E C D B A (2) E C B D A (2) D C A E B (2) D A C E B (2) C E D A B (2) C E A D B (2) C D E A B (2) C A E D B (2) C A E B D (2) B E A D C (2) B E A C D (2) B A E C D (2) B A D E C (2) E D C B A (1) D E B C A (1) D E A C B (1) D A C B E (1) D A B E C (1) C B E A D (1) C A B E D (1) B E D C A (1) B C E A D (1) A D C B E (1) A D B C E (1) A C B E D (1) Total count = 100 A B C D E A 0 10 -4 -6 8 B -10 0 0 2 -8 C 4 0 0 4 14 D 6 -2 -4 0 4 E -8 8 -14 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.202266 C: 0.797734 D: 0.000000 E: 0.000000 Sum of squares = 0.677290675758 Cumulative probabilities = A: 0.000000 B: 0.202266 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 10 -4 -6 8 B -10 0 0 2 -8 C 4 0 0 4 14 D 6 -2 -4 0 4 E -8 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591837111197 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=23 C=22 B=22 A=20 E=13 so E is eliminated. Round 2 votes counts: D=27 B=27 C=26 A=20 so A is eliminated. Round 3 votes counts: B=41 C=30 D=29 so D is eliminated. Round 4 votes counts: C=50 B=50 so C is eliminated. IRV winner is B compare: Computing Borda winner. C:211 A:204 D:202 B:192 E:191 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 10 -4 -6 8 B -10 0 0 2 -8 C 4 0 0 4 14 D 6 -2 -4 0 4 E -8 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591837111197 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 -4 -6 8 B -10 0 0 2 -8 C 4 0 0 4 14 D 6 -2 -4 0 4 E -8 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591837111197 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 -4 -6 8 B -10 0 0 2 -8 C 4 0 0 4 14 D 6 -2 -4 0 4 E -8 8 -14 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.285714 C: 0.714286 D: 0.000000 E: 0.000000 Sum of squares = 0.591837111197 Cumulative probabilities = A: 0.000000 B: 0.285714 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9909: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (11) A D E B C (11) C B E D A (10) A E C B D (10) E C B A D (5) E B C D A (5) C B D E A (5) A E D B C (4) A D B C E (4) E C B D A (3) E A B C D (3) D A C B E (3) D B C A E (2) C E B A D (2) A E B C D (2) A D E C B (2) E D B C A (1) E B C A D (1) E A B D C (1) D C B E A (1) D C B A E (1) D A B E C (1) D A B C E (1) C E B D A (1) C D B A E (1) C D A B E (1) C A B D E (1) B E C D A (1) B D C E A (1) B C E D A (1) A E B D C (1) A C E B D (1) A C B E D (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 -8 0 -4 B 4 0 2 12 -6 C 8 -2 0 10 -2 D 0 -12 -10 0 -6 E 4 6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -4 -8 0 -4 B 4 0 2 12 -6 C 8 -2 0 10 -2 D 0 -12 -10 0 -6 E 4 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 C=21 D=20 E=19 B=3 so B is eliminated. Round 2 votes counts: A=37 C=22 D=21 E=20 so E is eliminated. Round 3 votes counts: A=41 C=37 D=22 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. E:209 C:207 B:206 A:192 D:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 0 -4 B 4 0 2 12 -6 C 8 -2 0 10 -2 D 0 -12 -10 0 -6 E 4 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 0 -4 B 4 0 2 12 -6 C 8 -2 0 10 -2 D 0 -12 -10 0 -6 E 4 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 0 -4 B 4 0 2 12 -6 C 8 -2 0 10 -2 D 0 -12 -10 0 -6 E 4 6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999962 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9910: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) D B C E A (8) D C B A E (7) C D A E B (6) D B C A E (5) B E A D C (5) B D E A C (5) B D C E A (5) E A C B D (4) D C A E B (4) A E C D B (4) A E C B D (4) E A B D C (3) E A B C D (3) D C A B E (3) B E D A C (3) B D E C A (3) B C D E A (3) E B A D C (2) B E A C D (2) A E B D C (2) E B A C D (1) D B E A C (1) D A C B E (1) C D B A E (1) C D A B E (1) C A D E B (1) B E C A D (1) B C E A D (1) A E D C B (1) A E D B C (1) Total count = 100 A B C D E A 0 -6 -16 -12 0 B 6 0 8 -6 10 C 16 -8 0 -18 10 D 12 6 18 0 8 E 0 -10 -10 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999993 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -12 0 B 6 0 8 -6 10 C 16 -8 0 -18 10 D 12 6 18 0 8 E 0 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 B=28 C=18 E=13 A=12 so A is eliminated. Round 2 votes counts: D=29 B=28 E=25 C=18 so C is eliminated. Round 3 votes counts: D=38 E=34 B=28 so B is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:209 C:200 E:186 A:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -16 -12 0 B 6 0 8 -6 10 C 16 -8 0 -18 10 D 12 6 18 0 8 E 0 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -12 0 B 6 0 8 -6 10 C 16 -8 0 -18 10 D 12 6 18 0 8 E 0 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -12 0 B 6 0 8 -6 10 C 16 -8 0 -18 10 D 12 6 18 0 8 E 0 -10 -10 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999623 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9911: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B E A C (14) C E B A D (7) A D C B E (7) D A B E C (6) A C D E B (6) C A B E D (5) E C B D A (4) D B E C A (4) C A E B D (4) B E D C A (4) A C D B E (4) E C B A D (3) E B D C A (3) D B A E C (3) C E B D A (3) B D E C A (3) A D B E C (3) E B D A C (2) C E A B D (2) C B D E A (2) B E C D A (2) E B C D A (1) E A B C D (1) D E B A C (1) C A D B E (1) B C D E A (1) A D E C B (1) A D E B C (1) A D B C E (1) A C E B D (1) Total count = 100 A B C D E A 0 -14 2 -6 -14 B 14 0 0 -4 20 C -2 0 0 -6 -12 D 6 4 6 0 16 E 14 -20 12 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -14 2 -6 -14 B 14 0 0 -4 20 C -2 0 0 -6 -12 D 6 4 6 0 16 E 14 -20 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=24 A=24 E=14 B=10 so B is eliminated. Round 2 votes counts: D=31 C=25 A=24 E=20 so E is eliminated. Round 3 votes counts: D=40 C=35 A=25 so A is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:216 B:215 E:195 C:190 A:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -14 2 -6 -14 B 14 0 0 -4 20 C -2 0 0 -6 -12 D 6 4 6 0 16 E 14 -20 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -14 2 -6 -14 B 14 0 0 -4 20 C -2 0 0 -6 -12 D 6 4 6 0 16 E 14 -20 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -14 2 -6 -14 B 14 0 0 -4 20 C -2 0 0 -6 -12 D 6 4 6 0 16 E 14 -20 12 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999931 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9912: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D B A (11) B A D C E (10) E C D A B (9) B E A C D (9) C D E A B (6) A B D C E (6) D C A E B (5) B A E D C (5) A D B C E (5) E B C D A (4) D A C B E (4) A D C B E (4) C E D A B (3) B E C A D (3) E C B D A (2) E B C A D (2) B A D E C (2) E C A D B (1) E A C B D (1) D C E A B (1) D C A B E (1) D A C E B (1) C D A E B (1) B D A C E (1) B A E C D (1) A C E D B (1) A B E D C (1) Total count = 100 A B C D E A 0 0 2 2 -4 B 0 0 -2 -6 4 C -2 2 0 8 -2 D -2 6 -8 0 -6 E 4 -4 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.205604 B: 0.352802 C: 0.294396 D: 0.000000 E: 0.147198 Sum of squares = 0.275078501035 Cumulative probabilities = A: 0.205604 B: 0.558405 C: 0.852802 D: 0.852802 E: 1.000000 A B C D E A 0 0 2 2 -4 B 0 0 -2 -6 4 C -2 2 0 8 -2 D -2 6 -8 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.300000 D: 0.000000 E: 0.150000 Sum of squares = 0.275000000003 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.850000 D: 0.850000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=30 A=17 D=12 C=10 so C is eliminated. Round 2 votes counts: E=33 B=31 D=19 A=17 so A is eliminated. Round 3 votes counts: B=38 E=34 D=28 so D is eliminated. Round 4 votes counts: B=52 E=48 so E is eliminated. IRV winner is B compare: Computing Borda winner. E:204 C:203 A:200 B:198 D:195 Borda winner is E compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 0 2 2 -4 B 0 0 -2 -6 4 C -2 2 0 8 -2 D -2 6 -8 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.300000 D: 0.000000 E: 0.150000 Sum of squares = 0.275000000003 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.850000 D: 0.850000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 2 2 -4 B 0 0 -2 -6 4 C -2 2 0 8 -2 D -2 6 -8 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.300000 D: 0.000000 E: 0.150000 Sum of squares = 0.275000000003 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.850000 D: 0.850000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 2 2 -4 B 0 0 -2 -6 4 C -2 2 0 8 -2 D -2 6 -8 0 -6 E 4 -4 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.350000 C: 0.300000 D: 0.000000 E: 0.150000 Sum of squares = 0.275000000003 Cumulative probabilities = A: 0.200000 B: 0.550000 C: 0.850000 D: 0.850000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9913: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (7) B D E C A (7) A E C B D (7) A C D E B (7) E B A C D (6) D C A B E (6) D C B E A (5) D B C E A (5) B E D C A (5) E A B C D (4) D C B A E (4) C A D E B (4) B E D A C (4) B D C E A (4) A E B C D (4) A C E D B (4) A E C D B (3) E A C B D (2) E C A D B (1) E B C D A (1) E B C A D (1) E B A D C (1) D B C A E (1) C E D A B (1) C D B E A (1) C D A B E (1) B E A D C (1) A E B D C (1) A B E D C (1) A B D E C (1) Total count = 100 A B C D E A 0 8 -8 -4 2 B -8 0 -6 0 -8 C 8 6 0 8 0 D 4 0 -8 0 6 E -2 8 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.595903 D: 0.000000 E: 0.404097 Sum of squares = 0.518394731497 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.595903 D: 0.595903 E: 1.000000 A B C D E A 0 8 -8 -4 2 B -8 0 -6 0 -8 C 8 6 0 8 0 D 4 0 -8 0 6 E -2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 D=21 B=21 E=16 C=14 so C is eliminated. Round 2 votes counts: A=32 D=30 B=21 E=17 so E is eliminated. Round 3 votes counts: A=39 D=31 B=30 so B is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. C:211 D:201 E:200 A:199 B:189 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -8 -4 2 B -8 0 -6 0 -8 C 8 6 0 8 0 D 4 0 -8 0 6 E -2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -8 -4 2 B -8 0 -6 0 -8 C 8 6 0 8 0 D 4 0 -8 0 6 E -2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -8 -4 2 B -8 0 -6 0 -8 C 8 6 0 8 0 D 4 0 -8 0 6 E -2 8 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.500000 Sum of squares = 0.499999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9914: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C D A (7) B E C A D (6) E D B C A (5) E B D C A (4) D A C E B (4) C B E A D (4) C A B E D (4) B E D A C (4) B A C E D (4) B E D C A (3) B C A E D (3) A D C B E (3) A C D B E (3) A C B D E (3) E D C B A (2) E C D B A (2) D E C A B (2) D E B A C (2) D E A C B (2) D E A B C (2) D C E A B (2) D B E A C (2) D A E C B (2) C E B D A (2) C E B A D (2) C A D B E (2) B A D E C (2) A D C E B (2) A B D C E (2) E D B A C (1) E C B D A (1) E B D A C (1) D E C B A (1) D B A E C (1) D A B E C (1) C E D B A (1) C E D A B (1) C A E B D (1) B E A D C (1) B C E A D (1) B A E D C (1) A C B E D (1) Total count = 100 A B C D E A 0 -26 -12 -10 -22 B 26 0 6 14 16 C 12 -6 0 -4 -10 D 10 -14 4 0 -24 E 22 -16 10 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999306 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -12 -10 -22 B 26 0 6 14 16 C 12 -6 0 -4 -10 D 10 -14 4 0 -24 E 22 -16 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996337 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=32 D=21 C=17 E=16 A=14 so A is eliminated. Round 2 votes counts: B=34 D=26 C=24 E=16 so E is eliminated. Round 3 votes counts: B=39 D=34 C=27 so C is eliminated. Round 4 votes counts: B=57 D=43 so D is eliminated. IRV winner is B compare: Computing Borda winner. B:231 E:220 C:196 D:188 A:165 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -12 -10 -22 B 26 0 6 14 16 C 12 -6 0 -4 -10 D 10 -14 4 0 -24 E 22 -16 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996337 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -12 -10 -22 B 26 0 6 14 16 C 12 -6 0 -4 -10 D 10 -14 4 0 -24 E 22 -16 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996337 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -12 -10 -22 B 26 0 6 14 16 C 12 -6 0 -4 -10 D 10 -14 4 0 -24 E 22 -16 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996337 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9915: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (18) D E A C B (13) C B A E D (9) C A E D B (9) D E A B C (6) E D A C B (4) D B E A C (4) B D C E A (4) A E C D B (4) E A D C B (3) D E B A C (3) C A E B D (3) B D E A C (3) E A C D B (2) B D E C A (2) B C D A E (2) B C A D E (2) A E D C B (2) A C E D B (2) D B E C A (1) C E A D B (1) B D C A E (1) B D A E C (1) B D A C E (1) Total count = 100 A B C D E A 0 -2 -4 10 8 B 2 0 -4 -8 -4 C 4 4 0 4 4 D -10 8 -4 0 -14 E -8 4 -4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 -4 10 8 B 2 0 -4 -8 -4 C 4 4 0 4 4 D -10 8 -4 0 -14 E -8 4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 D=27 C=22 E=9 A=8 so A is eliminated. Round 2 votes counts: B=34 D=27 C=24 E=15 so E is eliminated. Round 3 votes counts: D=36 B=34 C=30 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:208 A:206 E:203 B:193 D:190 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -2 -4 10 8 B 2 0 -4 -8 -4 C 4 4 0 4 4 D -10 8 -4 0 -14 E -8 4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -4 10 8 B 2 0 -4 -8 -4 C 4 4 0 4 4 D -10 8 -4 0 -14 E -8 4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -4 10 8 B 2 0 -4 -8 -4 C 4 4 0 4 4 D -10 8 -4 0 -14 E -8 4 -4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9916: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A E D B (9) B D E A C (9) E C A B D (7) E B D A C (7) C A D B E (7) B D E C A (7) B D C A E (7) A C E D B (7) E A C B D (6) D B C A E (6) E A C D B (5) D B A C E (5) D B E A C (4) C A E B D (2) B E D C A (2) E D B A C (1) E B D C A (1) E B A D C (1) E A B D C (1) C B A D E (1) C A B E D (1) C A B D E (1) B E C D A (1) B D A C E (1) A C D B E (1) Total count = 100 A B C D E A 0 -6 -4 -2 -4 B 6 0 6 10 6 C 4 -6 0 -4 -4 D 2 -10 4 0 -2 E 4 -6 4 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999971 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -4 -2 -4 B 6 0 6 10 6 C 4 -6 0 -4 -4 D 2 -10 4 0 -2 E 4 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=27 C=21 D=15 A=8 so A is eliminated. Round 2 votes counts: E=29 C=29 B=27 D=15 so D is eliminated. Round 3 votes counts: B=42 E=29 C=29 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:214 E:202 D:197 C:195 A:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 -4 B 6 0 6 10 6 C 4 -6 0 -4 -4 D 2 -10 4 0 -2 E 4 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 -4 B 6 0 6 10 6 C 4 -6 0 -4 -4 D 2 -10 4 0 -2 E 4 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 -4 B 6 0 6 10 6 C 4 -6 0 -4 -4 D 2 -10 4 0 -2 E 4 -6 4 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9917: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (8) B C A D E (8) B C E A D (6) B C A E D (6) A E D B C (6) E A D C B (5) B A C D E (5) D C E B A (4) B C E D A (4) E D C A B (3) D E C A B (3) D E A C B (3) D A E C B (3) B A C E D (3) A E D C B (3) E D A C B (2) D C E A B (2) D C B A E (2) C D E B A (2) C D B E A (2) C B D A E (2) B A E C D (2) A D B C E (2) E D C B A (1) E A D B C (1) D C B E A (1) D B C A E (1) C E D B A (1) C D B A E (1) C B E D A (1) B E C A D (1) B C D E A (1) B C D A E (1) A E B D C (1) A D E C B (1) A B D E C (1) A B C D E (1) Total count = 100 A B C D E A 0 -26 -22 4 -2 B 26 0 0 2 18 C 22 0 0 10 28 D -4 -2 -10 0 8 E 2 -18 -28 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.630834 C: 0.369166 D: 0.000000 E: 0.000000 Sum of squares = 0.534235181067 Cumulative probabilities = A: 0.000000 B: 0.630834 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -26 -22 4 -2 B 26 0 0 2 18 C 22 0 0 10 28 D -4 -2 -10 0 8 E 2 -18 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=37 D=19 C=17 A=15 E=12 so E is eliminated. Round 2 votes counts: B=37 D=25 A=21 C=17 so C is eliminated. Round 3 votes counts: B=48 D=31 A=21 so A is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. C:230 B:223 D:196 A:177 E:174 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C , winner is: B compare: Computing GTS winners. A B C D E A 0 -26 -22 4 -2 B 26 0 0 2 18 C 22 0 0 10 28 D -4 -2 -10 0 8 E 2 -18 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -26 -22 4 -2 B 26 0 0 2 18 C 22 0 0 10 28 D -4 -2 -10 0 8 E 2 -18 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -26 -22 4 -2 B 26 0 0 2 18 C 22 0 0 10 28 D -4 -2 -10 0 8 E 2 -18 -28 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999995 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9918: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E C A D (12) D A C E B (9) A D C B E (6) E B C D A (5) D A C B E (5) D A E C B (4) C A D B E (4) B E A C D (4) A D E B C (4) E D B A C (3) E D A C B (3) E C B D A (3) E B C A D (3) D E A C B (3) E D C A B (2) E B D C A (2) E B A D C (2) D C A E B (2) C D A B E (2) C B E D A (2) C B A D E (2) B C E A D (2) B A E C D (2) B A C D E (2) E D A B C (1) E C D A B (1) E B D A C (1) E A D B C (1) C E D B A (1) C D A E B (1) C B D A E (1) C A B D E (1) B A C E D (1) A D B E C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 4 8 -2 -2 B -4 0 -4 -8 -2 C -8 4 0 -2 -14 D 2 8 2 0 -2 E 2 2 14 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999119 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 4 8 -2 -2 B -4 0 -4 -8 -2 C -8 4 0 -2 -14 D 2 8 2 0 -2 E 2 2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=27 D=23 B=23 C=14 A=13 so A is eliminated. Round 2 votes counts: D=35 E=27 B=24 C=14 so C is eliminated. Round 3 votes counts: D=42 B=30 E=28 so E is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:210 D:205 A:204 B:191 C:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 4 8 -2 -2 B -4 0 -4 -8 -2 C -8 4 0 -2 -14 D 2 8 2 0 -2 E 2 2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 -2 -2 B -4 0 -4 -8 -2 C -8 4 0 -2 -14 D 2 8 2 0 -2 E 2 2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 -2 -2 B -4 0 -4 -8 -2 C -8 4 0 -2 -14 D 2 8 2 0 -2 E 2 2 14 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999697 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9919: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B D E (9) A B C D E (8) E D C B A (7) E D B A C (5) C A D B E (5) C A B D E (5) E B D A C (4) C D A B E (4) A B D C E (4) E D B C A (3) E C D A B (3) C E D A B (3) B D A E C (3) B A D E C (3) B A D C E (3) A B C E D (3) E C A D B (2) D E B C A (2) D E B A C (2) C E A D B (2) B A E D C (2) E C D B A (1) E B C A D (1) E B A D C (1) D C B A E (1) D C A B E (1) D B E A C (1) D B A C E (1) C E D B A (1) C E A B D (1) C D E B A (1) C D E A B (1) C D A E B (1) C A E D B (1) C A E B D (1) C A D E B (1) B E A D C (1) A D B C E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 4 10 16 B -14 0 -2 0 12 C -4 2 0 10 16 D -10 0 -10 0 14 E -16 -12 -16 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 4 10 16 B -14 0 -2 0 12 C -4 2 0 10 16 D -10 0 -10 0 14 E -16 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C E , winner is: C compare: Computing IRV winner. Round 1 votes counts: E=27 C=27 A=26 B=12 D=8 so D is eliminated. Round 2 votes counts: E=31 C=29 A=26 B=14 so B is eliminated. Round 3 votes counts: A=38 E=33 C=29 so C is eliminated. Round 4 votes counts: A=58 E=42 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:222 C:212 B:198 D:197 E:171 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 4 10 16 B -14 0 -2 0 12 C -4 2 0 10 16 D -10 0 -10 0 14 E -16 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 4 10 16 B -14 0 -2 0 12 C -4 2 0 10 16 D -10 0 -10 0 14 E -16 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 4 10 16 B -14 0 -2 0 12 C -4 2 0 10 16 D -10 0 -10 0 14 E -16 -12 -16 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9920: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B A D C (9) A E B D C (8) C D B E A (7) A E B C D (7) E B D C A (6) D C B E A (6) D C A B E (5) B E C D A (5) A E D B C (5) A D C E B (5) A D E C B (4) E B A C D (3) D A C E B (3) C D B A E (3) B E A C D (3) D C A E B (2) C D A B E (2) A C D B E (2) E A B D C (1) D C B A E (1) D A C B E (1) C B D E A (1) C B D A E (1) C A D B E (1) C A B D E (1) B E D C A (1) B C E D A (1) B C A E D (1) A D E B C (1) A D C B E (1) A C D E B (1) A C B D E (1) A B E C D (1) Total count = 100 A B C D E A 0 4 12 10 14 B -4 0 4 0 -10 C -12 -4 0 -18 -8 D -10 0 18 0 -2 E -14 10 8 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 12 10 14 B -4 0 4 0 -10 C -12 -4 0 -18 -8 D -10 0 18 0 -2 E -14 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999309 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=36 E=19 D=18 C=16 B=11 so B is eliminated. Round 2 votes counts: A=36 E=28 D=18 C=18 so D is eliminated. Round 3 votes counts: A=40 C=32 E=28 so E is eliminated. Round 4 votes counts: A=56 C=44 so C is eliminated. IRV winner is A compare: Computing Borda winner. A:220 D:203 E:203 B:195 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 12 10 14 B -4 0 4 0 -10 C -12 -4 0 -18 -8 D -10 0 18 0 -2 E -14 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999309 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 12 10 14 B -4 0 4 0 -10 C -12 -4 0 -18 -8 D -10 0 18 0 -2 E -14 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999309 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 12 10 14 B -4 0 4 0 -10 C -12 -4 0 -18 -8 D -10 0 18 0 -2 E -14 10 8 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999309 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9921: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (7) E B A D C (6) E B A C D (6) D A C B E (5) C A D E B (5) E B D C A (4) E B D A C (4) E B C D A (4) C A D B E (4) B D E A C (4) A C B D E (4) E C B A D (3) E B C A D (3) C D A B E (3) C A E B D (3) B D A E C (3) E C A B D (2) D C B A E (2) D B E A C (2) D B C A E (2) D B A C E (2) D A B C E (2) C D E A B (2) C D A E B (2) B E D A C (2) B A E D C (2) B A D E C (2) E D C B A (1) E C D B A (1) D B A E C (1) C E D A B (1) C E A D B (1) C A E D B (1) A E B C D (1) A D C B E (1) A C B E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -2 -8 8 B 8 0 2 10 0 C 2 -2 0 -4 -2 D 8 -10 4 0 8 E -8 0 2 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.642283 C: 0.000000 D: 0.000000 E: 0.357717 Sum of squares = 0.540488722455 Cumulative probabilities = A: 0.000000 B: 0.642283 C: 0.642283 D: 0.642283 E: 1.000000 A B C D E A 0 -8 -2 -8 8 B 8 0 2 10 0 C 2 -2 0 -4 -2 D 8 -10 4 0 8 E -8 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500299 C: 0.000000 D: 0.000000 E: 0.499701 Sum of squares = 0.500000179185 Cumulative probabilities = A: 0.000000 B: 0.500299 C: 0.500299 D: 0.500299 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=34 D=23 C=22 B=13 A=8 so A is eliminated. Round 2 votes counts: E=35 C=27 D=24 B=14 so B is eliminated. Round 3 votes counts: E=39 D=33 C=28 so C is eliminated. Round 4 votes counts: D=54 E=46 so E is eliminated. IRV winner is D compare: Computing Borda winner. B:210 D:205 C:197 A:195 E:193 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 -2 -8 8 B 8 0 2 10 0 C 2 -2 0 -4 -2 D 8 -10 4 0 8 E -8 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500299 C: 0.000000 D: 0.000000 E: 0.499701 Sum of squares = 0.500000179185 Cumulative probabilities = A: 0.000000 B: 0.500299 C: 0.500299 D: 0.500299 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -2 -8 8 B 8 0 2 10 0 C 2 -2 0 -4 -2 D 8 -10 4 0 8 E -8 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500299 C: 0.000000 D: 0.000000 E: 0.499701 Sum of squares = 0.500000179185 Cumulative probabilities = A: 0.000000 B: 0.500299 C: 0.500299 D: 0.500299 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -2 -8 8 B 8 0 2 10 0 C 2 -2 0 -4 -2 D 8 -10 4 0 8 E -8 0 2 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500299 C: 0.000000 D: 0.000000 E: 0.499701 Sum of squares = 0.500000179185 Cumulative probabilities = A: 0.000000 B: 0.500299 C: 0.500299 D: 0.500299 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9922: compare: Profile of ballots (with multiplicities), in decreasing order by count: B A E D C (11) C E D B A (9) A B E D C (8) D E C A B (6) B A E C D (5) A B D E C (5) E B A D C (4) C E B A D (4) D C E A B (3) C D E B A (3) C D E A B (3) C D A B E (3) C B A E D (3) B A C E D (3) E D A B C (2) E B A C D (2) D A B E C (2) B C A E D (2) B A C D E (2) A D B E C (2) A B D C E (2) E D C B A (1) E D B A C (1) E B C A D (1) E A B D C (1) D E A B C (1) D C A B E (1) D A B C E (1) C D A E B (1) C B E D A (1) C B E A D (1) C B A D E (1) C A D B E (1) C A B D E (1) A E B D C (1) A D B C E (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 10 24 14 B 8 0 16 18 14 C -10 -16 0 -6 -6 D -24 -18 6 0 -20 E -14 -14 6 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 10 24 14 B 8 0 16 18 14 C -10 -16 0 -6 -6 D -24 -18 6 0 -20 E -14 -14 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=31 B=23 A=20 D=14 E=12 so E is eliminated. Round 2 votes counts: C=31 B=30 A=21 D=18 so D is eliminated. Round 3 votes counts: C=42 B=31 A=27 so A is eliminated. Round 4 votes counts: B=58 C=42 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:228 A:220 E:199 C:181 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -8 10 24 14 B 8 0 16 18 14 C -10 -16 0 -6 -6 D -24 -18 6 0 -20 E -14 -14 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 10 24 14 B 8 0 16 18 14 C -10 -16 0 -6 -6 D -24 -18 6 0 -20 E -14 -14 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 10 24 14 B 8 0 16 18 14 C -10 -16 0 -6 -6 D -24 -18 6 0 -20 E -14 -14 6 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9923: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (8) A C E B D (8) E A C B D (6) D B C A E (6) E C A D B (4) D E C B A (4) D E B C A (4) B D C A E (4) A C B E D (4) E D B C A (3) E B D A C (3) E A B C D (3) C D A B E (3) E D A C B (2) E A B D C (2) D C E B A (2) D B E C A (2) B D A C E (2) B A E D C (2) A E B C D (2) A B C E D (2) E D C B A (1) E D B A C (1) E B A D C (1) E A C D B (1) D C E A B (1) D C B A E (1) D C A B E (1) D B E A C (1) D B C E A (1) C E D A B (1) C B A D E (1) C A E D B (1) C A D E B (1) B E D A C (1) B E A D C (1) B D E A C (1) B D A E C (1) B C D A E (1) B A D C E (1) B A C D E (1) A E C B D (1) A C B D E (1) A B E C D (1) A B C D E (1) Total count = 100 A B C D E A 0 8 0 -10 -8 B -8 0 -4 2 -20 C 0 4 0 -14 -12 D 10 -2 14 0 -18 E 8 20 12 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999972 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 8 0 -10 -8 B -8 0 -4 2 -20 C 0 4 0 -14 -12 D 10 -2 14 0 -18 E 8 20 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 D=23 A=20 B=15 C=7 so C is eliminated. Round 2 votes counts: E=36 D=26 A=22 B=16 so B is eliminated. Round 3 votes counts: E=38 D=35 A=27 so A is eliminated. Round 4 votes counts: E=59 D=41 so D is eliminated. IRV winner is E compare: Computing Borda winner. E:229 D:202 A:195 C:189 B:185 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 8 0 -10 -8 B -8 0 -4 2 -20 C 0 4 0 -14 -12 D 10 -2 14 0 -18 E 8 20 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 0 -10 -8 B -8 0 -4 2 -20 C 0 4 0 -14 -12 D 10 -2 14 0 -18 E 8 20 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 0 -10 -8 B -8 0 -4 2 -20 C 0 4 0 -14 -12 D 10 -2 14 0 -18 E 8 20 12 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999627 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9924: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D E B (11) B E C D A (7) C E B A D (6) A D C B E (6) E C D A B (5) E C B D A (4) D A E C B (4) A D C E B (4) D A B E C (3) C E A B D (3) C A E D B (3) B E D A C (3) B D A E C (3) B A D C E (3) A D B C E (3) E B C D A (2) D E A B C (2) D B A E C (2) C E A D B (2) B E D C A (2) B C A E D (2) B A C D E (2) E D C A B (1) E D A C B (1) E D A B C (1) E C D B A (1) E C B A D (1) D A E B C (1) D A C E B (1) C B E A D (1) C A E B D (1) C A D B E (1) C A B E D (1) C A B D E (1) B E C A D (1) B D E A C (1) B C E A D (1) B C A D E (1) B A D E C (1) A C D E B (1) Total count = 100 A B C D E A 0 12 -16 12 10 B -12 0 -18 -6 -10 C 16 18 0 16 8 D -12 6 -16 0 2 E -10 10 -8 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -16 12 10 B -12 0 -18 -6 -10 C 16 18 0 16 8 D -12 6 -16 0 2 E -10 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=27 E=16 A=14 D=13 so D is eliminated. Round 2 votes counts: C=30 B=29 A=23 E=18 so E is eliminated. Round 3 votes counts: C=42 B=31 A=27 so A is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:229 A:209 E:195 D:190 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -16 12 10 B -12 0 -18 -6 -10 C 16 18 0 16 8 D -12 6 -16 0 2 E -10 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -16 12 10 B -12 0 -18 -6 -10 C 16 18 0 16 8 D -12 6 -16 0 2 E -10 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -16 12 10 B -12 0 -18 -6 -10 C 16 18 0 16 8 D -12 6 -16 0 2 E -10 10 -8 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999922 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9925: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E C B (11) B C E A D (11) A D E B C (9) C B E A D (7) E A D C B (5) B E C A D (5) C B E D A (4) D C A E B (3) D A E B C (3) B C A D E (3) E C D A B (2) D A C B E (2) D A B C E (2) C E D B A (2) C E D A B (2) C E B D A (2) C D E A B (2) C D A E B (2) C B D E A (2) B C D E A (2) B C D A E (2) E D A C B (1) E C A D B (1) E B A C D (1) E A D B C (1) E A B D C (1) D E C A B (1) D E A C B (1) D C B A E (1) D C A B E (1) D A B E C (1) C E B A D (1) B E A C D (1) B A E C D (1) B A D E C (1) B A C D E (1) A E D B C (1) A E B D C (1) Total count = 100 A B C D E A 0 6 -12 2 -10 B -6 0 -6 -8 -6 C 12 6 0 8 4 D -2 8 -8 0 0 E 10 6 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999677 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -12 2 -10 B -6 0 -6 -8 -6 C 12 6 0 8 4 D -2 8 -8 0 0 E 10 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=27 D=26 C=24 E=12 A=11 so A is eliminated. Round 2 votes counts: D=35 B=27 C=24 E=14 so E is eliminated. Round 3 votes counts: D=43 B=30 C=27 so C is eliminated. Round 4 votes counts: D=54 B=46 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:215 E:206 D:199 A:193 B:187 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -12 2 -10 B -6 0 -6 -8 -6 C 12 6 0 8 4 D -2 8 -8 0 0 E 10 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -12 2 -10 B -6 0 -6 -8 -6 C 12 6 0 8 4 D -2 8 -8 0 0 E 10 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -12 2 -10 B -6 0 -6 -8 -6 C 12 6 0 8 4 D -2 8 -8 0 0 E 10 6 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999982 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9926: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E D C B (15) B C D E A (12) E B A C D (9) B C D A E (8) A D C E B (8) E A B D C (7) D C A E B (7) E A D C B (5) B E C D A (4) E A B C D (3) C D B A E (3) C B D A E (3) B C E D A (3) D C A B E (2) B E A C D (2) A E B D C (2) E A D B C (1) D C B A E (1) D A C E B (1) D A C B E (1) C B D E A (1) B E C A D (1) A E D B C (1) Total count = 100 A B C D E A 0 6 10 8 4 B -6 0 6 10 -18 C -10 -6 0 -2 0 D -8 -10 2 0 -6 E -4 18 0 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999524 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 10 8 4 B -6 0 6 10 -18 C -10 -6 0 -2 0 D -8 -10 2 0 -6 E -4 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=26 E=25 D=12 C=7 so C is eliminated. Round 2 votes counts: B=34 A=26 E=25 D=15 so D is eliminated. Round 3 votes counts: B=38 A=37 E=25 so E is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:214 E:210 B:196 C:191 D:189 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 6 10 8 4 B -6 0 6 10 -18 C -10 -6 0 -2 0 D -8 -10 2 0 -6 E -4 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 10 8 4 B -6 0 6 10 -18 C -10 -6 0 -2 0 D -8 -10 2 0 -6 E -4 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 10 8 4 B -6 0 6 10 -18 C -10 -6 0 -2 0 D -8 -10 2 0 -6 E -4 18 0 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999983 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9927: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C B E D (11) B E D C A (8) A C B D E (7) D E B C A (6) B C E D A (6) B C A E D (6) A C D E B (6) D E B A C (5) E D B C A (3) D E A B C (3) C B A E D (3) C A B E D (3) B D E C A (3) B A C E D (3) A D E C B (3) A C E D B (3) C E D A B (2) C A E D B (2) B D E A C (2) A B C D E (2) E D C A B (1) D E A C B (1) D B E C A (1) D A E C B (1) D A E B C (1) C B E D A (1) C B E A D (1) C A E B D (1) B C E A D (1) B A C D E (1) A D C E B (1) A D B E C (1) A C E B D (1) Total count = 100 A B C D E A 0 0 4 12 12 B 0 0 4 20 20 C -4 -4 0 20 22 D -12 -20 -20 0 -12 E -12 -20 -22 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.571549 B: 0.428451 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.510238511624 Cumulative probabilities = A: 0.571549 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 4 12 12 B 0 0 4 20 20 C -4 -4 0 20 22 D -12 -20 -20 0 -12 E -12 -20 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999606 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 B=30 D=18 C=13 E=4 so E is eliminated. Round 2 votes counts: A=35 B=30 D=22 C=13 so C is eliminated. Round 3 votes counts: A=41 B=35 D=24 so D is eliminated. Round 4 votes counts: B=50 A=50 so B is eliminated. IRV winner is A compare: Computing Borda winner. B:222 C:217 A:214 E:179 D:168 Borda winner is B compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 0 4 12 12 B 0 0 4 20 20 C -4 -4 0 20 22 D -12 -20 -20 0 -12 E -12 -20 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999606 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 4 12 12 B 0 0 4 20 20 C -4 -4 0 20 22 D -12 -20 -20 0 -12 E -12 -20 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999606 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 4 12 12 B 0 0 4 20 20 C -4 -4 0 20 22 D -12 -20 -20 0 -12 E -12 -20 -22 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.500000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999606 Cumulative probabilities = A: 0.500000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9928: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) B E C D A (8) D A C E B (7) B E C A D (7) E C B A D (6) A D B C E (6) C E D B A (5) C E D A B (5) B A D E C (5) A D C E B (5) B D A E C (4) E C B D A (3) D A B C E (3) B A E C D (3) E B C D A (2) D C A E B (2) D B A E C (2) D A C B E (2) D A B E C (2) C E A B D (2) B E A C D (2) A D B E C (2) E C D B A (1) D C E A B (1) C E B D A (1) B E D C A (1) B E D A C (1) B E A D C (1) B D E A C (1) A D C B E (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 -4 -2 -10 B 6 0 2 -4 4 C 4 -2 0 6 -4 D 2 4 -6 0 -14 E 10 -4 4 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.181818 E: 0.181818 Sum of squares = 0.47107438 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.818182 E: 1.000000 A B C D E A 0 -6 -4 -2 -10 B 6 0 2 -4 4 C 4 -2 0 6 -4 D 2 4 -6 0 -14 E 10 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.181818 E: 0.181818 Sum of squares = 0.471074379689 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.818182 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=33 C=21 D=19 A=15 E=12 so E is eliminated. Round 2 votes counts: B=35 C=31 D=19 A=15 so A is eliminated. Round 3 votes counts: B=36 D=33 C=31 so C is eliminated. Round 4 votes counts: D=52 B=48 so B is eliminated. IRV winner is D compare: Computing Borda winner. E:212 B:204 C:202 D:193 A:189 Borda winner is E compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B C E , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -4 -2 -10 B 6 0 2 -4 4 C 4 -2 0 6 -4 D 2 4 -6 0 -14 E 10 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.181818 E: 0.181818 Sum of squares = 0.471074379689 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.818182 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -4 -2 -10 B 6 0 2 -4 4 C 4 -2 0 6 -4 D 2 4 -6 0 -14 E 10 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.181818 E: 0.181818 Sum of squares = 0.471074379689 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.818182 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -4 -2 -10 B 6 0 2 -4 4 C 4 -2 0 6 -4 D 2 4 -6 0 -14 E 10 -4 4 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.636364 C: 0.000000 D: 0.181818 E: 0.181818 Sum of squares = 0.471074379689 Cumulative probabilities = A: 0.000000 B: 0.636364 C: 0.636364 D: 0.818182 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9929: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (34) A B E D C (15) D E C B A (7) A B C E D (7) A C B D E (4) A B E C D (3) E D B C A (2) C B D E A (2) C A D B E (2) B E D A C (2) A E D B C (2) A E B D C (2) A C D E B (2) A C B E D (2) E D A C B (1) E B D C A (1) E A D C B (1) D E B C A (1) C D E A B (1) C D B E A (1) C D A E B (1) C B D A E (1) C B A D E (1) C A B D E (1) B E D C A (1) B A E D C (1) A E D C B (1) A B C D E (1) Total count = 100 A B C D E A 0 -8 -12 -10 -8 B 8 0 -24 -12 -12 C 12 24 0 26 20 D 10 12 -26 0 18 E 8 12 -20 -18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -8 -12 -10 -8 B 8 0 -24 -12 -12 C 12 24 0 26 20 D 10 12 -26 0 18 E 8 12 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=44 A=39 D=8 E=5 B=4 so B is eliminated. Round 2 votes counts: C=44 A=40 E=8 D=8 so E is eliminated. Round 3 votes counts: C=44 A=41 D=15 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:241 D:207 E:191 A:181 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -8 -12 -10 -8 B 8 0 -24 -12 -12 C 12 24 0 26 20 D 10 12 -26 0 18 E 8 12 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -12 -10 -8 B 8 0 -24 -12 -12 C 12 24 0 26 20 D 10 12 -26 0 18 E 8 12 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -12 -10 -8 B 8 0 -24 -12 -12 C 12 24 0 26 20 D 10 12 -26 0 18 E 8 12 -20 -18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999829 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9930: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B D E C (6) E B A C D (5) E B D C A (4) D C E B A (4) D B E C A (4) C E A B D (4) E C B A D (3) D A B C E (3) C E D B A (3) C E A D B (3) C D A E B (3) C A E B D (3) B A E D C (3) A B E C D (3) E C B D A (2) E B C A D (2) D C A E B (2) D C A B E (2) D B E A C (2) D B C A E (2) D B A C E (2) D A C B E (2) C E D A B (2) C D E A B (2) C A E D B (2) B D E A C (2) B D A E C (2) B A E C D (2) B A D E C (2) A E B C D (2) E C D B A (1) E B C D A (1) E A C B D (1) D E B C A (1) D C B A E (1) D B A E C (1) C E B A D (1) C D E B A (1) C A D E B (1) B E A C D (1) A D C B E (1) A D B C E (1) A C D B E (1) A C B D E (1) A B D C E (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 -8 2 2 B 4 0 8 6 -6 C 8 -8 0 4 0 D -2 -6 -4 0 2 E -2 6 0 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.391792 D: 0.000000 E: 0.608208 Sum of squares = 0.523418135548 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.391792 D: 0.391792 E: 1.000000 A B C D E A 0 -4 -8 2 2 B 4 0 8 6 -6 C 8 -8 0 4 0 D -2 -6 -4 0 2 E -2 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.571429 Sum of squares = 0.510204084137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.428571 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=25 E=19 A=18 B=12 so B is eliminated. Round 2 votes counts: D=30 C=25 A=25 E=20 so E is eliminated. Round 3 votes counts: D=34 C=34 A=32 so A is eliminated. Round 4 votes counts: C=52 D=48 so D is eliminated. IRV winner is C compare: Computing Borda winner. B:206 C:202 E:201 A:196 D:195 Borda winner is B compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 -8 2 2 B 4 0 8 6 -6 C 8 -8 0 4 0 D -2 -6 -4 0 2 E -2 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.571429 Sum of squares = 0.510204084137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.428571 E: 1.000000 GTS winners are ['C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -8 2 2 B 4 0 8 6 -6 C 8 -8 0 4 0 D -2 -6 -4 0 2 E -2 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.571429 Sum of squares = 0.510204084137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.428571 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -8 2 2 B 4 0 8 6 -6 C 8 -8 0 4 0 D -2 -6 -4 0 2 E -2 6 0 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.000000 E: 0.571429 Sum of squares = 0.510204084137 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.428571 D: 0.428571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9931: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (14) A D C B E (13) D C A E B (6) D C A B E (6) B E A D C (5) B E A C D (4) B A D C E (4) E C D B A (3) E C D A B (3) E B C A D (3) E B A C D (3) C D A E B (3) B E D C A (3) B D C A E (3) B D A C E (3) A E C D B (3) A C D E B (3) D C B E A (2) D A C B E (2) B A E D C (2) A B D C E (2) E A C D B (1) E A B C D (1) D C B A E (1) D A B C E (1) B E D A C (1) B D C E A (1) B A D E C (1) A E B C D (1) A D C E B (1) A D B C E (1) Total count = 100 A B C D E A 0 -6 4 -4 12 B 6 0 6 2 10 C -4 -6 0 -16 4 D 4 -2 16 0 6 E -12 -10 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 4 -4 12 B 6 0 6 2 10 C -4 -6 0 -16 4 D 4 -2 16 0 6 E -12 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 A=24 D=18 C=3 so C is eliminated. Round 2 votes counts: E=28 B=27 A=24 D=21 so D is eliminated. Round 3 votes counts: A=42 B=30 E=28 so E is eliminated. Round 4 votes counts: B=53 A=47 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:212 D:212 A:203 C:189 E:184 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 4 -4 12 B 6 0 6 2 10 C -4 -6 0 -16 4 D 4 -2 16 0 6 E -12 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 4 -4 12 B 6 0 6 2 10 C -4 -6 0 -16 4 D 4 -2 16 0 6 E -12 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 4 -4 12 B 6 0 6 2 10 C -4 -6 0 -16 4 D 4 -2 16 0 6 E -12 -10 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999847 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9932: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (13) B D C A E (12) E A C D B (9) A E C B D (9) E A D C B (6) D B E C A (5) A E C D B (5) B C D A E (4) A C E B D (4) D B E A C (3) C A E B D (3) E D C B A (2) E D A C B (2) E C A D B (2) C E A D B (2) C B A D E (2) C A B E D (2) B D C E A (2) B D A C E (2) A E D C B (2) A B C D E (2) E C D A B (1) E A D B C (1) D E B A C (1) C B D A E (1) B C A D E (1) B A C D E (1) A E B C D (1) Total count = 100 A B C D E A 0 2 -4 4 2 B -2 0 -4 -8 0 C 4 4 0 -2 2 D -4 8 2 0 -2 E -2 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999752 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -4 4 2 B -2 0 -4 -8 0 C 4 4 0 -2 2 D -4 8 2 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999954 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A E , winner is: A compare: Computing IRV winner. Round 1 votes counts: E=23 A=23 D=22 B=22 C=10 so C is eliminated. Round 2 votes counts: A=28 E=25 B=25 D=22 so D is eliminated. Round 3 votes counts: B=46 A=28 E=26 so E is eliminated. Round 4 votes counts: A=51 B=49 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:204 A:202 D:202 E:199 B:193 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C E , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -4 4 2 B -2 0 -4 -8 0 C 4 4 0 -2 2 D -4 8 2 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999954 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 4 2 B -2 0 -4 -8 0 C 4 4 0 -2 2 D -4 8 2 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999954 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 4 2 B -2 0 -4 -8 0 C 4 4 0 -2 2 D -4 8 2 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.200000 B: 0.000000 C: 0.400000 D: 0.400000 E: 0.000000 Sum of squares = 0.359999999954 Cumulative probabilities = A: 0.200000 B: 0.200000 C: 0.600000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9933: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (11) D B A E C (7) D B E A C (6) D A E B C (6) C A E B D (6) B E D A C (5) D A B E C (4) C B E A D (4) B D E C A (4) D A C E B (3) C A E D B (3) A D E B C (3) E B A C D (2) D C A E B (2) D B C E A (2) D A E C B (2) D A C B E (2) D A B C E (2) C A D E B (2) B E C D A (2) B E C A D (2) B E A C D (2) A D C E B (2) E C B A D (1) E B C A D (1) E A C B D (1) D C B E A (1) D C B A E (1) D B A C E (1) C E A B D (1) C D A B E (1) B E D C A (1) B D E A C (1) B C E D A (1) B C E A D (1) A E C D B (1) A D E C B (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 6 -8 2 B 12 0 6 -8 0 C -6 -6 0 -12 -4 D 8 8 12 0 8 E -2 0 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 6 -8 2 B 12 0 6 -8 0 C -6 -6 0 -12 -4 D 8 8 12 0 8 E -2 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 C=28 B=19 A=9 E=5 so E is eliminated. Round 2 votes counts: D=39 C=29 B=22 A=10 so A is eliminated. Round 3 votes counts: D=45 C=33 B=22 so B is eliminated. Round 4 votes counts: D=56 C=44 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:218 B:205 E:197 A:194 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 6 -8 2 B 12 0 6 -8 0 C -6 -6 0 -12 -4 D 8 8 12 0 8 E -2 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 6 -8 2 B 12 0 6 -8 0 C -6 -6 0 -12 -4 D 8 8 12 0 8 E -2 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 6 -8 2 B 12 0 6 -8 0 C -6 -6 0 -12 -4 D 8 8 12 0 8 E -2 0 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9934: compare: Profile of ballots (with multiplicities), in decreasing order by count: B C A E D (10) A C B D E (7) D E A C B (6) E D B A C (5) B A C E D (5) D A E C B (4) C A D B E (4) C A B D E (4) B C E A D (4) E B D A C (3) C D A E B (3) B E C D A (3) A D E B C (3) A C D B E (3) A B C D E (3) E D B C A (2) E D A B C (2) C D E A B (2) C B A E D (2) C B A D E (2) B A C D E (2) A D C E B (2) A C D E B (2) E D C B A (1) E C B D A (1) E B D C A (1) D E C A B (1) D E A B C (1) D C E A B (1) D C A E B (1) D A C E B (1) C B E D A (1) C A D E B (1) B E D C A (1) B E A D C (1) B A E D C (1) B A E C D (1) A E B D C (1) A D C B E (1) A B D C E (1) Total count = 100 A B C D E A 0 8 10 20 28 B -8 0 0 8 12 C -10 0 0 20 24 D -20 -8 -20 0 10 E -28 -12 -24 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999877 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 10 20 28 B -8 0 0 8 12 C -10 0 0 20 24 D -20 -8 -20 0 10 E -28 -12 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=28 A=23 C=19 E=15 D=15 so E is eliminated. Round 2 votes counts: B=32 D=25 A=23 C=20 so C is eliminated. Round 3 votes counts: B=38 A=32 D=30 so D is eliminated. Round 4 votes counts: A=54 B=46 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:233 C:217 B:206 D:181 E:163 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 8 10 20 28 B -8 0 0 8 12 C -10 0 0 20 24 D -20 -8 -20 0 10 E -28 -12 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 10 20 28 B -8 0 0 8 12 C -10 0 0 20 24 D -20 -8 -20 0 10 E -28 -12 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 10 20 28 B -8 0 0 8 12 C -10 0 0 20 24 D -20 -8 -20 0 10 E -28 -12 -24 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999961 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9935: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B C E A (8) B D E C A (8) A C E D B (7) E A C B D (6) C E A D B (5) B D A E C (4) A E C D B (4) E B C A D (3) D C A E B (3) D B C A E (3) D B A C E (3) C A E D B (3) B D A C E (3) A E B C D (3) A C D E B (3) E C B A D (2) E C A D B (2) E B A C D (2) D C A B E (2) D A C B E (2) C A D E B (2) B E D A C (2) B D C E A (2) E C D B A (1) E C D A B (1) E C A B D (1) E B D C A (1) E B C D A (1) E A B C D (1) D C B E A (1) C E D A B (1) C D E B A (1) C D E A B (1) C D A E B (1) B E D C A (1) B A D E C (1) A E C B D (1) A D C E B (1) A D C B E (1) A C D B E (1) A B D E C (1) Total count = 100 A B C D E A 0 6 -8 0 -2 B -6 0 -6 -14 -14 C 8 6 0 6 8 D 0 14 -6 0 4 E 2 14 -8 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999995 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -8 0 -2 B -6 0 -6 -14 -14 C 8 6 0 6 8 D 0 14 -6 0 4 E 2 14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A D , winner is: A compare: Computing IRV winner. Round 1 votes counts: D=22 A=22 E=21 B=21 C=14 so C is eliminated. Round 2 votes counts: E=27 A=27 D=25 B=21 so B is eliminated. Round 3 votes counts: D=42 E=30 A=28 so A is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. C:214 D:206 E:202 A:198 B:180 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -8 0 -2 B -6 0 -6 -14 -14 C 8 6 0 6 8 D 0 14 -6 0 4 E 2 14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -8 0 -2 B -6 0 -6 -14 -14 C 8 6 0 6 8 D 0 14 -6 0 4 E 2 14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -8 0 -2 B -6 0 -6 -14 -14 C 8 6 0 6 8 D 0 14 -6 0 4 E 2 14 -8 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999979 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9936: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C E B (8) B E C A D (7) E B A C D (5) D A E B C (5) C D A B E (5) C B E A D (5) A C E B D (5) E B D A C (4) C B E D A (4) E B A D C (3) C A D B E (3) A E B D C (3) D E A B C (2) D C B E A (2) D B E C A (2) D A E C B (2) C D B E A (2) C B D E A (2) C A B E D (2) B E D C A (2) B C E D A (2) A E D B C (2) A D E C B (2) A C E D B (2) E D B A C (1) E A B D C (1) D E B A C (1) D C A B E (1) D B E A C (1) D B C E A (1) C D B A E (1) C B A E D (1) C B A D E (1) C A B D E (1) B E D A C (1) B E C D A (1) B E A C D (1) B D E C A (1) B C D E A (1) A E B C D (1) A D E B C (1) A D C E B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 6 -4 -4 B 4 0 -2 8 0 C -6 2 0 6 2 D 4 -8 -6 0 -6 E 4 0 -2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.166667 B: 0.255568 C: 0.333333 D: 0.000000 E: 0.244432 Sum of squares = 0.263950896472 Cumulative probabilities = A: 0.166667 B: 0.422235 C: 0.755568 D: 0.755568 E: 1.000000 A B C D E A 0 -4 6 -4 -4 B 4 0 -2 8 0 C -6 2 0 6 2 D 4 -8 -6 0 -6 E 4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.250000 C: 0.333333 D: 0.000000 E: 0.250000 Sum of squares = 0.263888888886 Cumulative probabilities = A: 0.166667 B: 0.416667 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 D=25 A=18 B=16 E=14 so E is eliminated. Round 2 votes counts: B=28 C=27 D=26 A=19 so A is eliminated. Round 3 votes counts: C=35 B=33 D=32 so D is eliminated. Round 4 votes counts: C=51 B=49 so B is eliminated. IRV winner is C compare: Computing Borda winner. B:205 E:204 C:202 A:197 D:192 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B E , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 -4 -4 B 4 0 -2 8 0 C -6 2 0 6 2 D 4 -8 -6 0 -6 E 4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.250000 C: 0.333333 D: 0.000000 E: 0.250000 Sum of squares = 0.263888888886 Cumulative probabilities = A: 0.166667 B: 0.416667 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['A', 'B', 'C', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 -4 -4 B 4 0 -2 8 0 C -6 2 0 6 2 D 4 -8 -6 0 -6 E 4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.250000 C: 0.333333 D: 0.000000 E: 0.250000 Sum of squares = 0.263888888886 Cumulative probabilities = A: 0.166667 B: 0.416667 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 -4 -4 B 4 0 -2 8 0 C -6 2 0 6 2 D 4 -8 -6 0 -6 E 4 0 -2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.166667 B: 0.250000 C: 0.333333 D: 0.000000 E: 0.250000 Sum of squares = 0.263888888886 Cumulative probabilities = A: 0.166667 B: 0.416667 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9937: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E C A B (7) E C D A B (6) E C B A D (6) B D A C E (6) B A D C E (6) E C B D A (4) D A B C E (4) B A C E D (4) E D C A B (3) C A E B D (3) B D E A C (3) B A C D E (3) E C A D B (2) E B C A D (2) D E B C A (2) D B A E C (2) D B A C E (2) D A C E B (2) D A C B E (2) C E D A B (2) C D A E B (2) B D A E C (2) A B D C E (2) A B C D E (2) E D B C A (1) E C D B A (1) E C A B D (1) E B D C A (1) E B C D A (1) D E C B A (1) D E A C B (1) D E A B C (1) D B E A C (1) D A E C B (1) C E A D B (1) C E A B D (1) C A D E B (1) B E C A D (1) B E A C D (1) B D E C A (1) A D C E B (1) A D C B E (1) A D B C E (1) A C D E B (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 0 -18 -2 B 2 0 -2 2 -10 C 0 2 0 -8 -4 D 18 -2 8 0 16 E 2 10 4 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.357143 E: 0.071429 Sum of squares = 0.459183673466 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 0.928571 E: 1.000000 A B C D E A 0 -2 0 -18 -2 B 2 0 -2 2 -10 C 0 2 0 -8 -4 D 18 -2 8 0 16 E 2 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.357143 E: 0.071429 Sum of squares = 0.459183673292 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 0.928571 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=28 B=27 D=26 C=10 A=9 so A is eliminated. Round 2 votes counts: B=31 D=29 E=28 C=12 so C is eliminated. Round 3 votes counts: E=35 D=33 B=32 so B is eliminated. Round 4 votes counts: D=58 E=42 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:220 E:200 B:196 C:195 A:189 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 0 -18 -2 B 2 0 -2 2 -10 C 0 2 0 -8 -4 D 18 -2 8 0 16 E 2 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.357143 E: 0.071429 Sum of squares = 0.459183673292 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 0.928571 E: 1.000000 GTS winners are ['B', 'D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 0 -18 -2 B 2 0 -2 2 -10 C 0 2 0 -8 -4 D 18 -2 8 0 16 E 2 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.357143 E: 0.071429 Sum of squares = 0.459183673292 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 0.928571 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 0 -18 -2 B 2 0 -2 2 -10 C 0 2 0 -8 -4 D 18 -2 8 0 16 E 2 10 4 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.571429 C: 0.000000 D: 0.357143 E: 0.071429 Sum of squares = 0.459183673292 Cumulative probabilities = A: 0.000000 B: 0.571429 C: 0.571429 D: 0.928571 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9938: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A B C D (8) D A E B C (8) C B E A D (8) A E D B C (7) A E D C B (6) D A C B E (5) C B E D A (5) C B D E A (5) D B C E A (4) D A E C B (4) D C B A E (3) D A C E B (3) B C D E A (3) E B A C D (2) E A B D C (2) D C A B E (2) B D E C A (2) B D C E A (2) B C E A D (2) A E C B D (2) A D E C B (2) E D A B C (1) E B C A D (1) E B A D C (1) E A C B D (1) D C B E A (1) C E A B D (1) C D B A E (1) C B D A E (1) C A D E B (1) B E C D A (1) B E C A D (1) B C E D A (1) A E C D B (1) A E B D C (1) A E B C D (1) Total count = 100 A B C D E A 0 12 10 -4 -4 B -12 0 -4 2 -6 C -10 4 0 -8 -4 D 4 -2 8 0 -6 E 4 6 4 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999994 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 12 10 -4 -4 B -12 0 -4 2 -6 C -10 4 0 -8 -4 D 4 -2 8 0 -6 E 4 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=30 C=22 A=20 E=16 B=12 so B is eliminated. Round 2 votes counts: D=34 C=28 A=20 E=18 so E is eliminated. Round 3 votes counts: D=35 A=34 C=31 so C is eliminated. Round 4 votes counts: D=52 A=48 so A is eliminated. IRV winner is D compare: Computing Borda winner. E:210 A:207 D:202 C:191 B:190 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 12 10 -4 -4 B -12 0 -4 2 -6 C -10 4 0 -8 -4 D 4 -2 8 0 -6 E 4 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 10 -4 -4 B -12 0 -4 2 -6 C -10 4 0 -8 -4 D 4 -2 8 0 -6 E 4 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 10 -4 -4 B -12 0 -4 2 -6 C -10 4 0 -8 -4 D 4 -2 8 0 -6 E 4 6 4 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999147 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9939: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E B D (14) D B E C A (13) D B E A C (9) A D B E C (8) D A B E C (7) C E B A D (7) E B C D A (6) C E B D A (6) C A E B D (6) E B D C A (3) D B A E C (2) B D E C A (2) A D C E B (2) A D C B E (2) A D B C E (2) A C E D B (2) A C D E B (2) A C D B E (2) E D B C A (1) D E B C A (1) C E A B D (1) B E D C A (1) A C B E D (1) Total count = 100 A B C D E A 0 -2 6 -2 0 B 2 0 10 -6 -2 C -6 -10 0 -6 -6 D 2 6 6 0 4 E 0 2 6 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999991 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 6 -2 0 B 2 0 10 -6 -2 C -6 -10 0 -6 -6 D 2 6 6 0 4 E 0 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 D=32 C=20 E=10 B=3 so B is eliminated. Round 2 votes counts: A=35 D=34 C=20 E=11 so E is eliminated. Round 3 votes counts: D=39 A=35 C=26 so C is eliminated. Round 4 votes counts: D=51 A=49 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:209 B:202 E:202 A:201 C:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 6 -2 0 B 2 0 10 -6 -2 C -6 -10 0 -6 -6 D 2 6 6 0 4 E 0 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 6 -2 0 B 2 0 10 -6 -2 C -6 -10 0 -6 -6 D 2 6 6 0 4 E 0 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 6 -2 0 B 2 0 10 -6 -2 C -6 -10 0 -6 -6 D 2 6 6 0 4 E 0 2 6 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999763 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9940: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A D B (8) D B E A C (6) C B A E D (6) C A B E D (6) E D A B C (5) B D E A C (5) B C D A E (5) D E B A C (4) D E A B C (4) C B D E A (4) C A E D B (4) B D A E C (4) B C A D E (4) C A E B D (3) B C D E A (3) E A D B C (2) C E D A B (2) C B A D E (2) B D A C E (2) A E C D B (2) E D C A B (1) E D A C B (1) E C D A B (1) E A D C B (1) D E C B A (1) D E B C A (1) D C E A B (1) D B E C A (1) D B C E A (1) C D E B A (1) C D E A B (1) B D E C A (1) B D C A E (1) B A E D C (1) A E D B C (1) A E B D C (1) A C B E D (1) A B E D C (1) A B C E D (1) Total count = 100 A B C D E A 0 -6 -16 -12 -10 B 6 0 8 2 10 C 16 -8 0 8 12 D 12 -2 -8 0 4 E 10 -10 -12 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999965 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -16 -12 -10 B 6 0 8 2 10 C 16 -8 0 8 12 D 12 -2 -8 0 4 E 10 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=37 B=26 D=19 E=11 A=7 so A is eliminated. Round 2 votes counts: C=38 B=28 D=19 E=15 so E is eliminated. Round 3 votes counts: C=41 D=30 B=29 so B is eliminated. Round 4 votes counts: C=54 D=46 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:214 B:213 D:203 E:192 A:178 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 -16 -12 -10 B 6 0 8 2 10 C 16 -8 0 8 12 D 12 -2 -8 0 4 E 10 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -16 -12 -10 B 6 0 8 2 10 C 16 -8 0 8 12 D 12 -2 -8 0 4 E 10 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -16 -12 -10 B 6 0 8 2 10 C 16 -8 0 8 12 D 12 -2 -8 0 4 E 10 -10 -12 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997731 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9941: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (8) B E C D A (6) E B C A D (5) D A C E B (5) C B E D A (5) C B E A D (5) A D C B E (5) D E A B C (4) D A E C B (4) D A E B C (4) A D E B C (4) E D B A C (3) E B A C D (3) D C A B E (3) B E C A D (3) A C D B E (3) E B D C A (2) E B C D A (2) D E B C A (2) C D B A E (2) C B D E A (2) C B A E D (2) B C E A D (2) A E D B C (2) A C B E D (2) E B D A C (1) E B A D C (1) E A B D C (1) D E B A C (1) D E A C B (1) D C B E A (1) C B D A E (1) C A D B E (1) C A B E D (1) B C E D A (1) A E C B D (1) A D C E B (1) Total count = 100 A B C D E A 0 0 8 -16 -2 B 0 0 -6 -8 6 C -8 6 0 -6 0 D 16 8 6 0 4 E 2 -6 0 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999974 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 0 8 -16 -2 B 0 0 -6 -8 6 C -8 6 0 -6 0 D 16 8 6 0 4 E 2 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=19 E=18 A=18 B=12 so B is eliminated. Round 2 votes counts: D=33 E=27 C=22 A=18 so A is eliminated. Round 3 votes counts: D=43 E=30 C=27 so C is eliminated. Round 4 votes counts: D=52 E=48 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:217 B:196 C:196 E:196 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 8 -16 -2 B 0 0 -6 -8 6 C -8 6 0 -6 0 D 16 8 6 0 4 E 2 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 8 -16 -2 B 0 0 -6 -8 6 C -8 6 0 -6 0 D 16 8 6 0 4 E 2 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 8 -16 -2 B 0 0 -6 -8 6 C -8 6 0 -6 0 D 16 8 6 0 4 E 2 -6 0 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999998182 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9942: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (15) E C A D B (11) E D A C B (8) E C A B D (6) D A C B E (5) B A C D E (4) D A B C E (3) C A B E D (3) B C A D E (3) A D C B E (3) E D C A B (2) E D B A C (2) E B D C A (2) D E A C B (2) C B A E D (2) C A E D B (2) C A B D E (2) B D E A C (2) B C A E D (2) B A D C E (2) E D A B C (1) E C D A B (1) E B D A C (1) E B C A D (1) D E B A C (1) D B E A C (1) D B A E C (1) D B A C E (1) D A C E B (1) C E A B D (1) C A E B D (1) C A D E B (1) C A D B E (1) B E C D A (1) B E C A D (1) B D A E C (1) B C E A D (1) A D B C E (1) A C D E B (1) Total count = 100 A B C D E A 0 12 12 -2 10 B -12 0 -6 2 10 C -12 6 0 -10 10 D 2 -2 10 0 2 E -10 -10 -10 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593750000021 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 A B C D E A 0 12 12 -2 10 B -12 0 -6 2 10 C -12 6 0 -10 10 D 2 -2 10 0 2 E -10 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749997473 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 B=32 D=15 C=13 A=5 so A is eliminated. Round 2 votes counts: E=35 B=32 D=19 C=14 so C is eliminated. Round 3 votes counts: E=39 B=39 D=22 so D is eliminated. Round 4 votes counts: B=55 E=45 so E is eliminated. IRV winner is B compare: Computing Borda winner. A:216 D:206 B:197 C:197 E:184 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A D , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 -2 10 B -12 0 -6 2 10 C -12 6 0 -10 10 D 2 -2 10 0 2 E -10 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749997473 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 -2 10 B -12 0 -6 2 10 C -12 6 0 -10 10 D 2 -2 10 0 2 E -10 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749997473 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 -2 10 B -12 0 -6 2 10 C -12 6 0 -10 10 D 2 -2 10 0 2 E -10 -10 -10 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.125000 B: 0.125000 C: 0.000000 D: 0.750000 E: 0.000000 Sum of squares = 0.593749997473 Cumulative probabilities = A: 0.125000 B: 0.250000 C: 0.250000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9943: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B D C (10) C D B E A (8) A E C B D (6) E A C B D (5) D C B E A (5) D B C E A (5) B D C E A (4) A E C D B (4) A D B C E (4) E C A B D (3) D B C A E (3) B D E C A (3) A E B C D (3) A B E D C (3) A B D E C (3) E B D A C (2) E A B C D (2) C E D B A (2) C D E B A (2) C D A B E (2) B E D A C (2) A C E D B (2) A B D C E (2) E C D B A (1) E C B D A (1) E C A D B (1) E B C D A (1) E B A D C (1) D C B A E (1) D B A C E (1) C E B D A (1) C D B A E (1) C D A E B (1) C A D E B (1) B D E A C (1) A E D B C (1) A C D E B (1) A C D B E (1) Total count = 100 A B C D E A 0 10 8 6 0 B -10 0 2 6 -2 C -8 -2 0 -2 -6 D -6 -6 2 0 -2 E 0 2 6 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.468993 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.531007 Sum of squares = 0.501922926041 Cumulative probabilities = A: 0.468993 B: 0.468993 C: 0.468993 D: 0.468993 E: 1.000000 A B C D E A 0 10 8 6 0 B -10 0 2 6 -2 C -8 -2 0 -2 -6 D -6 -6 2 0 -2 E 0 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=40 C=18 E=17 D=15 B=10 so B is eliminated. Round 2 votes counts: A=40 D=23 E=19 C=18 so C is eliminated. Round 3 votes counts: A=41 D=37 E=22 so E is eliminated. Round 4 votes counts: A=53 D=47 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:212 E:205 B:198 D:194 C:191 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A E , winner is: A compare: Computing GTS winners. A B C D E A 0 10 8 6 0 B -10 0 2 6 -2 C -8 -2 0 -2 -6 D -6 -6 2 0 -2 E 0 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTS winners are ['A', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 10 8 6 0 B -10 0 2 6 -2 C -8 -2 0 -2 -6 D -6 -6 2 0 -2 E 0 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 10 8 6 0 B -10 0 2 6 -2 C -8 -2 0 -2 -6 D -6 -6 2 0 -2 E 0 2 6 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.500000 Sum of squares = 0.5 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 0.500000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9944: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C B A D (13) A D E C B (9) C E B D A (6) B C E D A (6) A D B E C (6) B D C E A (5) E C A B D (4) D A C E B (4) D A B C E (4) B E C A D (4) B C E A D (4) A E C D B (4) A B E C D (4) D C E B A (3) D C E A B (3) D A E C B (3) A E C B D (3) D B C E A (2) D B A C E (2) A D B C E (2) A B D E C (2) E C D B A (1) E C D A B (1) E C B D A (1) D B C A E (1) B A E C D (1) B A D C E (1) B A C E D (1) Total count = 100 A B C D E A 0 -2 -8 16 -6 B 2 0 -10 10 -10 C 8 10 0 6 -12 D -16 -10 -6 0 -6 E 6 10 12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999522 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -2 -8 16 -6 B 2 0 -10 10 -10 C 8 10 0 6 -12 D -16 -10 -6 0 -6 E 6 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=30 D=22 B=22 E=20 C=6 so C is eliminated. Round 2 votes counts: A=30 E=26 D=22 B=22 so D is eliminated. Round 3 votes counts: A=41 E=32 B=27 so B is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:217 C:206 A:200 B:196 D:181 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -2 -8 16 -6 B 2 0 -10 10 -10 C 8 10 0 6 -12 D -16 -10 -6 0 -6 E 6 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 -8 16 -6 B 2 0 -10 10 -10 C 8 10 0 6 -12 D -16 -10 -6 0 -6 E 6 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 -8 16 -6 B 2 0 -10 10 -10 C 8 10 0 6 -12 D -16 -10 -6 0 -6 E 6 10 12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999285 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9945: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B A D E (22) E D A B C (12) D E A B C (8) A B C D E (8) E D C B A (7) D E C B A (6) C B A E D (5) E A B C D (4) E D A C B (3) E D C A B (2) E C D B A (2) E C B D A (2) D C B E A (2) D C B A E (2) D A B C E (2) C B D A E (2) A B D C E (2) E C B A D (1) E A D B C (1) E A B D C (1) D B A C E (1) D A E B C (1) C B E A D (1) B C A D E (1) B A C D E (1) A B C E D (1) Total count = 100 A B C D E A 0 -10 -10 -4 -4 B 10 0 -14 2 0 C 10 14 0 0 0 D 4 -2 0 0 16 E 4 0 0 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.579395 D: 0.420605 E: 0.000000 Sum of squares = 0.51260721811 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.579395 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -10 -4 -4 B 10 0 -14 2 0 C 10 14 0 0 0 D 4 -2 0 0 16 E 4 0 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=35 C=30 D=22 A=11 B=2 so B is eliminated. Round 2 votes counts: E=35 C=31 D=22 A=12 so A is eliminated. Round 3 votes counts: C=41 E=35 D=24 so D is eliminated. Round 4 votes counts: E=50 C=50 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 D:209 B:199 E:194 A:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -10 -10 -4 -4 B 10 0 -14 2 0 C 10 14 0 0 0 D 4 -2 0 0 16 E 4 0 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -10 -4 -4 B 10 0 -14 2 0 C 10 14 0 0 0 D 4 -2 0 0 16 E 4 0 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -10 -4 -4 B 10 0 -14 2 0 C 10 14 0 0 0 D 4 -2 0 0 16 E 4 0 0 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9946: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (13) C E A B D (11) B D E C A (10) A C D E B (8) D A B C E (7) C A E B D (7) E C B A D (6) D B E A C (6) D B A E C (5) A C E B D (3) E C A B D (2) E B C A D (2) D B E C A (2) D B A C E (2) D A E C B (2) B E C D A (2) E C A D B (1) E B D C A (1) E B C D A (1) D E B A C (1) D A C E B (1) C B E A D (1) C A B E D (1) B E D C A (1) B D C A E (1) B C E D A (1) B C A D E (1) A D C E B (1) Total count = 100 A B C D E A 0 14 -2 14 4 B -14 0 -14 2 -20 C 2 14 0 20 16 D -14 -2 -20 0 -6 E -4 20 -16 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998049 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 14 4 B -14 0 -14 2 -20 C 2 14 0 20 16 D -14 -2 -20 0 -6 E -4 20 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 A=25 C=20 B=16 E=13 so E is eliminated. Round 2 votes counts: C=29 D=26 A=25 B=20 so B is eliminated. Round 3 votes counts: D=39 C=36 A=25 so A is eliminated. Round 4 votes counts: C=60 D=40 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:226 A:215 E:203 D:179 B:177 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 14 -2 14 4 B -14 0 -14 2 -20 C 2 14 0 20 16 D -14 -2 -20 0 -6 E -4 20 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 14 4 B -14 0 -14 2 -20 C 2 14 0 20 16 D -14 -2 -20 0 -6 E -4 20 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 14 4 B -14 0 -14 2 -20 C 2 14 0 20 16 D -14 -2 -20 0 -6 E -4 20 -16 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999985429 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9947: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A D B E (6) D B E A C (5) C D E A B (5) A C B D E (5) D C E B A (4) D C B A E (4) D B A C E (4) E D C B A (3) D E C B A (3) D B C A E (3) C D A B E (3) C A E B D (3) B A D C E (3) E D B C A (2) E D B A C (2) E C D A B (2) E C A B D (2) E A C B D (2) D E B C A (2) D B E C A (2) C E D A B (2) C E A D B (2) C D A E B (2) C A D E B (2) B E A D C (2) B D E A C (2) B A E D C (2) B A D E C (2) A E C B D (2) A B E C D (2) A B C D E (2) E C A D B (1) E B D A C (1) E B A D C (1) E A B C D (1) D B A E C (1) C A E D B (1) C A B E D (1) C A B D E (1) B E D A C (1) A E B C D (1) A C E B D (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 2 -12 -6 6 B -2 0 -16 -22 6 C 12 16 0 2 12 D 6 22 -2 0 22 E -6 -6 -12 -22 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999788 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 -12 -6 6 B -2 0 -16 -22 6 C 12 16 0 2 12 D 6 22 -2 0 22 E -6 -6 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C D , winner is: C compare: Computing IRV winner. Round 1 votes counts: D=28 C=28 E=17 A=15 B=12 so B is eliminated. Round 2 votes counts: D=30 C=28 A=22 E=20 so E is eliminated. Round 3 votes counts: D=39 C=33 A=28 so A is eliminated. Round 4 votes counts: C=51 D=49 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:224 C:221 A:195 B:183 E:177 Borda winner is D compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 2 -12 -6 6 B -2 0 -16 -22 6 C 12 16 0 2 12 D 6 22 -2 0 22 E -6 -6 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -12 -6 6 B -2 0 -16 -22 6 C 12 16 0 2 12 D 6 22 -2 0 22 E -6 -6 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -12 -6 6 B -2 0 -16 -22 6 C 12 16 0 2 12 D 6 22 -2 0 22 E -6 -6 -12 -22 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998166 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9948: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (14) B A D E C (9) C D E A B (7) E C A D B (6) E A C B D (6) E C A B D (5) B D A C E (5) A B E C D (4) D C B A E (3) C E D B A (3) B A E C D (3) A E B C D (3) E C D A B (2) E C B A D (2) E B C A D (2) D C E A B (2) D B A C E (2) D A B C E (2) B E A C D (2) A B E D C (2) E C D B A (1) E C B D A (1) E A B C D (1) D C E B A (1) D C B E A (1) D C A B E (1) D B C E A (1) D B C A E (1) D A C E B (1) C D E B A (1) B E A D C (1) B D A E C (1) B A D C E (1) A E C B D (1) A D B C E (1) A C E D B (1) Total count = 100 A B C D E A 0 18 -8 0 -18 B -18 0 -18 -2 -20 C 8 18 0 30 -4 D 0 2 -30 0 -20 E 18 20 4 20 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999883 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 -8 0 -18 B -18 0 -18 -2 -20 C 8 18 0 30 -4 D 0 2 -30 0 -20 E 18 20 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=25 B=22 D=15 A=12 so A is eliminated. Round 2 votes counts: E=30 B=28 C=26 D=16 so D is eliminated. Round 3 votes counts: C=35 B=35 E=30 so E is eliminated. Round 4 votes counts: C=59 B=41 so B is eliminated. IRV winner is C compare: Computing Borda winner. E:231 C:226 A:196 D:176 B:171 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 -8 0 -18 B -18 0 -18 -2 -20 C 8 18 0 30 -4 D 0 2 -30 0 -20 E 18 20 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 -8 0 -18 B -18 0 -18 -2 -20 C 8 18 0 30 -4 D 0 2 -30 0 -20 E 18 20 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 -8 0 -18 B -18 0 -18 -2 -20 C 8 18 0 30 -4 D 0 2 -30 0 -20 E 18 20 4 20 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999299 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9949: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A C D B (7) A E B C D (7) D C B E A (6) B D C A E (6) E A D C B (5) B C A D E (5) E D C B A (4) E C D B A (4) E A C B D (4) D B C E A (3) D B C A E (3) C D B E A (3) A E B D C (3) A B D E C (3) A B D C E (3) E D A C B (2) D E A B C (2) D C E B A (2) D B A E C (2) C D E B A (2) C B A E D (2) B A C D E (2) A B E D C (2) E D C A B (1) E D A B C (1) E C A D B (1) E C A B D (1) D E C B A (1) D E B A C (1) D C B A E (1) C E B A D (1) C E A B D (1) C B E A D (1) C B D E A (1) C B D A E (1) B A D C E (1) A E D B C (1) A D B E C (1) A B E C D (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -4 0 8 -8 B 4 0 -2 -6 -2 C 0 2 0 -8 -8 D -8 6 8 0 0 E 8 2 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.364923 E: 0.635077 Sum of squares = 0.536491461013 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.364923 E: 1.000000 A B C D E A 0 -4 0 8 -8 B 4 0 -2 -6 -2 C 0 2 0 -8 -8 D -8 6 8 0 0 E 8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 0.500202 Sum of squares = 0.50000008132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=30 A=23 D=21 B=14 C=12 so C is eliminated. Round 2 votes counts: E=32 D=26 A=23 B=19 so B is eliminated. Round 3 votes counts: D=34 E=33 A=33 so E is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. E:209 D:203 A:198 B:197 C:193 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -4 0 8 -8 B 4 0 -2 -6 -2 C 0 2 0 -8 -8 D -8 6 8 0 0 E 8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 0.500202 Sum of squares = 0.50000008132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 0 8 -8 B 4 0 -2 -6 -2 C 0 2 0 -8 -8 D -8 6 8 0 0 E 8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 0.500202 Sum of squares = 0.50000008132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 0 8 -8 B 4 0 -2 -6 -2 C 0 2 0 -8 -8 D -8 6 8 0 0 E 8 2 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 0.500202 Sum of squares = 0.50000008132 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.499798 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9950: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B E D C (12) B C D E A (9) A D C E B (9) B C E D A (8) B E C D A (7) C D E A B (6) B A E D C (6) A E D C B (5) A D E C B (5) D C A E B (4) C D E B A (4) D C E A B (3) D E C A B (2) C D B E A (2) B E A C D (2) B A E C D (2) A E D B C (2) A E B D C (2) E C D B A (1) D E A C B (1) C D A E B (1) C B D E A (1) B E C A D (1) B E A D C (1) A D C B E (1) A C D E B (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 12 2 2 4 B -12 0 8 6 8 C -2 -8 0 -8 2 D -2 -6 8 0 2 E -4 -8 -2 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999981 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 2 2 4 B -12 0 8 6 8 C -2 -8 0 -8 2 D -2 -6 8 0 2 E -4 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=39 B=36 C=14 D=10 E=1 so E is eliminated. Round 2 votes counts: A=39 B=36 C=15 D=10 so D is eliminated. Round 3 votes counts: A=40 B=36 C=24 so C is eliminated. Round 4 votes counts: A=56 B=44 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:210 B:205 D:201 C:192 E:192 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 2 2 4 B -12 0 8 6 8 C -2 -8 0 -8 2 D -2 -6 8 0 2 E -4 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 2 2 4 B -12 0 8 6 8 C -2 -8 0 -8 2 D -2 -6 8 0 2 E -4 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 2 2 4 B -12 0 8 6 8 C -2 -8 0 -8 2 D -2 -6 8 0 2 E -4 -8 -2 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996994 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9951: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A C B E (13) E B C A D (7) E B D C A (6) B E D A C (5) B D A C E (5) E C A D B (4) E B A C D (4) B E A C D (4) B D A E C (4) E D C A B (3) E C A B D (3) D C A E B (3) D B A C E (3) C A E D B (3) C A D E B (3) C A D B E (3) B A D C E (3) E C D A B (2) E B C D A (2) D C A B E (2) D A B C E (2) C E A D B (2) C D A E B (2) B E A D C (2) A C D B E (2) E B D A C (1) E B A D C (1) D E C A B (1) C D E A B (1) C A E B D (1) B A C E D (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 2 4 -10 4 B -2 0 2 0 2 C -4 -2 0 -10 2 D 10 0 10 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.657345 C: 0.000000 D: 0.342655 E: 0.000000 Sum of squares = 0.549514725028 Cumulative probabilities = A: 0.000000 B: 0.657345 C: 0.657345 D: 1.000000 E: 1.000000 A B C D E A 0 2 4 -10 4 B -2 0 2 0 2 C -4 -2 0 -10 2 D 10 0 10 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500119 C: 0.000000 D: 0.499881 E: 0.000000 Sum of squares = 0.500000028397 Cumulative probabilities = A: 0.000000 B: 0.500119 C: 0.500119 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=25 D=24 C=15 A=3 so A is eliminated. Round 2 votes counts: E=33 D=25 B=25 C=17 so C is eliminated. Round 3 votes counts: E=39 D=36 B=25 so B is eliminated. Round 4 votes counts: E=51 D=49 so D is eliminated. IRV winner is E compare: Computing Borda winner. D:209 B:201 A:200 E:197 C:193 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B D , winner is: B compare: Computing GTS winners. A B C D E A 0 2 4 -10 4 B -2 0 2 0 2 C -4 -2 0 -10 2 D 10 0 10 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500119 C: 0.000000 D: 0.499881 E: 0.000000 Sum of squares = 0.500000028397 Cumulative probabilities = A: 0.000000 B: 0.500119 C: 0.500119 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 4 -10 4 B -2 0 2 0 2 C -4 -2 0 -10 2 D 10 0 10 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500119 C: 0.000000 D: 0.499881 E: 0.000000 Sum of squares = 0.500000028397 Cumulative probabilities = A: 0.000000 B: 0.500119 C: 0.500119 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 4 -10 4 B -2 0 2 0 2 C -4 -2 0 -10 2 D 10 0 10 0 -2 E -4 -2 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.500119 C: 0.000000 D: 0.499881 E: 0.000000 Sum of squares = 0.500000028397 Cumulative probabilities = A: 0.000000 B: 0.500119 C: 0.500119 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9952: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (10) C D B E A (5) A E B D C (5) A B E C D (5) E D B A C (4) D E B A C (4) D C E B A (4) D C A E B (4) C D A B E (4) C B E D A (4) A D E B C (4) D C E A B (3) C A B E D (3) B C E A D (3) B A E C D (3) E B A D C (2) D E C B A (2) D E B C A (2) C D A E B (2) C B D A E (2) C B A E D (2) C A B D E (2) B E A D C (2) B E A C D (2) A C B E D (2) E D A B C (1) E B D A C (1) E A B D C (1) D E C A B (1) D A E C B (1) D A E B C (1) D A C E B (1) C D B A E (1) C B E A D (1) C A D E B (1) C A D B E (1) B E C D A (1) B C A E D (1) A E D B C (1) A B C E D (1) Total count = 100 A B C D E A 0 8 2 -16 -6 B -8 0 8 -14 -10 C -2 -8 0 -8 -6 D 16 14 8 0 10 E 6 10 6 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999945 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 2 -16 -6 B -8 0 8 -14 -10 C -2 -8 0 -8 -6 D 16 14 8 0 10 E 6 10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 C=28 A=18 B=12 E=9 so E is eliminated. Round 2 votes counts: D=38 C=28 A=19 B=15 so B is eliminated. Round 3 votes counts: D=39 C=33 A=28 so A is eliminated. Round 4 votes counts: D=54 C=46 so C is eliminated. IRV winner is D compare: Computing Borda winner. D:224 E:206 A:194 B:188 C:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 2 -16 -6 B -8 0 8 -14 -10 C -2 -8 0 -8 -6 D 16 14 8 0 10 E 6 10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 2 -16 -6 B -8 0 8 -14 -10 C -2 -8 0 -8 -6 D 16 14 8 0 10 E 6 10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 2 -16 -6 B -8 0 8 -14 -10 C -2 -8 0 -8 -6 D 16 14 8 0 10 E 6 10 6 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999607 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9953: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D E B A (11) D C A B E (9) A B E D C (6) D C A E B (4) C D B E A (4) B A E C D (4) E C B D A (3) E B A C D (3) E A B D C (3) D C E A B (3) D A C B E (3) B A C E D (3) A E B D C (3) E C D A B (2) E B C A D (2) D C B A E (2) D A E C B (2) D A C E B (2) C D E A B (2) C D B A E (2) B E C A D (2) B A E D C (2) A B D E C (2) A B D C E (2) E D A C B (1) E C D B A (1) E A D C B (1) E A B C D (1) D E A C B (1) C E D B A (1) C E B D A (1) C B D A E (1) B E A C D (1) B C E A D (1) B C D E A (1) B C D A E (1) B C A D E (1) B A C D E (1) A E D B C (1) A D E B C (1) A D C B E (1) A D B E C (1) A D B C E (1) Total count = 100 A B C D E A 0 4 -8 -14 10 B -4 0 -14 -12 2 C 8 14 0 -2 14 D 14 12 2 0 16 E -10 -2 -14 -16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999892 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 -8 -14 10 B -4 0 -14 -12 2 C 8 14 0 -2 14 D 14 12 2 0 16 E -10 -2 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 C=22 A=18 E=17 B=17 so E is eliminated. Round 2 votes counts: C=28 D=27 A=23 B=22 so B is eliminated. Round 3 votes counts: A=37 C=36 D=27 so D is eliminated. Round 4 votes counts: C=54 A=46 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:222 C:217 A:196 B:186 E:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 4 -8 -14 10 B -4 0 -14 -12 2 C 8 14 0 -2 14 D 14 12 2 0 16 E -10 -2 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 -8 -14 10 B -4 0 -14 -12 2 C 8 14 0 -2 14 D 14 12 2 0 16 E -10 -2 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 -8 -14 10 B -4 0 -14 -12 2 C 8 14 0 -2 14 D 14 12 2 0 16 E -10 -2 -14 -16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999994391 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9954: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) D A B E C (8) D E C B A (7) E C A B D (6) C E B A D (6) B A C E D (6) D E C A B (5) E C D B A (4) D B A C E (4) E C B A D (3) C E B D A (3) C B A E D (3) B C A E D (3) A B D C E (3) E D C A B (2) D E A B C (2) D A B C E (2) C B E A D (2) B A D C E (2) B A C D E (2) A B C D E (2) E D C B A (1) E C D A B (1) E C A D B (1) E A C B D (1) D E A C B (1) D C E B A (1) D C B E A (1) D B E A C (1) D B A E C (1) D A E B C (1) C E D B A (1) C E A B D (1) C B D A E (1) B C A D E (1) A D B E C (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -6 6 0 B 6 0 -2 10 6 C 6 2 0 12 6 D -6 -10 -12 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999973 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 6 0 B 6 0 -2 10 6 C 6 2 0 12 6 D -6 -10 -12 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 E=19 C=17 A=16 B=14 so B is eliminated. Round 2 votes counts: D=34 A=26 C=21 E=19 so E is eliminated. Round 3 votes counts: D=37 C=36 A=27 so A is eliminated. Round 4 votes counts: C=56 D=44 so D is eliminated. IRV winner is C compare: Computing Borda winner. C:213 B:210 A:197 E:197 D:183 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -6 -6 6 0 B 6 0 -2 10 6 C 6 2 0 12 6 D -6 -10 -12 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 6 0 B 6 0 -2 10 6 C 6 2 0 12 6 D -6 -10 -12 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 6 0 B 6 0 -2 10 6 C 6 2 0 12 6 D -6 -10 -12 0 -6 E 0 -6 -6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998711 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9955: compare: Profile of ballots (with multiplicities), in decreasing order by count: A D C B E (9) E C B D A (6) B E A D C (5) E C D B A (4) E B A C D (4) E A B D C (4) C E D B A (4) B E C D A (4) E C D A B (3) E B C D A (3) E B C A D (3) C D E A B (3) C D B E A (3) C D A E B (3) B C D A E (3) A D C E B (3) A B D E C (3) E C A D B (2) D C A B E (2) D A C B E (2) C E D A B (2) C E B D A (2) B E C A D (2) B C E D A (2) A D B C E (2) E C B A D (1) E A D C B (1) E A D B C (1) E A C D B (1) D C B A E (1) D C A E B (1) D B C A E (1) D A C E B (1) C D B A E (1) C B E D A (1) B D C A E (1) B D A C E (1) B A E D C (1) B A D E C (1) A E D C B (1) A D E C B (1) A B D C E (1) Total count = 100 A B C D E A 0 -8 -16 -8 -22 B 8 0 -16 -4 -8 C 16 16 0 14 -2 D 8 4 -14 0 -14 E 22 8 2 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.99999999866 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -8 -16 -8 -22 B 8 0 -16 -4 -8 C 16 16 0 14 -2 D 8 4 -14 0 -14 E 22 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 B=20 A=20 C=19 D=8 so D is eliminated. Round 2 votes counts: E=33 C=23 A=23 B=21 so B is eliminated. Round 3 votes counts: E=44 C=30 A=26 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:223 C:222 D:192 B:190 A:173 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -8 -16 -8 -22 B 8 0 -16 -4 -8 C 16 16 0 14 -2 D 8 4 -14 0 -14 E 22 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -8 -16 -8 -22 B 8 0 -16 -4 -8 C 16 16 0 14 -2 D 8 4 -14 0 -14 E 22 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -8 -16 -8 -22 B 8 0 -16 -4 -8 C 16 16 0 14 -2 D 8 4 -14 0 -14 E 22 8 2 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999996913 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9956: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A B E (8) B E A C D (7) D C A E B (6) E B A D C (5) D A C E B (5) C A B D E (5) E B D A C (4) E B A C D (4) B A C E D (4) A C B D E (4) E D A C B (3) E D A B C (3) E B D C A (3) D E A C B (3) C A D B E (3) B E D C A (3) B C E A D (3) D E C A B (2) C B A D E (2) B E C A D (2) A E B C D (2) A C D B E (2) A C B E D (2) E D B C A (1) E A D B C (1) E A B D C (1) E A B C D (1) D E C B A (1) D E B C A (1) D E A B C (1) D B C E A (1) D A C B E (1) C D A B E (1) B E C D A (1) B C A E D (1) B A E C D (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 12 12 4 -2 B -12 0 2 12 6 C -12 -2 0 -8 -2 D -4 -12 8 0 -6 E 2 -6 2 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999998 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 A B C D E A 0 12 12 4 -2 B -12 0 2 12 6 C -12 -2 0 -8 -2 D -4 -12 8 0 -6 E 2 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999708 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=29 E=26 B=22 A=12 C=11 so C is eliminated. Round 2 votes counts: D=30 E=26 B=24 A=20 so A is eliminated. Round 3 votes counts: D=36 B=36 E=28 so E is eliminated. Round 4 votes counts: B=56 D=44 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:213 B:204 E:202 D:193 C:188 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 12 12 4 -2 B -12 0 2 12 6 C -12 -2 0 -8 -2 D -4 -12 8 0 -6 E 2 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999708 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTS winners are ['A', 'B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 12 4 -2 B -12 0 2 12 6 C -12 -2 0 -8 -2 D -4 -12 8 0 -6 E 2 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999708 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 12 4 -2 B -12 0 2 12 6 C -12 -2 0 -8 -2 D -4 -12 8 0 -6 E 2 -6 2 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.300000 B: 0.100000 C: 0.000000 D: 0.000000 E: 0.600000 Sum of squares = 0.459999999708 Cumulative probabilities = A: 0.300000 B: 0.400000 C: 0.400000 D: 0.400000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9957: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (7) B C A D E (7) D E B A C (6) E D A C B (5) E D A B C (5) D A E B C (4) E D C A B (3) E D B C A (3) E D B A C (3) E B D C A (3) D E A B C (3) D B E A C (3) C A E B D (3) A D E C B (3) A C D E B (3) E C B D A (2) E A D C B (2) D B E C A (2) C B E A D (2) C A B E D (2) B D E A C (2) B D A C E (2) B C D A E (2) B A D C E (2) A C E D B (2) E C D B A (1) E C D A B (1) E C B A D (1) E C A D B (1) E B C D A (1) D B A E C (1) D A B E C (1) C E B D A (1) C E A B D (1) C B A E D (1) B E D C A (1) B D A E C (1) B C E D A (1) B A C D E (1) A E C D B (1) A D C B E (1) A C D B E (1) A C B D E (1) A B C D E (1) Total count = 100 A B C D E A 0 2 8 -14 -6 B -2 0 10 -10 -16 C -8 -10 0 -12 -18 D 14 10 12 0 8 E 6 16 18 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 2 8 -14 -6 B -2 0 10 -10 -16 C -8 -10 0 -12 -18 D 14 10 12 0 8 E 6 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=31 D=20 B=19 C=17 A=13 so A is eliminated. Round 2 votes counts: E=32 D=24 C=24 B=20 so B is eliminated. Round 3 votes counts: C=36 E=33 D=31 so D is eliminated. Round 4 votes counts: E=59 C=41 so C is eliminated. IRV winner is E compare: Computing Borda winner. D:222 E:216 A:195 B:191 C:176 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 2 8 -14 -6 B -2 0 10 -10 -16 C -8 -10 0 -12 -18 D 14 10 12 0 8 E 6 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 8 -14 -6 B -2 0 10 -10 -16 C -8 -10 0 -12 -18 D 14 10 12 0 8 E 6 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 8 -14 -6 B -2 0 10 -10 -16 C -8 -10 0 -12 -18 D 14 10 12 0 8 E 6 16 18 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999975 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9958: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B C D A (7) E C B D A (6) C E B D A (6) A D C B E (6) E B C A D (5) C A E D B (5) B E D A C (5) E C B A D (4) C A D E B (4) D B A E C (3) D A B E C (3) D A B C E (3) C E A B D (3) A D C E B (3) A D B E C (3) A D B C E (3) D B A C E (2) C D E B A (2) C B D E A (2) B E C D A (2) B D E C A (2) A E C B D (2) A E B D C (2) E B A C D (1) D B C E A (1) D B C A E (1) D A C B E (1) C E D B A (1) C E B A D (1) C E A D B (1) C D E A B (1) C D A E B (1) B E D C A (1) B E A D C (1) B D E A C (1) B D A E C (1) B A D E C (1) A E D B C (1) A E C D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -12 -12 -4 -6 B 12 0 -2 6 -16 C 12 2 0 12 -4 D 4 -6 -12 0 -10 E 6 16 4 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999784 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -12 -4 -6 B 12 0 -2 6 -16 C 12 2 0 12 -4 D 4 -6 -12 0 -10 E 6 16 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=27 E=23 A=22 D=14 B=14 so D is eliminated. Round 2 votes counts: A=29 C=27 E=23 B=21 so B is eliminated. Round 3 votes counts: A=36 E=35 C=29 so C is eliminated. Round 4 votes counts: E=53 A=47 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:218 C:211 B:200 D:188 A:183 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -12 -4 -6 B 12 0 -2 6 -16 C 12 2 0 12 -4 D 4 -6 -12 0 -10 E 6 16 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -12 -4 -6 B 12 0 -2 6 -16 C 12 2 0 12 -4 D 4 -6 -12 0 -10 E 6 16 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -12 -4 -6 B 12 0 -2 6 -16 C 12 2 0 12 -4 D 4 -6 -12 0 -10 E 6 16 4 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9959: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E A D (11) D A E C B (9) C D A E B (9) C B D A E (9) E A D B C (7) D A E B C (7) B E A D C (7) B C E A D (6) E A B D C (5) C D E A B (4) C B D E A (3) B A E D C (3) D E A C B (2) D A C E B (2) C B E D A (2) B D A E C (2) B C A E D (2) E C B A D (1) D C A E B (1) C D B A E (1) B E C A D (1) B E A C D (1) B C A D E (1) B A D E C (1) B A D C E (1) A E D B C (1) A D E B C (1) Total count = 100 A B C D E A 0 -4 -2 -2 0 B 4 0 -8 12 2 C 2 8 0 2 4 D 2 -12 -2 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999986 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -2 -2 0 B 4 0 -8 12 2 C 2 8 0 2 4 D 2 -12 -2 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=39 B=25 D=21 E=13 A=2 so A is eliminated. Round 2 votes counts: C=39 B=25 D=22 E=14 so E is eliminated. Round 3 votes counts: C=40 D=30 B=30 so D is eliminated. Round 4 votes counts: C=54 B=46 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:208 B:205 D:197 A:196 E:194 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -4 -2 -2 0 B 4 0 -8 12 2 C 2 8 0 2 4 D 2 -12 -2 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -2 -2 0 B 4 0 -8 12 2 C 2 8 0 2 4 D 2 -12 -2 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -2 -2 0 B 4 0 -8 12 2 C 2 8 0 2 4 D 2 -12 -2 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997636 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9960: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D A B (15) E C D A B (10) C E D B A (9) B A D E C (9) A B D E C (6) D C E A B (5) B A D C E (5) C E B A D (4) D E C A B (3) B A C E D (3) E D C A B (2) E C A D B (2) D A E B C (2) C E B D A (2) C B E A D (2) E D A C B (1) E C A B D (1) E A C D B (1) E A C B D (1) D C E B A (1) D C B A E (1) D B C A E (1) D B A E C (1) D B A C E (1) D A B E C (1) C E A B D (1) C D E B A (1) C D B E A (1) C B D E A (1) B D A C E (1) B C D A E (1) B C A E D (1) B A C D E (1) A E B C D (1) A D B E C (1) A B E D C (1) Total count = 100 A B C D E A 0 8 -28 -20 -26 B -8 0 -28 -18 -24 C 28 28 0 16 14 D 20 18 -16 0 -14 E 26 24 -14 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 8 -28 -20 -26 B -8 0 -28 -18 -24 C 28 28 0 16 14 D 20 18 -16 0 -14 E 26 24 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=36 B=21 E=18 D=16 A=9 so A is eliminated. Round 2 votes counts: C=36 B=28 E=19 D=17 so D is eliminated. Round 3 votes counts: C=43 B=33 E=24 so E is eliminated. Round 4 votes counts: C=64 B=36 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:243 E:225 D:204 A:167 B:161 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 8 -28 -20 -26 B -8 0 -28 -18 -24 C 28 28 0 16 14 D 20 18 -16 0 -14 E 26 24 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -28 -20 -26 B -8 0 -28 -18 -24 C 28 28 0 16 14 D 20 18 -16 0 -14 E 26 24 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -28 -20 -26 B -8 0 -28 -18 -24 C 28 28 0 16 14 D 20 18 -16 0 -14 E 26 24 -14 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9961: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C A E B (11) E A B D C (9) C B D E A (9) A E B D C (9) D C A B E (7) C D B E A (7) C D B A E (7) B E A C D (5) B C E D A (4) A E D B C (4) E B A D C (3) D C B E A (3) D E A C B (2) D A C E B (2) A D E C B (2) E B A C D (1) E A B C D (1) D E C A B (1) D C E A B (1) D A E C B (1) C D A B E (1) C B D A E (1) B E C D A (1) B E C A D (1) B C E A D (1) B C D E A (1) B A E C D (1) B A C E D (1) A E B C D (1) A D C B E (1) A B E C D (1) Total count = 100 A B C D E A 0 8 -12 -18 0 B -8 0 -12 0 4 C 12 12 0 -12 14 D 18 0 12 0 14 E 0 -4 -14 -14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.343796 C: 0.000000 D: 0.656204 E: 0.000000 Sum of squares = 0.548799652421 Cumulative probabilities = A: 0.000000 B: 0.343796 C: 0.343796 D: 1.000000 E: 1.000000 A B C D E A 0 8 -12 -18 0 B -8 0 -12 0 4 C 12 12 0 -12 14 D 18 0 12 0 14 E 0 -4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499607 C: 0.000000 D: 0.500393 E: 0.000000 Sum of squares = 0.500000309423 Cumulative probabilities = A: 0.000000 B: 0.499607 C: 0.499607 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=25 A=18 B=15 E=14 so E is eliminated. Round 2 votes counts: D=28 A=28 C=25 B=19 so B is eliminated. Round 3 votes counts: A=39 C=33 D=28 so D is eliminated. Round 4 votes counts: C=56 A=44 so A is eliminated. IRV winner is C compare: Computing Borda winner. D:222 C:213 B:192 A:189 E:184 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 8 -12 -18 0 B -8 0 -12 0 4 C 12 12 0 -12 14 D 18 0 12 0 14 E 0 -4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499607 C: 0.000000 D: 0.500393 E: 0.000000 Sum of squares = 0.500000309423 Cumulative probabilities = A: 0.000000 B: 0.499607 C: 0.499607 D: 1.000000 E: 1.000000 GTS winners are ['B', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 8 -12 -18 0 B -8 0 -12 0 4 C 12 12 0 -12 14 D 18 0 12 0 14 E 0 -4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499607 C: 0.000000 D: 0.500393 E: 0.000000 Sum of squares = 0.500000309423 Cumulative probabilities = A: 0.000000 B: 0.499607 C: 0.499607 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 8 -12 -18 0 B -8 0 -12 0 4 C 12 12 0 -12 14 D 18 0 12 0 14 E 0 -4 -14 -14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Terminated (singular KKT matrix). Optimal mixed strategy = A: 0.000000 B: 0.499607 C: 0.000000 D: 0.500393 E: 0.000000 Sum of squares = 0.500000309423 Cumulative probabilities = A: 0.000000 B: 0.499607 C: 0.499607 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9962: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (9) E A B C D (8) B D C E A (8) E B A C D (7) A E C D B (7) E A C D B (6) C A D E B (6) B E D A C (6) B E A D C (5) B D E C A (5) D C B A E (4) A C E D B (4) E A C B D (3) D C A B E (3) B D C A E (3) D C E A B (2) D C A E B (2) D B C E A (2) E B A D C (1) D C B E A (1) D B C A E (1) C D E A B (1) C D A B E (1) B E D C A (1) B E A C D (1) B A E D C (1) B A E C D (1) A E B C D (1) Total count = 100 A B C D E A 0 6 2 2 -14 B -6 0 2 2 -14 C -2 -2 0 10 -6 D -2 -2 -10 0 -4 E 14 14 6 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999938 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 6 2 2 -14 B -6 0 2 2 -14 C -2 -2 0 10 -6 D -2 -2 -10 0 -4 E 14 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 E=25 C=17 D=15 A=12 so A is eliminated. Round 2 votes counts: E=33 B=31 C=21 D=15 so D is eliminated. Round 3 votes counts: B=34 E=33 C=33 so E is eliminated. Round 4 votes counts: B=51 C=49 so C is eliminated. IRV winner is B compare: Computing Borda winner. E:219 C:200 A:198 B:192 D:191 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 6 2 2 -14 B -6 0 2 2 -14 C -2 -2 0 10 -6 D -2 -2 -10 0 -4 E 14 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 2 2 -14 B -6 0 2 2 -14 C -2 -2 0 10 -6 D -2 -2 -10 0 -4 E 14 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 2 2 -14 B -6 0 2 2 -14 C -2 -2 0 10 -6 D -2 -2 -10 0 -4 E 14 14 6 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999612 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9963: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (10) A E D B C (7) B C E D A (5) A E B C D (5) A D E C B (5) A D C E B (5) E D B A C (4) E A B D C (4) D C B E A (4) A E D C B (4) C D B A E (3) C B D A E (3) E D B C A (2) E B D A C (2) E B A D C (2) D E C B A (2) D E B C A (2) D E A C B (2) D E A B C (2) C D A B E (2) C A B D E (2) B C D E A (2) B C A E D (2) A C D B E (2) A C B E D (2) E B A C D (1) E A D B C (1) E A B C D (1) D C E B A (1) D B E C A (1) D A C E B (1) C D B E A (1) C B A D E (1) C A D B E (1) B E D C A (1) B E A C D (1) B C E A D (1) A E C D B (1) A D C B E (1) A C E B D (1) Total count = 100 A B C D E A 0 -2 8 0 -4 B 2 0 -8 -8 -10 C -8 8 0 -6 0 D 0 8 6 0 6 E 4 10 0 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.282183 B: 0.000000 C: 0.000000 D: 0.717817 E: 0.000000 Sum of squares = 0.594888088164 Cumulative probabilities = A: 0.282183 B: 0.282183 C: 0.282183 D: 1.000000 E: 1.000000 A B C D E A 0 -2 8 0 -4 B 2 0 -8 -8 -10 C -8 8 0 -6 0 D 0 8 6 0 6 E 4 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=33 C=23 E=17 D=15 B=12 so B is eliminated. Round 2 votes counts: C=33 A=33 E=19 D=15 so D is eliminated. Round 3 votes counts: C=38 A=34 E=28 so E is eliminated. Round 4 votes counts: A=54 C=46 so C is eliminated. IRV winner is A compare: Computing Borda winner. D:210 E:204 A:201 C:197 B:188 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -2 8 0 -4 B 2 0 -8 -8 -10 C -8 8 0 -6 0 D 0 8 6 0 6 E 4 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 8 0 -4 B 2 0 -8 -8 -10 C -8 8 0 -6 0 D 0 8 6 0 6 E 4 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 8 0 -4 B 2 0 -8 -8 -10 C -8 8 0 -6 0 D 0 8 6 0 6 E 4 10 0 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.000000 Sum of squares = 0.499999999994 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9964: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E D B A (8) B D C E A (8) C E A D B (6) E C A D B (5) C E B D A (5) B A D C E (5) A B D E C (5) C B E D A (4) B D A C E (4) A E C D B (4) E C D B A (3) D B A E C (3) B C D E A (3) A E C B D (3) A D B E C (3) D E C B A (2) D C E B A (2) D B E C A (2) C E D A B (2) C B E A D (2) C B D E A (2) B D C A E (2) A E D C B (2) A E D B C (2) E D C B A (1) E D C A B (1) E C D A B (1) E A C D B (1) D E B C A (1) D E B A C (1) C A E B D (1) B D A E C (1) B A D E C (1) A E B D C (1) A D E B C (1) A C B E D (1) A B C E D (1) Total count = 100 A B C D E A 0 -20 -22 -12 -20 B 20 0 -12 -2 -6 C 22 12 0 4 12 D 12 2 -4 0 -8 E 20 6 -12 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999222 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -20 -22 -12 -20 B 20 0 -12 -2 -6 C 22 12 0 4 12 D 12 2 -4 0 -8 E 20 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=30 B=24 A=23 E=12 D=11 so D is eliminated. Round 2 votes counts: C=32 B=29 A=23 E=16 so E is eliminated. Round 3 votes counts: C=45 B=31 A=24 so A is eliminated. Round 4 votes counts: C=56 B=44 so B is eliminated. IRV winner is C compare: Computing Borda winner. C:225 E:211 D:201 B:200 A:163 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -20 -22 -12 -20 B 20 0 -12 -2 -6 C 22 12 0 4 12 D 12 2 -4 0 -8 E 20 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -20 -22 -12 -20 B 20 0 -12 -2 -6 C 22 12 0 4 12 D 12 2 -4 0 -8 E 20 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -20 -22 -12 -20 B 20 0 -12 -2 -6 C 22 12 0 4 12 D 12 2 -4 0 -8 E 20 6 -12 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996707 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9965: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C D E B (6) C B A E D (5) B E D C A (5) B E C D A (5) B C A E D (5) E D C B A (4) E D B C A (4) B C E D A (4) B A D E C (4) B A C E D (4) A D E C B (4) E D C A B (3) D E C A B (3) C E D A B (3) A D E B C (3) A B C D E (3) D E B C A (2) D E B A C (2) D E A B C (2) C B E A D (2) C A D E B (2) B E D A C (2) B C E A D (2) B A C D E (2) A D C E B (2) A C B D E (2) A B D E C (2) E C D B A (1) E C D A B (1) D E A C B (1) D A E C B (1) D A E B C (1) C E A B D (1) C B E D A (1) C A E D B (1) C A E B D (1) C A B E D (1) B D A E C (1) A C B E D (1) A B D C E (1) Total count = 100 A B C D E A 0 -10 -12 8 4 B 10 0 8 8 4 C 12 -8 0 6 -2 D -8 -8 -6 0 -12 E -4 -4 2 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999955 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -10 -12 8 4 B 10 0 8 8 4 C 12 -8 0 6 -2 D -8 -8 -6 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=24 C=17 E=13 D=12 so D is eliminated. Round 2 votes counts: B=34 A=26 E=23 C=17 so C is eliminated. Round 3 votes counts: B=42 A=31 E=27 so E is eliminated. Round 4 votes counts: B=55 A=45 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:215 C:204 E:203 A:195 D:183 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -10 -12 8 4 B 10 0 8 8 4 C 12 -8 0 6 -2 D -8 -8 -6 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -10 -12 8 4 B 10 0 8 8 4 C 12 -8 0 6 -2 D -8 -8 -6 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -10 -12 8 4 B 10 0 8 8 4 C 12 -8 0 6 -2 D -8 -8 -6 0 -12 E -4 -4 2 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9966: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A E B (8) D C A B E (7) A D C E B (6) E B A C D (5) B E A D C (5) B A E D C (5) E B C A D (4) C D E B A (4) C D E A B (4) D C B E A (3) D C B A E (3) D C A E B (3) D A C B E (3) B E D C A (3) B E C D A (3) E A C B D (2) E A B C D (2) C E D B A (2) B E A C D (2) A E C B D (2) A D E C B (2) A D B C E (2) A C D E B (2) A B E D C (2) E C B A D (1) E C A D B (1) E B C D A (1) D B C A E (1) D B A C E (1) D A B C E (1) C E D A B (1) C A E D B (1) B D E C A (1) B D C E A (1) B D A E C (1) B D A C E (1) A E C D B (1) A E B D C (1) A E B C D (1) A D C B E (1) Total count = 100 A B C D E A 0 6 -4 -4 10 B -6 0 -14 -14 -8 C 4 14 0 -6 10 D 4 14 6 0 10 E -10 8 -10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999924 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -4 -4 10 B -6 0 -14 -14 -8 C 4 14 0 -6 10 D 4 14 6 0 10 E -10 8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=22 B=22 C=20 A=20 E=16 so E is eliminated. Round 2 votes counts: B=32 A=24 D=22 C=22 so D is eliminated. Round 3 votes counts: C=38 B=34 A=28 so A is eliminated. Round 4 votes counts: C=57 B=43 so B is eliminated. IRV winner is C compare: Computing Borda winner. D:217 C:211 A:204 E:189 B:179 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 -4 -4 10 B -6 0 -14 -14 -8 C 4 14 0 -6 10 D 4 14 6 0 10 E -10 8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -4 -4 10 B -6 0 -14 -14 -8 C 4 14 0 -6 10 D 4 14 6 0 10 E -10 8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -4 -4 10 B -6 0 -14 -14 -8 C 4 14 0 -6 10 D 4 14 6 0 10 E -10 8 -10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999946 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9967: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D C A (13) A C D B E (13) E B A D C (10) C D A B E (10) E B C D A (9) A D C B E (6) B E D C A (5) A C D E B (4) E B D A C (2) E A B C D (2) D C B E A (2) D B C E A (2) C D B E A (2) C A D B E (2) A E B D C (2) A D B C E (2) E C D B A (1) E C B D A (1) E B A C D (1) D C B A E (1) D C A B E (1) D B C A E (1) D A C B E (1) C D E A B (1) C D B A E (1) B D C E A (1) B D A C E (1) A E D B C (1) A E B C D (1) A B D E C (1) Total count = 100 A B C D E A 0 -6 -6 -10 -4 B 6 0 8 -2 4 C 6 -8 0 -4 2 D 10 2 4 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 -6 -10 -4 B 6 0 8 -2 4 C 6 -8 0 -4 2 D 10 2 4 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=39 A=30 C=16 D=8 B=7 so B is eliminated. Round 2 votes counts: E=44 A=30 C=16 D=10 so D is eliminated. Round 3 votes counts: E=44 A=32 C=24 so C is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. D:210 B:208 C:198 E:197 A:187 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -6 -6 -10 -4 B 6 0 8 -2 4 C 6 -8 0 -4 2 D 10 2 4 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 -6 -10 -4 B 6 0 8 -2 4 C 6 -8 0 -4 2 D 10 2 4 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 -6 -10 -4 B 6 0 8 -2 4 C 6 -8 0 -4 2 D 10 2 4 0 4 E 4 -4 -2 -4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999917 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9968: compare: Profile of ballots (with multiplicities), in decreasing order by count: C A B D E (12) A C B E D (11) D E B C A (9) E D B A C (8) D B E C A (8) A C E B D (8) C B A D E (6) E D A B C (5) E D B C A (3) D B C E A (3) B C D E A (3) A E C D B (3) A C B D E (3) E A D C B (2) E A D B C (2) C B D A E (2) C A D B E (2) B D C E A (2) B C D A E (2) A E C B D (2) E A C B D (1) E A B D C (1) D C B A E (1) A E D C B (1) Total count = 100 A B C D E A 0 6 -6 8 6 B -6 0 -8 6 10 C 6 8 0 10 10 D -8 -6 -10 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 8 6 B -6 0 -8 6 10 C 6 8 0 10 10 D -8 -6 -10 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=28 E=22 C=22 D=21 B=7 so B is eliminated. Round 2 votes counts: A=28 C=27 D=23 E=22 so E is eliminated. Round 3 votes counts: D=39 A=34 C=27 so C is eliminated. Round 4 votes counts: A=54 D=46 so D is eliminated. IRV winner is A compare: Computing Borda winner. C:217 A:207 B:201 D:191 E:184 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 8 6 B -6 0 -8 6 10 C 6 8 0 10 10 D -8 -6 -10 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 8 6 B -6 0 -8 6 10 C 6 8 0 10 10 D -8 -6 -10 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 8 6 B -6 0 -8 6 10 C 6 8 0 10 10 D -8 -6 -10 0 6 E -6 -10 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9969: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E A B D (9) E C B A D (8) B D A C E (8) D B A E C (7) A D B C E (7) D A B E C (5) E C A D B (4) D A B C E (4) B D A E C (4) A B D C E (4) E D C B A (3) E C B D A (3) C B E A D (3) E D B A C (2) D A E B C (2) C E B A D (2) C E A D B (2) C B A E D (2) B C A D E (2) B A D C E (2) A C D B E (2) E D A C B (1) E D A B C (1) E C D B A (1) E C D A B (1) E C A B D (1) D B A C E (1) C B A D E (1) C A E D B (1) C A E B D (1) C A B D E (1) B E C D A (1) B D E A C (1) B C E A D (1) A D C B E (1) A C B D E (1) Total count = 100 A B C D E A 0 -4 6 10 12 B 4 0 4 10 16 C -6 -4 0 -6 10 D -10 -10 6 0 6 E -12 -16 -10 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999581 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 6 10 12 B 4 0 4 10 16 C -6 -4 0 -6 10 D -10 -10 6 0 6 E -12 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999844 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=25 C=22 D=19 B=19 A=15 so A is eliminated. Round 2 votes counts: D=27 E=25 C=25 B=23 so B is eliminated. Round 3 votes counts: D=46 C=28 E=26 so E is eliminated. Round 4 votes counts: D=53 C=47 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:217 A:212 C:197 D:196 E:178 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 6 10 12 B 4 0 4 10 16 C -6 -4 0 -6 10 D -10 -10 6 0 6 E -12 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999844 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 6 10 12 B 4 0 4 10 16 C -6 -4 0 -6 10 D -10 -10 6 0 6 E -12 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999844 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 6 10 12 B 4 0 4 10 16 C -6 -4 0 -6 10 D -10 -10 6 0 6 E -12 -16 -10 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999844 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9970: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (6) E A D C B (5) B D C A E (5) B C E D A (5) A E D B C (5) A E B C D (5) A B E C D (5) E C D A B (4) C D B E A (4) B A D C E (4) A E D C B (4) A E B D C (4) E A C D B (3) E A C B D (3) C B D E A (3) A D B E C (3) E D A C B (2) E A B C D (2) D C E B A (2) D C B A E (2) D A B C E (2) A D E C B (2) A B E D C (2) A B D E C (2) E D C A B (1) E C D B A (1) E C B A D (1) E C A D B (1) E C A B D (1) D E A C B (1) C E D B A (1) C D E B A (1) B E A C D (1) B C E A D (1) B C D A E (1) B C A D E (1) B A C E D (1) A D E B C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 20 18 18 2 B -20 0 4 -4 0 C -18 -4 0 -10 -18 D -18 4 10 0 -16 E -2 0 18 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998634 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 20 18 18 2 B -20 0 4 -4 0 C -18 -4 0 -10 -18 D -18 4 10 0 -16 E -2 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=35 E=24 B=19 D=13 C=9 so C is eliminated. Round 2 votes counts: A=35 E=25 B=22 D=18 so D is eliminated. Round 3 votes counts: A=37 B=34 E=29 so E is eliminated. Round 4 votes counts: A=60 B=40 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:229 E:216 B:190 D:190 C:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 20 18 18 2 B -20 0 4 -4 0 C -18 -4 0 -10 -18 D -18 4 10 0 -16 E -2 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 20 18 18 2 B -20 0 4 -4 0 C -18 -4 0 -10 -18 D -18 4 10 0 -16 E -2 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 20 18 18 2 B -20 0 4 -4 0 C -18 -4 0 -10 -18 D -18 4 10 0 -16 E -2 0 18 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999967485 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9971: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E B C D (7) D C B E A (6) B A E C D (6) C E A B D (5) B E A C D (5) A B E C D (5) D C E B A (4) D B A E C (4) B D A E C (4) A E C B D (4) D A C E B (3) D A B E C (3) B A E D C (3) E B A C D (2) E A B C D (2) D C A E B (2) D A B C E (2) C E D B A (2) C E D A B (2) C E B A D (2) C E A D B (2) C A E B D (2) B E C A D (2) B D E C A (2) A D B E C (2) E C B A D (1) E A C B D (1) D B E A C (1) D B C E A (1) D B A C E (1) C D E A B (1) C D B E A (1) C D A E B (1) C B E D A (1) B E A D C (1) B D E A C (1) B D C E A (1) B C E A D (1) B A D E C (1) A E D B C (1) A E B D C (1) A B E D C (1) Total count = 100 A B C D E A 0 -6 22 14 6 B 6 0 20 22 10 C -22 -20 0 10 -20 D -14 -22 -10 0 -18 E -6 -10 20 18 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 22 14 6 B 6 0 20 22 10 C -22 -20 0 10 -20 D -14 -22 -10 0 -18 E -6 -10 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B D , winner is: B compare: Computing IRV winner. Round 1 votes counts: D=27 B=27 A=21 C=19 E=6 so E is eliminated. Round 2 votes counts: B=29 D=27 A=24 C=20 so C is eliminated. Round 3 votes counts: D=34 B=33 A=33 so B is eliminated. Round 4 votes counts: A=57 D=43 so D is eliminated. IRV winner is A compare: Computing Borda winner. B:229 A:218 E:211 C:174 D:168 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 22 14 6 B 6 0 20 22 10 C -22 -20 0 10 -20 D -14 -22 -10 0 -18 E -6 -10 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 22 14 6 B 6 0 20 22 10 C -22 -20 0 10 -20 D -14 -22 -10 0 -18 E -6 -10 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 22 14 6 B 6 0 20 22 10 C -22 -20 0 10 -20 D -14 -22 -10 0 -18 E -6 -10 20 18 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999979 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9972: compare: Profile of ballots (with multiplicities), in decreasing order by count: C E B A D (5) B E C A D (5) B A D E C (5) A D E C B (5) D A E C B (4) D A B E C (4) C E A D B (4) C D E A B (4) B D C A E (4) D C E A B (3) D B A C E (3) C B E D A (3) B D A E C (3) B D A C E (3) B C E A D (3) A E B D C (3) A D B E C (3) E C A D B (2) E C A B D (2) E A C D B (2) D A C E B (2) D A B C E (2) C E D A B (2) C B E A D (2) B C D E A (2) B A E D C (2) A E D C B (2) A B D E C (2) E C B A D (1) E B C A D (1) E B A C D (1) E A B C D (1) D C B E A (1) D C A E B (1) D B A E C (1) D A E B C (1) C D B E A (1) B E A C D (1) B A E C D (1) A E C D B (1) A D E B C (1) A B E C D (1) Total count = 100 A B C D E A 0 4 8 12 8 B -4 0 6 2 4 C -8 -6 0 -10 -10 D -12 -2 10 0 10 E -8 -4 10 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999919 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 8 12 8 B -4 0 6 2 4 C -8 -6 0 -10 -10 D -12 -2 10 0 10 E -8 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=29 D=22 C=21 A=18 E=10 so E is eliminated. Round 2 votes counts: B=31 C=26 D=22 A=21 so A is eliminated. Round 3 votes counts: B=38 D=33 C=29 so C is eliminated. Round 4 votes counts: B=51 D=49 so D is eliminated. IRV winner is B compare: Computing Borda winner. A:216 B:204 D:203 E:194 C:183 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 4 8 12 8 B -4 0 6 2 4 C -8 -6 0 -10 -10 D -12 -2 10 0 10 E -8 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 8 12 8 B -4 0 6 2 4 C -8 -6 0 -10 -10 D -12 -2 10 0 10 E -8 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 8 12 8 B -4 0 6 2 4 C -8 -6 0 -10 -10 D -12 -2 10 0 10 E -8 -4 10 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999863 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9973: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E B A (9) B E A C D (8) D C A B E (7) E B A D C (6) E B A C D (6) C D A B E (5) A B E C D (5) D A C B E (4) A D C B E (4) A B E D C (4) A B C E D (4) D C A E B (3) C D E B A (3) E D B A C (2) E B C D A (2) D E B A C (2) C E B D A (2) C E B A D (2) C D A E B (2) C A B E D (2) B E A D C (2) B A E C D (2) A C D B E (2) A C B D E (2) E D C B A (1) E D B C A (1) E B C A D (1) D C E A B (1) D A C E B (1) C D B A E (1) C B E A D (1) C B A E D (1) B A E D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 12 8 2 B 6 0 -6 2 12 C -12 6 0 2 12 D -8 -2 -2 0 -6 E -2 -12 -12 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999962 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 8 2 B 6 0 -6 2 12 C -12 6 0 2 12 D -8 -2 -2 0 -6 E -2 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999753 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=27 A=22 E=19 C=19 B=13 so B is eliminated. Round 2 votes counts: E=29 D=27 A=25 C=19 so C is eliminated. Round 3 votes counts: D=38 E=34 A=28 so A is eliminated. Round 4 votes counts: E=53 D=47 so D is eliminated. IRV winner is E compare: Computing Borda winner. A:208 B:207 C:204 D:191 E:190 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A B , winner is: A compare: Computing GTS winners. A B C D E A 0 -6 12 8 2 B 6 0 -6 2 12 C -12 6 0 2 12 D -8 -2 -2 0 -6 E -2 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999753 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 8 2 B 6 0 -6 2 12 C -12 6 0 2 12 D -8 -2 -2 0 -6 E -2 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999753 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 8 2 B 6 0 -6 2 12 C -12 6 0 2 12 D -8 -2 -2 0 -6 E -2 -12 -12 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.250000 B: 0.500000 C: 0.250000 D: 0.000000 E: 0.000000 Sum of squares = 0.374999999753 Cumulative probabilities = A: 0.250000 B: 0.750000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9974: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B D E A (9) E A D C B (7) E A C D B (6) C D E A B (5) B A E D C (5) B A D E C (5) A E B D C (5) D B C A E (4) B C D A E (4) A E D B C (4) E A D B C (3) D C E A B (3) B D C A E (3) B D A E C (3) B D A C E (3) D A E B C (2) D A B E C (2) C E D A B (2) C E A D B (2) C E A B D (2) C B E D A (2) C B E A D (2) C B D A E (2) B A E C D (2) E C A D B (1) E A B D C (1) D E A C B (1) D C B E A (1) D C A E B (1) C D B E A (1) B E C A D (1) B E A C D (1) B C E A D (1) B C A E D (1) B C A D E (1) B A C E D (1) A B E D C (1) Total count = 100 A B C D E A 0 -4 4 4 -2 B 4 0 6 10 10 C -4 -6 0 -8 0 D -4 -10 8 0 0 E 2 -10 0 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999789 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 4 4 -2 B 4 0 6 10 10 C -4 -6 0 -8 0 D -4 -10 8 0 0 E 2 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 C=27 E=18 D=14 A=10 so A is eliminated. Round 2 votes counts: B=32 E=27 C=27 D=14 so D is eliminated. Round 3 votes counts: B=38 C=32 E=30 so E is eliminated. Round 4 votes counts: B=53 C=47 so C is eliminated. IRV winner is B compare: Computing Borda winner. B:215 A:201 D:197 E:196 C:191 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 4 4 -2 B 4 0 6 10 10 C -4 -6 0 -8 0 D -4 -10 8 0 0 E 2 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 4 4 -2 B 4 0 6 10 10 C -4 -6 0 -8 0 D -4 -10 8 0 0 E 2 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 4 4 -2 B 4 0 6 10 10 C -4 -6 0 -8 0 D -4 -10 8 0 0 E 2 -10 0 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999942 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9975: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C B E A (10) A E B C D (9) B A E C D (7) B C D A E (5) E A D C B (4) E A D B C (4) E A C D B (4) D E A C B (4) D C E A B (4) B C A E D (4) D C B A E (3) D B E A C (3) C B D A E (3) B A C E D (3) E A C B D (2) D E C A B (2) D C E B A (2) D B C A E (2) D B A E C (2) C D E A B (2) C D B E A (2) C D B A E (2) B A E D C (2) A B E C D (2) E A B D C (1) E A B C D (1) D E B C A (1) D B E C A (1) C D E B A (1) C B A E D (1) C B A D E (1) B D C A E (1) B D A E C (1) B D A C E (1) B C A D E (1) A E C B D (1) A E B D C (1) Total count = 100 A B C D E A 0 -18 4 -4 4 B 18 0 4 -6 14 C -4 -4 0 2 -4 D 4 6 -2 0 8 E -4 -14 4 -8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888845 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 A B C D E A 0 -18 4 -4 4 B 18 0 4 -6 14 C -4 -4 0 2 -4 D 4 6 -2 0 8 E -4 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=34 B=25 E=16 A=13 C=12 so C is eliminated. Round 2 votes counts: D=41 B=30 E=16 A=13 so A is eliminated. Round 3 votes counts: D=41 B=32 E=27 so E is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. B:215 D:208 C:195 A:193 E:189 Borda winner is B compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -18 4 -4 4 B 18 0 4 -6 14 C -4 -4 0 2 -4 D 4 6 -2 0 8 E -4 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -18 4 -4 4 B 18 0 4 -6 14 C -4 -4 0 2 -4 D 4 6 -2 0 8 E -4 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -18 4 -4 4 B 18 0 4 -6 14 C -4 -4 0 2 -4 D 4 6 -2 0 8 E -4 -14 4 -8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.166667 C: 0.500000 D: 0.333333 E: 0.000000 Sum of squares = 0.388888888888 Cumulative probabilities = A: 0.000000 B: 0.166667 C: 0.666667 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9976: compare: Profile of ballots (with multiplicities), in decreasing order by count: C B E D A (14) C B E A D (9) E B C D A (7) A D E B C (7) E B D C A (5) C B D A E (5) A E D C B (5) D A E B C (4) B C E D A (4) A D C B E (3) E A D B C (2) D A C B E (2) D A B E C (2) C B A E D (2) C B A D E (2) C A B D E (2) B C D E A (2) A E D B C (2) A C D B E (2) A C B E D (2) E D B C A (1) E D B A C (1) E C B D A (1) E B C A D (1) D E B A C (1) D E A B C (1) D C B A E (1) D B C E A (1) D A B C E (1) C E B A D (1) C D B A E (1) C B D E A (1) C A B E D (1) B D E C A (1) A D E C B (1) A D B C E (1) A C B D E (1) Total count = 100 A B C D E A 0 -22 -24 -12 -6 B 22 0 -12 22 20 C 24 12 0 16 16 D 12 -22 -16 0 -16 E 6 -20 -16 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999985 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -22 -24 -12 -6 B 22 0 -12 22 20 C 24 12 0 16 16 D 12 -22 -16 0 -16 E 6 -20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=38 A=24 E=18 D=13 B=7 so B is eliminated. Round 2 votes counts: C=44 A=24 E=18 D=14 so D is eliminated. Round 3 votes counts: C=46 A=33 E=21 so E is eliminated. Round 4 votes counts: C=62 A=38 so A is eliminated. IRV winner is C compare: Computing Borda winner. C:234 B:226 E:193 D:179 A:168 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 -22 -24 -12 -6 B 22 0 -12 22 20 C 24 12 0 16 16 D 12 -22 -16 0 -16 E 6 -20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -22 -24 -12 -6 B 22 0 -12 22 20 C 24 12 0 16 16 D 12 -22 -16 0 -16 E 6 -20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -22 -24 -12 -6 B 22 0 -12 22 20 C 24 12 0 16 16 D 12 -22 -16 0 -16 E 6 -20 -16 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9977: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A E B C (8) C A E D B (6) B C E D A (6) A D E C B (5) E D A B C (4) C B D A E (4) C B A D E (4) C A D E B (4) C A B D E (4) A D C E B (4) E B D A C (3) E B C D A (3) D E A B C (3) C B E A D (3) C A D B E (3) B E D C A (3) B E C D A (3) A D E B C (3) E A D C B (2) D A B E C (2) C A E B D (2) B D E A C (2) E D B A C (1) E C B A D (1) E C A D B (1) E B D C A (1) D E B A C (1) D B E A C (1) D B A E C (1) D A C B E (1) C E B A D (1) C E A D B (1) C B E D A (1) B E D A C (1) B D C A E (1) B D A C E (1) B C D E A (1) B C D A E (1) A E D B C (1) A D C B E (1) A C D B E (1) Total count = 100 A B C D E A 0 12 -8 -6 14 B -12 0 2 -8 -10 C 8 -2 0 0 0 D 6 8 0 0 12 E -14 10 0 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.451999 D: 0.548001 E: 0.000000 Sum of squares = 0.504608108334 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.451999 D: 1.000000 E: 1.000000 A B C D E A 0 12 -8 -6 14 B -12 0 2 -8 -10 C 8 -2 0 0 0 D 6 8 0 0 12 E -14 10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 B=19 D=17 E=16 A=15 so A is eliminated. Round 2 votes counts: C=34 D=30 B=19 E=17 so E is eliminated. Round 3 votes counts: D=38 C=36 B=26 so B is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:213 A:206 C:203 E:192 B:186 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 12 -8 -6 14 B -12 0 2 -8 -10 C 8 -2 0 0 0 D 6 8 0 0 12 E -14 10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -8 -6 14 B -12 0 2 -8 -10 C 8 -2 0 0 0 D 6 8 0 0 12 E -14 10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -8 -6 14 B -12 0 2 -8 -10 C 8 -2 0 0 0 D 6 8 0 0 12 E -14 10 0 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.499999 D: 0.500001 E: 0.000000 Sum of squares = 0.499999999967 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.499999 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9978: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E A B C (7) D C A B E (6) D A C B E (5) E D C B A (4) D E C B A (4) D C E A B (4) C B E A D (4) B A E C D (4) A B E D C (4) A B D C E (4) A B C D E (4) E C B A D (3) D E C A B (3) D A E B C (3) C D E B A (3) A B E C D (3) E D A B C (2) E C D B A (2) D C E B A (2) C E B D A (2) C B E D A (2) C B A E D (2) A D B C E (2) A B C E D (2) E D B C A (1) E B D A C (1) E B C A D (1) E B A D C (1) E A D B C (1) E A B D C (1) D E A C B (1) D C B A E (1) D C A E B (1) D A E C B (1) D A B E C (1) C E D B A (1) C E B A D (1) C B D E A (1) C A B D E (1) B C E A D (1) B C A E D (1) B A C E D (1) A D B E C (1) Total count = 100 A B C D E A 0 14 -2 -16 -6 B -14 0 -8 -12 0 C 2 8 0 -22 2 D 16 12 22 0 10 E 6 0 -2 -10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999996 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 -2 -16 -6 B -14 0 -8 -12 0 C 2 8 0 -22 2 D 16 12 22 0 10 E 6 0 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=39 A=20 E=17 C=17 B=7 so B is eliminated. Round 2 votes counts: D=39 A=25 C=19 E=17 so E is eliminated. Round 3 votes counts: D=47 A=28 C=25 so C is eliminated. Round 4 votes counts: D=58 A=42 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:230 E:197 A:195 C:195 B:183 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 14 -2 -16 -6 B -14 0 -8 -12 0 C 2 8 0 -22 2 D 16 12 22 0 10 E 6 0 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 -2 -16 -6 B -14 0 -8 -12 0 C 2 8 0 -22 2 D 16 12 22 0 10 E 6 0 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 -2 -16 -6 B -14 0 -8 -12 0 C 2 8 0 -22 2 D 16 12 22 0 10 E 6 0 -2 -10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999925 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9979: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C E D (9) A C B D E (8) D E C B A (7) E D B C A (6) C D E B A (5) B E D C A (5) E D B A C (4) D E C A B (4) A B E D C (4) E D A C B (3) E B D A C (3) B C D E A (3) A D E C B (3) A C D E B (3) A C B E D (3) E A D B C (2) D E B C A (2) D C E A B (2) C A D E B (2) C A D B E (2) C A B D E (2) B C E D A (2) A E D C B (2) A E D B C (2) E D A B C (1) D A C E B (1) C D A E B (1) C B A D E (1) B E D A C (1) B E C D A (1) B E A C D (1) B C A E D (1) B A C E D (1) A E B D C (1) A C D B E (1) A B E C D (1) Total count = 100 A B C D E A 0 14 8 -2 -4 B -14 0 0 -6 -8 C -8 0 0 -6 -6 D 2 6 6 0 -6 E 4 8 6 6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999657 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 14 8 -2 -4 B -14 0 0 -6 -8 C -8 0 0 -6 -6 D 2 6 6 0 -6 E 4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=19 D=16 B=15 C=13 so C is eliminated. Round 2 votes counts: A=43 D=22 E=19 B=16 so B is eliminated. Round 3 votes counts: A=46 E=29 D=25 so D is eliminated. Round 4 votes counts: E=52 A=48 so A is eliminated. IRV winner is E compare: Computing Borda winner. E:212 A:208 D:204 C:190 B:186 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 14 8 -2 -4 B -14 0 0 -6 -8 C -8 0 0 -6 -6 D 2 6 6 0 -6 E 4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 8 -2 -4 B -14 0 0 -6 -8 C -8 0 0 -6 -6 D 2 6 6 0 -6 E 4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 8 -2 -4 B -14 0 0 -6 -8 C -8 0 0 -6 -6 D 2 6 6 0 -6 E 4 8 6 6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999999 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9980: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A C E (10) C E A D B (8) D C E B A (7) B D A E C (7) D B E C A (5) B D E C A (5) B A D E C (5) A E C B D (4) A C E B D (4) A B D C E (4) E C D B A (3) E C A B D (3) D B C E A (3) A B E C D (3) D C E A B (2) D B A E C (2) D A C B E (2) C D E A B (2) B A D C E (2) E C B D A (1) E C B A D (1) E B C A D (1) E A B C D (1) D C A E B (1) D B C A E (1) C D A E B (1) C A E B D (1) B E C D A (1) B E A C D (1) B D E A C (1) B D A C E (1) B A E D C (1) A E B C D (1) A D B C E (1) A C E D B (1) A B E D C (1) A B C E D (1) A B C D E (1) Total count = 100 A B C D E A 0 -16 8 -10 10 B 16 0 18 2 16 C -8 -18 0 -22 6 D 10 -2 22 0 26 E -10 -16 -6 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997772 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -16 8 -10 10 B 16 0 18 2 16 C -8 -18 0 -22 6 D 10 -2 22 0 26 E -10 -16 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=33 B=24 A=21 C=12 E=10 so E is eliminated. Round 2 votes counts: D=33 B=25 A=22 C=20 so C is eliminated. Round 3 votes counts: D=39 A=34 B=27 so B is eliminated. Round 4 votes counts: D=55 A=45 so A is eliminated. IRV winner is D compare: Computing Borda winner. D:228 B:226 A:196 C:179 E:171 Borda winner is D compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -16 8 -10 10 B 16 0 18 2 16 C -8 -18 0 -22 6 D 10 -2 22 0 26 E -10 -16 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -16 8 -10 10 B 16 0 18 2 16 C -8 -18 0 -22 6 D 10 -2 22 0 26 E -10 -16 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -16 8 -10 10 B 16 0 18 2 16 C -8 -18 0 -22 6 D 10 -2 22 0 26 E -10 -16 -6 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989502 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9981: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C B D (20) D B C E A (13) E A B D C (10) E A D B C (7) C B D A E (6) E A B C D (4) D C B A E (4) D B E C A (3) A C B D E (3) E D B A C (2) E B D C A (2) E B D A C (2) D E B C A (2) D B C A E (2) C D B A E (2) C A B D E (2) A E D B C (2) A C E B D (2) A C D B E (2) E A C B D (1) D E B A C (1) C D A B E (1) C A E B D (1) C A D B E (1) C A B E D (1) B E C D A (1) B C E D A (1) A E C D B (1) A D E B C (1) Total count = 100 A B C D E A 0 18 16 16 2 B -18 0 6 12 -16 C -16 -6 0 -2 -18 D -16 -12 2 0 -14 E -2 16 18 14 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999988 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 18 16 16 2 B -18 0 6 12 -16 C -16 -6 0 -2 -18 D -16 -12 2 0 -14 E -2 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=31 E=28 D=25 C=14 B=2 so B is eliminated. Round 2 votes counts: A=31 E=29 D=25 C=15 so C is eliminated. Round 3 votes counts: A=36 D=34 E=30 so E is eliminated. Round 4 votes counts: A=58 D=42 so D is eliminated. IRV winner is A compare: Computing Borda winner. A:226 E:223 B:192 D:180 C:179 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 18 16 16 2 B -18 0 6 12 -16 C -16 -6 0 -2 -18 D -16 -12 2 0 -14 E -2 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 16 16 2 B -18 0 6 12 -16 C -16 -6 0 -2 -18 D -16 -12 2 0 -14 E -2 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 16 16 2 B -18 0 6 12 -16 C -16 -6 0 -2 -18 D -16 -12 2 0 -14 E -2 16 18 14 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999987291 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9982: compare: Profile of ballots (with multiplicities), in decreasing order by count: D B A E C (7) C D B E A (7) B D C E A (7) D B C A E (5) C E B D A (5) C A E D B (5) E A C B D (4) C E A B D (4) B D E C A (4) E C A B D (3) D B C E A (3) D A B E C (3) A E C D B (3) A E C B D (3) A E B D C (3) A D B E C (3) E B A D C (2) D C B A E (2) D B A C E (2) C E A D B (2) C D A B E (2) A E D B C (2) A E B C D (2) E C B D A (1) E C B A D (1) E B C D A (1) E B A C D (1) E A B D C (1) E A B C D (1) D A C B E (1) D A B C E (1) C E B A D (1) C D B A E (1) C A D E B (1) B E A D C (1) B D E A C (1) A D E B C (1) A D B C E (1) A C E D B (1) A C D E B (1) Total count = 100 A B C D E A 0 -4 -10 -6 0 B 4 0 4 -8 2 C 10 -4 0 0 4 D 6 8 0 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.368406 D: 0.631594 E: 0.000000 Sum of squares = 0.534634088106 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.368406 D: 1.000000 E: 1.000000 A B C D E A 0 -4 -10 -6 0 B 4 0 4 -8 2 C 10 -4 0 0 4 D 6 8 0 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=28 D=24 A=20 E=15 B=13 so B is eliminated. Round 2 votes counts: D=36 C=28 A=20 E=16 so E is eliminated. Round 3 votes counts: D=36 C=34 A=30 so A is eliminated. Round 4 votes counts: D=50 C=50 so D is eliminated. IRV winner is C compare: Computing Borda winner. D:210 C:205 B:201 E:194 A:190 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -4 -10 -6 0 B 4 0 4 -8 2 C 10 -4 0 0 4 D 6 8 0 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTS winners are ['C', 'D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 -10 -6 0 B 4 0 4 -8 2 C 10 -4 0 0 4 D 6 8 0 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 -10 -6 0 B 4 0 4 -8 2 C 10 -4 0 0 4 D 6 8 0 0 6 E 0 -2 -4 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.500000 D: 0.500000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.500000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9983: compare: Profile of ballots (with multiplicities), in decreasing order by count: E B D A C (9) C A E D B (9) C A D B E (9) B D E A C (9) C E A D B (6) C A D E B (6) E B D C A (5) B E D A C (5) B D A E C (4) E B C D A (3) D B A C E (3) A D C B E (3) E D B A C (2) E C A D B (2) E C A B D (2) D A B E C (2) D A B C E (2) B D E C A (2) A D B C E (2) A C D E B (2) E C B D A (1) E C B A D (1) E A D C B (1) E A C D B (1) D B E A C (1) C E A B D (1) C B A D E (1) C A B D E (1) B E D C A (1) B D C E A (1) B D C A E (1) A C E D B (1) A C D B E (1) Total count = 100 A B C D E A 0 2 -4 -2 -6 B -2 0 4 -6 -4 C 4 -4 0 -6 -2 D 2 6 6 0 0 E 6 4 2 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.522422 E: 0.477578 Sum of squares = 0.501005476889 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.522422 E: 1.000000 A B C D E A 0 2 -4 -2 -6 B -2 0 4 -6 -4 C 4 -4 0 -6 -2 D 2 6 6 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: C , winner is: C compare: Computing IRV winner. Round 1 votes counts: C=33 E=27 B=23 A=9 D=8 so D is eliminated. Round 2 votes counts: C=33 E=27 B=27 A=13 so A is eliminated. Round 3 votes counts: C=40 B=33 E=27 so E is eliminated. Round 4 votes counts: B=52 C=48 so C is eliminated. IRV winner is B compare: Computing Borda winner. D:207 E:206 B:196 C:196 A:195 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D E , winner is: D compare: Computing GTS winners. A B C D E A 0 2 -4 -2 -6 B -2 0 4 -6 -4 C 4 -4 0 -6 -2 D 2 6 6 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 2 -4 -2 -6 B -2 0 4 -6 -4 C 4 -4 0 -6 -2 D 2 6 6 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 2 -4 -2 -6 B -2 0 4 -6 -4 C 4 -4 0 -6 -2 D 2 6 6 0 0 E 6 4 2 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 0.500000 Sum of squares = 0.499999999877 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.500000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9984: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E D C A (11) A C D E B (11) E D B C A (9) A C D B E (6) A C B D E (6) A B C D E (6) E B D C A (5) D E C A B (5) B E D A C (5) B A E C D (5) D C E A B (4) C A D E B (4) B E A C D (4) E D C B A (3) D C A E B (3) C D A E B (3) B A C E D (3) A C B E D (2) D E C B A (1) C E B D A (1) C A E B D (1) B E A D C (1) A B C E D (1) Total count = 100 A B C D E A 0 4 0 0 2 B -4 0 0 2 0 C 0 0 0 6 2 D 0 -2 -6 0 -2 E -2 0 -2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.603279 B: 0.000000 C: 0.396721 D: 0.000000 E: 0.000000 Sum of squares = 0.521332942037 Cumulative probabilities = A: 0.603279 B: 0.603279 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 4 0 0 2 B -4 0 0 2 0 C 0 0 0 6 2 D 0 -2 -6 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=32 B=29 E=17 D=13 C=9 so C is eliminated. Round 2 votes counts: A=37 B=29 E=18 D=16 so D is eliminated. Round 3 votes counts: A=43 B=29 E=28 so E is eliminated. Round 4 votes counts: A=52 B=48 so B is eliminated. IRV winner is A compare: Computing Borda winner. C:204 A:203 B:199 E:199 D:195 Borda winner is C compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A C , winner is: A compare: Computing GTS winners. A B C D E A 0 4 0 0 2 B -4 0 0 2 0 C 0 0 0 6 2 D 0 -2 -6 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 4 0 0 2 B -4 0 0 2 0 C 0 0 0 6 2 D 0 -2 -6 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 4 0 0 2 B -4 0 0 2 0 C 0 0 0 6 2 D 0 -2 -6 0 -2 E -2 0 -2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.500000 B: 0.000000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.499999999998 Cumulative probabilities = A: 0.500000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9985: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D B A C (6) C A D B E (5) B E D A C (5) B D E C A (5) D E B A C (4) C A B D E (4) A C B E D (4) E B D A C (3) D E C B A (3) D E C A B (3) D E B C A (3) D C E B A (3) D B C E A (3) C D A B E (3) C A D E B (3) B E A C D (3) A C E D B (3) A B E C D (3) A B C E D (3) E D A B C (2) E B A D C (2) E A B C D (2) D B E C A (2) C B A D E (2) B D E A C (2) B D C E A (2) A C E B D (2) E D A C B (1) E A D B C (1) D E A C B (1) D C E A B (1) D C B E A (1) D C A E B (1) D B E A C (1) C D B A E (1) C D A E B (1) C A E D B (1) B C D A E (1) B C A E D (1) B C A D E (1) B A E C D (1) B A C E D (1) Total count = 100 A B C D E A 0 -12 0 -16 -18 B 12 0 14 -6 8 C 0 -14 0 -10 -6 D 16 6 10 0 12 E 18 -8 6 -12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999795 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 0 -16 -18 B 12 0 14 -6 8 C 0 -14 0 -10 -6 D 16 6 10 0 12 E 18 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=26 B=22 C=20 E=17 A=15 so A is eliminated. Round 2 votes counts: C=29 B=28 D=26 E=17 so E is eliminated. Round 3 votes counts: D=36 B=35 C=29 so C is eliminated. Round 4 votes counts: D=53 B=47 so B is eliminated. IRV winner is D compare: Computing Borda winner. D:222 B:214 E:202 C:185 A:177 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 -12 0 -16 -18 B 12 0 14 -6 8 C 0 -14 0 -10 -6 D 16 6 10 0 12 E 18 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 0 -16 -18 B 12 0 14 -6 8 C 0 -14 0 -10 -6 D 16 6 10 0 12 E 18 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 0 -16 -18 B 12 0 14 -6 8 C 0 -14 0 -10 -6 D 16 6 10 0 12 E 18 -8 6 -12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999997 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9986: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C B A (6) E D C A B (6) E D B C A (5) D E A C B (5) B E D C A (5) B E C D A (5) B A C D E (5) C E D A B (4) B C A E D (4) C B E A D (3) B D E A C (3) A D C E B (3) E C D B A (2) D E B A C (2) D E A B C (2) D A E B C (2) C E D B A (2) C B A E D (2) C A E D B (2) B D E C A (2) B C E A D (2) B A D E C (2) A D E C B (2) A D B E C (2) A D B C E (2) A C E D B (2) A C D E B (2) A C B D E (2) A B C D E (2) E C B D A (1) D E C A B (1) D E B C A (1) D B E A C (1) D B A E C (1) D A E C B (1) C A B E D (1) B D A E C (1) B A C E D (1) A C B E D (1) A B D C E (1) A B C E D (1) Total count = 100 A B C D E A 0 -12 -8 -16 -16 B 12 0 4 -12 -2 C 8 -4 0 -12 -16 D 16 12 12 0 -10 E 16 2 16 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999977 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 -12 -8 -16 -16 B 12 0 4 -12 -2 C 8 -4 0 -12 -16 D 16 12 12 0 -10 E 16 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 E=20 A=20 D=16 C=14 so C is eliminated. Round 2 votes counts: B=35 E=26 A=23 D=16 so D is eliminated. Round 3 votes counts: E=37 B=37 A=26 so A is eliminated. Round 4 votes counts: E=51 B=49 so B is eliminated. IRV winner is E compare: Computing Borda winner. E:222 D:215 B:201 C:188 A:174 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 -12 -8 -16 -16 B 12 0 4 -12 -2 C 8 -4 0 -12 -16 D 16 12 12 0 -10 E 16 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -8 -16 -16 B 12 0 4 -12 -2 C 8 -4 0 -12 -16 D 16 12 12 0 -10 E 16 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -8 -16 -16 B 12 0 4 -12 -2 C 8 -4 0 -12 -16 D 16 12 12 0 -10 E 16 2 16 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999987891 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9987: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D A B C (7) E C D A B (7) B C A D E (7) B A D C E (6) E D A C B (5) C E D A B (5) E C B D A (4) A D B C E (4) D A E B C (3) D A C E B (3) C B E A D (3) C A D B E (3) E C D B A (2) E B D A C (2) E B C D A (2) C E B D A (2) C E B A D (2) C D A E B (2) C B A E D (2) C A D E B (2) B E A D C (2) A B D C E (2) E D C A B (1) E D B C A (1) E D B A C (1) E B D C A (1) D A E C B (1) D A C B E (1) C E A B D (1) C D E A B (1) C B A D E (1) C A B D E (1) B E D A C (1) B E C A D (1) B E A C D (1) B D A E C (1) B C E A D (1) B C A E D (1) B A E D C (1) B A D E C (1) B A C D E (1) A D C E B (1) A D C B E (1) A D B E C (1) A C D B E (1) Total count = 100 A B C D E A 0 6 -6 -6 -6 B -6 0 -4 -6 -12 C 6 4 0 6 8 D 6 6 -6 0 -12 E 6 12 -8 12 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 -6 -6 -6 B -6 0 -4 -6 -12 C 6 4 0 6 8 D 6 6 -6 0 -12 E 6 12 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=33 C=25 B=24 A=10 D=8 so D is eliminated. Round 2 votes counts: E=33 C=25 B=24 A=18 so A is eliminated. Round 3 votes counts: E=37 C=32 B=31 so B is eliminated. Round 4 votes counts: C=54 E=46 so E is eliminated. IRV winner is C compare: Computing Borda winner. C:212 E:211 D:197 A:194 B:186 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 6 -6 -6 -6 B -6 0 -4 -6 -12 C 6 4 0 6 8 D 6 6 -6 0 -12 E 6 12 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 -6 -6 -6 B -6 0 -4 -6 -12 C 6 4 0 6 8 D 6 6 -6 0 -12 E 6 12 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 -6 -6 -6 B -6 0 -4 -6 -12 C 6 4 0 6 8 D 6 6 -6 0 -12 E 6 12 -8 12 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998688 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9988: compare: Profile of ballots (with multiplicities), in decreasing order by count: D C E A B (15) E C D A B (10) B A E C D (6) A E B C D (6) D C E B A (5) B A D C E (5) D C B E A (4) B C D E A (4) A B E C D (4) E C D B A (3) E A B C D (3) C D E B A (3) C D E A B (3) C B D E A (3) B E A C D (3) E A C D B (2) D C A E B (2) B D C E A (2) B D C A E (2) A E D C B (2) A E C D B (2) A B E D C (2) E D C A B (1) E A C B D (1) D C B A E (1) D B C E A (1) B C E D A (1) B A C D E (1) A E B D C (1) A D B C E (1) A B D C E (1) Total count = 100 A B C D E A 0 12 -20 -20 -28 B -12 0 -14 -10 -18 C 20 14 0 10 8 D 20 10 -10 0 6 E 28 18 -8 -6 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999987 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 12 -20 -20 -28 B -12 0 -14 -10 -18 C 20 14 0 10 8 D 20 10 -10 0 6 E 28 18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 B=24 E=20 A=19 C=9 so C is eliminated. Round 2 votes counts: D=34 B=27 E=20 A=19 so A is eliminated. Round 3 votes counts: D=35 B=34 E=31 so E is eliminated. Round 4 votes counts: D=55 B=45 so B is eliminated. IRV winner is D compare: Computing Borda winner. C:226 E:216 D:213 B:173 A:172 Borda winner is C compare: Computing minimax winner. minimax winner is C compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: C , winner is: C compare: Computing GTS winners. A B C D E A 0 12 -20 -20 -28 B -12 0 -14 -10 -18 C 20 14 0 10 8 D 20 10 -10 0 6 E 28 18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 12 -20 -20 -28 B -12 0 -14 -10 -18 C 20 14 0 10 8 D 20 10 -10 0 6 E 28 18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is C (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 12 -20 -20 -28 B -12 0 -14 -10 -18 C 20 14 0 10 8 D 20 10 -10 0 6 E 28 18 -8 -6 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 1.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999944 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is C (randomly chosen according to balanced optimal mixed strategy). Trial 9989: compare: Profile of ballots (with multiplicities), in decreasing order by count: D A B E C (13) B D A C E (10) A D E C B (10) B C E D A (7) C E B A D (6) E C A D B (5) B D C E A (5) E C D A B (4) A E D C B (4) A D B E C (4) B D C A E (3) A D E B C (3) D B A E C (2) C E B D A (2) C E A B D (2) B C D E A (2) B A D C E (2) A E C D B (2) A B D C E (2) E C B D A (1) E A C D B (1) D E C B A (1) D E B C A (1) D E A C B (1) D B E A C (1) D A E C B (1) D A E B C (1) C B E D A (1) C B E A D (1) B D E C A (1) B C E A D (1) Total count = 100 A B C D E A 0 6 14 -14 14 B -6 0 16 -8 10 C -14 -16 0 -30 -12 D 14 8 30 0 26 E -14 -10 12 -26 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.999999999939 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 A B C D E A 0 6 14 -14 14 B -6 0 16 -8 10 C -14 -16 0 -30 -12 D 14 8 30 0 26 E -14 -10 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=31 A=25 D=21 C=12 E=11 so E is eliminated. Round 2 votes counts: B=31 A=26 C=22 D=21 so D is eliminated. Round 3 votes counts: A=42 B=35 C=23 so C is eliminated. Round 4 votes counts: A=53 B=47 so B is eliminated. IRV winner is A compare: Computing Borda winner. D:239 A:210 B:206 E:181 C:164 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 6 14 -14 14 B -6 0 16 -8 10 C -14 -16 0 -30 -12 D 14 8 30 0 26 E -14 -10 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTS winners are ['D'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 6 14 -14 14 B -6 0 16 -8 10 C -14 -16 0 -30 -12 D 14 8 30 0 26 E -14 -10 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 6 14 -14 14 B -6 0 16 -8 10 C -14 -16 0 -30 -12 D 14 8 30 0 26 E -14 -10 12 -26 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 0.000000 Sum of squares = 0.99999999722 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 1.000000 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9990: compare: Profile of ballots (with multiplicities), in decreasing order by count: B D A C E (9) E C A D B (8) E C A B D (7) E B A C D (6) C A E D B (6) B E D A C (6) B D E A C (6) D C A B E (5) A C D E B (5) E C D B A (3) D B C A E (3) A D B C E (3) E B C A D (2) D C B A E (2) D B C E A (2) D B A C E (2) D A C B E (2) C D A E B (2) B E D C A (2) B E A D C (2) B D C A E (2) A C E D B (2) E B A D C (1) E A C D B (1) E A C B D (1) D E B C A (1) D A B C E (1) C E D A B (1) C E A D B (1) C D E A B (1) C D A B E (1) B D E C A (1) A E C D B (1) A C D B E (1) A B D C E (1) Total count = 100 A B C D E A 0 0 0 -4 -4 B 0 0 0 -8 2 C 0 0 0 -2 4 D 4 8 2 0 0 E 4 -2 -4 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.768119 E: 0.231881 Sum of squares = 0.64377561822 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.768119 E: 1.000000 A B C D E A 0 0 0 -4 -4 B 0 0 0 -8 2 C 0 0 0 -2 4 D 4 8 2 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555556112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=29 B=28 D=18 A=13 C=12 so C is eliminated. Round 2 votes counts: E=31 B=28 D=22 A=19 so A is eliminated. Round 3 votes counts: E=40 D=31 B=29 so B is eliminated. Round 4 votes counts: E=50 D=50 so E is eliminated. IRV winner is D compare: Computing Borda winner. D:207 C:201 E:199 B:197 A:196 Borda winner is D compare: Computing minimax winner. minimax winner is D compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: D , winner is: D compare: Computing GTS winners. A B C D E A 0 0 0 -4 -4 B 0 0 0 -8 2 C 0 0 0 -2 4 D 4 8 2 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555556112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTS winners are ['D', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 0 0 -4 -4 B 0 0 0 -8 2 C 0 0 0 -2 4 D 4 8 2 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555556112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GTD winner is D (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 0 0 -4 -4 B 0 0 0 -8 2 C 0 0 0 -2 4 D 4 8 2 0 0 E 4 -2 -4 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 0.333333 Sum of squares = 0.55555556112 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.666667 E: 1.000000 GT winner is D (randomly chosen according to balanced optimal mixed strategy). Trial 9991: compare: Profile of ballots (with multiplicities), in decreasing order by count: A E C D B (9) E B A C D (6) B D A C E (6) D C B A E (4) D C A B E (4) B E A D C (4) B D C E A (4) A D C B E (4) E C D A B (3) E C A D B (3) E B C A D (3) E A C D B (3) E A B C D (3) B E D C A (3) B D C A E (3) B A D C E (3) A D C E B (3) A D B C E (3) C E D A B (2) C D B E A (2) B E D A C (2) B E C D A (2) A E B D C (2) A B D C E (2) E C B D A (1) E C B A D (1) E B C D A (1) E A C B D (1) D B C A E (1) D A C B E (1) C D E A B (1) C D B A E (1) C D A E B (1) C B E D A (1) B E A C D (1) B D E C A (1) B A E D C (1) A E D C B (1) A E D B C (1) A D E C B (1) A B E D C (1) Total count = 100 A B C D E A 0 -2 16 12 4 B 2 0 6 4 8 C -16 -6 0 -10 -8 D -12 -4 10 0 -10 E -4 -8 8 10 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999896 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -2 16 12 4 B 2 0 6 4 8 C -16 -6 0 -10 -8 D -12 -4 10 0 -10 E -4 -8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=30 A=27 E=25 D=10 C=8 so C is eliminated. Round 2 votes counts: B=31 E=27 A=27 D=15 so D is eliminated. Round 3 votes counts: B=39 A=33 E=28 so E is eliminated. Round 4 votes counts: B=51 A=49 so A is eliminated. IRV winner is B compare: Computing Borda winner. A:215 B:210 E:203 D:192 C:180 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 16 12 4 B 2 0 6 4 8 C -16 -6 0 -10 -8 D -12 -4 10 0 -10 E -4 -8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 16 12 4 B 2 0 6 4 8 C -16 -6 0 -10 -8 D -12 -4 10 0 -10 E -4 -8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 16 12 4 B 2 0 6 4 8 C -16 -6 0 -10 -8 D -12 -4 10 0 -10 E -4 -8 8 10 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999996512 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9992: compare: Profile of ballots (with multiplicities), in decreasing order by count: A B C D E (6) E D C B A (5) A C B E D (5) E D A B C (4) D E A B C (4) C B A D E (4) B C A D E (4) B A C D E (4) D E C B A (3) D E B C A (3) D E B A C (3) C B E D A (3) C B D E A (3) A D E B C (3) A B C E D (3) E C D B A (2) E C A D B (2) E C A B D (2) E A C D B (2) D C E B A (2) D B A C E (2) D A B E C (2) C E B D A (2) C B E A D (2) C B A E D (2) C A B E D (2) A E D B C (2) A B E C D (2) E C B D A (1) E A D C B (1) E A D B C (1) E A C B D (1) D B E A C (1) D B C A E (1) D B A E C (1) D A B C E (1) C E D B A (1) C E B A D (1) C E A B D (1) C D B E A (1) C B D A E (1) B C D A E (1) A E C B D (1) A E B D C (1) A D B E C (1) Total count = 100 A B C D E A 0 -6 2 6 -2 B 6 0 0 4 4 C -2 0 0 18 4 D -6 -4 -18 0 2 E 2 -4 -4 -2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.721261 C: 0.278739 D: 0.000000 E: 0.000000 Sum of squares = 0.597912553949 Cumulative probabilities = A: 0.000000 B: 0.721261 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 2 6 -2 B 6 0 0 4 4 C -2 0 0 18 4 D -6 -4 -18 0 2 E 2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=24 D=23 C=23 E=21 B=9 so B is eliminated. Round 2 votes counts: C=28 A=28 D=23 E=21 so E is eliminated. Round 3 votes counts: C=35 A=33 D=32 so D is eliminated. Round 4 votes counts: A=51 C=49 so C is eliminated. IRV winner is A compare: Computing Borda winner. C:210 B:207 A:200 E:196 D:187 Borda winner is C compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 2 6 -2 B 6 0 0 4 4 C -2 0 0 18 4 D -6 -4 -18 0 2 E 2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B', 'C'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 2 6 -2 B 6 0 0 4 4 C -2 0 0 18 4 D -6 -4 -18 0 2 E 2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 2 6 -2 B 6 0 0 4 4 C -2 0 0 18 4 D -6 -4 -18 0 2 E 2 -4 -4 -2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.500000 C: 0.500000 D: 0.000000 E: 0.000000 Sum of squares = 0.5 Cumulative probabilities = A: 0.000000 B: 0.500000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9993: compare: Profile of ballots (with multiplicities), in decreasing order by count: C D A B E (11) E B A D C (10) D C A E B (7) B E A C D (6) E A B D C (5) C D E A B (5) B E C A D (5) A E B D C (4) E B D A C (3) E A D B C (3) D C E A B (3) D A C E B (3) C D B E A (3) B C E D A (3) A D C B E (3) E B A C D (2) C D E B A (2) C D A E B (2) B A E C D (2) E D A C B (1) E D A B C (1) E B C A D (1) D C A B E (1) D A E C B (1) C D B A E (1) B E C D A (1) B E A D C (1) B A C E D (1) A E D B C (1) A D E C B (1) A D E B C (1) A D B E C (1) A C D B E (1) A C B D E (1) A B E D C (1) A B D C E (1) A B C D E (1) Total count = 100 A B C D E A 0 18 10 4 -10 B -18 0 8 -4 -12 C -10 -8 0 -4 -2 D -4 4 4 0 -2 E 10 12 2 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999988 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 A B C D E A 0 18 10 4 -10 B -18 0 8 -4 -12 C -10 -8 0 -4 -2 D -4 4 4 0 -2 E 10 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: E , winner is: E compare: Computing IRV winner. Round 1 votes counts: E=26 C=24 B=19 A=16 D=15 so D is eliminated. Round 2 votes counts: C=35 E=26 A=20 B=19 so B is eliminated. Round 3 votes counts: E=39 C=38 A=23 so A is eliminated. Round 4 votes counts: E=51 C=49 so C is eliminated. IRV winner is E compare: Computing Borda winner. E:213 A:211 D:201 C:188 B:187 Borda winner is E compare: Computing minimax winner. minimax winner is E compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: E , winner is: E compare: Computing GTS winners. A B C D E A 0 18 10 4 -10 B -18 0 8 -4 -12 C -10 -8 0 -4 -2 D -4 4 4 0 -2 E 10 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTS winners are ['E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 18 10 4 -10 B -18 0 8 -4 -12 C -10 -8 0 -4 -2 D -4 4 4 0 -2 E 10 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GTD winner is E (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 18 10 4 -10 B -18 0 8 -4 -12 C -10 -8 0 -4 -2 D -4 4 4 0 -2 E 10 12 2 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 Sum of squares = 0.999999999629 Cumulative probabilities = A: 0.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 1.000000 GT winner is E (randomly chosen according to balanced optimal mixed strategy). Trial 9994: compare: Profile of ballots (with multiplicities), in decreasing order by count: B E A C D (8) B C E D A (8) B C D E A (7) D C A E B (6) B C D A E (6) E A B D C (5) B E C A D (5) A E D B C (5) A E B D C (5) E A D B C (4) C B D E A (4) A E D C B (4) A D E C B (4) E B A C D (3) C D B E A (3) C D B A E (3) A B E D C (3) D C A B E (2) C D E A B (2) B E C D A (2) E D C A B (1) E C D B A (1) E C B D A (1) D A C E B (1) D A C B E (1) C B D A E (1) B C E A D (1) B A E D C (1) B A C E D (1) B A C D E (1) A D C B E (1) Total count = 100 A B C D E A 0 -12 -6 2 -10 B 12 0 30 24 16 C 6 -30 0 14 -4 D -2 -24 -14 0 -16 E 10 -16 4 16 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999992 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -12 -6 2 -10 B 12 0 30 24 16 C 6 -30 0 14 -4 D -2 -24 -14 0 -16 E 10 -16 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=40 A=22 E=15 C=13 D=10 so D is eliminated. Round 2 votes counts: B=40 A=24 C=21 E=15 so E is eliminated. Round 3 votes counts: B=43 A=33 C=24 so C is eliminated. Round 4 votes counts: B=56 A=44 so A is eliminated. IRV winner is B compare: Computing Borda winner. B:241 E:207 C:193 A:187 D:172 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -12 -6 2 -10 B 12 0 30 24 16 C 6 -30 0 14 -4 D -2 -24 -14 0 -16 E 10 -16 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -12 -6 2 -10 B 12 0 30 24 16 C 6 -30 0 14 -4 D -2 -24 -14 0 -16 E 10 -16 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -12 -6 2 -10 B 12 0 30 24 16 C 6 -30 0 14 -4 D -2 -24 -14 0 -16 E 10 -16 4 16 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.99999999969 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9995: compare: Profile of ballots (with multiplicities), in decreasing order by count: E D C A B (5) B C A D E (4) B A E D C (4) B A C D E (4) A E D C B (4) A B C D E (4) E D C B A (3) E D A C B (3) E B A D C (3) D C E A B (3) C D E B A (3) B E A D C (3) B C D E A (3) B A C E D (3) A E B D C (3) A C D E B (3) A C D B E (3) A B E D C (3) A B C E D (3) E D B A C (2) E B D C A (2) E A B D C (2) D E C B A (2) D C E B A (2) C D A E B (2) C B D E A (2) C B D A E (2) B C D A E (2) A B E C D (2) E D B C A (1) E D A B C (1) E B D A C (1) E A D C B (1) D E C A B (1) D C A E B (1) C D E A B (1) C D B E A (1) C D B A E (1) C D A B E (1) C A D E B (1) C A B D E (1) B E D C A (1) B E A C D (1) B A E C D (1) A C B D E (1) Total count = 100 A B C D E A 0 -2 10 8 6 B 2 0 6 10 0 C -10 -6 0 -2 2 D -8 -10 2 0 -4 E -6 0 -2 4 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 0.879622 C: 0.000000 D: 0.000000 E: 0.120378 Sum of squares = 0.788225130931 Cumulative probabilities = A: 0.000000 B: 0.879622 C: 0.879622 D: 0.879622 E: 1.000000 A B C D E A 0 -2 10 8 6 B 2 0 6 10 0 C -10 -6 0 -2 2 D -8 -10 2 0 -4 E -6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000803 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 LP and QP give different solutions to GT plurality potential winners: A B , winner is: A compare: Computing IRV winner. Round 1 votes counts: B=26 A=26 E=24 C=15 D=9 so D is eliminated. Round 2 votes counts: E=27 B=26 A=26 C=21 so C is eliminated. Round 3 votes counts: E=36 B=32 A=32 so B is eliminated. Round 4 votes counts: A=53 E=47 so E is eliminated. IRV winner is A compare: Computing Borda winner. A:211 B:209 E:198 C:192 D:190 Borda winner is A compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -2 10 8 6 B 2 0 6 10 0 C -10 -6 0 -2 2 D -8 -10 2 0 -4 E -6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000803 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTS winners are ['B', 'E'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -2 10 8 6 B 2 0 6 10 0 C -10 -6 0 -2 2 D -8 -10 2 0 -4 E -6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000803 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -2 10 8 6 B 2 0 6 10 0 C -10 -6 0 -2 2 D -8 -10 2 0 -4 E -6 0 -2 4 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 0.750000 C: 0.000000 D: 0.000000 E: 0.250000 Sum of squares = 0.625000000803 Cumulative probabilities = A: 0.000000 B: 0.750000 C: 0.750000 D: 0.750000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9996: compare: Profile of ballots (with multiplicities), in decreasing order by count: A C E D B (17) B D E A C (10) C A E D B (8) B D E C A (7) A E C D B (6) C D E A B (5) B D C E A (5) B A E D C (4) D C E A B (3) C D A E B (3) C A D E B (3) B E D A C (3) A E B D C (3) E D A C B (2) D B C E A (2) C B D E A (2) B A C E D (2) A C E B D (2) A B C E D (2) D E C B A (1) D E C A B (1) D E B A C (1) D B E C A (1) C D B E A (1) C B A D E (1) B D A C E (1) B C D E A (1) B C A D E (1) A E D C B (1) A E B C D (1) Total count = 100 A B C D E A 0 14 10 2 10 B -14 0 -12 -10 -14 C -10 12 0 10 18 D -2 10 -10 0 -2 E -10 14 -18 2 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999998873 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 14 10 2 10 B -14 0 -12 -10 -14 C -10 12 0 10 18 D -2 10 -10 0 -2 E -10 14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997335 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=34 A=32 C=23 D=9 E=2 so E is eliminated. Round 2 votes counts: B=34 A=32 C=23 D=11 so D is eliminated. Round 3 votes counts: B=38 A=34 C=28 so C is eliminated. Round 4 votes counts: A=57 B=43 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:218 C:215 D:198 E:194 B:175 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 14 10 2 10 B -14 0 -12 -10 -14 C -10 12 0 10 18 D -2 10 -10 0 -2 E -10 14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997335 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 14 10 2 10 B -14 0 -12 -10 -14 C -10 12 0 10 18 D -2 10 -10 0 -2 E -10 14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997335 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 14 10 2 10 B -14 0 -12 -10 -14 C -10 12 0 10 18 D -2 10 -10 0 -2 E -10 14 -18 2 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999997335 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9997: compare: Profile of ballots (with multiplicities), in decreasing order by count: E A D B C (11) A C E D B (11) A E C D B (9) E D B A C (7) C A B D E (7) B D E C A (7) A C E B D (7) C B D A E (6) E D B C A (5) D B E C A (5) A E D C B (5) B D C E A (4) B C D E A (4) E A D C B (2) C B A D E (2) A E C B D (2) A C B E D (2) E D A B C (1) D E B C A (1) B C D A E (1) A C D B E (1) Total count = 100 A B C D E A 0 16 16 18 6 B -16 0 -8 -16 -22 C -16 8 0 4 -10 D -18 16 -4 0 -24 E -6 22 10 24 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999968 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 16 16 18 6 B -16 0 -8 -16 -22 C -16 8 0 4 -10 D -18 16 -4 0 -24 E -6 22 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: A , winner is: A compare: Computing IRV winner. Round 1 votes counts: A=37 E=26 B=16 C=15 D=6 so D is eliminated. Round 2 votes counts: A=37 E=27 B=21 C=15 so C is eliminated. Round 3 votes counts: A=44 B=29 E=27 so E is eliminated. Round 4 votes counts: A=58 B=42 so B is eliminated. IRV winner is A compare: Computing Borda winner. A:228 E:225 C:193 D:185 B:169 Borda winner is A compare: Computing minimax winner. minimax winner is A compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: A , winner is: A compare: Computing GTS winners. A B C D E A 0 16 16 18 6 B -16 0 -8 -16 -22 C -16 8 0 4 -10 D -18 16 -4 0 -24 E -6 22 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['A'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 16 16 18 6 B -16 0 -8 -16 -22 C -16 8 0 4 -10 D -18 16 -4 0 -24 E -6 22 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is A (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 16 16 18 6 B -16 0 -8 -16 -22 C -16 8 0 4 -10 D -18 16 -4 0 -24 E -6 22 10 24 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 1.000000 B: 0.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 1.0 Cumulative probabilities = A: 1.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is A (randomly chosen according to balanced optimal mixed strategy). Trial 9998: compare: Profile of ballots (with multiplicities), in decreasing order by count: D E B A C (10) C A B E D (8) A B C E D (6) A B E D C (5) E B C A D (4) D E B C A (4) E B D C A (3) D B A E C (3) C E D B A (3) C D E A B (3) C A E B D (3) B E A D C (3) A B D E C (3) E C B D A (2) E B D A C (2) D E C B A (2) D B E A C (2) D A B E C (2) C A D B E (2) B A E D C (2) B A D E C (2) A C D B E (2) A C B E D (2) A C B D E (2) E D B C A (1) E D B A C (1) E C D B A (1) D E A B C (1) D C E B A (1) D C E A B (1) D A E B C (1) D A B C E (1) C E B D A (1) C E B A D (1) C E A B D (1) C D A E B (1) C B E A D (1) C B A E D (1) C A B D E (1) B E C A D (1) B A E C D (1) A D B C E (1) A B E C D (1) A B D C E (1) Total count = 100 A B C D E A 0 -4 8 8 2 B 4 0 22 14 6 C -8 -22 0 -4 -14 D -8 -14 4 0 -8 E -2 -6 14 8 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999911 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -4 8 8 2 B 4 0 22 14 6 C -8 -22 0 -4 -14 D -8 -14 4 0 -8 E -2 -6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989021 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: D , winner is: D compare: Computing IRV winner. Round 1 votes counts: D=28 C=26 A=23 E=14 B=9 so B is eliminated. Round 2 votes counts: D=28 A=28 C=26 E=18 so E is eliminated. Round 3 votes counts: D=35 C=34 A=31 so A is eliminated. Round 4 votes counts: D=52 C=48 so C is eliminated. IRV winner is D compare: Computing Borda winner. B:223 A:207 E:207 D:187 C:176 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -4 8 8 2 B 4 0 22 14 6 C -8 -22 0 -4 -14 D -8 -14 4 0 -8 E -2 -6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989021 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -4 8 8 2 B 4 0 22 14 6 C -8 -22 0 -4 -14 D -8 -14 4 0 -8 E -2 -6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989021 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -4 8 8 2 B 4 0 22 14 6 C -8 -22 0 -4 -14 D -8 -14 4 0 -8 E -2 -6 14 8 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999989021 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). Trial 9999: compare: Profile of ballots (with multiplicities), in decreasing order by count: E C D A B (8) B A D C E (8) B E A C D (6) B D A C E (6) E C A D B (5) A C D E B (5) E D C A B (4) E C D B A (4) E B C A D (3) D C E A B (3) B E A D C (3) B A C D E (3) A D C B E (3) A C D B E (3) A B D C E (3) E C B A D (2) E B C D A (2) D E C A B (2) D C A E B (2) D A C B E (2) B E D C A (2) B E D A C (2) B A E C D (2) E D C B A (1) E B D C A (1) D E C B A (1) D A C E B (1) D A B C E (1) C D A E B (1) B E C D A (1) B E C A D (1) B D E C A (1) B D E A C (1) B D A E C (1) B A E D C (1) B A C E D (1) A D C E B (1) A D B C E (1) A C E D B (1) A B C D E (1) Total count = 100 A B C D E A 0 -6 12 6 -6 B 6 0 2 2 6 C -12 -2 0 -2 -8 D -6 -2 2 0 0 E 6 -6 8 0 0 Using game_cvxopt.lp_solver (linear programming --> soln may be unbalanced) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999999643 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 A B C D E A 0 -6 12 6 -6 B 6 0 2 2 6 C -12 -2 0 -2 -8 D -6 -2 2 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 LP and QP give same solution to GT plurality potential winners: B , winner is: B compare: Computing IRV winner. Round 1 votes counts: B=39 E=30 A=18 D=12 C=1 so C is eliminated. Round 2 votes counts: B=39 E=30 A=18 D=13 so D is eliminated. Round 3 votes counts: B=39 E=36 A=25 so A is eliminated. Round 4 votes counts: B=53 E=47 so E is eliminated. IRV winner is B compare: Computing Borda winner. B:208 E:204 A:203 D:197 C:188 Borda winner is B compare: Computing minimax winner. minimax winner is B compare: Computing beatpath winner(s). (Variant based on `winning votes' ordering.) beatpath potential winners: B , winner is: B compare: Computing GTS winners. A B C D E A 0 -6 12 6 -6 B 6 0 2 2 6 C -12 -2 0 -2 -8 D -6 -2 2 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTS winners are ['B'] (candidates with positive probability in optimal mixed strategy). compare: Computing GTD winner. A B C D E A 0 -6 12 6 -6 B 6 0 2 2 6 C -12 -2 0 -2 -8 D -6 -2 2 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GTD winner is B (a candidate with max probability in optimal mixed strategy). compare: Computing GT winner. A B C D E A 0 -6 12 6 -6 B 6 0 2 2 6 C -12 -2 0 -2 -8 D -6 -2 2 0 0 E 6 -6 8 0 0 Using game_cvxopt.qp_solver (quadratic programming --> balanced soln) Optimal mixed strategy = A: 0.000000 B: 1.000000 C: 0.000000 D: 0.000000 E: 0.000000 Sum of squares = 0.999999992174 Cumulative probabilities = A: 0.000000 B: 1.000000 C: 1.000000 D: 1.000000 E: 1.000000 GT winner is B (randomly chosen according to balanced optimal mixed strategy). -------------------------------------------------------------------------------------- number of trials = 10000 number of profiles generated = 10000 number having Condorcet winner = 6428 number of times LP and QP gave same solution to GT = 7713 Nagree: plurality IRV Borda minimax beatpath GTS GTD GT plurality 10000 5557 4107 4356 4366 5515 4335 4262 IRV 5557 10000 5584 6047 6048 7299 5999 5802 Borda 4107 5584 10000 7854 7874 8913 7813 7193 minimax 4356 6047 7854 10000 9953 9915 8869 8232 beatpath 4366 6048 7874 9953 10000 9951 8895 8246 GTS 5515 7299 8913 9915 9951 10000 10000 10000 GTD 4335 5999 7813 8869 8895 10000 10000 8377 GT 4262 5802 7193 8232 8246 10000 8377 10000 Nprefs: plurality IRV Borda minimax beatpath GTS GTD GT plurality 0 210280 279121 266185 265636 0 267055 271911 IRV 234020 0 213726 189502 189410 0 192104 202964 Borda 310179 227874 0 105027 103973 0 106688 139089 minimax 298215 205798 109573 0 2321 0 55832 88313 beatpath 297764 205790 108627 2379 0 0 54549 87662 GTS 0 0 0 0 0 0 0 0 GTD 299445 207996 112012 57268 55951 0 0 81155 GT 301889 216836 141611 88487 87738 0 81145 0 Nmargins: plurality IRV Borda minimax beatpath GTS GTD GT plurality 0 -23740 -31058 -32030 -32128 0 -32390 -29978 IRV 23740 0 -14148 -16296 -16380 0 -15892 -13872 Borda 31058 14148 0 -4546 -4654 0 -5324 -2522 minimax 32030 16296 4546 0 -58 0 -1436 -174 beatpath 32128 16380 4654 58 0 0 -1402 -76 GTS 0 0 0 0 0 0 0 0 GTD 32390 15892 5324 1436 1402 0 0 10 GT 29978 13872 2522 174 76 0 -10 0 11:45:13 code $